Open Collections

UBC Theses and Dissertations

UBC Theses Logo

UBC Theses and Dissertations

An energy method for the analysis of structures subjected to earthquakes McKevitt, William Edward 1980

Your browser doesn't seem to have a PDF viewer, please download the PDF to view this item.

Notice for Google Chrome users:
If you are having trouble viewing or searching the PDF with Google Chrome, please download it here instead.

Item Metadata

Download

Media
831-UBC_1980_A1 M23.pdf [ 10.11MB ]
Metadata
JSON: 831-1.0062476.json
JSON-LD: 831-1.0062476-ld.json
RDF/XML (Pretty): 831-1.0062476-rdf.xml
RDF/JSON: 831-1.0062476-rdf.json
Turtle: 831-1.0062476-turtle.txt
N-Triples: 831-1.0062476-rdf-ntriples.txt
Original Record: 831-1.0062476-source.json
Full Text
831-1.0062476-fulltext.txt
Citation
831-1.0062476.ris

Full Text

AN ENERGY METHOD FOR THE ANALYSIS OF STRUCTURES  SUBJECTED TO EARTHQUAKES by WILLIAM EDWARD McKEVITT M.Sc. (Wales) A THESIS SUBMITTED IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY i n THE FACULTY OF GRADUATE STUDIES Department of C i v i l Engineering We accept t h i s thesis as conforming to the required standard: The U n i v e r s i t y of B r i t i s h Columbia ^ W i l l i a m Edward McKevitt, A p r i l , 1980 In presenting this thesis in partial fulfilment of the requirements for an advanced degree at the University of British Columbia, I agree that the Library shall make i t freely available for reference and study. I further agree that permission for extensive copying of this thesis for scholarly purposes may be granted by the Head of my Department or by his representatives. It is understood that copying or publication of this thesis for financial gain shall not be allowed without my written permission. C l \ / i t - E.h)£>\K)EE^)t^Gx, Department o f , The University of British Columbia 2075 Wesbrook Place Vancouver, Canada V6T 1W5 n i 23 APPZJU 0 8 0 . Date D E - 6 B P 75-51 1 E AN ENERGY METHOD FOR THE ANALYSIS OF STRUCTURES  SUBJECTED TO EARTHQUAKES ABSTRACT. Work toward d e v e l o p i n g a s i m p l e method f o r th e a s e i s m i c d e s i g n o f s t r u c t u r e s c o n s i d e r i n g e n e r g y d i s s i p a t i o n as t h e prime d e s i g n parameter i s r e p o r t e d . V i s c o u s damping i s used t o r e p r e s e n t t h e n o n - s t r u c t u r a l e n e r g y d i s s i p a t i n g elements i n t h e system, and h y s t e r e t i c e n ergy d i s s i p a t i o n i s c o n s i d e r e d e x p l i c i t l y . A d e t a i l e d p a r a m e t r i c s t u d y o f t h e e n e r g y d i s s i p a t i o n c h a r a c t e r i s t i c s o f s i n g l e degree o f freedom systems i s r e p o r t e d f i r s t . Here the amount o f e n e r g y d i s s i p a t e d by b o t h h y s t e r e t i c and v i s c o u s damping mechanisms i n each system i s d e t e r m i n e d f o r v a r i o u s ground m o t i o n s , v i s c o u s damping v a l u e s , system s t r e n g t h s e t c . The r e s u l t s o f t h i s s t u d y a r e p r e s e n t e d i n the form o f s p e c t r a , r e l a t i n g t o t a l e n e r g y d i s s i p a t e d , and e n e r g y d i s t r i b u t i o n between mechanisms t o known system p r o p e r t i e s . The r e s u l t s and i n s i g h t s g a i n e d f r o m t h e p a r a m e t r i c s t u d y a r e i n c o r p o r a t e d i n t o a d e s i g n method w h i c h a c c o u n t s e x p l i c i t l y f o r e n e r g y d i s s i p a t i o n . The i n c l u s i o n o f a system s t r e n g t h parameter i n the i n p u t o f the p r o p o s e d method i s f o u n d t o he most u s e f u l i n terms o f t h e l i m i t s t a t e p h i l o s o p h y employed i n the most r e c e n t e d i t i o n s o f b u i l d i n g codes. A l s o i f s p e c i a l damping d e v i c e s are t o he b u i l t i n t o t h e s t r u c t u r e , t h i s method w i l l f a c i l i t a t e such d e s i g n . F i n a l l y a p r e l i m i n a r y s t u d y o f the e n e r g y d i s s i p a t i o n c h a r a c t e r i s t i c s o f m u l t i - d e g r e e o f freedom systems i s r e p o r t e d . Here the d i s t r i b u t i o n o f e n e r g y i i d i s s i p a t e d by v i s c o u s damping and h y s t e r e s i s , t o g e t h e r w i t h t h e l o c a t i o n o f t h e e f f e c t i v e d i s s i p a t i n g mechanisms w i t h i n t h e s t r u c t u r e , i s s t u d i e d . In p a r t i c u l a r t h e p o s s i b i l i t y o f e x t r a p o l a t i o n from s i n g l e degree o f freedom s p e c t r a t o the m u l t i - d e g r e e o f freedom systems i s i n v e s t i g a t e d and shown t o g i v e e n c o u r a g i n g r e s u l t s . i i i TABLE OF CONTENTS PAGE A b s t r a c t . ', L i s t o f T a b l e s • v i i ; . L i s t o f F i g u r e s -x D e d i c a t i o n -.xvi- ' Acknowledgement x v i 5. CHAPTER 1 REVIEW OF CURRENT ANALYTIC METHODS IN EARTHQUAKE ENGINEERING. 1 1.1 Development o f t h e P r i n c i p a l Methods. 1 1.2 The D u c t i l i t y Method. 2 1.3 En e r g y Methods. 4 1.4 L i m i t S t a t e P h i l o s o p h y . 5 1.5 M u l t i - D e g r e e o f Freedom Energy S t u d i e s 6 2 THEORY AND PROGRAMMING CONSIDERATIONS FOR SINGLE DEGREE OF FREEDOM SYSTEMS 7 2.1 S i n g l e Degree o f Freedom Model 7 2.2 I n c r e m e n t a l E q u a t i o n s o f M o t i o n 10 2.3 A c c u r a c y o f t h e Time S t e p p i n g Scheme 11 2.4 En e r g y C a l c u l a t i o n s 12 2.5 N o n - D i m e n s i o n a l i s i n g the Earthquake 14 2.6 . An E s t i m a t e f o r t h e Permanent S et 15 CHAPTER PAGE 3 RESULTS OF SINGLE DEGREE OF FREEDOM STUDY 18 3 . 1 Scope o f S i n g l e Degree o f Freedom Study. 18 3 . 2 T o t a l E n e r g y D i s s i p a t e d i n System. 23 3 . 3 D i s t r i b u t i o n o f E n e r g y W i t h i n System. 46 3 . 4 S p e c t r a l D i s p l a c e m e n t s . 64 3 . 5 D i s p l a c e m e n t D u c t i l i t i e s . 75 3 . 6 Permanent S e t s . 87 3 . 7 C l a s s i f i c a t i o n o f S t r u c t u r e s by Y i e l d S t r e n g t h R a t i o . 98 3.8 A p p r o x i m a t i o n s f o r T o t a l . E n e r g y D i s s i p a t e d . 103 3 . 9 A p p r o x i m a t i o n s f o r D i s t r i b u t i o n o f E n e r g y W i t h i n System. 117 4 DEVELOPMENT AND VERIFICATION OF DESIGN METHOD FOR SINGLE DEGREE OF FREEDOM SYSTEMS 120 4 . 1 I n t r o d u c t i o n . 120 4 . 2 Requirements f o r an E a r t h q u a k e D e s i g n Method. • 121 4 . 3 Y i e l d S t r e n g t h R a t i o and Damage C o n t r o l . 122 4 . 4 Some n o t e s on E n e r g y D i s s i p a t i o n V e r s u s Time. 123 4 . 5 O u t l i n e o f t h e D e s i g n Method and Examples. 130 5 THEORY AND PROGRAMMING CONSIDERATIONS FOR MULTI-DEGREE OF FREEDOM SYSTEMS "" 137 5 . 1 M a t h e m a t i c a l Model f o r M u l t i - D e g r e e o f Freedom Systems. 137 5 . 2 E q u a t i o n s o f M o t i o n f o r M u l t i - D e g r e e o f Freedom Systems. 139 5 . 3 E n e r g y C a l c u l a t i o n s . 141 6 TOTAL ENERGY DISSIPATED IN MULTI-DEGREE OF FREEDOM SYSTEMS 142 6 . 1 I n t r o d u c t i o n 142 CHAPTER PAGE 6.2 S t r u c t u r e s Used i n t h i s S tudy. 143 6.3 T o t a l E n e r g y D i s s i p a t e d i n E l a s t o - P l a s t i c M u l t i - D e g r e e o f Freedom Systems. 146 6.4 T o t a l E n e r g y D i s s i p a t e d i n S t i f f n e s s D e g r a d i n g M u l t i - D e g r e e o f Freedom Systems. 148 7 DISTRIBUTION OF ENERGY WITHIN M.D.F. SYSTEMS 151 7.1 I n t r o d u c t i o n . - 151 7.2 D i s t r i b u t i o n o f E n e r g y between H y s t e r e t i c and V i s c o u s Damping Mechanisms. 151 7.3 D i s t r i b u t i o n o f E n e r g y i n E l a s t o - P l a s t i c M.D.F. Systems. 155 7.4 D i s t r i b u t i o n o f E n e r g y i n S t i f f n e s s D e g r a d i n g M.D.F. Systems ' 166 8 CONCLUSION 174 8.1 E n e r g y D i s s i p a t e d i n S i n g l e Degree o f Freedom Systems. 174 8.2 The Pr o p o s e d D e s i g n Method. 175 8.3 T o t a l E n e r g y D i s s i p a t e d by M u l t i - D e g r e e o f Freedom Systems. 176 8.4 The D i s t r i b u t i o n o f D i s s i p a t e d E n e r g y W i t h i n M u l t i - D e g r e e o f Freedom Systems. 177 8.5 F u t u r e Work. 178 BIBLIOGRAPHY 179 APPENDIX A RESULTS OF SINGLE DEGREE OF FREEDOM STUDY IN ORIGINAL FORM 181 APPENDIX B LISTING OF SUBROUTINES ENERGY AND SPDIS 218 APPENDIX C RESULTS OF MULTI-DEGREE OF FREEDOM STUDY IN ORIGINAL FORM 221 v i LIST OF'. TABLES TABLE TI T L E PAGE 3.1 Input C o m b i n a t i o n s used i n S i n g l e Degree o f Freedom S t u d y 22 4.1 E n e r g y D i s s i p a t i o n V e r s u s Time F o r S i n g l e Degree o f Freedom Systems 125 4.2 D i s p l a c e m e n t s and Permanent S e t s f o r Systems A n a l y z e d by t h e Method G i v e n i n Chapter 4 136 6.1 T o t a l E n e r g y D i s s i p a t e d b y E l a s t o - P l a s t i c M.D.F. Systems 147 6.2 T o t a l E n e r g y D i s s i p a t e d b y D e g r a d i n g S t i f f n e s s M.D.F. Systems 149 7.1 E n e r g y D i s s i p a t e d i n H y s t e r e s i s f o r E l a s t o - P l a s t i c M.D.F. Systems 153 7.2 Energy D i s s i p a t e d i n H y s t e r e s i s f o r S t i f f n e s s D e g r a d i n g M.D.F. Systems 154 7.3 E f f e c t o f V a r i o u s Forms o f V i s c o u s Damping on E n e r g y D i s t r i b u t i o n W i t h i n E l a s t o - P l a s t i c M.D.F. Systems 156 7.4 D i s t r i b u t i o n o f Energy i n S t r u c t u r e 1 E l a s t o - P l a s t i c Model.1% V i s c o u s Damping 158 7.5 D i s t r i b u t i o n o f Energy i n S t r u c t u r e 1 E l a s t o - P l a s t i c Model 10% V i s c o u s Damping 159 7.6 D i s t r i b u t i o n o f En e r g y i n S t r u c t u r e 2 E l a s t o - P l a s t i c Model \% V i s c o u s Damping 161 7.7 Energy D i s t r i b u t i o n i n S t r u c t u r e s 1,2,3,4 •& 5 E l a s t o - P l a s t i c Model, 1% V i s c o u s Damping S u b j e c t e d t o E l C e n t r o N.S. 163 V I 1 TABLE TITL E 7.8 D i s t r i b u t i o n o f En e r g y i n S t r u c t u r e 6 E l a s t o - P l a s t i c Model, 3% V i s c o u s Damping 7.9 D i s t r i b u t i o n o f E n e r g y i n S t r u c t u r e 1 S t i f f n e s s D e g r a d i n g Model 1% V i s c o u s Damping 7.10 D i s t r i b u t i o n o f En e r g y i n S t r u c t u r e 2 S t i f f n e s s D e g r a d i n g Model 1% V i s c o u s Damping 7.11 E n e r g y D i s t r i b u t i o n i n S t r u c t u r e s 1,2,3,4 & 5 S t i f f n e s s D e g r a d i n g Model 1% V i s c o u s Damping S u b j e c t e d t o E l C e n t r o N.S. 7.12 D i s t r i b u t i o n o f Energy i n S t r u c t u r e 6 S t i f f n e s s D e g r a d i n g Model 3% V i s c o u s Damping c 1 M.D.F. R e s u l t s Set 1 c 2 M.D.F. R e s u l t s Set 2 c 3 M.D.F. R e s u l t s Set 3 c 4 M.D.F. R e s u l t s Set '4 c 5 M.D.F. R e s u l t s Set 5 c 6 M.D.F. R e s u l t s Set 6 c 7 M.D.F. R e s u l t s Set 7 c 8 M.D.F. R e s u l t s Set 8 c 9 M.D.F. R e s u l t s Set 9 c 10 M.D.F. R e s u l t s Set 10 c 11 M.D.F. R e s u l t s Set 11 c 12 M.D.F. R e s u l t s Set 12 c 13 M.D.F. R e s u l t s Set 13 c 14 M.D.F. R e s u l t s Set 14 c 15 M.D.F. R e s u l t s Set 15 c 16 M.D.F. R e s u l t s Set 16 c 17 M.D.F. R e s u l t s Set 17 v i i i TABLE ' T I T L E ' PAGE C 18 M.D.F. R e s u l t s S e t 18 230 C 19 M.D.F. R e s u l t s S e t 19 230 C 20 M.D.F. R e s u l t s S e t 20 231 C 21 M.D.F. R e s u l t s S e t 21 232 C 22 M.D.F. R e s u l t s S e t 22 233 C 23 M.D.F. R e s u l t s S e t 23 233 C 24 M.D.F. R e s u l t s S e t 24 234 C 25 M.D.F. R e s u l t s S e t 25 234 C 26 M.D.F. R e s u l t s Set 26 235 C 27 M.D.F. R e s u l t s S e t 27 235 C 28 M.D.F. R e s u l t s Set 28 236 C 29 M.D.F. R e s u l t s S e t 29 236 C 30 M.D.F. R e s u l t s Set 30 237 C 31 M.D.F. R e s u l t s S e t 31 238 C 32 M.D.F. R e s u l t s S e t 32 239 i x LIST OF FIGURES FIGURE PAGE 2.1 S i n g l e Degree o f Freedom Model 9 2.2 Permanent S e t f o r T r i - L i n e a r System S u b j e c t e d t o Pacoima S16E 1% V i s c o u s Damping YSR = 0.085 P e r i o d = 0.8 Sec. 16 2.3 Permanent S e t f o r E l a s t o - P l a s t i c System S u b j e c t e d t o T a f t N69W 1% V i s e . Damping YSR = 0.638 P e r i o d = 0.8 Sec. 17 3.1 H y s t e r e s i s Loops 20 3.2 E f f e c t o f YSR on T o t a l E n e r g y D i s s i p a t e d Some T y p i c a l R e s u l t s 25 3 . 3 E f f e c t o f YSR on T o t a l E n e r g y D i s s i p a t e d Means f o r 18 Combinations 26 3 . 4 E f f e c t o f YSR on T o t a l E n e r g y D i s s i p a t e d Means f o r Ea c h E a r t h q u a k e R e c o r d 27 3.5 Means and D e v i a t i o n s i n T o t a l E n e r g y D i s s i p a t e d F o r a l l Systems S u b j e c t e d t o T a f t N69W 29 3.6 Means and D e v i a t i o n s i n T o t a l E n e r g y D i s s i p a t e d F o r a l l Systems S u b j e c t e d t o E l C e n t r o N.S. 30 3.7 Means and D e v i a t i o n s i n T o t a l E n e r g y D i s s i p a t e d F o r a l l Systems S u b j e c t e d t o P a r k f i e l d N65E 31 3.8 Means and D e v i a t i o n s i n T o t a l E n e r g y D i s s i p a t e d F o r a l l Systems S u b j e c t e d t o Pacoima S16E 32 3.9 E f f e c t o f Eart h q u a k e R e c o r d on T o t a l E n ergy D i s s i p a t e d E l a s t o - P l a s t i c Systems w i t h 1.0% V i s c o u s Damping 34 x FIGURES TITLE PAGE 3.10 E f f e c t of Earthquake Record on T o t a l Energy D i s s i p a t e d T r i - L i n e a r Systems w i t h 1.0% Viscous Damping 35 3.11 E f f e c t of M a t e r i a l H y s t e r e s i s on T o t a l Energy D i s s i p a t e d Systems Subjected to Ta f t N69W w i t h 1% Viscous Damping 37 3.12 E f f e c t of M a t e r i a l H y s t e r e s i s on T o t a l Energy D i s s i p a t e d Systems Subjected to E l Centro N.S. w i t h 1% Viscous Damping 38 3.13 E f f e c t of M a t e r i a l H y s t e r e s i s on T o t a l Energy D i s s i p a t e d Systems Subjected to P a r k f i e l d N65E w i t h 1% V i s e . Damping 39 3.14 E f f e c t of M a t e r i a l H y s t e r e s i s on T o t a l Energy D i s s i p a t e d Systems Subjected to P a r k f i e l d N65E w i t h 1% V i s e . Damping 40 3.15 E f f e c t of V i s e . Damping on T o t a l Energy D i s s i p a t e d For a l l Systems Subjected t o Ta f t N69W 42 3.16 E f f e c t of V i s e . Damping on T o t a l Energy D i s s i p a t e d For a l l Systems Subjected t o E l Centro N.S. 43 3.17 E f f e c t of Viscous Damping on T o t a l Energy D i s s i p a t e d For a l l Systems Subjected t o P a r k f i e l d N65E 44 3.18 E f f e c t of V i s e . Damping on T o t a l Energy D i s s i p a t e d For a l l Systems Subjected t o Pacoima S16E 45 3.19 E f f e c t of Y i e l d S t r e n g t h R a t i o on D i s t r i b u t i o n of Energy w i t h i n Systems. Some t y p i c a l R e s u l t s 47 3.20 E f f e c t o f YSR on D i s t r i b u t i o n of Energy W i t h i n Systems Means f o r a l l E l a s t o - P l a s t i c Systems w i t h 1% Damping 48 3.21 E f f e c t of YSR on D i s t r i b u t i o n of Energy W i t h i n Systems Means f o r a l l T r i - L i n e a r Systems w i t h 1% Damping 48 3.22 E f f e c t of Earthquake Record on D i s t r i b u t i o n of Energy W i t h i n E l a s t o - P l a s t i c Systems w i t h 1% V i s e . Damping 50 3.23 E f f e c t of Earthquake Record on D i s t r i b u t i o n of Energy W i t h i n T r i - L i n e a r Systems w i t h 1% V i s e . Damping 51 x i FIGURES TITLE PAGE 3.24 E f f e c t of M a t e r i a l H y s t e r e t i c P r o p e r t i e s on D i s t r i b u t i o n of Energy W i t h i n Systems f o r a l l Systems Subjected t o Taft N69W w i t h 1% Damping 53 3.25 E f f e c t of M a t e r i a l H y s t e r e t i c P r o p e r t i e s on D i s t r i b u t i o n of Energy w i t h i n Systems f o r a l l Systems Subjected to E l Centro N.S. w i t h 1% Damping 54 3.26 E f f e c t of M a t e r i a l P r o p e r t i e s on D i s t r i b u t i o n of Energy W i t h i n Systems f o r a l l Systems Subjected to P a r k f i e l d N65E w i t h 1% Damping 55 3.27 E f f e c t of M a t e r i a l H y s t e r e t i c P r o p e r t i e s on D i s t r i b u t i o n of Energy W i t h i n Systems f o r a l l Systems Subjected t o Pacoima S16E w i t h 1% Damping 56 3.28 E f f e c t of Viscous Damping on D i s t r i b u t i o n of Energy W i t h i n Systems Subjected to Taft N69E 58 3.29 E f f e c t of Visc o u s Damping on D i s t r i b u t i o n of Energy W i t h i n Systems Subjected t o E l Centro N.S. 59 3.30 E f f e c t of V i s e . Damping on D i s t r i b u t i o n of Energy W i t h i n Systems Subjected to P a r k f i e l d N65E 60 3.31 E f f e c t of V i s e . Damping on D i s t r i b u t i o n of Energy W i t h i n Systems Subjected t o Pacoima S16E 61 3.32 Schematic of Twin P a r a l l e l R e s i s t i n g Element Model 62 3.33 S p e c t r a l Displacements f o r Systems Subjected t o Taft N69W 67 3-34 S p e c t r a l Displacements f o r Systems Subjected t o E l Centro N.S. 69 3.35 S p e c t r a l Displacements f o r Systems Subjected t o P a r k f i e l d N65E . 71 3.36 S p e c t r a l Displacements f o r Systems Subjected t o Pacoima S16E 73 3.3V Displacement D u c t i l i t i e s f o r Systems Subjected to Taf t N69W 79 x n FIGURES TIT L E PAGE 3.38 D i s p l a c e m e n t D u c t i l i t i e s f o r Systems S u b j e c t e d t o E l C e n t r o N.S. 81 3.39 D i s p l a c e m e n t D u c t i l i t i e s f o r Systems S u b j e c t e d t o P a r k f i e l d N65E 83 3.40 D i s p l a c e m e n t D u c t i l i t i e s f o r Systems S u b j e c t e d t o Pacoima S16E 85 3.41 Permanent S e t s f o r Systems. S u b j e c t e d t o T a f t N69W 90 3.42 Permanent S e t s f o r Systems S u b j e c t e d t o E l C e n t r o N.S. 92 3.43 Permanent S e t s f o r Systems S u b j e c t e d t o P a r k f i e l d N65E 94 3.44 Permanent S e t s f o r Systems S u b j e c t e d t o Pacoima S16E 96 3.45 A p p r o x i m a t i o n s f o r T o t a l E n e r g y D i s s i p a t e d b y S t r o n g Systems 106 3.46 A p p r o x i m a t i o n s f o r Means o f T o t a l E n e r g y D i s s i p a t e d by a l l S t r o n g Systems 108 3.47 A p p r o x i m a t i o n s f o r T o t a l E n e r g y D i s s i p a t e d by Moderate Systems 112 3.48 A p p r o x i m a t i o n s f o r T o t a l E n e r g y D i s s i p a t e d b y Weak Systems 116 3-49 A p p r o x i m a t i o n s f o r D i s t r i b u t i o n o f E n e r g y W i t h i n Systems f o r E l a s t o - P l a s t i c Systems 118 3.50 A p p r o x i m a t i o n s f o r D i s t r i b u t i o n o f En e r g y W i t h i n Systems f o r T r i - L i n e a r Systems 119 4.1' Time H i s t o r y o f En e r g y D i s s i p a t e d f o r a T r i - L i n e a r System. YSR =1.0, P e r i o d = 1.4 s e c , 1% V i s e . Dmpg. S u b j e c t e d t o Pacoima S16E 126 4.2 Time H i s t o r y o f Energy D i s s i p a t i o n f o r a T r i - L i n e a r System. YSR = 0.5, P e r i o d = 1.4 sec , 1% V i s e . Dmpg. S u b j e c t e d t o T a f t N69W 127 x i i i FIGURES T I T L E PAGE 4.3 Time H i s t o r y o f En e r g y D i s s i p a t i o n f o r an E l a s t o - P l a s t i c System. YSR = 0.5, P e r i o d = 1.4 s e c , 1% V i s e Dmpg. S u b j e c t e d t o Pacoima S16E 128 4.4 Time H i s t o r y o f Energy D i s s i p a t i o n f o r an E l a s t o - P l a s t i c System. YSR = 0.5, P e r i o d = 0.8 s e c , 1% V i s e . Dmpg. S u b j e c t e d t o T a f t N69W 129 5.1 . M u l t i - D e g r e e o f Freedom Model 138 6.1 S t r u c t u r e s Used i n M.D.F. Stud y 145 6.2 Comparison o f T o t a l E nergy D i s s i p a t e d b y M u l t i - D e g r e e o f Freedom Systems w i t h S i n g l e Degree o f Freedom R e s u l t s 150 A . l R e s u l t s f o r C o m b i n a t i o n 1. TAFEP.01 182 A. 2 R e s u l t s f o r C o m b i n a t i o n 2. PACEP.01 184 A.3 R e s u l t s f o r C o m b i n a t i o n 3. ELCEP.01 186 A.4 R e s u l t s f o r C o m b i n a t i o n • 4. TAFTL.01 188 A. 5 R e s u l t s f o r C o m b i n a t i o n 5. PACTL.01 190 A.6 R e s u l t s f o r C o m b i n a t i o n 6. PKFEP.01 192 A.7 R e s u l t s f o r C o m b i n a t i o n 7. PKFTL.01 194 A.8 R e s u l t s f o r C o m b i n a t i o n 8. ELCTL.01 196 A.9 R e s u l t s f o r C o m b i n a t i o n 9. TAFEP.05 198 A. 10 R e s u l t s f o r C o m b i n a t i o n 10. PACTL.05 200 A.11 R e s u l t s f o r C o m b i n a t i o n 11. ELCTL.05 202 A.12 R e s u l t s f o r C o m b i n a t i o n 12. PKFEP.05 204 A.13 R e s u l t s f o r C o m b i n a t i o n 13. TAFTL.00 206 A.14 R e s u l t s f o r C o m b i n a t i o n 14. PACEP.00 208 A.15 R e s u l t s f o r C o m b i n a t i o n 15. ELCEP.00 210 x i v FIGURES TIT L E PAGE A. 16 R e s u l t s f o r C o m b i n a t i o n 16. PKFTL.00 212 A.17 R e s u l t s f o r Combination,17. TAFR0.01 214 A.18 R e s u l t s f o r Com b i n a t i o n 18. ELCR0.01 216 xv DEDICATED TO MY WIFE MAUREEN, MY PARENTS, AND MY CHILDREN, ELAINE GARETH MARK ANDREW BETHAN WHO HAVE SACRIFICED MUCH AND LOVED ME MORE TO MAKE THIS POSSIBLE. I f I have the g i f t o f p r o p h e c y and can fathom a l l m y s t e r i e s and a l l knowledge . . . . . . . b u t have n o t l o v e , I am n o t h i n g . I Cor. 13.: 2: x v i ACKNOWLEDGEMENT I would l i k e t o e x p r e s s my s i n c e r e a p p r e c i a t i o n and g r a t i t u d e t o P r o f s . N.D. Nathan, D.L. A n d e r s o n and S. C h e r r y f o r t h e i r encouragement, g u i d a n c e and c o n s t a n t a v a i l a b i l i t y f o r d i s c u s s i o n d u r i n g a l l s t a g e s o f t h e work p r e s e n t e d i n t h i s t h e s i s . S p e c i a l t h a n k s go t o Mrs. S h e i l a J o h n s t o n f o r h e r c a r e and p a t i e n t t y p i n g o f t h i s t h e s i s . The r e s e a r c h was f i n a n c e d b y t h e N a t i o n a l R e s e a r c h C o u n c i l o f Canada. x y i i 1 CHAPTER 1 REVIEW OF CURRENT ANALYTIC METHODS IN EARTHQUAKE ENGINEERING 1.1 Development o f t h e P r i n c i p a l Methods 1.1.1 E l a s t i c Response S p e c t r a : I n 194-0 B i o t i n t r o d u c e d t h e c o n c e p t o f an e l a s t i c r e s p o n s e spectrum, and o u t l i n e d a method f o r a p p l y i n g h i s spectrum t o the s e i s m i c a n a l y s i s o f s t r u c t u r e s . I t was i m m e d i a t e l y a p p a r e n t t h a t t h e d e s i g n f o r c e s t h e method p r o d u c e d were d i s p r o p o r t i o n a t e l y l a r g e compared t o t h e f o r c e s known t o have e x i s t e d i n s t r u c t u r e s w h i c h had s u r v i v e d a c t u a l e a r t h q u a k e s . S i m i l a r l y B i o t ' s e l a s t i c s p e c t r a l a n a l y s i s r e s u l t s were c o n s i d e r a b l y h i g h e r t h a n t h e f o r c e l e v e l s s p e c i f i e d b y d e s i g n c o d e s . The p r o v i s i o n s embodied i n t h e codes had been b a s e d on s t u d i e s o f s t r u c t u r e s s u r v i v i n g p r e v i o u s e a r t h q u a k e s . In a f i r s t attempt t o r e c o n c i l e t h e d i f f e r e n c e between the a n a l y t i c and o b s e r v e d f o r c e magnitudes B i o t p o i n t e d out t h e i m p o r t a n c e o f h y s t e r e t i c damping i n r e d u c i n g t h e r e s p o n s e . S i n c e 1945 a r c h i t e c t u r a l s t y l e s and c o n s t r u c t i o n methods have changed a p p r e c i a b l y from t h e t r a d i t i o n a l forms employed p r e v i o u s l y . These changes r e f l e c t t h e economic p r e s s u r e s on l a n d and c o n s t r u c t i o n c o s t s . The t r e n d toward l i g h t e r , t a l l e r s t r u c t u r e s u s i n g a minimum o f n o n - s t r u c t -u r a l m a t e r i a l s , made i t i m p e r a t i v e t h a t the means by which s t r u c t u r e s r e s i s t 2 earthquake e x c i t a t i o n he b e t t e r u n d e r s t o o d . B l o t ' s work found r e a d y a p p l i c a t i o n from t h e s t a r t i n e a r t h q u a k e 3 e n g i n e e r i n g r e s e a r c h . By 1953 Hcusner, M a t e l & A l f o r d had s t u d i e d numerous r e s p o n s e s p e c t r a and were a b l e t o c o n f i r m t h a t t h e d i f f e r e n c e between t h e e l a s t i c s p e c t r a l a n a l y s i s r e s u l t s , and o b s e r v a t i o n s made a f t e r r e a l e a r t h q u a k e s , c o u l d be a c c o u n t e d f o r by c o n s i d e r i n g t h e n o n - l i n e a r e f f e c t s i n t h e structural...response. . 1.1.2 Time S t e p I n t e g r a t i o n Methods The n e x t major development i n eart h q u a k e e n g i n e e r i n g came w i t h t h e i n t r o d u c t i o n o f time s t e p i n t e g r a t i o n p r o c e d u r e s f o r i n t e g r a t i n g t h e g o v e r n -4 5 i n g e q u a t i o n s o f motion. Newmark and Clough & W i l s o n p r o d u c e d t i m e s t e p p r o c e d u r e s which e n a b l e d r e s e a r c h e r s t o c a r r y out time h i s t o r y i n v e s t i g a t -i o n s o f s t r u c t u r a l r e s p o n s e , b o t h i n t h e l i n e a r and n o n - l i n e a r r a n g e . 1.1.3 • D u c t i l i t y and Energy Methods The second w o r l d c o n f e r e n c e on eart h q u a k e e n g i n e e r i n g t o o k p l a c e i n Tokyo i n I960. By t h i s time two main streams o f thought had emerged t o accou n t f o r the n o n - l i n e a r e f f e c t s on t h e s t r u c t u r a l r e s p o n s e . F i r s t l y , 6 7 V e l e t s o s and Newmark p i o n e e r e d the d u c t i l i t y c o n c e p t , and Housner , B e r g 8 9 and Thomaides and Ta n a b a s h i , pro d u c e d methods which c o n s i d e r e d energy d i s s i p a t i o n as a b a s i s f o r a n a l y s i s . 1.2 The D u c t i l i t y Method The d u c t i l i t y c o n c ept c o n t a i n s a number o f i m p o r t a n t a s s u m p t i o n s which c o n s t r a i n t h e range o f a p p l i c a t i o n o f t h e method. The l i m i t a t i o n s i n h e r e n t i n the method must be c l e a r l y u n d e r s t o o d i n o r d e r t o a v o i d 3 i n a p p r o p r i a t e a p p l i c a t i o n . The f i r s t o f t h e s e assumptions i s t h a t an i n e l a s t i c o r y i e l d i n g s t r u c t u r e w i l l have a peak d i s p l a c e m e n t e q u a l t o t h e peak d i s p l a c e m e n t o f an e l a s t i c s t r u c t u r e o f e q u a l p e r i o d , when b o t h s t r u c t u r e s have been s u b j e c t e d t o the same e a r t h q u a k e r e c o r d . T h i s a s s u m p t i o n i s b a s e d on s t u d i e s o f E l a s t o - P l a s t i c systems s u b j e c t e d t o f a r f i e l d e a r t h q u a k e r e c o r d s . The s e c o n d a s s u m p t i o n employs t h e f a c t t h a t s p e c t r a o b t a i n e d f o r e l a s t i c s t r u c t u r e s , have i n g e n e r a l , t h e same shape as s p e c t r a o b t a i n e d f o r s t r u c t u r e s w i t h v a r y i n g d e g r e e s o f n o n - l i n e a r i t y . F u r t h e r , t h e s p e c t r a l v a l u e s o f t h e n o n - l i n e a r r e s p o n s e v a r y i n d i r e c t p r o p o r t i o n t o t h e d u c t i l i t y f a c t o r o f t h e n o n - l i n e a r s t r u c t u r e . T h i s p e r m i t s a d e s i g n method b a s e d on an e l a s t i c a n a l y s i s and the n o n - l i n e a r e f f e c t s can be t a k e n i n t o a c c o u n t b y s i m p l y d i v i d i n g t h e e l a s t i c r e s u l t s by t h e - d u c t i l i t y f a c t o r . The t h i r d a s s u m p t i o n which i s i n h e r e n t i n t h e d u c t i l i t y method, b u t n e v e r d e f i n e d e x p l i c i t l y , i s t h a t t h e d u c t i l i t y i s assumed t o be d i s t r i b u t e d u n i f o r m l y t h r o u g h o u t t h e system. I n p r a c t i c e t h i s i s v e r y d i f f i c u l t t o a c h i e v e , and o f t e n i m p o s s i b l e due t o s e r v i c e a b i l i t y c o n s i d e r -a t i o n s . E x p e r i e n c e i n a number o f r e c e n t e a r t h q u a k e s has shown t h a t when t h i s i s not a p p r e c i a t e d t h e s t r u c t u r a l r e s p o n s e i s q u i t e d i f f e r e n t from t h a t p r e d i c t e d by a d u c t i l i t y a n a l y s i s . T h i s i s because t h e e l a s t i c v i b r a t i o n a l mode shape used^ i n t h e d e s i g n i s s i g n i f i c a n t l y d i f f e r e n t from the shape o f t h e f a i l u r e .mode. F u r t h e r m o r e , t h e r e i s always d i f f i c u l t y i n c a l c u l a t i n g t h e system d u c t i l i t y from t h e assemblage o f members even when the i n d i v i d u a l member d u c t i l i t i e s may be wel l defined. In review i t can be pointed out that the major weakness of the d u c t i l i t y method l i e s i n two main areas. (1) The l a t i t u d e of the assumptions on which the method i s based must be f u l l y understood i n order that t h i s deceptively simple method can be applied c o r r e c t l y . ( 2 ) The method i s not s u f f i c i e n t l y f l e x i b l e to take into account major concentrations of s t i f f n e s s or f l e x i b i l i t y w ithin the structure. 1. 3 Energy Methods The altern a t i v e methods of accounting f o r the non-linear effects i n the s t r u c t u r a l response considered energy d i s s i p a t i o n within the system. There are two groups of energy methods considered. 7 9 F i r s t l y , Housner and Tanabashi presented methods which e s s e n t i a l l y equated the maximum peak k i n e t i c energy of a l i n e a r system, to the energy to be dissipated by the i n e l a s t i c system during the earthquake. A value for the e l a s t i c peak k i n e t i c energy i s obtained by using the e l a s t i c spectral v e l o c i t y . Correlation studies supporting the assumption that the e l a s t i c peak k i n e t i c energy i s proportional to the t o t a l energy dissipated by an i n e l a s t i c structure during the earthquake e x c i t a t i o n could not be found i n the l i t e r a t u r e . An alternative energy method, produced at the same time, was 5 8 p r e s e n t e d by Berg & Thomaides . T h i s method c o n s i d e r s t h e t o t a l e n e r g y d i s s i p a t e d by t h e s t r u c t u r e f o r t h e d u r a t i o n o f t h e e a r t h q u a k e e x c i t a t i o n . By c o n s i d e r i n g the r e s p o n s e o f s i n g l e degree o f freedom systems h a v i n g e l a s t i c — p e r f e c t l y p l a s t i c s t i f f n e s s p r o p e r t i e s , a s e r i e s o f t o t a l e nergy s p e c t r a f o r v a r i o u s y i e l d l e v e l s can be p r o d u c e d . T h i s i s c a r r i e d out by p e r f o r m i n g a time s t e p a n a l y s i s , e v a l u a t i n g t h e t o t a l e n e r g y a t e v e r y s t e p , and a c c u m u l a t i n g t h e t o t a l e n ergy d i s s i p a t e d d u r i n g t h e e a r t h -quake r e c o r d . A s i m i l a r s t u d y was c a r r i e d out by J e n n i n g s 1 " u s i n g a more g e n e r a l Ramberg-Osgood h y s t e r e s i s l o o p t o model the n o n - l i n e a r r e s p o n s e . The energy t o be d i s s i p a t e d by. t h e s t r u c t u r e , o b t a i n e d by t h e above methods, becomes the main a n a l y s i s parameter, and t h e s t r u c t u r e i s d e s i g n e d t o ensure t h a t t h i s e n ergy can be d i s s i p a t e d s a f e l y . I l The Reserve E n e r g y Technique i s s i m i l a r t o t h e energy methods which equate t h e peak s p e c t r a l k i n e t i c e n e r g y t o t h e e n e r g y d i s s i p a t e d i n p l a s t i c work. U n f o r t u n a t e l y , i n p r e s e n t i n g t h i s method numerous unsub-s t a n t i a t e d assumptions were employed which draw a t t e n t i o n away from t h e energy b a l a n c e k e r n e l o f the method. C o n s e q u e n t l y t h i s method has n e v e r r e c e i v e d w i d e s p r e a d a c c e p t a n c e . 1.4 L i m i t S t a t e P h i l o s o p h y The main developments o f i d e n t i f y i n g t h e problems and p r o d u c i n g methods o f s o l u t i o n i n e a r t h q u a k e e n g i n e e r i n g had t a k e n p l a c e by I960. S i n c e , t h e i n t r o d u c t i o n , i n 1962, o f modal a n a l y s i s f o r m u l t i - d e g r e e o f 12 13 freedom systems by Merchant and Hudson , and Clough a l l t h e t e c h n i q u e s c u r r e n t l y employed i n a n a l y s i s have been a v a i l a b l e . 6 From t h a t t i m e methods have r e c e i v e d wide a p p l i c a t i o n i n a p r o l i f e r a t i o n o f e a r t h q u a k e e n g i n e e r i n g r e s e a r c h . The s i m u l t a n e o u s growth o f l a r g e c a p a c i t y d i g i t a l computers p e r m i t t e d s t u d i e s o f i n c r e a s i n g l y complex models t o he made. The knowledge g a i n e d from t h e s e endeavours has r e s u l t e d i n a f a r more d e t a i l e d u n d e r s t a n d i n g o f t h e way i n which s t r u c t u r e s r e s i s t e a r t h q u a k e s . However, s i n c e 1940 t h e most i m p o r t a n t unanswered q u e s t i o n i n earthquake e n g i n e e r i n g has been how t o produce a s i m p l e method o f a n a l y s i s w hich a d e q u a t e l y a c c o u n t s f o r t h e n o n - l i n e a r e f f e c t s i n t h e s t r u c t u r a l r e s p o n s e . The need f o r such a method o f a n a l y s i s i s now g r e a t e r t h a n e v e r due t o t h e n e c e s s i t y o f b r i n g i n g t h e e a r t h q u a k e d e s i g n o f s t r u c t u r e s w i t h i n 17,18,19 the framework o f t h e l i m i t s t a t e d e s i g n p h i l o s o p h y . In terms o f earthquake e n g i n e e r i n g t h i s r e q u i r e s a two l e v e l a n a l y s i s p r o c e d u r e . Such a method would p r o v i d e f o r an e l a s t i c a n a l y s i s , r e p r e s e n t i n g n o m i n a l damage l e v e l s , f o r an e a r t h q u a k e o f moderate i n t e n s i t y , c o r r e s p o n d i n g t o a r e t u r n p e r i o d o f say 10 - 20 y e a r s , and an u l t i m a t e p l a s t i c a n a l y s i s a v o i d i n g t o t a l c o l l a p s e i n a s e v e r e e a r t h q u a k e w i t h e x p e c t e d r e t u r n p e r i o d s o f s a y 50 - 100 y e a r s . 1.5 M u l t i - D e g r e e o f Freedom Energy S t u d i e s The o n l y o t h e r energy d i s s i p a t i o n s t u d i e s a v a i l a b l e i n t h e l i t e r a t u r e c o n c e r n t h e way i n w h i c h the e n e r g y i s d i s t r i b u t e d t h r o u g h o u t m u l t i - d e g r e e o f freedom systems. There seems t o be r e m a r k a b l y l i t t l e work done on t h i s a s p e c t o f t h e p r o b l e m t o d a t e , t h e o n l y r e f e r e n c e s found were by S p e n c e r 1 4 , i s 16 2 0 A r y a , G l u c k , and R u i z & P e n z i e n 7 CHAPTER 2 THEORY AND PROGRAMMING CONSIDERATIONS FOR SINGLE DEGREE OF FREEDOM SYSTEMS 2.1 S i n g l e Degree o f Freedom Model. The model f o r t h e s i n g l e degree o f freedom systems i n v e s t i g a t e d i n t h i s s t u d y i s shown i n f i g . 2.1. T h e w e 11 known g e n e r a l e q u a t i o n o f mo t i o n f o r such a system h a v i n g n o n - l i n e a r s t i f f n e s s e lements and s u b j e c t e d t o base e x c i t a t i o n i s g i v e n a s : where d o t s denote d i f f e r e n t i a t i o n w . r . t . t i m e . and F ( x ) r e p r e s e n t s t h e f o r c e i n t h e s t i f f n e s s elements which a r e a f u n c t i o n o f t h e d i s p l a c e m e n t ( x ) . The c h a r a c t e r i s t i c s o f n o n - l i n e a r s i n g l e degree o f freedom systems used i n t h i s s t u d y a r e d e f i n e d by t h e f o l l o w i n g t h r e e p a r a m e t e r s . ( i ) S t r e n g t h Parameter. M x + C x + F ( x ) = - M x 'g ( 2 . 1 ) ( Y i e l d L e v e l ) Y i e l d L e v e l F. (2.2) y M 8 where F = h o r i z o n t a l f o r c e , a p p l i e d a t t h e c e n t r e o f g r a v i t y o f t h e mass, which causes y i e l d i n g i n t h e n o n - l i n e a r s t i f f n e s s e l e m e n t s . M = Mass o f t h e system. Note: t h e parameter f i s e x p r e s s e d i n u n i t s o f a c c e l e r a t i o n . I t i s used t o r e d u c e r e a l system p a r a m e t e r s t o terms o f u n i t mass. ( i i ) S t i f f n e s s Parameter. o r F r e q u e n c y o f s m a l l a m p l i t u d e v i b r a t i o n s = wQ f i r s t d e f i n e K y = F y /'Xy ( 2 . 3 ) where Ky = e l a s t i c s t i f f n e s s o f system. Fy = h o r i z o n t a l y i e l d f o r c e . Xy = r e l a t i v e d e f l e c t i o n o f t h e system a t y i e l d p o i n t . hence (JJQ ( 2 . 4 ) ( i i i ) Damping Parameter. o r P e r c e n t a g e o f c r i t i c a l v i s c o u s damping. F o r t h i s s t u d y i t was assumed t h a t t h e p e r c e n t a g e o f v i s c o u s damping i s r e l a t e d t o the e l a s t i c p r o p e r t i e s o f t h e system a t a l l t i m e s . t h e r e f o r e n = p e r c e n t a g e o f c r i t i c a l damping = C (2.5) 2 \ZK7 M D i v i d i n g e q u a t i o n ( 2 . 1 ) by M and s u b s t i t u t i n g e q u a t i o n s ( 2 . 4 ) •and ( 2 . 5 ) g i v e s : -x + 2 n w n s + 0 3 n 2 F ( x ) - x, ( 2 . 6 ; ^y 9 F I G . 2.1 S I N G L E D E G R E E O F F R E E D O M M O D E L 10 2.2 Incremental Equations of Motion. In incremental form equation (2.6) becomes: Ax + 2n( jJ nAx + 2 AF( x ) = -Ax £ In the constant average time step scheme the incremental acceleration and v e l o c i t y are expressed i n terms of the incremental displacement and the conditions e x i s t i n g at the s t a r t of the time increment by: (2.7) Ax Ax where At x„ & x„ 4 Ax - 4 x Q _ 2 " AV At 2 Ax - 2x, At J (2.8) time i n c r e m e n t l e n g t h v e l o c i t y and acceleration at the s t a r t of the increment respectively. Substituting (2.8) into equations (2.7), noting that AF (x) = K T Ax, gives after some manipulation: ,4 + 4I"|<JJO + A t 2 At M Ax = -Ax; + x. 4 + 4n<jo. At + 2x, (2.9) 11 where K T = Tangent t o t h e s t i f f n e s s a t t h e s t a r t o f e a c h i n c r e m e n t . F o r c o m p u t a t i o n a l e f f i c i e n c y t h i s e q u a t i o n was programmed a s : ( F A C 1 + Krp * F A C 2 ) Ax = - A x g + x 0 ( F A C 3 ) + 2 x Q (2.10) where FAC1 = 4 + 4 n w 0 A t z A t FAC2 = __1_ M FAC3 4 + 4 n u ) 0 A t FAC1, FAC2 and FAC3 were c a l c u l a t e d once o n l y b e f o r e t h e s t a r t o f t h e time s t e p l o o p . 2.3 A c c u r a c y o f t h e Time S t e p p i n g Scheme. The c o n s t a n t average time s t e p p i n g scheme was u s e d f o r t h i s s t u d y because i t has t h e advantage o f n o t i n t r o d u c i n g n u m e r i c a l damping i n t o t h e c a l c u l a t i o n s . 2 1 T h i s was c o n s i d e r e d i m p o r t a n t b e cause t h e e f f e c t s o f s m a l l p e r c e n t a g e s o f v i s c o u s damping were t o be s t u d i e d . The c a l c u l a t i o n s were c a r r i e d out u s i n g time s t e p s o f 0.01 s e c , f o r systems w i t h p e r i o d s g r e a t e r t h a n 0.5 s e c , and u s i n g time s t e p s o f 0.005 s e c . f o r s h o r t e r p e r i o d systems. A t t h e end o f each time s t e p an e q u i l i b r i u m b a l a n c e check was made. Any e r r o r i n e q u i l i b r i u m a t t h e end o f t h e time s t e p was c o r r e c t e d f o r d u r i n g t h e f o l l o w i n g t i m e s t e p . C l o s e s t u d y o f t h e e q u i l i b r i u m b a l a n c e showed t h a t t h i s e r r o r was e x t r e m e l y s m a l l e x c e p t a t p o i n t s o f d i s c o n t i n u i t y o f s l o p e i n t h e h y s t e r e s i s r e l a t i o n s h i p . A t s u c h p o i n t s o f d i s c o n t i n u i t y t h e e r r o r was r e d u c e d t o e x t r e m e l y s m a l l amounts a f t e r t h e c o r r e c t i o n was a p p l i e d f o r one time s t e p . The f o l l o w i n g e r r o r s r e -mained i n s i g n i f i c a n t u n t i l t h e n e x t h y s t e r e t i c s l o p e d i s c o n t i n u i t y was e n c o u n t e r e d . The 'successive a p p l i c a t i o n o f t h e s e c o r r e c t i v e e r r o r f o r c e s m a i n t a i n e d dynamic e q u i l i b r i u m and p r e v e n t e d e r r o r s due t o t h e n o n - l i n e a r h y s t e r e s i s r e l a t i o n s h i p s from a c c u m u l a t i n g . An a l t e r n a t i v e method o f c o r r e c t i n g f o r t h e e q u i l i b r i u m i m b a l a n c e would be t o s u b d i v i d e the i n t e g r a t i o n time s t e p l e n g t h whenever a p o i n t o f n o n - l i n e a r i t y on t h e h y s t e r e s i s c u r v e I s r e a c h e d . However, such a method i s more d i f f i c u l t t o implement i n a g e n e r a l computer program and would p r o b a b l y t a k e more computer time t o p e r f o r m the c a l c u l a t i o n s . 2.4 Energy C a l c u l a t i o n s Energy c a l c u l a t i o n s were made a t the end o f e a c h time s t e p w i t h t h e f o l l o w i n g e n e r g i e s b e i n g c a l c u l a t e d i n d i v i d u a l l y . ( 1) E nergy D i s s i p a t e d b y H y s t e r e t i c S p r i n g s E n ergy was c o n s i d e r e d d i s s i p a t e d by t h e s p r i n g s when i t became c l e a r t h a t no p o r t i o n o f t h e p o t e n t i a l e nergy i n t h e s p r i n g c o u l d be r e c o v e r e d . 13 ( 2 ) P o t e n t i a l E nergy i n S p r i n g s E n ergy s t o r e d i n t h e s p r i n g s i s assumed t o be r e c o v e r a b l e u n l e s s t h e r e i s a change o f s i g n i n t h e f o r c e . ( 3 ) K i n e t i c E n ergy i n System C a l c u l a t e d a t t h e i n s t a n t o f t h e end o f each time s t e p . ( 4) E n e r g y D i s s i p a t e d i n V i s c o u s Damping C a l c u l a t e d as a s t e p f u n c t i o n f o r e a c h time s t e p b e i n g c o n t i n u o u s l y i n c r e a s e d f o r t h e d u r a t i o n o f m o t i o n . These e n e r g i e s a r e summed t o produce t h e t o t a l e n ergy s t o r e d w i t h i n the system a t t h e end o f e a c h time s t e p . The i n p u t e nergy f o r t h e system i s c a l c u l a t e d f o r each time s t e p u s i n g Newtons second law. Energy IN = Mass * t o t a l a c c e l e r a t i o n * Ground D i s p l a c e m e n t . i 4 In t h e s i t u a t i o n where t h e i n e r t i a f o r c e s and ground d i s p l a c e m e n t are o f o p p o s i t e s i g n energy i s b e i n g r a d i a t e d from t h e s t r u c t u r e back i n t o the ground. Hence a t t h e end o f c a l c u l a t i o n s , t h e t o t a l e n ergy i n p u t t o the system r e p r e s e n t s the net en e r g y s t o r e d ; o r d i s s i p a t e d w i t h i n t h e system. Comparison o f t h e sum o f t h e e n e r g i e s w i t h i n t h e system and t h e energy i n p u t t o the system p r o v i d e s a u s e f u l check on t h e a c c u r a c y o f c a l c u l a t i o n s . A p r a c t i c a l t o l e r a b l e l i m i t o f t>% e r r o r f o r t h i s check was placed... -However a t no p o i n t was t h i s r e a c h e d . L i s t i n g s o f t h e computer s u b r o u t i n e s w h i c h c a r r y o ut t h e s e c a l c u l a t i o n s a r e g i v e n i n appendix B. 2.5 N o n - D i m e n s i o n a l i s i n g t h e E a r t h q u a k e . I n t h i s s t u d y i t was n e c e s s a r y t o compare t h e r e s u l t s o f t h e res p o n s e o f systems s u b j e c t e d t o f o u r v e r y d i f f e r e n t ground m o t i o n r e c o r d s . . Because o f t h i s i t i s i m p o r t a n t t o n o n - d i m e n s i o n a l i s e t h e r e c o r d s t o one common b a s e l i n e s t r e n g t h i n o r d e r t h a t c o m p a r a t i v e s t u d i e s c o u l d be made. The maximum peak ground a c c e l e r a t i o n was used as a measure o f t h e earthquake s t r e n g t h f o r t h e f o l l o w i n g r e a s o n s . ( 1 ) The peak ground a c c e l e r a t i o n has been used as a s t a n d a r d p a r a m e t e r f o r s p e c i f y i n g e arthquake ground m o t i o n r e c o r d s f o r some t i m e , and i t s s i g n i f i c a n c e i s w e l l u n d e r s t o o d . ( 2 ) E s t i m a t e s o f peak ground a c c e l e r a t i o n a r e a v a i l a b l e i n e m p i r i c a l d a t a which can be u s e d t o p r e d i c t f u t u r e ground m o t i o n s a t a g i v e n s i t e . ( 3 ) A l t e r n a t i v e methods o f n o n d i m e n s i o n a l i s i n g t h e e a r t h q u a k e r e c o r d , R.M.S., s p e c t r a l a v e r a g e s e t c . , were t r i e d . However, no s i g n i f -i c a n t improvement i n the c o r r e l a t i o n o f t h e r e s p o n s e d a t a was o b s e r v e d . Because o f t h i s and i n v i e w o f t h e w e l l known d i f f i -c u l t y i n d e f i n i n g an average v a l u e , t h e s e n o r m a l i z i n g methods were n o t u s e d . . 2.6 An E s t i m a t e f o r t h e Permanent Set The r e s p o n s e c a l c u l a t i o n s were s t o p p e d a t the end o f t h e ground a c c e l e r a t i o n a f t e r 30 seconds o f e x c i t a t i o n . At t h i s p o i n t t h e s t r u c t u r e s a r e s t i l l v i b r a t i n g and a method o f e s t i m a t i n g t h e permanent s e t was r e q u i r e d i n o r d e r t o a v o i d t h e c o m p u t a t i o n a l expense o f w a i t i n g f o r t h e r e s i d u a l v i b r a t i o n a l e nergy t o ' d i s s i p a t e . I n a l l t h e ground a c c e l e r a t i o n r e c o r d s used, t h e most e n e r g e t i c p o r t i o n s o f t h e r e c o r d s had o c c u r r e d some time b e f o r e c a l c u l a t i o n s c e a s e d . The permanent s e t i s a r e s u l t o f e x c u r s i o n s i n t o the n o n - l i n e a r range o f the h y s t e r e s i s r e l a t i o n s h i p , w hich had t a k e n p l a c e b e f o r e t h e c a l c u l a t i o n s s t o p p e d . Hence the r e s i d u a l v i b r a t i o n s a r e t a k i n g p l a c e i n a p r e d o m i n a n t l y l i n e a r f a s h i o n c e n t r e d about the t r u e permanent s e t p o s i t i o n . An e s t i m a t e o f t h e permanent s e t was made by t a k i n g t h e a v e r age v a l u e o f t h e d i s p l a c e m e n t d u r i n g t h e l a s t t h r e e seconds o f t h e e a r t h q u a k e r e c o r d . A number o f t e s t r u n s were made t o i n v e s t i g a t e t h e v a l i d i t y o f t h i s a p p r o x -i m a t i o n and t h e r e s u l t s show good agreement w i t h permanent s e t v a l u e s o b t a i n e d 10 seconds a f t e r t h e e x c i t a t i o n s had s t opped. Some t y p i c a l r e s u l t s a r e shown i n f i g s . 2.2 and 2.3. F I G . 2.2 P E R M A N E N T S E T F O R T R I - L I N E A R S Y S T E M S U B J E C T E D T O P A C O I M A S16E 1% V i s c o u s D A M P I N G Y S R = 0.085 P E R I O D = 0.8 S E C . I—1 E N J P O F & A S E . E . X C I T A T I O M ID o FIG. 2.3 PERMANENT SET FOR ELASTO-PLASTIC SYSTEM SUBJECTED TO TAFT N69W 1% VISCOUS DAMPING YSR = 0.638 PERIOD = 0.8 SEC. CHAPTER 3 RESULTS OF SINGLE DEGREE OF FREEDOM STUDY 3.1 Scope o f S i n g l e Degree o f Freedom St u d y T h i s s t u d y employs t h e t h e o r y g i v e n i n c h a p t e r 2. I t t a k e s t h e form o f a b r o a d i n v e s t i g a t i o n t o i d e n t i f y the p r i n c i p a l p a r a m e t e r s a f f e c t i n g t h e energy d i s s i p a t i o n c h a r a c t e r i s t i c s and r e s p o n s e o f systems s u b j e c t e d t o earthquake ground m o t i o n s . C o r r e s p o n d i n g l y as wide a range o f i n p u t v a r i a b l e s as p o s s i b l e was employed i n o r d e r t o expose g e n e r a l t r e n d s and e f f e c t s . The f o l l o w i n g i n p u t v a r i a b l e s were used. Base e x c i t a t i o n s : T a f t N69W 1952, E l C e n t r o N.S. 1940, Pacoima Dam S16E 1971, P a r k f i e l d N65E 1956. i . e . two f a r f i e l d and two n e a r s o u r c e i m p u l s i v e e a r t h q u a k e s . The earthquake r e c o r d s were n o n - d i m e n s i o n a l i s e d by d i v i d i n g by the maximum peak a c c e l e r a t i o n . H y s t e r e s i s P r o p e r t i e s ( F i g . 3-1) E l a s t o - P l a s t i c . Ramberg-Qsgood 1 0 w i t h p a r a m e t e r s A = 0.1 R = 1; A = 0.1 R= 9 i n v a r i o u s r u n s . Both t h e s e h y s t e r e s i s l o o p s a r e r e p r e s e n t a t i v e o f t h e e x p e r i m e n t a l l y p r o d u c e d h y s t e r e t i c p r o p e r t i e s f o r s t e e l frame systems. : T r i - L i n e a r s t i f f n e s s d e g r a d i n g m o d e l s . 2 3 w i t h P B Y = 3 and ~ K 2 = 0.5 T h i s model r e p r e s e n t s t y p i c a l h y s t e r e t i c b e h a v i o u r o f r e i n f o r c e d c o n c r e t e frame s. V i s c o u s Damping Three v a l u e s were used i n c a l c u l a t i o n s : 0 % , 1% and 5%. o f c r i t i c a l damping. The v i s c o u s damping r e p r e s e n t s t h e n o n - s t r u c t u r a l damping i n the s t r u c t u r e . Y i e l d S t r e n g t h R a t i o At an e a r l y s t a g e i t was c l e a r t h a t t h e " Y i e l d S t r e n g t h R a t i o " d e f i n e d a s : YSR = H o r i z A c c e l 1 1 r e q ^ t o produce Y i e l d i n g I n e r t i a f o r c e Max Peak Ground A c c e l n has a marked e f f e c t on t h e n a t u r e and magnitude o f the r e s p o n s e o f a l l systems s t u d i e d . The s i g n i f i c a n c e o f t h i s p arameter w i l l become c l e a r i n t h e d i s c u s s -i o n o f t h e r e s u l t s which f o l l o w . A range o f v a l u e s from 0.1 t o 10.0 i s u s e d . 20 FIG, 3,1 HYSTERSIS LOOPS. I n i t i a l E l a s t i c P e r i o d Ten v a l u e s o f i n i t i a l e l a s t i c p e r i o d f o r s m a l l v i b r a t i o n s , i n seconds, a r e u s e d : 0.1, 0.3, 0.5, 0.8, 1.0, 1.2, 1.8, 2.0, 2.5. The i n p u t p a r a m e t e r s o f v i s c o u s damping, h y s t e r e t i c p r o p e r t i e s and e a r t h q u a k e r e c o r d were combined t o produce e i g h t e e n d a t a s e t s o f r e s u l t s . In k e e p i n g w i t h t h e o v e r v i e w p h i l o s o p h y o f t h i s s t u d y the s e t s c o v e r as wide a range o f c o m b i n a t i o n s as p o s s i b l e i n a l i m i t e d s t u d y s u c h as t h i s . However, i t i s a n t i c i p a t e d t h a t t h e wide range o f i n p u t v a r i a b l e s and c o m b i n a t i o n s employed w i l l p roduce r e s u l t s t h a t form a b o u n d i n g s e t i n t o which r e s u l t s o f more d e t a i l e d s t u d i e s w i l l f a l l . D e t a i l s o f t h e e i g h t e e n c o m b i n a t i o n s c o n s i d e r e d a r e g i v e n i n t a b l e 3.1. The r e s u l t s o f t h e e i g h t e e n c o m b i n a t i o n s s t u d i e d a r e p r e s e n t e d i n t h e form o f r e s p o n s e s p e c t r a c o v e r i n g t h e range o f p e r i o d s and y i e l d s t r e n g t h r a t i o s c o n s i d e r e d . The f o l l o w i n g r e s p o n s e p a r a m e t e r s a r e s t u d i e d i n d e t a i l i n t h e remain-der o f t h i s c h a p t e r . ( a ) T o t a l E n e r g y D i s s i p a t e d i n System, ( b ) D i s t r i b u t i o n o f Energy W i t h i n System, ( c ) S p e c t r a l D i s p l a c e m e n t s , ( d ) D i s p l a c e m e n t D u c t i l i t i e s , ( e ) Permanent S e t s . The complete r e s u l t s o f t h i s s t u d y a r e p r e s e n t e d i n o r i g i n a l form i n Appendix A. C o m b i n a t i o n Numb e r E a r t h q u a k e R e c o r d H y s t e r e t i c P r o p e r t i e s S e e F i g . 3-1 P e r c e n t a g e o f C r i t i c a l V i s c o u s D a m p i n g A b b r e v i a t i o n F o r S y s t e m 1 T a f t N69W E l a s t o - P l a s t i c 1 . 0% .TAFEP. 01 2 i P a c o i m a S 1 6 E E l a s t o - P l a s t i c 1.0% P A C E P . 0 1 3 E l C e n t r o N.S. E l a s t o - P l a s t i c 1.0% E L C E P . 0 1 4 T a f t N69W T r i - L i n e a r 1.0% T A F T L . 0 1 5 P a c o i m a S 1 6 E T r i - L i n e a r 1.0% P A C T L . 0 1 6 P a r k f i e l d N 65E E l a s t o - P l a s t i c 1.0% P K F E P . 0 1 • 7 P a r k f i e l d N 65E T r i - L i n e a r 1.0% P K F T L . 0 1 8 E l C e n t r o N.S. T r i - L i n e a r 1.0% E L C T L . 0 1 9 T a f t N69W E l a s t o - P l a s t i c 5.0% T A F E P . 0 5 10 P a c o i m a S 1 6 E T r i - L i n e a r 5.0% P A C T L . 0 5 11 E l C e n t r o N.S. T r i - L i n e a r 5.0% E L C T L . 0 5 12 P a r k f i e l d N 6 5 E E l a s t o - P l a s t i c 5.0% P K F E P . 0 5 13 T a f t N69W T r i - L i n e a r 0.0% T A F T L . 0 0 14 P a c o i m a S 1 6 E " E l a s t o - P l a s t i c 0.0% P A C E P . 0 0 15 E l C e n t r o N.S. E l a s t o - P l a s t i c 0.0% E L C E P . 0 0 16 P a r k f i e l d N 6 5 E T r i - L i n e a r 0.0% P K F T L . 0 0 17 T a f t N69W R a m b e r g - O s g o o d •' 1 . 0 * T A F R 0 . 0 1 18 E l C e n t r o N.S. R a m b e r g - O s g o o d 1.058 E L C R 0 . 0 1 T A B L E 3.1 INPUT COMBINATIONS USED IN S I N G L E DEGREE OF FREEDOM STUDY 3 . 2 T o t a l E n ergy D i s s i p a t e d i n System The t o t a l e n e r g y t o be d i s s i p a t e d by a s t r u c t u r e d u r i n g an e a r t h -quake e x c i t a t i o n i s t h e b a s i c d e s i g n parameter f o r a method b a s e d on e n e r g y d i s s i p a t i o n c o n s i d e r a t i o n s . In t h i s s e c t i o n t h e e f f e c t o f v a r y i n g the prime i n p u t p a r a m e t e r s o f y i e l d s t r e n g t h r a t i o , e arthquake r e c o r d , m a t e r i a l h y s t e r e s i s and v i s c o u s damping, a r e c o l l e c t i v e l y c o n s i d e r e d below. 3 . 2 . 1 E f f e c t o f Y i e l d S t r e n g t h R a t i o on T o t a l E n e r g y D i s s i p a t e d The most i m p o r t a n t e f f e c t o f r e d u c i n g t h e YSR i s t o l e s s e n the f r e q u e n c y dependence o f systems on t h e e n e r g y c o n t e n t o f t h e ground m o t i o n . See f i g . ( 3 - 2 ) and Appendix A. In a l l c a s e s t h e maximum e n e r g y d i s s i p a t e d o c c u r s i n systems w i t h YSR v a l u e s o f 1 .0 o r g r e a t e r a t f r e q u e n c i e s w h i c h c o i n c i d e w i t h the peak energy f r e q u e n c i e s o f t h e e a r t h q u a k e r e c o r d u s e d. S i m i l a r l y t h o s e s t r o n g e r systems d i s s i p a t e minimum e n e r g i e s a t minimum f r e -q u e n c i e s o f t h e r e c o r d . The v a r i a t i o n i n e n e r g y d i s s i p a t e d f o r d i f f e r e n t p e r i o d s i s g r e a t f o r t h e s e s t r o n g e r systems. However i n a l l s p e c t r a p r o d u c e d i t i s seen t h a t t h e s t r o n g e s t systems, w h i c h remained e l a s t i c t h r o u g h o u t t h e r e s p o n s e , do n o t f o r m an upper bound on the t o t a l e n ergy d i s s i p a t e d f o r a l l v a l u e s o f p e r i o d . T h i s p o i n t i s o f g r e a t p r a c t i c a l s i g n i f i c a n c e b e cause i t i n v a l i d a t e s a t t e m p t s t o e x t r a p o l a t e from. the. l i n e a r , .to"the n o n - l i n e a r - c a s e . As t h e v a l u e o f YSR i s r e d u c e d below 1 . 0 , i t i s seen t h a t i n g e n e r a l the t o t a l e n e r g y t o be d i s s i p a t e d i s r e d u c e d , and a l s o , t h e v a r i a t i o n i n t o t a l e n e rgy d i s s i p a t e d by systems w i t h d i f f e r e n t f r e q u e n c i e s i s r e d u c e d . T h i s v a r i a t i o n becomes a p p r o x i m a t e l y b i - l i n e a r f o r systems w i t h YSR = 0 . 5 and 0 . 3 and l i n e a r f o r systems w i t h YSR = 0 . 1 . These o b s e r v a t i o n s a r e a l s o t r u e when c o n s i d e r i n g mean v a l u e s o f t o t a l e n e r g y d i s s i p a t e d f o r v a r i o u s c o m b i n a t i o n s o f r e s u l t s . F i g . 3 . 3 shows mean v a l u e s o f t o t a l e n ergy d i s s i p a t e d f o r a l l e i g h t e e n d a t a s e t s i n t h i s s t u d y . In f i g . 3>A, means a r e p l o t t e d f o r systems s u b j e c t e d t o each i n d i v i d u a l e arthquake used i n t h e s t u d y . A n o t h e r Important f a c t o r a f f e c t e d b y t h e v a l u e o f YSR i s t h e d e v i a t i o n from t h e mean i n r e s u l t s o b t a i n e d f o r any g i v e n c o m b i n a t i o n s o f v a r i a b l e s . I n f i g . 3.5, 3.6, 3-7 and 3-8 i t can be seen t h a t t h i s d e v i a t i o n r e d u c e s w i t h YSR. Thus i t i s concluded' t h a t t h e e f f e c t s o f v a r y i n g m a t e r i a l t y p e o r damping d i m i n i s h e s as YSR i s r e d u c e d . 25 <x to CC CD UJ Z n Ujc £3 a: CD i n " a z; tr o — I 1 1 1 1 0.5 1.0 1.5 2.0 2.5 PERIOD (IN SECONDS) YSR 10.0 2.0 1.0 0.5 0.3 0.1 O -O & A *- * «^  e> * -1 1 1 1 1 0.5 1.0 1.5 2 J 2.S PERIOD (IN SECONDS) TAFT N69W ELRSTQ-PLA5TIC SYSTEMS ETA=0.01 PRCQIMA S16E. ELASTQ-PLRSTIC SYSTEMS ETfl=0.01 G.fl . MRX = 5.0493 FT/SEC/SEC G.R. MRX = 37.698 FT/SEC/SEC EL CENTRO N.S. TRI-LINERR SYSTEMS ETP=0.01 PARKFIELD N65E TRI-LINERR SYSTEMS ETfl=0.01 G.fl . MAX = 11.561 FT/SEC/SEC G.fl. MAX = 15.735 FT/SEC/SEC FIG. SOME 3,2 EFFECT OF YSR ON TOTAL ENERGY DISSIPATED. TYPICAL RESULTS. F I G , 3,3 E F F E C T OF YSR ON T O T A L ENERGY D I S S I P A T E D , MEANS FOR 18 COMBINATIONS. T O T A L E N E R G Y / U N I T M A S S ( / G . A . M A X O.Q 0.02 0.04 0.05 0.08 2 -n m >—• > o co > o m x T l -n m m > o 70 — l —I X o O T l cr > -< 7; co m TO >o o m z: o o H x> o o —\ - > m z m o -< co co > H m o Q 2 o - H 73 -Cr ft m r 0 Pi 0 I o _ TD m o — . Q u i • CO tn o o o Ln T J m 70 o CD m Q u i a in b ro In 0.1 T O T A L E N E R G Y / U N I T M A S S l / G . A . M A X ) 0.0 0.02 0.04 0.06 0.08 _L 0.1 LZ 3 - 4 ( C ) P A R I K F I E L O W<2>5>&. 3>-«* (d) ^ C O I M A Slc3>&. F I G . 3.4 CONTINUED (V) 0t> 29 cr LD YSR = 10.0 ex ZD >— U cr LD V. in •• tn" LD UJ — i 1 1 r 0.5 1.0 1.5 2.0 PERIOD (IN SECONDS) t YSR = 1.0 i 2.5 YSR =2.0 — i 1 1 1— 0.5 1.0 1.5 2.0 PERIOD (IN SECONDS) — i 2.5 0.0 I 1 1 1 0.5 1.0 1.5 2.0 PERIOD (IN SECONDS) 2.5 YSR =0.5 o.o CM YSR " 0.3 ? 8 cr o ~ ~ i 1 1 1— PERIOD (IN SECONDS) 2.S —1 1 1 1 0.5 1.0 1.5 2.0 PERIOD (IN SECONDS) YSR - 0.1 i 2.S — i 1 1 — r 1 0.5 1.0 1.5 2.0 2.5 PERIOD (IN SECONDS) F I G . 3.5 M E A N S A N D D E V I A T I O N S I N T O T A L E N E R G Y D I S S I P A T E D . F O R A L L S Y S T E M S S U B J E C T E D T O T A F T N69W. 30 FIG FOR . 3.6 MEANS AND DEVIATIONS IN TOTAL ENERGY DISSIPATED. ALL SYSTEMS SUBJECTED TO EL CENTRO N.S. YSR = 10.0 - YSR =2.0 F I G . 3,7 MEANS AND DEVIAT IONS IN TOTAL ENERGY D I S S I P A T E D . FOR A L L SYSTEMS S U B J E C T E D TO P A R K F I E L D Nbbt. F I G . 3.8 MEANS AND DEVIAT IONS IN TOTAL ENERGY D I S S I P A T E D , FOR A L L SYSTEMS S U B J E C T E D TO PACOIMA S16E. 3.2.2 E f f e c t o f E a r t h q u a k e R e c o r d on T o t a l E n ergy D i s s i p a t e d The q u a n t i t y o f t o t a l e n ergy d i s s i p a t e d b y t h e systems s t u d i e d i s found t o be e f f e c t e d b y two f a c t o r s ' a s s o c i a t e d w i t h t h e ground m o t i o n . These f a c t o r s a r e t h e s t r e n g t h and f r e q u e n c y c o n t e n t o f t h e ground m o t i o n . I n t h i s s t u d y t h e earthquake s t r e n g t h i s assumed p r o p o r t i o n a l t o t h e square Qf the- maximum peak ground a c c e l e r a t i o n -"The p e r i o d s a t which maximum and minimum v a l u e s o f t o t a l e n e r g y ' d i s s i p a t e d o c c u r depends on t h e e n e r g y / f r e q u e n c y c o n t e n t o f t h e eart h q u a k e r e c o r d used. F o r t h e T a f t , E l C e n t r o and P a r k f i e l d r e c o r d s t h e maxima o c c u r a t p e r i o d s between 0.5 and 1.0 Sec. Howdver, i n the Pacoima S16E • r e c o r d t h e maxima o c c u r a t p e r i o d s between 1.5 and 2.0 Sees. A l s o t h i s r e c o r d c o n t a i n s r e m a r k a b l y l i t t l e e n e r g y f o r p e r i o d s l e s s t h a n 1.0 Sec. .Because o f t h i s t h e Pacoima S16E s p e c t r a do n o t f i t i n t o t h e g e n e r a l p a t t e r n f o r s p e c t r a shape e x h i b i t e d b y t h e t h r e e o t h e r ground m o t i o n r e c o r d s . These e f f e c t s a r e shown i n f i g s . 3 - 9 and 3.10. * The square o f a maximum peak ground a c c e l e r a t i o n e n t e r s , s i n c e i n n o r m a l i z i n g t h e e a r t h q u a k e r e c o r d , b y d i v i d i n g b y t h e peak ground a c c e l e r a t i o n p r i o r t o c a l c u l a t i n g t h e r e s p o n s e o f t h e systems, a l l r e s p o n s e p a r a m e t e r s a r e s c a l e d by t h i s f a c t o r . The energy, b e i n g a p r o d u c t o f f o r c e and d i s p l a c e m e n t , o r v e l o c i t y s q u a r e d , t h u s b e-comes r e d u c e d by t h e peak ground a c c e l e r a t i o n s q u a r e d . 34 FIG, 3.9 EFFECT OF EARTHQUAKE RECORD ON TOTAL ENERGY DISSIPATED. ELASTO-PLASTIC SYSTEMS WITH 1.0% VISCOUS DAMPING. 35 F I G , 3,10 E F F E C T OF EARTHQUAKE RECORD ON T O T A L ENERGY D I S S I P A T E D , T R I - L I N E A R SYSTEMS WITH 1.0% VISCOUS DAMPING, 3.2.3 E f f e c t o f M a t e r i a l H y s t e r e t i c P r o p e r t i e s on T o t a l E n e r g y D i s s i p a t e d . The e f f e c t o f v a r y i n g m a t e r i a l H y s t e r e t i c p r o p e r t i e s on t o t a l e n e r g y d i s s i p a t e d by systems can be seen i n f i g s . 3.11, 3-12, 3-13 and 3-14. F o r t h e s t r o n g e s t systems c o n s i d e r e d (YSR = 10.0) t h e t o t a l e n ergy d i s s i p a t e d i s i d e n t i c a l f o r a l l m a t e r i a l t y p e s . T h i s i s because a l l t h e s e systems remained w i t h i n t h e e l a s t i c range and no n o n - l i n e a r e f f e c t s i n f l u e n c e d t h e r e s p o n s e . In the r e s p o n s e o f t h e weaker systems t h e e f f e c t o f t h e d i f f e r e n c e s i n h y s t e r e t i c b e h a v i o u r i s n o t i c e a b l e . The t o t a l e n e r g y d i s s i p a t e d can v a r y by up t o 50% f o r some i n d i v i d u a l systems. F o r t h e weakest systems, YSR = 0.1, the T r i - l i n e a r systems form t h e l o w e r bound. T r i - l i n e a r systems g e n e r a l l y seem t o be l e s s i n f l u e n c e d by t h e e n e r g y f r e q u e n c y c o n t e n t o f t h e e a r t h q u a k e r e c o r d . The v a r i a t i o n s caused by d i f f e r e n t h y s t e r e t i c b e h a v i o u r do not e f f e c t t h e g e n e r a l p a t t e r n and shape o f t h e s p e c t r a . F I G . 3.11 E F F E C T OF M A T E R I A L H Y S T E R S I S ON T O T A L ENERGY D I S S I P A T E D . SYSTEMS S U B J E C T E D TO T A F T WITH 1% V ISCOUS DAMPING. F I G . 3.12 E F F E C T OF M A T E R I A L H Y S T E R S I S ON T O T A L ENERGY D I S S I P A T E D . SYSTEMS S U B J E C T E D TO E L CENTRO N WITH 1% V ISCOUS DAMPING. YSR = 10.0 YSR =2.0 FIG. 3.13 EFFECT OF MATERIAL HYSTERSIS ON TOTAL ENERGY DISSIPATED. SYSTEMS SUBJECTED TO PARKFIELD WITH 1% VISCOUS DAMPING. YSR = 10.0 YSR = 2.0 • i 1 1 1 1 1 oH 1 1 1 1 1 0.0 0.5 1.0 1.5 2.0 2.5 0.0 0.5 1 .0 1.5 2.0 2.5 PERIOD (IN SECONDS) PERI00 (IN SECONDS) F I G , 3,14 E F F E C T OF M A T E R I A L H Y S T E R S I S ON T O T A L ENERGY D I S S I P A T E D , SYSTEMS S U B J E C T E D TO PACOIMA b. WITH 1% V ISCOUS DAMPING. 41 '3.2.4. E f f e c t o f V i s c o u s Damping on T o t a l E n e r g y D i s s i p a t e d The e f f e c t o f v a r i n g amounts o f v i s c o u s damping on th e t o t a l e n e r g y d i s s i p a t e d i s shown i n f i g s . 3-15, 3.16, 3.17 and 3.18. The g e n e r a l e f f e c t o f i n c r e a s i n g damping from 0 t o 5% i s seen t o be s m a l l and o f t h e o r d e r o f 105? t o 20% f o r t h e m a j o r i t y o f systems s t u d i e d . The e f f e c t o f damping i s more i m p o r t a n t f o r t h e s t r o n g e r systems. F o r one o r two i s o l a t e d systems i n t h i s s t r o n g range t h e e f f e c t o f i n c r e a s i n g damping from 0% t o 1% can i n c r e a s e t h e t o t a l e n e r g y d i s s i p a t e d b y a f a c t o r o f f o u r . However t h e s e a r e e x c e p t i o n s t h a t do n o t f i t i n t o t h e g e n e r a l p a t t e r n p r o d u c e d i n t h e m a j o r i t y o f systems. F o r systems h a v i n g YSR l e s s t h a n 1.0 t h e e f f e c t o f v a r y i n g damping i s v e r y s m a l l . F o r t h i s range a l s o t h e systems h a v i n g h i g h e r v a l u e s o f v i s c o u s damping form an upper bound on t h e t o t a l e n e r g y d i s s i p a t e d . 42 FIG, 3,15 EFFECT OF VISCOUS DAMPING ON TOTAL ENERGY DISSIPATED, FOR ALL SYSTEMS SUBJECTED TO TAFT N69W, 43 F I G . 3.16 E F F E C T OF V ISCOUS DAMPING ON T O T A L ENERGY D I S S I P A T E D . FOR A L L SYSTEMS S U B J E C T E D TO E L CENTRO N . S . 44 F I G , 3.17 E F F E C T OF V ISCOUS DAMPING 0 FOR A L L SYSTEMS S U B J E C T E D TO P A R K F I E L D N T O T A L ENERGY D I S S I P A T E D . % 5 E . 45 F I G i 3.18 E F F E C T OF V ISCOUS DAMPING ON T O T A L ENERGY D I S S I P A T E D . FOR A L L SYSTEMS S U B J E C T E D TO PACOIMA S16E. 46 3.3 Distribution of Energy Within the Systems Of equal importance to the amount of total energy to be dissipated within the system, i s the distribution of the energy to the various d i s s i -pating mechanisms which make up the system. There are two principal d i s s i -pating mechanisms i n the systems considered i n this present study. These are the energy dissipated by the hysteretic behaviour of the spring elements, and the energy dissipated by viscous dampers. The distribution of the to t a l energy dissipated to these mechanisms, and the manner and extent to which the distribution i s affected by changes i n the input parameters i s studied i n this section. 3.3.1. Effect of Yield Strength Ratio on Distribution of Energy Within Systems The general effect of reducing the YSR i s to increase the percent-age of energy dissipated by hysteretic behaviour. However this general trend needs to be qualified for systems with YSR less than 1.0 depending on hysteretic properties, damping and period. These are discussed further in sections 3.3.3 and 3.3-4. These trends can be seen for individual systems in f i g . 3.19 and appendix A. When considering means for a number of combinations the trends s t i l l hold good as can be seen in figs. 3-20 and 3.21. -< I—4 m r~ a CO —I TO m CD —I T. TO > ENERGY DISPTD BY SPRING(/OF TOTAL) 0.0 25.0 50.0 75.3 iOO'.O J 1 I ! ~0 m a. O I CO m CJln o / ____ X..*, <3»^  I : \ s S X-' 1 * -o x> o a in CD •c m i— D CO 3D CO o CO cn co 1! a ENERGY DISPTD BY SPRING(/OF T0TRL 0.0 25.0 50.0 75.0 J L ) 100.0 J o O O a \—4 CO H ;o W —I 1—4 O 2! O T l m : z m ?a CD X I—< CO -< CO —I m 2 co ENERGY o.o DISPTD BY SPRINGUQF TQTP.U 25.0 50.0 75.0 100.0 ENERGY DISPTD BY SPRING(/OF TOTAL) Q.Q 25.0 50.0 75.0 IQO.O rn ^3 8r$ CO m • a, a co T 4 JL _ i _ (3 cr^" \ i; \ • ii a a o — co In a a <3 \ \ t\) a eg : i \ 4> •f * ¥ 4> <!> * in m n m C3 z: co TO I rn ^3 CO -< CO —» n :^ co m —( D II o LI cr t— o c n c n cc°. UJo_| »—^ c n >— 00 o ,_,u->_| I— 0_ O >— CD Z o .... rse • o 1 r ' \ / / \ / \ / \ I \ \ \ \ \ \ \ V.. IQ-O , , 0.0 0.5 1.0 1.5 2.0 PERIOD (IN SECONDS) 2.5 F I G . 3,20 E F F E C T OF YSR ON D I S T R I B U T I O N OF ENERGY WITHIN S Y S T E M S . MEANS FOR A L L E L A S T O - P L A S T I C SYSTEMS WITH 1% DAMPING. o CX I— c n i — i c n U J UJa. h- m c n 00c t— CL a CD Z a Iff ^ '/ / / I O O o-l...—• 1 1 r— 0.5 1.0 1.5 2.0 PERIOD (IN SECONDS) 2.5 F I G . 3.21 E F F E C T OF YSR ON D I S T R I B U T I O N OF ENERGY WITHIN S Y S T E M S . MEANS FOR A L L T R I - L I N E A R SYSTEMS WITH 1% DAMPING. 49 3 . 3 . 2 E f f e c t of Earthquake Record on D i s t r i b u t i o n of Energy Within Systems The e f f e c t of d i f f e r e n t earthquake records on the d i s t r i b u t i o n of energy wi t h i n the systems i s shown i n f i g s . 3 . 2 2 and 3 . 2 3 . - The d i s t r i b u t i o n i s seen to be r e l a t i v e l y independent of the earthquake record used. In most instances i t i s observed that the r e s u l t s f o r the Pacoima S16E record form a lower bound. The curves representing the percentage of t o t a l energy d i s s i p a t e d i n h y s t e r e s i s are seen to f a l l i n t o w e l l defined patterns f o r each YSR and h y s t e r e t i c property used. This f a c t i s of great importance when considering the form of a design procedure based on energy consider-at i o n s , and i s elaborated upon i n s e c t i o n 3 . 9 . 50 TBFT N69U PfiCOHH 516E EL CENTRO N S PARKFIELD N65E YSR - io.Q => YSR " 2.0 3.5 YSR = 0.3 o YSR = 0.1 .5 F I G . 3.22 E F F E C T OF EARTHQUAKE RECORD ON D I S T R I B U T I O N OF ENERGY WITHIN E L A S T O - P L A S T I C SYSTEMS WITH 1% V ISCOUS DAMPING. TRFT N69W PfCOMfl S16E EL CENTRO N. S. PARKFIELD N65E YSR =10 .0 =! ' YSR = 2.0 YSR = 1 . 0 =: YSR = 0.5 F I G . 3.23 E F F E C T OF EARTHQUAKE RECORD ON D I S T R I B U T I O N OF EN WITHIN T R I - L I N E A R SYSTEMS WITH 1% V ISCOUS DAMPING. 52 3.3*3. E f f e c t o f M a t e r i a l H y s t e r e t i c P r o p e r t i e s on D i s t r i b u t i o n o f E n e r g y W i t h i n Systems A c o m p a r a t i v e s t u d y o f t h e p r o p o r t i o n s o f t o t a l energy d i s s i p a t e d b y systems w i t h d i f f e r e n t h y s t e r e t i c p r o p e r t i e s i s p r e s e n t e d i n f i g s . 3-24, 3.25, 3.26 and 3.27. F o r t h e s t r o n g e r systems, i e YSR = 2 . 0 and 1.0, t h e T r i - l i n e a r systems d i s s i p a t e more e n e r g y i n h y u t e r e s i s t h a n t h e E l a s t o - P l a s t i c and Ramberg - Osgood systems. T h i s d i f f e r e n c e r e d u c e s w i t h r e d u c t i o n i n YSR. F o r systems w i t h YSR = 0.5 and 0 .3 t h e T r i - l i n e a r systems f o r m t h e u p p e r bound f o r p e r i o d s g r e a t e r t h a n 1.0 - 1.5 s e e s . The r e v e r s e o f t h i s i s t r u e f o r systems w i t h p e r i o d s l e s s t h a n 1.0 - 1.5 s e e s , and t h e E l a s t o -P l a s t i c systems f o r m t h e -upper hound. ; The upper bound f o r t h e systems w i t h YSR = 0.1 i s seen t o be t h e E l a s t o - P l a s t i c and Ramberg - Osgood sy s t e m s . These o b s e r v a t i o n s a r e t r u e f o r t h e r e s u l t s o f a l l f o u r e a r t h -quake r e c o r d s s t u d i e d . The Ramberg - Osgood and E l a s t o - P l a s t i c systems d i s s i p a t e e s s e n t i a l l y t h e same p r o p o r t i o n s o f energy. T h i s i s e s p e c i a l l y t r u e f o r weaker systems where i t can be o b s e r v e d t h a t t h e two a r e f o r a l l p r a c -t i c a l p u r p o s e s i d e n t i c a l . . The d i f f e r e n c e between t h e r e s u l t s p r o d u c e d b y t h e d e g r a d i n g and n o n - d e g r a d i n g s t i f f n e s s systems i s o b s e r v e d t o be such t h a t any c o n s i d e r -a t i o n o f p e r c e n t a g e o f t o t a l e n e r g y d i s s i p a t e d must i n c l u d e t h e h y s t e r e t i c p r o p e r t i e s o f t h e s p r i n g elements as a v a r i a b l e . YSR = 2.0 C3 F I G . 3.24 E F F E C T OF M A T E R I A L H Y S T E R E T I C P R O P E R T I E S ON D I S T R I B U T I O N OF ENERGY WITHIN S Y S T E M S . FOR A L L SYSTEMS S U B J E C T E D TO T A F T N69W WITH 1% DAMPING. F I G . 3.25 E F F E C T OF M A T E R I A L H Y S T E R E T I C P R O P E R T I E S ON D I S T R I B U T I O N OF ENERGY WITHIN S Y S T E M S . FOR A L L SYSTEMS S U B J E C T E D TO E L CENTRO N . S . WITH 1% DAMPING. F I G , 3,26 E F F E C T OF M A T E R I A L H Y S T E R E T I C P R O P E R T I E S ON D I S T R I B U T I O N OF ENERGY WITHIN S Y S T E M S . FOR A L L SYSTEMS S U B J E C T E D TO P A R K F I E L D N65E WITH 1% DAMPING. YSR = 2.0 o PERIOD (IN SECONDS] F T G 3 27 EFFECT OF'MATERIAL HYSTERETIC PROPERTIES ON DISTRIBUTTON OF ENERGY WITHIN SYSTEMS. FOR ALL SYSTEMS SUBJECTED TO PACOIMA S16E WITH 1% DAMPING, 57 3.3»4- Effect of Viscous Damping on Distribution df Energy Within Systems The effect of varying amounts of viscous damping on the distribut-ion of the total energy to be dissipated within systems i s shown i n figs. 3.28, 3.29, 3-30 and 3-31-It i s clear that increased amounts of viscous damping reduce the proportion of total energy dissipated by hysteresis. The extent of the reduction depends also on the hysteretic properties used and the period of the system. The effect of variations i n damping i s considerably greater for the stronger systems having YSR = 1.0 or greater. Also the Tri-linear systems are more sensitive to changes i n damping ra t i o . 5 8 YSR = 2.0 FIG, 3.28 EFFECT OF VISCOUS DAMP ING ON DISTRIBUTION OF ENERGY WITHIN SYSTEMS SUBJECTED TO TAFT N69E. YSR = 2.0 F I G , 3,29 E F F E C T OF V ISCOUS DAMPING ON D I S T R I B U T I O N OF ENERGY WITHIN SYSTEMS S U B J E C T E D TO E L CENTRO N . S . FIG. 3.30 EFFECT OF VISCOUS DAMPING ON DISTRIBUTION OF ENERGY WITHIN SYSTEMS SUBJECTED TO PARKFIELD N65E. 61 FIG, 3,31 EFFECT OF VISCOUS DAMPING ON DISTRIBUTION OF ENERGY WITHIN SYSTEMS SUBJECTED TO PACOIMA S16E. 62 3 .3 .5 Implications of these Results The d i s t r i b u t i o n of the t o t a l energy d i s s i p a t e d by the system, between the h y s t e r e t i c and viscous mechanisms, i s s i g n i f i c a n t l y i n s e n s i t -ive to the i n d i v i d u a l earthquake record used, as discussed i n s e c t i o n 3 . 3 - 2 . This means that i t w i l l be p o s s i b l e to construct d i s t r i b u t i o n data which can be applied to a l l earthquakes. This i s an important r e s u l t i n terms of the design method being developed i n t h i s study. The r e s u l t s studied i n sections 3 -3 .3 and 3-3-4- provide valuable i n s i g h t i n t o the way i n which the systems r e s i s t the ground motions. The two energy d i s s i p a t i n g mechanisms can be thought of as two r e s i s t i n g elements i n p a r a l l e l , and are shown schematically i n f i g . 3-32. / f t ) HYSTERETIC -VWNA-E>AN/IR1KJ<3 v i s c o u s DAN/1PINJC3. V / FIG. 3.32 SCHEMATIC OF TWIN PARALLEL RESISTING ELEMENT MODEL, I f t h e h y s t e r e t i c element has l a r g e e nergy d i s s i p a t i n g c a p a c i t y and i t s YSR i s such t h a t i t i s t h e p r i n i c i p a l d i s s i p a t i n g d e v i c e t h e n t h e m a j o r i t y o f t h e energy w i l l be d i s s i p a t e d by h y s t e r e s i s . F o r t h e s t r o n g e r systems with. YSR >"1.0, whi c h r e m a i n e s s e n t i a l l y e l a s t i c d u r i n g t h e r e s p o n s e , t h e p r i n c i p a l d i s s i p a t i n g d e v i c e i s t h e v i s c o u s damping and hence the r e s p o n s e i s g o i n g t o be g r e a t l y i n f l u e n c e d by t h e p e r c e n t a g e o f c r i t i c a l damping used. The weaker systems, h a v i n g YSR < 1.0, p r o v i d e t h e h y s t e r e t i c mechanisms as p r i n c i p a l d i s s i p a t i n g d e v i c e s and t h e r e s p o n s e i s l e s s s e n s i t i v e t o v a r i a t i o n i n damping, p r o v i d e d t h a t t h e b a l a n c e between t h e v i s c o u s and h y s t e r e t i c dampers i s n o t a f f e c t e d g r e a t l y . However i n the case o f t h e T r i - l i n e a r s t i f f n e s s d e g r a d i n g model, t h e o r i g i n a l e n ergy d i s s i p a t i n g c a p a c i t y may w e l l be such t h a t a t the s t a r t o f the r e s p o n s e i t i s t h e p r i n c i p a l d i s s i p a t i n g d e v i c e . D u r i n g t h e r e s p o n s e , as the s t i f f n e s s degrades, t h i s c a p a c i t y i s r e d u c e d t o such an e x t e n t t h a t the v i s c o u s dampers take o v e r as the p r i n c i p a l d i s s i p a t i n g d e v i c e , and the p e r c e n t a g e o f e n e r g y d i s s i p a t e d b y h y s t e r e s i s i s r e d u c e d i n t h e s e systems. T l i i s becomes e v i d e n t i n t h e f a c t t h a t the E l a s t o - P l a s t i c and Ramberg-Osgood systems a r e l e s s s e n s i t i v e t o v a r i a t i o n i n damping t h a n t h e T r i - l i n e a r systems. 64 3.4 Spectral Displacements The displacement response of single degree of freedom systems to earthquake ground motions has been the subject of many exhaustive studies in the past and the results are well known. In this section the results obtained i n this study are discussed b r i e f l y , with particular emphasis placed on the effect of varying the YSR. The following effects are dis-cussed with reference to figs . 3-33, 3-34, 3-35 and 3.36. 3.4.1 Effect of Varying Yield Strength Ratio on Spectral Displacements The effect of varying yield strength ratio for a l l systems with YSR between 10.0 and 1.0 i s small. However as the YSR i s further reduced there i s an increasing difference between the elastic and non-linear dis-placement response. For systems with YSR = 0.5 to 0.3 i t i s seen that the non-linear response i s i n general similar to the elastic response for systems having periods greater than 0.5 sec. Ultimately at YSR = 0.1 there i s only agreement between the elastic and non-linear response i n the long period systems, ( i e . period equal to 1.5 Sec. or greater.) The shorter period systems having displacements far larger than would be predicted by the equal displacement criterion. The extent of these effects i s influenced by material type, earth-quake record type, and proportion of viscous damping used. However the effects of varying YSR are similar for a l l systems. 65 3.4.2 Effect of Earthquake Record on Spectral Displacement It i s observed that the equal displacement criterion i s defined more clearly i n the response of systems subjected to the far f i e l d ground motion records, ie Taft N69W and E l Centro N.S. In the response of systems subjected to the near source impulsive type of motions, Pacoima SI6E and Parkfield N65E, the difference between the non-linear and elastic displace-ment response i s greater than for far f i e l d records. 3.4.3 Effect of Material Hysteretic Properties on Spectral Displacements The effect of material hysteretic properties on the spectral displacements i s most pronounced i n the systems having YSR =0.5 and lower. For systems stronger than this the displacements are close to the elastic system displacements. In the systems having YSR = 0.5 and lower with periods less than 1.5 s e c , i t i s seen that the displacements for the Tri-linear systems are greater than the elastic system displacements. For these systems the deviation from the equal displacement.criterion i s greater than for systems with Elasto- Plastic or Ramberg-Osgood hysteresis properties. 66 3 . 4 . 4 E f f e c t of Viscous Damping on S p e c t r a l Displacements The e f f e c t of viscous damping on the s p e c t r a l displacements i s well known and the f o l l o w i n g observations are i n agreement with previous studies. Increasing viscous damping from 0% to 1% of c r i t i c a l had a general e f f e c t o f reducing displacements by 10% to 20%. Further i n c r e a s i n g viscous damping from 1% to 5% of c r i t i c a l reduced displacements by a f u r t h e r 20% to 30%. Another w e l l defined e f f e c t of i n c r e a s i n g damping was to improve the compliance of the non-linear systems to the equal displacement c r i t e r i o n . This was found to be the case f o r systems with a l l y i e l d strength values. TflFT N69V RflNBERG-OSGOOD SYSTEMS ETR=0.01 G.H. MRX = 5.0493 FT/SEC/SEC TRFT N69W ELRSTO-PLOSTIC SYSTEMS ETfl=0.01 G.Ft. MPX = 5.0493 FT/SEC/SEC -| 1 r — — — T 0.5 1.0 IS 2.0 PERIOD UN SEC0ND5) -i 1 1 r 0.5 1.0 l .S 2.0 PERIOD (IN SECONDS) TflFT N69W ELflSTO-PLflSIIC SYSTEMS ETH-0.05 G.R. MRX = 5.0493 BTT/SEC/SEC ^ - i 1 1 r 0.5 1.0 1.5 2.0 P E R I O D U N S E C O N D S ) F I G , 3.33 S P E C T R A L D ISPLACEMENTS FOR SYSTEMS S U B J E C T E D TO T A F T N69W. F I G . 3 ,34 S P E C T R A L D I S P L A C E M E N T S FOR SYSTEMS S U B J E C T E D TO E L CENTRO N . S . PARKFIELD N65E TRI-LINERR SYSTEMS ETfl=0.01 G.R. MAX = 15.735 FT/SEC/SEC PERIOD (IN SECONDS) F I G . 3 .35 C O N T I N U E D pflCOIMfl S16E TRI-LINERR SYSTEMS ETflrO.Ol G.fl. MAX = 37.698 FT/SEC/SEC F I G . 3.36 C O N T I N U E D 3. 5 D i s p l a c e m e n t D u c t i l i t i e s D i s p l a c e m e n t d u c t i l i t y o f n o n - l i n e a r s i n g l e degree o f freedom systems has been s t u d i e d i n g r e a t . d e p t h o v e r t h e l a s t twenty y e a r s , and t h e r e s u l t s a r e w e l l known. They a r e among the most i m p o r t a n t p r o d u c e d i n e a r t h q u a k e e n g i n e e r i n g t o d a t e . Here d i s p l a c e m e n t d u c t i l i t y i s d i s c u s s e d i n the c o n t e x t o f the y i e l d s t r e n g t h r a t i o parameter. The f o l l o w i n g e f f e c t s a r e d i s c u s s e d w i t h r e f e r e n c e t o f i g s . 3-37, 3.38, 3-39 and 3-40. 76 3.5.1 E f f e c t o f Y i e l d S t r e n g t h R a t i o on D i s p l a c e m e n t D u c t i l i t y R e d u c t i o n i n y i e l d s t r e n g t h r a t i o p roduces i n a l l systems s t u d i e d an i n c r e a s e i n d i s p l a c e m e n t d u c t i l i t y . T h i s e f f e c t i s g r e a t e r f o r t h e s h o r t p e r i o d systems t h a n the l o n g e r p e r i o d ones. D i s p l a c e m e n t d u c t i l i t i e s d e c r e a s e w i t h i n c r e a s i n g p e r i o d s . y . T h i s e f f e c t i s seen t o be t h e most c l e a r -l y d e f i n e d and c o n s i s t e n t o f a l l a s p e c t s o f r e s p o n s e i n v e s t i g a t e d i n t h i s s t u d y . The f o l l o w i n g g e n e r a l o b s e r v a t i o n s a r e found f o r e a c h y i e l d s t r e n g t h r a t i o . YSR = 10.0 No systems y i e l d e d . ( A l l d i s p l a c e m e n t d u c t i l i t i e s a r e < 1.0) YSR = .2.0 M a j o r i t y o f systems w i t h p e r i o d s l e s s t h a n 1.0 s e c . y i e l d e d . O t h e r s were c l o s e t o y i e l d d u c t i l i t y . No major, y i e l d e x c u r s i o n s o c c u r e d . Max. d i s p l a c e m e n t d u c t i l i t y i s o f o r d e r 2 o r 3. YSR = 1 . 0 Most systems w i t h p e r i o d s l e s s t h a n 1.5 s e c . y i e l d e d . Systems w i t h v e r y s h o r t p e r i o d s showed d i s p l a c e m e n t d u c t i l i t i e s o f t h e o r d e r 6 - 1 0 f o r p e r i o d s o f 0.1 s e c . YSR = 0 . 5 A l l systems y i e l d e d e x c e p t t h e T r i - l i n e a r systems w i t h p e r i o d s g r e a t e r t h a n 1.5 s e c . S h o r t p e r i o d systems w i t h T = 0.1 and 0.3 s e c . show e x t r e m e l y h i g h d i s p l a c e m e n t d u c t i l i t y i e >10. A l l o t h e r d u c t i l i t i e s were l e s s t h a n 10. YSR = 0 . 3 A l l systems y i e l d e d e x c e p t t h e T r i - l i n e a r systems w i t h p e r i o d s g r e a t e r t h a n 1.5 s e e s . Systems w i t h p e r i o d s l e s s t h a n 0.8 sec."show d i s p l a c e m e n t ' d u c t i l i t i e s g r e a t e r t h a n • 6. Some s h o r t p e r i o d , systems have huge d u c t i l i t i e s i e . > 100. YSR = 0 . 1 A l l systems y i e l d e d . Systems w i t h p e r i o d s l e s s t h a n 1.5 s e c . show e x c e s s i v e d i s p l a c e m e n t d u c t i l i t y . 3.5.2 E f f e c t o f Earthquake R e c o r d on D i s p l a c e m e n t D u c t i l i t y The v a r i a t i o n i n d i s p l a c e m e n t d u c t i l i t i e s f o r systems s u b j e c t e d t o the f o u r e a r t h q u a k e r e c o r d s used i n t h i s s t u d y was f o u n d t o be s m a l l . T h i s i n d i c a t e s t h a t when the e a r t h q u a k e r e c o r d and s t r u c t u r a l systems a r e n o n - d i m e n s i o n a l i s e d as i n t h i s s t u d y t h e d i s p l a c e m e n t d u c t i l i t y i s n o t s i g n i f i c a n t l y a f f e c t e d by d i f f e r e n t ground m o t i o n i n p u t s . 3.5.3 E f f e c t o f V a r y i n g M a t e r i a l H y s t e r e t i c P r o p e r t i e s on D i s p l a c e m e n t D u c t i l i t y D i s p l a c e m e n t d u c t i l i t i e s f o r t h e Ramberg-Osgood and E l a s t o - P l a s t i c systems a r e a l m o s t i d e n t i c a l e x c e p t f o r systems w i t h YSR = 0 . 3 and 0.1, and a l l systems w i t h p e r i o d l e s s t h a n 0.5 s e c . F o r t h e s e two groups o f systems t h e d i f f e r e n c e s can be o f t h e o r d e r 50 - 100$, w i t h t h e E l a s t o -P l a s t i c systems h a v i n g t h e l a r g e r d i s p l a c e m e n t d u c t i l i t i e s . The T r i - l i n e a r systems resp o n d e d t o t h e ground motions w i t h c o n s i d e r a b l y l ower d i s p l a c e m e n t d u c t i l i t i e s t h a n the systems w i t h E l a s t o -P l a s t i c h y s t e r e t i c p r o p e r t i e s . The d i s p l a c e m e n t d u c t i l i t i e s f o r the T r i - l i n e a r systems i s g e n e r a l l y o f the o r d e r o f 1/2 t o 1/4 o f t h e E l a s t o -P l a s t i c d i s p l a c e m e n t d u c t i l i t i e s . However, i t must be n o t e d t h a t t h e y i e l d d i s p l a c e m e n t f o r the T r i - l i n e a r systems i s g r e a t e r t h a n t h a t f o r t h e comparable E l a s t o - P l a s t i c systems. See f i g . 3.1. 78 3.5.4 E f f e c t o f ' V i s c o u s Damping on D i s p l a c e m e n t D u c t i l i t y I t i s seen t h a t i n c r e a s i n g t h e p r o p o r t i o n o f v i s c o u s damping r e d u c e s the d i s p l a c e m e n t o f a l l systems. The e x t e n t o f t h i s r e d u c t i o n i s dependent upon the h y s t e r e t i c p r o p e r t i e s , y i e l d s t r e n g t h r a t i o and p e r i o d o f t h e systems. The e f f e c t i s f o u nd t o be g r e a t e r f o r t h e s h o r t p e r i o d systems and f o r t h e E l a s t o - P l a s t i c s y stems. For E l a s t o - P l a s t i c systems an i n c r e a s e i n damping from 0 t o 1% o f c r i t i c a l p r o d u c e d a 20$ r e d u c t i o n i n d i s p l a c e m e n t d u c t i l i t i e s f o r systems w i t h YSR < 1.0 and 20 - 80% f o r systems w i t h YSR > 1.0 F u r t h e r i n c r e a s i n g damping f r o m 1% t o 5% shows g e n e r a l r e d u c t i o n o f 20 - 50% f o r systems w i t h p e r i o d > 1.0 and a r e d u c t i o n o f 50 - 100$ f o r systems w i t h p e r i o d < 1.0. F o r T r i - l i n e a r systems an i n c r e a s e i n damping from 0 t o 1% o f c r i t i c a l p r o d u c e d a r e d u c t i o n i n d i s p l a c e m e n t d u c t i l i t i e s o f t h e o r d e r 10% - 20$ f o r systems w i t h p e r i o d > 1.0 s e c . and 20 - 100$ f o r systems w i t h p e r i o d < 1.0 s e c . F u r t h e r i n c r e a s i n g damping from 1$ t o 5$ shows a g e n e r a l r e d u c t i o n o f d i s p l a c e m e n t d u c t i l i t y o f 80$ f o r systems w i t h p e r i o d 0.5. T h i s r e d u c t i o n r e d u c e s t o 20$ f o r systems w i t h p e r i o d s o f 2.5 s e c . t h e t r a n s i t i o n b e i n g a p p r o x i m a t e l y l i n e a r . The systems w i t h YSR < 1.0 and p e r i o d s < 0.5 s e c . showed r e d u c t i o n s o f t h e o r d e r 1/2 t o 1/3. 3"-UJ 1: UJ CJ (Z TflFT N69V ELASTO-PLASTIC SYSTEMS ETA=0.01 •f G.A. MAX = 5.0493 FT/SEC/SEC YSR 10.0 e-2.0 A— 1.0 +-0.5 X-0.3 0.1 -€) -r 0.5 1.0 1.5 2.0 PERIOD (IN SECONDS) * — i O 3 " U J £ o <x TAFT N69W RAMBERG-05GOOD SYSTEMS ETA=0.01 G.A. MAX = 5.0493 FT/SEC/SEC YSR JO.O O-2.0 A— 1.0 -4-0.5 X-0.3 0.1 - 0 •K A * — X « v y \ \ -| r 0.S 1.0 1.5 2.0 PERIOD (IN SECONDS) 3 U J o FHFT N69W ELASTO-PLHSTIC SYSTEMS FTH-0.05 G.H. MAX - 5.0493 FT/SEC/5EC -O YSR 10.0 o— 2.0 A— 1 .0 H h 0.5 X * 0.3 O © 0.1 * \ \ -x--x~. +--+ '•*—x-/ i) (D I'J i <D 1|J 0.5 1.0 1.5 2.0 P E R I O D I I N S E C O N D S ! ""•X F I G . 3.37 D ISPLACEMENT D U C T I L I T I E S FOR SYSTEMS S U B J E C T E D TO T A F T N69W. TRFT N69W TRI-LINEAR SYSTEMS ETR=0.01 G.R. MRX = 5.0493 FT/SEC/SEC YSR 10.0 © O ; 2 0 A — A 1.0 H 1-0.5 X X \ 0.3 O-- » • '•. 0.1 •+ %\ \ X \ 1 W—O © i © — © $ 0 0.5 1.0 1.5 2 0 2.5 PERIOD (IN SECONDS) F I G . 3 fed & : m -* • >— 1— \\\ • » =>„,- \ > z<n- \«> 27. UJ o tc _l CL. \\ TRFT N69W TRI-LINERR 5Y5TEMS ETR-D 00 G.R. MRX = 5.0493 FT/SEC/SEC YSR 10.0 © © 2.0 A A 1.0 -I r 0.5 X- X 0.3 0.1 • \ \ \ \ © a i © — ® -0.5 1.0 1.5 2.0 PERIOD (IN SECONDS) CONTINUED 00 o -i r 0.5 1.0 1.5 2.0 PERIOD (IN SECONDS) EL CENTRO N.S. G.R. MRX RRMBERG-05G000 SYSTEMS 11.561 FT/SEC/SEC YSR 10.0 © O 1.0 H 1-0.5 . X- * 0.3 « 1 0.1 ETfl=0J0l X ^ \ » ^ • f t _ A ^ \ > * - — H - H - K * H (--___ + 0.0 -i 1 en O (P 0.5 1.0 1.5 2.0 PERIOD (IN SECONDS) -i r 0.5 1.0 1.5 2.0 PERIOD (IN SECONDS) F I G . 3,38 D I S P L A C E M E N T D U C T I L I T I E S FOR SYSTEMS S U B J E C T E D TO E L CENTRO N . S . 0 0 h-1 DISPLACEMENT DUCTILITY 3 S 7 10' s s 7 in* 3 5 7 10* _ l • * t i i 11 3 S 7 10* ' • ' o CB // / f' ft * •* t i f ? .* r > + x <4 o o o — r\> o t/i a) — cu oi o o o f t f t i 4 * + cn n rn o m U 50 CO -< CO DISPLACEMENT DUCTILITY 3 5 7 10' 3 5 7 10" ' i i . i i . i l 1—i I i I i i I 5 7 ID" ' '9 5 « - CB 1 > r 4 X V x p / * * <f * t i x 4 i i ' ; J . 4 x 4 O O O — O U> XJ — w CJI o a o 4 0 x 4- ti- cs CD C/) m o 1—'-• t_> B" I— U J 3; UJ ( J cr _ j a.. too PflRKFJELO N65E ELRSTO-PLRSTIC SYSTEMS ETR=0.0| G.fl. MRX = 15.735 FT/SEC/SEC X I YSR 10.0 ®-" 2.0 A-1.0 0.5 0.3 0.1 H + X- X O «> ^ \ * ' \ ^ v. A—A-X-. --x ^ A --0—e 1 1 r 0 5 1.0 IS 2.0 PERIOD (IN SECONDS) PRRKFIELO N65F. ELRSIO-PI.RSTIC SYSTEMS trfl"0.05 G.fl. MAX - 15. 735 FT.'SEC/SEC YSR in .0 O O 2.0 A 1.0 H H 0 5 x- - - — • -X 0.3 <t> 0.1 -?• OS I II I.J 2.0 PERIOD UN SECONDS) F I G . 3.39 D I S P L A C E M E N T D U C T I L I T I E S FOR SYSTEMS S U B J E C T E D TO P A R K F I E L D N65E. 03 m 33 d y ? x' ? .•¥• 5 o) a CD DISPLACEMENT DUCTILITY . . , „ 5 1 Id 1 s s 7 IO* J s 7 10* ' • i . i . . i 1 • 1  5 7 10* • -D> 4- X »' 1 / / ' / rf rf t x 6 + 4 x 4- A X II cn a o o — N> o cn ui " to cn a a o LO m o CO m o CO cn m — ^ -J f~ r< CJ — 50 co -« CO —I m 2 CO L O O O m o 5 7 10" DISPLACEMENT DUCTILITY S 7 10' 3 S 7 I0» S 7 I F 5 7 10* ' i i i i i 78 bd 5" 5 o 5 a.. bd PRCOIMfl Sl€E ELRSTO-PLRSTIC SYSTEHS ETA=0.01 G.fl. MRX = 37.698 FT/SEC/SEC YSR 10.0 2.0 1.0 0.5 0.3 0.1 © - -O X- X O G •*> W V -K. \ ~«5 X -i o — i —t——O CP 0.5 1.0 1.5 2.0 PERIOD !IN SECONDS) ~9 2.5 PflCOIMfl S16E ELA5T0-PLASI1C SYSTEMS ETfl-0 00 G.fl. MAX = 37.698 FT/5EC/SEC YSR 10.0 2.0 1 .0 0.5 X X 0.3 $ <J> • 0.5 1 0 1 5 P E R I O D I I N S E C O N D S ) F I G , 3.40 D I S P L A C E M E N T D U C T I L I T I E S FOR SYSTEMS S U B J E C T E D TO PACOIMA S16E. PACOIMA 516E TRI-LINEAR SYSTEMS ETA=0.01 G.R. MAX = 37.698 FT/SEC/SEC YSR 10.0 © © 2.0 A A 1.0 H F 0.5 X- X 0.3 • * 0.1 •+ ^ ^ -O - © — 9 — 8 — © T © — © =^-f9 0.5 1.0 I.S 2.0 2.5 PERIOD (IN SECONDS) F I G . 3 . PACOIMA S16E TRI-LINEAR SYSItMS E.lfl^U.05 o =>ir Q UJ T. UJ G.A. MAX = 37.698 f^SEO/SEX YSR 10.0 2.0 1 .0 0.5 0.3 0.1 © -V ' \ X - K ' A X - - X -X h-x x. o o cp—o cp o o 0.5 1.0 I .S 2.1) PERIOD I IN SECONDS I X CONTINUED M 87 3.6 Permanent S e t s The r e s u l t s o f t h i s s t u d y f o r permanent s e t s showed t h a t t h e r e was no c l e a r p a t t e r n t o t h i s r e s p o n s e parameter as was the case f o r d i s p l a c e m e n t s o r d i s p l a c e m e n t d u c t i l i t i e s . As p o i n t e d out i n s e c t i o n 2 .6 , t h e permanent s e t can be thought o f as t h e n e t r e s u l t o f t h e y i e l d e x c u r s i o n e x p e r i e n c e d by t h e system d u r i n g t h e r e s p o n s e . The l a c k o f p a t t e r n i n t h e r e s u l t s i n d i c a t e s t h a t t h e n e t e f f e c t o f t h e y i e l d e x c u r s i o n s i s v e r y s e n s i t i v e t o the n a t u r e and h i s t o r y o f t h e r e s p o n s e . While t h e r e i s no d e t a i l e d p a t t e r n i n t h e r e s u l t s , on a more g e n e r a l l e v e l i t can be s a i d t h a t t h e permanent s e t r e s u l t s do f o l l o w a s i m i l a r p a t t e r n t o t h e d i s p l a c e m e n t d u c t i l i t y r e s u l t s . I n f a c t i t i s p o s s i b l e t o say t h a t t h e permanent s e t s i n g e n e r a l a r e o f t h e o r d e r o f about 1/4 o f t h e d i s p l a c e m e n t d u c t i l i t y r e s u l t s . However i t i s emphasised t h a t t h e r e a r e wide f l u c t u a t i o n s from t h i s norm. The f o l l o w i n g e f f e c t s a r e d i s c u s s e d w i t h r e f e r e n c e t o f i g s . 3.41, 3.42, 3.43 and 3-44. 3.6.1 E f f e c t o f Y i e l d S t r e n g t h R a t i o on Permanent S e t s . R e d u c t i o n i n YSR has a g e n e r a l e f f e c t o f i n c r e a s i n g t h e permanent s e t o f t h e systems. However t h i s e f f e c t i s n o t c l e a r l y d e f i n e d as i s t h e case f o r o t h e r p a r a m e t e r s s t u d i e d . The e f f e c t o f r e d u c i n g t h e YSR i s g r e a t e r f o r t h e s h o r t p e r i o d systems t h a n f o r l o n g p e r i o d s ystems. The f o l l o w i n g g e n e r a l o b s e r v a t i o n s a r e found f o r eac h y i e l d s t r e n g t h r a t i o : YSR = 10 A l l permanent s e t s a r e o f an o r d e r l e s s t h a n 10% o f y i e l d d i s p l a c e m e n t . YSR = 2 D i t t o e x c e p t f o r some s h o r t p e r i o d systems where permanent s e t can be 50 - 60% o f y i e l d d i s p l a c e m e n t , and i n one case 100%. YSR = 1 S i g n i f i c a n t permanent s e t s i n many systems w i t h p e r i o d l e s s t h a n 0.8 s e c . Some v a l u e s up t o 3.0 t i m e s y i e l d d i s p l a c e m e n t . YSR = 0.5 D i t t o f o r p e r i o d s l e s s t h a n 1.5 s e c . Some peak v a l u e s up t o 3 0 . YSR = 0.3 D i t t o f o r p e r i o d s l e s s t h a n 2.0 s e c . Some peak v a l u e s up t o 100. YSR = 0 . 1 S i g n i f i c a n t permanent s e t s i n a l l systems. Some peak v a l u e s up t o 1000. 3.6.2 E f f e c t o f E a r t h q u a k e Record on Permanent S e t s The e f f e c t o f v a r y i n g the e a r t h q u a k e ground m o t i o n on permanent s e t s was f o u nd t o he s m a l l . T h i s r e s u l t i s s i m i l a r t o t h a t f o u n d f o r d i s p l a c e m e n t d u c t i l i t y . In g e n e r a l terms i t can he c o n c l u d e d t h a t t h e permanent s e t s o f systems n o n - d i m e n s i o n a l i s e d as I n t h i s s t u d y a r e not. s i g n f i c a n t l y a f f e c t e d by d i f f e r e n t ground m o t i o n i n p u t s . 3 . 6 . 3 ' E f f e c t o f M a t e r i a l H y s t e r e t i c P r o p e r t i e s on Permanent S e t s Permanent s e t s f o r t h e E l a s t o - P l a s t i c and Ramberg-Osgood systems show no s i g n i f i c a n t d i f f e r e n c e . The permanent s e t s f o r t h e T r i - l i n e a r systems are i n g e n e r a l o f an o r d e r o f 1/2 t o 1/4 t h e v a l u e s o b t a i n e d f o r t h e E l a s t o - P l a s t i c systems. I t i s n o t e d t h a t t h e y i e l d d i s p l a c e m e n t f o r the T r i -l i n e a r systems i s g r e a t e r t h a n t h a t f o r t h e comparable E l a s t o - P l a s t i c systems. See f i g . 3.1. 3.6.4 E f f e c t o f V i s c o u s Damping on Permanent S e t s I n c r e a s i n g the p e r c e n t a g e o f v i s c o u s damping from 0% t o 1% o f c r i t i c a l showed o n l y a minor r e d u c t i o n i n t h e permanent s e t s . The most s i g -n i f i c a n t r e d u c t i o n was 40% i n a few c a s e s , b u t t h e m a j o r i t y o f systems showed r e d u c t i o n s c o n s i d e r a b l y l e s s than t h i s . F u r t h e r i n c r e a s i n g damping from 1% t o 5% p r o d u c e d a g e n e r a l r e d u c t i o n i n permanent s e t s o f a p p r o x i m a t e l y 40%. However, t h e r e was 'wide v a r i a t i o n s i n t h i s i n some i n s t a n c e s . One extreme case showed a r e d u c t i o n o f one t e n t h , and o t h e r s showed a 40% i n c r e a s e i n permanent s e t . PERMRNENT SET (/TLO DIS) 3 5 7 10' 3 5 7 10" 3 5 7 i I l I i I i 11 1 1—i I i I i 5 7 10* I l I l 11 O O © •— rvj o — ui cn o o o 4 * X | | Q + <> x 4- ri 1 cn o CO rn o CD CO < m CO 5 7 IB" PERMANENT SET l/TLD DIS) . . , , , , ,~ 3 5 7 10' 3 5 7 10» 3 S 7 ID* 3 5 7 10* I ' I i I i t i L _ l i , , l l I l 11 1 1—i I l I i 11 '9 9 CO rn o -C3„ ->X> " CD 1} ft CO CO m o CO m o 5 ffl ?! i • CO cn a a a m 3> 3 5 7 10" _J PERMRNENT SET l/TLD DIS) 3 S 7 ID' 5 S 7 I0» 3 5 7 10* I i I r I . i I I- i I I I I I I L_l 1 1 I I I I I K -o m 33 0 \ 5 9=r.\ r>-_ 4 o CO a 13 . 0 4 cn co 33 V. a T l CD -Cr TJ m 5 ! ! « S Si' PERMANENT SET l/TLO DIS) . . , ,„ , « , ,n. 5 7 10' 3 5 7 13* 3 5 7 10* 3 5 7 10* i , i , , i i I i I i I i 11 1 1—l I i I l ) I C7 H o o o — N> o co a — co cn o o o g t ? f t j i i * i i CO m o co m 9 £ O O m a 10- S 7 10* PERMANENT SET l/YLD DIS) . „ , „. . « • , , „ . 5 7 10' 3 5 7 10* 3 5 7 10* 3 5 7 10* ' . i . . i i . i . i . . i i I i I i I l l I 1 1—l I l I l l I \.--o /I .4 " . - -t>"' cn CD CO cn pn o CO m D 33 16 • G.fl. MAX = 11.561 FT/SEC/SEC -YSR 10.0 a © r- - 2.0 A A to - 1 .0 -1 h i n -n - 0.5 X X tn-* 0.3 o » 0.1 • S i a >-i — <n l— z UJ z cr . UJ EL CENTRO N.S. ELASTO-PLASTIC SYSTEMS ETA-0.01, t 1 * ». v •• x -A V 5 1 * * v \ o o cp—» ts a m i a—a 0.5 1.0 I.S 2.0 PERIOD (IN SECONDS) 2.5 O f b a UJ CL. JC=> C t -UJ,_ EL CENTRO N.S. ELflSTO-PLASIIC SYSTEMS ElflrOOO G.fl. MAX = 11.561 FT/SEC/SEC \ * 4>. V, \ / \ ' -tVA a a -©-0.5 1.0 I.S 2.0 P E R I O D ( I N S E C O N D S ) 2.5 in a 6 r *— s if. i?2 EL CENTRO N.S. RRMBERG-0SG300 SYSTEMS ETA=0.01 G.R. MAX = 11.561 FT/SEC/SEC <j> '. • \ 0.5 1.0 I.S 2.0 PERIOD (IN SECONDS) •9 2.3 F I G . 3.42 PERMANENT S E T S FOR SYSTEMS S U B J E C T E D TO E L CENTRO N . S . ro T l I — I CD s 7 io> PERMRNENT SET l/TLD DIS) . . , l l M 3 5 7 10' 3 5 7 10" 3 5 7 10* _ L J m 3) 5 7 10* ' o o o — rvj o co 30 — oj cn o o o f * ¥ t M ' 1 i i * 4- A ? -3 6 r~ n o z —t 30 a •C z 33 CO II — cn r-U) z m 3J 50 -n —i CO CO -< m co o —4 \ m CO ic rn CO o rn 33 II o o o o H tr m a PERMRNENT SET l/TLD DIS) 5 7 10' 3 5 7 10* 5 7 10' I l I i i I <S> _--X -•t~' -» x> co -< cn £6 t -1 aft, H O u v PARKFIELD N65E ELRSTO-PLASTIC SYSTEMS ETA^O.Ol G.A. MAX = 15.735 FT/SEC/SEC + tSR 10.0 © O 2.0 A — A 1.0 H F 0.5 X- X O '. 0.3 ^ » < • 0.1 •* i i \ X -\ <!> ••. \ i \ \ v \ \ \ / \ \ / a \ * a e cp—o\» ffi-^T—a—ffl ft .0 0.5 1.0 1.5 2.0 2.5 PERIOD (IN SECONDS) asb-1 PARKFIELD N65F ELASTO-PLASTIC 5YSTEM5 FIA G.A. MAX = 15.735 FT/SEC/SEC v. A ',X / v "X ' \ x- • '  v / / / \ / \ \ S a et)—a ip m m, m qj 0.5 1.0 1.5 2.0 PERIOD (IN SECONDS) F I G , 3.43 PERMANENT S E T S FOR SYSTEMS S U B J E C T E D TO P A R K F I E L D N65E. i-1 PARKFIELD N65E TRI-LINEAR STSTEMS ETA=0.01 G.A. MAX = 15.735 FT/SEC/SEC YSR 10.0 © © 2.0 A A 1.0 •) K 0.5 X X 0 3 • <*> cn o f c -Q _l<" >— UJ Z no QC-0.1 \ > 1>. \ \ \ I a — a — a — a — a - i — — f e - ^ .o o.s i.o i.s a.o PERIOD (IN SECONDS) F I G , 3 .43 PARKFIELD N65E TRI-LINEAR SYSTEMS ETA=0.00 G.A. HAX = 15.735 FT/SEC/SEC UJ z c r _ cc 3. 'a / \ M l , a a — a a 2.5 0.5 1.0 I.S 2.0 PERIOD (IN SECONDS I CONTINUED in 9 -UJ z tr. J E D PRC01MR S16E ELRSTO-PLRSTJC SYSTEMS ETR=0.01 FT/SEC/SEC G.R. MAX = 37.698 T5R 10.0 2.0 1.0 0.5 0.3 0.1 'rl -+ I \ A -o X- X * e> e $— a ffl i J.S 1.0 1.5 PERIOD !IN SECONDS! CO •& -O >— -t— Ii ><n IT) t— • Z UJ z d . H C J o:~ tn -PHCOIMR 516E ELRSTO-PLRSriC SYSTEMS ETR=0.00 t G.H. MRX 37.698 FT/SEC/SEC W A \ % \ \ \ * -\ \ \ \ • ^ \ / \ X . . it/- ffl )B I US >t< o s i n i .5 .'.o PERIOD (IN SLCONDbl F I G . 3.44 PERMANENT S E T S FOR SYSTEMS S U B J E C T E D TO PACOIMA S16E. PACOIMA S16E TRI-LINEAR SYSTEMS ETA=0.01 G.R. MRX = 37.69B FT/SEC/SEC YSR 10.0 © © i \ \ 2.0 A A 1.0 H F 0.5 X X 0.3 «• e> 0.1 *-i *^ - "• \ \ \ \ " \ \ 1 e » a — a a a »-i—• m 0.5 1.0 1.5 2.0 PERIOD (IN SECONDS) F I G , 3 ,44 PACOIMA S16E IRI-LINERR SYSTEMS ETA=0.05 G.fl. MAX = 37.698 FT/SEC/SEC • I \ i \ / i \ \ n n Tp tn n o os i o i.s ;'.o PERIOD I IN SECONDS) CONTINUED 98 3.7 C l a s s i f i c a t i o n o f S t r u c t u r e s b y Y i e l d S t r e n g t h R a t i o The r e s u l t s s t u d i e d i n t h e p r e v i o u s s e c t i o n s o f t h i s c h a p t e r c l e a r l y show t h a t t h i s p arameter has a marked e f f e c t upon t h e e x t e n t and t h e manner i n which, s i m i l a r systems r e s p o n d t o t h e same base e x c i t a t i o n s . There was a g e n e r a l p a t t e r n from a l l r e s u l t s t h a t was w e l l d e f i n e d and has l e d t o t h e f o l l o w i n g s t r e n g t h c l a s s i f i c a t i o n o f systems i n t o t h r e e g r o u p s . 1) S t r o n g Systems (Y.S.R. = 10.0 - 1.0) a) Response i s p r e d o m i n a n t l y e l a s t i c w i t h no major y i e l d e x c u r s i o n s . b ) E n e r g y i s v e r y s e n s i t i v e t o v a r i a t i o n s i n p e r i o d and v a r i e s w i t h f r e q u e n c y c o n t e n t o f E/Q. c) E q u a l d i s p l a c e m e n t c r i t e r i o n i s good f o r p e r i o d s g r e a t e r t h a n 0.5 s e c . f o r systems w i t h 1% damping and g r e a t e r . . d) D i s p l a c e m e n t d u c t i l i t i e s i n range 1 - 1 0 e x c e p t f o r one c a s e o f a T r i - l i n e a r system s u b j e c t e d t o T a f t (25 @ T = 0.1 s e c ) . e) Permanent s e t s a r e g e n e r a l l y l e s s t h a n y i e l d d i s p l a c e m e n t . f ) S t r u c t u r e s f a l l w i t h i n range o f code d e s i g n e d systems. 2 ) Moderate Systems (Y.S.R. ~ 1.0 - 0.3) a) M a j o r y i e l d e x c u r s i o n s o c c u r i n r e s p o n s e o f , a l l systems. RESPONSE IS SIGNIFICANTLY DIFFERENT FROM ELASTIC SYSTEMS. b) Energy v a r i e s w i t h p e r i o d , b u t i s n o t s t r o n g l y dependent on p e r i o d as i s t h e case f o r s t r o n g systems. c) E q u a l d i s p l a c e m e n t c r i t e r i o n i s good f o r p e r i o d s g r e a t e r t h a n 0.5 s e e s . T h i s t r e n d i s p a r t i c u l a r l y w e l l d e f i n e d f o r systems w i t h n o n - d e g r a d i n g s t i f f n e s s c h a r a c t e r i s t i c s . p o r -j^g d e g r a d i n g s t i f f n e s s models and systems w i t h s m a l l amounts o f damping t h e e q u a l d i s p l a c e m e n t c r i t e r i o n dees n o t h o l d . 99 D i s p l a c e m e n t d u c t i l i t i e s i n range 10 - 400 f o r p e r i o d s = 1.0 s e e s . However, f o r p e r i o d s g r e a t e r t h a n 1.0 s e c . t h e s e systems have demand d u c t i l i t i e s l e s s t h a n 10. Permanent s e t s a r e g e n e r a l l y about 1/4 o f d i s p l a c e m e n t d u c t i l i t y v a l u e s . These systems a r e w i t h i n t h e range o f code d e s i g n e d b u i l d i n g s . Weak Systems (Y.S.R. < 0.3) E x c e s s i v e d u c t i l i t i e s f o r a l l systems. E n e r g y demand i s c o n s t a n t f o r a l l p e r i o d s . E q u a l d i s p l a c e m e n t does n o t a p p l y . Systems a r e o u t s i d e t h e range o f code d e s i g n e d b u i l d i n g s . 1 0 0 3.7.1 C o r r e l a t i o n o f Y i e l d S t r e n g t h R a t i o t o Code I m p l i e d S t r e n g t h s In o r d e r t o g a i n some p h y s i c a l a p p r e c i a t i o n f o r t h i s s t r e n g t h parameter i t i s u s e f u l t o c o r r e l a t e i t w i t h t y p i c a l s t r e n g t h s i m p l i e d b y s t a n d a r d b u i l d i n g codes. A c o r r e l a t i o n was c a r r i e d out f o r c o m p a r i s o n w i t h the N a t i o n a l B u i l d i n g Code o f Canada ( 1 9 7 5 ) u s i n g the arguments and n o t a t i o n employed i n r e f e r e n c e 24. In r e f . 24 i t i s shown t h a t the maximum peak ground a c c e l e r a t i o n can be equated a s : G.A. max. = A S K I F Ay 3D and t h e y i e l d l e v e l base a c c e l e r a t i o n = A S K I F X YSR = YIELD ACCELN = 3D G.A. Max. u where: A, S, K, I, F a r e s t a n d a r d code symbols. X = Load f a c t o r u = D u c t i l i t y f a c t o r 3 = R a t i o o f the average a c c e l e r a t i o n o f a s t r u c t u r e t o the s p e c t r a l a c c e l e r a t i o n o f the f i r s t dominant mode o f the s t r u c t u r e . D = Response spectrum a m p l i f i c a t i o n f a c t o r . U s i n g f i g s . 1 and 2 o f r e f . 24 i t i s found t h a t : upper bound on s t r e n g t h r a t i o i m p l i e d by code .= 1.0 x 3.0 = 1 . 5 2.0 Lower bound on s t r e n g t h r a t i o i m p l i e d by code = 0.6 x 2.0 = 0.3 4.0 1 0 1 Average v a l u e f o r a t y p i c a l 10 s t o r e y frame s t r u c t u r e = 0.7$ x 3.0 = 0.56 4 o r = Q.75x 2.0 = 0 .4 4 From t h i s i t was c o n c l u d e d t h a t an u pper hound on t h e s t r e n g t h r a t i o i m p l i e d by t h e code i s a p p r o x i m a t e l y 1.5, w h i l s t a l o w e r bound o f 0 . 3 i s f o u n d . F o r a t y p i c a l 10 s t o r y frame s t r u c t u r e t h e y i e l d s t r e n g t h r a t i o i s o f t h e o r d e r o f 0.5. 102 3.7.2 Some Implications of These Results In order to ensure that d u c t i l i t y demands are kept within reason-able l i m i t s the following general conclusions can be expressed i n terms of the c l a s s i f i c a t i o n o u t l i n e d i n t h i s s e c t i o n . A l l structures with periods l e s s than 1.0 sec. should be designed as strong systems with y i e l d strength r a t i o s at l e a s t equal to the maximum peak ground a c c e l e r a t i o n . A l l structures with, periods greater than 1.0 sec. should be designed as moderate systems with y i e l d strength r a t i o s between 1.0 and 0.3 times the maximum peak ground a c c e l e r a t i o n . In f a c t the present codes probably do t h i s i m p l i c i t l y because of the way they have been developed, and the fac t that s t i f f n e s s and y i e l d strength of systems are inhe r e n t l y i n t e r - r e l a t e d . So whilst the type of r e s t r i c t i o n given above i s not checked e x p l i c i t l y by the present codes i t i s covered i n a combination of other i m p l i c i t f a c t o r s . 103 3.8 A p p r o x i m a t i o n s f o r T o t a l E n e r g y D i s s i p a t e d I n s e c t i o n 3.2 a d e t a i l e d s t u d y was made o f the f a c t o r s a f f e c t i n g t h e t o t a l e n e r g y t o be d i s s i p a t e d b y systems s u b j e c t e d t o e a r t h q u a k e ground m o t i o n s . There i t was s t a t e d t h a t t h i s p a r a m e t e r would be f u n d -amental t o a d e s i g n method b a s e d upon e n e r g y d i s s i p a t i o n c o n s i d e r a t i o n s . I n which c a s e i t . w o u l d be most v a l u a b l e t o be a b l e t o a p p r o x i m a t e t h e t o t a l e n e rgy t o be d i s s i p a t e d b y s u c h systems. I n t h i s s e c t i o n some f i r s t a t t e m p t s a t t h e s e a p p r o x i m a t i o n s a r e r e p o r t e d , and d i s c u s s e d i n t h e c o n t e x t o f t h e c l a s s i f i c a t i o n s ystem p r e s e n t e d i n s e c t i o n 3.7. 104 3.8.1 A p p r o x i m a t i o n s f o r T o t a l E n e r g y D i s s i p a t e d b y S t r o n g Systems I n c o n s i d e r i n g the r e s u l t s f o r t h e t o t a l e n ergy d i s s i p a t e d f o r a l l s t r o n g systems o f t h e e i g h t e e n d a t a s e t s p r o d u c e d , i t i s o b s e r v e d t h a t t h e s p r e a d i s t o o wide t o o b t a i n a u s e f u l average o r upper bound. T h e r e f o r e i t was e x p e d i e n t t o c o n s i d e r s t r o n g systems i n groups f o r t h e f o u r e a r t h -quake r e c o r d s s t u d i e d . . I n t h e s e f o u r groups the c a l c u l a t e d t o t a l e n ergy d i s s i p a t e d by the systems was compared t o p r e d i c t i o n s p r o d u c e d from e l a s t i c r e s p o n s e spectrum a n a l y s i s . I n t h i s c o m p a r i s o n e n e r g i e s were p r o d u c e d by c o n s i d e r i n g t h e e l a s t i c s p e c t r a l v e l o c i t y as a b a s i s f o r c a l c u l a t i n g t h e " E l a s t i c S p e c t r a l K i n e t i c Energy", as s u g g e s t e d by K o u s n e r 7 - 2 5 o 2 i e . E.S.K.E. = °v • p e r u n i t . m a s s . 2 I t was o b s e r v e d t h a t t h i s a p p r o x i m a t i o n formed a r e a s o n a b l e upper bound f o r a l l systems w i t h i n i t i a l p e r i o d g r e a t e r t h a n 0.8 s e c . However f o r s h o r t e r i n i t i a l p e r i o d s the E.S.K.E. u n d e r e s t i m a t e s the c a l c u l a t e d e n e r g y demands. T h i s i s p a r t i c u l a r l y t r u e f o r t h e case o f s h o r t p e r i o d systems w i t h d e g r a d i n g s t i f f n e s s p r o p e r t i e s . For t h e s e systems the t o t a l e n e r g y d i s s i p a t e d i s more c o n s i s t e n t w i t h l o n g e r p e r i o d e l a s t i c systems. A b i - l i n e a r upper bound f o r a l l systems i s p r o v i d e d by: For T < . . 5 s e c . E = 0.02 + 0.12 T F o r T > 0.5 s e c . E = 0.0925 - 0.025 T 1 0 5 T h i s upper bound i s r e a s o n a b l e f o r a l l s t r o n g systems s u b j e c t e d t o t h e T a f t , E l C e n t r o and P a r k f i e l d ground m o t i o n r e c o r d s . However, t h i s i s found t o be e x c e s s i v e l y c o n s e r v a t i v e f o r t h e Pacoima r e s u l t s where the f o l l o w i n g upper bound i s i n d i c a t e d . Ibr T « 1.8 s e c . E = 0.02 + 0.022 T For T > 1.8 s e c . E = 0.19 - 0.071 T These a p p r o x i m a t i o n s a r e shown i n f i g s . 3.-45. A b i - l i n e a r a p p r o x i m a t i o n f o r t h e mean energy o f a l l e i g h t e e n s e t s o f s t r o n g systems s t u d i e d i s g i v e n i n f i g . 3-4-6. (Q) S T E O r J C a S Y S T E M S S O & J E C T E D T O T A F T F I G . 3 .45 APPROXIMATIONS BY STRONG S Y S T E M S . (b) S T S O K J C S Y S T E M S . SU&JE.OTE.D T O E L C E M T C O Ki.S. FOR TOTAL ENERGY D I S S I P A T E D ^ o LOT 108 0.0 0.5 1.0 1.5 2.0 2.5 P E R I O D ( I N S E C O N D S ) F I G . 3.46 APPROXIMATIONS FOR MEANS OF TOTAL ENERGY D I S S I P A T E D BY A L L STRONG SYSTEMS. 109 3.8.2 Approximations for Total Energy Dissipated.byModerate' Systems For the re s u l t s of the moderate systems i t was found that the t o t a l energy dissipated for a l l hysteretic properties, damping values, and earth-quake ground motions form a reasonably narrow band and f a l l into a consist-ent pattern. Pbr these r e s u l t s i t i s possible to produce expressions f o r approximations to the mean energy by b i - l i n e a r regression. For systems with YSR = 1.0 a b i - l i n e a r regression gives the follow-ing r e s u l t s for mean energies. For T 4 0.5 sec. E = 0.0011 + 0.118 (T) For T > 0.5 sec. E = 0.0673 - 0.0197 (T) For systems with YSR =0.5 For T. < 0.5 sec. E = 0.022 + 0.056 (T) • For T > 0.5 sec. E = 0.062 - 0.016 (T) For systems with YSR = 0.3 For T. «0.5 sec. E = 0.026 + 0.312 (T) For T > 0.5 sec. E = 0.049 - 0.012 (T) 1 1 0 S i m i l a r l y b y c a r r y i n g out a b i - l i n e a r r e g r e s s i o n a n a l y s i s a t e n e r g y v a l u e s c o r r e s p o n d i n g t o t h e mean p l u s one s t a n d a r d d e v i a t i o n f o r d a t a a t each p e r i o d , i t i s p o s s i b l e t o c o n s t r u c t an upper bound which c o r r e s p o n d s a p p r o x i m a t e l y t o a c o n f i d e n c e l e v e l o f $3.7%. The r e s u l t s f o r t h i s upper bound r e g r e s s i o n a n a l y s i s a r e g i v e n below: Ibr systems w i t h YSR = 1 . 0 For T. « 0.5 s e c . E = 0.016 + 0.135 ( T ) Ibr T > 0.5 s e c . E = 0.094 - 0.028 ( T ) For systems w i t h YSR = 0 . 5 For T 4 0. 5 s e c . E = 0.042 + 0.056 (T) For T > 0.5 s e c . E = 0.079 - 0:021 ( T ) For systems w i t h YSR = 0 . 3 Ibr T. « 0 . : 5 s e c . E = 0.04 + 0.017 ( T ) For T > 0.5 s e c . E = 0.063 - 0.015 ( T ) The regression a n a l y s i s f o r a l l these systems leads to s i m i l a r expressions which suggests that one expression f o r the mean energies can be used f or a l l moderate systems. These r e s u l t s are shown i n f i g . 3.4-7. F I G , 3,47 APPROXIMATIONS FOR T O T A L ENERGY D I S S I P A T E D BY MODERATE S Y S T E M S . 113 Y 5 R - 0 - 5 1 1 1 1 : 1 : 1 0.0 0.5 i.O 1.5 2.0 2.5 P E R I O D ( IN SECONDS) (B) F I G i 3 .47 C O N T I N U E D YS£ = 0-3 115 3.8.3 Approximations for Total Energy Dissipated by Weak Systems • For these systems i t i s found that the t o t a l energy dissipated decreases l i n e a r l y with period, and i s e s s e n t i a l l y independent of viscous damping and earthquake e x c i t a t i o n used. The spread i n these r e s u l t s i s i n large measure a function of the way the earthquake records are non-dimensionalised. A simple l i n e a r regression analysis produced excellent c o r r e l a t i o n with the mean energy values as: E = 0.028 - 0.005 (T) for a l l periods. A s i m i l a r regression analysis for mean plus one standard deviation energy values produced an approximate 83.7$ confidence l e v e l upper bound as: E = 0.041 - 0.008 (T) for a l l periods. These results are shown i n f i g . 3.48. This energy represents the energy required to e f f e c t i v e l y destroy the systems. In terms of energy d i s s i p a t i o n capacity, these values can be considered the maximum energies that can be dissipated by the system during the earthquake. Additional energies from the earthquake can not be absorbed. 116 CM CL L D ca a CO-CO 0 CX >— C D ce LU LUfM a r— O a YSR = 0-1 P K O & A & 1 L I T Y L E V E L S , & 0 0 ° / o o-o -4 - i - o - o o S (T) UPPER &OUNJC 0 ' 0 2 8 - 0 ' 0 0 & ( T ) L O W E R 0.0 0.5 1.0 1.5 2.0 P E R I 0 0 ( IN SECONDS) 2.5 F I G . 3,48 APPROXIMATIONS FOR T O T A L ENERGY D I S S I P A T E D BY WEAK S Y S T E M S . 1 1 7 3.9 Approximations for Distribution of Energy Within Systems The results for the distribution of energy within systems were given i n section 3 . 3 . There i t was shown that the percentage of to t a l energy dissipated by hysteresis i s independent of earthquake record. Further, once these results are considered i n terms of hysteresis model and proportion of viscous damping, they f a l l into narrow well defined patterns. Because of the clear pattern produced by these results i t i s possi-ble to approximate the distribution of energy within the systems by simply taking the mean values of the results for the corresponding combinations of parameters. These results are shown i n f i g s . 3 . 4 9 and 3 . 5 0 for Elasto-Plastic and Tri-linear systems respectively. 118 .5 FIG. 3.49 APPROXIMATIONS FOR DISTRIBUTION OF ENERGY WITHIN SYSTEMS FOR ELASTO-PLASTIC SYSTEMS. 1 1 9 FIG, 3,50 APPROXIMATIONS FOR DISTRIBUTION OF ENERGY WITHIN SYSTEMS FOR TRI-LINEAR SYSTEMS. 120 CHAPTER 4 DEVELOPMENT AND VERIFICATION OF DESIGN METHOD FOR  SINGLE DEGREE OF FREEDOM SYSTEMS 4.1 Introduction In t h i s chapter the intention i s to incorporate the r e s u l t s and insight obtained i n chapter 3 into a design method, based on energy d i s s i p a t -ion requirements, which would be suitable for in c l u s i o n i n a l i m i t state design code. Due to the time l i m i t a t i o n s upon t h i s work i t was not possible to conduct as thorough an investigation into t h i s aspect of the research as was hoped. However, s u f f i c i e n t information i s available to enable prelimin-ary conclusions to be made, and a tentative form of a design method i s outlined. The r e s u l t s of the detailed study of chapter 3 give insight into the t o t a l energy to be dissipated by single degree of freedom systems. Also the manner i n which the t o t a l energy to be dissipated i s di s t r i b u t e d between the hysteretic and viscous damping mechanisms within the system was studied. To date the study has shown that the t o t a l energy and i t s d i s t r i b u t i o n within the system depends upon the following f i v e parameters, ( i ) y i e l d strength r a t i o ( i i ) period ( i i i ) hysteretic model ( i v ) viscous damping (v) earthquake type There i s a wide s p r e a d i n some o f the r e s u l t s w i t h r e s p e c t t o i n d i v i d u a l e arthquake r e c o r d s , and i t i s p o s s i b l e t h a t the i n f l u e n c e o f the o t h e r p a r a m e t e r s may n o t , i n comparison w i t h t h i s s p r e a d , need t o he g i v e n a v e r y r e f i n e d t r e a t m e n t i n a d e s i g n method. However, i n c h a p t e r 3> i t was shown t h a t i t was p o s s i b l e t o c o n s t r u c t s p e c t r a l c u r v e s o f t o t a l e n ergy and d i s t r i b u t i o n o f e n e r g y f o r systems i n terms o f a l l t h e s e p a r a m e t e r s . The p r e s e n t g e n e r a t i o n o f d e s i g n codes, b a s e d on a l i m i t s t a t e p h i l o s o p h y , r e q u i r e s a two l e v e l i n v e s t i g a t i o n o f e a r t h q u a k e r e s i s t a n c e . The i n t e n t i o n i s t o ensure minor damage o c c u r i n g i n a moderate e a r t h q u a k e , and t o p r e v e n t c o l l a p s e i n a s e v e r e e a r t h q u a k e . A l t h o u g h t h i s g e n e r a l i n -t e n t has l o n g been e s t a b l i s h e d i n the p r o v i s i o n s o f d e s i g n c o d e s, the e x p l i c i t n a t u r e o f the' l i m i t s t a t e r e q u i r e m e n t s p r e s e n t s a new c h a l l e n g e . In t h i s c o n t e x t t h e y i e l d s t r e n g t h r a t i o , and the s t r e n g t h c l a s s i f i c a t i o n system d i s c u s s e d i n c h a p t e r 3 a r e most u s e f u l . 4.2 Requirements f o r an E a r t h q u a k e D e s i g n Method In t h e d e s i g n o f b u i l d i n g s t h e s e r v i c e a b i l i t y and a r c h i t e c t u r a l r e q u i r e m e n t s p r e d e t e r m i n e t h e l a y o u t , and t o a l a r g e e x t e n t t h e s t r u c t u r a l f r a m i n g . Thus the s t i f f n e s s o f t h e s t r u c t u r e i s l a r g e l y f i x e d by t h e time a s e i s m i c c o n s i d e r a t i o n s a r e t a k e n up. Hence the i n i t i a l p e r i o d i s f i x e d and i s o n l y m o d i f i e d s l i g h t l y by e n g i n e e r i n g c h o i c e ' o f c o n s t r u c t i o n system and d e s i g n s t r e n g t h . The two main param e t e r s t h a t the e n g i n e e r can v a r y a r e t h e s t r u c t u r a l system t o be used and t h e y i e l d s t r e n g t h o f t h e members. The d e s i g n method s h o u l d c o n c e n t r a t e on the e f f e c t s o f v a r y i n g t h e ^ c o n s t r u c t i o n system and y i e l d s t r e n g t h and c h e c k i n g t h a t t h e r e s p o n s e and e n e r g y d i s s i p a t i n g c a p a c i t i e s a r e s a t i s f a c t o r y . 122 A most c r i t i c a l parameter i n m e a s u r i n g r e s p o n s e i s d i s p l a c e m e n t , because t h i s c a n be r e l a t e d d i r e c t l y t o damage. The s t r u c t u r e must, at a l l t i m e s d u r i n g t h e e a r t h q u a k e , be c a p a b l e of d i s s i p a t i n g t h e e nergy i m p a r t e d t o i t by t h e ground m o t i o n . A check f o r t h i s adequacy i s i m p e r a t i v e . T h i s means t h a t t h e s t r u c t u r e must be d e s i g n e d and d e t a i l e d t o ensure t h a t t h e R e q u i r e d E n e r g y C a p a c i t y i s p r o v i d e d . The d e s i g n method can t a k e t h e form o f an o p t i m i z a t i o n p r o c e s s i n which the m a t e r i a l s and s t r e n g t h s a r e v a r i e d and energy c a p a c i t y , peak d i s p l a c e m e n t s and permanent s e t s a r e checked f o r a c c e p t a b l e l e v e l s . 4 . 3 Y i e l d S t r e n g t h R a t i o and Damage C o n t r o l I t has l o n g been t h e p h i l o s o p h y i n d e s i g n i n g s t r u c t u r e s t o w i t h -s t a n d e a r t h q u a k e m o t i o n s t o p e r m i t l i m i t e d amounts o f damage c o n s i s t e n t w i t h economic c o n s i d e r a t i o n s . I t i s uneconomic t o t r y and d e s i g n f o r no damage, but a t t h e same tim e , e x c e s s i v e damage must be a v o i d e d . A s p e c t r a l a n a l y s i s u s i n g assumed d u c t i l i t y f a c t o r s t o p r o d u c e d e s i g n f o r c e s can a t b e s t o n l y meet the d e s i g n e r s p h i l o s o p h i c g o a l s i n a g e n e r a l sense. In o r d e r t o improve upon our a n a l y s i s t e c h n i q u e , we must i n c l u d e an a d d i t i o n a l parameter which i s based e x p l i c i t l y on s t r e n g t h c o n s i d e r a t i o n s . In t h e s i n g l e degree o f freedom s t u d y i t i s shown t h a t a s t r e n g t h v a r i a t i o n has marked e f f e c t s upon a l l s i g n i f i c a n t p a r a m e t e r s , i e . t o t a l e n ergy d i s s i p a t e d , d i s p l a c e m e n t d u c t i l i t y , d i s p l a c e m e n t and permanent s e t s . By i n c l u d i n g e x p l i c i t l y a s t r e n g t h parameter i n th e a n a l y s i s and d e s i g n p r o c e d u r e i t i s p o s s i b l e t o meet the g o a l s o f t h e d e s i g n p h i l o s o p h y i n a f a r more s p e c i f i c manner. H a v i n g - completed a d e s i g n w i t h a s p e c i f i c s t r e n g t h parameter, t h e d e s i g n e r w i l l be a b l e t o p r e d i c t w i t h r e a s o n a b l e a c c u r a c y t h e d i s p l a c e m e n t d u c t i l i t y and d i s p l a c e m e n t s t o be e x p e c t e d from a moderate o r a s e v e r e ground m o t i o n . The d i s p l a c e m e n t d u c t i l i t y and d i s p l a c e -ment p r e d i c t i o n s can t h e n be r e l a t e d t o e x p e c t e d damage l e v e l s which must be chosen t o comply w i t h economic r e q u i r e m e n t s . In t h i s way t h e d e s i g n e r has f a r more c o n t r o l o v e r t h e consquences he i n h e r e n t l y b u i l d s i n t o h i s d e s i g n and an optimum s o l u t i o n t o t h e p r o b l e m becomes p o s s i b l e . 4.4 Some Notes on Energy D i s s i p a t i o n V e r s u s Time F o r a d e s i g n method based upon e n e r g y d i s s i p a t i n g c h a r a c t e r i s t i c s i t i s i m p o r t a n t not o n l y t o know t h e amount o f e n e r g y t o be d i s s i p a t e d by each i n d i v i d u a l mechanism, b u t a l s o t h e manner and th e r a t e a t which t h i s e n ergy i s d i s s i p a t e d t h r o u g h o u t t h e e x c i t a t i o n . I n s i g h t i n t o t h i s a s p e c t o f t h e r e s p o n s e was g a i n e d from s t u d y i n g the time h i s t o r i e s o f t h e d i s p l a c e m e n t , energy i n p u t t o system, energy d i s s i p a t e d by v i s c o u s damping and energy d i s s i p a t e d by h y s t e r e s i s i n c o n j u n c t i o n w i t h t h e f o r c e / d i s p l a c e m e n t h i s t o r y o f e a c h system. From t h e s e s e t s o f d a t a f o r i n d i v i d u a l systems a g e n e r a l p i c t u r e o f the energy d i s s i p a t i n g mechanisms can be b u i l t up. Some t y p i c a l s e t s o f t h e s e r e s u l t s a r e g i v e n i n f i g s . 4.1, 4.2, 4.3 and 4.4. A summary o f the r e s u l t s o f t h i s s t u d y i s p r e s e n t e d i n t a b l e 4.1. From t h e s e s t u d i e s i t was f o u n d t h a t t h e most u s e f u l parameter f o r a p p l i c a t i o n i n a d e s i g n method i s t o c o n s i d e r t h e f r a c t i o n o f h y s t e r e t i c e n ergy d i s s i p a t e d i n t h e s p r i n g d u r i n g t h e l a r g e s t h a l f l o o p y i e l d e x c u r s i o n . In o r d e r t o be a b l e t o p r o p o r t i o n the amount o f h y s t e r e t i c e nergy t o be d i s s i p a t e d i n t h i s " p r i n c i p a l h a l f l o o p " a number o f systems were s t u d i e d . From t h e s e p r e l i m i n a r y s t u d i e s i t has been found t h a t t h e magnitude o f the p r i n i c i p a l h a l f l o o p s can be g e n e r a l l y c a t e g o r i z e d as f o l l o w s : ( i ) F o r a l l systems s u b j e c t e d t o Pacoima S16E, a ne a r s o u r c e i m p u l s i v e e a r t h q u a k e , and f o r s t r o n g systems s u b j e c t e d t o T a f t N69W, a f a r f i e l d e a r t h q u a k e , the l a r g e s t y i e l d e x c u r s i o n s d i s s i p a t e up t o 50% o f t h e t o t a l e n ergy d i s s i p a t e d i n h y s t e r e s i s , i e . s u ch systems e x p e r i e n c e 2 o r 3 major y i e l d e x c u r s i o n s . ( i i ) F o r moderate systems s u b j e c t e d t o T a f t N69W, a f a r f i e l d e a r t h q u a k e , t h e l a r g e s t y i e l d e x c u r s i o n s d i s s i p a t e from 20% t o 33% o f t h e h y s t e r e t i c energy. i e . s u ch systems e x p e r i e n c e 3 - 5 major e x c u r s i o n s . The r a t e a t which e n e r g y i s d i s s i p a t e d by v i s c o u s damping i s i n g e n e r a l e i t h e r a l i n e a r o r p a r a b o l i c f u n c t i o n o f t i m e . F o r some systems e n e r g y i s d i s s i p a t e d b y t h e v i s c o u s dampers i n a few i s o l a t e d p u l s e s , up t o 20%. However, f o r t h e v e r y weak systems s u b j e c t e d t o t h e Pacoima S16E r e c o r d t h e s e p u l s e s c a n be as l a r g e as 55%. 1 2 5 SYSTEM P e r c e n t a g e o f H y s t e r e t i c E n ergy In P r i n c i p a l H a l f Loop Shape o f V i s c o u s Damping Energy Vs. Time Curve Record HYST. YSR. P e r i o d V i s c o u s Damping T a f t E/P 1.0 0.8 1% 30% L i n e a r T a f t E/P 1.0 1.5 1% 40% L i n e a r T a f t E/P 0.5 0.8 1% 25% L i n e a r ( 1 0 l ? E u l s e ) T a f t E/P 0.5 1.5 1% 21% L i n e a r T a f t T/L 1.0 0.8 1% 14% L i n e a r ( 20% P u l s e ) T a f t T/L 1.0 1.4 1% 15% s m a l l s t e p s / L i n e a r T a f t T/L 0.5 0.8 1% 16% s m a l l s t e p s / L i n e a r T a f t T/L 0.5 1.4 1% 18% s m a l l s t e p s / L i n e a r Pacoima E/P 1.0 0.8 1% E l a s t i c P a r a b o l i c Pacoima E/P 1.0 1.4 1% 48% P a r a b o l i c Pacoima E/P 0.5 0.8 1% 41% L i n e a r Pacoima E/P 0.5 1.4 1% 54% P a r a b o l i c Pacoima T/L 1.0 0.8 1% 24% L i n e a r (15% P u l s e ) Pacoima T/L 1.0 1.4 1% 26% P a r a b o l i c Pacoima T/L 0.5 0.8 1% 47% P a r a b o l i c Pacoima T/L 0.5 1.4 1% 26% P a r a b o l i c T a f t E/P .64 0.8 1% 33% L i n e a r (10% P u l s e ) T a f t E/P .64 1.6 1% 25% L i n e a r T a f t E/P 1.3 1.6 1% 60% L i n e a r T a f t E/P 1.3 0.8 1% 55% L i n e a r Pacoima E/P 0.17 0.8 1% 42% 35% P u l s e Pacoima E/P 0.17 1.6 1% 56% 55% P u l s e Pacoima E/P 0.08 0.8 1% 38% 55% P u l s e T a f t E/P .64 0.8 5% 57% L i n e a r T a f t E/P .64 1.6 5% 30% P a r a b o l i c Pacoima E/P .25 0.8 5% 44% 28% ( P u l s e ) TABLE 4.1 ENERGY DISSIPATION VERSUS TIME FOR SINGLE DEGREE OF FREEDOM SYSTEMS 921 CO -< T l T l CZ CO O —• CO 70 70 CD m II > O CO • (—1 TJ o > 2 H TJ —• CD O oo -< -n TI C W O M td XI x> CD c_ -m ll > o -z. H O m - m -O vj-ii— Osl > H 00 O —I - O O H t) m i >-i o —• i— m o o > —• o oo x 2 H — • > II «—• OO O H OOI—> O I—'- 00 C D - C T - < -< m oo - OO —1 O m m n O 2 - m V . m r—1 111 xi CD -< < i—i a OO i — i o oo - oo o T3 > TJ —1 CD - O D I S P L A C E M E N T -0.06 -0.02 NG IXlCi-o .is CO -< Tl TT. C CO O 1 - 1 CO 70 70 CD rn || > a TJ 2 > TJ —I CD •—< O 4-5 O u t l i n e o f t h e D e s i g n Method and Examples With t h e i n f o r m a t i o n o b t a i n e d so f a r i t i s now p o s s i b l e t o o u t l i n e the form o f a d e s i g n method which c o u l d be used i n a l i m i t s t a t e d e s i g n code. The a n a l y s i s p r o c e d u r e can be d i v i d e d i n t o two b r o a d a r e a s . F i r s t , a check t h a t t h e R e q u i r e d E nergy C a p a c i t y i s p r o v i d e d forms a check upon the s a f e t y o f t h e s t r u c t u r e a g a i n s t c o l l a p s e . S e c o n d l y , checks upon t h e d i s p l a c e m e n t and d i s p l a c e m e n t d u c t i l i t y p r o v i d e i n f o r m a t i o n on t h e damage l e v e l s t o be e x p e c t e d from t h e e a r t h q u a k e . I t s h o u l d be n o t e d t h a t a l l t h e i n p u t p a r a m e t e r s r e q u i r e d f o r t h i s method can be c a l c u l a t e d o r e s t i m a t e d w i t h , r e a s o n a b l e c o n f i d e n c e . No assumption about d u c t i l i t y f a c t o r need be made, because i n t h i s method, the d i s p l a c e m e n t d u c t i l i t y i s g i v e n e x p l i c i t l y as an o u t p u t p a r a m e t e r . I t thus removes t h e u n c e r t a i n i t y common t o a l l p r e s e n t d e s i g n methods. In the c o n t e x t o f the two l e v e l l i m i t s t a t e p h i l o s o p h y o f t h e new g e n e r a t i o n o f b u i l d i n g codes, t h i s method i s o u t l i n e d as f o l l o w s : OUTLINE OF DESIGN METHOD 1 ( a ) E s t a b l i s h t y p e o f d e s i g n e a r t h q u a k e f o r a g i v e n s i t e . ( b ) E s t i m a t e maximum ground a c c e l e r a t i o n e x p e c t e d d u r i n g a moderate e a r t h q u a k e . (Say a 25 y e a r r e t u r n p e r i o d ) . ( c ) E s t i m a t e maximum ground a c c e l e r a t i o n e x p e c t e d d u r i n g a s e v e r e e a r t h q u a k e . (Say a 100 y e a r r e t u r n p e r i o d ) . ( d ) E s t i m a t e p r o p o r t i o n o f v i s c o u s damping i n system. ( e ) C a l c u l a t e p e r i o d o f f i r s t mode o f system. ( f ) Assuming y i e l d moments and c o l l a p s e mechanism, c a l c u l a t e y i e l d s t r e n g t h o f t h e system. N.B. The f o l l o w i n g p r o c e d u r e s , s t e p s 2 t h r o u g h 8, a r e c a r r i e d out t w i c e . Once f o r the moderate ea r t h q u a k e and once f o r t h e s e v e r e e a r t h q u a k e . 2 E s t a b l i s h Y i e l d S t r e n g t h R a t i o From spectrum f o r s t r o n g , moderate o r weak system o b t a i n : ( a ) T o t a l e n e r g y t o be d i s s i p a t e d by system. (From f i g s . 3-4-5, 3-46, 3-47 and 3-48 o r o t h e r ) . ( b ) P r o p o r t i o n o f e n e r g y d i s s i p a t e d by h y s t e r e s i s . (From f i g s . 3-49 and 3-50 o r o t h e r ) . 3 E s t i m a t e t h e e nergy t o be d i s s i p a t e d by h y s t e r e s i s i n t h e p r i n c i p a l h a l f l o o p . 4 C a l c u l a t e t h e d i s p l a c e m e n t r e q u i r e d t o d i s s i p a t e t h i s e n ergy i n one h a l f l o o p . 132 5 From s p e c t r a o b t a i n d i s p l a c e m e n t d u c t i l i t y and permanent s e t v a l u e s i n terms o f y i e l d d i s p l a c e m e n t . (See f i g s . 3-37 t o 3-44 and appen d i x A). 6 C a l c u l a t e y i e l d d i s p l a c e m e n t o f system. 7 C a l c u l a t e maximum d i s p l a c e m e n t . 8 Check i f r e s p o n s e i s w i t h i n d e s i g n c r i t e r i a . I t e r a t e t o o b t a i n a s a t i s f a c t o r y s o l u t i o n . I n t h e method o u t l i n e d above s t e p s 2, 3 and 4 can be c a t e g o r i z e d as s t r e n g t h and s a f e t y c h e c k s . S t e p s 5, 6 and 7 can be c o n s i d e r e d damage o r s e r v i c e a b i l i t y c h e c k s . By p e r f o r m i n g s e p a r a t e s e t s o f c a l c u l a t i o n s f o r eac h o f t h e ea r t h q u a k e s e v e r i t y l e v e l s , a d e t a i l e d u n d e r s t a n d i n g o f t h e r e s p o n s e and e x p e c t e d damage l e v e l s i s p r o v i d e d . The f o l l o w i n g example i l l u s t r a t e s t h e use o f t h e above method, and a summary o f s e v e r a l t y p i c a l c a l c u l a t i o n s f o r s i n g l e degree o f freedom systems i s g i v e n i n t a b l e 4.2. EXAMPLE •RIGID G I R D E R IOO KIPS.. w a « a>& COLS DETAILS OF FRAME USED IN EXAMPLE ( lA) IN GENERAL A DESIGN EARTHQUAKE WOULD BE ESTABLISHED BASED UPON STUDIES OF THE PARTICULAR SITE SEISMICITY. FOR PURPOSES OF THIS EXAMPLE, SPECTRA FOR THE TAFT N69W, AND PACOIMA S16E> RECORDS WILL BE USED. THIS PERMITS DIRECT COMPARISONS WITH TIME STEP CALCULATIONS. (1B) ESTIMATE USE TAFT MAXIMUM 6R0UND ACCELN. FOR 25 YEAR RETURN PERIOD. N69W. MAX GRND ACCELN = b.04 FT/SEC? (lc) ESTIMATE MAXIMUM GROUND ACCELN. FOR 100 YEAR RETURN PERIOD. 2 USE PACOIMA S16E MAX GRND ACCELN =37.7 FT/SEC. (ID) ASSUME 1% OF CRITICAL VISCOUS DAMPING TO REPRESENT THE NON -STRUCTURAL DAMPING IN THE SYSTEM. FXAMPIF CONTINUED ( l E ) FUNDAMENTAL PERIOD. K = 2 x 12EI = 2x12x30,000x126 = 15J5 K IPS/ INCH. 2n 100 386x 15.55 0.81 S E C ( IF ) YIELD STRENGTH OF SYSTEM. PLASTIC MOMENT IN W8 X 35 YIELD STRESS = 44 K . S . I Zxx = 34.7 I N 3 M P = 34.7 x 44 = 127 KIP F T . 12 HORIZONTAL SHEAR IN COLS AT YIELD = 127 X 2 = 16.95 KIPS. 15 TOTAL HORIZONTAL SHEAR AT YIELD = 33.9 KIPS, HORIZONTAL ACCELN OF MASS REQD TO PRODUCE THIS SHEAR. = 33.9 = 33.9% = 10.91 F T / S E C 2 g 100 M V o V o PESIGN FOR MODERATE EARTHQUAKE (2) YIELD STRENGTH RATIO = 10.91 = 2.16 5.05 .-. STRONG SYSTEM. THE TOTAL ENERGY TO BE DISSIPATED BY THE SYSTEM CAN BE APPROXIMATED FROM DESIGN SPECTRA SUCH AS FIGS 3.45 - 3.48. HOWEVER TO PERMIT DIRECT COMPARISON WITH TIME STEP CALCULATIONS THE SPECTRA FOR TAFT N69W OF FIG A.L WILL BE USED. (2A) TOTAL ENERGY TO BE DISSIPATED BY THE SYSTEM = o.i x 5.052 x m SPECTRAL X (G.A.MAX)" VALUE (2B) ENERGY TO BE DISSIPATED IN HYSTERSIS = 95 x 0.50 = 47.5 KIP INS (3) FROM SECTION 4.4 IT IS KNOWN THAT HYSTERETIC ENERGY WILL BE DISSIPATED IN TWO OR THREE MAJOR YIELD EXCURSIONS ENERGY TO BE DISSIPATED IN EACH MAJOR EXCURSION X 0.259 = 95 KIP INS X MASS = 47.5 = 23.8 KIP INS 2 EXAMPLE CONTINUED DESIGN FOR MODERATE EARTHQUAKE (4) ENERGY DEMAND PER JOINT WITH FOUR PLASTIC HINGES = 23.8 = 5.94 KIP INS/jOINT ~4~ JOINT ROTATION REQUIRED TO DISSIPATE THIS ENERGY = 5.94 = 0.0039 RADIANS 127 x 12 DISPLACEMENT AT TOP OF COLUMN DURING MAX YIELD EXCURSION = 0.0039 x 15 x 12 = 0.702 INS (5) DISPLACEMENT DUCTILITY (FROM SPECTRUM) = 1.5 (6) HORIZONTAL DISPLACEMENT OF BEAM AT YIELD IN COLS. = P Y =33.9 = 2J8 INS K 15.55 (7) MAXIMUM DISPLACEMENT = 1,5 X 2.18 = 3^27 INS (8) PERMANENT SET = 0.17 X 2.18 = 0.37 INS EXAMELE-CQNT1NUEP pFSIGN FOR SFVFRE EARTHQUAKE. (2) YIELD STRENGTH RATIO = 10.91 = 0.3 37.7 „\ MODERATE SYSTEM THE TOTAL ENERGY TO BE DISSIPATED BY THE SYSTEM CAN BE APPR^xj^ATEjj FROM DESIGN SPECTRA SUCH AS HOWEVER TO PERMIT DIRECT COMPARISON WITH TIME STEP CALCULATIONS THE SPECTRA FOR PACOIMA blbt OF FIG A . l WILL BE USED. (2A) TOTAL ENERGY TO BE DISSIPATED BY THE SYSTEM = 0.025 x 37.698 2 x M l x . 0.259 = 1330 KIP INS SPECTRAL X (G.A.MAX) 2 X MASS VALUE (2B) ENERGY TO BE DISSIPATED IN HYSTERESIS = 1330 x 0.8 = 1064 KIP INS. (3) FROM SECTION 4.4 IT IS KNOWN THAT HYSTERETIC ENERGY WILL BE DISSIPATED IN TWO OR THREE MAJOR YIELD EXCURSIONS. A ENERGY TO BE DISSIPATED IN EACH MAJOR EXCURSION = 1064 = 532 KIP INS. FXAMP1F CONTINUED DESIGN FOR SEVERE EARTHQUAKE (4) ENERGY DEMAND PER JOINT WITH FOUR PLASTIC HINGES = 532 = 133 KIP INS. JOINT ROTATION REQUIRED TO DISSIPATE THIS ENERGY. = 1 3 3 = 0.087 RADIANS. 127 x 12 DISPLACEMENT AT TOP OF COLUMN DURING MAX YIELD EXCURSION = 0.087 x 15 x 12 = 157 INS. (5) DISPLACEMENT DUCTILITY (FROM SPECTRUM) = 5.5 (6) HORIZONTAL DISPLACEMENT OF BEAM AT YIELD IN COLS. = P Y = 33.9 = 2 J8 INS. K 15.55 (7) MAXIMUM DISPLACEMENT = 5.5 X 2.18 = 12^ 0 INS. (8) PERMANENT SET =2.5 X 2.18 = 5.45 INS. SYSTEM P e r i o d Sec. YSR Type • MAX DISPLACEMENT (INS) PERMANENT SETS E a r t h q u a k e M a t e r i a l Damping E l a s t i c 1 P l a s t i c 2 Time S t e p 3 P l a s t i c 2 Time S t e p 3 Pacoima E/P 0.01 .8 • 3 Mod. 6.7 12.0 10.74 5.45 0.9 Pacoima E/P 0.05 .8 .3 Mod. 6.1 7.12 8.30 2.62 1.9 Pacoima E/P 0.05 1.0 .2 Weak 11.6 10.2 11.75 7.0 2.3 T a f t E/P 0.01 .8 1.85 S t r o n g 3.4 3.27 2.91 0.37 0.11 T a f t E/P 0.01 1.25 0.96 Mod. 2.9 2.42 2.50 0.6 1.23 T a f t E/P 0.01 1.41 0.8 Mod. 4.5 3-36 3.52 .1.86 1.86 Pacoima T/L 0.01 .8 .3 Mod. 6.7 18.8 10.91 5.64 0.73 T a f t T/L 0.01 .8 1.85 S t r o n g 3.4 2.64 2.86 .18 0.43 T a f t T/L 0.01 1.41 0.8 Mod. 4.5 4.35 3.72 .62 1.15 1. ELASTIC SPECTRAL ANALYSIS 5% DMPG. 2. METHOD GIVEN IN THIS CHAPTER 3. NON-LINEAR TIME STEP CALCS. MADE WITH DRAIN - 2 D 2 2 •f See example i n s e c t i o n 4.5 f o r d e t a i l e d c a l c u l a t i o n s . T A B L E 4-2 DISPLACEMENTS AND PERMANENT SETS FOR SYSTEMS ANALYZED BY THE METHOD GIVEN IN CHAPTER 4. h-1 VO 137 CHAPTER 5 THEORY AND PROGRAMMING CONSIDERATIONS  FOR MULTI-DEGREE OF FREEDOM SYSTEMS 5.1 M a t h e m a t i c a l Model f o r M u l t i - D e g r e e o f Freedom S t u d i e s The model u s e d f o r t h e m u l t i - s t o r e y s t r u c t u r e a n a l y s i s i s t h e s i m p l e s h e a r beam model. See f i g . 5.1. F o r each s t o r e y t h e r e i s one l a t e r a l •, degree o f freedom o n l y . The g i r d e r s a r e assumed t o have i n f i n i t e s t i f f n e s s , and t h e columns a r e assumed t o he f i x e d a t t h e f o u n d a t i o n s . 26 The i m p o r t a n t i n f l u e n c e o f g r a v i t y l o a d s on energy d i s s i p a t i o n i s n o t c o n s i d e r e d i n t h i s p r e l i m i n a r y s t u d y . The n u m e r i c a l p a r a m e t e r s o f t h e model a r e d e f i n e d f o r each s t o r e y by th e mass, i n i t i a l s t i f f n e s s , y i e l d s t r e n g t h r a t i o , and t h e v i s c o u s damping. In t h e computer program t h e i n p u t i s a r r a n g e d so t h a t t h e d a t a f o r each f l o o r i s d e f i n e d I n r e l a t i o n t o t h e v a l u e s f o r t h e f i r s t , o r l o w e s t , s t o r e y . The v i s c o u s damping can be d e f i n e d as mass o r s t i f f n e s s damping o r a c o m b i n a t i o n o f t h e two. I n t h e case o f s t i f f n e s s damping t h e damping f a c t o r s a r e r e l a t e d t o t h e i n i t i a l s t i f f n e s s o f t h e system t h r o u g h o u t t h e e n t i r e e x c i t a t i o n . I n t h e p r e s e n t s t u d y b o t h E l a s t o - P l a s t i c and t h e T r i - L i n e a r d e g r a d -i n g s t i f f n e s s models o f h y s t e r e s i s were used. These a r e s i m i l a r t o t h e models d e s c r i b e d i n s e c t i o n 3-1. T / M n K h Y s e n M K 3 Y S R : _M_ YSR 2 M Y S R , 7-1 r F I G . 5.1 L T I - D E G R E E OF FREEDOM MODEL 1 3 9 5.2 E q u a t i o n s o f M o t i o n f o r M u l t i - D e g r e e o f Freedom Systems. The e q u a t i o n s o f m o t i o n f o r t h e m u l t i - d e g r e e o f freedom systems were i n t e g r a t e d u s i n g t h e c o n s t a n t average a c c e l e r a t i o n time s t e p scheme. The development o f t h i s scheme f o r m u l t i - d e g r e e o f freedom systems i s s i m i l a r t o t h a t g i v e n i n C h a p t e r 2 f o r s i n g l e degree o f freedom systems. In m a t r i x n o t a t i o n t h e i n c r e m e n t a l e q u a t i o n o f m o t i o n f o r m u l t i - d e g r e e o f freedom systems can be w r i t t e n a s : 4 £ M ] + 2 £ c ] + [ K ] T A t 2 A t ' {Ax} [ M ] {1} A g a + [ M ] { 2 x Q + 4 k Q l + [ C ] { 2 x Q } A t . . . E q n . ( 5 . 1 ) where £ M J £ C J and £ K J a r e t h e mass, damping and s t i f f n e s s m a t r i c i e s f o r t h e system. 0 i s t h e ^ a c c e l e r a t i o n o f each node a t the s t a r t o f the time i n c r e m e n t . o i s t h e v e l o c i t y o f each node a t the s t a r t o f t h e time i n c r e m e n t . x Q i s t h e d i s p l a c e m e n t o f each node a t th e s t a r t o f the time i n c r e m e n t . A _ i s t h e change i n base a c c e l e r a t i o n f o r each i n c r e m e n t . &a A t i s t h e d u r a t i o n o f each i n c r e m e n t i n seconds. 1 4 0 I f i t i s assumed t h a t t h e v i s c o u s damping can be r e p r e s e n t e d as a •combination o f mass-dependent and i n i t i a l s t i f f n e s s - d e p e n d e n t e f f e c t s , t h e damping m a t r i x can be r e p l a c e d by: .Eqn..( 5.2) I n t r o d u c i n g t h i s i n t o t h e i n c r e m e n t a l e q u a t i o n o f m o t i o n and c o l l e c t i n g terms g i v e s : ....Eqn. ( 5 . 3 ) T h i s i s t h e form o f t h e e q u a t i o n u s ed i n t h e computer program f o r t h i s s t u d y . The e q u a t i o n was i n t e g r a t e d u s i n g time s t e p s o f 0.01 s e c . f o r systems w i t h p e r i o d s g r e a t e r t h a n 0.5 s e c . and 0.005 s e c . f o r s h o r t e r p e r i o d systems. E q u i l i b r i u m checks were made a t t h e end o f each time s t e p , and c o r r e c t i v e f o r c e s a p p l i e d when r e q u i r e d . T h i s p r o c e d u r e was s i m i l a r t o t h a t g i v e n i n d e t a i l i n s e c t i o n 2.3-c< U t : A t [ M ] • _2 * [KT_ A.t 1 4 1 5.3 Energy C a l c u l a t i o n s i The c a l c u l a t i o n o f e n e r g i e s w i t h i n t h e m u l t i - d e g r e e o f freedom systems i s s i m i l a r t o the s i n g l e degree o f freedom system c a l c u l a t i o n s . In t h e computer s u b r o u t i n e f o r m u l t i - d e g r e e o f freedom systems t h e c a l -c u l a t i o n s a r e made i n t h e f o l l o w i n g sequence a t t h e end o f e a c h time s t e p . 1. i F o r each degree o f freedom i n system. ( a ) C a l c u l a t e t h e e n e r g y s t o r e d and d i s s i p a t e d by each s p r i n g . ( b ) C a l c u l a t e t h e k i n e t i c e nergy f o r each mass. ( c ) C a l c u l a t e e nergy d i s s i p a t e d by mass p r o p o r t i o n a l v i s c o u s damping. ( d ) C a l c u l a t e e nergy d i s s i p a t e d b y i n i t i a l s t i f f n e s s p r o p o r t i o n a l v i s c o u s damping. ( e ) Sum component e n e r g i e s f o r each degree o f freedom. 2. Sum a l l e n e r g i e s s t o r e d w i t h i n t h e system t o t h i s p o i n t . 3. C a l c u l a t e the t o t a l e n ergy i n p u t t o t h e s t r u c t u r e t o t h i s p o i n t . T o t a l E n ergy Input NDF MASS i * TOTAL ACCEL. ± * GROUND DISPLACEMENT i = .1 The t o t a l a c c e l e r a t i o n f o r each mass i s t a k e n as t h e average t o t a l a c c e l e r a t i o n d u r i n g t h e t i m e s t e p . 4. E n e r g y b a l a n c e checks a r e made b y e q u a t i n g t h e s t o r e d and i n p u t e n e r g i e s . 142 CHAPTER 6 TOTAL ENERGY DISSIPATED IN MULTI-DEGREE OF FREEDOM SYSTEMS 6.1 INTRODUCTION D e t a i l s o f a p r e l i m i n a r y s t u d y o f energy d i s s i p a t i o n mechanisms i n m u l t i - d e g r e e o f freedom systems i s r e p o r t e d i n t h i s , and t h e f o l l o w i n g c h a p t e r . U l t i m a t e l y , t h e o b j e c t i v e o f such a s t u d y would be t o g a i n s u f f i c i e n t u n d e r s t a n d i n g o f t h e en e r g y d i s s i p a t i o n mechanisms i n v o l v e d I n such systems, t o e n a b l e t h e f o r m u l a t i o n o f a d e s i g n method b a s e d upon t h i s knowledge. O b v i o u s l y , such a t a s k would r e q u i r e a major r e s e a r c h e f f o r t , and i s beyond t h e r e s o u r c e s o f t h e p r e s e n t s t u d y . N e v e r t h e l e s s , i n t h i s s t u d y s u f f i c i e n t i n s i g h t i s g a i n e d about t h e n a t u r e o f t h e d i s s i -p a t i o n mechanisms t o j u s t i f y o p t i m i s m about t h e s u i t a b i l i t y o f such an approach f o r m u l t i - d e g r e e o f freedom systems. The p r e s e n t c h a p t e r d e a l s w i t h t h e t o t a l e n e r g y d i s s i p a t e d by t h e s e systems. In p a r t i c u l a r , t h e p o s s i b i l i t y o f o b t a i n i n g t h e t o t a l e n ergy t o be d i s s i p a t e d by m u l t i - d e g r e e o f freedom systems, by e x t r a p o l a t i o n o f the r e s u l t s f o r s i n g l e - d e g r e e o f freedom systems, r e p o r t e d i n c h a p t e r 3, i s s t u d i e d . A l s o t h e e f f e c t s o f d i f f e r e n t h y s t e r e t i c models and forms o f v i s c o u s damping, mass p r o p o r t i o n a l o r s t i f f n e s s p r o p o r t i o n a l o r c o m b i n a t i o n s o f b o t h , upon t h e t o t a l e n ergy d i s s i p a t e d i s i n v e s t i g a t e d . The d i s t r i b u t i o n o f t h e e n e r g y w i t h i n the m u l t i - d e g r e e o f freedom systems i s r e p o r t e d i n t h e n e x t c h a p t e r . D e t a i l e d r e s u l t s o f t h i s s t u d y a r e r e p r o d u c e d i n Appendix C. 6.2 S t r u c t u r e s Used i n T h i s S t u d y In t h i s s t u d y s i x d i f f e r e n t s t r u c t u r e s a r e u s e d . They a r e i n t e n d e d t o c o v e r as wide a v a r i e t y o f s t r u c t u r a l t y p e s as p o s s i b l e . A l l s i x s t r u c t u r e s were i n v e s t i g a t e d u s i n g d i f f e r e n t ground m o t i o n r e c o r d s , and b o t h E l a s t o - P l a s t i c and S t i f f n e s s D e g r a d i n g h y s t e r e t i c models. D e t a i l s o f the s t r u c t u r e s a r e g i v e n i n f i g 6.1 and d i s c u s s e d b r i e f l y below. STRUCTURE 1 T h i s i s a t h r e e s t o r e y s t r u c t u r e w i t h i n c r e a s i n g mass, s t i f f n e s s and y i e l d s t r e n g t h s from t o p t o bottom s t o r i e s . T h i s system i s i n t e n d e d t o r e p r e s e n t the' s t a n d a r d s t r u c t u r a l arrangement t h a t would be o b t a i n e d f r o m a b u i l d i n g code a n a l y s i s . STRUCTURE 2 A t h r e e s t o r e y s t r u c t u r e w i t h c o n s t a n t mass, s t i f f n e s s and y i e l d s t r e n g t h f o r each s t o r e y . STRUCTURE 3 A t h r e e s t o r e y s t r u c t u r e s i m i l a r t o s t r u c t u r e 2 h a v i n g i d e n t i c a l mass.and s t i f f n e s s b u t w i t h o n e - t h i r d t h e y i e l d s t r e n g t h o f s t r u c t u r e two. STRUCTURE L T h i s t h r e e s t o r e y s t r u c t u r e r e p r e s e n t s a s o f t f i r s t s t o r e y system. Each f l o o r i s assumed t o have e q u a l mass, t h e s t i f f n e s s and y i e l d s t r e n g t h 144 of the lowest storey are one-half of the values f o r the upper two f l o o r s . STRUCTURE 5 Another soft f i r s t storey structure s i m i l a r to structure four, but with the s t i f f n e s s and y i e l d values of the lowest f l o o r equal to one-third of the values f o r the upper f l o o r s . STRUCTURE 6 A ten storey structure with equal mass s t i f f n e s s and y i e l d strength 'at each storey. 145 K - c o o FY) E L D = 2 2 4 ^ X = I E O O K / M F Y I E L D " 377 K = / © O O K / M FYIE.LD = 5 1 3 K K = I 2 O 0 7 I P J F Y / E L D = 3 7 7 K = l 2 o o K / w FYIE.LD - 3 7 7 K = 1 2 0 0 K / i M F Y J E L O = 3 7 7 K K K S T R U C T U R E 1 T . - 0 - 4 3 3 S E C . T = 0 - 2 0 2 S E C . S T g U C T U E E . g Tj = © " 3 3 3 S E C . f 2 = O- | I 9 S E C . K= l 2 o o F Y I E L P 12 G> K K = 1 2 0 0 FY IELP= |2<2» K K - ! 2 o o ^ / J M F Y I E L D = I2<2>' o-fc&7 o<2>&>7 F Y / E L D ^ 3 7 7 K K = I Z O O ' V / W F Y ) E L D = 3 7 7 K K = <g .oo *7iw F Y J E L P * /8fi> K S T R U C T U R E 5> T ( = 0 . 3 3 3 S £ C . J Z - O-IO S E C . -|" = 0-47 S E C . J 2 = C 3 - I 7 S B C . 2. K = \ZOOK/IKJ F Y l E L e = 3 7 7 * K= l 2 o o K / w F Y ) E L D =377 K F Y I E L D = J 2£>K ft: > N T, = 0-E.& S £ C . T 2 a O - 2 | S E C . -F I G , K - i s o o / i u F Y l E L P s 513 K T Y P I C A L ALL F L O O D S T| = <=>-s>2> S E C . Ta= <=>-33 S E C . 6,1 STRUCTURES USED IN M . D . F . STUDY 1 4 6 6.3 T o t a l E n ergy D i s s i p a t e d i n E l a s t o - P l a s t i c  M u l t i - D e g r e e o f Freedom Systems The r e s u l t s o f t h i s s t u d y f o r t h e t o t a l energy d i s s i p a t e d by E l a s t o -P l a s t i c m u l t i - d e g r e e o f freedom systems i n v e s t i g a t e d a r e summarised i n t a b l e 6.1. From t h e s e r e s u l t s , i t can be seen t h a t the e s t i m a t e s o f t o t a l e nergy t o be d i s s i p a t e d , o b t a i n e d from s i n g l e degree o f freedom s p e c t r a , ( s e e Appendix A), a r e i n good agreement w i t h t h e c a l c u l a t i o n s o b t a i n e d u s i n g t h e m u l t i - d e g r e e o f freedom computer model. I t i s n o t e d t h a t t h e agreement i n r e s u l t s from t h e two methods was poor f o r t h e case o f s t r o n g systems where i n t e r p o l a t i o n o f t h e s p e c t r a l r e s u l t s was r e q u i r e d . However, when t h e r e s u l t s o f t h e s i n g l e degree o f freedom systems w i t h c o r r e s p o n d i n g p a r a m e t e r s t o t h e m u l t i - d e g r e e o f freedom systems a r e used, t h e agreement i s c l o s e a s . c a n be seen i n t h e t a b l e . T h i s shows t h a t the d i f f i c u l t y i s n o t i n e x t r a p o l a t i o n , b u t i n t h e c o a r s e d a t a mesh used f o r t h e s i n g l e degree o f freedom s p e c t r a . The g e n e r a l c l o s e agreement i n t h e two s e t s o f r e s u l t s s u g g e s t s t h a t t h e m u l t i - d e g r e e o f freedom systems r e s p o n d p r e d o m i n a n t l y i n t h e fundamental mode. The e f f e c t o f v a r i o u s forms o f v i s c o u s damping, mass o r s t i f f n e s s p r o p o r t i o n a l o r b o t h , upon the t o t a l e n e r g y d i s s i p a t e d by t h e systems i s seen t o be s m a l l , p r o v i d e d t h a t t h e p e r c e n t a g e o f c r i t i c a l damping i n t h e system remains c o n s t a n t . A comparison o f t o t a l energy d i s s i p a t e d by m u l t i - d e g r e e o f freedom systems w i t h t h e s i n g l e degree o f freedom system r e s u l t s i s shown i n f i g 6.2. 1 4 7 1 Structure Number of Stories Fund L Period (Sec. ) YSR Excitation Viscous Total Energy (Inch Kips) Damping M.D.F. Time Step S.D.F. Spectrum 1 3 0.43 1.9 Taft N69W 1% M. & S. 1520 1803 * 1 3 0.43 1.9 Taft N69W IX Mass 1517 1803 » 1 3 0.43 1.9 Taft N69W Xt S t i f f . 1498 1803 * 1 3 0.43 1.9 Taft N69W 10* S t i f f . 1260 1670 * 1 3 0.43 0.8 El Centro N.S. 1% M. & S. 3715 4330 1 3 0.43 0.8 El Centro N.S. 1% S t i f f . 3700 4330 1 3 0.43 0.8 El Centro N.S. 10% S t i f f . 3491 4330 1 3 0.43 0.26 Pacoima S16E 1? M. & S. 17,437 18,418 1 3 0.43 0.26 Pacoima S16E 1% S t i f f . 17,557 18,418 1 3 0.43 0.26 Pacoima S16E 10? S t i f f . 16,605 18,418 2 3 0.33 3.1 Taft N69W 1* M. & S. 702 785 * 2 3 C.33 1.4 El Centro N.S. 1% M. & S. 682 693 * 2 3 0.33 1.4 El Centro N.S. 1% Mass 622 693 * 2 3 0.33 0.43 Pacoima S16E 1% M. & S. 8222 8515 2 3 0.33 0.43 Pacoima S16E 1% S t i f f . 8297 8515 2 3 0.33 0.43 Pacoima S16E 156 Mass 8482 8515 3 3 0.33 0.45 El Centro N.S. 1% M. & S. 1413 1539 L. 3 0.47 0.68 El Centro N.S. IS M. & S. 2356 2502 5 3 0.57 0.45 El Centro N.S. 1% M. & S. 2152 2309 6 10 0.99 0.3V El Centro N.S. 35! Mass 9005 9622 6 10 0.99 0.11 Pacoima S16E 3% Mass 35,353 3J.000 Note: M. = Mass S.= Stiffness * STRONG SYSTEMS. VALUES NOT INTERPOLATED FROM SPECTRA, BUT OBTAINED FROM CORRESPONDING S.D.F. SYSTEM. T A B L E 6.1 TOTAL ENERGY D I S S I P A T E D BY E L A S T O - P L A S T I C M . D . F . SYSTEMS 6.4- T o t a l E n e r g y D i s s i p a t e d i n S t i f f n e s s D e g r a d i n g M u l t i - D e g r e e o f Freedom Systems The r e s u l t s o f t h i s s t u d y f o r t h e t o t a l e n e r g y d i s s i p a t e d by S t i f f - . n ess D e g r a d i n g m u l t i - d e g r e e o f freedom systems a r e summarised i n t a b l e 6 . 2 The agreement between t h e r e s u l t s o f m u l t i - d e g r e e o f freedom c a l c u l a t i o n s and e x t r a p o l a t i o n from t h e s i n g l e degree o f freedom r e s u l t s i s seen t o be good. The agreement i n r e s u l t s f o r s t r o n g systems i s good when i n t e r p o l a t i n g between s p e c t r a l c u r v e s . T h i s i n d i c a t e s t h a t t h e s t r o n g d e g r a d i n g s t i f f n e s s systems a r e l e s s s e n s i t i v e t o v a r i a t i o n s i n t h e s t r e n g t h parameter t h a n was t h e case f o r t h e s t r o n g E l a s t o - P l a s t i c systems. A c o m p a r i s o n o f t o t a l e n e r g y d i s s i p a t e d by m u l t i - d e g r e e o f freedom systems w i t h t h e s i n g l e degree o f freedom systems r e s u l t s i s shown i n f i g 6.2 S t r u c t u r e Number o f S t o r i e s F u n d i P e r i o d ( S e c . ) YSR E x c i t a t i o n V i s c o u s Damping T o t a l Energy ( I n c h K i p s ) M.D.F. Time S t e p S.D.F. Spectrum 1 3 0 .43 1.9 T a f t N69W 1% M.& S. 1052 .3 1156 1 3 0 .43 0.8 E l C e ntro N.S. 1% M. & S. 6,744-3 6000 1 3 0 .43 0.26 Pacoima S16E 1% M. & S. 22,367.1 18 ,420 2 3 0 .33 3.1 T a f t N69W 1% M. & S. 553-0 504 2 3 0 .33 1.4 E l C e n t r o N.S. 1% M. & S. 1308.5 1540 2 3 0 .33 0 .43 Pacoima S16E 1% M. & S. 5,773 .3 6960 3 3 0 .33 0.45 E l C e ntro N.S. 1% Mass 1,946.8 1925 4 3 0.47 0.68 E l C e n t r o N.S. 1% M. & S. 2,709.6 2700 5 3 0.57 0.45 E l C e ntro N.S. 1% M. & S. 1,908.5 1925 6 10 0.99 0.37 E l C e ntro N.S. 3% Mass 7,356.9 7300 .6 10 0.99 0.11 Pacoima S16E 3% Mass 33,493.4 33,000 Note: M = Mass S = S t i f f n e s s T A B L E 6.2 T O T A L ENERGY D I S S I P A T E D BY DEGRADING S T I F F N E S S M . D . F . SYSTEMS « <B S T O R E Y F R A M E S (l?EF S<=>) X E L A S T O - P L A S T I C M . C F. S Y S T E M 6 • DE6KADINJ<3 STIFFIOESS M. C2 F". S Y S T E M S &3>'~7 % UPPE.1Z. & O U W D FOR M O D E R A T E S - O . FT S Y S T E M S W I T H Y S P ? - l -o (>»(& 3 ^ 7 ( 0 ) ) X M E P I A K J F"<=>Ri M O D E R A T E . S . C=>. F. S Y S T E M S W I T H YsR = l-o ( F I £ > 3 - 4 - 7 (d)} - | .5 1.0 P E R I O D ( I N o . o S E C O N D S ) 2.0 2 .5 F IG . 6,2 COMPARISON OF TOTAL ENERGY DISSIPATED BY MULTI-DEGREE OF FREEDOM SYSTEMS WITH SINGLE DEGREE OF FREEDOM RESULTS .CHAPTER 7  DISTRIBUTION OF ENERGY WITHIN MULTI-DEGREE OF FREEDOM SYSTEMS 7.1 I n t r o d u c t i o n The d i s t r i b u t i o n o f energy w i t h i n m u l t i - d e g r e e o f freedom systems i s s t u d i e d i n t h i s c h a p t e r . The s t r u c t u r e s u s ed i n t h i s s t u d y a r e d e s c r i b -ed i n s e c t i o n 6.2, and d e t a i l e d r e s u l t s o f the c a l c u l a t i o n s a r e r e p r o d u c e d i n Appendix C. Two a s p e c t s o f t h e en e r g y d i s t r i b u t i o n w i t h i n t h e systems, a r e con-s i d e r e d . F i r s t l y , t h e d i s t r i b u t i o n o f energy between t h e h y s t e r e t i c and v i s c o u s damping mechanisms i n t h e s t r u c t u r e i s c o n s i d e r e d . P r i n c i p a l i n t e r e s t i s g i v e n t o the- p o s s i b i l i t y o f e x t r a p o l a t i n g from t h e s i n g l e degree o f freedom r e s u l t s t o the m u l t i - d e g r e e o f freedom c a s e . The second a s p e c t c o n s i d e r e d i s the d i s t r i b u t i o n o f t h e en e r g y d i s s i p a t e d by t h e two mechanisms a t v a r i o u s l o c a t i o n s w i t h i n t h e system; t h e r e t h e aim i s t o i d e n t i f y g e n e r a l p a t t e r n s and t r e n d s which c o u l d be s u i t a b l e f o r a d e s i g n method. 7.2 D i s t r i b u t i o n o f Energy Between H y s t e r e t i c and V i s c o u s Damping Mechanisms The d i s t r i b u t i o n o f e n e r g y between the h y s t e r e t i c and v i s c o u s 152 damping mechanisms i n the systems s t u d i e d i s summarized i n t a b l e 7.1 f o r the E l a s t o - P l a s t i c systems, and i n t a b l e 7.2 f o r t h e d e g r a d i n g s t i f f n e s s systems. I t i s seen from b o t h s e t s o f r e s u l t s t h a t t h e r e i s good agreement between the m u l t i - d e g r e e o f freedom c a l c u l a t i o n s , and t h e s i n g l e degree o f freedom r e s u l t s . The s i n g l e degree o f freedom r e s u l t s g i v e n i n the t a b l e s a r e t a k e n from f i g s . 3.49 and 3-50. In the case o f t h e s t r o n g E l a s t o - P l a s t i c systems s u b j e c t e d t o the T a f t N69W r e c o r d t h e s i n g l e degree o f freedom r e s u l t s o b t a i n e d from f i g 3-49, u n d e r - e s t i m a t e t h e p r o p o r t i o n o f e n e r g y t o be d i s s i p a t e d i n h y s t e r e s i s . T h i s d i f f i c u l t y can be overcome by u s i n g a f i n e r d a t a mesh when p r o d u c i n g s p e c t r a f o r s t r o n g E l a s t o - P l a s t i c systems. The r e s u l t s o b t a i n e d from s i n g l e degree o f freedom systems w i t h c o r r e s p o n d i n g p a r a m e t e r s t o t h e m u l t i -degree o f freedom systems compare w e l l w i t h the m u l t i - d e g r e e o f freedom r e s u l t s . Structure Number of Stories Fund1 Perlod (Sec.) YSR Excitation Viscous Damping % Energy dissipated in Hysteresis M.D.F. Time Step S.D.F. Spectrum + 1 3 0.43 1.9 Taft N69W 1% S & M 68 45 (70)* 1 3 0.43 1.9 Taft N69W 1% Mass 65 45 (70)* 1 3 0.43 1.9 Taft N69W 1% S t i f f . 61 45 (70)* 1 3 0.43 1.9 Taft N69W 10% S t i f f . 0 10 1 3 0.43 0.8 El Centro N.S. 1% S & M 65 75 1 3 0.43 0.8 El Centro N.S. 1% S t i f f . 75 75 1 3 0.43 0.8 El Centro N.S. 10% S t i f f . 22 50 1 3 0.43 0.26 Pacoima S16E 1% S & M 71 80 1 3 0.43 0.26 Pacoima S16E 1% S t i f f . 74 80 1 3 0.43 0.26 Pacoima S16E 10% S t i f f . 41 60 2 3 0.33 3.1 Taft N69W 1% S & M. 20 35 2 3 0.33 1.4 El Centro N.S. 1% S & M. 34 50 2 3 0.33 1.4 El Centro N.S. 1% Mass 43 50 2 3 0.33 0.43 Pacoima S16E 1% S & M. 70 ' 80 2 3 0.33 0.43 Pacoima S16E 1% S t i f f . 72 80 2 3 0.33 0.43 Pacoima S16E 1% Mass 74 80 3 3 0.33 0.45 El Centro N.S. 1% S & M. 90 80 3 0.47 0.68 El Centro N.S. 1% S & M. 73 80 5 3 0.57 0.45 El Centro N.S. It S 4 M. 77 80 6 10 0.99 0.37 El Centro N.S. 3% Mass 65 70 6 10 0.99 0.11 Pacoima S16E 3% Mass 64 70 Note: M = Mass S = Stiffness * Strong systems. Values obtained from corresponding S.D.F. system. + Values obtained from f i g . 3-49.. T A B L E 7,1 D I S S I P A T E D IN H Y S T E R E S I S FOR P L A S T I C M. D. F . S Y S T E M S . t—1 vn ENERGY E L A S T O -S t r u c t u r e Number o f F u n d i YSR E x c i t a t i o n V i s c o u s % Energy D i s s i p a t e d i n H y s t e r e s i s S t o r i e s P e r i o d ( S e c . ) Damping M.D.F. Time Step S.D.F. Spectrum + . 1 3 0 . 4 3 1.9 T a f t N69W 1% S & M 86 80 1 3 0.43 0.8 E l C e n t r o N.S. 1% S & M 86 80 1 3 0 . 4 3 0.26 Pacoima S16E 1% S & M 71 70 2 3 0.33 .3.1 T a f t N69W 1% S & M 69 80 2 3 0 . 3 3 1.4 E l C e n t r o N.S. 1% S & M 77 80 2 i 3 0 . 3 3 0.43 Pacoima S16E 156 S & M 57 70 3 3 0 . 3 3 0.45 E l C e n t r o N.S. 1% Mass 74 70 4 3 0.47 0.68 E l C e n t r o N.S. 1% S & M 70 75 5 3 0.57 0.45 E l C e n t r o N.S. 1% S & M 69 75 6 10 0.99 0.37 E l C e n t r o N.S. 3% Mass ' 70 65 6 10 0.99 0.11 Pacoima S16E 3% Mass 50 50 Note: M--= Mass S = S t i f f n e s s + V a l u e s o b t a i n e d from f i g . 3.50 T A B L E 7.2 ENERGY D I S S I P A T E D IN H Y S T E R E S I S FOR S T I F F N E S S DEGRADING M. D. F . S Y S T E M S . 155 7.3 D i s t r i b u t i o n o f Energy i n E l a s t o - P l a s t i c M u l t i - D e g r e e o f Freedom Systems 7 .3.1 E f f e c t o f D i f f e r e n t Forms o f V i s c o u s Damping on T o t a l E n e r g y  and Energy D i s t r i b u t i o n Some a p p r e c i a t i o n o f t h e e f f e c t o f d i f f e r e n t t y p e s o f v i s c o u s damping, s t i f f n e s s o r mass p r o p o r t i o n a l , o r a c o m b i n a t i o n o f b o t h , upon the t o t a l e n ergy d i s s i p a t e d by t h e systems under s t u d y , and t h e d i - s t r i b u t i o n o f the energy w i t h i n the systems, was o b t a i n e d from a p a r a m e t r i c s t u d y . Some t y p i c a l r e s u l t s f o r systems s u b j e c t e d t o the T a f t N69W and Pacoima S16E ground m o t i o n r e c o r d s a r e p r e s e n t e d i n t a b l e 7 .3-I t can be seen t h a t t h e e f f e c t o f t h e t h r e e t y p e s o f v i s c o u s damping i s o n l y m a r g i n a l on t h e t o t a l e n e r g y d i s s i p a t e d , and on the d i s t r i b u t i o n o f the energy w i t h i n the systems. The use o f average v a l u e s f o r t h e s e systems i s p r a c t i c a l . 156 STRUCTURE 1 ELASTO-PLASTIC 30 SECONDS TAFT N69W V i s c o u s T o t a l E n e r g y D i s s i p a t e d E n e r g y D i s t r i b u t i o n % Damping E n e r g y ( I n c h K i p s ) By H y s t e r e s i s % S t o r e y 1 S t o r e y 2 S t o r e y • 3 1% M & S 1520 68 41.5 16.8 9.7 135 S t i f f . 1495 60.6 • 33.1 21.7 5.8 1% Mass 1517 64.8 39.3 15.4 10.0 Mean. 1511 64.5 38.0 18.0 8.5 STRUCTURE 2 ELASTO-PLASTIC • 30 SECONDS PACOIMA S16E V i s c o u s T o t a l E n e rgy D i s s i p a t e d . E n e r g y D i s t r i b u t i o n % Damping E n e r g y ( I n c h K i p s ) By H y s t e r e s i s % S t o r e y 1 " S t o r e y 2 S t o r e y 3 1% M & S 8222.6 '69.3 61.9 7.3 0.1 1% S t i f f . 8297.0 71.3 63.4 7.8 O'.T 1% Mass 8483.0 73.8 64.8 8.5 0.5 Mean. 8334.2 71.5 63.4 7.9 0.23 T A B L E 7.3 E F F E C T OF VARIOUS FORMS OF V I S C O U S DAMPING ON ENERGY D I S T R I B U T I O N WITHIN E L A S T O - P L A S T I C M . D . F . S Y S T E M S . 157 7.3-2 Summary o f R e s u l t s f o r S t r u c t u r e 1  E l a s t o - P l a s t i c Model F o r t h e E l a s t o - P l a s t i c systems w i t h 1% o f c r i t i c a l v i s c o u s damping i t was found t h a t t h e d i s t r i b u t i o n o f e n e r g y between t h e h y s t e r e t i c and v i s c o u s mechanisms i s r e l a t i v e l y i n s e n s i t i v e t o d i f f e r e n t v a l u e s o f y i e l d s t r e n g t h r a t i o . F o r example i t can be seen from t a b l e 7.4 t h a t when t h i s s t r u c t u r e i s s u b j e c t e d t o t h e T a f t N69W r e c o r d i t responds as a s t r o n g s t r u c t u r e w i t h YSR = 1.9 and d i s s i p a t e s 67% o f t h e t o t a l e n e r g y i n . h y s t e r e s i s . The same s t r u c t u r e s u b j e c t e d t o t h e Pacoima S16E r e c o r d i s a weak s t r u c t u r e w i t h YSR = 0.26, and i n t h i s case 75% o f t h e t o t a l e n ergy i s d i s s i p a t e d i n h y s t e r e s i s . The r e s u l t s f o r t h e systems w i t h 10% o f c r i t i c a l v i s c o u s damping show t h a t t h e y i e l d s t r e n g t h r a t i o has a d i s t i n c t e f f e c t upon t h e e n e r g y d i s t r i b u t i o n w i t h i n the system. T h i s can be c l e a r l y seen i n t a b l e 7.5. In g e n e r a l i t i s n o t e d t h a t f o r t h e systems i n which t h e e n e r g y i s d i s s i p a t e d p r i n i c i p a l l y i n the h y s t e r e t i c mechanisms t h e m a j o r i t y o f t h e energy i s d i s s i p a t e d i n t h e l o w e s t s t o r e y . When the v i s c o u s damping forms the p r i n c i p a l d i s s i p a t i n g mechanism the m a j o r i t y o f t h e energy i s d i s s i p a t e d i n the upper s t o r i e s . These r e s u l t s a r e summarized i n t a b l e s 7.4 and 7.5. STRUCTURE 1 ELASTO-PLASTIC 1% VISCOUS DAMPING Earthquake YSR C a t e g o r y E nergy D i s s i p a t e d In H y s t e r e s i s {% ) Ene r g y D i s s i p a t e d In V i s c o u s Damping {%) T a f t N69W 1.9 S t r o n g 67 33 E l C e n t r o N.S. 6.8 Mod. 70 30 Pacoima S16E 0.26 Weak 75 25 DISTRIBUTION OF HYSTERETIC ENERGY WITHIN SYSTEM S t o r e y T a f t E l C e n t r o Pacoima ' 3 10 3 9 2 20 15 16 1 37 52 50 Z 67$ 70% 75% DISTRIBUTION OF ENERGY DISSIPATED BY VISCOUS DAMPING S t o r e y T a f t E l C e n t r o Pacoima 3 18 15 6 2 10 10 9 1 5 5 10 £ 33% 30% 25% T A B L E 7 .4 D I S T R I B U T I O N OF ENERGY IN STRUCTURE 1 E L A S T O - P L A S T I C MODEL 1% V ISCOUS DAMPING. S T R U C T U R E 1 E L A S T O - P L A S T I C 1056 VISCOUS DAMPING Ear t h q u a k e YSR C a t e g o r y E nergy D i s s i p a t e d E n e r g y D i s s i p a t e d In H y s t e r e s i s (5?) In V i s c o u s Damping {%) T a f t N69W 1.9 S t r o n g 0 100 E l C e n t r o N.S. 0.8 Mod. 22 78 Pacoima S16E 0.26 Weak 40 60 DISTRIBUTION OF HYSTERETIC ENERGY WITHIN SYSTEM S t o r e y T a f t E l C e n t r o Pacoima 3 0 0 2 2 0 4 12 1 0 18 26 I 05? 225? 405? DISTRIBUTION OF ENERGY DISSIPATED BY VISCOUS DAMPING S t o r e y T a f t E l C e n t r o Pacoima 3 54 38 20 2 34 27 20 1 12 13 20 E 1005? 785? 605? T A B L E 7 .5 D I S T R I B U T I O N OF ENERGY IN STRUCTURE 1 E L A S T O - P L A S T I C MODEL 10% V ISCOUS DAMPING. 7.3-3 Summary o f R e s u l t s f o r S t r u c t u r e 2  E l a s t o - P l a s t i c Model The r e s u l t s f o r t h i s s t r u c t u r e s u b j e c t e d t o d i f f e r e n t ground motions show t h a t the d i s t r i b u t i o n o f energy w i t h i n t h e systems i s g r e a t l y depend-ent upon-the y i e l d s t r e n g t h r a t i o . These r e s u l t s f o r systems w i t h 1% o f c r i t i c a l v i s c o u s damping a r e c o n s i d e r a b l y more s e n s i t i v e t o v a r i a t i o n s i n t h i s p arameter t h a n t h o s e r e p o r t e d f o r s t r u c t u r e 1 i n the p r e v i o u s s e c t i o n . The r e s u l t s f o r t h i s s t r u c t u r e a r e summarized i n t a b l e 7.6. STRUCTURE 2 ELASTO-PLASTIC 1$ VISCOUS DAMPING. Earth q u a k e YSR C a t e g o r y Energy D i s s i p a t e d E n e r g y D i s s i p a t e d In H y s t e r e s i s ( $) In V i s c o u s Damping ( $ ) T a f t N69W 3.1 S t r o n g 19 81 E l C e n t r o N.S. 1.4 S t r o n g 38 62 Pacoima S16E 0.43 Mod. 70 30 DISTRIBUTION OF HYSTERETIC ENERGY WITHIN SYSTEM S t o r e y T a f t E l C e n t r o Pacoima 3 0 . 0 0 2 0 0 8 1 19 38 62 E 19$ 38$ 70$ DISTRIBUTION OF ENERGY DISSIPATED BY VISCOUS DAMPING S t o r e y T a f t E l C e n t r o Pacoima 3 43 32 11 2 28 22 • 11 1 10 8 8 E 81$ 62$ 30$ T A B L E 7 .6 D I S T R I B U T I O N OF ENERGY IN STRUCTURE 2 E L A S T O - P L A S T I C MODEL 1% V ISCOUS DAMPING 162 7.3.4- Comparison o f E n e r g i e s i n S t r u c t u r e s 1, 2, 3, 4 and 5. E l a s t o - P l a s t i c Model T h i s comparison c o n s i d e r s t h e t o t a l energy and e nergy d i s t r i b u t i o n i n s t r u c t u r e s 1, 2, 3, 4 and 5, w i t h a l l systems h a v i n g 1% mass and s t i f f -n e s s p r o p o r t i o n a l v i s c o u s damping, and s u b j e c t e d t o 30 s e e s , o f E l C e n t r o N.S. The r e s u l t s a r e p r e s e n t e d i n t a b l e 7.7. The l a r g e v a r i a t i o n i n t h e amount o f t o t a l e n e r g y d i s s i p a t e d by t h e s e systems i s a r e s u l t o f v a r i a t i o n s i n p e r i o d and y i e l d s t r e n g t h r a t i o . In a l l c a s e s ( e x c e p t f o r s t r u c t u r e 2), h y s t e r e s i s forms t h e p r i n i c i p a l d i s s i p a t i n g mechanism, and the m a j o r i t y o f t h e h y s t e r e t i c energy i s d i s s i p a t -ed i n the l o w e s t s t o r e y . T h i s i s t r u e n o t o n l y f o r t h e systems h a v i n g a ' s o f t f i r s t s t o r e y ' , but a l s o when t h e r e i s a s t r o n g f i r s t s t o r e y as i n s t r u c t u r e 1. T h i s shows t h a t th e f i r s t s t o r e y i s a major f a c t o r i n d e t e r -m i n i n g the adequacy o f a system i n e a r t h q u a k e e x c i t a t i o n . I n a l l c a s e s t h e m a j o r i t y o f t h e e n e r g y d i s s i p a t e d b y v i s c o u s damp-i n g i s l o c a t e d i n t h e u p p er s t o r i e s . A l l f i v e o f the t h r e e s t o r e y s t r u c t u r e s c o n s i d e r e d i n t h i s s t u d y e x h i b i t a p r e d o m i n a n t l y f i r s t mode r e s p o n s e . The e f f e c t o f h i g h e r modes i s not s i g n i f i c a n t i n any o f t h e s e systems. S t r u c t u r e P e r i o d ( S e c . ) YSR T o t a l E n e rgy ( I n c h K i p s ) E n e r g y D i s s i p a t e d In H y s t e r e s i s (%) D i s t r i b u t i o n Energy D i s s i p a t e d i n V i s c o u s Damping (% ) D i s t r i b u t i o n 1 2 3 1 2 3 1 .43 .8 3715 65.5 48.9 13.9 2 6 34.5 6.2 11.0 16. 3 2 .33 1.4 682 34.0 34.0 0 0 65.0 8.2 22.0 34 8 3 .33 .45 1413 90.0 83.0 7.0 0 10.0 2.0 3.0 5 0 4 .47 .68 2356 73.0 73.0 0 0 27.0 7.0 8.0 12 0 5 .57 .45 2152 77.0 77.0 0 0 23.0 6.0 7.0 10 0 T A B L E 7 .7 ENERGY D I S T R I B U T I O N IN STRUCTURES 1, 2, 3 , 5 E L A S T O - P L A S T I C MODEL 1% V ISCOUS DAMPING S U B J E C T E D TO E L CENTRO N . S . O^  V-0 7.3.5 Summary o f R e s u l t s f o r S t r u c t u r e 6  E l a s t o - P l a s t i c Model A summary o f t h e r e s u l t s f o r t h i s s t r u c t u r e when s u b j e c t e d t o t h e E l C e n t r o N.S. and t h e Pacoima S16E ground m o t i o n r e c o r d s i s p r e s e n t e d i n t a b l e 7.8. The d i s t r i b u t i o n o f the energy d i s s i p a t e d ' w h e n t h e system i s s u b j e c t e d t o t h e E l C e n t r o r e c o r d i n wh i c h the system i s o f moderate s t r e n g -t h i s v e r y s i m i l a r t o t h e d i s t r i b u t i o n f o r t h e Pacoima r e c o r d i n which case the system i s weak. I t i s o b s e r v e d t h a t t h e p r i n c i p a l d i s s i p a t i n g mechanism i s h y s t e r e s i i n b o t h c a s e s . Almost a l l t h e h y s t e r e t i c e nergy i s d i s s i p a t e d i n t h e lower t h r e e s t o r i e s o f t h e s t r u c t u r e . The d i s t r i b u t i o n o f energy d i s s i p a t e d by v i s c o u s damping i s f o u n d t o be c o n s t a n t f o r t h e t o p h a l f o f t h e systems, showing t h e e f f e c t o f h i g h e r modes. 165 S T R U C T U R E 6 ELASTO-PLASTIC 3% Mass Damping E l C e n t r o N.S. Pacoima S16E S t o r e y E n e r g y D i s s i p a t e d By H y s t e r e s i s E n e r g y D i s s i p a t e d By V i s c o u s Damping S t o r e y E n e r g y D i s s i p a t e d By H y s t e r e s i s E n e r g y D i s s i p a t e d By V i s c o u s Damping 10' 0 6.6 10 0 4 . 3 9 0 5.8 9 0.2 4.1 8 0 5.0 8 0.9 s 4.0 7 0 4 .3 7 1 .5 3.8 6 1 .5 3-6 6 3.1 3.8 . 5 3.7 2.9 5 3.6 3.6 4 6.2 2 .5 4 ' 2.6 3.3 3 10.7 1.9 3.1 3.1 2 9 .3 1 .5 2 2..61 2.9 1 33.1 1.1 1 46.6 2.7 T o t a l 64.6 35.4 • T o t a l 64.1 35.9 T A B L E 7 . 8 D I S T R I B U T I O N OF ENERGY IN S T R U C T U R E 6. E L A S T O - P L A S T I C MODEL 3% V I S C O U S DAMPING 7.4- D i s t r i b u t i o n o f E n e r g y i n S t i f f n e s s D e g r a d i n g M u l t i - D e g r e e o f Freedom Systems 7.4-1 Summary o f R e s u l t s f o r S t r u c t u r e 1  S t i f f n e s s D e g r a d i n g Model A summary o f the r e s u l t s o b t a i n e d f o r t h i s s t r u c t u r e s u b j e c t e d t o t h r e e ground m o t i o n r e c o r d s i s p r e s e n t e d i n t a b l e 7.9. I t can be seen t h a t t h e systems w i t h s t r o n g e r y i e l d s t r e n g t h r a t i o s d i s s i p a t e s l i g h t l y l a r g e r p e r c e n t a g e s f o r e n e r g y i n h y s t e r e s i s . F o r example t h i s system when s u b j e c t e d t o the T a f t N69W r e c o r d i s a s t r o n g system w i t h YSR = 1.9 and d i s s i p a t e s 85.9$ o f the energy i n h y s t e r e s i s . The same system s u b j e c t e d t o the Pacoima S16E r e c o r d i s a weak system w i t h YSR = 0.26 and d i s s i p a t e s 70.9$ o f the e n e r g y i n h y s t e r e s i s . However t h i s d i f f e r e n c e i s c o n s i d e r e d -s m a l l i n comparison t o t h e change i n i n t e n s i t y o f the two r e c o r d s u s ed. An i n t e r e s t i n g t r e n d t h a t i s found i n t h e s e r e s u l t s i s t h e a p p a r e n t c o n c e n t r a t i o n o f h y s t e r e t i c e nergy d i s s i p a t i o n i n the l o w e s t s t o r e y as the y i e l d s t r e n g t h r a t i o o f t h e system i s r e d u c e d . In summary i t i s n o t e d t h a t t h e d i s t r i b u t i o n o f t h e t o t a l e n e r g y between t h e h y s t e r e t i c and v i s c o u s damping mechanisms i s n o t s i g n i f i c a n t l y a f f e c t e d by the changes i n YSR, but the l o c a t i o n o f t h e energy d i s s i p a t e d by t h e h y s t e r e t i c mechanisms i s v e r y s e n s i t i v e t o changes i n t h e YSR, f o r t h e s e s t i f f n e s s d e g r a d i n g models. STRUCTURE 1 STIFFNESS DEGRADING 15? VISCOUS DAMPING Eart h q u a k e YSR C a t e g o r y E nergy D i s s i p a t e d In H y s t e r e s i s ( 5?) Energy D i s s i p a t e d In V i s c o u s Damping {%) T a f t N69W 1.9 S t r o n g 85.9 14.1 E l C e n t r o N.S. 0.8 Mod. 86.4 13.6 Pacoima S16E 0.26 Weak 70.9 29.1 DISTRIBUTION OF HYSTERETIC ENERGY WITHIN SYSTEM S t o r e y T a f t E l C e n t r o Pacoima 3 19.1 6.2 1.2 2 33.3 29.4 3.2 1 33.6 50.8 66.5 Z 85.9% 86 . 45? 70.9% DISTRIBUTION OF ENERGY DISSIPATED BY VISCOUS DAMPING S t o r e y T a f t E l C e n t r o Pacoima 3 7.0 4.5 6.4 2 4.8 5.5 9.1 1 2 . 3 3.6 13.6 Z 13.6% 29.15? T A B L E 7.9 D I S T R I B U T I O N OF ENERGY IN STRUCTURE 1 S T I F F N E S S DEGRADING MODEL 1% V ISCOUS DAMPING 1 6 8 7.4-2 Summary o f R e s u l t s f o r S t r u c t u r e 2  S t i f f n e s s D e g r a d i n g Model The r e s u l t s f o r t h i s system a r e summarized i n t a b l e 7.10. F o r t h e s e systems i t was f o u n d t h a t the e f f e c t o f changes i n YSR i s o f the same o r d e r f o r b o t h t h e d i s t r i b u t i o n o f e n e r g y between the two mechanisms and the l o c a t i o n o f t h e e n e r g y d i s s i p a t e d i n the system. These e f f e c t s a r e much s m a l l e r t h a n was found f o r S t r u c t u r e 1. In a l l c a s e s the l o w e s t s t o r e y d i s s i p a t e s a p p r o x i m a t e l y t h e same p r o p o r t i o n o f e n e r g y i n h y s t e r e s i s . A l s o i t i s o b s e r v e d t h a t t h e l o w e s t s t o r e y d i s s i p a t e s t h e m a j o r i t y o f t h e energy i n v i s c o u s damping. The e x p l a n a t i o n f o r t h i s l i e s i n t h e l e n g t h e n i n g o f the funda m e n t a l p e r i o d caused by t h e s t i f f n e s s d e g r a d a t i o n . The energy spectrum f o r the Pacoima r e c o r d used has an u n u s u a l c o n c e n t r a t i o n o f e n e r g y at t h e l o n g p e r i o d end and the systems u s e d a r e i n e f f e c t moving i n t o t h e h i g h e r energy p e r i o d s as t h e s t i f f n e s s d e g r a d e s . STRUCTURE 2 STIFFNESS DEGRADING 135 VISCOUS DAMPING Earthquake YSR C a t e g o r y E nergy D i s s i p a t e d In H y s t e r e s i s ( $ ) Energy D i s s i p a t e d In V i s c o u s Damping ( $ ) T a f t N69W 3.1 S t r o n g 68.6 31.4 E l C e n t r o N.S. 1.4 S t r o n g 76.6 23.4 Pacoima S16E 0.43 Mod. 56.5 43.5 DISTRIBUTION OF HYSTERETIC ENERGY WITHIN SYSTEM S t o r e y T a f t E l C e n t r o Pacoima 3 0.0 2.4 0.9 2 8.8 22 . 3 5.5 1 59.9 51.9 50.1 E . 68.6$ 76.6$ 56.5$ DISTRIBUTION OF ENERGY DISSIPATED BY VISCOUS DAMPING S t o r e y T a f t E l C e n t r o Pacoima 3 15.5 10.6 12 . 3 2 10.2 8.5 12 . 3 1 5.7 4.3 18.9 E 31.4$ 23 . 4 $ 43.5$ T A B L E 7,10 D I S T R I B U T I O N OF ENERGY IN STRUCTURE 2 S T I F F N E S S DEGRADING MODEL 1% V ISCOUS DAMPING 170 7.A.3 Comparison o f E n e r g i e s i n S t r u c t u r e s 1, 2, 3, 4 and 5. S t i f f n e s s D e g r a d i n g Model. T h i s comparison i s s i m i l a r t o t h a t p r e s e n t e d f o r t h e E l a s t o - P l a s t i c model i n s e c t i o n 7.3-4. In t h i s s t u d y a l l t h e systems have 1% mass and . s t i f f n e s s p r o p o r t i o n a l v i s c o u s damping and a r e s u b j e c t e d t o t h i r t y seconds o f the E l C e n t r o N.S. r e c o r d . The r e s u l t s a r e p r e s e n t e d i n t a b l e 7.11. In a l l c a s e s i t i s found t h a t h y s t e r e s i s i s t h e p r i n c i p a l e n e r g y d i s s i p a t i n g mechanism. In s t r u c t u r e s 1 and 2 i t i s n o t e d t h a t the second s t o r e y d i s s i p a t e s a s i g n i f i c a n t p r o p o r t i o n o f t h e h y s t e r e t i c energy. T h i s i s n o t found t o be t h e case f o r t h e o t h e r s t r u c t u r e s s t u d i e d where t h e low-e s t s t o r e y d i s s i p a t e s almost a l l o f t h e h y s t e r e t i c e n ergy. F o r t h e s e systems the concept o f t h e ' s o f t f i r s t s t o r e y ' i s found t o be e f f e c t i v e i n concen-t r a t i n g the l o c a t i o n o f t h e h y s t e r e t i c and v i s c o u s e nergy d i s s i p a t i o n i n t o t h e l o w e s t s t o r e y . S t r u c t u r e P e r i o d ( S e c . ) YSR T o t a l E n e r g y ( I n c h K i p s ) Energy D i s s i p a t e d In H y s t e r e s i s (%) D i s t r i b u t i o n Energy D i s s i p a t e d In V i s c o u s Damping (%) D i s t r i b u t i o n 1 2 3 1 2 3 1 0.43 0.8 6744.3 86.4 50.8 29.4 6.2 13.6 3.6 5.5 4.5 2 0.33 1.4 1308.5 76.6 51.9 22.3 2.4 23-4 4.3 8.5 10.6 3 0.33 0.45 1946.8 73.4 71.0 1.9 0.5 26.6 8.4 8.9 9.3 4 0.47 0.68 2709.6 70.0 67.0 3.1 0.0 30.0 18.7 5.4 5.9 5 0.57 0.45 1908.5 69.1 68.7 0.4 0.0 30.9 19.7 5.2 6.0 T A B L E 7.11 ENERGY D I S T R I B U T I O N IN STRUCTURES 1, 2, 3, 4 & 5 S T I F F N E S S DEGRADING MODEL 1% V ISCOUS DAMPING S U B J E C T E D TO E L CENTRO N . S . 172 7.4--4 Summary o f R e s u l t s f o r S t r u c t u r e 6  S t i f f n e s s D e g r a d i n g Model The r e s u l t s o f t h i s s t u d y a r e p r e s e n t e d i n t a b l e 7.12. The r e s u l t s show t h a t i n t h e E l C e n t r o r e s p o n s e t h e h y s t e r e t i c mechanism i s the p r i n -c i p a l energy d i s s i p a t o r . However, when t h e system i s s u b j e c t e d t o t h e Pacoima r e c o r d t h e r e i s almost e q u a l d i s t r i b u t i o n o f energy between t h e two mechanisms. I t was a l s o be seen t h a t t h e r e d u c t i o n i n YSR t e n d s t o c o n c e n t r a t e t h e en e r g y d i s s i p a t e d i n h y s t e r e s i s i n t o t h e l o w e s t s t o r e y o f the s t r u c t u r e . The d i s t r i b u t i o n o f e n e r g y d i s s i p a t e d by t h e v i s c o u s damp-i n g mechanism i s a p p r o x i m a t e l y c o n s t a n t f o r t h e upper h a l f o f the s t r u c t u r e . 173 STRUCTURE 6 STIFFNESS DEGRADING 3% MASS DAMPING E l C e n t r o N.S. Pacoima S16E S t o r e y E n ergy D i s s i p a t e d By H y s t e r e s i s E n e r g y D i s s i p a t e d By V i s c o u s Damping S t o r e y E n e r g y D i s s i p a t e d By H y s t e r e s i s E n e r g y D i s s i p a t e d By V i s c o u s Damping 10 0.0 5.1 10 0.0 5.5 9 1.0 4.9 9 0.1 5.5 8 2.4 4.5 8 0 . 3 5.4 7 3 . 9 4.0 7 0.4 5.4 6 5.9 3.4 o 0.6 5.2 5 7.2 2.8 5 1.2 5.2 8.4 2.2 4 1.6 4.9 3 9.5 1.6 • 3 . 1.6 4.7 2 14.2 1.1 2 . iv. 3-8 4.5 1 17.2 0.6 1 39.8 4.2 . T o t a l 69.8 30.2 T o t a l 49.5 50.5 TABLE 7. 12 DISTRIBUTION OF ENERGY IN STIFFNESS DEGRADING MODEL STRUCTURE 6 3% VISCOUS DAMPING, 174 CHAPTER 8  CONCLUSION 8.1 Energy D i s s i p a t i o n i n S i n g l e Degree o f Freedom Systems The system s t r e n g t h parameter ( Y i e l d S t r e n g t h R a t i o , YSR) was f o u n d t o be o f g r e a t s i g n i f i c a n c e f o r a l l a s p e c t s o f r e s p o n s e measured i n t h i s s t u d y . The t o t a l e n ergy d i s s i p a t e d by systems which remained e l a s t i c d u r i n g the re s p o n s e does not form an upper bound f o r the t o t a l e n e r g y d i s s i p a t e d by systems which undergo s i g n i f i c a n t y i e l d e x c u r s i o n s d u r i n g t h e i r r e s p o n s e . In terms o f t h e YSR parameter t h e r e s p o n s e f o r a l l systems can be c l a s s i f i e d i n t o t h r e e b r o a d g r o u p s . 1. S t r o n g systems (YSR = 1 t o 10). These would i n c l u d e r e a l i s t i c b u i l d i n g s , w h i c h would, i n p r a c t i c e , t e n d t o have s h o r t p e r i o d s c h a r a c t e r i s t i c o f s h e a r w a l l systems. Response i s p r e d o m i n a n t l y e l a s t i c ; d u c t i l i t y demand i s i n t h e o r d e r o f 1 t o 3 when the p e r i o d i s l e s s t h a n 0.5 s e e s , but t h e r e a r e no major y i e l d e x c u r s i o n s . The e q u a l d i s p l a c e m e n t c r i t e r i o n i s good, p r o v i d e d t h a t t h e r e i s v i s c o u s damping a t 1% o f c r i t i c a l o r g r e a t e r , and t h e p e r i o d i s g r e a t e r t h a n 0.5 s e e s . The energy d i s s i p a t e d by t h e h y s t e r e s i s i n the members i s v e r y s e n s i t i v e t o the p e r i o d and f r e q u e n c y c o n t e n t o f t h e e a r t h q u a k e . 2. Moderate systems (YSR = 0.3 to 1 ) . These would a l s o i n c l u d e r e a l i s t i c b u i l d i n g s , w i t h l o n g e r p e r i o d s c h a r a c t e r i s t i c o f frame s t r u c t u r e s . 175 M a j o r y i e l d e x c u r s i o n s o c c u r , w i t h d u c t i l i t y demands up t o 10 f o r p e r i o d s o v e r 1 s e c . ( H i g h e r f o r l e s s r e a l i s t i c s t r u c t u r e s w i t h s h o r t p e r i o d s ) . The e q u a l d i s p l a c e m e n t c r i t e r i o n i s good f o r members w i t h n o n - d e g r a d i n g s t i f f n e s s , but does not h o l d when the s t i f f n e s s degrades u n l e s s t h e v i s c o u s damping i s h i g h . The h y s t e r e -t i c e n ergy d i s s i p a t e d i s l e s s s t r o n g l y p e r i o d dependent. 3. Weak systems (YSR < 0.3). Such systems would seldom be en c o u n t -e r e d i n p r a c t i c e ; t h e y would t e n d t o have v e r y l o n g p e r i o d s . a) D u c t i l i t y demand i s e x c e s s i v e f o r a l l systems. b ) R e q u i r e d E n e r g y C a p a c i t y i s v e r y weakly p e r i o d - d e p e n d e n t . c ) The e q u a l d i s p l a c e m e n t c r i t e r i o n does not a p p l y . In i n v e s t i g a t i n g t h e t i m e h i s t o r y o f en e r g y d i s s i p a t e d i n t h e h y s t e r e t i c mechanism i t was found t h a t the number o f e x c u r s i o n s i n t o t h e y i e l d range depends upon the t y p e o f eart h q u a k e and t h e Y i e l d S t r e n g t h R a t i o . F o r i m p u l -s i v e e a r t h q u a k e s , and f o r s t r o n g systems i n more d i s t a n t e a r t h q u a k e s , t h e r e a r e g e n e r a l l y 2 t o 3 such e x c u r s i o n s , w i t h t h e l a r g e s t d i s s i p a t i n g up t o 50% o f t h e t o t a l h y s t e r e t i c energy. F o r moderate systems i n more d i s t a n t e a r t h -quakes t h e r e a r e 3 t o 5 y i e l d e x c u r s i o n s , w i t h t h e l a r g e s t d i s s i p a t i n g up t o 30% o f t h e h y s t e r e t i c energy. 8.2 The Pr o p o s e d D e s i g n Method The p r e s e n t s t u d y has l e d t o t h e development o f a d e s i g n method f o r s e i s m i c d e s i g n o f s t r u c t u r e s , which c o n s i d e r s energy d i s s i p a t i o n as t h e prime d e s i g n parameter. In t h e . p r o p o s e d method t h e t o t a l e n ergy t o be a b s o r b e d by a s t r u c t u r e i s ap p r o x i m a t e d on the b a s i s o f t h e ' Y i e l d S t r e n g t h R a t i o (YSR) and p e r i o d ; t h e f r a c t i o n o f t h i s e n e r g y t h a t i s d i s s i p a t e d h y s t e r e t i c a l l y can, i n 176 t u r n , be p r e d i c t e d by means o f s p e c t r a t h a t depend upon t h e YSR, t h e p r o -p o r t i o n o f v i s c o u s damping and t h e h y s t e r e t i c model. Such s p e c t r a may be con-s t r u c t e d from t h e r e s p o n s e o f s i n g l e degree o f freedom systems. R e s u l t s so f a r suggest t h a t i t may be p o s s i b l e t o p r e d i c t i n advance t h e approximate d i s t r i b u t i o n o f h y s t e r e t i c e nergy d i s s i p a t i o n though t h e s t r u c t u r e ; i f t h i s , w i t h f u r t h e r s t u d y , p r o v e s t o be p r a c t i c a b l e , t h e b a s i s f o r a r a t i o n a l d e s i g n method e x i s t s . I t w i l l be p o s s i b l e t o p r e d i c t t h e amount o f h y s t e r e t i c energy t h a t must be a c c e p t e d by each l o c a t i o n i n t h e s t r u c t u r e , and t h e s i z e o f t h e h y s t e r e s i s l o o p f o r which t h e d e t a i l i n g must p r o v i d e . The form t h i s method t a k e s would make i t most u s e f u l f o r i n c o r p o r a t i o n i n t o t h e l i m i t s t a t e d e s i g n r e q u i r e m e n t s o f modern b u i l d i n g codes. 8.3 T o t a l E n ergy D i s s i p a t e d by M u l t i - D e g r e e o f Freedom Systems The r e s u l t s o f t h i s s t u d y i n d i c a t e t h a t a good e s t i m a t e o f t h e t o t a l e n e rgy t o be d i s s i p a t e d by m u l t i - d e g r e e o f freedom systems can be o b t a i n e d from a p p r o p r i a t e s p e c t r a p r o d u c e d w i t h a s i n g l e - d e g r e e o f freedom model. The e f f e c t s o f v a r i o u s forms o f v i s c o u s damping upon t h e t o t a l e n ergy d i s s i p a t -ed by t h e systems was found t o be s m a l l . The g e n e r a l l y c l o s e agreement i n t h e r e s u l t s s u g g e s t s t h a t t h e m u l t i -degree o f freedom systems r e s p o n d p r e d o m i n a n t l y i n the funda m e n t a l mode. F u r t h e r , the v a r i a b l e s w hich i n f l u e n c e e n e r g y d i s s i p a t i o n c h a r a c t e r i s t i c s i n s i n g l e - d e g r e e o f freedom systems a r e a l s o e f f e c t i v e f o r m u l t i - d e g r e e o f f r e e -dom systems. 8.4- The D i s t r i b u t i o n o f D i s s i p a t e d E n e r g y W i t h i n M u l t i - D e g r e e 177 o f Freedom Systems I t was shown i n c h a p t e r 7 t h a t the d i s t r i b u t i o n o f e n e r g y between t h e h y s t e r e t i c and v i s c o u s damping mechanisms i s a f u n c t i o n o f t h e m a t e r i a l p r o p e r t i e s , y i e l d s t r e n g t h r a t i o , p r o p o r t i o n o f v i s c o u s damping and p e r i o d . I t i s r e l a t i v e l y i n s e n s i t i v e t o d i f f e r e n t ground m o t i o n r e c o r d s , and can be e x t r a p o l a t e d w i t h r e a s o n a b l e a c c u r a c y from t h e s p e c t r a p r o d u c e d f o r s i n g l e degree o f freedom systems. The d i s t r i b u t i o n o f e n e r g y w i t h i n t h e s t r u c t u r e from s t o r e y t o s t o r e y i s f o u n d t o be dependent upon the m a t e r i a l h y s t e r e t i c p r o p e r t i e s u s e d and t h e mechanism c o n s i d e r e d ; (whether v i s c o u s o r h y s t e r e t i c ) . The d i s s i p a t i o n o f • energy by h y s t e r e s i s i s f o u n d t o t a k e p l a c e p r e d o m i n a n t l y i n t h e l o w e r s t o r i e s o f t h e s t r u c t u r e s s t u d i e d h e r e . F o r the E l a s t o - P l a s t i c model t h i s d i s t r i b u t i o n was f o u n d t o be r e l a t i v e l y i n d e p e n d e n t o f changes i n s t r e n g t h and s t i f f n e s s i n the l o w e r s t o r i e s . The s o f t f i r s t s t o r e y c o n c e p t d i d not work w e l l f o r t h e s e E l a s t o - P l a s t i c systems. T h i s was not found t o be the case f o r t h e s t i f f n e s s d e g r a d i n g model systems. Here i t was o b s e r v e d t h a t r e d u c t i o n i n t h e y i e l d s t r e n g t h r a t i o c a u s e d a c o n c e n t r a t i o n o f energy d i s s i p a t e d by h y s t e r e s i s i n t o t h e l o w e s t s t o r e y . In t h i s sense t h e s o f t f i r s t s t o r e y c o n c e p t was f o u n d t o be e f f e c t i v e f o r t h e s e systems. The e n e r g y d i s s i p a t e d by t h e v i s c o u s damping mechanism was f o u n d t o t a k e p l a c e m a i n l y i n t h e upper s t o r i e s o f t h e s t r u c t u r e s s t u d i e d . 1 7 8 8.5 F u t u r e Work The work r e p o r t e d h e r e s h o u l d he viewed as a p r e l i m i n a r y s t u d y f o r an energy based a n a l y s i s p r o c e d u r e . I t i s p r e s e n t e d as a f i r s t s t e p toward t h i s o b j e c t i v e . Throughout the s t u d y as wide a range o f v a r i a b l e s as p o s s i b l e was used i n o r d e r t o i d e n t i f y g e n e r a l t r e n d s and e f f e c t s . H aving i d e n t i f i e d t h e p r i n c i p a l p a r a m e t e r s , more d e t a i l e d s t u d i e s u s i n g more c l o s e l y spaced d a t a and a g r e a t e r number o f earthquake r e c o r d s can be made t o expand t h e r e s u l t s o b t a i n e d h e r e . Employing s i n g l e degree o f freedom models, d e s i g n s p e c t r a can be p r o d -uced f o r t o t a l energy d i s s i p a t e d and e nergy d i s t r i b u t i o n u s i n g t h e I n c r e a s e d i n p u t d a t a . T h i s would make t h e s p e c t r a s t a t i s t i c a l l y more r e p r e s e n t a t i v e o f the g e n e r a l earthquake problem. More e x t e n s i v e I n v e s t i g a t i o n and v e r i f i c a t i o n o f t h e d e s i g n a p p roach o u t l i n e d i n c h a p t e r 4 i s needed b e f o r e I t can be u s e d w i t h c o n f i d e n c e . The e x t e n s i o n o f t h e method t o m u l t i - d e g r e e o f freedom systems needs t o be i n v e s t i g a t e d f u r t h e r . Work on more d e t a i l e d m u l t i - d e g r e e o f freedom computer models i s r e q u i r e d t o c o n f i r m o r amend the f i n d i n g s p r o d u c e d f o r t h e shear beam model used i n th e p r e s e n t s t u d y . The use o f t h e method g i v e n h e r e f o r t h e d e s i g n o f systems which i n c o r p o r a t e s p e c i a l e n ergy d i s s i p a t i n g d e v i c e s o f known c h a r a c t e r i s t i c s i s a n o t h e r a r e a which c o u l d be u s e f u l l y i n v e s t i g a t e d i n t h e f u t u r e . The i n f l u e n c e o f a x i a l f o r c e combined w i t h f l e x u r e upon t h e energy d i s s i p a t i o n " c h a r a c t e r i s t i c s o f b o t h S.D.F. and M.D.F. systems c a n be s t u d i e d w i t h t h e i n c l u s i o n o f a s u i t a b l e h y s t e r e s i s r e l a t i o n s h i p . 179 BIBLIOGRAPHY 1. BIOT M.A. "A M e c h a n i c a l A n a l y s e r f o r t h e P r e d i c t i o n o f Ear t h q u a k e S t r e s s e s " , B u l l e t i n o f t h e S e i s m o l o g i c a l S o c i e t y o f America, A p r i l 194-0 2. BIOT M.A. " A n a l y t i c a l and E x p e r i m e n t a l Methods i n E n g i n e e r i n g S e i s m o l o g y " , T r a n s a c t i o n s A.S.C.E. Volume 108, 1943-3. Housner, G.W., M a r t e l , R.R., and A l f o r d , J . L., "Spectrum A n a l y s i s o f S t r o n g M o t i o n E a r t h q u a k e s " , B u l l e t i n Seism. S o c i e t y o f Am e r i c a , V o l . 4 3 , No. 2, A p r i l 1953. 4. Newmark, N.M., "A Method o f Computation f o r S t r u c t u r a l Dynamics", T r a n s .  A.S.C.E. V o l . 127, P a r t 1, 1962. 5. W i l s o n , E.L. and Clough, R.W., "Dynamic Response b y S t e p - b y - S t e p M a t r i x A n a l y s i s " , Symposium on Use o f Computers i n C i v i l E n g i n e e r i n g , - L i s b o n , P o r t u g a l , O c t o b e r , 1962. 6. V e l e t s o s A.S., and Newmark, N.M., " E f f e c t s o f I n e l a s t i c B e h a v i o u r on th e Response o f Simple Systems t o Eart h q u a k e M o t i o n s " , P r o c . 2ND.  W.C.E.E. Toyko, Japan I960. 7. Housner, G.W., "The P l a s t i c F a i l u r e o f Frames D u r i n g E a r t h q u a k e s " , P r o c . 2ND. W.C.E.E. Tokyo, Japan I960. 8. Berg, G.V., and Thomaides, S.S., "Energy Consumption b y S t r u c t u r e s i n S t r o n g M o t i o n E a r t h q u a k e s " , P r o c . 2ND. W.C.E.E. Tokyo, Japan I960 9 T a n a b a s h i , R., " S t u d i e s on t h e n o n - l i n e a r V i b r a t i o n s o f S t r u c t u r e s S u b j e c t e d t o D e s t r u c t i v e E a r t h q u a k e s " , P r o c . 1 s t . W.C.E.E. B e r k e l e y 1956. 10. J e n n i n g s , P . C , "Response o f Simple Y i e l d i n g S t r u c t u r e s t o Earthquake E x c i t a t i o n " , C a l i f o r n i a I n s t i t u t e o f Technology, E a r t h q u a k e E n g i n e e r - i n g R e s e a r c h L a b o r a t o r y R e p o r t , Pasadena, C a l i f o r n i a 1963. 11. Blume J.A., "A Re s e r v e E n e r g y Technique f o r the Earthquake'Deslgn and R a t i n g o f S t r u c t u r e s i n t h e I n e l a s t i c Range", P r o c . 2ND. W.C.E.E. Tokyo, Japan I960. ,12. Merchant, H.C. and Hudson, D.E., "Mode S u p e r p o s i t i o n i n M u l t i - D e g r e e o f F r e e d o n Systems u s i n g E a r t h q u a k e Response Spectrum D a t a " , B u l l , Seism. S o c i e t y o f A m e r i c a , V o l . 5 2 , No. 2, A p r i l 1962. 180 13. Clough, R.W., "Earthquake Analysis by Response Spectrum Superposition" Bulletin, Seisin. Soc. Amer. Volume 52, No. 3, July 1962. 14. Spencer, R.A.., "The Non-linear Response of a Multistorey Prestressed Concrete Structure to Earthquake Excitation", Proc. 4th. World  Conference on Earthquake Engineering, Chile, 1969. 15. Arya, A.S., "Inelastic and Reserve Energy Technique Analysis of Multi-Storied Buildings", Proc. 5th. W.C.E.E. Rome, Italy 1974. 16. Gluck, J., "An Energy Dissipation Factor as Structural Design Criterion for Strong Earthquake Motion", Proc. 5th. W.C.E.E., Rome, Italy, 1974. 17. Newark, N.M. and Hal l , W.J., "A Rational Approach to Seismic Design Standards for Structures", Proc. 5 t h . W.C.E.E., Rome, Italy 1974. 18. Blume, J.A., "Elements of a Dynamic-Inelastic Design Code", Proc. 5th. W.C.E.E., Rome, Italy 1974. 19. Biggs, J.M., Hansen, R.J., and Holley, M.J., "On Methods of Structural Analysis and Design for Earthquake", Structural and Geotechnical  Mechanics, W.J. Hall, Editor, Prentice Hall 1977. 20. Ruiz, P. and Penzien, J., "Probabilistic Study of the Behavior of Structures During Earthquakes", Earthquake Engineering Research Centre Report, No. E.E.R.C. 69-3, University of California, Berkeley, California. 21. Nickell, R.E., "On the Stability of Approximation Operators in Problems of Structural Dynamics", Int. J . of Solids and Structures, Vol. 7, 1971, pp. 301 - 319-22. Kanaan, A.E., and Powell, G.H., "Drain-2D, A General Purpose Computer Program for Dynamic Analysis of Inelastic Plane Structures", Earthquake  Engineering Research Center Report, No. E.E.R.C. 73-6 and E.E.R.C. 7 3 - 2 2 , University of California, Berkeley, California. 23. Murakami, M., and Penzien, J . , "Nonlinear Response Spectra for Probabilistic Seismic Design and Damage Assessment of Reinforced Concrete Structures", Earthquake Engineering Research Center Report No. E.E.R.C. 75-38 November 1975, University of California, Berkeley, California. 24. Anderson, D.L., Nathan, N.D., Cherry, S., "Correlation of Static and Dynamic Analysis By The National Building Code of Canada 1977", Proceedings Third Canadian Conference on Earthquake Engineering, Montreal, June 1979. 2 5 . Housner, G.W., "Limit Design of Structures to Resis Earthquakes", Proceedings 1 s t . World Conference on Earthquake Engineering, Earthquake' Engineering Research Institute, Berkeley, California 1956. 26. Anderson, J.C., and Bertero, V.V., "Seismic Behavior of Multistory Frames Designed by Different Philosophies," Report No. UBC/EERC  69-11 , Earthquake Engineering Research Center, University of California, Berkeley, 1969. APPENDIX A RESULTS OF SINGLE DEGREE OF FREEDOM  STUDY IN ORIGINAL FORM Complete results of the single degree of freedom study presented i n Chapter 3 are given here i n the o r i g i n a l form. For an explanation of the input parameters employed and the abbreviations used see section 3-1. 281 F I G . A . 2 . RESULTS FOR COMBINATION 2 P A C E P . 0 1 00 FIG. A.2 YSR 10.0 © © 2.0 A — A 1.0 -1 h 0 5 *-•- - X 0.3 « 0.1 PRC3IMA S16E ELflSTO-PLflSTIC SYSTEMS ETfl=0.01 \ / / \ \ / \ \ \ / - { £ — f f i — ® n — 3 . 5 1 .0 1 .5 2 . 0 2 . 5 PERIOD (IN SECONDS) CONTINUED 00 VJ1 FIG. A.3 RESULTS FOR COMBINATION 3 ELCEP.01 M 00 0 2 rn — i x> II LSI P o DISPLACEMENT DUCTILITY . , , , „ , . „ , , „ . 3 5 7 10' 3 S 7 10* 3 5 7 10* 3 5 7 10* . I . . I ' • l . I i •  l I l I l I l 1 I I l I l I l ll * " -* " * * * 3 o o o ro o tn TO <— co ut a o a ? f f t J -i> i * -r i II ai o co mn cn m —i •n II a o S 7 • • ' • Id* PERMANENT SET l/TLD DIS) , . 5 7 10' 3 5 7 10* 3 5 ' • I i I i I n I 1 1—i—L. 10» 5 i I 7 10* . —- — - " m 33 • • . X -SO D cn -< tn 681 FIG. A . 5 RESULTS FOR COMBINATION 5 PACTL .01 r—1 O F I G . A.7 RESULTS FOR COMBINATION 7 PKFTL .01 2 £61 F I G . A.8 RESULTS FOR COMBINATION 8 ELCTL .01 F I G . A.3 YSR 10.0 O © 2.0 A A t .O H »-0.5 X- X 0.3 * <> 0.1 EL CENTRO N.S. TRI-LINERR SYSTEMS ETR=0.01 cn t— CD C D o . in >— on a i— a_o •—'ru a CD CC UJ Z l 0.0 (D (TJ CD CP CD CD i CD CP 0.5 1.0 1.5 PERIOD (IN SECONDS) 2.0 4$ 2.5 CONTINUED FIG. A . 9 RESULTS FOR COMBINATION 9 TAFEP.05 FIG. A . 9 YSR 10 0 O 0 2 0 A A 1.0 -1 1-0.5 - X 0.3 » 0.1 •••••»> a a. TflFT N69W ELflSTO-PLASTIC SYSTEMS ETA=0.05 cr. y— o CD COo -in >— QQ CD CC UJ ¥ / \ / \ \ \ \ - e - -ffi—ffi-i ® — ® -0.0 0.5 1.0 1-5 2.0 PERIOD [IN SECONDS) 2.5 CONTINUED 002 TOc* zoz CD 3 » TOTAL ENERGY/UNIT MASS (/G.A.MAX ) D 0.02 0.04 0.06 0.08 J 1 L O.L J o o m a ENERGY DISPTD BY SPRINGUOF TOTAL) 0.0 25.0 50.0 75.0 LOO.O a o a rvj o t/> x — cu ut a o o t ? ^ t T ? + i * T £0Z o YSR 10.0 O O 2 0 A A 1.0 -I *• 0 5 X- * 0.3 • * 0.1 t-FIG, A , 1 2 CONTINUED FIG. A .13 RESULTS FOR COMBINATION 13 TAFTL .00 ro o TOTRL ENERGY/UNIT MASS ( /G.R.MRX ) 0.0 0.02 0.04 0.06 0.08 D.l J I 1 1 ' 1 o LOZ F I G , A ,15 RESULTS FOR COMBINATION 15 E L C E P . 0 0 ro h-1 o FIG. A.16 RESULTS FOR COMBINATION 16 PKFTL .00 ro M ro FIG. A.17 RESULTS FOR COMBINATION 17 TAFRO.01 ro t—1 4^ FIG. A.17 YSR 10 0 0 — © 2 0 A — A 1 0 H h 0 5 X- - X 0 3 «> 0 1 ^ d I— o »-< CD >—j 0 - ° C O a - l in V-CD a i— Q_<=> 1 — TM >-CD CC U J TRFT N69W RAMBERG-OSGOOD SYSTEMS ETfl=0.01 ft-'" 4 \ \ \ \ \ \ \ x - e -O Q CTJ ©-0 0.5 1.0 1.5 2.0 PERIOD (IN SECONDS) 2.5 CONTINUED ro M FIG. A . 1 8 YSR 10.0 0- o 2.0 A— A 1.0 4— r-0 5 X- -K 0.3 (•> 0.1 CD CD. E L CENTRO N . S . R A M B E R G - Q S G O Q D S Y S T E M S ETA^O.01 gPH ^ * * • LD ~*~ x \ X * £sH \ \ CG^  1 G o cp—CD ep CD a i a W -$ 0 . 0 3.5 1.0 1.5 2.0 2 - 5 PFRT0D f TN SFC0NDS1 .5 \ UJ CONTINUED 218 APPENDIX B  LISTINGS OF SUBROUTINES  ENERGY AND S P P I S In t h i s appendix l i s t i n g s o f t h e two s u b r o u t i n e s which p e r f o r m the energy c a l c u l a t i o n s f o r t h i s s t u d y a r e p r e s e n t e d . S u b r o u t i n e ENERGY c a l c u l a t e s a l l e n e r g i e s w i t h i n the system as g i v e n i n s e c t i o n 2.-4. S u b r o u t i n e SPDIS i s c a l l e d from ENERGY and c a l c u l a t e s the p o t e n t i a l energy s t o r e d i n t h e s p r i n g elements, and t h e energy d i s s i p a t e d by h y s t e r e s i s . STEP C SDBBOUTINE SPDIS(SPF,SPFO,OISP,DELI,TIH,DISPO) SUBROUTINE TO CALCULATE ENERGY DISSIPATED AND STORED IB PLASTTC SP8INGS B30TINE BASED OH ENERGY STOBED = AVERAGE FORCE*DIST TRAVELLED D DR I NG ciACU TINE I NPOT OUTPUT SPFO SPRING FORCE AT EM3 OF LAST INCREMENT SP F SPRING FORCE AT END 3 P THIS INCREMENT DISP DISPLACEMENT AT END OF THIS INCREMENT . DELX CHANGE IN DISPLAC EL ENT D0RIN3 T H I S I S C R B H E N I TIM TIRE STATION AT END OF THIS INCREMENT ENDISS ENBR3T D I S S I P A T E D I N P L A S T I C UORK BT S P R I H J ENSTOR ENERGY STORED IN SPRING AT EL D OF THIS INCRESIIMT (EITHER DISSIPATED DR RECOVERABLE OOHT KHOH Y2T) I M P L I C I T REAL*8(A-H,D-Z) COHHON/PLOTT/TITLE(23|,NPLOT,IPRINT,I3DT ,N30T COMMON/STBOCT/DAHP(33) , Y I E L D ( 3 0 | ,PERD(30| , NOAM P, H YI ELD, « ?T!R D C3NHON/GRN D/ACCOP.GHACC (3000) ,3ACC,3ACCO,DELGA,GV,GV0,GJ,GDO,SCAL *,CPS,NPTS,NPT2,NE3,INTTPE COMHON/HXST/FRIELD,YLDNEG,STIFYD, STIPKT,EH D I S S , E N S T 3 B , N J 2 J S 3 DIMENSION TIME (200) C HAVE HE CROSSED ZERO FORCE AXIS? CHECK=SPF»SPP3 IP(SPPO.EQ.0.3) GO TO 20 IF (CHECK. LE. 0. 3) 30 T3 30 20 DELHK = ((SPF* SPFO) /2.0)*DELI EMSTOF= ENS TOP* DELHK C HRITE(7,510| SPF,SPFO,DISP,DELX.OBLHK,CHECK,ENSTOR GO TO 63 30 CONTINDE C C HE HAVE CROSSED ZERO AND THE EI ERGY LEFT 18 ENST3R C MOST BE C3NSIDERED DISSIPATED C PORSUH=SPF*SPFO IF (FDBSno.EQ.O. 0) GO TO 40 DISPO=DISP-DELX ABSPF3=DABS(SPP3) ABSPP=DABS (SPF) DIST=ABSPP3*DELI/(ABSPF*ABSPFO) DELHKD=SPFO*DI ST/2.0 ENSTOB=ENSTOR*DELHKD ENDISS-ENDISS*ENSTOB DISTC=DABS (DELX) -DABS (DIST) ENSTOB-DABS (SPF«DISTC) /2. 0 GO TO 50 10 OIST=DBLI/2.0 DELHKD=SPFO*DIST/2.0 ENSTOR=ENSTOR* DELHK D ENDISS= ENDISS*ENSTOR DISTC=DIST ENSTOR = DABS (S PF* DIST C/2. 0) 50 NCHOSS=NCB0SS*1 C TIHE(NCROSS)=TIM C HRITE(7,505) NCROSS 505 FORMAT (' ZERO CROSSING NOHBER'.IH) C HRITE(7,510) SPP,SPFD,DISP,DELX,CHECK, C *ENSTOP,DIST,DISTC, DBLHKD,ENDISS,TIME(NCROSS) 510 FORMAT(13B12.») 60 CONTINUE IF (TOUT. NH. IPBIHT) GO TO 149 ISTEP=I3DT I=IOOT SPST3R=0.0 SPPEC=D. 0 C WRITE(7,500) ISTEP,TI8 C HRITE(7,531) C HRITE(7,502) I,SPSTOR,SPREC.ENSTOP,ENDISS,NCROSS 502 FORMAT (16, 7 X, El 2. 5,91, E12. 5,9X, El 2. 5, 5X.E12.5, 8X,12,14X,Fb.3) 500 F3BMAT(' DETAILS OF ENEB3T IN SHEAR BEAMS AT END OF TIHE STEP', I *5,' FOB TIME = '.P10.4,' SECS') 1M9 CONTINUE 501 F3 RM AT (//' 3EAM T3TAL SHEAR ENERGY SPF ENGY RECOYtdiJO SPF * ENGY REMAINING SPP ENERGY H3HBER 3F ZERO TIME 3t -AST',/, •131,'STORED SINCE LAST SINCE LAST',11X,•IN BEAM SINCi LAST D I •SSIPATED CBDSSIN3S IH ZERO CHOSSING' ,/, 10X , • Z EBO CH0SSIN3' •81,'ZERO CR05SIN3',8X,'ZERO CBOS5IH3',21X,'HYSTERESIS LOO?', •7X,' (SECS) •) PETORN END ro i—1 c c c c c c c c c c c c c c c c c c c c c c SUBROUTINE ENBB31(SPPBHD,SPPC,)ISP,DELI,TIN,DISPO,DELT,13ACC, *ABACCO,VEL,VISFAC,VELO,SOBVIS, EN S OH . SUB IN , EKIN) IMPLICIT REAL*B (A-H.O-Z) C0NN0N/PL3TT/TITLE(20) .NPLOT.IPRINT,IO0T ,HOOT COH«OH/STRUCT/DAnP(33) ,TIELD(33| ,PERD(30» , HDABP,HIIELD,I1P8B0 C3SH0H/GRBD/ACC0P,GBACC(3000),3ACC,GACC0,DBLGA.GV ,GVO,GD,*JDO, SCAL *,CPS,NPTS,NPT2.NE0,INTrPE COBBON/HIST/FTI ELD.rLDNEG, STIPTD, STIFKT.EBDISS.EHSTOR.HCaaSS «»*•««*•**< SPFEHD SPFC OISP DELI T I B DISPO DELT ABACC ABACCO V EL 7EL0 f I S F A C SOBVIS EN son SUBI I EK IH !«»»»»*«»••»» SPRING FORCE AT BHD OF PRESENT TIB B STEP SPRIN? PORCE AT END OF PREVIOUS TINE STEP DISPLACEBBNT AT BHD OF THIS TIDE STEP CHANGE IN DISPLACEBEHT DDRIN3 THIS TIBE STEP Tl HB 5 TATION AT END OF THIS INTERVAL DISPLACEMENT AT END OF PREVIOUS TIBE STEP TIBE STEP LENGTH (5 ECS) ABSOLUTE BASS ACCELERATION AT END OF THIS TIBE STeP ABSOL3TE BASS ACCELERATION AT END OF PREVIOUS 51a? VSL3CITT OF BASS AT END OF THIS TIBE STEP VELOCITI OF BASS AT END OF PREVIOUS TIBE STEP ?ISCOOS DAMPING PACTOB SOB OF ENERGI DISSIPATED IH VISC3DS DABPIN3 SOB OF ALL ENERGIES IH SISTBB SOB OF ENERGT INPOT TO STBOCTORE KINETIC ENBR3T OF BASS AT END OF TIBE STEP «««•»»»»»***«»•»•»•«« CALCOLATB ENERGT IN THE SPRING CALL SPDIS (SPPBND.SPPC. DISP, DELI,TIB, DISPO) OBTAIN GROUND ACCEL , V EL , AND DISPLACE1ENT CALL GHH(TIB,DELT) CALC. KINETIC ENERGI A BVEL = VEL • GV EK IN = ABVEL*ABVEL/2.0 NBITE (7,111) ABVEL, EKIN CALC ENERGT IN VISCOUS DAB PURS VISK = VISFAC«DEL*» (VEL»VBLO)/2.00 SOBVIS = SUBVIS • DABS(VISK) SUH ENERGIES HITHtH THE STSTRB EN SOB = ENDISS • ENST3R * EKIN • SOBVIS MBITE(7,111) VISK,SUBVIS,ENSON CALC ENERGT INPUT TO 3TRUCT0RE E NI N = (GD-r,DO) * (ABACC4-ABACC0)/2. 0 SUBIN = SUBIN • EN IN EHBAL = SDBIN - ENSUN IP(DABS(sniIN| .LT.0.0001) 30 T3 5C PERCE R = 100.3*ENBAL/S0HIN GO TO 55 50 PBBCEN=999.9 55 CONTINUE BRITE (7,111) ENIN,SUBIN,ENBAL.PERCEN IF(IOOT.NE.IPRINT) GO TO 119 «BITE(7,1D1) TIB 101 F3 RBAT (//• ENER3T BALANCE AT TIBE « * ,F10.«) NRITE (7,111) ENDISS, ENS TOR, EKIN, SOBVIS, ENSOS, SUBIN, ENBAL.PiiBCEN 111 FORBAT(10E12.5) 1U9 RETURN END ro ro o 221 APPENDIX C Result of Multi-Degree of Freedom  Study i n Original Form Complete results of the multi-degree of freedom study presented i n Chapters 6 and 7 are given here i n o r i g i n a l form. STRUCTURE 1 HYSTERESIS MODEL. E - P EXCITATION : 30 Sees. Taft N69W DAMPING: 1% Stiffness & Mass TOTAL ENERGY IN SYSTEM g END OF CALCULATIONS = 1 1520 INCH KIPS PERIOD 0.43 SECS. YSR = 1.9 STRONG SYSTEM Storey Energy Dissipated by Hysteresis In This Storey Residual Potential Energy In This Storey Residual Kinetic Energy In This Storey Energy Dissipated In Mass a Damping In This Storey Energy Dissipated In Stiffness a Damping In This Storey Total Energy Dissipated In This Storey 3 147.7 (9.7) 0.0 (0) 9.2 (0.6) 197.4 (13.0) 37.1 (2.4) 392.4 (25.8) 2 255.8 (16.8) 0.0 (0) 10.6 (0.7) 123.4 (8.1) 23.6 (1.6) 413.5 1 (27.2) 1 631.2 (41.5) o.o (0) 8.1 (0.5) 47.3 (3.1) 15.6 (1.0) 702.7 (46.3) Z 1034.7 0 27.9 368.1 76.3 1508.6 % Total (68) (0) (1.8) (24.2) (5.0) (99.3) ( ) = i of Total Energy in System. TABLE C l M.D.F. RESULTS SET 1 222 STRUCTURE 1 HYSTERESIS MODEL. E - P EXCITATION: 30 Sees. Taft N69W DAMPING: 1% Stiffness TOTAL ENERGY IN SYSTEM g END OF CALCULATIONS = 1495 INCH KIPS PERIOD 0.43 SECS. YSR = 1.9 STRONG SYSTEM Storey Energy Dissipated by Hysteresis In This Storey Residual Potential Energy In This Storey Residual Kinetic Energy In Thin Storey Energy Dissipated In Mas;; n Damping In This Storey Energy Dissipated In Stiffness a Damping In This Storey Total Energy Dissipated In This Storey 3 86.5 (5.8) 0.2 (0.0) 8.1 (0.5) 0.0 (0.0) 291.7 (19.5) 386.5 (25.8) 2 324.5 (21.7) 0.0 (0) 6.1 (0.4) 0.0 (0) 183.0 (12.2) 513.6 (34.3) 1 495.4 (33.1) 0.0 (0) 5.2 (0.3) 0.0 (0) 80.3 (5.4) 580.9 (38.9) T. 906.4 0.0 19.4 0.0 555.0 1481 % Total (60.6) (0) (1.3) (0) (37.1) (99.1) ( ) = % of Total Energy in System. TABLE C.2 M.D.F. RESULTS SET 2 STRUCTURE l HYSTERESIS MODEL. E - P EXCITATION: 30 Sees. Taft N69W DAMPING: 1% Mass TOTAL ENERGY IN SYSTEM g END OF CALCULATIONS = 1517 INCH KIPS PERTOD ° -43 SECS. YSR 1 - 9 STRONG SYSTEM Storey Energy Dissipated by Hysteresis In This Storey Residual Potential Energy In This Storey Residual Kinetic Energy In This Storey Energy Dissipated In Uar,r, a Damping In This Storey Energy Dissipated In Stiffness a Damping In This Storey Total Energy Dissipated In This Storey 3 152.9 (10.0) 1.8 (0.1) 5.3 (0.3) 265.8 (17.5) 0.0 (0) 425.8 (28.1) 2 234.0 (15.4) 0.0 (0) 7.8 (0.5) 166.3 (11.0) 0.0 (0) 408.0 (26.9) 1 595.9 (39.3) 0.5 (0) 7.2 (0.5) 67.9 (4.5) 0.0 (0) 671.5 (44.3) I 982.8 2.3 20.3 500.0 0.0 1505 % Total (64.8) (0.1) (1.3) (33.0) (0) (99.3) ( ) = % of Total Energy in System. TABLE C 3 M.D.F. RESULTS SET 3 223 STRUCTURE 1 HYSTERESIS MODEL. E - P EXCITATION: 30 Sees. Taft N69W DAMPING : 10i5 Stiffness TOTAL ENERGY IN SYSTEM 6 END OF CALCULATIONS - 1260 INCH KIPS PERIOD 0.43 SECS. YSR = 1.9 STRONG SYSTEM Storey Energy Dissipated by Hysteresis In This Storey Residual Potential Energy In This Storey Residual Kinetic Energy In This Storey Energy Dissipated In Macs <* Damping In This Storey Energy Dissipated In Stiffness a Damping In This Storey Total Energy Dissipated In This Storey 3 0.0 (0) 0.0 (0) 1.2 (0) 0.0 (0) 680 (54) 681.2 (54) 2 0.0 (0) 0.0 (0) 2.2 (0) -0.0 (0) 423-4 (33.8) 425.6 (33.8) 1 3.4 (0) 0.0 (0) 3.5 (0) 0.0 (0) 147.4 (12.2) 154.1 (12.2) Z 3.4 0.0 6.9 0.0 1250.8 1260.7 % Total (0.3) (0.0) (0.5) (0.0) (99.2) (100.0) ( ) = % of Total Energy in System. TABLE C.4 M.D.F. RESULTS SET 4 STRUCTURE 1 HYSTERESIS MODEL. E - P EXCITATION: 30 Sees. El Centro N. S. DAMPING: 1% Mass & Stiffness TOTAL ENERGY IN SYSTEM g END OF CALCULATIONS - 3715 INCH KIPS PERIOD 0.43 SECS. YSR = 0.8 MODERATE SYSTEM Storey Energy Dissipated by Hysteresis In This Storey Residual Potential Energy In This Storey Residual Kinetic Energy In This Storey Energy Dissipated In Uac.r- a Damping In This Storey Energy Dissipated In Stiffness a Damping In This Storey Total Energy Dissipated In This Storey 3 98.0 (2.6) 1.1 (0) 6.4 (0.1) 229.2 (6.2) 372.7 (10.0) 707.4 (19.0) 2 517.4 (13.9) 1.1 (0) 10.9 (0.3) 150.8 (4.0) 260.4 (7.0) 940.6 (25.3) 1 1816.7 (48.9) 1.3 (0) 17.6 (0.5) 54.9 (1.5) 157.6 (4.2) 2048.4 (55.1) Z 2432.1 3.5 34.9 434.9 790.7 3696.4 % Total (65.5) (0) (0.9) (11.7) (21.3) (99.5) ( ) = % of Total Energy in System. TABLE C.5 M.D.F. RESULTS SET 5 224 STRUCTURE l HYSTERESIS MODEL. E - P EXCITATION; 30 Sees. El Centro N.S. DAMPING: 1% Stiffness TOTAL ENERGY IN SYSTEM % END OF CALCULATIONS = 3701 INCH KIPS PERIOD 0.43 SECS. YSR = 0.8 MODERATE SYSTEM Storey Energy Dissipated by Hysteresis In This Storey Residual Potential Energy In This Storey Residual Kinetic Energy In This Storey Energy Dissipated In Mass a Damping In This Storey Energy Dissipated In Stiffness a Damping In This Storey Total Energy Dissipated In This Storey 3 111.8 (3.0) 2.2 (0) 9.0 (0.2) 0.0 (0) 406.7 (11.0) 529.7 (14.3) 2 645.8 (17.4) 2.6 (0) 13.8 (0.4) 0.0 (0) 286.9 (7.8) 949.1 (25.6) 1 2007.8 (54.2) 3.2 (0) 19.7 (0.5) 0.0 (0) 171.0 (4.6) 2201.7 (59.5) Z 2765.4 8.0 42.5 0.0 864.6 3680.5 % Total (74.7) (0) (1.1) (0) (23.4) (99.5) ( ) = % of Total Energy in System. TABLE C. 6 M.D.F. RESULTS SET 6 STRUCTURE 1 HYSTERESIS MODEL. E - P EXCITATION : 30 Sees of El Centro N.S. DAMPING: 10% Stiffness Damping TOTAL ENERGY IN SYSTEM 6 END OF CALCULATIONS = 3491 INCH KIPS PERIOD 0.43 SECS. YSR = 0.8 MODERATE SYSTEM Storey Energy Dissipated by Hysteresis In This Storey Residual Potential Energy In This Storey Residual Kinetic Energy In This Storey Energy Dissipated In Mass <* Damping In This Storey Energy Dissipated In Stiffness a Damping In This Storey Total Energy Dissipated In This Storey 3 0.0 (0) 0.2 (0) 5.9 (0.2) 0.0 (0) 1328.4 (38.0) 1334.6 (38.2) 2 143.2 (4.1) 0.4 (0) 10.2 (0.3) 0.0 (0) 917.6 (26.3) 1071.4 (30.7) 1 626.1 (17.9) 0.5 (0) 15.8 (0.5) 0.0 (0) 436.7 (12.5) 1079.1 (30.9) I 769.3 1.1 31.9 0.0 2682.7 3485.1 % Total (22.0) (0) (0.9) (0) (76.9) (99.8) ( ) = % of Total Energy in System. TABLE C.7 M.D.F. RESULTS SET 7 225 STRUCTURE 1 HYSTERESIS MODEL. E - P EXCITATION: 30 Sees. Pacoima S16E DAMPING: 1% Mass & Stiffness TOTAL ENERGY IN SYSTEM g END OF CALCULATIONS = 17,437.0 INCH KIPS PERIOD 0.43 SECS. YSR" 0.26 WEAK SYSTEM Storey Energy Dissipated by Hysteresis In This Storey Residual Potential Energy In This Storey Residual Kinetic Energy In This Storey Energy Dissipated In Mass a Damping In This Storey Energy Dissipated In Stiffness o Damping In This Storey Total Energy Dissipated In This Storey 3 1502.0 (8.6) 0.0 (0) 692.0 (4.0) 259.0 (1.5) 420.0 (2.4) 2874.0 (16.5) 2 2713.0 (15.5) 0.0 (0) 1054.0 (6.0) 190.0 (1.0) 313-0 (1.8) 4270.0 (24.5) 1 8135.0 (46.6) 0.0 (0) - 1400.0 (8.0) 104.0 (0.6) 477.0 (2.8) 10,116.0 (58.0) Z 12,350 0.0 3146.0 553.0 1210.0 17,260.0 % Total (70.8) - (0) (18.0) (3.0) (7.0) (99.1) ( ) = % of Total Energy in System. TABLE C.8 M.D.F. RESULTS SET 8 STRUCTURE 1 HYSTERESIS MODEL. E - P EXCITATION: 30 Sees. Pacoima S16E DAMPING: 1% Stiffness TOTAL ENERGY IN SYSTEM g END OF CALCULATIONS = 17,557.0 INCH KIPS PERIOD 0-43 SECS. YSR = 0.26 WEAK SYSTEM Storey Energy Dissipated by Hysteresis In This Storey Residual Potential Energy In This Storey Residual Kinetic Energy In This Storey Energy Dissipated In Mass m Damping In This Storey Energy Dissipated In Stiffness a Damping In This Storey Total Energy Dissipated In This Storey 3 1540.0 0.0 (a) 696.0 u.o) 0.0 439.0 r?.«n 2675.0 2 2886.0 (16.4) 0.0 (0) 1061.0 (6.0) 0.0 (0) 327.0 (2.0) 4274.0 (24.3) 1 8505.0 (48.5) 0.0 (0) 1405.0 (8.0) 0.0 (0) 512.0 (2.9) 10,422.0 (59.4) Z 12931.0 0.0 3162.0 0.0 1278.0 17,371.0 % Total (73.6) (0) (18.0) (0) (7.3) (98.9) ( ) = t of Total Energy in System. TABLE C.9 M.D.F. RESULTS SET 9 226 STRUCTURE: 1 HYSTERESIS MODEL. E - P EXCITATION: 30 Sees. Pacoima S16E DAMPING: 10% Stiffness TOTAL ENERGY IN SYSTEM 6 END OF CALCULATIONS - 16,605.0 INCH KIPS PERIOD 0.43 SECS. YSR = 0.26 WEAK SYSTEM Storey Energy Dissipated by Hysteresis In This Storey Residual Potential Energy In This Storey Residual Kinetic Energy In This Storey Energy Dissipated In Mass a. Damping In This Storey Energy Dissipated In Stiffness a Damping In This Storey Total Energy Dissipated In This Storey 3. 391.0 (2.3) 0.0 (0) 693-0 (4.2) 0.0 (0) 2724.0 (16.4) 3808.0 (22.9) 2 2107.0 (12.7) 0.0 (0) 1039.0 (6.2) 0.0 (0) 2281.0 (13.7) 5427.0 (32.7) 1 4321.0 (26.0) 0.0 (0) 1388.0 (8.3) 0.0 (0) 1566.0 (9.4) 7276.0 (43.8) T. 6819.0 0.0 3120.0 0.0 6571.0 16,510.0 % Total (41.0) (0) (18.8) (0) (39.6) (99.4) ( ) = % of Total Energy in System. TABLE CIO M.D.F. RESULTS SET 10 STRUCTURE 2 HYSTERESIS MODEL. E - P EXCITATION : 30 Sees. Taft N69W DAMPING : 1% Stiffness & Mass TOTAL ENERGY IN SYSTEM % END OF CALCULATIONS - 702.8 INCH KIPS r-TOTnn 0.33 suns. YSR = 3.1 STRONG SYSTEM Storey Energy Dissipated by Hysteresis In This Storey Residual Potential Energy In This Storey Residual Kinetic Energy In This Storey Energy Dissipated In Mass a. Damping In This Storey Energy Dissipated In Stiffness a Damping In This Storey Total Energy Dissipated In This Storey 3 0.0 (0) 0.0 (0) 0.5 (0) 143.7 (20.4) 162.8 (23.2) 307.0 (43.7) 2 0.0 (0) 0.0 (0) 0.6 (0) 92.2 (13.1) 103.5 (14.7) 196.4 (27.9) 1 131.5 (18.7) 0.1 (0) 0.9 (0) 28.9 (4.1) 35.8 (5.1) 197.1 (28.0) Z 131.5 0.1 2.0 264.8 302.1 700.5 % Total (18.7) (0) (0) (37.7) (43-0) (99.6) ( ) = * c jf Total Energy in System. TABLE C.ll M.D.F. RESULTS SET 11 227 STRUCTURE 2 HYSTERESIS MODEL. E - P EXCITATION : 30 Sees. El Centro N.S. DAMPING: 1% Stiffness & Mass TOTAL ENERGY IN SYSTEM 6 END OF CALCULATIONS = 681.9 INCH KIPS PERIOD 0.33 SECS. YSR = 1.4 STRONG SYSTEM Storey Energy Dissipated by Hysteresis In This Storey Residual Potential Energy In This Storey Residual Kinetic Energy In This Storey Energy Dissipated In Mass a Damping In This Storey Energy Dissipated In Stiffness a Damping In This Storey Total Energy Dissipated In This Storey 3 0.0 (0) 0.1 (0) 0.2 (0) 110.1 (16.1) 127.0 (18.6) 237.4 (34.8) 2 0.0 (0) 0.4 (0) 0.0 (0) 70.5 (10.3) 79.3 (11.6) 150.2 (22.0) 1 231.8 (34.0) 0.6 (0) 1.3 (0) 22.4 (3.3) 31.3 (4.5) 287.5 (42.2) I 231.8 1.1 1.5 203.0 237.6 675.1 % Total (34-0) (0) (0) (29.8) (34.8) (99.0) ( ) = % of Total Energy in System. TABLE C.12 M.D.F. RESULTS SET 12 STRUCTURE 2 HYSTERESIS MODEL, E - P EXCITATION: 30 Sees. El Centro N.S. DAMPING: 1% Mass TOTAL ENERGY IN SYSTEM 9 END OF CALCULATIONS = 622.3 INCH KIPS PERIOD 0.33 SECS. YSR = 1.4 STRONG SYSTEM Storey Energy Dissipated by Hysteresis In This Storey Residual Potential Energy In This Storey Residual Kinetic Energy In This Storey Energy Dissipated In Mass a Damping In This Storey Energy Dissipated In Stiffness a Damping In This Storey Total Energy Dissipated In This Storey 3 0.0 (0) 0.0 (0) 2.6 (0.4) 181.2 (29.1) 0.0 (0) 183.8 (29.5) 2 2.1 (0.3) 0.5 (0) 1.1 (0.2) 115.0 (18.5) 0.0 (0) 118.8 (19.1) 1 265.3 (42.6) 1.5 (0.2) 0.1 (0) 43.8 (7.0) 0.0 (0) 310.8 (49.9) I 267.4 2.0 3-8 340.0 0.0 613.4 % Total (43.0) (0.3) (0.6) (54.6) (0) (98.6) ( ) = i of Total Energy in System. TABLE C.13 M.D.F. RESULTS SET 13 228 STRUCTURE 2 HYSTERESIS MODEL. E - P EXCITATION: 30 Sees. Pacoima S16E DAMPING: 1% Stiffness & Mass TOTAL ENERGY IN SYSTEM 6 END OF CALCULATIONS = 8222.6 INCH KIPS PERIOD 0.33 SECS. YSR = 0.43 MODERATE SYSTEM Storey Energy Dissipated by Hysteresis In This Storey Residual Potential Energy In This Storey Residual Kinetic Energy In This Storey Energy Dissipated In Mass n Damping In This Storey Energy Dissipated In Stiffness a Damping In This Storey Total Energy Dissipated In This Storey 3 9.4 (0.1) 0.0 (0) 487.5 (5.9) 200.2 (2.4) 228.1 (2.8) 925.2 (11.?) 2 597.3 (7.3) 0.0 (0) 485.8 (5.9) 138.6 (1.7) 167.5 (2.0) 1389.3 (16.9) 1 5093.1 (61.9) 0.0 (0) 478.5 (5.8) 63.4 (0.8) 225.0 (2.7) 5860.1 (71.?) Z 5699.8 0.0 1451.8 402.2 620.6 8174.6 % Total (69.3) (0) (17.6) (4.9) (7.5) (99.4) ( ) = % of Total Energy in System. TABLE C.14 M.D.F. RESULTS SET 14 STRUCTURE 2 HYSTERESIS MODEL. E - P EXCITATION: 30 Sees. Pacoima S16E DAMPING: 1% Mass TOTAL ENERGY IN SYSTEM 6 END OF CALCULATIONS = 8483.0 INCH KIPS PERIOD 0.33 SECS. YSR = 0.43 MODERATE SYSTEM Storey Energy Dissipated by Hysteresis In This Storey Residual Potential Energy In This Storey Residual Kinetic Energy In This Storey Energy Dissipated In Mass oi Damping In This Storey Energy Dissipated In Stiffness a Damping In This Storey Total Energy Dissipated In This Storey 3 43.1 (0.5) 0.0 (0) 513.3 (6.0) 295.7 (3.5) 0.0 (0) 852.1 (10.0) 2 721.2 (8.5) 0.0 (0) 507.5 (6.0) 205.5 (2.4) 0.0 (0) 1434.2 (16.9) 1 5500.7 (64.8) 0.1 (0) 493-5 (5.8) 102.9 (1.2) 0.0 (0) 6097.2 (71.9) Z 6265.0 0.1 1514.3 604.1 0.0 8383.5 % Total (73.8) (0) (17.8) (7.1) (0) (98.8) ( ) = % of Total Energy in System. TABLE C.15 M.D.F. RESULTS SET 15 229 STRUCTURE: 2 HYSTERESIS MODEL. E - P EXCITATION: 30 Sees. Pacoima S16E DAMPING: 1% Stiffness TOTAL ENERGY IN SYSTEM 6 END OF CALCULATIONS = 8297.0 INCH KIPS PERTOD 0.33 SECS. YSR = 0.43 MODERATE SYSTEM Storey Energy Dissipated by Hysteresis In This Storey Residual Potential Energy In This Storey Residual Kinetic Energy In This Storey Energy Dissipated In Mass a Damping In This Storey Energy Dissipated In Stiffness a Damping In This Storey Total Energy Dissipated In This Storey 3 8.6 (0.1) 0.0 (0) 514.0 (6.2) 0.0 (0) 291.5 (3.5) 814.1 (9.8) 2 648.2 (7.8) 0.0 (0) 506.9 (6.1) 0.0 (0) 219.7 (2.6) 1374.8 (16.6) 1 5261.0 (63.4) 0.0 (0) 489.9 (5.9) 0.0 (0) 278.5 (3.3) 6029.5 (72.7) Z 5917.8 0.0 1510.8 0.0 789.7 8218.4 % Total (71.3) (0) (18.2) (0) (9.5) (99.0) ( ) = % of Total Energy in System. TABLE C.16 M.D.F. RESULTS SET 16 STRUCTURE 3 HYSTERESIS MODEL. E - p EXCITATION: 30 Sees. El Centro N .s. DAMPING: 1% Mass TOTAL ENERGY IN SYSTEM 6 END OF CALCULATIONS - 1412.8 INCH KIPS PERIOD 0.33 SECS. YSR = 0.45 MODERATE SYSTEM Storey Energy Dissipated by Hysteresis In This Storey Residual Potential Energy In This Storey Residual Kinetic Energy In This Storey Energy Dissipated In Mass « Damping. In This Storey Energy Dissipated In Stiffness a Damping In This Storey Total Energy Dissipated In This Storey 3 2.1 (0.1) 0.0 (0) 0.7 (0) 58.2 (4.1) 0.0 (0) 61.3 (4.3) 2 96.6 (6.8) 0.6 (0) 2.7 (0.1) 41.1 (2.9) 0.0 (0) 141.1 (10.0) 1 1172.2 (83.0) 0.5 (0) 3.2 (0.2) 23.9 (1.6) 0.0 (0) 1199.9 (84.9) Z 1270.9 1.1 6.6 123.2 0.0 1402.3 % Total (89.9) (0) (0.5) (8.7) (0) (99.3) ( ) = % of Total Energy in System. TABLE C.17 M.D.F. RESULTS SET 17 230 STRUCTURE 4 HYSTERESIS MODEL. E - P EXCITATION: 30 Sees. El Centro N.S. DAMPING: It Stiffness & Mass TOTAL ENERGY IN SYSTEM 6 END OF CALCULATIONS = 2356.6 INCH KIPS PERIOD 0.47 SECS. YSR = 0.68 MODERATE SYSTEM Storey Energy Dissipated by Hysteresis In This Storey Residual Potential Energy In This Storey Residual Kinetic Energy In This Storey Energy Dissipated In Mass ot Damping In This Storey Energy Dissipated In Stiffness a Damping In This Storey Total Energy Dissipated In This Storey 3 0.0 (0) 4.9 (0.2) 0.7 (0) 113.6 (4.8) 148.7 (6.3) 267.9 (11.4) 2 0.0 (0) 16.4 (0.7) 1.2 (0) 73.8 (3 - D 86.3 (3.7) 177.7 (7.5) 1 1722.9 (73.0) 26.3 (1.1) 2.8 (0.1) 32.3 (1.4) 112.4 (4.8) 1896.8 (80.5) Z 1722.9 47.6 4.7 219.7 347.4 2342.4 % Total (73) (2.0) (0.2) (9.3) (14.7) (99.4) ( ) = t of Total Energy in System. TABLE C.18 M.D.F. RESULTS SET 18 STRUCTURE 5 HYSTERESIS MODEL. E - P EXCITATION : 30 Sees, of El Centro N.S. DAMPING: 1% Stiffness & Mass TOTAL ENERGY IN SYSTEM % END OF CALCULATIONS = 2151.7 INCH KIPS PERIOD 0.57 SECS. YSR = 0.45 MODERATE SYSTEM Storey Energy Dissipated by Hysteresis In This Storey Residual Potential Energy In This Storey Residual Kinetic Energy In This Storey Energy Dissipated In Mass a Damping In This Storey Energy Dissipated In Stiffness a Damping In This Storey Total Energy Dissipated In This Storey 3 0.0 (0) 0.9 (0) 27.9 (1.3) 82.1 (3.8) 104.6 (4.9) 215.6 (10.0) 2 0.0 (0) 3.5 (0.2) 22.4 (1.0) 54.0 (2.5) 59.3 (2.7) 139.3 (6.5) 1 1648.5 (76.6) 6.0 (0.3) 14.5 (.0.7) 25.2 (1.2) 92.6 (4.3) 1786.9 (83.0) Z 1648.5 10.4 64.8 161.3 256.5 2141.5 % Total (76.6) (0.5) (3.0) (7.5) (11.9) (99.5) ( ) = % of Total Energy in System. TABLE C.19 M.D.F. RESULTS SET 19 231 STRUCTURE: D HYSTERESIS MODEL. E - p EXCITATION : 30 Sees E l C e n t r o N . S . DAMPING : 32 Mass TOTAL ENERGY IN SYSTEM g END OF CALCULATIONS = 9 0 0 5 . 0 INCH K I P S PERIOD 0.99 S E C S . YSR = 0 . 3 7 V [ODER ATE SYSTEM S t o r e y E n e r g y D i s s i p a t e d By H y s t e r e s i s In T h i s S t o r e y R e s i d u a l P o t e n t i a l E n e r g y I n T h i s S t o r e y R e s i d u a l K i n e t i c E n e r g y I n T h i s S t o r e y E n e r g y D i s s i p a t e d I n Mass ot Damping In T h i s S t o r e y E n e r g y D i s s i p a t e d I n S t i f f n e s s a Damping I n T h i s S t o r e y T o t a l E n e r g y D i s s i p a t e c I n T h i s Store;.-10 0 . 0 ( 0 ) 2 . 3 CO) 1 1 4 . 4 ( 1 . 3 ) 4 7 8 . 5 ( 5 . 3 ) 0 . 0 ( 0 ) 5 9 5 . 3 ( 6 . 6 ) 9 0 . 0 ( 0 ) 3 - 3 ( 0 ) 8 4 . 7 ( 1 . 0 ) 4 3 9 . 2 ( 4 . 9 ) 0 . 0 ( 0 ) 5 2 7 . 1 ( 5 . S ) S 0 . 0 CO) 3 .2 CO) 5 8 . 6 ( 0 . 6 ) 3 9 0 . 6 ( 4 . 3 ) 0 . 0 ( 0 ) 4 5 2 . 4 ( 5 . 0 ) 7 6 . 5 ( 0 ) 6 . 7 ( 0 ) 3 7 . 2 ( 0 . 4 ) 3 4 0 . 7 ( 3 . 8 ) 0 . 0 ( 0 ) 3 9 1 . 1 ( 4 . 3 ) 6 1 3 2 . 5 ( .1.5) 1 0 . 7 ( 0 . 1 ) 2 6 . 2 ( 0 . 3 ) 2 8 9 . 8 ( 3 . 2 ) 0 . 0 ( 0 ) 4 5 9 . 2 ( 5 . 1 ) C J 3 3 6 . 6 ( 3 . 7 ) 1 4 . 2 ( 0 . 1 ) 2 1 . 5 ( 0 . 2 ) 2 3 4 . 2 ( 2 . 6 ) 0 . 0 ( 0 ) 6 0 6 . 4 ( 6 . 7 ) 4 5 5 6 . 7 ( 6 . 2 ) 2 3 . 7 ( 0 . 3 ) 1 7 . 6 ( 0 . 2 ) 1 8 1 . 0 ( 2 . 0 ) 0 . 0 ( 0 ) 7 7 9 . 0 ( 8 . 6 ) 3 9 6 5 . 9 ( 1 0 . 7 ) 3 2 . 1 ( 0 . 3 ) 1 7 . 1 ( 0 . 2 ) 1 2 8 . 8 ( 1 . 4 ) 0 . 0 ( 0 ) 1 1 4 4 . 0 ( 1 2 . 7 ) 2 8 3 8 . 0 ( 9 . 3 ) 3 6 . 9 ( 0 . 4 ) 2 1 . 9 ( 0 . 2 ) 8 0 . 9 (0.9) 0 . 0 ( 0 ) 9 7 7 . 7 M n fi) 1 2 9 7 7 . 6 ( 3 3 . 1 ) 4 3 . 0 ( 0 . 5 ) 1 5 . 5 ( 0 . 2 ) 3 8 . 0 ( 0 . 4 ) 0 . 0 ( 0 ) 3 0 7 4 . 1 ( 11.1 ) I 5 8 1 3 . 8 1 7 6 . 1 4 1 4 . 7 2 6 0 1 . 7 0 . 0 9 0 0 6 . 3 % T o t a l ( 6 4 . 6 ) ( 1 . 9 ) ( 4 . 6 ) ( 2 8 . 9 ) ( 0 ) ( 1 0 0 ) ( ) = % o f T o t a l E n e r g y i n S y s t e m . TABLE C . 2 0 M . D . F . RESULTS SET 20 232 STRUCTURE: 6 HYSTERESIS MODEL. E - P EXCITATION : 30 Sees, o f P a c o i m a S16E DAMPING : 3% Mass TOTAL ENERGY IN SYSTEM g END OF CALCULATIONS = 3 5 , 4 3 0 . 3 INCH K I P S PERIOD 0 . 9 9 S E C S . YSR = 0 . 1 1 WEAK SYSTEM S t o r e y E n e r g y D i s s i p a t e d By H y s t e r e s i s I n T h i s S t o r e y R e s i d u a l P o t e n t i a l E n e r g y I n T h i s S t o r e y R e s i d u a l K i n e t i c E n e r g y I n T h i s S t o r e y E n e r g y D i s s i p a t e d In M a s - a Damping In T h i s S t o r e y E n e r g y D i s s i p a t e d In S t i f f n e s s d Dampine I n T h i s S t o r e y T o t a l Ene rgy D i s s i p a t e d Ir. T h i s S t o r e ; : 10 0 . 0 ( 0 ) 0 . 1 ( 0 ) 7 3 4 . 8 ( 2 . 1 ) 7 9 7 . 7 ( 2 . 3 ) 0 . 0 ( 0 ) 1 5 3 2 . 6 ( 4.3) 9 5 5 . 0 ( 0 . 2 ) 0 . 2 ( 0 ) 7 3 0 . 0 ( 2 . 1 ) 7 3 3 . 9 ( 2 . 1 ) 0 . 0 ( 0 ) 1519 .2 ( 4.3) 8 3 2 0 . 2 ( 0 . 9 ) 0 . 2 ( 0 ) 7 2 8 . 0 ( 2 . 1 ) 6 7 5 . 2 ( 1 . 9 ) 0 . 0 ( 0 ) 1 7 2 3 . 6 ( 4 . 9 ) 7 5 1 7 . 7 ( 1 . 5 ) 0 . 2 ( 0 ) 7 3 1 . 9 ( 2 . 1 ) 6 3 0 . 9 ( 1 . 8 ) 0.0 (0) 1 8 3 0 . 8 ( 5.3) 6 1106.3 ( 3 . 1 ) 0 . 2 ( 0 ) 7 4 0 . 1 ( 2 . 1 ) 5 8 0 . 9 ( 1 . 6 ) 0 . 0 ( 0 ) 2 4 2 7 . 4 ( 6 . 9 ) 5 1 2 9 1 . 8 ( 3 . 6 ) . 0 . 1 ( 0 ) 7 4 7 . 7 ( 2 . 1 ) 502 .2 ( 1 . 4 ) 0.0 ( 0 ) 2 5 4 1 . 8 ( 7 . 2 ) 4 9 2 2 . 4 ( 2 . 6 ) 0 .1 ( 0 ) 7 4 9 . 2 ( 2 . 1 ) 4 2 7 . 1 ( 1 . 2 ) 0 . 0 ( 0 ) 2 0 9 8 . 7 ( 5 . 9 ) 3 1 0 9 8 . 7 ( 3 - D 0 . 2 ( 0 ) 7 4 6 . 5 ( 2 . 1 ) 3 6 7 . 8 ( 1 . 0 ) 0 . 0 ( 0 ) 2 2 1 3 . 2 ( 6 . 2 ) 2 912.3 ( 2 . 6 ) 0.3 ( 0 ) 7 4 2 . 8 ( 2 . 1 ) 304 .2 ( 0 . 9 ) 0 . 0 ( 0 ) 1 9 5 9 . 6 ( 5 . 5 ) 1 16493-3 ( 4 6 . 6 ) 0 .4 ( 0 ) 7 2 3 . 8 ( 2 . 0 ) 238.3 ( 0 . 7 ) 0 . 0 ( 0 ) 1 7 4 5 5 . 7 ( 4 9.3) I 2 2 7 1 7 . 6 2.0 7 3 7 4 . 7 5258 .4 0 . 0 35352 .7 % T o t a l (64.1 ) ( 0 ) ( 2 0 . 8 ) ( 1 4 . 8 ) ( 0 ) ( 9 9 . 8 ) ( ) = % o f T o t a l E n e r g y i n S y s t e m . TABLE C .21 M . D . F . RESULTS SET 21 233 STRUCTURE 1 HYSTERESIS MODEL. T r i - L i n e a r EXCITATION: 30 S e e s . T a f t N69W DAMPING: 1% S t i f f n e s s & Mass TOTAL ENERGY IN SYSTEM % END OF CALCULATIONS = 1052.3 INCH K I P S PERIOD 0 . 4 3 S E C S . YSR = 1 - 9 STRONG SYSTEM S t o r e y E n e r g y D i s s i p a t e d b y H y s t e r e s i s I n T h i s S t o r e y R e s i d u a l P o t e n t i a l E n e r g y In T h i s S t o r e y R e s i d u a l K i n e t i c E n e r g y In T h i s S t o r e y Ene rgy D i s s i p a t e d In Mass a Damping In T h i s S t o r e y E n e r g y D i s s i p a t e d In S t i f f n e s s a Damping In T h i s S t o r e y T o t a l E n e r g y D i s s i p a t e d In T h i s S t o r e y 3 2 0 0 . 5 ( 1 9 . 1 ) 1.3 ( 0 . 1 ) 4 . 6 ( 0 . 4 ) 5 8 . 9 ( 5 . 6 ) 7 . 9 ( 0 . 8 ) 273.1 ( 2 5 . 9 ) 2 3 5 0 . 0 ( 3 3 - 3 ) 0 . 6 ( 0 ) 2 . 9 ( 0.3) 3 6 . 7 ( 3 . 5 ) 5 . 5 ( 0 . 5 ) 3 9 5 . 8 ( 3 7 . 6 ) 1 3 5 3 . 5 ( 3 3 . 6 ) 0 . 0 ( 0 ) 3 .2 ( 0.3) 13 . 1 ( 1 . 2 ) 3.1 ( 0.3) 3 7 2 . 9 ( 3 5 . 4 ) Z 9 0 4 - 0 1 .9 1 0 . 7 1 0 8 . 8 1 6 . 5 1 0 4 1 . 8 % T o t a l ( 8 5 . 9 ) ( 0 . 2 ) ( 1 . 0 ) (10.3) ( 1 . 6 ) ( 9 9 . 0 ) ( ) = % o f T o t a l E n e r g y i n S y s t e m . TABLE C .22 M . D . F . RESULTS SET 22 STRUCTURE 1 HYSTERESIS MODEL. T r i - L i n e a r E X C I T A T I O N : 30 S e e s . E l C e n t r o N . S . DAMPING : 1% S t i f f n e s s & Mass TOTAL ENERGY IN SYSTEM % END OF CALCULATIONS = 6 7 4 4 . 3 INCH K I P S PERTOD 0 . 4 3 S E C S . YSR = 0 . 8 MODERATE SYSTEM S t o r e y E n e r g y D i s s i p a t e d by H y s t e r e s i s I n T h i s S t o r e y R e s i d u a l P o t e n t i a l E n e r g y In T h i s S t o r e y R e s i d u a l K i n e t i c E n e r g y In T h i s S t o r e y E n e r g y D i s s i p a t e d In Mass ot Damping In T h i s S t o r e y E n e r g y D i s s i p a t e d In S t i f f n e s s a Damping In T h i s S t o r e y T o t a l E n e r g y D i s s i p a t e d In T h i s S t o r e y 3 4 1 9 . 3 ( 6 . 2 ) 4 . 0 ( 0 . 1 ) 1 6 . 0 ( 0 . 2 ) 2 7 0 . 0 ( 4 . 0 ) 1 5 . 3 ( 0 . 2 ) 7 2 4 . 6 ( 1 0 . 7 ) 2 1 9 8 4 . 1 ( 2 9 . 4 ) 1 6 . 8 ( 0 . 2 ) 2 8 . 7 ( 0 . 4 ) 2 8 2 . 7 ( 4 . 2 ) 4 1 . 7 ( 0 . 6 ) 2 3 5 4 . 0 ( 3 4 . 9 ) 1 3 4 2 3 . 3 ( 5 0 . 8 ) 9 . 5 ( 0 . 1 ) 2 5 . 6 ( 0 . 4 ) 1 4 0 . 5 ( 2 . 1 ) 4 3 . 2 ( 0 . 6 ) 3 6 4 2 . 1 ( 5 4 . 0 ) Z 5 8 2 6 . 7 3 0 . 3 7 0 . 3 6 9 3 . 2 1 0 0 . 2 6 7 2 0 . 7 % T o t a l ( 8 6 . 4 ) ( 0 . 4 ) ( 1 . 0 ) ( 1 0 . 3 ) ( 1 . 5 ) ( 9 9 . 6 ) ( ) = t o f T o t a l E n e r g y i n S y s t e m . TABLE C.23 M . D . F . RESULTS SET 23 234 STRUCTURE 1 HYSTERESIS MODEL. T r i - L i n e a r EXCITATION : 30 S e e s . P a c o i m a DAMPING: \% S t i f f n e s s & Mass TOTAL ENERGY IN SYSTEM % END OF CALCULATIONS = 2 2 3 6 7 . 1 INCH K I P S PERIOD 0.4"? S E C S . YSR = 0 . 2 6 WEAK SYSTEM S t o r e y E n e r g y D i s s i p a t e d b y H y s t e r e s i s I n T h i s S t o r e y R e s i d u a l P o t e n t i a l E n e r g y In T h i s S t o r e y R e s i d u a l K i n e t i c E n e r g y In T h i s S t o r e y Ene rgy D i s s i p a t e d In Mass a Damping In T h i s S t o r e y E n e r g y D i s s i p a t e d In S t i f f n e s s a Damping In T h i s S t o r e y T o t a l E n e r g y D i s s i p a t e d In T h i s S t o r e y 3 2 7 9 . 1 ( 1 . 2 ) 0 . 0 ( 0 ) 6 0 7 . 6 ( 2 . 7 ) 7 9 9 . 6 ( 3 . 6 ) 1 1 . 9 ( 0 . 1 ) 1 6 9 8 . 1 ( 7 . 6 ) 2 7 0 7 . 8 ( 3 . 2 ) 0 . 0 ( 0 ) 9 1 6 . 5 ( 4 . 1 ) 1 0 9 7 . 9 ( 4 . 9 ) 2 5 . 5 ( 0 . 1 ) 2 7 4 7 . 6 ( 1 2 . 3 ) 1 1 4 8 6 6 . 4 ( 6 6 . 5 ) 0 . 4 ( 0 ) 1 2 3 3 . 1 ( 5 . 5 ) 1 2 5 6 . 3 ( 5 . 6 ) 5 4 2 . 5 ( 2 . 4 ) 1 7 8 9 8 . 7 ( 8 0 . 0 ) Z 15853 .2 0 . 4 2 7 5 7 . 1 3 1 5 3 . 8 5 7 9 . 9 2 2 3 4 4 . 4 % T o t a l ( 7 0 . 9 ) ( 0 ) ( 1 2 . 3 ) ( 1 4 . 1 ) ( 2 . 6 ) ( 9 9 . 9 ) ( ) = % o f T o t a l E n e r g y i n S y s t e m . TABLE C.24 M . D . F . RESULTS SET 24 STRUCTURE 2 HYSTERESIS MODEL. T r i - L i n e a r EXCITATION : 30 S e e s . T a f t N69W DAMPING: 196 S t i f f n e s s i Mass TOTAL ENERGY IN SYSTEM 6 END OF CALCULATIONS = 5 5 3 . 0 INCH K I P S PERIOD 0 . 3 3 S E C S . YSR = 3 . 1 STRONG SYSTEM S t o r e y E n e r g y D i s s i p a t e d by H y s t e r e s i s I n T h i s S t o r e y R e s i d u a l P o t e n t i a l E n e r g y In T h i s S t o r e y R e s i d u a l K i n e t i c E n e r g y In T h i s S t o r e y E n e r g y D i s s i p a t e d In Mass rt Damping In T h i s S t o r e y E n e r g y D i s s i p a t e d In S t i f f n e s s a Damping In T h i s S t o r e y T o t a l E n e r g y D i s s i p a t e d In T h i s S t o r e y 3 0 . 0 ( 0 ) 0 . 0 ( 0 ) 0 . 7 ( 0 . 1 ) 3 7 . 7 ( 6 . 8 ) 4 5 . 6 ( 8 . 2 ) 8 4 . 0 ( 1 5 . 2 ) 2 48.5 ( 8 . 8 ) 0 . 0 ( 0 ) 0 . 8 ( 0 . 1 ) 2 4 . 7 ( 4 . 5 ) 2 9 . 5 ( 5.3) 103.4 ( 1 8 . 7 ) 1 331 . 1 ( 5 9 . 9 ) 0 . 1 ( 0 ) 1.0 ( 0 . 2 ) 9 . 0 ( 1 . 6 ) 2 0 . 7 ( 3 . 8 ) 3 6 1 . 8 ( 6 5 . 4 ) Z 3 7 9 . 5 0 . 1 2 . 5 7 1 . 3 9 5 . 8 5 4 9 . 3 % T o t a l ( 6 8 . 6 ) ( 0 ) ( 0 . 5 ) ( 1 2 . 9 ) ( 1 7.3) ( 9 9 . 3 ) ( ) = % o f T o t a l E n e r g y i n S y s t e m . TABLE C . 2 5 M . D . F . RESULTS SET 25 235 STRUCTURE 2 HYSTERESIS MODEL. T r i - L i n e a r EXCITATION : 30 S e e s . E l C e n t r o N . S . DAMPING: 1% S t i f f n e s s & Mass TOTAL ENERGY IN SYSTEM % END OF CALCULATIONS = 1 3 0 8 . 5 INCH K I P S PERIOD 0.33 S E C S . YSR = 1-4 STRONG SYSTEM S t o r e y E n e r g y D i s s i p a t e d b y H y s t e r e s i s I n T h i s S t o r e y R e s i d u a l P o t e n t i a l E n e r g y In T h i s S t o r e y R e s i d u a l K i n e t i c E n e r g y I n T h i s S t o r e y E n e r g y D i s s i p a t e d I n Mass n Damping I n T h i s S t o r e y E n e r g y D i s s i p a t e d I n S t i f f n e s s a Damping In T h i s S t o r e y T o t a l E n e r g y D i s s i p a t e d I n T h i s S t o r e y 3 31.2 ( 2 . 4 ) 0 . 1 ( 0 ) 1 .0 ( 0 . 1 ) 6 4 . 5 ( 4 . 9 ) 6 8 . 8 ( 5 . 3 ) 1 6 5 . 5 ( 1 2 . 7 ) 2 2 9 1 . 3 ( 2 2 . 3 ) 0 . 3 ( 0 ) 1 .8 ( 0 . 1 ) 4 4 . 6 ( 3 . 4 ) 6 2 . 7 ( 4 . 8 ) 4 0 0 . 6 ( 3 0 . 6 ) 1 6 7 9 . 5 ( 5 1 . 9 ) 0 . 5 ( 0 ) 3 - 3 ( 0 . 3 ) 1 5 . 7 ( 1 . 2 ) 3 3 . 0 ( 2 . 5 ) 7 3 2 . 0 ( 5 5 . 9 ) Z 1 0 0 2 . 0 0 . 9 6 . 1 1 2 4 . 7 1 6 4 . 5 1 2 9 8 . 1 % T o t a l ( 7 6 . 6 ) (0 .1 ) ( 0 . 5 ) ( 9 . 5 ) ( 1 2 . 6 ) ( 99.2) ( ) = % o f T o t a l E n e r g y i n S y s t e m . TABLE C . 2 6 M . D . F . RESULTS SET 26 STRUCTURE 2 HYSTERESIS MODEL. T r i - L i n e a r EXCITAT ION: 30 S e e s . P a c o i m a S16E DAMPING: 1% S t i f f n e s s & Mass TOTAL ENERGY IN SYSTEM 6 END OF CALCULATIONS = 5 , 7 7 3 . 3 INCH K I P S PERIOD 0 . 3 3 S E C S . YSR = 0 . 4 3 MODERATE SYSTEM S t o r e y E n e r g y D i s s i p a t e d b y H y s t e r e s i s I n T h i s S t o r e y R e s i d u a l P o t e n t i a l E n e r g y I n T h i s S t o r e y R e s i d u a l K i n e t i c E n e r g y In T h i s S t o r e y E n e r g y D i s s i p a t e d In Mass a Damping In T h i s S t o r e y E n e r g y D i s s i p a t e d I n S t i f f n e s s a Damping I n T h i s S t o r e y T o t a l E n e r g y D i s s i p a t e d In T h i s S t o r e y 3 5 0 . 4 ( 0 . 9 ) 0 . 0 ( 0 ) 463.5 ( 8 . 0 ) 1 7 3 . 0 (3 .0) 7 1 . 4 ( 1 . 2 ) 7 5 8 . 2 ( 1 3 . 1 ) 2 318.3 ( 5 . 5 ) 0 . 0 ( 0 ) 463.1 ( 8 . 0 ) 1 5 0 . 9 ( 2 . 6 ) 8 6 . 7 ( 1 . 5 ) 1 0 1 9 . 1 ( 1 7 . 7 ) 1 2 8 9 0 . 6 ( 5 0 . 1 ) 0 . 1 ( 0 ) 462.9 ( 8 . 0 ) 1 0 7 . 5 ( 1 . 9 ) 5 1 6 . 9 ( 9 . 0 ) 3 9 7 7 . 9 ( 6 8 . 9 ) Z 3259.3 0 . 1 1 3 8 9 . 4 4 3 1 . 4 6 7 5 . 0 5755 .2 % T o t a l ( 5 6 . 5 ) ( 0 ) ( 2 4 . 1 ) ( 7 . 5 ) ( 1 1 . 7 ) ( 9 9 . 7 ) ( ) = % o f T o t a l E n e r g y i n S y s t e m . TABLE C . 2 7 M . D . F . RESULTS SET 27 236 STRUCTURE 3 HYSTERESIS MODEL. T r i - L i n e a r EXCITATION: 30 S e e s . E l C e n t r o N . S . DAMPING: 1% Mass TOTAL ENERGY IN SYSTEM 6 END OF CALCULATIONS = 1 9 4 6 . 8 INCH K I P S PERIOD 0 . 3 3 S E C S . YSR = 0 . 4 5 MODERATE SYSTEM S t o r e y E n e r g y D i s s i p a t e d b y H y s t e r e s i s I n T h i s S t o r e y R e s i d u a l P o t e n t i a l E n e r g y In T h i s S t o r e y R e s i d u a l K i n e t i c E n e r g y In T h i s S t o r e y E n e r g y D i s s i p a t e d In Mass a Damping I n T h i s S t o r e y E n e r g y D i s s i p a t e d I n S t i f f n e s s a Damping In T h i s S t o r e y T o t a l E n e r g y D i s s i p a t e d In T h i s S t o r e y 3 9 . 2 ( 0 . 5 ) 0 . 0 ( 0 ) 8 . 9 ( 0 . 5 ) 1 7 0 . 4 ( 8 . 8 ) 0 . 0 ( 0 ) 1 8 8 . 6 ( 9 . 7 ) 2 3 7 . 8 ( 1 . 9 ) 0 . 2 ( 0 ) 8 . 5 ( 0 . 4 ) 1 6 3 . 4 ( 8 . 4 ) 0 . 0 ( 0 ) 2 0 9 . 9 ( 1 0 . 8 ) 1 1382 .1 ( 7 1 . 0 ) 5 .4 ( 0 . 3 ) 8 . 1 ( 0 . 4 ) 1 4 8 . 8 ( 7 . 6 ) 0 . 0 ( 0 ) 1 5 4 4 . 5 ( 7 9 . 3 ) Z 1 4 2 9 . 2 5 . 6 2 5 . 5 4 8 2 . 6 0 . 0 1 9 4 2 . 9 % T o t a l ( 7 3 . 4 ) ( 0 - 3 ) ( 1 . 3 ) ( 2 4 . 8 ) ( 0 ) ( 9 9 . 8 ) ( ) = % o f T o t a l E n e r g y i n S y s t e m . TABLE C . 2 8 M . D . F . RESULTS SET 28 STRUCTURE 4 HYSTERESIS MODEL. T r i - L i n e a r EXCITATION : 30 S e e s . E l C e n t r o N . S . DAMPING : K S t i f f n e s s & Mass TOTAL ENERGY IN SYSTEM % END OF CALCULATIONS = 2 7 0 9 . 6 I N C H K I P S PERIOD 0 . 4 7 S E C S . YSR = 0 . 6 8 MODERATE SYSTEM S t o r e y E n e r g y D i s s i p a t e d b y H y s t e r e s i s I n T h i s S t o r e y R e s i d u a l P o t e n t i a l E n e r g y I n T h i s S t o r e y R e s i d u a l K i n e t i c E n e r g y In T h i s S t o r e y E n e r g y D i s s i p a t e d In Mass a Damping In T h i s S t o r e y E n e r g y D i s s i p a t e d I n S t i f f n e s s a Damping In T h i s S t o r e y T o t a l E n e r g y D i s s i p a t e d In T h i s S t o r e y 3 0 . 0 ( 0 ) 0 . 0 ( 0 ) 3 . 0 ( 0 . 1 ) 1 1 5 . 9 ( 4 . 3 ) 3 8 . 1 ( 1 . 4 ) 1 5 7 . 0 ( 5 . 8 ) 2 8 3 . 1 ( 3 . 1 ) 0 . 1 ( 0 ) 3 .1 ( 0 . 1 ) 1 0 4 . 0 ( 3 . 8 ) 3 8 . 6 ( 1 . 4 ) 2 2 8 . 8 ( 8 . 4 ) 1 1 8 1 4 . 3 ( 6 7 . 0 ) 1 .0 ( 0 ) 3 .4 ( 0 . 1 ) 8 2 . 4 ( 3 . 0 ) 4 1 7 . 2 ( 1 5 . 4 ) 2 3 1 8 . 2 ( 8 5 . 6 ) Z 1 8 9 7 . 3 1 .2 9 . 4 3 0 2 . 3 4 9 3 . 8 2 7 0 4 . 0 t T o t a l ( 7 0 . 0 ) ( 0 ) ( 0 . 3 ) ( 1 1 . 2 ) ( 1 8 . 2 ) ( 9 9 . 8 ) ( ) = % of T o t a l E n e r g y i n S y s t e m . TABLE C . 2 9 M . D . F . RESULTS SET 29 237 STRUCTURE 5 HYSTERESIS MODEL. T r i - L i n . EXCITATION : 30 S e e s . E l C e n t r o N . S . DAMPING : 1* S t i f f n e s s & Mass TOTAL ENERGY IN SYSTEM 6 END OF CALCULATIONS = 1908.5 INCH K I P S PERIOD 0.57 S E C S . y S R = 0.45 MODERATE SYSTEM S t o r e y E n e r g y D i s s i p a t e d b y H y s t e r e s i s I n T h i s S t o r e y R e s i d u a l P o t e n t i a l E n e r g y I n T h i s S t o r e y R e s i d u a l K i n e t i c E n e r g y I n T h i s S t o r e y E n e r g y D i s s i p a t e d I n Mass a Damping I n T h i s S t o r e y E n e r g y D i s s i p a t e d I n S t i f f n e s s a Damping I n T h i s S t o r e y T o t a l E n e r g y D i s s i p a t e d I n T h i s S t o r e y 3 0.0 (0) 0.1 (0) 4.7 ro.?) 81.6 (1.1) 28.1 M .51 114.5 (6.0) 2 7.9 (0.4) 0.4 (0) 5.0 ' (0.3) 72.8 Ci.8) 20.2 (1.1) 106.3 (5.6) 1 13H.4 (68.7) 5.5 (0.3) 5.5 (0.3) 59.8 (3-D 302.7 (15.9) 1684.9 (88.3) Z 1319.3 6.1 15.2 214.1 351.0 1905.7 % T o t a l (69.1) (0.3) (0.8) (11.2 ) (18.4) (99.9) ( ) = % o f T o t a l E n e r g y i n S y s t e m . TABLE C.30 M . D . F . RESULTS SET 30 238 STRUCTURE: 6 HYSTERESIS MODEL. T r l _ L l r u EXCITATION : 30 S e e s . P a c o i m a S16E DAMPING: 3% Mass TOTAL ENERGY IN SYSTEM % END OF CALCULATIONS = 3 3 , 4 9 3 - 4 INCH K I P S PERIOD 0 . 9 9 S E C S . YSR = 0 . 1 1 WEAK SYSTEM S t o r e y E n e r g y D i s s i p a t e d By H y s t e r e s i s I n T h i s S t o r e y R e s i d u a l P o t e n t i a l E n e r g y In T h i s S t o r e y R e s i d u a l K i n e t i c E n e r g y I n T h i s S t o r e y E n e r g y D i s s i p a t e d In M a s " a Damping I n T h i s S t o r e y E n e r g y D i s s i p a t e d In S t i f f n e s s a Damping I n T h i s S t o r e y T o t a l E n e r g y D i s s i p a t e d In T h i s S t o r e y 10 1.2 ( 0 ) 0 . 0 ( 0 ) . 7 3 7 . 0 ( 2 . 2 ) 1 1 1 6 . 3 ( 3 . 3 ) 0 . 0 ( 0 ) 1 8 5 4 . 5 ( 5 . 5 ) 9 4 6 . 0 ( 0 . 1 ) 0 . 0 ( 0 ) 7 3 6 . 2 ( 2 . 2 ) 1 1 0 7 . 0 ( 3 . 3 ) 0 . 0 ( 0 ) 1 8 8 9 . 2 ( 5 . 6 ) 8 1 0 0 . 6 ( 0 . 3 ) 0 . 0 ( 0 ) 7 3 5 . 2 ( 2 . 2 ) 1 0 8 7 . 7 ( 3 . 2 ) 0 . 0 ( 0 ) 1 9 2 3 . 5 ( 5 . 7 ) 7 1 4 2 . 6 ( 0 . 4 ) 0 . 1 ( 0 ) 7 3 4 . 0 ( 2 . 2 ) 1 0 5 8 . 7 ( 3 . 2 ) 0 . 0 ( 0 ) 1 9 3 5 . 2 ( 5 . 8 ) 6 2 0 3 . 5 ( 0 . 6 ) 0 . 1 ( 0 ) 7 3 3 . 0 ( 2 . 2 ) 1 0 2 2 . 5 ( 3 . 1 ) 0 . 0 ( 0 ) 1 9 5 9 . 1 ( 5 . 8 ) 5 4 1 4 . 6 ( 1 . 2 ) 0 . 1 ( 0 ) 7 3 2 . 2 ( 2 . 2 ) 9 8 0 . 5 ( 2 . 9 ) 0 . 0 ( 0 ) 2 1 2 7 . 4 ( 6 . 4 ) 4 5 3 0 . 7 ( 1 . 6 ) 0 . 2 ( 0 ) 7 3 1 . 5 ( 2 . 2 ) 9 2 0 . 1 ( 2 . 7 ) 0 . 0 ( 0 ) 2182 .4 ( 6 . 5 ) 3 5 2 0 . 3 ( 1 . 6 ) 0 . 2 ( 0 ) 7 3 0 . 7 ( 2 . 2 ) 8 5 2 . 7 ( 2 . 5 ) 0 . 0 ( 0 ) 2 1 0 3 - 8 ( 6 . 3 ) 2 1 2 8 2 . 6 (3.8) 0 . 5 ( 0 ) 7 2 9 . 2 ( 2 . 2 ) 7 8 2 . 4 ( 2 . 3 ) 0 . 0 ( 0 ) 2 7 9 4 . 7 ( 8 . 3 ) 1 1 3 , 3 3 5 . 6 ( 3 9 . 8 ) 4 . 6 ( 0 ) 7 2 5 . 7 ( 2 . 2 ) 6 4 4 . 7 ( 1 . 9 ) 0 . 0 ( 0 ) 1 4 , 7 1 0 . 6 ( 4 3 . 9 ) T o t a l 1 6 , 5 7 7 . 9 5 . 7 7 3 2 4 . 6 9 5 7 2 . 4 0 . 0 3 3 , 4 8 0 . 6 ? T o t a l ( 4 9 . 5 ) (o) ( 2 1 . 9 ) ( 2 8 . 6 ) ( 0 ) ( 1 0 0 . o) ( ) = % o f T o t a l E n e r g y i n S y s t e m . TABLE C.3 .1 M . D . F . RESULTS SET 3 1 239 STRUCTURE: 6 HYSTERESIS MODEL. T r i - L i n . EXCITATION : 30 S e e s , o f E l C e n t r o N . S . DAMPING 3% Mass TOTAL ENERGY IN SYSTEM i END OF CALCULATIONS = 7356.9 INCH K I P S PERIOD 0-99 S E C S . YSR = 0.37 MODERATE SYSTEM S t o r e y E n e r g y D i s s i p a t e d By H y s t e r e s i s I n T h i s S t o r e y R e s i d u a l P o t e n t i a l E n e r g y In T h i s S t o r e y R e s i d u a l K i n e t i c E n e r g y I n T h i s S t o r e y E n e r g y D i s s i p a t e d I n Mass a Damping In T h i s S t o r e y E n e r g y D i s s i p a t e d In S t i f f n e s s a Damping I n T h i s S t o r e y T o t a l Ene rgy D i s s i p a t e c I n T h i s S t o r e y 10 0.0 (0) 0.0 (0) 3.7 (0.1) 374.2 (5.1) o.o-(0) 377.9 (5.1) 9 76.3 (1.0) 0.1 (0) 5.3 (0.1) 355.1 (4.8) 0.0 (0) 436.8 (5.9) e 175.6 (2.4) 1.7 (0) 6.7 (0.1) 323.6 (4.4) 0.0 (0) 507.6 (6.9) 7 289.6 (3.9) 2.6 (0) 8.1 (0.1) 283.8 (3.9) 0.0' (0) 584.2 (7.9) 6 434.0 (5.9) 1.8 (0) 9.6 (0.1) 238.7 (3.2) 0.0 (0) 684.2 (9.3) 5 529.4 (7.2) 1.3 (0) 10.4 (0.1) 191.3 (2.6) 0.0 (0) 732.3 (10.0) - 4 619.6 (8.4) 1.2 (0) 10.7 (0.1) 145.7 (2.0) 0.0 (0) 777.3 (10.6) 3 695.9 (9.5) 2.3 (0) 13.1 (0.2) 102.0 (1.4) 0.0 (0) 813.3 (11.1) 2 1045.2 (14.2) 2.9 (0) 15.4 (0.2) 60.7 (0.8) 0.0 (0) 1124.1 (15.3) 1 1268.3 (17.2) 3.4 (0) 11.4 (0.2) 20.6 (0.3) 0.0 (0) 1303.7 (17.7) T o t ' a l 5133.9 17.3 94.4 2095.6 0.0 7341.3 $ T o t a l (69.8 ) (0.2 ) ( 1.3 ) ( 28.5 ) ( o) ( 99.8 ) ( ) = % o f T o t a l E n e r g y i n S y s t e m . TABLE C.32 M . D . F . RESULTS SET 32 Fue>u C A T I O O S CD ! ^>Teu>CT(je.A.l_- ^6£»Pe>fOS£ To T e A O S t-ATfONJA U AMD o T l OfO S fJAj*&fO , 0. D . , MAe^&oiifi , J.J?, 4' M ct^&tTT, id. £ . £x>6.ose -£iN>& Vou JIL ISTAOS^L. T^et^y Sfifr 1376. S E I S M I C XrOPOT S . KJAT*A«O , to. D ; 6H«eY , S . 4 M ^ e v . T T , k J . £ . (T) *T©iOA£bs A SiMPLfe E.»oe£6iY K'UTHOP ^ e i & H t c Dfc-SI&O o f S>TEt>CTL>e£rS M ^ I T T , 1(0. E.. , A u D t e i o i o , . D . L , NUI^AA), K).D., GuttLy , €>. 

Cite

Citation Scheme:

        

Citations by CSL (citeproc-js)

Usage Statistics

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}]}"
                            data-media="{[{embed.selectedMedia}]}"
                            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:
https://iiif.library.ubc.ca/presentation/dsp.831.1-0062476/manifest

Comment

Related Items