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Modified substitute structure method for analysis of existing R/C structures Yoshida, Sumio 1979

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MODIFIED  SUBSTITUTE  ANALYSIS  S T R U C T U R E METHOD  OF E X I S T I N G  R/C  FOR  STRUCTURES  by SUMIO\YOSHIDA  B.  A.  A  Sc., U n i v e r s i t y  THESIS THE  of British  SUBMITTED IN P A R T I A L REQUIREMENTS MASTER  FOR  Columbia,  1976  FULFILLMENT  OF  THE DEGREE  OF A P P L I E D  OF  SCIENCE  • in" THE  FACULTY  (Department  We  accept  THE  of Civil  this  the  OF GRADUATE  thesis  required  UNIVERSITY  (c)  Engineering)  as conforming standard  OF B R I T I S H  March,  Sumio  STUDIES  COLUMBIA  19 79  Yoshida,  19 79  to  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.  Department of  Civil  Engineering  The University of British Columbia 2075 Wesbrook Place Vancouver, Canada V6T 1W5  n a t p  March,  1979  ABSTRACT  The m o d i f i e d s u b s t i t u t e s t r u c t u r e method i s developed the earthquake hazard  for  e v a l u a t i o n o f e x i s t i n g r e i n f o r c e d concrete  b u i l d i n g s c o n s t r u c t e d before the most r e c e n t advances i n seismic design codes.  The main c h a r a c t e r i s t i c of the proposed method i s  the use of m o d i f i e d l i n e a r a n a l y s i s f o r p r e d i c t i n g the i n c l u d i n g i n e l a s t i c response, j e c t e d to a given type and  behaviour,  of e x i s t i n g s t r u c t u r e s when sub-  i n t e n s i t y of earthquake motion,  represented by a l i n e a r response spectrum.  The procedure i n -  v o l v e s an e x t e n t i o n of the s u b s t i t u t e s t r u c t u r e method, which o r i g i n a l l y proposed by Shibata and  Sozen as a design  was  procedure.  With p r o p e r t i e s and s t r e n g t h s of a s t r u c t u r e known, the m o d i f i e d s u b s t i t u t e s t r u c t u r e method computes d u c t i l i t y demand of each member v i a an e l a s t i c modal a n a l y s i s , i n which reduced s t i f f n e s s and  s u b s t i t u t e damping f a c t o r s are used  flexural  iteratively.  As a r e s u l t o f the a n a l y s i s , i t i s p o s s i b l e to d e s c r i b e , i n g e n e r a l terms, the l o c a t i o n and extent of damage t h a t would occur i n a s t r u c t u r e s u b j e c t e d to earthquakes of d i f f e r e n t  intensity.  S e v e r a l r e i n f o r c e d concrete s t r u c t u r e s of d i f f e r e n t and  s t r e n g t h s were t e s t e d by the proposed method and the  compared w i t h a n o n l i n e a r dynamic a n a l y s i s . number o f i t e r a t i o n s was  sizes results  In g e n e r a l , a small  r e q u i r e d to o b t a i n an estimate o f damage  ratios.  The method appears t o work w e l l f o r s t r u c t u r e s i n which  y i e l d i n g i s not e x t e n s i v e and widespread.  Furthermore, i t  appears to work b e t t e r f o r those i n which y i e l d i n g occurs mainly i n beams and the e f f e c t of h i g h e r modes i s not predominant. Though f u r t h e r r e s e a r c h i s necessary, the m o d i f i e d s u b s t i t u t e s t r u c t u r e method can c o n s t i t u t e an i n t e g r a l p a r t o f the r a t i o n a l r e t r o f i t procedure.  iv TABLE OF CONTENTS Page ABSTRACT  i i  TABLE OF CONTENTS  iv  LIST OF TABLES  v i i  LIST OF FIGURES  ix  ACKNOWLEDGEMENTS  xii  CHAPTER 1.  INTRODUCTION 1.1  Background  1  1. 2  L i t e r a t u r e Survey  4  1.3 2.  (a)  ATC Report  4  (b)  Okada and B r e s l e r  6  (c)  Freeman, N i c o l e t t i , and T y r r e l l . . .  9  Purpose and Scope  SUBSTITUTE 2.1  STRUCTURE METHOD  Modal A n a l y s i s  14  (a)  Equation of Motion  14  (b)  Periods and Mode Shapes  15  (c)  Response S p e c t r a  16  (d)  Modal Forces  IV  (e)  Combination of Forces and Displacements  2.2  11  18  S u b s t i t u t e S t r u c t u r e Method (a)  Development  20  (b)  S u b s t i t u t e S t r u c t u r e Method  23  (c)  Computer Program  29  V  CHAPTER  Page 2.3  2.4  3.  4.  5.  Examples and Observations (a)  Frames w i t h F l e x i b l e Beams  31  (b)  S o f t - S t o r y Frame  33  (c)  2-Bay, 3-Story Frame  35  Equal-Area S t i f f n e s s Method (a)  Observation  37  (b)  Equal-Area S t i f f n e s s  38  (c)  Examples  39  (d)  Area f o r F u r t h e r S t u d i e s  40  MODIFIED SUBSTITUTE STRUCTURE METHOD 3.1  M o d i f i e d S u b s t i t u t e S t r u c t u r e Method  42  3.2  Computer Program  51  3.3  Convergence  54  3.4  A c c e l e r a t e d Convergence  60  EXAMPLES 4.1  Assumptions and Comments  4.2  Examples  65  (a)  2-Bay, 2-Story Frame  69  (b)  3-Bay, 3-Story Frame  72  (c)  1-Bay, 6-Story Frame  76  (d)  3-Bay, 6-Story Frame  79  (e)  Observations  83  FACTORS AFFECTING MODIFIED SUBSTITUTE STRUCTURE METHOD 5.1  E f f e c t o f Higher Modes  86  5.2  Spectrum  91  5.3  G u i d e l i n e s f o r Use o f Method  96  VI  CHAPTER  Page 5.4  6.  Further Studies  CONCLUSION  BIBLIOGRAPHY  99 101 16 7  APPENDIX A.  B.  M o d i f i c a t i o n of Damage R a t i o - S t r a i n Hardening Case  169  Computer Program  17 3  vii LIST OF TABLES  Table 2.1  Page N a t u r a l P e r i o d s and Smeared Damping R a t i o s f o r 3-, 5-, and 10-Story Frames  2.2  Computed Damage R a t i o s f o r 3-, 5-, and 10-Story Frames  2.3  10 4  Comparison o f Damage R a t i o s f o r 3-, 5-, and 10-Story Frames  2.4  103  105  Computed N a t u r a l P e r i o d s f o r 3-, 5-, and 10-Story Frames  106  3.1  N a t u r a l P e r i o d s f o r 2-Bay, 3-Story Frame A  106  3.2  Damage R a t i o s f o r 2-Bay, 3-Story Frame A  107  3.3  N a t u r a l P e r i o d s f o r 2-Bay, 3-Story Frame B  108  3.4  Number of I t e r a t i o n s - 2-Bay, 3-Story Frame B....  108  3.5  Damage R a t i o s f o r 2-Bay, 3-Story Frame B  109  4.1  N a t u r a l P e r i o d s f o r 2-Bay, 2-Story Frame  109  4.2  Displacements f o r 2-Bay, 2-Story Frame  110  4.3  N a t u r a l P e r i o d s f o r 3-Bay, 3-Story Frame  110  4.4  Displacements f o r 3-Bay, 3-Story Frame  110  4.5  N a t u r a l P e r i o d s f o r 1-Bay, 6-Story Frame  I l l  4.6  Displacements f o r 1-Bay, 6-Story Frame  I l l  4.7  N a t u r a l P e r i o d s f o r 3-Bay, 6-Story Frame  I l l  4.8  Displacements f o r 3-Bay, 6-Story Frame  5.1  N a t u r a l P e r i o d s f o r 3-Bay, 6-Story Frame A Spectrum B  5.2  112  112  Displacements f o r 3-Bay, 6-Story Frame A Spectrum B  112  viii Table 5.3  Page N a t u r a l P e r i o d s f o r 3-Bay, 6-Story Frame B Spectrum  5.4  ^  113  Displacements f o r 3-Bay, 6-Story Frame B Spectrum  5.5  B  B  113  N a t u r a l Periods f o r 3-Bay, 6-Story Frame B Spectrum A  5.6  Displacements f o r 3-Bay, 6-Story Frame B Spectrum  5.7  A  114  N a t u r a l P e r i o d s f o r 3-Bay, 6-Story Frame B E l Centro EW Spectrum  5.8  113  and T a f t S69E Spectrum  Displacements f o r 3-Bay, 6-Story Frame B E l Centro EW Spectrum  5.9  115  Displacements f o r 3-Bay, 6-Story Frame B T a f t S69E Spectrum  5.10  115  N a t u r a l Periods f o r 3-Bay, 6-Story Frame A E l Centro EW Spectrum  5.11  and T a f t S69E Spectrum  116  Displacements f o r 3-Bay, 6-Story Frame A E l Centro EW Spectrum  5.12  114  116  Displacements f o r 3-Bay, 6-Story Frame A T a f t S69E Spectrum  117  ix LIST OF FIGURES  Figure 1.1  Page Load-Deflection  Curve f o r E l a s t i c and  E l a s t o p l a s t i c Structure 2.1  118  I d e a l i z e d H y s t e r e s i s Loop f o r R e i n f o r c e d Concrete System  2.2  118  Force-Displacement Curve - D e f i n i t i o n o f Damage R a t i o  119  2.3  Flow Diagram f o r S u b s t i t u t e S t r u c t u r e Method  12 0  2.4  Member P r o p e r t i e s and Design Moments f o r 3-, 5-, 10-Story Frames  12 3  2.5  Smoothed Response Spectrum - Design Spectrum A...  124  2.6  S o f t Story Frame A - Member P r o p e r t i e s and Y i e l d Moments..  2.7  125  S o f t Story Frame A - Damage R a t i o s f o r I n d i v i d u a l Earthquakes  2.8  126  S o f t Story Frame B - Member P r o p e r t i e s and Y i e l d Moments  2.9  12 7  S o f t Story Frame B - Damage Ratios f o r I n d i v i d u a l Earthquakes  2.10  12 8  2-Bay, 3-Story Frame - Member P r o p e r t i e s and Y i e l d Moments..  2.11  2-Bay, 3-Story Frame - Damage Ratios f o r I n d i v i d u a l Earthquakes  2.12  129  *  Force-Displacement Curve - D e f i n i t i o n o f Equal-Area S t i f f n e s s  3.1  130  131  Moment-Rotation Curve - M o d i f i c a t i o n o f Damage Ratio...  132  X  Figure 3.2  Page Flow  Diagram  f o r Modified  Substitute  Structure  Method 3.3  2-Bay, and  3.4  3.5  Yield  3.8  2-Bay,  3-Story  2-Bay,  Frame  3-Story  Frame  of Periods vs. 137  A  - Plot  o f Damage  Ratios  B  138  - Member  P r o p e r t i e s and 139  of Iterations  2-Bay,  3-Story  and  A - Plot  Moments  Number  Frame  Frame  t h e E n d o f 4, 3-Story  Number  2-Bay,  Properties  of Iterations  3-Story  vs. 4.1  Frame  2-Bay,  2-Bay,  - Member  136  of Iterations  Number  A  Moments  Number  at 3.9  Frame  3-Story  Yield 3.7  3-Story  2-Bay,  vs. 3.6  13 3  of Periods vs. 140  B - Damage  1 2 , 2 0 , a n d 200  Frame  B - Plot  Ratios  Computed  Iterations  o f Damage  Frame  -  142 Member  Properties  Moments  14 3  4.2  2-Bay,  2-Story  Frame  -  Damage  Ratios  4.3  2-Bay,  2-Story  Frame  -  Damage  Ratios f o r  Individual 4.4  3-Bay, and  Earthquakes  3-Story  Yield  Frame  -  Member  Properties  Moments  145  3-Bay,  3-Story  Frame  -  Damage  Ratios  4.6  3-Bay,  3-Story  Frame  -  Damage  Ratios f o r  Earthquakes  14 3  144  4.5  Individual  141  Ratios  of Iterations  2-Story  Yield  B - Plot  146  14 7  xi Figure 4.7  Page 1-Bay, 6-Story Frame - Member P r o p e r t i e s and Y i e l d Moments  14 8  4.8  1-Bay, 6-Story Frame - Damage R a t i o s  149  4.9  1-Bay, 6-Story Frame - Damage R a t i o s f o r I n d i v i d u a l Earthquakes  4.10  150  3-Bay, 6-Story Frame - Member P r o p e r t i e s and Y i e l d Moments  151  4.11  3-Bay, 6-Story Frame - Damage R a t i o s  152  4.12  3-Bay, 6-Story Frame - Damage R a t i o s f o r I n d i v i d u a l Earthquakes  15 3  5.1  Smoothed Response Spectrum - Design Spectrum B...  154  5.2  3-Bay, 6-Story Frame A - Damage R a t i o s  155  5.3  3-Bay, 6-Story Frame A - P l o t o f Damage R a t i o s f o r Beams i n the E x t e r i o r Bay.  156  5.4  3-Bay, 6-Story Frame B - Damage R a t i o s  15 7  5.5  3-Bay, 6-Story Frame B - P l o t o f Damage R a t i o s f o r Beams i n the E x t e r i o r Bay.  158  5.6  3-Bay, 6-Story Frame B - Damage R a t i o s  159  5.7  3-Bay, 6-Story Frame B - Damage R a t i o s f o r I n d i v i d u a l Earthquakes  160  5.8  E l Centro EW Spectrum and Design Spectrum.A  161  5.9  T a f t S69E Spectrum and Design Spectrum A  162  5.10  3-Bay, 6-Story Frame B - Damage R a t i o s  16 3  5.11  3-Bay, 6-Story Frame B - Damage R a t i o s  16 4  5.12  3-Bay, 6-Story Frame A - Damage R a t i o s  16 5  5.13  3-Bay, 6-Story Frame A - Damage R a t i o s  166  A.l  Moment-Rotation Curve  I  7  2  xii ACKNOWLEDGEMENTS  The  author wishes t o express h i s s i n c e r e g r a t i t u d e t o h i s  s u p e r v i s o r s , Dr. N. D. Nathan, Dr. D. L. Anderson, and Dr. S. Cherry  f o r t h e i r advice and guidance d u r i n g the r e s e a r c h and  preparation of t h i s thesis.  Thanks are a l s o due t o Mr. R.  G r i g g , the C i v i l E n g i n e e r i n g  Department program l i b r a r i a n ,  f o r h i s advice and a s s i s t a n c e . The  f i n a n c i a l support  of the N a t i o n a l Research C o u n c i l of  Canada i n the form of Postgraduate S c h o l a r s h i p i s g r a t e f u l l y acknowledged.  March, 19 79 Vancouver, B r i t i s h Columbia  1  CHAPTER 1  1.1  INTRODUCTION  Background  During  the l a s t two  decades a g r e a t d e a l of progress  has  been made i n understanding  the behaviour of b u i l d i n g s d u r i n g  major earthquake motions.  The  new  knowledge r e s u l t i n g from  r e s e a r c h and o b s e r v a t i o n has been i n c o r p o r a t e d i n b u i l d i n g codes.  I t i s not reasonable  designed  to expect the m a j o r i t y of newly  b u i l d i n g s to be able to s u r v i v e a major earthquake  motion w i t h t o l e r a b l e damage. Unfortunately, which were designed i n s e i s m i c codes.  i n any  l a r g e c i t y there e x i s t many b u i l d i n g s  and c o n s t r u c t e d b e f o r e the r e c e n t advances The performance of these b u i l d i n g s are at  best u n c e r t a i n i f and when a s i z a b l e earthquake s t r i k e s the The  area.  c i t y a u t h o r i t i e s must assess the s e i s m i c r i s k s i n v o l v e d i n  such b u i l d i n g s from time to time.  T h i s p o i n t a r i s e s most o f t e n  when an owner of an o l d b u i l d i n g wishes to change the occupancy or do a s t r u c t u r a l a l t e r a t i o n . permit,  Before  i s s u i n g a new  the a u t h o r i t i e s must make a d e c i s i o n on how  p l i e s with c u r r e n t codes.  Unless  building w e l l i t com-  the b u i l d i n g i s judged to be  s a f e , they must decide on the m o d i f i c a t i o n s t h a t have to be made i n order to upgrade i t to a s a t i s f a c t o r y l e v e l .  Upon t h e i r  recommendations the owner can decide whether i t i s f e a s i b l e to  2 c a r r y on w i t h h i s p l a n o r whether i t i s more economical t o r e p l a c e the b u i l d i n g w i t h a new  one.  ' I t i s , t h e r e f o r e , necessary t o develop a methodology to screen and e v a l u a t e e x i s t i n g b u i l d i n g s a g a i n s t s e i s m i c hazards. Many i s s u e s are i n v o l v e d here, but the most d i f f i c u l t one i s how  to assess the degree o f compliance w i t h the c u r r e n t s e i s m i c  codes.  I t i s a p p r o p r i a t e here t o d e s c r i b e b r i e f l y the p h i l o s -  ophy behind the c u r r e n t codes, which should be borne i n mind when the e v a l u a t i o n of e x i s t i n g b u i l d i n g s i s d i s c u s s e d l a t e r . The c u r r e n t code procedure f o r the design of new i s based on the assumption  that a structure w i l l y i e l d  buildings in a  major earthquake, but t h a t i t s u l t i m a t e displacement w i l l be approximately equal to the displacement of the same s t r u c t u r e i f i t remained e l a s t i c d u r i n g the earthquake as i l l u s t r a t e d i n Fig.  1.1.  I t s h o u l d be noted t h a t the s t i f f n e s s  o f the s t r u c -  t u r e i s u s u a l l y predetermined by the l a y o u t and the design f o r g r a v i t y loads.  The combination o f d u c t i l i t y and s t r e n g t h must  be chosen such t h a t the s t r u c t u r e reaches i t s maximum l o a d maximum displacement r e l a t i o n s h i p w i t h o n l y a t o l e r a b l e  level  of damage. The code, such as the N a t i o n a l B u i l d i n g Code of Canada,^ achieves t h i s combination of s t r e n g t h and d u c t i l i t y by e s t i mating the a v a i l a b l e d u c t i l i t y f o r the p a r t i c u l a r  structural  system s e l e c t e d f o r the design of the b u i l d i n g , and the l o a d l e v e l i s set accordingly.  Thus a d u c t i l e system may  f o r a lower l o a d l e v e l than a more b r i t t l e system.  be designed The code  a l s o s p e c i f i e s the d e t a i l e d design requirements t o ensure t h a t t h i s d u c t i l e f a i l u r e mode occurs b e f o r e the b r i t t l e  failure  3 modes a s s o c i a t e d with shear, bond o r d e t a i l The  failure.  code a c t u a l l y gives a q u a s i - s t a t i c f o r c e such t h a t  the s t r u c t u r e i s s a t i s f a c t o r y i f i t can r e s i s t t h a t f o r c e , provided  t h a t i t i s d e t a i l e d p r o p e r l y t o ensure the a n t i c i p a t e d  d u c t i l i t y and t h a t i t i s a l s o d e t a i l e d c o r r e c t l y to ensure the d e s i r a b l e f l e x u r e f a i l u r e mode. I t should described  now be c l e a r t h a t without the philosophy  above the code s t a t i c f o r c e i s meaningless.  It i s  not the a c t u a l f o r c e which a s t r u c t u r e i s expected t o r e c e i v e d u r i n g a major earthquake i f i t i s designed and d e t a i l e d d i f f e r e n t l y from the c u r r e n t codes. obviously  The e x i s t i n g b u i l d i n g s were  designed w i t h a d i f f e r e n t philosophy  from the one  i m p l i e d i n the c u r r e n t codes, and merely a p p l y i n g s t a t i c load i s a questionable The  the q u a s i -  approach.  best way t o analyze e x i s t i n g b u i l d i n g s i s t o s u b j e c t  them t o a n o n l i n e a r  time-step a n a l y s i s .  Recent advances i n  computer technology have made t h i s approach p o s s i b l e .  But the  c o s t i n v o l v e d i n such a n a l y s i s i s s t i l l p r o h i b i t i v e l y h i g h and it The  r e q u i r e s very accurate  modelling  h i g h c o s t and tediousness  except i n very  o f the e n t i r e s t r u c t u r e .  make t h i s a n a l y s i s i m p r a c t i c a l  few cases.  S e v e r a l proposals  have been made t o f i n d a more p r a c t i c a l  way t o t r e a t the problem o f a n a l y z i n g  the e x i s t i n g b u i l d i n g s ,  which i s becoming known by the somewhat i n f e l i c i t i o u s "retrofit."  term,  4 1.2  Literature  Survey  The l i t e r a t u r e survey i n t h i s s e c t i o n i s i n t e n d e d t o be an i n t r o d u c t i o n t o the approaches t h a t must be f o l l o w e d i n o r d e r to i d e n t i f y the p o t e n t i a l l y hazardous b u i l d i n g s and t o e s t i m a t e an e x t e n t o f hazards and an a s s o c i a t e d damage.  Three papers are  discussed.  (a)  ATC Report The A p p l i e d Technology C o u n c i l i n the U n i t e d S t a t e s made a  f i r s t attempt a t a comprehensive  procedure f o r the s e i s m i c 2  hazard e v a l u a t i o n o f e x i s t i n g b u i l d i n g s .  The r e l e v a n t  section  o f ATC I I I , the r e p o r t o f the c o u n c i l , i s b r i e f l y d i s c u s s e d here. ATC I I I p o i n t s out t h a t there are probably thousands o f b u i l d i n g s i n the U n i t e d S t a t e s which are p o t e n t i a l l y hazardous. ically  earthquake  Since a thorough study o f a l l b u i l d i n g s i s econom-  i m p o s s i b l e , they suggest a graduated procedure.  They a r e ,  (1)  S e l e c t i o n t o i d e n t i f y p o t e n t i a l l y hazardous  buildings  (2)  E v a l u a t i o n t o e s t a b l i s h the p o s s i b l e e x t e n t o f hazards  (3)  C o r r e c t i o n t o ensure the e l i m i n a t i o n o f unacceptable hazards. The f i r s t s t e p i s t o screen the p o t e n t i a l l y  buildings.  hazardous  The s e i s m i c hazard i s r e l a t e d t o the s e v e r i t y o f  the ground motion and the usage o f b u i l d i n g s .  The s e v e r i t y o f  the ground motion i s i n d i c a t e d by the Seismic Hazard Index SHI c o r r e l a t e d w i t h ground motion.  SHI ranges from 1 t o 4, w i t h  the h i g h e r number i n d i c a t i n g g r e a t e r s e v e r i t y .  The usage o f  5 the b u i l d i n g s i s indexed by the Seismic SHE.  SHE  Hazard Exposure Group  ranges from I to I I I , w i t h the h i g h e r number i n d i c a t i n g  l e s s usage. The b u i l d i n g s i n the area where the Seismic  Hazard Index  i s l e s s than or equal to 3 are excluded from a n a l y s i s .  In  area where SHI  SHE-III  i s 4, the newer b u i l d i n g s and  b u i l d i n g s w i t h low h i s t o r i c a l values The  SHE-II and  occupancy are a l s o exempt. are subjected  The  the  b u i l d i n g s with  to the a l t e r n a t e procedure.  e v a l u a t i o n procedure may  tative.  A q u a l i t a t i v e evaluation  groups.  The  be q u a l i t a t i v e o r i s required  quanti-  f o r SHE-II and  procedure i s p r e s c r i b e d i n the r e p o r t .  -III  It involves  a judgement on the adequacy o f the primary s t r u c t u r a l system n o n s t r u c t u r a l elements, and SHE-I b u i l d i n g s and  i t can be  c a r r i e d out very r a p i d l y .  those judged u n c e r t a i n  i n the  previous  a n a l y s i s are s u b j e c t to more thorough a n a l y t i c a l s t u d i e s . aseismic The  design  procedure f o r new  and  constructions  procedure i n v o l v e s the determination  The  are s t i p u l a t e d .  of an earthquake capa-  c i t y r a t i o , R , which i s a r a t i o of a c t u a l l a t e r a l s e i s m i c c a p a c i t y of an e x i s t i n g system or element to the  force  capacity  r e q u i r e d to meet the p r e v a i l i n g s e i s m i c code p r o v i s i o n s f o r the design  of new  buildings.  The  occupancy p o t e n t i a l are a l s o used  to assess b u i l d i n g hazards. The  t o t a l l a t e r a l seismic  b u i l d i n g height and  and  f o r c e i s d i s t r i b u t e d over  the r e s u l t i n g a p p l i e d member moment, shear,  a x i a l f o r c e s are e v a l u a t e d  at p a r t i c u l a r sections.  The  member c a p a c i t i e s can be c a l c u l a t e d from the known s e c t i o n material properties. by d i v i d i n g the  the  The  and  earthquake c a p a c i t y r a t i o i s computed  s e c t i o n c a p a c i t y a v a i l a b l e f o r earthquake  loading  6 by the s e i s m i c a l l y induced l o a d .  The  r a t i o s are computed f o r  moments, shear, a x i a l f o r c e s , and  drift.  The  smallest  governs the earthquake c a p a c i t y of the b u i l d i n g .  ratio  In the  o p i n i o n , a d i s t i n c t i o n should be made i n f a i l u r e modes.  author's A  f a i l u r e i n bending i s much more p r e f e r a b l e to a f a i l u r e i n shear and  i t i s not proper to t r e a t them e q u a l l y i n choosing  governing earthquake c a p a c i t y  ratio.  Unless the earthquake c a p a c i t y r a t i o i s g r e a t e r equal t o one, b u i l d i n g and acceptable  there  i s a hazard which i s a f u n c t i o n o f  the occupancy p o t e n t i a l .  according  (b)  ATC  earthquake c a p a c i t y r a t i o s and  meet the requirements must be  than or the  s e t s the minimum those which f a i l  to  strengthened or demolished  to the schedule o u t l i n e d i n the  Okada and  Bresler  Okada and  B r e s l e r i n "Strength  E x i s t i n g Low-Rise R e i n f o r c e d  the  and  report.  D u c t i l i t y Evaluation  Concrete B u i l d i n g s -  of  Screening  3  Method"  describes  a procedure f o r e v a l u a t i n g the s e i s m i c  o f l o w - r i s e r e i n f o r c e d concrete  structures.  T h e i r method con-  s i s t s of s e r i e s of steps which are repeated i n s u c c e s s i v e w i t h more r e f i n e d modeling. Three s c r e e n i n g the  Each c y c l e r e p r e s e n t s  c y c l e s are proposed and  f i r s t execution  the f i r s t  a  I t a l s o shows how  cycles  "screening".  screening  of the b a s i c procedure, i s d e s c r i b e d  d e t a i l i n t h e i r paper.  safety  cycle,  in  t h i s procedure can  be  i s based on approximate e v a l u a t i o n  of  a p p l i e d to e x i s t i n g s c h o o l b u i l d i n g s . The  f i r s t screening  the l o a d - d e f l e c t i o n c h a r a c t e r i s t i c s of the story.  The  f i r s t or weakest  second i n v o l v e s a more p r e c i s e estimate of o v e r a l l  s t r u c t u r a l behaviour, and i n the t h i r d s c r e e n i n g n o n l i n e a r i t y of  each member i s modeled. In  d e s c r i b i n g the f i r s t s c r e e n i n g procedure, the authors  p o i n t out t h a t the c r i t e r i a which d e f i n e the p e r m i s s i b l e damage r e s u l t i n g from a s p e c i f i e d earthquake are the most important f a c t o r s which determine s t r u c t u r a l adequacy.  Two grades o f  earthquake motions and two c o r r e s p o n d i n g degrees o f b u i l d i n g damage are chosen.  Three types o f f a i l u r e modes, bending,  shear and shear bending are c o n s i d e r e d . The procedure c o n s i s t s o f f i v e major s t e p s , namely, Cl)  s t r u c t u r a l modeling  (2)  a n a l y t i c a l modeling  (3)  strength safety  (4)  d u c t i l i t y safety evaluation  (.5)  synthesis evaluation of safety.  evaluation  The s t r u c t u r a l modeling i s i n i t i a t e d by i d e n t i f y i n g the l o a d t r a n s m i s s i o n system o f the b u i l d i n g from examining design c a l c u l a t i o n s and f i e l d  investigations.  The main items t o  be determined are s t r u c t u r a l system, l o a d i n t e n s i t y , of  m a t e r i a l s , d e s i g n method, and o t h e r s p e c i a l  features.  drawings,  properties  structural  S e v e r a l models may have t o be c o n s i d e r e d .  The a n a l y t i c a l modeling i s done t o e v a l u a t e s t r u c t u r a l response under l a t e r a l f o r c e s . C  scl'  u  l  t  i  m  a  t  e  The shear c r a c k i n g  strength,  shear s t r e n g t h , C g , and bending s t r e n g t h , Cg ^, u l  i n terms o f base shear c o e f f i c i e n t s are computed.  The compar-  i s o n o f the three i d e n t i f i e s the type o f f a i l u r e .  The s t r e n g t h  i s e v a l u a t e d w i t h r e s p e c t t o shear c r a c k i n g , u l t i m a t e shear s t r e n g t h , and bending s t r e n g t h .  The c a p a c i t y w i t h r e s p e c t t o  8 each o f these three f a i l u r e modes and t h e i r r e l a t i v e v a l u e s are weighed h e a v i l y i n e v a l u a t i n g the s t r u c t u r e .  The  fundamental  p e r i o d and the modal p a r t i c i p a t i o n f a c t o r are computed i n an approximate manner. The s t r e n g t h s a f e t y e v a l u a t i o n determines the adequacy l a t e r a l strength.  of  For t h i s purpose a l i n e a r earthquake response  a n a l y s i s i s used w i t h a s t a n d a r d i z e d response spectrum.  In c a l -  c u l a t i n g the l i n e a r response i n terms o f base shear c o e f f i c i e n t , C, E  the b u i l d i n g assumed t o be a story-level-lumped-mass system  w i t h the number o f s t o r i e s equal t o the number of degrees o f freedom.  Only the f i r s t mode shape i s c o n s i d e r e d .  The d u c t i l i t y s a f e t y e v a l u a t i o n estimates the f i r s t  story  displacement u s i n g n o n l i n e a r displacement response s p e c t r a and m o d i f i e d modal p a r t i c i p a t i o n f a c t o r to i d e a l i z e the n o n l i n e a r behaviour of the b u i l d i n g . b u i l d i n g , which i s modeled  The response d u c t i l i t y of the as the e q u i v a l e n t one-mass system, i s  compared w i t h the s p e c i f i e d l i m i t  value.  The f i n a l step i s the s y n t h e s i s e v a l u a t i o n of s a f e t y .  The  assumptions and unknowns i n c o r p o r a t e d i n t o the s c r e e n i n g p r o c e s s and the need f o r m o d i f i c a t i o n of the e x i s t i n g b u i l d i n g are c a r e f u l l y analyzed.  Those b u i l d i n g s which f a i l e d to pass the  first  s c r e e n i n g are c l a s s i f i e d u n c e r t a i n and must go through the second and subsequent s c r e e n i n g procedure. The procedure s e t f o r t h by Okada and B r e s l e r r e p r e s e n t s a r a t i o n a l approach to the problem of e v a l u a t i n g e x i s t i n g b u i l d i n g s , and the p r e s e n t method of a n a l y s i s c o u l d be f i t t e d screening process.  into  their  9 (c)  Freeman, N i c o l e t t i , and T y r r e l l The  procedure d e s c r i b e d  f o r Seismic  i n "Evaluation of E x i s t i n g Buildings  Risk -- A Case Study o f Puget Sound Naval  Shipyard,  4  Bremerton, Washington," by Freeman e t a l .  i s intended  to f i l l  the gap between s t a t i s t i c a l procedures f o r l a r g e areas, and d e t a i l e d s t r u c t u r a l dynamic a n a l y s i s o f i n d i v i d u a l b u i l d i n g s . I t s main f e a t u r e i s a very r a p i d s c r e e n i n g a n a l y s i s w i t h minimum o f c a l c u l a t i o n . Sound Naval Shipyard  process and a simple  The s t r u c t u r e a t the Puget  a t Bremerton, Washington, was s t u d i e d and  the f i n d i n g s were r e p o r t e d .  A t o t a l of 9 6 buildings of d i f f e r e n t  s i z e , age, m a t e r i a l s , type o f c o n s t r u c t i o n and occupancy i s e v a l uated f o r the o v e r a l l v u l n e r a b i l i t y t o earthquake damage. The  study i s performed i n s i x phases, namely,  (1)  a v i s u a l survey o f 9 6 b u i l d i n g s  C2)  i n v e s t i g a t i o n o f two r e p r e s e n t a t i v e  (3)  determination  (4)  estimation  (5)  d e t a i l e d i n v e s t i g a t i o n of f i v e c r i t i c a l  (6)  estimation  buildings  o f the s e i s m i c i t y o f the area  o f probable damage f o r 80 b u i l d i n g s buildings  o f the average annual c o s t s o f expected  earthquake damage f o r 40 b u i l d i n g s . Phases  C D t o (3) need l i t t l e  explanation.  The f i n d i n g s i n the  second phase a r e used f o r the next phases o f study. phase response a c c e l e r a t i o n s p e c t r a a r e c o n s t r u c t e d seismic records the  In the t h i r d from the  i n the area and are used f o r the phase four o f  study. The  analyzing  f o u r t h phase i s the most r e l e v a n t t o t h i s r e p o r t .  In  the s t r u c t u r e s emphasis was p l a c e d on m i n i m i z a t i o n  of  the man-hours spent.  The l a t e r a l f o r c e s t r e n g t h  c a p a c i t i e s were  10 roughly looked  approximated and the n o n - s t r u c t u r a l m a t e r i a l s were a l s o at.  The base shear c a p a c i t i e s were used t o e s t a b l i s h the  y i e l d l i m i t and the u l t i m a t e the base shear represented  limit.  The former i s d e f i n e d as  by the f o r c e r e q u i r e d t o reach the  c a p a c i t y o f the most r i g i d l a t e r a l f o r c e - r e s i s t i n g system.  The  l a t t e r i s d e f i n e d as the base shear "required t o cause the most f l e x i b l e l a t e r a l f o r c e - r e s i s t i n g system t o y i e l d a f t e r the c o l l a p s e o r y i e l d of the more r i g i d ones.  These were converted  to s p e c t r a l a c c e l e r a t i o n c a p a c i t i e s by d i v i d i n g by the weight of structure.  The dynamic response c h a r a c t e r i s t i c s and the p e r i o d s  were estimated by approximate methods. Assumptions were made t o s i m p l i f y the e v a l u a t i o n o f damage. The damage l e v e l was assumed t o vary l i m i t t o 100% a t the u l t i m a t e nonlinear  limit.  l i n e a r l y from 0% at y i e l d In the i n e l a s t i c range  e f f e c t s were taken i n t o account by l i n e a r l y  the damping between the two l i m i t s .  varying  The procedure used f o r e s t i -  mating damage was based on r e c o n c i l i a t i o n o f the demand s p e c t r a l a c c e l e r a t i o n and the c a p a c i t y of the s t r u c t u r e i n r e l a t i o n to periods  and damping.  A graphical solution f o r estimating  centage damage was developed.  per-  The a n a l y s i s was done i n two  d i r e c t i o n s and a weighted average was  computed.  Sets o f response s p e c t r a were chosen t o r e p r e s e n t the earthquake motions w i t h d i f f e r e n t r e t u r n p e r i o d s .  From the  damage l e v e l s a s s o c i a t e d w i t h those r e t u r n p e r i o d s  the annual  costs were computed f o r the 80 b u i l d i n g s . The authors c l a i m t h a t the r e s u l t o f the procedure can be used to decide which b u i l d i n g s are most s u s c e p t i b l e to earthquake damage and t h a t the e f f e c t s o f m o d i f i c a t i o n on e x i s t i n g s t r u c t u r e s can be found.  11 1.3  Purpose and  The  Scope  three papers d i s c u s s e d  i n the previous  section  t r a t e the type of approach t h a t must be taken i n order  illus-  to a n a l y z e  a l a r g e number o f e x i s t i n g b u i l d i n g s which are p o t e n t i a l s e i s m i c hazards.  They a l l s e t up s c r e e n i n g  t i a l l y hazardous b u i l d i n g s and analysis.  procedures to s e l e c t poten-  then s u b j e c t them to  seismic  I t i s beyond the scope of t h i s t h e s i s to comment on  the s c r e e n i n g  procedure; the s t r u c t u r a l a n a l y s i s , however,  deserves a few  comments. 2  The seismic  ATC-III r e p o r t  suggests the use  of the q u a s i - s t a t i c  f o r c e s i n the c u r r e n t codes f o r the a n a l y s i s .  explained  i n the f i r s t  meaningless unless p r o p e r t i e s and mendations.  the  s e c t i o n of t h i s chapter,  are  s t r u c t u r e s were designed w i t h the d u c t i l e  the proper d e t a i l i n g i m p l i e d by the code recom-  Even i f a s t r u c t u r e can c a r r y o n l y a f r a c t i o n o f  because i n a c t u a l earthquakes the  tile  was  these f o r c e s  q u a s i - s t a t i c f o r c e s , c o l l a p s e or major damage may  and  As  not  the  occur,  f o r c e s w i l l be r e d i s t r i b u t e d  the b u i l d i n g w i l l respond d i f f e r e n t l y depending on i t s ducproperties. 3 B r e s l e r ' s methods  modelling  takes n o n l i n e a r i t y i n t o account  the s t r u c t u r e as a one-mass system and  of n o n l i n e a r  response s p e c t r a .  r i s e s t r u c t u r e s and,  The  through the  a n a l y s i s i s intended  f o r t h i s purpose, the assumptions  s i m p l i f i c a t i o n s t h a t the authors made are s a t i s f a c t o r y . extension  for  use low-  and An  of the method to the a n a l y s i s of medium- to h i g h - r i s e  b u i l d i n g s w i l l , however, i n v o l v e major m o d i f i c a t i o n s method.  by  to  their  12 Freeman's method to  4  i s a t b e s t approximate.  Their  approach  i n c l u s i o n o f n o n l i n e a r i t y i n t h e i r a n a l y s i s has many assump-  t i o n s and s i m p l i f i c a t i o n s .  The method i s probably e f f e c t i v e f o r  s c r e e n i n g many one- t o two-story, s i n g l e - b a y b u i l d i n g s , b u t the e x t e n s i o n of t h i s method t o l a r g e r b u i l d i n g s i s o f q u e s t i o n a b l e value. It  i s c l e a r t h a t a procedure f o r a n a l y s i s o f e x i s t i n g  b u i l d i n g s a g a i n s t s e i s m i c hazards must be developed, e s p e c i a l l y for  those b u i l d i n g s which are judged u n c e r t a i n a f t e r the i n i t i a l  screening process.  The procedure must be capable o f h a n d l i n g  medium- t o h i g h - r i s e s t r u c t u r e s without major assumptions and simplifications.  I t i s desirable that d i f f e r e n t  earthquake  motions can be used t o o b t a i n a good e s t i m a t e o f behaviour o f the  s t r u c t u r e and t h a t .the a n a l y s i s should i n c l u d e the e f f e c t s o f  n o n l i n e a r i t y a f t e r the y i e l d o f some o f the members.  A t the  same time the procedure must be simple and r e a s o n a b l y economical to use. Such a procedure i s developed and d e s c r i b e d i n the subsequent c h a p t e r s .  The m o d i f i e d s u b s t i t u t e s t r u c t u r e method i s  intended t o f i l l  the gap between s i m p l i f i e d  structural  and the f u l l - s c a l e , n o n l i n e a r time-step a n a l y s i s .  analysis  The proposed  method i s s u i t a b l e f o r r e i n f o r c e d concrete frame s t r u c t u r e s , but it  i s hoped t h a t i t can be used f o r s h e a r - w a l l type b u i l d i n g s  and s t e e l s t r u c t u r e s .  The procedure i s a m o d i f i e d e l a s t i c modal  a n a l y s i s , which i s developed from a design concept proposed by Shibata and Sozen.^ 5 . The design procedure proposed by S h i b a t a and Sozen described f i r s t  is  i n o r d e r t o d i s c u s s the theory and assumptions  which are e s s e n t i a l i n understanding  the proposed  b r i e f d i s c u s s i o n modal a n a l y s i s i s i n c l u d e d . d e s i g n procedure  are a l s o presented.  method.  A  Examples of the  An a l t e r n a t e approach i s  d e s c r i b e d and the f i n d i n g s are d i s c u s s e d . Then the m o d i f i e d s u b s t i t u t e s t r u c t u r e method i s presented i n the next chapter.  The  theory behind t h i s procedure  is dis-  cussed as w e l l as the development of the computer program. it  i s an i t e r a t i v e procedure,  Since  convergence c r i t e r i a are d i s c u s s e d .  A method to achieve f a s t e r convergence i s i n t r o d u c e d . In o r d e r to t e s t the v a l i d i t y of the m o d i f i e d  substitute  s t r u c t u r e method, frames of d i f f e r e n t type and h e i g h t are analyzed.  A comparison of r e s u l t s w i t h those of n o n l i n e a r dynamic  a n a l y s i s i s presented. described i n t h i s  A l l the assumptions are presented  and  section.  In the f i n a l chapter f a c t o r s which a f f e c t the r e s u l t s of the a n a l y s i s are d i s c u s s e d , and a p r e l i m i n a r y g u i d e l i n e i s presented f o r s u c c e s s f u l a p p l i c a t i o n s of the method. where f u r t h e r r e s e a r c h i s necessary are mentioned.  The  areas  14  CHAPTER 2  2.1  SUBSTITUTE STRUCTURE METHOD  Modal A n a l y s i s  Modal a n a l y s i s i s an approximate  dynamic a n a l y s i s t o s o l v e  the response o f a multi-degrees-of-freedom earthquake  motion.  t i c systems,  system t o a given  Although i t i s intended f o r a n a l y s i s o f e l a s -  a thorough knowledge o f t h i s method i s e s s e n t i a l f o r  the d i s c u s s i o n o f the subsequent  sections.  Since i t i s not the  i n t e n t i o n o f t h i s paper t o e x p l a i n the dynamics o f s t r u c t u r e s s u b j e c t e d t o the earthquake brief.  (a)  motion,  The s u b j e c t i s covered i n Clough and Penzien.  Equation o f Motion The b a s i c equation o f motion  system  the d i s c u s s i o n i s kept very  f o r a multi-degrees-of-freedom  i s given by  [m] (ii) + [c] (u) + [k] (u) = -x[m] (I)  where  [m] = mass matrix [c] = damping m a t r i x [k] = s t i f f n e s s m a t r i x (ii) , (u) , (u) = a c c e l e r a t i o n , v e l o c i t y , and displacement  corresponding  to each degree o f freedom.  (2.1)  15 x  = ground a c c e l e r a t i o n  (I) = i d e n t i t y v e c t o r where every e n t r y i s a unity The mass o f the system i s u s u a l l y lumped a t the modes f o r s i m p l i c i t y i n computation.  I f such an assumption i s made, the  mass m a t r i x becomes d i a g o n a l . , D i s c u s s i o n o f the damping m a t r i x i s beyond the scope o f t h i s paper.  Modal a n a l y s i s does not r e q u i r e an e v a l u a t i o n o f  t h i s m a t r i x , although the damping value i n each mode i s r e q u i r e d f o r s y n t h e s i s o f the r e s u l t s . The s t i f f n e s s m a t r i x i s formed by assembling the member s t i f f n e s s matrices. analysis.  The procedure i s i d e n t i c a l t o t h a t o f frame  The f u l l member m a t r i x w i t h three degrees o f freedom  at each member end i s 6 x 6.  I f only bending deformation i s o f  i n t e r e s t , i t s s i z e i s reduced t o 4 x 4.  (b)  P e r i o d s and Mode Shapes S o l u t i o n o f the f r e e , undamped system y i e l d s mode shapes  and n a t u r a l f r e q u e n c i e s .  The e q u a t i o n o f motion becomes,  [m] (ii) + [k] Cu) = CO)  (2.2)  The s o l u t i o n t o t h i s equation i s o f the form,  (2.3)  (u) = (A) s i n a)t with  Cu)  -  -ca  2  (A) s i n  031  (2.4)  16 S u b s t i t u t e equations  (2.3) and (2.4) i n t o  (2.2),  (2.5)  -a) [m] (A) + [k] (A) = (0) 2  For a n o n t r i v i a l  [k] -  solution,  OJ  2  (2.6)  [m] | = 0  T h i s i s an eigenvalue problem  o f the form,  (2.7)  [B] = X[C]  i n which  [B] i s a symmetric,  banded m a t r i c and fC] i s a d i a g o n a l  matrix.  Eigenvalues a s s o c i a t e d w i t h equation  (2.6) correspond  to the squares o f the angular f r e q u e n c i e s , w . 2  e i g e n v e c t o r s correspond t o the mode shapes. the mass m a t r i x ,  Associated  I f n i s the rank o f  [m], there are n n a t u r a l f r e q u e n c i e s and n mode  shapes.  (c)  Response S p e c t r a Given an earthquake  compute the response  r e c o r d , i t i s r e l a t i v e l y simple t o  spectra.  The peak a c c e l e r a t i o n ,  velocity,  or displacement o f a s i n g l e - d e g r e e - o f - f r e e d o m system w i t h a given value can be determined  from the response  modal a n a l y s i s o f multi-degree-of-freedom assumption response  spectra.  systems,  In the  w i t h the  t h a t a damping r a t i o f o r each mode i s known, a peak  f o r each mode can be read from the response s p e c t r a  when n a t u r a l p e r i o d s a r e known.  When a damping r a t i o i s s m a l l ,  17 with l i t t l e  e r r o r the peak a c c e l e r a t i o n , v e l o c i t y , and d i s -  placement a r e r e l a t e d i n the f o l l o w i n g manner,  Where  S  &  7  = peak a c c e l e r a t i o n c o r r e s p o n d i n g to the n a t u r a l frequency, oi.  S  v  = peak v e l o c i t y  S^ = peak displacement. The c h o i c e o f a damping r a t i o l e a v e s some room f o r a debate.  I t i s generally  taken t o be 5 t o 10% o f c r i t i c a l f o r  c o n c r e t e and 2 t o 5% o f c r i t i c a l  for steel.  S t r i c t l y speaking, the response earthquake  spectra  a r e v a l i d f o r one  o f known peak ground a c c e l e r a t i o n , b u t they can be  s c a l e d up o r down depending on the peak ground a c c e l e r a t i o n which i s a p p r o p r i a t e f o r a p a r t i c u l a r s i t e w i t h c e r t a i n assumpt i o n s on magnitude and p r o b a b i l i t y o f o c c u r r e n c e .  Cd)  Modal Forces Suppose t h a t the a c c e l e r a t i o n  spectrum  i s given and t h a t  the damping r a t i o s f o r a l l the modes are know o r estimated; then, it  i s a r e l a t i v e l y simple t o s e t up a f o r c e v e c t o r corresponding  to each mode. computed. vector.  Modal p a r t i c i p a t i o n f a c t o r s , a, must f i r s t be  L e t r denote the r t h mode and T the transpose o f a The modal p a r t i c i p a t i o n f a c t o r f o r the r t h mode can  be computed as f o l l o w s ,  18 CA ) [m] (T) r  a  =  (A ) [m](A )  r  where  (2.10)  T  r  T  r  (A ) = a v e c t o r r e p r e s e n t i n g the mode shape f o r the r t h mode [m]  = mass m a t r i x  (I)  = i d e n t i t y v e c t o r whose elements  are a l l u n i t y .  Then the f o r c e v e c t o r f o r the r t h mode becomes  ( F ) = (A )a S^[m] r  (2.11)  r  r  where  (F ) = f o r c e v e c t o r S  = peak a c c e l e r a t i o n c o r r e s p o n d i n g t o r t h mode  &  n a t u r a l frequency and damping. The modal displacements and response  f o r c e s can be computed  i n the i d e n t i c a l manner t o t h a t used i n the s t i f f n e s s method i n a plane frame a n a l y s i s .  That i s ,  ( F ) = [k] (A ) r  where  (2.12)  r  [k] = s t r u c t u r e s t i f f n e s s matrix (A ) = modal displacements i n g l o b a l c o o r d i n a t e s , r  With  r  (F ) known, ( A ) can be computed by simply i n v e r t i n g the  s t i f f n e s s matrix,  [k].  The member f o r c e s can be c a l c u l a t e d  from the displacement v e c t o r , (A ) . (e)  Combination  o f Forces and Displacements  These f o r c e s and displacements f o r each mode correspond t o the peak response.  I t i s n o t l i k e l y t h a t these i n d i v i d u a l maxima  19 occur a t the same time; t h e r e f o r e , summing up the a b s o l u t e v a l u e s o f these f o r c e s and displacements may r e s u l t i n overe s t i m a t i n g the response. (RSS)  approach  I t i s found t h a t the  root-sum-square  g i v e s a more reasonable estimate.  The i n d i v i d u a l  modal responses a r e combined by t a k i n g the square r o o t o f the sum o f the squares o f the responses. C o n t r i b u t i o n s from the h i g h e r modes d i m i n i s h very For  rapidly.  t h i s reason i t i s u s u a l l y s u f f i c i e n t t o take the f i r s t three  or f o u r modes f o r computation.  F o r l o w - r i s e s t r u c t u r e s o n l y the  f i r s t mode i s s u f f i c i e n t f o r a l l the p r a c t i c a l purposes.  For  h i g h - r i s e s t r u c t u r e s h i g h e r modes p l a y more dominant r o l e s , and, hence, cannot be n e g l e c t e d .  20 2.2  S u b s t i t u t e S t r u c t u r e Method  (a)  Development Gulkan and Sozen performed a s e r i e s o f experiments t o t e s t  the response of r e i n f o r c e d concrete s t r u c t u r e s to s e i s m i c g motions.  The t e s t s were r e s t r i c t e d to the s i n g l e - d e g r e e - o f -  freedom system.  They found t h a t the b a s i c c h a r a c t e r i s t i c s o f  r e i n f o r c e d concrete s t r u c t u r e which determine the response to earthquakes are a change i n s t i f f n e s s and a change i n energy d i s s i p a t i o n c a p a c i t y , both of which are r e l a t e d t o the maximum displacement.  During s t r o n g motions the s t i f f n e s s of r e i n f o r c e d  c o n c r e t e decreases because o f c r a c k i n g o f c o n c r e t e , s p a l l i n g o f concrete, and s l i p p i n g and r e d u c t i o n i n e f f e c t i v e modulus o f steel.  The r e s u l t o f t h i s i s t h a t the p e r i o d of the s t r u c t u r e  i n c r e a s e s as i t undergoes  i n e l a s t i c deformation.  The a r e a  w i t h i n a c y c l e o f the f o r c e - d i s p l a c e m e n t curve i s a measure o f the energy d i s s i p a t e d by the system.  They found t h a t the a r e a  w i t h i n the h y s t e r e t i c loop i n c r e a s e s w i t h i n c r e a s e i n d i s p l a c e ment i n t o the i n e l a s t i c range o f response. The e f f e c t of the h y s t e r e s i s loop and the change i n s t i f f ness i s s a i d t o l e a d to a q u a n t i t a t i v e , r e l a t i o n s h i p between l i n e a r response a n a l y s i s and i n e l a s t i c a n a l y s i s .  A concept o f  s u b s t i t u t e damping and e f f e c t i v e s t i f f n e s s are then i n t r o d u c e d i n order t o i n t e r p r e t the i n e l a s t i c response i n terms o f a l i n e a r response a n a l y s i s , u s i n g a s p e c t r a l response curve. Consider an i d e a l i z e d symmetrical h y s t e r e s i s loop as shown i n F i g . 2.1. I t f o l l o w s Takeda s h y s t e r e s i s loop which was 9 used as an a n a l y t i c a l model i n the experiment by Takeda e t a l . 1  I t i s assumed t h a t the s t r u c t u r e has a l r e a d y undergone s e v e r a l Let y be the o r i g i n a l  c y c l e s of i n e l a s t i c deformation.  then the slope of the unloading curve BC, d u c t i l i t y and a i s a c o n s t a n t .  i s Y[—)  stiffness;  where n i s the  The shape of the h y s t e r e s i s  curve i s such t h a t i t i s approximately r e p r e s e n t e d by a l i n e a r l y v i b r a t i n g system w i t h e q u i v a l e n t v i s c o u s damping."^ t h a t the energy  i n p u t i s e n t i r e l y d i s s i p a t e d by an  I t i s assumed imaginary  v i s c o u s damper a s s o c i a t e d w i t h the h o r i z o n t a l v e l o c i t y of the mass.  Using t h i s i d e a , the s u b s t i t u t e damping r a t i o ,  B  g  i s given  by,  B  where  2mto / ^ ( u ) d t o u 2  s  = -m/^  u  x u dt  (2.13)  m = mass u = velocity x = ground a c c e l e r a t i o n T = p e r i o d of v i b r a t i o n to = measured a b s o l u t e a c c e l e r a t i o n / m e a s u r e d o 2  absolute  displacement.  The l e f t - h a n d s i d e of the equation r e p r e s e n t s the energy pated per c y c l e and the r i g h t - h a n d s i d e r e p r e s e n t s the input per c y c l e .  On the h y s t e r e s i s loop diagram  dissi-  energy  i t can be  seen  that ^ area EBC area ABF  a i s taken as 0.5,  _ 1/2 1/2  ( h y s t e r e s i s loop area) (energy input)  then i t can be shown t h a t  (2.14)  22 (1  n  where  -  (2.15)  l/v^T)  = ductility.  From the .experimental data Gulkan and Sozen  8  gave the f o l l o w i n g B,  e x p r e s s i o n f o r the s u b s t i t u t e damping r a t i o ,  g  (2.16)  I t i s assumed i n equation of 0.02  at  =1.0.  The  (2.16) t h a t B  slope of the l i n e AE i s the  s t i f f n e s s and i s equal to y / n . to the e f f e c t i v e s t i f f n e s s i s Gulkan and Sozen  has a t h r e s h h o l d value  g  effective  The n a t u r a l p e r i o d corresponding T/n  proposed  a design procedure  for a rein-  f o r c e d concrete s t r u c t u r e which can be i d e a l i z e d as a s i n g l e degree-of-freedom  system.  The d e s i g n base shear can be  calcula-  t e d as f o l l o w s : (1)  assume an a d m i s s i b l e value of d u c t i l i t y , ri ,  (2)  c a l c u l a t e the s t i f f n e s s based on the cracked  (3)  determine  (4)  c a l c u l a t e the s u b s t i t u t e damping r a t i o ,  section,  the n a t u r a l p e r i o d , T, B, g  corresponding  to the assumed value of d u c t i l i t y , n , (5)  o b t a i n base shear and maximum displacement by e n t e r i n g a s p e c t r a l response  diagram w i t h an i n c r e a s e d n a t u r a l p e r i o d  of T >^T and a damping r a t i o equal t o Even though t h i s design procedure degree-of-freedom  B. g  i s intended f o r a s i n g l e -  system, the b a s i c concepts  are d i r e c t l y  t r a n s f e r r e d t o the s u b s t i t u t e s t r u c t u r e method, which i s a design method f o r m u l t i - s t o r y r e i n f o r c e d concrete  frames.  23 (b)  S u b s t i t u t e S t r u c t u r e Method The s u b s t i t u t e s t r u c t u r e method was c o n c e i v e d by Shbata and  Sozen.  5  I t i s an e x t e n s i o n o f the method by Gulkan and Sozen  which was d e s c r i b e d i n the p r e v i o u s s e c t i o n .  8  The method i s  intended f o r m u l t i - s t o r y r e i n f o r c e d c o n c r e t e frames and i s a design procedure t o e s t a b l i s h the minimum s t r e n g t h s t h a t the components must have so t h a t a t o l e r a b l e response displacement i s not l i k e l y to be exceeded.  The main c h a r a c t e r i s t i c s o f the sub-  s t i t u t e s t r u c t u r e method are the d e f i n i t i o n o f a s u b s t i t u t e frame, which i s a f i c t i t i o u s  frame w i t h i t s s t i f f n e s s and damping  d i f f e r i n g from the a c t u a l frame, and the c a l c u l a t i o n of the design f o r c e s from modal a n a l y s i s o f the s u b s t i t u t e frame u s i n g a l i n e a r response spectrum.  These c h a r a c t e r i s t i c s a r e chosen  such that the f o r c e s and the deformations from the a n a l y s i s agree w i t h these from the n o n l i n e a r dynamic 5 S h i b a t a and Sozen  analysis.  l i s t the f o l l o w i n g c o n d i t i o n s which must  be s a t i s f i e d i n order t o use the s u b s t i t u t e s t r u c t u r e method. (1)  The system can be analyzed i n one v e r t i c a l p l a n e .  (2)  There are no abrupt changes i n geometry o r mass along the h e i g h t o f the system.  (3)  Columns, beams and w a l l s may be designed w i t h d i f f e r e n t l i m i t s o f i n e l a s t i c response, but the l i m i t should be the same f o r a l l beams i n a given bay and a l l columns on a given a x i s .  (4)  A l l s t r u c t u r a l elements and j o i n t s a r e r e i n f o r c e d t o a v o i d s i g n i f i c a n t s t r e n g t h decay as a r e s u l t o f repeated r e v e r s a l s o f the a n t i c i p a t e d i n e l a s t i c d i s p l a c e m e n t s .  (5)  N o n s t r u c t u r a l components do not i n t e r f e r e w i t h s t r u c t u r a l response.  The  first  c o n d i t i o n i m p l i e s t h a t the method i s s u b j e c t  the l i m i t a t i o n s of plane frame a n a l y s i s . and b i a x i a l bending must be n e g l e c t e d .  Such e f f e c t s as t o r s i o n The  second c o n d i t i o n  r e s t r i c t s the use of t h i s method to s t r u c t u r e s of r e g u l a r w i t h uniform d i s t r i b u t i o n of mass and d i t i o n deserves the most a t t e n t i o n . columns may big  stiffness. The  The  third-con-  f a c t t h a t the beams and  method i n which the  t i l i t y of the e n t i r e s t r u c t u r e must be chosen to be T h i s p o i n t i s perhaps the b i g g e s t t u t e s t r u c t u r e method. beams to y i e l d and remain e l a s t i c .  s t o r y frames.  The  advantage i n u s i n g the s u b s t i -  absorb the b u l k of energy w h i l e the  The  duc-  uniform.  I t i s u s u a l l y d e s i r a b l e to allow  the  columns  t h i r d c o n d i t i o n does, however, exclude  p o s s i b i l i t y t h a t t h i s method may conditions  Before the design  be used f o r the design  (4) and  (5) need l i t t l e  of s o f t explanation.  As mentioned p r e v i o u s l y ,  s u b s t i t u t e frame i s a f i c t i t i o u s frame w i t h i t s s t i f f n e s s damping r e l a t e d but not  the  procedure i s presented, terms p a r t i c u l a r  to t h i s method must be e x p l a i n e d .  i d e n t i c a l t o the a c t u a l frame.  i s used i n s t e a d of d u c t i l i t y ,  n.  a  and  A damage  Consider a f o r c e -  displacement curve or a moment-rotation curve as i n F i g . Ductility  shapes  have d i f f e r e n t i n e l a s t i c deformation l i m i t s i s a  step forward from the c o n v e n t i o n a l  ratio, u,  to  i s u s u a l l y d e f i n e d as the r a t i o of u l t i m a t e  2.2.  displace-  ment to y i e l d displacement, or  n =  The  ^  damage r a t i o on the other  (2.17)  hand i s the r a t i o of the  s t i f f n e s s of the s u b s t i t u t e frame, or  initial  25 slope AB slope AC  = H  They are i d e n t i c a l f o r the e l a s t o - p l a s t i c  case, but i f the s t i f f -  ness a f t e r y i e l d has a p o s i t i v e s l o p e , the damage r a t i o i s always s m a l l e r than d u c t i l i t y .  Suppose s i s the r a t i o o f the s t i f f n e s s  a f t e r y i e l d to the i n i t i a l  s t i f f n e s s ; t h a t i s , the r a t i o of the  slope o f BC t o the slope o f AB i n F i g . 2.2.  Then the r e l a t i o n  between the damage r a t i o and d u c t i l i t y i s  (2.19)  1 + (n - l ) s  where  u = damage r a t i o ri = d u c t i l i t y s = r a t i o of s t i f f n e s s after y i e l d to i n i t i a l  stiffness  A s u b s t i t u t e damping r a t i o i s d e f i n e d and computed i n an i d e n t i cal  manner t o t h a t d e s c r i b e d i n the p r e v i o u s  section.  The damage  r a t i o , however, i s used i n s t e a d o f d u c t i l i t y ; hence,  3  s  = 0.2(1 - l//y") + 0.02  B = s u b s t i t u t e damping s  where  y The  (2.20)  ratio  = damage r a t i o .  design procedure w i l l now be d e s c r i b e d  .  A  necessary  assumption i s t h a t the p r e l i m i n a r y member s i z e s o f the a c t u a l s t r u c t u r e are known from g r a v i t y loads and other f u n c t i o n a l requirements. (1)  Then the f o l l o w i n g steps are i n v o l v e d .  Assume an a c c e p t a b l e group o f members.  value o f damage r a t i o , u, f o r each  (2)  Define the f l e x u r a l s t i f f n e s s of s u b s t i t u t e - f r a m e  elements  as  CEI)  where  s i  =  a u  (2.21)  i  i  (EI) . = f l e x u r a l si  s t i f f n e s s of i th  substitute-frame  element (EI)  . = flexural ai  s t i f f n e s s o f i t h element i n the  a c t u a l frame u. *i (3)  = t o l e r a b l e damage r a t i o f o r i t h element.  Compute n a t u r a l p e r i o d s , the undamped s u b s t i t u t e  (4)  mode shapes and modal f o r c e s f o r structure.  Compute an average o r a "smeared" damping r a t i o f o r each mode.  B  s i  = 0.2(1 - l / / y ) i  + 0.02  ^m  =\T^ ^si i i  where  P. l  =  and  6 . = s u b s t i t u t e dampinq r a t i o o f i t h member si  L.  <- > 2  (M . + M . + M . R . ) ai D I a i r>i 2  6 (EI)  . si  2  £>  = smeared s u b s t i t u t e damping f o r m t h mode  P^  = flexural  m  (2.22)  23  (2.24)  s t r a i n energy i n i t h element i n the  m t h mode = length o f frame element i (EI)  . = assumed si  s t i f f n e s s o f s u b s t i t u t e frame element  M ., M, . = end moments of s u b s t i t u t e frame element i ai bi f o r m t h mode. (5)  Repeat the modal a n a l y s i s u s i n g the smeared damping and compute the root-sum-square  (6)  Compute the design  F  where  i  = F  .v  ratios  (RSS) f o r c e s .  forces,  rss i rss 2v  +, v ,  abs  rss  = design f o r c e f o r i t h element F. = root-sum-square f o r c e s f o r i t h element 1 rss ^ v  rss  v , abs  = RSS base shear = maximum value f o r a b s o l u t e sum o f any two J  of the modal base shears. (7)  To avoid the r i s k o f e x c e s s i v e i n e l a s t i c a c t i o n i n the columns i n c r e a s e the design moments o f the columns by 20%. In the f i r s t step a designer can choose how much i n e l a s t i c  deformation can be allowed i n each element group.  Since the  t a r g e t damage r a t i o s are always g r e a t e r than o r equal t o one, i t i s c l e a r i n the second step t h a t the n a t u r a l p e r i o d s o f the subs t i t u t e frame a r e always g r e a t e r than these o f the a c t u a l frame. Steps 3 and 4 a r e necessary, because s u b s t i t u t e damping may be d i f f e r e n t f o r each element group. i s computed  ratios  A smeared damping  ratio  f o r each mode by assuming t h a t each element c o n t r i -  butes t o modal damping  i n proportion to i t s r e l a t i v e  s t r a i n energy a s s o c i a t e d w i t h the mode shape.  flexural  Elements w i t h  complex s t i f f n e s s can be used t o compute the smeared  damping  r a t i o s , but the f l e x u r a l energy approach i s e a s i e r t o use and has more p h y s i c a l meaning.  The s i x t h step i s an e x t r a f a c t o r o f  28 s a f e t y i n case any results.  The  combination of two  modes produces  undesirable  l a s t step i s d e s i r a b l e i n a design procedure,  because f a i l u r e i n a column p r i o r to f a i l u r e i n a d j o i n i n g beams may  l e a d to c a t a s t r o p h i c  f a i l u r e of a structure.  A linear  response spectrum i s used i n the a n a l y s i s ; the authors suggest t h a t a smoothed spectrum be used.  I t i s mentioned as a  critical  f e a t u r e of t h i s method t h a t i t becomes p l a u s i b l e only w i t h understanding t h a t the  f o r c e response decreases as the  becomes more f l e x i b l e ; t h e r e f o r e , r e l a t i o n to the n a t u r a l p e r i o d s  the  structure  the smoothed spectrum, i n  o f the  substitute  structure,  should have a shape such t h a t the s p e c t r a l a c c e l e r a t i o n response decreases w i t h an i n c r e a s e  i n period.  I m p l i c i t assumptions and s t r u c t u r e method are now  l i m i t a t i o n of the  discussed.  substitute  It is implicitly  t h a t the moment d i s t r i b u t i o n i n a l l the members are t h a t the p o i n t s of i n f l e c t i o n are p l a c e d p o i n t s of the member spans.  assumed  l i n e a r and  a t or near the mid-  With these assumptions, i t becomes  c l e a r t h a t the shape of f o r c e - d i s p l a c e m e n t curve i s i d e n t i c a l t o t h a t o f the moment-rotation curve.  Otherwise d i v i d i n g the  a c t u a l f l e x u r a l s t i f f n e s s by the damage r a t i o g r e a t e r  than  may  reasonable  not be a c o r r e c t approach.  These assumptions are  i n beams which are more l i k e l y to r e c e i v e but  they may  not be  inelastic  one  deformations  so v a l i d i n columns as shown by Blume et a l .  T h i s p o i n t , however, i s not an important f a c t o r as long  as  columns are designed w i t h a t a r g e t damage r a t i o of one,  which  i s d e s i r a b l e i n most p r a c t i c a l a p p l i c a t i o n s . In p r a c t i c e , unless  the design moments are known, the  s t i f f n e s s of a f u l l y cracked s e c t i o n t h a t must be  used to  c a l c u l a t e the s t i f f n e s s o f the s u b s t i t u t e frame cannot be d e t e r mined.  An educated guess i s r e q u i r e d and at the end o f the  c a l c u l a t i o n s , i t must be checked t h a t the guess was indeed reasonable.  The design moments correspond to e x t r a moment  c a p a c i t i e s r e q u i r e d over the c a p a c i t i e s f o r the g r a v i t y Two ends o f a member must be capable o f h a n d l i n g moment both i n p o s i t i v e and negative ment again  directions.  loads.  the same design This  require-  i s reasonable f o r beams, b u t may not be so f o r  columns. The  authors designed the t e s t frames u s i n g the s u b s t i t u t e  s t r u c t u r e method.  These t e s t frames were s u b j e c t e d  to nonlinear  time-step a n a l y s i s , and they s t a t e t h a t the frames behaved w e l l and  t h a t i n e l a s t i c deformation occurred  a t the p r e s c r i b e d  loca-  tions .  (c)  Computer Program Use  o f a computer i s almost as e s s e n t i a l i n the s u b s t i t u t e  s t r u c t u r e method, as i t i s i n the case o f r e g u l a r modal a n a l y s i s . A flow diagram i s shown i n F i g . 2.3. are r e q u i r e d t o convert  Only minor  modifications  an e x i s t i n g modal a n a l y s i s program t o  be used f o r the s u b s t i t u t e s t r u c t u r e method. A t a r g e t damage r a t i o f o r each member must be read s t o r e d when s t r u c t u r a l data are read  in.  i n and  A t t h i s stage i t may  be advantageous t o compute and s t o r e a s u b s t i t u t e damping r a t i o f o r each member.  When the s t r u c t u r e s t i f f n e s s m a t r i x i s  assembled from member s t i f f n e s s m a t r i c e s ,  f l e x u r a l components o f  the member s t i f f n e s s m a t r i x must be d i v i d e d by the a p p r o p r i a t e t a r g e t damage r a t i o .  The s t r u c t u r e s t i f f n e s s m a t r i x becomes  30 t h a t o f the s u b s t i t u t e frame, and t h i s m a t r i x i s used t o compute n a t u r a l p e r i o d s and a s s o c i a t e d mode shapes. C a l c u l a t i o n s o f modal responses a r e performed twice: the  first  on  c y c l e modal f o r c e s are computed f o r the undamped sub-  s t i t u t e s t r u c t u r e ; f l e x u r a l s t r a i n energy f o r each member i s computed and s t o r e d f o r each mode.  A smeared  damping r a t i o f o r  each mode i s computed a c c o r d i n g t o equation (2.23). smeared  With the  damping known the computation o f modal f o r c e s and d i s -  placements a r e repeated.  Root-sum-square f o r c e s and d i s p l a c e -  ments are computed on the second c y c l e , but s t r a i n energy c a l c u l a t i o n s a r e not r e q u i r e d .  From the modal base shears RSS  base shear and the maximum v a l u e o f the a b s o l u t e sum o f any two of  t h e modal base shears must be computed.  To compute t h e  design f o r c e s the RSS f o r c e s are m u l t i p l i e d by the f a c t o r i n equation  (2.25).  Furthermore, the column moments must be  i n c r e a s e d by 20%. If  a l i n e a r response spectrum i s chosen as was suggested 5  by S h i b a t a and Sozen,  only one i n v e r s i o n o f the s t r u c t u r e  ness m a t r i x i s necessary.  The program  stiff-  i s a very e f f i c i e n t one  that r e q u i r e s s m a l l storage and l i t t l e CPU time.  I f a regular  plane frame a n a l y s i s program i s t o be converted, s u b r o u t i n e s for  setup o f mass m a t r i x , response spectrum, and computation o f  n a t u r a l p e r i o d s , mode shapes, and modal p a r t i c i p a t i o n must be added.  factors  2.3  Examples  and Observations  (a)  Frames w i t h F l e x i b l e Beams In o r d e r to t e s t the computer  program  f o r the  substitute 5  s t r u c t u r e method, sample  frames from S h i b a t a and Sozen's  were chosen and the r e s u l t s were compared w i t h t h e i r s . frames are 3-, 5-, and 1 0 - s t o r i e s  paper The  h i g h and they c o n s i s t of  stiff  columns and f l e x i b l e beams. The data f o r the three frames are shown i n F i g . 2.4. width i n each case was  The  24 f e e t and the s t o r y h e i g h t was u n i f o r m  at 11 f e e t w i t h a weight o f 72 k i p s c o n c e n t r a t e d a t each s t o r y . The t a r g e t damage r a t i o s were one f o r columns and s i x f o r beams i n a l l three frames.  Since the moments o f i n e r t i a o f the c r a c k e d  s e c t i o n s were not known, the assumptions made by S h i b a t a and 5 Sozen were repeated; t h a t i s , 1/3 o f moment o f i n e r t i a o f the gross s e c t i o n was used f o r beams and 1/2 f o r the columns. The 5 desxgn spectrum A i n t h e i r paper acceleration  spectrum d e r i v e d  was used  ( F i g . 2.5).  I t i s an  from l i n e a r response s p e c t r a  of s i x  earthquake motions; namely, two components o f E l Centro 1940,  two  components o f T a f t 19 52, and two components o f Managua 19 72.  The  peak ground a c c e l e r a t i o n was n o r m a l i z e d at 0.5 t h a t the d e s i g n response a c c e l e r a t i o n  I t was  assumed  f o r any damping f a c t o r ,  c o u l d be r e l a t e d to the response f o r B = 0.02 Response a c c e l e r a t i o n , f o r B Response a c c e l e r a t i o n f o r B = 0.02  The n a t u r a l p e r i o d s and smeared  g.  _  8,  by using,  8 6 + 100  B  ,„ U.^b;  damping f a c t o r s of the  t h r e e frames are l i s t e d i n Table 2.1 Sozen's r e s u l t s .  5  along w i t h S h i b a t a and  The design moments are shown on F i g . 2.4.  The  design moments f o r the 3-story frame agreed w i t h those given by 5 Shibata and Sozen.  The design moments f o r 5- and  frames were not shown i n the paper.  One  may  10-story  conclude t h a t the  program was  capable of r e p r o d u c i n g the r e s u l t s shown i n S h i b a t a 5 and Sozen's paper. The three frames were then t e s t e d i n a s i m i l a r f a s h i o n t o 5 t h a t employed by S h i b a t a and Sozen.  An i n e l a s t i c dynamic program,  12 SAKE,  was  earthquake  used t o compute the response motions.  T h i s program was  s e l e c t e d , because i t was  w r i t t e n e x c l u s i v e l y f o r concrete frames. 13 r e p o r t e d by Otani and Sozen. quake was  I t s e f f e c t i v e n e s s was  A r e c o r d of Managua 19 72 e a r t h -  not a v a i l a b l e ; t h e r e f o r e , two  19 40 and two  h i s t o r y of each frame to  components of E l Centro  components of T a f t 19 52 were used.  These a c c e l e r a -  t i o n records were normalized so t h a t the peak ground a c c e l e r a t i o n was  0.5  g i n a l l four records.  design moments. initial  The y i e l d moments were s e t a t the  S t i f f n e s s beyond y i e l d was  stiffness.  The damping was  taken as 3% o f the  taken to be p r o p o r t i o n a l t o  s t i f f n e s s , corresponding to 2% damping f o r the f i r s t mode. puted damage r a t i o s of t h r e e frames are shown i n Table  Com-  2.2.  Comparison of some of the r e s u l t s w i t h those by S h i b a t a and 5 Sozen  i s shown i n Table  2.3.  The t h r e e - s t o r y frame behaved very w e l l .  None o f the  col-  umns y i e l d e d and the beam damage r a t i o s were s i x or l e s s i n a l l f o u r earthquakes.  Thus the s t r u c t u r e designed by the  s t r u c t u r e method behaved as expected. El  Centro EW  r e c o r d produced  substitute  In the f i v e - s t o r y  the worst r e s u l t .  The  frame,  columns  33 y i e l d e d a t three l o c a t i o n s and the damage r a t i o s of the beams, except the f i r s t - f l o o r beam, were about seven.  The frame, how-  ever, behaved very w e l l i n the o t h e r three earthquake motions. The columns remained i n the e l a s t i c range and the beam damage r a t i o s were l e s s than f i v e .  The t e n - s t o r y frame produced the  worst r e s u l t s o f the three frames.  L i k e the f i v e - s t o r y  frame  E l Centro EW motion produced the most u n f a v o r a b l e r e s u l t s . columns y i e l d e d a t many l o c a t i o n s .  The f i f t h  e x h i b i t e d a damage r a t i o o f about seven.  story  The  column  A l l the beams exceeded  the t a r g e t ^damage r a t i o o f s i x and some reached a damage r a t i o o f about t e n . quakes.  The r e s u l t s were much b e t t e r i n the other t h r e e e a r t h -  Although the columns y i e l d e d a t a few l o c a t i o n s i n two  earthquakes, i n e l a s t i c deformations were not e x c e s s i v e .  The beam  damage r a t i o s were a l l l e s s than s i x . These r e s u l t s agreed q u a l i t a t i v e l y w i t h those by S h i b a t a and Sozen, ~* but not q u a n t i t a t i v e l y  (Table 2.3).  The q u a n t i t a t i v e  d i f f e r e n c e was the s m a l l e s t f o r the t h r e e - s t o r y frame.  The b i g -  gest d i s c r e p a n c y o c c u r r e d i n the t e n - s t o r y frame, e s p e c i a l l y i n E l Centro EW motion.  The d i f f e r e n c e may be due t o modeling o f  elements i n the n o n l i n e a r dynamic program,  d u r a t i o n of earthquake  motion, o r d i f f e r e n c e i n earthquake r e c o r d s caused by d i g i t i z a t i o n o f the r e c o r d s o r f i l t e r i n g .  (b)  S o f t - S t o r y Frame 5 Shxbata and Sozen  r e s t r i c t e d a c h o i c e o f a t a r g e t damage  r a t i o f o r each element i n o r d e r t h a t the s u b s t i t u t e method may be used s u c c e s s f u l l y .  structure  They s t a t e d t h a t columns, beams,  and w a l l s may be designed w i t h d i f f e r e n t t a r g e t damage r a t i o s  f  but  34 t h a t the t a r g e t damage r a t i o s should be the a given bay implies the  and  a l l columns on a given a x i s -  t h a t a s o f t - s t o r y frame may  s u b s t i t u t e s t r u c t u r e method.  order to check the n e c e s s i t y Two vious Fig.  three-story  The  one  given  The  Two  examples were t e s t e d i n  was  a s s i g n e d to the  The  beams had  one  The  24-foot bay w i t h 11 f o o t s t o r y  3/4  The  design s p e c t r a  was  The  heights.  moment of  inertia  of t h a t of the columns above.  constant moment of i n e r t i a .  were then subjected  f i r s t - s t o r y columns  to the other beams.  The  computed by the s u b s t i t u t e s t r u c t u r e method and 2.6.  A  A t a r g e t damage r a t i o of s i x  to the f i r s t - f l o o r beam and  s t o r y columns was  pre-  designed as a " s o f t s t o r y " .  t o the r e s t of columns.  first  used i n the  f i r s t example are shown i n  f l o o r weight i s 72 k i p s f o r each l e v e l .  of the  by  for this r e s t r i c t i o n .  ground f l o o r was  frame c o n s i s t s of one  condition  not be designed p r o p e r l y  Data f o r the  t a r g e t damage r a t i o of two and  This  frames s i m i l a r t o the one  s e c t i o n were used. 2.6.  same f o r a l l beams i n  shown i n F i g . 2.5  design moments were are shown i n F i g .  were used.  The  frames  to f o u r earthquake motions, u s i n g the  non-  12 l i n e a r dynamic a n a l y s i s program, SAKE.  Each earthquake  record  was  normalized so that the maximum ground a c c e l e r a t i o n was  The  design moments were used as the y i e l d moments.  s t i f f n e s s - p r o p o r t i o n a l damping and assumed i n the n o n l i n e a r shown i n F i g . 2.7. designed t o : El  the  The  analysis.  Two  0.5  per  g.  cent  3% s t r a i n hardening were The  r e s u l t s of four runs  frame t r i e d to behave i n the way  are  i t was  f i r s t - s t o r y columns y i e l d e d i n a l l four cases.  Centro EW motion produced the worst r e s u l t ; the damage r a t i o  reached 2.8. which was  1.2.  T a f t S69E motion produced the s m a l l e s t The  damage r a t i o ,  r e s t of the columns remained e l a s t i c .  The  35 f i r s t - f l o o r beam y i e l d e d i n every case and the damage r a t i o from  3.7  t o 6.1.  The  s e c o n d - f l o o r beam remained more or l e s s  e l a s t i c except f o r one case. as w e l l as the other beams.  The t h i r d - f l o o r beam d i d not behave I t y i e l d e d i n a l l four cases, but  the damage r a t i o s were l e s s than 1.5 Although  ranged  except i n E l Centro EW  motion.  the t e s t frame d i d not perform very w e l l d u r i n g E l Centro  EW motion, the r e s u l t s from other motions seem to i n d i c a t e t h a t the s u b s t i t u t e s t r u c t u r e method produced  a s u c c e s s f u l design o f a  s o f t s t o r y frame i n t h i s example. In first The  the second example the s o f t s t o r y was  s t o r y t o the second  story.  same design spectrum was  method was  the  The data are shown i n F i g . 2.8.  used and the s u b s t i t u t e s t r u c t u r e  used t o compute the design moments.  moments are shown i n F i g . 2.8. the four earthquake  moved from  The  frame was  Those design again s u b j e c t e d t o  motions i n an i d e n t i c a l manner, w i t h the same  assumptions being made i n the n o n l i n e a r dynamic a n a l y s i s . r e s u l t s are shown i n F i g . 2.9. first all  They were not as good as the  example, s i n c e the second-story columns remained e l a s t i c i n  four cases, although they were designed to y i e l d .  columns remained e s s e n t i a l l y e l a s t i c . than the columns. earthquake; is  The second  The  other  The beams behaved b e t t e r  f l o o r beam d i d y i e l d i n every  w i t h the damage r a t i o ranging from 2.6  l e s s than the t a r g e t damage r a t i o of s i x .  e s s e n t i a l l y remained i n the e l a s t i c  (c)  The  t o 4.4  which  The o t h e r beams  range.  2-Bay, 3-Story Frame The r e s u l t s of the s o f t - s t o r y frames were i n c o n c l u s i v e .  The method worked w e l l i n the f i r s t  example, but o n l y a f a i r  36 r e s u l t was obtained  i n the second example.  A two-bay, t h r e e -  s t o r y frame was used t o t e s t whether the s u b s t i t u t e s t r u c t u r e method c o u l d be used f o r a frame w i t h randomly assigned damage r a t i o s .  target  The data f o r the s t r u c t u r e a r e shown i n F i g . 2.10.  The  design  spectrum was the same one used i n the p r e v i o u s  The  t a r g e t damage r a t i o s were randomly a s s i g n e d .  examples.  The s u b s t i t u t e  s t r u c t u r e was used t o compute the design moments, but the column moments were not i n c r e a s e d by 20%, because they c o u l d y i e l d  before  the beams. The  nonlinear  dynamic a n a l y s i s was c a r r i e d out i n an i d e n t -  i c a l manner as i n the previous quake records F i g . 2.11.  were used.  examples.  The same f o u r  earth-  The r e s u l t s o f four runs are shown i n  The s t r u c t u r e behaved q u i t e w e l l when the average  damage r a t i o s o f four earthquakes are compared w i t h the t a r g e t damage r a t i o s .  E l Centro EW motion produced the b i g g e s t  w h i l e T a f t motions produced the l e a s t .  In g e n e r a l ,  damage  the bottom-  s t o r y columns r e c e i v e d more damage than they were expected t o take, but the damage r a t i o s o f the second-story columns were very  c l o s e t o the t a r g e t damage r a t i o s .  The t h i r d - s t o r y columns  were damaged l e s s s e v e r e l y than they were designed f o r .  The same  t r e n d i s found i n the beams, but none o f the average damage r a t i o s were higher The  than the t a r g e t damage r a t i o s .  r e s u l t s o f t h i s example seem t o i n d i c a t e t h a t the sub-  s t i t u t e s t r u c t u r e method can be used t o design  a structure i n  which d i f f e r e n t t a r g e t damage r a t i o s a r e assigned the same bay and f o r columns on the same a x i s . beams work b e t t e r than columns.  f o r beams i n  I t appears t h a t  37 2.4  Equal-Area S t i f f n e s s Method  (a)  Observation As was d i s c u s s e d  i n the s e c t i o n 2.3(a), t h r e e  frames were  designed u s i n g the s u b s t i t u t e s t r u c t u r e method and they were subjected to nonlinear program was run, were obtained  dynamic a n a l y s i s .  When the dynamic a n a l y s i s  t i m e - h i s t o r y p l o t s o f displacements and moments  as a p a r t o f the output.  Upon o b s e r v a t i o n  p l o t s i t was p o s s i b l e t o p i c k up the p e r i o d s  of these  o f the most dominant  v i b r a t i o n , and i t was found t h a t these p e r i o d s were p e c u l i a r t o the frames, n o t t o the earthquake motions.  Furthermore, these  p e r i o d s were d i f f e r e n t from the n a t u r a l p e r i o d s  o f the a c t u a l  frames and from those o f the s u b s t i t u t e frames. Table 2.4 l i s t s and  the n a t u r a l p e r i o d s  o f the a c t u a l frames  s u b s t i t u t e frames f o r the f i r s t mode as w e l l as the observed  periods served  from the dynamic a n a l y s e s . p e r i o d s were longer  In a l l three cases the ob-  than the n a t u r a l p e r i o d s  frames, but s h o r t e r than the n a t u r a l p e r i o d s frames.  o f the a c t u a l  o f the s u b s t i t u t e  T h i s seemed t o imply t h a t the s u b s t i t u t e s t r u c t u r e  method d i d not give the c o r r e c t n a t u r a l p e r i o d s when i t underwent i n e l a s t i c  of a structure  deformation.  Some e f f o r t was made t o f i n d a method which would give a b e t t e r estimate o f the n a t u r a l p e r i o d s be  subjected  t o i n e l a s t i c deformation.  o f a s t r u c t u r e which would T h i s was f e l t  t o be  important, s i n c e modal a n a l y s i s was t o be used, i n which the response i s read a g a i n s t the p e r i o d .  38 (b)  Equal-Area  Stiffness  The p r e c e d i n g o b s e r v a t i o n supports the theory t h a t the s t i f f n e s s o f a system i s reduced when i t i s s u b j e c t e d t o s t r o n g motions  such t h a t i t s deformations exceed the e l a s t i c l i m i t .  At  the same time i t seems t o i n d i c a t e t h a t the s t i f f n e s s used i n the s u b s t i t u t e s t r u c t u r e method i s t o o s m a l l :  true e f f e c t i v e  stiff-  ness l i e s somewhere between the e l a s t i c s t i f f n e s s and the s t i f f ness o f the s u b s t i t u t e  structure.  Consider the l o a d - d e f l e c t i o n curve i n F i g . 2.12. t h a t i t i s an e l a s t o - p l a s t i c d u c t i l i t y a r e the same.  Assume  case so t h a t the damage r a t i o and  When a t a r g e t damage r a t i o i s chosen,  the maximum displacement i s i m p l i c i t l y s e l e c t e d .  The system i s  allowed t o undergo a deformation on the l o a d i n g curve up t o the p o i n t C.  The area under the curve i s equal t o the area o f the  t r a p e z o i d ABCD.  I t i s p o s s i b l e t o make up a f i c t i t i o u s  elastic  system which reaches the same u l t i m a t e displacement and has the same area under i t s l i n e a r  l o a d - d e f l e c t i o n curve AED as the area  of the b i l i n e a r curve ABCD, w h i l e both systems reach the same u l t i m a t e displacement, A , and absorb the same energy o f d e f o r mation F  i n doing so, the e l a s t o - p l a s t i c  system has the y i e l d  , as maximum f o r c e and the f i c t i t i o u s e l a s t i c system  force,  reaches  y F^, which i s g r e a t e r than the y i e l d f o r c e .  The s l o p e o f the l i n e  AE i s the s t i f f n e s s o f t h i s e l a s t i c system, which the author c a l l s an "equal-area s t i f f n e s s " . area s t i f f n e s s  By equating the two areas, the e q u a l -  can be expressed i n terms o f the i n i t i a l  stiffness  and the t a r g e t damage r a t i o ,  1  v  li  z ;  (2.27)  39 where  k  q  = equal-area  stiffness  k  = inital  stiffness  u  = t a r g e t damage r a t i o .  The y i e l d f o r c e i s unknown, but i t i s expressed i n terms o f the maximum f o r c e , F^,  F  where  y  = F_ (V .} 1 2y - 1  y  = yield •*  (2.28)  y  F  L  J  force  F^ = maxium f o r c e and  y  = t a r g e t damage r a t i o .  I f the moment-curvature curve o f an element has the same shape as t h a t o f the l o a d - d e f l e c t i o n durve, the f l e x u r a l  stiff-  ness o f the element can be reduced a c c o r d i n g t o equation  (2.26).  This s t i f f n e s s the system.  can be used t o s o l v e f o r the n a t u r a l p e r i o d s o f  T h i s approach  i s , o f course, very h y p o t h e t i c a l and  there i s no experimental data t o support i t .  The concept o f  s u b s t i t u t e damping l o s e s much o f i t s meaning, because i t was derived  from the s i m p l i f i e d  forced concrete.  h y s t e r e s i s loop o f degraded  But t h i s h y p o t h e s i s can be t e s t e d  rein-  analytically  by modifying the s t i f f n e s s p a r t o f the s u b s t i t u t e s t r u c t u r e program. (c)  Examples The same three frames used i n s e c t i o n 2.2(a) were used to  t e s t the equal-area s t i f f n e s s method.  The t a r g e t damage r a t i o s  were s e t a t one f o r the columns and s i x f o r the beams. f l e x u r a l components o f the member s t i f f n e s s e s  When the  were assembled, they  40 were reduced  a c c o r d i n g to the equation  (EI)  (EI)  where  (EI)  (EI)  ex  that i s ,  2p. - 1  ax  .  equal-area  ex  (2.27);  a  (2.29)  . . .  s t i f f n e s s of element i  s t i f f n e s s of i th element of a c t u a l frame  ax  t a r g e t damage r a t i o of i t h element. The n a t u r a l p e r i o d s of the three frames were computed u s i n g the equal-area  stiffness.  The p e r i o d s corresponding  to the f i r s t mode  are l i s t e d on Table 2.4.  Those p e r i o d s agreed  very w e l l with  dominant p e r i o d s observed  i n the n o n l i n e a r a n a l y s i s .  the  Therefore,  as f a r as the n a t u r a l p e r i o d s are concerned, t h i s approach g i v e s a more r e a l i s t i c  (d)  estimate.  Area f o r F u r t h e r  Studies  The design f o r c e s computed by the s u b s t i t u t e s t r u c t u r e method were used as the y i e l d moments i n the n o n l i n e a r dynamic analysis.  I f a method to o b t a i n the same design f o r c e s c o u l d be  developed,  t h i s equal-area  attractive.  An e f f o r t was  f o r c e s t h a t are s i m i l a r method, but i t was  s t i f f n e s s method would become more made to f i n d a way  t o those  to compute design  from the s u b s t i t u t e s t r u c t u r e  not p o s s i b l e to o b t a i n a s a t i s f a c t o r y  F u r t h e r s t u d i e s may  be worthwhile, because the agreement  i n p e r i o d s i s too good to i g n o r e .  Any  further research  be s t a r t e d w i t h a s i n g l e - d e g r e e - o f freedom system. support t h i s hypothesis  result.  should  A theory to  needs to be e s t a b l i s h e d along  with  41 experimental data.  I f a l i n e a r response spectrum i s t o be used,  a new method o f computing developed.  s u i t a b l e damping p r o p e r t i e s  must be  42  CHAPTER 3  3.1  MODIFIED SUBSTITUTE STRUCTURE METHOD  M o d i f i e d S u b s t i t u t e S t r u c t u r e Method  The term,  "retrofit",  i s d e f i n e d i n the f i r s t  d e s c r i b e s the problem o f e v a l u a t i n g the performance b u i l d i n g s a g a i n s t s e i s m i c hazards.  chapter.  It  of existing  A r e t r o f i t procedure i s , then,  a procedure f o r a n a l y z i n g e x i s t i n g b u i l d i n g s .  I t i s inevitable  t h a t almost a l l the s t r u c t u r e s y i e l d and s u f f e r i n e l a s t i c  deforma-  t i o n under a s t r o n g earthquake motion; such a procedure, t h e r e f o r e , must perform some s o r t o f i n e l a s t i c a n a l y s i s .  I t must be  capable o f i d e n t i f y i n g the l o c a t i o n s and e x t e n t o f damage a s s o c i a t e d with a p a r t i c u l a r earthquake motion. fail,  I f a structure i s to  the mode o f f a i l u r e must be i d e n t i f i e d .  I t i s desirable  t h a t a method be f l e x i b l e enough t o handle earthquakes o f d i f f e r ent nature and magnitude.  A t the same time i t must be reasonably  economical and easy t o use i n o r d e r to become a p r a c t i c a l t o o l f o r average e n g i n e e r s . because  The use o f a computer i s probably i n e v i t a b l e  of the nature o f the problem, b u t a program t o run such  an a n a l y s i s must be easy to w r i t e and economical t o operate. The m o d i f i e d s u b s t i t u t e s t r u c t u r e method f u l f i l l s the aforementioned requirements.  As the name suggests, i t was d e v e l 5  oped from the s u b s t i t u t e s t r u c t u r e method by S h i b a t a and Sozen. At present i t s use i s r e s t r i c t e d to r e i n f o r c e d c o n c r e t e s t r u c t u r e s  as i s the case f o r the s u b s t i t u t e w i t h proper m o d i f i c a t i o n s  s t r u c t u r e method i t s e l f , but  the method may be used f o r a n a l y s i s o f  s t e e l and other s t r u c t u r e s .  I t i s a modified e l a s t i c analysis i n  which the s t i f f n e s s and damping p r o p e r t i e s  are changed f o r use  w i t h modal a n a l y s i s so that the f o r c e s and deformations agree with nonlinear  dynamic a n a l y s i s .  A l i n e a r response spectrum i s  used t o compute the i n e l a s t i c response.  The concepts o f s u b s t i -  t u t e damping, damage r a t i o , and s u b s t i t u t e from the s u b s t i t u t e The  s t i f f n e s s are borrowed  s t r u c t u r e method.  d i f f e r e n c e between a design procedure and a r e t r o f i t  procedure i s worth n o t i n g .  In a seismic  i n i t i a l s t i f f n e s s o f the s t r u c t u r e o t h e r requirements.  design procedure the  i s known approximately from  A d e s i g n e r can choose and s p e c i f y the amount  of i n e l a s t i c deformation each element i s allowed t o undergo i n a given earthquake motion.  I t i s the design f o r c e s o r y i e l d  forces  t h a t must be determined.  In the s u b s t i t u t e s t r u c t u r e method, the  s t i f f n e s s o f the a c t u a l frame i s known o r i t can be estimated fairly  precisely.  by a d e s i g n e r .  Target damage r a t i o s a r e s e l e c t e d  Hence, the s u b s t i t u t e  s t i f f n e s s and s u b s t i t u t e  damping r a t i o s o f the elements are p r e s c r i b e d . associated be  f o r elements  Natural  periods,  mode shapes, and modal p a r t i c i p a t i o n f a c t o r s need to  computed only once.  A f t e r computation o f a smeared damping  r a t i o f o r each mode, modal f o r c e s are c a l c u l a t e d and combined as specified.  No i t e r a t i o n i s r e q u i r e d  during  computation.  In a  r e t r o f i t procedure the i n i t i a l s t i f f n e s s and the y i e l d moments and  other s t r e n g t h  properties  of a structure  are known o r they  can be found from design c a l c u l a t i o n s , drawings, and f i e l d igations.  What i s known i s the amount o f i n e l a s t i c  invest-  deformation;  t h a t i s , the damage r a t i o f o r each member must be computed given an earthquake  motion.  a s u i t a b l e combination y i e l d forces.  In the m o d i f i e d s u b s t i t u t e s t r u c t u r e method of modal f o r c e s must agree with the known  To achieve t h i s the damage r a t i o s of a l l the  elements must be estimated p r e c i s e l y so t h a t c o r r e c t  substitute  s t i f f n e s s and s u b s t i t u t e damping r a t i o s can be used.  This, of  course, i s i m p o s s i b l e to do; otherwise t h e r e would be no need to perform an a n a l y s i s .  I t i s , t h e r e f o r e , i n e v i t a b l e t h a t an  t i v e process must be used.  itera-  A f t e r each i t e r a t i o n damage r a t i o s  must be m o d i f i e d to approach nearer t o the c o r r e c t v a l u e s . i s c e r t a i n l y a disadvantage, because more computations q u i r e d and hence more c o s t s .  This  are r e -  But i f the number of i t e r a t i o n s are  s m a l l , i t i s s t i l l an economical  a l t e r n a t i v e t o f u l l - s c a l e non-  l i n e a r dynamic a n a l y s i s . Before the procedure  f o r the m o d i f i e d s u b s t i t u t e s t r u c t u r e  method i s d e s c r i b e d i n d e t a i l , s e v e r a l c o n d i t i o n s are  listed.  They must be s a t i s f i e d i n order to apply the m o d i f i e d s u b s t i t u t e s t r u c t u r e method p r o p e r l y .  These c o n d i t i o n s a r e :  Cl)  the system can be analyzed i n one v e r t i c a l  plane,  (2)  there i s no abrupt change i n geometry and p r e f e r a b l y i n mass along the h e i g h t of the system,  (3)  reinforcement of a l l members and j o i n t s are known such t h a t t h e i r a b i l i t y t o withstand repeated r e v e r s a l s of  inelastic  deformation without s i g n i f i c a n t s t r e n g t h decay can estimated, (4)  be  and  n o n s t r u c t u r a l components do not i n t e r f e r e w i t h  structural  response. The aforementioned  c o n d i t i o n s are s i m i l a r to those l i s t e d  by  45 Shibata it  and Sozen  i n the s u b s t i t u t e s t r u c t u r e method.  should be noted t h a t , a f t e r convergence, the f i n a l  of the m o d i f i e d design and  5  In f a c t , iteration  s u b s t i t u t e s t r u c t u r e method i s i d e n t i c a l to the  procedure, and t h e r e f o r e has e x a c t l y the same r e s t r i c t i o n s  validity. The  f o l l o w i n g i s the step-by-step d e s c r i p t i o n o f the proce-  dure f o r the m o d i f i e d  s u b s t i t u t e s t r u c t u r e method.  I t must be  remembered t h a t the y i e l d f o r c e cannot be exceeded a t any time. CD  Perform a modal a n a l y s i s on the assumption o f e l a s t i c behaviour.  Damping r a t i o s must be chosen so t h a t they are  appropriate  f o r the given earthquake.  Compute the r o o t -  sum-square (RSS) f o r c e s . C2)  F i n d the members i n which RSS moments exceed the y i e l d moments.  Note t h a t the b i g g e r  o f the two end moments i s  used. C3)  In such members modify the damage r a t i o s a c c o r d i n g formula t h a t w i l l be d e s c r i b e d  l a t e r on.  The other  t o the members  w i l l have a damage r a t i o o f one. C4)  Follow steps  (.2) t o C5) f o r the s u b s t i t u t e s t r u c t u r e method  which was d e s c r i b e d  on pages 26 and 27 i n Chapter 2.2(b).  Compute the RSS moments. (5)  Compare the RSS moments with the y i e l d moments. damage r a t i o s  (6)  according  Repeat the steps  Modify the  t o the formula t o be d i s c u s s e d  later  (4) and (5) u n t i l a l l the computed moments,  except i n those members f o r which the damage r a t i o s are one, (7)  are equal t o the r e s p e c t i v e y i e l d moments.  The members i n which the damage r a t i o s are g r e a t e r w i l l r e c e i v e i n e l a s t i c deformation.  than one  Check i f each member  46 can take such deformation. It  I f not, such a member w i l l  fail.  i s now p o s s i b l e t o make an estimate o f the l o c a t i o n s and  extent o f damage i n the whole s t r u c t u r e .  S i m i l a r checks  can be made f o r other components o f i n t e r n a l  force.  An o r d i n a r y e l a s t i c modal a n a l y s i s i s performed i n the f i r s t i t e r a t i o n , because a t t h i s stage i t i s not c l e a r i f a s t r u c t u r e w i l l go through quake. critical  i n e l a s t i c deformations  i n a g i v e n earh-  A value f o r damping must be chosen; a r a t i o o f 10% o f i s a p p r o p r i a t e f o r a r e i n f o r c e d concrete s t r u c t u r e sub-  j e c t e d to a strong earthquake motion.  Since i t i s i m p o s s i b l e to  exceed the y i e l d moments, those members i n which the computed moments a r e g r e a t e r than t h e i r y i e l d moments w i l l y i e l d .  In the  t h i r d step the f i r s t estimate o f damage r a t i o s i s made.  Starting  from the second c y c l e o f i t e r a t i o n , the s u b s t i t u t e s t r u c t u r e method i s used t o compute the n a t u r a l p e r i o d s , mode shapes, and modal f o r c e s .  Damage r a t i o s c a l c u l a t e d a t the end o f the p r e v i o u s  i t e r a t i o n are used t o compute the s u b s t i t u t e s t i f f n e s s and subs t i t u t e damping r a t i o s . throughout  The root-sum-square moments a r e used  the i t e r a t i o n s .  In the s u b s t i t u t e s t r u c t u r e method  they were i n c r e a s e d a c c o r d i n g t o equation  (2.25) and the column  moments were f u r t h e r i n c r e a s e d by 20% t o o b t a i n the design moments. T h i s approach i s acceptable i n a design procedure, would provide an e x t r a margin o f s a f e t y .  because i t  But i n order t o be on  the c o n s e r v a t i v e s i d e i t i s a d v i s a b l e t o use the root-sum-square moments and ignore the f a c t o r i n equation  (2.25) .  I n c r e a s i n g the  column moments by 20% here i s , o f course, t o t a l l y absurd.  Unless  c o r r e c t damage r a t i o s are o b t a i n e d i n the p r e v i o u s i t e r a t i o n , the computed moments do not agree w i t h the y i e l d moments except f o r  47 the  members which remain e l a s t i c .  The damage r a t i o s  m o d i f i e d arid another i t e r a t i o n must be made. the  A t some stage a l l  damage r a t i o s w i l l converge t o the c o r r e c t v a l u e s and the  i t e r a t i o n process w i l l be stopped.  Then an e v a l u a t i o n of the  performance of the s t r u c t u r e can be c a r r i e d the  must be  l a s t step.  out as o u t l i n e d i n  I t must be noted t h a t the e f f e c t  of s t r a i n  hardening i s ignored i n the d i s c u s s i o n above, b u t i t can be i n c l u d e d w i t h only a s l i g h t It ratios  modification.  i s now a p p r o p r i a t e t o e x p l a i n a way t o modify damage  a t the end o f each i t e r a t i o n .  case shown i n F i g . 3.1.  Consider the e l a s t o p l a s t i c  Suppose a t the end o f the f i r s t  itera-  t i o n , which i s an o r d i n a r y modal a n a l y s i s , the computed moment which i s g r e a t e r than the y i e l d moment, M^.  was  Since the  member was assumed t o behave e l a s t i c a l l y , i t f o l l o w e d the l i n e OA and reached the p o i n t B w i t h the moment, M^, and the r o t a t i o n , <J)^.  Since a computed moment cannot exceed M^, the s t i f f n e s s , k,  must be reduced i n the next i t e r a t i o n . rotation,  cj>^, was c o r r e c t .  I t i s assumed t h a t the  A p o i n t B' i s l o c a t e d on the p l a s t i c  p a r t o f the moment-rotation curve and the slope o f the l i n e OB' i s used as the s t i f f n e s s  f o r the next i t e r a t i o n .  r a t i o c o r r e s p o n d i n g t o t h i s new s t i f f n e s s the  geometry.  The damage  can be c a l c u l a t e d from  The damage r a t i o a t the end o f the f i r s t  iteration  i s given by, M.  y2  where  2 M.  1  1  M  > 1  (3.1)  y  damage r a t i o t o be used i n the second computed moment i n the f i r s t  iteration  iteration  48 M  = y i e l d moment  y  J  Suppose t h a t a t the end o f the second i t e r a t i o n moment, M , 2  still  exceeded  the y i e l d moment, M •; t h a t i s , i t  reached the p o i n t C on the curve. s t i f f n e s s was s t i l l increased.  the computed  I t means t h a t the assumed  too b i g and t h a t the damage r a t i o must be  T h i s time a p o i n t C  slope of the l i n e OC  i s l o c a t e d on the curve and the  i s used t o d e f i n e the new s t i f f n e s s .  1  A new  damage r a t i o corresponding t o the new s t i f f n e s s can be o b t a i n e d from the geometry. M  1-U =  ^3  =  (3.2)  M y  J  where  2  damage r a t i o t o be used i n the t h i r d  1^2 = damage r a t i o used i n the second  iteration  iteration  = computed moment a t the end o f the second  itera-  tion M = yield y It  i s p o s s i b l e t h a t the computed moment, M^, was l e s s  the y i e l d moment, M . is,  moment.  The s t i f f n e s s must now be i n c r e a s e d ;  the damage r a t i o must be decreased.  than that  The new damage r a t i o can  be computed from the geometry i n a s i m i l a r t i o n as i n equation (3.2) can be o b t a i n e d .  way and the same r e l a A t t e n t i o n must be  p a i d t h i s time, s i n c e i f the new damage r a t i o i s l e s s than one, i t must be s e t a t one. In g e n e r a l , a t the end o f the n t h i t e r a t i o n r a t i o can be computed by the f o l l o w i n g e q u a t i o n .  the new damage  49  y  _  n+l  y  n  M ... n  -C3.31  >1  M y  u , . = damage r a t i o t o be used i n the n+l t h n+l  where  J  iteration y  = damage r a t i o used i n n t h i t e r a t i o n  n  M  = computed moment i n n t h i t e r a t i o n  n  M  = y i e l d moment. y  If  V 2_ n+  equals y  n  f o r a l l the members, the i t e r a t i o n p r o c e s s i s  complete. When the moment-rotation curve a f t e r y i e l d e x h i b i t s hardening, the s i t u a t i o n the  i s a l i t t l e more complex.  I f such i s  case, the y i e l d moment i s not the a b s o l u t e l i m i t .  The com-  puted moment can be and w i l l be g r e a t e r than the y i e l d p r o v i d e d t h a t the damage r a t i o i s g r e a t e r than one. of the formula f o r the new damage r a t i o s  strain  moment  Derivation  i s shown i n Appendix A.  It i s ,  n' n M (1 - s) + s.y .M y n n M  y  where  n+l  y  >1  (3.4)  ^n+l ~ m o d i f i e d damage r a t i o t o be used i n n+l t h iteration = damage r a t i o used i n n t h i t e r a t i o n M  = computed moment i n n t h i t e r a t i o n  M  = y i e l d moment y  s  = r a t i o of s t i f f n e s s stiffness.  after  y i e l d to i n i t i a l  50 Inherent l i m i t a t i o n s o f the m o d i f i e d s u b s t i t u t e method are now d i s c u s s e d .  structure  The moment-rotation curve o f each  member must be such t h a t i t can be approximated by a b i l i n e a r curve.  Furthermore, i t must have the same shape as t h a t o f the  l o a d - d e f l e c t i o n curve.  I f l i n e a r l y d i s t r i b u t e d moment w i t h a  p o i n t o f i n f l e c t i o n i n the mid-span condition i s s a t i s f i e d .  o f a member i s assumed, t h i s  The moment c a p a c i t y o f each member i s  assumed t o be the same f o r both ends and f o r both p o s i t i v e and negative moments.  I f the computed moment a t one end o f the  member i s g r e a t e r than a t the o t h e r end, the b i g g e r moment i s chosen t o compute the damage r a t i o .  51 3.2  Computer Program  The  use o f a computer i s e s s e n t i a l f o r p r a c t i c a l a p p l i c a -  t i o n s o f the m o d i f i e d  s u b s t i t u t e s t r u c t u r e method.  The i t e r a t i v e  process t h a t i s r e q u i r e d i n the method can be i n c o r p o r a t e d program q u i t e e a s i l y . little  difficulty.  The program i t s e l f can be w r i t t e n w i t h  I f an e l a s t i c modal a n a l y s i s program i s  a v a i l a b l e , r e l a t i v e l y few m o d i f i c a t i o n s The  flow diagram o f the m o d i f i e d  gram i s shown i n F i g . 3.2.  a r e necessary. s u b s t i t u t e s t r u c t u r e pro-  Data f o r s t r u c t u r a l d e f i n i t i o n ,  member p r o p e r t i e s , and j o i n t l o c a t i o n s a r e read the  i n the  f i r s t p a r t o f the program.  i n and s t o r e d i n  The damage r a t i o s o f a l l the  members should be i n i t i a l i z e d a t one.  Then the mass matrix  should be s e t up; i t remains unchanged throughout the i t e r a t i o n process.  The s t r u c t u r e s t i f f n e s s m a t r i x i s assembled from member  matrices. according  The f l e x u r a l p a r t o f the member s t i f f n e s s i s m o d i f i e d t o the damage r a t i o u s i n g equation  the damage r a t i o s are s e t a t one i n the f i r s t  (2.21).  Since a l l  i t e r a t i o n the s t r u c -  ture s t i f f n e s s m a t r i x i s the same as i n the e l a s t i c a n a l y s i s . T h i s m a t r i x and the mass m a t r i x are used t o s o l v e f o r n a t u r a l periods, Since  a s s o c i a t e d mode shapes, and modal p a r t i c i p a t i o n f a c t o r s .  i t i n v o l v e s a r e g u l a r e i g e n v a l u e problem, a l i b r a r y sub-  routine i s usually available.  I n i t i a l l y a s u i t a b l e set of  damping r a t i o s should be given by the user. for  a l l the modes was used by the author.  Ten per cent A spectrum  i s c a l l e d and a peak ground a c c e l e r a t i o n i s r e t u r n e d . load v e c t o r  damping  subroutine Then a  i s s e t up and the s t i f f n e s s m a t r i x i s i n v e r t e d t o  solve f o r d e f l e c t i o n s .  Modal f o r c e s can be computed i n the u s u a l  52 manner.  T h i s process i s repeated f o r a l l the modes and RSS f o r c e s  and displacements are computed.  A t the end RSS moments are com-  pared with r e s p e c t i v e y i e l d moments.  I f the y i e l d moment o f any  member i s exceeded, i t e r a t i o n i s necessary.  The damage r a t i o o f  such a member i s m o d i f i e d a c c o r d i n g to equation From the second  (3.3) o r (3.4).  i t e r a t i o n the s u b s t i t u t e s t r u c t u r e method  i s used to compute modal f o r c e s and displacements. s t i f f n e s s m a t r i x i s reassembled and the computation participation all  The s t r u c t u r e  u s i n g the new s e t o f damage r a t i o s  o f n a t u r a l p e r i o d s , mode shapes, and modal  f a c t o r s i s repeated.  S u b s t i t u t e damping r a t i o s o f  the members should be c a l c u l a t e d a t t h i s stage using equation  (2.20).  Modal f o r c e s a r e c a l c u l a t e d twice.  F o r c e s f o r the  undamped case are computed f i r s t t o c a l c u l a t e the f l e x u r a l energy s t o r e d i n each member.  Smeared damping r a t i o s  modes are computed u s i n g equations  (2.23) and (2.24).  strain  f o r a l l the They are  used t o g e t the peak ground a c c e l e r a t i o n s from the spectrum. Modal f o r c e s and displacements are recomputed and RSS f o r c e s and displacements a r e obtained a t the end. i s used t o modify  the damage r a t i o s .  necessary u n t i l a l l the damage r a t i o s very many i t e r a t i o n s t i c a l convergence iterations  Equation  (3.3) o r (3.4)  F u r t h e r i t e r a t i o n s are stop changing.  In p r a c t i c e ,  are necessary t o achieve t h i s and more p r a c -  c r i t e r i a must be used t o keep the number o f  a t a reasonable l e v e l .  The program used by the author  i s l i s t e d i n Appendix B. The c o s t o f running the program depends d i r e c t l y on the number o f i t e r a t i o n s .  I f the convergence  can be a c c e l e r a t e d , the  saving i n CPU time and hence c o s t can be s u b s t a n t i a l . was made t o achieve a c c e l e r a t e d convergence  An attempt  and a method i s  described  i n a subsequent s e c t i o n o f t h i s chapter.  Obviously  the  proposed method i s more c o s t l y than an o r d i n a r y modal a n a l y s i s because of the amount of computation i n v o l v e d , but requirement i s roughly t h i s method i s s t i l l  the  this analysis i s s t i l l  time r e q u i r e d f o r  Therefore,  o v e r a l l c o s t of running  small compared to the c o s t of running  dynamic a n a l y s i s .  advantage of the modified nonlinear  the CPU  storage  a f r a c t i o n of t h a t f o r the f u l l - s c a l e non-  l i n e a r dynamic a n a l y s i s .  nonlinear  same and  the  the  Coupled w i t h ease of data setup,  s u b s t i t u t e s t r u c t u r e method over  dynamic a n a l y s i s i s s u b s t a n t i a l .  the  the  54 3.3  Convergence  In order t o t e s t whether the m o d i f i e d  substitute  method a c t u a l l y works, t e s t frames a r e r e q u i r e d .  structure  The damage  r a t i o s o f a l l the members i n such frames must be known f o r a given  l i n e a r response spectrum.  Since the method u t i l i z e s the  s u b s t i t u t e s t r u c t u r e method, i t i s p o s s i b l e t o design a frame by the  s u b s t i t u t e s t r u c t u r e method and then s u b j e c t  by the m o d i f i e d  s u b s t i t u t e s t r u c t u r e method.  i t to analysis  When the i t e r a t i o n  procedure i s complete the computed damage r a t i o s should be equal to the t a r g e t damage r a t i o s a s s i g n e d i n the d e s i g n method.  Since  the RSS f o r c e s a r e used as the computed f o r c e s , the design  forces  i n the s u b s t i t u t e s t r u c t u r e method must a l s o be the RSS f o r c e s , not the f o r c e s which a r e i n c r e a s e d (2.25).  by the f a c t o r i n equation  Two frames were t e s t e d t h i s way.  The  first  t e s t frame i s a 2-bay, 3-story  are shown on F i g . 3.3.  frame.  The s u b s t i t u t e s t r u c t u r e method was used  to compute the y i e l d moments and n a t u r a l p e r i o d s shown on F i g . 3.3. The  The data  which are a l s o  RSS moments were taken as the d e s i g n moments.  t a r g e t damage r a t i o s were one f o r the columns and s i x f o r the  beams. was used  The same response spectrum as i n the p r e v i o u s examples ( F i g . 2.5). T h i s frame was then s u b j e c t e d  t o the modi-  f i e d s u b s t i t u t e s t r u c t u r e a n a l y s i s t o t e s t the convergence o f periods  and damage r a t i o s .  The i t e r a t i o n was c a r r i e d out 20  times and the n a t u r a l p e r i o d s the end o f each i t e r a t i o n .  and damage r a t i o s were p r i n t e d a t  The damping r a t i o s f o r a l l three  modes were taken as 10% o f the c r i t i c a l i n the f i r s t c y l c e of iteration.  The three n a t u r a l p e r i o d s  computed i n each i t e r a t i o n  are  l i s t e d on Table 3.1.  To i l l u s t r a t e the speed o f convergence,  each p e r i o d i s normalized t o t h a t computed i n the s u b s t i t u t e s t r u c t u r e method and the p l o t o f the normalized p e r i o d s versus the  number o f i t e r a t i o n s i s shown on F i g . 3.4.  As can be seen  from the p l o t , the n a t u r a l p e r i o d f o r the f i r s t mode converged very r a p i d l y .  I t took only f i v e i t e r a t i o n s f o r the f i r s t mode  p e r i o d s t o be w i t h i n 1% o f the c o r r e c t p e r i o d . of  The convergence  the second mod p e r i o d and the t h i r d mode p e r i o d were slower;  they were w i t h i n 1% o f the c o r r e c t p e r i o d s a f t e r 13 i t e r a t i o n s . The second mode p e r i o d s approached the c o r r e c t value more r a p i d l y d u r i n g the f i r s t  few i t e r a t i o n s than the t h i r d mode p e r i o d .  The damage r a t i o s o f s e l e c t e d columns and beams are l i s t e d i n Table 3.2 and the p l o t i s shown i n F i g . 3.5.  The damage  r a t i o s of column 1 and beam 1 converged very r a p i d l y .  Only 6  i t e r a t i o n s were necessary b e f o r e they were w i t h i n 1% o f t h e i r r e s p e c t i v e t a r g e t damage r a t i o s .  Convergence of damage r a t i o i n  beam 2 was slower and i t took 15 i t e r a t i o n s t o be w i t h i n 1% o f the of  t a r g e t damage r a t i o . the four members.  Column 2 had the slowest convergence  I t s damage r a t i o was w i t h i n 1% o f the  t a r g e t damage r a t i o a t the end o f 20 i t e r a t i o n s . As can be seen from the two p l o t s , the p e r i o d s converged f a s t e r than the damage r a t i o s .  Among the n a t u r a l p e r i o d s , the  lowest mode p e r i o d converged a t the f a s t e s t r a t e , and the h i g h e s t mode the slowest.  As f a r as the convergence o f the damage r a t i o s  i s concerned those o f the members i n the lower s t o r y converged f a s t e r than i n the upper s t o r y .  T h i s i s l o g i c a l , because the  response o f the members i n the lower s t o r y i s governed by the lower mode and the convergence o f the n a t u r a l p e r i o d s and hence  56 the mode shapes i s f a s t e r f o r the lower mode. The example.  same 2-bay, 3-story frame was used i n the second The member p r o p e r t i e s were the same as i n the f i r s t  frame, the t a r g e t target  damage r a t i o s were changed.  damage r a t i o s o f two, one, and t h r e e .  The columns had The same damage  r a t i o s were assigned t o a l l columns on the same a x i s . get  damage r a t i o s f o r beams were s i x i n one bay and two i n the  other bay. the  The s u b s t i t u t e  structure  method was used t o compute  y i e l d moments, which were RSS moments.  natural  p e r i o d s are shown i n F i g . 3.6.  yzed by the m o d i f i e d s u b s t i t u t e The  natural  i n F i g . 3.7. very r a p i d .  The frame was then  structure  method. 20 i t e r a t i o n s a r e  The p l o t o f normalized p e r i o d s a r e shown  The convergence o f the f i r s t two p e r i o d s was again  I t was w i t h i n 4.3% o f the c o r r e c t  v a l u e a f t e r 20  The damage r a t i o s converged very s l o w l y .  shows the damage r a t i o s a t the end o f s e l e c t e d tions.  anal-  The p e r i o d f o r the t h i r d mode, however, was r e l a -  t i v e l y slow. iterations.  Those f o r c e s and the  p e r i o d s computed i n the f i r s t  t a b u l a t e d i n Table 3.3.  F i g . 3.8  numbers o f i t e r a -  At the end o f 20 i t e r a t i o n s the damage r a t i o s o f the  third-story the  The t a r -  target  columns and beams were s t i l l q u i t e d i f f e r e n t ones.  from  The i t e r a t i o n was c a r r i e d out 200 times and by  then they d i d converge to the c o r r e c t  values.  The p l o t o f damage  r a t i o s a g a i n s t the number o f i t e r a t i o n s i s shown i n F i g . 3.9. The  r a t e o f convergence o f damage r a t i o s were much slower i n the  second example than i n the f i r s t example.  F i g . 3.9 shows t h a t  about 100 i t e r a t i o n s were necessary t o achieve reasonable estimate of damage r a t i o s . applies  The same c o n c l u s i o n i n the p r e v i o u s example  i n the second example.  57 The (1)  r e s u l t s o f these two examples showed the f o l l o w i n g .  The n a t u r a l p e r i o d s  converge a t a f a s t e r r a t e than the  damage r a t i o s . (2)  The n a t u r a l p e r i o d s  f o r the lower modes converge f a s t e r  than those i n the h i g h e r modes. C3)  In g e n e r a l ,  the damage r a t i o s o f the upper s t o r y columns  and beams converge a t a slower r a t e than those o f the lower stories. (4)  Both the damage r a t i o s and the n a t u r a l p e r i o d s verge m o n o t o n i c a l l y . the  (5)  first  This point i s p a r t i c u l a r l y true i n  few c y c l e s o f i t e r a t i o n s .  The r a t e o f convergence slows down as the number o f i t e r a tions increases. first  The most r a p i d changes occur during the  few c y c l e s o f i t e r a t i o n .  These o b s e r v a t i o n s be  do not con-  were confirmed i n the other examples t h a t w i l l  shown l a t e r on. It  i s , i n p r a c t i c e , impossible  t o c a r r y out the i t e r a t i o n  process u n t i l a l l the damage r a t i o s cease t o f l u c t u a t e . as a good estimate o f damage r a t i o s i s obtained, procedure should be stopped. for  t h i s purpose.  As soon  the i t e r a t i o n  Some c r i t e r i o n must be e s t a b l i s h e d  I t i s p o s s i b l e , but not p r a c t i c a l , t o keep  t r a c k o f every damage r a t i o a t the end o f each i t e r a t i o n .  It is  a l s o i m p r a c t i c a l t o s e t the l i m i t on the number o f i t e r a t i o n s a t a c e r t a i n number.  The two examples i n t h i s s e c t i o n  t h i s p o i n t very c l e a r l y . 30, in  illustrated  I f the number o f i t e r a t i o n s i s s e t a t  say, then the l a s t 10 t o 15 i t e r a t i o n s i s t o t a l l y unnecessary the f i r s t example.  On the other hand, i n a c c u r a t e  damage r a t i o s r e s u l t s i n the second example.  estimate o f  Two approaches  58 seem p o s s i b l e as s u i t a b l e convergence c r i t e r i a .  One approach i s  to compare the v a l u e s o f the damage r a t i o o f each member a t the end o f the i t e r a t i o n w i t h t h a t o f the p r e v i o u s i t e r a t i o n .  The  f o l l o w i n g formula may be used.  ( V n - l C y  where  i n-1  ^ i ^ n  ^  =  a m a <  ?  e  (3.5)  < 6  }  r a t i o o f i t h element a t the end o f  n th iteration (y^) _ n  = damage r a t i o o f the same element a t the end  1  of n-1 t h i t e r a t i o n 6 = constant I f t h i s i s t r u e f o r a l l the elements i n the s t r u c t u r e , the i t e r a t i o n i s complete and the f o r c e s , d i s p l a c e m e n t s , and damage r a t i o s can  be p r i n t e d .  An a l t e r n a t i v e approach i s t o compare the com-  puted moments w i t h the y i e l d moments.  The f o l l o w i n g formula i s  s u i t a b l e f o r t h i s purpose. (M. ) - (M . ) i n yi (M .) yi where  ^ i^n M  =  c o m  (3.6)  < e  P t e d RSS moment i n i t h element u  during n th i t e r a t i o n (M .) = y i e l d moment f o r i t h element yl J  £ = constant. I f t h i s i n e q u a l i t y i s s a t i s f i e d f o r a l l the elements w i t h damage r a t i o s g r e a t e r than one, no more i t e r a t i o n i s n e c e s s a r y . A definite direct  advantage o f the f i r s t method i s t h a t i t i s a  comparison o f the damage r a t i o s  computed i n the l a t e s t two  iterations.  The second method i s an i n d i r e c t comparison o f the  damage r a t i o s . ratios. the  I t i s not c l e a r  how much change i s made on damage  The f i r s t approach has a d e f i n i t e  disadvantage, because  denominator changes a t every i t e r a t i o n .  Because o f t h i s  reason the second approach was adopted by the author.  It i s  hoped t h a t t h i s c r i t e r i o n produces a more uniform r e s u l t f o r different  types o f s t r u c t u r e s .  With a l i t t l e e x p e r i e n c e a s u i t -  able value f o r e can be s p e c i f i e d . it  In running a computer program  i s d e s i r a b l e t o s e t the l i m i t on the number of i t e r a t i o n s ,  because no output would be o b t a i n e d i f a value f o r e was too s m a l l and CPU time exceeded the l i m i t s e t by the user. In s p i t e o f the f o r e g o i n g d i s c u s s i o n , i t should be noted that, i n practice,  because of the i n a c c u r a c i e s i n modeling the  structure, i n predicting  the earthquake, and i n c o r r e l a t i n g  damage r a t i o w i t h a c t u a l damage, the r e s u l t s to a h i g h degree o f p r e c i s i o n .  a r e not s i g n i f i c a n t  60 3.4  Ac eele r a t e d C onverge nce  The  c o s t o f running the m o d i f i e d  method i s roughly  substitute structure  p r o p o r t i o n a l to the number o f i t e r a t i o n s  i s necessary t o meet the convergence c r i t e r i o n .  I f there  that isa  way t o a c c e l e r a t e the convergence, the method becomes a more powerful t o o l .  An e f f o r t was made t o achieve t h i s goal and the  f o l l o w i n g procedure was developed. I t was observed i n the two examples i n the l a s t  section  t h a t the most r a p i d changes i n the damage r a t i o occurred the f i r s t  s e v e r a l c y c l e s o f the i t e r a t i o n process and then the  damage r a t i o s  g r a d u a l l y approached the f i n a l v a l u e s .  r a t i o s are m o d i f i e d equation  The damage  a t the end o f each i t e r a t i o n by the use o f  (3.3) or (3.4).  I t appeared p o s s i b l e t o make over-  c o r r e c t i o n s on the damage r a t i o s vergence.  during  i n order  t o speed up the con-  I t i s easy t o keep t r a c k o f the d i f f e r e n c e between the  new damage r a t i o o f an element, and the damage r a t i o o f the same element i n the previous  iteration.  The f o l l o w i n g formula was  proposed f o r o v e r c o r r e c t i o n of damage r a t i o s .  ( y . ) ' = (y.) + a (y.) - (y.) . i n i n i n l n-1 p  where  (3.7)  (y • ) ' o v e r c o r r e c t e d damage r a t i o of i t h element i n used f o r n t h i t e r a t i o n damage r a t i o o f i t h element computed a t the end  o f n-1 t h i t e r a t i o n u s i n g equation (3.3)  or (3.4) (u-)  in-  = damage r a t i o o f i t h element used i n n-1  61 th  iteration  a = positive  constant.  What i s proposed i n equation  (3.7) i s t h a t some f r a c t i o n o f the  d i f f e r e n c e between the m o d i f i e d damage r a t i o and the p r e v i o u s damage r a t i o be added t o the m o d i f i e d damage r a t i o .  Since a  i s a p o s i t i v e constant, the o v e r c o r r e c t e d damage r a t i o i s s m a l l e r than the m o d i f i e d one when the damage r a t i o i s a l t e r e d t o have a lower value than the previous one, but o v e r c o r r e c t e d damage r a t i o s cannot be l e s s than one.  I t was found t h a t a p p l y i n g t h i s  o v e r c o r r e c t i o n from the beginning c o u l d l e a d t o an unexpected r e s u l t , because the damage r a t i o s change q u i t e r a p i d l y d u r i n g the f i r s t stage o f the i t e r a t i o n procedure.  The damage r a t i o s may  f l u c t u a t e up and down v i o l e n t l y from one i t e r a t i o n t o another. I t i s s t r o n g l y a d v i s a b l e t h a t the constant, a , be s e t t o zero during the f i r s t t i o n procedure  f i v e t o t e n i t e r a t i o n s , so t h a t the o v e r c o r r e c -  i s a p p l i e d when the damage r a t i o s change a t a  I f such a p r e c a u t i o n i s taken, the value o f a may  small r a t e .  be s e t a t as h i g h as one t o achieve f a s t e r but s t i l l  smooth  convergence. The procedure.  f o l l o w i n g example i l l u s t r a t e s the u s e f u l n e s s o f the I t a l s o shows how c l o s e l y the damage r a t i o s approach  the exact values when d i f f e r e n t l i m i t s a r e used as convergence criteria.  The second example i n the p r e v i o u s s e c t i o n was used.  A l l the r e l e v a n t i n f o r m a t i o n i s shown i n F i g . 3.6.  R e c a l l the  convergence c r i t e r i o n proposed i n the p r e v i o u s s e c t i o n .  CM..). . - CM .) l n y i (M .) yi v  v  <  e  I t was  C3.6)  where e i s a c o n s t a n t . the  e was s e t a t 10  , 10  , and 10  .  When  r e l a t i o n i n (3.6) was s a t i s f i e d f o r a l l the members, the  i t e r a t i o n procedure was stopped.  S i x runs were made i n t o t a l .  In the f i r s t three runs no o v e r c o r r e c t i o n was made and the numbers o f i t e r a t i o n s r e q u i r e d t o achieve the three convergence c r i t e r i a were recorded.  In the next three runs the same t h r e e -2  convergence c r i t e r i a were used; t h a t i s , e was s e t a t 10  -3 ,10  ,  -4 and 10  , but the o v e r c o r r e c t i o n of damage r a t i o s was  a was s e t a t 1.0 a t the end of the f i f t h  i t e r a t i o n and the number  of i t e r a t i o n s r e q u i r e d was r e c o r d e d f o r each run. are  applied,  The r e s u l t s  g i v e n i n Table 3.4. -2 When e was s e t a t 10  , i t took 29 i t e r a t i o n s  to s a t i s f y  t h i s c r i t e r i o n without o v e r c o r r e c t i o n of damage r a t i o s .  When the  damage r a t i o s were o v e r c o r r e c t e d , the number of i t e r a t i o n s -3 reduced t o 18 f o r a saving of 11 i t e r a t i o n s .  was  At £ = 10  158 i t e r a t i o n s were r e q u i r e d without o v e r c o r r e c t i o n t e c h n i q u e . With i t , the number was reduced t o only 81 f o r a s a v i n g of 77 -4 iterations.  At e = 10  the convergence c r i t e r i o n was not met  a f t e r 200 i t e r a t i o n s when o v e r c o r r e c t i o n s were not made, but i t was met a f t e r 12 4 i t e r a t i o n s when they were made.  Clearly  technique a c c e l e r a t e d the convergence o f the damage r a t i o s .  this The  number of i t e r a t i o n s was reduced by one t h i r d t o almost one h a l f . The saving i n computation i s s u b s t a n t i a l when the convergence i s slow i n a case such as the example used here.  The g a i n i s not so  s i g n i f i c a n t when the convergence i s f a s t , as i t i s i n the f i r s t example i n the p r e v i o u s s e c t i o n .  Since i t i s i m p o s s i b l e t o p r e -  d i c t the r a t e of convergence beforehand, t h i s technique should be used a l l the time.  On r a r e o c c a s i o n s the method produced bad  r e s u l t s i n which the damage r a t i o s o s c i l l a t e d .  In order to a v o i d  t h i s p o s s i b i l i t y , a may be s e t a t a constant l e s s than one o r the a p p l i c a t i o n o f the technique may be delayed u n t i l more than 10 i t e r a t i o n s are completed. Table 3.5 shows how c l o s e l y the damage r a t i o s o f a l l the members approached the exact value when d i f f e r e n t e v a l u e s were specified.  O v e r c o r r e c t i o n s were made i n a l l cases.  The same  -2 a p p l i e s t o the n a t u r a l p e r i o d s .  When an e o f 10  was  reached,  some o f the damage r a t i o s were s t i l l q u i t e f a r from the exact ones; the t h i r d - s t o r y columns and beams f a l l  i n this  category.  The n a t u r a l p e r i o d f o r the t h i r d mode d i f f e r s the most from the exact one but the d i f f e r e n c e i s l e s s than three p e r c e n t .  At  -3 e = 10 almost a l l the damage r a t i o s are very c l o s e t o the exact v a l u e s . The n a t u r a l p e r i o d s are even c l o s e r t o the exact -4 values than the damage r a t i o s . A t e = 10 both the damage r a t i o s -2 and the n a t u r a l p e r i o d s a r e p r a c t i c a l l y exact. e s e t a t 10 is -3 probably too c o a r s e . e should be s e t a t somewhere between 10 -2 -3 and 10 . I t was found from other runs t h a t e s e t at 10 produced s a t i s f a c t o r y r e s u l t s . However, i f the m o d i f i e d s u b s t i t u t e s t r u c t u r e method i s used t o o b t a i n a rough estimate e may be s e t -2 at  a value a l i t t l e  t h a t i s warranted  s m a l l e r than 10  ; and t h i s g e n e r a l l y i s a l l  i n practice.  I t may be p o s s i b l e t o i n c o r p o r a t e the o v e r c o r r e c t i o n o f damage r a t i o s i n t o the formula f o r modifying the damage r a t i o s at  the end o f the i t e r a t i o n .  When equations  (3.3) and (3.4) were  d e r i v e d , i t was assumed t h a t the same r o t a t i o n would be o b t a i n e d in  the next i t e r a t i o n .  The s u b s t i t u t e s t i f f n e s s and hence the  damage r a t i o was i n c r e a s e d o r decreased a c c o r d i n g l y t o s a t i s f y  t h i s assumption.  But t h i s assumption i s not a b s o l u t e l y n e c e s s a r y .  Another assumption i s p o s s i b l e and w i t h such an assumption a new formula may be d e r i v e d t o achieve f a s t e r convergence.  Further  study i s p o s s i b l e i n t h i s area. As a f i n a l remark i n t h i s chapter i t i s worth n o t i n g the  two examples  that  i n the p r e v i o u s s e c t i o n , even though they were  i d e n t i c a l frames, except f o r the y i e l d moments, l a y on the two extreme s i d e s as f a r as the r a t e o f convergence was concerned. I t was very r a r e t h a t the damage r a t i o s o f a s t r u c t u r e converged at  a f a s t e r r a t e than they d i d i n the f i r s t example,  slower r a t e than i n the second example.  or at a  Even when the s i z e o f a  s t r u c t u r e i n the f i r s t example was c o n s i d e r a b l y g r e a t e r than the s t r u c t u r e i n the second example, to  fewer i t e r a t i o n s were r e q u i r e d  s a t i s f y the same convergence c r i t e r i o n .  In g e n e r a l ,  less  than 20 i t e r a t i o n s a r e necessary t o o b t a i n a good e s t i m a t e on damage r a t i o s f o r most o f the s t r u c t u r e s i n p r a c t i c e .  65  CHAPTER 4  4.1  EXAMPLES  Assumptions and Comments  The goal of the m o d i f i e d s u b s t i t u t e s t r u c t u r e  analysis i s  to p r e d i c t the behaviour o f an e x i s t i n g r e i n f o r c e d concrete t u r e under a g i v e n earthquake motion. f i n d out whether the method f u l f i l l s  T e s t s must be performed t o this intent.  impossible to do an a c t u a l experiment. analytically.  struc-  I t i s almost  The t e s t must be done  Among many a n a l y t i c a l methods, a n o n l i n e a r  dynamic  a n a l y s i s produces the most accurate p r e d i c t i o n o f the behaviour o f a s t r u c t u r e which i s subjected therefore,  t o an earthquake motion.  e s s e n t i a l t h a t the m o d i f i e d s u b s t i t u t e  It i s ,  s t r u c t u r e method  produce a r e s u l t which i s comparable t o t h a t o b t a i n e d from the nonlinear  dynamic  analysis.  A s e r i e s o f t e s t frames were analyzed by the m o d i f i e d subs t i t u t e s t r u c t u r e method. analyses using  The same frames were a l s o subjected t o  the n o n l i n e a r  dynamic a n a l y s i s program.  r e s u l t s from the two analyses were compared.  The  The e x t e n t of damage  represented by damage r a t i o s , l o c a t i o n s o f damage and the d i s placements are the q u a n t i t i e s o f i n t e r e s t . are d e s c r i b e d  i n d e t a i l , a l l the r e l e v a n t  t i o n s w i l l be d i s c u s s e d  Before the r e s u l t s information  and assump-  i n this section.  A t o t a l o f four frames were t e s t e d .  They were not modeled  from a c t u a l e x i s t i n g b u i l d i n g s , but sent  they were intended to  repre-  s m a l l - to medium-sized r e i n f o r c e d c o n c r e t e s t r u c t u r e s .  t e s t on a l a r g e r s t r u c t u r e was l i m i t a t i o n s of the n o n l i n e a r cost involved  not p o s s i b l e mainly due  to  dynamic a n a l y s i s program.  i n the a n a l y s i s was  the  The  another reason to l i m i t  high the  s i z e of a t e s t frame.  In order to s a t i s f y the second  listed  they were a l l r e g u l a r frames w i t h no  i n s e c t i o n 3.2,  change i n geometry.  The  condition abrupt  dimensions o f a frame were determined  that they would represent Member s i z e s and  A  so  an a c t u a l b u i l d i n g of comparable s i z e .  p r o p e r t i e s were chosen somewhat a r b i t r a r i l y  are not n e c e s s a r i l y completely r e a l i s t i c .  and  Since the method would  be used i n p r a c t i c e f o r a n a l y s i s of b u i l d i n g s t h a t may  not have  been designed to r e s i s t earthquakes, the member p r o p e r t i e s were d e l i b e r a t e l y chosen i n an a r b i t r a r y f a s h i o n . the m o d i f i e d  f e l t that i f  s u b s t i t u t e s t r u c t u r e method worked f o r these t e s t  frames, i t would work f o r more r e a l i s t i c t e s t was  I t was  a n a l y t i c a l , there was  structures.  Since  the  no r e s t r i c t i o n on the choice  of  these parameters. The  f o l l o w i n g assumptions were made i n the modeling of  frames f o r use with the m o d i f i e d Beams and  columns were modeled as l i n e members.  deformations were ignored. be  f i x e d a t ground l e v e l .  The  P-A  modeled as a p o i n t .  ends of a member were taken to  e f f e c t i n the columns were not  Upon running a program o v e r c o r r e c t i o n a p p l i e d a f t e r the f i r s t the equation  (3.7)  was  Their a x i a l  bottom columns were assumed to  A j o i n t was  Moment c a p a c i t i e s a t the two equal.  s u b s t i t u t e s t r u c t u r e program.  included. o f damage r a t i o s  ten c y c l e s of i t e r a t i o n was s e t a t 0.95.  be  Equation  (3.6)  over. was  was  a in  used  as  a convergence c r i t e r i o n and e was  s e t a t 10  .  Iteration  stopped as soon as t h i s convergence c r i t e r i o n was A n o n l i n e a r dynamic a n a l y s i s program was  satisfied.  f o r frames, SAKE,  used t o compute the response h i s t o r y o f each frame.  ness a f t e r y i e l d was  was  taken as 2% o f the i n i t i a l  12  The  stiffness.  stiffThe  a n a l y s i s was made w i t h v i s c o u s damping p r o p o r t i o n a l to s t i f f n e s s , corresponding t o a damping r a t i o o f 2% f o r the f i r s t mode. J o i n t s were modeled as i n f i n i t e l y r i g i d beams, w i t h s i z e s p r o p o r t i o n e d a c c o r d i n g t o the member s i z e s . to  1/50  A time step c o r r e s p o n d i n g to  of the s m a l l e s t p e r i o d was  1/30  used f o r n u m e r i c a l i n t e r a t i o n .  Response c a l c u l a t i o n s were done a t every f i v e to ten time steps. Choosing a proper response spectrum i s beyond the scope of 5 this thesis.  The d e s i g n spectrum A i n Shibata and Sozen's paper  was used f o r the m o d i f i e d s u b s t i t u t e s t r u c t u r e a n a l y s i s . mentioned i n s e c t i o n 2.3, six  earthquake motions  i t was  As  d e r i v e d from response s p e c t r a of  ( F i g . 2.5).  Equation (2.26) was  used to  compute the response a c c e l e r a t i o n when the damping r a t i o was f e r e n t from 2%. of  The maxium ground a c c e l e r a t i o n was  0.5 g.  difFour  the s i x earthquake r e c o r d s , from which the d e s i g n spectrum  was  made, were used t o compute the response h i s t o r i e s .  They were E l  Centro EW,  Each r e c o r d  E l Centro NS,  T a f t S69E, and T a f t N21E.  was normalized to g i v e a peak ground a c c e l e r a t i o n of 0.5 g. d u r a t i o n o f each earthquake r e c o r d was  chosen such t h a t each  frame r e c e i v e d the maximum damage d u r i n g t h a t d u r a t i o n . otherwise noted, the f i r s t were used f o r computation.  The  Unless  15 seconds of each earthquake r e c o r d CPU time f o r running the two programs  i s given t o i l l u s t r a t e the d i f f e r e n c e i n c o s t , but i t should be  68 noted t h a t the c o s t f o r storage was much h i g h e r f o r the n o n l i n e a r dynamic a n a l y s i s program, because i t r e q u i r e d more memory. The damage r a t i o s and displacements son o f the two a n a l y s e s .  Since the design spectrum was the  average spectrum o f the s i x earthquakes, ified  were used f o r compari-  the r e s u l t s o f the mod-  s u b s t i t u t e s t r u c t u r e a n a l y s i s should be viewed as  of the f o u r n o n l i n e a r dynamic a n a l y s e s .  "average"  69 4.2  Examples  (a)  2-Bay, 2-Story Frame The two-bay, two-story frame of F i g . 4.1 was  frame. was  The widths o f both bays were 30 f e e t .  used as a t e s t  The ground  12 f e e t i n h e i g h t and the second s t o r y was  story  10 f e e t high.  f l o o r weights f o r the f i r s t and second s t o r y were 120 k i p s 1Q0  kips respectively.  i n t e r i o r columns.  and  The e x t e r i o r columns were b i g g e r than the  T h e i r cracked transformed moments of i n e r t i a  were taken as approximately one-half of the gross s e c t i o n . moments o f i n e r t i a f o r beams were about o n e - t h i r d section.  An e l a s t i c a n a l y s i s was  periods.  As shown i n Table 4.1,  were 0.50  sec. and 0.13  period  structure.  The  The  of the gross  run to compute the  natural  the p e r i o d s f o r the two modes  sec. r e s p e c t i v e l y , r e p r e s e n t i n g  a  short  The y i e l d moments were assigned randomly  t h a t each member was  expected to r e c e i v e  such  a d i f f e r e n t amount of  i n e l a s t i c deformation. In the m o d i f i e d s u b s t i t u t e s t r u c t u r e a n a l y s i s i t took i t e r a t i o n s t o s a t i s f y the convergence on the Amdahl V/6-II computer was of the s u b s t i t u t e sec.  0.91  sec.  The CPU  time  The n a t u r a l p e r i o d s  frame computed i n the l a s t i t e r a t i o n were  f o r the f i r s t mode and 0.18  Table 4.1).  criterion.  24  0.76  sec. f o r the second mode (See  The f l o o r displacements were computed as the  root-  sum-square of the modal displacements and are shown i n Table The displacement of the f i r s t second f l o o r was  3.8  in.  q u i t e random as expected columns y i e l d e d .  f l o o r was  1.8  i n . and t h a t of the  The d i s t r i b u t i o n of damage r a t i o s (See F i g . 4.2).  4.2.  A l l the  was  first-story  The damage r a t i o s f o r those columns were  4.2,  70 2.6,  and 1.4 r e s p e c t i v e l y .  y i e l d v e r y much.  The second-story columns d i d not  One o f the e x t e r i o r  A l l the four beams y i e l d e d .  columns remained e l a s t i c .  The f i r s t - f l o o r beam i n the l e f t bay  had the b i g g e s t damage r a t i o a t 4.8. Response h i s t o r i e s o f the t e s t frame t o f o u r earthquake 12 motions were computed by the computer program, SAKE.  The f i r s t  15 seconds o f earthquake r e c o r d s were used f o r response computation.  0.003 sec. was chosen as the time step f o r numerical i n t e -  gration. sec.  CPU time was 12.9 s e c . f o r E l Centro EW motion, 12.2  f o r E l Centro NS, 11.8 s e c . f o r T a f t S69E, and 11.4 sec. f o r  T a f t N21E.  R e s u l t s o f the n o n l i n e a r a n a l y s e s are shown i n F i g .  4.3 and Table 4.2. resulted  i n more damage t o the t e s t frame than the two components  of T a f t earthquake. 4.3.  The two components o f the E l Centro earthquake  The displacements and damage r a t i o s i n F i g .  and Table 4.2 were the maximum v a l u e s r e c o r d e d i n the r e -  sponse h i s t o r i e s .  The displacement o f the f i r s t - f l o o r  1.3 i n . to 2.8 i n . f o r d i f f e r e n t motions. placement was 2.1 i n .  The mean maximum d i s -  The second f l o o r displacement ranged from  2.7 i n . t o 5.3 i n . w i t h a mean o f 4.2 i n . Fig.  ranged from  The damage r a t i o s i n  4.3 correspond to the b i g g e r o f the two damage r a t i o s f o r  each member. In the E l Centro EW motion a l l o f the f i r s t - s t o r y columns suffered 9.6.  e x t e n s i v e damage w i t h damage r a t i o s r a n g i n g from 3.3 t o  On the o t h e r hand, none o f the columns on the second  yielded.  story  A recorded damage r a t i o l e s s than one i n F i g . 4.3 im-  p l i e s t h a t the maximum computed moment was t h a t y i e l d moment.  The l e f t e x t e r i o r  f r a c t i o n o f the  column had the l e a s t damage. A l l  four beams y i e l d e d w i t h t h e i r damage r a t i o s r a n g i n g from 2.8 t o  71 6.4.  In the E l Centro NS motion  the f i r s t - s t o r y columns s u f f e r e d  approximately the same amount o f damage as i n the p r e v i o u s case. The damage r a t i o s f o r the beams were a l s o approximately the same as those i n the E l Centro EW motion.  Two o f the second-story  columns, however, y i e l d e d w i t h damage r a t i o s o f 1.3. The T a f t S69E motion produced two p r e v i o u s cases the f i r s t  the l e a s t damage.  As i n the  s t o r y columns y i e l d e d , b u t the  damage r a t i o s were roughly a h a l f o f those with E l Centro. same a p p l i e s t o the beam damage r a t i o s . second s t o r y remained  The  The t h r e e columns on the  e l a s t i c , but the maximum moments were com-  p a r a b l e t o those found i n the E l Centro EW motion.  The T a f t N21E  motion was more severe, but i t was not s t r o n g enough f o r the second-story columns t o y i e l d .  The damage r a t i o s f o r the other  columns ranged from 2.0 t o 6.5 and those f o r the beams from 2.0 to  4.6.  The members which remained  e l a s t i c reached roughly the  same maximum moments i n a l l four motions, but those which y i e l d e d s u f f e r e d d i f f e r e n t amounts o f damage i n each When the average  motion.  f l o o r displacements from the n o n l i n e a r  dynamic analyses a r e compared w i t h those from the m o d i f i e d subs t i t u t e s t r u c t u r e a n a l y s i s as i n Table 4.2, i t i s found t h a t the l a t t e r p r e d i c t e d s m a l l e r displacements i n both s t o r i e s .  The d i f -  ference was g r e a t e r f o r the f i r s t - f l o o r displacement which was about 20% o f f than f o r the s e c o n d - f l o o r displacement which was about 10% o f f . Fig. of  N e v e r t h e l e s s the estimate was reasonable.  4.2 shows the comparison  o f the average damage r a t i o s  the four motions w i t h the p r e d i c t e d v a l u e s .  sense the m o d i f i e d s u b s t i t u t e s t r u c t u r e a n a l y s i s p r e d i c t e d t h a t the columns on the f i r s t  In a q u a l i t a t i v e correctly  s t o r y would y i e l d and  t h a t the extent column and  of damage would be g r e a t e s t f o r the l e f t e x t e r i o r  l e a s t f o r the r i g h t e x t e r i o r column.  damage r a t i o s were about 60% f o u r motions.  The  But  the  predicted  of the average damage r a t i o s of  p r e d i c t i o n f o r the second-story columns  good except f o r the r i g h t e x t e r i o r column.  The m o d i f i e d  the  was  substi-  t u t e s t r u c t u r e method p r e d i c t e d t h a t t h i s column would y i e l d s l i g h t l y , but  i t d i d not happen.  moment c a p a c i t y . was  q u i t e good.  The  nonlinear  of i t s  p r e d i c t i o n of damage r a t i o s f o r the beams  Although they were s l i g h t l y underestimated, they  were a l l w i t h i n 20% one  I t only reached 60%  of the average v a l u e s .  a n a l y s i s was  The  c o s t of running  about 13 times t h a t of the  modified  s u b s t i t u t e s t r u c t u r e a n a l y s i s i n t h i s example.  (b)  3-bay, 3-story The  example.  Frame  three-bay, t h r e e - s t o r y  Data i s shown i n F i g . 4.4.  f e e t f o r the e x t e r i o r bays and f i r s t s t o r y was  As  f l o o r , 200  second, and  180  k i p s f o r the k i p s f o r the than  floors.  i n e r t i a l of the gross s e c t i o n was formed s e c t i o n .  The  third. inter-  columns  the same dimension.  t h i r d - s t o r y columns were made s m a l l e r than the o t h e r s . s i z e s were reduced at higher  One  The  third stories  In each group o f columns the f i r s t - s t o r y  the second-story columns were given  30  i n t e r i o r bay.  the second and  f l o o r weights were 240  k i p s f o r the  second  The width of bays was  i n the l a s t example, e x t e r i o r columns were b i g g e r  i o r columns. and  The  t e s t e d i n the  2 0 f e e t f o r the  15 f e e t high and  were 12 f e e t h i g h . first  frame was  The  The beam  h a l f o f the moment of  used f o r the cracked  trans-  r i g h t e x t e r i o r column on the second s t o r y  much s m a l l e r moment of i n e r t i a than i t s c o u n t e r p a r t .  had  The moment  73 of  i n e r t i a o f each beam was taken as one t h i r d o f t h a t o f the  gross s e c t i o n .  The y i e l d moments o f the columns were s e t a t h i g h  v a l u e s , e s p e c i a l l y i n the f i r s t s t o r y , so t h a t the columns would not y i e l d too much.  The beam y i e l d moments were s m a l l e r i n the  l e f t bay than i n the other bays. An e l a s t i c a n a l y s i s was performed p r i o r t o the t e s t t o compute the n a t u r a l p e r i o d s o f the e l a s t i c frame.  As shown i n  Table 4.3, they were 0.94 s e c , 0.30 s e c , and 0.14 s e c .  These  p e r i o d s were much longer than those i n the p r e v i o u s example. The  r e s u l t s o f the m o d i f i e d s u b s t i t u t e s t r u c t u r e a n a l y s i s  are shown i n F i g . 4.5, Table 4.3, and Table 4.4. t i o n s were necessary  to s a t i s f y the convergence c r i t e r i o n .  took 0.92 s e c . o f CPU time t o do a l l the necessary The  Only 14 i t e r a It  computation.  three n a t u r a l p e r i o d s o f the s u b s t i t u t e frame were 1.22 s e c ,  0.36 s e c , and 0.16 s e c . The i n c r e a s e i n n a t u r a l p e r i o d s  over  those from the e l a s t i c a n a l y s i s was l e s s f o r t h i s frame than the p r e v i o u s frame.  The h o r i z o n t a l displacements  were 2.2 i n . f o r the f i r s t , for  o f the t h r e e  floors  5.0 i n . f o r the second, and 8.0 i n .  the t h i r d , i n d i c a t i n g a f a i r l y  uniform p a t t e r n o f d i s p l a c e -  ments (See Table 4.4). P r e d i c t e d damage r a t i o s a r e shown i n Fig.  4.5.  A damage r a t i o l e s s than one i s e q u i v a l e n t t o the  r a t i o o f the computed moment t o the y i e l d moment.  I f the two  end moments were d i f f e r e n t i n a member, the b i g g e r o f the two was used. All  the columns on the f i r s t  s t o r y had damage r a t i o s of  1.1, i n d i c a t i n g t h a t t h e i r y i e l d moment c a p a c i t i e s were exceeded.  slightly  One o f the columns on the second s t o r y y i e l d e d t o a  damage r a t i o o f 1.5, but the other three remained i n the e l a s t i c  range.  The moment c a p a c i t i e s of the t h i r d - s t o r y columns were  f u l l y u t i l i z e d , as t h e i r damage r a t i o s were almost 1.0 e x a c t l y . Two beams i n the r i g h t e x t e r i o r bay remained e s s e n t i a l l y  elastic;  others had damage r a t i o s r a n g i n g from 2.0 t o 5.2. N o n l i n e a r dynamic a n a l y s e s were run t o compute the response h i s t o r i e s o f the frame i n the four earthquake motions. 15 seconds o f the r e c o r d s were used.  The f i r s t  Since t h i s was a b i g g e r  frame than the p r e v i o u s one, a c o n s i d e r a b l y l o n g e r time on the Amdahl V/6-II computer was r e q u i r e d f o r computation.  The average  CPU time of one run was about 28 seconds, double the time r e q u i r e d i n the p r e v i o u s example.  A time increment o f 0.003 sec.  was s e l e c t e d f o r numerical i n t e g r a t i o n . The r e s u l t s o f f o u r runs a r e shown i n F i g . 4.6. ments are shown i n Table 4.4. from one earthquake t o another. the  They e x h i b i t e d a l a r g e  Displacevariation  E l Centro EW component produced  b i g g e s t displacements, twice as b i g as those i n T a f t N21E  component.  E l Centro NS produced the second b i g g e s t  ment and T a f t S69E motion f o l l o w e d . were 2.2 i n . f o r the f i r s t 7.5 i n . f o r the t h i r d .  displace-  The average displacements  f l o o r , 4.7 i n . f o r the second, and  The t h i r d - f l o o r displacement, f o r example,  ranged from 5.2 i n . i n T a f t N31E t o 10.6 i n . i n E l Centro EW. The same t r e n d was found i n damage r a t i o s . The damage r a t i o s were the h i g h e s t i n the E l Centro EW motion.  A l l the columns on the f i r s t  around 1.8.  s t o r y had damage r a t i o s  The r i g h t i n t e r i o r columns on the second and t h i r d  s t o r i e s y i e l d e d as w e l l , but the r e s t o f the columns remained elastic.  The two e x t e r i o r columns on the t h i r d s t o r y had the  lowest computed moments.  A l l the beams y i e l d e d w i t h damage  75 r a t i o s r a n g i n g from 1.5 t o 6.4. experienced the l e a s t damage. in  The beams i n the r i g h t bay Inelastic  deformations o c c u r r e d  the same beams and columns i n the E l Centro NS motion.  The  damage r a t i o s o f these members, however, were lower i n t h i s motion than i n EW motion.  The moment c a p a c i t i e s  the two columns on the second s t o r y , In the T a f t  were reached i n  but they d i d not y i e l d .  S69E motion o n l y one column underwent i n e l a s t i c  deformation, the r i g h t i n t e r i o r column on the second s t o r y , damage r a t i o o f 1.2.  A l l the columns on the f i r s t  story  with,  and two  on the t h i r d s t o r y had computed moments equal t o o r a l i t t l e than t h e i r r e s p e c t i v e y i e l d moments.  less  Two o f the beams d i d not  y i e l d , although t h e i r damage r a t i o s were almost one.  Damage  r a t i o s o f the o t h e r beams ranged from 1.8 to 4.6 which were much lower than the values found i n the E l Centro EW motion. damage r a t i o s were the lowest i n the T a f t N21E motion. the columns and two beams remained  The A l l of  e l a s t i c and those which  yielded  had damage r a t i o s r a n g i n g from 1.5 t o 4.0. Average damage r a t i o s a r e shown i n F i g . 4.5.  When these  v a l u e s are compared w i t h those p r e d i c t e d by the m o d i f i e d substitute structure analysis, prediction  t h e r e i s a remarkable  o f beam damage r a t i o s  i s excellent  Even the worst one was o f f by o n l y 15%.  agreement.  without e x c e p t i o n .  The p r e d i c t i o n  damage r a t i o s was a l i t t l e worse than f o r the beams. damage r a t i o s o f the e x t e r i o r  modified substitute At  o f column  Only the  columns on the t h i r d s t o r y were  s l i g h t l y o f f , but o t h e r s were i n good agreement. displacements a l s o  The  The average  agreed very w e l l w i t h those p r e d i c t e d by the structure analysis,  as shown i n Table 4.4.  l e a s t f o r t h i s example i t i s safe t o t o say t h a t the m o d i f i e d  76 substitute structure the n o n l i n e a r  analysis correctly predicted  dynamic a n a l y s i s .  This  the r e s u l t s of  i s a remarkable achievement  when the d i f f e r e n c e i n CPU time i s concerned.  (c)  1-Bay, 6-Story Frame Fig.  4.7 shows the data f o r the one-bay, s i x - s t o r y frame  t h a t was used as the t h i r d t e s t frame.  The width o f the frame  was 35 f e e t , and the s t o r y h e i g h t was constant a t 13 f e e t f o r an o v e r a l l h e i g h t of 78 f e e t . to the f i f t h  s t o r y a t 100 k i p s , but a t the top s t o r y i t was  reduced t o 90 k i p s . second s t o r y . a smaller  The f l o o r weight was constant up  The column s i z e s were decreased a t every  Beam s i z e was constant up t o the f i f t h  beam b e i n g used at the top f l o o r .  f l o o r , with  The moment of i n e r t i a  o f a l l the members were taken as approximately o n e - h a l f of the v a l u e s based on gross s e c t i o n . were reduced p r o g r e s s i v e l y  The y i e l d moments o f the columns  up the h e i g h t o f the frame.  The y i e l d  moments o f the beams were l a r g e , except a t the top, compared t o those of the columns.  I t was hoped t h a t columns would r e c e i v e a  f a i r amount of damage. E l a s t i c p e r i o d s were computed f o r a l l s i x modes and the v a l u e s are shown i n Table 4.5.  The n a t u r a l p e r i o d s f o r the f i r s t  two modes were 1.1 sec. and 0.37 sec. r e s p e c t i v e l y .  The  smallest  p e r i o d was 0.0 8 sec. The in  m o d i f i e d s u b s t i t u t e s t r u c t u r e a n a l y s i s was c a r r i e d out  the u s u a l manner.  I t was necessary t o perform 9 6 i t e r a t i o n s  to achieve the convergence.  CPU time was 2.3 s e c .  The n a t u r a l  p e r i o d s o f the s u b s t i t u t e frame, as shown i n Table 4.5, were considerably  longer than the p e r i o d s of the a c t u a l frame.  The  p e r i o d f o r the f i r s t mode was 1.85 sec. and the second mode p e r i o d was 0.84 sec. U s u a l l y the n a t u r a l p e r i o d f o r the h i g h e s t mode o f the s u b s t i t u t e s t r u c t u r e i s not much d i f f e r e n t from t h a t o f the a c t u a l frame, but they were q u i t e d i f f e r e n t i n t h i s example. former was 0.13 sec. and the l a t t e r was 0.0 8 sec. ment p a t t e r n was a l s o q u i t e unique  The  The d i s p l a c e -  (See Table 4.6). The d i s p l a c e -  ment o f the second f l o o r was much g r e a t e r  than the f i r s t  floor.  There was a b i g d i f f e r e n c e between the f o u r t h - f l o o r displacement and  the f i f t h - f l o o r The  displacement.  damage r a t i o s a r e shown i n F i g . 4.8.  The a n a l y s i s  pre-  d i c t e d t h a t the damage r a t i o s would vary w i d e l y among the members. The  column i n the f i r s t  respectively.  three  s t o r i e s were 2.5, 6.6, and 2.9  The l a r g e damage r a t i o f o r the second-story  column i s the reason f o r the b i g jump i n displacement between the second and t h i r d f l o o r .  Two columns, the one i n the f o u r t h  s t o r y and the one i n the s i x t h s t o r y , d i d not y i e l d .  A large  i n e l a s t i c deformation was p r e d i c t e d i n the f i f t h - s t o r y column w i t h a damage r a t i o o f 16.6.  The f i r s t - f l o o r beam had a damage  r a t i o o f 1.5, i n d i c a t i n g a small amount of i n e l a s t i c The  beams on the next three  f l o o r s d i d not y i e l d .  the f i f t h and s i x t h f l o o r s were given and  deformation.  The beams on  l a r g e damage r a t i o s o f 9.5  6.8 r e s p e c t i v e l y . Response h i s t o r i e s of the frame were computed by the non-  l i n e a r dynamic a n a l y s i s program, u s i n g the f i r s t f o u r earthquake r e c o r d s .  The time increment f o r numerical  g r a t i o n was s e t a t 0.004 sec. The Fig.  20 s e c . of the inte-  The average CPU time was 42.6 s e c .  damage r a t i o s f o r i n d i v i d u a l earthquake motions a r e shown i n 4.9.  E l Centro EW motion produced by f a r the worst  result.  78 The  damage due t o other motions were s i m i l a r t o each o t h e r i n  magnitude. In  E l Centro EW motion a l l the members except the t o p - s t o r y  column s u f f e r e d severe damage and the f l o o r displacements were l a r g e , as shown i n Table 4,6. the f i r s t story.  f i v e s t o r i e s ranged  Damage r a t i o s o f the columns i n from 6.3 t o 14.4 i n the f i r s t  The t h i r d - s t o r y column was a l s o damaged badly w i t h a  damage r a t i o o f over 10.  A l l the beams experienced a l a r g e  amount o f i n e l a s t i c deformation, w i t h damage r a t i o s ranging  from  6.3 to 10.8, w i t h the h i g h e s t v a l u e i n the f i f t h - f l o o r beam. El  Centro NS motion a l l s i x columns y i e l d e d .  r a t i o was 1.3 and the h i g h e s t was 5.2. f o u r t h and f i f t h All  In  The s m a l l e s t damage  The columns on the t h i r d ,  s t o r i e s were damaged more than the o t h e r t h r e e .  the beams a l s o y i e l d e d .  The damage r a t i o s i n c r e a s e d up the  h e i g h t of the b u i l d i n g except a t the f i f t h  f l o o r where the damage  r a t i o o f the beam was the h i g h e s t a t 8.1.  The displacements were  s m a l l compared t o those found i n E l Centro EW motion. placement o f the f i r s t  f l o o r was p a r t i c u l a r l y  The d i s -  s m a l l (See Table  4.6). In  T a f t S69E motion every member o f the frame y i e l d e d .  Among the columns those i n the f i r s t  four s t o r i e s r e c e i v e d the  most damage, with damage r a t i o s about s i x .  The damage r a t i o s o f  the beams on the f i r s t three f l o o r s were approximately at  about 3.5.  the same  The other three were damaged t o a g r e a t e r extent.  The damage r a t i o o f the f i f t h - f l o o r beam was the h i g h e s t a t 9.5, w h i l e the other two beams had damage r a t i o s o f about seven. i n c r e a s e i n displacements was q u i t e uniform i n the f i r s t floors.  The  four  T a f t N21E motion r e s u l t e d i n q u i t e a d i f f e r e n t p a t t e r n  79 of  damage r a t i o s .  Most of the damage i n the columns was  concen-  t r a t e d i n the second-story column and the t h i r d - s t o r y column w i t h damage r a t i o s of 7.9  and  6.6  respectively.  the other f o u r columns were s m a l l . c o n c e n t r a t e d i n the f i r s t were 5.5  and  3.2.  two  The  The  damage r a t i o s o f  damage i n the beams  was  f l o o r beams, where the damage r a t i o s  The other f o u r beams escaped with minor damage.  The displacements  above the t h i r d f l o o r d i d not i n c r e a s e s i g n i -  ficantly. The average displacements damage r a t i o s i n F i g . 4.8.  are shown i n Table 4.6  and  average  They were very d i f f e r e n t from the  f i g u r e s computed by the m o d i f i e d s u b s t i t u t e s t r u c t u r e a n a l y s i s . The  displacement  p a t t e r n s were q u i t e d i f f e r e n t .  The  prediction  by the m o d i f i e d s u b s t i t u t e s t r u c t u r e a n a l y s i s r e s u l t e d i n an underestimate  of the displacements  p r e d i c t i o n of damage r a t i o s was  of the f i r s t  a l s o poor.  four f l o o r s .  The damage was  c o n c e n t r a t e d i n a p a r t i c u l a r column or a beam, but was over the whole s t r u c t u r e .  The  not  spread  The m o d i f i e d s u b s t i t u t e s t r u c t u r e  method f a i l e d i n t h i s t e s t frame.  (d)  3-Bay, 6-Story Frame The  3-bay, 6-story frame shown i n F i g . 4.10  f o u r t h t e s t frame.  Each bay was  constant a t 11 f t .  A weight of 200  story.  2 4 f t . wide and k i p s was  was  story height  was  c o n c e n t r a t e d a t each  Members s i z e s were uniform along the h e i g h t .  24 i n . by 24 i n . f o r columns and 18 i n . by  used as the  They were  30 i n . f o r beams.  h a l f of the moment o f i n e r t i a of the gross s e c t i o n was compute the i n i t i a l s t i f f n e s s of columns, and one  One  used to  t h i r d f o r beams.  In t h i s example a l l the columns were intended to remain e l a s t i c .  80 For t h i s purpose the s u b s t i t u t e s t r u c t u r e method was used t o compute design moments.  These moments were used as a guide t o  e s t a b l i s h the y i e l d moments. The p e r i o d s are summarized i n Table 4.7.  The computed  p e r i o d s f o r the f i r s t two modes were 1.1 s e c . and 0.34 sec. r e s p e c t i v e l y , w h i l e the p e r i o d f o r the h i g h e s t mode was 0.0 75 sec.  These e l a s t i c p e r i o d s were comparable t o those of the l a s t  t e s t frame. The m o d i f i e d s u b s t i t u t e s t r u c t u r e a n a l y s i s was c a r r i e d out i n the u s u a l manner; 16 i t e r a t i o n s were r e q u i r e d t o s a t i s f y the convergence c r i t e r i o n . V/6-II computer. the f i r s t  The CPU time was 2.30 sec. on the Amdahl  As shown i n Table 4.7, the n a t u r a l p e r i o d s f o r  two modes were 1.66 sec. and 0.48 sec. r e s p e c t i v e l y ,  w h i l e the s h o r t e s t p e r i o d was 0.0 76 s e c . of  The f i r s t  two p e r i o d s  the s u b s t i t u t e frame were much longer than the c o r r e s p o n d i n g  p e r i o d s of the e l a s t i c frame, but the other p e r i o d s were r e l a t i v e l y unchanged.  As f a r as the displacements,  4.8, were concerned,  the s e c o n d - f l o o r displacement was q u i t e  l a r g e compared t o the f i r s t - f l o o r displacement. displacement  shown i n Table  from the f i f t h  The i n c r e a s e i n  f l o o r to the s i x t h f l o o r was s m a l l .  The r e l a t i v e displacement was q u i t e uniform f o r the o t h e r  floors,  the top d e f l e c t i o n b e i n g 8.8 i n . Damage r a t i o s are shown i n F i g . 4.11.  Those of the columns  were roughly constant a t around 0.8; t h a t i s , the computed moments o f a l l the columns were about 80% o f the y i e l d moments. All  the y i e l d i n g took p l a c e i n the beams.  The beams i n the  e x t e r i o r bays had higher damage r a t i o s than those i n the i n t e r i o r bay.  In both bays the bottom beams had the h i g h e s t damage r a t i o s .  81 They decreased a t an i n c r e a s i n g r a t e w i t h h e i g h t i n the frame. For  the beams i n the e x t e r i o r bays the damage r a t i o s ranged  3.5 t o 4.5.  from  F o r those i n the i n t e r i o r bay they ranged from 2.0  to 2.7. Response h i s t o r i e s of the frame t o the four motions were computed. i n each run. sec.  The f i r s t  earthquake  15 sec. of r e c o r d s were used  Numerical i n t e g r a t i o n was performed every 0.002  and the response c a l c u l a t i o n was done a f t e r every f i v e  steps.  time  Each n o n l i n e a r dynamic a n a l y s i s was expensive, as i t  r e q u i r e d , on the average,  120 sec. o f CPU time.  A summary o f  r e s u l t s i s shown i n F i g . 4.12. In E l Centro EW motion yielded.  t h r e e o f the e x t e r i o r columns  They were the third.-, f o u r t h - and f i f t h - s t o r y columns  and t h e i r damage r a t i o s were about  1.5.  None o f the i n t e r i o r  columns y i e l d e d , but the maximum moments o f the t h r e e columns were equal t o or j u s t below the y i e l d moments. y i e l d e d t o some e x t e n t . h i g h e s t damage r a t i o s .  A l l the beams  The s e c o n d - f l o o r beams r e c e i v e d the The f i r s t - f l o o r beams and the t h i r d -  f l o o r beams were damaged t o the same extent as the s e c o n d - f l o o r beams.  Damage r a t i o s decreased r a p i d l y w i t h h e i g h t above the  t h i r d story. elastic.  The top beam i n the i n t e r i o r bay almost  remained  The top d e f l e c t i o n was 9.8 i n .  Response o f the frame t o E l Centro NS motion was moderate. None o f the columns y i e l d e d with t h e i r damage r a t i o s r a n g i n g from 0.5 8 t o 0.96.  In both the i n t e r i o r bay and the e x t e r i o r bays the  h i g h e s t damage r a t i o was found i n the f i r s t - f l o o r beams. 3.2 f o r the i n t e r i o r bay and 5.0 f o r the e x t e r i o r bay. damage r a t i o s decreased s t e a d i l y w i t h h e i g h t .  I t was  The  The top beam i n  82 the i n t e r i o r bay d i d not y i e l d . The  The top d e f l e c t i o n was 6.3 i n .  f l o o r displacement d i d not i n c r e a s e much above the t h i r d -  story. T a f t S69E motion was more severe than E l Centro NS The columns on the f i f t h s t o r y y i e l d e d .  motion.  The damage r a t i o o f the  e x t e r i o r column was 1.5 and the i n t e r i o r column 1.1.  The maxi-  mum moments o f s e v e r a l columns were very c l o s e t o the y i e l d moments.  In the e x t e r i o r bay the maximum damage r a t i o was 5.5  at the bottom beam.  The damage r a t i o s o f the beams on the next  t h r e e ' f l o o r s were about the same a t 4.8. lowest damage r a t i o a t 2.8.  The top beam had the  The same t r e n d was found i n the  beams i n the i n t e r i o r bay, but the damage r a t i o s were s m a l l e r . The h i g h e s t damage r a t i o was 3.5 and the lowest was 1.6.  The  displacement a t the top was 7.3 i n . T a f t N21E motion produced Centro NS motion.  s i m i l a r r e s u l t s t o those i n E l  A l l the columns remained  The damage r a t i o s ranged  from 0.58 t o 0.89.  columns were q u i t e f a r from y i e l d i n g .  i n the e l a s t i c  range.  The f i f t h - s t o r y  Damage r a t i o s o f the  beams decreased w i t h h e i g h t i n each bay.  In the e x t e r i o r bay  they were 4.4 a t the bottom and 1.0 a t the top. bay they were 2.8 and 0.61 r e s p e c t i v e l y .  In the i n t e r i o r  The displacement o f  the top f l o o r was 5.4 i n . which was the s m a l l e s t f o r the f o u r records. The average  damage r a t i o s and displacements are shown i n  F i g . 4.11 and Table 4.8.  The p r e d i c t i o n by the m o d i f i e d sub-  s t i t u t e s t r u c t u r e method was compared w i t h the average v a l u e s o f the four n o n l i n e a r a n a l y s i s r e s u l t s . p r e d i c t e d reasonably w e l l .  Column damage r a t i o s were  Those o f the three e x t e r i o r columns  83 were s l i g h t l y underestimated, b u t they were not bad.  The damage  r a t i o s o f the beams on the f i r s t two f l o o r s were o v e r e s t i m a t e d . Those on the top three f l o o r s were underestimated.  In more  g e n e r a l terms, the m o d i f i e d s u b s t i t u t e s t r u c t u r e a n a l y s i s pred i c t e d f a i r l y uniform damage r a t i o s i n the beams up the h e i g h t of  the frame w i t h a s m a l l decrease towards the top f l o o r , but  the average  damage r a t i o s were h i g h e r a t the bottom and decreased  quite rapidly with height.  The p r e d i c t i o n was s t i l l  e s p e c i a l l y when the two top beams were excluded. displacements were concerned, up t o the t h i r d f l o o r .  reasonable,  As f a r as  the two methods agreed very w e l l  The m o d i f i e d s u b s t i t u t e s t r u c t u r e a n a l y s i s  overestimated the displacements above the t h i r d f l o o r , but the d i f f e r e n c e was not s u b s t a n t i a l .  In t h i s example the m o d i f i e d  s u b s t i t u t e s t r u c t u r e method worked reasonably w e l l .  (e)  Observations Four t e s t frames were analyzed by the m o d i f i e d s u b s t i t u t e  s t r u c t u r e method.  The r e s u l t s were compared w i t h those by the  n o n l i n e a r dynamic a n a l y s i s . three-bay, t h r e e - s t o r y frame. placements  The method worked very w e l l i n the Average damage r a t i o s and d i s -  o f the four earthquake  motions agreed w i t h those pre-  d i c t e d i n the m o d i f i e d s u b s t i t u t e s t r u c t u r e a n a l y s i s .  The method  was l e s s s u c c e s s f u l i n the two examples, the two-bay, two-story frame and the three-bay, s i x - s t o r y frame.  But i t was  still  p o s s i b l e t o o b t a i n good estimates o f damage r a t i o s and d i s p l a c e ments. rapidly. in  For these three frames damage r a t i o s converged  very  The d i f f e r e n c e i n the CPU time was enormous, e s p e c i a l l y  the three-bay, s i x - s t o r y frame.  When t h i s p o i n t i s taken  into  84 consideration,  i t i s reasonable to c l a s s i f y the r e s u l t s of these  three examples as The frame.  success.  method d i d not work w e l l f o r the s i n g l e - b a y ,  The  damage r a t i o s p r e d i c t e d by the method were q u i t e  d i f f e r e n t from those computed i n the n o n l i n e a r I t should be p o i n t e d that excessive The m o d i f i e d such badly  six-story  out t h a t the  frame was  y i e l d i n g took p l a c e  dynamic a n a l y s i s .  badly  i n every member i n the  s u b s t i t u t e s t r u c t u r e method does not  designed s t r u c t u r e s .  designed  But  and frame.  seem to work i n  at l e a s t the method was  to p r e d i c t t h a t the frame would behave very p o o r l y .  able  In p r a c t i c e  i t w i l l be r a r e t h a t such a s t r u c t u r e e x i s t s i n an area where a strong earthquake i s l i k e l y to occur.  Most i m p o r t a n t l y ,  however,  i t must be observed t h a t the a c t u a l behaviour of t h i s  structure,  as determined by the f u l l  unpredict-  able.  That i s to say,  dynamic a n a l y s i s , was  truly  i t behaved d i f f e r e n t l y i n d i f f e r e n t e a r t h -  quakes, so i t i s not s u r p r i s i n g t h a t the m o d i f i e d s t r u c t u r e method was ily.  unable to p r e d i c t the behaviour s a t i s f a c t o r -  I t i s suspected t h a t a s t r u c t u r e i n which there  spread and and  substitute  extensive  should,  unsafe, even i f damage r a t i o s  acceptable.  r e s u l t s found i n the two-bay, two-story frame and  three-bay, s i x - s t o r y frame may modified  e x h i b i t t h i s type of behaviour  t h e r e f o r e , be c o n s i d e r e d  would be otherwise The  y i e l d i n g may  i s wide-  be considered  s u b s t i t u t e s t r u c t u r e method.  frame were h i g h l y h y p o t h e t i c a l and  as t y p i c a l of  Considering  the the  t h a t these  t h a t no p a r t i c u l a r e f f o r t  was  made to c o n t r o l the behaviour of the s t r u c t u r e , the method would be  l i k e l y to work a t l e a s t as w e l l i n a r e a l s t r u c t u r e ,  behaviour of which i s l i k e l y t o be more c o n t r o l l e d .  the  85 Since the m o d i f i e d s u b s t i t u t e s t r u c t u r e a n a l y s i s i s so much cheaper to run than the n o n l i n e a r repeatedly  dynamic a n a l y s i s , i t can be used  t o see the e f f e c t o f m o d i f i c a t i o n s .  From the r e s u l t s  of such analyses a recommendation can be made on what steps can be  taken to upgrade the performance o f a b u i l d i n g t o a s a t i s -  factory  level.  86  CHAPTER 5  FACTORS AFFECTING MODIFIED SUBSTITUTE STRUCTURE METHOD  5.1  E f f e c t o f Higher Modes  Design Spectrum A i n Ref. 5 was spectrum i n the p r e v i o u s chapter. r e f e r e n c e s was  used as a smoothed response  Spectrum B from the same  d e r i v e d from the 8244 O r i o n , San Fernando  r e c o r d and i s shown i n F i g . 5.1.  Among the f o u r t e s t frames, the  3-bay, 6-story frame shown i n F i g . 4.10 frame was  19 71  was  s e l e c t e d and the  analyzed by the m o d i f i e d s u b s t i t u t e s t r u c t u r e method  i n the same manner as b e f o r e , except t h a t Design Spectrum B used as a smoothed response spectrum.  was  The m o d i f i e d s u b s t i t u t e  method had worked reasonably w e l l f o r t h i s frame when Design Spectrum A i n F i g . 2.5 was was  used.  The purpose of t h i s  analysis  to see i f the method c o u l d work e q u a l l y w e l l f o r a d i f f e r e n t  type o f earthquake motion, r e p r e s e n t e d by a d i f f e r e n t  response  spectrum. The p r o p e r t i e s of the t e s t frame were unchanged and the a n a l y s i s was  carried  out w i t h the same assumptions as i n Chap. 4.  The maximum ground a c c e l e r a t i o n s was new  taken as 0.5  g.  With the  response spectrum i t took 27 i t e r a t i o n s t o s a t i s f y the con-  vergence c r i t e r i o n s e t i n Chap. 4.  N a t u r a l p e r i o d s f o r the  a c t u a l frame and the s u b s t i t u t e frame are shown i n Table 5.1  and  87 displacements  i n Table 5.2.  Damage r a t i o s are shown i n F i g . 5.2.  Most of the i n e l a s t i c deformations which y i e l d e d .  o c c u r r e d i n the beams, a l l of  Damage r a t i o s i n the beams i n a given bay  i n c r e a s e d w i t h h e i g h t up the frame.  Only the  second-story  columns y i e l d e d . A n o n l i n e a r dynamic a n a l y s i s was  done, u s i n g the f i r s t  seconds of the 8244 Orion 19 71 r e c o r d to compute the h i s t o r y of the frame. a l i z e d at 0.5 Table 5.2  response  The maximum ground a c c e l e r a t i o n was  g as b e f o r e .  Maximum displacements  and damage r a t i o s i n F i g . 5.2.  The  20  norm-  are shown i n  r e s u l t s of the non-  l i n e a r a n a l y s i s were q u i t e d i f f e r e n t from those of the m o d i f i e d substitute structure analysis. i n the e x t e r i o r bay  A p l o t of damage r a t i o s f o r beams  i s shown i n F i g . 5.3.  I t i s c l e a r t h a t the  m o d i f i e d s u b s t i t u t e s t r u c t u r e method g r o s s l y overestimated damage r a t i o s of upper-story beams.  A s i m i l a r t r e n d was  the p r e v i o u s example, though i t was  much l e s s n o t i c e a b l e .  Although t h i s f i n d i n g was made t o f i n d out the reason why t e s t frame with t h i s response  the  seen i n  very d i s a p p o i n t i n g , an e f f o r t  was  the method f a i l e d t o work f o r the  spectrum.  The  f l o o r weights  of the  t e s t frame were changed t o see i f the n a t u r a l p e r i o d s of the frame had any e f f e c t .  They were reduced  to 130 k i p s per f l o o r to decrease  from 2 00 k i p s per  the n a t u r a l p e r i o d s .  of the p r o p e r t i e s were the same as shown i n F i g . 4.10.  floor  The  rest  The modi-  f i e d s u b s t i t u t e s t r u c t u r e a n a l y s i s and the n o n l i n e a r dynamic a n a l y s i s were c a r r i e d out i n an i d e n t i c a l manner.  Natural periods  of the a c t u a l frame and the s u b s t i t u t e frame are l i s t e d 5.3.  Displacements  i n the two  and damage r a t i o s i n F i g . 5.4.  i n Table  analyses are shown i n Table The r e s u l t s of the two  5.4  analyses  88 agreed very w e l l t h i s time. modified all  The  displacements computed i n the  s u b s t i t u t e s t r u c t u r e a n a l y s i s were almost i d e n t i c a l at  l e v e l s to those i n the n o n l i n e a r  dynamic a n a l y s i s .  damage r a t i o s agreed very w e l l as shown i n F i g . 5.5 damage r a t i o s f o r beams i n the e x t e r i o r bay y i e l d e d s l i g h t l y a t three  i n which  are p l o t t e d .  l o c a t i o n s i n the n o n l i n e a r  a n a l y s i s , though the m o d i f i e d  Beam  Columns  dynamic  s u b s t i t u t e s t r u c t u r e method pre-  d i c t e d t h a t a l l the columns would remain i n the e l a s t i c Nevertheless,  the  range.  column damage r a t i o s agreed very w e l l i n g e n e r a l .  Thus n a t u r a l p e r i o d s  d i d a f f e c t the accuracy of the  modified  substitute structure analysis. There are two  ways to e x p l a i n why  the a n a l y s i s of  two  frames, i d e n t i c a l except f o r the f l o o r weights, r e s u l t e d i n f a i l u r e i n one  case and  success i n another.  One  a t i o n i s t h a t an a c t u a l response spectrum i s very many peaks and  troughs.  rugged w i t h  When a smoothed spectrum i s used,  response a c c e l e r a t i o n at a c e r t a i n p e r i o d may w h i l e t h a t at another p e r i o d may periods  p o s s i b l e explan-  be  the  overestimated,  be underestimated.  The  natural  of the s u b s t i t u t e frame i n the f i r s t case were such t h a t  c o r r e c t response a c c e l e r a t i o n s were not obtained response spectrum. of h i g h e r modes. responses due  The The  other  explanation  from a smoothed  i s based on the e f f e c t  shape of a response spectrum i s such that  t o higher modes p l a y a more prominent r o l e f o r a  s t r u c t u r e with longer p e r i o d s .  For a t y p i c a l s t r u c t u r e  the  longest p e r i o d , and p o s s i b l e the second l o n g e s t p e r i o d , may  cor-  respond to the downward s l o p i n g p a r t o f the response spectrum. As  the n a t u r a l p e r i o d s  of a s u b s t i t u t e frame i n c r e a s e ,  a c c e l e r a t i o n s f o r the lower modes become s m a l l e r and  response  less  89 s i g n i f i c a n t compared to those f o r the h i g h e r modes.  Since the  modal damping r a t i o s of s u b s t i t u t e s t r u c t u r e s decrease f o r h i g h e r modes, h i g h e r modes a f f e c t response  c a l c u l a t i o n s even more.  T h e r e f o r e , i t i s p o s s i b l e t h a t the s u b s t i t u t e s t r u c t u r e method overestimates the e f f e c t of h i g h e r modes and t h a t t h i s p o i n t shows up more c l e a r l y i n a s t r u c t u r e w i t h l o n g e r p e r i o d s . In order to see which e x p l a n a t i o n was t r u e , a t e s t frame w i t h s h o r t e r p e r i o d s was o r i g i n a l smoothed response i n F i g . 2.5 was was  repeated.  reduced  used. The  spectrum; t h a t i s , Design Spectrum A  f l o o r weight of the 3-bay, 6-story frame  was  The response h i s t o r i e s of the frame  motions were a l s o computed by the n o n l i n e a r  dynamic a n a l y s i s program.  N a t u r a l p e r i o d s of the a c t u a l frame  and the s u b s t i t u t e frame are i n Table 5.5  Fig.  i n S e c t i o n 4.2(d)  to 130 k i p s a t a l l l e v e l s , but the r e s t of the p r o p e r t i e s  f o u r earthquake  listed  analyzed u s i n g the  The a n a l y s i s procedure  were as shown i n F i g . 4.10. to  more l i k e l y t o be  i n Table 5.6.  5.7.  and displacements  Damage r a t i o s are shown i n F i g . 5.6  are  and  Average damage r a t i o s from the f o u r n o n l i n e a r analyses  agreed w e l l w i t h those i n the m o d i f i e d s u b s t i t u t e s t r u c t u r e analysis.  B e t t e r agreement was  observed  i n the response  of upper  s t o r i e s f o r t h i s frame than the t e s t frame used i n the chapter.  Thus, although the d i f f e r e n c e was  last  l e s s apparent  i n the  case of Design Spectrum A, the frame with s h o r t e r p e r i o d s again worked b e t t e r . T h i s seems to support the second e x p l a n a t i o n . d i f f e r e n c e between a smoothed response response  spectrum  may  a f f e c t response  spectrum  Although  the  and an a c t u a l  computations  i n the modi-  f i e d s u b s t i t u t e s t r u c t u r e method, the r e s u l t s d e s c r i b e d i n t h i s  90 s e c t i o n favours the argument t h a t the m o d i f i e d s u b s t i t u t e s t r u c t u r e method works b e t t e r f o r a s t r u c t u r e w i t h s h o r t e r p e r i o d s . Or c o n v e r s e l y , the method overestimates h i g h e r modes. accordance  the c o n t r i b u t i o n from  The s u b s t i t u t e damping r a t i o i s c a l c u l a t e d i n  w i t h equation  (2.20) i n Chapter  2, and modal damping  r a t i o s are computed on the assumption t h a t each element c o n t r i butes  t o the modal damping i n p r o p o r t i o n t o the s t r a i n energy  a s s o c i a t e d with i t i n each mode shape.  T h i s has the e f f e c t o f  making modal damping r a t i o s h i g h e r i n the lower modes.  In terms  o f energy i t i m p l i e s t h a t lower modes d i s s i p a t e more energy. T h i s i s probably t r u e , but when response  c a l c u l a t i o n s are made,  t h i s works a g a i n s t the o r i g i n a l i n t e n t i o n .  Since  response  a c c e l e r a t i o n s i n h i g h e r modes w i t h s m a l l e r damping r a t i o s a r e much g r e a t e r , responses  i n h i g h e r modes are probably g i v e n more  weight than they should have.  When the response a c c e l e r a t i o n  i s c a l c u l a t e d from the design s p e c t r a i n Ref. 5, lower  damping  r a t i o s do i n f a c t have a p r o p o r t i o n a l l y g r e a t e r e f f e c t ;  this  should have the e f f e c t o f s l i g h t l y de-emphasizing h i g h e r modes, which tend t o have lower damping, but the evidence  here i n d i c a t e s  t h a t t h i s e f f e c t should be i n c r e a s e d t o de-emphasize them further.  still  91 5.2  Spectrum  A smoothed response spectrum d e v i a t e s spectrum a t many p l a c e s . way  i s minimized, a s i z -  occur a t c e r t a i n p e r i o d s .  This p o i n t  o f t e n when a smoothed response spectrum i s d e r i v e d one response spectrum.  response  Although the curve i s drawn i n such a  t h a t the d i f f e r e n c e i n the two s p e c t r a  able d i f f e r e n c e may  from a r e a l  arises  from more than  R e c a l l t h a t i n the examples i n Chapter 4  the damage due t o E l Centro EW motion was  c o n s i s t e n t l y h i g h e r than  t h a t a n t i c i p a t e d i n the m o d i f i e d s u b s t i t u t e  structure  analysis.  On the o t h e r hand, the response h i s t o r i e s of t e s t frames to T a f t S69E motion agreed reasonably w e l l w i t h the m o d i f i e d structure  analysis.  These r e s u l t s may  substitute  p a r t l y be caused by the  d i s c r e p a n c y between a smoothed response spectrum and an response  actual  spectrum.  A computer program was  used to generate the response  spectra  14 f o r E l Centro EW motion and T a f t S69E motion. between a smoothed spectrum and E l Centro EW t r a t e d i n F i g . 5.8.  The two s p e c t r a  The  difference  spectrum i s i l l u s -  are reasonably s i m i l a r in.  shape and magnitude at 2% damping r a t i o except a t a few where peaks i n the a c t u a l spectrum are c o n s i d e r a b l y smoothed spectrum.  places  above the  At 10% damping r a t i o , however, E l Centro  EW  spectrum i s c o n s i s t e n t l y above the smoothed spectrum f o r a p e r i o d greater  than 0.4  sec.  The response a c c e l e r a t i o n  spectrum i s 50% to 100% g r e a t e r  from the a c t u a l  than the smoothed spectrum.  appears t h a t the b i g d i f f e r e n c e i n the two s p e c t r a damping r a t i o s e x p l a i n s  It  at high  i n a q u a l i t a t i v e manner the d i s c r e p a n c y  i n the r e s u l t s of the m o d i f i e d s u b s t i t u t e  structure  analysis  and  the n o n l i n e a r and  dynamic a n a l y s i s .  The smoothed response spectrum  T a f t S69E spectrum are p l o t t e d i n F i g . 5.9.  F o r Both damping  r a t i o s the smoothed spectrum i s reasonably c l o s e t o the a c t u a l spectrum.  This seems t o e x p l a i n q u a l i t a t i v e l y why the r e s u l t s of  the two a n a l y s i s were not very f a r apart. From these o b s e r v a t i o n s  i t seemed p o s s i b l e t h a t a b e t t e r  estimate o f damage r a t i o s and displacement could be o b t a i n e d i f an a c t u a l response spectrum was used i n s t e a d o f the smoothed spectrum.  Response a c c e l e r a t i o n s were computed a t a short  ment o f p e r i o d s  incre-  f o r s e v e r a l damping r a t i o s ranging from 3 = 0.0  to 8 = 0.20 from E l Centro EW r e c o r d t o t a l 50 p e r i o d s  In  f o r both cases were chosen t o complete a t a b l e  of response s p e c t r a . l i z e d a t 0.5 g.  and T a f t S69E r e c o r d .  The maximum ground a c c e l e r a t i o n was norma-  The m o d i f i e d  s u b s t i t u t e s t r u c t u r e a n a l y s i s was  performed i n the same way as b e f o r e except f o r the f o l l o w i n g change.  The s p e c t r a l a c c e l e r a t i o n was read d i r e c t l y o r i n t e r -  polated  from the t a b l e .  then the damping.  The p e r i o d was i n t e r p o l a t e d f i r s t and  Suppose t h a t the p e r i o d , T, and the damping  8, were known and t h a t the s p e c t r a l a c c e l e r a t i o n  ratio,  ponding to t h i s p e r i o d and damping was t o be computed. periods,  Then two damping r a t i o s ,  8^ and  were  found from the t a b l e such t h a t 3 was between 3-^ and Q^linear interpolation, spectral accelerations ar  Two  and 1^> were l o c a t e d i n the t a b l e such t h a t T l a y  between T^ and 1^•  at 3 j  corres-  *d T at  were c a l c u l a t e d .  Using a  corresponding t o T  A l i n e a r i n t e r p o l a t i o n was  again performed t o compute the a c c e l e r a t i o n a t 6. Several  frames were analyzed by the m o d i f i e d  substitute  s t r u c t u r e method, using E l Centro EW spectrum and T a f t S69E  spectrum.  Although the response s p e c t r a were no l o n g e r smooth,  the damage r a t i o converged.  In o t h e r v/ords, i t was p o s s i b l e t o  f i n d a s u b s t i t u t e s t r u c t u r e such t h a t the computed moments were equal t o the y i e l d moments f o r a l l the members which y i e l d e d . The number o f i t e r a t i o n s  i n c r e a s e d i n many cases.  I t was found  that the o v e r c o r r e c t i o n s o f damage r a t i o s r e s u l t e d , i n some cases, i n u n s t a b l e behaviour; the damage r a t i o s o s c i l l a t e d from one i t e r a t i o n t o another. The  3-bay, 6-story frame i n F i g . 4.10, w i t h f l o o r weights  taken as 130 k i p s a t a l l l e v e l s , was analyzed by the m o d i f i e d s u b s t i t u t e s t r u c t u r e method, u s i n g E l Centro EW spectrum. i t e r a t i o n s were necessary t o achieve convergence. response spectrum i t took 13 i t e r a t i o n s convergence  criterion.  The r e s u l t s  With a smoothed  t o s a t i s f y the i d e n t i c a l  from t h i s a n a l y s i s were com-  pared w i t h those from the p r e v i o u s a n a l y s e s . are  Twenty  The n a t u r a l p e r i o d s  summarized i n Table 5.7, the displacements i n Table 5.8, and  the damage r a t i o s  i n F i g . 5.10.  The n a t u r a l p e r i o d s o f the sub-  s t i t u t e frame were longer w i t h E l Centro EW spectrum than with the smoothed spectrum.  The displacements agreed a l i t t l e b e t t e r w i t h  those from the n o n l i n e a r dynamic a n a l y s i s .  The damage r a t i o s  were h i g h e r w i t h the r e a l response spectrum than w i t h the smoothed response spectrum.  They were c l o s e r t o the damage r a t i o s  found  i n the n o n l i n e a r dynamic a n a l y s i s , but the m o d i f i e d s u b s t i t u t e s t r u c t u r e method s t i l l underestimated the damage r a t i o s a t lower l e v e l s and overestimated those at upper l e v e l s .  The r e s u l t s  firmed the o b s e r v a t i o n t h a t the smoothed spectrum was  con-  unconserva-  t i v e f o r E l Centro EW motion. The a n a l y s i s was repeated, u s i n g T a f t S69E spectrum.  It  took 14 i t e r a t i o n s f o r the damage r a t i o s t o converge, i t e r a t i o n s were r e q u i r e d w i t h the smooth response  while 13  spectrum.  The  comparison o f n a t u r a l p e r i o d s i s shown i n Table 5.7, the d i s placements i n Table 5.9, and damage r a t i o s i n F i g . 5.11. same t r e n d observed  i n the a n a l y s i s w i t h E l Centro EW  The  spectrum  was present, but the two m o d i f i e d s u b s t i t u t e s t r u c t u r e analyses d i d not d i f f e r s i g n i f i c a n t l y . response  spectrum  I t i n d i c a t e s t h a t the smoothed  r e p r e s e n t e d T a f t S69E motion w e l l .  with the n o n l i n e a r dynamic a n a l y s i s , the r e a l response produced  Compared spectrum  s l i g h t l y b e t t e r r e s u l t s than the smoothed spectrum, but  the improvement was m a r g i n a l . The same frame, except t h a t the f l o o r weight was s e t a t 200 k i p s , was next t e s t e d . i n S e c t i o n 4.2(d).  T h i s i s the i d e n t i c a l frame used  The a n a l y s i s w i t h E l Centro EW spectrum  done i n the same manner.  The p e r i o d s , displacements,  was  and damage  r a t i o s are shown i n Table 5.10, Table 5.11, and F i g . 5.12 respectively.  T h i r t y - t h r e e i t e r a t i o n s were r e q u i r e d , w h i l e i t took 16  i t e r a t i o n s w i t h the smoothed spectrum. disappointing.  The r e s u l t s were very  The m o d i f i e d s u b s t i t u t e s t r u c t u r e method with  E l Centro EW spectrum  badly overestimated the displacements and  the damage r a t i o s , e s p e c i a l l y f o r the beams.  The displacements  were too l a r g e a t a l l ; l e v e l s , but the d e v i a t i o n from the nonl i n e a r dynamic a n a l y s i s r e s u l t s became p r o g r e s s i v e l y l a r g e r a t upper l e v e l s .  Some y i e l d i n g i n the columns was observed, b u t  those columns d i d not y i e l d i n the n o n l i n e a r a n a l y s i s w h i l e o t h e r s did.  The beam damage r a t i o s i n c r e a s e d w i t h h e i g h t when E l Centro  EW spectrum was used. t r e n d was  observed.  But i n the n o n l i n e a r a n a l y s i s the o p p o s i t e  95 The a n a l y s i s was  repeated w i t h T a f t S69E spectrum.  number o f i t e r a t i o n s was  23, an i n c r e a s e of 7 i t e r a t i o n s over the  a n a l y s i s w i t h the smoothed spectrum. i n Table 5.10,  The  Table 5.12,  The r e s u l t s are summarized  and F i g . 5.13.  They compared more f a v -  o r a b l y t h i s time w i t h those from the n o n l i n e a r a n a l y s i s .  The mod-  i f i e d s u b s t i t u t e s t r u c t u r e method w i t h T a f t S6 9E spectrum, a g a i n , overestimated the displacements  and damage r a t i o s , but not as badly  as i n the l a s t example. The r e s u l t s f o r the two r e a l response  spectrum  t e s t frames i n d i c a t e t h a t u s i n g a  does not guarantee  a b e t t e r estimate of  damage r a t i o s and displacements.  T h i s o b s e r v a t i o n was  i n the analyses o f o t h e r frames.  A marginal improvement  achieved w i t h the use of a r e a l response  spectrum  imate of damage r a t i o s r e s u l t e d i n some cases. i f any, was  so s m a l l and the i n c r e a s e i n computation  u s e f u l t o make a smoothed response response  spectrum  spectrum  was  w h i l e a bad  The  i t would not be p r a c t i c a l t o employ t h i s approach.  confirmed  est-  improvement, so b i g t h a t I t i s more  c l o s e r t o the  real  and perform the m o d i f i e d s u b s t i t u t e s t r u c t u r e  a n a l y s i s w i t h the smoothed spectrum.  The d i f f e r e n c e i n r e s u l t  between t h i s a n a l y s i s and the n o n l i n e a r dynamic a n a l y s i s should be regarded as an i n h e r e n t e r r o r due this  t o the approximate nature of  analysis. I t must, of course  earthquake either.  , a l s o be borne i n mind t h a t the f u t u r e  w i l l not have a r e c o r d i d e n t i c a l to those of the p a s t ,  Thus the smoothed spectrum  r e p r e s e n t s the f u t u r e e a r t h -  quake j u s t as w e l l as does the " r e a l " spectrum quake.  from a p a s t e a r t h -  However, the f o r e g o i n g d i s c u s s i o n does i n d i c a t e t h a t one  source of " e r r o r " i n the m o d i f i e d s u b s t i t u t e s t r u c t u r e method l a y i n the smoothing and averaging of the spectrum.  96 5.3  Guidelines  f o r Use o f Method  As was i l l u s t r a t e d i n the example i n Chapter 4, the modif i e d s u b s t i t u t e s t r u c t u r e works very w e l l f o r some s t r u c t u r e s , w h i l e i t works p o o r l y  f o r others.  An e f f o r t was made to estab-  l i s h the c o n d i t i o n s which must be s a t i s f i e d i n order t o apply the method s u c c e s s f u l l y f o r a n a l y s i s o f e x i s t i n g b u i l d i n g s .  The  author, however, has so f a r been unable to s e t f i r m g u i d e l i n e s . More r e s e a r c h  i s necessary t o achieve t h i s g o a l ; t h e r e f o r e , the  f o l l o w i n g comments should be i n t e r p r e t e d with The modified  caution.  s u b s t i t u t e s t r u c t u r e method i s an e x t e n t i o n o f  the s u b s t i t u t e s t r u c t u r e method.  Therefore,  the success of the  former depends g r e a t l y on the success o f the l a t t e r .  As  described  i n Chapter 2, c e r t a i n c o n d i t i o n s must be s a t i s f i e d i n order f o r the method t o work.  They are a l s o a p p l i c a b l e t o the m o d i f i e d  s u b s t i t u t e s t r u c t u r e method w i t h the e x c e p t i o n The p r e l i m i n a r y  of one c o n d i t i o n .  r e s u l t s i n d i c a t e t h a t the damage r a t i o s o f beams  i n a given bay or the damage r a t i o s of columns on a given  axis  need not be the same. The modified small s t r u c t u r e s .  s u b s t i t u t e s t r u c t u r e method works w e l l f o r The 2-bay, 2-story  frame and the 3-bay, 3-  s t o r y frame i n the l a s t chapter can be used t o support t h i s argument.  Although t h e i r member p r o p e r t i e s  and strengths  were not  very uniform, the r e s u l t s agreed very w e l l w i t h those from the nonlinear  dynamic a n a l y s i s .  I t appears t h a t any s t r u c t u r e up t o  f o u r - s t o r y h i g h can be analyzed by the m o d i f i e d  substitute struc-  ture method q u i t e s u c e s s f u l l y . Some c a u t i o n  i s necessary t o i n t e r p r e t the r e s u l t s f o r  medium-rise s t r u c t u r e s .  Although  the method works reasonably  w e l l f o r most o f the s t r u c t u r e , there are i n s t a n c e s when i t produces erroneous  results.  When a s t r u c t u r e i s badly  underdesigned  f o r a given ground motion and y i e l d i n g takes p l a c e i n almost a l l the members, the m o d i f i e d s u b s t i t u t e s t r u c t u r e method may work very p o o r l y .  The 6-story frame i n Chapter  4 i s a good example.  Though the method can show q u a l i t a t i v e l y t h a t a s t r u c t u r e i s behaving  p o o r l y , the damage r a t i o s and displacements may be q u i t e  d i f f e r e n t from the n o n l i n e a r dynamic a n a l y s i s .  I n t u i t i o n should  be used t o judge whether the r e s u l t s are r e a s o n a b l e .  In t h i s  par-  t i c u l a r 6-story frame, however, i t was noted t h a t the " a c t u a l " behaviour was e r r a t i c :  the dynamic a n a l y s i s l e d t o a very  f e r e n t answer from the d i f f e r e n t earthquake  records.  dif-  Thus one  reason why the m o d i f i e d s u b s t i t u t e s t r u c t u r e method was unable t o g i v e a good answer was t h a t t h e r e was no " r e a l " answer.  One may  conclude t h a t when there are few l o a d paths and e x t e n s i v e y i e l d i n g the behaviour o f the s t r u c t u r e i n f u t u r e earthquakes i s e s s e n t i a l l y u n p r e d i c a b l e , and the m o d i f i e d s u b s t i t u t e s t r u c t u r e method w i l l , of course,  fail.  As long as the damage r a t i o s are not very h i g h , say, l e s s than f i v e i n any member, the r e s u l t s can be r e c e i v e d with dence.  confi-  The method seems t o work b e t t e r when y i e l d i n g i s concen-  t r a t e d i n beams.  The method may overestimate the damage r a t i o s  f o r upper-story beams, but they are u s u a l l y not very f a r from those i n the n o n l i n e a r a n a l y s i s .  A l l o f the 3-bay, 6-story  frames can be used as evidence f o r t h i s argument.  A multi-bay  s t r u c t u r e seems t o work b e t t e r w i t h the method. H i g h - r i s e s t r u c t u r e s , g r e a t e r than 10 s t o r i e s ,  say, have  98  not been t e s t e d .  They can be analyzed  by the m o d i f i e d  s t r u c t u r e method a t a r e l a t i v e l y s m a l l c o s t .  substitute  The damage r a t i o s  converge q u i t e r a p i d l y , but t h e i r accuracy has not been compared w i t h the n o n l i n e a r  dynamic a n a l y s i s , mainly because of h i g h  i n v o l v e d i n such an a n a l y s i s .  I t i s hoped t h a t the method  as w e l l f o r h i g h - r i s e s t r u c t u r e s as i t does f o r medium-rise structures.  cost works  99 5.4  Further Studies  The  m o d i f i e d s u b s t i t u t e s t r u c t u r e method was  proposed f o r  a n a l y s i s of e x i s t i n g r e i n f o r c e d c o n c r e t e s t r u c t u r e s . s i s of the r e s e a r c h  by  the author was  placed  of the procedure f o r the proposed method. t e s t frames were analyzed and nonlinear  on the  empha-  development  Although a s e r i e s of  the r e s u l t s were compared w i t h  dynamic a n a l y s i s , the  f i n d i n g s are s t i l l  More researches are needed to e s t a b l i s h the t r u e and  The  the  preliminary.  effectiveness  the l i m i t a t i o n s of the m o d i f i e d s u b s t i t u t e s t r u c t u r e method.  Some of the areas f o r f u r t h e r s t u d i e s  are d i s c u s s e d  in this  section. A multi-bay, h i g h - r i s e s t r u c t u r e has  not been t e s t e d ,  and  the performance of the method f o r such a frame i s not known p r e cisely.  The  r e s u l t s from the m o d i f i e d s u b s t i t u t e  a n a l y s i s should be Though the  compared w i t h the n o n l i n e a r  c o s t f o r the n o n l i n e a r  structure  dynamic a n a l y s i s .  a n a l y s i s w i l l be  undoubtedly  h i g h , the c a r e f u l c h o i c e of an earthquake r e c o r d may  h e l p keep  it  at a reasonable l e v e l .  should  be  tested.  Actual  More r e a l i s t i c  r e i n f o r c e d c o n c r e t e s t r u c t u r e s may  t e s t frames f o r t h i s purpose. h e l p s e t up b e t t e r g u i d e l i n e s it  structures  The  be  also  used  r e s u l t s of such a n a l y s i s  as  will  f o r a p p l i c a b i l i t y of the method as  stands at the present time. An  cedure.  attempt should a l s o be made to improve the present proThe  method becomes more f l e x i b l e and,  p r a c t i c a l i f some of the r e s t r i c t i o n s are  hence, more  removed.  For  example,  at present a s i n g l e value f o r the y i e l d moment i s a s s i g n e d each member.  I f the moment c a p a c i t i e s of' the  two  end  to  of a member  100 d i f f e r e n t , the method cannot be a p p l i e d c o r r e c t l y without a able s i m p l i f i c a t i o n i n the modeling of such a member. procedure should be m o d i f i e d  to handle t h i s case.  The  suitcurrent  It i s also  d e s i r a b l e to i n c l u d e the e f f e c t of a x i a l f o r c e s i n the a n a l y s i s . Behaviour of columns can be estimated more p r e c i s e l y i f such modif i c a t i o n s are made. As was  d i s c u s s e d b r i e f l y i n the f i r s t  t e r , the present  s e c t i o n of t h i s chap-  method f o r computation of "smeared" or average  modal damping r a t i o s may  not be the b e s t way:  e f f e c t o f higher modes are overemphasized.  i t appears t h a t Perhaps a new  way  the to  combine the damping r a t i o f o r each member can be developed to  give  more r e a l i s t i c modal damping r a t i o s . So  f a r only r e i n f o r c e d concrete  frame s t r u c t u r e s were t e s t e d .  In p r a c t i c e , i t i s very r a r e to f i n d r e i n f o r c e d concrete t u r e s without shear w a l l s .  The  a p p l i c a b i l i t y of the m o d i f i e d  s t i t u t e s t r u c t u r e method to shear w a l l s should be I f the present d i f f e r e n t way to be  strucsub-  investigated.  method d i d not work w e l l w i t h shear w a l l s , a of modifying s t i f f n e s s and  damping r a t i o s would have  developed. I t i s p o s s i b l e t h a t the m o d i f i e d  s u b s t i t u t e s t r u c t u r e method  can be a l t e r e d to handle s t r u c t u r e s made o f other m a t e r i a l s , as s t e e l .  I f s u i t a b l e r u l e s to modify s t i f f n e s s and  such  damping  r a t i o s are developed f o r s t e e l s t r u c t u r e s , the method can be  used  i n a s i m i l a r manner f o r a n a l y s i s of e x i s t i n g s t e e l b u i l d i n g s .  It  probably i s not very d i f f i c u l t to study the h y s t e r e s i s loop of a s t e e l s t r u c t u r e a f t e r s e v e r a l c y c l e s of i n e l a s t i c The  s t i f f n e s s and  damping p r o p e r t i e s may  s i m i l a r manner to t h a t used by Gulkan and  deformation.  be determined i n a Sozen.  g  101  CHAPTER 6  The  CONCLUSION  m o d i f i e d s u b s t i t u t e s t r u c t u r e method has been presented  f o r determining damage r a t i o s i n an e x i s t i n g r e i n f o r c e d building. and  These values are r e q u i r e d  concrete  f o r e s t a b l i s h i n g the l o c a t i o n  extent of damage which would occur i n an earthquake. I t i s  obvious t h a t they cannot be p r e d i c t e d f u t u r e seismic  p r e c i s e l y f o r uncertain  events; thus, i n s p i t e o f i t s i m p r e c i s i o n , the  method may c o n s t i t u t e a u s e f u l p a r t o f the r a t i o n a l r e t r o f i t procedure. At present i t i s not always p o s s i b l e t o p r e d i c t the accuracy o f the m o d i f i e d s u b s t i t u t e s t r u c t u r e a n a l y s i s , but the method appears to work w e l l f o r s t r u c t u r e s e x t e n s i v e and widespread.  i n which y i e l d i n g i s not  In a d d i t i o n the p r e l i m i n a r y  i n d i c a t e t h a t i t works b e t t e r  findings  i f y i e l d i n g occurs mainly i n beams.  There i s an i n d i c a t i o n t h a t the e f f e c t o f h i g h e r modes i s overemphasized.  I t i s hoped t h a t  further research  would c l a r i f y  requirements f o r s u c c e s s f u l a p p l i c a t i o n o f the method. Although not p e r f e c t e d ,  the m o d i f i e d s u b s t i t u t e  method o f f e r s a cheap and e f f e c t i v e way o f e s t i m a t i n g r a t i o s or d u c t i l i t y activity.  damage  demands under one o r more l e v e l o f s e i s m i c  Though l e s s p r e c i s e ,  scale nonlinear  structure  i t i s much cheaper than a f u l l -  dynamic a n a l y s i s and, as an a d d i t i o n a l advantage,  an a n a l y s i s can be done on a smaller  s i z e d computer.  Its  102 advantage over a l i n e a r e l a s t i c a n a l y s i s i s t h a t i t takes of the r e d i s t r i b u t i o n o f f o r c e s as members begin t o y i e l d . s l i g h t l y h i g h e r c o s t o f computation  i s amply rewarded w i t h  account A this  a d d i t i o n a l i n f o r m a t i o n on i n e l a s t i c behaviour o f a s t r u c t u r e , which cannot be o b t a i n e d by a c o n v e n t i o n a l modal a n a l y s i s .  103  N a t u r a l P e r i o d s i n sec  Smeared Damping R a t i o s  Mode  Computed  10-Story Frame  1 2 3 4 5 6 7 8 9 10  3.1807 0.8763 0.3945 0.2172 0.1358 0.0930 0.0681 0.0531 0.0442 0.0397  3.18 0.87 0 . 39 0.22 0 .14 0.093 0.068 0.053 0.044 0.040  0.1061 0.0805 0.0525 0.0383 0.0312 0.0272 0.0244 0.0224 0.0211 0.0204  0.106 0.081 0.053 0.038 0.032 0.027 0.024 0.022 0.021 0.020  5-Story Frame  1 2 3 4 5  1.5868 0.4101 0.1751 0.0967 0.0670  1.58 0.41 0. 18 0.097 0 . 067  0.0991 0.0680 0.0409 0.0283 0.0218  0.099 0.068 0.041 0.028 0.022  3-Story Frame  1 2 3  0.8525 0.1883 0.0784  0 .85 0 .19 0.078  0.0852 0.0454 0.0245  0 .086 0.045 0.025  S & S*  Computed  S & S*  * Shibata and Sozen Table 2.1  N a t u r a l P e r i o d s and Smeared Damping R a t i o s f o r 3-, 5-, and 10-Story Frames  104 Damage R a t i o s El  Centro EW  El  Centro NS  T a f t S69E  T a f t N21E  Average  2.0 4.4 4.8 2.5 6.9 1.0 1.1 1.8 0.96 4.0  0.85 0.90 0.97 0.91 0.94 0.81 0.72 0.90 0.89 0.92  0.98 1.1 0.90 0.88 0.97 0.95 0.98 0.95 0.90 0.90  0.95 1.4 1.1 0.92 1.0 1.1 1.2 1.7 0 . 96 0.92  1.2 2.0 1.9 1.3 2.5 0.98 0. 99 1. 35 0.93 1.7  6.5 7.6 8.3 8.1 8.6 9.3 9.8 9.9 9.9 9.9  4.1 4.3 4.6 4.7 4.7 4.8 4.7 4.4 4.1 3.9  5.0 5.0 4.9 4.5 4.1 4.2 4.1 3.9 3.6 3.4  4.9 5.0 5.1 5.1 5.0 4.8 4.6 4.2 4.0 4.0  5.1 5.5 5.7 5.6 5.6 5.8 5.8 5.6 5.4 5.3  1.1 3.9 0.97 1.1 1.0  0.98 0.86 0.97 1.1 0.89  0.87 0.84 0.89 0.93 0.87  0.84 0.70 0.78 0.88 0.85  0.95 1.6 0.90 1.0 0.90  5.4 7.1 7.1 6.7 6.7  4.8 4.6 4.7 4.4 4.2  4.4 4.4 4.1 4.1 3.9  3.7 3.3 3.2 2.8 2.4  4.6 4.8 4.8 4.5 4.3  . 1 fe 0 2 u 3 >i  0.95 0.89 0.91  0.90 0.94 0. 89  0.64 0.61 0.84  0.77 0.65 0.86  0.82 0.73 0.88  , 1 £ 2 3  6. 3 6.1 6.0  5.8 6. 0 6.3  4.0 3.3 2.7  4.9 4.5 4.1  5.3 5.0 4.8  1 2 3 co 4 6 3 6 0 7 8 9 10  10-Story Frame  b  u  1 2 3 4 «* 6 CD  D  m 7 8 9 10  5-Story Frame  1  §  2  13 4  O 5 1 co 2 3 cu 4 5 e  00  3-Stor  u  ,H  z  ffl  Table 2.2 Computed Damage R a t i o s f o r 3-, 5-, and 10-Story Frames  105 Damage R a t i o s E l Centro EW  10-Story Frame  5-Story Frame  3-Story Frame  T a f t S69E  Computed  S & S*  Computed  S & S*  Column 1 2 3 4 5 6 7 8 9 10  2.0 4.4 4.8 2.5 6.9 1.0 1.1 1.8 0.96 4.0  0.95 1.2 1.0 0.98 2.8 1.2 0.96 0.98 0. 85 1.7  0.98 1.1 0.90 0.88 0.97 0.95 0.98 0.95 0.90 0.90  0.58 0.80 0.70 0.80 0.90 0.80 0.80 0.85 0 . 80 0 . 80  Beam  1 2 3 4 5 6 7 8 9 10  6.5 7.6 8.3 8.1 8.6 9. 3 9.8 9.9 9.9 9.9  5.0 5.0 4.9 4.5 4 .1 4.2 4.1 3.9 3.6 3.4  5.5 5.5 5.0 4.9 4.8 4.6 4.8 3.8 3.0 2.2  Column 1 2 3 4 5  1.1 3.9 0.97 1.1 1.0  0.90 2.2 0.94 2.3 0.96  0.87 0.84 0.89 0.93 0.87  0.70 0 . 70 0.80 0.80 0. 90  Beam  1 2 3 4 5  5.4 7.1 7.1 6. 7 6.7  7.0 8.3 8.4 7.3 6.9  4.4 4.4 4.1 4.1 3.9  4.4 4.3 3.6 2.5 1.5  Column 1 2 3  0.95 0. 89 0.91  0.97 0.90 0.90  0.64 0.61 0.84  0.65 0.61 0.90  Beam  6. 3 6.1 6.0  6. 8 6. 3 6.0  4.0 3.3 2.7  4.5 3.7 3.0  1 2 3  6.9 7.2 7.5 7.8 7.5 8.8 9.6 9.9 9.8 10.0  * S h i b a t a and Sozen'  Table 2.3  Comparison of Damage R a t i o s f o r 3-, 5-, 10-Story Frame's  106 N a t u r a l P e r i o d s f o r the F i r s t Modes i n sec Initial Elastic  Substitute Structure  Nonlinear Analysis Average  Equal-Area Stiffness  3-Story  Frame  0.50  0.85  0.65  0.72  5-Story  Frame  0. 85  1.58  1.20  1.29  10-Story  Frame  1.58  3.18  2 . 50  2.55  Table 2.4  Computed N a t u r a l P e r i o d s f o r 3-, 5-, and Frames  N a t u r a l P e r i o d s i n sec  No. of Iterations  Mode 1  Mode 2  Mode 3  1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20  1.0679 1.3701 1.7655 1.7810 1.7945 1.8004 1.8066 1.8076 1.8073 1.8069 1.8067 1.8060 1.8052 1.8046 1.8046 1.8041 1.8036 1.8035 1.8036 1.8036  0 . 3233 0. 3632 0.4484 0.4486 0.4513 0.4505 0.4496 0.4476 0.4455 0.4439 0.4431 0.4423 0.4414 0.4405 0.4397 0.4390 0.4386 0.4383 0.4381 0.4380  0.1804 0.1917 0.2231 0.2129 0.2074 0.2033 0 .2009 0.1990 0.1975 0.1964 0.1960 0.1956 0.1952 0.1948 0.1944 0.1940 0.1937 0.1936 0.1934 0.1933  1.8036  0.4377  0.1932  Subst.  ( a )  10-Story  (a) N a t u r a l p e r i o d s computed i n the s u b s t i t u t e s t r u c t u r e analysis  Table  3.1  N a t u r a l P e r i o d s f o r 2-Bay, 3-Story  Frame A  107  No. o f Iterations 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 Subst.  ( a )  Damage R a t i o s Column 1  Column 2  Beam 1  Beam 2  1.000 1.205 1.079 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000  1.155 1.848 1.964 2.030 1.986 1.881 1.749 1.621 1.508 1.409 1. 324 1.250 1.188 1.134 1.087 1.057 1.038 1.025 1. 017 1.011  2.853 6.084 6 . 382 6 . 281 6 .116 6.021 5.982 5.975 5.981 5.988 5.992 5.996 5.999 6 .002 6.004 6 .006 6.006 6.006 6.006 6.006  1. 344 2.538 3.281 4.119 4.758 5.195 5 . 453 5.612 5 .716 5. 785 5.827 5. 857 5.882 5.905 5.926 5.945 5.961 5.973 5.988 5.992  1.000  1.000  6.000  6 . 000  (a) Target damage r a t i o s i n the s u b s t i t u t e s t r u c t u r e  Table 3.2  Damage R a t i o s f o r 2-Bay, 3-Story  Frame A  analysis  108  _ _  NO.  N a t u r a l P e r i o d s i n sec  J-  Of  Iterations  Mode 1  Mode 2  Mode 3  1 2 3 4 5 6 7 8 9 10 12 14 16 18 20  1.0674 1.2606 1.6682 1.6338 1.6371 1.6379 1.6382 1.6375 1.6366 1.6360 1.6350 1.6339 1.6331 1.6325 1.6320  0.3233 0.3694 0.4758 0.4609 0.4601 0.4605 0.4620 0.4636 0.4650 0.4663 0.4682 0.4692 0,4697 0 .4699 0.4700  0.1804 0 .2062 0.2666 0.2579 0.2568 0.2563 0.2560 0.2556 0.2552 0,2546 0.2534 0.2518 0.2503 0 .2489 0.2476  1.6307  0.4633  0.2 37 5  Subst.  ( a )  (a) N a t u r a l p e r i o d s computed i n the s u b s t i t u t e s t r u c t u r e analysis  Table 3.3  N a t u r a l P e r i o d s f o r 2-Bay, 3-Story  Frame B  Number o f I t e r a t i o n s  t =10"  Table 3.4  2  t  = i o "  3  i =10  0.0 1.0  29 18  158 81  200 124  Diff  11  77  76  -4  Number of I t e r a t i o n s - 2-Bay, 3-Story  Frame B  109  Damage R a t i o s Member  _2 £ =10  =10  £=10  3  4  A f t e r 100 Iterations  Exact  Col. 1 2 3  1.969 1.489 3. 476  1.998 2.002 2.017  2.001 2 .003 1.996  2.000 2.003 2.002  2. 2. 2.  Col. 4 5 6  l.ooo 1.000 1.496  1.000 1.000 1.036  1.000 1.000 1.005  1.000 1.000 1.013  1. 1. 1.  Col. 7 8 9  2.973 3.143 3.582  2.999 3.013 3.049  3.002 3.003 3.003  3.001 3.005 3.019  3. 3. 3.  Beam 1 2 3  6 . 016 6.160 4.675  5.993 6.000 5.956  5.995 5.999 5.991  5.995 5.999 5.981  6. 6. 6.  Beam 4 5 6 No. o f Iterations  1.992 1.968 1.496  2.001 1.999 1.964  2 .002 2 .001 1.995  2 .001 2.000 1.987  2. 2. 2.  124  100  18  81  i  Table 3. 5  Damage R a t i o s f o r 2-Bay, 3-Story Frame B  N a t u r a l P e r i o d s i n sec Mode 1 2  Table 4.1  Initial Elastic 0.50 0.13  Substitute 0.76 0.18  N a t u r a l P e r i o d s f o r 2-Bay, 2-Story Frame  110  Displacements Level  Centro EW  1 2  2.8 5.3  Table 4.2  Centro NS 2.6 5.1  Displacements  i n inches  Taft S69E  Taft N21E  Average  Subst.  1.3 2.7  1.9 3.6  2.1 4.2  1.8 3.8  f o r 2-Bay, 2-Story Frame  N a t u r a l P e r i o d s i n sec Mode  Initial Elastic  1 2 3  Table 4.3  Substitute  0.94 0.30 0.14  1.22 0. 36 0.16  1 2 3  Table 4.4  1.04  N a t u r a l P e r i o d s f o r 3-Bay, 3-Story Frame  Displacements Level  Nonlinear Average  Centro EW 3.0 6.7 10 .6  Centro NS 2.4 5.2 7.9  Displacements  •  i n inches  Taft S69E  Taft N21E  Average  Subst  1.8 3.8 6.2  1.6 3.0 5.2  2.2 4.7 7.5  2.2 5.0 8.0  f o r 3-Bay, 3-Story Frame  Ill  N a t u r a l P e r i o d s i n sec Initial Elastic  Mode 1 2 3 4 5 6  1.08 0. 37 0.21 0.15 0 .10 0.077  Table 4.5  Nonlinear Average  Substitute 1.85 0.84 0. 38 0.28 0.17 0 .13  1.65  N a t u r a l Periods f o r 1-Bay, 6-Story  Displacements Level  Centro EW  1 2 3 4 5 6  3.7 8.2 12.0 14.5 17.0 19. 3  Table 4.6  Centro NS  Taft S69E  0.74 1.7 3.0 4.5 6.5 8.4  1.4 3.3 4.8 6.7 9.4 11.6  Displacements  in  Frame  inches Average  Taft N21E  2.1 4.5 6.5 8.1 10.0 11.6  2.4 4.8 6. 1 6.6 6.9 7.2  f o r 1-Bay, 6-Story  Frame  N a t u r a l P e r i o d s i n sec Mode 1 2 3 4 5 6 Table 4.7  Initial Elastic 1.07 0.34 0.19 0.12 0.090 0.075  Substitute 1.66 0.48 0.24 0.14 0 .096 0.076  Nonlinear Average  1.25  N a t u r a l P e r i o d s f o r 3-Bay, 6-Story  Frame  Sub s t 0. 71 2.1 2.9 3. 3 6.8 8.6  112  Displacements i n inches Level  Centro EW  Centro NS  1 2 3 4 5 6  1.3 3.5 5.9 7.9 9.2 9.8  1.1 2.9 4.5 5.5 6 .1 6.3  Table 4.8  Taft . S69E  Taft N21E  1.3 3.1 4.5 5.8 6.6 7.3  0.98 2.5 3.7 4.6 5.1 5.4  Average  Subst  1.2 3.0 4.7 6.0 6. 8 7.2  Displacements f o r 3-Bay, 6-Story Frame  N a t u r a l P e r i o d s i n sec Mode 1 2 3 4 5 6  Table 5.1  Initial Elastic 1.07 0.34 0.19 0.12 0.090 0 .075  Substitute 2.24 0.63 0.29 0 .16 0 .11 0.078  N a t u r a l P e r i o d s f o r 3-Bay, 6-Story Frame A Spectrum B  Displacements i n inches Level 1 2 3 4 5 6  Table 5.2  Substitute 1.4 5.2 9.4 13. 3 16.8 19.7  Nonlinear 1.6 4.5 8.0 10.9 13.1 14.0  Displacements f o r 3-Bay, 6-Story Frame A Spectrum B  1.1 3.0 5.0 6. 7 7.9 8.8  113  N a t u r a l P e r i o d s i n sec Mode 1 2 3 4 5 6  Table 5.3  Initial Elastic 0 . 86 0.27 0.15 0.099 0.073 0.060  Substitute 1.20 0 .34 0 .17 0 .11 0.076 0.061  N a t u r a l P e r i o d s f o r 3-Bay, 6-Story Frame B Spectrum B  Displacements i n inches Level 1 2 3 4 5 6  Table 5.4  Substitute 1.1 3.3 5.5 7.4 8.7 9.4  Nonlinear 1.2 3.3 5.5 7.4 8.5 9.1  Displacements f o r 3-Bay, 6-Story Frame B Spectrum B  N a t u r a l P e r i o d s i n sec Mode 1 2 3 4 5 6  Table 5.5  Initial Elastic 0.86 0.27 0.15 0.099 0.073 0.060  Substitute "  1.20 0. 34 0 .17 0.11 0 .076 0.061  N a t u r a l P e r i o d s f o r 3-Bay, 6-Story Frame B - Spectrum A  114 Displacements Level  Centro EW  Centro NS  1 2 3 4 5 6  1. 3 3.4 5.4 6.9 7.7 8.0  1.1 2.8 4.3 5.4 6.0 6. 3  Table 5.6  Displacements Spectrum A  i n inches  Taft S69E  Taft N21E  1.1 2.9 4.5 5.6 6.2 6.4  0.95 2.6 4.1 5.2 5.9 6.1  Average 1.1 2.9 4.6 5.8 6.4 6. 7  Subst 0.94 2.6 4.3 5.5 6.4 6.8  f o r 3-Bay, 6-Story Frame B -  N a t u r a l P e r i o d s i n sec  Table 5.7  Mode  Initial Elastic  1 2 3 4 5 6  0.86 0.27 0.15 0.099 0.073 0.060  Modified Smooth Spectrum 1.20 0.34 0.17 0.11 0.076 0 . 061  Subst. S t r . A n a l y s i s Centro EW T a f t S69E Spectrum Spectrum 1.32 0 . 36 0.18 0.11 0.076 0.061  1.23 0.34 0.18 0.11 0.076 0.061  N a t u r a l P e r i o d s f o r 3-Bay, 6-Story Frame B E l Centro EW Spectrum and T a f t S69E Spectrum  115  Displacements i n inches Substitute Structure Level  Smooth Spectrum  1 2 3 4 5 6  0.94 2.6 4. 3 5.5 6.4 6. 8  Table 5.8  E l Centro EW Spectrum  Nonlinear E l Centro EW  1.1 3.2 5.4 7.2 8.4 9.0  1.3 3.4 5.4 6.9 7.7 8.0  Displacements f o r 3-Bay, 6-Story Frame B E l Centro EW Spectrum  Displacements i n inches Substitute Structure Level  Smooth Spectrum  T a f t S69E Spectrum  1 2 3 4 5 6  0.94 2.6 4.3 5.5 6.4 6. 8  0.98 2.8 4.5 5.8 6.8 7.3  Table 5.9  Nonlinear . T a f t S69E  1.1 2.8 4.3 5.4 6.0 6. 3  Displacements f o r 3-Bay, 6-Story Frame B T a f t S69E Spectrum  116  N a t u r a l P e r i o d s i n sec M o d i f i e d Subst. Str.. A n a l y s i s Mode  Initial Elastic  1 2 3 4 5 6  1.07 0. 34 0.19 0.12 0.090 0.075  Table 5.10  Smooth Spectrum  Centro EW Spectrum  T a f t S69E Spectrum  2.04 0.58 0.28 0.16 0.11 0.082  1.82 0.52 0.25 0.15 0.098 0.076  1.66 0.48 0.24 0.14 0.096 0.076  N a t u r a l P e r i o d s f o r 3-Bay, 6-Story Frame A E l Centro EW Spectrum and T a f t S69E Spectrum  Displacements i n inches Substitute Structure Level 1 2 3 4 5 6  Table 5.11  Smooth . i E l Centro EW Spectrum Spectrum 1.1 3.0 5.0 6.7 7.9 8.8  1.5 4.8 8.4 11.6 14.3 16 .4  Nonlinear E l Centro EW  1.3 3.5 5.9 7.9 9.2 9.8  Displacements f o r 3-Bay, 6-Story Frame A E l Centro EW Spectrum  117  Displacements i n inches Substitute Structure Level 1 2 3 4 5 6  Table 5.12  Smooth Spectrum 1.1 3.0 5.0 6.7 7.9 8.8  T a f t S69E Spectrum 1.3 3.9 6.7 9.2 11.2 12.4  Nonlinear T a f t S69E  1.3 3.1 4.5 5.8 6.6 7.3  Displacements f o r 3-Bay, 6-Story Frame A T a f t S69E Spectrum  118 Max. l o a d - d e f l e c t i o n relationship for hypothetical structure which remains e l a s t i c  Max. l o a d - d e f l e c t i o n relationship for actual structure which y i e l d s  Deflection  F i g . 2.1  I d e a l i z e d H y s t e r e s i s Loop f o r R e i n f o r c e d Concrete System  119  120 Start  Read:  1. s t r u c t u r a l  information  2. j o i n t i n f o r m a t i o n 3. member i n f o r m a t i o n target  including  damage r a t i o s  Compute: 1. number o f unknowns 2. h a l f bandwidth 3. member s u b s t i t u t e  damping  ratios  •  Assemble the mass m a t r i x  1  1. Compute member s t i f f n e s s m a t r i c e s . Modify the f l e x u r a l p a r t o f s t i f f n e s s e s a c c o r d i n g to the t a r g e t  damage r a t i o s .  2. Assemble the s t r u c t u r a l s t i f f n e s s m a t r i x .  i == 1  n == 0  © g. 2.3  Flow Diagram f o r S u b s t i t u t e S t r u c t u r e  Method  0  121  No  Yes Set (3 = 0 f o r a l l the modes  R e c a l l the smeared damping  ratios  Compute the response acceleration for n th mode  Set up the load  vector  Compute the f l e x u r a l  strain  energy s t o r e d i n each member  N = number o f modes  Yfes Compute the f l e x u r a l energy  strain  s t o r e d i n each member  © g. 2 . 3  Flow Diagram  f o r S u b s t i t u t e S t r u c t u r e Method  122  Compute the smeared  damping  r a t i o f o r n t h mode n = n + 1 N= Number o f modes  Yes i  = 2  1  Compute RSS displacements ' and  RSS f o r c e s  Compute the design jp _ design  forces  liabs ^rss rss V +  r  2  r  s  s  Increase the column moments by 2 0%  •  Stop  Fig.  2.3  Flow Diagram f o r S u b s t i t u t e S t r u c t u r e  Method  123 M =173 k - f t J£ 216  Beams 216  186 366  366 19 5  404  404  Size  3-Story Frame  18"x 30"  13,500 i n  5-Story Frame  18"x 30"  10-Story Frame  18"x 30"  13,500 i n ^ 4 13,500 i n  3-Story Frame  24"x 24"  13,824 i n '  5-Story Frame  24"x 24"  13,824 i n '  10-Story Frame  30"x 30"  33,750 in  Columns  206 448  448  E = 3,600 k s i  219 F l o o r weight i s 72 k i p s a t a l l l e v e l s 428  428 M =199 k - f t  228 401  y  401  239  228 446  226 446  313  217 496  328  19 7  328  254  h  407  183 1165  831  24 '  Member P r o p e r t i e s  407 189  831  24 '  254 231  417 •  417  128  F i g . 2.4  M =212 k - f t _,y_  233 695  1165  313 237  496  695  239  850  850  24 '  and Design Moments f o r  3-, 5-, 10-Story Frames  124  0.0 0.5 12  10  8  6  4  1.0  2  1.5  2.0  Period  2.5  3.0  i n sec.  Frequency i n h e r t z  Response A c c e l e r a t i o n f o r Response A c c e l e r a t i o n f o r (i=0.02  F i g . 2.5  _  8 6 + 100p  Smoothed Response Spectrum - Design Spectrum A  125  /*• = 1  W = 72 k i p s /A = 1  1 /A -1  W = 72 k i p s 1  yU = 1  /A = 6  /*-  W = 72 k i p s fA = 2  2  yU = Target damage  24  E = 3600 k s i  M =325 k - f t y 390  390 669  736  736  Yield  239 728  Moments  728  Size 1st StoryColumns  2nd  Story  3rd Story Beams  F i g . 2.6  24"  I  x 24"  10,368  24" x 24"  13,824  24"  x 24"  13,824  18"  x 30"  13,500  S o f t Story Frame A - Member P r o p e r t i e s and Yield  Moments  ratio  126  0.84  1.2  1.5  3.1  0.79  0.79  1.9  0.93  1.1  0.70  0.80  0.88  3.7  4.4  6.1  1.2  1.8  2.8  E l Centro EW  T a f t S69E  E l Centro NS  1.8  1.3  0.81  0.79  1.2  0.96  0.77  0.72  4.5  3.9  1.3  Average  T a f t N21E  F i g . 2.7  S o f t Story Frame A - Damage R a t i o s f o r Individual  Earthquakes  127  /*=  1  W = 72 k i p s yu= 1  = 1  /U = 6  W = 72 k i p s /U = 2  >U = 1 W = 72 k i p s = 1  = 1  yU = Target Damage R a t i o  24  M =583 k - f t y 699  Size Columns 699  238 610  610  1  24"x24"  13,824 i n  2  24"x24"  10,368  3  24"x24"  13,824 i n ^  18"x30"  13,500 i n  Beams  707 902  902  Design Moments  F i g . 2.8  S o f t Story Frame B - Member P r o p e r t i e s and Y i e l d Moments  in  1  128 0.96  0.82  0 .78  0 .78  4.4  4.1  4.2  0.79  0 .84  0.82  1.2  1.2  1.1  1.1  0.96  1.2  E l Centro EW  T a f t S69E  E l Centro NS  0 .89  0.60  0 .71  0.48  3.8  2.6  0 .74  0.53  1.0  0.70  0.97  0.66  Average  T a f t N21E  F i g . 2.9  1.0  0.96  S o f t Story Frame B - Damage R a t i o s f o r Individual  Earthquakes  129  /i=  }K=  6  2  W = 600 k i p s y(A= 2  1  fK=  //= 2  fk= 3  yU= 2 W = 600 k i p s yU =  yU = 4 yU = 6  2  JJi= 3  yU = 2  W = 600 k i p s /A- 3  CM  E = 3,600 k s i 50  50  M =339 k - f t y  478  745  614  1061 962  1069  524  1323  2513  1127  Y i e l d Moments  823  1288  610  938  Size  F i g . 2.10  Columns  21" x 21'  16,000 i n  Beams  20"  40,000 i n  x 36'  2-Bay, 3-Story Frame - Member and  Y i e l d Moments  4  Properties  130 4.2  0.92  0.70  3.2  1.6 2.8  1.8 2.2  5.0 2.7  5.6  1.7  0.87  4.2  4.7  1.9  1.1  1.6  3.2  7.2 4.4  0.70  2.8  2.0  2.0 6.7  3.3  1.5  3.6  1.0  E l Ce rttro EW  E l Centro NS  4.4  3.1  1.6 0 .90  1.7 1.8  4.2  0 .90  1.8  3.8  0.89  F i g . 2.11  0.76  2.6  2.2  1.8 2.5  1.7  Average  2.1  5.1 2.6  1.6  1.3  1.7 3.7  1.3  T a f t N21E  T a f t S69E  0 .69  1.5  4.0 2.6  1.2  0.95  2.3  1.6 0.77  1.8  1.3  1.9 0.90  1.7  0.85  0.67  3.0  0.96  1.1  3.6  2-Bay, 3-Story Frame - Damage R a t i o s f o r I n d i v i d u a l Earthquakes  131  F i g . 2.12  Force-Displacement Curve - D e f i n i t i o n of Equal-Area  Stiffness  132  ig.  3.1  Moment-Rotation Curve - M o d i f i c a t i o n of Damage Ratio  133 Start  Read:  1. s t r u c t u r a l  information  2. j o i n t i n f o r m a t i o n 3. member i n f o r m a t i o n . Compute:  1. number of unknowns 2. h a l f bandwidth.  " Set the damage r a t i o s a t one V  Assemble the mass matrix f k = 1  Compute member s t i f f n e s s m a t r i c e s . Modify the f l e x u r a l  stiffnesses  a c c o r d i n g to the damage r a t i o s . Assemble the s t r u c t u r a l  stiffness  matrix.  Yes =  1 ^ >  No Compute member s u b s t i t u t e damping ratios. }  Compute:  1. n a t u r a l p e r i o d s 2. mode shapes 3. modal p a r t i c i p a t i o n  F i g . 3.2  factors.  Flow Diagram f o r M o d i f i e d S u b s t i t u t e S t r u c t u r e Method  134  1 i  = 1 and n = 0  Yes  Set (b a t appropriate  s Set  (3=0 f o r  values R e c a l l smeared damping  a l l modes  ratios  Compute the response a c c e l e r a t i o n f o r n t h mode  Set up the l o a d  vector  Compute modal displacements and modal  forces  No  Compute the f l e x u r a l  s t r a i n energy  s t o r e d i n each member. Compute the smeared damping  ratio  f o r n t h mode  g. 3.2  Flow Diagram f o r M o d i f i e d  S u b s t i t u t e S t r u c t u r e Method  i  = 2 Compute RSS and RSS  Write:  displacements  forces  1. RSS  displacements  2. RSS  forces  3. damage r a t i o s  Y Stop  Flow Diagram f o r M o d i f i e d  Substitute Structure  Method  136 /A = 6  /A = 1  6  1  yU=  6  A=  CN  /A =  M=  yU= 1 6  yU = 1  yU = 1 /U= 6  W = 600 k i p s  W = 600 k i p s Damage  Ratio  /A= 6 /A:  /U = Target  yU= 1  W = 600 k i p s JU=  i  E = 3,600 k s i  Columns 50  50 M =461 k - f t Y 722 501  722 501  921 443  1612  I  461  917  746  Size  21" x 21" 20" x 36" 16,000 i n  Natural  3.3  0.4 38 s e c .  Mode 3  0.19 3 s e c .  443  Y i e l d Moments  Periods  Mode 2  1612  2-Bay, 3-Story Frame A - Member P r o p e r t i e s and  40,000 i n  1.804 s e c .  Y i e l d Moments  Fig.  4  Mode 1  746  1701  Beams  4  Number F i g . 3.4  of  Iterations  2-Bay, 3-Story Frame A - P l o t of Periods v s . Number of I t e r a t i o n s  F i g . 3.5  2-Bay, 3-Story Frame - P l o t of Damage Ratios v s . Number o f I t e r a t i o n s  139 /A. =  CN rH  /U = 2  6  2  yU =  /A =  jX =  /A=  1  JU. =  6  W = 600 k i p s 3  2  W = 600 k i p s  i  CN rH  /A=  CN rH  jU =  2  JU=  }k =  6  1  /A=  1  Target Damage Ratio  3  2  /A=  2  /A=  W = 600 k i p s yU=  3 E = 3,600 k s i  1r  *  Size  50  50 ' M =365 k - f t y  8  6  6  1212  368  Columns  21"x21"  16,000 i n  Beams  20"x36"  40,000 i n '  751 1299  487  Natural 761  1389  517  1349  525 1171  2513  Mode 1  1.6307 s e c .  Mode 2  0.4633 s e c .  Mode 3  0.2375 s e c .  967  Y i e l d Moments  Fig.  3.6  2-Bay, 3-Story Frame B - Member P r o p e r t i e s and Y i e l d Moments  Periods  Period 3  Period 2  -Period 2  iod I  _1  |  5  !  !  1  I  1  10  I  1  1  1  •  15  1  '  1  1  1  20  Number of Iterations 2-Bay, 3-Story Frame B - P l o t o f Periods v s . Number of I t e r a t i o n s  141 3.48  1.41 1.34  3.35 5.47 2.50  2.13  2.17  1.76 1.45  6.17  3.48 3.63  1.34  5. 47 3.45  2.03 1.09  1.41  1. 76  2.11  1.45  6 .]7 3.20  2.17  3.45  2. 03  1.94  1.09  3.20  <  After 4 iterations  4.28 3.63  1.42 1.47  6.07 1.84  1.95  5.98 3.49  1.92 1.15  6.01  A f t e r 12 i t e r a t i o n s  6.00 3.43  1.98 1.00  3.8  2.00 6.00  2.97  A f t e r 20 i t e r a t i o n s  Fig.  2.00  2.00  1.99 1.01 2.00 1.00 .  3.01  2.00 1.00  A f t e r 200 i t e r a t i o n s  2-Bay, 3-Story Frame B - Damage R a t i o s Computed a t the End o f 4, 12, 20, and 200 I t e r a t i o n s  3.02  3.00  143  M =115 k - f t Y  60 W = 100 k i p s 200  140  110 210  90  W = 120 k i p s  200  19 5  600 E = 4,320 k s i 30 '  30 '  Size E x t e r i o r Columns  21" x 21"  8,100 i n  I n t e r i o r Columns  18" x 18"  4,375 i n '  Beams 1st Story  18"  x 21"  4,630 i n '  15" x 18"  2,430 i n '  2nd Fig.  4.1  Story  2-Bay, 2-Story Frame - Member P r o p e r t i e s and Y i e l d Moments  2.3  4.2 0.61  2.3  Modified  Substitute  Structure Analysis  F i g . 4.2  Average of 4 N o n l i n e a r Dynamic A n a l y s i s  2-Bay, 2-Story Frame - Damage R a t i o s  144  0 .74  0 . 59  3.3  0.94  0.82  4.4 -to  2.5 ^sv  T a f t S69E  Fig.  4.3  0.94  0.78  0.56  6.5  1.1 •**  •«*  3.7  *«  T a f t N21E  2-Bay, 2-Story Frame - Damage R a t i o s f o r I n d i v i d u a l Earthquakes  0 .56  2.1  4.6  1.7  3.6  3.7  2.0  3.0  1.6  2.0  *^  145  M =60 k - f t 2  90  250  205  225 110  385  240  410  430 240  170  30  325  600  20  895  Size  I n t e r i o r Columns  Beams  W = 240 k i p s  E = 3,600 k s i  30  1  E x t e r i o r Columns  W = 200 k i p s  480  530  780  355  125  305  W = 180 k i p s  I  1  24"  X  24"  2  24"  X  24"  3  21"  X  21"  1  21"  X  21"  2  21"  X  21"  3  18"  X  18"  1  18"  X  24"  2  18"  X  21"  3  18"  X  18"  . 4 13,800 i n . 4 13,800 i n . 4 8,100 i n . 4 8,100 i n . 4 8,100 i n . 4 4,375 i n . 4 6,910 i n . 4 4,630 i n . 4 2,920 i n  (a) I f o r the r ght-hand-side i column i s 4,375 i n 4  F i g . 4.4  3-Bay, 3-Story Frame - Member P r o p e r t i e s and Y i e l d Moments  146  5.2  1.0  4.5  1.1  1.0 4.9  0.70  1.1 2.1  2.0  0 . 86 3.6  1.1  1.5  3.3  1.0  1.1  Modified  1.1  Substitute  Structure  Analysis  0.62  0.94 4. 3  0.80  2.4  1.6 3.7  1.2  Average of 4 Nonlinear  0.68 1.2  1.3  1.2  *  F i g . 4.5  0.64 2.1  0.91 3.8  1.3  1.2  •<  Dynamic A n a l y s i s  3-Bay, 3-Story Frame - Damage Ratios  147  6.4 0.54 5.7 0.81 5.0 1.9  6.2 0.86  1.5 1.8  3.3 0.95  0.68  2.8 2.1  5.2 1.7  5.5  4.5 0.80  0.99  1.9  1.4  E l Centro EW  4.6  0 . 79 3.7  0.73 3.1  1.0  0.98  0.94  2.0  0.84  0.64  1.2  0.61  0.97  0.95  4.6  0.60 3.2  0.66 2.4  0.99  T a f t S69E  Fig.  0.67  2.2  2.5  1.8  0.78  1.3  4.2 1.3  4.0  1.8  3.1  0.97  1.2  1.4  1.4  E l Centro NS  4.4  1.0  0.94  1.0  4.1  1.6 1.8  0 .56  1.3  5.3  0.91  3.7  0.93  0.84  0 .84  1.7  0.84  1.5  1.0  2.4  0.89  0.85  T a f t N21E  Earthquakes  0.53  0.78  3-Bay, 3-Story Frame - Damage R a t i o s f o r Individual  0.55  0.84  148 M - 110 k - f t V  130  Size  I  (in)  (in )  1  21x30  24 ,000  2  21x30  24,000  3  21x27  19,200  4  21x27  19,200  5  21x21  9 ,400  6  21x21  9 ,400  1  15x36  33,700  2  15x36  33,700  3  15x36  33,700  4  15x36  33,700  5  15x36  33,700  6  15x31  11,600  130 235  180  180 363  235  235 458  262  262 518  296  4  296 487 E = 3,600 k s i 422  422  35  1-Bay, 6-Story Frame - Member P r o p e r t i e s and Y i e l d Moments  149  5.7  6.8 0.85  1.1 7.5  9.5  16 .6  3.7 5.0  1.0  5.5  0.97 0.94  3.8 7.2  2.9  3.9  1.0 6.2  6.6  4.7  1.5  6.9  2.5  Modified  Substitute  Structure Analysis  F i g . 4.8  Average of 4 N o n l i n e a r Dynamic A n a l y s i s  1-Bay, 6-Story Frame - Damage Ratios  150 8.4  6.1  0.96  1.3 10.8  8.1  6.3  4.3 7.1  8.1  4.7 5.2  6.3  10.4  5.2 7.4  2.2  .5  3.3 8.2  1.7  14.4  3.7  E l Centro EW  E l Centro NS  7.2  1.1  1.1  0.84 1.7  9.5 3.1  1.1 6.8  7.0  6.5  1.3 1.7  3.8  6.6  3.0  5.1  1.3  3.2  7.9 3.4  5.5  6.7  2.8  T a f t S69E  F i g . 4.9  3.7  T a f t N21E  1-Bay, 6-Story Frame - Damage R a t i o s f o r Individual  Earthquakes  151 M =153 k - f t  228  y  183  447 19 8  309  447 317  471  330  471  550  267  438 661  228  276  Columns  24" x 24"  Beams  18" x 30"  416 228  356  1020  24  Size  330  661  1020  24  330  550  276  1020  237  423 550  416  309  550  267 330  183 198  376  237  rH rH  153  1020  24 •  I 13,824 i n ' 4 13,500 i n  E = 3,600 k s i F l o o r weight i s 200 k i p s a t a l l l e v e l s  F i g . 4.10  3-Bay, 6-Story frame - Member P r o p e r t i e s and Yield  Moments  152  3.5 0.84  2.0 0.83  3.9  2.3  4.2  0.83 2.5  0.81  ,  s  0 . 78  0 .80 4.3  0 .80  2.6  Modified  Substitute  Structure Analysis 1  0.83 4.5  2.7  4.5  0.79 2.7  0.77  .  i  0.72  0 . 78  1.7 0.86  0.95  2.7  0.70 1.6  3.6 .  0.91 2.2  4.3  0.88 2.4  1.1  1.0  ,  Average o f 4 N o n l i n e a r •• s  1.1 4.7 0.70  F i g . 4.11  i  0.75 5.2  0.81  0.93 3.0  Dynamic Analyses  3.3  1  0.88  3-Bay, 6-Story Frame - Damage Ratios  153 2.0  1.0  3.2  2.0  ,  0.99  1.5 4.7  3.0  5.7  )  2.0  0.58  2.6  1.1 0.76 1.5  0.70 , s  0.99  1.4  0.59 . )  0.82  0.81  0 .89  1.0  i  >  0 . 86 3.5  3.7  s  0.83 2.2 5  0.96  1.0  1.5 6.0  3.8  4.5  ,  0.91 2.8  , >  0.59  0.85  0.86 5.9  5.0  3.8  0 .69 3.2  Jf  E l Cenlbro  E l Cen t r o NS  EW  2.8  1.6  3.6  1.0  , 0.82  0.81  0.90  0.82  0.76  0.93  0.86  1.9  2.3  0.61 . 0.58 1.1  1  ) 1.1  1.5 4.6  4.8  1.5  3.1  0.79 1.9  3.7  0 .89 2.3  4.4  0.68 2.8  3.0  5.5  0.63 3.5  ,  Taft  N21E  3-Bay, 6-Story Frame - Damage R a t i o s f o r Individual  Earthquakes  i  i  it  l 0.78  0 .71  0.98  T a f t S69E  F i g . 4.12  2.5  0.88  0.76  0.91  0 .77  0.77  0.89  0.71  0.69  3.1  4.9 0.94  )  2.9 0.91  1.0  <  154  2.5 -  Frequency i n Hertz  Response A c c e l e r a t i o n Response A c c e l e r a t i o n  F i g . 5.1  for  ft  f o r p=0.02  _  8 6 + 100p>  Smoothed Response Spectrum - Design Spectrum B  155  12.4 0.97  8.0 0.84  11.0  6.7  , >  0.96  0.91 10.3  1.0  6.4 0.90  9.9 0.92  Modified  6.1  Structure  3.1  5.8  >  2.3 6.3  3.9  0.90  i  W = 200 k i p s / f l o o r  0.96 4.2 0.94 4.6 5.6  s 0.85 3.7  7.5 1.9  \  {  —4  1.3  Nonlinear  5.1 Jf  3.6  2.4 8.0  2.7  Dynamic  Analysis ( 8244 Orion  5.2 1.9  7.1  F i g . 5.2  3.0 2.5 5.0  7.8  1.1  Analysis  ( Spectrum B )  0.91 9.5  Substitute  4.6 if 1.2  3-Bay, 6-Story Frame A - Damage Ratios  1971 )  F i g . 5.3  3-Bay, 6-Story Frame A - P l o t of Damage R a t i o s f o r Beams i n the E x t e r i o r Bay  157 2.1  1.1 0.82  0.85 3.1 0.91  1.9 0.91  4.1 0.87  2.4 0 .88  4.6 0.82  2.8 0.87  4.8 0.84  Modified Structure  Substitute Analysis  ( Spectrum B )  2.9 0.85  4.7 0.75  2.9 0 .82 W = 130 k i p s / f l o o r  1.7 0.90  0.84 0.73  2.9 1.5  1.1 4.6  1.2  0.92  3.4 0 .89  5.6 0.78  Nonlinear  Dynamic  Analysis ( 8244 Orion  3.5 0.80  5.6  F i g . 5.4  2.9 0.95  5.4  0.80  1,  3.6 0.81  3-Bay, 6-Story Frame B - Damage R a t i o s  1971 )  158 *—  Modified  Subst.  Str. M e t h o d  ( Spectrum  B)  Nonlinear A n a l y s i s 18244 Orion)  N  \\ \\ \\ \\  \ \ • i  \  \  \ 3  V  •  1 i i  1  i i  (i  O.O  2.0  4.0 e  1 1 1 I  6.0  Ratios  3-Bay, 6-Story Frame B - P l o t of Damage R a t i o s f o r Beams i n the E x t e r i o r Bay  159 0.71  1.3  0.84  i  0.67  2.1  1.2  J i  0.75  0.83  2.8  1.7  0 .75  0.79  1  (  3.2  Modified  Substitute  Structure  0.76  0.83 3.6  ( Spectrum  Analysis A )  2.1 l  0.67  0.74  3.8  2.3 0.71  0 .64  i  {  W =  130  kips/floor  0 .52  0.87  \  0.73  0.51 1.8  1.0  0.78  0.83 3.0  1.8  0 .94  0.83 3.9  f  \  Average  of 4  Dynamic  Analyses  0.87 4.6  2.9  0.63  i  0 .70 5.0  3-2  •TO  3-Bay,  ,  0.82  0.75  5.6  \  2.4  0.90  Fig.  \ {  '  6-Story  Frame B  - Damage  Ratios  Nonlinear  160 0.85  0.52  0.72  0.50 1.8  0.94  ,  0.56  0 .79  1.0  0.55 2.0  1.1  >  0.84  0.77  0.87 3.3  2.0  1.2 4.7  2.9  0.95  3.5  , )  0 .86 3.7  1  2.3  0.91  0.91 5.4  1.8  0.87  0.93  j  0 .83 3.0  >  ,  .  0.87 4.3  2.7  >  4  0.72  0.77 5.8  0.60  3.7  4.9  1  0.83  0.90  •*>  0.67 s  0.74  0.81 v*  i  E l Centro EW  E l Cen t r o NS 0.85  <,  0.69 1.7  0.49 0.97  0 . 70  2.8 0.85  1.7  3.6 0 .89  2.2  2.8  *  0.78 2.7  0.89  1.6  i  0.87 3.6  2.2  1 (  0 .84  0.87 4.5  0.96  t {  0.83  ,  0 .49  0.74  0.82  0.51  1.7  \t  0.77  3.1  >  0.83 4.2  2.6  i  1  0 .59  0.60  0.68 5.0  0.75  4.4  3.2  2.8 >  0.82  0.68  T a f t S69E  F i g . 5.7  0.69  0 .74  T a f t N21E  3-Bay, 6-Story Frame B - Damage R a t i o s f o r Individual  Earthquakes  Fig.  I  O.O  I  I  1  I  I  0.5  1  I  I  I  !  I  1.0 Period i n  E l Centro EW Spectrum and Design Spectrum A  5.8  1  I  1  I  1.5 Sec  •  1  1  1  2.0  L  Response Acceleration o  in  in CJ to ©  o  —i—  o o  — i —  o  - o n> —s  O  4  H-  o  vQ  a  -3  fD Cu cn hh H- r t  3  c n c n  c n T j  vo M  (D o  c n  rj fD C O  3 rt  > S  b  Z9T  16 3 0.85  a r - ^ u .  t  0.85  0 .72  0.76 2.7  0.89  1.6  0.88  2  1.0 , >  0.84 '  3  0.87 3.3  ti  0.89 4.3  0.50 1.8  s  0.91 3.7  0.52 ,  S  2.0  1.2  2 7  .  1  0.93 4.7  2.9 Ji  0.82  0.88 4.5  0.95  2.8  5.4  1  J  0.81 4.5 0.74  0.85  1  0.64 5.8  3.7 }{  )  MSSA^ ) ( E l Centro EW Spectrum) 1  1.3  3.5  0.72  2.8 0.81  0.91  0.83  0.90  (2) NDA^ J  ( E l Centro EW Motion)  0.71 >  0.84  0.67 2.1  1.2 S  0.75 2.8 0.75  0.83 1.7 0.75  3.2 0.76  \  f  (1) M o d i f i e d Structure  1.9  (2) Nonlinear  0.83 3.6  W = 130 k i p s / f l o o r  Substitute Analysis Dynamic  Analysis  2.1 i  0.67 3.8 0.64  MSSA > (1  F i g . 5.10  0.74 2.3 0.71  1  f  (Smooth Spectrum)  3-Bay, 6-Story Frame B - Damage Ratios  16 4  1.5  0.79 0.71  0.85 2.3  0.85  , 0.70  1.7  1.4  ( ) 0.78 3.0 0.77  1.8  3.4 0.78  2.1  0.70  2.3  3.6  0.87 2.2  i  0.83  0 .84  4.2  2.6  >  0.69  0 .60  0 .76  3.8  1.6  0 .89  0.84  3.7  2.7 s i  0.81  0 .49 0.96 0.78  0 .74  0.85  0.51  2.8  4.4  2.4 \ f  0.74  0.67  MSSA  (Taft S69E Spectrum)  (1)  1.3 0.84 2.1 0.75 2.8 0.75 3.2 0.76 3.6  0.67 3.8 0.64  0 .74  0.68 NDA  (2)  ' ( T a f t S69E Motion)  0.71 0.67 1.2  W = 130 k i p s / f l o o r  0.83 1.7 0.83 1.9  (1) M o d i f i e d  Substitute  Structure Analysis (2) Nonlinear  Dynamic Analy  0.83 2.1 0.74 2.3  0.71  MSSA (1) (Smooth Spectrum) V  F i g . 5.11  3-Bay, 6-Story Frame B - Damage R a t i o s  165 7.7 0.97  5.1 0.94  7.4  1.0  2.0 0.89  4.8  0.81 3.2  2.0  {I  1.0  0.97 7.3  1.5  4.7  0.99 4.7  3.0  <  1.0  0.93 7.1  1.0  1.4  4.6  2.2  4  1.5 -  5  1.0 6.0  1  1.75.8  3.7  5.7  t  0.97 7.1  0.99  0.86  3.8  3.8 0.85  5.9  3.8  >  0.94  1.0  0.86  M S S A ^ ( E l Centro EW Spectrum) 3.5 0.84  2.0  NDA  0.93  (2) ' ( E l Centro EW Motion)  -5  0.83 3.9  0.81  2.3  W = 200 k i p s / f l o o r  0.83 4.2  0.78  2.5 0.80  4.3 0.80  2.6 0.83  4.5 0.77  (1) M o d i f i e d Structure (2) Nonlinear  Substitute Analysis Dynamic  Analysis  2.7 0.79  4.5 0.72  2.7 0.78  (1) MSSA' (Smooth Spectrum)  F i g . 5.12  3-Bay, 6-Story Frame A - Damage R a t i o s  166 4.7  2.9  2.8  1.6 >  0.90  0 .88 5.0  0.97  0.81  3.1 0.95  5.4  0.90 3.6  2.3 i  1.5  1.1  3.4  4.6  2.9 l  0.95  0.92 5.6  0.87  1.0  0.86  MSSA  4.9 0.94  3.6 0.96  5.3  0.91  3.6 0.89  5.6  1.0  >  0 .89 4.8  3.0  , i  0.71  3.4 0.93  3.1  0.76 5.5  0.91  3.5  j  0.98  (1) ( T a f t S69E Spectrum)  V  ;  3.5 0.84  0.83 3.9  0.81  2.3 0.83  4.2 0.78  W = 200 k i p s / f l o o r  2.5 0.80  4.3 0.80  2.6 0.83  4.5 0.77  2.7  (1) M o d i f i e d  Substitute  Structure (2) Nonlinear Analysis  0.79 4.5  0.72 MSSA  2.0  (1)  F i g 5.13  2.7 0.78  (Smooth Spectrum)  3-Bay, 6-Story Frame A - Damage R a t i o s  Analysis Dynamic  167 BIBLIOGRAPHY  1.  A s s o c i a t e Committee on the N a t i o n a l B u i l d i n g Code 19 75, Supplement No.  4 to the N a t i o n a l B u i l d i n g Code of Canada,  N a t i o n a l Research C o u n c i l of Canada, Ottawa,  Ontario,  1975. 2.  A p p l i e d Technology C o u n c i l , " T e n t a t i v e P r o v i s i o n s f o r the Development of Seismic ATC  3-06  NSF  78-8,  C a l i f o r n i a , June, 3.  Regulations  for Buildings",  A p p l i e d Technology C o u n c i l , Palo A l t o , 1978.  Okada, T. and B r e s l e r , B.,  "Strength  and  Ductility  E v a l u a t i o n of E x i s t i n g Low-Rise R e i n f o r c e d B u i l d i n g s - Screening Engineering Berkeley, 4.  Method", EERC 76-1,  Concrete Earthquake  Research Center, U n i v e r s i t y of  C a l i f o r n i a , February,  Freeman, S. A.,  California,  1976.  N i c o l e t t i , J . P.,  and  Tyrrell, J.  " E v a l u a t i o n of E x i s t i n g B u i l d i n g s f o r Seismic A Case Study of Puget Sound Naval Shipyard,  Risk  V., —  Bremerton,  Washington", Proceedings of the U. S. N a t i o n a l Conference on Earthquake E n g i n e e r i n g , 1975, 5.  pp.  Ann  Shibata, A. and  Sozen, M. A.,  Design i n R/C",  Clough, R. W.  and  McGraw-Hill, New Hudson, D. E.,  " S u b s t i t u t e - S t r u c t u r e Method  J o u r n a l of the S t r u c t u r a l  D i v i s i o n , ASCE, V o l . 10 2, No.  7.  June,  113-122.  f o r Seismic  6.  Arbor, I l l i n o i s ,  S T l , January, 19 76,  Penzien, J . , Dynamics of York, 1975,  pp.  pp.1-18.  Structures,  545-610.  "Some Problems i n the A p p l i c a t i o n of Spectrum  Technique to Strong-Motion Earthquake A n a l y s i s " , B u l l e t i n  16 8 of  the S e i s m o l o g i c a l S o c i e t y o f America, V o l . 52, No. 2,  April, 8.  1962, pp. 417-430.  Gulkan, P. and Sozen, M. A., " I n e l a s t i c Response o f R e i n f o r c e d Concrete S t r u c t u r e s t o Earthquake Motions", J o u r n a l o f the American Concrete I n s t i t u t e , V o l . 71, No. 12, December, 1974, pp. 604-610.  9.  Takeda,  T., Sozen, M. A., and N i e l s e n , N. N.,  "Reinforced  Concrete Response t o Simulated Earthquakes", J o u r n a l of S t r u c t u r a l D i v i s i o n , ASCE, V o l . 96, No. ST12, December, 1970, pp. 2557-2573. 10.  Jenning, P. C ,  " E q u i v a l e n t V i s c o u s Damping f o r Y i e l d i n g  S t r u c t u r e s " , J o u r n a l of the E n g i n e e r i n g Mechanics  Division,  ASCE, V o l . 94, No. EMl, February, 1968, pp. 103-116. 11.  Blume, J . A., Newmark, N. M., and Corning, L. H., Design of M u l t i s t o r y R e i n f o r c e d Concrete B u i l d i n g s f o r Earthquake Motions, P o r t l a n d Cement A s s o c i a t i o n , Chicago, 1961, pp. 73-86.  12.  Otani, S., "SAKE.  A Computer Program f o r I n e l a s t i c  Response o f R/C Frames t o Earthquakes", S t r u c t u r a l Research S e r i e s No. 413, C i v i l E n g i n e e r i n g S t u d i e s , U n i v e r s i t y o f Illinois, 13.  Urbana,  Illinois,  November,  1974.  O t a n i , S. and Sozen, M. A., "Behaviour of M u l t i s t o r y R e i n f o r c e d Concrete Frames d u r i n g Earthquakes", S t r u c t u r a l Research S e r i e s No. 39 2, U n i v e r s i t y of I l l i n o i s ,  14.  C i v i l Engineering Studies, Urbana,  Illinois,  Nigam, N. C. and Jennings, P. C ,  November,  1972.  " D i g i t a l Calculation of  Response Spectra from Strong Motion Earthquake  Records",  Earthquake E n g i n e e r i n g Research L a b o r a t o r y , C a l i f o r n i a I n s t i t u t e of Technology, Pasadena,  C a l i f o r n i a , June, 1968.  169 Appendix A  M o d i f i c a t i o n o f Damage R a t i o - S t r a i n Hardening Case  Consider the b i l i n e a r moment-rotation curve shown i n Fig. A . l . Let  k  = initial  stiffness,  s  = r a t i o of s t i f f n e s s a f t e r y i e l d to i n i t i a l  stiffness,  = damage r a t i o used i n n t h i t e r a t i o n , j^ -^ n+  = damage r a t i o t o be used i n n+1 t h i t e r a t i o n ,  M y  = y i e l d moment,  M  = computed moment i n n t h i t e r a t i o n ,  n  <p  - yield  rotation,  and c£ = r o t a t i o n corresponding t o M on l i n e OC. M' and 6 ' a r e the moment and r o t a t i o n a t B, which i s an i n t e r n n n  n  s e c t i o n of l i n e s OC and AC'. Assume t h a t the damage r a t i o , j x ^ ,  used i n the n t h i t e r a t i o n  was too s m a l l ; p o i n t C i s o f f the b i l i n e a r curve. the  damage r a t i o must be i n c r e a s e d i n the next i t e r a t i o n . 0 ,  i s assumed t h a t the r o t a t i o n , of l i n e O C JJ[ -^r n+  Therefore, It  i s c o r r e c t and t h a t the s l o p e  n  i s used as the new s t i f f n e s s .  The new damage r a t i o ,  i s d e r i v e d i n a f o l l o w i n g manner. k  Slope o f l i n e OC*:  Slope o f l i n e OC:  From (A.2),  M _  M  n  _  <p = yW n  n+1  (A.l)  /M . I n_  (A.2)  { A n  '  3 )  170 Substitute  equation  (A.3)  M  into  (A.l).  n+l  ' n+1  k  ^  M  M  A n+l 1  /  n  V k M  Z^n+l  /*n \  Slove  for M  M  , , i n terms n+l  , n+l  = M  1  n  rn  +  s • k • ( d> 'n  1  +  s-k  n  = M' n  M  Now  solve  . , = M' n+l n  for  M  1  n  S M  M' n  n  ( 1 -  -  s•  ) + M  U  Pn  s  k//A  n  U n /n l  ,  - d> • ) "n  k/yW  +  (A.4)  n+1  and U  of M y  M = M  M  n  M' n/^n  1  n  U f*n  • s •  M, n 1  = M  M  M  y  y  y  +  s- k • ( <£> n  +  s• k  1  r  - -i "y  M' n  k//X  n  )  M n  ( l - S ) + M ' - S y W n / n  y  \  (A. 5)  171  M  1 - s 1  n  V  S u b s t i t u t e equation  M  M  (A.6)  = M "  1  (A.6)  = M  y  S u b s t i t u t e equation  M  n  into  ( 1  1 - s  " /*n s  ( 1 - s ) + s.^ -M n  (A.7)  into  / n n+l  (A.5)  1 - s  , , = M n+l y  n+1  s  M  y  )  +  S  '/V n M  (A.7)  n  (A.4).  n  ( 1 - s ) + s.^  (3.4) n  M  n  172  Fig. A . l  Moment-Rotation Curve  173 Appendix B  Computer Program  The FORTRAN IV program f o r the m o d i f i e d s u b s t i t u t e s t r u c t u r e method i s l i s t e d i n the f o l l o w i n g i n t h i s appendix. The subroutine, MOD3, i s w r i t t e n f o r an e l a s t o - p l a s t i c Important v a r i a b l e s are e x p l a i n e d i n each s u b r o u t i n e .  case.  174 DIMENSION KL (50) , KG (50) , ABEA (50) ,CBMOM (50) ,BMCAP(100) , 1 DA L I B AT (50) ,ND(3,50) , HP (6,50) ,XM (50) ,¥M (50) ,DM (50) ,S (500)  C C C  2F(100) ,TITLE(20) ,SDAMP(5Q) ,AV (50) DIMENSION 11 (300) DIMENSION AMASS (50) ,EVAL(2Q) ,EVEC(50,20) SAMPLE MAIN PROGRAM  IUNIT=7 CALL CONTBL (TITLE,N1J, NBH, E, G, 7) CALL SETUP (NRJ,NRtt,E,G,XM,YM,DM,ND,NP,ABEA,CBMOM,DAMEAT, AV,KL,KG, 1NU,NB,SDAHP,BHCAP,IUNIT,0) NMODES=10 ICOUNT=0 AMAX=.5 IFLAG=0 CALL MASS (NU,ND,AMASS,IUNIT,NBJ) IUNIT=6 IMAX=200 IM=IMAX-1 1=0  BETA=0. EBRQB=1.E-3 10 CONTINUE 1=1+ 1 CALL BUILD (NU,NB,XM,YM,DH,NP,AREA,CEMOM,A7,E,G,DAMRAT,KL , KG, NRM,S, 1500) CALL EIGEN (NU, NB ,S,500,AMASS,EVAL,EVEC,NMODES,IUNIT) IF (I .GE. .10) BETA=. 95 CALL MOD3 (ICOONT,2,NBJ,NRM,NU,NB,NMODES,S,500,ND,NP,XH,Y 3,DS,AREA, 1CBMOM,DAMBAT,KL,KG,SDAMP,BMCAP,E,AMASS,EVEC,EVAL,AMAX, IS IGN, 21 UNIT, BETA,ERROR , 1) I1(I)=ISIGN WRITE (8,201) (DAMRAT (II) ,11=1, NBM) 201 FORM AT (• »,15F8. 3) IF (IPLAG. EQ. 1 . AND. I. EQ.IMAX) GO TO 40 IF(IFLAG.EQ. 1) GO TO 20 IF(I.EQ. 1 .AND. ISIGN.EQ.O) GO TO 46 IF(I.EQ.IM .OB. ISIGN.EQ.O) GO TO 35 GO TO 10 35 CONTINUE IFLAG=1 IUNIT=7 GO TO 10 20 CONTINUE WRITE(IUNIT,30) I 30 FOBS AT(*- *,5X,* NO. OF ITERATIONS =»,I5///) GO TO 50 40 CONTINUE WRITE (IUNIT,45) I 45 FOBMAT(*-•,5X,'DOES NOT CONVERGE AFTER*,15,* ITERATION  175 sv//)  GO TO 50 46 CONTINUE ICOUNT=0 IFLAG=1 IUNIT=7 WHITE (IBMIT,48) 18 FORK AT ('-* ,5X, 'MEMBERS DO NOT YIELD •///) GO TO 10 50 CONTINUE WRITE(IUNIT,60) BETA,ERROR 60 FORMAT('-* ,5X, 'BETA =»,F5.3,///5X,•ERROR = «,F8.6///) JJJ=I-1 WRITE (7,200) (11 C U ) , I J = 1 * J J J ) 200 FORM AT (• »,2016) STOP END  176 SUBROUTINE CONTRL(TITLE, NBJ,NRM, E,G,IUNIT) DIMENSION TITLE(20) C C C C C C C C C C  1 2 3 4 5 6 »,I5)  BEAD IN TITLE BEAD  (5, 1) {TITLE (I) ,1=1,2 0)  BEAD IN NEJ,NBM,E,G NBJ = NUMBEB OF JOINTS NRM = NUMBEB OF MEMBERS E ELASTIC MODULUS IN KSI G SHEAfi MODULUS IN KSI BEAD (5,2) NBJ, NRM, E, G HBITE (IUNIT,3) (TITLE(I) ,1=1,20) WRITE (IUNIT,4) E, G WRITE (I0NIT,5) WBITE (IUNIT,6) NBJ, NRM RETURN FOBMAT(20A4) FORMAT(2I5,2F10. 0) FOBMAT(» 1 * ,20'A4) FOBHAT(»-»,5X,«E =» , F8.3, 5X , • G =* ,F8. 3) FOBM AT 10 {»*»)) FOBM AT (*-* , ' NO* OF JOINTS*, • = » , 15, 10X, • NO. OF MEMBERS = END  177 SUBROUTINE SETUP (NRJ,NRM,£,G,XM,YM,DM,ND,NP,AREA,CRMOM,D AMRAT,A?, 1 KL,KG,NU,NB,SDAMP,B MCAP,IUNIT,IFL AG) C C SET UP TBE FRAME DATA FOR MODIFIED SUBSTITUTE C STRUCTURE METHOD C DIMENSION KL (NRM) , KG (NRM), AREA (NRM) , CRMOM (NRM) , SDAMP (NRM) , 1 DAMRAT (NRM) , AV (NRM) , ND(3,NRJ), NP(6,NRM), XH (NRM), 2 YM (NRM), DM(NRM) DIMENSION X(100), Y(100), JNL(IOO), JNG(100), BMCAP (NRM) C JOINT NUMBER JN C ND(1,JN) JOINT DEGREE OF FREEDOM IN X-•DIRECTION C = JOINT DEGREE OF FREEDOM IN Y-•DIRECTION C ND(2,JN) JOINT DEGREE OF FREEDOM IN ROTATION C ND (3, JN) = X-COORDINATE OF JN IN FEET X(JN) c Y(JN) Y-COOEDINATE OF JN IN FEET c = MEMBER NUMBER MN c = LESSER JOINT NUMBER JNL(MN) c JNG (MN) GREATER JOINT NUMBER c = KL (MN) MEMBER TYPE AT LESSER JOINT c • = KG(MN) MEMBER TYPE AT GREATER JOINT c = AREA (MN) AREA IN IN**2 c = CRMOM (MN) MOMENT OF INERTIA IN IN**4 c DAMRAT (MN) DAMAGE RATIO FOR MN c AV(MN) c - SHEAR AREA IN IN**2 BMCAP (MN) = YIELD MOMENT IN K-FT c c c c c c c c c  XM (MN) YM (MN) DM(MN) SDAMP (MN) NP(I,MN) NU NB WRITE WRITE  C C C  MEMBER LENGTH IN X-DIRECTION  - MEMBER LENGTH IN Y-DIRECTION  MEMBER LENGTH SUBSTITUTE DAMPING RATIO FOR MN = MEMBER DEGREE OF FREEDOM NUMBER OF UNKNOWNS = HALF BANDWIDTH  (IUNIT, 1) (IUNIT, 2)  READ IN JOINT DATA AND COMPUTE NO- OF DEGREES OF FREEDOM NU=1  C  DO 50 1=1,NRJ READ (5,3) JN, ND(1,I), HD(2,I), ND(3,I), X ( I ) , Y ( I  )  C 10 20  DO 40 -8=1,3 IF(ND(K,I)-1) 30,10,20 ND(K,I)=NU NU=NU+1 GO TO 40 JNN=ND(K,I)  178 ND (K,I)=ND(K, JNN) GO TO 40 CONTINUE ND (K,I) =0 CONTINUE  30 40  C C C  PHINT JOINT DATA WRITE (IUNIT,4) I , X (I) , 1(1)-, ND(1,I), ND(2,I), ND  (3,1)  50 CONTINUE C NU=NU-1 WRITE (I0NIT,5) WRITE (IUNIT,6) WRITE (IUNIT,7)  C C C  READ IN MEMBER DATA AND COMPUTE THE HALF BANDWIDTH NB=0  C  )  DO 180 1=1,NRM READ (5,8) MN, JNL(I) , JNG (I) , K L ( I ) , KG(I), AREA (I  ,  1  60 70 80  C  C C  90 100 110  CBMOH(I), DAMBAT (I) , AV (I) , BMCAP(I) IF (IFLAG. NE. 1) GO TO 70 IF(DAHRAT (I).NE.Q.) GO TO 60 DAMR AT (I) = 1GO TO 80 DAMBAT (I) = 1. CONTINUE JX.=JNL(I) JG=JNG(I) XM (I)=X(JG)-X(JL) YH(I)=Y(JG)-Y<JL) DM (I)=SQBT ( (XM (I) ) **2 + (YM (I) ) **2) DAMAGE=DAMBAT(I) ROOT=SQBT(DAMAGE) SDAMP (I) =0. 0 2*0- 2* ( 1 1 . /BOOT) NP(1,I)=ND(1,JL) NP(2,I)=ND(2,JL) HP(3,I)=ND(3,JL) NP(4,I)=ND(1,JG) NP(5,I)=ND(2,JG) NP{6/I)=HD(3,JG) MAX=0 DO 110 K=1,6 IF (NP (K ,1) -MAX) MAX=NP(K,I) CONTINUE CONTINUE MIN=1Q0Q DO 150 K=1,6  100,100,90  179 120 130 140 150  IF(NP(K,I)) 140,140,120 IF (NP (K,I)-HIN) 130,140, 140 MIN=NP(K,I) CONTINUE CONTINUE  160 170  NBB=MAX-MIN•1 IF (NBB-NB) 170,170,160 NB=NBB CONTINUE  C  C C C  PEINT MEMBER DATA  CD ,  WRITE  (IUNIT,9) I , J N L ( I ) , JNG(I), DM ( I ) , XM(I), YM  1 NP(1,I), NP(2,I), NP(3,I), NP(4,I), NP(5,I), NP(6,I) , 2 AREA ( I ) , CRMOH(I), DAMBAT (I) , AV(I) , BMCAP(I) , KL{I), 3 KG (I) C CHANGE THE LENGTHS FBOM FEET TO INCHES XM (I)=XH(I) *12. YM (I)=YM(I) *12. DH(I)=DM(I) *12. 180 CONTINUE C C PRINT THE NO. OF DEGREES OF FREEDOM AND THE HALF BANDWID TH C WRITE (IUNIT,11) NU WRITE (IUNIT,12) NB RETURN 1 FORM AT (*~ * , * JOINT DATA') 2 FOBMAT(* 0 *,7X, JN ,3X,'X(FEET)»,3X,» Y(FEET)•,4X,» NDX«,2X , NDY', 1 2X, NDB«) 3 FORMAT(4I5,2F10.5) 4 FOBMAT (* »,5X,I4,2F10.3,2X,3I5) 5 FOBMAT{»-•,«MEMBER DATA') 6 FORMAT( 0» ,7X,»MN JNL JNG LENGTH XM (FT) YH (FT) NP1 NP 2 NP3 NP4 1HP5 NP6 AREA I (CRACKED) DAMAGE A? * ,4X, * MOMENT *, 2 4X,»KL«,3X,»KG») 7 FORM AT (* *,19X,» (FEET) • «1X^« (SQ.IN) *,2X,« (IN**4) •,6X,«R ATIO», 1 2 X , ( S Q . I N ) « CAPACITY*) 8 FOBMAT(5I5,5F10.5) 9 FOBS AT (* »,5X,3I4,3F8.2,6I4,F8. 1,F12. 1 ,2F8. 3 ,F10. 2, 215) 11 FOBM AT(*- *,•NO.OF DEGREES OF FREEDOM OF STRUCTURE =«,I5) 12 FOBMAT( 0 ,»HALF BANDWIDTH OF STIFFNESS MATRIX =',I5) END ,  ,  1  ,  #  #  f  ,  i  180 SUBROUTINE MASS(NU,ND,AMASS,IUNIT,NRJ) C C THIS SUBROUTINE SETS UP THE MASS MATRIX C C ND(J,I)=DEGBEES OF FREEDOM OF I TH JOINT C WTX,WTY,8TR=X-MASS,Y-MASS,ROT.MASS IN FORCE UNITS(KIPS 0 R IN-KIPS) C AMASS (I) =MASS MATRIX C NMASS=NO.OF MASS POINTS C C MASSES ARE LUMPED AT NODES.. THE MASS MATRIX IS DIAGONAL IZED. C DIMENSION ND(3,NRJ), AMASS (NU) C C READ IN NO. OF NODES WITH MASS C READ (5,1) NMASS WRITE (IDNIT,2) WRITE (IUNIT,3) NMASS WRITE (IUNIT,4) WRITE (ION IT, 5) C C ZERO MASS MATRIX C DO 10 1=1,NU AMASS (I)=0. 10 CONTINUE C C READ IN X-HASS,Y-MASS AND SOT. MASS (IN UNITS OF WEIGHT ) C DO 50 1=1,NMASS READ (5,6) JN, WTX, HTY, WTR WRITE (IUNIT,7) JN, WTX, WTY, WTR N1=ND(1, JN) N2=ND(2,JN) N3=ND(3,JN) IF(N1.EQ. 0) GO TO 20 AMASS (N 1) =AH ASS (N 1) + (WTX/386. 4) 20 IF(N2.EQ. 0) GO TO 30 AMASS (N2) =A8ASS (N2) * (WTY/386.4) 30 IF(N3.EQ. 0) GO TO 40 AMASS (N3) =AMASS (H3) > (WTR/386.4) 40 CONTINUE 50 CONTINUE C RETURN 1 FORM AT (15) 2 FORMAT (///110 («*»)) 3 FORM AT (' — *,• NO. OF NODES WITH MASS*,* = ',I5) 4 FORMAT( 0«,7X,'JN*,3X, X-MASS*,'*X, "Y-MASS* ,2X, ' ROT. MASS ' ,  )  ,  5 FORMAT (• «,12X,' (KIPS) *,4X,» (KIPS) • ,2X,» (IN-KIPS) ') 6 FORM AT (15, 3F 10.0) 7 FORM AT (• ' ,5X,I4,3F10.3)  181 END  182 BAT, C C C C C C C C C C C  DIMENSION XM (NRM) , YM (NRM) , DM (NRM) , NP(6,NRM), ABEA (NRM  C  CRMOM(NRM), AV(NRM) , DAMRAT (NRM) , KL (NRM), KG{  DIMENSION S(IDIM), SM(21)  DO 10 1=1,IDIM S(I)=0. 10 CONTINUE BEGIN MEMBER LOOP DO 200 1=1,NRM ZERO MEMBER STIFFNESS NATRIX  20  30 C C C  1  ZERO STRUCTURE STIFFNESS MATRIX  C C C  C  KL,KG,NBM,S,IDIM)  DAMRAT (I) = DAMAGE RATIO FOR I TH MEMBER S(I) = STRUCTURE STIFFNESS MATRIX SM(I) = MEMBER STIFFNESS MATRIX  NRM)  C C C  1  THIS SUBROUTINE CALCULATES THE STIFFNESS MATBIX OF EACH MEMBER AND ADDS IT INTO THE STRUCTURE STIFFNESS MATRIX. FLEXURAL STIFFNESSES OF MEMBERS ABE MODIFIED ACCORDING TO THE DAMAGE RATIOS. THE FINAL STIFFNESS MATBIX S IS RETURNED.  ) ,  C C  SUBROUTINE BUILD (NU,NB,XM,YM,DM,NP, AREA,CRMOM, AV,E,G,DAM  DO 20 J=1,21 SM (J)=0. CONTINUE DM2= DM (I) *DM (I) XM2=XM(I) *XH (I) YM2= YM (I) *YM (I) XMYM=XH (I) *YM(I) F=AREA (I) *E/(DM (I) *DM2) H=0. IF(AV(I).EQ.O. .OB.G.EQ.O.) GO TO 30 H=12-*E*CRMOM(I)/(AV (I) *G*DM2) XM2F=XM2*F ¥M2F=YM2*F XMYMF=XM¥M*F FILL IN PIN-PIN SECTION OF MEMBER STIFFNESS MATRIX SM (1)=XM2F SM (2)=XHYMF SM(1)=-XM2F SM (5)=-XMYMF SM(7)=YM2F SH(9)=-XMYHF  183  40 C C C C  C C C  C C C  C C C  50  SM (10)=-YM2F SM (16)=XM2F SM (17)=XMYMF SH(19)=YM2F IF (KL (I) +KG (I) -1) 100,40,50 F=3. *E*CRHOM (I) /(DM2*DM2*DM (I) * ( 1- «/H/4. ) ) /DAMRAT (I) GO TO 60 F=12.*E*CBMGM(I)/(DM2*DM2*DM(I)*(1.*H))/DAMBAT(I) FILL IN TEBMS WHICH ABE COBaON TO PIN-FIX,FIX-PIN,AND FIX-FIX a EMBERS  60  XH2F=XM2*F YM2F=YM2*F XMYMF=XMYM*F DM2F=DH2*F SH(1)=SM(1) *YM2F SH (2) =SM (2)-XMYMF SM (4)=SM (4)-YM2F SM(5)=SM(5) +XHYMF SM (7)=SM (7) +XM2F SB (9)=SB(9) +XMYHF Sa(10)=SM (10)-XM2F SM(16)=SM(16) +YM2F SM (17) =SH (17)-XMYMF SM(19) = SH(19) + XH2F IF(KL(I) -KG (I)) 70,80,90 FILL IN REMAINING PIN-FIX TERMS  70  SM (6) =-YH (I) *DM2F SM (11)=XM (I) *DM2F SH(18)=-SM(6) SM (20) =-SN (11) SM(21) = DH2*DM2F GO TO 100 FILL IN REMAINING FIX-FIX TEEMS  80  SM (3) =- YM (I) *DM2F*. 5 SM (6)*SM{3) SM(8)=XH(I)*DM2F*-5 SM(11)=SM<8) SM(12)=DH2*DM2F*(4.+H)/12. SH (13)=-SM (3) SH (14) =-SM (8) SM(15)=DM2*DM2F*(2.-H)/12SM (18)=-SM (3) SM(20)=-SM (8) SM (21)=SM (12) GO TO 100 FILL IN REMAINING FIX-PIN TERMS  90  SH (3)=-YM (I) *DM2F SM (8)=XM(I) *DM2F  184  100 C C C C  SM (12)=DM2*DM2F SM (13)=-SM (3) SM (14)=-SM{8) CONTINUE ADD THE MEMBER STIFFNESS MATRIX SM INTO THE STROCTORE STIFFNESS MATRIX SNB1=NB-1  C 110  DO 190 3=1,6 I F ( N P ( J , I ) ) 190,190,110 J1= (J-1)*(12-J)/2  C  DO 180 L=J,6 I F ( N P ( L , I ) ) 180, 180,120 IF(NP(J,I)-NP(L,I)) 150,130, 160 I F ( L - J ) 140,150, 140 K=(NP (L,I)-1) •NBUNP (J,I) H-J1+L S (K)=S(K) *2.*SM (N) GO TO 180 K= ( H P < J , I) -1) * NB 1 + NP ( L , I) GO TO 170 K=(NP{L,I)-1)*NBH-NP(J,I) N=J1*1 S(K)=S (K) *SM(N) CONTINUE  120 130 140  150 160 170 180 C C C  190  CONTINUE  200 CONTINUE RETURN END  185 SUBROUTINE EIGEN (NU,NB,S,IDIM,AHASS,EVAL,EVEC,NMODES,IUN IT) C C THIS SUBROUTINE COMPUTES A SPECIFIED NO- OF NATURAL FSEQ UENCIES C AND ASSOCIATED MODE SHAPES C C NU=NQ. OF DEGREES OF FREEDOM C NB=HALF BANDWIDTH C NMODES=NO. OF MODE SHAPES TO BE COMPUTED C AMASS(I)=MASS MATRIX M=RANK OF MASS MATRIX C S(I)=STIFFNESS MATRIX STORED BY COLUMNS C EVAL (I)—NATURAL FREQUENCIES C EVEC (I,J)=MODE SHAPES C DIMENSION S(IDIM), AMASS (NU) , EVAL(NMODES), EVEC (50,20) , 1 SCR (900) DIMENSION CMASS(100), SS(500) C C COMPUTE THE RANK OF MASS MATRIX C M=0 C DO 10 1=1,NU CMASS (I) =AMASS (I) IF (AMASS (I).EQ.O. ) GO TO 10 M=M*1 10 CONTINUE C IF(NMODES.GT.M) NMODES=M IF(NMODES.EQ.0) NMODES-M WRITE (IUNIT,1) NMODES C C CALL RVPOW TO COMPUTE EIGENVALUES AND EIGENVECTORS C DO 20 1=1,500 SS(I)=S(I) 20 CONTINUE C EPS=0. EPSV=0. CALL RVPOWR(SS,CMASS NU,NB,EVEC 50 EVAL,NMODES,EPS,EPSV, 100, 1 SCR,M) C C PRINT EIGENVALUES AND EIGENVECTORS (MODE SHAPES) C WRITE (IUNIT, 2) WRITE (IUNIT, 3) WRITE (IUNIT, H) C DO 30 1=1,NMODES EVAL1=EVAL(I) EVAL (I) =SQRT (EVAL 1) FEEQ=EVAL (I)/6. 283185308 PERIOD=6.283185308/£VAL(I) #  #  #  186 WRITE 30 CONTINUE C C  WRITE WRITE  (IUNIT,5) I , EVAL1, EVAL ( I ) , FREQ, PERIOD  (IDNIT,6) NMODES (IUNIT,7) (1,1=1,NMODES)  DO 40 1=1,M WRITE (IONIT,8) (EVEC (I, J) ,J= 1 , NMODES) 40 CONTINUE  C  ///)  RETORN 1 FGRHAT{*-*,*NQ. OF MODES TO BE ANALIZED = » ,I5///11 0 {» *• )  2 FORM AT (///110 (»*•)) 3 FORMAT(*Q*,5X,•MODES',4X,» EIGENVALUES*,6X,* NATURAL FREQU ENCIES* , 1 13X,*PERIODS*) 4 FORMAT {• •,30X,• (RAD/SEC) •,5X,*(CYCS/SEC) *,8X,* (SECS) •) 5 FORM AT {' *,5X,I5,4F15.4) 6 FORMAT{*0«,5X,*MODE SHAPES CORRESPONDING TO FIRST*,15,* FREQUENCIE IS') 7 FORMAT{*0•,10112) 8 FORMAT {* *,10F12. 6) END  187 SUBROUTINE MOD3(ICOUNT,ISPEC,NRJ,NRM,NU,NB,NMODES,S,IDIM ,ND,NP,XM, 1 YM,DH, AREA,CRMOM,DAMRAT,KL,KG SDAMP,BMCAP,E,& MASS,EVEC, 2 EVAL , AM AX, I SIGN, I UNIT, BET A , ERROR, XBASE) C C MODIFIED SUBSTITUTE STRUCTURE METHOD C THIS SUBROUTINE COMPUTES JOINT DISPLACEMENTS AND MEMBER FORCES C NEW DAMAGE RATIOS WILL BE CALCULATED AND RETURNED. C C ICOUNT = 0 IF DAMPING IS SET AT 10% AND ELASTIC C ANALYSIS IS TO BE CARRIED OUT C ISPEC = 1 FOR SPECTRUM A, 2 FOR B, AND 3 FOR C C IDIM = DIMENSION OF S ( I ) C ISIGN = NUMBER OF MEMBERS FOR WHICH DAMAGE RATIOS C ARE MODIFIED C IUNIT = OUTPUT DEVICE UNIT C BETA = CONSTANT FOR ACCELERATED CONVERGENCE C ERROR = CONSTANT FOR CONVERGENCE CRITERION C IBASE = 1 IF BASE SHEAR IS TO BE PRINTED #  C c )  DIMENSION ND(3,NRJ), NP(6,NRH), XM (NRM) , YM(NRM), DM (NRM  .  1 G (NRM) , 2  AREA (NRM), CRMOM (NRM) , DAMRAT (NRM), KL (NRM) , K AMASS (NRM) , EVEC (50, 20), EVAL (NMODES) , S (IDIM)  3  SDAMP (NRM), ZETA(10), PI (100) DIMENSION BMASS(40), IDOF(100), ALPHA (20), RMS (7, 100), 1 F(100) , D(6) DIMENSION BMCAP (1)  C C C  C  C  CALCULATE THE MODAL PARTICIPATION FACTOR JJ=1 DO 10 1=1,NU IF (AMASS (I) . EQ-O. ) 30 TO 10 BMASS (JJ) =AMASS (I) IDOF (JJ)=I JJ=JJ*1 10 CONTINUE JJ=JJ-1  C  DO 3 0 1=1,NMODES AMT=0. AMB=0. ALPHA (I) =0.  20  DO 20 J=1,JJ AMT=AMT*BMASS (I) *EVEC (J , I) AM B=AMB*BMASS(I)*EVEC(J,I)**2 CONTINUE  188 C C C  ALPH &(I) = AMT/AMB 30 CONTINUE WRITE  (IUNIT,1)  DO 4 0 1=1,NMODES WRITE (IUNIT,2) I , ALPHA (I) 40 CONTINUE  C C WHEN KK=1, MODAL FORCES FOR UNDAMPED SUBSTITUTE STRUCTUR E ARE C COMPUTED. THEY ARE USED TO COMPUTE * SMEARED * DAMPING VA LUES, C WHICH ARE USED TO CALCULATE THE ACTUAL RESPONSE OF THE S UBSTITUTE C STRUCTURE C INDEX=1 C DO 420 KK=1,2 IF(ICOUNT-I) 400,70,50 50 CONTINUE IF (KK.NE. 1) GO TO 70 C DO 60 K=1,NMODES ZETA<K)=0. 60 CONTINUE C 70 CONTINUE SHRMS=0. C C ZERO ABSO(J,I) AND RMS (J,I) C DO 90 1=1,100 C DO 80 J=1,7 RMS (J,I)=0. 80 CONTINUE C 90 CONTINUE C C CALCULATE THE MODAL DISPLACEMENT VECTOR C DO 290 K=1,NMODES C C CALCULATE NATURAL PERIOD AND CALL SPECTA C WN = 6. 2831 85308/EVAL (K) DAMP=ZETA (K) CALL SPECTS(ISPEC,DAMP,WN,AMAX,SA) C C ZERO LOAD VECTOR C DO 100 J=1,NU F(J)=0.  189 C  100  FF=0.  C C C  COMPUTE LOAD VECTOR FAC=SA*ALPHA(K)*386.4  C  C C C  C C C C C C  110  DO 110 J=1,JJ I1=IDOF(J) F (I1)=EVEC (J,K) *FAC*AMASS (11) FF=FF+F(I1) CONTINUE CALCULATE THE BASE SHEAR  120  IF(KK.NE.2) GO TO 120 SHRMS=SHRHS+FF**2 IF(K.LT.NMODES) GO TO 120 SHRMS=SQBT (SHRMS) CONTINUE COMPUTE DEFLECTIONS CALL SUBROUTINE FBAND RATIO=1.E-7 CALL FBAND(S,F,NU,NB,INDEX,RATIO,DET,JEXP,0,0,0.) INDEX=INDEX+1  C  DO  130  140  150 160 C C C  CONTINUE  160 1=1,NRJ DX=0DY=0. DR=0. N1=ND(1,I) N2=ND(2,I) N3=ND(3,I) IF(NI-EQ-O) GO TO 130 DX=F(N1) BBS(1,1)=RMS(1,I)*DX**2 CONTINUE IF(N2.EQ.0) GO TO 140 DY=F(N2) RMS (2,1)-BBS(2,1)*DY**2 CONTINUE IF(N3.EQ.O) GO TO 150 DR=F (N3) RMS(3,1)-BBS(3,1)+DR**2 CONTINUE CONTINUE  COMPUTE MEMBER FORCES SIGPI=0.  190 DO 260 1=1,NBM DO 200 J=1,6 N1 = NP(J,I) IF{N1) 180,180,170 D(J)=F(N1) GO TO 190 D(J)=0. CONTINUE CONTINUE  170 180 190 200  XL=XM(I) ¥L=YM (I) DL=DM (I) &XIAL= (ABEA (I) *E/DL**2) * (D (4) *XL + D (5) *¥L-D ( 1)*X L-D{2) *YL)  210  220  230 240 C C C  C C C  COMPUTE THE RELATIVE FLEXU8AL STBAIN ENEBGY  250  C  IF(KK.NE.1) GO TO 250 PI (I) = (BML**2*BMG**2*BML*BMG) /6./AK SIGPI=SIGPI*PI(I) CONTINUE ACCUMULATE ABSOLUTE SUM AND BMS SUM  260 C C C  IF (KL (I) .EQ. 0 . AND. KG (I) .EQ.O) GO TO 230 DV= (D (2) *XL—D (1) *YL+D (4) *YL-D (5) *XL)/DL AK=CBMOM (I) *E/DL/DAMBAT (I) BHL=-AK*(6.*DV/DL*4.*D(3) +2. *D(6) )/12. SHEAB=AK*6. * (2- *DV/DL*D (3) *D (6) ) /DL BMG=BHL+SHEAB*DL/12. IF (KL (I) -KG (I)) 210,240,220 BHG=BMG+BML*.5 SHEAB=SHEAB•1.5*BHL/(DL/12.) BML=0. GO TO 240 BML=BML*BMG*.5 SHEAB=SHEAB-1.5*BMG/(DL/12.) BHG=0. GO TO 240 BMG=0. BEL=0., SHEAB=0. CONTINUE  BMS (4,1) =BMS(4,I) *AXIAL**2 BMS(5,I)=BHS(5,1)+SHEAB**2 BMS (6,I)=BMS(6,I) +BSL**2 BMS (7,1)=BMS(7,1)*BMG**2 CONTINUE COMPUTE THE SMEABED DAMPING FOB EACH MODE IF(KK.NE.I) GO TO 280 DO 270 1=1,NBM  191 270 C 280 290  ZETA (K) = ZETA (K) + PI(I) *SDAMP(I) CONTINUE ZETA (K) =ZETA (K) /SIGPI CONTINUE CONTINUE  C I F (KK. EQ. 1) GO TO 420  C C C  PRINT RMS DISPLACEMENTS AND FORCES WRITE WRITE WRITE  C  #  DO 310 1=1,NBJ  C  300 C C C C  (IUNIT,4) (IUNIT,5) (IUNIT 3)  310  DO 300 J=1,3 SCRAT=RaS{J,I) RMS (J, I) =SQRT{SCRAT) CONTINUE WRITE (IUNIT,6) I , (BMS (J,I) , J= 1,3) CONTINUE MODIFY DAMAGE RATIOS  320 C 330 340 350  360 370 C 380  WRITE (10NIT,7) IF(IBASE.NE. 1) GO TO 320 WRITE (IUNIT,8) SHBHS CONTINUE WRITE (IUNIT,9) ISIGN=0 DO 390 1=1,NRM I F (RMS (6,I)-RMS (7,1) ) 330,330,340 BIG=RMS(7,I) GO TO 350 BIG=RMS(6,I) CONTINUE BM=SQBT (BIG) DAM0LD=DAMBAT (I) DAMRAT (I) =BM/BMCAP (I) *DAMRAT (I) DAMBAT (I) =DAMRAT (I) +BETA* (DAMRAT (I) —D AMOLD) IF (DAMBAT (I) . LT. 1.0) GO TO 360 CHECK=ABS (BM-BMCAP (I) ) /BMCAP (I) I F (CHECK. GT.ERROR) ISIGN=ISIGN+1 . GO TO 370 CONTINUE DAMRAT (I) = 1. CONTINUE SDAMP (I)=0.02+0.2*(1.-1-/SQBT(DAMRAT(I))) DO 380 J=4,7 RMS (J,I) =SQBT (BMS (J,I) ) CONTINUE  192 C AMRAT (I) 390 C C  C  C  WRITE (I0NIT,11) I , (RHS(J,I) ,J=4,7) , BMCAP(I), D CONTINUE  400  GO TO 420 CONTINUE  410  DO 4 10 1=1,NMODES ZETA (I) =. 1 CONTINUE  ICOUNT=ICOUNT*1 WRITE {IHNIT,12) 420 CONTINUE  ICOUNT=ICOUNT+1 RETURN 1 FORM AT {•-',* MODAL PARTICIPATION FACTOR*,/) 2 FORM AT(* »,5X,•MODE»,I5,5X,F10.5) 3 FORMAT (*-' ,7X,*JOINT NO. »,10X, 'X-DISP(IN) ' , 10X, • Y-DISP (I N)',7X, 1 'ROTATION (RAD)') 4 FORMAT (* —• , 110 (**') ) 5 FORM AT('-',*ROOT MEAN SQUARE DISPLACEMENTS *) 6 FORMAT {* • ,6X,I10,3F20.4) 7 FORMAT(*-*,* ROOT MEAN SQUARE FORCES') 8 FORM AT(1H0,7X,»RSS BASE SHEAR = *,F10.3) 9 FORMAT(* — * ,8X,* MN *,1QX,* AXIAL*,10X,•SHEAR*,11X,*BML*,12X ,* BMG', 1 9X,'MOMENT*,1 OX,'DAMAGE*/21X,•KIPS»,12X,»KIPS',2( 9X, » (K—FT) 2*), 8X, * CAPACITY' ,9X, * RATIO *) 11 FORMAT {* • ,5X,I5,6F15.3) 12 FORMAT(*~* ,110 (•*•)) END  193 C C C C C C C C C C C C  C C C  C C C  SUBROUTINE SPECTR (ISPEC,DAMP,WN,AMAX,SA) ISPEC=1 I F SPECTRUM A IS USED =2 I F SPECTRUM B IS USED =3 I F SPECTRUM C IS USED DAMP=DAMPING FACTOR (FRACTION OF CRITICAL DAMPING) WN =NATURAL PERIOD IN SECONDS AMAX=MAXIMUH GROUND ACCELERATION (FRACTION OF G) SA =RESPONSE ACCELERATION (FRACTION OF G) I F (ISPEC. EQ. 2) GO TO 10 IF (ISPEC. EQ. 3) GO TO 60 SPECTRUM A IF (WN.LT. 0. 15) SA=25.*AMAX*WN IF (WN.GE-0. 15 .AND. WN.LT.0.4) SA=3.75*AMAX IF(WN.GT-0-4) SA=1.5*AMAX/WN GO TO 90 SPECTRUM B 10 CONTINUE IF (WN.LT.0.1875) GO TO 20 IF (WN.LT. 0-53333333) GO TO 30 IF (WN.LT. 1. 6666667) GO TO 40 IF(WN.LT.1.81666667) GO TO 50 SA=2.*AMAX/(WN-0.75) GO TO 90 20 SA=20.*AMAX*WN GO TO 90 30 SA=3.75*AMAX GO TO 90 40 SA=2.*AMAX/WN GO TO 90 50 SA=1.875*AMAX GO TO 90 SPECTRUM C 60 CONTINUE IF(WN.LT.O. 15) GO TO 70 IF (WN.LT. 0.38333333) GO TO 80 SA=0. 5*AHAX/ (WN-0. 25) GO TO 90 70 SA=25.*AMAX*WN GO TO 90 80 SA=3.75*AMAX 90 CONTINUE SA=SA*8./(6.*100.*DAMP) RETURN END  

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