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Pseudo inelastic torsional seismic analysis utilizing the modified substitute structure method Tam, Ken Sau Kuen 1985

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PSEUDO  INELASTIC TORSIONAL SEISMIC ANALYSIS U T I L I Z I N G 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  by KEN B.A.Sc,  SAU KUEN  The U n i v e r s i t y  TAM  Of B r i t i s h  Columbia,  1980  A T H E S I S S U B M I T T E D I N P A R T I A L F U L F I L M E N T OF THE R E Q U I R E M E N T S FOR THE DEGREE MASTER OF A P P L I E D  OF  SCIENCE  in THE F A C U L T Y OF GRADUATE S T U D I E S (Department  We  accept to  Of C i v i l  this  Engineering)  thesis  the required  as  standard  THE U N I V E R S I T Y OF B R I T I S H April  ©  conforming  COLUMBIA  1985  KEN S A U KUEN TAM,  1985  THE  In  presenting  requirements  this f o r an  Columbia,  I  available  for  permission p u r p o s e s may or  her  of  without  Department  of  26  April  fulfilment  the  Library  shall  reference  and  study.  I  extensive granted  by  this thesis my  written  Civil  1985  copying the  of  Head of  It for  is  further  financial  Engineering Columbia  gain  the  of  British  it  freely  agree for  Department  understood  permission.  make  this thesis my  of  University  the  The U n i v e r s i t y of B r i t i s h 2075 Wesbrook P l a c e V a n c o u v e r , Canada V6T 1W5  Date:  partial  that  representatives.  publication allowed  be  in  advanced degree at  agree  for  thesis  that  that  scholarly  or  by  copying  shall  not  his or be  ii Abstract A  computational  procedure  inelastic  r e s p o n s e s and d u c t i l i t y  buildings  subjected  presented.  modal  involves  a  damping  fictitious  is  line,  is  linear  building  interconnected  individual It  but  evaluation  three  earthquake  structure an  structure  excitations i s  method  iterative  is by  idealized rigid  not n e c e s s a r y  so t h a t  f o r a l l storey  structures  which  stiffness  of p l a n e  Coupling  i s taken  and  demands.  an a s s e m b l y  diaphragms.  f r a m e s t h r o u g h common c o l u m n s  i s b a s e d on  procedure  whose  as  of  dimensional  c h a r a c t e r i s t i c s a r e r e l a t e d t o the d u c t i l i t y  The frames  substitute  analysis  the  demands of  t o multi-component  The m o d i f i e d  elastic  for  into  of  account.  masses t o l i e on a v e r t i c a l  with varying  floor  dimensions  can  be  modelled.  Two r e i n f o r c e d moment the  resisting  method.  structures  frames and c o u p l e d  The a c c u r a c y  demands and l a t e r a l averaged  concrete  o f t h e method  deflections  results obtained  wall  cores,  the time  method  fills  step  types,  are tested  for predicting  i s determined  from  of d i f f e r e n t  with  ductility  by c o m p a r i s o n analysis  with  program  DRAIN-TABS. The  proposed  modal a n a l y s i s  which cannot  behaviour  the  which  of  provide  structure,  i s complex and e x p e n s i v e  the void information  and f u l l to use.  between  scale  time  on step  elastic  inelastic analysis  iii  TABLE OF CONTENTS  Page  ABSTRACT  i i  TABLE OF CONTENTS  i i i  L I S T OF TABLES  vi  L I S T OF FIGURES  v i i  DEDICATION  x  ACKNOWLEDGEMENTS  xi  CHAPTER 1 . INTRODUCTION 1.1  Background  1.2  Review o f P r e v i o u s  1.3  P u r p o s e a n d Scope  1.4  A s s u m p t i o n s and L i m i t a t i o n s  2. STRUCTURAL  1 Work  5 7 10  IDEALIZATION  2.1  Introduction  2.2  Inter-frame  2.3  C o n d e n s e d Frame  2.4  Assembling Stiffness  ,  11  Compatibilities  and  Stiffness  14  Matrix  Condensing  of  16 Structure  Matrix  3. MODIFIED SUBSTITUTE  18  STRUCTURE METHOD  3.1  Introduction  3.2  Substitute  3.3  Modified  3.4  Convergence Routine  Structure  Substitute  '. Method S t r u c t u r e Method  21 22 27 32  iv  3.5  Two Damage R a t i o s  P e r Member  34  4. EARTHQUAKE ANALYSIS 4.1  Introduction  4.2  Dynamic  4.3  Mode Shapes  4.4  Spectrum  Analysis  4.5  Complete  Quadratic  4.6  Multi-component  5. COMPUTER  .36  Equilibrium  Equation  39  and F r e q u e n c i e s  42 43  C o m b i n a t i o n Method  Ground  46  Motions  49  PROGRAM  5.1  Program  Concepts  52  5.2  Program  Organization  53  5.3  Design Spectra  55  6. PROGRAM TESTING 6.1  Testing 6.1.1  for Elastic  Comparison Program  6.1.2  6.2  Analysis  57  W i t h A n o t h e r Modal  Analysis  - ETABS  Numerical  57  Examples  (a)  Five-storey  Frame S t r u c t u r e  (b)  Five-storey  Coupled Wall S t r u c t u r e  Testing 6.2.1  for Inelastic  Assumptions  6.2.2  Numerical  Program  W i t h A Time  - DRAIN-TABS ....  61  Examples  Five-storey  Frame S t r u c t u r e  (b)  Five-storey  Frame S t r u c t u r e  (c)  59 60  (a)  Times  ...  Analysis  f o r Comparison  Step A n a l y s i s  58  64 With Four  The Mass  Five-storey  Coupled Wall S t r u c t u r e  66 ...  67  V  (d)  Five-storey With  6.2.3  Four  Coupled  Wall  Times The Mass  Costs of Execution  BIBLIOGRAPHY  APPENDIX B - Sample  68 69 7 1  7. C O N C L U S I O N S  APPENDIX A - Program  Structure  114 User's  Manual  Input and Output  116 128  v i  L I S T OF TABLES  Table  Page  2.1  Member  4.1(a)  Amplification Factors  S t i f f n e s s Matrix  Recommended (b)  Recommended Basic  6.1  Elastic  Building  Elastic  Spectrum 6.3  6.4  Bounds 74  Bounds  75  Results  for  Five-storey  ( F i g . 6 . 1 ) Subjected  To 76  Building  Results  for  ( F i g . 6.2)  Five-storey  Subjected  To  'A'  77  of L a t e r a l Displacements  Five-storey  Comparison for  f o r Ground Motion  Modal A n a l y s i s  Comparison for  Bounds  ' A'  Coupled Wall  Frame 4  Frame S t r u c t u r e  78  of L a t e r a l Displacements  Five-storey  of  Frame  Structure  of  Frame 4  With  Revised  Mass 6.5  6.6  79  Comparison for  Coupled  Wall  Coupled  Mass  C o s t s of E x e c u t i o n  of  Wall  4  Structure  of L a t e r a l Displacements  Five-storey  Revised 6.7  of L a t e r a l Displacements  Five-storey  Comparison for  73  74  Modal A n a l y s i s  Spectrum 6.2  f o r Ground Motion  ....  By NBCC  Ground M o t i o n  Frame  R i g i d Ends  By Newmark  Amplification Factors  4.2  Including  Wall  of  ' 80 Wall  Structure  4  With 81 82  vii  L I S T OF  FIGURES  Figure 1.1  Page Floor  1.2  Plan  Shear  Configuration  Wall Wall  Showing  and North  Building,  This  Shows The  Wall  and  North  East  East  After  Penney The  C o r n e r of The  in  Eccentric 83  Corner The  of The  Building  Highly  of  1964  Complete C o l l a p s e  Portions  Roof  The  J.  C.  Earthquake.  of  The  and F l o o r s  ,Shear at  The  Building  84  G r o s s Frame Degrees of Freedom  85  (b)  Condensed  of Freedom  85  (c)  Gross Structure  Degrees of Freedom  86  (d)  Condensed Plan  Frame D e g r e e s  Structure  View  of  and Diaphragm  Degrees of Freedom  n-th  Physical  4.1  Newmark's  Idealized  ( 0 . 5 g Max.  Ground  Plot  Floor  Horizontal  3.1  4.2  J . C.  Alaska,  Penney  2.2  The  Anchorage,  East  2.1(a)  of  Interpretation  Showing  86 Frame  and  Displacements  87  o f Damage R a t i o  88  Elastic  Design  Acceleration,  of Cross-modal C o e f f i c i e n t s  Spectrum  5% D a m p i n g ) vs.  Ratio  Periods  89 of ' 90  4.3  Five  Storey  4.4  Periods  4.5  Comparison of Modal  5.1  General  Conceptual  5.2  Program  (PITSA) O r g a n i z a t i o n  and  Building  Example  Directions  o f Mode S h a p e s  C o m b i n a t i o n Methods Outline  of PITSA  91 92 93 94 95  viii  5.3.  Spectrum  ' A'  5.4  Acceleration Spectrum  5.5  Spectra  N69W  of  Taft  S21W  Spectra  and  o f E l C e n t r o EW and  'A*  99  Acceleration  Spectra  of E l C e n t r o NS and  'A'  100  Motion  Bound  Tripartite  E a r t h q u a k e s Which Make Up Spectrum 6.1  and  98  Acceleration  Ground  Taft  'A*  Spectrum 5.8  of  97  Acceleration  Spectrum 5.7  Spectra  'A'  Spectrum 5.6  96  Dimensions  and  Properties  Plot  of Four  'A' (0.5g)  of  ... 101  Five-storey  Frame B u i l d i n g 6.2  102  Dimensions  and  Coupled Wall  Building  6.3  Damage R a t i o s  6.3.1  Beam  Properties  Five-storey 103  for Five-storey  Damage  of  Ratios  for  Frame S t r u c t u r e  Five-storey  ... 104  Frame  Structure 6.4  105  Damage R a t i o s With Revised  6.4.1  Beam  Damage  Frame  Structure  Mass  Damage  Structure 6.5  for Five-storey  106  Ratios  for Five-storey  With Revised  Ratios  for  Frame  Mass  Five-storey  107 Coupled  Wall  Structure 6.5.1  Beam Damage R a t i o s Structure  108 for Five-storey  Coupled  Wall 109  6.6  Damage  Ratios  Structure 6.6.1  Beam Damage Structure  6.7  Damage With  6.7.1  With  Beam  Ratios  With  Strong  Coupled  Wall  Mass  111  for Five-Storey.  Ratios  Wall 110  for Five-storey  Revised  Coupled  Mass  Beams a n d Weak  Damage  Structure  Five-storey  Revised  Ratios  With  Strong  for  for  Frame  Structure  Columns  112  Five-Storey  Beams a n d Weak  Frame  Columns  . . . . 113  MY  MOTHER  MY  WIFE  MY  SON  STELLA NICHOLAS  xi  ACKNOWLEDGEMENT  The Dr.  D. L .  their is  author  Anderson,  encouragement,  indebted  to  D r . N. D. support  suggestions.  proofreading  thank  Nathan,  and  advisors,  comments.  The  and D r . N. D. Nathan  forh i sassistance  in  John  author  valuable  Stevens  converting  for  for their  f o r making e x t r e m e l y  T h a n k s a r e a l s o due t o Mr.  Centre  h i s three  and D r . S. C h e r r y  and c r i t i c a l  t o D r . D. L . A n d e r s o n  conscientious  Computing  wishes  of  the  the  program  DRAIN-TABS. The  financial  support  Engineering  Research  Assistantship  i s gratefully  Finally without Frank  this  the cooperation  Lam, and W i l l i a m  Council  of  the  Natural  in  the  form  a  Research  have been  completed  of f e l l o w graduate  students,  acknowledged.  thesis  would  and a d v i c e Tong.  of  S c i e n c e s and  not  1  CHAPTER 1 INTRODUCTION  1.1  Bac k q r o u n d Current  the  selection  behave but  behaviour in  the  cannot,  s m a l l or  is  response  i n which  medium  i n many s t r u c t u r e s ,  to  based  estimate  of a  Methods  elastic  members.  techniques  in  before  earthquakes. applied and The  If  o t h e r s - under the  design  the  result action  philosophy  i t i s necessary to e s t a b l i s h  which  of is  of t h e s e  yielded  members  to  members  must  on  the  some members strong-motion  t h e m a g n i t u d e of t h e c o r r e s p o n d i n g i n e l a s t i c capacities  of  p r o v i d e i n f o r m a t i o n on t h e i n e l a s t i c demands  yielding  in  assumptions  o r on t h e d u c t i l i t y  design  is  i s inevitable in  behaviour of the s t r u c t u r e , Present  it  structure  earthquakes.  on l i n e a r  the  results  However,  behaviour  strong  collapse,  probably  small shocks.  yet i n e l a s t i c  analysis  easy  will  disturbances  catastrophic  the response  permits  the elements  seismic  without  under  subjected to  response  practice  i s assumed, which  accurately  range;  design  relatively  behaviour  to predict  many s t r u c t u r e s dynamic  It  i f elastic  inelastic  system  inelastically,  shocks.  satisfactory  difficult  under  deform  major  resistant  of a s t r u c t u r a l  elastically  will  under  earthquake  be c o r r e c t l y will  yield,  deformations. be  checked  to  2  ensure  that  the d e s i r e d  ductility  High d u c t i l i t y large is  inelastic  a very  greatly  reduces  earthquake  and  deflections  without  the f o r c e s  that The  a ductile  of  a  building  sustain  must  i s an  by  This  since  resist  due  i s s e t i n motion  of t h e e n e r g y  structure  to  or c o l l a p s e .  structure  building  the d i s s i p a t i o n  of  the  of a b u i l d i n g  failure  characteristic  excitation.  earthquake  be a c h i e v e d .  i s the a b i l i t y  deflections  important  can  large  by  it to the  inelastic  important part  of  the  seismic design. Three design  alternatives  l o a d s due  (i)  The  to  (ii)  available  to  estimate  the  earthquake:  equivalent  National  are  static  Building  loading  as  Code of Canada  A dynamic a n a l y s i s  based  on  s u g g e s t e d by  the  (NBCC);  the  response  spectrum  technique; (iii)  loads of  the  soil  A  complete  analysis  r e s p o n s e of t h e p r o p o s e d  The  Building  National  t o be a p p r o x i m a t e d l o a d s depends on condition,  lateral  the  first  and  mode shape. that  over  structure:  by  static  the s e i s m i c  lateral  yielding  assumed  i s widespread  i e . , that  i n a d v a n c e of t h e o t h e r s and  and  earthquake  The  magnitude  fundamental The  period,  distribution  building  response  the  structure.  loads.  zone,  t h e h e i g h t o f the The  to obtain  Code of Canada a l l o w s  usage of t h e b u i l d i n g .  loads over  assumption the  response  time h i s t o r y  the  far  dynamic  of  approximates  i s predicated  on  the  distributed uniformly  t h e r e a r e no members t h a t are thus s u b j e c t  to  yield  excessive  3  ductility  demands.  In a  more  complex  accurate  However, it  more  i s  distribution  in applying again  that  simultaneously  and that  members  be  doubtful  validity  be  used  in elastic  difficult. support Most  There  available  modal  test  relatively  results  tests,  accurately  reflect  damping  large  amplitude  A dynamic  is  point  often  response  accurately major  motions  how  a given  earthqakes.  Even  not p o s s i b l e to predict  movement analysis  at  a  given  i s t h e need  ensure  covering  future  earthquake.  Field  with the  site.  the  recent  assumption  most  damping very  might  important  be  data  probably  do n o t f o rthe  earthquake.  even  time  step  expected  to  site  respond  advances  drawback of  to  amplitude  be e x p e c t e d  i s that  and  of a s t r u c t u r e . small  a severe  for a suite  following  i s of  coefficients  characteristics  characteristics  observations  yielding  of damping  on a g i v e n  A major  to analyze  the  less  pronounced  on  with  cannot  building  more  with  i s  that  overlook  analysis  on  and the r e s u l t s  associated  being  or  place  few a p p l i c a b l e t e s t  a r e based  o r on c o m p a c t the  take  This  of the true  distortions  forces. design,  structures  analysis  estimate  a n d member  give  an a n a l y s i s t o  uniform.  many  analysis will  demand  the determination  are  an a c c u r a t e  will  the d u c t i l i t y  in  Also  of such  yielding  essentially  irregularities. to  of the loads  the r e s u l t s  assumed  will  s t r u c t u r e s , a modal  will  predict to  i n seismology, of  the  i t  ground  of the time  step  earthquakes  to  of the n o n - d e t e r m i n i s t i c  earthquakes  have  often  4  shown  structural  example by  of  the  what  J.C.  torsion  Penney  1964  earthqake.  Fig.  1.2  The  main  was  failure  eccentricity  wall  are  accidental  in  subsequent the  f.or a n y  simple to  inelastic method, proposed  use  by  on  f o r the  the and  demolished afterwards.  by  or  was  the  elastic  highly  other  ground  Buildings may  elements  load  due  to  of w a l l s ,  and  performance  of  cracking  and  become  Furthermore,  i n dead  the  from  rotational  axis.  side.  affect  Lateral  the  structural  uncertainties  severely  arise  response  i f the  material,  may  from  vertical  the  to earthquake  t o r s i o n a l motions  motions  may  be  of  coupled  reasons. there  is a  computer  Sozen  three inelastic  need  programs  1 6  for practical, efficient, which  This thesis  linear analysis  S h i b a t a and  General programs  before  torsional behaviour.  based  during  i n F i g . 1.1  performance  a  response  in a earthquake.  the above  shown  structures  during  caused  Therefore, and  in  workmanship and  subjectd  of  dismal  about  a l t e r a t i o n s may  building  buildings  motion  yield  torsion  variations  i t was  to asymmetrical layout,  nonlinear  side  that  Alaska,  suffered  configuration.  eccentric  during  one  i t s  building  A classic  i s damage  i n Anchorage,  responses  ground  not  eccentric on  due  of  to a building  Building  for  Torsional  that  do  so b a d l y damaged  shear  component  can  to t o r s i o n a l motion.  This five story  reason  eccentric  due  of a  can will  substitute  account develop  for  such  structure,  a as  .  dimensional analysis  of  finite complex  element  computer  structures  are  5  available, because: simple floor  but  input  they data  geometry  of  are  with  assumption  degrees  are  often  horizontal  i s not  freedom;  a  rigid  not  available,  cannot  be  used  to a c c u r a t e l y  a  restricted  number  of  include  ETABS  buildings,  for 2  buildings,  and  DRTABS  The  two  provide  require on  the  large other  analysis  1.2  or  Review  hand, can f o r modal  of  has  been  digital  building  analysis  of  analysis  for  inelastic  three time  memory and be  used  are  only  a  analysis  of  for  analysis  of  a  loading limited programs  two  dimensional  dimensional  expensive  sized  dimensional  history  either  integration  at  of  usually  the  three  inelastic  complete  finite  structure;  applied  rigid  number  dimensions  for  section of  on  reviews  analysis.  response to  run.  modal  linear  but  ETABS, spectrum  structures.  the  works w h i c h a r e  dynamic  structures. analytical  after  the  nonlinear  concrete done  especially  are  the  of  P r e v i o u s Work  development  reinforced  loads  the  large  to model  centerline  system  buildings  members;  a  option  Existing  a  computer  This the  3  in  building  for  vertical  describe  elastic  DRAIN-2D  latter  and  form, and  locations. 1  and  arm  for a  complex  made, r e s u l t i n g  is  of  suited  unnecessarily  joints  is  not  A considerable  models  development  analysis  and  of  methods amount  numerical  high  related  speed  of  to on  work  methods, electronic  computers. For  estimating  plane the  frame  inelastic  structures,  the  deformations  difficult is  the  part  of  modeling  of  6  inelastic used  material  because  of  representation to  account  Then  of  the  freedom with  stiffness in  found  that  and  ductility  stiffness of  axial is  model members  loads.  of  Sozen to  in  structures  observed  was  and  good  model  the B a u s c h i n g e r  in studies  6  an  of  type  fails  effect.  strain  hardening of model  to  degree  of  single  requirements  elasto-plastic  structures,  short  and  period  Nielson  flexural  reproducing  the  in shaking  with  relationship.  They  influence  the  but  increase  did  7  used  the  a more c o m p l i c a t e d  load  reversals  was  of  and  behaviour  of  tests  reinforced  w i t h or  proposed  Gulkan  table  ductility  structures.  behaviour  model  i n systems  relationships  the  t o as T a k e d a ' s m o d e l .  successful  this  Ramberg-Osgood  reproduce  under  But  force-displacement  This hysteresis  referred  and  d e g r a d a t i o n d i d not  requirements  hysteresis  steel.  the d u c t i l i t y  long period  Takeda,  concrete  with  model"  simplicity  to s i m u l a t e the a  Johnston  degrading  elasto-plastic  effect.  compared  systems  requirements  used  adopted  5  of  hardening  was  Bauschinger  systems  those  behaviour  model  Clough  The  mathematical  strain  Jennings  simulate  its  the  for  a bilinear  effect.  behaviour.  by Sozen  without  Takeda  and  have  been  8  concrete  with  t h e use  frame of 'this  mode1.  For and  three dimensional  torsional  subjected  to  motions  seismic  lateral-torsional  are  ground  response  structures, usually motions.  problem  both  present Most assume  translational when  studies linearly  they  are  on  the  elastic  7  force-displacement are  not  relationships  applicable  designed  t o deform  very  intense  study  concerned  9  , 0  buildings  because  beyond  the y i e l d  limit  with  coupled in  They  indicated  that  yielding,  and  more  affects degree Tso  than  and  Purpose  and  The  i s larger  purpose  researchers  response  of s t r u c t u r e s  on  than  previously  and  of t h i s  thesis  concerned to  behaved  of freedom  systems  of  with  procedure  Yoshida " 1  computer PITSA,  1 3  on t h e members  handling  simplifications. MacKenzie  more system,  to a  lesser  However,  of e c c e n t r i c i t y  on  reported.  the  the  A  provides  It  i s  high-rise  structures  a  of  hybrid  and  Metten  program  1 5  on t h e i n e l a s t i c  put together  stands  elastic  for  2  in s  e  u  d  o  this  and  inelastic  on  the  procedure  without  major  developed  problem,  problem.  thesis  simple,  information  This  methods  i s  of  torsional  excitation. which  work  inelastic  i s presented.  on t h e t h r e e d i m e n s i o n a l  which  a  coupling  systems.  i s to further  earthquake  economical  demands  capable  of  ranges.  torsional  the influence  out a  response  the t h r e e d i m e n s i o n a l behaviour of the s t r u c t u r e  ductility is  Thus,  to  Scope  previous  effective,  degree  in inelastic  that  carried  and i n e l a s t i c  i n corresponding elastic found  1 2  demand  in translation.  usually  moderate  have  1 1  studies  are  the structure  single  deformations  i t does  Sadek  the e l a s t i c  inelastic  primarily  maximum  ductility  1.3  an  they  lateral-torsional  structure after  of these  during  Kan a n d C h o p r a  single . story  responding  Results  to  earthquakes.  like  .  and  by by  T h e FORTRAN code  torsional  named seismic  8  analysis.  The the  three dimensional  stiffness  freedom frame  of  for  is  plane  each  condensed  to  levels,  w e l l as  as  frames  floor.  obtained  those  vertical  compatibility  An  overall  structure  frames a r e  assembled  translations  and  matrix  assembling degrees  of  each  and  of  is further  rotation  at  freedom  floor  required frames.  i s obtained a f t e r  condensed  the  plane  at  a t common columns of a d j a c e n t matrix  of  statically  translations  degrees  stiffness  and  one  horizontal  by  three  techniques  additional  for  only  stiffness  standard  only  i s handled  using  The  using  retain  problem  center  to r e t a i n  only  of  of  mass  the two each  floor.  The substitute Metten " 1  1 5  inelastic structure  .  It elastic  analysis.  T h i s method  evolves  Sozen' .  It  6  because  In  direction  a  to modify  been d e v e l o p e d  structure  is a  to  bridge  full  scale  modified  concept  stiffness  and  by by  the  gap  time  modal  proposed  by  and  between  inelastic  concrete  modified  Yoshida  the  elastic-  for reinforced  step  analysis, Shibata  and  s t r u c t u r e s only  damping m a t r i c e s  have  not  of  the  for other m a t e r i a l s . asymmetrical  i n two is  a  design  i s intended  rules  and  handled  developed  intended  analysis  from  is  method  is  simplified  which  problem  buildings,  directions  affected  direction,  but  also  direction.  Therefore,  i s coupled;  not by  only  ground  both  by  the  response  i e . , the ground  response motion  in  in  one that  motions p e r p e n d i c u l a r to that  horizontal  components  of  ground  9  motion  should  rotation to  be  about  some t y p e s  a vertical  attempt  were i m p o s e d  make i t  more  altered  to  the  height  mass  of  each was  ends  a  Yoshida "  i s made t o the  A third  component,  a l s o cause unexpected  damage  of  remove some of  the  restrictions  components of  the  procedure  original The  elastic  torsional  for  a change  i n the  the  structure.  As  floor  need  developed member,  and  1  a x i s , can  flexible.  along  of  on  allow  technique  simultaneously.  of s t r u c t u r e .  An that  considered  Metten  not  to  to account  whereas t h e 1 5  only  center  program  of mass  a result,  the  previous  considered  centers  largest  of  line.  demand a t  inelastic  the  was  location  l i e in a straight  for d u c t i l i t y  to  A  both  a n a l y s i s of  ductility  in  e a c h member.  This  thesis  a s s u m p t i o n s made t o analysis.  Then  presented as  discussed.  in earthquake with of  more  the  dimensional  substitute  a  this  chapter  analysis.  a p p l i c a t i o n of  ground the  elastic  final  the  Some  are  describing  i s presented excitation version  I t i s shown t h a t ductility  modal  and  some  the  of  method and  also  for the  is discussed. of  the  ductility  demand  results  presented  t e s t e d under the  and  s t r u c t u r e method i s  method  w a l l b u i l d i n g s are  importantly,  theory  A modal c o m b i n a t i o n  spaced p e r i o d s  earthquake e x c i t a t i o n s . and,  three  using  multi-component  shear  discussing  fundamental p r i n c i p l e s .  is  closely  frame and  by  modified  previously  Then comes t h e where  the  Following  structures question  perform  t o o u t l i n e the  reported  concepts  begins  method  different factors  patterns  are  10  consistent to  with  expect  feel  fullscale  "exact"  time  step analysis^  answers, but t o p r o v i d e  f o r the behaviour  members a r e u n d e r g o i n g In t h e f i n a l  of h i s s t r u c t u r e .  chapter,  presented.  1.4  A s s u m p t i o n s and L i m i t a t i o n s  some common respect with  mathematical  to  A l s o , bad s p o t s ,  out  i n the program  and  plane  t o be r i g i d  negligible  deformations.  on  with  stiffness  The t o r s i o n a l  rotation  r e l a t i o n s h i p a t a member end i s assumed t o be b i l i n e a r  assumed  for  relatively along as  the  columns.  height  No a x i a l - f l e x u r e  load  test  The  is  not  is  structures  are  Combination possible  moment-  interaction  l a y o u t and s t i f f n e s s  of t h e s t r u c t u r e s .  p l u s dead  neglected.  Furthermore,  regular in overall  earthquake  stage.  degradation.  is  i s based  of  no s t i f f n e s s  members  of  study  rigidity  with  individual  of  where  c a n be l o c a t e d .  F l o o r s a r e assumed  to in-plane deformations  respect  with a  c o n c l u s i o n s of the p r e s e n t  model u s e d  approximations.  i s not  the designer  excessive deformations,  are  The  The i d e a  at  distribution of loads the  such  present  CHAPTER 2 STRUCTURAL  IDEALT ZATION  2 . 1 Introduction For common  to  the purpose  reduce  of s t a t i c  a building  into  lateral  c a n be s o l v e d i n two d i m e n s i o n s .  method  not  building  acceptable  i s symmetrical  mass a n d s t i f f n e s s symmetrical stiffness,  in  f o r earthquake  the  or having  i t i s necessary  height.  is  f r a m e s so t h a t approximate  analysis  unless the  d i s t r i b u t i o n of  For  buildings  uneven d i s t r i b u t i o n  t o model them  i t  This  i n shape a n d h a s u n i f o r m  throughout  shape  analysis,  separate plane  the problem is  load  as  not  o f mass o r  three  dimensional  structures.  However, analysis  an  exact  three  i s not required f o r the  majority  approximations  c a n be made t o s i m p l i f y  data  computer  each  and reduce floor  a c t s as a r i g i d  the h o r i z o n t a l uniquely  only  displacements  determined  the v e r t i c a l normal  time.  axis  dimensional  The  assumption  i n i t s own p l a n e ,  is  floor.  t o t h e diaphragm,  Two  and one r o t a t i o n  Consequently  the  a n d have no a x i a l  that  so t h a t  of a l l p o i n t s i n t h e diaphragm  by two t r a n s l a t i o n s  of each  buildings.  the p r e p a r a t i o n of i n p u t  first  diaphragm  of  structural  beams  are about bend  deformations.  .  The  second assumption  lumped  into  diaphragms are  2  i s that  t h e masses of e a c h  translational  and  for horizontal' inertia  transferred  motions  1  of  t o t h e frames  diaphragms.  rotational effects.  Also  the t o r s i o n a l  can  inertias The  and s h e a r w a l l s  floor  of  inertia  be the  loads  through r i g i d stiffnesses  body  for a l l  members a r e i g n o r e d . The plane Each  structure  frame  must  and as  associated  wall  frames  beams  plan.  properties.  rigid  Isolated  of two column ends  diaphragm  diaphragms. arbitrarily walls  are  line  having the  shear w a l l s a r e  considered  lines  representing  and one bay o f c o u p l i n g  the j o i n t s .  substructure,  and a t j o i n t s  be  shear  o f one column  Coupled  of a s e r i e s of  horizontal  p l a n e , b u t may  consisting  a s an i n d e p e n d e n t  by a r i g i d  Each  frame i s  connected a t each  which  floor  a r e common t o more t h a n  frame.  Column reference columns  lines  centerlines used  rigid  ends,  Each stiffness Table  floor  i n the d e s c r i p t i o n  F o r beams w i t h r i g i d  set equal to half  with  and  and beams c a n have r i g i d  allowed. are  in  frames  consisting  with  treated  be i n t h e v e r t i c a l  oriented  considered  one  a s composed  f r a m e s , c o n n e c t e d t o g e t h e r by r i g i d  located  as  i s considered  ends  ends,  of  each  below  the average depth of g i r d e r s  Axial,  member c o n t r i b u t e s  flexural,  and  frame.  Both  where no d e f o r m a t i o n s a r e  w i d t h of t h e columns  individual  form t h e b a s i c  the l e n g t h s of  m a t r i x v i a t h e 6 by 6 member  2.1.  levels  ends  and f o r columns  on e i t h e r  side.  to the t o t a l  stiffness shear  these  frame  m a t r i x shown i n  deformations  are  13  included. to  the r i g i d  Note  extra  ends.  terms  in this  member s t i f f n e s s  matrix  due  14  2.2 I n t e r - f r a m e The planes  is  building beams floor  Compatibilities  assumption  made r o u t i n e l y  structures.  in  that  each  each  joint:  vertical  lateral  translation  are  axial  assumed  frame has  displacement  In t h i s  are  rigid  t o be z e r o .  two  and  per s t o r y .  degrees in-plane  frames  Thus  i n a column which  This  requirement  bending  mode r a t h e r  coupling  between  buildings. motion resulting transferred  into  through  When angle, the  two  when  intersecting frames  stiffer t h e frames  the r i g i d  freedom  per  and one  to  two  the  frames  i s common be  such  is  as an  to  Lateral  t o two same.  deforms  there  resistance  structure.  the  frame  perpendicular to the  intersecting  ina  strong  tube-type earthquake  that  motion  forces  p e r p e n d i c u l a r t o the motion  are  by s h e a r  beams.  two frames  the r o t a t i o n  rotation  where  a  to  t h a n a s h e a r mode o r when  contribute  in  common  deformations w i l l  important  Consequently  will  transfer  the a x i a l  is  the  with regard to v e r t i c a l  are  frames,  of  rotation;  which  intersecting  m o d e l s of  With of  own  ( F i g . 2.1(a))  program, c o m p a t i b i l i t y  enforced.  their  deformations  displacements at j o i n t s is  in  i n f o r m u l a t i n g mathematical  As a r e s u l t ,  frame  assumption,  floors  meet a t some a n g l e o t h e r t h a n a r i g h t  of e x t e r i o r  beams i n one frame  will  o f t h e a d j a c e n t beams i n t h e o t h e r f r a m e . frames  are in line  desirable  t o have b o t h  imposed  on  the  or c l o s e  vertical  common  joints.  and  to being  rotational  However,  affect In c a s e s  in line,  i t  is  compatibility  frames  typically  . 1 5 .  intersect  each  compatibility  other  at  requirement  right t h e n would  beam and t h e t o r s i o n of a n o t h e r . torsional  resistance  rotational  compatibility With  place,  the  with l a t e r a l degrees  displacements  at  frame  The  condensed  structure  frame  such t h a t  per  floor:  "additional" columns. process freedom matrix.  freedom  with  associated The  inertia  matrix  forces,  of  to form  ( F i g . 2.1(d))  eliminate the  with  and  vertical of  are eliminated  from  c o n d e n s a t i o n . ( F i g . 2.1(b))  and  freedom  Finally,  in  degrees  the assumption  translational  degrees  levels  m a t r i c e s a r e then assembled on  the  a r e t h e ones  inactive  there are three generalized  ( F i g . 2.1(c))  to  floor  colums.  that  compatibility  displacements at the  stiffness  two  i s used  vertical  frame  m a t r i x by s t a t i c  stiffness  rigid  and  o f one  i s negligible, this  i n each  common  stiffness  the bending  i s usually ignored.  of freedom  of  not a s s o c i a t e d  the  involve  rotational  S i n c e i t i s assumed  requirement  horizontal  "additional"  freedom,  The  of any member s e c t i o n  the a c t i v e degrees  associated  angles.  that  degrees one  these desired  same  f l o o r s are of  freedom  rotational  associated  the  into a  with  static  "additional" structure  plus common  condensation degrees  of  stiffness  16  2.3:Condensed Frame S t i f f n e s s  The obtained  by  subjected  condensed  first  stiffness  writing  to a given  Matrix  the  s e t of  matrix  stiffness  for  equation  each for  frame i s any  frame  loads:  { P  } = [ K  ]• { p  }  (2.1)  { P  } = frame  force  [ K  ] = frame  stiffness  { p  } = frame d i s p l a c e m e n t  where  Let  Eqn.(2.1)  the  first  one  "additional"  generalized  partitioned  involves  degrees  displacements  {0}  be  vector  of  along  into  {x},  remaining  forces corresponding  since only horizontal.forces  r  A  K and  21  {x}  and  the  degrees {A}  second of  will  {h}, one  freedom  be  a  that plus  involves {A}.  null  The  vector  are c o n s i d e r e d .  } =  { 0 } =  K2  2  t o g e t h e r and KM  Ki 2  K ,  K 2  of E q n . ( 2 . 3 ) { F  displacements  such  2.2)  2  expansion  to  sub-equations  -  0 {H}  two  K,  H  x  grouping  vector  horizontal  freedom  the  matrix  calling  {F} (2.3)  2  yields: 2  ] { A }  (2 .4a)  2 2  ] { A }  (2.4b)  [ K,,  ] { f } + [ K,  [ K  ] { f ] +  2 I  it  [ K  1 7  rearranging give:  Eqn.(2.4b)  and -  { F } = [ K,, { A Eqn.(2.5a)  } = -[ K "  K,  substituing K "  2  K ,  1  2 2  Eqn.(2.4a)  ] { f }  2  will  (2.5a)  ] { f }  2  can be w r i t t e n  K ,  1  2 2  into  (2.5b)  as:  K  Khx  (2.6)  K xh where K hh  -  K xh The  K  [ K, ,  K  K " K , 1  1 2  2 2  (2.7)  ]  2  xx  stiffness  matrix  {H},  defined  relates  generalized  forces  {h} , {x} and  i s known a s t h e .condensed frame s t i f f n e s s  The m a t r i x major  reasons.  generally storage for  inversion  of  process  its  Furthermore, Eqn.(2.7)  becomes  ]  and would  a  matrix  [L L ] . T  [K  [K  Then  [K  ]=[R , ], 2  for [K  large  _ 1 2 2  by  and  so, the  ]  - 1 2 2  the  i s written right  T  2  time.  overcomes  i n banded  triangular  form' and  matrix as  hand  ] is  required  Choleski  i s stored  two  amounts of  amounts o f computer  [K,,-(L"'K ,) (L"'K ,)]. 2  banded,  i s considerable  ]  matrix.  multiplications  of a l o w e r  T  1 2  2 2  is  the  displacements  computed  require  of s o l u t i o n d e v e l o p e d The m a t r i x  since  2 2  t h e number of  i n t o the product  transpose,  [K  would use up l a r g e  problems.  decomposed  i s not a c t u a l l y  but f u l l  such  A method these  to the g e n e r a l i z e d  although  Second,  inversion  ~ ] 1  2 2  First  not banded  space.  [K  {Xj  by E q n . ( 2 . 7 )  times  [L" L~ ]. T  side  1  of  18  2.4  A s s e m b l i n g and Condensing In  from  order  the condensed  transformed global of r is  from  which  with  directional  the center  o f mass  1^  of  a  and  I  respectively,  ( s e e F i g . 2.2)  by  [I  of  the matrix mass  to  direction If  the  which  t h e frame  ],  e  system  this  frame  is  in  the  ,  r  the perpendicular  moving  The  line,  rotation  a  iocation  of each  q and  The  origin  leads  to a  mass  by  the  q and r d i r e c t i o n s  of a frame from  is first  positive  about  Since  the  the centers  diagonal  frame  consists  defined  distance  frame  in i t s defined  as p o s i t i v e .  be  floor:  i s defined  the  given  matrix  relative  direction  center of mass  [ I  e  of  to the centers  positive cosines. causes  mass, I  rarely  ] i s used  centers  a  p r e s c r i b e s the s i g n of the d i r e c t i o n a l  vertical  floor  because  must  t o a common  i s rotation.  The p o s i t i o n  frame.  counter-clockwise defined  frame  matrix  terms.  orientation cosines,  of each  o f mass o f e a c h  and 6 which  no o f f - d i a g o n a l  The  the l a t t e r  The g l o b a l c o o r d i n a t e  at the center  are translations, at  matrices,  coordinates  system.  Matrix  the s t r u c t u r e s t i f f n e s s  stiffness  the l o c a l  coordinates  located  matrix  assemble  frame  coordinate  three  to  of S t r u c t u r e S t i f f n e s s  a e  l i e in  is a  t o define the  o f mass  at  each  level. The  coordinate  transformation  system  arbitrarily  i s obtained  in plan.  The  from  the  local  as follows f o r  rigid  floor  a  assumption  to  the  frame  global oriented  results  i n the  equation: {h}  =  I^{q}  +  I,{r}  +  [I ){6} e  (2.8)  19  giving  the system a s e t of v i r t u a l {6h} {H} = {6q} {Q} T  substitution  T  of eqn.(2.8)  into  displacements  + {6r} {R} T  leads t o :  + {89} {Q}  (2.9)  T  eqn.(2.9)  gives,  I <^ { 6 q } { H } + I { 8 r } { H } + [ l ] {6c9} {H} = { 5 q } { Q } { 5 r } { R } { 5 f 9 } T { 0 } T  T  T  r  T  +  T  +  e  (2.10) since them  the v i r t u a l may  vectors  are a r b i t r a r y ,  any two of  be s e t t o z e r o ,  I,  M  [ I ]H e  putting  displacement  Eqn.(2.8)  (2.11  = / R 0 v. y i n t o Eqn.(2.6) K Kh  K  gives: I^{q}+I  r  {r} + [ l  e  ]{dV (2.12  K, or K (2.13)  X  v. - y inserting  eqn.(2.11)  fs1 R 0  eqn.(2.13)  into  <\, r  l  l  K  leads t o :  hh Ir [ I * 3'Khh  \ -  f 9£l \  (  _  I  <), K xh  1  r %h K  3Kxh  (2.14)  20  The by  a  overall  direct  matrices  in  set  zero  to  frames these  form when  i s completed,  as  of shown  the  stiffness  the  no  i s now  transformed  in Eqn.(2.14).  summation  since  matrix  of  external  the  obtained  frame The  stiffness  forces  {x}  are  of  all  applied  at  contributions  forces  are  joints.  condensed  such  summation  the  The  This  structure  in  process  structure the  stiffness  same m a n n e r  yields  the  as  reduced  the  matrix  in Eqn.(2.14)  frame  stiffness  structure  stiffness  is  then  matrices. matrix  [K],  that,  (2.15)  21  CHAPTER 3 MODIFIED SUBSTITUTE STRUCTURE METHOD  3. 1  Introduction The  inelastic and  modified  method t h a t  extends  that  name s u g g e s t e d , structure  design  for  the  concrete analysis.  buildings  medium the  hazard  The m o d i f i e d  and  an  the modified  before  iterative  substitute  appears  The  substitute  so t h a t  advent  procedure with  damping  factors.  with to  nonlinear  work  well  a  exceeded.  of e x i s t i n g the  minimum  spectrum, f o r  method was  frames o f d i f f e r e n t s i z e s and s t r e n g t h s  method  substitute  structure t o be  As t h e  the  of  reduced Several  dynamic demand  of  flexural small  to  tested  and  dynamic a n a l y s i s .  The  for  were  created  reinforced  method computes t h e d u c t i l i t y  r e s u l t s compared w e l l  modified  concrete  evaluation  constructed  .  range.  response  i s not l i k e l y  1  1 6  analysis  to e s t a b l i s h the  to a given  thesis \  i s a pseudo  spectral  from  Sozen  procedure  response  earthquake  e a c h member u s i n g stiffness  and  members of a r e i n f o r c e d  In Y o s h i d a ' s  modal  developed  corresponding  inelastic  method  i n t o the i n e l a s t i c  Shibata  i s a design  forces,  tolerable  by  structure  elastic  t h e method was  method  individual  utilizes  technique  method  structure  substitute  frames  in  which  22  yielding  is  work b e t t e r and  not  e x t e n s i v e and  f o r those  the e f f e c t  Metten  1 5  to .  tested  seismic  demands  in this  is  of d u c t i l i t y  Substitute In  Structure order  features  structure Sozen ,  method which  8  dissipation  system  method  broadens  using  The the  coupled  the a c t u a l  frame,  analysis  shown  structural five  to  and  the  modified  to  both  methods.  sixteen good  substitute  The  the  effects  linear  of  by  response  of  the  properties  related  substitute  frame  of  of  substitute  substitute  calculation  Gulkan  frame  and  energy  degree  spectrum.  concepts a  substitute  for a single  multi-degree  of  because  inelastic  to  the  of the  to  an  walls  provide  procedure  main  damping  for predicting  deflections.  the d e s i g n f o r c e  definition  and  by  i t s predecessor i s h e l p f u l  the o r d i n a r y  this  walls  o b t a i n e d from  i s an e x t e n s i o n o f a p r o c e d u r e  stiffness  modal  common  incorporates  freedom  method a r e  results  was  understand  to determine  structures.  The  and  Method  to  are  beams  refined  structural  of t h e method  r e q u i r e m e n t s and  s t r u c t u r e " method, a r e v i e w o f many  method was  coupled  method  in  to  predominant.  evaluated using  modified  appears  occurs mainly  modified  analysis.  It also  s t u d y were of h e i g h t r a n g i n g from  The  estimates  the  resistant  time h i s t o r y  storeys.  3.2  after,  Again, the e f f e c t i v e n e s s  ductility inelastic  yielding  of h i g h e r modes i s n o t  Shortly extended  i n which  widespread.  of The  freedom structure with  its  t o but d i f f e r i n g  from  design  using  forces  a linear  from  response  23  spectrum. actual  It  is  assumed  t h a t p r e l i m i n a r y member s i z e s  s t r u c t u r e a r e known t o t h e d e s i g n e r  and  other  the  response  tolerable  requirements. spectrum  damage  only  At t h e d i s c r e t i o n for  the  design  to  Shibata  and  (2) No a b r u p t  (3)  changes  earthquake  and  the  6  t h e method can be  the f o l l o w i n g :  i n one v e r t i c a l  i n geometry  plane.  o r mass a l o n g  the  height  the system.  Columns,  beams,  different should  limits  and  may  be  response,  designed but  be t h e same f o r a l l beams i n a g i v e n  A l l structural avoid  walls  of i n e l a s t i c  c o l u m n s on a g i v e n (4)  loads  of t h e d e s i g n e r a r e  Sozen'  to structures satisfying  (1) The s y s t e m c a n be a n a l y z e d  of  gravity  f o r t h e members.  According applied  from  of t h e  the  limit  bay and a l l  axis.  elements  significant  repeated  with  reversals  and j o i n t s  strength of  decay  are r e i n f o r c e d to as  a  result  the  anticipated  do  not  of  inelastic  displacements. (5) N o n s t r u c t u r a l structural Details stiffness  components  interfere  with  response.  o f t h e method a r e now d e s c r i b e d .  The  flexural  o f s u b s t i t u t e frame members a r e d e f i n e d a s , (EI ) . = Sc  (3.1)  24  where  ( E I ) ^=  cross-sectional member  (EI) -  member selected  l  stiffness  i n the substitute  = cross-sectional  a  M =  flexural  tolerable  i-th  of t h e  i-th  frame  flexural  i n the actual  of the  stiffness  frame  damage  ratio  f o r the  i-th  member  The section.  The  stiffness the  term,  plotted  ratio  rotation,  always yield  y  ,  moment.  rotation and  I t should than  be  the  the  where  relation  moment,  caused  ratio  by  the  implies  damage  based  case. the  between  M,  of end is  flexural that  c?y i s t h e r o t a t i o n  ratio  are  initial  interpretation  applied  that  hardening  of the  cracked  t o a n t i symmet r i c a l  8,  ductility  ratios  The  subjected  noted  f o r the s t r a i n  case.  Physical  T h e damage  reached  the f u l l y  the r a t i o  rotation,  the span. be  using  is  where  end  ductility  elastoplastic ductility  the  will  smaller  u,  stiffness.  i n F i g . 3.1  within n8  damage  ratio,  f o r . a beam  against  deformation  yield  damage  i s shown  i s calculated  a  t o the reduced  damage  moments  (EI) ,  a  at the  ratio  is  o n maximum t o  Quantitatively,  same  only  damage  for ratio  the and  ratio is  u  =  (3.2) 1+ ( T J - 1 ) S  where  u = damage  ratio  TJ = d u c t i l i t y s  = strain  Natural  ratio  hardening  periods,  mode  ratio shapes,  and modal  forces  f o r the  25  undamped  substitute  analysis.  From  structure  the  are obtained  member  moments, a smeared damping  forces,  ratio  from a l i n e a r  in particular  i s computed  dynamic  the bending  f o r e a c h mode.  m gm=  1  I (P.  \ I  0 .=  where  (3.3)  IP. L  0.02  + 0.2  (  1 -  (3.4)  1/V/T )  St,  P%  and  [( M  A  L  )  2+  (M[. )  2 +  (l%L  (3.5)  ) ]  6 (EI ) . SL  /3 = s u b s t i t u t e sL  P- =  relative  L  for L  flexural  factor strain  of i - t h  M  b e n d i n g moments a t t h e ends of s u b s t i t u t e  =  formula  on d y n a m i c t e s t s  to  smeared  the e f f e c t  is  to  strain  done  by  step  for  mode  assuming  i s based storey  is  required  that  factor  A  damping.  in proportion  single  f o r modal  each  element  to i t s relative  w i t h e a c h mode.  producing  t h e undamped c a s e .  e a c h mode a r e c o m b i n e d u s i n g  factor  e l e m e n t s and one  energy  i s t o repeat  ratios,  damping  f o r t h e v i s c o u s damping  each  energy a s s o c i a t e d  s m e a r e d damping those  for  t h e modal damping  The n e x t  from  concrete  of h y s t e r e t i c  ratio  This  contributes  the  f o r the s u b s t i t u t e  of r e i n f o r c e d  damping  analysis.  flexural  i f o r m-th mode  I t p r o v i d e s an e s t i m a t e  simulate  i n i - t h member  member  frame member  frames.  energy  m  a.L) bi.  The  f o r i - t h member  m-th mode  = length to  M  damping  the  modal  analysis  member f o r c e s  which  Then t h e member  using differ  forces for  t h e r o o t - s u m - s q u a r e method,  unless  26  t h e m a g n i t u d e o f t h e two l a r g e s t which  case  the  RSS  t e r m s o f t h e base structure by  failure  to  prevent  and  s t r u c t u r e method used ranging  from  to determine method,  specified  failure  one-bay  forces using  storey  the  calculated  A l l frames were  frame p r o d u c e d  substitute f o r columns  column  test  found  values  chosen.  The f i v e  greater  than  column  prior  to  frames  with  substitute a  height  T h e i r method o f t e s t i n g the  substitute to these  to analyze  forces.  the  on  of  ten  the  results  with  a  the  a n a l y s i s being storey  values  frames,  target The  average  c l o s e t o and  frame had one c o l u m n  For the t e n  storey  ratios frame,  were above u n i t y , w h i l e t h e  o f 5.5 and were a l l below  were f a v o r a b l e i n t h a t a v e r a g e  of a l l motions c o n s i d e r e d  A  to the i n i t i a l l y  based  results  was  structure  one, and a l l beam damage  of s i x .  beams had an a v e r a g e damage r a t i o summary,  the  designed  i n the time-step  only  three  testing  were compared  the best  value  values.  in  given i n  o f s i x f o r t h e beams and one f o r t h e c o l u m n s .  a damage r a t i o  ratios  the  moments a  the frames a c c o r d i n g  were below t h e d e s i g n  In  in  in  when  to ten s t o r e y s .  to design  ones.  ratio  below  step  a n a l y s i s program SAKE was u s e d  damage r a t i o s  with  three  t h e damage r a t i o s  three  last  Sozen,  three  the design  then  time-step  damage  similar  i n t h e beams. Shibata  and  The  i s to increase the design  twenty p e r c e n t  are  f o r c e s a r e a m p l i f i e d by a f a c t o r  shears.  method  contributions  were a l l l e s s  than  the  6.  damage target  27  3.3  Modified  S u b s t i t u t e S t r u c t u r e Method  The procedure  modified  substitute  for determining  corresponding represented  to a given by  structure.  the  type  design  The method  objective  the  location  and  and i n t e n s i t y spectrum,  is explicity  i s to identify  structure  inelastic  extent  which  that  ratio  but  modified  data.  stiffness  the  modified  procedure  deformation  damage r a t i o s  the  patterns  Its  f o r the  b u i l d i n g s and  i s performed  are calculated  mode s h a p e s o f t h e s y s t e m . smeared damping  in  value  time-step  the  analysis.  substitute  i s an i t e r a t i v e  from b e f o r e capacities  combination yield  so The  structure one.  i n that the  are s p e c i f i e d  initial as i n p u t  i s the output. of modal  moments,  Also  In t h e  forces  must  and e v e n t u a l l y t h e  the c o r r e c t values.  modal a n a l y s i s , i n the  analysis  properties are altered  f o r e a c h member  approach  spectral  s u b s t i t u t e damping, and damage  procedure  moment  specified  A elastic  eigenvectors  used  a suitable  will  elastic  inelastic  differs  damage r a t i o  process,  exceed  unity,  motion  f o r a reinforced concrete  a n d damping  agree with  and t h e y i e l d  The  iterative not  the  a r e t h e same as t h o s e  stiffness  damage  an a n a l y s i s p r o c e d u r e .  i s a modified  of s u b s t i t u t e s t i f f n e s s ,  method, this  procedure  the responses  concepts  of  a  a i d f o r new b u i l d i n g s .  The in  is  of earthquake  p u r p o s e s o f e v a l u a t i n g t h e p e r f o r m a n c e of e x i s t i n g as a d e s i g n  method  first  with  iteration.  t o determine A value  t h e damage r a t i o  Eigenvalues  and  t h e n a t u r a l p e r i o d s and  f o r damping  i s not a v a i l a b l e  set to  at t h i s  i s c h o s e n as t h e point.  Since the  28  chosen to  damping  two p e r c e n t  Modal  forces  v a l u e does n o t a f f e c t of c r i t i c a l  The  CQC method  Chapter their  4.  and  damage r a t i o s  a r e computed  In  (3.6a)  n  n  i n t h e n+1 t h i t e r a t i o n i n the n th i t e r a t i o n  M  = damage r a t i o  M  n  = t h e CQC moment  My  = the y i e l d  s  .= t h e s t r a i n there  formula:  1  = damage r a t i o  where  have  uM  M  the case  manner.  yield  modified according to the following  y  n M  in  i sdescribed in  members whose CQC moments e x c e e d  M (l-s)+s y  iteration.  and a r e combined  modal c o n t r i b u t i o n s  *  where  i t i s set  in complete-quadratic-combination  of combining Those  results  f o r a l l modes i n t h e f i r s t  and d i s p l a c e m e n t s  root-sum-square  the f i n a l  i n the n th i t e r a t i o n  moment hardening  ratio  i s no s t r a i n  hardening,  Eqn.(3.6a)  becomes y  h + 1  =  *  M The because initial  those  y  new damage r a t i o s members  stiffness.  structure account increased  that  The  for closely by t w e n t y  method  spaced  have have  final  method a r e o m i t t e d .  Starting' structure  (3.6b)  1  two There  a not  lower  limit  yielded  steps  in  of  still the  unity  have t h e substitute  i s no i n c r e a s e i n f o r c e s t o  p e r i o d s a n d column  moments a r e n o t  percent.  from  the  i s used  second  t o compute  iteration,  the  substitute  smeared damping  r a t i o s and  29  modal  f o r c e s which  in turn  produces  new damage r a t i o s ,  another  iteration  the  calculation  further  refined  forces  are  yield  values,  the  a  new  s e t of  either  below  or within  are a c c e p t a b l e i n r e l a t i o n  Yoshida plane  testing  and  Metten  structures  Yoshida that  tested  four  used by S h i b a t a  storey  2-bay  frame;  and a  program N69W,  6-storey  Taft  S21W,  method  seconds. predicted  that  from  frame.  eleven  average v a l u e s . with  t h e columns  damage r a t i o s  slightly  structure analysis  structures frame; The  3-storey  a l l beam  frame,  members  but 3-bay  'A' a s  were:  a  2-  a 6 - s t o r e y 1-bay analysis  earthquakes  (Taft  The c o m p a r i s o n f o r the  t o o v e r one h u n d r e d  the  method  correctly  storey  would  y i e l d , but  ratios  for  a l l were w i t h i n frame  within  on  used.  60% o f t h e a v e r a g e  o f damage  underestimated, The  seconds  on t h e f i r s t  The p r e d i c t i o n  been  time r e d u c t i o n s  were a b o u t  method,  programs  time-step  of four  w i t h computer  I t then  t h e same s p e c t r u m  These  F o r t h e 2 - s t o r e y 2-bay  four motions.  results  results  ratios  of the s t r u c t u r e .  E l C e n t r o NS, E l C e n t r o EW).  ranging  the p r e d i c t e d  were  under  3 - s t o r e y 3-bay 3-bay  of t h e i r  i f t h e s e damages  t h e m o d i f i e d method had  SAKE was used w i t h t h e r e c o r d s  showed v e r y good MSS  a  limit  values.  to determine  t i m e - s t e p dynamic  and S o z e n .  with  member  Then t h e damage  substitute  structures  frame;  a tolerable  to the d u c t i l i t y  f o r which  starting  a l l the  t o the c o r r e c t  the m o d i f i e d used  When  i s stopped.  designer's responsibility  With the  m a t r i x and e n d i n g w i t h a  ratios.  t o have c o n v e r g e d  In  i s performed,  stiffness  damage  the i t e r a t i o n  are considered is  of  new damage r a t i o s .  15%  showed  of the  the  beams  20% o f t h e the  best  of the p r e d i c t e d  30  values.  The columns on t h e t o p s t o r e y  exceeded  this  amount,  MSS method e s s e n t i a l l y frame,  i n that  damage  ratio  However  for this  produced  failed  predictions  step a n a l y s i s . method  b u t were c o n s e r v a t i v e l y  Finally, fairly  t h e h e i g h t of t h e frame  and  decreased  analysis.  quite  Otherwise,  the  u n i f o r m damage r a t i o s  damage r a t i o s  rapidly  with  records  using  3-bay  decrease  1-bay  v a l u e of the  earthquake  6-storey  The  step analyses.  when c a l c u l a t e d  with a small  But t h e a v e r a g e  time  different  results  for  that  predicted.  t o the average  several  type of s t r u c t u r e  predicted  storey.  from  different  ones  i n the case of t h e 6 - s t o r e y  i t d i d n o t come c l o s e  widely  were t h e o n l y  a time  frame,  the  i n t h e beams up  towards  the  top  were h i g h e r a t t h e b o t t o m height  t h e method worked  according  reasonably well  t o SAKE in  this  example.  A l t h o u g h a smoothed 'A'  deviates  that  using  estimate thesis  a real of  damage  spectrum  was  and  by  a real  and  s u c h as s p e c t r u m  spectra,  does  i t was o b s e r v e d  not g u a r a n t e e a  displacements.  with spectral  f o r some o f t h e  n o t be w o r t h w h i l e .  I t was  These  spectrum.  A  noted  fifty  marginal  i n computation  i s so  that  the  h i g h e r than  r e s u l t s may p a r t l y  t h e d i s c r e p a n c y between a smoothed response  In Y o s h i d a ' s  tests.  due t o E l C e n t r o EW were c o n s i s t e n t l y motions.  better  values evaluated at  a c h i e v e d but t h e i n c r e a s e  i t would  ratios  response  spectrum  ratios  those of the other ground caused  real  p e r i o d s was u s e d  improvement that  the  response  damage  a jagged  different  great  from  response spectrum  response  be  spectrum  31  The coupled under  time-step  capacity,  percent mass  greater  than  was  acceleration results tests  were  estimate  of  results  were  The  a  factor  to  a  part  has  in  uncoupled  wall.  a  much  effect  of on  structural  a  series a  given  wall  method c a n be used using  t h e MSS  managable  of  and and  sets beam  ground  t e s t s were 50 t o 100  by i n c r e a s i n g t h e the  fundamental such that the  in period.  The  next  a  very  The  set  of  Again, the reasonable  Much  the  same  of the s i x t e e n - s t o r e y  wall  that  ductility  s c a t t e r when d i f f e r e n t r e c o r d s I t was n o t p o s s i b l e  records  will  are  to  predict  t h e most  dramatic  a  sixteen-storey  u s e d t o d e m o n s t r a t e how  T h e r e was a t o t a l  complete  s e t o f damage r a t i o s .  produce  Finally,  example was  to  peak  values  The t e s t s showed  in practice.  method  a  ductility.  the case  used f o r  three  structural wall. provided  The  of c o u p l i n g  four,  better.  structure.  design  of  w i t h an i n c r e a s e  e x a m i n e d t h a n does d e f l e c t i o n . which  was  of the spectrum  were  greater  Yoshida.  at  Also,  to  structures  r e s u l t s of t h e s e  of  on a t e n - s t o r e y  obtained  values  damage r a t i o  by  deflection  by  consisted  tested  applied  three  DRAIN-2D  Two  100 K - F t , were  conservative  both  w i t h an e x t r a demand  walls.  frequencies  was p e r f o r m e d  results  program  response decreased  f o r these  used  t h e DRAIN-2D r u n s .  shifted  and  He t e s t e d  that  with p r e d i c t e d  of t h e s t r u c t u r e  period  as  of 20% of g r a v i t y .  conservative,  refined  test structures  structural  60 K - F t and  acceleration  'A'  analysis  The f i r s t  five-storey  further  by M e t t e n .  t h e same s p e c t r u m  comparison.  all  method was  s t r u c t u r a l walls  inelastic  of  MSS  the  design  of seven and  Some of t h e f a c t o r s  the runs  obtain  a  juggled in  32  the  series  of  runs  were t h e moment c a p a c i t i e s  columns,  a n d Young's m o d u l u s .  computer  runs  were p e r f o r m e d  that  the  MSS  method  ratio  and d e f l e c t i o n .  3.4 C o n v e r g e n c e The stop  the  certain  be  five  percent  is  of  t h e computed a  yielded  bending  is  to require  sucessive iterations in the current  the  case  the absolute  of  difference  iterations  summarize  t h e convergence  E q n . ( 3 . 7 b ) must  this  to  This  difference  The f i r s t  members. that  be l i m i t e d  is  criterion is i s less The  t h e change  iteration.  than  second i n damage  t o one p e r c e n t of  This  last  than  small value, the c r i t e r i o n  i sto  of  criteria.  ratio  condition less  0.1.  be s a t i s f i e d  involves  moment and t h e b e n d i n g  moment e r r o r  a l l  to  has r e a c h e d a  criterion  member.  for  criteria  solution  first  i f the bending  sucessive  and  showed  of b o t h damage  two  i n t h e c a s e o f a member w i t h damage  In  design,  The r e s u l t s  on  the  moment e r r o r .  capacity  criterion  between  waived  limit  The  of  damage r a t i o s  five.  accuracy.  satisfied  convergence  final  predictor  i s based  when  t o as the bending  to  the  scheme  process  between  said  ratios  of  capacity  referred  u s i n g DRAIN-2D.  i s i n d e e d a good  convergence  level  moment  of t h e  Routine  iterative  comparison  As a c h e c k  of l i n t e l s and  the  damage  The  following  Note t h a t  ratios  both  between  inequalities Eqn.(3.7a)  f o r convergence.  M - M n  CAP  —:  < 0.05 M  i f (i > 1  (3.7a)  33  n-l  n  < 0.01.  i f A< > 5  or  (3.7b) |  ^ n "  " V i ^  It stricter  is  condition  greater  than  ratios  greater  satisfied  two.  it  1  worthwhile  noting  and g o v e r n s Since  than  before As  •  0  most  two,  a  this  by  convergence  and save u n n e c e s s a r y  damage  previous  subtracting  ratios lies for  consistently a portion  rationale  direction  i s that  toward  oscillated, between  to  Those  f o r damage  have  routine  accelerate  r a t i o s that  to  The  plus  which  damage  those  two v a l u e s .  iterations,  ratio  have  uses  from t h e  decreased  by a p p r o p r i a t e l y  or  adding or  two  values.  t h e damage r a t i o s would move f a s t e r values.  be  was  routine  o f t h e d i f f e r e n c e of t h e l a s t  the m o d i f i c a t i o n  of m o d i f i c a t i o n  will  ratios  i s halted.  one  are modified  i s the  damage  are designed  iterations.  current  the c o r r e c t  the l a s t  two c o n s e c u t i v e  degree  the  two i t e r a t i o n s .  increased  The  ratios,  for  speeding  used  5  Eqn.(3.7b)  criterion  process  developed  three  is  that  structures  convergence  Metten  <  convergence  the i t e r a t i v e  well,  Id  in  a  In t h o s e c a s e s which t h e  results  in  a  value  which  I f t h e r a t i o s d i d n o t change  no m o d i f i c a t i o n  c a n be c o n t r o l l e d d u r i n g  is  made.  program  ' The  input.  34  3.5 Two Damage R a t i o s Per Member One o f t h e l i m i t a t i o n s that  members u n d e r g o r e v e r s e b e n d i n g  inflection the  a r e near  midspan.  force-displacement  rotation by  curve.  dividing  the a c t u a l  equal  yielding,  In and  flexural  flexural  this  point  previous  member  In  stiffness  matrix  i s not  the  t h e damage r a t i o  flexural by t h i s  e a c h end o f t h e member  effect.  As  a result,  are  calculated  bit  more c o m p l e x .  used  derivation repeated  ends s h o u l d  developed  damage r a t i o s  strain is here.  Also affected t h e member energy  covered The  i n the  ratio.  to  Basically,  account  this  substitute  matrices  ( R e f . 17),  improvements  brought  member a  formulas  and t h e r e l a t i v e  ( E q n . ( 3 . 4 ) and E q n . ( 3 . 5 ) ) . elsewhere  this  becomes  change a r e t h e damping  hinge  for  a t t h e ends o f e a c h  by t h i s  It  be c o n s i d e r e d  A new model w i t h a p l a s t i c  and m o d i f i c a t i o n o f s t i f f n e s s  to calculate  flexural  is  kept  i s calculated  damage  t h e end moments a r e n o t c l o s e t o e a c h o t h e r .  at  of both  .  are  components o f t h e  if  1 7  moments  o f t h e two end moments  are modified  been done by H u i  ratio.  important.  clear  has  obtained  end  columns  is  work  is  by t h e damage  ifall  iterations,  t h a t t h e damage r a t i o  of  e q u a l , but i s not so v a l i d f o r  programs,  subsequent  points  i s t h e shape o f  stiffness  beams where  However,  m o d i f i e d by u s i n g t h e b i g g e r  member.  the  assumption  t o t h a t o f t h e moment-  stiffness  f o r most  beams.  that  identical  or very c l o s e t o being  c o l u m n s and e x t e r i o r from  curve  is valid  and  i s the  Then and o n l y t h e n  The s u b s t i t u t e  T h i s assumption are  o f t h e method  it  Since will  about  not by  the be this  35  alteration offset the  a r e shown  by  program.  the added  t o be  significant,  complexity  and  although  higher  cost  this  is  partly  in execution  of  36  CHAPTER 4 EARTHQUAKE ANALYSIS  4.1  Introduct ion The  an  f o r c e s , which a s t r u c t u r e  earthquake,  motion. and  r e s u l t from t h e d i s t o r t i o n i n d u c e d  displacements and  character  of  displaced,  inertia  are influenced  during  by t h e g r o u n d  the  surrounding  the ground motion.  by t h e p r o p e r t i e s foundation  height  of both the  well  t h e base o f t h e s t r u c t u r e moves w i t h  t o undergo a d i s t o r t i o n .  the  as  forces  as  the  One c a n i m a g i n e a s t h e g r o u n d  o f t h e s t r u c t u r e mass r e s i s t s  structure along  to  The m a g n i t u d e and d i s t r i b u t i o n o f t h e r e s u l t i n g  structure  is  i s subjected  this  This  of the s t r u c t u r e  motion  i t .  But  the  and c a u s e s t h e  d i s t o r t i o n wave  and o s c i l l a t e s  travels  i n a complex  manner.  An loading  is  structure. exposed loads  important  distinction  the  in  way  which  Whereas wind l o a d s  surface resulting  the  magnitude  i s then a f u n c t i o n  surface.  Thus, the  stiffer  these  loads  are external  of t h e s t r u c t u r e , from  between wind  earthquake  and  earthquake  are induced  loads  i n the  applied  t o the  are  inertia  loads  d i s t o r t i o n of the s t r u c t u r e . of t h e mass r a t h e r and  heavier  than  structure  Their  i t s exposed does  not  37  necessarily  mean t h e s a f e r  There an  earthquake  difficulty and  i s a great  the d i f f i c u l t y  exact  However, of  a  continuing  of  any  structural  Since  earthquake  way  to  ascertain  and  site. geology  e s t i m a t e s of the expected parameters  such  c h a r a c t e r i s t i c s and d u r a t i o n  as of  pulses. any p a r t i c u l a r p o i n t , by h o r i z o n t a l  directions,  a  rotational  vertical  vertical  component  and  the  amplification. dimensional assuming  a  is  of  A further  response  of  usually  resistance  the s t r u c t u r e  s i m p l i f i c a t i o n of the  structure  components  to  means  While  gravity dynamic  actual  three-  sometimes made by  to act nonconcurrently I t i s assumed  acceleration  the  i s strong in  small  i n these  by t h i s a p p r o a c h w i l l  resultant  component.  carry  the is  could  perpendicular  negligible.  stiffness  designed  against, the  two  i t s requirement  of each p r i n c i p a l a x i s . structure  along  and a r o t a t i o n a l  significant  large  the h o r i z o n t a l  direction  component,  is  the ground a c c e l e r a t i o n  components  component  d i r e c t i o n because  loads;  that  earthquake  the  history  acceleration  frequency  a r e the  earthquake at a given  valuable  ground  be d e s c r i b e d  this  i s no  of the s e i s m i c  yield  acceleration,  At  The  future  ones  of  response.  there  i n performing  of t h e d e s i g n  the values  dynamic  studies  should  important  the character  the  r a n g e of s i g n i f i c a n t  large  most  in character,  nature  region  maximum  The  of e s t i m a t i n g  affecting  m o t i o n s a r e random the  number o f u n c e r t a i n t i e s  analysis.  of p r e d i c t i n g  parameters  design.  i n the cases  have  adequate  acting  i n any  38  direction.  The dynamic  properties  response  and  as a r e s u l t  the  onset  of  of y i e l d i n g .  of v i b r a t i o n  not  received  Convenience  in  viscous-type cases to  can  Yielding  systematic  modelling  damping  has  However,  the  during  the  members,  study  i n p l a c e of the a c t u a l  of  the  i s a problem  that the  even  increases the  evaluation  o f damping  required  estimates of the e q u i v a l e n t  in  further  The  ductility  mass and i n i t i a l  change  occurring  of t h e s t r u c t u r e .  the  the  affect i t s  and  easily.  damping,  cracks  which  damping  of  quite  n a t u r e , m a g n i t u d e , and d i s t r i b u t i o n has  structure  determination  stiffness  earthquake  period  The  p r o p e r t i e s c a n be made  effective  before  the  a r e i t s mass, s t i f f n e s s ,  characteristics. stiffness  of  i t  deserves.  assumption  mechanism.  v i s c o u s damping  that  of  a  In most  range  from  5%  10% o f c r i t i c a l .  A  convenient  way of s t u d y i n g t h e dynamic  structures  i s by c o n s i d e r i n g  component  modal  may be t h o u g h t modes  of  vibration vibrate each  of  considered response  total  that  response  The r e s p o n s e  of as the s u p e r p o s i t i o n  as degrees  other.  period  responses.  vibration.  such  the  freedom.  vibration.  In  with  each  mode  Each  mode  of  response  is  of  range  of  the  h a s a s many modes o f  each  as a s i n g l e - d e g r e e - o f - f r e e d o m total  terms  of the responses  mode,  t h e y m a i n t a i n t h e same p o s i t i o n s  Associated  the  in  i n the e l a s t i c  G e n e r a l l y , a system of  response of  is  a  the  masses  relative to  characteristic  vibration  then  c a n be  system.  For  seismic  made up p r e d o m i n a n t l y  of the  39  first  few  portion,  modes, w i t h except  t h e h i g h e r modes c o n t r i b u t i n g  perhaps  at  the  top  of  only a  relatively  small  flexible  buildings. Another involves This  the  at  conditions approach,  use  of  the  can  end  be  of  used  costs.  reliable  the  where  interval.  The  few  the  of  integration  the  equations However,  projects  because  t h e o n l y means of  of members  substitute  the  initial  analysis.  of as  the purposes  the  direct  important  thought  of  as  uncoupling  corresponding  of t h i s  structure  n o n l i n e a r dynamic  Equilibrium  The  motion  of  analysis  study  method  i s to  provides  results  at  a  cost.  Dynamic  behaviour  interval  requirements  modified  e s t i m a t e s of  much l o w e r  time  is  One  earthquake.  i n s t e p - b y - s t e p manner u s i n g  for a  It  analysis  integration  i n n o n l i n e a r dynamic only  dynamic  to a p a r t i c u l a r  r e q u i r e the  the d u c t i l i t y  that  with  numerical  one  succeeding  is justifiable high  response  proceeding  a design earthquake.  show  4.2  the  associated  direct  w h i c h does not  determining to  employs  f o r the  motion,  its  history  of m o t i o n ,  conditions  of  time  analysis  equations  approach  of  basic three  Equation differential  dimensional  equation  structures  g o v e r n i n g ' the  subjected  to  base  is, [M]  {ii} +  [M]  = mass  [C]  {u}  +  [K]  {u}  = -[M]  {Og}  matrix  [C] = damping m a t r i x  = a  [M]  + 0  [K]  (4.1)  40  [K]  = stiffness  {u} = r e l a t i v e  matrix displacement  vector {1}  {u }  = ground  q  {I}  = un i t v e c t o r  a  /3 . = s c a l a r  The into  inertia  the force  ordinary  since mass.  and r o t a t i o n a l effects.  system.  of t h e system  It  is  of motion  comprised  three global  9 sub-matrix The  generalized partitioned  which  would  structure  vector  degrees  are simply  {u},  instead be  the  of of  required mass m a t r i x  sub-matrices  the  a  set  the  for  the  i s diagonal centers  mass  The while  inertia.  displacements  t o the ground,  of  corresponding to  t h e masses of t h e system,  representing  of  partial  o f f r e e d o m : q, r , and 9.  to the r o t a t i o n a l  coordinates relative  the f o r m u l a t i o n  terms  a r e w r i t t e n about  of t h r e e  corresponds  in  t o be lumped  of t h e d i a p h r a g m s f o r  T h i s makes p o s s i b l e  Furthermore,  the equations  a r e assumed  inertias  equations .  equations  q and r s u b - m a t r i c e s the  constants  differential  continuous  vector  of the s t r u c t u r e  equilibrium  differential  the  masses  translational  horizontal of  acceleration  of  the  i s expressed  form.  (4.2)  Similar  relationships  are expressed  f o r the r e l a t i v e  velocity right from  and r e l a t i v e a c c e l e r a t i o n  hand base  side  of E q n . ( 4 . 1 ) i s  motion  vectors.  the  The q u a n t i t y  effective  load  i n the  resulting  and i s e x p r e s s e d a s  [M] {u } = -[M] ^ u 5  (4.3)  \  3 | r  -3,6 where  each  term  of  u  is  a a  t h e ground  acceleration  i n the q  direction. For  a viscous  response  of  a  possible  only  condition forms  damped  structure if  so t h a t  the  into  independent  in  stiffness  which  it  is  m a t r i x , or i s a  which  linear  uncouple  the e q u a t i o n s of motion  performed, required. is  On  implies  t h e o t h e r hand, the s c a l a r  dependent  progressively  would  be  desirable  higher  modes.  Among t h e  condition  to either  are  t h e mass or t h e  which  of  the  modal  a  of  undamped time-step  damping  i f full  appears  that  certain  by t h e same mode shapes  h i g h e r damping in  a  The  constants  damping  the  these.  the f r a c t i o n of c r i t i c a l  performed,  Stiffness  or  this  of  responses i s  modes e x i s t .  combination  uncoupled  analysis  modal  satisfies  satisfy  proportional  matrix  spectrum  matrix  uncoupled  damping  response  i s then  the decomposition  i t s component  damping  of t h e damping m a t r i x  those  system,  scale and  in  system.  If  analysis  is  each  mode  is  time-step analysis /3  are  specified.  t o be more r e a s o n a b l e as i t i n t h e h i g h e r modes.  i t limits  the c o n t r i b u t i o n s  This from  42  4.3  Mode S h a p e s and F r e q u e n c i e s  The  coupled  simultaneously However,  a  uncouples  with  the  Eqn.(4.1),  appropriate  approach  equations  numerical  i s to find  so  that  This transformation  may be s o l v e d technique.  a transformation  they  may  be  which solved  i s w e l l known and makes use  t h e e i g e n v e c t o r s o r mode s h a p e s o f t h e s y s t e m . The  free  mode s h a p e s r e p r e s e n t  vibration  problem  [M] The  an  simpler  independently. of  s e t of e q u a t i o n s ,  {u} +  eigenvalue  [  {A  K  {u} = {0}  ]  (4.4)  t o be s o l v e d becomes, [M]  } ~ co?  r  o f t h e undamped  g i v e n by,  problem  [ K ]  the s o l u t i o n  { A " }  =  {0}  (4.5)  or |  [ K  {A  where  R  co  co? •  -  [M] | = 0  } = mode shape  f o r t h e r - t h mode  mode s h a p e s a r e n o r m a l i z e d {A" } {h {A  (4.6)  = r - t h n a t u r a l frequency  r  The  ]  T  } R  T  }  T  such  i n rad/sec  that,  [M] { A ' } = 1  (4.7a)  [K]  (4.7b)  { A " } = co?  [C] { A  F  } = 2/3 co h  where /3 r e p r e s e n t s t h e f r a c t i o n h  (4.7c)  h  of c r i t i c a l  damping  of t h e ' r - t h  mode. The linear  combination {u}  where  actual  «  [  displacements,  o f t h e mode A  ]  {u}, a r e now e x p r e s s e d  as  shapes.  {4>}  {0] r e p r e s e n t s t h e a m p l i t u d e  (4.8) o f t h e modes.  a  43  4.4 S p e c t r u m  Analysis  A response  spectrum  maximum v a l u e o f a p a r t i c u l a r of a l i n e a r forcing the  single  function.  expected  degree  i s a graphical  response parameter  of freedom  The f o r c i n g  ground  relationship  with the p e r i o d  (SDF) s y s t e m  function  displacement.  subjected  i n earthquake A  point  on  i s o b t a i n e d by a n a l y z i n g  defined  by  record.  The maximum v a l u e o f t h e r e s p o n s e p a r a m e t e r  generates a point obtained yield for  by  on a  a given  maximum  response  varying  a curve that  the  an  spectrum.  period  ratio.  (while  The  displacement,  maximum p s e u d o a c c e l e r a t i o n . gives  subjected  r e p r e s e n t t h e maximum  damping  relative  and damping,  indication  SDF  Other  fixing  points  pseudo  displacement  i n the s t r u c t u r e ,  while the  stored  a n d t h e maximum pseudo  to  the  lateral  force  coefficient  energy  acceleration  found  are  v e l o c i t y , and  p r o v i d e s a measure of t h e e l a s t i c  related  are  f o r any p e r i o d  maximum p s e u d o v e l o c i t y i n the s t r u c t u r e  as  of i n t e r e s t  responses  The maximum r e l a t i v e  of the s t r a i n s  system,  t h e damping) t o  useful  maximum  response  t o an e a r t h q u a k e  response  most  to a  studies i s a  spectrum  i t s period  a particular  of the  is  i n most d e s i g n  codes. If displacement  t h e maximum is  denoted  or  spectral  value  by Sd, t h e s p e c t r a l  of  the  relative  pseudo v e l o c i t y i s  defined as, Sv and  = w Sd  the s p e c t r a l  pseudo a c c e l e r a t i o n a s ,  (4.9)  44  Sa  = to Sd  The of  response  the s t r u c t u r e  ratio,  as w e l l  the  same  the  response  differ  Furthermore,  plots  as g i v e n by i t s n a t u r a l  spectra  to  i s dependent  as t h e c h a r a c t e r  provide  strongly  spectrum  v a l u e o f damping  expected spectra  (4.10)  2  a  means  because  influenced  of  motion.  ground  other.  properties  and i t s . d a m p i n g  and t h e same range  different  from e a c h  the  period  of t h e ground  ratio  for  on  Thus  of p e r i o d s ,  motions  may  T h i s means t h a t  characterizing  ground  the b e h a v i o u r of m u l t i s t o r e y  by t h e f u n d a m e n t a l  mode  be  response motions.  buildings i s  response,  p r o v i d e a c o n v e n i e n t means o f a s s e s s i n g  for  spectral  the response of a  structure.  Actual sharp This  peaks  purposes,  varying  based  evident  for  idealized  smoothed  relationship  (4.10),  and  velocity i t is  tripartite  plot,  quantities  with  that  the  lines.  with  showing the  have the  frequency  a  of the  of both  motions spectra  spectra  the the  logarithms  single  variation or  For  are  r e g i o n s of the spectrum  Because  logarithms  average  Response  and f r e q u e n c y as i n d i c a t e d to  systems.  by u s i n g  the various  with  i n the system.  damped  records.  acceleration  possible  irregular,  t o allow f o r earthquake  characteristics  straight  between  displacement spectral  to  so  quite  lightly  i t i s customary  frequency  are  the resonant behaviour  on a number o f e a r t h q u a k e  further are  spectra  reflecting  is especially  design of  response  linear spectral of  by E q n s . ( 4 . 9 ) plot,  called  the and the  of a l l t h r e e response  period.  The  simplified  45  Newmark's e l a s t i c Fig.  4.1  dashed  with  design  the  broken  maximum  line. as  amplification  factors  displacement less  than  factors are  is less  in  Table  are  In  spaced  structure  can  motion.  and was are  4.2  MacKenzie .  of  that  significant.  the  velocity,  which  the  by et  during  Further,  requires  torsional  a  when  damping by  the more  due  San  to  by  seismic  damage.  5%  accidental  horizontal and and  has  Nathan of  earthquake  ground  motions  corner  columns  The  National  e c c e n t r i c i t y but  included.  a  from  behaviour  Fernando  why  of  in  also  18  explain  is  and  Newmark ,  rotational  to  input  response  observed  from  motion  1 9  the  ground  derived  of  excited by  real  Rosenbleuth ,  1971  t h i s may  ground  of  are  base e x c i t a t i o n  the  response  spectra  Torsional  studied  1  is  amplification  and  components  discussed  2  of for  generally  both design  Newmark and a l  by  turn  recommended are  be  4.1(b).  shown  first  of  amounts  those  may  region  in  in the  factor  which  Values  various  structure  was  frequency  amplification  Canada,  p a r t i c u l a r l y vulnerable  doubtful  the  horizontal  the  buildings  Code  on  t o r s i o n a l component  Hart  2 0  Building  multiplied  while  are  discussed  concluded  maxima  accelerometers.  topic  further  instrumented  4.1(a),  in Table  in  This  motion  for  come from t o r s i o n a l  eccentricities  line)  acceleration. Newmark  bounds shown as  (solid  for  shown  records  closely  that  of  The  motion  the  Code  Table  bounds.  time h i s t o r y  and  for  Building  conservative,  motion  than  i n s u c h a manner  values  w h i c h depend  Generally,  that  shown  ground  ground  recommended by  National  been  the  spectrum.  is plotted  The. s p e c t r a l  interpreted  the  spectrum  i t is  46  4.5  Complete Q u a d r a t i c It  excitation place  is the  Combination  reasonable maximum  simultaneously,  total  and  response.  does  provide  This  estimate  much  an  practical  r e s p o n s e can  so  dimensional  quite  well  However  poor  results  The  directly  to the  and  more r e a s o n a b l e the  The  RSS  i n some of of  random  sum  or  the  p r o p o s e d a method w h i c h duration  of  values.  simpler  and  A similar  more p r a c t i c a l  method  p r o p o s e d by  as  obtain  modal  total  maxima  response.  of  not  of  root-sum-square  (RSS)  method  yields  good  For are  the  most  two-  usually  well  frequencies  been  shown  (Ref.  are 22)  cases. v i b r a t i o n have been u s e d  method.  has  earthquake  damping  (CQC)  RSS  to  response c a l c u l a t i o n ' s for  d e r i v e a l t e r n a t e methods w h i c h e l i m i n a t e absolute  individual  total  method has these  take  the  frequencies  RSS  not  is therefore  a n a l y s i s many of  the  of  frequencies.  natural  in  estimate  using  separated  theories  earthquake  superimposed  maximum of  to t i m e - h i s t o r y  and  for  superposition  conservative  f o r 3D  close together  to g i v e  be  obtained  analyses,  that  maximum r e s p o n s e s  superposition.  with  separated.  too  u s u a l l y be  the  direct  A  when compared  buildings  not  upper bound  use.  method of modal results  The  i s often  assume  c o n t r i b u t i o n s of a l l modes do  modes of v i b r a t i o n s h o u l d the  to  Method  inherent  errors  Newmark and  in  to the  Rosenbleuth  1 9  cross-modal  terms  involving  the  well  modal  frequencies  and  as  the  method of modal c o m b i n a t i o n i s the  Complete Q u a d r a t i c  Wilson  and  Der  Kiureghian  which  is  Combination 2 2  .  The  CQC  47  method  requires  following  a l l modal  =  ( I E p . Q.Q.  combined  by  the  (4.11)  which  p..=  (1-r ) 2  Q.=  Note  that  of  0-=  damping  = ratio  through well  periods,  a l l cross-modal  terms.  mode r e a c h e s  the i n s t a n t s  the cross-modal  i s a complete  cross-modal  natural  when  the  coefficients  become s i g n i f i c a n t . assume  positive  corresponding accounts  for  responses  taking  The f a c t  or  modal the  tend  negative  responses  are  to unity that  have  possibilities  into  t h e RSS method.  t o each  cross-modal depending or  strongly  p l a c e w i t h phase a n g l e s c l o s e  are  ( a s shown  o t h e r , the  and t h e c r o s s - m o d a l  t h e same of  periods  are small  close  each  values  associated  i n Eqn. ( 4 . 1 1 )  When n a t u r a l  coefficients  form  of s t o c h a s t i c  response  i s reflected  coefficients.  periods  quadratic  The i n f l u e n c e  F i g . 4 . 2 ) a n d t h e CQC method d e g e n e r a t e s when  2  /Tj  i t s maximum  s e p a r a t e d , the cross-modal  However,  2  i n t h e i - t h mode  formula  with  2  coefficients  of modal  between  + 4(0. + 0. )r  o f t h e i - t h mode t o t h e  contribution  each  '(4.12)  interest  ratio  correlation  2  '—J.  maximum c o n t r i b u t i o n  cross-modal  including  3  2  p..=  this  (0 + r 0 ) r /  + 40.0 . r d + r )  2  response  r  1 2  ^  tj  where  180  be  )P  8 (00.)  in  responses  equation: Q  in  that  terms  term  may  on whether t h e opposite  signs  correlated  modal  to  either  0  or  degrees. The  superior  performance  o f t h e CQC method o v e r t h e  48  absolute  sum and t h e  application building four  t o the f i v e  i s symmetrical  feet  from  directions. natural are  RSS  the  of  i n F i g . 4.4.  which  past  components.  rotational  components  evident  i s that to  distribution  and  asymmetrical  shears  i n both  applied  the  the  in  The  comparison  motion,  o f t h e mode  the  mode  shapes  shapes  of  mass  with  have no  orthogonal t o these  have  and y components.  Also  x  from  the f i r s t group.  shapes  i s tested  group  is  very  T h i s type of p e r i o d  are  very  common  u s i n g the response  'A' a s t h e g r o u n d  four  exterior  F i g . 4.5.  response  to  motion.  frames  are  in  purposes.  analysis  only  spectrum the  plotted  modal  base  by t h r e e  These  results  shears a r e not  methods. i s then  'A'  Taft  The maximum base  are also  spectrum  combined  CQC.  1952, N69W e a r t h q u a k e  Although  motions,  The  The s i g n s o f t h e b a s e  of the s t r u c t u r e  the T a f t ,  a n a l y s i s program. earthquake  earthquake  center  as  i n any o f t h e s e c o m b i n a t i o n  subjected  mass  directions  methods: RSS, a b s o l u t e sum, and  shown  retained  structure  spectrum  for  different  mode  of  The  c e n t e r of t h e b u i l d i n g  the second  coupled  i n F i g . 4.3.  buildings.  This method w i t h  well  by i t s  located  However, t h o s e  from  shown  One n o t e s  through  as  illustrated  is  the  one modal p e r i o d  another  be  the center  geometric  rotational  history  except  p e r i o d s and t h e p r i n c i p a l  resultants  are  can  storey building  The d i r e c t i o n  illustrated  close  method  calculated  using a time-step  indues  one  is  shears  i n F i g . 4.5.  when  several used  from  here this  real for time  49  For  t h i s case  i t is clear  good a p p r o x i m a t i o n of t h e greatly  overestimates  motion.  The  suggested  sum  the  For  t h i s example, The  the  'exact' r e s u l t s  Based  RSS  where  frames CQC  from  on  the  method  forces  case  o v e r e s t i m a t e s a l l four  the  the  i n the  CQC  and  which  i s a method  but  to the  normally  close,  i s worse t h a n  the RSS  preceding  greatly method.  the b e s t a p p r o x i m a t i o n to analysis.  example  i s t o be  i n a l l response  normal  are  time h i s t o r y  method  method g i v e s a  frames  periods  method g i v e s  the  t h e RSS  i n t h e d i r e c t i o n of m o t i o n  of a b s o l u t e v a l u e s ,  for  discussion,  forces  that  used  spectrum  and  the  above  as a r e p l a c e m e n t f o r  calculations  in  this  study.  4.6  Multi-Component In  simultaneous taken  the a n a l y s i s action  into account.  technique subjected  to  the well  and  The  A procedure based  o t h e r two  ground  response in that  and  i n one  the response  one  direction  the  The  response  unidirectional  is  is affected  excitation,  be  structures  needed. in  two  direction  is  not o n l y  by  by g r o u n d  spectrum  must  spectrum  response  rotational  d i r e c t i o n , but a l s o  directions. for  on  excitations  structures,  directions  established  structures,  t h e r e s p o n s e of a s y m m e t r i c a l  asymmetrical  motions  d e s i g n of a s y m m e t r i c a l  of a l l components of t h e e a r t h q u a k e  determine  translational  ground  Motions  to multi-component In  coupled.  Ground  motions  technique but  in is  not f o r  50  multi-component stationary  Gaussian  uncorrelated, the  square  with  excitation.  processes  the expected  root  If  a  and  taken  squared  components.  Consider  the  maximum v a l u e  of t h e sum o f  the v a r i o u s  a l l  components as  completely  of any r e s p o n s e expectations  are  will  be  associated  (Newmark and R o s e n b l e u t h ) 1 9  structure  subjected  to  two  orthogonal  components o f g r o u n d a c c e l e r a t i o n s i n t h e q and r d i r e c t i o n s and rotational structure  component  i n the 8 d i r e c t i o n .  g e n e r a l i z e d degrees  From S e c t i o n  4.2,  The  o f freedom v e c t o r i s ,  (4.2)  {u} =/ u ^ He  where  £ ^} u  centers is  a n <  iv }  3  o f mass  about  t o the displacement  vector  of t h e  i n . t h e q a n d r d i r e c t i o n s r e s p e c t i v e l y , and {u^}  the r o t a t i o n a l  of m o t i o n  refer  r  vector  of the f l o o r  the c e n t e r s  diaphragms.  The  equation  o f mass c a n be w r i t t e n a s ,  (4.14)  [M] {u} + [C] {u} + [ K ] {u} = -IM]  Making  t h e n o r m a l mode  transformations,  {u} = [A] {</>} and  premultiplying  by  (4,8) [ A ] , the uncoupled T  equation  of m o t i o n  becomes,  [M*]  {.0} + [ C * ] { 0 } + [ K * ] {0.} = ~ [ A ]  T  [M]/ U 3 A(4.15) u 3,  e  where  [M ] a n d [K ] a r e g e n e r a l i z e d mass and s t i f f n e s s  diagonal  51  matrices  respectively,  matrix  assumed  mode  is,  i  to  0.=  L  0 is  mode  i .  t h e damping  L  i%'  a  a  For necessary  ir'  a  n  a.  f t  the  u ^(t) o. +  3  > r  r a t i o and c ^ i s d  a  i,e  a  r  e  m o <  generalized  The t y p i c a l modal  3al  the  multi-component  to obtain right  known  spectral  the  total  individual modal  u  3 i r  .(t)+a  i /  damping  equation in  ,u (t)  the natural  w  (4.16)  frequency  participation  where  hand  side  values  u  L Lt  =  {u  from L,%  =  Sd.q=  m  for  factors for  excitations, of the time  of Eqn. ( 4 . 1 6 ) .  can  be  u  g r  (t),  and  approximated  u  g e  by  sum s q u a r e manner.  a l l three  components  U  i/  °de  +  u  Le )' 2  excitation  is  given  directly,  combining That  is,  the the  i s g i v e n by, (4.17).  r e s p o n s e due t o q d i r e c t i o n  direction  series  (t)  / 2  displacement  i t  But t o make use o f t h e  i n a root  spectral q  etc.  +  values  of u ^ ^ ( t ) ,  response  responses  ground  the s p e c t r a l  responses  U  and  is  i i n t h e q, r , a n d 8 d i r e c t i o n s .  mode  by  [C ]  be d i a g o n a l .  h2(S.co ^ where  and  value  excitation  f o r i - t h mode due t o  52  CHAPTER 5 COMPUTER  5,1  Program  Concepts  A analyses  very  of  substitute plane  effective  plane  1 5  1 7  eccentricities,  frame p r o g r a m . modify  .  has  proven Based  pseudo  program  to on  As w e l l ,  be  t h e program  so t h a t  the  modified  method,  various  capabilities  s t r u c t u r e s with  address an  for nonlinear  be  this  nonlinear  should  and n o t m e r e l y  or extend  method  However, t o a n a l y z e  the  three-dimensional  approximate  method.  frame p r o g r a m s w i t h 1  a  frames  structure  been w r i t t e n "  to  PROGRAM  large  the s t r u c t u r e as  extension  should  have  of  be f l e x i b l e  a  plane  and e a s y  i t would n o t become o b s o l e t e  within  few y e a r s .  A special this  study  in  multi-storey,  provides  the  occur  a  limitations general  order  to estimate  three-dimensional  multi-component  in  p u r p o s e computer  earthquake  during  defined  conceptual  by  outline  the i n e l a s t i c  motions.  a  the  is  developed behaviour  reinforced concrete  maximum d e f o r m a t i o n  member  program  This  program  earthquake  assumptions  of t h e program  (PITSA)  demand t h a t  motion,  of the program. i s shown  of a  structure to  and t h e d u c t i l i t y  given  in  with The  i n F i g . 5.1.  53  The frames.  The  Stiffness Each  structure  i s idealized  stiffness  matrix  Method, w i t h t h e  node  has  up  to  a s an a s s e m b l a g e  is  nodal  obtained  by  displacements  three degrees  of  planar  the  Direct  as  of freedom.  unknowns.  However, t h e  horizontal  d i s p l a c e m e n t s o f a l l nodes on t h e same l e v e l  specified  to  degree  o f freedom  frame  is  contributions  values,  statically  degree  which.case  o f freedom  condensed  such  that  per f l o o r  plus  those  compatibility  from a l l frames  a way a s t o y i e l d  in  remain  a r e combined  three s t r u c t u r a l  process permits the s i z e  of  degrees the  Each  only  one  required  active.  Finally,  and c o n d e n s e d of freedom  problem  be  o n l y one  i s a s s i g n e d t o a l l these displacements.  vertical  This  identical  then  translational for  have  may  to  i n such  per f l o o r . be  greatly  reduced.  The elastic  spectral  substitute  the  effective of  stiffness  which  information,  substitute  within  each  structure  stiffness  and damping  i t e r a t i v e l y by stiffness  iteration. stiffness  matrix  into  If a matrix  is rebuilt.  p r o p e r t i e s a r e thus  has gone  and  a  the i n e l a s t i c  The  function  range.  Organization  The routine  with  determined  a r e made t o t h e element  how f a r t h e s t r u c t u r e  5.2 Program  is  calculated  changes  overall  response  analysis,  damping  member y i e l d s , and  dynamic  program  is  dimensions checks  controlled  variables,  f o r compliance  by reads of  the in  main global  global  or d r i v e r control  dimensioning  54  restrictions, and  checks  are shown  called  sets  up t h e i t e r a t i o n  f o r convergence inside  loop for i n e l a s t i c  of r e s u l t s .  t h e main  routine  Three  major  analysis,  subroutines  (subroutine organization i s  in F i g . 5 . 2 ) : 1) MAINF - The b a s i c  structure  generator  data  completes  The n e x t  step involves  of member s t i f f n e s s each  form and  frame  a  of the  the formation  m a t r i c e s and summing  i n the  member  the rest  structure.  The  of t h e frame s t i f f n e s s m a t r i x  them up reduced  i s determined  stored.  2) MAINR - The  individual  together  frames  t o form  variables  compatibility  degrees  columns a r e c o n d e n s e d stiffness 3) MAIND - The  matrix  mass  of  acceleration dimensional  the  natural  storey  of  joint  freedom  out.  are  The  factors  displacements  and  in  and  are  read.  are  The  three-  mode  shapes,  evaluated.  The  associated  with  F o r each then  frame  the  are evaluated.  combined  structure  and t h e e a r t h q u a k e  frame d i s p l a c e m e n t s t h e member f o r c e s evaluated  corner  and s t o r e d .  displacements  the reduced  matrix. with  net  frequencies,  e a c h mode a r e d e t e r m i n e d . structure  assembled  associated  structure  spectra  then  stiffness  i s determined  participation  maximum  are  the s t r u c t u r e  Displacement  and  and  subroutine  structure.  for  i s read  i n root  i n the  unreduced From are  these also  sum s q u a r e  55  and  complete quadratic  damage r a t i o s updated with if  and  smeared  from t h e  inelastic they  combination  last  the  The  ratios  are  damping  iteration.  deformations  exceed  manners.  are  Those members  checked  tolerable  limits  to  see  of moment  capacities.  5.3  Design  Spectra The  building  will  condition, of  selection  and  public  depend the  intensity  given  site  earthquakes  motion soil has  be  recorded fails  recordings  strong  accelerograms. Jennings  site.  et a l  spectrum  smoothed s p e c t r a  5  .  simulated  from a  is  same peak a c c e l e r a t i o n . . Figure  5.3  shows  soil types  a design  from  records  not  be  areas.  But  that  i s by  generating  deriving  earthquake  records  real  strong  even  when record  particular simulated described  spectrum chosen  by  a  choosing  earthquake  of  for  to  of  enough  only  design  (see Ref.  local  Certain  justify  individual  acquired  s e t of  the  e a r t h q u a k e s have been  However, t h e  'A',  may  similar  A n o t h e r method  Eight  area,  particular  T h i s method of  there  characteristic  a  spectra appropriate  choosing  each  for  building.  sites.  that  similar,  features and  by  in g e o l o g i c a l l y are  the  Response  similar in  of  hospitals will  derived at  spectrum  geographical  use  earthquake.  conditions  study,  the  intended  may  input  earthquake  by  on  b u i l d i n g s u c h as  larger  dynamic  of a d e s i g n  in  averaged s c a l e d to  this or the  16)  spectrum  'A'  d e s c r i b e d by  simple  56  expressions  and the  assumed  relationship  response  a c c e l e r a t i o n f o r any damping  Because  of  spectra  Figures  of l o c a l  5.4  recorded  to  the jagged  motion,  El  Centro  other  EW  design  tuning i s very  w h i c h may  work  with  averaged  represented  response  average  eliminate  spectra  each  undue  for  normalized  that  relatively  spectral  for periods  four  t o 0.5g,  higher  spectra  of from  i s  damage  than  than  about  0.7  'A'.  shows  of  the  seconds.  i s p o s s i b l e , but unless the  ratio  maximum than  from a l l  of  design four  of  another  acceleration,  E l  analysis  for practical  results  ratios.  those  i s the p o s s i b i l i t y  time-step  time-step  particular  F i g . 5.8 w h i c h  c o n c e n t r a t i n g on c o m p a r i n g  justified  by s p e c t r u m  one  l a r g e damage  values  t h e same c h a r a c t e r i s t i c  i n larger  i t  damping.  As shown by t h e p l o t s i n  revealed  conservative, there  those  earthquakes,  to  be e x p l a i n e d by e x a m i n i n g  instead  to  produces  of the design  result  Therefore,  EW,  with  a n d f o r 2%  design  nature.  motions  ground motion  PITSA  may  the  t h e smoothed  help  components,  t o have h i g h e r  three  Further  acceleration  s t u d i e s have  E l Centro  tendency  will  i n response.  motion  Previous  This  peaks  5.7,  ground  indicate  factor  t h e randomness of earthquakes,  of s e v e r a l earthquakes  influence  between  Centro  EW.  results  from  individual purposes to earthquakes  57  CHAPTER 6 PROGRAM TESTING  6.1  Testing  for E l a s t i c  The evaluation three  principal  of  the  s t r u c t u r e method. and  analysis.  capabilities,  After  great  iteration  behaviour  section  program, of  PITSA,  frame and  established  the  intended  to  proposed  order  program,  to a  verify previously  the  evaluates For for  the  analysis  is  an  properties.  the this  that  the  elastic purpose,  comparison.  Program - ETABS  elastic  tested  dynamic  modal a n a l y s i s .  demonstrate  used  a  inelastic  damping  buildings.  computer p r o g r a m was  produce  to e s t a b l i s h that  and  correctly  shear w a l l  will  inelastic  stiffness  the  substitute  time-step  results for e l a s t i c pseudo  is  multi-storey  program's  C o m p a r i s o n W i t h A n o t h e r Modal A n a l y s i s  In  of  modified  to  importance  i n the  is  research  research  alternative  a n a l y s i s with modified  proposed  6.1.1  of  the  the  considering  produce v a l i d  This  a well  is  this  response  using  i s hoped t h a t  before it  a l l , each  elastic  It  of  seismic  buildings  economical  But  program c o u l d  objective  inelastic  dimensional  reliable  Analysis  and  c a p a b i l i t y of proven  the  computer  58  program was  was  used  t o compare  chosen' b e c a u s e  profession. California of  of  This  its  results.  widespread  program  was  For use  the a s s e m b l y  capability  is  not  ETABS was that  translation enforced  frame  wall  on  c o r e as  analysis  ground  involving 0.42  was  and  storey the  Frame  It  should  i n the sense  freedom,  the h o r i z o n t a l  be that  vertical  axis,  i s not  frames.  of  the  structures.  second force  two  The  structure  resisting  programs  first  i s one  were  structure  is  with a coupled  system.  Structure  structure  properties  a n a l y z e d was  as shown  in  c a r r i e d ,out w i t h s p e c t r u m motion  applied  the d i r e c t i o n  seconds.  extended  Examples  first  and  about  into  allows for planar  inferior of  of  extension  this  available.  degrees  rotation  the l a t e r a l  Five-Storey  dimensions  joint  f o r comparison  five  building  i t was  1  engineering  i s an  Although  PITSA o n l y  is actually  intersecting  two  The  the  the  tests  6 . 1 . 2- N u m e r i c a l  (a)  the  and  The  a  the  ETABS  of t h r e e d i m e n s i o n a l frames  since  because  program  of  between  performed  used  this  compatibility  required  purpose  developed at the U n i v e r s i t y  a general three dimensional structure.  noted  by  a t B e r k e l e y i n t h e m i d - s e v e n t i e s and  TABS t o e n a b l e  frames,  this  The  of  location  a frame  F i g . 6.1. 'A'  1 S  ,  q,  had  of the diaphragm  The  scaled  i n the q d i r e c t i o n . interest,  b u i l d i n g 'with  The  a natural mass  elastic  t o 0.5g  as  l o w e s t mode period  of  i n the top  two  59  storeys  was  moved  program's a b i l i t y vertical  line.  four  to handle  The  first  and  2% damping assumed  root  sum  square  responses this  natural  in  agreement.  t h e o r d e r o f one  the  RSS  bending  displacements  produced The  shown  examined.  At  this  p r o g r a m can  indeed produce  for  PITSA and  stage  The  elastic  analysis  0.2g  as  lowest  the ground mode  was  method.  In  since  of  and  and  periods  and  the  member the  results t h e two  seconds.  of was  proposed  for  frame  programs a r e  The  sec.  severe  purpose  each  t o say t h a t  3.7  were  t h e most  For for  results  CPU  time  respectively.  Structure analyzed  properties  carried  out  applied the  natural  running  sec.  Wall  p r o g r a m s were i n  i n the  elastic  process unit)  motion  involving  modal  unnecessary  two  6.1.  moment  correct  structure  with dimensions  the  i t i s safe  ETABS were 5.5  building  a  reported are  combine  frame u n d e r g o i n g Table  cost  (central  next  is  elastic  bending  (b) F i v e - S t o r e y C o u p l e d The  not  differences  The  in  largest  i n CPU  by  slight  the  measured  values  on  were c o n s i d e r e d  q u a d r a t i c combination  moments f o r t h e  structures.  does  not  the  separated.  comparison,  type  masses  A l l results  ETABS  CQC  percent.  are  with  s i x modes o f v i b r a t i o n  of  p e r i o d s are well  excellent  structures  the c o m p l e t e  results  the p l a n c e n t r e t o t e s t  f o r e a c h mode.  calculation  The  from  v a l u e s because  using  case,  feet  was  shown  a  direction  the  wall  i n F i g . 6.2.  with spectrum in  coupled  q  'A'  scaled  direction.  of a p p l i e d  The  loading  to The  had  a  60  natural their  period mass  building The  to  first  of  0.38  seconds.  centres simulate  moved a  Again,  two  the  feet  structure  top  from  the  with varying  s i x modes were c o n s i d e r e d  two  storeys middle  plan  and  2%  two  programs  had  of  the  dimensions.  damping  assumed  for  e a c h mode.  The the  same.  r e s u l t s produced The  moments f o r the are 2.3  shown sec.  elastic wall  in Table and  1.8  6.2.  sec.  Testing  for  The the  elastic  The  wall  Inelastic results  the  behaviour  multi-storey  carry  out  ability  method  i s t o compare on  from an  inelastic  patterns  of  elastic  chosen  two  so  method. that  RSS  This  example  i n the  bending  displacements  f o r PITSA and  structures  of of  previous PITSA.  the  ETABS were shows  elastic  real  program  have  next  that  range.  Since  basis  ductility  is  the  to  inelastic  it  is  the  best a n a l y t i c a l  with  The  established  step  to p r e d i c t  structures,  a numerical  by  section The  structures.  storey  section  the  time  the  almost  impractical  results  degree  factors,  of  obtained inelastic  location  and  examined.  five  The  severe  step analysis.  damage a r e  testing  structure  time  represented  The  most'  were  Analysis  e x p e r i m e n t s on  deformation  and  the CPU  i n the  investigate  to  periods  respectively.  capability  of  the  natural  undergoing  PITSA works f o r c o u p l e d  6.2  by  are  structures  analyzed  by  the  moment c a p a c i t i e s  of  considered modified beam  maximum damage r a t i o i s a r o u n d  in  the  substitute  members six.  are These  61  structures time  are a l s o  s t e p program  scaled  to  modified results  the  same  the  frame  strength  method  the  viable  test  to  designed of  should  determine structures,  weak  columns  indicate  for structures  with  W i t h A Time S t e p  earthquake response  buildings,  DRAIN-TABS , 3  of t e s t  University  horizontal  structures.  of C a l i f o r n i a  motions plus v e r t i c a l  motion.  A  building  interconnected  enforcement  whether  irregular  Analysis  of  structural  element  axial-bending  a c c o u n t , and extension  m o t i o n may  interaction  be  was  as a s e r i e s  and can  and  hysteretic  effects  model  the  of p l a n e with  no  rotational Each  a variety  of  beam-column e l e m e n t s i n can  be  beam e l e m e n t s w i t h d e g r a d i n g s t i f f n e s s ,  of T a k e d a ' s  as  frames.  include  the  written  specified  more  three  independent  diaphragms,  or  include  for  t o compute  Two  vertical  two  geometry These  rigid  for  common t o  types.  used  T h i s program  is idealized  compatibility  can be of a r b i t r a r y  was  program  at Berkeley.  by h o r i z o n t a l  displacements at j o i n t s  which  This  from t h e  to the average  Furthermore,  with a storey  For Comparison  inelastic  response h i s t o r y  frame  results  motions  - DRAIN-TABS  dimensional  frames  is  The  inelastic  ground  method a r e compared  can h a n d l e p o o r l y  beams.  the  different  acceleration.  i s analyzed  of s t r o n g  Assumptions  ground  with four  using  patterns.  The  at  to analyses  ground m o t i o n s .  structure  proposed  Program  peak  p r o p o s e d program  and a s t o r e y  3  structure  of t h e f o u r  the  6.2.1  DRAIN-TABS  substitute  if  the  subjected  i s used.  taken  into  i n which  an  62  Since  DRAIN-TABS  was  developed  f o r the Berkeley  6400 computer,  t h e program had t o be. c o n v e r t e d  work  UBC  on  FORTRAN core  the  during  execution.  yet c r i t i c a l  difficulties the  Also  files  being  using  stored  real  variables,  was  said  to  single  and  frame  be e x t r e m e l y  was  precision  analyzed  In  process.  t o open and  addition,  I t should  Because  precision.  be n o t e d  that  data  which  blocks  o f i n t e g e r and  for  the  Amdahl  no a t t e m p t t o  However  u s i n g DRAIN-TABS, and then i n t o double  some  the  Therefore,  v e r s i o n was made.  caused  a n a l y s i s problems  precision  difficult.  several  computers  contain mixtures  t o double  setting  using  a  plane  DRAIN-2D  The r e s u l t s  2  of  two p r o g r a m s showed no s i g n i f i c a n t d i f f e r e n c e s .  The requires the  dependent  the l e n g t h of the  i t was n o t n e c e s s a r y  precision.  w h i c h has been c o n v e r t e d the  System  l a r g e number i n s t e a d o f  transferred  reduce  memory.  f o r t h e CDC o r s i m i l a r  conversion  produce a double  to  was n o t n e c e s s a r y  a d e q u a t e word l e n g t h t o s o l v e s t r u c t u r a l  accurately  written in  i n order  d i f f e r e n c e s i n t h e two c o m p i l e r s  i n the conversion  would  technique  i n our s y s t e m .  program was d e v e l o p e d  have  virtual  t o be a r e l a t i v e l y  random a c c e s s  subtle  This  i t  Although  w h i c h were r e w r i t t e n i n c l u d e s e t t i n g  common b l o c k  close  ample  computer.  into overlays  requirements.  because of t h e  features  it  V/8  IV, i t was s u b d i v i d e d  storage  here  Amdahl  before  CDC  dynamic a n a l y s i s o f r e i n f o r c e d c o n c r e t e  t h e a p p r o p r i a t e member  modified  substitute  which the c r a c k e d  moment  structures  p r o p e r t i e s as input d a t a .  structure  of i n e r t i a  Since  method employs a scheme i n i s assumed t o be h a l f  of the  63  g r o s s moment of i n e r t i a third be  used  in  the  to  chosen  time  be  t h e program  stiffness  Shibata  and S o z e n ' s analysis.  spectrum  'A'  histories.  of  were  were  term  ductility  in  the  which  i n t h e member.  determined  i n terms  DRAIN-TABS  outputs  member, t h e member adopted arbitrary  in  this  study.  assumption  This  to  'A'  in  from  which  the response  moment  give  approach  peak  members,  but  i s in order.  of the c u r v a t u r e  at  i s commonly u s e d .  It  o f t h e c o n c r e t e and  t h e member d u c t i l i t y  terms  a  acceleration.  of d u c t i l i t i e s  Since  rotations in  i s beyond t h e  spectrum  flexural  characteristics  ductility  and  index t o d e s c r i b e the  i s the r a t i o  hinge  value  critical,  records,  o f beam end r o t a t i o n s . the  elastic  S21W, E l C e n t r o NS, and E l  in  Alternatively,  would  The damping  t o compute  i s a useful  to the curvature at y i e l d  a  f o r the modified substitue  peak g r o u n d  definitions  depends on t h e s t r e s s - s t r a i n steel  design  a l l scaled  deformation  the  spectrum  used  N69W, T a f t  and  damping.  of s i x earthquake  records  curvature d u c t i l i t y ,  ultimate  was u s e d  1 6  derived,  inelastic  clarification The  response  equal to the d e s i r e d  The amount  paper Four  was  The  acceleration  insure  t o be 2% o f  Therefore, the  They were T a f t  EW.  to  proportional  proper  thesis.  structure  Centro  a  present  t h e same a s s u m p t i o n s  analysis  DRAIN-TABS was t a k e n  as t a n g e n t  of t h i s  step  is  t h e same i n b o t h a n a l y s e s .  Choosing scope  compression  o t h e r w i s e , i t was d e c i d e d t h a t  stiffness for  i f axial  the  may be program  f o r e a c h end o f e a c h of  end  does  on t h e h i n g e l e n g t h a s would  rotations  is  n o t r e q u i r e an have been  the  64  case  if  rotation  curvature  is  rotation  defined  rotation  rotation  which  is  moments  i s defined  computed giving  advantage  used  i n the modified  The the  substitute  end  structure  The y i e l d  i s subjected a t the ends.  under  positive  values, This  equal  yield  to The and  the l a r g e s t of definition  has  t o t h e 'damage r a t i o ' method  f o r the e l a s t o -  Examples  building  Frame  The  height  were  o f 10%.  were  on  of  was  frames  of three  feet,  weights  were d i s p l a c e d  in the  a total  for  each  four  feet  perpendicular  represented  a  uniform  The c r a c k e d moments o f i n e r t i a  of  columns  of t h e i r  This  used  20-feet  giving  i n the d i r e c t i o n  motion.  gross  vibration  diaphragm  and t h e c e n t r o i d s  one-half  of t h e i r  12  6.1  The  consist  was  The  of the s t r u c t u r e  of F i g .  structure.  height  o f 60 f e e t .  eccentricity  one-third  test  storey  earthquake  based  frame s t r u c t u r e  ground e x c i t a t i o n  130 k i p s ,  from t h e c e n t r e the  Structure  inelastic  of applied  bays.  period  bending  possible  numerically  five-storey  first  direction  floor  each  the  cases.  Five-Storey  wide  at  by  factor.  when t h e member  four  of b e i n g  6.2.2 N u m e r i c a l  (a)  divided  a s t h e member d u c t i l i t y .  the  plastic  The sum of t h e y i e l d  a s t h e member d u c t i l i t y  i s the angle developed  negative  to  used.  y i e l d moment under a n t i - s y m m e t r i c  hinge  as  was  and t h e maximum h i n g e  rotation  its  ductility  gross  sections.  sections,  The  elastic  and t h e beams fundamental  i n the d i r e c t i o n of t h e earthquake,  the q  65  direction,  was  contributed  determined  predominantly  direction,  with a very  motion.  The y i e l d  ft  200  and  strain  to to  small  acceleration  In  contribution  lowest frame  the modified  Fig. of  mode  4, w h i c h  Table  6.3. 6.3.  4.4.  was assumed.  substitute  was  period  undergoes The  damage  The s e c o n d  time  ratios  histories  used.  4  f o r each motion  in  T a b l e 6.3. and  the q  rotational  step  of  The maximum  analysis,  criterion. seconds.  0.01 ground  to  The f i r s t  10 s e c o n d s  of  The maximum l a t e r a l  in  in  a r e shown i n damage  ratio  range.  these  100 s e c .  records  of the four damage  F i g . 6.3.1.  12  f o r each  d i s p l a c e m e n t s of  frame  motions  a r e shown  ratios  for  t h e a v e r a g e v a l u e s a r e shown i n F i g . 6.3.  damage r a t i o s a r e p l o t t e d  shown  where t h e f i r s t  The CPU t i m e was a b o u t  S i m i l a r l y , t h e maximum  f o r the  e a r t h q u a k e m o t i o n s were  o f E l C e n t r o EW,  and t h e a v e r a g e  are  frame  i n the e l a s t i c  four  last  d i s p l a c e m e n t s of  deformation, this  time  the  structure  The l a t e r a l  for  i t took  The CPU For  f l o o r beams had t h e l a r g e s t  by DRAIN-TABS.  step a n a l y s i s .  motion  most  A l l t h e columns r e m a i n e d  were  the  A time  of the s u b s t i t u t e  the  were u s e d w i t h t h e e x c e p t i o n seconds  to  structure  18.6  was 0.66 s e c o n d s .  Response computed  in  was 0.5g.  the n a t u r a l  q  mode  motion  f o r numerical i n t e g r a t i o n .  t h e Amdahl V/8 computer  iteration  This  F o r r e a s o n s o f s i m p l i c i t y , no  5 i t e r a t i o n s t o s a t i s f y the convergence on  seconds.  translational  k - f t respectively.  was u s e d  0.42  moments o f t h e c o l u m n s and beams were 1200 le-  hardening a f t e r y i e l d  seconds  be  The  El  each  The beam Centro  EW  66  earthquake three  resulted  i n much h i g h e r  damage  r a t i o s than  motions. The  substitute  results are  structure  r e q u i r e m e n t s and  slightly  on  the  very  encouraging,  method p r e d i c t e d  ductility  deflections.  unconservative  side,  r e s u l t i n g from EL C e n t r o EW.  Five-Storey The  the  last  kips  more  members.  period  to  a  of  the  level,  k-ft for  the  their beams  Nevertheless,  while results  Table  The o r i g i n a l so t h a t  mass  of  from  shows  as i n  of  130  the r e v i s e d  mass  was t o e m u l a t e a  without  and  6.4 and 6.4.1  6.4  they  taller  and  a d d i n g more j o i n t s and  shifting part  the  fundamental  of t h e s p e c t r u m .  0.42 s e c .  t o 0.84  The  sec.  the  To  ratios  moment c a p a c i t i e s were i n c r e a s e d  maximum g r o u n d a c c e l e r a t i o n was  Figs.  large  was t h e same s t r u c t u r e  different  was d o u b l e d  because  to the  y i e l d i n g i n t h e c o l u m n s and t o keep t h e beam damage  to a reasonable 2000  structure  i t had t h e e f f e c t  fundamental p e r i o d avoid  structure  by a f a c t o r o f f o u r  flexible  compared  are  W i t h F o u r Times The Mass  The p u r p o s e o f t h i s  Also  undamaged  test  modified  values.  w i t h one e x c e p t i o n .  increased  was 520 k i p s . thus  Frame S t r u c t u r e  second  section  was  mainly  20% of t h e a v e r a g e  the  predictions  when  damage  were a l l w i t h i n  runs,  The  time  ratios  step  as  c o r r e c t l y the p a t t e r n of  average of four  (b)  the other  333 k - f t f o r t h e c o l u m n s .  to The  unchanged.  show frame  a r e n o t t o o d i f f e r e n t from  the  damage  ratio  results  deflection results. before,  providing  These  evidence  67  that  the  modified  deflection  and  substitute  ductility  structure  requirements  can  for  a  predict  range  both  of  frame  structures.  (c)  Five-Storey The  wall of  uniform the  the  dimensions  the  was  time  step  weight 6.5  predicted  by  comparison  shows t h a t  the  are  time  runs.  step  conservative modified  The  side.  substitute 6.5.  In  feet  the  elastic per  in Table  The  a  f l o o r d e f l e c t i o n of  The  18 by  size  deflection  these  maximum  a v e r a g e of  to  0.38  sec.  the  inches  and  are  four on  both  from DRAIN-TABS  time s t e p ,  from  while  The  analysis  a v e r a g e of  predictions  for  results  method.  structure  estimates  the  The  ground  the  structure  with  analysis  of  The  feet.  was  a  joined  feet. 60  period  substitute  agreement  r  floor.  substitute  1.1.4  the  show damage r a t i o r e s u l t s  compare  structure  by  and  most members, the  The  in  shape s e c t i o n s  kip-ft  modified  in excellent  given top  and  centres  represented  height.  feet  kips  coupled,  The  feet  This  100  6.5.1  modified the  30  2  were  300  and  the  channel  initial  was  analyses  predictions  two  100  five-storey  structure.  throughout  was  the  in t h i s chapter.  measured  0.2g.  diaphragm  the  of  on  were d i s p l a c e d  diaphragms  beam c a p a c i t y  Figures four  of  3%  beams,  of  acceleration the  level  which c o n s i s t e d  coupling  coupling  earlier  center  e c c e n t r i c i t y of  Structure  performed  described  from  core, two  t e s t was  each diaphragm  direction  and  next  structure  mass a t  by  Coupled Wall  the the are  runs  indicates  the  proposed  68  method test,  predicts the T a f t  ductility  (d)  this  deflection  N69W m o t i o n  to  produced  be 1.49 inches..  the l a r g e s t  In t h i s  deflections  and  demands.  Five-Storey  Coupled  Wall  S t r u c t u r e With  Four  Times The  Mass As  i n t h e c a s e of t h e frame s t r u c t u r e ,  t h e mass of t h e  structure  The  coupled  wall  elastic  period  capacity a  was  reasonable  structure ratios 17%. for runs  doubled  increased  average to  method  are  0.76  fold.  seconds.  The  fundamental  coupling  beam  t o 130 k i p - f t  t o keep t h e damage r a t i o t o  damage r a t i o s  from  those  predicted  in Figs.  t h e beams.  the  predicted  lateral  predicted  deflection  was 3.42 i n .  deflection  f o r the four  are  Again,  substitute  t h e beam damage  w i t h the worst  one o f f by worse  d i s p l a c e m e n t s from  shown  in  f o r t h e method w h i l e time  step analyses  modified  on column moments was a l i t t l e  The a v e r a g e  those  by  the time  6.6 and 6.6.1.  conservatively  The p r e d i c t i o n  and  to  four  level.  The a r e compared  was i n c r e a s e d  time  T a b l e 6.6. the  than step  The t o p  average  top  s t e p a n a l y s e s was 2.44 i n .  6.2.3 C o s t s o f E x e c u t i o n The test  structures  charges are  computing  do  time  a r e . summarized  not i n c l u d e  f o r low p r i o r i t y  and  of the four  i n T a b l e 6.7.  the cost  batch  costs  of p r i n t i n g  In a l l  inelastic cases  the outputs.  jobs run u s i n g r e s e a r c h  rates.  the They The  69  same j o b could  running  cost  as  mind, s a v i n g s structure  real  the  one  A  thirty  time  of  made o f  frame  this  analysis  method can  pick  the  to  out  the  the  analyses.  The  damage r a t i o the  four  predicted damage  modified  f o r the  time value  ratios  agreement,  columns  but  step of of  from  see  those  the  this in  substitute  be s i g n i f i c a n t . analysis  test  the  program  structure  weak and  with  k - f t to  the  the  k-ft.  structure  3.1.  2.4 The  averaged floor  Otherwise of  structure  structure.  and  6.7.1.  yielded  in  As both  method p r e d i c t e d while  the  average  e x t e r i o r column had value  beams  fourth  fifth  purpose  i n the  floor  third  the  substitute  6.7  original  the  k - f t , and  spots  third  the  The  and  structure.  with  666  modified  column of  fifth of  500  beams  the  times  tested  strong  the  was  of  were u n c h a n g e d .  substitute  and  four  shown i n F i g s .  on  strong  height  with  333  i f the  analyses  the  to  2000 k - f t t o  interior  1.8  step  6.2.2(b) was  Damage r a t i o s a r e expected,  can  priority  With  modified  e f f e c t s of  over  structure  was  high  Weak Columns  structure  strengthened of  the  time  the  areas  in section  properties  at  t i m e s more.  analysis  a  Beams and  in selected  beams  step  and  record.  c o l u m n s weakened f r o m  floor  five  i f i t i s decided  Strong  mass d e s c r i b e d  account  with  using  earthquake  five-storey  floor  commercial  dollars  costs  s t u d y was  weak c o l u m n s  the  as  in  E f f e c t s of  The  much  accumulates  more t h a n  6.3  a  method o v e r a  Furthermore, quickly  using  was  were floor  4.4. in  a of a The  excellent  beams  were  70  overestimated. Although  the predictions  structures  of  reasonable.  The m e t h o d  pattern  of  regular  damage.  were not as good as t h o s e f o r  strength was  at least  patterns, able  they  to predict  were the  still general  71  CHAPTER 7 CONCLUSIONS This method  thesis  presents  a s an a n a l y s i s  seismic  response  concrete  buildings.  analysis  technique  reference  to  stiffness  and  function  of  objective  procedure  of  a  The  extends  to  structure are  gap between  inelastic  retains  t h e s i m p l i c i t y and economy of a modal  The  response a n a l y s i s .  direct  step  other cases,  range by member as  information  principal  The  analysis on  a  the  modal method  while at ductility  incurred  with a  analysis. The  storeys  modal  determined  demands on t h e members a t a f r a c t i o n of t h e c o s t time  elastic  linear elastic  and t i m e  same t i m e p r o v i d i n g  reinforced  whose  factor.  analysis  the  step  linear  the  the  inelastic  i t i n t o the i n e l a s t i c  ductility  bridge  structure  the  dimensional  utilizes  characteristics  maximum  substitute  determining three  method  and  damping  is  for  multi-storey  fictitious  the  the modified  two  structures  in height, coupled the  wall  cores  analyses  increased  four  structure  on  preliminary  one u s e s  times a  moment  as l a t e r a l  are to  different  findings,  tested  as  t h i s study a r e f i v e -  resisting  frames  and ' t h e  resisting  systems.  In b o t h  repeated place  in  with  the  storey  weights  the fundamental p e r i o d  portion determined  of  the by  spectrum.  comparison  of the The  o f damage  72  ratios  with  using  four  well  the average  time  indicate  step  analysis  t h e method  works  f o r a l l cases.  investigated columns. predict  five-storey with  a storey  The method the general  i s hoped  that  on a s t r u c t u r e  structure torsional large  pattern  further  method  is a  responses  and  clarify  further o f weak  still  able  to  the structure. the requirements  of t h e method.  the  modified  effective  ductility  The method  b u t was  throughout  application  and  was  beams a n d a s t o r e y  would  perfected,  fast  structure  as w e l l  o f damage  research  not  a n d c a n be m o d i f i e d , offices.  of strong  for successful  eccentricities.  design  frame  d i d n o t work  Although  use  of i n e l a s t i c  d i f f e r e n t ground motions,  The  It  values  way  demands  i s simple  and  of  substitue estimating  of buildings  with  inexpensive  to  t o r u n on m i c r o - c o m p u t e r s  found  i n many  EAcos'0  •+ 12EIs1n'e (1+a)L 3  EAcos8s<nfl L  EAsin'0  12EIcosesine (1+a)L'  + 12EIcos fl (1+a)L ;  J  - 12EIsin0Li (1+a)L 3  - 6EIs1n0 (1+a)L' 1  12EIcosgLi ( 1+a)L'  +  + 6EIcosfl (1+a)L  + (4+a)EI (1+o)L  !  ]  EAcos0s1n0  EAcos'fl L +  12EIcos0sin0 ( 1+a)L 5  (SYMMETRICAL)  EAcos'O L  12EIslneLi (1+a)L'  !  12EIs1n'6 (1+a)L'  (1+a)L 12EILi (1+a)L'  +  6EIs1ng (1+a)L'  + l2EIsin e (1+a)L ;  3  EAcos6s1n0  EAsin'0  12EIcoseL (1+a)l'  + 12EIcosfl5ine (1+a)L'  12EIcos'fl (1+a)L  - 6EIcosg ' (1+a)L  - 12EIsinflL; (1+a)L'  12EIcoseLi ( 1+a)L'  12EIL.L? + 6EIL. (1+a)L' (1+a)L'  12EIsin6L; ( 1+a)L  6EIsin0 (1+a)L'  + 6EIcosQ ( 1+a)L'  + 6EIL; +(2-a)EI (1+a)L' (1+a)L  + 6EIsin0 ( 1+a)L  Table  5  2.1  Member  stiffness  EAcosgs i ne  t  EAsln'S  12EIcosesine ( 1 + a)L'  !  ]  12EIcoseL; ( 1+a)L  ]  matrix  12EIcos'6 (1+a)L  ]  !  including  12EIL;' (1+a)L  6EI cose (1+a)L !  rigid  ends  +  3  .+ 12EIL, (l+a)L  (4+a)EI (1+a)L  !  74  D a m p i n g ,  % of  Amplification  Factor  for Spectral  Bounds  Critical Acceleration  Veloc i t y  0,  6.4  4.0  2.5  .5;  5.8  3.6  2.2  1  5.2  3.2  2.0  2  4.3  2.8  1 .8  5  2.6-  1 .9  1 .4  7'  1.9  1.5  1 .2  10  1 .5  1 .3  1. 1  20  1 .2  1 .1  1 .0  T a b l e 4.1(a) A m p l i f i c a t i o n f a c t o r s bounds recommended by Newmark.  Damping, \ % of C r i t i c a l  Amplification A c c e l e r a t ion  Displacement  f o r ground  Factor  motion  for Spectral  Veloc i t y  Di splacement  .5  5.8  3.3  3.0  2  4.2  2.5  2.5  3  3.8  2.4  2.4  5  3.0  2.0  2.0  .10  2.2  1 .7  1 .7  T a b l e 4.1(b) A m p l i f i c a t i o n f a c t o r s bounds recommended by NBC.  Bounds  f o r ground  motion  EARTHQUAKE  GROUND  MOTION  ACCELERATION BOUND  NEWMARK S 1  IDEALI ZED  NBC  BOUNDS  VELOCITY BOUND  DISPLACEMENT  BOUND  g*se.c**2 ( i n )  g  g*sec  (in/sec)  1.0  0.1246  (48.14)  0.0934  (36.10)  1.0  0. 1035  (40.00)  0.0828  (.32.00)  EL CENTRO  1940 NS  0.348  0.0340  (13.14)  0.0111  (4.29)  EL CENTRO  1940 EW  0.214  0.0376  (14.53)  0.02016  (7 .79)  TAFT 1952 N69W  0.179  0.0180  (6.96)  0.00937  (3.62)  TAFT 1952 S21W  0. 1554  0.0160  (6.18)  0.00662  (2.64)  CAL.  TECH. D1  0.485  0.0278  (10.74)  0.0049  (1.89)  CAL.  TECH. D2  0.492  0.0299  (11.55)  0.0072  (2.78)  g/ft ' EL CENTRO 1940  TORSION  (g*sec)/ft  (g*sec**2)/ft  0.00347  0.000422  0.00445  TAFT 1952 TORSION  0.00128  0.000128  0.000381  CAL.  0.004152  0.000336  0.0000474  TECH. D.TORSION  Table  4.2 B a s i c  ground  motion  bounds  76  ELASTIC MODE 1 2 3 4 5 6  PERIODS  (SECONDS)  PITSA  ETABS  0.5008 0.4191 0.2688 0.1545 0.1291 0.0836  0.5006 0.4189 0.2686 0.1544 0.1290 0.0836  FRAME 4 - RSS BENDING MOMRENTS  MEMBER  (kip-ft)  PITSA  ETABS  INTERIOR BEAMS  5th F L . 4th 3rd 2nd 1 St  228 473 678 808 742  229 476 683 814 747  INTERIOR COLUMNS  5th F L . 4th 3rd 2nd 1St  457 690 858 955 1 1 72  460 695 864 962 1181  T a b l e 6 . 1 . E l a s t i c modal a n a l y s i s r e s u l t s f o r f i v e - s t o r e y b u i l d i n g ( F i g . 6.1) s u b j e c t e d t o s p e c t r u m 'A'  frame  77  ELASTIC MODE 1 2 3 4 5 6  PERIODS  (SECONDS)  PITSA  ETABS  1.0958 0.5840 0.3810 0.2221 0.0948 0.0915  1 .0950 0.5839 0.3812 0.2216 0.0950 0.0914  WALL 4 - RSS BENDING MOMRENTS  MEMBER  PITSA  COUPLED BEAMS  5th F L . 4th 3rd 2nd 1 St  COUPLED WALLS  5th F L . 4th 3rd 2nd 1 St  265.9 317.2 360 . 5 352.2 250.6 960.1 1234.2 1168.9 2270.3 3858.0  (kip-ft)  ETABS 271 . 1 322.2 363 . 3 352.1 249.2 978.8 1229.8 1196.8 2269.7 3825.2  T a b l e 6.2 E l a s t i c modal a n a l y s i s r e s u l t s f o r f i v e - s t o r e y c o u p l e d w a l l b u i l d i n g ( F i g . 6.2) s u b j e c t e d t o s p e c t r u m 'A*  78  LATERAL  DISPLACEMENT  (in.)  DRAIN-TABS  FLOOR  NO. N69W  TAFT S21W  3.32 3.05 2.45 1 .50 0.50  5 4 3 2 1  2.81 3.55 2.71 1 .63 0.55  EL CENTRO NS EW  3.10 2.88 2.36 1 .50 0.53  5.03 4.52 3.50 2.08 0.68  DRAIN-TABS FLOOR  Table  NO,  PITSA  AVERAGE  6.3 C o m p a r i s o n o f l a t e r a l d i s p l a c e m e n t f o r f i v e - s t o r e y frame s t r u c t u r e  o f frame 4  79  LATERAL  DISPLACEMENT ( i n . )  DRAIN -TABS  FLOOR  NO. N69W  5 4 3 2 1  6.49 5.64 4.22 2.41 0.77  TAFT S21W  4.96 4.44 3.66 2.36 0.85  EL CENTRO NS EW  6.24 5.81  4.75 2.98 1.02  8.95 7.68 5.65 3.28 1.07  DRAIN-TABS  FLOOR  5 4 3 2 1  Table  NO.  PITSA  7.97 6.95 5.30 3.19 1 .09  AVERAGE  6.66 5.89 4.57 2.76 0.93  6.4 C o m p a r i s o n o f l a t e r a l d i s p l a c e m e n t o f frame 4 f o r f i v e - s t o r e y frame s t r u c t u r e w i t h r e v i s e d mass  80  LATERAL  DISPLACEMENT ( i n . )  DRAIN -TABS FLOOR NO.  5 4 3 2 1  N69W  1.37 1.01 0.66 0.34 0.10  TAFT S7 1W  1.03 0.78 0.52 0.26 0.08  EL CENTRO NS EW  0.86 0.66 0.44 0.24 0.07  1.30 0.97 0.64 • 0.32 0.10  DRAIN-TABS FLOOR NO.  5 4 3 2 1  Table  PITSA  1 .49 1.10 0.71 0.36 0.10  AVERAGE  1.14 0.86 0. 56 0.29 0.09  6.5 C o m p a r i s o n o f l a t e r a l d i s p l a c e m e n t o f w a l l for f i v e - s t o r e y coupled wall structure  4  81  LATERAL  DISPLACEMENT ( i n . )  DRAIN-TABS FLOOR  NO.  N69W  TAFT S21W  1 .98 1 .45 0.95 0.49 0.14  5 4 3 2 1  09 50 0.94 0.50 0.16  EL NS  CENTRO EW  2.66 1 .98 1.31 0. 68 0.20  3 . 0 1  2.16 1 . 3 4  0.67 0 . 1 9  DRAIN-TABS FLOOR  NO.  PITSA  AVERAGE  3.42 2.49 1 .60 0.82 0.24  Table  6.6 C o m p a r i s o n o f l a t e r a l d i s p l a c e m e n t o f w a l l for f i v e - s t o r e y coupled wall structure w i t h r e v i s e d mass  4  PITSA NO.  (ONE  ANALYSIS)  OF  ITERATIONS  FRAME STRUCTURE  DRAIN-TABS  5  CPU  TIME  18.6 s e c .  COMPUTER  $1 .30  $  TIME  STEP  1000 @  CPU  TIME  COMPUTER $  98.9 s e c .  $5.15  71.2 s e c .  $6.18  30.0 s e c .  $2.60  29.5 s e c .  $2.55  0.01 s e c .  FRAME STRUCTURE REVISED MASS  8  WALL STRUCTURE  5  20.7 s e c .  $2. 38  1000 @ 0.01 s e c .  5.6 s e c .  $0.63  1000 @ 0.01 s e c .  WALL STRUCTURE  5  6.9  sec .  REVISED MASS  $0.81  1000 @ 0.01  Table  6.7 C o s t s  of e x e c u t i o n  sec .  83  F i g u r e 1.1 F l o o r p l a n o f t h e J . C . in Anchorage, A l a s k a , showing the shear wall c o n f i g u r a t i o n .  Penney highly  Building eccentric-  84  Figure 1.2 E a s t w a l l and n o r t h e a s t c o r n e r of the J . C . Penney B u i l d i n g , a f t e r the 1964 e a r t h q u a k e . T h i s shows t h e c o m p l e t e c o l l a p s e of t h e s h e a r w a l l and p o r t i o n s of t h e r o o f and f l o o r s at the north e a s t c o r n e r of the b u i l d i n g .  85  7777  Figure  2.1(a) G r o s s  7777  frame d e g r e e s  7777  Figure  2.1(b) C o n d e n s e d  7777  of -freedom  7777  frame d e g r e e s  7777  of  freedom  86  Figure  2.1(c)  Gross  structure  degrees  of  freedom  >7r  Figure  2.1(d)  Condensed  structure  degrees  of  freedom  87  F i g u r e 2.2 P l a n view of n f l o o r showing and d i a p h r a g m h o r i z o n t a l d i s p l a c e m e n t s .  frame  88  6 F i g u r e 3.1 P h y s i c a l  Interpretation  of  damage  ratio  SPECTRAL  O  VELOCITY  U3 i-l fD  3  Cu  z  >-| CD O « C D CD  3  n ui o CD  M-  H-> a  CD CD r( (3J Qj h-' rt i-" N O CD 3 Di  MCD  c n i—• o\° CD  cn D-j nOJ M 3 o TJ M- a r> co i£> in  - M' D  U)  T5 CD  n rr  i-t  C 3  v  50  M.  • iQ cn C  U3  S ,  .  t  68  IN/SEC 100  200  500  lOOO  90  o  0.2  0.4  0.6  0.8  RATIO OF PERIODS, r Figure  4.2 P l o t o f c r o s s - m o d a l c o e f f i c i e n t s vs. r a t i o of p e r i o d s  1.0  91  20  20  o II CM  < CO X u « o Eto  tn 777  777  777  ELEVATION T Y P I C A L FRAME  PLAN  Figure  4.3  Five  storey  building  example  PERIODS(SEC)  0.5199 0.5008 0.3522 0.1604 0.1545 0.1089  Figure  4.4 P e r i o d s  and d i r e c t i o n s  o f mode s h a p e s  93  324 . 9  •=}< CN CO  ro  CM  237.0 376. 1  CO  TIME HISTORY  co CO  5 5 3 .4  CN  ro CN "3<  ABSOLUTE SUM  CO  m in  255.0  CN  o oo  CM  283. 1 TIME-STEP ANALYSIS USING TAFT 1952 N69W  Figure  4.5  C o m p a r i s o n of  RESPONSE SPECTRUM ANALYSIS USING SPECTRUM 'A'  modal c o m b i n a t i o n  methods  94  PITSA  Read  in.data  Builds  - structure,  earthquake  [ K ], [ M ]  Calculate  periods,  mode  shapes  •* Calculate Perform  Substitute  smeared  spectrum  Evaluate  force  damping  Structure  ratio  analysis  levels  M o d i f y damage r a t i o s  Modify  Check  E I , member damping  f o r convergence — YES  Figure  5.1 G e n e r a l c o n c e p t u a l o u t l i n e  of.PITSA  Method  95  MAIN  MEMGEN MMATRX BUILDM MAINF  REDUCE  GETAR  PART SYMM MAINR  BANDED BANDEF REDUCE  GETAR  FORCEV MAIND  FREQ SPECTR FORCES  LOWTRI DSYMFU PARFAC  DEFLNS FORSP  Figure  5.2 Program  (PITSA)  SRUM  organization  DSTURM  96  O.O  1  1  1  1  1  1  0.5 12  10  8  Frequency, Spectral Spectral  6  4  1  1.0  2  1  1.5  i  2.0  i  2.5  P e r i o d , sec  hertz Accel, Accel,  for ft = 8 f o r 0=0.02 6+100-0  Figure  5.3 S p e c t r u m  'A'  3.0  3.0  Figure  5.4  Acceleration  P e r i o d , sec s p e c t r a of T a f t N69W and  spectrum  'A'  3.0  I  1  0.0  Figure  5.5  1.0  Acceleration  P e r i o d , sec s p e c t r a of T a f t  L_  I  1  0.5  1.5  S21W  and  spectrum  2.0  'A  1  Figure  5.6  Acceleration  P e r i o d , sec s p e c t r a of E l C e n t r o  EW  and  spectrum  'A'  3.0 El  C e n t r o NS  Spectrum a n d S p e c t r u m A  .01  .02  .05  .1  .2  .5  1  PERIOD,  2  5  10  20  50  100  SEC  F i g u r e 5.8 Ground m o t i o n bound t r i p a r t i t e p l o t e a r t h q u a k e s which make up Spectrum 'A' ( s c a l e d  of to  four 0.5g)  1 02  COLUMNS 30x30  A=900  1=67500  l(cr)=33750  BEAMS  A=648  1=69984  Kcr)=23328  18x36  20'  20*  20'  20'  20'  STOREY = 130 k  o  >  WEIGHT  o I  LJJ  MASS CENTR  130  <  130 k  k  130  k  130  k  t  Q  \  O I  I 1  CN  / > \ >  E< CO X  w o  CO  ELEVATION OF FRAMES 1 & 4  ELEVATION OF FRAMES 2 & 3  FRAME 4 D-  -a ro  CN  LJ <  E/Q MOTION -a-  - o  FRAME 1  PLAN  Figure  6.1 D i m e n s i o n s and p r o p e r t i e s o f f i v e - s t o r e y  frame  building  103 12'-0" STOREY WEIGHT  12'-0"  18'-0"  = 300k  0 MASS CENTRE 300  k  o 10  300  k  300  k  >  ^  300 k  BEAMS  12x36  A =432 1=5184 I(cr)=1728  CHANNELS A=6664 1=1536444  ELEVATION OF WALLS 1 t 4  I(cr)=768222  ELEVATION OF WALLS 2 & 3  WALL 4  u  r  CM  a <  E/.Q MOTION  i  co  FLOOR AREA -  CORE AREA - 3 0 ' X 1 8 '  21'-8' 4'-2'  14  4 ' -2'  ,4+  WALL 1 C  PLAN  Figure  6.2 D i m e n s i o n s coupled  100'X60'  and p r o p e r t i e s  wall  building  of f i v e - s t o r e y  104  O 0  o  12  o  10  0 0  22 19  b 0  23 2 1  0 0  23  0  49  0  48  72 56  2 00 1 41  20  3 3  00  5 4  OO 25  4  08 78  3  85  0 0  21 17  0 0  30 28  0 0  31 29  0 0  33 30  0  52  0  51  0 0  52  1 89 1 27 3 2  86  7 1  4 4  86 1 1  3 3  85 55  0 0  12 17  0  21  0  30  0 0  23 26  0 0  23 31  0 0  52  0.25  2.25  EL EL  0.22 0.25  0.32  3.74 4 . 44 3.63  0.30 0.35  2.15 3.62 4.33 3.46  0.56  0.52  3 6  58 3 1  4 7  8 1 38  4  15 69  0 0  2 1 33  0 0  30 37  0 0  31 34  0 0  33 4 1  0 0  55 66  1 69  4.19  0.23  5.36  0.24  65  3 6  44  4 7  67 25  3 5  92 45  15  0.71 0.23  2 .23  0.23  3  CENTRO NS CENTRO EW  0.88 0.13  0 62 1 06  0.31 0.31 0.34  4.43  2.12 4. 04 5.22 4.19'  0.56  0. 53  AVERAGE OF FOUR TIME STEP ANALYSIS  PITSA Figure  68  5  0.72  0.88  0.24  1 82 3  63  TAFT N69W TAFT S21W  0.15  0 70 1 52  64  6.3 Damage r a t i o s  for five-storey  frame  structure  1 05  TAFT  N69W  TAFT  S21W  -0  E L CENTRO  NS  -•  E L CENTRO  EW  AVERAGE -  PITSA  O 2  o o  3  4  DUCTILITY  Figure  6.3.1  Beam damage r a t i o s  for five-storey  frame  structure  1 06  0 0  17 17  0 25 0 2 1 0 24 0 21 0 0  27 22  0 0  44 51 |  1 67 1 01  0 34 0 29  3 43 2 64  0 34 0 29  4 98 3 67  0 32 0 30  5 40 4 38  0 37 0 31  3 78 3 98  0 0  48 54  1 19 0 86 3 4 1 2 57 4 80 3 52 5 27 4 22 3 55 3 '75  0. 23 0. 28  0.34  3. 62  0.31  4. 77 5. 1 6  4. 23  0.30 0.38 0.65  0. 62  PITSA  Figure  4 30 6 96  0 26 0 26  5 88 7 23  0 26 0 34  4 96 5 30  0 58 0 61  EL EL  1 .92  2. 29  0. 25  2 26 4 68  0 24 0 33  TAFT N69W TAFT S21W  0. 1 7  0 89 2 83  0 15 0 17  3. 57 4. 63 5. 04 4. 06  0 26 0 34 0 33 0 39 0 34 0 34 0 36 0 44 0 62 0 65  3.25  0. 26  4.98  0. 24  5.72  0. 27  2 13 4 64 4 15 6 80 5 73 7 or 4 73 5 06  CENTRO NS CENTRO EW  1 .29  1 .60 0. 1 7  0 77 2 36  4.51  0 .31  3. 1 9  0 . 34 4 .82 0 . 33 0 .37  5 . 56 4 .27  0 . 57  0. 54  AVERAGE OF FOUR TIME STEP ANALYSIS  6.4 Damage r a t i o s f o r f i v e - s t o r e y w i t h r e v i s e d mass  frame  structure  1 0 7  -*  TAFT N69W  +  TAFT  O-  -O  EL CENTRO  NS  B-  -Q  EL CENTRO  EW  S21W  AVERAGE PITSA  O 2  «  o o  3  4  DUCTILITY  Figure  6 . 4 . 1 Beam damage r a t i o s with  revised  mass  for five-storey  frame  structure  108  3.74 2 .67  08 08  0 08 0 08  3 . 85 2 . 83  12 10  0 0  3 . 70  2.80  o. 23 o. 18  2.42  10 10 '  1 .52  2 .05 2 . 99  EL EL  3.37  0.24 0.38  0.08  3.32  0.10  2.84  0.19  1 .86  0.31  3.13 3.08 2.64 1 .66  0.46  0.54  AVERAGE OF FOUR TIME STEP ANALYSIS  PITSA  Figure  CENTRO CENTRO  2.98  3.44  0.13  1 . 34 1 . 87  0 38 0 5 1  TAFT N69W TAFT S21W  0.07  2.31 3 . 50  0 24 0 34  1 .92  52 42  2 . 26 3 . 57  0 14 0 20  ' 3 .09  o. 35 o. 29  2 .09 3 .43  6.5 Damage  ratios  for five-storey  coupled  wall  structure  109  TAFT N69W TAFT S21W -e  EL  -Q  EL CENTRO EW  CENTRO NS  AVERAGE •-  PITSA  o z  o o J  2  -  3 4 DUCTILITY  Figure  6.5.1  Beam damage r a t i o s structure  for five-storey  coupled  wall  11 0  4 .70 4 88  07 09  4 66 4 88  18 2 1  4 13 4 42  25 27  3 44 3 43  0 . 32 0 . 33  2 13 2 26  0 . 46 0.52  0 08 0 10 0 17 0 20 0 27 0 34 0 42 0 46 0 66 0 64  TAFT N69W TAFT S21W  0.29 0.37 0.51 0. 77  5.99 7 .02 5 . 57 6 38 4 . 76 4 95  3.05 , 2,93  EL CENTRO NS EL CENTRO EW  6.03 0.15  5.99 7 .05  5.65  5.97  0.08  5.52  0.19  4.61  0.29  3.03  0.38  5.64 5.13 4.14 2.59  0.57  PITSA  Figure  AVERAGE OF FOUR TIME STEP ANALYSIS  6.6 Damage r a t i o s f o r f i v e - s t o r e y s t r u c t u r e w i t h r e v i s e d mass  coupled  wall  111  TAFT  N69W  TAFT  S21W  o-  -O  E L CENTRO  NS  Q-  -Q  E L CENTRO  EW  AVERAGE PITSA  O z CC  o o j  Ci-.  2  —  3  4  DUCTILITY  Figure  6.6.1  Beam  damage  structure  ratios for  with  five-storey  r e v i s e d mass  coupled  wall  11 2  0 0  17 18  0 35 0 40 3 4 1 5 56 O 43 0 45 0 77 0 . 85  2 04 3 16  2 .03 • O 34 3 05 0 35 1 28 1 16 1 73 0 56 1 60 0 62 1 77 1 73 2 15 2 13 2 26 4 19 3 78 3 15 5 34 0 60 4 70 0 62 5 66 5 44 6 73 0 . 80 6 50 0 . 87  0 0  1 80 1 26  18 17  1 17 0 93  0 34 0 29  1 63 1 39  4 49 4 2 1  4 47 4 38  0 43 0 43  6 34  6 05  0 85 0 80  TAFT N69W TAFT S21W  1 . 30 0.16  1 .29  0.36  4. 48  1 .82  5.75  0.31  4 .70  1 . 0.32 0.52 2.40 0.41 0.71  0.68  PITSA  Figure  EL EL  1 . 4. 5. 4.  0 35 0 31 0 54 0 48 3 17 2 92 0 59 0 60 0 88 0 82  1  0. 34  0.43  .28  1 . 7 4  4 . 42  1 04 0 85 1 51 1 23 3 85 3 77 6 12 5 83  CENTRO NS CENTRO EW  2.05  2.07 0.17  1 87 1 24  0.34 1.17 °3  4 . 4 9  6.19  5 5  •  10  0.60  1.68 3.87 5  >  9  7  0.84  0.82  AVERAGE OF FOUR TIME STEP ANALYSIS  6.7 Damage r a t i o s f o r f i v e - s t o r e y frame w i t h s t r o n g beams and weak c o l u m n s  structure  11 3  T A F T N69W TAFT  321W  O-  EL  CENTRO  NS  O-  EL  CENTRO  EW  AVERAGE PITSA  O z  o o j  2  3  4  DUCTILITY  Figure  6.7.1  Beam d a m a g e r a t i o s f o r f i v e - s t o r e y f r a m e w i t h s t r o n g b e a m s a n d weak c o l u m n s  structure  114  BIBLIOGRAPHY 1.  W i l s o n , E. L . , H o l l i n g s , J . P. a n d D o v e y , H. H., m e n s i o n a l A n a l y s i s of B u i l d i n g Systems ( E x t e n d e d R e p o r t No. E E R C 7 5 - 1 3 , U n i v e r s i t y o f C a l i f o r n i a , April 1975.  "Three D i Version)," Berkeley,  2.  K a n a a n , A., a n d P o w e l l , G. H., " G e n e r a l P u r p o s e C o m p u t e r Program f o r I n e l a s t i c Dynamic Response of P l a n e S t r u c t u r e s , " R e p o r t No. EERC 7 3 - 6 , U n i v e r s i t y o f C a l i f o r n i a , B e r k e l e y , 1973.  3.  G u e n d e l m a n - I s r a e l , R., a n d P o w e l l , G. H., " D R A I N - T A B S : A Computer Program f o r I n e l a s t i c Earthquake Response of Three D i m e n s i o n a l B u i l d i n g s , " R e p o r t No. 7 7 - 8 , U n i v e r s i t y o f C a l i f o r n i a , B e r k e l e y , March 1977.  4.  P e n z i e n , J . , " E l a s t o - P l a s t i c Response of I d e a l i z e d M u l t i storey Structures Subjected to a Strong Motion Earthquake," P r o c , 2 n d WCEE, T o k y o a n d K y o t o , J a p a n , 1 9 6 0 , V o l . I I , S e s s i o n 11, pp. 739-60.  5.  J e n n i n g s , P. C , "Response of Simple Y i e l d i n g S t r u c t u r e s t o Earthquake E x c i t a t i o n s , " t h e s i s submitted to the C a l i f o r n i a I n s t i t u t e of Technology at Pasadena in partial fulfillment of t h e r e q u i r e m e n t s o f t h e d e g r e e of D o c t o r of P h i l o s o p h y , 1 963.  6.  C l o u g h , R. W., a n d J o h n s t o n , S. B., " E f f e c t o f S t i f f n e s s D e g r a d a t i o n on E a r t h q u a k e D u c t i l i t y R e q u i r e m e n t s , " P r o c . Japan E a r t h q u a k e E n g i n e e r i n g Symposium, Tokyo, October 1966 , pp. 227-32.  7.  T a k e d a , T., S o z e n , M. A., a n d N i e l s e n , N. N., " R e i n f o r c e d C o n c r e t e Response t o S i m u l a t e d E a r t h q u a k e s , " 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 . 96, December 1970, pp. 2557 -73.  8.  G u l k a n , P., a n d S o z e n , M. A., " R e s p o n s e a n d E n e r g y - D i s s i p a t i o n of R e i n f o r c e d C o n c r e t e Frames S u b j e c t e d t o S t r o n g Base Motions," C i v i l Engineering Studies, S t r u c t u r a l Research S e r i e s No. 3 7 7 , U n i v e r s i t y o f I l l i n o i s , U r b a n a , May 1971.  9.  T s o , W. K., a n d D e m p s e y , K. M., "Seismic Torsional Provisi o n s f o r Dynamic E c c e n t r i c i t y , " E a r t h q u a k e E n g i n e e r i n g and S t r u c t u r a l D y n a m i c s , V o l . 8, 1 9 8 0 , p p . 2 7 5 - 2 8 9 .  10.  K a n , C. L . , a n d C h o p r a , A. K., " E f f e c t o f T o r s i o n a l C o u p l i n g on E a r t h q u a k e f o r c e s i n B u i l d i n g s , " J o u r n a l o f t h e S t r u c t u r a l D i v i s i o n , ASCE, V o l . 103, A p r i l 1977, pp. SOSS i 9.  115  11.  Kan, C. L . , a n d C h o p r a , A. K., "Torsional Coupling and E a r t h q u a k e R e s p o n s e of S i m p l e E l a s t i c and I n e l a s t i c S y s t e m s ," J o u r n a l o f t h e S t r u c t u r a l D i v i s i o n , A S C E , 1 9 8 1 , p p . 1569 -1588.  12.  T s o , W. K., a n d S a d e k , A. W . , " I n e l a s t i c R e s p o n s e o f E c c e n t r i c S t r u c t u r e s , " 4 t h CCEE, V a n c o u v e r , C a n a d a , June 1 9 8 3 , pp. 261-270.  13.  M a c K e n z i e , J . R., " T o r s i o n a l S t r u c t u r a l Response During E a r t h q u a k e E x c i t a t i o n s , " M a s t e r ' s t h e s i s U n i v e r s i t y of B r i t i s h Columbia, A p r i l 1974.  14.  Y o s h i d a , Sumio, " 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 o f E x i s t i n g R/C Structures," Master's U.B.C, March 1979.  15.  M e t t e n , A n d r e w W. F., "The Modified Substitute Structure M e t h o d As a D e s i g n A i d f o r S e i s m i c R e s i s t a n t Coupled S t r u c t u r a l W a l l s , " M a s t e r ' s t h e s i s U.B.C., M a r c h 1 9 8 1 .  16.  S h i b a t a , A k e n o r i , and S o z e n , Mete M e t h o d f o r S e i s m i c D e s i g n i n R/C," D i v i s i o n , A S C E , J a n u a r y 1976, pp.  17.  Hui, Lawerence, Master's thesis  18.  N e w m a r k , N. M. , " T o r s i o n i n S y m m e t r i c a l B u i l d i n g s , " 4 t h WCEE, S a n t i a g o , C h i l e , 1 9 6 9 , pp. A3.19-A3.32  19.  N e w m a r k , N. M., a n d R o s e n b l u e t h , E., " F u n d a m e n t a l s of Earthquake Engineering," P r e n t i c e - H a l l Inc., Englewood C l i f f s , N.J., pp. 243.  20.  N a t h a n , N. D., a n d M a c K e n z i e , J . R., "Rotational o f E a r t h q u a k e M o t i o n , " C a n a d i a n J o u r n a l of C i v i l ing, V o l . I I , 1975, pp. 430-436.  21.  H a r t , G. C , D i J u l i o , M., a n d Lew, M., of H i g h - r i s e B u i l d i n g s , " J o u r n a l of the , P r o c , A S C E , 101, ST2, pp. 397-416.  22.  W i l s o n , E. L . , D e r K i u r e g h i a n , A., a n d B a y o , E. P., "A Rep l a c e m e n t f o r t h e SRSS M e t h o d i n S e i s m i c A n a l y s i s , " Earthquake Engineering and S t r u c t u r a l D y n a m i c s , V o l . 9, 1981, pp. 187-194.  Method thesis  for  A., "Substitute-Structure J o u r n a l of the S t r u c u r a l 1-18.  "Pseudo N o n - l i n e a r Seismic U.B.C., O c t o b e r 1984.  Analysis,"  Proc.  Components Engineer-  " T o r s i o n a l Response Structural Division  1 1 6  APPENDIX A  PROGRAM USER'S  MANUAL  IDENTIFICATION PITSA:  PSEUDO INELASTIC  TORSIONAL  SEISMIC ANALYSIS  Computer p r o g r a m f o r t h e i n e l a s t i c s e i s m i c a n a l y s i s of three-dimensional buildings under earthquake exc i t a t i o n .  DISCLAIMER: The C i v i l E n g i n e e r i n g D e p a r t m e n t , F a c u l t y and S t a f f do not g u a r a n t e e n o r i m p l y t h e a c c u r a c y o r r e l i a b i l i t y of t h i s program or related documentation. As such,, t h e y can n o t be held responsible f o r i n c o r r e c t r e s u l t s o r damages r e s u l t i n g from t h e use o f t h i s program. I t i s the r e s p o n s i b i l i t y of the user t o determine the usefulness and t e c h n i c a l a c c u r a c y of t h i s program i n h i s o r h e r own e n v i r o n m e n t . This  program  may n o t be s o l d  to a third  party.  1  1 7  PITSA  PROGRAM  UPDATES  HISTORY  MODIFICATIONS  1974  MACK.FRAME  program  1975  Spectral analysis t o MACK.FRAME  198 1  EDAM  1984  PITSA  program program  PROGRAMMER  written added-  written written  J.R. M a c K e n z i e Bill  McKevitt  Andrew W . F . M e t t e n K e n S.K. Tarn  1  18  PURPOSE The program utilizes the modified substitute structure method to determine the inelastic dynamic r e s p o n s e of threedimensional b u i l d i n g s c o n s i s t of a r b i t r a r y l o c a t e d frames and/or shear walls due to earthquake motions. Two independent horizontal motions plus r o t a t i o n a l motion may be s p e c i f i e d . The p r o g r a m u s e s s p e c t r a l a n a l y s i s i t e r a t i v e l y i n w h i c h s t i f f n e s s and d a m p i n g p r o p e r t i e s a r e s u b s t i t u t e d .  THEORY Details theses: J.R.  the  MacKenzie:  Andrew  Ken  of  W.F.  S.K.  Metten:  Tarn:  theory  can  be  "Torsional Earthquake  found  i n the  following  master's  Structural Response During Excitations" April 1974  "The M o d i f i e d Substitute Structure Method As A Design Aid For Seismic Resistant Coupled S t r u c t u r a l W a l l s " March 1981 "Pseudo I n e l a s t i c T o r s i o n a l Seismic U t i l i z i n g The M o d i f i e d S u b s t i t u t e Method" A p r i l 1985  PROGRAM  Analysis Structure  RESTRICTIONS  ( 1 )  T h e r e a r e t o be no abrupt or geometry t h r o u g h o u t the  changes h e i g h t of  (2)  Non-structural not a f f e c t the  a r e t o be s u c h the s t r u c t u r e .  (3)  Members a r e r e i n f o r c e d t o of i n e l a s t i c d e f o r m a t i o n s decay.  (4)  Diaphragms are the c e n t e r s of  (5)  Inelastic analysis p o r t i o n of the p r o g r a m can o n l y be used f o r concrete construction because development of the s t i f f n e s s r e d u c t i o n and s u b s t i t u t e damping f o r m u l a s was d o n e on c o n c r e t e members.  ( 6 )  The m e m b e r s a r e assumed n e g a t i v e moment c a p a c i t y  components r e s p o n s e of  assumed mass of  i n mass, s t i f f n e s s the s t r u c t u r e . that  they  do  withstand repeated reversals without significant strength  rigid and each storey  masses are level.  lumped  at  t o h a v e t h e same p o s i t i v e a n d a t b o t h e n d s o f t h e member.  119  DIMENSIONING The  program  i s dimensioned  150  joints  300  members  10  frames  20  storeys  30  modes  per per  6 additional  LIMITS  f o r the  following:  frame frame  degrees  of freedom  per  frame  1 20 •  Input  grouped  and  Output  The i n p u t d a t a f o r t h e a n a l y s i s i n t o the f o l l o w i n g f i v e sets:  1) structure  control  2)  frame  member  3)  number  4)  mass  5)  earthquake  and of  data  modes  of  a  structure  can  be  cards data  cards  card  card spectrum  cards  Input data m u s t be p r e p a r e d and a r r a n g e d i n a f i x e d order. The u n i t s y s t e m is chosen t o ' be Imperial thus problems with metric m e a s u r e m e n t s must be converted f i r s t . A l l input values need not follow t h e f i x e d 'format, a s l o n g a s t h e y a r e s e p a r a t e d by commas. The c o n t e n t a n d f o r m a t of i n d i v i d u a l data c a r d s are d e s c r i b e d i n the next section. The joint numbering system of a frame used i n the program s t a r t s w i t h 1 f o r each frame, s e q u e n t i a l l y from l e f t to r i g h t a t a g i v e n f l o o r l e v e l , and c o n t i n u o u s l y from the base to the top. The l a s t j o i n t number o f an N - s t o r e y M-bay f r a m e must be e q u a l t o ( N + 1 ) ( M + 1 ) . Members o f a frame a r e numbered i n the same m a n n e r . Columns are numbered f i r s t , s t a r t i n g with 1 for each frame, s e q u e n t i a l l y from the bottom to the top at a given column l i n e , and from l e f t t o r i g h t u n t i l a l l column l i n e s are accounted for. Beams are numbered, continuous with column numbers, s e q u e n t i a l l y from the b o t t o m t o the top a t a g i v e n bay, and from l e f t . to r i g h t u n t i l a l l bays are accounted f o r . The l a s t c o l u m n number o f an N - s t o r e y M-bay f r a m e m u s t be equal to N ( M + 1 ) , a n d t h e l a s t beam n u m b e r m u s t be e q u a l t o N ( 2 M + 1 ) . ;  The program reads and p r i n t s a l l i n p u t i n f o r m a t i o n , such t h a t the user can examine the p r i n t e d i n f o r m a t i o n to ensure t h a t a s t r u c t u r e d e f i n e d by t h e u s e r a n d t h a t i n t e r p r e t e d by t h e p r o g r a m be t h e s a m e . A u s e r may o f t e n be a b l e t o f i n d an error s u c h a s w r o n g d a t a f o r m a t i n d a t a c a r d s by c h e c k i n g t h e printed information. In a d d i t i o n t o a p r i n t - o u t of a l l i n p u t d a t a , ' t h e program also provides for the complete structure: natural period, mode shape, participation f a c t o r s f o r e a c h mode, a n d d i a p h r a g m d i s p l a c e m e n t s f o r e a c h mode a n d e a c h e a r t h q u a k e excita -tion. While for each frame, the program provides joint displacements and member f o r c e s f o r e a c h mode a n d e a c h e a r t h quake excitation. Total r e s p o n s e s ..are o b t a i n e d by combining modal responses from a g i v e n e x c i t a t i o n i n a c o m p l e t e - q u a d r a t i c c o m b i n a t i o n m a n n e r , a n d t h e n by c o m b i n i n g c o n t r i b u t i o n s from a l l excitations in a root-sum-square manner. Also printed and s t o r e d a r e t h e d a m a g e r a t i o s o f e v e r y member i n e a c h iteration. !  121  INPUT DATA CONTROL  INFORMATION  STRUCTURE CONTROL  CARD  >NOFR,NSTOR,NOCOR,NDISPL,NFORCE,MASSVT,INELAS,I SPEC, AMAX,DAMPIN,STRHRD (8I4,3F8.4) NOFR=  number o f  frames  NSTOR=  number o f s t o r e y s  NOCOR=  number o f a d d i t i o n a l structure  NDISPL=  number o f modes of d i s p l a c e m e n t s t o be  NFORCE=  number o f modes o f f o r c e s  MASSVT=  0 i f c e n t e r s o f mass do n o t l i e i n a s t r a i g h t line  degrees  of freedom i n printed  t o be p r i n t e d  1 i f c e n t e r s o f mass do l i e i n a s t r a i g h t INELAS=  0 i f elastic  analysis  line  i s requested  1 or g r e a t e r i f i n e l a s t i c a n a l y s i s i s r e q u e s t e d INELAS i s t h e maximum number of i n e l a s t i c i t e r a t i o n s t o be p e r f o r m e d b e f o r e t h e program stops. A v a l u e o f 25 i s s u g g e s t e d . ISPEC= =  t y p e of s p e c t r u m  requested  1 spectrum  'A'  2 spectrum  'B' (  "  )  3 spectrum  'C  "  )  4 National  Building  5 Newmark 6 C.I.T.  type  (Shibata  ( Code  and Sozen)  spectrum  spectrum  s i m u l a t e d spectrum, ground  acceleration  C-2  AMAX=  maximum gravity  DAMPIN=  damping r a t i o t o be used i n t h e e l a s t i c a n a l y s i or i n t h e f i r s t i t e r a t i o n o f t h e i n e l a s t i c analysis  STRHRD=  strain  hardening  ratio  as a f r a c t i o n  of  1 22  TITLE  CARD  >DESCRIPTIVE TITLE  (20A4)  FRAME DATA - One s e t o f d a t a FRAME CONTROL  m u s t be e n t e r e d  f o r each  frame  CARD  >NFR,NDF,INPUT,IQ,IR,(IO(J),J=1,NSTOR) f o r MASSVT=0 >NFR,IQ,IR,10,NDF,INPUT f o r MASSVT=1  '  (3I4,22F8.0)  (14,3E8.0,214)  NFR=  frame  number  NDF=  number o f a d d i t i o n a l  INPUT=  0 i f member g e n e r a t i n g  routine  i s not  1 i f member g e n e r a t i n g  routine  i s requested  degrees of freedom  IQ=  direction  cosine  in Q  direction  IR=  direction  cosine  in R  direction  10=  perpendicular mass  distance  MEMBER DATA CARDS - f o r I N P U T = 0 for >NRJ,NRM,E,G  from frame  (2I4,2F8.0) number o f  NRM=  number  joints  E=  modulus  G=  shear modulus, k s i  o f members  requested  t o c e n t e r of  only  INPUT=1, go t o n e x t  NRJ=  i n frame  . i  of e l a s t i c i t y , k s i  section  1 23  >JN,X,Y,ND(1),ND(2),ND(3) one c a r d f o r e a c h j o i n t  (14,2F8.0,314)  JN=  joint  X=  x-coordinate, f t  Y=  y-coordinate, f t  ND(1)=  fixity 0  number  code  in x-direction  i f t h e node  i s fixed  1 i f t h e node  i s free;  +N  i f t h e node node N  i s t o h a v e t h e same m o t i o n  -M  i f t h e node i s t o h a v e an a d d i t i o n a l d e g r e e f r e e d o m , M i s a t t a c h e d t o NDF, n o t NCOR  ND(2)=  fixity  code  for y-direction  ND(3)=  fixity  code  for rotation  >NC0R(I) (614) omit t h i s card i f t h i s of f r e e d o m NCOR(I)=  global frame,  f r a m e h a s no a d d i t i o n a l  a d d i t i o n a l degree of freedom up t o s i x p e r f r a m e  as  degree number i n  >MN,JNL,JNG,KL,KG,AREA,AI,AV,BMCAP,EXTL,EXTG (5I4,6F12.0) one c a r d  f o r each  member  MN=  member  number  JNL=  lesser  joint  JNG=  greater  KL=  fixity  =  joint  number number  :  c o d e o f member! a t l e s s e r  joint  0 i f pinned 1 i f fixed  KG=  fixity  AREA=  cross  c o d e o f member section  area  at greater  o f member,  joint  in**2  1 24  c.  AI=  cracked  AV=  shear  BMCAP=  bending  EXTL=  rigid  extension  o f member  at lesser  EXTG=  rigid  extension  o f member  at greater  MEMBER  DATA  moment  area,  CARDS  of i n e r t i a  in**4  in**2  moment  -  o f member,  capacity,k - f t joint, f t joint, f t  f o r INPUT=1 o n l y f o r INPUT=0, go t o p r e v i o u s  section  >NCOR(I),NLINE(I),NDIR(I), 1=1,NDF (2014) o m i t t h i s c a r d i f t h i s f r a m e h a s no a d d i t i o n a l d e g r e e o f freedom NCOR(I)=  global  NLINE(I)=1ine NDIR(I)=  a d d i t i o n a l degree  i n which  direction  >NOB,NOD,E,G,IPIN  a d d i t i o n a l degree  of a d d i t i o n a l degree  number  of  bays  NOD=  number  of  diagonals  E=  modulus  G=  shear  IPIN=  0 i f frame  i s fixed-at  1 i f frame  i s pinned  of freedom of  freedom  of e l a s t i c i t y , k s i  modulus, k s i  (I4,F8.0) cards as required,  NX=  number  XX=  bay s p a c i n g , f t  >NY,YY a s many  number  (214,2F8.0,14)  NOB=  >NX,XX a s many  of freedom  omit  of r e p e t i t i o n s  base  a t base  i f NOB=0 of bay  spacing  (I4,F8.0) cards as required  NY=  number  of r e p e t i t i o n s  YY=  storey  spacing, f t  of storey  spacing  occurs  125  >ICBD,LINE1 ,LINE2,LEVEL 1 ,LEVEL2,KL,KG,AREA,AI,AV, BMCAP,EXTL,EXTG (71 4,F8.2,5F10.2) as many c a r d s a s r e q u i r e d ICBD=  1 f o r column 2 f o r beam two d i g i t number f o r d i a g o n a l , f i r s t d i g i t i s t h e no. of bays a c r o s s , s e c o n d d i g i t i s t h e no. of s t o r e y s up, n e g a t i v e s l o p e i s d e f i n e d by minus s i g n a t t a c h e d t o LINE1  LINE1=  line/bay diagonal  LINE2=  line/bay last diagonal  LEVEL1=  level  first  LEVEL2=  level  last  KL=  f i x i t y code o f l e s s e r end  =  first  member  starts  f o r column/beam or  member ends f o r column/beam or  member  starts  member ends  0 i f pinned 1 if  f i xed  KG=  f i x i t y code o f g r e a t e r end  AREA=  cross  AI=  cracked  AV=  shear  BMCAP=  b e n d i n g moment c a p a c i t y ,  EXTL=  rigid, extension  o f member a t l e s s e r  EXTG=  rigid  of member a t g r e a t e r  section  area  o f member,  moment o f i n e r t i a  area,  in**2  of member,  in**4  in**2  extension  k-ft joint, f t joint, ft  1 26  3.  NUMBER OF MODES >MM one  (14) card required  MM=  number of modes t o be  MASS DATA CARDS  - one c a r d  >NF,FQ,FR,FO  5.  (I 4,2F8.0,F10 . 0 )  :  storey  number  FQ=  weight  in Q-direction , k ips  FR=  weight  in R-direction , • k ips  FO=  rotational inertia,  kip-ft**2  SPECTRUM  NEWMARK TYPE SPECTRUM CARDS >AFAC,VFAC,DFAC 3 cards required,  b.  per s t o r e y  NF =  EARTHQUAKE a.  considered  - f o r ISPEC=4 o r 5 o n l y  (3F12.6) one f o r e a c h o f Q,R,0 d i r e c t i o n s  AFAC=  ground  acceleration limit,  VFAC=  ground  v e l o c i t y limit!,  DFAC=  ground d i s p l a c e m e n t l i m i t ,  ACCELERATION SPECTRUM CARDS >AMAXF(I), 1=1,3 1 card required,  g ;  g-sec  g/ft  ; g-sec/ft  g-sec**2  ; g-sec**2/ft  - f o r ISPEC=1 o r 2 o r 3 o r 6 on 1 y  (3F12.6) one v a l u e f o r e a c h o f Q,R,0 d i r e c t i o n s  AMAXF (I ) =magn i f a c a t i on f a c t o r :for maximum AMAX, d e f i n e d i n s e c t i o n l a  acceleration,  1 27  OPERATING INSTRUCTIONS The FORTRAN IV s o u r c e v e r s i o n o f t h e program i s i n t h e f i l e P I T S A . S , and t h e c o m p i l e d v e r s i o n of the program i s i n t h e f i l e PITSA. The f o l l o w i n g command w i l l r u n t h e p r o g r a m : $RUN  PITSA  1=-1 8 = -8  2=-2 3=-3 4=-4 5=INPUT 6=-OUTPUT 10 = - 10 11=-11 1 2 = - 12 1 3 = - 13  7=*DUMMY*  U n i t 7 c o n t a i n s o u t p u t from a l l i n t e r m e d i a t e i t e r a t i o n s and is assigned t o *DUMMY* i n order t o reduce c o s t . I f t h e user wants t o view o r p r i n t o u t t h i s f i l e , a t e m p o r a r y f i l e -7 s h o u l d be a s s i g n e d .  LOGICAL I/O UNIT  ASSIGNMENT  T h i s i s a g e n e r a l l i s t of t h e d e f a u l t a c t i o n taken some o f t h e u n i t s a r e n o t a s s i g n e d on; t h e r u n command.  by MTS  and  When r u n n i n g BATCH u n i t s 5 a n d 6: d e f a u l t s t o t h e c a r d printer respectively.  the  When r u n n i n g from a t e r m i n a l u n i t s 5 a n d 6 b o t h terminal screen.  LOGICAL I/O UNIT 1 -4  f i l e s used i n t e r n a l l y by t h e p r o g r a m . *SOURCE*  f i l e from which a l l input i s read. f i l e t o which a l l input data i s echoed and t o which f i n a l i t e r a t i o n output i s w r i t t e n . i f i l e t o which a l l intermediate i t e r a t i o n output i s w r i t t e n . f i l e used i n t e r n a l l y : by t h e p r o g r a m . 10  11-13  f i l e t o which damage r a t i o s from e a c h i t e r a t i o n are w r i t t e n . :  f i l e s used i n t e r n a l l y by t h e p r o g r a m .  reader  defaults  DEFAULTS  DESCRIPTION  *SINK*  if  to  1 28  APPENDIX  SAMPLE  B'  I N P U T AND  OUTPUT  FOR F I V E - S T O R E Y C O U P L E D WALL S T R U C T U R E S E E C H A P T E R 6 . 2 . 2 ( c ) FOR D E T A I L S  4,5,O,O.0,0.25,1,.2,  .02,0.,  F I V E STOREYS STRUCTURE WITH COUPLING BEAMS WITH SPECTRUM  1 , 0 , 1 , 1. , 0 . . 1 1 . . 1 1 . , 1 1. , 11. , 1 1 . , 1 ,0.3600.,0.,0. 1,21.667. 5,12.. 1 , 1 , 2 , 1 . 5 , 1 , 1 , 6 6 6 4 . ,768241. .5553. , 5 0 0 0 . , 0 . . 0 . . 2, 1 , 1 , 1 , 5 , 1 , 1 , 4 3 2 . , 1 7 2 8 . , 3 6 0 . , 1 0 0 . , 7 . 8 3 3 , 7 . 8 3 3 , 2.0,1,0.,1..-10.833,-10.833,-10.833,-10.833,-10.833. 0,0,3600.,0.,0, 5,12., 1 , 1 , 1 , 1 , 5 , 1 , 1 , 6 6 6 4 . , 2 7 1 9 3 5 6 . . 5 5 5 3 . . 10000. . 0 . , 0 . , 3.0.1.0..1..10.833,10.833,10.833,10.833,10.833, 0,0,3600..0.,0, 5,12., 1,1,1,1,5,1,1,6664.,2719356.,5553.,10000.,0..0., 4,0,1,1.,0..-7.,-7.,-7.,-7.,-7., 1.0. 3 6 0 0 . . 0 . . 0 . 1,21.667, 5,12., 1 , 1 , 2 , 1 , 5 , 1 , 1 ,6664.,768241. ,5553. , 5 0 0 0 . , 0 . , 0 . , 2 . 1 , 1, 1 . 5 , 1 , 1 , 4 3 2 . , 17 28. . 3 6 0 . , 1 0 0 . , 7 . 8 3 3 , 7 . 8 3 3 , 6, 1,300.,300.,340000.. 2,300.,300.,340000., 3,300.,300..340000., 4,300.,300.,340000., 5.300.,300.,340000.. 1.,0.,0.,  1PITSA  NOV. 84  ***********************************************************^ FIVE  STOREYS  STRUCTURE  WITH  COUPLING  BEAMS  WITH  SPECTRUM  'A'(0.2G)  * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * 4 ^ * * * * * * * * *  NUMBER NUMBER  OF OF  FRAMES= STOREYS*  4 5  N U M B E R O F D E G R E E S OF F R E E D O M I N A D D I T I O N T O H O R I Z O N T A L N U M B E R O F M O D E S O F D I S P L A C E M E N T S TO B E P R I N T E D 0 N U M B E R O F M O D E S O F F O R C E S TO B E P R I N T E D * ' 0 C E N T E R OF MASS C O D E * 0 MAXIMUM N O . OF I N E L A S T I C A N A L Y S I S I T E R A T I O N S * 25 SPECTRUM TYPE = 1 • MAXIMUM A C C E L E R A T I O N * 0 . 2 0 0 0 TIMES GRAVITY I N I T I A L DAMPING RATIO* 0.0200 STRAIN HARDENING RATIO* 0.0  DEGREES  OF  FREEDOM*  0  1  CA)  O  --INITIAL MODES  ELASTIC  PERIOD NATURAL  EIGENVALUES  1  (RAD/SEC) 5.7339 10.7589  32.8775 115.7543  2 3 4  272.0156  16.4929 28.2874  FREQUENCIES (CYCS/SEC) 0.9126 1 .7123 2.6249  PERIODS (SECS) 1.0958 0.5840 0.3810 0.2221  4.5021 800.1800 5 4390.7263 66.2625 0.0948 10.5460 10.9337 6 4719.5053 68.6987 0.0915 * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * *.* * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * *  INELASTIC  RESULTS  ************************************************************************************************************** -ITERATION  NO.  NO.  ABOVE  CAPACITY  DAMERR  1  10  0.275  2  4  -0.071  3  O  0.002  - I T E R A T I O N NUMBER 4 1 FRAME N O . ADD DOF 1 BAYS 1  0 STOREYS 5  BAYS 1  SPACING 21.67  STOREYS 5  SPACING 12.00  C=1 , B = 2 , D = 3 1 2  LINE 1 1 1  INPUT  10  1  IR  1.00  0.0  LEVEL 1 1 1  LEVEL2 5 5  IO(N) 11.00  N=1,NST0R 11.00  11.00  11.00  11.00  PINS 0  LINE2 2 1  KL 1 1  KG 1 1  EXTL 0.0 7 .83  EXTG 0.0 7 .83  AREA 6664.00 432.00  I(CRACKED) AV 76824 1 . 0 0 5553 .00 1728.00 360 .00  MOM.  CAP.  5000.00 100.00  ************************************************************************************************************** NF R NRJ NRM E • G 1 12 15 3600.0 0.0 **************************************************************************************************************  JOINT JN  INFORMATION X  Y  ND1  ND2  ND3  1 "2  0 21  0 6G7  3 4  0 21  0 667  5 6 7  0 21  0 667  0 21  0 667  0 21  0 667  36 48 48  0 21  0 667  60 60  8 9 10 1 1 12  0 0 12 12 24 24 36  0  0  0  0  0 000 000  0 1 3 1  0  0  000 000 000  5 1  000 000  7 1  000 000 000  9 1 1 1  *********************>  MEMBER MN 1  JNL 1  2 3 4  3 5 7  5 6  9 2 4  7 8 9 10 1 1' 12 13 14 15  10  5 7  6 8  9 1 1  10 12  MEMBER 1 2 3 4 5 6 7  EXTG  0  12  00  0  0  -0  0  12  00  KL 1  KG 1  0 0 0  0 0  12 12 12  00 00  0 0  0 0  1 1  0 0 0  12 12 12 12 12  1 1  0  0 0 0 0 0  00 00  00 00 00 00 00  -0 -0 -0 -0 -0  00 00 00  -0 -0 -0  0 0  12 0 0  0 0 0  12 12 12  00 00 00  1 1 1 1  0 0  1 1 1 1 1  12 0 0 6 00  0 7  0 83  -0 6  0 00  12  00  00 00  7 7 7 7  6 6 6 6  00 00 00 00  -0 -0 -0 -0 -0  o•  6 00 6 00  83 83 83 83  0 0 0 0 0  6 8 '12 4  DM  0  EXTL  5 7 9 1 1 4  10 3  c*****  INFORMATION  JNG 3  6 8  *********************!  0 0 0  12 12 12 12  0 0  0 0 0 7 83 7 83 7 83 7 83 7 83  NP 1 0 1 6 1 1 16  6 6  NP2  NP3  0 2 7  0 3 8 13  • 12 17  8 9  0 1 6 11  10 1 1 12  16 1 6  9 14 19 2 7  13  1 1  12  18  0 0 0  NP4 1 6 1 1 16 2 1 1 6 1 1  XM  0  NP5 2 7  NP6  12 17  13 18  22 4  23 5  9 14  10 15 20 25 5 10 15  0 4  0 5 10 15 20 .. 3 • 8  16 21 1 6  19 24 4 9  13  1 1  14  3 8  YM  0 0 0 0  1 1  1 1 1  1 1 1 1 1  1 1 1 1 1  AREA 6664 6664 6664 6664  00 00 OO  6664 6664  00 00 00  6664 6664 6664  00 00 00  6664  00 00 00 00 00 00  432 432 432 432 432  I(CRACKED) 76824 1 00 768241 768241 768241 76824 1 76824 1 768241 76824 1 76824 1 768241 1728 1728 1728 1728 1728  00 00 00 00 00 00 00 00 00 00 00 00 00 00  AV  MOM.  CAP.  5553  00  5000.00  5553 5553  00 00  5553 5553  00 00 00  5000.00 5000.00 5000.00  5553 5553 5553 5553 5553 360 360 360 360 360  00 00 00 00 00 00 00 00 00  50O0.00 5000.00 5000.00 5000.00 5000.00 5000.00 100.00 100.00 1O0.O0 100.00 100.00  i  14 15  16 21  17 22  18 23  16 21  19 24  20 25  N O . O F D E G R E E S OF F R E E D O M O F S T R U C T U R E 25 H A L F B A N D WIDTH= 10 ********************************************************** 1  1  FRAME  NO.  ADD DOF  2 BAYS  INPUT  0 STOREYS  0  10  1 PINS  5  BAYS  SPACING  STOREYS  SPACING  5  12.00  C=1,B =2,D =3 1  LINE 1 1  IR  0 . 0  IO(N)  1.00  N=1.NST0R  -10.83  -10.83  -10.83  -10.83  -10.83  0  LINE2 1  LEVEL 1 1  LEVEL2 5  KL 1  KG EXTL EXTG 1 0 . 0 0 . 0  AREA 6664.00  I(CRACKED) 2719356.00  A V MOM. C A P . 5553.00 10000.00  ************************************************************************************************************* NFR  NRJ  NRM  E  G  2 6 5 3600.0 0 . 0 *************************************************************************************************************  JOINT UN 1 2 3 4 5 6  INFORMATION  0.0 0.0 0.0 0.0 0.0 0.0  Y  ND 1  0.0 12.000 24.000 36.000 48.000 60.000  0  ND3  NO 2 0  0  A * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * *  ****************** V******************************************M  MEMBER MN 1 2 3 4 5  JNL 1 2 3 4 5  JNG 2 3 4 5 6  INFORMATION EXTL 0.0 0.0 0.0 0.0 0.0  DM 12 12 1212 12  00 00 00 00 00  EXTG 0.0 0.0 0.0 0.0 0.0  XM -0 -0 -0 -0 -0  0 0 0 0 0  YM 12 12 12 12 12  00 00 00 00 00  KL 1 1 1 1 1  KG 1 1 1 1 1  AREA 6664.00 6664.OO 6664.00 6664.00 6664.00  I (CRACKED) 27 1 9 3 5 6 . 0 0 27 1 9 3 5 6 . 0 0 2719356.00 2719356.00 2719356.00  AV 5553 5553 5553 5553 5553  00 00 00 00 00  MOM.  CAP.  10O00.00 10000.00 10000.00 10000.00 10000.00  NP 1  NP2  NP3  0 1  0 2  0 3  3 4  4  6 9  5  10  5 . 8 1 1  MEMBER 1 2  7  NP4  NP5  NP6  2 5  6  1 4 7  8 1 1 14  10. 13  12  3 9 12 15  N O . OF D E G R E E S OF FREEDOM OF S T R U C T U R E * H A L F BAND W I D T H * 6  15  ********************************************* 1  FRAME  NO.  ADD D O F  3  0  BAYS  10  1  STOREYS  0  5  BAYS  SPACING  STOREYS 5  SPACING 12.00  C=1.B = 2,D = 3  INPUT  IR  0.0  10.83  10.83  10.83  10.83  10.83  PINS 0  LINE 1  LINE2  LEVEL 1  LEVEL2  1  1  1  5  1  I O ( N ) N=1,NST0R  1.00  KL  KG  1  1  0  EXTL .  0  0  ' .  EXTG 0  AREA 6S64.00  I(CRACKED) 2719356.00  AV 5553.00  MOM.  CAP.  10000.00  * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * *~*"* *"* * * * * V* * * *'* V* * * * * * * * * * * * * * * * * * * * * * * * * * * ' * * * * * * * * * * * * * * * * * * * * * * * * * • * * * * * * * * * * * NFR  NRJ  3  6  NRM 5  E  G  3600.0  0.0  **************************************************************************************  JOINT JN 1 2 3 4 5 6  INFORMATION ND 1 0.0 0.0 0.0 0.0 0.0 0.0  0.0 12.000 24.000 36.000 48.000 60.000  0  ND2  ND3  0  0  * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * **********************************************>  1  MEMBER  INFORMATION  ***********  MN 1 2 3 4 5  JNL .1 2 3 4 5  JNG 2 3 4 5 6  EXTL 0.0 0.0 0.0 0.0 0.0  NP 1 0 1 4 7 10  MEMBER 1 2 3 4 5  12 12 12 12 12  DM 00 00 00 00 00  NP2 NP3 0 0 2 3 5 6 8 9 1 1 12  EXTG 0.0 0.0 0.0 0.0 0.0  NP4 1 4 7 10 13  -0 -0 -0 -0 -0  XM 0 0 0 0 0  SPACING 21.67  STOREYS 5  SPACING 12.00  C=1,B = 2,D = 3 1 2  LINE 1 1 1  LINE2 2 1  YM KL 1 00 1 00 . 1 00 1 00 1 00  KG 1 1 1 1 1  AREA 6664.00 6664.00 6664.00 6664.00 6664.00  I(CRACKED) 2719356.00 2719356.00 2719356.00 2719356.00 2719356.00  5553 5553 5553 5553 5553  AV 00 00 00 00 00  MOM. CAP: 10000.00 10000.00 10000.00 10000.00 10000.00  NP5 NP6 2 3 5 6 8 9 1 1 12 14 15  NO. OF DEGREES OF FREEDOM OF STRUCTURE* HALF BAND WIDTH* 6 ************************************* 1 FRAME NO. ADD DOF INPUT 10 IR 4 0 1 1.00 0.0 BAYS STOREYS PINS 1 5 0 BAYS 1  12 12 12 12 12  LEVEL 1 1 1  LEVEL2 5 5  15  IO(N) N=1,NST0R -7.00 -7.00  -7.00  KL KG EXTL EXTG 1 1 0 . 0 0 . 0 1 1 7.83 7.83  -7.00  AREA 6664.00 432.00  -7.00  I(CRACKED) AV 76824 1.00 5553.00 1728.00 360.00  MOM. CAP. 5000.00 100.00  *************************************************************************************************************  NFR NRJ NRM E G 4 12 15 3600.0 0.0 *************************************************************  JOINT JN 1 2 3 4  INFORMATION  X 0.0 21.667 0.0 21.667  Y 0.0 0.0 12.000 ' 12.000  ND1 0 0 1 3  ND2 0 0 1 1  ND3 0 0 1 1  5 6  0.0 21.667  24.000 24.000  1 5  7 8 9 10  0.0 2 1.667 0.0 21.667  36.000 36.000 48.000 48.000  1 7 1 9  11 12  0.0 21.667  60.000 60.000  1 11  **********************************************************  MEMBER  INFORMATION  MN 1 2 3 4  UNL .1  5 6 7  9 2 4  1 1 4  8 9 10 1 1 12 13 14 15  3 5 7  UNG 3 5 7 9  EXTL 0 0 0 0  0 0 0  0  0 0  6  0 0  0 0  6 8  8 10  0 0  0 0  10 3  12 4  0 0 7 83  5 7  6 8  9 1 1  MEMBER 1 2 3 4 5 6 7 8 9 10 1 1 12 13 14 15  DM 12 12 12 12  00  12 12  00 00  12 12 12  00 OO 00  12 6  12 0 0 -0 0  83 83  6 6  00 00  -0 -0  83 83  6' 0 0 6 00  -0  -o  0 0  0 0  0 0  -0 -0  00 00  0 7  0 83  00 00  7 7  00  7 7  NP2  NP3  0 2 7 12 17  0 3 8 13 18  0 4 9 14  0 5  16 21  0 00  -0  0 1  12 17 22  -0 6  0 0  NP 1  6 1 1  0 0  12 0 0 12 0 0 12 0 0 12 0 0  0  83  19 2 7  1  1 1 1 1 1  -o  7  6 1 1 16 1  1  0 0 0  00 00 00  6 6 6 6  0 1  KG 1 1 1  0 0 0 0  83 83 83  6 1 1 16  KL 1 1 1  -0 -0 -0 -0  7 7 7  10 12  XM  EXTG 0 0  00  10 15 20 3 8 13 18 -23  0 0 0 0  NP4  0 0 0  NP5  NP6  2 7  3 8  12 17 22 4  13 18 23 5  9 14  10 15 20 25 5  6 1 1  19 24 4 9 14  16 21  19 24  20 25  1 6 1 1 16 21 1 6 1 1 16 21 1  10 15  YM 12 0 0 12 0 0 12 0 0 12 0 0 12 0 0  -0  AREA  I (CRACKED) 76824 1 00 76824 1 00 76824 1 00 76824 1 00  AV  MOM.  CAP.  5553  00  5553 5553 5553  00 00 00  5553  00 00 00  5000 5000 5000  00  5553 5553 5553 5553  00 00  5000 5000  oo  00 00  5553 360  00 00  5000 100  1728 1728  00 OO  360 360  00 00  100 100  oooo  1728 1728  00  360 360  00 00  100 100  6664 6664 6664 6664  00 00 00 00  1 1 1 1 1  6664 6664 6664  00 00 00  6664 6664  00 00  76824 1 768241 76824 1 76824 1  1 1  1 1  6664 432  00 00  768241 1728  0 0  1 1  1 1  432 432  00 00  0 0  1 1 "  1  432  1  432  00 00  768241  00 OO 00 00 00  00  5000 5000 5000 5000  OO 00 00 00  oo 00  00  00 00 00 00  NO. OF DEGREES OF FREEDOM OF STRUCTURE* HALF BAND WIDTH= 10  25  ****************************************#*********************i 1MASS VECTOR FLOOR NO. Q-WEIGHT(KIPS) R-WEIGHT(KIPS) 300.0 1 300.0 300.0 2 300.0 300.0 3 300.0 300.0 4 300.0 300.0 5 300.0 M EXCEEDS MM. M= 15 MM= 6 MODAL ANALYSIS OF STRUCTURE FOR FIRST MODES THE PERIODS IN SECONDS ARE: 1.376672 0.583998  0.564295  0.236085  INERT!Al 340000.0 340000.0 340000.0 340000.0 340000.0  0.112311  THE PARTICIPATION FACTORS ARE: LISTED PHI 10,PHI1R.PHI10,PHI20.PHI2R,PHI20. 3 FOR EACH MODE 386739 0.000000 -189.681620 000000 5.625168 0. 000000 682874 -0.000000 12 .903042 219915 0.000000 102 .878972 974102 -0.000000 7 ,569023 000000 -3.101507 . 0.000000  MODE SHAPE FOR MODE 1 0.001081 0.003868 0.000000 0.OOOOOO -0.001657 -0.0OO461  0.007734 0.OOOOOO -0.003330  0.012131 0.OOOOOO -0.005251  0.016663• 0.OOOOOO -0.007250  MODE SHAPE FOR MODE 0.OOOOOO 0.OOOOOO 0.015039 0.054672 0.OOOOOO 0.OOOOOO  0.OOOOOO 0.110921 0.OOOOOO  0.OOOOOO 0.176597 0.OOOOOO  0.OOOOOO 0.246052 0.OOOOOO  MODE SHAPE FOR MODE 0.016637 0.058534 -O.OOOOOO -0.000000 0.000030 0.000109  O. 1153 18 -0.000000 0.000223  0.178084 -0.000000 0.000358  0.240896 -0.OOOOOO 0.000501  0.091460  MODE  SHAPE  Q  0.005692  -0.013758  -0.014049  -0.003577  0.013490  R  0.000000 0.002232  0.000000 0.005424  0.000000 0.005600  0.000000 0.001566  .0.000000 -0.005087  0  FOR  MODE  SHAPE  0:  0.076053  0.182935  0.186292  0.047972  -0.174290  R: 0:  0.000000 0.000164  0.000000 0.000404  0.000000 0.000424  -0.000000 0.000121  -0.000000 -0.000396  -0.OOOOOO -0.190006 O.OOOOOO  -0.000000 -0.054912 0.OOOOOO  0.OOOOOO 0.170317 -0.000000  MODE  SHAPE  FOR  MODE  FOR  -0.000000 -0.075010 -0.OOOOOO  -MODE 1  SMEARED  MODE  5  MODE -0.000000 -0.183015 0.OOOOOO  DAMPING 0.03857  2 3  0.02000 0.06065  4 5 6  0.02409 0.03280 0.02000  DETAILS  OF  SPECTRUM  RATIO  "A"  APPLIED  BASE  ACCELERATION  LIMIT  IN  0  DIRECTION*  0.200000  BASE  ACCELERATION  LIMIT  IN  R  DIRECTION*  0.0  BASE  ACCELERATION  LIMIT  IN  0  DIRECTION*  0.0  i MODE NO. 1  SMEARED DAMPING RATIO 0.038566  NATURAL FREO. (RAD/SEC) 4.564039  SPECTRAL DISPLACEMENTS MODAL A M P L I T U D E S Q-DIRECTION R-DIRECTION ROTATION Q-DIRECTION R - D I R E C T I ON (G-SEC**2) (G-SEC**'2) (G-SEC* *2/FT) (UNITLESS) 0.008491 0.0 0.0 0.105652 0.0  ROTATION 0.0  2 3 4 5 6  0..020000 0..060653 0..024086 0..032797 0,.020000  16.. 758919 , 134583 11 .  26..614123 55..944599 68..698656  0..004438 0..002843 0 .001007 0..000155 0,.000097  0 .0 0..0 0..0 0..0 0..0  0..oooooo 0..519866 -0..007128 0..014801 -0. OOOOOO  0 .0 0 .0 0 .0 0 .0 0..0  0 .0 0..0 0..0 0 .0 0..0  0..0 0..0 0..0 0..0 0..0  **************************************  DISPLACEMENTS IN FEET AND RADIANS  FORCES IN KIPS AND KIP-FT  **************************************************************************************************************  MODE NUMBER  1 DISPLACEMENTS--  FLOOR NO. FLOOR NO. FLOOR NO . FLOOR NO . FLOOR NO.  1 (0) (R) (0) 2 (0) (R) (0) 3 (0) (R) (0) 4 (0) (R) (0) 5 (0) (R) (0)  MODE NUMBER  Q-DIRECTION 0 0001142 0 0 0 0 .0004086 0 .0 0 .0 0 .0008171 0 .0 0 .0 0 ,0012816 0 ,0 0 .0 0 .0017605 0 .0 0 .0  -DIRECTION 0.0000000 0..0 0..0 0..0000000 0..0 0.0 O.0000000 0..0 0..0 0..0000000 0..0 0..0 0..0000000 0..0 0..0  -FORCES-ROTATION -0.0000487 0.0 0..0 -0..0001751 0..0 0..0 -0..0003519 .0 O. 0..0 -0..0005548 .0 O. .0 O. -0.0007660 0.0 0.0  Q-DIRECTION 0..022 0..0 0 .0 0..079 0..0 0..0 0.. 159 0..0 0. 0 0. 249 0..0 ' 0. 0 0. 342 0. 0 0. 0  FLOOR NO. FLOOR NO. FLOOR NO.  ROTATION -10 .714 0 .0 0 .0 -38 .541 0 .0 0 .0 -77 .454 . 0..0 0..0 - 122 . 127 0..0 0..0 -168..610 0..0 0..0  2 DISPLACEMENTS-  FLOOR NO.  R-DIRECTION -0 .000 0 .0 0 .0 0..000 0..0 0..0 0..000 0..0 0..0 O..000 0,,0 0. 0 -0. 000 0. 0 0. 0  1 (0) (R) (0) 2 (0) (R) (0) 3 (0) (R) (0) 4 (0)  0-DIRECTION 0000000 0 0 0000000 O 0 0000000 0 0 0000000  R-DIRECTION 0.0000000 0.0 0.0  0.ooooooo o..0  0..0 0..ooooooo o..0 0..0 0.. ooooooo  -FORCESROTATION OOOOOOO 0 0  ooooooo 0 0  ooooooo 0 0  ooooooo  Q-DIRECTION 0.000 0..0 O. .0 0..000 O. .0 O. .0  0.000 0.0 0.0 0.000  R-DIRECTION 0.000 0..0 0..0 o..000 0..0 0..0 0.000 0.0 0..0 0..000  ROTATION  -0.000 0.0 0.0 -0.000 0.0 0.0 0.000 0.0 0.0 0.000  CO  FLOOR NO.  (R) (0) 5 (0) (R) (0)  0.0 0.0 0.0000000 0.0 0.0  0.0 0.0 0.0000000 0.0 0.0  *********** **********************************  MODE NUMBER  0.0 0.0 0.000 0.0 0.0  1  FLOOR NO.  2  FLOOR NO.  3  FLOOR NO.  4  FLOOR NO .  5  3  (0) (R) (0) (0) (R) (0) (0) (R) (0) (0) (R) (0) (0) (R) (0)  DIRECTION 0 0086492 0 0 0 0 0 0304300 0 0 0 0 0 0 0 0 0925797 0 0 0 0 0  R-DIRECTION -O.OOOOOOO 0..0 0..0 -0..OOOOOOO 0..0 0..0 -0.0000000 0.0 0..0 -0..OOOOOOO 0..0 0..0 -0..OOOOOOO 0..0 . 0.0  *********************************************  MODE NUMBER  ---FORCES--0-DIRECTION 9.. 999 0 .0 0..0 35,. 178 0..0 0..0 69.. 303 0..0 0..0 107..024 0. 0 0 .0 144 .772 0..0 0. 0  1  FLOOR NO .  2  FLOOR NO .  3  FLOOR NO .  4  FLOOR NO .  5  R-DIRECTION -o .000 0 .0 0 .0 -0 .000 0 .0 0 .0 -o .000 0..0 0 0 -0 000 0 .0 0..0 -0 .000 0..0 0. 0  ROTATION 20 . 264 0 .0 0 .0 74 . 286 0 0. 0 .0 151 .971 0 .0 0 .0 243 .716 0 .0 0 .0 34 1 . 395 0 .0 0..0  ****************************************  4 FORCES  ---DISPLACEMENTSFLOOR NO.  0.0 0.0 0.000 0.0 0.0  ****************************************  DISPLACEMENTSFLOOR NO.  0.0 0.0 0.000 0.0 0.0  (0) (R) (0) (0) (R) (0) (0) (R) (0) (0) (R) (0) (0) (R)  Q-DIRECTION 0 0000406 0 0 0 0 0000981 0 O 0 0 0 0001001 0 0 0 0 0 0000255 0 0 -0 0  R-DIRECTION -0.0000000 0..0 0..0 -0..OOOOOOO 0..0 0..0 -0..OOOOOOO 0..0 0..0 -0..OOOOOOO 0..0 0..0 -0.0000000 0.0  Q-DIRECTION 0.268 0.0 0.0 0.648 0.0 0.0 0.661 0.0 0.0 0.168 0.0 O.O -0.635 0.0  R-DIRECTION 0.000 0.0 0.0 • -0.000 0.0 0.0 -0.000 0.0 ' 0.0 -0.000 0.0 O.O 0.000 0.0  ROTATION -119.096 0.0 0.0 -289.389 0.0 0.0 -298.757 O.O O.O -83.570 0.0 0.0 271.398 0.0  (0)  0.0  0.0  0.0  ******************************************************  MODE NUMBER  0.0 ********  *********************************  1 (0)  FLOOR NO.  2  FLOOR NO.  3  FLOOR NO.  4  FLOOR NO.  5  0.0 ********  5 DISPLACEMENTS  FLOOR NO.  0.0  (R) (0) (0) (R) (0) (0) (R) (0) (0) (R) (0) (0) (R) (0)  Q-DIRECTION 0 0 0 0 0 0 0 0 0 0 0 0 -0 0 0  R-DIRECTION 0.OOOOOOO 0..0 0..0 0..OOOOOOO 0..0 0..0 0..OOOOOOO o..0 0..0 -0..OOOOOOO 0..0 0..0 -0.OOOOOOO 0.0 0.0  -FORCESROTATION 0.0000024 0.0 0.0 0.0000060 0.0 0..0 0..0000063 0..0 0..0 0..0000018 0..0 0..0 -0.0000059 0.0 0.0  -DIRECTION 32.850 0.0 0.0 . 79.017 0.0 0.0 80.467 0.0 0.0 20.721 0.0 0.0 -75.283 0.0 0.0  -DIRECTION 0. OOO 0.0 0.0 -0.000 0..0 O. .0 0..000 0..0 0..0 0..000 0..0 0..0 -0..000 0..0 o..0  ROTATION 80.039 0..0 0..0 197 . .684 O. .0 O. .0 207 . 770 0..0 0.0 59.206 0.0 0.0 -194.070 0.0 0.0  *************************** *'* * * * * *'* *************************************************************************** MODE NUMBER  6 DISPLACEMENTS-  FLOOR NO. FLOOR NO. FLOOR NO. FLOOR NO . FLOOR NO .  1 (0) (R) (0) 2 (0) (R) (0) 3 (0) (R) (0) 4 (0) (R) (0) 5 (0) (R) (0)  Q-DIRECTION 0.. OOOOOOO 0.0 0.0 0.. OOOOOOO 0,.0 0..0 0..OOOOOOO 0..0 0..0 0..OOOOOOO 0 .0 0 .0 -0 .OOOOOOO 0 .0 0 .0  R-DIRECTION 0..OOOOOOO 0..0 0..0 0 .OOOOOOO 0 .0 0..0 0 ,OOOOOOO 0..0 0,,0 0 .OOOOOOO 0 .0 0..0 -0 . OOOOOOO 0..0 0 .0  -FORCESROTATION 0 .OOOOOOO 0 .0 0 .0 -0 .OOOOOOO' 0 .0 0 .0 -0 .OOOOOOO 0 .0 0 .0 -0 .OOOOOOO 0 .0 0 .0 0 .OOOOOOO 0 .0 0 .0  Q-DIRECTION 0 .000 0 .0 0 .0 0 .000 0 .0 0 .0 0 .000 0 .0 0 .0 -0 .000 0 .0 0 .0 -0 .000 0 .0 0 .0  R-DIRECTION 0 .000 0 .0 0..0 0 .000 0 .0 0 .0 0..000 0 .0 0 .0 0..000 0 .0 0 .0 -0 .000 0 .0 0 .0  ROTATION 0.000 0.0 0.0 0.000 0.0 0.0 -0.000 0.0 0.0 0.000 0.0 0.0 0.000 0.0 0.0  **************************************************************************************************************  ROOT SUM SOUARE DISPLACEMENTS  AND FORCES  ***********************************  -FORCES-  -DISPLACEMENTS  ROTATION 145.31 1 0..0 0..0 145 . .311  86. 496 O.0 O.0 86. 496  000 O O 000  360. 317 0. 0 0. 0 360.317  .0003727 0 0 0003727  106 ,200 0, 0 O.O 106. 2O0  000 O 0 000  401 .894 0. O 0. 0 401 .894  0005853 .0 0 .0005853  109.012 0.0 O.O 109 .012  0..000 0..0 0.0 0.000  291 .208 0. O 0. 0 29 1 . 208  . 0008099 .0 .0 .0008099  163 .178 O.0 0. 0 163 .178  000 .0 0 .000  506 .260 0. 0 0. 0 506 .260  R-DIRECTION 0.OOOOOOO 0..0 0..0 0..OOOOOOO  ROTATION 0.0000535 0..0 0..0 0..0000535  0305531 0 0 0305531  OOOOOOO O 0 OOOOOOO  0001881 0 0 0001881  3 (0) (R) (0) (RSS)  0 0600190 0 0 0 0 0 .0600190  OOOOOOO 0 0  4 (0) (R) (0) (RSS)  0 .0925913 0 ,0 0 ,0 0 .0925913  5 (0) (R) (0) (RSS)  0 . 1252726 0 .0 0 .0 0 .1252726  1 (0) (R) (0) (RSS)  FLOOR NO.  2 (0) (R) (0) (RSS)  0 0 0 0  FLOOR NO.  FLOOR NO.  FLOOR NO .  -DIRECTION 0.000 0..0 0..0 0.000  Q-DIRECTION 0 0087230 0 0 0 0 0 0087230  FLOOR NO.  ooooooo ooooooo 0 0  ooooooo ooooooo 0 0  Q-DIRECTION 34.339 0..0 O. .0 34 . 339  ooooooo  ***********************************************************  COMPLETE QUADRATIC COMBINATION DISPLACEMENTS  AND FORCES  ************************************************************************************************************* FORCES  -DISPLACEMENTSFLOOR NO.  FLOOR NO.  1 (Q) (R) (0) (RSS)  Q-DIRECTION 0.0087263 0.0 0.0 0.0087.263  R-DIRECTION 0.OOOOOOO 0.0 0.0 0.OOOOOOO  2 (Q)  0.0305624  O.OOOOOOO  ROTATION 0.0000534 0.0 0.0 0.0000534O.0001875  Q-DIRECTION 34.356 0.0 0.0 34.356 86.551  R-DIRECTION  0.000 0.0 O.O  ROTATION  144.860  0.0 0.0  0.000  144.860  O.OOO  359.085  143  oo O O O O O CT) m o co  oi cn - O O O O O O O  rr  rr  O  LO CD  OCT)  O O  CT) co  10 CD  0) CO  CO CN  O O CN in O O co O O CT) O O in to co  CM  CN  O  O O O O O  O O O O O O O O  O  OOO  O O O O  OOOO  OOOO  OOOO coOOoi O O  OOOO co o O ro  in  O O co O O CD CO  co CN  co O O rg  CD  O O  o  CD  o  o o o oo o  O O O O O O O O  CD  CD  rr  o <  CD  cc  OOOO OOOO in r~  1  o o  ro O O OOO O O O O  oo O OOO  r-  »-  t~  CO  o o o  co ro CO  ro  CO CO  10  O O OOO OOO  in O O O O  CM  O co O O o OOOO O O O O  o o o o o o  O  OOO  o o o o o  OOO  O O O O  O O O O  O  o o o o o o  O  o o o o o  o o o o o  OOOO  OOO OOOO  o 6 6 6  OOOO  O O OOOO  O O OOOO  OOOO  OOOO  OOOO  o o  * * CO  CC  o o o o o o  rr CD  o CC  CN r~  O co O  rr  CD  * -H l CL  m O  O O O O  ooo o  00  CO CT) CT)  CT) 01 CT) CO o o o o  OOOO  OOO  OOOO  OOOO  o o  OOOO OOOO  o o  m  LU 3  <  >  CC  <  CN CP  in O CO OOO OOO  or  w  o  w  m  (/>  DC  CT)  CN  ro O O CO O O O O O O  O  0L  v i w  ro  CT) CN  ro O O CD O O  O to  > - L0 QC w  rCD  CD  CD CM CT)  CD CN 0)  O  o  OOOO O O O O  O  w  rf  oc  o  ' w  LO  oo in CN  co CN in <N  CN  «- o o -  OOOO  in  (/)  CC  w  in  w L0  w  w  o in  s o  rr  co  co  CD  CD  CD  CL  oOO O O O O O o in  O  rf  CD  O O - O O -  OOOO OOOO  oooo  CD  CD  CO  CO  CD  CD  O O - O o OOOO  O)  in  cc  m CC w  in  < z  <  O  cc  o  in  O  CC  www CN  oc  o  in  OKOB1  to  cc  w  DC  ro  w  OCC www  rr  D  1/1  tn cc  w  O w  m  (R) (0). (RSS)  0.0 ' 0.0 0. 374591  0,.0 0..0 0,.001797  0 .0 0 .0 0 .002244  6 (0) (R) (0) (RSS) "  0. 374591 0.0 0.0 0. 374591  0..001797 0 .0 0 .0 0,.001797  0 .002244 0 .0 0 .0 0 .002244  7 (0) (R) (0) (RSS)  0. 736417 0.0 0.0 0..736417  0..002397 0.,0 0.,0 0..002397  0..002694 0 .0 0 .0 0 .002694  8 (0) (R) (0) (RSS)  0..736417 0..0 0..0 0. 736417  0..002397 0.:0 0,.0 0.,002397  0,.002694 0 .0 0,.0 0 ,002694  9 (0) (R) (0) (RSS)  1,,137018 0..0 0..0 1.. 137018  0..002797 0..0 0..0 0..002797  0 .002814 0..0 0,.0 0 ,002814  10 (0) (R) (0) (RSS)  1..137018 0..0 0..0 1 . 137018  0..002797 0..0 0..0 0..002797  0,,002814 0 ,0 0..0 0 .002814  11 (0) (R) (0) (RSS)  1..539609 0,.0 0..0 1.539609  0,,002997 0..0 0..0 0.002997  0 .002755 . 0,.0 0,.0 0,,002755  12 (0) (R) (0) (RSS)  1 539609 0 .0 0 .0 1..539609  0.002997 0..0 0..0 0..002997  0 ,002755 0 0 0..0 0..002755  ***********************************  MEMBER NO. 1 (0) (R) (0) (RSS)  AXIAL (KIPS) 166.216 0.0 0.0 166.216-  SHEAR (KIPS) 98.906 0.0 0.0 98.906  BML (KIP-FT) 2761.863 0.0 0.0 2761.863  BMG (KIP-FT) 1616.254 0.0 0.0 1616.254  2 (0)  133.208  93.854  1958.073  915.001  MOMENT CAPACITY  5000.OOO  DAMAGE RATIO  (R) (0) (RSS)  0..0 0 .0 133,.208  0..0 0..0 93,.854  0 .0 0 .0 1958 .073  0.0 915.001  5000.000  3 (0) (R) (0) (RSS)  100..030 0..0 0..0 100..030  81 ,361 , 0..0 0,,0 81 .361 ,  1231 .854 0 .0 0..0 1231 .854  400.442 0.0 0.0 400.442  5000.000  '4 (0) (R) (0) (RSS)  66..725 0,.0 0..0 66 .725  65.. 148 0,.0 0 .0 65 . 148  651 .839 0 .0 0 .0 651 .839  265.302 0.0 0.0 265.302  5000.000  5 (Q) (R) (0) (RSS)  33..373 0..0 0 .0 33 .373  4 1. ,462 0..0 0 .0 41 .462 .  229..500 0..0 0 .0 229 .500  324 . 378 0.0 0.0 324.378  5000.000  6 (0) (R) (0) (RSS)  166 .216 0 .0 0 .0 166 .216  98 .906 0 .0 0 .0 98 .906  2761 .863 0 .0 0 .0 2761 .863  1616.254 0.0 0.0 1616.254  5000.000  7 (0) (R) (0) (RSS)  133 .208 0 .0 0 .0 133 .208  93,.854 0..0 0 .0 93,.854  1958 .073 . 0..0 0..0 1958 .073  915.001 0.0 0.0 915.001  5000.000  8 (0) (R) (0) (RSS)  100 .030 0 .0 0 .0 100 .030  81 .361 0 .0 0 .0 81 . , 361  1231 .854 0..0 0 .0 1231 .854 .  400.442 0.0 0.0 400.442  5000.000  9 (0) (R) (0) (RSS)  66 .725 0 .0 0 .0 66 .725  65,. 148 0 .0 0,.0 65 . 148  65 1.839 . 0..0 0,.0 651 .839 ,  265.302 0.0 0.0 265.302  5000.000  10 (0) (R) (0) (RSS)  33 .373 0 .0 0 .0 33 .373  41 .462 0 .0 0 .0 4 1.462 .  229,.500 0,,0 0,.0 229. 500  324.378 0.0 0.0 324.378-  5000.000  33 . 339 0 .0 0 .0 33 . 339  100. 033 0. 0 0. 0 100..033  100.033 0.0 0.0 100 033  33.348  100.061  100.061  11 (0) (R) (0) (RSS)  0 .0 0 .0 0 .0 0 .0  12 (0)  0.0  "  t  100.000  (R) (0) (RSS)  0,.0 0,.0 0,.0  0..0 0..0 33..348  0 .0 0 .0 100 .061  0 .0 0 .0 100 .061  100 .000  13 (0) (R) (0) (RSS)  0 .0 0,.0 0 .0 0 .0  33,.352 0,.0 0..0 33., 352  100.. 074 0 .0 0 ,0 100,,074  100..074 0.,0 0,.0 100..074  100..000  14 (0) (R) (0) . (RSS)  0 .0 0..0 0 .0 0 .0  33 . 355 0..0 0,.0 33.. 355  100 .081 0 ,0 0 .0 100,,081  100,,081 0..0 0..0 100,,081  100..000  15 (0) (R) (0) (RSS)  0 .0 0 .0 0..0 0 .0  33.. 374 0.,0 0..0 33..374  100.. 138 0..0 0..0 100.. 138  100., 138 0..0 0..0 100., 138  100.,000  MODAL ANALYSIS  COMPLETE QUADRATIC COMBINATION VALUES  FRAME  1  **************************************************** cn JOINT NO. 1 (Q) (R) (0) (RSS)  X-DISP(IN) 0.,0 0..0 0..0 0,.0  Y-DISP(IN) 0 .0 0 .0 0 .0 0 .0  ROTATION(RAD) 0 .0 0 .0 0..0 0,,0  2 (Q) (R) (0) (RSS)  O..0 0..0 0..0 0..0  0 .0 0 .0 0 .0 0 .0  0.,0 0,.0 . 0.,0 0..0  3 (0) (R) (0) (RSS)  0.. 106819 0..0 0..0 0,,106819  0 ,000997 0 .0 0.,0 0 .000997  0. 001364 0..0 0. 0 0. 001364  4 (Q) (R) (0) (RSS)  0.. 106819 0, O 0..0 0 . 1(56819  0 .000997 0 .0 0..0 0 .000997  0. 001364 0. O 0. 0 0. 001364  5 (Q)  0 . 374415  0 .001796  0. 002242  0. 0 0. 0 0. 3744 15  0. 0 0. 0 0. 001796  0,,0 0..0 0..002242  0. 3744 15. 0. 0 0. 0 0. 374415  0. 001796 0..0 0..0 0.,001796  0..002242 0 .0 0,.0 0,.002242  0. 736038 0. 0 0. 0 0. 736038  0..002395 0.,0 0,,0 0.,002395  0 .002692 0..0 0,.0 0 .002692  (RSS)  0..736038 0..0 0.,0 0.,736038  0.,002395 0.,0 0.,0 0,,002395  0 .002692 0 .0 0 .0 0 .002692  9 (0) (R) (0) (RSS)  1..136384 0,,0 0..0 1.. 136384  0..002795 0.,0 0..0 0..002795  0 .002812 0 .0 0 .0 0 .002812  10 ( 0 ) (R) (RSS)  1..136384 0 .0 0 .0 1 . 136384  0..002795 0 .0 0..0 0 .002795  0 .002812 0 .0 0 .0 . 0.002812  11 ( 0 ) (R) (0) (RSS)  1 .538696 0 .0 0 .0 1 .538696  0 .002995 0 .0 0 .0 0 .002995  0 .002753 0 .0 0 .0 0 .002753  12 (0) (R) (0) (RSS)  1 .538696 0 .0 0 .0 1 .538696  0 .002995 0 .0 0 .0 0..002995  0 .002753 0 .0 0 .0 0 .002753  (R) (0)  (RSS) 6 (0)  (R) (0)  (RSS) 7 (0) (R) (0)  (RSS) 8(0) (R) (0)  (0)  A * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * *  MEMBER NO. 1 (0) (R) (0) (RSS)  AXIAL (KIPS) 166.119 0.0 0.0 166.119'  SHEAR (KIPS) 98.888 0.0 0.0 98.888  BML (KIP-FT) 2760.797 0.0 0.0 2760.797  BMG (KIP-FT) 1615.357 0.0 0.0 1615.357  2 (0)  133.125  93.826  1957.024  914.223  MOMENT CAPACITY  DAMAGE RATIO  5000.OOO  0.552  (R) (0) (RSS)  0. 0 0. 0 133 .125  0.,0 0. 0 93 . ,826  0 0 0 0 1957 024  0.0 0.0 914.223  5000.000  0.391  3 (0) (R) (0) (RSS)  99. 964 0. 0 0. 0 99. 964  81 .324 0.,0 0. 0 81 ,324 .  1230 0 0 1230  906 0 0 • 906  399.986 0.0 0.0 399.986  5000.000  0.246  4 (0) (R) (0) (RSS)  66. 679 0..0 0. 0 66..679  65., 101 0.,0 0,.0 65.. 101  651 0 0 651  136 0 0 136  265.342 0.0 0.0 265.342  5000.000  0.130  5 (0) (R) (0) (RSS)  33.,350 0..0 0.,0 33,,350  41 .413 . 0.,0 0..0 41 ,413 ,  229 0 0 229  204 0 0 204  324.151 0.0 0.0 324.151  5000.000  0.065  6 (0) (R) (0) (RSS)  166.. 1 19 0,,0 0..0 166 .119 .  98 ,888 , 0..0 0,.0 98 .888 .  2760 0 0 2760  797 0 0 797  1615.357 0.0 0.0 1615.357  5000.000  0.552  7 (0) (R) (0) (RSS)  133.. 125 0,.0 0..0 133 . 125  93,,826 0,,0 0,,0 93 .826  1957 0 0 1957  024 0 0 024  914.223 0.0 0.0 914.223  5000.000  0.391  8 (0) (R) (0) (RSS)  99 .964 0 .0 0 .0 99 .964  81 .324 , 0,.0 0..0 81 . 324  1230 0 0 1230  906 0 0 906  399.986 0.0 0.0 399.986  5000.000  0.246  9 (0) (R) (0) (RSS)  66 .679 0 .0 0 .0 66 .679  65.. 101 0..0 0..0 65.. 101  651 0 0 651  136 0 0 136  265.342 0.0 0.0 265.342  5000.000  0.130  10 (0) (R) (0) (RSS)  33 .350 0 .0 0 .0 33 . 350  4 1.413 , 0,.0 0 .0 41 .413 ,  229 0 0 229  204 0 0 204  324. 151 0.0 0.0 324.151  5000.000  0.065  33 .324 0 .0 0 .0 33 .324  99 988 0 0 0 0 99 988 100 009  99.988 0.0 0.0 100.000  1.909  11 (0) (R) (0) (RSS)  0 .0 0 .0 0 .0 0 .0  12 (0)  0 .0  -  33 . 331  .  99.988 100.009  (R) (0) (RSS)  0.0 0.0 0.0  0.,0 0..0 33..331  0 .0 0..0 100,,009  0 .0 0 .0 100 .009  100 .000  2 .924  13 (0) (R) (0) (RSS)  0.0 0.0 0.0 0.0  33 .332 0 .0 0 .0 33..332  100..014 0,.0 0..0 100,,014  100 .014 0 .0 0 .0 100..014  100 .000  3 .424 .  14 (0) (R) (0) (RSS)  0.0 0.0 0.0 0.0  33,.332 0..0 0,.0 33 .332  100.,014 0 .0 0..0 100..014  100..014 0 .0 0..0 100..014  100..000  3 .554 .  15 (0) (R) (0) (RSS)  0.0 0.0 0.0 0,0  33 .350 0 .0 0 .0 33 .350  100..068 0..0 0..0 100..068  100..068 0. 0 0..0 100. 068  100. 000  3. 487  MODAL ANALYSIS  ROOT SUM SOUARE VALUES  FRAME  2  _^  ************************************************************ Y-DISP(IN) 0..0 0..0 0..0 0..0  ^0  JOINT NO. 1 (0) (R) (0) (RSS)  X-DISP(IN) 0..0 0..0 0..0 0,,0  ROTATION(RAD) 0 .0 0..0 0 .0 0 .0  2 (0) (R) (0) (RSS)  0..006961 0,,0 0,.0 0,.006961  0.,0 0..0 0..0 0..0  0 .000091 0 .0 0 .0 0 .000O91  3 (0) (R) (0) (RSS)  0.,024459 0,,0 0,.0 0..024459  0. 0 0..0 0..0 0..0  0..000149 0..0 0. 0 0..000149  4 (0) (R) (0) (RSS)  0,.048447 0..0 0 .0 0 .048447  0..0 0..0 0..0 0 .0  0..000185 0..0 0..0 0. 000185  5 (0)  0..076082  0. 0  0. 000204  (R) (0) (RSS)  0. 0 ' 0, 0 0.,076082  0..0 0 .0 0..0  0 .0 0 .0 0..000204  6 (0) (R) (0) (RSS)  0..105285 0.,0 0..0 0..105285  0,.0 0..0 0,.0 0..0  .0.,000208 0,,0 0,.0 0..000208  *************************************  1 (0) (R) (0) (RSS)  AXIAL (KIPS) , 0.0 0.0 0.0 0.0  SHEAR (KIPS) 21 .668 . 0..0 0.,0 21 .668 .  BML (KIP--FT) 620,. 703 0,.0 ' 0 .0 620 . 703  2 (0) (R) (0) (RSS)  0..0 0..0 0.,0 0..0  17 .859 , 0..0 0,.0 17 .859 ,  417 .648 0,.0 0..0 417..648  300 . 760 0 .0 0 .0 300 .760  10000.000  3 (0) (R) (0) (RSS)  0..0 0..0 0,.0 0..0  1 1.600 , 0..0 0..0 1 1.600 ,  300.. 763 0..0 0..0 300,. 763  224 . 344 0 .0 0.,0 224 , . 344  10000.000'  4 (0) (R) (0) (RSS)  0..0 0..0 0.,0 0..0  10,.607 0..0 0..0 10,,607  224 .335 0,.0 0.0 224 . 335  108 . ,181 0. 0 0..0 108 . , 181  10000.000  5 (0) (R) (0) (RSS)  0,.0 0..0 0..0 0..0  9,.015 0,.0 0,.0 9,.015  108. 176 0..0 0..0 108 . .176  • MODAL ANALYSIS  COMPLETE QUADRATIC COMBINATION VALUES  BMG (KIP--FT) 417 .649 0 .0 0 .0 417 .649  0..01 1 0..0 0.0 0.011  FRAME  2  ************************************************ JOINT NO. 1 (Q)  X-DISP(IN) 0.0  Y-DISP(IN) 0.0  ROTAT I ON(RAD)  0.0  MOMENT CAPACITY  10000.000  10000.000  DAMAGE RATIO  (R) (0) (RSS)  0.,0 0..0 0.,0  2 (0) (R) (0) (RSS)  '  0 .0 0 .0 0 .0  0.0 0.0 0.0  0..006937 0..0 0..0 0..006937  0 .0 0 .0 0,.0 0,,0  0.000090 0.0 0.0 0.000090  3 (0) (R) (0) (RSS)  0..024375 0..0 0..0 0..024375  0,.0 0,,0 0,.0 0.,0  0.000148 0.0 0.0 0.000148  4 (Q) (R) (0) (RSS)  0..048282 0.,0 0. 0 0.,048282  0..0 0.,0 0. 0 0. 0  0.000184 0.0 0.0 0.000184  5 (0) (R) (0) (RSS)  0.,075826 0.,0 0.,0 0.,075826  0. 0 0. 0 0. 0 0. 0  0.000203 0.0 0.0 0.000203  6 (0) (R) (0) (RSS)  0. 104936 0.,0 0.,0 0.,104936  0. 0 0. 0 0. 0 0. 0  0.000208 0.0 0.0 0.000208  **********************  1 (0) (R) (0) (RSS)  AXIAL (KIPS) 0.0 0.0 0.0 0.0  SHEAR (KIPS) 21 .626 0.0 0.0 21 .626  BML (KIP--FT) 618 .691 0 .0 O .O 618 .691  BMG (KIP--FT) 4 16 . 227 0 .0 0 .0 416,.227  MOMENT CAPACITY  DAMAGE RATIO  10000.000  0.062  2 (0) (R) (0) (RSS)  0.0 0.0 0.0 0.0  17.817 0.0 0.0 17.817  416 . 225 0 .0 0 .0 416 .225  300,.015 0,.0 0,.0 300..015  10000.000  0.042  3 (0) (R) (0) (RSS)  0.0 0.0 0.0 0.0  11.563 0.0 0.0 1 1 . 563  300 .018 . 0.0 0,,0 300..018  224 .069 . 0 .0 0..0 224 .069  10000.000  0.030  4 (0)  0.0  10.587  224 ,060 .  108 .124  -  (R) (0) (RSS)  0..0 0,.0 0, 0  0..0 0 .0 10..587  5 (0) OR) (0) (RSS)  0,.0 0..0 0 .0 0 .0  9 .010 0 .0 0..0 9..010  MODAL ANALYSIS  0.,0 0..0 224.,060  ROOT SUM SOUARE VALUES  108., 119 0.,0 0..0 108.. 119  FRAME  0,.0 0..0 108.. 124  100O0,.OOO  0 .022  0..01 1 0..0 0. 0 0.,011  10000,.000  0 .01 1  3  ********************************************************  JOINT NO. 1 (0) (R) (0) (RSS)  X-DISP(IN) 0..0 0.,0 0,,0 0..0  Y-DISP(IN) 0 .0 0..0 0 .0 0 .0  ROTATION(RAD) 0 .0 0 .0 0 .0 0 .0  2 (0) (R) (0) (RSS)  0..006961 0,,0 0..0 0..006961  0..0 0..0 0 .0 0 .0  0 .000091 0 .0 0 .0 0 .000091  3(0) (R) CO) (RSS)  0..024459 0,.0 0.,0 0..024459  0 .0 0 .0 0 .0 0 .0  0 .000149 0 .0 o .0 0 .000149  4 (0) (R) (0) (RSS.)  0,.048447 0..0 0..0 0,.048447  0 .0 0 .0 0 .0 0 .0  0 .000185 0..0 0 .0 0..000185  5 (0) (R) (0) (RSS)  0..076082 0,.0 0.,0 0..076082  0..0 0 .0 0..0 0 .0  0..000204 0..0 0..0 0..000204  6 (0) (R) (0) (RSS)  0..105285 0,.0 0. 0 o..105285  0 .0 0 .0 0 .0 0 .0  0..000208 0,.0 0..0 0..000208  •  MEMBER NO. 1 (0) (R) (0) (RSS)  SHEAR (KIPS) 21 .668 . 0..0 0..0 21 .668 .  BML (KIP--FT) 620.,703 0..0 0..0 620.,703  2 (0) (R) (0) (RSS)  0.0 0.0 0.0 0.0  17 .859 0..0 0 .0 17 .859 .  417 ,648 . 0,.0 0,.0 417 ,648 .  300 .760 0 .0 0 .0 300,. 760  10000.000  3 (0) (R) (0) (RSS)  0.0 0.0 0.0 0.0  1 1.600 0 .0 0 .0 1 1.600  300.,763 0.,0 0..0 300..763  224,. 344 0..0 0,.0 224 .344 ,  10000.000  4 (0) (R) (0) (RSS)  0.0 0.0 0.0 0.0  10..607 0 .0 0 .0 10 .607  224..335 0..0 0.,0 224 .335  108,. 181 0,.0 0,.0 108,. 181  10000.000  5 (0) (R) (0) (RSS)  0.0 0.0 0.0 0.0  9 .015 0 .0 0 .0 9 .015  108. 176 0. 0 0. 0 108. 176  0,.011 0,.0 0..0 0. 01 1  10000.000  MODAL ANALYSIS  COMPLETE QUADRATIC COMBINATION VALUES  BMG (KIP--FT) 417 .649 0 .0 0 .0 4 17 .649  MOMENT CAPACITY  AXIAL (KIPS) 0.0 0.0 0.0 0.0  FRAME  3  *************************************** Y-DISP(IN) 0.0 0.0 0.0 0.0  ROTATION(RAD) 0.0 0.0 0.0 0.0  1 (0) (R) (0) (RSS)  X-DISP(IN) 0.0 0.0 0.0 0.0  2 (0) (R) (0) (RSS)  0.006937 0.0 0.0 0.006937  0.0 0.0 0.0 0.0  0.000090 0.0 0.0 0.000090  3 (0)  0.024375  0.0  0.000148  10000.000  DAMAGE RATIO  ,,00  OOO'OOOO,  l ,00 00 00 i.  ssoo  oeoo  ZPO'O  OOO'OOOO,  i.o-o  ooo-oooo.  ouva 3DVWVQ  OOO'OOOO,  AllOVdVO 1N3W0W  0 0 0 0  0 0 0 0  (ssa)  0 0 0 0  0 0 0 0  (ssa)  (0)  (a)  (0) s  PZl "801. 0 0 O'O frCl 801  090'PZZ O'O 00 090 t>Sc:  690 « S 00 00 690 t-Z2  81000E O'O 00 8iO"ooe  E9S'ii 00 00 E99'11  0 0 0 0  0 0 0 0  (ssa)  gio'ooe oo oo  9SS' 9iP 0' 0 0' 0 SS2"9,f  U8'U 00 00 LIB'Li  00 O'O 00 0 0  (ssa)  LZZBiV 00 00 LZZ'9iV (ld-dlX)  i69'8i9 O'Oi  9S9I2 00 00 939'iZ (Sdl») aV3HS  00 00 00 00 (SdIX)  (ssa) (o) (a)  S1000E Z90 0  0,0'6 0 0 00 0,06 LBS'Oi 00 00 £89 01  -  ooo oooo.  801 0 0 80,  t 0 0 6i i 61  owa  -  oo  169 819 (Id-dlX) -  -  (0)  (a)  (0) 17 (0)  (a)  (0)  e  (0)  (a)  (0) z  (0) i  ivixv  ON aaawaw *************  80S000 0 00 00 802000 O  00 00 00 00  9E6KH 0 0 986*01  0 0 0 0  (ssa)  eoeoooo  00 00 00  9589i0 0 00 00 9589*0 O  (ssa)  00 00  -  (0)  (a)  (0) 9 (0)  (a)  eoeoooo  op  fr8l000'0 00. O'O fr8t0000  00 00 00 00  2828KTO O'O 00  (ssa)  Z8Z8W>*0  (0)  BP 1000 0 00 00  00 00 00  SL£PZ0'0 0 0 0 0  (0) 9 (0)  (a)  v  (ssa) (0)  (a)  MODAL ANALYSIS  ROOT SUM SQUARE VALUES  FRAME  *****************  JOINT NO  1 (0) (R) (0)  (RSS)  X-DISP(IN) 0 0 0 0 0 0 0 0  4 ^*********************************.********.******************  Y-DISP(IN) 0 0 0 0 0 0 0 0  ROTATION(RAD) 0 0 0 0 0 0 0 0  (RSS)  0 0 0 0  b. 0 0 0  0 0 0 0  0 0 0 0  0 0 0 0  0 0 0 0  3 (0) (R) (0) (RSS)  0 0 0 0  103512 0 0 103512  0 0 0 0  000997 0 0 000997  0 0 0 0  001320 0 0 001320  4 (0)  (R) (0) (RSS)  0 0 0 0  103512 0 0 103512  0 0 0 0  000997 0 0 000997  0 0 0 0  001320 0 0 001320  5 (Q) (R) (0) (RSS)  0 0 0 0  362372 0 0 362372  0 0 0 0  001795 0 0 001795  0 0 0 0  002167 0 0 002167  6 (Q) (R) (0) (RSS)  0 0 0 0  362372 0 0 362372  0 0 0 0  001795 0 0 001795  0 0 0 0  002167 0 0 002167  7 (Q) (R) (0) (RSS)  0 0 0 0  711507 0 0 711507  0 0 0 0  002395 0 0 002395  0 002595 0 0 0 0 b 002595  8 (Q) (R) (0) (RSS)  0 711507 0 0 0.0, 0 711507  0 0 0 0  002395 0 0 002395  0 0 0 0  9 (Q)  1 0971 15  0 002794  2 (Q) (R) (0)  •  002595 0 0 002595  0 0027O4  Co)  (RSS)  0,o • 0 .0 1..097115  0.,0 0..0 0.,002794  0.0 0.0 0.002704  10 (0) (R) (0) (RSS)  1..097115 0..0 0..0 1,.097115  0..002794 0..0 0..0 0..002794  0.002704 0.0 6.0 0.002704  11 (0) (R) (0) (RSS)  1..483697 0 .0 0..0 1 ,483697  0.,002994 0..0 0.,0 0..002994  0.002642 0.0 0.0 0.002642  12 (0) (R) (0) (RSS)  1.,483697 0 .0 0 ,0 1 .483697  0..002994 0..0 0..0 0..002994  0.002642 0.0 0.0 0.002642  (Ri-  *********************************************  NO. 1 (0) (R) (0) (RSS)  AXIAL (KIPS) 166..047 0..0 0..0 166..047  SHEAR (KIPS) 97,. 105 0..0 0..0 97.. 105  BML (KIP-•FT) 2680 .675 0..0 0..0 2680..675  BMG (KIP-FT) 1554.561 0.0 0.0 1554.581  2 (0) (R) (0) (RSS)  133..072 0..0 0..0 133..072  92..172 0..0 0..0 92.. 172  1896..919 0..0 0..0 1896..919  871:108 0.0 0.0 871.108  500O.O00  3 (0) (R) (0) (RSS)  99..926 0..0 0..0 99..926  79..914 0..0 0..0 79..914  . 1188.,355 0.,0 0.,0 ,355 1 188.  375.806 0.0 0.0 375.806  50OO.000  4 (Qj (R) (0) (RSS)  66..655 0,.0 0..0 66..655  63..857 0..0 0..0 63..857  624.. 173 0,.0 0.0 624 . , 173  268.475 0.0 0.0 268.475  5000.OOO  5 (0) (R) (0) (RSS)  33 .337 0,.0 0..0 33..337*  40..530 0..0 0..0 40..530  218., 143 0.0 0..0 218. 143  325.136 0.0 0.0 325.136  5000.000  6 (0)  166.047  97.105  2680.675  ' 1554.581  MOMENT CAPACITY  5000.OOO  DAMAGE RATIO  cn cr.  (R) (0) (RSS)  0. 0 0. o 166. 047  0. 0 0.,0 97., 105  0..0 o.,0 2680.,675  0.,0 0.,0 1554,,581  5000.000  7 (0)  133. 072 0. 0 0. 0 133.,072  92. 172 0.,0 0..0 92., 172  1896..919 0.,0 0.,0 1896.,919  871 . , 108 0..0 ' 0.,0 871 ., 108  5000,. OOO  99..926 0.,0 0..0 99..926  79.,914 0. 0 0..0 79.,914  1188..355 0,.0 0,.0 1188..355  375,.806 0,.0 0,,o 375,.806  5000.000  66..655 0..0 0..0 66..655  63..857 0..0 0..0 63..857  624.. 173 0,.0 0..0 624,. 173  ' 268..475 0..0 0,.0 268..475  5000.000  33..337 0..0 0 .0 33..337  40.,530 0.,0 0..0 40..530  218.. 143 0,.0 0.,0 218.. 143  0,.0 0 .0 0 .0 0 .0  33..305 0,.0 0..0 33..305  99..932 0..0 0. O • 99.,932  99..932 0..0 0,.0 99,.932  100.000  0 .0 0 .0 0 .0 0 .0  33..314 0..0 0..0 33,.314  99.,960 0..0 0.,0 99.,960  99,.960 0,.0 0,.0 99,.960  100.000  0 .0 0 .0 0 .0 0 .0  33,.319 0,.0 0,.0 33,.319  99..973 0..0 0,.0 99..973  99 .973 0,.0 0 .0 99,.973  100.000  0 .0 0 .0 0 .0 0 .0  33,.321 0,.0 0,.0 33,.321  99..980 0..0 0,.0 99.,980  99 .980 0,.0 0,.0 99 .980  100.ooo  0 .0 0 .0 0 .0 0 .0 *  33,.337 0,.0 0,.0 33,.337  100,.029 0,,0 0..0 100,,029  100 .029 0 .0 0 .0 100 .029  100.ooo  (R) (0) (RSS)  8  (Q)  (R) (0) (RSS)  9 (0)  (R) (0) (RSS)  10 (0)  (R)  (0)  (RSS)  11 (0)  (R)  (0)  (RSS)  12 (0)  (R)  (0)  (RSS)  13 (0)  (R)  (0)  (RSS)  14 (0)  (R)  (0)  (RSS)  15 (0)  (R)  (0)  (RSS)  325,. 136 0..0 0,.0 325.. 136  5000.000  MODAL ANALYSIS  COMPLETE QUADRATIC COMBINATION VALUES  FRAME  4  ******************************************* JOINT NO. 1 (Q) (R) (0) (RSS) 2 (Q)  X-DISP(IN) 0..0 0..0 0..0 0..0  Y-DISP(IN) 0..0 0 .0 0 .0 0 .0  ROTATION*RAD) 0.0 0.0 0.0 0.0  (R) . (0) (RSS)  0..0 0..0 0..0 0..0  0 .0 0 .0 0 .0 0 .0  0.0 .0.0 0.0 0.0  3 (Q) (R) (0) (RSS)  0.. 103604 0..0 0..0 0.. 103604  0,.000997 0 .0 0 .0 0..000997  0.001321 0.0 0.0 0.001321  4 (Q) (R) (0) (RSS)  0..103604 0..0 0..0 0.. 103604  0 .000997 0,.0 0..0 0 .000997  0.001321 0.0 0.0 0.001321  5 (Q)  (R) (0) (RSS)  0..362666 0..0 0..0 0..362666  0,.001796 0..0 0,.0 0..001796  0.002168 0.0 0.0 0.002168  6 (Q) (R) (0) (RSS)  0,.362666 0,.0 0..0 0..362666  0..001796 0,.0 0..0 0..001796  0.002168 0.0 0.0 0.002168  7 (Q) (R) (0) (RSS)  0..712021 0..0 0,.0 0..712021  0..002396 0..0 0..0 0,.002396  0.002597 0.0 • . 0.0 0.002597  8 (Q) (R) (0) (RSS)  0..712021 0,.0 0..0. 0..712021  0..002396 0..0 0..0 0..002396  0.002597 0.0 O.O 0.002597  9 (Q)  1..097822  0,.002796  0.002705  (R) (0) (RSS)  0 .0 " 0 .0 1 .097822  0. 0 0. 0 0. 002796  0 .0 0..0 0..002705  10 (0) (R) (0) (RSS)  1 .097822 . 0 .0 0 .0 1 .097822  0..002796 0..0 0,.0 0..002796  0 .002705 0..0 0..0 0, .002705  11 (0) (R) (0) (RSS)  1 .484571 0 .0 0 .0 1 .484571  0..002995 0..0 0 .0 0,.002995  0..002643 0..0 0..0 0 .002643  12 (Q)  1 .484571 0::0 0 .0 1 .484571  0..002995 0..0 0..0 0..002995  0..002643 0..0 • 0.0 . 0..002643  (R) (0) (RSS)  ******************************* IO. MEMBER NO. 1 (0) (R) (0)  (RSS)  AXIAL (KIPS) 166.147 . 0.0  0.0 . 166.147  SHEAR (KIPS) 97.231 0.0 0.0 97.231  BML (KIP-•FT) 2683..238 0,.0 0,.0 2683..238  BMG (KIP--FT) 1555 .621 0 .0 0 .0 1555..621  MOMENT CAPACITY  DAMAGE RATIO  5000.OOO  0.537  1898..251 0,.0 0,.0 1898..251  871 . 154 0 .6 0 .0 .871 . 154  5000.000  0. 380  2 (Q) (R) (0) (RSS)  133.143 0.0 133.143  92.284 0.0 0.0 92.284  3 (0) (R) (0) (RSS)  99.974 0.0 0.0 99.974  79.975 0.0 0.0 79.975  .613 1 188. 0,.0 0..0 1188..613  375..716 0 .0 0..0 375 .716  5000.000  0. 238  4 (0) (R) (0) (RSS)  66.684 0.0 0.0 66.684  63.865 0.0 0.0 63.865  624.,019 0..0 0..0 624..019  268..863 0..0 0..0 268..863  5000.000  0. 125  5 (0) (R) (0) (RSS)  33.352 0.0 0.0 33.352-  40.529 0.0 0.0 40.529  218., 148 0..0 0,,0 218.. 148  325..274 0..0 0..0 325..274  5000.000  0.065  2683.. 238  1555..621  6 (0)  0.0  166.147  97.231 '  (R) (0) (RSS)  0. 0 0. 0 166. 147  0..0 0..0 97..231  0 0 0 0 2683 238  7  133., 143 0. 0 0. 0 133.. 143  92..284 0..0 0..0 92..284  1898 0 0 1898  (RSS)  99. 974 0. 0 0. 0 99..974  79..975 0..0 0..0 79..975  9 (0) (R) (0) (RSS)  66..684 0..0 0..0 66.,684  63..865 0.,0 0..0 63..865  624 0 0 624  10 (0) (R) (0) (RSS)  33..352 0..0 0..0 33..352  40.,529 0..0 0.,0 40..529  (0)  (R) (0)  (RSS) 8  (0)  (R) (0)  0.0 0.0 1555.621  5000.OOO  0.537  251 0 0 251  871.154 0.0 0.0 871.154  5000.000  0.380  1 188 613 0 0 0 0 1 188 613  375.716 0.0 0.0 375.716  5000.OOO  0.238  019 0 0 019  268.863 0.0 0.0 • 268.863  5000.000  0.125  218 0 0 218  148 0 0 148  325.274 0.0 0.0 325.274  5000.000  0.065  11 (0) (R) (0) (RSS)  0..0 0..0 0..0 0..0  33.,334 0.,0 0..0 33..334  100 0 0 100  018 0 0 018  . 100.018 0.0 0.0 100.018  100.000  1.857  12 (0) (R) (0) (RSS)  0..0 0..0 0 .0 0..0  33..338 0..0 0..0 33. 338  100 0 0 100  031 0 O 031  100.031 0.0 0.0 100.031  100.000  2.840  13 (0) (R) (0) (RSS)  0,.0 0 .0 0 .0 0 .0  33..337 0.,0 0..0 33.,337  100 0 0 100  029 0 0 029  100.029 0.0 0.0 100.029  100.000  3.318  14 (0) (R) (0) (RSS)  0 .0 0 .0 0 .0 0 .0  33.,336 0..0 0.,0 33. 336  100 0 0 100  026 O 0 026  100.026 0.0 0.0 100.026  100.OOO  3.436  15 (0) (R) (0) (RSS)  0 .0 0 .0 0 .0 0 .0  33.,351 0.,0 0.,0 33.,351  100 0 0 100  071 0 0 07 1  100.071 0.0 0.0 100.07 1  100.OOO  3.365  4  0  '  0.000  ,  NO. OF ITERATIONS  BETA=0.0  BENDING MOMENT ERROR=  0.050  DAMAGE RATIO ERROR= 0.010 T=0.566 DR=0 $.43, $.44T  SIG  

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