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Minimum seismic requirements for masonry walls Chung, Rufus Wai-Kwong 1978

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MINIMUM SEISMIC REQUIREMENTS FOR MASONRY WALLS  by RUFUS WAI-KWONG CHUNG B.A.Sc. (1975) The U n i v e r s i t y o f B r i t i s h Columbia  A THESIS SUBMITTED IN PARTIAL FULFILMENT OF THE REQUIREMENTS FOR THE DEGREE OF  MASTER OF APPLIED SCIENCE  in THE FACULTY OF GRADUATE STUDIES Department o f C i v i l E n g i n e e r i n g We a c c e p t t h i s t h e s i s as c o n f o r m i n g to the required  standard  The U n i v e r s i t y o f B r i t i s h Columbia A p r i l 1978  © Rufus Wai-Kwong Chung, 1978  In p r e s e n t i n g  this thesis  an advanced degree at the  Library  in p a r t i a l  the U n i v e r s i t y  s h a l l make i t f r e e l y  I f u r t h e r agree t h a t p e r m i s s i o n f o r s c h o l a r l y purposes may representatives.  his  of  this thesis for financial permission.  The  of B r i t i s h  University  2075 Wesbrook Place Vancouver, Canada V6T 1W5  A  Pril  28  1978  requirements f o r  f o r reference  the Head of my  Columbia  shall  that  not  I agree and  f o r e x t e n s i v e copying o f  gain  C i v i l Engineering  Date  available  It i s understood  Department of  the  of B r i t i s h Columbia,  be granted by  by  written  fulfilment of  study.  this  thesis  Department  copying or  that  or  publication  be allowed without  my  ABSTRACT  A study by computer a n a l y s i s was the minimum p e r c e n t a g e o f s t e e l r e q u i r e d masonry  walls with respect  masonry l o a d - b e a r i n g  to t r a n s v e r s e  shear w a l l and  were s t u d i e d .  The  Linear  s p e c t r a l analyses  elastic  conducted to e s t a b l i s h  i n the s e i s m i c forces.  design  Buildings  masonry i n f i l l - f r a m e  of  of  construction  b u i l d i n g s were m o d e l l e d as p l a n e frame s t r u c t u r e s . were performed by u s i n g  the  modified  computer program DYNAMIC.  By  examining the r e s p o n s e of the models to a peak ground  a c c e l e r a t i o n of 1.0  g, a c o r r e l a t i o n between the  of a masonry w a l l s t r u c t u r e and d i f f e r e n t height  fundamental p e r i o d  i t s response a c c e l e r a t i o n s at  i s found, i . e . , envelopes of dynamic  f a c t o r are developed.  A new  e m p i r i c a l formula f o r estimating  fundamental p e r i o d of a s t r u c t u r e w i t h masonry w a l l s The  the  i s recommended.  minimum s t e e l i s e s t a b l i s h e d i n t y p i c a l w a l l p a n e l s d e s i g n e d  f o r the i n e r t i a Zone  amplication  3.  f o r c e s imposed by a major earthquake i n  seismic  iii  TABLE OF CONTENTS  ABSTRACT  i i  TABLE OF CONTENTS . . ..  i i i  LIST OF TABLES  vi  LIST OF FIGURES  vii  ACKNOWLEDGEMENTS  , .. ..  x  CHAPTER 1.  2.  3.  INTRODUCTION  1  1.1  Background  1  1.2  Purpose and Scope  3  BACKGROUND ON DYNAMIC ANALYSIS  ..  5  2.1  Introduction  5  2.2  Spectral Analysis  6  2.3  Method of S t r u c t u r a l Modelling  13  2.4  Wide Column Model f o r Shear Walls  14  2.5  E f f e c t i v e Width of Slabs  15  2.6  I n f i l l Walls  16  MODELLING OF BUILDINGS FOR DYNAMIC ANALYSIS  18  3.1  Introduction  18  3.2  Load-Bearing Shear Wall Building i n Vancouver..  19  3.3  Park Lane Towers  23  3.4  I n f i l l - F r a m e Building  25  iv  Page 4.  5.  6.  DISCUSSION OF THE  28  4.1  Computer Program  28  4.2  D e s i g n Spectrum  29  4.3  Dynamic A n a l y s i s and R e s u l t s  31  4.3.1  Load-Bearing  4.3.2  Infill-Frame Building  RESULTS-ANALYSIS AND  Shear W a l l B u i l d i n g s  ....  35 45  RECOMMENDATIONS  55  5.1  Observations  55  5.2  Level Coefficient  56  5.3  Fundamental P e r i o d of B u i l d i n g s  .. ..  64  MINIMUM STEEL REQUIREMENTS 6.1  7.  ANALYTICAL EFFORT  68  Introduction  .. ..  68  6.1.1  Racking F o r c e s  68  6.1.2  Transverse Forces  69  6.2  D e s i g n Earthquake  6.3  D e s i g n Formula  6.4  Wall Panel Designing  70  f o r Masonry W a l l s i n Bending  6.4.1  Load-Bearing  6.4.2  I n f i l l Wall  SUMMARY, CONCLUSIONS AND  ..  73 74  Shear W a l l  75 77  RECOMMENDATIONS  FOR  FUTURE RESEARCH  79  7.1  Summary and C o n c l u s i o n s  79  7.2  Recommendations f o r F u t u r e Research  81  V  Page  REFERENCES APPENDIX A  83 BEAM STIFFNESS MATRIX FOR WIDE  .., COLUMN FRAME APPENDIX B  MODELLING OF INFILL WALLS  APPENDIX C  COMPUTER LISTING OF SUBPROGRAM SPECTR  APPENDIX D  INPUT AND RESULT DATA OF MODEL ANALYSES  86 93 ....  96  ..  98  LIST OF TABLES  Table  Page  1  COMPARISON OF FUNDAMENTAL PERIODS  66  2  MEMBER PROPERTIES OF STRUCTURE 1A  100  3  MEMBER PROPERTIES OF STRUCTURE IB  102  4  MEMBER PROPERTIES OF STRUCTURE 2  104  5  DYNAMIC ANALYSIS RESULTS OF STRUCTURE 1A  ....  6  DYNAMIC ANALYSIS RESULTS OF STRUCTURE IB  .....  7  DYNAMIC ANALYSIS RESULTS OF STRUCTURE 2  ....  105 109 112  vii  LIST OF FIGURES  Number  —  1  Single-degree-of-freedom  system  6  2  Response spectra for e l a s t i c systems, 1940 Centro earthquake (from Ref. 3)  EL 10  3  Wide column frame model  15  4  T y p i c a l f l o o r plan of load-bearing shear w a l l b u i l d i n g i n Vancouver  20  5  T y p i c a l f l o o r plan of Park Lane Towers  24  6  T y p i c a l f l o o r plan of i n f i l l - f r a m e b u i l d i n g  7  Peak ground motion bounds and e l a s t i c average response spectrum for 1.0 g max. ground acceleration (from Ref. 30)  8  T o r s i o n a l induced a c c e l e r a t i o n  9a  Structure 1A: R-S-S  storey displacements  9b  Structure 1A: R-S-S  storey displacements  10a 10b  10c  lOd  lOe  ....  26  ..  30 34  .......  37 38  Structure 1A: Storey accelerations of 5-storey model  39  Structure 1A: Storey accelerations of model  10-storey 40  Structure 1A: Storey accelerations of model  15-storey  Structure 1A: Storey accelerations of model  20-storey  Structure 1A: Storey accelerations of model  25-storey  41  42 Li  viii  F  i  g  u  r  Page ___s—  e  Number 11  V a r i a t i o n of response a c c e l e r a t i o n with fundamental period at l e v e l s h/H = 0.2, 0.4, 0.6, 0.8, 1.0 of Load-Bearing Shear Wall Buildings ..  44  12a  Structure 2: R-S-S  storey displacements  47  12b  Structure 2: R-S-S  storey displacements  13a  Structure 2: Storey accelerations of 5-storey model  49  Structure 2: Storey accelerations of 8-storey model  50  Structure 2: Storey accelerations of 10-storey model  51  Structure 2: Storey accelerations of 15-storey model  52  Structure 2: Storey accelerations of 20-storey model  53  V a r i a t i o n of response acceleration with fundamental period at l e v e l s h/H = 0.2, 0.4, 0.8, 1.0 of Infill-Frame Buildings  54  13b 13c 13d 13e 14  15a  15b  15c  15d  15e  16  ......  48  0.6,  V a r i a t i o n of response acceleration with fundamental period at h/H = 0.2  57  V a r i a t i o n of response a c c e l e r a t i o n with fundamental period at h/H =0.4  58  V a r i a t i o n of response a c c e l e r a t i o n with fundamental period at h/H = 0.6  59  V a r i a t i o n of response acceleration with fundamental period at h/H = 0.8  60  V a r i a t i o n of response acceleration with fundamental period at h/H = 1.0  61  Comparison of fundamental periods by computer and by Code empirical formula  67  ix  F  i  g  u  r  Page — ° —  e  Number 17  E f f e c t of u and X on the design load  18  Models f o r ( i ) shear wall panel and ( i i ) i n f i l l  72  panel  76  19  Beam element f o r wide column frame  87  20  Equivalent strut model of i n f i l l wall  94  21  Plane frame model of Structure 1A  99  22  Plane frame model of Structure IB  101  23  Plane frume model of Structure 2  103  ACKNOWLEDGEMENTS  I would l i k e t o e x p r e s s my g r a t i t u d e t o D r . N.D. Nathan who served  as my major a d v i s e r f o r t h i s study.  p a r t i c u l a r h i s a d v i c e and s u g g e s t i o n s  during the preparation  of my t h e s i s a r e v e r y much a p p r e c i a t e d . thank D r . S. C h e r r y  In  I would a l s o l i k e t o  f o r h i s a d v i c e and s u g g e s t i o n s  L. Anderson f o r many h e l p f u l d i s c u s s i o n s .  and D r . D.  A p p r e c i a t i o n i s due  t o N a t i o n a l R e s e a r c h C o u n c i l and P o r t l a n d Cement A s s o c i a t i o n f o r permissions  to reprint copyright  figures.  F i n a l l y , I l i k e t o thank my w i f e , Yvonne, f o r t y p i n g this thesis.  A l s o h e r p a t i e n c e and u n d e r s t a n d i n g  my study a r e d e e p l y  appreciated.  a t times o f  CHAPTER 1  INTRODUCTION  1.1  Background  Masonry has been used as a b u i l d i n g m a t e r i a l e a r l i e s t days o f man.  since the  T r a d i t i o n a l l y masonry s t r u c t u r e s were  d e s i g n e d and b u i l t w i t h no s t e e l r e i n f o r c e m e n t and s u p p o r t e d only gravity loads.  These s t r u c t u r e s had t h i c k w a l l s and heavy  f l o o r s and were u s u a l l y one- o r t w o - s t o r y b u i l d i n g s .  The heavy  g r a v i t y l o a d tended t o s t a b i l i z e t h e s t r u c t u r e a g a i n s t f o r c e s due t o minor wind o r s e i s m i c a c t i o n . were n o t c a p a b l e o f r e s i s t i n g the low t e n s i l e s t r e n g t h  lateral  These s t r u c t u r e s  l a r g e l a t e r a l f o r c e s because o f  o f masonry.  They were e s p e c i a l l y r i s k y  i n e a r t h q u a k e Zone 3 where major earthquakes would most  likely  occur.  In t h e l a s t introduced.  few decades, r e i n f o r c e d masonry has been  S t e e l r e i n f o r c e m e n t i n masonry a l l o w s a b e t t e r u s e  o f t h e masonry m a t e r i a l .  M u l t i s t o r y r e i n f o r c e d masonry  buildings  have been b u i l t .  These b u i l d i n g s a r e found t o p r o v i d e a b e t t e r  r e s i s t a n c e t o l a t e r a l f o r c e s and t o behave v e r y w e l l d u r i n g earthquakes.  R e i n f o r c e d masonry i s now r e c o g n i z e d  as another  h i g h - r i s e b u i l d i n g m a t e r i a l and i s v e r y p o p u l a r l y used.  R e i n f o r c e d masonry i s a r e l a t i v e l y new m a t e r i a l t o which v e r y l i m i t e d r e s e a r c h has been devoted. simply  Present  designers  often  f o l l o w t h e b u i l d i n g codes, b u t many c l a u s e s i n t h e b u i l d i n g  codes a r e n o t f u l l y j u s t i f i e d .  Only one o f these w i l l be mentioned  i n the f o l l o w i n g .  The N a t i o n a l B u i l d i n g Code o f Canada  1975^"^specifies:  "4.4.3.30. (1) Reinforced masonry walls shall be reinforced horizontally and vertically with steel having a total cross-sectional area not less than 0.002 times the gross cross-sectional area of the wall so that not less than 1/3 of the recruired steel is either vertical or horizontal." These minima appear t o have t h e i r o r i g i n s i n t r a d i t i o n than t o be based on e x p e r i m e n t a l  evidence.  They have  economic impact on the use o f masonry, and cause  rather considerable  considerable  controversy  among d e s i g n e r s  and c o n t r a c t o r s .  The presumed purpose o f t h i s s t e e l i s t w o f o l d : 1.  To p r e v e n t t h e masonry from f a l l i n g o u t due to t r a n s v e r s e  f o r c e s normal t o t h e p l a n e o f  the w a l l . 2.  To l e n d  some d u c t i l i t y  t o the w a l l with  respect  to r a c k i n g d e f o r m a t i o n s c o r r e s p o n d i n g t o i n plane  1.2  loading.  Purpose and Scope  The  purpose o f t h i s t h e s i s i s t o e s t a b l i s h t h e minimum  percentages o f s t e e l f o r the seismic  design  o f masonry w a l l s .  These minima w i l l be e s t a b l i s h e d w i t h r e s p e c t force  (out-of-plane  to the tranverse  b e n d i n g ) ; the e f f e c t o f t h e r a c k i n g  plane shearing) i s a l s o recognized  force ( i n -  and w i l l be d i s c u s s e d  Two common t y p e s of b u i l d i n g s w i t h masonry w a l l s t r u c t i o n w i l l be i n v e s t i g a t e d . buildings,  and  (b)  Infilled  (a)  Load- bearing  frame b u i l d i n g s .  briefly.  con-  shear w a l l  Typical buildings  of each type w i l l be m o d e l l e d as p l a n e frame s t r u c t u r e s and then subjected  t o dynamic a n a l y s i s by the r e s p o n s e spectrum approach.  Response w i l l be o b t a i n e d f o r a range o f b u i l d i n g s w i t h d i f f e r e n t numbers o f s t o r i e s so t h a t t h e r e l a t i o n s h i p between t h e fundamental p e r i o d and t h e s t o r e y a c c e l e r a t i o n a t each l e v e l o f t h e b u i l d i n g can be d e t e r m i n e d .  That i s t o say, envelopes o f t h e dynamic  a m p l i f i c a t i o n f a c t o r w i l l be sought f o r . panel w i l l  An i n d i v i d u a l w a l l  t h e n be d e s i g n e d t o prevent i t s f a l l i n g  inertia force arising  from i t s own mass.  Code-type  out under t h e recommendations  w i l l a l s o be d e v e l o p e d .  In  t h e dynamic a n a l y s i s , t h e s t r u c t u r e s w i l l be assumed  to behave a s l i n e a r systems. first  The e l a s t i c modal response o f t h e  t h r e e modes o f v i b r a t i o n w i l l be o b t a i n e d and superposed.  I t i s recognized that i n p r a c t i c e , n o n l i n e a r behaviour of the structures w i l l  o c c u r i n s e v e r e earthquakes.  Thus, the t r u e  dynamic b e h a v i o u r o f t h e s t r u c t u r e s can be determined a l l o w a n c e i s made f o r n o n l i n e a r i t y i n t h e a n a l y s i s .  only i f Nevertheless,  p l a s t i c d e f o r m a t i o n s o f i n e l a s t i c s t r u c t u r e s absorb l a r g e amounts of  energy from earthquake-generated motions and s t i f f n e s s  d e g r a d a t i o n s have t h e e f f e c t o f r e d u c i n g t h e response o f t h e systems.  T h e r e f o r e , t h e maximum r e s p o n s e a c c e l e r a t i o n s o b t a i n e d  from t h e l i n e a r e l a s t i c dynamic a n a l y s i s w i l l be c o n s e r v a t i v e and c a n be c o n s i d e r e d as t h e upper  limits.  CHAPTER 2  BACKGROUND ON DYNAMIC ANALYSIS  2.1  Introduction  In r e c e n t y e a r s , t h e r e has been a growing of  acceptance  t h e dynamic a n a l y s i s o f s t r u c t u r e s s u b j e c t e d t o earthquake  loading.  The response o f a s t r u c t u r e can be determined by  e i t h e r t h e t i m e - h i s t o r y ground m o t i o n approach o r t h e r e s p o n s e spectrum  approach.  In t h e t i m e - h i s t o r y ground motion approach, t h e r e s p o n s e o f a s t r u c t u r e i s o b t a i n e d by n u m e r i c a l s t e p - b y - s t e p i n t e g r a t i o n s i n t h e time domain. the  An a c c u r a t e r e s p o n s e , w i t h i n  m a t h e m a t i c a l model, c a n be o b t a i n e d f o r a p a r t i c u l a r  motion.  T h i s method i s o f t e n proven t o be u n s a t i s f a c t o r y  a d i f f e r e n t ground motion w i l l  because  cause a d i f f e r e n t response o f a  s t r u c t u r e , and t h e r e s p o n s e i s a l s o dependent one has chosen.  ground  I t i s extremely d i f f i c u l t  on t h e time s t e p  to predict the detailed  c h a r a c t e r i s t i c s o f an earthquake f o r a g i v e n s i t e  and a c h o i c e o f  a s h o r t time s t e p w i l l make t h e a n a l y s i s v e r y  costly.  I n most a n a l y s e s , where o n l y t h e maximum response o f a s t r u c t u r e i s r e q u i r e d , i t i s o f t e n found r e s p o n s e spectrum  approach.  s i m p l e r t o use t h e  However, i t p r o v i d e s o n l y  r e s u l t s f o r structures with multi-degrees-of-freedom,  approximate and i s  a p p l i c a b l e o n l y t o l i n e a r s t r u c t u r e s w i t h damping t h a t the uncoupled modal response  2.2  satisfies  equations.  Spectral Analysis  x mass m  1  /  columns, stiffness k  / / / /  / /  /  J44 1  Figure 1  Single-degree-of-freedom  x y g  system  Figure 1 shows a l i n e a r one-storey single-degree-offreedom system with mass m, column s t i f f n e s s k and damping c o e f f i c i e n t c.  The displacements x , x and y are related by the  expression:  x  where  =  x  +  g  y  (2.1)  x  =  absolute displacement of the mass,  y  =  mass displacement r e l a t i v e to the ground,  Xg =  ground displacement.  The d i f f e r e n t i a l equation of motion f o r the system may be written as:  my  +  cy  +  ky  =  -mx  o  (2.2)  where the dots denote d i f f e r e n t i a t i o n s with respect to time. The s o l u t i o n f o r Equation (2.2) may be obtained by the a p p l i c a t i o n of the Duhamel i n t e g r a l .  By neglecting the s l i g h t difference  between the damped and undamped frequencies, the response of a . (5) damped structure a t any time t i s :  y(t)  =  - -  0) O  x ( T ) e x p [ - 5 u ) ( t - T ) ] s i n [ t o ( t - T ) ] dx o  .  (2.3a)  y(t)  =  X  g  ( T )  exp[-c>(t-T)]  cos[to(t-r)] dx - fc>y(t) (2.3b)  x(t)  where  =  - o) y(t) - 2?wy(t)  (2.3c)  2  co = undamped natural c i r c u l a r frequency of v i b r a t i o n of the structure = v^kTm £ = f r a c t i o n of c r i t i c a l damping -  c 2v^k  In most p r a c t i c a l problems where only the maximum numerical values of the response are of i n t e r e s t , i t i s convenient to introduce the spectral v e l o c i t y :  S  v  =  | f Q  X  g  ( x )  exp[-£u)(t-r)]  sin[co(t-T)]  dx  1^  (2.4)  For systems with small damping (£<0.2) and short period (T<1 s e c ) , as are normally encountered i n b u i l d i n g structures, i t can be shown that the maximum quantities of the r e l a t i v e displacement, r e l a t i v e v e l o c i t y and absolute acceleration can be expressed r e s p e c t i v e l y as  y(t) J  = max  S, d  =  - S to v  .  (2.5a) \ /  X ( t ) max  where S  d  =  Sa  =  coSv  i s t h e s p e c t r a l d i s p l a c e m e n t and S  (2.5c) \*..-»w  the s p e c t r a l  a c c e li e r a t^ion d*)  From E q u a t i o n (2.4), i t i s apparent t h a t  the s p e c t r a l  v e l o c i t y i s a f u n c t i o n o f 1) t h e c h a r a c t e r i s t i c s o f t h e ground motion and 2) t h e c h a r a c t e r i s t i c s o f t h e s t r u c t u r e d e f i n e d by i t s natural period  o f v i b r a t i o n and damping.  recorded a c c e l e r a t i o n s idealized  By s t u d y i n g t h e  o f t h e p a s t e a r t h q u a k e s , Newmark^^  the ground m o t i o n o f a n earthquake by a p o l y g o n made  up o f t h r e e upper bounds as shown i n F i g u r e  2.  The average  e l a s t i c r e s p o n s e spectrum can be o b t a i n e d by m u l t i p l y i n g t h e peak ground motion bounds by a m p l i f i c a t i o n f a c t o r s which a r e functions  of the period  and damping o f a s t r u c t u r e .  peak ground a c c e l e r a t i o n , damping and p e r i o d the maximum response of a s t r u c t u r e r  The N a t i o n a l  Building C o d e ^ ^  dynamic a n a l y s i s .  With the  of vibration  given,  S,, S and S can be d e t e r m i n e d . d' v a  has adopted t h i s p r o c e d u r e f o r  10  Undamped natural period T, sec.  Figure 2  Response s p e c t r a f o r e l a s t i c systems, 1940 E I Centro earthquake, (from Ref.3)  11  The dynamic analysis of a multi-degree-of-freedom system i s more complex.  However, for l i n e a r structures with (9)  small damping, the normal mode theory simplify the problem.  can be applied to  According to the normal mode concept, the  p r i n c i p a l modes of v i b r a t i o n of a multi-degree-of-freedom system are uncoupled.  Each mode responds to the base e x c i t a t i o n as an  independent single-degree-of-freedom system.  The modal maxima  of a structure can be determined by the s p e c t r a l analysis as was discussed f o r single-degree-of-f reedom systems.  For the i * " * mode, 1  the modal maximum response of the 3^ mass can be found by the following equations: i  y  j max  <)1 >. 3. =  S  y. i ]. max  =  X. .  =<)>.. y. S . 1 j 1 ai  1  where  * i j ^1 d i  =  3 max  ( 2  <J> . . y. S . 1 j 1 vi  v  th r e l a t i v e displacement of the j mass i n 1  =  =  modal p a r t i c i p a t i o n factor of the i E m. <)>. .  ]  2 2 m. (j>. . J J 1  .  . .  J = 1  n  6 a )  (2.6b) \ • /  the i * " ^ mode Y^  -  t  n  mode  (2.6c) '  12  The actual response of the structure can then be obtained by superposing the modal maxima.  Since the modal maxima probably  would not occur simultaneously, the root-sum-square method w i l l normally give a better estimate of the actual response.  The  root-sum-square method consists of taking the square root of the sum of the modal maxima squared.  Since the effect of higher modes  on a building i s usually i n s i g n i f i c a n t , only the f i r s t few modes need to be included.  From Equation (2.6c), i t i s apparent that the maximum absolute acceleration i s d i r e c t l y proportional to the r e l a t i v e displacement, of a structure.  I t i s clear that the r e l a t i v e  displacement i s zero at the base of a structure.  This i n turn  w i l l give zero absolute acceleration at the base by Equation (2.6c) which contradicts the fact that the base, which i s attached to the ground must have the same a c c e l e r a t i o n as the ground.  In order to  r e c t i f y t h i s s i t u a t i o n , a separate response mode of constant acceleration with a magnitude equal to the peak ground a c c e l e r a t i o n w i l l be superposed with the other normal modes to obtain the actual acceleration by the root-sum-square method, i . e . :  X. j max  =  / /  2 2 2 2 x * + X, / + X_ . + x» . + g max 1 j max 2 j max 3 j max (2.7)  13  T h i s w i l l n o r m a l i z e t h e a c c e l e r a t i o n a t t h e base o f a s t r u c t u r e t o t h e ground a c c e l e r a t i o n .  The response a c c e l e r a t i o n s so  o b t a i n e d w i l l be r e f e r r e d t o as " A d j u s t e d A c c e l e r a t i o n s "  i n the  future.  2.3  Method of S t r u c t u r a l  When a n a l y z i n g  Modelling  a b u i l d i n g f o r earthquake l o a d i n g , i t  i s a common p r a c t i c e t o s i m p l i f y t h e a n a l y s i s by assuming t h e s t r u c t u r e t o be r e p r e s e n t e d by a s e t o f p l a n e frames  perpendicular  t o each o t h e r i n t h e d i r e c t i o n s o f t h e p r i n c i p a l axes o f t h e building.  Assuming t h a t  each component o f t h e earthquake  i s r e s i s t e d o n l y by frames i n t h a t d i r e c t i o n , independent i s required  f o r each p r i n c i p a l d i r e c t i o n .  forces analysis  I t i s assumed t h a t  approach w i l l p r o v i d e adequate r e s i s t a n c e t o h o r i z o n t a l  this  accelerations  i n any o t h e r d i r e c t i o n .  In case o f a s y m m e t r i c a l s t r u c t u r e where t h e t o r s i o n a l e f f e c t i s i n s i g n i f i c a n t , a two-dimensional computer program can be used f o r t h e a n a l y s i s . act as r i g i d  I t i s u s u a l l y assumed t h a t t h e f l o o r s  diaphragms, so t h a t t h e b u i l d i n g c a n be m o d e l l e d as  a s e r i e s o f two-dimensional frames connected by a x i a l l y  rigid  s t r u t s which e n f o r c e e q u a l l a t e r a l d e f l e c t i o n s a t each s t o r y  level.  IA  Each frame c a n be r e p r e s e n t e d by a p l a n a r assemblage of beams and  columns.  The degrees o f freedom c o n s i d e r e d i n t h e p l a n e o f  each frame a r e the v e r t i c a l and r o t a t i o n a l d i s p l a c e m e n t s beam-column j o i n t , and a s i n g l e h o r i z o n t a l displacement floor level.  o f each f o r each  I n t h e b u i l d i n g s c o n s i d e r e d h e r e i n , which have  r i g i d basement s t r u c t u r e s , columns a r e assumed t o be f i x e d a t t h e ground  2.A  level.  Wide Column Model f o r Shear W a l l s  I n t h e wide column frame analogy,  i t i s recognized that  the e f f e c t o f t h e f i n i t e j o i n t w i d t h s due t o column depths w i l l c o n t r i b u t e s i g n i f i c a n t l y t o the s t i f f n e s s o f a b u i l d i n g . common d e v i c e t o account  A  f o r t h e f i n i t e w i d t h s o f the shear w a l l s  or columns i n frame a n a l y s e s i s t o c o n s i d e r t h a t p o r t i o n o f a member from t h e f a c e t o the n e u t r a l a x i s o f the w a l l as a segment w i t h i n f i n i t e f l e x u r a l and a x i a l s t i f f n e s s e s .  Thus a beam i n t h e  frame a n a l y s i s can be r e p r e s e n t e d by a member w i t h t h r e e s e c t i o n s . The m i d d l e  s e c t i o n i s the o r i g i n a l beam l e n g t h taken as t h e c l e a r  d i s t a n c e between w a l l s and columns.  There i s one i n f i n i t e l y  s e c t i o n a t t a c h e d t o each end o f t h e m i d d l e the h a l f - w i d t h s of t h e w a l l s or columns.  rigid  section, representing The w a l l s c a n be r e p r e s e n t e d  by l i n e members a t t h e i r n e u t r a l axes s i n c e t h e beams a r e u s u a l l y s h a l l o w and t h e i r depths can be n e g l e c t e d .  The member  stiffness  ACTUAL STRUCTURE  I =  oo  A =  oo  Figure 3  matrix  I =  oo  A = °°  Wide column frame model.  f o r a t h r e e - s e c t i o n e d beam member c a n be f o r m u l a t e d  s t i f f n e s s method.  The d e r i v a t i o n i s p r e s e n t e d  by t h e  i n Appendix A.  The  e f f e c t o f shear d e f l e c t i o n s c a n be i n c l u d e d i n a l l members; such e f f e c t i s e s p e c i a l l y important  2.5  f o r shear w a l l s .  E f f e c t i v e Width o f S l a b s  In r e c e n t y e a r s , very popular  s l a b - s h e a r w a l l b u i l d i n g s have become  i n N o r t h America.  T h i s system c o n s i s t s o f  continuous  16  slabs spanning between load-bearing of such s t r u c t u r a l systems subjected  shear walls.  In the analysis  to l a t e r a l loads, i t i s  often convenient to assume an e f f e c t i v e width of the slab to act as a beam, coupling the shear walls.  A study of the e f f e c t i v e  width of coupling slabs has been c a r r i e d out by many p e o p l e ^ * ^ ^ ^ , but t h e i r studies have been r e s t r i c t e d to some simple plan configurations.  For many other configurations, useful r e s u l t s are  not a v a i l a b l e .  In a t y p i c a l frame consisting of beams and columns, with the f l o o r slab monolithic with the beams, the e f f e c t i v e width of the slab working as a T-beam flange i s s p e c i f i e d by the ACI (12) Standard 318-71  . The same slab e f f e c t i v e width i s assumed  for slab-shear wall buildings i n t h i s a n a l y s i s .  I t i s found,  however, that inaccuracy i n estimating the e f f e c t i v e widths of the coupling slabs w i l l generally not a f f e c t the o v e r a l l behaviour of the structure s i g n i f i c a n t l y . 2.6  I n f i l l Walls  In many frame type structures, walls and p a r t i t i o n s are made of masonry u n i t s . considered  Although such elements are often  to be non-structural, they contribute s i g n i f i c a n t l y  to the l a t e r a l s t i f f n e s s of the structure.  When the e f f e c t of  17  i n f i l l s i s included i n an analysis, the frequency of v i b r a t i o n of  the structure w i l l be increased and i t s mode of response  will  be changed.  The f i r s t major i n v e s t i g a t i o n of i n f i l l e d  frames sub-  jected to racking loads was started by Polyakov  i n 1948.  Polyakov found that i n a l l h i s t e s t s , the i n f i l l s f i r s t from the frames, except near the compression around the perimeters.  separated  areas, by cracking  He also found that the usual mode of  f a i l u r e of the complete i n f i l l following the separation was by cracking of the mortar j o i n t i n g along the compression  diagonal.  This l e d to Polyakov's proposal f o r analysing i n f i l l s as diagonal bracing s t r u t s .  In the present analysis, due to the two way a c t i o n i n earthquake loading, i t i s appropriate to represent each i n f i l l by two diagonal bracing s t r u t s crossing each other.  The diagonal  struts are assumed to have the same e l a s t i c properties as the beams and columns i n the frame. i s determined  The s e c t i o n a l area of the struts  by equating the l a t e r a l displacements f o r the i n f i l l e d  frame and the braced model f o r the same applied l a t e r a l force, a (31) procedure which i s similar to Hrennikoff's framework method The d e t a i l derivation i s shown i n Appendix B.  CHAPTER 3  MODELLING OF BUILDINGS FOR DYNAMIC ANALYSIS  3.1  Introduction  Masonry w a l l s a r e commonly used i n two types o f construction  l o a d - b e a r i n g shear w a l l s , and i n f i l l  f o r frame b u i l d i n g s .  walls  A t y p i c a l b u i l d i n g o f each type w i l l be  a n a l y z e d by dynamic a n a l y s i s t o determine t h e i r d i f f e r e n c e i n response when s u b j e c t e d t o earthquake  loadings.  The b u i l d i n g s a r e m o d e l l e d as two-dimensional structures. all  elements.  frame  Shear and b e n d i n g d e f o r m a t i o n s a r e c o n s i d e r e d f o r One t y p i c a l s t o r e y i s s t u d i e d f o r each b u i l d i n g ;  the s e c t i o n a l a r e a and moment o f i n e r t i a a r e o b t a i n e d f o r each beam and column i n t h a t s t o r e y based on t h e i r uncracked  sections.  Members i n o t h e r s t o r i e s a r e assumed t o be t h e same a s t h o s e i n t h e t y p i c a l s t o r e y which i s an a p p r o p r i a t e assumption f o r r e i n f o r c e d c o n c r e t e and masonry b u i l d i n g s .  For  t h e p u r p o s e of the dynamic a n a l y s i s , the mass of  the b u i l d i n g i s assumed to be  concentrated  a t the f l o o r  Only the dead l o a d i s used f o r d e r i v i n g the load, since i t i s l i k e l y earthquake and  s t o r e y masses.  only i n part during  i s g e n e r a l l y not p o s i t i v e l y attached  b u i l d i n g , w i l l not structure.  to be p r e s e n t  levels.  the  c o n t r i b u t e t o the i n e r t i a f o r c e s on  the  the s e i s m i c  design  method i n the N a t i o n a l B u i l d i n g Code, which i m p l i e s t h a t the dead l o a d s h o u l d  be used i n o b t a i n i n g  only  l a t e r a l forces.  E x i s t i n g b u i l d i n g s w i l l be used as our The  an  to  T h i s assumption i s based on  Live  object  of  study.  b u i l d i n g s w i l l be m o d e l l e d as c l o s e l y t o the r e a l ones as  possible.  However, i t s h o u l d  be  clarified  t h a t i t i s not  our  i n t e n t i o n to a c c u r a t e l y a n a l y z e a s p e c i f i c b u i l d i n g , so changes and  assumptions w i l l be made to s i m p l i f y our  a n a l y s i s and  meet  our r e q u i r e m e n t s under the c r i t e r i o n t h a t i t s t i l l remains a realistic  3.2  building.  L o a d - B e a r i n g Shear W a l l B u i l d i n g i n Vancouver  A t y p i c a l apartment b u i l d i n g w i t h r e i n f o r c e d masonry load-bearing The  shear w a l l s , l o c a t e d i n Vancouver, B.C.,  b u i l d i n g has  a mixed beams and  f l a t plate  i s studied.  construction  20  Figure 4  T y p i c a l f l o o r p l a n of l o a d - b e a r i n g  shear w a l l b u i l d i n g i n Vancouver  supported by shear walls.  The shear walls are constructed with  reinforced and grouted 8" thick concrete blocks with f = 3,000 m psi.  High t e n s i l e r e i n f o r c i n g s t e e l of 60,000 p s i y i e l d strength  i s used throughout.  Floor to f l o o r heights are 8' - 8" and a  t y p i c a l f l o o r area i s about 10,000 sq. f t . The dead load of one storey i s calculated to be about 1,680,000 l b s . which averages out to 168 l b s . per sq. f t . of f l o o r area.  The e l a s t i c and shear moduli of the reinforced masonry are assumed to be 3,000,000 p s i and 1,200,000 p s i r e s p e c t i v e l y based on the formulae:  E  =  1000 f m  G  =  400  f m  (3.1)  (3.2)  given by the Commentaries on Part 4 of the National Building Code 1975  .  They agree with the values used by Kenneth Medearis.  Associates i n their analysis of a masonry load-bearing shear w a l l b u i l d i n g , the Park Lane Towers, located i n Denver, Colorado (20)  I n dynamic a n a l y s i s , the l a t e r a l s t i f f n e s s c o n t r i b u t i o n of walls perpendicular  to the r e s p o n s e d i r e c t i o n can be  neglected  and w a l l s p a r a l l e l t o the r e s p o n s e can be grouped i n t o frames. The  shear w a l l s a r e r e p r e s e n t e d  by columns w i t h f i n i t e w i d t h .  A n a l y s i s i s c a r r i e d out o n l y f o r the N-S which t h e w a l l s a r e grouped i n t o seven frames.  Due  direction in t o symmetry,  o n l y the west h a l f of t h e b u i l d i n g , as shown i n F i g u r e 4, t o be  analyzed.  Frame 1 c o n s i s t s of seven w a l l s . neglected  Frame 3 i s r e p r e s e n t e d  l a r g e column.  a d j a c e n t w a l l s by s t i f f n e s s i n order  combined i n t o one  elevator shaft to act  I t i s assumed t o be connected t o i t s  f i c t i c i o u s beams w i t h n e g l i g i b l e bending to complete a r e g u l a r g r i d .  I n the  i d e n t i c a l t o the o t h e r ,  cases they a r e .  column so as to reduce the degrees of freedom.  Frame 4 c o n s i s t s of two of the b u i l d i n g .  The  the s t a i r c a s e s a r e c o n s i d e r e d  where one w a l l i s p a r a l l e l and  halved.  s l e n d e r ones a r e  Frame 2 c o n s i s t s of s i x w a l l s .  by f o u r columns.  the w a l l s s u r r o u n d i n g  as one  Two  s i n c e t h e i r c o n t r i b u t i o n t o the l a t e r a l s t i f f n e s s of  the b u i l d i n g w i l l be n e g l i g i b l e .  and  needs  l o n g w a l l s and  i s shared  by b o t h  halves  Thus the member s t i f f n e s s e s i n t h i s frame a r e  3.3  Park Lane Towers  Another masonry load-bearing shear wall building, the Park Lane Towers, with a typical floor plan shown i n Figure 5, i s analyzed so as to compare with the results of the previous building.  The towers had been dynamically analyzed by Kenneth  Medearis Associates, and a comparison of the two studies can be used to check our accuracy in building modelling.  The Park Lane Towers complex, located in Denver, Colorado, consists of three essentially identical high-rise towers.  Each of the towers i s about 206' high, having twenty  main levels above the ground plus a two-level penthouse. The storey heights are 9' - 2".  Each storey has a floor area of  approximately 7,800 sq. f t . The towers are of reinforced masonry load-bearing shear wall construction. The masonry walls are 10" thick. are used.  Both reinforced grouted brick and block masonry  The floor and roof are constructed with 12" precast,  prestressed concrete double tees with 2h" poured-in-place concrete topping.  The elastic and shear moduli of the walls are  taken as 3,000,000 psi and 1,200,000 psi respectively.  Analysis i s performed only for the E-W direction.  Only  24  Figure 5  T y p i c a l f l o o r p l a n o f P a r k Lane Towers  the north h a l f w i l l be analyzed due  to symmetry.  Again only  those walls a l i g n i n g with the response are assumed e f f e c t i v e i n r e s i s t i n g l a t e r a l loads. frames i n the model.  The walls are grouped into three  Frame 1 consists of f i v e walls, frame 2  consists of two and frame 3 has one.  The e f f e c t of the  finite  width of the shear walls i s accounted for i n the analysis.  3.4  Infill-Frame Building  Great d i f f i c u l t y was  encountered i n finding a high-  r i s e b u i l d i n g with pure frames and  i n f i l l s construction.  Most  frame buildings designed nowadays consist of some shear walls or elevator shafts which act as the p r i n c i p a l l a t e r a l load r e s i s t i n g members.  Dynamic response of t h i s type of  structure  i s expected to be very s i m i l a r to that of a load-bearing shear wall b u i l d i n g .  A pure f r a m e - i n f i l l b u i l d i n g i s desirable so  that a comparison can be made.  The design of a reinforced concrete o f f i c e b u i l d i n g located i n Vancouver, B.C.  i s altered to a r e a l i s t i c frame  building with masonry i n f i l l s .  Each storey i s 12' high and  a f l o o r area of approximately 7,000 sq. f t . shown i n Figure 6, i s 30' i n the E-W  has  The framing, as  and 25' i n the N-S  directions.  N  CN  LU  UJ  LL.  Figure 6  T y p i c a l f l o o r p l a n of i n f i l l - f r a m e  building  The f l o o r s are 5" thick. 24" x 24".  The corner and i n t e r n a l columns are  The external or face columns are 24" x 30".  The  beam sizes are 19" wide and 22" deep (including the thickness of the 5" s l a b ) .  The columns, beams and f l o o r slabs are of reinforced concrete with an average f ^ of 4,000 p s i . The  corresponding  e l a s t i c and shear moduli are 3,600,000 p s i and 1,440,000 p s i respectively.  The i n f i l l walls are of masonry m a t e r i a l .  A  shear modulus of 500,000 p s i i s adopted based on Benjamin's (21) tests  on unreinforced masonry panels and supported by the (22) test r e s u l t s obtained by Blume  Only the N-S  d i r e c t i o n w i l l be analyzed.  There i s a  t o t a l of four frames a l i g n i n g with the response i n t h i s d i r e c t i o n . Each frame c o n s i s t s of four columns.  Due to symmetry, only the  two frames on the east side, representing one-half of the b u i l d i n g , w i l l be analysed. p l e t e l y with w a l l panels.  The outside frame i s f i l l e d comThe other has panels only i n one  bay.  The masonry i n f i l l walls are represented by diagonal s t r u t s . The depth e f f e c t of the members i n the frame ( i . e . the f i n i t e s i z e of the j o i n t s ) i s ignored.  CHAPTER 4  DISCUSSION OF THE ANALYTICAL EFFORT  4.1  Computer  Program  The e x i s t i n g computer analyzing  the b u i l d i n g s .  program DYNAMIC  DYNAMIC  i s used f o r  i s t h e c o n c a t e n a t i o n o f programs  w r i t t e n by s e v e r a l p e o p l e a t t h e U n i v e r s i t y o f B r i t i s h I t performs l i n e a r e l a s t i c s m a l l d e f l e c t i o n dynamic  Columbia.  analysis of  s t r u c t u r e problems; p i n - p i n and f i x - f i x members may be used.  There a r e t h r e e major p a r t s t o t h e program DYNAMIC STRDYN, FREQ and DYNAM.  STRDYN assembles t h e g l o b a l  and mass m a t r i c e s by t h e s t i f f n e s s method. frequencies  FREQ f i n d s t h e n a t u r a l  and mode shapes o f t h e s t r u c t u r e .  r e s p o n s e o f t h e s t r u c t u r e t o earthquake ground  DYNAMIC has been m o d i f i e d c o n t r i b u t i o n o f member d e p t h s . three  sectioned  stiffness  DYNAM f i n d s t h e accelerations.  t o account f o r t h e s t i f f n e s s  A l l members a r e assumed t o be  w i t h the middle s e c t i o n representing  the c l e a r  29  l e n g t h o f the member and  the r i g i d  the h a l f - w i d t h s of the j o i n t s . a n a l y z i n g shear w a l l b u i l d i n g s .  The  s e c t i o n s a t each end  above model i s n e c e s s a r y  for  I f i t i s desired to neglect a  j o i n t w i d t h , the l e n g t h o f t h a t r i g i d equal to zero.  representing  s e c t i o n can s i m p l y be  E f f e c t s of shear d e f l e c t i o n s as w e l l as  set  flexural  d e f l e c t i o n s a r e i n c l u d e d i n f o r m u l a t i n g t h e member s t i f f n e s s m a t r i x (Appendix A ) .  The  o r i g i n a l program uses the Newmark Spectrum f o r  s p e c t r a l a n a l y s i s . A d i f f e r e n t spectrum i s used f o r the  present  analysis.  listing  Thus t h e s u b r o u t i n e SPECTR i s r e w r i t t e n and a  i s a t t a c h e d i n A p p e n d i x G.  4.2  D e s i g n Spectrum  The  dynamic a n a l y s i s p r o c e d u r e d e s c r i b e d i n the N a t i o n a l  B u i l d i n g Code 1975  makes use of a d e s i g n spectrum i n w h i c h t h e  h o r i z o n t a l peak ground m o t i o n bounds a r e g i v e n by t h r e e i n t e r s e c t i n g straight lines. unrealistic.  The  However, t h e r e s p o n s e s p e c t r u m i s found t o 1975  Code has i g n o r e d the f a c t t h a t t h e r e s p o n s e  o f a s t r u c t u r e w i l l be e q u a l t o t h e ground m o t i o n a t h i g h (or  low p e r i o d s ) .  be  frequencies  T h i s w i l l i n c r e a s e s i g n i f i c a n t l y the c o n t r i b u t i o n  o f h i g h e r modes t o the a c t u a l r e s p o n s e of a s t r u c t u r e .  30  PERIOD T. SEC  'Figure 7  Peak ground motion bounds and e l a s t i c average response spectrum f o r 1.0 g max. ground a c c e l e r a t i o n (from Ref. 30)  31  A more r e a l i s t i c e l a s t i c average response spectrum, which appears i n the proposed r e v i s i o n to the National Building Code 1975 i s used i n t h i s study.  The response i s assumed to be the same as  the acceleration ground motion bound for periods lower than 0.03 sec. For periods within the range of 0.03 sec. and 0.1 sec.j the response i s assumed to vary l i n e a r l y on the logarithmic p l o t .  The peak ground  motion bounds and the e l a s t i c response spectrum for 1.0 g maximum ground acceleration are shown i n Figure 7.  4.3  Dynamic Analysis and Results  The three b u i l d i n g s , which have been modelled as plane frame structures, w i l l be subjected e l a s t i c behaviour i s assumed.  to dynamic analysis.  For convenience, the  Linear  load-bearing  shear w a l l b u i l d i n g i n Vancouver and the Park Lane Towers w i l l be referred to as Structures 1A and IB respectively and the i n f i l l frame b u i l d i n g as Structure 2 i n future discussions.  The response  of the buildings i s calculated by the s t r u c t u r a l analysis  program  DYNAMIC.  The buildings w i l l be subjected ground acceleration of 1.0 g.  to a horizontal peak  Storey accelerations and r e l a t i v e  displacements due to the f i r s t three response modes and t h e i r  32  root-sum-square values w i l l be found. as discussed  i n section 2.2,  The adjusted  accelerations,  w i l l also be obtained so as to  r a t i o n a l i z e the response acceleration at the base of a structure. Due  to the l i n e a r r e l a t i o n between the response and the ground  acceleration, the response can l a t e r be scaled l i n e a r l y for any value of peak ground acceleration desirable.  The m u l t i p l i e r s  for the spectral bounds, that are used i n the spectral a n a l y s i s , are based on 10% of c r i t i c a l damping as recommended by the  National  Building Code for masonry construction and concrete frame with masonry walls  structures.  The e f f e c t s of. v e r t i c a l ground accelerations are not considered since they are usually smaller than the horizontal accelerations.  Besides, since the v e r t i c a l v i b r a t i o n  frequencies  of a building are generally higher than the horizontal ones, coupling of the v e r t i c a l and horizontal modes of v i b r a t i o n w i l l unlikely.  For buildings with small e c c e n t r i c i t e s , the t o r s i o n a l  modes of v i b r a t i o n can also be assumed uncoupled from the horizontal ones for s i m i l a r reasons.  Because of the symmetrical design i n the s t i f f n e s s and mass of the buildings, computed e c c e n t r i c i t y between the center of mass and  the center of r i g i d i t y at each storey l e v e l i s  be  negligible.  N e v e r t h e l e s s , the N a t i o n a l  B u i l d i n g Code  requires  a minimum d e s i g n e c c e n t r i c i t y , e, of 0.05 times t h e p l a n  dimension,  D , t o account f o r t h e a c c i d e n t a l t o r s i o n , which i s i n t e n d e d t o n ' account f o r t h e p o s s i b l e a d d i t i o n a l t o r s i o n a r i s i n g from v a r i a t i o n s i n the estimates o f the r e l a t i v e r i g i d i t i e s , u n c e r t a i n of dead and l i v e l o a d s and  estimates  a t the f l o o r l e v e l s , a d d i t i o n of w a l l  panels  p a r t i t i o n s a f t e r completion of the b u i l d i n g , v a r i a t i o n of the  s t i f f n e s s w i t h t i m e , and i n e l a s t i c o r p l a s t i c a c c i d e n t a l t o r s i o n a l moment tends t o i n c r e a s e or a c c e l e r a t i o n s  a c t i o n . T h e the i n e r t i a  forces  on one s i d e o f the b u i l d i n g and d e c r e a s e t h e i n e r t i a  forces or a c c e l e r a t i o n s  on t h e o t h e r s i d e .  The i n e r t i a  acceleration  induced by t h e moment c a n be approximated by assuming the f l o o r diaphragm a t each s t o r e y as a r i g i d r e c t a n g u l a r s t o r e y mass and t h i c k n e s s ,  and by s t u d y i n g  For a square p l a t e w i t h d i m e n s i o n D  n  p l a t e with uniform  the k i n e t i c s of the p l a t e .  and s t o r e y mass  m, i t s mass  moment o f i n e r t i a about t h e v e r t i c a l c e n t r o i d a l a x i s i s :  (4.1)  I t i s shown i n F i g u r e  8 t h a t t h e t o r s i o n a l a c c e l e r a t i o n caused by  the minimum d e s i g n t o r s i o n a l moment, v a r i e s l i n e a r l y a c r o s s t h e width, D , i n the d i r e c t i o n perpendicular n  This  t o the earthquake  loading.  t o r s i o n a l a c c e l e r a t i o n i s maximum a t t h e two ends w i t h magnitude  34  Figure  8  T o r s i o n a l induced  acceleration  xa which i s found to be 0.15 times the t r a n s l a t i o n a l i n e r t i a acceleration, x.  To account f o r the e f f e c t of accidental t o r s i o n ,  an adjustment factor of 1.15 can be used to r a i s e the i n e r t i a forces or response accelerations obtained from the t r a n s l a t i o n a l dynamic analysis by 15% to a r r i v e at an upper bound value.  In order to study the c o r r e l a t i o n between the response and the fundamental period of a structure, buildings of d i f f e r e n t heights are analyzed.  This i s achieved by adding to or reducing  the number of storeys of the b u i l d i n g .  Since a l l structure  s t i f f n e s s e s and storey masses (except the roof) are assumed the same f o r each storey, only the mass of the top storey needs to be changed.  For ease of comparison, the storey l e v e l i n each model  w i l l be represented by the dimensionless  r a t i o , h/H, i n which h i s  the height of the n*"* storey and H i s the height of the e n t i r e 1  building.  4.3.1  Load-Bearing Shear Wall Buildings  For Structure 1A, 5 models are analyzed.  Computer runs  are made for models with 5, 10, 15, 20 and 25 storeys.  In each  model, the mass of the top storey i s assumed to be h a l f of the others.  The f i r s t three natural periods are as follows:  36  T-Csec)  T (sec)  T„(sec)  5  0.115  0.034  0.018  10  0.313  0.084  0.041  15  0.592  0.150  0.071  20  0.960  0.230  0.106  25  1.417  0.321  0.145  No. of Storey  9  A l l r e s u l t s are plotted for observation. displacements  The R-S-S r e l a t i v e  are p l o t t e d i n Figures 9a - 9b.  The R-S-S along  with the adjusted accelerations are p l o t t e d i n Figures 10a -lOe. The adjusted accelerations are plotted against the fundamental periods at h/H = 0.2, 0.4, 0.6, 0.8, 1.0 r e s p e c t i v e l y i n Figure 11.  For Structure IB, the o r i g i n a l tower i s analyzed as a 20-storey structure, with the two-level penthouse included i n the top storey.  Results are compared with those obtained by Kenneth  Medearis Associates (KMA).  KMA had analyzed the tower experimentally  by taking measurements of the tower micro v i b r a t i o n s at various l e v e l s and t h e o r e t i c a l l y by model a n a l y s i s .  Both the KMA and the  present r e s u l t s are tabulated below for comparison.  37  1.0  0.8  ^  0.6  H—t  LU X _l < r-  o  U_  °  0.4  o M  rO  0.2  0.0 0.0  0.1  A  5-STOREY MODEL  •  10-STOREY MODEL  0.2 DISPLACEMENT ( f t )  Figure 9a Structure 1A: R-S-S storey displacements  0.3  DISPLACEMENT ( f t ) Figure 9b  Structure 1A: R-S-S storey displacements  39  40  41  0.0  2.0  4.0  ACCELERATION (g) Figure 10c  Structure 1A: Storey accelerations of 15-storey model  42  43  F i g u r e lOe  S t r u c t u r e 1A:  Storey  a c c e l e r a t i o n s of 2 5 - s t o r e y  model  Figure 11  V a r i a t i o n of response acceleration with fundamental period at l e v e l s h/H = 0.2, 0.4, 0.6,0.8,1.0 of Load-Bearing Shear Wall Buildings  45  T^sec)  T (sec) 2  T (sec) 3  KMA. (experimental)  0. 83  0.27  0.14  KMA(theoretical)  0. 95  0.25  0.12  Present study  1. 12  0.28  0.13  Despite the limited information on the Park Lane Towers, which meant that several d e t a i l s were asssumed, our r e s u l t s compare very c l o s e l y with those of KMA.  One may conclude that our procedure  f o r modelling the b u i l d i n g was reasonably s i m i l a r to that used by KMA.  Computer runs are also made for models with 15, 10 and 5  storeys.  This time the mass of the top storey i s l e f t to be  the same as the others.  The adjusted accelerations versus the  fundamental periods are also plotted i n Figure 11.  4.3.2  Infill-Frame Building  For Structure 2, models with 5, 8, 10, 15 and 20 storeys are analyzed.  As f o r the type 1 structures, i n each model the mass  of the top storey i s assumed to be half of the others. three natural periods of the models are as follows:  The f i r s t  Storey  T^sec)  T (sec)  5  0.210  0.070  0.042  8  0.380  0.119  0.065  10  0.519  0.156  0.082  15  0.960  0.260  0.130  20  1.537  0.381  0.184  The R-S-S r e l a t i v e displacements  0  T (sec 3  are plotted i n Figures 12a - 12b.  The accelerations are plotted i n Figures 13a - 13e. The adjusted accelerations versus the fundamental periods at h/H = 0.2, 0.4, 0.6, 0.8, 1.0 are plotted i n Figure 14.  DISPLACEMENT ( f t ) Figure 12a  Structure 2: R-S-S storey displacements  48  1.0  •——IB  u  / m 0.6  x  *  O  / /  / /  e  // / / // /// 'A' / /  X  o  LU X  h-  l  |  /  £  D  c5  i  /  0.8 h  _J <  »A  0.4  u_ o z: o  i—i  I-  u  •  /  •  0.2 h  B A  •  A  •  0  0  h80  0.0 « 0.0  1  F i g u r e 12b  —I  •  10-STOREY MODEL  A  15-STOREY MODEL  •  20-STOREY MODEL  1  I  1.0 DISPLACEMENT ( f t )  2.0  S t r u c t u r e 2: R-S-S s t o r e y  displacements  49  Figure 13a  Structure 2: Storey accelerations of 5-storey model  0.0  2.0  4.0  ACCELERATION (g) Figure 13b  Structure 2: Storey accelerations of 8-storey model  51  0.0  2.0  4.0  ' ACCELERATION (g) Figure 13c  Structure 2: Storey accelerations bf 10-storey model  52  53  0.0  2.0 ACCELERATION (g)  Figure 13e  Structure 2: Storey accelerations of 20-storey model  4.0  Figure 14  V a r i a t i o n of response acceleration with fundamental period at l e v e l s h/H = 0.2, 0.4, 0.6, 0.8, 1.0 of Infill-Frame Buildings  CHAPTER 5  RESULTS-ANALYSIS AND RECOMMENDATIONS  5.1  Observations  From the storey displacements p l o t s , i t i s observed that both the shear wall buildings and the i n f i l l - f r a m e buildings have a combined shear beam and c a n t i l e v e r behaviour.  Cantilever action  i s seen to predominate; e s p e c i a l l y at the lower l e v e l s of the buildings.  Some p o r t a l action i s observed at the upper  levels.  As f o r the i n f i l l - f r a m e b u i l d i n g s , t h i s indicates that the i n f i l l walls contribute s i g n i f i c a n t l y to the l a t e r a l s t i f f n e s s of a frame building.  The frames are changed from a supposedly shear beam  behaviour to a predominantly c a n t i l e v e r behaviour due to the i n f i l l stiffnesses.  It  i s also observed that the higher modes become more  s i g n i f i c a n t i n buildings of long periods.  Their e f f e c t i s more  e a s i l y recognized at the upper l e v e l s , h/H =0.6 and 0.8, from the R-S-S storey acceleration p l o t s .  From the a c c e l e r a t i o n v e r s u s p e r i o d that For  the  r e s p o n s e a c c e l e r a t i o n of a s t r u c t u r e  a l l t h r e e types of  to be  structures  p l o t s , i t i s seen i s period  dependent.  t h a t were a n a l y z e d , t h e r e seems  a s i m i l a r c o r r e l a t i o n between the r e s p o n s e a c c e l e r a t i o n  g i v e n l e v e l s and  the p e r i o d .  l e v e l s versus period  Separate a c c e l e r a t i o n at d i f f e r e n t  curves are p l o t t e d  r e s u l t s from a l l t h r e e S t r u c t u r e s  i n Figures  p l o t t e d on  i s seen t h a t a l l d a t a f a l l r o u g h l y on  the  the  15a  - 15e  i n t e r p r e t e d as  the  with  same graph.  same c u r v e .  An  It  upper  bound i s drawn f o r each c u r v e to i n c l u d e a l l d a t a p o i n t s . c u r v e s can be  at  These  e n v e l o p e s of dynamic a m p l i f i c a t i o i  f a c t o r f o r masonry b u i l d i n g s .  5,2  Level  Coefficient  The  National  B u i l d i n g Code recommends the  V  f o r the  =  e q u a t i o n .  ASKIFW  (5.1)  c a l c u l a t i o n of the base shear of a s t r u c t u r e , where A  t h e a s s i g n e d h o r i z o n t a l d e s i g n ground a c c e l e r a t i o n g i v e n i n T a b l e of C l i m a t i c Data i n P a r t then d i s t r i b u t e d a l o n g the  2 of  h e i g h t of  the Code.  The  the  base shear i s  the b u i l d i n g by  l a t e r a l i n e r t i a force d i s t r i b u t i o n that  is  assuming a  i s approximately  triangular  1.0  0.1 °-6  0.8  1.0  2.0  3.0  PERIOD (sec) F i g u r e 15b  V a r i a t i o n of  response a c c e l e r a t i o n w i t h fundamental  p e r i o d a t h/H  =0.4  CO  6.0  T  r  i  s  r  4.0  «  r  T  r  _L  J  L  i — i — i o  STRUCTURE IA  Q  STRUCTURE IB  A  STRUCTURE 2  3.0  o LU Lu  2.0  CJ  <  Q  LU 00  1.0 0.1  J  I 0,2  I  L 0.3  0,4  J 0.6  i  « '» 0.8  1.0  2.0  PERIOD (sec) Figure 15c  V a r i a t i o n of response acceleration with fundamental period at h/H =0.6  3.0  6.0 ® .•  4.0  A 60  3.0  STRUCTURE 1A STRUCTURE  IB  STRUCTURE 2  • 0  2  LU _! LU  (_)  2.0  U < Q  LU LO ID "2 r-  Q <  0.1  °'  2  °'  3  0.4- '  0.6  0.8  1.0  2.0  3.0  PERIOD ( s e c ) F i g u r e 15d  V a r i a t i o n o f response a c c e l e r a t i o n w i t h fundamental  p e r i o d a t h/H = 0 . 8  O  6.0  1  1  ~T  i  r  i—j—i  4.0  r  ©  STRUCTURE IA  e  STRUCTURE IB  A  STRUCTURE 2  T  r  J 2.0  L  © 60  3.0  LU _J LU U  o <  2.0  Q  LU H-  00  J  1.0 0.1  0.2  I _L 0.3  0.4  ±-—I 0.6  I  I I L 0.8 1.0  3.0  PERIOD ( s e c ) F i g u r e 15e  V a r i a t i o n o f response a c c e l e r a t i o n w i t h fundamental  period  a t h/H = 1.0  62  i n shape with the apex at the base f o r stubby structures with fundamental periods less than about 1 second.  For more slender  structures, the Code suggests a r e d i s t r i b u t i o n of forces by applying part of the base shear as a concentrated force at the top of the structure.  These procedures are intended f o r developing  a shear force envelope, but the i n e r t i a forces so obtained give, in e f f e c t , forces at any f l o o r which can be used f o r the design of masonry walls.  For the design of parts of buildings, the Code gives a l a t e r a l force, V • P  ( 1 0 )  •  V  The values of  P  =  AS W P P  (5.2) '  are given as 10.0 for cantilever walls and 2.0 for  other exterior or i n t e r i o r walls. (5.2) appears to be a b i t crude.  The design force given by Equation It should be graded, at l e a s t  p a r t i a l l y , f o r height as implied by the Code suggested shear force envelope.  With the a v a i l a b l e response r e s u l t s on hand, l a t e r a l  forces at any f l o o r f o r use i n design of masonry walls can be obtained by the introduction of a l e v e l c o e f f i c i e n t (or l e v e l dynamic amplif i c a t i o n f a c t o r ) , a.  S i m i l a r l y , a more accurate d i s t r i b u t i o n of  63  shear or i n e r t i a forces f o r masonry buildings can also be determined.  The i n e r t i a acceleration at the i  a.  where  a  ±  =  t  h  a. A S K I F  =  level coefficient  f l o o r can b e expressed  ( 5  .  3 )  at l e v e l i .  With the i n e r t i a acceleration known, the i n e r t i a forces at the .th x r l o o r can be obtained by:  V F.  i  where  V  =  =  a  -  ip  (5.4)  W  a. W. x i  t r  c\  (->.5)  l a t e r a l force on a part of the structure (eg. walls) at l e v e l i ,  W^  =  weight of a part of a structure,  F. x  =  i n e r t i a force at l e v e l i , '  W^  =  the portion of dead load which i s located at l e v e l i .  For s i m p l i c i t y ,  a b u i l d i n g can be divided into f i v e equal sections  along i t s height.  The values of a  at h/H = 0.2, 0.4, 0.6, 0.8,  64  1.0  can be o b t a i n e d from F i g u r e s 15a - 15e.  be i n c r e a s e d by 15% For  i n t e r m e d i a t e h/H  t o i n c l u d e the e f f e c t of a c c i d e n t a l  torsions.  v a l u e s , the a can be assumed c o n s t a n t  e q u a l t o t h a t c o r r e s p o n d i n g t o the next  5.3  These v a l u e s should  l a r g e r h/H  and  ratio.  Fundamental P e r i o d of B u i l d i n g s  In  l i e u of more a c c u r a t e e s t i m a t e s , the Code has  recommended  . . , , (10) the empxrxcal f o r m u l a : f  T  for  (5.6)  the d e t e r m i n a t i o n o f the fundamental p e r i o d f o r b u i l d i n g s .  I t i s assumed i n the f o r m u l a t h a t the p e r i o d v a r i e s l i n e a r l y the t o t a l h e i g h t of a b u i l d i n g assumption  may  be  f o r the same dimension  D.  with  This  true for s t e e l buildings, i t i s d e f i n i t e l y  u n t r u e f o r r e i n f o r c e d c o n c r e t e or masonry l o a d - b e a r i n g shear w a l l and  infill-frame  buildings.  The p e r i o d s o b t a i n e d by  the e m p i r i c a l f o r m u l a , T  , are  compared to those o b t a i n e d by computer a n a l y s e s , T , f o r b o t h S t r u c t u r e s IA and w a l l and  infill  2 (see T a b l e 1 ) . Both the l o a d - b e a r i n g  p a n e l systems a r e found  to be s t i f f e r  shear  than  expected from the empirical formula.  The.empirical period exceeds  the computed value by as much as .140% i n the worst p r e d i c t i o n ; the e r r o r i s greatest with the lowest structures. p l o t t e d against T  £  i n Figure 16.  The r a t i o T /T i s c e  The following formula i s derived  to predict the fundamental period of reinforced concrete or masonry load-bearing shear wall and i n f i l l frame b u i l d i n g s :  T  =  (0.35 + 0.5T )T~ e e  T  =  0.0175 — fi  2 or  +  0.00125 ~  (5.7)  D  where D i s the dimension of the b u i l d i n g i n feet i n a d i r e c t i o n p a r a l l e l to the applied  forces.  • Table 1 COMPARISON OF FUNDAMENTAL PERIODS  No. of S t o r i e s  Height of Bldg. (H - f t . )  Period by Computer (T - sec) c  Period by Formula ,rp 0.05H IT - sec)  Percentage Error in T ( % ) e  /D  T Ratio - ~ T e  Structure 1A: D = 60 f t . 5 10 15 20 25 Structure 2: 5 8 10 15 20  43.3 86.7 130.0 173.3 216.7  0.115 0.313 0.593 0.959 1.417  0.280 0.559 0.839 1.119 1.399  + 143 + 79 + 41 + 17 1  0.41 0.56 0.71 0.86 1.01  0.210 0.380 0.519 0.961 1.537  0.346 0.554 0.693 1.039 1.386  + + + + -  0.61 0.69 0.75 0.92 1.11  D = 75 f t . 60.0 96.0 120.0 156.0 192.0  65 46 34 8 10  CHAPTER 6  MINIMUM STEEL REQUIREMENTS  6.1  Introduction  A major earthquake i s the most s e v e r e l o a d i n g to which a b u i l d i n g might ever be  subjected.  b u i l d i n g s located i n seismic most l i k e l y o c c u r . reinforcement The  For  i s required  to r e s i s t .  have to be  6.1.1  Zone 3 where major earthquakes w i l l  t a l l b u i l d i n g s w i t h masonry w a l l s , to s a f e l y guard the w a l l s  amount of s t e e l r e q u i r e d  w a l l has  This i s e s p e c i a l l y true for  depends on  There a r e  two  from  steel  failure.  the earthquake f o r c e s  the  major t y p e s of f o r c e s which  considered:  1.  Racking f o r c e s  (In-plane  2.  Transverse forces  shearing),  and  (Out-of-plane bending).  Racking F o r c e s  i It i s realized w a l l s or i n f i l l  panels,  that w a l l s , e i t h e r load-bearing are  the dominant l a t e r a l l o a d  shear resisting  69  elements i n a b u i l d i n g .  Unreinforced masonry subjected to i n -  plane shear forces i s presumed to f a i l i n a b r i t t l e manner, and the minimum s t e e l as required by the Code i s intended to give r i s e to a d u c t i l e behaviour.  In  the minimum earthquake  force design formula, Equation  (5.1), the c o e f f i c i e n t K r e f l e c t s the damping and expected of  a s t r u c t u r a l system.  Types of construction that are recognized  to have performed well i n earthquakes of  K.  ductility  are assigned lower values  For reinforced masonry shear wall b u i l d i n g s and frame  buildings with masonry i n f i l l s , the assigned K value i s 1.3 which can be shown to correspond to an expected d u c t i l i t y of about 2. So  the key to masonry seismic design requirements  f o r in-plane  shear forces i s to design a w a l l panel which has a d u c t i l e behaviour consistent with the s p e c i f i e d K c o e f f i c i e n t .  In order to  achieve t h i s , experimental e f f o r t i s required by t e s t i n g w a l l panels with various amounts of s t e e l reinforcement.  Only then can the  amount of s t e e l necessary to provide the required degree of d u c t i l i t y for a p a r t i c u l a r seismic zone be determined. i s not attempted  Such e f f o r t  i n this thesis.  j  6.1.2  Transverse Forces  thquake-induced  response accelerations of a b u i l d i n g  70  produce i n e r t i a The  inertia  f o r c e s i n elements, a r i s i n g from t h e i r own masses.  f o r c e on a w a l l p a n e l can be a n a l y z e d  as a u n i f o r m  l o a d a c t i n g normal t o the f a c e o f the w a l l by assuming t h a t t h e whole w a l l has the same a c c e l e r a t i o n . response already  obtained  With t h e maximum dynamic  f o r t h e masonry w a l l b u i l d i n g s , t h e  i n e r t i a f o r c e s a c t i n g on t h e w a l l s can be determined by E q u a t i o n (5.4)  given  t h e i n p u t peak ground a c c e l e r a t i o n a p p r o p r i a t e t o  s p e c i f i c zones and s i t e s . of-plane  These i n e r t i a f o r c e s w i l l  bending i n the w a l l s .  tensile strength, tension areas.  S i n c e masonry has a f a i r l y low  s t e e l reinforcement  i s needed.in the high  The b e h a v i o u r o f r e i n f o r c e d masonry i s q u i t e  s i m i l a r to that of r e i n f o r c e d concrete.  By assuming t h a t t h e  masonry w a l l s a r e coherent w i t h bonded r e i n f o r c e m e n t edge c o n d i t i o n s p r o v i d e  and t h a t t h e  adequate s u p p o r t , the same p r i n c i p l e s  a p p l i c a b l e to r e i n f o r c e d concrete  may be employed f o r meeting t h e  minimum s t e e l r e q u i r e m e n t s f o r o u t - o f - p l a n e  6.2  cause o u t -  bending i n masonry w a l l s .  D e s i g n Earthquake  It i s generally recognized g e n e r a t e a much g r e a t e r  input  t h a t a major earthquake  than t h e Code s p e c i f i e s .  will  An e x p r e s s -  i o n f o r the peak ground a c c e l e r a t i o n s t h a t a r e a c t u a l l y i m p l i e d by  71  the q u a s i - s t a t i c p r o c e d u r e o f the N a t i o n a l  B u i l d i n g Code has been  (23) developed by Anderson, Nathan and C h e r r y  .  Their  interpretation  of the Code p r o c e d u r e can be summarized as f o l l o w i n g : Working l e v e l l o a d  =  ASKIFW  l e v e l load  =  XASKIFW  f o r major earthquake  =  uXASKIFW  =  uXASKIF  Y i e l d or u l t i m a t e E l a s t i c load  Average s t r u c t u r a l r e s p o n s e a c c e l e r a t i o n as a f r a c t i o n o f g , Spectral acceleration  =  average a c c e l e r a t i o n — — •— 2  p  yXASKIF „ . ' Ground a c c e l e r a t i o n  =  spectral acceleration — • c  uXASKIF BD  where  (6.1)  X  =  load  factor,  u  =  d u c t i l i t y r a t i o corresponding to the K c o e f f i c i e n t ,  6  =  0.6 f o r a u n i f o r m c a n t i l e v e r i n t h e f i r s t  mode,  D  =  dynamic a m p l i f i c a t i o n f a c t o r i n t h e f i r s t  mode,  E l a s t i c Load  =  i n e r t i a f o r c e on h y p o t h e t i c a l  building  w i t h same i n i t i a l s t i f f n e s s which remains '  elastic.  By u s i n g t h e developed e x p r e s s i o n the ground a c c e l e r a t i o n  implied  by the q u a s i - s t a t i c procedure was found to vary with the fundamental period of a b u i l d i n g and the type of structure.  The  implied ground acceleration tends to increase with the period of a structure, T (since the r a t i o S/D i n Equation (6.1) increases with T ) . The Code appears to be conservative for structures with long periods.  By applying Equation (6.1) to the seismic Zone 3  where A = 0.08 with y = 2, X = 1.8, K =  1.3, I = 1 , F = 1, 8 = 0.6,  S = 0.5 and D = 1.1, i t can be shown that .the ground acceleration i s equal to 0.28 of gravity.  The values of S and D correspond to  a damping r a t i o of 10% and a period of 1 second, which represent a t y p i c a l medium h i g h - r i s e masonry construction.  LOAD  ELASTIC LOAD = yXV  /  ULTIMATE LOAD = XV  WORKING LOAD = V = ASKIFW  I  7i"~1 /  I I DISPLACEMENT  Figure 17  E f f e c t of y and X on the design  load  6  -3  Design Formula f o r Masonry Walls i n Bending  Face loading perpendicular  to the masonry surface has (24)  received much research attention f o r unreinforced walls Combined bending and compression loading of unreinforced walls (25) has also been researched  .  Scrivener, on the other hand, has  conducted a s e r i e s of face loading t e s t s on r e i n f o r c e d b r i c k Of.\ walls  .  In the t e s t s , i t was found that the ultimate load  could be predicted to within a few percent by considering the b r i c k w a l l as a l i g h t l y reinforced wide beam and applying the ultimate moment theory as f o r reinforced concrete.  The s t r e s s -  s t r a i n curve f o r the b r i c k masonry was assumed to be the same as that f o r concrete i n order that the concrete constant 0.59 could be used.  Thus the Whitney Formula:  \  where  =  A  f s  y  [ d  " °-  5 9  (  A  f s  y  / f  c  b ) ]  M u  =  ultimate moment, '  A  =  cross-sectional area of s t e e l ,  =  y i e l d stress of s t e e l ,  d  =  depth to center of gravity of s t e e l ,  b  =  beam width,  f^  =  concrete compressive strength.  s  f y  ( 6  -  2 )  I  74  with  replaced by the masonry compressive strength, f ^ , can be  used for the design of masonry walls.  The Whitney Formula i s  s a t i s f a c t o r y for very l i g h t l y reinforced masonry because the second term within the bracket, which contains 0.59 and f , c a f f e c t s the r e s u l t s by only a small percentage.  6.4  Wall Panel  Designing  The Code-implied  ground a c c e l e r a t i o n corresponding to  A - 0.08 (Zone 3) for masonry buildings with fundamental period ol 1 second has been found to be 0.28 g. With an amplication factor obtained from Figure 15e at T = 1 second, the maximum response a c c e l e r a t i o n (including the a c c i d e n t a l t o r i s o n e f f e c t ) i s calculated to be 0.9 g at the top storey of a b u i l d i n g . The corresponding  uniform face load, q , on a w a l l panel can be  assessed with the w a l l density known.  By assuming i s o t r o p i c  properties and appropriate boundary conditions of the walls, the maximum bending moment due to the face load can be obtained. assuming that the s t e e l reinforcement the wall can then be designed  By  i s placed along the center,  for t h i s maximum bending moment  according to the ACI code by using Equation  (6.2), and applying  the appropriate load and capacity reduction f a c t o r s .  75  By applying the Code-given  formula, Equation (5.2), i n  the design of parts of b u i l d i n g s for l a t e r a l forces, the design i n e r t i a accelerations corresponding to A = 0.08 are computed to be 0.8 g and 0.16 g f o r c a n t i l e v e r walls and other i n t e r i o r or exterior walls r e s p e c t i v e l y .  The value for cantilever walls compares c l o s e l y  with the maximum 0.9 g from the present analysis but the 0.16 g f o r exterior or i n t e r i o r walls i s very much below our present value and seems to be u n f i t f o r design a p p l i c a t i o n .  6.4.1  Load-Bearing  Shear Wall  A continuous shear wall can be assumed f i x e d to the slabs at f l o o r l e v e l s and attached at some points to perpendicular w a l l s . In general, each shear w a l l panel can be modelled as a p l a t e with three edges fixed and the fourth edge free as shown i n Figure 1 8 ( i ) . A w a l l panel of dimension a x b i s assumed to be 8" t h i c k and has a density of 90 psf. From Reference  (28), the bending moment i s  found to be maximum when a/b = 0.75, where:  IM I 1  y 'max  =  0.107 q a o  2  (6.3)  This bending moment i s compared to those f o r walls with other possible boundary conditions and i s proved to be the highest.  The  t  (i)  (ii)  FIXED  S.S.  /  / / / / / / > • /  thickness t  /  UJ LU LL  /  to oo  thickness t  / / / / / /  FIXED  S.S.  b  F i g u r e 18  oo 00  Models f o r ( i ) shear w a l l p a n e l and ( i i ) i n f i l l  panel  77  panel i s then designed to the maximum moment.  The s t e e l area  required i s found to be dependent on the w a l l dimensions.  It i s  approximately proportional to the square of the height to thickness r a t i o , a/t, of the w a l l .  The v e r t i c a l s t e e l area i s calculated to  be 0.0006 bt f o r a/t = 12, where bt i s the gross c r o s s - s e c t i o n a l area of the w a l l .  The v e r t i c a l compressive  load on the w a l l i s  ignored i n the c a l c u l a t i o n because compressive  loads w i l l increase  the ultimate bending capacity of a wall as only f a i l u r e s i n i t i a t e d by y i e l d i n g of the tension s t e e l are of concern. thus obtained i s on the conservative side.  The s t e e l area  The maximum bending  moment i n the other d i r e c t i o n f o r the same panel i s :  |M I ' x 'max  =  0.05 q a o  (6.4)  2  n  which gives a d e s i g n h o r i z o n t a l s t e e l area of 0.0002 bt f o r a/t = 12. The t o t a l s t e e l area adds up to 0.0008 bt.  For an a/t r a t i o of 18,  the t o t a l area of s t e e l required i s 0.0018 b t .  6.4.2  I n f i l l Wall  An i n f i l l wall panel of height a and width b can be modelled as a plate with four sides simply supported. i s assumed to be 8" thick and has a density of 60 psf.  A w a l l panel For a square  plate (a/b = 1.0), the maximum bending moment i s :  =  l I ' x 'max M  1  IM I y 'max  =  0.046 q  o  H  a  2  s (6.5) K  The corresponding design s t e e l i s 0.0004 bt i n both the v e r t i c a l and h o r i z o n t a l d i r e c t i o n s which gives a t o t a l s t e e l area of 0.0008 bt f o r a / t = 18. slab, Reference  For a/b = 0.2, which i s the same as a one-way  (29) gives:  |M  I  =  0.125 q a ^o  =  0.0375 q  y 'max  |M  x  [  'max  M  o  (6.6)  2  a  2  /  .  N  V.O./; F  I  7  The v e r t i c a l s t e e l i s calculated to be 0.001 bt and the h o r i z o n t a l s t e e l 0.0001 bt which give a t o t a l of 0.0011 bt f o r a/t = 18.  CHAPTER 7  SUMMARY, CONCLUSIONS AND RECOMMENDATIONS FOR FUTURE RESEARCH  7.1  Summary and  Conclusions  In o r d e r for  t o e s t a b l i s h t h e minimum p e r c e n t a g e s of s t e e l  the s e i s m i c d e s i g n o f masonry w a l l s , t h e r a c k i n g and t r a n -  s v e r s e f o r c e s caused by a major earthquake on t h e w a l l s have t o be c o n s i d e r e d . i s decided  The minimum s t e e l w i t h r e s p e c t t o r a c k i n g  forces  by t h e d u c t i l i t y r e q u i r e m e n t s as i m p l i e d by t h e  B u i l d i n g Code.  The r e i n f o r c e m e n t  forces i s decided  with respect  to t r a n s v e r s e  by t h e w a l l ' s a b i l i t y t o r e s i s t  inertia  forces  due t o i t s own mass.  A study has been c a r r i e d out t o e s t a b l i s h t h e minimum s t e e l r e q u i r e d i n masonry w a l l s f o r a major earthquake w i t h to the t r a n s v e r s e f o r c e s .  B u i l d i n g s w i t h masonry  load-bearing  shear w a l l s and frame b u i l d i n g s w i t h masonry i n f i l l m o d e l l e d as p l a n e DYNAMIC w i t h  frame s t r u c t u r e s .  respect  p a n e l s were  An e x i s t i n g computer program  some m o d i f i c a t i o n s was used f o r t h e l i n e a r  elastic  dynamic analysis.  The response of the structure was  response spectrum approach.  found by  the  I t i s assumed that the response  accelerations so obtained provide the upper l i m i t s and are conservative.  Findings based on the r e s u l t s of the dynamic analysis  are summarized as  (a)  followings:  The base response acceleration of a structure obtained by the s p e c t r a l analysis i s zero which i s obviously Equation (2.7)  unrealistic.  i s recommended to adjust the accelerations so  as to normalize the base a c c e l e r a t i o n to the ground acceleration. (b)  A l l models analyzed, including the i n f i l l - f r a m e s , are observed to behave predominantly l i k e cantilevers with some p o r t a l action at the top of the buildings.  The  i n f i l l panels contribute s i g n i f i c a n t l y to the s t i f f n e s s of a frame b u i l d i n g . (c)  The r e s u l t s of the Park Lane Towers are compared with those reported by Kenneth Medearis Associates.  They agree very  c l o s e l y with each other. (d)  The buildings analyzed are found to be s t i f f e r than expected from Equation (5.6) given by the Code.  A more appropriate  empirical formula, Equation (5.7), i s recommended for estimating  the fundamental periods of masonry buildings.  81  (e)  A c o r r e l a t i o n between the response acceleration and the fundamental period of v i b r a t i o n of a structure i s developed for l e v e l s h/H = 0.2, 0.4, 0.6, 0.8, 1.0.  (f)  The l a t e r a l load design formula, Equation ( 5 . 2 ) , f o r parts of b u i l d i n g s , given by the National Building Code, appears to be very unconservative.  A l e v e l c o e f f i c i e n t i s recommended  for finding the i n e r t i a acceleration envelope. (g)  In order to account f o r the accidental t o r s i o n , the response forces or accelerations obtained  (h)  should be increased by 15%.  The maximum response acceleration of the top storey of a t y p i c a l masonry high-rise building i n Zone 3 i s found.  The  minimum s t e e l required i n the masonry walls to r e s i s t the i n e r t i a forces can be determined by applying the Whitney Formula.  For a height to thickness r a t i o a/t = 18, the  t o t a l s t e e l cross-sectional area required i s computed to be 0.0018 and 0.0011 times the gross c r o s s - s e c t i o n a l area of the wall r e s p e c t i v e l y f o r the load-bearing i n f i l l wall.  shear w a l l and the  The amount of s t e e l required i n other l e v e l s or  zones can be established s i m i l a r l y .  7.2  Recommendations for Future Research  The minimum seismic requirement f o r masonry walls  82  established  i n t h i s t h e s i s considers only transverse forces.  c c r . p l e t e study l o g i c a l next  should  include racking forces.  Therefore,  A  the  s t e p i n the development of t h e minimum s e i s m i c  s t e e l i s to t e s t w a l l p a n e l specimens w i t h v a r i o u s amount of reinforcement  subjected  to r a c k i n g l o a d s and  s i n g l e out  the  t h a t meets the d u c t i l i t y r e q u i r e m e n t s .  Preliminary tests  be undertaken w i t h  i t i s envisaged  static  v e r y s i m i l a r s e t and  l o a d i n g s , but  full  should  that a  number of specimens would be t e s t e d under a  dynamic c y c l i c l o a d i n g , s i m u l a t i n g earthquake r e s p o n s e . t e s t s on a few  one  Further  s c a l e r e i n f o r c e d w a l l s w i l l be h e l p f u l  c o n f i r m the h y p o t h e s e s from the specimen t e s t i n g .  Furthermore,  tin; arrangement of s t e e l such as the maximum s p a c i n g , the amount i n the d i r e c t i o n normal to the span, and be e s t a b l i s h e d e x p e r i m e n t a l l y .  to  necessary  so f o r t h , have to  83  REFERENCES  1.  Sahlin, S. S t r u c t u r a l Masonry. P r e n t i c e - H a l l , Inc., Englewood C l i f f s , N.J., 1971.  2.  Amrhein, J.E. Reinforced Masonry Engineering Handbook. 2nd Ed., Masonry I n s t i t u t e of America, Los Angeles, C a l i f o r n i a , 1973.  3.  Blume, J.A., Newmark, N.M. and Corning, L.H. Design of M u l t i story Reinforced Concrete Buildings for Earthquake Motions. Portland Cement Association, U.S.A., 1961.  4.  Wiegel, R.L., Ed., Earthquake Engineering. Inc., Englewood C l i f f s , N.J., 1970.  5-  Newmark, N.M. and Rosenblueth, E. Fundamentals 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., 1971.  6.  F i n t e l , M., Ed., Response of Multistory Concrete Structures to L a t e r a l Forces. Publication SP-36, American Concrete I n s t i t u t e , D e t r o i t , 1973.  7.  Beaufait, F.W., Ed., Proceedings of the Symposium on T a l l Buildings - Planning, Design & Construction. Nashville, Tennessee, November 14-15, 1974.  8.  C o u l l , A. and Smith, B.S., Ed., T a l l Buildings. Oxford, N.Y., Symposium Publications D i v i s i o n , Pergamon Press, 1966.  9.  Timoshenko, S., Young, D.H. and Weaver, W. J r . V i b r a t i o n Problems i n Engineering. 4th Ed., John Wiley & Sons, Inc., U.S.A., 1974.  Prentice-Hall,  10.  National Building Code of Canada 1975. National Research Council of Canada. Associate Committee on the National Building Code, Ottawa, NRCC No.13982.  11.  Commentaries on Part 4 of the National Building Code of Canada 1975. National Research Council of Canada. Associate Committee on the National Building Code, Ottawa, NRCC No. 13989.  12.  Building Code Requirements for Reinforced Concrete (ACI 318-71). American Concrete I n s t i t u t e , D e t r o i t , Michigan.  84  13.  Cherry, S. "Basic P r i n c i p l e s of Response of Linear Structures to Earthquake Ground Motions", Proceedings of the Symposium on Earthquake Engineering, University of B r i t i s h Columbia, September 8-11, 1965.  2-.  Hudson, D.E. "Some Problems i n the A p p l i c a t i o n of Spectrum Techniques to Strong Motion Earthquake Analysis" Bui. Seismological Society of America, V o l . 52, No.2, 1962.  15.  Newmark, N.M. and H a l l , W.J. "Seismic Design C r i t e r i a f o r Nuclear Reactor F a c i l i t i e s " Proceedings of the Fourth World Conference on Earthquake Engineering, V o l . 2, Santiago De C h i l e , January 13-18, 1969.  16.  Pecknold, D.A. "Slab E f f e c t i v e Width f o r Equivalent Frame Analysis". ACI Journal, A p r i l 1975, pp. 135-137.  17.  Tso, W.K. and Mahmond, A.A. " E f f e c t i v e Width of Coupling Slabs i n Shear Wall Buildings". Journal of Structural D i v i s i o n , ASCE, March 1977, pp. 573-586.  18.  Polyakov, S.V. Masonry i n Frame Buildings: An Investigation into the Strength and S t i f f n e s s of Masonry I n f i l l i n g . (English Translation) Moscow, 1957.  19.  Polyakov, S.V. "On the Interaction Between Masonry F i l l e r Walls and Enclosing Frame when loaded i n the Plane of the Walls". Translations i n Earthquake Engineering, Earthquake Engineering Research I n s t i t u t e , San Francisco, 1960.  20.  Kenneth Medearis Associates "An Investigation of the Dynamic Response of t h e Park Lane Towers to Earthquake Loadings". Report to Colorado Masonry I n s t i t u t e , November 1973.  21.  Benjamin, J.R. and Williams, H.A. "The Behaviour of One-Story Brick Shear Walls". Proceedings of ASCE, S t r u c t u r a l D i v i s i o n , N.Y., July 1958.  22.  Blume, J . and Associates "Shear i n Grouted B r i c k Masonry Wall Elements". Report to Western States Clay Products Association, San Francisco, C a l i f o r n i a , 1968.  23.  Anderson, D.L., Nathan, N.D. and Cherry S. " C o r r e l a t i o n of S t a t i c and Dynamic Earthquake Analysis of the National Building Code of Canada 1975". Department of C i v i l Engineering, University of B r i t i s h Columbia, 1977.  85  24.  Cox, F.W. and Ennenga, J . L . " T r a n s v e r s e S t r e n g t h of C o n c r e t e Block Walls". ACI J o u r n a l , May 1958 P r o c . 54, pp. 951-960.  25.  Y o k e l , F.Y., Mathey, R.G. and D i k k e r s , R.D. " S t r e n g t h of Masonry W a l l s under Compressive and T r a n s v e r s e Loads". N a t i o n a l Bureau of Standards Report, U.S. Dept. of Commerce, Washington D . C , B u i l d i n g S c i e n c e S e r i e s 34, March 1971  2c.  S c r i v e n e r , J.C. Non-load-bearing p p . 215-220.  27.  S c r i v e n e r , J.C. " R e i n f o r c e d Masonry - S e i s m i c Behaviour and Design". B u l l e t i n of N.Z. S o c i e t y f o r Earthquake E n g i n e e r i n g , V o l . 5, No.4, December 1972, pp. 143-155.  28.  Bares, R. T a b l e s f o r the A n a l y s i s of P l a t e s , S l a b s and Diaphragms Based on the E l a s t i c Theory. T r a n s l a t e d by C a r e l van Amerongen, B a u v e r l a g Gmbh., Wiesbaden und B e r l i n g (Germany), 1969.  29.  Timoshenko, S. and Woinowski - K r i e g e r , S. Theory of P l a t e s and S h e l l s , 2nd E d i t i o n , McGraw - H i l l Book Company, U.S.A., 1959.  30.  Commentaries on P a r t 4 of the N a t i o n a l B u i l d i n g Code of Canada 1977. A s s o c i a t e Committee on the N a t i o n a l B u i l d i n g Code, Ottawa, NRCC No. 15558.  31.  H r e n n i k o f f , A. " S o l u t i o n of Problems of E l a s t i c i t y by the Framework Method", J o u r n a l of A p p l i e d M e c h a n i c s , ASME, December 1941, p.A160.  "Face Load T e s t s on R e i n f o r c e d H o l l o w - b r i c k Walls" New Zealand E n g i n e e r i n g , J u l y 1969,  APPENDIX A  BEAM STIFFNESS MATRIX FOR WIDE COLUMN FRAME  walls  Figure  1 9 ( i ) shows a beam spanning between two shear  (columns).  P o i n t s A and B a r e on t h e c e n t r o i d a l axes o f  t h e w a l l s a t beam l e v e l and a r e t h e node p o i n t s . element c a n be m o d e l l e d a s shown i n F i g u r e  The beam  19(ii).  N o r m a l l y t h e member s t i f f n e s s f o r a beam element i s w i i t t e n i n terms o f t h e d e f o r m a t i o n s a t i t s ends.  (i)  A  .B  .  b  (ii)  F i g u r e 19  Beam element f o r wide column frame  88  For  c o n v e n i e n c e , t h e s t i f f n e s s m a t r i x can be broken up i n t o two  p a r t s , the a x i a l  terms  and t h e bending terms ( K ^ ) •  K  For  =  K  +  a  (A.2)  an o r d i n a r y beam o f l e n g t h £' w i t h b o t h ends  '^N_6  lis  7  F =  fixed,  K  A  s t i f f n e s s m a t r i c e s w i t h shear d e f o r m a t i o n s i n c l u d e d a r e :  K  AE  EI £ (l+g)  1  0  0  -1  0  0  0  0  0  0  0  0  0  0  0  0  0  0  -1  0  0  1  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  12  61  0  -12  61  0  61  £ (4+g)  0  -61  0  0  0  0  0  0  -12  -61  0  12  0  61  0  -61  2  * (2-g) 2  3  where  g  14.4EI 2 AG I  £ (2-g) 2  0 -6£ £ (4+g) 2  89  Since  the e q u i l i b r i u m e q u a t i o n s of t h e s t r u c t u r e a r e  w r i t t e n i n terms o f the f o r c e s and d e f o r m a t i o n s a t the node p o i n t s A and B, a t r a n s f o r m a t i o n  i s necessary.  2  ?  1,2,3  i — \ t  4,5,6  A  w  '1  .(—  For  M K  L _  F = K A  ' 31  .small d e f o r m a t i o n s , the f o r c e s a t the ends of t h e beam, f , a r e  related  to the n o d a l f o r c e s , f , by the  F  =  1  expression:  F  (A. 3)  where  T  1  =  1  0  0  0  0  0  0  1  0  0  0  0  0  a  1  0  0  0  0  0  0  1  0  0  0  0  0  0  1  0  0  0  0  0  -b  0  I t can be shown t h a t : K  where  = transpose of  K 1  (A.4)  90  In g e n e r a l ,  the structure co-ordinates  w i t h the member c o - o r d i n a t e s .  do n o t c o i n c i d e  I t i s necessary to define a s t i f f n e s s  n a t r i x i n terms o f t h e n o d a l f o r c e s and d e f o r m a t i o n s i n t h e s t r u c t u r e co-crdinates  (F, A ) .  F = KA  The r e l a t i o n between t h e n o d a l f o r c e s i n s t r u c t u r e and  co-ordinates  t h o s e i n member c o - o r d i n a t e s a r e :  (A. 5)  TF X  where  T=  -y  0  0  0  0  y  X  0  0  0  0  0  0  L  0  0  0  0  0  0  X  0  0  0  y  X  0  0  0  0  0  0  L  -y  0  I t c a n be shown t h a t  K where T  = transpose of T  =  = * T KT  (A.6)  Substituting  (A.4)  into  K  or  By  K  c a r r y i n g out  a  (A.6)  =  T ?1  =  T T.  gives  K T.J*  K  1  T*  T,  1  a  (A.7)  T  the m a t r i x m u l t i p l i c a t i o n s , K  &  and  are  -xy  0  found  as  following:  2  2  xy AE  K L  0 2  I  2  -x  0  12y  2  -12xy EI L  3 2 £  (1+g)  y  0  0  0  -xy  0  2  -xy  -\y -W  2  0  xy  X  -x -xy  .0 2 X  -y  0  xy  0  0  0  -12xy  12x  2  k^x  k^x  k  12xy  k  l  Y  -12x  -k^x  -k y  k x  k  2  2  -y  0  0  0  xy  0  2 y  0  0  0  -W  12xy  12xy  -12x  3  12xy  2  2  4  W -12xy  -k y 2  2  k x 2  -k^x  k  -12xy  k y  2 12x -k x 2  4 2  -k x 2  k  5  92  where  1^  =  6L(2a + £)  k  2  =  6L(2b + £)  k  3  =  12L a(a + £) + L £ ( 4 + g)  k^  =  12L ab + 6L £(a + b) + L £ ( 2 - g)  k  =  12L b(b + £) + L £ ( 4 + g)  5  2  2  2  2  2  2  2  2  2  2  The complete beam s t i f f n e s s matrix, K , for the wide column frame, in the structure co-ordinates can be obtained by adding K  V  q  and  APPENDIX B  MODELLING OF INFILL WALLS  An i n f i l l w a l l which i s assumed t o r e s i s t f o r c e s can be r e p r e s e n t e d  by two d i a g o n a l s t r u t s .  only  shear  A w a l l panel  oi: dimension a x b and t h i c k n e s s t i s s h o w n i n F i g u r e 2 0 ( i ) . The p a n e l has a shear modulus G.  The e q u i v a l e n t s t r u t model r e p r e s e n t -  i n g the w a l l p a n e l i s shown i n F i g u r e 2 0 ( i i ) .  Each s t r u t has a  c r o s s - s e c t i o n a l a r e a A and an e l a s t i c modulus E.  A constant wall panel.  shear  s t r e s s , T , i s assumed a c t i n g on t h e  I t i s the same as h a v i n g  concentrated  forces acting  a t the f o u r c o r n e r s of the p a n e l as shown i n F i g u r e 2 0 ( i ) .  The  same f o r c e s and shear d e f l e c t i o n a r e assumed i n the s t r u t model. From F i g u r e 2 0 ( i ) ,  T  From F i g u r e  = G a  (B.l)  20(ii), strut elongation 6 ' =  strain force  F  <5  = M L  (B.2)  6  1  94  F i g u r e 20  Equivalent  s t r u t model of i n f i l l  wall  From the free body at point 0 i n Figure 20(111),  rtb  T~  By substituting  (B.l)  a  n  d  ( B  =  . )  L  A  L  =  £ E  tb  G  L  ~ a  6  (B.4)  into (B.4),  3  - • M b  0  6  2 ~  =  0  tL^ 2ab  (B.5)  The equivalent strut area i s calculated f o r a masonry wall with a  =  12 f t .  b  =  25 f t .  t  =  4 in.  G  =  500,000 p s i  The t = 4" corresponds to a wall thickness of 8" since the net crosss e c t i o n a l area of a hollow masonry w a l l i s approximately 50% of the gross area.  The strut i s assumed to have an e l a s t i c modulus  E  =  3,600,000 p s i  By substituting the above values into equation (B.5), the crosss e c t i o n a l area of a strut i s computed to be 1.645 f t .  96  APPEND I X C  COMPUTER  SUBROUTINE C C C C C C C C C C C  LISTING  OF  SUBPROGRAM  SPECTR  S P E C T R ( ITYPE,MODE,DAMP,SD,SA,AE,WN, I V . D A F )  Mf.DE I S THE MODE NUMBER Sn I S THE VECTOR OF S P E C T R A L D I S P L A C E M E N T S SA I S THE VECTOR OF S P E C T R A L A C C E L E R A T I O N S I S THE VECTOR OF MODAL N A T U R A L F R E Q U E N C I E S ( R A C S / S E C ) !•« I S THE NIJVBER CF yCDES »,',i TS THE DYNAMIC A M P L I F I C A T I O N FACTOR M i C T R READS WHATEVER D A T A I T NEEDS ON THE F I R S T C A L L , WHICH I S N'ADF WITH NCDE=G I' MODE.NF.0 THEN S O , S A , ANO DA F ARE RETURNED  f  D I M E N S I O N UNI I I , SC( I t , S A l 1 ) I F {MODE.NE.0 ) GO TO 1 0 1 R E A D ! 5 , 1 0 0 ) I T Y E , C 1 » C? »C3 »C4,PGA 100 FORMAT!I 8 , 5 P 8 . 4 ) F0G=1.25664 F1G=9.66644 GA=PGA*C1 V=GA/FIG D=V/FOG A=GA*C2 V=V*C3 D=D*C4 FO=V/D F1=A/V <=2 = 6 2 . 8 3 2 D  F3=209.44  D2=A/F2/F? 03=GA/F3/i= 3 WRITE(6,107) GA,C1,C2,C3,C4,A,V,D  97  107 P ORM A T ( / • N B C ' S I D E A L I Z E D SPFCTRUM A P P L I E D ' / / *' PEAK GROUND ACC =»,F8.4/« SOIL C O E F F = ' , F 8 . 4 • ACC CCEFF=«, *F8.4,» VEL COEF F = ' » F8 » 4» ' DISP COEFF='» F B . 4 / ' ACC L I M I T = *F8.4,« VEL L I M I T =»,F8.4,» D I S P L I M I T =»,F8.4/) f  ~-TURN  i::  20! 202 203  R  EO^MMCDF) l'i FREQ.GE.FC) GO Tf) 201 S C I MODE) = D "V Tn 2 0 5 I - ( F P . E O . G F . F l ) GO TO 202 S T ! ••'CCF )=V/ PREQ GO ?05 I F ( F R E 0 . G E . P 2 5 GO TO 203 SC(MODE)=A/FREQ/FREQ GO T O 2 0 5 I P C R E 0 . G F . F 3 ) GO TO 204 SD( ODE ) -D?*EXP ( AL0G(D3/D2)*AL0G( FP.E0/F2 )/ALOG { F 3 / F 2 ) ) GC TO ?05 SD( T O E » = GA/FREO/FREO 5A{M CDE 5 = SD(MODE) #WN{MODE)*WN(MODE ) C A P =•' S A (MODE) /GAAF = 0 .0 RF TU N FND C  M  204 205  r  <  APPENDIX D  INPUT AND RESULT DATA OF MODEL ANALYSES  FRAME 2  23.6 13.8' 10.8' 23.8'  A3.2' ,19.4' ,16.9' 14.5' 10.8'  1  r  10 F i g u r e 21  FRAME H-  FRAMI *,  FRAME 1  11  P l a n e frame model of S t r u c t u r e  13.9' 14.r 12 1A  13  13.0'  14  15  32.0' 16  17  Table 2 IA  MEMBER PROPER'IIi  Columns  I (ft )  A (ft )  1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17  25.07 28.44 1.50 16.49 10.13 3.56 227.56 202.11 96.00 73.94 3.56 371.51 3.00 1220.00 162.03 571.41 571.41  5.11  4  2  5.33  2.00 4.45 3.78 2.67 10.67 10.25 8.00 7.33 2.67 12.56 4.00 46.00 15.12 9.13 9.13  F i c t i c i o u s Beam, p i n n e d - p i n n e d  a(ft)  b (ft)  2.67 2.67 2.67 2.67  3.84 4.00 1.50 3.34  4.00 1.50 3.34 2.84  7.18 0.26 0.26 0.26 7.18  5.78 7.11 7.11 7.11 5.78  2.00 8.00 7.92 6.00 5.50  8.00 7.92 6.00 5.50 2.00  12-13 13-14 14-15  0.26  7.11  9.42 (pin-pin) (pin-pin)  1.50  16-17  0.38  4.11  13.70  13.70  Beams  I (ft )  1-2 2-3 3-4 4-5  0.1 0.1 0.1 0.1  6-7 7-8 8-9 9-10 10-11  endsj I =  4  ;'c  (ft ) 2  co  •k co  A =  FRAME 1  n  FRAMh:  7  >-  ON  -V 2  , 15.50'  , 17.85'  14.85'  ,  17.50'  •  Figure  22  18.50'  P l a n e frame model o f S t r u c t u r e IB  FRAME  3  MEMBER PROPERTIES OF STRUCTURE IB  Columns  I  (ft ) 4  A (ft ) 2  Beams  I  (ft ) 4  A (ft ) 2  a (ft)  b (ft)  1  203.07  11.92  2  85.07  8.92  1-2  1.50  3.00  7.15  5.35  3  23.82  5.83  2-3  1.50  3.00  5.35  3.50  4  220.59  12.25  3-4  1.50  3.00  3.50  7.35  5  203.07  11.92  4-5  1.50  3.00  7.35  7.15  6  347.24  14.25  7  156.12  10.92  6-7  1.50  3.00  8.55  6.65  8  41.16  7.00  FRAME 1  FRAME  Figure 23  2  Plane frame model of Structure 2  MEMBER PROPERTIES OF STRUCTURE 2  Columns  I  (ft ) 4  A (ft ) 2  Beams  I  (ft ) 4  A (ft ) 2  1  1.33  4.00  2  1.67  5.00  1-2  1.21  3.94  3  1.67  5.00  2-3  1.21  3.94  4  1.33  4.00  3-4  1.21  3.94  5  2.60  5.00  6  1.33  4.00  5-6  1.57  5.68  7  1.33  4.00  6-7  1.57  5.68  8  2.60  4.00  7-8  1.57  5.68  Bracing Strut:  A = 1.65 f t  2  Table 5 DYNAMIC ANALYSIS RESULTS OF STRUCTURE 1A (a^  5-Storey  = 0.115 s e c 2 = 0.034 s e c 3 = 0.018 s e c  Tl i  T  S t o r e y Mass (lbs) 0.2 0.4  0.6 0.8 1.0  (b)  840,000 840,000  840,000 840,000 420,000  Rel. X-Displ. (ft)  0.005 0.012 0.021 0.028 0.034  X-Accel. (g)  Adjusted A c c e l , (g)  0.652 1.289 - 1.949 2.605 3.253  1.194 1.632 2.191 2.790 3.403  Rel. X-Displ. (ft)  X-Accel.  Adjusted A c c e l , (g)  0.011 0.032 0.058 0.087 0.118 0.149 0.179 0.208 0.234 0.258  0.398 0.880 1.280 1.571 1.804 2.029 2.284 2.626 3.070 3.550  10- S t o r e y T l = 0.313 sec 2 = 0.084 sec 3 = 0.041 sec T T  h/H  0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0  S t o r e y Mass (lbs) 840,000 840,000 840,000  840,000 840,000 840,000 840,000 840,000 840,000 420,000  (g)  1.076 1.332 1.624 1.862 2.063 2.262 2.493 2.810 3.228 3.689  Table 5 (Continued)  (c)  15-Storey Ti = 0.592 sec T = 0 . 1 5 0 sec T3 = 0 . 0 7 1 sec 2  h/H 0.067 0.1 n  o.:>.  0.21.7 0..TI3 0./,  0.4b/'  o.' ;; 0.6 0.667 0.733 0.8 0.867 0.933 1.0  Storey Mass (lbs)  Rel. X-Displ. (ft)  X-Accel. - (g)  Adjusted Accel, (g)  840,000 840,000 840,000 840,000 840,000 840,000 840,000 840,000 840,000 840,000 840,000 840,000 840,000 840,000 420,000  0.016 0.047 0.088 0.136 0.191 0.250 0.312 0.375 0.439 0.503 0.567 0.628 0.688 0.745 0.800  0.267 0.638 0.997 1.281 1.473 1.593 1.678 1.754 1.829 1.910 2.029 2.229 2.531 2.904 3.297  1.035 1.186 1.412 1.625 1.780 1.881 1.954 2.019 2.085 2.156 2.262 2.443 2.721 3.071 3.446  107  Table 5 (Continued) 23-Storey ~l = 0.960 sec ~2 = 0.230 sec T q = 0.106 sec  h/H 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5 0.55 0.6 0.65 0.7 0.75 0.8 0.85 0.9 0.95 1.0  Storey Mass (lbs)  Rel. X-Displ. (ft)  X-Accel. (g)  840,000 840,000 840,000 840,000 840,000 840,000 840,000 840,000 840,000 840,000 840,000 840,000 840,000 840,000 840,000 840,000 840,000 840,000 840,000 420,000  0.015 0.044 0.084 0.133 0.189 0.250 0.317 0.387 0.460 0.536 0.614 0.693 0.773 0.853 0.933 1.011 1.089 1.165 1.240 1.313  0.194 0.482 0.789 1.062 1.270 1.403 1.467 1.483 1.478 1.469 1.459 1.438 1.399 1.351 1.336 1.408 1.599 1.886 2.222 2.567  Adjusted Accel, (g) 1.019 1.110 1.274 1.459 1.617 1.723 1.776 1.789 1.784 1.777 1.769 1.752 1.719 1.681 1.669 1.727 1.886 2.135 2.437 2.755  108  Table 5 (Continued) ie)  25-Storey 7i = 1.417 sec 12 = 0.321 sec T 3 = 0.145 sec  h/H  0.04 0.08 0.] ' 0.16 0.2  0.  ^1  0.  0. .! 0.3(  0.4 0.44 0.48 0.52 0.56 0.6 0.64 0.68 0.72 0.76 0.8 0.84 0.88 0.92 0.96 1.0  S t o r e y Mass (lbs)  Rel. X-Displ. (ft)  840,000 840,000 840,000 840,000 840,000 840,000 840,000 840,000 840,000 840,000 840,000 840,000 840,000 840,000 840,000  0.015 0.043 0.083 0.132 0.189 0.252 0.321 0.395 0.473 0.555 0.640 0.728 0.818 0.910 1.004 1.099 1.195 1.292 1.389 1.483 .1.581 1.677 1.771 1.864 1.957  840,000 840,000  840,000 840,000 840,000 840,000 840,000 840,000 840,000 420,000  X-Accel. (g)  * 0.137 0.354 0.600 0.844 1.060 1.234 1.357 1.429 1.459 1.458 1.440 1.413 1.383 1.346 1.293 1.218 1.124 1.028 0.967 0.994 1.134 1.368 1.655 1.960 2.263  Adjusted A c c e l , (g)  1.009 1.061 1.166 1.309 1.458 1.588 1.685 1.744 1.769 1.768 1.753 1.731 1.707 1.677 1.634 1.576 1.505 1.434 1.391 1.410 1.512 1.695 1.934 2.201 2.475  109  Table 6 DYNAMIC ANALYSIS RESULTS OF STRUCTURE I B  (a  5-Storey l ! = 0.153 sec 72 = 0.045 sec 7 q = 0.024 sec  N  h/H  Storey Mass (lbs)  Rel.X-Displ. (ft)  0.2 0.4 0.6 0.8 1.0  660,000 660,000 660,000 660,000 660,000  0.008 0.021 0.034 0.046 0.056  (b)  X-Accel. (g) 0.675 - 1.298 1.861 2.424 3.019  Adjusted Accel, (g) 1.206 1.638 2.113 2.622 3.180  10-Storey T i = 0.378 sec T = 0.105 sec T3 = 0.052 sec 2  h/H  Storey Mass (lbs)  Rel.X-Displ. (ft)  X-Accel. (g)  Adjusted Accel, (g)  0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0  660,000 660,000 660,000 660,000 660,000 660,000 660,000 660,000 660,000 660,000  0.018 0.047 0.083 0.123 0.166 0.208 0.251 0.291 0.328 0.363  0.453 0.976 1.385 1.657 1.852 2.027 2.224 2.508 2.916 3.392  1.098 1.397 1.708 1.936 2.105 2.260 2.439 2.700 3.083 3.536  110  Table 6 (Continued)  (c)  I5-Storey Ti = 0.677 sec T = 0.179 sec T3 = 0.086 sec 2  h/H 0.067 0.1 M 0.2 0.2 iv/ 0. : r n 0.4 O.'n  ;  0.511 0.6 0.667 0.733 0.8 0.867 0.933 1.0  Storey Mass (lbs) 660,000 660,000 660,000 660,000 660,000 660,000 660,000 660,000 660,000 660,000 660,000 660,000 660,000 660,000 660,000  Rel. X-Displ. (ft) 0.021 0.057 0.104 0.159 0.220 0.285 0.353 0.422 0.492 0.562 0.630 0.697 0.761 0.823 0.882  X-Accel, .  (8) 0.301 0.689 1.039 1.295 1.447 1.527 1.580 1.632 1.684 1.727 1.781 1.901 2.132 2.461 2.828  Adjusted Accel, (g) 1.044 1.215 1.442 1.636 1.759 1.826 1.870 1.914 1.959 1.995 2.042 2.148 2.355 2.656 3.000  Table 6 (Continued)  (c'i  20-Storey = 1.121 sec T = 0.280 sec T3 = 0.131 sec TT 2  h/H  0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5 0.55 0.6 0.65 0.7 0.75 0.8  0.85 0.9 0.95 1.0  Storey Mass (lbs) 660,000 660,000 660,000 660,000 660,000 660,000 660,000 660,000 660,000 660,000 660,000 660,000 660,000 660,000 660,000 660,000 660,000 660,000 660,000 1,200,000  Rel. X-Displ. (ft) 0.020 0.055 0.101 0.156 0.219 0.287 0.360 0.437 0.517 0.599 0.683 0.768 0.854 0.940 1.025 1.110 1.193 1.275 1.356 1.435  * X-Accel. (g) 0.204 0.490 0.783 1.040 1.236 1.363 1.425 1.441 1.431 1.417 1.404 1.383 1.343 1.277 1.205 1.173 1.240 1.430 1.711 2.031  Adjusted A c c e l , (g) 1.021 1.114 1.270 1.443 1.590 • 1.691 1.741 1.754 1.746 1.734 1.723 1.707 1.674 1.622 1.566 1.542 1.593 1.745 1.982 2.264  Table 7 DYNAMIC ANALYSIS RESULTS OF STRUCTURE 2  (a)  5-Storey T l = 0.210 sec T 2 = 0.070 sec 1< = 0.042 sec  S t o r e y Mass (lbs)  h/H 0.2 0.4 0.6  600,000 600,000 600,000 600,000 300,000  o.»  1.0  (b)  Rel.  X-Displ. (ft)  0.024 0.051 0.075 0.095 0.107  X-Accel. (g) * 0.947 1.631 2.152 2.669 3.119  Adjusted A c c e l , (g) 1.377 1.913 2.373 2.851 3.275  ii-Storey T i = 0.380 sec T 2 = 0.119 sec T 3 = 0.065 sec  h/H  S t o r e y Mass (lbs)  0.125 0.25 0.375 0.5 0.625 0.75 0.875 1.0  600,000 600,000 600,000 600,000 600,000 600,000 600,000 300,000  Rel.  X-Displ. (ft)  0.040 0.089 0.142 0.195 0.247 0.294 0.335 0.368  X-Accel. (g) 0.748 1.345 1.686 1.934 2.188 2.518 2.974 3.413  Adjusted A c c e l , (g) 1.249 1.676 1.961 2.177 2.405 2.709 3.138 3.556  113  Table 7 (Continued) (c)  10-Storey T i = 0.519 sec  7 = 0.156 sec 7, = 0.082 sec 2  Storey Mass (lbs)  h/H  600,000 600,000 600,000 600,000 600,000 600,000 600,000 600,000 600,000 300,000  0.1 0.2 0.3 0.4 0.5 0 . 6  0.7 0.8  0.9 1.0  (d)  Rel. X-Displ. (ft) 0.049 0.112 0.183 0.259 0.337 0.415 0.490 0.560 0.624 0.679  X-Accel. (g)  0.633 1.181 ' 1.513 1.709 1.880 2.063 2.277 2.595 3.024 3.437  Adjusted Accel, (g) 1.183 1.547 1.814 1.980 2.130 2.293 2.487 2.781 3.185 3.580  15-Storey T i = 0.960 sec T 2 = 0.260 sec T 3 = 0.130 sec  h/H 0.067 0.133 0.2 0.267 0.333 | 0.4 j 0.467  0.533 0.6 0.667  0.733 0.8 0.867 0.933 1.0  Storey Mass (lbs) 600,000 600,000 600,000 600,000 600,000 600,000 600,000 600,000 600,000 600,000 600,000 600,000 600,000 600,000 300,000  Rel. X-Displ. (ft) 0.043 0.102 0.172 0.251 0.337 0.429 0.525 0.625 0.725 0.827 0.927 1.026 1.121 1.213 1.299  X-Accel. (g)  0.434 0.863 1.187 1.382 1.462 1.471 1.457 1.435 1.393 1.331 1.310 1.430 1.716 2.083 2.428  Adjusted Accel, (g) 1.090 1.321 1.552 1.706 1.771 1.779 1.767 1.749 1.715 1.665 1.648 1.745 1.986 2.310 2.626  Table 7 (Continued)  (e)  20-Storey T = 1.537 sec T = 0.381 sec T3 = 0.184 sec 1  2  h/H  S t o r e y Mass (lbs)  o.ns 0.1 0. 1 '. o.:' 0. 0. •• 0. v, 0. . 0.45 0.5 0.55 0.6 0.65 0.7 0.75 0.8 0.85 0.9 0.95 1.0  600,000 600,000 600,000 600,000 600,000 600,000 600,000 600,000 600,000 600,000 600,000 600,000 600,000 600,000 600,000 600,000 600,000 600,000 600,000 300,000  Rel. X-Displ. (ft)  0.042 0.101 0.171 0.252 0.341 0.437 0.540 0.649 0.762 0.879 1.000 1.123 1.248 1.374 1.500 1.627 1.752 1.875 1.995 2.113  X-Accel. (g)  0.312 0.647 0.945 1.182 1.344 1.432 1.460 1.448 1.416 1.375 1.319 1.235 1.117 0.981 0.890 0.938 1.154 1.472 1.813 2.127  Adjusted A c c e l , (g)  1.047 1.191  1.376 1.548 1.676 1.747 1.769 1.759 1.734 1.700 1.655 1.589 1.499 1.401 1.339 1.371 1.527 1.779 2.070 2.351  

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