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The modified substitute structure method as a design aid for seismic resistant coupled structural walls Metten, Andrew W. F. 1981

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THE MODIFIED SUBSTITUTE STRUCTURE METHOD AS A DESIGN AID FOR SEISMIC RESISTANT COUPLED STRUCTURAL WALLS by ANDREW W. F. METTEN B . A . S c , The U n i v e r s i t y of B r i t i s h Columbia, 1978 A THESIS SUBMITTED IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF MASTER OF APPLIED SCIENCE i n THE FACULTY OF GRADUATE STUDIES (Department of 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 co n f o r m i n g t o the r e q u i r e d s t a n d a r d THE UNIVERSITY OF BRITISH COLUMBIA March, 1981 (c) Andrew W. F. M e t t e n , 1981 In p r e s e n t i n g t h i s t h e s i s i n p a r t i a l f u l f i l m e n t o f t h e r e q u i r e m e n t s f o r an advanced degree a t t h e U n i v e r s i t y o f B r i t i s h C o l u m b i a , I a g r e e t h a t t h e L i b r a r y s h a l l make i t f r e e l y a v a i l a b l e f o r r e f e r e n c e and s t u d 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 e x t e n s i v e c o p y i n g o f t h i s t h e s i s f o r s c h o l a r l y p u r p o s e s may be g r a n t e d by t h e head o f my department o r by h i s o r h e r r e p r e s e n t a t i v e s . I t i s u n d e r s t o o d t h a t c o p y i n g o r p u b l i c a t i o n o f t h i s t h e s i s f o r f i n a n c i a l g a i n s h a l l n o t be a l l o w e d w i t h o u t my w r i t t e n p e r m i s s i o n . Department o f Cm , f_ E^?&tA. E F f i ; ^ The U n i v e r s i t y o f B r i t i s h C o l u m b i a 2075 Wesbrook P l a c e Vancouver, Canada V6T 1W5 n n c n /mi i i ABSTRACT The m o d i f i e d s u b s t i t u t e s t r u c t u r e method i s p r e s e n t e d as a d e s i g n a i d which e v a l u a t e s d u c t i l i t y r e q u i r e m e n t s and d e f l e c t i o n s t o dete r m i n e the s u i t a b i l i t y of a r e i n f o r c e d c o n c r e t e s t r u c t u r a l w a l l t o w i t h s t a n d an a n t i c i p a t e d s i e s m i c d i s t u r b a n c e . The p r o c e d u r e i s analogous t o e l a s t i c modal a n a l y s i s but i s an i t e r a t i v e t e c h n i q u e which t a k e s account of the s t i f f n e s s l o s s of those members a t t e m p t i n g t o c a r r y moments i n .excess of t h e i r u l t i m a t e moment c a p a c i t y . U n l i k e the e l a s t i c modal a n a l y s i s p r o c e d u r e , the method i s c a p a b l e of p r e d i c t i n g the d u c t i l i t y demand i n i n d i v i d u a l members of a g i v e n s t r e n g t h . T h i s i s the a p p r o p r i a t e form of the problem f o r c o u p l e d w a l l s . The m o d i f i e d s u b s t i t u t e s t r u c t u r e method i s a p p l i e d through a computer program and the t e s t i n g of t h i s program i s d e s c r i b e d . The e f f e c t i v e n e s s of the method f o r p r e d i c t i n g d u c t l i t y demands and o t h e r parameters r e l a t i n g t o s t r u c t u r e s u n d e r g o i n g i n e l a s i c b e h a v i o r i s e v a l u a t e d by c o m p a r i s i o n w i t h r e s u l t s o b t a i n e d from the time s t e p a n a l y s i s program DRAIN-2D. The m o d i f i e d s u b s t i t u t e s t r u c t u r e method i s a proce d u r e which i s i n e x p e n s i v e t o use and c o u l d be a p p l i e d e a s i l y i n a n o n - r e s e a r c h d e s i g n environment. i i i TABLE OF CONTENTS Page ABSTRACT i i TABLE OF CONTENTS i i i LIST OF TABLES v i LIST OF FIGURES v i i ACKNOWLEDGEMENTS X CHAPTER 1. INTRODUCTION 1.1 Coupled W a l l s as Earthquake R e s i s t i n g Elements 1 1.2 Purpose of t h i s T h e s i s 5 1.3 E x a m i n a t i o n of S t r u c t u r a l A n a l y s i s Methods f o r S e i s m i c Design 7 1.4 Scope 13 2. INTRODUCTION TO THE SUBSTITUTE STRUCTURE METHOD AND THE MODIFIED SUBSTITUTE STRUCTURE METHOD. 2.1 The s u b s t i t u t e s t r u c t u r e method 14 2.2 The m o d i f i e d s u b s t i t u t e s t r u c t u r e method 22 3. ALTERATIONS TO THE METHOD FOR THE ANALYSIS OF STRUCTURAL WALLS.' 3.1 Convergence schemes 30 3.2 Convergence sp e e d i n g r o u t i n e 33 3.3 The e f f e c t of u s i n g z e r o smeared damping r a t i o a t the s t a r t of each 3.4 R i g i d beam e x t e n s i o n TESTING THE PROGRAM FOR ELASTIC CAPABILITIES 4.1 T e s t i n g the s t i f f n e s s m a t r i x f o r m u l a t i o n and e i g e n v a l u e p r o d u c t i o n . ... 4.2 C o m p a r i s i o n w i t h a n o t h e r modal a n a l y s i s program 4.3 C o m p a r i s i o n w i t h e l a s t i c time s t e p r e s u l t s TESTING THE INELASTIC PROPERTIES OF THE PROGRAM 5.1 L i t e r a t u r e comparison of damage p a t t e r n s 5.2 Assumptions f o r comparison w i t h a time s t e p a n a l y s i s program 5.3 R e s u l t s and comparisons w i t h time s t e p programs (a) F i v e - S t o r y s t r u c t u r a l w a l l (b) T e n - S t o r y s t r u c t u r a l w a l l (c) S i x t e e n - S t o r y s t r u c t u r a l w a l l w i t h an e x t r a uncoupled w a l l 5.4 c o s t s of e x e c u t i o n APPLICATION OF THE METHOD THROUGH A DESIGN EXAMPLE 6.1 A n a l y s i s f o r the d e s i g n of a s i x t e e n -S t o r y s t r u c t u r a l w a l l 6.2 E x a m i n a t i o n of the e f f e c t of cha n g i n g maximum ground a c c e l e r a t i o n V 7. CONCLUSIONS ^ BIBLIOGRAPHY 137 APPENDIX (A) Program Manual 138 (B) Program L i s t i n g 171 I V 1 LIST OF TABLES Tab l e Page 3.1 A d d i t i o n a l Member S t i f f n e s s M a t r i x t o Account For R i g i d Arms 94 4.1 A n a l y t i c R e s u l t s of V e r t i c a l and H o r i z o n t a l Pendulums 95 4.2 E l a s t i c Modal A n a l y s i s R e s u l t s For S h i b a t a and Sozen's 5-Story S t r u c t u r e ( F i g u r e 4.2) u s i n g t h e i r Spectrum 'A' 96 5.1 Earthquake r e c o r d s used i n DRAIN-2D Computer Runs 97 5.2 P r o p e r t i e s of Test S t r u c t u r e s ' 98 6.1 R e s u l t s of Computer Runs on 16-Story D e s i g n v i i LIST OF FIGURES F i g u r e Page 2.1 F l o w c h a r t f o r the S u b s t i t u t e S t r u c t u r e Method ... 100 2.2 F l o w c h a r t f o r the M o d i f i e d S u b s t i t u t e S t r u c t u r e Method 101 3.1 F l o w c h a r t f o r Convergence Speeding R o u t i n e 102 3.2 Comparison Of R e s u l t s U s i n g D i f f e r e n t Convergence Schemes on One-Bay, S i x - S t o r y Frame 103 3.3 Graph of Damage R a t i o vs I t e r a t i o n For One Column of a S i x S t o r y One Bay Frame 104 3.4 Diagram of M o d i f i e d Member t o I n c l u d e R i g i d E x t e n s i o n s 105 4.1 S t r u c t u r a l I d e a l i z a t i o n Of V e r t i c a l And H o r i z o n t a l Pendulums 106 4.2 S h i b a t a And Sozen's F i v e - S t o r y S t r u c t u r e 107 4.3 C o n f i g u r a t i o n Of Test S t r u c t u r e 'A' 108 4.4 F i v e - S t o r y S t r u c t u r e w i t h R i g i d Arms Showing Bending Moments Produced from E l a s t i c Modal and E l a s t i c Time S t e p A n a l y s i s 109 5.1 G e n e r a l Test S t r u c t u r e C o n f i g u r a t i o n 110 5.2a Angle Used For C a l c u l a t i o n Of Member D u c t i l i t y .. 110 5.2b Member D u c t i l i t y As G i v e n By P a u l a y 110 5.3 D u c t i l i t y Demand of the C o u p l i n g Beams For The 5 - S t o r y W a l l (Test S e r i e s 'B') I l l 5.4 D i s p l a c e m e n t E n v e l o p e s f o r the 5-Story W a l l (Test S e r i e s *B' ) 112 v i i i 5.5 D u c t i l i t y Demand Of The C o u p l i n g Beams For The 5-Story W a l l (Mass = 4 Times O r i g i n a l Run) 113 5.6 Di s p l a c e m e n t E n v e l o p e s For The 5-Story W a l l (Mass=4 Times O r i g i n a l Run) 114 5.7 S h i b a t a And Sozen's Spectrum 'A' Showing Fundamental P e r i o d s Of Te s t S t r u c t u r e s From T h i s T h e s i s 115 5.8 D e f l e c t i o n E n v e l o p e s For The 10 - S t o r y W a l l 116 5.9 C o u p l i n g Beam Damage R a t i o s For The 1 0 - S t o r y w a l l 117 5.10 D u c t i l i t y Demand Of C o u p l i n g Beams For 16-S t o r y Coupled W a l l w i t h E x t r a Uncoupled W a l l .... 118 5.11 D e f l e c t i o n E n v e l o p e s For The 16-S t o r y Coupled W a l l With A t t a c h e d Uncoupled W a l l 119 5.12 Damage R a t i o s For The 16 - S t o r y Coupled W a l l W i t h A t t a c h e d Uncoupled W a l l 120 5.13 Average D u c t i l i t y Demand of C o u p l i n g Beams f o r The 1 6 - S t o r y Coupled W a l l w i t h A t t a c h e d Uncoupled W a l l 121 5.14 T y p i c a l E x e c u t i o n And P r i n t i n g C o s t s For A S i n g l e Computer Run 122 6.1 C o n f i g u r a t i o n of 1 6 - S t o r y Coupled W a l l For De s i g n Example 123 6.2 Damage R a t i o s from the F i r s t Run on the 16-S t o r y Design Example 124 6.3 Damage R a t i o s from t h e Second Run on the 16-S t o r y D esign Example 125 .4 Damage R a t i o s from the T h i r d Run on the 16-S t o r y D e s i g n Example 126 1 . 5 Damage R a t i o s from the F o r t h Run on the 16-S t o r y Design Example 127 6.6 Damage R a t i o s from the F i f t h Run on the 16-S t o r y D e s ign Example 128 6.7 Damage R a t i o s from the S i x t h Run on the 16-S t o r y D e s ign Example 129 6.8 Damage R a t i o s ' from the Seventh Run on the 16-S t o r y D e s i g n Example 130 6.9 Damage R a t i o s of C o u p l i n g Beams from DRAIN-2D Runs On The 16-S t o r y Design Example . 131 6.10 D e f l e c t i o n E n v e l o p e s from DRAIN-2D Runs On 16-S t o r y D e s ign Example 132 6.11 C o u p l i n g Beam Damage R a t i o s of 1 6 - S t o r y Example For V a r i o u s V a l u e s Of Maximum Ground A c c e l e r a t i o n 133 6.12 Damaged P e r i o d as a F u n c t i o n Maximum Ground A c c e l e r a t i o n f o r the 1 6 - S t o r y Example 134 6.13 Smeared Damping R a t i o as a F u n c t i o n Of Maximum Ground A c c e l e r a t i o n f o r the 1 6 - S t o r y Example .... 135 6.14 H o r i z o n t a l D i s p l a c e m e n t s of 1 6 - S t o r y D e s i g n Example At V a r i o u s Ground A c c e l e r a t i o n s . .' 136 X ACKNOWLEDGEMENT I wish t o acknowledge and thank my t h r e e a d v i s o r s , Dr. S. C h e r r y , Dr. N. D. Nathan and Dr. D. L. Anderson f o r t h e i r s upport and a d v i c e . I p a r t i c u l a r l y w i s h t o thank Dr. Nathan f o r h i s h e l p f u l and c o n s t r u c t i v e a d v i c e a f t e r r e a d i n g the o r i g i n a l m a n u s c r i p t . I a l s o w i s h t o thank Dr. S. K. Ghosh of the P o r t l a n d Cement A s s o c i a t i o n i n S k o k i e I l l i n o i s f o r h i s h e l p f u l a s s i s t a n c e i n o b t a i n i n g i n i t i a l r e s u l t s from t'he program DRAIN-2D. T h i s t h e s i s was made p o s s i b l e by the f i n a n c i a l a s s i s t a n c e of the N a t i o n a l R e s e a r c h C o u n c i l of Canada i n the form of a r e s e a r c h a s s i s t a n t s h i p . F i n a l l y I wish t o thank those p e o p l e who have made graduate s c h o o l such a e n j o y a b l e and w o r t h w h i l e e x p e r i e n c e f o r me: the f a c u l t y and s t a f f of the C i v i l E n g i n e e r i n g Department a t UBC, Mr. Steve Ramsay and o t h e r f e l l o w g r a d u a t e - s u d e n t s , my mother, Joan D. M e t t e n , and o t h e r members of my f a m i l y and Ann-Marie D e r r i c k . 1 CHAPTER 1 INTRODUCTION AND BACKGROUND 1.1 Coupled W a l l s as Earthquake R e s i s t i n g Elements The e x c e l l e n t b e h a v i o r of s t r u c t u r a l w a l l s b oth i n eart h q u a k e s and under s e r v i c e l o a d c o n d i t i o n s has been r e p o r t e d i n the l i t e r a t u r e from s t u d i e s performed i n f a r - r a n g i n g l o c a l i t i e s . W h i l e the news cameras and r e s e a r c h e r s have been p h o t o g r a p h i n g f a l l e n and s e v e r e l y damaged d u c t i l e frames, examples of the good b e h a v i o r of s t r u c t u r a l w a l l s o f t e n goes u n n o t i c e d . F i n t e l 1 r e p o r t s examples of the s u c c e s s f u l s u r v i v a l of s t r u c t u r a l w a l l b u i l d i n g s i n ear t h q u a k e s o c c u r r i n g i n San Fernando, C a l i f o r n i a (1971), C a r a c a s , Venezuela (1967) and Sk o p j e , Y u g o s l a v i a (1963). D e s p i t e the cases of e x c e l l e n t b e h a v i o r , the s t r u c t u r a l w a l l system cannot be ex p e c t e d t o behave w e l l i f the b u i l d i n g i s d e t a i l e d i n a manner t h a t does not f u l l y t a k e account of the f o r c e s on the s t r u c t u r e . Examples of t h i s a r e the infamous O l i v e View h o s p i t a l (1971) i n which the f i r s t f l o o r columns y i e l d e d b e f o r e the s t r u c t u r a l w a l l s above had a chance t o a c t , and the Mount M c k i n l e y apartments i n Anchorage (1964) which s u f f e r e d d i a g o n a l shear f a i l u r e s i n the l i n t e l beams of the c o u p l e d s t r u c t u r a l w a l l . 2 The numerous s u p p o r t e r s of the s t r u c t u r a l w a l l system c i t e i t s b e n e f i c i a l energy a b s o r p t i o n p a t t e r n s and the way the system can cope w i t h e a r t h quake f o r c e s w i t h o u t u n d e r g o i n g l a r g e d e f o r m a t i o n s which damage the o f t e n d e l i c a t e n o n - s t r u c t u r a l elements and c o n t e n t s of the b u i l d i n g . The s t r u c t u r a l w a l l system was o r i g i n a l l y thought of as a n o n - d u c t i l e system l a r g e l y because of a s e r i e s of s e i s m i c f a i l u r e s of i m p r o p e r l y d e t a i l e d w a l l s . T h i s l e d t o the r equirement of h i g h 'k' f a c t o r s when u s i n g s t a t i c l a t e r a l l o a d a n a l y s i s of s t r u c t u r a l w a l l e d b u i l d i n g s . R esearch such as t h a t performed by P a u l a y 5 has shown t h a t i t i s p o s s i b l e t o o b t a i n d u c t i l i t y w i t h p r o p e r d e t a i l i n g of the w a l l s . The Canadian St a n d a r d s A s s o c i a t i o n b u i l d i n g code r e a l i z e s t h i s by making s p e c i a l p r o v i s i o n s t o a s c e r t a i n t h a t the w a l l s remain d u c t i l e . These p r o v i s i o n s attempt t o p r e c l u d e non-d u c t i l i t y by making shear f a i l u r e and o t h e r u n d e s i r a b l e b e h a v i o r more u n l i k e l y . However, w h i l e much of the r e s e a r c h has d e a l t w i t h the b e h a v i o r of the w a l l s , t h e i r c a p a c i t i e s and d u c t i l e c a p a b i l i t i e s , the d e s i g n of s t r u c t u r a l w a l l s under dynamic l o a d i n g has o f t e n been a somewhat i r r a t i o n a l p r o c e s s . The s t r u c t u r a l w a l l system i s a d u a l l o a d p a t h system and i t i s i m p o r t a n t t o a p p r e c i a t e the l a t e r a l l o a d c a r r y i n g methods of c o u p l e d s t r u c t u r a l w a l l s i f t h e i r a n a l y s i s i s t o be u n d e r s t o o d c o r r e c t l y . A l a r g e p r o p o r t i o n of the l a t e r a l l o a d a c t i n g upon a c o u p l e d shear w a l l i s t aken e s s e n t i a l l y as two independant c a n t i l e v e r s -would t a k e the l o a d - b y . f l e x u r a l b ending. Compounding the s i t u a t i o n , though, are the c o u p l i n g beams which ar e bent i n r e v e r s e c u r v a t u r e . Examining a f r e e b o d y diagram of the d i s t o r t e d l i n t e l beam shows t h a t the moments c a u s i n g the 3 r e v e r s e c u r v a t u r e must be accompanied by a shear t o m a i n t a i n e q u i l i b r i u m . T h i s shear produces an a x i a l f o r c e i n the w a l l s , t e n s i o n i n one and compression i n the o t h e r , i n such a manner t h a t i t c r e a t e s a c o u p l e which a i d s i n c o u n t e r a c t i n g the o v e r t u r n i n g moment caused by the l a t e r a l f o r c e s . The p r o p o r t i o n of l a t e r a l l o a d c a r r i e d by each method i s t h e r e f o r e d etermined by the member p r o p e r t i e s . As the l i n t e l s t r e n g t h and r i g i d i t y i n c r e a s e s , the r e s u l t i n g a x i a l c o u p l e i n c r e a s e s and the moment c a r r i e d by the w a l l s as i n d i v i d u a l c a n t i l e v e r s d e c r e a s e s . Making t h i s s i t u a t i o n even more complex i s the i n e l a s t i c b e h a v i o r of the elements of the w a l l . The c o u p l i n g beams, b e i n g s u b j e c t t o h i g h r e v e r s e c u r v a t u r e over t h e i r s h o r t l e n g t h , bear the brunt of t h i s b e h a v i o r and w i l l be e x p e c t e d t o pass w e l l i n t o t h e i r p o s t - e l a s t i c range d u r i n g a major s e i s m i c d i s t u r b a n c e . W h i l e a member s h o u l d not be e x p e c t e d t o c a r r y more than i t s u l t i m a t e moment i t i s up t o the d e s i g n e r t o a s c e r t a i n t h a t the e x c u r s i o n s i n t o the u l t i m a t e moment range w i l l not r e s u l t i n l a r g e s t r e n g t h d e g r a d a t i o n s . To a c h i e v e t h i s the d e s i g n e r must know the l e v e l s of d u c t i l i t y demand t h a t he can expect from the d e s i g n e a r t h q uake a c c e l e r a t i o n of the s t r u c t u r e . T h e r e f o r e the d e s i g n p r o c e d u r e must ta k e account of the i n e l a s t i c b e h a v i o r of the c o u p l i n g beams as they c a r r y u l t i m a t e moment, but s t i l l undergo a d e f l e c t i o n which i s c o m p a t i b l e w i t h the remainder of the s t r u c t u r e as i t a t t e m p t s - t o r e s i s t the earthquake f o r c e s i n a s i m i l a r manner. T h i s s h a r i n g of the method of f o r c e - c a r r y i n g i l l u s t r a t e s how the d e s i g n p r o c e s s must a r r i v e a t a proper r e l a t i o n s h i p between the moment c a p a c i t y of the w a l l s , and the c a p a c i t i e s , s t r e n g t h s and d u c t i l i t y demands of the members' w h i l e 4 a t the same time g i v i n g an i n d i c a t i o n of the d e f l e c t i o n s t o be e x p e c t e d . From the energy a b s o r p t i o n s t a n d p o i n t the system i s e x c e l l e n t as the l i n t e l beams a r e u s u a l l y the f i r s t t o undergo energy a b s o r b i n g i n e l a s t i c d e f o r m a t i o n w h i l e the w a l l s which a c t as the main l o a d c a r r y i n g p a t h a c t e l a s t i c a l l y . By t h e i r v e r y n a t u r e as low l o a d c a r r y i n g members under normal c i r c u m s t a n c e s and by v i r t u e of t h e i r e a s i l y r e p a i r a b l e l o c a t i o n , the c o u p l i n g beams p r o v i d e a good p l a c e f o r energy a b s o r p t i o n t o occur w i t h o u t r i s k i n g s e r i o u s damage t o the e n t i r e s t r u c t u r e . T h i s i n e l a s t i c a c t i o n of the c o u p l i n g beams does not u s u a l l y endanger the s t r u c t u r e as the w a l l s c a r r y l o a d i n both a x i a l c o u p l i n g and f l e x u r a l b e n d i n g , the mechanism f o r t h e i r c o l l a p s e can t h e r e f o r e o n l y o c c u r when both l o a d p a t h s a r e d e s t r o y e d . For t h i s t o occur a s e r i e s of h i n g e s must form, one i n e i t h e r end of the c o u p l i n g beams and one i n each of the two w a l l s . The e a r l y b e l i e f t h a t d u c t i l e frames were the i d e a l energy d i s s i p a t i n g system i s s l o w l y l o s i n g f a v o r . T h i s r e f l e c t s the growing b e l i e f t h a t i t i s no l o n g e r s u f f i c i e n t merely t o save the b u i l d i n g d u r i n g a s e i s m i c d i s t u r b a n c e o n l y t o have i t p u l l e d down s u b s e q u e n t l y due t o i r r e p a r a b l e damage. The d u c t i l e frame i s a l s o l o s i n g f a v o r as the d i s p l a c e m e n t s n e c e s s a r y f o r proper energy a b s o r p t i o n w i l l cause e x t e n s i v e damage t o the c o n t e n t s and a r c h i t e c t u r a l f i n i s h of the s t r u c t u r e which f r e q u e n t l y exceeds the c o s t of the frame. P a s t e a r t h q u a k e s have shown t h a t the d u c t i l e frame o f t e n a b sorbs energy i n d i s c r e t e l o c a t i o n s i n s t e a d of u n i f o r m l y throughout the s t r u c t u r e as a n t i c i p a t e d . When t h i s o c c u r s the damaged l o c a t i o n s f r e q u e n t l y undergo more 5 deformation than the designer would a n t i c i p a t e or d e s i r e . T h i s was the case with the E l Centro county o f f i c e b u i l d i n g i n the 1979 earthquake in which l a r g e deformations o c c u r r e d i n the base of some of the columns r e q u i r i n g subsequent d e m o l i t i o n of the b u i l d i n g . One advantage in the seismic design of s t r u c t u r a l w a l l s over d u c t i l e frames i s that the former w i l l f r e q u e n t l y behave more l i k e the mathmatical model than w i l l the l a t t e r . T h i s i s a r e s u l t of the frequent n e g l e c t of the e f f e c t s that p a r t i a l i n f i l l w a l l s and other 'cosmetic' b u i l d i n g components w i l l have on the behavior of frame s t r u c t u r e s . Some examples of t h i s were seen in the O l i v e View h o s p i t a l i n which some e f f e c t i v e column lengths had been reduced by a r c h i t e c t u r a l i n f i l l . T h i s has the e f f e c t of c o n c e n t r a t i n g any i n e l a s t i c a c t i o n i n t o a s h o r t e r s e c t i o n of the column as w e l l as i n c r e a s i n g the shear f o r c e i n the member. I t i s f a r l e s s l i k e l y that the behavior of a w a l l w i l l be a c c i d e n t a l l y m o d i f i e d . 1.2 Purpose of T h i s T h e s i s The a n a l y s i s of s t r u c t u r e s f o r the purpose of seismic design can be done with v a r i o u s l e v e l s of s o p h i s t i c a t i o n : ( i ) For small s t r u c t u r e s a q u a s i - s t a t i c a n a l y s i s using the e q u i v a l e n t f o r c e s d e f i n e d by a b u i l d i n g code i s an a p p r o p i a t e procedure, ( i i ) For medium s i z e s t r u c t u r e s - f o r example, r e s i d e n t i a l b u i l d i n g s i n the 10 to 25 s t o r y r a n g e — a n e l a s t i c modal a n a l y s i s based on a design spectrum i s g e n e r a l l y used. 6 The root-sum-square f o r c e s from t h i s a n a l y s i s are then d i v i d e d by the a v a i l a b l e s t r u c t u r e d u c t i l i t y a s s o c i a t e d w i t h the p a r t i c u l a r s t r u c t u r a l system, t o g i v e the y i e l d l e v e l f o r c e s f o r which the b u i l d i n g s h o u l d be d e s i g n e d , ( i i i ) For l a r g e r o r more complex s t r u c t u r e s , an i n e l a s t i c time s t e p a n a l y s i s based on an a p p r o p i a t e earthquake r e c o r d or r e c o r d s s h o u l d be a p p l i e d . The p r o c e d u r e d e s c r i b e d under ( i i ) above i s a p p l i c a b l e t o the case of frame s t r u c t u r e s , where the a v a i l a b l e d u c t i l i t y a s s o c i a t e d w i t h the s t r u c t u r a l system i s known and the y i e l d l e v e l moments are the d e s i r e d q u a n t i t i e s . In the case of r e s i d e n t i a l b u i l d i n g s c o n s i s t i n g of c o u p l e d s t r u c t u r a l w a l l s however, the pr o c e d u r e i s not r e a l l y a p p l i c a b l e . In these b u i l d i n g s the c o u p l i n g beams are g e n e r a l l y s l a b s or s h o r t l i n t e l s of minimum c r o s s - s e c t i o n a l d i m e n s i o n s — t y p i c a l l y 18 i n c h e s deep by 8 i n c h e s wide. I t i s not p o s s i b l e t o r e i n f o r c e such members i n the manner i n d i c a t e d by P a u l a y t o g i v e the optimum l e v e l s of y i e l d moment and d u c t i l i t y . I n s t e a d , one can r e i n f o r c e the members t o g i v e the maximum p o s s i b l e moment and then a n a l y s e the system t o see whether the d u c t i l i t y demand and shear c a p a c i t y of the members can be met. I f i t cannot, some change i n the s t r u c t u r a l l a y o u t i s r e q u i r e d . To r e p e a t : i n the case of frames, the d e s i r e d d u c t i l i t y l e v e l i s known, and the c o r r e s p o n d i n g y i e l d s t r e n g t h i s r e q u i r e d ; the l i n e a r e l a s t i c s p e c t r a l a n a l y s i s d e s c r i b e d under ( i i ) above g i v e s the d e s i r e d r e s u l t . In the case of c o u p l e d w a l l s the maximum a v a i l a b l e s t r e n g t h i s known, and the d u c t i l i t y demand i s t o be e v a l u a t e d t o t e s t the s t r u c t u r a l l a y o u t . The aim 7 of t h i s t h e s i s i s t o p r o v i d e a method of d o i n g t h i s f o r s m a l l e r s t r u c t u r e s , f o r which a f u l l s c a l e i n e l a s t i c a n a l y s i s i s not f e a s i b l e from an economic or d e s i g n time p o i n t of view . 1.3 E x a m i n a t i o n of S t r u c t u r a l A n a l y s i s Methods f o r S e i s m i c D esign B e f o r e d e s c r i b i n g the proposed method, i t i s w o r t h w h i l e t o p o i n t out the f a u l t s of the p r e s e n t methods of a n a l y s i n g c o u p l e d w a l l s . They f a l l i n t o t h r e e broad c a t a g o r i e s . The f i r s t of these i n v o l v e s a code s p e c i f i e d s t a t i c l a t e r a l l o a d a p p l i e d t o v a r i o u s models of the s t r u c t u r e i n c l u d i n g a ' l a m i n a r ' model i n which the p r o p e r t i e s of the l i n t e l s have been smeared throughout the h e i g h t of the s t r u c t u r e . T h i s s u f f e r s from not f u l l y r e f l e c t i n g the dynamic n a t u r e of l o a d and s t r u c t u r e . The second c a t e g o r y i s the e l a s t i c modal a n a l y s i s method which has been used many times i n p r a c t i c e f o r earthquake d e s i g n but has the d i s a d v a n t a g e t h a t i t does not r e f l e c t the c o n s i d e r a b l e e f f e c t s t h a t the i n e l a s t i c b e h a v i o r w i l l have on the the s t r u c t u r a l w a l l . T h i s method a l s o does not p r e d i c t the d u c t i l i t y demands of the w a l l e l e m e n t s . The f i n a l c a t e g o r y i s the time s t e p i n e l a s t i c method. I t s h a n d i c a p i s t h a t i t i s e x p e n s i v e t o use and would f r e q u e n t l y o n l y be a p p l i e d t o l a r g e r b u i l d i n g s i n c i r c u m s t a n c e s where an o u t s i d e c o n s u l t a n t i s brought i n f o r the earthquake a n a l y s i s . The advantages of i n e l a s t i c programs t o model the b e h a v i o r of b u i l d i n g s under earthquake l o a d s have been known f o r y e a r s . Most of the arguments put f o r w a r d i n t h e i r f a v o r e x t o l e the v i r t u e s of b e i n g a b l e t o d e t e r m i n e much b e t t e r the t r u e 8 performance of the s t r u c t u r e as w e l l as the d u c t i l t y demands of the members. While t h e s e p o i n t s a r e v a l i d , the programs t h a t have been used m o s t l y to. date t o model i n e l a s t i c p r o p e r t i e s are time s t e p programs. Though these programs can f r e q u e n t l y reproduce the e f f e c t of a g i v e n e a r t h q u a k e on a g i v e n s t r u c t u r e they do so a t a c o s t t h a t i s o f t e n p r o h i b i t i v e f o r many s t r u c t u r e s . There are two reason f o r t h i s : the f i r s t of which i s of g r e a t p r a c t i c a l s i g n i f i c a n c e -- t ime s t e p programs a r e not cheap t o r u n . There w i l l n a t u r a l l y be a n e c e s s i t y f o r s e v e r a l computer runs as the s t r u c t u r e i s a l t e r e d to i t e r a t e i n on the r e q u i r e d s t r e n g t h and s t i f f n e s s , and a l s o t o meet the demands of the a r c h i t e c t and owner. For a s u c c e s s f u l i n e l a s t i c a n a l y s i s of a s t r u c t u r e u t i l i z i n g time s t e p methods, the m a t h e m a t i c a l model of the s t r u c t u r e must undergo t e s t i n g w i t h a v a r i e t y of a p p r o p r i a t e e a rthquakes i f i t i s t o be d e s i g n e d p r o p e r l y . T e s t i n g w i t h a s e l e c t i o n of e a r t h q u a k e s i s the o n l y way t h a t time s t e p a n a l y s i s can r e f l e c t the u n c e r t a i n t y a s s o c i a t e d w i t h the m o t ions of a f u t u r e d i s t u r b a n c e . T h i s i t e m i t s e l f n e c e s s i t a t e s s e v e r a l r u n s . Time s t e p a n a l y s i s programs may a l s o r e q u i r e an i n i t i a l run on a modal a n a l y s i s type program i n o r d e r t h a t the f r e q u e n c i e s can be d e t e r m i n e d t o i n p u t t o parameters of the time s t e p damping. T h i s a l s o i n c r e a s e s the c o s t of program use. The o t h e r expense of time s t e p a n a l y s i s programs i s t h a t t h ey a r e somewhat removed from the realm of the average s t r u c t u r a l e n g i n e e r . The a v a i l a b i l i t y and i n p u t r e q u i r e m e n t s i m p l i e s t h a t they w i l l o n l y be c o n s i d e r e d i n somewhat s p e c i a l i z e d c o n s u l t i n g s i t u a t i o n s . T h i s a l s o i m p l i e s t h a t the 9 earthquake a n a l y s i s of the 'average' s t r u c t u r e w i l l be conducted u s i n g at be s t e l a s t i c modal a n a l y s i s or code s p e c i f i e d l a t e r a l l o a d s a p p l i e d t o an e l a s t i c s t r u c t u r e . W h i l e t h e s e e l a s t i c methods have much m e r i t i n t h e i r own r i g h t , l a r g e earthquake d i s t u r b a n c e s p r e s e n t a v i o l e n t l y n o n - s t a t i c l o a d on a s t r u c t u r e d u r i n g which v e r y few b u i l d i n g s can be e x p e c t e d to remain t o t a l l y i n t h e i r e l a s t i c range. F a i l u r e t o i n c l u d e these i n e l a s t i c a c t i o n s i n the d e s i g n a n a l y s i s procedure i s a s e r i o u s drawback when c o n s i d e r i n g the s t r u c t u r a l w a l l . E a r l y s t u d i e s of c o u p l e d shear w a l l s a t t e m p t e d t o model t h e i r p r o p e r t i e s by r e p l a c i n g the d i s c r e t e c o u p l i n g beams by c o n t i n u o u s l a m i n a e . The t e c h n i q u e was advanced t o take account of i n e l a s t i c b e h a v i o r but s t i l l had d i f f i c u l t y i n d e a l i n g w i t h w a l l s w i t h p r o p e r t i e s t h a t were not c o n s t a n t over the h e i g h t of the s t r u c t u r e . The method a l s o s u f f e r e d from the o b j e c t i o n t h a t i t was not one t h a t t a k e s account of the dynamic response of the s t r u c t u r e . A r t i c l e s d e m o n s t r a t i n g the method showed t h e i r l o a d i n g as a s t a t i c , o f t e n t r i a n g u l a r l o a d a c t i n g on the w a l l w i t h o n l y one d e f l e c t e d shape c o n s i d e r e d , t h a t b e i n g the one t h a t c o u l d be d e s c r i b e d as a p p r o x i m a t i n g the f i r s t mode of the w a l l . The o m i s s i o n , i n an earthquake a n a l y s i s method, of the dynamic i n t e r a c t i o n between the l o a d , s t r u c t u r e and response i s too g r e a t a s i m p l i f i c a t i o n t o be a c c e p t e d when methods e x i s t t h a t do ta k e account of the problem. I t has been f o r t u n a t e t h a t t h e i n c r e a s e i n e l e c t r o n i c c o m p u t a t i o n a l c a p a b i l i t i e s have i n c r e a s e d a t the same time t h a t our n e c e s s i t y f o r r a t i o n a l earthquake a n a l y s i s has i n c r e a s e d , for- w i t h o u t the computer the t a s k v e r g e s on the i m p o s s i b l e . T h i s 10 i s e s p e c i a l l y the case i f i n e l a s t i c and dynamic e f f e c t s are taken i n t o account d u r i n g the a n a l y s i s . D u r i n g the p r o g r e s s of r e s e a r c h on t h i s t h e s i s a computer program was d e v e l o p e d by m o d i f i c a t i o n of a program w r i t t e n d u r i n g some e a r l i e r r e s e a r c h on the M o d i f i e d S u b s t i t u t e S t r u c t u r e Method by Sumio Y o s h i d a 1 0 . In d e v e l o p i n g a computer program which c o u l d a p p l y the method t o a s t r u c t u r e , c o n s i d e r a b l e e f f o r t was a p p l i e d t o make i t one t h a t c o u l d be used by the p r a c t i s i n g e n g i n e e r f o r b u i l d i n g d e s i g n . The d a t a i n p u t , w h i l e p o s s i b l y v a r y i n g i n format from o t h e r e x i s t i n g s t a t i c a n a l y s i s programs, does not demand i n p u t t h a t i s g r e a t l y d i f f e r e n t e i t h e r i n type or amount from one of those programs. The e n g i n e e r who can a p p l y a s t a t i c l a t e r a l l o a d t o a computer model of a s t r u c t u r e w i l l not f i n d i t much of a t a s k t o determine the d u c t i l i t y r e q u i r e m e n t s and i n e l a s t i c d e f l e c t i o n s used i n t h i s method. A v e r y i m p o r t a n t advantage t h a t comes w i t h the i n c r e a s e d a v a i l a b i l i t y of computer a i d e d e a r t h q u a k e d e s i g n i s t h a t i t a l l o w s a w i d e r range i n the s i z e of s t r u c t u r e s t h a t can e c o n o m i c a l l y be c o n s i d e r e d r a t i o n a l l y f o r earthquake l o a d i n g s . As the c o s t of e x e c u t i n g the programs d e c r e a s e s , i t becomes p r a c t i c a l e c o n o m i c a l l y t o i n c l u d e i n the d e s i g n of the cheaper s t r u c t u r e a more complete c o n s i d e r a t i o n of i t s dynamic c h a r a c t e r i s t i c s than i s p e r m i t t e d w i t h the l a t e r a l l o a d method. Another i m p o r t a n t advance i n the f i e l d of earthquake e n g i n e e r i n g i s our i m p r o v i n g a b i l i t y t o p r e d i c t the e x p e c t e d i n t e n s i t i e s of ground d i s t u r b a n c e s a t a g i v e n l o c a t i o n . T h i s has r e s u l t e d from a c o m b i n a t i o n . of improved and more numerous measuring s t a t i o n s and b e t t e r c o m p u t a t i o n a l means t o i n t e r p r e t 11 the r e c o r d e d d a t a . At p r e s e n t the a b i l i l t y t o p r e d i c t e x a c t l y the time and motions of a d i s t u r b a n c e i s n o n e x i s t a n t . Yet i t i s p o s s i b l e f o r the s e i s m o l o g i s t t o make good e s t i m a t e s of both the maximum a c c e l e r a t i o n and s p e c t r a l c o n t e n t t h a t can be ex p e c t e d i n most l o c a t i o n s . Time s t e p a n a l y s i s makes use of t h i s i n f o r m a t i o n by f i n d i n g e a r t h q u a k e s which approximate the a n t i c i p a t e d spectrum, s c a l i n g these t o the p r e d i c t e d a c c e l e r a t i o n and a p p l y i n g them t o a computer model of the s t r u c t u r e . For e l a s t i c a n a l y s i s the problem of c h o o s i n g a p p r o p r i a t e e a r t h q u a k e s was c i r c u m v e n t e d by the use of modal a n a l y s i s based on the d e s i g n spectrum d i r e c t l y . I t was soon r e a l i z e d by those u s i n g b o t h schemes t h a t the modal method had s e v e r a l o t h e r advantages i n terms of s a v i n g s of computer time and ease of programming. Indeed, i f a computer program i s a v a i l a b l e f o r p e r f o r m i n g s t r u c t u r a l a n a l y s i s u s i n g the s t i f f n e s s m a t r i x method, and i f t h i s i s used on a system t h a t has an e i g e n v a l u e f i n d i n g r o u t i n e , then i t i s a f a i r l y s i m p l e problem t o combine the two t o produce a modal a n a l y s i s program. W h i l e the modal method has been q u i t e w i d e l y used f o r e l a s t i c a n a l y s i s , i n e l a s t i c dynamic a n a l y s i s has had t o r e l y on time s t e p a n a l y s i s programs which examine the s t a t e of the s t r u c t u r a l members a t d i s c r e t e i n t e r v a l s of the e x c i t a t i o n p e r i o d , t o determine s t r e n g t h and s t i f f n e s s d e g r a d a t i o n u s i n g i d e a l i z e d h y s t e r e s i s d i agrams. What the M o d i f i e d S u b s t i t u t e S t r u c t u r e Method does i s e x t e n d the modal method w i t h a l l i t s i n h e r e n t advantages, i n t o the i n e l a s t i c range. No n u m e r i c a l t e c h n i q u e i s e x a c t : t h e r e always remains a 12 t r a d e o f f between the c o m p l e x i t y , and t h e r e f o r e the e x e c u t i o n c o s t of the method, and the d e s i r e d a c c u r a c y of the answer. I t must be r e a l i z e d t h a t the i n p u t p r o p e r t i e s t o most s t r u c t u r a l and e s p e c i a l l y earthquake a n a l y s e s a re s u b j e c t t o v a r i a t i o n and e x p e r i m e n t a l e r r o r . I t makes l i t t l e sense t o become a g i t a t e d over d i f f e r e n c e s i n the second or t h i r d f i g u r e of a d i s p l a c e m e n t or d u c t i l i t y v a l u e when the i n p u t a c c e l e r a t i o n i s a t best a c c u r a t e i n o n l y i t s f i r s t f i g u r e . I t a l s o makes l i t t l e sense t o a c h i e v e t h i s e x t r a a c c u r a c y when i t r e q u i r e s an o r d e r of magnitude c o s t i n c r e a s e . What i s im p o r t a n t i n a n u m e r i c a l method to be a p p l i e d i n a d e s i g n s i t u a t i o n i s t h a t i t g i v e good, r e a s o n a b l e answers, and t h a t i t can be used t o p r e d i c t the d i r e c t i o n s e l e c t e d changes i n the s t r u c t u r a l or e x c i t a t i o n p r o p e r t i e s w i l l have upon those r e s u l t s . While i n some p l a c e s the p r e s e n t work makes comparisons w i t h those r e s u l t s o b t a i n e d from i n e l a s t i c time s t e p a n a l y s i s , i t i s done not i n the b e l i e f t h a t t hey p r e s e n t the i n d i s p u t a b l e t r u t h i n terms of the b e h a v i o r of a s t r u c t u r e under e a r t h q u a k e l o a d s , but r a t h e r t h a t the method i s p r e s e n t l y a c c e p t e d as one of the b e t t e r n u m e r i c a l a n a l y s i s t e c h n i q u e s t h a t can be a p p l i e d t o the problem. 13 1.4 Scope T h i s t h e s i s proceeds by d e s c r i b i n g the M o d i f i e d S u b s t i t u t e S t r u c t u r e Method, i t s development and l i m i t a t i o n s . I t then moves on t o d i s c u s s some of the improvements, developments and o b s e r v a t i o n s made w h i l e a t t e m p t i n g t o modify and a p p l y a computer program t o a n a l y z e s t r u c t u r a l w a l l s u s i n g the method. The t e s t i n g of the program f o r e l a s t i c modal a n a l y s i s i s then d i s c u s s e d , p a r t l y because t h i s p r o c e s s took f a r l o n g e r than e x p e c t e d and r e s u l t e d i n some unexpected changes b e i n g made t o the program. F o l l o w i n g t h i s i s perhaps the c h a p t e r of most s i g n i f i c a n c e and c oncern i n which the t e s t s of the program's i n e l a s t i c c a p a b i l i t i e s a r e r e l a t e d . T h i s i n v o l v e s a two s t e p d e m o n s t r a t i o n i n which i t i s shown t h a t , f i r s t l y , the d u c t i l i t y demand p a t t e r n s such as t h o s e r e p o r t e d by P a u l a y 4 can be p r e d i c t e d ; and, s e c o n d l y , t h a t the n u m e r i c a l v a l u e s c o n s i s t a n t w i t h time s t e p a n a l y s i s methods can be r eproduced s a t i s f a c t o r l y . The a p p l i c a t i o n of the method t o the a n a l y s i s of a s i x t e e n - s t o r y s t r u c t u r a l w a l l i n a d e s i g n example a c t s as a f u r t h e r t e s t of the method and i s r e l a t e d i n a c h a p t e r of i t s own as a r e the c o n c l u s i o n s which f o l l o w . I t i s hoped t h a t i n r e a d i n g the pages t h a t f o l l o w , r e s e a r c h e r s and e n g i n e e r s w i l l be a b l e t o see a d e s i g n method t h a t can be a p p l i e d t o s t r u c t u r e s t a k i n g account of i n e l a s t i c b e h a v i o r and the dynamic n a t u r e of b oth e a r thquake and s t r u c t u r e , i n a r a t i o n a l , s a f e , y e t e a s i l y a p p l i e d manner. 14 CHAPTER 2 INTRODUCTION TO THE SUBSTITUTE STRUCTURE METHOD AND  THE MODIFIED SUBSTITUTE STRUCTURE METHOD. The m o d i f i e d s u b s t i t u t e s t r u c t u r e method i s a n u m e r i c a l method c l o s e l y a k i n t o modal a n a l y s i s but e x t e n d i n g t h a t t e c h n i q u e i n t o the i n e l a s t i c range. The method and i t s developments a re d i s c u s s e d here so t h a t the reader can g a i n a b e t t e r i n s i g h t i n t o the a p p l i c a t i o n t o s t r u c t u r a l w a l l s . 2.1 The S u b s t i t u t e S t r u c t u r e Method. As can be expected from i t s t i t l e , the m o d i f i e d s u b s t i t u t e s t r u c t u r e method was de v e l o p e d from a d a p t a t i o n s made t o the s u b s t i t u t e s t r u c t u r e method and an e x a m i n a t i o n of t h i s e a r l i e r method can g i v e i n s i g h t t o the l a t e r one. The s u b s t i t u t e s t r u c t u r e method was proposed by S h i b a t a and Soz e n 8 as" a d e s i g n procedure f o r r e i n f o r c e d c o n c r e t e s t r u c t u r e s which c o u l d be used t o e s t a b l i s h t he y i e l d f o r c e s t h a t s h o u l d be p r o v i d e d f o r i n the d e s i g n , assuming t h a t the i n i t i a l s t i f f n e s s and a v a i l a b l e d u c t i l i t y a r e known. T h i s i s g e n e r a l l y the case by the time a s e i s m i c d e s i g n i s approached. The method was developed as a means of e s t a b l i s h i n g the member p r o p e r t i e s n e c e s s a r y t o 15 a c h i e v e an a c c e p t a b l e s t r u c t u r a l response under earthquake l o a d i n g . As w i t h any t e c h n i q u e the method i s s u b j e c t t o some r e s t r i c t i o n s which d e f i n e the type of problem t o which i t i s a p p l i c a b l e . For the s u b s t i t u t e s t r u c t u r e method, these a re as f o l l o w s : (1) The system must be c a p a b l e of a n a l y s i s i n one v e r t i c a l p l a n e . T h i s l i m i t s the method t o p l a n e frame a n a l y s i s and e l i m i n a t e s problems which i n v o l v e t o r s i o n and b i a x i a l b e n d i n g . (2) The s t r u c t u r e must be one i n which a b r u p t changes i n geometry or mass do not occur over the h e i g h t . T h i s l i m i t s the a n a l y s i s t o r e g u l a r - s h a p e d s t r u c t u r e s . A l t h o u g h i t i s p o s s i b l e t h a t some s t r u c t u r e s o u t s i d e t h i s c l a s s c o u l d be a n a l y z e d w i t h s u c c e s s , they a r e p r o b a b l y the e x c e p t i o n and a l l s t r u c t u r e s not of t h i s c l a s s s h o u l d be e l i m i n a t e d . Systems f a i l i n g t o meet t h i s r e s t r i c t i o n f r e q u e n t l y cause problems i n dynamic a n a l y s i s r e g a r d l e s s of the t e c h n i q u e i n use. (3) Columns, w a l l s r e p r e s e n t e d as columns, and beams may be d e s i g n e d w i t h d i f f e r e n t damage r a t i o s but a l l beams i n a g i v e n bay or the columns on a g i v e n a x i s s h o u l d have the same v a l u e . J u s t why t h i s i s a c r i t e r i o n f o r the s u b s t i t u t e s t r u c t u r e method was not e x p l a i n e d i n the o r i g i n a l papers c o n c e r n i n g the method, but i t may w e l l have been imposed s i m p l y because o n l y s t r u c t u r e s of t h i s type were t e s t e d by the o r i g i n a l a u t h o r s . As w i l l be shown l a t e r t h i s r e s t r i c t i o n i s not n e c e s s a r y i n the m o d i f i e d s u b s t i t u t e s t r u c t u r e method. 16 (4) A l l s t r u c t u r a l elements and j o i n t s must be r e i n f o r c e d t o a v o i d s i g n i f i c a n t s t r e n g t h decay as a r e s u l t of re p e a t e d r e v e r s a l s of the a n t i c i p a t e d i n e l a s t i c d i s p l a c e m e n t s . I t i s assumed i n the method t h a t the s t i f f n e s s of the members i n v o l v e d w i l l be reduced when they y i e l d and s t i f f n e s s l o s s e s are c a l c u l a t e d on the b a s i s of g i v e n 'damage r a t i o s ' . What the method does not a l l o w f o r i s a f a i l u r e of the member b e f o r e i t reaches the s p e c i f i e d d u c t i l i t y ; the r e s p o n s i b i l i t y f o r s e l e c t i n g t h i s d u c t i l i t y l i e s w i t h the d e s i g n e r . I t s h o u l d a l s o be noted t h a t i t i s presumed t h a t members do not f a i l i n shear or b u c k l i n g b e f o r e r e a c h i n g the d e s i r e d f l e x u r a l l o a d . T h i s n e c e s s a r y c h a r a c t e r i s t i c of any a s e i s m i c s t r u c t u r e i s a g a i n the r e s p o n s i b i l i t y of the d e s i g n e r . (5) The n o n - s t r u c t u r a l elements must not i n t e r f e r e s i g n i f i c a n t l y w i t h the dynamic response of the s t r u c t u r e . T h i s i s an o b v i o u s r e s t r i c t i o n a p p l y i n g t o any method of dynamic a n a l y s i s i n which s p e c i a l elements have not been i n c l u d e d i n the model t o account f o r items such as i n f i l l w a l l s . For many s i m p l e s t r u c t u r e s t h e s e f i v e r e s t r i c t i o n s a r e not a s e r i o u s drawback and the method p r o v i d e s an i n e x p e n s i v e d e s i g n a i d . The s t e p s i n v o l v e d i n the method w i l l be d e s c r i b e d b r i e f l y here and are a l s o shown i n the f l o w c h a r t of f i g u r e 2.1. B e f o r e s t a r t i n g t o use t h i s method i t i s assumed t h a t the d e s i g n e r , t h r o u g h e v a l u a t i o n of the wind and g r a v i t y l o a d s , and a i d e d by e x p e r i e n c e , has a l r e a d y d e t e r m i n e d the g r o s s s i z e s of the c o n c r e t e members i n v o l v e d . I t i s a l s o assumed t h a t a smoothed response a c c e l e r a t i o n spectrum has been o b t a i n e d f o r the d e s i g n 17 ea r t h q u a k e , and t h a t the d e s i g n e r has chosen t o l e r a b l e 'damage r a t i o s ' f o r the members. The f i r s t s t e p i s then the e v a l u a t i o n of the s t i f f n e s s of the members on the b a s i s of t h e i r e x p e c t e d 'damage r a t i o s ' ; t h i s i s done t h r o u g h the f o l l o w i n g f o r m u l a : (EZ\l = (2.1) where ( E I ) ^ i s the s t i f f n e s s of the element i n the r e a l s t r u c t u r e ( E I ) c ^ i s the s t i f f n e s s of the element i n the s u b s t i t u t e s t r u c t u r e ^JL^ i s the damage r a t i o of the element. The c o n c e p t of 'damage r a t i o ' i s c e n t r a l t o the a p p l i c a t i o n of the method: i t i s comparable t o d u c t i l i t y ; but w h i l e c u r v a t u r e d u c t i l i t y i s the r a t i o of the c u r v a t u r e of the member under the a p p l i e d moment t o the c u r v a t u r e a t y i e l d moment, the damage r a t i o i s a number d e s i g n e d s p e c i f i c a l l y t o g i v e the e q u i v a l e n t l i n e a r member s t i f f n e s s , which may be used as though the moment were l i n e a r l y r e l a t e d t o c u r v a t u r e from i n i t i a l t o f i n a l l o a d . The damage r a t i o g i v e s a s t i f f n e s s by form u l a 2.1 which i m p l i e s t h a t under e q u a l and o p p o s i t e end moments an end r o t a t i o n of ^ J L ^ w o u l d be a c h i e v e d where i s the r o t a t i o n a t the onset of y i e l d i n g . Under these c o n d i t i o n s , where the end moments of the beam under c o n s i d e r a t i o n a r e e q u a l , and where the mome n t - r o t a t i o n curve f o r the r e a l beam i s t r u l y e l a s t o - p l a s t i c then the numeric v a l u e s of damage r a t i o and c u r v a t u r e d u c t i l i t y f o r the member w i l l be e q u a l . W h i l e not g i v i n g any r e a l 18 i n d i c a t i o n of what v a l u e s s h o u l d be used i n g e n e r a l f o r the damage r a t i o s , S h i b a t a and Sozen i n t h e i r a n a l y s i s used a v a l u e of 6 f o r the beams and 1 f o r the columns f o r those s t r u c t u r e s w i t h f l e x i b l e beams. Knowing these ' s u b s t i t u t e ' s t i f f n e s s e s and o t h e r s t r u c t u r e i n f o r m a t i o n such as j o i n t l o c a t i o n s and member l e n g t h s , the s t r u c t u r e s t i f f n e s s m a t r i x i s c o n s t r u c t e d . A mass m a t r i x w i t h the masses c o n c e n t r a t e d a t the j o i n t s , which l e a d s t o a d i a g i o n a l mass m a t r i x and d y n a m i c a l l y uncouples the response e q u a t i o n s , must a l s o be c o n s t r u c t e d . From mass and s t i f f n e s s m a t r i c e s , the modal f r e q u e n c i e s and mode shapes a re determined as u s u a l from the f o l l o w i n g e q u a t i o n : ([Kl - CO* M l *0 (2.2) where (K) i s the s t r u c t u r e s t i f f n e s s m a t r i x [Tnlis the mass m a t r i x and k> i s the a n g u l a r f r e q u e n c y . W i t h the a n g u l a r f r e q u e n c i e s e v a l u a t e d from e q u a t i o n 2.2 the v a l u e of the s p e c t r a l a c c e l e r a t i o n can be dete r m i n e d from the response spectrum f o r the chosen e a r t h q u a k e . In the manner of s t a n d a r d modal a n a l y s i s the m a t r i x of a p p l i e d s e i s m i c f o r c e s (F*) i s now c a l c u l a t e d f o r each mode by use of the f o l l o w i n g f o r m u l a : (F 1)- (A') r 5! t*>3 (2 . 3 ) uhere ( A 1 ) i s the r t h mode shape v e c t o r and s u p e r s c r i p t T 19 denotes t r a n s p o s e . ( I ) i s the i d e n t i t y m a t r i x and ( S p J i s the s p e c t r a l a c c e l e r a t i o n f o r the r t h mode computed from the z e r o damped d e s i g n spectrum u s i n g the n a t u r a l frequency of t h a t mode. In the above f o r m u l a the e x p r e s s i o n i n the c u r l e d b r a c k e t s i s f r e q u e n t l y c a l l e d the modal p a r t i c i p a t i o n f a c t o r and i s c a l c u l a t e d s e p a r a t e l y . At t h i s s t a g e i n the pro c e d u r e the f o r c e s on the s t r u c t u r e have been c a l c u l a t e d and i t i s now n e c e s s a r y t o compute the r e s u l t i n g d i s p l a c e m e n t s ; t h e s e a re c a l c u l a t e d u s i n g the s t a n d a r d s t i f f n e s s method. From the member f o r c e s a r i s i n g from the s e i s m i c l o a d s , i n p a r t i c u l a r the member end bending moments, the smeared damping r a t i o i s computed f o r each mode. The damping f a c t o r f o r the i n d i v i d u a l members i s c a l c u l a t e d f i r s t u s i n g a formu l a from l a b o r a t o r y t e s t s by Gulkan and Sozen p u b l i s h e d i n 1974, which f o l l o w s : %i = o.oa. + o.a f\ - i ] ( 2 # 4 ) where i s the s u b s t i t u t e damping f a c t o r , a v a l u e of v i s c o u s damping t o r e p r e s e n t the h y s t e r e t i c energy d i s s i p a t e d by the member. The f o r m u l a was de v e l o p e d t o a d j u s t the a n a l y t i c r e s u l t s of one-s t o r y frames a n a l y z e d by means of a l i n e a r spectrum t o match more c l o s e l y the observ e d e x p e r i m e n t a l b e h a v i o r of the frames. A s i n g l e damping v a l u e i s r e q u i r e d f o r each mode so t h a t the damping r a t i o s f o r the i n d i v i d u a l members must be combined 2 0 to form a composite value f o r the s t r u c t u r e . T h i s 'smearing'of the s t r u c t u r e damping i s based on the f l e x u r a l energy of deformation of the members, computed by the f o l l o w i n g formula: P*. L i _ [( f * (Wj - rtli rt*] < 2 . 5 , i n which P* i s the energy of deformation f o r element i and M^ -j are the moments at the ends of s u b s t i t u t e frame element i , f o r the r t h ' mode. Using t h i s formula f o r the f l e x u r a l energy of deformation of the i n d i v i d u a l members, the smeared damping r a t i o i s expressed as: ( 2 . 6 ) L y i s the r a t i o of c r i t i c a l damping f o r the r mode. T h i s procedure gi v e s unique damping r a t i o s f o r each mode. The smeared damping r a t i o r e c e i v e s i t s major c o n t r i b u t i o n s from those members with the l a r g e s t element damping r a t i o s and those members with the l a r g e s t bending moments, two groups which do not n e c e s s a r i l y c o i n c i d e . With the damping known f o r each mode the s o l u t i o n is' r e c a l c u l a t e d . As no r e v i s i o n has been made to the damage r a t i o s the s t r u c t u r e s t i f f n e s s matrix remains the same, as does the make-up of the mass matrix. With damping values i n the e i g h t to f i f t e e n percent range common to concrete frames, the mode shapes and f r e q u e n c i e s do not change and t h e r e f o r e do not need 21 r e c a l c u l a t i n g . What does change, however, i s the a c c e l e r a t i o n f o r c e c a l c u l a t e d from the response spectrum. Hence form u l a 2.3 must be r e - e v a l u a t e d , p r o d u c i n g member f o r c e s which d i f f e r from the i n i t i a l v a l u e s f o r the undamped s t r u c t u r e . As o n l y one ' i t e r a t i o n ' i s performed i t i s unnecessary t o r e c a l c u l a t e damping r a t i o s from the new f o r c e s . The member f o r c e s which have been c a l c u l a t e d f o r each mode are now combined i n the u s u a l manner by the Root-Sum-Square (RSS) method, w i t h a m o d i f i c a t i o n suggested by by S h i b a t a and Sozen: they m u l t i p l y a l l the f o r c e s by a common f a c t o r which w i l l i n c r e a s e them i f the magnitude of the two l a r g e s t c o n t r i b u t o r s a r e s i m i l a r . T h i s r e f l e c t s t he h i g h e r p r o b a b i l i t y of c o i n c i d e n c e of the maximum modal f o r c e s i n any two modes compared t o t h e i r p r o b a b l e c o i n c i d e n c e i n s e v e r a l modes. The m u l t i p l y i n g f a c t o r i s d e t e r m i n e d u s i n g the base shear of the s t r u c t u r e i n the f o l l o w i n g f o r m u l a : (^-(wi V r ; s v 4 V , k dVR<-<_ where: (F^ ) = .the i t h d e s i g n f o r c e (F i P i S S) = the Root-Sum-Square f o r c e Xy&= t n e R S S base shear Vftft1= the maximum v a l u e of the sum of the a b s o l u t e v a l u e s of any two,base s h e a r s . T h i s f a c t o r w i l l i n c r e a s e a l l the d e s i g n f o r c e s by s l i g h t l y o v e r twenty p e r c e n t i n the ca s e s where o n l y two modes a r e a n a l y z e d and they have e q u a l base s h e a r s . Under any o t h e r c o n d i t i o n i t w i l l i n c r e a s e the RSS f o r c e s by between z e r o and 22 twenty p e r c e n t . The f i n a l s tep in the s u b s t i t u t e s t r u c t u r e method i s to i n c r e a s e the des i gn moments p e r t a i n i n g to the columns by twenty percent to prevent the u n d e s i r a b l e r e s u l t s of p l a s t i c h inges in these members. Thus the f i n a l aim of the s u b s t i t u t e s t r u c t u r e method i s a c h i e v e d : the des i gn f o r c e s fo r se i sm ic l oad ing are produced. 2 ; 2 The M o d i f i e d S u b s t i t u t e S t r u c t u r e Method. A l though the s u b s t i t u t e s t r u c t u r e method was intended e x p l i c i t l y as a des ign method and not an a n a l y s i s method, the m o d i f i e d s u b s t i t u t e s t r u c t u r e method was deve loped for the a n a l y s i s of e x i s t i n g r e i n f o r c e d c o n c r e t e b u i l d i n g s . T h i s was done to p r e d i c t the extent and l o c a t i o n of damage fo r ' r e t r o f i t ' purposes . In t h i s method the input data d i f f e r s from that of the s u b s t i t u t e s t r u c t u r e method in that the y i e l d moments, which presumably would be known f o r the members of an e x i s t i n g s t r u c t u r e , are read in as pa r t of the input d a t a , toge ther with i n i t i a l s t i f f n e s s e s . The damage r a t i o s are the sought fo r q u a n t i t i e s . Dur ing the e x e c u t i o n of the method the members are not a l l owed to c a r r y moments which exceed the s p e c i f i e d y i e l d moments. Much the same procedure i s used as in the s u b s t i t u t e s t r u c t u r e method, but the techn ique i s an i t e r a t i v e one. The s t r u c t u r e s t i f f n e s s matr ix i s set up in the same manner u s i ng the damage r a t i o s to modify the member s t i f f n e s s e s ; though these may be set to u n i t y fo r the f i r s t i t e r a t i o n . An 23 a l t e r n a t i v e procedure i n v o l v e s the d e s i g n e r e s t i m a t i n g the damage r a t i o s f o r the members p r i o r t o the a n a l y s i s ; a l t h o u g h t h i s does not a f f e c t the f i n a l damage r a t i o s produced i t w i l l o f t e n reduce the number of i t e r a t i o n s t h a t a re performed b e f o r e convergence i s a c h i e v e d . E i g e n v a l u e s and e i g e n v e c t o r s a re then c a l c u l a t e d t o f i n d n a t u r a l f r e q u e n c i e s and mode shapes as b e f o r e . D u r i n g the f i r s t t r i a l the smeared damping v a l u e s a c c o u n t i n g f o r h y s t e r e t i c energy l o s s a re unknown, so the member f o r c e s a re c a l c u l a t e d u s i n g ' a p p r o p r i a t e ' damping v a l u e s which can be s p e c i f i e d by the program u s e r , i n s t e a d of a c a l c u l a t e d v a l u e . Subsequent i t e r a t i o n s use the same procedure as the s u b s t i t u t e s t r u c t u r e method t o c a l c u l a t e the damping r a t i o s . Knowing the damping r a t i o s , r e v i s e d f o r c e s and d i s p l a c e m e n t s a re computed, as w e l l as root-sum-square f o r c e s . Those members whose RSS moments exceed y i e l d have t h e i r damage r a t i o s m o d i f i e d a c c o r d i n g t o the f o l l o w i n g f o r m u l a : M„ >A <2-8> M 8 where: j L L w l i s the damage r a t i o f o r the n-1 i t e r a t i o n / J L n i s the damage r a t i o f o r the nth i t e r a t i o n M n i s the l a r g e r RSS end moment from the n t h i t e r a t i o n M u i s the y i e l d moment f o r the member. The l i m i t of u n i t y i s s e t s i n c e those members t h a t have not y i e l d e d c l e a r l y s t i l l have the i n i t i a l s t i f f n e s s . The f i n a l two s t e p s from the s u b s t i t u t e s t r u c t u r e method are o m i t t e d : t h e r e i s no i n c r e a s e i n the RSS f o r c e s t o account f o r c o n c i d e n c e of modes 24 and the moments i n the columns a r e not i n c r e a s e d by twenty p e r c e n t . The e l i m i n a t i o n of these two s t e p s r e f l e c t s the d i f f e r e n c e i n p h i l o s o p h y when the pr o c e d u r e i s used f o r a n a l y s i s r a t h e r than d e s i g n . W i t h the new damage r a t i o s , and the smeared damping r a t i o s from the . p r e v i o u s i t e r a t i o n , a n o t h e r i t e r a t i o n i s performed, commencing w i t h the c a l c u l a t i o n of a new s t i f f n e s s m a t r i x and f i n i s h i n g w i t h a f u r t h e r r e f i n e d s e t of damage r a t i o s . When a l l the member f o r c e s a re e i t h e r below o r w i t h i n a t o l e r a b l e l i m i t of t h e i r y i e l d v a l u e , the c y c l i n g i s h a l t e d . At t h i s s tage the damage r a t i o s t h a t have been d e t e r m i n e d by i t e r a t i o n a r e p r i n t e d . A d i a g r a m a t i c d e m o n s t r a t i o n of the program s t e p s can be seen i n f l o w c h a r t form i n f i g u r e 2.2. A l t h o u g h the program ends w i t h the p r i n t i n g of the c a l c u l a t e d damage r a t i o s , the f i n a l s t e p r e q u i r e d i s an i n t e r p r e t a t i o n of the o u t p u t . In the r e t r o f i t p r o c e d u r e f o r which t h i s method was o r i g i n a l l y i n t e n d e d , t h i s i n v o l v e s the e n g i n e e r ' s d e t e r m i n a t i o n of the a c c e p t a b i l i t y of t h e s e r a t i o s i n . r e l a t i o n t o the d e t a i l i n g of the s t r u c t u r e under a n a l y s i s . A l t h o u g h most of the r e s t r i c t i o n s which a p p l y a l s o t o the o r i g i n a l s u b s t i t u t e s t r u c t u r e method a p p l y t o the m o d i f i e d method, t h e r e a re some o t h e r s i m p l i f i c a t i o n s which are a c c e p t e d i n most computer a n a l y s i s of s t r u c t u r e s . Beams and columns are m o d e l l e d as l i n e members, the P - d e l t a e f f e c t i s i g n o r e d and f o r purposes of d i a g o n a l i z i n g the mass m a t r i x , the s t r u c t u r e mass i s assumed c o n c e n t r a t e d a t the nodes. Only one mass per f l o o r seems t o be n e c e s s a r y or d e s i r a b l e . Members i n v o l v e d i n the a n a l y s i s s h o u l d be symmetric as the damage r a t i o s a re based o n l y on the 25 l a r g e s t r o o t mean square moment on the member concerned and no d i f f e r e n t i a t i o n i s made between p o s i t i v e and n e g a t i v e bending moment. Changing a x i a l and shear f o r c e s are not c o n s i d e r e d i n d e t e r m i n i n g the y i e l d s t a t e of the members. Account i s taken of a x i a l s h o r t e n i n g g e n e r a t e d from the l a t e r a l earthquake l o a d s , but the s t a t i c f o r c e s t h a t would be g e n e r a t e d by the dead weight or o t h e r g r a v i t y l o a d s a r e not c o n s i d e r e d e i t h e r i n the d e t e r m i n a t i o n of the damage r a t i o s or of the r o o t mean square f o r c e s . One of the c h i e f advantages of both the s u b s t i t u t e and the m o d i f i e d s u b s t i t u t e s t r u c t u r e methods over time s t e p a n a l y s i s i s , as a l r e a d y mentioned, the use of a smoothed response spectrum. W h i l e d i s c u s s i n g r e s t r i c t i o n s on the methods i t i s perhaps w o r t h w h i l e t o d i s c u s s the r e s t r i c t i o n s t h a t a r e and a r e not p l a c e d upon t h i s spectrum. The use of a l i n e a r spectrum i s un n e c e s s a r y . S h i b a t a and Sozen" g i v e as one of t h e i r r e q u i r e m e n t s f o r the s u b s t i t u t e s t r u c t u r e method t h a t any i n c r e a s e i n p e r i o d r e s u l t s i n a de c r e a s e i n the s p e c t r a l a c c e l e r a t i o n . I n t h e i r t h r e e t e s t s t r u c t u r e s t h i s i s the case f o r a t l e a s t the fundamental mode. The m o d i f i e d s u b s t i t u t e s t r u c t u r e method removes any r e s t r i c t i o n s imposed by r e q u i r e m e n t s of the spectrum, t o a l a r g e e x t e n t , t h r o u g h i t s i t e r a t i v e p r o c e d u r e . In Y o s h i d a ' s t h e s i s a spectrum i n v o l v i n g f i f t y i n c r e m e n t s was used i n t a b u l a r form f o r some of the r u n s . A l t h o u g h i t was found t h a t the damage r a t i o s d i d not converge w i t h o u t some o s c i l l a t i o n and up t o a 100 p e r c e n t i n c r e a s e i n the number of i t e r a t i o n s was r e q u i r e d , a s u c c e s s f u l convergence was found i n the t r i a l s . These t e s t s , 26 w h i l e not showing t h a t b e t t e r r e s u l t s c o u l d be o b t a i n e d by u s i n g a n o n - l i n e a r response spectrum, d i d prove t h a t such a spectrum was not an impediment t o convergence of the damage r a t i o s . To d e t e r m i n e the a p p l i c a b i l i t y of t h e i r methods S h i b a t a and Sozen i n t h e i r paper, and Y o s h i d a i n h i s t h e s i s , used time s t e p dynamic a n a l y s i s programs on the s t r u c t u r e s f o r which the methods had been used. S h i b a t a and Sozen, when t e s t i n g the s u b s t i t u t e s t r u c t u r e method used t h r e e one-bay t e s t frames w i t h a h e i g h t r a n g i n g from t h r e e t o ten s t o r i e s . T h e i r method of t e s t i n g was t o f i n d the d e s i g n f o r c e s u s i n g the s u b s t i t u t e s t r u c t u r e method, then t o d e s i g n the frames on the b a s i s of th e s e f o r c e s . The frames were then a n a l y s e d u s i n g the time s t e p a n a l y s i s program SAKE and a comparison of the damage r a t i o s so o b t a i n e d w i t h the i n i t i a l l y s p e c i f i e d v a l u e s was made. The r e s u l t s were f a v o u r a b l e f o r a l l t h r e e frames where the d e s i g n f o r c e s had been c a l c u l a t e d on the b a s i s of a damage r a t i o of s i x f o r the beams and one f o r the columns. For t h e ' t e n - s t o r y frame, w h i l e the column v a l u e s showed some s c a t t e r , w i t h o n l y t h r e e of the t en s t o r i e s p r e d i c t e d c o n s e r v a t i v e l y , the beams had an average damage r a t i o of 5.5 and were a l l c o n s e r v a t i v e . The f i v e -s t o r y frame had o n l y one damage r a t i o l a r g e r than u n i t y i n the columns, w h i l e the beams averaged a damage r a t i o of 4.6, and a l l were below the d e s i g n v a l u e of 6, which was t h e r e f o r e c o n s e r v a t i v e . The t h r e e - s t o r y frame produced the b e s t r e s u l t s w i t h a l l average damage r a t i o s found i n the time s t e p a n a l y s i s b e i n g c l o s e t o but below the v a l u e s chosen when d o i n g the s u b s t i t u t e s t r u c t u r e a n a l y s i s . As was ex p e c t e d i n u s i n g a d e s i g n spectrum t h a t comprised 27 f o u r e a r t h q u a k e s ( i n the case of S h i b a t a and Sozen's d e s i g n spectrum 'A': E l C e n t r o E.W., E l C e n t r o N.S., Kern County S.69E., and Kern County N.21E.), some r e c o r d s produced damage r a t i o s and d i s p l a c e m e n t s t h a t were c o n s i d e r a b l y above the average w h i l e o t h e r s were below. To d e s i g n a s t r u c t u r e so t h a t damage r a t i o s s h o u l d be below the s p e c i f i e d v a l u e s f o r s p e c t r a c o r r e s p o n d i n g t o a l l e a r t h q u a k e s s c a l e d t o a g i v e n a c c e l e r a t i o n would produce an o v e r l y c o n s e r v a t i v e d e s i g n . As a t e s t of the M o d i f i e d S u b s t i t u t e S t r u c t u r e Method, Yo s h i d a t e s t e d f o u r s t r u c t u r e s under the same spectrum 'A' as t h a t used by S h i b a t a and Sozen. These s t r u c t u r e s o f f e r e d a v a r i e t y of s t r u c t u r a l c o n f i g u r a t i o n s c o r r e s p o n d i n g t o s m a l l and medium s t r u c t u r e s . They were: a t w o - s t o r y , two-bay frame; a t h r e e - s t o r y , t h r e e - b a y frame; a s i x - s t o r y , one-bay frame; and a s i x - s t o r y , t h r e e - b a y frame. For comparison purposes the damage r a t i o s were c a l c u l a t e d by time s t e p a n a l y s i s u s i n g the program SAKE, w i t h the r e c o r d s of the f o u r i n d i v i d u a l e a r t h q u a k e s t h a t had gone i n t o the spectrum. The comparison showed v e r y f a v o u r a b l e r e s u l t s i n a l l c a s e s . The CPU time r e d u c t i o n f o r the m o d i f i e d s u b s t i t u t e s t r u c t u r e method ranged from over one hundred seconds i n the case of the l a r g e s t s t r u c t u r e (120 sec f o r the time s t e p a n a l y s i s as opposed t o 2.3 sec f o r the proposed method) 'to e l e v e n seconds f o r the s m a l l e s t s t r u c t u r e (12.1 sec t o 0.91 s e c ) . To summarize Y o s h i d a ' s r e s u l t s they can be regarded as g i v i n g an e x c e l l e n t i n d i c a t i o n t o a d e s i g n e r of ' t r o u b l e s p o t s ' i n h i s s t r u c t u r e . The t h r e e - b a y , t h r e e - s t o r y s t r u c t u r e showed the b e s t r e s u l t s w i t h a l l members except t h r e e w i t h i n f i f t e e n 28 p e r c e n t of what would be p r e d i c t e d by time s t e p a n a l y s i s . The t h r e e members o u t s i d e t h i s group were a l l columns on the top s t o r y ; t h e i r damage r a t i o was p r e d i c t e d c o n s e r v a t i v e l y by the method. The two-bay, t w o - s t o r y s t r u c t u r e showed e x c e s s i v e y i e l d i n g i n the bottom s t o r y columns i n both a n a l y s i s p r o c e d u r e s but the method d i d not p r e d i c t as much y i e l d i n g here as d i d time s t e p a n a l y s i s . A l l o t h e r members i n the s t r u c t u r e were w i t h i n f i f t e e n p e r c e n t or c o n s e r v a t i v e l y p r e d i c t e d . The t h r e e - b a y , s i x -s t o r y frame showed a l l members w i t h i n t h i r t y p e r c e n t of the t r u e v a l u e or c o n s e r v a t i v e l y p r e d i c t e d , over h a l f of the members were w e l l w i t h i n f i f t e e n p e r c e n t of the average f o r the n o n - l i n e a r a n a l y s i s . The s i x - s t o r y , one-bay frame, when a n a l y z e d by the m o d i f i e d s u b s t i t u t e s t r u c t u r e method, showed n u m e r i c a l r e s u l t s which p r e d i c t e d e x c e s s i v e y i e l d i n g throughout the s t r u c t u r e , but d i d not produce a c l o s e n u m e r i c a l f o r e c a s t of the damage r a t i o s . I t was c o n c l u d e d t h a t the method was a poor numeric p r e d i c t o r i n ca s e s where t h e r e was e x t e n s i v e and e x c e s s i v e y i e l d i n g of the members thro u g h o u t the s t r u c t u r e . I t s h o u l d be noted t h a t i n the t e s t s t r u c t u r e s , the columns of one l i n e or the beams of one bay sometimes d i d not have e q u a l c a p a c i t i e s . A l t h o u g h the s u b s t i t u t e s t r u c t u r e method was r e s t r i c t e d by S h i b a t a and Sozen t o s t r u c t u r e s which d i d have e q u a l c a p a c i t i e s i n these c i r c u m s t a n c e s t h e s e t e s t s show i t not t o be a n e c e s s a r y r e s t r i c t i o n f o r the m o d i f i e d s u b s t i t u t e s t r u c t u r e method. Our p r e s e n t knowledge of p r e d i c t i n g the e x a c t e x c i t a t i o n p a t t e r n of a f u t u r e e a r t h q uake a t a g i v e n s i t e i s a t best l i m i t e d . The spectrum approach makes c o n c e s s i o n s t o t h i s by u s i n g an envelope of e f f e c t s from past e v e n t s , t h u s e x p r e s s i n g 29 the f u t u r e e a r t hquake i n a more g e n e r a l manner than can be c o n s i d e r e d when u s i n g d i r e c t l y the i n d i v i d u a l e x c i t a t i o n r e c o r d s of former e a r t h q u a k e s . The m o d i f i e d s u b s t i t u t e s t r u c t u r e method has been shown t o o f f e r the d e s i g n e r a good a l t e r n a t i v e t o time s t e p a n a l y s i s f o r p r e d i c t i o n of the damage r a t i o s i n r e i n f o r c e d c o n c r e t e frame members. The method becomes even more a t t r a c t i v e s h o u l d the d e s i g n e r wish t o d e s i g n h i s s t r u c t u r e on the b a s i s of ' m i x i n g ' the e x c i t a t i o n r e s u l t s from s e v e r a l p a s t e a r t h q u a k e s t o b e t t e r e s t i m a t e the damage r a t i o s caused by f u t u r e s e i s m i c e v e n t s . W h i l e the method has been found e f f e c t i v e f o r normal r e i n f o r c e d c o n c r e t e frame elements i t i s the purpose of t h i s t h e s i s t o examine the e f f e c t i v e n e s s of the method when a p p l i e d t o s t r u c t u r a l w a l l s . 30 CHAPTER 3 ALTERATIONS TO THE METHOD FOR THE ANALYSIS OF STRUCTURAL WALLS Under i t s o r i g i n a l f o r m u l a t i o n the m o d i f i e d s u b s t i t u t e s t r u c t u r e method was i n t e n d e d t o be used i n the a n a l y s i s of r e i n f o r c e d c o n c r e t e frames. T h i s chaper d i s c u s s e s the changes made t o the method t o adapt i t t o the a n a l y s i s of s t r u c t u r a l w a l l s . o 3.1 CONVERGENCE SCHEMES As w i t h any i t e r a t i v e p r o c e d u r e , some c r i t e r i o n must be used f o r d e t e r m i n i n g when the s o l u t i o n has reached a l e v e l of a c c u r a c y such t h a t the p r o c e s s can be h a l t e d . F o r the m o d i f i e d s u b s t i t u t e s t r u c t u r e method t h i s c r i t e r i o n can be based on e i t h e r a maximum change between the damage r a t i o s of s u c c e s s i v e i t e r a t i o n s , or on the c l o s e n e s s of y i e l d e d members t o t h e i r moment c a p a c i t y . By ex a m i n i n g the fo r m u l a f o r m o d i f y i n g the damage r a t i o s a t the end of each i t e r a t i o n , i t can be shown t h a t f o r a member which remains above a u n i t damage r a t i o , the damage r a t i o a t the end of the n t h i t e r a t i o n i s g i v e n by the f o l l o w i n g f o r m u l a : 31 where M n i s the l a r g e s t end moment f o r the member a t the end of the n t h i t e r a t i o n and M C A P i s the bending moment c a p a c i t y of the member. D u r i n g the p r o g r e s s of the i t e r a t i v e p r o c e d u r e , i f the damage r a t i o s a r e t o converge, the r a t i o of the member end moment t o c a p a c i t y must converge t o u n i t y . The o r i g i n a l convergence c r i t e r i o n of the m o d i f i e d s u b s t i t u t e s t r u c t u r e method was deemed t o be a c h i e v e d when none of the members w i t h damage r a t i o s above u n i t y were o u t s i d e a s p e c i f i e d t o l e r a n c e from t h e i r c a p a c i t y . T h i s d e v i a t i o n of the damaged members from t h e i r c a p a c i t y i s r e f e r r e d t o here as the bending moment e r r o r . To ensure t h a t the damage r a t i o s c onverged, a v e r y s t r i c t t o l e r a n c e was imposed on the bending moment e r r o r r e q u i r i n g the maximum moments c a r r i e d by the members t o be almost e x a c t l y the c a p a c i t y of the member. These t o l e r a n c e s were i n the or d e r of 1 0 " 3 , i m p l y i n g t h a t damaged members s h o u l d be w i t h i n a t e n t h of a p e r c e n t of t h e i r c a p a c i t y . In p r a c t i c a l terms t h i s i s an e x c e s s i v e l y s m a l l t o l e r a n c e t o p l a c e on the moments. With v a r i a t i o n s i n member and m a t e r i a l p r o p e r t i e s i t i s u n l i k e l y t h a t member c a p a c i t i e s would be known t o anywhere near t h i s a c c u r a c y . I t was observed d u r i n g some runs t h a t damage r a t i o s , when c o n v e r g i n g t o meet t h i s c r i t e r i o n , would o f t e n v a r y o n l y i n the second or t h i r d d e c i m a l p l a c e s d u r i n g a l l but the i n i t i a l i t e r a t i o n s . As damage r a t i o s c a n n o t , under even the best c i r c u m s t a n c e s , be r e g a r d e d as more ' a c c u r a t e ' than a s i n g l e 32 d e c i m a l p l a c e , t h e s e e x t r a i t e r a t i o n s a r e un n e c e s s a r y . A l t h o u g h the CPU time f o r the m o d i f i e d s u b s t i t u t e s t r u c t u r e method i s dependent upon the degrees of freedom of the s t r u c t u r e and the h a l f - b a n d w i d t h of the banded s t i f f n e s s m a t r i x , i t i s most h e a v i l y i n f l u e n c e d by the number of i t e r a t i o n s t o a c h i e v e convergence and any s a v i n g of unnecessary i t e r a t i o n s w i l l be r e f l e c t e d i n a s a v i n g of computation c o s t s . For t h i s reason a r e v i s i o n of the o r i g i n a l convergence c r i t e r i o n was un d e r t a k e n . The r e v i s e d convergence c r i t e r i o n was based on two c o n d i t i o n s r a t h e r than one. The f i r s t of th e s e was t o r e q u i r e t h a t the bending moment e r r o r be l e s s than f i v e p e r c e n t of the member c a p a c i t y f o r a l l the members of the s t r u c t u r e . T h i s i s a r a d i c a l change from the p r e v i o u s c r i t e r i o n of a t e n t h of a p e r c e n t on t h i s e r r o r . The second convergence c r i t e r i o n was t o r e q u i r e t h a t the l a r g e s t change i n damage r a t i o s between s u c c e s s i v e i t e r a t i o n s be one p e r c e n t . T h i s l a s t c o n d i t i o n was o v e r r i d d e n , i n the case of members w i t h damage r a t i o s of l e s s than f i v e , which would be unduly r e f i n e d by t h i s r e q u i r e m e n t . In the case of the s e s m a l l e r v a l u e s the c r i t e r i o n was t h a t the a b s o l u t e d i f f e r e n c e i n the r a t i o s f o r the i n d i v i d u a l members be l e s s than 0.1. In a l g e b r a i c terms the new convergence c r i t e r i a can be d e s c r i b e d as f o l l o w s : For convergence both 3.2(a) and 3.2(b) must be s a t i s f i e d , < 0.05 (3.2a) 33 i f / i n — < o.o\ (3.2b) i f / J U ^ 5 Ti - < o. (3.2b) The r e s u l t s of u s i n g these r e v i s e d convergence c r i t e r i a w i l l be d i s c u s s e d a f t e r a t e c h n i q u e i s i n t r o d u c e d t o f u r t h e r save unnecessary i t e r a t i o n s - a convergence s p e e d i n g r o u t i n e . 3.2 CONVERGENCE SPEEDING ROUTINE In e a r l y computer runs u s i n g the m o d i f i e d s u b s t i t u t e s t r u c t u r e method i t was obser v e d t h a t f o r some s t r u c t u r e s , the damage r a t i o s would converge v e r y s l o w l y t o the f i n a l answer or o s c i l l a t e around t h i s p o i n t . The o r i g i n a l m o d i f i e d s u b s t i t u t e s t r u c t u r e method program c o n t a i n e d a r o u t i n e which proved e f f e c t i v e i n a r r i v i n g a t the f i n a l answer f o r tho s e cases i n which the cha n g i n g damage r a t i o s were e i t h e r d e c r e a s i n g ' or i n c r e a s i n g s t e a d i l y . T h i s r o u t i n e o p e r a t e d on the b a s i s of ad d i n g t o the damage r a t i o s t h a t were t o be r e t u r n e d t o the main program a f a c t o r m u l t i p l i e d by the change i n the damage r a t i o s over the l a s t i t e r a t i o n . In t h i s manner, cha n g i n g damage r a t i o s were moved f a s t e r i n a d i r e c t i o n which h o p e f u l l y was toward the t r u e answer. The r o u t i n e a c h i e v e d good r e s u l t s i n many cases by c u t t i n g down c o n s i d e r a b l y the number of i t e r a t i o n s w h i l e s t i l l a c h i e v i n g the same s o l u t i o n upon convergence. U n f o r t u n a t e l y , i n 34 those c a s e s where the damage r a t i o s were o s c i l l a t i n g a t each i t e r a t i o n the r o u t i n e a c t u a l l y was a d e t e r r e n t t o convergence. In t h e s e c a s e s the i t e r a t i o n p r o c e d u r e u s u a l l y c o n t i n u e d u n t i l the maximum number of i t e r a t i o n s had been exceeded. The s o l u t i o n t o t h i s problem was t o b e t t e r e s t a b l i s h the damage r a t i o t r e n d s by keeping t r a c k of damage r a t i o s from more than j u s t the l a s t i t e r a t i o n . However, p r a c t i c a l i t y d i c t a t e s t h a t s t o r a g e of a l l damage r a t i o s i s u n d e s i r a b l e . For the case of a c o u p l e d t e n -s t o r y s t r u c t u r a l w a l l w i t h a maximum i t e r a t i o n count of two hundred, t h i s would r e q u i r e an a r r a y t o s t o r e a p o s s i b l e s i x thousand damage r a t i o s . The r e q u i r e d a r r a y space would r a p i d l y become l a r g e r as the number of s t o r i e s or c o u p l e d w a l l s i n c r e a s e d . J u s t e x a c t l y what t o do w i t h t h i s p o t e n t i a l l y v a s t c o l l e c t i o n of damage r a t i o s when s t o r e d would a l s o be a problem of c o n s i d e r a b l e p r o p o r t i o n . The adopted s o l u t i o n t o t h e s e problems was a convergence r o u t i n e which s t o r e d and used the damage r a t i o s from the p a s t two i t e r a t i o n s . Hence, t h r e e v a l u e s a r e known, these b e i n g the damage r a t i o produced i n the c u r r e n t i t e r a t i o n and those from the p r e v i o u s two i t e r a t i o n s . What o c c u r s i n the r o u t i n e i s a p o s s i b l e m o d i f i c a t i o n of the l a t e s t damage r a t i o b e f o r e r e t u r n i n g i t t o the program. By r o t a t i n g the r a t i o s by d i s c a r d i n g the o l d e s t v a l u e d u r i n g the i t e r a t i o n p r o c e d u r e , a m i n i m a l amount of e x t r a a r r a y space i s r e q u i r e d . D u r i n g the deployment of the r o u t i n e , by e x e c u t i n g a maximum of two ' a r i t h m e t i c i f ' c o m p a risons, the n i n e p o s s i b l e t r e n d s i n the r a t i o can be d e t e r m i n e d . In t h i s manner those r a t i o s t h a t seem t o be c o n s i s t e n t l y d e c r e a s i n g o r i n c r e a s i n g can have the damage 35 r a t i o m o d i f i e d by a p p r o p r i a t e l y a d d i n g or s u b t r a c t i n g a f a c t o r m u l t i p l i e d by the d i f f e r e n c e of the l a s t two v a l u e s as i n the o r i g i n a l program. On the o t h e r hand, i n the case of o s c i l l a t i n g damage r a t i o s , the o s c i l l a t i o n s a r e damped i n t o p r o d u c i n g an answer l y i n g between the l a s t two v a l u e s . I t was d e c i d e d t h a t i n those c a s e s i n which the r a t i o s d i d not change f o r two c o n s e c u t i v e i t e r a t i o n s , no m o d i f i c a t i o n s h o u l d be made by the convergence speeding r o u t i n e . A more d e t a i l e d view of the workings of t h i s r o u t i n e can be o b t a i n e d by l o o k i n g at the f l o w c h a r t of the r o u t i n e , shown i n f i g u r e 3.1, which a l s o shows i n s chematic form, the n i n e p o s s i b l e c ases f o r the r e l a t i v e p o s i t i o n i n g of the t h r e e damage r a t i o s . From the f l o w c h a r t i t can be seen t h a t the r o u t i n e i s c o n t r o l l e d by a f a c t o r beta ( P ) , which i s a p o s i t i v e number l e s s than u n i t y . A v a l u e of ze r o f o r be t a e f f e c t i v e l y s h u t s o f f the convergence speeding r o u t i n e . T h i s i s done d u r i n g the f i r s t few w i l d l y changing i t e r a t i o n s t o l e t the mo d i f i e d - s u b s t i t u t e s t r u c t u r e method program n a t u r a l l y home c l o s e r t o the f i n a l answers. As the new convergence r o u t i n e was f o r m u l a t e d t o work w i t h the r e v i s e d convergence c r i t e r i a b o t h were i n c l u d e d i n the m o d i f i e d s u b s t i t u t e s t r u c t u r e method computer program b e f o r e f u r t h e r t e s t i n g was c a r r i e d o u t . Hence any r e d u c t i o n i n the number of- i t e r a t i o n s r e q u i r e d cannot be s o l e l y a t t r i b u t e d t o e i t h e r the new r o u t i n e or the r e v i s e d c r i t e r i a . S i x t e s t s were made on v a r i o u s s t r u c t u r e s which had been run a l r e a d y under t h e o r i g i n a l convergence scheme. D u r i n g t h e s e t e s t s the v a l u e of be t a i n the new convergence s p e e d i n g r o u t i n e was kept a t an a r b i t r a r i l y chosen v a l u e of 0.25. With o u t e x c e p t i o n the r e s u l t s 3 6 showed a c o n s i d e r a b l e d e c rease i n the number of i t e r a t i o n s r e q u i r e d t o a c h i e v e convergence. Some showed a decrease of n e a r l y e i g h t y p e r c e n t and o t h e r s a more modest twenty p e r c e n t . To d e t e r m i n e the v a l i d i t y of the pr o c e d u r e a comparison must be made between the r e s u l t s o b t a i n e d f o r damage r a t i o s produced under the new and o l d convergence schemes. In those c a s e s i n which the number of i t e r a t i o n s had been s m a l l ( i . e . l e s s than about f i f t e e n ) under the o l d convergence scheme the new method produced almost i d e n t i c a l r e s u l t s , changing o n l y i n s i g n i f i c a n t f i g u r e s i n a l l q u a n t i t i e s of c o n c e r n . One would hope t h a t s u c c e s s f u l convergence c r i t e r i a would o n l y change the i n s i g n i f i c a n t d e c i m a l p l a c e s as the t o l e r a n c e s were made more s t r i c t . Indeed the i n i t i a l r e s u l t s show t h i s t o be t r u e i n tho s e s t r u c t u r e s f o r which time s t e p and m o d i f i e d s u b s t i t u t e s t r u c t u r e method answers have p r e v i o u s l y been c l o s e s t . Those s t r u c t u r e s t h a t had d i f f i c u l t y c o n v e r g i n g under the o r i g i n a l convergence scheme showed t h i s t r e n d a g a i n under the new scheme. The r e s u l t s of the s i x - s t o r y frame under the two schemes a r e shown i n f i g u r e 3.2. A l t h o u g h the r e s u l t s a r e somewhat d i f f e r e n t under the new c r i t e r i a they s t i l l r e f l e c t t he t r e n d s t h a t emerge from the time s t e p a n a l y s i s runs performed by Sumio Y o s h i d a . The v a l u e of beta used t o o b t a i n the convergence i n the p r e v i o u s r e s u l t s was s e t at 0.25. T h i s number has been chosen q u i t e a r b i t r a r i l y but meets the c r i t e r i o n as i t l i e s between z e r o and one. To de t e r m i n e an o p t i m a l v a l u e of beta f o r a l l s t r u c t u r e s t h a t c o u l d ever be c o n s i d e r e d i s beyond the scope of t h i s work. As d i f f e r e n t s t r u c t u r e s converge upon t h e i r f i n a l 37 answer i n a v a r i e t y of ways a 'be s t ' v a l u e of b e t a w i l l not be u n i q u e l y d e f i n e d . To t e s t the e f f e c t of v a r y i n g b e t a on one s t r u c t u r e , the t o l e r a n c e demand of the bending moment e r r o r was t e m p o r a r i l y a l t e r e d t o make convergence more d i f f i c u l t and t o a c c e n t u a t e the e f f e c t of the convergence s p e e d i n g r o u t i n e . S e v e r a l computer runs were then made on a s i x - s t o r y s t r u c t u r e u s i n g d i f f e r e n t v a l u e s of beta t o a c h i e v e convergence. The damage r a t i o s at the end of each i t e r a t i o n were then p l o t t e d f o r one of the members of the s t r u c t u r e as a f u n c t i o n of the i t e r a t i o n number. T h i s graph can be seen i n f i g u r e 3.3. E x a m i n a t i o n of t h i s f i g u r e shows t h a t as beta approaches one the convergence i s a c c e l e r a t e d . The f i n a l v a l u e of the damage r a t i o i s independent of the v a l u e of beta as lo n g as b e t a l i e s i n i t s a d m i s s i b l e range. As was e x p e c t e d , a v a l u e of b e t a i n exce s s of one causes d i v e r g e n c e . E x a m i n a t i o n of t h i s graph and the r e d u c t i o n i n e x e c u t i o n t i m e s f o r the cases s t u d i e d l e a v e s no doubt t h a t the new convergence scheme i s a v i a b l e method of r e d u c i n g the r e q u i r e d number of i t e r a t i o n s and s a v i n g CPU t i m e . 38 3.3 THE EFFECT OF USING ZERO SMEARED DAMPING RATIO AT THE START OF EACH ITERATION As has been o u t l i n e d i n an e a r l i e r s e c t i o n d e a l i n g w i t h the t h e o r y of the m o d i f i e d s u b s t i t u t e s t r u c t u r e method, each i t e r a t i o n i n v o l v e s two major s e c t i o n s . To r e v i e w : f i r s t the n a t u r a l f r e q u e n c i e s of the s t r u c t u r e are found, u s i n g z e r o damping, t o o b t a i n i n e r t i a f o r c e s from the spectrum. These f o r c e s are then a p p l i e d t o the s t r u c t u r e , g i v i n g the i n t e r n a l member f o r c e s and a smeared damping r a t i o . The second major i t e r a t i o n s t e p i s t o use t h i s smeared damping f o r c e t o r e c a l c u l a t e the member f o r c e s and hence the new damage r a t i o s . The use of the z e r o damping r a t i o a t the s t a r t of a l l i t e r a t i o n s , i s a v e s t i g e of the s u b s t i t u t e s t r u c t u r e method i n w h i c h , w i t h o u t an i t e r a t i v e p r o c e d u r e , any b e t t e r e s t i m a t e of the smeared damping r a t i o i s a guess. In the m o d i f i e d s u b s t i t u t e s t r u c t u r e method one knows a p p r o x i m a t e damping r a t i o s f o r the d i f f e r e n t modes of the s t r u c t u r e by l o o k i n g a t the r a t i o d e t e r m i n e d i n the l a s t i t e r a t i o n . I t was b r i e f l y thought t h a t convergence would be improved by u s i n g t h e s e l a t e s t v a l u e s when c a l c u l a t i n g the s p e c t r a l a c c e l e r a t i o n f o r d e t e r m i n a t i o n of the smeared damping r a t i o of the c u r r e n t i t e r a t i o n . T h i s would r e p l a c e the p r o c e s s of r e t u r n i n g t o z e r o damping f o r the f i r s t h a l f of each c y c l e of the i t e r a t i o n . T h i s p r o c e d u r e was adopted i n t e s t runs u s i n g those s t r u c t u r e s which had undergone t e s t i n g i n Y o s h i d a ' s t h e s i s . The t h r e e - b a y , t h r e e - s t o r y t e s t s t r u c t u r e was run as was the s i x -s t o r y , one-bay s t r u c t u r e . Convergence c r i t e r i a and schemes f o r 39 these t e s t s were i n a l l c a s e s those of the o r i g i n a l m o d i f i e d s u b s t i t u t e s t r u c t u r e method. S u r p r i s i n g l y , the number of i t e r a t i o n s r e q u i r e d f o r b oth s t r u c t u r e s remained p r e c i s e l y the same. The t h r e e - s t o r y s t r u c t u r e s t i l l r e q u i r e d t h i r t e e n i t e r a t i o n s and the s i x - s t o r y frame r e q u i r e d s i x t y t h r e e b e f o r e a c h i e v i n g convergence. When the answers were examined i t was found t h a t the damage r a t i o s v a r i e d o n l y i n the t h i r d d e c i m a l p l a c e and t h e r e f o r e i n s u f f i c i e n t l y t o be of c o n c e r n . Hence i t was d e c i d e d t o r e f r a i n from z e r o i n g the smeared damping r a t i o s a t - t h e s t a r t of each i t e r a t i o n and t o use those a l r e a d y i n the damping a r r a y . As the damping m a t r i x i s s m a l l t h i s a c h i e v e s o n l y a m i n i m a l s a v i n g i n s t o r a g e and e x e c u t i o n r e q u i r e m e n t s . A f t e r e l i m i n a t i n g the n e c e s s i t y t o r e p e a t e d l y z e r o the damping m a t r i x , the next s t e p i n t h i s e v a l u a t i o n was t o compute the damping t w i c e i n each i t e r a t i o n : once from the i n i t i a l pass (which p r e v i o u s l y was the z e r o damped p a s s ) ; then s u b s e q u e n t l y from the pass i n which the damped f o r c e s were a p p l i e d . G i v i n g the subsequent i t e r a t i o n a more ' a c c u r a t e ' damping r a t i o t o s t a r t was thought t o l e a d t o a q u i c k e r convergence. In Y o s h i d a ' s o r i g i n a l method the damping had not been c a l c u l a t e d i n the second pass of each i t e r a t i o n as i t s e r v e s no purpose i f t h i s damping r a t i o i s not t o be used i n the f o l l o w i n g i t e r a t i o n . A f t e r m o d i f i c a t i o n of the computer program the same two s t r u c t u r e s used above were r e t e s t e d t o d e t e r m i n e the e f f e c t of t h i s measure. The r e s u l t s showed s i m i l a r t r e n d s t o those of the p r e v i o u s t e s t s , f o r w h i l e a few i n s i g n i f i c a n t d e c i m a l p l a c e s had been changed, the number of i t e r a t i o n s remained e x a c t l y the same. I t was c o n c l u d e d , t h e r e f o r e , t h a t the second c a l c u l a t i o n 40 of the modal damping v a l u e s a t the end of an i t e r a t i o n was not a w o r t h w h i l e use of CPU time and was as unnecessary as z e r o i n g the modal damping v a l u e s a t the s t a r t of each i t e r a t i o n . R e m o d i f i c a t i o n of the computer program s l i c e d t he double c a l c u l a t i o n of damping r a t i o s from the i t e r a t i o n sequence. 3.4 RIGID BEAM EXTENSIONS The o r i g i n a l m o d i f i e d s u b s t i t u t e s t r u c t u r e method was devel o p e d f o r a n a l y s i n g frames r a t h e r than s t r u c t u r a l w a l l s . In the former case beam and column l e n g t h s a re c o n s i d e r a b l y g r e a t e r than the j o i n t d i mensions and hence the c o n s i d e r a t i o n of the j o i n t s as a s i n g l e p o i n t i s an a c c e p t a b l e d e t a i l of the s t r u c t u r e m o d e l l i n g . In s t r u c t u r a l w a l l systems t h i s i s not the case and f a i l u r e t o i n c l u d e measures t o model the j o i n t w i d t h w i l l l e a d t o s e r i o u s e r r o r s . I f a j o i n t i s t o be m o d e l l e d , say at the c e n t r e of a f i f t e e n - f o o t w a l l , then the j o i n t can be c o n s i d e r e d as h a v i n g a w i d t h of seven and a h a l f f e e t b e f o r e c o n n e c t i n g t o a t y p i c a l f o u r - f o o t l o n g l i n t e l beam. In t h i s manner the w i d t h of the beam-wall j o i n t reduces the e f f e c t i v e l e n g t h of the l i n t e l beam and i n c r e a s e s i t s s t i f f n e s s . S e v e r a l s o l u t i o n s a r e p o s s i b l e t o s o l v e t h i s problem, p r o b a b l y the c r u d e s t i s t o i n c l u d e i n the model e x t r a members which are r i g i d and i n e x t e n s i b l e . These e x t r a members would be r i g i d l y c o n n e c t e d a t the c e n t r e of the w a l l and a t the f a c e of the l i n t e l beam. T h i s would g i v e the t r u e end of the l i n t e l beam the same r o t a t i o n and l a t e r a l d i s p l a c e m e n t as the c e n t r e l i n e of the w a l l . Two problems a r e i n h e r e n t i n t h i s s o l u t i o n . The f i r s t 41 of these i s that e x t r a degrees of freedom w i l l be r e q u i r e d f o r the extra j o i n t s at the i n t e r f a c e of the l i n t e l beam and r i g i d member. The e x t r a j o i n t s w i l l a l s o i n c r e a s e the half-bandwidth of the s t i f f n e s s matrix. Both these f a c t o r s i n c r e a s e CPU requirements and the co s t of running the program. The second problem i s that to use ' r i g i d ' members r e q u i r e s the use of a very l a r g e moment of i n e r t i a for the c r o s s s e c t i o n of these members. T h i s can tend somewhat to dominate the s t i f f n e s s matrix and i f too l a r g e a value i s used i t can reduce the accuracy of the r e s u l t s . On the other hand, a lower value of the moment of i n e r t i a , while being more s a t i s f a c t o r y f o r use i n the s t i f f n e s s matrix, d e f e a t s the aims of a r i g i d member. When the two t e s t s t r u c t u r e s were analyzed using e x t r a beams and a s i n g l e p r e c i s i o n s t i f f n e s s matrix i t was found that the best compromise f o r t h i s s i t u a t i o n was to use a r i g i d beam moment of i n e r t i a of approximately t h i r t y times that of the w a l l to which i t was connected. Another p o s s i b l e s o l u t i o n to the problem of non-zero j o i n t s i z e l i e s i n the conception of a new member. T h i s member along with the a s s o c i a t e d member degrees of freedom i s i l l u s t r a t e d i n f i g u r e 3.4. The member displacements can be f u l l y d e s c r i b e d by the degrees of freedom at the center of the w a l l . T h i s element has a member s t i f f n e s s matrix composed of three p a r t s : an a x i a l p o r t i o n s i m i l a r to that of a member of length L; a bending p o r t i o n a l s o s i m i l a r t o that of a member of l e n g t h L; and an e x t r a s t i f f n e s s i n c u r r e d from the r i g i d ends. T h i s e x t r a s t i f f n e s s matrix which corresponds to the unprimed degrees of freedom of f i g u r e 3.4 i s shown i n t a b l e 3.1. T h i s matrix does 42 not r e q u i r e t h a t the r i g i d ends of the member be of eq u a l l e n g t h which a l l o w s f o r the p o s s i b i l i t y of a c o u p l e d s t r u c t u r a l w a l l w i t h unequal w a l l d e p t h s . I t s h o u l d a l s o be noted t h a t , as e x p e c t e d , t h e m a t r i x goes t o z e r o when the member has no r i g i d e x t e n s i o n s and hence r e v e r t s back t o the case of a frame element. D u r i n g the program i n p u t f o r the s t r u c t u r e the e x t e n s i o n s on each end a r e read i n ; i f e i t h e r i s non-zero the e x t r a s t i f f n e s s m a t r i x due t o r i g i d e x t e n s i o n i s c a l c u l a t e d and added t o the s t r u c t u r e s t i f f n e s s m a t r i x . A f t e r the d i s p l a c e m e n t s have been c a l c u l a t e d f o r the j o i n t s , the d i s p l a c e m e n t s a r e computed f o r the ends' of the f l e x i b l e member by u s i n g the r e l a t i o n s i p s between the d i s p l a c e m e n t s a t each end of the r i g i d beam. That i s , t h a t h o r i z o n t a l d i s p l a c e m e n t and r o t a t i o n a re the same a t both ends of the r i g i d s e c t i o n and the v e r t i c a l d i s p l a c e m e n t i s e q u a l t o t h a t of the c e n t e r w a l l j o i n t m o d i f i e d by the a p p r o p r i a t e a d d i t i o n or s u b t r a c t i o n of the p r o d u c t of r i g i d arm r o t a t i o n and l e n g t h . W i t h the end d i s p l a c e m e n t s of the f l e x i b l e r e g i o n known, the member f o r c e s can be c a l c u l a t e d i n a manner s i m i l a r t o t h a t of any normal member of l e n g t h L. At t h i s stage the program i s s e t up t o handle r i g i d e x t e n s i o n s on o n l y h o r i z o n t a l members w i t h f i x e d ends as t h e s e a r e the o n l y ones of c o n c e r n f o r s t r u c t u r a l w a l l systems. A f t e r i n c l u s i o n of the p r o v i s i o n s f o r r i g i d e x t e n s i o n s t e s t i n g was performed t o dete r m i n e the a c c u r a c y and e f f e c t i v e n e s s of the i n c l u s i o n . A o n e - s t o r y , one-bay t e s t frame was used t o remove 'bugs' from the r o u t i n e s . The o n e - s t o r y , one-bay frame o f f e r s an e x c e l l e n t means of t e s t i n g . As t h e r e i s o n l y 43 one mode and a l i m i t e d number of members and j o i n t s , hand c a l c u l a t i o n s can e a s i l y be performed as a check. The next s t r u c t u r e t e s t e d was a s i x - s t o r y , one-bay s t r u c t u r e w i t h r i g i d e x t e n s i o n s on both ends of the beams. A run was made w i t h t h i s s t r u c t u r e u s i n g e x t r a members f o r the r i g i d beams and a n o t h e r run u s i n g the method of a d d i n g the r i g i d e x t e n s i o n s by the use of the t h r e e segment element a l r e a d y d e s c r i b e d . Upon convergence the v a l u e of a l l f a c t o r s of concern was found t o be e q u a l t o a l l r e a s o n a b l e s i g n i f i c a n t f i g u r e s . The run i n which e x t r a members were used t o model the r i g i d arms r e q u i r e d 39 per cen t more degrees of freedom (50 v s . 36) and a 45 per cent i n c r e a s e i n the h a l f - b a n d w i d t h (13 v s . 9) when compared t o the same s t r u c t u r e m o d e l l e d by use of the t h r e e segment e l e m e n t s . For t h i s s t r u c t u r e the use of the composite member f o r h a n d l i n g j o i n t s of f i n i t e w i d t h saved 38 per cen t of the CPU time r e q u i r e m e n t s (3.91 seconds v s . 6.348 seconds) over u s i n g e x t r a ' r i g i d ' members t o model the j o i n t s . 44 CHAPTER 4 TESTING THE PROGRAM FOR ELASTIC CAPABILITIES. The w r i t i n g of any computer program t o s o l v e a g i v e n problem i s always s u b j e c t t o i n a c c u r a c i e s caused by r o u n d o f f e r r o r or i n c o r r e c t l o g i c . Even i f the program i s w r i t t e n w i t h extreme c a r e , minor e r r o r s may c r e e p i n t h a t can produce r e s u l t s i n d i c a t i n g the s o l u t i o n a l o g r i t h m i s not v a l i d when the d i f f i c u l t i e s may l i e w i t h the programming of t h a t a l o g r i t h m . The computer program which was de v e l o p e d t o a p p l y the m o d i f i e d s u b s t i t u t e s t r u c t u r e method t o shear w a l l s underwent a s e r i e s of t e s t s t o examine i t s e l a s t i c c a p a b i l i t i e s b e f o r e b e i n g r e g a r d e d as a c c e p t a b l e . Some of these t e s t s w i l l now be r e l a t e d as t h e y s e r v e t o demonstrate some of the p r a c t i c a l c o n c e r n s f o r a f u n c t i o n i n g e l a s t i c modal a n a l y s i s program. 4.1 TESTING THE STIFFNESS MATRIX FORMULATION AND EIGENVALUE PRODUCTION. The f i r s t i t e m of con c e r n w i t h the a n a l y s i s of a program i s to a s c e r t a i n t h a t a l l i n p u t d a t a i s b e i n g r e a d c o r r e c t l y by the computer. T h i s i s a c h i e v e d t h r o u g h a complete 'echo p r i n t i n g ' of 45 the i n p u t d a t a b e f o r e f u r t h e r o p e r a t i o n s o c c u r . A l t h o u g h t h i s i s s t a n d a r d programming p r a c t i c e and not p a r t i c u l a r t o a modal a n a l y s i s program i t i s a p o i n t too f r e q u e n t l y o v e r l o o k e d . H a ving e s t a b l i s h e d t h a t the i n p u t data i s c o r r e c t , the programmer must then check the b u i l d i n g of the s t i f f n e s s m a t r i x . T h i s i s a c c o m p l i s h e d by h a v i n g the s t i f f n e s s m a t r i x o utput i n a f i l e where i t can be examined s e p a r a t e l y . In the case of s m a l l s t r u c t u r e s t h i s i s performed by d o i n g hand c a l c u l a t i o n s w h i l e f o r l a r g e r s t r u c t u r e s by comparing w i t h s t i f f n e s s m a t r i c e s produced from computatio n s performed on an i d e n t i c a l s t r u c t u r e by a proven s t a t i c a n a l y s i s program. A f u r t h e r t e s t t h a t can be performed t o check the s t i f f n e s s m a t r i x p r o d u c t i o n w h i l e a l s o ' c h e c k i n g the e i g e n v a l u e r o u t i n e i s t h r o u g h the e x a m i n a t i o n of the f r e q u e n c i e s and mode shapes from a s i m p l e s t r u c t u r e . A p a i r of v e r y e l e m e n t a r y examples f o r t h i s which w i l l h e l p t o p i n p o i n t e r r o r s a t an e a r l y stage a r e the h o r i z o n t a l and v e r t i c a l pendulums. These s t r u c t u r e s though almost t r i v i a l i n n a t u r e p r o v i d e examples which can be e a s i l y c o n f i r m e d by hand c a l c u l a t i o n . T h i s i s a c o n s i d e r a t i o n of noteworthy importance i n the c h o o s i n g of s t r u c t u r e s t o t e s t d u r i n g the e a r l y a n a l y s i s s tage of a computer program. The a b i l i t y t o t e s t s t r u c t u r e s which can be v e r i f i e d by knowledge of the ' c o r r e c t ' answer a v o i d s the c o m p l i c a t i o n of t r y i n g t o r a t i o n a l i z e the d i f f e r e n c e s a r i s i n g from the use of two independent programs n e i t h e r of which may be c o r r e c t . These s i m p l e s t r u c t u r e s a r e shown i n f i g u r e 4.1 (a and b ) . T a b l e 4.1 g i v e s the a l g e b r a i c e x p r e s s i o n s f o r r e l e v a n t p r o p e r t i e s such as f r e q u e n c y , d i s p l a c e m e n t and f o r c e s t o be e x p e c t e d when the the program i s o p e r a t i n g 46 c o r r e c t l y . These f o r m u l a s may be v e r i f i e d by r e a l i z i n g t h a t the h o r i z o n t a l pendulum i s analogous t o the s t a n d a r d c a r t on f r i c t i o n l e s s r o l l e r s a t t a c h e d to a s p r i n g of s t i f f n e s s c o n s t a n t k as shown i n f i g u r e 4.1c. In the case of the pendulum though the a x i a l s t i f f n e s s i s d e t e r m i n e d from the e x t e n s i o n a l s t i f f n e s s of the u n i f o r m r o d . I f the mass i s o n l y a t t a c h e d t o the h o r i z o n t a l degree of freedom or i f the mass i s a l s o a t t a c h e d t o the v e r t i c a l degree of freedom but w i t h the pendulum p r o p e r t i e s chosen so t h a t the bending mode frequ e n c y i s w e l l s e p a r a t e d from t h a t of the a x i a l mode, then the modal p a r t i c i p a t i o n f a c t o r f o r the h o r i z o n t a l mode s h o u l d be p l u s or minus u n i t y . The u n c e r t a i n t y i n s i g n i s a r e s u l t of the f a c t t h a t w h i l e e i g e n v e c t o r s can be n o r m a l i z e d on one a r b i t r a r i l y chosen d i s p l a c e m e n t the magnitude or s i g n i s never known except i n r e l a t i v e terms. I f the program i s w o r k i ng c o r r e c t l y w i t h a x i a l and bending modes w e l l s e p a r a t e d , the a x i a l mode s h o u l d not produce d i s p l a c e m e n t s or f o r c e s i n the v e r t i c a l or r o t a t i o n a l d i r e c t i o n s . W ith the modal p a r t i c i p a t i o n f a c t o r b e i n g u n i t y the a x i a l f o r c e would c o r r e c t l y be the v a l u e of the s p e c t r a l a c c e l e r a t i o n m u l t i p l i e d by the h o r i z o n t a l mass. When hand c a l c u l a t i o n s a r e performed the s p e c t r a l a c c e l e r a t i o n i s t aken from a g r a p h i c a l r e p r e s e n t a t i o n of the spectrum. I f the s p e c t r a l a c c e l e r a t i o n v a l u e i s g i v e n i n terms of a f r a c t i o n of g r a v i t y i n s t e a d of an a b s o l u t e a c c e l e r a t i o n then the s p e c t r a l a c c e l e r a t i o n v a l u e w i l l have t o be m u l t i p l e d by the a c c e l e r a t i o n of g r a v i t y t o make the p r e c e d i n g e q u a l i t y t r u e . W i t h the f o r c e s i n the r o d known, the d i s p l a c e m e n t s can be e a s i l y d e t e r m i n e d 47 from e l e m e n t a r y s t r e n g t h of m a t e r i a l s . In the same manner t e s t i n g of the v e r t i c a l pendulum checks the bending mode of the b a r . The s t i f f n e s s i n t h i s mode can be equated t o t h a t of a c a n t i l e v e r w i t h a p o i n t l o a d l o c a t e d a t the t i p a c t i n g p e r p e n d i c u l a r t o the a x i s of the c a n t i l e v e r . T h i s mode s h o u l d not produce any a x i a l f o r c e i n the member though shear s h o u l d a r i s e as w e l l as a bend i n g moment a t the base. Shear d e f l e c t i o n s are u s u a l l y not c o n s i d e r e d i n normal modal a n a l y s i s as the frames under c o n s i d e r a t i o n a re u s u a l l y made up of l o n g s l e n d e r members f o r which d e f l e c t i o n s due t o shear a r e i n s i g n i f i c a n t when compared t o those due t o bending. With the use of the more s t o c k y members found i n s t r u c t u r a l w a l l s i t becomes d e s i r a b l e t o i n c l u d e i n the program the c a p a b i l i t y of computing shear d e f l e c t i o n s . T h i s must a l s o be r e f l e c t e d i n the c o n s t r u c t i o n of the s t i f f n e s s m a t r i x b e f o r e d e t e r m i n a t i o n of the n a t u r a l f r e q u e n c i e s , s i n c e the a l l o w a n c e of shear d e f l e c t i o n s w i l l make the s t r u c t u r e more f l e x i b l e , r e s u l t i n g i n l o n g e r p e r i o d s than would o t h e r w i s e be the c a s e . The shear d e f l e c t i o n p r o v i s i o n can be t e s t e d d u r i n g the e l a s t i c t e s t i n g of the program i n the same manner as bending d e f l e c t i o n . The v e r t i c a l c a n t i l e v e r a g a i n forms a good t e s t s t r u c t u r e and the a l g e b r a i c e x p r e s s i o n s f o r the p e r t i n e n t r e s u l t s a r e shown i n t a b l e 4.1. As shear d e f l e c t i o n i s so f r e q u e n t l y i g n o r e d i n a n a l y s i s i t i s d e s i r a b l e t h a t a program h a v i n g the a b i l i t y t o c a l c u l a t e i t a l s o have p r o v i s i o n s by which the c a l c u l a t i o n of shear d e f l e c t i o n can be bypassed. T h i s i s a c c o m p l i s h e d i n t h i s program by a p l a c i n g a z e r o v a l u e f o r e i t h e r the shear modulus of the s t r u c t u r e or of the shear a r e a of members f o r which the 48 shear d e f l e c t i o n i s not d e s i r e d . In t h i s manner shear d e f l e c t i o n can be c o n s i d e r e d i n i n d i v i d u a l members and not i n o t h e r s or by the change of one number i n the d a t a f i l e can be t o t a l l y i g n o r e d f o r the whole s t r u c t u r e . The d i s c u s s i o n of the pendulums used f o r t e s t purposes r a i s e d the problem of whether or not masses s h o u l d be a s s o c i a t e d w i t h the v e r t i c a l degrees of freedom as w e l l as those r e p r e s e n t i n g h o r i z o n t a l m o t i o n . The pendulums a r e r a t h e r s p e c i a l i z e d t e s t s t r u c t u r e s and t h i s p o i n t i s of more i n t e r e s t i n the l a r g e r s t r u c t u r e s t h a t are of a more r e a l i s t i c n a t u r e . A problem a r i s e s because i f masses a r e a t t a c h e d t o v e r t i c a l as w e l l as h o r i z o n t a l degrees of freedom then computation c o s t s can i n c r e a s e by as much as a f a c t o r of two. The " v e r t i c a l " masses c r e a t e e x t r a mode shapes which may cause an unwanted c o n t r i b u t i o n t o the v e c t o r sum of f o r c e s t h a t are e x c i t e d by a h o r i z o n t a l spectrum. In the case of one of the t e s t s t r u c t u r e s examined w i t h masses a t t a c h e d t o v e r t i c a l degrees of freedom, th e s e produced almost pure a x i a l column l e n g t h e n i n g as one of the h i g h e r modes. Though the h o r i z o n t a l d i s p l a c e m e n t s were a l l v e r y s m a l l and of v a r y i n g s i g n f o r t h i s mode the v e r t i c a l d i s p l a c e m e n t s were a l l i n the same d i r e c t i o n p r o d u c i n g a l a r g e modal p a r t i c i p a t i o n f a c t o r f o r t h i s h i g h numbered mode. I f a program i s d e s i g n e d t o a n a l y s e s t r u c t u r e s f o r which i t i s i m p o r t a n t t h a t v e r t i c a l i n e r t i a f o r c e s be i n c l u d e d then t h a t program must keep t r a c k of which masses a r e a s s o c i a t e d w i t h h o r i z o n t a l f o r c e s i n o r d e r t h a t they o n l y have the a c c e l e r a t i o n from the h o r i z o n t a l spectrum a p p l i e d t o them. When masses are s e p a r a t e d a c c o r d i n g t o the d i r e c t i o n of motion which they oppose 49 and the a p p r o p r i a t e s p e c t r a l a c c e l e r a t i o n v a l u e s are a p p l i e d a c c o r d i n g l y , then s t r u c t u r e modes which are p r i m a r i l y v e r t i c a l w i l l . not induce s i g n i f i c a n t f o r c e s from the h o r i z o n t a l a c c e l e r a t i o n . The n e c e s s i t y of a t t a c h i n g v e r t i c a l masses t o a s t r u c t u r e can o f t e n be d e t e r m i n e d from an e x a m i n a t i o n of the a m p l i t u d e s when o n l y h o r i z o n t a l masses are a t t a c h e d . The a m p l i t u d e of any degree of freedom i n any one mode i s g i v e by f o r m u l a 4.1: X= A S i n o J t (4.1) where A i s the maximum a m p l i t u d e . I f t h i s i s d i f f e r e n t i a t e d t w i c e then e q u a t i o n 4.2 g i v e s the a c c e l e r a t i o n of the same p o i n t : X=-A co^Sin co t . (4.2) Hence the maximum a c c e l e r a t i o n of a p o i n t on the s t r u c t u r e w i l l be Aco a . For any g i v e mode the v a l u e of to 2 , w i l l be the same f o r a l l p o i n t s and hence the a c c e l e r a t i o n of the nodes w i l l be d i r e c t l y p r o p o r t i o n a l t o t h e i r d i s p l a c e m e n t s . T h e r e f o r e , an e x a m i n a t i o n of the r e l a t i v e magnitudes of the h o r i z o n t a l and v e r t i c a l d i s p l a c e m e n t s w i l l show i f t h e r e i s a l a r g e component of v e r t i c a l a c c e l e r a t i o n t h a t s h o u l d have an i n e r t i a f o r c e a s s o c i a t e d w i t h i t . E x a m i n a t i o n of s e v e r a l t r i a l s t r u c t u r e s t h a t were used i n t e s t i n g the frame a n a l y s i s program has shown t h a t 50 the v e r t i c a l a c c e l e r a t i o n of the column l i n e nodes i s i n the range of two o r d e r s of magnitude lower than the h o r i z o n t a l a c c e l e r a t i o n . The low p r o p o r t i o n of v e r t i c a l a c c e l e r a t i o n r e f l e c t s the l a r g e a x i a l s t i f f n e s s p r e s e n t i n the columns r e l a t i v e t o t h e i r bending s t i f f n e s s . The c o r r e c t modal a n a l y s i s of s t r u c t u r e s which have masses a t t a c h e d o f f the column l i n e s c o u l d q u i t e e a s i l y form the t o p i c f o r a s e p a r a t e t h e s i s ; as t h i s i s not a problem i n the shear w a l l s t r u c t u r e s t h a t a re of conc e r n here a l l f u t u r e r e f e r e n c e s t o masses i n t h i s paper w i l l r e f e r s o l e l y t o masses a s s o c i a t e d w i t h h o r i z o n t a l i n e r t i a forces,. 4.2 COMPARISON WITH ANOTHER ELASTIC MODAL ANALYSIS PROGRAM. Another method of c h e c k i n g the r e s u l t s of a new program i s by c o m p a r i s o n - w i t h r e s u l t s of an e x i s t i n g and p r e v i o u s l y t e s t e d r o u t i n e . One such program t h a t was a v a i l a b l e f o r t h i s purpose was the program 'DYNAMIC. T h i s program had been w r i t t e n i n the e a r l y s e v e n t i e s and w h i l e i t s l o g i c and language i s somewhat d a t e d i n terms of modern programming s t y l e i t has a v a r i e t y of o p t i o n s t h a t make i t a p o w e r f u l e l a s t i c a n a l y s i s program which i s known t o have produced v a l i d r e s u l t s on s e v e r a l o c c a s i o n s . The f i r s t t e s t s f o r c o m p a r i s i o n of the two programs was performed on a f i v e s t o r y frame s t r u c t u r e shown i n f i g u r e 4.2. T h i s s t r u c t u r e was the same f i v e s t o r y s t r u c t u r e t e s t e d by S h i b a t a and S o z e n 7 and r e p o r t e d i n t h e i r 1975 paper on the s u b s t i t u t e s t r u c t u r e method. R e s u l t s o t h e r than the n a t u r a l p e r i o d s are not l i s t e d i n t h e i r paper f o r e l a s t i c a n a l y s i s , but 51 the p e r i o d s they l i s t agree w e l l w i t h those o b t a i n e d from the two programs under e x a m i n a t i o n h e r e . In t h i s s t r u c t u r e the r e s u l t s produced by the two programs were v e r y c l o s e . S l i g h t d i f f e r e n c e s ( m o s t l y i n the or d e r of one p e r c e n t ) i n the r e s u l t s l i s t e d f o r f o r c e s and d i s p l a c e m e n t s were a t t r i b u t e d t o d i f f e r e n c e s i n i n p u t d a t a . These r e s u l t s a r e shown i n t a b l e 4.2. The i n p u t d a t a f o r the two programs v a r i e d s l i g h t l y as one program r e q u i r e d i n p u t i n f o o t u n i t s w h i l e the o t h e r program r e q u i r e d t h a t the d a t a be i n i n c h u n i t s . As the i n p u t p r o p e r t i e s a r e o n l y g i v e n t o t h r e e f i g u r e s t h i s causes s l i g h t d i f f e r e n c e s i n the output produced. Another source of d i f f e r e n c e was i n the spectrum used; w h i l e the m o d i f i e d s u b s t i t u t e s t r u c t u r e method program was u s i n g a N a t i o n a l B u i l d i n g Code spectrum d i r e c t l y , 'DYNAMIC used a Newmark-Beta spectrum which had been a d j u s t e d t o r e p r e s e n t an NBC spectrum. As both 'DYNAMIC and the e l a s t i c component of the m o d i f i e d s u b s t i t u t e s t r u c t u r e method program o p e r a t e c o m p l e t e l y i n d e p e n d e n t l y , t h i s agreement was judged t o be a v a l i d i n d i c a t i o n t h a t both programs were a b l e t o produce a c c u r a t e r e s u l t s when t e s t i n g t h i s s i z e and s t y l e of s t r u c t u r e . At t h i s time i t i s a p p r o p r i a t e t o d i s c u s s the u n i t s t h a t go i n t o the makeup of the s t i f f n e s s m a t r i x . In u s i n g the I m p e r i a l system the j o i n t c o o r d i n a t e s t h a t produce member l e n g t h s a re f r e q u e n t l y i n p u t i n f e e t w h i l e the member p r o p e r t i e s are i n square i n c h e s and i n c h e s t o the f o u r t h . A common u n i t of l e n g t h must be chosen t o c o n s t r u c t the s t i f f n e s s m a t r i x . At f i r s t e x a m i n a t i o n the c h o i c e would seem t o be an a r b i t r a r y one w i t h the i n c h e s b e i n g f a v o r e d as f i n a l d e f l e c t i o n s a r e perhaps b e t t e r 52 ' f e l t ' i n i n c h e s and the use of i n c h e s i n the s t i f f n e s s m a t r i x would save the n e c e s s i t y f o r c o n v e r s i o n l a t e r . A l t h o u g h the use of i n c h e s i n the s t i f f n e s s m a t r i x would be c o r r e c t the use of a common u n i t of f e e t produces a b e t t e r c o n d i t i o n e d s t i f f n e s s m a t r i x . T h i s i s because the terms making up the s t i f f n e s s m a t r i x do not c o n t a i n a l e n g t h f a c t o r t o a u n i f o r m power and the use of the l a r g e r l e n g t h f a c t o r tends t o e q u a l i z e the magnitude of terms i n the s t i f f n e s s m a t r i x . While s t r u c t u r a l p r o p e r t i e s can be imagined f o r which t h i s i s not t r u e e x a m i n a t i o n of some s t r u c t u r e s such as the f i v e s t o r y s t r u c t u r e shown i n f i g u r e 4.2 shows t h a t f o o t u n i t s do reduce the r a t i o of l a r g e s t t o s m a l l e s t elements l y i n g on 'the s t i f f n e s s m a t r i x d i a g o n a l . For example i n the f i v e s t o r y frame a r a t i o of the l a r g e s t t o s m a l l e s t d i a g o n a l element i s 527 when i n c h u n i t s a re used i n the c o n s t r u c t i o n of the s t i f f n e s s m a t r i x but when f o o t u n i t s a r e used the r a t i o d rops t o 4.5. T h i s r e i n f o r c e s the theme t h a t i n t e r n a l use of f o o t u n i t s p r o v i d e s a b e t t e r c o n d i t i o n e d s t i f f n e s s m a t r i x than i n t e r n a l use of i n c h u n i t s . The adequate t e s t i n g of some s u b r o u t i n e s may r e q u i r e t h a t they be c o p i e d t o t a l l y from the program i n t o a second program whose s o l e purpose i s the c a l l i n g of the s u b r o u t i n e under a l o g i c a l v a r i e t y of c i r c u m s t a n c e s . T h i s proved t o be the case f o r the s u b r o u t i n e t h a t was used t o c a l c u l a t e the s p e c t r a l a c c e l e r a t i o n from an i n p u t of n a t u r a l p e r i o d and damping. Though the s t a n d a r d t e s t runs produced s a t i s f a c t o r y answers i t was not u n t i l v e r y low damping v a l u e s were t e s t e d t h a t i t was found t h a t the spectrum r o u t i n e was i n e r r o r and c o r r e c t i o n s c o u l d be made. A thorough e x a m i n a t i o n showed t h a t t h i s e r r o r o c c u r r e d o n l y 53 d u r i n g one of the more r a r e l y summoned l o g i c a l paths of the s u b r o u t i n e . Under these c i r cums tances the on ly c e r t a i n method of check ing the sub rou t ine was to use a ' d r i v e r ' program which l o g i c a l l y went through d i f f e r e n t va lue s of damping and p e r i o d wh i le c a l l i n g the s p e c t r a l a c c e l e r a t i o n and p r i n t i n g out a l l t h ree va lues to be checked by hand. It i s on ly through ted ious e f f o r t and check ing such as t h i s that any s o r t of r e a l c o n f i d e n c e can be deve loped in the program's a b i l i t y to produce a ccu ra te r e s u l t s . In many cases the use of double or extended p r e c i s i o n w i l l be regarded as an ext ravagant waste of CPU time to ach ieve a l e v e l of accuracy that i s u n n e c e s s a r i l y h i g h . In the a n a l y s i s of a sma l l s t r u c t u r e wi th member s t i f f n e s s e s approx imate l y equa l to each other the use of extended p r e c i s i o n i s p robab ly not n e c e s s a r y . However in the a n a l y s i s of l a r ge s t r u c t u r e s and in the cases where through l a r g e v a r i a t i o n s in s e c t i o n p r o p e r t i e s a wide range of va lue s e x i s t s i n the s t i f f n e s s matr ix then the e x t r a accuracy i s r e q u i r e d . One such s t r u c t u r e that proved to r e q u i r e double p r e c i s i o n was encountered in t h i s t e s t i n g program and w i l l be r e f e r r e d to as ' s t r u c t u r e A' shown in f i g u r e 4 .3 . T h i s s t r u c t u r e has s e v e r a l f e a t u r e s which d i d not a i d in i t s a n a l y s i s . For example i t i n c o r p o r a t e d s h o r t , h i gh moment of i n e r t i a , ' r i g i d ' beams and the top four members were of a d i f f e r e n t m a t e r i a l and r i g i d i t y than the remainder of the s t r u c t u r e . A l though i t d i d not seem to be the case wi th ' s t r u c t u r e A ' , i t i s not d i f f i c u l t to conce i ve of s t r u c t u r e s in which a f l e x i b l e top s e c t i o n a c t s as a ' f r e e v i b r a t i o n damper' g r e a t l y a f f e c t i n g the modal r e s u l t s . T h i s danger becomes acute 54 when a f l e x i b l e r e g i o n of a s t r u c t u r e has an independent fundamental p e r i o d which c o i n c i d e s c l o s e l y t o t h a t of one of the lower modes of the whole s t r u c t u r e . T h i s i s not the case w i t h s t r u c t u r e 'A', as can be r e a l i z e d when the t o p s e c t i o n i s s e p a r a t e d and a n a l y s e d as a s e l f c o n t a i n e d s t r u c t u r e . The e i g e n v a l u e s produced show the top s e c t i o n t o have a frequency p l a c i n g i t a r e s p e c t a b l e d i s t a n c e from any of the lower p e r i o d s of the t o t a l s t r u c t u r e . Another f e a t u r e of ' s t r u c t u r e A' which makes i t d i f f i c u l t t o a n a l y s e i s the presence of the s h o r t stubby beams. They were i n c l u d e d i n the model to r e p r e s e n t an o f f s e t i n a column c e n t e r -l i n e and had t o t r a n s f e r the r e s u l t i n g moments and downward f o r c e s w i t h o u t e x h i b i t i n g l a r g e d i f f e r e n c e s i n d e f l e c t i o n between t h e i r ends. Whi l e i t i s p o s s i b l e t o ' j u g g l e ' the degrees of freedom i n a s t r u c t u r e t o make the d e f l e c t i o n s of one p o i n t agree w i t h those of a n o t h e r , i t i s d i f f i c u l t t o do so w i t h o u t d e s t r o y i n g some of the e q u i l i b r i u m e q u a t i o n s f o r the s t r u c t u r e . A l t h o u g h i t was t e m p t i n g t o a s s i g n the same v e r t i c a l and r o t a t i o n a l d egrees of freedom t o c o r r e s p o n d i n g ends of the stubby beams, t h i s would have e l i m i n a t e d the c o r r e s p o n d i n g moment caused by the o f f s e t of the column l i n e . In view of the c o n s i s t e n c y of the r e s u l t s found when t e s t i n g the f i v e s t o r y s t r u c t u r e under the two modal programs, t h e r e was some c o n s i d e r a b l e s u r p r i s e and puzzlement when the r e s u l t s of a n a l y s i n g ' s t r u c t u r e A' showed the two programs t o d i f f e r by up t o one hundred p e r c e n t f o r some of the me.mber f o r c e s . At t h i s stage i t was not c e r t a i n which i f e i t h e r program was p r o d u c i n g the ' c o r r e c t ' answers and a l e n g t h y s e a r c h f o r the 55 cause of the d i f f e r e n c e s r e s u l t e d . The f i r s t thought was t h a t one of the programs was not dimensioned l a r g e enough f o r the s t r u c t u r e . The m o d i f i e d s u b s t i t u t e s t r u c t u r e method program was checked f o r t h i s by r u n n i n g on ' I n t e r a t i v e F o r t r a n ' . T h i s i s a F o r t r a n c o m p i l e r a v a i l a b l e on the UBC system which performs more e x t e n s i v e e r r o r c h e c k i n g than the s t a n d a r d f o r t r a n c o m p i l e r , i n c l u d i n g c h e c k i n g f o r d i m e n s i o n i n g e r r o r s . For t h i s reason i t i s more e x p e n s i v e t o run and i s used p r i m a r i l y f o r the 'debugging' of programs. The m o d i f i e d s u b s t i t u t e s t r u c t u r e method program passed the I n t e r a t i v e F o r t r a n t e s t and as 'DYNAMIC had a n a l y s e d s t r u c t u r e s of f a r g r e a t e r s i z e d i m e n s i o n i n g was e l i m i n a t e d as a cause of the d i f f e r e n c e s . I t was noted t h a t the f i r s t major d i s c r e p a n c i e s i n the a n a l y s e s appeared i n those v a l u e s p r i n t e d from the e i g e n v a l u e f i n d i n g r o u t i n e . Hence i n t e r e s t s h i f t e d t o the comparison of the i n f o r m a t i o n and more i m p o r t a n t l y the a r r a y s e n t e r i n g t h i s r o u t i n e . T e s t i n g and comparison of s t i f f n e s s and mass m a t r i c e s was performed by h a v i n g the programs m o d i f i e d t o p r i n t these a r r a y s on s e q u e n t i a l f i l e s . Other computer programs were then w r i t t e n which used t h e s e f i l e s as t h e i r i n p u t d a t a . The f i r s t of these a u x i l i a r y programs compared the s t i f f n e s s m a t r i x terms from the two dynamic a n a l y s i s programs on a term f o r term b a s i s . As the magnitudes of the terms v a r i e s c o n s i d e r a b l y w i t h i n the m a t r i x t h i s was done by computing a r a t i o between the elements r a t h e r than t r y i n g t o c a l c u l a t e a n u m e r i c a l d i f f e r e n c e between any two c o r r e s p o n d i n g terms. P r o v i s i o n s were made i n t h i s program t o a s c e r t a i n t h a t z e r o v a l u e d elements c o r r e s p o n d e d 56 w i t h o u t p r o d u c i n g i n f i n i t e v a l u e d r a t i o s d u r i n g t h i s c o mparison. The mass m a t r i c e s were a l s o c o p i e d t o t h e i r own s e q u e n t i a l f i l e s i n the same manner as the s t i f f n e s s m a t r i c e s and underwent s i m i l a r element t o element c o m p a r i s i o n s . A second a u x i l i a r y program p r o v i d e d the o p p o r t u n i t y t o c h e a p l y t e s t the s t i f f n e s s m a t r i c e s i n an e i g e n v a l u e r o u t i n e under c o n t r o l l e d c o n d i t i o n s . T h i s program was w r i t t e n such t h a t the o n l y i n p u t g i v e n t o i t was a s t i f f n e s s m a t r i x and a mass m a t r i x , each i n a s e p a r a t e s e q u e n t i a l f i l e . T h i s r o u t i n e produced e i g e n v a l u e s i n a manner which e l i m i n a t e d d i f f e r e n c e s not a t t r i b u t a b l e p u r e l y t o d i f f e r e n c e s i n the m a t r i c e s e n t e r i n g the e i g e n v a l u e f i n d i n g r o u t i n e . The o p e r a t i o n of the program c o n s i s t e d m o s t l y of t r a n s f e r r i n g the d a t a from the mass and s t i f f n e s s s e q u e n t i a l f i l e s i n t o a r r a y s , c a l l i n g on the e i g e n v a l u e f i n d i n g r o u t i n e s and p r i n t i n g the r e s u l t i n g e i g e n v a l u e s . By v a r y i n g the assignment of the i n p u t f i l e s i t was p o s s i b l e t o f i n d the e i g e n v a l u e s t h a t would be produced when the mass m a t r i x t h a t would be used i n one modal a n a l y s i s program i s p l a c e d i n t o the r o u t i n e accompanied by the s t i f f n e s s m a t r i x from a s e p a r a t e modal a n a l y s i s program. In t h i s manner the causes of the e i g e n v a l u e d i s c r e p a n c i e s c o u l d be u n i q u e l y d e t e r m i n e d . I t was t h r o u g h the use of these two a u x i l i a r y programs t h a t the n e c e s s i t y of extended p r e c i s i o n was a p p r e c i a t e d f o r the c o r r e c t a n a l y s i s of ' s t r u c t u r e A' At the time t h e s e t e s t s were b e i n g performed the m o d i f i e d s u b s t i t u t e s t r u c t u r e method program had been m o d i f i e d t o p e r m i t e l a s t i c modal a n a l y s i s but was s t i l l o p e r a t i n g c o m p l e t e l y i n 57 s i n g l e p r e c i s i o n . However, 'DYNAMIC c o n s t r u c t e d i t s s t i f f n e s s m a t r i x , computed e i g e n v a l u e s and v e c t o r s and c a r r i e d out most major o p t i o n s i n extended p r e c i s i o n . U s i n g the f i r s t a u x i l i a r y program i t was de t e r m i n e d t h a t the d i f f e r e n c e i n the s t i f f n e s s and mass m a t r i c e s was ve r y s l i g h t . When compared on an element f o r element b a s i s the members of the s t i f f n e s s m a t r i x produced by s i n g l e p r e c i s i o n were minus 0.126 p e r c e n t t o p l u s 0.028 p e r c e n t d i f f e r e n t from those produced by double p r e c i s i o n . However the e i g e n v a l u e f o r the f i r s t mode f o r these m a t r i c e s d i f f e r e d by almost ten p e r c e n t . T h i s d i f f e r e n c e g r a d u a l l y d e c r e a s e d w i t h i n c r e a s i n g mode number and the e i g e n v a l u e c o r r e s p o n d i n g t o the t e n t h mode was d i f f e r e n t by l e s s than a t e n t h of a p e r c e n t . Exchanging the mass m a t r i c e s used i n the e i g e n v a l u e r o u t i n e s d i d l i t t l e t o change the e i g e n v a l u e s produced by each of the s t i f f n e s s m a t r i c e s . . T h i s l e d t o the c o n c l u s i o n t h a t the d i f f e r e n c e l a y i n the- use of a s i n g l e p r e c i s i o n r o u t i n e or a double p r e c i s i o n r o u t i n e t o c o n s t r u c t the s t i f f n e s s m a t r i x . T h i s c o n c l u s i o n was v e r i f i e d by f u r t h e r t e s t s a f t e r the r o u t i n e which u t i l i z e d s i n g l e p r e c i s i o n was c o n v e r t e d t o double p r e c i s i o n by r e a s s i g n i n g the s t i f f n e s s m a t r i x f o r m a t i o n a r r a y s and v a r i a b l e s used t o double p r e c i s i o n . A p a r t from c h a n g i n g s i n g l e p r e c i s i o n r e a l c o n s t a n t s and v a r i a b l e s t o double p r e c i s i o n c o n s t a n t s , no changes were made i n the e x e c u t a b l e s t a t e m e n t s i n the r o u t i n e . Once these s t e p s had been implemented the e i g e n v a l u e s produced were e s s e n t i a l l y the same as those from the o r i g i n a l double p r e c i s i o n r o u t i n e . Minor d i f f e r e n c e s i n the o r d e r of one or two p e r c e n t c o u l d 58 now e a s i l y be t o l e r a t e d . These d i f f e r e n c e s were a t t r i b u t e d t o the d i f f e r e n t u n i t s of i n p u t and the v a r i a t i o n s i n a c c e l e r a t i o n spectrum as d e s c r i b e d e a r l i e r . These d i f f e r e n c e s cannot be regarded as s i g n i f i c a n t due t o the e x p e r i m e n t a l e r r o r s which a r e i n h e r e n t i n the p h y s i c a l measuring of the in p u t p r o p e r t i e s . A f t e r t h e s e changes were performed the program t h a t had been produced t o perfor m e l a s t i c modal a n a l y s i s and compute the damage r a t i o s e x p e c t e d by the m o d i f i e d s u b s t i t u t e s t r u c t u r e method was renamed 'EDAM' . T h i s stands f o r E l a s t i c and/or Damage A f f e c t e d Modal a n a l y s i s and d i f f e r e n t i a t e s the program from any o t h e r s u s i n g the method. 4.3 COMPARISON WITH ELASTIC TIME STEP RESULTS. Perhaps the most r i g o r o u s way t o check the e l a s t i c c a p a b i l i t e s of a modal a n a l y s i s program i s t o compare the r e s u l t s w i t h those produced by time s t e p a n a l y s i s u s i n g an earthquake which has a spectrum which matches t h a t used i n the modal a n a l y s i s . T h i s method of program e x a m i n a t i o n not o n l y d e t e r m i n e s i f the a l o g r i t h m i s o p e r a t i n g but a l s o a s c e r t a i n s the v i a b i l i t y of modal a n a l y s i s and the a p p r o p r i a t e n e s s of the spectrum. To c a r r y out the s e t e s t s a time s t e p program must be a c c e s s a b l e . At l e a s t two such o p t i o n s were a v a i l a b l e a t UBC w i t h the program chosen b e i n g DRAIN-2D. T h i s program has been de v e l o p e d a t the U n i v e r s i t y of C a l i f o r n i a at B e r k e l e y 3 and i t s use has been r e p o r t e d i n s e v e r a l s t u d i e s i n v o l v i n g time s t e p a n a l y s i s 7 •>2. The p r o p e r t i e s of the program w i l l be d i s c u s s e d more f u l l y i n s e c t i o n s of t h i s work d e a l i n g w i t h the i n e l a s t i c 59 t e s t i n g of the m o d i f i e d s u b s t i t u t e s t r u c t u r e method program. At t h i s time i t i s s u f f i c i e n t t o s t a t e t h a t DRAIN-2D has the c a p a b i l i t y t o compute the f o r c e and d i s p l a c e m e n t e n v e l o p e s f o r a s t r u c t u r e of f a i r l y a r b i t r a r y c o n f i g u r a t i o n and member p r o p e r t i e s when undergoing a s e t of a c c e l e r a t i o n s which are p a r t of the i n p u t d a t a . To a c h i e v e r e s u l t s u n a f f e c t e d by i n e l a s t i c a c t i o n i t i s noted t h a t DRAIN-2D performs e l a s t i c a n a l y s i s when the y i e l d moment of the members of the s t r u c t u r e i s not exceeded; t h i s i s e a s i l y p r e v e n t e d e i t h e r by s p e c i f y i n g a low a c c e l e r a t i o n or by s e t t i n g the y i e l d l e v e l of the members a t a h i g h v a l u e . In o r d e r t o a p p l y time s t e p a n a l y s i s t e s t s were undertaken u s i n g the program DRAIN-2D and the f i r s t t e n seconds of f o u r r e c o r d s , these b e i n g two components of the Kern County ( T a f t ) 1952 e arthquake and two components of the E l Ce n t r o 1940 e v e n t . The a c c e l e r a t i o n s were s p e c i f i e d f o r t h i s earthquake a t i n t e r v a l s c o r r e s p o n d i n g t o 0.02 Seconds and u s i n g a l i n e a r i n t e r p o l a t i o n between a c c e l e r a t i o n p o i n t s , a time s t e p i n t e r v a l c o r r e s p o n d i n g t o 100 h e r t z was used. In t e s t i n g the program EDAM a g a i n s t DRAIN-2D e l a s t i c runs were performed on a f i v e s t o r y frame, t h i s b e i n g S h i b a t a and Sozen's f i v e s t o r y frame m o d i f i e d by the a r b i t r a r y a d d i t i o n of 9 f o o t r i g i d arms on the beam ends. In the modal a n a l y s i s Spectrum 'A' from S h i b a t a and So z e n 7 was used as i t i s an a p p r o p r i a t e spectrum f o r the r e c o r d s chosen. T h i s spectrum i s shown i n F i g u r e 5.7. F i v e p e r c e n t damping was used i n both DRAIN-2D and modal a n a l y s i s r u n s . For the purpose of compa r i s o n , the l a r g e s t bending moment f o r each member was examined. As 'spectrum A' i s 60 an average spectrum f o r the earthquake r e c o r d s used, the r e s u l t s of the f o u r DRAIN-2D runs were averaged b e f o r e comparing w i t h those of modal a n a l y s i s . These r e s u l t s a r e p r e s e n t e d i n f i g u r e 4.4. The r e s u l t s can o n l y be d e s c r i b e d as e x c e l l e n t , w i t h the spectrum moments a l l b e i n g w i t h i n 6 p e r c e n t of those p r e d i c t e d by the average of the time s t e p r u n s . They form an almost t e x t book example of the v i a b i l i t y of the modal-spectrum approach t o e l a s t i c a n a l y s i s . I t s h o u l d be noted t h a t not o n l y were the comp u t a t i o n c o s t f o r modal a n a l y s i s an o r d e r of magnitude below those of the time s t e p a n a l y s i s but the da t a f i l e p r e p a r a t i o n f o r the modal a n a l y s i s was c o n s i d e r a b l y e a s i e r and l e s s time consuming. I t was thus through t e s t i n g s e v e r a l s t r u c t u r e s of v a r y i n g s i z e , c o m p l e x i t y and f e a t u r e s on two c o m p l e t e l y independent modal a n a l y s i s programs and a time s t e p program t h a t i t was e s t a b l i s h e d t h a t the program under e x a m i n a t i o n c o u l d t r u l y produce v a l i d r e s u l t s f o r e l a s t i c modal a n a l y s i s . T h i s i s an i m p o r t a n t s t e p i n d e t e r m i n i n g t h a t the program can produce v a l i d i n e l a s t i c r e s u l t s by a m o d i f i c a t i o n of the e l a s t i c method. 61 CHAPTER 5 TESTING THE INELASTIC PREDICTIONS OF THE METHOD. Wit h the e l a s t i c c a p a b i l i t y of the program e s t a b l i s h e d , we are i n a p o s i t i o n t o a s s e s s the a c c u r a c y of the method w i t h r e s p e c t t o i n e l a s t i c b e h a v i o r . However, t h i s i s not as s i m p l e as the t e s t f o r e r r o r s i n the e l a s t i c range. W h i l e e l a s t i c modal a n a l y s i s i s a w e l l e s t a b l i s h e d p r a c t i c e , the use of the m o d i f i e d s u b s t i t u t e s t r u c t u r e method t o p r e d i c t i n e l a s t i c a c t i o n s i s t r e a d i n g on much newer ground. The v i a b i l i t y of the method was a s s e s s e d i n two ways. The f i r s t was t o examine the t r e n d s i n d u c t i l i t y demand p r e d i c t e d by the program, comparing these t o those t r e n d s r e p o r t e d by o t h e r r e s e a r c h e r s i n the l i t e r a t u r e . The second approach was t o examine s e v e r a l t e s t s t r u c t u r e s which c o u l d be compared on a n u m e r i c a l b a s i s w i t h r e s u l t s o b t a i n e d from the i n e l a s t i c time s t e p a n a l y s i s program DRAIN-2D. 5.1 L i t e r a t u r e Comparison of Damage P a t t e r n s Many r e s e a r c h e r s have shown t h a t the d u c t i l i t y demand on the c o u p l i n g beams of w a l l systems i s h i g h e s t i n the a r e a of o n e - t h i r d the d i s t a n c e up the h e i g h t of the s t r u c t u r e . T h i s 62 caused c o n c e r n d u r i n g the f i r s t a t t e m p t s t o compare damage p a t t e r n s from the m o d i f i e d s u b s t i t u t e s t r u c t u r e method w i t h those of p u b l i s h e d p a p e r s . A l l the i n i t i a l t e s t s t r u c t u r e s t h a t were m o d e l l e d , a l t h o u g h a p p a r e n t l y r e a s o n a b l e i n t h e i r p r o p e r t i e s , showed the h e a v i e s t damage r a t i o s t o occur i n the c o u p l i n g beams at the top of the w a l l s . Causes f o r t h i s d i s c r e p e n c y a re r e l a t e d below. The reason t h a t the maximum d u c t i l i t y demand o c c u r s below the t o p of the s t r u c t u r e , as found by o t h e r r e s e a r c h e r s , l i e s i n the d u a l method of l a t e r a l l o a d c a r r y i n g by the w a l l . The l a t e r a l f o r c e i m p a r t s a f l e x u r a l d e f l e c t i o n t o the w a l l s which, i f the l i n t e l s had a low moment c a p a c i t y , would put the l a r g e s t damage r a t i o s f o r the s t r u c t u r e i n the top l i n t e l . T h i s e f f e c t i s o f f s e t , however, because when the shear i n the l i n t e l causes a x i a l f o r c e s i n the w a l l s , the r e s u l t i n g a x i a l d e f o r m a t i o n s r e l i e v e some of the f l e x u r a l s t r e s s i n the c o u p l i n g beams. The e f f e c t i s much more d r a m a t i c towards the top of the s t r u c t u r e as the a x i a l w a l l d e f o r m a t i o n s a re c u m u l a t i v e from the base. When t e s t s were performed on the program EDAM i n v o l v i n g a s i x t e e n - s t o r y s t r u c t u r e w i t h w a l l and beam s e c t i o n p r o p e r t i e s s i m i l a r t o the e i g h t e e n - s t o r y b u i l d i n g which P a u l a y 4 had a n a l y s e d by the l a m i n a r method, i t was found t h a t the maximum c o u p l i n g beam damage r a t i o s p r e d i c t e d by the program o c c u r r e d i n the range of o n e - t h i r d t o o n e - h a l f of the height- of the s t r u c t u r e , c o n f i r m i n g P a u l a y ' s p r e d i c t i o n s . To examine the e f f e c t t h a t t h i s a x i a l s h o r t e n i n g of the w a l l s has on the damage r a t i o s , a n o ther run was performed i n which the w a l l a r e a was m u l t i p l i e d by a f a c t o r of ten w h i l e a l l o t h e r s t r u c t u r a l d e t a i l s 63 were h e l d c o n s t a n t . As expec ted , the a x i a l de format ions of the w a l l s were reduced by one order of magnitude, and the c e n t e r of major c o u p l i n g beam damage s h i f t e d towards the top of the s t r u c t u r e . The a x i a l d e f o r m a t i o n s decrease the damage r a t i o s of the c o u p l i n g beams in the s t r u c t u r e and , in t h i s c a s e , the l a r g e r area w a l l s l e d to damage r a t i o s 30 percent g r e a t e r than those of the o r i g i n a l r u n . The s h i f t i n g of the l a r g e s t damage r a t i o downward from the top of the s t r u c t u r e can o n l y be expected to occur when the a x i a l de format ions of the w a l l s are s i g n i f i c a n t r e l a t i v e to the d i s p l a c e m e n t s of the c o u p l i n g beam ends . Hence i t w i l l be l e s s pronounced in s t r u c t u r e s wi th wide w a l l s , as t h i s tends to i n c r e a s e the d i sp lacement of the l i n t e l beam ends . I t w i l l a l s o be l e s s pronounced when the l i n t e l s are more f l e x i b l e or have lower y i e l d moments, s i n c e each of these reduces the shear in the l i n t e l and hence the a x i a l f o r c e and d e f o r m a t i o n caused in the w a l l s . F i n a l l y , the e f f e c t w i l l be l e s s prominent i n those s t r u c t u r e s which have w a l l s w i th a h i g h r a t i o of c r o s s - s e c t i o n a l a r e a to c r o s s - s e c t i o n a l moment of i n e r t i a . The s t r u c t u r e t h a t was t e s t e d by Paulay was mode l l ed from an e l e v a t o r or s t a i r sha f t w a l l and was composed of two channe l s connected by c o u p l i n g beams. Compared to the s t r u c t u r e s wi th s i m p l e p l a n a r w a l l s t e s t e d h e r e , t h i s s t r u c t u r e had a much lower r a t i o of area to moment of i n e r t i a , and narrower j o i n t s (model led by s h o r t e r r i g i d a r m s ) . I t shou ld be noted that P a u l a y ' s s t r u c t u r e shows up one of the f a i l i n g s of the m o d i f i e d s u b s t i t u t e s t r u c t u r e method as p r e s e n t l y f o r m u l a t e d : i t i s on ly t r u l y a p p l i c a b l e to members w i th symmetric s e c t i o n s . T h i s i s a 64 r e s u l t of the assumption that a l l members w i l l have the same ul t i m a t e moment r e g a r d l e s s of which sid e i s i n compression. In a channel s e c t i o n , t h i s assumption i s not v a l i d , as the moment c a p a c i t y i s as unsymmetrical as the concrete d i s t r i b u t i o n about the n e u t r a l a x i s . The a n a l y s i s of such a coupled channel s e c t i o n by the method w i l l only be v a l i d i f any i n e l a s t i c behavior i s r e s t r i c t e d to the c o u p l i n g beams. These comments on the i n f l u e n c e of a x i a l deformations on n o n l i n e a r behavior, poi n t to one of the problems that would be encountered with 'lumping' w a l l s of a s t r u c t u r e to reduce computation c o s t s . While i t may o f t e n be p o s s i b l e to combine w a l l s that are e x a c t l y s i m i l a r by m u l t i p l y i n g the s t r u c t u r a l p r o p e r t i e s and loads of the f i r s t w a l l by the number of s i m i l a r w a l l s , t h i s procedure may l e a d to d i f f i c u l t i e s with d i s s i m i l a r w a l l s . I f the two w a l l s that are 'lumped' together have d i f f e r e n c e s i n s t i f f n e s s p r o p e r t i e s , then the damage r a t i o s so determined w i l l be i n c o r r e c t . Another p o i n t f o r p r a c t i c a l c o n s i d e r a t i o n i s that during the l a t e r a l a n a l y s i s of a s t r u c t u r e i t i s common to ignore the columns, although they are awarded a f i x e d p r o p o r t i o n of the v e r t i c a l l o a d . While t h i s might be a v a l i d assumption where the w a l l i s undergoing small v e r t i c a l deformations, the columns would i n t e r a c t to c a r r y d i f f e r e n t v e r t i c a l loads i f the v e r t i c a l deformations of the w a l l s should get too l a r g e . As the Paulay s t r u c t u r e shows damage p a t t e r n s q u i t e t y p i c a l of the f i n d i n g s of other r e s e a r c h e r s , i t was concluded a f t e r examining t h i s s t r u c t u r e that the m o d i f i e d s u b s t i t u t e s t r u c t u r e method was capable of reproducing the general damage p a t t e r n s 65 c o r r e c t l y . 5.2 Assumptions f o r Comparison w i t h a Time Step A n a l y s i s Program A f t e r e s t a b l i s h i n g t h a t the p a t t e r n p r e d i c t i o n was r e a s o n a b l e i t was then n e c e s s a r y t o examine n u m e r i c a l p r e d i c t i o n s of i n e l a s t i c b e h a v i o r by comparison w i t h t i m e - s t e p r e s u l t s . The r e q u i r e m e n t s and c h o i c e of a time s t e p a n a l y s i s program t h a t i s v i a b l e f o r the a n a l y s i s of s t r u c t u r a l w a l l s i s governed by both m a t e r i a l , s t r u c t u r a l and g e o m e t r i c a l c o n s i d e r a t i o n s . A n a l y s i s of s t r u c t u r a l w a l l s c o n s t r u c t e d of c o n c r e t e r e q u i r e s a h y s t e r e s i s l o o p t h a t i s a p p r o p r i a t e f o r t h a t m a t e r i a l . T h i s c o n s i d e r a t i o n e l i m i n a t e s many f i n i t e element programs which, w h i l e s a t i s f a c t o r y i n a l l o t h e r r e s p e c t s , c o n s i d e r a c o n c r e t e member o n l y i n terms of pure e l a s t o p l a s t i c i t y , not d i f f e r e n t i a t i n g between l o a d i n g and u n l o a d i n g s t i f f n e s s c u r v e s or o t h e r items which a r e i m p o r t a n t i n the post y i e l d a n a l y s i s of c o n c r e t e members. I t i s a l s o n e c e s s a r y , t h a t any method used t o check the assumptions af a n other method do so i n a manner t h a t t a k e s a thorough account of the f a c t o r s most l i k e l y t o i n f l u e n c e the r e s u l t s . In the a n a l y s i s of c o u p l e d w a l l s i t i s i m p o r t a n t t h a t t h e r e be no r e s t r i c t i n g assumptions c o n c e r n i n g the l o c a t i o n of i n f l e c t i o n p o i n t s i n the members, s i n c e t h e s e p o i n t s w i l l be v e r y d i f f e r e n t l y l o c a t e d i n the c o u p l i n g beams and i n the w a l l s . In the fundamental response mode, the w a l l s w i l l a c t l i k e two c a n t i l e v e r s w i t h a l a r g e base moment. Thus, most, i f not a l l , 66 the w a l l segments w i l l have no i n f l e c t i o n p o i n t s , whereas in the c o u p l i n g beams, there w i l l g e n e r a l l y be a c e n t r a l p o i n t of i n f e c t i o n . Another s t r u c t u r a l f a c t o r worthy of c o n s i d e r a t i o n i s the i n t e r a t i o n between the a x i a l load and the y i e l d moment of the w a l l s . The w a l l s w i l l be subjected to a l t e r n a t i n g t e n s i l e and compressive loads; while the l a t t e r w i l l i n c r e a s e the moment c a p a c i t y of the w a l l , t e n s i l e loads may lower the c a p a c i t y to the p o i n t where y i e l d i n g occurs. Thus the nodes should be allowed three degrees of freedom to permit a x i a l deformation of the w a l l s , to show the reduced d u c t i l i t y demands on the c o u p l i n g beams, and r e f l e c t the a x i a l f o r c e imparted to the w a l l s and consequent change in y i e l d moment. It i s these c o n d e r a t i o n s , as w e l l as the d e s i r e to use a reputable time step a n a l y s i s program, which l e d to the choice of DRAIN-2D. Through other s t u d i e s 7 , i n c l u d i n g experimental work, the program has been demonstrated to have the c a p a c i t y to handle s t r u c t u r a l w a l l s and to produce reasonable r e s u l t s . DRAIN-2D was w r i t t e n at the U n i v e r s i t y of C a l i f o r n i a at B e r k e l e y 3 . The program uses a step-by-step dynamic a n a l y s i s procedure in which an a c c e l e r a t i o n , s p e c i f i e d as part of the input data, acts upon a s t r u c t u r e of a r b i t r a r y c o n f i g u r a t i o n . The program handles the d egradation of concrete s t i f f n e s s with the use of an extended v e r s i o n of Takeda's model, and i s capable of r e f l e c t i n g the e f f e c t of a x i a l f o r c e on the y i e l d moment of c o n c r e t e s e c t i o n s . Test s t r u c t u r e s were chosen to t e s t the method i n a v a r i e t y of s i t u a t i o n s c o v e r i n g a comprehensive range of the r e l e v a n t parameters, while attempting to reduce the s t r u c t u r e s t e s t e d to 67 a reasonable number. The s t r u c t u r e s were modelled by a set of l i n e members connected by j o i n t s l o c a t e d at each f l o o r l e v e l to which the members were r i g i d l y connected at each end.. Hence each wa l l i s broken i n t o a number of segments equal to the number of s t o r i e s i n the s t r u c t u r e . The j o i n t s d e s c r i b i n g the l o c a t i o n of the w a l l s were placed on the n e u t r a l a x i s of the uncracked s e c t i o n . S t r u c t u r a l p r o p e r t i e s used i n the t e s t s t r u c t u r e s were based on member s i z e s and p r o p e r t i e s approximating those used i n p r a c t i c e . Thus, member s e c t i o n s with reasonable m a t e r i a l p r o p e r t i e s , s t e e l q u a n t i t i e s and l o c a t i o n s were analysed to get the member p r o p e r t i e s . Although the area and i n i t i a l moment of i n e r t i a of the wa l l s were u s u a l l y h e l d constant throughout the height of the s t r u c t u r e , the y i e l d moment of the wal l was assumed to vary l i n e a r l y throughout the height of the s t r u c t u r e . T h i s was to r e f l e c t the f a c t that the moment c a p a c i t y of a w a l l i n c r e a s e s with i n c r e a s i n g a x i a l l o a d .toward the base of the s t r u c t u r e . T h i s l a t t e r p o i n t turned out to be somewhat academic f o r the s t r u c t u r e s t e s t e d , s i n c e when hin g i n g o c c u r r e d in the w a l l s i t always took p l a c e in the bottom s t o r y . For the s t r u c t u r e s used i n t h i s study, the re d u c t i o n of the moment c a p a c i t y because of d e c r e a s i n g dead l o a d at g r e a t e r heights i n the w a l l always had a much smal l e r e f f e c t than the re d u c t i o n i n the a p p l i e d moment as a f u n c t i o n of h e i g h t . The u l t i m a t e moment-axial f o r c e d i s t r i b u t i o n f o r the member was obtained on the b a s i s of standard concrete s e c t i o n a n a l y s i s . A l i n e a r s t r a i n r e l a t i o n s h i p was used with a maximum compressive s t r a i n of 0.003 in the c o n c r e t e . The Whitney s t r e s s block with ACI code p r o v i s i o n s , was used to compute the c o n t r i b u t i o n of the 68 c o n c r e t e t o the c a p a c i t y of the s e c t i o n . C o n s i s t e n t w i t h these p r o v i s i o n s , no s t r e n g t h was g i v e n t o the c o n c r e t e i n t e n s i o n . The s t e e l , assumed t o be p l a c e d i n d i s c r e t e l a y e r s , was mo d e l l e d as p e r f e c t l y e l a s t i c - p e r f e c t l y p l a s t i c i n both t e n s i o n and com p r e s s i o n . The l a y e r s of s t e e l f r e q u e n t l y regarded as 'temperature s t e e l ' were i n c l u d e d i n these a n a l y s e s as t h e i r l a r g e l e v e r arms produce a s i z a b l e c o n t r i b u t i o n t o the moment c a p a c i t y of the member. Should the e n g i n e e r view t h i s as an e s o t e r i c e x e r c i s e a p p l i c a b l e o n l y t o the r e s e a r c h e r w i t h a c c e s s t o l a r g e computer funds, i t i s w o r t h w h i l e commenting t h a t the c a l c u l a t i o n of the u l t i m a t e moment-axial c u r v e s f o r s e c t i o n s w i t h up t o 19 l a y e r s of s t e e l were a l l performed on a programmable pocket c a l c u l a t o r . A t y p i c a l c urve take a p p r o x i m a t l y o n e - h a l f hour t o c a l c u l a t e and p l o t , i n c l u d i n g the i n p u t of s e c t i o n d a t a . The w a l l s were connected by a s e r i e s of c o u p l i n g beams, whose s e c t i o n a l p r o p e r t i e s and c a p a c i t y were kept c o n s t a n t throughout the w a l l h e i g h t . The c o u p l i n g beams were mo d e l l e d as a member w i t h t h r e e s e c t i o n s , a deformable c e n t r a l r e g i o n e q u a l i n l e n g t h t o the c l e a r s p a n of the member, and two r i g i d ends s t r e t c h i n g from the fa c e of the w a l l t o i t s c e n t e r - l i n e . The' method of i n c l u d i n g t h i s member i n the m o d i f i e d s u b s t i t u t e s t r u c t u r e program has been d e s c r i b e d i n s e c t i o n 3.4. I t might w e l l occur i n p r a c t i c e t h a t d u r i n g the r e s i s t a n c e of the s e i s m i c f o r c e s , the n e u t r a l a x i s of the w a l l s h i f t s away from i t s l o c a t i o n i n the uncracked w a l l . T h i s would have the e f f e c t of changing the l e n g t h of the r i g i d arms and the r e s u l t i n g f o r c e s a p p l i e d t o the c o u p l i n g beams, u s u a l l y i n an u n c o n s e r v a t i v e 69 manner. T h i s p o i n t seems to be i g n o r e d in time s t ep a n a l y s i s of s t r u c t u r a l w a l l s and the e f f e c t i s a l s o not c o n s i d e r e d in the m o d i f i e d s u b s t i t u t e s t r u c t u r e a n a l y s i s . Shear d e f l e c t i o n was not i n c l u d e d i n the c a l c u l a t i o n of member f o r c e s and d i s p l a c e m e n t s . The v a l i d i t y of t h i s assumption w i l l be demonstrated in an example l a t e r in t h i s c h a p t e r . To de termine the masses that should be a p p l i e d i t was assumed tha t the w a l l s were spaced at f i f t y fee t normal to t h e i r own p lane and that the l o a d on each f l o o r was 150 l b / s q f t . , t h i s be ing a combinat ion of dead and l i v e l o a d . Each w a l l was assumed to have a t r i b u t a r y area equa l to i t s l e n g t h p l u s h a l f the span of the l i n t e l beam t imes f i f t y f e e t . I t was a l s o assumed t h a t whi le the s t r u c t u r e c o u l d be imagined as hav ing columns t a k i n g up about f i f t y percent of t h i s l o a d in the v e r t i c a l d i r e c t i o n , the h o r i z o n t a l mass sh ou ld comprise the t o t a l l o a d on the t r i b u t a r y a r e a . Due to the g r e a t e r s t i f f n e s s of the w a l l s they would ac t to take the h o r i z o n t a l f o r c e long be fore the columns took any h o r i z o n t a l l o a d . The v e r t i c a l force on the w a l l s was used o n l y to determine the u l t i m a t e moment c a p a c i t y of the w a l l from i t s u l t i m a t e moment-axia l curve when u s i n g the program EDAM and no v e r t i c a l f o r c e s were p l a c e d on the s t r u c t u r e d u r i n g dynamic r u n s . In the program DRAIN-2D, the c a p a b i l i t y e x i s t s to reproduce the moment-ax ia l curve for the member and to p l a c e s t a t i c p r e l o a d s on the s t r u c t u r e before the dynamic a n a l y s i s b e g i n s . Hence, the v e r t i c a l f o r c e s which had been used in c a l c u l a t i n g the u l t i m a t e c a p a c i t y of members for the program EDAM were p l a c e d as p r e d e f i n e d s t a t i c loads for the time s tep a n a l y s i s . 70 I n i t i a l t e s t s of the program EDAM had shown t h a t , with the exception of a x i a l f o r c e s in the l i n t e l s , the r e s u l t s were s i m i l a r r e g a r d l e s s of whether one or two masses per f l o o r were attached as long as the t o t a l mass was kept c o n s t a n t . A l s o , computation c o s t s i n c r e a s e d as the number of a t t a c h e d masses i n c r e a s e d . T h e r e f o r e , i t was decided that only one mass per f l o o r should be assumed. When using the DRAIN-2D a n a l y s i s however, two masses per f l o o r were attached, p a r t l y to see i f the a x i a l f o r c e s generated in the l i n t e l s were as low as expected, and p a r t l y to check that these f o r c e s c o u l d be assumed to have a n e g l i g i b l e e f f e c t on the moment c a p a c i t i e s . In p r a c t i c e any a x i a l f o r c e i n the l i n t e l beams would be p a r t i a l l y d i s s i p a t e d i n the f l o o r s l a b s which, though weak in f l e x u r e , provide a good a x i a l c o n n e c t i o n . The dynamic a n a l y s i s of r e i n f o r c e d concrete s t r u c t u r e s f r e q u e n t l y provokes debate on the a p p r o p r i a t e member p r o p e r t i e s . In the a n a l y s i s for s t a t i c loads, gross moments of i n e r t i a and c r o s s - s e c t i o n a l areas are f r e q u e n t l y used, p a r t l y because the r e s u l t s w i l l be l i t t l e a f f e c t e d by other refinements as long as a l l members are t r e a t e d c o n s i s t e n t l y , but a l s o because b e t t e r estimates are o f t e n not a v a i l a b l e i n the a n a l y s i s stages. In the a n a l y s i s of dynamic loads t h i s assumption cannot be made so l i g h t l y . I f the cracked moment of i n e r t i a i s used i n s t e a d of the gross moment of i n e r t i a , then the f l e x i b i l i t y w i l l be a f f e c t e d and hence, the p e r i o d and dynamic loads a c t i n g on the s t r u c t u r e . Shibata and Sozen's 8 o r i g i n a l development of the S u b s t i t u t e S t r u c t u r e method took t h i s i n t o account by proposing that the gross moment of i n e r t i a be used, but that c r a c k i n g be accounted 71 f o r by d i v i d i n g by 2 i f a x i a l compression i s p r e s e n t or o t h e r w i s e by 3. T h i s scheme was used i n the work on the m o d i f i e d s u b s t i t u t e s t r u c t u r e method performed by Y o s h i d a 1 0 . The a s s i g n i n g of s t i f f n e s s v a l u e s i n the use of the program DRAIN-2D i s not as s i m p l e a p r o c e d u r e . The manual f o r the program s u g g e s t s u s i n g the f l e x u r a l r i g i d i t y v a l u e f o r the c r a c k e d s e c t i o n , though i t notes t h a t " c o n s i d e r a b l e e x p e r i e n c e and e x p e r i m e n t a t i o n w i l l be needed b e f o r e the element p r o p e r t i e s can be s p e c i f i e d w i t h c o n f i d e n c e " 3 . The use of the c r a c k e d s e c t i o n i s i m p o r t a n t s i n c e the h y s t e r e s i s r u l e s employed i n the program DRAIN-2D use the same s e c t i o n modulus up t o f i r s t y i e l d i n g ; u s i n g the g r o s s s e c t i o n modulus throughout t h i s range would c l e a r l y i n v o l v e too l a r g e a s t i f f n e s s . I t was d e c i d e d t o base the c r a c k e d s e c t i o n modulus on the same assumptions t h a t were used i n the M o d i f i e d S u b s t i t u t e S t r u c t u r e method. T h i s i n s u r e s t h a t u n y i e l d e d members have the same p r o p e r t i e s i n both a n a l y s e s . In any c a s e , more d e t a i l e d a p p r o x i m a t i o n s of the c r a c k e d s e c t i o n modulus a r e are u s u a l l y beyond the scope of a d e s i g n method. For a x i a l s t i f f n e s s , the t o t a l s e c t i o n a rea was i n p u t i n both c a s e s because, a l t h o u g h c r a c k i n g would be e x p e c t e d t o de c r e a s e the s e c t i o n modulus, i t would not be expected t o a f f e c t response t o a c o m p r e s s i v e a x i a l l o a d s i g n i f i c a n t l y . Were the member t o be i n t e n s i o n , i t would be ex p e c t e d t h a t the g r o s s a r e a would be too g r e a t , but n e i t h e r DRAIN-2D nor the program EDAM reduce the a r e a s of members t o ta k e account of c r a c k i n g , and r e s u l t s f o r s t r u c t u r e s w i t h c o n c r e t e members i n t e n s i o n s h o u l d be viewed w i t h c a u t i o n . 72 I t was always assumed t h a t the s t r u c t u r e s t e s t e d were r i g i d l y c o n n e c t e d t o an u n y i e l d i n g f o u n d a t i o n . T h i s would r e p r e s e n t the commoner case where shear w a l l s t e r m i n a t e i n more massive basement w a l l s . I t i s not n e c e s s a r y t o choose a v a l u e of damping f o r the m o d i f i e d s u b s t i t u t e s t r u c t u r e method as damping i s d e t e r m i n e d d u r i n g e x e c u t i o n . However, i t i s n e c e s s a r y t o d etermine such a v a l u e f o r use w i t h the program DRAIN-2D, and 2 p e r c e n t of c r i t i c a l was chosen as r e f l e c t i n g normal e l a s t i c damping; i t was i n c l u d e d as s t i f f n e s s p r o p o r t i o n a l damping. The.energy l o s t i n h y s t e r e t i c damping by a s t r u c t u r e undergoing i n e l a s t i c a c t i o n i s a u t o m a t i c a l l y accounted f o r w i t h the program DRAIN-2D. I t was n e c e s s a r y t o choose an a p p r o p r i a t e s e t of e a r t h q u a k e s f o r the time s t e p a n a l y s i s and a matching spectrum f o r the m o d i f i e d s u b s t i t u t e s t r u c t u r e a n a l y s i s . T h i s was r e s o l v e d by u s i n g the same earthquakes and spectrum t h a t had been used i n the e a r l y e x a m i n a t i o n of the s u b s t i t u t e s t r u c t u r e method by S h i b a t a and S o z e n 8 . The e a r t h q u a k e s used, i n c l u d i n g a p p r o p r i a t e d e t a i l s , may be found i n t a b l e 5.1. The r e c o r d s were s c a l e d l i n e a r l y t o g i v e a peak a c c e l e r a t i o n e q u a l t o the d e s i r e d maximum ground a c c e l e r a t i o n f o r the s t r u c t u r e . The spectrum used i n the m o d i f i e d s u b s t i t u t e s t r u c t u r e method was spectrum 'A' which had been developed by S h i b a t a and Sozen". Most of the s t r u c t u r e s t e s t e d were of a form shown i n f i g u r e 5.1 and r e p r e s e n t a s i n g l e p a i r of c o u p l e d w a l l s . R e l e v a n t s t r u c t u r e p r o p e r t i e s of these w a l l s can be found i n t a b l e 5.2. In comparing d u c t i l i t y v a l u e s i t i s v i t a l t o a s c e r t a i n t h a t 73 a s i m i l a r d e f i n i t i o n i s used f o r t h i s term i n a l l c a s e s . A l o g i c a l d e f i n i t i o n used f o r both time s t e p a n a l y s i s and the m o d i f i e d s u b s t i t u t e s t r u c t u r e method i s t h a t of member d u c t i l i t y . T h i s can be d e f i n e d w i t h r e s p e c t t o the a n g l e between the tangent t o the member at i t s end and the c h o r d j o i n i n g the ends of the member: i t i s the v a l u e of t h i s a n g l e a t response d i v i d e d by the v a l u e a t f i r s t y i e l d . The measurement of t h i s a n g l e a l o n g w i t h a more f a m i l i a r view of i t from a paper by P a u l a y ' i s shown i n f i g u r e 5.2. T h i s i s e q u i v a l e n t t o the term 'damage r a t i o ' used i n the m o d i f i e d s u b s t i t u t e s t r u c t u r e method. A l t h o u g h t h i s can be measured a t two ends of any member under both p o s i t i v e and n e g a t i v e moment g i v i n g f o u r p o s s i b l e v a l u e s of d u c t i l i t y , some of which may be e q u a l , the l a r g e s t d u c t i l i t y demand d e t e r m i n e d i s the one of concern and the one t h a t i s used i n the comparisons t h a t f o l l o w . 5.3 R e s u l t s and Comparisons w i t h Time Step Programs  (a) F i v e S t o r y S t r u c t u r a l W a l l The f i r s t i n e l a s t i c t e s t s t r u c t u r e s c o n s i s t e d of t h r e e s e t s of f i v e s t o r y s t r u c t u r a l w a l l s , used t o examine the a p p l i c a b i l i t y of the method t o s m a l l s t r u c t u r a l w a l l s . Two v a l u e s of c o u p l i n g beam c a p a c i t y , 60 K i p - F t and 100 K i p - F t were t e s t e d a t a maximum ground a c c e l e r a t i o n of 20 p e r c e n t of g r a v i t y . The h i g h e r beam c a p a c i t y was a l s o t e s t e d a t a ground a c c e l e r a t i o n of 50 p e r c e n t of g r a v i t y . A l t h o u g h c h a n g i n g the c a p a c i t y of the l i n t e l - beams and maximum a c c e l e r a t i o n a l t e r e d the amount of i n e l a s t i c t y i n the s t r u c t u r e s , none of the changes 74 a l t e r e d the i n i t i a l e l a s t i c p e r i o d of the s t r u c t u r e . The r e s u l t s of tnese t e s t s were a l l v e r y s i m i l a r - - t h e m o d i f i e d s u b s t i t u t e s t r u c t u r e method p r e d i c t e d c o r r e c t l y the p a t t e r n of d u c t i l i t y r e q u i r e m e n t s and d e f l e c t i o n s but was v e r y c o n s e r v a t i v e , p r e d i c t i n g v a l u e s 50 t o 100 p e r c e n t g r e a t e r than DRAIN-2D runs. The r e s u l t s f o r ' s e r i e s B' t e s t s on the f i v e s t o r y w a l l (which used 100 K i p - F t l i n t e l s and 20 p e r c e n t g r a v i t y ) a re shown i n f i g u r e 5.3 and 5.4 f o r d u c t i l i t y and d e f l e c t i o n . The f i v e - s t o r y w a l l examined i n the o r i g i n a l t e s t s was very s t i f f and w i t h the mass used, had a fundamental p e r i o d of o n l y 0.22 seconds. For a g i v e n damping, S h i b a t a and Sozen's spectrum 'A' i s c o n s t a n t between 0.15 and 0.4 seconds so t h a t any s o f t e n i n g of s t r u c t u r e s f a l l i n g i n t h i s p e r i o d range w i l l not r e s u l t i n a l o w e r i n g of the s p e c t r a l a c c e l e r a t i o n r e s p onse. As noted i n c h a p t e r 2 t h i s c o n t r a v e n e s one of the r e s t r i c t i o n s on the s u b s t i t u t e s t r u c t u r e method. To examine the e f f e c t of i n c r e a s i n g the fundamental undamaged p e r i o d t o more than 0.4 seconds, the o r i g i n a l mass used i n the f i v e - s t o r y w a l l a n a l y s i s was m u l t i p l i e d by a f a c t o r of 4. T h i s changed the fundamental p e r i o d t o 0.45 seconds. The s t r u c t u r e w i t h t h i s r e v i s e d mass was then a n a l y s e d by both the m o d i f i e d s u b s t i t u t e s t r u c t u r e method and by DRAIN-2D. R e s u l t s i n terms of d e f l e c t i o n and d u c t i l i t y demand a r e shown i n f i g u r e s 5.5 and 5.6; they a r e c o n s i d e r a b l y more e n c o u r a g i n g as they i n d i c a t e r e s u l t s f o r ' the m o d i f i e d s u b s t i t u t e s t r u c t u r e method more a k i n t o the average of the fo u r t i me s t e p r e s u l t s . From th e s e r e s u l t s i t was c o n c l u d e d t h a t the m o d i f i e d s u b s t i t u t e s t r u c t u r e method, w h i l e g i v i n g q u a l i t a t i v e l y c o r r e c t 75 damage and d e f l e c t i o n p a t t e r n s , may g i v e r e s u l t s t h a t are n u m e r i c a l l y c o n s e r v a t i v e when the a c c e l e r a t i o n response does not decrease w i t h p e r i o d . F i g u r e 5.7 shows spectrum 'A' a l o n g w i t h fundamental p e r i o d s of the s t r u c t u r e s examined i n t h i s s t u d y . The undamaged fundamental p e r i o d s h o u l d be g r e a t e r than 0.4 seconds f o r a c c u r a t e r e s u l t s t o be produced w i t h t h i s spectrum. (b) T e n - s t o r y w a l l The next s e t of t e s t s was performed on a t e n - s t o r y c o u p l e d w a l l . F i g u r e 5.9 shows the d e f l e c t i o n r e s u l t s f o r these t e s t s w h i l e f i g u r e 5.10 shows g r a p h i c a l l y the d u c t i l i t y demand of the c o u p l i n g beams. A l t h o u g h the r e s u l t s of the t e s t s show the m o d i f i e d s u b s t i t u t e s t r u c t u r e method p r o v i d e s a c o n s e r v a t i v e v a l u e s f o r the r e c o r d s used, both d e f l e c t i o n and d u c t i l i t y e s t i m a t e s a re v e r y r e a s o n a b l e . W h i l e the m o d i f i e d s u b s t i t u t e s t r u c t u r e method p r e d i c t s a d e f l e c t i o n f o r the s t r u c t u r e of 3.75 i n c h e s the d e f l e c t i o n e n v e l o p e s produced from the DRAIN-2D computer runs i n d i c a t e a t o p d e f l e c t i o n of 2.5 t o 3.5 i n c h e s . In terms of d u c t i l i t y demand, the m o d i f i e d s u b s t i t u t e s t r u c t u r e method p r e d i c t s the l a r g e s t c o u p l i n g beam damage r a t i o t o be 7.55 w h i l e DRAIN-2D runs i n d i c a t e t h a t i t l i e s between 5.05 and 7.25. When u n c e r t a i n t i e s of the s t r u c t u r e and earthquake parameters a r e c o n s i d e r e d the r e s u l t s f o r t h i s t e n - s t o r y w a l l a r e v e r y e n c o u r a g i n g . 76 (c) S i x t e e n - s t o r y w a l l w i t h an e x t r a uncoupled w a l l . The next t e s t s were performed on a s i x t e e n - s t o r y w a l l which had been p r e v i o u s l y r e p o r t e d by F i n t e l and G o s h 2 . The i n i t i a l r e s u l t s f o r t h i s w a l l a re shown i n f i g u r e 5.10 which shows the d u c t i l i t y demand of the c o u p l i n g beams e s t i m a t e d w i t h f o u r d i f f e r e n t s e t s of s t r u c t u r a l parameters u s i n g the program DRAIN-2D w i t h the f i r s t 10 seconds of the E l C e n t r o East-West r e c o r d . The f i r s t of these i s cu r v e 'A' which c o r r e s p o n d s t o the d u c t i l i t y demand e s t i m a t e d by F i n t e l f o r the l a r g e s t p o s s i b l e earthquake f o r the s t r u c t u r e . A l t h o u g h t h e s e r e s u l t s were o b t a i n e d from the U n i v e r s i t y of B r i t i s h Columbia v e r s i o n of DRAIN-2D they agree w e l l w i t h those r e s u l t s p u b l i s h e d by F i n t e l . These r e s u l t s c o r r e s p o n d t o damping, e x c l u s i v e of h y s t e r e t i c damping, of ten p e r c e n t . However, i t was our f e e l i n g t h a t non-h y s t e r e t i c damping, r e p r e s e n t i n g the e f f e c t of n o n - s t r u c t u r a l components, s h o u l d be l e s s than t h i s s i n c e a l l the s t r u c t u r a l damping would be r e f l e c t e d i n the h y s t e r e t i c e f f e c t s . In a program such as DRAIN-2D any i n e l a s t i c a c t i o n w i l l r e s u l t i n h y s t e r e t i c damping and i t i s not n e c e s s a r y t o d u p l i c a t e t h i s by e x t r a s t i f f n e s s p r o p o r t i o n a l damping. Curve 'B' of f i g u r e 5.10 shows the d u c t i l i t y demand of the c o u p l i n g beams when the s t i f f n e s s p r o p o r t i o n a l damping i s lowe r e d t o 2 p e r c e n t . T h i s has a c o n s i d e r a b l e e f f e c t on the damage e x p e r i e n c e d i n the c o u p l i n g beams w i t h maximum d u c t i l i t y demands r i s i n g from 9.8 t o 17.5. At t h i s v a l u e of d u c t i l i t y demand the 5 p e r c e n t s t r a i n h a r d e n i n g r a t i o on the c o u p l i n g beams causes them t o re a c h a moment almost t w i c e t h e i r o r i g i n a l 77 c a p a c i t y . Hence, run 'C was performed i n which the s t r a i n h a r d e n i n g r a t i o was dropped t o 0.5 p e r c e n t thus p l a c i n g i t c l o s e r to the e l a s t i c - p e r f e c t l y p l a s t i c i d e a l i z a t i o n . As s t r a i n s o f t e n i n g r a t h e r than s t r a i n h a r d e n i n g may o c c u r , e s p e c i a l l y a t h i g h d u c t i l i t y demands, the use of a v e r y low v a l u e of s t r a i n h a r d e n i n g i s an a p p r o p i a t e a s s u m p t i o n . Curve 'C shows the r e s u l t s t h a t a re o b t a i n e d u s i n g 0.5 p e r c e n t s t r a i n h a r d e n i n g and 2 p e r c e n t s t i f f n e s s p r o p o r t i o n a l damping. Note, of co u r s e t h a t the d i f f e r e n c e s between the a n a l y s i s and t h a t of F i n t e l and Ghosh do not r e s u l t from the methods, but s i m p l y from the c h o i c e of s t r u c t u r a l p arameters. The damping v a l u e s used here c o r r e s p o n d w i t h the smeared damping v a l u e s proposed by S h i b a t a and Sozen, but the l a t t e r can e a s i l y be changed i n the m o d i f i e d s u b s t i t u t e s t r u c t u r e method to agree t h a t those of F i n t e l and Ghosh i f d e s i r e d . S i m i l a r l y , i f s t r a i n h a r d e n i n g i s f e l t t o be a p p r o p r i t e t h a t can be i n p u t t o the m o d i f i e d s u b s t i t u t e s t r u c t u r e method. Curve 'D' was performed t o c o n f i r m the c o n t e n t i o n t h a t shear d e f l e c t i o n s need not be i n c l u d e d i n s t r u c t u r a l w a l l a n a l y s i s . In run 'D' no shear d e f l e c t i o n s were i n c l u d e d , p r o d u c i n g r e s u l t s almost i n d i s t i n g u i s h a b l e from run 'C i n which the shear d e f l e c t i o n s have been i n c l u d e d . T h i s a l s o r e f l e c t s t h a t the predominant b e h a v i o r of s t r u c t u r a l w a l l s i s f l e x u r a l r a t h e r than s h e a r . F i g u r e s 5.11 and 5.12 r e s p e c t i v e l y show r e s u l t s of d e f l e c t i o n s and d i s p l a c e m e n t s f o r f o u r e a r t h q u a k e s when run on DRAIN-2D and compared t o the r e s u l t s p r e d i c t e d by the m o d i f i e d s u b s t i t u t e s t r u c t u r e method. The d e f l e c t i o n e s t i m a t e s f o r t h i s 78 s t r u c t u r e are v e r y c o n s i s t e n t f o r a l l DRAIN-2D runs and the m o d i f i e d s u b s t i t u t e s t r u c t u r e method. While the l a t t e r method p r e d i c t s a t o p d e f l e c t i o n 2.88 i n c h e s , the time s t e p runs p l a c e t h i s d e f l e c t i o n between 2.82 and 3.28 i n c h e s . The e s t i m a t e s of d u c t i l i t y demand show much g r e a t e r s c a t t e r w i t h v a l u e s h a v i n g a range of e i g h t . F i g u r e 5.13 shows g r a p h i c a l l y the average of the f o u r DRAIN-2D r e s u l t s and m o d i f i e d s u b s t i t u t e s t r u c t u r e method. Both i n terms of d i s t r i b u t i o n and n u m e r i c a l agreement, the m o d i f i e d s u b s t i t u t e s t r u c t u r e method g i v e s an e x c e l l e n t e s t i m a t e of the average of f o u r time s t e p r u n s . I t s h o u l d be noted t h a t a l t h o u g h these t e s t s i n d i c a t e damage r a t i o s f o r which i t may not be p o s s i b l e t o d e s i g n , the purpose of t h e s e t e s t s i s t o examine the a b i l i t y of the m o d i f i e d s u b s t i t u t e s t r u c t u r e method t o e s t i m a t e the r e s u l t s t h a t would be o b t a i n e d from time s t e p i n e l a s t i c a n a l y s i s g i v e n t h a t the same assumptions are used i n each a n a l y s i s . The r e s u l t s of the t e s t s on the s i x t e e n - s t o r y w a l l demonstrate t h a t even w i t h l a r g e d u c t i l i t y demands, the method i s c a p a b l e of r e p r o d u c i n g time s t e p r e s u l t s . T h i s s i x t e e n - s t o r y s t r u c t u r e forms a good t e s t as i t c o n t a i n s many a t t r i b u t e s which might g i v e the m o d i f i e d s u b s t i t u t e s t r u c t u r e method d i f f i c u l t y : the w a l l s have a s t i f f n e s s change at m i d h e i g h t , the mass i s not c o n s t a n t throughout the h e i g h t of the s t r u c t u r e and h i n g i n g o c c u r s i n the base of the w a l l s . Examining the r e s u l t s p r e s e n t e d i n t h i s c h a p t e r produces at l e a s t two o b s e r v a t i o n s worth n o t i n g . Without c a l c u l a t i n g the spectrum f o r a s e r i e s of i n d i v i d u a l e a r t h q u a k e s , i t i s not p o s s i b l e t o p r e d i c t which of a s e r i e s of r e c o r d s w i l l produce 79 the most d r a m a t i c e f f e c t on a g i v e n s t r u c t u r e . For example the E l C e n t r o East-West r e c o r d produces the l a r g e s t d e f l e c t i o n s and d u c t i l i t y demands f o r the t e n and f i v e - s t o r y w a l l s but the Kern S69E r e c o r d shows the l a r g e s t v a l u e s f o r the s i x t e e n - s t o r y w a l l . The r e s u l t s a l s o show t h a t d u c t i l i t y demand has a much g r e a t e r s c a t t e r when d i f f e r e n t r e c o r d s a r e examined than does d e f l e c t i o n and a t t e m p t s t o determine d u c t i l i t y demands to t h r e e s i g n i f i c a n t f i g u r e s i s a f u t i l e e f f o r t . 5.4 C o s t s of E x e c u t i o n As a f i n a l item of c o n c e r n , computing c o s t s s h o u l d be examined t o determine the economic v i a b i l i t y of the method. F i g u r e 5.14 shows the c o s t s of a s i n g l e run f o r v a r i o u s s i z e d s t r u c t u r e s on e l a s t i c modal a n a l y s i s , the m o d i f i e d s u b s t i t u t e s t r u c t u r e method and DRAIN-2D. In a l l c ases the c h a r g e s i n c l u d e the c o s t of p r i n t i n g the i n p u t d a t a and s u f f i c i e n t output f o r e v a l u a t i o n of the r e s u l t s . A l s o the s t r u c t u r e s r e p r e s e n t e d on t h i s f i g u r e are a l l s i n g l e p a i r s of c o u p l e d w a l l s conneced by l i n t e l beams a t each f l o o r . The graph shows c o s t s f o r normal p r i o r i t y b a t c h j o b s i n a n o t - f o r - p r o f i t computing c e n t e r , and the f i g u r e s are o n l y r e p r e s e n t a t i v e of r e l a t i v e c o s t s . Commercial charges c o u l d be a t l e a s t f o u r times the c o s t s shown i n f i g u r e 5.14. S a v i n g s w i t h the m o d i f i e d s u b s t i t u t e s t r u c t u r e over the DRAIN-2D a n a l y s i s a r e o n l y i n d i c a t i v e of the c o s t of a s i n g l e r u n ; they i n c r e a s e s i g n i f i c a n t l y i f i t i s d e c i d e d to t e s t the s t r u c t u r e w i t h more than one earthquake r e c o r d . For runs u s i n g a s p e c i f i c e a r t h q uake or s e r i e s of e a r t h q u a k e s u s i n g a 80 program such as DRAIN-2D, i t i s necessary f i r s t to determine the fr e q u e n c i e s of the s t r u c t u r e f o r c a l c u l a t i o n of the damping parameters. Even under these circumstances, where i t has been f i r m l y decided to use a program such as DRAIN-2D, i t would be worthwhile to run a program such as EDAM which in a d d i t i o n to determining the i n t i a l p e r i o d s of the s t r u c t u r e give the designer an e x c e l l e n t i n d i c a t i o n of the d u c t i l i t y demands to be expected. 81 CHAPTER 6 APPLICATION OF THE METHOD THROUGH A DESIGN EXAMPLE 6.1 A n a l y s i s f o r the d e s i g n of a s i x t e e n - s t o r y s t r u c t u r a l w a l l . H aving examined the a p p l i c a b i l i t y and l i m i t a t i o n s of the m o d i f i e d s u b s t i t u t e s t r u c t u r e method i t i s now a p p r o p r i a t e t o demonstrate how i t can be used i n a h y p o t h e t i c a l d e s i g n . The example chosen i s a s i x t e e n - s t o r y s t r u c t u r a l w a l l , of a t y p i c a l h e i g h t f o r r e s i d e n t i a l or o f f i c e b u i l d i n g s u s i n g t h i s system f o r l a t e r a l f o r c e r e s i s t a n c e . In t h i s example, the maximum l a t e r a l d e s i g n a c c e l e r a t i o n f o r the s i t e i s g i v e n as 0.3 tim e s t h a t of g r a v i t y w i t h the spectrum of the 1940 E l C e n t r o . The assumptions c o n c e r n i n g f l o o r l o a d i n g , s e c t i o n p r o p e r t i e s , and o t h e r such d e t a i l s a r e s i m i l a r t o those d i s c u s s e d i n s e c t i o n 5.2 f o r the s t r u c t u r e s t h a t underwent i n e l a s t i c t e s t i n g . These assumptions s h o u l d not be regarde d as ne c e s s a r y r e s t r i c t i o n s , but s i m p l y as a b a s i s by which r e a s o n a b l e v a l u e s can be chosen. For example, the use of an i n p u t f l o o r l o a d of 150 l b / f t 2 would o b v i o u s l y be the d e s i g n e r ' s c h o i c e . I t s s e l e c t i o n i n t h i s a n a l y s i s s h o u l d have no e f f e c t on the v a l i d i t y of the method. The b u i l d i n g under c o n s i d e r a t i o n has s t r u c t u r a l w a l l s of a symmetric d e s i g n as shown i n f i g u r e 6.1. 82 These w a l l s must be d e s i g n e d t o c a r r y the l a t e r a l l o a d of the s t r u c t u r e . The f i r s t s t e p i n a p p l y i n g the method i s t o determine t h a t the s t r u c t u r e under e x a m i n a t i o n s a t i s f i e s the n e c e s s a r y r e s t r i c t i o n s . In the case of the s i x t e e n - s t o r y s t r u c t u r a l w a l l i n the example, t h i s i s a f a i r l y s i m p l e p r o c e d u r e . The w a l l i s c o n s i d e r e d a component of a r e s i d e n t i a l b u i l d i n g w i t h o u t f l a n g e s on w a l l ends, so the element i s symmetric. Thus, no d i f f i c u l t i e s w i l l be e n countered as would have o c c u r r e d i f the w a l l had a g r e a t e r c a p a c i t y i n the p o s i t i v e h o r i z o n t a l d i r e c t i o n than i n the o p p o s i n g d i r e c t i o n . The system i s t o be a n a l y z e d as a p l a n e frame s t r u c t u r e and i s of such a n a t u r e t h a t t o r s i o n i s . not a problem. As the w a l l s are c o n t i n u o u s t o the ground, no abrupt changes i n mass or s t i f f n e s s a re apparent over t h e i r h e i g h t . U s i n g l i g h t p a r t i t i o n i n g w a l l s , or i s o l a t i n g those w a l l s which might i n t e r f e r e w i t h response and a r e not c o n s i d e r e d i n the model of the b u i l d i n g , the s t r u c t u r e meets the c r i t e r i o n of non-i n t e r f e r e n c e of n o n - s t r u c t u r a l e l e m e n t s . We assume t h a t a l l the j o i n t s and elements w i l l be r e i n f o r c e d as n e c e s s a r y f o r d u c t i l i t y ; i n fact., the main purpose of t h i s a n a l y s i s i s t o determine the d u c t i l i t y demands so t h a t proper d e s i g n can p r e v e n t c a t a s t r o p h i c f a i l u r e . From t h i s b r i e f e x a m i n a t i o n i t i s d e t e r m i n e d tha xt the w a l l i s one t h a t can be a n a l y z e d by the m o d i f i e d s u b s t i t u t e s t r u c t u r e method. Having d e c i d e d t h a t the b u i l d i n g meets the r e s t r i c t i o n c r i t e r i a f o r the method, i t i s now n e c e s s a r y t o model the s t r u c t u r e . I t i s a t t h i s s tage t h a t the d e s i g n e r uses h i s judgement t o make assumptions r e g a r d i n g such f a c t o r s as the 83 v a l u e s of c r a c k e d moment of i n e r t i a and h o r i z o n t a l mass. As the l a t e r a l f o r c e a n a l y s i s u s u a l l y f o l l o w s t h a t of the v e r t i c a l f o r c e r e s i s t a n c e and a r c h i t e c t u r a l l a y o u t , the g r o s s s i z e of members and the l o c a t i o n s of j o i n t c e n t e r s would a l r e a d y have been d e t e r m i n e d . The next s t e p i s c o d i n g of the s t r u c t u r e and an i n i t i a l run of the program. D u r i n g t h i s p r o c e d u r e the d e s i g n e r w i l l a p p r e c i a t e the v i r t u e s of d a t a g e n e r a t o r s which can be a p p l i e d to the s t r u c t u r e type he most f r e q u e n t l y e n c o u n t e r s . For example most of the s t r u c t u r e s used i n t h i s s t u d y were m o d e l l e d as two w a l l s and t h e i r c o n n e c t i n g c o u p l i n g beams. A data g e n e r a t o r which can e a s i l y produce a d a t a f i l e f o r a s t r u c t u r e w i t h two column l i n e s was used. However, data g e n e r a t o r s f o r reasons of g e n e r a l i t y have not been i n c l u d e d i n the program and i n t h i s study were w r i t t e n and used s e p a r a t e l y . I t may be the case t h a t a s t r u c t u r e under c o n s i d e r a t i o n by the d e s i g n e r cannot be m o d e l l e d by o n l y one c o u p l e d w a l l but must be m o d e l l e d by a l a r g e r s e t of w a l l s c o n n e c t e d by i n e x t e n s i b l e h i n g e d l i n k s which r e p r e s e n t the e f f e c t of a f l o o r diaphragm. Such a case was i l l u s t r a t e d i n the s i x t e e n - s t o r y t e s t s t r u c t u r e of Chapter 5. With the i n p u t d a t a g e n e r a t e d , which i n t h i s case t a k e s up about one hundred l i n e s , an i n i t i a l run can be made. The damage r a t i o r e s u l t s of the f i r s t run a r e shown i n f i g u r e 6.2, w h i l e p e r t i n a n t r e s u l t s such as f r e q u e n c y a r e shown i n T a b l e 6.1. Here, damage at the base of the w a l l s and i n the upper l i n t e l s i s deemed t o be u n a c c e p t a b l e f o r the d e s i g n e a r t h q u a k e , and changes i n some of the s t r u c t u r e p r o p e r t i e s are n e c e s s a r y t o r e a l i z e a r e d u c t i o n i n damage. 84 The f i r s t change i s t o i n c r e a s e the moment c a p a c i t y of the l i n t e l s . A d o u b l i n g of t h i s v a l u e i s made b e f o r e the e x e c u t i o n of the second run, here an i n c r e a s e from 40 t o 80 K i p - F t . The damage r a t i o s w i t h these i n c r e a s e d c a p a c i t y l i n t e l s a r e shown i n f i g u r e 6.3. The changes made b e f o r e the s t a r t of t h i s run cause a c o n s i d e r a b l e r e d u c t i o n i n the damage r a t i o s of the c o u p l i n g beams as w e l l as a s l i g h t r e d u c t i o n i n the damage r a t i o at the base of the w a l l s . T h i s i s t o be expected as the c o u p l i n g a c t i o n of the w a l l s i s i n c r e a s e d by a s t r e n g t h e n i n g of the c o u p l i n g beam. A t h i r d t e s t of the s t r u c t u r e was performed t o examine the e f f e c t of i n c r e a s i n g the v a l u e of Young's modulus on the damage r a t i o s of the s t r u c t u r e . The v a l u e t y p i c a l of 5 K s i c o n c r e t e chosen f o r t h i s run r e p l a c e s the v a l u e r e p r e s e n t i n g 4 K s i c o n c r e t e used i n e a r l i e r r u n s . T h i s change has the e f f e c t of i n c r e a s i n g the modulus from 3600 K s i t o 4030 K s i . A l t h o u g h the use of i n c r e a s e d c o n c r e t e s t r e n g t h would a l s o a l t e r the c a p a c i t y of the members somewhat, t h i s was i g n o r e d and no change was made t o the c a p a c i t y or g e o m e t r i c a l p r o p e r t i e s of the members from the p r e v i o u s t e s t . Other t e s t s i n the s e r i e s are performed t o examine the e f f e c t s of c h a n g i n g member s t r e n g t h s on the response of the s t r u c t u r e . I t i s up t o the d e s i g n e r t o d e t e r m i n e i f he needs i n c r e a s e d c o n c r e t e s t r e n g t h s t o a c h i e v e the d e s i r e d member c a p a c i t i e s . The 12 p e r c e n t i n c r e a s e i n the modulus r e s u l t e d i n a 6 p e r c e n t decrease i n the root-mean-square d i s p l a c e m e n t a t the t o p of the s t r u c t u r e but a l s o a 4 t o 12 p e r c e n t i n c r e a s e i n the damage r a t i o s of the members. An i n c r e a s e i n Young's modulus w i l l have a s i m i l a r e f f e c t on the f l e x u r a l r i g i d i t y v a l u e f o r 85 the c o u p l i n g beams. T h i s s t i f f e n i n g has the e f f e c t of a t t r a c t i n g l a r g e r l o a d s and hence more i n e l a s t i c a c t i o n which produces h i g h e r damage r a t i o s . Under th e s e c i r c u m s t a n c e s i t i s p r o b a b l y not w o r t h w h i l e t o pay f o r i n c r e a s e d c o n c r e t e s t r e n g t h s o l e l y t o i n c r e a s e the v a l u e of Young's modulus t o a c h i e v e a decrease i n the d e f l e c t i o n s , as the r e s u l t i s m i n i m a l . A l l f u r t h e r t e s t s t h a t a re performed on t h i s s t r u c t u r e w i l l use a v a l u e of Young's modulus t h a t c o r r e s p o n d s t o t h a t of 4 K s i c o n c r e t e . The d e s i g n e r may wi s h t o reduce the i n e l a s t i c a c t i o n i n the w a l l s t o the p o i n t t h a t they a v o i d any e x c u r s i o n s past t h e i r y i e l d v a l u e and t h e r e f o r e have damage r a t i o s below u n i t y . Such a d e c i s i o n would be c o n s i s t e n t w i t h the b e l i e f t h a t i n e l a s t i c i t y i n columns i s u n d e s i r a b l e as i t o f t e n o c c u r s i n a l e s s d u c t i l e manner than when the i n e l a s t i c t y i s c o n c e n t r a t e d o n l y i n members w i t h o u t a x i a l l o a d . Hence, the f o u r t h t e s t was performed f o l l o w i n g the c a l c u l a t i o n of a new moment-axial c u r v e f o r the w a l l s w i t h the same c r o s s - s e c t i o n used i n the p r e v i o u s t e s t s but w i t h i n c r e a s e d s t e e l c o n t e n t . Based on our assumption of the c r a c k e d moments of i n e r t i a b e i n g dependant o n l y on the g r o s s s i z e and presence or absence of a x i a l l o a d , t h i s change i n s t e e l a r e a w i l l have no e f f e c t on the moment of i n e r t i a used and w i l l o n l y i n f l u e n c e the moment c a p a c i t y of the w a l l s . For the i n i t i a l run w i t h t h i s new w a l l member the c o u p l i n g beams were g i v e n the reduced c a p a c i t y of 40 k i p - f t t o make the run comparable t o the f i r s t t e s t . Compared w i t h t h a t t e s t , the r e s u l t i n g damage r a t i o s a t the base of the w a l l s have now been reduced t o below u n i t y (see f i g u r e 6.5), but the l i n t e l beams now i n c u r much h i g h e r damage r a t i o s . The i s due i n p a r t t o the damping i n t e s t number 86 4 b e i n g lower than i n the f i r s t t e s t , r e f l e c t i n g the lower damage encountered by the major members. The r e s u l t s of t e s t number 4 when compared w i t h t e s t number 1 a l s o show t h a t r e d u c i n g the damage r a t i o s of the w a l l s may not reduce the d i s p l a c e m e n t s . Indeed i n t h i s c a s e , they show a 22 p e r c e n t i n c r e a s e . The f i f t h t e s t c o r r e s p o n d s t o the second t e s t , as i n both c a s e s the l i n t e l c a p a c i t y i s dou b l e d from the p r e v i o u s r u n . With the e x c e p t i o n of i n c r e a s i n g the l i n t e l c a p a c i t y , the i n p u t f o r t h i s run was o t h e r w i s e unchanged from the f o u r t h t e s t . T h i s r u n , as d i d the second t e s t , showed c l e a r l y the d r a m a t i c e f f e c t of i n c r e a s i n g the l i n t e l beam c a p a c i t y i n r e d u c i n g the damage r a t i o s , both i n those members and i n the w a l l s (see f i g u r e 6.6). A l t h o u g h the v a l u e s o b t a i n e d a r e p o s s i b l y w i t h i n our a b i l i t y t o d e s i g n i n terms of d u c t i l i t y r e q u i r e m e n t s , f u r t h e r t e s t s were needed t o reduce the h i g h e r damage r a t i o s and t o examine some of the p r o p e r t i e s of t h i s s i x t e e n - s t o r y w a l l . The s i x t h t e s t was performed a f t e r e x amining the maximum a l l o w a b l e shear c a p a c i t y of the l i n t e l u s i n g the p r o v i s i o n s of the ACI code but i g n o r i n g the component of shear c a r r i e d by the c o n c r e t e . From the a n a l y s i s i t was found t h a t the l i n t e l s c o u l d approach a moment c a p a c i t y of 300 k i p - f t w i t h o u t f i r s t f a i l i n g i n ' s h e a r . T h i s v a l u e was then used f o r the u l t i m a t e moment c a p a c i t y of the l i n t e l beams. The run showed t h a t even w i t h t h i s member s t r e n g t h some damage had t o be ex p e c t e d i n the c o u p l i n g beams (see f i g u r e 6.7). The r e s u l t s a l s o showed t h a t w i t h so h i g h a l i n t e l c a p a c i t y one of the w a l l s would be i n , or v e r y c l o s e t o , a s t a t e of t e n s i o n . T h i s was viewed as b e i n g 87 u n d e s i r a b l e f o r these r e i n f o r c e d c o n c r e t e elements and hence a sev e n t h run was performed, l o w e r i n g the l i n t e l c a p a c i t y t o the p o i n t where a r e d u c t i o n of o n l y 50% of the v e r t i c a l l o a d would occur i n a w a l l . Computations of the c a p a c i t y of the c o u p l i n g beams t o s a t i s f y t h i s c r i t e r i o n can be performed by hand: s i n c e the beams a r e going t o be v e r y c l o s e t o i f not a t , t h e i r y i e l d l e v e l , the shear c a r r i e d by the beam i s c a l c u l a t e d by d i v i d i n g t w i c e the moment c a p a c i t y of the beam by i t s l e n g t h . The a x i a l f o r c e s i n the w a l l s a re i n c r e a s e d or reduced by the accumulated t o t a l of the s e beam s h e a r s , and the a p p r o p r i a t e v a l u e s r e q u i r e d t o cause a s p e c i f i e d r e d u c t i o n i n the a x i a l f o r c e due t o v e r t i c a l l o a d s are e a s i l y computed. A rough ' i d e a of the d e s i r a b l e c a p a c i t y of the c o u p l i n g beams i s thus d e t e r m i n e d . The s e v e n t h and f i n a l run was performed w i t h the v a l u e of the moment c a p a c i t y of the l i n t e l s reduced t o 130 K i p - F t f o r the reasons o u t l i n e d i n the p r e v i o u s p a r a g r a p h . The r e s u l t s a re q u i t e a c c e p t a b l e f o r a l l v a r i a b l e s examined. The l a r g e s t damage r a t i o , as shown i n f i g u r e 6.8, i s 4.7, a f i g u r e e a s i l y w i t h s t o o d by p r o p e r d e t a i l i n g . The l a r g e s t d e f l e c t i o n when compared w i t h the h e i g h t of the s t r u c t u r e a t t h a t p o i n t i s 1/190. The a x i a l f o r c e s i n the w a l l s induced by the earthquake are 45 p e r c e n t of the s t a t i c a x i a l l o a d c a r r i e d by those members so they are s a f e l y away from a s t a t e of t e n s i o n . The damage r a t i o s i n the base of the w a l l a re 0.64 which, i n terms of economy of- s e c t i o n , i s p r o b a b l y too low. By e x a m i n a t i o n of s e c t i o n s w i t h the same c r o s s - s e c t i o n a l shape but d i f f e r e n t s t e e l a r e a s , a s t e e l q u a n t i t y and d i s t r i b u t i o n can be chosen which p l a c e s the w a l l c l o s e r t o y i e l d i n a more e c o n o m i c a l manner. I t s h o u l d be noted 88 t h a t c o n s i s t e n t w i t h the d e s i r e t o a v o i d h i n g e s i n column members t h i s c a p a c i t y s h o u l d be t h a t of the base of the w a l l under minimum e x p e c t e d a x i a l f o r c e . The s e v e n t h run completes our a n a l y s i s of the the c o u p l e d w a l l as f a r as the m o d i f i e d s u b s t i t u t e s t r u c t u r e method i s concerned. The f i n a l s t a g e s of the d e s i g n i n v o l v e e n s u r i n g t h a t members a r e d e t a i l e d t o p r o v i d e s u f f i c i e n t d u c t i l i t y t o s u s t a i n the damage r a t i o s p r e d i c t e d f o r the s t r u c t u r e . As a check of the r e s u l t s p r e d i c t e d by the m o d i f i e d s u b s t i t u t e s t r u c t u r e method f o r t h i s example ( s t r u c t u r e number 7 ) , computer runs were performed u s i n g the program DRAIN-2D. These runs were made u s i n g 2 p e r c e n t of c r i t i c a l damping and the f i r s t t e n seconds of the same f o u r earthquake r e c o r d s o u t l i n e d i n Chapter 5. The r e s u l t s of these runs i n terms of d u c t i l i t y r e q u i r e m e n t s of the c o u p l i n g beams are shown i n f i g u r e 6.9 w i t h d e f l e c t i o n e s t i m a t e s shown i n f i g u r e 6.10. The r e s u l t s show t h a t the m o d i f i e d s u b s t i t u t e s t r u c t u r e method i s a good p r e d i c t o r of both damage r a t i o and d e f l e c t i o n . As the spectrum i s an average f o r t he f o u r r e c o r d s used and not an e n v e l o p e , some d u c t i l i t y demands and d e f l e c t i o n s from the program DRAIN-2D are g r e a t e r than those p r e d i c t e d by the m o d i f i e d s u b s t i t u t e s t r u c t u r e method. Indeed, f o r t h i s example the r e s u l t s p r e d i c t e d by the method a r e a v e r y r e a s o n a b l e e s t i m a t e of the average of the r e s u l t s from the f o u r i n e l a s t i c time s t e p r u n s . E x c l u d i n g the c o s t of the run n e c e s s a r y t o e s t a b l i s h the f r e q u e n c y f o r i n p u t of damping t o DRAIN-2D the c o s t of e x e c u t i n g the fo u r runs i s over twenty t i m e s the c o s t of the s i n g l e run of the m o d i f i e d s u b s t i t u t e s t r u c t u r e method. As the spectrum method r e q u i r e s 89 o n l y i n p u t of maximum a c c e l e r a t i o n and spectrum type r a t h e r than an e x t e n s i v e s t r i n g of a c c e l e r a t i o n s , and as the program EDAM o u t p u t s damage r a t i o s d i r e c t l y , b o th i n p u t d a t a p r e p a r a t i o n and program output i n t e r p r e t a t i o n a re c o n s i d e r a b l y e a s i e r when u s i n g the m o d i f i e d s u b s t i t u t e s t r u c t u r e method. 6.2 E x a m i n a t i o n of the e f f e c t of changing maximum ground a c c e l e r a t i o n . One of the many advantages of a t e c h n i q u e such as the m o d i f i e d s u b s t i t u t e s t r u c t u r e method i s t h a t p a r a m e t r i c s t u d i e s can be performed q u i c k l y and c h e a p l y . An example of t h i s i s an e x a m i n a t i o n of the e f f e c t t h a t maximum ground a c c e l e r a t i o n changes have on the v a r i o u s s t r u c t u r a l response parameters. Knowledge of the b e h a v i o r of the s t r u c t u r e under a c c e l e r a t i o n s which d i f f e r from the d e s i g n maximum may be of i n t e r e s t when i t i s c o n s i d e r e d how u n c e r t a i n t h i s maximum i s , and a s e r i e s of t e s t s was performed on a s i x t e e n - s t o r y s t r u c t u r e s i m i l a r t o the f i n a l one o b t a i n e d i n s e c t i o n 6.1. The l i n t e l beams were made s i x f o o t l o n g r a t h e r than e i g h t f o o t , and t h e r e was a c o r r e s p o n d i n g two f o o t d e c r e a s e i n the w a l l c e n t e r l i n e s p a c i n g . There were no o t h e r changes t o i n p u t d a t a . Damping, c a l c u l a t e d by the program, n a t u r a l l y i n c r e a s e s a t h i g h e r a c c e l e r a t i o n s as the members undergo more damage. Hence, i t i s n e c e s s a r y o n l y t o change the maximum a c c e l e r a t i o n f i g u r e i n the d a t a f i l e b e f o r e p e r f o r m i n g a t e s t run from t h i s s e r i e s . In d o i n g t h i s a n a l y s i s the use of the c r a c k e d moment of i n e r t i a might appear t o render the r e s u l t s i n v a l i d f o r those 90 s t r u c t u r e s where some members were b e i n g s t r e s s e d i n s u f f i c i e n t l y t o cause c r a c k i n g . To examine t h i s s i t u a t i o n the s i x t e e n - s t o r y frame used i n t h e s e a c c e l e r a t i o n parameter s t u d i e s was recoded, a s s i g n i n g u n c r a c k e d moments of i n e r t i a t o a l l members h a v i n g a damage r a t i o e q u a l t o or l e s s than 0.25. These r e s u l t s were then compared w i t h a run i n which c r a c k e d s e c t i o n s had been used t h r o u g h o u t . In r e c o d i n g i t had been n e c e s s a r y t o change the moment of i n e r t i a v a l u e s f o r the f i v e t o p s t o r i e s of the w a l l s , and changes i n fundamental f r e q u e n c y , damage r a t i o s and d i s p l a c e m e n t s were i n the o r d e r of 1 p e r c e n t and were t h e r e f o r e judged t o be i n s i g n i f i c a n t . On the b a s i s of these r e s u l t s , u s i n g the c r a c k e d s e c t i o n f o r an e n t i r e s t r u c t u r e appears t o be an a c c e p t a b l e p r o c e d u r e . T h i s p r o c e s s s h o u l d , however, be r e -examined i f the damage r a t i o s i n the bottom s t o r y a re low enough t o suggest t h a t c r a c k i n g has not o c c u r r e d i n t h i s r e g i o n . The i n d i v i d u a l r e s u l t s of the s e t e s t s w i l l not be reproduced h e r e , but f i g u r e s 6.11 t o 6.13 show the t r e n d s . F i g u r e 6.11 shows the damage r a t i o s i n the c o u p l i n g beams a t t h r e e v a l u e s of maximum a c c e l e r a t i o n . As e x p e c t e d , the damage r a t i o s were h i g h e r a t i n c r e a s e d v a l u e s of a c c e l e r a t i o n , but what i s a l s o a p parent i n t h i s f i g u r e , i s t h a t the l o c a t i o n of h i g h e s t damage moves up the s t r u c t u r e as the a c c e l e r a t i o n i n c r e a s e s . T h i s can be e x p l a i n e d w i t h r e f e r e n c e t o the d u a l l o a d paths p r e s e n t i n c o u p l e d s t r u c t u r a l w a l l s : i n the lowest a c c e l e r a t i o n as shown i n f i g u r e 6.11, w i t h the l o w e s t ground a c c e l e r a t i o n , a l l but t h r e e of the c o u p l i n g beams have a l r e a d y y i e l d e d and are c a r r y i n g the maximum sh e a r . Hence, the maximum a x i a l d e f o r m a t i o n of the w a l l s i s p r e s e n t , g i v i n g maximum r e l i e f t o the damage i n 91 the t o p c o u p l i n g beams. Higher ground a c c e l e r a t i o n s d e c r ease the r e l a t i v e importance of the e f f e c t as the a x i a l s t r a i n s remain almost c o n s t a n t w h i l e w a l l bending i n c r e a s e s . F i g u r e 6.12 shows the e f f e c t on the f i r s t two p e r i o d s of i n c r e a s i n g the maximum a c c e l e r a t i o n . T h i s f i g u r e r e f l e c t s a f i n d i n g r e p o r t e d i n the l i t e r a t u r e from many s h a k i n g t a b l e and f r e e v i b r a t i o n t e s t s of damaged r e i n f o r c e d c o n c r e t e s t r u c t u r e s . T h i s o b s e r v a t i o n i s t h a t the p e r i o d i n c r e a s e s as h i g h e r v a l u e s of a c c e l e r a t i o n cause more damage and a l o s s i n s t i f f n e s s . T h i s f i g u r e shows the fundamental p e r i o d t o be much more a f f e c t e d than t h a t of the second mode. The t r e n d c o n t i n u e s t o h i g h e r modes, so t h a t by the t e n t h mode the d i f f e r e n c e between the damaged and e l a s t i c p e r i o d s i s i n d i s t i n g u i s h a b l e f o r t h i s s i x t e e n - s t o r y w a l l u ndergoing a maximum a c c e l e r a t i o n of f i f t y p e r c e n t of g r a v i t y . The reason t h a t the fundamental mode i s more a f f e c t e d i s t h a t i t i s more dependant on the s t i f f n e s s of the f i r s t f l o o r w a l l s . I f h i n g e s were t o form h i g h e r up the b u i l d i n g , the h i g h e r modes would be more a f f e c t e d . F i g u r e 6.13 shows the e f f e c t of i n c r e a s i n g , ground a c c e l e r a t i o n on the v a l u e of the smeared damping c a l c u l a t e d f o r the f i r s t t h r e e modes. The graph shows an i n c r e a s e i n damping f o r the fundamental mode w i t h i n c r e a s i n g a c c e l e r a t i o n i n the t e n t o t h i r t y p e r c e n t of g r a v i t y range. Above t h i s range of ground m o t i o n , damping i s somewhat c o n s t a n t i n the 5.5% of c r i t i c a l range. For t h i s s t r u c t u r e i n c r e a s e d v a l u e s of e x c i t a t i o n have l i t t l e e f f e c t on the damping of the second and t h i r d mode. F i g u r e 6.14 i l l u s t r a t e s the e f f e c t on the h o r i z o n t a l 92 d i s p l a c e m e n t s of the s t r u c t u r e of i n c r e a s i n g the i n p u t ground a c c e l e r a t i o n . The i n s e r t graph shows t h a t d e s p i t e the n o n - l i n e a r b e h a v i o r of the c o u p l i n g beams and, e v e n t u a l l y , the f o r m a t i o n of h i n g e s a t the base of the w a l l s , the f i n a l t o p d e f l e c t i o n i s almost l i n e a r w i t h i n c r e a s i n g a c c e l e r a t i o n . I t can a l s o be seen t h a t the d e f l e c t i o n i s caused m o s t l y by c u r v a t u r e i n the lower r e g i o n s of the w a l l s as h i g h e r segments show l i t t l e c u r v a t u r e . These l a s t t e s t s a r e a s i m p l e example of how the method can be used t o determine the e f f e c t s on response parameters of changes i n a s i n g l e i n p u t v a r i a b l e . For t h e s e t e s t s , most changes r e q u i r e o n l y minor e d i t i n g of the d a t a f i l e t o modify the i n p u t from one run t o the n e x t . The c o m p u t a t i o n and output c o s t s f o r a t y p i c a l run are i n the o r d e r of $3.50 f o r a run performed on normal p r i o r i t y on a n o n - p r o f i t b a s i s on the U n i v e r s i t y of B r i t i s h Columbia computing system. Thus the m o d i f i e d s u b s t i t u t e s t r u c t u r e method i s demonstrated t o be an e c o n o m i c a l and p r a c t i c a l approach -to p a r a m e t r i c s t u d i e s of the s e i s m i c response of c o u p l e d s t r u c t u r a l w a l l s . 93 CHAPTER 7 CONCLUSIONS The m o d i f i e d s u b s t i t u t e s t r u c t u r e method has been p r e s e n t e d as a d e s i g n a i d f o r the s e i s m i c d e s i g n of c o u p l e d s t r u c t u r a l w a l l s . The method extends the e l a s t i c modal a n a l y s i s t e c h n i q u e i n t o the i n e l a s t i c range and has been shown t o p r o v i d e good e s t i m a t e s of the d u c t i l i t y r e q u i r e m e n t s and d e f l e c t i o n s of c o u p l e d s t r u c t u r a l w a l l s r e s i s t i n g l a t e r a l f o r c e s which p l a c e some of the members i n t o t h e i r i n e l a s t i c range. The c o u p l e d s t r u c t u r a l w a l l s t e s t e d i n t h i s study were of h e i g h t r a n g i n g from 5 t o 16 s t o r i e s . The method has been shown t o g i v e good r e s u l t s i n a l l c a s e s e x c e p t where the fundamental p e r i o d of the s t r u c t u r e p l a c e s i t on a c o n s t a n t p o r t i o n of the i n p u t spectrum. The a c c u r a c y of the r e s u l t s , as determined by comparison w i t h i n e l a s t i c time s t e p a n a l y s i s , appears to improve as the fundamantal p e r i o d of the s t r u c t u r e i n c r e a s e s . The method i s i n e x p e n s i v e t o use and can be performed w i t h a computer program u s i n g a d a t a f i l e h a v i n g o n l y minor changes from t h a t used i n s t a t i c a n a l y s i s . I t i s t h e r e f o r e a method t h a t c o u l d be used i n the p r a c t i c a l d e s i g n of s e i s m i c r e s i s t a n t c o u p l e d s t r u c t u r a l w a l l s . 94 k= 12EI o 0 0 0 0 0 (Symmetric) - L T Y L1*X L1*L*(L+L1) L1*Y -L1*X -L2*Y L2*X L1+L2 0 0 0 L2*Y -L2*X L 2*L2+L*L2 X= H o r i z o n t a l p r o j e c t i o n o f member Y= V e r t i c a l p r o j e c t i o n of member L= Length of e l a s t i c p o r t i o n of member. L1= Length of R i g i d arm a t l e s s e r j o i n t end L2= Length of R i g i d arm a t g r e a t e r j o i n t end. Table 3.1 : A d d i t i o n a l Member S t i f f n e s s M a t r i x t o Account f o r R i g i d Arms. P e r i o d FORCES A r i o l Shear Free End Bending Moment F i x e d End Bending Moment DISFLACFMFNTS ( f r e e end) H o r i z o n t a l V e r t i c a l R o t a t i o n HORIZONTAL PENDULUM 21/mL AE (S<J(m) ( D 0 0 0 ( S J ( m ) ( 1 ) ( L ) AE 0 0 VERTICAL PENDULUM No Shear D e f l e c t i o n With Shear D e f l e c t i o n 3 " 21V mL-3ET 21/m 0 ( S a ) ( r n ) ( 1 ) 0 ( S 0 H m ) ( 1 ) ( L ) ( S « Q ( m ) ( 1 ) L 3 3EI 0 (ScQ(m ) (1)L 2 2EI L,3 + _ L _ EI AVG 0 ( S „ ) ( m ) ( 1 J 0 (S a.)(m ) ( 1)(L) ( S j ( m ) ( 1 ) ( L 3 ) + (5<0(m)(1 )(L) " 5EI ~T (SoJ(m)(l)L £ 2EI T a b l e i | . 1 A n a l y t i c r e s u l t s of V e r t i c a l and H o r i z o n t a l pendulums. (Free end p e r m i t t e d 3 degrees of freedom but mass only- opposes h o r i z o n t a l motion) vo cn 96 P e r i o ^ a r ^ ^ T _ t ± c ± r , B . t i - Q r i f a c t o r s E l £ s t i c. _ F e r i o d s Mode EDAM DYNAMIC S h i b a t a and P a r t i c i p a t i o n F a c t o r Sozen 1 0 . 8 5 3 0 . 8 5 8 0 . 8 5 1.286 0.254 2 0.262 0.262 0.26 0 . 4 5 2 0.545 3 0.137 0.137 0.14 0.253 0.500 h 0.055 0.088 ' 0.057 0.21 1 0.321 .0.067 0.067 0.065 -0.1 1 1 0.245 ROOT MEAN SQUARE FORCES MN 1 2 3 4 5 6 7 B 9 10 11 12 13 14 15 AXIAL ^KIPS> 14.929 9.449 9.449 11.325 26.229 26.229 11.163 49.014 49.014 10.843 75.272 75.272 7.183 99.229 99. 229 SHEAR (KIPS) 9.449 14.926 14.961 16.853 24.004 24.032 23.308 30.551 30.573 27.420 35.841 35.861 25.317 38.972 39.025 BML (K-FT) 113.309 54 .429 54.739 202.164 108.358 108.598 279.633 153.003 153.175 328.989 207.414 207.525 303.748 328.082 328.494 BMG (K-FT) 113.472 113.309 113.472 202.313 160.067 160.218 279.749 186.991 187 .131 329.093 188.921 189.084 303.869 102.058 102.267 NOTE: E n t i r e mass f o r each f l o o r i s a t t a c h e d t o r i g h t column L e s s e r j o i n t end f o r beams i s l e f t end. L e s s e r j o i n t end f o r columns i s lower end. 0.2 t i m e s g r a v i t y , 5% Damping, F i r s t 5 modes used. 'Table 4.2 E l a s t i c Modal A n a l y s i s r e s u l t s ' f o r S h i b a t a and Sozen' 5 - S t o r y s t r u c t u r e ( F i g u r e 4 . 2 ) u s i n g t h e i r Spectrum ' EARTHQUAKE E l Centre- (NS) E l C e ntro (EW) Kern County (S69E) Kern County (N21E) DATE May 18, 1%0 May 18, 1%0 J u l y 21, 1952 J u l y 21, 1952 AM AX RECORDING STATION 0.348 E l Centro s i t e I m p e r i a l V a l l e y I r r i g a t i o n D i s t r i c t 0.182 E l Centro s i t e I m p e r i a l V a l l e y I r r i g a t i o n D i s t r i c t 0.179 T a f t L i n c o l n S c h o o l Tunnel 0.156 T a f t L i n c o l n S c h o o l Tunnel NOTE: AMAX = Maximum A c c e l e r a t i o n of o r i g i n a l r e c o r d d u r i n g segment of r e c o r d used. F i r s t ten seconds of each r e c o r d used. Table 5.1 Earthquake r e c o r d s used i n DRAIN-2D computer runs. Fundamental P e r i o d ( S e c . ) 2% Damping F a c t o r W e i g h t / F l o o r ( K i p ) Young<s Modulus ( K u i ) S t r u c t u r e Height ; F t ) Maximum Ground A c c e l e r a t i o n L i n t e l C a p a c i t y ( K i p - F t ) C l e a r e p a n ( F t ) Moment o f . I n e r t i a ( I n 4 ) Area ( i n 2 ) L e f t coupled w a l l Moment of , I n e r t i a ( I n 4 ) Area ( i n ? ) R i g i d Arm ( F t ) n i f h t Coupled B a l l Moment of . I n e r t i a ( I n 4 ) Area ( i n ? ) R i g i d Arm ( F t ) Uncoupled' Wall Nurut ti I ul I n e r t i a ( I n 4 ) Area (in'') ^ - K t o r y W a l l 5-Story W a l l 1 0-Story W a l l 1 6-Story (Ueriuti 15) .221*8 .0011*3 270 3600. 1*1 -75 (Mas;u"l|) .'*i*96 .00286 1080 3600. f * l . 7 5 .8:51*6 .00531 270 36oo . 8'* . 2 5 ( l luc lg l l ) I .'i'l85 .00858 270 3600 . 155 .25 1 6-Story W a l l w i t h e x t r a Uncoupled w a l l . F l o o r 0 -8 F l o o r 9-16 .8538 .00551* 1050 11*75 ( t o p f l o o r ) 3600 . l ' * 6 .93 0 . 2 g O.Pg 0 . 2 g 0 . 3 g . 2 2 7 1 g 100 3 . 5 1021* 11*1* 100 3-5 1021* 11*1* 60 6 . 0 1021* 11*1* 130 8 . 0 •1296 11*|* 375 3.591* 1*87.8 2187000 1620 7 . 5 2187000 1620 7 . 5 2187000 1620 7 . 5 5181*000 2160 1 0 . 0 1*9560000 6308 12 .8 1*01*70000 5150 12.80 2187000 2187000 2187000 5181*000 1 3290000 1 0850000 1620 1620 1620 2160 1*067 3 3 ' 9 7 . 5 7 . 5 7 . 5 1 0 . " 8 - p £ 8 , 2 6 95/17OOO 951*7000 Table 5-2 P r o p e r t i e s of T e s t S t r u c t u r e s No V e r t i c a l degrees of Freedom on Uncoupled w a l l . 99 Kaximum Accelerator Lintel Capacity (Kip-Ft) Hall Ease Capacity (Kip-Ft) Wall Top Capacity (Kip-Ft) Young's Modulus (Ksi) Results Period Kode (1 ) (Laiaged) ^ ) ( 3 ) (4) (5) Par.ring Mode (1 ) ( 2 ; ( 3 ) (4) ( 5 ) Kayiir.uir. RKS Displacement (Inches) KuKber of Iterations Etectrel Acceleration Kode ( 1 ) ( 2 ) ( 3 ) Participation Factor Kode ( 1 ) ( 2 ) ( 3 ) RKS Axial Force at Base (Kips) Pun #1 Run #2 Run #3 Run #4 Run #5 Run #6 Pun #7 0.3g 0 . 3 g - 0 . 3 g 0 . 3 g 0.3g 0 . " 3g 0 . 3 g 4 0 8 0 80 4 0 80 3 0 0 1 3 0 1 6 6 0 0 1 6 6 0 0 1 6 6 0 0 34711 34711 34711 34711 1 4 4 0 0 1 4 4 0 0 1 4 4 0 0 2 7 8 3 0 2 7 8 5 0 2 7 8 3 0 2 7 8 3 0 3600 3600 4 0 3 0 3 6 0 0 3 6 0 0 3 6 0 0 3 6 0 0 2 . 3 3 3 0 . 3 7 5 0 . 1 3 5 0 . 0 7 2 0 . 0 4 7 0 . 0 7 2 0 . 0 4 1 0 . C 3 2 0 . 0 2 8 0 . 0 2 6 9 . 3 4 11 0 . 1 1 7 0 . 8 9 1 0 . 8 8 4 1 . 4 9 - 0 . 7 3 0 . 3 7 1 0 8 . 8 2 . 0 6 8 . 0.360 0 . 1 3 2 0.072 0.047 0.069 0.0^0 0 . 0 3 1 0.027 0.025 8 . 5 0 7 0 . 1 3 5 0 . 9 0 3 0.872 1.49 -0.73 0.38 1 . 9 9 9 0 . 3 4 4 0 . 1 2 6 0.068 0.044 0.'072 0 . 0 4 1 0 . 0 3 2 0 . 0 2 8 0 . 0 2 6 7 . 9 7 6 O . 1 3 6 0 . 8 8 9 0 . 8 2 4 1 . 4 9 - 0 . 7 3 0 . 3 8 2 . 0 3 5 0 . 3 4 0 0 . 1 2 6 0.069 0 . 0 4 6 0 . 0 3 5 C . 0 2 3 0 . 0 2 1 0 . 0 2 0 0 . 0 2 0 11 . 4 0 0 . 1 8 7 1 . 0 8 5 0 . 9 3 4 1 . 5 0 - 0 . 7 4 0 . 3 9 1 . 9 1 3 0 . 3 3 5 0 . 1 2 6 0 . 0 o 9 0 . 0 4 5 0 . 0 4 5 0 . 0 2 5 0 . 0 2 2 0 . 0 2 1 0 . 0 2 0 9 . 6 8 5 0.18C 1 . 0 5 3 0 . 9 2 0 1 . 5 0 - 0 . 7 4 0 . 3 9 3 0 6 . 7 1.479 0 . 3 1 1 0 . 1 2 3 0.068 C . 0 4 5 0 . 0 3 7 0 . 0 2 c 0 . C 2 2 C . 0 2 1 C .021 7.99 4 C . 2 52 1 . 0 4 8 0 . 8 9 4 1 . 4 8 -0.72 0 . 3 9 1 0 7 0 1.775 0 . 3 2 9 0.125 0.069 0.045 0.049 0.027 C.022 w • \_> n 1 0 . 0 2 1 £.58 c. 186 1 . 0 3 0 0 . 9 0 5 1 . 4 9 -0.7k 0 . 3 9 485 T a b l e 6.1 R e s u l t s o f Computer Runs on 1 6 - S t o r y Design Example 100 I n p u t member s i z e s , c a p a c i t i e s and damage r a t i o s , j o i n t l o c a t i o n s and o t h e r s t r u c t u r e d a t a .  M o d i f y moment of i n e r t i a of member* a c c o r d i n g t o i n p u t damage r a t i o s . Assemble mass and s t i f f n e s s ma t r i c e s F i n d mode shapes and f r e q u e n c i e s of s u b s t i t u t e s t r u c t u r e . i F i n d smeared damping f o r each mode based on s t r a i n energy i n members and member damping P e r f o r m modal a n a l y s i s w i t h smeared damping u s i n g f r e q u e n c i e s and mode shapes of undamped s t r u c t u r e . F i n d root-sum-square d e s i g n f o r c e s A m p l i f y d e s i g n f o r c e s u s i n g base shear C I n c r e a s e column moments by 20% r Output d e s i g n f o r c e s S top F i g u r e 2.1: F l o w c h a r t f o r the S u b s t i t u t e S t r u c t u r e Method. 101 Input member s i z e s , c a p a c i t i e s j o i n t l o c a t i o n s and o t h e r s t r u c t u r e d a t a .  Assemble mass and s t i f f n e s s m a t r i c e s P e r f o r m e l a s t i c modal a n a l y s i s u s i n g ' a p p r o p r i a t e ' damping F i n d members where Root-Sum-Square moments exceed y i e l d  — T M o d i f y damage r a t i o s  Reset a l l damage r a t i o s below u n i t y t o u n i t y .  T Check f o r convergence C a l c u l a t e s t i f f n e s s m a t r i x 1 t Compute n a t u r a l p e r i o d s and shapes mode \ Compute smeared damping t P e r f o r m modal a n a l y s i s u s i n g the . smeared damping r a t i o s .  i 1 F i g u r e 2.2: F l o w c h a r t f o r the M o d i f i e d S u b s t i t u t e S t r u c t u r e • Method 102 CM>ES i •I DAKOLD = Damage P a t i o f o r i - 2 I t e r a t i o n DAMB = Damage R a t i o f o r i - 1 I t e r a t i o n DAKRAT = Damage R a t i o f o r i I t e r a t i o n DAMDIF = DAMRAT-DAMB DR = Damage R a t i o r e t u r n e d t o program. F i g u r e 3.1; F l o w c h a r t f o r convergence s p e e d i n g r o u t i n e 103 7.k 0 . 8 6 ( 6 . 1 ) 0 . 86 (0.85! 10.^ (0 . 85 ) 15.1 (15 .6) ( 8 . 5 3 ) 0s32 15.1 (15.6) 0 . 94 (0 . 9 6 ) (0 . 9 9 ) 0 . 9 3 -0 . 9 4 (0 . 96 ) (0.9k) 3 . 7 (3.2) 1.2k 3 . 7 ( 3 .2 ) (0 . 99 ) 3 . 7 ( 6 . ^ ) 1 .76 3 . 7 (6. i+) 3 .0 (2 . 5 ) (1 .k9) 3 .0 (2 . 5 ) 3_.2- Comparison of Damage R a t i o s u s i n g a i f f e r e Convergence schemes on one-bay, s i x - s t o r y frame. ( R e s u l t s of o l d scheme shown i n p a r e n t h e s i s ) . i F i g u r e 3 . 3 Graph of Damage R a t i o v s . I t e r a t i o n f o r one Column of a s i x - s t o r y , one-bay frame. t \ . E M £ H T O F -—| C O N C E R N O IO Z O 3 0 + 0 S O 7 0 3 0 l O O ' 2 . 0 I t e r a t i o n dumber 14-0 IbO /so o 105 3> 1 > 3'. W.W77/7/777/777A Ll —4 L e s s e r J o i n t G r e a t e r J o i n t F i g u r e J>.k' Diagram of M o d i f i e d Member t o I n c l u d e R i g i d E x t e n s i o n s . •AE m L -•EI 1 0 6 F i g u r e U s ) H o r i z o n t a l Pendulum F i g u r e 4.1(b) V e r t i c a l Pendulum F i g u r e 4 . 1 ( c ) C a r t on f r i c t i o n l e s s r o l l e r s 1 0 7 <„k Beams: 18" x 30" 1=13,300 in' Columns: 24" x 24" 1=13,824 i n Young's Modulus= 3600 K s i . F l o o r Weight = 72 K i p s per f l o o r . 1 2 4 3 5 7 6 8 10 9 11 13 12 14 7 15 ^ 7 11 11« 11 11 11 • Not to Scale 24 F i g u r e 4.2 S h i b a t a and Sozen's f i v e - s t o r y s t r u c t u r e (showing member numbering used to designate Root-Sum-Square f o r c e s i n t a b l e s ) 1 1 10 F E E T •Figure k.^ C o n f i g u r a t i o n of T e s t S t r u c t u r e 'A' 132 (126) 241 (239) 326 (334) 390 (411) 431 (461) 7*77 33 ( 3 D 93 ( 9 D 141 (143) 178 (186) 201 (214) 132 (126) 241 (239) 326 (334) 390 (411) 431 (461) r*7 1^ Note: A c c e l e r a t i o n = 0.2 times g r a v i t y . A l l Bending moments i n K i p - F t . DRAIN-2D r e s u l t s shov;n i n p a r e n t h e s i s , F i g u r e 4.4 F i v e - S t o r y S t r u c t u r e w i t h r i g i d arms showing bending moments produced from e l a s t i c modal and e l a s t i c time s t e p a n a l y s i s . 110 R I G I D F i g u r e 5.1 G e n e r a l Test S t r u c t u r e C o n f i g u r a t i o n F i g u r e 5.2a Angle Used For C a l c u l a t i o n Of Member D u c t i l i t y Member F i g u r e 5.2h p a , , i U t i l i t y As Given By P a u l a y ( F i g u r e From Ref. 6) — 6 U L T I l l D u c t i l i t y Demand 5 - S t o r y W a l l F i g u r e 5.3 OV the C o u p l i n g (Test Beams S e r i e s 'B') f o r the 0) X> E D Z u o o i-H o I 1 3 4-D u c t i l i t y Demand of C o u p l i n g Beams 1 1 2 F i g u r e 5.4 D i s p l a c e r e n t Envelopes f o r the 5-Story W a l l (Test S e r i e s 'B') F i g u r e 5.5 D u c t i l i t y Demand Of the Coup l i n g Beams fo r the 5 -S tory Wa l l (Mass=4 Times O r i g i n a l Run) £ 3 - 4 - 5 * 7 8 9 IO II 12. D u c t i l i t y Demand of Coup l i n g Beams 114 F i g u r e 5.6 D i s p l a c e m e n t Envelopes f o r the 5-Story W a l l (Mass=4 Times O r i g i n a l Run) .4 .6 .8 1.0 ».ZL 1.4- 1.* 1.8 e.O 2.2. H o r i z o n t a l D i s p l a c e m e n t ( I n c h e s ) 5 - S T O R Y *WAV-L C O R I G I N M - ttKSS^ S - S T O R Y W M L ( O R T G X H A L t A A S S * - ^ , \ Q - S T O R Y W A L L I f c - S T O R X V A L L W I T H W f c L L C/l 3* o tfe S T O R Y rq I - I . c cr 3 OJ Cb rr Oj 0) 3 "I(D 0) 1 - 13 3 O ri- CL 3 o> "1 t-" CO *3 O zr u N c M - fD fD B I D fD (-•• -H O Ul cn • <D cn co O T J l-h fD cn o • >-3 rt ro n Ul c ri" 3 CO rt C o rr C •n fD Ul F i g u r e 5.8 D e f l e c t i o n E nvelopes For The 1 0 - s t o r y W a l l 1.0 t 3 4- S Horizontal Displacement (Inches) Figure 5.9 Coupling Beam Damage Ra t i o s For The Ten Story W a l l . Coupling Beam Damage R a t i o \ B \ V \ \ A \ / v c\\ D F i g u r e ^.10 \ D u c t i l i t y Demand of C o u p l i n g Beams \ f o r 16-Story W a l l w i t h E x t r a Uncoupled W a l l V \ w » K E Y ( M /^S DONE BY pc * i ( 0 % DAMPING S ' / o S T R A I N H A R D E N I N G f B l 2.% D/\MPING 5 % s. H . (p) AS> 'C' B U T W I T H O U T SH£APv D E F L . ,o ia 14- i£ is 20 e& z4- a t ee 3o 3 a D u c t i l i t y Demand of Coupling- Beams 1 1 9 F i q u r e 5.11 c r u r a s s s ^ i c K s s : ^ a ° K u H o r i z o n t a l D e f l e c t i o n ( Inches) 1 2 0 F i g u r e 5.12 Damage R a t i o s For The 1 6 - s t o r y C oupled W a l l With A t t a c h e d Uncoupled W a l l 7 5 6 8 JO IZ 14- It, 18 &0 2a 24- U 26 30 32 34-C o u p l i n g Beam Damage R a t i o 1 2 1 F i g u r e 5.13 Average D u c t i l i t y Demand of C o u p l i n g Beams f o r the 1 6 - S t o r y Coupled W a l l w i t h A t t a c h e d Uncoupled W a l l . 122 IB 17 — ' 16 4-> W o IS u c M-- r - i a 1 3 e o u 12 x: u II ro 03 " 10 4-> • rH V J 9 o .1-1 Pr 8 r-l (0 E 7 Ui o z 6 4-> • r-i s o IX 1 4 c o z F i g u r e 5.14 T y p i c a l E x e c u t i o n and P r i n t i n g C o s t s f o r a S i n g l e Computer Run / / / / / / / / / / / O R M N - I O E L A S T I C M O D M -\\ .12 »3 I* i<» 4 - 5 6 7 8 3 >0 Number of S t o r i e s i n S t r u c t u r e 123 B.50 Til 1 - 2 0 ' -T 7.75' ( 135.25' -He. T IB* 1 S E C T I O N ! A.-t i_ I F T 10 FEET APPROX SCALE ZO' -F i g u r e 6.1 C o n f i g u r a t i o n of 16- S t o r y Coupled W a l l f o r Design Example 16.3 0.05 0.14 0.25 0.36 0.45 0.53 0.53 0.61 0.62 0.62 0.61 0.63 0.70 0.75 0.87 2.87 16,3 16.3 16.1 15-8 15.3 14 .8 1L .2 1 3.6 12.8 12.0 11 .2 10.1 8 . 9 7.4 5 . 5 xz 0.06 0.15 0.26 0.36 0.46 0.54 0.59 0.62 0.63 0.63 0.62 0.64 0.71 0 . 7 5 0 . 87 2 . 8 7 Floor 1 6 1 5 1 4 1 3 1 2 11 1 0 . 9 8 7 6 5 4 3 2 1 Figure 6.2 Damage Ratios from the F i r s t Run on the 16-Story Design Example 125 0 . 0 5 0 . 1 4 0.24 0.34 0.44 0.51 0.56 0.59 0 . 6 0 0 . 6 0 0.59 0.62 0.68 0.78 0.91 2.29 7.4. rtr, 7.4 7.3 7.2 7.0 0.05 0.14 0.25 0.36 0.45 6.7 === L*_8_ 5.5 5.0 4.5 3.9 3.1 0.52 0.57 0 . 6 0 0.61 0 . 5 9 0.59 0 .62 0 . 6 8 2 . 2 0.78 0.91 2.29 F l o o r 1 6 1 5 14 1 3 1 2 11 1 0 9 8 7 6 . 5 4 3 2 1 Damage R a t i o s from the S e c o n d ^ on'the 1 6 - S t o r y D e s i g n Example • 7.7 F l o o r 16 15 7.7 0.05 0.14 7 7 0.14 14 0.24 7.6 0.25 13 0.34 7.5 0.35 12 0.43 7.3 0.41' 11 0.50 7.1 0.52 10 0.56 6.8 C.57 9 0.59 6.5 0.60 8 C.60 c t T 1 . 0.61 7 0.60 5.7 , 0.61 6 0.61 grr— . —• 1— 0.61 5 0.63 0.63 4 0.69 4.1 0.69 3 0.78 3.4 0.78 2 0.91 2 .5 R  0.90 1 2.54 , : i_ X 2.54 F i g u r e 6 .4 Damage R a t i o s from the T h i r d Run on the 1 6 r S t o r y D e s i g n Example 127 21 .2 0.04 0.09 0.16 0.23 0.29 0.33 0.37 2 i .2 21 .1 20.8 0.04 0 . 0 9 0.17 20.4 19.8 19.1 0.23 0.29 0.34 0.38 18.1 0.39 0.41 0.42 0.44 0.48 0.53 0.61 0 . 6 9 0.78 17.1 15.9 12.9 11 , n 8.8 6.2 . 3.2 — 0.40 0.41 0.43 0.45 0.46 0.53 0.61 0 . 6 9 0.78 F l o o r 1 6 1 5 14 13 12 11 10 9 8 7 6 5 4 3 2 1 F i g u r e 6.5 Damage R a t i o s from the F o r t h Run on the 1 6 - S t o r y D e s i g n Example 128 0.03 0.09 0.15 0.21 0.26 0.3C 0.3. 0.33 0.36 0.37 0.33 0.41 0.46 0.53 0.61 0.69 8.9 8.8 8.7 8.3 8.0 1A. 7.2 6.7 6.2 5.5 4.7 2.7 1.4 —i 0.03 0.09 0.15 0 . 22 0 . 2 7 0.31 0.34 0.36 0.36 0.37 0.33 0.41 0.46 0.53 0.61 0.69 F l o o r 16 15 14 13 12 11 10 9 8 7 6 5 4 3 2 1 F i g u r e 6.6 Damage R a t i o s from the F i f t h Run on the 1 6 - S t o r y D e s i g n Example 1 2 9 1.6 0.07 0.1 1 0.16 0.20 C.23 0.26 1.6 1.6 1 . 7 0.?7 0.28 0.29 0.32 0.35 0.41 0.66 1 .7..-^= i .6 0.04 0 .07 0.11 0.16 0.21 0.24 0.26 1.6 1 .5 l .4 1 .3 1.2 1.1 0 . 9 0.6 7 0.3 —i-—-0.28 0.28 0.29 0.32 0.35 0.41 0.48 0.57 0.66 F l o o r 16 15 14 13 12 11 10 9 8 7 6 5 4 3 2 1 F i g u r e 6.7 , , Damage R a t i o s from the S i x t h Run on the 16-Story Design Example 130 0.03 0.08 0.13 0 . 2 0 0.21+ 0.28 0.30 0.32 0.32 0.33 0.34 0.37 0 . 4 2 0.48 0 . % 0 .64 4 . 7 4 .7- ' 4 . 7 4.6 4 .A 4.1 3.9 3 .6 3.4 3.0 2.1 J_*_5_ 0.8 ri7 0.03 0.08 0.14 0 . 2 0 0.25 0.28 0.30 0.32 0.32 0.33 0.34 0.37 0 . 42 0 . 4 8 0 . 56 0 .64 F l o o r 16 15 14 13 12 11 . 10 9 8 7 6 5 4 3 2 1 F i g u r e 6.8 Damage R a t i o s from the Seventh Run on the 1 6 - S t o r y D e s i g n Example 1 3 1 1 3 2 F i g u r e 6.10 D e f l e c t i o n E n v e l o p e s from DRAIN-2D Runs On 1 6 - S t o r y Design Example u 0> X) e z u o o r H lb • 5 13 «a 8 S 4-3 O t£ r-Z <J</> 2 Z Cf Z f LU Ul / o of H Z 4 3 4 - 5 6 7 8 9 tO U H o r i z o n t a l D i s p l a c e m e n t ( I n c h e s ) 12 «3 t 0 . 3 q Maximum ground acceleration M A X I M U M D A . M A 6 E R A T I O 0.5 Figure 6.11 , Coupling Beam Damage Ratio Of 16-story Example for Various Values of Maximum Ground Acceleration — _ 1 % 4 5 - 6 7 Coupling Beam Damage Ratio 10 F i g u r e 6.12 Damage P e r i o d as a F u n c t i o n of Maximum Ground A c c e l e r a t i o n f o r the 16-St o r y Example. F U N D A M E N T A L PERIOD C . E C O N D r A O D E .1 •» ° - S Maximum Ground A c c e l e r a t i o n as a F r a c t i o n of G r a v i t y F i g u r e 6.13 Speared Damping R a t i o as a F u n c t i o n of Maximum Ground A c c e l e r a t i o n f o r the 1 6 - S t o r y Example. r— i — 1 — 1 " TV O., 0.2 0.3 0.4- 0 . 5 Maximum Ground A c c e l e r a t i o n as a F r a c t i o n of G r a v i t y 0.5G u OJ x> 6 Z >i Li o 4J 1 0 Maximum ground a c c e l e r a t i o n F i g u r e H o r i z o n t a l Displacement Of 1 6 - s t o r y D e s i g n Example At V a r i o u s Ground A c c e l e r a t i o n s j»i 12. % o 0- B if) 8 H D * 6 ? 4 y R M S " T O P D I S P L A C E M E M T Vs M A X . G R O U N D . A C C E L , 7\ 1 1 ^ (*6> M A X . G R O U N D ACCLL.ERA.TI0W 5 6 7 6 - 9 H o r i z o n t a l D e f l e c t i o n (Inches) l 4 I 5 17 M CO 137 BIBLIOGRAPHY (1) F i n t e l , Mark. " D u c t i l e Shear W a l l s i n Earthquake R e s i s t a n t M u l t i s t o r y B u i l d i n g s . " ACI J o u r n a l , June 1974, pp 96-305. (2) F i n t e l , Mark and Ghosh, S.K. S e i s m i c R e s i s t a n c e of a 16- S t o r y C o u p l e d - W a l l S t r u c t u r e : A Case Study U s i n g I n e l a s t i c  Dynamic A n a l y s i s . P o r t l a n d Cement A s s o c i a t i o n , 1980. (3) Kanaan, A.E. And P o w e l l , Graham H. DRAIN-2D A G e n e r a l  Purpose Computer Program f o r Dynamic A n a l y s i s of I n e l a s t i c P l a n e  S t r u c t u r e s . W i t h U s e r ' s Guide and Supplement. Report No. EERC 73-6 and EERC 73-22 R e v i s e d August 1975. U n i v e r s i t y of C a l i f o r n i a , B u r k e l e y . (4) P a u l a y , Thomas "An E l a s t o P l a s t i c A n a l y s i s of Coupled Shear W a l l s . " ACI J o u r n a l , November 1970, pp 915-922. (5) P a u l a y , Thomas "Design A s p e c t s of Shear W a l l s f o r S e i s m i c A r e a s " . Canadian J o u r n a l of C i v i l E n g i n e e r i n g 2,321 (1975). (6) P a u l a y , T and Uzumeri, S.M. "A C r i t i c a l Review of the S e i s m i c D e s i g n P r o v i s i o n s f o r D u c t i l e Shear W a l l s of the Canadian Code and Commentary". Canadian J o u r n a l of C i v i l  E n g i n e e r i n g . 2, 592, (1975) pp 592-601. (7) S a a t c i o g l u , Murat, Derecho, A r n a l d o T., C o r l e y , w. Gene Coupled W a l l s i n E a r t h q u a k e - R e s i s t a n t B u i l d i n g s . M o d e l i n g  Techniques and Dynamic A n a l y s i s . PCA R/D s e r . 1628 C o n s t r u c t i o n Technology L a b o r a t o r i e s , P o r t l a n d Cement A s s o c i a t i o n . June 1980. (8) S h i b a t a , A k e n o r i and Sozen, Mete. A. " S u b s t i t u t e - S t r u c t u r e Method f o r S e i s m i c D e s i gn i n R/C. J o u r n a l of the S t r u c t u r a l  D i v i s i o n ASCE Jan 1976, pp 1-18. (9) S t r u c t u r a l A n a l y t i c a l S e c t i o n , E n g i n e e r i n g Development Department P o r t l a n d Cement A s s o c i a t i o n i n s t r u c t i o n s f o r  p r e p a r i n g i n p u t data f o r DRAIN-2D. May 1978. (10) Y o s h i d a , Sumio M o d i f i e d S u b s t i t u t e S t r u c t u r e Method f o r  A n a l y s i s of E x i s t i n g R/C S t r u c t u r e s . M a s t e r ' s t h e s i s U n i v e r s i t y of B r i t i s h Columbia 1979. 138 USER'S MANUAL ELASTIC AND/OR DAMAGE AFFECTED MODAL ANALYSIS ( U t i l i z i n g the M o d i f i e d S u b s t i t u t e S t r u c t u r e Method.) Program Name: EDAM DISCLAIMER: The C i v i l E n g i n e e r i n g Department, F a c u l t y and S t a f f do not guarantee nor im p l y the a c c u r a c y or r e l i a b i l i t y of t h i s program or r e l a t e d d o c u m e n t a t i o n . As suc h , they can not be h e l d r e s p o n s i b l e f o r i n c o r r e c t r e s u l t s or damages r e s u l t i n g from the use of t h i s program. I t i s t h e r e s p o n s i b i l i t y of the user t o determine the u s e f u l n e s s and t e c h n i c a l a c c u r a c y of t h i s program i n h i s or her own environment. T h i s program may not be s o l d t o a t h i r d p a r t y . 139 EDAM PROGRAM HISTORY UPDATES 1978 Aug 1979 1980-1981 Sept-Oct 1980 MODIFICATIONS MSSM.S program w r i t t e n MSSM.S manual w r i t t e n MSSM.S t o EDAM EDAM manual w r i t t e n PROGRAMMER Sumio Y o s h i d a Ron G r i g Andrew W. F. Metten Andrew W. F. Metten, NOTE the program was renamed from i t s o r i g i n a l t i t l e of MSSM.S t o d i s t i n g u i s h i t from the o r i g n a l program i n the C i v i l E n g i n e e r i n g Program L i b r a r y and t o r e f l e c t t he d i f f e r e n t c a p a b i l i t i e s and modus o p e r a n d i of the new program. EDAM i s 1646 l i n e s l o n g w h i l e MSSM.S i s 824 l i n e s l o n g . TABLE OF CONTENTS Program H i s t o r y 139 T a b l e of C o n t e n t s 140 I n t r o d u c t i o n 141 Theory 142 Program R e s t r i c t i o n s 142 D i m e n s i o n i n g l i m i t s 143 Input D e t a i l s 144 Output Assignments 148 O p e r a t i n g I n s t r u c t i o n s 148 L i b r a r y S u b r o u t i n e s C a l l e d 149 T i m i n g 150 Other I n s t i t u t i o n s 150 Appendix 151 sample problem 141 INTRODUCTION EDAM s t a n d s f o r E l a s t i c and/or Damage A f f e c t e d Modal a n a l y s i s . T h i s program performs e l a s t i c and e l a s t o p l a s t i c a n a l y s i s of pl a n e frames. The e l a s t o p l a s t i c a n a l y s i s i s performed u s i n g the M o d i f i e d S u b s t i t u t e S t r u c t u r e Method. The program i s c a b a b l e of h a n d l i n g a l l c o m b i n a t i o n s of f i x e d and p i n ended beams as w e l l as f i x e d ended beams which have r i g i d e x t e n s i o n s . EDAM was o r i g i n a l l y w r i t t e n by Sumio Y o s h i d a and c a l l e d MSSM.S upon c o m p l e t i o n . The program was renamed t o EDAM a f t e r b e i n g e x t e n s i v e l y m o d i f i e d d u r i n g a subsequent master's t h e s i s r e s e a r c h p r o j e c t d u r i n g 1980. E v e n t u a l l y both EDAM and MSSM.S w i l l be found i n the C i v l E n g i n e e r i n g Program L i b r a r y . In d e v e l o p i n g the computer program which c o u l d a p p l y the m o d i f i e d s u b s t i t u t e s t r u c t u r e method t o a s t r u c t u r e , c o n s i d e r a b l e e f f o r t was a p p l i e d t o make i t a method t h a t c o u l d be used by the p r a c t i s i n g e n g i n e e r i n a d e s i g n s i t u a t i o n . At the expense of s l i g h t l y i n c r e a s i n g t h e s t o r a g e r e q u i r e m e n t s and e x e c u t i o n time 'common b l o c k s ' were not u t i l i z e d i n w r i t i n g of the code. T h i s was done i n the b e l i e f t h a t a computer program i s o n l y complete when i t i t i s found t o be too un w i e l d y t o modify t o t a c k l e a problem v a r y i n g i n some form from the o r i g i n a l l y d e s i g n e d t a s k . A program w i t h o u t l a r g e common b l o c k s w i l l o f t e n be e a s i e r t o modify as each s u b r o u t i n e has more autonomy from the remainder of the program. By e l i m i n a t i n g common b l o c k s the m o d i f i c a t i o n of the program i s made e a s i e r . Should anyone t r y t o modify the program EDAM they w i l l f i n d a l a r g e c o l l e c t i o n of 'comment c a r d s ' i n the program o u t l i n i n g what i s b e i n g a t t e m p t e d a t each s t a g e as w e l l as d e t a i l i n g what many of the v a r i a b l e names s t a n d f o r . When m o d i f y i n g the program the concept throughout was t o e l i m i n a t e a l l u nnecessary c o m p l e x i t y . W h i l e i t i s p o s s i b l e t o b u i l d s t r u c t u r e d a t a g e n e r a t o r s i n t o a program t h i s o f t e n c r e a t e s unnecessary c o m p l e x i t y . Data g e n e r a t i o n programs a r e a v a l u a b l e a s s e t t o the speedy computer a n a l y s i s of a s t r u c t u r e ; however i t was the b e l i e f i n m o d i f y i n g the program t h a t they a r e b e t t e r kept s e p e r a t e from the program t h a t i s t o use the d a t a . B e s i d e s making t h e main program s i m p l e r , t h i s system has s e v e r a l o t h e r a d v a n t a g e s . I f the g e n e r a t o r i s s e p a r a t e from the e x e c u t i o n of t h a t d a t a , e d i t i n g of the data can occur b e f o r e i t s e x e c u t i o n which e x t e n d s the c a p a b i l i t i e s of the g e n e r a t o r . Another advantage i s t h a t r e c o m p i l a t i o n c o s t s are reduced i f the source code i s a l t e r e d when t h e r e a r e fewer e x e c u t a b l e l i n e s . 142 THEORY The program uses the spectrum t e h n i q u e f o r c a l c u l a t i n g f o r c e s and d i s p l a c e m e n t s . Modal a n a l y s i s i s c a r r i e d out t o produce the r e s u l t s t h a t a r e p r i n t e d . The user s h o u l d be f a m i l a r w i t h modal a n a l y s i s t o f u l l y a p p r e c i a t e the o u t p u t . F u r t h e r d e t a i l s may be found i n Sumio Y o s h i d a ' s master's t h e s i s "MODIFIED SUBSTITUTE STRUCTURE METHOD FOR ANALYSIS OF EXISTING  R/C STRUCTURES" March 1979 and Andrew W.F.Metten's t h e s i s : ' The M o d i f i e d S u b s t i t u t e S t r u c t u r e Method As A Design A i d For S e i s m i c  R e s i s t a n t Coupled S t r u c t u r a l W a l l s , March 1981. PROGRAM RESTRICTIONS Most of the program r e s t i c t i o n s t h a t w i l l a f f e c t the user can be found i n Sumio Y o s h i d a ' s t h e s i s . These p o i n t s are r e p r o d u c e d here as a reminder. The f o l l o w i n g r e s t r i c t i o n s a p p l y t o both e l a s t i c and damage a f f e c t e d r u n s . (1) The system can be a n a l y z e d i n one v e r t i c a l p l a n e . (2) There a r e t o be no a b r u p t changes i n mass, s t i f f n e s s or geometry throughout the h e i g h t of the s t r u c t u r e . (3) N o n s t r u c t u r a l components a r e t o be such t h a t they do not a f f e c t the response of the system as modeled. (4) The program cannot handle more than one type of m a t e r i a l d i r e c t l y . S h o u l d the u s e r d e s i r e t o t e s t a s t r u c t u r e t h a t c o n t a i n s more than one type of m a t e r i a l the members c o n s t r u c t e d of the second m a t e r i a l t ype s h o u l d have the a r e a and i n e r t i a m u l t i p l i e d by the modular r a t i o (E f o r type 1 d i v i d e d by E f o r type 2) (5) The program a p p l i e s the same a c c e l e r a t i o n t o both h o r i z o n t a l and v e r t i c a l masses. T h e r e f o r e v e r t i c a l masses s h o u l d not. be i n c l u d e d ; s h o u l d masses be a t t a c h e d t o some of the nodes then o n l y mode shapes and f r e q u e n c i e s w i l l be computed c o r r e c t l y . T h i s r e s t r i c t i o n l i m i t s the program t o the a n a l y s i s of s t r u c t u r e s f o r which v e r t i c a l a c c e l e r a t i o n of the nodes i s not a s i g n i f i c a n t f a c t o r . For most 143 s t r u c t u r e s w i t h masses on v e r t i c a l column l i n e s t h i s r e s t r i c t i o n w i l l not be a l i m i t i n g c o n s i d e r a t i o n . (6) The s t r u c t u r e must comply w i t h the d i m e n s i o n i n g r e q u i r e m e n t s of EDAM. The f o l l o w i n g r e s t r i c t i o n s a p p l y o n l y t o damage a f f e c t e d a n a l y s i s . (1) The m a t e r i a l s used f o r the c o n s t r u c t i o n of the s t r u c t u r e must be c o n c r e t e . The development of the s t i f f n e s s d e g r e d a t i o n and damping f o r m u l a was done . c o m p l e t l y on c o n c r e t e members and the r e s e a r c h does not a p p l y t o s t e e l or o t h e r n o n - c o n c r e t e m a t e r i a l s . (2) The members must be d e s i g n e d such t h a t they can w i t h s t a n d the damage r a t i o s imposed w i t h o u t u n d e r g o i n g b r i t t l e f a i l u r e . (3) The members a r e assumed t o be symmetric and have the same moment c a p a c i t y under both p o s i t i v e and n e g a t i v e bending moments. (4) The i n i t i a l fundamental e l a s t i c p e r i o d s h o u l d be such t h a t i t p l a c e s the s t r u c t u r e on a segment of the spectrum used which causes a d e c r e a s e i n the s p e c t r a l a c c e l e r a t i o n when the p e r i o d of the s t r u c t u r e i n c r e a s e s . DIMENSIONING LIMITS The maximum di m e n s i o n s of the s t r u c t u r e a r e . 100 members 100 j o i n t s 50 a s s i g n e d masses 10 e i g e n v a l u e s ( t o t a l degrees of f r e e d o m ) * ( h a l f bandwidth) i s l e s s than 2000. These d i m e n s i o n s can be e a s i l y i n c r e a s e d by i n t e r n a l adjustment of the program s h o u l d t h i s be d e s i r e d . 144 INPUT Input t o the program c o n s i s t s of program o p t i o n s and s t r u c t u r e d a t a . The u n i t s a r e B r i t i s h . The j o i n t c o o r d i n a t e s t o be i n p u t i n f e e t and d e c i m a l s of a f o o t . Weights a r e i n k i p s . M a t e r i a l c o n s t a n t s i n k i p s p e r square i n c h . The i n e r t i a of the s e c t i o n s t o be i n s e r t e d i n i n c h e s t o the f o u r t h and the c r o s s s e c t i o n a l a r e a s i n square i n c h e s . A l l i n p u t d a t a i s echo p r i n t e d by the program. CARD 1; INELAS,NMODES,NPRINT,I SPEC,AMAX,DAMPIN Format(4l5,2F10.5) (one c a r d ) INELAS =0 i f e l a s t i c a n a l y s i s i s r e q u e s t e d . =1 or g r e a t e r , i f i n e l a s t i c a n a l y s i s i s r e q u e s t e d . INELAS s h o u l d be s e t t o the maximum number of i n e l a s t i c i t e r a t i o n s t h a t may be performed b e f o r e the program h a l t s . A v a l u e of 50 s h o u l d be s u f f i c i e n t . NMODES = the number of modes t o be i n c l u l d e d i n the a n a l y s i s and s h o u l d be l e s s than or e q u a l t o 10. NPRINT = the number of modes f o r which p r i n t e d d i s p l a c e m e n t s and f o r c e s a re r e q u e s t e d . Mode 1 t o mode NPRINT i n c l u s i v e w i l l be p r i n t e d . I f NPRINT i s g r e a t e r than NMODES, then NPRINT w i l l be s e t e q u a l t o NMODES. I f NPRINT e q u a l s z e r o then o n l y r o o t -mean-square f o r c e s and d i s p l a c e m e n t s w i l l be p r i n t e d . ISPEC =the spectrum type t h a t i s r e q u i r e d . =1 spectrum 'A' from the work of S h i b a t a and Sozen =2 spectrum 'B' from Y o s h i d a =3 spectrum 'C from Y o s h i d a =4 N a t i o n a l B u i l d i n g Code spectrum. N o t e : f i g u r e s showing the spectrums may be found i n the appendix of t h i s manual. AMAX =maximum ground a c c e l e r a t i o n as a f r a c t i o n of g r a v i t y . DAMPIN = the f r a c t i o n of c r i t i c a l damping t h a t i s t o be used i n the e l a s t i c a n a l y s i s or i n the f i r s t i t e r a t i o n of the i n e l a s t i c a n a l y s i s . Card 2 TITLE Format(20A4) (one c a r d ) Any a p p r o p i a t e t i t l e composed of l e s s than 80 l e t t e r s , numbers and sp a c e s . 145 Card 3 NRJ,NRM, E, G Format(215,2F10.0) (one c a r d ) NRJ =number of j o i n t s i n the s t r u c t u r e NRM =number of members i n the s t r u c t u r e E=Young's modulus G=Shear modulus. I f shear modulus i s i n p u t as z e r o , then no shear d e f l e c t i o n s w i l l be c a l c u l a t e d . C ard 4: JN,NDX,NDY,NDR,X,Y Format(415,2F10.3) (NRJ c a r d s : 1 c a r d / j o i n t ) JN = the node number NDX =0 i f the node cannot move i n the x d i r e c t i o n =1 i f the node can move i n the x d i r e c t i o n . =N i f the node i s t o have the same motion as node N. NDY =0 i f the node cannot move i n the y d i r e c t i o n =1 i f the node can move i n the y d i r e c t i o n . =N i f the node i s t o have the same y motion as node N. NDR =0 i f the node cannot move i n the r o t a t i o n d i r e c t i o n =1 i f the node can r o t a t e =N i f the node i s t o have the same r o t a t i o n as node N. Card 5: MN,JNL,JNG,KL,KG,AREA,CRMOM,AV,BMCAP,EXTL,EXTG FORMAT(5I5,F8.2,F12.3,2F10.3,2F6.3) (NRM CARDS: 1-CARD/MEMBER) MN =the member number. JNL =the l e s s e r j o i n t number JNG =the g r e a t e r j o i n t number Note: The o r d e r i n g of the j o i n t numbers w i l l not a f f e c t the r e s u l t s produced. For eve r y member t h e r e i s a j o i n t numbering t h a t w i l l cause e i t h e r x or y d i s p l a c e m e n t s t o be p r i n t e d as a n e g a t i v e number. The p r i n t i n g of n e g a t i v e d i s p l a c e m e n t s of the member s h o u l d not d i s t u r b the user.. KL =1 i f the member i s f i x e d a t the l e s s e r j o i n t number. =0 i f the member i s p i n n e d a t the l e s s e r j o i n t number KG =1 i f the member i s f i x e d a t the g r e a t e r j o i n t number. =0 i f the member i s p i n n e d a t the g r e a t e r j o i n t number AREA =cross s e c t i o n a l a r e a of the member. CRMOM = the c r a c k e d moment of i n e r t i a of the member. For e l a s t i c a n a l y s i s the number used here i s the i n e r t i a 146 d e s i r e d f o r a c t u a l a n a l y s i s . AV =Shear a r e a of the member. At t h i s stage AV s h o u l d e q u a l z e r o as t e s t i n g i s i n c o m p l e t e on the program's a b i l i t y t o h a n d l e shear d e f l e c t i o n s . =0.0 then shear d e f l e c t i o n s w i l l not be computed. EXTG = r i g i d e x t e n s i o n on the l e s s e r j o i n t end of member EXTL = r i g i d e x t e n s i o n on the g r e a t e r j o i n t end of the member. NOTE; f o r p r o p e r use of the r i g i d e x t e n s i o n s they must be p o s i t i v e and w i t h a l e n g t h t h a t has a sum t o t a l f o r b o t h r i g i d e x t e n s i o n s of l e s s than the s p a c i n g between the j o i n t s which the member i s s p a n n i n g . In o t h e r words f o r the program t o execute t h e r e cannot be a member w i t h z e r o or n e g a t i v e e l a s t i c l e n g t h . Note a l s o t h a t a t t h i s s t a g e of program development the r i g i d arms a r e assumed t o be a t t a c h e d o n l y t o h o r i z o n t a l members. The attachment of r i g i d arms t o n o n - h o r i z o n t a l members w i l l r e s u l t i n the p r i n t i n g of an e r r o r message. Card 6. NMASS Format(15) (one c a r d ) NMASS =The number of nodes t o which a weight i s a t t a c h e d . T h i s i s independant of the number of w e i g h t s which a r e a t t a c h e d t o those nodes. I f t h e r e a r e l e s s than NMODES degrees of freedom t o which masses a r e a t t a c h e d then NMODES w i l l be s e t e q u a l t o the number of degrees of freedom t o which masses a r e a t t a c h e d . Card 7. JN,WTX,WTY,WTR For m a t ( I 5,3F10.0) ( NMASS c a r d s : 1 c a r d / j o i n t w i t h mass) JN = J o i n t t o which the weight i s a p p l i e d . WTX = Weight i n x d i r e c t i o n . WTY = Weight i n y d i r e c t i o n . . WTR = R o t a t i o n a l w e i g h t . NOTE t h a t w e i g h t s must be i n s e r t e d as such. The program c o n v e r t s them t o mass by d i v i d i n g by the s t a n d a r d v a l u e f o r the a c c e l e r a t i o n of g r a v i t y (32.2 f t / s e c 2 ) . A l s o note t h a t once the masses have been a s s i g n e d t o the a p p r o p i a t e degrees of freedom the program does not d i s t i n g u i s h between masses opposing motion i n the h o r i z o n t a l d i r e c t i o n and t h o s e o p p o s i n g motion i n the v e r t i c a l or r o t a r y d i r e c t i o n s . T h i s means t h a t the . same s p e c t r a l a c c e l e r a t i o n w i l l be a p p l i e d t o both d i r e c t i o n s . The u s e r i s c a u t i o n e d t h a t masses o p p o s i n g motion i n 1 4 7 d i r e c t i o n s o t h e r than the h o r i z o n t a l d i r e c t i o n a re i n most c a s e s u n n e c e s s a r y . For f u r t h e r d e t a i l s see a l s o 'program r e s t r i c t i o n s ' and 'approximate e x e c u t i o n t i m e s ' i n t h i s manual. 148 OUTPUT U n i t 6 T h i s f i l e i s f o r d a t a from i n t e r m e d i a t e i t e r a t i o n s from i n e l a s t i c a n a l y s i s . I t s h o u l d not be needed u n l e s s an e r r o r o c c u r s , or i t i s wished t o examine the p r o g r e s s of convergence a t t h e c o n c l u s i o n of a r u n . N o t h i n g of use i s w r i t t e n on u n i t 6 d u r i n g e l a s t i c a n a l y s i s . The user i s c a u t i o n e d t h a t f i l e 6 may become q u i t e l e n g t h y d u r i n g runs u s i n g the M o d i f i e d S u b s t i t u t e S t r u c t u r e Method and t h a t i t i s w o r t h w h i l e d e t e r m i n i n g f i r s t l y i f what i s on the f i l e i s d e s i r e d and s e c o n d l y i f the f i l e i s un r e a s o n a b l y l e n g t h y p r i o r t o p r i n t o u t . UNIT 7 T h i s f i l e c o n t a i n s the m a j o r i t y of the u s e f u l o utput from both e l a s t i c and i n e l a s t i c a n a l y s i s . Input member d a t a as w e l l as output f o r c e s and d i s p l a c e m e n t s appear on t h i s f i l e i n a manner t h a t s h o u l d make them r e a s o n a b l y s t r a i g h t f o r w a r d t o un d e r s t a n d . Note t h a t f o r e l a s t i c a n a l y s i s as the i n p u t moment c a p a c i t i e s have l i t t l e purpose n e i t h e r w i l l the output damage r a t i o s . S h o u l d the program s t o p u n e x p e c t e d l y i t i s i n p o r t a n t t h a t u n i t 7 be p r i n t e d out t o a i d i n the debugging p r o c e s s . U n i t 7 w i l l c o n t a i n any e r r o r messages t h a t a r e g e n e r a t e d by the e r r o r c h e c k i n g r o u t i n e s i n s i d e the program i t s e l f . UNIT 8 T h i s f i l e c o n t a i n s the damage r a t i o s f o r each member a t the c o n c l u s i o n of each i t e r a t i o n . I t s h o u l d not be r e q u i r e d t o be p r i n t e d a f t e r p e r f o r m i n g e l a s t i c a n a l y s i s . The user s h o u l d not a s s i g n a f i l e t o c o n t a i n output i f he has no i n t e r e s t i n ever v i e w i n g o r p r i n t i n g out t h a t f i l e . F o r example s h o u l d e l a s t i c a n a l y s i s be run then f i l e 8 w i l l not be r e q u i r e d . Under th e s e c i r c u m s t a n c e s the program w i l l run more e f f i c i e n t l y and v i r t u a l memory c o s t s w i l l be l e s s i f the ou t p u t f i l e i s a s s i g n e d t o *DUMMY*. OPERATING INSTRUCTIONS I t i s assumed f o r t h i s d i s c u s s i o n t h a t the user has a c o m p i l e d v e r s i o n of EDAM i n h i s f i l e COMPILED and t h a t the d e s i r e d d a t a f i l e i s the f i l e DATA. The f o l l o w i n g command w i l l run the program. $RUN COMPILED 5=DATA 6=~6 7=-7 8 = ~8 Sho u l d e l a s t i c a n a l y s i s be performed then the f o l l o w i n g command would be a p r e f e r a b l e command. $RUN COMPILED 5=DATA 6=*DUMMY* 7=-OUT 8=*DUMMY* 1 4 9 LIBRARY SUBROUTINES CALLED The program c a l l s two main s u b r o u t i n e s from the U n i v e r s i t y of B r i t i s h Columbia (UBC) s u b r o u t i n e l i b r a r y . These main s u b r o u t i n e s c a l l o t h e r r o u t i n e s d u r i n g t h e i r e x e c u t i o n . The complete w r i t e u p of the programs c a l l e d can be found i n the book UBC MATRIX a v a i l a b l e from the UBC computing c e n t e r . The two main s u b r o u t i n e s c a l l e d a r e PRITZ and DFBAND both of t h e s e s u b r o u t i n e s work i n extended p r e c i s i o n and have them s e l v e s undergone r i g o r u s t e s t i n g b e f o r e b e i n g a l l o w e d g e n e r a l a c c e s s . By u s i n g the 'canned' programs EDAM ta k e s advantage of t h i s t e s t i n g . By c a l l i n g the s u b r o u t i n e s r a t h e r than k e e p i n g them i n s o u r c e , c o m p i l a t i o n and s t o r a g e c o s t s a re a l s o saved. Both r o u t i n e s r e q u i r e the s t i f f n e s s m a t r i x t o be i n p u t i n the same manner, t h i s b e i n g the lower h a l f of the m a t r i x i n c l u d i n g the d i a g o n a l t o be s t o r e d by columns one h a l f -bandwidth a f t e r a n o t h e r . In t h i s manner the l a r g e d o u b l y s u b s c r i p t e d s t i f f n e s s m a t r i x i s s t o r e d as a s m a l l e r one d i m e n s i o n a l a r r a y . PRITZ i s an e i g e n v a l u e and e i g e n v e c t o r f i n d i n g r o u t i n e . At the time of w r i t i n g i t appears t o be the be s t r o u t i n e p u b l i c l y a v a i l a b l e a t UBC f o r t h i s p u rpose. The program i s e f f i c i e n t and a l s o c h e c k s t h a t the e i g e n v a l u e s g i v e n a r e those r e q u e s t e d . T h i s means t h a t i f the program i s a n a l y s i n g t e n modes t h a t the e i g e n v a l u e f i n d i n g r o u t i n e w i l l r e t u r n w i t h the t e n lowe s t e i g e n v a l u e s , not the lowe s t n i n e and the e l e v e n t h . PRITZ p r i n t s out a s e l e c t i o n of o p e r a t i o n a l i n f o r m a t i o n d u r i n g i t s e x e c u t i o n which i n c l u d e s the number of s i g n i f i c a n t f i g u r e s t o be e x p e c t e d i n each e i g e n v a l u e and e i g e n v e c t o r and a statement c o n f i r m i n g t h a t a l l those e i g e n v a l u e s r e q u e s t e d have been l o c a t e d as d e s c r i b e d . The program EDAM s u p r e s s e s t h i s i n f o r m a t i o n f o r a l l but the l a s t i t e r a t i o n . I f the progam i s b e i n g e x e c u t e d from a t e r m i n a l s c r e e n then t h i s i n f o r m a t i o n w i l l appear on the s c r e e n s h o r t l y b e f o r e the c o m p l e t i o n of the r u n . I f the program i s b e i n g e x e c u t e d from b a t c h , the i n f o r m a t i o n w i l l be p r i n t e d on the same sheet as the e x e c u t i o n c o s t i n f o r m a t i o n . The user s h o u l d note t h a t the manual UBC MATRIX has o m i t t e d t o i n f o r m t h a t the m a t r i x e n t e r i n g PRITZ from which the e i g e n v a l u e s a r e t o be computed i s d e s t r o y e d d u r i n g the e x e c u t i o n of the s u b r o u t i n e . T h i s problem i s c i r c u m v e n t e d by d u p l i c a t i n g the m a t r i x b e f o r e s e n d i n g i t t o the s u b r o u t i n e and r e t a i n i n g the copy which i s r e q u i r e d l a t e r f o r s o l v i n g the d i s p l a c e m e n t s caused by the f o r c e s on the s t r u c t u r e . T h i s o m i s s i o n has been p o i n t e d out t o the a p p r o p i a t e a u t h o r i t i e s i n the computing c e n t e r and a note s h o u l d appear i n f u t u r e e d i t i o n s of UBC MATRIX . DFBAND s o l v e s the m a t r i x problem Ax=B where A i s a symmetric banded m a t r i x and B i s a column m a t r i x . DFBAND was used i n EDAM t o r e p l a c e the s i n g l e p r e c i s i o n e q u i v a l e n t FBAND used i n the o r i g i n a l program MSSM.S. T h i s saves c o n v e r t i n g the s t i f f n e s s m a t r i x from double t o s i n g l e p r e c i s i o n b e f o r e s o l v i n g f o r the d i s p l a c e m e n t s . As an added bonus the e x e c u t i o n times l i s t e d f o r DFBAND a r e l e s s than t h o s e f o r FBAND, though DFBAND o f f e r s the o p t i o n of i t e r a t i v e improvement t h i s i s not 150 undertaken i n EDAM b e l i e v i n g i t u n n e c e s s a r y . TIMING The t i m i n g r e l a t i o n s h i p f o r the program depends s t r o n g l y on the f o l l o w i n g i t e m s : (a) the number of degrees of freedom i n the s t r u c t u r e . (b) the h a l f b a n d w i d t h of the s t r u c t u r e . (c) the number of masses a t t a c h e d . ( T h i s appears t o be an almost d i r e c t r e l a t i o n s h i p - d o u b l i n g the number of i n p u t masses w i l l a p p r o x i m a t l y double the e x e c u t i o n c o s t ) . (d) the number of modes i n c l u d e d i n the a n a l y s i s . (e) the number of i t e r a t i o n s when d o i n g an MSSM a n a l y s i s . Examples of e x e c u t i o n c o s t s a r e g i v e n i n Metten's t h e s i s . OTHER INSTITUTIONS The program EDAM has been w r i t t e n t o o p e r a t e on the M i c h i g a n T e r m i n a l System (MTS) of the U n i v e r s i t y of B r i t i s h Columbia u s i n g IBM s t y l e f o r t r a n and two main UBC canned s u b r o u t i n e s . I t i s e x p e c t e d t h a t t r a n s f e r t o o t h e r i n s t i t u t i o n s of t h i s program would i n v o l v e u s i n g the s u b r o u t i n e s of t h a t i n s t i t u t i o n t o c a l c u l a t e the e i g e n v a l u e s and s o l v e the s t a n d a r d m a t r i x problem. As t h e s e a r e both problems of f r e q u e n t occurence i t i s e x p e c t e d t h a t s o l u t i o n t o them w i l l e x i s t a t many o t h e r i n s t i t u t i o n s . I t w i l l be n e c e s s a r y t o change the c a l l i n g command i n EDAM t o match the s u b r o u t i n e of the i n s t i t u t i o n . A sample problem i s i n c l u d e d i n the appendix of t h i s manual which s h o u l d be an a i d i n d e t e r m i n i n g , i f the program i s e x e c u t i n g c o r r e c t l y as w e l l as d e m o n s t r a t i n g i n p u t and output s t y l e s . • APPENDIX Sample problem ( I n p u t and ouput) / 30 7 7 1 0 .20000 0 .05000 STORY TEST WALL TYPE C WALLS 6FT 60 K IP -FT LINTEL 22 30 3600. 1200. 1 0 0 0 0 .00000 0 .00000 2 0 o o 21 .00000 0 .ooooo 3 1 1 1 0 .00000 7 .75000 4 1 1 1 21 .00000 7 .75000 5 1 1 1 0 .00000 16 .25000 6 1 1 1 21 .00000 16 .25000 . 7 1 1 1 0 .00000 24 .75000 8 1 1 1 21 .00000 24 .75000 9 1 1 1 0 .00000 33 .25000 10 1 1 1 21 .00000 33 .25000 11 1 1 1 0 .00000 41 .75000 12 1 1 1 21 .00000 41 .75000 13 1 1 1 0 .00000 50 .25000 14 1 1 1 21 .00000 50 .25000 15 1 1 1 0 .00000 58 .75000 16 1 1 1 21 .ooooo 58 .75000 17 1 1 1 0 .ooooo 67 .25000 18 1 1 1 21 .ooooo 67 .25000 19 1 1 1 0 .ooooo 75 .75000 20 1 1 1 21 .ooooo 75 .75000 21 1 1 1 0 .ooooo 84 .25000 22 1 1 1 21 .ooooo 84 .25000 1 21 22 1 1 144 00 1024 000 0 000 60 000 7 2 19 20 1 1 144 00 1024 000 0 000 60 000 7 3 17 18 1 1 144 00 1024 000 0 000 60 000 7 4 15 16 1 1 144 00 1024 000 0 000 60 000 7 5 13 14 1 1 144 00 1024 000 0 000 60 000 7 6 1 1 12 1 1 144 00 1024 000 0 000 60 000 7 7 9 10 1 1 144 00 1024 000 0 000 60 000 7 8 , 7 8 1 1 144 00 1024 000 0 000 60 OOO 7 9 5 6 1 1 144 00 1024 000 0 000 60 000 7 10 3 4 1 1 144 00 1024 000 0 000 60 000 7 1 1 1 3 1 1 1620 00 2187000 ooo 0 000 13000 000 0 12 2 4 1 1 1620 00 2187000 000 0 000 13000 000 0 13 3 5 1 1 1620 00 2187000 000 0 000 12666 660 0 14 4 6 1 1 1620 00 2187000 000 0 000 12666 660 0 15 5 7 1 1 1620 00 2187000 000 0 000 12333 330 0 16 6 8 1 1 1620 00 2187000 000 0 000 12333 330 0 17 7 9 1 1 1620 00 2187000 000 0 000 12000 000 0 18 8 10 1 1 1620 00 2187000 000 0 000 12000 000 0 19 9 1 1 1 1 1620 00 2187000 000 0 000 1 1666 660 0 20 10 12 1 1 1620 00 2187000 000 0 000 1 1666 660 0 21 1 1 13 1 1 1620 00 2187000 000 0 000 1 1333 330 0 22 12 14 1 1 1620 00 2187000 000 0 000 1 1333 330 0 23 13 15 1 1 1620 00 2187000 000 0 .ooo 1 1000 000 0 24 14 16 1 1 1620 00 2187000 000 0 .000 1 1000 000 0 25 15 17 1 1 1620 00 2187000 000 0 .000 10666 660 0 SAMPLE 50 50 50 50 50 50 .50 50 50 50 00 00 00 00 00 00 00 00 .00 .00 00 00 7.50 7 . 50 7 . 50 7.50 50 50 50 50 50 50 O.OO 0 .00 00 00 00 00 00 00 00 00 0 .00 0 .00 O.OO 0 .00 O.OO M u j - ^ ^ - . ^ U M (O IO U r o o o o c n - < * r o o o o c n - e . o o t o o o - j c n M O ID 00 -1 01 M U M U U U U I O I O U O O O O O O O O O O ro ro ro • • • • M -» O 10 CD O O O O O O O O O O O O O O O O O O O O cn cn cj) cn cn ro ro ro to ro O O O O O O O O O O O O O O O ro ro ro ro ro oo co cu co cn -4 -1 -J -J -J O O O O O O O O O O O O O O O o o b b o o o o o o o o o o o o o o o o o o b o o o o o o o o o o o o o o o o o O O U U 01 o O CJ co cn O O cj CJ cn O O CJ CJ cn O O CJ CJ cn O O O O O O O O O O O O O O o o o o o o o o o o o o o o o o o o o o o * * * * * * * P R O G R A M O P T I O N S * * * * * * * M A X I M U M NUMBER OF MODES IN A N A L Y S I S 7 I N E L A S T I C A N A L Y S I S M A X I M U M I T T E R A T I O N S = I N I T I A L D A M P I N G R A T I O = 0 . 0 5 0 N U M B E R O F M O D E S T O H A V E O U T P U T P R I N T E D = - S E I S M I C I N P U T - M A X I M U M A C C E L E R A T I O N S . 2 0 0 T I M E S G R A V I T Y S P E C T R U M A U S E D 1 T E N S T O R Y T E S T W A L L T Y P E C W A L L S 6 F T 6 0 K I P - F T L I N T E L E = 3 G 0 0 . 0 K S I G = 1 2 0 0 . 0 K S I * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * ^ - N O . OF J O I N T S = 2 2 N O . OF M E M B E R S = 3 0 - J O I N T D A T A N X ( F E E T ) Y ( F E E T ) NDX NDY NDR 1 0 . 0 0 . 0 0 0 0 2 21 . 0 0 0 0 . 0 0 0 0 3 O . O 7 . 7 5 0 1 2 3 4 21 . OOO 7 . 7 5 0 4 5 6 5 0 . 0 16 . 2 5 0 7 8 9 6 21 . . 0 0 0 16 . 2 5 0 10 1 1 12 7 0 . . 0 24 . . 7 5 0 13 14 15 8 21 . . 0 0 0 24 . . 7 5 0 16 17 18 9 0 . 0 33 . 2 5 0 . 19 2 0 21 10 21 . 0 0 0 33 . 2 5 0 22 23 24 1 1 0 . 0 41 . 7 5 0 25 26 27 12 21 . 0 0 0 41 . 7 5 0 28 29 3 0 13 0 . 0 5 0 . 2 5 0 31 32 33 14 21 . 0 0 0 5 0 . 2 5 0 34 35 36 15 0 . 0 58 . 7 5 0 37 38 39 16 21 . 0 0 0 58 . 7 5 0 4 0 41 42 17 0 . 0 67 . 2 5 0 4 3 44 45 18 21 . 0 0 0 67 . 2 5 0 46 47 48 19 0 . o 7 5 . 7 5 0 4 9 5 0 51 2 0 21 . O O O 75 . 7 5 0 52 5 3 54 21 0 . o 84 . 2 5 0 55 56 57 SAMPLE P R O B L E M O U T P U T 7 22 -MEMBER DATA 21 .000 84.250 58 59 60 MN JNL JNG EXTL LENGTH EXTG XM(FT) YM(FT) 1 2 3 4 5 6 7 8 9 10 1 1 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 -NO.OF 21 19 17 15 13 1 1 9 , 7 5 3 1 2 3 4 5 6 7 8 9 10 1 1 12 13 14 15 16 17 18 19 20 22 20 18 16 14 12 10 8 6 4 3 4 5 6 7 8 9 10 1 1 12 13 14 15 16 17 18 19 20 21 22 500 500 500 500 500 500 500 500 500 500 O 0 O 0 0 O 0 0 0 O 0 O 0 O 0 0 0 0 o o (FEET) 6.OOOO .0000 .0000 .0000 .0000 .0000 6.OOOO 6 . OOOO 6 . OOOO 6 . OOOO 7 . 7500 7 . 7500 8 . 5000 8 . 5000 8 . 5000 8 .5000 8 .5000 8.5000 8 .5000 8 .5000 8 . 5000 8 . 5000 8 . 5000 8 . 5000 8 .5000 8 .5000 8 .5000 8 .5000 8 . 5000 8 . 5000 7 . 7 . 7 . 7 . 7 . 7 . 7 . 7 . 7. 7 . 0. O. O. 0. O. O. O. O. 0 .0 0 .0 500 500 500 500 500 500 500 500 500 500 0 0 0 0 0 0 0 0 .0000 .0000 .0000 .0000 .0000 .0000 .0000 .0000 .0000 .0000 .0 .0 .0 .0 .0 .0 .0 .0 .0 .0 .0 .0 .0 .0 .0 .0 .0 .0 .0 .0 0 .0 0 .0 0 .0 0 .0 8 0 0 0 7500 7500 8.5000 8.5000 8.5000 8.5000 8.5000 8.5000 8.5000 5000 8.5000 8.5000 5000 5000 5000 5000 5000 5000 5000 5000 AREA (SO.IN) 144 .0 144 144 144 144 144 144 144 144 144 1620 1620 1620 1620 1620 1620 1620 1620 1620 1620 1620 1620 1620.0 1620.0 1620 1620 1620 1620 1620 1620 I(CRACKED) (IN**4 ) 1024 .0 1024 1024 1024 1024 1024 1024 1024 .0 1024 .0 1024 2187000 2187000 2187000 2187000 2187000 2187000 2187000.0 2187000.0 2187000.0 2187000.0 2187000.0 2187000.0 2187000 2187000 2187000 2187000 2187000 2187000 2187000 2187000 .0 .0 .0 .0 .0 .0 .0 .0 .0 .0 .0 .0 .0 .0 AV (SO 0 . 0 0 0 0 0 0 0 0 0 . 0 0 . 0 0 . 0 0 . 0 0 . 0 0 .0 0 . 0 0 . 0 MOMENT KL KG IN) CAPACITY 60 .00 60 .00 60 .00 60. OO 60 .00 60 .00 60 .00 60 .00 60 .00 60 .00 13000.00 13000.00 12666.66 12666.66 12333.33 12333.33 12000.00 12000.00 11666.66 11666.66 11333.33 1 1333.33 11000.00 11000.00 10666.66 10666.66 10333.33 10333.33 10000.00 10000.00 DEGREES OF FREEDOM OF STRUCTURE = 60 HALF BANDWIDTH OF STIFFNESS MATRIX = 9 MEMBER NP 1 NP2 NP3 NP4 NP5 NP6 1 55 56 57 58 59 60 2 49 50 51 52 53 54 3 43 44 45 46 47 48 4 37 38 39 40 41 42 5 31 32 33 34 35 36 6 25 26 27 28 29 30 7 19 . 20 21 22 23 24 8 13 14 15 16 17 18 9 7 8 9 10 1 1 12 10 1 2 3 4 5 6 1 1 0 0 0 1 2 3 12 13 14 15 16 17 18 19 20 2 1 22 23 24 25 26 27 28 29 30 0 1 4 •7 10 13 16 19 22 25 28 31 34 37 40 43 46 49 52 0 2 5 8 1 1 14 17 20 23 26 29 32 35 38 4 1 44 47 50 53 0 3 6 9 12 15 18 21 24 27 30 33 36 39 42 45 48 51 54 4 7 10 13 16 19 22 25 28 31 34 37 40 43 46 49 52 55 58 5 8 1 1 14 17 20 23 26 29 32 35 38 41 44 47 50 53 56 59 6 9 12 15 18 21 24, 27 30 33 36 39 42 45 48 51 54 57 60 * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * NO. OF NODES WITH MASS = 10 JN 4 6 8 10 12 14 16 18 20 22 S NO. DOF 1 4 2 10 3 16 4 22 5 28 6 34 7 40 8 46 9 52 10 58 -MASS Y -MASS ROT.MASS (KIPS) (KIPS) ( IN-KIPS) 270.000 0 .0 0 .0 270.000 0 .0 0 .0 270.000 0 .0 0 .0 270.000 0.0 O.O 270.000 0 .0 0 .0 270.000 0 .0 0 .0 270.000 0 .0 0 .0 270.000 0.0 0 .0 270.000 0.0 0 .0 270.000 0 .0 0 .0 ASSIGNED MASS (K I P * SEC* *2 /FT ) 8 . 38509 8 . 38509 8 . 38509 8 . 38509 8 . 38509 8 . 38509 8 . 38509 8 .38509 8 . 38509 8 . 38509 I N I T I A L E L A S T I C P E R I O D MODES E I G E N V A L U E S 5 6 . 6 7 1 5 -1 1 4 1 . 1 0 9 9 6 6 9 7 . 1 8 7 5 2 0 1 9 1 . 7 5 3 9 4 6 2 9 9 . 0 7 4 2 9 3 2 7 7 . 6 2 5 0 1 6 9 3 6 6 . 6 2 5 0 N A T U R A L F R E Q U E N C I E S ( R A D / S E C ) 7 . 5 2 8 0 3 3 . 7 8 0 3 81 . 8 3 6 3 1 4 2 . 0 9 7 7 2 1 5 . 1722 3 0 5 . 4 1 3 8 41 1 . 5 4 1 7 ( C Y C S / S E C ) 1 . 1 9 8 1 ' 5 . 3 7 6 3 1 3 . 0 2 4 7 2 2 . 6 1 5 7 3 4 . 2 4 5 9 4 8 . 6 0 8 4 6 5 . 4 9 9 3 P E R I O D S SA ( S E C S ) ( 2 P E R C E N T D A M P I N G ) 0 . 8 3 4 6 0 . 3 5 9 4 0 . 1860 0 . 7 5 0 0 0 . 0 7 6 8 0 . 3 8 3 9 0 . 0 4 4 2 0 . 2 2 1 1 0 . 0 2 9 2 0 . 1 4 6 0 0 . 0 2 0 6 0 . 1 0 2 9 0 . 0 1 5 3 0 . 0 7 6 3 I N E L A S T I C R E S U L T S - I T E R A T I O N N O . 1 2 3 4 5 N O . A B O V E C A P A C I T Y 10 10 10 0 0 D A M D I F S M A T R I X R A T I O 0 . 3 0 5 0 . 7 8 7 E + 0 2 0 . 4 3 8 0 . 7 8 8 E + 0 2 0 . 1 2 1 0 . 7 8 7 E + 0 2 0 . 0 2 8 0 . 7 8 7 E + 0 2 0 . 0 0 6 • 0 . 7 8 7 E + 0 2 - I T E R A T I O N NUMBER 6 A L L E L E M E N T S OF M A I N D I A G O N A L OF S T I F F N E S S M A T R I X ARE P O S I T I V E D E F I N I T E R A T I O OF L A R G E S T TO S M A L L E S T D I A G O N A L S T I F F N E S S M A T R I X E L E M E N T IS 0 . 7 8 7 E + 0 2 - N O . OF M O D E S TO B E A N A L I Z E D = 7 ****** ************************************************************************' * + + + + + * **************************** .************+********+********************+****************** T O T A L MODE S H A P E S C O R R E S P O N D I N G TO F I R S T 7 F R E Q U E N C I E S DOF 1 2 3 4 5 6 '7 1 0 . 0 1 4 4 8 0 - 0 . . 0 8 9 3 6 6 0 . 1 8 0 7 5 9 0 . 1 8 5 1 7 6 0 . . 1 3 6 7 0 4 - 0 . 0 9 1 3 7 3 - 0 . . 0 5 8 7 8 8 2 0 . 0 0 0 8 3 2 0 . 0 0 0 0 6 2 0 . 0 0 1 6 2 2 0 . 0 0 0 2 0 7 0 . . 0 0 1 1 5 5 0 . 0 0 0 1 1 8 - 0 . . 0 0 0 4 9 9 3 - 0 . 0 0 3 6 3 8 0 . 0 2 1 1 0 6 - 0 . 0 3 9 4 9 9 - 0 . 0 3 6 1 4 3 - 0 . 0 2 2 4 9 6 0 . 0 1 1 3 6 0 0 . 0 0 4 3 1 7 4 0 . 0 1 4 5 0 0 - 0 . 0 9 3 4 4 9 0 . 2 4 3 9 3 1 0 . . 4 3 4 2 8 1 0 . 6 4 2 2 6 1 - 0 . . 8 4 4 9 5 6 - 0 . . 9 8 3 0 1 1 5 - 0 . 0 0 0 8 3 2 - 0 . 0 0 0 0 6 2 - 0 . 0 0 1 6 2 2 - 0 . 0 0 0 2 0 7 - 0 . 0 0 1 1 5 5 - 0 . O O O 1 1 8 0 . 0 0 0 4 9 9 6 - 0 . 0 0 3 6 4 4 0 . 0 2 2 0 7 4 - 0 . . 0 5 3 3 5 6 - o 0 8 5 0 4 2 - 0 . 1 0 6 6 0 5 0 . 1 0 7 2 3 0 0 . 0 7 5 4 7 0 7 0. 060413 - 0 . 323009 0. 543219 0. 426354 0. 212601 - 0 . 073798 - 0 . 006932 8 0. 001657 0. 000667 0. 002252 - 0 . 001070 0. 000822 0. 001720 - 0 . 00O058 9 - 0 . 007052 0. 031594 - 0 . 038389 - 0 . 011338 0. 010394 - 0 . 015855 - 0 . 011988 10 0. 060498 - 0 . 337775 0. 733097 1 . 000000 1 . 000000 - 0 . 686251 - 0 . 118837 1 1 - 0 . 001657 - 0 . 000667 - 0 . 002252 0. 001070 - 0 . 000822 - 0 . 001720 0. 000058 12 - b . 007061 0. 033039 - 0 . 051814 -o. 026563 0. 049026 - 0 . 147799 - 0 . 204711 13 0. 132796 - 0 . 592181 0. 74 1 1 10 0. 3 14341 - 0 . 012295 0. 090194 0. 056461 14 0. 002391 0. 001695 0. 002296 - 0 . 002595 0. 000878 0. 002255 - 0 . 001032 15 - 0 . 009864 0. 029747 - 0 . 004390 0. 036044 0. 033295 - 0 . 01 1436 0. 003732 16 0. 132983 - 0 . 619245 1 . 000000 0. 736926 - 0 . 055688 0. 829688 0. 948003 17 - 0 . 00239 1 - 0 . 001695 - 0 . 002296 0. 002595 - 0 . 000878 - 0 . 002255 0. 001032 18 - 0 . 009878 0. 031107 - 0 . 005905 0. 084618 0. 156614 - 0 . 106476 0. 062653 19 0. 226589 - 0 . 800561 0. 595616 - 0 . 092702 - 0 . 207787 0. 050825 - 0 . 040850 20 0. 003034 0. 003013 0. 002293 - 0 . 003550 0. 001837 0. 002232 - 0 . 001692 21 - 0 . 012092 0. 017805 0. 037481 0. 050040 0. 004301 0. 017922 0. 008860 22 0. 226908 - 0 . 837141 0. 803557 -o. 217068 - 0 . 972834 0. 471274 - 0 . 673248 23 - 0 . 003034 - 0 . 003013 - 0 . 002293 0. 003550 - 0 . 001837 - 0 . 002232 0. 001692 24 - 0 . 012110 0. 018618 0. 050594 0. 117323 o. 020299 0. 165483 0. 149593 25 0. 336946 - 0 . 876319 0. 155338 - 0 . 394789 - 0 . 066773 - 0 . 102618 - 0 . 024142 26 0. .003584 0. 004476 0. 002637 - 0 . 003852 0. 002893 0. 002929 - 0 . 001733 27 - 0 . .013768 - 0 . 000762 0. 061014 o. 013716 - 0 . 031325 0. 007404 - 0 . 011063 28 0. . 337421 - 0 . .916357 0. 209492 - 0 . 924509 - 0 . 313806 - 0 . 939801 - 0 . 400681 29 - 0 .003584 - 0 . ,004476 - 0 . .002637 0. 003852 - 0 . .002893 - 0 . 002929 0. 001733 30 - 0 .013787 - 0 .000797 0. .082336 0. .032111 - 0 . . 146764 0. 068657 - 0 . 185375 31 0 .459345 - 0 .780754 - 0 . 348535 - 0 . .287772 0. . 182996 - 0 . 028460 0. .060082 32 0 .004042 0 .005937 0 .003477 - 0 .003998 0. .003336 0 .003890 -0 . .002453 33 -0 .014935 - 0 .021745 0 .051414 - 0 .036140 - 0 .016588 -0 .019700 0. .000139 34 0 .459992 - 0 .816428 - 0 .470298 - 0 .673617 0 .853931 -0 .263258 1 .000000 35 - 0 .004042 -o .005937 -0 .003477 0 .003998 -0 .003336 -0 .003890 0 .002453 36 - 0 .014956 - 0 .022738 0 .069378 - 0 .0847 19 - 0 .077633 -0 .181016 0 .001738 37 0 .589724 - 0 .511407 -0 .636027 0 .112152 0 .140570 0 .109676 -0 .025701 38 0 .004406 0 .007259 0 .004704 - 0 .004531 0 .003482 0 .0042 16 -0 .003169 39 - 0 .015658 - 0 .040973 0 .012418 -0 .047861 0 .024660 -0 .003182 0 .011053 40 0 .590555 - 0 .534783 - 0 .858189 0 .263109 0 .654825 1 .000000 -0 .416777 41 - 0 .004406 - 0 .007259 -0 .004704 0 .004531 - 0 .003482 -0 .004216 0 .003169 42 - 0 .015680 - 0 .042845 0 .016765 - 0 . 1 12175 0 . 1 15596 - 0 .028864 0 .184100 43 0 .724649 - 0 .097985 - 0 .531 185 0 .380906 -0 .122555 0 .006524 -0 .040948 44 0 .004679 0 .008334 0 .006022 - 0 .005548 0 .004052 0 .004448 -0 .003298 45 - 0 .016020 - 0 .055200 - 0 .036672 - 0 .008149 0 .025556 0 .020131 -0 .009016 46 0 .725670 - 0 .102482 - 0 .716832 0 .892665 - 0 .575976 0 .051683 -0 .664422 47 - 0 .004679 - 0 .008334 -0 .006022 0 .005548 - 0 .004052 -0 .004448 0 .003298 48 - 0 .016043 - 0 .057723 - 0 .049475 -0 .019113 0 . 1 19679 0 . 18537 1 -0 .151033 49 0 .861487 0 .409079 -o .048502 0 .211016 - 0 .170379 -0 .103605 0 .055257 50 0 .004861 0 .009082 0 .OO7091 -0 .006649 0 .005050 0 .005284 -0 .003898 51 - 0 .016127 - 0 .062921 -0 .072953 0 .04561 1 -0 .017371 -0 .002399 -0 .003307 52 0 .862701 o .427760 -0 .065562 0 .494502 -0 .797822 -0 . 95382 1 0 .924039 53 - 0 .004861 - 0 .009082 - 0 .007091 0 .006649 - 0 .005050 -0 .005284 0 .003898 54 - 0 .016150 - 0 .065801 - 0 .098475 0 .107006 -0 .081857 -0 .023383 -0 .052926 55 0 .998592 0 .956252 0 . 64 1494 -0 .307975 0 .124753 0 .049196 -0 .018785 56 0 .004952 0 .009462 . 0 .007666 -o .007311 0 .005763 0 .006049 -0 .004663 57 58 59 60 -0 .016106 1.000000 •-0.004952 -0 .016129 -0 .064993 1.000000 -0 .009462 -0 .067976 MODES EIGENVALUES 1 29. .0268 2 920. . 2505 3 6249. .08 59 4 19637 .9102 5 45667 . 3555 6 92555 .0000 7 168589 .8125 -0.085127 0.865994 -0.007666 -0.114993 0.068598 -0.723500 0.007311 O. 161251 -0.043200 O. 590963 -0.005763 -O.203948 NATURAL FREQUENCIES (RAD/SEC) 5.3877 30.3356 79.0512 140.1353 213.6992 304.2285 410.5969 (CYCS/SEC) 0.8575 4 12 22 34 48 65 8281 5815 3033 0115 4197 3489 -0.025556 0.469507 -0.006049 -0.239265 PERIODS (SECS) 1.1662 0.207 1 0.0795 0.0448 0.0294 0.0207 0.0153 0.014511 -0.345542 0.004663 0.250288 SA (2 PERCENT DAMPING) 0. 2572 0. 7500 O. 3974 0. 2242 0.1470 0.1033 0.0765 . MASS MODE SHAPES CORRESPONDING TO FIRST MASS 1 2 3 1 0. 014500 - 0 . 093449 0. 243931 2 0. 060498 - 0 . 337775 0. .733097 3 0. 132983 - 0 . 619245 1 . 000000 4 0. .226908 - 0 . 837141 0. .803557 5 0. 337421 - 0 . 916357 0. .209492 6 0. .459992 - 0 . 816428 - 0 . 470298 7 O. . 590555. - 0 . 534783 - 0 . .858189 8 0 . 725670 - 0 . 102482 - 0 . .716832 9 .0 .862701 0. 427760 - 0 . .065562 10 1 . .000000 1 . 000000 0. .865994 -MODAL PARTICIPATION FACTOR 7 FREQUENCIES 4 5 6 7 0. 434281 0. 642261 - 0 . 844956 - 0 . 983011 1. 000000 1 . .000000 - 0 . 686251 - 0 . 1 18837 0. 736926 - 0 . .055688 0. 829688 0. 948003 0. 217068 - 0 . .972834 0. 471274 - 0 . 673248 0. 924509 - 0 . .313806 - 0 . 939801 - 0 . 400681 0. .673617 0. .853931 - 0 . .263258 1 . .000000 0. .263109 0 .654825 1 . .000000 - 0 . .416777 0. .892665 - 0 . 575976 0. .051683 - 0 . , 664422 0. .494502 - 0 .797822 -0 . .953821 0 .924039 0. .723500 0 .590963 0. .469507 - 0 .345542 MODE 1 1 . 46159 MODE 2 - 0 . 67494 MODE 3 0. 38676 MODE 4 0. 27170 MODE 5 0. 20783 MODE 6 - 0 . 16714 MODE 7 - 0 . 14376 MODE 1 CONTRIBUTION FACTOR^ 0. 28314 MODE 2 CONTRIBUTION FACTOR^ - 0 . 47085 MODE 3 CONTRIBUTION FACTOR= 0. 15008 MODE 4 CONTRIBUTION FACTOR= 0. .06028 MODE 5 CONTRIBUTION FACTOR= 0. 03040 MODE 6 CONTRIBUTION FACTORS - 0 . .01721 MODE 7 CONTRIBUTION FACTOR= - 0 . .01098 -MODE SMEARED DAMPING RATIO 1 0.04618 2 0.02599 3 0.02192 4 0.02083 5 0.02040 6 0.02021 7 0.02012 _**•***+ *.* * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * *+* * * * * * * * * * * * MODE NUMBER 1 MODAL FORCES AND DISPLACEMENTS JOINT NO. X-D ISP(FT) 1 0 .0 2 0 .0 3 0.0046 4 0.0046 5 0 .0190 6 0.0190 7 0.0417 8 0.04 18 9 0 .07 12 10 0.0713 11 0.1059 12 0.1060 13 0.1444 14 0.1446 15 0.1853 16 0.1856 17 0.2277 18 0.2280 19 0.2707 20 0.27 11 21 0.3138 22 0.3143 * * * * * * * * * * * * Y-DISP(FT) 0 .0 0 .0 0 -0 0 -O 0 -0 0 -0 O 0003 0003 0005 0005 0008 0008 0010 0010 001 1 -0.001 1 0 . 001 3 0013 0014 0014 0015 0015 0015 0015 0016 0016 - 0 . 0. -O. 0. -O. 0. - 0 . 0. -O. ROTAT 10N(RAD) 0 .0 0 -0 -0 -0 -0 - 0 -O -O -O -O - 0 - 0 -O -0 -0 -0 -0 - 0 -0 -0 -O 0 001 1 001 1 0022 0022 0031 0031 0038 0038 004 3 0043 0047 0047 0049 0049 0050 0050 0051 005 1 0051 0051 MN 1 2 3 4 5 6 7 8 9 10 1 1 12 13 14 15 16 17 18 19 20 AXIAL KIPS 38.232 32.959 27 . 740 22 .577 17.582 12.895 8 .672 5.082 2.311 0. 553 196.839 - 196 .839 177 .775 -177 .775 158.341 ' - 158 .341 138.593 - 138.593 118.639 - 118.639 SHEAR KIPS -19.558 -19.588 - 19.683 -19.827 - 19.963 -20.022 - 19 .954 -19.748 - 19 . 434 -19.064 168.589 168.795 168.030 168.239 165.712 165.915 160.604 160.824 151.961 152.130 BML (K -FT ) 58.669 58.759 59.045 59.477 59.884 60.061 59.857 59.240 58.299 57.188 -8719.730 -8731.918 -7613 . 270 -7623.887 -6388 .94 1 -6397.809 -5187.566 -5194.801 -4031.718 -4037.229 BMG (K -FT ) -58 .677 -58 .767 -59 .053 -59 .485 -59.892 -60.069 -59.865 -59.248 -58.307 -57.196 -7413 . 168 -7423.758 -6185.016 -6193.852 -4980.383 -4987.527 -3822.430 -3827.795 -2740.051 -2744. 124 o 21 98. 618 139. 021 22 -98 . 618 139. . 224 23 78 , 655 121 . 464 24 -78 . .655 121 .667 25 58 . 828 98 .967 26 -58 . .828 99 .035 27 39 . 145 71 . 194 28 -39 . 145 71 . 244 29 19 .557 38 . 193 30 -19 .557 38 . 227 MODE • 1 CONTRIBUTION FACTOR= 0.28328 DAMPING=0.0462 PERI0D=1.1662 SEC. SA=0.194 _ » * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * MODE NUMBER 2 MODAL FORCES AND DISPLACEMENTS JOINT NO. X-D ISP(FT) 1 0 . 0 2 0 .0 3 0.0015 4 0.0015 5 0 .0053 6 0.0056 7 0.0098 8 0.0102 9 0.0132 10 0.0138 11 0.0144 12 0.0151 13 0.0129 14 0.0135 15 0.0084 16 0.0088 17 0.0016 18 0.0017 19 -0 .0067 20 - 0 . 0 0 7 0 21 -0 .0158 22 -0 .0165 MN AXIAL SHEAR KIPS KIPS 1 -62 .288 4.299 2 - 2 6 . 5 9 7 4.161 3 6.403 3.687 4 33.282 2.810 5 50.791 1.558 6 57.005 0.028 7 52.082 - 1 .639 8 38.533 -3 .285 9 21.022 ' - 4 .783 10 5,813 -6.061 -2950. 104 -1768. 424 -2954 . 274 - 1770. 868 -1977. 663 -945. 220 -1980. 538 -946. 370 -1153. 405 -312 . . 182 -1154. .747 -312 .949 -519, . 121 86 .025 -519 .648 85 .929 -119 .428 205 . 214 -119 .716 205 .213 ******************************* Y-DISP(-FT) ROTATI0N(RAD) 0. 0 0. 0 0. 0 0. 0 - 0 . OOOO - 0 . 0003 0. OOOO - 0 . 0004 - 0 . OOOO - 0 . 0005 0. OOOO - 0 . 0005 - 0 . OOOO - 0 . 0005 0. OOOO - 0 . 0005 - 0 . OOOO - 0 . 0003 0. OOOO - 0 . 0003 - 0 . 0001 0, .0000 0. ,0001 0 .0000 - 0 , .0001 0. .0004 0, .0001 0. .0004 -0 .0001 0 .0007 0 .0001 0 .0007 -0 .0001 0 .0009 0 .0001 0 .0010 -0 .0001 0 .0010 0 .0001 0 .001 1 -0 .0002 0 .001 1 0 .0002 0 .001 1 BML BMG (K -FT ) (K -FT ) - 12 . 869 12 .925 - 12 .455 12.509 -11 .036 11.083 -8 .413 8 . 449 -4 .665 4 .685 -0 .082 0.083 4 .906 -4 .926 9 .834 -9 .875 14 .319 - 14.380 18 . 145 - 18.223 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 MOD E «£ ^ U I N ' I A I ss i w > DAMPING=s0.0260 PERI0D=0. 207 1 SEC . SA=0.698 _ * * * * * * * * * * * * * * * « * * * * * * * * * * * * * * * * * * * * * * * * * * * * * MODE NUMBER 3 MODAL FORCES AND DISPLACEMENTS - 0 . 7 7 4 176.046 0.774 183.794 - 6 . 8 3 6 170.230 6.836 177.717 - 1 1 . 6 1 9 149.205 11.619 155.797 -14 .904 110.678 14.904 115.590 -16 .542 58.593 16.542 61.222 - 1 6 . 5 1 5 1.593 16.515 1.704 -14 .956 - 49 .200 14.956 -51 .315 -12 .146 -82 .479 12.146 -86.031 - 8 . 4 5 9 -88 .872 8.459 -92.652 - 4 . 2 9 9 -62 .277 4.299 -64 .858 CO TRIBUTION FACTOR=-0.47095 -3135. -3278. -1835. -1917 . -438 . -457 . 795. 832 . 1719. 1797 . 2217 . 2318 . 2247 . 2349. 1858 1942 1 196 1250 484 506 883 397 135 -637 34 1 265 446 495 .019 . 786 358 .474 . 261 . 343 .549 .669 .115 .075 .286 . 166 •1771 . -1853. -388. -407 . 829. 867 . 1736. 1815. 2217. 2318 . 2230. 2332 . 1829. 1913 . 1 157 . 1211. 440 462 -45 -45 528 993 182 046 901 ,013 . 206 .012 .062 177 .900 .959 .058 168 .479 . 406 .707 .534 .073 . 127 tt******************************* JOINT NO. 1 2 3 4 5 6 7 8 9 10 1 1 12 13 14 15 16 17 18 19 20 21 22 MN X-D ISP(FT) 0 .0 0 . 0 0.0001 0.0002 0.0004 0.0006 0.0006 O.0008 0.0005 0.0006 0.0001 0.0002 -0 .0003 -0 .0004 -0 .0005 -0 .0007 -0 .0004 -0 .0006 -0.OOOO -0.0001 0.0005 0.0007 -D ISP(FT) 0 .0 0 .0 0. OOOO -0 .0000 0. OOOO -0 .0000 0 . o o o o - 0 .0000 0 . o o o o - 0 .0000 0 . o o o o - 0 . o o o o ' 0 . o o o o - 0 .0000 0 . o o o o -0 .0000 0 . o o o o .0000 .0000 .0000 .0000 .0000 - 0 . 0. - o . 0. - o . R0TATION(RAD) 0 .0 0 .0 - 0 .0000 - o . o o o o -0 .0000 -0 .0000 - 0 . o o o o - o . o o o o 0 . o o o o 0 . o o o o 0.0000 O.0001 0 . o o o o 0.0001 .0000 . o o o o . .0000 . o o o o .0001 .0001 .0001 0. o . - 0 . - 0 . - 0 . - 0 . - 0 . -0.0001 AXIAL KIPS SHEAR KIPS BML (K -FT) BMG (K -FT ) 1 15 . 001 - 0 . 306 0 . 904 2 -1 . 140 - 0 . 262 0 . 773 3 - 1 2 . 4 0 5 - 0 . 132 0 . 391 4 - 1 4 . 845 0 . 048 - 0 . 142 5 - 8 . 136 0 . 205 - 0 . 606 6 3 . 6 1 9 0 . 264 - 0 . 780 7 13 . 8 9 5 0 . 184 - 0 . 544 8 17 . 2 9 9 - 0 . 0 2 5 0 . 073 9 12 . 6 8 8 - 0 . 311 0 . 9 2 0 10 4 . 221 - 0 . 6 1 0 1 . 803 1 1 0 . .944 3 0 . 197 - 3 3 2 . 524 12 - 0 . . 944 4 0 . 527 - 4 4 8 . 161 13 0 . 334 2 5 . 975 - 104 . 872 14 - 0 . 334 34 . . 863 - 1 4 0 . 499 15 0 . 0 2 3 13. . 287 112. 665 16 - 0 . 0 2 3 17 .842 152 . , 556 17 - 0 . 0 0 2 -4 . 0 1 2 225 . .347 18 0 . 0 0 2 - 5 . 383 303 .951 19 0 . 182 - 1 7 . 9 0 6 193 . 170 2 0 - 0 . 182 - 2 4 . 0 5 3 2 6 0 . 130 21 0 . 446 -21 . 5 2 5 43 . 7 2 5 22 - 0 . 4 4 6 - 2 8 . 9 2 3 58 .461 23 0 . 651 - 13 . 389 - 1 3 7 . 0 9 6 24 - 0 . 6 5 1 - 1 8 .001 - 1 8 5 .231 25 0 . 6 9 9 1 . 4 5 6 - 2 5 0 . 3 9 7 26 - 0 . 6 9 9 1 . 9 3 2 - 3 3 7 . 735 27 0 . 5 6 7 13 .861 - 2 3 9 . 4 0 0 28 - 0 . 5 6 7 18 . 5 7 6 - 3 2 2 . 7 0 7 29 0 . 306 15 .001 - 1 2 4 . 3 1 3 3 0 - 0 . 306 20 . 0 9 2 - 1 6 7 . 5 6 6 MODE -3 C O N T R I B U T I O N FACT0R= 0 . 1 5 0 0 9 D A M P I N G = 0 . 0 2 1 9 P E R I 0 D = O . 0 7 9 5 SEC.. S A = 0 . 3 8 8 _*****************************++*•******************** MODE NUMBER 4 MODAL F O R C E S AND D I S P L A C E M E N T S J O I N T N O . X - D I S P ( F T ) Y - D I S P ( F T ) 1 0 . 0 0 . 0 2 0 . 0 0 . 0 3 0 . OOOO 0. . 0 0 0 0 4 0 . OOOO - 0 . . 0 0 0 0 5 0 . OOOO - 0 , . 0 0 0 0 6 0 . 0001 0 . 0 0 0 0 7 0 , OOOO - 0 . 0 0 0 0 8 0 . ,0001 0 . 0 0 0 0 9 - 0 . 0 0 0 0 - 0 . 0 0 0 0 10 - 0 . 0 0 0 0 0 . 0 0 0 0 1 1 - 0 . 0 0 0 0 - 0 . 0 0 0 0 12 - 0 .0001 0 . 0 0 0 0 13 - 0 . 0 0 0 0 - 0 . 0 0 0 0 14 - 0 .0001 0 . 0 0 0 0 15 O . 0 0 0 0 - 0 . 0 0 0 0 - 0 . 9 3 0 - 0 . 7 9 6 - 0 . 4 0 2 0 . 146 0 . 6 2 3 0 . 803 0 . 5 6 0 - 0 . 0 7 6 - 0 . 9 4 7 -1 . 8 5 5 - 9 8 . 5 0 0 1 3 4 . 0 7 6 1 1 5 . 9 1 6 1 5 5 . 8 3 4 2 2 5 . 6 0 7 3 0 4 . 2 1 4 1 9 1 . 2 4 9 2 5 8 . 1 9 2 4 0 . 9 7 0 5 5 . 6 8 3 1 3 9 . 2 3 7 1 8 7 . 3 8 9 2 5 0 . 9 0 1 338 . 243 2 3 8 . 0 2 0 321 . 3 1 7 121 . 583 1 6 4 . 8 1 5 3 . 193 3 . 220 R O T A T I O N ! R A D ) 0 . 0 0 . 0 -0.OOOO - 0 . 0 0 0 0 - 0 . 0 0 0 0 -0.OOOO 0.OOOO 0.OOOO 0.OOOO 0.OOOO O.OOOO 0.OOOO -0.OOOO - 0 . 0 0 0 0 -0.OOOO 16 0 .0000 0.0000 17 0 .0000 -O.OOOO 18 0.0001 0.0000 19 0.0000 -0 .0000 20 0 . o o o o o.OOOO 21 -O.OOOO -0 .0000 22 -0.0001 0.0000 MN AXIAL SHEAR BML KIPS KIPS (K- FT) 1 -3 . 549 0. 045 - 0 . 130 2 2 . 421 0. 030 - 0 . 086 3 4 . 371 - 0 . 006 0. 016 4 1 . 289 - 0 . 033 0. 095 5 -3 . 295 - 0 . 026 0. 076 6 -4 . 524 0. 01 1 - 0 . 030 7 -1 . 062 0. 044 - 0 . 128 8 3 . 609 0. 039 - 0 . 112 9 4 . 899 - 0 . ,017 0. 049 10 2 . 127 - 0 . , 102 0. ,294 1 1 0. 015 6 , 286 -49 . ,564 12 - 0 . 015 14 , .593 -115, ,851 13 - 0 . .087 4 . 159 -1 . ,904 14 0, ,087 9 .652 -3 .838 15 - 0 . . 103 - 0 .740 33, . 272 16 0. . 103 - 1 .726 78 .023 17 - 0 .065 -4 . 349 27 . 382 18 0 .065 -10 .111 63 . 766 19 - 0 .020 -3 .287 -9 . 126 20 0 .020 -7 .640 -21 .708 21 - 0 .010 1 . 237 -36 .955 22 ' 0 .010 2 .884 -86 .537 23 - 0 .036 4 .532 -26 .714 24 0 .036 10 .552 -62 . 304 25 - 0 .069 3 . 243 1 1 . 467 26 0 .069 7 .559 27 .043 27 - 0 .075 -1 . 128 38 .973 28 0 .075 -2 .600 91 .236 29 - 0 .045 -3 .549 29 .697 30 0 .045 -8 .227 69 .455 CONTRIBUTION FACTOR= 0.06028 DAMPING=0.0208 PERI0D=6.O448 SEC. SA=0.222 _ * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * + * + + * * * * * * * * * * * * * * * * MODE NUMBER 5 MODAL FORCES AND DISPLACEMENTS JOINT NO. X-D ISP(FT) Y-D ISP(FT) 1 O.O 0 .0 2 0 .0 0 .0 3 O.OOOO 0.OOOO 4 0.OOOO -0.OOOO 5 0.OOOO 0.OOOO -0 .0000 - 0 . o o o o - 0 .0000 0 . o o o o 0 . o o o o 0 . o o o o 0 . o o o o BMG (K -FT ) 0. 140 0.093 -0 .017 -0 .102 -0 .082 0.033 0. 138 0.121 -0 .053 -0 .318 -0 .845 -2 .756 33.447 78.202 26.980 63.356 -9 .585 -22.178 -37 .065 -86.649 -26.442 -62.026 11.809 27.392 39.032 91.296 29.389 69.140 -0 .466 -0 .477 ROTATION(RAD) 0 .0 0 .0 -O.OOOO -0 .0000 0.OOOO g> 3 0 ( o S ^ a i u i * » e d M ^ O < o c o ^ a > u i A U M - 0 < o a » ^ i f l > a i A U M - » M - » O ( 0 0 J ~ j < J ) o i ^ c j t o - ' O « J 0 0 - ~ ) c n O O O O O O O O O O O O O O O O O O O O O - O - O - O O - O X X 2 2 2 5 8 8 8 8 8 8 2 2 2 2 8 8 8 8 2 ° 8 SlSi 5 SSS^SS? u O - ' O U - ' O O i i i i i I I i i </i H - U O U O I ' O M O I I I ' O O O O O O O O O O ^ I i o i » c j M i o - c o - . ( f l O ~ i u i ~ ' i ' > - a | a , u , l o * 0 0 0 2 S 2 2 2 2 0 ^ 5 n S S o u m m o S o o t o o o - . i ) i ( f l i » o i M t o u O - 0 0 0 0 0 0 -™ U & l O C » C a U l < J > I O ^ * . l O U I C D M ^ ooooooooooooooooo bbbbboobobbbooogo ooooooooooooooooo ooooooooooooooooo ooooooooooooooooo ^ i S ^ o ^ ^ U o ^ S ^ i o o o o o o o o o o ^ c a ooooooooooooooooo ^ u c o m o « H J i > u i > o » ~ i " n ' I I I ! i " < I " o 0 0 0 S 0 0 0 0 0 S ] O O O O O O O O O O O O O O O O O ooooooooooooooooo ooooooooooooooooo ooooooooooooooooo O O CJ g ^ i u t O ^ « " ^ O M O w O O O O O O O O O O ^ B ^ ^ ^ ' - . c D ^ c D C D U c ^ M C j f f l & o a a i t ' C D ^ u O O O O O O O O O O - n i r ) ^ O u o 3 0 - o i c o c n N J O c n c 3 D ^ c n c ^ c n - ^ c o ^ ^ & O f o ^ j o u - * c j ^ M O T ^ c n S u i c D - - i ^ i c j c j i c n i o & i o < X 3 ^ c n 0 ^ i o ' C D i o a i c D t > . O O c j c j w O O O O O O O O O O O O O O O O O boooboboboboooooo O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O 99T MODE 5 CONTRIBUTION FACTOR 3 0 .03040 0AMPING=0.0204 PERI0D=O.O294 SEC. SA=0.146 -*******************************+***+********************+* MODE NUMBER 6 MODAL FORCES AND DISPLACEMENTS JOINT NO. X-D ISP(FT) Y-D ISP(FT) ROTATION(RAD) 1 0 . 0 0 .0 0 .0 2 0 .0 0 .0 0 .0 3 0 .0000 -0 .0000 -0 .0000 4 0 .0000 0.0000 -0 .0000 5 0 .0000 -0 .0000 0.0000 G 0 .0000 0.0000 0.0000 7 - 0 . 0 0 0 0 -0 .0000 0.0000 8 - 0 . 0 0 0 0 0.0000 0.0000 9 -O.OOOO -0 .0000 -0 .0000 10 - 0 . 0 0 0 0 0.0000 -0 .0000 11 O.OOOO -O.OOOO -O.OOOO 12 0.0000 0.0000 -O.OOOO 13 0.0000 -0 .0000 0.0000 14 O.OOOO 0.0000 0.0000 15 - 0 . 0 0 0 0 -0 .0000 0.0000 16 - 0 . 0 0 0 0 0.0000 0.0000 17 - 0 . 0 0 0 0 -0 .0000 -0 .0000 18 -O.OOOO 0.0000 -O.OOOO 19 " 0 .0000 -0 .0000 0.0000 0.OOOO 0.OOOO 0.OOOO 0.0000 -0 .0000 0.0000 22 - 0 .0000 0.0000 0.0000 MN AXIAL SHEAR BML BMG KIPS KIPS (K -FT ) (K -FT) 1 - 0 . 2 1 7 0.003 -0 .009 0.010 2 0.440 0 .000 -0.001 0.001 3 -0.023 -0 .002 0.007 -0 .008 4 -0.461 0 .000 -0.001 0.001 5 0.121 0.003 -0 .007 0.008 6 0.433 -0.001 0.003 -0 .004 7 - 0 .218 -0 .003 0.008 -0 .010 8 -0.383 0.002 -0 .006 0.007 9 0.317 0.004 -0 .012 0.014 10 0. 390 -0 .006 0.0.1 7 -0 .020 1 1 -0.001 0.400 -2 .029 1 .069 12 0.001 3.625 -18 .577 9.516 13 -0.007 0.010 1 .007 1 .090 14 0.007 0.088 9.451 10.197 15 - 0 . 0 0 2 -0 .307 1 . 135 -1 .475 16 0.002 -2 .785 10.244 - 13 .428 17 0 .000 0.076 -1 .452 -0 .810 18 - 0 . 0 0 0 0.689 -13.403 -7 .550 19 - 0 . 0 0 3 0.293 -0 .840 1 .651 20 0.003 2 .661 -7.581 15.04 1 21 -0 .004 - 0 . 1 4 0 1.640 0. 449 22 0.004 -1 .273 15.030 4 . 206 23 -0 .001 -0.2G2 0.475 -1 .748 24 0.001 -2 .375 4 . 234 - 15.957 25 -0.001 0. 199 -1 .744 -0 .052 2G 0.001 1.812 -15.952 -0 .553 27 - 0 . 0 0 3 0.222 -0 .077 1.813 28 0 .003 2.029 • - 0 . 5 8 0 16.663 29 • - 0 . 0 0 3 -0 .217 1 .816 -0 .032 30 0.003 -1 .965 16 .666 -0 .034 CONTRIBUTION FACT0R=-O. .01721 MODE 6 DAMPING=0.0202 PERIOD=0.0207 SEC. SA=0.103 _ * * « * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * ********************************** MODE NUMBER JOINT 7 MODAL FORCES AND DISPLACEMENTS MN 1 2 3 4 5 6 7 8 9 10 NO. X -D ISP(FT) Y-D ISP(FT) ROTATION(RAD) 1 0 .0 0. 0 0. 0 2 0 .0 0. 0 0. 0 3 0.OOOO 0. o o o o - 0 . OOOO 4 0.OOOO - o . o o o o - 0 . OOOO 5 0.OOOO 0. o o o o 0. OOOO 6 0.OOOO - 0 . o o o o 0. OOOO 7 - 0 .0000 0. o o o o - 0 . OOOO' 8 - 0 . o o o o - o . o o o o - 0 . o o o o 9 0 . o o o o 0. o o o o - o . .0000 10 0.0000 - 0 . o o o o - 0 . .0000 1 1 0 . o o o o 0. o o o o 0 .0000 12 0 . o o o o - 0 . o o o o 0 .0000 13 - 0 . o o o o 0. o o o o -0 .0000 14 - 0 .0000 - 0 . o o o o -0 .0000 15 0 . o o o o 0. .0000 -0 .0000 16 0 . o o o o - 0 . .0000 -0 .0000 17 0 . o o o o 0. .0000 0 . o o o o 18 0 . o o o o -0 .0000 0 .0000 19 - 0 . o o o o 0 .0000 0 .0000 20 -0 .0000 -0 .0000 0 .0000 21 0 . o o o o 0 .0000 -0 .0000 22 0 . o o o o - 0 .0000 - 0 .0000 AXIAL SHEAR BML BMG KIPS KIPS (K--FT) (K -FT ) 0 .059 -0.001 0 .003 -0 .004 - 0 . 1 5 7 0 .000 -0 .001 0.001 0 .113 0.001 -0 .002 0.002 0.071 -0.001 0 .002 -0 .003 - 0 . 1 7 0 - 0 . 0 0 0 0 .000 - 0 .000 0 .068 0.001 -0 .003 0.003 0.115 -0.001 0 .002 -0 .003 -0 .162 - 0 .000 0 .001 -0.001 0 .020 0.002 -0 .006 0.007 0. 168 -0.001 0 .004 -0 .005 11 0. 0 0 1 0 . 124 - 0 . 546 12 - 0 . 0 0 1 2 . 0 4 2 - 9 . 0 2 8 13 - 0 . 0 0 1 - 0 . 0 4 3 0 . 4 0 3 14 0. 001 - 0 . 706 6 . 7 8 0 15 0 . .001 - 0 . 0 6 3 0 . 0 5 8 16 - 0 . .001 - 1 . 0 3 8 0 . 8 0 3 17 0 . .001 0 . 0 9 8 - 0 . 4 8 6 18 - 0 . . 001 1. 6 1 2 - 8 . 0 2 3 19 0 . 0 0 0 - 0 . 0 1 6 0 . 3 3 9 2 0 - 0 . 0 0 0 - 0 . , 2 7 0 5. . 6 6 7 21 0 .001 - 0 . , 0 8 5 0 , . 2 0 9 22 - 0 . 001 . - 1 . , 3 9 0 3 . 3 8 2 23 0 . 001 0 . 0 8 6 -0 .511 24 - 0 . 001 1 . 4 0 5 -8 . 4 3 2 2 5 0 . 0 0 0 0 . 0 1 5 0 . 2 0 8 26 - 0 . 0 0 0 0 . 2 4 0 3 . 5 0 2 27 0 . 001 -0 . 0 9 8 0 . 3 4 0 28 - 0 . 0 0 1 -1 . 6 1 7 5 . 5 4 9 2 9 0 . 0 0 1 0 . 0 5 9 -0 . 4 9 2 3 0 . - 0 . 0 0 1 0 . 9 6 5 -8 . 194 MODE ' 7 C O N T R I B U T I O N F A C T O R = - 0 . 0 1 0 9 8 D A M P I N G = 0 . 0 2 0 1 P E R I 0 D = O . 0 1 5 3 S E C . S A = 0 . 0 7 6 . • • • • i t * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * -ROOT MEAN SQUARE D I S P L A C E M E N T S J O I N T N O . X - D I S P ( F T ) Y - D I S P ( F T ) 1 0 . 0 0 . 0 2 0 . 0 0 . 0 3 0 . 0 0 4 8 0 . 0 0 0 3 4 0 . 0 0 4 8 0 . 0 0 0 3 5 0 . 0 1 9 7 0 . 0 0 0 5 6 0 . 0 1 9 8 . 0 . 0 0 0 5 7 0 . 0 4 2 9 0 . 0 0 0 8 8 0 . 0 4 3 0 0 . 0 0 0 8 9 0 . . 0 7 2 4 0 . 0 0 1 0 10 0 . , 0 7 2 6 0 . . 0 0 1 0 1 1 0 . 1069 0 ,001 1 12 0 . 1 0 7 1 0 .001 1 13 0 . 1449 0 . 0 0 1 3 14 0 . 1452 0 . 0 0 1 3 15 0 . 1855 0 . 0 0 1 4 16 0 . 1858 0 . 0 0 1 4 17 0 . 2 2 7 7 0 . 0 0 1 5 18 0 . 2 2 8 1 0 . 0 0 1 5 19 0 . 2 7 0 8 0 . 0 0 1 5 2 0 0 . 2 7 1 2 0 . 0 0 1 5 21 0 . 3 1 4 2 0 . 0 0 1 6 22 0 . 3 147 0 . 0 0 1 6 -ROOT MEAN SQUARE F O R C E S 0 RSS B A S E SHEAR = 4 9 8 . 8 5 3 K I P S MN A X I A L SHEAR BML 0 . 4 1 8 6 . 7 9 5 0 . 0 3 7 0 . 7 8 1 - 0 . 4 8 2 - 8 . 0 1 8 0 . 348 5 . 6 7 7 0 . 199 3 . 3 7 2 - 0 . 5 1 1 - 8 . 4 3 2 0 . 2 1 7 3 . 5 1 1 0 . 334 5 . 5 4 2 - 0 . 4 9 5 -8 . 197 0 . 0 1 1 0 . 0 1 2 *********************** R O T A T I 0 N ( R A D ) 0 . 0 0 . 0 0 . 0 0 1 2 0 . 0 0 1 2 0 . 0 0 2 3 0 . 0 0 2 3 0 . 0 0 3 1 0 . 0 0 3 1 0 . 0 0 3 8 0 . 0 0 3 8 0 . 0 0 4 3 0 . 0 0 4 3 0 . 0 0 4 7 0 . 0 0 4 7 0 . 0 0 5 0 0 . 0 0 5 0 O . 0 0 5 1 0 . 0 0 5 1 0 . 0 0 5 2 0 . 0 0 5 2 0 . 0 0 5 2 0 . 0 0 5 2 BMG MOMENT €89 MN AXIAL SHEAR t S M L B M W 690 KIPS KIPS (K- FT) ( K - F T ) 691 1 0. 059 -0.001 0. 003 -0 .004 692 2 - 0 . 157 O.OOO - 0 . 001 0.001 693 3 0. 113 0.001 - 0 . 002 0.002 694 4 0. 07 1 -0.001 0. 002 -0 .003 695 5 - 0 . 170 - 0 . 0 0 0 ' 0 . 000 - 0 . 0 0 0 696 6 0. 068 0.001 - 0 . 003 0.003 697 7 0. 1 15 -0.001 0. 002 -0 .003 698 8 - 0 . 162 - 0 . 0 0 0 0. 001 -0.001 699 9 0. 020 0.002 - 0 . 006 0.007 700 10 0. 168 -0.001 0. 004 - 0 .005 701 1 1 0. 001 0. 124 - 0 . 546 0.4 18 702 12 - 0 . 001 2 .042 - 9 . 028 6 . 795 703 13 - 0 . 001 -0 .043 0. 403 0.037 704 14 0. 001 -0 .706 6 . 780 0.781 705 15 0. 001 - 0 . 0 6 3 0. 058 -0 .482 .706 16 - 0 . ,001 - 1.038 0. 803 -8 .018 707 17 0. .001 0.098 - 0 . 486 0. 348 708 18 - 0 ,001 1.612 - 8 . 023 5.677 709 19 0. .000 - 0 . 0 1 6 0. 339 0. 199 710 20 -O. .000 - 0 . 2 7 0 5 . 667 3 . 372 71 1 21 0 .001 - 0 . 0 8 5 0. 209 -0.511 712 22 - 0 .001 - 1 . 3 9 0 3. 382 . -8 .432 713 23 0 .001 0.086 - 0 . 511 0.217 714 24 - 0 .001 1 .405 -8 . 432 3.511 715 25 0 .000 0 .015 0. 208 O. 334 716 26 - 0 .000 0. 240 3. .502 5.542 717 27 0 .001 - 0 .098 0. 340 - 0 .495 718 28 - 0 .001 -1 .617 5 , 549 - 8 . 1 9 7 719 29 0 .001 0 .059 - 0 , 492 0.01 1 720 30 - 0 .001 0 .965 -8 . 194 0.012 721 MODE 7 CONTRIBUTION FACTOR=-0.01098 722 DAMPING=0.0201 PERIOD' =0.0153 1 SEC. SA=0.076 723 _ * » * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * ' 724 -ROOT MEAN SQUARE DISPLACEMENTS 725 JOINT NO. X-D ISP(FT) Y -D I SP (FT ) ROTATION(RAD) 726 1 0 .0 0 .0 0. 0 727 2 O.O 0 .0 0. 0 728 3 0.0048 0 .0003 0. 0012 729 4 O.0048 0 .0003 0. 0012 730 5 0.0197 0 .0005 0. ,0023 731 6 0.0198 0 .0005 0 0023 732 7 0.0429 0 .0008 0 ,0031 733 8 0.0430 0 .0008 0 .0031 734 9 0.0724 0 .0010 0 .0038 735 10 0.0726 0 .0010 0 .0038 736 1 1 0.1069 0 .001 1 0 .0043 737 12 0.1071 0 .001 1 0 .004 3 738 13 0. 1449 0 .0013 0 .0047 739 14 0.1452 0 .0013 0 .0047 740 15 0.1855 0 .0014 0 .0050 74 1 16 0.1858 0 .0014 0 .0050 742 17 0.2277 0 .0015 0 .0051 743 18 0.2281 0 .0015 0 .0051 744 19 0.2708 0 .0015 O .0052 745 20 0.2712 0 .0015 0 .0052 746 2 1 0.3142 0 .0016 O .0052 747 22 0.3147 O .0016 0 .0052 748 -ROOT MEAN SQUARE FORCES ******* CTl KIPS KIPS (K-•FT) (K- FT) CAPACITY RATIO 1 74.699 20.027 60. 070 60. 091 60. 000 7 . 520 2 42.455 20.026 60. 069 60. 089 60. 000 7 . 522 3 31 . 373 20.026 60. 069 60. 086 60. 000 7 . 443 4 42.902 20.025 60. 069 60. 082 60. 000 7 . 228 5 54.475 20.024 60. 068 60. 078 60. 000 6 . 855 6 58.735 20.023 60. 066 60. 074 60. 000 6 . 306 7 54.626 20.022 60. 060 60. 070 60 000 5 . 563 B 42.697 20.019 60. 051 60. 066 60. 000 4 . 588 9 25.189 20.017 60. .039 60. 062 60. .000 3 . 335 10 7 .583 20.014 60. .025 60. 058 60. ,000 1 . 755 1 1 196.843 245.699 9272 . .570 . 7622 . 535 13000. .000 0.713 12 196.843 253.363 9338. .672 7652 . .949 13000. .000 0.7 18 13 177.907 240.634 7832 .023 6198. . 359 12666 .660 0.618 14 177.907 247.393 7862 .629 6209 .750 12666 .660 0.621 15 158.767 223.385 6405 .039 5054 . 160 12333 . 328 0.519 16 158.767 228.360 6416 .504 5071 .875 12333 .328 0 .520 17 - 139.392 195. 138 5253 . 102 4202 .629 1 2000 .000 0.4 38 18 139.392 198.429 5270 .266 4244 . 375 12000 .000 0.439 19 119.787 163.882 4387 . 164 3525 .095 1 1666 .660 0. 376 20 119.787 165.962 4427 .262 3593 .762 1 1666 .660 0. 379 2 1 99.992 140.696 3690 .945 2850 . 327 1 1333 . 328 0. 326 22 99.992 142.333 3756 .903 2935 .752 1 1333 .328 0.331 23 80.067 131.811 2996 .812 2074 . 128 1 1000 .000 0.272 24 80.067 133.715 3079 . 146 2161 .430 1 1000 .OOO 0. 280 .25 26 60.073 128.885 2201 .682 1222 .870 10666 .660 0.206 60.073 131.535 2285 .387 1295 . 195 10666 .660 0.2 14 27 40.053 114.718 1326 .283 466 . 180 10333 . 328 0. 128 28 40.053 118.409 1394 .880 504 . 726 10333 . 328 0. 135 29 20.027 74.670 514 .963 210 . 130 10000 .000 0.051 30 20.027 78.485 552 . 199 210 . 142 10000 .000 0.055 6 0 0.002 0.787E+02 NO. OF ITERATIONS = 6 BETA=0.0 BENDING MOMENT ERROR=0.050000 DAMAGE RATIO ERROR= 0 .010 1 2 3 4 C 5 C 6 C 7 C 8 9 10 C 1 1 C 12 13 . 14 15 16 17 18 19 c 20 c 21 c 22 c 23 c 24 c 25 c 26 c 27 c 28 c 29 c 30 c 31 c 32 c 33 34 c 35 c 36 c 37 c 38 39 40 c 41 c 42 43 c 44 c 45 c 46 c 47 48 49 50 c MODAL ANALYSIS PROGRAM 'EDAM' MARCH 1981 (ELASTIC AND/OR DAMAGE AFFECTED MODAL ANALYSIS) PROGRAM ORIGINALY WRITTEN BY SUMIO YOSHIDA TITLED MSSM EXTENSIVELY REWRITTEN AND EXPANDED BY ANDREW W.F. METTEN PROGRAvM L I S T I N G DOUBLE PRECISION STIFFNESS MATRIX ROUTINE REAL*8 S(2000) DIMENSION KL(100),KG(100).AREA(100),CRMOM(100),BMCAP(100). 1 DAMRA T (100 ) ,ND(3 , 100),NP(6, 100),XM(100).YM( 100).DM(100), 2 F (300 ) ,EXTL (100 ) .EXTG(100 ) ,T ITLE (20 ) , SDAMP(100 ) ,AV (100 ) DIMENSION DAMB(IOO), MD0F(50) DIMENSION AMASS(300) ,EVAL(10) ,EVEC(50.10) DIMENSION BMY(75), BETAM(10) PROGRAM DIMENSIONED FOR A MAXIMUM OF 100 MEMBERS 100 JOINTS 50 ASSIGNED MASSES 10 EIGENVALUES 300 UNKNOWNS (NUMBER OF UNKNOWNS)*(HALF BANDWIDTH) IS LESS THAN 2000 ' IUNIT DEFINES THE INPUT AND OUTPUT F ILES IUNIT=5 IS DATA SOURCE F ILE IUNIT=6 IS TEMPORARY STORAGE FOR INTERMEDIATE DATA IUNIT=7 IS FINAL OUTPUT FILE IUNIT=8 IS DAMAGE RATIO F ILE THIS IS SEPARATE FROM OTHER FINAL OUTPUT FILE TO MAKE PLOTTING OF RESULTS EASIER. IUNIT=7 SUBROUTINE CONTRL READS IN DATA SUCH AS THE NUMBER OF JOINTS AND THE TITLE OF THE STRUCTURE, AND PROGRAM OPTIONS. SUBROUTINE CONTRL IS INDEXED FROM 1001 CALL CONTRL(TITLE.NRJ.NRM,E,G,7,AMAX,I SPEC,DAMP IN. 1 INELAS,NMODES.NPRINT) IDIM DIMENSIONS STRUCTURE AND MATRICES FOR SUBROUTINES IDIM=2000 SUBROUTINE SETUP READS AND ECHO PRINTS THE MEMBER AND JOINT DATA ITEMS SUCH AS HALF BANDWIDTH AND NUMBER OF UNKNOWNS ARE CALCULATED SUBROUTINE SETUP IS INDEXED FROM 2001. IF LAG = 0 CALL SETUP(NRJ.NRM.E,G,XM.YM,DM,ND,NP,AREA,CRMOM,DAMRAT,AV.KL,KG. 1 NU,NB,SDAMP,BMCAP,IUNIT,EXTL,EXTG ) 51 C 52 C 53 54 55 10 56 57 58 C 59 C 60 C 61 62 20 63 C 64 C 65 C 66 C 67 68 C 69 C 70 . C 71 C 72 C 73 74 C 75 C 76 77 78 79 80 30 81 C 82 83 C 84 C 85 C 86 C 87 88 89 90 91 92 70 93 C 94 C 95 C 96 97 98 C 99 C 100 CHECK IF IDIM HAS BEEN ASSIGNED LARGE ENOUGH LSTM=LENGTH OF STIFFNESS MATRIX LSTM=NU*NB IF(LSTM.GT.IDIM) WRITE(7,10) LSTM,IDIM FORMAT (/// '.PROGRAM STOPPED' . / / ' L E N G T H OF STIFFNESS MATR IX ' ' , 1 16,/ 'PROVIDED STORAGE ( ID IM)= ' .16) IF (LSTM.GT.IDIM) STOP ASSIGN TEMPORARY VARIABLE BMY EQUAL TO THE YIELD MOMENT (BMCAP) DO 20 MEMBN=1,NRM BMY(MEMBN) = BMCAP(MEMBN) ICOUNT IS THE NUMBER OF TIMES MAIN MSSM SUBROUTINE IS CALLED ICOUNT IS INITIALIZED TO ZERO HERE. ICOUNT'O SUBROUTINE MASS READS AND ASSIGNS MASSES TO NODES TO DETERMINE THE MASS MATRIX. SUBROUTINE MASS HAS INDEX NUMBERS STARTING AT 4001 CALL MASS(NU,NO,AMASS,IUNIT,NRJ,NMASS,MDOF) CALCULATE IF IDIM HAS BEEN SUFFICIENTLY DIMENSIONED IVAR1=(NU*NB)+NMASS IVAR2=NMASS*(NMODES+3) IF( IVAR1 .GE.IDIM) WRITE(7.30) IF( IVAR2.GE. ID IM) WRITE(7,30) FORMAT(' ' , ' T H E VALUE OF IDIM IS SMALLER THAN RVPOW REQUIRES') REASSIGN OUTPUT TO TEMPORARY F ILE 6 IUNIT=6 IF ELASTIC ANALYSIS ONLY IS REQUIRED: RESET CONTROL FLAGS SET FLAG TO INDICATE ONLY ONE ITERATION REQUIRED IF( INELAS.NE.O) GO TO 70 WRITE(7,110) IUNIT=7 I F L AG= 1' WRITE(7,110) CONTINUE SET THE MAXIMUM NUMBER OF ITERATIONS. I F ( INELAS.NE.0 ) IMAX = INE LAS IM=IMAX-1 I IS A PROGRAM LOCATION VARIABLE (SEE FLOWCHART) IT SIGNIFIES NUMBER OF ITERATIONS PERFORMED. 101 C BETA IS A NUMBER USED IN SPEEDING CONVERGENCE. SHOULD BE A POSITIVE 102 C NUMBER LESS THAN ONE. 103 C A VALUE OF BETA OF ZERO EFFECTIVELY SHUTS OFF CONVERGENCE SPEEDING 104 C ROUTINE 105 BETA=0. 106 C SET ERROR RATIO OF MOMENTS OF YIELDED MEMBERS (BMERR). 107 C A VALUE OF 0.05 HERE ENSURES YIELDED MEMBERS ARE WITHIN 108 C 5 PERCENT OF THEIR CAPACITY. 109 BMERR=0.05 1 10 C 111 C SET STOPPING VALUE FOR MINIMUM DAMAGE RATIO CHANGE BETWEEN SUCCESSIVE 112 C ITERATIONS. DAMERRE=0.01 ENSURES THAT THE MAXIMUM DAMAGE RATIO 113 C CHANGE IN THE FINAL ITERATION IS ONE PERCENT FOR DAMAGE RATIOS 114 C ABOVE 5.0 115 C THOSE DAMAGE RATIOS BELOW 5.0 WILL HAVE A STOPPING CRITERION OF THEIR 116 C ABSOLUTE VALUE DIFFERENCE BEING TEN TIMES THE RATIO. 117 C 118 DAMERR'O.01 119 C INITIALIZE ARRAY USED IN SPEEDING OF CONVERGENCE. 1'20 DO 80 MEM= 1 , NRM 121 DAMB(MEM)=DAMRAT(MEM) 122 80 CONTINUE 123 C 124 C 125 C FINISHED INPUT OF DATA AND INITIAL ACTIVITIES. 126 C BEGIN LOOP FOR MSS METHOD. 127 C 128 C 129 100 CONTINUE 130 C INCREMENT ITERATION COUNTER. 131 1=1+1 132 WRITE(IUNIT,110) 133 110 FORMAT(' '.110('-')) -134 WRITE(IUNIT,120) I 135 120 FORMAT('-','ITERATION NUMBER',14) 136 C 137 C SUBROUTINE BUILD COMPUTES THE MEMBER AND GLOBAL STIFFNESS MATRIX 138 C SUBROUTINE BUILD IS INDEXED STARTING AT' LINE 3001 139 C WITH THE CALLING BELOW STIFFNESS MATRIX CANNOT HAVE GREATER THAN 140 C 1500 ENTRIES. 141 C 142 C CRMOM IS THE CRACKED MOMENT OF INERTIA OF THE SECTION. 143 CALL BUILD(NU.NB,XM,YM,DM,NP.AREA.CRMOM,AV,E,G.DAMRAT,KL,KG.NRM.S. 144 1 IDIM,EXTL,EXTG) 145 C 146 C CALL SUBROUTINE TO CHECK ON THE CONDITIONING AND STABILITY OF 147 C THE STIFFNESS MATRIX. 148 CALL SCHECK(S.NU,NB.IDIM,IUNIT,SRATIO) 149 C 150 C SUBROUTINE EI GEN COMPUTES THE FREQUENCIES AND MODES FOR THE CO 151 C SUBSTITUTE STRUCTURE. 152 C 153 CALL EIGEN(NU.NB,S,IDIM,AMASS.EVAL.EVEC,NMODES.IUNIT,I SPEC, 154 . 1 AMAX,ICOUNT,MDOF,INELAS) 155 C INSERT HEADINGS FOR ITERATION PROGRESS OUTPUT AND TO 156 C DIFFERENTIATE INELASTIC OUTPUT. 157 IF( INELAS.EO.O.OR. ICOUNT.NE.O) GO TO 105 158 WRITE(7,110) 159 WRITE(7.115) 160 115 FORMAT(' ' , / / 25X, ' INELAST IC RESULTS ' / / ) 161 WRITE(7,110) 162 WRITE(7.90) 163 90 F O R M A T ( ' - ' , ' I T E R A T I O N N O . ' , 2 X , ' N O . ABOVE CAPAC ITY ' ,2X , 'DAMD IF ' , 164 1 3X , ' S MATRIX RATIO ' ) 165 105 CONTINUE 166 C AFTER 10 ITERATIONS BETA IS REASSIGNED FROM 0 . 0 TO 0.25 167 IF ( I .GE. 9) BETA=0.80 168 C ISIGN IS A COUNT OF THE NUMBER OF MEMBERS UNTOLERABLY ABOVE ULTIMATE. 169 C 170 C FIND THE MEMBER WITH THE LARGEST DIFFERENCE IN DAMAGE RATIOS 171 C BETWEEN THIS AND THE LAST ITERATION, USE VARIABLE 'OVARY'. 172 C IN IT IAL IZE DRFIFF TO ZERO HERE. 173 DVARY=0.0 174 C 175 C M0D3 IS THE MAIN SUBROUTINE FOR THE MSSM. IT IS INDEXED FROM 6001. 176 C 77 CALL M0D3(ICOUNT,I SPEC,NRJ.NRM,NU,NB,NMODES,S.500.NO,NP.XM.YM DM. 1 7 o * AREA AV CRMOM,DAMRAT.KL.KG,SDAMP.BMCAP,E.G.AMASS,EVECEVAL. 2 AMAX ', ISIGN, IUNIT. BETA. BMERR , IF L AG , EXTL . EXTG, BET AM . D AMB . 3 DVARY.INELAS, DAMPIN,NPRINT) 178 179 180 181 C 182 C IF ONLY DOING ELASTIC ANALYSIS THEN STOP PROGRAM 183 IF ( INELAS.EO.O) GO TO 250 184 C 185 C OUTPUT DAMAGE RATIOS ON UNIT 8 186 C THESE ARE OUTPUT FOR EACH MEMBER AT EACH ITERATION. 187 C 188 C OUTPUT NUMBER OF MEMBER IN EXCESS OF CAPACITY AND LARGEST 189 C DIFFERENCE FROM PREVIOUS ITERATIONS DAMAGE RATIOS. 190 C ALSO OUTPUT RATIO OF LARGEST TO SMALLEST MEMBER OF STIFFNES 191 C MATRIX DIAGONAL (SRATIO) 192 WRITE(7,130) I.I SIGN,DVARY,SRAT10 193 130 FORMAT(' ' , 5 X , I 4 , 9 X , I 4 , 1 2 X . F 7 . 3 , 1 0 X . E 1 0 . 3 ) WRITE(8,140) (DAMRAT(MEMBRJ),MEMBRJ=1,NRM) 194 195 140 FORMAT(' ' , 1 5 F 8 . 3 ) 196 C 197 C I FLAG IS A FLAG USING INTEGER VALUES 1 AND 0. MODIFIED q 8 C FROM 0 TO 1 WHEN NO MEMBERS ARE ABOVE CAPACITY. IF ALL MEMBERS 99 C ARE BELOW OR AT CAPACITY ONE FINAL ITERATION IS PERFORMED. ^ 200 C THE FOLLOWING LINES CHECK FOR YIELDING OF ALL MEMBERS AND THE ^ 201 C 202 C 203 204 205 206 207 208 209 C 210 211 150 212 C 213 C 214 215 216 217 160 218 219 170 220 221 180 222 223 190 224 225 200 226 227 228 229 • 230 210 231 232 220 233 234 230 235 236 240 237 250 238 1001 C 1002 1003 C 1004 1005 1006 C 1007 1008 C 1009 1010 C 101 1 C 1012 C MAXIMUM NUMBER OF ITERATIONS. I F ( I FLAG.EO. 1 .AND. I.GE.IMAX) GO TO 180 IF ( I FLAG.EO. 1) GO TO 160 I F ( I . EQ .1 .AND. IS IGN.EO.O) GO TO 200 IF ( I .GE . IM) GO TO 150 ADERR=ABS(OVARY) IF ( I SIGN.EO.0.AND.ADERR.LT.DAMERR) GO TO 150 GO TO 100 CONTINUE IFLAG= 1 IUNIT=7 GO TO 100 CONTINUE WRITE! IUNIT,170) I F O R M A T ( ' - ' , 5 X , ' N O . GO TO 220 CONTINUE WRITE(IUNIT,190) I F O R M A T C - ' , 5 X , 'DOES NOT CONVERGE A F T E R ' . 1 5 . ' GO TO 220 CONTINUE ICOUNT=0 IFLAG=1 IUNIT=7 WRITE(IUNIT.210) FORMATC' - ' ,5X. 'MEMBERS DO NOT YIELD ' / / / ) GO TO 100 CONTINUE• WRITE(IUNIT,230) BETA,BMERR F O R M A T ( ' - ' , 5 X , ' B E T A = ' , F 5 . 3 . / / / 5 X . WRITEUUNIT.240) DAMERR FORMAT( ' ' , 'DAMAGE RATIO ERROR' ' . STOP END OF ITERATIONS = ' .15/ / / ) ITERAT IONS ' / / / ) 'BENDING MOMENT ERROR'' . F 8 . 6 / / / ) F6 .3 ) SUBROUTINE CONTRL(TITLE,NRJ,NRM,E.G,IUNIT.AMAX.I SPEC,DAMP IN, 1 INELAS,NMODES,NPRINT) DIMENSION T ITLE (20 ) READ IN PROGRAM OPTIONS —1 1013 1014 1015 1016 1017 1018 1019 1020 1021 1022 1023 1024 1025 1026 1027 1028 1029 1030 1031 1032 1033 1034 1035 1036 1037 1038 1039 1040 1041 1042 1043 1044 1045 1046 1047 1048 1049 1050 1051 1052 1053 1054 1055 1056 1057 1058 1059 1060 1061 1062 10 C C C C C C C C C C C C C C 20 30 40 50 60 70 80 C READ(5,10) INELAS,NMODES.NPRINT,ISPEC,AMAX,DAMPIN FORMAT(4I5,2F10.5) DAMPIN IS THE PROPORTION OF CRITICAL DAMPING USED IN ELASTIC ANALYSIS OR THE FIRST ITERATION OF THE MSSM. NPRINT IS A FLAG SET IF MODAL FORCES AND DISPLACEMENTS ARE REQUIRED IF NPRINT'O ONLY RMS FORCES AND DISPLACEMENTS WILL BE PRINTED. IF NPRINT IS GREATER THAN ZERO THAT NUMBER OF MODES (UP TO NMODES) WILL HAVE THEIR FORCES AND DISPLACEMENTS PRINTED. INELAS IS A FLAG INDICATING IF ONLY AN ELASTIC ANALYSIS IS REQUIRED IF INELAS'O THEN ELASTIC ANALYSIS ONLY WILL BE PERFORMED. IF INELAS IS GREATER THAN ZERO THEN THIS IS THE MAXIMUM NUMBER OF ITERATIONS THAT WILL BE PERFORMED DURING INELASTIC ANALYSIS. • ECHO PRINT PROGRAM OPTIONS WRITE(IUNIT,20) FORMAT( ' ' ,// ' *******PROGRAM OPTIONS*******'/) WRITE(IUNIT,30)NMODES FORMAT( ' '.'MAXIMUM NUMBER OF MODES IN ANALYSIS', IF(INELAS.EQ.O) WRITE(I UNIT,40) FORMAT( ' ','ELASTIC ANALYSIS REQUESTED') IF(INELAS.NE.O) WRITE(IUNIT,50) INELAS FORMAT( ' ', 'INELASTIC ANALYSIS MAXIMUM ITTERAT IONS''.14 ) IF(INELAS,EQ.O) WRITE(IUNIT,60) DAMPIN FORMAT(' ','FRACTION OF CRITICAL DAMPING'',F6.4) IF(INELAS.GT.O). WRITE(IUNIT,70) DAMPIN FORMAT(' ','INITIAL DAMPING RATIO' '.F6.3) WRITE(IUNIT,80) NPRINT FORMATC ','NUMBER OF MODES TO HAVE OUTPUT PRINTED'',13) 14) (IUNIT,120) ( IUNIT. 130) ( IUNIT,140) WRITE(IUNIT,90) WRITE(IUNIT,100) AMAX 90. FORMAT('-','SEISMIC INPUT') 100 FORMAT( ' -', 'MAXIMUM ACCELERATION' 110 F0RMAT(///11O('-')) IF(I SPEC.EQ. 1) WRITE IFUSPEC.EQ.2) WRITE IF(ISPEC.EQ.3) WRITE .IF(ISPEC.EQ.4) WRITE(IUNIT,150) IF(ISPEC.GE.5) WRITE(IUNIT,160) WRITE(IUNIT.110) 120 FORMAT( ' 130 FORMAT( ' 140 FORMAT( ' 150 FORMAT( 160 FORMAT( .F5.3.' TIMES GRAVITY') ISPEC SPECTRUM A USED')• ' SPECTRUM B USED' ) 'SPECTRUM C USED') 'NATIONAL BUILDING CODE SPECTRUM USED') 'ERROR-SPECTRUM TYPE',13,' IS NOT VALID') IF(ISPEC.NE.4) GO TO 200 DPCNT =100.0*DAMPIN 1063 1064 C 1065 1066 170 1067 1068 180 1069 1070 190 1071 C 1072 C 1073 C 1074 200 1075 C 1076 C 1077 C 1078 1079 1080 1081 1082 1083 1084 C 1085 C < 1086 1087 1088 C 1089 1090 210 1091 220 1092 230 1093 240 1094 250 1095 260 1096 2001 C 2002 2003 C 2004 2005 2006 C 2007 2008 C 2009 C 2010 C 201 1 C 2012 2013 2014 2015 2016 c CALL SPECTRfI SPEC.DAMPIN,1.0.AMAX,SA,6.283,SABND,SVBND.SDBND) WRITE(IUNIT,170) DPCNT, SABND FORMAT( ' ',F5.2,'% DAMPING SPECTRAL ACCEL. B0UND=',F6.3.' *G') WRITE(IUNIT,180) SDBND FORMAT( ' ',' DISPLACEMENT BOUND3' ,F6.3. ' IN') WRITE(IUNIT,190) SVBND FORMAT' ' '.' VELOCITY BOUND= ' , F6 . 3 . ' IN/SEC) READ IN TITLE READ (5,210)(TITLE(I),1=1.20) REAO IN NRJ,NRM,E,G READ (5,220) NRJ, NRM, E, G WRITE (IUNIT.230)(TITLE(I).1=1,20) WRITE (IUNIT.240) E. G WRITE (IUNIT,250) WRITE (IUNIT,260) NRJ, NRM WRITE(IUNIT,110) E=E*144.0 G=G* 144.0 RETURN FORMAT(20A4 ) F0RMAT(2I5,2F10.0) FORMAT('1',20A4) FORMAT( ' - ' ,5X, 'E ='.F8.1.' KSI F0RMAT(///110('*')) FORMAT( ' -', 'NO. OF JOINTS'.' = END , 5X,'G =' ,F8. 1 .' KSI') , 15.10X, 'NO. OF MEMBERS .15) SUBROUTINE SETUP(NRJ.NRM,E.G.XM,YM,DM,NO.NP,AREA.CRMOM.DAMRAT.AV, 1 KL,KG,NU,NB,SDAMP,BMCAP,IUNIT,EXTL,EXTG) SET UP THE FRAME DATA DIMENSION KL(NRM). KG(NRM). AREA(NRM), CRMOM(NRM), SDAMP(NRM), 1 DAMRAT(NRM), AV(NRM). ND(3.NRJ), NP(6.NRM). XM(NRM), 2 YM(NRM ) ,EXTL(NRM).EXTG(NRM), DM(NRM) DIMENSION X(100). Y(100). JNL(IOO). JNG(100). BMCAP(NRM) 2017 C 2018 C 2019 C 202O C 2021 C .2022 2023 2024 C 2025 C 202G C 2027 2028 C 2029 2030 2031 C 2032 2033 2034 10 2035 2036 2037 20 2038 2039 2040 30 2041 2042 40 2043 C 2044 C 2045 C 2046 2047 50 2048 C 2049 . 2050 2051 2052 2053 C 2054 C 2055 C 2056 C 2057 . C 2058 2059 C 2060 2061 2062 2063 2064 C 2065 C 2066 C E AND G IN KSF X ( I ) AND Y( I ) IN FEET MEMBER EXTENSIONS EXTG AND EXTL ARE IN FEET. AREA( I ) IN SO. INCHES: CRM0M(I) IN INCHES*M CONVERTED TO FOOT UNITS IN ROUTINE WRITE ( IUNIT,230) WRITE ( IUNIT,240) READ IN JOINT DATA AND COMPUTE NO. OF DEGREES OF FREEDOM NU»1 DO 50 I" 1 ,NRJ READ (5 .250) JN , ND( 1 ,I ) , ND(2 . I ) . ND (3 . I ) . X ( I ) , Y ( I ) DO 40 K=1,3 IF(ND(K, I ) - 1 ) 30. 10.20 ND(K,I ) =NU NU=NU+1 GO TO 40 JNN=ND(K,I) ND(K,I)=ND(K,JNN) GO TO 40 CONTINUE ND(K,I ) =0 CONTINUE PRINT JOINT DATA WRITE ( IUNIT.260) I, X ( I ) , Y ( I ) . ND (1 . I ) . ND (2 . I ) , ND(3. I ) CONTINUE NU=NU- 1 WRITE ( IUNIT.270) WRITE ( IUNIT,280) WRITE ( IUNIT,290) READ IN MEMBER DATA AND COMPUTE THE HALF BANDWIDTH (NB) HALF BANDWIOTH=MAX DEGREE OF FREEDOM-MIN DEGREE OF FREEDOM •»• 1 NB=0 DO 190 MBR=1,NRM READ (5 .300) MN,JNL(MBR),JNG(MBR),KL(MBR),KG(MBR), 1 AREA(MBR), CRMOM(MBR).AV(MBR),BMCAP (MBR), 2 EXTL(MBR).EXTG(MBR) IF DAMAGE RATIOS ARE LESS THAN ONE SET EQUAL TO ONE >1 CD 2067 2068 C 2069 2070 2071 2072 -2073 2074 2075 2076 2077 C 2078 2079 2080 2081 C 2082 C 2083 C 2084 C 2085 2086 2087 60 2088 2089 C 2090 70 2091 C 2092 2093 C 2094. C 2095 C 2096 2097 80 2098 2099 C 2100 C 2101 2102 2103 90 2104 C 2105 C 2106 2107 C 2108 C 2109 21 10 2111 2112 2113 2114 2115 C 2116 C DAMRAT(MBR) = 1 .0 COMPUTE MEMBER LENGTH (DM)=LENGTH BETWEEN JOINTS-RIGID EXTENSIONS JL=JNL(MBR) JG=JNG(MBR ) XM(MBR)=X(JG)-X(JL) YM(MBR)=Y(JG)-Y(JL) DM(MBR)=SQRT((XM(MBR))**2+(YM(MBR))++2) EXTSUM=EXTL(MBR)+EXTG(MBR) XM(MBR)=XM(MBR)*(1.0-EXTSUM/DM(MBR)) YM(MBR ) =YM(MBR)*(1 .0-EXTSUM/DM(MBR)) RESET NEGATIVE VALUES OF ZERO TO ZERO 01.AND.YM(MBR).LT.0.01) YM(MBR)=0.0 AND.XM(MBR).LT.0.01) XM(MBR)=0.0 IF(YM(MBR).GT IF(XM(MBR).GT 01 DM(MBR)=DM(MBR)-EXTSUM CHECK FOR NEGATIVE LENGTHS OF MEMBER (PROBABLY CAUSED BY INCORRECT USE OF MEMBER EXTENSIONS) IF(DM(MBR).GT.0.0) GO TO 70 WRITE(7,60) MBR FORMATC ' . / / / ' PROGRAM HALTED:ZERO OR -VE LENGTH FOR MEMBER',16) STOP CONTINUE YLEN=YM(MBR) PRINT ERROR MESSAGE IF ATTEMPT TO HAVE RIGID EXTENSIONS ON VERTICAL MEMBERS. IF (EXTSUM.NE.O.O.AND.YLEN.GT.0.2) WRITE(7,80) I FORMATC ' , 'ERROR-HAVE END EXTENSIONS ON NON-HORIZONTAL 1 MEMBER N O . ' , 1 3 ) PRINT ERROR MESSAGE IF ATTEMPT TO HAVE RIGIND EXTENSIONS ON A NON F IX -F IX TYPE MEMBER KLSUM=KL(MBR )+KG(MBR) IF(EXTSUM.NE.0.O.AND.KLSUM.NE.2) WRITE(7.90) MBR FORMAT(' ' , 'ERROR-HAVE RIGID EXTENSIONS ON HINGED MEMBER' .14) GIVE MEMBERS INITIAL ELASTIC DAMPING SDAMP(MBR)=0.02 ASSIGN MEMBER DEGREES OF FREEDOM NP(1,MBR)=ND(1.JL) NP(2,MBR)=ND(2,JL) NP(3,MBR)=ND(3,JL) NP(4,MBR)=ND(1.JG) NP(5,MBR)=ND(2,JG) NP(6,MBR)=ND(3,JG) DETERMINE THE HIGHEST DEGREE OF FREEDOM FOR EACH MEMBER STORING THE RESULT IN 'MAX' VD 2 117 2118 C 21 19 2120 2121 100 2122 1 10 2123 120 2124 C 2125 C 2126 C 2127 C 2128 C 2129 2130 c 2131 2132 2133 130 2134 140 2135 150 2136 160 2137 C 2138 2139 2 140 170 2 141 180 2142 C 2143 C 2144 C 2145 2146 2 147 2148 2149 C 2150 2151 2152 2153 C 2 154 190 2155 C 2 156 c 2157 c 2158 2159 2160 c 2161 2162 200 2163 C 2164. 2165 210 2166 C MAX=0 DO 120 K= 1 .6 IF(NP(K,MBR)-MAX) 110.110,100 MAX=NP(K,MBR) CONTINUE CONTINUE DETERMINE THE MINIMUM DEGREE OF FREEDOM FOR EACH MEMBER,NOTE THAT FOR STRUCTURES WITH GREATER THAN 330 JOINTS INITIAL VALUE OF MIN WILL HAVE TO BE INCREASED FROM ITS PRESTENT POINT OF 1000. MIN=1000 00 160 K=1,6 IF(NP(K,MBR)) 150.150,130 IF(NP(K,MBR)-MIN) 140.150.150 MIN=NP(K,MBR) CONTINUE CONTINUE NBB=MAX-MIN+1 IF(NBB-NB) 180,180.170 NB=NBB CONTINUE PRINT MEMBER DATA AND CONVERT TO FOOT UNITS. WRITE ( IUNIT.310) MBR.JNL(MBR).JNG(MBR).EXTL(MBR).DM(MBR) 1 EXTG(MBR),XM(MBR),YM(MBR), 2 AREA(MBR).CRMOM(MBR) .AV(MBR),BMCAP(MBR),KL(MBR), 3 KG(MBR) AREA(MBR) = AREA(MBR )/ 144.0 AV(MBR ) = AV(MBR)/144.0 CRMOM(MBR)=CRMOM(MBR)/20736.0 CONTINUE PRINT THE NO. OF DEGREES OF FREEDOM AND THE HALF BANDWIDTH WRITE ( IUNIT,320) NU WRITE ( IUNIT,330) NB OUTPUT THE ASSIGNED DEGREES OF FREEDOM. WRITE(IUNIT,200) FORMAT(' ' , ' MEMBER NP1 NP2 NP3 NP4 NP5 NP6 ' ) DO 210 MEMBR=1,NRM WRITE(IUNIT,220) MEMBR,(NP(IVAR,MEMBR),IVAR=1,6) CO o 2167 220 2168 C 2169 C 2170 2171 230 2172 240 2173 2174 250 2175 260 2176 270 2177 280 2178 2179 2180 290 2181 2182 300 2183 310 2184 . 320 2185 330 2186 3001 C 3002 3003 C 3004 3005 3006 C 3007 C = = 3008 C 3009 C 3010 C 301 1 C 3012 C 3013 C 3014 C 3015 c 3016 c 3017 c 3018 c 3019 c 3020 c 3021 c 3022 3023 3024 3025 3026 3027 3028 3029 3030 c FORMATC ' . 2X . I 4 .2X .6 I4 ) RETURN F O R M A T ( ' J O I N T DATA ' ) FORMAT(/7X. ' J N ' ,3X, ' X ( FEET ) 1 2X , 'NDR ' ) F0RMAT(4 I5.2F10.5) FORMAT(' FORMAT( ' -FORMAT(/ ' 1 5X 2 3X FORMAT(' 1 3X ,3X. ' Y ( F E E T ) ' ,4X, 'NDX' ,2X, 'NDY ' . 5 X . 1 4 , 2 F 1 0 . 3 , 2 X . 3 I 5 ) ,'MEMBER DATA' ) MN JNL JNG EXTL LENGTH ' AREA I(CRACKED ) A V . 4 X ' K L ' , 1X, ' KG ' ) .19X. ' (FEET) ' .29X, ' ( S O . I N ) ' -..,'( SQ . IN ) ' , 3X , 'CAPACITY ' ) F 0 R M A T ( 5 I 5 , F 8 . 2 . F 1 2 . 3 . 2 F 1 0 . 3 , 2 F 6 . 3 ) FORMAT(' ' , I 3 , 2 I 4 , F 7 . 3 , F 9 . 4 , F 7 . 3 , 2 F 9 . 4 , F 8 . 1 , F 1 2 . 1 FORMAT( ' - ' , ' ' NO . OF DEGREES OF FREEDOM. OF STRUCTURE FORMAT(/ ' HALF BANDWIDTH OF STIFFNESS MATRIX END EXTG XM(FT) , 'MOMENT' , , 3X. ' ( I N * * 4 ) ' , F8 Y M ( F T ) ' . . 3 . F 1 0 . 2 . 2 I 3 ) , 15) 15) SUBROUTINE BUILD(NU.NB.XM.YM,DM,NP,AREA.CRMOM.AV,E.G,DAMRAT, 1 KL,KG.NRM.S. IDIM.EXTL,EXTG) THIS SUBROUTINE WORKS IN DOUBLE PRECISION THIS SUBROUTINE CALCULATES THE STIFFNESS MATRIX OF EACH MEMBER AND ADDS IT INTO THE STRUCTURE STIFFNESS MATRIX. THE FINAL STIFFNESS MATRIX S IS RETURNED. THIS SUBROUTINE IS SIMILAR TO ONE THAT WOULD BE USED IN NORMAL FRAME ANALYSIS. DIFFERENCES INCLUDE USING CRACKED MOMENT OF INERTIA INSTEAD OF THE GROSS SECTION. DAMAGE RATIOS ARE USED AND FLEXTURAL STIFFNESSES MODIFIED ACCORDING TO THESE RATIOS. IDIM IS THE DIMENSIONING SIZE OF THE STRUCTURE STIFFNESS MATRIX INTERNAL FOOT UNITS FOR STIFFNESS MATRIX REAL*8 SM(21),S( IDIM) DIMENSION XM(NRM), YM(NRM), DM(NRM), NP(6.NRM), AREA(NRM). 1 CRMOM(NRM), AV(NRM). DAMRAT(NRM). KL(NRM). KG(NRM) DIMENSION EXTL(NRM), EXTG(NRM) REAL*8 RF,GMOD.CMOMI,DRAT I,F ,H REAL *8 LONE,LONEX.LONEY.LTWO,LTWOX.LTWOY.AVI REAL *8 YMI,DMI,DM2.XM2,YM2,XMI,ARE AI,EMOD.XM2F,YM2F,XMYMF REAL*8 DBLE CO 3031 C ZERO STRUCTURE STIFFNESS MATRIX 3032 C 3033 DO 10 1= 1 , IDIM 3034 S(I)=O.ODOO 3035 10 CONTINUE 3036 C REASSIGN YOUNGS MODULUS TO DOUBLE PRECISION VARIABLE EMOD 3037 C 3038 EMOD=DBLE(E) 3039 GMOD=DBLE(G) 3040 C 3041 C BEGIN MEMBER LOOP 3042 C 3043 DO 200 1=1.NRM 3044 C 3045 C ZERO MEMBER STIFFNESS NATRIX 3046 C 3047 DO 20 J=1 .21 3048 SM(J)=O.ODOO 3049 •20 CONTINUE 3050 C ASSIGN MEMBER PROPERTIES TO DOUBLE PRECESION VARIABLES 3051 C 3052 C 3053 LONE=DBLE(EXTL( I ) ) 3054 LTWO=DBLE(EXTG(I)) 3055 YMI=DBLE(YM(I)) 3056 DMI=DBLE(DM(I ) ) .3057 XMI=DBLE(XM(I )) 3058 AREAI=DBLE(AREA(I)) 3059 CMOMI=DBLE(CRMOM(I)) . 3060 DRATI=DBLE(DAMRAT(I ) ) 3061 AVI=AV(I) 3062 DM2=DMI*DMI 3063 XM2=XMI*XMI 3064 YM2 = YMI* YMI 3065 XMYM=XMI*YMI 3066 F = AREA I * EMOD/(DMI*DM2) 3067 H=O.ODOO 3068 C SHEAR DEFLECTIONS ARE IGNORED WHENEVER G OR AV IS ZERO. 3069 IF(AV( I ) .EO.O.O.OR.G.EO.O. ) GO TO 30 3070 H=12.ODOO*EM0D*CM0MI/(AVI*GM0D*DM2) 307 1 30 XM2F=XM2*F 3072 YM2F = YM2 * F 3073 XMYMF = XMYM* F 3074 C F I LL IN PIN-PIN SECTION OF MEMBER STIFFNESS MATRIX 3075 C 3076 C 3077 SM(1)=XM2F 3078 SM(2)=XMYMF 3079 SM(4)=-XM?F 3080 SM(5)=-XMYMF 3081 SM(7)=YM2F 3082 SM(9)=-XMYMF 3083 SM(10)=-YM2F 3084 SM(16)=XM2F 3085 SM(17)=XMYMF 3086 SM(19)=YM2F 3087 I F (KL ( I ) +KG( I ) - 1 ) 100,40.50 3088 C 3089 C VALUES OF F CALCULATED HERE DIFFER FROM STANDARD BUILD SUBROUTINE 3090 C BY DEVIDING BY THE DAMAGE RATIOS. 309 1 3092 c 40 F=3.0DOO*EMOD*CMOMI/(DM2*DM2*DMI*(1.ODOO+H/4.ODOO))/DRATI 3093 GO TO 60 3094 50 F=12.0DOO*EMOD*CMOMI/(DM2+DM2 + DMI*(1.ODOO+H ) )/DRAT I 3095 C RF IS A FACTOR COMMON TO THE ENTIRE MATRIX FOR ADDITION OF STIFFNESS 3096 C DUE TO RIGID BEAM END EXTENSIONS. 3097 RF=12.ODOO*EM0D*CM0MI/(DM2 *DM2)/DRAT I 3098 C F ILL IN TERMS WHICH ARE COMMON TO PIN-FIX,F IX-P IN,AND 3099 C 3100 C F IX -F IX MEMBERS 3101 C 3102 60 XM2F=XM2*F 3103 YM2F = YM2 * F 3104 XMYMF = XMYM* F 3105 DM2F =DM2 *F 3106 LONEY = LONE *YMI*RF 3107 LONEX=LONE*XMI*RF 3108 LTWOY=LTWO*YMI*RF 3109 LTWOX=LTWO*XMI*RF 31 10 C 3111 SM(1)=SM(1)+YM2F 3112 SM(2)=SM(2)-XMYMF 3113 SM(4)=SM(4)-YM2F 3114 SM(5)=SM(5)+XMYMF 3115 SM(7)=SM(7)+XM2F 3116 SM(9)-SM(9)+XMYMF 3117 SM(10)=SM(10)-XM2F 31 18 SM(16)=SM(16)+YM2F 3119 SM(17)=SM(17)-XMYMF 3120 SM(19)=SM( 19) + XM2F 3121 I F ( K L ( I ) - K G ( I ) ) 70 .80 ,90 3122 C 3123 C F ILL IN REMAINING PIN-FIX TERMS 3124 . C 3125 70 SM(6)=-YMI*DM2F 3126 SM(11)=XMI*DM2F 3127 SM(18)=-SM(6) 3128 SM(20)=-SM(11) 3129 SM(21)=DM2*DM2F 3 130 GO TO 100 3131 C 3132 C 3133 C 3134 80 3135 3136 3137 3138 3139 3140 3141 3142 3143 3144 3145 C 3146 3147 3148 3149 3150 3151 ' 3152 3153 3154 3155 3156 3157 3158 C 3159 C 3160 C 3161 90-3162 3163 3164 3 165 3166 100 3167 C 3168 C 3169 C 3170 C 3171 3172 C 3173 3174 3175 1 10 3176 c 3177 3178 3179 120 3 180 130 FILL IN REMAINING FIX-FIX TERMS .0D00+H)/12.0000 .0D00-H)/12.ODOO SM(3)=-YMI*DM2F*0.5D00 SM(6)=SM(3) SM(8)=XMI*DM2F*0.5D00 SM(11)=SM(8) SM(12 )=DM2*DM2F*(4, SM(13)=-SM(3) SM(14)=-SM(8) SM(15)=DM2*DM2F*(2 SM(18)=-SM(3) SM(20)=-SM(8) SM(21)=SM(12) ADD IN TERMS FOR RIGID END EXTENSIONS. SM(3)=SM(3)-(L0NEY) SM(6)=SM(6)-(LTW0Y) SM(8)=SM(8)+L0NEX SM(11)=SM( 1 1 ) + LTWOX SM(12)=SM(12)+(L0NE+DMI*(DMI+L0NE)*RF) SM(13)=SM(13)+L0NEY SM( 14) = SM( 14 )-LONEX SM(15)=SM(15)+((LONE*LTWO*DMI ) + (DM2*(LONE +LTWO)/2.ODOO))*RF SM(18)=SM(18)+LTW0Y SM(20)=SM(20)-LTWOX SM(21)=SM(21)+(DM2*LTW0+(DMI*(LTW0*LTW0)))*RF GO TO lOO FILL IN REMAINING FIX-PIN TERMS SM(3)=-YMI*DM2F SM(8)=XMI+DM2F SM(12)=DM2*DM2F SM(13)=-SM(3) SM( 14) = -SM(8) CONTINUE ADD THE MEMBER STIFFNESS MATRIX SM INTO THE STRUCTURE STIFFNESS MATRIX S. NB 1 =NB-1 DO 190 0=1.6 IF(NP(J,I)) 190.190.110 J1=(d-1)*(12-J)/2 DO 180 L=J.6 IF(NP(L.I)) 180,180.120 IF(NP(J.I )-NP(L.I)) 150.130.160 IF(L - d ) 140.150.140 CO 3181 140 3182 3183 3184 3185 150 3186 3187 160 3188 170 3189 3190 180 3191 C 3192 190 3193 C 3194 200 3195 C 3196 3197 4001 C 4002 4003 C 4004 4005 c 4006 c= = 4007 c 4008 c 4009 c 4010 c 401 1 c 4012 c 4013 c 4014 c 4015 c 4016 c 4017 c 4018 4019 c 4020 c 4021 c 4022 4023 4024 4025 4026 4027 c 4028 c 4029 c 4030 4031 4032 10 4033 c K « ( N P ( L , I ) - 1 ) * N B 1 + N P ( 0 . 1 ) N=J1+L S(K)=S(K)+2.0D00*SM(N) GO TO 180 K = ( N P ( J . I ) - 1)*NB1+NP(L.I ) GO TO 170 K=(NP(L, I ) -1)*NB1+NP(d, I ) N=J1+L S(K)=S(K)+SM(N) CONTINUE CONTINUE CONTINUE RETURN END SUBROUTINE MASS(NU,ND.AMASS.IUNIT.NRd.NMASS.MDOF) THIS SUBROUTINE SETS UP THE MASS MATRIX ND(J,I)=DEGREES OF FREEDOM OF I TH JOINT WTX,WTY,WTR=X-MASS,Y-MASS,ROT.MASS IN FORCE UNITS(KIPS OR IN-K.IPS) AMASS(I)=MASS MATRIX, I IS THE DEGREE OF FREEDOM OF APPLIED MASS NMASS=NO.OF MASS POINTS MASSES ARE LUMPED AT NODES. THE MASS MATRIX IS DIAGONAL I ZED. DIMENSION N D O . N R J ) . MD0F(5O). AMASS(NU) READ IN NO. OF NODES WITH MASS READ (5 ,90) NMASS WRITE ( IUNIT.100) WRITE ( IUNIT.110) NMASS WRITE ( IUNIT.120) WRITE ( IUNIT,130) ZERO MASS MATRIX DO 10 1=1,NU AMASS( I ) '0 . CONTINUE 00 O l 4034 C 4035 C 4036 4037 4038 4039 4040 404 1 4042 4043 4044 20 4045 4046 30 4047 4048 40 4049 50 4050 C 4051 C 4052 C 4053 4054 4055 • C 4056 4057 4058 4059 4060 4061 4062 60 4063 C 4064 C 4065 70 4066 80 4067 4068 90 4069 100 4070 1 10 4071 120 4072 130 4073 140 4074 150 4075 5001 C 5002 5003 C 5004 5005 5006 C 5007 5008 C READ IN X-MASS,Y-MASS AND ROT. MASS (IN UNITS OF WEIGHT) DO 50 1=1,NMASS READ ( 5 , 1 4 0 ) J N . WTX. WTY. WTR WRITE ( IUNIT ,150 ) J N . WTX, WTY, WTR N1=ND(1,JN) N2=ND(2,JN) N3=ND(3,JN) I F ( N 1 . E O . O ) GO TO 20 AMASS(N1)=AMASS(N1 ) + (WTX/32.2 ) I F ( N 2 . E Q . 0 ) GO TO 30 AMASS(N2)=AMASS(N2)+(WTY/32.2) I F ( N S . E Q . O ) GO TO 40 AMASS(N3)=AMASS(N3)+(WTR/32.2) CONTINUE CONTINUE OUTPUT THE DEGREES OF FREEDOM WITH MASS AND ASSIGNED MASS. JCNT=1 WRITE( IUNIT,70) DO 60 ID0F=1,NU RMASS = AMASS(IDOF ) I F ( R M A S S . E O . O . O ) GO TO 60 MDOF(JCNT)=IDOF WRITE( IUNIT,80) JCNT.MDOF(JCNT).RMASS JCNT=JCNT+1 CONTINUE F O R M A T ( ' - ' , ' M A S S FORMAT( ' ' .2X . 13 RETURN FORMAT(15) F 0 R M A T ( / / / 1 1 0 ( ' • F O R M A T ( ' - ' , ' N O F 0 R M A T ( / 7 X , ' J N FORMAT( ' ' , 12X F0RMAT( I5 ,3F1O FORMA T ( ' ' , 5X . END NO 3X DOF ' 13.9X 2X. F10 ASSIGNED 5) MASS (KIP + S E C * " 2 / F T ) ' ) ' ' )) . OF NODES WITH M A S S ' . ' = ' , 3 X , ' X - M A S S ' , 4 X , ' Y - M A S S ' , ' ( K I P S ) ' , 4X , ' ( K I P S ) ' ,2X, .0) I 4 . 3 F 1 0 . 3 ) ' , 15) , 2 X , ' R O T . M A S S ' ' ( I N - K I P S ) ' ) SUBROUTINE E IGEN(NU,NB,S , ID IM .AMASS.EVAL.EVEC.NMODES. IUNIT . 1 ISPEC.AMAX. ICOUNT.MDOF, INELAS) I—' CO cn 5009 C 5010 C THIS SUBROUTINE COMPUTES A SPECIFIED NO. OF NATURAL FREQUENCIES 5011 C AND ASSOCIATED MODE SHAPES 5012 C 5013 C NU=NO. OF DEGREES OF FREEDOM 5014 C NB=HALF BANDWIDTH 5015 C N M 0 0 £ S * N 0 . OF MODE SHAPES TO BE COMPUTED 5016 C IF NMODES IS ZERO OR IS GREATER THAN THE NUMBER OF STRUCTURE 5017 C MASSES THEN NMODES WILL BE ASSIGNED THE NUMBER OF STRUCTURE 5018 C MASSES. 5019 C AMASS(I ) =MASS MATRIX MCOUNT=NUMBER OF NONZERO MASSES 5020 C S( I )=STIFFNESS MATRIX STORED BY COLUMNS 502 1 C E V A L ( I ) » N A T U R A L FREQUENCIES 5022 C E V E C ( I . J ) =MODE SHAPES 5023 C 5024 REAL*8 DVEC(300, 10) ,DVAL( 10) .CMASS(300).SD(2000) 5025 REAL*8 S(IDIM) 5026 DIMENSION AMASS(NU). EVAL(NMODES), EVEC(50,NMODES), 5027 1 MOOF(50) 5028 REAL*8 DBLE 5029 C 5030 C ZERO DUMMY MASS MATRIX CMASS 5031 DO 10 ITRY=1,100 5032 10 CMASS(ITRY)=0.0 5033 C 5034 C DEBUG ON-OFF SWITCH FOLLOWS. 5035 IOFF=0 5036 I0N=1 5037 IDEBUG=ION 5038 C 5039 C COMPUTE THE NUMBER OF NONZERO MASS MATRIX ENTRIES 5040 C 504 1 MC0UNT=0 5042 C 5043 DO 20 1 = 1 ,NU 5044 CMASS(I)=DBLE(AMASS(I)) 5045 IF(AMASS(I ) .EQ.O. ) GO TO 20 5046 MC0UNT=MC0UNT+1 5047 20 CONTINUE 5048 C 5049 IF(NMODES.GT.MCOUNT ) NMODES = MCOUNT 5050 IF(NMODES.EQ.O) NMODES=MCOUNT 5051 IF( IUNIT.EQ.6.AND. ICOUNT.GT.25) GO TO 30 5052 WRITE ( IUNIT,160) NMODES 5053 30 CONTINUE 5054 C 5055 C CALL PRITZ TO COMPUTE EIGENVALUES AND EIGENVECTORS 5056 C CREATE A DUPLICATE STRUCTURE MATRIX (SD) (DESTROYED IN PRITZ) 5057 ' C 5058 C CALCULATE USEFUL LENGTH OF STIFFNESS MATRIX (LSTM) J"* - 1 5059 LSTM=(NU)*NB • 5060 C 5061 DO 40 1 = 1 , LSTM 5062 SD( I )=S( I ) 5063 40 CONTINUE 5064 C SET CONVERGENCE CRITERIA FOR PRITZ. MAKE NEGATIVE IF RESIDUALS NOT 5065 C DESIRED. 5066 C 5067 D E P S » 1 . 0 D - 1 0 5068 IF ( IUN IT .NE.7 ) DEPS = (- 1.ODOO)*DEPS 5069 C 5070 C 5071 C CALL EIGENVALUE FINDING ROUTINE 5072 CALL PRITZ(SD.CMASS,NU.NB.1.OVAL,DVEC.300.NMODES,DEPS.&140) 5073 C 5074 C CONVERT MATRICES TO SINGLE PRECESION 5075 C 5076 C PRINT EIGENVALUES AND EIGENVECTORS(MODE SHAPES) 5077 C EIGENVALUES (EVAL) ARE THE VALUES OF OMEGA SQUARED. 5078 C 5079 C SKIP PRINTING INTERMEDIATE DATA AFTER SEVERAL CYCLES. 5080 IF ( ICOUNT.GT.3.AND. IUNIT.EQ.6) GO TO 70 5081 WRITE ( IUNIT.170) 5082 WRITE ( IUNIT,210) NMODES 5083 WRITE ( IUN IT .230) ( I . I = 1.NMODES) 5084 C 5085 DO 60 I D « 1 , N U 5086 WRITE(IUNIT,50) ID , (DVEC ( ID ,J ) , 0=1,NMODES) 5087 50 FORMAT(' ' , 1 3 , 1 O F 11.6) 5088 60 CONTINUE 5089 C 5090 70 CONTINUE 5091 C ALSO CONVERT MEMBERS OF EVAL FROM OMEGA SQUARED TO OMEGA 5092 C 5093 C CONVERT EIGENVECTORS TO ONLY INCLUDE DEGREES OF FREDOM WITH MASS 5094 C ASSIGNED TO THEM 5095 • DO 90 MAS=1,MC0UNT 5096 IVAR=MDOF(MAS ) 5097 C 5098 DO 80 M0D=1,NMODES 5099 EVEC(MAS,MOD)=SNGL(DVEC(IVAR.MOD)) 5100 80 CONTINUE 5101 C 5102 90 CONTINUE 5103 C 5104 IF( ICOUNT.EQ.O) WRITE(7,900) 5105 900 FORMAT( ' ' , / / ' INITIAL ELASTIC PERIOD ' ) 5106 IF( ICOUNT.EQ.O) IUNIT=7 5107 WRITE ( IUNIT,180) 5108 WRITE ( IUNIT,190) 5 1 0 9 C 51 10 C 5 1 1 1 5 1 1 2 1 0 0 51 13 C 5 1 1 4 51 15 51 16 51 17 51 18 5 1 1 9 5 1 2 0 5 1 2 1 5 1 2 2 5 1 2 3 1 10 5 1 2 4 5 1 2 5 C 5 1 2 6 5 1 2 7 5 1 2 8 5 1 2 9 C 5 1 3 0 5 1 3 1 5 1 3 2 1 2 0 5 1 3 3 C 5 1 3 4 1 3 0 5 1 3 5 C 5 1 3 6 5 1 3 7 140 5 1 3 8 1 5 0 5 1 3 9 1 6 0 5 1 4 0 1 7 0 5 1 4 1 . 1 8 0 5 1 4 2 5 1 4 3 1 9 0 5 1 4 4 5 1 4 5 2 0 0 5 1 4 6 2 1 0 5 1 4 7 5 1 4 8 2 2 0 5 1 4 9 5 1 5 0 2 3 0 '5151 ' 2 4 0 5 1 5 2 2 5 0 5 1 5 3 5 1 5 4 6 0 0 1 C 6 0 0 2 6 0 0 3 C 6 0 0 4 C O M P U T E F R E Q U E N C I E S AND P E R I O D S DO 1 0 0 J U I C E = 1 , N M O D E S E V A L ( J U I C E ) = S N G L ( D V A L ( J U I C E ) ) DO 110 1 = 1 . N M O D E S E V A L 1 = E V A L ( I ) E V A L ( I ) = SQRT ( E V A L 1 ) W N = E V A L ( I ) P E R I 0 D = 6 . 2 8 3 1 5 3 / W N F R E Q = 1 / P E R I O D I F ( I C O U N T . G T . 2 5 . A N D . I U N I T . E Q . 6 ) GO TO 110 C A L L S P E C T R ( I S P E C , 0 . 0 2 , P E R I O D . A M A X , S A , W N , S A E S N D , S V B N D , S D B N D ) W R I T E ( I U N I T . 2 0 0 ) I , E V A L 1 , E V A L ( I ) , F R E Q . P E R I O D , SA C O N T I N U E I F ( I C O U N T . E Q . O . A N D . I N E L A S . N E . 0 ) I U N I T = 6 I F ( I C O U N T . G T . 5 . A N D . I U N I T . E Q . 6 ) GO TO 130 W R I T E ( I U N I T , 2 2 0 ) NMODES W R I T E ( I U N I T , 2 4 0 ) ( I , 1 = 1 . N M O D E S ) DO 120 I = 1 , M C 0 U N T W R I T E ( I U N I T . 5 0 ) I , ( E V E C ( I , J ) , J = 1 , N M O D E S ) C O N T I N U E C O N T I N U E R E T U R N W R I T E ( I U N I T , 1 5 0 ) F O R M A T ( ' ' , ' C R A P O U T I N P R I T Z ' ) F O R M A T ( ' - ' , ' N O . OF M O D E S TO B E A N A L I Z E D F 0 R M A T ( / / / 1 10( ' * ' ) )• F 0 R M A T ( / 5 X ; ' M O D E S ' , 4 X , ' E I G E N V A L U E S ' , 6 X . ' N A T U R A L F R E Q U E N C I E S ' 1 1 3 X , ' P E R I O D S ' , 1 0 X , ' S A ' ) F O R M A T ( ' ' , 3 0 X . ' ' ( R A D / S E C ) ' , 5 X , ' ( C Y C S / S E C ) ' , 8 X . ' ( S E C S ) ' . 1 4 X , ' ( 2 P E R C E N T D A M P I N G ) ' ) F O R M A T C ' , 5 X , I 5 . 5 F 1 5 . 4 ) F O R M A T C ' T O T A L MODE S H A P E S C O R R E S P O N D I N G TO F I R S T ' , 1 5 , 1 1 X , ' F R E Q U E N C I E S ' ) F O R M A T ( / ' M A S S MODE S H A P E S C O R R E S P O N D I N G TO F I R S T ' . 1 5 , 1 X , 1 ' F R E Q U E N C I E S ' ) F O R M A T ( / ' DOF ' . 1 8 , 9 1 1 1 ) F O R M A T ( / ' M A S S ' , 1 0 1 1 1 ) F O R M A T ( ' ' . 1 0 F 1 2 . 6 ) R E T U R N END S U B R O U T I N E M O D 3 ( I C O U N T . I S P E C . N R J . N R M , N U , N B , N M O D E S . S , I D I M , N D , N P . X M , . I 5 / / / 1 1 0 C * ' ) / / / ) 00 VD 6005 6006 6007 6008 C 6009 6010 C 601 1 c 6012 c 6013 c 6014 c 6015 6016 c 6017 6018 6019 6020 602 1 6022 6023 6024 6025 c 6026 c 6027 c 6028 10 6029 20 6030 6031 c 6032 6033 6034 6035 6036 6037 30 6038 C 6039 6040 C 6041 6042 6043 6044 C 6045 C 6046 C 6047 6048 6049 6050 60 6051 C 6052 6053 70 6054 C YM DM,AREA.AV,CRMOM,DAMRAT,KL,KG.SDAMP.BMCAP.E.G.AMASS, EVEC EVAL, AMAX,IS IGN,IUNIT,BETA,BMERR,IFLAG.EXTL.EXTG, BETAM DAMB.DVARY, INELAS.DAMP IN.NPRINT) SUBSTITUTE STRUCTURE METHOD FOR RETROFIT THIS SUBROUTINE COMPUTES JOINT DISPLACEMENTS AND MEMBER FORCES NEW DAMAGE RATIOS WILL BE CALCULATED AND RETURNED. REAL*8 S( ID IM),DF(100) DIMENSION ND(3,NRJ) , NP(6.NRM), XM(NRM), YM(NRM), DM(NRM), 1 AREA(NRM), CRMOM(NRM). DAMRAT(NRM), KL(NRM), KG(NRM) 2 AMASS(NU),SUMDAM(100),EVEC(50.NMODES), EVAL(NMODES ) . 3 SDAMP(NRM), AV(NRM), ZETA( IO) , PI( IOO) DIMENSION BMASS(50), ID0F(50) , ALPHA(20), RMS(7.100). 1 F(300),EXTL(NRM),EXTG(NRM). D(6) DIMENSION BMCAP(NRM).DAMB(NRM), BETAM(NMODES) REAL*8 DRATIO,DET CALCULATE THE MODAL PARTICIPATION FACTOR JJ= TEMPORARY VARIBLE USED IN NEXT LOOP ONLY. FORMAT(' CONTINUE JJ=1 ICOUNT=', 13) DO 30 JD0F=1,NU IF(AMASS(JDOF).EO.O.) GO TO 30 BMASS(JJ)=AMASS(JDOF) IDOF(JJ)=JOOF JJ=JJ+1 CONTINUE MC0UNT=JJ-1 DO 70 MODEY=1.NMODES AMT=0. AMB=0. EIGEN VALUES ARE STORED AS FOLLOWS EVEC(MASS NO.,MODE NO.) DO 60 JAM=1.MCOUNT AMT=AMT+BMASS(JAM)*EVEC(JAM,MODEY) AMB=AMB+BMASS(JAM)•((EVEC(JAM,MODEY))**2) CONTINUE ALPHA(MODE Y)=AMT/AMB CONTINUE VD O 6055 6056 6057 C 6058 6059 6060 80 6061 C 6062 90 6063 C 6064 C 6065 C 6066 C 6067 C 6068 C 6069 6070 C 6071 6072 C 6073 C S 6074 6075 6076 6077 6078 6079 C 6080 C 6081 C 6082 6083 C 6084 6085 6086 100 6087 C 6088 1 10 6089 c 6090 c 6091 6092 6093 c 6094 6095 ' c 6096 6097 6098 120 6099 C 6100 130 6101 140 6102 150 6103 C 6104 C IF ( ICOUNT.GT.25.AND. IUNIT.EQ.6) GO TO 90 WRITE ( IUNIT,810) DO 80 MODEX=1,NMODES WRITE ( IUNIT.820) MODEX, ALPHA(MODEX) CONTINUE CONTINUE WHEN KK=1, MODAL FORCES FOR UNDAMPED SUBSTITUTE STRUCTURE ARE COMPUTED. THEY ARE USED TO COMPUTE 'SMEARED' DAMPING VALUES, WHICH ARE USED TO CALCULATE THE ACTUAL RESPONSE OF THE SUBSTITUTE STRUCTURE INDEX=1 DO 800 KK=1,2 INTPR-1 IF (KK.EQ. I ) INTPR=0 IF( IFLAG.EQ.O.OR.NPRINT.EQ.O) INTPR=0 IF( ICOUNT.EQ.O) GO TO 780 SHRMS=0. ZERO RMS(J. I ) 00 110 1=1,100 DO 100 J=1.7 RMS(J, I )=0. CONTINUE CONTINUE OUTPUT THE SMEARED DAMPING RATIOS (FOR DAMPED CASES) IF ( IUNIT.E0.6.AND. ICOUNT.GT.25) GO TO 130 I F ( K K . L T . 2 ) GO TO 130 WRITE(IUNIT,140) DO 120 MODEC=1,NMODES WRITE!IUNIT,150) MODEC.BETAM(MODEC) CONTINUE CONTINUE FORMAT( ' - ' , 'MODE ' ,2X , ' SMEARED DAMPING RATIO ' ) FORMAT(' ' . I X . I3 .7X.F 10.5) CALCULATE THE MODAL DISPLACEMENT VECTOR 6105 6106 6107 6108 6109 61 10 6111 6112 6113 6114 6115 61 16 6117 6118 61 19 6120 6121 6122 6123 6124 6125 6126 6127 6128 6129 6130 6131 6132 6133 6134 6135 6136 6137 6138 6139 6140 6141 6142 6143 6144 6145 6146 6147 6148 6149 6150 6151 6152 6153 6154 C FIRST ZERO TEMPORARY VARIABLE ZETA USED IN CALCULATING DAMPING. DO 160 MODEd=1.NMODES ZETA(MODEJ)=0.0 160 CONTINUE C DO 570 MODEN=1.NMODES C L IST MEMBER FORCES IF DOING ELASTIC ANLYSIS ONLY C I F ( I N T P R . E O . O ) GO TO 180 IF(NPRINT.LT.MODEN) GO TO 180 WRITE( IUNIT.840) WRITEUUNIT . 170) MODEN FORMAT(' ' . ' M O D E N U M B E R ' , 1 3 , ' MODAL FORCES AND DISPLACEMENTS 170 1') WRITEUUNIT .830 ) CONTINUE 180 C C CHECK IF MODAL PARTICIPATION FACTOR IS ZERO C IF ALPHA IS ZERO MODAL FORCES AND DISPLACEMENTS WILL ALSO BE ZERO C IF (ALPHA(MODEN) .NE.0 .0 ) GO TO 200 WRITE( IUNIT,190) 190 FORMAT(/ ' MODAL PARTICIPATION .FORCES AND DISPL.=ZERO' ) GO TO 570 CONTINUE 200 C C C C C C 210 C C C C c c c c c c CALCULATE NATURAL PERIOD AND CALL SPECTA TN=6.28318531/(EVAL(MODEN)) WN=EVAL(MODEN) DAMP=BETAM(MODEN) CALL SPECTR(I SPEC.DAMP,TN,AMAX,SA,WN,SABND,SVBND.SDBND) ZERO LOAD VECTOR DO 210 J=1,NU F (d )=0 . CONTINUE FF=0. COMPUTE LOAD VECTOR FAC=SA*ALPHA(MODEN)*32.2 NOTE THAT AS THESE FORCES ARE BEING GENERATED FROM A LATERAL EXCITATION SPECTRUM THAT ONLY 'X MASSES' SHOULD BE USED. IN OTHER WORDS LATERAL ACCELERATION SHOULD NOT CAUSE NON HORIZONTAL INERTIA FORCES DIRECTLY. VD 6155 6156 6157 6158 6159 6160 6161 6162 6163 6164 6165 6166 6167 6168 6169 6170 6171 6172 6173 6174 6175 6176 6177 6178 6179 6180 6181 6182 6183 6184 6185 6186 6187 6188 6189 6190 6191 6192 6193 6194 6195 6196 6197 6198 6199 6200 6201 6202 6203 6204 DO 220 J=1.MCOUNT I1=ID0F(d) F(I 1)=EVEC(J,MODEN)*FAC*AMASS(I 1) FF-FF + F( I 1 ) 220 CONTINUE C C CALCULATE THE BASE SHEAR C IF(KK.NE.2) GO TO 230 SHRMS=SHRMS+FF**2 IF(MODEN.LT.NMODES) GO TO 230 SHRMS=SQRT(5HRMS) 230 CONTINUE C CONVERT SINGLE PRECISION FORCE MATRIX TO DOUBLE PRECISION DO 240 IFREE=1.100 DF(IFREE)=DBLE(F(IFREE)) 240 CONTINUE C C COMPUTE DEFLECTIONS BY CALLING SUBROUTINE DFBAND LSTM«NU*NB C NOTE THAT NO SOLUTION IMPROVING ITERATIONS WILL BE PERFORMED. C SCALING WILL BE PERFORMED TO IMPROVE THE SOLUTION WHEN NSCALE.NE.O C NSCALE=1 C DRATI0=1.OD-16 CALL DFBAND(S.DF.NU.NB,INDEX,DRAT10,DET,JEXP.NSCALE) C DFBAND EXITS WITH F BEING THE DISPLACEMENT MATRIX C CONVERT DOUBLE PRECISION DISPLACEMENTS TO SINGLE PRECISION DO 250 JFREE=1,100 F(JFREE)=SNGL(DF(JFREE)) 250 CONTINUE C INDEX=INDEX+1 C C C CALCULATE RMS DISPLACEMENTS. DO 290 JNT= 1,NRJ 0X=O. DY=0. DR = 0. N1=ND( 1 ,vJNT) N2=ND(2.JNT) N3=ND(3,JNT) IF(N1.EO.O) GO TO 260 DX=F(N1) RMS(1,JNT)=RMS(1,JNT)+DX**2 260 CONTINUE IF(N2.EQ 0) GO TO 270 co 6205 DY=F(N2) 6206 RMS(2,JNT) =RMS(2,JNT)+DY * + 2 6207 270 CONTINUE 6208 IF(N3.EQ.O) GO TO 280 6209 DR=F(N3) 6210 RMS(3,JNT)=RMS(3.JNT)+DR**2 621 1 280 CONTINUE 6212 IF( INTPR.EQ.O) GO TO 290 6213 • IF(NPRINT.LT.MODEN) GO TO 290 6214 C OUTPUT MODAL DEFLECTIONS FOR REQUIRED MODES 6215 IF(N1.EQ.O) DX=0.0 6216 IF(N2.EQ.O) DY=0.0 6217 IF(N3.EQ.O) DR=0.0 6218 WRITE(IUNIT,860) JNT,DX.DY,DR 6219 C 6220 290 CONTINUE 622 1 6222 C AT THIS STAGE RMS(1,JNT)=(RMS DISPLACEMENT)SQUARED OF X DISPLACEMENT 6223 C COMPUTE MEMBER FORCES USING DISPLACEMENTS FROM INDIVIDUAL MODES 6224 C NOTE THAT 'ENGINEERING' SIGN CONVENTION IS USED HERE. 6225 6226 SIGPI=0. 6227 C INSERT MODAL MEMBER FORCE HEADINGS BEFORE STARTING MEMBER FORCE LOOP 6228 C IF(INTPR.NE.O.AND.NPRINT.GE.MODEN) WRITE(IUNIT,300) 6229 6230 ' 300 FORMAT(' ' , / 8 X , ' M N ' , 1 0 X , ' A X I A L ' , ' 1 0 X , ' S H E A R ' , 1 1 X , ' B M L ' , 1 2 X , 6231 1 'BMG ' , /21X. 'K IPS ' , 12X. 'K IPS ' , 2 ( 9 X . ' ( K - F T ) ' ) ) 6232 C 6233 6234 C 6235 DO 460 I=1,NRM 6236 C 6237 6238 C 623,9 • C XL AND YL =X AND Y COMPONENTS OF MEMBER LENGTH RESPECTIVELY 6240 C DL IS TRUE LENGTH OF MEMBER 6241 C BMG IS THE BENDING MOMENT AT GREATER JOINT NO. END OF MEMBER. 6242 C BML IS THE BENDING MOMENT AT THE LESSER JOINT NO. END. 6243 C 6244 XL=XM(I) 6245 YL=YM(I) 6246 DL = DM(I ) 6247 AVI=AV(I ) 6248 C 6249 DO 340 MEMDOF = 1 ,6 6250 N1=NP(MEMDOF,I) 6251 IF(N1) 320.320,310 6252 310 D(MEMDOF) = F(N1 ) 6253 GO TO 330 6254 320 D(MEMDOF)=0. 6255 330 6256 340 6257 C 6258 C 6259 C 6260 6261 6262 6263 350 6264 6265 6266 6267 360 6268 C 6269 6270 6271 370 6272 380 6273 6274 C 6275 6276 C 6277 C 6278 C 6279 .6280 6281 6282 390 6283 C 6284 C t 6285 C 6286 6287 6288 6289 6290 6291 6292 6293 6294 C 6295 400 6296 6297 6298 6299 C 6300 4 10 6301 6302 6303 6304 C CONTINUE CONTINUE MODIFY END DISPLACEMENTS FOR HORIZONTAL MEMBERS WITH END EXTENSIONS FORMULA ONLY WORKS FOR HORIZONTAL MEMBERS N3=NP(3,I ) I F (N3.E0 .0 ) GO TO 350 D(2)=D(2) + (F(N3) ) + EXTL( I ) CONTINUE N6=NP(6.I) IF (N6.EQ.O) GO TO 360 D (5 )=D(5 ) - (F (N6 ) ) *EXTG( I ) CONTINUE PRINT OUT MEMBER END DISPLACEMENTS FOR .DEBUG, IF ( ICOUNT.GT.1) GO TO 380 WRITE(6.370) I.(D(M).M=1,6) FORMAT(' ' , 'MEMB NO. = ' , 1 3 , ' D I SPL= ' ,6F10 .5 ) CONTINUE AXIAL=(AREA(I ) *E /DL**2 ) * (D (4 ) *XL+D(5 ) *YL -D ( 1 ) * X L - 0 ( 2 ) * Y L ) EISI=ASSUMEO STIFFNESS IN SUBSTITUTE FRAME ELEMENT I EISI=CRMOM(I)*E/DAMRAT(I) GFACT=FACTOR TO COMPUTE EFFECT OF SHEAR DEFL. ON MEMBER FORCES GFACT=0.0 IMPLIES THAT NO SHEAR DEFLECTION INCLUDED. GFACT>=0.0 IF (AV I .EO.O.O.OR.G.EO.O.O) GO TO 390 GFACT=12.0*E IS I / (AV I t G*DL*DL) CONTINUE ASSIGN DISPLACEMENTS TO THEIR RESPECTIVE MEMBER DEGREES OF FREEDOM CHECK FOR PIN-PIN MEMBERS I F ( K L ( I ) . E O . 0 .AND. KG( I ) .EO.O) GO TO 420 DELT= ( (D (5 ) -D (2 ) ) *XL+ (D (1 ) -D (4 ) ) *YL ) /DL BML=(2 .0*E I S I / (DL* (1 .O+GFACT) ) ) * ( (3•0*DELT/DL) 1 - ( D ( 6 ) * ( 1 . O - G F A C T / 2 . 0 ) ) - ( 2 . 0 + D ( 3 ) * ( 1 . O + G F A C T / 4 . 0 ) ) ) SHEAR=(6.0*E IS I / (DL*DL) )*((D( 3 )+D(6 ) - (2 .0*DELT/DL ) ) / ( 1.0+ 1 GFACT ) ) BMG=BML+SHEAR*DL I F ( K L ( I ) - K G ( I ) ) 400.430,410 ADJUST PIN-FIX MEMBER FORCES. BMG=BMG+BML*(1.O-GFACT/2.O)/(2.O*(1.O+GFACT/4.0)) SHEAR=SHEAR+1.5*BML/(DL) BML=0. GO TO 430 ADJUST F IX-PIN MEMBER FORCES. BML=BML+BMG*(1.O-GFACT/2.0)/ (2.0*(1.O+GFACT/4.0)) SHEAR=SHEAR-1 .5 *BMG/(DL) BMG=0. GO TO 430 F ILL IN MEMBER FORCES FOR PIN-PIN MEMBERS. VO 6305 420 6306 6307 6308 430 6309 C . 63J0 C 631 1 C 6312 6313 6314 6315 440 6316 C 6317 C P 6318 6319 6320 450 632 1 C 6322 C 6323 C 6324 6325 6326 6327 6328 460 6329 C 6330 C 6331 C 6332 6333 . C 6334 C 6335 6336 6337 . 6338 470 6339 C 6340 C 634 1 6342 ' C 6343 C 1 6344 C 6345 . 6346 6347 480 6348 6349 6350 . C 635 1 6352 490 6353 6354 BMG=0. BML =0. SHEAR=0. CONTINUE COMPUTE THE RELATIVE FLEXURAL STRAIN ENERGY IF (KK.NE.1) GO TO 440 PI ( I)=(BML**2+BMG**2+BML*BMG)*DL/(6.*EISI ) SIGPI=SIGPI+PI( I) CONTINUE IF( INTPR.EQ.O) GO TO 450 IF(NPRINT.GE.MODEN) WRITE(IUNIT,900) I,AX IAL.SHEAR,BML,BMG CONTINUE ACCUMULATE ABSOLUTE SUM AND RMS SUM RMS(4,I) = RMS(4,1)+AXIAL+*2 RMS(5.I)=RMS(5,I)+SHEAR**2 RMS(6,I)=RMS(6.I)+BML**2 RMS(7.I)=RMS(7,I)+BMG* *2 CONTINUE COMPUTE THE SMEARED DAMPING FOR EACH MODE IF (KK.NE.1) GO TO 540 SUMDAM= THE PRODUCT OF MEMBER STRAIN ENERGY*MEMBER DAMPING. DO 470 1=1,NRM SUMDAM(I)=PI(I)* SDAMP(I) ZETA(MODEN)=ZETA(MODEN)+SUMDAM(I) CONTINUE BETAM=SMEARED SUBSTITUTE DAMPING FOR THE M TH MODE. BETAM(MODEN)=ZETA(MODEN)/SIGPI PRINT DAMPING INFORMATION FROM FINAL ITERATION. IF ( IFLAG.NE. 1 ) GO TO 520 WRITE(6,480)SIGPI,MODEN.BETAM(MODEN) FORMAT( ' ' . ' T O T A L FLEX. STR . ENERGY=' ,F10.3 12.3X, 'SMEARED DAMPING FACTOR=' . F7 .5 ) WRITE(6,490) , 3X, 'MODE NUMBER' DO 510 MEMB=1,NRM FORMATC ' , 'MEMBER NO. ' ,3X, 'STRAIN ENERGY',3X, 'MEMBER DAMPING'. 3X, 'MEMBER DAMP ING* STRAIN ENERGY ) WRITE(6.500) MEMB.PI(MEMB) ,SDAMP(MEMB).SUMDAM(MEMB) VD 6355 6356 6357 6358 6359 6360 6361 6362 6363 6364 6365 6366 6367 6368 6369 6370 6371 6372 6373 6374 6375 6376 6377 6378 6379 6380 6381 6382 6383 6384 6385 6386 6387 6388 6389 6390 6391 6392 6393 6394 6395 6396 6397 6398 6399 6400 6401 6402 6403 6404 500 510 C 520 C 530 FORMAT(' CONTINUE CONTINUE ' ,4X ,12. 10X ,E10 .3 .8X .E10 .3 , 13X.F 1 1 .7) 1G ' ) IF (S IGP I .EO.O.O) WRITE(IUNIT,530) FORMATC ' , 'ERROR-ZERO DEVIDE WHILE CALCULATING SMEARED DAMPIN CONTINUE 540 C C COMPUTE AND WRITE MODAL CONTRIBUTION FACTOR CONMOD = SA *ALPHA(MODEN) WRITE(IUNIT.550) MODEN, CONMOD 550 FORMAT( ' ' , 'MODE ' .I 3,3X. ' CONTRIBUTION FACTOR=' , F8 .5 ) C OUTPUT SPECTRAL ACCELERATION. C IF(INTPR.EO.0.OR.MODEN.GT.NPRINT) GO TO 570 WRITE(IUNIT,560) DAMP, TN.SA FORMAT( ' ' , 'DAMPING=' , F 6 . 4 , ' P E R I O D ' ' . F 6 . 4 , ' SEC. SA= ' . F5 .3 ) CONTINUE 560 570 C C 580 C C C 590 C C 600 C 610 C IF (KK.EQ.1.AND. ICOUNT.LT.2) GO TO 580 IF (KK.EO.1) GO TO 800 CONTINUE PRINT RMS DISPLACEMENTS AND FORCES IF( IUNIT.EO.6.AND. ICOUNT.GT.25) GO TO 590 WRITE ( IUNIT,840) OUTPUT THE COUNT OF ENTRANCES INTO M0D3 WRITE(6.10) ICOUNT WRITE ( IUNIT,850) WRITE ( IUNIT,830) CONTINUE CONVERT SQUARE OF RMS DISPLACEMENTS TO RMS DISPLACEMENTS. DO 610 1=1,NRJ DO 600 J=1,3 SCRAT=RMS(J,I) RMS (<J , I ) = SQRT(SCRAT) CONTINUE IF( ICOUNT.GT.25.AND. IUNIT.EQ.6) GO TO 610 WRITE ( IUNIT.860) I, (RMS(J. I ) .J= 1 . 3 ) CONTINUE MODIFY DAMAGE RATIOS 6406 IF( IC0UNT.GT.25.AND. IUNIT.EQ.6) GO TO 630 6407 WRITE ( IUNIT,870) 6 4 0 8 - WRITE ( IUNIT,880) SHRMS 6409 620 CONTINUE 64 10 IF( ICOUNT.GT.25.AND. IUNIT.EO.6) GO TO 630 6411 WRITE ( IUNIT,890) 6412 630 CONTINUE 6413 C 6414 C ISIGN IS A COUNT OF THE NUMBER OF MEMBERS WITH WHICH THE RATIO OF 64 15 C THE ABSOLUTE VALUE OF THE DIFFERENCE BETWEEN THE LARGEST RMS 6416 • C BENDING MOMENT AND ULTIMATE MOMENT TO ULTIMATE MOMENT IS IN 6417 C EXCESS OF 'BMERR'. 6418 C ISIGN IS INITIALIZED TO ZERO HERE. 6419 C 6420 ISIGN^O 6421 C 6422 DO 770 MEM=1,NRM 6423 C FIND THE BIGGEST OF THE SQUARE OF THE RMS BENDING MOMENT(=BIG) 6424 IF(RMS(6,MEM)-RMS(7,MEM))640,640,650 6425 G40 BIG=RMS(7,MEM) 6426 GO TO 660 6427 650 BIG=RMS(6,MEM) 6428 660 CONTINUE 6429 IF (KK.EQ.1)G0 TO 750 6430 C TAKE SQUARE ROOT TO GIVE RMS BENDING MOMENT. 6431 BMBIG=SQRT(BIG) 6432 C 6433 C SET DAMOLD AS THE DAMAGE RATIO IN THE ( I - 2 ) T H ITERATION 6434 C DAMB AS THE DAMAGE RATIO IN THE ( I - 1 )TH ITERATION. 6435 C 6436 DAMOLD=DAMB(MEM) 6437 DAMB(MEM)=DAMRAT(MEM) 6438 C CALCULATE NEW OAMAGE RATIO 6439 • C 6440 DAMRAT(MEM)=BMBIG/BMCAP(MEM)*DAMRAT(MEM) 6441 C DO NOT ALTER DAMAGE RATIOS OF LESS THAN UNITY. AS THEY ARE RESET AT 6442 C END OF ROUTINE. 6443 IF(DAMRAT(MEM).LT.1.0) GO TO 730 6444 C 6445 C 6446 C CONVERGENCE SPEEDING ROUTINE FOLLOWS. 6447 IF(DAMRAT(MEM).LT.5.0) DERROR=(DAMRAT(MEM)-DAMB(MEM))/10.0 6448 IF(DAMRAT(MEM).GE.5.0) DERROR=(DAMRAT(MEM)-DAMB(MEM))/DAMRAT( 6449 1 MEM) 6450 ADIFF=ABS(DERROR) 6451 IF(ADIFF.GT.OVARY) DVARY = DERROR 6452 C 6453 DAMDIF = DAMRAT(MEM)-DAMB(MEM) 645'4 C CO 6455 6456 670 6457 6458 680 6459 6460 690 6461 6462 700 6463 6464 710 6465 6466 6467 720 6468 6469 730 6470 6471 C 6472 C D 6473 C I 6474 C 6475 6476 6477 6478 740 6479 C 6480 6481 C 6482 750 6483 C 6484 C 6485 6486 6487 760 6488 C 6489 C 6490 6491 6492 6493 770 6494 C 6495 6496 780 6497 C 6498 C 6499 6500 6501 790 6502 . C 6503 6504 IF(DAMOLD-DAMB(MEM)) 670,730,700 CONTINUE IF(DAMDIF) 690 ,730 ,680 DAMRAT(MEM)=DAMRAT(MEM)+BETA*(DAMDIF) GO TO 730 DAMRAT(MEM)=DAMRAT(MEM)-BETA*(DAMDIF ) GO TO 7 30 CONTINUE I F (DAMDIF) 720 .730.710 CONTINUE DAMRAT(MEM)'DAMRAT(MEM)-BETA*(DAMDIF) GO TO 730 CONTINUE DAMRAT(MEM)'DAMRAT(MEM)+BETA+(DAMDIF) CONTINUE IF(DAMRAT(MEM).LT. 1.0.AND.I FLAG:NE. 1) DAMRAT(MEM) = 1.0 AMAGE RATIOS CANNOT BE LESS THAN 1.0 N LAST ITERATION SKIP RESETTING DAMAGE- RATIOS LESS THAN UNITY IF(DAMRAT(MEM).LE.1.0) GO TO 740 CHECK = ABS(BMBIG-BMCAP(MEM))/BMCAP(MEM) IF(CHECK.GT.BMERR) ISIGN'ISIGN+1 CONTINUE COMPUTE DAMPING VALUE FOR THE MEMBER SDAMP(MEM)=0.02+0.2*(1.-1./SORT(DAMRAT(MEM))) CONTINUE CONVERT SQUARE OF RMS AXIAL, SHEAR AND MOMENT TO RMS VALUE. DO 760 J ' 4 , 7 RMS(J.MEM)=SQRT(RMS(J.MEM)) CONTINUE OUTPUT THE RMS AXIAL SHEAR AND MOMENT. IF( ICOUNT.GT.25.AND. IUNIT.EQ.6) GO TO 770 WRITE ( IUNIT.900) MEM,(RMS(J.MEM),J=4,7),BMCAP(MEM), 1 DAMRAT(MEM) CONTINUE GO TO 800 CONTINUE SET DAMPING RATIOS TO 'APPROPIATE' VALUES FOR INITIAL TRIAL. DO 790 MODEA=1,NMODES BETAM(MODEA )=DAMPIN CONTINUE IC0UNT=IC0UNT+1 IF( ICOUNT.GT.25.AND. IUNIT.EQ.6) GO TO 800 » VD VD 6505 6506 6507 6508 6509 6510 651 1 6512 6513 6514 6515 6516 6517 6518 6519 6520 6521 6522 6523 7001 7002 7003 . 7004 7005 7006 7007 7008 7009 7010 701 1 7012 7013 7014 7015 7016 7017 7018 7019 7020 7021 7022 7023 7024 7025 7026 7027 7028 7029 7030 7031 WRITE ( IUNIT,840) 800 CONTINUE C IC0UNT=IC0UNT+1 RETURN 810 FORMAT( ' - ' , 'MODAL PARTICIPATION FACTOR ' . / ) 820 FORMAT(' ' ,5X, 'MODE' . 1 5 . 5 X , F 1 0 . 5 , 5 X , F 1 0 . 5 ) 830 FORMAT( ' - ' , 7 X , 'JOINT NO. ' , 10X. ' X - D l S P ( F T ) ' . 10X, ' Y - D I S P ( F T ) ' ,7X , 1 'ROTATION(RAD)' ) 840 F O R M A T ( ' - ' , 1 1 0 ( ' * ' ) ) 850 FORMAT( ' - ' , ' ROOT MEAN SQUARE DISPLACEMENTS') 860 FORMAT( ' ' ,6X,I 10.3F20.4 ) 870 F O R M A T ( ' - ' , ' R O O T MEAN SQUARE FORCES' ) 880 FORMAT(1H0,7X, 'RSS BASE SHEAR = ' , F 1 0 . 3 , ' K I P S ' ) 890 F O R M A T ( ' - ' , 8 X , 'MN' , 10X, ' A X I A L ' , 10X, ' SHEAR ' , 11X, ' BML ' , 12X, 'BMG' 1 9X, 'MOMENT' .10X. ' DAMAGE ' /21X . ' K I PS ' . 12X . ' K I P S ' , 2 ( 9 X , 2 ' ( K - F T ) ' ) , 8X, 'CAPACITY' ,9X, 'RATIO' ) 900 FORMATC ' , 5X . I5 .6F 15. 3) END C C = = = == = == = = = == = »» = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = C SUBROUTINE SPECTRfI SPEC,DAMP,TN,AMAX,SA.WN,SABND.SVBND,SDBND) C C=============oc==================================================== c c C ISPEC=1 IF SPECTRUM A IS USED C =2 IF SPECTRUM B IS USED C =3 IF SPECTRUM C IS USEO C =4 IF NBC SPECTRUM IS USED C DAMP=DAMPING FACTOR (FRACTION OF CRITICAL DAMPING) C TN ^NATURAL PERIOD IN SECONDS C AMAX = MAXIMUM GROUND ACCELERATION (FRACTION OF G) C SA ^RESPONSE ACCELERATION (FRACTION OF G) C WN =NATURAL FREQUENCY IN RADIANS PER SECOND. C I F ( I S P E C . E 0 . 2 ) GO TO 10 IF ( I SPEC.EQ.3 ) GO TO 60 IF ( I SPEC.EQ.4 ) GO TO 100 C C SPECTRUM A C I F ( T N . L T . 0 . 1 5 ) SA=25.*AMAX*TN I F (TN .GE .0 .15 .AND. T N . L T . 0 . 4 ) SA=3.75*AMAX I F ( T N . G T . 0 . 4 ) SA=1.5*AMAX/TN GO TO 90 C C SPECTRUM B C O o • 7032 10 CONTINUE 7033 I F ( T N . L T . 0 . 1875 ) GO TO 20 7034 I F (TN .LT .0 .53333333 ) GO TO 30 7035 I F (TN .LT .1 .6666S67 ) GO TO 40 7036 I F ( T N . L T . 1 .81666667) GO TO 50 7037 SA=2.*AMAX/(TN-0.75) 7038 GO TO 90 7039 20 SA=20.*AMAX*TN 7040 GO TO 90 7041 30 SA=3.75*AMAX 7042 GO TO 90 7043 40 SA=2.*AMAX/TN 7044 GO TO 90 7045 50 SA=1.875*AMAX 7046 GO TO 90 7047 C 7048 C SPECTRUM C 7049 C 7050 60 CONTINUE 7051 I F ( T N . L T . 0 . 1 5 ) GO TO 70 7052 I F (TN .LT .0 .38333333 ) GO TO 80 7053 SA=0.5*AMAX/(TN-0.25) 7054 GO TO 90 7055 70 SA=25.*AMAX*TN 7056 GO TO 90 7057 80 SA=3.75*AMAX 7058 . 90 CONTINUE 7059 SA=SA*8./(6.+100.*DAMP) '7060 RETURN 7061 C 7062 C NBC SPECTRUM • 7063 C 7064 100 CONTINUE 7065 SV=40.0*AMAX 7066 SD=32.0*AMAX 7067 SACC= 1 .O'AMAX 7068 C PRINT OUT A CAUTION NOTE SHOULD DAMPING BE LESS THAN 0.5% 7069 IF (DAMP.LT.0.005) WRITE(7.110) 7070 110 FORMAT(' ' , 'CAUTION-DAMPING LESS THAN 0 . 5 % ' ) 7071 C 7072 C COMPUTE MULTIPLICATION FACTOR FOR ACCELERATION AT DESIRED DAMPING 7073 IF (DAMP.LE.0.02) AML=4.2+((0.02-DAMP)/O.015)•1.6 7074 IF(DAMP.GT. .02.AND.DAMP.LE. .05 )AML = 3.0+(( .05-DAMP)/.03)•1 .2 7075 ' IF (DAMP.GT.0.05.AND.DAMP.LE.O.1)AML '2 .2+( (0 .1 -DAMP)/O.05)*0.8 7076 IF(DAMP.GT.0.10) AML=1.0 + ( (1 .00 -DAMP) /0 .90 ) * 1.2 7077 C 7078 C COMPUTE MULTIPLICATION FACTOR FOR VELOCITY AT DESIRED DAMPING. 7079 IF (DAMP.LE.0.02) VML = 2.5+( (0 .02-DAMP)/O.015)*0.8 7080 IF(DAMP .GT. .02.AND.DAMP.LE. .05)VML = 2.0+(( .05-DAMP) / .03) *0 .5 7081 IF(DAMP GT..05.AND.DAMP.LF.0.1)VML=1.7+( (O.1-DAMP)/O.05)*0.3 P 7082 7083 7084 7085 7086 7087 7088 7089 7090 7091 7092 7093 7094 7095 7096 7097 7098 7099 7100 7101 7102 7103 7 104 7105 7106 7107 7 108 7109 7 1 10 8001 8002 8003 8004 8005 8006 8007 8008 8009 8010 801 1 8012 8013 8014 8015 8016 8017 8018 8019 8020 . 802 1. IF(DAMP.GT.O.10) VML=1.0+((1.00-DAMP)/0.90)*0.7 C C C C COMPUTE MULTIPLICATION FACTOR FOR DISPLACEMENT AT DESIRED DAMPING. IF (DAMP.LE.0.02) DML=2.5+((0.02-DAMP)/O.015)*0.5 IF(DAMP.GT.0.02) DML=VML COMPUTE BOUNDS USING DAMPING FACTORS COMPUTED ALREADY SDBND=SD+DML SABND=SACC*AML SVBND=SV*VML COMPUTE WHICH IS THE APPROPIATE BOUND. CONVERT FROM IN/SEC**2 TO FRACTION OF G BY DEVI DING BY 386.4 SAATAP = SVBND*WN/386 . 4 IF(SAATAP.GT.SABND) SA=SABND IF(SAATAP.GT.SABND) GO TO 120 SDATCP=SVBND/WN IF(SDATCP.GT.SDBND) SA=SDBND*WN*WN/386.4 IF(SDATCP.GT.SDBND) GO TO 120 C C IF HAVE NOT YET GONE TO STEP 180 THEN NATURAL FREQUENCY LIES ON C VELOCITY BOUND. C SA=SVBND*WN/386.4 C SA IS RETURNED AS A FRACTION OF GRAVITY, G C 120 RETURN C END C C = C C C = C C C C c c c c SUBROUTINE SCHECK(S,NU.NB.IDIM.IUNIT,SRATIO) THIS SUBROUTINE CHECKS THAT ALL DIAGONAL STIFFNESS MATRIX ELEMENTS ARE POSITIVE NUMBERS GREATER THAN ZERO. IT ALSO DETERMINES THE RATIO BETWEEN THE LARGEST AND SMALLEST MEMBERS ON THE DIAGONAL THIS WILL GIVE SOME INDICATION AS TO THE CONDITIONING OF THE STIFFNESS MATRIX MATRIX REAL *8 S(IDIM) REAL*8 SMIN.SMAX.DIAG.RATIO THE STIFFNESS MATRIX IS STORED AS A COLUMN VECTOR. ONLY THE LOWER TRIANGLE ELEMENTS BEING STORED (BY COLUMNS) S (1 ) IS ON THE DIAGONAL AS IS S(1+NB),S(1+2*NB).ETC. THE to O to 8022 C NB IS THE HALF BANDWIDTH OF THE STIFFNESS MATRIX 8023 C INITIALIZE THE LARGEST AND SMALLEST VALUES OF DIAGONAL (SMAX.SMIN) 8024 C 8025 C 8026 SMIN=1.0D45 8027 SMAX=-1.ODOO 8028 C 8029 DO 50 ID0F=1,NU 8030 IELEM=((IDOF-1)*NB)+1 8031 DlAG=S(IELEM) 8032 C COMPUTE IF DIAGONAL ELEMENT IS ZERO OR NEGATIVE 8033 IF(DIAG.NE.0.ODOO) GO TO 20 8034 WRITE(7,10) IDOF 8035 10 FORMAT(III' PROGRAM HALTED-A ZERO IS ON THE DIAGONAL OF STIFFNE 8036 1SSMATRIX'.//'EXAMINE DEGREE OF FREEDOM '.14) 8037 STOP 8038 C 8039 20 CONTINUE 8040 IF(DIAG.GT.0.0) GO TO 40 8041 WRITE(7.30) IDOF 8042 30 FORMAT(///' PROGRAM HALTED-NEGATIVE ELEMENT ON DIAGONAL OF ', 8043 1 'STIFFNESS MATRIX'.//' EXAMINE DEGREE OF FREEDOM'.14) 8044 STOP 8045 40 CONTINUE 8046 C DETERMINE IF THE DIAGONAL ELEMENT UNDER EXAMINATION IS THE LARGEST OR 8047 C 8048 C SMALLEST OF THE DIAGONAL ELEMENTS. 8049 IF(DIAG.GT.SMAX) SMAX=DIAG 8050 IF(DIAG.LT.SMIN) SMIN=DIAG 8051 C 8052 50 CONTINUE 8053 C 8054 WRITE(IUNIT.60) 8055 60 FORMAT(/' ALL ELEMENTS OF MAIN DIAGONAL OF STIFFNESS MATRIX'. 8056 1 ' ARE POSITIVE DEFINITE') 8057 8058 c c COMPUTE AND PRINT RATIO OF LARGEST TO SMALLEST DIAGONAL ELEMENTS 8059 c 8060 RATIO=SMAX/SMIN 8061 SRATIO=SNGL(RATIO) 8062 WRITEUUNIT,70) SRATIO 8063 70 FORMAT( ' '..'RATIO OF LARGEST TO SMALLEST DIAGONAL STIFFNESS', 8064 1 'MATRIX ELEMENT IS',E10.3) 8065 C 8066 RETURN 8067 END E n d o f F i l e o CO 

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