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

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PSEUDO INE L A S T I C TORSIONAL SEISMIC ANALYSIS U T I L I Z I N G THE MODIFIED SUBSTITUTE STRUCTURE METHOD by KEN SAU KUEN TAM B . A . S c , The U n i v e r s i t y Of B r i t i s h C o l u m b i a , 1980 A THESIS SUBMITTED IN PARTIAL FULFILMENT OF THE REQUIREMENTS FOR THE DEGREE OF MASTER OF APPLIED SCIENCE i n THE FACULTY OF GRADUATE STUDIES ( D e p a r t m e n t 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 a s c o n f o r m i n g t o t h e r e q u i r e d s t a n d a r d THE UNIVERSITY OF B R I T I S H COLUMBIA A p r i l 1985 © KEN SAU KUEN TAM, 1985 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 of the requirements f o r an advanced degree at the U n i v e r s i t y of B r i t i s h Columbia, I agree that the 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 study. I f u r t h e r agree that p e r m i s s i o n f o r e x t e n s i v e copying of t h i s t h e s i s f o r s c h o l a r l y purposes may be granted by the Head of my Department or by h i s or her r e p r e s e n t a t i v e s . I t i s understood that copying or p u b l i c a t i o n of t h i s t h e s i s f o r f i n a n c i a l gain s h a l l not be allowed without my w r i t t e n p e r m i s s i o n . Department of C i v i l E n g i n e e r i n g The U n i v e r s i t y of B r i t i s h Columbia 2075 Wesbrook Place Vancouver, Canada V6T 1W5 Date: 26 A p r i l 1985 i i A b s t r a c t A computational procedure f o r the e v a l u a t i o n of i n e l a s t i c responses and d u c t i l i t y demands of three dimensional b u i l d i n g s s u b j e c t e d to multi-component earthquake e x c i t a t i o n s i s presented. 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 i s based on 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 procedure which i n v o l v e s a f i c t i t i o u s l i n e a r s t r u c t u r e whose s t i f f n e s s and damping c h a r a c t e r i s t i c s are r e l a t e d to the d u c t i l i t y demands. The b u i l d i n g i s i d e a l i z e d as an assembly of plane frames i n t e r c o n n e c t e d by r i g i d diaphragms. Coupling of i n d i v i d u a l frames through common columns i s taken i n t o account. It i s not necessary f o r a l l s t o r e y masses to l i e on a v e r t i c a l l i n e , so that s t r u c t u r e s with v a r y i n g f l o o r dimensions can be modelled. Two r e i n f o r c e d concrete s t r u c t u r e s of d i f f e r e n t types, moment r e s i s t i n g frames and coupled w a l l c o r e s , are t e s t e d with the method. The accuracy of the method f o r p r e d i c t i n g d u c t i l i t y demands and l a t e r a l d e f l e c t i o n s i s determined by comparison with averaged r e s u l t s o b t a i n e d from the time step a n a l y s i s program DRAIN-TABS. The proposed method f i l l s the v o i d between e l a s t i c modal a n a l y s i s which cannot provide i n f o r m a t i o n on i n e l a s t i c behaviour of the s t r u c t u r e , and f u l l s c a l e time step a n a l y s i s which i s complex and expensive to use. 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 DEDICATION x ACKNOWLEDGEMENTS x i CHAPTER 1 . INTRODUCTION 1 . 1 Background 1 1.2 Review of Previous Work 5 1.3 Purpose and Scope 7 1.4 Assumptions and L i m i t a t i o n s 10 2. STRUCTURAL IDEALIZATION 2.1 I n t r o d u c t i o n , 11 2.2 Inter-frame C o m p a t i b i l i t i e s 14 2.3 Condensed Frame S t i f f n e s s Matrix 16 2.4 Assembling and Condensing of S t r u c t u r e S t i f f n e s s Matrix 18 3. MODIFIED SUBSTITUTE STRUCTURE METHOD 3.1 I n t r o d u c t i o n '. 21 3.2 S u b s t i t u t e S t r u c t u r e Method 22 3.3 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 27 3.4 Convergence Routine 32 i v 3.5 Two Damage Ra t i o s Per Member 34 4. EARTHQUAKE ANALYSIS 4.1 I n t r o d u c t i o n .36 4.2 Dynamic E q u i l i b r i u m Equation 39 4.3 Mode Shapes and Frequencies 42 4.4 Spectrum A n a l y s i s 43 4.5 Complete Quadratic Combination Method 46 4.6 Multi-component Ground Motions 49 5. COMPUTER PROGRAM 5.1 Program Concepts 52 5.2 Program O r g a n i z a t i o n 53 5.3 Design Spectra 55 6. PROGRAM TESTING 6.1 T e s t i n g f o r E l a s t i c A n a l y s i s 57 6.1.1 Comparison With Another Modal A n a l y s i s Program - ETABS 57 6.1.2 Numerical Examples (a) F i v e - s t o r e y Frame S t r u c t u r e 58 (b) F i v e - s t o r e y Coupled Wall S t r u c t u r e ... 59 6.2 T e s t i n g f o r I n e l a s t i c A n a l y s i s 60 6.2.1 Assumptions f o r Comparison With A Time Step A n a l y s i s Program - DRAIN-TABS .... 61 6.2.2 Numerical Examples (a) F i v e - s t o r e y Frame S t r u c t u r e 64 (b) F i v e - s t o r e y Frame S t r u c t u r e With Four Times The Mass 66 (c) F i v e - s t o r e y Coupled Wall S t r u c t u r e ... 67 V (d) F i v e - s t o r e y C o u p l e d W a l l S t r u c t u r e W i t h F o u r T i m e s The Mass 68 6.2.3 C o s t s o f E x e c u t i o n 69 7. CONCLUSIONS 7 1 BIBLIOGRAPHY 114 APPENDIX A - P r o g r a m U s e r ' s M a n u a l 116 APPENDIX B - Sample I n p u t a nd O u t p u t 128 v i LIST OF TABLES Table Page 2 .1 Member S t i f f n e s s Matrix I n c l u d i n g R i g i d Ends .... 73 4 . 1 ( a ) A m p l i f i c a t i o n F a c t o r s f o r Ground Motion Bounds Recommended By Newmark 74 (b) A m p l i f i c a t i o n F a c t o r s for Ground Motion Bounds Recommended By NBCC 74 4.2 Basic Ground Motion Bounds 75 6 .1 E l a s t i c Modal A n a l y s i s R e s u l t s f o r F i v e - s t o r e y Frame B u i l d i n g ( F i g . 6 . 1 ) Subjected To Spectrum ' A' 76 6.2 E l a s t i c Modal A n a l y s i s R e s u l t s f o r F i v e - s t o r e y Coupled Wall B u i l d i n g ( F i g . 6.2) Subjected To Spectrum 'A' 77 6.3 Comparison of L a t e r a l Displacements of Frame 4 for F i v e - s t o r e y Frame S t r u c t u r e 78 6.4 Comparison of L a t e r a l Displacements of Frame 4 fo r F i v e - s t o r e y Frame S t r u c t u r e With Revised Mass 79 6.5 Comparison of L a t e r a l Displacements of Wall 4 fo r F i v e - s t o r e y Coupled Wall S t r u c t u r e ' 80 6.6 Comparison of L a t e r a l Displacements of Wall 4 fo r F i v e - s t o r e y Coupled Wall S t r u c t u r e With Revised Mass 81 6.7 Costs of E x e c u t i o n 82 v i i L I S T OF FIGURES F i g u r e Page 1.1 F l o o r P l a n o f The J . C. Penney B u i l d i n g i n A n c h o r a g e , A l a s k a , S h o w i n g The H i g h l y E c c e n t r i c S h e a r W a l l C o n f i g u r a t i o n 83 1.2 E a s t W a l l a n d N o r t h E a s t C o r n e r of The J . C. Penney B u i l d i n g , A f t e r The 1964 E a r t h q u a k e . T h i s Shows The C o m p l e t e C o l l a p s e o f The , S h e a r W a l l a n d P o r t i o n s o f The Roof and F l o o r s a t The N o r t h E a s t C o r n e r of The B u i l d i n g 84 2 . 1 ( a ) G r o s s Frame D e g r e e s of F r e e d o m 85 (b) C o n d e n s e d Frame D e g r e e s o f Freedom 85 ( c ) G r o s s S t r u c t u r e D e g r e e s o f Freedom 86 (d) C o n d e n s e d S t r u c t u r e D e g r e e s o f Freedom 86 2.2 P l a n V i e w o f n - t h F l o o r S h o w i n g Frame and and D i a p h r a g m H o r i z o n t a l D i s p l a c e m e n t s 87 3.1 P h y s i c a l I n t e r p r e t a t i o n o f Damage R a t i o 88 4.1 Newmark's I d e a l i z e d E l a s t i c D e s i g n S p e c t r u m ( 0 . 5 g Max. G r o u n d A c c e l e r a t i o n , 5% Damping) 89 4.2 P l o t o f C r o s s - m o d a l C o e f f i c i e n t s v s . R a t i o o f P e r i o d s ' 90 4.3 F i v e S t o r e y B u i l d i n g E x a m p l e 91 4.4 P e r i o d s a n d D i r e c t i o n s o f Mode Shapes 92 4.5 C o m p a r i s o n o f M o d a l C o m b i n a t i o n M e t h o d s 93 5.1 G e n e r a l C o n c e p t u a l O u t l i n e o f PITSA 94 5.2 P r o g r a m ( P I T S A ) O r g a n i z a t i o n 95 v i i i 5.3. Spectrum ' A' 96 5.4 A c c e l e r a t i o n Spectra of T a f t N69W and Spectrum 'A' 97 5.5 A c c e l e r a t i o n Spectra of T a f t S21W and Spectrum 'A* 98 5.6 A c c e l e r a t i o n Spectra of E l Centro EW and Spectrum 'A* 99 5.7 A c c e l e r a t i o n Spectra of E l Centro NS and Spectrum 'A' 100 5.8 Ground Motion Bound T r i p a r t i t e P l o t of Four Earthquakes Which Make Up Spectrum 'A' (0.5g) ... 101 6.1 Dimensions and P r o p e r t i e s of F i v e - s t o r e y Frame B u i l d i n g 102 6.2 Dimensions and P r o p e r t i e s of F i v e - s t o r e y Coupled Wall B u i l d i n g 103 6.3 Damage R a t i o s f o r F i v e - s t o r e y Frame S t r u c t u r e ... 104 6.3.1 Beam Damage R a t i o s f o r F i v e - s t o r e y Frame S t r u c t u r e 105 6.4 Damage R a t i o s f o r F i v e - s t o r e y Frame S t r u c t u r e With Revised Mass 106 6.4.1 Beam Damage R a t i o s f o r F i v e - s t o r e y Frame S t r u c t u r e With Revised Mass 107 6.5 Damage Ra t i o s f o r F i v e - s t o r e y Coupled Wall S t r u c t u r e 108 6.5.1 Beam Damage R a t i o s f o r F i v e - s t o r e y Coupled Wall S t r u c t u r e 109 6.6 Damage R a t i o s f o r F i v e - s t o r e y C o u p l e d W a l l S t r u c t u r e W i t h R e v i s e d Mass 110 6.6.1 Beam Damage R a t i o s f o r F i v e - s t o r e y C o u p l e d W a l l S t r u c t u r e W i t h R e v i s e d Mass 111 6.7 Damage R a t i o s f o r F i v e - S t o r e y . Frame S t r u c t u r e W i t h S t r o n g Beams and Weak Columns 112 6.7.1 Beam Damage R a t i o s f o r F i v e - S t o r e y Frame S t r u c t u r e W i t h S t r o n g Beams and Weak Columns .... 113 MY MOTHER MY WIFE STELLA MY SON NICHOLAS x i ACKNOWLEDGEMENT The author wishes to thank h i s three a d v i s o r s , Dr. D. L. Anderson, Dr. N. D. Nathan, and Dr. S. Cherry f o r t h e i r encouragement, support and c r i t i c a l comments. The author i s indebted to Dr. D. L. Anderson and Dr. N. D. Nathan f o r t h e i r c o n s c i e n t i o u s p r o o f r e a d i n g and f o r making extremely v a l u a b l e s u g g e s t i o n s . Thanks are a l s o due to Mr. John Stevens of the Computing Centre f o r h i s a s s i s t a n c e i n c o n v e r t i n g the program DRAIN-TABS. The f i n a n c i a l support of the N a t u r a l Sciences and En g i n e e r i n g Research C o u n c i l in the form of a Research A s s i s t a n t s h i p i s g r a t e f u l l y acknowledged. F i n a l l y t h i s t h e s i s would not have been completed without the c o o p e r a t i o n and advic e of f e l l o w graduate students, Frank Lam, and W i l l i a m Tong. 1 CHAPTER 1 INTRODUCTION 1.1 Bac kqround Current earthquake r e s i s t a n t design p r a c t i c e permits the s e l e c t i o n of a s t r u c t u r a l system in which the elements w i l l behave e l a s t i c a l l y under small or medium seismic d i s t u r b a n c e s but w i l l deform i n e l a s t i c a l l y , without c a t a s t r o p h i c c o l l a p s e , under major shocks. It i s r e l a t i v e l y easy to estimate the behaviour i f e l a s t i c response i s assumed, which probably r e s u l t s in s a t i s f a c t o r y behaviour under small shocks. However, i t i s d i f f i c u l t to p r e d i c t a c c u r a t e l y the response of a s t r u c t u r e in the i n e l a s t i c range; yet i n e l a s t i c behaviour i s i n e v i t a b l e in many s t r u c t u r e s s u bjected to strong earthquakes. Methods of dynamic response a n a l y s i s based on l i n e a r e l a s t i c assumptions cannot, in many s t r u c t u r e s , p rovide information on the i n e l a s t i c behaviour of the s t r u c t u r e , or on the d u c t i l i t y demands on the members. Present design techniques r e s u l t in some members y i e l d i n g before o t h e r s - under the a c t i o n of strong-motion earthquakes. If the design philosophy i s to be c o r r e c t l y a p p l i e d i t i s necessary to e s t a b l i s h which members w i l l y i e l d , and the magnitude of the corres p o n d i n g i n e l a s t i c deformations. The c a p a c i t i e s of these y i e l d e d members must be checked to 2 ensure that the d e s i r e d d u c t i l i t y can be achieved. High d u c t i l i t y i s the a b i l i t y of a b u i l d i n g to s u s t a i n l a r g e i n e l a s t i c d e f l e c t i o n s without f a i l u r e or c o l l a p s e . T h i s i s a very important c h a r a c t e r i s t i c of a b u i l d i n g s i n c e i t g r e a t l y reduces the f o r c e s that the s t r u c t u r e must r e s i s t due to earthquake e x c i t a t i o n . The b u i l d i n g i s set in motion by the earthquake and the d i s s i p a t i o n of the energy by lar g e i n e l a s t i c d e f l e c t i o n s of a d u c t i l e s t r u c t u r e i s an important part of the seismic d e s i g n . Three a l t e r n a t i v e s are a v a i l a b l e to estimate the design loads due to earthquake: ( i ) The e q u i v a l e n t s t a t i c l o a d i n g as suggested by the N a t i o n a l B u i l d i n g Code of Canada (NBCC); ( i i ) A dynamic a n a l y s i s based on the response spectrum technique; ( i i i ) A complete dynamic response a n a l y s i s to o b t a i n the time h i s t o r y response of the proposed s t r u c t u r e . The N a t i o n a l B u i l d i n g Code of Canada al l o w s earthquake loads to be approximated by s t a t i c l a t e r a l l o a d s . The magnitude of the loads depends on the seismic zone, fundamental p e r i o d , s o i l c o n d i t i o n , and usage of the b u i l d i n g . The d i s t r i b u t i o n of the l a t e r a l loads over the height of the b u i l d i n g approximates the f i r s t mode shape. The assumed response i s p r e d i c a t e d on the assumption that y i e l d i n g i s widespread and d i s t r i b u t e d u n i f o r m l y over the s t r u c t u r e : i e . , that there are no members that y i e l d f a r i n advance of the o t h e r s and are thus s u b j e c t to e x c e s s i v e 3 d u c t i l i t y demands. In more complex s t r u c t u r e s , a modal a n a l y s i s w i l l g i v e a more a c c u r a t e d i s t r i b u t i o n of t h e l o a d s and member f o r c e s . However, i n a p p l y i n g t h e r e s u l t s of su c h an a n a l y s i s t o d e s i g n , i t i s a g a i n assumed t h a t y i e l d i n g w i l l t a k e p l a c e more o r l e s s s i m u l t a n e o u s l y and t h a t t h e d u c t i l i t y demand on t h e y i e l d i n g members w i l l be e s s e n t i a l l y u n i f o r m . T h i s a s s u m p t i o n i s of d o u b t f u l v a l i d i t y i n many s t r u c t u r e s w i t h p r o n o u n c e d i r r e g u l a r i t i e s . A l s o t h e d e t e r m i n a t i o n of damping c o e f f i c i e n t s t o be us e d i n e l a s t i c modal a n a l y s i s i s most i m p o r t a n t and d i f f i c u l t . T h e r e a r e r e l a t i v e l y few a p p l i c a b l e t e s t d a t a t o s u p p o r t an a c c u r a t e e s t i m a t e o f t h e t r u e damping of a s t r u c t u r e . Most a v a i l a b l e t e s t r e s u l t s a r e b a s e d on v e r y s m a l l a m p l i t u d e d i s t o r t i o n s o r on compact t e s t s , and t h e r e s u l t s p r o b a b l y do not a c c u r a t e l y r e f l e c t t h e damping t h a t might be e x p e c t e d f o r t h e l a r g e a m p l i t u d e m o t i o n s a s s o c i a t e d w i t h a s e v e r e e a r t h q u a k e . A p o i n t o f t e n b e i n g o v e r l o o k i s t h a t even time s t e p d y n a m i c r e s p o n s e a n a l y s i s c a n n o t be e x p e c t e d t o p r e d i c t a c c u r a t e l y how a g i v e n b u i l d i n g on a g i v e n s i t e w i l l r e s p o n d t o ma j o r e a r t h q a k e s . Even w i t h r e c e n t a d v a n c e s i n s e i s m o l o g y , i t i s n o t p o s s i b l e t o p r e d i c t t h e c h a r a c t e r i s t i c s of t h e g r o u n d movement a t a g i v e n s i t e . A major drawback of t h e t i m e s t e p a n a l y s i s i s t h e need t o a n a l y z e f o r a s u i t e of e a r t h q u a k e s t o e n s u r e c o v e r i n g t h e c h a r a c t e r i s t i c s of t h e n o n - d e t e r m i n i s t i c f u t u r e e a r t h q u a k e . F i e l d o b s e r v a t i o n s f o l l o w i n g e a r t h q u a k e s have o f t e n 4 shown s t r u c t u r a l f a i l u r e due t o t o r s i o n a l m o t i o n . A c l a s s i c e x a m p l e o f what t o r s i o n c a n do t o a b u i l d i n g i s damage s u f f e r e d by t h e J.C. Penney B u i l d i n g i n A n c h o r a g e , A l a s k a , d u r i n g t h e 1964 e a r t h q a k e . T h i s f i v e s t o r y b u i l d i n g shown i n F i g . 1.1 and F i g . 1.2 was s o b a d l y damaged t h a t i t was d e m o l i s h e d a f t e r w a r d s . The m a i n r e a s o n f o r i t s d i s m a l p e r f o r m a n c e was t h e h i g h l y e c c e n t r i c s h e a r w a l l c o n f i g u r a t i o n . T o r s i o n a l r e s p o n s e s i n s t r u c t u r e s may a r i s e f r o m e c c e n t r i c i t y due t o a s y m m e t r i c a l l a y o u t , o r f r o m t h e r o t a t i o n a l c o mponent o f g r o u n d m o t i o n a b o u t a v e r t i c a l a x i s . B u i l d i n g s t h a t a r e n o t e c c e n t r i c d u r i n g e l a s t i c r e s p o n s e may become e c c e n t r i c d u r i n g n o n l i n e a r r e s p o n s e i f t h e s t r u c t u r a l e l e m e n t s on one s i d e y i e l d b e f o r e t h e o t h e r s i d e . F u r t h e r m o r e , a c c i d e n t a l t o r s i o n c a u s e d by u n c e r t a i n t i e s i n d e a d l o a d due t o v a r i a t i o n s i n w o r k m a n s h i p and m a t e r i a l , c r a c k i n g o f w a l l s , and s u b s e q u e n t a l t e r a t i o n s may s e v e r e l y a f f e c t t h e p e r f o r m a n c e o f t h e b u i l d i n g i n a e a r t h q u a k e . L a t e r a l and t o r s i o n a l m o t i o n s of b u i l d i n g s s u b j e c t d t o e a r t h q u a k e g r o u n d m o t i o n s may be c o u p l e d f.or any o f t h e a b o v e r e a s o n s . T h e r e f o r e , t h e r e i s a need f o r p r a c t i c a l , e f f i c i e n t , a n d s i m p l e t o u s e c o m p u t e r p r o g r a m s w h i c h c a n a c c o u n t f o r i n e l a s t i c t o r s i o n a l b e h a v i o u r . T h i s t h e s i s w i l l d e v e l o p s u c h a m e t h o d , b a s e d on l i n e a r a n a l y s i s o f a s u b s t i t u t e s t r u c t u r e , a s p r o p o s e d by S h i b a t a and S o z e n 1 6 . G e n e r a l t h r e e d i m e n s i o n a l f i n i t e e l e m e n t c o m p u t e r p r o g r a m s f o r t h e i n e l a s t i c a n a l y s i s o f c o m p l e x s t r u c t u r e s a r e 5 a v a i l a b l e , but they are o f t e n not s u i t e d for a b u i l d i n g system because: input data are u n n e c e s s a r i l y complex for b u i l d i n g s of simple geometry with h o r i z o n t a l and v e r t i c a l members; the r i g i d f l o o r assumption i s not made, r e s u l t i n g in a l a r g e number of degrees of freedom; a r i g i d arm option to model f i n i t e s i z e d j o i n t s i s not a v a i l a b l e , and c e n t e r l i n e dimensions u s u a l l y cannot be used to a c c u r a t e l y d e s c r i b e the s t r u c t u r e ; the loading i s of a r e s t r i c t e d form, and loads are a p p l i e d at only a l i m i t e d number of l o c a t i o n s . E x i s t i n g b u i l d i n g a n a l y s i s programs in c l u d e ETABS 1 for e l a s t i c a n a l y s i s of three dimensional b u i l d i n g s , DRAIN-2D 2 f o r i n e l a s t i c a n a l y s i s of two dimensional b u i l d i n g s , and DRTABS3 f o r i n e l a s t i c three dimensional a n a l y s i s . The l a t t e r two provide a complete time h i s t o r y response but r e q u i r e l a r g e computer memory and are expensive to run. ETABS, on the other hand, can be used e i t h e r for a modal spectrum a n a l y s i s or f o r modal i n t e g r a t i o n a n a l y s i s of l i n e a r s t r u c t u r e s . 1.2 Review of Previous Work T h i s s e c t i o n reviews the works w h i c h a r e r e l a t e d to the development of n o n l i n e a r dynamic a n a l y s i s methods on r e i n f o r c e d concrete s t r u c t u r e s . A c o n s i d e r a b l e amount of work has been done on a n a l y t i c a l models and numerical methods, e s p e c i a l l y a f t e r the development of high speed e l e c t r o n i c d i g i t a l computers. For plane frame s t r u c t u r e s , the d i f f i c u l t part of e s t i m a t i n g the i n e l a s t i c deformations i s the modeling of 6 i n e l a s t i c m a t e r i a l behaviour. The e l a s t o - p l a s t i c model" was used because of i t s mathematical s i m p l i c i t y and good r e p r e s e n t a t i o n of the behaviour of s t e e l . But t h i s model f a i l s to account f o r s t r a i n hardening and the Bauschinger e f f e c t . Then a b i l i n e a r model was used to simulate the s t r a i n hardening e f f e c t . J e n n i n g s 5 adopted a Ramberg-Osgood type of model to s i m u l a t e the Bauschinger e f f e c t . Clough and J o h n s t o n 6 in s t u d i e s of s i n g l e degree of freedom systems compared the d u c t i l i t y requirements in systems with s t i f f n e s s degrading force-displacement r e l a t i o n s h i p s with those in systems with an e l a s t o - p l a s t i c r e l a t i o n s h i p . They found that s t i f f n e s s degradation d i d not i n f l u e n c e the d u c t i l i t y requirements of long p e r i o d s t r u c t u r e s , but d i d increase the d u c t i l i t y requirements of short p e r i o d s t r u c t u r e s . Takeda, Sozen and N i e l s o n 7 used a more complicated h y s t e r e s i s model to reproduce the behaviour of r e i n f o r c e d c o n c r e t e members under f l e x u r a l load r e v e r s a l s with or without a x i a l l o a d s . T h i s h y s t e r e s i s model was proposed by Takeda and i s r e f e r r e d to as Takeda's model. Gulkan and Sozen 8 have been s u c c e s s f u l i n reproducing the behaviour of concrete frame s t r u c t u r e s observed i n shaking t a b l e t e s t s with the use of 'this mode1. For three dimensional s t r u c t u r e s , both t r a n s l a t i o n a l and t o r s i o n a l motions are u s u a l l y present when they are s u b j e c t e d to s e i s m i c ground motions. Most s t u d i e s on the l a t e r a l - t o r s i o n a l response problem assume l i n e a r l y e l a s t i c 7 f o r c e - d i s p l a c e m e n t r e l a t i o n s h i p s 9 , 0 . R e s u l t s o f t h e s e s t u d i e s a r e n o t a p p l i c a b l e t o b u i l d i n g s b e c a u s e t h e y a r e u s u a l l y d e s i g n e d t o d e f o r m b e y o n d t h e y i e l d l i m i t d u r i n g m o d e r a t e t o v e r y i n t e n s e e a r t h q u a k e s . Kan a n d C h o p r a 1 1 have c a r r i e d o u t a s t u d y c o n c e r n e d w i t h c o u p l e d l a t e r a l - t o r s i o n a l r e s p o n s e o f a s i n g l e . s t o r y s t r u c t u r e i n t h e e l a s t i c and i n e l a s t i c r a n g e s . They i n d i c a t e d t h a t a f t e r y i e l d i n g , t h e s t r u c t u r e b e h a v e d more an d more l i k e an i n e l a s t i c s i n g l e d e g r e e of f r e e d o m s y s t e m , r e s p o n d i n g p r i m a r i l y i n t r a n s l a t i o n . T h u s , t o r s i o n a l c o u p l i n g a f f e c t s maximum d e f o r m a t i o n s i n i n e l a s t i c s y s t e m s t o a l e s s e r d e g r e e t h a n i t d o e s i n c o r r e s p o n d i n g e l a s t i c s y s t e m s . However, T s o a nd S a d e k 1 2 f o u n d t h a t t h e i n f l u e n c e o f e c c e n t r i c i t y on d u c t i l i t y demand i s l a r g e r t h a n p r e v i o u s l y r e p o r t e d . 1.3 P u r p o s e a nd Scope The p u r p o s e o f t h i s t h e s i s i s t o f u r t h e r t h e work o f p r e v i o u s r e s e a r c h e r s c o n c e r n e d w i t h t h e i n e l a s t i c t o r s i o n a l r e s p o n s e o f s t r u c t u r e s t o e a r t h q u a k e e x c i t a t i o n . A s i m p l e , e f f e c t i v e , a n d e c o n o m i c a l p r o c e d u r e w h i c h p r o v i d e s i n f o r m a t i o n on t h e t h r e e d i m e n s i o n a l b e h a v i o u r o f t h e s t r u c t u r e and on t h e d u c t i l i t y demands on t h e members i s p r e s e n t e d . T h i s p r o c e d u r e i s c a p a b l e o f h a n d l i n g h i g h - r i s e s t r u c t u r e s w i t h o u t m a j o r s i m p l i f i c a t i o n s . I t i s a h y b r i d o f methods d e v e l o p e d by M a c K e n z i e 1 3 on t h e t h r e e d i m e n s i o n a l e l a s t i c p r o b l e m , and by Y o s h i d a 1 " a n d M e t t e n 1 5 on t h e i n e l a s t i c p r o b l e m . The FORTRAN c o m p u t e r p r o g r a m p u t t o g e t h e r i n t h i s t h e s i s i s code named P I T S A , w h i c h s t a n d s f o r 2 s e u d o i n e l a s t i c t o r s i o n a l s e i s m i c 8 a n a l y s i s . The three dimensional problem i s handled by assembling the s t i f f n e s s of plane frames using only three degrees of freedom fo r each f l o o r . The s t i f f n e s s matrix of each plane frame i s obtained using standard techniques and s t a t i c a l l y condensed to r e t a i n only h o r i z o n t a l t r a n s l a t i o n s at f l o o r l e v e l s , as w e l l as those a d d i t i o n a l degrees of freedom r e q u i r e d for v e r t i c a l c o m p a t i b i l i t y at common columns of adjacent frames. An o v e r a l l s t r u c t u r e s t i f f n e s s matrix i s obtained a f t e r the frames are assembled and i s f u r t h e r condensed to r e t a i n only two t r a n s l a t i o n s and one r o t a t i o n at the center of mass of each f l o o r . The i n e l a s t i c problem i s handled 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 developed by Yoshida and M e t t e n 1 " 1 5 . It i s intended to bridge the gap between s i m p l i f i e d e l a s t i c a n a l y s i s and a f u l l s c a l e i n e l a s t i c time step a n a l y s i s . T h i s method i s a m o d i f i e d e l a s t i c - modal a n a l y s i s , which e v o l v e s from a design concept proposed by Shibata and Sozen' 6. I t i s intended f o r r e i n f o r c e d concrete s t r u c t u r e s only because r u l e s to modify s t i f f n e s s and damping matrices have not been developed f o r other m a t e r i a l s . In asymmetrical b u i l d i n g s , the response of the s t r u c t u r e i n two d i r e c t i o n s i s coupled; i e . , the response in one d i r e c t i o n i s a f f e c t e d not only by ground motion in that d i r e c t i o n , but a l s o by ground motions p e r p e n d i c u l a r to that d i r e c t i o n . T h e r e f o r e , both h o r i z o n t a l components of ground 9 motion should be considered s i m u l t a n e o u s l y . A t h i r d component, r o t a t i o n about a v e r t i c a l a x i s , can a l s o cause unexpected damage to some types of s t r u c t u r e . An attempt i s made to remove some of the r e s t r i c t i o n s that were imposed on the o r i g i n a l components of the procedure to make i t more f l e x i b l e . The e l a s t i c t o r s i o n a l program was a l t e r e d to a l l o w for a change in the center of mass l o c a t i o n along the height of the s t r u c t u r e . As a r e s u l t , the c e n t e r s of mass of each f l o o r need not to l i e i n a s t r a i g h t l i n e . A technique was developed to account fo r d u c t i l i t y demand at both ends of a member, whereas the previous i n e l a s t i c a n a l y s i s of Y o s h i d a 1 " and M e t t e n 1 5 only c o n s i d e r e d the l a r g e s t d u c t i l i t y i n each member. T h i s t h e s i s begins by d i s c u s s i n g the theory and assumptions made to perform three dimensional e l a s t i c modal a n a l y s i s . Then the m o d i f i e d s u b s t i t u t e s t r u c t u r e method i s presented to o u t l i n e the fundamental p r i n c i p l e s . Some r e s u l t s as r e p o r t e d p r e v i o u s l y using t h i s method are presented and d i s c u s s e d . F o l l o w i n g i s a chapter d e s c r i b i n g some of the concepts in earthquake a n a l y s i s . A modal combination method for s t r u c t u r e s with c l o s e l y spaced p e r i o d s i s presented and a l s o the q u e s t i o n of multi-component ground e x c i t a t i o n i s d i s c u s s e d . Then comes the a p p l i c a t i o n of the f i n a l v e r s i o n of the method where frame and shear w a l l b u i l d i n g s are t e s t e d under d i f f e r e n t earthquake e x c i t a t i o n s . I t i s shown that the d u c t i l i t y f a c t o r s and, more imp o r t a n t l y , the d u c t i l i t y demand p a t t e r n s are 10 c o n s i s t e n t with f u l l s c a l e time step a n a l y s i s ^ The idea i s not to expect "exact" answers, but to provide the designer with a f e e l f o r the behaviour of h i s s t r u c t u r e . A l s o , bad spots, where members are undergoing e x c e s s i v e deformations, can be l o c a t e d . In the f i n a l chapter, c o n c l u s i o n s of the present study are presented. 1.4 Assumptions and L i m i t a t i o n s The mathematical model used i n the program i s based on some common approximations. F l o o r s are assumed to be r i g i d with respect to in-plane deformations and of n e g l i g i b l e s t i f f n e s s with respect to out of plane deformations. The t o r s i o n a l r i g i d i t y of i n d i v i d u a l members i s n e g l e c t e d . The moment-r o t a t i o n r e l a t i o n s h i p at a member end i s assumed to be b i l i n e a r with no s t i f f n e s s d e gradation. No a x i a l - f l e x u r e i n t e r a c t i o n i s assumed f o r columns. Furthermore, t e s t s t r u c t u r e s are r e l a t i v e l y r e g u l a r in o v e r a l l l ayout and s t i f f n e s s d i s t r i b u t i o n along the height of the s t r u c t u r e s . Combination of loads such as earthquake p l u s dead load i s not p o s s i b l e at the present stage. CHAPTER 2 STRUCTURAL IDEALT ZATION 2 . 1 I n t r o d u c t i o n For the purpose of 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 , i t i s common to reduce a b u i l d i n g i n t o separate plane frames so that the problem can be s o l v e d i n two dimensions. T h i s approximate method i s not a c c e p t a b l e f o r earthquake a n a l y s i s unless the b u i l d i n g i s symmetrical i n shape and has uniform d i s t r i b u t i o n of mass and s t i f f n e s s throughout the h e i g h t . For b u i l d i n g s not symmetrical i n shape or having uneven d i s t r i b u t i o n of mass or s t i f f n e s s , i t i s necessary to model them as three dimensional s t r u c t u r e s . However, an exact three dimensional s t r u c t u r a l a n a l y s i s i s not r e q u i r e d f o r the m a j o r i t y of b u i l d i n g s . Two approximations can be made to s i m p l i f y the p r e p a r a t i o n of input data and reduce computer time. The f i r s t assumption i s that each f l o o r a c t s as a r i g i d diaphragm in i t s own plane, so that the h o r i z o n t a l displacements of a l l p o i n t s i n the diaphragm are uniquely determined by two t r a n s l a t i o n s and one r o t a t i o n about the v e r t i c a l a x i s of each f l o o r . Consequently the beams bend only normal to the diaphragm, and have no a x i a l deformations. . 1 2 The second assumption i s that the masses of each f l o o r can be lumped i n t o t r a n s l a t i o n a l and r o t a t i o n a l i n e r t i a s of the diaphragms f o r h o r i z o n t a l ' i n e r t i a e f f e c t s . The i n e r t i a loads are t r a n s f e r r e d to the frames and shear w a l l s through r i g i d body motions of diaphragms. A l s o the t o r s i o n a l s t i f f n e s s e s f o r a l l members are ignored. The s t r u c t u r e i s c o n s i d e r e d as composed of a s e r i e s of plane frames, connected together by r i g i d h o r i z o n t a l diaphragms. Each frame must be in the v e r t i c a l plane, but may be a r b i t r a r i l y l o c a t e d and o r i e n t e d i n p l a n . I s o l a t e d shear w a l l s are c o n s i d e r e d as frames c o n s i s t i n g of one column l i n e having the a s s o c i a t e d w a l l p r o p e r t i e s . Coupled shear w a l l s are c o n s i d e r e d as frames c o n s i s t i n g of two column l i n e s and one bay of c o u p l i n g beams with r i g i d ends r e p r e s e n t i n g the j o i n t s . Each frame i s t r e a t e d as an independent s u b s t r u c t u r e , connected at each f l o o r by a r i g i d diaphragm and at j o i n t s which are common to more than one frame. Column c e n t e r l i n e s and f l o o r l e v e l s form the b a s i c r e f e r e n c e l i n e s used in the d e s c r i p t i o n of each frame. Both columns and beams can have r i g i d ends where no deformations are allowed. For beams with r i g i d ends, the lengths of these ends are set equal to h a l f width of the columns below and f o r columns with r i g i d ends, the average depth of g i r d e r s on e i t h e r s i d e . Each i n d i v i d u a l member c o n t r i b u t e s to the t o t a l frame s t i f f n e s s matrix v i a the 6 by 6 member s t i f f n e s s matrix shown i n Table 2 . 1 . A x i a l , f l e x u r a l , and shear deformations are 13 i n c l u d e d . Note e x t r a terms in t h i s member s t i f f n e s s matrix due to the r i g i d ends. 14 2.2 Inter-frame C o m p a t i b i l i t i e s The assumption that f l o o r s are r i g i d i n t h e i r own planes i s made r o u t i n e l y in f o r m u l a t i n g mathematical models of b u i l d i n g s t r u c t u r e s . As a r e s u l t , a x i a l deformations of the beams in each frame are assumed to be zero. With the r i g i d f l o o r assumption, each frame has two degrees of freedom per j o i n t : v e r t i c a l displacement and in-plane r o t a t i o n ; and one l a t e r a l t r a n s l a t i o n per s t o r y . ( F i g . 2.1(a)) In t h i s program, c o m p a t i b i l i t y with regard to v e r t i c a l displacements at j o i n t s which are common to two i n t e r s e c t i n g frames i s en f o r c e d . Thus i n a column which i s common to two i n t e r s e c t i n g frames, the a x i a l deformations w i l l be the same. T h i s requirement i s important when the frame deforms i n a bending mode r a t h e r than a shear mode or when there i s strong c o u p l i n g between i n t e r s e c t i n g frames such as tube-type b u i l d i n g s . Consequently frames p e r p e n d i c u l a r to an earthquake motion w i l l c o n t r i b u t e to the r e s i s t a n c e to that motion r e s u l t i n g i n a s t i f f e r s t r u c t u r e . L a t e r a l f o r c e s are t r a n s f e r r e d i n t o the frames p e r p e n d i c u l a r to the motion by shear t r a n s f e r through beams. When two frames meet at some angle other than a r i g h t angle, the r o t a t i o n of e x t e r i o r beams in one frame w i l l a f f e c t the r o t a t i o n of the adjacent beams i n the other frame. In cases where two frames are in l i n e or c l o s e to being in l i n e , i t i s d e s i r a b l e to have both v e r t i c a l and r o t a t i o n a l c o m p a t i b i l i t y imposed on the common j o i n t s . However, frames t y p i c a l l y . 1 5 . i n t e r s e c t each other at r i g h t a n g l e s . The r o t a t i o n a l c o m p a t i b i l i t y requirement then would i n v o l v e the bending of one beam and the t o r s i o n of another. Since i t i s assumed that the t o r s i o n a l r e s i s t a n c e of any member s e c t i o n i s n e g l i g i b l e , t h i s r o t a t i o n a l c o m p a t i b i l i t y requirement i s u s u a l l y ignored. With the h o r i z o n t a l and v e r t i c a l c o m p a t i b i l i t y in p l a c e , the a c t i v e degrees of freedom in each frame are the ones a s s o c i a t e d with l a t e r a l displacements at the f l o o r l e v e l s and " a d d i t i o n a l " degrees of freedom a s s o c i a t e d with v e r t i c a l displacements at common colums. The i n a c t i v e degrees of freedom, not a s s o c i a t e d with i n e r t i a f o r c e s , are e l i m i n a t e d from the frame s t i f f n e s s matrix by s t a t i c condensation. ( F i g . 2.1(b)) The condensed frame s t i f f n e s s m a t r i c e s are then assembled i n t o a s t r u c t u r e s t i f f n e s s matrix on the assumption that f l o o r s are r i g i d such that there are three g e n e r a l i z e d degrees of freedom per f l o o r : two t r a n s l a t i o n a l and one r o t a t i o n a l plus " a d d i t i o n a l " degrees of freedom a s s o c i a t e d with common columns. ( F i g . 2.1(c)) F i n a l l y , the same s t a t i c condensation process i s used to e l i m i n a t e these " a d d i t i o n a l " degrees of freedom to form the d e s i r e d s t r u c t u r e s t i f f n e s s m a t r i x . ( F i g . 2.1(d)) 1 6 2.3:Condensed Frame S t i f f n e s s Matrix The condensed s t i f f n e s s matrix f o r each frame i s obtained by f i r s t w r i t i n g the s t i f f n e s s equation for any frame s u b j e c t e d to a given set of loads: { P } = [ K ]• { p } (2.1) where { P } = frame f o r c e vector [ K ] = frame s t i f f n e s s matrix { p } = frame displacement v e c t o r Let Eqn.(2.1) be p a r t i t i o n e d i n t o two sub-equations such that the f i r s t one i n v o l v e s h o r i z o n t a l displacements {h}, plus " a d d i t i o n a l " degrees of freedom {x}, and the second one i n v o l v e s displacements along the remaining degrees of freedom {A}. The g e n e r a l i z e d f o r c e s corresponding to {A} w i l l be a n u l l vector {0} s i n c e only h o r i z o n t a l . f o r c e s are c o n s i d e r e d . r A H x -0 K, K 2 1 K 2 2 grouping {H} and {x} together and c a l l i n g i t {F} K M K 2 , K i 2 K 2 2 expansion of Eqn.(2.3) y i e l d s : { F } = [ K,, ] { f } + [ K,2 ] { A } { 0 } = [ K 2 I ] { f ] + [ K 2 2 ] { A } 2.2) (2.3) (2 .4a) (2.4b) 1 7 r e a r r a n g i n g Eqn.(2.4b) and s u b s t i t u i n g i n t o Eqn.(2.4a) w i l l g i v e : { F } = [ K,, - K , 2 K 2 2" 1 K 2, ] { f } { A } = -[ K 2 2" 1 K 2, ] { f } Eqn.(2.5a) can be w r i t t e n as: (2.5a) (2.5b) K K xh where K hh K xh K xx Khx - [ K , , K 1 2 K 2 2 " 1 (2.6) K 2, ] (2.7) The s t i f f n e s s matrix d e f i n e d by Eqn.(2.7) r e l a t e s the g e n e r a l i z e d f o r c e s {H}, {Xj to the g e n e r a l i z e d displacements {h} , {x} and i s known as the .condensed frame s t i f f n e s s m atrix. The matrix [ K 2 2 ~ 1 ] i s not a c t u a l l y computed f o r two major reasons. F i r s t although [ K 2 2 ] i s banded, [ K 2 2 _ 1 ] i s g e n e r a l l y not banded but f u l l and would r e q u i r e l a r g e amounts of storage space. Second, the number of m u l t i p l i c a t i o n s r e q u i r e d for i n v e r s i o n of such a matrix i s c o n s i d e r a b l e and so, the i n v e r s i o n process would use up l a r g e amounts of computer time. A method of s o l u t i o n developed by C h o l e s k i overcomes these problems. The matrix [ K 2 2 ] i s s t o r e d in banded form' and decomposed i n t o the product of a lower t r i a n g u l a r matrix times i t s transpose, [L L T ] . Then [ K 2 2 - 1 ] i s w r i t t e n as [ L " T L ~ 1 ] . Furthermore, s i n c e [ K 1 2 ] = [ R 2 , T ] , the r i g h t hand s i d e of Eqn.(2.7) becomes [ K , , - ( L " ' K 2 , ) T ( L " ' K 2 , ) ] . 18 2.4 A s s e m b l i n g a nd C o n d e n s i n g o f 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 n o r d e r t o a s s e m b l e t h e s t r u c t u r e s t i f f n e s s m a t r i x f r o m t h e c o n d e n s e d f r a m e s t i f f n e s s m a t r i c e s , t h e l a t t e r must be t r a n s f o r m e d f r o m t h e l o c a l c o o r d i n a t e s o f e a c h frame t o a common g l o b a l c o o r d i n a t e s y s t e m . The g l o b a l c o o r d i n a t e s y s t e m c o n s i s t s of t h r e e c o o r d i n a t e s a t t h e c e n t e r o f mass of e a c h f l o o r : q and r w h i c h a r e t r a n s l a t i o n s , and 6 w h i c h i s r o t a t i o n . The o r i g i n i s l o c a t e d a t t h e c e n t e r o f mass b e c a u s e t h i s l e a d s t o a mass m a t r i x w i t h no o f f - d i a g o n a l t e r m s . The o r i e n t a t i o n o f a frame i s d e f i n e d by t h e d i r e c t i o n a l c o s i n e s , 1^  and Ir , i n t h e q and r d i r e c t i o n s r e s p e c t i v e l y , ( s e e F i g . 2 .2 ) The p o s i t i o n o f a f r a m e i s d e f i n e d by t h e m a t r i x [Ie ], t h e p e r p e n d i c u l a r d i s t a n c e f r o m t h e c e n t e r s o f mass t o t h e f r a m e . The fr a m e i s f i r s t g i v e n a p o s i t i v e d i r e c t i o n w h i c h p r e s c r i b e s t h e s i g n o f t h e d i r e c t i o n a l c o s i n e s . I f t h e f r a m e m o v i n g i n i t s d e f i n e d p o s i t i v e d i r e c t i o n c a u s e s a c o u n t e r - c l o c k w i s e r o t a t i o n a b o u t t h e c e n t e r o f mass, I e i s d e f i n e d a s p o s i t i v e . S i n c e t h e c e n t e r s o f mass r a r e l y l i e i n a v e r t i c a l l i n e , a d i a g o n a l m a t r i x [ I e ] i s u s e d t o d e f i n e t h e i o c a t i o n o f e a c h f r a m e r e l a t i v e t o t h e c e n t e r s o f mass a t e a c h f l o o r l e v e l . The t r a n s f o r m a t i o n f r o m t h e l o c a l t o t h e g l o b a l c o o r d i n a t e s y s t e m i s o b t a i n e d a s f o l l o w s f o r a f r a m e o r i e n t e d a r b i t r a r i l y i n p l a n . The r i g i d f l o o r a s s u m p t i o n r e s u l t s i n t h e e q u a t i o n : {h} = I ^ { q } + I , { r } + [Ie){6} ( 2 . 8 ) 19 g i v i n g the system a set of v i r t u a l displacements leads t o : { 6h} T{H} = {6q} T{Q} + {6r} T{R} + { 8 9 } T { Q } (2.9) s u b s t i t u t i o n of eqn.(2.8) i n t o eqn.(2.9) g i v e s , I <^  { 6 q } T{H}+I r {8r} T{H} + [ l e ] {6c9}T{H} = {5q} T{Q} + {5r} T{R} + { 5 f 9 } T { 0 } (2.10) s i n c e the v i r t u a l displacement v e c t o r s are a r b i t r a r y , any two of them may be set to zero, I , M = / R (2.11 [ I e ]H 0 v. y p u t t i n g Eqn.(2.8) i n t o Eqn.(2.6) g i v e s : I^{q}+I r {r} + [ l e ]{dV K Kh K K, (2.12 or K (2.13) X v. - y i n s e r t i n g eqn.(2.11) i n t o eqn.(2.13) leads t o : fs1 l<\, l r Khh f 9 l R \ - Ir [ I * 3'Khh £ 0 ( \ _ I <), K xh 1rK%h 3Kxh (2.14) 20 The o v e r a l l 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 now o b t a i n e d by a d i r e c t s u m m a t i o n of t h e t r a n s f o r m e d frame s t i f f n e s s m a t r i c e s i n t h e f o r m a s shown i n E q n . ( 2 . 1 4 ) . The f o r c e s {x} a r e s e t t o z e r o when t h e summation of t h e c o n t r i b u t i o n s of a l l f r a m e s i s c o m p l e t e d , s i n c e no e x t e r n a l f o r c e s a r e a p p l i e d a t t h e s e j o i n t 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 n E q n . ( 2 . 1 4 ) i s t h e n c o n d e n s e d i n t h e same manner as t h e f r a m e s t i f f n e s s m a t r i c e s . T h i s p r o c e s s y i e l d s t h e r e d u c e d s t r u c t u r e s t i f f n e s s m a t r i x [ K ] , s u c h t h a t , ( 2 . 1 5 ) 21 CHAPTER 3 MODIFIED SUBSTITUTE STRUCTURE METHOD 3. 1 I n t r o d u c 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 i s a pseudo i n e l a s t i c method t h a t u t i l i z e s e l a s t i c modal s p e c t r a l a n a l y s i s and extends that technique i n t o the i n e l a s t i c range. As the name suggested, the method was developed from the s u b s t i t u t e s t r u c t u r e method by Shibata and S o z e n 1 6 . The s u b s t i t u t e s t r u c t u r e method i s a design procedure to e s t a b l i s h the minimum design f o r c e s , corresponding to a given response spectrum, f o r i n d i v i d u a l members of a r e i n f o r c e d concrete s t r u c t u r e so that a t o l e r a b l e i n e l a s t i c response i s not l i k e l y to be exceeded. In Yoshida's t h e s i s 1 \ the m o d i f i e d method was c r e a t e d f o r the earthquake hazard e v a l u a t i o n 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 c o n s t r u c t e d before the advent of dynamic a n a l y s i s . The m o d i f i e d method computes the d u c t i l i t y demand of each member using an i t e r a t i v e procedure with reduced f l e x u r a l s t i f f n e s s and s u b s t i t u t e damping f a c t o r s . S e v e r a l small to medium frames of d i f f e r e n t s i z e s and s t r e n g t h s were t e s t e d and the r e s u l t s compared w e l l with n o n l i n e a r dynamic a n a l y s i s . The mo d i f i e d method appears to work w e l l f o r frames in which 22 y i e l d i n g i s not e x t e n s i v e and widespread. I t a l s o appears to work b e t t e r f o r those i n which y i e l d i n g occurs mainly in beams and the e f f e c t of higher modes i s not predominant. S h o r t l y a f t e r , the modified method was r e f i n e d and extended to seismic 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 by M e t t e n 1 5 . Again, 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 i l i t y demands i s e v a l u a t e d using r e s u l t s o b t ained from an i n e l a s t i c time h i s t o r y a n a l y s i s . The coupled 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 height ranging from f i v e to s i x t e e n s t o r e y s . The m o d i f i e d method was shown to provide good e s t i m a t e s of d u c t i l i t y requirements and d e f l e c t i o n s . 3.2 S u b s t i t u t e S t r u c t u r e Method In order to understand 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 review of i t s predecessor i s h e l p f u l because many f e a t u r e s are common to both methods. The s u b s t i t u t e s t r u c t u r e method i s an e x t e n s i o n of a procedure by Gulkan and S o z e n 8 , which i n c o r p o r a t e s the e f f e c t s of i n e l a s t i c energy d i s s i p a t i o n to determine the design fo r c e f o r a s i n g l e degree of freedom system using the o r d i n a r y l i n e a r response spectrum. The method broadens t h i s procedure to multi-degree of freedom s t r u c t u r e s . The main concepts of the s u b s t i t u t e s t r u c t u r e method are the d e f i n i t i o n of a s u b s t i t u t e frame with i t s s t i f f n e s s and damping p r o p e r t i e s r e l a t e d to but d i f f e r i n g from the a c t u a l frame, and the c a l c u l a t i o n of design f o r c e s from modal a n a l y s i s of the s u b s t i t u t e frame using a l i n e a r response 23 spectrum. I t i s assumed that p r e l i m i n a r y member s i z e s of the a c t u a l s t r u c t u r e are known to the designer from g r a v i t y loads and other requirements. At the d i s c r e t i o n of the designer are the response spectrum f o r the design earthquake and the t o l e r a b l e damage for the members. According to Shibata and Sozen' 6 the method can be a p p l i e d only to s t r u c t u r e s s a t i s f y i n g the f o l l o w i n g : (1) The system can be analyzed in one v e r t i c a l plane. (2) No abrupt changes in geometry or mass along the height of the system. (3) Columns, beams, and w a l l s may be designed with d i f f e r e n t l i m i t s of i n e l a s t i c response, but the l i m i t should be the same f o r a l l beams i n a given bay and a l l columns on a given a x i s . (4) A l l s t r u c t u r a l elements and j o i n t s are r e i n f o r c e d to 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 repeated 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 lacements. (5) N o n s t r u c t u r a l components do not i n t e r f e r e with s t r u c t u r a l response. D e t a i l s of the method are now d e s c r i b e d . The f l e x u r a l s t i f f n e s s of s u b s t i t u t e frame members are d e f i n e d as, (EI ) . = (3.1) Sc 24 where ( E I ) ^ = c r o s s - s e c t i o n a l f l e x u r a l s t i f f n e s s of t h e i - t h member i n t h e s u b s t i t u t e frame ( E I ) a - = c r o s s - s e c t i o n a l f l e x u r a l s t i f f n e s s of t h e i - t h member i n t h e a c t u a l frame M l= s e l e c t e d t o l e r a b l e damage r a t i o f o r t h e i - t h member The term, ( E I ) a , i s c a l c u l a t e d u s i n g t h e f u l l y c r a c k e d s e c t i o n . The damage r a t i o , u, i s t h e r a t i o of the i n i t i a l s t i f f n e s s t o the r e d u c e d s t i f f n e s s . P h y s i c a l i n t e r p r e t a t i o n of th e damage r a t i o f o r . a beam s u b j e c t e d t o a n t i symmet r i c a l end moments i s shown i n F i g . 3.1 where t h e a p p l i e d moment, M, i s p l o t t e d a g a i n s t t h e end r o t a t i o n , 8, c a u s e d by f l e x u r a l d e f o r m a t i o n w i t h i n t h e s p a n . The damage r a t i o i m p l i e s t h a t a r o t a t i o n , n8y , w i l l be r e a c h e d where c?y i s t h e r o t a t i o n a t t h e y i e l d moment. I t s h o u l d be n o t e d t h a t t h e damage r a t i o i s a l w a y s s m a l l e r t h a n t h e d u c t i l i t y r a t i o b a s e d on maximum t o y i e l d r o t a t i o n f o r t h e s t r a i n h a r d e n i n g c a s e . Q u a n t i t a t i v e l y , damage and d u c t i l i t y r a t i o s a r e t h e same o n l y f o r t h e e l a s t o p l a s t i c c a s e . The r e l a t i o n between damage r a t i o and d u c t i l i t y r a t i o i s u = (3.2) 1 + ( T J - 1 ) S where u = damage r a t i o TJ = d u c t i l i t y r a t i o s = s t r a i n h a r d e n i n g r a t i o N a t u r a l p e r i o d s , mode s h a p e s , and modal f o r c e s f o r t h e 25 undamped s u b s t i t u t e s t r u c t u r e are obtained from a l i n e a r dynamic a n a l y s i s . From the member f o r c e s , in p a r t i c u l a r the bending moments, a smeared damping r a t i o i s computed for each mode. m I (P. gm= 1 \ ( 3 . 3 ) IP. I L where 0 .= 0 . 0 2 + 0 . 2 ( 1 - 1/V/T ) ( 3 . 4 ) St, and P % [ ( M A L ) 2 + (M[. ) 2 + ( l % L ) ] ( 3 . 5 ) 6 (EI ) . S L /3sL= s u b s t i t u t e damping f a c t o r for i - t h member P-L= r e l a t i v e f l e x u r a l s t r a i n energy in i - t h member for m-th mode L = length of i - t h member to m Ma.L) M bi. = bending moments at the ends of s u b s t i t u t e frame member i f o r m-th mode The formula f o r the s u b s t i t u t e damping f a c t o r i s based on dynamic t e s t s of r e i n f o r c e d c o n c r e t e elements and one storey frames. I t pr o v i d e s an estimate f o r the v i s c o u s damping f a c t o r to s i mulate the e f f e c t of h y s t e r e t i c energy damping. A s i n g l e smeared damping r a t i o f o r each mode i s r e q u i r e d for modal a n a l y s i s . T h i s i s done by assuming that each element c o n t r i b u t e s to the modal damping in p r o p o r t i o n to i t s r e l a t i v e f l e x u r a l s t r a i n energy a s s o c i a t e d with each mode. The next step i s to repeat the modal a n a l y s i s using the smeared damping r a t i o s , producing member f o r c e s which d i f f e r from those f o r the undamped case. Then the member f o r c e s for each mode are combined u s i n g the root-sum-square method, unless 26 the magnitude of the two l a r g e s t c o n t r i b u t i o n s are s i m i l a r i n which case the RSS f o r c e s are a m p l i f i e d by a f a c t o r given i n terms of the base shears. The l a s t step i n the s u b s t i t u t e s t r u c t u r e method i s to i n c r e a s e the design moments f o r columns by twenty percent to prevent f a i l u r e i n a column p r i o r to f a i l u r e in the beams. Shibata 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 three one-bay t e s t frames with a he i g h t ranging from three to ten s t o r e y s . T h e i r method of t e s t i n g was to determine the design 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 to design the frames a c c o r d i n g to these f o r c e s . A time-step a n a l y s i s program SAKE was used to analyze the frames, and the damage r a t i o s c a l c u l a t e d were compared to the i n i t i a l l y s p e c i f i e d ones. A l l frames were designed based on a t a r g e t damage r a t i o of s i x f o r the beams and one f o r the columns. The three s t o r e y frame produced the best r e s u l t s with the average damage r a t i o s found i n the time-step a n a l y s i s being c l o s e to and below the valu e s chosen. The f i v e storey frame had one column with a damage r a t i o g r e a t e r than one, and a l l beam damage r a t i o s were below the design value of s i x . For the ten storey frame, only three of ten column values were above u n i t y , while the beams had an average damage r a t i o of 5.5 and were a l l below 6. In summary, the r e s u l t s were f a v o r a b l e in that average damage r a t i o s of a l l motions c o n s i d e r e d were a l l l e s s than the t a r g e t v a l u e s . 27 3.3 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 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 procedure f o r determining the l o c a t i o n and extent of damage corresp o n d i n g to a given type and i n t e n s i t y of earthquake motion represented by the design spectrum, f o r a r e i n f o r c e d concrete s t r u c t u r e . The method i s e x p l i c i t y an a n a l y s i s procedure. I t s o b j e c t i v e i s to i d e n t i f y i n e l a s t i c deformation p a t t e r n s f o r the purposes of e v a l u a t i n g the performance of e x i s t i n g b u i l d i n g s and as a design a i d f o r new b u i l d i n g s . The procedure i s a m o d i f i e d e l a s t i c s p e c t r a l a n a l y s i s in which the s t i f f n e s s and damping p r o p e r t i e s are a l t e r e d so that the responses agree with i n e l a s t i c time-step a n a l y s i s . The concepts of s u b s t i t u t e s t i f f n e s s , s u b s t i t u t e damping, and damage r a t i o are the same as those used i n the s u b s t i t u t e s t r u c t u r e method, but the mo d i f i e d procedure i s an i t e r a t i v e one. A l s o t h i s m o d i f i e d procedure d i f f e r s from before i n that the i n i t i a l s t i f f n e s s and the y i e l d moment c a p a c i t i e s are s p e c i f i e d as input data. The damage r a t i o f o r each member i s the output. In the i t e r a t i v e process, a s u i t a b l e combination of modal f o r c e s must not exceed the s p e c i f i e d y i e l d moments, and e v e n t u a l l y the damage r a t i o s w i l l approach the c o r r e c t v a l u e s . A e l a s t i c modal a n a l y s i s , with the damage r a t i o set to u n i t y , i s performed i n the f i r s t i t e r a t i o n . Eigenvalues and e i g e n v e c t o r s are c a l c u l a t e d to determine the n a t u r a l p e r i o d s and mode shapes of the system. A value f o r damping i s chosen as the smeared damping value i s not a v a i l a b l e at t h i s p o i n t . Since the 28 chosen damping value does not a f f e c t the f i n a l r e s u l t s i t i s set to two percent of c r i t i c a l f o r a l l modes i n the f i r s t i t e r a t i o n . Modal f o r c e s and displacements are computed and are combined in root-sum-square and in complete-quadratic-combination manner. The CQC method of combining modal c o n t r i b u t i o n s i s d e s c r i b e d in Chapter 4. Those members whose CQC moments exceed y i e l d have t h e i r damage r a t i o s m o d ified a c c o r d i n g to the f o l l o w i n g formula: * 1 ( 3 . 6 a ) M y ( l - s ) + s unMn where y n M = damage r a t i o i n the n+1 th i t e r a t i o n MM = damage r a t i o i n the n th i t e r a t i o n M n = the CQC moment in the n th i t e r a t i o n My = the y i e l d moment s .= the s t r a i n hardening r a t i o In the case where there i s no s t r a i n hardening, Eqn.( 3 . 6 a ) becomes y h + 1 = * 1 ( 3 . 6 b ) M y The new damage r a t i o s have a lower l i m i t of uni t y because those members that have not y i e l d e d 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 steps in the s u b s t i t u t e s t r u c t u r e method are omitted. There i s no i n c r e a s e in fo r c e s to account f o r c l o s e l y spaced p e r i o d s and column moments are not in c r e a s e d by twenty percent. S t a r t i n g ' from the second i t e r a t i o n , the s u b s t i t u t e s t r u c t u r e method i s used to compute smeared damping r a t i o s and 29 modal f o r c e s which in turn produces new damage r a t i o s . With the new damage r a t i o s , another i t e r a t i o n i s performed, s t a r t i n g with the c a l c u l a t i o n of a new s t i f f n e s s matrix and ending with a f u r t h e r r e f i n e d set of damage r a t i o s . When a l l the member fo r c e s are e i t h e r below or 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 s , the i t e r a t i o n i s stopped. Then the damage r a t i o s are c o n s i d e r e d to have converged to the c o r r e c t v a l u e s . I t then i s the d e s i g n e r ' s r e s p o n s i b i l i t y to determine i f these damages are a c c e p t a b l e i n r e l a t i o n to the d u c t i l i t y of the s t r u c t u r e . In t e s t 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, Yoshida and Metten used t i m e - s t e p dynamic a n a l y s i s programs on plane s t r u c t u r e s for which the m o d i f i e d method had been used. Yoshida t e s t e d four s t r u c t u r e s under the same spectrum 'A' as that used by Shibata and Sozen. These s t r u c t u r e s were: a 2-store y 2-bay frame; a 3-storey 3-bay frame; a 6-storey 1-bay frame; and a 6-storey 3-bay frame. The time-step a n a l y s i s program SAKE was used with the records of four earthquakes (Taft N69W, T a f t S21W, E l Centro NS, E l Centro EW). The comparison showed very good r e s u l t s with computer time r e d u c t i o n s f o r the MSS method ranging from e l e v e n seconds to over one hundred seconds. For the 2-storey 2-bay frame, the method c o r r e c t l y p r e d i c t e d that the columns on the f i r s t s torey would y i e l d , but the p r e d i c t e d damage r a t i o s were about 60% of the average of the four motions. The p r e d i c t i o n of damage r a t i o s f o r the beams were s l i g h t l y underestimated, but a l l were w i t h i n 20% of the average v a l u e s . The 3-storey 3-bay frame showed the best r e s u l t s with a l l beam members w i t h i n 15% of the p r e d i c t e d 30 va l u e s . The columns on the top stor e y were the only ones that exceeded t h i s amount, but were c o n s e r v a t i v e l y p r e d i c t e d . The MSS method e s s e n t i a l l y f a i l e d i n the case of the 6-storey 1-bay frame, i n that i t d i d not come c l o s e to the average value of the damage r a t i o p r e d i c t i o n s from s e v e r a l time step a n a l y s e s . However f o r t h i s type of s t r u c t u r e d i f f e r e n t earthquake records produced widely d i f f e r e n t r e s u l t s when c a l c u l a t e d using a time step a n a l y s i s . F i n a l l y , f o r the 6-storey 3-bay frame, the method p r e d i c t e d f a i r l y uniform damage r a t i o s in the beams up the height of the frame with a small decrease towards the top s t o r e y . But the average damage r a t i o s were higher at the bottom and decreased q u i t e r a p i d l y with height a c c o r d i n g to SAKE a n a l y s i s . Otherwise, the method worked reasonably w e l l i n t h i s example. Although a smoothed response spectrum such as spectrum 'A' d e v i a t e s from the r e a l response s p e c t r a , i t was observed that using a r e a l response spectrum does not guarantee a b e t t e r estimate of damage r a t i o s and displacements. In Yoshida's t h e s i s a jagged spectrum with s p e c t r a l values evaluated at f i f t y d i f f e r e n t p e r i o d s was used f o r some of the t e s t s . A marginal improvement was achieved but the in c r e a s e i n computation i s so great that i t would not be worthwhile. It was noted that the damage r a t i o s due to E l Centro EW were c o n s i s t e n t l y higher than those of the other ground motions. These r e s u l t s may p a r t l y be caused by the d i s c r e p a n c y between a smoothed response spectrum and a r e a l response spectrum. 31 The MSS method was f u r t h e r r e f i n e d and a p p l i e d to coupled s t r u c t u r a l w a l l s by Metten. He t e s t e d three s t r u c t u r e s under the same spectrum 'A' as that used by Yoshida. The i n e l a s t i c time-step a n a l y s i s program DRAIN-2D was used f o r comparison. The f i r s t t e s t s t r u c t u r e s c o n s i s t e d of three sets of f i v e - s t o r e y s t r u c t u r a l w a l l s . Two values of c o u p l i n g beam c a p a c i t y , 60 K-Ft and 100 K-Ft, were t e s t e d at a peak ground a c c e l e r a t i o n of 20% of g r a v i t y . The r e s u l t s of these t e s t s were a l l c o n s e r v a t i v e , with p r e d i c t e d damage r a t i o values 50 to 100 percent g r e a t e r than the DRAIN-2D runs. A l s o , by i n c r e a s i n g the mass of the s t r u c t u r e by a f a c t o r of four, the fundamental p e r i o d was s h i f t e d to a part of the spectrum such that the a c c e l e r a t i o n response decreased with an increase in p e r i o d . The r e s u l t s f o r these f r e q u e n c i e s were b e t t e r . The next set of t e s t s was performed on a t e n - s t o r e y s t r u c t u r a l w a l l . Again, the r e s u l t s were c o n s e r v a t i v e and provided a very reasonable estimate of both d e f l e c t i o n and d u c t i l i t y . Much the same r e s u l t s were obtained in the case of the s i x t e e n - s t o r e y w a l l with an extra uncoupled w a l l . The t e s t s showed that d u c t i l i t y demand has a much grea t e r s c a t t e r when d i f f e r e n t records are examined than does d e f l e c t i o n . I t was not p o s s i b l e to p r e d i c t which of a s e r i e s of records w i l l produce the most dramatic e f f e c t on a given s t r u c t u r e . F i n a l l y , a s i x t e e n - s t o r e y s t r u c t u r a l w a l l design example was used to demonstrate how the method can be used i n p r a c t i c e . There was a t o t a l of seven runs using the MSS method to complete the design and o b t a i n a managable set of damage r a t i o s . Some of the f a c t o r s juggled i n 32 the s e r i e s of runs were the moment c a p a c i t i e s of l i n t e l s and columns, and Young's modulus. As a check of the f i n a l d esign, computer runs were performed using DRAIN-2D. The r e s u l t s showed that the MSS method i s indeed 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 . 3.4 Convergence Routine The convergence scheme i s based on two c r i t e r i a to stop the i t e r a t i v e process when the s o l u t i o n has reached a c e r t a i n l e v e l of accuracy. The f i r s t c r i t e r i o n i n v o l v e s comparison between the computed bending moment and the bending moment c a p a c i t y of a y i e l d e d member. T h i s d i f f e r e n c e i s r e f e r r e d to as the bending moment e r r o r . The f i r s t c r i t e r i o n i s s a i d to be s a t i s f i e d i f the bending moment e r r o r i s l e s s than f i v e p ercent of c a p a c i t y f o r a l l members. The second convergence c r i t e r i o n i s to r e q u i r e that the change in damage r a t i o s between s u c e s s i v e i t e r a t i o n s be l i m i t e d to one percent of the damage r a t i o s in the c u r r e n t i t e r a t i o n . T h i s l a s t c o n d i t i o n i s waived i n the case of a member with damage r a t i o l e s s than f i v e . In the case of t h i s small value, the c r i t e r i o n i s to l i m i t the a b s o l u t e d i f f e r e n c e of the damage r a t i o s between s u c e s s i v e i t e r a t i o n s to 0.1. The f o l l o w i n g i n e q u a l i t i e s summarize the convergence c r i t e r i a . Note that both Eqn.(3.7a) and Eqn.(3.7b) must be s a t i s f i e d f o r convergence. M - M n CAP — : < 0.05 i f (i > 1 (3.7a) M 33 n n-l < 0.01. i f A< > 5 or (3.7b) | ^ n " " V i ^ 0 • 1 it Id < 5 I t i s worthwhile n o t i n g that Eqn.(3.7b) i s the s t r i c t e r c o n d i t i o n and governs convergence f o r damage r a t i o s g r e a t e r than two. Since most s t r u c t u r e s are designed f o r damage r a t i o s g r e a t e r than two, t h i s c r i t e r i o n w i l l have to be s a t i s f i e d before the i t e r a t i v e process i s h a l t e d . As w e l l , a convergence speeding r o u t i n e which was developed by Metten i s used to a c c e l e r a t e damage r a t i o convergence and save unnecessary i t e r a t i o n s . The r o u t i n e uses three damage r a t i o s , the c u r r e n t one plus those from the pr e v i o u s two i t e r a t i o n s . Those r a t i o s that have decreased or i n c r e a s e d c o n s i s t e n t l y are m o d i f i e d by a p p r o p r i a t e l y adding or s u b t r a c t i n g a p o r t i o n of the d i f f e r e n c e of the l a s t two valu e s . The r a t i o n a l e i s that the damage r a t i o s would move f a s t e r in a d i r e c t i o n toward the c o r r e c t v a l u e s . In those cases which the r a t i o s o s c i l l a t e d , the m o d i f i c a t i o n r e s u l t s in a value which l i e s between the l a s t two v a l u e s . If the r a t i o s d i d not change for 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 i s made. ' The degree of m o d i f i c a t i o n can be c o n t r o l l e d d u r i n g program input. 34 3.5 Two Damage R a t i o s Per Member One of the l i m i t a t i o n s of the method i s the assumption that members undergo reverse bending and that the p o i n t s of i n f l e c t i o n are near midspan. Then and only then i s the shape of the force-displacement curve i d e n t i c a l to that of the moment-r o t a t i o n curve. The s u b s t i t u t e f l e x u r a l s t i f f n e s s i s obtained by d i v i d i n g the a c t u a l f l e x u r a l s t i f f n e s s by the damage r a t i o . T h i s assumption i s v a l i d f o r most beams where the end moments are equal or very c l o s e to being equal, but i s not so v a l i d f o r columns and e x t e r i o r beams. However, i f a l l columns are kept from y i e l d i n g , t h i s p o i n t i s not important. In p r e v i o u s programs, the damage r a t i o i s c a l c u l a t e d and m o d i f i e d by using the bigger of the two end moments i n the member. In subsequent i t e r a t i o n s , f l e x u r a l components of the member s t i f f n e s s matrix are m o d i f i e d by t h i s damage r a t i o . I t i s c l e a r that the damage r a t i o of both ends should be c o n s i d e r e d i f the end moments are not c l o s e to each other. B a s i c a l l y , t h i s work has been done by H u i 1 7 . A new model with a p l a s t i c hinge at each end of the member i s developed to account f o r t h i s e f f e c t . As a r e s u l t , damage r a t i o s at the ends of each member are c a l c u l a t e d and m o d i f i c a t i o n of s t i f f n e s s m a t r i c e s becomes a b i t more complex. A l s o a f f e c t e d by t h i s change are the formulas used to c a l c u l a t e the member s u b s t i t u t e damping and the r e l a t i v e f l e x u r a l s t r a i n energy (Eqn.(3.4) and Eqn.(3.5)). Since the d e r i v a t i o n i s covered elsewhere (Ref. 17), i t w i l l not be repeated here. The improvements brought about by t h i s 35 a l t e r a t i o n a r e shown t o be s i g n i f i c a n t , a l t h o u g h t h i s i s p a r t l y o f f s e t by t h e a d d e d c o m p l e x i t y a nd h i g h e r c o s t i n e x e c u t i o n o f t h e p r o g r a m . 3 6 CHAPTER 4 EARTHQUAKE ANALYSIS 4.1 Introduct ion The f o r c e s , which a s t r u c t u r e i s subjected to du r i n g an earthquake, r e s u l t from the d i s t o r t i o n induced by the ground motion. The magnitude and d i s t r i b u t i o n of the r e s u l t i n g f o r c e s and displacements are i n f l u e n c e d by the p r o p e r t i e s of both the s t r u c t u r e and the surrounding foundation as w e l l as the c h a r a c t e r of the ground motion. One can imagine as the ground i s d i s p l a c e d , the base of the s t r u c t u r e moves with i t . But the i n e r t i a of the s t r u c t u r e mass r e s i s t s t h i s motion and causes the s t r u c t u r e to undergo a d i s t o r t i o n . T h i s d i s t o r t i o n wave t r a v e l s along the height of the s t r u c t u r e and o s c i l l a t e s i n a complex manner. An important d i s t i n c t i o n between wind and earthquake l o a d i n g i s the way in which these loads are induced i n the s t r u c t u r e . Whereas wind loads are e x t e r n a l loads a p p l i e d to the exposed s u r f a c e of the s t r u c t u r e , earthquake loads are i n e r t i a loads r e s u l t i n g from the d i s t o r t i o n of the s t r u c t u r e . T h e i r magnitude i s then a f u n c t i o n of the mass rather than i t s exposed s u r f a c e . Thus, the s t i f f e r and heavier s t r u c t u r e does not 37 n e c e s s a r i l y mean the s a f e r d e s i g n . There i s a great number of u n c e r t a i n t i e s in performing an earthquake a n a l y s i s . The most important ones are the d i f f i c u l t y of p r e d i c t i n g the c h a r a c t e r of the design earthquake and the d i f f i c u l t y of e s t i m a t i n g the values of the s t r u c t u r a l parameters a f f e c t i n g the dynamic response. Since earthquake motions are random in c h a r a c t e r , there i s no way to a s c e r t a i n the exact nature of any f u t u r e earthquake at a given s i t e . However, c o n t i n u i n g s t u d i e s of the seismic h i s t o r y and geology of a re g i o n should y i e l d v a l u a b l e estimates of the expected range of s i g n i f i c a n t ground a c c e l e r a t i o n parameters such as maximum a c c e l e r a t i o n , frequency c h a r a c t e r i s t i c s and d u r a t i o n of l a r g e p u l s e s . At any p a r t i c u l a r p o i n t , the ground a c c e l e r a t i o n c o u l d be d e s c r i b e d by h o r i z o n t a l components along two pe r p e n d i c u l a r d i r e c t i o n s , a v e r t i c a l component, and a r o t a t i o n a l component. The r o t a t i o n a l component i s u s u a l l y n e g l i g i b l e . While the v e r t i c a l component i s s i g n i f i c a n t the s t r u c t u r e i s strong in t h i s d i r e c t i o n because of i t s requirement to c a r r y g r a v i t y l o a d s ; and the l a r g e s t i f f n e s s means small dynamic a m p l i f i c a t i o n . A f u r t h e r s i m p l i f i c a t i o n of the a c t u a l t h r e e -d i m ensional response of the s t r u c t u r e i s sometimes made by assuming the h o r i z o n t a l components to act nonconcurrently in the d i r e c t i o n of each p r i n c i p a l a x i s . I t i s assumed in these cases that a s t r u c t u r e designed by t h i s approach w i l l have adequate r e s i s t a n c e against, the r e s u l t a n t a c c e l e r a t i o n a c t i n g i n any 38 d i r e c t i o n . The p r o p e r t i e s of the s t r u c t u r e which a f f e c t i t s dynamic response are i t s mass, s t i f f n e s s , damping and d u c t i l i t y c h a r a c t e r i s t i c s . The determination of the mass and i n i t i a l s t i f f n e s s p r o p e r t i e s can be made q u i t e e a s i l y . However, the e f f e c t i v e s t i f f n e s s and damping, can change during the earthquake as a r e s u l t of cracks o c c u r r i n g i n members, even before the onset of y i e l d i n g . Y i e l d i n g f u r t h e r increases the p e r i o d of v i b r a t i o n of the s t r u c t u r e . The e v a l u a t i o n of the nature, magnitude, and d i s t r i b u t i o n of damping i s a problem that has not r e c e i v e d the systematic study that i t deserves. Convenience i n mode l l i n g has r e q u i r e d the assumption of a v i s c o u s - t y p e damping i n p l a c e of the a c t u a l mechanism. In most cases estimates of the e q u i v a l e n t v i s c o u s damping range from 5% to 10% of c r i t i c a l . A convenient way of studying the dynamic response of s t r u c t u r e s i s by c o n s i d e r i n g the t o t a l response i n terms of component modal responses. The response i n the e l a s t i c range may be thought of as the s u p e r p o s i t i o n of the responses of the modes of v i b r a t i o n . G e n e r a l l y , a system has as many modes of v i b r a t i o n as degrees of freedom. In each mode, the masses v i b r a t e such that they maintain the same p o s i t i o n s r e l a t i v e to each other. A s s o c i a t e d with each mode i s a c h a r a c t e r i s t i c p e r i o d of v i b r a t i o n . Each mode of v i b r a t i o n then can be co n s i d e r e d as a single-degree-of-freedom system. For seismic response the t o t a l response i s made up predominantly of the 39 f i r s t few modes, with the higher modes c o n t r i b u t i n g only a small p o r t i o n , except perhaps at the top of r e l a t i v e l y f l e x i b l e b u i l d i n g s . Another approach a s s o c i a t e d with dynamic a n a l y s i s i n v o l v e s the time h i s t o r y response to a p a r t i c u l a r earthquake. This a n a l y s i s employs d i r e c t numerical i n t e g r a t i o n of the equations of motion, proceeding in s t e p - b y - s t e p manner using the c o n d i t i o n s at the end of one time i n t e r v a l as the i n i t i a l c o n d i t i o n s f o r the succeeding i n t e r v a l . The d i r e c t i n t e g r a t i o n approach, which does not r e q u i r e the uncoupling of the equations of motion, can be used i n n o n l i n e a r dynamic a n a l y s i s . However, i t s use i s j u s t i f i a b l e only f o r a few important p r o j e c t s because of the high c o s t s . I t i s thought of as the only means of determining the d u c t i l i t y requirements of members corresponding to a d esign earthquake. One of the purposes of t h i s study i s to show 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 p r o v i d e s r e l i a b l e estimates of n o n l i n e a r dynamic a n a l y s i s r e s u l t s at a much lower c o s t . 4.2 Dynamic E q u i l i b r i u m Equation The b a s i c d i f f e r e n t i a l equation governing ' the behaviour of three dimensional s t r u c t u r e s subjected to base motion i s , [M] {ii} + [C] {u} + [K] {u} = -[M] {Og} (4.1) where [M] = mass matrix [C] = damping matrix = a [M] + 0 [K] 40 [K] = s t i f f n e s s matrix {u} = r e l a t i v e displacement v e c t o r {1} {u q} = ground a c c e l e r a t i o n v e c t o r {I} = un i t v e c t o r a /3 . = s c a l a r c o n s t a n t s The masses of the s t r u c t u r e are assumed to be lumped i n t o t r a n s l a t i o n a l and r o t a t i o n a l i n e r t i a s of the diaphragms for h o r i z o n t a l i n e r t i a e f f e c t s . T h i s makes p o s s i b l e the for m u l a t i o n of the f o r c e e q u i l i b r i u m of the system i n terms of a set of o r d i n a r y d i f f e r e n t i a l equations . i n s t e a d of the p a r t i a l d i f f e r e n t i a l equations which would be r e q u i r e d for the continuous system. Furthermore, the mass matrix i s di a g o n a l s i n c e the equations of motion are w r i t t e n about the ce n t e r s of mass. It i s comprised of three sub-matrices corresponding to the three g l o b a l s t r u c t u r e degrees of freedom: q, r, and 9. The q and r sub-matrices are simply the masses of the system, while the 9 sub-matrix corresponds to the r o t a t i o n a l mass i n e r t i a . The v e c t o r {u}, r e p r e s e n t i n g displacements of the g e n e r a l i z e d c o o r d i n a t e s r e l a t i v e to the ground, i s expressed p a r t i t i o n e d form. (4.2) S i m i l a r r e l a t i o n s h i p s are expressed f o r the r e l a t i v e v e l o c i t y and r e l a t i v e a c c e l e r a t i o n v e c t o r s . The q u a n t i t y in the r i g h t hand s i d e of Eqn.(4.1) i s the e f f e c t i v e load r e s u l t i n g from base motion and i s expressed as [M] {u5 } = -[M] ^  u 3 | r \ ( 4 . 3 ) -3,6 where each term of u a a i s the ground a c c e l e r a t i o n in the q d i r e c t i o n . For a v i s c o u s damped system, the decomposition of the response of a s t r u c t u r e i n t o i t s component modal responses i s p o s s i b l e only i f the damping matrix s a t i s f i e s a c e r t a i n c o n d i t i o n so that independent uncoupled modes e x i s t . Among the forms of the damping matrix which s a t i s f y t h i s c o n d i t i o n are those i n which i t i s p r o p o r t i o n a l to e i t h e r the mass or the s t i f f n e s s matrix, or i s a l i n e a r combination of these. The damping matrix i s then uncoupled by the same mode shapes which uncouple the equations of motion of the undamped system. If response spectrum a n a l y s i s or modal time-step a n a l y s i s i s performed, the f r a c t i o n of c r i t i c a l damping in each mode i s r e q u i r e d . On the other hand, i f f u l l s c a l e time-step a n a l y s i s i s performed, the s c a l a r c o n s t a n t s a and /3 are s p e c i f i e d . S t i f f n e s s dependent damping appears to be more reasonable as i t i m p l i e s p r o g r e s s i v e l y higher damping in the higher modes. T h i s would be d e s i r a b l e i n that i t l i m i t s the c o n t r i b u t i o n s from higher modes. 42 4.3 Mode Shapes and Frequencies The coupled set of equations, Eqn.(4.1), may be solved s i m u l t a n e o u s l y with an a p p r o p r i a t e numerical technique. However, a simpler approach i s to f i n d a t r a n s f o r m a t i o n which uncouples the equations so that they may be solved independently. T h i s t r a n s f o r m a t i o n i s w e l l known and makes use of the e i g e n v e c t o r s or mode shapes of the system. The mode shapes represent the s o l u t i o n of the undamped f r e e v i b r a t i o n problem given by, [M] {u} + [ K ] {u} = {0} (4.4) The ei g e n v a l u e problem to be so l v e d becomes, [ K ] {Ar } ~ co? [M] { A " } = {0} (4.5) or | [ K ] - co? • [ M ] | = 0 (4.6) where { A R } = mode shape f o r the r - t h mode cor = r - t h n a t u r a l frequency in rad/sec The mode shapes are normalized such t h a t , { A " } T [M] { A ' } = 1 (4.7a) {h } T [ K ] { A " } = co? (4.7b) { A R } T [C] { A F } = 2/3hcoh (4.7c) where /3h r e p r e s e n t s the f r a c t i o n of c r i t i c a l damping of the 'r-th mode. The a c t u a l displacements, {u}, are now expressed as a l i n e a r combination of the mode shapes. {u} « [ A ] {4>} (4.8) where {0] r e p r e s e n t s the amplitude of the modes. 43 4.4 Spectrum A n a l y s i s A response spectrum i s a g r a p h i c a l r e l a t i o n s h i p of the maximum value of a p a r t i c u l a r response parameter with the p e r i o d of a l i n e a r s i n g l e degree of freedom (SDF) system subjected to a f o r c i n g f u n c t i o n . The f o r c i n g f u n c t i o n in earthquake s t u d i e s i s the expected ground displacement. A point on a response spectrum i s obtained by a n a l y z i n g a p a r t i c u l a r SDF system, as de f i n e d by i t s p e r i o d and damping, subjected to an earthquake r e c o r d . The maximum value of the response parameter of i n t e r e s t generates a po i n t on a response spectrum. Other p o i n t s are obtained by v a r y i n g the p e r i o d (while f i x i n g the damping) to y i e l d a curve that represent the maximum response f o r any p e r i o d for a given damping r a t i o . The most u s e f u l responses are maximum r e l a t i v e displacement, maximum pseudo v e l o c i t y , and maximum pseudo a c c e l e r a t i o n . The maximum r e l a t i v e displacement giv e s an i n d i c a t i o n of the s t r a i n s i n the s t r u c t u r e , while the maximum pseudo v e l o c i t y p r o v i d e s a measure of the e l a s t i c energy s t o r e d i n the s t r u c t u r e and the maximum pseudo a c c e l e r a t i o n i s r e l a t e d to the l a t e r a l f o r c e c o e f f i c i e n t found in most design codes. If the maximum or s p e c t r a l value of the r e l a t i v e displacement i s denoted by Sd, the s p e c t r a l pseudo v e l o c i t y i s d e f i n e d as, Sv = w Sd (4.9) and the s p e c t r a l pseudo a c c e l e r a t i o n as, 44 Sa = to2 Sd (4.10) The response spectrum i s dependent on the p r o p e r t i e s of the s t r u c t u r e as given by i t s n a t u r a l p e r i o d and its.damping r a t i o , as w e l l as the c h a r a c t e r of the ground motion. Thus f o r the same value of damping r a t i o and the same range of p e r i o d s , the response s p e c t r a f o r d i f f e r e n t ground motions may be expected to d i f f e r from each other. T h i s means that response s p e c t r a p rovide a means of c h a r a c t e r i z i n g ground motions. Furthermore, because the behaviour of m u l t i s t o r e y b u i l d i n g s i s s t r o n g l y i n f l u e n c e d by the fundamental mode response, s p e c t r a l p l o t s p r ovide a convenient means of a s s e s s i n g the response of a s t r u c t u r e . A c t u a l response s p e c t r a are q u i t e i r r e g u l a r , with sharp peaks r e f l e c t i n g the resonant behaviour i n the system. T h i s i s e s p e c i a l l y e vident f o r l i g h t l y damped systems. For design purposes, i t i s customary to allow f o r earthquake motions of v a r y i n g frequency c h a r a c t e r i s t i c s by using average s p e c t r a based on a number of earthquake r e c o r d s . Response sp e c t r a are f u r t h e r i d e a l i z e d so that the v a r i o u s regions of the spectrum are smoothed to s t r a i g h t l i n e s . Because of the l i n e a r r e l a t i o n s h i p between the logarithms of both the s p e c t r a l displacement and a c c e l e r a t i o n with the logarit h m s of the s p e c t r a l v e l o c i t y and frequency as i n d i c a t e d by Eqns.(4.9) and (4.10), i t i s p o s s i b l e to have a s i n g l e p l o t , c a l l e d the t r i p a r t i t e p l o t , showing the v a r i a t i o n of a l l three response q u a n t i t i e s with the frequency or p e r i o d . The s i m p l i f i e d 45 Newmark's e l a s t i c design spectrum i s p l o t t e d i n such a manner in F i g . 4.1 with the maximum ground motion bounds shown as the dashed broken l i n e . The. s p e c t r a l v a l u e s ( s o l i d l i n e ) may be i n t e r p r e t e d as the ground motion maxima m u l t i p l i e d by a m p l i f i c a t i o n f a c t o r s which depend on the frequency region of the spectrum. G e n e r a l l y , the a m p l i f i c a t i o n f a c t o r f o r displacement i s l e s s than that f o r v e l o c i t y , which in turn i s l e s s than that f o r a c c e l e r a t i o n . Values of a m p l i f i c a t i o n f a c t o r s recommended by Newmark for v a r i o u s amounts of damping are shown i n Table 4.1(a), while those recommended by the N a t i o n a l B u i l d i n g Code of Canada, which are g e n e r a l l y more c o n s e r v a t i v e , are shown in Table 4.1(b). In Table 4.2 are shown both design and r e a l ground motion bounds. The t o r s i o n a l component s p e c t r a are d e r i v e d from time h i s t o r y records of h o r i z o n t a l components of motion of c l o s e l y spaced a c c e l e r o m e t e r s . T o r s i o n a l response in a s t r u c t u r e can come from t o r s i o n a l base e x c i t a t i o n and a l s o from e c c e n t r i c i t i e s in the s t r u c t u r e when e x c i t e d by h o r i z o n t a l motion. T h i s t o p i c was f i r s t d i s c u s s e d by Newmark 1 8, and has been f u r t h e r d i s c u s s e d by Newmark and R o s e n b l e u t h 1 9 , and Nathan and M a c K e n z i e 2 0 . Hart et a l 2 1 s t u d i e d the observed behaviour of instrumented b u i l d i n g s d u r i n g the 1971 San Fernando earthquake and concluded that the response due to r o t a t i o n a l ground motions was s i g n i f i c a n t . F u r t h e r , t h i s may e x p l a i n why corner columns are p a r t i c u l a r l y v u l n e r a b l e to seismic damage. The N a t i o n a l B u i l d i n g Code r e q u i r e s a 5% a c c i d e n t a l e c c e n t r i c i t y but i t i s d o u b t f u l the t o r s i o n a l ground input i s i n c l u d e d . 46 4.5 Complete Quadratic Combination Method It i s reasonable to assume that f o r earthquake e x c i t a t i o n the maximum c o n t r i b u t i o n s of a l l modes do not take place simultaneously, and so the maximum responses in i n d i v i d u a l modes of v i b r a t i o n should not be d i r e c t l y superimposed to o b t a i n the t o t a l response. The d i r e c t s u p e r p o s i t i o n of modal maxima does provide an upper bound to the maximum of t o t a l response. T h i s estimate i s often too c o n s e r v a t i v e and i s t h e r e f o r e not of much p r a c t i c a l use. A more reasonable estimate of the t o t a l response can u s u a l l y be o b t a i n e d using the root-sum-square (RSS) method of modal s u p e r p o s i t i o n . The RSS method y i e l d s good r e s u l t s when compared to t i m e - h i s t o r y response c a l c u l a t i o n ' s f o r b u i l d i n g s with well separated f r e q u e n c i e s . For most two-dimensional analyses, n a t u r a l f r e q u e n c i e s are u s u a l l y w e l l separated. However for 3D a n a l y s i s many of the f r e q u e n c i e s are q u i t e c l o s e together and the RSS method has been shown (Ref. 22) to give poor r e s u l t s in some of these cases. The t h e o r i e s of random v i b r a t i o n have been used to d e r i v e a l t e r n a t e methods which e l i m i n a t e inherent e r r o r s i n the absolute sum or the RSS method. Newmark and R o s e n b l e u t h 1 9 proposed a method which has cross-modal terms i n v o l v i n g the d u r a t i o n of earthquake as w e l l as the modal fr e q u e n c i e s and damping v a l u e s . A s i m i l a r method of modal combination which i s simpler and more p r a c t i c a l i s the Complete Quadratic Combination (CQC) method proposed by Wilson and Der K i u r e g h i a n 2 2 . The CQC 4 7 method r e q u i r e s that a l l modal responses be combined by the f o l l o w i n g e q u a t i o n : Q = ( I E p . Q.Q. )P ( 4 . 1 1 ) 8 ( 0 0 . ) 1 2 ( 0 + r 0 ) r 3 / 2 p..= ^ '—J. ' ( 4 . 1 2 ) t j ( 1 - r 2 ) 2 + 4 0 . 0 . r d + r 2 ) + 4 ( 0 . 2 + 0 . 2 ) r 2 Q.= maximum c o n t r i b u t i o n of the i - t h mode to the response of i n t e r e s t p..= cross-modal c o e f f i c i e n t s 0 - = damping r a t i o in the i - t h mode r = r a t i o of modal p e r i o d s , /Tj Note that t h i s c o n t r i b u t i o n formula i s a complete q u a d r a t i c form i n c l u d i n g a l l cross-modal terms. The i n f l u e n c e of s t o c h a s t i c c o r r e l a t i o n between the i n s t a n t s when the response a s s o c i a t e d with each mode reaches i t s maximum i s r e f l e c t e d in Eqn. ( 4 . 1 1 ) through the cross-modal c o e f f i c i e n t s . When n a t u r a l p e r i o d s are we l l separated, the cross-modal c o e f f i c i e n t s are small (as shown in F i g . 4 . 2 ) and the CQC method degenerates i n t o the RSS method. However, when n a t u r a l p e r i o d s are c l o s e to each other, the cross-modal c o e f f i c i e n t s tend to u n i t y and the cross-modal terms become s i g n i f i c a n t . The f a c t that each cross-modal term may assume p o s i t i v e or negative v a l u e s depending on whether the cor r e s p o n d i n g modal responses have the same or opposite si g n s accounts f o r the p o s s i b i l i t i e s of s t r o n g l y c o r r e l a t e d modal responses t a k i n g p l a c e with phase angles c l o s e to e i t h e r 0 or 1 8 0 degrees. The s u p e r i o r performance of the CQC method over the in which where 48 absolute sum and the RSS method can be i l l u s t r a t e d by i t s a p p l i c a t i o n to the f i v e storey b u i l d i n g shown i n F i g . 4.3. The b u i l d i n g i s symmetrical except the ce n t e r of mass i s l o c a t e d four f e e t from the geometric c e n t e r of the b u i l d i n g in both d i r e c t i o n s . The d i r e c t i o n of the a p p l i e d earthquake motion, n a t u r a l p e r i o d s and the p r i n c i p a l d i r e c t i o n s of the mode shapes are i l l u s t r a t e d in F i g . 4.4. One notes the mode shapes with r e s u l t a n t s which past through the center of mass have no r o t a t i o n a l components. However, those orthogonal to these have r o t a t i o n a l components as w e l l as x and y components. A l s o evident i s that one modal p e r i o d from the f i r s t group i s very c l o s e to another from the second group. T h i s type of p e r i o d d i s t r i b u t i o n and coupled mode shapes are very common i n asymmetrical b u i l d i n g s . T h i s s t r u c t u r e i s t e s t e d u sing the response spectrum method with spectrum 'A' as the ground motion. The modal base shears f o r the four e x t e r i o r frames are combined by three d i f f e r e n t methods: RSS, absolute sum, and CQC. These r e s u l t s are shown i n F i g . 4.5. The s i g n s of the base shears are not r e t a i n e d i n any of these combination methods. The response of the s t r u c t u r e i s then c a l c u l a t e d when subjected to the T a f t , 1952, N69W earthquake using a time-step a n a l y s i s program. Although spectrum 'A' i n d u e s s e v e r a l r e a l earthquake motions, only the T a f t one i s used here f o r comparison purposes. The maximum base shears from t h i s time h i s t o r y a n a l y s i s are a l s o p l o t t e d in F i g . 4.5. 49 For t h i s case i t i s c l e a r that the RSS method g i v e s a good approximation of the f o r c e s in the d i r e c t i o n of motion but g r e a t l y overestimates the f o r c e s i n the frames normal to the motion. The sum of a b s o l u t e values, which i s a method normally suggested f o r the case where p e r i o d s are c l o s e , g r e a t l y o verestimates a l l four frames and i s worse than the RSS method. For t h i s example, The CQC method g i v e s the best approximation to the 'exact' r e s u l t s from time h i s t o r y a n a l y s i s . Based on the preceding example and the above d i s c u s s i o n , the CQC method i s to be used as a replacement f o r the RSS method in a l l response spectrum c a l c u l a t i o n s i n t h i s study. 4 . 6 Multi-Component Ground Motions In the a n a l y s i s and design of asymmetrical s t r u c t u r e s , simultaneous a c t i o n of a l l components of the earthquake must be taken i n t o account. A procedure based on the response spectrum technique to determine the response of asymmetrical s t r u c t u r e s s u b j e c t e d to multi-component ground e x c i t a t i o n s i s needed. In asymmetrical s t r u c t u r e s , the response in two t r a n s l a t i o n a l d i r e c t i o n s and one r o t a t i o n a l d i r e c t i o n i s coupled. The response i n one d i r e c t i o n i s a f f e c t e d not only by ground motions in that d i r e c t i o n , but a l s o by ground motions in the other two d i r e c t i o n s . The response spectrum technique i s w e l l e s t a b l i s h e d f o r u n i d i r e c t i o n a l e x c i t a t i o n , but not f o r 50 multi-component e x c i t a t i o n . I f a l l the components are s t a t i o n a r y Gaussian p r o c e s s e s and taken as completely u n c o r r e l a t e d , the expected maximum value of any response w i l l be the square root of the sum of squared e x p e c t a t i o n s a s s o c i a t e d with the v a r i o u s components. (Newmark and R o s e n b l e u t h 1 9 ) Consider a s t r u c t u r e subjected to two orthogonal components of ground a c c e l e r a t i o n s in the q and r d i r e c t i o n s and r o t a t i o n a l component in the 8 d i r e c t i o n . From S e c t i o n 4.2, The s t r u c t u r e g e n e r a l i z e d degrees of freedom vector i s , {u} =/ u^ He (4.2) where £ u^} a n <3 ivr} r e f e r to the displacement v e c t o r of the c e n t e r s of mass in.the q and r d i r e c t i o n s r e s p e c t i v e l y , and {u^} i s the r o t a t i o n a l v e c t o r of the f l o o r diaphragms. The equation of motion about the c e n t e r s of mass can be w r i t t e n as, [M] {u} + [C] {u} + [K] {u} = -IM] (4.14) Making the normal mode t r a n s f o r m a t i o n s , {u} = [A] {</>} (4,8) and p r e m u l t i p l y i n g by [ A ] T , the uncoupled equation of motion becomes, [M*] {.0} + [C*] {0} + [K*] {0.} = ~ [ A ] T [M]/ U 3 A(4.15) u 3,e where [M ] and [K ] are g e n e r a l i z e d mass and s t i f f n e s s d i a g o n a l 51 m a t r ices r e s p e c t i v e l y , and [C ] i s the g e n e r a l i z e d damping matrix assumed to be d i a g o n a l . The t y p i c a l modal equation in mode i i s , h2(S.coL^ 0.= a . f t u 3 ^ ( t ) + o . > r u 3 i r . ( t ) + a i / , u w ( t ) (4.16) where 0Lis the damping r a t i o and c ^ i s the n a t u r a l frequency for mode i . ai%' air' a n d ai,e a r e m o < 3 a l p a r t i c i p a t i o n f a c t o r s f o r mode i in the q, r, and 8 d i r e c t i o n s . For multi-component ground e x c i t a t i o n s , i t i s necessary to o b t a i n the s p e c t r a l v a l u e s of the time s e r i e s given by the r i g h t hand s i d e of Eqn. (4.16). But to make use of the known s p e c t r a l v a l u e s of u ^ ^ ( t ) , u g r ( t ) , and u g e ( t ) d i r e c t l y , the t o t a l response can be approximated by combining the i n d i v i d u a l responses i n a root sum square manner. That i s , the modal responses from a l l three components i s given by, U L = {uL,% + U i / + u L e 2 ) ' / 2 (4.17). where u L t = m ° d e response due to q d i r e c t i o n e x c i t a t i o n and Sd.q= s p e c t r a l displacement value for i - t h mode due to q d i r e c t i o n e x c i t a t i o n e t c . 52 CHAPTER 5 COMPUTER PROGRAM 5,1 Program Concepts A very e f f e c t i v e approximate method for n o n l i n e a r a n a l y s e s of plane frames has proven to be 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. Based on t h i s method, v a r i o u s plane frame programs with pseudo n o n l i n e a r c a p a b i l i t i e s have been w r i t t e n 1 " 1 5 1 7 . However, to analyze s t r u c t u r e s with l a r g e e c c e n t r i c i t i e s , the program should address the s t r u c t u r e as thre e - d i m e n s i o n a l and not merely be an e x t e n s i o n of a plane frame program. As w e l l , the program should be f l e x i b l e and easy to modify or extend so that i t would not become obsolete w i t h i n a few years. A s p e c i a l purpose computer program i s developed i n t h i s study i n order to estimate the i n e l a s t i c behaviour of a m u l t i - s t o r e y , three-dimensional 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 to multi-component earthquake motions. T h i s program (PITSA) p r o v i d e s the maximum deformation and the d u c t i l i t y demand that occur i n a member durin g a given earthquake motion, with l i m i t a t i o n s d e f i n e d by the assumptions of the program. The gen e r a l conceptual o u t l i n e of the program i s shown i n F i g . 5.1. 53 The s t r u c t u r e i s i d e a l i z e d as an assemblage of planar frames. The s t i f f n e s s matrix i s obtained by the D i r e c t S t i f f n e s s Method, with the nodal displacements as unknowns. Each node has up to three degrees of freedom. However, the h o r i z o n t a l displacements of a l l nodes on the same l e v e l may be s p e c i f i e d to have i d e n t i c a l v a l u e s, in which.case only one degree of freedom i s assigned to a l l these displacements. Each frame i s then s t a t i c a l l y condensed such that only one t r a n s l a t i o n a l degree of freedom per f l o o r plus those r e q u i r e d f o r v e r t i c a l c o m p a t i b i l i t y remain a c t i v e . F i n a l l y , c o n t r i b u t i o n s from a l l frames are combined and condensed i n such a way as to y i e l d three s t r u c t u r a l degrees of freedom per f l o o r . T h i s process permits the s i z e of the problem to be g r e a t l y reduced. The dynamic response i s determined i t e r a t i v e l y by e l a s t i c s p e c t r a l a n a l y s i s , with s u b s t i t u t e s t i f f n e s s and s u b s t i t u t e damping c a l c u l a t e d w i t h i n each i t e r a t i o n . If a member y i e l d s , changes are made to the element s t i f f n e s s matrix and the o v e r a l l s t r u c t u r e s t i f f n e s s matrix i s r e b u i l t . The e f f e c t i v e s t i f f n e s s and damping p r o p e r t i e s are thus a f u n c t i o n of how f a r the s t r u c t u r e has gone i n t o the i n e l a s t i c range. 5.2 Program O r g a n i z a t i o n The program i s c o n t r o l l e d by the main or d r i v e r r o u t i n e which dimensions v a r i a b l e s , reads in g l o b a l c o n t r o l i n f o r m a t i o n , checks f o r compliance of g l o b a l dimensioning 54 r e s t r i c t i o n s , s e t s up the i t e r a t i o n loop for i n e l a s t i c a n a l y s i s , and checks f o r convergence of r e s u l t s . Three major subroutines are c a l l e d i n s i d e the main r o u t i n e (subroutine o r g a n i z a t i o n i s shown in F i g . 5 . 2 ) : 1) MAINF - The b a s i c s t r u c t u r e data i s read and a member generator subroutine completes the r e s t of the s t r u c t u r e . The next step i n v o l v e s the formation of member s t i f f n e s s matrices and summing them up for each frame i n the s t r u c t u r e . The reduced form of the frame s t i f f n e s s matrix i s determined and s t o r e d . 2) MAINR - The i n d i v i d u a l frames are then assembled together to form the s t r u c t u r e s t i f f n e s s matrix. Displacement v a r i a b l e s a s s o c i a t e d with c o m p a t i b i l i t y degrees of freedom in corner columns are condensed out. The net s t r u c t u r e s t i f f n e s s matrix i s determined and s t o r e d . 3) MAIND - The mass of the s t r u c t u r e and the earthquake a c c e l e r a t i o n s p e c t r a are read. The t h r e e -dimensional n a t u r a l f r e q u e n c i e s , mode shapes, and p a r t i c i p a t i o n f a c t o r s are e v a l u a t e d . The maximum store y displacements a s s o c i a t e d with each mode are determined. For each frame i n the s t r u c t u r e the reduced and then the unreduced j o i n t displacements are eval u a t e d . From these frame displacements the member f o r c e s are a l s o e v a l u a t e d and are combined i n root sum square 55 and complete q u a d r a t i c combination manners. The damage r a t i o s and smeared damping r a t i o s are updated from the l a s t i t e r a t i o n . Those members with i n e l a s t i c deformations are checked to see i f they exceed the t o l e r a b l e l i m i t s of moment c a p a c i t i e s . 5.3 Design Spectra The s e l e c t i o n of a design spectrum fo r a p a r t i c u l a r b u i l d i n g w i l l depend on the g e o g r a p h i c a l area, the l o c a l s o i l c o n d i t i o n , and the intended use of the b u i l d i n g . C e r t a i n types of p u b l i c b u i l d i n g such as h o s p i t a l s w i l l j u s t i f y a design f o r l a r g e r i n t e n s i t y earthquake. Response s p e c t r a a p p r o p r i a t e to a given s i t e may be d e r i v e d by choosing from records of r e a l earthquakes recorded at s i m i l a r s i t e s . T h i s method of choosing dynamic input f a i l s i n that there may not be enough strong motion r e c o r d i n g s i n g e o l o g i c a l l y s i m i l a r areas. But even when s o i l c o n d i t i o n s are s i m i l a r , each i n d i v i d u a l earthquake r e c o r d has strong f e a t u r e s c h a r a c t e r i s t i c only of that p a r t i c u l a r earthquake and s i t e . Another method i s by g e n e r a t i n g simulated accelerograms. E i g h t simulated earthquakes have been d e s c r i b e d by Jennings et a l 5 . However, the design spectrum chosen in t h i s study, spectrum 'A', i s a c q u i r e d by d e r i v i n g averaged or smoothed s p e c t r a from a set of earthquake records s c a l e d to the same peak a c c e l e r a t i o n . . (see Ref. 16) F i g u r e 5.3 shows spectrum 'A' d e s c r i b e d by simple 56 e x p r e s s i o n s a n d t h e assumed r e l a t i o n s h i p b e t ween t h e d e s i g n r e s p o n s e a c c e l e r a t i o n f o r any damping f a c t o r and f o r 2% d a m p i n g . B e c a u s e o f t h e r a n d o m n e s s o f e a r t h q u a k e s , t h e smoo t h e d a v e r a g e s p e c t r a o f s e v e r a l e a r t h q u a k e s w i l l h e l p t o e l i m i n a t e undue i n f l u e n c e o f l o c a l p e a k s i n r e s p o n s e . As shown by t h e p l o t s i n F i g u r e s 5.4 t o 5.7, a c c e l e r a t i o n r e s p o n s e s p e c t r a f o r f o u r r e c o r d e d g r o u n d m o t i o n c o m p o n e n t s , e a c h n o r m a l i z e d t o 0.5g, i n d i c a t e t h e j a g g e d n a t u r e . P r e v i o u s s t u d i e s have r e v e a l e d t h a t one p a r t i c u l a r m o t i o n , E l C e n t r o EW, p r o d u c e s r e l a t i v e l y l a r g e damage r a t i o s . T h i s t e n d e n c y may be e x p l a i n e d by e x a m i n i n g F i g . 5.8 w h i c h shows E l C e n t r o EW t o have h i g h e r s p e c t r a l v a l u e s t h a n t h o s e o f t h e o t h e r t h r e e m o t i o n s f o r p e r i o d s h i g h e r t h a n a b o u t 0.7 s e c o n d s . F u r t h e r t u n i n g o f t h e d e s i g n s p e c t r a i s p o s s i b l e , b u t u n l e s s t h e d e s i g n i s v e r y c o n s e r v a t i v e , t h e r e i s t h e p o s s i b i l i t y o f a n o t h e r g r o u n d m o t i o n w i t h t h e same c h a r a c t e r i s t i c maximum a c c e l e r a t i o n , w h i c h may r e s u l t i n l a r g e r damage r a t i o t h a n E l C e n t r o EW. T h e r e f o r e , i n s t e a d o f c o n c e n t r a t i n g on c o m p a r i n g r e s u l t s f r o m P I T S A t o t h o s e f r o m t i m e - s t e p a n a l y s i s o f 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 j u s t i f i e d f o r p r a c t i c a l d e s i g n p u r p o s e s t o work w i t h a v e r a g e d t i m e - s t e p r e s u l t s f r o m a l l f o u r e a r t h q u a k e s r e p r e s e n t e d by s p e c t r u m 'A'. 57 CHAPTER 6 PROGRAM TESTING 6 . 1 T e s t i n g f o r E l a s t i c A n a l y s i s The p r i n c i p a l o b j e c t i v e of t h i s r e s e a r c h i s the e v a l u a t i o n of the i n e l a s t i c s e i s m i c response of m u l t i - s t o r e y three dimensional b u i l d i n g s using 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 hoped that the research w i l l produce a r e l i a b l e and economical a l t e r n a t i v e to time-step dynamic a n a l y s i s . But before c o n s i d e r i n g the program's i n e l a s t i c c a p a b i l i t i e s , i t i s of great importance to e s t a b l i s h that the program c o u l d 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 . A f t e r a l l , each i t e r a t i o n in the pseudo i n e l a s t i c a n a l y s i s i s an e l a s t i c a n a l y s i s with modified s t i f f n e s s and damping p r o p e r t i e s . T h i s s e c t i o n i s intended to demonstrate that the proposed program, PITSA, c o r r e c t l y e v a l u a t e s the e l a s t i c behaviour of frame and shear w a l l b u i l d i n g s . For t h i s purpose, a w e l l e s t a b l i s h e d computer program was used f o r comparison. 6 . 1 . 1 Comparison With Another Modal A n a l y s i s Program - ETABS In order to v e r i f y the e l a s t i c c a p a b i l i t y of the proposed program, a p r e v i o u s l y t e s t e d and proven computer 58 program was used to compare r e s u l t s . For t h i s purpose ETABS 1 was chosen' because of i t s widespread use by the engineering p r o f e s s i o n . T h i s program was developed at the U n i v e r s i t y of C a l i f o r n i a at Berkeley in the mid-seventies and i s an extension of TABS to enable the assembly of three dimensional frames i n t o a g e n e r a l three dimensional s t r u c t u r e . Although t h i s extended c a p a b i l i t y i s not r e q u i r e d s i n c e PITSA only a l l o w s for planar frames, ETABS was used because i t was a v a i l a b l e . I t should be noted that t h i s program i s a c t u a l l y i n f e r i o r in the sense that c o m p a t i b i l i t y of the j o i n t degrees of freedom, v e r t i c a l t r a n s l a t i o n and the r o t a t i o n about the h o r i z o n t a l a x i s , i s not e n f o r c e d between i n t e r s e c t i n g frames. The t e s t s f o r comparison of the two programs were performed on two f i v e s t o r e y s t r u c t u r e s . The f i r s t s t r u c t u r e i s a frame b u i l d i n g and the second s t r u c t u r e i s one with a coupled w a l l core as the l a t e r a l f o r c e r e s i s t i n g system. 6 . 1 . 2- Numerical Examples (a) F i v e - S t o r e y Frame S t r u c t u r e The f i r s t s t r u c t u r e analyzed was a frame b u i l d i n g 'with dimensions and p r o p e r t i e s as shown in F i g . 6.1. The e l a s t i c a n a l y s i s was c a r r i e d ,out with spectrum 'A' 1 S , s c a l e d to 0.5g as the ground motion a p p l i e d i n the q d i r e c t i o n . The lowest mode i n v o l v i n g the d i r e c t i o n of i n t e r e s t , q, had a n a t u r a l p e r i o d of 0.42 seconds. The l o c a t i o n of the diaphragm mass in the top two 59 s t o r e y s was moved four f e e t from the plan c e n t r e to t e s t the program's a b i l i t y to handle s t r u c t u r e s with masses not on a v e r t i c a l l i n e . The f i r s t s i x modes of v i b r a t i o n were c o n s i d e r e d and 2% damping assumed f o r each mode. A l l r e s u l t s r e p o r t e d are root sum square values because ETABS does not combine modal responses using the complete q u a d r a t i c combination method. In t h i s case, c a l c u l a t i o n of CQC v a l u e s i s unnecessary s i n c e n a t u r a l p e r i o d s are w e l l separated. The r e s u l t s produced by the two programs were i n e x c e l l e n t agreement. The s l i g h t d i f f e r e n c e s i n the r e s u l t s were in the order of one percent. The e l a s t i c n a t u r a l p e r i o d s and the RSS bending moments f o r the frame undergoing the most severe displacements are shown i n Table 6.1. For the purpose of comparison, the l a r g e s t bending moment for each member was examined. At t h i s stage i t i s safe to say that the proposed program can indeed produce c o r r e c t e l a s t i c r e s u l t s f o r frame type s t r u c t u r e s . The c o s t of running the two programs are measured i n CPU ( c e n t r a l p rocess u n i t ) seconds. The CPU time f o r PITSA and ETABS were 5.5 sec. and 3.7 sec. r e s p e c t i v e l y . (b) F i v e - S t o r e y Coupled Wall S t r u c t u r e The next s t r u c t u r e analyzed was a coupled w a l l b u i l d i n g with dimensions and p r o p e r t i e s shown in F i g . 6.2. The e l a s t i c a n a l y s i s was c a r r i e d out with spectrum 'A' s c a l e d to 0.2g as the ground motion a p p l i e d i n the q d i r e c t i o n . The lowest mode i n v o l v i n g the d i r e c t i o n of a p p l i e d l o a d i n g had a 60 n a t u r a l p e r i o d of 0.38 seconds. Again, the top two s t o r e y s had t h e i r mass centres moved two f e e t from the middle of the b u i l d i n g to simulate a s t r u c t u r e with v a r y i n g plan dimensions. The f i r s t s i x modes were c o n s i d e r e d and 2% damping assumed f o r each mode. The r e s u l t s produced by the two programs were almost the same. The e l a s t i c n a t u r a l p e r i o d s and the RSS bending moments for the w a l l undergoing the most' severe displacements are shown i n Table 6.2. The CPU time for PITSA and ETABS were 2.3 sec. and 1.8 sec. r e s p e c t i v e l y . T h i s example shows that PITSA works for coupled w a l l s t r u c t u r e s in the e l a s t i c range. 6.2 T e s t i n g f o r I n e l a s t i c A n a l y s i s The r e s u l t s i n the p r e v i o u s s e c t i o n have e s t a b l i s h e d the e l a s t i c c a p a b i l i t y of PITSA. The next step i s to i n v e s t i g a t e the a b i l i t y of the program to p r e d i c t the i n e l a s t i c behaviour of m u l t i - s t o r e y s t r u c t u r e s . Since i t i s i m p r a c t i c a l to c a r r y out experiments on r e a l s t r u c t u r e s , the best a n a l y t i c a l method i s to compare on a numerical b a s i s with r e s u l t s obtained from an i n e l a s t i c time step a n a l y s i s . The degree of i n e l a s t i c deformation represented by d u c t i l i t y f a c t o r s , l o c a t i o n and p a t t e r n s of damage are examined. The two f i v e s t o r e y s t r u c t u r e s c o n s i d e r e d in the e l a s t i c t e s t i n g s e c t i o n are analyzed by the m o d i f i e d s u b s t i t u t e s t r u c t u r e method. The moment c a p a c i t i e s of beam members are chosen so that the maximum damage r a t i o i s around s i x . These 61 s t r u c t u r e s are a l s o subjected to analyses using the i n e l a s t i c time step program DRAIN-TABS3 with four d i f f e r e n t ground motions s c a l e d to the same peak a c c e l e r a t i o n . The r e s u l t s from the m o d i f i e d s u b s t i t u t e s t r u c t u r e method are compared to the average r e s u l t s of the four ground motions. Furthermore, to determine i f the proposed program can handle p o o r l y designed s t r u c t u r e s , the frame s t r u c t u r e i s analyzed with a sto r e y of weak columns and a sto r e y of strong beams. T h i s t e s t should i n d i c a t e whether the proposed method i s v i a b l e for s t r u c t u r e s with i r r e g u l a r s t r e n g t h p a t t e r n s . 6.2.1 Assumptions For Comparison With A Time Step A n a l y s i s  Program - DRAIN-TABS The i n e l a s t i c earthquake response program f o r three dimensional b u i l d i n g s , DRAIN-TABS 3, was used to compute the response h i s t o r y of t e s t s t r u c t u r e s . T h i s program 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 Berkeley. Two independent h o r i z o n t a l motions plus v e r t i c a l motion may be s p e c i f i e d as the ground motion. A b u i l d i n g i s i d e a l i z e d as a s e r i e s of plane frames i n t e r c o n n e c t e d by h o r i z o n t a l r i g i d diaphragms, with no enforcement of c o m p a t i b i l i t y f o r v e r t i c a l and r o t a t i o n a l displacements at j o i n t s common to two or more frames. Each frame can be of a r b i t r a r y geometry and can i n c l u d e a v a r i e t y of s t r u c t u r a l element types. These i n c l u d e beam-column elements i n which a x i a l - b e n d i n g i n t e r a c t i o n e f f e c t s can be taken i n t o account, and beam elements with degrading s t i f f n e s s , i n which an extension of Takeda's h y s t e r e t i c model i s used. 62 Since DRAIN-TABS was developed f o r the Berkeley CDC 6400 computer, the program had to be. converted before i t would work on the UBC Amdahl V/8 computer. Although w r i t t e n in FORTRAN IV, i t was subdivided i n t o o v e r l a y s i n order to reduce core storage requirements. T h i s technique was not necessary here because of the ample v i r t u a l memory. System dependent f e a t u r e s which were r e w r i t t e n i n c l u d e s e t t i n g the l e n g t h of the common block to be a r e l a t i v e l y l a r g e number i n s t e a d of s e t t i n g i t during e x e c u t i o n . A l s o i t was not necessary to open and c l o s e random access f i l e s i n our system. In a d d i t i o n , s e v e r a l s u b t l e yet c r i t i c a l d i f f e r e n c e s i n the two compilers caused some d i f f i c u l t i e s i n the conver s i o n process. I t should be noted that the program was developed f o r the CDC or s i m i l a r computers which have adequate word length to solve s t r u c t u r a l a n a l y s i s problems a c c u r a t e l y using s i n g l e p r e c i s i o n . Because the data blocks being s t o r e d and t r a n s f e r r e d c o n t a i n mixtures of i n t e g e r and r e a l v a r i a b l e s , c o n v e r s i o n to double p r e c i s i o n f o r the Amdahl was s a i d to be extremely d i f f i c u l t . T h e r e f o r e , no attempt to produce a double p r e c i s i o n v e r s i o n was made. However a plane frame was analyzed using DRAIN-TABS, and then using DRAIN-2D2 which has been converted i n t o double p r e c i s i o n . The r e s u l t s of the two programs showed no s i g n i f i c a n t d i f f e r e n c e s . The dynamic 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 s t r u c t u r e s r e q u i r e s the a p p r o p r i a t e member p r o p e r t i e s as input data. Since the modified s u b s t i t u t e s t r u c t u r e method employs a scheme i n which the cracked moment of i n e r t i a i s assumed to be h a l f of the 63 gross moment of i n e r t i a i f a x i a l compression i s present and a t h i r d otherwise, i t was decided that the same assumptions would be used in the time step a n a l y s i s to i n s u r e the e l a s t i c s t i f f n e s s to be the same i n both a n a l y s e s . The damping value f o r the program DRAIN-TABS was taken to be 2% of c r i t i c a l , and chosen as tangent s t i f f n e s s p r o p o r t i o n a l damping. Choosing a proper response spectrum i s beyond the scope of t h i s t h e s i s . T h e r e f o r e , the design spectrum 'A' in Shibata and Sozen's p a p e r 1 6 was used f o r the m o d i f i e d s u b s t i t u e s t r u c t u r e a n a l y s i s . Four of s i x earthquake r e c o r d s , from which spectrum 'A' was d e r i v e d , were used to compute the response h i s t o r i e s . They were T a f t N69W, T a f t S21W, E l Centro NS, and E l Centro EW. The records were a l l s c a l e d to give a peak a c c e l e r a t i o n equal to the d e s i r e d peak ground a c c e l e r a t i o n . The term d u c t i l i t y i s a u s e f u l index to d e s c r i b e the amount of i n e l a s t i c deformation in f l e x u r a l members, but c l a r i f i c a t i o n i n the d e f i n i t i o n s of d u c t i l i t i e s i s i n o r d e r . The c u r v a t u r e d u c t i l i t y , which i s the r a t i o of the curv a t u r e at u l t i m a t e to the cu r v a t u r e at y i e l d moment i s commonly used. I t depends on the s t r e s s - s t r a i n c h a r a c t e r i s t i c s of the concrete and s t e e l in the member. A l t e r n a t i v e l y , the member d u c t i l i t y may be determined i n terms of beam end r o t a t i o n s . Since the program DRAIN-TABS outputs the hinge r o t a t i o n s f o r each end of each member, the member d u c t i l i t y i n terms of end r o t a t i o n s i s adopted i n t h i s study. T h i s approach does not r e q u i r e an a r b i t r a r y assumption on the hinge l e n g t h as would have been the 64 case i f curva t u r e d u c t i l i t y was used. The sum of the y i e l d r o t a t i o n and the maximum hinge r o t a t i o n d i v i d e d by the y i e l d r o t a t i o n i s d e f i n e d as the member d u c t i l i t y f a c t o r . The y i e l d r o t a t i o n i s the angle developed when the member i s subjected to i t s y i e l d moment under anti-symmetric bending at the ends. The hinge r o t a t i o n i s computed at each end under p o s i t i v e and negative moments g i v i n g four p o s s i b l e v a l u e s , the l a r g e s t of which i s d e f i n e d as the member d u c t i l i t y . T h i s d e f i n i t i o n has the advantage of being n u m e r i c a l l y equal to the 'damage r a t i o ' used i n the modified s u b s t i t u t e s t r u c t u r e method f o r the e l a s t o -p l a s t i c cases. 6.2.2 Numerical Examples (a) F i v e - S t o r e y Frame S t r u c t u r e The f i v e - s t o r e y frame s t r u c t u r e of F i g . 6.1 was used as 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 . The frames in the d i r e c t i o n of a p p l i e d ground e x c i t a t i o n c o n s i s t of three 20-feet wide bays. The st o r e y height was 12 f e e t , g i v i n g a t o t a l b u i l d i n g height of 60 f e e t . The diaphragm weights f o r each f l o o r were 130 k i p s , and the c e n t r o i d s were d i s p l a c e d four f e e t from the centre of the s t r u c t u r e in the d i r e c t i o n p e r p e n d i c u l a r to the earthquake motion. T h i s represented a uniform e c c e n t r i c i t y of 10%. The cracked moments of i n e r t i a of columns were based on one-half of t h e i r gross s e c t i o n s , and the beams o n e - t h i r d of t h e i r gross s e c t i o n s . The e l a s t i c fundamental p e r i o d of v i b r a t i o n i n the d i r e c t i o n of the earthquake, the q 65 d i r e c t i o n , was determined to be 0.42 seconds. T h i s mode c o n t r i b u t e d predominantly to t r a n s l a t i o n a l motion in the q d i r e c t i o n , with a very small c o n t r i b u t i o n to the r o t a t i o n a l motion. The y i e l d moments of the columns and beams were 1200 le-f t and 200 k - f t r e s p e c t i v e l y . For reasons of s i m p l i c i t y , no s t r a i n hardening a f t e r y i e l d was assumed. A time step of 0.01 seconds was used f o r numerical i n t e g r a t i o n . The maximum ground a c c e l e r a t i o n was 0.5g. In the modified s u b s t i t u t e s t r u c t u r e a n a l y s i s , i t took 5 i t e r a t i o n s to s a t i s f y the convergence c r i t e r i o n . The CPU time on the Amdahl V/8 computer was 18.6 seconds. For the l a s t i t e r a t i o n the n a t u r a l p e r i o d of the s u b s t i t u t e s t r u c t u r e f o r the lowest q mode was 0.66 seconds. The l a t e r a l displacements of frame 4, which undergoes the most deformation, are shown i n Table 6.3. The damage r a t i o s f o r t h i s frame are shown i n F i g . 6.3. The second f l o o r beams had the l a r g e s t damage r a t i o of 4.4. A l l the columns remained i n the e l a s t i c range. Response h i s t o r i e s to four earthquake motions were computed by DRAIN-TABS. The f i r s t 10 seconds of these r e c o r d s were used with the exception of E l Centro EW, where the f i r s t 12 seconds were used. The CPU time was about 100 sec. f o r each time step a n a l y s i s . The maximum l a t e r a l displacements of frame 4 f o r each motion and the average of the four motions are shown in Table 6.3. S i m i l a r l y , the maximum damage r a t i o s f o r each motion and the average values are shown i n F i g . 6.3. The beam damage r a t i o s are p l o t t e d i n F i g . 6.3.1. The E l Centro EW 66 earthquake r e s u l t e d i n much higher damage r a t i o s than the other three motions. The r e s u l t s are very encouraging, as 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 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 requirements and d e f l e c t i o n s . The p r e d i c t i o n s are s l i g h t l y on the unconservative s i d e , when compared to the average of four time step runs, mainly because of the l a r g e damage r a t i o s r e s u l t i n g from EL Centro EW. N e v e r t h e l e s s , they were a l l w i t h i n 20% of the average v a l u e s . (b) F i v e - S t o r e y Frame S t r u c t u r e With Four Times The Mass The second t e s t s t r u c t u r e was the same s t r u c t u r e as i n the l a s t s e c t i o n with one ex c e p t i o n . The o r i g i n a l mass of 130 kips was i n c r e a s e d by a f a c t o r of four so that the r e v i s e d mass was 520 k i p s . The purpose of t h i s was to emulate a t a l l e r and thus more f l e x i b l e s t r u c t u r e without adding more j o i n t s and members. A l s o i t had the e f f e c t of s h i f t i n g the fundamental undamaged p e r i o d to a d i f f e r e n t part of the spectrum. The fundamental p e r i o d was doubled from 0.42 sec. to 0.84 sec. To avo i d y i e l d i n g i n the columns and to keep the beam damage r a t i o s to a reasonable l e v e l , t h e i r moment c a p a c i t i e s were i n c r e a s e d to 2000 k - f t f o r the beams and 333 k - f t for the columns. The maximum ground a c c e l e r a t i o n was unchanged. F i g s . 6.4 and 6.4.1 show the damage r a t i o r e s u l t s while Table 6.4 shows the frame d e f l e c t i o n r e s u l t s . These r e s u l t s are not too d i f f e r e n t from before, p r o v i d i n g evidence 67 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 can p r e d i c t both d e f l e c t i o n and d u c t i l i t y requirements for a range of frame s t r u c t u r e s . (c) F i v e - S t o r e y Coupled Wall S t r u c t u r e The next t e s t was performed on the f i v e - s t o r e y coupled, w a l l s t r u c t u r e d e s c r i b e d e a r l i e r in t h i s c h a pter. The c e n t r e s of mass at each diaphragm l e v e l were d i s p l a c e d 2 f e e t in the r d i r e c t i o n from the center of the s t r u c t u r e . T h i s represented a uniform e c c e n t r i c i t y of 3% throughout the h e i g h t . The s i z e of the core, which c o n s i s t e d of two channel shape s e c t i o n s j o i n e d by two c o u p l i n g beams, measured 30 feet by 18 f e e t . The dimensions of the diaphragms were 100 feet by 60 f e e t . The c o u p l i n g beam c a p a c i t y was 100 k i p - f t and the maximum ground a c c e l e r a t i o n was 0.2g. The i n i t i a l e l a s t i c p e r i o d was 0.38 sec. and the diaphragm weight was 300 k i p s per f l o o r . F i g u r e s 6.5 and 6.5.1 show damage r a t i o r e s u l t s f o r four time step analyses and compare these to 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 comparison shows 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 a n a l y s i s p r e d i c t i o n s are i n e x c e l l e n t agreement with the average of four time step runs. In most members, the p r e d i c t i o n s are on the c o n s e r v a t i v e s i d e . The d e f l e c t i o n estimates from 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 a n a l y s i s and from DRAIN-TABS are given in Table 6.5. The average of the time step runs i n d i c a t e s a top f l o o r d e f l e c t i o n of 1.1.4 inches , while the proposed 68 method p r e d i c t s t h i s d e f l e c t i o n to be 1.49 inches.. In t h i s t e s t , the T a f t N69W motion produced 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. (d) F i v e - S t o r e y Coupled Wall S t r u c t u r e With Four Times The Mass As i n the case of the frame s t r u c t u r e , the mass of the coupled w a l l s t r u c t u r e was i n c r e a s e d four f o l d . The fundamental e l a s t i c p e r i o d doubled to 0.76 seconds. The c o u p l i n g beam c a p a c i t y was i n c r e a s e d to 130 k i p - f t to keep the damage r a t i o to a reasonable l e v e l . The average damage r a t i o s from the time step analyses are compared to those p r e d i c t e d by 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 i n F i g s . 6.6 and 6.6.1. Again, the beam damage r a t i o s are c o n s e r v a t i v e l y p r e d i c t e d with the worst one o f f by 17%. The p r e d i c t i o n on column moments was a l i t t l e worse than f o r the beams. The average l a t e r a l displacements from time step runs and those p r e d i c t e d are shown i n Table 6.6. The top d e f l e c t i o n was 3.42 i n . f o r the method while the average top d e f l e c t i o n f o r the four time step analyses was 2.44 i n . 6.2.3 Costs of Execution The computing time and c o s t s of the four i n e l a s t i c t e s t s t r u c t u r e s are . summarized in Table 6.7. In a l l cases the charges do not in c l u d e the cost of p r i n t i n g the outputs. They are f o r low p r i o r i t y batch jobs run using r e s e a r c h r a t e s . The 69 same job running using a commercial account and at high p r i o r i t y c o u l d cost as much as t h i r t y f i v e times more. With t h i s i n mind, savings i n r e a l d o l l a r s with 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 over a time step a n a l y s i s can be s i g n i f i c a n t . Furthermore, the c o s t s of u s i n g a time step a n a l y s i s program q u i c k l y accumulates i f i t i s decided to t e s t the s t r u c t u r e with more than one earthquake r e c o r d . 6.3 E f f e c t s of Strong Beams and Weak Columns A study was made of the e f f e c t s of strong beams and weak columns in s e l e c t e d areas over the height of the s t r u c t u r e . The f i v e - s t o r e y frame s t r u c t u r e with four times the o r i g i n a l mass d e s c r i b e d i n s e c t i o n 6.2.2(b) was t e s t e d with the t h i r d f l o o r columns weakened from 2000 k - f t to 500 k - f t , and the f i f t h f l o o r beams strengthened from 333 k - f t to 666 k - f t . Otherwise the p r o p e r t i e s of the s t r u c t u r e were unchanged. The purpose of t h i s a n a l y s i s was to see i f the m o d i f i e d s u b s t i t u t e s t r u c t u r e method can p i c k out the weak and strong spots i n the s t r u c t u r e . Damage r a t i o s are shown in F i g s . 6.7 and 6.7.1. As expected, the columns on the t h i r d f l o o r y i e l d e d in both a n a l y s e s . The modified 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 a damage r a t i o f o r the i n t e r i o r column of 2.4 while the average of the four time step a n a l y s e s was 3.1. The e x t e r i o r column had a p r e d i c t e d value of 1.8 and the averaged value was 4.4. The damage r a t i o s of the f i f t h f l o o r beams were in e x c e l l e n t agreement, but those of the f o u r t h f l o o r beams were 70 o v e r e s t i m a t e d . A l t h o u g h t h e p r e d i c t i o n s were n o t a s good a s t h o s e f o r s t r u c t u r e s of r e g u l a r s t r e n g t h p a t t e r n s , t h e y were s t i l l r e a s o n a b l e . The method was a t l e a s t a b l e t o p r e d i c t t h e g e n e r a l p a t t e r n o f damage. 71 CHAPTER 7 CONCLUSIONS Thi s t h e s i s p r e s e n t s the modified s u b s t i t u t e s t r u c t u r e method as an a n a l y s i s procedure for determining the i n e l a s t i c s e ismic response of m u l t i - s t o r e y three dimensional 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 . The method u t i l i z e s the e l a s t i c modal a n a l y s i s technique and extends i t i n t o the i n e l a s t i c range by re f e r e n c e to a f i c t i t i o u s l i n e a r s t r u c t u r e whose member s t i f f n e s s and damping c h a r a c t e r i s t i c s are determined as a f u n c t i o n of the maximum d u c t i l i t y f a c t o r . The p r i n c i p a l o b j e c t i v e i s to bridge the gap between l i n e a r e l a s t i c modal a n a l y s i s and time step i n e l a s t i c response a n a l y s i s . The method r e t a i n s the s i m p l i c i t y and economy of a modal a n a l y s i s while at the same time p r o v i d i n g d i r e c t information on the d u c t i l i t y demands on the members at a f r a c t i o n of the co s t i n c u r r e d with a time step a n a l y s i s . The two s t r u c t u r e s t e s t e d in t h i s study are f i v e -s t o r e y s in height, one uses moment r e s i s t i n g frames and ' the other coupled w a l l cores as l a t e r a l r e s i s t i n g systems. In both cases, the analyses are repeated with the storey weights i n c r e a s e d four times to p l a c e the fundamental p e r i o d of the s t r u c t u r e on a d i f f e r e n t p o r t i o n of the spectrum. The p r e l i m i n a r y f i n d i n g s , as determined by comparison of damage 72 r a t i o s w i t h t h e a v e r a g e v a l u e s o f i n e l a s t i c t i m e s t e p a n a l y s i s u s i n g f o u r d i f f e r e n t g r o u n d m o t i o n s , i n d i c a t e t h e method w o r k s w e l l f o r a l l c a s e s . The f i v e - s t o r e y f r a m e s t r u c t u r e was f u r t h e r i n v e s t i g a t e d w i t h a s t o r e y o f s t r o n g beams a n d a s t o r e y of weak c o l u m n s . The method d i d n o t work a s w e l l b u t was s t i l l a b l e t o p r e d i c t t h e g e n e r a l p a t t e r n o f damage t h r o u g h o u t t h e s t r u c t u r e . I t i s hoped t h a t f u r t h e r r e s e a r c h w o u l d c l a r i f y t h e r e q u i r e m e n t s on a s t r u c t u r e f o r s u c c e s s f u l a p p l i c a t i o n o f t h e method. A l t h o u g h n o t p e r f e c t e d , t h e m o d i f i e d s u b s t i t u e s t r u c t u r e method i s a f a s t a n d e f f e c t i v e way of e s t i m a t i n g t o r s i o n a l r e s p o n s e s a n d d u c t i l i t y demands o f b u i l d i n g s w i t h l a r g e e c c e n t r i c i t i e s . The method i s s i m p l e a n d i n e x p e n s i v e t o use a n d c a n be m o d i f i e d , t o r u n on m i c r o - c o m p u t e r s f o u n d i n many d e s i g n o f f i c e s . EAcos'0 •+ 12EIs1n'e (1+a)L 3 EAcos8s<nfl L 12EIcosesine (1+a)L' - 12EIsin0Li (1+a)L 3 - 6EIs1n0 1 (1+a)L' EAcos'fl L ! 12EIs1n'6 (1+a)L' EAcos6s1n0 + 12EIcosfl5ine (1+a)L' EAsin'0 + 12EIcos ;fl (1+a)L J 12EIcosgLi ( 1+a)L' + 6EIcosfl (1+a)L ! EAcos0s1n0 + 12EIcos0sin0 ( 1+a)L 5 EAsin'0 12EIcos'fl (1+a)L 5 - 12EIsinflL; (1+a)L' 6EIsin0 (1+a)L' 12EIcoseLi ( 1+a)L' + 6EIcosQ ( 1+a)L' (1+a)L ] + 12EILi (1+a)L' + (4+a)EI (1+o)L 12EIslneLi (1+a)L' + 6EIs1ng (1+a)L' 12EIcoseL t (1+a)l' - 6EIcosg ' (1+a)L ! 12EIL.L? + 6EIL. (1+a)L' (1+a)L' + 6EIL; +(2-a)EI (1+a)L' (1+a)L EAcos'O L + l2EIsin ;e (1+a)L 3 EAcosgs i ne 12EIcosesine ( 1 + a)L' 12EIsin6L; ( 1+a)L ] + 6EIsin0 ( 1+a)L! (SYMMETRICAL) EAsln'S 12EIcos'6 (1+a)L ] 12EIcoseL; ( 1+a)L ] 6EI cose (1+a)L ! 12EIL;' .+ 12EIL, (1+a)L 3 (l+a)L ! + (4+a)EI (1+a)L T a b l e 2.1 Member s t i f f n e s s m a t r i x i n c l u d i n g r i g i d ends 74 D a m p i n g , % of C r i t i c a l A m p l i f i c a t i o n F a c t o r f o r S p e c t r a l Bounds A c c e l e r a t i o n Veloc i t y Displacement 0 , 6.4 4.0 2.5 .5; 5.8 3.6 2.2 1 5.2 3.2 2.0 2 4.3 2.8 1 .8 5 2.6- 1 .9 1 .4 7 ' 1.9 1.5 1 .2 10 1 .5 1 .3 1 . 1 20 1 .2 1 .1 1 .0 Table 4.1(a) A m p l i f i c a t i o n f a c t o r s f o r ground motion bounds recommended by Newmark. Damping, \ % of C r i t i c a l A m p l i f i c a t i o n F a c t o r f o r S p e c t r a l Bounds A c c e l e r a t ion Veloc i t y Di splacement .5 5.8 3.3 3.0 2 4.2 2.5 2.5 3 3.8 2.4 2.4 5 3.0 2.0 2.0 .10 2.2 1 .7 1 .7 Table 4.1(b) A m p l i f i c a t i o n f a c t o r s f o r ground motion bounds recommended by NBC. EARTHQUAKE GROUND MOTION BOUNDS ACCELERATION BOUND g VELOCITY BOUND g*sec (in/sec ) DISPLACEMENT BOUND g*se.c**2 (in) NEWMARK1S IDEALI ZED 1.0 0.1246 (48.14) 0.0934 (36.10) NBC 1.0 0. 1035 (40.00) 0.0828 (.32.00) EL CENTRO 1940 NS 0.348 0.0340 (13.14) 0.0111 (4.29) EL CENTRO 1940 EW 0.214 0.0376 (14.53) 0.02016 (7 .79) TAFT 1952 N69W 0.179 0.0180 (6.96) 0.00937 (3.62) TAFT 1952 S21W 0. 1554 0.0160 (6.18) 0.00662 (2.64) CAL. TECH. D1 0.485 0.0278 (10.74) 0.0049 (1 . 8 9 ) CAL. TECH. D2 0.492 0.0299 (11.55) 0.0072 (2.78) g / f t ( g * s e c ) / f t ( g * s e c * * 2 ) / f t ' EL CENTRO 1940 TORSION 0.00347 0.000422 0.00445 TAFT 1952 TORSION 0.00128 0.000128 0.000381 CAL. TECH. D.TORSION 0.004152 0.000336 0.0000474 T a b l e 4.2 B a s i c g r o u n d m o t i o n bounds 76 ELASTIC PERIODS (SECONDS) MODE PITSA ETABS 1 0.5008 0.5006 2 0.4191 0.4189 3 0.2688 0.2686 4 0.1545 0.1544 5 0.1291 0.1290 6 0.0836 0.0836 FRAME 4 - RSS BENDING MOMRENTS ( k i p - f t ) MEMBER PITSA ETABS INTERIOR 5th FL. 228 229 BEAMS 4th 473 476 3rd 678 683 2nd 808 814 1 S t 742 747 INTERIOR 5th FL. 457 460 COLUMNS 4th 690 695 3rd 858 864 2nd 955 962 1 S t 1 1 72 1181 Table 6 . 1 . E l a s t i c modal a n a l y s i s r e s u l t s f o r f i v e - s t o r e y frame b u i l d i n g ( F i g . 6.1) su b j e c t e d t o spectrum 'A' 77 ELASTIC PERIODS (SECONDS) MODE PITSA ETABS 1 1.0958 1 .0950 2 0.5840 0.5839 3 0.3810 0.3812 4 0.2221 0.2216 5 0.0948 0.0950 6 0.0915 0.0914 WALL 4 - RSS BENDING MOMRENTS ( k i p - f t ) MEMBER PITSA ETABS COUPLED 5th FL. 265.9 271 . 1 BEAMS 4th 317.2 322.2 3rd 360 . 5 363 . 3 2nd 352.2 352.1 1 S t 250.6 249.2 COUPLED 5th FL. 960.1 978.8 WALLS 4th 1234.2 1229.8 3rd 1168.9 1196.8 2nd 2270.3 2269.7 1 S t 3858.0 3825.2 Table 6.2 E l a s t i c modal a n a l y s i s r e s u l t s f o r f i v e - s t o r e y coupled w a l l b u i l d i n g ( F i g . 6.2) subjected to spectrum 'A* 78 LATERAL DISPLACEMENT ( i n . ) FLOOR NO. DRAIN-TABS TAFT N69W S21W EL CENTRO NS EW 5 4 3 2 1 3.32 2.81 3.10 5.03 3.05 3.55 2.88 4.52 2.45 2.71 2.36 3.50 1 .50 1 .63 1 .50 2.08 0.50 0.55 0.53 0.68 DRAIN-TABS FLOOR NO, PITSA AVERAGE Table 6.3 Comparison of l a t e r a l displacement of frame 4 f o r f i v e - s t o r e y frame s t r u c t u r e 79 LATERAL DISPLACEMENT ( i n . ) DRAIN -TABS FLOOR NO. TAFT EL CENTRO N69W S21W NS EW 5 6.49 4.96 6.24 8.95 4 5.64 4.44 5 . 8 1 7.68 3 4.22 3.66 4.75 5.65 2 2.41 2.36 2.98 3.28 1 0.77 0.85 1.02 1.07 DRAIN-TABS FLOOR NO. PITSA AVERAGE 5 7.97 6.66 4 6.95 5.89 3 5.30 4.57 2 3.19 2.76 1 1 .09 0.93 Table 6.4 Comparison of l a t e r a l displacement of frame 4 fo r f i v e - s t o r e y frame s t r u c t u r e with r e v i s e d mass 80 LATERAL DISPLACEMENT ( i n . ) DRAIN -TABS FLOOR NO. TAFT N69W S7 1W EL CENTRO NS EW 5 4 3 2 1 1.37 1.03 1.01 0.78 0.66 0.52 0.34 0.26 0.10 0.08 0.86 1.30 0.66 0.97 0.44 0.64 • 0.24 0.32 0.07 0.10 DRAIN-TABS FLOOR NO. PITSA AVERAGE 5 4 3 2 1 1 .49 1.10 0.71 0.36 0.10 1.14 0.86 0. 56 0.29 0.09 Table 6.5 Comparison of l a t e r a l displacement of wall 4 for f i v e - s t o r e y coupled w a l l s t r u c t u r e 8 1 LATERAL DISPLACEMENT ( i n . ) FLOOR NO. DRAIN-TABS TAFT N69W S21W EL CENTRO NS EW 5 4 3 2 1 1 .98 1 .45 0.95 0.49 0.14 09 50 0.94 0.50 0.16 2.66 1 .98 1.31 0. 68 0.20 3 . 0 1 2.16 1 . 3 4 0.67 0 . 1 9 DRAIN-TABS FLOOR NO. PITSA AVERAGE 3.42 2.49 1 .60 0.82 0.24 Table 6.6 Comparison of l a t e r a l displacement of w a l l f o r f i v e - s t o r e y coupled w a l l s t r u c t u r e with r e v i s e d mass 4 PITSA DRAIN-TABS (ONE ANALYSIS) NO. OF ITERATIONS CPU TIME COMPUTER $ TIME STEP CPU TIME COMPUTER $ FRAME STRUCTURE 5 18.6 sec. $1 .30 1000 @ 0.01 sec. 98.9 sec. $5.15 FRAME STRUCTURE REVISED MASS 8 20.7 sec. $2. 38 1000 @ 0.01 sec. 71.2 sec. $6.18 WALL STRUCTURE 5 5.6 sec. $0.63 1000 @ 0.01 sec. 30.0 sec. $2.60 WALL STRUCTURE REVISED MASS 5 6.9 sec . $0.81 1000 @ 0.01 sec . 29.5 sec. $2.55 Table 6.7 Costs of execution 83 F i g u r e 1.1 F l o o r p l a n of t h e J . C . Penney B u i l d i n g i n A n c h o r a g e , A l a s k a , showing t h e h i g h l y e c c e n t r i c -s h e a r w a l l c o n f i g u r a t i o n . 84 F i g u r e 1 .2 E a s t w a l l and n o r t h e a s t c o r n e r of the J . C . Penney B u i l d i n g , a f t e r t h e 1 9 6 4 e a r t h q u a k e . T h i s shows the c o m p l e t e c o l l a p s e of t h e s h e a r w a l l and p o r t i o n s of t h e r o o f and f l o o r s a t t h e n o r t h e a s t c o r n e r of t h e b u i l d i n g . 8 5 7777 7777 7777 F i g u r e 2.1(a) Gross frame degrees of -freedom 7777 7777 7777 F i g u r e 2.1(b) Condensed frame degrees of freedom 86 F i g u r e 2 . 1 ( c ) Gross s t r u c t u r e degrees of freedom >7r F i g u r e 2 . 1 ( d ) Condensed s t r u c t u r e degrees of freedom 87 F i g u r e 2.2 Plan view of n f l o o r showing frame and diaphragm h o r i z o n t a l displacements. 88 F i g u r e 3.1 P h y s i c a l 6 I n t e r p r e t a t i o n of damage r a t i o SPECTRAL VELOCITY S v , IN/SEC O M. • iQ cn C U3 i-l fD 3 Cu U3 z >-| CD O « C 3 D CD n ui o CD M-H-> a CD CD r( (3J Qj h-' r t M-i-" N O CD 3 D i CD cn i—• o\° CD cn D-j n-OJ M-3 o TJ M- a r> co i£> in - M' D U) T5 CD n r r i-t C . 3 50 100 200 500 lOOO t 6 8 90 o 0.2 0.4 0.6 0.8 1.0 RATIO OF PERIODS, r F i g u r e 4.2 P l o t of cross-modal c o e f f i c i e n t s v s . r a t i o of p e r i o d s 91 20 777 777 20 777 o II C M < CO X u « o E-to tn ELEVATION  TYPICAL FRAME PLAN F i g u r e 4.3 F i v e s t o r e y b u i l d i n g example PERIODS(SEC) 0.5199 0.5008 0.3522 0.1604 0.1545 0.1089 Figure 4.4 Periods and d i r e c t i o n s of mode shapes 93 ro CM 3 7 6 . 1 324 .9 2 3 7 . 0 553 .4 •=}< CN CO CO co CO TIME HISTORY CN 2 5 5 . 0 ro CN "3< CN o oo ABSOLUTE SUM 283. 1 CO m in CM TIME-STEP ANALYSIS USING TAFT 1952 N69W RESPONSE SPECTRUM ANALYSIS USING SPECTRUM 'A' F i g u r e 4.5 Comparison of modal combination methods 94 PITSA Read i n . d a t a - s t r u c t u r e , earthquake B u i l d s [ K ], [ M ] C a l c u l a t e p e r i o d s , mode shapes •* S u b s t i t u t e S t r u c t u r e Method C a l c u l a t e smeared damping r a t i o Perform spectrum a n a l y s i s E v a l u a t e f o r c e l e v e l s Modify damage r a t i o s Modify E I , member damping Check f o r convergence — YES F i g u r e 5.1 General conceptual o u t l i n e of.PITSA 95 MAIN MEMGEN MMATRX BUILDM MAINF REDUCE GETAR MAINR PART SYMM BANDED BANDEF REDUCE GETAR MAIND FREQ SPECTR FORCES DEFLNS FORSP FORCEV LOWTRI DSYMFU DSTURM PARFAC SRUM F i g u r e 5.2 Program (PITSA) o r g a n i z a t i o n 96 O . O 1 1 1 1 1 1 1 1 i i 0.5 1.0 1.5 2.0 2.5 3.0 12 10 8 6 4 2 Pe r i o d , sec Frequency, h e r t z S p e c t r a l A c c e l , f o r ft = 8 S p e c t r a l A c c e l , f o r 0=0.02 6+100-0 F i g u r e 5.3 Spectrum 'A' 3.0 Period, sec Figure 5.4 A c c e l e r a t i o n spectra of T a f t N69W and spectrum 'A' 3.0 I 1 1 I L _ 0.0 0.5 1.0 1.5 2.0 P e r i o d , sec Figure 5.5 A c c e l e r a t i o n spectra of T a f t S21W and spectrum 'A1 P e r i o d , sec Figure 5.6 A c c e l e r a t i o n s p e c t r a of E l Centro EW and spectrum 'A' 3.0 E l Centro NS Spectrum and Spectrum A .01 .02 .05 .1 .2 .5 1 2 5 10 20 50 100 PERIOD, SEC Figure 5.8 Ground motion bound t r i p a r t i t e p l o t of four earthquakes which make up Spectrum 'A' ( s c a l e d to 0.5g) 1 02 COLUMNS 30x30 A=900 1=67500 l ( c r)=33750 BEAMS 18x36 A=648 1=69984 K c r ) = 2 3 3 2 8 S T O R E Y W E I G H T = 130 k 130 k 130 k 130 k 130 k 20' 20* 20' Q 20' 20' MASS C E N T R > < LJJ t I \ / 1 > \ > o I o O I CN E-< CO X w o CO ELEVATION OF FRAMES 1 & 4 ELEVATION OF FRAMES 2 & 3 D -CN E/Q MOTION FRAME 4 - o - a -FRAME 1 - a ro LJ < PLAN F i g u r e 6.1 Dimensions and p r o p e r t i e s of f i v e - s t o r e y frame b u i l d i n g 103 1 2 ' - 0 " S T O R E Y W E I G H T = 3 0 0 k 3 0 0 k 3 0 0 k 3 0 0 k 3 0 0 k C H A N N E L S A = 6 6 6 4 1 = 1 5 3 6 4 4 4 I ( c r ) = 7 6 8 2 2 2 E/.Q MOTION 4 ' - 2 ' 14 CM a i < ,4+ 1 2 ' - 0 " 1 8 ' - 0 " 0  ^ B E A M S 1 2 x 3 6 A = 4 3 2 1 = 5 1 8 4 I ( c r ) = 1 7 2 8 ELEVATION OF WALLS 1 t 4 WALL 4 r u 2 1 ' - 8 ' WALL 1 PLAN co 4 ' - 2 ' C MASS C E N T R E ELEVATION OF WALLS 2 & 3 o 10 > FLOOR AREA - 100'X60' CORE AREA - 30'X18' F i g u r e 6.2 Dimensions and p r o p e r t i e s of f i v e - s t o r e y coupled w a l l b u i l d i n g 104 O 72 0 64 o 12 0 5 6 0 21 0 52 o 10 2 0 0 0 17 1 8 9 0 22 1 41 0 3 0 1 27 0 19 3 8 5 0 28 3 7 1 b 23 3 0 0 0 31 2 8 6 0 2 1 5 OO 0 2 9 4 8 6 0 2 3 4 2 5 0 33 4 1 1 0 2 0 4 0 8 0 3 0 3 8 5 0 4 9 3 78 0 5 2 3 5 5 0 4 8 0 51 TAFT N69W TAFT S21W 0 7 0 0 62 0 12 1 52 0 2 1 1 0 6 0 17 1 82 0 33 1 6 9 0 21 3 68 0 3 0 3 65 0 3 0 3 58 0 37 3 44 0 23 6 3 1 0 31 6 15 0 26 4 8 1 0 34 4 67 0 23 7 38 0 33 7 25 0 31 4 15 0 4 1 3 92 0 52 5 6 9 0 5 5 5 45 0 6 3 0 66 EL CENTRO NS EL CENTRO EW 0.88 0.72 0.88 0.71 0.15 2.25 0.25 2.15 0.13 2 .23 0.23 2.12 0.24 3.74 0.32 3.62 0.23 4 . 1 9 0.31 4. 04 0.22 4 . 44 0.30 4.33 0.23 5.36 0.31 5.22 0.25 3.63 0.35 3.46 0.24 4.43 0.34 4.19' 0.52 0.56 0. 53 0.56 PITSA AVERAGE OF FOUR TIME STEP ANALYSIS F i g u r e 6.3 Damage r a t i o s for f i v e - s t o r e y frame s t r u c t u r e 1 05 TAFT N69W TAFT S21W -0 EL CENTRO NS -• EL CENTRO EW AVERAGE - PITSA O 2 o o 3 4 DUCTILITY F i g u r e 6.3.1 Beam damage r a t i o s f o r f i v e - s t o r e y frame s t r u c t u r e 1 0 6 1 67 1 19 0 17 1 01 0 34 0 86 0 17 3 43 0 29 3 4 1 0 25 2 64 0 34 2 57 0 2 1 4 98 0 29 4 80 0 24 3 67 0 32 3 52 0 21 5 40 0 30 5 27 0 27 4 38 0 37 4 22 0 22 3 78 0 31 3 55 0 44 3 98 0 48 3 '75 0 51 | 0 54 TAFT N69W TAFT S21W 2. 29 1 . 92 0. 1 7 3. 62 0.34 3. 57 0. 25 4. 77 0.31 4. 63 0. 23 5. 1 6 0.30 5. 04 0. 28 4. 23 0.38 4. 06 0. 62 0.65 PITSA 0 89 0 77 0 15 2 83 0 26 2 36 0 17 2 26 0 34 2 13 0 24 4 68 0 33 4 64 0 33 4 30 0 39 4 15 0 26 6 96 0 34 6 80 0 26 5 88 0 34 5 73 0 26 7 23 0 36 7 o r 0 34 4 96 0 44 4 73 0 58 5 30 0 62 5 06 0 61 0 65 EL CENTRO NS EL CENTRO EW 1 .60 1 .29 0. 1 7 3.25 0 .31 3 . 1 9 0. 26 4.98 0 . 34 4 .82 0. 24 5.72 0 . 33 5 . 56 0. 27 4.51 0 .37 4 .27 0. 54 0 . 57 AVERAGE OF FOUR TIME STEP ANALYSIS F i g u r e 6.4 Damage r a t i o s f o r f i v e - s t o r e y frame s t r u c t u r e with r e v i s e d mass 1 0 7 -* TAFT N69W + TAFT S21W O- -O EL CENTRO NS B- -Q EL CENTRO EW AVERAGE PITSA O 2 « o o 3 4 DUCTILITY F i g u r e 6 . 4 . 1 Beam damage r a t i o s f o r f i v e - s t o r e y frame s t r u c t u r e with r e v i s e d mass 108 o. o. o. o. 3.74 2 .09 08 2 .67 08 3 .43 0 08 3 . 85 0 08 2 . 26 12 2 . 83 0 10 3 . 57 10 3 . 70 0 10 ' 2.31 23 2 . 8 0 14 3 . 50 0 18 ' 3 .09 0 20 2 .05 35 2.42 0 24 2 . 99 29 1 .92 0 34 1 . 34 52 1 .52 0 38 1 . 87 42 0 5 1 TAFT N69W EL CENTRO TAFT S21W EL CENTRO 3.37 0.07 0.13 0.24 0.38 0.54 3.44 3.32 2.84 1 .86 PITSA 2.98 0.08 0.10 0.19 0.31 0.46 3.13 3.08 2.64 1 .66 AVERAGE OF FOUR TIME STEP ANALYSIS F i g u r e 6.5 Damage r a t i o s f o r f i v e - s t o r e y coupled wall s t r u c t u r e 109 o z o o J 2 --e TAFT N69W TAFT S21W EL CENTRO NS -Q EL CENTRO EW AVERAGE •- PITSA 3 4 DUCTILITY F i g u r e 6.5.1 Beam damage r a t i o s f o r f i v e - s t o r e y coupled wall s t r u c t u r e 1 1 0 07 09 18 2 1 25 27 0 . 32 0 . 33 0 . 46 0 .52 4 .70 5 .99 4 88 08 7 .05 0 4 66 0 10 5.99 4 88 17 7 .02 0 4 13 0 20 5 . 57 4 42 6 38 0 27 3 44 0 34 4 . 76 3 43 4 95 0 42 2 13 0 46 3.05 2 26 , 2,93 0 66 0 64 TAFT N69W EL CENTRO NS TAFT S21W EL CENTRO EW 0.15 0.29 0.37 0.51 0. 77 6.03 5.97 5.52 4.61 3.03 PITSA 0.08 0.19 0.29 0.38 0.57 5.65 5.64 5.13 4.14 2.59 AVERAGE OF FOUR TIME STEP ANALYSIS F i g u r e 6.6 Damage r a t i o s f o r f i v e s t r u c t u r e with r e v i s e d - s t o r e y coupled w a l l mass 111 o-Q-TAFT N69W TAFT S21W -O EL CENTRO NS -Q EL CENTRO EW AVERAGE PITSA O z CC o o j Ci-. 2 — 3 4 DUCTILITY F i g u r e 6 . 6 . 1 Beam damage r a t i o s f o r s t r u c t u r e w i t h r e v i s e d f i v e - s t o r e y c o u p l e d w a l l mass 1 1 2 2 04 2 .03 0 17 3 16 • O 34 3 05 0 18 0 35 1 28 1 16 0 35 1 73 0 56 1 60 0 40 0 62 1 77 1 73 3 4 1 2 15 2 13 2 26 5 56 4 19 3 78 3 15 O 4 3 5 34 0 60 4 70 0 45 0 62 5 66 5 44 0 77 6 73 0. 80 6 50 0. 85 0 . 87 1 80 1 87 0 18 1 26 0 35 1 24 0 17 1 17 0 31 1 04 0 34 0 93 0 54 0 85 0 29 1 63 0 48 1 5 1 4 49 1 39 3 1 7 1 23 4 2 1 4 47 2 92 3 85 0 43 4 38 0 59 3 77 0 4 3 6 34 0 60 6 12 0 85 6 05 0 88 5 83 0 80 0 82 TAFT N69W TAFT S21W EL CENTRO NS EL CENTRO EW 1 . 30 1 . 0 . 1 6 1 .29 0.32 1 . 0.36 4. 48 0.52 4. 1 .82 5.75 2.40 5. 0.31 4 .70 0.41 4. 0.68 0.71 2.07 2.05 0.17 0. 34 4 . 42 0.43 0.82 1 .28 0.34 1.17 1 . 7 4 ° - 5 5 1.68 4 . 4 9 3 • 1 0 3.87 6 . 1 9 0.60 5 > 9 7 0.84 PITSA AVERAGE OF FOUR TIME STEP ANALYSIS F i g u r e 6.7 Damage r a t i o s f o r f i v e - 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J a p a n E a r t h q u a k e E n g i n e e r i n g Symposium, T o k y o , O c t o b e r 1966 , pp. 2 2 7-32. 7. T a k e d a , T., S o z e n , M. A., and N i e l s e n , N. N., " R e i n f o r c e d C o n c r e t e R e s p o n s e t o S i m u l a t e d E a r t h q u a k e s , " J o u r n a l o f t h e S t r u c t u r a l D i v i s i o n , ASCE, V o l . 96, December 1970, pp. 2557 - 7 3 . 8. G u l k a n , P., and S o z e n , M. A., " R e s p o n s e and E n e r g y - D i s s i p a -t i o n o f R e i n f o r c e d C o n c r e t e F rames S u b j e c t e d t o S t r o n g Base M o t i o n s , " C i v i l E n g i n e e r i n g S t u d i e s , S t r u c t u r a l R e s e a r c h S e r i e s No. 377, U n i v e r s i t y o f I l l i n o i s , U r b a n a , May 1971. 9. T s o , W. K., and Dempsey, K. M., " S e i s m i c T o r s i o n a l P r o v i s -i o n s f o r D y n a m ic E c c e n t r i c i t y , " E a r t h q u a k e E n g i n e e r i n g and S t r u c t u r a l D y n a m i c s , V o l . 8, 1980, pp. 2 7 5 - 2 8 9 . 10. K a n , C. L., and C h o p r a , A. K., " E f f e c t o f T o r s i o n a l C o u p l -i n g on E a r t h q u a k e f o r c e s i n B u i l d i n g s , " J o u r n a l o f t h e S t r u c t u r a l D i v i s i o n , ASCE, V o l . 103, A p r i l 1977, pp. SOS-S i 9. 1 1 5 1 1 . Kan, C. L., and C h o p r a , A. K., " T o r s i o n a l C o u p l i n g and E a r t h q u a k e R e s p o n s e of S i m p l e E l a s t i c and I n e l a s t i c S y s t e m s ," J o u r n a l o f t h e S t r u c t u r a l D i v i s i o n , ASCE, 1 9 8 1 , pp. 1569 - 1588. 1 2 . T s o , W. K., and Sadek, A. W . , " I n e l a s t i c R e s p o n s e of E c c e n -t r i c S t r u c t u r e s , " 4 t h CCEE, V a n c o u v e r , C a n a d a , J u n e 1 9 8 3 , pp. 2 6 1 - 2 7 0 . 13. M a c K e n z i e , J . R., " T o r s i o n a l S t r u c t u r a l R e s p o n s e D u r i n g E a r t h q u a k e E x c i t a t i o n s , " M a s t e r ' s t h e s i s U n i v e r s i t y of B r i t i s h C o l u m b i a , A p r i l 1974. 14. Y o s h i d a , S u m i o , " 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 M e t h o d f o r A n a l y s i s o f 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 . B . C , M a r c h 1979. 15. M e t t e n , Andrew W. F., "The M o d i f i e d S u b s t i t u t e S t r u c t u r e M e t h o d As a D e s i g n A i d f o r S e i s m i c R e s i s t a n t C o u p l e d S t r u c t u r a l W a l l s , " M a s t e r ' s t h e s i s U.B.C., M a r c h 1 9 8 1 . 16. S h i b a t a , A k e n o r i , and S o z e n , Mete A., " S u b s t i t u t e - S t r u c t u r e M e t h o d f o r S e i s m i c D e s i g n i n R/C," J o u r n a l of t h e S t r u c u r a l D i v i s i o n , ASCE, J a n u a r y 1976, pp. 1-18. 17. H u i , L a w e r e n c e , "Pseudo N o n - l i n e a r S e i s m i c A n a l y s i s , " M a s t e r ' s t h e s i s U.B.C., O c t o b e r 1984. 1 8 . Newmark, N. M. , " T o r s i o n i n S y m m e t r i c a l B u i l d i n g s , " P r o c . 4 t h WCEE, S a n t i a g o , C h i l e , 1969, pp. A 3 . 1 9 - A 3 . 3 2 19. Newmark, N. M., and R o s e n b l u e t h , E., " F u n d a m e n t a l s of E a r t h q u a k e E n g i n e e r i n g , " P r e n t i c e - H a l l I n c . , E n g l e w o o d C l i f f s , N . J . , pp. 243. 20. N a t h a n , N. D., and M a c K e n z i e , J . R., " R o t a t i o n a l Components o f E a r t h q u a k e M o t i o n , " C a n a d i a n J o u r n a l of C i v i l E n g i n e e r -i n g , V o l . I I , 1975, pp. 4 3 0 - 4 3 6 . 21. H a r t , G. C , D i J u l i o , M., and Lew, M., " T o r s i o n a l R e s p o n s e o f H i g h - r i s e B u i l d i n g s , " J o u r n a l of t h e S t r u c t u r a l D i v i s i o n , P r o c , ASCE, 101, ST2, pp. 397-416. 22. W i l s o n , E. L., Der K i u r e g h i a n , A., and B a y o , E. P., "A Re-p l a c e m e n t f o r t h e SRSS M e t h o d i n S e i s m i c A n a l y s i s , " E a r t h q u a k e E n g i n e e r i n g a n d S t r u c t u r a l D y n a m i c s , V o l . 9, 1981, pp. 187-194. 1 1 6 APPENDIX A  PROGRAM USER'S MANUAL IDENTIFICATION PITSA: PSEUDO INELASTIC TORSIONAL SEISMIC ANALYSIS Computer program for the i n e l a s t i c seismic a n a l y s i s of t hree-dimensional b u i l d i n g s under earthquake exc i t a t i o n . DISCLAIMER: The C i v i l E n g i n e e r i n g Department, F a c u l t y and S t a f f do not guarantee nor imply the accuracy 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 documentation. As such,, they can not be hel 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 the r e s p o n s i b i l i t y of the user to determine the u s e f u l n e s s and t e c h n i c a l accuracy of t h i s program in h i s or her own environment. T h i s program may not be s o l d to a t h i r d p a r t y . 1 1 7 PITSA PROGRAM HISTORY UPDATES MODIFICATIONS PROGRAMMER 1974 MACK.FRAME program w r i t t e n J.R. M a c K e n z i e 1 9 7 5 S p e c t r a l a n a l y s i s added- B i l l M c K e v i t t t o MACK.FRAME 198 1 EDAM program w r i t t e n Andrew W . F . M e t t e n 1984 PITSA p r o g r a m w r i t t e n Ken S.K. Tarn 1 1 8 PURPOSE The p r o g r a m u t i l i z e s 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 m e t hod t o d e t e r m i n e t h e i n e l a s t i c d y n a m i c r e s p o n s e of t h r e e -d i m e n s i o n a l b u i l d i n g s c o n s i s t of a r b i t r a r y l o c a t e d f r a m e s a n d / o r s h e a r w a l l s due t o e a r t h q u a k e m o t i o n s . Two i n d e p e n d e n t h o r i z o n t a l m o t i o n s p l u s r o t a t i o n a l m o t i o n may be s p e c i f i e d . The p r o g r a m u s e s s p e c t r a l a n a l y s i s i t e r a t i v e l y i n w h i c h s t i f f n e s s a nd d a m p i n g p r o p e r t i e s a r e s u b s t i t u t e d . THEORY D e t a i l s o f t h e t h e o r y c a n be f o u n d i n t h e f o l l o w i n g m a s t e r ' s t h e s e s : J.R. M a c K e n z i e : " T o r s i o n a l S t r u c t u r a l R e s p o n s e D u r i n g E a r t h q u a k e E x c i t a t i o n s " A p r i l 1974 Andrew W.F. M e t t e 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 M e t h o d As A D e s i g n A i d F o r S e i s m i c R e s i s t a n t C o u p l e d S t r u c t u r a l W a l l s " M a r c h 1 9 8 1 Ken S.K. Tarn: " P s e u d o I n e l a s t i c T o r s i o n a l S e i s m i c A n a l y s i s 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 M e t h o d " A p r i l 1985 PROGRAM RESTRICTIONS T h e r e a r e t o be no a b r u p t c h a n g e s i n mass, s t i f f n e s s o r g e o m e t r y t h r o u g h o u t t h e h e i g h t o f t h e s t r u c t u r e . N o n - s t r u c t u r a l c o m p o n e n t s a r e t o be s u c h t h a t t h e y do n o t a f f e c t t h e r e s p o n s e o f t h e s t r u c t u r e . Members a r e r e i n f o r c e d t o w i t h s t a n d r e p e a t e d r e v e r s a l s of i n e l a s t i c d e f o r m a t i o n s w i t h o u t s i g n i f i c a n t s t r e n g t h d e c a y . D i a p h r a g m s a r e assumed r i g i d and m asses a r e lumped a t t h e c e n t e r s o f mass of e a c h s t o r e y l e v e l . I n e l a s t i c a n a l y s i s p o r t i o n of t h e p r o g r a m c a n o n l y be u s e d f o r c o n c r e t e c o n s t r u c t i o n b e c a u s e d e v e l o p m e n t o f t h e s t i f f n e s s r e d u c t i o n and s u b s t i t u t e damping f o r m u l a s was done on c o n c r e t e members. The members a r e a s s u m e d t o have t h e same p o s i t i v e and n e g a t i v e moment c a p a c i t y a t b o t h e n d s of t h e member. ( 1 ) ( 2 ) ( 3 ) (4) (5) ( 6 ) 119 DIMENSIONING L I M I T S The p r o g r a m i s d i m e n s i o n e d f o r t h e f o l l o w i n g : 150 j o i n t s p e r frame 300 members p e r frame 10 f r a m e s 20 s t o r e y s 30 modes 6 a d d i t i o n a l d e g r e e s o f f r e e d o m p e r frame 1 20 • I n p u t and O u t p u t The i n p u t d a t a f o r t h e a n a l y s i s of a s t r u c t u r e can be g r o u p e d i n t o t h e f o l l o w i n g f i v e s e t s : 1 ) s t r u c t u r e c o n t r o l c a r d s 2) frame and member d a t a c a r d s 3) number of modes c a r d 4) mass d a t a c a r d 5) e a r t h q u a k e s p e c t r u m c a r d s I n p u t d a t a must be p r e p a r e d and a r r a n g e d i n a f i x e d o r d e r . The u n i t s y s t e m i s c h o s e n to' be I m p e r i a l t h u s p r o b l e m s w i t h m e t r i c measurements must be c o n v e r t e d f i r s t . A l l i n p u t v a l u e s need not f o l l o w t h e f i x e d 'format, as l o n g as t h e y a r e s e p a r a t e d by commas. The c o n t e n t and f o r m a t of i n d i v i d u a l d a t a c a r d s a r e d e s c r i b e d i n t h e n e x t s e c t i o n . The j o i n t numbering s y s t e m of a frame used i n t h e p r o g r am s t a r t s w i t h 1 f o r e a c h f r a m e , s e q u e n t i a l l y from l e f t t o r i g h t a t a g i v e n f l o o r l e v e l , and c o n t i n u o u s l y from t h e base t o t h e t o p . The l a s t j o i n t number o f an ; N - s t o r e y M-bay frame must be e q u a l t o (N+1)(M+1). Members o f a frame a r e numbered i n t h e same manner. Columns a r e numbered f i r s t , s t a r t i n g w i t h 1 f o r e a c h frame, s e q u e n t i a l l y from t h e b o t t o m t o t h e t o p a t a g i v e n column l i n e , and f r om l e f t t o r i g h t u n t i l a l l column l i n e s a r e a c c o u n t e d f o r . Beams a r e numbered, c o n t i n u o u s w i t h column numbers, s e q u e n t i a l l y from t h e b o t t o m t o t h e t o p a t a g i v e n bay, and from l e f t . t o r i g h t u n t i l a l l b ays a r e a c c o u n t e d f o r . The l a s t column number of an N - s t o r e y M-bay frame must be e q u a l t o N(M+1), and t h e l a s t beam number must be e q u a l t o N(2M+1). The p r o g r a m r e a d s and p r i n t s a l l i n p u t i n f o r m a t i o n , s u c h t h a t t h e u s e r can examine t h e p r i n t e d i n f o r m a t i o n t o e n s u r e t h a t a s t r u c t u r e d e f i n e d by t h e u s e r and t h a t i n t e r p r e t e d by t h e p r o g r a m be t h e same. A u s e r may o f t e n be a b l e t o f i n d an e r r o r s u c h as wrong d a t a f o r m a t i n d a t a c a r d s by c h e c k i n g t h e p r i n t e d i n f o r m a t i o n . In a d d i t i o n t o a p r i n t - o u t of a l l i n p u t d a t a , ' t h e p r o g r a m a l s o p r o v i d e s f o r t h e c o m p l e t e s t r u c t u r e : n a t u r a l p e r i o d , mode shape, p a r t i c i p a t i o n f a c t o r s f o r e a c h mode, and d i a p h r a g m d i s p l a c e m e n t s f o r e a c h mode and each e a r t h q u a k e e x c i t a - t i o n . W h i l e f o r e a c h frame, the ! program p r o v i d e s j o i n t d i s p l a c e m e n t s and member f o r c e s f o r each mode and e a c h e a r t h -quake e x c i t a t i o n . T o t a l r e s p o n s e s ..are o b t a i n e d by c o m b i n i n g modal r e s p o n s e s from a g i v e n e x c i t a t i o n i n a c o m p l e t e - q u a d r a t i c -c o m b i n a t i o n manner, and t h e n by c o m b i n i n g c o n t r i b u t i o n s from a l l e x c i t a t i o n s i n a r o o t - s u m - s q u a r e manner. A l s o p r i n t e d and s t o r e d a r e t h e damage r a t i o s of e v e r y member i n e a c h i t e r a t i o n . 121 INPUT DATA CONTROL INFORMATION STRUCTURE CONTROL CARD >NOFR,NSTOR,NOCOR,NDISPL,NFORCE,MASSVT,INELAS,I SPEC, AMAX,DAMPIN,STRHRD (8I4,3F8.4) NOFR= number of frames NSTOR= number of s t o r e y s NOCOR= number of a d d i t i o n a l degrees of freedom in s t r u c t u r e NDISPL= number of modes of displacements to be p r i n t e d NFORCE= number of modes of fo r c e s to be p r i n t e d MASSVT= 0 i f c e n t e r s of mass do not l i e in a s t r a i g h t l i n e 1 i f c e n t e r s of mass do l i e in a s t r a i g h t l i n e INELAS= 0 i f e l a s t i c a n a l y s i s i s requested 1 or g r e a t e r i f i n e l a s t i c a n a l y s i s i s requested INELAS i s the maximum number of i n e l a s t i c i t e r a t i o n s to be performed before the program stops. A value of 25 i s suggested. ISPEC= type of spectrum requested = 1 spectrum 'A' (Shibata and Sozen) 2 spectrum 'B' ( " ) 3 spectrum 'C ( " ) 4 N a t i o n a l B u i l d i n g Code spectrum 5 Newmark type spectrum 6 C.I.T. s i m u l a t e d spectrum, C-2 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= damping r a t i o to be used in the e l a s t i c a n a l y s i or in 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 STRHRD= s t r a i n hardening r a t i o 1 22 T I T L E CARD >DESCRIPTIVE T I T L E (20A4) FRAME DATA - One s e t o f d a t a must be e n t e r e d f o r e a c h frame FRAME CONTROL CARD >NFR,NDF,INPUT,IQ,IR,(IO(J),J=1,NSTOR) ' ( 3 I 4 , 2 2 F 8 . 0 ) f o r MASSVT=0 >NFR,IQ,IR,10,NDF,INPUT ( 1 4 , 3 E 8 . 0 , 2 1 4 ) f o r MASSVT=1 NFR= frame number NDF= number o f a d d i t i o n a l d e g r e e s o f f r e e d o m i n frame INPUT= 0 i f member g e n e r a t i n g r o u t i n e i s n o t r e q u e s t e d 1 i f member g e n e r a t i n g r o u t i n e i s r e q u e s t e d IQ= d i r e c t i o n c o s i n e i n Q d i r e c t i o n IR= d i r e c t i o n c o s i n e i n R d i r e c t i o n 10= p e r p e n d i c u l a r d i s t a n c e f r o m f r a m e t o c e n t e r of mass MEMBER DATA CARDS - f o r INPUT=0 o n l y f o r INPUT=1, go t o n e x t s e c t i o n >NRJ,NRM,E,G ( 2 I 4 , 2 F 8 . 0 ) NRJ= number o f j o i n t s NRM= number o f members . i E= modulus o f e l a s t i c i t y , k s i G= s h e a r m o d u l u s , k s i 1 23 >JN,X,Y,ND(1),ND(2),ND(3) ( 1 4 , 2 F 8 . 0 , 3 1 4 ) one c a r d f o r e a c h j o i n t JN= j o i n t number X= x - c o o r d i n a t e , f t Y= y - c o o r d i n a t e , f t ND(1)= f i x i t y c o d e i n x - d i r e c t i o n 0 i f t h e node i s f i x e d 1 i f t h e node i s f r e e ; +N i f t h e node i s t o have t h e same m o t i o n as node N -M i f t h e node i s t o have an a d d i t i o n a l d e g r e e f r e e d o m , M i s a t t a c h e d t o NDF, n o t NCOR ND(2)= f i x i t y c o d e f o r y - d i r e c t i o n ND(3)= f i x i t y c o d e f o r r o t a t i o n >NC0R(I) ( 6 1 4 ) o m i t t h i s c a r d i f t h i s frame h a s no a d d i t i o n a l d e g r e e of f r e e d o m NCOR(I)= g l o b a l a d d i t i o n a l d e g r e e o f f r e e d o m number i n f r a m e , up t o s i x p e r f r a m e >MN,JNL,JNG,KL,KG,AREA,AI,AV,BMCAP,EXTL,EXTG ( 5 I 4 , 6 F 1 2 . 0 ) one c a r d f o r e a c h member MN= member number JNL= l e s s e r j o i n t number JNG= g r e a t e r j o i n t number : KL= f i x i t y c o d e o f member! a t l e s s e r j o i n t = 0 i f p i n n e d 1 i f f i x e d KG= f i x i t y c o d e o f member a t g r e a t e r j o i n t AREA= c r o s s s e c t i o n a r e a o f member, i n * * 2 1 24 AI= c r a c k e d moment of i n e r t i a of member, i n * * 4 AV= s h e a r a r e a , i n * * 2 BMCAP= b e n d i n g moment c a p a c i t y , k - f t EXTL= r i g i d e x t e n s i o n of member a t l e s s e r j o i n t , f t EXTG= r i g i d e x t e n s i o n of member a t g r e a t e r j o i n t , f t c. MEMBER DATA CARDS - f o r INPUT=1 o n l y f o r INPUT=0, go t o p r e v i o u s s e c t i o n > N C O R ( I ) , N L I N E ( I ) , N D I R ( I ) , 1=1,NDF (2014) omit t h i s c a r d i f t h i s frame has no a d d i t i o n a l d e g r e e of freedom NCOR(I)= g l o b a l a d d i t i o n a l d e g r e e of freedom number N L I N E ( I ) = 1 i n e i n wh i c h a d d i t i o n a l d e g r e e of freedom o c c u r s NDIR(I)= d i r e c t i o n of a d d i t i o n a l d e g r e e of freedom >NOB,NOD,E,G,IPIN (214,2F8.0,14) NOB= number of bays NOD= number of d i a g o n a l s E= modulus of e l a s t i c i t y , k s i G= s h e a r modulus, k s i IPIN= 0 i f frame i s f i x e d - a t base 1 i f frame i s p i n n e d a t base >NX,XX (I4,F8.0) as many c a r d s a s r e q u i r e d , o m i t i f NOB=0 NX= number of r e p e t i t i o n s of bay s p a c i n g XX= bay s p a c i n g , f t >NY,YY ( I 4 , F 8 . 0 ) as many c a r d s a s r e q u i r e d NY= number of r e p e t i t i o n s of s t o r e y s p a c i n g YY= s t o r e y s p a c i n g , f t 125 >ICBD,LINE1 ,LINE2,LEVEL 1 ,LEVEL2,KL,KG,AREA,AI,AV, BMCAP,EXTL,EXTG (71 4,F8.2,5F10.2) as many cards as r e q u i r e d ICBD= 1 for column 2 for beam two d i g i t number f o r d i a g o n a l , f i r s t d i g i t i s the no. of bays a c r o s s , second d i g i t i s the no. of stor e y s up, negative slope i s d e f i n e d by minus sign attached to LINE1 LINE1= l i n e / b a y f i r s t member s t a r t s for column/beam or diagonal LINE2= l i n e / b a y l a s t member ends for column/beam or diagonal LEVEL1= l e v e l f i r s t member s t a r t s LEVEL2= l e v e l l a s t member ends KL= f i x i t y code of l e s s e r end = 0 i f pinned 1 i f f i xed KG= f i x i t y code of g r e a t e r end AREA= c r o s s s e c t i o n area of member, in**2 AI= cracked moment of i n e r t i a of member, in**4 AV= shear area, in**2 BMCAP= bending moment c a p a c i t y , k - f t EXTL= r i g i d , e x tension of member at l e s s e r j o i n t , f t EXTG= r i g i d extension of member at grea t e r j o i n t , f t 1 26 3. NUMBER OF MODES >MM (14) one card r e q u i r e d MM= number of modes to be c o n s i d e r e d MASS DATA CARDS - one card per storey >NF,FQ,FR,FO (I 4,2F8.0,F10 . 0 ) : NF = s t o r e y number FQ= weight in Q - d i r e c t i o n FR= weight in R - d i r e c t i o n , k ips , • k ips FO= r o t a t i o n a l i n e r t i a , k i p - f t * * 2 5. EARTHQUAKE SPECTRUM a. NEWMARK TYPE SPECTRUM CARDS - f o r ISPEC=4 or 5 only >AFAC,VFAC,DFAC (3F12.6) 3 cards r e q u i r e d , one for each of Q,R,0 d i r e c t i o n s AFAC= ground a c c e l e r a t i o n l i m i t , g ; g / f t VFAC= ground v e l o c i t y limit!, g-sec ; g - s e c / f t DFAC= ground displacement l i m i t , g-sec**2 ; g- s e c * * 2 / f t b. ACCELERATION SPECTRUM CARDS - for ISPEC=1 or 2 or 3 or 6 on 1 y >AMAXF(I), 1=1,3 (3F12.6) 1 car d r e q u i r e d , one value f o r each of Q,R,0 d i r e c t i o n s AMAXF (I ) =magn i f acat i on f a c t o r :for maximum a c c e l e r a t i o n , AMAX, d e f i n e d in s e c t i o n l a 1 27 OPERATING INSTRUCTIONS The FORTRAN IV source v e r s i o n of the program i s in the f i l e PITSA.S, and the compiled v e r s i o n of the program i s in the f i l e PITSA. The f o l l o w i n g command w i l l run the program: $RUN PITSA 1=-1 2=-2 3=-3 4=-4 5=INPUT 6=-OUTPUT 7=*DUMMY* 8 = -8 10 = - 10 11=-11 1 2 = - 12 1 3 = - 13 Un i t 7 c o n t a i n s output from a l l intermediate i t e r a t i o n s and i s a s s i g n e d to *DUMMY* in order to reduce c o s t . If the user wants to view or p r i n t out t h i s f i l e , a temporary f i l e -7 should be a s s i g n e d . LOGICAL I/O UNIT ASSIGNMENT T h i s i s a general l i s t of the d e f a u l t a c t i o n taken by MTS i f some of the u n i t s are not assigned on; the run command. When running BATCH u n i t s 5 and 6: d e f a u l t s to the card reader and p r i n t e r r e s p e c t i v e l y . When running from a te r m i n a l u n i t s 5 and 6 both d e f a u l t s to the t e r m i n a l screen. LOGICAL  I/O UNIT 1 -4 10 DESCRIPTION f i l e s used i n t e r n a l l y by the program. f i l e from which a l l input i s read. f i l e to which a l l input data i s echoed and to which f i n a l i t e r a t i o n output i s w r i t t e n . i f i l e to which a l l intermediate i t e r a t i o n output i s w r i t t e n . f i l e used i n t e r n a l l y : by the program. f i l e to which damage : r a t i o s from each i t e r a t i o n are w r i t t e n . DEFAULTS *SOURCE* *SINK* 11-13 f i l e s used i n t e r n a l l y by the program. 1 28 APPENDIX B' SAMPLE INPUT AND OUTPUT FOR FIVE-STOREY COUPLED WALL STRUCTURE SEE CHAPTER 6.2.2(c) FOR DETAILS 4 , 5 , O , O . 0 , 0 . 2 5 , 1 , . 2 , . 0 2 , 0 . , FIVE STOREYS STRUCTURE WITH COUPLING BEAMS WITH SPECTRUM 1 , 0 , 1 , 1. , 0 . . 1 1 . . 1 1 . , 1 1. , 11. , 1 1 . , 1 , 0 . 3 6 0 0 . , 0 . , 0 . 1 , 2 1 . 6 6 7 . 5 , 1 2 . . 1 , 1 , 2 , 1 . 5 , 1 , 1 , 6 6 6 4 . ,768241. .5553. , 5 0 0 0 . , 0 . . 0 . . 2, 1 , 1 , 1,5, 1 , 1 , 4 3 2 . , 1 7 2 8 . , 3 6 0 . , 1 0 0 . , 7 . 8 3 3 , 7 . 8 3 3 , 2 . 0 , 1 , 0 . , 1 . . - 1 0 . 8 3 3 , - 1 0 . 8 3 3 , - 1 0 . 8 3 3 , - 1 0 . 8 3 3 , - 1 0 . 8 3 3 . 0 , 0 , 3 6 0 0 . , 0 . , 0 , 5 , 1 2 . , 1 , 1 , 1 , 1 , 5 , 1 , 1 , 6 6 6 4 . ,2719356. .5553. . 10000. . 0 . , 0 . , 3 . 0 . 1 . 0 . . 1 . . 1 0 . 8 3 3 , 1 0 . 8 3 3 , 1 0 . 8 3 3 , 1 0 . 8 3 3 , 1 0 . 8 3 3 , 0 , 0 , 3 6 0 0 . . 0 . , 0 , 5 , 1 2 . , 1 , 1 , 1 , 1 , 5 , 1 , 1 , 6 6 6 4 . , 2 7 1 9 3 5 6 . , 5 5 5 3 . , 1 0 0 0 0 . , 0 . . 0 . , 4 , 0 , 1 , 1 . , 0 . . - 7 . , - 7 . , - 7 . , - 7 . , - 7 . , 1 .0. 3 6 0 0 . . 0 . . 0 . 1 , 2 1 . 6 6 7 , 5 , 1 2 . , 1 , 1 , 2 , 1 , 5 , 1 , 1 , 6 6 6 4 . , 7 6 8 2 4 1 . ,5553. , 5 0 0 0 . , 0 . , 0 . , 2 . 1 , 1, 1.5, 1 , 1 ,432 . , 17 28. .360. ,100. , 7 . 8 3 3 , 7 . 8 3 3 , 6, 1 , 3 0 0 . , 3 0 0 . , 3 4 0 0 0 0 . . 2 , 3 0 0 . , 3 0 0 . , 3 4 0 0 0 0 . , 3 , 3 0 0 . , 3 0 0 . . 3 4 0 0 0 0 . , 4 , 3 0 0 . , 3 0 0 . , 3 4 0 0 0 0 . , 5 . 3 0 0 . , 3 0 0 . , 3 4 0 0 0 0 . . 1 . , 0 . , 0 . , 1 P I T S A N O V . 84 ***********************************************************^ F I V E S T O R E Y S S T R U C T U R E WITH C O U P L I N G BEAMS WITH SPECTRUM ' A ' ( 0 . 2 G ) * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * 4 ^ * * * * * * * * * NUMBER OF FRAMES= 4 NUMBER OF S T O R E Y S * 5 NUMBER OF D E G R E E S OF FREEDOM IN A D D I T I O N TO H O R I Z O N T A L D E G R E E S OF FREEDOM* 0 NUMBER OF MODES OF D I S P L A C E M E N T S TO BE P R I N T E D 1 0 NUMBER OF MODES OF F O R C E S TO BE P R I N T E D * ' 0 C E N T E R OF MASS C O D E * 0 MAXIMUM N O . OF I N E L A S T I C A N A L Y S I S I T E R A T I O N S * 25 S P E C T R U M T Y P E = 1 • MAXIMUM A C C E L E R A T I O N * 0 . 2 0 0 0 T I M E S G R A V I T Y I N I T I A L D A M P I N G R A T I O * 0 . 0 2 0 0 S T R A I N H A R D E N I N G R A T I O * 0 . 0 C A ) O - - 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 NATURAL F R E Q U E N C I E S P E R I O D S ( R A D / S E C ) ( C Y C S / S E C ) ( S E C S ) 1 3 2 . 8 7 7 5 5 . 7 3 3 9 0 . 9 1 2 6 1 . 0 9 5 8 2 1 1 5 . 7 5 4 3 1 0 . 7 5 8 9 1 . 7 1 2 3 0 . 5 8 4 0 3 2 7 2 . 0 1 5 6 1 6 . 4 9 2 9 2 . 6 2 4 9 0 . 3 8 1 0 4 8 0 0 . 1 8 0 0 2 8 . 2 8 7 4 4 . 5 0 2 1 0 . 2 2 2 1 5 4 3 9 0 . 7 2 6 3 6 6 . 2 6 2 5 1 0 . 5 4 6 0 0 . 0 9 4 8 6 4 7 1 9 . 5 0 5 3 6 8 . 6 9 8 7 1 0 . 9 3 3 7 0 . 0 9 1 5 * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * *.* * * * * * * * * ** * * * * * * * * * * * * * * * * * * * * * * * * * 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 . NO . ABOVE C A P A C I T Y DAMERR 1 10 0 . 2 7 5 2 4 - 0 . 0 7 1 3 O 0 . 0 0 2 - I T E R A T I O N NUMBER 4 1 FRAME N O . ADD DOF INPUT 10 IR I O ( N ) N = 1 , N S T 0 R 1 0 1 1 . 0 0 0 . 0 1 1 . 0 0 1 1 . 0 0 1 1 . 0 0 1 1 . 0 0 1 1 . 0 0 B A Y S S T O R E Y S P I N S 1 5 0 B A Y S S P A C I N G 1 2 1 . 6 7 S T O R E Y S S P A C I N G 5 1 2 . 0 0 C=1 , B = 2 , D = 3 L I N E 1 L I N E 2 L E V E L 1 L E V E L 2 KL KG E X T L EXTG AREA I ( C R A C K E D ) AV MOM. C A P . 1 1 2 1 5 1 1 0 . 0 0 . 0 6 6 6 4 . 0 0 7 6 8 2 4 1 . 0 0 5 5 5 3 . 0 0 5 0 0 0 . 0 0 2 1 1 1 5 1 1 7 . 8 3 7 . 8 3 4 3 2 . 0 0 1 7 2 8 . 0 0 3 6 0 . 0 0 1 0 0 . 0 0 ************************************************************************************************************** NF R N R J NRM E • G 1 12 15 3 6 0 0 . 0 0 . 0 * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * J O I N T INFORMATION J N X Y ND1 ND2 ND3 1 0 0 0 0 0 "2 21 6G7 0 0 0 3 0 0 12 0 0 0 1 4 21 6 6 7 12 0 0 0 3 5 0 0 24 0 0 0 1 6 21 6 6 7 24 0 0 0 5 7 0 0 36 0 0 0 1 8 21 6 6 7 36 0 0 0 7 9 0 0 48 0 0 0 1 10 21 6 6 7 48 0 0 0 9 1 1 0 0 6 0 0 0 0 1 12 21 6 6 7 6 0 0 0 0 1 1 0 0 0 0 * * * * * * * * * * * * * * * * * * * * * > * * * * * * * * * * * * * * * * * * * * * ! c * * * * * i MEMBER INFORMATION MN J N L J N G E X T L DM E X T G XM YM KL KG AREA I ( C R A C K E D ) AV MOM. C A P . 1 1 3 0 0 12 0 0 0 0 - 0 0 12 0 0 1 1 6 6 6 4 0 0 7 6 8 2 4 1 0 0 5 5 5 3 0 0 5 0 0 0 . 0 0 2 3 5 0 0 12 0 0 0 0 - 0 0 12 0 0 1 1 6 6 6 4 0 0 7 6 8 2 4 1 0 0 5 5 5 3 0 0 5 0 0 0 . 0 0 3 5 7 0 0 12 0 0 0 0 - 0 0 12 0 0 1 1 6 6 6 4 OO 7 6 8 2 4 1 0 0 5 5 5 3 0 0 5 0 0 0 . 0 0 4 7 9 0 0 12 0 0 0 0 - 0 0 12 0 0 1 1 6 6 6 4 0 0 7 6 8 2 4 1 0 0 5 5 5 3 0 0 5 0 0 0 . 0 0 5 9 1 1 0 0 12 0 0 0 0 - 0 0 12 0 0 1 1 6 6 6 4 0 0 7 6 8 2 4 1 0 0 5 5 5 3 0 0 5 0 O 0 . 0 0 6 2 4 0 0 12 0 0 0 0 - 0 0 12 0 0 1 1 6 6 6 4 0 0 7 6 8 2 4 1 0 0 5 5 5 3 0 0 5 0 0 0 . 0 0 7 4 6 0 0 12 0 0 0 0 - 0 0 12 0 0 1 1 6 6 6 4 0 0 7 6 8 2 4 1 0 0 5 5 5 3 0 0 5 0 0 0 . 0 0 8 6 8 0 0 12 0 0 0 0 - 0 0 12 0 0 1 1 6 6 6 4 0 0 7 6 8 2 4 1 0 0 5 5 5 3 0 0 5 0 0 0 . 0 0 9 8 10 0 0 12 0 0 0 0 - 0 0 12 0 0 1 1 6 6 6 4 0 0 7 6 8 2 4 1 0 0 5 5 5 3 0 0 5 0 0 0 . 0 0 10 10 '12 0 0 12 0 0 0 0 - 0 0 12 0 0 1 1 6 6 6 4 0 0 7 6 8 2 4 1 0 0 5 5 5 3 0 0 5 0 0 0 . 0 0 1 1' 3 4 7 83 6 0 0 7 83 6 0 0 - 0 o • 1 1 432 0 0 1728 0 0 3 6 0 0 0 1 0 0 . 0 0 12 5 6 7 83 6 0 0 7 83 6 0 0 - 0 0 1 1 432 0 0 1728 0 0 3 6 0 0 0 1 0 0 . 0 0 13 7 8 7 83 6 0 0 7 8 3 6 0 0 - 0 0 1 1 432 0 0 1728 0 0 3 6 0 0 0 1 O 0 . O 0 14 9 10 7 83 6 0 0 7 8 3 6 0 0 - 0 0 1 1 432 0 0 1728 0 0 3 6 0 0 0 1 0 0 . 0 0 15 1 1 12 7 83 6 0 0 7 8 3 6 0 0 - 0 0 1 1 432 0 0 1728 0 0 3 6 0 0 0 1 0 0 . 0 0 MEMBER NP 1 NP2 NP3 NP4 NP5 NP6 1 0 0 0 1 2 3 2 1 2 3 6 7 8 3 6 7 8 1 1 12 13 4 1 1 • 12 13 16 17 18 5 16 17 18 2 1 22 23 6 0 0 0 1 4 5 7 1 4 5 6 9 10 8 6 9 10 1 1 14 15 9 11 14 15 16 19 2 0 10 16 19 20 21 24 25 1 1 1 2 .. 3 1 4 5 12 6 7 • 8 6 9 10 13 1 1 12 13 1 1 14 15 14 16 17 18 16 19 20 15 21 22 23 21 24 25 N O . OF D E G R E E S OF FREEDOM OF S T R U C T U R E 1 25 H A L F BAND WIDTH= 10 ********************************************************** 1 FRAME N O . ADD DOF INPUT 10 IR I O ( N ) N = 1 . N S T 0 R 2 0 1 0 . 0 1 . 0 0 - 1 0 . 8 3 - 1 0 . 8 3 - 1 0 . 8 3 - 1 0 . 8 3 - 1 0 . 8 3 B A Y S S T O R E Y S P I N S 0 5 0 B A Y S S P A C I N G S T O R E Y S S P A C I N G 5 1 2 . 0 0 C = 1 , B = 2 , D = 3 L I N E 1 L I N E 2 L E V E L 1 L E V E L 2 KL KG E X T L E X T G AREA I ( C R A C K E D ) AV MOM. C A P . 1 1 1 1 5 1 1 0 . 0 0 . 0 6 6 6 4 . 0 0 2 7 1 9 3 5 6 . 0 0 5 5 5 3 . 0 0 1 0 0 0 0 . 0 0 * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * NFR NRJ NRM E G 2 6 5 3 6 0 0 . 0 0 . 0 * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * J O I N T INFORMATION UN 1 2 3 4 5 6 0 . 0 0 . 0 0 . 0 0 . 0 0 . 0 0 . 0 Y 0 . 0 1 2 . 0 0 0 2 4 . 0 0 0 3 6 . 0 0 0 4 8 . 0 0 0 6 0 . 0 0 0 ND 1 0 NO 2 0 ND3 0 A * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * V******************************************M * * * * * * * * * * * * * * * * * * MEMBER INFORMATION MN J N L J N G EXTL DM E X T G XM YM KL KG AREA I ( C R A C K E D ) AV MOM. C A P . 1 1 2 0 . 0 12 0 0 0 . 0 - 0 0 12 0 0 1 1 6 6 6 4 . 0 0 27 1 9 3 5 6 . 0 0 5 5 5 3 0 0 1 0 O 0 0 . 0 0 2 2 3 0 . 0 12 0 0 0 . 0 - 0 0 12 0 0 1 1 6 6 6 4 . O O 27 1 9 3 5 6 . 0 0 5 5 5 3 0 0 1 0 0 0 0 . 0 0 3 3 4 0 . 0 12- 0 0 0 . 0 - 0 0 12 0 0 1 1 6 6 6 4 . 0 0 2 7 1 9 3 5 6 . 0 0 5 5 5 3 0 0 1 0 0 0 0 . 0 0 4 4 5 0 . 0 12 0 0 0 . 0 - 0 0 12 0 0 1 1 6 6 6 4 . 0 0 2 7 1 9 3 5 6 . 0 0 5 5 5 3 0 0 1 0 0 0 0 . 0 0 5 5 6 0 . 0 12 0 0 0 . 0 - 0 0 12 0 0 1 1 6 6 6 4 . 0 0 2 7 1 9 3 5 6 . 0 0 5 5 5 3 0 0 1 0 0 0 0 . 0 0 MEMBER NP 1 NP2 NP3 NP4 NP5 NP6 1 0 0 0 1 2 3 2 1 2 3 4 5 6 3 4 5 6 7 8 9 4 7 . 8 9 10 . 1 1 12 5 10 1 1 12 13 14 15 N O . OF D E G R E E S OF FREEDOM OF S T R U C T U R E * 15 H A L F BAND WIDTH* 6 ********************************************* 1 FRAME NO . ADD DOF INPUT 10 IR I O ( N ) N = 1 , N S T 0 R 3 0 1 0 . 0 1 . 0 0 1 0 . 8 3 1 0 . 8 3 1 0 . 8 3 1 0 . 8 3 1 0 . 8 3 B A Y S S T O R E Y S P I N S 0 5 0 B A Y S S P A C I N G S T O R E Y S S P A C I N G 5 1 2 . 0 0 C = 1 . B = 2 , D = 3 L I N E 1 L I N E 2 L E V E L 1 L E V E L 2 KL KG E X T L ' EXTG AREA I ( C R A C K E D ) AV MOM. C A P . 1 1 1 1 5 1 1 0 . 0 0 . 0 6 S 6 4 . 0 0 2 7 1 9 3 5 6 . 0 0 5 5 5 3 . 0 0 1 0 0 0 0 . 0 0 ********************** * * * * * * * * * *~*"* *"* * * * * V* * * *'* V* * * * * * *********************'*************************•*********** NFR NRJ NRM E G 3 6 5 3 6 0 0 . 0 0 . 0 ************************************************************************************** J O I N T INFORMATION J N 1 2 3 4 5 6 0 . 0 0 . 0 0 . 0 0 . 0 0 . 0 0 . 0 0 . 0 1 2 . 0 0 0 2 4 . 0 0 0 3 6 . 0 0 0 4 8 . 0 0 0 6 0 . 0 0 0 ND 1 0 ND2 0 ND3 0 *********************************************** **********************************************> * * * * * * * * * * * 1 MEMBER INFORMATION MN JNL JNG EXTL DM EXTG XM YM KL KG AREA I(CRACKED) AV MOM. CAP: 1 .1 2 0.0 12 00 0.0 -0 0 12 00 1 1 6664.00 2719356.00 5553 00 10000.00 2 2 3 0.0 12 00 0.0 -0 0 12 00 . 1 1 6664.00 2719356.00 5553 00 10000.00 3 3 4 0.0 12 00 0.0 -0 0 12 00 1 1 6664.00 2719356.00 5553 00 10000.00 4 4 5 0.0 12 00 0.0 -0 0 12 00 1 1 6664.00 2719356.00 5553 00 10000.00 5 5 6 0.0 12 00 0.0 -0 0 12 00 1 1 6664.00 2719356.00 5553 00 10000.00 MEMBER NP 1 NP2 NP3 NP4 NP5 NP6 1 0 0 0 1 2 3 2 1 2 3 4 5 6 3 4 5 6 7 8 9 4 7 8 9 10 1 1 12 5 10 1 1 12 13 14 15 NO. OF DEGREES OF FREEDOM OF STRUCTURE* 15 HALF BAND WIDTH* 6 ************************************* 1 FRAME NO. ADD DOF INPUT 10 IR IO(N) N=1,NST0R 4 0 1 1.00 0.0 -7.00 -7.00 -7.00 -7.00 -7.00 BAYS STOREYS PINS 1 5 0 BAYS SPACING 1 21.67 STOREYS SPACING 5 12.00 C=1,B = 2,D = 3 LINE 1 LINE2 LEVEL 1 LEVEL2 KL KG EXTL EXTG AREA I(CRACKED) AV MOM. CAP. 1 1 2 1 5 1 1 0 . 0 0 . 0 6664.00 76824 1.00 5553.00 5000.00 2 1 1 1 5 1 1 7.83 7.83 432.00 1728.00 360.00 100.00 * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * NFR NRJ NRM E G 4 12 15 3600.0 0.0 ************************************************************* JOINT INFORMATION JN X Y ND1 ND2 ND3 1 0.0 0.0 0 0 0 2 21.667 0.0 0 0 0 3 0.0 12.000 1 1 1 4 21.667 ' 12.000 3 1 1 5 0 . 0 2 4 . 0 0 0 1 6 2 1 . 6 6 7 2 4 . 0 0 0 5 7 0 . 0 3 6 . 0 0 0 1 8 2 1 . 6 6 7 3 6 . 0 0 0 7 9 0 . 0 4 8 . 0 0 0 1 10 2 1 . 6 6 7 4 8 . 0 0 0 9 11 0 . 0 6 0 . 0 0 0 1 12 2 1 . 6 6 7 6 0 . 0 0 0 11 * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * MEMBER I N F O R M A T I O N MN UNL UNG E X T L DM E X T G XM YM KL KG AREA I ( C R A C K E D ) AV MOM. C A P . 1 .1 3 0 0 12 0 0 0 0 - 0 0 12 0 0 1 1 6 6 6 4 0 0 7 6 8 2 4 1 0 0 5 5 5 3 0 0 5 0 0 0 OO 2 3 5 0 0 12 0 0 0 0 - 0 0 12 0 0 1 1 6 6 6 4 0 0 7 6 8 2 4 1 0 0 5 5 5 3 0 0 5 0 0 0 0 0 3 5 7 0 0 12 0 0 0 0 - 0 0 12 0 0 1 1 6 6 6 4 0 0 7 6 8 2 4 1 0 0 5 5 5 3 0 0 5 0 0 0 0 0 4 7 9 0 0 12 0 0 0 0 - 0 0 12 0 0 1 1 6 6 6 4 0 0 7 6 8 2 4 1 0 0 5 5 5 3 0 0 5 0 0 0 0 0 5 9 1 1 0 0 12 0 0 0 0 -o 0 12 0 0 1 1 6 6 6 4 0 0 7 6 8 2 4 1 0 0 5 5 5 3 0 0 5 0 0 0 0 0 6 2 4 0 0 12 0 0 0 0 - 0 0 12 0 0 1 1 6 6 6 4 0 0 7 6 8 2 4 1 OO 5 5 5 3 0 0 5 0 0 0 oo 7 4 6 0 0 12 0 0 0 0 -o 0 12 0 0 1 1 6 6 6 4 0 0 7 6 8 2 4 1 0 0 5 5 5 3 0 0 5 0 0 0 0 0 8 6 8 0 0 12 OO 0 0 - 0 0 12 0 0 1 1 6 6 6 4 0 0 7 6 8 2 4 1 0 0 5 5 5 3 0 0 5 0 0 0 oo 9 8 10 0 0 12 0 0 0 0 - 0 0 12 0 0 1 1 6 6 6 4 0 0 7 6 8 2 4 1 0 0 5 5 5 3 0 0 5 0 0 0 0 0 10 10 12 0 0 12 0 0 0 0 - 0 0 12 0 0 1 1 6 6 6 4 0 0 7 6 8 2 4 1 0 0 5 5 5 3 0 0 5 0 0 0 oo-1 1 3 4 7 83 6 0 0 7 8 3 6 0 0 - 0 0 1 1 4 3 2 0 0 1728 0 0 3 6 0 0 0 100 oo 12 5 6 7 8 3 6 0 0 7 83 6 0 0 - 0 0 1 1 4 3 2 0 0 1728 0 0 3 6 0 0 0 100 0 0 13 7 8 7 83 6 0 0 7 8 3 6 0 0 - 0 0 1 1 4 3 2 0 0 1728 OO 3 6 0 0 0 100 0 0 14 9 10 7 83 6 0 0 7 8 3 6' 0 0 - 0 0 1 1 4 3 2 0 0 1728 0 0 3 6 0 0 0 100 0 0 15 1 1 12 7 83 6 0 0 7 8 3 6 0 0 - 0 0 1 " 1 432 0 0 1728 0 0 3 6 0 0 0 100 0 0 MEMBER NP 1 NP2 NP3 NP4 NP5 NP6 1 0 0 0 1 2 3 2 1 2 3 6 7 8 3 6 7 8 1 1 12 13 4 1 1 12 13 16 17 18 5 16 17 18 21 22 23 6 0 0 0 1 4 5 7 1 4 5 6 9 10 8 6 9 10 1 1 14 15 9 1 1 14 15 16 19 2 0 10 16 19 2 0 21 24 25 1 1 1 2 3 1 4 5 12 6 7 8 6 9 10 13 1 1 12 13 1 1 14 15 14 16 17 18 16 19 2 0 15 21 22 -23 21 24 25 NO. OF DEGREES OF FREEDOM OF STRUCTURE* HALF BAND WIDTH= 10 25 ****************************************#*********************i 1MASS VECTOR FLOOR NO. Q-WEIGHT(KIPS) 1 300.0 2 300.0 3 300.0 4 300.0 5 300.0 M EXCEEDS MM. M= 15 MM= 6 MODAL ANALYSIS OF STRUCTURE FOR FIRST R-WEIGHT(KIPS) 300.0 300.0 300.0 300.0 300.0 MODES INERT!Al THE PERIODS IN SECONDS ARE: 1.376672 0.583998 0.564295 0.236085 0.112311 THE PARTICIPATION FACTORS ARE: LISTED PHI 10,PHI1R.PHI10,PHI20.PHI2R,PHI20. 3 FOR EACH MODE 386739 000000 682874 219915 974102 000000 0.000000 -189.681620 5.625168 -0.000000 0.000000 -0.000000 -3.101507 0. 12 . 102 . 7 , . 0. 000000 903042 878972 569023 000000 MODE SHAPE FOR MODE 1 0.001081 0.OOOOOO -0.0OO461 0.003868 0.000000 -0.001657 0.007734 0.OOOOOO -0.003330 0.012131 0.OOOOOO -0.005251 0.016663• 0.OOOOOO -0.007250 MODE SHAPE FOR MODE 0.OOOOOO 0.015039 0.OOOOOO 0.OOOOOO 0.054672 0.OOOOOO 0.OOOOOO 0.110921 0.OOOOOO 0.OOOOOO 0.176597 0.OOOOOO 0.OOOOOO 0.246052 0.OOOOOO MODE SHAPE FOR MODE 0.016637 -O.OOOOOO 0.000030 0.058534 -0.000000 0.000109 O. 1153 18 -0.000000 0.000223 0.178084 -0.000000 0.000358 0.240896 -0.OOOOOO 0.000501 340000.0 340000.0 340000.0 340000.0 340000.0 0.091460 MODE SHAPE FOR MODE Q R 0 0 . 0 0 5 6 9 2 - 0 . 0 1 3 7 5 8 - 0 . 0 1 4 0 4 9 - 0 . 0 0 3 5 7 7 0 . 0 1 3 4 9 0 0 . 0 0 0 0 0 0 0 . 0 0 0 0 0 0 0 . 0 0 0 0 0 0 0 . 0 0 0 0 0 0 . 0 . 0 0 0 0 0 0 0 . 0 0 2 2 3 2 0 . 0 0 5 4 2 4 0 . 0 0 5 6 0 0 0 . 0 0 1 5 6 6 - 0 . 0 0 5 0 8 7 MODE SHAPE FOR MODE 5 0 : 0 . 0 7 6 0 5 3 0 . 1 8 2 9 3 5 0 . 1 8 6 2 9 2 0 . 0 4 7 9 7 2 - 0 . 1 7 4 2 9 0 R: 0 . 0 0 0 0 0 0 0 . 0 0 0 0 0 0 0 . 0 0 0 0 0 0 - 0 . 0 0 0 0 0 0 - 0 . 0 0 0 0 0 0 0 : 0 . 0 0 0 1 6 4 0 . 0 0 0 4 0 4 0 . 0 0 0 4 2 4 0 . 0 0 0 1 2 1 - 0 . 0 0 0 3 9 6 MODE SHAPE FOR MODE - 0 . 0 0 0 0 0 0 - 0 . 0 7 5 0 1 0 - 0 . O O O O O O - 0 . 0 0 0 0 0 0 - 0 . 1 8 3 0 1 5 0 .OOOOOO - 0 . O O O O O O - 0 . 1 9 0 0 0 6 O .OOOOOO - 0 . 0 0 0 0 0 0 - 0 . 0 5 4 9 1 2 0 .OOOOOO 0 .OOOOOO 0 . 1 7 0 3 1 7 - 0 . 0 0 0 0 0 0 -MODE SMEARED DAMPING R A T I O 1 0 . 0 3 8 5 7 2 0 . 0 2 0 0 0 3 0 . 0 6 0 6 5 4 0 . 0 2 4 0 9 5 0 . 0 3 2 8 0 6 0 . 0 2 0 0 0 D E T A I L S OF SPECTRUM " A " A P P L I E D B A S E A C C E L E R A T I O N L I M I T IN 0 D I R E C T I O N * 0 . 2 0 0 0 0 0 B A S E A C C E L E R A T I O N L I M I T IN R D I R E C T I O N * 0 . 0 B A S E A C C E L E R A T I O N L I M I T IN 0 D I R E C T I O N * 0 . 0 i S P E C T R A L D I S P L A C E M E N T S MODAL A M P L I T U D E S MODE SMEARED N A T U R A L F R E O . Q - D I R E C T I O N R - D I R E C T I O N R O T A T I O N Q - D I R E C T I O N R - D I R E C T I ON R O T A T I O N N O . DAMPING R A T I O ( R A D / S E C ) ( G - S E C * * 2 ) ( G - S E C * * ' 2 ) ( G - S E C * * 2 / F T ) ( U N I T L E S S ) 1 0 . 0 3 8 5 6 6 4 . 5 6 4 0 3 9 0 . 0 0 8 4 9 1 0 . 0 0 . 0 0 . 1 0 5 6 5 2 0 . 0 0 . 0 2 0. .020000 16. . 758919 0. .004438 0 .0 0 .0 0. .oooooo 0 .0 0. .0 3 0. .060653 1 1 . ,134583 0. .002843 0. .0 0 .0 0. .519866 0. .0 0. .0 4 0. .024086 26. .614123 0 .001007 0. .0 0 .0 -0. .007128 0. .0 0. .0 5 0. .032797 55. .944599 0. .000155 0. .0 0 .0 0. .014801 0 .0 0. .0 6 0, .020000 68. .698656 0, .000097 0. .0 0. .0 -0. OOOOOO 0. .0 0. .0 * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * DISPLACEMENTS IN FEET AND RADIANS FORCES IN KIPS AND KIP-FT ************************************************************************************************************** MODE NUMBER 1 DISPLACEMENTS-- -FORCES--FLOOR NO. 1 (0) 0 (R) 0 (0) 0 FLOOR NO. 2 (0) 0 (R) 0 (0) 0 FLOOR NO . 3 (0) 0 (R) 0 (0) 0 FLOOR NO . 4 (0) 0 (R) 0 (0) 0 FLOOR NO. 5 (0) 0 (R) 0 (0) 0 Q-DIRECTION 0001142 0 .0004086 .0 .0 .0008171 .0 .0 ,0012816 ,0 .0 .0017605 .0 .0 -DIRECTION 0.0000000 .0 .0 .0000000 .0 0.0 O.0000000 .0 .0 .0000000 .0 .0 .0000000 .0 .0 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. ROTATION -0.0000487 0.0 .0 .0001751 .0 .0 .0003519 .0 .0 .0005548 .0 .0 -0.0007660 0.0 0.0 Q-DIRECTION R-DIRECTION ROTATION 0. -0. 0. 0. -0. O  0. -0. O  O  0. .022 -0 .000 -10 .714 0. .0 0 .0 0 .0 0 .0 0 .0 0 .0 0. .079 0. .000 -38 .541 0. .0 0. .0 0 .0 0. .0 0. .0 0 .0 0. . 159 0. .000 -77 . 454 0. .0 0. .0 0. .0 0. 0 0. .0 0. .0 0. 249 O. .000 - 122 . 127 0. .0 ' 0, ,0 0. .0 0. 0 0. 0 0. .0 0. 342 -0. 000 -168. .610 0. 0 0. 0 0. .0 0. 0 0. 0 0. .0 CO MODE NUMBER 2 DISPLACEMENTS- -FORCES-0-DIRECTION FLOOR NO. 1 (0) (R) (0) FLOOR NO. 2 (0) (R) (0) FLOOR NO. 3 (0) (R) (0) FLOOR NO. 4 (0) 0000000 0 0 0000000 O 0 0000000 0 -0 0000000 R-DIRECTION 0.0000000 0.0 0.0 0.ooooooo .0 .0 .ooooooo .0 .0 . ooooooo o. 0. 0. o. 0. 0. ROTATION OOOOOOO 0 0 ooooooo 0 0 ooooooo 0 0 ooooooo Q-DIRECTION 0.000 R-DIRECTION 0.000 0. O. 0. O. O. .0 0 .000 0 0 0.000 0.0 0.0 0.000 .0 .0 .000 .0 .0 0.000 0.0 .0 .000 0. 0. o. 0. 0. 0  0  ROTATION -0.000 0.0 0.0 -0.000 0.0 0.0 0.000 0.0 0.0 0.000 (R) (0) FLOOR NO. 5 (0) (R) (0) 0.0 0.0 0.0000000 0.0 0.0 0.0 0.0 0.0000000 0.0 0.0 *********** ********************************** MODE NUMBER 3 DISPLACEMENTS-FLOOR NO. 1 (0) 0 (R) 0 (0) 0 FLOOR NO. 2 (0) 0 (R) 0 (0) 0 FLOOR NO. 3 (0) 0 (R) 0 (0) 0 FLOOR NO. 4 (0) 0 (R) 0 (0) 0 FLOOR NO . 5 (0) 0 (R) 0 (0) 0 DIRECTION 0086492 0 0 0304300 0 0 0925797 R-DIRECTION -O.OOOOOOO .0 .0 .OOOOOOO .0 .0 -0.0000000 0.0 .0 .OOOOOOO .0 .0 .OOOOOOO .0 . 0. 0. -0. 0. 0. 0. -0. 0. 0. -0. 0. 0.0 ********************************************* MODE NUMBER 4 ---DISPLACEMENTS-FLOOR NO. 1 (0) 0 (R) 0 (0) 0 FLOOR NO . 2 (0) 0 (R) 0 (0) 0 FLOOR NO . 3 (0) 0 (R) 0 (0) 0 FLOOR NO . 4 (0) 0 (R) 0 (0) 0 FLOOR NO . 5 (0) -0 (R) 0 Q-DIRECTION 0000406 0 0000981 O 0 0001001 0 0 0000255 R-DIRECTION -0.0000000 .0 .0 .OOOOOOO .0 .0 .OOOOOOO .0 .0 .OOOOOOO .0 .0 -0.0000000 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 0.000 0.000 0.000 0.0 0.0 0.0 0.0 0.0 0.0 * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * ---FORCES---0-DIRECTION R-DIRECTION ROTATION 9. . 999 -o .000 20 . 264 0 .0 0 .0 0 .0 0. .0 0 .0 0 .0 35, . 178 -0 .000 74 . 286 0. .0 0 .0 0 0. 0. .0 0 .0 0 .0 69. . 303 -o .000 151 .971 0. .0 0. .0 0 .0 0. .0 0 0 0 .0 107. .024 -0 000 243 .716 0. 0 0 .0 0 .0 0 .0 0. .0 0 .0 144 . 772 -0 .000 34 1 . 395 0. .0 0. .0 0 .0 0. 0 0. 0 0. .0 **************************************** FORCES Q-DIRECTION R-DIRECTION ROTATION 0.268 0.000 -119.096 0.0 0.0 0.0 0.0 0.0 • 0.0 0.648 -0.000 -289.389 0.0 0.0 0.0 0.0 0.0 0.0 0.661 -0.000 -298.757 0.0 0.0 O.O 0.0 ' 0.0 O.O 0.168 -0.000 -83.570 0.0 0.0 0.0 O.O O.O 0.0 -0.635 0.000 271.398 0.0 0.0 0.0 (0) 0.0 0.0 0.0 0.0 0.0 0.0 ****************************************************** * * * * * * * * ********************************* * * * * * * * * MODE NUMBER 5 DISPLACEMENTS -FORCES-Q-DIRECTION FLOOR NO. 1 (0) 0 (R) 0 (0) 0 FLOOR NO. 2 (0) 0 (R) 0 (0) 0 FLOOR NO. 3 (0) 0 (R) 0 (0) 0 FLOOR NO. 4 (0) 0 (R) 0 (0) 0 FLOOR NO. 5 (0) -0 (R) 0 (0) 0 R-DIRECTION 0.OOOOOOO .0 .0 .OOOOOOO .0 .0 .OOOOOOO .0 .0 .OOOOOOO .0 .0 -0.OOOOOOO 0.0 0.0 0. 0. 0. 0. 0. 0. o. 0. -0. 0. 0. ROTATION 0.0000024 0.0 0.0 0.0000060 0.0 .0 .0000063 .0 .0 .0000018 .0 .0 -0.0000059 0.0 0.0 0. 0. 0. 0. 0. 0. 0. -DIRECTION 32.850 0.0 0.0 . 79.017 0.0 0.0 80.467 0.0 0.0 20.721 0.0 0.0 -75.283 0.0 0.0 -DIRECTION 0. OOO 0.0 0.0 -0.000 0. O. 0. 0. 0. 0. 0. 0. -0. 0. o. .0 .0 .000 .0 .0 .000 .0 .0 .000 .0 .0 ROTATION 80.039 0. 0. 197 . O. O. 207 . 0. 0.0 59.206 0.0 0.0 -194.070 0.0 0.0 .0 .0 684 0 .0 770 .0 *************************** *'* * * * * *'* *************************************************************************** MODE NUMBER 6 DISPLACEMENTS- -FORCES-Q-DIRECTION R-DIRECTION ROTATION Q-DIRECTION R-DIRECTION ROTATION FLOOR NO. 1 (0) 0. . OOOOOOO 0. .OOOOOOO 0 .OOOOOOO 0 .000 0 .000 0.000 (R) 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 FLOOR NO. 2 (0) 0. . OOOOOOO 0 .OOOOOOO -0 .OOOOOOO' 0 .000 0 .000 0.000 (R) 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 FLOOR NO. 3 (0) 0. .OOOOOOO 0 ,OOOOOOO -0 .OOOOOOO 0 .000 0. .000 -0.000 (R) 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 FLOOR NO . 4 (0) 0. .OOOOOOO 0 .OOOOOOO -0 .OOOOOOO -0 .000 0. .000 0.000 (R) 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 FLOOR NO . 5 (0) -0 .OOOOOOO -0 . OOOOOOO 0 .OOOOOOO -0 .000 -0 .000 0.000 (R) 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 ************************************************************************************************************** ROOT SUM SOUARE DISPLACEMENTS AND FORCES *********************************** -DISPLACEMENTS -FORCES-FLOOR NO. 1 (0) 0 (R) 0 (0) 0 (RSS) 0 FLOOR NO. 2 (0) 0 (R) 0 (0) 0 (RSS) 0 FLOOR NO. 3 (0) 0 (R) 0 (0) 0 (RSS) 0 FLOOR NO. 4 (0) 0 (R) 0 (0) 0 (RSS) 0 FLOOR NO . 5 (0) 0 (R) 0 (0) 0 (RSS) 0 Q-DIRECTION 0087230 0 0 0087230 0305531 0 0 0305531 0600190 0 0 .0600190 .0925913 ,0 ,0 .0925913 . 1252726 .0 .0 .1252726 R-DIRECTION 0.OOOOOOO .0 .0 .OOOOOOO 0. 0. 0. OOOOOOO O 0 OOOOOOO OOOOOOO 0 0 ooooooo ooooooo 0 0 ooooooo ooooooo 0 0 ooooooo ROTATION 0.0000535 .0 .0 .0000535 0. 0. 0. 0001881 0 0 0001881 .0003727 0 0 0003727 0005853 .0 0 .0005853 . 0008099 .0 .0 .0008099 Q-DIRECTION 34.339 0. O. 34 . .0 .0 339 86. O. O. 86. 106 , 0, O. 106. 496 0 0 496 200 0 O 2O0 109.012 0.0 O. 109 . 163 . O. 0. 163 . O 012 178 0 0 178 -DIRECTION 0.000 0. 0. .0 .0 0.000 000 O O 000 000 O 0 000 0. 0. .000 .0 0.0 0.000 000 .0 0 .000 ROTATION 145.31 1 0. 0. 145 . .0 .0 311 360. 0. 0. 317 0 0 360.317 401 . 0. 0. 401 . 291 . 0. 0. 29 1 . 506 . 0. 0. 506 . 894 O 0 894 208 O 0 208 260 0 0 260 *********************************************************** COMPLETE QUADRATIC COMBINATION DISPLACEMENTS AND FORCES ************************************************************************************************************* FORCES FLOOR NO. 1 (Q) (R) (0) (RSS) Q-DIRECTION 0.0087263 0.0 0.0 0.0087.263 -DISPLACEMENTS-R-DIRECTION 0.OOOOOOO 0.0 0.0 0.OOOOOOO ROTATION 0.0000534 0.0 0.0 0.0000534-Q-DIRECTION 34.356 0.0 0.0 34.356 R-DIRECTION 0.000 0.0 O.O 0.000 ROTATION 144.860 0.0 0.0 144.860 FLOOR NO. 2 (Q) 0.0305624 O.OOOOOOO O.0001875 86.551 O.OOO 359.085 143 oo oi cn O O O - O O -O O C T ) 0) LO 10 CO CO CD O O CD CN O O CN O O CT) O O O O m o O co rr rr CT) O O CT) co co CM C N in O O co O O in to O O O O O in O O co O O O O O O O O O O o o o o o o co co C N O O rg 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 CD CO CD O O CD o o c o O O o i co o O ro O O CD CD o < cc rr CD rr CD O O O O O O O O O O O O O O O O O O O O O O rr CD O O O O O O o o O O O O O O O O in 1 co ro CM CN o r~ »- r- ro CO r~ CC oo t~ CO CO O O CO ro 10 in co co 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 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 O o 6 6 6 CC m LU 3 < > * l-H CL m O O O O O o o o o O O O O O O O O CO CT) CT) o o O O O 00 01 CO o o CO CT) CT) o o O O O O O O O O O O O O o o CT) CT) r- in LO CN CN CN CD CD CP ro ro O o oo co in O O CD CD CN CN O O O CM CN in in CO CO CD CT) 0) CN <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 O O or o m O 0L O to O oc o in w w (/> v i w > - L0 w ' w (/) w w w m DC QC CC CC ro w r f w L0 w CC < o in s O o cc in < z < CL in o O O O O O O O o O O O O O O O O rf rr CD CD co co CD CD O O - O O -CD CD CO CO CD CD O O - O o -O) in o o o o O O O O O cc o in CC O oc o in www to cc CN w OKOB1 DC ro w OCC D 1/1 O www tn w cc rr w m (R) 0. 0 ' 0, .0 0 .0 (0). 0. 0 0. .0 0 .0 (RSS) 0. 374591 0, .001797 0 .002244 6 (0) 0. 374591 0. .001797 0 .002244 (R) 0. 0 0 .0 0 .0 (0) 0. 0 0 .0 0 .0 (RSS) " 0. 374591 0, .001797 0 .002244 7 (0) 0. 736417 0. .002397 0. .002694 (R) 0. 0 0. ,0 0 .0 (0) 0. 0 0. ,0 0 .0 (RSS) 0. .736417 0. .002397 0 .002694 8 (0) 0. .736417 0. .002397 0, .002694 (R) 0. .0 0. :0 0 .0 (0) 0. .0 0, .0 0, .0 (RSS) 0. 736417 0. ,002397 0 ,002694 9 (0) 1, ,137018 0. .002797 0 .002814 (R) 0. .0 0. .0 0. .0 (0) 0. .0 0. .0 0, .0 (RSS) 1. . 137018 0. .002797 0 ,002814 10 (0) 1. .137018 0. .002797 0, ,002814 (R) 0. .0 0. .0 0 ,0 (0) 0. .0 0. .0 0. .0 (RSS) 1 . 137018 0. .002797 0 .002814 11 (0) 1. .539609 0, ,002997 0 .002755 (R) 0, .0 0. .0 . 0, .0 (0) 0. .0 0. .0 0, .0 (RSS) 1. 539609 0. 002997 0, ,002755 12 (0) 1 539609 0. 002997 0 ,002755 (R) 0 .0 0. .0 0 0 (0) 0 .0 0. .0 0. .0 (RSS) 1. .539609 0. .002997 0. .002755 * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * MEMBER NO. AXIAL SHEAR BML BMG MOMENT DAMAGE (KIPS) (KIPS) (KIP-FT) (KIP-FT) CAPACITY RATIO 1 (0) 166.216 98.906 2761.863 1616.254 (R) 0.0 0.0 0.0 0.0 (0) 0.0 0.0 0.0 0.0 (RSS) 166.216- 98.906 2761.863 1616.254 5000.OOO 2 (0) 133.208 93.854 1958.073 915.001 (R) 0. .0 0. .0 0 .0 (0) 0 .0 0. .0 0 .0 (RSS) 133, .208 93, .854 1958 .073 3 (0) 100. .030 81 , 361 1231 .854 (R) 0. .0 0. .0 0 .0 (0) 0. .0 0, ,0 0. .0 (RSS) 100. .030 81 , .361 1231 .854 '4 (0) 66. .725 65. . 148 651 .839 (R) 0, .0 0, .0 0 .0 (0) 0. .0 0 .0 0 .0 (RSS) 66 .725 65 . 148 651 .839 5 (Q) 33. .373 4 1 , . 462 229. .500 (R) 0. .0 0. .0 0. .0 (0) 0 .0 0 .0 0 .0 (RSS) 33 .373 41 . 462 229 .500 6 (0) 166 .216 98 .906 2761 .863 (R) 0 .0 0 .0 0 .0 (0) 0 .0 0 .0 0 .0 (RSS) 166 .216 98 .906 2761 .863 7 (0) 133 .208 93, .854 1958 . 073 (R) 0 .0 0. .0 0. .0 (0) 0 .0 0 .0 0. .0 (RSS) 133 .208 93, .854 1958 .073 8 (0) 100 .030 81 .361 1231 .854 (R) 0 .0 0 .0 0. .0 (0) 0 .0 0 .0 0 .0 (RSS) 100 .030 81 , . 361 1231 . 854 9 (0) 66 .725 65, . 148 65 1 . .839 (R) 0 .0 0 .0 0. .0 (0) 0 .0 0, .0 0, .0 (RSS) 66 .725 65 . 148 651 , .839 10 (0) 33 .373 41 .462 229, .500 (R) 0 .0 0 .0 0, ,0 (0) 0 .0 0 .0 0, .0 (RSS) 33 .373 4 1 . .462 229. 500 11 (0) 0 .0 33 . 339 100. 033 (R) 0 .0 0 .0 0. 0 (0) 0 .0 0 .0 0. 0 (RSS) 0 .0 " 33 . 339 100. .033 12 (0) 0.0 33.348 100.061 0.0 915.001 400.442 0.0 0.0 400.442 265.302 0.0 0.0 265.302 324t. 378 0.0 0.0 324.378 1616.254 0.0 0.0 1616.254 915.001 0.0 0.0 915.001 400.442 0.0 0.0 400.442 265.302 0.0 0.0 265.302 324.378 0.0 0.0 324.378-100.033 0.0 0.0 100 033 100.061 5000.000 5000.000 5000.000 5000.000 5000.000 5000.000 5000.000 5000.000 5000.000 100.000 (R) 0, .0 0. .0 0 .0 0 .0 (0) 0, .0 0. .0 0 .0 0 .0 (RSS) 0, .0 33. .348 100 .061 100 .061 100 .000 13 (0) 0 .0 33, .352 100. . 074 100. .074 (R) 0, .0 0, .0 0 .0 0. ,0 (0) 0 .0 0. .0 0 ,0 0, .0 (RSS) 0 .0 33. , 352 100, ,074 100. .074 100. .000 14 (0) 0 .0 33 . 355 100 .081 100, ,081 (R) 0. .0 0. .0 0 ,0 0. .0 (0) 0 .0 0, .0 0 .0 0. .0 . (RSS) 0 .0 33. . 355 100, ,081 100, ,081 100. .000 15 (0) 0 .0 33. . 374 100. . 138 100. , 138 (R) 0 .0 0. ,0 0. .0 0. .0 (0) 0. .0 0. .0 0. .0 0. .0 (RSS) 0 .0 33. .374 100. . 138 100. , 138 100. ,000 MODAL ANALYSIS COMPLETE QUADRATIC COMBINATION VALUES FRAME 1 **************************************************** cn JOINT NO. X-DISP(IN) Y-DISP(IN) ROTATION(RAD) 1 (Q) 0. ,0 0 .0 0 .0 (R) 0. .0 0 .0 0 .0 (0) 0. .0 0 .0 0. .0 (RSS) 0, .0 0 .0 0, ,0 2 (Q) O. .0 0 .0 0. ,0 (R) 0. .0 0 .0 0, .0 (0) 0. .0 0 .0 . 0. ,0 (RSS) 0. .0 0 .0 0. .0 3 (0) 0. . 106819 0 ,000997 0. 001364 (R) 0. .0 0 .0 0. .0 (0) 0. .0 0. ,0 0. 0 (RSS) 0, ,106819 0 .000997 0. 001364 4 (Q) 0. . 106819 0 .000997 0. 001364 (R) 0, O 0 .0 0. O (0) 0. .0 0. .0 0. 0 (RSS) 0 . 1(56819 0 .000997 0. 001364 5 (Q) 0 . 374415 0 .001796 0. 002242 (R) 0. 0 0. 0 0, ,0 ( 0 ) 0. 0 0. 0 0. .0 (RSS) 0. 3744 15 0. 001796 0. .002242 6 ( 0 ) 0. 3744 15. 0. 001796 0. .002242 (R) 0. 0 0. .0 0 .0 ( 0 ) 0. 0 0. .0 0, .0 (RSS) 0. 374415 0. ,001796 0, .002242 7 (0) 0. 736038 0. .002395 0 .002692 (R) 0. 0 0. ,0 0. .0 ( 0 ) 0. 0 0, ,0 0, .0 (RSS) 0. 736038 0. ,002395 0 .002692 8 ( 0 ) 0. .736038 0. ,002395 0 .002692 (R) 0. .0 0. ,0 0 .0 ( 0 ) 0. ,0 0. ,0 0 .0 (RSS) 0. ,736038 0, ,002395 0 .002692 9 (0) 1. .136384 0. .002795 0 .002812 (R) 0, ,0 0. ,0 0 .0 (0) 0. .0 0. .0 0 .0 (RSS) 1. . 136384 0. .002795 0 .002812 10 ( 0 ) 1. .136384 0. .002795 0 .002812 (R) 0 .0 0 .0 0 .0 ( 0 ) 0 .0 0. .0 0 .0 (RSS) 1 . 136384 0 .002795 . 0 .002812 11 ( 0 ) 1 .538696 0 .002995 0 .002753 (R) 0 .0 0 .0 0 .0 (0) 0 .0 0 .0 0 .0 (RSS) 1 .538696 0 .002995 0 .002753 12 (0) 1 .538696 0 .002995 0 .002753 (R) 0 .0 0 .0 0 .0 (0) 0 .0 0 .0 0 .0 (RSS) 1 .538696 0. .002995 0 .002753 A * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * MEMBER NO. AXIAL SHEAR BML BMG MOMENT DAMAGE (KIPS) (KIPS) (KIP-FT) (KIP-FT) CAPACITY RATIO 1 (0) 166.119 98.888 2760.797 1615.357 (R) 0.0 0.0 0.0 0.0 (0) 0.0 0.0 0.0 0.0 (RSS) 166.119' 98.888 2760.797 1615.357 5000.OOO 0.552 2 (0) 133.125 93.826 1957.024 914.223 (R) 0. 0 0. ,0 0 (0) 0. 0 0. 0 0 (RSS) 133 . 125 93 . ,826 1957 3 (0) 99. 964 81 . 324 1230 (R) 0. 0 0. ,0 0 (0) 0. 0 0. 0 0 (RSS) 99. 964 81 . ,324 1230 4 (0) 66. 679 65. , 101 651 (R) 0. .0 0. ,0 0 (0) 0. 0 0, .0 0 (RSS) 66. .679 65. . 101 651 5 (0) 33. ,350 41 . 413 229 (R) 0. .0 0. ,0 0 (0) 0. ,0 0. .0 0 (RSS) 33, ,350 41 , 413 229 6 (0) 166. . 1 19 98 , 888 2760 (R) 0, ,0 0. .0 0 (0) 0. .0 0, .0 0 (RSS) 166 . 119 98 . 888 2760 7 (0) 133. . 125 93, ,826 1957 (R) 0, .0 0, ,0 0 (0) 0. .0 0, ,0 0 (RSS) 133 . 125 93 .826 1957 8 (0) 99 .964 81 , .324 1230 (R) 0 .0 0, .0 0 (0) 0 .0 0. .0 0 (RSS) 99 .964 81 . 324 1230 9 (0) 66 .679 65. . 101 651 (R) 0 .0 0. .0 0 (0) 0 .0 0. .0 0 (RSS) 66 .679 65. . 101 651 10 (0) 33 .350 4 1 , .413 229 (R) 0 .0 0, .0 0 (0) 0 .0 0 .0 0 (RSS) 33 . 350 41 , .413 229 11 (0) 0 .0 33 .324 99 (R) 0 .0 0 .0 0 (0) 0 .0 0 .0 0 (RSS) 0 .0 - 33 .324 99 12 (0) 0 .0 33 . 331 100 0 0.0 0 . 0.0 024 914.223 5000.000 0.391 906 399.986 0 0.0 0 • 0.0 906 399.986 5000.000 0.246 136 265.342 0 0.0 0 0.0 136 265.342 5000.000 0.130 204 324.151 0 0.0 0 0.0 204 324.151 5000.000 0.065 797 1615.357 0 0.0 0 0.0 797 1615.357 5000.000 0.552 024 914.223 0 0.0 0 0.0 024 914.223 5000.000 0.391 906 399.986 0 0.0 0 0.0 906 399.986 5000.000 0.246 136 265.342 0 0.0 0 0.0 136 265.342 5000.000 0.130 204 324. 151 0 0.0 0 0.0 204 324.151 5000.000 0.065 988 99.988 0 0.0 0 0.0 988 99.988 100.000 1.909 009 100.009 (R) 0.0 0. ,0 0 .0 0 .0 (0) 0.0 0. .0 0. .0 0 .0 (RSS) 0.0 33. .331 100, ,009 100 .009 100 .000 2 .924 13 (0) 0.0 33 .332 100. .014 100 .014 (R) 0.0 0 .0 0, .0 0 .0 (0) 0.0 0 .0 0. .0 0 .0 (RSS) 0.0 33. .332 100, ,014 100. .014 100 .000 3 . 424 14 (0) 0.0 33, .332 100. ,014 100. .014 (R) 0.0 0. .0 0 .0 0 .0 (0) 0.0 0, .0 0. .0 0. .0 (RSS) 0.0 33 .332 100. .014 100. .014 100. .000 3 . 554 15 (0) 0.0 33 .350 100. .068 100. .068 (R) 0.0 0 .0 0. .0 0. 0 (0) 0.0 0 .0 0. .0 0. .0 (RSS) 0,0 33 .350 100. .068 100. 068 100. 000 3. 487 MODAL ANALYSIS ROOT SUM SOUARE VALUES FRAME 2 _^  ************************************************************ 0^ JOINT NO. X-DISP(IN) Y-DISP(IN) ROTATION(RAD) 1 (0) 0. .0 0. .0 0 .0 (R) 0. .0 0. .0 0. .0 (0) 0. .0 0. .0 0 .0 (RSS) 0, ,0 0. .0 0 .0 2 (0) 0. .006961 0. ,0 0 .000091 (R) 0, ,0 0. .0 0 .0 (0) 0, .0 0. .0 0 .0 (RSS) 0, .006961 0. .0 0 .000O91 3 (0) 0. ,024459 0. 0 0. .000149 (R) 0, ,0 0. .0 0. .0 (0) 0, .0 0. .0 0. 0 (RSS) 0. .024459 0. .0 0. .000149 4 (0) 0, .048447 0. .0 0. .000185 (R) 0. .0 0. .0 0. .0 (0) 0 .0 0. .0 0. .0 (RSS) 0 .048447 0 .0 0. 000185 5 (0) 0. .076082 0. 0 0. 000204 (R) 0. 0 ' 0. .0 0 .0 (0) 0, 0 0 .0 0 .0 (RSS) 0. ,076082 0. .0 0. .000204 6 (0) 0. .105285 0, .0 .0. ,000208 (R) 0. ,0 0. .0 0, ,0 (0) 0. .0 0, .0 0, .0 (RSS) 0. .105285 0. .0 0. .000208 ************************************* AXIAL SHEAR BML BMG (KIPS) (KIPS) (KIP--FT) (KIP--FT) 1 (0) , 0. 0 21 . 668 620, . 703 417 .649 (R) 0. 0 0. .0 0, .0 ' 0 .0 (0) 0. 0 0. ,0 0 .0 0 .0 (RSS) 0. 0 21 . 668 620 . 703 417 .649 2 (0) 0. .0 17 , .859 417 .648 300 . 760 (R) 0. .0 0. .0 0, .0 0 .0 (0) 0. ,0 0, .0 0. .0 0 .0 (RSS) 0. .0 17 , .859 417. .648 300 .760 3 (0) 0. .0 1 1 , .600 300. . 763 224 . 344 (R) 0. .0 0. .0 0. .0 0 .0 (0) 0, .0 0. .0 0. .0 0. ,0 (RSS) 0. .0 1 1 , .600 300, . 763 224 . , 344 4 (0) 0. .0 10, .607 224 . 335 108 . ,181 (R) 0. .0 0. .0 0, .0 0. 0 (0) 0. ,0 0. .0 0. 0 0. .0 (RSS) 0. .0 10, ,607 224 . 335 108 . , 181 5 (0) 0, .0 9, .015 108. 176 0. .01 1 (R) 0. .0 0, .0 0. .0 0. .0 (0) 0. .0 0, .0 0. .0 0. 0 (RSS) 0. .0 9, .015 108 . 176 0. 011 • MODAL ANALYSIS COMPLETE QUADRATIC COMBINATION VALUES FRAME 2 ************************************************ JOINT NO. X-DISP(IN) Y-DISP(IN) ROTAT I ON(RAD) 1 (Q) 0.0 0.0 0.0 MOMENT CAPACITY DAMAGE RATIO 10000.000 10000.000 10000.000' 10000.000 10000.000 (R) 0. ,0 ' 0 .0 0.0 (0) 0. .0 0 .0 0.0 (RSS) 0. ,0 0 .0 0.0 2 (0) 0. .006937 0 .0 0.000090 (R) 0. .0 0 .0 0.0 (0) 0. .0 0, .0 0.0 (RSS) 0. .006937 0, ,0 0.000090 3 (0) 0. .024375 0, .0 0.000148 (R) 0. .0 0, ,0 0.0 (0) 0. .0 0, .0 0.0 (RSS) 0. .024375 0. ,0 0.000148 4 (Q) 0. .048282 0. .0 0.000184 (R) 0. ,0 0. ,0 0.0 (0) 0. 0 0. 0 0.0 (RSS) 0. ,048282 0. 0 0.000184 5 (0) 0. ,075826 0. 0 0.000203 (R) 0. ,0 0. 0 0.0 (0) 0. ,0 0. 0 0.0 (RSS) 0. ,075826 0. 0 0.000203 6 (0) 0. 104936 0. 0 0.000208 (R) 0. ,0 0. 0 0.0 (0) 0. ,0 0. 0 0.0 (RSS) 0. ,104936 0. 0 0.000208 ********************** AXIAL SHEAR BML BMG MOMENT DAMAGE (KIPS) (KIPS) (KIP--FT) (KIP--FT) CAPACITY RATIO 1 (0) 0.0 21 .626 618 .691 4 16 . 227 (R) 0.0 0.0 0 .0 0 .0 (0) 0.0 0.0 O .O 0 .0 (RSS) 0.0 21 .626 618 .691 416, .227 10000.000 0.062 2 (0) 0.0 17.817 416 . 225 300, .015 (R) 0.0 0.0 0 .0 0, .0 (0) 0.0 0.0 0 .0 0, .0 (RSS) 0.0 17.817 416 .225 300. .015 10000.000 0.042 3 (0) 0.0 11.563 300 .018 224 . 069 (R) 0.0 0.0 . 0 .0 0 .0 (0) 0.0 0.0 0, ,0 0. .0 (RSS) 0.0 - 1 1 . 563 300. .018 224 . 069 10000.000 0.030 4 (0) 0.0 10.587 224 . ,060 108 . 124 (R) 0. .0 0. .0 0. ,0 0, .0 (0) 0, .0 0 .0 0. .0 0. .0 (RSS) 0, 0 10. .587 224. ,060 108. . 124 100O0, .OOO 0 .022 5 (0) 0, .0 9 .010 108. , 119 0. .01 1 OR) 0. .0 0 .0 0. ,0 0. .0 (0) 0 .0 0. .0 0. .0 0. 0 (RSS) 0 .0 9. .010 108. . 119 0. ,011 10000, .000 0 .01 1 MODAL ANALYSIS ROOT SUM SOUARE VALUES FRAME 3 ******************************************************** JOINT NO. X-DISP(IN) Y-DISP(IN) ROTATION(RAD) 1 (0) 0. .0 0 .0 0 .0 (R) 0. ,0 0. .0 0 .0 (0) 0, ,0 0 .0 0 .0 (RSS) 0. .0 0 .0 0 .0 2 (0) 0. .006961 0. .0 0 .000091 (R) 0, ,0 0. .0 0 .0 (0) 0. .0 0 .0 0 .0 (RSS) 0. .006961 0 .0 0 .000091 3 ( 0 ) 0. .024459 0 .0 0 .000149 (R) 0, .0 0 .0 0 .0 CO) 0. ,0 0 .0 o .0 (RSS) 0. .024459 0 .0 0 .000149 4 (0) 0, .048447 0 .0 0 .000185 (R) 0. .0 0 .0 0. .0 (0) 0. .0 0 .0 • 0 .0 (RSS.) 0, .048447 0 .0 0. .000185 5 (0) 0. .076082 0. .0 0. .000204 (R) 0, .0 0 .0 0. .0 (0) 0. ,0 0. .0 0. .0 (RSS) 0. .076082 0 .0 0. .000204 6 (0) 0. .105285 0 .0 0. .000208 (R) 0, .0 0 .0 0, .0 (0) 0. 0 0 .0 0. .0 (RSS) o. .105285 0 .0 0. .000208 MEMBER NO. AXIAL SHEAR BML BMG (KIPS) (KIPS) (KIP--FT) (KIP--FT) 1 (0) 0.0 21 . 668 620. ,703 417 .649 (R) 0.0 0. .0 0. .0 0 .0 (0) 0.0 0. .0 0. .0 0 .0 (RSS) 0.0 21 . 668 620. ,703 4 17 .649 2 (0) 0.0 17 .859 417 . ,648 300 .760 (R) 0.0 0. .0 0, .0 0 .0 (0) 0.0 0 .0 0, .0 0 .0 (RSS) 0.0 17 . 859 417 . ,648 300, . 760 3 (0) 0.0 1 1 .600 300. ,763 224, . 344 (R) 0.0 0 .0 0. ,0 0. .0 (0) 0.0 0 .0 0. .0 0, .0 (RSS) 0.0 1 1 .600 300. .763 224 , .344 4 (0) 0.0 10. .607 224. .335 108, . 181 (R) 0.0 0 .0 0. .0 0, .0 (0) 0.0 0 .0 0. ,0 0, .0 (RSS) 0.0 10 .607 224 . 335 108, . 181 5 (0) 0.0 9 .015 108. 176 0, .011 (R) 0.0 0 .0 0. 0 0, .0 (0) 0.0 0 .0 0. 0 0. .0 (RSS) 0.0 9 .015 108. 176 0. 01 1 MODAL ANALYSIS COMPLETE QUADRATIC COMBINATION VALUES FRAME 3 *************************************** X-DISP(IN) Y-DISP(IN) ROTATION(RAD) 1 (0) 0.0 0.0 0.0 (R) 0.0 0.0 0.0 (0) 0.0 0.0 0.0 (RSS) 0.0 0.0 0.0 2 (0) 0.006937 0.0 0.000090 (R) 0.0 0.0 0.0 (0) 0.0 0.0 0.0 (RSS) 0.006937 0.0 0.000090 3 (0) 0.024375 0.0 0.000148 MOMENT CAPACITY DAMAGE RATIO 10000.000 10000.000 10000.000 10000.000 10000.000 , , 0 0 OOO'OOOO, l ,00 0 0 0 0 i. i.o-o 61 t 0 0 6i i 801 0 0 80, 0,0'6 0 0 0 0 0,06 0 0 0 0 0 0 0 0 (ssa) (0) (a) (0) s ssoo oeoo OOO'OOOO, ooo oooo. PZl "801. 0 -0 O'O frCl 801 690 « S 0 0 0 0 690 t-Z2 090'PZZ O'O 0 0 090-t>Sc: 81000E O'O 0 0 8iO"ooe LBS'Oi 0 0 0 0 £89 01 E9S'ii 0 0 0 0 E99'11 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 (ssa) (0) (a) (0) 17 (ssa) (0) (a) (0) e ZPO'O ooo-oooo. gio'ooe o o o o S1000E 9SS' 0' 0' 9iP 0 0 SS2"9,f U8'U 0 0 0 0 LIB'Li 0 0 O'O 0 0 0 -0 (ssa) (0) (a) (0) z Z90 0 ouva 3DVWVQ OOO'OOOO, AllOVdVO 1N3W0W LZZBiV 0 0 0 0 LZZ'9iV (ld-dlX) owa i69'8i9 O'Oi o o 169-819 (Id-dlX) 9S9I2 0 0 0 0 939'iZ (Sdl») aV3HS 0 0 0 0 0 0 0 0 (SdIX) ivixv (ssa) (o) (a) (0) i ON aaawaw ************* 80S000 0 0 0 0 0 802000 O eoeoooo 0 0 0 0 eoeoooo fr8l000'0 0 0 . O'O fr8t0000 0 0 0 0 0 0 0 0 0 0 0 0 0 0 o p 0 0 0 0 0 0 0 0 9E6KH 0 0 0 0 0 986*01 0 9589i0 -0 0 0 0 0 9589*0 O 2828KTO O'O 0 0 Z 8 Z 8 W > * 0 (ssa) (0) (a) (0) 9 (ssa) (0) (a) (0) 9 (ssa) (0) (a) (0) v BP 1000 0 0 0 0 0 0 0 0 0 0 0 SL£PZ0'0 0 0 0 0 (ssa) (0) (a) MODAL ANALYSIS ROOT SUM SQUARE VALUES FRAME 4 ***************** ^*********************************.********.****************** JOINT NO X-DISP(IN) Y-DISP(IN) ROTATION(RAD) 1 (0) 0 0 0 0 0 0 (R) 0 0 0 0 0 0 (0) 0 0 0 0 0 0 (RSS) 0 0 0 0 0 0 2 (Q) 0 b. 0 0 0 0 (R) 0 0 0 0 0 0 (0) 0 0 0 0 0 0 (RSS) 0 0 0 0 0 0 3 (0) 0 103512 0 000997 0 001320 (R) 0 0 0 0 0 0 (0) 0 0 0 0 0 0 (RSS) 0 103512 0 000997 0 001320 4 (0) 0 103512 0 000997 • 0 001320 (R) 0 0 0 0 0 0 (0) 0 0 0 0 0 0 (RSS) 0 103512 0 000997 0 001320 5 (Q) 0 362372 0 001795 0 002167 (R) 0 0 0 0 0 0 (0) 0 0 0 0 0 0 (RSS) 0 362372 0 001795 0 002167 6 (Q) 0 362372 0 001795 0 002167 (R) 0 0 0 0 0 0 (0) 0 0 0 0 0 0 (RSS) 0 362372 0 001795 0 002167 7 (Q) 0 711507 0 002395 0 002595 (R) 0 0 0 0 0 0 (0) 0 0 0 0 0 0 (RSS) 0 711507 0 002395 b 002595 8 (Q) 0 711507 0 002395 0 002595 (R) 0 0 0 0 0 0 (0) 0.0, 0 0 0 0 (RSS) 0 711507 0 002395 0 002595 9 (Q) 1 0971 15 0 002794 0 0027O4 (Ri- 0, o • 0. ,0 Co) 0 .0 0. .0 (RSS) 1. .097115 0. ,002794 10 (0) 1. .097115 0. .002794 (R) 0. .0 0. .0 (0) 0. .0 0. .0 (RSS) 1, .097115 0. .002794 11 (0) 1. .483697 0. ,002994 (R) 0 .0 0. .0 (0) 0. .0 0. ,0 (RSS) 1 ,483697 0. .002994 12 (0) 1. ,483697 0. .002994 (R) 0 .0 0. .0 (0) 0 ,0 0. .0 (RSS) 1 .483697 0. .002994 NO. AXIAL SHEAR BML (KIPS) (KIPS) (KIP-•FT) 1 (0) 166. .047 97, . 105 2680 .675 (R) 0. .0 0. .0 0. .0 (0) 0. .0 0. .0 0. .0 (RSS) 166. .047 97. . 105 2680. .675 2 (0) 133. .072 92. .172 1896. .919 (R) 0. .0 0. .0 0. .0 (0) 0. .0 0. .0 0. .0 (RSS) 133. .072 92. . 172 1896. .919 3 (0) 99. .926 79. .914 . 1188. ,355 (R) 0. .0 0. .0 0. ,0 (0) 0. .0 0. .0 0. ,0 (RSS) 99. .926 79. .914 1 188. ,355 4 (Qj 66. .655 63. .857 624. . 173 (R) 0, .0 0. .0 0, .0 (0) 0. .0 0. .0 0. 0 (RSS) 66. .655 63. .857 624 . , 173 5 (0) 33 .337 40. .530 218. , 143 (R) 0, .0 0. .0 0. 0 (0) 0. .0 0. .0 0. .0 (RSS) 33. .337* 40. .530 218. 143 6 (0) 166.047 97.105 2680.675 0.0 0.0 0.002704 0.002704 0.0 6.0 0.002704 0.002642 0.0 0.0 0.002642 0.002642 0.0 0.0 0.002642 ********************************************* BMG MOMENT DAMAGE (KIP-FT) CAPACITY RATIO 1554.561 0.0 0.0 1554.581 5000.OOO 871:108 0.0 0.0 871.108 500O.O00 375.806 0.0 0.0 375.806 50OO.000 268.475 0.0 0.0 268.475 5000.OOO 325.136 0.0 0.0 325.136 5000.000 cn cr. ' 1554.581 (R) 0. 0 0. 0 0. .0 0. ,0 (0) 0. o 0. ,0 o. ,0 0. ,0 (RSS) 166. 047 97. , 105 2680. ,675 1554, ,581 5000.000 7 (0) 133. 072 92. 172 1896. .919 871 , . 108 (R) 0. 0 0. ,0 0. ,0 0. .0 (0) 0. 0 0. .0 0. ,0 ' 0. ,0 (RSS) 133. ,072 92. , 172 1896. ,919 871 , . 108 5000,. OOO 8 (Q) 99. .926 79. ,914 1188. .355 375, .806 (R) 0. ,0 0. 0 0, .0 0, .0 (0) 0. .0 0. .0 0, .0 0, ,o (RSS) 99. .926 79. ,914 1188. .355 375, .806 5000.000 9 (0) 66. .655 63. .857 624. . 173 ' 268. .475 (R) 0. .0 0. .0 0, .0 0. .0 (0) 0. .0 0. .0 0. .0 0, .0 (RSS) 66. .655 63. .857 624, . 173 268. .475 5000.000 10 (0) 33. .337 40. ,530 218. . 143 325, . 136 (R) 0. .0 0. ,0 0, .0 0. .0 (0) 0 .0 0. .0 0. ,0 0, .0 (RSS) 33. .337 40. .530 218. . 143 325. . 136 5000.000 11 (0) 0, .0 33. .305 99. .932 99. .932 (R) 0 .0 0, .0 0. .0 0. .0 (0) 0 .0 0. .0 0. O 0, .0 (RSS) 0 .0 33. .305 • 99. ,932 99, .932 100.000 12 (0) 0 .0 33. .314 99. ,960 99, .960 (R) 0 .0 0. .0 0. .0 0, .0 (0) 0 .0 0. .0 0. ,0 0, .0 (RSS) 0 .0 33, .314 99. ,960 99, .960 100.000 13 (0) 0 .0 33, .319 99. .973 99 .973 (R) 0 .0 0, .0 0. .0 0, .0 (0) 0 .0 0, .0 0, .0 0 .0 (RSS) 0 .0 33, .319 99. .973 99, .973 100.000 14 (0) 0 .0 33, .321 99. .980 99 .980 (R) 0 .0 0, .0 0. .0 0, .0 (0) 0 .0 0, .0 0, .0 0, .0 (RSS) 0 .0 33, .321 99. ,980 99 .980 100.ooo 15 (0) 0 .0 33, .337 100, .029 100 .029 (R) 0 .0 0, .0 0, ,0 0 .0 (0) 0 .0 0, .0 0. .0 0 .0 (RSS) 0 .0 * 33, .337 100, ,029 100 .029 100.ooo MODAL ANALYSIS COMPLETE QUADRATIC COMBINATION VALUES FRAME 4 ******************************************* JOINT NO. X-DISP(IN) Y-DISP(IN) ROTATION*RAD) 1 (Q) 0. .0 0. .0 0.0 (R) 0. .0 0 .0 0.0 (0) 0. .0 0 .0 0.0 (RSS) 0. .0 0 .0 0.0 2 (Q) 0. .0 0 .0 0.0 (R) 0. .0 0 .0 .0.0 . (0) 0. .0 0 .0 0.0 (RSS) 0. .0 0 .0 0.0 3 (Q) 0. . 103604 0, .000997 0.001321 (R) 0. .0 0 .0 0.0 (0) 0. .0 0 .0 0.0 (RSS) 0. . 103604 0. .000997 0.001321 4 (Q) 0. .103604 0 .000997 0.001321 (R) 0. .0 0, .0 0.0 (0) 0. .0 0. .0 0.0 (RSS) 0. . 103604 0 .000997 0.001321 5 (Q) 0. .362666 0, .001796 0.002168 (R) 0. .0 0. .0 0.0 (0) 0. .0 0, .0 0.0 (RSS) 0. .362666 0. .001796 0.002168 6 (Q) 0, .362666 0. .001796 0.002168 (R) 0, .0 0, .0 0.0 (0) 0. .0 0. .0 0.0 (RSS) 0. .362666 0. .001796 0.002168 7 (Q) 0. .712021 0. .002396 0.002597 (R) 0. .0 0. .0 0.0 (0) 0, .0 0. .0 • . 0.0 (RSS) 0. .712021 0, .002396 0.002597 8 (Q) 0. .712021 0. .002396 0.002597 (R) 0, .0 0. .0 0.0 (0) 0. .0. 0. .0 O.O (RSS) 0. .712021 0. .002396 0.002597 9 (Q) 1. .097822 0, .002796 0.002705 (R) 0 .0 " 0. 0 0 .0 (0) 0 .0 0. 0 0. .0 (RSS) 1 .097822 0. 002796 0. .002705 10 (0) 1 . .097822 0. .002796 0 .002705 (R) 0 .0 0. .0 0. .0 (0) 0 .0 0, .0 0. .0 (RSS) 1 .097822 0. .002796 0, .002705 11 (0) 1 .484571 0. .002995 0. .002643 (R) 0 .0 0. .0 0. .0 (0) 0 .0 0 .0 0. .0 (RSS) 1 .484571 0, .002995 0 .002643 12 (Q) 1 .484571 0. .002995 0. .002643 (R) 0: :0 0. .0 0. .0 (0) 0 .0 0. .0 • 0. .0 (RSS) 1 .484571 0. .002995 0. .002643 ******************************* MEMBER NO. IO. AXIAL SHEAR BML BMG MOMENT DAMAGE (KIPS) (KIPS) (KIP-•FT) (KIP--FT) CAPACITY RATIO 1 (0) 166.147 . 97.231 2683. .238 1555 .621 (R) 0 . 0 0.0 0, .0 0 .0 (0) 0.0 0.0 0, .0 0 .0 (RSS) . 166.147 97.231 2683. .238 1555. .621 5000.OOO 0.537 2 (Q) 133.143 92.284 1898. .251 871 . 154 (R) 0 . 0 0.0 0, .0 0 .6 (0) 0.0 0.0 0, .0 0 .0 (RSS) 133.143 92.284 1898. .251 .871 . 154 5000.000 0. 380 3 (0) 99.974 79.975 1 188. 613 375. .716 (R) 0.0 0.0 0, .0 0 .0 (0) 0.0 0.0 0. .0 0. .0 (RSS) 99.974 79.975 1188. .613 375 .716 5000.000 0. 238 4 (0) 66.684 63.865 624. ,019 268. .863 (R) 0.0 0.0 0. .0 0. .0 (0) 0.0 0.0 0. .0 0. .0 (RSS) 66.684 63.865 624. .019 268. .863 5000.000 0. 125 5 (0) 33.352 40.529 218. , 148 325. .274 (R) 0.0 0.0 0. .0 0. .0 (0) 0.0 0.0 0, ,0 0. .0 (RSS) 33.352- 40.529 218. . 148 325. .274 5000.000 0.065 6 (0) 166.147 97.231 ' 2683. . 238 1555. .621 (R) 0. 0 0. .0 0 (0) 0. 0 0. .0 0 (RSS) 166. 147 97. .231 2683 7 ( 0 ) 133. , 143 92. .284 1898 (R) 0. 0 0. .0 0 ( 0 ) 0. 0 0. .0 0 (RSS) 133. . 143 92. .284 1898 8 ( 0 ) 99. 974 79. .975 1 188 (R) 0. 0 0. .0 0 ( 0 ) 0. 0 0. .0 0 (RSS) 99. .974 79. .975 1 188 9 (0) 66. .684 63. .865 624 (R) 0. .0 0. ,0 0 (0) 0. .0 0. .0 0 (RSS) 66. ,684 63. .865 624 10 (0) 33. .352 40. ,529 218 (R) 0. .0 0. .0 0 (0) 0. .0 0. ,0 0 (RSS) 33. .352 40. .529 218 11 (0) 0. .0 33. ,334 100 (R) 0. .0 0. ,0 0 (0) 0. .0 0. .0 0 (RSS) 0. .0 33. .334 100 12 (0) 0. .0 33. .338 100 (R) 0. .0 0. .0 0 (0) 0 .0 0. .0 0 (RSS) 0. .0 33. 338 100 13 (0) 0, .0 33. .337 100 (R) 0 .0 0. ,0 0 (0) 0 .0 0. .0 0 (RSS) 0 .0 33. ,337 100 14 (0) 0 .0 33. ,336 100 (R) 0 .0 0. .0 0 (0) 0 .0 0. ,0 0 (RSS) 0 .0 33. 336 100 15 (0) 0 .0 33. ,351 100 (R) 0 .0 0. ,0 0 (0) 0 .0 0. ,0 0 (RSS) 0 .0 ' 33. ,351 100 4 0 0.000 0 0.0 0 0.0 238 1555.621 5000.OOO 0.537 251 871.154 0 0.0 0 0.0 251 871.154 5000.000 0.380 613 375.716 0 0.0 0 0.0 613 375.716 5000.OOO 0.238 019 268.863 0 0.0 0 0.0 019 • 268.863 5000.000 0.125 148 325.274 0 0.0 0 0.0 148 325.274 5000.000 0.065 018 . 100.018 0 0.0 0 0.0 018 100.018 100.000 1.857 031 100.031 0 0.0 O 0.0 031 100.031 100.000 2.840 029 100.029 0 0.0 , 0 0.0 029 100.029 100.000 3.318 026 100.026 O 0.0 0 0.0 026 100.026 100.OOO 3.436 071 100.071 0 0.0 0 0.0 07 1 100.07 1 100.OOO 3.365 NO. OF ITERATIONS BETA=0.0 BENDING MOMENT ERROR= 0.050 DAMAGE RATIO ERROR= 0.010 T=0.566 DR=0 $.43, $.44T SIG 

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