UBC Theses and Dissertations

UBC Theses Logo

UBC Theses and Dissertations

Pseudo non-linear seismic analysis Hui, Ho Yin Lawrence 1984

Your browser doesn't seem to have a PDF viewer, please download the PDF to view this item.

Item Metadata

Download

Media
831-UBC_1984_A7 H83.pdf [ 10.04MB ]
Metadata
JSON: 831-1.0062849.json
JSON-LD: 831-1.0062849-ld.json
RDF/XML (Pretty): 831-1.0062849-rdf.xml
RDF/JSON: 831-1.0062849-rdf.json
Turtle: 831-1.0062849-turtle.txt
N-Triples: 831-1.0062849-rdf-ntriples.txt
Original Record: 831-1.0062849-source.json
Full Text
831-1.0062849-fulltext.txt
Citation
831-1.0062849.ris

Full Text

PSEUDO NON-LINEAR SEISMIC ANALYSIS by HO YIN LAWRENCE ^HUI B . E . S c , The U n i v e r s i t y of W e s t e r n O n t a r i o , 1982 A THESIS SUBMITTED IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF MASTER OF APPLIED SCIENCE i n THE FACULTY OF GRADUATE STUDIES (Department o f C i v i l E n g i n e e r i n g ) We a c c e p t t h i s t h e s i s 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 BRITISH COLUMBIA O c t o b e r , 1984 © Ho Y i n Lawrence H u i , 1984 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 a t the U n i v e r s i t y o f B r i t i s h Columbia, I agree t h a t 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 t h a t p e r m i s s i o n f o r e x t e n s i v e copying o f 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 o r by h i s o r her r e p r e s e n t a t i v e s . I t i s understood t h a t copying or p u b l i c a t i o n o f t h i s t h e s i s f o r f i n a n c i a l g a i n s h a l l 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 Engineering The U n i v e r s i t y of B r i t i s h Columbia 1956 Main Mall Vancouver, Canada V6T 1Y3 Date 1 2 t h ° c t . , 1984 ABSTRACT A method of e v a l u a t i n g t h e damage p a t t e r n and d e f l e c t i o n s of p l a n a r s t r u c t u r e s under e a r t h q u a k e e x c i t a t i o n i s p r e s e n t e d . The p r o p o s e d method i s an i t e r a t i v e p r o c e d u r e b a s e d on o r d i n a r y e l a s t i c modal a n a l y s i s , b ut i t i s e x t e n d e d t o t h e i n e l a s t i c r a n g e by u s i n g a s p e c i a l t e c h n i q u e t o t a k e i n t o a c c o u n t t h e r e d u c t i o n i n e f f e c t i v e s t i f f n e s s and t h e v a r i a t i o n i n damping. The t h e o r e t i c a l d e v e l o p m e n t o f t h e method i s r e v i e w e d i n t h i s s t u d y and some improvements a r e a l s o made. Th e s e new d e v e l o p m e n t s a r e t h e n i n c o r p o r a t e d i n an e x i s t i n g computer p r o g r a m f o r t h e method and t e s t e d w i t h d i f f e r e n t k i n d s of i d e a l i z e d s t r u c t u r e s . T e s t r e s u l t s show good agreement w i t h r e s u l t s o b t a i n e d from a t i m e - s t e p a n a l y s i s p r o g r a m DRAIN-2D, p r o v i d e d t h a t t h e f u n d a m e n t a l r e s p o n s e p e r i o d of t h e s t r u c t u r e d o e s not o s c i l l a t e n e a r t h e s t e e p d e s c e n d i n g b r a n c h o f t h e smooth s p e c t r u m and no e x t e n s i v e y i e l d i n g o c c u r s i n t h e c o l u m n s . The p r o p o s e d method i s l e s s e x p e n s i v e t h a n an 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 and can p r o d u c e f a r more s u p e r i o r r e s u l t s t h a n t h e e l a s t i c modal a n a l y s i s ; t h e r e f o r e i t i s c o n s i d e r e d as a good a l t e r n a t i v e t o t h e two c o n v e n t i o n a l t y p e s of a n a l y s i s. i i i TABLE OF CONTENTS Page ABSTRACT i i TABLE OF CONTENTS i i i L I S T OF TABLES v i L I S T OF FIGURES v i i ACKNOWLEDGEMENTS x i CHAPTER 1.0 INTRODUCTION 1.1 G e n e r a l B a c k g r o u n d 1 1.2 O b j e c t i v e o f S t u d y 8 1 .3 Scope 11 2.0 THE PRESENT METHOD 2.1 B r i e f Review of t h e P r e s e n t Method 13 2.1.1 The S u b s t i t u t e S t r u c t u r e Method .... 13 2.1.2 P r o p o s e d M o d i f i c a t i o n s o f t h e S u b s t i t u t e S t r u c t u r e Method 18 2.2 The Development of t h e Computer Program .. 20 2.3 I n t r o d u c t i o n t o t h e S t r u c t u r e o f t h e Computer Program 22 3.0 CONVERGENCE 3.1 C o n v e r g e n c e Schemes 27 3.2 E f f e c t o f I n c l u d i n g S t r a i n H a r d e n i n g 30 3.3 F a c t o r s t h a t H i n d e r C o n v e r g e n c e 31 i v 3.4 The S o l u t i o n S e a r c h i n g R o u t i n e 36 3.5 T e s t i n g t h e S o l u t i o n S e a r c h i n g r o u t i n e ... 44 4.0 SUBSTITUTE DAMPING AND SMEARED DAMPING 4.1 G e n e r a l 46 4.2 S u b s t i t u t e Damping F a c t o r 46 4.3 Smeared Damping F a c t o r 50 5.0 ALTERATION TO THE DETERMINATION OF DAMAGE RATIOS 5.1 Member S t i f f n e s s M o d i f i c a t i o n U s i n g a S i n g l e Damage R a t i o 52 5.2 Member S t i f f n e s s M o d i f i c a t i o n U s i n g Damage R a t i o a t Two Ends 54 5.3 A l t e r a t i o n i n t h e C a l c u l a t i o n o f S t r a i n E n e r g y and S u b s t i t u t e Damping 61 5.3.1 S t r a i n E n e r g y 61 5.3.2 S u b s t i t u t e Damping 63 5.4 Computer Program 63 6.0 THE PRESENT METHOD VS. INELASTIC TIME-STEP ANALYSIS 6. 1 G e n e r a l 65 6.2 C o m p a r i s o n o f t h e Two Methods 66 6.2.1 A d v a n t a g e s and D i s a d v a n t a g e s o f t h e Two Methods 66 6.2.2 C o m p a r i s o n o f t h e Two Programs 68 6.2.3 L i m i t a t i o n i n t h e C o m p a r i s o n of R e s u l t s 69 6.3 C h o i c e o f E a r t h q u a k e R e c o r d 70 6.4 T e s t R e s u l t s and D i s c u s s i o n s 72 V 6.4.1 G e n e r a l 72 6.4.2 T e s t Frame No. 1 73 6.4.3 T e s t Frame No. 2 75 6.4.4 T e s t Frame No. 3 77 6.4.5 T e s t Frame No. 4 78 6.4.6 T e s t Frame No. 5 79 6.4.7 T e s t Frame No. 6 80 6.4.8 A C o m p a r i s o n o f R e s u l t s Between EDAM, EDAM2 and DRAIN-2D 81 6.5 C o s t of E x e c u t i o n 83 7.0 CONCLUSIONS 85 REFERENCES 87 APPENDIX : A. U s e r ' s Manual A-1 B. Sample I n p u t / O u t p u t B-1 C. P r o g r a m L i s t i n g C-1 v i L I S T OF TABLES T a b l e Page 5.1 V a l u e s f o r t h e E f f e c t i v e Damage R a t i o , M e 89 6.1 S t r u c t u r a l P r o p e r t i e s f o r A l l T e s t Frames 90 6.2 Summary of R e s u l t s 91 6.3 Summary o f Program E x e c u t i o n C o s t 92 v i i L I S T OF FIGURES F i g u r e Page 2.1 D i a g r a m s f o r D e f i n i n g D u c t i l i t y R a t i o and Damage R a t i o 93 2.2 F l o w c h a r t f o r t h e Main Program 94 2.3 F l o w c h a r t f o r t h e Main S u b r o u t i n e - M0D3 95 3.1 A Smoothed R e s p o n s e S p e c t r u m w i t h A l l T y p i c a l B r a n c h e s 96 3.2 An Example t o Show t h e F l u c t u a t i o n i n Respon s e P e r i o d v s . I t e r a t i o n Number 96 3.3 G r a p h Showing t h e P r o g r e s s of E a c h I t e r a t i o n P l o t t e d w i t h t h e Respo n s e S p e c t r u m 97 3.4 A P l o t of S p e c t r a l A c c e l e r a t i o n v s . P e r i o d Showing t h e C a p a c i t y and Demand C u r v e (Freeman's Method) 98 3.5 R e s u l t s f o r t h e F o u r t h and F i f t h I t e r a t i o n 99 3.6 A G r a p h i c a l S o l u t i o n A c c o r d i n g t o Freeman's Method 100 3.7 A P l o t o f Smeared Damping F a c t o r v s . P e r i o d .... 100 3.8 A R e v i s e d F l o w c h a r t f o r S u b r o u t i n e MOD3 I n c l u d i n g a New S u b r o u t i n e STACHK 101 3.9 F l o w c h a r t f o r t h e S o l u t i o n S e a r c h i n g R o u t i n e - STACHK 102 3.10 G r a p h Showing t h e P r o g r e s s o f t h e S o l u t i o n S e a r c h i n g R o u t i n e 103 v i i i 4.1 (a) S i n g l e - d e g r e e of Freedom System (b) Moment v s . R o t a t i o n D i a g r a m ( c ) V i s c o u s Damping F o r c e v s . D i s p l a c e m e n t 104 4.2 A P l o t o f S u b s t i t u t e Damping F a c t o r v s . Damage R a t i o 105 5.1 (a) A x i a l S t i f f n e s s M a t r i x (b) B e n d i n g S t i f f n e s s M a t r i x ( c ) R i g i d Arm S t i f f n e s s M a t r i x • 106 5.2 A T y p i c a l Member w i t h Two H i n g e s 107 5.3 D i a g r a m s f o r t h e C o n j u g a t e Beam Method 108 5.4 D i f f e r e n t Components of t h e D i s t r i b u t e d E l a s t i c L o a d 109 5.5 A P l o t of t h e E f f e c t i v e Damage R a t i o v s . M A and M & 1 1 0 5.6 S t i f f n e s s M a t r i x w i t h B e n d i n g S t i f f n e s s e s W r i t t e n i n Terms of k 3 3 , k 6 6 and k 6 3 111 5.7 A T r a n s f o r m e d S t i f f n e s s M a t r i x 112 5.8 B e n d i n g S t i f f n e s s M a t r i x w i t h S i x M o d i f i c a t i o n F a c t o r s 112 5.9 A R e v i s e d F l o w c h a r t f o r t h e Main Program 113 5.10 A r e v i s e d F l o w c h a r t f o r S u b r o u t i n e MOD3 I n c l u d i n g A l l Changes 114 6.1 C . I . T . S i m u l a t e d E a r t h q u a k e Type C2 S p e c t r u m ... 115 6.2 San F e r n a n d o S90W S p e c t r u m 116 6.3 D i m e n s i o n s and Y i e l d Moments f o r T e s t Frame No. 1 117 6.4 T e s t Frame No. 1 - Maximum Base S h e a r , Damage R a t i o s and H o r i z o n t a l D i s p l a c e m e n t s 118 i x 6.5 Beam Damage R a t i o on T e s t Frame No. 1 119 6.6 D i m e n s i o n s and Y i e l d Moments f o r T e s t Frame No. 2 120 6.7 T e s t Frame No. 2 - Maximum Base S h e a r , Damage R a t i o s and H o r i z o n t a l D i s p l a c e m e n t s 121 6.8 Beam Damage R a t i o on T e s t Frame No. 2 122 6.9 D i m e n s i o n s and Y i e l d Moments f o r T e s t Frame No. 3 123 6.10 T e s t Frame No. 3 - Maximum Base S h e a r , Damage R a t i o s 124 6.11 (a) Beam Damage R a t i o on T e s t Frame No. 3 125 6.11 (a) Beam Damage R a t i o on T e s t Frame No. 3 ( c o n t ' d ) (b) H o r i z o n t a l D i s p l a c e m e n t s 126 6.12 T e s t Frame No. 4 - Maximum Base S h e a r , Damage R a t i o s and H o r i z o n t a l D i s p l a c e m e n t s 127 6.13 Beam Damage R a t i o on T e s t Frame No. 4 128 6.14 D i m e n s i o n s and Y i e l d Moments f o r T e s t Frame No. .5 129 6.15 T e s t Frame No. 5 - Maximum Base S h e a r , Damage R a t i o s and H o r i z o n t a l D i s p l a c e m e n t s 130 6.16 Column Damage R a t i o on T e s t Frame No. 5 131 6.17 T e s t Frame No. 6 - Maximum Base S h e a r , Damage R a t i o s and H o r i z o n t a l D i s p l a c e m e n t s 132 6.18 Beam Damage R a t i o on T e s t Frame No. 6 133 6.19 A C o m p a r i s o n o f Beam Damage R a t i o s f r o m EDAM, EDAM2 and DRAIN-2D - T e s t Frame No. 1 134 X 6.20 A C o m p a r i s o n of Beam Damage R a t i o s f r o m EDAM, EDAM2 and DRAIN-2D - T e s t Frame No. 2 135 6.21 A C o m p a r i s o n of Beam Damage R a t i o s f r o m EDAM, EDAM2 and DRAIN-2D - T e s t Frame No. 3 136 6.21 A C o m p a r i s o n of Beam Damage R a t i o s f r o m EDAM, EDAM2 and DRAIN-2D - T e s t Frame No. 3 ( c o n t ' d ) 137 x i ACKNOWLEDGEMENT The a u t h o r would l i k e t o e x p r e s s h i s s i n c e r e g r a t i t u d e t o h i s a d v i s o r , D r . N.D. Natha n , f o r h i s v a l u a b l e a d v i c e and g u i d a n c e t h r o u g h o u t t h e r e s e a r c h and p r e p a r a t i o n o f t h i s t h e s i s . Thanks a r e a l s o e x t e n d e d t o Dr. S. C h e r r y , f e l l o w g r a d u a t e s t u d e n t s , Ken Tarn and F r a n k Lam f o r t h e i r s u p p o r t and a d v i c e d u r i n g t h e r e s e a r c h . The a s s i s t a n c e of M i r a n d a Mak f o r t y p i n g t h i s t h e s i s has been g r e a t l y a p p r e c i a t e d . The f i n a n c i a l s u p p o r t f r o m t h e N a t i o n a l R e s e a r c h C o u n c i l o f Canada i n t h e form o f a R e s e a r c h 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 a c k n o w l e d g e d . September, 1984 VANCOUVER, BRITISH COLUMBIA 1 1.0 INTRODUCTION 1.1 G e n e r a l B a c k g r o u n d A l t h o u g h e a r t h q u a k e s o f d e s t r u c t i v e i n t e n s i t y o c c u r i n f r e q u e n t l y and o n l y i n c e r t a i n a r e a s of t h e w o r l d , a d e q u a t e p r o c e d u r e s t o e n s u r e s a f e t y o f s t r u c t u r e s i n t h o s e a r e a s a r e i m p o r t a n t . B u t , d e s i g n i n g e a r t h q u a k e - r e s i s t a n t s t r u c t u r e s i s n o t a s i m p l e t a s k . The d i f f i c u l t i e s i n t h i s a r e a of s t u d y a r e i n h e r e n t i n t h e dynamic n a t u r e o f t h e p r o b l e m and t h e random v a r i a t i o n s i n e a r t h q u a k e m o t i o n . When a s t r u c t u r e i s s u b j e c t e d t o s e v e r e e a r t h q u a k e e x c i t a t i o n , some members o f t h e s t r u c t u r e w i l l e x p e r i e n c e s t r a i n s t h a t go beyond t h e i r e l a s t i c c a p a c i t y . As members r e s p o n d i n t h e i n e l a s t i c r a n g e , t h e s t i f f n e s s of t h e s t r u c t u r e w i l l d e c r e a s e . The r e d u c t i o n of s t i f f n e s s a s s o c i a t e d w i t h t h e l a r g e - s c a l e a b s o r p t i o n of e n e r g y , i n e f f e c t , c h a n g e s t h e p e r i o d of v i b r a t i o n and t h e damping o f t h e s t r u c t u r e d u r i n g t h e c o u r s e o f t h e e a r t h q u a k e . W i t h t h e s e v a r i a t i o n s , i t i s d i f f i c u l t t o d e t e r m i n e s t r u c t u r a l p a r a m e t e r s , s u c h as s t i f f n e s s and damping, i n a dynamic a n a l y s i s . In t h e c a s e o f r e i n f o r c e d c o n c r e t e s t r u c t u r e s , t h e s e v a r i a t o n s a r e more p r o n o u n c e d , due t o t h e change i n s e c t i o n a l p r o p e r t i e s a s c o n c r e t e c r a c k s d u r i n g e a r t h q u a k e l o a d i n g . 2 A n o t h e r c o n t r i b u t i o n w h i c h f u r t h e r c o m p l i c a t e s t h e m a t t e r but i s o f t e n n e g l e c t e d when d o i n g dynamic a n a l y s i s stems from t h e p r o p e r t i e s of t h e s o i l and f o u n d a t i o n . S o i l -s t r u c t u r e i n t e r a c t i o n i s i m p o r t a n t , e s p e c i a l l y i n t h e d e s i g n o f s t r u c t u r e s s i t u a t e d on r e l a t i v e l y s o f t s o i l s . I t s e f f e c t on 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 depends on t h e p r o p e r t i e s of t h e s o i l and t h a t of t h e s t r u c t u r e , as w e l l as on t h e c h a r a c t e r i s t i c s of t h e e a r t h q u a k e . To 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 o f f u t u r e e a r t h q u a k e s t o w h i c h a s t r u c t u r e may be s u b j e c t e d i n i t s l i f e t i m e i s a n o t h e r u n c e r t a i n t y a s s o c i a t e d w i t h t h e p r o b l e m . F i r s t o f a l l , e a r t h q u a k e s a r e g e n e r a l l y random i n c h a r a c t e r . S e c o n d l y , t h e m a g n i t u d e o f g r o u n d a c c e l e r a t i o n t h a t a s t r u c t u r e may e x p e r i e n c e depends on t h e m a g n i t u d e o f t h e e a r t h q u a k e i t s e l f and t h e d i s t a n c e o f t h e s t r u c t u r e from t h e e p i c e n t e r . F u r t h e r , i t a l s o depends on t h e p r o p e r t i e s o f t h e s u r r o u n d i n g s o i l and n e a r b y c o n s t r u c t i o n s . F o r t u n a t e l y , due t o t h e a d v a n c e s i n s e i s m o l o g i c a l s t u d y and t h e i n c r e a s i n g number of s e i s m o g r a p h i c s t a t i o n s , a l a r g e number of e a r t h q u a k e r e c o r d s a r e a v a i l a b l e f o r p e r f o r m i n g s e i s m i c a n a l y s i s . And t h r o u g h s t u d i e s o f t h e s e r e c o r d s , e s t i m a t e s o f t h e e x p e c t e d r a n g e of g r o u n d a c c e l e r a t i o n i n a p a r t i c u l a r s e i s m i c zone a r e p o s s i b l e . However, b e c a u s e o f t h e random n a t u r e of e a r t h q u a k e m o t i o n , t h e e x a c t c h a r a c t e r o f f u t u r e e a r t h q u a k e s i s s t i l l u n p r e d i c t a b l e . R e c o r d s s u c h a s t h a t of t h e 1940 E l C e n t r o e v e n t a r e s t i l l u s e d w i d e l y i n p e r f o r m i n g a n a l y s e s d e s p i t e t h e f a c t t h a t t h e y may n o t be s u i t a b l e f o r t h e a r e a o f i n t e r e s t . 3 D u r i n g t h e p a s t d e c a d e s , c a t a s t r o p h i c f a i l u r e of some b u i l d i n g s t h a t went t h r o u g h m a j o r e a r t h q u a k e s d i d o c c u r . But i n s t u d y i n g t h e s e b u i l d i n g s and o t h e r s t h a t s u r v i v e d t h e e a r t h q u a k e s , more u n d e r s t a n d i n g o f t h e a c t u a l b e h a v i o u r o f s t r u c t u r e s i n e a r t h q u a k e s i s g a i n e d . O b s e r v a t i o n s combined w i t h a n a l y t i c a l s t u d i e s o f t h e r e s p o n s e of some of t h e s e s t r u c t u r e s have p r o v e n t h a t i t i s p o s s i b l e t o p r e d i c t t h e p e r f o r m a n c e of a s t r u c t u r e a n a l y t i c a l l y t o a c e r t a i n d e g r e e o f a c c u r a c y f o r a g i v e n e a r t h q u a k e . B e c a u s e of t h e s e d e v e l o p m e n t s , a t l e a s t t h r e e k i n d s o f s e i s m i c a n a l y s i s , e a c h s u i t a b l e f o r d i f f e r e n t s i t u a t i o n s , have e v o l v e d and a r e commonly p r a c t i s e d i n t h e a n a l y s i s and d e s i g n o f e a r t h q u a k e - r e s i s t a n t s t r u c t u r e s . The f o l l o w i n g i s an o v e r v i e w of t h e s e methods. Of t h e t h r e e methods, t h e s i m p l e s t one i s c a l l e d t h e e q u i v a l e n t l a t e r a l f o r c e p r o c e d u r e . I t i s s u i t a b l e f o r d e s i g n i n g s t r u c t u r e s w i t h r e g u l a r d i s t r i b u t i o n o f mass and s t i f f n e s s o v e r h e i g h t , o r s t r u c t u r e s t h a t a r e not l o c a t e d i n h i g h l y s e i s m i c a r e a s . I t i s e a s y t o use b e c a u s e i t o n l y r e q u i r e s t h e v a l u e of t h e f u n d a m e n t a l p e r i o d of v i b r a t i o n , w h i c h c a n be e s t i m a t e d by u s i n g a s i m p l e f o r m u l a , and an e s t i m a t i o n o f t h e damping. B a s e d on t h e s e d a t a , w i t h an a p p r o p r i a t e d e s i g n s p e c t r u m , d e r i v e d f r o m t h e o b s e r v e d b e h a v i o u r o f b u i l d i n g s i n e a r t h q u a k e s , a s e t o f e q u i v a l e n t l a t e r a l f o r c e s i s g e n e r a t e d . W i t h t h i s s e t o f e q u i v a l e n t l a t e r a l f o r c e s , t h e r e m a i n i n g p r o c e d u r e i s no more t h a n an o r d i n a r y s t a t i c a n a l y s i s . T h e r e f o r e , t h i s method i s e s p e c i a l l y u s e f u l f o r t h e p u r p o s e of d o i n g p r e l i m i n a r y 4 d e s i g n . A more r i g o r o u s a n a l y t i c a l method i s 'modal a n a l y s i s ' . In c o n t r a s t t o t h e e q u i v a l e n t f o r c e method, t h e p e r i o d s and t h e mode shapes o f a t l e a s t t h e f i r s t few modes of v i b r a t i o n a r e needed. The ma g n i t u d e of t h e l a t e r a l f o r c e s f o r e a c h mode a r e t h e n d e t e r m i n e d from t h e r e s p o n s e s p e c t r u m . The r o o t - s u m - s q u a r e o f t h e r e s p o n s e f o r e a c h v i b r a t i o n mode i s t a k e n a s t h e e n v e l o p e , w h i c h r e p r e s e n t s t h e maximum r e s p o n s e t h e s t r u c t u r e w o u l d e x h i b i t d u r i n g t h e e a r t h q u a k e , i f i t were t o remain e l a s t i c . I t has been o b s e r v e d by Newmark and H a l l 1 t h a t t h e d i s p l a c e m e n t r e s p o n s e of a s i n g l e d e g r e e o f fr e e d o m s y s t e m i s e s s e n t i a l l y t h e same, whether o r not i t y i e l d s . The geometry o f t h e f o r c e - d e f o r m a t i o n d i a g r a m t h e n shows t h a t , i f t h e r e s p o n s e i s e l a s t i c - p e r f e c t l y p l a s t i c , t h e y i e l d f o r c e w i l l be t h e f o r c e o b t a i n e d from t h e e l a s t i c a n a l y s i s d i v i d e d by t h e d u c t i l i t y . ( D u c t i l i t y i s d e f i n e d a s t o t a l d i s p l a c e m e n t d i v i d e d by y i e l d d i s p l a c e m e n t . ) I t i s assumed, t h e r e f o r e , t h a t t h e modal a n a l y s i s j u s t d e s c r i b e d g i v e s a p p r o x i m a t e l y c o r r e c t d i s p l a c e m e n t s , and t h a t y i e l d -l e v e l f o r c e s f o r d e s i g n p u r p o s e s c a n be o b t a i n e d by d i v i d i n g t h o s e f o u n d i n t h e a n a l y s i s by t h e a v a i l a b l e d u c t i l i t y i n h e r e n t i n t h e s t r u c t u r a l s y s t e m . C l e a r l y , t h e s e a s s u m p t i o n a r e i n v a l i d f o r t h e m u l t i - d e g r e e - o f - f r e e d o m s y s t e m u n l e s s i t u n d e r g o e s u n i f o r m l y d i s p e r s e d y i e l d i n g , w h i c h w i l l o n l y o c c u r i f t h e d i s t r i b u t i o n s o f s t i f f n e s s , mass, and s t r e n g t h a r e a l s o u n i f o r m . The two p r e c e d i n g methods a r e s i m p l e t o a p p l y , b u t n e i t h e r i s a p p l i c a b l e i f t h e s t r u c t u r e i s not s i m p l e , 5 u n i f o r m , and ' w e l l - b e h a v e d ' . T h e r e f o r e when d e a l i n g w i t h l a r g e r o r more complex s t r u c t u r e s , t h e more s o p h i s t i c a t e d 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 i s r e q u i r e d . T i m e - s t e p a n a l y s i s i s b a s e d on a d i f f e r e n t c o n c e p t from t h e two p r e v i o u s methods. An e a r t h q u a k e r e c o r d , i n s t e a d o f a r e s p o n s e s p e c t r u m , i s u s e d t o d e t e r m i n e 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 a t any i n s t a n t d u r i n g t h e e a r t h q u a k e . The e n e r g y d i s s i p a t i n g mechanism of t h e s y s t e m and t h e n o n - l i n e a r c h a r a c t e r i s t i c s o f s t r u c t u r a l e l e m e n t s a r e more a c c u r a t e l y r e p r e s e n t e d by m o d e l l i n g t h e h y s t e r e s i s e f f e c t s . T h i s a n a l y s i s , i s , t h u s f a r , t h e most r e l i a b l e a v a i l a b l e method of dynamic a n a l y s i s . I t not o n l y y i e l d s t h e maximum r e s p o n s e o f t h e s t r u c t u r e , and t h e a p p r o x i m a t e t i m e a t w h i c h t h i s maximum o c c u r s , but a l s o makes t h e whole r e s p o n s e t i m e h i s t o r y a v a i l a b l e f o r e x a m i n a t i o n . However, t h e c o m p l e x i t y o f t i m e - s t e p a n a l y s i s makes i t i m p o s s i b l e t o use w i t h o u t t h e a i d of a l a r g e c omputer; and r u n n i n g s u c h a p r o gram i s b o t h e x p e n s i v e and t i m e - c o n s u m i n g . The r e s u l t s o b t a i n e d from s u c h an a n a l y s i s a r e o n l y v a l i d f o r one s e t of s t r u c t u r a l p r o p e r t i e s and one e a r t h q u a k e r e c o r d . Changes i n s t r u c t u r a l p r o p e r t i e s , w h i c h f r e q u e n t l y o c c u r as a d e s i g n p r o c e e d s , mean a d d i t i o n a l r u n s o f t h e p r o g r a m and a d d i t i o n a l c o s t t o t h e d e s i g n e r . Sometimes, i n o r d e r t o g a i n t h o r o u g h u n d e r s t a n d i n g of t h e b e h a v i o u r o f t h e d e s i g n e d s t r u c t u r e , more t h a n one i n p u t e a r t h q u a k e r e c o r d i s u s e d . T h e s e f a c t o r s make t h e a n a l y s i s so e x p e n s i v e t h a t i t i s g e n e r a l l y out of t h e r e a c h of a v e r a g e e n g i n e e r i n g p r a c t i c e and o n l y u s e d i n l a r g e p r o j e c t s where enough f u n d s 6 c a n be j u s t i f i e d . In v i e w o f t h e s e comments, t h e hope o f o b t a i n i n g b e t t e r and more r e l i a b l e dynamic a n a l y s i s a p p e a r s t o r e s t on s e v e r a l d i f f e r e n t f a c t o r s . One p o s s i b i l i t y of c o u r s e , i s t h a t r a p i d d e v e l o p m e n t i n t h e f i e l d of computer t e c h n o l o g y w i l l l e a d t o more p o w e r f u l c o m p u t e r s , w h i c h w i l l l o w e r t h e c o s t of e x e c u t i n g complex a n a l y s e s . In t h a t c a s e , t h e 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 a p p e a r s t o be t h e u l t i m a t e answer. However, i t w i l l s t i l l be a l o n g t i m e b e f o r e t h e c o s t of p r e p a r i n g and r u n n i n g s u c h p r o g r a m s c a n be b r o u g h t down t o a l e v e l t h a t i s a t t a i n a b l e f o r t h e a v e r a g e e n g i n e e r i n g o f f i c e f o r s m a l l t o medium b u i l d i n g s . T h e r e f o r e , i t i s d e s i r a b l e t o improve modal a n a l y s i s so t h a t s o l u t i o n s c an r e p r e s e n t more a d e q u a t e l y t h e r e s p o n s e o f s t r u c t u r e s i n t h e i n e l a s t i c r a n g e . In o r d e r t o make modal a n a l y s i s work i n t h e i n e l a s t i c r a n g e , a t t e m p t s have been made t o d e v e l o p an i n e l a s t i c r e s p o n s e s p e c t r u m from t h e e l a s t i c r e s p o n s e s p e c t r u m . In t h i s method t h e o r d i n a r y modal a n a l y s i s c a n be p e r f o r m e d i n t h e u s u a l way, but t h e i n e l a s t i c s p e c t r u m i s u s e d t o d e t e r m i n e t h e s p e c t r a l a c c e l e r a t i o n f o r e a c h mode. The i n e l a s t i c r e s p o n s e s p e c t r u m i s d e r i v e d f r o m t h e e l a s t i c s p e c t r u m w i t h an a l l o w a b l e d u c t i l i t y f a c t o r , w h i c h i s d e t e r m i n e d from t h e s t r u c t u r a l f o r m and t h e p r o p e r t i e s of t h e m a t e r i a l s u s e d . U n f o r t u n a t e l y , i t s u f f e r s f r o m t h e same s h o r t c o m i n g a s t h e e q u i v a l e n t l a t e r a l f o r c e p r o c e d u r e , and c a n o n l y be a p p l i e d t o s t r u c t u r e s where d u c t i l i t y demand i s e x p e c t e d t o be d i s t r i b u t e d u n i f o r m l y . 7 A r e l a t i v e l y s i m p l e method, a l s o b a s e d on modal a n a l y s i s , w h i c h i s c a l l e d t h e s u b s t i t u t e s t r u c t u r e method, was i n t r o d u c e d by S h i b a t a and S o z e n 2 . T h i s method p r o v i d e s a s i m p l e means o f t a k i n g i n t o a c c o u n t t h e i n e l a s t i c p r o p e r t i e s of t h e s t r u c t u r e . I t i s a d e s i g n p r o c e d u r e whereby t h e d e s i g n e r can a s s i g n 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 r e s p o n s e t o e a c h member. Then d i f f e r e n t v a l u e s o f damping and s t i f f n e s s f o r a l l t h e members a r e c a l c u l a t e d and m o d i f i e d by t h e c o r r e s p o n d i n g p r e d e f i n e d l i m i t s , i n s u c h a way as t o g i v e e q u i v a l e n t l i n e a r members w i t h v i s c o u s damping r e p r e s e n t i n g t h e h y s t e r e t i c damping of t h e r e a l n o n - l i n e a r members. One e q u a t i o n i s a l s o p r o v i d e d t o d e t e r m i n e t h e s y s t e m damping from v a l u e s o f member damping. F i n a l l y , t h e d e s i g n y i e l d f o r c e s o f t h e members a r e o b t a i n e d as t h e r e s u l t s o f an o r d i n a r y modal a n a l y s i s . The p r e s e n t method i s an a n a l y s i s b a s e d on t h e same t h e o r y b u t w i t h t h e above p r o c e d u r e s r e v e r s e d . In o t h e r words, t h e d e s i g n c a p a c i t i e s a r e p r e d e f i n e d and t h e i n e l a s t i c l i m i t s a r e t r e a t e d as unknowns. S i n c e t h e s t i f f n e s s and damping o f t h e s t r u c t u r e a r e a l s o f u n c t i o n s of t h e s e l i m i t s , an i t e r a t i v e p r o c e d u r e must be u s e d . D e t a i l s and d e v e l o p m e n t s o f t h i s method a r e d i s c u s s e d i n R e f . 3, 4 and a l s o i n t h e f o l l o w i n g c h a p t e r s . I t i s p r o p o s e d as a method o f i d e n t i f y i n g anomalous b e h a v i o u r i n s t r u c t u r e s w h i c h have been d e s i g n e d by an e q u i v a l e n t l a t e r a l f o r c e p r o c e d u r e . 8 1.2 O b j e c t i v e Of S t u d y The t h r e e methods o f s e i s m i c a n a l y s i s commonly u s e d i n p r a c t i c e were i n t r o d u c e d i n t h e l a s t s e c t i o n . I t was p o i n t e d o u t t h a t a l t h o u g h t h e e q u i v a l e n t l a t e r a l f o r c e p r o c e d u r e or modal a n a l y s i s c an p r o v i d e s i m p l e means of p e r f o r m i n g dynamic a n a l y s e s , b o t h methods a r e r e s t r i c t e d by t h e i r i n a b i l i t y t o p r e d i c t a c c u r a t e l y t h e i n e l a s t i c b e h a v i o u r of s t r u c t u r e s . 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 i s w e l l - k n o w n t o be t h e most p o w e r f u l and i n f o r m a t i v e method i n t h e a r e a of s e i s m i c a n a l y s i s . However, t h e a p p l i c a t i o n of s u c h p r o c e d u r e s i s l i m i t e d by t h e h i g h c o s t i n v o l v e d i n b o t h d a t a p r e p a r a t i o n and e x e c u t i o n . The p r e s e n t method w h i c h was d e v e l o p e d i n r e c e n t y e a r s , i s i n t e n d e d t o f i l l t h e gap between t h e r e l a t i v e l y s i m p l e e l a s t i c modal a n a l y s i s and t h e more c o m p l i c a t e d 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 . The a d v a n t a g e s o f t h e p r o p o s e d method a r e : 1. The i n e l a s t i c r e s p o n s e of a s t r u c t u r e can be d e t e r m i n e d by a l i n e a r modal a n a l y s i s u s i n g an e l a s t i c r e s p o n s e s p e c t r u m ; 2. The r e l i a b i l i t y of s o l u t i o n s o b t a i n e d i s c o m p a r a b l e t o 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 ; and 3. The e x e c u t i o n c o s t o f s u c h a p r ogram i s s u b s t a n t i a l l y l o w e r t h a n t h a t o f t h e t i m e - s t e p a n a l y s i s . T h ese a d v a n t a g e s were v e r i f i e d t h r o u g h t h i s s t u d y . However, t h e p r o p o s e d method was a l s o s u b j e c t t o a 9 number o f l i m i t a t i o n s , w h i c h o r i g i n a l l y a p p l i e d t o t h e s u b s t i t u t e s t r u c t u r e method s u g g e s t e d by S h i b a t a and S o z e n 2 . They impose t h e f o l l o w i n g r e q u i r e m e n t s on t h e s t r u c t u r e : 1. The s y s t e m can be a n a l y s e d i n one v e r t i c a l p l a n e . 2. T h e r e a r e no a b r u p t c h a n g e s i n geometry o r mass a l o n g t h e h e i g h t o f t h e s y s t e m . 3. The l i m i t s of i n e l a s t i c r e s p o n s e f o r c o l u m n s , beams and w a l l s c a n be d i f f e r e n t , b ut t h e s e l i m i t s s h o u l d be t h e same f o r a l l beams i n a g i v e n bay and a l l columns on a g i v e n v e r t i c a l a x i s . 4. The f o r c e r e s p o n s e s h o u l d d e c r e a s e a s t h e s t r u c t u r e becomes more f l e x i b l e . 5. A l l s t r u c t u r a l e l e m e n t s and j o i n t s a r e r e i n f o r c e d t o a v o i d s i g n i f i c a n t s t r e n g t h d e c a y as a r e s u l t o f r e p e a t e d r e v e r s a l s o f t h e a n t i c i p a t e d i n e l a s t i c d i s p l a c e m e n t s . 6. N o n s t r u c t u r a l components do not i n t e r f e r e w i t h s t r u c t u r a l r e s p o n s e . S i n c e t h e d i s c u s s i o n i n t h i s t h e s i s i s r e s t r i c t e d t o two d i m e n s i o n a l a n a l y s i s , t h e f i r s t l i m i t a t i o n s t i l l a p p l i e s . L i m i t a t i o n (4) can be overcame by t h e method s u g g e s t e d by S h i b a t a and S o z e n 2 ; t h a t i s , i f i n t h e range of t h e l o w e r modes o f t h e s t r u c t u r e t h e s p e c t r a l a c c e l e r a t i o n r e s p o n s e i n c r e a s e s w i t h an i n c r e a s e i n p e r i o d , a c o n s t a n t a c c e l e r a t i o n r e s p o n s e , up t o t h e p e r i o d a t w h i c h t h e r e s p o n s e s t a r t s d e c r e a s i n g , c a n be assumed. L i m i t a t i o n s (5) and (6) a p p l y t o most dynamic a n a l y s e s , even i n t i m e - s t e p methods. However, no e x p l a n a t i o n was g i v e n f o r t h e l i m i t a t i o n s s t a t e d i n ( 2 ) , and (3) even i n t h e o r i g i n a l 1 0 p a p e r of S h i b a t a and S o z e n 2 ; t h e r e f o r e , one o b j e c t i v e of t h i s s t u d y i s t o show when r e s t r i c t i o n s (2) and (3) a r e u n n e c e s s a r y , a t l e a s t i n t h e p r e s e n t method. The p r o g r a m f o r t h e p r o p o s e d method was f i r s t d e v e l o p e d by S. Y o s h i d a 3 i n 1979. At t h a t t i m e , a v a r i e t y of s m a l l f r a m e s were t e s t e d u s i n g t h e f i r s t v e r s i o n of t h e p r o g r a m and r e s u l t s were p r e s e n t e d i n R e f . 3. S u b s e q u e n t l y , t h e p r o g r a m was m o d i f i e d by A.W.F. M e t t e n " t o extend- i t s c a p a b i l i t y t o i n c l u d e t h e a n a l y s i s of s t r u c t u r a l w a l l s . A g a i n , a number of d i f f e r e n t c o u p l e d s h e a r - w a l l s t r u c t u r e s were examined and r e s u l t s were p r e s e n t e d i n R e f . 4. As an e x t e n s i o n t o t h e p r e v i o u s s t u d i e s , t h e o b j e c t i v e o f t h i s s t u d y i n c l u d e s f u r t h e r improvement t o t h e p r o g r a m . A r e a s t o be i m p r o v e d i n c l u d e : 1. s t u d y and r e s o l v e p r o b l e m s t h a t h i n d e r p r o p e r c o n v e r g e n c e o f t h e p r o g r a m d u r i n g t h e c o u r s e of t h i s s t u d y ; 2. i n v e s t i g a t e t h e a d e q u a c y o f t h e e l e m e n t damping f o r m u l a ; and 3. remove s i m p l i f i c a t i o n s t h a t were imposed on t h e p r o p o s e d method i n p r e v i o u s s t u d i e s . I t was m e n t i o n e d i n t h e l a s t s e c t i o n t h a t modal a n a l y s i s i s most s u i t a b l e f o r a n a l y s i n g medium s i z e s t r u c t u r e s ; t h e r e f o r e i t i s i m p o r t a n t t o know whether t h e p r o p o s e d method i s a d e q u a t e f o r a n a l y s i s o f s u c h s y s t e m s . A l i m i t e d number of p a r a m e t r i c s t u d i e s w i l l be c o n d u c t e d t o i n v e s t i g a t e t h e e f f e c t o f v a r y i n g c e r t a i n p a r a m e t e r s on t h e r e l i a b i l i t y o f t h e method. T h e s e r e s u l t s w i l l be compared t o 11 a s t a n d a r d 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 . 1 .3 Scope S i n c e a d e t a i l e d d e s c r i p t i o n o f t h e p r e s e n t method has been p r o v i d e d i n each o f t h e p r e v i o u s s t u d i e s ( R e f . 3 and 4 ) , o n l y a b r i e f summary o f t h e method i s p r e s e n t e d i n t h i s t h e s i s . R e a d e r s a r e a d v i s e d t o r e f e r t o one of t h e above r e f e r e n c e s when more d e t a i l s o f t h e method a r e r e q u i r e d . However, a d e e p e r l o o k i n t o t h e s t r u c t u r e o f t h e p r o g ram i s p r o v i d e d h e r e i n -the hope o f a i d i n g f u r t h e r improvement t o t h e e x i s t i n g program i n f u t u r e s t u d i e s . In t h e b e g i n n i n g o f t h i s s t u d y , a number o f s t r u c t u r e s a n a l y s e d by t h e m o d i f i e d v e r s i o n of t h e p r o g r a m r e v e a l e d some c o n v e r g e n c e p r o b l e m s . C a r e f u l e x a m i n a t i o n of t h e s e p r o b l e m s showed t h a t t h e r e a r e o t h e r l i m i t a t i o n s t o t h e method t h a t were not r e a l i z e d i n t h e p r e v i o u s s t u d i e s . T h e s e c o n v e r g e n c e p r o b l e m s and t h e l i m i t a t i o n s t h a t t h e y imposed on t h e method, as w e l l as a p r o p o s e d r o u t i n e t o e n s u r e c o n v e r g e n c e , a r e d i s c u s s e d i n c h a p t e r 3. D u r i n g t h e c o u r s e o f s t u d y i n g t h e above t o p i c , t h e a d e q u a c y o f t h e member damping f o r m u l a was a l s o i n v e s t i g a t e d . The r e s u l t s a r e p r e s e n t e d i n C h a p t e r 4. In p r e v i o u s s t u d i e s , a number o f s i m p l i f i c a t i o n s were i n c l u d e d i n t h e p r o p o s e d method. In t h i s s t u d y , t h e s e s i m p l i f i c a t i o n s were removed i n o r d e r t o i n c r e a s e t h e a p p l i c a b i l i t y and r e l i a b i l i t y of t h e method. T h e s e improvements a r e d i s c u s s e d i n d e t a i l i n C h a p t e r 5. 1 2 F i n a l l y , no s t u d y i s c o m p l e t e d w i t h o u t p e r f o r m i n g t e s t s on t h e p r o g r a m . A number of medium s i z e s t r u c t u r e s was d e s i g n e d and e a c h of them was s p e c i f i c a l l y u s e d t o t e s t c e r t a i n f u n c t i o n s of t h e p r o g r a m . The r e s u l t s o b t a i n e d a r e compared w i t h a s t a n d a r d 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 . To p r o v e t h a t improvements were o b t a i n e d t h r o u g h t h e above m e n t i o n e d v a r i a t i o n s , a c o m p a r i s o n of r e s u l t s w i t h t h e p r e v i o u s p r o g r a m was a l s o made. T h e s e r e s u l t s a r e p r e s e n t e d i n c h a p t e r 6, a l o n g w i t h a f i n a l n o t e on t h e e x e c u t i o n c o s t o f t h e p r o g r a m . 1 3 2.0 THE PRESENT METHOD 2.1 B r i e f Review Of The P r e s e n t Method The p r e s e n t method i s an i t e r a t i v e p r o c e d u r e f o r a n a l y s i n g t h e r e s p o n s e o f s t r u c t u r e s u nder e a r t h q u a k e e x c i t a t i o n . The method was d e v e l o p e d from t h e s u b s t i t u t e s t r u c t u r e method, w h i c h i s a d e s i g n p r o c e d u r e b a s e d on modal a n a l y s i s , p r o p o s e d by S h i b a t a and S o z e n . In o r d e r t o u n d e r s t a n d t h e m o d i f i e d method, i t i s e a s i e s t t o s t a r t w i t h t h e o r i g i n a l s u b s t i t u t e s t r u c t u r e method; t h e f o l l o w i n g i s an i n t r o d u c t i o n t o t h a t p r o c e d u r e . 2.1.1 The S u b s t i t u t e S t r u c t u r e Method The b a s i c i d e a s o f t h e s u b s t i t u t e s t r u c t u r e method a r e as f o l l o w s : 1. The a c c e p t a b l e l i m i t o f t h e i n e l a s t i c r e s p o n s e f o r e a c h member i s e s t a b l i s h e d a c c o r d i n g t o t h e m a t e r i a l s and t h e d e s i r e d l e v e l of d e t a i l i n g . 2. The r e d u c e d s t i f f n e s s o f an e q u i v a l e n t l i n e a r member w h i c h would r e s p o n d i n a s i m i l a r way i s c a l c u l a t e d . 3. A f i c t i t i o u s l e v e l o f v i s c o u s damping w h i c h d i s s i p a t e s an amount o f e n e r g y e q u i v a l e n t t o t h a t l o s t i n t h e h y s t e r e t i c l o o p o f t h e r e a l i n e l a s t i c member i s d e d u c e d . 1 4 4. T h i s f i c t i t i o u s damping f o r e a c h of t h e members i s combined i n t o a s t r u c t u r e damping f o r e a c h mode. 5. A modal a n a l y s i s i s made f o r t h e e q u i v a l e n t l i n e a r s t r u c t u r e w i t h t h e r e d u c e d s t i f f n e s s e s and t h e c a l c u l a t e d v i s c o u s damping s u b s t i t u t i n g f o r h y s t e r e t i c e f f e c t s i n t h e r e a l s y s t e m . In m o d i f y i n g t h e s t i f f n e s s , t h e 'damage r a t i o ' i s u s e d ; t h i s q u a n t i t y i s r a t h e r s i m i l a r t o t h e member d u c t i l i t y r a t i o . I t i s i m p o r t a n t t o d i s t i n q u i s h damage r a t i o f r o m member d u c t i l i t y r a t i o w h i c h i s more commonly u s e d i n p r a c t i c e . Member d u c t i l i t y r a t i o , w h i c h w i l l be r e f e r r e d t o as t h e d u c t i l i t y r a t i o f r o m h e r e on, i s d e f i n e d a s t h e r a t i o o f u l t i m a t e d i s p l a c e m e n t t o y i e l d d i s p l a c e m e n t . T h i s c o n c e p t ca n be i l l u s t r a t e d by t h e example shown i n F i g . 2.1. The f i g u r e shows a t y p i c a l frame member d i s p l a c e d f r o m i t s o r i g i n a l p o s i t i o n i n F i g . 2.1(a) t o t h e p o s i t i o n a t F i g . 2 . 1 ( b ) . Assuming t h a t t h e end moments a r e e q u a l and o p p o s i t e , t h e moment d i a g r a m of t h e member w i l l be a s i n F i g . 2 . 1 ( c ) . F i g . 2.1(d) i s t h e moment v s . r o t a t i o n d i a g r a m f o r t h e two ends of t h e member. In t h i s c a s e , t h e d u c t i l i t y o f t h e member i s s i m p l y . e V = (2.1) 8y where rj i s t h e d u c t i l i t y r a t i o 6 i s t h e end r o t a t i o n of t h e member d e f i n e d by t h e a n g l e shown i n F i g . 2 . 1 ( a ) . 1 5 On t h e o t h e r hand, damage r a t i o i s d e f i n e d a s t h e r a t i o of t h e i n i t i a l s t i f f n e s s of t h e el e m e n t t o t h e s e c a n t s t i f f n e s s o f t h e e l e m e n t i n i t s f i n a l c o n f i g u r a t i o n . T h i s s e c a n t s t i f f n e s s i s t h e s t i f f n e s s f o r t h e s u b s t i t u t e s t r u c t u r e . U s i n g t h e same i l l u s t r a t i o n i n F i g . 2 . 1 ( d ) , t h e s l o p e of l i n e OA i s t h e i n i t i a l s t i f f n e s s of t h e e l e m e n t , s l o p e o f l i n e OB i s t h e a p p a r e n t s t i f f n e s s of t h e e l e m e n t w h i c h i s use d i n t h e s u b s t i t u t e s t r u c t u r e method. The damage r a t i o i s s l o p e of OA u = (2.2) s l o p e of OB A l t h o u g h d u c t i l i t y r a t i o and damage r a t i o a r e d e f i n e d d i f f e r e n t l y , t h e y a r e r e l a t e d by t h e f o l l o w i n g e q u a t i o n , V u = (2.3) 1 + ( TJ " 1 ) S where s i s t h e s t r a i n h a r d e n i n g r a t i o i n p r o p o r t i o n t o t h e i n i t i a l s t i f f n e s s . E q u a t i o n ( 2 . 3 ) , shows t h a t t h e d u c t i l i t y r a t i o and damage r a t i o a r e e q u a l o n l y when s t r a i n h a r d e n i n g r a t i o i s z e r o , w h i c h i s t r u e f o r e l a s t o - p l a s t i c r e s p o n s e . In any o t h e r s i t u a t i o n , t h e damage r a t i o would be 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 . A f t e r t h e damage r a t i o s a r e a s s i g n e d t o a l l t h e e l e m e n t s o f t h e s t r u c t u r e , t h e n e x t s t e p i s t o o b t a i n t h e s t i f f n e s s o f t h e s u b s t i t u t e s t r u c t u r e : 16 EI a, E I s ; = (2.4) Mi where EIai i s the s e c t i o n a l s t i f f n e s s of the element i n the a c t u a l s t r u c t u r e E I s ; i s the s e c t i o n a l s t i f f n e s s of the element i n the s u b s t i t u t e s t r u c t u r e M; i s the damage r a t i o assigned to element i . In order to perform the modal a n a l y s i s , an estimate of the damping f a c t o r f o r each mode i s needed. An equation to c a l c u l a t e " damping f o r concrete elements was developed j u s t fo r t h i s purpose by Gulkan and Sozen 5. The s u b s t i t u t e damping i s approximated by the f o l l o w i n g equation, 1 j3-, = 0.02 + 0.2 ( 1 - ) (2.5) / M 7 where /3; i s the s u b s t i t u t e damping f a c t o r f o r element i . To o b t a i n a damping f a c t o r f o r the e n t i r e s t r u c t u r e , the i n d i v i d u a l values of 0; have to be combined by a p p l y i n g some kind of averaging technique. In the s u b s t i t u t e s t r u c t u r e method, the s t r u c t u r a l damping i s obtained from the weighted average of a l l the element damping f a c t o r s , and the weight f a c t o r used i s the s t r a i n energy generated from each element. T h i s s t r a i n energy i s c a l c u l a t e d as f o l l o w s 1 7 n ( = ( NI*, + M2Bi - MAi M B i ) (2.6) 6 (EL.. ) where II; i s t h e s t r a i n e n e r g y of e l e m e n t i E I s i i s o b t a i n e d from Eqn. (2.4) and M A i , M & i a r e end moments of e l e m e n t i ( r e f e r t o F i g . 2.1(b) ) and t h u s t h e s t r u c t u r a l damping f a c t o r i s , I ( II; 0; ) L II; (2.7) where |3S i s t h e 'smeared' damping f a c t o r f o r t h e s u b s t i t u t e s t r u c t u r e . To s i m p l i f y t h e p r o c e s s o f p i c k i n g t h e s p e c t r a l a c c e l e r a t i o n f o r a g i v e n damping f a c t o r f r o m a smooth r e s p o n s e s p e c t r u m , w h i c h w o u l d u s u a l l y show c u r v e s c o r r e s p o n d i n g t o c e r t a i n s e l e c t e d damping v a l u e s o n l y , t h e f o l l o w i n g e x p r e s s i o n i s u s e d : 8 S a B = Sa (@ 2% damping) x (2.8) ( 6 + 100 0 S ) where S a s i s t h e s p e c t r a l a c c e l e r a t i o n , d e r i v e d from t h e v a l u e a t 2% damping. To s t a r t t h e p r o c e d u r e , f i r s t , t h e s t i f f n e s s m a t r i x and t h e mass m a t r i x a r e a s s e m b l e d . The modal p e r i o d s and mode shapes a r e d e t e r m i n e d u s i n g any s t a n d a r d E i g e n - v a l u e s o l v e r . S i n c e t h e e q u a t i o n f o r c a l c u l a t i n g damping ( E q n . ( 2 . 6 ) ) i n v o l v e s t h e end moments, w h i c h a r e t h e r e q u i r e d q u a n t i t i e s , 18 an i n i t i a l e s t i m a t i o n of t h e smeared damping f a c t o r f o r e a c h mode i s r e q u i r e d i n o r d e r t o d e t e r m i n e t h e s p e c t r a l a c c e l e r a t i o n f r o m t h e r e s p o n s e s p e c t r u m . B u t , a l t e r n a t i v e l y , an a r b i t r a r y v a l u e of s p e c t r a l a c c e l e r a t i o n c an be assumed. Then t h e modal a n a l y s i s can be p e r f o r m e d as u s u a l w h i c h y i e l d s a s e t of r e s u l t i n g end moments. W i t h t h e s e r e s u l t s , i t i s p o s s i b l e t o r e f i n e t h e v a l u e s of smeared damping u s i n g Eqn. (2.6) and (2.7) f o r e a c h mode. A f t e r t h a t , a new v a l u e o f s p e c t r a l a c c e l e r a t i o n c an be d e t e r m i n e d f o r e a c h mode and by r e p e a t i n g t h e modal a n a l y s i s t h e f i n a l d e s i g n moments and d i s p l a c e m e n t s a r e o b t a i n e d . An i t e r a t i v e p r o c e d u r e i s not needed b e c a u s e t h e m a g n i t u d e s of t h e end moments us e d i n Eqn. (2.6) can be p r o - r a t e d f o r d i f f e r e n t v a l u e s of s p e c t r a l a c c e l e r a t i o n , due t o t h e f a c t t h a t t h e s t i f f n e s s e s of t h e members a r e f i x e d d u r i n g t h i s p r o c e s s . 2.1.2 P r o p o s e d M o d i f i c a t i o n s Of The 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 an a n a l y s i s , w h i c h means t h a t a l l s t r u c t u r a l p r o p e r t i e s , s u c h as i n i t i a l s t i f f n e s s and member c a p a c i t i e s , a r e known q u a n t i t i e s . In o t h e r words, m o m e n t - r o t a t i o n d i a g r a m s s u c h as t h a t i n F i g . 2 . 1 ( c ) a r e d e f i n e d f o r e a c h member. What i s not known i s t h e s t a t u s o f t h e members a t maximum r e s p o n s e o f t h e s t r u c t u r e i n a g i v e n e a r t h q u a k e m o t i o n . In t h i s method, t h e s t a t u s o f e a c h member w i l l , a g a i n , be d e f i n e d by t h e damage r a t i o a s i n t h e s u b s t i t u t e s t r u c t u r e method. However, t h e member r e s p o n s e s a r e known, and t h e damage r a t i o s a r e 19 t h e r e q u i r e d q u a n t i t i e s , whereas, i n t h e o r i g i n a l p r o p o s a l of S h i b a t a and S o z e n 2 , t h e t o l e r a b l e damage r a t i o s a r e known and t h e y i e l d v a l u e s of t h e member f o r c e s a r e t h e r e q u i r e d q u a n t i t i e s . S i n c e t h e y a r e n o t known, a f i r s t g u e s s a t t h e s e damage r a t i o s i s r e q u i r e d . W i t h t h e s e t r i a l v a l u e s , t h e p r o c e d u r e i n t h e s u b s t i t u t e s t r u c t u r e method c a n be p e r f o r m e d e x a c t l y a s d e s c r i b e d i n t h e l a s t s e c t i o n . The end r e s u l t w i l l be a s e t of end moments, a x i a l f o r c e s , s h e a r f o r c e s and d i s p l a c e m e n t s ; t h e end moments of e a c h member must c o n f o r m t o i t s own c a p a c i t y . I f t h e end moment o f a member i s d i f f e r e n t f r o m i t s c a p a c i t y , i t s i m p l y means t h a t t h e t r i a l v a l u e o f t h e damage r a t i o i s i n c o r r e c t and must be a d j u s t e d . An i m p r o v e d t r i a l v a l u e c a n be o b t a i n e d f r o m , Mn = Mn-i * 1 (2.9) where Mr, i s t h e new damage r a t i o Mn-i i s t h e damage r a t i o f r o m t h e l a s t c a l c u l a t i o n M n i s t h e l a r g e r o f t h e two end moments of a member and My i s t h e y i e l d moment o f t h e member. W i t h s t r a i n h a r d e n i n g , t h e e x p r e s s i o n would be Mn Mn-t Mn = = * 1 (2.10) My ( 1 - S ) + S Mn-«Mn where s , a g a i n , i s t h e s t r a i n h a r d e n i n g r a t i o a s d e f i n e d p r e v i o u s l y . 20 I n e v i t a b l y , t h i s i s an i t e r a t i v e p r o c e d u r e where t h e damage r a t i o s have t o be r e f i n e d t h r o u g h e a c h i t e r a t i o n u n t i l moments on e v e r y member match t h e moment c a p a c i t y . In r e f i n i n g t h e t r i a l v a l u e o f t h e damage r a t i o s , i t must be n o t e d t h a t t h e s e t r i a l v a l u e s c a n n o t be l e s s t h a n one i n b e g i n n i n g o f any i t e r a t i o n b e c a u s e t h e s t i f f n e s s o f t h e s u b s t i t u t e s t r u c t u r e i s n o t a l l o w e d t o be l a r g e r t h a n t h a t o f t h e a c t u a l s t r u c t u r e ( r e f e r t o Eqn. ( 2 . 4 ) ) i n any s i t u a t i o n . A l t h o u g h i t i s t h e o r e t i c a l l y p o s s i b l e t o o b t a i n a s e t of damage r a t i o s f o r w h i c h t h e moments match e x a c t l y w i t h t h e c a p a c i t y o f t h e members, i t i s p r a c t i c a l l y u n f e a s i b l e b e c a u s e of t h e v a s t number o f i t e r a t i o n s r e q u i r e d . The a c c u r a c y of t h e r e s u l t s i s a l s o l i m i t e d by o t h e r f a c t o r s . T h e r e f o r e , somewhat r e l a x e d c r i t e r i a a r e employed. T h e s e c r i t e r i a w i l l be' i n t r o d u c e d i n t h e f o l l o w i n g c h a p t e r . In g e n e r a l , a r e l a t i v e t o l e r a n c e between r e s u l t i n g moment and member c a p a c i t y o f l e s s t h a n f i v e p e r c e n t w i l l p r o d u c e r e a s o n a b l e r e s u l t s . F o r t h e c r i t e r i a a d o p t e d i n t h e p r o g r a m , c o n v e r g e n c e i s u s u a l l y a t t a i n e d i n fewer t h a n t e n i t e r a t i o n s , w h i c h i s s i g n i f i c a n t f o r c o m p u t a t i o n a l economy. 2.2 The Development Of The Computer Program A p r o g r a m f o r e x e c u t i n g t h e s u g g e s t e d p r o c e d u r e c a n be o b t a i n e d e a s i l y by m o d i f y i n g an e l a s t i c modal a n a l y s i s p r o g r a m . One way t o a c c o m p l i s h t h i s i s t o i n c o r p o r a t e 21 E q n s . (2.4) t h r o u g h (2.8) i n t o any e l a s t i c modal a n a l y s i s p r o g r a m so t h a t i t c a n p e r f o r m t h e s u b s t i t u t e s t r u c t u r e method. A f t e r t h a t . i s done, t o i n t r o d u c e t h e n e c e s s a r y m o d i f i c a t i o n , a l o o p must be i n s e r t e d i n t o t h e program t o make t h e p r o c e d u r e an i t e r a t i v e one. E q u a t i o n (2.9) o r (2.10) must be i n c l u d e d i n t h e p r o g ram so t h a t t h e damage r a t i o s c a n be m o d i f i e d i n e a c h i t e r a t i o n . Then the f i n a l s t e p i s t o impose a c o n v e r g e n c e scheme i n o r d e r t h a t t h e p r o g r a m c a n be s t o p p e d once t h e s o l u t i o n has r e a c h e d t h e r e q u i r e d l i m i t of a c c u r a c y . These a r e e s s e n t i a l l y a l l t h e i n g r e d i e n t s t h a t a r e c o n t a i n e d i n t h e f i r s t v e r s i o n o f t h e p r o g r a m t h a t was d e v e l o p e d by S. Y o s h i d a 3 . The s e c o n d v e r s i o n , e d i t e d by A.W.F. M e t t e n " , came as a r e f i n e m e n t t o t h e f i r s t v e r s i o n o f t h e p r o g r a m . A number of f e a t u r e s was added t o t h e program t o make i t more e f f i c i e n t . One of t h e s e f e a t u r e s i s t h e ' c o n v e r g e n c e s p e e d i n g r o u t i n e ' w h i c h s p e e d s up c o n v e r g e n c e by i m p r o v i n g t h e t r i a l v a l u e o f t h e damage r a t i o s a f t e r e a c h i t e r a t i o n b a s e d on some e x t r a p o l a t i o n t e c h n i q u e . However, t h i s i s e f f e c t i v e o n l y a f t e r t h e e i g h t h i t e r a t i o n i s p e r f o r m e d . A n o t h e r f e a t u r e i s t h e r e l a x a t i o n of t h e c o n v e r g e n c e l i m i t s t h a t e l i m i n a t e s u n n e c e s s a r y i t e r a t i o n s w h i c h do not improve t h e a c c u r a c y of t h e s o l u t i o n . The s e c o n d v e r s i o n of t h e p r o g r a m a l s o a l l o w s f o r s t r u c t u r a l members w i t h r i g i d e n d s . T h i s f e a t u r e opened t h e p o s s i b i l i t y o f m o d e l l i n g s t r u c t u r a l w a l l s . The l a t e s t v e r s i o n o f t h e p r o g ram i s , a g a i n , a r e f i n e m e n t t o t h e p r e v i o u s one. Improvements made i n t h i s v e r s i o n were a l r e a d y m e n t i o n e d b r i e f l y i n s e c t i o n 1.2 and 22 w i l l be d i s c u s s e d i n d e t a i l i n t h e f o l l o w i n g c h a p t e r s as w e l l . 2.3 I n t r o d u c t i o n To The S t r u c t u r e Of The Computer Program The i n t e n t i o n o f t h i s s e c t i o n i s t o e x p l a i n t h e b a s i c s t r u c t u r e of t h e computer program so t h a t f u r t h e r improvement t o t h e program c a n be made w i t h e a s e . The pro g r a m t o be i n t r o d u c e d i n t h e f o l l o w i n g p a r a g r a p h s w i l l have t h e s t r u c t u r e o f t h e t h i r d v e r s i o n but t h e c a p a b i l i t y of t h e s e c o n d v e r s i o n o n l y ; t h e a d d i t i o n a l f e a t u r e s f o r t h e t h i r d v e r s i o n w i l l be added o n t o t h e f l o w c h a r t a f t e r t h e y have been i n t r o d u c e d i n n e x t few c h a p t e r s . One major change i n s t r u c t u r e of t h e s e c o n d v e r s i o n i s i n t h e main s u b r o u t i n e MOD3. MOD3, whi c h i s a t e n and a h a l f pages l o n g s u b r o u t i n e i n t h e s e c o n d v e r s i o n , has been s h o r t e n e d t o h a l f of i t s o r i g i n a l l e n g t h . The f u n c t i o n s o f t h e r e m a i n i n g h a l f a r e s h a r e d by two new s u b r o u t i n e s , FORCE and DAMOD. I t i s b e l i e v e d t h i s change w i l l make t h e program much e a s i e r t o u n d e r s t a n d and a l s o e a s i e r t o m o d i f y i n f u t u r e . In F i g . 2.2, a f l o w c h a r t o f t h e main p r o g r a m i s shown. T h i s f l o w c h a r t f o l l o w s t h e same se q u e n c e i n w h i c h s u b r o u t i n e s and i n s t r u c t i o n s a r e e x e c u t e d i n t h e program. The f i r s t s u b r o u t i n e CONTRL i s use d t o r e a d i n non-s t r u c t u r a l i n f o r m a t i o n , e.g. number o f modes t o be a n a l y s e d , maximum g r o u n d a c c e l e r a t i o n , i n i t i a l damping e t c . , and s t r u c t u r a l i n f o r m a t i o n t h a t i s common t o t h e whole 23 s t r u c t u r e , e . g . t h e Young's modulus, s t r a i n h a r d e n i n g r a t i o e t c . The s e c o n d s u b r o u t i n e SETUP i s a l s o f o r i n p u t p u r p o s e s where s t r u c t u r a l i n f o r m a t o n s u c h as j o i n t c o o r d i n a t e s , member s i z e , s e c t i o n a l s t i f f n e s s e s , and moment c a p a c i t i e s a r e r e a d i n . T h e s e d a t a a r e r e o r g a n i z e d w i t h i n t h e s u b r o u t i n e and t h e h a l f b a n d w i d t h of t h e m a t r i x i s d e t e r m i n e d . The damage r a t i o s a r e a l s o i n i t i a l i z e d t o one i n t h e s u b r o u t i n e , t o p r o v i d e t h e f i r s t s e t o f t r i a l v a l u e s . D a t a i n p u t i n t o t h e program t h r o u g h t h e s e two s u b r o u t i n e s a r e a l l e c ho p r i n t e d so t h a t t h e u s e r c a n c h e c k whether i n f o r m a t o n i s b e i n g r e a d i n c o r r e c t l y . In t h e n e x t s t e p , t h e mass m a t r i x i s a s s e m b l e d i n t h e s u b r o u t i n e MASS. S i n c e t h i s m a t r i x d o e s n o t v a r y t h r o u g h o u t t h e i t e r a t i v e p r o c e d u r e , i t i s o n l y r e q u i r e d t o be c a l c u l a t e d o n c e . A f t e r a l l t h i s i n f o r m a t i o n i s r e t u r n e d t o t h e main p r o g r a m , t h e n e x t s t e p i s t o d e f i n e some c o n t r o l v a r i a b l e s s u c h as c o n v e r g e n c e l i m i t s , i t e r a t i o n number e t c . T h e s e f i n a l i z e t h e p r e p a r a t i o n s t a g e o f t h e p r o g r a m and t h e f o l l o w i n g s t e p i s t o p r o c e e d t o t h e i t e r a t i o n p r o c e s s . In t h e b e g i n n i n g of e a c h i t e r a t i o n , s u b r o u t i n e BUILD i s c a l l e d t o a s s e m b l e t h e s t i f f n e s s m a t r i x o f t h e s u b s t i t u t e s t r u c t u r e . T h i s w i l l i n v o l v e t h e use o f e q u a t i o n 2.4, i n w h i c h t h e s t i f f n e s s o f t h e s u b s t i t u t e s t r u c t u r e i s d e f i n e d as a f u n c t i o n of damage r a t i o . A f t e r t h a t , t h e program w i l l c h e c k f o r i l l - c o n d i t i o n i n g o f t h e s t i f f n e s s m a t r i x u s i n g s u b r o u t i n e SCHECK. The r a t i o of l a r g e s t t o s m a l l e s t d i a g o n a l e l e m e n t s i s c a l c u l a t e d and p r i n t e d o u t from t h e s u b r o u t i n e . However, i t i s up t o t h e u s e r t o d e t e r m i n e whether t h e r a t i o 24 i s a c c e p t a b l e o r n o t . I f t h e s t i f f n e s s m a t r i x does n o t c o n t a i n d i a g o n a l e l e m e n t s s m a l l e r t h a n z e r o , t h e p r o g ram w i l l p r o c e e d t o t h e n e x t s u b r o u t i n e EIGEN. The f u n c t i o n of s u b r o u t i n e EIGEN i s t o d e t e r m i n e t h e modal p e r i o d s and mode s h a p e s . T h i s i s done t h r o u g h t h e E i g e n - v a l u e s o l v e r SPRIT w h i c h i s c a l l e d i n s i d e EIGEN. A f t e r EIGEN comes the main s u b r o u t i n e of t h e program, M0D3, w h i c h a c t u a l l y p e r f o r m s t h e modal a n a l y s i s t o come up w i t h t h e r e s u l t i n g f o r c e s , d i s p l a c e m e n t s and damage r a t i o s . I t a l s o p r e p a r e s t h e r e s u l t s f o r t h e c o n v e r g e n c e c h e c k . When t h i s i n f o r m a t i o n i s r e t u r n e d , i n s t r u c t i o n s i n t h e main program " w i l l c h e c k whether c o n v e r g e n c e has been a c q u i r e d . I f t h e c h e c k i s p o s i t i v e , t h e p r o g r a m w i l l s t o p a f t e r p e r f o r m i n g one more i t e r a t i o n and p r i n t i n g r e s u l t s a t t h e same t i m e . O t h e r w i s e , a s e c o n d i t e r a t i o n w i l l be p e r f o r m e d a g a i n s t a r t i n g w i t h s u b r o u t i n e BUILD, where t h e s t i f f n e s s m a t r i x w i l l be m o d i f i e d by t h e new s e t of damage r a t i o s . S i n c e s u b r o u t i n e M0D3 i s t h e h e a r t o f t h e computer program, i t d e s e r v e s f u r t h e r e x p l a n a t i o n . As a r e m i n d e r , M0D3 i s e x e c u t e d a f t e r t h e modal p e r i o d s and t h e mode sh a p e s have been f o u n d . F i g . 2.3 i s a f l o w c h a r t of t h e s u b r o u t i n e M0D3. The s u b r o u t i n e s t a r t s o u t w i t h t h e d e t e r m i n a t i o n o f t h e modal 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. Then t h e p r o c e d u r e , w h i c h i s e n c l o s e d by d o t t e d l i n e s i n F i g . 2.3, i s r e p e a t e d t w i c e f o r a l l modes w i t h i n t h e s u b r o u t i n e . The f u n c t i o n o f t h i s p r o c e d u r e i s t o compute t h e r e s u l t i n g f o r c e s and d i s p l a c e m e n t s . In t h e f i r s t t i m e t h r o u g h t h e p r o c e d u r e , t h e 25 end moments o b t a i n e d w i l l be u s e d t o c a l c u l a t e t h e smeared damping f a c t o r s u s i n g E q n s . ( 2 . 5 ) , (2.6) and ( 2 . 7 ) . In t h e s e c o n d t i m e , t h e a c t u a l r e s p o n s e of t h e s t r u c t u r e i s d e t e r m i n e d base on t h e s e damping f a c t o r s . T h i s i s done e x a c t l y a s d e s c r i b e d i n s e c t i o n 2.1.1. F o l l o w i n g t h e p r o c e d u r e i n F i g . 2.3, f i r s t , t h e s p e c t r a l a c c e l e r a t i o n i s d e t e r m i n e d i n s u b r o u t i n e SPECTR. E q u a t i o n (2.8) i s a p p l i e d t o s c a l e t h e s p e c t r u m a c c o r d i n g t o th e damping f a c t o r s t h a t a r e o b t a i n e d f r o m t h e l a s t i t e r a t i o n . W i t h t h e s p e c t r a l a c c e l e r a t i o n known, modal f o r c e s a p p l i e d t o t h e s t r u c t u r e c a n be c a l c u l a t e d . Next, t h e m a t r i x s u b r o u t i n e s SDFBAN and DSBAND a r e u s e d t o compute t h e modal d i s p l a c e m e n t s from t h e a l r e a d y a s s e m b l e d s t i f f n e s s m a t r i x and modal f o r c e m a t r i x . The d i s p l a c e m e n t s o f e a c h mode a r e t h e n s q u a r e d and summed up. A f t e r t h a t , a s u b r o u t i n e FORCE i s c a l l e d t o d e t e r m i n e t h e r e s u l t i n g f o r c e s base on t h o s e modal d i s p l a c e m e n t s . I f t h i s i s t h e f i r s t t i m e t h r o u g h t h e l o o p , i . e . when KK i s s e t e q u a l t o one, t h e smeared damping f a c t o r f o r t h a t mode w i l l a l s o be c a l c u l a t e d . O t h e r w i s e , t h e p r o g r a m w i l l move on t o c a l c u l a t e t h e r o o t - s u m - s q u a r e d i s p l a c e m e n t s , and t h e n go d i r e c t l y t o t h e n e x t s u b r o u t i n e DAMOD. In DAMOD t h e r e s u l t i n g end moments a r e use d t o m o d i f y t h e damage r a t i o s a c c o r d i n g t o Eqn. (2.9) o r ( 2 . 1 0 ) . I t w i l l a l s o c a l c u l a t e t h e r e l a t i v e d i f f e r e n c e i n damage r a t i o s between c u r r e n t and l a s t i t e r a t i o n i n p r e p a r i n g f o r t h e c o n v e r g e n c e c h e c k i n t h e main p r o g r a m . F u r t h e r m o r e , t h e c o n v e r g e n c e s p e e d i n g r o u t i n e t h a t was m e n t i o n e d i n t h e l a s t s e c t i o n i s a l s o c o n t a i n e d i n t h i s 26 s u b r o u t i n e . D e t a i l s o f t h e c o n v e r g e n c e s p e e d i n g r o u t i n e a r e d i s c u s s e d e l s e w h e r e , i n R e f . 4. F i n a l l y , t h e i n f o r m a t i o n from MOD3 i s r e t u r n e d t o t h e main program, and t h e c o n v e r g e n c e c h e c k w i l l be c a r r i e d out i m m e d i a t e l y t h e r e a f t e r . 27 3.0 CONVERGENCE 3.1 C o n v e r g e n c e Schemes S i n c e t h e p r o p o s e d method i s an i t e r a t i v e p r o c e d u r e , some c o n v e r g e n c e c r i t e r i a must be imposed t o h a l t t h e p r o c e s s when an a c c e p t a b l e l e v e l of a c c u r a c y i s a c q u i r e d . B a s i c a l l y , t h i s has done by p r e v i o u s w o r k e r s t h r o u g h t h e f o l l o w i n g c r i t e r i a ( R e f . 3 and 4) M n - M cap M cap < 0.05 f o r M n > 1 (3.1 ) and Mn-1 < 0. 1 f o r 1 < Mn < 5 (3.2a) or Mn - Mn-1 Mr, < 0.01 f o r Mn * 5 (3.2b) R e g a r d i n g t h e s e c r i t e r i a , t h e r e a r e a few p o i n t s w o r t h n o t i n g . F i r s t of a l l , Eqn (3.1). i s u s e d t o e n s u r e t h a t t h e moment o f e a c h y i e l d e d member would c o n f o r m t o i t s c a p a c i t y . The number of members t h a t have end moments d e v i a t i n g from t h e i r c a p a c i t y by o v e r 5 p e r c e n t w i l l be c o u n t e d and t h e n p r i n t e d o ut under t h e h e a d i n g 'no. above c a p a c i t y ' a f t e r 28 e a c h i t e r a t i o n . T h i s i s sometimes q u i t e m i s l e a d i n g , b e c a u s e t h e use of t h e a b s o l u t e s i g n t e n d s t o o v e r - c o u n t t h e number of members by i n c l u d i n g t h o s e w h i c h a r e t e m p o r a r i l y below c a p a c i t y b u t w i t h damage r a t i o l a r g e r t h a n one. T h i s s i t u a t i o n f r e q u e n t l y o c c u r s when t h e s p e c t r a l a c c e l e r a t i o n d r o p s i n t h e s e c o n d i t e r a t i o n and when a c o n v e r g e n c e p r o b l e m i s e n c o u n t e r e d . B e f o r e making any a d j u s t m e n t , i t may be w o r t h w h i l e f i r s t t o u n d e r s t a n d some i m p l i c a t i o n s o f t h e s e c r i t e r i a . S i n c e Eqn. (2.9) r e l a t e s moments t o damage r a t i o s , t h e r e f o r e from M n Mn = Mn-1 (2.9) M y Eqn. (3.1) c a n be r e w i t t e n a s Mn - Mcap M n ~ M n-1 = (3.3) I f Mn-i i s e q u a l t o two, t h e n o b v i o u s l y c r i t e r i o n (3.1) i s t h e same as c r i t e r i o n ( 3 . 2 a ) . In a n o t h e r word, i t means t h a t c r i t e r i o n ( 3 . 1 ) , i n f a c t , w i l l g o v e r n o n l y when Mn-i i s l e s s t h a n two. W i t h t h e s e i n mind, t h e a f o r e m e n t i o n e d c r i t e r i a c a n be r e d e f i n e d a s f o l l o w s : When damage r a t i o i s i n c r e a s i n g , M n - Mca$> Mn ~ Mn-i = < o . 0 5 f o r M n > 1 and un-\ * 2 (3.4) Mcap Mn-1 29 Mn-1 < 0. 1 f o r 1 < Mn < 5 and Mnn ^  2 (3.5a) Mn - Mn-1 Mn < 0.01 f o r Mn * 5 (3.5b) When damage r a t i o i s d e c r e a s i n g , M n _ Mn-i < 0. 1 f o r 1< M n < 5 (3.5a) Mn _ Mn-1 < 0.01 f o r M n ^ 5 (3.5b) From t h e c o n d i t i o n s o f t h e above c r i t e r i a , i t i s shown t h a t c r i t e r i a (3.5a and b) c o v e r e d t h e l a r g e s t range of damage r a t i o s . S i n c e t e s t s c o n d u c t e d t h r o u g h o u t t h i s s t u d y i n v o l v e d l a r g e damage r a t i o s , t h e r e s u l t s a l s o i n d i c a t e d t h a t t h e s e two c r i t e r i a g o v e r n e d c o n v e r g e n c e o f t h e program a l l t h e t i m e . T h e r e f o r e , c r i t e r i o n (3.4) would o n l y c o n t r o l t h e p r o c e s s i f most of t h e damage r a t i o s a r e below two. A l s o b e c a u s e c r i t e r i o n (3.4) i s much t o u g h e r t h a n ( 3 . 5 a ) , d r o p p i n g t h e a b s o l u t e s i g n would t e n d t o g i v e more c o n s e r v a t i v e r e s u l t s under c e r t a i n s i t u a t i o n s , w h i c h i n f a c t i s d e s i r a b l e when damage r a t i o s a r e c l o s e t o one. So, i n t h i s v e r s i o n of t h e pro g r a m , t h e a b s o l u t e s i g n i s d r o p p e d . I t c a n be shown l a t e r on, t h a t t h e s e c r i t e r i a c a n be u s e d as i n d i c a t o r s o f p r o g r e s s t o w a r d s c o n v e r g e n c e as w e l l . A n o t h e r p o i n t c o n c e r n i n g t h e s e c r i t e r i a i s , as A.W.F. M e t t e n had p o i n t e d o u t i n h i s t h e s i s ( R e f . 4 ) , t h e 30 main r e a s o n f o r a d o p t i n g t h o s e l i m i t s was t h a t t h e damage r a t i o s c a l c u l a t e d by u s i n g t h i s method i s o n l y s i g n i f i c a n t t o t h e f i r s t d e c i m a l d i g i t . T e s t r e s u l t s i n t h i s s t u d y have f u r t h e r p r o v e n t h a t t h i s argument i s i n f a c t c o r r e c t . A l t h o u g h i t i s b e l i e v e d t h a t even g r e a t e r r e l a x a t i o n t o t h e above l i m i t s i s s t i l l p o s s i b l e , t h e amount of s a v i n g r e s u l t i n g from d o i n g so i s i n s i g n i f i c a n t . As a l a s t comment t o t h i s s e c t i o n , t h e above c r i t e r i a p r o v i d e o n l y one o f many ways t o d e t e r m i n e when t h e p r o g ram s h o u l d s t o p . O t h e r a l t e r n a t i v e s a r e p o s s i b l e ; one example i s t o use c r i t e r i o n (3.2b) a l o n e and s e t up a c o u n t e r , t h a t c o u n t s t h e number o f members w h i c h d o e s n o t s a t i s f y t h i s c r i t e r o n , t o i n d i c a t e t h e p r o g r e s s of c o n v e r g e n c e . However, i t i s i m p o r t a n t t h a t t h e c o n v e r g e n c e l i m i t t o any c r i t e r i o n must be a p p r o p r i a t e so t h a t r e l i a b l e r e s u l t s c a n be p r o d u c e d t h r o u g h t h e i t e r a t i v e method. 3.2 E f f e c t Of I n c l u d i n g S t r a i n H a r d e n i n g As a s t r a i n h a r d e n i n g r a t i o i s i n c l u d e d i n t h e program, c r i t e r i o n (3.4) must be a l t e r e d t o i n c o r p o r a t e t h e i n c r e a s e i n m a t e r i a l s t r e n g t h due t o t h i s f a c t o r . T h i s i s done by r a i s i n g t h e moment c a p a c i t y u s i n g t h e f o l l o w i n g e q u a t i o n My ( 1 - s ) M c a p = (3.7) Y ( 1 - S M ) where M y i s t h e i n p u t moment c a p a c i t y 31 s i s t h e s t r a i n h a r d e n i n g r a t i o i n p r o p o r t i o n t o t h e i n i t i a l s t i f f n e s s and v i s t h e damage r a t i o . T h i s e q u a t i o n was d e r i v e d f r o m t h e same m o m e n t - r o t a t i o n d i a g r a m u s e d i n d e r i v i n g t h e damage r a t i o m o d i f i c a t i o n e q u a t i o n ( E q n . ( 2 . 1 0 ) ) , t h e r e f o r e t h e r e s u l t s s h o u l d be c o n s i s t e n t . 3.3 F a c t o r s T h a t H i n d e r C o n v e r g e n c e A l m o s t any a n a l y t i c a l method has i t s weaknesses and l i m i t a t i o n s , and t h e p r e s e n t method i s no e x c e p t i o n . In t h e e a r l y s t a g e o f t h i s s t u d y , numerous c o n v e r g e n c e p r o b l e m s were e n c o u n t e r e d . In o r d e r t o f i n d o ut t h e o r i g i n o f t h e s e c o n v e r g e n c e p r o b l e m s , t h e p r o g r a m was a l t e r e d so t h a t v a r i a b l e s s u c h as t h e p e r i o d , s p e c t r a l a c c e l e r a t i o n , smeared damping, number o f members above c a p a c i t y ( a c c o r d i n g t o c r i t e r i o n ( 3 . 4 ) ) and maximum v a r i a t i o n i n damage r a t i o ( a c c o r d i n g t o c r i t e r i o n (3.5a and b) f o r e a c h i t e r a t i o n would be p r i n t e d o u t t o t h e t e r m i n a l w h i l e t h e program was r u n n i n g . By d o i n g s o , t h e whole p r o c e s s , was p u t under t h e s u p e r v i s i o n o f t h e u s e r . The u s e r was a b l e t o d e t e r m i n e whether p r o p e r c o n v e r g e n c e was a c q u i r e d d u r i n g t h e p r o c e s s . In c a s e o f any c o n v e r g e n c e p r o b l e m s , t h e p r o g r a m c o u l d be s t o p p e d i m m e d i a t e l y from t h e t e r m i n a l t o p r e v e n t f u r t h e r l o s s i n computer money. 32 W i t h t h i s a d d i t i o n a l a i d , i t was f o u n d t h a t t h e r e were g e n e r a l l y two c a s e s t h a t would c a u s e t r o u b l e i n c o n v e r g e n c e . T h e s e c a s e s and t h e method o f d e t e c t i n g them a r e d i s c u s s e d i n t h e f o l l o w i n g p a r a g r a p h s . In t h e b e g i n n i n g of t h i s s t u d y , i t was i n t e n d e d t o t e s t s t r u c t u r e s t h a t were s u p p o s e d t o be p o o r l y d e s i g n e d . So an 8-bay, 7 - s t o r e y frame was d e s i g n e d u s i n g o r d i n a r y e l a s t i c modal a n a l y s i s . The moment c a p a c i t y f o r a l l members was c a l c u l a t e d b a s e d on t h e r e s u l t s of t h e a n a l y s i s d i v i d e d by a d u c t i l i t y o f f o u r . O b v i o u s l y , a d u c t i l i t y e q u a l t o f o u r f o r e v e r y member i s a v i o l a t i o n of p r e s e n t d e s i g n p h i l o s o p h y . In p r a c t i c e , t h e d u c t i l i t y g i v e n t o beams i s u s u a l l y much g r e a t e r t h a n t o c o l u m n s , so t h a t y i e l d i n g w i l l o c c u r i n t h e beams r a t h e r t h a n i n t h e c o l u m n s , t h u s p r o v i d i n g t h e e s s e n t i a l s t a b i l i t y t h a t i s r e q u i r e d t o s u r v i v e an e a r t h q u a k e . However, t h i s s t r u c t u r e might r e p r e s e n t some of t h e o l d e r b u i l d i n g s t h a t were c o n s t r u c t e d under t h e o l d d e s i g n p h i l o s o p h y . So i t was d e s i r e d t o see how t h e method would p r e d i c t t h e r e s p o n s e of s u c h a s t r u c t u r e . When t h e p r o g r a m f o r t h i s s t r u c t u r e was r u n n i n g , t h e number of members above c a p a c i t y s t a y e d c o n s t a n t , and t h e maximum change i n damage r a t i o a l s o s t a y e d above t h e r e q u i r e d l i m i t s f o r a l a r g e number of i t e r a t i o n s . However, t h e p r o g r a m d i d manage t o c o n v e r g e a f t e r more t h a n 20 i t e r a t i o n s . The r e s u l t s i n d i c a t e d t h a t most o f t h e members were no t damaged, e x c e p t t h a t t h e columns o f one f l o o r had damage r a t i o s a l l above t e n . To v e r i f y t h e r e s u l t s , a r e p u t a b l e t i m e - s t e p a n a l y s i s was used t o a n a l y s e t h e same 33 s t r u c t u r e . U n f o r t u n a t e l y , t h e r e s u l t s o b t a i n e d f r o m t h i s a n a l y s i s p r o v e d t h a t t h e p r e v i o u s r e s u l t s were i n c o r r e c t . S i n c e t h e n , two more s t r u c t u r e s , i n c l u d i n g one s h e a r -w a l l frame and one o t h e r frame s i m i l a r t o t h e f i r s t one, were d e s i g n e d u s i n g t h e same i d e a and t e s t e d u s i n g t h e two p r o g r a m s . The s h e a r - w a l l frame c o n v e r g e d w i t h i n t e n i t e r a t i o n s and showed good c o m p a r i s o n w i t h t h e t i m e - s t e p a n a l y s i s , e x c e p t f o r a few members, but t h e same p r o b l e m was e n c o u n t e r e d f o r t h e s e c o n d f r a m e . So, i t was s u s p e c t e d t h a t t h e p r o p o s e d method would have a t e n d e n c y t o i n d i c a t e a s o f t s t o r e y f o r s t r u c t u r e s t h a t c o n s i s t m a i n l y o f s t r o n g beams and weak c o l u m n s . The' s h e a r - w a l l frame i s an e x c e p t i o n , p r o b a b l y b e c a u s e t h e h i g h moment c a p a c i t y of t h e s h e a r - w a l l p r e v e n t e d t h e s o f t s t o r e y f r o m o c c u r r i n g . T h i s k i n d of drawback, would c e r t a i n l y a f f e c t t h e u s e f u l n e s s o f t h e method f o r ' r e t r o f i t ' p u r p o s e s . But t h e method i s s t i l l a p o w e r f u l a n a l y t i c a l t o o l t o a n a l y s e s t r u c t u r e s t h a t a r e d e s i g n e d a c c o r d i n g t o t h e p r e s e n t c o d e . B e s i d e s , i t i s a l s o a r g u a b l e whether t h e p e r f o r m a n c e o f s t r u c t u r e s , w hich c a u s e t h i s k i n d of c o n v e r g e n c e p r o b l e m , would be s a t i s f a c t o r y under e a r t h q u a k e m o t i o n . T h e r e f o r e , f u r t h e r work on t h i s p r o b l e m i s r e q u i r e d . A n o t h e r f a c t o r t h a t would c a u s e t r o u b l e i n c o n v e r g e n c e i s r e l a t e d t o . t h e i n p u t r e s p o n s e s p e c t r u m . A l m o s t a l l smoothed e a r t h q u a k e r e s p o n s e s p e c t r u m s have a s i m i l a r shape ( s e e F i g . 3 . 1 ) ; t h e y s t a r t w i t h an a s c e n d i n g b r a n c h a t low p e r i o d ( s p e c t r a l a c c e l e r a t i o n i n c r e a s e s w i t h p e r i o d ) ; a c o n s t a n t b r a n c h f o l l o w s , where t h e r e i s no change i n 34 s p e c t r a l a c c e l e r a t i o n o v e r a s h o r t range of p e r i o d ; t h e n t h e r e i s a d e s c e n d i n g b r a n c h , where s p e c t r a l a c c e l e r a t i o n d e c r e a s e s w i t h i n c r e a s i n g p e r i o d . These b r a n c h e s r e f l e c t t h e f o r c e s i n d u c e d i n a s t r u c t u r e o f a g i v e n v i b r a t i o n p e r i o d . The i n e l a s t i c r e s p o n s e p e r i o d of s t r u c t u r e s v a r i e s w i d e l y d e p e n d i n g on t h e c o n s t r u c t i o n m a t e r i a l s u s e d and t h e h e i g h t o f t h e s t r u c t u r e , a s w e l l a s t h e t y p e of s t r u c t u r a l s y s t e m c o n s i d e r e d . T h i s r e s p o n s e p e r i o d a l s o v a r i e s i n t h e c o u r s e o f an e a r t h q u a k e . However, i t i s b e l i e v e d t h a t t h e r e e x s i s t s one u n i q u e v i b r a t i o n p e r i o d , i n a g i v e n e a r t h q u a k e e x c i t a t i o n , t h a t would y i e l d t h e maximum r e s p o n s e t h a t c a u s e s t h e l a r g e s t damage. I n t h e s u b s t i t u t e s t r u c t u r e method, t h e r e s p o n s e p e r i o d f o r e a c h mode i s a v a r i a b l e t h a t c h a n g e s f r o m i t e r a t i o n t o i t e r a t i o n . In n o r m a l c o n v e r g e n c e , t h i s p e r i o d w i l l i n c r e a s e , o r d e c r e a s e , a t a d i m i n i s h i n g r a t e , a s r e s u l t s a p p r o a c h t h e f i n a l s o l u t i o n a f t e r e a c h i t e r a t i o n . T h i s k i n d o f c o n v e r g e n c e i s u s u a l l y a t t a i n a b l e p r o v i d e d t h a t t h e s p e c t r a l a c c e l e r a t o n d o e s not v a r y s i g n i f i c a n t l y i n m a g n i t u d e d u r i n g s u c c e s s i v e i t e r a t i o n s . U n f o r t u n a t e l y , t h i s v a r i a t i o n i s u n a v o i d a b l e once t h e i t e r a t i o n p e r i o d f a l l s i n t o t h e r a p i d l y d e s c e n d i n g b r a n c h o f th e r e s p o n s e s p e c t r u m , w h i c h u s u a l l y o c c u r s a t a r o u n d t h e p e r i o d o f 0.5 s e c . In an a t t e m p t t o e x p l a i n t h e p r o b l e m , s u p p o s e t h e r e i s an a b r u p t r e d u c t i o n i n s p e c t r a l a c c e l e r a t i o n due t o a s l i g h t i n c r e a s e i n p e r i o d d u r i n g a s u b s e q u e n t i t e r a t i o n . The immediate e f f e c t of t h i s w i l l be a l a r g e d e c r e a s e i n damage r a t i o s t h a t l e a d s t o l o w e r damping and v i b r a t i o n p e r i o d . 35 T h i s i s r e f l e c t e d from t h e t e r m i n a l o u t p u t as a sudden d e c r e a s e i n t h e number of members above c a p a c i t y and a n e g a t i v e d i f f e r e n c e i n damage r a t i o s . B e c a u s e o f t h e s h o r t e n e d v i b r a t i o n p e r i o d , t h e s p e c t r a l a c c e l e r a t i o n i n t h e f o l l o w i n g i t e r a t i o n w i l l r e t u r n t o a h i g h e r l e v e l . Such a v a r i a t i o n w i l l r e v e r s e t h e p r o c e s s of t h e l a s t i t e r a t i o n , and r e t u r n t o h i g h e r damping and p e r i o d a g a i n . From t h e t e r m i n a l o u t p u t , i t w i l l show an i n c r e a s e i n t h e number of members above c a p a c i t y and a p o s i t i v e d i f f e r e n c e i n damage r a t i o . T h i s mechanism w i l l go on f o r a l a r g e number of i t e r a t i o n s b e f o r e t h e p r o g r a m f i n a l l y c o n v e r g e s . F o r t u n a t e l y , t h i s p r o b l e m r a r e l y o c c u r s . When i t does o c c u r , i t i s l i k e l y t h a t e i t h e r t h e s t r u c t u r e i s s m a l l , o r t h e s t r u c t u r e i s r a t h e r s t i f f so t h a t i t s i n e l a s t i c p e r i o d f a l l s i n t o t h e v i c i n i t y o f t h e r a p i d l y d e s c e n d i n g b r a n c h . F i g . 3.2 i s one example o f t h e l a t t e r c a s e . In t h i s c a s e t h e p r o g r a m o n l y went t h r o u g h f i f t e e n i t e r a t i o n s t o c o n v e r g e . F i g . 3.3 i s t h e same example p l o t t e d w i t h t h e r e s p o n s e s p e c t r u m . The r e a s o n t h a t t h e r e s u l t s seem t o be c o n v e r g i n g t o a p o i n t below t h e s p e c t r u m i s t h a t t h e damping has i n c r e a s e d above 2 p e r c e n t , and t h e s p e c t r u m has been m o d i f i e d t h e r e b y t o a l o w e r c u r v e , n o t shown. One i n t e r e s t i n g p o i n t t o n o t e h e r e i s t h e e f f e c t of t h e c o n v e r g e n c e s p e e d i n g r o u t i n e on t h e p r o b l e m o f c o n v e r g e n c e . The f i g u r e shows t h a t a f t e r t h e c o n v e r g e n c e s p e e d i n g r o u t i n e became e f f e c t i v e i n t h e n i n t h i t e r a t i o n , t h e o s c i l l a t i o n p a t t e r n was b r o k e n , and t h e r e s u l t s c o n v e r g e d . However, by o b s e r v i n g t h e r e s u l t s o f t h e example, i t was f o u n d t h a t r e a l 36 c o n v e r g e n c e i s a c t u a l l y n o t a c q u i r e d , but r a t h e r , t h e r e s u l t s have s a t i s f i e d t h e c o n v e r g e n c e c r i t e r i a t e m p o r a r i l y ; i n t h e l a s t i t e r a t o n , t h e e r r o r b e g i n s t o i n c r e a s e a g a i n . I t was a l s o f o u n d t h a t t h e c o n v e r g e n c e s p e e d i n g r o u t i n e does not n e c e s s a r i l y b r i n g a b o u t r a p i d c o n v e r g e n c e ; i n some c a s e s i t i n c r e a s e d t h e m a g n i t u d e of f l u c t u a t i o n , w h i c h means t h a t i t m i g h t c r e a t e more p r o b l e m s t h a n i t s o l v e s . T h i s k i n d of f a l s e c o n v e r g e n c e i s l e s s l i k e l y t o be o b t a i n e d w i t h o u t t h e c o n v e r g e n c e s p e e d i n g r o u t i n e , b u t a v a s t number of i t e r a t i o n s w i l l , of c o u r s e , be r e q u i r e d i n s u c h a c a s e . A d i f f e r e n t and b e t t e r p r o c e d u r e i s d e s c r i b e d i n t h e r e m a i n d e r of t h i s c h a p t e r . 3.4 The S o l u t i o n S e a r c h i n g R o u t i n e From p a s t e x p e r i e n c e , i t was o b s e r v e d t h a t even f o r c a s e s where smooth c o n v e r g e n c e was a c q u i r e d , t h e r e a r e g e n e r a l l y some f l u c t u a t i o n s i n r e s p o n s e p e r i o d i n t h e f i r s t f o u r t o f i v e i t e r a t i o n s , a f t e r w h i c h t h e p e r i o d s t a r t s t o a p p r o a c h s m o o t h l y t o t h e f i n a l r e s p o n s e p e r i o d . But i n t h e c a s e s w i t h c o n v e r g e n c e p r o b l e m s , t h e f l u c t u a t i o n w i l l c a r r y on a f t e r t h e f i f t h i t e r a t i o n . T h i s c h a r a c t e r i s t i c p r o v i d e s an i m p o r t a n t c l u e f o r d e t e c t i o n o f c o n v e r g e n c e p r o b l e m s and i n d i c a t e s when t h e s e a r c h r o u t i n e s h o u l d p u t i n t o e f f e c t . A c o n c e p t t h a t has h e l p e d t h e d e v e l o p m e n t o f t h e s e a r c h r o u t i n e came from an a n a l y t i c a l p r o c e d u r e p r o p o s e d by S.A. F r e e m a n 6 . A l t h o u g h Freeman's method i s a g r a p h i c a l one, 37 th e b a s i c t h e o r y b e h i n d i t i s a c t u a l l y t h e same as t h e m o d i f i e d s u b s t i t u t e s t r u c t u r e method. W i t h F i g . 3.4, i t i s p o s s i b l e t o e x p l a i n some f u n d a m e n t a l i d e a s b e h i n d h i s method. F i g . 3.4 i s a p l o t of s p e c t r a l a c c e l e r a t i o n v e r s u s r e s p o n s e p e r i o d . L i n e U K r e p r e s e n t s t h e c a p a c i t y of t h e s t r u c t u r e , c a l l e d t h e ' c a p a c i t y ' c u r v e . F o r s p e c t r a l a c c e l e r a t i o n up t o j o i n t J , t h e s t r u c t u r e r e m a i n s e l a s t i c , t h e r e f o r e t h e p e r i o d i s j u s t e q u a l t o t h e i n i t i a l e l a s t i c p e r i o d . W i t h s p e c t r a l a c c e l e r a t i o n i n c r e a s e d a f t e r p o i n t J , th e s t r u c t u r e b e g i n s t o y i e l d and t h e p e r i o d becomes l o n g e r . T h i s i s r e f l e c t e d by t h e l i n e JK. S i n c e t h e s t r u c t u r e has e n t e r e d i n t o i t s i n e l a s t i c r a n g e , damping of t h e s t r u c t u r e s t a r t s t o i n c r e a s e a s w e l l . As a r e s u l t , a t r a n s i t i o n o c c u r s between t h e r e s p o n s e s p e c t r u m c o r r e s p o n d e d i n g t o t h e i n i t i a l damping o f t h e s t r u c t u r e and s p e c t r u m c o r r e s p o n d i n g t o h i g h e r damping v a l u e s . The shape o f t h i s t r a n s i t i o n c u r v e , c a l l e d t h e 'demand' c u r v e , w i l l depend on t h e change i n damping o f t h e s t r u c t u r e a s s o c i a t e d w i t h t h e change i n r e s p o n s e p e r i o d and a l s o on t h e shape of t h e r e s p o n s e s p e c t r u m i t s e l f . A c c o r d i n g t o Freeman's method, t h e i n t e r s e c t i o n between t h e c a p a c i t y c u r v e and t h e demand c u r v e shows t h e maximum r e s p o n s e of t h e s t r u c t u r e f o r a p a r t i c u l a r e a r t h q u a k e . T h i s s u p p o r t s t h e p r e v i o u s c o n t e n t i o n t h a t a u n i q u e s o l u t i o n e x i s t s . What t h e s e a r c h r o u t i n e s h o u l d do i s s e a r c h f o r t h e i n t e r s e c t i o n p o i n t a l o n g t h e c a p a c i t y l i n e once t h e d i r e c t i t e r a t i v e p r o c e d u r e has f a i l e d . (Note t h a t Freeman's method d o e s n o t i n d i c a t e t h e d i s t r i b u t i o n o f damage 38 t h r o u g h o u t t h e s t r u c t u r e ; t h e p r e s e n t method i s i n t e n d e d t o do t h i s . ) One a d v a n t a g e of t h e e x i s t i n g p r o g r am i s t h a t i t i s e c o n o m i c a l f o r use i n s p e c i a l s t u d i e s . P r i o r t o t h e w r i t i n g o f t h e s e a r c h r o u t i n e , a s p e c i a l s t u d y on Freeman's method was c o n d u c t e d . I t was done t h r o u g h t h e same example t h a t was u s e d i n F i g . 3.3 and 3.4. In F i g . 3.5, r e s u l t s of t h e f i r s t mode from i t e r a t i o n 4 and 5 a r e p l o t t e d i n t e r m s of s p e c t r a l a c c e l e r a t i o n and v i b r a t i o n p e r i o d . The s p e c t r u m c o r r e s p o n d i n g t o t h e damping o f e a c h i t e r a t o n i s a l s o p l o t t e d . Then t h e pr o g r a m was a l t e r e d so t h a t a . c o n s t a n t f i r s t mode s p e c t r a l a c c e l e r a t i o n c a n be f e d i n t o be use d i n s t e a d of t h a t r e a d from t h e s p e c t r u m . So t h e s p e c t r a l a c c e l e r a t i o n a t 2 p e r c e n t damping f o r t h e f o u r t h and f i f t h i t e r a t i o n was c a l c u l a t e d and t h e n f e d i n t o t h e m o d i f i e d p r o g r a m . In t h e f o r m e r c a s e , t h e p r o g r a m c o n v e r g e d t o a p o i n t a t A and i n t h e l a t t e r c a s e , t h e p r o g r a m c o n v e r g e d t o a n o t h e r p o i n t a t B. D u r i n g c o n v e r g e n c e , t h e damping changed t o 5.2 p e r c e n t a t A and 6.6 p e r c e n t a t B. (Note t h a t i n i t e r a t i o n s 4 and 5 above t h e damping was 5.3 p e r c e n t and 6.4 p e r c e n t r e s p e c t i v e l y . ) I t i s i n t e r e s t i n g t o o b s e r v e t h e r e s u l t s c o r r e s p o n d i n g from o t h e r v a l u e s o f c o n s t a n t s p e c t r a l a c c e l e r a t i o n . T h e s e r e s u l t s a r e p l o t t e d i n F i g . 3.6 a l o n g w i t h t h e s p e c t r u m c o r r e s p o n d i n g t o t h e f i n a l smeared damping f a c t o r f o r e a c h c a s e . P o i n t s 2 and 4 a r e t h e same as A and B, r e s p e c t i v e l y , i n F i g . 3.5. A c c o r d i n g t o Freeman's method, t h e c a p a c i t y c u r v e i s s i m p l y a l i n e w h i c h goes t h r o u g h t h e s e p o i n t s . The demand c u r v e c an a l s o be o b t a i n e d by j o i n i n g 39 p o i n t s on t h e s p e c t r u m l i n e s t h a t c o r r e s p o n d t o t h e f i n a l p e r i o d s and smeared damping v a l u e s of e a c h of t h e s e sub-i t e r a t i o n s . Then t h e s o l u t i o n s u g g e s t e d by Freeman's method would be t h e i n t e r s e c t i o n of t h e two c u r v e s , i . e . 0.579 s e c . and s p e c t r a l a c c e l e r a t o n o f 0.292g. I t i s i n t e r e s t i n g t o f i n d f r o m F i g . 3.6 and F i g . 3.7 t h a t t h e smeared damping t e n d s t o v a r y l i n e a r l y o v e r t h e p e r i o d s w h i l e t h e t r a n s i t i o n of t h e s p e c t r u m does n o t . However, t h e most i m p o r t a n t p o i n t o f t h i s e x e r c i s e i s t o show t h a t t h e s o l u t i o n o f t h e p r o b l e m i s a c t u a l l y bounded by the v a l u e s of s p e c t r a l a c c e l e r a t i o n c o r r e s p o n d i n g t o t h e e a r l y s t a g e s of t h e i t e r a t i v e p r o c e d u r e , l i k e t h a t f o r t h e f o u r t h and f i f t h i t e r a t i o n . The s i g n i f i c a n c e of t h i s i s t h a t t h e s e v a l u e s of s p e c t r a l a c c e l e r a t i o n c an be u s e d as t h e upper and l o w e r bounds t h a t w i l l l i m i t t h e ran g e of t h e c a p a c i t y c u r v e where t h e s o l u t i o n has t o be f o u n d . (Note t h a t t h e v a l i d i t y of t h e upper and l o w e r bounds c a n e a s i l y be c h e c k e d , i f n e c e s s a r y , s i n c e t h e upper bound w i l l g i v e a s o l u t i o n below t h e a c t u a l s p e c t r u m whereas t h e l o w e r bound w i l l g i v e a s o l u t i o n above t h e a c t u a l s p e c t r u m , see F i g . 3.6.) Because of t h i s , t h e f o l l o w i n g p r o c e d u r e c an be u s e d t o s e t up a s e a r c h r o u t i n e : 1 . To d e t e c t a c o n v e r g e n c e p r o b l e m : a f t e r t h e f i r s t f i v e i t e r a t i o n s , s t a r t t o compare t h e c h a n g e s i n f i r s t mode r e s p o n s e p e r i o d w i t h t h e p r e v i o u s two i t e r a t i o n s and d e t e r m i n e whether r e v e r s a l h as o c c u r r e d i n t h e l a s t t h r e e i t e r a t i o n s . 2. S t o r e t h e v a l u e s of f i r s t mode s p e c t r a l a c c e l e r a t i o n f o r 40 t h e p r e v i o u s i t e r a t i o n . T h i s w i l l be u s e d as an upper o r low e r bound f o r t h e s e a r c h once a c o n v e r g e n c e p r o b l e m i s d e t e c t e d . 3. When a c o n v e r g e n c e p r o b l e m i s d e t e c t e d , use a c o n t r o l f l a g t o t u r n on t h e s e a r c h r o u t i n e i n p l a c e of p a r t s o f th e n o r m a l p r o c e d u r e . 4. Use t h e b i n a r y s e a r c h method t o s e a r c h f o r t h e s o l u t i o n . D u r i n g e a c h t r i a l , a c o n s t a n t f i r s t mode s p e c t r a l a c c e l e r a t o n i s us e d u n t i l t h e program c o n v e r g e s i n t h e n o r m a l way. 5. Then c h e c k t h i s a c c e l e r a t i o n a g a i n s t t h e a c c e l e r a t i o n f r o m t h e s p e c t r u m , f o r t h e same p e r i o d and t h e same l e v e l of damping. I f t h e v a l u e s do not a g r e e t h e n r e p e a t s t e p 4 w i t h a n o t h e r t r i a l v a l u e of c o n s t a n t a c c e l e r a t i o n . I f t h e two v a l u e s do a g r e e w i t h i n a c e r t a i n l i m i t of t o l e r a n c e t h e n t h e p r o g r a m has a c t u a l l y c o n v e r g e d . In d e s i g n i n g t h e s e a r c h r o u t i n e f o r t h e program, i t was a l s o i n t e n d e d t o m i n i m i z e c h a n g e s t o t h e p r o g r a m as a whole. T h e r e f o r e , o n l y a c o n t r o l f l a g , 'LOCK', was added t o t h e c o n v e r g e n c e c h e c k i n g r o u t i n e i n t h e main program and a s u b r o u t i n e 'STACHK' was i n c o r p o r a t e d i n t o t h e main s u b r o u t i n e MOD3. The f l o w c h a r t i n F i g . 3.8 shows how STACHK i s u s e d i n t h e main s u b r o u t i n e . I t s h o u l d be n o t e d t h a t STACHK o n l y works f o r t h e f i r s t mode of t h e a n a l y s i s . I t w i l l d e t e c t any r e v e r s a l o f t h e r e s p o n s e p e r i o d a f t e r t h e f i f t h i t e r a t i o n . I f t h e r e v e r s a l i s l a r g e r t h a n a p r e d e f i n e d l i m i t , t h e c o n t r o l f l a g w i l l be s e t t o 1. T h e r e a f t e r , 41 s u b r o u t i n e STACHK w i l l t a k e o v e r s u b r o u t i n e SPECTR and f e e d i n t o t h e p r o g r a m a t r i a l v a l u e o f s p e c t r a l a c c e l e r a t i o n . On t h e o t h e r hand, i f no r e v e r s a l i n r e s p o n s e p e r i o d i s d e t e c t e d , t h e p r o g ram w i l l go t o s u b r o u t i n e SPECTR and c o l l e c t t h e v a l u e of s p e c t r a l a c c e l e r a t i o n i n t h e n o r m a l way. F i g . 3.9 i s a d e t a i l f l o w c h a r t of t h e s e a r c h r o u t i n e STACHK. B e g i n n i n g from t h e f o u r t h i t e r a t i o n , t h e r e s p o n s e p e r i o d and t h e s p e c t r a l a c c e l e r a t i o n w i l l be s t o r e d i n memory. S t a r t i n g f r o m t h e s i x t h i t e r a t i o n , p e r i o d s from t h e l a s t two i t e r a t i o n s w i l l be compared u s i n g t h e f o l l o w i n g c o n d i t o n s T n-2 " Tn-, < -0.005 and Tn_, - T n > 0.005 T n_ 2 - Tn_, > 0.005 and Tn_, - T„ < -0.005 where T i s t h e p e r i o d i n s e c o n d s and n i s t h e i t e r a t i o n number. I f e i t h e r of t h e above c o n d i t i o n s i s met, t h e n t h e b i n a r y s e a r c h r o u t i n e w i l l be u s e d . Upper and l o w e r bounds w i l l be s e t up u s i n g t h e v a l u e s o f s p e c t r a l a c c e l e r a t i o n , a t 2 p e r c e n t damping, from t h e l a s t and c u r r e n t i t e r a t i o n . The f i r s t t r i a l v a l u e i s t h e mean of t h e upper and l o w e r bounds. T h i s t r i a l v a l u e w i l l be u s e d t h r o u g h o u t t h e p r o c e s s u n t i l t h e p r o g r a m c o n v e r g e s , w h i c h w i l l be i n d i c a t e d by t h e v a l u e of IFLAG b e i n g s e t e q u a l t o one i n t h e main p r o g r a m . However, t h i s does n o t n e c e s s a r i l y mean t h a t t h e s o l u t i o n has been f o u n d , b e c a u s e t h e s p e c t r a l a c c e l e r a t i o n , a t t h e (3.8a) (3.8b) 42 c u r r e n t l e v e l of damping, may not be e q u a l t o t h a t of the spectrum, a t the same l e v e l of damping. T h e r e f o r e , the f o l l o w i n g c r i t e r i o n i s used t o check the c o n d i t i o n . SADIF = Sa cal Sa Sa sp < 0.015 (3.9) where Sa c a| i s the t r i a l v a l u e of s p e c t r a l a c c e l e r a t i o n a t the c u r r e n t l e v e l of damping Sasp i s the s p e c t r a l a c c e l e r a t i o n d e t e r m i n e d from the spectrum a t the same l e v e l of damping. I f t h i s c r i t e r i o n i s not met, then the upper or the lower bound has t o be a d j u s t e d a c c o r d i n g t o these r u l e s . : S a n - Sa n_ 1 S a n - Sa n_ 1 f o r S a ^ i > Sa f o r Sa^gt < Sa (3.10a) (3.1 Ob) where S a n H i s the t r i a l v a l u e of s p e c t r a l a c c e l e r a t i o n , a t 2% damping, from the l a s t i t e r a t i o n Sa" i s the new upper bound v a l u e , a t 2% damping and SaJ; i s the new lower bound v a l u e , a l s o a t 2% damping. Then the next t r i a l v a l u e w i l l , a g a i n , be the mean of these new bounds. T h i s w i l l c a r r y on u n t i l c r i t e r i o n (3.9) i s s a t i s f i e d . Once i t i s s a t i s f i e d , the c o n t r o l f l a g w i l l be s e t e q u a l t o 2 and I FLAG remains e q u a l t o 1. T h i s w i l l f i n a l i z e the s e a r c h p r o c e d u r e . However, one e x t r a i t e r a t i o n , as i n the normal p r o c e s s , must s t i l l be performed i n o r d e r t o get the output of the r e s u l t s . 43 As a f i n a l comment, i t must n o t e d t h a t t h e s e a r c h r o u t i n e i s not p e r f e c t i n a l l c a s e s . F i r s t , t h e l a s t i t e r a t i o n may sometimes i n c r e a s e t h e d i f f e r e n c e between t r i a l and a c t u a l s p e c t r a l a c c e l e r a t i o n . S e cond, t h e r o u t i n e may r e q u i r e more ' a c c u r a c y ' i n t h e c o n v e r g e d r e s u l t s . T h i s would mean t i g h t e r l i m i t s t o c r i t e r i a (3.4) and ( 3 . 5 ) . In t h e program t h e o r i g i n a l l i m i t s a r e r e d u c e d by h a l f once t h e s e a r c h r o u t i n e i s i n e f f e c t . T h i s w i l l r e q u i r e more i t e r a t i o n s t h a n b e f o r e , b u t f o r t u n a t e l y t h e program c o n v e r g e s more r a p i d l y as t h e upper and l o w e r bounds g e t c l o s e r t o e a c h o t h e r . T h i r d l y , i f t h e s e a r c h r o u t i n e f a i l s , o t h e r t h a n t h e 'BREAK' key on t h e t e r m i n a l , t h e maximum number o f i t e r a t i o n s d e f i n e d i n t h e i n p u t d a t a w i l l be t h e o n l y means t o s t o p t h e program; t h e r e f o r e c a u t i o n i s needed when c h o o s i n g t h i s number. F i n a l l y , t h e l i m i t s i n c r i t e r i a (3.8a and b) may n o t be s u i t a b l e f o r v e r y s m a l l s t r u c t u r e s , b e c a u s e t h e s e t e n d t o have m i n u t e f l u c t u a t i o n s i n p e r i o d when c o n v e r g e n c e f a i l s . I t i s u n c e r t a i n whether t h e l i m i t s i n c r i t e r i a (3.8a and b) s h o u l d be r e d u c e d t o s u i t t h e s e s t r u c t u r e s s i n c e l a r g e s t r u c t u r e s m i g h t be a b l e t o c o n v e r g e w i t h i n t h e s e l i m i t s w i t h a few more i t e r a t i o n s . Due t o t h e r a r i t y of t h i s c o n v e r g e n c e p r o b l e m , s t u d i e s have been l i m i t e d t o o n l y two o r t h r e e c a s e s . 44 3.5 T e s t i n g The S o l u t i o n S e a r c h i n g R o u t i n e The g r a p h i n F i g . 3.10 i s u s e d t o i l l u s t r a t e t h e p r o g r e s s of t h e s e a r c h r o u t i n e i n f i n d i n g t h e s o l u t i o n f o r t h e example m e n t i o n e d b e f o r e . S i n c e t h e r o u t i n e u s e d was an e a r l i e r v e r s i o n of what was p r e s e n t e d a b o v e , t h e p r o c e s s s t a r t e d from t h e f i f t h i t e r a t i o n r a t h e r t h a n t h e s i x t h and t h e l i m i t s s t a t e d i n c r i t e r i a (3.4) and (3.5) were u s e d . In t h i s p a r t i c u l a r c a s e , t h e s p e c t r a l a c c e l e r a t i o n o f t h e f o u r t h i t e r a t i o n i s t h e u p p e r bound and t h a t of t h e f i f t h i t e r a t i o n i s t h e l o w e r bound. The f i r s t g u e s s i s t h e mean . v a l u e , o f t h e l o w e r and upper bounds. The s o l u t i o n f o r t h a t l e v e l o f c o n s t a n t a c c e l e r a t i o n i s p o i n t number 1 i n F i g . 3.10, w h i c h i s s i m i l a r t o p o i n t number 3 i n F i g . 3.6. So, c l e a r l y , t h e c o n s t a n t a c c e l e r a t i o n i s l a r g e r t h a n t h a t o f t h e s p e c t r u m , and a c c o r d i n g t o Eqn. ( 3 . 1 0 a ) , th e l a s t t r i a l v a l u e i s t r e a t e d as t h e new upper bound. The s e a r c h goes on u n t i l c r i t e r i o n (3.9) i s s a t i s f i e d . In t h i s c a s e , t h r e e t r i a l s f o r t h e s e a r c h r o u t i n e and a t o t a l of e i g h t i t e r a t i o n s were needed t o c o m p l e t e t h e s e a r c h . The f i n a l r e s u l t i s a t p o i n t number t h r e e i n F i g . 3.10 where t h e p e r i o d i s 0.5779 s e c . and t h e s p e c t r a l a c c e l e r a t i o n i s 0.2922g, as compared t o 0.579 s e c . and 0.292g i n Freeman's method. The c a l c u l a t e d s p e c t r a l a c c e l e r a t i o n i s d i f f e r e n t from t h a t o f t h e s p e c t r u m by o n l y 0.8 p e r c e n t . The r e s u l t s o f t h i s c a s e a r e good, s i n c e t h e f l u c t u a t i o n i n p e r i o d i s r a t h e r s m a l l . I t i s b e l i e v e d t h a t r e a s o n a b l e r e s u l t s s h o u l d 45 a t t a i n a b l e i n o t h e r c a s e s as w e l l . 46 4.0 SUBSTITUTE DAMPING AND SMEARED DAMPING 4.1 G e n e r a l The d e t e r m i n a t i o n of s t r u c t u r a l damping i s one o f t h e most d i f f i c u l t p r o b l e m s i n dynamic a n a l y s i s ( R e f . 7 ) . I t s v a l u e depends on a number of f a c t o r s , s u c h a s t h e s t r u c t u r a l t y p e , d e g r e e of damage t h a t t h e s t r u c t u r e u n d e r g o e s , t y p e of b u i l d i n g m a t e r i a l s u s e d , as w e l l a s t h e p a r t i c i p a t i o n o f n o n - s t r u c t u r a l e l e m e n t s . T e s t s r e l a t e d t o t h i s a r e a of s t u d y a r e g e n e r a l l y l i m i t e d t o s m a l l f r a m e s o r components and as s u c h do n o t n e c e s s a r i l y r e p r e s e n t a c c u r a t e l y t h e damping of r e a l s t r u c t u r e s . In p r a c t i c e , damping i s u s u a l l y e s t i m a t e d on t h e b a s i s of e x p e r i e n c e , b e a r i n g i n mind t h e above f a c t o r s . 4.2 S u b s t i t u t e Damping F a c t o r In t h e s u b s t i t u t e - s t r u c t u r e method, t h e f o r m u l a f o r s u b s t i t u t e damping (Eqn. 2.5) s u g g e s t e d by P. G u l k a n and M.A. S o z e n 5 was d e v e l o p e d f r om t h e r e s u l t s o f a s e r i e s o f t e s t s o f r e i n f o r c e d c o n c r e t e f r a m e s . A l l t h e f r a m e s t e s t e d 47 were p o r t a l f r a m e s i n w h i c h a lumped mass was a t t a c h e d t o t h e beam s u c h t h a t t h e u n i t s would r e p r e s e n t e d 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 s . The d i m e n s i o n s o f . t h e s e frames were a p p r o x i m a t e l y 3 f e e t by 5 f e e t . The d e r i v a t i o n o f t h e f o r m u l a i s b a s e d on t h e a s s u m p t i o n t h a t t h e h y s t e r e s i s l o o p of t h e s y s t e m can be a p p r o x i m a t e d by a l i n e a r s y s t e m w i t h e q u i v a l e n t v i s c o u s damping. A c c o r d i n g t o L.S. J a c o b s o n 8 , t h i s a s s u m p t i o n would o n l y be a c c e p t a b l e when d e a l i n g w i t h s y s t e m s t h a t a r e a l m o s t l i n e a r and have s m a l l damping v a l u e s ; f o r s t r o n g l y n o n - l i n e a r s y s t e m s w i t h n o n - v i s c o u s f r i c t i o n , t h e use o f an e q u i v a l e n t v i s c o u s damping would be i n a d e q u a t e . However, L.S. J a c o b s o n 8 has a l s o p o i n t e d out t h a t most i n v e s t i g a t o r s g e n e r a l l y p r e f e r t o r e p r e s e n t t h e damping p r o p e r t i e s of any v i b r a t i n g s y s t e m i n t h i s manner, r e g a r d l e s s o f whether t h e s y s t e m i s l i n e a r o r n o t . The above a s s u m p t i o n was a l s o a d o p t e d t h r o u g h o u t t h i s s t u d y m a i n l y b e c a u s e of i t s s i m p l i c i t y . N e v e r t h e l e s s , an a t t e m p t was made t o d e r i v e an a n a l y t i c a l f o r m u l a f o r s u b s t i t u t e damping t h a t does not r e l y on t e s t r e s u l t s . F i g . 4.1(a) shows t h e s y s t e m b e i n g c o n s i d e r e d f o r t h e d e r i v a t i o n of t h e f o r m u l a . In F i g . 4 . 1 ( b ) , a d i a g r a m s i m i l a r t o F i g . 2.1(d) i s shown. The s l o p e of OB r e p r e s e n t s t h e a p p a r e n t s t i f f n e s s of t h e r e a l s t r u c t u r e , w h i c h i s a l s o t h e a c t u a l s t i f f n e s s o f t h e s u b s t i t u t e s t r u c t u r e . By l i n e a r i z i n g t h e p r o b l e m i n t h i s way, t h e shaded a r e a t h a t i s bounded by p o i n t s OAB i s n e g l e c t e d . A s s u m i n g t h a t t h i s s h a d e d a r e a i s t h e e n e r g y l o s t by a member a t maximum d i s p l a c e m e n t , i t i s p o s s i b l e t o e x p r e s s t h e amount of e n e r g y l o s t by t h e 48 f o l l o w i n g e q u a t i o n , [ s U-1 ) + 1 ] (M-1 ) My By E = (4.1) 2 ( 1 - u s ) where E i s t h e e n e r g y l o s t by t h e member dy i s t h e r o t a t i o n a t y i e l d s i s t h e s t r a i n h a r d e n i n g r a t i o and M i s t h e damage r a t i o . F o r t h e c a s e of l i n e a r v i s c o u s damping w h i c h has been assumed i n t h i s s t u d y , t h e p l o t o f damping v e r s u s d i s p l a c e m e n t a p p e a r s as an e l l i p s e , s u c h as i s shown i n F i g . 4 . 2 ( c ) , a s s u g g e s t e d by L.S. J a c o b s e n 7 , and R.W. C l o u g h and J . P e n z i e n 9 . At r e s o n a n c e , t h e work b e i n g done p e r c y c l e by t h i s damping f o r c e c an be c a l c u l a t e d as W c = TT c co(\ A 2 (4.2) where W& i s t h e work done by t h e damping f o r c e c i s t h e v i s c o u s damping c o e f f i c i e n t con i s t h e n a t u r a l f r e q u e n c y and A i s t h e d i s p l a c e m e n t of t h e p o r t a l f r a m e . The a b s o l u t e damping f a c t o r , c , can a l s o be e x p r e s s e d i n t e r m s of a s u b s t i t u t e damping c o e f f i c i e n t , P, , and t h e c r i t i c a l damping f a c t o r , C . T h u s , c = C o r i t 0i • (4.3) but C C H t = 2 m c j n (4.4) 49 12 EI and m u>n 2 = (4.5) M L 3 T h e r e f o r e Eqn. (4.2) can be r e w r i t t e n as 24 EI Wc = ir A 2 Bi (4.6) M L 3 Now e q u a t e t h e amount o f e n e r g y l o s t f r o m t h e s y s t e m t o t h e work done by t h e damping f o r c e . T h i s i m p l i e s t h a t t h e s h a d e d a r e a i n F i g . 4.1(b) i s e q u a l t o one q u a r t e r of t h e a r e a of t h e e l l i p s e i n F i g . 4 . 1 ( c ) . Thus E = Wc / 4 (4.7) By s u b s t i t u t i n g My L 0 y= (4.8) 6 EI M Y L 2 M and A = — = ' (4.8) 6 EI i n t o b o t h s i d e s o f t h e e x p r e s s i o n i n ( 4 . 7 ) , a f o r m u l a f o r j3-( i s d e r i v e d . The e q u a t i o n i s 1 [ s U-1 ) + 1 ] (M-1 ) Pi = 0.02 + [ ] (4.10) 2 7T ju ( 1 - ju s ) A f a c t o r of 0.02, w h i c h r e p r e s e n t s t h e v i s c o u s damping b e f o r e y i e l d , i s added t o t h e e q u a t i o n . I f t h e s t r a i n h a r d e n i n g r a t i o , s, i s z e r o , t h e n t h e above e q u a t i o n becomes 50 1 1 Pi = 0.02 + [ 1 ] (4.11) 2 77 M w h i c h i s v e r y s i m i l a r t o Eqn. (2.5) t h a t i s s u g g e s t e d by P. G u l k a n and M.A. S o z e n 5 . A c o m p a r a t i v e p l o t of Eqn. (2.5) and Eqn. (4.11) i s shown i n F i g . 4.2. From t h e f i g u r e , i t i s c l e a r t h a t t h e damping v a l u e s o b t a i n e d from Eqn. (4.11) a r e h i g h e r t h a n t h o s e o b t a i n e d f r o m Eqn. ( 2 . 5 ) . The a p p l i c a t i o n of t h e new f o r m u l a t h e r e f o r e r e s u l t s i n a r e d u c t i o n of t h e m a g n i t u d e o f t h e s p e c t r a l a c c e l e r a t i o n and t h u s o f t h e r e s p o n s e of s t r u c t u r e . The v a l u e s f o r s u b s t i t u t e damping a t 2 p e r c e n t s t r a i n h a r d e n i n g , as o b t a i n e d from Eqn. ( 4 . 1 0 ) , a r e a l s o p l o t t e d on F i g . 4.2. The f i g u r e i n d i c a t e s t h a t damping i n c r e a s e s more r a p i d l y w i t h damage r a t i o when s t r a i n h a r d e n i n g i s i n c l u d e d . 4.3 Smeared Damping F a c t o r A f t e r t h e s u b s t i t u t e damping f a c t o r i s d e t e r m i n e d f o r e a c h member, t h e smeared damping f a c t o r of t h e s t r u c t u r e f o r e a c h mode c a n be c a l c u l a t e d u s i n g E q n s . (2.6) and ( 2 . 7 ) . T h e s e e q u a t i o n s a r e s t i l l v a l i d even t h o u g h t h e f o r m u l a f o r s u b s t i t u t e damping has been c h a n g e d . However, a comment r e g a r d i n g t h e r e l i a b i l i t y of t h e s e e q u a t i o n s seems i n o r d e r . In t h e d e r i v a t i o n o f Eqn. ( 2 . 7 ) , i t i s assumed t h a t e a c h member c o n t r i b u t e s t o t h e modal damping i n p r o p o r t i o n t o 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 e n e r g y . The r e a s o n f o r t h i s 51 a s s u m p t i o n i s n o t s t a t e d i n t h e p a p e r w r i t t e n by A. S h i b a t a and M.A. S o z e n 2 . Even i f t h e f l e x u r a l s t r a i n e n e r g y i s t h e c o r r e c t w e i g h t i n g f a c t o r f o r d e t e r m i n i n g t h e smeared damping, i t i s s t i l l d o u b t f u l whether t h e magnitude of t h e w e i g h t i t s e l f i s g e n e r a l enough f o r a l l t y p e s of s t r u c t u r e s . A l s o as m e n t i o n e d by P. G u l k a n and M.A. S o z e n 5 , i t i s n o t j u s t i f i a b l e t o s p e c i f y v a l u e s of 0 S t o more t h a n two d e c i m a l p l a c e s ; t h i s i m p l i e s t h a t i t i s r a t h e r i m p r a c t i c a l t o demand a h i g h d e g r e e of a c c u r a c y from t h e smeared damping e q u a t i o n s . I t i s q u i t e p o s s i b l e t h a t t h i s i s a s o u r c e of e r r o r i n t h e p r e s e n t method. F u r t h e r r e s e a r c h i n t h i s area, m i ght p r o v e f r u i t f u l . 52 5.0 ALTERATION TO THE DETERMINATION OF DAMAGE RATIOS 5.1 Member S t i f f n e s s M o d i f i c a t i o n U s i n g A S i n g l e Damage  R a t i o 'Damage r a t i o ' i s t h e key f a c t o r i n t h e i d e a of u s i n g a n . e l a s t i c modal a n a l y s i s t o d e a l w i t h an i n e l a s t i c p r o b l e m . B e c a u s e o f t h e use o f t h i s damage r a t i o , t h e c o m p l i c a t i o n o f f o r m i n g i n e l a s t i c s t i f f n e s s m a t r i c e s and l o a d i n c r e m e n t i n g i s a v o i d e d . T h i s i m p l i e s t h a t t h e s t i f f n e s s m a t r i c e s o r f o r m u l a s s u i t a b l e f o r t h e u s u a l s t r u c t u r a l a n a l y s i s would a l s o be a p p l i c a b l e t o t h e p r e s e n t method. I n t h e program, t h r e e s t a n d a r d member s t i f f n e s s m a t r i c e s a r e u s e d ( F i g . 5 . 1 ) . They a r e t h e a x i a l s t i f f n e s s m a t r i x ( a ) , t h e b e n d i n g s t i f f n e s s m a t r i x ( b ) , and t h e r i g i d arm s t i f f n e s s m a t r i x ( c ) . S i n c e t h e use o f t h e s e m a t r i c e s r e q u i r e s t h a t t h e members be l i n e a r e l a s t i c , a l l s t i f f n e s s e s i n v o l v i n g t h e ter m EI have t o be d i v i d e d by t h e damage r a t i o a s shown i n F i g . 5.1. The o b j e c t o f t h i s i s t o r e d u c e t h e s t i f f n e s s of t h e member i n o r d e r t h a t t h e l o s s o f s t i f f n e s s due t o p l a s t i c a c t i o n c an be t a k e n i n t o a c c o u n t . In t h e p r e v i o u s p r o g r a m , t h e 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 i n s u b s e q u e n t i t e r a t i o n s by u s i n g t h e b i g g e r of t h e two end moments i n t h e member. Then t h e s e c t i o n a l 53 r i g i d i t y of t h e member i s m o d i f i e d by t h i s damage r a t i o . In d o i n g t h i s , i t i s i m p l i e d t h a t t h e whole member i s damaged by an amount e q u a l t o t h i s b i g g e r damage r a t i o a t one end. T h i s argument i s a c c e p t a b l e o n l y i f t h e end moments a r e e q u a l o r v e r y c l o s e t o e a c h o t h e r , w h i c h i n f a c t i s t r u e f o r most members i n a t y p i c a l frame, e x c e p t t h e b a s e columns and t h e edge beams. However, i t can be shown l a t e r t h a t t h i s argument i s o n l y a r o u g h a p p r o x i m a t i o n even i f t h e end moments a r e e q u a l . F o r o t h e r t y p e s o f s t r u c t u r e s , s u c h as c o u p l e d frame s h e a r - w a l l s t r u c t u r e s , t h e v a r i a t i o n i n b e n d i n g moment a l o n g t h e s t r u c t u r a l w a l l c o u l d be q u i t e d i f f e r e n t t o what i s assumed a b o v e . In s u c h c a s e s , l a r g e d i s c r e p a n c i e s would o c c u r a t t h e base of t h e s t r u c t u r a l w a l l s i n c e t h e s t i f f n e s s f o r members a t t h e base i s g r o s s l y m i s -r e p r e s e n t e d by t h e above a s s u m p t i o n . I t i s b e l i e v e d t h a t t h e r e m o v a l of s u c h a s i m p l i f i c a t i o n c o u l d improve t h e r e l i a b i l i t y o f t h e r e s u l t s o b t a i n e d from t h e p r e s e n t method. T h i s c a n be done by a p p l y i n g one damage r a t i o t o e a c h end of t h e member. The t r a d e - o f f i n d o i n g so would be added c o m p l e x i t y i n t h e p r o g r a m and t h u s h i g h e r c o s t i n r u n n i n g t h e p r o g r a m . However, t h e improvements b r o u g h t a b o u t by t h e a l t e r a t i o n a r e r a t h e r s i g n i f i c a n t , as c a n be shown by t h e r e s u l t s i n C h a p t e r 6. 54 5.2 Member S t i f f n e s s M o d i f i c a t i o n U s i n g Damage R a t i o A t Two Ends In o r d e r t o remove t h e drawback of t h e s i m p l i f i e d p r o c e d u r e , a new model i s i n t r o d u c e d so t h a t t h e damage r a t i o of b o t h ends c a n be t a k e n i n t o c o n s i d e r a t i o n . The new model w i l l i n v o l v e h i n g e s a t e a c h end of t h e member. A t y p i c a l member i s shown i n F i g . 5.2. The s e c t i o n a l r i g i d i t y a l o n g t h e l e n g t h of t h e h i n g e w i l l be a f u n c t i o n of t h e damage r a t i o a t b o t h e n d s . A ssuming t h a t t h e s e c t i o n a l r i g i d i t y of t h e h i n g e a t j o i n t A and t h a t a t j o i n t B i s E I A = A EI (5.1) and E I B = B EI (5.2) r e s p e c t i v e l y , where EI i s t h e i n i t i a l s e c t i o n a l r i g i d i t y of t h e member; A and B a r e f a c t o r s t h a t a r e u s e d t o r e f l e c t t h e damage o f t h e j o i n t by r e d u c i n g t h e i n i t i a l s e c t i o n a l s t i f f n e s s a s damage o c c u r s . F a c t o r s A and B w i l l a l w a y s be l e s s t h a n one when damage r a t i o s a r e g r e a t e r t h a n one. To d e r i v e a r e l a t i o n s h i p between t h e two f a c t o r s and t h e damage r a t i o o f t h e member ends, t h e f o l l o w i n g a s s u m p t i o n s become n e c e s s a r y : 1. In a c c o r d a n c e w i t h b a s i c s t r u c t u r a l t h e o r y , when a u n i t r o t a t i o n i s a p p l i e d t o t h e p i n n e d end of a member w h i l e t h e o t h e r end i s h e l d f i x e d , t h e moment t h u s c r e a t e d a t t h e p i n n e d end would be 4 E I / L . I f t h e damage r a t i o a t 55 t h a t end i s M, t h e n t h e c o r r e s p o n d i n g moment o r s t i f f n e s s w i l l be 4 E I / ( M L ) . 2. When damage o c c u r s t o a j o i n t , a p l a s t i c h i n g e w i t h h i n g e l e n g t h e q u a l t o one t w e n t i e t h o f t h e member l e n g t h w i l l d e v e l o p . ( I t i s t h o u g h t t h a t t h i s w i l l be a p p r o x i m a t e l y e q u a l t o t h e member d e p t h . ) 3. The s e c t i o n a l s t i f f n e s s a l o n g t h e l e n g t h of t h e h i n g e i s c o n s t a n t . From t h e f i r s t a s s u m p t i o n , i t i s c l e a r t h a t , f o r j o i n t A, t h e moment due t o a u n i t r o t a t i o n a t t h e same j o i n t would be 4 E I / ( M A L ) , and f o r j o i n t B, i t would be 4 E I / ( M B L ) , where M A and M & a r e t h e damage r a t i o s a t j o i n t A and j o i n t B, r e s p e c t i v e l y . T hus, 4 EI k 3 3 = (5.3) M A L 4 EI k 6 6 = (5.4) MB L In many n o n - l i n e a r dynamic a n a l y s i s p r o g r a m s , a p l a s t i c h i n g e w i t h z e r o h i n g e l e n g t h i s a d o p t e d . The use of su c h an a s s u m p t i o n would g e n e r a l l y s i m p l i f y t h e p r o c e d u r e i n e s t a b l i s h i n g n o n - l i n e a r s t i f f n e s s m a t r i c e s . However, t h i s a s s u m p t i o n i s u n r e a l i s t i c due t o t h e f a c t t h a t p l a s t i c h i n g e s a r e known t o have a c e r t a i n l e n g t h . In n o r m a l c a s e s , t h e p l a s t i c h i n g e would d e v e l o p from t h e end of t h e member t o a d i s t a n c e e q u a l t o t h e d e p t h of t h e beam. So f o r a t y p i c a l beam w i t h d e p t h t o l e n g t h r a t i o o f 0.05, t h e h i n g e 56 l e n g t h would be a p p r o x i m a t e l y e q u a l t o t h e d e p t h o f t h e member. F o r an o r d i n a r y 6 by 6 s t i f f n e s s m a t r i x , t h e r e s h o u l d be 36 d i f f e r e n t damage r a t i o s . However, a l l t h e s e s t i f f n e s s e s c a n be e x p r e s s e d i n terms of k 3 3 , k 6 g , and k 6 3 ( R e f . 1 1 ), t h e r e f o r e o n l y t h e damage r a t i o f o r k 6 3 has t o be f o u n d . T h i s i s done by u s i n g t h e c o n j u g a t e beam method. A d i a g r a m showing t h e r e a l and c o n j u g a t e beam i s g i v e n i n F i g . 5.3 t o h e l p e x p l a i n t h e method. F i g . 5.3(a) i s t h e r e a l beam w i t h a u n i t r o t a t i o n a t j o i n t A. The moment d i a g r a m o f t h e beam i s shown i n F i g . 5 . 3 ( b ) . The c o n j u g a t e beam c o r r e s p o n d i n g t o t h e r e a l beam w i l l be a s shown i n F i g . 5 . 3 ( c ) . To p r o d u c e t h e e l a s t i c l o a d o f t h e c o n j u g a t e beam i n F i g . 5.3(d) and a l s o t h e e q u a t i o n s o f e q u i l i b r u m , t h e f o l l o w i n g r u l e s a r e a p p l i e d a c c o r d i n g t o t h e c o n j u g a t e beam method: 1. The c u r v a t u r e of t h e r e a l beam i s e q u a l t o t h e d i s t r i b u t e d l o a d of t h e c o n j u g a t e beam. 2. The s l o p e of t h e r e a l beam i s e q u a l t o t h e s h e a r i n t h e c o n j u g a t e beam. 3. The d e f l e c t i o n of t h e r e a l beam i s e q u a l t o t h e moment i n t h e c o n j u g a t e beam. To d e t e r m i n e t h e d i s t r i b u t e d l o a d i n t h e c o n j u g a t e beam, s u p e r p o s i t i o n of t h e d i f f e r e n t component p a r t s i s a p p l i e d . T h e s e components a r e shown i n F i g . 5.4. The f i r s t two e l a s t i c l o a d components a r e c a u s e d by t h e moments a t t h e two ends, n e g l e c t i n g t h e h i n g e s . The n e x t two components a c c o u n t f o r t h e e f f e c t of t h e h i n g e a t j o i n t A and t h e l a s t two 57 c o r r e s p o n d t o h i n g e a t j o i n t B. The sum of a l l t h e s e p r o d u c e s t h e t o t a l d i s t r i b u t e d e l a s t i c l o a d a t t h e b o t t o m of t h e f i g u r e . Two e q u a t i o n s o f e q u i l i b r i u m c an be o b t a i n e d by t a k i n g moments a b o u t j o i n t A a n d j o i n t B. A c c o r d i n g t o r u l e 3, t h e sum of moments a t t h e two ends s h o u l d be z e r o , s i n c e t h e end d e f l e c t i o n s o f t h e r e a l beam a r e z e r o . A f t e r r e o r g a n i z i n g t h e e q u a t i o n s , t h e f o l l o w i n g e q u a t i o n s , w r i t t e n i n t e r m s of f a c t o r s A and B, a r e o b t a i n e d EI [ 0.2858AB + 0.000417B + 0.04754A ] M A = k 3 3 = = : (5.5) L f (A, B) EI [ 0.1643AB + 0.01208 (A+B) ] M B = k 6 3 = (5.6) L f (A, B) where f(A,B) = A 2+B 2 0.00226 + 0.0547AB + 0.0132(A+B) + 5(10)~ 7 (5.7) AB From t h e c o n f i g u r a t i o n shown i n F i g . 5 . 3 ( a ) , i t i s c l e a r t h a t M A i s t h e same as k 3 3 , t h e r e f o r e from Eqn. (5.3) and ( 5.5), t h e f o l l o w i n g e q u a t i o n i s d e r i v e d 4 f (A, B) „ = (5.8) [ 0.2858AB + 0.0000417B + 0.04754A ] A s i m i l a r e q u a t i o n c a n a l s o be d e r i v e d f o r M b by i n t e r c h a n g i n g j o i n t A and j o i n t B i n F i g . 5 . 3 ( a ) . T h u s , 58 4 f (A , B) M B = (5.9) [ 0.2858AB + 0.0000417A + 0.04754B ] By a s s u m i n g t h a t 2 EI k 6 3 = (5.10) M e L Eqn. (5.6) g i v e s 2 f ( A , B ) Me = (5.11) [ 0.1643AB + 0.01208 (A+B) ] where ne i s t h e unknown damage r a t i o and w i l l be r e f e r r e d t o as t h e ' e f f e c t i v e ' damage r a t i o f r o m h e r e o n . The n e x t s t e p i s t o d e r i v e an e q u a t i o n f o r M e i n terms of M a and He,. A l t h o u g h t h e r e a r e enough e q u a t i o n s f o r t h e d e r i v a t i o n , i t i s u n l i k e l y t h a t t h i s can be done d i r e c t l y f r o m t h e above e q u a t i o n s . T h e r e f o r e , an i t e r a t i v e method was a d o p t e d t o d e t e r m i n e A and B f r o m Eqn. (5.8) and ( 5 . 9 ) ; t h e n , t h e v a l u e s f o r Me can be o b t a i n e d f r o m Eqn. ( 5 . 1 1 ) . S i n c e M a and MB would be known q u a n t i t i e s d u r i n g t h e a n a l y s i s , a range of v a l u e s f o r M a and M& c a n be assumed h e r e . The range of damage r a t i o s u s e d i s f r o m one t o s i x w i t h an i n c r e m e n t of one, f o r b o t h M a and M&» The r e s u l t s a r e t a b u l a t e d i n T a b l e 5.1. From t h e f i r s t s i x l i n e s of t h e t a b l e , i t i s i n t e r e s t i n g t o f i n d t h a t j o i n t A becomes more r i g i d as t h e damage r a t i o i n t h e o t h e r j o i n t i n c r e a s e s . T h i s has c o n t r i b u t e d some e r r o r t o t h e v a l u e s i n k 3 3 , w h i c h s h o u l d be e q u a l t o 4. However, t h e e r r o r i s s m a l l and 59 t h e r e f o r e c a n be a c c e p t e d . O t h e r v a l u e s f o r k 3 3 a r e i n c l o s e agreement w i t h Eqn. ( 5 . 3 ) . F i g . 5.5 i s a p l o t of \xe v e r s u s M a and M&. I t shows t h a t M E i s an i n c r e a s i n g f u n c t i o n of b o t h M a and M b . A c o m p a r i s o n w i t h t h e s i m p l i f i e d method u s e d p r e v i o u s l y i s a l s o made i n F i g . 5.5. The d o t t e d l i n e s a t t h e b o t t o m of t h e g r a p h w i t h M a r a n g i n g f r o m one t o s i x r e p r e s e n t t h e v a l u e s o b t a i n e d f r o m t h e s i m p l i f i e d method. The d i f f e r e n c e between t h e p r e v i o u s s i m p l i f i c a t i o n and t h e p r e s e n t model i s o b v i o u s , and t h e d i f f e r e n c e becomes more p r o n o u n c e d as b o t h M A and M& i n c r e a s e . T h e r e f o r e , t h e r e l i a b i l i t y o f r e s u l t s g e n e r a t e d by t h e p r e v i o u s method i s r a t h e r l i m i t e d a s damage r a t i o s o f t h e s t r u c t u r e become l a r g e r . The f i n a l s t e p o f d e r i v i n g t h e e q u a t i o n i s t o f i t t h e l i n e s i n F i g . 5.5 by an a p p r o p r i a t e f o r m u l a . F o r t u n a t e l y , t h e r e l a t i o n s h i p among M A , M B and n& i s l i n e a r , h e n c e , i t c a n e a s i l y be done by u s i n g t h e l e a s t - s q u a r e method. F i r s t , t h e s i x l i n e s i n t h e f i g u r e a r e f i t t e d w i t h s t r a i g h t l i n e s w h i c h w i l l p r o d u c e s i x l i n e e q u a t i o n s r e l a t i n g M B T O M E f o r d i f f e r e n t v a l u e s o f MA. Then by f i t t i n g s t r a i g h t l i n e s t h r o u g h t h e s i x c o n s t a n t s and s l o p e s , s e p a r a t e l y , an e q u a t i o n f o r t h e c o n s t a n t s and an e q u a t i o n f o r t h e s l o p e s , b o t h w r i t t e n i n terms o f MA> i s o b t a i n e d . P u t t i n g t h e s e e q u a t i o n s back i n t o t h e e x p r e s s i o n f o r M & and \x& w i l l y i e l d an e q u a t i o n f o r M s i n t e r m s of M a and MB. However, t h i s e q u a t i o n must be a d j u s t e d s l i g h t l y , due t o e r r o r s i n l i n e f i t t i n g , so t h a t n& would be e q u a l t o one when b o t h M A and M B a r e u n i t y . The f i n a l e x p r e s s i o n i s 6 0 M E = 0 . 0 9 5 + 0 . 2 ( M A + M s ) + 0 . 5 0 5 M A M & ( 5 . 1 2 ) T h i s e q u a t i o n i s s y m m e t r i c a l i n t e r m s of M a and M&, w h i c h has e l i m i n a t e d any p o s s i b i l i t y of h a v i n g d i r e c t i o n a l e f f e c t s . The f i n a l s t e p i s t o put t h e s e t h r e e damage r a t i o s i n t o t h e s t i f f n e s s m a t r i x . T h i s can be done, f i r s t , by s u b s t i t u t i n g Eqn. ( 5 . 3 ) , ( 5 . 4 ) , and (5.10) i n t o t h e m a t r i x ( R e f . 11) shown i n F i g . 5.6 where a l l f l e x u r a l t e r m s a r e e x p r e s s e d i n t e r m s of k 3 3 , k 6 6 and k 6 3 . However, t h e m a t r i x i s formed w i t h r e s p e c t t o t h e member l o c a l a x e s o n l y . So t h e n e x t s t e p would be t o t r a n s f o r m t h e m a t r i x so t h a t i t c a n be u s e d i n any a r b i t r a r y s e t of r e f e r e n c e a x e s . The t r a n s f o r m e d s t i f f n e s s m a t r i x c a n be d e t e r m i n e d by s u b s t i t u t i n g t h e r e s u l t s o b t a i n e d from t h e l a s t s t e p i n t o t h e m a t r i x i n F i g . 5.7. The f i n a l m a t r i x i s a c o m b i n a t i o n o f t h e b e n d i n g and a x i a l s t i f f n e s s e s . By c o m p a r i n g i t w i t h t h e m a t r i x i n F i g . 5.1(a) and ( b ) , damage r a t i o s f o r a l l t h e t e r m s a r e o b t a i n e d . T h e r e a r e a t o t a l of s i x damage r a t i o s , b u t , of c o u r s e , f o u r o f them w i l l be f u n c t i o n s o f MA and M&. F o r e a s e of programming, t h e r e c i p r o c a l o f t h e s e damage r a t i o s i s u s e d as a m u l t i p l i c a t i o n f a c t o r f o r e a c h s t i f f n e s s t e r m . The f o l l o w i n g a r e t h e two b a s i c f a c t o r s , Mi = 1/MA (5.13) M 2 = 1/M B (5.14) w h i c h o n l y a p p l y t o 4 E I / L t e r m s . The f a c t o r f o r 2 E I / L t e r m s i s 61 M 3 = (5.15) t h e n , t h e o t h e r t h r e e f a c t o r s a r e 1 2 1 MA = [ + ] (5.16) 3 M A M e 1 2 1 M 5 = [ + ] (5.17) 3 M & Me 1 1 1 1 Me = [ + + ] (5.18) 3 M A M e M & The s t i f f n e s s t e r m s where t h e s e f a c t o r s a p p l y a r e shown i n t h e b e n d i n g s t i f f n e s s m a t r i x g i v e n i n F i g . 5.8. The f a c t o r , h, w h i c h a c c o u n t s f o r s h e a r d e f l e c t i o n , i s not m o d i f i e d b e c a u s e o f t h e d i f f i c u l t i e s i n o b t a i n i n g a p r o p e r m o d i f i c a t i o n f a c t o r f o r i t . T h e r e f o r e , i t i s assumed t h a t s h e a r d e f l e c t i o n o f t h e s t r u c t u r e i s not a f f e c t e d by f l e x u r a l damage. F i n a l l y , terms i n t h e r i g i d arm s t i f f n e s s m a t r i x a r e m o d i f i e d i n t h e same manner as a p p e a r e d i n t h e b e n d i n g s t i f f n e s s m a t r i x . 5.3 A l t e r a t i o n In The C a l c u l a t i o n Of S t r a i n E n e r g y And  S u b s t i t u t e Damping 5.3.1 S t r a i n E n e r g y In t h e p r e s e n t method, t h e member s t r a i n e n e r g y i s used as a w e i g h t f a c t o r so t h a t t h e smeared damping of t h e 62 s t r u c t u r e c a n be d e t e r m i n e d from t h e s u b s t i t u t e damping f a c t o r o f t h e members. Eqn. (2.6) was u s e d t o c a l c u l a t e t h e s t r a i n e n e r g y i n t h e p r e v i o u s method ( R e f . 2, 3 ) , where n = 6 ( E I a / M ) [ M? + M? _ M A M B ] (2.6) where E I a i s t h e s e c t i o n r i g i d i t y f o r t h e e l e m e n t i n t h e a c t u a l s t r u c t u r e . T h i s e q u a t i o n i s d e r i v e d f r o m t h e s t r a i n e n e r g y , w h i c h i s d e f i n e d a s n * 0.5 [M] [ f ] [M] (5.19) where and [M] = [ f ] = EI 1/3 -1/6 -1/6 1/3 w h i c h i s t h e f l e x i b i l i t y m a t r i x . By a n a l o g y , a s i m i l a r f o r m u l a can be d e r i v e d f o r t h e new m o d e l . The f i r s t s t e p i s t o i n v e r t p a r t of t h e s t i f f n e s s m a t r i x t h a t was f o u n d i n t h e l a s t s e c t i o n t o y i e l d t h e f l e x i b i l i t y m a t r i x . By i n v e r t i n g EI [ k ] t h i s w i l l g i v e 4/M A 2 A e 2/ne 4/Me, (5.20) [ f ] = 16 M A Me. M i ] EI 4 / M & • 2 / M e • 2/Me 4 / M A (5.21) 63 Then, p u t t i n g [ f ] i n t o Eqn. (5.19), a new e x p r e s s i o n f o r s t r a i n e n e r g y i s d e r i v e d . T h i s g i v e s 2 L M A Ml M A M & n = 16 4 [ + ] (5.22) [ ] EI M& M A Me M A MB M J T h i s e q u a t i o n i s much more complex t h a n Eqn. (2.6) b e c a u s e of t h e i n v o l v e m e n t of a n o t h e r damage r a t i o . The above e q u a t i o n , i f b o t h M a and M& a r e e q u a l t o one, w i l l d e g e n e r a t e t o Eqn. ( 2 . 6 ) . 5.3.2 S u b s t i t u t e Damping The e q u a t i o n d e r i v e d i n C h a p t e r 4 i s s t i l l v a l i d i n t h e new mod e l . The o n l y change t o be made i s t o r e p l a c e t h e s i n g l e damage r a t i o by t h e a v e r a g e o f t h e damage r a t i o s a t th e two h i n g e s . T h e r e f o r e , 1 [ s U'-1 ) • + 1 ] (M'-1 ) 0; = 0.02 + [ ] (5.23) 2TT M' ( 1 - M ' S ) where M' i s t h e a v e r a g e o f M a and M B . 5.4 Computer Program To i n c o r p o r a t e a l l t h e s e m o d i f i c a t i o n i n t o t h e computer p r o g r a m w o u l d r e q u i r e s e v e r a l c h a n g e s . F i r s t o f a l l , t h e s u b r o u t i n e DAMOD has t o be a l t e r e d so t h a t i t can m o d i f y two 64 damage r a t i o s f o r e a c h member. B o t h damage r a t i o s have t o be c h e c k e d by c r i t e r i o n (3.5a and b) . However, o n l y t h e b i g g e r moment of t h e two ends i s c h e c k e d by c r i t e r i a (3.4) i n o r d e r t o a v o i d d o u b l e c o u n t i n g of t h e number o f members above c a p a c i t y . M a j o r c h a n g e s have t o be made t o s u b r o u t i n e BUILD and FORCE, where t h e s t i f f n e s s m a t r i x i s a s s e m b l e d and f o r c e s a r e c a l c u l a t e d from d i s p l a c e m e n t s . C a l c u l a t i o n s f o r f a c t o r s i n E q n s . (5.13) t h r o u g h (5.18) a r e p e r f o r m e d i n a new s u b r o u t i n e DAMCAL. F i g . 5.9 and 5.10 a r e t h e f i n a l r e v i s e d f l o w c h a r t s f o r t h e main p r o g r a m and s u b r o u t i n e MOD3, r e s p e c t i v e l y , showing a l l t h e m a j o r c h a n g e s made t o t h e program i n t h i s s t u d y . 65 6.0 THE PRESENT METHOD VS. INELASTIC TIME-STEP ANALYSIS 6.1 G e n e r a l T h e r e a r e s e v e r a l ways o f t e s t i n g t h e p r e s e n t method f o r i t s a b i l i t y t o p r e d i c t t h e i n e l a s t i c r e s p o n s e o f s t r u c t u r e s . . T h e b e s t way t o do t h i s i s , of c o u r s e , t o f e e d i n t o t h e p r o g r a m t h e model o f a r e a l s t r u c t u r e w h i c h has s u r v i v e d a s e v e r e e a r t h q u a k e and f o r w h i c h b o t h t h e g r o u n d e x c i t a t i o n and t h e r e s p o n s e of t h e s t r u c t u r e have been r e c o r d e d . Then t h e p r e d i c t e d r e s u l t s can be compared w i t h t h e a c t u a l measurements t o d e t e r m i n e t h e a c c u r a c y o f t h e p r o g r a m . In f a c t , r e c o r d s of t h i s t y p e a r e a v a i l a b l e from r e a l l i f e e v e n t s o r can be o b t a i n e d f r o m s m a l l s c a l e e x p e r i m e n t s . A l t h o u g h t h e f o r m e r p r o v i d e s t h e b e s t means of c o m p a r i s o n , t h e r e s p o n s e r e c o r d s a r e u s u a l l y l i m i t e d t o o n l y a few p o i n t s i n t h e s t r u c t u r e . T h e r e f o r e , • t h e r e l i a b i l i t y o f t h e p r o g r a m may n o t be j u s t i f i e d by t h i s k i n d o f c o m p a r i s o n w h i l e , a t t h e same t i m e , a c o n s i d e r a b l e amount o f e f f o r t has t o be s p e n t on m o d e l l i n g t h e r e a l s t r u c t u r e . S m a l l s c a l e t e s t s p r o v i d e more i n f o r m a t i o n under c o n t r o l l e d l a b o r a t o r y c o n d i t i o n s , but t h e r e s u l t i s p r o b a b l y i n a d e q u a t e when a p p l i e d t o l a r g e s c a l e s t r u c t u r e s . B e s i d e s , i t i s r a t h e r 66 i m p r a c t i c a l , e s p e c i a l l y when o n l y t h e peak r e s p o n s e i s r e q u i r e d , t o c o n d u c t t h i s k i n d o f e x p e r i m e n t . A t i m e - s t e p a n a l y s i s can p r o d u c e s u f f i c i e n t l y a c c u r a t e r e s u l t s f o r the-p u r p o s e of t h i s s t u d y , and s u c h a p r o gram was c h o s e n t o c a l i b r a t e t h e p r e s e n t method. 6.2 C o m p a r i s o n Of The Two Methods 6.2,1 A d v a n t a g e s And D i s a d v a n t a g e s Of The Two Methods T i m e - s t e p a n a l y s i s i s c a p a b l e of p r o d u c i n g a c o m p l e t e r e s p o n s e t i m e h i s t o r y f o r a g i v e n s t r u c t u r e and a g i v e n e a r t h q u a k e r e c o r d . I t i s t h e c o m p l e t e n e s s and t h e a c c u r a c y o f t h e a n a l y s i s t h a t g i v e i t t h e a d v a n t a g e s o v e r a l l t h e o t h e r methods. N e v e r t h e l e s s , t h e c o s t o f e x e c u t i o n imposes a major s e t b a c k f o r t h i s k i n d o f a n a l y s i s . The e x e c u t i o n t i m e (CPU t i m e ) of a t i m e - s t e p a n a l y s i s depends l a r g e l y on t h e l e n g t h of t h e d i g i t i z e d e a r t h q u a k e r e c o r d , t h e s i z e o f e a c h i n t e g r a t i o n t i m e - s t e p and t h e number o f d e g r e e s o f f r e e d o m t o be a n a l y s e d . The a c t u a l ^ r u n n i n g t i m e would, of c o u r s e , be much l o n g e r t h a n t h e e x e c u t i o n t i m e . F o r some t e s t s t r u c t u r e s i n t h i s s t u d y t h a t were a n a l y s e d by t h e t i m e - s t e p a n a l y s i s p rogram, DRAIN-2D, u s i n g an e i g h t s e c o n d r e c o r d and 0.0125 s e c o n d t i m e - s t e p , t o o k more t h a n an hour t o c o m p l e t e i n a h i g h p r i o r i t y j o b , y e t EDAM2, t h e p r o g r a m f o r t h e p r o p o s e d method, c a n c o m p l e t e 67 t h e a n a l y s i s f o r t h e same s t r u c t u r e i n l e s s t h a n t e n m i n u t e s . T h i s h as made t i m e - s t e p a n a l y s i s i n f e r i o r e s p e c i a l l y when d e a l i n g w i t h p a r a m e t r i c s t u d i e s and s y s t e m o p t i m i z a t i o n . As was m e n t i o n e d b e f o r e , t h e c h o i c e o f r e p r e s e n t a t i v e e a r t h q u a k e r e c o r d s added a n o t h e r d i s a d v a n t a g e t o t h i s t y p e o f a n a l y s i s . In p r e p a r i n g t h e i n p u t d a t a , a modal a n a l y s i s i s r e q u i r e d t o d e t e r m i n e t h e f u n d a m e n t a l p e r i o d of t h e s t r u c t u r e . T h i s p e r i o d may be u s e d t o c a l c u l a t e t h e damping f a c t o r f o r t h e i n p u t s t r u c t u r e . Hence, d a t a p r e p a r a t i o n f o r t h e a n a l y s i s i s a l s o r a t h e r t i m e c o n s u m i n g . A l t h o u g h t i m e - s t e p a n a l y s i s c a n p r o v i d e a c o m p l e t e r e s p o n s e t i m e h i s t o r y of a s t r u c t u r e , t h e i n t e r p r e t a t i o n of th e r e s u l t s i s l e f t open t o e a c h i n d i v i d u a l who u s e s t h e pro g r a m . In p r a c t i c e , t h e r e s u l t i n g s e t s of f o r c e s and d i s p l a c e m e n t s a r e u s u a l l y c o n v e r t e d t o d u c t i l i t y r a t i o s . However, d i f f e r e n t d e f i n i t i o n s of d u c t i l i t y r a t i o l e a d t o d i f f e r e n t i n t e r p r e t a t i o n of t h e r e s u l t s ; t h i s i s a l s o a d i s a d v a n t a g e o f t h e a n a l y s i s . In c o n t r a s t , t h e a d o p t i o n of t h e damage r a t i o i n t h e p r e s e n t method has removed t h e a m b i g u i t y of u s i n g d u c t i l i t y r a t i o s . A l s o , due t o t h e a d v a n t a g e s i n e x e c u t i o n s p e e d and e x e c u t i o n c o s t of t h e p r e s e n t method, i t i s more s u i t a b l e f o r p a r a m e t r i c s t u d i e s . However, t h e r e a r e a l s o d i s a d v a n t a g e s i n t h e p r e s e n t method m a i n l y r e l a t e d t o modal a n a l y s i s . F i r s t l y , t h e r e i s no e x a c t way o f c o m b i n i n g t h e r e s p o n s e o f e a c h i n d i v i d u a l mode t o y i e l d t h e o v e r a l l r e s p o n s e . S e c o n d l y , some o b j e c t i o n 68 has been r a i s e d a g a i n s t modal a n a l y s i s owing t o t h e f a c t t h a t t h e d u r a t i o n o f t h e e a r t h q u a k e i s not b e i n g t a k e n i n t o a c c o u n t 7 . 6.2.2 C o m p a r i s o n Of The Two Programs DRAIN-2D i s a g e n e r a l p u r p o s e t i m e - s t e p a n a l y s i s p r o g r a m d e v e l o p e d a t t h e U n i v e r s i t y o f C a l i f o r n i a a t B e r k e l e y 1 2 . The p r o g r a m has been w e l l t e s t e d and documented. I t has been u s e d w i d e l y t o p e r f o r m dynamic a n a l y s i s of a l a r g e v a r i e t y of s t r u c t u r e s and i s known t o have p r o d u c e d good r e s u l t s . The f o l l o w i n g i s a c o m p a r i s o n between DRAIN-2D and EDAM2. In g e n e r a l , b o t h t h e code of t h e program and t h e r e q u i r e d i n p u t f o r DRAIN-2D a r e more complex t h a n t h a t o f EDAM2. However, DRAIN-2D o f f e r s more f l e x i b i l i t y o v e r t h e a v a i l a b l e member t y p e s and t h e ways t h a t member p r o p e r t i e s a r e s p e c i f i e d . In DRAIN-2D, t h e r e a r e s i x t y p e s of e l e m e n t s t o c h o o s e from w h i c h i n c l u d e a t r u s s e l e m e n t , two t y p e s o f beam column e l e m e n t s , an i n f i l l p a n e l e l e m e n t , s e m i - r i g i d c o n n e c t i o n e l e m e n t and beam e l e m e n t w i t h d e g r a d i n g s t i f f n e s s . The l a s t e l e m e n t t y p e was u s e d i n t h i s s t u d y . The c h a r a c t e r i s t i c s of t h i s e l e m e n t a r e r e p r e s e n t e d by m o d e l l i n g t h e h y s t e r e s i s l o o p w i t h s t i f f n e s s d e g r a d i n g t h r o u g h e a c h c y c l e of r e s p o n s e . In u s i n g t h i s t y p e of e l e m e n t , t h e u s e r has a c h o i c e o f a d o p t i n g t h e o r i g i n a l T akeda model or t h e m o d i f i e d Takeda m o d e l . The o r i g i n a l Takeda model was c h o s e n i n t h i s s t u d y , f o r s i m p l i c i t y . A t t h i s t i m e , EDAM2 can o n l y s u p p o r t two b a s i c t y p e s o f 69 e l e m e n t s , a t r u s s e l e m e n t and a frame e l e m e n t . On t h e o t h e r hand, DRAIN-2D a l s o a l l o w s : members t o have d i f f e r e n t moment c a p a c i t y and s t i f f n e s s a t t h e two ends, t h e i n c l u s i o n of P-A e f f e c t , and i n p u t o f s t a t i c l o a d s t h a t a p p l y t o t h e s t r u c t u r e . In o r d e r t h a t t h e two pr o g r a m s can be more c o m p a r a b l e , t h e s e f u n c t i o n s a r e b y - p a s s e d i n p e r f o r m i n g t h e t i m e - s t e p a n a l y s e s . However, some o f t h e s e d i f f e r e n c e s a r e c o n s i d e r e d as p o t e n t i a l a r e a s f o r i m p r o v i n g EDAM2 and can be i n c o r p o r a t e d i n t o t h e p r o g r a m l a t e r on, i f n e c e s s a r y . 6.2.3 L i m i t a t i o n s I n The C o m p a r i s o n Of R e s u l t s S i n c e t h e b a s e s o f t h e two methods a r e d i f f e r e n t , d i f f i c u l t i e s i n o b t a i n i n g a f a i r c o m p a r i s o n between t h e r e s u l t s a r e e v i d e n t . The f o l l o w i n g a r e some major d i f f e r e n c e s between t h e two pr o g r a m s t h a t a r e w o r t h n o t i n g . F i r s t l y , t h e ways of h a n d l i n g damping a r e d i f f e r e n t between t h e two methods. Even s o , no a t t e m p t was made t o match t h e damping o b t a i n e d f r o m two d i f f e r e n t m o d e l s , s i n c e t h e a c c u r a c y of t h e T a k e d a Model i t s e l f i s l i m i t e d . S e c o n d l y , t h r o u g h a c a r e f u l s t u d y of t h e d o c u m e n t a t i o n 1 2 f o r t h e e l e m e n t w i t h d e g r a d i n g s t i f f n e s s , a major d i f f e r e n c e i n t h e d e f i n i t i o n o f s t r a i n h a r d e n i n g r a t i o was u n v e i l e d . I t was f o u n d t h a t t h e s t r a i n h a r d e n i n g r a t i o d e f i n e d i n EDAM2 i s e q u i v a l e n t t o 3/4 of t h a t a p p l y i n g t o DRAIN-2D. At t h e same t i m e , an e x p r e s s i o n u s e d t o c a l c u l a t e damage r a t i o from t h e r e s u l t s o f DRAIN-2D i s a l s o d e v e l o p e d . T h i s e x p r e s s i o n i s 70 4 M n " My My M - ( ) + (6.1) 3 s M n M„ where t h e y i e l d moment, maximum moment and s t r a i n h a r d e n i n g r a t i o a r e t h e o n l y r e q u i r e d q u a n t i t i e s . T h e r e f o r e , o n l y t h e e n v e l o p e of t h e maximum d i s p l a c e m e n t s and f o r c e s from t h e o u t p u t i s needed, whereas t h e l e n g t h y r e s p o n s e t i m e h i s t o r y c an be n e g l e c t e d i n c a l c u l a t i n g damage r a t i o s . F i n a l l y , a z e r o p l a s t i c h i n g e l e n g t h i s assumed i n DRAIN-2D, w h i l e n o n - z e r o h i n g e l e n g t h i s a d o p t e d i n EDAM2. T h i s r e p r e s e n t s a major d i f f e r e n c e i n t h e m o d e l l i n g o f an a c t u a l s t r u c t u r e . 6.3 C h o i c e Of E a r t h q u a k e R e c o r d Nowadays, a l a r g e number o f d i g i t i z e d e a r t h q u a k e r e c o r d s a r e a v a i l a b l e f o r use i n dynamic a n a l y s i s . E l C e n t r o , 1940, i s one of t h e p o p u l a r r e c o r d s t h a t i s us e d w i d e l y i n s t r u c t u r a l d e s i g n and a n a l y s i s - . However, t h e h i g h l y i r r e g u l a r shape o f t h e s p e c t r u m f o r t h i s and many o t h e r r e c o r d s imposes g r e a t d i f f i c u l t y i n o b t a i n i n g a smooth s p e c t r u m w h i l e r e t a i n 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 r e a l s p e c t r u m . In o r d e r t o o b t a i n a b e t t e r c o m p a r i s o n between t h e two methods, an e a r t h q u a k e r e c o r d t h a t w i l l p r o d u c e a smooth s p e c t r u m i s e s s e n t i a l . One s u c h r e c o r d was f o u n d i n a r e s e a r c h r e p o r t c o n c e r n i n g s i m u l a t e d e a r t h q u a k e m o t i o n s 1 3 . A t o t a l o f e i g h t d i g i t i z e d r e c o r d s were g e n e r a t e d u s i n g t h e 71 method p r e s e n t e d i n t h i s p a p e r , where C2 i s one of them. The d u r a t i o n of t h e r e c o r d i s 12 s e c o n d s and i t s s p e c t r u m , shown i n F i g . 6.1, i s r a t h e r smooth e s p e c i a l l y i n t h e r e g i o n of l o n g e r p e r i o d . As can be seen from F i g . 6.1, t h e s p e c t r a l a c c e l e r a t i o n v a r i e s s i g n i f i c a n t l y from t h e p e r i o d of 0.2 t o 0.5 s e c . , w h i c h i s t h e a p p r o x i m a t e l y c o n s t a n t b r a n c h of t h e s p e c t r u m . Smoothening t h a t r e g i o n o f t h e s p e c t r u m i s i m p o r t a n t t o t h e a c c u r a c y of t h e h i g h modes. S i n c e i t i s not p o s s i b l e t o have p i n p o i n t a c c u r a c y i n t h e c a l c u l a t i o n o f modal r e s p o n s e p e r i o d s f r o m EDAM2, . some a v e r a g i n g of t h e s p e c t r a l a c c e l e r a t i o n must be imposed i n t h a t r e g i o n , so t h a t a s l i g h t e r r o r i n r e s p o n s e p e r i o d would not c r e a t e a l a r g e d i f f e r e n c e i n t h e v a l u e of s p e c t r a l a c c e l e r a t i o n . In f i t t i n g t h e e a r t h q u a k e s p e c t r u m , t h e c u r v e c o r r e s p o n d i n g t o 2 p e r c e n t damping was u s e d . T h e s e l i n e s a r e t h e n i n p u t i n t o t h e computer p r o g r a m . I t i s f o u n d t h a t t h e s c a l i n g e q u a t i o n ( E q n . ( 2 . 8 ) ) i s n o t a l w a y s a c c u r a t e ; t h e e f f e c t of s c a l i n g a s p e c t r u m from 2 p e r c e n t damping t o 10 p e r c e n t damping i s shown i n F i g . 6.1 and F i g . 6.2. On t h e o t h e r hand, t h e l e n g t h of t h e e a r t h q u a k e r e c o r d t o be i n p u t i n t o t i m e - s t e p a n a l y s i s must a l s o be c h o s e n a p p r o p r i a t e l y . The t i m e when t h e e a r t h q u a k e a c c e l e r a t i o n i s h i g h does n o t n e c e s s a r i l y c o i n c i d e w i t h maximum r e s p o n s e of t h e s t r u c t u r e b e c a u s e o f t h e d i f f e r e n c e i n p e r i o d between t h e e a r t h q u a k e m o t i o n and t h e s t r u c t u r e . T h e r e f o r e t h e e a r t h q u a k e r e c o r d i n p u t t o t h e t i m e - s t e p program must be l o n g enough t o c o v e r t h e t i m e where t h e maximum r e s p o n s e 72 o c c u r s . Or e l s e , t h e r e s p o n s e s p e c t r u m must be g e n e r a t e d f r o m t h e same e a r t h q u a k e r e c o r d w i t h t h e same d u r a t i o n t h a t i s u s e d i n t h e t i m e - s t e p a n a l y s i s . To i l l u s t r a t e t h e u s e f u l n e s s o f t h e p r o g r a m w i t h o t h e r e a r t h q u a k e r e c o r d s , a t e s t was c o n d u c t e d u s i n g t h e 1971 San F e r n a n d o e a r t h q u a k e . The e a r t h q u a k e r e c o r d o b t a i n e d i n t h e g r o u n d f l o o r of 8244 O r i o n Ave., Van Nuys, H o l i d a y Inn was u s e d . I t s s p e c t r u m i s shown i n F i g . 6.2. I t i s o b v i o u s t h a t t h e s p e c t r a l a c c e l e r a t i o n v a r i e s s i g n i f i c a n t l y w i t h t h e r e s p o n s e p e r i o d , w h i c h i s t y p i c a l i n many r e a l e a r t h q u a k e r e c o r d s . In s p i t e of t h a t , r e a s o n a b l e r e s u l t s were o b t a i n e d from t h e p r o p o s e d method; t h e r e s u l t s w i l l be' shown i n S e c t i o n 6.4.5. 6.4 T e s t R e s u l t s And D i s c u s s i o n s 6.4.1 G e n e r a l B a s i c a l l y , t h r e e d i f f e r e n t s t r u c t u r e s were d e s i g n e d so t h a t t h e r e s u l t s from t h e p r e s e n t method arid from t i m e - s t e p a n a l y s i s c o u l d be compared. The f i r s t t e s t s t r u c t u r e i s an 8-bay, 7 - s t o r e y frame; t h e s e c o n d t e s t s t r u c t u r e i s a 5-bay, 7 - s t o r e y s h e a r - w a l l frame and t h e l a s t t e s t s t r u c t u r e i s an i r r e g u l a r frame r e s e m b l i n g t h e f i r s t t e s t frame but w i t h t h e l e f t p o r t i o n of t h e uppe r 3 s t o r e y s c u t o f f t o g i v e a s e t b a c k . T e s t s on t h e s e s t r u c t u r e s were d i v i d e d i n t o t h r e e 73 p h a s e s . In t h e f i r s t phase ( S e c t i o n 6.4.2 t o 6.4.4), a l l t h r e e s t r u c t u r e s were a n a l y s e d once by b o t h programs u s i n g t h e C . I . T . s i m u l a t e d e a r t h q u a k e , C2, r e c o r d . In t h e s e c o n d phase ( S e c t i o n 6.4.5 t o 6.4.7), some p a r a m e t e r c h a n g e s were made so t h a t t h e e f f e c t s o f t h e s e p a r a m e t e r s and t h e r e l i a b i l i t y o f t h e p r e s e n t method c o u l d be i n v e s t i g a t e d . In t h e f i n a l phase ( S e c t i o n 6.4.8), a c o m p a r i s o n between r e s u l t s f r o m t h e o r i g i n a l p r o g r am (EDAM), t h e new p r o g r a m (EDAM2) and DRAIN-2D was made. T h i s was done s i m p l y by r u n n i n g t h e f i r s t p h a s e t e s t s a g a i n on EDAM. T h i s c o m p a r i s o n shows t h e e f f e c t of a p p l y i n g t h e new s e t of damping f o r m u l a s i n C h a p t e r 4 and a l s o t h e e f f e c t of u s i n g d i f f e r e n t damage r a t i o a t two ends o f a member. S i n c e smooth c o n v e r g e n c e was a t t a i n e d i n a l l o f t h e s i x t e s t s w i t h i n t h e f i r s t and s e c o n d p h a s e , t h e s o l u t i o n s e a r c h i n g r o u t i n e m e n t i o n e d i n C h a p t e r 3 was not t e s t e d i n t h i s s t u d y . 6.4.2 T e s t Frame No. 1 T e s t frame no. 1 i s an 8-bay, 7 - s t o r e y frame r e s e m b l i n g t h e H o l i d a y Inn d e s c r i b e d i n R e f . 14. The d i m e n s i o n s o f t h e t e s t frame a r e g i v e n i n F i g . 6.3. The g r o u n d s t o r e y i s 13.5' i n h e i g h t , s e c o n d t o s i x t h s t o r e y a r e 8.708' h i g h , and t h e t o p s t o r e y i s 8.67' h i g h . Columns a r e s p a c e d e q u a l l y a t 18.75'. The p r o p e r t i e s o f t h i s t e s t frame, as w e l l a s t h e o t h e r t e s t s t r u c t u r e s , a r e t a b u l a t e d i n t a b l e 6.1. F o r t e s t frame no. 1, t h e f u n d a m e n t a l p e r i o d i s 0.78 s e c . F l o o r w e i g h t d e c r e a s e s from t h e f i r s t t o t h e t o p f l o o r . The 74 e x t e r i o r columns a r e s m a l l e r t h a n t h e i n t e r i o r c o l u m n s . S e c t i o n a l p r o p e r t i e s f o r columns and beams were o b t a i n e d by summing t h e p r o p e r t i e s o f f o u r i n d i v i d u a l f r a m e s t o g e t h e r i n t h e t r a n s v e r s e d i r e c t i o n ; t h e r e f o r e , t h e numbers w i l l a p p e a r t o be l a r g e r t h a n any o r d i n a r y s i n g l e frame. S i n c e i t was assumed t h a t t h e f l o o r s l a b c an a c t a s a d i a p h r a g m , an a r b i t r a r y l a r g e number was a s s i g n e d t o t h e beam a r e a s . F i r s t , an e l a s t i c modal a n a l y s i s was p e r f o r m e d on t h e t e s t frame t o d e t e r m i n e t h e e a r t h q u a k e l o a d f o r e a c h member. In d o i n g t h e a n a l y s i s , t h e NBC s p e c t r u m a t 5 p e r c e n t damping and a g r o u n d a c c e l e r a t i o n of 0.25g was u s e d . A d u c t i l i t y o f 6 was a p p l i e d t o a l l beams and a d u c t i l i t y o f 2 t o a l l col u m n s i n t h e i n i t i a l d e s i g n ; t h i s was t o e n s u r e t h a t beams y i e l d b e f o r e c o l u m n s . Then t h e moment c a p a c i t y o f e a c h member, shown i n F i g . 6.3, was o b t a i n e d a c c o r d i n g t o d e s i g n c o d e e q u a t i o n s . In t h i s t e s t , t h e C . I . T . s i m u l a t e d e a r t h q u a k e , C2, m o t i o n was a p p l i e d a t a maximum a c c e l e r a t i o n of 0.3g. A s t r a i n h a r d e n i n g r a t i o o f 1.5 p e r c e n t was a l s o u s e d i n EDAM2; t h i s was e q u i v a l e n t t o 2 p e r c e n t s t r a i n h a r d e n i n g i n DRAIN-2D. The r e s u l t s o f t h e t e s t were p r e s e n t e d i n F i g . 6.4, F i g . 6.5 and summarized i n T a b l e 6.2. F i g . 6.4 shows t h a t damage o n l y o c c u r r e d i n beams as p r e d i c t e d . The d i s t r i b u t i o n of damage r a t i o s was h i g h l y n o n - u n i f o r m . From F i g . 6.5, i t i s c l e a r t h a t t h e maximum damage of t h e frame a p p e a r e d on t h e f i r s t f l o o r and damage d e c r e a s e s as f l o o r l e v e l i n c r e a s e s . R e s u l t s p r e d i c t e d by EDAM2 were r a t h e r c o n s e r v a t i v e a t t h e upper l e v e l s , e s p e c i a l l y on t h e f i f t h 75 f l o o r , but were u n s a f e a t t h e l o w e r l e v e l s . On t h e r i g h t hand s i d e o f t h e f i g u r e i s a p l o t of h o r i z o n t a l d i s p l a c e m e n t v e r s u s f l o o r l e v e l . The maximum d i s p l a c e m e n t p r e d i c t e d was 3.27" from EDAM2 and 3.94" f r o m DRAIN-2D. A l t h o u g h t t h e d i s c r e p a n c y i n h o r i z o n t a l d i s p l a c e m e n t was q u i t e h i g h , t h e p r e s e n t method was a b l e t o p r e d i c t t h e d e f l e c t e d shape o f t h e t e s t frame. T a b l e 6.2 shows a summary o f t h e above r e s u l t s . The i n e l a s t i c r e s p o n s e p e r i o d was 1.14 s e c . The maximum e r r o r i n b a s e s h e a r was 7.7 p e r c e n t w h i c h was t h e l a r g e s t among a l l t h e t e s t s c o n d u c t e d i n t h i s s t u d y . The maximum e r r o r i n h o r i z o n t a l d i s p l a c e m e n t was 36.1 p e r c e n t and t h e maximum a b s o l u t e e r r o r i n damage r a t i o was 1.67, o c c u r r i n g i n t h e f i r s t f l o o r . The maximum p e r c e n t a g e e r r o r i n damage r a t i o , however, was on t h e f i f t h f l o o r . Even t h o u g h t h e e r r o r s i n damage r a t i o seem l a r g e , t h e p r e s e n t method was c a p a b l e of p i c k i n g o u t t h e t r o u b l e s p o t s i n t h e t e s t frame, w h i c h i s r e g a r d e d as one o f t h e g o a l s i n d e v e l o p i n g t h e method. 6.4.3 T e s t Frame No. 2 T e s t frame no. 2 i s a s h e a r - w a l l f r a m e . I t s d i m e n s i o n s and t h e y i e l d moment f o r e a c h member a r e shown i n F i g . 6.6. The s h e a r w a l l was i d e a l i z e d as l i n e members on t h e w a l l c e n t r e - l i n e , c o n n e c t e d t o t h e beams by r i g i d arms. T h e s e c o l u m n s had t h e c a p a c i t y and s e c t i o n a l p r o p e r t i e s of t h e a c t u a l s h e a r w a l l . The a t t a c h e d frame had t h e same d i m e n s i o n s as t e s t frame no. 1, e x c e p t 5 bays were u s e d i n s t e a d o f 8. The s t r u c t u r a l p r o p e r t i e s of t h e t e s t frame 76 a r e g i v e n i n T a b l e 6.1. The f u n d e r m a n t a l p e r i o d f o r t h e t e s t frame was 0.51 s e c . The p r o p e r t i e s shown i n t h e t a b l e c o r r e s p o n d t o one s i n g l e frame. The d u c t i l i t y a s s i g n e d t o columns and beams was, a g a i n , 2 and 6, r e s p e c t i v e l y . The C.I. T . S i m u l a t e d e a r t h q u a k e , C2, m o t i o n was u s e d a t a maximum a c c e l e r a t i o n of 0.3g. O n l y a v e r y s m a l l amount of s t r a i n h a r d e n i n g was a p p l i e d t o s i m u l a t e an e l a s t o - p l a s t i c c o n d i t i o n so t h a t t h e e f f e c t of s t r a i n h a r d e n i n g c an be i s o l a t e d . The r e a s o n f o r a p p l y i n g n o n - z e r o s t r a i n h a r d e n i n g was b e c a u s e , f o r t h e d e g r a d i n g s t i f f n e s s e l e m e n t , DRAIN-2D w i l l p r o d u c e an i n f i n i t e d i s p l a c e m e n t r e s p o n s e a s s t r a i n h a r d e n i n g goes t o z e r o . The r e s u l t s o b t a i n e d from EDAM2 and DRAIN-2D a r e shown i n F i g . 6.7 and 6.8. A g a i n , a l l t h e beams y i e l d e d whereas a l l t h e col u m n s r e m a i n e d e l a s t i c , e x c e p t a t t h e b a s e of t h e s h e a r w a l l , w h i c h was t h e p l a s t i c h i n g e a r e a of t h e w a l l . I t was a l s o t h e a r e a where t h e maximum e r r o r i n damage r a t i o was f o u n d . F i g . 6.8 shows t h a t t h e damage r a t i o s p r e d i c t e d by EDAM2 were g e n e r a l l y on t h e low s i d e , however, t h e t r e n d i n d i s t r i b u t i o n o f t h e damage was p r e d i c t e d s u c c e s s f u l l y . On th e o t h e r hand, t h e h o r i z o n t a l d i s p l a c e m e n t s o b t a i n e d from b o t h methods were i n c l o s e agreement as can be seen from F i g . 6.7. The maximum h o r i z o n t a l d i s p l a c e m e n t p r e d i c t e d by EDAM2 and DRAIN-2D was 2.17" and 2.48", r e s p e c t i v e l y . A summary of t h e t e s t r e s u l t s i s g i v e n i n T a b l e 6.2. I t can be se e n t h a t t h e change i n r e s p o n s e p e r i o d was s m a l l e r i n m a g n i t u d e t h a n t h e l a s t t e s t frame even when t h e s t r a i n h a r d e n i n g r a t i o i s l o w e r . T h i s i n d i c a t e d t h e s t i f f e n i n g 77 e f f e c t of u s i n g a s h e a r w a l l i n s t e a d of a b a r e frame. The e r r o r s shown i n T a b l e 6.2 were low f o r t h i s t e s t ; t h e l a r g e s t e r r o r i n h o r i z o n t a l d i s p l a c e m e n t and damage r a t i o was g e n e r a t e d a t t h e base of t h e s h e a r w a l l . 6.4.4 T e s t Frame No. 3 T e s t frame no. 3 i s a 7 - s t o r e y frame w i t h t h e upper l e f t p o r t i o n c u t o f f . T h i s t e s t frame i s u s e d t o examine t h e n e c e s s i t y f o r r e s t r i c t i n g a b r u p t c h a n g e s i n ge o m e t r y i n t h e h e i g h t of t h e s t r u c t u r e . The d i m e n s i o n s and y i e l d moments a r e shown i n F i g . 6.9 and t h e s t r u c t u r a l p r o p e r t i e s a r e shown i n T a b l e 6.1. The s t r u c t u r a l p r o p e r t i e s o f t h i s frame were e x a c t l y t h e same as t e s t frame no. 1. However, t h e f u n d a m e n t a l p e r i o d o f t h i s t e s t frame was l o w e r t h a n t h e f i r s t t e s t frame due t o t h e p a r t b e i n g c u t o f f . The r e s u l t i n g damage r a t i o s of t h i s t e s t a r e shown i n F i g . 6.10 and F i g . 6 . 1 1 ( a ) . In g e n e r a l , r e s u l t s o b t a i n e d from EDAM2 were c o n s e r v a t i v e e s p e c i a l l y a t t h e base o f t h e s t r u c t u r e . DRAIN-2D p r e d i c t s t h a t a l l columns s h o u l d r e m a i n e l a s t i c , but EDAM2 p r e d i c t s t h a t t h e t o p l e v e l columns would s l i g h t l y e x c e e d t h e i r e l a s t i c l i m i t . D e s p i t e t h e d i s c r e p a n c y between t h e r e s u l t s , t h e t r e n d a s t h e damage r a t i o v a r i e s from f l o o r t o f l o o r was f a i t h f u l l y r e p r o d u c e d by EDAM2, as can be seen from F i g . 6 . 1 1 ( a ) . F i g . 6.11(b) shows t h e h o r i z o n t a l d i s p l a c e m e n t s of t h e t e s t f r a m e . The t i p d i s p l a c e m e n t was 3.25" and 2.76" as p r e d i c t e d by EDAM2 and DRAIN-2D, r e s p e c t i v e l y . From T a b l e 6.2, t h e maximum e r r o r s were a b o u t n o r m a l , 78 a t l e a s t n o t as h i g h a s t h e f i r s t t e s t f r a m e . T h e s e e r r o r s o c c u r r e d m a i n l y i n t h e v a l u e s a t t h e base of t h e s t r u c t u r e . As shown by t h e r e s u l t s o f t h i s t e s t f r a m e, as w e l l a s th e l a s t two f r a m e s , i t i s a p p a r e n t t h a t t h e weakness of EDAM2 i s t h e i n a b i l i t y t o p r e d i c t t h e r e s p o n s e a t t h e base of a s t r u c t u r e a c c u r a t e l y . B u t , on t h e o t h e r hand, EDAM2 i s v e r y s u c c e s s f u l i n p r e d i c t i n g t h e p a t t e r n o f damage t h a t o c c u r s i n a s t r u c t u r e and i t i s a l s o a b l e t o p r e d i c t base s h e a r a c c u r a t e l y w i t h i n 10 p e r c e n t e r r o r . T h i s ends t h e f i r s t p h a s e of c a l i b r a t i n g EDAM2 a g a i n s t DRAIN-2D w i t h t h e t h r e e b a s i c t e s t f r a m e s . 6.4.5 T e s t Frame No. 4 T e s t frame no. 4 i s a c t u a l l y t h e same as t e s t frame no. 1. The d i f f e r e n c e i n t h i s t e s t was i n t h e e a r t h q u a k e r e c o r d , where t h e C . I . T . s i m u l a t e d e a r t h q u a k e r e c o r d was r e p l a c e d by t h e San F e r n a n d o e a r t h q u a k e . A maximum gro u n d a c c e l e r a t i o n o f 0.2g was a p p l i e d . T h i s i s t o t e s t t h e a b i l i t y o f EDAM2 t o p r o d u c e c o n s i s t e n t r e s u l t s w i t h d i f f e r e n t e a r t h q u a k e s p e c t r a . S i n c e t h e San F e r n a n d o e a r t h q u a k e r e c o r d p r o d u c e s a v e r y j a g g e d r e s p o n s e s p e c t r u m , a number o f d i f f e r e n t smooth s p e c t r a c a n be c h o s e n . T h e r e f o r e , i t can a l s o t e s t t h e s e n s i t i v i t y o f t h e method t o d i f f e r e n t smooth s p e c t r a r e p r e s e n t i n g t h e same r e c o r d . F i g . 6.12 shows t h e damage r a t i o s and h o r i z o n t a l d i s p l a c e m e n t s , and F i g . 6.13 shows a p l o t o f t h e beam damage r a t i o v e r s u s f l o o r l e v e l . The agreement of t h e r e s u l t s was v e r y good, a l t h o u g h r e s u l t s o b t a i n e d f r o m EDAM2 were 79 s l i g h t l y c o n s e r v a t i v e . V a l u e s of maximum e r r o r g i v e n i n T a b l e 6.2 a l s o r e f l e c t e d s i g n i f i c a n t improvement o v e r t h e f i r s t t e s t . T h e r e f o r e , c o n s i s t e n t r e s u l t s f r o m t h e p r e s e n t method a r e a t t a i n a b l e i f t h e i n p u t r e s p o n s e s p e c t r u m i s a p p r o p r i a t e . 6.4.6 T e s t Frame No. 5 T e s t frame no. 5 i s t h e s h e a r - w a l l frame t h a t was m e n t i o n e d i n S e c t i o n 3.3. A d u c t i l i t y of 4 was a s s i g n e d t o b o t h beams and c o l u m n s . T h i s w i l l a f f e c t t h e y i e l d moments of t h e t e s t frame whereas o t h e r p r o p e r t i e s , i n c l u d i n g t h e e l a s t i c r e s p o n s e p e r i o d , w i l l r e m a i n t h e same as t e s t frame no. 2. The y i e l d moment of e a c h member i s shown i n F i g . 6.15. The C . I . T . s i m u l a t e d e a r t h q u a k e m o t i o n was u s e d a g a i n i n t h i s t e s t . However, t h e maximum g r o u n d a c c e l e r a t i o n a p p l i e d was 0.2g s i n c e t h e t e s t frame was n o t a b l e t o s u s t a i n 0.3g w i t h o u t a l a r g e amount of damage. The t e s t r e s u l t s i n f i g . 6.15 show t h a t most of t h e c o lumns had y i e l d e d . A l a r g e p o r t i o n of t h e f o r c e a t t h e b ase was a b s o r b e d by t h e s h e a r w a l l and t h u s y i e l d i n g was a v o i d e d i n some base c o l u m n s . Beams c o n n e c t e d t o t h e s h e a r w a l l a l s o y i e l d e d . As seen from F i g . 6.15 and F i g . 6.16, t h e agreement between t h e r e s u l t s was o u t s t a n d i n g . However, p a s t e x p e r i e n c e has shown t h a t t h e p r e s e n t method t e n d s t o y i e l d b e t t e r r e s u l t s as l o n g as t h e damage of t h e i n p u t s t r u c t u r e s t a y s r e l a t i v e l y low; t h e r e f o r e , i t was not s u r p r i s i n g t h a t t h e s e r e s u l t s were o b t a i n e d . In t h i s t e s t , t h e b e s t agreement was i n t h e p r e d i c t i o n 80 of base s h e a r , where t h e match was a l m o s t e x a c t . The w o r s t p o i n t was a g a i n a t t h e base of t h e s h e a r w a l l . A l t h o u g h t h e d i f f e r e n c e i n damage r a t i o a t t h e base of t h e s h e a r w a l l was 0.8, t h e m a g n i t u d e of t h e damage r a t i o was o n l y about 3, compared t o ab o u t 6 i n t e s t frame no. 2; h e n c e , i n terms o f r e l a t i v e e r r o r , t h i s d i s c r e p a n c y was a c t u a l l y c o m p a r a b l e t o t h a t of t e s t frame no. 2. In c o n c l u d i n g t h i s p a r t i c u l a r t e s t , i t must be n o t e d t h a t t h i s t e s t frame i s an e x c e p t i o n a l c a s e , i n t h a t EDAM2 c o n v e r g e d s u c c e s s f u l l y . S i m i l a r c h a n g e s made t o t h e o t h e r two f r a m e s l e d t o t h e d e v e l o p e m e n t o f a s o f t s t o r e y , t h a t c a u s e d a c o n v e r g e n c e p r o b l e m . 6.4.7 T e s t Frame No. 6 T e s t frame no. 6 i s t h e same as t e s t frame no. 2. A l l t h e d e t a i l s were t h e same e x c e p t t h a t t h e s t r a i n h a r d e n i n g r a t i o f o r t h e t e s t frame was c h a n g e d from 0.075 p e r c e n t t o 1.5 p e r c e n t i n EDAM2 (from. 0.1 t o 2.0 p e r c e n t i n DRAIN-2D) i n o r d e r t o o b s e r v e t h e e f f e c t of i n c o r p o r a t i n g s t r a i n h a r d e n i n g i n t h e method. F i g s . 6.17 and 6.18 show t h e r e s u l t s o f t h i s t e s t . When c o m p a r i n g t h e s e two f i g u r e s w i t h F i g . 6.7 and 6.8, s e v e r a l d i f f e r e n c e s a r e n o t i c e d . F i r s t l y , t h e damage r a t i o s and t h e h o r i z o n t a l d i s p l a c e m e n t s a r e n o t i c e a b l y s m a l l e r w h i l e t h e base s h e a r , on t h e o t h e r hand, i s l a r g e r . S e c o n d l y , t h e agreement between t h e r e s u l t s i s not as c l o s e a s i n t h e s e c o n d t e s t , even t h o u g h EDAM2 was s t i l l a b l e t o p r e d i c t t h e d i s t r i b u t i o n of damage r a t i o s . F i n a l l y , t h e d i s c r e p a n c y 81 between beam damage r a t i o s i n t h e f i r s t f l o o r was more p r o n o u n c e d . T a b l e 6.2 i n d i c a t e s t h a t t h e i n e l a s t i c r e s p o n s e p e r i o d was s m a l l e r t h a n t e s t frame no. 2. I t a l s o shows a g e n e r a l i n c r e a s e i n maximum e r r o r . A l t h o u g h t h e c o m p a r i s o n was not as f a v o u r a b l e a s t h a t w i t h o u t s t r a i n h a r d e n i n g , t h e r e s u l t s s h o u l d be a c c e p t a b l e . 6.4.8 A C o m p a r i s o n Of R e s u l t s Between EDAM, EDAM2 And DRAIN-2D The f i n a l p hase o f t e s t i n g was t o a n a l y s e t h e t h r e e b a s i c t e s t f r a m e s a g a i n u s i n g t h e o r i g i n a l EDAM program and t o compare t h e r e s u l t s w i t h t h e new v e r s i o n , EDAM2, and DRAIN-2D. In o r d e r t o o b t a i n a f a i r c o m p a r i s o n , t h e e f f e c t o f s t r a i n h a r d e n i n g was i n c o r p o r a t e d i n t o EDAM. The r e s u l t s i n beam damage r a t i o s of t e s t frame no. 1 a r e g i v e n i n F i g . 6.19. T h e r e was a g e n e r a l s h i f t i n damage r a t i o , where r e l a t i v e l y h i g h e r damage r a t i o s were p r e d i c t e d by EDAM. The r e s u l t s f r o m EDAM compare more f a v o r a b l y t o DRAIN-2D t h a n EDAM2 on t h e f i r s t two f l o o r s , b u t i t was t o o c o n s e r v a t i v e i n o t h e r a r e a s . When t h e damage r a t i o s i n columns ( n o t shown) were compared, i t was o b s e r v e d t h a t t h e v a l u e s f r o m EDAM were a p p r o x i m a t e l y 4 p e r c e n t h i g h e r t h a n t h o s e f r o m EDAM2 a t t h e ba s e o f t h e t e s t frame and a b o u t t h e same i n o t h e r a r e a s . The n e x t t e s t was t o r e - a n a l y s e t e s t frame no. 2 by EDAM. F i g . 6.20 shows a p l o t o f beam damage r a t i o s o b t a i n e d from t h e t h r e e p r o g r a m s . A g a i n , r e s u l t s f r o m EDAM were a b i t 82 b e t t e r t h a n t h o s e o f EDAM2 a t t h e b a s e , but n o t as good a s EDAM2 anywhere e l s e on t h e f r a m e . A c o m p a r i s o n of column damage r a t i o s r e v e a l e d t h a t v a l u e s from EDAM were s i g n i f i c a n t l y l o w e r i n t h e g r o u n d f l o o r s . I t was p a r t i c u l a t e l y i m p o r t a n t t o n o t e t h e d i f f e r e n c e i n damage r a t i o s a t t h e base of t h e s h e a r w a l l . EDAM p r e d i c t s a damage r a t i o of 3.37 w h i c h i s v e r y low compare t o 5.67 from EDAM2 and 7.17 f r o m DRAIN-2D. T h i s d i s c r e p a n c y was caused- by a s s u m i n g e q u a l moment a t t h e two ends of t h e base column. F i n a l l y , t e s t frame no. 3 was a n a l y s e d by EDAM. R e s u l t s were shown i n F i g . 6.21. T h e r e was, a g a i n , a s h i f t i n beam damage r a t i o , as o c c u r r e d i n t h e l a s t two t e s t s , b ut t h i s t i m e t h e damage r a t i o s were t o o c o n s e r v a t i v e , e s p e c i a l l y a t t h e base of t h e t e s t f r a m e . Column damage r a t i o s were r a t h e r s i m i l a r t o t h o s e o b t a i n e d from EDAM2. T a b l e 6.2 shows t h e summary of t e s t r e s u l t s f r o m EDAM f o r t e s t frame no. 1', 2' and 3'. I t i s shown i n t h e t a b l e t h a t , i n u s i n g t h e new v e r s i o n , g e n e r a l l y , t h e r e s u l t i n g i n e l a s t i c r e s p o n s e p e r i o d i s s h o r t e r , whereas t h e smeared damping i s h i g h e r , t h a n t h a t o b t a i n e d from t h e p r e v i o u s v e r i s o n . The f o r m e r i s l i k e l y c a u s e d by t h e i n t r o d u c t i o n o f two damage r a t i o s t o a member, w h i c h c h a n g e s t h e s t i f f n e s s m a t r i x t h a t d e t e r m i n e s t h e r e s p o n s e p e r i o d , whereas t h e l a t t e r i s t h e e f f e c t o f u s i n g t h e new s e t o f damping e q u a t o n s ( s e e F i g . 4 . 2 ) . A l t h o u g h t h e e r r o r s i n base s h e a r a r e s m a l l f o r EDAM, t h e e r r o r s i n h o r i z o n t a l d i s p l a c e m e n t s and damage r a t i o a r e h i g h . A l s o from t h e s e c o m p a r i s o n s , EDAM2 seems t o p r o d u c e more c o n s i s t e n t r e s u l t s t h a n EDAM. 83 6 . 5 C o s t Of E x e c u t i o n The computer u s e d f o r a l l o f t h e above a n a l y s e s was an Amdahl V/6 I I w h i c h i s c o m p a r a b l e w i t h an IBM 360/370. The c o s t of e x e c u t i o n d i s c u s s e d i n t h e f o l l o w i n g p a r a g r a p h s r e f e r s t o t h e same c o m p u t e r . In o r d e r t o i s o l a t e o t h e r e f f e c t s , s u c h as d i f f e r e n t r a t e s c h a r g e d a t d i f f e r e n t t i m e s o f t h e day e t c . , t h e CPU t i m e i s u s e d f o r t h e f o l l o w i n g d i s c u s s i o n . The e x e c u t i o n c o s t o f EDAM2 depends on a number of f a c t o r s , s u c h as t h e number o f modes, number of d e g r e e s o f freedom, h a l f b a n d w i d t h of s t i f f n e s s m a t r i x and t h e number of i t e r a t i o n s r e q u i r e d f o r t h e program t o c o n v e r g e . F o r DRAIN-2D, t h e f a c t o r s a r e q u i t e s i m i l a r , b u t i n s t e a d of number o f modes and number o f i t e r a t i o n s , i t depends on t h e number o f t i m e s t e p s r e q u i r e d t o go t h r o u g h t h e c o m p l e t e d u r a t i o n of t h e i n p u t e a r t h q u a k e r e c o r d . In o t h e r words, a l o n g e r r e c o r d w i l l c o s t more, i f t h e s t e p s i z e i s b e i n g h e l d c o n s t a n t . A l s o , a s more members go i n t o t h e i n e l a s t i c r a n g e , t h e c o s t i n r u n n i n g DRAIN-2D i n c r e a s e s , b e c a u s e o f t h e more complex r o u t i n e u s e d t o d e a l w i t h i n e l a s t i c a c t i o n . Most o f t h e above f a c t o r s a r e summarized i n T a b l e 6.3. In u s i n g t h e C . I . T . s i m u l a t e d e a r t h q u a k e r e c o r d , o n l y t h e f i r s t 5 s e c o n d s out of t h e 12-second r e c o r d i s r e q u i r e d , s i n c e t h e maximum r e s p o n s e u s u a l l y o c c u r s w i t h i n t h e s e c o n d and t h e f o u r t h s e c o n d o f t h e r e c o r d . T h i s has g i v e n DRAIN-2D an a d v a n t a g e i n t h e c o s t c o m p a r i s o n . However, when t h e San 84 F e r n a n d o e a r t h q u a k e r e c o r d i s u s e d , t h e f i r s t 15 s e c o n d s of the r e c o r d a r e r e q u i r e d , w h i c h i n f a c t i s v e r y common among p o p u l a r e a r t h q u a k e r e c o r d s , s u c h a s E l C e n t r o o r T a f t . The CPU t i m e r e q u i r e d f o r EDAM2 t o c o m p l e t e t h e f o u r t h t e s t was 10.58 s e c . and i t took 159.05 s e c . f o r DRAIN-2D t o c o m p l e t e t h e a n a l y s i s on t h e same s t r u c t u r e . In a n o t h e r words, u s i n g t h e p r e s e n t method i n s t e a d o f a t i m e - s t e p a n a l y s i s would save up t o a p p r o x i m a t e l y 15 t i m e s i n c o m p u t i n g c o s t . By c o m p a r i n g t e s t s on fra m e s no. 3 and 6, i n T a b l e 6.3, i t i s seen t h a t DRAIN-2D i s v e r y s e n s i t i v e t o t h e number of d e g r e e s o f f r e e d o m i n t h e i n p u t s t r u c t u r e , w h i l e EDAM2 i s not a s s e n s i t i v e . The t a b l e a l s o shows t h a t t h e c o s t of r u n n i n g EDAM2 i s s l i g h t l y h i g h e r , by a p p r o x i m a t e l y 3 p e r c e n t , t h a n t h e c o s t of r u n n i n g t h e o r i g i n a l EDAM. As a f i n a l comment on e x e c u t i o n c o s t , i t i s a l s o n o t i c e d t h a t i f a c o n v e r g e n c e p r o b l e m a r i s e s , t h e c o s t of r u n n i n g EDAM2 might t r i p l e , d e p e n d i n g on t h e m a g n i t u d e of o s c i l l a t i o n o c c u r r i n g i n t h e r e s p o n s e p e r i o d ; a s m a l l o s c i l l a t i o n t a k e s fewer e x t r a i t e r a t i o n s t o c o n v e r g e , and w i l l r e s u l t i n a s m a l l e r i n c r e a s e i n e x e c u t i o n c o s t . 85 7.0 CONCLUSIONS A method, b a s e d on e l a s t i c modal a n a l y s i s , o f a n a l y s i n g t h e n o n - l i n e a r r e s p o n s e of 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 has been p r e s e n t e d . S i n c e t h e method i s c a p a b l e of t a k i n g i n t o a c c o u n t t h e r e d i s t r i b u t i o n o f f o r c e s due t o r e d u c t i o n i n member s t i f f n e s s , i t i s a b l e t o p r e d i c t t h e p a t t e r n o f damage t h a t a p l a n a r s t r u c t u r e would undergo 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 . A n a l y s e s o f t h r e e d i f f e r e n t t e s t s t r u c t u r e s , i n c l u d i n g two s t r u c t u r a l f r a m e s and a s h e a r - w a l l frame, u s i n g t h e p r e s e n t method have shown good agreement when compared w i t h t h e r e s u l t s o b t a i n e d from t h e t i m e - s t e p a n a l y s i s p r o g r am DRAIN-2D. In a l l c a s e s , t h e method i s a b l e t o i d e n t i f y t h e weak s p o t s and p r e d i c t t h e d e f l e c t e d shape o f t h e s t r u c t u r e . A t e s t frame w i t h a d r u p t change i n geometry a l o n g i t s h e i g h t does not c a u s e any p a r t i c u l a r p r o b l e m i n t h e method. I t i s a l s o f o u n d t h a t t h e method t e n d s t o work b e t t e r 'when t h e damage on t h e s t r u c t u r e i s s m a l l and when damage o c c u r s m a i n l y i n beams. A l t h o u g h i t shows a c e r t a i n weakness i n p r e d i c t i n g t h e m a g n i t u d e o f damage a t t h e b a s e of a s t r u c t u r e , i t i s v e r y good i n t h e p r e d i c t i o n of b a s e s h e a r . However, t h e r e a r e a l s o some o t h e r w e aknesses i n t h e method t h a t d e s e r v e s p e c i a l a t t e n t i o n . F i r s t l y , r e s u l t s d e t e r m i n e d u s i n g t h e p r e s e n t method a r e v e r y s e n s i t i v e t o t h e shape of 86 t h e smooth s p e c t r u m , i . e . , a s l i g h t change i n s p e c t r a l a c c e l e r a t i o n w i l l c a u s e s i g n i f i c a n t change t o t h e end r e s u l t s . S e c o n d l y , o s c i l l a t i o n s i n s p e c t r a l a c c e l e r a t i o n between i t e r a t i o n s w i l l c r e a t c o n v e r g e n c e p r o b l e m s . F i n a l l y , s t r u c t u r e s w i t h weak c o l u m n s w i l l a l s o c a u s e c o n v e r g e n c e p r o b l e m s ; h e n c e , i t i s recommended t h a t t h e method not be u s e d f o r t h i s t y p e o f s t r u c t u r e . In s p i t e of t h e s e w e a k n e s s e s , t h e p r o p o s e d method does p r o v i d e an e f f i c i e n t way of p e r f o r m i n g s e i s m i c a n a l y s i s . I t i s r e l a t i v e l y i n e x p e n s i v e t o u s e , and y e t , t h e r e s u l t s p r o d u c e d a r e c o m p a r a b l e t o t h o s e o f an 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 . I t i s a l s o a p r a c t i c a l s u b s t i t u t e f o r e l a s t i c modal a n a l y s i s i n t h e d e s i g n o f medium s i z e s t r u c t u r e s . 87 REFERENCES 1. Newmark, N.M., and H a l l , W.J. . 'A R a t i o n a l A p p r o a c h t o S e i s m i c D e s i g n S t a n d a r d s f o r S t r u c t u r e s ' , P r o c e e d i n g s , 5 t h W o r l d C o n f e r e n c e on E a r t h q u a k e E n g i n e e r i n g , Rome, 2, 1973, P.2266-2275. 2. S h i b a t a , A., and Sozen, M.A. ' S u b s t i t u t e - s t r u c t u r e Method f o r S e i s e 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 t u r a l D i v i s i o n , J a n u a r y 1976. 3. Y o s h i d a , S. ' M o d i f i e d S u b s t i t u t e S t r u c t u r e Method f o r A n a l y s i s of E x i s t i n g R/C S t r u c t u r e s ' , M a s t e r ' s t h e s i s , U n i v e r s i t y of B r i t i s h C o l u m b i a , 1979. 4. M e t t e n , A.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 Method 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 ' , M a s t e r ' s t h e s i s , U n i v e r s i t y o f B r i t i s h C o l u m b i a , 1981. 5. G u l k a n , P., and Soz e n , M.A. ' I n e l a s t i c R e s p o n s e 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 t o E a r t h q u a k e M o t i o n s ' , ACI J o u r n a l , December 1974. 6. Freeman, S.A. ' P r e d i c t i o n o f Response of C o n c r e t e B u i l d i n g s t o S e v e r e E a r t h q u a k e M o t i o n ' , SP 55, The D o u g l a s McHenry Symposium, 1978. 7. G r e e n , N.B. ' E a r t h q u a k e R e s i s t a n t B u i l d i n g D e s i g n and C o n s t r u c t i o n ' , Van N o s t r a d R e i n h o l d L t d . , 1981. 8. J a c o b s e n , L.S. 'Damping i n C o m p o s i t e S t r u c t u r e s ' , P r o c e e d i n g s , 2nd W o r l d C o n f e r e n c e on E a r t h q u a k e E n g i n e e r i n g (Tokyo,1960) S c i e n c e C o u n c i l of J a p a n , Tokyo, 1960, P.1029-1044. 9. C l o u g h , R.W., and P e n z i e n , J . 'Dynamics o f S t r u c t u r e s ' , M c G r a w - H i l l , I n c . , 1975. 88 10. B e a u f a i t , F.W. ' B a s i c C o n c e p t s of S t r u c t u r a l A n a l y s i s ' , P r e n t i c - H a l l , I n c . , 1977. 11. B e a u f a i t , F.W., Rowan, Jr.W.H., H o a d l e y , P.G., and H a c k e t t , R.M. 'Computer Methods o f S t r u c t u r a l A n a l y s i s ' , P r e n t i c e -H a l l , I n c . , 1970. 12. Kanaan, A.E. And Graham, H.P. ' D r a i n - 2 D A G e n e r a l P u r p o s e Computer Program f o r Dynamic A n a l y s i s o f I n e l a s t i c P l a n e S t r u c t u r e s . W i t h U s e r ' s G u i d e and S u p p l e m e n t ' , R e p o r t No. EERC 73-6 and EERC 73-22, U n i v e r s i t y of C a l i f o r n i a , B u r k e l e y , A u g u s t 1975. 13. J e n n i n g s , P.C., H o u s n e r , G.W., and T s a i , N.C. ' S i m u l a t e d E a r t h q u a k e M o t i o n s ' , C a l i f o r n i a I n s t i t u t e o f T e c h n o l o g y , P a s a d e n a , A p r i l . 1968. 14. Murphy, L.M. 'San F e r n a n d o , C a l i f o r n i a , E a r t h q u a k e o f F e b r u a r y 9, 1971', U.S. Department of Commerce, 1973, P.359-422. 89 M B A B k 3 3 k 6 3 Me 1 1 1 0000 1 0000 4 .0000 2.OOOO 1 0000 1 2 3 4755 0 1516 .3.9997 1.1516 1 7367 1 3 38 7880 0 0826 3.9950 0 .8298 2 4103 1 4 537 4277 0 0570 3.9085 0 .6472 3 0904 1 5 764 3048 0 0435 3.8459 0 .5357 3 7335 1 6 876 8327 0 0352 3.8039 0 .4614 4 3350 2 1 0 1516 3 4755 2 .0000 1.1516 1 7367 2 2 0 1755 0 1755 2 .0000 0 .6393 3 1284 2 3 0 1861 0 0903 2 .0000 0.4532 4 4 132 2 4 0 1921 0 0608 2 .0000 0 .3566 5 6085 2 5 0 1959 0 0458 2 .0000 0.2974 6 7244 2 6 0 1987 0 0368 2.0000 0.2574 7 7691 3 1 0 0826 38 7882 1.3333 0.8298 2 4103 3 2 0 0903 0 1861 1.3333 0.4532 4 4132 3 3 0 0934 0 0934 1.3333 0 .3187 6 2755 3 4 0 0950 0 0623 1.3333 0.2496 8 0138 3 5 0 0961 0 0468 1.3333 0.2074 9 64 12 3 6 0 0968 0 0374 1 .3333 0:1791 1 1 1682 4 1 0 0570 537 8186 1.0000 0 .6472 3 0904 4 2 0 0608 0 1921 1.0000 0 .3566 5 6085 4 3 0 0623 0 0950 1 .0000 0.2496 8 0138 4 4 0 0631 0 0631 1.0000 0.1948 10 2651 4 5 0 0636 0 0473 1 .0000 0.. 1616 12 3774 4 6 0 0640 0 0378 1.0000 0.1392 14 3636 5 1 O 0435 765 6 156 0 .8000 0 .5357 3 7335 5 2 0 0458 0 1959 0 .8000 0.2974 6 7244 5 3 0 0468 0 096 1 0.8000 0.2074 9 64 12 5 4 0 0473 0 0636 0 .8000 0 .1616 12 3774 5 5 0 0476 0 0476 0 .8000 0. 1338 14 9501 5 6 0 0478 0 0380 0 .8000 0.1151 17 3738 6 - 1 0 0352 879 0687 0 .6667 0.4614 4 3350 6 2 0 0368 0 1987 O.6667 0.2574 7 769 1 6 3 0 0374 0 0968 0 .6667 0.1791 1 1 1682 6 4 0 0378 0 0640 0 .6667 0.1392 14 3633 6 5 0 0380 0 0478 0 .6667 0.1151 17 3737 6 6 0 0381 0 0381 0 .6667 0 .0989 20 2 144 EI assume = 1 L T a b l e 5.1 V a l u e s f o r t h e e f f e c t i v e damage r a t i o , ve 90 TEST FRAME NO • 1,4 2 ,5 ,6 3 FUNDERMENTAL ( s e c . ) PERIOD 0.778 0.508 0.672 FLOOR WEIGHT ( k i p ) LEVEL LEVEL LEVEL LEVEL 1 2-4-6-4 6 7 1830.0 1460.0 1460.0 1410.0 393.4 340.6 340.6 259.5 1830.0 1460.0 730.0 705.0 YOUNG'S MODULUS ( k s i ) - 3300 3300 3300 COLUMN : EXT. INT. EXT. INT. EXT. INT. AREA ( i n . ) LEVEL LEVEL LEVEL o-1 -2-1 2 7 1264 1534 1186 1279 1120 1208 316 316 297 297 297 297 1264 1534 1186 1279 1120 1208 MOMENT OF INERTIA ( i n 4 ) LEVEL LEVEL LEVEL o-1-2-1 2 7 43943 64089 43434 47653 41020 45004 7847 7847 7756 7756 7756 7756 43943 64089 43434 47653 41020 45004 DUCTILITY 2 2 (2 ,6 )/4 (5 ) 2 BEAM : AREA ( i n . ) LEVEL 1 -7 900000 900000 900000 MOMENT OF INERTIA ( i n " ) LEVEL LEVEL LEVEL 1 2-7 6 190174 89534 79251 69409 29874 2721 0 190174 89534 79251 DUCTILITY 6 6 (2 ,6 )/4 (5 ) 6 SHEAR WALL : AREA ( i n . ) LEVEL o- 7 - 2250 -MOMENT OF INERTIA ( i n 4 ) LEVEL o- 7 - 4746000 -T a b l e 6.1 S t r u c t u r a l p r o p e r t i e s f o r a l l t e s t f r a m e s TEST FRAME NO. 1 2 3 4 5 6 1 ' 2' 3' MAX. ACCELARATION (g) 0.3 0.3 0.3 0.2 0.2 0.3 0.3 0.3 0.3 STRAIN HARDENING (%) 1 .50 0.075 0.075 1 .50 0.075 1 .50 1 .50 0.075 0.075 RESPONSE PERIOD ( s e c . ) 1.141 0.725 0.892 1 .238 0.578 0.689 1.180 0.767 0.918 SMEARED DAMPING 0.10 0.093 0.081 0.11 0.053 0.092 0.091 0.089 0.079 SPECTRAL ACCELARATION 0.208 0.295 0.288 0.228 0.286 0.304 0.214 0.294 0.285 MAXIMUM ERROR : BASE SHEAR (%) 7.7 3.1 6.4 4.6 0.02 4.3 4.8 3.6 5.6 DISPLACEMENT (%) 36. 1 20.8 30.3 8.7 14.0 30.4 27.4 47.3 42.7 DAMAGE RATIO ( a b s . ) 1 .67 1 .50 1 .44 0.67 0.8 1 .64 1 .82 3.80 2.62 T a b l e 6.2 Summary of r e s u l t s TEST FRAME NO. 1 2 3 4 5 6 1 ' 2' 3' NUMBER OF MODES 4 4 4 4 4 4 4 4 4 NUMBER OF DEGREES OF FREEDOM 1 33 91 109 133 91 91 133 91 109 HALF BANDWIDTH OF STIFFNESS MATRIX 38 26 38 38 26 26 38 26 38 NO. OF ITERATIONS (EDAM, EDAM2) 8 10 7 9 6 7 8 9 9 NO. OF TIME STEPS (DRAIN-2D) 400 400 400 1500 480 400 400 400 400 EXECUTION TIME : (CPU s e c . ) EDAM - - - - - - 9.22 5.80 8.20 EDAM2 9.51 6.57 6.64 10.58 4.09 4.70 - - -DRAIN-2D 48.90 26.41 33.84 159.05 32.97 26.55 48.90 26.41 33.84 T a b l e 6.3 Summary of program e x e c u t i o n c o s t 93 94 START ± MASS CONTRL SETUP s e t c o n t r o l v a r i a b l e s MAIN PROGRAM • r e a d i n p r o g r a m s p e c i f i c a t i o n s , e . g . maximum number o f i t e r a t i o n s , i n i t i a l damping, maximum a c c e l e r a t i o n , t i t l e e t c . • r e a d i n and o r g a n i z e s t r u c t u r a l i n f o r m a t i o n • c a l c u l a t e number of D.O.F. and h a l f b a n d w i t h • i n i t i a l i z e damage r a t i o s t o 1.0 • s e t up mass m a t r i x r — BUILD • s e t up s t i f f n e s s m a t r i x SCHECK c h e c k t h e c o n d i t i o n of t h e s t i f f n e s s m a t r i x EIGEN SPRIT compute modal p e r i o d s and mode s h a p e s • p e r f o r m modal a n a l y s i s : - d e t e r m i n e smeared damping f a c t o r s - c a l c u l a t e RSS f o r c e s and d i s p l a c e m e n t s - m o d i f i e d damage r a t i o s ( d e t a i l r e f e r t o F i g . 2.3 ) F i g . 2.2 F l o w c h a r t f o r t h e main p r o g r a m 95 Ma i n S u b r o u t i n e - M0D3 START c a l c u l a t e modal p a r t i c i p a t i o n f a c t o r s r ^ c a l c u l a t e smeared v damping f a c t o r s s e t KK = 1 I mode no. = 1 SPECTR • 1 c a l c u l a t e modal f o r c e s modal p e r i o d SDFBAN DSBAND I • c a l c u l a t e d i s p l a c e m e n t s f o r e a c h mode n e x t mode sum t h e s q u a r e o f d i s p l a c e m e n t s 1 FORCE • c a l c u l a t e f o r c e s from d i s p l a c e m e n t s f o r e a c h mode • f o r KK=1 : c a l c u l a t e smeared damping f a c t o r compute RSS d i s p l a c e m e n t s / c a l c u l a t e > (RSS f o r c e s / s e t KK = 2 DAMOD • m o d i f y damage r a t i o s • c a l c u l a t e member damping f a c t o r s • p r e p a r e f o r c o n v e r g e c h e c k • compute RSS f o r c e s RETURN F i g . 2.3 F l o w c h a r t f o r t h e main s u b r o u t i n e - M0D3 96 PERIOD F i g . 3.1 A smoothed r e s p o n s e s p e c t r u m w i t h a l l t y p i c a l b r a n c h e s ( s e c . 0.5 J F i g . 3.2 An example t o show t h e f l u c t u a t i o n i n r e s p o n s e p e r i o d v s . i t e r a t i o n number 1 1 1 1 1 1 1 1 1 I 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 PERIOD ( s e c . ) F i g . 3.3 G r a p h s h o w i n g t h e p r o g r e s s of e a c h i t e r a t i o n p l o t t e d w i t h t h e r e s p o n s e s p e c t r u m 9 8 F i g . 3.4 A p l o t of s p e c t r a l a c c e l e r a t i o n v s . p e r i o d s h o wing t h e c a p a c i t y and demand c u r v e ( Freeman's method ) 0.5 0.6 U./ PERIOD ( s e c . ) F i g . 3.5 . R e s u l t s f o r the f o u r t h and f i f t h i t e r a t i o n M i Q i B ) -—> 0) 0) 3 t a n 3 (1 001 101 START Main S u b r o u t i n e - MOD3 c a l c u l a t e modal p a r t i c i p a t i o n f a c t o r s n e x t mode no s e t KK = 1 1 mode no. = 1 y e s STACHK SPECTR y e s • d e t e c t o s c i l l a t i o n s v i b r a t i o n p e r i o d • s e a r c h f o r s o l u t i o n c a l c u l a t e modal f o r c e s SDFBAN DSBAND sum t h e s q u a r e of d i s p l a c e m e n t s FORCE s e t KK = 2 DAMOD RETURN F i g . 3.8 A r e v i s e d f l o w c h a r t f o r s u b r o u t i n e MOD3 i n c l u d i n g a new s u b r o u t i n e STACHK 102 I s e t upper and lower bound Sa (@> 2% damping) s e t I FLAG = 0 c a l c u l a t e mean Sa as guess (@ 2% damping) modify Sa w i t h appropr i a t e damp i ng RETURN check convergence compare Sa w i t h Sa from spectrum yes c a l c u l a t e the di f f e r e n c e between two Sa's (SADIF) PRINT t r i a l no. i n b i n a r y s e a r c h 5 SAD IF no I s \ n o / d i f f e r e n c e i n ^ > Sa < 1 i m i ' a d j u s t lower or u p p e r b o u n d Sa ( 2 % damp i ng ) yes LOCK : 0 = f o l l o w normal r o u t i n e 1 = f o l l o w s e a r c h r o u t i n e 2 = s e a r c h completed I FLAG : 0 = normal i t e r a t i o n 1 = l a s t i t e r a t i o n PRINT c a l c u l a t e Sa a c t u a l Sa & SADIF ICOUNT set I FLAG = 0 set LOCK = 2 i t e r a t i o n no. RETURN F i g . 3.9 F l o w c h a r t f o r t h e s o l u t i o n s e a r c h i n g r o u t i n e - STACHK 103 0.55 0.60 0.65 ( s e c . ) F i g . 3.10 G r a p h showing t h e p r o g r e s s of t h e s o l u t i o n s e a r c h i n g r o u t i n e 104 (a) P o r t a l Frame Under C o n s i d e r a t i o n MOMENT (b) ROTATION 6 VISCOUS DAMPING FORCE (c ) DISPLACEMENT A F i g . 4.1 (a) S i n g l e - d e g r e e o f fr e e d o m s y s t e m (b) Moment v s . r o t a t i o n d i a g r a m ( c ) V i s c o u s damping f o r c e v s . d i s p l a c e m e n t 105 " 0.15 H 0.10 1 0.05 i E q u a t i o n (4.10) ( w i t h 2% s t r a i n h a r d e n i n g ) ^ ' / E q u a t i o n (4.11) 0.02 0.00 5 6 DAMAGE RATIO u F i g . 4.2 A p l o t of s u b s t i t u t e damping f a c t o r v s . damage r a t i o AE K = xy 0 -x 2 -xy 0 o -xy - y 2 0 0 0 0 (a) A x i a l s t i f f n e s s matrix ~ y 2 -xy 12 EI L 5(i+h)tf SYMMETRICAL -y 2 xy xy 0 y 0 -yL 2/2 xL 2/2 L«(1+h/4)/3 xy yL 2/2 -xL 2/2 -yL 2/2 xL 2/2 L"(1-h/2)/6 SYMMETRICAL y 2 -xy x 2 yL 2/2 -xL 2/2 L"(1+h/4)/3 (b) Bending s t i f f n e s s matrix F i g . 5.1 (a) A x i a l s t i f f n e s s matrix (b) Bending s t i f f n e s s matrix 1 07 12 EI 0 0 -ay 0 0 -by 0 SYMMETRICAL ax aL(a+L) 0 ay 0 0 -ax 0 0 bx a b L + L 2 ( a + b ) / 2 by -bx b L 2 + b 2 L j F i g . 5.1 (c) R i g i d arm s t i f f n e s s m a t r i x J o i n t A J o i n t B 0.05L F i g . 5.2 A t y p i c a l member w i t h two h i n g e s 108 1 . (a) R e a l beam M, (b) Moment d i a g r a m f o r t h e r e a l beam (c ) C o n j u g a t e beam (d) E l a s t i c l o a d f o r t h e c o n j u g a t e beam F i g . 5.3 D i a g r a m s f o r t h e c o n j u g a t e beam method 109 T o t a l D i s t r i b u t e d E l a s t i c L o a d I I • • 0 0.05L 0.95L L F i g . 5.4 D i f f e r e n t components of t h e d i s t r i b u t e d e l a s t i c l o a d 1 10 1 2 3 4 5 6 DAMAGE RATIO uB F i g . 5.5 A p l o t of t h e e f f e c t i v e damage r a t i o v s . M a and MB 111 " 3 3 k 2 3 0 0 k 6 6 k 2 6 0 0 k 2 2 k 5 2 0 0 SYMMETRICAL 0 k, , 0 ^ ft i ^ ft ft 4EI M A L 2EI Me L K 3 3 + ^ 6 3 L k 3 3 + k 6 3 4EI M B L ^ 6 3 + ^ 6 6 L k 6 3 + k 6 6 SYMMETRICAL k 3 3 + 2 k 6 3 + k 6 6 k 3 3 + 2 k 6 3 + k 6 6 k 3 3 + 2 k 6 3 + k 6 6 AE L AE AE F i g . 5.6 S t i f f n e s s m a t r i x w i t h t e r m s of k 3 3 , k 6 6 and b e n d i n g s t i f f n e s s e s w r i t t e n k 6 3 i n + k b ] = 3 6 2 5 1 4 k 6 3 SYMMETRICAL Cyk 2 3 2 6 C x 2 k 2 2 + C y 2 k 1 1 C x k 5 3 C x 2 ^  5 2 + C y 2 k (, ! C x 2 k 5 5+C y 2 k (, u C yk 2 3 ~Cyk 2 6 C x C y k j i - C x C y k 2 2 C^Cykq j ~C X C y k 5 2 C x 2 k , i + C y 2 k 2 2 C y k 5 3 ~Cy k 5 6 C x C y k a , - C x C y k 5 2 C x C y k i, _ C x C y k 5 5 C x 2 k a , + C y 2 k 5 2 C x 2 k , t « + C y 2 ^  5 5 C„ = x/L c y = y/L F i g . 5. 7 A t r a n s f o r m e d s t i f f n e s s m a t r i x 1 2 3 4 5 6 ~ M 6 ( y 2 ) M 6 ( - x y ) M 6 ( X 2 ) SYMMETRICAL 1 2 EI M « ( - y L 2 / 2 ) M « ( x L 2 / 2 ) M , ( L V 3 ) L 5 M 6 ( - y 2 ) d 6 ( x y ) ju<,(yL 2/2) M 6 ( y 2 ) M 6 ( x y ) M 6 ( - x 2 ) M « ( - x L 2 / 2 ) U 6 ( - x y ) M 6 ( x 2 ) M 5 ( - y L 2 / 2 ) M 5 ( X L 2 / 2 ) <i 3(LV6) 5 ( y L 2 / 2 ) M 5 ( - X L 2 / 2 ) M 2 ( 3 L " ) F i g . 5. 8 B e n d i n g s t i f f n e s s m a t r i x w i t h s i x m o d i f i c a t i o n f a c t o r s 1 1 3 START MAIN PROGRAM CONTRL SETUP MASS s e t c o n t r o l v a r i a b l e s • DAMCAL - c a l c u l a t e f a c t o r s from Eqn. (5.13) t o Eqn. (5.18) • BUILD - a s s e m b l e b e n d i n g s t i f f n e s s m a t r i x i n F i g . 5.8 FI G . 5.9 A r e v i s e d f l o w c h a r t f o r t h e main p r o g r a m 1 14 START Main S u b r o u t i n e - M0D3 I c a l c u l a t e modal p a r t i c i p a t i o n f a c t o r s I s e t KK = 1 mode no. = 1 yes • STACHK • d e t e c t o s c i l l a t i o n s v i b r a t i o n p e r i o d s e a r c h f o r s o l u t i o n in y e s c a l c u l a t e modal f o r c e s SDFBAN DSBAND sum t h e s q u a r e of d i s p l a c e m e n t s n e x t mode FORCE DAMCAL • DAMCAL • FORCE no - c a l c u l a t e f a c t o r s f r o m Eqn. (5.13) t o (5.18) - r e p l a c e Eqn. (2.6) w i t h Eqn. (5.22) compute RSS d i s p l a c e m e n t s s e t KK = 2 DAMOD • r e p l a c e E qn. (4.10) w i t h E q n . ( 5 . 2 3 ) • m o d i f y two damage r a t i o s f o r e a c h member RETURN F i g . 5.10 A r e v i s e d f l o w c h a r t f o r s u b r o u t i n e MOD3 i n c l u d i n g a l l c h a n g e s 1 16 1.0" (0 CO z o n E-< < w o u < < « EH (J W CU co 0.5 0.0 ( n o r m a l i z e d t o 0.2g ) = 0.02 1.0 h0. 5 •0.0 0.0 0.4 0.8 1 .2 1.6 2.0 PERIOD (SEC.) SAN FERNANDO EARTHQUAKE 8244 O r i o n Ave. H o l i d a y I n n , g r o u n d f l o o r F i g . 6.2 San F e r n a n d o S90W s p e c t r u m 8 @ 225.0" = 1800" symmet r l ea l 440 530 350 520 350 520 350 520 520 O o 3 560 750 530 530 530 870 870 870 870 cn <a> O 4* cn 3 cn M to cn 3 740 1050 670 670 670 1380 1380 1380 1380 740 1050 670 670 670 1380 1380 1380 1380 840 1290 760 760 760 1830 1830 1830 1830 840 1290 760 760 760 1830 1830 1830 1830 1 150 2590 1000 1000 1000 3040 3040 3040 3040 TEST FRAME NO. 1 to o 3 ( YIELD MOMENT IN KIP-FT ) F i g . 6.3 D i m e n s i o n s and y i e l d moments f o r T e s t Frame No. 1 RSS BASE SHEAR (EDAM2) 2096.7 k i p s MAX. BASE SHEAR (DRAIN-2D) 2259.0 k i p s C.I.T. - TYPE C2 SIMULATED EARTHQUAKE MAXIMUM GROUND ACCELARATION 0.3g SYMMETRICAL 0.44 (0.35) 0. 37 (0.29) 0.64 (0.55) 0. 72 (0.60) 0.57 (0.50) 0.78 (0.68) 0. 58 (0.50) 0.77 (0.67) 0.77 (0.67) 0.97 (0.GG) 0.61 (0.39) 1 .05 (0.77) 0.84 (0.57) 1 .03 (0.74) 0.84 (0.60) 1 .02 (0.74) 0.84 (0.60) 0.84 (0.60) 1 . 78 (0.91) 0.57 (0.53) 1 .86 (0.93) 0.70 (0.64) 1 .86 (0.92) 0.68 (0.64) 1 .86 (0.92) 0.68 (0.64) 0.68 (0.64) 2.94 (2.37) 0.56 (0.72) 3.11 (2.46) 0.64 (0.84) 3.11 (2.46) 0.67 (0.67) 3.11 (2.46) 0.67 (0.67) 0.67 (0.67) 3 . 24 (3.56) 0.47 (0.64) 3 . 37 (3.72) 0.55 (0.68) 3.37 (3.72) 0.53 (0.66) 3.37 (3.72) 0.53 (0.66) 0.53 (0.66) 3.61 (4.56) 0.49 (0.57) 3.79 (4.85) 0.62 (0.69) 3.79 (4.85) 0. 57 (0.65) 3.79 (4.85) 0.57 (0.65) 0.57 (0.65) 4.83 (6.32) 0. 57 (0.67) 5.09 (6.77) 0. 75 (0.88) 5.09 (6.76) 0. 74 (0.87) 5.09 (6.76) 0.74 (0.87) 0.74 (0.87) u > w J K O O J (X4 7 n 6" 77777777777777777777777777777777777777 TEST FRAME NO. F i g . 6.4 DAMAGE RATIOS EDAM2 (DRAIN-2D) 3-2 -1 -+ EDAM2 • DRAIN-2D 0.0 Test Frame No. 1 1.0 2.0 3.0 4.0 HORIZONTAL DISPLACEMENT (IN.) -maximum base s h e a r , damage r a t i o s and h o r i z o n t a l d i s p l a c e m e n t s TEST FRAME NO. 1 ( S T A R T I N G FROM THE LEFT ) + EDAM2 • DRAIN-2D -7 FLOOR LEV •+ • + -6 5" • + • + • + • + -5 4- • + • + • + - 4 3- +• + • + • - 3 2- + • + • + • - 2 1 - + 1ST BAY (8TH BAY) • -2ND (7TH + BAY BAY) • 3RD (6TH BAY BAY) + •-4TH BAY (5TH BAY) + 4 • - 1 -0 0 H ( i l i i i 3 1 2 3 4 5 6 ( 3 1 i 1 2 3 4 5 6 ( ) 1 2 3 4 5 i 6 D < 1 i 1 1 2 3 4 1 5 i 6 DAMAGE RATIO H F i g . 6.5 Beam damage r a t i o on T e s t Frame No. 1 120 180 160 180 160 180 170 180 170 180 160 180 160 180 5 @ 225.0" = 1125" 1 10 270 160 270 160 270 150 270 150 270 140 270 140 270 1 10 270 160 270 160 270 150 270 150 270 140 270 140 270 1 10 290 160 290 160 290 150 290 150 290 140 290 140 290 230 400 300 400 300 400 310 400 310 400 4 IO 400 4 10 400 225.0" SHEAR WALL 6200 7100 8000 8800 9600 10500 1 1000 ( YIELD MOMENT IN KIP-FT ) TEST FRAME NO. 2 F i g . 6.6 D i m e n s i o n s and y i e l d moments f o r T e s t Frame No. 2 RSS BASE SHEAR (EDAM2) 635.9 k i p s MAX. BASE SHEAR (DRAIN-2D) 655.6 k i p s C.I.T. - TYPE C2 SIMULATED EARTHQUAKE MAXIMUM GROUND ACCELARATIN 0.3g 1.81 (2.31) O.G7 (0.67) 1 . 28 ( 1 .68) 0.83 (0.86) 1 . 26 (1.65) 0.81 (0.83) 1 .43 (1.83) 0.75 (0.77) 2.70 (2.96 ) 0.68 (0.80) 2 . 27 (2.46) 0.45 (0.51) 2.21 (2.39) 0.59 (0.61) 2 . 20 (2.37) 0.58 (0.63) 2.37 (2.58) 0.54 (0.56) 2.47 (2.59) 0.49 (0.58) 2 . 20 (2.42) 0. 50 (0.51) 1 .99 (2.18) 0.63 (0.64) 1 .97 (2 . 16) 0.61 (0.63) 2. 13 (2.37) 0.57 (0.58) 2.55 (2.68) 0.52 (0.60) 2 . 35 (2.53) 0.49 (0.54) 2.41 (2.59) 0.61 (0.62) 2.40 (2.57) 0.57 (0.59) 2.53 (2.77) 0. 54 (0.58) 2.53 (2.65) 0.50 (0.59) 2 . 32 (2.54) 0.48 (0.52) 2 . 24 (2.47) 0.58 (0.59) 2 . 23 (2.45) 0.54 (0.56) 2.37 (2.63) 0.50 (0.53) 2.54 (2.66) 0.60 (0.66) 2.44 (2.64) 0. 49 (0.64) 2.51 (2.72) 0.67 (0.73) 2.51 (2.70) 0.62 (0.70) 2.49 (2.86) 0.58 (0.63) 1 .85 (2.05) 0.67 (0.84) 4 . 49 (5.41) 0.69 (0.81) 3 . 40 (4.51 ) 0.55 (0.63) 3 . 40 (4.49) 0.54 (0.62) 3 . 42 (4.67) 0.50 (0.57) 3.63 (4.02) 0.41 (0.52) 7-TEST FRAME NO. 2 0.24 (0.28) 0.34 (0.33) 0.41 (0.36) 0.44 (0.33) 0.47 (0.46) 0.54 (0.61) 5.67 (7.17) DAMAGE RATIOS EDAM2 (DRAIN-2D) w > « O O t J 7 n 6-5-4-3-1-+ EDAM2 • DRAIN-2D F i g . 6.7 Test Frame No. 2 - maximum base s h e a r , damage r a t i o s and h o r i z o n t a l d i s p l a c e m e n t s 1 0.0 1.0 2.0 HORIZONTAL DISPLACEMENT (IN.) TEST FRAME NO. 2 ( STARTING FROM THE LEFT ) w « o o •J DM 64 5H 4H 3H 24 + EDAM2 • DRAIN-2D o + + • 1ST BAY T • + • + • + 2 3 4 2ND BAY T +• + • + 0 1 3 3RD BAY 4 0 1 ~T 2 3 + < 4TH BAY -r 0 2 + • 5TH BAY F i g . 6.8 Beam damage r a t i o on T e s t Frame No. 2 -i 1 1 1 1 1 i 3 4 0 1 2 3 4 DAMAGE RATIO \-6 r5 h4 h3 r2 to to 8 @ 225.0" = 1800" 510 510 510 510 390 700 700 700 710 700 710 1 140 710 1 140 710 1 140 880 1 135 880 1395 880 1395 880 1395 665 815 585 1350 585 1350 700 1350 755 1 135 755 1395 755 1395 820 1395 855 815 775 1350 775 1350 775 1350 775 1350 775 1395 775 1395 825 1395 925 995 825 1610 825 1610 825 1610 825 1610 825 1610 825 1610 825 1610 1305 1820 1115 2785 1115 2785 1115 2785 1115 2785 1115 2785 1115 2785 1305 2785 TEST FRAME NO. 3 .( YIELD MOMENT IN KIP-FT ) F i g . 6 . 9 D i m e n s i o n s and y i e l d moments f o r T e s t Frame No. 3 RSS BASE SHEAR (EDAM2) 2162.6 k i p s MAX. BASE SHEAR (DRAIN-2D) 2031.6 k i p s C . I . T . - TYPE C2 SIMULATED EARTHQUAKE MAXIMUM GROUND ACCELARATION 0.3g 0. 77 ( 0 .71 ) 1 .56 ( 0 . 9 2 ) 0. 70 ( 0 . 6 4 ) 1 . 28 ( 0 . 9 0 ) 0 .74 ( 0 . 6 9 ) 1 . 33 ( 0 . 9 3 ) 0. 77 ( 0 . 6 2 ) 1 .25 ( 0 . 8 7 ) 2 . 22 ( 1 .54) 0 .92 ( 0 .89 ) 2 .02 ( 1 .39) 0 .95 ( 0 . 96 ) 2 .02 ( 1 39) 0 .95 ( 0 . 94 ) 2 . 15 ( 1 .46 ) 0 .95 ( 0 .95 ) 2 .80 ( 2 .24 ) 0 .86 ( 0 . 87 ) 2.44 ( 1 . 9 1 ) 0.81 ( 0 . 8 3 ) 2 . 45 ( 1 . 9 2 ) 0.81 ( 0 . 8 3 ) 2 .65 ( 2 . 0 9 ) 0 .82 ( 0 .85 ) 1.81 (1 .54) 0 .82 ( 0 . 8 5 ) 1 . 53 (1 .26) 0 .93 ( 0 . 92 ) 1 .50 ( 1 26) 0 .88 ( 0 . 89 ) 2 . 35 (2 .04 ) 0 .97 (0 .95 ) 2 .40 ( 1 .97 ) 0. 55 ( 0 .71 ) 2.48 ( 2 . 0 3 ) 0 . 4 0 ( 0 . 5 1 ) 2.48 ( 2 . 0 4 ) 0 . 4 0 ( 0 . 5 1 ) 2 .40 ( 1 95) 0 .42" ( 0 . 54 ) 2 . 47 ( 2 .04 ) 0 . 72 ( 0 . 7 7 ) 2 . 54 (2 .10 ) 0. 70 ( 0 . 7 7 ) 2.58 ( 2 .13 ) 0 .67 ( 0 . 73 ) 2 . 59 ( 2 .15 ) 0 .66 (0 .73 ) 2.28 ( 1 .86) 0 .76 ( 0 .77 ) 2 . 29 ( 1 .85) O. 75 ( 0 .76 ) 2 . 29 ( 1 .85) 0 . 75 ( 0 . 7 6 ) 2 .25 ( 1 92) 0 .78 ( 0 . 78 ) 2 . 76 ( 2 . 0 6 ) 0.61 ( 0 . 6 7 ) 2 .84 (2 .10 ) 0 .69 ( 0 . 7 4 ) 2.84 ( 2 .10 ) 0 .63 ( 0 . 67 ) 2 .90 (2 . 14) 0 .63 (0 .67 ) 2.87 ( 2 .10 ) 0 .62 (0 .67 ) 2 .89 (2 .13 ) 0 .62 ( 0 . 6 7 ) 2 .89 ( 2 . 1 3 ) 0 .62 ( 0 .67 ) 2 . 77 ( 2 .05 ) , 0 . 6 9 ( 0 .73 ) 3.91 ( 2 .63 ) 0 .79 ( 0 . 6 7 ) 4.12 ( 2 .69 ) 0 . 8 0 ( 0 . 6 9 ) 4 .13 ( 2 .69 ) 0 .79 ( 0 .67 ) 4.14 ( 2 .70 ) 0 .79 ( 0 .67 ) 4 . 10 (2 .66 ) 0 .79 ( 0 . 67 ) 4.11 ( 2 .67 ) 0. 79 ( 0 . 6 7 ) 4.11 ( 2 .68 ) 0 . 7 9 ( 0 . 6 8 ) 3.88 ( 2 . 60 ) 0 . 8 0 ( 0 . 69 ) 1 .42 ( 0 . 8 2 ) 0 .96 ( 0 . 9 3 ) 0 .88 ( 0 . 8 8 ) 0 .37 ( 0 . 4 5 ) 0 . 8 0 ( 0 . 80 ) 0.61 ( 0 .66 ) 0 . 8 0 ( 0 . 67 ) TEST FRAME NO. 3 F i g . 6.10 T e s t Frame No. DAMAGE RATIOS 3 - maximum b a s e s h e a r and damage r a t i o s EDAM2 (DRAIN-2D) TEST FRAME NO. 3 •J > w K o o j 7 J ( STARTING FROM THE LEFT ) 5 H 2 H 1 -J + EDAM2 • DRAIN-2D • + • + 1ST BAY + + i i • + • + 2ND BAY + + T 1 1 P • + • + 3RD BAY • + • + • + • + 4TH BAY i i i i • + • + • + • + • + 5TH BAY i i h 6 h 5 h 4 r 3 h 2 0 1 2 3 4 0 1 2 3 4 0 1 2 3 4 0 1 2 3 4 0 1 2 3 4 DAMAGE RATIO M F i g . 6.11(a) Beam damage r a t i o on T e s t Frame No. 3 to TEST FRAME NO. 3 0 1 2 3 4 0 1 2 3 4 0 1 2 3 4 0.0 0.1 0.2 0.3 DAMAGE RATIO UL HORIZONTAL DISPLACEMENT (IN.) F i g . 6.11(a) Beam damage r a t i o on T e s t Frame No. 3 ( c o n t ' d ) (b) H o r i z o n t a l d i s p l a c e m e n t s RSS BASE SHEAR (EDAM2) 2117.9 k i p s MAX. BASE SHEAR (DRAIN-2D) 2214.4 k i p s SAN FERNANDO S90W EARTHQUAKE MAXIMUM GROUND ACCELARATION 0.2g SYMMETRICAL 0 . 2 9 ( 0 . 3 2 ) 0 . 2 4 ( 0 . 2 6 ) 0 . 5 1 ( 0 . 5 0 ) 0 . 5 2 ( 0 . 5 4 ) 0 . 4 4 ( 0 . 4 4 ) 0 . 6 1 ( 0 . 6 1 ) 0 . 4 4 ( 0 . 4 4 ) 0 . 6 0 ( 0 . 6 0 ) 0 . 6 0 ( 0 . 6 0 ) 0 . 7 0 ( 0 . 7 2 ) 0 . 4 7 ( 0 . 4 7 ) 0 . 8 2 ( 0 . 8 1 ) 0 . 6 9 ( 0 . 6 8 ) 0 . 7 8 ( 0 . 7 7 ) 0 . 7 0 ( 0 . 6 9 ) 0 . 7 8 ( 0 . 7 8 ) 0 . 7 0 ( 0 . 6 9 ) 0 . 7 0 ( 0 . 6 9 ) 1 . 4 9 ( 1 . 3 1 ) 0 . 6 2 ( 0 . 6 1 ) 1 . 5 5 ( 1 . 2 8 ) 0 . 7 5 ( 0 . 7 3 ) 1 . 5 5 ( 1 . 2 8 ) 0 . 7 2 ( 0 . 7 1 ) 1 . 5 5 ( 1 . 2 8 ) 0 . 7 2 ( 0 . 7 2 ) 0 . 7 2 ( 0 . 7 2 ) 3 . 3 0 ( 2 . 7 0 ) 0 . 6 9 ( 0 . 7 1 ) 3 . 5 0 ( 2 . 8 4 ) 0 . 8 1 ( 0 . 8 3 ) 3 . 5 0 ( 2 . 8 4 ) 0 . 7 9 ( 0 . 8 0 ) 3 . 5 0 ( 2 . 8 4 ) 0 . 7 9 ( 0 . 8 0 ) 0 . 7 9 ( 0 . 8 0 ) 4 . 2 1 ( 3 . 5 9 ) 0 . 5 4 ( 0 . 6 1 ) 4 . 4 3 ( 3 . 7 6 ) 0 . 6 1 ( 0 . 6 7 ) 4 . 4 3 ( 3 . 7 6 ) 0 . 5 9 ( 0 . 6 5 ) 4 . 4 3 ( 3 . 7 6 ) 0 . 5 9 ( 0 . 6 5 ) 0 . 5 9 ( 0 . 6 5 ) 4 . 7 9 ( 4 . 3 4 ) 0 . 5 8 ( 0 . 5 7 ) 5 . 0 9 ( 4 . 6 0 ) 0 . 7 0 ( 0 . 6 9 ) 5 . 0 9 ( 4 . 6 0 ) 0 . 6 6 ( 0 . 6 5 ) 5 . 0 8 ( 4 . 60.) 0 . 6 6 ( 0 . 6 5 ) 0 . 6 6 ( 0 . 6 5 ) 6 . 0 8 ( 5 . 7 0 ) 0 . 6 2 ( 0 . 6 3 ) 6 . 4 6 ( 6 . 0 7 ) 0 . 8 2 ( 0 . 8 3 ) 6 . 4 6 ( 6 . 0 6 ) 0 . 8 1 ( 0 . 8 2 ) 6 . 4 6 ( 6 . 0 6 ) 0 . 8 1 ( 0 . 8 2 ) 0 . 8 1 ( 0 . 8 2 ) w > w J K O O i J TEST FRAME NO. 4 F i g . 6.12 T e s t Frame No DAMAGE RATIOS ( D R A I N ? 2 D ) + EDAM2 • DRAIN-2D i 1 1 1 1 r 1.0 2.0 3.0 4.0 HORIZONTAL DISPLACEMENT (IN.) maximum base s h e a r , damage r a t i o s and h o r i z o n t a l d i s p l a c e m e n t s TEST FRAME NO. 4 ( STARTING FROM THE LEFT ) + EDAM2 • DRAIN-2D T • + • • + • + • + • + + 1ST BAY (8TH BAY) T * 4- • • + • + • + •+ + 2ND BAY (7TH BAY) -r • + • • + • + • + 3RD BAY (6TH BAY) • + • + • + 4TH BAY (5TH BAY) r-7 • + - i 1 i i i 2 3 4 5 6 DAMAGE RATIO u r 5 r 4 h3 V2 V 1 0 1 1 ' i i i 2 3 4 5 6 0 1 ~2 3^  4 5 6* 0 1 2 3 4 5 6 0 1 F i g . 6.13 Beam damage r a t i o on Te s t Frame No. 4 ~2 !5 I ^ ~ 142 8 9 2 1 8 8 9 2 1 8 8 9 2 1 8 8 9 2 1 8 8 9 2 0 4 8 9 2 0 4 8 9 5 @ 225.0" = 1125" 132 135 201 135 201 135 2 1 2 135 2 1 2 135 173 135 173 135 132 135 201 135 201 135 2 1 2 135 2 1 2 135 173 135 173 135 132 142 2 0 1 142 201 142 2 1 2 142 2 1 2 142 173 142 173 142 3 1 7 197 41 1 197 4 1 1 197 4 2 8 197 4 2 8 197 5 7 3 197 5 7 3 197 225.0' SHEAR WALL I I 1 6 2 0 0 7 100 8 0 0 0 8 8 0 0 9 6 0 0 1 0 5 0 0 1 1 0 0 0 T e s t Frame No. 5 ( YIELD MOMENT IN KIP-FT ) F i g . 6.14 D i m e n s i o n s and y i e l d moments f o r T e s t Frame No. 5 RSS BASE SHEAR (EDAM2) 599.0 k i p s MAX. BASE SHEAR (DRAIN-2D) 599.1 k i p s C.I.T. - TYPE C2 SIMULATED EARTHQUAKE MAXIMUM GROUND ACCELARATION 0.2g 0 .63 ( 0 . 65 ) 1.31 ( 1 .54) 0 .59 ( 0 .59 ) 1 .29 ( 1 • 40) 0 .50 ( 0 .54 ) 1 .33 ( 1 43) 0.63 (0 .63 ) 1 . 30 ( 1 .39) 1 .03 (1 .18 ) 1 .34 (1 .43 ) 0 .82 ( 0 . 8 4 ) 1 .03 ( 1 .23) 0.71 ( 0 .74 ) 1 .20 ( 1 .30) 0 .67 ( 0 .68 ) 1 . 20 (1 .30 ) 0.81 (0 .83 ) 1.21 ( 1 .29) 1 .01 (1 . 12) 1 . 17 (1 .26 ) 0 .82 ( 0 .84 ) 1.15 ( 1 32) 0.71 (0 .74 ) 1 . 27 ( 1 .35) 0 .68 ( 0 .69 ) 1 .27 ( 1 35) 0 .80 (0 .82 ) 1 .25 ( 1 .32) 1 .08 (1 .18 ) 1 . 24 (1 .32 ) 0 .83 ( 0 . 83 ) 1.21 ( 1 .36) 0 .67 ( 0 . 69 ) 1 . 30 ( 1 36) 0 .65 (0 .68 ) 1 .29 (1 .35 ) 0. 74 (0 .75 ) 1 . 24 (1 .31 ) 1 .08 • (1 .18 ) 1 .26 ( 1 32) 0 .84 ( 0 . 83 ) '1 . 16 ( 1 .31 ) 0 .66 ( 0 . 78 ) 1 .25 ( 1 .30) 0 .66 (0 .84 ) 1 . 23 ( 1 .29) 0 .72 (0 .73 ) 1 . 16 (1 .21 ) 1 . 10 ( 1 . 18) 1 .20 (1 .28 ) 0 . 9 0 ( 0 .88 ) 1.51 ( 1 .66 ) 0 . 8 0 ( 0 .78 ) 1 .31 ( 1 38) 0.84 (0 .84 ) 1 .31 (1 .39 ) 0 .85 (0 .84 ) 1 . 19 ( 1 .27) 0. 78 (0 .82 ) 1 . 26 (1 .21 ) 0 .79 ( 0 . 83 ) 0 .92 (1 .11 ) 0 .64 ( 0 .67 ) 0 .66 ( 0 .79 ) 0.75 (0 .80 ) 0 .66 (0 .79 ) 0 .57 (0 .58 ) 0 .62 (0 .73 ) 1 .42 ( 1 .60) 0 .59 (0 .69 ) 0. 17 ( O . 1 9 ) O . 25 ( 0 .27 ) 0. 27 ( 0 .29 ) 0. 29 ( 0 .28 ) 0 .39 ( 0 . 46 ) 0. 55 ( 0 .63 ) 2 . 74 (3 .54 ) TEST FRAME NO. 5 DAMAGE RATIOS EDAM2 (DRAIN-2D) > »j a o o j F i g . 6.15 Test Frame No. 5 - maximum base s h e a r , damage r a t i o s and h o r i z o n t a l d i s p l a c e m e n t s + EDAM2 • DRAIN-2D 1 1 0.0 1.0 2.0 HORIZONTAL DISPLACEMENT (IN.) TEST FRAME NO. 5 ( STARTING FROM THE LEFT ) + EDAM2 • DRAIN-2D 6-7- + • + • + • -j -Ed g5-6-o (J +• +• -(• • * 4-5- +• •f* • • • 3-4- + • • 4- • 2-3- + • • -»• • 1-2- •+• 0-1- + • + • + • +• + • + • 1ST ROW 2ND ROW 3RD ROW 4TH ROW 5TH ROW SHEAR WALL 4 , 1 , 1 , 1 , 1 , 1 1 T T 1 0 1 0 1 0 1 0 1 0 1 0 1 2 3 4 DAMAGE RATIO H F i g . 6.16 Column damage r a t i o on T e s t Frame No. 5 RSS BASE SHEAR (EDAM2) 646.9 k i p s MAX. BASE SHEAR (DRAIN-2D) 674.7 k i p s C . I . T . - TYPE C2 SIMULATED EARTHQUAKE MAXIMUM GROUND ACCELARATION 0.3g 1 .63 ( 2 . 2 9 ) 0 .66 ( 0 . 68 ) 1.17 ( 1 70) 0 . 8 0 ( 0 . 86 ) 1.14 (1 .67 ) 0 . 8 0 ( 0 .85 ) 1 . 30 ( 1 .85) 0 .74 ( 0 .77 ) 2 . 49 1 ( 2 .91 ) 0 .67 (0 .81 ) 2.08 ( 2 .44 ) 0 .46 ( 0 .51 ) 1 .99 (2 .35 ) 0 . 6 0 ( 0 . 61 ) 1 .98 (2 .34 ) 0. 58 (0 .61 ) 2 . 14 (2 .54 ) 0 .55 (0 .56 ) 2 .30 ' 1 ( 2 .57 ) 0 .50 ( 0 .58 ) 2 .02 ( 2 . 3 8 ) 0. 50 (0 .52 ) 1 . 82 ( 2 .16 ) 0 .63 ( 0 . 64 ) 1 .80 (2 .14 ) 0.61 (0 .62 ) 1 .96 (2 .33 ) 0 .57 (0 .58 ) 2 . 37 (2 .63 ) 0 .53 ( 0 .61 ) 2. 15 ( 2 .50 ) 0 . 5 0 " ( 0 . 55 ) 2.17 ( 2 .54 ) 0 .62 (0 .63 ) 2.17 (2 .52 ) 0. 59 (0 .59 ) 2 . 29 (2 .71 ) 0 .55 (0 .58 ) 2 . 36 ' ( 2 .62 ) 0.51 (0 .60 ) 2 . 10 ( 2 . 4 7 ) 0 .48 (0 .52 ) 2.01 (2.41 ) O. 58 (0 .59 ) 2.01 (2 .39 ) 0. 55 (0 .57 ) 2.13 ( 2 .56 ) 0.51 (0 .53 ) 2 . 35 (2 .61 ) 0 . 6 0 (0 .67 ) 2 . 23 (2 .57 ) 0.54 ( 0 .65 ) 2 . 26 (2 .63 ) 0.71 ( 0 .74 ) 2 . 26 (2 .61 ) 0 .67 (0 .70 ) 2.21 (2 .75 ) 0.62 (0 .63 ) 1.71 (2 .00 ) 0 .69 ( 0 .84 ) 3 . 54 ( 4 .78 ) 0 .63 ( 0 . 79 ) 2 . 42 ( 3 . 91 ) 0.51 (0 .61 ) 2.43 ( 3 .89 ) 0 .50 (0 .60 ) 2 . 34 (3 .99 ) 0 .47 ( O .56) 3. 19 (3 .78 ) 0 .38 ( 0 .50 ) 0. 23 ( 0 . 2 8 ) O. 33 ( 0 .33 ) 0 .39 ( 0 . 36 ) 0 .43 ( 0 . 32 ) 0 .48 ( 0 . 47 ) 0 .57 ( 0 . 61 ) 4 . 64 ( 6 .28 ) TEST FRAME NO. 6 F i g . 6.17 > u a o o >-j b 7 n 6 -5 -4 -3 " 2 -1 -DAMAGE RATIOS E D A M 2 (DRAIN-2D) T e s t Frame No. 6 - maximum base s h e a r , damage r a t i o s and h o r i z o n t a l d i s p l a c e m e n t s + EDAM2 • DRAIN-2D —> 1 1 1 0.0 1.0 2.0 HORIZONTAL DISPLACEMENT (IN.) TEST FRAME NO. 6 ( STARTING FROM THE LEFT ) + EDAM2 • DRAIN-+ • + • r + • r + • r + • -7 + • + • +• +• -6 + • + • +• +• - 5 + • +• + • +• -4 + • + • + • + • + • - 3 + • + • + • -2 + + • + • + • - 1 1ST BAY 2ND BAY 3RD BAY 4TH BAY 5TH BAY i i i i 1 -0 I I I I I I I I p I 1 1 1 1 1 I I > I 0 1 2 3 4 0 1 2 3 4 0 1 2 3 4 0 1 2 3 4 0 1 2 3 4 DAMAGE RATIO H F i g . 6.18 Beam damage r a t i o on T e s t Frame No. 6 TEST FRAME NO. 1 ( STARTING FROM THE LEFT ) a1-> i- «*A - «tA A EDAM -7 + EDAM2 w j • DRAIN-2D • + A - •+ A • •+ A - •+ A -6 FLO< 5" • + A • + A - • + A • + A -5 4 - • +• A • + A • + A • + A - 4 3- + • A + • A + • A + • A - 3 2- + « A + «A - 2 1 - + A - + A + «& + A - 1 1ST BAY 2ND BAY 3RD BAY 4TH BAY 0-(8TH I i BAY) i 1 i 1— (7TH 1 r BAY) ' i i i (6TH ' i i BAY) i I i (5TH 1 r BAY) r — 1 1 i ' -0 0 1 2 3 4 5 6 0 1 2 3 4 5 6 0 1 2 3 4 5 6 0 1 2 3 4 5 6 DAMAGE RATIO u F i g . 6.19 A c o m p a r i s o n o f beam damage r a t i o s f r o m EDAM, EDAM2 and DRAIN-2D - T e s t Frame No. 1 TEST FRAME NO. 2 ( STARTING FROM THE LEFT ) W > co j § 6H o j ELI 5H 3^ 24 + • A -r» A A + A« 1ST BAY - f o A +• A +• A +• A A + + 4 + 2ND BAY + • A 4* A +• A A t 3RD BAY + o A +• A +* A +• A + • A + +A •+ 4TH BAY 1 1 1 1 r +• is. A H» A A h7 r6 r5 h4 + • A 5TH BAY A EDAM + EDAM2 • DRAIN-2D r2 -i 1 1 1 1 " 1 1 1 1 ' r n 1 2 3 4 5 0 1 2 3 4 0 1 2 3 4 0 1 2 3 4 0 1 2 3 4 DAMAGE RATIO n F i g . 6.20 A c o m p a r i s o n of beam damage r a t i o s f r o m EDAM, EDAM2 and DRAIN-2D - T e s t Frame No. 2 co cn TEST FRAME NO. 3 w j a o o j 4-1 -O-l ( STARTING FROM THE LEFT ) • + A • + A • + A 1ST BAY i i i •+ A • + A • + A + A 2ND BAY • - 4 * • + A • + A + A - • 3RD BAY A EDAM + EDAM2 • DRAIN-2D •+ A • + A • + A • + A 4TH BAY -i 1 i i -4 -3 -2 -1 0 1 2 3 4 0 1 2 3 4 5 0 1 2 3 4 5 0 1 2 3 4 5 r „, > * , , , DAMAGE RATIO M F i g . 6.21 A c o m p a r i s o n of beam damage r a t i o s from EDAM, EDAM2 and DRAIN-2D - T e s t Frame No. 3 0 TEST FRAME NO. 3 ( STARTING FROM THE L E F T ) « - * • # - -7 • + A • + A • + A • + A - 6 • + A • + A • + A • +A - 5 •4- A • + A • + A • + A - 4 • + A • + 4 • + A • + A -3 • + A • + A • + A • + A -2 • + A • • + A - • + A • + A - 1 5TH BAY 6TH BAY 7TH BAY 8TH BAY A EDAM + EDAM2 • DRAIN-2D -0 f I I I I I > I I I I I I I I I I • 1 • • . I 0 1 2 3 4 5 0 1 2 3 4 5 0 1 2 3 4 5 0 1 2 3 4 6.21 A c o m p a r i s o n of beam damage r a t i o s f r o m EDAM, DAMAGE RATIO u EDAM2 and DRAIN-2D - T e s t Frame No. 3 ( c o n t ' d ) A-1 APPENDIX A USER'S MANUAL ELASTIC OR DAMAGE AFFECTED MODAL ANALYSIS VER. 2.0 Program Name : EDAM2 Programmed by : Lawrence H.Y. H u i Andrew W.F. M e t t e n Sumio Y o s h i d a The U n i v e r s i t y o f B r i t i s h C o l u m b i a , 1984 DISCLAIMER THE C I V I L ENGINEERING DEPARTMENT, FACULTY AND STAFF DO NOT GUARANTEE NOR IMPLY THE ACCURACY OR R E L I A B I L I T Y OF THIS PROGRAM OR RELATED DOCUMENTATION. AS SUCH, THEY CAN NOT BE HELD RESPONSIBLE FOR INCORRECT RESULTS OR DAMAGES RESULTING FROM THE USE OF THIS PROGRAM. IT IS THE RESPONSIBILITY OF THE USER TO DETERMINE THE USEFULNESS AND TECHNICAL ACCURACY OF THIS PROGRAM IN HIS OR HER OWN ENVIRONMENT. THIS PROGRAM MAY NOT BE SOLD TO A THIRD PARTY. PURPOSE The p u r p o s e of t h e p r o gram i s t o d e t e r m i n e t h e dynamic r e s p o n s e o f i n e l a s t i c t w o - d i m e n s i o n a l frame or s h e a r - w a l l s t r u c t u r e s u s i n g a m o d i f i e d modal a n a l y s i s . R e s u l t s of t h e a n a l y s i s w i l l i n c l u d e t h e p r e d i c t i o n o f a damage p a t t e r n i n terms of damage r a t i o s , a l s o t h e d i s p l a c e m e n t s and f o r c e s o f t h e m o d e l l e d s t r u c t u r e under a s p e c i f i c e a r t h q u a k e e x i t a t i o n . The p r o g r a m can p e r f o r m o r d i n a r y modal a n a l y s i s on any t w o - d i m e n s i o n a l s t r u c t u r e s as w e l l . PROGRAM RESTRICTIONS The f o l l o w i n g r e s t r i c t i o n s a p p l y t o BOTH e l a s t i c and damage a f f e c t e d a n a l y s i s . 1. The s y s t e m c a n be a n a l y s e d i n one v e r t i c a l p l a n e . 2. N o n - s t r u c t u r a l components a r e t o be d e s i g n e d 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 of t h e s y s t e m a s m o d e l l e d . 3. The p r o g r a m c a n n o t h a n d l e more t h a n one t y p e of m a t e r i a l d i r e c t l y . S h o u l d t h e u s e r d e s i r e t o t e s t a s t r u c t u r e t h a t c o n t a i n s more t h a n one t y p e o f m a t e r i a l , t h e members c o n s t r u c t e d o f t h e s e c o n d m a t e r i a l t y p e s h o u l d have t h e a r e a and i n e r t i a m u l t i p l i e d by t h e m o d u l a r r a t i o (E f o r t y p e 1 d i v i d e d by E f o r t y p e 2 ) . 4. The p r o g r a m a p p l i e s t h e same a c c e l e r a t i o n t o b o t h h o r i z o n t a l and v e r t i c a l masses. T h e r e f o r e v e r t i c a l masses s h o u l d not be i n c l u d e d ; s h o u l d masses be a t t a c h e d t o some o f t h e nodes t h e n o n l y mode shapes and f r e q u e n c i e s w i l l be computed c o r r e c t l y . T h i s r e s t r i c t i o n l i m i t s t h e p r o g ram t o t h e a n a l y s i s o f s t r u c t u r e s f o r w h i c h v e r t i c a l a c c e l e r a t i o n of t h e nodes i s n o t a s i g n i f i c a n t f a c t o r . F o r most s t r u c t u r e s w i t h masses on v e r t i c a l column l i n e s t h i s r e s t r i c t i o n w i l l n ot be a l i m i t i n g c o n s i d e r a t i o n . 5. The s t r u c t u r e must comply w i t h t h e d i m e n s i o n i n g r e q u i r e m e n t s o f EDAM2. A-3 The f o l l o w i n g r e s t r i c t i o n s a p p l y ONLY t o damage a f f e c t e d a n a l y s i s . 1. The m a t e r i a l s u s e d f o r t h e c o n s t r u c t i o n of t h e s t r u c t u r e must be c o n c r e t e . The d e v e l o p m e n t of t h e s t i f f n e s s d e g r e d a t i o n and damping f o r m u l a was done c o m p l e t e l y on c o n c r e t e members and t h e r e s e a r c h does not a p p l y t o s t e e l o r o t h e r n o n - c o n c r e t e m a t e r i a l s . 2. The members must be d e s i g n e d s u c h t h a t t h e y c a n w i t h s t a n d t h e damage r a t i o s imposed w i t h o u t under g o i n g b r i t t l e f a i l u r e . 3. The members a r e assumed t o be symmetric and have t h e same moment c a p a c i t y under b o t h p o s i t i v e and n e g a t i v e b e n d i n g moments. 4. The f u n d a m e n t a l p e r i o d s h o u l d be s u c h t h a t i t p l a c e s t h e s t r u c t u r e on a segment o f t h e s p e c t r u m w h i c h c a u s e s a d e c r e a s e i n t h e s p e c t r a l a c c e l e r a t i o n when t h e p e r i o d of t h e s t r u c t u r e i n c r e a s e s . 5. The s t r u c t u r e s h o u l d be so d e s i g n e d t h a t m a j o r i t y o f t h e damage w i l l o c c u r a t t h e beams w h i l e o n l y minor damage w i l l o c c u r a t t h e c o l u m n s . The p r o g ram has been d e v e l o p e d f o r Amdahl 470 V / 8 - I I , w h i c h i s IBM 360/370 c o m p a t i b l e , o p e r a t i n g under t h e M i c h i g a n T e r m i n a l System (MTS) of t h e 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 . I n o r d e r t o r u n t h e p r o g r a m i n a n o t h e r i n s t i t u t i o n w h i c h i s u s i n g a d i f f e r e n t c o m p u t i n g s y s t e m , t h e E i g e n v a l u e s o l v e r and t h e m a t r i x r o u t i n e s i n t h e p r o g r a m might have t o be r e p l a c e d . To o b t a i n t e r m i n a l o u t p u t of i n t e r m e d i a t e r e s u l t s d u r i n g program e x e c u t i o n , some F o r t r a n commands, w h i c h a r e u s e d o n l y i n MTS, have t o be c h a n g e d a l s o . CAPACITY LIMITATIONS The maximum d i m e n s i o n s of t h e s t r u c t u r e a r e : 150 members 150 j o i n t s 100 a s s i g n e d masses 10 modes ( t o t a l d e g r e e s o f freedom) x ( h a l f b a n d w i d t h ) < 8000 Th e s e d i m e n s i o n s c a n be change by a d j u s t i n g t h e DIMENSION s t a t e m e n t s i n t h e pr o g r a m , as w e l l a s t h e l i m i t s s t a t e d i n t h e main p r o g r a m . EXECUTION TIME The e x e c u t i o n t i m e f o r t h e p r o g r a m depends on t h e f o l l o w i n g i t e m s : 1. t h e number o f d e g r e e s of f r e e d o m i n t h e s t r u c t u r e 2. t h e h a l f - b a n d w i d t h of t h e s t i f f n e s s m a t r i x 3. t h e number of lumped masses 4. t h e number of modes i n c l u d e d i n t h e a n a l y s i s 5. t h e number o f i t e r a t i o n s r e q u i r e d t o c o m p l e t e t h e a n a l y s i s A-5 INPUT DATA PROBLEM INITIATION :. INELAS,NMODES,NPRINT,I SPEC,AMAX,DAMPIN FORMAT (4I 5 , 2 F 1 0 . 5 ) - ONE CARD INELAS: Maximum number o f i t e r a t i o n f o r i n e l a s t i c a n a l y s i s (NOTE 1) 0 = E l a s t i c modal a n a l y s i s NMODES: Number of modes (< 10) t o be i n c l u d e d i n t h e a n a l y s i s (NOTE 2) NPRINT: Number of modes f o r w h i c h d i s p l a c e m e n t s and f o r c e s w i l l be p r i n t e d I SPEC: I n p u t s p e c t r u m t y p e -1 = S p e c t r u m 'A' from S h i b a t a and Sozen 2 = S p e c t r u m 'B' from Y o s h i d a 3 = S p e c t r u m ' C from Y o s h i d a 4 = N a t i o n a l B u i l d i n g Code s p e c t r u m 5 = San F e r n a n d o e a r t h q u a k e S90W s p e c t r u m 6 = C . I . T . S i m u l a t e d e a r t h q u a k e t y p e C2 s p e c t r u m AMAX: Maximum g r o u n d a c c e l a r a t i o n (g) DAMPIN: E l a s t i c or i n i t i a l damping e x p r e s s e d as a f r a c t i o n of c r i t i c a l damping I I . T I T L E : T I T L E FORMAT (20A4) - ONE CARD Maximum l e n g t h i s 80 c h a r a c t e r s I I I . GENERAL STRUCTURAL INFORMATION : NRJ,NRM,E,G,HARD FORMAT (2I5,2F10.3,F10.8) - ONE CARD NRJ: Number o f j o i n t s i n t h e s t r u c t u r e NRM: Number of members i n t h e s t r u c t u r e E: Young's modulus ( k s i ) G: She a r modulus ( k s i ) 0 = n e g l e c t s h e a r d e f l e c t i o n HARD: S t r a i n h a r d e n i n g r a t i o , as a p r o t o r t i o n of i n i t i a l s t i f f n e s s A-6 IV. JOINT INFORMATION : JN,NDX,NDY,NDR,X,Y FORMAT (415,2F10.3) - ONE CARD/JOINT JN: Node number NDX: 0 = J o i n t i s f i x e d i n t h e x - d i r e c t i o n 1 = J o i n t i s f r e e t o move i n t h e x - d i r e c t i o n N = Same m o t i o n i n x - d i r e c t i o n a s node N NDY: 0 = J o i n t i s f i x e d i n t h e y - d i r e c t i o n 1 = J o i n t i s f r e e t o move i n t h e y - d i r e c t i o n N = Same m o t i o n i n y - d i r e c t i o n a s node N NDR: 0 = J o i n t i s not a l l o w t o r o t a t e 1 = J o i n t i s f r e e t o r o t a t e N = Same r o t a t i o n a s node N X: X - c o o r d i n a t e of t h e j o i n t ( f t . ) Y: Y - c o o r d i n a t e of t h e j o i n t ( f t . ) V. MEMBER PROPERTIES : MN,JNL,JNG,KL,KG,AREA,CRMOM,AV,BMCAP,EXTL,EXTG FORMAT (5I5,F8.2,F12.3,2F10.3,2F6.3) - ONE CARD/MEMBER MN: Member number JNL: The l e s s e r j o i n t number JNG: The g r e a t e r j o i n t number (NOTE 3) KL: 1 = Member i s f i x e d a t t h e l e s s e r j o i n t 0 = Member i s p i n n e d a t t h e l e s s e r j o i n t KG: 1 = Member i s f i x e d a t t h e g r e a t e r j o i n t 0 = Member i s p i n n e d a t t h e 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 l a r e a o f t h e member ( i n 2 ) CRMOM: Moment of i n e r t i a of t h e member ( i n " ) AV: Shear a r e a of t h e member ( i n 2 ) (NOTE 4) BMCAP: Y i e l d moment of t h e member ( k i p - f t . ) EXTG: R i g i d e x t e n s i o n on t h e l e s s e r j o i n t end of member ( f t . ) EXTL: R i g i d e x t e n s i o n on t h e g r e a t e r j o i n t end of member ( f t . ) (NOTE 5) A-7 V I . NUMBER OF JOINTS WITH LUMPED MASSES : NMASS FORMAT (15) - ONE CARD NMASS: Number of nodes t o w h i c h a w e i g h t i s a t t a c h e d V I I . SPECIFICATION FOR LUMPED MASSES : JN,WTX,WTY,WTR FORMAT (I5,3F10.0) " ONE CARD/JOINT WITH MASS JN: J o i n t number WTX: Weight i n x - d i r e c t i o n ( k i p ) WTY: Weight i n y - d i r e c t i o n ( k i p ) WTR: R o t a t i o n a l w e i g h t ( s e e P o i n t 4 i n p r o g r a m r e s t r i c t i o n s ) NOTES 1. ' INELAS i s t h e maximum number of i t e r a t i o n s t h a t t h e pro g r a m w i l l p e r f o r m i f c o n v e r g e n c e i s not a t t a i n e d w i t h i n t h i s l i m i t . A v a l u e o f 40 i s u s u a l l y s u f f i c i e n t . 2. I f t h e r e a r e l e s s t h a n NMODES d e g r e e s o f freedom t o w h i c h masses a r e a t t a c h e d , t h e n NMODES w i l l be s e t e q u a l t o t h e number of d e g r e e s o f f r e e d o m t o w h i c h masses a r e a t t a c h e d . 3. The o r d e r i n g o f t h e j o i n t numbers w i l l n o t a f f e c t t h e r e s u l t s p r o d u c e d . F o r e v e r y member t h e r e i s a j o i n t n u mbering t h a t w i l l c a u s e e i t h e r x or y d i s p l a c e m e n t s t o be p r i n t e d a s a n e g a t i v e number. The p r i n t i n g o f n e g a t i v e d i s p l a c e m e n t s o f t h e member s h o u l d n ot d i s t u r b t h e u s e r . 4. S i n c e t e s t i n g i s i n c o m p l e t e on t h e program's a b i l i t y t o h a n d l e s h e a r d e f l e c t i o n s , a z e r o v a l u e s h o u l d be a s s i g n e d t o AV a t t h i s s t a g e so t h a t s h e a r d e f l e c t i o n s w i l l n o t be computed. 5. At t h i s s t a g e of p r o g r a m d e v e l o p m e n t , t h e r i g i d arms can o n l y be a t t a c h e d t o h o r i z o n t a l members. The a t t a c h m e n t of r i g i d arms t o n o n - h o r i z o n t a l members w i l l r e s u l t i n t h e p r i n t i n g of an e r r o r message. A-8 OUTPUT UNITS UNIT 6 T h i s f i l e i s f o r d a t a o u t p u t from i n t e r m e d i a t e i t e r a t i o n s from i n e l a s t i c a n a l y s i s . N o t h i n g o f i m p o r t a n c e i s w r i t t e n i n o u t p u t u n i t 6, t h e r e f o r e p r i n t e d o u t p u t f o r t h i s f i l e i s g e n e r a l l y n o t r e q u i r e d . A l s o , p r i n t i n g of d a t a from u n i t 6 may r e s u l t i n a v e r y l e n g t h y o u t p u t . UNIT 7 I n p u t d a t a and r e s u l t s f o r e i t h e r e l a s t i c o r i n e l a s t i c a n a l y s i s a r e p r i n t e d t h r o u g h o u t p u t u n i t 7. I t a l s o c o n t a i n s t h e e r r o r message when p r o b l e m a r i s e s . Note t h a t f o r e l a s t i c a n a l y s i s , t h e i n p u t moment c a p a c i t i e s have l i t t l e p u r p o s e n e i t h e r w i l l t h e o u t p u t damage r a t i o s . UNIT 99 U n i t 99 i s a s s i g n e d by a F o r t r a n command s t a t e m e n t i n s i d e t h e program t o I/O u n i t SPRINT, from w h i c h e s s e n t i a l r e s u l t s o b t a i n e d a f t e r e a c h i t e r a t i o n a r e p r i n t e d on t h e t e r m i n a l d u r i n g p r o g r a m e x e c u t i o n . In t h e p r o c e s s , t h e u s e r s h o u l d be a b l e t o d e t e r m i n e whether c o n v e r g e n c e p r o b l e m has o c c u r r e d . OPERATING INSTRUCTION The f o l l o w i n g s t a t e m e n t c a n be u s e d t o c o m p i l e t h e s o u r c e p r o gram $RUN *FTN SCARDS=EDAM2 The o b j e c t code o f t h e p r o g r a m w i l l be s t o r e i n a t e m p o r a r y f i l e -LOAD and can be c o p i e d t o a permanent d i s k f i l e f o r l a t e r use i f n e c e s s a r y . The p r o g r a m can be e x e c u t e d by i s s u i n g t h e f o l l o w i n g command $RUN -LOAD 5=INPUT 6=~6 7=-7 where INPUT i s t h e i n p u t f i l e , and -6, -7 a r e t e m p o r a r y o u t p u t f i l e s . I f o u t p u t from u n i t 6 i s n o t r e q u i r e d by t h e u s e r , a dummy f i l e s h o u l d be a s s i g n e d t o t h e u n i t so t h a t t h e p r o g r a m c a n run more e f f i c i e n t l y and a l s o s a v e t h e c o s t o f u s i n g more v i r t u a l memory. T h i s c a n be done by t h e f o l l o w i n g command $RUN -LOAD 5=INPUT 6=*DUMMY* 7=-7 B-1 APPENDIX B SAMPLE INPUT/OUTPUT F i g . B.1 I l l u s t r a t i v e example B-3 I . SAMPLE INPUT 20 4 1 6 0 .20000 0 .02000 SAMPLE INPUT/OUTPUT 18 23 3300 000 0 .0 0.0200O0OO 1 0 0 0 0 .0 0 .0 2 0 0 0 18 .000 0 .0 3 0 0 0 36 .000 0 .0 4 0 0 0 54 .000 0 .0 5 0 0 0 81 .ooo 0 .0 6 1 1 1 0 .0 10 .000 7 6 1 1 18 .000 10 .000 8 6 1 1 36 .000 10 .000 9 6 1 1 54 .000 10 .000 * 10 6 1 1 81 .000 10 .000 1 1 1 1 1 0 .0 20 .000 12 1 1 1 1 18 .000 20 .000 13 1 1 1 1 36 .000 20 .000 14 1 1 1 1 54 .000 20 .000 15 1 1 1 1 81 .000 20 .000 16 1 1 1 36 .000 30 .000 17 16 1 1 54 .000 30 .000 18 16 1 1 81 .000 30 .000 1 1 6 1 1 320 00 7850 000 0 0 40 000 0 0 0 0 2 2 7 1 1 320 00 7850 000 0 0 50 000 0 0 0 0 3 3 8 1 1 320 00 7850 000 0 0 50 000 0 0 0 0 4 4 9 1 1 320 00 7850 000 0 0 50 000 0 0 0 0 5 5 10 1 1 2000 00 4000000 000 0 0 9000 000 0 0 0 0 6 6 1 1 1 1 290 00 7750 000 0 0 40 000 0 0 0 0 7 7 12 1 1 290 00 7750 000 0 0 50 000 0 0 0 0 8 8 13 1 1 290 00 7750 000 0 0 50 000 0 0 0 0 9 9 14 1 1 290 00 7750 000 0 0 50 000 0 0 0 0 10 10 15 1 1 2000 00 4000000 000 0 0 8500 000 0 0 0 0 1 1 13 16 1 1 290 oo 7750 000 0 0 50 000 0 0 0 0 12 14 17 1 1 290 00 7750 000 0 0 60 000 0 0 0 0 13 15 18 1 1 2O00 00 4000000 000 0 0 8000 000 0 0 0 0 14 6 7 1 199999 00 68000 000 0 0 40 000 0 0 0 0 15 7 8 1 199999 00 68000 000 0 0 30 000 0 0 0 0 16 8 9 1 199999 00 68000 000 0 0 30 000 0 0 0 0 17 9 10 1 199999 00 68000 000 0 0 40 000 0 0 0 0 18 1 1 12 1 199999 00 30000 ooo 0 0 30 000 0 o 0 0 19 12 13 1 199999 00 30000 000 0 0 30 000 0 0 0 0 20 13 14 1 199999 00 30000 000 0 0 40 000 0 0 0 0 21 14 15 1 199999 00 30000 000 0 0 40 000 0 0 0 0 22 16 17 1 199999 00 30000 000 0 0 40 000 0 0 0 0 23 17 18 1 199999 00 30000 000 0 0 40 000 0 0 0 0 J 8 330. 0. 0. 13 280. 0. 0. 17 220. 0. 0. B-4 I I . OUTPUT FROM TERMINAL $RUN EDAM2RC 5=SAMPLE 6 = *DUMMY:> 7 = -7 E x e c u t i o n b e g i n s ITERATION NO. ABOVE DAMDIF CAPACITY NO . 1 2 3 4 5 6 7 8 9 10 S MATRIX RATIO 17 5 9 9 9 5 2 O 0 O 0. 803 O. 107 0.078 0.059 0.055 0.039 0.021 0.007 0.757E+03 O.762E+03 0.761E+03 0.760E+03 0.057 0.760E+03 0.035 0.761E+03 0.761E+03 O.761E+03 0.761E+03 0.761E+03 SMEARED DAMPING < 0.02000 < 0.03545 < 0.03823 < 0.03913 < 0.03963 < 0.03993 < 0.04013 < 0.04026 < 0.04033 < 0.04040 PERIOD PERIOD PERIOD PERIOD PERIOD PERIOD PERIOD PERIOD PERIOD PERIOD 0.172 SA 0.187 SA 0.188 SA 0.188 SA 0.189 SA 0.189 SA 0.189 SA 0.189 SA 0.189 SA 0.189 SA 0.887 > 0.743 > 0.722 > 0.716 > 0.712 > 0.710 > 0.709 > 0.708 > 0.707 > 0.707 > E I GEN RESIDUAL APPROX. NUMBER OF SIG. FIGURES IN EIGENVALUE IN EIGENVECTOR 1 3.477126E-32 2 1.138719E-29 3 7.389004E-31 15 .O 15.0 15.0 7 . 5 7 . 5 7 . 5 * * * STURM SEQUENCE TEST INDICATES ALL 3 SMALLEST EIGENVALUES FOUND < PERIOD = 0.189 SA = 0.707 > 11 0 0.004 0.761E+03 0.04043 E x e c u t i o n t e r m i n a t e d B-5 I I I . OUTPUT FROM UNIT 7 • PROGRAM OPTIONS MAXIMUM NUMBER OF MODES IN ANALYSIS 4 INELASTIC ANALYSIS MAXIMUM ITERATIONS' 20 INITIAL DAMPING RATIO* 0.020 NUMBER OF MODES TO HAVE OUTPUT PRINTED' 1 SEISMIC INPUT MAXIMUM ACCELERATIONS. 200 TIMES GRAVITY CIT/SIMULATED EARTHQUAKE TYPE C-2 SPECTRUM B-6 SAMPLE INPUT/OUTPUT E = 3300.0 KSI G = 0.0 KSI STRAIN HARDENING RATIO = 0. 020000 •»•» ... NO. OF JOINTS = 18 NO . OF MEMBERS = 23 JOINT DATA JN X(FEET) Y(FEET) NDX NDY NDR 1 0.0 0.0 0 0 0 2 18.000 0.0 0 0 0 3 36.OOO 0.0 0 0 0 4 54.000 0.0 0 0 0 5 81.000 0.0 0 0 0 e 0.0 10.000 1 2 3 7 18.000 10.OOO 1 4 5 8 36.000 10.000 1 6 7 9 54.OOO 10. OOO 1 8 9 10 8 1.000 10.000 1 10 1 1 1 1 0.0 20. OOO 12 13 14 12 18.OOO 20.OOO 12 15 16 13 36.000 20.000 12 17 18 14 54.000 20.OOO 12 19 20 15 8 1 .000 20 . OOO 12 21 22 16 36 . OOO 30. OOO 23 24 25 17 54. OOO 30.000 23 26 27 18 8 1 . OOO 30 . OOO 23 28 29 MEMBER DATA MN JNL JNG EXTL LENGTH EXTG XM(FT ) YM(FT) AREA I(CRACKED) AV MOMENT KL KG (FEET ) (SO.IN) (IN--4 ) (SO.IN) CAPACITY 1 1 6 0.0 10.0000 0 0 0 0 10.0000 320.0 7850.0 0.0 40.00 1 1 2 2 7 0.0 10.0000 0 0 0 0 10.0000 320.0 7850.0 0.0 50.00 1 1 3 3 8 0.0 10.0000 0 0 0 0 10.0OOO 320.0 7850.0 0.0 50.00 1 1 4 4 9 0.0 10.0000 0 0 0 0 10.0OOO 320.0 7850.0 0.0 50.00 1 1 5 5 10 0.0 10.0000 0 0 0 10.0000 2000.0 4000000.0 0.0 9000.00 1 1 E 6 11 0.0 10.0000 0 0 0 0 10.0OO0 290.0 7750.0 0.0 40.OO 1 1 7 7 12 0.0 10.0000 0 0 0 0 10.0000 290.0 7750.0 0.0 50.00 1 1 8 8 13 0.0 10.0000 0 0 0 0 290.0 7750.0 0.0 50.00 1 1 9 9 14 0.0 10.0000 0 0 0 0 10.0000 290.0 7750.0 0.0 50.00 1 1 10 10 15 0 0 2000.0 4000000.0 0.0 8500.00 1 1 11 13 16 0 0 10.0000 0 0 0 0 10.0000 290.0 7750.0 0.0 50 00 1 1 12 14 17 0.0 10.0000 0 0 0 10.0000 290.0 7750.0 0.0 60.00 1 1 13 15 18 0 0 0 10.0000 2000.0 4000000.0 0.0 80OO .00 1 1 14 6 7 0.0 18.0000 0 0 18 0000  99999.0 68000.0 0.0 40.00 1 1 B -7 15 7 8 0 0 18 0000 0 0 18 0000 0 0 99999 0 68000 0 0 0 30 00 16 8 9 0 0 18 OO  0 0 18 OO  0 0 99999 0 68000 0 0 0 30 00 17 9 10 0 0 27 0000 0 0 27 0000 0 0 99999 0 68000 0 0 0 40 00 18 1 1 12 0 0 18 0000 0 0 18 OO  0 0 99999 0 30000 0 0 0 30 00 19 12 13 0 0 18 0000 0 0 18  0 0 99999 0 30000 0 0 0 30 00 20 13 14 0 0 18 0000 0 0 18 OO  0 0 99999 0 30000 0 0 0 40 00 2 1 14 15 0 0 27 0000 0 0 27  0 0 99999 0 30000 0 0 0 40 00 22 16 17 0 0 18 0000 0 0 18 OO  0 0 99999 0 30000 0 0 0 40 00 23 17 18 0 0 27 0000 0 0 27  0 0 99999 0 30000 0 0 0 40 00 NO.OF DEGREES OF FREEDOM OF STRUCTURE = 29 HALF BANDWIDTH OF STIFFNESS MATRIX = 22 NO. OF NODES WITH MASS » 3 JN X-MASS V-MASS ROT.MASS (KIPS) (KIPS) (IN-KIPS) 8 330. OOO 0.0 0.0 13 280.OOO 0.0 0.0 17 220.000 0.0 0.0 S NO. OOF ASSIGNED MASS (KIP•SEC••2/FT) 1 1 10.24845 2 12 8.69565 3 23 6.83230 -INITIAL ELASTIC PERIOD-EIGENVALUES NATURAL FREQUENCIES PERIODS (RAD/SEC) (CVCS/SEC) (SECS). 1329.0283 36.4558 5.8022 0.1723 39721.2188 199.3018 31.7200 0.0315 256996.5625 506.9482 80.6837 0.0124 SA (2 PERCENT DAMPING) O.8870 O.2658 O.1799 INELASTIC RESULTS DAMERR = 0.01 ITERATION NO. ABOVE DAMDIF S MATRIX SMEARED NO. CAPACITY RATIO DAMPING < PERIOD •= O. 172 1 17 O 803 0.757E+03 0.02000 B-8 < PERIOD = 0 187 SA = 0. 743 > 2 5 O.107 0.762E*03 0.03545 < PERIOD = 0 188 SA = 0. 722 > 3 9 0.078 0.761E*03 0.03823 < PERIOD = 0 188 SA = 0 7 16 > 4 9 0.059 0.760E->03 0.03913 < PERIOD = 0. 189 SA = 0. 7 12 > 5 9 0.057 0.760E*03 0.03963 < PERIOD = 0. 189 SA = 0. 710 > 6 5 0.035 0.761E+03 0.03993 < PERIOD = 0. 189 SA •= 0. 709 > 7 2 0.055 0.761E+03 0.04013 < PERIOD = 0. 189 SA * 0. 708 > 8 0 0.039 0.761E*03 0.04026 < PERIOD • 0. 189 SA = 0. 707 > 9 0 0.021 0.7G1E+03 0.04033 < PERIOD ' 0. 189 SA = 0. 707 > 10 0 0.007 0.761E*03 0.04040 ITERATION NUMBER 11 ALL ELEMENTS OF MAIN DIAGONAL OF STIFFNESS MATRIX ARE POSITIVE DEFINITE RATIO OF LARGEST TO SMALLEST DIAGONAL STIFFNESSMATRIX ELEMENT IS 0.7G1E+03 NO. OF MODES TO BE ANAL I ZED = 3 * »*•>»• • .... • + * # * .... MODES EIGENVALUES NATURAL FREQUENCIES PERIODS SA (RAD/SEC) (CYCS/SEC) (SECS) (2 PERCENT DAMPING) 1 1102.3137 33.2011 5.2841 0.1892 0.8870 2 36625.6016 191.3782 30.4589 0.0328 0.2716 3 248227.5000 498.2244 79.2953 0.0126 0. 1809 MODAL PARTICIPATION FACTOR MODE 1 1.37 195 MDDE 2 0.49626 MODE 3 0.27532 MODE SMEARED DAMPING RATIO 1 0.04043 2 O.02632 3 0.02611 *•*•••«***••••••••*•*••**»*** ***************************** » • * * ********** B-9 MODE NUMBER 1 MODAL =ORCES AND DISPLACEMENTS JOINT NO . X -DISPIFT ) V-DISP(FT) ROTATION!RAD) 1 0.0 0 0 0 0 2 0.0 0 0 0 0 3 0.0 0 0 0 0 4 0.0 0 0 0 0 5 0 0 0 0 O 0 6 0.0049 0 0001 -0 0006 7 0.0049 -0 0000 -0 0004 8 0.0049 0 0001 -0 0005 9 0.0049 -0 0000 -0 0004 10 0.0049 -0 0000 -0 0008 1 1 0.0155 0 0001 -0 0008 12 0 0155 -0 oooo -0 OOOO 13 0.0155 0 0001 -0 0009 14 0.0155 -0 0001 -0 0010 15 0.0155 -o oooo -0 0012 16 0.0283 0 0002 -0 0OO9 17 0.0283 -0 0OO1 -0 OOOO 18 0.0283 -o oooo -0 0013 MN AXIAL SHEAR BML BMG KIPS KIPS (K •FT) (K -FT) 1 7 946 3.577 -29 499 6 274 2 - 1 019 6 . 395 -38 891 25 059 3 5 488 5.451 -35 745 18 767 4 -3 466 6 374 -38 82 1 24 918 5 -8 949 394.812 -9037 145 -S089 02 3 6 3 123 6 .978 -37 886 31 89 1 7 0 142 10.120 -50 339 50 860 8 5 488 8.018 -47 521 32 659 9 -2 969 8 .047 -49 772 30 699 10 -5 784 328 253 -5132 547 - 1850 01S 1 1 4 022 8 .402 -42 061 4 1 962 12 - 1 344 11.442 -53 601 60 820 13 -2 678 193.404 - 1892 138 41 901 14 -0 0 -4.823 44 160 -42 654 15 -0 0 -3.G62 32 744 -33 17 1 16 -0 0 -3.662 33 117 -32 798 17 -0 0 -3.165 4 1 891 -43 569 18 -0 0 -3.123 31 891 -24 315 19 -0 0 -3.265 26 546 -32 219 20 -0 0 -4.730 42 502 -42 639 2 1 -0 0 -3.105 4 1 66 1 -42 1B5 22 -0 0 -4.022 4 1 962 -30 439 23 -0 0 -2.678 30 381 -41 927 ROOT MEAN SQUARE DISPLACEMENTS JOINT NO . X -DISPt FT) Y-DISP(FT) ROTATION!RAD) 1 0.0 0 0 0 0 2 0.0 0 0 o 0 3 0.0 0 0 0 0 B-1 0 4 0.0 0 0 0 0 5 0.0 0 0 0 0 6 0.0O49 0 0001 0 0006 7 0.0049 0 OOOO 0 0004 8 0.0049 0 0001 0 0005 9 0.004 9 0 OOOO 0 0004 10 0.0049 0 OOOO 0 0008 1 1 0.0155 0 0OO1 0 0008 12 0.0155 0 OOOO 0 OOOO 13 0.0155 0 0001 0 0009 14 0.0155 0 0001 0 0010 15 0.0155 0 oooo 0 0012 16 0.0283 0 0002 0 0009 17 0.0283 0 0OO1 0 OOOO 18 0.0283 0 OOOO 0 0013 ROOT MEAN SOUARE FORCES RSS BASE SHEAR 4 19 603 KIPS MN AXIAL SHEAR BML BMG MOMENT DAMAGE KIPS KIPS (K FT ) (K -FT) CAPACITY RATIO 1 7 946 3 581 29 514 6 318 40 000 0 738 2 1 019 6 398 38 905 25 075 50 OOO 0 778 3 5 4B8 5 454 35 759 18 784 50 000 0 7 15 4 3 466 6 377 38 834 24 932 50 000 0 777 5 8 949 397 873 9042 859 5091 938 90O0 000 1 233 6 3 123 6 978 37 888 31 891 40 000 0 947 7 0 144 10 120 60 339 50 861 50 000 1 840 8 5 488 8 018 47 522 32 662 50 000 0 950 9 2 969 8 047 49 772 30 704 50 000 1 027 10 5 784 328 465 5 135 4 14 1867 891 8500 000 0 604 1 1 4 023 8 404 42 076 4 1 969 50 000 0 842 12 1 344 1 1 445 53 618 60 832 60 000 1 672 13 2 679 195 131 1909 675 4 1 909 8000 OOO 0 239 14 0 0 4 823 44 162 42 656 40 000 5 667 15 0 0 3 662 32 746 33 173 30 000 5 676 16 0 0 3 662 33 120 32 800 30 000 5 599 17 0 0 3 165 4 1 894 43 572 40 OOO 5 014 18 0 0 3 123 31 891 24 315 30 000 3 953 19 0 0 3 265 26 546 32 220 30 000 4 379 20 0 0 4 730 42 504 42 64 1 40 OOO 4 048 2 1 0 0 3 106 4 1 663 42 188 40 OOO 3 548 22 0 0 4 02 3 4 1 969 30 445 40 000 3 276 23 0 0 2 679 30 387 4 1 936 40 000 3 276 1 1 0 0.004 0. 761E+03 0.0404 3 NO. OF ITERATIONS - 1 1 BETA=0.800 BENDING MOMENT ERROR" 3.050OOO DAMAGE RATIO ERROR = 0 010 C-1 APPENDIX C PROGRAM LISTING 1 C 2 C 3 C 4 C **** MODAL ANALYSIS PROGRAM ' EDAM2 ' ****** 5 6 C c ORIGINAL PROGRAM TITLED MSSM.S BY SUMIO YOSHIDA 1979 7 c FIRST EDITION TITLEO EDAM BY ANDREW W.F. METTEN 1981 8 c SECOND EDITION TITLED EDAM2 BY LAWRENCE H.Y. HUI 1984 9 to c c . • • » * » • » • • * * « * * * * * • * » * • • * • * * * • * • « • » * * * * « • * * * * • « * * * * * * * 1 1 12 c PROGRAM DIMENSIONED FOR A MAXIMUM OF :-13 c 14 c 150 MEMBERS 15 c 150 JOINTS 16 c 100 ASSIGNED MASSES 17 c 10 EIGENVALUES 18 c 300 UNKNOWNS 19 c (NUMBER OF UNKNOWNS)*(HALF BANDWIDTH) IS LESS THAN 8000 20 c 21 c * NOTE - PRITZ IS A UBC:MATRIX LIBRARY SUBROUTINE 22 c FOR SOLVING EIGENVALUES 23 c 24 c VARIABLE DEFINITIONS:-25 c 26 c KL.KG - JOINT TYPE : FIXED JOINT = 1 27 c PINNED JOINT - 0 28 c AREA • CROSS-SECTIONAL AREA 29 c CRMOM - MOMENT OF INERTIA OF CRACKED SECTION 30 c BMCAP * BENDING MOMENT CAPACITY OF SECTION 31 c DAMRAT = DAMAGE RATIO OF MEMBER 32 c NO = OOF. NO. IDENTIFIED BY JOINT NO. 33 c ND(K.I) - K = 1 (X-DOF). 2 (Y-DOF). 3 (R-DOF) 34 c I = JOINT NO. 35 c NP - D.O.F. NO. IDENTIFIED BY MEMBER NO. 36 c NP(K.I) - K = OOF 1 TO 6 FOR STANDARD MEMBER 37 c I => MEMBER NO. 38 c XM = LENGTH OF FLEXIBLE PORTION OF BEAM IN X-DIRECTION 39 c YM = LENGTH OF FLEXIBLE PORTION OF BEAM IN Y-DIRECTION 40 c DM = TRUE LENGTH OF FLEXIBLE PORTION OF BEAM 4 1 c F = LOAD VECTOR 42 c EXTL.EXTG - LENGTH OF RIGID END 43 c TITLE = TITLE (80 CHARACTERS) 44 c SDAMP = STRUCTURAL DAMPING 45 c AV = SHEAR AREA 46 c DAMB = DAMAGE RAIO IN THE (I-I)TH ITERATION 47 c MDOF = OOF. NO. FOR MASSES IDENTIFIED BY MASS NO. 48 c AMASS = LUMPED MASS (IN UNITS OF WEIGHT) INDENTIFY BY 49 c D.O.F. NO. SO c EVOL = EIGENVALUE FOR EACH MODE 5 1 c EVEC - MODE SHAPE 52 c EVECIK.I) - K = MASS NO. 53 c I * MODE NO. 54 c BE! AM ' SMEARED SUBSTITUTE DAMPING FOR EACH MODE 55 c LOCK * CONTROL DIFFERENT STAGE OF ITERATION. CORRESPOND TO 56 c SUBROUTINE STACHK WHICH IS USED 10 STABILIZE CONVERGENCE 57 c LOCK - 0 = USUAL CONVERGENCE PROCEDURE 58 c 1 * BINARY SEARCH ROUTINE IN EFFECT 59 c 2 = PROGRAM CONVERCFJ) 60 c 01.DTN = STORE THE PERIOD OF THE LAST ITERATION 6 1 c 01 DSA = STORE UPPER AND LOWER EOUND SA VALUES 63 64 c DOUBLE PRECISION STIFFNESS MATRIX 65 c 66 REAL'S S(80001 67 C 68 DIMENSION KL( 150),KG<150),AREA( 150) ,CRMOM(150),BMCAP(150). 69 1 ND(3,150),NP(6.150).XM( 150),YM( 150).OM( 150), 70 2 F(300).EXTL( 150),EXTG( 150).TITLE 120).SDAMP(150),AV( 150) 7 1 DIMENSION DAMB(2.150).MDOF(100).0LDTN(2).0LDSA(3),DAMRAT(2.150) 72 DIMENSION AMASS(300).EVAL( 10).EVECt 100. 10).BETAM(10) 73 CALL FTNCMD('EQUATE 99=SPRINT;') 74 C 75 C IUNIT DEFINES THE INPUT AND OUTPUT FILES :-76 C 77 C IUNIT=5 IS DATA SOURCE FILE 78 C IUNIT»6 IS TEMPORARY STORAGE FOR INTERMEDIATE DATA 79 C IUNIT=7 IS FINAL OUTPUT FILE 80 C IUNIT=8 IS DAMAGE RATIO FILE ( SEPARATE FROM OTHER FINAL 81 C OUTPUT FILE TO MAKE PLOTTING OF RESULTS EASIER ) 82 IUNIT=7 83 C 84 C CALL CONTRL TO READ IN DATA OF STURCTURE . TITLE AND PROGRAM 85 C OPTIONS. 86 C 87 CALL CONTRL(TITLE.NRJ.NRM.E,G.7.AMAX.I SPEC,DAMP IN. 88 1 INELAS.NMODES,NPRINT.HARD) 89 C 90 C IDIM DIMENSIONS THE STIFFNESS MATRIX FOR SUBROUTINES 91 c 92 IDIM=8000 93 c 94 c CALL SETUP TO READ AND TO ECHO PRINT MEMBER AND JOINT DATA 95 c -HALF BANDWIDTH AND NUMBER OF UNKNOWNS ARE CALCULATED 96 c 97 CALL SE TUP ( NRM, E ,G. XM. YM, DM. ND.NP. AREA .CRMOM. DAMRAT . 98 1 NRJ.A V,K L,KG,NU,NB,SOAMP.BMCAP,IUNIT.EXTL.EXTG) 99 c 100 c SET I FLAG EOUAL TO 1 IF ONLY ONE ITERATION IS REQUIRED 101 c HERE IFLAG IS SET EOUAL TO 0 102 c 103 1FLAG=0 104 c 105 c CHECK IF IDIM HAS BEEN ASSIGNED LARGE ENOUGH 106 c LSTM = LENGTH OF ONE-DIMENSIONAL STIFFNESS MATRIX 107 c 108 LSTM=NU*NB 109 IFUSTM GT. IDIM) WRITE(7.10) LSTM. IDIM 1 10 10 FORMAT(/// 'PROGRAM STOPPED'.//'LENGTH OF STIFFNESS MATRIX3', 1 1 1 1 16./'PROVIDED STORAGE (IDIM)='.I6) 1 12 IF (LSTM.GT.IDIM) STOP 1 13 c 1 14 c ICOUNT IS THE NUMBER OF TIMES MAIN MSSM SUBROUTINE IS CALLEO 1 15 c ICOUNT IS INITIALIZED TO ZERO HERE. 1 16 c 1 17 IC0UNT=O 1 18 c 1 19 c CALL MASS TO READ AND ASSIGN MASSES TO NODES 120 c -ASSEMBLE THE MASS MATRIX : AMASS 12 1 c 122 CALL MASS!NU.ND.AMASS.IUNIT.NRJ.NMASS.MDOF) 123 c 124 c CHECK IF IDIM HAS BEEN SUFFICIENTLY DIMENSIONED 125 c 126 IVAR1MNU*NB)*NMASS 127 IVAR2=NMASS-(NMODES+3) 128 IF(IVAR1.GE.IDIM) WRITE(7.30) 129 IF(IVAR2 GE.IOIM) WRITEI7.30) 130 30 FORMAT(' THE VALUE OF IDIM IS SMALLER THAN "PRITZ" REQUIRES') 13 1 c 132 c REASSIGN OUTPUT TO TEMPORARY FILE 6 133 C 134 IUNIT=6 135 C 136 C 137 C IF ONLY ELASTIC ANALYSIS IS REQUIRED: RESET CONTROL FLAGS 138 C SET IFLAG=1 TO INDICATE ONLY ONE ITERATION IS REQUIRED 139 C 140 IF(INELAS NE.0) GO TO 70 14 1 WRITEI7.110) 142 IUNIT'7 143 IFLAG=1 144 WRITE(7,110) 145 70 CONTINUE 146 C 147 C 148 C SET THE MAXIMUM NUMBER OF ITERATIONS. 149 C 150 IMAX"1 151 1F(INELAS.NE.0) IMAX=INELAS 152 IM-1MAX-1 153 C 154 C I •= THE NUMBER OF ITERATIONS PERFORMED 155 C 156 1=0 157 C 158 C SET LOCK TO 0 FOR NORMAL CONVERGENCE PROCEDURE 159 C 160 LOCK=0 161 C 162 C BETA IS A FACTOR USED IN SPEEDING CONVERGENCE (0 <BETA < 1). 163 C BETA = 0. EFFECTIVELY SHUTS OFF CONVERGENCE SPEEDING ROUTINE 164 C 165 BETA-O. 166 C 167 c SET ERROR RATIO OF MOMENTS OF YIELDED MEMBERS (BMERR). 168 c A VALUE OF 0.05 HERE ENSURES YIELDED MEMBERS ARE WITHIN 169 c 5 PERCENT OF THEIR CAPACITY. 170 c 171 BMERR-0.05 172 c 173 c - SET CONVERGENCE LIMIT FOR CHANGE IN DAMAGE RATIO 174 c 175 c DAME RR = 0.01 ENSURES T HAT THE MAXIMUM DAMAGE RATIO CHANGE 176 c !N THE FINAL ITERATION IS ONE PERCENT - FDR DAMAGE RATIOS 177 c ABOVE 5.0 178 c - 1 HOSE DAMAGE RATIOS BELOW 5.0 WILL CONVERGE TO THEIR 179 c ABSOLUTE VALUE DIFFERENCE BEING TEN TIMES THE RATIO 180 c 18 1 DAMERR=0.01 182 c INITIALIZE ARRAY USED IN SPEEDING OF CONVERGENCE. 183 DO 80 MEM=1.NRM 184 OAMB!t.MEM)=OAMRAT(1.MEM) 185 DAMB(2,MEM)=OAMRAT(2.MEM) 186 80 CONTINUE 187 c 188 c 189 c FINISHED INPUT OF DATA AND INITIAL ACTIVITIES. 190 c BEGIN LOOP FOR MSS METHOD. 19 1 c 192 c 193 c 194 c INCREMENT ITERATION COUNTER :-195 c 196 100 1 = 1+1 197 WRITEI I UN IT. 1 10) 198 1 10 FORMAT!' ' , 1 10! '-' ) ) 199 WRITEIIUNIT,120) I 200 120 FORMAT!'-','ITERATION NUMBER',14) 201 C 202 C CALL BUILD TO COMPUTE THE MEMBER AND GLOBAL STIFFNESS MATRIX 203 C 204 CALL BUILOlNU.NB.XM.YM,DM.NP.AREA.CRMOM.AV.E.G.DAMRAT.KL.KG, 205 1 NRM.S.IDIM.EXTL.EXTG) 206 C 207 C CALL SCHECK TO CHECK THE CONDITION OF THE STIFFNESS MATRIX 208 C 209 CALL SCHECMS.NU.NB. IDIM, I UNI T , SRAT10 ) 2 10 C 21 1 C CALL EIGEN TO COMPUTE THE FREQUENCIES AND MODE SHAPES FOR 2 12 C THE SUBSTITUTE STRUCTURE 213 C 214 CALL EIGEN(NU,NB.S.IDIM.AMASS.EVAL.EVEC,NMODES.IUNIT,I SPEC. 215 1 AMAX,ICOUNT,M0OF,INELAS) 216 C 217 C INSERT HEADINGS FOR ITERATION PROGRESS (FOR INELASTIC 218 C ANALYSIS ONLY) 219 c 220 IF(INELAS.EO.0.OR.ICOUNT.NE.0) GO TO 105 221 WRITE(7.110) 222 WRITE(7.115) 223 1 15 FORMAT!' '.// 25X,'INELASTIC RESULTS'//) 224 WRITE17.110) 225 WRITEI7.85) DAMERR 226 WRITE(7.90) 227 WRITEO9.90) 228 85 FORMAT!/.' OAMEHR « '.F5.2./) 229 90 FORMAT!'-','ITERATION '.IX.'NO. ABOVE DAMDIF',3X. 230 1 'S MATRIX ',2X,'SMEARED'/' NO.',5X.'CAPACITY', 231 2 14X.'RATIO '.2X,'DAMPING') 232 105 CONTINUE 233 c 234 C AFTER 9 ITERATIONS BETA IS REASSIGNED FROM 0.0 TO 0.8 235 c IF NO. ABOVE CAPACITY = 0. SET BETA=0.0 236 c 237 IF(I .GE. 9) BETA=0.SO 238 C IF!ISIGN.EO.O) BETA=0.0 239 C 240 C DVARY = THE LARGEST DAMAGE RATIO DIFFERENCE BETWEEN THIS AND 24 1 C THE LAST ITERATION 242 C 243 DVARY=0.0 244 c 245 c CALL M0D3 - THE MAIN SUBROUTINE FOR THE MSSM 246 c 247 CALL MODS!ICOUNT.I SPEC.NRJ,NRM,NU,NB.NMODES.S,IDIM,ND.NP.XM,YM 248 1 DM.AREA,AV,CRMOM,DAMRAT.KL.KG.SDAMP.BMCAP.E.G.AMASS. 249 2 EVEC.EVAL,AMAX.ISIGN.IUNIT.BETA.BMERR.I FLAG.EXTL. 250 3 EXTG.BETAM.DAMB.DVARY.INELAS.OAMPIN,NPRINT.HARD, 251 4 OLDTN.OLDSA.LOCK) 252 IF (LOCK.EQ.1) DAMERR=0.005 253 c 254 c IF ONLY DOING ELASTIC ANALYSIS THEN STOP PROGRAM 255 IF!INELAS.EO.O) GO TO 250 256 c 257 c - OUTPUT DAMAGE RATIOS ON UNIT 8 258 c - OUTPUT NUMBER OF MEMBER IN EXCESS OF CAPACITY AND LARGEST 259 c DIFFERENCE FROM PREVIOUS ITERATION DAMAGE RATIOS 260 c - OUTPUT RATIO OF LARGEST TO SMALLEST NUMBER IN DIAGONAL 261 c OF STIFFNESS MATRIX (SRATIO) 262 c 263 WRITE(7 . 130) I . I SIGN.DVARY,SRAT10.BETAM! 1) 264 WRITE(99. 130) I.ISIGN.DVARV.SRAT10.BETAMI 1) 265 130 FORMAT(• '.I4.7X.I4.5X.F7.3.2X.E10.3.3X.F7.5) 266 C WRITEI8.140) (OAMRATIM), M= 1,NRM) 267 C 140 FORMAT I ' '.15FB.3I 268 C 269 C I FLAG IS MODIFIED FROM 0 TO 1 WHEN NO MEMBER IS ABOVE CAPACITY 270 C ONE FINAL ITERATION IS PERFORMED 271 C THE FOLLOWING LINES CHECK FOR YIELDING OF ALL MEMBERS AND THE 272 C MAXIMUM NUMBER OF ITERATIONS 273 c 274 IFI I FLAG.EO.1 AND. I.GE.1MAXI GO TO 180 275 IF(I FLAG.EO. 1) GO TO 160 276 IFU.EQ 1 AND. ISIGN.EO.O) GO TO 200 277 I F I I.GE.IM) GO TO 150 278 ADERR = ABS(DVARY ) 279 IFUSIGN. EO.O. AND. ADERR.LT.OAMERR) GO TO 150 280 GO TO 100 281 c 282 150 CONTINUE 283 IFLAG-1 284 IF ( I GE . IM. AND.LOCK.EO. 1 ) L0CK*2 285 IF (LOCK.NE.1) IUNIT=7 286 GO TO 100 287 C 288 160 CONT INUE 289 WRITE (IUNIT,170) I 290 170 FORMAT('-',5X,'NO. OF ITERATIONS "'.15///) 291 GO TO 220 292 c 293 180 CONTINUE 294 WRITE1IUNIT.1901 I 295 190 FORMAT I '-', 5X. 'DOES NOT CONVERGE AFTER',15,' ITERATIONS'/// 296 GO TO 220 297 C 298 200 CONTINUE 299 ICOUNT'O 300 IFLAG'1 301 IUNIT'7 302 WRITE!IUNIT,210) 303 2 10 FORMAT!'-',5X,'MEMBERS DO NOT YIELD '///) 304 GO TO 100 305 C 306 220 CONTINUE 307 WRITE!IUNIT.230) BETA.BMERR 308 2 30 FORMAT!'-'.5X.'BETA='.F5.3.///5X.'BENDING MOMENT ERROR='. 309 1 FB.6///) 310 WRITE!IUNIT,240) OAMERR 31 1 240 FORMAT I ' '.DAMAGE RATIO ERROR='. F6.3) 312 250 STOP 3 13 END 314 C 315 C ft****.*.************************************.**,************* 3 16 C 317 SUBROUTINE CONTRL1 TITLE.NRJ,NRM.E.G,IUNIT.AMAX.I SPEC.DAMPIN. 318 1 INELAS.NMODES.NPRINT.HARD) 3 19 C 320 C 321 C 322 DIMENSION TITLEI20) 323 c 324 C READ IN PROGRAM OPTIONS 325 C 326 READ!5. 10) INELAS,NMODES,NPRINT,I SPEC.AMAX.DAMP IN 327 10 F0RMATI4I5.2F1O.5) 328 C DAMPIN IS THE PROPORTION OF CRITICAL DAMPING USED IN ELASTIC 329 C ANALYSIS OR THE FIRST ITERATION OF THE MSSM. 330 C 331 C NPRINT IS A FLAG SET IF MODAL FORCES AND DISPLACEMENTS ARE REQUIRED 332 C IF NPRINT'O ONLY RMS FORCES AND DISPLACEMENTS WILL BE PRINTED. 333 C IF NPRINT IS GREATER THAN ZERO THAT NUMBER OF MODES (UP TO NMODES) 334 C WILL HAVE THEIR FORCES AND DISPLACEMENTS PRINTED. 335 c 336 c INELAS IS A FLAG INDICATING IF ONLY AN ELASTIC ANALYSIS IS REQUIRED 337 c IF INELAS'O THEN ELASTIC ANALYSIS ONLY WILL BE PERFORMED. 338 c IF INELAS IS GREATER THAN ZERO THEN THIS IS THE MAXIMUM NUMBER OF 339 c ITERATIONS THAT WILL BE PERFORMED DURING INELASTIC ANALYSIS. 340 c 34 1 c ECHO PRINT PROGRAM OPTIONS 342 WRITE!IUNIT.20) 343 20 FORMAT!' './/'**•••••PROGRAM OPTIONS•••••••'/) 344 WRITE!IUNIT,3OINM00ES 345 30 FORMAT 1 ' '.'MAXIMUM NUMBER OF MODES IN ANALYSIS'. 14) 346 IF!INELAS.EO 0) WRITE 1IUNIT,40) 347 40 FORMAT(' '.'ELASTIC ANALYSIS REQUESTED') 348 IF!INELAS.NE.0) WRITE(IUNIT,50) INELAS 349 50 FORMAT!' '.'INELASTIC ANALYSIS MAXIMUM ITERATIONS''.14) 350 1FI INELAS.EO.O) WRITEIIUNIT.60) DAMP 1N 35 1 60 FORMAT! ' '.'FRACTION OF CRITICAL DAMPING''.F6.4) 352 IF! INELAS GT 0) WR I {E ( I UNI T . 70) OAMPIN 353 70 FORMAT(' '.'INITIAL DAMPING RATIO' '.F6.3) 354 WRITE(IUNIT.801 NPRINT 355 80 FORMAT! ' '.'NUMBER OF MODES TO HAVE OUTPUT PRINTED'' .13) 356 C 357 WRITE!IUNIT.90) 358 WRITE(IUNIT.100) AMAX 359 90 FORMAT!'-','SEISMIC INPUT') 360 too FORMAT!'-'. 'MAXIMUM ACCELERATION''.F5.3.' TIMES GRAVITY') 361 1 10 FORMAT!///1101'-')) 362 IF(ISPEC.EQ.1) WRITE (IUNIT.120) 363 IF(I SPEC.EQ.2) WRITE (IUNIT. 130) 364 IF!ISPEC.EQ.3) WRITE (IUNIT,140) 365 IF<ISPEC.EQ.4) WR!TE(IUNIT.150) 366 IF(ISPEC.EO.5) WRITE!IUNIT.155) 367 IF!ISPEC EQ.6) WRITE(IUNIT,156) 368 IF(ISPEC.GE.7) WRITE!IUNIT.160) ISPEC 369'. WRITE!IUNIT.110) 370 120 FORMAT!' '. 'SPECTRUM A USED) 371 130 FORMAT(' '. 'SPECTRUM B USED') 372 140 FORMAT ( ' ', ' SPECTRUM C USED') 373 150 FORMAT(' '.'NATIONAL BUILDING CODE SPECTRUM USED') 374 155 FORMAT)' ,'SAN FERNANDO E/0. HOLIDAY INN. LONGITUDINAL DIRN.') 375 156 FORMAT!' ','CIT/SIMULATED EARTHQUAKE TYPE C-2 SPECTRUM') 376 160 FORMAT!' ', 'ERROR-SPECTRUM TYPE.13.' IS NOT VALID') 377 IFUSPEC.NE.4) GO TO 200 378 DPCNT =100.0*DAMPIN 379 C 380 CALL SPECTR!I SPEC.DAMP IN.1.0.AMAX.SA,6.283,SABND.SVBND.SDBND) 381 C 382 WRITE!IUNIT,170) DPCNT, SABNO 383 170 FORMAT ( ' '. F5 . 2 .'*/. DAMPING SPECTRAL ACCEL. BOUND* ' , F6 . 3 , ' *G') 384 WRITEfIUNIT.180) SOBND 385 180 FORMAT(' '.' DISPLACEMENT BOUND'' .F6.3.' IN") 386 WRITE!IUNIT. 190) SVBND 387 190 FORMAT(' '.' VELOCITY BOUND'',F6.3,' IN/SEC) 388 C 389 C READ IN TITLE 390 C 391 200 READ (5,210)(TITLE(I).I'1.20) 392 C 393 C READ IN NRJ.NRM.E.G 394 C 395 READ (5.220) NRJ, NRM. E. G. HARO 396 WRITE (IUNIT.230)(TITLE(I).I'1.20) 397 WRITE (IUNIT,240) E. G, HARO 398 WRITE (IUNIT.250) 399 WRITE (IUNIT.260) NRJ. NRM 400 WRITE)IUNIT.110) 401 C 402 C CONVERT E AND G FROM KSI TO KSF. 403 E'E*144.0 404 G = G*144 .0 405 C 406 RETURN 407 2 10 TORMATI20A41 408 220 FORMA r(2I5.2F10.3.F10.8l 409 2-?0 FORMAT( - 1 ' .20A4 1 4 10 240 FORMAT( ' -' .5X. ' E »'.F8.1.' KSI'.5X.'G ='.F8.1.' KSI'. 4 1 1 1 5X.'STRAIN HARDENING RATIO = '.FB.6I 4 12 250 F0RMAT(///1 10( "' )) 4 13 260 FORMAT( '-'. 'NO. OF JOINTS'.' = '.15, 10X, 'NO. OF MEMBERS ='.I5) 4 14 END 4 15 C 4 16 C 4 17 C 418 SUBROUTINE SETUP(NRM.E,G.XM.YM,DM.ND,NP.AREA.CRMOM.DAMRAT, 4 19 1 NRJ.AV.KL.KG.NU.NB.SDAMP.BMCAP.IUNIT.EXTL.EXTG) 420 C 421 C 422 C 423 C 424 C SET UP THE FRAME DATA 425 C 426 DIMENSION KL(NRM) . KG(NRM). AREA(NRM). CRMOM(NRM). SDAMP(NRM). 427 1 DAMRAT(2.NRM), AV(NRM). NDI3.NRJ). NPI6.NRM). XM(NRM) 428 2 YM(NRM),EXTL(NRM).EXTG(NRM). DM(NRM) 429 DIMENSION X(150). Y(150). JNLI150). JNGI150). BMCAP(NRM) 430 C 43 1 C E AND G IN KSF 432 C XI I ) AND Y( I) IN FEET 433 C MEMBER EXTENSIONS EXTG AND EXTL ARE IN FEET. 434 C AREA!I) IN SO. INCHES: CRMOMII) IN INCHES»*4 435 C CONVERTED TO FOOT UNITS IN ROUTINE 436 C 437 WRITE (IUNIT.230) 4 38 WRITE (IUNIT.2401 439 C 440 C READ IN JOINT DATA AND COMPUTE NO. OF DEGREES OF FREEOOM 441 C 442 NU = 1 443 C 444 DO 50 1=1,NRJ 445 READ (5.250) JN, ND(1.I). ND(2.I). ND(3.I). X(I). Y(I) 446 c 447 DO 40 K=1,3 448 IF(ND(K,I)- 1) 30. 10.20 449 10 ND(K.I)=NU 4 50 NU=NU*1 451 GO TO 40 452 20 JNN=N0(K.I) 453 ND(K.I > =ND(K,JNN) 454 GO TO 40 455 30 CONTINUE 456 ND(K.I)=0 457 40 CONTINUE 458 C 459 C PRINT JOINT DATA 460 C 46 1 WRITE (IUNIT.260) I. X(I). Y(I). ND(I.I). ND(2.I). ND(3.I) 462 50 CONTINUE 463 C 464 NU=NU-1 465 WRITE (IUNIT.2701 466 WRITE (IUNIr.2801 467 WRIT E ( IUNIT.2901 468 C 469 C READ IN MEMBER DATA AND COMPUTE THE HALF BANDWIDTH (NB) 470 C HALF BANDWIDTH=MAX DEGREE OF FREEDOM-MIN DEGREE OF FREEDOM +1 47 1 C 472 C 473 NB=0 474 C 475 DO 190 M8R=1,NRM 476 READ (5.300) MN,JNL(MBR).JNG(MBR).KL(MBR).KG(MBR). 477 1 AREA(MBR), CRM0M(MBR),AV(MBR).BMCAPIMBR). 478 2 EXTLIMBR).EXTG(MBR) 479 C 480 C IF DAMAGE RATIOS ARE LESS THAN ONE SET EOUAL TO ONE 4S 1 C 482 DAMRAT(1,MBR)=1.0 483 DAMRAT!2.MBR)=1.0 484 C COMPUTE MEMBER LENGTH (DM)=LENGTH BETWEEN JOINTS-RIGID EXTENSIONS 485 JL=JNL(MBR) 486 JG-JNGI MBR) 487 XM(M8R)=X(JG)-X(JL) 488 YM ( MBR ) = Y ( JG) - Y ( JL ) 489 DM(MBR)=SORT((XM(MBR))••2*(YM( MBR))••2) 490 EXTSUM = EX TL(MBR) + EXTG(MBR) 49 1 XM(MBR)=XM(MBR)M1 0-EXTSUM/DM(MBR)) 492 YMIMBR) = YM(MBR)*(1 .0-EXTSUM/DM(MBR)) 493 C RESET NEGATIVE VALUES OF ZERO TO ZERO 494 IF(YMIMBR) GT.-0.01.AND.YMIMBR).LT.0.01) YM(MBR)»0.0 495 IF(XM(MBR) GT.-0.01.AND.XM(MBR1.LT.0.01) XM(MBR1=0.0 496 DM(MBR)=0M(MBR)-EXTSUM 497 C 498 C CHECK FOR NEGATIVE LENGTHS OF MEMBER 499 C (PROBABLY CAUSED BY INCORRECT USE OF MEMBER EXTENSIONS) 500 C 501 IF(DM(MBR1 GT.0.0) GO TO 70 502 WRITE(7.60) MBR 503 60 FORMAT*' './//'PROGRAM HALTED:ZERO OR -VE LENGTH FOR MEMBER',16) 504 STOP 505 C 506 70 CONTINUE 507 C 508 YLEN=YM(MBR) 509 C 510 C PRINT ERROR MESSAGE IF ATTEMPT TO HAVE RIGID EXTENSIONS 51 1 C ON VERTICAL MEMBERS. 512 IF(EXTSUM.NE.0.0.AND.YLEN.GT.0.2) WRITEI7.80) 1 513 80 FORMAT(' '.'ERROR-HAVE END EXTENSIONS ON NON-HORIZONTAL 514 1 MEMBER NO.'.13) 515 C PRINT ERROR MESSAGE IF ATTEMPT TO HAVE RIGID EXTENSIONS ON 516 C A NON FIX-FIX TYPE MEMBER 517 KLSUM=KL(MBR)+KG(MBR) 518 IF(EXTSUM NE.0.0.AND KLSUM.NE.2) WRITE(7,90) MBR 5 19 90 FORMAT(' '.'ERROR-HAVE RIGID EXTENSIONS ON HINGED MEMBER'.14) 520 C 52 1 C GIVE MEMBERS INITIAL ELASTIC DAMPING 522 SDAMP(MBR)=0.02 5 2 3 r: 524 c ASSIGN MEMBER DEGREES DF FREEDOM 5 2 5 NP( 1.MBR}=ND( 1 .OL ) 526 NP(2.MBR1 =ND(2.UL) 527 NP( 3.MBR ) =ND( 3. JL ) 528 NP(4.MBR)=ND( 1.JG) 529 NP(5.MBR)=ND(2.JG) 530 NP(6.MBR)=ND(3.JG) 531 c DETERMINE THE HIGHEST DEGREE OF FREEDOM FOR EACH MEMBER STORING 532 c THE RESULT IN 'MAX' 533 MAX=0 534 c 535 DO 120 K=1.6 536 IFINP(K.MBR)-MAX) 110.110.100 537 100 MAX=NP(K,MBR ) 538 1 10 CONTINUE 539 120 CONTINUE 540 c 54 1 c DETERMINE THE MINIMUM DEGREE OF FREEDOM FOR EACH MEMBER.NOTE THAT 542 c FOR STRUCTURES WITH GREATER THAN 330 JOINTS INITIAL VALUE OF MIN 543 c WILL HAVE TO BE INCREASED FROM ITS PRESTENT POINT OF 1000. 544 c 545 MIN=1000 546 c 547 DO 160 K =1.6 548 IF(NP(K.MBR)1 150,150.130 549 130 IF(NP(K,MBR)-MIN) 140.150.150 550 140 MlN=NPIK.MBR ) 55 1 150 CONTINUE 552 160 CONTINUE 553 C 554 NBB=MAX-MIN+1 555 IF(NBB-NB) 180,180.170 556 170 NB'NBB 557 180 CONTINUE 558 C 559 C PRINT MEMBER DATA AND CONVERT TO FOOT UNITS. 560 C 561 WRITE (IUNIT.310) MBR.JNL(MBR),JNG(MBR),EXTL(MBR).DMIMBR) 562 1 EXTG(MBR).XM(MBR).YMIMBR). 563 2 AREA(MBR).CRMOMIMBR).AVIMBR).BMCAP(MBR),KL(MBR). 564 3 KG(MBR) 565 C 566 AREA IMBR)=AREA(MBR)/144.0 567 AV(MBRl=AV(MBR1/144 0 568 CRMOMI MBR1=CRMOM(MBR)/20736.0 569 C 570 190 CONTINUE 571 C 572 C PRINT THE NO. OF DEGREES OF FREEDOM AND THE HALF BANDWIDTH 573 C 574 WRITE ( IUNIT.320) NU 575 WRI TE ( IUNIT . 330) NB 576 c 577 c OUTPUT THE ASSIGNED DEGREES OF FREEDOM. 578 c WRITE!IUNIT.200) 579 200 FORMAT!' '. MEMBER NP1 NP2 NP3 NP4 NP5 NP6 ' ) 580 C 5 8 1 C 0 0 2 If) MEMBR = 1 . NRM 5 8 2 C 2 1 0 WRI TE 1 IUNI 1 , 2 2 0 ) MEMBR . 1 NP 1 I VAR , MEMBR ) . I VAR = 1 . 6 ) 5 8 3 c 58-1 2:") FORMA r ( ' '.2X.I4.2X.6I4) '585 C 586 C 587 RETURN 58B 2 30 FORMAT( '-' . 'JO I NT DATA') 5B9 240 FORMATI/7X. ' JN' .3X . 'X(FEET ) ',3X, ' V(FEET ) ',4X. 'NDX' ,2X, 'NDY', 590 1 2X,'NDR') 591 250 FORMAT(415.2F10.5) 592 260 FORMAT(' '.5X,14,2F10.3.2X.315) 593 270 FORMAT!'-', MEMBER DATA') 594 280 FORMAT!/' MN JNL JNG EXTL LENGTH EXTG XM!FT) YM(FT)'. 595 1 3X.AREA I(CRACKED) AV',7X,'MOMENT', 596 2 2X, 'KL' . IX, 'KG' ) 597 290 FORMAT! ' ' , 19X , '(FEET)',29X, ' ( SO . IN)' , 3X . ' < IN"4 ) ' . 598 1 2X.'<SO.IN)'.3X.'CAPACITY') 599 300 FORMAT!515.F8 2,F12.3.2F10.3,2F6.3) 600 3 10 FORMAT!' '.I3.2I4.F7.3.F9.4.F7.3.2F9.4,F8.1.F12.1.F8.3,F10.2.2I3) 601 320 FORMAT!'-'.'NO OF DEGREES OF FREEDOM OF STRUCTURE ='.I5) 602 330 FORMAT!/' HALF BANDWIDTH OF STIFFNESS MATRIX =',I5) 603 END 604 C 605 C 606 C 607 SUBROUTINE BUILD(NU.NB.XM.YM.DM.NP.AREA.CRMOM,AV.E.G,DAMRAT, 608 1 KL.KG.NRM,S.IDIM.EXTL.EXTG) 609 C 610 C 61 1 C 612 C 613 C - THIS SUBROUTINE WORKS IN DOUBLE PRECISION 614 C - THIS SUBROUTINE CALCULATES THE STIFFNESS MATRIX OF EACH 615 C MEMBER AND ADOS IT INTO THE STRUCTURE STIFFNESS MATRIX. 616 C - THE FINAL STIFFNESS MATRIX S IS RETURNED 617 C - THIS SUBROUTINE IS SIMILAR TO ONE THAT WOULD BE USED IN 618 C NORMAL FRAME ANALYSIS. 619 C - DIFFERENCES INCLUDE USING CRACKED MOMENT OF INERTIA INSTEAD 620 C OF THE GROSS SECTION DAMAGE RATIOS ARE USED AND FLEXURAL 62 1 C STIFFNESSES MODIFIED ACCORDING TO THESE RATIOS. 622 C IDIM IS THE DIMENSIONING SIZE OF THE STRUCTURE STIFFNESS MATRIX. 623 c INTERNAL FOOT UNITS FOR STIFFNESS MATRIX 624 c 625 REAL'8 SMI21).SIIDIM) 626 DIMENSION XM(NRM). YM(NRM), DM!NRM), NP(6.NRM), AREA(NRM), 627 1 CRMOM(NRM). AV(NRM). OAMRAT(2.NRM). KL(NRM). KG(NRM) 628 DIMENSION EXTL(NRM), EXTG!NRM) 629 REAL-8 RF,GMOO.CMOMI.DRATI (6).F , H 630 REAL'8 LONE,LONEX,LONEY.LTWO.LTWOX.LTWOY.AVI 63 1 REAL'S YMI.DMI.DM2.XM2.YM2,XMI,AREA I,EMOD,XM2F.YM2F.XMYMF 632 REAL•8 DBLE 633 c 634 c ZERO STRUCTURE STIFFNESS MATRIX 635 c 636 DO 10 I=1.IDIM 637 S(I)=0.ODOO 638 10 CONTINUE 639 c 640 c RF.ASSIGN YOUNGS MODULUS TO DOUBLE PRECISION VARIABLE EMOD 64 1 r.MOD = OBI.E( E ) 642 <JM0O = OP,LE ( G ) 64 3 c 644 c BEGIN MEMBER LOOP 645 c 646 DO 200 1=1.NRM 647 c '648 C ZERO MEMBER STIFFNESS NATRIX 7 14 YM2F=YM2'F*DR»TI(6) 619 C 7 15 XMYMF=XMYM*F 'DRAT1(6) 650 DO 20 J= 1 ,21 7 16 DM2F=DM2*F 651 SMIJ) =0.0000 7 17 C 652 20 CONTINUE 718 SMI 1)=SM( 1) + YM2F 653 C 719 SMI 2) = SM(2)-XMYMF 654 C ASSIGN MEMBER PROPERTIES TO DOUBLE PRECESION VARIABLES 720 SM(4)=SM(4)-YM2F 655 C 72 1 SM(5)=SM(5)»XMYMF 656 LONE =DBLE(EXTL(I)) 722 SM(7)=SM(7)+XM2F 657 I.TWO=OBLE(EXTG< I 1 1 723 SM(9)=SM(9)*XMYMF 658 YMI=DBLE< YMI I)) 724 SMI10)=SM(10I-XM2F 659 DMI=DBLEI DM!I 1) 725 SMI16)'SM(I6)+YM2F .660 XMI=DBLEIXM(I)) 726 SMI17)-SM(17)-XMYMF 66 1 AREAI=DBLE(AREA! I )) 727 SM(19)=SM(19I+XM2F 662 CMOMI=DBLE I CRMOM! I )) 728 IF(KL(I)-KG(I)) 70.80.90 663 AVI=DBLE(AV(I)) 729 C 664 C 730 C FILL IN REMAINING PIN-FIX TERMS 665 c OBTAIN EFFECTIVE DAMAGE RATIO FROM 'DAMCAL' 731 C 666 c 732 70 SM(6)=-YMI«DM2F'DRATI(5) 667 CALL DAMCAL!OAMRAT,DRAT I,1 ) 733 SMI 11)=XMI'OM2F*DRATI(5) 668 c 734 SM(18)=-SM(6) 669 DM2 =DMI-DMI 735 SMI20)=-SM(11) 670 XM2=XMI-XMI 736 SMI 21)=DM2'DM2F'DRATI(2) 67 1 YM2=YMI*YMI 737 GO TO 100 672 XMYM=XMI'YMI 738 C 673 F =AREA 1•EMOD/(DMI-DM2) 739 C FILL IN REMAINING FIX-FIX TERMS 674 H=O.ODOO 740 C 675 c SHEAR DEFLECTIONS ARE IGNDRED WHENEVER G OR AV IS ZERO. 74 1 80 SMI 3) =-YMI*DM2F'0.SDOO'DRATI(4) 676 IF! AVi 1 ) . EO.O.O OR GEO 0. ) GO TO 30 742 SM(6)=SM(3)/DRATI(4)'DRATI<5) 677 H=12 .ODOO'EMOD'CMOMI/IAVI'GM0D*0M2) 743 SM(8)»XMI'DM2F*0 SDOO'DRATI(4) 678 30 XM2F=XM2'F 744 SM(11)=SM(8)/DRAT I(4)'DRAT I(5) * 679 YM2F=YM2'F 745 SM(12 >=DM2'DM2F'(4 ODOO+H)/12 ODOO*DRATI(1) 680 XMYMF=XMYM'F 746 SM(13)=-SM(3) 68 1 C 747 SM(14)=-SM(8) 682 C FILL IN PIN-PIN SECTION OF MEMBER STIFFNESS MATRIX 748 SM(15)=DM2 *DM2F•(2.ODOO-H)/12.OOOO'DRAT1(3) 68 3 C 749 SMI18)=-SM(6) 684 SM( 1)=XM2F 750 SM(20)=-SM(11) 685 SM(2)=XMYMF 751 SM( 2 1 ) = SM( 12 )/DRATI ( 1 I'ORATK 2 ) 686 SM(4)=-XM2F 752 C ADD IN TERMS FOR RIGID END EXTENSIONS. 687 SMI5(=-XMYMF 753 SM(3)=SM(3)-(L0NEY) 688 SM(7)=YM2F 754 SMI 6)=SM(6)-(LTWOY) 689 SM(9)=-XMYMF 755 SMI 8 ) = SMI8 HLONEx 690 SM(10)=-YM2F 756 SMI 1 1)=SM( 11I'LTWOX 691 SM(16)=XM2F 757 SMI 12)=SM( 12 )M LONE *0M1*(OMI+LONE)*RF)'DRATI(1) 692 SM( 17)=XMYMF 758 SMI 13) = SM( 13l+LONEY 693 SM(19)=YM2F 759 SMI 14) = SM( 14)-LONEX 694 IF(KL(I )+KG(1)-1) 100.40.50 760 SMI 151»SM( 15) + ( ( LONE * LTWO *DMI )•(DM2•(LONE*LTWO 1/2.ODOO))•RF 695 C 761 1 '0RATK3I 696 C VALUES OF F CALCULATED HERE DIFFER FROM STANDARD BUILD 762 SM(18)=SM(18)+LTW0Y 697 c SUBROUTINE BY DIVIDING BY THE DAMAGE RATIOS. 763 SM(20)=SM(20)-LTWOX 698 c 764 SMI 2 1) = SM( 2 1)+I DM2-LTWO*IDMI*ILTWO*LTWO)))'RF'DRATI(2) 699 40 F =3.ODOO* EMOD* CMOMI/(DM2 *DM2 *OMI•( 1.OOOO+H/4.ODOO)) 765 GO TO 100 700 GO TO 60 766 C 701 50 F =12.ODOO * EMOD'CMOMI/I DM2 *DM2'DM I•1 1 OOOO + H)) 767 C FILL IN REMAINING FIX-PIN TERMS 702 C RF IS A FACTOR COMMON TO THE ENTIRE MATRIX FOR ADDITION OF 76B C 703 C STIFFNESS DUE TO RIGID BEAM END EXTENSIONS 769 90 SMI 3) =-YMI*DM2F'DRAT 1(4) 704 RF=12.ODOO* EMOD* CMOMI/(DM2'DM2)/[ 1.DO+H) 770 SM(8) = XMI•DM2F'DRATI ( 4) 705 c 77 1 SMI 12)=0M2 *DM2F'DRATI(1) 706 c FILL IN TERMS WHICH ARE COMMON 10 PIN-FIX,FIX-PIN.AND 772 SMI13)=-SM(3) 707 c FIX-FIX MEMBERS 773 SMI14)=-SM(8) 708 c 774 100 CONTINUE 709 LONEY=LONE'YMI'RF'ORAT1(4) 775 C 710 LONEX=LONE•XMI'RF *ORATI(4) 776 C ADD THE MEMBER STIFFNESS MATRIX SM INTO THE STRUCTURE 7 1 1 LTWOY = L TWO * YMI'RF'DRATI(5 ) 777 C STIFFNESS MATRIX S. 7 12 LTWOX = LTWO'XMI'RF'DRATI(5 I 778 C 7 13 60 XM2F=XM2'F'DRATI(6) 779 NB1=NB- 1 o I 780 C 78 1 00 190 J»1,6 782 IFINP(J.l)) 190,190.110 783 1 10 J1=IJ-1 )•( 12-J)/2 784 C 785 DO 180 L-J.6 786 IF(NPIL.l)) 180,180,120 787 120 IFINPIJ,Il-NPIL.I>I 150.130.160 788 130 IF(L-J) 140.150.140 789 140 K=!NP<L.I)-1)*NB1+NP(J.I) 790 N=J1+L 791 S(K)=S(K)*2.0000'SM<N) 792 GO TO 180 793 150 K=(NP<J.I)-1 )'NB1+NP(L. I ) 794 GO TO 170 795 160 K=INP(L.I)-1)'NB1+NP(J.I) 796 170 N-JHL 797 S(K)=S(K)»SM(N) 798 180 CONTINUE 799 C 800 190 CONTINUE 801 C 802 200 CONTINUE 803 C 804 RETURN 805 END 806 C 807 C 808 C 809 SUBROUTINE DAMCAL(DAMRAT,ORATI.NOM) 810 C 8 12 C 813 C EFFECTIVE DAMAGE RATIO CALCULATION 8 14 C 815 DIMENSION OAMRAI(2,NOM) 816 REAL '8 DRAT 1(6 I .DBLE 8 17 C 818 DRATI ( 1 ) =DBLE (UAMRAK 1 .NOM) ) 819 DRATI(2)=DBLE(DAMRAT(2.NOM)) 820 DRATI(3 ) = 1 . D0/( .09500+.2D0*(DRAT I( 1 ) + DRATI(2))+.50500*DRATI( 1) 821 1 •DRATU2I) 822 DRATI( 1 ) = 1 D0/DRAT1( 1) 823 DRAT I(2)=1.DO/DRAT 1(2) 824 DRATI(4 ) = ( 2 DO'DRAT I( 1 )+ DRATI(3)I/3.00 825 OR ATI < 5 )M 2 .00'OR ATI ( 2 l+DRATI ( 3 ) )/3 .00 826 DRATI(6> = (DRAT II 1 )+DRATI(2)+DRATI(3))/3.DO 827 RETURN 828 END 829 C 830 C 83 I C 832 SUBROUTINE MASS'NU.ND.AMASS.IUNIT,NRJ.NMASS.MDOF) 833 C 834 C 835 C 836 C 837 C THIS SUBROUTINE SETS UP THE MASS MATRIX 838 C 839 C ND(J,I)'DEGREES OF FREEDOM OF I TH JOINT 840 C WTX.WTY,WTR=X-MASS,Y-MASS.ROT.MASS IN FORCE UNITSIKIPS OR IN-KIPS) 84 1 C AMASS!II'MASS MATRIX. I IS THE DEGREE OF FREEDOM OF APPLIED MASS 842 C 843 C MASSES ARE LUMPED AT NODES. THE MASS MATRIX IS DIAGONAL I ZED. 844 C 845 DIMENSION ND(3.NRJ). MOOF(IOO), AMASS(NU) 846 C 847 C REAO IN NO. OF NODES WITH MASS 848 C 849 READ 15.90) NMASS 850 WRITE IIUNIT.100) 85 1 WRITE IIUNIT.110) NMASS 852 WRITE IIUNIT.120) 853 WRITE IIUNIT.130) 854 C 855 C ZERO MASS MATRIX 856 C 857 DO 10 I=1.NU 858 AMASS!I)=0. B59 10 CONTINUE 860 C B6 1 C READ IN X-MASS.V-MASS AND ROT. MASS (IN UNITS OF WEIGHT) B62 C B63 DO 50 1=1,NMASS 864 READ (5.140) JN. WTX, WTV, WTR 865 WRITE (IUNIT.150) JN, WTX, WTV. WTR 866 N1=ND(1,JN) 867 N2=ND(2.JN) 868 N3=ND(3.JN) 869 IF(N1.EO.O) GO TD 20 870 AMASS!N1)=AMASS!N1l+IWTX/32.2) 87 1 20 IFIN2.F.0.OI GO TO 30 872 AMASS!N2l=AMASS(N2)+(WTY/32.2> 873 3n IF IN3.EO.O) GO TO 40 874 AMASS!N3)=AMASS(N3) + (WTR/32. 2 I 875 •10 CONTINUE 876 50 CONTINUE 877 C 878 C OUTPUT THE DEGREES OF FREEOOM WITH MASS AND ASSIGNED MASS. 879 C 880 JCNT=1 88 1 WRITE!IUNIT,70) 882 C 883 DO 60 IDOF=1,NU 884 RMASS = AMASS( IDOF) 885 IF(RMASS.EO.0.0) GO TO 60 886 MDOF(JCNT)=IDOF 887 WRITE!IUNIT.80) JCNT.MDOF(JCNT).RMASS 888 JCNT=JCNT+1 889 60 CONTINUE 890 C 891 70 FORMAT! '-'. 'MASS NO DOF' .2X, 'ASS IGNED MASS (KIP*SEC'«2/FT ) ') 892 80 FORMAT!' ',2X.13.3X.13.9X.Flo.5) 893 RETURN 894 90 FORMAT!15) 895 100 FORMAT!///I 10! '•' 11 896 1 10 FORMAT! -'.'NO. OF NODES WITH MASS',' =',!5) 897 120 FORMAT(/7X. 'JN' ,3X . 'X-MASS' ,4X, 'Y-MASS' ,2X. 'ROT.MASS' ) 898 130 FORMAT(' ' . 12X , (KIPS)'.4X.'(KIPS)'.2X. (IN-KIPS)') 899 140 FORMAT(15,3F10.0) 900 150 FORMAT!' '.5X.14.3F10.3) 901 END 902 C 904 C 905 SUBROUTINE EIGEN(NU,NB,S,IDIM.AMASS.EVAL.EVEC,NMODES,IUNIT, 90G 1 I SPEC,AMAX.I COUNT,MDOF,INELAS) 907 C 908 C •••»••****•«*•»•»»•»»•*«**•«**•»»**•*»«• 909 C 910 C 911 C THIS SUBROUTINE COMPUTES A SPECIFIED NO. OF NATURAL FREQUENCIES 912 C AND ASSOCIATED MODE SHAPES 9 13 C 914 C NU*NO. OF DEGREES OF FREEDOM 915 C NB=HALF BANDWIDTH 916 C NMODES=NO. OF MODE SHAPES TO BE COMPUTED 917 C IF NMODES IS ZERO OR IS GREATER THAN THE NUMBER OF STRUCTURE 918 C MASSES THEN NMODES WILL BE ASSIGNED THE NUMBER OF STRUCTURE 919 C MASSES. 920 C AMASS(I)=MASS MATRIX MCOUNT=NUMBER OF NONZERO MASSES 921 C SI I )=STIFFNESS MATRIX STORED BY COLUMNS 922 C EVALII("NATURAL FREQUENCIES 923 C EVEC(I.J)=MODE SHAPES 924 C 925 REAL'S OVECI300.10).OVAL(10).CMASSI3001.SD(8000) 926 REAL'S SIIDIM) 927 DIMENSION AMASS(NU). EVALINMODES), EVECI 100,NMODES), 928 1 MDDF(100) 929 REAL'S DBLE 930 C 93 1 C ZERO DUMMY MASS MATRIX CMASS 932 DO 10 IT R Y =1 . 100 933 10 CMASSI11RY)=0.000 934 C 935 C COMPUTE THE NUMBER OF NONZERO MASS MATRIX ENTRIES 936 C 937 MC0UNT=O 938 C 939 DO 20 1=1,NU 940 CMASSI I )=DBLEI AMASS I I)) 941 IFIAMASSII ) EO.O. ) GO TO 20 942 MC0UNT=MC0UNT+1 943 20 CONTINUE 944 IF(NMODES.GT.MCOUNT) NMODES=MCOUNT 945 IF(NMODES.EO.O) NMODES=MCOUNT 946 IF( IUNIT.EQ 6.AND.ICOUNT GT.25) GO TO 30 947 WRITE (IUNIT,160) NMODES 948 30 CONTINUE 949 C 950 C CALL PRITZ TO COMPUTE EIGENVALUES AND EIGENVECTORS 951 C CREATE A DUPLICATE STRUCTURE MATRIX (SD) (DESTROYED IN PRITZ) 952 C 953 C CALCULATE USEFUL LENGTH OF STIFFNESS MATRIX (LSTM) 954 LSTM=(NU)'NB 955 C 956 DO 40 1=1.LSTM 957 SD(I )=S(I I 958 40 CONTINUE 959 C 960 C SET CONVERGENCE CRITERIA FOR PRITZ. MAKE NEGATIVE IF 961 C RESIDUALS NOT DESIRED 962 C 963 DEPS=1.0D-I0 964 IF(IUNIT NE.7 ) DEPS = (- 1.ODO>'DEPS 965 C 966 C CALL EIGENVALUE FINDING ROUTINE 967 CALL PRITZISD.CMASS.NU.NB, 1.DVAL.DVEC.300.NMODES.DEPS.& 140) 96a C 969 C CONVERT MATRICES TO SINGLE PRECESION 970 C 97 1 c PRINT EIGENVALUES AND EIGENVECTORS(MODE SHAPES) 972 c EIGENVALUES (EVAL) ARE THE VALUES OF OMEGA SQUARED. 973 c 974 c SKIP PRINTING INTERMEDIATE DATA AFTER SEVERAL CYCLES. 975 IF(ICOUNT.GT.3.AND.IUNIT.EQ.6) GO TO 70 976 WRITE (IUNIT,170) 977 c WRITE (IUNIT,210) NMODES 978 c WRITE (IUNIT. 230X1. 1 = 1. NMODES) 979 c DO 60 ID=1,NU 980 c WRITE(IUNIT,50) ID.(DVEC(ID.J). J=1.NMODES) 981 50 FORMAT(' '.13.10F11.6) 982 60 CONTINUE 983 C 984 70 CONTINUE 98S C CONVERT MEMBERS OF EVAL FROM OMEGA SOUAREO TO OMEGA 986 C 987 C CONVERT EIGENVECTORS TO ONLY INCLUDE DEGREES OF FREEDOM 988 C WITH MASS ASSIGNED TO THEM 989 DO 80 MAS=1.MCOUNT 990 IVAR =MDOF(MA S 1 991 DO 00 MOD=1.NMODES 992 EVEC(MAS.MOD)=SNGL(DVEC(IVAR.MOD)) 993 80 CONTINUE 994 C 995 IF(ICOUNT.EQ.O) WRITEI7.90) 996 90 FORMAT ( ' '.// ' INITIAL ELASTIC PERIOD ') 997 IF(ICOUNT.EQ.O) IUNIT=7 998 WRITE (IUNIT.180) 999 WRITE (IUNIT.190) 1000 C 1001 C COMPUTE FREQUENCIES AND PERIODS 1002 DO 100 JUICE=1.NMODES 1003 100 EVALfJUICE)=SNGL(DVAL(JUICE)) 1004 C 1005 DO 110 1=1.NMODES 1006 EVAL1=EVAL(I) 1007 EVAL(I)=SQRT(EVAL1) 1008 WN=EVAL(I) 1009 PER10D=6 283153/WN 1010 FREO=l/PERIOD 101 1 IF(ICOUNT GT.25.AND IUNIT.EO.G) GO TO 110 1012 CALL SPECTR( I SPEC.0.02.PERIOD.AMAX.SA.WN.SABNO.SVBND.SDBND) 1013 WRITE (IUNIT.200) I. EVAL1. EVAL(I). FREO. PERIOD, SA 1014 1 10 CONTINUE 1015 IF(ICOUNT.EQ.O.AND.INELAS NE.0) I UNIT=6 1016 C 1017 IF(ICOUNT.GT.5 AND.IUNIT EO.6) GO TO ISO 1018 C WRITE (IUNIT.220) NMODES 1019 C WRITE ( IUNIT,240X I . 1 = 1 .NMODES) 1020 C DO 120 1=1,MCOUNT 102 1 c WRITE IIUNIT.50) I.IEVEC(I.J).J=1,NMODES) 1022 120 CONTINUE 1023 C 1024 130 CONTINUE 1025 C 1026 RETURN 1027 140 WRITE!IUNIT.150) 1028 150 FORMAT!- •."CRAPOUT IN PRITZ) 1029 160 FORMAT('-','NO. OF MOOES TO BE ANALIZED ='.15///110! ' " )///) 1030 170 F0RMAT(///110!'")) 1031 1B0 FORMAT(/5X.'MODES'.4X.'EIGENVALUES',6X,'NATURAL FREQUENCIES'. 1032 1 13X . 'PERIODS' . 10X , ' SA ' ) t033 190 FORMAT I ' '.SOX, 'I RAD/SEC)',5X, '(CYCS/SEC I '.8X. '(SECS)'. 1034 1 4X.'(2 PERCENT DAMPING 1 ' ) 1035 200 FORMATC " , 5X . 15 . 5F 15 . 4 ) 1036 210 FORMAT(/'TOTAL MODE SHAPES CORRESPONDING TO FIRST'.15. 1037 1 1X.'FREQUENCIES') 1038 220 FORMAT(/'MASS MODE SHAPES CORRESPONDING TO FIRST',15.1X, 1039 1 'FREQUENCIES') 1040 230 FORMAT)/' DOF' .18.91 1 1 ) 1041 240 FORMAT)/'MASS'.10111) 1042 250 FORMAT)' '.10F12.6) 1043 RETURN 1044 END 1045 C 1046 C 104 7 C 1048 SUBROUTINE M0D3(ICOUNT.I SPEC.NRJ,NRM.NU.NB.NMODES.S.IDIM.NO.NP. 1049 1 XM.YM,DM.AREA.AV,CRMOM.DAMRAT.KL.KG.SDAMP,BMCAP.E.G. 1050 2 AMASS.EVEC.EVAL.AMAX,1SIGN,IUNIT.BETA.BMERR.IFLAG, 1051 3 EXTL.EXTG.BETAM.DAMB.DVARY.INELAS.DAMP IN,NPRINT.HARO, 1052 4 0L01N.0LDSA.LOCK) 1053 C 1055 C 1056 C SUBSTITUTE STRUCTURE METHOD FOR RETROFIT 1057 C 1058 C THIS SUBROUTINE COMPUTES JOINT DISPLACEMENTS AND MEMBER FORCES 1059 C NEW DAMAGE RATIOS WILL BE CALCULATED AND RETURNED. 1060 C 1061 REAL*8 S(IDIMI.DFI3O0) 1062 C 1063 DIMENSION ND)3.NRJ). NP(6,NRM), XM)NRM), YM(NRM). OM(NRM). 1064 1 AREA(NRM). CRMOM)NRM). DAMRAT(2.NRM). 1065 2 KLINRM). KG)NRM). EVEC)100.NMODES). EVAL(NMODES). 1066 3 SDAMP(NRM). AV)NRM), AMASS(NU) 1067 DIMENSION BMASS) 100) . IDOFUOO), ALPHA (20) , RMS(7.150). 1068 1 F(300). EXTL(NRM), EXTG)NRM). 1069 2 BMCAP)NRM).DAMB)2.NRM).BETAM)NMDDE S),OLDTNl 1),0LDSA( 1) 1070 REAL'S DRATIO.DET 107 1 C 1072 C CALCULATE THE MODAL PARTICIPATION FACTOR :-1073 C 1074 C JJ = TEMPORARY VARIABLE USED IN THE FOLLOEWING LOOP ONLY 1075 JJ=1 1076 C 1077 DO 30 JDOF=1 .NU 1078 IF(AMASS)JDOF).EO.O.) GO TO 30 1079 BMASS(JJ)=AMASS(JDOF) 1080 IOOF!JJ)=JDOF 1081 JJ=JJ+1 1082 30 CONTINUE 1083 C 1084 MC0UNT=JJ-1 1085 C 1086 00 70 MODE =1,NMODES 1087 AMT =0. 1088 A MB =0. 1089 C 1090 C EIGEN VALUES ARE STORED AS FOLLOWS EVEOMASS NO,MODE NO.) 1091 C 1092 DO 60 J-1.MCOUNT 1093 AMT =AMT'BMASS(J)* EVEC(J.MODE) 1094 AMB = AMB*8MASS ( J) * ( ) EVEC) J.MODE ) )"2 ) 1095 60 CONTINUE 1096 ALPHAIMODE)=AMT/AMB 1097 70 CONTINUE 1098 C 1099 IF)ICOUNT.GT.25.AND.IUNIT.£0.6) GO TO 90 1 100 WRITE (IUNIT.8101 1 101 C 1 102 DO 80 MODEM .NMODES 1 103 WRITE (IUNIT.820) MODE. ALPHA(MODE) 1 104 80 CONTINUE 1 105 90 CONTINUE 1 106 C 1 107 C WHEN KK=1, MODAL FORCES FOR UNDAMPED SUBSTITUTE STRUCTURE ARE 1 108 C COMPUTED THEY ARE USED TO COMPUTE 'SMEARED' DAMPING VALUES. 1 109 C WHICH ARE USED TO CALCULATE THE ACTUAL RESPONSE OF THE 1 1 10 C SUBSTITUTE STRUCTURE 1111 C 1112 INDEX'1 1 1 13 C 1 1 14 DO 800 KK- 1 .2 1 1 15 C 1 1 16 C SET PRINT FLAG FOR MODAL OUTPUT IO=OFF) 1 1 17 INTPR=1 1 1 18 IF(KK.EQ.1) 1NTPR=0 1 1 19 IF(IFLAG.EO.O.OR.NPRINT-EO.O) INTPR'O 1 120 IF ( I COUNT . NE . 0 ) GO TO 100 112 1 C 1 122 C SET DAMPING RATIOS TO "APPROPIATE' VALUES FOR INITIAL TRIAL 1 123 DO 790 MODEAM .NMODES 1 124 BETAMI MODEA)=DAMPIN 1 125 790 CONTINUE 1 126 C 1 127 ICOUNT=ICOUNT+1 1 128 WRITE (IUNIT.840) 1 129 GO TO 80O 1 130 100 SHRMS=0. 1131 C 1 132 C ZERO RMS)J.I) 1 133 c 1 134 DO 110 1=1.150 1 135 DO 110 J=1.7 1 136 RMS)J.I)=0. 1 137 1 10 CONTINUE 1 138 c 1 139' c OUTPUT THE SMEARED DAMPING RATIOS (FOR DAMPED CASES) 1 140 IF(IUNIT.EQ.6.AND.IC0UNT.GT.25( GO TO 150 1 14 1 IF(KK.EO.1) GO TO 150 1 142 c 1 143 WRITE!IUNIT.130) 1 144 c 1 145 DO 120 MODE=1.NMODES 1 146 WRITEIIUNIT.140) MODE.BETAM!MODE) 1 147 120 CONTINUE 1 148 C 1 149 130 FORMAT!•-.'MODE'.2X. SMEARED DAMPING RATIO) 1 150 140 FORMAT!' ',IX,I3.7X.F10.5) 1 151 C 1 152 C 1 153 C CALCULATE THE MODAL DISPLACEMENT VECTOR 1 154 C 1 155 150 DO 570 MODEN=1,NMODES 1 156 C 1 157 C CALCULATE NATURAL PERIOD AND CALL SPECTA 1 158 C 1 159 TN=6.28318531/(EVAL!MODEN)) 1 160 WN=EVALI MODEN) 1161 DAMP=BETAM(MODEN 1 1162 IF (MODEN.NE.1.OR.LOCK.EO.O) 1163 1 CALL 5PEC1R11SPEC.DAMP.TN.AMAX.SA.WN.SABND.SVBND.SDBND) 164 IF (MOOEN.EO. 1 I 165 1 CALL STACHK(OLOSA.SA.OLDTN.TN.ISPEC.LOCK.ICOUNT,IFLAG. 166 2 IUNIT.AMAX.DAMP.KK I 167 IF (MOOEN.EO.1.AND.KK.EO.2) WRITE(99.205) TN.SA 168 IF ( MODEN.EQ. 1 AND.KK.EQ.2.AND. IFLAG.NE. 1) WRITEI7.205) TN,SA 169 205 FORMAT (50X.'< PERIOD •'.F6.3.2X.'SA ='.F6.3.' >') 170 C 171 C LIST MEMBER FORCES IF DOING ELASTIC ANLYSIS ONLY 172 C 173 IF(INTPR.EQ.O) GO TO 180 174 IFINPRINT.LT.MODEN) GO TO 180 175 WRITE!IUNIT.840) 176 WRITE!IUNIT.170) MODEN , 177 170 FORMAT!' '.'MODE NUMBER'.13.' MODAL FORCES ANO DISPLACEMENTS') 178 WRITE!IUNIT.830) 179 180 CONTINUE 180 C 181 C CHECK IF MODAL PARTICIPATION FACTOR IS ZERO 182 C IF ALPHA IS ZERO MODAL FORCES AND DISPLACEMENTS WILL BE ZERO 183 C 184 IF!ALPHA(MODEN).NE.0.0) GO TO 200 185 WRITE(IUNIT.190) 186 190 FORMAT!/ ' MODAL PARTICIPATION ,FORCES AND DISPL.=ZERO') 187 GO TO 570 188 200 CONTINUE 1B9 C 190 C ZERO LOAD VECTOR 191 C 192 DO 2 10 0= 1 .NU 193 210 F(J)=0. 194 C 195 C COMPUTE LOAD VECTOR 196 C 197 FAC=SA*ALPHA(M00EN)>32.2 198 C 199 C NOTE THAT AS THESE FORCES ARE BEING GENERATED FROM A 200 C LATERAL EXCITATION SPECTRUM THAT ONLY 'X MASSES' SHOULD 201 C BE USED. IN OTHER WORDS LATERAL ACCELERATION SHOULD NOT 202 C CAUSE NON HORIZONTAL INERTIA FORCES DIRECTLY. 203 C 204 FF=0. 205 DO 220 0=1,MCOUNT 206 II-IDOF(O) 207 F(I 1)=EVEC(J,MODEN)'FAC*AMASS< I 1) 208 FF=FF*F(I1) 209 220 CONTINUE 210 C 211 C CALCULATE THE BASE SHEAR 2 12 C 213 IF1KK.E0.1) GO TO 230 214 SHRMS=SHRMS+FF»'2 215 IF(MODEN LT NMODES) GO TO 230 216 SHRMS=SORT(SHRMS) 217 230 CONTINUE 2 18 C CONVERT SINGLE PRECISION FORCE MATRIX TO DOUBLE PRECISION 219 DO 21(1 IFREE=1.NU 220 240 DF(ITREEl=DBLE(F(IFREEI) 22 1 C 222 C COMPUTE DEFLECTIONS BY CALLING SUBROUTINE SDFBAN 223 C • NOTE THAT NO SOLUTION IMPROVING ITERATIONS WILL BE PERFORMED. 224 C SCALING WILL RE PERFORMEO TO IMPROVE THE SOLUTION WHEN NSCALE.NE.O 225 C 226 NSCALE=1 1227 C 1228 DRAT 10=1 OD-16 1229 CALL SDFBANIS.DF.NU.NB.INDEX.DRAT 10.DET.OEXP.NSCALE) 1230 C 1231 c SDFBAN EXITS WITH DF BEING THE DISPLACEMENT MATRIX 1232 c 1233 c CONVERT OOUBLE PRECISION DISPLACEMENTS TO SINGLE PRECISION 1234 DO 250 OFREE =1.NU 1235 F1 OF REE)=SNGL(DF(OFREE)) 12 36 250 CONTINUE 1237 c 1238 INDEX=INDEX+1 1239 c 1240 c CALCULATE RMS DISPLACEMENTS 124 1 c 1242 DO 290 JNT= 1.NRJ 1243 DX=0. 1244 DY=0. 1245 DR=0. 1246 N1=ND(1.JNT) 1247 N2=ND(2.JNI ) 1248 N3=ND(3.JNT) 1249 IF(N1.EO.O) GO TO 260 1250 DX=F(NI) 1251 RMS!1.JNT)=RMS(1,JNT)+DX**2 1252 260 CONTINUE 1253 IF(N2.E0.0) GO TO 270 1254 DY=F(N2) 1255 RMSI 2.JNT)= RMS(2,JNT)+DY *•2 1256 270 CONTINUE 1257 IF(N3.EO.O) GO TO 280 1258 DR=F(N3) 1259 RMS(3.JNT)=RMS(3,JNT)*DR**2 1260 280 CONTINUE 1261 IF(INTPR.EQ.O) GD TO 290 1262 IF(NPRINT.LT.MODEN) GO TO 290 1263 C OUTPUT MODAL DEFLECTIONS FOR REOUIRED MOOES 1264 WRITE!IUNIT.860) JNT,DX,DY.OR 1265 290 CONTINUE 1266 C 1267 C CALL FORCE TO CALCULATE MEMBER FORCES AND SMEARED 1268 C DAMPING RATIOS 1269 C 1270 CALL FORCE (NRM,XM.YM,DM,AV.NP,F,EXTL,EXTG, 1271 1 AREA,E.G,NPRINT.CRMOM,DAMRAT,INTPR. 1272 2 KL,KG,KK,SDAMP.NMODES.IUNIT.IFLAG. 1273 3 MODEN,ICOUNT,RMS.BETAM) 1274 C 1275 C COMPUTE AND WRITE MODAL CONTRIBUTION FACTOR 1276 CONMOD=SA*ALPHA(MODEN) 1277 C WRITE(IUNIT.5501 MODEN. CONMOD 1278 550 FORMAT!' '.'MOOE ' .I 3.3X. 'CONTRIBUT ION FACTOR*'.F8.5) 1279 C OUTPUT SPECTRAL ACCELERATION. 1280 C 128 1 C IF!INTPR.EQ.O.OR MODEN.GT NPRINT) GO TO 570 1282 C WRITE!IUNIT.560) DAMP. TN,SA 1283 560 FORMAT 1' '. 'OAMPING=',F6.4. ' PER100='.F6.4, ' SEC. SA='.F5 1284 570 CONTINUE 1285 1286 C 1287 IF(KK.EO.1) GO TO 800 1288 C 1289 C PRINT RMS DISPLACEMENTS AND FORCES 1290 C 129 1 IF(IUNIT.EO.G.AND.ICOUNT.GT.25) GO TO 590 1292 WRITE (IUNIT,840) 1293 C OUTPUT THE COUNT OF ENTRANCES INTO M0D3 1294 WRITE(6.10) ICOUNT 1295 10 FORMAT(' ', 'ICOUNT*'. 13) 1296 WRITE (IUNIT,850) 1297 WRITE (IUNIT.830) 1298 590 CONTINUE 1299 C 1300 C CONVERT SOUARE OF RMS DISPLACEMENTS TO RMS DISPLACEMENTS. 1301 C 1302 DO 610 1=1.NRJ 1303 DO 600 J= 1 . 3 1304 RMS(J.I> = SORT(RMS(J.I)) 1305 600 CONTINUE 1306 IF(ICOUNT GT.25.AND.IUNIT.EO.6) GO TO 610 1307 WRITE (IUNIT.860) I. (RMS(J.I),J=1.3) 1308 610 CONTINUE 1309 IF!ICOUNT GT.25.AND IUNIT.EO.6) GO TO 630 1310 WRITE (IUNIT.870) 1311 WRITE (IUNIT.880) SHRMS 1312 IF!ICOUNT GT.35 AND.IUNIT.EQ.6) GO TO 630 1313 WRITE (IUNIT,890) 1314 630 CONTINUE 1315 C 1316 C CALL DAMOD TO MODIFY DAMAGE RATIOS 1317 C 1318 CALL DAMOD (RMS.NRM,DAMB.BMCAP.DVARY,I FLAG,BETA,HARD, 1319 1 ICOUNT,IUNIT.BMERR.DAMRAT.ISIGN.SDAMP( 1320 C 1321 800 CONTINUE 1322 C .•• = = »» = .. = = *•..... = »..».». = = .•«. = = = »•• = . = . = ...«. = . = .•.»=. 1323 C 1324 I COUNT = ICOUNT•1 1325 RETURN 1326 810 FORMAT!'-'. MODAL PARTICIPATION FACTOR'./) 1327 820 FORMAT( ' '.5X, 'MODE' .15.5X.F10.5.5X,F10.5) 1328 830 FORMAT('-' ,7X . 'JOINT NO. '.10X. 'X-01SP(FT ) '. 10X.'Y-DISP(FT('.7X 1329 1 'ROTATION(RAO)') 1330 840 FORMAT('-'.110('*')) 1331 850 FORMAT('-'.'ROOT MEAN SOUARE DISPLACEMENTS') 1332 860 FORMAT( ' ' .6X.I 10.3F20.4) 1333 870 FORMAT!'-'.'ROOT MEAN SOUARE FORCES') 1334 880 FORMAT!1H0.7X.-RSS BASE SHEAR ='.F10.3.' KIPS') 1335 890 FORMAT( '- ' . 8X . 'MN' . 10X . 'AXIAL' . IOX. 'SHEAR' . 11X. 'BML', 12X.'BMG' 1336 1 9X.'MOMENT'.10x.'DAMAGE'/2IX. KIPS'.12X.'KIPS'.2(9X. 1337 2 ' I K-FT I ' l . 8X, 'CAPACITY' ,9X. 'RATIO' I 1338 END 1339 . C 1340 ' C ' 134 1 C 1342 SUBROUTINE FORCE INRM.XM.YM,DM.AV.NP,F.EXTL.EXTG . 1343 1 AREA,E.G.NPRINT.CRMOM.DAMRAT . 1NTPR. 1344 2 KL.KG.KK.SDAMP.NMODES. I UNIT.I FLAG. 1345 3 MODEN,I COUNT.RMS.BETAM) 1346 C 1347 C *•• •• 1348 C 1349 C THIS SUBROUTINE CALCULATES AXIAL.SHEAR FORCES 1350 C BENDING MOMENT (RETURN AS RMSt 4-7,JO I NT NO.)). 1351 C AND SMEARED DAMPING FACTOR (BETAMI 1352 C 1353 C 1354 C •*• NOTE *'* 1355 C AT THIS STAGE RMS I 1.JNT) = (RMS DISPLACEMENT(SQUARED OF X DISPLACEMENT 1356 C COMPUTE MEMBER FORCES USING DISPLACEMENTS FROM INDIVIDUAL MODES 1357 C NOTE THAT ENGINEERING' SIGN CONVENTION IS USED HERE. 1358 C 1359 DIMENSION XM(NRM).YM(NRM).OM(NRM).AV(NRM),NP(6,NRM). 1360 1 D(6).F(300).EXTL(NRM).EXTG(NRM).KL(NRM).KG(NRM). 136 1 2 PI I 150).RMSI 7.150).SDAMP!NRM).BETAM!NMODES). 1362 3 SUMDAM!150).2ETA(101.AREA(NRM).CRMOM(NRM),DAMRAT!2.NRM) 1363 REAL *8 DRATI(6) 1364 C 1365 SIGPI-O. 1366 C 1367 C INSERT MODAL MEMBER FORCE HEADINGS 1368 C 1369 IF(INTPR NE.0.AND NPRINT.GE.MODEN) WRITE (IUNIT.10) 1370 10 FORMAT( ' './8X. 'MN' .10X. 'AXIAL'. I0X. 'SHEAR' . 1 IX. 'BML'. 12X. 137 1 1 'BMG', /21X.'KIPS'.12X, KIPS'.2(9X.'(K-FT)')) 1372 C 1373 C 1374 C 1375 DO 170 1=1.NRM 1376 c 1377 c 1378 c 1379 c XL.YL = X.Y COMPONENTS DF MEMBER LENGTH RESPECTIVELY 1380 c OL = TRUE LENGTH OF MEMBER 13B1 c BMG = BENDING MOMENT AT GREATER JOINT NO. END OF MEMBER. 1382 c BML = BENDING MOMENT AT THE LESSER JOINT NO. END. 1383 c 1384 XL=XM(I) ' 1385 YL = YM(I ) 1386 DL=DM(I) 1387 AVI=AV<I) 1388 c 1389 DO 40 MEMDOF =1,6 1390 N1=NP(MEMDOF.I) 1391 IF(N1) 30,30.20 1392 20 D(MEMDOF)=F(N1) 1393 GO TO 40 1394 30 DIME MDOF)=0. 1395 •10 C0N1 INUF. 1396 c 1397 c MODIFY END DISPLACEMENTS FOR HORIZONTAL MEMBERS WITH END 1398 c EXTENSIONS FORMULA ONLY WORKS FOR HORIZONTAL MEMBERS 1399 N3=NPI3.1) 1400 IFIN3.EQ.0) GO TO 50 1401 DI2)=D(2)+(FIN3)I'EXTLlI) 1402 50 CONTINUE 1403 N6=NP(6,I ) 1404 IF!N6.EO.O) GO TO 60 1405 0(5)=D(5)-(F(N6))*EXTG(I) 1406 60 CONTINUE 1407 c PRINT OUT MEMBER END DISPLACEMENTS FOR DEBUG 1408 t IF(ICOUNT.GT.1) GO TO 80 1409 C WRITEI6.70) I.(DIM),M=1,6) 14 10 70 FORMAT( ' ', 'MEMB NO. = ' .I 3, 'DISPL='.6F10.5) 14 11 80 CONTINUE 14 12 AXIALM AREA( I )»E/DL>-2)'(D(4 )'XL+D(5)'YL-D( 1 ) • XL-D( 2 ) • YL ) 14 13 C GET EFFECTIVE DAMAGE RATIO 14 14 CALL DAMCAL(DAMRAT.DRATI,I ) 14 15 EISI=CRMOM(I)'E 14 16 C 14 17 C GFACT=FACTOR TO COMPUTE EFFECT OF SHEAR DEFL. ON MEMBER FORCES 14 18 C GFACT=0.0 IMPLIES THAT NO SHEAR DEFLECTION INCLUDED. 14 19 GFACT=0.0 1420 IFIAVI EO.O.0.OR.G.EO.O.0) GO TO 90 1421 GFACT=12.O*EISI/(AVI"G*DL*DL)*0RATI(6) 1422 90 CONTINUE 1423 C 1424 C ASSIGN DISPLACEMENTS TO THEIR RESPECTIVE MEMBER DEGREES OF 1425 C FREEDOM CHECK FOR PIN-PIN MEMBERS 1426 IFIKLII).EQ.O AND. KG(I).EQ.O) GO TO 120 1427 DELT=((D(5)-0(2))'XL+(D(1)-D(4) ) - VL VOL 1428 BML=(2 0'EISI/(DL*(1 .O+GFACT)))•((3.O'DELT'DRATI(4)/DL) 1429 1 -<D(6)*(1.0-GF ACT/2.0)*DRATI(3))-(2.0'D(3).'(1 . O+GFACT/4 1430 2 'DRAT 1(1))) 1431 BMG=-(2.0*EISI/(DL*<1 O+GFACT)))*(( 3.O'DELT'DRAT I(5 )/DL ) 1432 1 -(D(3)'( 1 O-GFACT/2.0)'DRAT1(3) )-(2.0'D(6)•( 1 O+GFACT/4 1433 2 *DRATI(2))) 1434 SHEAR=(6.0*EISI/(DL'DL))'((D(3)'DRAT I(41+D(6)'DRAT I(5) 1435 1 -(2.0'DELT'DRATI(6)/DL))/(1.O+GFACT)) 1436 C BMG=BML+SHEAR'DL 1437 IF(KL(I)-KG(I)) 100.130.110 1438 C ADJUST PIN-FIX MEMBER FORCES. 1439 100 BMG-BMG+BML•(1.O-GFACT/2.0)/(2.0'<1.0+GFACT/4.0)) 1440 SHEAR=SHEAR+1.5*BML/(DL) 1441 BML=0. 1442 GO TO 130 1443 c ADJUST FIX-PIN MEMBER FORCES. 1444 1 10 BML=BML+BMG*(1.O-GFACT/2.0)/(2.0*<1.O+GFACT/4.0)) 1445 SHEAR=SHEAR-1.5 *BMG/(DL) 1446 BMG=0. 1447 GO TO 130 1448 c FILL IN MEMBER FORCES FOR PIN-PIN MEMBERS. 1449 120 BMG=0. 1450 BML =0. 1451 SHEAR=0. 1452 130 CONTINUF 1453 C 1454 C COMPUTE THE RELATIVE FLEXURAL STRAIN ENERGY 1455 C 1456 IF(KK.EO.2 ) GO TO 140 1457 PI ( I ) = IBML*BML *DRATI(2)+BMG•BMG•DRAT I(11 + 1458 1 BML'BMG'DRA1 I(3 ) )'2.'DL/EISI/ 1459 2 ( 16 . 'DRAT I ( 1 ) 'DRAT I ( 2 ) - 4 • (DRAT I ( 3 ) " 2 . ) ) 1460 SIGPI=SIGPI+PI( I ) 1461 140 CONTINUE 1462 C 1463 C PRINT OUT FORCES FOR EACH MEMBER IF ELASTIC CASE DESIRED. 1464 IF(INTPR EO.O) GO TO 150 1465 IF(NPRI NT.GE.MODEN) WRITE(IUNIT. 160) I.AX IAL.SHEAR.BML.BMG 1466 C IF(I.EO.I) WRITE! 13. 160) I.AX IAL,SHEAR.BML.BMG 1467 160 FORMAT (6X.I5.6F15.3) 1468 ISO CONTINUE 1469 C 1470 c ACCUMULATE ABSOLUTE SUM AND RMS SUM 147 1 C 1472 RMS(4. I )=RMS(4. I)+AXIAL"2 1473 RMS( 5 . I )=RMS(5. I )• SHEAR"2 1474 RMS(6. I )=RMS(6. I )+BML"2 1475 RMS(7,1)=RMS(7.1)+BMG* *2 , 1476 170 CONTINUE 1477 C 1478 c 1479 c COMPUTE THE SMEARED DAMPING FOR EACH MODE 1480 c 1481 IFIKK.EQ 2) GO TO 240 1482 c 1483 c SUMDAM= THE PRODUCT OF MEMBER STRAIN ENERGY'MEMBER DAMPING. 1484 ZET A(MODEN > =0. 1485 00 180 1=1.NRM 1486 SUMDAMII)=PI(I)'SDAMP(I) 1487 ZETA(MODEN)=ZETA(MODEN)+SUMDAM(I) 1488 180 CONTINUE 1489 IF(SIGPI EO 0.0) WRITE(IUNIT.190) 1490 190 FORMAT!' ','ERROR-DIVIDED BY ZERO WHILE CALCULATING SMEARED 1491 1 DAMPING') 1492 C 1493 C BETAM=SMEARED SUBSTITUTE DAMPING FOR THE M TH MODE. 1494 BETAM!MODEN)=ZETA(MODEN)/SIGPI 1495 C 1496 C PRINT DAMPING INFORMATION FROM FINAL ITERATION. 1497 C 1498 IF(IFLAG.EO.O) GO TO 240 1499 WR I TE ( 6 . 200) SIGPI . MODEN. BETAM( MODEN) 1500 200 FORMAT(' '.'TOTAL FLEX. STR. ENERGY-'.F10.3,3X.'MODE NUMBER'. 1501 1 12,3X, SMEARED DAMPING FACTOR=',F7.5) 1502 WRITE(6.210) 1503 C 1504 DO 230 MEMB*1.NRM 1505 2 10 FORMAT(' ','MEMBER NO.',3X,'STRAIN ENERGY',3X. 1506 1 'MEMBER DAMPING', 3X.'MEMBER DAMPING'STRAIN ENERGY') 1507 WRITE(6.220) MEMB.PI(MEMB),SDAMP!MEMB),SUMDAM!MEMB) 1508 220 FORMAT!' '.3X.13.10X,E10.3.8X.E10.3.13X,F11.7) 1509 2 30 CONTINUE 15 10 2-10 CONTINUE 15 11 RETURN 1512 END 1513 C 15 14 C •it.....»....*..*.****•*•«*•***«**•*»***•****•*•****•»*•••««•* 1515 C 1516 SUBROUTINE DAMOD ( RMS,NRM.DAMB.BMCAP.OVARY,I FLAG.BETA.HARD. 1517 1 ICOUNT.IUNIT.BMERR.DAMRAT.I SIGN.SDAMP) 1518 C 1519 C 1520 C 152 1 DIMENSION RMS(7.150).DAMBI2.NRM ).BMCAP(NRM),DAMRAT(2.NRM), 1522 1 SDAMP!NRM),DAM0LD(2) 1523 C 1524 C MODIFY DAMAGE RATIOS 1525 C 1526 C ISIGN IS A COUNT OF THE NUMBER OF MEMBERS WITH WHICH THE RATIO 1527 C OF THE ABSOLUTE VALUE OF THE DIFFERENCE BETWEEN THE LARGEST RMS 1528 C BENDING MOMENT AND ULTIMATE MOMENT TO ULTIMATE MOMENT IS IN 1529 c EXCESS OF 'BMERR'. 1530 c ISIGN IS INITIALIZED TO ZERO HERE. 153 1 c 1532 ISIGN=0 1533 c 1534 DO 200 MEM=1,NRM 1535 c 1536 c CONVERT SQUARE OF RMS AXIAL. SHEAR AND MOMENT TO RMS VALUE. 1537 DO 10 d=4,7 1538 •RMS!J.MEM)=SORT(RMS(J.MEM)) 1539 10 CONTINUE 1540 DO 30 L=1.2 154 1 c 1542 c SET DAMOLD AS THE DAMAGE RATIO IN THE (I-2)TH ITERATION 1543 c DAMB AS THE DAMAGE RATIO IN THE (I-1)TH ITERATION. 1544 c 1545 DAMOLDIL)=DAMB(L.MEM) 1546 DAMBIL.MEM)=DAMRAT(L.MEM) 1547 c 1546 C CALCULATE NEW DAMAGE RATIO 1549 C 1550 IF (DAMRAT!L.MEM).GT.1.0) GO TO 20 1551 DAMRAT!L.MEM)"RMS!5+L.MEM(/BMCAP(MEM I-DAMRAT!L.MEM) 1552 GO TO 30 1553 20 TEMP-RMS(5+L.MEM)'DAMRAT(L.MEM) 1554 DAMRAT(L,MEM)"TEMP/(BMCAP(MEM)•(1-HARD)+HARD'TEMP) 1555 30 CONTINUE 1556 C 1557 c OUTPUT THE RMS AXIAL SHEAR AND MOMENT 155S c 1559 IF(ICOUNT GT.35.AND.IUNIT.E0.6) GO TO 50 1560 DAMAX*DAMRAT(I.MEM) 1561 IF (DAMRAT(2.MEM).GT.DAMRAT(1.MEM)) DAMAX=DAMRAT(2.MEM) 1562 WRITE (IUNIT.40) MEM.(RMS(0.MEM).J"4.7).BMCAP(MEM).DAMAX 1563 40 FORMAT ( 6X.I5.6F15.3) 1564 C 1565 50 DO 130 L=1,2 1566 C 1567 c DO NOT ALTER DAMAGE RATIOS OF LESS THAN UNITY 1568 c 1569 IFIOAMRATIL.MEMI.LT.1.0) GO TO 120 1570 c 157 1 c CONVERGENCE SPEEDING ROUTINE FOLLOWS. 1572 c 1573 IFIDAMRATIL.MEM).LT.5.0) 1574 1 DERROR=(DAMRAT(L.MEM)-DAMBIL.MEM))/10.0 1575 IF(DAMRAT!L.MEM) GE 5.0) 1576 1 DERROR"1DAMRAT!L.MEM(-OAMB! L.MEM))/DAMRAT(L.MEM) 1577 IF(ABS< DERRDR1 GT.ABS<DVARY)) DVARY"DERROR 1578 c 1579 DAMDIF=DAMRAT(L.MEM)-DAMB(L.MEM) 1580 c 1581 I F ( DAMOLD( D-DAMBt L.MEM) ) 60. 120.90 1582 60 CONTINUE 1583 IF(DAMDIF ) 80.120.70 1584 70 DAMRAT(L.MEM)"DAMRAT(L.MEMl+BETA' (DAMDIF) 1585 GO TO 120 1586 80 DAMRAT(L.MEM)"DAMRAT(L.MEM)-BETA'(DAMDIF) 1587 GO TO 120 1588 90 CONTINUE 1589 IF(DAMDIF ) 110. 120.100 1590 100 CONTINUE 159 1 DAMRAT!L.MEM)"DAMRAT!L,MEM)-BETA*(DAMDIF) 1592 GO TO 120 1593 1 10 CONTINUE 1594 DAMRAT(L.MEM)=OAMRAT(L.MEMl+BETA'(DAMDIF) 1595 120 CONTINUE 1596 IF(DAMRAT!L.MEM).LT.1.0.AND.I FLAG NE. 1) DAMRAT(L.MEM) = 1.0 1597 130 CONTINUE 1598 C 1599 C DAMAGE RATIOS CANNOT BE LESS THAN 1.0 1600 C IN LAST ITERATION SKIP RESETTING DAMAGE RATIOS LESS THAN UNITY 1601 C 1602 IFIOAMRATI1.MEM). LE.1.0.AND.DAMRAT(2.MEM) LE.1.0) GO TO 180 1603 C F I NO THE BIGGEST OF THE SOUARE OF THE RMS BENDING MOMENT(=BIG) 1604 IF I RMS(6.MEMl-RMS!7.MEM))140. 140, ISO 1605 140 BMBIG=RMS(7.MEM) 1606 DAM"DAMRAT(2.MEM) 1607 GO TO 160 1608 150 BMBIG=RMS(6.MEM) 1609 DAM=DAMRAT(1.MEM) 1610 160 CONTINUE 16 1 1 C 1612 C BMSH = INCREASED MOMENT CAPACITY DUE TO STRAIN HARDENING 1613 C 1614 BMSH"BMCAP1 MEM)*( 1-HARD)/(1-HARD'DAM) 1615 CHECK"(BMBIG-BMSH)/BMSH 1616 IF(CHECK.GT BMERR) ISIGN=I SIGN+1 1617 C COMPUTE DAMPING VALUE FOR THE MEMBER 1618 SDAMPIMEM)"0.0 1619 DO 170 L"1.2 1620 DAM'OAMRAT(L.MEM)-1 162 1 170 SDAMPIMEM)=SDAMP(MEM) + ((HARD'OAM+1)*DAM)/ 1622 1 ( 1-DAMRAT(L.MEM)'HARD) 1623 SDAMP(MEM)"0.15915494'SDAMP(MEM)/(DAMRAT(1.MEM)+DAMRAT(2.MEM))+. 02 1624 GO TO 190 1625 180 S0AMP(MEMI-0.02 1626 190 CONTINUE 1627 200 CONTINUE 1628 RE TURN 1629 END 1630 C 163 1 C . ....... ... ..».*...**....*.*•.•*.......... ............. « *...*** * * * 1632 C 1633 SUBROUTINE STACHK(OLDSA.SA,OLDTN.TN.ISPEC.LOCK.ICOUNT. IFLAG,IUNIT, 1634 1 AMAX,DAMP.KK) 1635 C 1636 C ...*.*.....***»...*........*...**..*.«.*......*....*..**..*..**** * * » 1637 C 1638 C THE INTENTION OF THIS SUBROUTINE IS TO DEAL WITH CONVERGENCE 1639 C INSTABILITY CAUSED BY STEEP SPECTRUM CONTOUR 1640 C 164 1 DIMENSION OLOTNI1). OLDSA(I) 1642 C 1643 IF (LOCK.GT.O) GO TO 70 1644 C 1645 IF (ICOUNT.LT.4) RETURN 1646 IF (ICOUNT EO.4.AND KK EO.1) GO TO 20 1647 IF <ICOUNT.EO.5.AND KK.EQ.1) GO TO 10 1648 IF (KK.E0.2I GO TO 30 1649 DIF1=OLDTN(2)-OLDTNI1) 1650 DIF 2 = 0LDTN( 1)-TN 1651 IF (DIF1.LT.-0.0O5.AND.DIF2.GT.0.005) GO TO 40 1652 IF (DIF1.GT.0.005.AND.DIF2.LT.-0.005) GO TO 40 1653 10 OLDTNf 2)=OLDTN( 1) 1654 20 DLDTN(1)=TN 1655 RETURN 1656 C 1657 30 OLDSA(2)=SA'(6.+100.'DAMP1/8. 1658 RETURN 1659 C 1660 C INSTABILITY DETECTED : START BINARY SEARCH ROUTINE 1661 C 1662 40 WRITE (99.130) 1663 WRITE (7.130) 1664 WRITE (99.50) I COUNT 1665 WRITE (7,50) I COUNT 1666 50 FORMAT (/3X,'CONVERGENCE PROBLEM OCCURRED AT ITERATION NO . 1667 1 12.' :'/3X.'SPECIAL CONVERGENCE ROUTINE IS NOW IN EFFECT '/ 1668 2 3X,'START BINARY SEARCH PROCEDURE -') 1669 KCOUNT=0 1670 LOCK" 1 167 1 OLOSA( 1 )"SA'I 6 * 100.*DAMP)/8. 1672 C 1673 C FIND UPPER BOUND AND LOWER BOUND SA(» 2% DAMPING) 1674 C - SET OLDSA! 1)*UPPER BOUND, OLDSAI2 ) = LOWER BOUND 1675 C OLDSA! 3 1 "TRIAL SA(«> 2% DAMPING) 1676 C 1677 IF (0LDSA(2).LT.OLDSA(1)) GO TO 52 1678 TEMP'OLDSA! 1 ) 1679 OLDSAl1)-0LDSA(2) 1680 0LDSA!2)'TEMP 1681 52 WRITE (99.55) OLOSA(1),OLDSA(2) 1682 WRITE (7,55) OLDSA(1).OLDSAl2) 1683 WRITE (99.120) 1684 WRITE I 7. 120) 1685 55 FORMAT I/3X,'UPPER BOUND SA = '.F7 5.4X.•LOWER BOUND SA 1686 60 OLDSAl3 I'(OLDSA! 1 )+pLDSA(2))/2. 1687 C 1688 C CALCULATE SA AND CHECK FOR CONVERGENCE (IFLAG=1 ) 1689 C 1690 70 SA'OLDSAI 3 )'8 . / (6. «-100. -DAMP ) 1691 IF ( I FLAG.EO. 1.AND.KK.EO.2) GO TO 80 1692 RETURN 1693 C 1694 C CHECK FOR REAL CONVERGENCE 1695 C 1696 80 CALL SPECTRIISPEC.DAMP.TN.AMAX.SAA.O .0..0..0.) 1697 SADIF=ABS(SA-SAAl/SAA 1698 IF (L0CK.E0.2) GO TO 150 1699 KCOUNT'KCOUNT*1 1700 WRITE (99.90) KCOUNT.SADIF 1701 WRITE (7.90) KCOUNT.SADIF 1702 90 FORMAT (/.' ',12.55('-').'SADIF = .F6.4) 1703 C 1704 C CONVERGENCE LIMIT FOR SA IS 0.015 1705 C 1706 IF (SADIF.LE.0.015) GO TO 100 1707 GO TO 140 1708 C 1709 100 WRITE(99.110) 17 10 WRITE(7. 1 10) 17 11 1 10 FORMAT (/50X. PROGRAM CONVERGED I SAERR'0.015>' ) 17 12 120 FORMAT('-'.'ITERATION '.IX.'NO. ABOVE DAMDIF'.3X. 17 13 1 'S MATRIX ',2X. 'SMEAREO'/' NO. '.5X. 'CAPACI TV' 17 14 2 14X.'RATIO ',2X,'DAMPING') 17 15 130 FORMAT!/80I'-')) 17 16 IFLAG'O 1717 L0CK»2 17 18 RETURN 17 19 C 1720 140 IF (SA.GT.SAA) OLDSA!1)=0LDSA(3) 172 1 IF (SA.LT.SAA) OLDSA!2)'OLDSAI 31 1722 IFLAG=0 1723 GO TO 60 1724 C 1725 150 WRITEI99,160) SA.SAA.SADIF 1726 WRITE(7.160) SA.SAA.SADIF 1727 160 FORMAT (/' '. 'SA = ' .F7.5, 1728 1 3X.'SA(ACTUAL) ' '.F7.5.3X,'SADIF ' '.F7.5/) 1729 RETURN 1730 END 1731 C 1732 C 1733 C 1734 SUBROUTINE SPECTRII SPEC.DAMP.TN.AMAX.SA,WN.SABND.SVBND.: 1735 C 1736 C 1737 C 1738 C 1739 C ISPEC'1 IF SPECTRUM A IS USED 1740 C =2 IF SPECTRUM B IS USED 1741 C =3 IF SPECTRUM C IS USED 1742 C =4 IF NBC SPECIRUM IS USED 1743 C =5 IF SAN FERNANDO E/Q RECORD 143 IS USED 1744 C =6 IF SIMULATED E/Q C-2 IS USED 1745 C DAMP=DAMPING FACTOR (FRACTION OF CRITICAL DAMPING) 1746 C TN N^ATURAL PERIOD IN SECONDS 1747 C AMAX =MAXIMUM GROUND ACCELERATION (FRACTION OF G) 1748 C SA 'RESPONSE ACCELERATION (FRACTION OF G) 1749 C WN 'NATURAL FREQUENCY IN RADIANS PER SECOND. 1750 C 1751 CALL FTNCMDI'EQUATE 99'SPRINT:') 1752 GO TO (5,10.60.100.130.135).ISPEC 1753 C 1754 C SPECTRUM A 1755 C 1756 5 IFITN.LT.O.15) SA=25.*AMAX*TN 1757 IFITN.GE 0.15 AND. TN.LT.0.4) SA'3.75»AMAX 1758 IF!TN.GT.0.4) SA=1.5'AMAX/TN 1759 GO TO 90 1760 C 176 1 C SPECTRUM B 1762 C 1763 10 CONTINUE 1764 IFITN.LT.O.1875) GO TO 20 1765 IF(TN. LT .0. 53333333 ) GO TO 30 1766 IF!TN.LT.1.6666667> GO TO 40 1767 IF(TN.LT.1.81666667) GO TO 50 1768 SA=2.*AMAX/(TN-0.75) O 1769 GO TO 90 I 1770 20 SA=20.*AMAX*TN _» 177 1 GO TO 90 (Jl 1772 30 SA'3.75*AMAX 1773 GO TO 90 1774 40 SA = 2.* AMAX/TN 1775 GO TO 90 1776 50 SA'1.875*AMAX 1777 GO TO 90 1778 C 1779 C SPECTRUM C 1780 C 1781 60 CONTINUE 1782 IFITN.LT.O.15) GO TO 70 1783 IF(TN.LT.0.38333333) GO TO 80 1784 SA'O.5*AMAX/(TN-O.25) 1785 GO TO 90 1786 70 SA=25.*AMAX*TN 1787 GO TO 90 1788 80 SA=3.75*AMAX 1789 90 CONTINUE 1790 SA=SA*8./(6.+100.*DAMP) 1791 RETURN 1792 C 1793 C NBC SPECTRUM 1794 C 1795 100 CONTINUE 1796 SV«40.0*AMAX 1797 SD = 32.0*AMAX 1798 SACC'1 0«AMAX 1799 C PRINT OUT A CAUTION NOTE SHOULD DAMPING BE LESS THAN 0.5% 1800 I F'l DAMP . LT . 0.0O5 I WRITEI7.110) 1801 1 10 FORMAT! ' ' . ' C AUT I ON-DAMP I NG LESS THAN 0.5"/.') 1802 C 1803 C COMPUTE MULTIPLICATION FACTOR FOR ACCELERATION AT DESIRED DAMPING 1804 I FIDAMP.LE.0.02) AMLC4.2-M (0.02-DAMP 1/0.0151*1.6 1805 IF1DAMP.GT. .02 .AND.DAMP.LE. .051AML = 3.0»(( 05-OAMP1/.03)* 1.2 180G IF(DAMP GT.0.05.AND DAMP LE.0.1IAML=2.2+( (0.1-DAMP)/0.05)*0.8 1807 IF(DAMP GT 0 10) AML=1 .0+1(1 00-DAMPI/O.901*1.2 1808 C 1809 C COMPUTE MULTIPLICATION FACTOR FOR VELOCITY AT DESIRED DAMPING. 1810 IFI DAMP LE.0.02) VML=2.5+1 10.02-DAMP)/0.015)'0.8 181 1 IF(DAMP.GT. 02 .AND DAMP.LE..051VML=2.0+((.05-DAMP)/.03)-0.5 1812 IF(DAMP GT..05.AND DAMP.LE.0.1)VML=1.7+1(0.1-OAMP)/0.05)»0.3 1813 IF1 DAMP.GT.0 10) VML=1.0+1(1.00-DAMPI/O.90)'0.7 18 14 c 1815 c COMPUTE MULTIPLICATION FACTOR FOR DISPLACEMENT AT DESIREO DAMPING 18 16 IF! DAMP.LE.0.02) DML = 2.5+((0.02-DAMP I/O.015)'0.5 1817 IFIDAMP.GT.0.021 DML=VML 1818 c 1819 c COMPUTE BOUNDS USING DAMPING FACTORS COMPUTED ALREADY 1820 SDBND=SD'DML 182 1 SABND=SACC*AML 1822 SVBND=SV*VML 1823 c COMPUTE WHICH IS THE APPROPIATE BOUND. 1824 c CONVERT FROM IN/SEC**2 TO FRACTION OF G BY DEVIDING BY 386.4 1825 c 1826 SAATAP=SVBND'WN/386.4 1827 IF(SAATAP.GT.SABND) SA-SABND 1828 IF<SAATAP.GT.SABND) GO TO 120 1829 SDATCP»SVBND/WN 1830 IFISDATCP.GT SDBND) SA=SDBND'WN'WN/386.4 1831 IF(SDATCP.GT.SDBND) GO TO 120 1832 c 1833 c IF HAVE NOT YET GONE TO STEP 180 THEN NATURAL FREQUENCY LIES ON 1834 c VELOCITY BOUND. 1835 c 1836 SA=SVBND'WN/386.4 1837 c SA IS RETURNED AS A FRACTION OF GRAVITY, G 1838 c 1839 120 RETURN 1840 C 184 1 C SAN FERNANDO E/Q. HOLIDAY INN. LONGITUDINAL DIRECTION 1842 c 1843 130 IF (TN.LE.0.2) 1844 1 SAM 1 .013+1 1 605*TN)*AMAX 1845 IF (TN GT.0 2 AND TN.LE 0.62) 1846 1 SA=3.334'AMAX 1847 IF ITN.GT.0.62.AND.TN.LE.0.82) 1848 1 SAM5 . 750-3 896'TN)*AMAX 1849 IF (TN.GT.0.82 AND.TN.LE.1.7) 1850 1 SAM 2 . 772-0. 265'TN)'AMAX 1851 IF ITN.GT.1.7.AND.TN.LE.2.4) 1852 1 SAM3.263-0.554'TN)'AMAX 1853 IF (TN.GT.2.4) GO TO 140 1B54 GO TO 90 1855 C 1856 C CIT/SIMULATED EARTHQUAKE C-2 1857 C 1858 i :.ir ir I IN.LT.0.17) 1859 1 SAM 0 62 16*22 432•TN)'AMAX 1860 IF HIJ.GE.O 17. AND TN.LT.0.511 186 1 1 5A»4 . .I35'AMA> 1862 IF (TN.GE.0.51.AND.TN.LT.0.58) 1863 1 SAM 2 1 .83 1 -34 . 1 1 'TNI'AMAX 1864 IF (TN.GE.0.58.AND.TN.LT.1.8) 1865 1 SAM 2 . 723- 1 . 164 • TN) *AMAX 1866 IF (TN.GE. 1.8.AND.TN.LE.2.4 > 1867 1 SAM 1 . 457-0. 461 *TN) 'AMAX 1868 IF (TN.GT.2.4) GO TO 140 1869 GO TO 90 1870 140 WRITEI99.150) TN 1871 150 FORMAT (/.5X,'PERIOD = '.F10.3.' IS OUT OF THE SPECTRUM') 1872 RETURN 1873 END 1874 C 1875 C 1876 C 1877 SUBROUTINE SCHECKIS.NU.N8.IDIM.IUNIT.SRAT10) 1878 C 1879 C 1880 C 1881 C THIS SUBROUTINE CHECKS THAT ALL DIAGONAL STIFFNESS MATRIX 1882 C ELEMENTS ARE POSITIVE NUMBERS GREATER THAN ZERO. IT ALSO DETERMINES 1883 C THE RATIO BETWEEN THE LARGEST AND SMALLEST MEMBERS ON THE DIAGONAL 1884 C THIS WILL GIVE SOME INDICATION AS TO THE CONDITIONING OF THE 1885 C STIFFNESS MATRIX 1886 C MATRIX 1887 C 1888 REAL'S SIIDIM) 1889 REAL'S SMIN.SMAX,DIAG.RATIO 1890 C 1891 C 1892 C THE STIFFNESS MATRIX IS STORED AS A COLUMN VECTOR. ONLY THE 1893 C THE LOWER TRIANGLE ELEMENTS BEING STORED (BY COLUMNS) 1894 C S(D IS ON THE DIAGONAL AS IS SI 1+NB ) .S(1+2'NB),ETC. 1895 C NB IS THE HALF BANDWIDTH OF THE STIFFNESS MATRIX 1896 C 1897 C INITIALIZE THE LARGEST AND SMALLEST VALUES OF DIAGONAL (SMAX.SMIN) 1898 C 1899 SMIN=1.0D45 1900 SMAX = - 1 ODOO 1901 c 1902 DO 50 IDOF=1,NU 1903 IELEM=((IDOF-1)"NB )+1 1904 OIAG=S(IELEMI 1905 c COMPUTE IF DIAGONAL ELEMENT IS ZERO OR NEGATIVE 1906 IF(DIAG NE O.ODOO) GO TO 20 1907 WRITEI7.10) IDOF 1908 10 FORMAT(///' PROGRAM HALTEO-A ZERO IS ON THE DIAGONAL OF STIFFNE 19091 1SSMATRIX ',.//' EXAMINE DEGREE OF FREEDOM ',14) 19 10 STOP 191 1 c 1912 20 CONTINUE 1913 IF(DIAG.GT.0.0) GO TO 40 1914 WRITEI7.30) IDOF 19 15 30 FORMAT!///' PROGRAM HALTED-NEGATIVE ELEMENT ON DIAGONAL OF '. 19 16 1 STIFFNESS MATRIX'.//' EXAMINE DEGREE OF FREEDOM'.14) 19 17 STOP 19 18 JO CONTINUE 19 19 c 1920 c DETERMINE IF THE DIAGONAL ELEMENT UNDER EXAMINATION IS THE LARGEST OR 192 1 c SMALLESI OF THE DIAGONAL ELEMENTS 1922 IF(DIAG.GT.SMAX) SMAX=OIAG 1923 IFIDIAG.LT SMIN) SMIN = DI AG 1924 c 1925 50 CONTINUE 1926 C 1927 WRITE!IUNIT.60) 1928 60 FORMAT!/' ALL ELEMENTS OF MAIN DIAGONAL OF STIFFNESS MATRIX', 1929 1 ' ARE POSITIVE DEFINITE') 1930 C 193 1 C COMPUTE AND PRINT RATIO OF LARGEST TO SMALLEST DIAGONAL ELEMENTS 1932 C 1933 RATIO=SMAX/SMIN 1934 SRATIO=SNGLIRATIO) 1935 WRITE!IUNIT.70) SRATIO 1936 70 FORMAT(' '.-RATIO OF LARGEST TO SMALLEST DIAGONAL STIFFNESS'. 1937 1 'MATRIX ELEMENT IS.E10.3) 1938 RETURN 1939 ENO 1940 SUBROUTINE SDFBAN(A,B.N,M.LT.RAT 10.DET.NCN,NSCALE) 1941 C 1942 C THIS ROUTINE SOLVES SYSTEM OF EONS. AX=B WHERE A IS + TVE DEFINITE 1943 C SYMMETRIC BAND MATRIX. BY CHOLESKY'S METHOD. 1944 C LOWER HALF BAND ONLY (INCLUDING THE DIAGONAL) OF A IS STORED 1945 C COLUMN BY COLUMN IN A 1 DIMENSIONAL ARRAY. 1946 C SOLUTIONS X ARE RETURNED IN ARRAY B. 1947 C OPTIONAL SCALING OF MATRIX A IS AVAILABLE 1948 C N - ORDER OF MATRIX A. 1949 C M - LENGTH DF LOWER HALF BAND. 1950 C DETERMINANT OF A = DET *( 10* *NCN). 1 .E- 15<|DET|<1 E15 1951 C LT = 1 IF ONLY 1 B VECTOR OR IF FIRST OF SEVERAL. LT NOT = 1 FOR 1952 C SUBSEQUENT B VECTORS. 1953 C RATIO = SMALLEST RATIO OF 2 ELEMENTS ON MAIN DIAGONAL OF 1954 C TRANSFORMED A >1.E-7. 1955 C NSCALE-0 IF SCALING NOT REOUIREO. 1956 C 1957 C 1958 IMPLICIT REAL'S (A-H.O-Z) 1959 DIMENSION A(1).B(1) 1960 REAL *8 MULT(200) 1961 1FIM.EQ.1) GO TO 101 1962 MM=M-1 1963 NM=N*M 1964 NM 1 =NM-MM 1965 C 1966 C DUMMY STATEMENT INSERTED FOR COMPATIBILITY WITH ASSEMBLER VERSION. 1967 C IF(LT.LE.O) RETURN 1968 C 1969 IFILT.NE.1) GO TO 55 1970 IF(NSCALE.EO.O) GO TO 3001 1971 DO 3000 1=1.N 1972 C 1973 C MATRIX SCALED BY DIVIDING ROW I AND COLUMN I BY SORT(A(I.I)). SUCH 1974 C THAT DIAGONAL ELEMENTS All.I I ARE I. 1975 C 1976 1 I=( I - 1 ) *M»1 1977 1 F(A(I I I.LE.0.0) GO TO 104 1978 3000 MULTII) = 1 0/OSORTI A(II)) 1979 KK=1 1980 DO 301 1=1.N 198 1 I I = I I - 1)* M+ 1 1982 JEND=II+MM 1983 IMN=(I - 1 )*M-N 1984 IF(IMN.GT.O) JEND = JEND-IMN 1985 DO 302 J=I I .JEND 1986 A(J)=A(JI'MULTII) 1987 302 CONTINUE 1988 DO 306 J = KK,II.MM 1989 306 A(J)=AIJ)* MULT(I> 1990 IFIKK.GE.M) GO TO 307 1991 KK=KK*1 1992 GO TO 301 1993 307 KK = KK+M 1994 301 CONTINUE 1995 3001 MP=M+1 1996 C 1997 C TRANSFORMATION OF A. 1998 C A IS TRANSFORMED INTO A LOWER TRI ANGULAR MATRIX L SUCH THAT A-L.LT 1999 C (L T = TRANSPOSE OF L ) . IF Y=LT X THEN L.Y=B. 2000 C ERROR RETURN TAKEN IF RATI0<1.E-7 2001 C 2002 KK=2 2003 NCN=0 2004 DET=0. 2005 FAC=RATI0 2006 IF(A(1) GT O.) GO TO 15 2007 NROW-1 2008 RATIO-AO ) 2009 GO TO 60 2010 15 0ET=A(1) 2011 A ( 1 ) = 1 . /DSQRT ( AO ) ) 2012 BIGL=A(1) 2013 SML = AO) 2014 A(2)=A(2)«A(1) 2015 TEMP=A(MP)-A(2)*A(2) 2016 IFITEMP.LT 0.0) RATIO=TEMP 2017 IF(TEMP.EQ.O.O) RATIO=0.0 2018 IF(TEMP.GT.0.0) GO TO 21 2019 NR0W-2 2020 GO TO 60 2021 101 0ET=1.D0 2022 NCN-0 2023 DO 102 1-1,N 2024 DET=DET*A( I ) (-} 2025 IFIA(I).EO.O.O) GO TO 104 | 2026 IFCDET GT.1.E-15) GO TO 1144 _. 2027 0ET=DET*1.E+15 ^ 2028 NCN-NCN-15 2029 GO TO 1145 2030 1144 IF(DET.LT. 1 .E+15) GO TO 1145 200 1 DE T = DE T ' 1 . E - 15 2032 NCN=NCN*(5 2033 1145 CONTINUE 2034 102 BI I ) =EI I )/A( I ) 2035 RETURN 2036 104 RATIO = A( I ) 2037 103 NROW=I 2038 GO TO 60 2039 21 A(MP)=1.0/0SQRT(TEMP) 2040 DET = DET * TEMP 2041 IF(A(MP) GT BIGL)BIGL=A(MP) 2042 IF(AI MP) L T.5ML)SML =A(MP) 2043 IFIN.EQ 2) GO TO 52 2044 MP=MP+M 2045 DO 62 J=MP.NM1.M 2046 JP=J-MM 2047 MZC-0 2048 IFIKK.GE.M) GO TO 1 2049 KK=KK+1 2050 11=1 2051 JC=1 2052 GO TO 2 2053 1 KK=KK+M 2054 1I=KK-MM 2055 JC=KK-MM 2056 2 DO 65 I=KK.JP.MM 2057 IF I A(I).EQ.0. ) GO TO 64 2058 GO ID 66 2059 64 JC=JC+M 2060 65 MZC=MZC+1 2121 C THE FOLLOWING STATEMENTS SOLVE FOR L.V-B BY A FORWARDS 2061 ASUM1=0 00 2122 C HENCE FOR X FROM LT.X-Y BY A BACKWARDS SUBSTITUTION. 2062 GO TO 61 2123 C IF SCALING OPTION USED. B IS SCALED AND NORMALISED BEFORE 2063 66 MMZC=MM*MZC 2124 C SUBSTITUTION BEGINS. 2064 II-II+MZC 2125 C 206S KM-KK+MMZC 2126 55 SUM-O.DO 2066 A(KM)=A(KM)*A(UC) 2127 IF( NSCALE .EO.O) GO TO 3010 2067 IFIKU.GE.JP) GO TO 6 2128 DO 303 I-1.N 2068 KO=KM+MM 2129 B(I)=B(I)*MULT<I) 2069 00 5 I=K0.OP,MM 2130 SUM»SUM+B(I)«B(I) 2070 ASUM2=0.D0 2131 303 CONTINUE 207 1 IM=I-MM 2132 ELENB=DS0RT( SUM) 2072 11=11+1 2133 DO 304 I=1.N 207 3 KI-II+MMZC 2 134 304 B(I)=B(I)/ELENB 2074 DO 7 K = KM.IM.MM 2135 3010 B( 1 )=B(1)*A(1) 2075 ASUM2=ASUM2+A(KI)*A(K) 2136 KK- 1 2076 7 KI=KI+MM 2 137 K1 = 1 2077 5 A(1 > = (A(I )-ASUM2)'A(KI) 2 138 0=1 2078 6 CONTINUE 2139 DO 8 L-2.N 2079 ASUM1=0.DO 2140 BSUM1-0.D0 2080 00 4 K=KM.UP.MM 2141 LM=L-1 208 1 4 ASUM1= ASUM1 + A(K)* A(K) 2142 d-d+M 2082 6 1 5=A(U)-ASUM1 2143 IF (KK . GE . M )G0 TO 12 2083 IF(S.LT.0.)RATIO=S 2144 KK-KK+1 2084 IFIS.EQ.O.>RAT10=0. 2145 GO TO 13 2085 IF(S GT.0.IGO TO 63 2146 12 KK=KK+M 2086 NROW=< 0+MM)/M 2 147 K1=K1+1 2087 GO TO 60 2 148 13 dK=KK 2088 63 A(0 > =1./DSORTIS) 2 149 DO 9 K=K1.LM 2089 DET=DET'S 2 150 BSUM1=BSUM1+A1JK)'B(K) 2090 IF(OET.GT.1.E- 15) GO TO 144 2151 OK = dK + MM 2091 DET-DET'I.E+15 2 152 9 CONTINUE 2092 NCN=NCN-15 2153 8 B(L)=(B(L)-BSUM1)'A(d) 2093 GO TO 145 2 154 100 B(N)=B(N)*A(NM1) 2094 144 IF(DET.LT. 1 . E+15) GO TO 145 2155 NMM=NM1 2095 DET=DET*1 E-15 2 156 NN=N-1 2096 NCN=NCN+15 2 157 ND=N 2097 145 CONTINUE 2 158 DO 10 L=1.NN 2098 lF(A(d) GT BIGL)BIGL-A(d) 2 159 BSUM2=0.D0 2099 IF(A(d).LT.SML)SML=A(J) 2160 NL=N-L 2 100 62 CONTINUE 2161 NL1=N-L+1 2101 52 IF(SML.LE.FAC*BIGL) GO TO 54 2162 NMM=NMM-M 2102 GO TO 53 2163 NO 1=NMM 2 103 54 RATIO=0. 2 164 I F ( L.GE.M)ND=ND- 1 2104 RETURN 2 165 DO 11 K=NL1.N0 2 105 60 WRITE(6.99) NROW 2 166 N01=N01+1 2 106 99 FORMAT* 0*'*SVSTEM IS NOT POSITIVE DEFINITE'. 2167 BSUM2=BSUM2+A(NU1)»B(K) 2107 C ' ERROR CONDITION OCCURRED IN ROW'.14) 2 168 11 CONTINUE 2108 RETURN 2169 10 B(NL)=(B(NL)-BSUM2)*A(NMM) 2 109 53 RATIO=SML/BIGL 2 170 IF(NSCALE.EO.O) GO TO 3015 2 1 10 55 CALL DSBAND(A.MULT .B.N.M.NSCALE ) 2171 DO 305 I-I.N 2 111. RETURN 2 172 305 B(I ) =B(I )*ELENB*MULT(I ) 21 12 END 2 173 3015 RETURN 2 174 END 2113 SUBROUTINE DSBANDIA.MULT.B.N.M.NSCALE) 2 114 IMPLICIT REAL'S (A-H.O-Z) 2115 DIMENSION A( 1 ).B( 1) 2116 REAL'S MULT( 1 I 2 117 MM = M-1 2 1 18 NM=N+M 2 1 19 NM1=NM-MM 2120 C 

Cite

Citation Scheme:

        

Citations by CSL (citeproc-js)

Usage Statistics

Share

Embed

Customize your widget with the following options, then copy and paste the code below into the HTML of your page to embed this item in your website.
                        
                            <div id="ubcOpenCollectionsWidgetDisplay">
                            <script id="ubcOpenCollectionsWidget"
                            src="{[{embed.src}]}"
                            data-item="{[{embed.item}]}"
                            data-collection="{[{embed.collection}]}"
                            data-metadata="{[{embed.showMetadata}]}"
                            data-width="{[{embed.width}]}"
                            async >
                            </script>
                            </div>
                        
                    
IIIF logo Our image viewer uses the IIIF 2.0 standard. To load this item in other compatible viewers, use this url:
http://iiif.library.ubc.ca/presentation/dsp.831.1-0062849/manifest

Comment

Related Items