UBC Theses and Dissertations

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UBC Theses and Dissertations

Reliability of floors under impact vibration Gupta, Ashwani 1985

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R E L I A B I L I T Y OF FLOORS UNDER IMPACT V IBRATION by ASHWANI GUPTA B . T E C H , INDIAN I N S T I T U T E OF TECHNOLOGY, DELHl j l I N D I A , 198 A T H E S I S SUBMITTED IN P A R T I A L F U L F I L M E N T OF THE REQUIREMENTS FOR THE DEGREE OF MASTER OF A P P L I E D S C I E N C E i n THE F A C U L T Y OF GRADUATE S T U D I E S DEPARTMENT OF C I V I L ENGINEERING 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 ^ - s t a n d a r d THE U N I V E R S I T Y OF B R I T I S H COLUMBIA A P R I L , 1 9 8 5 © ASHWANI G U P T A , A P R I L , 1 9 8 5 I n p r e s e n t i n g t h i s t h e s i s i n p a r t i a l f u l f i l m e n t o f t h e r e q u i r e m e n t s f o r an advanced degree a t t h e U n i v e r s i t y o f B r i t i s h C o l u m b i a , I agree t h a t t h e L i b r a r y s h a l l make i t f r e e l y a v a i l a b l e f o r r e f e r e n c e and s t u d y . I f u r t h e r a g r e e t h a t p e r m i s s i o n f o r e x t e n s i v e c o p y i n g o f t h i s t h e s i s f o r s c h o l a r l y p u rposes may be g r a n t e d by t h e head o f my department o r by h i s o r h e r r e p r e s e n t a t i v e s . I t i s u n d e r s t o o d t h a t c o p y i n g o r p u b l i c a t i o n o f t h i s t h e s i s f o r f i n a n c i a l g a i n s h a l l n o t be a l l o w e d w i t h o u t my w r i t t e n p e r m i s s i o n . Department o f CIVIL ENGINEERING 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 1956 Main Mall V a ncouver, Canada V6T 1Y3 Date APRIL,1985 /RI ABSTRACT F l o o r v i b r a t i o n i s a s e r v i c e a b i l i t y l i m i t s t a t e p r o b l e m a s o p p o s e d t o l i m i t s t a t e o f c o l l a p s e . P e o p l e f i n d e x c e s s i v e v i b r a t i o n s a n n o y i n g a n d i t l e a d s t o u n a c c e p t a b l e f l o o r p e r f o r m a n c e . I m p r o v e m e n t s i n d e s i g n t e c h n i q u e s a n d u s e o f h i g h e r s t r e n g t h m a t e r i a l s h a v e made f l o o r s l i g h t e r a n d more s e n s i t i v e t o v i b r a t i o n s . T h e e x i s t i n g code" c r i t e r i a f o r l i m i t i n g f l o o r d e f l e c t i o n a r e n o t s a t i s f a c t o r y a s t h e y do n o t r e l a t e t o human p e r c e p t i o n o f v i b r a t i o n s . A f i n i t e e l e m e n t c o m p u t e r p r o g r a m i s d e v e l o p e d f o r a r a t i o n a l a n a l y s i s o f f l o o r v i b r a t i o n s due t o human f o o t f a l l i m p a c t . The a p p r o a c h f o l l o w e d i n t h e s o l u t i o n i s t o u s e a F o u r i e r s e r i e s e x p a n s i o n o f d i s p l a c e m e n t f u n c t i o n s a l o n g t h e s p a n a n d a f i n i t e e l e m e n t a p p r o x i m a t i o n i n t h e t r a n s v e r s e d i r e c t i o n . A t i m e s t e p a n a l y s i s i s f o l l o w e d t o o b t a i n t h e f l o o r ' s d y n a m i c r e s p o n s e a t a n y p o i n t on t h e f l o o r due t o f o o t f a l l i m p a c t e l s e w h e r e . A s e n s i t i v i t y a n a l y s i s i s done t o s t u d y t h e e f f e c t o f v a r i o u s f l o o r p a r a m e t e r s on f l o o r p e r f o r m a n c e . A r e l i a b i l i t y s t u d y o f f l o o r p e r f o r m a n c e a s a f f e c t e d by v a r i a b i l i t y o f j o i s t s t i f f n e s s i s made by s i m u l a t i o n . F l o o r s s i m u l a t e d by r a n d o m l y s e l e c t i n g j o i s t s f r o m a p o p u l a t i o n a r e a n a l y z e d a n d v i b r a t i o n r e s p o n s e r a t i n g s a r e o b t a i n e d by r e l a t i n g t h e r e s p o n s e t o human p e r c e p t i o n c r i t e r i a . The p r o b a b i l i t y o f p e o p l e r a t i n g t h e f l o o r a s o f a c e r t a i n p e r f o r m a n c e l e v e l i s p r e d i c t e d . B a s e d on t h e r e l i a b i l i t y a n a l y s i s , a l l o w a b l e s p a n s a n d a l l o w a b l e d e f l e c t i o n s f o r a i i c e r t a i n l e v e l o f f l o o r p e r f o r m a n c e a n d a t a r g e t r e l i a b i l i t y a r e p r o p o s e d . i i i T a b l e o f C o n t e n t s ABSTRACT i i L I S T OF T A B L E S v i L I S T OF F IGURES v i i ACKNOWLEDGEMENTS i x 1. INTRODUCTION 1 1 .1 T h e P r o b l e m 1 1.2 B a c k g r o u n d a n d P r e v i o u s R e s e a r c h 2 1.3 O b j e c t i v e a n d C o n t e n t o f t h e T h e s i s 5 2 . T H E O R E T I C A L FORMULATION 8 2.1 T h e M o d e l 8 2 . 2 F o r m u l a t i o n o f t h e E q u a t i o n s o f M o t i o n 13 2 . 2 . 1 D e r i v a t i o n o f S t i f f n e s s M a t r i x 16 2 . 2 . 2 D e r i v a t i o n o f M a s s M a t r i x 24 2 . 2 . 3 D e r i v a t i o n o f D a m p i n g M a t r i x 28 2 . 2 . 4 D e r i v a t i o n o f a C o n s i s t e n t L o a d V e c t o r . . . . 3 0 2 . 2 . 5 D y n a m i c E f f e c t s o f P e o p l e on F l o o r 31 2 . 2 . 6 F i n a l E q u a t i o n s o f M o t i o n 35 2 . 3 S o l u t i o n o f t h e E q u a t i o n s o f M o t i o n 35 3 . THE COMPUTER PROGRAM 40 3.1 P r o g r a m F e a t u r e s 40 3 . 2 P r o g r a m S t r u c t u r e 41 4 . HUMAN PERCEPTION OF V IBRATIONS 46 4.1 Human D i s c o m f o r t C r i t e r i o n 46 4 . 2 Human P e r c e p t i o n a n d F l o o r R e l i a b i l i t y 50 5 . R E S U L T S AND DISCUSSIONS 53 5.1 N u m e r i c a l C o n v e r g e n c e 53 5 . 2 S e n s i t i v i t y A n a l y s i s 54 5 . 3 R e l i a b i l i t y A n a l y s i s a n d D e s i g n C r i t e r i o n 64 6 . CONCLUSIONS 75 REFERENCES 76 APPENDIX I 78 APPENDIX I I 80 APPENDIX I I I 96 v L I S T O F T A B L E S T a b l e Page 5.1 R e f e r e n c e F l o o r 54 5 .2 P e r f o r m a n c e o f R e g r e s s i o n E q u a t i o n s 64 5 . 3 F l o o r t y p e s a n a l y z e d 65 5 . 4 A l l o w a b l e s p a n s i n f t f o r R 0 =3 w i t h r e c e i v e r 69 a n d i m p a c t e r on a d j a c e n t j o i s t s 5 . 5 A l l o w a b l e d e f l e c t i o n i n mm f o r R 0 =3 w i t h 70 r e c e i v e r a n d i m p a c t e r on a d j a c e n t j o i s t s 5 . 6 A l l o w a b l e s p a n i n f t f o r R 0 = 3 w i t h r e c e i v e r 71 a n d i m p a c t e r on same j o i s t 5 .7 A l l o w a b l e d e f l e c t i o n i n mm f o r R 0 =3 w i t h 71 r e c e i v e r a n d i m p a c t e r on same j o i s t 5 . 8 D e f l e c t i o n p r o v i s i o n s i n d i f f e r e n t c o u n t r i e s 73 5 . 9 A l l o w a b l e d e f l e c t i o n b a s e d on d i s t r i b u t e d 74 l o a d w i t h r e c e i v e r a n d i m p a c t e r on a d j a c e n t j o i s t s v i L I S T OF F IGURES F i g . Page 1 . 1 P r o b l e m of i n t e r e s t 6 2 . 1 F l o o r A s s e m b l y 8 2 . 2 N o d e s f o r T - b e a m f i n i t e e l e m e n t 9 2 . 3 J o i s t d e g r e e s o f f r e e d o m 10 2 . 4 F l a n g e a r e a o f f i n i t e e l e m e n t 18 2 . 5 V e l o c i t y c o m p o n e n t s o f j o i s t p a r t i c l e 26 2 . 6 Human B e i n g M o d e l 32 2 . 7 F l o o r w i t h P e o p l e 33 2 . 8 L i n e a r a c c e l e r a t i o n v a r i a t i o n 36 4 . 1 V i b r a t i o n s i g n a l u s e d by W i s s & P a r m e l e e 46 4 . 2 W i s s & P a r m e l e e ' s m o d e l c u r v e s 48 4 . 3 T y p i c a l f l o o r r e s p o n s e t o f o o t f a l l i m p a c t 49 4 . 4 R e l i a b i l i t y i n d e x a n d P r o b a b i l i t y o f f a i l u r e 51 5 . 1 E f f e c t o f " n " on f l o o r r e s p o n s e 53 5 . 2 E f f e c t o f f l o o r d a m p i n g on r e s p o n s e 57 5 . 3 E f f e c t o f mass d a m p i n g on r e s p o n s e 57 5 . 4 E f f e c t o f s p a n on r e s p o n s e 58 5 . 5 E f f e c t o f j o i s t d e p t h on r e s p o n s e 58 5 . 6 E f f e c t o f j o i s t s p a c i n g on r e s p o n s e 59 5 . 7 E f f e c t o f n a i l s p a c i n g on r e s p o n s e 59 5 . 8 E f f e c t o f p e o p l e on r e s p o n s e 60 5 . 9 E f f e c t o f t o r s i o n a l r e s t r a i n t on r e s p o n s e 60 5 . 10 P l o t f o r f l o o r t y p e A 67 5 . 1 1 P l o t f o r f l o o r t y p e B 67 5 . 12 P l o t f o r f l o o r t y p e C 68 v i i 5 . 1 3 P l o t f o r f l o o r t y p e D 68 5 . 1 4 R e c e i v e r and I m p a c t e r on same j o i s t 70 v i i i ACKNOWLEDGEMENTS The a u t h o r t a k e s t h i s o p p o r t u n i t y t o g r a t e f u l l y a c k n o w l e d g e t h e g u i d a n c e o f h i s a d v i s o r D r . R . O . F o s c h i t h r o u g h o u t t h i s r e s e a r c h a n d t h e s i s p r e p a r a t i o n . H i s s u g g e s t i o n s a n d a d v i c e d u r i n g o u r d i s c u s s i o n s p r o v e d i n v a l u a b l e . The a u t h o r a l s o e x p r e s s e s h i s s i n c e r e g r a t i t u d e t o D r . N . D . N a t h a n f o r r e a d i n g t h i s t h e s i s a n d s u g g e s t i n g i m p r o v e m e n t s . The f i n a n c i a l s u p p o r t i n t h e f o r m o f a R e s e a r c h A s s i s t a n t s h i p f r o m N a t u r a l S c i e n c e and E n g i n e e r i n g R e s e a r c h C o u n c i l o f C a n a d a i s g r a t e f u l l y a c k n o w l e d g e d . i x 1 . INTRODUCTION 1.1 THE PROBLEM F l o o r v i b r a t i o n i s an i m p o r t a n t s e r v i c e a b i l i t y l i m i t s t a t e i n f l o o r d e s i g n a s p e o p l e f i n d e x c e s s i v e v i b r a t i o n s a n n o y i n g . V i b r a t i o n s c a u s e d by i m p a c t s due t o human f o o t f a l l i s a common s o u r c e o f t h e s e v i b r a t i o n s . More e f f i c i e n t d e s i g n m e t h o d s a n d u s e o f h i g h e r s t r e n g t h m a t e r i a l s t e n d t o make t h e f l o o r s l i g h t e r a n d more s e n s i t i v e t o v i b r a t i o n s . T h e e x i s t i n g c o d e c r i t e r i a o f l i m i t i n g s t a t i c f l o o r d e f l e c t i o n a r e g e n e r a l l y n o t s a t i s f a c t o r y b e c a u s e t h e y a r e n o t r e l a t e d t o t h e v i b r a t i o n o f t h e f l o o r w h i c h t h e p e r s o n f e e l s . H e n c e t h e r e e x i s t s a n e e d f o r a s y s t e m a t i c s t u d y o f t h e f l o o r v i b r a t i o n s c a u s e d by f o o t f a l l i m p a c t s a n d t o e s t a b l i s h a d e s i g n c r i t e r i o n f o r a c c e p t a b l e f l o o r p e r f o r m a n c e . We n e e d t o f o c u s o u r a t t e n t i o n on t h e f o l l o w i n g a s p e c t s o f t h e p r o b l e m : What i s t h e p e a k d i s p l a c e m e n t a n d f r e q u e n c y o f t h e v i b r a t i o n s w h i c h a p e r s o n s e n s e s a s f o o t f a l l i m p a c t i s a p p l i e d t o t h e f l o o r by a n o t h e r p e r s o n ; How do v a r i o u s f l o o r p a r a m e t e r s a f f e c t t h e f l o o r ' s d y n a m i c r e s p o n s e ; How t o c l a s s i f y t h e f l o o r ' s r e s p o n s e a s s a t i s f a c t o r y o r d i s t u r b i n g a s p e r c e i v e d by human b e i n g s ; What d e s i g n c r i t e r i a t o e m p l o y t o e n s u r e a c c e p t a b l e f l o o r p e r f o r m a n c e ; 1 2 1.2 BACKGROUND AND PREVIOUS RESEARCH F l o o r s t r u c t u r e s h a v e t r a d i o n a l l y b e e n d e s i g n e d w i t h r e s p e c t t o s t a t i c l o a d c a r r y i n g c a p a c i t y . T h e s t i f f n e s s r e q u i r e m e n t h a s u s u a l l y b e e n e x p r e s s e d i n t e r m s o f m i d s p a n d e f l e c t i o n g i v e n a s a f r a c t i o n o f a c t u a l s p a n l e n g t h . A more r a t i o n a l a n a l y s i s o f t h e p e r f o r m a n c e o f f l o o r s y s t e m s h a s b e e n made p o s s i b l e w i t h t h e d e v e l o p m e n t o f t h e f i n i t e e l e m e n t m e t h o d . Two e x c e l l e n t w o r k s w h i c h e m p l o y t h i s s o p h i s t i c a t e d t e c h n i q u e t o a n a l y z e t i m b e r f l o o r s a r e q u o t e d h e r e . T h e f i r s t one by T h o m p s o n , Goodman a n d V a n d e r b i l t {13} d e s c r i b e s a f i n i t e e l e m e n t m e t h o d f o r t h e a n a l y s i s o f l a y e r e d wood s y s t e m s . T h e m e t h o d i n c l u d e s t h e e f f e c t s o f s l i p b e t w e e n l a y e r s due t o f a s t e n e r d e f o r m a t i o n s , o r t h o t r o p i c m a t e r i a l p r o p e r t i e s a n d g a p s i n l a y e r s . F o r t h e p u r p o s e s o f t h e f i n i t e e l e m e n t a n a l y s i s , t h e f l o o r i s i d e a l i z e d a s a s e t o f c r o s s i n g b e a m s . The beams i n t h e d i r e c t i o n o f t h e j o i s t s c o n s i s t o f t h e j o i s t p l u s t h e c o m p o s i t e f l a n g e o f one o r more l a y e r s o f s h e a t h i n g , e q u a l i n w i d t h t o t h e j o i s t s p a c i n g . The beams i n t h e p e r p e n d i c u l a r d i r e c t i o n c o n s i s t o f s h e a t h i n g s t r i p s o f one o r more l a y e r s o f a r b i t r a r y w i d t h . T h e T - b e a m s a n d s h e a t h i n g s t r i p s a r e s u b d i v i d e d i n t o f i n i t e e l e m e n t s a n d t h e i r d e f o r m a t i o n s a r e m a t c h e d a t p o i n t s o f i n t e r s e c t i o n . The t o r s i o n a l s t i f f n e s s o f T - b e a m s a n d s h e a t h i n g i s n e g l e c t e d . B a s e d upon t h e a b o v e d e s c r i b e d m o d e l , a c o m p u t e r p r o g r a m F E A F L O {14} was d e v e l o p e d by Thompson e t a l . w h i c h c a n 3 a n a l y z e wood f l o o r s s u b j e c t e d t o any p a t t e r n of l o a d i n g p e r p e n d i c u l a r t o the f a c e of the s h e a t h i n g . The second and more r e c e n t paper i s by F o s c h i {7} where a combined F o u r i e r s e r i e s and f i n i t e element a n a l y s i s of wood f l o o r s i s d i s c u s s e d . The method i n c l u d e s the l a t e r a l and t o r s i o n a l d e f o r m a t i o n of j o i s t s as degrees of freedom. For t h e purposes of a n a l y s i s , the f l o o r i s i d e a l i z e d as c o n s i s t i n g of T-beam f i n i t e e l e m e n t s . Each element c o n s i s t s of the j o i s t s and the s h e a t h i n g s t r i p e q u a l i n w i d t h t o the s p a c i n g of the j o i s t s . A l o n g the d i r e c t i o n of the span of the j o i s t s a F o u r i e r s e r i e s a p p r o x i m a t i o n i s employed. Thus the f l o o r needs t o be broken up i n t o f i n i t e e l ements o n l y i n the t r a n s v e r s e d i r e c t i o n , making the model s u i t a b l e f o r a computer w i t h s m a l l c o r e memory. Based on the above d e s c r i b e d model a computer program FAP ( F l o o r A n a l y s i s Program) was d e v e l o p e d and e f f i c i e n t l y implemented i n a PDP 11/60 m i n i c o m p u t e r . The FAP program does a s t a t i c a n a l y s i s of a f l o o r system. As w i l l be d e s c r i b e d i n t h i s t h e s i s , t h i s program has been m o d i f i e d t o c a r r y out a dynamic a n a l y s i s of f l o o r s . Human response t o v i b r a t i o n s i s v e r y s u b j e c t i v e and d i f f i c u l t t o g e n e r a l i z e . A s u b s t a n t i a l amount of e x p e r i m e n t a l r e s e a r c h has been done i n t h i s a r e a and the p e r t i n e n t c o n t r i b u t i o n s a r e q u o t e d h e r e . One of the e a r l y works i s by R e i h e r and M e i s t e r ( 1 2 ) . In t h e i r work, human s u b j e c t s were s u b j e c t e d t o s t e a d y s t a t e v e r t i c a l v i b r a t i o n s and were asked t o c l a s s i f y v i b r a t i o n s i n one of the 4 f o l l o w i n g c a t e g o r i e s : (1) S l i g h t l y P e r c e p t i b l e ; (2) D i s t i n c t l y P e r c e p t i b l e ; (3) S t r o n g l y P e r c e p t i b l e ; ( 4 ) D i s t u r b i n g ; a n d ( 5 ) V e r y D i s t u r b i n g . P l o t s o f peak d i s p l a c e m e n t v e r s u s f r e q u e n c y w e r e made a n d t h e a b o v e m e n t i o n e d r e g i o n s i d e n t i f i e d . H o w e v e r , i t was f o u n d by A l l e n a n d R a i n e r {1} t h a t t r a n s i e n t v i b r a t i o n s c a u s e d by f o o t s t e p s a r e l e s s a n n o y i n g t h a n c o n t i n u o u s v i b r a t i o n s . O h l s s o n {8} e x p e r i m e n t a l l y e v a l u a t e d t h e human f o o t f a l l l o a d i n g . He s t u d i e d t h e human r e s p o n s e t o f l o o r v i b r a t i o n s a n d s u g g e s t e d d e s i g n c r i t e r i a b a s e d upon m o d a l p a r a m e t e r s l i k e e i g e n f r e q u e n c i e s , m o d a l m a s s e s a n d d a m p i n g . In an a n o t h e r p a p e r p u b l i s h e d by A l l e n a n d R a i n e r {2} on t h e v i b r a t i o n o f l o n g s p a n f l o o r s , a h e e l i m p a c t t e s t was u s e d f o r e v a l u a t i n g f l o o r p e r f o r m a n c e . T h e y p r e s e n t e d an a n n o y a n c e c r i t e r i o n f o r f l o o r v i b r a t i o n i n t e r m s o f f l o o r a c c e l e r a t i o n a n d d a m p i n g . T h e y c o n c l u d e d t h a t d a m p i n g i s e f f e c t i v e i n r e d u c i n g a n n o y i n g w a l k i n g v i b r a t i o n s . P o l e n s e k {10} h a s s t u d i e d t h e human r e s p o n s e t o v i b r a t i o n o f w o o d - j o i s t f l o o r s y s t e m s . He u s e d a r a t i n g s c a l e more o r l e s s on t h e same l i n e s a s R e i h e r a n d M e i s t e r a n d i n v e s t i g a t e d t h e v i b r a t i o n c a u s e d by an i m p a c t o f a 70 p o u n d s t e e l w e i g h t d r o p p e d o n t o f l o o r c e n t r o i d . P l o t s o f maximum d e f l e c t i o n v e r s u s f r e q u e n c y were made a n d s m o o t h c u r v e s r e p r e s e n t e d by m a t h e m a t i c a l e q u a t i o n s were p r e d i c t e d . A t h e r t o n , P o l e n s e k a n d C o r d e r {3} f o u n d t h a t a m p l i t u d e was t h e b e s t i n d i c a t o r o f human r e s p o n s e , b u t r e c o m m e n d e d t h a t a v a r i a b l e c o n t a i n i n g a m p l i t u d e , d a m p i n g a n d f r e q u e n c y w o u l d be more a p p r o p r i a t e f o r p r e d i c t i n g human 5 r e s p o n s e t o f l o o r v i b r a t i o n s . W i s s a n d P a r m e l e e {15} s t u d i e d e x p e r i m e n t a l l y t h e human r e s p o n s e t o t r a n s i e n t v e r t i c a l v i b r a t i o n s i n t e r m s o f t h e i r f r e q u e n c y , maximum d i s p l a c e m e n t a n d d a m p i n g . T h e y a s s u m e d a p p r o x i m a t e l y t h e same r a t i n g s c a l e a s u s e d by R e i h e r a n d M e i s t e r b u t a s s i g n e d n u m e r i c a l v a l u e s t o t h e r a t i n g s t o i n c o r p o r a t e t h o s e i n t o t h e a n a l y s i s . F o r e a c h r a t i n g v a l u e p l o t s o f t h e p r o d u c t o f f r e q u e n c y a n d a m p l i t u d e (FA) v e r s u s d a m p i n g c o e f f i c i e n t (D) were made on a l o g - l o g p l o t . A r e g r e s s i o n a n a l y s i s was a l s o c a r r i e d o u t on t h e e x p e r i m e n t a l d a t a a n d a m a t h e m a t i c a l e q u a t i o n p r e d i c t i n g r a t i n g a s a f u n c t i o n o f a m p l i t u d e , f r e q u e n c y a n d d a m p i n g was s u g g e s t e d . In t h i s t h e s i s , t h e s e p l o t s a n d e q u a t i o n s a r e u s e d , a s w i l l be d e s c r i b e d f u l l y i n s u b s e q u e n t c h a p t e r s . 1.3 O B J E C T I V E AND CONTENT OF THE T H E S I S The a i m o f t h e work p r e s e n t e d i n t h i s t h e s i s i s t o d e v e l o p a r a t i o n a l a n a l y t i c a l m e t h o d f o r a n a l y z i n g f l o o r v i b r a t i o n s due t o i m p a c t . I t was f o u n d by O n y s k o {9} t h a t t h e p r e s e n t d e f l e c t i o n c r i t e r i o n b a s e d on d e f l e c t i o n c a u s e d by u n i f o r m l o a d i n g d o e s n o t a d e q u a t e l y i d e n t i f y . " g o o d " f r o m " p o o r " f l o o r p e r f o r m a n c e . An a t t e m p t i s made h e r e t o e s t a b l i s h a d e s i g n c r i t e r i o n . f o r s a t i s f a c t o r y f l o o r p e r f o r m a n c e a s p e r c e i v e d by human b e i n g s . T h e s p e c i f i c p r o b l e m on w h i c h we f o c u s o u r a t t e n t i o n i n t h i s work i s shown i n F i g . ( 1 . 1 ) , a f l o o r i n s t a t i c e q u i l i b r i u m u n d e r d e a d a n d l i v e l o a d s . T h e l i v e l o a d c o u l d 6 b e p e o p l e o c c u p y i n g t h e f l o o r . We a r e i n t e r e s t e d i n t h e v i b r a t i o n f e l t b y a p e r s o n ( i d e n t i f i e d h e r e a f t e r a s " r e c e i v e r " ) a t p o i n t 1 o f t h e f l o o r w h e n s o m e o t h e r p e r s o n ( i d e n t i f i e d h e r e a f t e r a s " i m p a c t e r " ) a p p l i e s a f o o t f a l l i m p a c t a t p o i n t 2 o f t h e f l o o r . S t a t e d o t h e r w i s e , we w i s h t o e s t i m a t e t h e f l o o r ' s d y n a m i c r e s p o n s e ( d i s p l a c e m e n t , v e l o c i t y a n d a c c e l e r a t i o n ) a t p o i n t 1 d u e t o i m p a c t a t p o i n t 2 . A f i n i t e e l e m e n t c o m p u t e r p r o g r a m i s d e v e l o p e d h e r e f o r t h i s p u r p o s e . T h i s i s a m o d i f i e d v e r s i o n o f t h e F l o o r A n a l y s i s P r o g r a m ( F A P ) a s d e v e l o p e d b y F o s c h i { 7 } . T h i s p r o g r a m d o e s a t i m e s t e p a n a l y s i s a n d i s c a p a b l e o f p r e d i c t i n g t h e d y n a m i c r e s p o n s e a t a n y p o i n t o n t h e f l o o r d u e t o a n i m p a c t a t s o m e o t h e r p o i n t . T h e p r o g r a m i s a l s o u s e d f o r a s e n s i t i v i t y a n a l y s i s o f t h e e f f e c t o f P l a t e — M I I J o ' i s t F i g . 1.1. P r o b l e m o f i n t e r e s t 7 v a r i o u s f l o o r p a r a m e t e r s on f l o o r v i b r a t i o n r e s p o n s e . A m u l t i p l e l i n e a r r e g r e s s i o n a n a l y s i s i s a l s o d o n e and a p r e d i c t i o n e q u a t i o n f o r e s t i m a t i o n o f peak d i s p l a c e m e n t i n t e r m s o f maximum s t a t i c d i s p l a c e m e n t i s s u g g e s t e d . T o p r e d i c t f l o o r p e r f o r m a n c e d e t e r m i n e d by human p e r c e p t i o n o f v i b r a t i o n , W i s s a n d P a r m e l e e ' s r e l a t i o n b e t w e e n r a t i n g , f r e q u e n c y , a m p l i t u d e a n d d a m p i n g i s a s s u m e d . T h e e f f e c t o f v a r i a b i l i t y i n m o d u l u s o f e l a s t i c i t y f o r a p o p u l a t i o n o f j o i s t s on t h e f l o o r ' s r e s p o n s e i s s t u d i e d by s i m u l a t i o n a n d t h e d i s t r i b u t i o n o f t h e r e s p o n s e r a t i n g s i s e s t a b l i s h e d f o r t h e p o p u l a t i o n o f s i m u l a t e d f l o o r s . The r e s u l t s a r e u s e d t o e s t i m a t e t h e r e l i a b i l i t y t h a t t h e f l o o r w o u l d f a l l i n t o a p a r t i c u l a r r a t i n g c a t e g o r y . F i n a l l y , a d e s i g n c r i t e r i o n i s e s t a b l i s h e d f o r o b t a i n i n g a l l o w a b l e s p a n s c o r r e s p o n d i n g t o a t a r g e t r e l i a b i l i t y . 2. THEORETICAL FORMULATION 2.1 THE MODEL The type of f l o o r c o n s i d e r e d i s shown i n F i g . (2.1). The f l o o r cover (sheathing) i s assumed f a s t e n e d ( n a i l e d ) to the wood j o i s t s to produce an assembly capable of composite a c t i o n and behaving under lo a d as a s t i f f e n e d p l a t e . The f l o o r cover may have gaps a c r o s s the span. However, i n t h i s work no gaps are c o n s i d e r e d . Although the program developed here would be s u i t a b l e f o r wood f l o o r s , i t can a l s o be used f o r concrete or s t e e l f l o o r s by making minor m o d i f i c a t i o n s . F i g . 2.1. F l o o r Assembly In g e n e r a l , i t i s assumed that the cover has d i f f e r e n t e l a s t i c p r o p e r t i e s i n the d i r e c t i o n s p a r a l l e l and p e r p e n d i c u l a r to the j o i s t s and i s c o n s i d e r e d as an o r t h o t r o p i c p l a t e f o r the purposes of the a n a l y s i s . The 8 9 f l o o r i s m o d e l l e d as c o n s i s t i n g of T - b e a m - f i n i t e elements i n the Y - d i r e c t i o n as shown i n F i g . (2.1). The nodes c o n s i d e r e d f o r each f i n i t e element are shown i n F i g . (2.2). The d e f o r m a t i o n s of the p l a t e i n the Y - d i r e c t i o n a re ap p r o x i m a t e d by a f i n i t e element approach and matched a t nodes 1 and 3 as shown i n F i g . (2.2). In the l o n g i t u d i n a l d i r e c t i o n ( X - d i r e c t i o n ) a F o u r i e r S e r i e s a p p r o x i m a t i o n i s employed. * 1 •2 n 3 f • F i g . 2.2. Nodes f o r T-beam F i n i t e Element U s i n g the model, the d e f l e c t i o n s of the p l a t e and the j o i s t s can be e x p r e s s e d a s : P l a t e D i s p l a c e m e n t s ( m i d d l e s u r f a c e ) : N V e r t i c a l D i s p l a c e m e n t : w ( x , y ) = I F. ( y ) S i n ( n ^ x - ) n=1 1 n L ( z - d i r e c t i o n ) N A x i a l D i s p l a c e m e n t :u(x,y)=Z F_ (y)Cos(—=—) ( x - d i r e c t i o n ) n = l N 2n L a t e r a l D i s p l a c e m e n t : v ( x , y ) = I F ^ ( y ) S i n ( — j — ) n= 1 ( y - d i r e c t i o n ) where, N i s the number of terms i n the F o u r i e r S e r i e s c o n s i d e r e d t o approximate the d i s p l a c e m e n t . S i n c e w=v=0 a t x=0 & x=L, the assumed f u n c t i o n s imply s i m p l y (2.1a) (2.1b) (2.1c) 10 s u p p o r t e d b o u n d a r y c o n d i t i o n s a t e n d s . The f u n c t i o n s F l n ( y ) / F 2 n ^ a n d F 3 n ^ a r e d e t e r m i n e d f r o m a f i n i t e e l e m e n t p o l y n o m i a l a p p r o x i m a t i o n i n Y - d i r e c t i o n . T h e s e f u n c t i o n s a r e e x p r e s s e d i n t e r m s o f t h e n o d a l d e g r e e s o f f r e e d o m a t p o i n t s 1, 2 , 3 a n d 4 shown i n F i g . ( 2 . 2 ) . J o i s t D i s p l a c e m e n t s : The d i s p l a c e m e n t f u n c t i o n s f o r j o i s t s a r e e x p r e s s e d a s f o l l o w s : N V e r t i c a l D i s p l a c e m e n t : W(x)= I W S i n ( n - ^ x - ) ( 2 . 2 a ) n=l n L ( z - d i r e c t i o n ) N A x i a l D i s p l a c e m e n t : U ( x ) = I U C o s ( i ^ ) ( 2 . 2 b ) n=l n ( x - d i r e c t i o n ) N L a t e r a l D i s p l a c e m e n t : V ( x ) = Z VSin(^±) ( 2 . 2 c ) n-l n L ( y - d i r e c t i o n ) N R o t a t i o n : 0 ( x ) = I 6 S i n ( ^ ) ( 2 . 2 d ) n=l n L w h e r e W , U , V_ a n d 8 a r e j o i s t d e g r e e s o f f r e e d o m a s n n n n J . 3 shown i n F i g . ( 2 . 3 ) . F i g . 2 . 3 . J o i s t d e g r e e s o f f r e e d o m F o r e a c h e l e m e n t a s shown i n F i g . ( 2 . 2 ) , a t o t a l o f 19 d e g r e e s o f f r e e d o m a r e i d e n t i f i e d a s s o c i a t e d w i t h e a c h 11 n( t e r m i n F o u r i e r s e r i e s ) . The e l e m e n t a l n o d a l degree of freedom v e c t o r i s e x p r e s s e d by Eq. ( 2 . 3 ) . w 1n w 1 n s u m u l n s 1 n V 1 n s u 2n v 2 n w 2 n s w u. V V s w 3n w 3 n S u 3n u 3 n s 3n v 3 n s (2.3) There a r e s i x degrees of freedom each a t nodes 1 and 3, t h r e e a t node 2 and f o u r a t node 4. For example, w 1 n * s = t n e component d e n o t i n g the d e r i v a t i v e of w(x,y) w i t h r e s p e c t t o y ( ' ) a t node 1, m u l t i p l i e d by the j o i s t s p a c i n g " s " , t o make i t d i m e n s i o n a l l y u n i f o r m . 1 2 U s i n g t h e s h a p e f u n c t i o n s shown i n A p p e n d i x I, a n d t h e n o n d i m e n s i o n a l v a r i a b l e £ = 2 y / s , t h e f u n c t i o n s F 1 n ( y ) » F 2 n ^ a n d F_ (y ) c a n be w r i t t e n i n t e r m s o f {6 }: j n n F 1 n ( * ) = { M 0 U ) } T { 6 n } ( 2 - 4 a ) F 2 n ( ^ ) = { M 3 ( U } T { 5 n } ( 2 . 4 b ) F 3 n U ) = { M 5 U ) } T U n } ( 2 . 4 c ) F u r t h e r m o r e , t h e s e e q u a t i o n s p e r m i t u s t o o b t a i n t h e f o l l o w i n g d e r i v a t i v e s : d F l n ( « > f ,T d£ d 2 F 1 w , U ) = {M 1 (0) ( S n } ( 2 . 5 a ) = { M 0 U ) } T { 5 „ } ( 2 . 5 b ) d £ 2 ' n dF (I) { M,U ) } T { 6 n } ( 2 . 5 c ) dt d F , I t ) d£ - = { M , ( ^ ) } i { 6 n } ( 2 . 5 d ) D n T h u s w i t h t h e d i s p l a c e m e n t f u n c t i o n s d e f i n e d , t h e s t r a i n e n e r g y , k i n e t i c e n e r g y a n d t h e d i s s i p a t i v e e n e r g y c a n be c o m p u t e d a n d s y s t e m m a t r i c e s c a n be d e r i v e d a s e x p l a i n e d i n n e x t s e c t i o n . 13 2 . 2 FORMULATION OF THE EQUATIONS OF MOTION In o r d e r t o f o r m u l a t e t h e e q u a t i o n s o f m o t i o n f o r t h e f l o o r s y s t e m , t h e V a r i a t i o n a l A p p r o a c h i s e m p l o y e d . T h i s a p p r o a c h makes u s e o f s c a l a r e n e r g y q u a n t i t i e s i n a v a r i a t i o n a l f o r m . T h i s p r o c e s s a v o i d s e s t a b l i s h i n g t h e v e c t o r i a l e q u a t i o n s o f e q u i l i b r i u m w h i c h b e c o m e s v e r y c o m p l e x h e r e . M o r e o v e r , a s o p p o s e d t o v i r t u a l work a n a l y s i s no d i r e c t u s e o f t h e i n e r t i a l o r e l a s t i c f o r c e s a c t i n g on t h e s y s t e m i s m a d e . The e f f e c t s o f t h e s e f o r c e s a r e r e p r e s e n t e d i n s t e a d by t h e v a r i a t i o n s o f t h e k i n e t i c a n d p o t e n t i a l e n e r g y o f t h e s y s t e m . The m o s t g e n e r a l l y a p p l i c a b l e v a r i a t i o n a l c o n c e p t i s H a m i l t o n ' s P r i n c i p l e , w h i c h s t a t e s t h a t t h e v a r i a t i o n o f t h e k i n e t i c a n d p o t e n t i a l e n e r g y p l u s t h e v a r i a t i o n o f t h e work d o n e by t h e n o n c o n s e r v a t i v e f o r c e s c o n s i d e r e d d u r i n g a n y t i m e i n t e r v a l t , t o t 2 must e q u a l . z e r o . M a t h e m a t i c a l l y , t h e p r i n c i p l e i s e x p r e s s e d a s : t 2 t 2 ; 6 ( T - v ) d t + r 6w dt=o t , t . ( 2 . 6 ) w h e r e : T t o t a l k i n e t i c e n e r g y o f t h e s y s t e m V p o t e n t i a l e n e r g y o f t h e s y s t e m , i n c l u d i n g b o t h s t r a i n e n e r g y a n d p o t e n t i a l o f a n y c o n s e r v a t i v e e x t e r n a l f o r c e s . W nc work done by n o n c o n s e r v a t i v e f o r c e s a c t i n g on s y s t e m , i n c l u d i n g d a m p i n g a n d any a r b i t r a r y e x t e r n a l l o a d s . 14 6 = v a r i a t i o n t a k e n d u r i n g i n d i c a t e d t i m e i n t e r v a l . F o r t h e f l o o r s y s t e m we a r e d e a l i n g w i t h , t h e k i n e t i c e n e r g y c a n be e x p r e s s e d i n t e r m s o f t h e g e n e r a l i z e d c o o r d i n a t e s a n d t h e i r f i r s t t i m e d e r i v a t i v e s , a n d t h e p o t e n t i a l e n e r g y c a n be e x p r e s s e d i n t e r m s o f t h e g e n e r a l i z e d c o o r d i n a t e s a l o n e . In a d d i t i o n , t h e v i r t u a l work w h i c h i s p e r f o r m e d by t h e n o n c o n s e r v a t i v e f o r c e s a s t h e y a c t t h r o u g h t h e v i r t u a l d i s p l a c e m e n t s c a u s e d by an a r b i t r a r y s e t o f v a r i a t i o n s i n t h e g e n e r a l i z e d c o o r d i n a t e s c a n be e x p r e s s e d a s a l i n e a r f u n c t i o n o f t h o s e v a r i a t i o n s . In m a t h e m a t i c a l t e r m s , we c a n w r i t e T = T ( q , , q 2 , , q n , q , , q 2 , <3n) ( 2 . 7 a ) V = V ( q 1 f q 2 , , q n ) ( 2 , 7 b ) 5 W n c = Q ^ q , + Q 2 5 q 2 + + Q _ n S q n ( 2 . 7 c ) where t h e c o e f f i c i e n t s Q 1 f Q 2 , ,Q_ n a r e t h e g e n e r a l i z e d f o r c i n g f u n c t i o n s c o r r e s p o n d i n g t o t h e g e n e r a l i z e d c o o r d i n a t e s q , , q 2 , , q n r e s p e c t i v e l y . I n t r o d u c i n g E q . ( 2 . 7 ) i n t o H a m i l t o n ' s P r i n c i p l e ( E q . 2 . 6 ) a n d c a r r y i n g o u t t h e m a t h e m a t i c s l e a d s t o t h e w e l l known L a g r a n g e ' s E q u a t i o n Of M o t i o n w h i c h i s e x p r e s s e d a s 3t 9 q i 3 q i d q i l In t h e a b o v e e q u a t i o n Q i i n c l u d e s c o n t r i b u t i o n f r o m e x t e r n a l f o r c e s a s w e l l a s d a m p i n g f o r c e s i e . 1 5 Q. = Q . + Q. , v i v i e i d T h e p a r t o f c a n be c o m p u t e d by u s i n g R a l e i g h ' s D i s s i p a t i v e F u n c t i o n (D) a n d i s e x p r e s s e d a s U s i n g E q . (.2.9) we g e t t h e f o l l o w i n g c o n v e n i e n t f o r m o f L a g r a n g e ' s e q u a t i o n o f m o t i o n : lr< V W VQ'« (2-,0> w h e r e : Q ^ e = e x t e r n a l g e n e r a l i z e d f o r c e s a c t i n g on t h e s y s t e m . D = W^/2 , b e i n g t h e work d o n e by d i s s i p a t i v e f o r c e s p e r u n i t t i m e . F o r t h e f l o o r s y s t e m , w h i c h i s m o d e l l e d by f i n i t e e l e m e n t s , n o d a l d e g r e e s o f f r e e d o m a s g i v e n by E q . ( 2 . 3 ) r e p l a c e t h e g e n e r a l i z e d c o o r d i n a t e s q^ a n d t h e n o d a l f o r c e s r e p l a c e t h e g e n e r a l i z e d f o r c e s . On s u b s t i t u t i n g t h e k i n e t i c e n e r g y , p o t e n t i a l e n e r g y a n d d i s s i p a t i v e e n e r g y o f t h e s t r u c t u r e i n t o E q . ( 2 . 1 0 ) we d i r e c t l y o b t a i n t h e e q u a t i o n o f m o t i o n o f t h e s y s t e m e x p r e s s e d a s : [M]{X} + [C]{X} + [K]{X} = {R} ( 2 . 1 1 ) w h e r e : [ M ] = g l o b a l mass m a t r i x ; [ C ] = g l o b a l d a m p i n g m a t r i x ; [ K ] = g l o b a l s t i f f n e s s m a t r i x ; { R } = g l o b a l n o d a l f o r c e v e c t o r ; 16 { X } = g l o b a l n o d a l d i s p l a c e m e n t v e c t o r ; { X } = g l o b a l n o d a l v e l o c i t y v e c t o r ; " ' " d e n o t i n g t h e d i f f e r e n t i a t i o n w i t h r e s p e c t t o t i m e ; { X } = g l o b a l n o d a l a c c e l e r a t i o n v e c t o r ; T h e s y s t e m m a t r i c e s a n d v e c t o r s a s d e f i n e d a b o v e a r e o b t a i n e d by a s s e m b l i n g t h e m a t r i c e s a n d v e c t o r s o f e a c h f i n i t e e l e m e n t , a s shown i n d e t a i l i n t h e f o l l o w i n g s e c t i o n s . 2 . 2 . 1 DERIVATION OF S T I F F N E S S MATRIX T h e s t i f f n e s s m a t r i x i s o b t a i n e d f r o m t h e t o t a l s t r a i n e n e r g y o f t h e s y s t e m . In o r d e r t o o b t a i n t h e t o t a l s t r a i n e n e r g y , we h a v e t o c o n s i d e r c o n t r i b u t i o n s f r o m t h e v a r i o u s c o m p o n e n t s o f t h e f l o o r s y s t e m i e . t h e p l a t e , j o i s t s a n d t h e c o n n e c t o r s b e t w e e n t h e j o i s t s and t h e p l a t e . S t r a i n E n e r g y i n t h e P l a t e : A s t h e f l o o r i s b e i n g m o d e l l e d by f i n i t e e l e m e n t s , we c o n s i d e r h e r e s t r a i n e n e r g y i n t h e f l a n g e o f one T - b e a m e l e m e n t . T h i s w i l l l e a d t o t h e s t i f f n e s s m a t r i x o f one f i n i t e e l e m e n t . The t o t a l s y s t e m s t i f f n e s s m a t r i x c a n be o b t a i n e d by a s s e m b l i n g a l l t h e e l e m e n t s t i f f n e s s m a t r i c e s . U s i n g s m a l l d e f l e c t i o n o r t h o t r o p i c p l a t e t h e o r y , t h e s t r a i n e n e r g y o f t h e f l a n g e p e r u n i t a r e a c a n be e x p r e s s e d a s : 17 Auf = ^ x - ( ^ ) 2 + ^ ( ^ ) 2 + K ii£ i i * f 2 3 x 2 2 9 y 2 " 9 x 2 3 y 2 • « o < lay >* + r' IS >! + ^< I? >* w h e r e : K = f l e x u r a l s t i f f n e s s i n x - d i r e c t i o n 12(1-1/ v ) xy yx K = f l e x u r a l s t i f f n e s s i n y - d i r e c t i o n = K (E / E ) y 2 x y ' x K = c o e f f i c i e n t a s s o c i a t e d w i t h P o i s s o n ' s r a t i o = v K p xy x Kg = t o r s i o n a l s t i f f n e s s = G d 3 / 1 2 D = a x i a l s t i f f n e s s i n x - d i r e c t i o n (\-v V ) xy yx D = a x i a l s t i f f n e s s i n y - d i r e c t i o n = D (E / E ) y x y x D = c o e f f i c i e n t a s s o c i a t e d w i t h P o i s s o n ' s r a t i o = v D v xy x Dg = i n p l a n e s h e a r s t i f f n e s s = Gd a n d E = m o d u l u s o f e l a s t i c i t y i n x - d i r e c t i o n ; E = x y m o d u l u s o f e l a s t i c i t y i n y - d i r e c t i o n ; v = P o i s s o n ' s xy r a t i o , s t r a i n i n x - d i r e c t i o n when s t r e s s i s a p p l i e d i n 18 y - d i r e c t i o n ; v = P o i s s o n ' s r a t i o , s t r a i n i n yx y - d i r e c t i o n when s t r e s s i s a p p l i e d i n x - d i r e c t i o n ; a n d G = s h e a r m o d u l u s i n x - y p l a n e . T h e t o t a l s t r a i n e n e r g y i n t h e f l a n g e c a n be o b t a i n e d by i n t e g r a t i n g o v e r t h e a r e a a s shown i n F i g . ( 2 . 4 ) T h u s , t h e f l a n g e s t r a i n e n e r g y i s e x p r e s s e d a s : s / 2 L U f = / / A U f d x d y £ - s / 2 0 £ ( 2 . 1 3 ) On s u b s t i t u t i n g t h e e x p r e s s i o n s f o r d i s p l a c e m e n t s a s g i v e n by E q . ( 2 . 1 ) i n t o E q . ( 2 . 1 2 ) a n d i n t e g r a t i n g o v e r s p a n l e n g t h , we o b t a i n f o r e a c h t e r m L K 3 2 w K N n«ir" j- -JL ( !L-!L ) 2 dx = — I F 2 ( y ) 0 2 3 x 2 2 n=1 I n 2 L 3 L K , 2 K N _ ( £ _ Z )2 d x = -2- Z ( F . " (y ) ) 2 (h) 0 2 3 y 2 2 n=1 l n Z ( 2 . 1 4 a ) ( 2 . 1 4 b ) F i g . 2 . 4 . F l a n g e A r e a o f F i n i t e E l e m e n t 19 I f l Tl d X = - K » B " , p l n ( » ) F i ; « » > T E 1 ( 2 - , 4 c ) 0 3 x z 3 y z n=1 2KG J < life >' d X = ^ j / ^ n ' y " ' ( 2 - ' 4 d ) L D D N 2 2 / — ( | £ ) 2 dx = -2. z ( F - ( y ) ) 2 ° ^ f - ( 2 . 1 4 e ) 0 2 dx 2 n = 1 2n ZL L D D N J ( ) 2 dx = Z ( F - ' ( y ) ) 2 ( £ ) ( 2 . 1 4 f ) 0 2 ° Y 2 n=1 J n ^ { D, ( I i ) ( I j > d x = "DvJ1F2n<y>F3i(») ? ( 2' U9> < If + lx" >' d x = rS^2^ ^ F 3 n ( y ) P i ( 2 . 1 4 h ) In o r d e r t o o b t a i n t h e s t r a i n e n e r g y i n t h e f l a n g e o f one e l e m e n t we s u b s t i t u t e t h e e x p r e s s i o n s f o r f u n c t i o n s F 1 n ^ ' F 2 n ^ a n d F 3 n ^ a n d t h e ^ r d e r i v a t i v e s a s g i v e n by E q . ( 2 . 4 ) a n d ( 2 . 5 ) i n t o E q . ( 2 . 1 4 ) , i n t e g r a t e o v e r j o i s t s p a c i n g , a n d we g e t f 1 n=1 2 4 L 3 -1 n 0 0 n ( 2 . 1 5 a ) 7 J " f « n } T { M 2 U ) } { M 2 U ) } T { 8 n } d ^ N K = Z x : lN K = Z -I . n=1 2 N K = Z (- -n=1 2 N = Z 2K_ • n=1 G N D = Z X n=1 2 ( 2 . 1 5 b ) v , 2 n 2 7 r 2 r { 8 n} T{M 0(UHM 2U ) } T { 5 n}dS ( 2 . 1 5 c ) n 27T 2 l f e , T , 3 ^ I I I I I I ( 2 . l 5 d ) ' f 5 \ 2 . ^ ^ _ j { 5 n } T { M 3 ( U } { M 3 ( U } T { 6 n } d i 20 (2.15e) N D v L 1 T T (2.15f) (2.15g) f6 „ 1 n s ; n J 6 s 6 * n' n= i z -1 U f 7 = S ( - D , ^ ) } { 6 n } T { M 3 ( U } { M 6 ( 0 } T { 6 n } d ^ n= 1 -1 U f 8 =J,7 ^ _ { { 5 n } T ( l { M 4 ( « ) } + I > 5 { * ) } ) ( | ( M 4 ( ^ ) } T + n ^ { M 5 ( ^ ) } T ) { 5 n } d ^ (2.15h) Thus s t r a i n energy i n the f l a n g e of one element i s : u f = u f 1 + U f 2 + u f 3 + u f 4 + U f 5 + U f 6 + U f 7 + U f 8 ( 2 ' 1 6 ) S t r a i n Energy i n the J o i s t : S t r a i n energy i n the j o i s t c o n s i s t s of two f l e x u r a l components(one b e i n g the bending i n v e r t i c a l p l a n e and o t h e r l a t e r a l b e n d i n g ) , an a x i a l component and a t o r s i o n a l component. The s t r a i n energy i s e x p r e s s e d as o . . } ( < ^ )• ax • f k . ; < # 1 d x 3 2 0 d x 2 2 0 d x 2 L G I L + ; ( g >2 d x + j ( | | >2 d x (2.17) z 0 2 0 where: I = (BH 3)/12; I g = (HB 3)/12; A = BH; I t = 0(H/B)HB 3; and "B" b e i n g the j o i s t w i d t h , "H" the j o i s t depth and "0" the t o r s i o n a l c o n s t a n t . 21 U s i n g t h e e x p r e s s i o n s f o r W ( x ) , U ( x ) , V ( x ) a n d 0 (x ) a s g i v e n by E q . ( 2 . 2 ) we o b t a i n f o r s t r a i n e n e r g y i n t h e j o i s t a s : . E I N . 4 EI N * „ u. = — Z - Z W 2 + — ^ - Z V 2 S-^LI 3 2 n=1 n 2 L 3 2 n=1 n 2 L 3 N , 2 G l . N , 2 + | ^ Z U 2 ^ f - + — Z 0 2 ( 2 . 1 8 ) 2 n=1 n 2 L 2 n-1 n 2 L S t r a i n E n e r g y i n t h e C o n n e c t o r s : I t i s a s s u m e d h e r e , t h a t p l a t e i s c o n n e c t e d t o t h e j o i s t s by u n i f o r m l y s p a c e d n a i l s a l o n g t h e c e n t r e l i n e o f t h e t o p s u r f a c e o f t h e j o i s t . T h e m o d e l a s s u m e s t h r e e t y p e s o f s l i p s b e t w e e n t h e p l a t e a n d t h e j o i s t w h i c h c a n be e x p r e s s e d a s : S l i p p a r a l l e l t o t h e j o i s t = A u = [ u 0 - § ( ) ] - [ U + | ( j g ) ] ( 2 . 1 9 a ) S l i p p e r p e n d i c u l a r t o t h e j o i s t = A v 3w 0 R o t a t i o n a l s l i p = <p = - 6 3y w h e r e d= the c o v e r t h i c k n e s s ; a n d t h e d i s p l a c e m e n t s u 0 , v 0 a n d w 0 c o r r e s p o n d t o t h e p o i n t i n t h e p l a t e ' s m i d d l e p l a n e d i r e c t l y o v e r t h e j o i s t , a s shown i n F i g . ( 2 . 1 ) . In t h e a b o v e E q . ( 2 . 1 9 ) i t h a s b e e n a s s u m e d t h a t t h e = [ v 0 - § ( — - [ v + § e] )] ( 2 . 1 9 b ) ( 2 . 1 9 c ) 22 vertical displacement of the plate, w0, and the displacement, W, of the joist are identical. If we assume that K n x , represent the stiffness for the slips in x and y-direction respectively, and represents the stiffness for the rotational s l i p , converting the discreet nailing pattern into a continuous connector, the strain energy can be expressed as: K L K L U = / (Au) 2 dx + -22 / (Av) 2 dx c 2e 0 2e 0 Knf? L + -J21 ; ^ 2 dx (2.20) 2e 0 where e is the nail spacing. On substituting the expressions for plate and joist displacements from Eq. (2.1) and (2.2) we obtain: ° c - " , i i r [ F 2 " l l = 0 > " " n " w " £ ( H t d > p n= 1 2e • I T 2 ' F3n<«=°> - v n - e n < I + ! " 2 Now we can express scalar nodal degrees of freedom in the above Eq. (2.21) in terms of the vector {5n } as given in the Eq. (2.3). We define the following (19x1) vectors: 23 { e 7 } T = ( 0 0 . . . . 1 . . . . 0 0 ) , t h e s e v e n t h e l e m e n t i s u n i t y . { e B } T = (0 0 1 0 0 ) , t h e e i g h t h e l e m e n t i s u n i t y . S i m i l a r l y v e c t o r s { e , } , { e 2 } , , { e 1 3 } a r e d e f i n e d . T h u s we g e t s t r a i n e n e r g y i n t h e c o n n e c t o r s a s : U = I { § — ( 6 n } T [ e 7 - e M - e 1 0 (H+d) ] n=1 2 2e n 2 L [ e 7 - e i 1 - e 1 0 g (H+d) ] T {5 n } + L ^Bl f x i T r o o o H « d l 2 { 5 n } [ e 8 - e i 2 " e , 3 5 i - e , ^ ] [ e e - e 1 2 - e 1 3 ^ - e . ^ f ] T U n ) + \ — ( 6 n } T t — - — ][ — - — ] T ^ 2e n s s s s . ( 2 . 2 2 ) The T o t a l s t r a i n e n e r g y i n one T - b e a m e l e m e n t i s g i v e n by U ( n ) = U f ( n ) + U j ( n ) + U j n ) ( 2 . 2 3 ) On t a k i n g t h e f i r s t v a r i a t i o n o f U ( n ) w i t h r e s p e c t t o { 6 „ } we g e t e x p r e s s i o n o f t h e f o r m [K ]{6^} i n w h i c h n ne n [K ] i s t h e e l e m e n t s t i f f n e s s m a t r i x o f s i z e ( 1 9 x 1 9 ) . ne T h e g l o b a l s t i f f n e s s m a t r i x i s o b t a i n e d by a s s e m b l i n g t h e s t i f f n e s s m a t r i c e s o f e a c h e l e m e n t . By c o n s i d e r i n g t h e v a r i a t i o n o f U w . r . t d i f f e r e n t {& n ) g l o b a l s t i f f n e s s m a t r i x c a n be o b t a i n e d f o r e a c h t e r m i n t h e F o u r i e r s e r i e s . 24 2 . 2 . 2 DERIVATION OF MASS MATRIX A c o n s i s t e n t mass m a t r i x i s o b t a i n e d f r o m t h e t o t a l k i n e t i c e n e r g y o f t h e s y s t e m . T h e t o t a l k i n e t i c e n e r g y c o n s i s t s o f c o n t r i b u t i o n s f r o m t h e p l a t e a n d t h e j o i s t s . K i n e t i c e n e r g y i n P l a t e : T h e v e l o c i t y c o m p o n e n t s o f any p l a t e p a r t i c l e c a n be e x p r e s s e d a s : u p = ii - z ( | £ ) ( 2 . 2 4 a ) v p = v - z ( | | ) ( 2 . 2 4 b ) w p = w ( 2 . 2 4 c ) where u »v and w a r e t h e v e l o c i t y c o m p o n e n t s a t any tr tr r p o i n t z b e l o w t h e m i d d l e s u r f a c e o f t h e p l a t e . T h u s t h e k i n e t i c e n e r g y o f t h e p l a t e i s g i v e n by T = —*- J" ( u 2 + v 2 + w 2 ) dV ( 2 . 2 5 ) P 2 V p P P i n w h i c h i s t h e mass d e n s i t y o f t h e p l a t e . On s u b s t i t u t i n g t h e e x p r e s s i o n s g i v e n by E q . ( 2 . 2 4 ) we o b t a i n , s / 2 L T = p_ § J / ( i i 2 + v 2 + w 2 ) d x d y p p 2 - s / 2 0 P J S S / 2 L _ . + T2 ' / [(S)2 +(lv)2 ]dxdv (2'26) 2 1 2 - s / 2 0 9 x 9 y i n w h i c h d i s t h e p l a t e t h i c k n e s s , s i s t h e j o i s t s p a c i n g a n d L i s t h e s p a n o f t h e f l o o r . 25 Now u s i n g E q . ( 2 . 1 ) , ( 2 . 4 ) a n d ( 2 . 5 ) we o b t a i n : n=1 2 2 2 2 z -{ 8 n } T { M 3 ( U } { M 3 ( U } T { 6 n } d« { 6 n } T { M 5 ( U H M 5 U ) } T { 6 n } at + J B h § | / { 6 „ } T { M N ( U } { M N ( U } T { 6 „ } d{ ' n J l l I 0 + ^ fi \ T T U ^ n } T { M 0 ( U } { M n ( n } T { 6 n } d* + Pflik-2i { ^ n ^ M 1 ^ H M ^ ) } T { 6 n } d * } 2 S - 1 ( 2 . 2 7 ) K i n e t i c E n e r g y i n t h e J o i s t : T h e v e l o c i t y c o m p o n e n t s o f a n y j o i s t p a r t i c l e a s shown i n F i g . ( 2 . 5 ) c a n be e x p r e s s e d a s : * b - 6 " 2 i " v i ( 2 - 2 8 a ) L z -1 1 n -v f a = V + dr C o s a = V - dz ( 2 . 2 8 b ) w. = W + 0r S i n a = W + 6y ( 2 . 2 8 c ) T h u s t h e k i n e t i c e n e r g y o f t h e j o i s t i s g i v e n by T . = - i / ( ii. 2 + v . 2 + w 2 ) d V ( 2 . 2 9 ) 2 V i n w h i c h P j i s t h e mass d e n s i t y o f t h e j o i s t . U s i n g E q . 2 6 F i g . 2 . 5 . V e l o c i t y C o m p o n e n t s o f J o i s t P a r t i c l e ( 2 . 2 ) a n d ( 2 . 2 8 ) we o b t a i n p. N T. » - 1 B H T Z ( U 2 + V * + W 2 ) D 2 2 n = 1 n n n 12 2 2 n=1 L -2 n n I X . ' PA u n 3 r. N n 2 . 2 + 11 H | l L z ( n^ll v 2 + Q 2 ) 12 2 * 2 n n n= 1 L z ( 2 . 3 0 ) U s i n g t h e v e c t o r s { e , } , { e 2 } { e 1 3 } a s d e f i n e d 27 b e f o r e the k i n e t i c energy in the j o i s t can be e x p r e s s e d a s : PA T N . T T T . = -1 BH \ L { ( { 5 n } T e 1 i e i 1 T {6 }) + ( { 8 n } e , 2 e 1 2 T { 6 n } ) + ( { 6 n } T e , o e 1 0 T ( 5 n } ) } f j BH 3 L y , n 2 7 T 2 ,T a T r A •> \ + ^ T2" 2 Z , { " T T " ( { 6 n } 6 1 0 6 1 0 { 6 n } ) 2 n=1 _ „ + ( { 5 n } T e , 3 e 1 3 T { 6 n } ) } . HB 3 L ? r n 2 7 T 2 n k ,T o T ,* n + , — 2 Z , { "TT ( { 6 n } e ' 2 6 1 2 { 6 n } ) 2 n=1 _ _ + ( i t j e 1 3 e 1 3 T { 4 } ) } (2.31) The t o t a l k i n e t i c energy i n one T-beam element i s g i v e n by: T(n) = T + T. (2.32) P 3 The k i n e t i c energy i s o n l y a f u n c t i o n of { 6 n } and 7)T t h e r e f o r e the term i n Eq. (2.10) v a n i s h e s and the 9q i 7\ 3T term -|^ -( ) y e i l d s an e x p r e s s i o n of the form f M n e H 5 n ) i n which [M ] i s the element mass m a t r i x of s i z e ne (19x19). The g l o b a l mass m a t r i x i s _ o b t a i n e d by a s s e m b l i n g mass m a t r i c e s of a l l the ele m e n t s . By t a k i n g the v a r i a t i o n o t "T" w i t h r e s p e c t t o d i f f e r e n t { 6 n }, mass m a t r i x f o r each term i n the F o u r i e r S e r i e s can be o b t a i n e d . 28 2 . 2 . 3 D E R I V A T I O N OF DAMPING MATRIX T h e d a m p i n g m a t r i x i s o b t a i n e d f r o m t h e R a l e i g h ' s d i s s i p a t i v e f u n c t i o n . A s R a l e i g h ' s d i s s i p a t i v e f u n c t i o n i s d e f i n e d a s h a l f t h e work done by t h e d i s s i p a t i v e f o r c e s p e r u n i t t i m e , we n e e d t o c o m p u t e t h e work done by t h e d i s s i p a t i v e f o r c e s . We d e f i n e t h e f o l l o w i n g d a m p i n g c o e f f i c i e n t s : C = p l a t e d a m p i n g c o e f f i c i e n t h a v i n g u n i t s o f f o r c e p e r P u n i t v e l o c i t y p e r u n i t v o l u m e . = j o i s t d a m p i n g c o e f f i c i e n t h a v i n g u n i t s o f f o r c e p e r u n i t v e l o c i t y p e r u n i t v o l u m e . C f i x = d a m p i n g c o e f f i c i e n t f o r s l i p a t t h e c o n n e c t o r s p a r a l l e l t o t h e j o i s t h a v i n g u n i t s o f f o r c e p e r u n i t v e l o c i t y . = d a m p i n g c o e f f i c i e n t f o r s l i p a t t h e c o n n e c t o r s p e r p e n d i c u l a r t o t h e j o i s t h a v i n g u n i t s o f f o r c e p e r u n i t v e l o c i t y . = d a m p i n g c o e f f i c i e n t f o r r o t a t i o n a l s l i p a t t h e c o n n e c t o r s h a v i n g u n i t s o f f o r c e p e r u n i t a n g u l a r v e l o c i t y . Now we c a n e x p r e s s t h e R a l e i g h ' s d i s s i p a t i v e f u n c t i o n a s : D = D + D, + D p b i c ( 2 . 3 4 ) w h e r e : D = d i s s i p a t i v e e n e r g y i n t h e p l a t e 2 V ( 2 . 3 5 a ) 29 >k = d i s s i p a t i v e e n e r g y i n t h e j o i s t = — / ( u, 2 + v, 2 + w 2 ) dV ( 2 . 3 5 b ) 2 v b b b D = d i s s i p a t i v e e n e r g y i n t h e c o n n e c t o r s C L C L C f l L = -Si / ( A u ) 2 dx + -SI ; ( A v ) 2 dx + -S£ ; 0 2 dx 2e 0 2e 0 2e 0 ( 2 . 3 5 c ) i n w h i c h u . \> a n d W a r e v e l o c i t i e s o f a n y p l a t e p p p 1 p a r t i c l e ; , v ^ a n d w^ a r e v e l o c i t i e s o f a n y j o i s t p a r t i c l e ; a n d A i i , Av a n d a r e s l i p v e l o c i t i e s . T h e E q s . ( 2 . 3 5 a ) a n d ( 2 . 3 5 b ) a s a b o v e a r e s i m i l a r t o E q s . ( 2 . 2 5 ) a n d ( 2 . 2 9 ) e x c e p t t h a t C p r e p l a c e s p p a n d C ^ r e p l a c e s P j . H e n c e t h e d i s s i p a t i v e e n e r g y i n t h e p l a t e ( C p ) c a n be e x p r e s s e d by E q . ( 2 . 2 7 ) i f p p i s r e p l a c e d by C p a n d t h e d i s s i p a t i v e e n e r g y i n t h e j o i s t ( C ^ ) c a n be e x p r e s s e d by E q . ( 2 . 3 1 ) i f P j i s r e p l a c e d by C b . The E q . ( 2 . 3 5 c ) i s v e r y s i m i l a r t o E q . ( 2 . 2 0 ) e x c e p t t h a t s t i f f n e s s c o e f f i c i e n t s h a v e b e e n r e p l a c e d by t h e d a m p i n g c o e f f i c i e n t s a n d s l i p v e l o c i t i e s a r e s u b s t i t u t e d f o r s l i p s . T h u s d i s s i p a t i v e e n e r g y a s s o c i a t e d w i t h t h e s l i p s ( D c ) b e t w e e n t h e c o v e r a n d t h e j o i s t s c a n be e x p r e s s e d by E q . ( 2 . 2 2 ) i f K n x , K n y a n d a r e r e p l a c e d b y C n x , C n y a n d C n g r e s p e c t i v e l y a n d { 6 n } r e p l a c e d by On t a k i n g t h e v a r i a t i o n o f t h e t o t a l d i s s i p a t i v e e n e r g y i n one e l e m e n t (D) w i t h r e s p e c t t o { $ n } a s g i v e n b y t h e t e r m i n E q . ( 2 . 1 0 ) we o b t a i n a n e x p r e s s i o n o f 30 t h e t y p e [C ] { 6 „ } i n w h i c h [C ] i s t h e e l e m e n t d a m p i n g •* ne n ne m a t r i x o f s i z e ( 1 9 x 1 9 ) . T h e g l o b a l d a m p i n g m a t r i x i s o b t a i n e d by a s s e m b l i n g t h e d a m p i n g m a t r i c e s o f a l l t h e e l e m e n t s . By t a k i n g t h e v a r i a t i o n o f D w . r . t . d i f f e r e n t { $ n ) d a m p i n g m a t r i x f o r e a c h t e r m i n t h e F o u r i e r s e r i e s c a n be o b t a i n e d . 2 . 2 . 4 DERIVATION OF A CONSISTENT LOAD VECTOR A c o n s i s t e n t l o a d v e c t o r i s d e r i v e d f r o m t h e l o a d p o t e n t i a l . The l o a d p o t e n t i a l p e r j o i s t c o r r e s p o n d i n g t o t h e a p p l i e d l o a d f u n c t i o n p ( x , y ) i s g i v e n by s / 2 L U T = / f p ( x , y ) « w ( x , y ) d x d y L - s / 2 0 A p p l y i n g t h e p r i n c i p l e o f v i r t u a l work a n d u s i n g E q , ( 2 . 1 ) we g e t N T 5 / 2 L N n f f x Z ( 6 n } T {R n } = J f p ( x , y ) Z F 1 n ( y ) S i n ( ^ f x - ) d x d y n=1 n n - s / 2 0 n=1 1 n L S i n c e ( 6 n ) i s an a r b i t r a r y v i r t u a l d i s p l a c e m e n t e q u a l i t y h o l d s f o r e a c h n a n d { R n ) c a n be o b t a i n e d . F o r e x a m p l e f o r u n i f o r m l y d i s t r i b u t e d l o a d p ( x , y ) = p we g e t { 5 n } T {R n } = } / p { M 0 ( U } T { 5 n } S i n ( ^ ) d x d * 1 L o r {R n } = J / p { M n ( ? ) } S i n ( ^ x - ) d x d ^ ( 2 . 3 6 ) -1 0 31 T h e g l o b a l c o n s i s t e n t l o a d v e c t o r c a n be o b t a i n e d by c o m b i n i n g a l l t h e e l e m e n t l o a d v e c t o r s { R n ) o f s i z e ( 1 9 x 1 ) . Now we h a v e a l l t h e m a t r i c e s a n d v e c t o r s i n E q . ( 2 . 1 1 ) a n d we c a n f o r m u l a t e t h e c o m p l e t e e q u a t i o n o f m o t i o n o f t h e f l o o r s y s t e m . H o w e v e r , we y e t h a v e t o i n c l u d e t h e e f f e c t o f t h e p e o p l e s t a n d i n g on t h e f l o o r . T h i s i s d o n e i n t h e n e x t s e c t i o n . 2 . 2 . 5 DYNAMIC E F F E C T S OF P E O P L E ON FLOOR R e s e a r c h e r s h a v e p r o p o s e d v a r i o u s m o d e l s w i t h s p r i n g - d a s h p o t c o m b i n a t i o n s t o s t u d y t h e m e c h a n i c a l b e h a v i o u r o f t h e human b o d y . T h e I n t e r n a t i o n a l O r g a n i s a t i o n f o r S t a n d a r d i z a t i o n h a s p r e s e n t e d a two d e g r e e o f f r e e d o m m o d e l i n t h e s t a n d a r d I S O : 5 9 8 2 . T h e m o d e l i n c l u d i n g t h e p r o p o s e d p a r a m e t e r v a l u e s i s shown i n F i g . ( 2 . 6 a ) . T h e b i o d y n a m i c m o d e l l i n g w o r k i n g g r o u p ( I S O / T C 1 0 8 / S C 4/WG 5) h a s p r o p o s e d a d r a f t addendum t o I S O : 5 9 8 2 w h i c h i n c l u d e s an i m p r o v e d a n d more c o m p l e x m o d e l o f , human b e i n g . H e r e , f o r s i m p l i c i t y we w i l l c o n s i d e r j u s t a s i n g l e d e g r e e o f f r e e d o m s p r i n g - d a s h p o t 32 model as shown i n F i g . ( 2 . 6 b ) . ki' |±I e, (2.6a) m 2=13Kg k2=80,OOON/m c 2=930Ns/m m,=62Kg k ,=62,OOON/m c,=1460Ns/m (2.6b) F i g . 2.6. Human Bein g Model The n u m e r i c a l v a l u e s f o r the s p r i n g c o n s t a n t and damping c o e f f i c i e n t have been chosen as the average of the c o r r e s p o n d i n g v a l u e s i n the model shown i n F i g . ( 2 . 6 a ) . In some of the f u t u r e d i s c u s s i o n s , the peopl e on the f l o o r would be r e f e r r e d t o j u s t as masses on the f l o o r . Now c o n s i d e r the f l o o r w i t h p e o p l e on i t as shown i n F i g . ( 2 . 7 ) . The degrees of freedom of the masses a r e shown by y^ ,i=1,2...n. The k i n e t i c energy, p o t e n t i a l energy and the d i s s i p a t i v e energy of f l o o r system i n c l u d i n g t he masses can be e x p r e s s e d as T = 1 { X } T [M]{X} + I I m.y. 2 (2.37) 33 V 1 T N 1 = ^{X} T [K]{X} +Z [ ^ . ( y . - w . ) 2 ] i=1 N " 2 m.gy. n=1 (2.38) D = 1 {X } T [ C ] { i } + Z [ i c , ( Y : - w , ) 2 ] (2.39) i = 1 i n which, [M], [K] and [C] are r e s p e c t i v e l y the g l o b a l mass, s t i f f n e s s and damping m a t r i c e s of the f l o o r . The v e c t o r s {X} and {X} are the g l o b a l displacement and v e l o c i t y v e c t o r s f o r the f l o o r and w^  denotes the v e r t i c a l d e f l e c t i o n of the f l o o r at the l o c a t i o n of mass m.. Now from Eqs. (2.1a) and (2.4a) we have * F l o o r w. l F i g . 2.7. F l o o r with people 34 w = L { M 0 ( ^ ) } T { X n } S i n ( n ^ x - ) i = 1 = < { M 0 ( ^ ) } T S i n ( £ * - ) { M n ( U } T S i n ( 2™ ) . . { M 0 ( U } T S i n ( n ^ ) > { X } i n w h i c h {X } T = { ( X , ) ( X 2 ) ( X n ) } and v e c t o r { M Q U ) } T h a s b e e n e x t e n d e d t o t h e g l o b a l s i z e . F o r e x a m p l e i n c a s e o f two f i n i t e e l e m e n t s t o t a l g l o b a l d . o . f . a r e c o m p o n e n t s . T h e s h a p e f u n c t i o n (MQ ( £ ) } r e l a t e s t o t h e n o d a l d e g r e e o f f r e e d o m o f o n l y one e l e m e n t a n d g i v e s t h e v a r i a t i o n o f d i s p l a c e m e n t o v e r t h a t e l e m e n t . H e n c e i f t h e d i s p l a c e m e n t u n d e r f i r s t e l e m e n t i s d e s i r e d , t h e f i r s t 19 c o m p o n e n t s o f t h e e x t e n d e d ( M Q ( £ ) } v e c t o r c o r r e s p o n d i n g t o t h e d e g r e e s o f f r e e d o m o f f i r s t e l e m e n t s h o u l d be t h e u s u a l s h a p e f u n c t i o n {MQ ( i - ) } . O t h e r c o m p o n e n t s o f t h e e x t e n d e d ( M Q ( £ ) } v e c t o r w i l l be z e r o . T h u s d e p e n d i n g on t h e e l e m e n t number u n d e r w h i c h d e f l e c t i o n i s d e s i r e d , t h e e x t e n d e d v e c t o r w o u l d c h a n g e . T h u s we c a n e x p r e s s w. a s On s u b s t i t u t i n g E q . ( 2 . 4 0 ) i n t o E q s . ( 2 . 3 7 ) , ( 2 . 3 8 ) a n d ( 2 . 3 9 ) we o b t a i n (2x13+6=32) . E a c h o f ( X , ) , ( X 2 ) , • • • , ( X n ) w i l l h a v e 32 w. = { L j } 7 {X} ( 2 . 4 0 ) i = 1 ( 2 . 4 1 ) 35 N V = 1 { X } T [K]{X} + Z [ l k . ( y . - { L . } T { X } ) 2 ] N - Z m . g y . ( 2 . 4 2 ) i = 1 N D = 1 { X } T [C]{X} + Z [ ^ c . ( y . - { L . } T { X } ) 2 ] ( 2 . 4 3 ) z i = 1 z 2 . 2 . 6 F I N A L EQUATIONS OF MOTION F i n a l l y by u s i n g t h e L a g r a n g e ' s e q u a t i o n o f m o t i o n a s g i v e n by E q . ( 2 . 1 0 ) a n d s u b s t i t u t i n g f o r T , V a n d D f r o m E q s . ( 2 . 4 1 ) , ( 2 . 4 2 ) a n d ( 2 . 4 3 ) , we o b t a i n : m i y i + c i y i " c i * L i * T * x } + - k i { L i } T { X } = n u g ( 2 . 4 4 ) N N _ [M]{X} - Z [ c . C L . J l y . + [ Z c . { L . } { L . } T + [ C ] ] { X } i = 1 i = 1 - Z [ k . { L . } ] y . + [ Z k . { L . } { L . } T + [ K ] ] { X } = {R} i=1 1 i=1 ( 2 . 4 5 ) 2 . 3 SOLUTION OF THE EQUATIONS OF MOTION T h e d y n a m i c e q u a t i o n s o f e q u i l i b r i u m a s g i v e n by E q s . ( 2 . 4 4 ) & ( 2 . 4 5 ) a r e a s e t o f s e c o n d o r d e r l i n e a r d i f f e r e n t i a l e q u a t i o n s . A n u m e r i c a l s t e p by s t e p d i r e c t i n t e g r a t i o n m e t h o d i s e m p l o y e d t o o b t a i n t h e s o l u t i o n . In t h i s p r o c e d u r e , d y n a m i c e q u i l i b r i u m i s s a t i s f i e d o n l y a t 36 d i s c r e e t t i m e p o i n t s w i t h i n t h e t o t a l i n t e r v a l o f t h e s o l u t i o n . A v a r i a t i o n o f d i s p l a c e m e n t , v e l o c i t y a n d a c c e l e r a t i o n w i t h i n e a c h t i m e i n t e r v a l i s a s s u m e d a n d t h i s d e f i n e s t h e a c c u r a c y , s t a b i l i t y a n d t h e c o s t o f t h e s o l u t i o n p r o c e d u r e . Of t h e many a v a i l a b l e n u m e r i c a l i n t e g r a t i o n m e t h o d s , an u n c o n d i o n a l l y s t a b l e i m p l i c i t d i r e c t i n t e g r a t i o n m e t h o d i s u s e d h e r e . W i l s o n 0 M e t h o d : T h i s m e t h o d i s an e x t e n s i o n o f t h e l i n e a r a c c e l e r a t i o n m e t h o d . In t h i s m e t h o d a c c e l e r a t i o n i s a s s u m e d t o h a v e a l i n e a r v a r i a t i o n f r o m t i m e t t o ( t + 0 A t ) , where 0 ^ 1 . 0 . When 0=1, t h i s m e t h o d r e v e r t s b a c k t o t h e l i n e a r a c c e r a t i o n - m e t h o d b u t f o r 0>1 .37 i t becomes u n c o n d i t i o n a l l y s t a b l e . U s u a l l y a v a l u e o f 0=1.4 i s u s e d . The a l g o r i t h m i s d e r i v e d h e r e . By l o o k i n g a t F i g . ( 2 . 8 ) we c a n w r i t e t h e l i n e a r a c c e l e r a t i o n a s s u m p t i o n a s £ ( t + 0 A t ) 0At F i g . 2 . 8 . L i n e a r A c c e l e r a t i o n V a r i a t i o n 37 k t + r " x t + ek < X t + 0 A t " X t > ( 2 ' 4 6 ) on i n t e g r a t i o n : Xt + r = *t + V + 2 M ( \ + 6At " Xt ) ( 2 ' 4 7 ) o n c e a g a i n on i n t e g r a t i o n : Xt + r " Xt + V + \ T- + 6 W ( X t + t7At ' Xt > ( 2 ' 4 8 ) Now f r o m E q s . ( 2 . 4 7 ) a n d ( 2 . 4 8 ) , a t t i m e ( t + 0 A t ) o r r=0At we g e t X t + 0 A t - X t + ^ ( X t + 0 A t + X t > ( 2 . 4 9 ) X t + 0 A t - X t * + V A t + ^ f ^ ( X t + 0 A t + 2 X t > ( 2 . 5 0 ) F r o m E q s . ( 2 . 4 9 ) a n d ( 2 . 5 0 ) we s o l v e f o r ^ T + Q ^ T a n d x t + 0 A t a n d we g e t X t + 0 A t " 6 2 ^ t 2 { X t + c?At " X t } 0 A T X t " 2 X t ( 2 . 5 1 ) X t + 0At = 0 A T ( X t + 0 A t X t ) " 2 X t ^ X t ( 2 . 5 2 ) In o r d e r t o o b t a i n t h e r e s p o n s e a t t i m e ( t + A t ) , e q u i l i b r i u m i s c o n s i d e r e d a t t i m e ( t + 0 A t ) i e . 38 [ M ] { X t + 0 A t } + [ C ] { X t + 0 A t } + [ K ] { X t + 0 A t } - { R t + 0 A t } { 2 ' 5 3 ) i n w h i c h R f c + g ^ t = R f c + ^ R t + A t ~ R t ^ * s ! i n e a r l y p r o j e c t e d l o a d v e c t o r . On s u b s t i t u t i n g E q s . ( 2 . 5 1 ) a n d ( 2 . 5 2 ) i n t o ( 2 . 5 3 ) we o b t a i n [ ^ [ M ] + ^ [ C ] - [ K ] ] X t + , A t { R t + * A t + ( ^ 7 [ M ] + M t [ c ] ) x t + ( _ | _ [ M ] + 2 [ C ] ) X t + (2[M] + i | t [ C ] ) X t } ( 2 . 5 4 ) T h e c o e f f i c i e n t m a t r i x on t h e l e f t h a n d s i d e o f E q . ( 2 . 5 4 ) i s c a l l e d t h e e f f e c t i v e s t i f f n e s s m a t r i x a n d t h e v e c t o r on t h e r i g h t h a n d s i d e i s c a l l e d t h e e f f e c t i v e l o a d v e c t o r . T h i s e q u a t i o n c a n be s o l v e d f o r x t + ^ ^ t « T h e n f r o m E q . ( 2 . 5 1 ) X t + 0 A t i s o b t a i n e d « F i n a l l y , E q s . ( 2 . 4 6 ) , ( 2 . 4 7 ) a n d ( 2 . 4 8 ) a r e u s e d t o o b t a i n r e s p o n s e a t t i m e ( t + A t ) by u s i n g r = A t . T h e s o l u t i o n a t t i m e ( t + A t ) i s g i v e n by 6 •( X _ f l A , - X , ) - - £ — X f c + ( I - ! ) * . ( 2 . 5 5 ) t + A t 0 3 A t 2 V t+0At ~t d 2 A t t 6' t X t + A t * X t + f{ X t + A t + X t } ( 2 * 5 6 ) X t + A t - X t + A t ' X t + ^ ( X t + A t + 2 X t > ( 2 ' 5 7 ) 39 Thus, by using t h i s stepwise procedure, s o l u t i o n at any time (t+At) can be obtained i f the response at time t i s known. We need to know the displacement, v e l o c i t y and the a c c e l e r a t i o n v e c t o r s at the beginning to s t a r t t h i s marching procedure. The displacement and v e l o c i t y v e c t o r s are known from the i n i t i a l c o n d i t i o n s of the problem and the i n i t i a l a c c e l e r a t i o n v e c t o r i s computed from Eq. (2.11) T h i s method can be e a s i l y programmed. Even though t h i s method i s u n c o n d i t i o n a l l y s t a b l e , the accuracy of the s o l u t i o n depends on the c h a r a c t e r i s t i c s of the dynamic l o a d i n g and the s i z e of the time step. 3. THE COMPUTER PROGRAM 3.1 PROGRAM FEATURES The computer program p e r f o r m s a dynamic a n a l y s i s o f t h e f l o o r s w i t h e q u i d i s t a n t s t i f f e n e r s i n one d i r e c t i o n . The program e v a l u a t e s t h e r e s p o n s e o f t h e f l o o r s u b j e c t e d t o imp a c t l o a d i n g . The s o u r c e s o f im p a c t l o a d i n g c o u l d be d r o p p i n g o f heavy o b j e c t s , human a c t i v i t y s u c h as d a n c i n g , j u m ping o r w a l k i n g . In t h i s work, t h e p r o g r a m has been a p p l i e d t o e v a l u a t e t h e f l o o r r e s p o n s e a s a r e s u l t o f s i n g l e human f o o t f a l l i m p a c t . T h i s t y p e o f im p a c t o c c u r s .when a p e r s o n w i t h h i s h e e l s r a i s e d a p p r o x i m a t e l y 2 i n . ( 5 c m ) r e l a x e s and a l l o w s h i s h e e l s t o impact t h e f l o o r . S i n c e t h e human b e i n g h as been m o d e l l e d as a s p r i n g - d a s h p o t s y s t e m , t h e f o o t f a l l i m p a c t i s s i m u l a t e d by g i v i n g an i n i t i a l v e l o c i t y (computed f r o m c o n s i d e r a t i o n s o f f r e e f a l l u n d e r g r a v i t y ) t o t h e mass. F o r example, f o r an h e e l impact o f 2 i n c h e s , t h e i n i t i a l v e l o c i t y w ould be i/2x386.4x2=39.31 i n c h e s p e r s e c o n d . The p r o g r a m p e r f o r m s a t i m e s t e p a n a l y s i s and g i v e s i n f o r m a t i o n a b o u t t h e d i s p l a c e m e n t , v e l o c i t y and a c c e l e r a t i o n a t any t i m e , t>0 a n d a t any p o i n t on t h e f l o o r . The p r o g r a m p r o v i d e s t h e o p t i o n t o r a n d o m l y s e l e c t t h e modulus o f e l a s t i c i t y of t h e j o i s t s and t o c a r r y o u t s i m u l a t i o n s t o e v a l u a t e t h e s y s t e m v a r i a b i l i t y p r o d u c e d by v a r i a b i l i t y i n t h e p o p u l a t i o n o f j o i s t s . The p r o g r a m c o n s i d e r s damping and t h e damping m a t r i x h a s been d e r i v e d f r o m e n e r g y c o n s i d e r a t i o n s a s e x p l a i n e d i n t h e s e c o n d 4 0 41 c h a p t e r . T h e mass m a t r i x u s e d i n t h e p r o g r a m i s a c o n s i s t e n t mass m a t r i x d e r i v e d f r o m e n e r g y c o n s i d e r a t i o n s a s e x p l a i n e d i n t h e s e c o n d c h a p t e r a n d n o t a lumped mass m a t r i x . The d i m e n s i o n s t a t e m e n t s i n t h e p r o g r a m i m p o s e t h e f o l l o w i n g l i m i t a t i o n s : 1. T h e number o f j o i s t s i n t h e f l o o r a r e l i m i t e d t o 1.0. 2 . T h e number o f t h e F o u r i e r s e r i e s t e r m s t h a t c a n be c o n s i d e r e d f o r s o l u t i o n a r e l i m i t e d t o 3 . 3 . T h e number o f p e r s o n s o r t h e m a s s e s t h a t c a n be c o n s i d e r e d t o o c c u p y t h e f l o o r a r e l i m i t e d t o 5 . 4 . T h e number o f f l o o r p o i n t s a t w h i c h r e s p o n s e c a n be e v a l u a t e d a r e l i m i t e d t o 10 . T h e s e l i m i t a t i o n s c a n be e a s i l y o v e r c o m e by c h a n g i n g t h e d i m e n s i o n s t a t e m e n t s f o r t h e d i f f e r e n t m a t r i c e s a n d v e c t o r s . 3 . 2 PROGRAM STRUCTURE T h e p r o g r a m c o n s i s t s o f a number o f s u b r o u t i n e s w h i c h r e a d a n d p r i n t t h e s t r u c t u r e a n d l o a d d a t a , c a r r y o u t n u m e r i c a l i n t e g r a t i o n , a s s e m b l e m a s s , s t i f f n e s s a n d d a m p i n g m a t r i c e s , s o l v e s y s t e m o f e q u a t i o n s o f d y n a m i c e q u i l i b r i u m a n d g e n e r a t e r e s u l t s . T h e s e s u b r o u t i n e s c o m b i n e d w i t h t h e m a i n p r o g r a m where t h e y a r e c a l l e d p r o d u c e t h e c o m p l e t e p r o g r a m . A b r i e f d i s c u s s i o n o f t h e m a i n s u b r o u t i n e s a n d t h e o p e r a t i o n s t h e y c a r r y o u t i s p r e s e n t e d h e r e . S u b r o u t i n e GENMTX: T h i s s u b r o u t i n e e v a l u a t e s t h e i n t e g r a l s a s shown i n E q . ( 2 . 1 5 ) r e q u i r e d i n t h e d e v e l o p m e n t o f t h e s t i f f n e s s , mass a n d t h e d a m p i n g m a t r i c e s . T h e r o u t i n e 42 c o m p u t e s t h e i n t e g r a l s n u m e r i c a l l y u s i n g t h e s i x p o i n t G a u s s Q u a d r a t u r e r u l e . S u b r o u t i n e DATA & P R I N T : The s u b r o u t i n e DATA r e a d s i n a l l t h e i n p u t f o r t h e s t r u c t u r e , s u c h a s m a t e r i a l p r o p e r t i e s , s i z e s a n d d i m e n s i o n s , b o u n d a r y c o n d i t i o n s e t c . a n d t h e l o a d i n g d a t a . T h e n s u b r o u t i n e PRINT p r i n t s o u t t h e r e a d d a t a w h i c h s e r v e s a s a c h e c k on t h e r e a d d a t a . S u b r o u t i n e ASMBLY: T h i s r o u t i n e a s s e m b l e s t h e g l o b a l f l o o r s t i f f n e s s , mass a n d t h e d a m p i n g m a t r i c e s . I t a l s o a s s e m b l e s t h e g l o b a l l o a d v e c t o r . T h i s r o u t i n e c a l l s a s u b r o u t i n e STFMAS w h i c h c o m p u t e s t h e e l e m e n t s t i f f n e s s , mass a n d t h e d a m p i n g m a t r i c e s i e . S T I F F , DMASS a n d DAMP r e s p e c t i v e l y o f s i z e ( 19x19 ) a n d t h e e l e m e n t l o a d v e c t o r V E C T R . T h i s r o u t i n e a s s e m b l e s t h e e l e m e n t m a t r i c e s . The g l o b a l m a t r i c e s a r e s y m m e t r i c a n d t h u s o n l y t h e l o w e r t r i a n g u l a r p a r t s o f t h e m a t r i c e s a r e s t o r e d i n t o one d i m e n s i o n a l v e c t o r s . T h i s i s r e p e a t e d f o r e a c h n ( t e r m i n F o u r i e r S e r i e s ) a n d a l l t h e v e c t o r s a r e s t o r e d i n G S T I F , GMASS a n d GDAMP. S i n c e t h e r e a r e 19 d e g r e e s o f f r e e d o m p e r e l e m e n t a n d 12 a r e s h a r e d w i t h t h e a d j o i n i n g e l e m e n t s ( 6 a t p o i n t 1 a n d 6 a t p o i n t 2 o f F i g . 2 . 2 ) , t h e g l o b a l d e g r e e s o f f r e e d o m w o u l d be ( 1 3 N J T + 6 ) , where N J T a r e t h e number o f j o i s t s . T h e g l o b a l m a t r i c e s w o u l d be b a n d e d m a t r i c e s w i t h a b a n d w i d t h o f 1 9 . T h u s , e a c h o f t h e v e c t o r s i n t h e m a t r i c e s G S T I F , GMASS a n d GDAMP w o u l d h a v e l 9 ( l 3 N J T + 6 ) c o m p o n e n t s . By a s s e m b l i n g t h e e l e m e n t l o a d v e c t o r s V E C T R t h e g l o b a l v e c t o r FORCE 1 o f s i z e (13NJT+6) a s s o c i a t e d w i t h e a c h n i s o b t a i n e d . T h i s r o u t i n e a l s o 43 i m p o s e s t h e b o u n d a r y c o n d i t i o n s on t h e s t i f f n e s s , mass a n d t h e d a m p i n g m a t r i c e s o n c e t h e y a r e f o r m e d . T o do t h i s t h e rows a n d c o l u m n s o f t h e m a t r i c e s c o r r e s p o n d i n g t o t h e r e s t r a i n e d d e g r e e s o f f r e e d o m a r e made z e r o . A l s o t h e e l e m e n t i n t h e l o a d v e c t o r c o r r e s p o n d i n g t o t h e r e s t r a i n e d d e g r e e o f f r e e d o m i s made z e r o . T h i s r o u t i n e a l s o c a l l s a s u b r o u t i n e DECMP w h i c h c a r r i e s o u t a C h o l e s k y d e c o m p o s i t i o n o f t h e g l o b a l mass m a t r i x a n d s t o r e s i t i n t o m a t r i x DCPMAS. S u b r o u t i n e A X L R T N : T h i s r o u t i n e c o m p u t e s t h e i n i t i a l a c c e l e r a t i o n v e c t o r i e . t h e a c c e l e r a t i o n i n d u c e d i n t h e f l o o r s y s t e m a t t h e i n s t a n t o f i m p a c t . T h e i n i t i a l d i s p l a c e m e n t v e c t o r o f t h e f l o o r s y s t e m a t t h e i n s t a n t o f i m p a c t i s t a k e n a s z e r o a s t h e f l o o r i s c o n s i d e r e d t o be i n s t a t i c e q u i l i b r i u m p r i o r t o i m p a c t . T h e r o u t i n e c o m p u t e s t h e i n i t i a l a c c e l e r a t i o n v e c t o r d i r e c t l y f r o m t h e e q u a t i o n o f m o t i o n g i v e n by E q . ( 2 . 1 1 ) . S u b r o u t i n e E F M A T : T h i s r o u t i n e c o m p u t e s t h e e f f e c t i v e s t i f f n e s s m a t r i x o f E q . ( 2 . 5 4 ) . T h e e f f e c t i v e s t i f f n e s s m a t r i x d e p e n d s n o t o n l y on t h e b a s i c s t i f f n e s s , mass a n d t h e d a m p i n g m a t r i c e s b u t a l s o on t h e v a l u e o f 6 a n d t h e s i z e o f t h e t i m e s t e p . T h i s r o u t i n e a l s o c a r r i e s o u t a C h o l e s k y d e c o m p o s i t i o n o f t h e e f f e c t i v e s t i f f n e s s m a t r i x by c a l l i n g a s u b r o u t i n e DECMP. T h i s i s n e e d e d when s o l v i n g f o r t h e unknown d i s p l a c e m e n t v e c t o r . T h i s d e c o m p o s e d e f f e c t i v e s t i f f n e s s m a t r i x i s s t o r e d i n E F S T I F . S u b r o u t i n e E F L O A D ; T h i s r o u t i n e c o m p u t e s t h e e f f e c t i v e l o a d v e c t o r o f E q . ( 2 . 5 4 ) . T h i s a l s o l i k e e f f e c t i v e s t i f f n e s s 44 m a t r i x depends on the b a s i c s t i f f n e s s , mass and the damping m a t r i c e s and on 6 and the time s t e p s i z e . S u b r o u t i n e SOLVE: T h i s r o u t i n e s o l v e s f o r the v e c t o r { X t i „ . J t+0At J i n Eq. (2.54) and computes the d i s p l a c e m e n t , v e l o c i t y and the a c c e l e r a t i o n v e c t o r s a t time (t+At) g i v e n by Eqs. (2 . 5 5 ) , (2.56) and ( 2 . 5 7 ) . An i t e r a t i v e p r o c e d u r e i s employed t o o b t a i n the s o l u t i o n . S c h e m a t i c a l l y , Eq. (2.54) can be w r i t t e n as S ( 1 , 1 ) S (1,k) S(1,N) x , R i S ( k , 1) S ( k , k ) S(k,N) X k = R k S(N,1) S(N,k) S(N,N) _ X N _ _ R N _ i n which S ( k , k ) = [ S ( k , k ) ] i s a g l o b a l m a t r i x of s i z e (13NJT+6)x(13NJT+6) ; X k={X k} i s a g l o b a l v e c t o r of s i z e (13NJT+6) and N i s the no of terms i n the F o u r i e r S e r i e s . Now the s o l u t i o n f o r each v e c t o r (X^} can be w r i t t e n as N . {X . } 1 = [ S ( k , k ) ] - 1 {{R. }- I [ S ( k , n ) ] { R n } 1 - 1 } (3.1) n= 1 n i = ( 1 , 2 , ) n*k i n which i denotes the i t e r a t i o n number. For the s t a r t i n g v e c t o r f o l l o w i n g a p p r o x i m a t i o n i s used {X k}° = [ S ( k , k ) ] - 1 { R k } (3.2) 45 The i t e r a t i o n s a r e stopped when ' V 1 - < V 1 " ' < e (3.3) f o r k=1,2,...,N and e a s m a l l number ( e . g . , 0.001). In the above e q u a t i o n s the terms a s s o c i a t e d w i t h the masses have not been i n c l u d e d f o r s i m p l i c i t y . The s o l u t i o n f o r y^ (degrees of freedom a s s o c i a t e d w i t h the masses) i s o b t a i n e d by u s i n g the same i t e r a t i v e p r o c e d u r e . In the above d e s c r i b e d p r o c e d u r e o n l y (13NJT+6) e q u a t i o n s need be s o l v e d a t each s t e p of the i t e r a t i o n . The convergence i s u s u a l l y o b t a i n e d w i t h i n 5 i t e r a t i o n s . S u b r o u t i n e RESULT: The s o l u t i o n v e c t o r s which a r e o b t a i n e d by r o u t i n e SOLVE are the complete s o l u t i o n v e c t o r s g i v i n g i n f o r m a t i o n about a l l the d e f o r m a t i o n s a s s o c i a t e d w i t h each degree of freedom. However, we a r e o n l y i n t e r e s t e d i n the v e r t i c a l out of p l a n e d e f o r m a t i o n of the f l o o r . T h i s s u b r o u t i n e p i c k s up the r e l e v a n t components of the s o l u t i o n v e c t o r and p r i n t s out the v e r t i c a l d i s p l a c e m e n t , v e l o c i t y and the a c c e l e r a t i o n of the f l o o r . Appendix I I i n c l u d e s the "User's Manual" and i l l u s t r a t i v e i n p u t / o u t p u t f i l e s . A comprehensive l i s t i n g of the computer program i s p r o v i d e d i n Appendix I I I . 4. HUMAN PERCEPTION OF VIBRATIONS 4.1 HUMAN DISCOMFORT CRITERION Depending on the f l o o r c o n f i g u r a t i o n , a person may f i n d v i b r a t i o n s caused by f o o t f a l l impacts during normal walking o b j e c t i o n a b l e . Wiss and Parmelee {15} conducted an experimental study of human response to t r a n s i e n t v e r t i c a l v i b r a t i o n s i n terms of t h e i r frequency, maximum displacement and damping. The wave-form which was used -for these v i b r a t i o n s i s shown i n F i g . (4.1). The v i b r a t i o n s las-ted not more than 5 seconds and the damping r a t e v a r i e d between 0.01 to 0.20 of c r i t i c a l damping. During the course of Wiss & Parmelee's s t u d y , s e v e r a l persons were su b j e c t e d to v i b r a t i o n s of v a r i o u s combinations of frequency, peak amplitude and damping and were asked to 1A < 5 F i g . 4.1. V i b r a t i o n S i g n a l used by Wiss & Parmelee 46 4 7 c l a s s i f y e a c h o f t h e v i b r a t i o n s a c c o r d i n g t o t h e f o l l o w i n g c l a s s i f i c a t i o n : ( 1 ) i m p e r c e p t i b l e ; ( 2 ) b a r e l y p e r c e p t i b l e ; ( 3 ) d i s t i n c t l y p e r c e p t i b l e ; ( 4 ) s t r o n g l y p e r c e p t i b l e ; a n d ( 5 ) s e v e r e . A t o t a l o f 40 s u b j e c t s were t e s t e d . N u m e r i c a l v a l u e s w e r e a s s i g n e d t o e a c h r a t i n g a s f o l l o w s : R a t i n g ( R ) C l a s s i f i c a t i o n 1 I m p e r c e p t i b l e 2 B a r e l y P e r c e p t i b l e 3 D i s t i n c t l y P e r c e p t i b l e 4 S t r o n g l y P e r c e p t i b l e 5 S e v e r e S i n c e r a t i n g s 1 a n d 5 a r e u n b o u n d e d a t t h e l o w e r a n d u p p e r r e g i o n s r e s p e c t i v e l y , i t i s h a r d t o e s t a b l i s h t h e t r u e mean l i n e f o r t h e s e r a t i n g s a n d t h u s t h e s e a r e n o t c o n s i d e r e d i n t h e a n a l y s i s . F o r r a t i n g s R = 2 , 3 a n d 4 , a n d d a m p i n g c o e f f i c i e n t s l a r g e r t h a n 0.01 m u l t i p l e r e g r e s s i o n a n a l y s i s were p e r f o r m e d a n d a r e l a t i o n s h i p b e t w e e n t h e t h r e e v a r i a b l e s R, FA a n d D was o b t a i n e d . The p r e d i c t i o n e q u a t i o n i s g i v e n by * " 5 - 0 8 < F ^ 2 T 7 > 0 - 2 6 5 (4-'> w h e r e : F = f r e q u e n c y , i n c y c l e s p e r s e c o n d A=peak d i s p l a c e m e n t i n i n c h e s D=damping r a t i o 48 R=response r a t i n g In t h e i r study Wiss and Parmelee a l s o suggested some a l t e r n a t e m a t h e m a t i c a l models by p e r f o r m i n g somewhat d i f f e r e n t s t a t i s t i c a l a n a l y s e s . The p r e d i c t i o n e q u a t i o n s from two such models a r e R = 2.624( FA ,0.0463 0.1445 (4.2) R = 3.830( FA ,0.311 0.155 (4.3) P l o t of pr o d u c t of fr e q u e n c y & a m p l i t u d e (FA) v s . damping (D) f o r response r a t i n g s R=2,3 & 4 and based upon Eqs. (4.1) i s shown i n F i g . ( 4 . 2 ) . 1 .000 a 0 .300 "i o . i o o 0 .030 ST 0 . 0 1 0 0 .003 0.001 R»4 R-3 H»2 0.01 0 .03 0 . 1 0 0 .20 D a m p i n g r a t i o The F i g . 4.2. Wiss & Parmelees model c u r v e s p l o t i l l u s t r a t e s t h a t v i b r a t i o n s of a p a r t i c u l a r 49 f r e q u e n c y a n d a m p l i t u d e a r e p r o g r e s s i v e l y l e s s p e r c e p t i b l e a s d a m p i n g i s i n c r e a s e d . A l s o n o t e t h a t f o r a g i v e n r a t i n g t h e p r o d u c t o f f r e q u e n c y a n d a m p l i t u d e may be a p p r o x i m a t e l y t w i c e a s much when d a m p i n g i s i n c r e a s e d f r o m 0 . 0 2 t o 0 .20 o f c r i t i c a l . Now l o o k a t a t y p i c a l f l o o r r e s p o n s e t o f o o t f a l l i m p a c t a s shown i n F i g . 4 . 3 . 4 E E c c CD CD O _o CL CO Q 200 250 300 Time in millisec 500 F i g . 4 . 3 . T y p i c a l f l o o r r e s p o n s e t o f o o t f a l l i m p a c t T h e f l o o r v i b r a t i o n h a s a d a m p i n g r a t e o f 0 . 1 5 o f c r i t i c a l . I t d i e s much e a r l i e r t h a n 5 s e c o n d s a n d t h u s h a s c h a r a c t e r s t i c s s i m i l a r t o t h e ^ a v e - f o r m u s e d by W i s s & P a r m e l e e . H e n c e i n t h i s work E q . ( 4 . 1 ) h a s b e e n u s e d a s a c r i t e r i o n f o r c l a s s i f y i n g human r e s p o n s e t o f l o o r v i b r a t i o n s . 50 4 . 2 HUMAN PERCEPTION AND FLOOR R E L I A B I L I T Y L i m i t s t a t e s m e t h o d s o f d e s i g n a r e b e i n g i n t r o d u c e d i n t o d e s i g n r e c o m m e n d a t i o n s by v a r i o u s c o d e s . T h i s p r o b a b i l i t y b a s e d a p p r o a c h i s more r e a l i s t i c a s t h e v a r i a b i l i t y o f s e v e r a l d e s i g n p a r a m e t e r s i s t a k e n i n t o a c c o u n t r a t h e r t h a n a s s u m i n g them t o be d e t e r m i n i s t i c . F o s c h i {6} h a s p r e s e n t e d an i n t e r e s t i n g d i s c u s s i o n on t h i s d e s i g n c o n c e p t . F l o o r v i b r a t i o n s a n d human p e r c e p t i o n o f a n n o y i n g m o t i o n i s a " s e r v i c e a b i l i t y l i m i t s t a t e " . T h a t i s , i t r e f e r s t o p e r f o r m a n c e a c c e p t a n c e r a t h e r t h a n t o c a t a s t r o p h i c f a i l u r e . H a v i n g a means o f c l a s s i f y i n g human r e s p o n s e t o f l o o r v i b r a t i o n s , we c a n u s e r e l i a b i l i t y t h e o r y t o p r e d i c t t h e p r o b a b i l i t y t h a t f l o o r s w o u l d be c l a s s i f i e d i n t o a p a r t i c u l a r r a t i n g c a t e g o r y by p e o p l e . S u p p o s e we a r e i n t e r e s t e d i n k n o w i n g t h e p r o b a b i l i t y t h a t p e o p l e w o u l d r a t e t h e f l o o r s a s h a v i n g a r e s p o n s e r a t i n g o f R 0 o r l e s s . We d e f i n e t h e f o l l o w i n g f u n c t i o n : G = R 0 - R ( 4 . 4 ) i n w h i c h R i s t h e r a t i n g g i v e n t o e a c h f l o o r by p e o p l e . T h e d i s t r i b u t i o n o f r a t i n g s i s o b t a i n e d f o r a p o p u l a t i o n o f s i m u l a t e d f l o o r s c o n s i d e r i n g t h e v a r i a b i l i t y i n . t h e m o d u l u s o f e l a s t i c i t y o f t h e j o i s t s . Now G>0 f o r R < R 0 ; G=0 f o r R = R 0 ; a n d G<0 f o r R > R 0 . T h u s , p o s i t i v e v a l u e s o f G i m p l y t h a t t h e f l o o r s w o u l d q u a l i f y t o be i n a s p e c i f i e d c a t e g o r y o r i n o t h e r s w o r d s w o u l d s u r v i v e t h e t e s t a n d n e g a t i v e v a l u e s o f G i m p l y t h a t f l o o r w o u l d f a i l t h e t e s t . H e n c e , G c a n be 51 i n t e r p r e t e d a s a F a i l u r e F u n c t i o n . S t u d y o f t h e P r o b ( G < 0 ) t h u s l e a d s t o an e s t i m a t e o f t h e P r o b a b i l i t y o f F a i l u r e , P^ w i t h r e s p e c t t o t h e p a r t i c u l a r l i m i t s t a t e o f e x c e e d i n g t h e d e s i r e d r a t i n g R 0 . The p r o b a b i l i t y o f f a i l u r e c a n be r e l a t e d t o t h e R e l i a b i l i t y I n d e x , 0 w h i c h f a c i l i t a t e s t h e u n d e r s t a n d i n g a n d t h e d e t e r m i n a t i o n o f t h e p r o b a b i l i t y o f f a i l u r e . T h e r e l i a b i l i t y i n d e x , (J c a n be w r i t t e n a s : & = G / a G ( 4 . 5 ) R e l i a b i l i t y i n d e x , /3 i s a m e a s u r e o f t h e number o f s t a n d a r d d e v i a t i o n s (oG ) , G i s f r o m t h e f a i l u r e e v e n t G = 0 . The p r o b a b i l i t y o f f a i l u r e , P^ i s g i v e n by t h e a r e a o f t h e s h a d e d r e g i o n i n F i g . ( 4 . 4 ) . P R O B A B I L I T Y D E N S I T Y 0 G G F i g . 4 . 4 . R e l i a b i l i t y i n d e x a n d P r o b a b i l i t y o f f a i l u r e I f G h a s a n o r m a l d i s t r i b u t i o n t h e p r o b a b i l i t y o f f a i l u r e c a n be d i r e c t l y d e t e r m i n e d f r o m t h e ' n o r m a l t a b l e s a s : P f = ( 4 . 6 ) i n w h i c h $ i s t h e c u m u l a t i v e d i s t r i b u t i o n f o r t h e s t a n d a r d i z e d n o r m a l p r o b a b i l i t y f u n c t i o n . 52 Thus, r e l i a b i l i t y index and t h e p r o b a b i l i t y of f a i l u r e can be determined w i t h o u t any d i f f i c u l t y i f G i s a l i n e a r c o m b i n a t i o n of n o r m a l l y d i s t r i b u t e d v a r i a b l e s . I f the v a r i a b l e s a r e non-normal or G i s a n o n - l i n e a r f u n c t i o n , the Eq. ( 4 . 6 ) would g i v e o n l y an approximate v a l u e of P^. However, the e r r o r can be m i n i m i z e d i f the b a s i c v a r i a b l e s a r e t r a n s f o r m e d i n t o a s e t of normal v a r i a b l e s b e f o r e computing the r e l i a b i l i t y i n d e x . R a c k w i t z F i e s s l e r A l g o r i t h m {11} does t h i s t r a n s f o r m a t i o n . 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 t o l o c a t e the most l i k e l y f a i l u r e p o i n t and g i v e s an e s t i m a t e of the r e l i a b i l i t y i ndex which can be r e l a t e d t o the normal t a b l e s t o get P^. I f the v a r i a b l e s x , , x 2 , . . . . , x n a r e dependent on one a n o t h e r , t h e y s h o u l d be u n c o r r e l a t e d b e f o r e u s i n g the R a c k w i t z F i e s s l e r A l g o r i t h m . A computer program has been w r i t t e n based on t h i s a l g o r i t h m and i s q u i t e s i m p l e t o use. 5. RESULTS AND DISCUSSIONS 5.1 NUMERICAL CONVERGENCE Since a F o u r i e r s e r i e s approximation i s employed along the f l o o r span ( x - d i r e c t i o n ) , the e f f i c i e n c y of the a n a l y s i s depends on the number of F o u r i e r s e r i e s terms (n) needed f o r convergence of the s o l u t i o n . The f l o o r c o n s i d e r e d c o n s i s t s of nine j o i s t s supported on a l l four s i d e s (the usual boundary c o n d i t i o n s of simple supports at j o i s t ends p l u s r e s t r a i n t on the v e r t i c a l d e f l e c t i o n of the f i r s t and the l a s t j o i s t ) . The convergence was t e s t e d by a n a l y z i n g the f l o o r response to , impact f o r two values of "n", Only the 8 7- O n=1 • n=3 -1 o 50 100 150 200 250 300 Time in millisec 350 400 450 500 F i g . 5.1. E f f e c t of "n" on F l o o r response 53 54 f i r s t t e r m i n t h e F o u r i e r s e r i e s was c o n s i d e r e d f o r one c a s e a n d f i r s t t h r e e t e r m s were c o n s i d e r e d f o r t h e o t h e r c a s e . T h e r e s u l t s a r e shown i n F i g . ( 5 . 1 ) . T h e c o n v e r g e n c e i s a p p a r e n t a n d t h e r e s u l t s s u g g e s t t h a t u s i n g o n l y t h e f i r s t t e r m i n t h e F o u r i e r s e r i e s i s s a t i s f a c t o r y . T h i s was e x p e c t e d a s t h e f l o o r was e x c i t e d by a s i n g l e p o i n t i m p a c t on i t s c e n t r o i d . 5 . 2 S E N S I T I V I T Y A N A L Y S I S A s e n s i t i v i t y a n a l y s i s was d o n e t o s t u d y t h e e f f e c t o f v a r i o u s f l o o r p a r a m e t e r s on t h e f l o o r r e s p o n s e . A r e f e r e n c e f l o o r w i t h t y p i c a l p a r a m e t e r v a l u e s a s shown i n t a b l e ( 5 . 1 ) was c h o s e n . P a r a m e t e r S y m b o l R e f e r e n c e v a l u e Range J o i s t s p a c i n g S 1 6 i n 1 2 -20 R e c e i v e r ' s w e i g h t W R 2001b 140 -200 I m p a c t e r ' s w e i g h t . WI 2001b 140-200 M a s s d a m p i n g RC 6 1 b . s / i n 3 -12 S p a n RL 1 4 4 i n 96-192 J o i s t MOE E 1 . 5 5 d 0 6 ) p s i 1 . 2-1 . 8 F l o o r d a m p i n g CP 6 1 b . s / i n 3 -12 J o i s t d e p t h H 7 . 2 5 i n 7 . 2 5 - 1 1 . 2 5 N a i l s p a c i n g ENL 8 i n 0 . 5 - 1 6 N a i l s t i f f n e s s KN 6 8 0 0 1 b / i n 1 0 3 - 1 0 5 T a b l e 5.1. R e f e r e n c e F l o o r 55 T h e r a n g e s o f p a r a m e t e r v a l u e s s t u d i e d a r e a l s o shown i n t a b l e 5 . 1 . In a d d i t i o n t h e r e f e r e n c e f l o o r h a d a p l a t e t h i c k n e s s o f 5 / 8 i n a n d was s u p p o r t e d on a l l f o u r s i d e s . To s t u d y t h e e f f e c t o f a p a r t i c u l a r p a r a m e t e r , t h e v a l u e o f t h a t p a r a m e t e r was v a r i e d a r o u n d t h e r e f e r e n c e v a l u e w h i l e k e e p i n g o t h e r p a r a m e t e r s u n c h a n g e d a n d t h e r e s p o n s e a n a l y z e d . T h e e f f e c t o f t h e s e p a r a m e t e r s on t h e f l o o r r e s p o n s e a r e i l l u s t r a t e d by f i g u r e s ( 5 . 2 ) t o ( 5 . 9 ) . F o l l o w i n g i s a b r i e f d i s c u s s i o n o f t h e p l o t s . F i g u r e s ( 5 . 2 ) a n d ( 5 . 3 ) show t h e e f f e c t o f f l o o r d a m p i n g a n d mass d a m p i n g . W h i l e t h e i n f l u e n c e o f mass d a m p i n g i s s i g n i f i c a n t , t h e e f f e c t o f f l o o r d a m p i n g i s n e g l i g i b l e . I t c a n n o t be c o n c l u d e d f r o m h e r e t h a t f l o o r d a m p i n g i s u n i m p o r t a n t s i n c e , v a l u e s o f CP were a s s u m e d a r b i t r a r i l y . H o w e v e r , t h e s e p l o t s c l e a r l y i l l u s t r a t e t h a t i n c o n s i d e r i n g t o t a l s y s t e m d a m p i n g mass d a m p i n g c a n n o t be i g n o r e d . F i g u r e s ( 5 . 4 ) a n d ( 5 . 5 ) i l l u s t r a t e t h a t a s s p a n i s r e d u c e d o r j o i s t d e p t h i s i n c r e a s e d , t h e f l o o r becomes s t i f f e r , t h e r e b y r e d u c i n g t h e p e a k d i s p l a c e m e n t a n d i n c r e a s i n g t h e f r e q u e n c y o f v i b r a t i o n s . T h e i n i t i a l k i n k o b s e r v e d i n t h e c u r v e s f o r s t i f f e r f l o o r s i s due t o t h e f a c t t h a t t h e a c c e l e r a t i o n v a r i a t i o n a t t h e i n s t a n t o f i m p a c t i s much more i r r e g u l a r f o r s t i f f e r f l o o r s t h a n f o r r e l a t i v e l y f l e x i b l e f l o o r s . S i n c e we a r e a s s u m i n g a l i n e a r a c c e l e r a t i o n v a r i a t i o n w i t h i n t h e t i m e s t e p , i t means t h a t t h e s i z e o f t h e t i m e s t e p i n t h e b e g i n n i n g i s t o o b i g f o r s t i f f e r 56 f l o o r s . F i g . ( 5 . 6 ) i l l u s t r a t e s t h a t j o i s t s p a c i n g h a s v e r y l i t t l e i n f l u e n c e on f l o o r r e s p o n s e . As j o i s t s p a c i n g i s i n c r e a s e d , t h e f l e x i b i l i t y o f t h e f l o o r i n c r e a s e s due t o w h i c h t h e r e s p o n s e t e n d s t o go u p b u t a t t h e same t i m e t h e d i s t a n c e b e t w e e n t h e i m p a c t e r a n d t h e r e c e i v e r i n c r e a s e s due t o w h i c h t h e r e s p o n s e t e n d s t o d r o p . T h u s t h e c o m b i n e d e f f e c t i s t h a t t h e r e s p o n s e r e m a i n s p r a c t i c a l l y u n c h a n g e d . F i g . ( 5 . 7 ) i l l u s t r a t e s t h a t a s n a i l s p a c i n g i s d e c r e a s e d , t h e r e l a t i v e movement be tween t h e p l a t e a n d t h e j o i s t i s r e d u c e d , t h e r e b y m a k i n g t h e f l o o r s t i f f e r a n d t h u s a l o w e r p e a k d i s p l a c e m e n t i s o b s e r v e d . F i g . ( 5 . 8 ) i l l u s t r a t e s t h a t t h e w o r s t i m p a c t c o n d i t i o n i s when t h e r e i s a h e a v y i m p a c t e r a n d a l i g h t r e c e i v e r . T h i s i m p a c t c o n d i t i o n i s u s e d f o r a l l t h e a n a l y s i s . F i g . ( 5 . 9 ) i l l u s t r a t e s t h e e f f e c t o f t o r s i o n a l a n d l a t e r a l r e s t r a i n t on t h e j o i s t s . I t i s o b s e r v e d t h a t t o r s i o n a l r e s t r a i n t i s q u i t e e f f e c t i v e i n r e d u c i n g t h e p e a k d i s p l a c e m e n t . 57 3.5-3-CP=3lb.see/ln  CP=6lb.sec/in CP=12lb.sec/in E E CD E CD O _o CL (/) Q -0.5--1--50 100 150 1 i 1 1 1 1 I 1 1 1 1 I 1 1 1 1 i • • • • i • • • • i 200 250 300 350 400 450 500 Time in millisec F i g . 5 . 2 . E f f e c t o f f l o o r d a m p i n g on r e s p o n s e E E _c ~c CD E CD O _o CL </) Q 3.5-2.5-RC=5lb. sec/ in RC=6lb.sec/in  RC=12lb.sec/in 1.5-0.5--0.5--1-50 100 150 200 250 300 Time in millisec 350 400 450 500 F i g . 5 . 3 . E f f e c t o f mass d a m p i n g on r e s p o n s e 58 F i g . 5 . 5 . E f f e c t o f j o i s t d e p t h on r e s p o n s e 59 4-, _ , , 0.5 H "I I 1 1 1 1 I 1 1 1 1 I 1 1 1 1 I ' 1 1 1 I 1 ! 1 ' I 1 1 1 1 I 1 1 1 1 I 1 1 1 1 I 1 1 1 1 I 1 1 1 1 I 0 50 100 150 200 250 300 350 400 450 500 Time in millisec F i g.5.6. E f f e c t of j o i s t spacing on response 4-1 1 0.5 A I ' 1 1 1 I 1 1 1 1 I 1 1 1 ' I ' 1 1 1 I ! 1 ' I . I I I I ' • ' ' I ' ' ' 1 I ' • • I 0. 50 100 150 200 250 300 350 400 450 500 Time in mill isec F i g . 5 . 7 . E f f e c t o f n a i l s p a c i n g on r e s p o n s e 6 0 E E E CD O _D C L V) Q 3.5-2.5-2-1.5-0.5 -0.5 -1-Wi=200lb;Wr=U0lb Wi=200lb;Wr=200lb Wi=140lb;Wr=200lb 50 100 150 200 250 300 350 400 450 500 Time in millisec F i g . 5 . 8 . E f f e c t of weight of people on response c CD E CD O JO CL V) Q 3.5 2.5 2-1.5 0.5--0.5 -1-50 NO TORSIONAL RESTRAINT TORSIONALLY RESTRAINED 100 150 200 250 300 Time in millisec 350 400 450 500 Fig.5.9. E f f e c t of t o r s i o n a l r e s t r a i n t on response 61 Regression A n a l y s i s ; A m u l t i p l e . l i n e a r r e g ression a n a l y s i s was done to come up with expressions f o r the peak displacement and the frequency of v i b r a t i o n s sensed by the r e c e i v e r i n terms of the f l o o r parameters. An equation of the f o l l o w i n g form was proposed. X p/6 = A 0 + A,(S/16) + A 2(W R/200) + A 3(W I/200) + A,(RC/6) + A 5(RK/400) + A 6(RL/144) + A 7(E/1.55) + A 8(CP/6) + A 9(H/7.25) + A 1 0(ENL/8) + A^KN /eSOO) (5.1) subject to c o n s t r a i n t (X p/5)o = A 0 + A, + A 2 + + A,, (5.2) in which: A 0, A 1 ( . . . . f A n are the c o e f f i c i e n t s to be determined by r e g r e s s i o n . X p i s the peak f l o o r displacement at the r e c e i v e r ' s point made nondimensional by d i v i d i n g i t by the s t a t i c d e f l e c t i o n , 6=WjL 3/48EI. Note that instead of using the absolute f l o o r parameters t h e i r r a t i o s to the reference values are used. Values of the f l o o r parameters used i n Eq. (5.1) should be of the same u n i t s as the reference values. The c o n s t r a i n t i n Eq. (5.2) ensures that the r e g r e s s i o n equation i s s a t i s f i e d f o r the reference f l o o r . (Xp / 5 ) 0 corresponds to the reference f l o o r . This r e l a t i o n w i l l be a p p l i c a b l e f o r the range of parameters l i s t e d i n t a b l e (5.1). In order to b r i e f l y e x p l a i n how the c o e f f i c i e n t s were obtained, the f l o o r parameter r a t i o s are denoted by x and (X D / 6 ) by y and the problem i s expressed as a minimization 62 p r o b l e m . k k M i n i m i z e : Z d? = Z [ ( A o + A ^ , * + A x ) . - y . ] 2 ( 5 . 3 ) • i -1 1 i = l n n i i i n w h i c h k>n . C o n s t r a i n t : A 0 + A , + + A n = y 0 ( 5 . 4 ) U s i n g L a g r a n g e ' s m u l t i p l i e r E q s . ( 5 . 3 ) a n d ( 5 . 4 ) c a n be e x p r e s s e d a s : M i n i m i z e : e = Z [{K0+h,Xi+ . . . . + A x ) . - y . ] 2 n n I •* I + X ( A 0 + A , + . + A n " y o ) ( 5 , 5 ) • | | — ( i = 1 ,2 , . . . , n) , l e a d s t o a s y s t e m of e q u a t i o n s w h i c h a l l o w us t o d e t e r m i n e t h e c o e f f i c i e n t s A 0 , A , , . . . . , A n . F i n a l l y , t h e s e c o e f f i c i e n t s c a n be s u b s t i t u t e d i n t o E q . ( 5 . 1 ) t o g e t t h e e q u a t i o n f o r X p / S . S i m i l a r l y , a r e l a t i o n f o r t h e f r e q u e n c y r a t i o , f / f 0 » i s o b t a i n e d i n w h i c h : f ° = 2T^/pF' fc^e f r e q u e n c y o f a s i m p l y s u p p o r t e d b e a m . The R e g r e s s i o n E q u a t i o n s : F o r a f l o o r o f n i n e beams a n d s u p p o r t e d on a l l f o u r s i d e s t h e f o l l o w i n g e q u a t i o n s a r e o b t a i n e d f o r a f o o t f a l l d r o p o f 2 i n c h e s : X p / 6 = 0 . 4 8 0 0 - 0 . 0 2 7 8 ( S / 1 6 ) - 0 . 0 6 0 0(W R / 2 0 0 ) - 0 . 1 8 7 0 ( W i / 2 0 0 ) - 0 . 0 l 9 2 ( R C / 6 ) + 0 . 1 9 2 0 ( R K / 4 0 0 ) - 0 . 0 8 9 7 ( R L / 1 4 4 ) + 0 . 1 3 7 0 ( E / 1 . 5 5 ) - 0 . 0 l 4 4 ( C P / 6 ) + 0 . 0 8 l 6 ( H / 7 . 2 5 ) + 0 . 1 2 7 0 ( E N L / 8 ) - 0 . 0 1 4 4 ( K N / 6 8 0 0 ) 63 ( 5 . 6 ) f / f o = 0 . 0 7 8 5 + 0 . 0 0 0 1 7 ( S / 1 6 ) - 0 . 0 1 6 7 ( W R / 2 0 0 ) - 0 . 0 6 0 0 ( W I / 2 0 0 ) - 0 . 0 0 2 4 ( R C / 6 ) + 0 . 0 4 8 8 ( R K / 4 0 0 ) + 0 . 2 4 9 0 ( R L / 1 4 4 ) - 0 . 0 6 6 2 ( E / 1 . 5 5 ) - 0 . 0 0 0 4 ( C P / 6 ) - 0 . 0 8 8 8 ( H / 7 . 2 5 ) - 0 . 0 0 1 0 ( E N L / 8 ) + 0 . 0 0 0 1 5 ( K N / 6 8 0 0 ) ( 5 . 7 ) T h e r e g r e s s i o n a n a l y s i s f o r 2 i n . f o o t f a l l d r o p was r e p e a t e d f o r f l o o r h a v i n g a d d i t i o n a l c o n s t r a i n t s on t o r s i o n a l a n d l a t e r a l d e g r e e s o f f r e e d o m o f t h e j o i s t s . T h e e q u a t i o n s f o r t h i s c a s e a r e a s f o l l o w s : X p / 5 = 0 . 0 8 0 2 + 0 . 1 4 5 0 ( S / 1 6 ) - 0 . 0 1 0 0 ( W R / 2 0 0 ) - 0 . 0 9 3 4 ( W I / 2 0 0 ) + 0 . 0 0 9 6 ( R C / 6 ) + 0 . 0 9 7 2 ( R K / 4 0 0 ) - 0 . 4 3 2 0 ( R L / 1 4 4 ) + 0 . 1 5 1 0 ( E / 1 . 5 5 ) - 0 . 0 0 6 4 ( C P / 6 ) + 0 . 2 9 5 0 ( H / 7 . 2 5 ) + 0 . 0 3 6 5 ( E N L / 8 ) - 0 . 0 0 4 0 7 ( K N / 6 8 0 0 ) ( 5 . 8 ) f / f O = 0 . 0 9 1 1 - 0 . 0 0 0 9 8 ( S / 1 6 ) - 0 . 0 2 0 0 ( W R / 2 0 0 ) - 0 . 0 9 6 6 ^ / 2 0 0 ) - 0 . 0 0 7 6 ( R C / 6 ) + 0 . 0648 ( R K / 4 0 0 ) + 0 . 2 9 5 0 ( R L / 1 4 4 ) - 0 . 0 7 9 6 ( E / 1 . 5 5 ) - 0 . 0 0 0 0 ( C P / 6 ) - 0 . 0 9 7 8 ( H / 7 . 2 5 ) - 0 . 0 0 2 9 ( E N L / 8 ) + 0 . 0 0 0 4 4 ( K N / 6 8 0 0 ) ( 5 . 9 ) T o s e e t h e p e r f o r m a n c e o f t h e r e g r e s s i o n e q u a t i o n s , a f l o o r w i t h f o l l o w i n g p a r a m e t e r s d i f f e r e n t f r o m t h o s e o f t h e r e f e r e n c e f l o o r was s e l e c t e d . R L = 1 6 8 i n S = 2 0 i n E=1 . 7 5 d 0 6 ) p s i 64 C P = 1 O l b . s / i n W R = l 4 0 1 b E N L = 2 i n T h i s f l o o r was a n a l y z e d w i t h a n d w i t h o u t t o r s i o n a l r e s t r a i n t s . A c t u a l c o m p u t e r r e s u l t s a r e c o m p a r e d w i t h t h o s e o b t a i n e d f r o m t h e r e g r e s s i o n e q u a t i o n s a n d a r e shown i n t a b l e ( 5 . 2 ) . F l o o r C o m p u t e r R e g r e s s i o n C o m p u t e r R e g r e s s i o n 1 2 2.77mm 1.48mm 3.09mm 1.3 5mm 4 . 1 7 H z 4 . 1 7 H z 3 . 9 2 H z 4 . 2 0 H z N o t e s : F l o o r 1 i s w i t h o u t t o r s i o n a l r e s t r a i n t s on j o i s t s . F l o o r 2 i s w i t h t o r s i o n a l r e s t r a i n t on j o i s t s . T a b l e 5 . 2 . P e r f o r m a n c e o f R e g r e s s i o n E q u a t i o n s . We o b s e r v e f a i r l y g o o d a g r e e m e n t b e t w e e n t h e c o m p u t e d v a l u e s a n d t h o s e o b t a i n e d f r o m t h e r e g r e s s i o n e q u a t i o n s . H e n c e , f o r t h e r a n g e o f p a r a m e t e r s s t u d i e d , r e g r e s s i o n ' e q u a t i o n s c a n be u s e d t o a p p r o x i m a t e l y d e t e r m i n e t h e peak d i s p l a c e m e n t a n d f r e q u e n c y . D a m p i n g r a t e c a n be a s s u m e d 0 . 1 5 o f t h e c r i t i c a l f o r t h e p a r a m e t e r r a n g e c o n s i d e r e d a n d t h u s t h e f l o o r r a t i n g c a n be e s t i m a t e d by u s i n g E q . ( 4 . 1 ) . 5 . 3 R E L I A B I L I T Y A N A L Y S I S AND DESIGN CRITERION T h e r e s u l t s o f a s t u d y o f t i m b e r f l o o r b e h a v i o u r a s i n f l u e n c e d by t h e v a r i a b i l i t y o f j o i s t s t i f f n e s s e s a r e . p r e s e n t e d h e r e . F o s c h i {7} r e p o r t s t e s t s on D o u g l a s - F i r . l u m b e r ( N o . 2 g r a d e & b e t t e r ) . T h e m o d u l i i o f e l a s t i c i t y o f 65 the j o i s t s obey the f o l l o w i n g 3 parameter W e i b u l l d i s t r i b u t i o n : E = E 0 + m [ - l n ( 1 - p ) ] 1 / k (5.10) i n w h i ch: E o=0.7718X10 6 p s i m=0.8777xl0 6 p s i k=2.7l98 p=uniform random number; 0^p^1 Four t y p e s of f l o o r s as shown i n t a b l e (5.3) were a n a l y z e d . The f l o o r s have d i f f e r e n t j o i s t t y p e s and boundary c o n d i t i o n s . Other f l o o r parameters were the same as i n the r e f e r e n c e f l o o r . F l o o r J o i s t B H. T o r s i o n a l type type i n i n r e s t r a i n t t A 2x8 . 1 .5 7.25 No B 2x8 1 .5 7.25 Yes C 2x1 0 1 .5 9.25 No D 2x1 2 1 .5 1 1 .25 No f i n a d d i t i o n t o s u p p o r t s on a l l f o u r s i d e s . T a b l e 5.3. F l o o r t y p e s a n a l y z e d For each f l o o r t y p e , t h r e e spans ( 8 f t , 12ft and 1 6 f t ) were c o n s i d e r e d . For each span, 300 f l o o r s were s i m u l a t e d and they were a n a l y z e d f o r 1 i n & 2 i n f o o t f a l l d r o p s . T h i s y i e l d e d the f l o o r response r a t i n g f i l e s f o r the two f o o t f a l l drops each c o n t a i n i n g 300 v a l u e s . The r e l i a b i l i t y i n d i c e s (0) were d e t e r m i n e d , w i t h r e s p e c t t o t h r e e d i f f e r e n t 66 r e f e r e n c e r a t i n g s (R 0=2, 3 and 4) and two f o o t f a l l drops ( l i n and 2 i n ) , using the Rackwitz F i e s s l e r a l g o r i t h m . Then, f o r each f l o o r type smooth c u r v e s , /3 vs. L were p l o t t e d as shown i n F i g . (5.10) to (5.13). The 0 vs. L p l o t s show higher r e l i a b i l i t y f o r 1in f o o t f a l l drop, which means that f o r a lower impact there i s a higher r e l i a b i l i t y of not exceeding a c e r t a i n r e f e r e n c e r a t i n g . A l s o f o r a p a r t i c u l a r r e f e r e n c e r a t i n g , as f l o o r span i s i n c r e a s e d , the r e l i a b i l i t y index drops, which means th a t the f l o o r s of longer span have a lower r e l i a b i l i t y of not exceeding a c e r t a i n r e f e r e n c e r a t i n g . For t a r g e t r e l i a b i l i t i e s of 2(corresponds t o a p r o b a b i l i t y of f a i l u r e of approximately 20/1000) and 3 ( p r o b a b i l i t y of f a i l u r e 1/1000) and a r e f e r e n c e r a t i n g of 3 ( d i s t i n c t l y p e r c e p t i b l e v i b r a t i o n s ) a l l o w a b l e spans are obtained f o r each f l o o r type and are l i s t e d i n t a b l e 5.4. , IIN FOOTFALL DROP ,2IN FOOTFALL DROP - \ n . M-J-r . 1 1 1 1 i • 1 • •—— 1 4 8 12 16 20 Span in ft F i g , 5.10. Plot for floor type A — - UN FOOTFALL DROP —21N FOOTFALL DROP 20 Span in ft F i g . 5.11. Plot for floor type B Reliability Index i i i . -o ui o cn o K> C*J ui o ui Rel iab i l i t y Index 89 69 D r o p F l o o r A F l o o r B F l o o r C F l o o r D 0=2 1 i n . 8 . 2 8 . 8 1 0 . 3 1 3 . 8 2 i n . 7 . 7 7 . 9 9 . 3 1 2 . 2 0 = 3 1 i n . 7 . 4 7 . 6 9 . 5 1 3 . 5 2 i n . 6 . 9 6 . 9 8 . 6 1 1 . 7 T a b l e 5 . 4 . A l l o w a b l e s p a n s i n f t . f o r R 0 =3 w i t h r e c e i v e r a n d i m p a c t e r on a d j a c e n t j o i s t s . T o e x p l a i n what i s i m p l i e d by an a l l o w a b l e s p a n o f 8 . 2 f t f o r f l o o r t y p e A , i t means t h a t i f f l o o r t y p e A i s r e s t r i c t e d t o a s p a n o f 8 . 2 f t , t h e n f o r a f o o t f a l l d r o p o f 1 i n , t h e r e l i a b i l i t y i n d e x f o r n o t e x c e e d i n g a r e s p o n s e r a t i n g o f 3 i s 2 o r , i n o t h e r w o r d s , t h e r e a r e o n l y 20 o u t o f 1000 c h a n c e s t h a t t h e v i b r a t i o n s w o u l d be d i s t i n c t l y p e r c e p t i b l e o r more s e v e r e . To d e v e l o p a d e s i g n c r i t e r i o n i n t e r m s o f a l l o w a b l e s t a t i c d e f l e c t i o n s , d e f l e c t i o n s o f s i n g l e j o i s t s l o a d e d by a 1KN c o n c e n t r a t e d l o a d a t i t s c e n t r e and w i t h a l l o w a b l e s p a n s g i v e n i n t a b l e 5 . 4 a r e c o m p u t e d a n d l i s t e d i n t a b l e 5 . 5 . 70 0= = 2 0= = 3 D r o p 1 i n . 2 i n . 1 i n . 2 i n . F l o o r A 1 . 3 6 6 1 .1308 1 . 0037 0 .8137 F l o o r B 1 .688 1 .2212 1 . 0873 0 . 8 1 3 7 F l o o r C 1 . 3032 0 . 9 5 9 3 1 . 0 2 2 5 0 . 7 5 8 6 F l o o r D 1 .7422 1 .2038 1 .631 1 1 .0618 T a b l e 5 . 5 . A l l o w a b l e d e f l e c t i o n i n mm f o r R 0 =3 w i t h r e c e i v e r a n d i m p a c t e r on a d j a c e n t j o i s t s . T h e m o d u l u s o f e l a s t i c i t y f o r c o m p u t i n g t h e d e f l e c t i o n i s t a k e n a s 1 . 5 5 ( l O s ) p s i w h i c h i s t h e mean o f t h e d i s t r i b u t i o n o f E - v a l u e s c o n s i d e r e d . N o t e t h a t t a b l e s 5 .4 a n d 5 . 5 a r e f o r t h e c a s e when t h e r e c e i v e r ' a n d t h e i m p a c t e r a r e on t h e a d j a c e n t j o i s t s . T h e a n a l y s i s was r e p e a t e d f o r t h e c a s e when t h e r e c e i v e r was on t h e same j o i s t a s t h e i m p a c t e r a n d 2 f e e t away f r o m h i m as- shown i n F i g . 5 . 1 4 . F i g . 5 . 1 4 . R e c e i v e r a n d i m p a c t e r on same j o i s t T h e r e s u l t s o f t h e a n a l y s i s a r e shown i n t a b l e s 5 . 6 and 5 . 7 . 71 0= = 2 0= = 3 D r o p 1 i n . 2 i n . 1 i n . 2 i n . F l o o r A 5 . 7 5 5 . 4 5 . 4 5.1 F l o o r B 6 . 5 5 . 8 5 5 .8 5 .3 F l o o r C 7 . 5 5 6 . 7 6 . 7 5 6.1 F l o o r D 8 . 5 7 . 8 5 7 . 6 7 . 1 5 T a b l e 5 . 6 . A l l o w a b l e s p a n s i n f t . f o r R 0 =3 w i t h r e c e i v e r a n d i m p a c t e r on same j o i s t s . 0=2 0=3 D r o p F l o o r A F l o o r B F l o o r C F l o o r D 1 i n . 0 . 5 2 8 0 . 7 6 3 0 . 5 7 6 0 . 4 5 7 2 i n . 0 . 4 3 7 0 . 5 5 6 0 . 4 0 2 0 . 3 6 0 1 i n . 0 . 4 3 7 0 . 5 4 2 0.411 0 . 3 2 6 2 i n . 0 . 368 0 . 4 1 3 0 . 3 0 4 0 . 2 7 2 T a b l e 5 . 7 . A l l o w a b l e d e f l e c t i o n i n mm f o r R 0 = 3 w i t h r e c e i v e r a n d i m p a c t e r on same j o i s t s . On l o o k i n g a t t a b l e s 5 . 5 a n d 5 .7 we n o t e t h a t t h e c a s e when t h e r e c e i v e r a n d i m p a c t e r a r e on t h e same j o i s t i s more s e v e r e . H o w e v e r , i t i s v e r y u n l i k e l y i n r e a l p r a c t i c e t h a t t h e r e c e i v e r a n d i m p a c t e r w o u l d be j u s t 2 f e e t a p a r t . A t g r e a t e r d i s t a n c e s , t h e r e c e i v i n g p o i n t w o u l d be c l o s e t o t h e . s u p p o r t a n d t h e v i b r a t i o n s much l e s s . On t h i s b a s i s , t h e c a s e o f r e c e i v e r a n d i m p a c t e r on a d j a c e n t j o i s t s g o v e r n s a n d 7 2 we base our d e s i g n c r i t e r i o n on t a b l e 5 . 5 . Design C r i t e r i o n : - I f the d e f l e c t i o n of a j o i s t l o a d e d by 1KN c o n c e n t r a t e d l o a d a t i t s c e n t r e i s l i m i t e d t o a p p r o x i m a t e l y 1.0mm t h e n , f o r the f l o o r made w i t h t h e s e j o i s t s , the r e l i a b i l i t y i ndex f o r not e x c e e d i n g a response r a t i n g of 3 i s 2 o r , i n o t h e r words, t h e r e would be o n l y 20 out of 1000 chances t h a t the v i b r a t i o n s would be d i s t i n c t l y p e r c e p t i b l e or more s e v e r e , under f o o t f a l l s of upto 2 i n c h e s . I f t he d e f l e c t i o n of a j o i s t l o a d e d by 1KN c o n c e n t r a t e d l o a d a t i t s c e n t r e i s l i m i t e d t o a p p r o x i m a t e l y 0.8mm th e n , f o r the f l o o r made w i t h t h e s e j o i s t s , t he r e l i a b i l i t y i ndex f o r not e x c e e d i n g a response r a t i n g of 3 i s 3 o r , i n o t h e r words, t h e r e would be o n l y 1 out of 1000 chances t h a t t h e v i b r a t i o n s would be d i s t i n c t l y p e r c e p t i b l e or more s e v e r e , under f o o t f a l l s of upto 2 i n c h e s . I t i s i n t e r e s t i n g t o compare t h i s d e s i g n c r i t e r i o n t o the l i m i t i n g d e f l e c t i o n v a l u e s imposed by the codes of d i f f e r e n t c o u n t r i e s as l i s t e d i n t a b l e 5.8. 73 C o u n t r y L o a d D e f l e c t i o n E (MPa) C a n a d a 1 . 9 K N / m 2 L / 3 6 0 8400 Norway 1 KN 0 . 9mm 10800 Sweden 1 KN 1 . 5mm 1 1 700 Denmark 1 KN 0 . 9mm 1 2600 UK D L + 1 . 5 K N / m m 2 L / 3 3 3 6900 Germany DL+2KN/mm 2 L / 3 0 0 10000 F i n l a n d D L + 1 . 5 K N / m m 2 L / 3 0 0 7000 T a b l e 5 . 8 . D e f l e c t i o n p r o v i s i o n s i n d i f f e r e n t C o u n t r i e s . We o b s e r v e t h a t o u r p r o p o s e d d e s i g n c r i t e r i o n a g r e e s q u i t e c l o s e l y w i t h t h o s e u s e d i n S c a n d i n a v i a , p a r t i c u l a r l y Norway a n d D e n m a r k . Our s t u d y s u g g e s t s t h a t S w e d i s h d e f l e c t i o n p r o v i s i o n s c o r r e s p o n d t o a somewhat l o w e r j3. I t i s a l s o w o r t h w h i l e t o c o m p a r e o u r d e s i g n c r i t e r i o n w i t h t h e one u s e d i n C a n a d a . F o r t h i s we c o m p u t e " t h e d e f l e c t i o n s o f t h e i s o l a t e d j o i s t s w i t h s p a n s a s l i s t e d i n t a b l e 5 .4 a n d l o a d e d by a u n i f o r m l y d i s t r i b u t e d l o a d o f 1 . 9 K N / m 2 ( 4 0 1 b / f t 2 ) . T h e s e a r e l i s t e d i n t a b l e 5 . 9 a s a f r a c t i o n o f t h e s p a n l e n g t h . 74 0=2 0 = 3 D r o p 1 i n . 2 i n . 1 i n . 2 i n . F l o o r A L / 1 3 3 9 L / 1 6 1 7 L / 1 8 2 2 L / 2 2 4 8 F l o o r B L / 1 0 8 3 L / 1 4 9 8 L / 1 6 8 2 L / 2 2 4 8 F l o o r C L / 1 4 0 3 L / 1 9 0 6 L / 1 7 8 9 L /2411 F l o o r D L / 1 0 5 0 L / 1 5 1 9 L /1121 L / 1 7 2 2 T a b l e 5 . 9 . A l l o w a b l e d e f l e c t i o n b a s e d on d i s t r i b u t e d l o a d w i t h r e c e i v e r a n d i m p a c t e r on a d j a c e n t j o i s t s . We o b s e r v e t h a t t h e d e f l e c t i o n l i m i t s i m p o s e d by o u r d e s i g n c r i t e r i o n a r e c o n s i d e r a b l y l e s s t h a n t h e L / 3 6 0 l i m i t i m p o s e d by t h e C a n a d i a n c o d e . T h i s means t h a t L / 3 6 0 a n d 1 . 9 K N / m 2 a r e u n s a t i s f a c t o r y a s a means o f d e s i g n i n g f o r t h e l i m i t s t a t e o f f l o o r v i b r a t i o n s . 6. CONCLUSIONS A r a t i o n a l a n a l y t i c a l study of the f l o o r v i b r a t i o n problem can be made using the computer program developed in t h i s work. Since a time step method of a n a l y s i s i s employed, complete v a r i a t i o n of f l o o r displacement, v e l o c i t y and the a c c e l e r a t i o n can be t r a c e d . A r e l i a b i l i t y based design c r i t e r i o n r e l a t e d to human p e r c e p t i o n of v i b r a t i o n was proposed. Our design c r i t e r i o n g i v e s an i n d i c a t i o n of the amount of r i s k undertaken in design and thus enables the designer to reach a judgment on the l i m i t i n g value of d e f l e c t i o n i n a case where he may be r e q u i r e d to undertake a g r e a t e r or a l e s s e r r i s k . The design c r i t e r i o n proposed here agrees w e l l with the one used i n Norway, Denmark and Sweden, where the s t a t i c d e f l e c t i o n produced by a concentrated l o a d of 1KN at midspan of a j o i s t i s r e q u i r e d not to exceed an a b s o l u t e maximum independent of span l e n g t h . T h i s maximum i s approximately 1mm. The study c l e a r l y shows that the Canadian code c r i t e r i o n f o r l i m i t i n g s t a t i c d e f l e c t i o n s under a u n i f o r m l y d i s t r i b u t e d l o a d i s u n s a t i s f a c t o r y f o r the l i m i t s t a t e of f l o o r v i b r a t i o n s . As a p r o p o s a l f o r f u t u r e r e s e a r c h , f l o o r v i b r a t i o n response r e s u l t i n g from a walking person, which comprises of a sequence f o o t f a l l impacts, should be s t u d i e d . F u r t h e r , some experimental work needs t o be done i n the area of f l o o r system damping, to improve the input assumptions f o r the program. 75 REFERENCES 1. A l l e n , D . E . and R a i n e r , J . H . , " F l o o r V i b r a t i o n " , Canadian Bui Idi ng Digest , Sept.,1975, pp.173 t o 173-4. 2. A l l e n , D . E . , and R a i n e r , J . H . , " V i b r a t i o n C r i t e r i a f o r Long Span F l o o r s " , Canadian Journal of Civil Engineering, V o l . 3 , No.2, June,1976, pp.165-173. 3. Atherton,G.H., P o l e n s e k , A . , and Co r d e r , S . E . , "Human Response t o Walk i n g and Impact V i b r a t i o n of Wood F l o o r s " , Forest Products Journal, V o l . 2 6 , No.10, Oct.,1976. 4. B a t h e , K . J . and W i l s o n , E . L . , "Numerical Methods in Finite Element Analysis", P r e n t i c e - H a l l , Englewood C l i f f s , N.J., 1976. 5. Clough,R.W. and . P e n z i e n , J . , "Dynamics of Structures", McGraw H i l l book company, New York, 1975. 6. Foschi,R.O., "A D i s c u s s i o n on the A p p l i c a t i o n of the S a f e t y Index concept t o Wood S t r u c t u r e s " , Canadian Journal of Civil Engineering, V o l . 6 , No.1, 1979, pp.51-58. 7. Foschi,R.O., " S t r u c t u r a l A n a l y s i s of Wood F l o o r Systems", Journal of Structural Division, ASCE, V o l . 1 0 8 , NO.ST7, J u l y , 1 9 8 2 , pp.1557-1574. 8.0h l s s o n , S . , " F l o o r V i b r a t i o n s and Human D i s c o m f o r t " , Ph.d. T h e s i s , Chalmers U n i v e r s i t y of Technology, Div. of Steel and Timber structures, Gothenburg, Sweden, 1982. 9.0nysko,D.M. and R i j n , G . J . , " F l o o r Performance and A c c e p t a b i l i t y " , Paper p r e s e n t e d a t f i r s t Canadian Symposium on Wood F l o o r T r u s s e s h e l d a t A i r p o r t H i l t o n h o t e l , T o r o n t o , June 24, 1982. 10. P o l e n s e k , A . , "Human Response t o V i b r a t i o n of Wood J o i s t F l o o r Systems", Wood Science, V o l . 3 , No.2, Oct.,1970, pp.111-119. 11. R a c k w i t z , R . , and F i e s s l e r , B . , " S t r u c t u r a l R e l i a b i l i t y under Combined Random Load Sequences", Computers and Structures, V o l . 9 , 1978, pp.489-494. 12. R e i h e r , H . , and M e i s t e r , F . J . , "The E f f e c t of V i b r a t i o n on P e o p l e " ( i n German), t r a n s l a t i o n : Report No. F-TS-616-RE, Headquaters a i r m a t e r i a l command, Wright f i e l d O h i o , 1946. 13. Thompson,E.G., Goodman,J.R. and V a n d e r b i l t , M . D . , " F i n i t e 76 77 E l e m e n t A n a l y s i s o f L a y e r e d Wood S y s t e m s " , Journal of Structural Division, A S C E , V o l . 1 0 1 , N O . S T 1 2 , D e c . , 1 9 7 5 , p p . 2 6 5 9 - 2 6 7 2 . 14 . T h o m p s o n , E . G . , V a n d e r b i l t , M . D . a n d G o o d m a n , J . R . , " F E A F L O : A P r o g r a m f o r t h e A n a l y s i s o f L a y e r e d Wood S y s t e m s " , Computers and St ruct ures, V o l . 7 , 1 9 7 7 , p p . 2 3 7 - 2 4 8 . 1 5 . W i s s , J . F . a n d P a r m e l e e , R . A . , "Human P e r c e p t i o n o f T r a n s i e n t V i b r a t i o n s " , Journal of Structural Di visi on, A S C E , V o l . 1 0 0 , S T 4 , A p r i l , 1 9 7 4 , p p . 7 7 3 - 7 8 7 . APPENDIX I Shape F u n c t i o n s : V e c t o r {M 0} M 0 ( 1 ) = £ 2 - 5/4£ 3 - 1/2£" + 3/4£ 5 M 0(2) = U 2 - V ~ + £ 5)/8.0 M 0 (10) = 1.0 - 2.0£ 2 + M 0 (9 ) = U - 2.0$ 3 + £ 5)/2.0 M 0(14) = £ 2 + 5/4£ 3 - 1 / 2 ^ - 3/4£ 5 M 0(15) = (-£2 - V + £ft + £ 5)/8.0 A l l o t h e r components, M o(k)=0.0, k=1,2,....,19. V e c t o r {M 3} M 3(3) = l/4(-3£ + 4£ 2 + £ 3 - 2V) M 3(4) = (-$ + £2 + S3 - «")/8.6 M 3 (7) = 1 . 0 - 2.0? 2 + M 3(16) = 1/4(3^ + 4£ 2 - V - f ) M 3(17) = (-^  - £ 2 + £ 3 + $")/8.0 A l l o t h e r components, M 3(k)=0.0, k=1,2,....,19. V e c t o r {M 5} M 5 ( 5 ) = M 3 ( 3 ) ; M 5 ( 6 ) = M 3 ( 4 ) ; M 5 ( 8 ) = M 3 ( 7 ) ; M 5 ( 1 8 ) = M 3 ( 1 6 ) ; M 5(19)=M 3(17) A l l o t h e r components, M 5(k)=0.0, k=1,2,....,19. 78 V e c t o r s {M,} , {M 2 } , {M,} a n d {M 6} d M 0 ( k ) d 2 M 0 ( k ) M , ( k ) = ; M 2 ( k ) = d£ d£ 2 d M 3 ( k ) d M 5 ( k ) M 4 ( k ) = ; M 6 ( k ) = d£ d£ i n w h i c h : k=1,2,....,19. APPENDIX II U S E R ' S MANUAL  DYNAMIC FLOOR A N A L Y S I S PROGRAM I N P U T : A l l d a t a t o be e n t e r e d i n f r e e f o r m a t . 1. N F L O R , NM, N J T , I S Y M , NPROP, I N P T E , TOL w h e r e : NFLOR = Number o f f l o o r s t o be a n a l y z e d . NM = Maximum o r d e r o f s i n e / c o s i n e t e r m s i n t h e F o u r i e r s e r i e s . T h e r e c a n be a maximum o f t h r e e t e r m s , s o t h a t f o r s y m m e t r i c a l p r o b l e m s a l o n g t h e j o i s t s , NM c a n be u p t o 5 ( t h r e e t e r m s w i l l be o f o r d e r 1, 3 a n d 5 ) . F o r n o n - s y m m e t r i c a l p r o b l e m NM i s e q u a l t o maximum o r d e r o f s i n e / c o s i n e t e r m s e g . NM=3 means t e r m s w i t h o r d e r 1, 2 a n d 3 . N J T = Number o f j o i s t s i n t h e f l o o r . ISYM = 0 , f o r n o n - s y m m e t r i c a l p r o b l e m a l o n g t h e j o i s t s ; a n d e q u a l t o 1, f o r s y m m e t r i c a l p r o b l e m a l o n g t h e j o i s t s . NPROP = 1, i f a l l j o i s t s h a v e u n i f o r m p r o p e r t i e s ; a n d e q u a l t o 0 , i f j o i s t s h a v e d i f f e r e n t p r o p e r t i e s . I N P T E = 1, i f d o i n g s i m u l a t i o n ; a n d e q u a l t o 2 , i f n o t a s i m u l a t i o n p r o b l e m . I f d o i n g s i m u l a t i o n u s e NFLOR=1, e x e c u t i o n f o r t h e w h o l e s a m p l e c o n t r o l l e d by v a r i a b l e " N R E P L " a s g i v e n i n i t e m 4 . When n o t d o i n g s i m u l a t i o n a n d a n a l y z i n g more t h a n one f l o o r , s u b r o u t i n e " U P D A T E " 80 81 s h o u l d be r e w r i t t e n t o s u i t s p e c i f i c r e q u i r e m e n t s . As i l l u s t r a t e d i n t h e p r o g r a m l i s t i n g , s u b r o u t i n e UPDATE i s s o w r i t t e n t h a t 20 f l o o r s c a n be a n a l y z e d , e a c h h a v i n g o n l y c e r t a i n p a r a m e t e r s d i f f e r e n t f r o m a r e f e r e n c e f l o o r . T h e p a r a m e t e r v a l u e s f o r t h e r e f e r e n c e f l o o r a r e r e s e t e a c h t i m e UPDATE i s c a l l e d . TOL = T o l e r a n c e t o s t o p i t e r a t i o n s w h i l e s o l v i n g t h e s y s t e m o f e q u a t i o n s . T h e v a l u e f o r TOL may be t a k e n a s 0 .001 . T H E T A , NTMZON w h e r e : THETA = P a r a m e t e r f o r W i l s o n - 0 t i m e s t e p m e t h o d o f s o l u t i o n . I t s v a l u e i s t a k e n a s 1.4 f o r u n c o n d i t i o n a l s t a b i l i t y . NTMZON = Number o f t i m e z o n e s . T I M E ( I , J ) , J = 1 , 3 ; 1=1,NTMZON w h e r e : T I M E ( I , 1 ) = S t a r t i n g t i m e o f t i m e z o n e . T I M E ( I , 2 ) = E n d i n g t i m e o f t i m e z o n e . T I M E ( I , 3 ) = I n c r e m e n t a l s t e p f o r t h e t i m e z o n e . I f , INPTE=2 ( n o t a s i m u l a t i o n p r o b l e m ) , go t o 7 . N R E P L , S E E D w h e r e : 82 NREPL = Sample size for simulation. SEED = Random number i n i t i a l i z e r . 5. EMIN, EMAX where: EMIN = Minimum value of modulas of elasticity of joist. EMAX = Maximum value of modulas of elasticity of joist. 6. EO, EM, EK where: EO, EM and EK = Weibull distribution parameters. 7. S, RL, D where: S = Joist spacing RL = Joist span D = Cover or plate thickness 8. RKX, RKY, RKV, RKG, RHOC where: RKX = Bending stiffness of the cover in the x-direction, parallel to the joists. For plywood, this is usually perpendicular to the face grain. RKY = Bending-stiffness of the cover in the y-direction, perpendicular to the joists. For plywood this is usually parallel to the face grain. RKV, RKG = Parameters for plate bending, connected 8 3 r e s p e c t i v e l y t o P o i s s o n ' s e f f e c t . a n d t o r s i o n . RHOC = M a s s d e n s i t y o f t h e p l a t e . 9 . D X , D Y , D V , DG DX = I n - p l a n e ( a x i a l ) s t i f f n e s s o f t h e c o v e r i n t h e x - d i r e c t i o n . DY = I n - p l a n e ( a x i a l ) s t i f f n e s s o f t h e c o v e r i n t h e y - d i r e c t i o n . D V , DG = I n - p l a n e ( a x i a l ) s t i f f n e s s , r e l a t e d r e s p e c t i v e l y t o P o i s s o n ' s e f f e c t a n d i n - p l a n e s h e a r . I f NPROP=0, n o n - u n i f o r m p r o p e r t i e s ; go t o 12 . 10 . B J T , H J T , R H O J T , CBT w h e r e : B J T = W i d t h o f j o i s t s . H J T = D e p t h o f j o i s t s . RHOJT = M a s s d e n s i t y o f t h e j o i s t s . CBT = D a m p i n g c o - e f f i c i e n t f o r t h e j o i s t s , p e r u n i t v o l u m e o f t h e j o i s t ; u n i t s o f f o r c e / v e l o c i t y / v o l u m e . I f I N P T E = 1 , s i m u l a t i o n c a s e , go t o 14 . 1 1 . E J T w h e r e : E J T = M o d u l a s o f e l a s t i c i t y o f t h e j o i s t . 84 Go t o 14 . 12 . F o r e a c h j o i s t , i n p u t c o n s e c u t i v e l y , B J ( I ) , I = 1 , N J T ( w i d t h o f j o i s t s ) H J ( I ) , I = 1 , N J T ( d e p t h o f j o i s t s ) I f I N P T E = 1 , go t o 13 Y M O D ( I ) , I = 1 , N J T ( E - v a l u e o f j o i s t s ) 13 . C B ( I ) , I = 1 , N J T ( d a m p i n g c o e f f i c i e n t o f j o i s t s ) 14 . B E T A , A L P H A , REG w h e r e : BETA = T o r s i o n c o n s t a n t , f u n c t i o n o f r a t i o , H J T / B J T . ALPHA = S h e a r d e f l e c t i o n c o n s t a n t f o r r e c t a n g u l a r c r o s s - s e c t i o n s . REG = R a t i o o f Y o u n g ' s m o d u l a s o f j o i s t t o s h e a r m o d u l a s 1 5 . E N L , X I N , R K P A L , R K P E R , RKROT, NDISCR w h e r e : ENL = N a i l s p a c i n g a l o n g t h e j o i s t s . X IN = D i s t a n c e b e t w e e n t h e s u p p o r t a n d t h e f i r s t n a i l a l o n g t h e j o i s t . R K P A L , RKPER = N a i l l o a d - s l i p m o d u l i i , r e s p e c t i v e l y i n d i r e c t i o n s p a r a l l e l a n d p e r p e n d i c u l a r t o t h e j o i s t . RKROT = S t i f f n e s s f o r r e l a t i v e r o t a t i o n o f t h e p l a t e and t h e j o i s t . ( E / G ) . 85 NDISCR = 0 , i f a c t u a l n a i l i n g p a t t e r n i s t r a n s f o r m e d i n t o an e q u i v a l e n t c o n t i n u o u s c o n n e c t o r f o r a n a l y s i s and e q u a l t o 1, i f n a i l s a r e t r e a t e d a s d i s c r e t e . 16 . C P , C N P A L , C N P E R , CNROT w h e r e : CP = D a m p i n g c o - e f f i c i e n t f o r t h e p l a t e , p e r u n i t v o l u m e o f t h e f l a n g e o f one T - b e a m e l e m e n t ; u n i t s f o r e e / v e l o c i t y / v o l u m e . C N P A L , CNPER = Damping c o - e f f i c i e n t f o r n a i l s l i p p a r a l l e l a n d p e r p e n d i c u l a r t o t h e j o i s t s r e s p e c t i v e l y ; u n i t s f o r c e / v e l o c i t y . CNROT = D a m p i n g c o - e f f i c i e n t f o r r o t a t i o n a l s l i p ; u n i t s f o r c e / a n g u l a r v e l o c i t y . 17 . NGAPS w h e r e : NGAPS = Number o f g a p s p r e s e n t i n t h e c o v e r . I f NGAPS=0, go t o 2 0 . 18 . NGT w h e r e : NGT = 0 , i f g a p s a r e a s e n t e r e d i n 19 a n d e q u a l t o 1, i f g a p s a r e a d j u s t e d a u t o m a t i c a l l y , t o t h e n a i l i n g p a t t e r n . 1 9 . F o r e a c h g a p , e n t e r : G A P X ( I ) , G A P ( I ) w h e r e : 86 G A P X ( I ) = x - c o - o r d i n a t e o f t h e b e g i n n i n g o f g a p . G A P ( I ) = W i d t h o f t h e g a p . 20. NLOAD, N L U , PLOAD w h e r e : NLOAD = Number o f l o a d e d a r e a s ; NLOAD=0 f o r u n i f o r m l y d i s t r i b u t e d l o a d . NLU = F o r NLOAD=0 , NLU=1, means t h a t e v e n o u t e r f l a n g e s o f o u t s i d e j o i s t s a r e l o a d e d a n d NLU=0 means t h e y a r e u n l o a d e d . F o r N L O A D *0 , NLU=1, means a l l j o i s t s h a v e same l o a d d i s t r i b u t i o n a n d NLU=0, means o t h e r w i s e . PLOAD = M a g n i t u d e o f t h e u n i f o r m l y d i s t r i b u t e d l o a d . 21. NDRMAS w h e r e : NDRMAS = Number o f p e o p l e o r m a s s e s on t h e f l o o r . 22. F o r e a c h m a s s , e n t e r c o n s e c u t i v e l y , R M ( I ) , 1 = 1 , NDRMAS ( m a g n i t u d e o f t h e m a s s ) R C ( I ) , 1 = 1 , NDRMAS ( d a m p i n g c o - e f f i c i e n t o f t h e mass ) R K ( I ) , 1 = 1 , NDRMAS ( s t i f f n e s s o f t h e m a s s ) V E L ( I ) , 1 = 1 , NDRMAS ( i n i t i a l i m p a c t v e l o c i t y o f t h e m a s s ) N J D R ( I ) , 1 = 1 , NDRMAS ( f l o o r e l e m e n t n o . on w h i c h mass i s l o c a t e d ) R X O ( I ) , R Y O ( I ) , 1 = 1 , NDRMAS ( x / y c o - o r d i n a t e s o f t h e mass ) 87 TIMIMP(I), 1 = 1 , NDRMAS ( t i m e f o r w h i c h mass i s on t h e f l o o r ) R F O R C ( I ) , 1 = 1 , NDRMAS ( W e i g h t o f t h e mass ) I f NLOAD=0, go t o 2 5 . 2 3 . F o r e a c h l o a d e d a r e a , e n t e r : N L J O ( I ) , S T 0 R E ( I , 5 ) , 1 = 1 , NLOAD w h e r e : N L O A D ( I ) = Number o f j o i s t l o a d e d . S T O R E ( I , 5 ) = M a g n i t u d e o f t h e l o a d , u n i f o r m l y d i s t r i b u t e d o v e r t h e l o a d e d a r e a . 2 4 . S T O R E ( 1 , 1 ) , S T O R E ( I , 2 ) , S T O R E ( 1 , 3 ) , S T O R E ( I , 4 ) w h e r e : S T O R E ( 1 , 1 ) t o S T O R E ( I , 4 ) = C o - o r d i n a t e s o f t h e l o a d e d a r e a , X 1 , X 2 , Y 1 , Y2 r e s p e c t i v e l y a s shown i n f i g . ( A . 1 ) . X1 a n d X2 a r e t h e g l o b a l c o - o r d i n a t e s , w h i l e Y1 a n d Y2 a r e l o c a l t o t h e j o i s t e l e m e n t . 2 5 . GRVITY w h e r e : GRVITY = A c c e l e r a t i o n due t o g r a v i t y . 2 6 . NBC w h e r e : NBC = Number o f b o u n d a r y c o n d i t i o n s i m p o s e d . 88 Floor Span 1 1 yyy / / s / / Y > -S /2 Y ' ELEMENT F i g . A . 1 . L o a d i n g a r e a - c o o r d i n a t e s I f NBC=0, go t o 2 8 . F o r e a c h b o u n d a r y c o n d i t i o n , (1=1 ,NBC) e n t e r : I B C ( I , 1 ) , I B C ( I , 2 ) w h e r e : I B C ( I , 1 ) = J o i s t n o . where b o u n d a r y c o n d i t i o n e x i s t s . I B C ( I , 2 ) = B o u n d a r y c o n d i t i o n c o d e n u m b e r . I t i s e q u a l t o t h e s e q u e n c e number o f t h e d i s p l a c e m e n t c o m p o n e n t i n t h e v e c t o r {5 n } a s g i v e n by E q . ( 2 . 3 ) i n c h a p t e r 2 e g . , i f t h e v e r t i c a l d i s p l a c e m e n t o f t h e j o i s t (W n ) i s 89 r e s t r i c t e d , the code i s 10. 28. NPNTS where: NPNTS = Number of p o i n t s on f l o o r where response i s d e s i r e d . 29. For each p o i n t , (1=1,NPNTS), e n t e r : N D E F J T ( I ) , DEFM(I,1), DEFM(I,2) where: NDEFJT(I) = J o i s t number on which p o i n t i s l o c a t e d . DEFM(I,1), DEFM(I,2) = x/y c o - o r d i n a t e s of the p o i n t . 30. SCALE(1), SCALE(2), SCALE(3) where: SCALE(1) = C o n v e r s i o n f a c t o r f o r time i n seconds t o m i l l i s e c o n d s . SCALE(2) = C o n v e r s i o n f a c t o r f o r d i s p l a c e m e n t i n i n c h e s t o mm. SCALE(3) = C o n v e r s i o n f a c t o r f o r l b / i n t o KN/mm. NOTE: The program works w i t h any s e t of c o n s i s t e n t u n i t s . E x e c u t i o n Command: a ) S i m u l a t i o n c a s e : $Run o b j e c t p r o g 1=-1 2=-2 3=-3 4=-4 5=-5 6=-6 7=-7 8=-8 9=-9 90 b ) O t h e r t h a n s i m u l a t i o n p r o b l e m : $Run o b j e c t p r o g 1=-1 2=-2 3=-3 4=-4 5=-5 6=-6 7=-7 8=-8 i n w h i c h : 1. F i l e a t t a c h e d t o u n i t 1 i s t h e i n p u t f i l e t o t h e p r o g r a m . 2 . F i l e a t t a c h e d t o u n i t 2 c o n t a i n s t h e c o m p r e h e n s i v e o u t p u t . 3 . F i l e s a t t a c h e d t o u n i t s 3 , 4 a n d 5 c o n t a i n t h e summary o f d i s p l a c e m e n t s , v e l o c i t i e s a n d a c c e l e r a t i o n s r e s p e c t i v e l y a t e a c h t i m e s t e p . 4 . F i l e a t t a c h e d t o u n i t 6 l i s t s t h e number o f i t e r a t i o n s u s e d t o s o l v e t h e s y s t e m o f e q u a t i o n s a t e a c h t i m e s t e p . 5 . F i l e a t t a c h e d t o u n i t 7 c o n t a i n s t h e f o r c e i m p a r t e d t o t h e f l o o r by t h e i m p a c t i n g m a s s . 6 . F i l e a t t a c h e d t o u n i t 8 l i s t s t h e f i r s t two peak d i s p l a c e m e n t s a n d t h e t i m e o f t h e i r o c c u r e n c e . 7 . F i l e a t t a c h e d t o u n i t 9 l i s t s t h e m o d u l i i o f e l a s t i c i t y o f t h e j o i s t s u s e d f o r s i m u l a t i o n . S a m p l e I n p u t / O u t p u t : T h e f o l l o w i n g p a g e s i l l u s t r a t e t h e i n p u t / o u t p u t f i l e s p r o d u c e d by t h e p r o g r a m . I n p u t f i l e A i s f o r t h e s i m u l a t i o n c a s e a n d i n p u t f i l e B i s f o r n o r m a l p r o b l e m . T h e o u t p u t f i l e s c o r r e s p o n d t o t h e s i m u l a t i o n p r o b l e m . O u t p u t f i l e " - 2 " , l i s t s t h e s e c t i o n a n d m a t e r i a l p r o p e r t i e s . A l s o t h e d i s p l a c e m e n t s , v e l o c i t i e s a n d t h e 91 a c c e l e r a t i o n s w i t h a p p r o p r i a t e h e a d i n g s a r e l i s t e d . O u t p u t f i l e s "-3", "-4" a n d "-5" s u m m a r i z e t h e d i s p l a c e m e n t s , v e l o c i t i e s a n d a c c e l e r a t i o n s r e s p e c t i v e l y . O n l y f i l e "-3" i s i l l u s t r a t e d h e r e , o t h e r s a r e s i m i l a r . The f i r s t c o l u m n i s t h e t i m e i n m i l l i s e c o n d s , s e c o n d t h e f l o o r d i s p l a c e m e n t a t r e c e i v e r ' s p o i n t i n mm, t h i r d t h e d i s p l a c e m e n t o f t h e f l o o r a t i m p a c t e r ' s p o i n t i n mm, f o u r t h t h e d i s p l a c e m e n t o f t h e r e c e i v i n g mass i n mm a n d t h e f i f t h i s t h e d i s p l a c e m e n t o f t h e i m p a c t i n g mass i n mm. The f o r c e s i n f i l e "-7" a r e i n K N . P e a k d i s p l a c e m e n t s i n o u t p u t f i l e "-8" c a n be v e r i f i e d f r o m f i l e "-3". g STJTJ mdui '>>itsL•i'»'tr'o•ooot o? o • e ' o • ZL ' t ic • o c ' o ct1 • f c •t LC 'LI 'S t C ' O I ' • IC •i'8 tt ' I I ' I cc 'OL ' 1 CC 1' 1 IC ' < oc > «tc (C 'o•OOC'O'O 1c •oi 0'OS o ci 'o•o'o•o sc ' 0 * ZL ' O ' Ei. SC ' s ' t »c 'II'II't'0 cc ' O ' O O * ' O ' O O * cc ' O • 9 • O • 1 I C '8Is•O'CSC ' o oc ' C S 1 ' O " O ' I ' O 8 t ' 0 11 o s o s ' o s ' c - a i t t «i ' O ' 0 O O 0 O l ' ' O O « S ' ' O O S S ' 0 " 0 ' 0 - 8 St ' o • L \ ' oc ' L ' eetz • o »i • sass i ci ' c-ics • c ' s-ao • s ' sc 1 1 • s • i ci ' 'OOOOS ' 'OOOt L ' ' O O O O C * ' 'OOOOVC I I 's-ao'S''oo«i'*oos*'ooofC'ooos oi 'SCS ' 0 • 0 t»I '0 8 I S ' I O ' 0 ' S ' O 'C ' O » * SOO•O'OC'0'»0"O L ' zoo • o' »o • 0 ' CO ' 0 s ioo•0'Co•O'IO'o s ' S O O O ' C ' I O ' O ' Ioo * o t ' scooo ' o ' too ' o ' o • o c '8 ' » • I C 'C-BO-t'C'l'l'C't't I ' * - assi't'»'sc'o•ooot ct .'O'O'O'CL'S IV 'O'O'O'Ct'S o» ' c sc 'll'S BC '01'S LZ 'c's sc 't I ' I sc 'OI'I fc ' i ' I cc ' s cc I l i t IC 'O'OOC'o•o oc • s • 0 ' s • o sc •o•o'o•o se 'OZL'OZL LZ •ft sc ' I t l t ' O 1 sc 'O'OOt'O'OO* tc • o•S 'O's cc ' 11S'O'ESC'O CC •z IC 'O'O'I'O OC ' O SI ' O " s 'o • s 'o • s ' c-a*. I ' * St ' o ' • OOOOOI ''OOBS''OOSS'0'0'0-S i.1 'O•Lt'OC'I'CSSC•O SI i i c i t ' i - i i i i : i i i si ' * OOOOS * •0008 I ' 'OOOOC»' 'OOOO8C * I 's-ao «' OOSt ' '008 ' 'OOOSZ' 'OOOS CI 'SCS •0'0 »»I 'O•8 I Cl 'ft ' S1SS ' O ' tILl O II ' sao•s'o'o oi 'SC I s 'too'soto s 'SOO•0'Z"0'»0'O L 'ZOO•0'*© * O'EO'O S 'too•0'co•0't o•o s 'SOOO'O'IC'O'lOO'O 8 'SEOOO'o'too•O'0'o c ' 8 ' » I C * t-ao•i* t•i•t•s't'i i 93 D Y N A M I C A N A L Y S I S O P F L O O R F O R I M F A C T L 0 A 0 1 N 6 N U M B E R O F F L O O R S • 1 mm l NO OF E L E M E N T S B 9 NO OF G A P S a O • NO OF T1MC Z O N E S a 9 10 ZONE 1 S T A R T i O . O 11 ZONE : S T A R T B O . O O f O O 12 ZONE 3 S T A R T s O . 0 1 O O O 13 ZONE 4 S T A R T B O . 0 2 0 0 0 14 ZONE I S T A R T B O . 0 4 0 0 0 15 Z O N E S S T A R T B O . 2 0 0 0 0 I S F R O F E R T I E S AND D I M E N S I O N S FOR F I A N C E 17 S • O. 1SOOOOE+02 L • 0. 1 4 4 © 0 0 E * 0 3 11 K ( X ) a 0 . t 0 0 0 O 0 C * O 4 K ( Y ) B 0 , 2 I O O O O « * 0 » 19 0 ( X ) a O . 3 4 0 0 0 0 E + 0 * D l Y J a 0.4?OOOOE+09 2 0 C F a 0 . 4 1 7 0 O O E - 0 2 R M O C » 0 . S 0 O 0 O 0 E - 0 4 TMCTAB 1.40 BNOSO.OO1OO S N D « 0 . O I O O O E N 0 » 0 . 0 2 0 0 0 B N D B O . 0 4 0 0 0 ENDBO.2OO0O BNDBO.SOOOO I N C R E M E N T B O . 0 0 0 2 S I N C R E M E N T B O . O O O S O I N C R E M E N T B O . O O I O O I N C R E M E N T B O . 0 0 2 0 0 I N C R E M E N T B O . O O S O O I N C R E M E N T B O . O I O O O 0 " 0 . C 2 S 0 0 0 E * 0 0 K ( V | a 0 . S O O O O O E * 0 3 K ( C ) B D t Y j a 0 . I S O O O O E + O S 0 ( C ) * . 1aOOOOE * 0 4 . « o o o o o e * o s 2 2 2 3 24 2 5 2 1 2 7 2 4 F R O F E R T 1 E S FOR CONNECTORS S T I F F N E S S P A R A L L E L TO J O I S T o.g»ooooe*o4 S T I F F N E S S F E R F E N D I C U L A R TO J O I S T B 0.S4O0OOE*O4 R O T A T I O N A L S T I F F N E S S F L A N G E / J O I S T • 0 . 1 0 0 0 0 0 8 * 0 4 D A M F I N C C O E F F I C I E N T S C N P A L B O. 1 0 0 0 0 0 1 * 0 1 C N F E RB 0 . S O 0 0 O 0 E + O 1 S P A C I N G B E T W E E N C O N N E C T O R S • D . 9 O O 0 0 O B * O 1 e»ROTa 0.80OOOOE*01 3 0 3 1 3 2 3 3 3 4 3S 3 9 37 CONSTANTS FOR J O I S T S BETA a 0 . 2 4 9 3 0 0 6 * 0 0 A L P H A a O.120OOOE+O1 E/C * 0.17OOOOfi*O2 4 3 44 5 2 S 3 5 7 5 4 L O A D I N C J O 1 S T O . 0 O . 0 O . O i O NLU V2 1 4 4 0 0 0 9 * 0 3 . 1 4 4 0 0 0 E * 0 3 . 1 4 4 0 0 0 E * 0 3 . 1 4 4 0 0 0 E * 0 3 . 1 4 « O O O E * 0 3 . 1 4 4 O O 0 E * 0 3 . 1 4 4 0 O O E * 0 3 . 1 4 4 0 0 0 9 * 0 3 . 1 4«000fi*03 COORD I N ATES I N I T I A L V E L O C I T Y * NO OF MASSES ON FLOOR* MASS NO• 1 M« O . 3 6 2 0 0 0 e * 0 0 MASS ON J O I S T NO* 6 X* 0 . 7 2 0 0 0 0 E * 0 2 T 1 . 8 0 0 0 0 0 E * 0 1 . S O O O O O E * 0 1 . 4 0 0 0 0 0 E + 0 1 . S O O O O O E * O i . 8 O O O O O E * 0 1 . S O O O O O E * 0 1 . f l o o o o o e * o i . a o o o o o E + o i . a o o o o o e * o i C * 0 . 8 0 0 0 0 0 E * 0 1 V2 . 8 O 0 O O O E * O 1 . 4 o o o o o e * o i . 4 O O O O O E * O i . S O O O O O E * O I . 8 0 0 0 0 0 E * 0 1 . 4 0 0 0 O O 6 + O 1 .lOOOOOC^OI . S O O O O O E * O ' . . S O O O O O E * O i 0 . o o . o 0 . o K* 0 . 4 0 0 0 0 0 6 * 0 3 O . 0 CONTACT T I M E * O.SOOOOOE*00 S 9 9 0 6 5 S 4 7 9 7 7 7 9 MASS NO* 2 Ma O . S l 9 0 0 0 E * 0 0 MASS ON J O I S T N C COORD I MATES C* 0 . 8 0 0 0 0 0 E * 0 1 K a o . 400000E+-03 I N I T I A L V E L O C I T Y * 0 . 3 9 3 1 0 0 E + 0 2 J O I S T P R O P E R T I E S ? 2 0 0 0 O E * O 7 e« o . Es 0 . E B 0 . E * 0 . E * O . E » O . E * O . e* o. E « O . N O O F E.C. B.C. E.C. B.C. 9 . C . 1 S 0 3 1 S E + 0 7 6* 1 5 9 9 6 4 E * 0 7 C» 1 2 2 3 4 1 E * 0 7 Ca 1 3 3 4 0 0 E + 0 7 Ca 1 1 7 7 5 0 E * 0 7 CB t 3 0 4 7 0 E * 0 7 Ca 1 5 4 S 6 2 E * 0 7 Ca 1 3 8 l S 3 E * 0 7 C a t 7 4 4 4 6 6 * 0 7 G* BOUNOARV CONO ON J O I S T NOa ON J O I S T NOa ON J O I S T NOa ON J O I S T NO a ON J O I S T NOa ON J O I S T NOa O . 1 O 6 0 S 9 E + 0 S O . 9 4 0 9 9 1 E * O S 0 . 7 1 9 8 8 7 6 * 0 5 O . 7 S 4 7 0 4 E * 0 5 O. B9 2 8 4 9 1 * 0 5 O . 7 S 7 4 7 0 E * 0 5 0 . 9 0 9 1 47E*OS O . 41 2 B 8 S E + OS O. I O I 2 0 3 e * O I I T I O N S * 4 1 6 1 9 C O N T A C T T I M E * 0 . 5 0 0 0 0 0 6 * 0 0 RHOa RHOa RHOa RMO* RHOa RH 0 * RHOa RHO* R HO * 0 SOOOOOE -04 Ca 0 3 S 3 0 0 0 E -02 8* 1 . s o o Ha 7 2 5 0 o SOOOOOE • 04 Ca o 3 8 3 0 0 0 E - 02 9* 1 . s o o H a 7 2 5 0 0 SOOOOOE - 04 Ca 0 3 8 3 O O 0 E -02 B* ! . 5 0 0 Ha 7 2 5 0 0 SOOOOOE -04 C» 0 3 8 3 0 0 0 E -02 B> 1 . SOO HB 7 2 5 0 0 SOOOOOE • 04 C* 0 3 8 3 0 0 0 E •02 8 a 1 . SOO ' H* 7 2SO 0 SOOOOOE -04 z * 0 3 8 3 0 0 0 E - 0 2 8a t . 5 0 0 HB 7 2 5 0 o SOOOOOE -04 Ca 0 383DOOE -02 B » 1 . SOO H * 7 2 5 0 0 SOOOOOE •04 Ca 0 3 B 3 0 O 0 E - 02 Ba 1 . 5 0 0 HB 7 2 5 0 0 SOOOOOE -04 Ca 0 2 4 3 0 0 0 6 - 0 2 B* 1 .SOO h a 7 2SO c o o t N O . C O O E N O . ease M O . C O D E N O * cooe NOB C O D E N O * NO OF R E P L I C A T I O N S FOR S I M U L A T I O N . E M I N . O.O [MBIIs O.SOOOOOE.07 «7 * • • • I O I 1 9 2 S 3 9. 3 5 S S t t S « S t 1 oo 101 1 0 2 1 0 I 1 04 IOS l o t . l O T IOS l O t 1 IO I 1 1 1 I I W I I t U L L D I S T R I B U T I O N P A R A M E T E R S EO. 0 . 7 7 0 0 0 0 E . 0 6 EM. 0 . t t O O O O E * O t S O L U T I O N AT S T A R T E K . 0 . 2 7 0 0 0 0 E . O 1 S S L N AT FLOOR F O I NT D I S P L A C E M E N T O . O SOLN AT FLOOR P O I N T D I S P L A C E M E N T O . 0 S O L U T I O N FOR DROPPED MASS D I S P L A C E M E N T O.O S O L U T I O N FOR OROPPED MASS D I S P L A C E M E N T » I L O C I T T O . 0 V E L O C I T Y 0 . O V E L O C I T Y 0.0 V E L O C I T Y > . 3 S 3 1 0 0 E . 0 3 A C C E L E R A T I ON -O.* I 4 A 1 3 1 . 0 2 A C C E L E R A T I O N O . < S 2 3 3 t E . O < A C C E L E R A T ! O N A C C E L E R A T I O N - O . • S 2 2 7 t S . O I A p a r t of o u t p u t f i l e "-2" 1 o . 9 o. 0 O . 0 0 . 0 O . o 2 o . 2SOOO -O . SOI 11-OS 0 . 34 tfE-02 0 . 3 S 4 S E - 0 7 0. 3 49 9 E * 0 0 3 e. SOOOO • o . 2 S 1 S H - 0 8 0 . I 2 7 9 8 - 0 1 0 . 1 0 1 4 E - 0 4 0 . 49906 + 00 4 ©. 7SOOO • o . 2 4 1se - 0 3 o . 2 7 2 6 6 - O 1 - o . 4 2 0 8 E - O S 0 . 7 * » 4 £ + o O 1 i . O O O O O - o . t O S 2 E - 0 2 0 . 4 I04S-O1 - o . 3 4 7 3 B - O S 0 . 9977 6+ 0 0 • i . SOOOO - 0 . 3 4 S 1 E * 0 2 0. 94406*01 • o . 2 2 0 0 E - 0 4 0 . 149 86*01 T 2 . OOOOO • o . 4 4 0 S 6 - 0 2 o . 11736+00 - 0 . S 0 0 2 E - 0 4 0 . 1 9946*01 • 2 . SOOOO -0 . 1 S 3 S E - 0 2 0 . 23 1 1 E * 0 0 - o . S 8 0 S S - 0 4 o . 2 4 9 1 E * 0 1 • 3 . OOOOO o. S 4 0 3 E - 0 2 0 . 3 12SE + 00 - o . 3 S 6 4 E - 0 4 o . 29446*01 lO 3 . SOOOO o . 1 8 2 4 E - 0 1 o . 3947B+00 0 . 7 7 9 0 E - 0 * o . 3 4 4 S E * 0 1 1 1 4 . O O O O O o . 3 S SOC-O1 0 . 4 4 4 2 6 * 0 0 0 . 324 11-03 0 . 3 9 4 1 E * 0 1 12 4 . SOOOO 0 . s isie - o i 0 . 54046+0O *. 7 8 3 3 6 - 0 3 0 . 44786*01 13 S OOOOO 0 . 9 3 1 3 E - 0 1 0 . S 7 S 2 E * 0 0 0 . 1 4 4 8 E - 0 2 o . 4871E*01 14 S SOOOO o . 13OBE*O0 0 . 7 7 1 3 E * 0 0 0. 2 4 3 8 B - 0 2 0. 84446*01 IS • OOOOO 0 . 172»e *oo 0 . 4S4SB+O0 0. 3 7 4 0 1 - 0 2 o 99896*01 I t ( SOOOO 0 . 2 1S4fi * 0 0 0 18706+00 0 S S 2 4 E - 0 2 0 •4S26+01 17 7 O O O O O 0 2 8 1 3 E * 0 O 0 10476*01 0 7 7 0 9 6 - 0 2 0 • 8 4 3 E * 0 1 1 a 7 SOOOO 0 3 2 1»e - o o 0 1 1 8 7 6 * 0 1 0 1037E-01 o 7 4 3 4 E * 0 1 11 4 O O O O O 0 3 7 9 9 6 * 0 0 0 1288E+01 0 13136 -O I 0 7 9 2 4 E * 0 1 20 a SOOOO 0 4304 E * O O 0 13896*01 0 1 7 2 1 E - O I o 8 4 1 3 E * 0 1 21 9 O O O O O 0 44446+00 0 1 4 7 0 E * O 1 0 2 1 4 4 E - 0 1 o 41006*01 22 • SOOOO o S 4 2 S E * 0 0 0 1 S 7 0 B * O 1 0 2 S 2 1 E - O 1 0 9 3 9 6 E * 0 1 23 i o OOOOO 0 S S S S E * O O 0 1 470E + 01 0 3 1 S S E - O I 0 9871E+OI 24 1 T OOOOO o 7 103E*OO 0 1S8SE*01 0 4 3 1 5 E - 0 1 0 10446 * 0 2 2S 12 OOOOO 0 4 2 2 0 1 * 0 0 0 20886*01 0 5 8 7 1 6 - 0 1 0 1 1 806*02 24 13 OOOOO 0 9 3 3 6 E * 0 0 o 224SE+01 0 7 8 8 8 6 * 0 1 0 1 2 7 S E * 0 2 27 1 4- OOOOO 0 1 0 4 S E * 0 1 0 2 4 2 S E * 0 1 0 S S S O E - 0 1 o 1 3 7 0 6 * 0 2 28 1S O O O O O 0 1 i s 4 E * o i 0 2 s s t e * o i 0 1 1 76E + 00 o 14131*02 29 1 < O O O O O o 1 2 S 2 E * 0 1 0 2 7 4 4 E « Q 1 o 1 4 2 3 6 * 0 0 0 1 8 9 8 6 * 0 2 30 17 O O O O O 0 t 3 8 7 E * 0 1 o 2 1 2 1 B * 0 1 0 1 1 9 8 6 * 0 0 o 1 4446+02 31 IS OOOOO 0 1 * § 4 E + 01 0 3 0 7 1 E * 0 1 0 1 8 8 2 8 * 0 0 0 1 7 4 0 E * 0 2 32 1 s O O O O O 0 1 S 8 S E * 0 1 0 32131+01 0 2 3 1 4 B - 0 0 0 1 4 3 0 E * 0 2 33 20 OOOOO 0 18B7E*01 0 334SE+01 0 2 8 8 0 E * 0 0 0 1 9 1 9 E * 0 2 34 22 O O O O O 0 t 4276+01 o 38976*01 0 3 4 2 4 C * O 0 0 2 0 9 4 E * 0 2 36 24 . O O O O O 0 1 8746+01 0 3414E+01 0 4 2 7 1 6 * 0 0 0 . 2 2 B 3 E * 0 2 34 2S . O O O O O 0 . 2104E+01 0 4 0 1 S E * 0 1 0 . 8 2 1 S E * 0 0 0 . 24246+02 37 24 .OOOOO 0 . 2 2 1 7 E * 0 1 0 .4181e+oi o . 8 2 3 4 6 * 0 0 o . 2 S 4 7 E * 0 2 34 30 . O O O O O 0 . 2 3 0 7 B * O 1 0 4 3 4 * E * 0 1 0 . 7 3 1 9 E * 0 0 0 , 2 7 4 0 E * 0 2 31 32 . O O O O O o . 2 3 4 4 E * 0 1 0 . 4 4 7 1 E * 0 1 o . 44 44 6 * 0 0 o . 2 S 8 6 E * 0 2 40 34 . O O O O O 0 . 2 4 8 7 E * 0 1 0 . 41408*01 0 . 9 1 1 6 6 * 0 0 0 . 30296+02 4 1 34 . O O O O O 0 . 2 S S O E * 0 1 0 .488 1 E»01 0 . t 0 9 2 E * 0 1 0 . 3 1446*02 4 2 34 . O O O O O 0 . 2 6 3 0 6 * 0 1 0 . 4733E + 01 0 . 12236*01 0 . 3 2 8 S B * 0 2 43 40 . O O O O O 0 .2701 E* 0 1 o . 49076 * 0 1 o . 1 3S76 + 01 o . 3 4 0 4 6 * 0 2 * 4 4S . O O O O C 0 . 2 8 2 S e * 0 1 0 . S02Se + O1 0 . 1 7 0 8 6 * O 1 0 . 3 S 6 6 e *02 4S S O . O O O O O o . 28306+01 0 . S 2 1 S E * 0 1 0 . 20696*01 0 . 3840E+02 44 S S . O O O O O 0 . 3 0 2 4 E * 0 1 0 . S320E + 01 0 . 2 4 3 4 6 * 0 1 0 . 4 0 4 4 6 * 0 2 47 so . O O O O O 0 . 3 0 9 1 E * O 1 0 . 5343E + 01 0 . 279 t e*o t 0 .415 7 E * 0 2 44 S S O O O O O 0 . 3 I 2 0 E * 0 1 0 . S304E + 01 0 . 3 1 3 1 6 * 0 1 0 . 4 2 2 0 6 * 0 2 4S 70 . O O O O O o . 3 1146 * 01 0 . 52231 + 01 0 . 34456 * 0 1 0 . 4 2 3 4 6 * 0 2 S O 75 . O O O O O 0 . 3 0 7 4 E * 0 1 o . S 0 S 9 6 * O 1 0 . 3 7 2 3 E * 0 1 0 .420 1e*o2 S 1 40 . O O O O O 0 . 3 0 0 2 6 * 0 1 o . 49306+01 o . 39 8 4 6 * 0 1 0 . 4 I 2 4 6 * 0 2 A p a r t of output f i l e "-3" 1 N O O F I T E R A T I O N S F O R S O L N A T T I M E . O 0 0 0 2 S A R E 2 2 N O or I T E R A T I O N S F O R J O I N A T T I M E . O O O O S O A R E 2 3 NO ar I T E R A T I O N S F O R S O L N A T T I M E . O 0 O O 7 S A R E 2 4 N O or I T E R A T I O N S F O R S O L N A T T I M B . O O O I O O A R E 2 S N O or I T E R A T I O N S F O R S O L N A T T I M E . O OO 1 S O A R E 2 • N O or 1 T E R A T I O N S F O R S O L N A T T I M E . O O 0 2 0 O A R E 2 7 NO or I T E R A T I O N S F O R S O L N A T T I M E . O 0 0 2 S 0 A R E 2 » N O or I T E R A T I O N S F O R S O L N A T T I M E . O 0 O 3 O O A R E 2 t N O ar I T E R A T I O N S F O R S O L N A T T I M E . O 0 O 3 S O A R E 2 1 0 NO or I T E R A T I O N S F O R S O L N A T T I M E . O 0 0 4 0 0 A R E 2 1 1 N O or I T E R A T I O N S F O R S O L N A T T I M E » 0 0 O 4 S O A R E 2 11 N O or I T E R A T I O N S F O R S O L N A T T I M E . O O O S O O A R E 2 1 1 N O ar I T E R A T I O N S F O R S O L N A T T I M E . O O O S S O A R E 2 1 4 N O or I T E R A T I O N S F O R S O L N A T T I M E . O O O S O O A R E 2 1 S N O or I T E R A T I O N S F O R S O L N A T T I M E ' 0 O O S S O A R E 2 1 < NO or I T E R A T I O N S F O R S O L N A T T I M E " 0 0 0 7 0 0 A R C 2 1 7 N O or I T E R A T I O N S F O R S O L N A T T I M E . O O O 7 S 0 A R E 2 1 A N O ar I T E R A T I O N S F O R S O L N A T T I M E . O O O A O O A R E 2 1 s NO or I T E R A T I O N S F O R S O L N A T T I M E " 0 O O S S O A R E 2 2 0 NO or I T E R A T I O N S F O R S O L N A T T1 M E » 0 O O S O O A R E 2 21 N O ar I T E R A T I O N S F O R S O L N A T T I M E » 0 t>0950 A R E 2 2 2 NO ar I T E R A T 1 O N S F O R S O L N A T T I M E . O 0 1 O O O A R E 2 2 2 NO or I T E R A T 1 O N S F O R S O L N A T T I M E . O 0 1 1 OO A R E 2 4 N O or 1 T E R A T I O N S F O R S O L N A T T I M E . O 0 1 2 0 0 A R E 2 S NO or I T E R A T I O N S F O R S O L N A T T I M E . O 0 1 3 0 0 A R E 2 S NO or I T E R A T I O N S F O R S O L N A T T I M E . O 0 1 4 0 0 A R E 2 7 N O ar I T E R A T I O N S F O R S O L N A T T I M E . O 0 1 9 O O A R E 2 4 N O or I T E R A T I O N S F O R S O L N A T T I M E . O 0 1 S O O A R E 2 1 NO or I T E R A T I O N S F O R S O L N A T T I M E . O O 1 7 0 0 A R E 2 0 N O or I T E R A T I O N S F O R S O L N A T T I M E . O 0 1 1 0 0 A R E 31 N O or I T E R A T I O N S F O R S O L N A T T I M E . O 0 1 loo A R E 3 2 N O or 1 T E R A T I O N S F O R S O L N A T T I M B * 0 0 2 O O O A R E 3 3 N O or I T E R A T I O N S F O R S O L N A T T I M E . O 0 2 2 O O A R E 3 4 N O or I T E R A T I O N S F O R S O L N A T T I M E . O 0 2 4 O O A R E 3 S NO or I T E R A T I O N S F O R S O L N A T T I M E . O 0 2 1 0 0 A R E 3 1 N O ar I T E R A T I O N S F O R S O L N A T T I M E . O 0 2 4 O O A R E 3 7 N C or I T E R A T I O N S F O R S O L N A T T I M E . O 0 3 0 0 0 A R E 3a N O or I T E R A T I O N S F O R S O L N A T T I M E . O 0 3 2 0 0 A R B 3 1 N O or I T E R A T I O N S F O R S O L N A T T I M E . O 0 3 4 O O A R E 4 0 N O or 1 T E R A T I O N S F O R S O L N A T T I M E . O 0 3 S O O A R E 4 1 N O or I T E R A T I O N S F O R S O L N A T T I M E . O 0 3 S O O A R E 4 2 N O or I T E R A T I O N S F O R S O L N A T T I M E . O 0 4 0 0 0 A R E 4 3 N O or I T E R A T I O N S F O R S O L N A T T I M E . O 0 4 S O O A R E 4 4 N O or I T E R A T I O N S F O R S O L N A T T I M E . O O S O O O A R E 4 S N O or I T E R A T I O N S F O R S O L N A T T I M E . O O S S O O A R E 4 1 N O or I T E R A T I O N S F O R S O L N A T T I M E . O O S O O O A R E 4 7 NQ or I T E R A T I O N S F O R S O L N A T T I M E . O O S S O O A R E 4 1 N O or I T E R A T I O N S F O R S O L N A T T I M E . O 0 7 0 O O A R E 4 1 N O or I T E R A T I O N S F O R S O L N A T T I M E . O 0 7 S 0 0 A R E I O N O or I T E R A T I O N S F O R S O L N A T T I M E . O oaooo A R E 1 1 N O ar I T E R A T I O N S F O R S O L N A T T I M E . O oisoo A R E A pa r t of output f i l e 95 1 O o 0 . 1 0 9 2 6 * 0 1 2 o 2 S O O O 0 . 1 0 4 i E * O I 3 o S O O O O o 1 0 3 3 6 * 0 1 * 0 7 S O O O 0 1 0 3 1 E * O I * 1 O O O O O 0 1 0 3 0 6 * 0 1 • 1 S O O O O 0 1 0 3 0 6 * 0 1 7" 2 O O O O O 0 1 0 3 3 6 * 0 1 * 2 S O O O O 0 1 0 4 2 6 * 0 1 • 3 O O O O O o 1 0 6 4 6 * 0 1 1 0 3 S O O O O 0 1 0 7 7 6 * 0 1 4 O O O O O 0 1 0 B 8 6 * O 1 4 S O O O O 0 1 1 2 0 6 * 0 1 1 3 S O O O O O o 1 1 4 3 6 * 0 1 1 4 S S O O O O •o 1 1 8 7 6 * 0 1 1 S B O O O O O 0 1 1 soe+oi 1 B 6 S O O O O o 1 2 1 4 6 * 0 1 t 7 7 O O O O O 0 1 2 3 7 6 * 0 1 I 8 7 S O O O O 0 1 2 6 1 6 * 0 1 1 9 8 O O O O O o 1 2 6 S e * 0 1 2 0 8 S O O O O 0 1 3 1 1 e * o i 2 1 B O O O O O 0 1 3 3 9 6 * 0 1 2 2 t S O O O O 0 1 3 8 1 6 + 0 l 2 3 t o O O O O O 0 1 3 8 S 6 * O t 2* 1 1 O O O O O 0 1 4 4 0 6 * 0 1 21 1 2 O O O O O 0 1 4 9 2 6 * 0 1 2 9 1 3 O O O O O 0 1 5 4 S e * 0 1 2 7 1 4 O O O O O 0 1 5 1 6 6 * 0 1 2 8 1 S O O O O O 0 1 6 5 1 6 * 0 1 21 1 B O O O O O o 1 7 0 4 6 * 0 1 3 0 1 7 O O O O O 0 1 7 s s e*oi 3 1 1 8 O O O O O 0 1 6 0 7 E * 0 1 3 2 1 9 O O O O O 0 1 4 5 4 6 * 0 1 3 3 2 0 O O O O O o 1 1 0 7 e * 0 1 3 4 2 2 O O O O O 0 2 0 0 2 8 * 0 1 3 S 2 4 O O O O O 0 2 0 8 3 6 * 0 1 3 8 28 O O O O O 0 2 1 7 8 6 * 0 1 3 7 2 8 O O O O O 0 2 2 5 7 6 * 0 1 3 8 3 0 O O O O O o 2 3 3 3 6 * 0 1 38 3 2 O O O O O 0 2 4 0 8 6 * 0 1 4 0 3 4 O O O O O o 3 4 7 S E * 0 1 4 1 3 6 O O O O O 0 2 5 3 4 E * 0 1 4 2 38 O O O O O o 2 S 8 1 E * 0 1 4 3 4 0 O O O O O o 2 8 1 5 E * 0 1 4 4 4 S O O O O O 0 2 6 7 9 6 * 0 1 4 S 3 0 O O O O O 0 2 7 2 S E * 0 1 4 8 S S O O O O O 0 2 7 4 3 6 * 0 1 4 7 S O O O O O O 0 2 7 3 2 E * 0 1 4 8 6 5 O O O O O o 2 6 8 5 E * 0 1 4 9 7 0 O O O O O 0 2 6 0 8 B * 0 1 S O 7 5 O O O O O 0 2 5 O 8 E * 0 1 S 1 S O O O O O O 0 2 3 6 S 6 * 0 1 A p a r t o f o u t p u t f i l e " - 7 " 1 0 . 3 1 2 0 6 * 0 1 9 8 . O O 0 . 1 S S 0 B * 0 1 3 1 0 . O O O u t p u t f i l e "-8" 1 0 . 1 8 > O 6 * 0 7 0 . 1 9 0 6 * 0 7 0 . 1 2 2 6 * 0 7 0 . 1 3 3 6 * 0 7 0 . 1 1 8 6 * 0 7 0 . 1 3 0 6 * 0 7 0 . 1 5 5 6 * 0 7 0 . 1 3 * 6 * 0 7 0 . 1 7 9 6 * 0 7 O u t p u t f i l e " - 9 " APPENDIX I I I DYNAMIC FLOOR ANALYSIS PROGRAM PROGRAM L I S T I N G 96 PROGRAM FOR DYNAMIC A N A L Y S I S OF FLOORS THE NUMBER OF S T I F F E N E R S ARE L IMITED TO 10 THE NUMBER OF FOURIER S E R I E S TERMS WHICH CAN BE CONSIDERED FOR SOLUTION ARE L I M I T E D TO 3 S 10 1 3 1 3 1 S IT 1 S 1 9 30 2 1 22 23 24 2S 30 3 1 32 33 34 35 3S 31 3< 33 40 Z) R L . 0 . A L P H A , B E T A . H J T M . R K P A L . R K P E R . X I N . N C A P S , I M P L I C I T REAL * S ( A - H.O ' COMMON / B 4 / E T A I S ) , H IS) COMMON / B S / NM, N S T E P , NMAX. N J T , S . B J I 1 0 ) , H J ( I O ) , NF L OR, INPTE / B E / RKX. R K V , RKV, R K C , D I , DV, DV, DC, RKROT, RHOC. C P . CNPAL , CNPER, CNROT / B 7 / CAP XOI 5) , C A P O I S ) , C A P X I S ) , CAP I S ) , E N L , N A I . NOISCR / B I O / R C I S I . R K I S ) , R M I S I , R F O R C I ( S ) . NDRMAS, N J D R I S ) . R X O I S ) . R V O I S I . V E L I S ) . T IM IMPIS ) / B 1 1 / R S T I F I 1S, 13S) , ROAMP I IS , 138) / B 1 2 / V M 0 D I 1 O ) , 0 N S T V I 1 0 ) , C B ( I O ) , T H E T A , G R V I T Y , SM00 I1O) / B 1 3 / CMASSI 3 , 2 S S 4 ) , C O A M P ( 3 . 2 S 8 4 | . ADAMPI 3 , S 0 3 2 ) , G S T I F I 3 . 2 S S 4 ) . A S T I F ( 3 , S 0 3 2 ) . OCPMASI 3 , 2 S » 4 I , FORCE 1 I 3 . 1 3S ) / B 1 4 / P I , P I N 2 . P I N 4 , NEO, L H B . NA, HA 1 , A J T I 1 0 ) , R I Y I I O ) , R I I I 1 0 1 . R I T I 1 0 ) / B I S / NLOAD, N L U , P L O A O , S T 0 R E ( 1 0 , 5 ) . N L J 0 I 1 0 ) / B I B / SVEC1 I 3 . I l l I . S V E C 2 I 3 , I3S) , RV EC 1 IS) , R V E C 2 I S ) , S V E C 3 I 3 , 1 3 6 ) , R V E C 3 I S ) / B I T / E F 5 T 1 F ( 3 . 2 S S 4 1 . A F S T I F I 3 . 8 0 3 2 1 , R E F S T F ( S ) , R E F E C T I 15. 13S) COMMON / S I S / F0RCE2 I 3 . 138) . RFQRC 2 ( S I ISYM, N TMZON, T I M E I 1 0 . 3 I . REG , TOL E O , E M . E K , EMIN , EMAX, S E E D . NREPL NBC . I BCI S O . 2 I N 0 E F J T I 1 O ) . D E F M ( 1 0 , 2 ) . NPNTS , A2 , A 3 , A 4 , A S , A 6 , A T , A S , A9 3 ) , AMP L1 , O C C U R ) , AMP L 2 , 0CCUR2 • O E F I N I N G V A R I A B L E S ' COMMON COMMON COMMON COMMON COMMON COMMON COMMON COMMON COMMON COMMON COMMON COMMON COMMON COMMON COMMON COMMON / B I S / / B 2 0 / / B 2 1 / / B 2 2 / / B 2 3 / A O , / B 2 S / S C A L E I SO S 1 S2 S3 S4 SO S 1 72 73 79 80 82 S3 84 SS SS 97 SS S9 100 101 102 103 104 lOS lOS 107 i o a 109 110 AO . . . . A A F S T I F ADAMP ALPHA AMP L 1 AMP L 2 BETA BJI I ) CB CP CNPAL CNPER CNROT D DNS TV I I ) DXSDY DVSOC DCPMAS ENL FORCE 1 F0RCE2 GDAMP CST I F CRVITV HJ I I I 1SVM IBCI I I B C I 1 , 2 ) I NPTE LHB NM NMAX NJT NCAPS NA 1 NDISCR . 1 ) NDRMAS NJORI I I NLOAO NLU NL JO I I ) NTM20N NBC NPNTS N D E F J T | I ) NEO NREPL NPROP OC CUR 1 0CCUR2 PLOAD RL RKXBRKV RKVSRKC :CONSTANT FACTORS USED FDR E F F E C T I V E S T I F F N E S S MATRIX AND E F F E C T I V E LOAD VECTOR : O F F - D I A G O N A L S UBMA TRICES DF E F F E C T I V E S T I F F N E S S MATRIX STORED COLUMNWISE : O F F - D I A G O N A L SUBMATRICES OF S T I F F N E S S MATRIX STORED COLUMNWISE :OFF - 0 I AGONAL S UBMA T R I C E S OF DAMP INC MATRIX STORED COL UMNWIS E 'SHEAR D E F L E C T I O N CONSTANT FOR RECTANGULAR C . S . : F I R S T D ISPLACEMENT PEAK :SECOND D ISPL AC E MEN T PEAK :TORSION CONSTANT :WIOTM OF J O I S T I :DAMPINC C O E F F I C I E N T OF J O I S T :DAMPINC C O E F F I C I E N T OF SLAB ^DAMPING C O E F F I C I E N T FOR S L I P P A R A L L E L TO CONNECTORS :DAMPINC C O E F F I C I E N T FOR S L I P PERPENDICULAR TO CONNECTORS :DAMPINC C O E F F I C I E N T FOR ROTATIONAL S L I P AT CONNECTORS . T H I C K N E S S OF SLAB :MASS DENSITY OF J O I S T I : A X I A L S T I F F N E S S OF S LAS IN X4Y D IRECTIONS R E S P E C T I V E L Y P A R A M E T E R S FOR AXIAL S T I F F N E S S RELATED TO P O I S S O N ' S E F F E C T ANO SHEAR R E S P E C T I V E L Y :OECOMPOSED OIACONAL SUBMATRICES OF MASS MATRIX STORED COLUMNWISE :DE C 0MP05 E D DIAGONAL SUBMATRICES OF E F F E C T I V E S T I F F N E S S MATRIX STORED COLUMNWISE : S P A C I N G OF CONNECTORS : L 0AD VECTOR . E F F E C T I V E LOAD' VECTOR AND ALSO USED AS WORKING LOAD VECTOR :OIAGONAL SUBMATRICES OF DAMP 1N C MATRIX :DI AGONAL SUBMATRICES OF S T I F F N E S S MATRIX : A C C E L C R A T I ON DUE TO GRAVITY :HE IGHT DF J O I S T I : « 1 , IF SYMMETRIC PROBLEM; I * 0 IF NON-SVMMETRIC : N 0 . OF J O I S T ON WHICH B . C . 1 IS IMPOSED : COOE NO. FOR B . C . I : F 0 R S IMULATION • 1 : OTHERWISE ' 2 :HAL F BAND WIDTH :MAXM ORDER OF S I N E / C O S I N E TERMS IN FOURIER S E R I E S :MAX M NO. OF TERMS IN FOURIER S E R I E S ~ : N 0 . OF J O I S T S :N 0. OF GAPS IN FLOOR : N O . OF CONNECTORS BETWEEN ONE J O I S T AND SLAB IF ACTUAL CONNECTORS TRANSFORMED TO AN E O U I V A L E N T CONTINUOUS CONNECTOR FOR ANALYSIS IF CONNECTORS T R E A T E D AS D I S C R E T E OF MASSES ON FLOOR OF J O I S T UNDER MASS I OF LOADEO AREAS, ' I F . O . LOAD UNIFORMLY D I S T R I B U T E D IF ALL J O I S T S HAVE SAME LOAD D I S T R I B U T I O N • O . I F D I F F E R E N T J O I S T S HAVE D I F F E R E N T LOADINGS ND. OF J O I S T LOADED WITH LOADING I OF TIME ZONES OF BOUNDARY CONDITIONS -OF POINTS ON FLOOR WHERE RESPONSE DESIRED OF J O I S T ON WHICH POINT 1 IS LOCATED OF UNKNOWNS IN PROBLEM OR NO. OF EOUATIONS NO OF R E P L I C A T I O N S FOR S IMULATION P l . I F ALL BEAMS HAVE SAME P R O P E R T I E S ; BO,OTHERWISE TIME OF OCCURENCE OF F I R S T DISPLACEMENT PEAK TIME OF OCCURENCE OF SECOND DISPLACEMENT PEAK VALUE OF UNIFORMLY D I S T R I B U T E D LOAD SPAN OF FLOOR BENDING S T I F F N E S S OF SLAB IN XSY DIRECTIONS R E S P E C T I V E L Y PARAMETERS FOR P L A T E BEN01NC CONNECTED R E S P E C T I V E L Y : * 0 , : NO . : NO . : • 1 . : NO . ; NO . : NO . : NO . ; NO . 111 C TO P O I S S O N ' S E F F E C T AND SHEAR ST TORSION 1 1 2 C R X O ! I ) :X COORDINATE OF MASS I t 1 3 c RT0 I I1 :T COORDINATE OF MASS I 1 1 4 c RS T I F :VECTOR IN S T I F F N E S S MATRIX DUE TD INTERACTION OF 1 IS c MASS AND FLOOR 1 1 S c ROAMP :SAME AS ABOVE EXCEPT FOR DAMPING MATRIX t 1 7 c R E F E C T :SAME AS ABOVE EXCEPT FOR E F F E C T I V E S T I F F N E S S MATR I 1 18 c RF0RC1 :L OAD VECTOR FOR MASSES 1 19 c RF0RC2 : E F F E C T I V E LOAD VECTOR FOR MASSES 6 ALSO USED AS 130 c WORKING LOAD VECTOR 131 c R V E C 1 , : D I S P L A C E M E N T , V E L O C I T Y AND A C C E L E R A T I O N 1 23 c RVEC2A R E S P E C T I V E L Y OF MASSES 123 RVEC3 1 24 c R M . R C . R K : V A L U E S OF MASS.OAMPING C O E F F I C I E N T I S T I F F N E 5 S FOR 1 2S c MASSES 128 c R E F S T F : E F F E C T I V E S T I F F N E S S FOR MASSES 1 27 REG : E / C RATIO 128 c RKPAL TCONNECTOR L O A O - S L I P MODULUS P A R A L L E L TO J O I S T 1 29 c RKPER ^CONNECTOR L O A O - S L I P MODULAS PERPENDICULAR TO J O I S T 1 30 c RKROT :ROTATIONAL S L I P MODULAS AT CONNECTORS 131 c RHOC :MASS DENSITY OF SLAB 132 c S : S P A C I N C OF J O I S T S 133 c SMOD :SH EAR MODULUS 134 STREN :STRENGTH DF J O I S T S 135 c S C A L E . -MULTIPLYING FACTORS ; S E E ROUTINE RESULT 138 SEED :RANDOM NUMBER I N I T I A L I Z E R 1 37 c TIMIMP :CONTACT TIME OF MASS WITH FLOOR 136 c • VEL r V E L O C I T V OF F A L L OF MASSES 139 c YMOD : Y O U N C ' S MOOULAS OF E L A S T I C I T Y 1 40 c 1 4 1 c 142 c 143 PI s 4 .DO • DATANI 1 .DO ) 1 44 E T A ( I ) • 0 . 9 324 896 1420315200 145 E T A [ 2 ) • 0 . 6 612 0 9 3 6 6 4 6 6 2 6 5 0 0 146 ETA{3 ) a O . 2 3 6 6 1 9 1 8 6 0 8 3 3 1 9 0 O 147 H(1 ) » 0 .17132449237917O0O 146 HI2) a 0 . 3 9 0 7 615 7 3 0 4 8 1 3 9 0 0 149 H(3) • 0 . 4579 1393457269 1 DO 1 60 DO 10 I a 1 , 3 1 S 1 ETA t 7 - I) . - E T A ( I ) 152 H(7 - 11 • H i l l 1 S3 10 CONTINUE 154 CALL GENMTX 155 CALL OA TA 156 IF 1 INPTE . E O . 1 1 GO TO 20 157 CO TO 40 156 20 NCOUNT 8 0 159 X a R A N D t S E E D l 1 60 30 CALL E P I C K ( N J T ) 1 6 1 40 00 170 NF B 1 , NFLOR 1 62 IF (NF . E O . 11 GO TO 50 1 63 CALL UPOATEINF I 1 84 50 DO 60 1 B 1 , NJT 1 65 AJTI I 1 B BJ1 I ) • HJ1 1 1 1 66 R I Y ( I ) s B J ( I ] • H J ( I ) • « 3 / 12.OO 1 67 R I Z I I I B H J I I I • B J I I I •> 3 / 1 2 . 0 0 1 66 R I T I I ) B BETA « H J I I I • B J I I I • • 3 1 69 SMOD(I ) s YMOOII) / REG 1 70 60 CONTINUE 1 7 1 PIN2 = PI • • 3 / 1 2 . D O * R L ) 172 P1N4 B PI mm 4 / | 2 . D O » R L * « 3 ) 173 NEO B NJT • 13 • 8 174 L HB B I S 1 75 NA B t 9 m NEO 176 NAI B 37 • NEO 1 77 NSTEP * 1 176 IP ( ISYM . E O . 1) NSTEP B 2 179 NMAX B NM 1 60 IF INSTEP . E O . 2] NMAX a (NM . 1 1 / 2 1 6 1 CALL PRINT 163 CALL ASMBLY 183 DO 70 I a 1 , NMAX 1 84 DO 70 J a 1, NEO 185 S V E C 1 I I . J ) ' 0 . 0 0 186 70 SVEC21 I . J I a 0 . D O 1 87 IF (NORMAS . E O . 0) CO TO 90 186 DO SO I a 1, NORMAS 1 69 RVEC1 1 I 1 a O O O 190 80 RVEC2 III a VEL 1 1 I 1 9 1 PERIOD a TIMIMP1NORMAS1 192 SO CONTINUE 1 93 AMP LI a 0 . D O 194 AMP L 2 B 0 . 0 0 195 OCCUR 1 a O.DO 196 0CCUR2 a O.DO 197 CALL A X L R T N I N E O . L H B , NAI 196 WRITE I 2 . 1OO1 196 100 FORMAT 1/. 10X. ' S O L U T I O N AT S T A R T ' , / ) 300 E L T I M a 0 . D O 201 CALL RESULT I P E R I O D , E L T I M , NDRMAS. P I I 202 DO 15 0 I a 1, NTMZON 203 E L T I M a T I M E ( I . I ) 204 AO a 6.ODO / I 1 T H E T A « T I ME 1 I . 3 1 I • • 2 1 205 A l a 3 . D O / I T H E T A 8 T I M E I I . 3 1 1 206 A2 a 2 . D O a A l 207 A3 a THETA • TI M E ( 1 , 3 ) / 2 . 0 0 206 A4 a AO / THETA 209 AS a - A 2 / T H E T A 210 AS a 1 . D O • 3 . D O / THETA 21 1 A7 a TI M E ( 1 , 3 ) / 2 . 00 212 AS a T 1 M E I I . 3 ) * * 3 / 6 .DO 213 AB a TIME 1 1 . 3 ) 2 14 C A L L EFMATINMAX, NA, N A I , L H B , NEO) 215 E L T I M a E L T I M * TI M E ( 1 . 3 ) 216 1 10 C A L L EPLOAD1NMAX, NEO, LHB) 217 CALL S O L V E I N M . NMAX. NEO, L H B . T O L , E L T I M . NA. NORMAS) 2 1 8 WR ITE ( 2 , 120) E L T I M 219 130 FORMAT ( / , 10X, ' S O L U T I O N AT T I M E a ' . F 7 . 5 , / ) 220 CALL R E S U L T ( P E R I O D , E L T I M , NORMAS. P I ) 121 ELTIM • ELTIM • TIMEI1.3) 222 CI • O.OOOIDO * TIME(I,3) 223 C2 • EITIM - TIMEI1,2) 224 C3 • ELTIM - TI Ml MP(NDRMAS) 225 C4 • TIMEI1.3) » CI 22B IF IC3 .CT. CI .AND. C3 LT. C4 ) CO TO 130 227 IF (C2 .CT. CI 1 CO TO 1S0 228 CO TO 1IO 229 130 IF (1 .80- NTMZON .ANO. C2 .CT. CI) CO TO 150 230 NDRMAS • NDRMAS • 1 231 CALL ASMBLV 232 CALL AXLRTNINEO, L HB. NA) 233 CALL EFMATINMAX. NA. NA1. LMB. NEO) 234 WRITE 12.140) 23S 1 40 FORMAT 1/, 10X, 'SOLUTION AT SEPARATION OF MASS', /) 236 SEPTIM a ELTIM - TI ME I I ,3 I 237 CALL RESULT I PERI 00, SEPTIM, NORMAS. PI) 236 IF (C2 . CT . CI I GO TO ISO 23> CO TO 1 10 240 1 BO CONTINUE 24 1 WRITE (8.1101 AMP L1 , OCCUR 1 , AMP L 2, 0CCUR2 242 1 60 FORMAT IE11.4, IX, F6.2, IX, Ell.4, 1X, F8.2) 243 170 CONTINUE 244 IF [INPTE .60. 2) 60 TO ISO 24S NCOUNT • NCOUNT • 1 241 IF (NCOUNT .80. NREPL) GO TO 1 SO , 247 CO TO 30 244 - 1 BO CONTINUE 249 STOP 250 END 25 1 C 252 SUBROUTINE DATA 253 IMPLICIT REAL*8(A - H,0 • Z) 2S4 COMMON /B5/ NM, NSTEP, NMAX, NJT, S, RL, 0, ALPHA, BETA, HJTM, 255 1 BJI10I, HJI10), NFLOR. INPTE 255 COMMON /B6/ RKX , RKT, RKV . RKC , OX . DY, DV , DG , RKPAL . RKPER. 257 1 RKROT, RKOC, CP, CNPAL, CNPER, CN RO T 258 COMMON /B7/. CAPXOIS). GAPO(5), CAPXIS), CAP|5), ENL, XIN, , NGAPS 2S5 1 NAI . HO I SCR ' 250 COMMON /BIO/ RCISI, RKIS). RMI5I. RF0RC1I5I. NORMAS, NJOR15 1. 26 1 1 RXOI 5) . RTO 15) . VELI5), TIMIMP(S) 282 COMMON. /B12/ YMO0I1O), DNSTT(IO), CBI10). THETA, CRVITY 283 COMMON /B15/ NLOAD, NLU, PLOAD. ST0RE(1O,S). NLJOMO) 284 COMMON /BIS/ ISYM. NTM20N. TI ME 1 10.3) . REG. TOL 265 COMMON /B20/ EO, EM. EK, EMIN, EMAX. SEED, NREPL 268 COMMON /B21/ NBC, IBCI50.2) 267 COMMON /B22/ N0EFJTI1O). DEFM(10.2I, HPNTS 268 COMMON /B25/ SCALEI3I 289 INTEGER LISTI1I /•»•/ 270 READ (1.LIST) NFLOR. NM, NJT, ISYM, NPROP, INPTE, TOL 271 READ 11,LIST] THETA, NTMZON 272 DO 1O I • 1 , NTMZON 273 10 READ (1.LISTI (TIME(I,J).J»1.31 274 IF (INPTE .EO. 2) CO TO 20 275 READ 11,LIST! NREPL. SEED 276 READ (1.LIST) EMIN, EMAX 277 REAO I 1 , L1ST ) EO, EM, EK 278 20 CONTINUE 279 READ (1.LIST) S, RL, D 260 READ (1.LIST) RKX. RKY, RKV, RKC. RHOC 28 1 READ (1.L1ST) DX. DY. DV, DG 282 IF (HPROP EO. Ol GO TO 60 283 READ ( 1 . LIST) BJT, HJT, RHOJT, CBT 264 IF (INPTE .EO. 11 GO TO 40 286 REAO 1 1 ,L1ST I E JT 266 DO 30 I « 1. NJT 287 30 VMOD(I) * EJT 266 40 DO SO I a 1, NJT 269 BJ(I) B BJT 290 HJII) B HJT 29 1 ONSTYI) B RHOJT 292 60 CB( I I B CBT 293 CO TO 80 294 60 READ 11.LIST) IBJ1 I ) . IB 1 .NJT) 296 REAO 11.LIST) 1HJ1 1 ) . IB1 ,NJT) 296 IF I INPTE .EO. 1 ) GO TO 70 297 READ (1.LIST) I VMOD 1 1 )'. i a 1 . NJT ) 298 70 READ 11.LIST) ( DlfSTY (,f 1 , I a 1 , NJT ] 299 READ (1.LIST1 tCBI I ) . IB1 ,NJT1 300 80 READ 1 1 . L 1ST I BETA. ALPHA. REG 30 1 HJTM B 0.ODO . 302 DO 90 I B 1 , NJT 303 90 HJTM B HJTM • HJI I 1 304 HJTM B HJTM / NJT 306 READ (1.LIST) ENL, XIN, RKPAL, RKPER, RKROT, NDISCR 306 READ (1.LIST) CP, CNPAL. CNPER. CNROT 307 NAI B (RL - 2.000'XIN) / ENL 308 NAI B NAI » 1 309 READ I 1 . L1ST I NGAPS 310 IF (NGAPS .EO. 01 CD TO 120 31 1 REAO I 1 .L1ST I NOT 312 DO 110 I B 1, NGAPS 3 1 3 READ 11,LIST) GAPXII), GAP I I ) 3 1 4 GAPXOI I ) B CAPXI I ) 316 CAPOI I ) a CAP ( I ) 316 IF (NOT .EO. 0) GO TO 110 317 NG B (GAPXII) - XIN) / ENL 318 X G1 B XIN • NG • ENL 319 XG2 B XC1 • ENL 320 IF (XG1 . EO . GAPXII)) GO TD 10O 321 XG3 • CAPXII) • GAPI1) 322 GAPX(I I B XC1 323 CAP! I) B 2.00 • ENL 324 IF IXG2 .CT. XG3) CAP(I) a ENL 321 GO TO 110 326 100 GAPXI) a IC1 - ENL 327 GAP II) a 2.OOO • ENL 328 1 10 CONTINUE 329 120 CONTINUE 330 READ 11,LIST) NLOAD. NLU, PLOAD 331 READ 1 1 , L I S T ) NDRMAS 333 IF INDRMAS . S O . OI GO TO 130 333 READ I I . L I S T ) I R M l I ) . I s 1 . N O R M A S ) 334 REAO I I . L I S T ) I RC I I ) . I > 1 . NDRMAS ) 335 REAO I I . L I S T ) I RK I I ) . I > 1 , NDRMAS ) 33S REAO 1 1 . L I S T ) I VE L I I I . I • 1 . NDRMAS ) 337 READ I I . L I S T ) INJDRI I ) . I •1 .NORMASI 33S REAO I I . L I S T ) I R >0 I I ) . I > 1 . NDRMAS ) 339 READ ( 1 . L I S T ) IRVOI I I , I •1 ,NDRMASI 340 REAO ( 1 . L 1 S T ) ITIMI MP 1 I ) . I •1 ,NDRMAS) 341 READ M . L I S T ) | RF DRC 1 I I ) , I • 1 , ND RMAS ) 342 130 CONTINUE 343 IF [NLOAD . N C . 0) CO TO ISO 344 00 140 I a 1, NJT 345 STORE I I . 1 ) • O.DO 34S STORE 1 1 , 2 ) » RL 347 S T 0 R E ( I . 3 1 • -S / 2 . 0 D 0 34S S T 0 R E I I . 4 ) • S / 2 .OOO 349 S T O R E I I . S ) • PLOAO 350 140 CONTINUE 351 GO TO 170 352 ISO 00 1S0 I - I . NLOAD 353 READ ( 1 . L I S T ) N L J O I I ) . S T O R E I I . S ) 354 READ I I . L I S T ) (STORE I I , J ) . J « 1 , 4 ) 355 ISO CONTINUE 35S 1TO CONTINUE 357 READ I I , L I S T ) CRVITV 35 B READ I 1 . L I S T ) NBC 359 IF INBC . E O . 0) CO TO 190 350 DD ISO I • 1, NBC 351 ISO READ 1 1 . L I S T ) ( I SC I I . J I . J » 1 . 2 ) 352 1S0 CONTINUE 363 READ I I . L I S T ) NPNTS 394 IF (NPNTS . E O . 0) CO TO 210 355 DO 200 I a 1, NPNTS 3 6 6 200 REAO I I . L I S T ) H O E F J T I I ) . D E P M I I . I ) . D E F M I I . 2 ) 367 210 CONTINUE 368 REAO ( I . L I S T ) I S C A L E I I ) . I •1 , 3 ) 369 RETURN 370 END 371 C 372 SUBROUTINE PRINT 373 I M P L I C I T R E A L ' S I A - H.O - 21 374 COMMON / B S / NM, N S T E P . NMAX. N J T . S . R L , D, A L P H A , B E T A , H J T M , 375 1 B J I 1 0 ) . H J I 1 0 I , NF L OR, INPTE 376 COMMON / B B / RKX, RKY. RKV, RKG, DX, DY, DV, DC, R K P A L , RKPER, 377 1 RKROT, RHOC, C P , C N P A L , CNPER. CNROT 378 COMMON / B 7 / G A P X 01 5) , G A P O I S ) . G A P X I 5 ) , G A P I 5 ) , E N L , X I N . NGAPS, 379 1 NAI . NO I SCR 3S0 COMMON / B I O / RC I 5 I . R M S ) . R M I 5 ) . R F 0 R C 1 I 6 ) . NORMAS. N J 0 R I 5 I . 381 1 R X 0 I 5 ) , R V O I S ) . V E L I 5 ) . T I M I M P I 5 ) 382 COMMON / B 1 2 / Y M 0 D I 1 O ) . 0 N S T Y I 1 O ) , C B I 1 0 ) , T H E T A , G R V I T Y , SMO0I1O) 353 COMMON / S I S / NLOAD. N L U . P L O A D . S T 0 R E I 1 O . S I . NL JO I 10) 3>4 COMMON / B 1 9 / ISYM, NTM20N, TI ME I 1 0 . 3 ) . R E G , TOL 365 COMMON / B 2 0 / E O , EM, EK. E M I N , EMAX, S E E D . NREPL 356 COMMON / B 2 1 / NBC. I B C I S 0 . 2 ) 361 W R 1 T E I 2 . 1 0 ) 366 10 FORMAT I / . 'DYNAMIC ANALYSIS OF FLOOR FOR IMPACT L O A D I N G ' . / / ) 389 WRITE ( 2 , 2 0 ) NFLOR. N J T . NM, NGAPS 390 20 FORMAT I ' NUMBER OF FLOORS a ' , 15 , IOX, 'NO OF E L E M E N T S . ' , 15 , / . 391 1 IOK. ' N i ' , I S , 10X, ' NO OF GAPS a ' , I S , / / ) 392 WRITE ( 2 . 3 0 ) NTMION. THETA 393 30 FORMAT ( ' N O OF TIME ZONES a ' , 13 . IOX, ' T N E T A a ' , F 5 . 2 ) 394 DO 4 0 I a 1, NTMZON 395 40 WRITE ( 2 , 5 0 ) I , tTI ME( 1 , J ) . J a 1 , 3 ) 396 SO FORMAT ( ' 2 0 N E ' , 12, SX. ' S T A R T S ' , F 7 . 5 , 5 X , ' E N D S ' , F 7 . 5 . 5X. 397 1 ' I N C R E M E N T S ' , F 7 . 5 ) 396 WRITE ( 2 . S O I 399 60 FORMAT (' P R O P E R T I E S AND DIMENSIONS FOR F L A N G E ' ) 400 WRITE ( 2 . 7 0 I S, R L . D 401 70 FORMAT ( ' S s ' , E 1 3 . 6 , IOX, ' L a ' , B 1 3 . 6 , IOX, 'D a ' , E 1 3 . 6 ) 402 WRITE ( 2 , 8 0 ) RKX, RKY, RKV, RKG 403 SO FORMAT I ' K ( X ) a ' , E 1 3 . 6 , 5X . ' K ( Y ) a ' , E 1 3 . 6 . E X . _ 404 1 ' K I V I s ' , B 1 3 . 6 , SX, ' K ( G ) s ' , E 1 3 . 6 ) 405 WRITE ( 2 , 9 0 1 DX, DY. DV. DG 40S 90 FORMAT I ' O i l ) ' ' , E 1 3 . 6 , S X . ' D I Y ) s ' , E 1 3 . 6 , S X . 407 1 ' D ( V ) a ' , E 1 3 . 6 . SX, ' D ( C ) > ' , E 1 3 . 6 ) 408 WRITE 1 2 , S S ) C P , RHOC 409 SS FORMAT I ' C P a ' , E 1 3 . 8 . S X , ' R H O C s ' , E 1 J . I I 4 10 WR I T E 12 ,1 OO )• 411 lOO FORMAT ( / / . * PROPERTIES FOR C O N N E C T O R S ' ) 412 WRITE ( 2 , 1 1 0 ) R K P A L , RKPER. RKROT 413 110 FORMAT ( ' S T I F F N E S S P A R A L L E L TO J O I S T a ' . E 1 3 . 8 , IOX, ' S T I F F N E S S ' 414 1 ' PERPENDICULAR TO J O I S T a ' , E 1 3 . 6 , / , 415 2 ' ROTATIONAL S T I F F N E S S F L A N G E / J O I S T a ' , E 1 3 . E I 416 WRITE 1 2 . 1 2 0 ) C N P A L . CNPER. CNROT 417 120 FORMAT ( ' D A M P I N G C O E F F I C I E N T S ' , / , ' C N P A L " ' , E 1 3 . 6 . S X . 'CNP ERa ' , 418 1 E 1 3 . 8 , SX, * CNROT a ' , E 1 3 . 6 ) 4 19 WRITE I 2 , 1 30 I ENL 420 130 FORMAT I ' SPACING BETWEEN CONNECTORS a ' , E 1 3 . 6 , / / ) 421 WRITE ( 2 , 1 4 0 ) B E T A , A L P H A , REC 422 140 FORMAT (• CONSTANTS FOR J O I S T S ' / / . ' BETA a ' , E 1 3 . S , / , 423 1 ' ALPHA a ' , E 1 3 . 6 . / . ' E / G a ' . E 1 3 . S . / / ] 424 WRITE ( 2 , ISO) NLOAD. NLU 425 150 FORMAT ( / / , ' L O A D I N G ' , 1BX, ' N L O A D a ' , 13 , S X , ' N L U a ' , 13) 426 WRITE I 2 . 1 SO) 427 ISO FORMAT I ' J O I S T ' . 11X. " E l " . 14X, ' X 2 ' . 14X. ' V ! ' , 14X, ' V 2 ' . 428 1 1 2 X . ' L O A D ' ) 42S IP (NLOAD . N E . OI GO TO ISO 430 DO 170 I a I, NJT 431 WRITE ( 2 . 2 0 0 ) I , ( S T O R E ( I . J ) , J > 1 . 5 ) 432 170 CONTINUE 433 GO TO 210 434 ISO DO ISO I • 1. NLOAD 435 ISO WRITE ( 2 . 2 0 0 1 N L J O I I ) , I S T O R E ( I , J ) . J a I , 5 ) 438 200 FORMAT ( 3 X , 12, 3X, S ( 3 X , E 1 3 . S ) 1 437 210 WRITE I 2 . 2 2 0 ) NDRMAS 436 220 FORMAT ( / , IOX, 'NO DF MASSES ON F L O O R S ' , 13 , / ) 438 00 270 I a 1 , NDRMAS 440 WRITE I 2 . 2 3 0 ) I 101 44 1 230 FORMAT I 1 0 K , 'MASS HOt ' , 13) 442 WRITE 1 2 . 2 4 0 ) R M ( I ) . RC1 I I . R K I I ) 443 240 FORMAT ( 1 0 X . • M « • , E I 3 . S , 10X. ' C « ' . E 1 3 . S , l O X , ' K ' ' , E 1 3 . S ) 444 WRITE ( 2 . 2 5 0 ) N J O R ( I ) , R X O ( l ) , R V O ( I ) 445 2S0 FORMAT ( l O X , 'MASS ON J O I S T N O ' ' . 12, / , ' C O O R D I N A T E S ' , l O X , 445 1 • X " ' . 613.6. 5X . ' » " ' , E 1 3 . 6 ) 447 WRITE ( 2 , 2 6 0 ) V E L I I ) . T I M I M F ( I ) 444 2SO FORMAT ( ' I N I T I A L V E L O C I T Y * ' , E 1 3 . 6 . 5 X . ' C O N T A C T T I M E " ' . E 1 3 . 6 ) 449 270 CONTINUE 4S0 IF (NCAFS . E O . O) GO TO 310 45 1 WRITE ( 2 , 2 S O ) 452 280 FORMAT ( / / , ' G A P S ' . SX . * X ' . 7X, ' W I O T H ' ) 453 DO 290 I • 1, NGAPS 454 2SO WRITE 1 2 . 3 0 0 ) G A P X I I ) , CAP 1 I 1 4SS 300 FORMAT [ I X , 4 X . F 8 . 3 . 2 X , F 6 . 3 ) 456 310 CONTINUE 457 WRITE ( 2 . 3 2 0 ) 456 320 FORMAT ( 1 0 X , ' J O I S T P R O P E R T I E S ' , / ) 459 00 330 1 * 1 , NJT 460 330 WRITE 1 2 , 3 4 0 ) V M O D I I ) . SMOOI 1 ) , D N S T V ( I ) . CB( I I , B J I 1 ) . H J I I ) 46 1 340 FORMAT I ' E ' ' . 5 1 3 . 6 , 2 X , ' C . E 1 3 . 6 , 2 X , ' R H O a ' , E 1 3 . 6 , 2 X . ' C , 462 1 E 1 3 . 6 , 2X . ' B ' ' . F 7 . 3 . 2X . ' H • ' , F 7 . 3 ) 463 WRITE ( 2 , 3 5 0 1 NBC 464 3S0 FORMAT [ 'NO OF BOUNDARY C O N D I T I O N S ' ' . 13) 466 IF (NBC E O . 0) CO TO 380 496 DO 3SO I • 1 , NBC 467 350 WRITE ( 2 . 3 7 0 ) ( IBC( I . J I . J ' 1 . 2) 466 3 7 0 FORMAT ( ' B . C . ON J O I S T N O ' ' , 13 , SX , ' B . C . COOE N O ' ' , 12) 469 380 CONTINUE 470 IF I I N P T E E O . 2) CO TO 420 471 WRITE 1 2 , 3 6 0 ) NREPL 472 390 FORMAT I / , 'NO OF R E P L I C A T I O N S FOR S I M U L A T I O N ' ' , 17) 473 WRITE 1 2 , 4 0 0 ) E M I N , EMAX 474 4O0 FORMAT I S X . ' E M I N ' ' , E 1 3 . S . 10X, ' E M A X ' ' . E13.6, /) 475 WRITE 1 2 , 4 1 0 ) E O , E M , EK 476 4 1 0 FORMAT ( / , ' W E I B U L L D I S T R I B U T I O N P A R A M E T E R S ' , / , S X , ' E O « ' , E 1 3 . S 477 1 I 5X , ' E M ' ' . E13.6. 5X , ' E K ' ' . E 1 3 . 6 . / ) 476 420 CONTINUE 479 RETURN 460 END 46 1 C 462 SUBROUTINE ASMBLY 463 I M P L I C I T REAL * 61 A - H . O - 2) 444 COMMON / B 4 / E T A I 6 ) , HI6 ) 465 COMMON / B 5 / NM. N S T E P , NMAX, N J T , S, R L , 0 . A L P H A . B E T A , H J T M , 466 1 BJI 10 I . HJI 10 I 467 COMMON / B S / RKX, R K Y , RKV. R K C , OX. DY. OV. OC, R K P A L , RKPER, 466 I RKROT, RHOC, C P , C N P A L , CNPER, CN RO T 486 COMMON / S 7 / C A P X 0 I 5 I . G A P O I S I . C A P X l S ) , GAP I 5 I , E N L . X I N , NGAPS , 490 I N A I , NDISCR 49 1 COMMON / S S / DMASS I 19 . 19 ) , S T I F F ( 1 9 . 1 9 I . 0 A M P ( 1 9.19I 492 COMMON / B 9 / E J T . C J T . RHOJT, P L D , X I , X 2 . Y l , Y2 493 COMMON / B I O / R C ( S ) , R K I 5 ) , R M I S ) . R F 0 R C 1 I 5 ) , NDRMAS. N J 0 R I 5 I , 494 1 R X O 1 5 ) , R Y 0 I 6 ) 495 COMMON / B 1 1 / R S T 1 F I 1 5 , 1 3 6 ) , R O A M P I 1 5 , 1 3 6 ) 49 8 COMMON / B 1 2 / Y M 0 D I 1 O I , D N S T Y I I O I . C B I l O I . T H E T A . C R V I T Y , S M O D110) 497 COMMON / B 1 3 / GMASS( 3, 2584) , COAMP 13,2564 ) , AOAMP 13 .5032) , 496 1 GST IF 1 3 . 2 5 8 4 ) , A S T I F1 3 .5032 ) . DCPMA5 ( 3,2584 ) , F 0 R C E 1 I 3 . 1 3 6 499 COMMON / B 1 4 / P I , P I N 2 , P I N 4 , N E C L HB, NA, N A I . A J T I I O ) , R I Y ( I O ) . SOO 1 R I 2 I 1 0 ) . R I T I 1 0 ) . SO 1 COMMON / B I S / NLOAD, N L U . PLOAD,' S T 0 R E I 1 0 . 5 ) , N L J 0 I 1 O ) 502 COMMON / B 2 1 / NBC. I B C I 5 0 , 2 I S03 DIMENSION D M I 2 S 8 4 I , D K I 2 S 6 4 ) . D D I 2 S B 4 ) , V E C T R I 1 9 ) , RDRP1 I 1 9 . 19 I , S04 1 R 0 R P 2 I 1 9.19I S05 IF 1NORMAS . E O . O) GO TO 10 506 CALL DRPMASINEO. NM, NMAX, N S T E P , R L , P I , 5) 507 10 CONT1NUE SOS DO 4 10 IK ' 1 . NM, NSTEP 509 DO 410 IN • 1 , I K , NSTEP 5 10 I KM ' IK 5 1 1 INM • IN 612 IF INSTEP . E O . 1) CO TO 20 5 13 I KM • ( IK • 1 ) / 2 5 1 4 INM * | I N • 1 ) / 2 5 1 5 20 IF ( IN . E O . IK) CO TO 40 5 1 6 IKO * ( INM - 1) » NMAX * 1 KM - INM * (INM * 1) / 2 6 1 7 DO 30 I ' 1 . NA 1 5 1 6 ADAMP1 I K O . I ) ' 0 . D O SIS 30 AST IFI I K O , I 1 ' 0 . 0 0 S20 IF (NDRMAS . E O . 0) GO TO 410 52 1 GO TO 300 S22 40 IKO B I KM S23 DO 50 I B 1 , NA S24 DMI I ) B 0 . O D O 525 DO 1 I ) B 0 . 0 0 0 526 SO OK 1 I 1 B 0 . 0 0 0 527 DO 60 I B 1 , NEO S26 SO F0RCE1 I IKO , 1 ) ' O.OOO 529 00 220 IE ' 1, NJT 530 I J s I 1 E - 1) • 13 53 1 DO TO I » 1, IS 532 DO TO J • 1 , 16 533 R D R P 1 ( I , J ) • 0 . 0 0 534 70 RDRP2I I , J1 • 0 . D O S35 IF INDRMAS . 8 0 . O) CO TO IIO S38 1 DR > 1 S37 SO IF I N J D R I I D R ) . 6 0 . IE) GO TO 90 538 1DR • IDR • 1 539 IF IIDR . G T . NDRMAS) GO TO I I O S40 GO TO SO 54 1 so ID • ( IDR - 11 • NMAX • I KM S42 00 IOO I • 1, 19 S43 DO IOO J • 1 , 19 S44 R D R P I ( I . J ) • R D R P 1 I I . J ) • R S T I F I I O . I J «• I ) * R S T I F I I D . 546 1 I J • J ) / R K I I D R ) S46 i o o RDRP2I I , J ) • R D R P 2 I I . J ) • R O A M P I I D . I J • I ) • RDAMPI I D , I J • S47 1 J ) / R C I I D R ) S4S IDR • IDR • 1 549 IF IIDR C T . NDRMAS) GO TO 110 SSO CO TO SO 102 SSI 1 10 CONTINUE SS 2 E JT • TM0DI1E) S S I C JT • S M O D I I E ) SS 4 RHOJT • ONSTV( IE ) BBS KLO a 0 sss I LO • 1 8 5 7 I' (NLOAD N E . 01 SO TO 130 SSS XI B STORE 1 I E , 1 1 s s s X2 B STORE I I E . 2 ) s s o V1 B S T O R E ( I E . 3 ) SSI V2 B S T O R E ! I E . 4 ) SS2 IF (NLU . E O . 1) SO TO 120 S S3 IF ( I E . E O . 1) Y l B 0 . O D O S 6 4 IF ( I E . E O . N J T ) Y2 B O .ODO SSS 1 20 PLD B S T 0 R E I I E . 5 ) BBS CO TD ISO 8 S 7 130 IF ( N L J O I I L O I E O . IE . O R . NLU . E O . 1) EO TO 140 SSS ILO B IL 0 • 1 SSS IF ( I L O . C T . NLOAD) 60 TO ISO S70 CO TO 130 S7 1 1 40 11 B S T O R E ( I L D . I ) S72 X 2 B STORE( I L O , 2 1 B73 Y1 B STORE 1 I L O , 3 1 S74 V2 B STORE I 1 1 0 , 4 ) S75 PLD B STORE( I L O , 5 I 8 7 S 1 SO CALL S T F M A S 1 V E C T R . I E . K L O , I N , I K , P I ) 577 SO TO 170 67 8 1 80 IF (KLO . E O . 1) GO TO 2 l O S78 PLD B O.ODO 860 CALL S T F M A S 1 V E C T R , I E , K L O . I N . I K , P I ) SSI 1 70 I J B (IE - 1) » 13 S 6 2 DD 200 J B 1 , 19 SS3 IF (KLO to. 11 GO TO ISO 8 6 4 DO 1 6 0 K B 1. J SSS JK B LHB • ( I J • K - 11 • J - K • 1 S66 D K I J K ) B O K I J K ) • S T I F F I J . K ) • R D R P I I J . K ) 8 8 7 OM(JK) B DMIJK) * D M A S S ( J . K ) 888 D O ( J K ) a DD1JK) * DAMP 1 J ,K ) » RDRP 2 ( J , K ) 589 1 SO CONTINUE 590 190 FORCE 1 1 1KO, I J • J ) a F0RCE1 ( 1K0, I J » J ) * V E C T R ( J ) 89 1 200 CONTINUE 592 KLO a 1 593 IF (NLOAD . E O . 0) GO TO 210 594 ILO a ILO • 1 595 IF ( I L O G T . NLOAD) CO TO 210 59 6 CO TO ISO S87 210 JK a LHB • I I J » 9 ) » 1 888 O K I J K ) a O K ( J K ) • E JT ' R I Y I I E ) • PIN4 • IK 4 588 DM(JK) a DMIJK) • RHOJT • 1 1 AJT1 I E ) • R L / 2 . 0 0 0 ) • I R I Y I I E ) ' 600 1 PIN2-IK>>2 ) ) SO 1 D O ( J K ) a O D ( J K ) * C B ( I E ) • I I A J T I I E ) » R L / 2 . D O ) * ( R I Y ( I E ) * 602 1 P 1 N 2 « I K " » 2 ) 1 603 JK a LHB • 1 1 J * 10] * 1 604 O K I J K ) a D K I J K ) • E J T • A J T I I E ) • P1N2 • IK 2 60S DMIJK] a DMIJK) • RHOJT • A J T I I E ) • RL / 2 . 0 0 0 606 D D ( J K ) a D D I J K ) * C S I I E ) • A J T I I E ) • RL / 2 .ODO 607 JK a LHB • ( I J * 11) * 1 606 D K I J K ) a D K I J K ) » E J T » R I 2 I I E ] • PIN4 » IK • • 4 609 DMIJK) a DMIJK) ' RHOJT • 1 1 AJT1 I E ) • R L / 2 . O D O 1 * ( R I Z I I E ) ' 6IO 1 P I N 2 » I K " 2 1 ] 8 1 1 D D I J K ) a O D f J K ) * C B ( I E ) * I I A J T ( I E ) > R L / 2 . OOO) * ( R I Z ( I E ) * 6 1 2 1 P I N 2 « I K " 2 ) ) 6 1 3 J K a LHB • 1 I J « 12] • 1 6 1 4 O K I J K ) a D K I J K ) * CJT « R I T I I E ) • PIN2 * IK « • 2 / ( S » " 2 I 6 1 5 DMIJK) a DMIJK) • RHOJT • I R I Y I I E ] • R I Z I I E ) ] • RL / 12.ODO-6 1 6 1 S<>2 ) 6 1 7 D D I J K ] a D D I J K ) • C B I I E ) » ( R I Y I I E ) • R I Z I I E ) ) » RL / 12. 6 1 6 1 O 0 O ' S ' * 2 ) 6 1 9 220 CONTINUE 620 IF (NBC E O . 0) CO TO 270 62 1 DO 2B0 I a 1, NBC 622 NE a I B C I 1 , 1 ) 623 NOOF a IBC1 I ,21 624 M a (HE • 1) * 1 3 * NOOF 625 J1 a 1 626 MM a M - 1 627 IF IM . S E . IS) J1 a M • 16 626 DO 230 J a J 1 , MM 629 JK a | LHB - 1 1 - 1J - 1 ) * M 630 DMIJK) a O .DO 63 1 O D I J K ) a O . D O 632 230 D K I J K ) a 0 . 0 0 0 633 Ml a M • 1 834 MS a M • 18 635 IF IM6 S T . NEO) M6 a NEO 836 DO 240 J a Ml , M8 637 JK a | L H B - 1) « IM - 1) • J 636 DMIJK) a O . D O 638 D D I J K ) a 0 . D O 640 240 D K ( J K ) a O.ODO 64 1 J K a {L HB • 1) a (M - 1) • M 642 DMIJK) a 1.DO 643 D O ( J K ) a 1.DO 644 D K I J K ) a 1.OO 648 FORCE 1 ( I K O , M l a O.ODO 64 6 IF (NORMAS E O . 0) GO TO 2SO 847 DO 2S0 IM a 1, NDRMAS 646 IKOR a ( IM • 1) • NMAX • IKO 84 8 R S T I F 1 1 K 0 R , M l a O . D O SSO R D A M P I I K D R . M ) s O . D O 651 250 CONTINUE 882 260 CONTINUE 883 270 CONTINUE 684 DO 2SO I • 1. NA 885 G S T I F I I K O , I ) a OKI I ) 888 G M A S S I I K O , I ) a OMII ) 887 2SO BOAMPI I K O . I ) a OD1 I ) 888 CALL DECMP (NEO . L H B , DM) 88 8 DO 290 I a 1, NA SSO 2SO D C P M A S ( I K O , I ) • DMI I ) « • 1 GO TO 410 162 300 CONTINUE 663 00 310 IE • 1, NJT 66< I J « ( IE - 1 1 » 13 665 DO 310 I • 1, IS 666 00 3 10 J • 1 . IS 667 ROUP 1 i I , J 1 • 0 00 666 310 RDRP2 1 I , J I > O DO 666 IP INDRMAS . E O . 0 | CO TO 350 670 I DR * 1 • 71 320 IP ( N J D R I I D R I . E O . IE) CO TO 330 872 IDR • IDR • 1 673 IF (IDR . C T . NORMAS) GO TO 350 874 CO TO 320 876 330 ID1 • 1 1 OR - 1) • NMAX • I KM 875 ID2 a (IDR - 1) • NMAX • INM 677 DO 340 l n l , IS 678 DO 340 J » 1, IS 679 R D R P I ( I . J ) « R D R P I ( I . J ) - R S T I F I I D 1 . I J • I) « R S T l F t 1 D 2 , 660 1 I J • J l / R K I I O R ] 66 1 340 R D R P 2 I I . J ) > R 0 R P 2 I 1 . J ) • ftOAMPMDI.IJ » I ) • R D A M P I 1 D 2 . 1 J * 682 1 J l / R C I I D R ) 663 IDR • IDR • 1 684 IF (IDR . G T . NDRMAS) GO TO 350 688 CO TD 320 666 360 CONTINUE 887 00 3 80 J • 1, IB 666 DO 3S0 K • 1 . 19 668 JK • ( I J » K - 1] « ( 2 - L H B - 1) • 1MB * J - K 690 A S T I F ( I K D . J K 1 • AST IF1 I K O . J K ) * R D R P 1 I J . K ) 66 1 ADAMP1 I K O . J K ) • ADAMP1 I K O . J K ) • R 0 R P 2 I J . K I 692 3BO CON TINUE 693 370 CONTINUE 694 IF INSC E O . 01 CC TO 400 695 DD 380 I • 1. NBC 696 NE • IBCI 1 .1) 697 NOOF > IBC( 1 , 2 1 698 M a INE - 1 ) • 13 + ND OF 688 Ml a M - 16 700 M2 • M • 18 70 1 IF (Ml . I E . 0) Ml • 1 702 IF IM2 . G T . NEO) M2 • NEO 703 00 380 J • M l , M2 704 JK s IM - 1) • I 2 * L HB • 1) • J • M • LHB 705 ADAMPI I K O , J K ) a O .DO 706 AST IF( I K O . J K ) a 0 . D O 707 JK a ( J - l l » ( 2 * L HB - 1 ( + M - J • LHB 706 AOAMPI I K O . J K 1 m O.DO 708 3BO AST IF1 I K O , J K ) * O .DO 7 IO 390 CONTINUE 7 1 1 400 CONTINUE 7 1 2 4 1 O CONTINUE 7 1 3 RETURN 7 1 4 END 7 1 5 C 7 1 G SUBROUTINE A X L R T N I N E O , L H B . NA1 7 1 7 I M P L I C I T R E A L * 6 ( A - H.O - Z) 7 1 6 COMMON / B 5 / NM. N S T E P . NMAX 7 1 9 COMMON / B I O / R C ( S ) , R K I 5 ) , R M 1 5 ) , R F 0 R C 1 I S ) . NDRMAS 720 COMMON / B 1 1 / R S T I P I 15 , 1361 , ROAMPI 15, 136) 72 1 COMMON / B 1 3 / GMASSI 3 , 2 5 6 4 ) , C0AMPf 3 . 2 6 6 4 ) , ADAMPI 3 , 5 0 3 2 ) , 722 1 GST I F ( 3 , 2S64 1 , A S T 1 F 1 3 . 5 0 3 2 1 . 0CPMAS13 . 2 5 64 | , F0RCE1 ( 3 , 136] 723 COMMON / l i t / S V E C 1 ( 3 , 1 3 6 ) , S V E C 2 ( 3 , 1 3 6 ) . R V E C 1 I 5 ) , R V E C 2 I S I . 724 1  S V E C 3 I 3 . 1 3 6 I , R V E C 3 I 5 ) 725 DIMENSION F 0 R C E 2 1 3 . 136 ] . DM1 2564 ) , X I 1 3 6 ) 728 DO 160 IK a 1 . NM, NSTEP 727 I KM a IK 729 IF (NSTEP . E O . 2) I KM a I I KM • 1) / 2 729 00 IO I a 1, NEO 730 IO F0RCE2( 1 KM, 1 ) a FORCE1 1 I KM, I ) 731 DO 11O IN a 1, NM, NSTEP 732 INM a IN 733 IF INSTEP . E O . 2) INM a ( INM • 1 I / 2 734 IF (INM C T . I KM) GO TO 20 736 IKO a (INM - 1) a NMAX * 1 KM - INM * ( INM • 1) / 2 736 GO TO 30 737 20 IKO a ( IKM - 1) • NMAX » INM - I KM • 1 1 KM • 1) / 2 736 30 DO 100 I a 1, NEO 738 J 1 a I - 16 740 J2 a I • 16 74 1 IF ( J l . L E . O) J l a 1 742 IF <J2 C T . NEO) J2 a NEO 743 SUM a O.ODO 744 DO SO J m J l , J2 745 TEMPI m S V E C l t l N M . J ) 748 TEMP2 « S V E C 2 I 1 N M . J ) 747 IF (INM .BO - IKM) CO TO SO 746 IF (INM G T . IKM) GO TO 40 748 I J a ( J • 1 J • ( 2 * L HB - 1) • LHS . I - J 760 GO TO 60 78 1 40 I J a | I - i ) • | 2 • L H B - 1) * LHB -» J - I 752 GO TO SO 753 SO CON T1NUE 754 IF I I . CT . J I CO TO 60 755 JK a (I . 1| a LHB • J • I • 1 786 GO TO 70 757 SO JK a ( J • 1| • LHB • I - J • 1 766 70 BUM a SUM • GST IF1 I K M , J K ) • TEMPI * GDAMPI I K M . J K ) > TBMP2 788 CO TO SO 760 • O SUM a SUM • AST I F ! I K O , 1JI ' TEMPI • ADAMPI I K O , I J ) ' TEMP2 761 SO CONTINUE 763 PDRCE2I IKM, I ) • F 0 R C E 2 ( I KM, I ) - SUM 783 too CONTINUE 764 1 IO CONTINUE 768 IF (NDRMAS . 1 0 . 0) BO TO 1 SO 766 DO 130 I a I, NEO 767 SUMI • O .DO 766 DO 120 J a 1. NDRMAS 768 JM a ( J • 1) « NMAX • I KM 770 120 SUM1 a SUM1 » R D A M P I J M . I I • R V E C 2 I J ) * R S T I F I J M . I ) a R V E C 1 I J ) 104 771 P0RCE2IIKM,I) • P0RCE2( I KM, I ) - SUM1 773 ISO CONTINUE 773 140 CONTINUE 774 00 1 SO 1 a 1 , NA 77« ISO 0M(I) • DCPMASIKM,1) 77« 00 1S0 I « 1. NEO 777 1 SO Kill • P0RCE2IIKM,I) 77» CALL SOLVINEO. LHB, DM, X) 779 DO 170 I • 1. NEO 7»0 170 SVEC3IIKM.II • III) 7»1 1 SO CONTINUE 782 IP (NDRMAS .EO. O) 60 TO 220 793 DO 210 I a 1 . NORMAS 794 SUM a BCIII a RVEC2II) • RKII) ' RVEC1III 799 00 2O0 IK a 1, NMAX 799 IKM a [1 • 1) • NMAX • IK 797 00 ISO J a 1, NEO 799 1 to SUM a SUM * RDAMPIKM.J) > SVEC2IIK.J) * RS T I F ( I KM , J ) • SVECM 799 1 I K . J 1 790 2O0 CONTINUE 79 1 RVEC3III a IRP0RC1III • SUM) / RMIII 792 210 CONTINUE 793 220 CONTINUE 794 RETURN 789 END 794 C 797 SUBROUTINE EPMATINMAX, NA, NAI, LHB. NEOI 798 IMPLICIT REAL*tlA - H.O • Z) 799 COMMON /BIO/ RCI5). RKI5I, RMISI, RP0RC1IS), NDRMAS aoo COMMON / I I I / RSTIPI15,136), RDAMP[15,136) 801 COMMON /B13/ CMASS13,2S64) , COAMP(3,2584) , ADAMP13,5032 I , 802 1 CSTIP I 3.2544) , AST IP(3,5032) 803 COMMON /BIT/ EPSTIF 1 3.2664 ) , APSTIFI 3.5032 1 , REPS TP|5) . 904 1 REFECT(15.1361 (OS COMMON /B23/ AO, Al 806 DIMENSION 0KI2564) 807 DO 60 IK a 1, NMAX 808 00 60 1N a 1 , IK 809 IF (IN .EO. 1K1 CO TO 10 8 1 O IKO a (IN • 11 • NMAX • IK - IN < (IN • l l / 2 8 1 1 60 TO 40 8 1 2 10 00 30 I a 1 , NA 8 1 3 20 DKII) a CSTIF(IK.I) • AO • 6MASSIIK.I) • A l • CDAMP(IK.I) 8 1 4 CALL DE CMP(NEO, LHB. DK) 91S DO 30 I a 1, NA 8 1 9 30 EFSTIFI IK. I ) a DK1 I 1 817 CO TO 60 8 1 8 40 00 50 I a 1, NA1 819 SO AFST1PIIKO,1) a A5T1FIIK0.1) * A l » ADAMPIKO.I) 820 to CONTINUE 82 1 IF (NORMAS .EO. 0) CO T 0 IOO 822 DO SO I a 1 , NDRMAS 823 REFSTPUI a RKIII .AO a RMl I ) • A1 » RCI1) 824 DO 60 J a 1, NMAX 825 IM a (I - 1) - NMAX • J 825 DO 70 K a 1, NEO 827 70 REFECTI 1M.K] a RSTIFIIM.Kl * Al * RO AMP I 1M,K1 828 to CONTINUE 829 to CONTINUE 830 IOO CONTINUE 83 1 RETURN 833 END 833 c 834 SUSROUTINE EPLOAO(NMAX, NEO, LHt) ass IMPLICIT REAL*6(A - H . O • Z l 838 COMMON /BIO/ RCI5). RKI6). RMIS), RFORC115), NDRMAS 837 COMMON /B11/ RSTIF( 15. 1 361 , RDAMP( 15,J36) 838 COMMON /B13/ CMASS(3,3584), GDAMP(3,2S841, ADAMP13.5032), 839 1 CSTIF 1 3,2584 | , ASTIF13.S0321 , OC PMAS13,2584 ) , F0RCE1 ( 3 , 136) 840 COMMON /B16/ SVEC113.1361, SVEC213.136). RVEC1IS), RVEC2I5), 84 1 1 SVEC3(3,136I. R'EC3IS) 842 COMMON / B I B / F0RCE213,136), RF0RC2I5) 843 COMMON /B23/ AO, A l , A2. A3 •• -844 00 10 I a 1 , NMAX 84 5 DO 10 J a 1, NEO 84 6 IO F0RCE2II,JI a P0RCE1II.J) 847 DO 120 IKM a 1. NMAX 844 00 110 INM a 1, NMAX 949 IF | INM CT IKM) CO TO 20 850 IKO a (INM - 1) » NMAX * IKM - INM * 1 INM - l | / 2 85 1 60 TO 30 852 20 IKO a ( 1 KM - 1) a NMAX . INM - IKM • (IKM • 1) / 2 853 30 DO IOO I a 1, NEO 854 Jl a I - 18 855 J2 a I • 18 asc IF (Jl LE. Ol Jl a 1 957 IF IJ2 .CT. NEO) J2 a NEO ass SUM a O.DO ass 00 SO J a Jl , J2 sso TEMPI a SVECI(INM.J) • AO * SVEC3(INM,J) a A3 • SVEC3(INM SS 1 1 J) " 2.OO 882 TEMP 2 a SVECl(INM.J) • Al • SVEC2IINM.J) • 2.0O.SVEC3I INM 883 1 J ) • A3 894 IP (INM .EO. IKMI CO TO SO aas IF (INM CT. I KM| CO TO 40 sss IJ a (J - 1) • I2*LHB - 1) • LHB • I - J 887 60 TO SO ssa 40 IJ * II - 1) a (2s L HB • 1) • LHB • J - I aaa 60 TO SO 870 SO CONTINUE 871 IF II CT. J) 60 TO SO 872 IJ a (I - 1 | • LHB • J • I • 1 873 GO TO 70 874 ao I J • U • il • m » • I • j • 1 476 70 SUM a SUM • GMASSI IKM, IJI a TEMPI » GDAMP(I KM. IJ) * TBMP2 875 60 TO 80 877 SO SUM a SUM * ADAMP( IKO, IJ) a TEMP 2 876 to CONTINUE 879 FORCE 2( I KM, I ) a P0RCE2I I KM, I ) • SUM aao 10O CONTINUE a s i 1 10 CONTINUE 6 8 2 130 CONTINUE a s s IF (NORMAS . E O . 0) CO TO 18 0 a a < DO ISO IK • 1 , NMAX a a s 00 140 I • 1 , NEO 886 SUM • O .DO a a i DD 130 J • 1, NDRMAS a a a J H > ( J - 1) » NMAX • IK a a a TEMP » At • R V E C l t J ) • 2 . DO • R « E C 2 I J ) » A3 • R V E C 3 I J ) a a o 130 SUM > SUM • R D A M P ( J M . I ) • TEMP 8 8 1 140 P 0 R C E 2 I I K . I ) • P 0 R C E 2 I I K . I ) • SUM 892 1 80 CONTINUE 883 00 1 SO I * 1 . NORMAS 688 SUM • O . D O 88S 00 170 IK • 1 , NMAX 88 6 I J > (I - 1 ) » NMAX • IK 697 DO 1 60 J • 1 . NEO 896 1 80 SUM « SUM - R D A M P I I J . J ) « (A 1 • S V E C 1 ( I K , J ) • 2 . D O » S V E C 2 ( I K . J ) * 698 1 A 3 « S V E C 3 I I K . J ) ) 900 1 70 CONTINUE 60 1 SUM • SUM » RMII ) • ( A 0 » R V E C 1 ( I ) * A 2 ' R V E C 2 I I I • 2 . 0 0 * R V E C 3 ( I ) ) 902 1 * R C ( I ) • ( A 1 > R V E C 1 ( I ) » 2 . D O P R V E C 2 1 I 1 • A 3 * R V E C 3 ( I I ) 903 R P 0 R C 2 I I ) • R P O R C I ( I ) * SUM « 0 4 1 80 CONT1NUE 906 1S0 CONT1NUE 905 RETURN 907 END 906 c 909 SUBROUTINE S O L V E I N M , NMAX. NEO. L H B . T O L , E L T I M , NA, NORMAS) 910 I M P L I C I T REA L * 6 (A - H . O • Z) 9 1 1 COMMON / B 1 6 / 5 V E C 1 1 3 , 1 3 6 ) . S V E C 2 1 3 , 136) , R V E C I ( S ) . RVEC215 1. 913 1 S V E C 3 1 3 . 1 3 6 ) , R V E C 3 1 51 9 1 3 COMMON / B 1 7 / E P S T I P ( 3 , 2 S 6 4 ) . APSTIP I 3 , 5 0 3 3 1 . R E F S T F ( S ) , 9 1 4 1 R E F E C T ( 1 5 , 1 3 5 1 9 1 5 COMMON / B 1 6 / F0RCE2I 2 , 13 6 I , R F 0 R C 2 ( 6 ) 9 1 6 COMMON / B 2 3 / AO, A l , A 2 , A 3 , A 4 , A S . A 6 . A 7 . A t . A9 9 1 7 DIMENSION S V E C 4 1 3 . 1361 . S V E C 5 [ 3 , 1 36 ) , R V E C 4 ( 5 ) , R V E C 5 I 5 ) . X X I 1 3 6 ) . 9 1 6 1 1 X1 13 6 ) , DKI 2584) a 18 DO 10 I • 1. NMAX 820 DO 10 J • 1. NEO 821 10 S V EC 4( I ,J) s O .DO 822 DO 60 IK • 1, NMAX 823 00 20 I 4 1, NA 824 JO DK1 I 1 • E F S T I F ( IK. I ) 926 DO 30 I « 1. NEO 828 30 X I 1 ) • FORCE2 t 1K , 1 ) 927 CALL S O L V ( N E O . LHB. OK. X) 926 DD 40 I » 1 , NEO 929 40 5 V E C 4 ( I K , I ) a X I I ) 930 SO CONTINUE 93 1 NITER • O 9 3 3 IF (NORMAS . E O . 0 . A N D . NM . EO . 1 ) 60 TO 320 933 IP (NDRMAS . E O . O) CO TO 70 934 00 60 I » 1 , NORMAS 935 s o R V E C 4 I I ) • R F 0 R C 2 I I I / R E F S T F I I I 935 70 CONTINUE 931 80 NF L AC ' 1 938 DO 230 I KM o 1 , NMAX 9 3 9 DD 90 I • 1, NEO 840 SO X I I I ) • F 0 R C E 2 I I K M , I ) 94 1 DO 1 SO INN * 1 , NMAX 942 IF (INM . E O . IKMI CO TO ISO 943 IF I INM . C T . IKM) GO TO 100 944 IKO * (INM - 1 } * NMAX • IKM - INM * (INM * 1 ) / 2 945 CO TO 1 1 O 94 6 i o o IKO • ( IKM - 1 ) • NMAX * INM - IKM • ( I KM * 1 1 / 2 941 1 10 DO ISO 1 • 1 , NEO 946 J 1 • I - 1 a 9 4 9 J2 » I • 16 950 IF ( J l . L E . 0) J l • 1 95 1 IF ( J2 C T . NEO) J2 • NEO 952 SUM * O .DO 863 DO 140 J • J 1 , J2 954 TEMP * S V E C 4 ( 1 N M , J ) 955 IF [ IN . C T . IK) CO TO 120 866 I J " ( J • 1 ) • I 2 * L HB - 1 1 » LHB * I * J 9B1 CD TD 130 956 120 I J B II - 1) « ( 2 * L H B - 1) * LHB • J • I 9 5 9 1 30 SUM • SUM • A F S T I F ( I K D , I J ) • TEMP 990 1 40 CONTINUE 96 1 X X I I I • X X I I ) - SUM 992 ISO CONTINUE 963 1 SO CONTINUE 9 64 IF (NDRMAS . E O . 0) GO TO ISO 865 00 18 0 I « 1 , NEO 966 SUM 1 • O . D O 881 DO 170 J • 1, NORMAS 8 6 6 JM • ( J - 1 ) • NMAX * I KM 968 1 70 SUM 1 • SUM1 • R E F E C T I J M , I ) • R V E C 4 I J ) 670 X X I I ) » XX ( I ) - SUMI 87 1 1 80 CONTINUE 672 190 CONTINUE 873 " DO 200 I > 1, NA 574 200 OK I I I • E F S T I F ( IKM, I ) S7S DO 210 I • 1 , NEO 876 210 I I I ) • X X I I I 877 C A L L S O L V I N E O , L H B , OK. I I 876 DO 220 I - 1. NEO 876 220 SVEC51 IKM, I ) • X1 I) SSO 230 CONTINUE SSI IF 1 NORMAS . E O . 0) CO TO 270 a a 2 DO 280 I • 1 , NDRMAS 883 SUM • O . O O 8 8 4 DO 2SO IK • 1. NMAX s a s I M P II - 1 ) • NMAX • I K • t e 00 240 J • 1. NEO s a l 240 SUM • SUM • R E F E C T ( I M , J ) « < V E C 4 ( I K . J 1 t a t 280 CONTINUE a a a R V E C S I I ) • ( R F 0 R C 2 I I ) - SUM) / R E P S T F I I ) a a o 280 CONTINUE 106 • t l 270 CONTINUE • • 2 DO 290 IK a 1, NMAX 993 TEMPI a O .DO 994 TEMP3 a O . D O S9S DO 2S0 J a 1, NEO 999 TEMPI a TEMPI • S V E C M I K . J I aa 2 997 TEMP2 a TEMP 2 • S V E C 4 1 I K . J ) aa 2 99 t 2 t O S VEC4 I I K . J ) a S V E C S I I K . J ) 999 TEMPI a DSORT t TEMP 1 ) lOOO T BMP 2 a D S O R T I T E M P 2 ) lOOt EPS1 • OABS( (T BMP 1 - T E M P 2 ) / T E M P 1 I 1002 NAC a 1 1003 IF (EPS 1 . C T . T O L ) NAC a o 1004 NF L AC a NF L AC a NAC tOOS 290 CONTINUE 1005 NITER a NITER • 1 1007 IF INFLAC E C . 1) CO TO 300 100S CO TO 80 1009 300 IF [NDRMAS . E O . O) CO TO 320 1010 T BMP 3 a O .DO 1011 TEMP 4 a O . D O 1012 DO 310 I a 1. NORMAS 1013 TEMP 3 a TEMP 3 • R V E C S ( I ) a . 2 1014 TEMP 4 a TEMP 4 + RVEC4(1) aa 2 1015 310 R V EC 4 ( I ) a R V E C 5 ( I ) 1019 TEMPS a D S 0 R T ( T E M P 3 ) 1017 TE MP 4 a D S O R T ( T E M P 4 ) 1018 EPS 2 a 0 A B S I I T E M P 3 • T E M P 4 ) / T E M P 3 I 1019 NAC a 1 1020 IF ( EPS 2 . C T . T O L ] NAC a o 1021 NF L AC a NF LAC « NAC 1022 IF I N F L A C . E O . 1) CO TO 320 1023 CO TO SO 1024 320 CONTINUE 1025 WRITE [ 6 , 3 3 0 ] E L T I M , NITER 1026 330 FORMAT ( 'NO OF I T E R A T I O N S FOR SOLN AT T1 ME a ' , F 7 . 5 , 3 X , ' A R E ' , 13) 1027 DO 340 I a 1, NMAX 1026 DO 340 J a 1. NEO 1029 TEMPI a S V E C l l l . J ) 1030 TEMP 2 a S V E C 2 I I . J ) 1031 TEMP3 a SVEC3I I . J ) 1032 TEMP 4 a S V E C 4 I I . J ) 1033 S V E C 3 I I . J ) a A4 > (TEMP4 - T E M P I ) + AS " TEMP2 • A8 » TEMP 3 1034 S V E C 2 I I . J ) a TEMP 2 • A7 • I S V E C 3 I I . J I » TEMP3) 1035 S V E C l l l . J ) a TEMPI . AS • TEMP2 • A 8 • I S V E C 3 ( I , J ) • 2 .DOa 1036 1 TEMP3 ) 1037 340 CONTINUE 1038 IF [N D RMAS . E C OI CO TO 360 1038 DO 350 I a 1, NDRMAS 1040 TEMP 1 a RV E C1 I I ) 104 1 TEMP2 a RVEC2I I ) 1042 TEMPS a RVEC3I I 1 1043 TEMP 4 a R V E C 4 I I ) 1044 R V E C 3 I 1 ) a A4 • ( T E M P * - T E M P I ) * AS • TBMP2 + A6 • TEMP3 1045 R V E C 2 I 1 ) a TEMP2 * AT • ( R V E C 2 ( I ) * TEMPS) 1046 R V E C I ( I ) a TEMPI • AS » T BMP 2 • A t • ( R V E C 3 I I ) * 2 .DO>TEMP3) 1047 3S0 CONTINUE 1046 360 CONTINUE 1049 RETURN 1050 END 105 1 C 1052 SUBROUTINE R E S U L T ( P E R I O D , E L T I M , NORMAS, P I ) 1053 I M P L I C I T REA L * 8(A - H , 0 - Z) 1054 COMMON / B 5 / NM. N S T E P . NMAX. N J T . S. RL 1055 COMMON / B I O / RCI 5 I . R K I S ) 1056 COMMON / B 1 6 / SVEC 1 I 3 . 136) . S V E C 2 I 3 , 136) , R V E C 1 ( 5 ) , R V E C 2 ( 5 ) , 1057 1 S V E C 3 1 3 , 136) , R V EC 3 ( 5 ) 1058 COMMON / B 2 2 / N D E F J T ( I O ) , 0 E F M ( 1 0 , 2 ) , NPNTS 1059 COMMON / B 2 4 / W l , W2, W3 1060 COMMON / B 2 5 / S C A L E I 3 ) . A M P L 1 . OC CUR 1 , AMP L 2, 0CCUR2 1061 DIMENSION GRAPH 1 ( 1 6 ) , G R A P H 2 I 1 6 ) , CRAP H3I 16) 1062 CRAPH1 I 1 ) a E L T I M • S C A L E 1 1 ) 1063 C R A P H 2 I I ) a E L T I M • S C A L E I 1 ) 1064 GRAPHS! 1) a E L T I M a S C A L E ! 1 ) 1065 DO 10 I a 1, NPNTS 1068 IE a N O E F J T I I ) 1067 XO a DEFMI1,1) 1066 ' VO a O E F M I 1 , 2 ) 1069 XIO a VO * 2 . D O / S 1070 CALL V E R O I S d , NM, N S T E P , I E , IO , X I O , RL , P I ) 1071 GRAPH 1 ( I » 1) a Wl a S C A L E I 2 ) 1072 C R A P H 2 ( I • 1) a W2 a S C A L E I 2 ) 1073 GRAPH 3 I I • 1 ) a W3 a S C A L E I 2 ) 1074 IO CONTINUE 1075 L a NPNTS • 1 1076 IF INORMAS . E O . O) GO TO SO 1077 DO 30 1 a 1, NORMAS 1076 WRITE 13 ,201 I , R V E C M I ) , R V E C 2 I I 1 . R V E C 3 I I ) 1078 20 FORMAT ( / , ' S O L U T I O N FOR DROPPED M A S S ' . SX, 13, / , S I , 1060 1 ' D I S P L A C E M E N T ' , I S X , ' V E L O C I T Y ' , 1SX, ' A C C E L E R A T I O N ' , / , l O t l 2 S X , 3 ( 5 1 3 . 6 , 1 2 1 ) , / ) 1082 CRAPH1(L * I) a R V E C 1 I I ) • S C A L E I 2 ) 1083 CRAPH2IL » I) a R V E C 2 I I ) a S C A L E I 2 ) 1084 G R A P H S ! L • II » R V E C 3 I I ) a S C A L E I 2 ) 1086 30 CONTINUE 1066 IF ( E L T I M . C T . P E R I O D ) GO TO 50 1067 F a RK[NDRMAS) a S C A L E I 3 ) a (GRAPH 1 IL * NDRMAS) - C R A P H I ( L I ) * RCI 1056 1NORMAS) a S C A L E I 3 ) a ( C R A P H 2 I L • NORMAS) - C R A P H 2 ( L ) I 1088 WRITE 17 ,401 C R A P H I ( I ) , F 1090 40 FORMAT I F 8 . B , I X . E l l . 4 ) 1081 SO CONTINUE 1082 K a NPNTS • NORMAS * 1 1063 WRITE ( 3 . 6 0 ) C R A P H I ( I ) . I GRAPH 1 (NI . N » 2 . K I 1084 WRITE ( 4 , 6 0 1 G R A P H 2 ( 1 ) , ( C R A P H 2 ( N 1 . N a 2 , K ) 1085 WRITE 1 5 , S O ) GRAPHS( 1 ) , (GRAPHS INI . N a 2 . K ) 1098 SO FORMAT I P S . 5 . IX , S I E 1 1 . 4 . 1 X ) ) 1057 IF ( GRAPH 1 (2 ) . GT . AMP L 1 ) GO TO SO l O B t IF I AMP L1 . G T . O . D O ) 60 TO 70 1088 CO TO 1O0 1 100 70 IF (GRAPH 111) . L T . ( 2 . B O O " O C C U R 1 ) ) CO TO 10O 107 I10 t IP ( G R A P H 1 I 2 ) G T . AMP 1.2] GO TO AO 1102 CO TO 100 1 103 AO AMP L 2 • G R A P H 1 ( 2 ) 1104 0CCUR2 > GRAPH1I1 ) 1105 CO TO 100 1106 IO AMP LI • GRAPH 1 (2 ) I I O ? OCCUR 1 a GRAPH 111) 1106 100 CONT1NUE 1 106 RETURN 1110 ENO 1111 C 1112 SUBROUTINE O M A T I I I ) 1111 I M P L I C I T REA L a 8 I A - H . O - Z) 1114 COMMON / B 2 / A I I 6 . 1 6 ) 1115 COMMON / B 3 / R M ( 6 , 1 8 ) . R M 1 ( « , 1 9 ) 1116 COMMON / B 4 / E T A I 6) , HI6 ) 1117 DO 10 I * 1 , 19 1118 00 10 J • 1, I t 1116 A l I . J ) a o . D O 1120 IO CONTINUE 112 1 DO 40 I a I . 6 1 122 00 30 J • I . 19 1 123 P1 • RMl 1 . J1 1124 IP ( I I . E O . 2 . O R . II E O . 4) PI • R M 1 I I . J ) 1125 IP (P1 . E O . O . O O O ) CO TO 30 1126 00 20 K a 1, 19 1 127 P2 " RMl ( I . K I 1126 IP ( I I E O . 1 OR. II E O . 4) P2 « R M I I . K ) 1129 IP IP2 . E O . O .OOO) CO TO 20 1130 A l J . K ) • A I J . K ) • PI » P2 • H I I ) 113 1 20 CONTINUE 1132 30 CONTINUE 1133 40 CONTINUE 1 I 34 RETURN 1135 END 1136 C 1137 SUBROUTINE O E C M P I N . L H S . A) 1136 I M P L I C I T R E A L * 8 (A • H.O - Z) 1139 DIMENSION A I 2 S 6 4 ) 1140 C * A IS STOREO COLUMNWISE* 114 1 K6 a LHB • 1 1142 C • D E C O M P O S I T I O N * 1 143 TEMP B A( 1 ) 1144 TEMP 8 DSORTITEMP) 1145 A l t ) 8 TEMP 1146 00 10 I 8 2, LHB 1147 10 A ( I ) s A l l ) / TEMP 1 148 00 80 J 8 2 , N 1149 J1 a J - 1 1 150 I JO * LHB * J • KB 115 1 SUM a A I I JO ] 1152 KO a 1 1153 IP I J . G T LHB) KO a J - KB 1 154 00 20 K a KO, J l 1155 JK a KB • K * J - KB 115 6 TEMP a A ( J K) 1 157 20 5 UM a SUM - TEMP * • 2 t ! 5 6 A I 1 J D ) 8 DSORT(SUM) 1159 DO 50 I a 1 , KB 1180 I I a J • I 116 1 KO a 1 1162 IF I I I G T . LHB) KO a II KB 1 163 SUM a A l 1 JO • II 1164 IF II . E O . KB I GO TO 40 1165 00 30 K a KO, J1 1166 JK a KB • K * J " KB 1167 IK a KB • K • 11 - KB 1168 TEMP a A I J K ) 1169 30 SUM a SUM - A I I K I • TEMP 1170 40 A I I J D • II a SUM / A I I J D I 117 1 SO CONTINUE 1172 SO CONTINUE 1 1 73 RETURN 1174 END 1 176 C 1176 SUBROUTINE S O L V l N , L H B , A , B) 1177 I M P L I C I T RE A L * 8 I A • H . O • Z) 1176 DIMENSION A I 2 S 8 4 ) , B I 1 3 6 I 1179 C 'FORWARD S U B S T I T U T I O N * 1160 KB 8 LHB • 1 118 1 TEMP a A l l ) 1 182 BI 1 ) a BI 1 ) / TEMP 1 163 00 20 I a 2 . N 1184 I 1 a I • 1 1185 KO a 1 1186 IF (I C T . LHB) KO a I - KB 1167 SUM a BI I I 1166 I I a LHB • I • KB 1189 DO 10 K a K O , 11 1 1 BO IK a KB * K * I - KB 119 1 TEMP a A ( I K I 1192 10 SUM a SUM • TEMP * B IKI 1193 B1 1 ) " SUM / A | I I | 1184 30 CONTINUE 1116 C • S A C KWARD S U B S T I T U T I O N * 1196 NI a N - 1 119? LB a LHB • N - KB 1 198 TEMP a A l LB I 1199 B(NI a B IN) / TEMP 1200 00 40 I a 1, N1 1201 I 1 a N - I • 1 1202 N1 a N • 1 1203 KO a N 1204 IF (I G T . K l ) KO a NI • KB 130S SUM a B I N I ) 1206 II a LHB * NI • KB 1207 DO 30 K a 11 , KO 1206 IK a KB * NI * K - KB 1209 TEMP » A l I K ) 1210 30 SUM ' SUM - TEMP a B I K ) 108 121 1 B I N ! I a SUM / A l 1 I ) 1212 40 CONTI NUB 12 13 RETURN 1214 END 1215 C 1216 SUBROUTINE GE NMT X 1217 I M P L I C I T REAL * 8 IA - H . O - I ) 1216 COMMON / B I / R M O O I 1 S . 1 9 ) . R M 2 2 I 1 9 . 1 9 ) , R M 0 2 ( 1 9 , 1 9 I . R M 2 0 I 1 S . 1 S ) . 1216 1 R M 3 3 I 1 9 . 1 9 ) . R M S 9 I 1 9 . 1 9 ) . RMS 6 ( 1 9 . 1 9 ) , RM63( 19, 191 , 1 220 3 R M 4 4 I 1 9 . 1 9 ) . R M 4 S I 1 9 . 1 9 ) . R M 5 4 I 1 S . 1 9 ) , R M S S ( 1 9 , 1 9 ) , 1 22 1 3 RMl 1 ( 19 , 19) 1222 COMMON / 8 2 / A » ( 1 9 . 1 9 ) 1223 COMMON / B 4 / E T A I 6 I , HIS ) 1326 COMMON / B 3 / R M I S . I S ) , R M 1 ( S , 1 9 ) 1225 CALL ZERO 1226 c MO AND M3 MATRICES 1227 DO 10 I • 1, 6 1 226 ETA2 • ETA 1 I ) • • 2 1229 ET A3 • ETA I I ) • • 3 1230 ETA4 a ETA 1 I) a . 4 1231 ETAS • ETAI1 ) • • S 1233 R M I I . 1 I • E T A 2 . - S . O O • E T A 3 / 4 . 0 0 - E T A 4 / 2 . 0 0 * 3 . 0 0 • ETAS / 4 . 1233 1 00 123* R M I I . 1 4 ) • E T A 2 • 5 . O O • E T A3 / 4 . D O - E T A 4 / 2 . 0 0 * 3 . 0 0 • ETAS / 1 235 1 4 . DO 1 236 R M I I . 1 0 ) • 1 . 0 0 - 2 . 0 0 • ETA2 * ETA4 1337 R M I I . 2 ) « I E T A 2 - ET A3 • ETA4 . ETAS I / 6 . DO 1236 R M I I . 1 B I • ( - E T A 2 - ET A3 • ETA4 • E T A S ) / S . O O 1 239 R M I I . 9 ) • I ETA I I ) - 2 . D O . E T A S * E T A S ) / 2 . D O 1 2 * 0 RMl I I , 1 ) * 2 . 0 O - 1 S . D O » ETAI 1 ) / 2 . D O - 9 . 0 O • ETA2 * 1 5 . 0 0 • ETAS 12*1 R M 1 I I . 1 4 ) • 2 . D O * I S . D O . E T A I I ) / 2 . 0 O - 9 . D O > ETA2 - 1 S . 0 0 i • 12*2 1 ETA3 12*3 R M 1 1 I . 1 0 ) • - 4 . D O . 1 2 . D O * ETA2 1 3*4 • M i l l , 2 ) . 1 2 . D O - 6 . D O ' E T A 1 I ) - 1 2 D 0 ' E T A 2 * 2 0 . D O . E T A 3 ) / 6 . DO 1 2 *5 R M 1 I I . 1 5 ) • I - 2 . O O - 6 . D O * E T A 1 I ) - 1 2 . 0 O . E T A 2 • 2 0 . D O . E T A 3 ) / 8 . DO 1 2 * 6 R M 1 I I . 9 ) • 1 - 1 2 . D O ' E T A | I | . 2 0 . D O . E T A 3 ) / 2 . D O 1 2 * 7 1 o CONTINUE 1246 CAL L D M A T I I ) 1249 DO 20 I . 1 . 19 1 2SO 00 20 J . 1 . 19 1251 20 RMOOI I , J) • AA1 I , J | 1253 CALL 0MATI2 ) 1 253 00 30 I » 1, 19 1 254 DO 3 0 J . 1, 19 1 255 30 RM221 I . J I • AA1 I . J I 1256 CALL 0MATI3 ) 1 257 0 0 4 0 1 . 1 . 19 1 256 00 40 J . 1 , 19 1 259 40 RM021 1 . J ) ' AA1 I . J ) 1260 CALL 0MATI4 ) 1261 DO SO I . 1. 19 1262 DO SO J . 1. 19 1263 SO RM20I I . J ) a A A ( I . J ) 1264 CALL ZERO 1265 c M3 AND M6 MATRICES 1266 DO BO 1 . 1 . 6 1267 ETA2 • ETAI I 1 " 2 1266 ETA3 * E TA( I 1 » " 3 1269 ETA4 a E TA( I ) * • 4 1 270 R M I I . 3 ) « - 3 . 0 0 • E T A ( I ) / 4 . D O . E T A 2 . E T A3 / 4 . D O - E T A 4 / 2 . DO 1 27 1 R M ( I . I S ) . 3 . OO . E T A ( I ) / 4 . D O . E T A 2 - ETAS / 4 . D O - E T A 4 / 2 . DO 1 272 R M ( I , 7 ) B 1 . D O - 2 . D O • ETA2 * ET A 4 1 273 R M I 1 . 4 ) B ( - E T A I I ) . ETA2 . E T A3 - E T A 4 ) / 6 . 0 0 1 274 R M I I . I 7 ) 8 l - E T A I I ) • ETA2 * E TA3 » E T A 4 I / 8 . 0 0 1 275 R M 1 I I . 5 ) 8 - 3 . D O / 4 . D O . 2 . D O • E T A I I I * 3 . D O • E T A 2 / 4 . D O - 2 . DO 1276 1 * ETAS 1 277 R M 1 I I . 1 8 ) B 3 . 0 0 / 4 . D O . 2 . D O • E T A ( I ) - 3 . D O • ETA2 / « . D 0 - 2 . DO 1 278 1 . E TA3 1 279 R M 1 I I . 8 ) B - 4 . D O . E T A ( I ) . 4 . D O * ETAS 1 290 R M 1 I I . 6 ) B | • 1 . 0 O . 2 . D O . E T A | I ) » 3 . D 0 . E T A 2 - 4 . D O . E T A 3 1 / 8. DO 1 26 1 R M 1 I I . 1 9 ) • 1 - 1 . D O - 2 . D O ' E T A 1 I ) - 3 . 0 O . E T A 2 • 4 . D O . E T A 3 ) / 8 . 00 1 282 40 CONTINUE 1 263 CAL L DMATI 1 ) 1 284 DO 70 I 8 1 ( 19 1 285 00 70 J B 1, 19 1 286 70 RM33 1 I . J I B AA I I . J I 1287 CALL 0MATI2I 1288 DO 80 I B I , 19 1 289 DO 80 J B 1, 19 1 290 SO RM661 I . J1 8 A A ( I . J ) 1 29 1 CALL DMAT(3) 1292 0 0 6 0 1 8 1 , 19 1 293 DO 90 J 8 1, 19 1394 90 RM39I I . J ) B AAI I , J ) 1 295 CALL DMAT(4) 1296 DO 1OO 1 B 1, 19 1297 DO IOO J B 1, 19 1 298 too RM93I 1 , J l B AA( I , J ) 1 299 c MS AND M4 M A T R I C E S 1300 DO 1IO I 8 1, 6 1301 RMl I , S ) 8 R M ( 1 , 3 ) 1 302 R M l I , 3 ) 8 0 . O D O 1 303 RMl I , 14) 8 RMI 1 , IS) 1 304 R M l I , 1 6 ) 8 0 . 0 0 0 1 305 R M l 1 . 8 ) 8 R M l 1 , 7 ) 1 306 R M l I , 7 ) 8 0 . O D O 1 307 RMl I . 8) 8 R M ( 1 , 4 ) 1306 RMl I . 4 I • 0 . O D O 1309 R M l I , 1 S ) ' R M l I , 1 7 ) 1310 R M ( I , 1 7 ) 8 0 . O D O 131 1 R M l 1 1 , 3 1 8 R M 1 ( I , S ) 1312 R M l ( I , S 1 ' 0 . O D O 1313 RMl I I . IS 1 a RMl ( 1 . 1 6 ) 1314 RMl ( 1 . 1 6 1 a o . ODO 1315 RMl 1 I . 7 1 a R M l I I , 8 1 13 19 R M 1 ( 1 , 8 ) a 0 . O D O 1317 RMl ( I .4 I a RM1 I I . S) 1318 RMl I I , S ) ' O . O D O 1319 R M l 1 1 , 1 7 ) a R M l 1 1 . 1 9 ) 1320 RMl 1 I . IB 1 a O . ODO 109 1321 1 10 CONTINue 1322 CALL 0 M A T I 2 I 1323 00 120 I • 1, 19 1324 00 120 J • 1 , 1S 132S 120 RM44( 1 , J ) • AA1 I . J ) 1326 CALL 0MATI4 ) 1327 DO 130 I • 1. 19 1326 DO 130 J • 1 . 19 1326 130 RM4S ( I , J 1 • AA( I . J ) 1330 CALL DMATI3) 1331 DO 140 I • 1, 19 1332 DO 140 J • 1, 19 1333 1 « 0 RMS 4( I . J ) • A A ( I . J ) 1336 CAL L DMA T ( 1 | 1336 00 ISO I • 1. 19 1336 DO 160 J a 1 , 19 1337 1 60 RMS 5( I . J ) • A A ( I . J ) 1338 CALL ZERO 1338 C Ml MATRIX 1340 BO I I O I • 1, 6 136 1 ETA2 « ET A( II 2 13*2 E T A S • E T A ( I I * * 3 13*3 ETA4 • E T A I 1 ) 4 1388 R M I I . l l • 2 . D O • E T A I I ) - I S . D O * ETA2 / 4 . D O - 2 . D O • ETA3 * 1! 1386 1 DO • ETA4 / 4 . D O 13*6 R M I I . 1 4 ) > 2 . 0 O • E T A I I ) « 1 5 . 0 0 • ETA2 / 4 . 0 O - 2 . 0 O • ETA3 -1347 1 I DO * E T A 4 / 4 .DO 13*6 R M ( I . I O ) s - 4 . D O * E T A I I ) * 4 . D O * ETA3 1348 R M I I . 2 ) s I 2 . D O S E T A I I ) • 3 . D O S E T A 2 - 4 . D O B E T A 3 • 8 . 0 0 B E T A 4 ) / 13SO 1 I 00 135 1 R M I I . 1 S ) • ( - 2 . D O a E T A 1 I ) • 3 . 0 O » E T A 2 • 4 D 0 « E T A 3 • S . O O ' E T A A ) 1352 1 1 8 . 0 0 1353 RMt 1 .6 I * I 1 . D O - 6 . D 0 * E T A 2 * 5 . 0 O B E T A 4 ) / 2 . D O 1354 1 S O CONTINUE 1355 CALL 0MATI1 ) 1356 DO 170 I < 1, I S 1357 DO 110 J • 1 , IB 1358 1 70 R M 1 1 ( I , J I 4 A A ( I . J ) 1 358 RETURN 1360 END 136 1 C 1362 SUBROUTINE S T F M A S I V . I E . K L O , I N , I K . P I ) 1363 I M P L I C I T REAL * 6 IA - M.O - Z) 1364 DIMENSION V M S ) , AD ( 1 9 , 1 9 ) 1365 COMMON / B I / R M O O I 1 9 . 1 9 ) , R M 2 2 I 1 9 . 1 9 ) , R M 0 2 ( 1 9 , 1 9 ) , R M 2 0 I 1 8 . 1 8 ) 1366 1 R M 3 3 I 1 9 . I B ) . R M 6 6 I 1 9 . 1 9 ) , R M 3 6 I 1 9 . I 9 ) . R M 6 3 I 1 S . 1 9 ) . 1367 2 RM4 4 ( 1 9 , 1 9 ) . R M 4 5 I 1 9 . 1 9 ) , R M S 4 I 1 9 . 1 9 ) , R M S 5 ( 1 9 . 1 9 ) . 1366 3 R M t 1 1 1 9 , 1 9 ) 1369 COMMON / B 4 / E T A I 6 I . H | 6 | 1370 COMMON / B 6 / NM. N S T E P , NMAX, N J T . S , R L . 0 , A L P H A . B E T A , H J T M , 137 1 1 B J I 1 O ) , HJ 1 1 0 ) 1372 COMMON / B 6 / RKX, RKV, RKV, RKC , OX. DV , DV , DC, R K P A L , RKPER, 1373 1 RKROT, RHOC, C P . CNPAL . CNPER. CNROT 1 374 COMMON / B 7 / C A P X O I S ) , C A P O I S ) . C A P X I S ) . CAP 15) , E N L , X I N . NGAPS 1375 1 NAI , NDI S C R 1376 COMMON / B 6 / D M A S S ( 1 9 , 1 9 ) . S T I P F I 19 , 191 , D A M P I 1 9 . 1 9 ) 1 377 COMMON / B 9 / E J T , C J T , R H O J T , P L D , X I , X 2 , V I , V2 1376 IF I IE . CT . 1 ) GO TO 240 1 378 IF IKLO . C O . 1) CO TO 240 1380 P2 » P I 2 138 1 P4 a PI > * 4 1362 c • COVER • 1383 DO 1O I a 1, 19 1364 DO 10 J a 1 , 19 1365 DMASSI I , J 1 a 0 . 0 0 0 1366 DAMP I 1 , J ) B O.ODO 1387 10 S T I F F I I . J ) B 0 . 0 0 0 1388 F A C 1 B RKX * ( 1 K > * 4 ) » P4 a S / I 4 . 0 O B R L B » 3 ) 1368 FAC2 B RHOC * ( ( D * R L * S / 4 . 0 0 0 ) * I 0 s a 3 « I K • » 2 • P 2 • S / 1 4 8 . O D O * R L 1 ) ] 1 1390 PAC3 B F AC 2 • C P / RHOC 139 1 DO 20 I 8 1 , . 1 9 1392 DO 20 J B 1, 19 1393 20 AO 1 I , J1 B RMOO1 I . J I 1394 CALL ADD ( AD , FAC 1 . PAC2 , FAC3 ) 1395 PAC1 B RKV • 4 . 0 0 0 • R L / ( S * « 3 ) 1396 FAC2 B 0 . 0 0 0 1397 FAC3 B O .DO 1398 DO 30 I B 1 , I S 1 399 DO 30 J B 1 . 19 1 400 30 AD ( 1 . J1 B R M 2 2 ( I . J ) 140 1 CALL ADD I A D , F A C I . P A C 2 , F A C 3 ) 1 402 FAC1 B -RKV « P2 • IK s » 2 / ( S s R L ) 1403 FAC 2 B O .ODO 1404 FAC3 s O . D O 1406 DO 40 I s 1 , 19 1 406 DO 40 J s 1 . 19 1407 40 AD 1 I . J I s RM021 I . J ) 1 406 CALL ADO 1 A D , F A C I . P A C 2 , PAC3I 1409 DO SO I B 1, I B 14 10 DO 60 J s 1 , 19 14 11 50 A O ( I . J 1 s R M 2 0 I I . J 1 1412 CALL ADD 1 A D , F A C I . P A C 2 . F A C 3 ) 1413 FACI s RKC * 4 . O D O • P2 " IK *> 2 / I S s R L ) 14 14 FAC2 B RHOC • D ' • 3 • RL / 1 1 2 . D O B S ) 14 15 FAC3 s PAC 2 s CP / RHOC 14 16 00 SO I a 1, 19 1*17 DO BO J s 1 . 16 14 18 S O AD 1 1 ,J) s RM1 1 I I , J ) 1*18 CALL ADD 1 A D , F A C I . P A C 2 . FAC3I 1430 FACI s D X • S s P2 s IK SB 2 / I 4 . 0 D 0 B R L ) 1*21 FAC 2 B RHOC s 0 • RL * S / 4 . 0 0 0 1422 PAC3 B FAC2 s CP / RHOC 1*23 00 70 I s 1. 16 1424 00 70 J B 1 , 19 1*25 70 AO 1 I . J I s R M 3 3 I I . J ) 1426 CALL A D D I A D , PAC1 , P A C 2 . PAC3) 1*27 PAC2 s O.ODO 1 *2 * PAC3 a O.DO 1*28 FACI a D V a RL / S 1 « 3 0 DO S O I a 1, 19 110 1431 DO ( 0 J • 1, 19 1432 00 A O I l . J ) a R M S 8 I I . J ) 1433 C A L L A O D I A D . F A C I . F A C 2 . F A C 3 ) -1434 FAC2 • O.ODO 143S FAC3 • O .DO 1*36 FAC1 • -OV • IK • P l / 3 .ODO 1437 DO 90 I • 1. 16 1*36 DO IO J • I , II 1*39 90 A O I l . J I • R M 3 6 ( 1 , J ) 14AO C A L L A O O I A D . F A C I , FAC 2 , FAC31 1*4 1 00 1OO I • 1 . 19 1442 DO 100 J • 1, 19 1 * * 3 1O0 A D I I . J I • R M 6 3 I I . J ) 1 * * * CALL A D D ( A D , F A C I , F A C 2 , FAC3 I 1**6 FAC2 • 0 . 0 0 0 14 *9 FAC3 • O.DO 1447 FACI • DC • RL / S 1446 DO 110 I B 1 , 19 1449 DO 110 J B 1 , 1 1 1450 110 A O I l . J I B R M 4 4 I 1 . J I 1411 CALL A O O I A D . F A C I , F A C 2 , F A C 3 I 1*52 FACI B | K • PI / 2 . O O O • DC 1453 FAC2 B O.ODO 1*5* F A C 3 B O .DO 1455 DO 120 I B 1. 1 1 1*56 DO 120 J s 1. 11 1457 120 A D I I . J I B R M 4 5 I I . J ) 1*56 CALL A O O I A D , F A C 1 , F A C 2 . FAC3I 1459 DO 130 1 B 1 . 19 1*60 DO 130 J B I , 19 1461 130 A D I I . J I B R M 5 4 I I . J I 1482 CALL ADD I A O , F A C I , F A C 2 , FAC3 I 1463 FAC1 B (D C B R L * S / 4 . 0 0 0 I • IK BB 2 a P2 / | « l t . J | 1464 FAC2 B RHOC • D a RL • S / 4 . 0 0 0 1466 FAC 3 B FAC2 « CP / RHOC 1466 DO 140 I B 1 , 16 1467 DO 140 J B 1 . 19 1466 140 A O I l . J I B R M S B I I . J ) 1469 CALL ADO I A D , F A C I , F A C 2 , FAC 3 I 1470 C * N A I L I N G * 147 1 DO 1 BO I B I , 19 1472 A D I 1 , I ) B 0 . O O O 1473 AD I 2 , I I s o .ODO 1474 AO I 3 , I I B O.ODO 1475 AD 1 4 , 1 ) 4 0 . O D O 1476 A D ( 5 . 1 ) 3 0 . 0 0 0 1477 150 A D ( 6 , 1 ) B O.ODO 1476 A D I 1 . 7 ) B 1 . O O O 1 4 7 6 AD I 1 . 1 1 ) B - 1 .ODO 14BO A D I I . I O ) B - IK • PI » IHJTM • 0) / ( 2 . D O * R L ) 1461 A D I 2 . 7 ) B 1 . O O O 1462 A D ( 2 , 1 1 ) B - 1 . 0 0 0 1483 A D I 2 . 1 0 ) B - I N • P I * (HJTM • D) / ( 2 . D O * R L ) 1 4 8 4 A O ( 3 , 6 ) B 1 . ODO 1 465 AO 1 3 , 1 2 ) B • 1 . ODO 1 4 8 6 A D I 3 . 9 ) • -0 / l 2 . D O * S I 1467 AD I 3 . 13) B - H J T M / 12.D O B S ) I486 A D ( 4 , 8 ) s 1 . O D O 1469 A D ( 4 , 1 2 ) a - 1 . O D O 1 4 9 0 A D I 4 , 9 I B -0 / 1 2 . D O B S ) 1491 A D M . 1 3 ) B - H J T M / ( 2 . 0 O > S ) 1462 A D I S , 9 | B | . D 0 / S 1493 AO I 5 , 13) B - 1 .DO / S 1494 A O I 6 , S ) s l . O O / S 1 4 9 5 A D ( 6 . 1 3 ) B - I . 0 O / S 1496 IF INDISCR . 5 0 O) GO TO 200 1497 S P A R B O . O D O 1496 S P E R a 0 . 0 0 0 1499 DO 190 I B I , N A1 1 SOO X a X IN * I I - 1 ) * E N L 1501 IF INCAPS . E O . O) CO TO 180 1502 DD ISO J B 1, NCAPS 1 5 0 3 X 1 1 B G A P X ( J ) 1 5 0 4 X12 B C A P X ( J ) * C A P ( J ) 1605 IF ( X . C E . X I I . A N D . X . L E . X12) CO TO 170 1506 ISO CONTINUE 1807 GO TO 180 1608 170 CO TD 18 0 1505 ISO X B X • PI / RL 1810 SPAR B SPAR • D C D S ( I N B X ) • D C O S ( I K B X ) 1511 S P E R B S P E R • D S I N I I N B X ) • D S I N I I K s X ) 1512 190 CONTINUE 1513 CO TO 220 1514 200 I F ( IN . N E . IK) CD TD 210 1 5 1 5 SPAR B RL / ( 2 . 0 B E N L ) 1816 S P E R B RL / ( 2 . 0 B E N L I 1817 CO TO 220 1818 210 SPAR B O . O D O 1519 SPER B O . O D O 1520 220 SPAR B SPAR « RKPAL 1521 SPER B SPER > RKPER 1822 SROT B SPER • RKROT 1823 DO 230 1 8 1 , 1 9 1524 DO 230 J B 1 , 19 1525 S T I F F ( 1 , J ) B S T I F F ( I . J ) • SPAR a AO( 1 , I ) a AO 1 2 . J I * SPER a 1S2S 1 A D O . I I a A D I 4 . J ) • SROT * AO I S . I I a AO ( 6 . J I 1527 D A M P I I . J I a DAMP I I . J I * SPAR a CNPAL / RKPAL a AD I 1 , I ) • AD I 2 , 1528 1 J ) • SPER * CNPER / RKPER a AO I 3 . I ) a AO I 4 , J ) » SROT a CNROT / IS2S 2 RKROT a AD I 8 . I ) a A O ( 8 , J I 1S30 230 CONTINUE 1831 C a LOAD VECTOR a 1832 240 IF ( IN N E . IKI RETURN 1833 DD 3S0 I a 1, IS 1834 250 V I I ) a O.ODO 1S3S IF IPLD E O . O . D O ) RETURN 1838 PINL a PI a IK / RL 1837 PACTR a PLD a | V 2 • V I I a ( D C O SI P I N L aX t ) - DCOS 1 P I H L • X 2 ) ) / (2.DOB 1S38 1PINL) 1838 X I X B 2 . 0 0 B V 1 / S 18*0 XIV • Tl.OO a V2 / S 1S41 DO 2SO I • 1. 6 1 S42 XI • XIX • [XIV - X I X ) • ( 1 . O O . E T A I I I | / 2 . DO 1*43 112 • XI 2 1 ( 4 4 XI3 » XI • « 3 1 445 XI4 • XI 4 1S46 XIS » XI • • s 1547 V I I ) • V I I ) • ( X I 2 - 5 . O O a X I 3 / 4 . D O - X I 4 / 2 . D O » 3 . D O a X I S / 4 . D O ) • HI 1 544 I I ) 1549 V I 1 4 ) « V I 1 4 ) * (XI2 • Sax I 3 / 4 . D O - X 1 4 / 2 . D O - 3 . D O a X I 5 / 4 . D O I • H I I ) 1550 V I 1 0 ) • V I 1 0 I • 1 1 . D O - 3 . D 0 * X I 2 • XI4) • H I I ) 1(5 1 V I 2 I • V I 2 ) • IXI2 - X13 - XI4 • XIS ) • H I I ) / 6 . D O 1552 V | 1 S ) • V M S ) • I - X I 2 - XI3 • XI4 • X I S ) • H I I ) / 6 . DO 1553 250 V ( 9 ) • V I 9 I • IXI - 2 . D O » X I 3 • X I 5 ) > H I I ) / 2 . D O 1554 V I I I • V I I I » PACTR 1555 V | 1 4 ) • V I 1 4 ) • PACTR 1855 V | 1 0 ) • V I 1 0 ) > PACTR 1 557 V | 2 I • V | 2 I • PACTR 1658 V M S ) • V M S ) « PACTR 1 559 V I 5 I • V I S ) • PACTR 1560 RETURN 1561 END 1562 C 1663 SUBROUTINE A D O I A O . P AC 1 , P A C 2 , PAC3) 1564 I M P L I C I T REAL * 91 A - H . O - Z) 1 565 0 I ME NS ION AD I 1 9 . 1 9 ) 1666 COMMON / B 4 / D M A S S 1 1 9 , 1 9 ) . S T I P P ( 1 9 , 1 9 ) , D A M P M 9 . 1 S ) 1567 DO 10 I a 1 , 19 1568 DO 10 J • 1. 19 1569 10 S T I P P I I . J I • S T I P P I I . J ) . PAC1 " A O I I . J ) 1570 IP I DABS 1PAC2 - O . D O ) . L T . 1 . 0 E - 0 8 ) RETURN 1571 DO 20 I > 1, 19 1 672 00 20 J • 1 . 19 1673 D A M P ( l . J ) * O A M P t l . J ) * P AC 3 > A D ( I . J ) 1574 20 DMASS1 I . J ) • DMASS1 I . J I » PAC 2 > A O I I . J ) 1575 RETURN 1576 END 1877 c 1578 SUBROUTINE V E R 0 1 S I J , NM. N S T E P , I E . XO. X I O . R L , P I ) 1579 I M P L I C I T R E A L * 6 ( A - H.O • 21 1580 COMMON / B 1 6 / S V E C I ( 3 , 1 3 6 ) . S V E C 3 1 3 . 1 3 6 1 , R V E C 1 I 5 I . R V E C 2 I 5 ) . 158 1 1 S V E C 3 1 3 , 1 3 6 1 . R V E C 3 I 51 1582 COMMON / B 2 4 / Wl . W2 , W3 1583 DIMENSI ON RMO119) 1 564 I J > ( I E - 1) > 13 1565 J 1 « 1J • 1 1 566 J2 * I J * 2 1 587 J3 a I J • 6 1586 J4 • I J • 10 1689 J5 a I J • 14 1 S90 J 6 a I J * 15 159 1 CALL S H A P P N I X I O , RMO) 1592 Wl a O .OO 1593 W2 a 0 . D O 1594 W3 * O .DO 1595 DO 20 I tt 1 . NM. NSTEP 1596 IK s I 1587 1P (NSTEP . E O . 1 I CO TO 10 1596 IK « 1 I • 1 ) / 2 1599 10 W1 a wl • | S V E C 1 ( I K , J 1 )>RMO1 1 1 - SVEC1 1 I K . J 2 1 » R M O 1 2 I * S V E C 1 I I K , 1 900 1 J3 )>RMOI9) + S V E C 1 ( I K , J 4 I * R M O ( 1 O I * S V E C I ( I K , J 5 ) B R M 0 ( 1 4 } * 1 SO 1 2 SVECI 1 IK,J6)->RM0( 151 ) • DSINI I * P I * X O / R L ) 1 502 W2 a W2 • ISVEC2C I K . J 1 I * RMO( 1 I • SVEC21 I K . J 2 1 • R M O 1 2 ) * S V E C 2 I I K , 1 603 1 J3 )>RM0(9 ) * SVEC2I I K , J 4 I « R M O ( 10 I * SVEC2I I K , J 5 ) * R M 0 ( 14) * 1 604 2 SVEC2I I K , J 6 ) * R M 0 [ IS) ) » DSIN( 1 * P I * X O / R L ) 1 605 W3 a W3 • I S V E C 3 I I K . J l ) •RMO( 1 ) * SVEC3( I K , J 2 ) » R M O I 2) * S V E C 3 1 I K . 1 606 1 J 3 I * R M 0 I 8 ) * SVEC3I I K , J 4 ) * RMOI IO) • SVEC31 I K , J 5 1 • R M O I 14) * 1 607 2 S V E C 3 ( I K , J S I ' R M O ( 1 5 1 1 > D S I N I l * P I * X p / R L 1 1 608 20 CONTINUE 1 809 WRITE 1 2 , 3 0 1 J . W l . W2, W3 19 10 30 PORMAT ( / , ' S O L N AT FLOOR P O I N T ' , SX . 13 . / . 5X, ' D I S P L A C E M E N T ' , 19 11 1 15X, ' V E L O C I T Y ' , 15X, ' A C C E L E R A T I O N ' , / , S X , 3 1 E 1 3 . 6, 12X) , 16 12 2 / ) 1 S 1 3 RETURN 18 14 END 16 15 c 18 16 SUBROUTINE S H A P P N I X I O , RMO) 16 17 I M P L I C I T REAL * 6 (A - H.O - Z) 16 18 DIMENS I ON RMO119) 18 19 DO IO I ' 1 . 19 1 920 10 RMOII) • O . 0 0 0 182 1 XI02 • I I O • • 2 1 522 XI03 • XIO aa 3 1 823 XI04 a XIO * * 4 1 6 2 * XI05 B XIO « • 5 1 625 RM0I1) B XI02 • 5 .OOO • X I 0 3 / 4 . O 0 O - X 1 O 4 / 2 . O D O . 3 . O D O • XIOS / 1826 14.ODO 1527 RM0I2I B I X I 0 2 • X103 - XI04 • I I O S I / 6 . 0 0 0 1 528 RM0I9I a (XIO • 2 . 0 0 0 - X I 0 3 • XIOS) / 2 .OOO 1526 R M O M O ) a 1 . O D O - 2 . O D 0 • X I02 * XI04 1 S30 R M 0 M 4 ) a X I02 * S .OOO • XIOS / 4 . O 0 O - X I O 4 / 2 . O D O - 3 . O 0 O > XIOS / 1 S31 14.ODO 1 632 RMO(15) a [ -X102 • XI03 * X104 • XIOS) / 6 .ODO 1833 RETURN 1 S 3 * END 1 535 c 1 536 SUBROUTINE ORPMAS(NEO, NM, NMAX, N S T E P , R L . P I , S ) 1 537 I M P L I C I T REAL a 8 (A - H . O - Z) 1 536 COMMON / B I O / R C ( B ) , R K 1 5 ) , R M I S ) , R P 0 R C 1 I 5 ) . NDRMAS, N J D R I S ) , 1 638 1 R X O I S 1 . RVOI5) 1 640 COMMON / B I ! / RSTI F I IS . 135 I , RD AMP 1 1 5 , 1 3 5 ) 164 1 0 I ME NS ION ORM( 1 9 ) 1 S43 DO 40 I a 1, NDRMAS 1 543 IE a NJORI I ) 1 544 XO a RXOI I ) ' 1 S46 XIO a RVOI I | > 2 . DO / S 1 546 J1 a ( I - 1 | a NMAX • 1 1 S47 J2 a I a NMAX 1 S48 DO IO J a J l , J J 1 S49 00 10 K a 1, NEO 1 (SO R I T I F I J . K I a O .DO 1 • G 1 10 R D A M P ( J , K ) • O . D O 1152 CALL S H A P P N I X I O . DRM) 1553 I J • I IE • 1 ) • 13 1554 DO 30 L • 1, NM, NSTEP 1 555 1LM • L 1 656 IP INSTEP . E O . 2) ILM • | L • 1) / 2 1557 I M P II • 1| • NMAX • ILM 1 658 PAC1 • - R K I I ) • D S I N I L ' P 1 B X O / R L ) 1 558 PAC2 « - R C ( I ) • D S I N I L « P l > X O / R L ) 1 560 DO 20 M • 1 , 19 156 1 R S T I P I I M . I J * Ml • PAC1 » ORMIMI 1862 20 RDAMP! I M . 1 J • M) • PAC 2 « DRMIMI 1663 30 CONTINUE 1 664 40 CONTINUE 1865 RETURN 1666 ENO 1 687 C 1666 SUBROUTINE ZERO 1 688 I M P L I C I T REAL * 91 A • H . O • Z) 1 670 COMMON / B 3 / R M I 5 . 1 9 I , R M 1 I 6 . 1 9 ) 167 1 DO 10 I B 1, 8 1 872 DO 10 J B 1 , 19 1 673 R M l I , J ) B 0 . O D O 1 674 10 RMl I 1 , J | B 0 . O D O 1 675 RETURN 1 676 END 1 877 C 1 676 SUBROUTINE E P I C K I N J T ) 1679 I M P L I C I T R E A L * 61 A • H . O • Z l 1 680 COMMON / B 1 2 / YMOD(IO) 1 68 1 COMMON / B 2 0 / E O , EM, EK 1682 00 10 I B 1 , NJT 1683 P B PRANDI 1 . O I 1 684 P B DLOC1 1 . 0 0 - P I 1685 P B DLOC 1 -P ) / EK 1686 Z » O E X P I P I 1 887 IO VMOD1 I I B z » EM » EO 1 668 WRITE 19 ,201 1 V M O D 1 J 1 .JB1 , N J T 1 1 689 20 FORMAT I 1 0 I E 1 0 . 3 , 1 X 1 ) 1 990 RETURN 169 1 ENO 1692 c 1 993 SUBROUTINE U P D A T E I I I 1 694 I M P L I C I T R E A L B 8 I A • H . O - Z l 1695 COMMON / B 5 / NM, N S T E P , NMAX, N J T , S, R L , 0 , A L P H A . B E T A , HJTM, 1696 1 1 B J I 1 0 I . H J I 1 0 ) , NFLOR, INPTE 1 697 COMMON / B 6 / RKX, RKV, RKV, RKC, DX. DY. DV, DC, R K P A L , RKPER, 1 698 1 1 RKROT, RHOC. C P , C N P A L , CNPER, CNROT 1699 COMMON / B ? / C A P X O I S I , C A P 0 I 5 I , C A P X I S I . C A P ( 5 1 , E N L . XIN . NCAPS 1 7O0 1  N A I , N01SCR 1 701 COMMON / B I O / R C I 5 I , R K I 5 ) , A M I S ) . R F O R C I ( B ) , NORMAS. N J D R I 6 ) . 1 702 1  R X O I S ) , R Y 0 I 5 I , V E L I 5 I . T I M I M P I 5 ) 1703 COMMON / B 1 2 / Y M O D ( I O ) , D N S T Y ( I O ) , C B I I O I . T H E T A . C R V I T V 1 704 COMMON / B I S / NLOAO. N L U . P L O A D . S T 0 R E I 1 O . S I . N L J O I I O l 1 70S COMMON / B 1 9 / ISYM. NTMZON, T I M E ( 1 0 . 3 I , R E C . TOL 1 706 COMMON / B 2 0 / E O . EM. E K . E M I N . EMAX, S E E D . NREPL 1 707 COMMON / B 2 1 / NBC. I B C I 5 0 . 2 ) 1 708 COMMON / 5 2 2 / NDEFJTI 10) , 0 E F M I 1 O . 2 ) , NPNTS 1 709 COMMON / B 2 5 / S C A L E I 3 ) 1710 S B I 6 . O 17 11 RL « 1 4 4 . 0 17 12 DO 10 N B 1 , NDRMAS 17 13 R M lN ) B 0 . S 1 8 17 14 RC1N) B 6 . 0 17 15 RKIN) B 4 0 0 . O 1716 10 RXO1N) B 7 2 . O 17 17 R P O R C 1 ( 2 ) s 2 0 0 . 0 17 16 DO 20 N B 1, NPNTS 17 19 20 D E F M IN , 1 1 B 7 2 . O 1 720 DO 30 N B 1 , NJT 1721 VMODINI B 1 . SSE6 1722 HJ(N1 B 7 , 2 5 1723 SO CB I N | B 3 . S3E -3 1 724 CP B 4 . 1 7 E - 3 1 726 CNPAL B 5 . 0 1726 CNPER B S . O 1727 CNROT B 6 . 0 1 728 RKPAL B S S O O . O 1729 RKPER B 8 6 0 0 . 0 1730 ENL B 8 . 0 1731 c MODIFY VALUES OF PARAMETERS FOR D I F F E R E N T FLOORS 1732 CO TO ( 4 0 , SO , SO, 7 0 , 60 . SO, 110 , 130 , 1SO, 170 , 2 0 0 . 2 3 0 . 1 733 1250, 2 7 0 , 2 S O . 3 1 0 . 3 3 0 . 3 4 0 , 3 5 0 , 3 6 0 ) , I 1 734 40 CONTINUE 1735 60 5 B 1 2 . O 1736 CO TO 370 1737 60 S B 2 0 . 0 1736 CO TO 370 1 739 70 R M l 1 ) B O . 3 6 2 1740 CO TO 370 1 74 1 SO RMl21 B o . 362 1742 R P 0 R C 1 I 2 ) B 1 4 0 . O 1743 CO TO 370 1 744 90 DO IOO J B 1. NORMAS 1 745 1 00 RC1J1 B 1 2 . O 1748 CD TO 370 1747 1 10 00 120 J B 1, NDRMAS 1 748 120 RC1J) B 3 .o 1 749 00 TO 370 1760 130 DO 140 J a 1. NORMAS 17S1 1 40 R K ( J ) B S O O . O 1 752 CO TO 370 17S3 1 SO DD 1SO J B 1, NDRMAS 1754 1 so R K U ) B 2 0 0 . 0 1755 CO TO 370 1758 1 70 RL B 1 S 2 . 0 1 7S7 DD ISO J a 1, NDRMAS 1758 ISO R X O I J I a 9 5 . 0 1 759 DO 1 9 0 K B 1 , NPNTS 17 SO 150 O E F M I K , 1 ) • B S . O 1 13 1781 CO TO 370 1782 20O RL • 9 6 . 0 1783 DO 2 10 J • 1 , NORMAS 1784 210 RKOt J1 • 4 6 . 0 1788 DD 220 K • 1 . NPNTS 1788 220 D E F M I K . 1 I • 4 8 . 0 1787 CO TO 370 1788 230 DO 240 J • 1 , NJT 1768 2 * 0 HJ1J1 • 1 1 . 2 5 1770 GO TO 370 1771 280 DO 260 J > 1 . NJT 1773 260 VMOO(J) • 1 .6ES 1773 GO TO 370 1 774 270 DO 280 J « 1 . NJT 1 778 280 VMOOIJ) • 1 .2E6 1776 CO TO 370 1777 290 DO 300 J • 1 . NJT 1 778 300 CSI J l • 7 . 668 -3 1778 CP > 6 . 3 4 8 - 3 1 760 CNPAL • 1 2 . 0 1 78 1 CNPER 4 1 2 . O 1782 CNROT • 1 2 . O 1783 CD TO 170 1764 310 DO 320 J • 1 . NJT 1785 320 CB( J1 < 1 .626 - 3 1786 CP « 2 . 0 6 E - 3 1 757 CNPAL • 3 . 0 1 788 CNPER * 3 . 0 1 788 CNROT • 3 . 0 1780 CO TO 370 179 1 330 ENL » 1 6 . 0 1 782 CD TD 370 1793 3 * 0 ENL • 0 . 5 1794 GO TO 370 1795 360 RKPAL • 10O0. 0 1796 RKPER • 1000 . 0 1797 CO TO 370 1 798 360 RKPAL B l O O O O O . O 1799 RKPER « l O O O O O . O 1 600 370 CONT1NUE 1 80 1 RETURN 1602 END 

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