UBC Theses and Dissertations

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UBC Theses and Dissertations

Analysis of cable structures by Newton's method Miller, Ronald Ian Spencer 1971

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ANALYSIS OF CABLE STRUCTURES by NEWTON'S METHOD by RONALD IAN SPENCER MILLER B.A. (1965) B.A.Sc. (1967) 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 A THESIS SUBMITTED IN PARTIAL FULFILMENT OF THE REQUIREMENTS FOR THE DEGREE OF MASTER OF APPLIED SCIENCE I n t h e D e p a r t m e n t o f 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 d s t a n d a r d 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 A p r i l 1971 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 a n a d v a n c e d d e g r e e a t t h e U n i v e r s i t y o f B r i t i s h C o l u m b i a , I a g r e e t h a t t h e L i b r a r y s h a l l make i t f r e e l y a v a i l a b l e f o r r e f e r e n c e and s t u d y . I f u r t h e r 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 r p o s e s may be g r a n t e d by t h e Head o f my D e p a r t m e n t o r b y h i s 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 . D e p a r t m e n t o f C I V I L 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 , V a n c o u v e r 8, C a n a d a . A b s t r a c t The a n a l y s i s o f s t r u c t u r e s w h i c h c o n t a i n c a t e n a r y c a b l e s i s made d i f f i c u l t by t h e n o n - l i n e a r f o r c e - d e f o r m a t i o n r e l a t i o n s h i p s o f t h e c a b l e s . F o r a l l b u t t h e s m a l l e s t d e f l e c t i o n s i t i s n o t p o s s i b l e t o l i n e a r i z e t h e s e r e l a t i o n s h i p s w i t h o u t c a u s i n g s i g n i f i c -a n t i n a c c u r a c i e s . Newton's Method s o l v e s n o n - l i n e a r e q u a t i o n s by s o l v i n g a s u c c e s s i o n o f l i n e a r i z e d p r o b l e m s , t h e ' a n s w e r c o n v e r g i n g t o t h e s o l u t i o n o f t h e n o n - l i n e a r p r o b l e m . Newton's Method so u s e d t o a n a l y z e c a b l e - c o n t a i n i n g s t r u c t u r e s r e s u l t s i n a s u c c e s s i o n o f l i n e a r s t i f f n e s s a n a l y s i s p r o b l e m s . As a r e s u l t , c o n v e n t i o n a l s t i f f n e s s a n a l y s i s computer programs may be m o d i f i e d w i t h o u t g r e a t d i f f i c u l t y t o s o l v e c a b l e s t r u c t u r e s by Newton's Method. The use o f Newton's Method t o s o l v e c a b l e s t r u c t u r e s f o r m s t h e body o f t h i s t h e s i s . The two b a s i c i n n o v a t i o n s n e c e s s a r y , w h i c h a r e t h e p r o v i s i o n o f methods f o r c a l c u l a t i n g t h e e n d - f o r c e s o f a c a b l e i n an a r b i t a r y p o s i t i o n , and f o r e v a l u a t i n g t h e s t i f f n e s s m a t r i x o f a c a b l e , a r e p r e s e n t e d . A l s o d i s c u s s e d a r e t h e c o - o r d i n a t e t r a n s f o r m a t i o n s n e c e s s a r y t o d e s c r i b e t h e c a b l e s t i f f n e s s m a t r i x and c a b l e end f o r c e s i n a G l o b a l C o - o r d i n a t e S y s t e m . The v i r t u e s o f t h e method a r e d e m o n s t r a t e d i n two example p r o b l e m s , and t h e t h e o r e t i c a l b a s i s f o r Newton's Method i s e x a m i n e d . F i n a l l y , t h e v a l u e o f t h e method p r e s e n t e d i s b r i e f l y d i s c u s s e d . TABLE OF CONTENTS A b s t r a c t L i s t o f F i g u r e s Acknowledgements Page 1. The P rob l em 1 2. The Method 5 3. C a b l e End F o r c e s 12 4 . The C a b l e S t i f f n e s s M a t r i x 21 5. The C a b l e C o - o r d i n a t e System 26 6 . Advanced T o p i c s 1. N o n - l i n e a r B e h a v i o u r o f Non-Cab le S t r u c t u r a l Components 31 2. S p e c i f i e d C a b l e Ten s i on s 33 3. M i s c e l l a n e o u s P rob lems 3^ 4 . C a b l e Loads 3^ 7. Examples Example 1 36 Example 2 38 8. D i s c u s s i o n 44 B i b l i o g r a p h y 46 Append i x 1 i Append i x 2 v i i Append i x 3 x LIST OF FIGURES. Page Fi g . 2.1 Example Problem 7 F i g . 2.2 Path of Solution to Example 10 F i g . 3.1a The Cable Co-ordinate System 12 F i g . 3.1b Forces i n the Cable Plane 13 F i g . 3«lc Dimensions i n the Cable Plane 13 F i g . 3.2 Element of a Catenary Cable 14 F i g . 4.1 Degrees of Freedom i n the Cable Plane 22 F i g . 4.2 Degrees of Freedom for a General Cable 25 F i g . 5*1 Cable and Global Co-ordinate Systems 30 F i g . 7.1 Guyed Tower 39 F i g . 7.2 A x i a l Force at 750' Level Versus I n i t i a l Cable Stress 40 Fi g . 7.3 Bending Moment at 750' Level Versus I n i t i a l Cable Stress 40 Fi g . 7.4 Lateral Deflection at 1,000* Level Versus I n i t i a l Cable Stress 41 F i g . 7.5 Lateral Deflection at 750' Level Versus I n i t i a l Cable Stress 41 Fi g . 7.6 Stress i n Higher Windward Cable 42 F i g . 7.7 Stress i n Lower Windward Cable 42 AC KNOWLEDGEMENTS I should l i k e to express my a p p r e c i a t i o n f o r the guidance and h e l p g i v e n by my s u p e r v i s o r , Dr. R.F. Hooley throughout the w r i t i n g of t h i s t h e s i s . I should a l s o l i k e to thank the N a t i o n a l Research C o u n c i l of Canada f o r t h e i r f i n a n c i a l support, and the U n i v e r s i t y o f B r i t i s h Columbia Computing Centre f o r the use of t h e i r f a c i l i t i e s . F i n a l l y , I should l i k e to thank Miss Sarah Fenning f o r her p a i n s t a k i n g e f f o r t s i n t y p i n g t h i s t h e s i s . A p r i l , 1971 Vancouver, B r i t i s h Columbia. C h a p t e r 1. The P r o b l e m . T h e r e a r e many s t r u c t u r e s w h i c h i n v o l v e c a b l e s - s u s p e n s i o n b r i d g e s , guyed t o w e r s , t r a n s m i s s i o n l i n e s , a e r i a l tramways, c a b l e - s u p p o r t e d r o o f s and numerous o t h e r s . F o r some o f t h e s e p r o b l e m s t h e c a b l e s a r e so t a u t t h a t t h e y may be t r e a t e d a s b a r s : f o r o t h e r s t h e y a r e so t h i c k t h e y must be t r e a t e d a s beams: f o r s t i l l o t h e r s , t h e y a r e so c l o s e l y s p a c e d t h e y may be t r e a t e d as membranes. T h e r e r e m a i n , however, a l a r g e number o f s t r u c t u r e s w h e r e i n t h e c a b l e s may be a n a l y z e d u n d e r t h e a s s u m p t i o n s o f c a t e n a r y b e h a v i o u r : t h a t c a b l e s a r e s u b j e c t e d t o a l o a d i n g p e r u n i t l e n g t h w h i c h i s c o n s t a n t i n i n t e n s i t y and d i r e c t i o n , and t h a t t h e y a r e c o m p l e t e l y f l e x i b l e i n b e n d i n g . T h e s e p r o b l e m s a r e d i f f i c u l t t o s o l v e , f o r u n l i k e many o f t h e p r o b l e m s e n c o u n t e r e d i n s t r u c t u r a l a n a l y s i s , t h e i r l o a d -d e f o r m a t i o n r e l a t i o n s h i p s a r e m a r k e d l y n o n - l i n e a r . As a c a b l e i s s t r e t c h e d , i t becomes s t i f f e r , and as i t i s r e l a x e d i t becomes more f l e x i b l e . M o r e o v e r , t h i s n o n - l i n e a r i t y i s s i g n i f i c a n t f o r a l l b u t t h e s m a l l e s t d e f l e c t i o n s . The f r i e n d l y a s s u m p t i o n s r e -q u i r e d f o r s t i f f n e s s a n a l y s i s c a n n o t be made, f o r e v e n i f we d e t -e r m i n e d t h e s t i f f n e s s m a t r i x f o r t h e s t r u c t u r e i n i t s i n i t i a l c o n f i g u r a t i o n , t h a t s t i f f n e s s w o u l d change so m a r k e d l y as t h e d e f o r m a t i o n s i n c r e a s e d t h a t t h e answer we f o u n d w o u l d be q u i t e u n r e l i a b l e . We may be a b l e t o s i m p l i f y t h e f o r m a t i o n o f t h e s t i f f n e s s m a t r i x b y m a k i n g f u r t h e r a s s u m p t i o n s : f o r i n s t a n c e , we may - 2 -assume t h a t t h e c a b l e s a r e i n e x t e n s i b l e a x i a l l y , o r t h a t t h e y a r e r e a s o n a b l y t a u t ( i n w h i c h c a s e t h e c a b l e s f o l l o w homely p a r a b o l a s , i n s t e a d o f e s o t e r i c c a t e n a r i e s ) . N e v e r t h e l e s s , t h e c e n t r a l p r o b l e m w i l l r e m a i n - t h e c a b l e s a r e n o n - l i n e a r . T h i s t h e s i s p r e s e n t s a method f o r t h e a n a l y s i s o f s t r u c t u r e s c o n t a i n i n g c a t e n a r y c a b l e s . The method may be s i m p l i f i e d (and r e s t r i c t e d ) b y t h e a s s u m p t i o n o f i n e x t e n s i b l e b e h a v i o u r a n d / o r p a r a b o l i c c a b l e s . As p r e s e n t e d , i t s o l v e s t h o s e p r o b l e m s where c a b l e s may be t r e a t e d as s u b j e c t e d t o c o n s t a n t l o a d i n g p e r u n i t l e n g t h , c o m p l e t e l y f l e x i b l e i n b e n d i n g , and l i n e a r - e l a s t i c a x i a l l y . N o n - c a b l e components o f t h e s t r u c t u r e a r e a n a l y z e d by c o n v e n t i o n a l s m a l l s t r a i n - s m a l l r o t a t i o n t h e o r y s t i f f n e s s a n a l y s i s . I n t h e d e v e l o p m e n t o f t h i s t h e s i s , t h r e e methods f o r t h e a n a l y s i s o f t h e p r o b l e m s d e s c r i b e d i n t h e p r e v i o u s p a r a -g r a p h were i n v e s t i g a t e d . The t h r e e methods a r e d i s c u s s e d b r i e f l y b e l o w . The f i r s t method was s i m p l y t o t r e a t e a c h c a b l e as a s e r i e s o f p i n - e n d e d b a r s . L i v e and d e a d l o a d s were a p p l i e d a t t h e j o i n t s , and t h e s t r u c t u r e was a n a l y z e d u s i n g l a r g e d e f l e c t i o n t h e o r y f o r t h e b a r s . R e s u l t s were o f t e n s a t i s -f a c t o r y , b u t two d i s a d v a n t a g e s were a p p a r e n t : f o r c a b l e s w i t h l i t t l e s a g t h e s t i f f n e s s m a t r i x was p o o r l y c o n d i t i o n e d s and t h e amount o f c o m p u t a t i o n i n v o l v e d i n t h e method was q u i t e h i g h . The s e c o n d method was t o t r e a t e a c h c a b l e as one member and t o a p p l y t h e l o a d i n i n c r e m e n t s , t r e a t i n g t h e c a b l e s t i f f n e s s as l i n e a r f o r e a c h i n c r e m e n t . T h i s method was a l s o - 3 -o f t e n s a t i s f a c t o r y , b u t i t , t o o , had a drawback: t h e a c c u r a c y o f t h e s o l u t i o n depended on t h e s i z e o f t h e l o a d i n c r e -ment c h o s e n . The o n l y way t o e n s u r e an a c c u r a t e answer was t o p e r f o r m a n a l y s e s w i t h s u c c e s s i v e l y s m a l l e r l o a d i n c r e m e n t s i z e s , u n t i l t h e s o l u t i o n s c o n v e r g e d . I f o n l y one s o l u t i o n was made, t h e a c c u r a c y was i n d e t e r m i n a t e . F o r some s t r u c t u r e s , w h i c h b e h a v e d a l m o s t l i n e a r l y , o n l y a few i n c r e m e n t s were r e q u i r e d , whereas f o r h i g h l y n o n - l i n e a r s t r u c t u r e s t h e l o a d had t o be b u i l t up i n many s m a l l i n c r e m e n t s . The t h i r d method i n v e s t i g a t e d , w h i c h i s t h e method d e s c r i b e d i n t h e r e m a i n d e r o f t h i s t h e s i s , had none o f t h e drawbacks o f t h e f i r s t two methods. Any d e s i r e d d e g r e e o f a c c u r a c y c o u l d be o b t a i n e d , and t h e amount o f c o m p u t a t i o n r e q u i r e d was r e l a t i v e l y m o d e s t . I n t h e i n i t i a l c o n f i g u r a t i o n , t h e ' u n b a l a n c e d f o r c e s ' a c t i n g on t h e s t r u c t u r e were e v a l u a t e d ( t h e u n b a l a n c e d f o r c e s a r e s i m p l y t h e e x t e r n a l l o a d s minus t h e i n t e r n a l f o r c e s r e s i s t -i n g t h e m ) . The l i n e a r b e h a v i o u r o f t h e s t r u c t u r e was t h e n r e p r e s e n t e d by a s t i f f n e s s m a t r i x ( e a c h c a b l e , l i k e e a c h beam, b e i n g r e p r e s e n t e d b y a s i n g l e member m a t r i x ) and t h e l i n e a r d e f l e c t i o n s due t o t h e u n b a l a n c e d f o r c e s were c a l c u l a t e d . I n t h i s new d e f o r m e d p o s i t i o n a new s e t o f u n b a l a n c e d f o r c e s was d e t e r m i n e d , and a new s t i f f n e s s m a t r i x was f o u n d ( s i n c e t h e s t i f f n e s s o f t h e s t r u c t u r e was n o t t h e same i n t h e d e f o r m e d p o s i t i o n as i t was i n t h e i n i t i a l p o s i t i o n ) . The d e f l e c t i o n s due t o t h e new s e t o f u n b a l a n c e d f o r c e s were c a l c u l a t e d and added t o t h e p r e v i o u s d e f l e c t i o n s . T h i s p r o c e d u r e was c a r r i e d o u t u n t i l t h e u n b a l a n c e d f o r c e s were e f f e c t i v e l y z e r o . _ 4 _ M a t h e m a t i c a l l y , t h i s i s Newton's method. A good d e s c r i p -t i o n may be found i n L i v e s l e y (Ch. 10.3. p. 241) ( 1 ) . The procedure w i l l be d i s c u s s e d i n more d e t a i l i n the next chapter, and the mathematical i m p l i c a t i o n s of Newton's method are r e -viewed i n Appendix 3« - 5 -C h a p t e r 2. The Method. The method p r o p o s e d has much i n common w i t h s t i f f n e s s a n a l y s i s as i t c o n s i s t s o f a s e r i e s o f e v e r f i n e r a p p r o x i m -a t i o n s t o t h e s o l u t i o n , t h e change f r o m one a p p r o x i m a t i o n t o t h e n e x t b e i n g f o u n d by s o l v i n g a l i n e a r s t i f f n e s s p r o b l e m . As i s u s u a l i n s t i f f n e s s a n a l y s i s , t h e p o s i t i o n o f t h e s t r u c t u r e i s d e f i n e d b y a s e t o f g e n e r a l i z e d c o - o r d i n a t e s , one d e g r e e o f f r e e d o m b e i n g a s s i g n e d t o e a c h p o s s i b l e d e f o r m -a t i o n d i r e c t i o n o f e a c h j o i n t . W h erever t h e r e i s a p o i n t l o a d on a c a b l e , o r w h e r e v e r t h e r e i s a change i n t h e d i r e c t i o n o r i n t e n s i t y o f t h e u n i f o r m l o a d on a c a b l e , a j o i n t must be d e f i n e d . As d e s c r i b e d i n C h a p t e r 1, t h e method p r o p o s e d i s an i t e r a t i v e p r o c e d u r e c o n s i s t i n g o f t h e f o l l o w i n g b a s i c s t e p s : (0) Choose a d e f l e c t e d shape w h i c h w i l l s e r v e a s t h e s t a r t i n g p o i n t o f t h e i t e r a t i o n . The most c o n -v e n i e n t i n i t i a l p o s i t i o n w i l l be t h a t a t w h i c h a l l t h e n o n - c a b l e components o f t h e s t r u c t u r e a r e u n s t r e s s e d , when s u c h a p o s i t i o n e x i s t s . (1) I n t h e d e f l e c t e d shape, c a l c u l a t e t h e u n b a l a n c e d l o a d s (UBL), w h i c h a r e j u s t t h e e x t e r n a l l o a d s minus (a) t h e c a b l e end f o r c e s , and (b) t h e end f o r c e s d e v e l o p e d by t h e n o n - c a b l e members, i n c l u d i n g t h e e f f e c t s o f member l o a d i n g i f p r e s e n t . A l s o c a l c u l a t e t h e s t i f f n e s s m a t r i x f o r t h e s t r u c t u r e i n t h i s d e f o r m -ed p o s i t i o n . - 6 -(2) S o l v e f o r t h e i n c r e m e n t a l d e f l e c t i o n s due t o t h e u n b a l a n c e d l o a d s , u s i n g t h e s t i f f n e s s m a t r i x j u s t f o u n d . Add t h e s e i n c r e m e n t a l d e f l e c t i o n s t o t h e p r e v i o u s d e f l e c t i o n s . (3) R e p e a t s t e p s (1) and (2) u s i n g t h e new d e f l e c t e d shape, u n t i l t h e u n b a l a n c e d l o a d s a r e n e g l i g i b l e . The u n b a l a n c e d l o a d s r e p r e s e n t t h e amount b y w h i c h t h e s t r u c t u r e i s o u t o f e q u i l i b r i u m . When we e v a l u a t e t h e un-b a l a n c e d l o a d s i n s t e p ( 1 ) , we have i n f a c t p e r f o r m e d a n e x a c t s o l u t i o n o f t h e s t r u c t u r e , b u t u n d e r a d i f f e r e n t l o a d i n g f r o m t h a t i n w h i c h we a r e i n t e r e s t e d . We have a n e x a c t s o l u t i o n f o r t h e l o a d i n g w h i c h c o n s i s t s o f t h e a p p l i e d l o a d s minus t h e un-b a l a n c e d l o a d s . When we a p p l y t h e u n b a l a n c e d l o a d s t o t h e s t r u c t u r e , and add t o t h e p r e s e n t d e f l e c t i o n s t h e i n c r e m e n t a l d e f l e c t i o n s t h e y c a u s e , t h e new u n b a l a n c e d l o a d s a r e much s m a l l e r , r e p r e s e n t i n g o n l y t h e e r r o r i n t h e s o l u t i o n . As t h e u n b a l a n c e d l o a d s a p p r o a c h z e r o , t h e l o a d i n g f o r w h i c h o u r d e f o r m e d p o s i t i o n i s a n e x a c t s o l u t i o n a p p r o a c h e s t h e l o a d i n g whose e f f e c t s we w i s h t o s t u d y . F u r t h e r m o r e , t h e c l o s e r t h e u n b a l a n c e d l o a d s a r e t o z e r o t h e more r a p i d l y do t h e two l o a d -i n g s a p p r o a c h c o i n c i d e n c e . We c a n t h u s f i n d a s o l u t i o n t o w i t h i n any* a r b i t r a r y s m a l l t o l e r a n c e f o r e r r o r , l i m i t e d o n l y by o u r c o m p u t a t i o n a l t e c h n i q u e s . * T h i s i s p e r h a p s more v a l o r o u s t h a n d i s c r e e t . See C h a p t e r 8. - 7 -L e t us c o n s i d e r a s i m p l e s i n g l e d e g r e e o f f r e e d o m p r o b l e m w h i c h i l l u s t r a t e s t h e m a j o r f e a t u r e s o f t h e method. I n f i g u r e 2.1 a 1000 f o o t l o n g i n e x t e n s i b l e c a b l e w h i c h w e i g h s one pound p e r f o o t i s shown a t t a c h e d a t i t s l e f t end t o a s u p p o r t and a t i t s r i g h t end t o a s p r i n g . The ends a r e c o n s t r a i n e d t o r e m a i n a t t h e same h e i g h t . The s p r i n g has a s t i f f n e s s o f 1 k i p / f o o t and i s u n s t r a i n e d when t h e c a b l e s p a n , L, i s 1000 f e e t . One k i p i s a p p l i e d t o t h e r i g h t hand end o f t h e c a b l e , and i t i s d e s i r e d t o f i n d t h e e q u i l i b r i u m p o s i t i o n . F o r t h i s c a b l e , t h e h o r i z o n t a l component o f t e n s i o n , H, i s r e l a t e d t o t h e s p a n , L, b y e q u a t i o n ( 3 « 5 ) i w h i c h may be s i m p l i f i e d t o : IOOO' long inextensible cable weight* 1 ibyft. E x a m p l e Problem F i g . 2.1. Hsinhy = . 5 2.1 Where • Q005L H 2.2 - 8 -D i f f e r e n t i a t i n g e quation 2.1, we f i n d : dH _ H c o s h Y 2 3 dL Lcoshy-^inhy The g e n e r a l i z e d degree of freedom a c t s to the r i g h t on the r i g h t hand end of the c a b l e . Then at any span L, the un-balanced l o a d equals the e x t e r n a l l o a d minus the r e s i s t i n g f o r c e s due to the cable and the s p r i n g : UBL = 1 - [ H + ( 1000- L )] (kips) 2.4 Where H i s found by s o l v i n g e q u a t i o n 2.1. The s t i f f n e s s , k, equals the cable s t i f f n e s s p l u s the s p r i n g s t i f f n e s s : k - -^r-* 1 ( k/ft.) 2 , 5 d L The s o l u t i o n now proceeds as f o l l o w s : (0) We cannot s t r e t c h the cable to the p o s i t i o n L = 1000 f e e t without c a u s i n g an i n f i n i t e f o r c e , so we choose as the i n i t i a l p o s i t i o n L = 999 f e e t , c o r r e s p o n d i n g to a d e f l e c t i o n § of -1 f o o t . C y c l e 1. (1) At L = 999 f e e t (S-l'). UBL = - 4.4463 kips k= 4.2318 k/ft. (2) The i n c r e m e n t a l d e f l e c t i o n i s : " \ • 6 3 = -1.0507' 4.2 318 The new d e f l e c t i o n i s : -|.0-1.0507 = - 2.0507' C y c l e 2. (1) At § =-2.0507', UBL*-1.4445 kips . k = 2.10 21 k/ft. I 4 4 4 « S (2) The new d e f l e c t i o n i s : -2.0507- g |Q2| = ~ 2 7 3 7 8 - 9 -Cycle 3. (1) At § =- 2.7378', UBL = - . 1489 kips k= 1. 7151 k/ft. (2) The new d e f l e c t i o n i s : - 2.73 78- = -2.8246 Cycle 4. (1) At 8 =-2.8246*. UBL =-.001435 kips k = 1.6824 k/ft. (2) The new d e f l e c t i o n i s : - 2.8246- .001435 1.6824 = -2.8255 Cycle 5. (1) At § =-2.8255', UBL= .00000013 kips Which i s small enough for most p r a c t i c a l purposes. So our solution i s L = 1000'-§ = 997.1745' Note that t h i s procedure exhibits quadratic convergence -as the d e f l e c t i o n approaches the solution, the incremental d e f l e c t i o n approaches the true error i n d e f l e c t i o n . In other words, over the small deflections calculated as the solution i s neared, the structure remains almost l i n e a r . In the f i r s t solu-t i o n , the error i n the d e f l e c t i o n was cut by a factor of 2.36, from 1.8255 feet to 0.7748 feet. In the second solution i t was cut by a factor of 8.8 (from .7748 to .0877) i n the t h i r d by 103, and i n the fourth by 10,900. The path followed i n the solution i s shown graphically i n Figure 2.2. - 10 -4 3 2 I 0 9- -i -3 -4 -5 L , feet 9 9 9 l i near stiffness © start 1 o S T A R T 9 9 8 L 5 ? - L 3,4 9 9 7 P a t h of S o l u t i o n to E x a m p l e F i g . 2.2. The example c h o s e n was v e r y s i m p l e , b u t i t e x h i b i t s a l l t h e i m p o r t a n t a s p e c t s o f t h e method. F o r more g e n e r a l p r o b l e m s t h e r e may be many c a b l e s and many n o n - c a b l e e l e m e n t s , t h e c a b l e s w i l l n o t be i n e x t e n s i b l e , and t h e l o a d i n g w i l l be q u i t e c o m p l i c a t e d ; b u t t h e same method w i l l c o n v e r g e r e a d i l y t o a s o l u -t i o n f o r a l m o s t any s t a r t i n g p o i n t . A f u r t h e r p o i n t may be o b s e r v e d f r o m t h i s example -b e c a u s e o f t h e q u a d r a t i c c o n v e r g e n c e o f t h e method, we c a n s a v e t i m e by u s i n g a s t a r t i n g p o i n t as c l o s e as p o s s i b l e t o t h e s o l u -t i o n . I f we have s o l v e d a s t r u c t u r e f o r some l o a d i n g , and now w i s h t o s o l v e f o r a d i f f e r e n t b u t s i m i l a r l o a d i n g , we w i l l be w e l l a d v i s e d t o s t a r t t h e s o l u t i o n p r o c e d u r e a t t h e d e f l e c t e d - 11 -shape r e s u l t i n g f r o m t h e e a r l i e r s o l u t i o n . The u n b a l a n c e d l o a d s i n t h i s p o s i t i o n w i l l be j u s t t h e d i f f e r e n c e b e t w e e n t h e two l o a d -i n g s , and w i l l be q u i t e s m a l l i f t h e two l o a d i n g s a r e s i m i l a r . F o r i n s t a n c e , i f we w i s h t o s o l v e t h e example j u s t comp-l e t e d w i t h o u t t h e one k i p l o a d a p p l i e d a t t h e r i g h t end, t h e s o l u t i ( L = 9 9 6 . 5 4 f e e t ) i s c o n s i d e r a b l y c l o s e r t o t h e p r e v i o u s s o l u t i o n ( L= 9 9 7 . 1 7 f e e t ) t h a n i t i s t o t h e i n i t i a l p o i n t (L= 9 9 9 f e e t ) . T h u s , as m i g h t be v i s u a l i z e d f r o m F i g . 2 . 2 . s t a r t i n g a t t h e p r e -v i o u s s o l u t i o n p o i n t w i l l r e s u l t i n c o n s i d e r a b l y f a s t e r c o n v e r g e n c e t h a n s t a r t i n g a t t h e i n i t i a l p o i n t . By s o l v i n g t h e n o n - l i n e a r p r o b l e m as a s e r i e s o f l i n e a r p r o b l e m s , we a r e a b l e t o u t i l i z e f a m i l i a r methods t o s o l v e e a c h l i n e a r p r o b l e m - i n p a r t i c u l a r , we use t h e methods o f s t i f f n e s s a n a l y s i s . I n o r d e r t o m o d i f y a s t i f f n e s s a n a l y s i s c o mputer p r o g r a m t o s o l v e c a b l e p r o b l e m s we need make o n l y two m a j o r mod-i f i c a t i o n s ! a t e a c h s t a g e o f t h e i t e r a t i o n we need ( 1 ) t o e v a l -u a t e t h e v e c t o r o f u n b a l a n c e d l o a d s , and ( 2 ) t o f i n d t h e s t i f f -n e s s m a t r i x . M o r e o v e r , t h e c o n t r i b u t i o n s o f t h e n o n - c a b l e e l e m e n t s t o t h e u n b a l a n c e d l o a d v e c t o r and t h e s t i f f n e s s m a t r i x a r e a l r e a d y known. A s s u m i n g t h a t t h e s e e l e m e n t s behave l i n e a r l y , t h e i r s t i f f n e s s m a t r i c e s a r e c o n s t a n t and t h e f o r c e s t h e y d e v e l o p a r e s i m p l y t h e p r o d u c t s o f t h e i r s t i f f n e s s m a t r i c e s w i t h t h e i r d e f l e c t i o n s . F o r t h e c a b l e s , however, t h i n g s a r e n o t so s i m p l e . I t i s n e c e s s a r y t o f i n d t h e e n d - f o r c e s d e v e l o p e d by e a c h c a b l e ; and t o f i n d t h e i r d e r i v a t i v e s , w h i c h c o m p r i s e t h e c a b l e s t i f f n e s s m a t r i x . T h e s e p r o b l e m s a r e d i s c u s s e d i n t h e n e x t two c h a p t e r s . - 12 -C h a p t e r 3» C a b l e End F o r c e s . S i n c e t h e l o a d i n g on e a c h c a b l e i s c o n s t a n t i n d i r e c t i o n , t h e c a b l e l i e s i n a p l a n e . T h i s p l a n e , h e r e i n a f t e r c a l l e d t h e " c a b l e p l a n e " i s r e a d i l y f o u n d s i n c e (1) i t c o n t a i n s b o t h ends o f t h e c a b l e , and (2) i t c o n t a i n s t h e v e c t o r r e p r e s e n t i n g t h e l o a d on t h e c a b l e . I t i s c o n v e n i e n t t o use a c o - o r d i n a t e s y s t e m i n t h e c a b l e p l a n e when c a l c u l a t i n g t h e c a b l e end f o r c e s and s t i f f n e s s m a t r i x : t h i s c o - o r d i n a t e s y s t e m w i l l be c a l l e d t h e " c a b l e c o - o r d i n a t e s y s t e m " . As shown i n F i g . 3.1a, t h e y - a x i s i s o p p o s i t e t o t h e d i r e c t i o n o f l o a d i n g , and t h e x - a x i s i s p e r p e n d i c u l a r t o i t . F o r c e s and d i m e n s i o n s a r e shown i n F i g . 3«lt>, and i n F i g . 3»lc. The a c t u a l d i r e c t i o n s o f t h e x and y a x e s a r e d i s c u s s e d i n C h a p t e r 5 - f o r now, s u f f i c e i t t o s a y t h a t t h e y a r e r e a d i l y f o u n d . yj direction of loading on cable The Cable Coordinate Sy s tem F i g . ' 3.1a. - 13 -F o r c e s in the C a b l e P l a n e Fig. 3.1b. a = Cable area E= effective modulus of elasticity USL = unstressed length C - curve (stressed ) length D i m e n s i o n s in the C a b l e P l a n e Fig. 3.1c - 14 -H ( y ' + d y ' ) Element of a Catenary Cable F i g . 3 - 2 . The relationships between the forces of F i g . 3«lb and the dimensions and properties of the cable as shown i n F i g . 3.1c may be r e a d i l y derived. Summation of the y forces of F i g . 3 .2 y i e l d s : H y ' + w d C = H ( y 1 + dy ' ) so d y ' = -77-dC n y H d x solving , y' = Sjnh(^x-+ A) 3.1 y = "S-cosh(^ - + A ) + B 3.2 - 15 -We have two boundary c o n d i t i o n s @ x = 0 y = 0 @ x = L y = h Which g i v e : A = s i n h " T w h w L 2Hsinh L 2H -I wL 2H 3.3 B = "-^-coshA 3.^ Knowing the shape of the ca b l e , and knowing t h a t the component of cable t e n s i o n i n the x - d i r e c t i o n i s the constant value Hf i t i s r e l a t i v e l y s t r a i g h t f o r w a r d to c a l c u l a t e the f o r c e s at any p o i n t a l o n g the cable and the l e n g t h of the cable i t s e l f . The s t r e s s e d l e n g t h of the cable i s : C = |V+^sinh ] 2 3.5 The y-components of the end t e n s i o n s a r e : w _ wh , . t u wL , Cw _ , Vo --~2~ c o t n 2TT ~2~ 3* A n d u I n \i - wh .. wL . Cw _ _ V, - -ycoth -grr + 3.7 The end t e n s i o n s themselves are simply: To * | H"+ V 0 | ' 2 3.8 And S [ H 2 + v . ' ] ' .3-9 * In f a c t , x was chosen normal to the load i n order to y i e l d t h i s s i m p l i f i c a t i o n . - 16 -And i n t e r e s t i n g l y : T, - T 0 = w h The e l a s t i c e l o n g a t i o n o f t h e c a b l e , A» I s f o u n d f r o m : 0 o A = H L r ^ c o t h v ^ + ± + J ^ s i n h ^ l 3.10 " a E[_2HL 2H 2 2wL H j The u n s t r e s s e d l e n g t h o f t h e c a b l e i s o f c o u r s e t h e a c t u a l l e n g t h minus t h e e l a s t i c e l o n g a t i o n : U S L = C - A 3 . H I n g e n e r a l , we w i l l know t h e c a b l e p r o p e r t i e s w . a , and E , and we w i l l know t h e p o s i t i o n s o f t h e ends o f t h e c a b l e (and so h and L ). I f we a l s o know t h e v a l u e o f H i we c o u l d d i r e c t l y s o l v e e q u a t i o n s 3.6 and 3.7 f o r t h e y-components o f t h e end t e n s i o n s . U n f o r t u a n a t e l y we s e l d o m know t h e v a l u e o f H » h u t i n s t e a d know some o t h e r q u a n t i t y : t h e u n s t r e s s e d l e n g t h o f t h e c a b l e , USL: o r t h e s a g a t some v a l u e o f x i o r one o f t h e end t e n s i o n s To o r T, . I n t h i s c a s e we use t h e known q u a n t i t y t o f i n d H , and t h e n use H t o f i n d V 0 and V\ . - 17 -J u s t as we a r e g o i n g t o u s e Newton's method t o s o l v e t h e s e t o f s i m u l t a n e o u s n o n - l i n e a r e q u a t i o n s w h i c h d e f i n e t h e e q u i l -i b r i u m p o s i t i o n o f t h e e n t i r e s t r u c t u r e , so we w i l l now use i t t o s o l v e f o r H '> H i s r e l a t e d t o o u r known q u a n t i t y (USL, s a g , T 0 o r T i ) by a n o n - l i n e a r e q u a t i o n , and Newton's method s o l v e s n o n -l i n e a r e q u a t i o n s . I f we l e t K s t a n d f o r t h e known q u a n t i t y , t h e f o l l o w i n g p r o c e d u r e ( s i m i l a r m a t h e m a t i c a l l y t o t h e example d i s c u s s e d i n C h a p t e r 2) w i l l f i n d H . (1) Guess a v a l u e o f H , (2) C a l c u l a t e K*, t h e v a l u e t h a t t h e q u a n t i t y K wo u l d have i f i t were b a s e d on t h e g u e s s e d v a l u e o f H . (3) D e f i n e f ( H ) = K - K * , t h e e r r o r i n K . (4) C a r r y o u t t h e N e w t o n i a n s e q u e n c e : H i+ 1 H' + f ( H ' ) V dH J The g u e s s e d v a l u e o f H w i l l c o n v e r g e q u a d r a t i c a l l y t o t h e c o r r e c t v a l u e , a t w h i c h t i m e f l H ) = 0 . We c a n s a y t h a t c o n v e r -gence has o c c u r r e d when t h e a b s o l u t e v a l u e o f f(H) i s l e s s t h a n some a r b i t r a r i l y s m a l l f u n c t i o n o f K , f o r i n s t a n c e : | f ( H ) | < . 0 0 0 0 0 1 K - 18 -rl K I t i s o n l y n e c e s s a r y , t h e n , t o d e t e r m i n e - ^ — . T h i s t a s k On i s s i m p l i f i e d b y u s i n g t h e f o l l o w i n g f u n c t i o n s : oE o -JL a-f 3 - t A - W J L + A -n - sinh2T sinh2Y A ' H M '/ ' 2 € y €^ 2 =j3y?c sc n 2 U s i n g t h e s e f u n c t i o n s , we c a n r e - w r i t e t h e • g o v e r n i n g e q u a t i o n s : And s a g 1 y = sinhX 3.1 y = -tL ( coshX - cosh A ) 3.2 A = 3.3 c = 3.5 Vo = aySL(€-^ gcothy ) 3.6 v, = aySL(€ + /Scothy ) 3.7 A • L§(iev^ hr + T + ^ y r ) 3.10 sag = £ * - y 3.12 - 1 9 -We can now d i f f e r e n t i a t e to f i n d 4 5 * T h e r e s u l t s a r e : an Case 1 . I f the known v a r i a b l e K i s the USL: Case 2 . 3 . 1 3 I f the known v a r i a b l e K i s the sag at x dsoq -i - \ L I"*. ^ J-2Y* dA \ coshA HA + ——+sinhA-dH H d H Where: dA xr, - ff( l-ycothX) Case 3^  . I f the known v a r i a b l e K i s t e n s i o n T0 3.14 3 . 1 5 Where: 3 . 1 6 3 - 1 7 Case 3i-» I f the known v a r i a b l e K i s t e n s i o n T, dT, dH H dV, d H Where: dV d -4>-rv 3.18 3.19 T h i s method i s not i n f a l l i b l e : there are two ways i n which i t can f a i l . F i r s t l y , the cable equations are not s o l v a b l e f o r any g i v e n c a b l e . We have d e f i n e d w as the loa d per u n i t - 20 -l e n g t h o f c a b l e : t h u s i f we s t r e t c h a c a b l e t o d o u b l e i t s o r i g i n a l l e n g t h we a r e c o n s t r a i n e d t o d o u b l e t h e t o t a l l o a d on i t ( w h i c h i s q u i t e r e a s o n a b l e f o r , s a y , i c e l o a d i n g ) . T h i s e x t r a l o a d i n t u r n p r o d u c e s f u r t h e r s t r e t c h i n g , w h i c h p r o d u c e s f u r t h e r l o a d i n g e t c . T h i s e f f e c t i s c o m p l e t e l y n e g l i g i b l e e x c e p t f o r v e r y h e a v i l y l o a d e d v e r y f l e x i b l e c a b l e s , i n w h i c h i t c a n g e t o u t o f h a nd. F o r t u n a t e l y s u c h c a b l e s e x i s t o n l y i n t h e i m a g i n a t i o n . The o t h e r d a n g e r i s more r e a l : i t i s p o s s i b l e t h a t a t some s t a g e i n t h e s o l u t i o n t h e v a l u e o f H w i l l become p a t h -o l o g i c a l l y s m a l l , o r e v e n n e g a t i v e , i n w h i c h c a s e t h e p r o c e d u r e w i l l succumb t o n u m e r i c a l a i l m e n t s . The a n t i d o t e i s s i m p l y , a t e a c h i t e r a t i o n , t o l i m i t t h e new v a l u e o f H t o be no l e s s t h a n h a l f t h e p r e v i o u s v a l u e . I n p r a c t i c e , t h e s o l u t i o n f o r t h e c a b l e end f o r c e s i s e a s i e r done t h a n s a i d . I n A p p e n d i x 1 a s u b r o u t i n e w h i c h f i n d s c a b l e end f o r c e s ( w r i t t e n i n G - l e v e l /360 F o r t r a n ) i s r e p r o d u c e d , a l o n g w i t h t h r e e m i n o r s u b r o u t i n e s w h i c h c a l c u l a t e — j - . dH F o r c a b l e s w h i c h a r e d e f i n e d b y known s a g o r t e n s i o n v a l u e s i n t h e i n i t i a l p o s i t i o n o f t h e s t r u c t u r e , t h e s o l u t i o n f o r t h e i n i t i a l c a b l e end f o r c e s a l s o g i v e s t h e u n s t r e s s e d l e n g t h s , upon w h i c h t h e c a l c u l a t i o n s o f c a b l e end f o r c e s a r e b a s e d a t s u b s e q u e n t s t a g e s i n t h e s o l u t i o n . 0 - 21 -Chapter 4. The Cable S t i f f n e s s M a t r i x . As was d i s c u s s e d i n Chapter 1, the s t i f f n e s s of a cable changes as i t i s deformed. When we r e f e r to the s t i f f n e s s matrix of a c a b l e , we mean the s e t of d e r i v a t i v e s o f cable end f o r c e s with r e s p e c t to cable end movements evaluated i n the present p o s i t i o n of the cable-. T h i s i s c a l l e d a "tangent" s t i f f n e s s matrix, and i s analagous to a tangent modulus. W i t h i n i t s own cable plane, each cable has f o u r degrees of freedom: two at each end. I f we a s s i g n these degrees of f r e e -dom as shown by the numbered arrows i n F i g . 4.1, then, i g n o r i n g f o r the moment the p o s s i b i l i t y of displacements out of the plane, the c a b l e s t i f f n e s s matrix i s : Hr 3F,, 3 5 , 3F, 3S2 3F , 3 5 3 9F, 354 3 F 2 3 5 , dFz 3 5 2 8 F 2 3 S 3 3 F 2 3 5 4 3 F 3 35 , 3 F 3 35a 8 F 3 3 5 3 a F 3 3S4 3 F 4 3 5 , 3 F 4 3 5 2 . 3 F 4 3 5 3 3F 4 3 5 4 4.1 Where F, i s the f o r c e i n d i r e c t i o n 1 , 5 , »• I s d e f l e c t -i o n i n d i r e c t i o n 1 , e t c . - 22 -I f we now make t h e s u b s t i t u t i o n : aS3 a8< ah =• 3F,=-aF2 = aF3 = 8F 4 = •a§, s§2 aH a v o aH av, Our m a t r i x w i l l become: aH aH - a H - a H aL ah aL ah -avo -a v.. avo avo a L ah 3 L ah -aH -aH an aH aL ah aL ah -av, - a v , av. av, aL ah aL ah 4.2 Degrees of Freedom in the Cable Plane F i g . 4.1. - 23 -Upon e v a l u a t i n g t h e d e r i v a t i v e s , i t i s f o u n d t h a t i n g e n e r a l t h i s m a t r i x i s n o t s y m m e t r i c , and t h a t t h e s e t o f f o r c e s r e p r e s e n t e d b y e a c h c o l u m n i s n o t s e l f - e q u i l i b r a t i n g . T h i s un-u s u a l b e h a v i o u r i s due t o t h e f a c t t h a t we a r e a p p l y i n g a non-c o n s e r v a t i v e l o a d t o t h e c a b l e . By d e f i n i n g t h e l o a d p e r u n i t l e n g t h o f c a b l e a s t h e c o n s t a n t v a l u e w , we have assumed t h a t i f t h e c a b l e i s s t r e t c h e d t h e t o t a l l o a d i n c r e a s e s . T h i s d o e s n o t mean t h a t i n p r a c t i c e o u r c a b l e e q u a t i o n s a r e i n a d e q u a t e : t h e e l a s t i c e l o n g a t i o n i s s m a l l compared t o t h e l e n g t h o f t h e c a b l e , and t h e i n c r e a s e i n l o a d i n g on t h e c a b l e i s i n t h e same s m a l l r a t i o t o t h e t o t a l l o a d . We do f a c e a p r o b l e m , however: c o n v e n t i o n a l s t i f f n e s s a n a l y s i s p r o g r a m s use s y m m e t r i c m a t r i c e s , and i f we w i s h t o m o d i f y s u c h a c o n v e n t i o n a l p r o g r a m t o h a n d l e c a b l e s , we w i l l s a v e a l o t o f t r o u b l e by u s i n g s y m m e t r i c m a t r i c e s f o r o u r c a b l e s . Now t h e asymmetry i n t h e c a b l e m a t r i x i s s m a l l , and i n f a c t i s v i r t u a l l y n e g l i g i b l e i n most c a s e s . A c c o r d i n g l y , we w i l l make m i n o r m o d i f i c a t i o n s t o t h e m a t r i x w h i c h w i l l r e n d e r i t , t h o u g h no l o n g e r s t r i c t l y e x a c t , s y m m e t r i c . T h i s i s r e a d i l y a c h i e v e d b y : R e p l a c i n g : avo 3L by - 8 H 8 h a v i by aH aL ah - 24 -And: Where: av. 8h av* ah by av* an av. ah a Vc ah T h i s g i v e s us a new approximate matrix: [K"] = aH an -aH -aH aL ah ah aH av* - aH - av* ah ah ah ah -aH -aH aH aH aL ah aL ah -aH -av* aH av* a h ah ah ah h.3 Using the approximate matrix does not hinder the s o l u t i o n procedure. F i r s t l y , i t t u r n s out t h a t f o r c a b l e s w i t h i n (and somewhat beyond) the range of e n g i n e e r i n g usage, the approxim-a t i o n s are s m a l l . Secondly, Newton's method does not r e q u i r e the c o r r e c t m a t r i x : a c l o s e one w i l l do. ( 1 ) , ( 2 ) , ( 3 ) « (In the example i n Chapter 2 , f o r i n s t a n c e , any p o s i t i v e f i n i t e value could have been used f o r the cable s t i f f n e s s , ^ ^ , and Newton's method d L would have i n e v i t a b l y l e d to the c o r r e c t s o l u t i o n , though conver-gence might have been slow). I t i s of course necessary to e v a l -uate the unbalanced l o a d e x a c t l y , but t h i s i s independant of cable s t i f f n e s s . Let us now t u r n back to what was ignored at the s t a r t of t h i s chapter: the p o s s i b i l i t y of cable displacements out of the - 25 -cable plane. Two more degrees of freedom are required to describe these displacements, and are numbered 3 and 6 i n F i g . 4.2. As for a pin-ended bar i n tension, the s t i f f n e s s i n u these di r e c t i o n s i s simply y- , so the approximate matrix becomes: 3H 3 L 3H 3 h 0 -3-H 8 L - 3 H ah 0 3 H 8 h 3 V* 3 h 0 ah -av* ah 0 0 0 X 0 O " \ - 3 H 3 L - 3 H 8 h 0 an 8 L 8 H ah 0 - 8 H a h 0 dH ah 3 V * ah 0 0 0 -\ 0 0 \ 4.4 The terms i n the matrices of equations 4.2., 4.3.., and 4.4., are derived i n Appendix 2. In the remainder of t h i s thesis i t w i l l be assumed that the approximate matrix Kca i - s used. Degrees of Freedom for a General Cable F i g . 4.2. - 2 6 -C h a p t e r 5. The C a b l e C o - o r d i n a t e Sy s tem. A t t he end o f C h a p t e r 2 we s e t out t o f i n d t he two new f e a t u r e s w h i c h wou ld enab l e us t o c o n v e r t an o r d i n a r y s t i f f -ness a n a l y s i s program i n t o an improved v e r s i o n c a p a b l e o f hand -l i n g c a b l e s t r u c t u r e s . These two f e a t u r e s were the c a p a c i t i e s t o f i n d t he c a b l e end f o r c e s and the c a b l e s t i f f n e s s m a t r i x , and t h e y have been p r o v i d e d i n C h a p t e r s 3 and 4 . L i k e many m o d i f i c a t i o n s , t h e y do not f i t d i r e c t l y i n t o the e x i s t i n g f ramework , f o r t h e y work i n terms o f c a b l e c o -o r d i n a t e s , and s t i f f n e s s a n a l y s i s programs work i n g l o b a l c o -o r d i n a t e s . To adapt them we need a t r a n s f o r m a t i o n m a t r i x , and t o ge t the t r a n s f o r m a t i o n m a t r i x we need t o know the d i r e c t i o n s o f t he c a b l e c o - o r d i n a t e a x e s . The l o a d p e r u n i t l e n g t h o f c a b l e , W , may be s p l i t i n t o components p a r a l l e l t o t he g l o b a l X , Y , and Z a x e s : wx i W y i and r e s p e c t i v e l y . Thus the c a b l e l o a d i n g may be r e p r e s e n t e d v e c t o r i a l l y a s : W = Wy 5.1 Now y , t h e d i r e c t i o n o f the y - a x i s i n t h e c a b l e c o -o r d i n a t e sys tem must be o p p o s i t e t o t he d i r e c t i o n o f l o a d i n g . A v e c t o r i n t h i s d i r e c t i o n i s t h u s : y = w, - w. - 2 ? -L e t C be the l i n e from the end o f the c a b l e a t the o r i g i n o f the c a b l e c o - o r d i n a t e system t o the o t h e r end o f the c a b l e . ( F i g . 5 . 1 ) . C i s r e p r e s e n t e d by i t s t h r e e components: c = c c 2 5 - 3 Now the c r o s s p r o d u c t o f two v e c t o r s i s p e r p e n d i c u l a r t o b o t h o f them. The c a b l e z - a x i s i s , o f c o u r s e , p e r p e n d i c u l a r t o the c a b l e p l a n e , and s i n c e b o t h y" and C l i e i n the c a b l e p l a n e , Z must be p e r p e n d i c u l a r t o each o f them. Thus we w r i t e : z = C X y 5 . 4 The c a b l e X - a x i s i s , o f c o u r s e , p e r p e m d i c u l a r t o the y and Z -axes, and so i s found by: x = y X z 5 . 5 E q u a t i o n s 5 . 2 . , 5 . 4 . , and 5 . 5 . , d e f i n e v e c t o r s p a r a l l e l t o the c a b l e c o - o r d i n a t e a x e s . I t i s c o n v e n i e n t t o n o r m a l i z e t h e s e v e c t o r s by d i v i d i n g each term by the l e n g t h o f the v e c t o r , A s u b s c r i p t 1 w i l l denote a n o r m a l i z e d v e c t o r . (The components o f a n o r m a l i z e d v e c t o r a r e , o f c o u r s e , the d i r e c t i o n c o s i n e s o f t h e v e c t o r ) . For example, the l e n g t h o f the W v e c t o r ( e q u a t i o n 5 . 1 . ) i s w , where: w = 2 ^ 2 2 Wx + Wy + wz V / , 5 . 6 - 28 -So t h e n o r m a l i z e d c a b l e l o a d i n g v e c t o r would be 'w x /w ^ V l y / W w z/w 5 . 7 The q u a n t i t i e s L and h a r e s i m p l y t h e x and y components o f C , and a r e r e a d i l y f o u n d b y : L = X i - C h =y,-c 5 . 8 5 . 9 Where • r e p r e s e n t s d o t p r o d u c t . I n F i g . 4 . 2 a r e shown t h e s i x d e g r e e s o f f r e e d o m o f a c a b l e . I n t e r m s o f them, t h e c a b l e end f o r c e s ( F i g . 3 . 1 6 ) a r e r e p r e s e n t e d b y t h e v e c t o r : - H V o 0 H V , 0 5 . 1 0 and t h e s t i f f n e s s m a t r i x i s as i n e q u a t i o n ( 4 . 4 ) . To t r a n s f o r m them t o g l o b a l c o - o r d i n a t e s we w i l l u se a 6 x 6 t r a n s f o r m a t i o n - 29 -m a t r i x , J_TJ . T h i s m a t r i x i s composed o f two i d e n t i c a l 3 x 3 s u b -m a t r i c e s ft] a r r a n g e d on t h e d i a g o n a l : o o p 0 0 0 0 0 0! 0 0 0 0 0 0 0 0 0 5 . 1 1 Where t h e t h r e e c olumns o f t h e s u b - m a t r i x a r e t h e v e c t o r s x\, y t , and Z t : [T.] = [ xx y x z t ] I n g l o b a l c o - o r d i n a t e s , t h e c a b l e end f o r c e s a r e 5 . 1 2 q I ' I • c And t h e s t i f f n e s s m a t r i x i s : 5 . 1 3 K CO 9 T c T 5.14 We now know a l l we need i n o r d e r t o c o n v e r t a s t i f f n e s s a n a l y s i s p r o g ram t o s o l v e c a b l e s t r u c t u r e s . I n t h e n e x t c h a p t e r we w i l l c o n s i d e r some ways t o e x t e n d t h e v e r s a t i l i t y o f t h e method. - 30 -Cable and Global Coordinate Systems F i g . 5 . 1 . - 31 -C h a p t e r 6 . A d v a n c e d T o p i c s . W i t h what has b e e n d i s c u s s e d i n t h e p r e v i o u s f i v e c h a p t e r s we a r e now a b l e t o a n a l y z e many c a b l e s t r u c t u r e s . I n t h i s c h a p t e r we w i l l c o n s i d e r some r e f i n e m e n t s w h i c h w i l l make t h e method more g e n e r a l . 1. N o n - l i n e a r B e h a v i o u r o f Non-Cable S t r u c t u r a l Components. I n C h a p t e r 2 i t was assumed t h a t t h e n o n - c a b l e components o f t h e s t r u c t u r e were l i n e a r - t h e i r s t i f f n e s s m a t r i c e s w o u l d n o t change as t h e y d e f l e c t e d . T h e r e a r e many c a s e s where t h i s a s s u m p t i o n i s n o t j u s t i f i e d , f o r i n s t a n c e : i f t h e m a t e r i a l s t r e s s - s t r a i n r e l a t i o n s h i p i s n o n - l i n e a r , o r i f t h e member u n d e r -goes l a r g e r o t a t i o n s , o r i f t h e r e i s a n i n t e r a c t i o n b e t w e e n a x i a l f o r c e and b e n d i n g s t i f f n e s s . I n p r i n c i p l e , t h e s e n o n - l i n e a r i t i e s pose no p r o b l e m - we c a n s i m p l y h a n d l e t h e s e e l e m e n t s j u s t as we h a n d l e t h e c a b l e s , by u s i n g a t a n g e n t s t i f f n e s s m a t r i x , and e v a l u a t i n g t h e member end f o r c e s a t e a c h s u c c e s s i v e d e f o r m e d p o s i t i o n o f t h e s t r u c t u r e . I n p r a c t i c e , t h i s a p p r o a c h c a n be q u i t e d i f f i c u l t , b u t f o r a t l e a s t one o f t h e n o n - l i n e a r i t i e s m e n t i o n e d above t h e r e i s a s i m p l e r p r o c e d u r e . I n frame a n a l y s i s , t h e e f f e c t o f a x i a l f o r c e on t h e b e n d i n g s t i f f n e s s o f a beam i s g e n e r a l l y h a n d l e d by a d i f f e r e n t k i n d o f m a t r i x : a " s e c a n t " m a t r i x . " S t a b i l i t y f u n c t i o n s " (4) d e f i n e t h e s e c a n t m a t r i x i n t e r m s o f t h e a n t i c i p a t e d a x i a l f o r c e . U s i n g t h i s scheme, a s o l u t i o n i s p e r f o r m e d u s i n g t h e l i n e a r m a t r i x . T h i s s o l u t i o n g i v e s a n e s t i m a t e o f t h e a x i a l f o r c e s , and a s e c a n t m a t r i x i s b u i l t b a s e d on t h i s e s t i m a t e . A s o l u t i o n i s now p e r -- 32 -f o r m e d b a s e d on t h i s new m a t r i x , and a b e t t e r e s t i m a t e o f t h e a x i a l f o r c e s r e s u l t s . The p r o c e d u r e i s c o n t i n u e d u n t i l s u c c e s s -i v e s o l u t i o n s c o n v e r g e , w h i c h i s u s u a l l y a c h i e v e d a f t e r j u s t two o r t h r e e s o l u t i o n s . Now t h i s s e c a n t m a t r i x i s n o t t h e t a n g e n t m a t r i x we want, b u t i t i s a l o t c l o s e r t o i t t h a n t h e l i n e a r m a t r i x i s , and as was m e n t i o n e d i n C h a p t e r 4, i t i s n o t n e c e s s a r y t o have t h e c o r r e c t m a t r i x : a c l o s e one w i l l do. M o r e o v e r , t h e s e c a n t m a t r i x l e t s us f i n d t h e member end f o r c e s : i n i t s d e f o r m e d p o s i t i o n we c a l c u l a t e t h e a x i a l f o r c e i n t h e member ( w h i c h i s no p r o b l e m s i n c e t h e a x i a l s t i f f n e s s i s c o n s t a n t ) and t h e n t h e s e c a n t m a t r i x b a s e d on t h e a x i a l f o r c e . M u l t i p l y i n g t h e s e c a n t m a t r i x by t h e member d e f l e c t i o n s now g i v e s t h e t r u e ( e x a c t ) n o n - l i n e a r b e n d i n g moments and s h e a r s . The p o i n t t o n o t e a b o u t t h e s e c a n t m a t r i x i s t h i s : i f t h e g u e s s e d v a l u e o f a x i a l f o r c e i s c o r r e c t , t h e n t h e s e c a n t m a t r i x i s l i n e a r , w i t h r e s p e c t t o b e n d i n g and s h e a r d e f o r m a t i o n s . The a x i a l f o r c e s i n t h e members t e n d t o c o n v e r g e r a p i d l y on t h e i r f i n a l v a l u e s , so t h e beams behave q u i t e l i n e a r l y by t h e t i m e t h e f i n a l s o l u t i o n i s a p p r o a c h e d . T h e r e i s one d a n g e r i n t h i s method: i f t h e a x i a l l o a d s a r e t o o g r e a t , t h e s t i f f n e s s m a t r i x w i l l become s i n g u l a r . I t i s p o s s i b l e t h a t i f a s t r u c t u r e were l o a d e d a l m o s t t o i t s c r i t i c a l l o a d t h e N e w t o n i a n s e q u e n c e w o u l d wander beyond t h e c r i t i c a l l o a d and s u f f e r a p r e m a t u r e d e m i s e . T h i s p r o b l e m o c c u r s r a r e l y , and c a n be o b v i a t e d b y a p p l y i n g t h e l o a d i n s t e p s : f o r s m a l l s t e p s t h e e r r o r i n e a c h s t a g e o f t h e i t e r a t i o n i s r e d u c e d , and t h e - 33 -d e f l e c t i o n s n e v e r d e v i a t e f a r f r o m t h e s o l u t i o n . I n o t h e r words we w i l l f i r s t s o l v e t h e p r o b l e m f o r , s a y , 50 p e r c e n t o f t h e r e -q u i r e d l o a d , t h e n f o r 75 p e r c e n t u s i n g t h e p r e v i o u s s o l u t i o n as t h e s t a r t i n g p o i n t , t h e n f o r 85 p e r c e n t s t a r t i n g a t t h e s o l u t i o n f o r 75 p e r c e n t , and so on. Beam-column e f f e c t s a r e o f t e n s i g n i f i c a n t i n t h e b e h a v i o u r o f guyed t o w e r s . S t a b i l i t y f u n c t i o n s were u s e d , as d e s c r i b e d above, i n t h e s e c o n d example o f C h a p t e r 7. 2. S p e c i f i e d C a b l e T e n s i o n s . I f we know t h e u n s t r e s s e d l e n g t h o f a c a b l e , we c a n f i n d i t s end f o r c e s f o r any d e f o r m e d p o s i t i o n . I f , i n t h e i n i t i a l p o s i t i o n o f t h e s t r u c t u r e , we know t h e s a g o f t h e c a b l e o r t h e t e n s i o n a t e i t h e r end, we c a n use t h e methods o f C h a p t e r 3 "to f i n d t h e u n s t r e s s e d l e n g t h ( b y f i r s t f i n d i n g H). F r e q u e n t l y , however, t h e c a b l e s i n a s t r u c t u r e a r e t e n s i o n e d t o p r e d e t e r m i n e d v a l u e s a f t e r t h e s t r u c t u r e i s e r e c t e d . I n t h i s c a s e we do n o t know t h e p o s i t i o n o f t h e s t r u c t u r e , f o r i t d e f o r m s f r o m i t s un-s t r e s s e d p o s i t i o n a s t h e c a b l e s a r e t e n s i o n e d . The s o l u t i o n t o t h i s p r o b l e m i s s u r p r i s i n g l y s i m p l e : a t e a c h s u c c e s s i v e d e f o r m e d p o s i t i o n f o u n d d u r i n g t h e s o l u t i o n p r o -c e s s , we r e - t e n s i o n t h e c a b l e s t o t h e i r s p e c i f i e d t e n s i o n s ( b y c h a n g i n g t h e i r u n s t r e s s e d l e n g t h s ) . When c o n v e r g e n c e i s a c h i e v e d , we c a l c u l a t e t h e t r u e u n s t r e s s e d l e n g t h s o f t h e c a b l e . T h i s a n a l y s i s , o f c o u r s e , c o n s i d e r s o n l y t h e l o a d s a p p l i e d a t t h e t i m e o f t h e c a b l e t e n s i o n i n g . Once t h e u n s t r e s s e d l e n g t h s o f t h e c a b l e s a r e known, o t h e r l o a d i n g c a s e s ( w i n d , snow, e t c . ) - 34 -a r e r e a d i l y h a n d l e d i n t h e u s u a l f a s h i o n . The s e c o n d example c o n s i d e r e d i n C h a p t e r 7 employed t h i s method o f s p e c i f y i n g c a b l e t e n s i o n s . 3. M i s c e l l a n e o u s P r o b l e m s . The e f f e c t s on c a b l e s o f t e m p e r a t u r e c h a n g e s , end s l i p p a g e , and t u r n b u c k l e a d j u s t m e n t a r e r e a d i l y h a n d l e d by c h a n g i n g t h e i r u n s t r e s s e d l e n g t h s . 4. C a b l e L o a d s . The l o a d s on a c a b l e due t o i t s own w e i g h t and t h e w e i g h t o f a c c u m u l a t e d i c e a r e r e a d i l y e v a l u a t e d . F o r w i n d l o a d i n g t h e e v a l u a t i o n i s more d i f f i c u l t . F i r s t l y , t h e wind l o a d i n g a c t s p e r p e n d i c u l a r t o t h e c a b l e . S i n c e t h e c a b l e i s c u r v e d , t h e d i r e c t i o n o f t h e wind l o a d i n g v a r i e s a l o n g t h e l e n g t h o f t h e c a b l e , w h i c h i s c o n t r a d i c t a r y t o t h e t h e o r y o f c a t e n a r y c a b l e s . We a v o i d t h i s e m b a r r a s s e m e n t i n a r a t h e r d i r e c t manner: i f a c a b l e i s r e a s o n a b l y t a u t , we t r e a t t h e d i r e c t i o n o f e a c h e l e m e n t o f t h e c a b l e as b e i n g t h e same as t h a t o f t h e l i n e b e t w e e n t h e ends o f t h e c a b l e : C i and a p p l y t h e w i n d l o a d n o r m a l t o C , ( F i g . 5»1«) I f t h e c a b l e has a l a r g e s a g , we s i m p l y t r e a t i t as a s e r i e s o f s h o r t e r c a b l e s , e a c h o f w h i c h has a low s a g . Now f o r w i n d a c t i n g p e r p e n d i c u l a r t o a c a b l e , t h e d r a g f o r c e p e r u n i t l e n g t h i s : w Lf T P < l v 2 C a 6.1 - 35 -Where P i s t h e d e n s i t y o f t h e a i r , d i s t h e c a b l e d i a m -e t e r , v i s t h e wi n d v e l o c i t y , and C d i s t h e c o e f f i c i e n t o f d r a g f o r t h e c a b l e . A r e a s o n a b l e v a l u e f o r C d i s 1.2. A i r a t s . t . p . w e i g h s .08071 l b s / f t . 3 I t has b e e n shown (5) t h a t i f t h e wi n d d i r e c t i o n i s a t an a n g l e 7) t o t h e p l a n e p e r p e n d i c u l a r t o t h e c a b l e ( t h a t i s , t h e p l a n e p e r p e n d i c u l a r t o C ) t h e d r a g i s s t i l l p e r p e n d i c u l a r t o t h e c a b l e and has m a g n i t u d e : The w i n d v e l o c i t y may be r e p r e s e n t e d by i t s t h r e e g l o b a l components V x , v y , and V 2 , so t h a t v e c t o r i a l l y i t i s : 6.2 V 6.3 The d i r e c t i o n o f t h e d r a g , wrf » i s f o u n d b y : wd = C X V I C 6.4 And when n o r m a l i z e d i s w r i t t e n W, 'dl Cos Tj i s s i m p l y : 6.5 So t h e c a b l e l o a d i n g v e c t o r due t o wi n d i s : 6.6 wind = W d P r o g C O s 2 7 ? W < " - 36 -Chapter 7. Examples.  Example 1. This simple example has been solved by others (6), (7). A cable spans 1,000 feet h o r i z o n t a l l y between fixed supports, the midspan sag being 100 feet. The cable weighs 3.16 l b s / f t , i s 2 6 0.85 i n area, and has an e f f e c t i v e modulus of 19x10 p s i . A v e r t i c a l load of o i s then placed 400 feet from the l e f t support. As solved by Frances and O'Brien (7) the loaded point moves from x = 400', y = 96.0495' to x = 397.180', y = 114.509'. The problem was solved by the methods presented herein, using a number of d i f f e r e n t i n i t i a l p o sitions. The convergence c r i t e r i o n used was that the unbalanced forces should a l l be less than one pound. The procedure converged to the same f i n a l p o s i t i o n as that found by Frances and O'Brien, regardless of the i n i t i a l p o s i t i o n chosen. The number of i t e r a t i o n s required to achieve convergence for each i n i t i a l p o s i t i o n i s shown i n Table 7.1. It i s apparent upon examining the range of s t a r t i n g points used that the method i s not p a r t i c u l a r l y vulnerable. For the case where the s t a r t i n g point was x = 400', y = -50' the v e r t i c a l s t i f f n e s s i n the i n i t i a l p o s i t i o n was so small that the f i r s t s o l u t i o n led to a value of y which was about 700' too low! Never-theless, the correct solution was eventually found, though twenty i t e r a t i o n s were required. For an i n i t i a l p o s i t i o n which was i n any sense reasonable, only f i v e or six i t e r a t i o n s were required. - 37 -CONVERGENCE OF EXAMPLE 1. CASE IN IT IAL POSITION NUMBER OF ITERATIONS REQUIRED TO ACHIEVE CONVERGENCE. X y 1 400 100 10 2 400 0 15 3 400 -50 20 4 400 -96.0495 7 5 400 -110 5 6 400 -120 5 7 400 -200 6 8 400 -300 6 9 350 -110 6 10 390 -110 6 11 410 -110 5 12 450 -110 8 T a b l e . 7.1. - 38 -Example 2. I n t h i s example t he e f f e c t s o f v a r y i n g t he i n i t i a l c a b l e t e n s i o n s and t he b e n d i n g s t i f f n e s s i n t he mast o f a guyed tower a re i n v e s t i g a t e d . The tower i s 1,000' h i g h ( u n s t r e s s e d ) and i s ancho red a t i t s t o p and m i d - p o i n t s by f o u r c a b l e s a t each l e v e l . The c a b l e s a r e ancho red 700' f rom the t o w e r . A r e a s , w e i g h t s , and l o a d i n g a r e as shown i n F i g . 7.1. Four d i f f e r e n t s l e n d e r n e s s r a t i o s f o r t he mast were c o n s i d -e r e d : l / r = 310, l / r = 269, l / r = 240 and l / r = 219, where 1 = t o t a l tower h e i g h t . The c a b l e s were s e t t o t he p r e d e t e r m i n e d t e n s i o n s under t h e i n f l u e n c e o f c a b l e and mast dead l o a d s o n l y . The a n t e n n a l o a d s a t the t o p o f the mast and t he w ind l o a d s were t h e n a p p l i e d , and t he deformed shape and member end f o r c e s f o u n d . Beam - co lumn e f f e c t s on t he mast were c o n s i d e r e d as d e s c r i b e d i n C h a p t e r 6. The c a b l e s were i n i t i a l l y t e n s i o n e d , f o r each s l e n d e r n e s s r a t i o o f t he mas t , t o 10, 20, 30 and 40 k s i a t t h e i r b o t t o m s , g i v i n g a t o t a l o f 16 c a se s c o n s i d e r e d . Some t y p i c a l r e s u l t s under t h e t o t a l l o a d a r e shown i n F i g s . 7.2 t o 7.7. The a x i a l f o r c e i n the mast was u n a f f e c t e d by c h a n g i n g the s l e n d e r n e s s r a t i o , bu t i n c r e a s e d m a r k e d l y w i t h i n c r e a s i n g c a b l e p r e t e n s i o n , as shown i n F i g . 7.2. - 39 -Note: the values of l/r = 310,269, 240,219 correspond to r 2( inches2) = 1500,2000,2500,3000 Note: similar cables out-of-plane G u y e d Tower F i g . 7.1. - 4 0 -< 2 5 0 I 1 1 1 1 1_ 10 20 3 0 4 0 50 Initial c o b l e s t ress (ksi) F i g . 7 . 2 . The bending moments i n the mast i n c r e a s e d as the b u c k l i n g l o a d was approached. Thus i n F i g . 7.3 we see the moment i n c -r e a s i n g as the p r e t e n s i o n or the slenderness r a t i o i n c r e a s e d . F i g . 7 . 3 . - 4 1 -The d e f l e c t i o n s f o l l o w e d a p r e d i c t a b l e p a t t e r n : h i g h e r i n i t i a l guy t e n s i o n s d e c r e a s e d t h e d e f l e c t i o n s a t t h e guy a t t a c h -ment p o i n t s , w h i c h were r e l a t i v e l y u n a f f e c t e d by t h e t o w e r s l e n d e r -n e s s r a t i o , b u t i n c r e a s e d t h e d e f l e c t i o n s a t t h e 750' l e v e l as t h e b u c k l i n g l o a d was a p p r o a c h e d . The d e f l e c t i o n s a t t h e 250' l e v e l were q u i t e s m a l l due t o t h e h i g h e r s t i f f n e s s o f t h e l o w e r h a l f o f t h e m a s t . The d e f l e c t i o n s a t t h e 1,000' and 750' p o i n t s a r e p l o t -t e d i n F i g s . 7.4 and 7.5. to > 1 6 O O 2 5 _ +-- a » o o » o o v O 10 l/r = 310, 2 6 9 , 2 4 0 , 2 1 9 20 30 4 0 In i t i a l c a b l e s t r e s s ( k s i ) F i g . 7.4. o m N 2 0 5 *" 1 5 o a > v o •*-- 10 • 4 -a a JL l /r = 310 l/r = 2 6 9 l/r = 2 4 0 l/r : 219 10 20 3 0 4 0 In i t i a l c a b l e s t r e s s ( K s i ) F i g . 7.5. - 42 -F i n a l l y , t h e guy t e n s i o n s on t h e windward s i d e i n c r e a s e d w i t h i n i t i a l t e n s i o n , as m i g h t w e l l have b e e n e x p e c t e d . The t e n s i o n i n t h e h i g h e r c a b l e was a l m o s t u n a f f e c t e d by t h e v a r y i n g s t i f f n e s s e s o f t h e mast, w h i l e t h a t i n t h e l o w e r c a b l e i n c r e a s e d w i t h i n c r e a s i n g s l e n d e r n e s s o f t h e mast, as i n d i c a t e d i n F i g s . 7. and 7.7. 6 0 a. o Q. O ^ 3 — •o in e *• * tt T3 tt c <D — £ X 50 4 0 10 " r • 219 l/r = 3 1 0 20 30 4 0 I n i t i a l c a b l e t e n s i o n s t r e s s (ks i ) F i g . . 7 . 6 . . F i g . 7.7. - 43 -T h i s example i s n o t i n t e n d e d t o be a c a s e s t u d y , b u t r a t h e r t o p o i n t o u t t h e f a c i l i t y w i t h w h i c h t h e method c a n h a n d l e o t h e r w i s e i n t r a c t a b l e p r o b l e m s . Over t h e 16 s o l u t i o n s p e r f o r m e d , on a v e r a g e o f 2.5 i t e r a t i o n s were r e q u i r e d t o s e t t h e p r e t e n s i o n s i n t h e c a b l e s , and on a v e r a g e o f 3*25 i t e r a t i o n s were r e q u i r e d t o s o l v e t h e l i v e - l o a d c o n d i t i o n . The f u l l s e t o f 16 s o l u t i o n s were p e r f o r m e d i n a b o u t 70 s e c o n d s on a n IBM 360-67. _ 44 -C h a p t e r 8. D i s c u s s i o n . I n g e n e r a l , t h e Method p r o p o s e d h e r e i n w i l l c o n v e r g e f r o m t h e i n i t i a l p o s i t i o n t o t h e n e a r e s t s t a b l e e q u i l i b r i u m p o s i t i o n . The c l o s e r t h e i n i t i a l and f i n a l p o s i t i o n s , t h e f a s t e r t h e Method w i l l c o n v e r g e . I f t h e s t r u c t u r e s t a r t s a t , o r e r r o n e o u s l y wanders i n t o , an u n s t a b l e c o n f i g u r a t i o n , t h e s t i f f n e s s method w i l l b r e a k down. I t i s r a t h e r h a r d t o make a r e l i a b l e p r e d i c t i o n a s t o w h e t h e r t h e Method w i l l c o n v e r g e o r n o t f o r a g i v e n s t r u c t u r e . We may* s a y , however, t h a t i f a s t r u c t u r e h as a w e l l - d e f i n e d p o s i t i o n o f s t a b l e e q u i l i b r i u m , and i s n o t u n s t a b l e c l o s e t o t h i s p o s i t i o n , t h e n f o r an i n i t i a l p o s i t i o n r e a s o n a b l y c l o s e t o t h i s p o i n t t h e Method w i l l c o n v e r g e t o i t . G i v e n t h e p r e s e n t s t a t e o f m a t h e m a t i c a l knowledge r e g a r d i n g Newton's Method, we c a n s a y no more. More o p t i m i s t i c a l l y , we c a n p o i n t o u t t h a t , i n t h e p r o b l e m s so f a r p r e s e n t e d t o i t , t h e Method has n o t f a i l e d t o c o n v e r g e on a s t a b l e s o l u t i o n i f one e x i s t e d . The a d v a n t a g e s o f t h e Method may be summarized: ( 1 ) The r e s u l t i s an e x a c t s o l u t i o n o f t h e e q u a t i o n s c h o s e n t o d e s c r i b e t h e b e h a v i o u r o f t h e s t r u c t u r e . (2) The Method c a n be a p p l i e d t o s t r u c t u r e s w h i c h c o n t a i n many c a b l e s ( s l a c k o r t a u t ) and many n o n - c a b l e e l e m e n t s . * By t r a n s l a t i n g t h e m a t h e m a t i c a l p r o o f o f c o n v e r g e n c e g i v e n by G o l d s t e i n (3) ( A p p e n d i x 3) i n t o s t r u c t u r a l t e r m i n o l o g y . - 45 -(3) A l r e a d y e x i s t i n g s t i f f n e s s a n a l y s i s p r o g r a m s c a n be a d a p t e d t o t h e Method w i t h o u t g r e a t d i f f i c u l t y . And i t s d i s a d v a n t a g e s : (1) B e c a u s e o f t h e g e n e r a l i t y o f t h e Method, i t d o e s n o t s o l v e c e r t a i n r e s t r i c t e d t y p e s o f p r o b l e m s as e f f i c -i e n t l y as more s p e c i f i c m e thods. (2) The Method may b r e a k down ( t h o u g h t h i s a p p e a r s t o be a v e r y r a r e o c c u r r e n c e ) . - 46 -B i b l i o g r a p h y . L i v e s l e y R.K., " M a t r i x Methods o f S t r u c t u r a l A n a l y s i s " , Pergamon P r e s s , 1964. J o h n F., " L e c t u r e s on N u m e r i c a l A n a l y s i s " , G o r d e n and B r e u c h , I967. K a n t o r o v i t c h L.V. and A k i l o v G.P., " F u n c t i o n a l A n a l y s i s i n Normed S p a c e s " , M a c M i l l a n , New Y o r k , 1964. H o m e and M e r c h a n t . "The S t a b i l i t y o f Frames", Pergamon P r e s s , I965. R e l f E . F . and P o w e l l C.H., " T e s t s on Smooth and S t r a n d e d W i r e s I n c l i n e d t o t h e Wind D i r e c t i o n , and a C o m p a r i s o n o f R e s u l t s on S t r a n d e d W i r e s i n A i r and W a t e r " . A s s o c i a t e d R e s e a r c h Committee, R & M 30?, London, J a n . 1917. R e s u l t s r e p r o d u c e d i n : N a t i o n a l R e s e a r c h C o u n c i l o f Canada, R e p o r t MER-1 "The A n a l y s i s o f t h e S t r u c t u r a l B e h a v i o u r o f Guyed A n t e n n a M a s t s Under Wind and I c e L o a d i n g " , Ottawa, 1956. M i c h a l o s J . and B i r n s t i e C , "Movements o f a C a b l e Due t o Changes i n L o a d i n g " , T r a n s . . ASCE, V127, 1962. P a r t 11. O ' B r i e n W.T. and F r a n c i s A . J . , " C a b l e Movements Under Two D i m e n s i o n a l L o a d s " , J . S t r . D i v . ASCE, V 9 0 , No. ST3• June 1964. P a r t 1. G o l d s t e i n A.A., " C o n s t r u c t i v e R e a l A n a l y s i s " , H a r p e r and Row, I967. - i -Appendix 1. L i s t i n g of FORTRAN Subroutine to C a l c u l a t e  Cable-End F o r c e s . The f o l l o w i n g i s a l i s t i n g of a F o r t r a n Subroutine which uses Newton's Method to c a l c u l a t e the end-forces of a cable as a f u n c t i o n of known USL, sag at some p o i n t x, or end t e n s i o n T or T. . o 1 A l s o i n c l u d e d are three f u n c t i o n sub-programs to c a l c u l a t e the d e r i v a t i v e o f the known f u n c t i o n w i t h r e s p e c t to the h o r i -z o n t a l components of cable t e n s i o n (H). - i i -APPENDIX 1 C-**** SUBROUTINE TO DETERMINE CABLE END FORCES SUBROUTINE CABPOS<W,EL,V,USL,AREA,E,H,A,B,T0,Tl,SAG,X) IMPLICIT REAL*8tA-H,0-Z) C***» FOR GIVEN : w = LOAD IN POUNDS PER LINEAR FOOT c** *« EL HORIZONTAL LENGTH C **** V = VERTICAL LENGTH c « * * * USL = UNSTRETCHED LENGTH OF CABLE c * * * * AREA AREA OF CABLE SAG SAG OF CABLE AT X c * * * * E = MODULUS OF CABLE TO = TENSION AT BEGINNING c * * « * T l TENSION AT END c* * *« C**«* NOTE: IF SAG.NE.O CALCS WILL BE BASED ON SAG C***« I F SAG.EQ.O AND TO OR T l NE.O CALCS WILL BE BASED ON C***« TENSION, OTHERWISE CALCS WILL BE BASED ON USL C**** C***« THIS PROGRAM USES A NEWTON-RAPHSON METHOD TO CALCULATE THE C***« POSITION OF THE CABLE, AND RETURNS VALUES OF: C**** USL C*«** H = HORIZONTAL TENSION C**«* T0,T1 {NOTE: IF VERT COMP < 0, T IS SET < 0) C**«* SAG AT X C**«* A,B = CONSTANTS IN THE CABLE EQUATIONS: C***« C«*** Y = H/W*COSH(W*X/H+A)+B C**** Y* = SINHtWX/H+A) C**#* NITER = 0 C * * « * GUESS A VALUE OF H, IF NECESSARY IFtH.EQ.O.>H=W*EL/2. IFC H.LT.(W*EL/20.))H=W*EL/20. AE=AREA*E 1 CONTINUE IFtSAG.NE.O.)G0 TO 7 IFCTO.NE.O.O.OR.Tl.NE.0.0)G0T011 CALL DUSLDHtW,EL,V,AE,H,USL$,DERI V) F=USL-USL$ C**«* F = ERROR IN CALCULATED USL IF{DABS(F/USL).LT.1.D-8)G0T02 G0T08 7 CALL DSAGDH(W,EL,V,AE,H,X,A,B,SAG$tDERIV> F=SAG-SAG$ C***# F = ERROR IN CALCULATED SAG IF(DABS(F/SAG).LT.1.D-6)G0T02 GOT 08 - I l l -11 I F (TO.HE.0.) CALL DTODH (W, EL, V, AE,H, T$ , DERIV, & 1 2, & 1 3) CALL DT1DH (W,EL,V,AE,H,T$,DERI V,& 12,&13) 12 F=ro-rf C * * * * F = ERROR IH CALCULATED TENSION AT BEGINNING I F (DABS (F/TO) . LT. 1.D-6) G0T02 G0T08 13 F=T1~T$ c * * * * F = ERROR IN CALCULATED TENSION RT ESD I F (DABS (F/T1) . LT. 1.D-6) GDT02 8 DELTAH-F/DERIV IF{(DELTAH+H/2.).LT.O.)DELTAH=-H/2. H=H+DELTAH KITEB=BITER+1 IF ( N I T E R . L T . 1 3 ) G 0 T 0 1 WRITE(6,100) H,F 100 FORMAT(' CABPOS: NO SOLS AFTER 12 ITERATIONS. H=•,F9 * • F=',F9.6)) C**** WRAP UP - GET UNKNOWNS 2 IF(SAG.NE.0.)GO TO 9 IF(TO.HE.0.0.OR.T1.NE.0.0)GOT015 CALL DSAGDH (W,EL,V,fcE,H,X,A,B,SAG$,DERIV) SAG=SAG$ GO TO 10 9 CALL DUSLDH(W fEL,V,AE,H,USL$,DERIV) USL=USL$ GOTO10 15 CALL DSAGDH (W,EL,V,AE,H,X,A, B, SAG, DERIV) CALL DUSLDH(W,EL,V,AE,H,USL,DERIV) 10 CONTINUE S0=DSIHH(A) S 1 = D S I » H (W*EL/H+A) T0=H*DSQRT (1. +S0*S0) *DSIGN (1. DO,SO) T 1=H*DSQRT (1. +S 1*S1)*DSIGN (1. D0,S1) WRITE(7,101)NITER,H 101. FORMAT(13,' ITERATIONS. H=',F13.5) RETURN END - iv -SUBROUTINE DUSLDH (W , EL, V, AE, H , US L$, DERIV) IMPLICIT REAL*8 (A-H,0-Z) S/R TO FIND USL AMD D(USL)/DL FOR k CABLE BE=V/EL GA=W*EL/2./H SHGA=DSINH (GA) CHGA=DCOSH (GA) DE=H/AE EP=DSQST(BE*BE+SHGA*SHGA/GA/3A) rH=DE*(BE*BE*GA*CHGA/SHGA+0.5+SHGft*CHGA/2./GA) OSL$= (EP-TH)*EL EIA=SHGA*CHGA/EP/G&-SHGA*SHGA/GA/GA/EP PSI=BE*BE*GA* (CHGA/SHGA-S ft/SHGA/SHGA)-SHGA*CHGA/2./GA 1+(1.+2.*SHGA*SBGA)/2. DERIV = EL/R* (- ET A - T E* D E* PSI) RETURN END - V -SUBROUTINE DSAGDH(W,EL,V,AE,H,X,A,B,SAG$, DE81V) IHPLICIT REAL*8 (A-H,0-Z) c * * « * s/R TO FIND SAG, D (SAG) /DH, A AND B FOR A CABLE C**** NOTE THAT THE EQUATIONS ARE SLIGHTLY REWORKED C**** FROH THOSE IN THE THESIS IN ORDER TO INCREASE C**** COMPUTATIONAL EFFICIENCY AL=AE/EL BE=V/EL GA=W*EL/2./H SHGA=DSINH (GA) CHGA=DCOSH {GA) A=DRSINH(GA*BE/SHGA) -GA B=-EL/2./GA*DC03H (A) SAG$=BE*X-EL/2./GA*DC0SH (2. *GA*X/EL*A) -B DE=H/& E DGADH=-GA/H DADH=1./DSQRT (1.+GA*GA*BE* BE/SHGA/SHGA) *(BE/SHGA-GA*BE 1/SHGA/SHGA*CHGA)*DGADH-DGADH DBDH=EL/2./GA/GA*DCOSH(A)*DGADH-EL/2./GA*DSINH(A)*DADH DERIV=-1./W*DC0SH(2.*GA*X/EL+A)-EL/2./GA*DSINH( 12.*GA*X/EL+A)* (2,*X/EL*DGADH+DADH)-DBDH RETURN END - v i -SUBROUTINE DTODH (W , EL , V, AE, H, T $, DER IV , *, * ) IMPLICIT REAL*8 (A~H,0-Z) c * * « * S / B T 0 F I H D TENSION AND D (TENSION)/DH FOR A CABLE TMOLT=-1. GO TO 1 ENTRY DT1DH(W,EL,V,AE,H,T$,DERIV,*,*) TMULT=1. 1 CONTINUE AL=AE/EL BE=V/EL GA=W*EL/2./H SHGA = DSINH{GA) CHGA = DCOSH (GA) DE=H/AE EP=DSQRT(BE*BE+S HGft*SHGA/G A/3A) ETA=SHGA*CHGA/EP/GA-SHGA*SHGA/GA/G&/EP PHI=BE*GA*GA/SHGA/SHGA VV=AL*GA*DE*EL*(BE*THULT*CHGA/SHGA+EP) T$=DSQRT (H*rI + VV*VV) DVDH=TMULT*PHI-GA*ETA DERIV=1./T$* (H + VV*DVDH) I F (TMULT.LT. 0.0) RETURN 1 RETURN 2 END - v i i -A p p e n d i x 2. D e r i v a t i o n o f Terms i n t h e C a b l e M a t r i x . I n e q u a t i o n s (4.2), (4.3) and (4.4) t h e s t i f f n e s s m a t r i x f o r a c a b l e was d e f i n e d i n t e r m s o f s e v e r a l d e r i v a t i v e s . I n t h i s A p p e n d i x t h e s e d e r i v a t i v e s a r e e v a l u a t e d . I n a d d i t i o n t o t h e s y m b o l s d e f i n e d i n C h a p t e r 3t w e w i l l r e q u i r e a n a d d i t i o n a l f u n c t i o n : The u n s t r e s s e d l e n g t h o f a c a b l e i s c o n s t a n t , s o : a^SL . ai£ iA ) , 0 A 2 a a h dh So : a c . a A ah " a h A2.2 From e q u a t i o n (3.5) -3C - Ot V BH_ ah ' p / € og - ah A2.3 And f r o m e q u a t i o n (3.10) A2.4 - v i i i -S o l v i n g e q u a t i o n s ( A 2 . 3 ) and (A2.4) f o r ^tL 3h | y - ^ [ 2 y S c o , h y - ± ] D i f f e r e n t i a t i n g e q u a t i o n s (3 . 7 ) and (3.6) we f i n d : |^=a[ r8(coth r +/9/€)] + «t>-yr)) | i i 3h 2 y S c o t h * / j - 2<j> 3 H 3Vt 3h + 3 h As d e f i n e d i n C h a p t e r 4 , t h e a p p r o x i m a t e term: 3h " 2 3h 3h So 3 C 3A 3 L " 3 L A2.5 A2.6 A2.7 A2.8 Or = a/Scothy + ^> A2.9 As i n e q u a t i o n (A2.1) 3USL 8 ( C - A ) 3 L 3 L ' U A 2 - i o A 2.ll - i x -From eq u a t i o n (3.5) 8 L € € + 7> o £ 8 l _ And from e q u a t i o n (3.10) A2.12 3A a S o l v i n g e q u a t i o n (A2.12) and (A2.13) f o r dH dL A2.13 dH dL -a Or dL " " ^ a F A2.14 A2.15 And d i f f e r e n t i a t i n g equations (3.7) and (3.6) we f i n d : U-a[y8(€-% + i,-*v ) ] + ( « - ^ ) |H And 9 H , 3V, A2.16 A2.17 Knowing H . equation (A2.5), (A2.9) and (A2.15) are r e a d i l y e v a l u a t e d to p r o v i d e the cable s t i f f n e s s matrix o f equations (4.3) or (4.4) - X -A p p e n d i x 3. Newton's Method. Newton's Method ( a l s o c a l l e d t h e Newton-Raphson method) i s one o f t h e o l d e s t , s i m p l e s t , and b e s t p r o c e d u r e s f o r t h e s o l u t i o n o f n o n - l i n e a r e q u a t i o n s . I t i s somewhat s u r p r i s i n g t h e r e f o r e , t h a t i t was n o t u n t i l c o m p a r a t i v e l y r e c e n t l y t h a t m a t h e m a t i c i a n s were a b l e t o come t o g r i p s w i t h i t . Theorems d e s c r i b i n g t h e c o n v e r g e n c e o f Newton's Method may be f o u n d i n K a n t o r o v i t c h (3), J o h n (2) and G o l d s t e i n (3)» Q u a n t i t -a t i v e l y t h e s e t h e o r e m s a r e o f l i t t l e use t o u s , b u t q u a l i t a t i v e l y t h e y a r e most v a l u a b l e i n t h a t t h e y show t h a t Newton's Method w i l l c o n v e r g e t o a s t a b l e s o l u t i o n ( a s s u m i n g one e x i s t s ) f o r a s t r u c t u r e p r o v i d e d o n l y t h a t t h e i n i t i a l p o i n t i s c l o s e enough t o t h e s o l u t i o n . C o n s i d e r t h e f o l l o w i n g t h e o r e m s * ( b a s e d on G o l d s t e i n ' s Theorem 1, C h a p t e r C-4, page 143, w h i c h i n i t s t u r n was b a s e d on K a n t o r o v i t c h ' s w o r k * * ) . I n i t l e t : be d e f l e c t e d s h a p e s o f t h e s t r u c t u r e . be t h e s t i f f n e s s m a t r i x a t X be t h e u n b a l a n c e d l o a d s a t X K(x) UBL, . (x) Which we w i l l m e r e l y s t a t e , and n o t p r o v e . The i n s i s t e n t ' r e a d e r may v e r i f y f o r h i m s e l f i t ' s i s o m o r p h i s m t o G o l d s t e i n ' s t h e o r e m . Some r e a d e r s m i g h t f i n d K a n t o r o v i t c h somewhat t o o h i r s u t e t o be r e a d i l y d i g e s t e d , hence t h e r e f e r e n c e s t o G o l d s t e i n and J o h n , - x i -L e t a p o i n t X 0 ( t h e i n i t i a l p o i n t ) L e g i v e n f o r w h i c h K_1(x0) e x i s t s . S e t 7)o = || K'Hxo) U B L ( X o ) l l= t h e l e n g t h o f t h e i n c r e -m e n t a l d e f l e c t i o n v e c t o r c a l c u l a t e d a t X 0 . D e f i n e t h e s p h e r e S s u c h t h a t i t c o n t a i n s a l l X where: II x- X o 11$ 27?0 ( S i s t h u s a s p h e r e i n d e f l e c t i o n s p a c e c e n t e r e d a t X o and h a v i n g a r a d i u s e q u a l t o 27] 0). D e f i n e ^ 0 = || K ^ ( X o ) ! ! = t h e i n v e r s e o f t h e s m a l l e s t e i g e n v a l u e o f K (x0) . ( I n a h i g h l y s t a b l e p o s i t i o n o f t h e s t r u c t u r e j 3 o i s s m a l l and p o s i t i v e , i t becomes l a r g e r as a n u n s t a b l e p o s i t i o n i s a p p r o a c h e d , and i s i n f i n i t e a t a p o s i t i o n o f n e u t r a l e q u i l i b r i u m ) . I f a numberk e x i s t s s u c h t h a t : II K(x)- K ( y ) l l > k l l x ^y ' ! f o r a l l x and y i n S A n d ! /§o7? 0 k$ l/2 Then t h e s t r u c t u r e has a p o s i t i o n o f s t a b l e e q u i l i b -r i u m i n S and t h e N e w t o n i a n s e q u e n c e d e f i n e d by: x,+1= X j - K ' ^ X j ) U B L ( X j ) c o n v e r g e s q u a d r a t i c a l l y t o i t . - x i i -Note t h a t t h e c l o s e r t h e i n i t i a l p o s i t i o n X 0 i s t o t h e s o l u -t i o n , t h e s m a l l e r w i l l be7) 0and hence t h e s p h e r e S . The s m a l l e r t h e s p h e r e S i s , t h e more n e a r l y w i l l t h e s t i f f n e s s m a t r i x v a r y l i n e a r l y w i t h d e f l e c t i o n s a c r o s s S and t h e s m a l l e r w i l l k b e . Thus t h e c l o s e r t h e i n i t i a l p o i n t i s t o t h e s o l u t i o n , t h e b e t t e r t h e c h a n c e s (and t h e f a s t e r t h e r a t e ) o f c o n v e r g e n c e . What i s t h e e f f e c t o f u s i n g an a p p r o x i m a t e m a t r i x ? J o h n shows ( C h a p t e r 2.12) t h a t i f t h e e r r o r i n t h e a p p r o x i m a t e m a t r i x i s bounded t h e n t h e c o n v e r g e n c e c r i t e r i a become h a r s h e r , b u t t h e same g e n e r a l s t a t e m e n t c a n be made: i f t h e i n i t i a l p o i n t i s c l o s e enough t o t h e ( s t a b l e ) s o l u t i o n t h e n c o n v e r g e n c e i s a s s u r e d . 

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