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Simplified calculation of cable tension in suspension bridges Richmond, Kenneth Marvin 1963

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S I M P L I F I E D C A L C U L A T I O N O F C A B L E T E N S I O N I N S U S P E N S I O N B R I D G E S b y K E N N E T H M A R V I N R I C H M O N D B.A.Sc. ( C i v i l E n g . ) 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 , 1959 A T H E S I S S U B M I T T E D I N P A R T I A L F U L F I L L M E N T OF T H E R E Q U I R E M E N T S F O R T H E D E G R E E O F M A S T E R O F A P P L I E D S C I E N C E i n t h e D e p a r t m e n t o f . C I V I L E N G I N E E R I N G We a c c e p t t h i s t h e s i s a s c o n f o r m i n g t o t h e r e q u i r e d s t a n d a r d 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 S e p t e m b e r , 1963 In presenting th i s thesis in p a r t i a l fulf i lment of the requirements for an advanced degree at the Univers i ty of B r i t i s h Columbia, I agree that the Library s h a l l make i t free ly avai lable for reference and study. I further agree that per-mission for extensive copying of this thesis for scholarly purposes may be granted by the Head of my Department or by his representativeso It i s understood that copying, or p u b l i -cation of this thesis for f i n a n c i a l gain sha l l not be allowed without my written permission. The Univers i ty of B r i t i s h Columbia-, Vancouver 8 , Canada. Department i i ABSTRACT T h i s t h e s i s p r e s e n t s a method w h i c h f a c i l i t a t e s r a p i d d e t e r m i n a t i o n of the c a b l e t e n s i o n i n s u s p e n s i o n b r i d g e s . A s e t o f t a b l e s and c u r v e s i s i n c l u d e d f o r use i n the a p p l i c a t i o n of the method. The method i s v a l i d f o r s u s p e n s i o n b r i d g e s w i t h s t i f f e n i n g g i r d e r s o r t r u s s e s e i t h e r ' h i n g e d a t the s u p p o r t s o r c o n t i n u o u s . A m o d i f i e d s u p e r p o s i t i o n method i s d i s c u s s e d and the use o f i n f l u e n c e l i n e s f o r c a b l e t e n s i o n i n n o n - l i n e a r s u s p e n -s i o n b r i d g e s i s d e m o n s t r a t e d . A d e r i v a t i o n o f the s u s p e n s i o n b r i d g e e q u a t i o n s i s i n c l u d e d and v a r i o u s r e f i n e m e n t s i n the t h e o r y a r e d i s c u s s e d . A computer p r o g r a m t o a n a l y s e s u s p e n s i o n b r i d g e s was w r i t t e n as an a i d i n the r e s e a r c h and f o r the purpose of t e s t i n g the manual method p r o p o s e d . A d e s c r i p t i o n of the p r o g r a m i s i n c l u d e d a l o n g w i t h I t s F o r t r a n l i s t i n g . v i i . ACKNOWLEDGEMENTS The a u t h o r i s i n d e b t e d t o Dr. R. P. Hooley f o r the a s s i s t a n c e , guidance and encouragement g i v e n d u r i n g the r e s e a r c h and i n the p r e p a r a t i o n of t h i s t h e s i s . A l s o , the a u t h o r i s g r a t e f u l t o the N a t i o n a l R e s e a r c h C o u n c i l of Canada f o r making money a v a i l a b l e f o r a r e s e a r c h a s s i s t a n t s h i p , and t o the B r i t i s h Columbia E l e c t r i c Company f o r the d o n a t i o n of $500 i n the form of a s c h o l a r s h i p . K. M. R. September 1 6 , ±9&3 Vancouver, B r i t i s h Columbia i i i TABLE OF CONTENTS Page CHAPTER 1. INTRODUCTION 1 CHAPTER 2. THEORY AND REFINEMENTS 5 G e n e r a l 5 C a b l e E q u a t i o n 8 G i r d e r E q u a t i o n 12 S o l u t i o n of E q u a t i o n s ( l l ) and (35) 19 E f f e c t o f R e f i n e m e n t s i n Theo ry on A c c u r a c y 21 CHAPTER 3. COMPUTER PROGRAM . 25 S o l u t i o n of the G i r d e r E q u a t i o n 27 I n t e g r a t i o n of the C a b l e E q u a t i o n 32 Program L i n k a g e 33 I n p u t Da ta f o r the Program 3^ F i n a l No tes on the Computer P rogram 36 CHAPTER 4. DETERMINATION OF H 37 G e n e r a l 37 S u p e r p o s i t i o n o f P a r t i a l L o a d i n g Cases 38 S i n g l e Span 39 T h r e e - S p a n B r i d g e w i t h H i n g e d S u p p o r t s 44 T h r e e - S p a n B r i d g e w i t h C o n t i n u o u s G i r d e r 45 V a r i a b l e E I 50 CHAPTER 5. CONCLUSIONS 52 APPENDIX 1 . BLOCK DIAGRAM AND FORTRAN LISTING FOR COMPUTER PROGRAM 55 APPENDIX 2 . TABLES OF CONSTANTS 60 APPENDIX 3. NUMERICAL EXAMPLES OF CALCULATION OF H 68 BIBLIOGRAPHY 8 l i v TABLE OF SYMBOLS Geometry L = L e n g t h of span B = D i f f e r e n c e i n e l e v a t i o n of c a b l e s u p p o r t s x = A b s c i s s a o f u n d e f l e c t e d c a b l e y = O r d i n a t e o f u n d e f l e c t e d c a b l e measured f r o m c h o r d j o i n -i n g u n d e f l e c t e d c a b l e s u p p o r t s dx = Inc rement i n x dy = Inc rement i n y ds = . I n c r e m e n t a l l e n g t h o f c a b l e c o r r e s p o n d i n g t o dx and dy LT = I f 1 -^1 d x f o r a l l spans J o Wj L e = I J L ^ | j 3 d x f o r a l l spans D e f l e c t i o n s v = V e r t i c a l d e f l e c t i o n o f c a b l e and g i r d e r h = H o r i z o n t a l d e f l e c t i o n of c a b l e h& - H o r i z o n t a l d e f l e c t i o n of l e f t c a b l e s u p p o r t .hg =' H o r i z o n t a l d e f l e c t i o n of r i g h t c a b l e s u p p o r t A = E q u i v a l e n t s u p p o r t d i s p l a c e m e n t f o r i n e x t e n s i b l e c a b l e ( i n c l u d e s e f f e c t . o f t e m p e r a t u r e and s t r e s s e l o n g a t i o n of c a b l e ) V F o r c e s w = U n i f o r m l y d i s t r i b u t e d dead l o a d of b r i d g e p = D i s t r i b u t e d l i v e l o a d on b r i d g e q = D i s t r i b u t e d l o a d e q u i v a l e n t t o suspende r f o r c e s = G i r d e r s u p p o r t r e a c t i o n a t l e f t end of span Rj3 = G i r d e r s u p p o r t r e a c t i o n a t r i g h t end of span H = T o t a l h o r i z o n t a l component o f c a b l e t e n s i o n Hp = H o r i z o n t a l component o f dead l o a d c a b l e t e n s i o n H-^ = H o r i z o n t a l component o f c a b l e t e n s i o n due t o l i v e l o a d , t e m p e r a t u r e change and s u p p o r t d i s p l a c e m e n t H L ' = H o r i z o n t a l component o f c a b l e t e n s i o n due t o l i w e l o a d on e q u i v a l e n t b r i d g e w i t h i n e x t e n s i b l e c a b l e and immovable s u p p o r t s 8H = C o r r e c t i o n t o H-^' t o a c c o u n t f o r e x t e n s i o n o f c a b l e and s u p p o r t movement B e n d i n g Moments M'i = B e n d i n g moment i n g i r d e r = B e n d i n g moment i n g i r d e r a t l e f t s u p p o r t Mg = B e n d i n g moment i n g i r d e r a t r i g h t s u p p o r t M' = B e n d i n g moment i n e q u i v a l e n t g i r d e r w i t h no c a b l e E l a s t i c and T h e r m a l P r o p e r t i e s 6 = C o e f f i c i e n t o f t h e r m a l e x p a n s i o n f o r c a b l e t = Tempera tu re r i s e A. = C r o s s - s e c t i o n a l a r e a o f c a b l e E = Y o u n g ' s Modulus I = Moment o f i n e r t i a o f g i r d e r m = C o e f f i c i e n t o f s h e a r d i s t o r t i o n f o r g i r d e r o r t r u s s v i A = C r o s s - s e c t i o n a l a r e a o f g i r d e r web w G = Shea r modulus A ^ = C r o s s - s e c t i o n a l a r e a o f t r u s s d i a g o n a l ( s ) 0 = A n g l e measured f r o m t r u s s v e r t i c a l t o d i a g o n a l ( s ) Computer P rogram a - A = C o e f f i c i e n t s o f d i f f e r e n c e e q u a t i o n s a p p r o x i m a t i n g g i r d e r e q u a t i o n D = 1 f o r D e f l e c t i o n T h e o r y s o l u t i o n = 0 f o r E l a s t i c Theo ry s o l u t i o n F h = 1 t o i n c l u d e e f f e c t o f h o r i z o n t a l d e f l e c t i o n = 0 t o d e l e t e e f f e c t o f h o r i z o n t a l d e f l e c t i o n P s = " 1 t o i n c l u d e change i n c a b l e s l o p e i n c a b l e e q u a t i o n = 0. t o d e l e t e e f f e c t o f c a b l e s l o p e change M i s c e l l a n e o u s V E I E_ I a = R a t i o o f s i d e span l e n g t h ' L g t o main span l e n g t h L b = f L s f E I 7 E I S I M P L I F I E D CALCULATION OF CABLE TENSION IN SUSPENSION BRIDGES . CHAPTER 1 INTRODUCTION T h i s t h e s i s adds a few new words , and pe rhaps a few new t h o u g h t s t o an a r e a o f s t u d y w h i c h has a l r e a d y been the s u b j e c t o f a c o n s i d e r a b l e amount o f s t u d y and l i t e r a t u r e . The a n a l y s i s of s u s p e n s i o n b r i d g e s i s a p r o b l e m somewhat d i f f e r e n t f r o m the u s u a l ' p r o b l e m s e n c o u n t e r e d by the s t r u c t u r a l e n g i n e e r and somewhat more d i f f i c u l t t o s o l v e . I t i s because of the d i f f e r e n c e s and the d i f f i c u l t i e s t h a t so much work has been done b o t h t o e x p l o r e e x t e n s i v e l y the p r o b l e m s i n v o l v e d and t o o v e r -come the d i f f i c u l t i e s i n a n a l y s i n g and d e s i g n i n g s u s p e n s i o n b r i d g e s . The p r o b l e m i n a n a l y s i s of s u s p e n s i o n b r i d g e s I s a r e s u l t o f t h e i r r e l a t i v e f l e x i b i l i t y and t h e i r d e s i r a b l e a b i l i t y t o d e f l e c t i n such a manner as t o m i n i m i z e the b e n d i n g s t r e s s e s i n the s t i f f e n i n g g i r d e r . D o u b l i n g the l o a d a p p l i e d t o a s u s -p e n s i o n b r i d g e does n o t n e c e s s a r i l y d o u b l e the b e n d i n g moments. T h e r e f o r e , s u s p e n s i o n b r i d g e s a r e s a i d t o be n o n - l i n e a r s t r u c -t u r e s . Tha t i s , , t h e r e i s a n o n - l i n e a r r e l a t i o n s h i p be tween l o a d 1 2 and r e s u l t a n t s t r e s s e s . A d i r e c t r e s u l t o f t h i s n o n - l i n e a r i t y i s t h a t s u p e r p o s i t i o n o f r e s u l t s o f p a r t i a l l o a d i n g s , and methods of a n a l y s i s dependent on s u p e r p o s i t i o n a r e n o t a p p l i c -a b l e i n the a n a l y s i s o f s u s p e n s i o n b r i d g e s . I t w i l l be shown h e r e t h a t a m o d i f i e d s u p e r p o s i t i o n method can be a d a p t e d t o the s o l u t i o n of s u s p e n s i o n b r i d g e p r o b l e m s . I n v e s t i g a t i o n of s u s p e n s i o n b r i d g e s has been i n s p i r e d by t w o . o b j e c t i v e s and a t l e a s t two main t h e o r i e s have been d e v e l o p e d . One g o a l o f i n v e s t i g a t o r s has been the deve lopment o f an e x a c t t h e o r y of a n a l y s i s . As i n most e n g i n e e r i n g p r o b -l e m s , so i n the a n a l y s i s o f s u s p e n s i o n b r i d g e s , a c o m p l e t e l y e x a c t t h e o r y i s v i r t u a l l y i m p o s s i b l e t o d e v e l o p and w o u l d be e x t r e m e l y cumbersome t o use f o r d e s i g n p u r p o s e s . However , r e a s o n a b l y a c c u r a t e s o l u t i o n s can be o b t a i n e d by the use o f the D e f l e c t i o n T h e o r y , w h i c h t a k e s a c c o u n t o f the n o n - l i n e a r b e h a -v i o r o f s u s p e n s i o n b r i d g e s . A n o t h e r g o a l o f i n v e s t i g a t o r s has been the s i m p l i f i c a t i o n o f s u s p e n s i o n b r i d g e t h e o r y i n o r d e r t o r educe the l a b o r r e q u i r e d f o r a n a l y s i s and d e s i g n . A r e s u l t has been the E l a s t i c T h e o r y , w h i c h i g n o r e s the changes i n g e o - ' me t ry r e s u l t i n g f r o m d e f l e c t i o n , o f a s u s p e n s i o n b r i d g e under l i v e l o a d and t e m p e r a t u r e c h a n g e s . Thus', the E l a s t i c T h e o r y i s a l i n e a r r e l a t i o n s h i p between l o a d and s t r e s s and the u s u a l methods of s u p e r p o s i t i o n can be u s e d . As migh t be e x p e c t e d , the two t h e o r i e s g i v e d i f f e r e n t r e s u l t s w h i c h can v a r y w i d e l y d e p e n d i n g on the f l e x i b i l i t y of the b r i d g e . . C h a p t e r 2 i s d e v o t e d t o a deve lopment o f the D e f l e c - . t i o n Theo ry or forms o f i t . There seems t o be no u n i v e r s a l l y a c c e p t e d s t a n d a r d D e f l e c t i o n T h e o r y . E a c h o f the many e x p e r t s i n the f i e l d f a v o r s a s l i g h t l y d i f f e r e n t v e r s i o n . V a r i o u s 3 r e f i n e m e n t s i n the t h e o r y may be i n c l u d e d to improve the a c c u -r a c y o f the c a l c u l a t e d r e s u l t s and thus the D e f l e c t i o n T h e o r y e q u a t i o n s may t ake d i f f e r e n t forms depend ing on the a c c u r a c y d e s i r e d . Some of these r e f i n e m e n t s a r e d i s c u s s e d and the e f f e c t on the e q u a t i o n s i s shown. The E l a s t i c T h e o r y i s shown as a s i m p l i f i e d v e r s i o n o f the D e f l e c t i o n Theory..., A l s o i n -c l u d e d i s a q u a n t i t a t i v e i n d i c a t i o n o f the e f f e c t s on a c c u r a c y w h i c h migh t be e x p e c t e d as a r e s u l t o f i n c l u s i o n o r n e g l e c t o f some of the r e f i n e m e n t s . . No new t h e o r y i s t o be found i n C h a p t e r 2 b u t the deve lopment has been i n c l u d e d he re t o p r o v i d e a f ramework of r e f e r e n c e f o r the f o l l o w i n g c h a p t e r s . I t s h o u l d be n o t e d t h a t t h r o u g h o u t t h i s work c o n s i d e r a -t i o n i s c o n f i n e d t o s t a t i c c o n d i t i o n s and s t a t i c l o a d i n g s . No a t t e n t i o n i s g i v e n he re t o the more complex c o n s i d e r a t i o n s o f dynamic l o a d i n g s on s u s p e n s i o n b r i d g e s . I t w i l l be seen i n deve lopment o f the t h e o r y t h a t s o l u t i o n of a s u s p e n s i o n b r i d g e p r o b l e m i n v o l v e s the s i m u l t a n e o u s s o l u t i o n o f a d i f f e r e n t i a l e q u a t i o n and an i n t e g r a l e q u a t i o n . I n the more g e n e r a l and more e x a c t s o l u t i o n s , i t i s n e c e s s a r y t o r e s o r t t o n u m e r i c a l methods f o r the s o l u t i o n of each of t he se e q u a t i o n s . The s i m u l t a n e o u s s o l u t i o n i s found by a c u t and t r y method. Hence , s o l u t i o n o f a n u m e r i c a l example can become an e x t r e m e l y l e n g t h y and t e d i o u s p r o c e d u r e by hand c a l c u l a t i o n s . F o r t u n a t e l y , because o f the e x i s t e n c e of computers i t i s no l o n g e r n e c e s s a r y t o p e r f o r m a l l c a l c u l a t i o n s by h a n d . The p r o b l e m of s u s p e n s i o n b r i d g e a n a l y s i s i s w e l l s u i t e d f o r s o l u t i o n on a compu te r . C h a p t e r 3 d e s c r i b e s a p rog ram w h i c h was w r i t t e n f o r the IBM 1620 d i g i t a l computer I n o r d e r t o i n v e s t i g a t e s u s p e n s i o n b r i d g e a n a l y s i s . I t i s b e l i e v e d t h a t 4 the methods employed I n the p rog ram a re w e l l s u i t e d f o r computer a n a l y s i s . F o r t h a t r e a s o n , a l i s t i n g o f the F o r t r a n p rogram has been i n c l u d e d i n the hopes t h a t i t may s e r v e as a g u i d e i n the p r e p a r a t i o n of s u s p e n s i o n b r i d g e p r o g r a m s . The k e y t o a s i m p l i f i e d s o l u t i o n t o s u s p e n s i o n b r i d g e p r o b l e m s i s a r a p i d d e t e r m i n a t i o n o f the v a l u e o f the c a b l e t e n s i o n . I n the more e x a c t methods , c a b l e t e n s i o n i s found by a c u t and t r y method. C h a p t e r 4 d e s c r i b e s a method, b e l i e v e d t o be new, whereby H , the h o r i z o n t a l component o f c a b l e t e n s i o n can be found e x t r e m e l y q u i c k l y . The p r i n c i p l e s upon w h i c h the method depends a r e shown and the method i s d e v e l o p e d . Use o f the method r e q u i r e s the use o f t a b l e s o r c u r v e s r e l a t i n g c e r t a i n d i m e n s i o n l e s s r a t i o s . These a r e i n c l u d e d , a l o n g w i t h n u m e r i c a l examples i l l u s t r a t i n g the a p p l i c a t i o n o f the method. The method employs a f o r m o f s u p e r p o s i t i o n , w h i c h i s shown t o be v a l i d , p r o v i d i n g the t o t a l v a l u e o f H i s known. S i n c e H i s the v a l u e t o be d e t e r m i n e d and i s t h e r e f o r e unknown, an e s t i m a t e i s . r e -q u i r e d t o i n i t i a t e c a l c u l a t i o n s . The i n i t i a l e s t i m a t e o f H i s improved by a r a p i d l y c o n v e r g i n g i t e r a t i v e p r o c e d u r e t o g i v e an a c c u r a t e v a l u e of H . The method may be a p p l i e d t o s u s p e n s i o n b r i d g e s e i t h e r w i t h c o n t i n u o u s s t i f f e n i n g g i r d e r s o r w i t h g i r d e r s h i n g e d a t the s u p p o r t s . 5 CHAPTER 2 THEORY AND REFINEMENTS G e n e r a l The f o l l o w i n g d e r i v a t i o n i s c o n c e r n e d w i t h the case of a l o a d e d g i r d e r , o r e q u i v a l e n t p l a n e t r u s s , o f known r i g i d i t y , suspended by v e r t i c a l suspende r s f r o m a p e r f e c t l y c a b l e , w h i c h i s a n c h o r e d a t tower t ops o r a n c h o r a g e s . I n the a n a l y s i s , the f o l l o w i n g s i m p l i f y i n g a s s u m p t i o n s a r e made: 1. The suspende r s a r e I n e x t e n s i b l e . 2. The suspende r s a r e so c l o s e t o g e t h e r t h a t t h e y may be r e p l a c e d by a c o n t i n u o u s f a s t e n i n g . 3. The dead l o a d o f the b r i d g e i s d i s t r i b u t e d a l o n g the g i r d e r s . 4 . The g i r d e r i s i n i t i a l l y s t r a i g h t under the a c t i o n o f dead l o a d a l o n e , and c a r r i e s no b e n d i n g moment. 5. The dead l o a d i s c o n s t a n t f o r each span , and hence the c a b l e i s i n i t i a l l y p a r a b o l i c . The above a s s u m p t i o n s a r e u s u a l f o r the s o - c a l l e d " D e f l e c t i o n T h e o r y " o r more e x a c t t h e o r y o f s u s p e n s i o n b r i d g e a n a l y s i s . L e s s e x a c t forms o f the D e f l e c t i o n T h e o r y i n common use u s u a l l y make the f o l l o w i n g a d d i t i o n a l a s s u m p t i o n s : 6. The h o r i z o n t a l d e f l e c t i o n s o f the c a b l e a r e v e r y s m a l l compared w i t h the v e r t i c a l d e f l e c t i o n s , and can be n e g l e c t e d . 6 7 . Deflections of the cable are very small compared with cable ordinates, and their effect on cable slope can be neglected in calculation of cable extension. 8. Shear deflections in the girders are very small compared with bending deflection and can be neglec-ted . Assumptions 6, 1 and 8 may be excluded with little difficulty in the derivation, and may even be excluded in an analysis by digital computer. Therefore, the effects of hori-zontal deflections, cable slope change, and shear deflection are included here and discussed briefly. It is'not to be thought that their inclusion results in a complete theory, but perhaps these are some of the more important refinements which can be made. Others* have discussed the effect of the above refine-ments,- and in addition have introduced, or at least mentioned, other refinements such as tower horizontal force, tower shorten-ing cable lock at midspan, effect of loads between hangers, temperature differentials between girder flanges, finite hanger spacing, weight of cable and hangers, variation of horizontal component of cable tension with hanger inclination, and so forth. The Deflection Theory of suspension bridge analysis results in a non-linear relationship between forces and deflec-tions and hence the principle of superposition and methods dependent on superposition are not applicable in the usual manner. In order to simplify the force-deflection relationship "Into a linear one, it is necessary:to make a further simplifying * Reference ( 12 ) 7 a s s u m p t i o n . I t i s t h i s f u r t h e r a s s u m p t i o n w h i c h i s t h e b a s i s f o r t h e " E l a s t i c T h e o r y " . I t m a y b e s t a t e d a s f o l l o w s : 9. T h e d e f l e c t i o n s o f t h e c a b l e a n d g i r d e r a r e s o s m a l l a s t o h a v e a n e g l i g i b l e e f f e c t o n t h e g e o -m e t r y o f t h e c a b l e a n d h e n c e o n t h e m o m e n t a r m o f t h e c a b l e f o r c e . I t i s w e l l k n o w n t h a t t h e E l a s t i c T h e o r y r e s u l t s i n e r r o r s w h i c h a r e t o o l a r g e t o s a t i s f y e c o n o m y o f d e s i g n . W e r e i t n o t f o r t h e l e n g t h i n e s s o f D e f l e c t i o n T h e o r y c a l c u l a t i o n s , t h e E l a s t i c T h e o r y w o u l d h a v e l o n g s i n c e p a s s e d o u t o f u s e f u l -n e s s . M u c h e n e r g y h a s b e e n e x p e n d e d i n a t t e m p t s t o s i m p l i f y t h e D e f l e c t i o n T h e o r y t o y i e l d r e s u l t s o f h i g h a c c u r a c y w i t h a n e a s e a p p r o a c h i n g t h a t o f t h e E l a s t i c T h e o r y ; a n d i t i s t o t h a t e n d t h a t t h i s t h e s i s i s d e v o t e d . S o l u t i o n b y t h e D e f l e c t i o n T h e o r y c o n s i s t s o f t h e s i m u l t a n e o u s s o l u t i o n o f t w o e q u a t i o n s . T h e f i r s t I s r e f e r r e d t o h e r e a s t h e c a b l e e q u a t i o n , a n d r e l a t e s c a b l e d e f l e c t i o n s t o c a b l e l o a d s . T h e s e c o n d e q u a t i o n i s t h e d i f f e r e n t i a l e q u a t i o n r e l a t i n g g i r d e r d e f l e c t i o n t o g i r d e r l o a d s a n d c a b l e t e n s i o n . T h i s s e c o n d e q u a t i o n i s r e f e r r e d t o h e r e a s t h e g i r d e r e q u a t i o n . F i g u r e 1 s h o w s a s i n g l e s p a n s u s p e n s i o n b r i d g e w i t h a p p l i e d l o a d s . A l l d i s t a n c e s , f o r c e s a n d d e f l e c t i o n s a r e p o s i -t i v e a s s h o w n . B o t h c a b l e a n d g i r d e r a r e i n i t i a l l y s u p p o r t e d a t A a n d B s e p a r a t e d b y a d i s t a n c e e q u a l t o t h e s p a n l e n g t h L . T h e g i r d e r i s l o a d e d w i t h a c o n s t a n t d e a d l o a d w , a l i v e l o a d p , e n d r e a c t i o n s a n d R g , a n d e n d m o m e n t s M ^ a n d M g , w h i c h m a y b e e n d m o m e n t s a p p l i e d t o a h i n g e d g i r d e r o r t h e r e s u l t o f c o n t i n u i t y a t t h e s u p p o r t . I n a d d i t i o n , t h e g i r d e r i s s u b j e c t t o t h e d i s t r i b u t e d l o a d q e q u i v a l e n t t o t h e s u s p e n d e r f o r c e s . T h e TO FOLLOW PAGE Figure 2. cable is connected to the girder by vertical suspenders and carries the distributed load q. The cable is in tension, the horizontal component of which is constant and is equal to H. At the supports, the vertical components of the cable tension are y A' and Vg'. Under the action of live load and temperature changes, the cable and girder deflect from the positions shown in solid lines to the positions indicated by dashed lines. The cable supports deflect horizontally the distances h^ and hg. The original cable position is given by co-ordinates x and y measured horizontally from A and vertically from the chord join-ing the undeflected cable supports at A and B. A point P on the cable deflects from its initial position to a point P' horizontally a distance h and vertically a distance v. A point Q, on the girder deflects from Its initial position vertically below P to a position Q,' vertically a distance v. Cable Equation Figure 2 shows an elemental length of the cable at point P. Its undeflected position is shown as a solid line, while its deflected position is shown as a dashed line. The length of the element in the undeflected position is given by (ds)2 (dx)2 (dy Bdx\ 2 •~ + \ -I J - • ( 1 > Under the action of live loads the cable deflects as shown and the length of the same element of cable in the deflected posi-tion is given by (ds + Sds)2 (dx + dh)2 /dy Bdx dv\ 2 + + . . . (2) ds S'ds fl 1 6ds - — + - — dx dx \ 2 dx 1 1 dh\ dv /dy 2 dx/ dx Vdx B 1 dv - + ] . . . ( 3 ) L 2 dx. Subtracting (l) from (2) and rearranging terms, it is found that dh dx Since ^ ds and ^ h a r e both extremely small compared with unity, dx dx they may be dropped from the terms 1 + i ^ s and 1 + 1 ^ 1 . 2 dx 2 dx The term i is generally small compared with - ^  over 2 dx dx li most of the span, but may be significant, especially in very flat cables. Expression ( 3 ) then reduces to ds 6ds dh dv /dy B 1 dv' dx dx dx dx \dx L 2 dx> The extension of the cable 8ds as caused by temperature expansion and stress is given by 8ds 6 t d s ds dx dx AE dx 1 2 ( 5 ) where: 6 = coefficient of thermal expansion t = temperature rise . H-^  = change in horizontal component of cable tension due to application of live load, temperature changes, support movement, etc. A = cross-sectional area of cable E = Young's Modulus for cable material Again, since fUl is extremely small compared with unity, it may d X / \2 be deleted from- the term ( 1 + <^±\ . Then, the binomial theorem V dx/ can be applied to expand the bracketed term, two terms can be neglected, giving All but the first 8 d.s £ tds H L ds dx dx AE dx 1 1 /dy 2 Vdx B' L, + dv /dy B 1 dv\ dx Vdx L 2 dx/ (6) 10 or, since' ds 1 1 /dy B> dx 2 \dx L, 2 (7) then 6ds etds + HL ds ds dv / dy B 1 dv> dx dx V dx L 2 dxy . (8) dx dx AE dx A combination of equation (4) representing cable geometry and equation (8) representing Hooke's Law gives the cable equation as *~ 2 dh et /ds\ 2 HT fas\ 3 + — dx d^xy AE VdxJ H L ds) AE Vdxi dv / dy B 1 dvA dx I dx L 2 dx, (9) The above cable equation may be simplified significantly if it is observed that the term [OS- - — + In expression (5) is \dx L dx/ normally less than .2, and £LY_ is generally small compared with dx dy - IL Hence, dv_--j_-s n o ^ v e r y significant in the total expres-dx L dx sion and can reasonably be neglected. Since — has already dx been neglected compared with unity in the same expression, this amounts, to neglect of the effect of deflections on cable slope and expression (5) becomes 6ds etds HT 'dsV dx - + - H dx AE \dx/ (5a) When (5a) is combined with ( 4 ) , the simplified cable equation is dh et /ds\ 2 HL /ds\ 3 dv (dy B I d + dx idxy AE \dx> dx \dx v' L 2 dx, (9a) It can be seen that neglect of the change of cable slope is reflected in expression (9) by neglect of the term ^  (4J AE \dxI 11 compared with unity. _J± is usually of the order .001 and — AE d x Is normally not much larger than 1, so a term of order .001 has been neglected compared with 1. On this basis, Timoshenko argues that it is negligible. However, it is not difficult to see that a given percentage error in one term of expression (9) could be magnified by subtracting that term from another of similar magnitude to give a larger percentage error in ^Jl. Expression (9) can be further simplified if — — is 2.Mx neglected compared with - 5. in expression (4). Then the dx L cable equation becomes dh et /ds\ 2 HT Ids\ 3 dv /dy B\ — = — + - - ( 9 b ) dx \dx/ AE \dx/ dx \dx L / This final expression gives a linear relationship between hori-zontal and vertical deflections. It should be noted that the above linear relationship between horizontal and vertical deflections does not imply that the structure is linear. The cable equation has been reduced to a linear equation, but a non-linear relationship can and does s t i l l exist between stresses and applied loads. If the cable equation is integrated over the span length and the horizontal displacements of the supports are in-serted as constants of integration, the following expression results: h B - h A r L Gt /ds\ 2 dx J o r L o H L ds AE Vdx, dx (10) 12 or, if L|ds_^2 dx i s denoted by L, and L f d _^ .^ 3 d x is denoted by L.Q, then hB " hA = € t L t + % Le AE _^  o \ y C L HL /ds v 2 AE \dx', dv /dy B 1 dv dx \dx L 2 dx dx ( l i ) } If the change in cable slope can be neglected, then - hn G tL<_ HT L_ f L dv / dy B 1 dv \ dx AE + o dx \ dx 2 dx (•lla) If the term ^ can be neglected compared with Q*L - ^  then 2 dx, dx L h - h„ G tL H L • r L B A = t + L e _ AE . .. (lib) o dv / dy B \ dx dx \dx L J The cable equation (ll) or the simplified forms (lla) and (lib) relate cable displacements to cable loads. It will be seen that the refinement represented in equation (ll) is difficult to justify, considering the additional accuracy attained and the effort required to solve- the equation. Equation (lla) requires considerably less effort to solve, but is also difficult to justify. Equation (lib) will be found to be sufficiently accurate for most purposes. The cable equation is one of the two equations to be solved in analysis of suspension bridges. Cable deflections must be consistent with support movements and with girder deflections as given by the girder equation. Girder Equation In order to derive the girder equation, it is necessary to consider separately the two main components of the bridge, TO FOLLOW PAGE Figure 4. 13 the cable and the girder. Figure 3 shows a free-body diagram of the girder under the action of applied loads. The reaction RA can be found from RA MB - MA + (p + w - q) (L - a) da ... (12) Then, the bending moment in the girder at x, denoted by M Is given by L(p+w-q)(L-a)da fx(p+w-q)(x-a)da... (13) M MA (MB-MA)x x + — + L o For simplicity define a quantity M' equal to the bending moment produced by all loads except those applied by the cable. This is given by r L (p+w)(L-a)da r x o (p+w)(x-a)da ... (14) ... (15) M' MA (MB-MA)x x = + — + L Then M M1 x ^ L q(L-a) da T x q(x-a) da -Jo L Jo Now, it is necessary to consider the static equilibrium of the cable under the applied loads. Figure 4 shows the forces acting on the deflected cable, indicated by a dashed line. The cable tensions at the supports are resolved into components in the direction of the chord joining the deflected points of support of the cable, and vertical components V^  and Vg. The vertical component V^  is given by V A r L q ( L + h B - (a+h)) da L + h B - h A . (16) It will be noted that the forces in the direction of the closing chord have horizontal components equal to H, the horizontal com-ponent of cable tension.' Then, for the equilibrium of the 14 cable H(y-8]_+52) = VA(x-hA) - x q(x - (a+h))da .. (17) . (18) where y - S]_ + S 2 is the vertical distance from the deflected position of the closing chord to the deflected cable. It is clear from geometry that 8 1 B|h A (hB-hA)x> L V L V Reference to Figure 2 will show that 8 2 v h /dy B' Idx L The term VA(x-hA)in equation (17). now becomes .. (19) VA(x-hfl) x A' L q(L-a)da r L o ^ hB"hA + q(hg-h) da o hA VA.h A A (20) When it is observed that the horizontal deflections are very small compared with the dimensions L, a and x, it can be seen that the terms x f (^hB~h) da a n ( ^ vflha are very small compared Jo L+ho-hfl R R with x I ^ QA-k-a) da. and can be approximated with negligible jo L+hB^A error in the total term VA(x-hA) as follows: x ^ Lq(hB-h) da x o ^ B ^ A V Ah A = hAH dy> VdXy Lq(hB-h) da A Then, the term x. f L q(L~a) d a of equation (20) Jo L+hB'hA neglecting all but the first two terms to give ... (21) ... (22) can be expanded, x ^  ^  q(L-a) da x o L + hB" hA Lq(L-a) da x(hA-hfi) f L q(L-a) da + ... (23) L L Note that the second term on the right hand side of expression (23) is much smaller than the first term. It is therefore per-15 mlssible to approximate it as follows x(hA-hB) r L q(L-a) da x(hA-hB) H / dyi dx i ... (24) o ~ \ ~ v A If substitutions from expressions ( 2 1 ) , ( 2 2 ) , (23) and (24) are made in expression ( 2 0 ) , it is found that VA(x~hA) xf Lq(L-a)da x(hffi-hB) H / dy^  L \ dx, A x r L + q(hB-h) da hAH /dy^  . . ( 2 5 ) L \dxJ A If substitutions from expressions ( l 8 ) , (19) and (25) are made in expression (17) and the result is combined with expression (15)j the following expression results: M W H y v r x + hqda hAH /dy) x + H /dy\ h A r h B r L + q(hB-h) da ... (26) .dxj A \dx/AL Jo L Since terms involving the horizontal deflections are small, it is permissible to make some approximations in these terms. Specifically, it is permissible to approximate the suspender forces q by -H-d y dx' ... (27) If the above is substituted in (26) the expression for bending moment in the girder becomes M M-< H y v + B L h A + (hB-hA)x h /dy B \ r x hd^ y da d^x L o dx 2 h + A x L 'dy\ (h A-h B) \dx/ A d^y(hB-h)da o dx 2~ . (28) 16 It can be shown that, If all horizontal displacements are increased by a constant amount h* , the bending moment in the girder at all points will be unchanged. Hence, if h 0 is set equal to -h^ , h^ may be replaced by zero in expression ( 2 8 ) , hg may be replaced by hg - hA, and h may be replaced by h - h A. Expression (28) may be differentiated twice to give d2M -p -w - H dx' d2y d2v d2h /dy B\ dh d2y dx2 dx2 dx2 \dx L j dx dx2 ... (29) If horizontal deflections are neglected in the girder equation, expressions (28) and (29) reduce to M - M-' - H(y + v) ... (28a) d2M -p -w -H /d2y d2v\ T = \~? + ~?~\ •" ( 2 9 a ) dx2 \dx2 dx2/ If vertical deflections are neglected in the girder equation, expressions (28) and (29) reduce to the Elastic Theory expressions M = M"'1 - Hy ... (28b) d2M -p -w -Hd2y — ~ = ... (29b) dx dx'-Expression (13) can be differentiated twice to give q - p - w ... (30) d2M dx' Then equations (29b) and (30) can be combined to give expression ( 2 7 ) , an approximate relationship which was used earlier. From elementary strength of materials, the basic differential equation relating deflection to bending moment and shear in a girder is EId2v -M + EImd2M ... (31) dx' dx< 17 where: E I m Young's Modulus for the girder material moment of inertia of the girder coefficient of shear distortion m = m = ,. 1 for a girder ADE sin 0 cos 0 for a plane truss Aw = G = AT, = D cross-sectional area of girder web shear modulus cross-sectional area of the diagonal member (s) angle measured from vertical to diagonal member (s) The deflections due to shear are small compared with the deflections due to bending. Therefore, negligible error 2 results if %—^ is represented by the approximate expression (29a), dx^  If expressions (28) and (29a) are substituted in expression ( 3 0 ) , 'and the resulting equation is differentiated twice, the following fourth-order differential equation is found: (1 + Hm) Eld^v 2d(El) d 3 v d2(El) d 2 v dx 4 + + dx dx3 dx^ dx< p + w H + d2y d2v + dx^ dx^ H d^h /dy , B d x 2 Idx L dh d2y dx d x 2 d2(El) dx' m p + w + Hd y dx' .. (32) The above differential equation reduces in the Elastic Theory to EId4v 2d(El) d 3 v d2(El) d2v dx 4 + + dx dx- dx' dx 2 p w + + Hd2y dx' m d 2 ( E l ) dx' p w + + Hd2y dx' ... (33) S u b s t i t u t i o n f r o m the c a b l e e q u a t i o n can be made, i n e x p r e s s i o n 18 (32) to give the horizontal deflections in terms of the vertical deflections. Cable equation (9b) is sufficiently accurate here. Remember that horizontal deflections, have a very small effect on bending moments. It is a simple matter to prove that 0 d h / dy dx2 \dx + HL d^ y ds AE dx2 dx p d v / dy p dx \ dx ... (34) Expression (34) can be substituted in equation (32) and terms may be collected to give the following fourth-order linear differ-ential equation El(l + Hm) h d^ v dx 4 dx H 2d(El)(l + Hm) d2v + dx' d 2(El)(l + Hm) dx 2 -2Hd y /dy B^  dx^  [dx L/ p w Hd y dx^  HGtd2y md2(El) dx HHL ds 'dry AE dx dx2 2 p w Hd y + + dx 2 (35) Equation (35) is the girder equation.in a general form. It includes the effect of variable girder stiffness, horizontal ^ cable deflection and shear deflection. Unknowns present•in the equation are H and v. The cable equation (ll) also relates unknowns H and v. Therefore, simultaneous solution of equations 19 (ll) and (35) will yield the values of H and v. The ordinates of the cable in the undeflected position can be found by considering the equilibrium of the cable and girder under the action of dead load alone. No bending moment is present anywhere in the girder and the applied moment M' is given by M' wx(L - x) ... (36) 2 From equation (28) Hrjy" wx(L - x) (37) If the cable ordinate at mid-span is denoted by f, then HD wL2 8f and y 4f x(L - x) = L 2 dy = 4f(L - 2x) (39) dx L 2 d2y • 8f dx2 L 2 (40) (41) (42) Solution of Equations (ll) and (35) Solutions to equations (ll) and (35) individually can-not be evaluated in terms of simple functions except in a few very special cases. Therefore, s.ome method of numerical analysis must be resorted to in most cases. whether the evaluation is by algebraic or numerical methods, the general approach is the same. First, some estimate 20 must be made of H. Then, equation (35) must be solved to give values of v corresponding to the estimated value of H. Then, equation (ll) can be solved to give a value of hg - h^ which must be equal to a value consistent with external conditions (for example, zero for fixed cable anchorages). In general, the first estimate of H will result in an error in the computed value of h-Q - h^; and so a second estimate of H must be made, and the procedure repeated to give a second error in the com-puted value of hg - h^. Subsequent estimates of H can be made by interpolation between previous estimates to give finally a tolerable error in h - h , and a sufficiently accurate value of ri A H. The last solution of equation (35) thus obtained can be used to evaluate bending .moments at all points, and shearing forces at all points. is similar to that for a single span. Equation (35) must be solved to give values of v for all spans, corresponding to an estimated value of H. Then equation (ll) must be evaluated for all spans to give the total value of h - h where h and h. 13 A D A here represent the horizontal displacements of the cable anchor-ages at the ends of the bridge. ' In doing this, L e and L-j-respectively, for all spans. If numerical analysis is resorted to in the solution of equations (ll) and (35)* no mathematical difficulties are horizontal deflection of the cable, and shear deflection in the girder. Except that the calculations are a little longer, it is as convenient to include these refinements as to leave them In multiple span bridges, the procedure for analysis represent the sums of values of encountered due to inclusion of the effects; of cable slope change, 21 out. Equations (ll) and (35) are suitable for solution by numerical analysis on a digital computer, and In the program outlined in the next chapter, the refinements mentioned above may be included or left out. at will. It is not to be thought that the reduction in labor due to simplification of equations (ll) and (35) is insignificant. In-fact, If the error introduced is tolerable, considerable saving in labor is possible by the use of equation (lib) in place of (ll) and by the omission of horizontal deflection and shear deflection from the girder equation. The girder equation may then be used in the form EId2v Hv Hy M' — -- = - ... (43) dx^  Equation (43) is usually considered to be the basic differential equation for suspension bridges. Solutions to equation (43) are to be found tabulated in Steinman's text*and elsewhere for loading conditions commonly encountered in design. Numerical analysis is unnecessary except for programming a digital computer. Effect of Refinements in Theory on Accuracy It may be of some value to indicate here briefly the order of magnitude of error introduced by neglect of the effects of change in slope of the cable, horizontal deflection of the cable and shear deflection in the girder. It has been shown that the error resulting from neglect of the change of cable slope in the calculation of cable exten-sion is largely dependent on the ratio Mi. Test calculations AE were performed with the aid of the computer program outlined in TT the next chapter. In the test case, the value of _L w a s # Q 2 3 l Reference (l) 22 considerably larger than the usual range of values of ^L. The AE test bridge was a three-span bridge, loaded over the entire main span. The resultant error in H L was 0.1%, and the maximum error in bending moment was 0.11%. Therefore, it is reasonable to expect that neglect of the effect of cable slope change will never result in errors larger than 1% in cable tension and bending moments. Shear deflections have been investigated by C. D. Crosthwaite.* It was found that.the largest errors in bending moments and shears occur near the supports and that the effect of shear deformation diminishes toward mid-span. Neglect of shear deformation was shown to account for errors as large as 9.5% In shears and 6.5% in bending moments. The writer has compared the results of calculations for the case of a three-span bridge loaded over one half of the main span. The magni-tude of the percentage error depends on the ratio of shear flexi-bility to bending flexibility, in the girder, and in addition depends to some extent on the cable tension and cable area. Figure 5 indicates the relationship between percentage error and given dimension ratios. The computed quantity under con-sideration is the- bending moment at the quarter point on the loaded portion of the main span. Within the normal range of values of errors in computed values of the bending moment L 2 can be as high as 15%. For all values of the dimensionless ratios within the range of normal values shown in Figure 5, it was found that the errors in the value of H determined by i j neglecting shear deflection were less than 1%. Reference ( l 8 ) TO FOLLOW PAGE 22 Figure 5; 23 The. effect of longitudinal deflection has been inves-tigated by H. H. Rode * and he has reported that the reduction in bending moment will usually be between 5$ and 2$. The writer has examined a single case of a three-span bridge loaded over half the main span and the adjacent side span. The bridge TJ T2 under consideration was very flexible, having a ratio D of EI 100. The errors' in bending moments due to neglect of the horizontal deflection were found to be as high as 8.5$ of the maximum bending moment in the girder. The error in was 0.32$. The writer has compared also the effect of using cable equation,(lib) for a particular example calculation. The bridge examined was a three-span bridge having ratio of 100 for • EI the main span. It carried a live load of .4 times the dead load over•one half of the main span. It was found that approximation of the cable equation by equation (lib) Instead of equation (lla) resulted in. an error in H-^  of 3.7$.' This was reflected in the girder moments by an error of 1.1$ of the maxi-mum moment. It was observed that any approximation of the cable equation resulted in an error in H-^  which was reflected by an error of reduced significance in the girder bending moments. On the other hand, approximation of the girder equation by neglecting shear deflection or horizontal cable deflection results in small errors in but larger errors in bending moments. These larger errors in bending moment are due not to the inaccuracy in H but to the omission of terms in the girder equation. Hence, it would seem reasonable to determine H-^  by Reference (l6) 24 using approximate equations (43) and either (lla) or (lib). Then, the value of the H found by these approximate equations can be used with the more exact girder equation (35) or a simpli-fied version of it to yield bending moments of required accuracy. Usually/ the value of HL found by the approximate equations will be in error by less than 1% due to the neglect of cable slope change, shear deformation and horizontal deflection. Choice of equation (lla) or (lib) will depend on the flexibility of the girder and the ratio of live load to dead load; and on the accuracy desired. It is suggested that equation (lib) will be satisfactory for all but very flexible girders and heavy loads. 25 CHAPTER 3 COMPUTER PROGRAM The analysis of suspension bridges is a problem especially suitable for solution on a digital computer. Each of the equations (ll) and (35) may be readily programmed for ' numerical solution, and their simultaneous solution by a cut and try method reduces simply to an Iteration procedure. It is a simple matter to delete terms in either equation to test the effect of neglecting refinements In the theory. Also the elastic theory solution can be readily obtained from the deflection theory equations merely by deleting terms which apply only to the deflection theory. Therefore, the program has been written to give, for each loading case, both a deflection theory solution and an elastic theory solution. The computer used is an IBM 1620 desk-size digital computer, at the University of British Columbia Computing Centre. The core-storage memory holds up to 40,000 digits of data and instructions. Available input devices are the console type-, writer and a card-reader. Output is by console typewriter, card punch or on-line printer. It was found to be convenient to write the program ' using Fortran, the simplified scientific language available for IBM computers. The particular version of Fortran used is desig-nated by the U.B.C. Computing Centre as Fortran lA. 26 All input data required for the program, and all out-put are in dimensionless form. It is convenient, to think of the unit of length as being L, the length of the main span, and the unit of load or force is wL, the total uniformly distributed dead load on the main span. The program was written originally for the purpose of investigating the relationship between elastic theory bending moments and deflection theory bending moments. It analyses a three-span suspension bridge using first the elastic theory, and then the deflection theory, and compares the computed bending moments. In both cases, a cut and try method is used to deter-mine H, the horizontal component of cable tension. In general, each trial value of H produced an error In the computed value of the relative support movement. Each trial value of HL and its accompanying value of error Is printed out. The actual values printed out are Eli and A error^ i n dimensionless form. When wL L 2 2 2 the error is practically zero, the values of HD L , H L L and H L EI EI EI are printed and the bending moments at several points on the girder are computed. When the bending moment has been computed by both methods, the program prints the deflection theory bending moment, the elastic theory bending moment, the ratio between the two, and the deflection.at points along the girder. The program analyses a three-span suspension bridge with constant or variable girder stiffness. The girder may be either continuous at the towers or hinged. In the numerical procedures used, the main span is divided into twenty equal partial lengths equal in length to the main span sections. The bending moments, deflections and so forth are computed for the points separating these partial lengths of girder. The points are 27 numbered c o n s e c u t i v e l y f r o m l e f t t o r i g h t on the g i r d e r w i t h p o i n t s a t the l e f t and r i g h t tower numbered 17 and 37} r e s p e c t -i v e l y . I n i t s p r e s e n t f o r m , the p rogram r e q u i r e s t h a t the s i d e spans be d i v i d e d i n t o an even number o f p a r t s , and s t o r a g e l i m i t a t i o n s f u r t h e r l i m i t the r a t i o o f s i d e span l e n g t h t o main span l e n g t h t o a. maximum of .7. T h e r e f o r e , a c c e p t a b l e r a t i o s wou ld be .7, .6,. .5,. .3, .2 o r .1. The method of s o l u t i o n i s i l l u s t r a t e d i n the g e n e r a l b l o c k d i a g r a m f o r the p r o g r a m shown i n F i g u r e 6. I t can be seen t h a t the h e a r t o f the p rog ram i s the s o l u t i o n of the d i f f e r e n t i a l e q u a t i o n f o r the g i r d e r and the i n t e g r a l e q u a t i o n f o r the c a b l e . These e q u a t i o n s a r e i n c l u d e d i n the p r o g r a m i n as e x a c t a f o r m as i t i s deemed p r a c t i c a l , i n c l u d i n g such r e f i n e -ments as shea r d e f l e c t i o n , h o r i z o n t a l d e f l e c t i o n of c a b l e and change i n s l o p e o f c a b l e . Terms w h i c h a r e found i n the d e f l e c -t i o n t h e o r y e q u a t i o n s b u t n o t i n the e l a s t i c t h e o r y e q u a t i o n s a r e m u l t i p l i e d by a f a c t o r D, w h i c h i s made z e r o t o g i v e an e l a s t i c t h e o r y s o l u t i o n , o r u n i t y t o g i v e a d e f l e c t i o n t h e o r y s o l u t i o n . The a b o v e - m e n t i o n e d r e f i n e m e n t s i n the d e f l e c t i o n t h e o r y a r e i n c l u d e d o r d e l e t e d by the same t e c h n i q u e . S o l u t i o n o f the G i r d e r E q u a t i o n Two approaches t o n u m e r i c a l s o l u t i o n o f the g i r d e r e q u a t i o n were c o n s i d e r e d . The e q u a t i o n c o u l d be t r e a t e d as an " i n i t i a l v a l u e " p r o b l e m , and some a s s u m p t i o n as t o unknown i n i t i a l c o n d i t i o n s c o u l d be made i n o r d e r to f i n d a s o l u t i o n by the R u n g e - K u t t a method. Then , f o r each unknown i n i t i a l c o n -d i t i o n , a d i f f e r e n t v a l u e o f the i n i t i a l c o n d i t i o n c o u l d be assumed t o g i v e an a d d i t i o n a l s o l u t i o n t o the e q u a t i o n . The TO FOLLOW PAGE 27 R E A D PRINT INPUT V INPUT v DATA s D = 0 INITIAL ESTIMATE H L = O.lwL Note •• D = 0 for Elastic Theory. D= I for Deflection Theory. / "K = 0 M SWITCH OFF), SOLVE GIRDER EQUATION (SEE FIG. 7 ) I SOLVE C A B L E EQUATION FOR ERROR IN h B " h A PRINT H. ERROR NEW E S T I M A T E H L = H L+0.lwL AV K = 0 K = l (Switch on). NO FIND NEW TRIAL VALUE H|_ BY INTERPOLATION COMPUTE BENDING MOMENTS D = 0 D=l COMPUTE PRINT BM'S END MAGNIFICATION MAG. FACT. OF F A C T O R S DEFLECTION JOB Figure 6. 28 solutions thus found could then be superimposed so as to satisfy all boundary equations. Another approach is to deal with the equation directly as a "boundary condition" problem, by '.-substit-uting for the differential equation a system of difference equations, which may be solved to give the deflections at points along the girder. The latter method was thought to be most suitable for this problem and was adopted. At a point i on the girder, the derivatives in the girder equation may be approximated by finite differences as given below: 1 (vi_ 2 - 4 v ± _ 1 + 6v± - 4 v 1 + 1 + v 1 + 2, d 4 1 / 1Y±_2 V±_± V 1 + 1 l V 1 + 2 dxJ / i d 3 \ 2 2, + - + (44) dx2 / I d 2 1 f v ^ - 2 v ± + vi+l) _ 1 / _ l v i - l l v i + l i d V 2 2 'where v i - 2, Vj__]_, v-j_, v i + 1, v^ + 2 are deflections at points separated by a small distance d. If substitution from equations (44) is made in equation (35)* given in the preceding chapter, it is possible at each point on the girder to write a difference equation of the form: a i l v i - 2 + a i 2 v i - l + a i 3 v i + a i 4 v i + l + a i 5 v i + 2 = b i ••• ( 45) 29 where a i l E I 1 d ( E l ) d4 d3 dx (1 + DHm) a 12 4 E I 2 d ( E l ) 1 d (E I ) + ^ d^ d3 dx d 2 d x 2 ( l + DHm) DH + ,2 d 2 Vdx B \ 2 P h d 2 y / d y B ' Ll d d x 2 \ d x a 13 = 6EI 2 d ( E I ) d 4 d 2 d x 2 (1 + DHm) DH + d ' 2 P h /dy d 2 Vdx B L a . 14 4EI 2 d ( E l ) 1 d 2 ( E l ) F ^ + d H d J dx d2 d x 2 ( l + DHm) (46) DH + 1 _ \ / d y d 2 d 2 I d x B 2 P h d 2 y /dy d d x 2 Vdx B a . c E I 1 d ( E l ) d 3 dx ( l + DHm) b i (p + w) m d2 ( E l ) dx 2 H + d 2 y / 1 dx ' m d 2 ( E l ) \ d x 2 J HP + h G t d 2 y (1 3 / d y B x 2 dx ' + ,dx H L ds d 2 y I 1 4 / d y B N 2> + AE dx dx \ \ dx L I n e q u a t i o n s (46) a b o v e , D may be e i t h e r one or z e r o , f o r d e f l e c t i o n t h e o r y or e l a s t i c t h e o r y , as p r e v i o u s l y d e s c r i b e d . I n a d d i t i o n P ^ may be one o r z e r o t o I n c l u d e o r d e l e t e the e f f e c t . 30 of h o r i z o n t a l d e f l e c t i o n of the c a b l e . The e f f e c t o f s h e a r d e f o r m a t i o n may be d e l e t e d - s i m p l y by a s suming a z e r o v a l u e f o r m, the shea r f l e x i b i l i t y o f the g i r d e r . D e r i v a t i v e s o f E I , the f l e x u r a l r i g i d i t y o f the g i r d e r , may be r e p l a c e d by f i n i t e d i f f e r e n c e s , as I n e q u a t i o n s (44). I n the case o f a g i r d e r o f c o n s t a n t E I , d e r i v a t i v e s o f E I v a n i s h . I n o r d e r t o o b t a i n a s o l u t i o n t o the d i f f e r e n t i a l e q u a t i o n , i t i s n e c e s s a r y t o e x p r e s s the boundary c o n d i t i o n s i n s i m i l a r f o r m t o e q u a t i o n (45). The case o f z e r o d e f l e c t i o n a t the s u p p o r t s i s o b v i o u s l y e x p r e s s e d i n the f o r m V i = o ... (47) F o r the case of z e r o moment a t a p o i n t , use i s made of the f o l l o w i n g e q u a t i o n , d e r i v e d f r o m e q u a t i o n s (29a) and ( 3 1 ) . M d 2 v ( E l ( l + DHm)\ E l m / p + w + H d 2 y V (48) d x 2 * / \ d x 2 I f the b e n d i n g moment i s z e r o a t a p o i n t i , t h e n the d i f f e r e n c e e q u a t i o n becomes 1 v . ., 2 v . 1 v . ,-, -m ( p + w + H d 2 y \ "2 -"2 + ~ 2 = : 1 - 2 "...(49) t\d &d L\d 1 + DHm \ d x 2 / i . I f the g i r d e r i s d i v i d e d i n t o a f i n i t e number N o f e q u a l p a r t i a l l e n g t h s , a t each of the N - 1 i n t e r i o r p o i n t s on the g i r d e r , i t i s . p o s s i b l e t o w r i t e an e q u a t i o n o f the f o r m (45) i n v o l v i n g the d e f l e c t i o n a t the p o i n t under c o n s i d e r a t i o n and a t two a d j a c e n t p o i n t s on each s i d e o f the p o i n t under c o n s i d e r a t i o n . Note t h a t the e q u a t i o n f o r the i n t e r i o r p o i n t s n e a r e s t the e x t e r i o r s u p p o r t s i n v o l v e the d e f l e c t i o n s a t the s u p p o r t s and a t f i c t i t i o u s p o i n t s e x t e r i o r , t o the s u p p o r t s . T h e r e f o r e , i t I s p o s s i b l e t o w r i t e N - 1 t y p i c a l i n t e r i o r e q u a t i o n s i n v o l v i n g N + 3 unknown d e f l e c t i o n s . The f o u r a d d i t i o n a l e q u a t i o n s 31 r e q u i r e d a r e p r o v i d e d a c c o r d i n g t o the boundary c o n d i t i o n s j t h a t I s , z e r o d e f l e c t i o n at. each e x t e r i o r s u p p o r t , and z e r o b e n d i n g moment a t each e x t e r i o r s u p p o r t . I n the case of a con-t i n u o u s g i r d e r , a t one or more of the i n t e r i o r p o i n t s t h e r e i s a s u p p o r t . I n t h a t , c a s e , the t y p i c a l i n t e r i o r e q u a t i o n a t t h a t p o i n t i s s i m p l y r e p l a c e d by an e q u a t i o n f o r z e r o d e f l e c t i o n a t t h a t p o i n t . A n . a r r a y of c o e f f i c i e n t s f o r a s e t of e q u a t i o n s t o be w r i t t e n f o r a g i r d e r w i t h no i n t e r i o r s u p p o r t s i s i n d i c a t e d below. C o e f f i c i e n t s d i f f e r e n t f r o m z e r o a r e i n d i c a t e d by an x. C o e f f i c i e n t s of Condensed - A s y s t e m a t i c method of s o l v i n g the e q u a t i o n s i s e v i d e n t f r o m the f o r m of the a r r a y . The f i r s t c o e f f i c i e n t of the t h i r d e q u a t i o n can be reduced t o z e r o by s u b t r a c t i n g a m u l t i p l e of the f i r s t e q u a t i o n . Then, the f i r s t non-zero c o e f f i c i e n t s i n the t h i r d and f o u r t h e q u a t i o n s can be reduced t o z e r o by s u b t r a c t i n g a m u l t i p l e of the second e q u a t i o n . T h i s p r o c e s s of e l i m i n a t i o n of c o e f f i c i e n t s can be c o n t i n u e d u n t i l the l a s t e q u a t i o n i s reduced t o an- e q u a t i o n i n v o l v i n g a s i n g l e unknown d e f l e c t i o n w h i c h i s r e a d i l y d e t e r m i n e d . Then, a l l o t h e r d e f l e c t i o n s can be q u i c k l y d e t e r m i n e d by s u b s t i t u t i o n of v a l u e s as t h e y a r e dete r m i n e d i n t o p r e c e d i n g e q u a t i o n s . The pr o c e d u r e d e s c r i b e d above i s known as t r i a n g u l a r i z a t i o n and back s u b s t i t u t i o n . I t i s r e l a t i v e l y s i m p l e . a n d f a s t here s i n c e t h e r e i s a band w i d t h 32 of o n l y f i v e n o n - z e r o c o e f f i c i e n t s . I t i s i m p o r t a n t t o no te t h a t the v a l u e s o f the c o e f f i c i e n t s i n the shaded p a r t o f the a r r a y r e m a i n z e r o t h r o u g h o u t the e n t i r e s o l u t i o n . I n p r o g r a m -m i n g , use i s made of t h i s k n o w l e d g e , and no memory space i s a l l o t t e d I n the computer f o r t hese z e r o c o e f f i c i e n t s . F o r s t o r a g e i n the compute r , the a r r a y i s condensed t o the f o r m shown t o the r i g h t o f the f u l l a r r a y . I n the s o l u t i o n of a s u s p e n s i o n b r i d g e p r o b l e m , i t i s n e c e s s a r y t o s o l v e the g i r d e r e q u a t i o n . s e v e r a l t i m e s , once f o r each each new t r i a l v a l u e o f H . The e q u a t i o n s (46) f o r a_ .. r e p r e s e n t t i m e - c o n s u m i n g c a l c u l a t i o n s , even on a compute r . T h e r e f o r e , i t i s w o r t h w h i l e t o f a c t o r out terms i n v o l v i n g H i n each o f the c o e f f i c i e n t s a^j and b ^ . Then i t i s p o s s i b l e t o compute and s t o r e v a l u e s o f c o n s t a n t s a b . . , s u c h t h a t f o r each new t r i a l v a l u e . o f H , the c o e f f i c i e n t s a^^ and bj_ a r e computed as f o l l o w s : a . I , = a b . . (1 + DHm) 1 i i v a . „ = ab . „ + DHab . .. i 2 l 3 14 a i 3 = a b i 5 + D H a b i 6 a . . = ab._ + D H a b . Q . . . (50) i 4 17 10 a^ - ab^g ( l + DHm) b . = ab . . , ^ + Hab.-,-, + D H a b . n o + DHH T ab . n ^ 1 l l O i l l i l 2 L 113 The c o m p o s i t i o n o f the terms a b . . i s o b v i o u s f r o m an e x a m i n a t i o n of e q u a t i o n s (4.6) and need n o t be w r i t t e n h e r e . I n t e g r a t i o n of the C a b l e E q u a t i o n I n t e g r a t i o n o f the c a b l e e q u a t i o n i s p e r f o r m e d by a p p l y i n g S i m p s o n ' s R u l e , w h i c h may be s t a t e d as f o l l o w s : 33 r B A f ( x ) dx . N E 1=2,4/6 f . _ + 4 f . + f . , _ i - l i i + i ... (51) dh where , I n the case o f the c a b l e e q u a t i o n , f i s — and the l i m i t s dx o f i n t e g r a t i o n a r e the two ends o f the span under c o n s i d e r a t i o n . dh I n the p r o g r a m , the e q u a t i o n f o r — i s dx dh H L / d s \ 3 G t (<ls\ 2 dy dv dx AE Vdx/ dx dx dx D + F H / d s \ 2 ( d y dv 1 / d v x 2 s l i / \ I _|_ AE \ dx dx dx 2 \ dx 1 / d v ' 2 Vdx , 2 ,. ... (52) where F i s one or z e r o t o i n c l u d e . o r d e l e t e the e f f e c t o f change s • • •. o f s l o p e i n the c a b l e . E q u a t i o n (52) i s e q u i v a l e n t t o e q u a t i o n (9) when the f a c t o r s F a and D a re made u n i t y . I t s h o u l d be n o t e d t h a t S i m p s o n ' s R u l e r e q u i r e s t h a t each span be, d i v i d e d i n t o an even number o f e q u a l p a r t i a l l e n g t h s . S i n c e the l e n g t h d has been made e q u a l t o o n e - t w e n t i e t h o f the main s p a n , i t i s r e q u i r e d t h a t the r a t i o o f s i d e span l e n g t h t o main span l e n g t h be .1, .2, .3, e t c . P rogram L i n k a g e • F o u r main p a r t s o f the p rog ram w h i c h a r e used more t h a n once a r e w r i t t e n i n the f o r m o f s u b r o u t i n e s and t h e y a r e l i n k e d up l a r g e l y by the use o f "computed GO TO" s t a t e m e n t s . One o f these s u b r o u t i n e s computes the a r r a y a b . . f o r a s i n g l e _L J s p a n , f r o m the l e f t s u p p o r t a t I I t o the r i g h t s u p p o r t a t I E . A n o t h e r s u b r o u t i n e i n t e g r a t e s the c a b l e e q u a t i o n f o r a s i n g l e span f r o m I I t o I E . A t h i r d s u b r o u t i n e s o l v e s the g i r d e r e q i i a t i o n , e i t h e r f o r a s i n g l e span f r o m I I t o I E , o r f o r a 34 c o n t i n u o u s g i r d e r f r o m I I t o IE w i t h i n t e r m e d i a t e s u p p o r t s a t 1=37 and 1=17. Each of the above s u b r o u t i n e s r e q u i r e s the . l o c a t i o n of I I , the i n i t i a l p o i n t I of the c a l c u l a t i o n and I E , the end p o i n t I of the c a l c u l a t i o n . Two of the s u b r o u t i n e s a l s o r e q u i r e the span l e n g t h L of the span under c o n s i d e r a t i o n and the d i f f e r e n c e i n e l e v a t i o n of the anchorages B. T h e r e f o r e , t h e r e i s a s u b r o u t i n e which s p e c i f i e s the l o c a t i o n of I I and IE f o r s u c c e s s i v e spans as w e l l as the v a l u e s of B and L f o r each span f o r use i n the o t h e r s u b r o u t i n e s d e s c r i b e d above. B e f o r e e n t e r i n g a s u b r o u t i n e , i t i s n e c e s s a r y t o s p e c i f y the v a l u e of • a c o n s t a n t which i s t o be used i n a "computed GO TO" statement t o d etermine the r o u t e of the program upon e x i t f r o m the sub-r o u t i n e . F i g u r e 7 shows how the l i n k a g e i s e f f e c t e d t o s o l v e the g i r d e r e q u a t i o n f o r e i t h e r a c o n t i n u o u s g i r d e r or f o r h i n g e s a t the towers. I n p u t Data f o r the Program I t has been s t a t e d t h a t c e r t a i n r e f i n e m e n t s I n the t h e o r y may be d e l e t e d or i n c l u d e d a t w i l l by s e t t i n g the v a l u e s of F^ and F g e q u a l t o z e r o or u n i t y . F u r t h e r , the g i r d e r may be e i t h e r h i n g e d a t the towers or c o n t i n u o u s , of c o n s t a n t r i g i d i t y or v a r i a b l e . I n o r d e r t o s p e c i f y any c o m b i n a t i o n of the above c h o i c e s , a s i n g l e code number i s r e a d i n from the f i r s t two columns of the f i r s t d a t a c a r d . The meaning of the code number i s shown by the f o l l o w i n g t a b l e : TO FOLLOW PAGE 34 PORTION OF MAIN LINE PROGRAM CONTINUOUS N4 = 2 ^CONTINUOUS GIRDER) SPECIFY II AND IE FOR CONTINUOUS GIRDER HINGED N4 = l HINGES N2 = 3 (GIRDER CALC. ) CONTINUATION OF MAIN PROGRAM SUBROUTINE 4 SOLVE GIRDER EQUATION II TO I E Determine v at all points SUBROUTINE 2 SPECIFY I I , I E , L , B L E F T SIDE SPAN SPECIFY 11, IE , L , B MAIN SPAN SPECIFY I I , IE, L , B RIGHT SIDE S P A N Figure 7. 35 CODE C o n s t a n t E I V a r i a b l e E I £ h F, 11 01 1 1 12 02 0 1 13 03 1 0 14 04 0 0 15 05 1 1 16 06 0 1 • 17 07 1 . 0 18 08 0 0 C o n t i n u o u s H i n g e d The n e x t t h r e e d a t a c a r d s each c o n t a i n the v a l u e of a c o n s t a n t H L 2 r e l a t e d t o the e l a s t i c p r o p e r t i e s of the b r i d g e . They a r e D E I 2 * a n < 3 ^Y?T> where I i s the g i r d e r moment o f i n e r t i a a t m i d -y A E L s p a n . The above v a l u e s each occupy the f i r s t 14 colums o f a d a t a c a r d and a r e i n E14.7 f o r m a t . There f o l l o w t h r e e d a t a c a r d s s p e c i f y i n g the geomet ry of the b r i d g e . They a r e £. f o r ) 'li the main s p a n , the l e n g t h of the s i d e spans as a r a t i o o f the main span l e n g t h , and the r i s e o f the s i d e span as a r a t i o - o f the main span l e n g t h . They a r e i n F6.4 f o r m a t and occupy the f i r s t s i x columns o f the c a r d s . F o l l o w i n g a r e two d a t a c a r d s s p e c i f y i n g , i n E14.7 f o r m a t , the v a l u e s o f € t and the anchorage s l i p as a r a t i o o f the main span l e n g t h . I f the g i r d e r r i g i d i t y i s v a r i a b l e , t h e r e i s r e q u i r e d a s e t o f d a t a c a r d s g i v i n g the v a l u e o f the g i r d e r s t i f f n e s s E I a t each p o i n t on the g i r d e r f r o m the l e f t anchorage t o the r i g h t a n c h o r a g e . The v a l u e s g i v e n a r e i n F7.4 f o r m a t , one t o a c a r d , - ' a n d a r e the r a t i o o f the g i r d e r r i g i d i t y a t t h a t p o i n t t o the g i r d e r r i g i d i t y a t the c e n t r e o f the main s p a n . I f the g i r d e r has c o n s t a n t E I , t h i s s e t o f d a t a c a r d s i s o m i t t e d . F i n a l l y , a-:-set o f c a r d s i s r e a d 36 g i v i n g the v a l u e o f the l i v e l o a d on the g i r d e r as a r a t i o P . w One c a r d i s r e a d f o r each p o i n t on the g i r d e r f r o m . t h e l e f t anchorage t o the r i g h t a n c h o r a g e . The r a t i o s a r e g i v e n i n F6.4 f o r m a t . F i n a l Notes I n A p p e n d i x 1 t h e r e i s a comple t e b l o c k d i a g r a m and a l i s t i n g o f the p rogram as i t was w r i t t e n and used i n i n v e s t i g a -t i o n s f o r t h i s t h e s i s . No c l a i m i s made t h a t the p rog ram i s the b e s t one t h a t c o u l d have been w r i t t e n f o r the s t u d i e s u n d e r -t a k e n . However , i t d i d g i v e s a t i s f a c t o r y r e s u l t s . The a c c u r a c y was good and the amount o f i n f o r m a t i o n y i e l d e d by the p rogram was e n t i r e l y , a d e q u a t e . I t may be t h a t the p r o g r a m w i l l be o f l i t t l e f u r t h e r v a l u e I n i t s p r e s e n t f o r m , and t h e r e f o r e i t was n o t c o n s i d e r e d w o r t h w h i l e t o r e v i s e i t t o the more s o p h i s -t i c a t e d f o r m u s u a l l y r e q u i r e d of a l i b r a r y p r o g r a m . However , i t may have some v a l u e as a g u i d e t o o t h e r s i n the p r e p a r a t i o n of s i m i l a r p r o g r a m s , and f o r t h a t r e a s o n i t has been p r e s e r v e d h e r e . The major p o r t i o n of the comput ing t ime was t a k e n t o s o l v e the s e t o f e q u a t i o n s r e p r e s e n t i n g the g i r d e r e q u a t i o n . R u n n i n g t ime f o r each t r i a l s o l u t i o n was a p p r o x i m a t e l y two m i n u t e s . S i n c e t h r e e t r i a l s were r e q u i r e d f o r an E l a s t i c T h e o r y s o l u t i o n and f o u r or f i v e t r i a l s were r e q u i r e d f o r a D e f l e c t i o n Theory s o l u t i o n , the t o t a l compu t ing t ime was a p p r o x i m a t e l y f o u r t e e n t o s i x t e e n m i n u t e s . 37 CHAPTER 4 DETERMINATION OF H G e n e r a l I t has been shown t h a t a n a l y s i s o f s u s p e n s i o n b r i d g e s i n v o l v e s the s i m u l t a n e o u s s o l u t i o n of two e q u a t i o n s i n H and v . S i n c e the method of s o l u t i o n must be a t r i a l and e r r o r p r o c e d u r e , the c a l c u l a t i o n s a r e l e n g t h y . However , once the v a l u e of H i s d e t e r m i n e d , , i t i s a s t r a i g h t f o r w a r d m a t t e r t o compute a l l d e f l e c t i o n s , bending•moments and s h e a r s i n the g i r d e r . I t has f u r t h e r been shown t h a t , f o r most p u r p o s e s , i t i s s u f f i c i e n t l y a c c u r a t e t o use e q u a t i o n s ( l i b ) and (4-3) i n the d e t e r m i n a t i o n of H . The e q u a t i o n s a r e r e p e a t e d h e r e f o r c o n v e n i e n c e of r e f e r e n c e . h B - h A e t L t H l L e r L + AE j dv / d y B \ dx . . . ( l i b ) o dx V dx L E I d 2 v Hv Hy M' — p - ... (43) dx^ I t w i l l be n o t e d i m m e d i a t e l y t h a t e x t e n s i o n of the c a b l e due t o t e m p e r a t u r e r i s e and s t r e s s e l o n g a t i o n has an e f f e c t e x a c t l y e q u i v a l e n t t o a s m a l l r e l a t i v e s u p p o r t movement. I f the t e r m A i s d e f i n e d as h-o h A 6 t L . H T B A t L e / s AE t h e n A may be t h o u g h t o f as the e q u i v a l e n t s u p p o r t d i s p l a c e m e n t s 38 ... (54) of an i n e x t e n s i b l e c a b l e . E q u a t i o n ( l i b ) r e d u c e s to r L dv / d y B \ dx J o dx \ d x L J A method based on e q u a t i o n s (43) and (54) i s p r e s e n t e d h e r e , whereby a d e s i g n e r can d e t e r m i n e v e r y q u i c k l y the v a l u e of H f o r a s i n g l e span o r f o r a m u l t i p l e span b r i d g e e i t h e r h i n g e d a t the s u p p o r t s o r c o n t i n u o u s . S u p e r p o s i t i o n of P a r t i a l L o a d i n g C o n d i t i o n s S i n c e e q u a t i o n (43) i s a 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 , i t i s p e r m i s s i b l e t o supe r impose a number o f s o l u t i o n s t o (43) f o r v a r i o u s r i g h t hand s i d e e x p r e s s i o n s . . I n p a r t i c u l a r , i t i s p e r m i s s i b l e t o r e p l a c e H on the r i g h t hand s i d e o f the e q u a t i o n by H Q + H-p- t h e n e q u a t i o n (43) may be w r i t t e n E l d ^ v Hv H Q y H x y + M' dx 2 The s o l u t i o n t o (55) may be g i v e n by V = V Q + V ] _ where E I d 2 v Q H v 0 H Q y M' dx" ... (.55) ... (57) E I d 2 v - Hv-dx £ Then H ny ... (58) A may be r e p l a c e d by. A + A ^ i n e q u a t i o n (54) t o g i v e 0 dv, 0 'dy B \ dx ... (59) o dx \ dx L A r L dv 1 / dy B \ dx ... (60) o dx \ dx 39 The p h y s i c a l s i g n i f i c a n c e of e q u a t i o n s (57) t o (60) i s d i f f i c u l t t o d e s c r i b e s i n c e the s u p e r p o s i t i o n i s r e a l l y o n l y a s u p e r p o s i t i o n o f m a t h e m a t i c a l s o l u t i o n s and n o t the sum o f two p h y s i c a l s t a t e s . However , i t m igh t be c o n v e n i e n t t o t h i n k o f a b r i d g e w i t h movable anchorages r e s t r a i n e d by a c o n s t a n t f o r c e H . Then the d e f l e c t i o n s v Q and A Q a r e a r e s u l t of the a p p l i e d l o a d moments M' and a p o r t i o n of the c a b l e t e n s i o n r e p r e s e n t e d by H Q . The d e f l e c t i o n s v^ and A c a n be a t t r i b u t e d t o the a c t i o n o f the p a r t i a l c a b l e t e n s i o n r e p r e s e n t e d by H ^ . When H Q and H ^ t o t a l H and when A 0 and A ]_ t o t a l A , t h e n c o m p a t i -b i l i t y e x i s t s and the s o l u t i o n f o r v I s g i v e n by the sum of V Q and V ] _ f o r the two p a r t i a l l o a d i n g c a s e s . I t w i l l be shown t h a t i t i s advan tageous t o make A e'qual t o z e r o . Then H Q i s the p o r t i o n of c a b l e t e n s i o n r e s u l t -i n g f r o m the a p p l i e d l o a d a c t i n g on a b r i d g e w i t h i n e x t e n s i b l e c a b l e and immovable s u p p o r t s . Then , H-^ i s the p o r t i o n o f c a b l e t e n s i o n r e s u l t i n g f r o m A , w h i c h , i t w i l l be remembered, I s the e f f e c t o f c a b l e s t r e t c h and s u p p o r t d i s p l a c e m e n t . B o t h H Q and H-^ a r e s t i l l f u n c t i o n s o f the t o t a l v a l u e o f H , b u t i t w i l l be shown t h a t d e t e r m i n a t i o n o f H ^ i s n o t s e n s i t i v e t o e r r o r i n an e s t i m a t e d v a l u e of H . S i n g l e Span S i n c e s u p e r p o s i t i o n o f r e s u l t s has been shown t o be v a l i d i f a p p l i e d i n the manner o u t l i n e d , i t i s p e r m i s s i b l e to make use o f the R e c i p r o c a l Theorem i n d e t e r m i n a t i o n of c a b l e t e n s i o n . F i g u r e 8 shows, the two cases r e q u i r e d f o r a p p l i c a t i o n of the theo rem. Case 1 i l l u s t r a t e s the d e f l e c t i o n s v-^ and A ]_ a t t r i b u t a b l e to the p a r t i a l c a b l e t e n s i o n H 1 # T h i s c o r r e s p o n d s TO FOLLOW PAGE CASE I Figure 8. 4o l i v e l o a d t e n s i o n due t o a u n i t l o a d on the span t o the s o l u t i o n of e q u a t i o n s (58) and ( 6 0 ) . Case 2 shows the I n case 2 the c a b l e i s assumed t o be i n e x t e n s i b l e and suspended f r o m immovable s u p p o r t s . T h i s c o r r e s p o n d s t o a s o l u t i o n of e q u a t i o n s (57) and (59) f o r the case o f a s i n g l e u n i t l o a d on the s p a n , where H Q i s the sum of H L and the dead l o a d t e n s i o n H ^ . A c c o r d i n g t o the r e c i p r o c a l t heo rem, the e q u a t i o n o f the i n f l u e n c e l i n e f o r H£ i s g i v e n by V v- (61) A E q u a t i o n (58) can be r e a d i l y s o l v e d t o g i v e v1 H x f L 2 4 E I (CL) 2 L 2 x 2 - + , L (CL) ' 2 ( ( l - e - C L ) e C x + ( e C L - l ) e - C x ) ( C L ) ^ ( e U i j - e - U i j ) (62) .. (63) E q u a t i o n (62) can be w r i t t e n i n a s i m p l e r f o r m as v 1 H x f L 2 E I . (64) where v . 4 ( C L ) ' x \ 2 x L , + ( C L ) ' 2 ( ( i - e - C L ) e C x + ( e C L - i ) e - C x ) ( C L ) 2 ( e C L - e - C L ) (65) When the e x p r e s s i o n f o r v^ i s d i f f e r e n t i a t e d and sub-s t i t u t i o n i s made i n e q u a t i o n ( 6 0 ) , i t I s found t h a t 2 H vr L A E I 1 ... (66) 41 where T± 64 ( C L ) ' • 1 4 + e C L ( C L - 2 ) - e " C L ( C L + 2 ) 12 ( C L ) 3 ( e C L - e - C L ) ... (67) Then the i n f l u e n c e l i n e f o r H £ i s g i v e n by H L - L / v± \ Note t h a t v and A ^  a r e d i m e n s i o n l e s s and a r e f u n c -t i o n s o f the d i m e n s i o n l e s s q u a n t i t y C L , where C i s d e f i n e d by e q u a t i o n ( 6 3 ) . i s , o f c o u r s e , a l s o a f u n c t i o n of the r e l a -t i v e p o s i t i o n on the s p a n . V a l u e s o f - V l , r e p r e s e n t i n g the i n f l u e n c e l i n e f o r on a s i n g l e span i n d i m e n s i o n l e s s f o r m a re t a b u l a t e d i n T a b l e 1 o f A p p e n d i x 2. A l s o , F i g u r e 1 0 : ' • shows i n f l u e n c e l i n e s f o r v a l u e s o f ( C L ) 2 = 1 and ( C L ) 2 = 100. To f a c i l i t a t e d e t e r m i n a t i o n of H f o r d i s t r i b u t e d l o a d s , the a r e a under one o f the i n f l u e n c e c u r v e s i n F i g u r e 10 has been p l o t t e d i n the same f i g u r e . The c u r v e shows the p a r t i a l a r e a A-^ t o the l e f t o f p o i n t x , where A n f x v n dx 1 = ~ ... (69) J o A 1 T a b u l a t e d v a l u e s a r e a l s o i n c l u d e d l i n T a b l e 1. By the use o f the c u r v e s o r t a b l e s d e s c r i b e d above i t i s p o s s i b l e t o d e t e r m i n e H £ and hence H Q , the t e n s i o n t h a t wou ld e x i s t i n an i n e x t e n s i b l e c a b l e w i t h immovable s u p p o r t s . Then a c o r r e c t i o n 8H must be added f o r the e f f e c t o f c a b l e s t r e t c h and s u p p o r t d i s p l a c e m e n t . F i g u r e 13 shows a p l o t o f A ^  a g a i n s t ( C L ) 2 w h i c h can be used t o f i n d the c o r r e c t i o n 8H . T a b u l a t e d v a l u e s a r e a l s o i n c l u d e d i n T a b l e 1. E q u a t i o n (53) shows t h a t A i s a f u n c t i o n o f H the 42 unknown l i v e l o a d t e n s i o n . I t i s p o s s i b l e t o a v o i d e s t i m a t i n g t h i s v a l u e i n i t i a l l y by r e w r i t i n g (53) i n the f o r m • 6H L. A = A where h B h A e AE € t i n (70) H L ' ^ e AE .., (71) The c o r r e c t i o n S r l ' I s , i n the case of a s i n g l e s pan SH AH-. (72) where i t has been shown i n e q u a t i o n (66) t h a t H - 1 ,2 T — A X r - L A E I (73). E q u a t i o n s (73) and ( 7 l ) can be s u b s t i t u t e d i n e q u a t i o n (72) t o g i v e • 8H LC 8H A - _ AE_ f 2 L A~l ... (74) E I E q u a t i o n (74) can t hen be s o l v e d f o r :8'H t o g i v e 8H A!-f 2 L A 1 L 1 + -1 . (75) E I AE I n o r d e r t o d e t e r m i n e oE, I t i s n e c e s s a r y t o compute A ' once f o r H-^  and e s t i m a t e the v a l u e o f H t o f i n d TJT^  f r o m F i g u r e 13. Then 8H and hence H can be computed f r o m e q u a t i o n ( 7 5 ) . I f the computed v a l u e of H does n o t ag ree w i t h the e s t i m a t e d v a l u e , a n o t h e r c o m p u t a t i o n must be made w i t h a d i f f e r e n t v a l u e o f 43 N u m e r i c a l examples i n A p p e n d i x 3 show t h a t the i t e r a t i o n •converges r a p i d l y . i D e t e r m i n a t i o n o f H£ depended on an i n i t i a l e s t i m a t e of H and so an i t e r a t i v e p r o c e d u r e I s i m p l i e d f o r the d e t e r m i n a -t i o n o f H£. However , i t e r a t i o n i s u s u a l l y u n n e c e s s a r y because the i n f l u e n c e l i n e s a r e r e l a t i v e l y i n d e p e n d e n t o f the v a l u e o f C L . T h i s i s i l l u s t r a t e d by a s t u d y of the ext reme v a l u e s o f C L . A t one ext reme as CL approaches z e r o , the e l a s t i c t h e o r y becomes v a l i d and the i n f l u e n c e l i n e i s g i v e n by f 8 x L 2 / x 3 4' + L ... (76) A t the o t h e r ext reme as CL becomes i n f i n i t e l y l a r g e , - - t h e g i r d e r i s so f l e x i b l e as t o o f f e r no r e s i s t a n c e t o d e f l e c t i o n , and i t can be shown t h a t the i n f l u e n c e l i n e e q u a t i o n ' i s 2 ' V_ L 3 f 4 x L x (77) F i g u r e 9 shows i n f l u e n c e l i n e s f o r H ' f o r the ext reme Li v a l u e s o f C L . F u r t h e r i n v e s t i g a t i o n shows t h a t the i n f l u e n c e l i n e o r d i n a t e s f o r a l l v a l u e s o f CL l i e w i t h i n the range d e f i n e d by the ext reme v a l u e s e x c e p t i n the r e g i o n where the c u r v e s i n t e r s e c t . I t i s a p p a r e n t t h a t no s i g n i f i c a n t e r r o r i n Hi^ w i l l be i n t r o d u c e d by i n a c c u r a t e e s t i m a t e s o f the v a l u e o f H . F o r T c-a l l v a l u e s o f CL the a r e a under the c u r v e must be — . T h i s -2 87' p o i n t becomes c l e a r e r when i t i s r e a l i z e d t h a t a u n i f o r m l o a d p c o v e r i n g the e n t i r e span i n t r o d u c e d no g i r d e r b e n d i n g moment and r e s u l t s i n a c a b l e t e n s i o n w i t h a h o r i z o n t a l component e q u a l t o pL 8f TO FOLLOW PAGE 43 .2 C L = CL = cx 0 — « -• / .2 .4 .6 .8 x 17 X L CL= 0 CL= oo .05 .0311 .0356 .10 .0613 .0675 .1 5 .0899 .0956 .20 .1160 .1200 .25 .1392 . 1406 .30 .1588 .1575 .35 .1745 .1706 .40 .I860 .1800 . .45 .1926 .1856 .50 .1953 .1875 Figure 9. 44 T h r e e - S p a n B r i d g e w i t h H i n g e d S u p p o r t s I t i s a s i m p l e m a t t e r t o e x t e n d the a p p l i c a t i o n of the method d e s c r i b e d above t o the case o f a t h r e e - s p a n b r i d g e w i t h h i n g e s a t the g i r d e r s u p p o r t s . E q u a t i o n s (58) and (60) must be s o l v e d t o f i n d a t o t a l A ^ f o r a l l the spans w h i c h i s E I 1 2 a 3 b ( A . ) + 1 s A ... (78) where: a = r a t i o o f s i d e span l e n g t h L g t o main span l e n g t h L b L s2 E I £ 2 E X ( A l l s = "S-! ( b C L ) F i g u r e 14(a) shows c u r v e s o f b ( " ^ l ) s a g a i n s t ( C L ) 2 f o r s e l e c t e d v a l u e s o f b . The I n f l u e n c e l i n e f o r H£ i n the main span i s t h e n H L ' f v- X (79) where X 1 2a3b(A ) + L- a-... (80) The i n f l u e n c e l i n e f o r H£ i n the s i d e span I s L f v-(81) 1 / s The s u b s c r i p t s i n the t e r m - v l i n d i c a t e s t h a t the i n f l u e n c e A 1 / s c u r v e f o r the s i d e span i s g e n e r a l l y d i f f e r e n t f r o m t h a t f o r the main s p a n , due t o the d i f f e r e n t v a l u e of C L . The m u l t i p l i e r X g f o r the s i d e span I s g i v e n by 45 2 a b( A ]_) s x s ..: A~ l t 1 " x> 1 2 a 3 b ( A i ) . + '• ... (82) I n e q u a t i o n s (79) and ( 8 l ) , L and f a r e the v a l u e s o f span l e n g t h and sag f o r the main s p a n . F i g u r e 14(b) shows X and X g p l o t t e d a g a i n s t b ( A 1^s f o r s e l e c t e d v a l u e s o f a . I t i s i n t e n d e d t h a t F i g u r e s 14(a) and 3.4(b) be used t o g e t h e r i n o r d e r t o d e t e r m i n e X g and X , the m u l t i p l i e r s f o r the i n f l u e n c e ^ l i n e o r d i n a t e s f rom F i g u r e 10. I n the case o f the m u l t i p l e span b r i d g e , i t c an be shown t h a t the e q u a t i o n f o r 8H i s g i v e n by S H . / IL \ 1 ... (83) A l I X As i n the case o f the s i n g l e s p a n , s u b s t i t u t i o n can be made f r o m e q u a t i o n s ( 7 l ) and (73) and t h e r e s u l t i n g e q u a t i o n can be s o l v e d f o r 8H t o g i v e 8H A 1f 2L A -, L a 1 , e (84) — + E I X AE V a l u e s o f b ( ^ 1^s a r e t a b u l a t e d i n T a b l e 2, A p p e n d i x 2, and A~ l A p p e n d i x 3 c o n t a i n s a n u m e r i c a l example showing how the t a b l e s o r F i g u r e s 10 and 14 can be u sed t o d e t e r m i n e H f o r a t h r e e -span b r i d g e w i t h h i n g e d s u p p o r t s . I t i s c l e a r t h a t the g e n e r a l p r o c e d u r e i s the same as t h a t f o r a s i n g l e s p a n . T h r e e - S p a n B r i d g e w i t h C o n t i n u o u s G i r d e r Up t o t h i s p o i n t , c o n s i d e r a t i o n has been r e s t r i c t e d 46 t o s i n g l e and m u l t i p l e span b r i d g e s ' w i t h h i n g e s a t a l l s u p p o r t s . The p r o b l e m becomes somewhat more c o m p l i c a t e d i f the g i r d e r i s c o n t i n u o u s a t the s u p p o r t s , b u t i t may be s o l v e d i n a s i m i l a r manner. E q u a t i o n (43) can be r e p l a c e d by e q u a t i o n (57) and by e q u a t i o n s (85) t o (87) b e l o w : Eld^v-L H v 1 H j y dx ' E l d v 2 H v 2 dx ,2 2 MgX L E I d ^ v 3 H V 3 M 3 ( L - x ) (85) . (86) ... (87) dx^ L E q u a t i o n s (85) t o (87) r e p r e s e n t a s i n g l e span w i t h no a p p l i e d l o a d moments bu t w i t h end moments Mg and M^ r e s u l t i n g f r o m the a c t i o n of the p a r t i a l t e n s i o n H]_ a c t i n g on a c o n t i n u o u s g i r d e r . The t h r e e e q u a t i o n s t o g e t h e r a r e e q u i v a l e n t t o the s i n g l e e q u a t i o n (58) f o r a s i n g l e s p a n . I n a n a l y s i s o f a c o n t i n u o u s s t r u c t u r e , I t i s n e c e s s a r y t o e q u a l i z e end s l o p e s where c o n t i n -u i t y e x i s t s . E q u a t i o n (85) has been s o l v e d f o r the case of a s i n g l e span and i t can be shown t h a t the s l o p e s a r e d v 1 lijfL / dv 1> ... (88) dx where dv E I \ d x 1 dx / o 4 4 .+ e C L ( C L - 2 ) e " C L ( C L + 2 ) ( C L)3 E q u a t i o n (86) can be s o l v e d t o g i v e o — v 2 M 2 Ir V 2 E I where CL - C L e - e (89) . (90) 47 Vr x e C x - e _ C x L " e C L - e ~ C L ( C L ) 2 Then the end s l o p e s can be found f r o m ... (91) dVg Mg L dVg dx E I dx . (92) where / d V 2 \ 1 1 •o CL CL / d v ^ 1 1 \ d x j L CL CL OL - C L e - e e C L + e - C L ~ C L ^ C L e - e From symmetry of the g i r d e r , i t i s c l e a r t h a t ( v q ) x = ( v 2 ) L _ x . (93) ... (94) . (95) (96) ... (97) I n the case o f a s y m m e t r i c a l t h r e e - s p a n b r i d g e , w i t h no a p p l i e d l o a d moment, the b e n d i n g moments a t the towers a r e e q u a l . The unknown moment M can be found by e q u a l i z i n g end s l o p e s a t the towers and i s g i v e n by ab / dv-M H l f ,dx / Ls dv^ dx + b / dv ... (98) 2 a Vdx j Ls I t can be shown t h a t i n the case o f an e x t r e m e l y s t i f f g i r d e r where the e l a s t i c t h e o r y i s v a l i d , e q u a t i o n (98) can be r e d u c e d t o 48 H l f 1 + ab: 3 b — + — 2 a ... (99) I n the i t e r a t i v e p r o c e d u r e r e q u i r e d t o compute H , i t w i l l be n e c e s s a r y t o compute M a number o f t i m e s , and i t w i l l be shown t h a t i t i s advan tageous t o compute M e w h i c h i s i n d e p e n d e n t of H and c o r r e c t by means of a m u l t i p l i e r K w h i c h must be d e t e r -mined f o r each new t r i a l v a l u e o f H . V a l u e s o f K a r e t a b u l a t e d i n A p p e n d i x 2 f o r s e l e c t e d v a l u e s o f a , b and (CL) . K i s a l s o p l o t t e d a g a i n s t t he se pa rame te r s i n F i g u r e 16. When M g i s com-p u t e d and K i s found f r o m the t a b l e s or c u r v e s , M i s - f o u n d f r o m M = K JVL ... (100) From the s o l u t i o n s t o e q u a t i o n s (86) and ( 8 7 ) , i t i s f ound t h a t A 2 Mg f IL X 2 E I A 3 M 3 . . f L X 3 E I where X 2 ~A~3 4 ... (101) ... (102) (CL) "3 4 + e C L ( C L - 2 ) - e " C L ( C L+2) , C L e -CL (103) The t o t a l v a l u e o f the s u p p o r t movement c o r r e s p o n d i n g t o the s o l u t i o n o f e q u a t i o n s (85) t o (87) f o r a l l t h r e e spans i s g i v e n by A t H-^f L T A 2. E I . . . (10-4) where T 1 2 a 3 b ( A 1 ) , M + A 2 A 2 2 a b ( A 2 ) . + : A 1 ... (105) t 49 where M K 1 + ab 3 b — + — 2 a ... (106) V a l u e s o f b ( ^ l ^ s a r e the same as t hose used f o r the case o f a t h r e e - s p a n b r i d g e w i t h h i n g e d s u p p o r t s and a r e found i n A p p e n -d i x 2 and F i g u r e 14 ( a ) . V a l u e s , of A 2 and b ^ A 2^s a r e p l o t t e d ,2 A A l a g a i n s t (CL) i n F i g u r e 15 f o r s e l e c t e d v a l u e s o f b and a r e a l s o i n c l u d e d i n A p p e n d i x 2. I n f l u e n c e l i n e s f o r H ' i n a c o n t i n u o u s s u s p e n s i o n li b r i d g e must be found by s u p e r i m p o s i n g two i n f l u e n c e l i n e s . The f i r s t i s the same i n f l u e n c e l i n e used f o r h i n g e d g i r d e r s . The second i s a c o r r e c t i o n f o r c o n t i n u i t y . I n the case o f the main span the I n f l u e n c e l i n e , i s g i v e n by L f X v 1 + Y v 2 + v 3 A i A 2 + A3 (107) where X 1 T Y M 2 X 2 ... (108) (109) T C u r v e s o f v 2 + v 3 a r e shown p l o t t e d i n F i g u r e 12 and t a b u l a t e d A ~ 2 + A ~ 3 v a l u e s a r e found i n A p p e n d i x 2. F o r the l e f t s i d e s p a n , the i n f l u e n c e l i n e f o r H-^  i s g i v e n by L X s / V l + Y s f ^2 \ A ii A 2/ ... (110) 50 where . . . ( i l l ) T Y Mab ( A o ) s , S = ^ ... (112) F i g u r e 11 shows c u r v e s o f v 2 and A p p e n d i x 2 has t a b u l a t e d v a l u e s f o r s e l e c t e d v a l u e s o f ( C L ) 2 . The i n f l u e n c e l i n e f o r the r i g h t s i d e span i s o f c o u r s e s i m i l a r to t h a t f o r the l e f t s i d e span b u t o p p o s i t e hand . The n u m e r i c a l example i n A p p e n d i x 3 shows t h a t the i t e r a t i o n p r o c e d u r e f o r d e t e r m i n a t i o n of H c o n v e r g e s r a p i d l y and use o f the t a b l e s or c u r v e s makes the c a l c u l a t i o n s s i m p l e . V a r i a b l e EI . . . I t can be seen t h a t the s o l u t i o n s t o e q u a t i o n s ( 5 8 ) , (86) and (87) a r e g i v e n f o r the s p e c i a l case i n w h i c h the g i r d e r r i g i d i t y E I w i t h i n the span i s c o n s t a n t . T h e r e f o r e , use o f the c o n s t a n t s t a b u l a t e d i n A p p e n d i x 2 depends on the a s s u m p t i o n o f c o n s t a n t E I w i t h i n each s p a n . I t i s p o s s i b l e t o s o l v e the e q u a t i o n s , a t l e a s t n u m e r i c a l l y , f o r o t h e r p a r t i c u l a r v a r i a t i o n s i n g i r d e r r i g i d i t y and t a b u l a t e s i m i l a r d a t a f o r use i n a n a l y s i s . A s u i t a b l e a p p r o a c h . m i g h t be t o d e t e r m i n e a t y p i c a l mode of v a r i a t i o n of g i r d e r s t i f f n e s s such t h a t most o r a l l s u s p e n s i o n b r i d g e g i r d e r s have ,a s t i f f n e s s v a r i a t i o n w h i c h l i e s w i t h i n a r ange d e f i n e d by two s e t s o f t a b u l a t e d d a t a . A n a l y s i s c o n s t a n t s migh t t h e n be d e t e r m i n e d by i n t e r p o l a t i o n be tween the t a b u l a t e d v a l u e s . However , i t I s s u g g e s t e d t h a t the a s s u m p t i o n of c o n s t a n t 51 g i r d e r r i g i d i t y i s a r e a s o n a b l e and d e s i r a b l e one f o r hand c a l -c u l a t i o n s . Some n u m e r i c a l examples were s o l v e d u s i n g the computer p rogram d e s c r i b e d i n the p r e c e d i n g c h a p t e r . A t h r e e -span b r i d g e was a n a l y s e d as a c o n t i n u o u s g i r d e r and w i t h h i n g e s a t the s u p p o r t s . The main epan g i r d e r s t i f f n e s s v a r i e d f r o m a minimum of .5 t imes the midspan s t i f f n e s s a t the towers t o a maximum o f 1.5 t i m e s t h e . m i d - s p a n s t i f f n e s s a t . the q u a r t e r p o i n t s . The same b r i d g e was a n a l y s e d a s suming a c o n s t a n t g i r d e r r i g i d i t y e q u a l t o the ave rage v a l u e . The r e s u l t s o f the s t u d y i n d i c a t e d t h a t a r e a s o n a b l y a c c u r a t e v a l u e of H can be d e t e r m i n e d by a s suming an ave rage v a l u e f o r the g i r d e r s t i f f n e s s . E r r o r s i n H-^ e n c o u n t e r e d were l e s s t h a n 2 p e r c e n t . T h e r e f o r e , i t i s recommended t h a t f o r a l l hand c a l c u l a t i o n s , a c o n s t a n t g i r d e r r i g i d i t y E I e q u a l t o the ave rage v a l u e be assumed f o r each span i n o r d e r t o d e t e r m i n e the v a l u e f o r H . TO FOLLOW PAGE 51 Figure 13 52 CHAPTER 5 CONCLUSIONS I t i s d o u b t f u l t h a t D e f l e c t i o n T h e o r y a n a l y s i s o f s u s p e n s i o n b r i d g e s by hand c a l c u l a t i o n s w i l l e v e r be a s i m p l e p r o c e d u r e o r one t h a t the s t r u c t u r a l e n g i n e e r w i l l a p p r o a c h e n t h u s i a s t i c a l l y . However , i t has been f o u n d , t h a t E l a s t i c T h e o r y s o l u t i o n s a r e t oo i n a c c u r a t e even f o r p r e l i m i n a r y d e s i g n i n many c a s e s . T h e r e f o r e , the need f o r D e f l e c t i o n T h e o r y s o l u t i o n s does e x i s t . I t wou ld be e x p e d i e n t t o t u r n the t a s k o v e r t o a computer and a v o i d a l l hand c a l c u l a t i o n s , bu t t h a t i s n o t a l w a y s a p r a c t i c a l p r o c e d u r e . D u r i n g the e a r l y s t a g e s of a d e s i g n , t h e r e w i l l a l w a y s be a n e c e s s i t y f o r some hand c a l c u l a -t i o n s , and i t i s hoped t h a t t he se w i l l be made somewhat e a s i e r w i t h the methods p r e s e n t e d h e r e . I t was a p p a r e n t i n the c h a p t e r on T h e o r y and R e f i n e -ments t h a t the s i m p l e r D e f l e c t i o n Theo ry r e p r e s e n t e d by e q u a t i o n s (43) and ( l i b ) g i v e s r e s u l t s o f h i g h a c c u r a c y . I t i s q u e s t i o n -a b l e whe the r the a d d i t i o n a l a c c u r a c y a t t a i n e d by f u r t h e r r e f i n e -ment i s j u s t i f i e d i n hand c a l c u l a t i o n s , and i t i s recommended t h a t the more r e f i n e d v e r s i o n s o f the D e f l e c t i o n Theory be r e s e r v e d f o r computer a n a l y s i s . E q u a t i o n (43) i s r e l a t i v e l y a t t r a c t i v e f o r use i n hand c a l c u l a t i o n s s i n c e s o l u t i o n s a r e t o be found t a b u l a t e d i n S t e i n m a n ' s t e x t * on s u s p e n s i o n b r i d g e s . R e f e r e n c e ( l ) 53 Once the c a b l e t e n s i o n f o r a t o t a l l o a d i n g case i s known i t i s p o s s i b l e t o supe r impose s o l u t i o n s f o r p a r t i a l l o a d i n g c o n d i t i o n s as g i v e n i n S t e i n m a n ' s t e x t . E q u a t i o n s (43) and ( l i b ) f o r m the b a s i c t h e o r y f o r the method p r e s e n t e d of d e t e r m i n i n g the t o t a l v a l u e o f c a b l e t e n s i o n . No a p p r o x i m a t i o n n o t i n h e r e n t i n the above e q u a t i o n s i s made f o r the method g i v e n and hence the v a l u e of c a b l e t e n s i o n c a l c u l a t e d by t h i s method has a r e l a t i v e l y h i g h a c c u r a c y . I t c an be seen i n the sample c a l c u l a t i o n s g i v e n i n A p p e n d i x 3 t h a t the method i s e x t r e m e l y easy t o a p p l y , e s p e c i a l l y i n the case o f a b r i d g e w i t h g i r d e r s h i n g e d a t the s u p p o r t s . A c o n t i n u o u s g i r d e r p r e s e n t s some a d d i t i o n a l d i f f i c u l t y b u t no more t h a n s h o u l d be e x p e c t e d . C o n t i n u i t y i n any s t r u c t u r e i s p a r t l y p a i d f o r by e f f o r t i n . a n a l y s i s . The f i r s t s t e p toward an a c c u r a t e , s i m p l i f i e d method o f a n a l y s i n g s u s p e n s i o n b r i d g e s must be an a c c u r a t e , s i m p l e method of d e t e r m i n i n g the c a b l e t e n s i o n . I t i s b e l i e v e d t h a t the method d e v e l o p e d i n C h a p t e r 3 and i l l u s t r a t e d i n sample c a l c u l a t i o n s i n A p p e n d i x 3 meets the o b j e c t i v e s o f a c c u r a c y and s i m p l i c i t y . T h e r e f o r e , the method s h o u l d be u s e f u l as p a r t o f • a t o t a l method o f a n a l y s i s . One a p p r o a c h t o a s i m p l i f i e d method o f a n a l y s i s m i gh t be a d e t e r m i n a t i o n and t a b u l a t i o n o f i n f o r m a t i o n on a m p l i f i c a t i o n f a c t o r s i n a manner s i m i l a r t o t h a t d e s c r i b e d by A . F r a n k l i n i n h i s t h e s i s on n o n - l i n e a r a r c h e s . whatever methods a r e used t o comple t e the a n a l y s i s , i t i s i m p o r t a n t t o r e c o g n i z e t h a t methods of s u p e r p o s i t i o n a r e v a l i d i n s u s p e n s i o n b r i d g e a n a l y s i s , d e s p i t e the n o n - l i n e a r b e h a v i o r o f s u s p e n s i o n b r i d g e s . So l o n g as the t o t a l . v a l u e of the c a b l e 54 tension is known and applied in the equations for the partial loadings, the bending moments and deflections for the partial loading cases may be superimposed to give the total values. The key, then, is the determination of the total cable tension, and a simple, accurate method of determining the cable tension has been presented In this work. 55 APPENDIX 1 COMPUTER PROGRAM C KEN RICHMOND CIVIL ENGINEERING THESIS C PROGRAM TO ANALYSE SUSPENSION BRIDGES AND COMPUTE MAGNIFICATION C FACTORS. C C MAIN LINE PROGRAM C DIMENSION E l (53) , P(53) , A B ( 1 3 , 5 3 ) ,A (53 ,5 ) DIMENSION E(53) , B ( 5 3 ) , BM(2,53) 43 READ 1 , KODE , C , R , S 1 FORMAT ( 12 / ( E l 4 .7)) READ 2 , F , SIDE , RISE , T , SLIP 2 FORMAT ( F 6.4 / F6.4 / F 6 . 4 / ( E 1 4 . 7 ) ) PRINT 3 , KODE , R , S 3 FORMAT(//6H KODE= 14 ,3H R= E14.7 ,3H S= E14.7 / ) PRINT 4 , F , SIDE , RISE 4 FORMAT ( 3H F= F7.4 , 6H SIDE= F11.4 , 6H RISE= F l 1 . 4 / ) PRINT 5 , T , SLIP 5 FORMAT ( 3H T= El 7.7 , 6H SLIP= El 7.7 ) EIO =1.0 / (8.0 * F * C ) IA= 17.0 - SIDE * 20.0 ID= 37 + 17 - IA IF ( KODE - 10 ) 8 , 8 , 6 6 KODE = KODE - 10 DO 7 I = IA , ID , 1 7 E l ( I ) = EIO GO TO 11 8 DO 9 I = IA, ID, 1 READ 10 , E l ( I ) 10 FORMAT ( F7.4) 9 E l ( I ) = EIO * E l ( I ) 11 PRINT 12 12 FORMAT(/23H I P El / ) DO 13 I = IA,ID,1 READ 14 , P( I ) 14 FORMAT ( F8.4) 13 PRINT 1 5 , 1 , P ( D , E l ( l ) 15 FORMAT ( 13 , F9 .4 , E17.7 ) DF = 0.0 SF = 0.0 GO TO ( 16 ,17 ,18 ,19 ,16 ,17 ,18 ,19 ), KODE 56 16 DF = 1.0 17 SF = 1.0 GO TO 19 18 DF =1.0 19 N2 = 1 GO TO 200 20 D = 0.0 HD = 0.125 / F HL = 0.1 DO 36 N = 1,2 K = 0 21 HT = HL + HD IF (KODE - k ) 22,22 ,23 22 Hk = 2 I I = IA IE = ID GO TO 400 23 Hk = 1 N2 = 3 GO TO 200 2k ERROR = 0.0 N2 = 2 GO TO 200 25 ERROR = ERROR + SLIP PRINT 26 , HL , ERROR 26 FORMAT ( A H HL= E17.7 , 7H ERROR= El 7.7 ) IF (K- I ) 27,28,27 27 HL1 = HL ERR = ERROR HL = HL + .1 * HD K = 1 GO TO 21 28 IF(ABS(ERR0R)-1.0E-5) 30,30,29 29 DELH = ERROR *(HL - HL1)/(ERROR - ERR) ERR = ERROR HL1 = HL HL = HL - DELH GO TO 21 30 E( IA-1) = -E(IA+1) DO 31 l=IA,ID,1 BM(N,I) = ( 400 .0* (2 .0*E( I ) -E ( I -1 ) -E( I+1) ) ) * (1 .0+D*HT*SM) 31 BM(N,I) = (BM (N, I ) -SM*(P( I )+1.0+HT*D2Y))*EI ( I ) BM(N,IA) = 0.0 BM(N,ID) = 0.0 IF ( KODE -k ) 33,33,32 32 BM(N,17) = 0.0 BM(N,37) = 0.0 33 SUM = 0 Q = ID - IA + 1 DO 3k I = IA,ID,1 3k SUM = SUM + E l ( I ) 57 35 36 41 39 38 37 42 C C c 100 AVG = SUM / Q CDL = HD / AVG CLL = HL / AVG CTOT = HT /AVG PRINT 35 , CDL , CLL , CTOT FORMAT (/ 5H CDL= El 7.7 , 5H CLL= El 7.7 , 6H CTOT= El 7.7) D = 1.0 PRINT 41 F0RMAT(/48H I DEFLECTION BM ELASTIC BM PHI / ) DO 37 I = IA,ID,1 IF ( B M ( 1 , U ) 38 ,39 ,38 PHI = 1.0 GO TO 37 PHI = BM(2,I) / BM(1,1) PRINT 42 , I , B M ( 2 , I ) , B M ( 1 , I ) , P H I , E ( I ) FORMAT( I3#F14 .8 ,F17 .8 ,F17 .8 ,E20 .7 ) GO TO 43 SUBROUTINE 1 E I ( IA - I ) = EI(IA+1) EI(ID+1) = E l ( I D - 1 ) SM = S / EIO D2Y = - 8 . 0 * F RAE = R * R / EIO DO 101 I =1 I , IE,1 DEI = 10.0 * ( E l O + 1 ) D2EI = 400.0 X = I - I I X = .05 * X DY = 4 .0 * F DS = SQR(1.0 AB(1 , AB(3 , 101 C C C 200 EK * ( E l ( 1 - 1 ) - 2, - 1 ) ) 0 * El ( I) + E l (1+1) ) AB(4 , AB(5 , AB(6 , AB(7 , AB(8 , AB(9 , AB(10,I AB(11,1 AB(12,1 AB(13,I GO TO * (AL - 2 .0 * X ) - BL / AL + DY * DY ) = 1.6E5 * E l ( I ) - 8.0E3 * DEI = - 6 . 4 E 5 * E l ( l ) + 1.6E4 * DEI + 400.0 * D2EI = SM*AB(3 ,D + 400 .0* ( -1 .0 -DF*DY*DY)+20 .0*DF*DY*D2Y = 9.6E5 * E l ( I ) - 800.0 * D2EI = SM*AB(5 ,D + 800.0 * (1 .0 + DF*DY*DY ) = - 6 . 4 E 5 * E l ( I ) - 1.6E4 * DEI + 400.0 * D2EI = SM*AB(7 , I ) + 400.0 * ( - 1 ,0 -DF*DY*DY) -20 .0 *DF*DY*D2Y = 1.6E5 * E l ( I ) + 8.0E3 * DEI = (P ( l )+1 .0 ) * (1 .0 - SM * D2EI) = D2Y * (1 .0 - SM * D2EI) = - D F * T * D 2 Y * ( 3.0*DY*DY + 1.0 ) = -DF*RAE*D2Y*DS* ( 4 .0*DY*DY + 1.0 ) 201,202,203) ,N1 SUBROUTINE 2 II = IA IE = 17 AL = SIDE BL = RISE N1 = 1 GO TO 204 201 I I = 17 IE = 37 AL = 1.0 BL = 0.0 N1 = 2 GO TO 204 202 I I = 37 IE = ID AL = SIDE BL = -R ISE N1 = 3 204 GO TO (100,300,400) ,N2 203 GO TO (20, 25 , 24),N2 C C SUBROUTINE 3 C 300 DO 301 I = I I , IE , 1 X = I - I I X = .05 * X DY = 4 .0 * F * (AL - 2 .0 * X ) - BL / AL DS = SQR(1.0 + DY * DY) RAE = R * R / EIO IF(I - I I ) 302 , 302 , 303 302 DE = 20.0 * E( l+1) GO TO 307 303 IF (I - IE) 305 , 304 , 304 304 DE = - 2 0 . 0 * E ( l - 1 ) GO TO 307 305 DE = 10.0 * (E( l+1) - E ( l - 1 ) ) 307 B( I )=HL*RAE*SF*DS*DS* (DY*DE+.5*DE*DE) - .5 *DE*DE 301 B( l )= D*B(I )+RAE*HL*DS**3+T*DS*DS-DY*DE IS = IE -2 DO 306 I = I I , IS , 2 306 ERROR = ERROR +(.05/3.0) * ( B ( l ) + 4 .0 * B(l+1) + B( l+2) ) GO TO ( 201, 202 , 203 ) , N1 C C SUBROUTINE 4 C 400 DO 401 I = I I , IE , 1 A ( l , 1 ) = A B ( 1 , I ) * ( 1.0 + SM*D*HT ) A ( l , 2 ) = AB(3,D + D * HT * AB(4,1 ) A ( l , 3 ) = AB(5,D + D * HT * AB(6 , I ) A ( l , 4 ) = AB(7,D + D * HT * AB(8,1 ) A ( l , 5 ) = AB(9,D * (1 .0 + SM*D*HT ) 401 B ( l ) =AB(10, I )+HT*AB(11,1)+D*HT*AB(12,1)+D*HT*HL*AB(13,1) IM = II -1 IN = IE +1 B(IM) = - ( P ( l l + 1 ) + 1.0 + HT *D2Y) * ( SM / (1 .0 + HT * SM oo # O A s—s ,—^ _ in < + U J o • < S N ' L A 1 — - — . — —^ < C O 1 U J C M '—* »—• + —• C M — — >- + + C M —> —> L U O * - * 4 * — — 1 ->v. L U 1— < < — .—, X * N * — * — < * + U J —_ C M C O — * 0 a: a: < • •—•> C Q ' — -*— I I ' — s t—- C M — + — C A O A ctL . K U J — ' «» 5 - 5 — « — L U ' MD o o - 4 - 4 — » Z O A ^ « 1 r- » I — Q ; + l ^ - 0 »— r— O O —*—'—' «— » C M <C I O • C M I «• O • O O • O •»< < * + O A + I — -4 •» L U L A » 0 • • O • S Z - ^ ^ - J - — >— » — C M ^ . ^ L U '—* CM — » 0 O O O O O O O O O O O O O — — » w + C M " — ' + '—- L U I -4" O — ' 1 — • O OO O O 0 0 O CM O A < I — < C — ^ - - Z — ^ CM CM Cu O O O O ' - ' - ^ ^ J - I 4 4 I 4 N II »•» — ' II — C M ' —. SI »•> C O O O ^ II r- CM II II CO II OQ » w CO — — C O • • • • I II II II II II II II II II II II II II II I — + + " J Z O Q ^ ^ O O 0 0 0 0 - 3 —5 ' — II —» l l C Q I -4 CM || || || || || ^ - . ^ - s ^ ^ ^ - , ^ ^ ^ ^ ^ ^ - > ^ - > ^ ^ Z ^*>_ - , - 3 w || || ^ CM - 3 -> " J —) 0 A 0 A O A 0 A O A - 4 L A * — CM O A — O A < < -4" »'—* L A > <t | | — *———* —^v <—O n * * * * * * * * * * * * * O O '— *— O CM CM — I O O - N M i J Z 4 - N M i l - N M i l I S 2 Z Z Z 11 -4 II II -4 + + - 4 + + II Z 1 1 1 — ' — - — ' I — H -' — ^—'O — - — —~—'—-——-——^—'iz: O C M O ' O—"—*—— ' — ' II L L O O Z 0 D C D C D C D C O Q < < < < < < < < < < < < < < — Q 0 t D i Q < C O Q < C D c C U J L l J — U J O C J L U CM -4" L A O A vD r-~-co O 0 0 0 0 0 0 -4 -4 - 4 - 4 -4 - 4 - 4 TO FOLLOW PAGE 59 MAIN LINE PROGRAM \ READ: KODE, C , R , S , F, SIDE,RISE,T, SLIP 1 PRINT: K(j)DE, R,S F, SIDE, RISE T, SLIP COMPUTE EIO, IA, ID K<!)DE> 10 KODE < 10 1= IA ,ID READ EI (I) EIO COMPUTE E I ( I) I = IA. ID READ P(I) PRINT I,P(I),EI(I) SHEET "I OF 6 TO FOLLOW PAGE TO F O L L O W PAGE ) INITIALIZE ERROR=0 I , \ ERRORl < I O 5 L COMPUTE BM ( N , I A ) TO BM (N,ID) N2 = 2 ,200) 25 ERROR = ERROR + SLIP P R I N T : HL,ERROR S W I T C H X K S ° / S W I T C H K y \ K=I V NEW ESTIMATE H L = H L + .1 H D I N T E R P O L A T E NEW ESTIMATE H, COMPUTE A V E R A G E E I CDL.CLL.CTOT P R I N T CDL, C L L , CT(|)T N = 1,2 -<3 S H E E T .3 OF 6 TO FOLLOW PAGE 59 1= IA , ID SUBROUTINE I. Compute and store constants AB( I , I ) to AB( I3 , I ) BM(I , I ) * 0 PRINT 1, BM(I , I ) , BM(2, I ) , P H I , E(I) COMPUTE SM, D2Y, • R A E I = I I , IE COMPUTE AB ( 1,1) TO AB ( 13,1) S H E E T 4 OF 6 TO FOLLOW PAGE SUBROUTINE 2 (200) SPECIFY II, I E , L , B L E F T SIDE SPAN © SPECIFY I I , I E , L , B MAIN SPAN (202) SPECIFY I I , IE, L , B RIGHT SIDE S P A N © SUBROUTINE 3 Integrate cable equation by Simpson's Rule < 1= I I , I E - 2 ,2 ERROR = ERROR + 3 I B(I) -h 4 B ( I + I) + B (I +2)J @ S H E E T 5 OF 6 TO FOLLOW P A G E 59 SUBROUTINE 4 Compute deflections at all points I I - l to IE + I and store in E ( I ) I = I I , IE C O M P U T E A (1,1) TO A(T,5) a B ( I ) C O M P U T E B O U N D A R Y CONDITIONS I = I I - I, I E - I REDUC A ( I + 1 aA( 1 + TO Z :E ,2) 2,1 ) ERO <! s ? R E D U C E A ( I E + 1,2) TO Z E R O COMPUTE E ( I E +1), E ( I E ) S H E E T 6 OF 6 6o APPENDIX 2 TABLES F-P. m TABLE 1, ll v 2 + v 3 T 2 + ~A~ INPLUENCE CURVES A-A r L -V]_ dx A m A1 L r x v 2 dx ' o X Q L r x ^ V o + V-: A 2 + A 3 dx L HL2 EI X L Pl Ai As Am 1. ."4843 . 0 5 . 0 3 1 1 . 0 2 4 5 . 0 3 6 0 . 0 0 0 7 . 0 0 0 6 . 0 0 0 9 . 1 0 . 0 6 1 3 . 0 4 8 7 . 0 6 8 1 . 0 0 3 0 . 0 0 2 4 . 0 0 3 5 . 1 5 . 0 8 9 9 . 0 7 2 2 . 0 9 6 1 . 0 0 6 8 . 0 0 5 4 . 0 0 7 6 . 2 0 . . 1 1 6 0 . 0 9 4 6 . 1 2 0 3 . 0 1 2 0 . 0 0 9 6 . 0 1 3 0 . 2 5 . 1 3 9 1 . 1 1 5 6 . 1 4 0 7 . 0 1 8 4 . 0 1 4 9 . 0 1 9 6 . 3 0 . . 1 5 8 8 . 1 3 4 8 . 1 5 7 3 . 0 2 5 9 . 0 2 1 1 . 0 2 7 1 . 3 5 . 1 7 4 4 . 1 5 1 9 . 1 7 0 2 . 0 3 4 2 . 0 2 8 3 . 0 3 5 3 .40 . 1 8 5 9 . 1 6 6 5 . 1 7 9 4 . 0 4 3 2 . 0 3 6 3 . 0 4 4 0 . 4 5 ' . 1 9 2 8 . 1 7 8 2 . 1 8 4 8 . 0 5 2 7 . 0 4 4 9 . 0 5 3 1 • 50 ' . 1 9 5 2 . 1 8 6 7 . 1 8 6 7 . 0 6 2 5 - . 0 5 4 1 . 0 6 2 5 . 5 5 . 1 9 2 8 . 1 9 1 5 . 1 8 4 8 . 0 7 2 2 . 0 6 3 5 . 0 7 1 8 . 6 0 . 1 8 5 9 . 1 9 2 2 . 1 7 9 4 . 0 8 1 7 . 0 7 3 2 . 0 8 0 9 . 6 5 . 1744 . 1 8 8 4 • . . 1 7 0 2 . 0 9 0 7 . 0 8 2 7 . 0 8 9 6 . 7 0 . 1 5 8 8 . 1 7 9 8 . 1 5 7 3 . 0 9 9 0 . 0 9 1 9 . 0 9 7 8 . 7 5 . 1 3 9 1 . . 1 6 5 8 . 1 4 0 7 . 1 0 6 5 . 1 0 0 6 . 1 0 5 3 . 8 0 . 1 1 6 0 . 1 4 6 1 . 1 2 0 3 . 1 1 2 9 . 1 0 8 4 . . 1 1 1 9 . 8 5 . 0 8 9 9 - .1201 . 0 9 6 1 . . 1 1 8 1 . 1 1 5 1 . 1 1 7 3 . 9 0 . 0 6 1 3 . 0 8 7 5 . 0 6 8 1 . 1 2 1 9 . 1 2 0 3 . 1 2 1 4 . 9 5 . 0 3 1 1 . 0 4 7 6 . 0 3 6 0 . 1 2 4 2 . 1 2 3 7 . 1240 61 TABLE 1 (CONTINUED) H L 2 x EI A l L F l p s P m A l A 2 . . 4 4 3 5 . 0 5 . 0 3 1 1 . 0 2 4 1 . 0 3 6 5 . 0 0 0 ? . 0 0 0 6 . 0 0 0 9 . 1 0 . 0 6 l 4 . 0 4 7 9 . 0 6 8 7 . 0 0 3 1 . 0 0 2 4 . 0 0 3 5 . 1 5 . 0 8 9 9 . 0 7 1 1 . 0 9 6 7 . 0 0 6 8 . 0 0 5 3 . 0 0 7 7 ••.20 . 1 1 6 1 . 0 9 3 3 . 1 2 0 7 . 0 1 2 0 . 0 0 9 5 . 0 1 3 1 . 2 5 . 1 3 9 2 . 1 1 4 1 . 1 4 0 9 . 0 1 8 4 . 0 1 4 7 ' . 0 1 9 7 . 3 0 . 1 5 8 8 . 1 3 3 3 . 1 5 7 2 . 0 2 5 9 . 0 2 0 9 . 0 2 7 2 . 3 5 • . 1 7 4 4 . 1 5 0 4 . 1 6 9 8 . 0 3 4 2 . 0 2 8 0 . 0 3 5 4 . 4 0 . 1 8 5 8 . 1 6 5 2 . 1 7 8 8 . 0 4 3 2 • . 0 3 5 9 . 0 4 4 1 . 4 5 . 1 9 2 8 . 1 7 7 1 . 1 8 4 1 . 0 5 2 7 . 0 4 4 4 . 0 5 3 2 . 5 0 . 1 9 5 1 . . 1 8 5 9 . 1 8 5 9 . 0 6 2 4 . 0 5 3 5 . 0 6 2 5 . 5 5 . 1 9 2 8 . 1 9 1 2 . 1 8 4 1 . 0 7 2 2 . 0 6 3 0 . 0 7 1 7 . 6 0 . 1 8 5 8 . 1 9 2 4 . 1 7 8 8 . 0 8 1 7 . 0 7 2 6 . 0 8 0 8 . 6 5 . 1 7 4 4 . 1 8 9 2 . 1 6 9 8 . 0 9 0 7 . 0 8 2 1 . 0 8 9 5 • . 7 0 . 1 5 8 8 . 1 8 1 1 . 1 5 7 2 . 0 9 9 0 . 0 9 1 4 . 0 9 7 7 . 7 5 . . 1 3 9 2 . . 1 6 7 6 . 1 4 0 9 . 1 0 6 5 . 1002 . 1 0 5 2 . 8 0 . 1 1 6 1 . 1 4 8 2 . 1207 . 1 1 2 9 . 1 0 8 1 . 1 1 1 8 . 8 5 . 0 8 9 9 . 1 2 2 3 . 0 9 6 7 . 1 1 8 1 . 1 1 4 9 . 1 1 7 2 . 9 0 . 0 6 l 4 . 0 8 9 4 . 0 6 8 7 . 1 2 1 8 . 1 2 0 2 . 1 2 1 4 95 . 0 3 1 1 . 0 4 8 8 . 0 3 6 5 . 1242 . 1 2 3 7 . 1 2 4 0 3 . . 4 0 9 1 . 0 5 . 0 3 1 1 . 0 2 3 8 . 0 3 6 9 . 0 0 0 7 . 0 0 0 5 ' . 0 0 0 9 . 1 0 . . 0 6 1 4 ^ . 0 4 7 3 • . 0 6 9 2 . 0 0 3 1 . 0 0 2 3 . 0 0 3 6 . 1 5 . 0 9 0 0 . 0 7 0 1 . 0 9 7 2 . 0 0 6 9 . 0 0 5 3 . 0 0 7 8 . 2 0 . 1 1 6 1 . 0 9 2 1 . 1 2 1 1 . 0 1 2 0 . 0 0 9 3 . 0 1 3 2 . 2 5 . 1 3 9 2 . 1 1 2 8 . l 4 i o . 0 1 8 4 . 0 1 4 5 . 0 1 9 8 . 3 0 . 1 5 8 8 . 1 3 1 8 . 1 5 7 1 . 0 2 5 9 . 0 2 0 6 . 0 2 7 3 . 3 5 . 1 7 4 4 • . 1 4 9 0 . 1 6 9 4 . 0 3 4 2 . 0 2 7 6 . 0 3 5 4 . 4 0 . 1 8 5 8 . 1 6 3 9 . 1 7 8 2 . 0 4 3 3 . 0 3 5 4 . 0 4 4 2 . 4 5 . 1 9 2 7 . 1 7 6 1 . 1 8 3 5 . 0 5 2 7 . 0 4 4 0 . 0 5 3 2 . 5 0 . 1 9 5 0 . 1 8 5 2 . 1 8 5 2 . 0 6 2 5 . 0 5 3 0 . 0 6 2 5 . 5 5 . 1 9 2 7 . 1 9 0 8 . 1 8 3 5 . 0 7 2 2 . 0 6 2 4 . 0 7 1 7 . 60 . 1 8 5 8 . 1 9 2 6 . 1782 . 0 8 1 6 . 0 7 2 0 . 0 8 0 7 . 6 5 . 1 7 4 4 . 1 8 9 9 . 1 6 9 4 . 0 9 0 7 . 0 8 1 6 . 0 8 9 5 . 7 0 ; . 1 5 8 8 . 1 8 2 3 . 1 5 7 1 . 0 9 9 0 . 0 9 0 9 . 0 9 7 6 . 7 5 . 1 3 9 2 . 1 6 9 2 . 1410 . 1 0 6 5 . 0 9 9 8 . 1 0 5 1 . 8 0 . 1 1 6 1 . 1 5 0 1 . 1 2 1 1 . 1 1 2 9 . 1 0 7 8 . 1 1 1 7 . 8 5 . 0 9 0 0 . 1 2 4 4 . 0 9 7 2 . 1 1 8 0 . 1 1 4 7 . 1171 . 9 0 . 0 6 1 4 . 0 9 1 2 . 0 6 9 2 . 1 2 1 8 . 1 2 0 1 . 1 2 1 3 . 9 5 . 0 3 1 1 . 0 5 0 0 . 0 3 6 9 . 1242 . 1 2 3 7 .1240 62 T A B L E 1 ( C O N T I N U E D ) H L 2 AT X FT F AT A „ A E I . 1 L 1 s m 1 m 5. .35^2 .05 .0312 .0231 .0377 .0007 ..0005 .0009 .10 .0615 .0460 .0704 .0031 - .0023 .0036 .15 .0901 .0683 .0983 .0069 .0051 .0079 .20 .1162 .0898 .1218 .0120 .0091 .0134 .25 .' .1392 .1102 .1412 .0184 .0141 .0200 .30 .1587 .1291 .1568 .0259 .0201 .0275 .35 • .1743 .1463 .1687 .0343 .0270 .0356 .40 .1856 .1614 .1771 .0433 .0347 .0443 .45 .1925 .1740 .1821 .0527 .0431 .0533 .50 .1948 .1838 .1838 .0625 .0521 .0625 .55 .1925 . 1902 .1821 .0722 .0614 .0716 .60 .1856 .1928 .1771 .0816 .0710 .0806 .65 .1743 .1911 .1687 ' .0906 .0806 .0893 .70 .1587 .1845 . 1568 .0990 .0900 .0974 .75 .1392 .1723 .1412 .1065 .0990 . 1049 .80 .1162 .1538 .1218 .1129 .1072 . 1115 .85 .0901 .1283 .0983 . 1180 . 1 1 4 3 .1170 .90 .0615 .0948 .0704 .1218 .1199 .1213 .95 .0312 .0524 .0377 . 1 2 4 2 .1236 .1240 7. .3122 .05 .0313 -.0225 .0386 .0007 .0005 .0009 .10 .0616 .0448 .0715 .0031 .0022 .0037 .15 .0902 .0667 .0993 .0069 .0050 .0080 .20 .1163 .0878 .1225 .0121 .0089 .0136 .25 .1393 .1079 .1415 .0185 .0138 .0202 .30 .1587 .1267 .1566 .0259 .0196 .0277 .35 .1742 .1439 .1680 .0343 .0264 .0358 .40 .1855 .1592 .1761 .0433 ' .0340 .0444 .45 .1924 .1722 .1809 .0528 .0423 .0534 .50 .1947 .1824 .1824 .0625 .0512 .0625 .55 .1924 .1895 .1809 .0721 .0605 .0715 .60 . 1855 .1930 . 1761 .0816 .0701 .0805 • .65 .1742 .1921 .1680 .0906 .0797 .0891 .70 .1587 .1864 . 1566 .0990 .0892 .0972 .75 .1393 .1750 .1415 .1064 .0983 .1047 .80 • .1163 .1572 .1225 .1128 .1066 .1113 • .85 .0902 .1319 .0993 .1180 .1139 .1169 .90 .0616 .0981 .0715 .1218 .1197 .1212 .95 .0313 .0546 ..0386 . 1242 .1235 . 1240 63 ' TABLE 1 (CONTINUED) HL 2 E I A l X L p l x s F m A l Am 1 0 . . . 2 6 5 2 . 0 5 . 1 0 . 1 5 . 2 0 . 2 5 . 3 0 . 3 5 . 4 0 . 4 5 . 5 0 • . 5 5 . 6 0 . 6 5 • . 7 0 . 7 5 . 8 0 . 8 5 . 9 0 . 9 5 . 0 3 1 4 . 0 6 1 8 . 0 9 0 4 . 1164 . 1 3 9 4 . 1 5 8 7 . 1 7 4 1 . 1 8 5 4 . 1 9 2 2 . 1 9 4 4 . 1 9 2 2 - . 1 8 5 4 . 1 7 4 1 . 1 5 8 7 . 1 3 9 4 . 1 1 6 4 . 0 9 0 4 . 0 6 1 8 . 0 3 1 4 . 0 2 1 8 . . 0 4 3 4 • . 0 6 4 5 . 0 8 5 1 . 1 0 4 8 . 1 2 3 4 . 1 4 0 6 . 1 5 6 2 . 1 6 9 6 . 1 8 0 5 . 1 8 8 6 . 1 9 3 1 . 1 9 3 4 . 1 8 8 9 . 1 7 8 7 . 1 6 1 8 . 1 3 6 9 . 1 0 2 8 . 0 5 7 7 . 0 3 9 7 . 0 7 3 1 . 1 0 0 7 . 1 2 3 4 . 1 4 1 8 . 1 5 6 2 . 1 6 7 0 . 1 7 4 6 . 1 7 9 1 . 1 8 0 5 . 1 7 9 1 . 1 7 4 6 . 1 6 7 0 . 1 5 6 2 . 1 4 1 8 . 1 2 3 4 . 1 0 0 7 . 0 7 3 1 . 0 3 9 7 . 0 0 0 7 . 0 0 3 1 . 0 0 6 9 . 0 1 2 1 . 0 1 8 5 . 0 2 6 0 . 0 3 4 3 . 0 4 3 3 . 0 5 2 8 . 0 6 2 5 . 0 7 2 1 . 0 8 1 6 . 0 9 0 6 . 0 9 8 9 . 1 0 6 4 . 1 1 2 8 . 1 1 8 0 . 1 2 1 8 . 1 2 4 2 . 0 0 0 5 . 0 0 2 1 . 0 0 4 8 . 0 0 8 6 . 0 1 3 3 . 0 1 9 0 . 0 2 5 7 . 0 3 3 1 . 0 4 1 2 . 0 5 0 0 . 0 5 9 3 . 0 6 8 8 . 0 7 8 5 ' . 0 8 8 1 . 0 9 7 3 . 1 0 5 8 . 1 1 3 3 . 1 1 9 4 . 1 2 3 4 . 0 0 1 0 . 0 0 3 8 . 0 0 8 2 . 0 1 3 8 . 0 2 0 5 . 0 2 7 9 . 0 3 6 0 . 0 4 4 6 . 0 5 3 4 . 0 6 2 5 . 0 7 1 5 . 0 8 0 3 . 0 8 8 9 . 0 9 7 0 . 1 0 4 4 . 1 1 1 1 . 1 1 6 7 . 1 2 1 1 . 1 2 3 9 2 0 . . 1 7 6 6 f . 0 5 . 1 0 . 1 5 . 2 0 . 2 5 . 3 0 . 3 5 . 4 0 . 4 5 ' . 5 0 • . 5 5 . 6 0 . 6 5 . 7 0 . 7 5 . 8 0 . 8 5 . 9 0 . 9 5 . 0 3 1 6 . 0 6 2 2 . 0 9 0 8 . 1 1 6 8 . 1 3 9 6 . 1 5 8 7 . 1 7 3 9 . 1 8 4 9 . 1 9 1 5 . 1 9 3 8 • . 1 9 1 5 . 1 8 4 9 . 1 7 3 9 . 1 5 8 7 . 1 3 9 6 . 1 1 6 8 . 0 9 0 8 . 0 6 2 2 . 0 3 1 6 . 0 1 9 9 . 0 3 9 7 . 0 5 9 2 . 0 7 8 4 . 0 9 7 1 . 1 1 5 1 . 1 3 2 2 . 1 4 8 1 . 1 6 2 5 . 1 7 5 1 . 1 8 5 3 . 1 9 2 5 . 1 9 6 1 . 1 9 4 9 . 1 8 8 0 . 1 7 3 8 . 1 5 0 5 . 1 1 5 7 . 0 6 6 8 . 0 4 3 3 . 0 7 7 7 . 1 0 4 9 . 1 2 6 1 . 1 4 2 6 . 1 5 5 0 . 1 6 4 1 . 1 7 0 3 . 1 7 3 9 . 1 7 5 1 . 1 7 3 9 . 1 7 0 3 . 1 6 4 1 . 1 5 5 0 . 1 4 2 6 . 1 2 6 1 . 1 0 4 9 . 0 7 7 7 . 0 4 3 3 . 0 0 0 7 . 0 0 3 1 . 0 0 6 9 . 0 1 2 1 . 0 1 8 6 . 0 2 6 0 . 0 3 4 4 . 0 4 3 4 . 0 5 2 8 . 0 6 2 5 . 0 7 2 1 . 0 8 1 5 . 0 9 0 5 . 0 9 8 9 . 1063' . 1 1 2 8 . 1 1 8 0 . 1 2 1 8 . 1 2 4 2 , o o o 4 . 0 0 1 9 . 0 0 4 4 . 0 0 7 9 . 0 1 2 3 . 0 1 7 6 . 0 2 3 8 . 0 3 0 8 . 0 3 8 5 . 0 4 7 0 . 0 5 6 0 . 0 6 5 5 . 0 7 5 2 . 0 8 5 0 . 0 9 4 6 . 1 0 3 7 . 1 1 1 9 . 1 1 8 6 . 1 2 3 2 . 0 0 1 1 . o o 4 i . 0 0 8 7 . 0 1 4 5 . 0 2 1 3 . 0 2 8 7 . 0 3 6 7 . 0 4 5 1 . 0 5 3 7 . 0 6 2 5 . 0 7 1 2 . 0 7 9 8 . 0 8 8 2 . 0 9 6 2 . 1 0 3 6 . 1 1 0 4 . 1 1 6 2 . 1 2 0 8 . 1 2 3 8 64 TABLE 1 (CONTINUED) HL 2 E I X V;L F l p s P m A l A s A 3 0 . . 1 3 2 4 . 0 5 . 1 0 . 1 5 . 2 0 . 2 5 . 3 0 . 3 5 . 4 0 . 4 5 . 5 0 . 5 5 . 6 0 . 6 5 -.70 • 7 5 . 8 0 . 8 5 . 9 0 . 9 5 - . 0 3 1 9 . 0 6 2 6 . 0 9 1 2 . 1 1 7 1 . 1 3 9 7 . 1 5 8 6 . 1 7 3 6 . 1 8 4 5 . 1 9 1 0 . 1 9 3 2 . 1910 . 1 8 4 5 . 1 7 3 6 . 1 5 8 6 . 1 3 9 7 . 1 1 7 1 . 0 9 1 2 . 0 6 2 6 . 0 3 1 9 . 0 1 8 6 . 0 3 7 3 . 0 5 5 7 . 0 7 4 0 . 0 9 1 9 . 1 0 9 4 . 1 2 6 2 • . 1 4 2 2 . 1 5 7 2 . 1 7 0 7 . 1 8 2 3 . 1 9 1 3 . 1 9 7 1 . 1 9 8 5 . 1 9 4 3 . 1 8 2 4 . 1 6 0 8 . 1 2 6 1 . 0 7 4 2 . 0 4 6 4 . 0 8 1 7 . 1 0 8 2 . 1 2 8 2 . 1 4 3 1 . 1 5 4 0 . 1 6 1 7 ' . 1 6 6 8 . 1 6 9 7 . 1 7 0 7 . 1 6 9 7 . 1 6 6 8 . 1 6 1 7 . 1 5 4 0 . 1 4 3 1 . 1 2 8 2 . 1 0 8 2 . 0 8 1 7 . . 0 4 6 4 . 0 0 0 8 . 0 0 3 1 . 0 0 7 0 . 0 1 2 2 . 0 1 8 6 . 0 2 6 1 . 0 3 4 4 . 0 4 3 4 • . 0 5 2 8 •' . 0 6 2 5 . 0 7 2 1 • . 0 8 1 5 . 0 9 0 5 . 0 9 8 8 . 1 0 6 3 . 1 1 2 7 . 1 1 7 9 . 1 2 1 8 . 1 2 4 1 . o o o 4 . 0 0 1 8 . 0 0 4 1 . 0 0 7 4 . 0 1 1 5 . 0 1 6 6 . . 0 2 2 5 . 0 2 9 2 . 0 3 6 7 . 0 4 4 9 . 0 5 3 7 . 0 6 3 1 . 0 7 2 8 . 0 8 2 7 . 0 9 2 6 . 1 0 2 0 . 1 1 0 7 . 1 1 7 9 . 1 2 3 0 . 0 0 1 2 . 0 0 4 4 . 0 0 9 2 . 0 1 5 1 . 0 2 1 9 . 0 2 9 4 . 0 3 7 3 . 0 4 5 5 . 0 5 3 9 . 0 6 2 5 . 0 7 1 0 . 0 7 9 4 . 0 8 7 6 . 0 9 5 5 . 1 0 3 0 . 1 0 9 8 . 1 1 5 7 . 1 2 0 5 . 1 2 3 7 . 5 0 . . 0 8 8 2 . 0 5 . 1 0 . 1 5 . 2 0 . 2 5 . 3 0 . 3 5 . 4 0 . 4 5 . 5 0 . 5 5 . 6 0 • . 6 5 . 7 0 . 7 5 . 8 0 . 8 5 • . 9 0 . 9 5 . 0 3 2 2 . 0 6 3 2 . 0 9 1 9 . 1 1 7 6 . i 4 o o . 1 5 8 6 . 1 7 3 2 . 1 8 3 8 . 1 9 0 2 • . 1 9 2 3 . 1 9 0 2 . 1 8 3 8 . 1 7 3 2 . 1 5 8 6 . i 4 o o . 1 1 7 6 . 0 9 1 9 . 0 6 3 2 . 0 3 2 2 . 0 1 7 2 . 0 3 4 3 . 0 5 1 5 . 0 6 8 5 . 0 8 5 4 . 1 0 2 0 . 1 1 8 4 . 1 3 4 3 . 1 4 9 6 .l64o . 1 7 7 1 . 1 8 8 4 . 1 9 7 1 . 2 0 2 0 . 2 0 1 8 . 1 9 3 9 . 1 7 5 4 . 1 4 1 7 . 0 8 6 3 . 0 5 1 7 . 0 8 8 0 . 1 1 3 5 . 1 3 1 2 . . 1 4 3 6 . 1 5 2 0 . 1 5 7 7 . 1 6 1 3 . 1 6 3 3 . 1640 . 1 6 3 3 . 1 6 1 3 . 1 5 7 7 . 1 5 2 0 . 1 4 3 6 . 1 3 1 2 . 1 1 3 5 . 0 8 8 0 . 0 5 1 7 . 0 0 0 8 . 0 0 3 2 . 0 0 7 0 . 0 1 2 3 . 0 1 8 8 . . 0 2 6 2 . 0 3 4 5 . 0 4 3 5 . 0 5 2 9 . 0 6 2 5 . 0 7 2 0 . 0 8 1 4 . 0 9 0 4 . 0 9 8 7 . 1 0 6 1 -. 1 1 2 6 . 1 1 7 9 . 1 2 1 7 . 1 2 4 1 . 0 0 0 4 . 0 0 1 7 . 0 0 3 8 . 0 0 6 8 . 0 1 0 7 . 0 1 5 4 . 0 2 0 9 . 0 2 7 2 . 0 3 4 3 . 0 4 2 1 . 0 5 0 7 . 0 5 9 8 . 0 6 9 5 . 0 7 9 5 . 0 8 9 6 . 0 9 9 5 . 1 0 8 8 . 1 1 6 8 . 1 2 2 6 . 0 0 1 3 . 0 0 4 9 . 0 0 9 9 . 0 1 6 1 . 0 2 3 0 . 0 3 0 4 . 0 3 8 1 . 0 4 6 1 . 0 5 4 3 . 0 6 2 5 ' . 0 7 0 6 . 0 7 8 8 . 0 8 6 8 . 0 9 4 5 . 1 0 1 9 . 1 0 8 8 . 1 1 5 0 . 1200 . 1 2 3 6 65 TABLE 1 (CONTINUED) HL 2 A -, X P P A n A A EI a l • L 1 s- • m 1 s m 7 0 . . 0 6 6 2 . 0 5 . 0 3 2 5 . 0 1 6 3 . 0 5 6 1 . . 0 0 0 8 . o o o 4 . 0 0 1 5 . 1 0 . 0 6 3 6 . 0 3 2 7 . 0 9 3 0 . 0 0 3 2 . 0 0 1 6 . 0 0 5 2 . 1 5 ' . . 0 9 2 3 . 0 4 9 0 . 1 1 7 2 . 0 0 7 1 .. . 0 0 3 6 . 0 1 0 5 . 2 0 . 1 1 8 0 . 0 6 5 3 . 1 3 3 2 . 0 1 2 4 . 0 0 6 5 . 0 1 6 8 . 2 5 • . l 4 o i . 0 8 1 5 . 1 4 3 6 . 0 1 8 8 . 0 1 0 2 . 0 2 3 8 . 3 0 . 1 5 8 5 . . . 0 9 7 6 . 1 5 0 4 . 0 2 6 3 . 0 1 4 6 . 0 3 1 1 . 3 5 . 1 7 3 0 . 1 1 3 5 . 1 5 4 7 . 0 3 4 6 . 0 1 9 9 . 0 3 8 8 . 4 0 . 1 8 3 3 . 1 2 9 2 . 1 5 7 3 . 0 4 3 6 . 0 2 6 0 . 0 4 6 6 . 4 5 . 1 8 9 6 . 1 4 4 5 . 1 5 8 7 . 0 5 2 9 . 0 3 2 8 . 0 5 4 5 . 5 0 . 1 9 1 7 . 1 5 9 2 . 1 5 9 2 . 0 6 2 5 . 0 4 0 4 - . 0 6 2 5 • . 5 5 . 1 8 9 6 . 1 7 3 0 . 1 5 8 7 . 0 7 2 0 . 0 4 8 7 . 0 7 0 4 . 6 0 . 1 8 3 3 . 1 8 5 5 . 1 5 7 3 . 0 8 1 3 . 0 5 7 7 . 0 7 8 3 • . 6 5 . 1 7 3 0 . 1 9 5 9 . 1 5 4 7 . 0 9 0 3 . 0 6 7 3 . 0 8 6 1 . 7 0 . 1 5 8 5 . 2 0 3 2 . 1 5 0 4 . 0 9 8 6 . 0 7 7 3 . 0 9 3 8 . 7 5 . 1 4 0 1 . 2 0 5 8 . . 1 4 3 6 . 1 0 6 1 . 0 8 7 5 ' . 1 0 1 1 . 8 0 . 1 1 8 0 . 2 0 1 1 . 1 3 3 2 . 1 1 2 5 , . 0 9 7 7 . 1081 . . 8 5 . 0 9 2 3 . 1 8 5 5 . 1 1 7 2 . 1 1 7 8 • . 1 0 7 4 . 1 1 4 4 . 9 0 . 0 6 3 6 . 1 5 3 3 • . 0 9 3 0 . 1 2 1 7 . 1 1 6 0 . 1 1 9 7 . 9 5 . 0 3 2 5 . 0 9 5 8 . 0 5 6 1 : . 1 2 4 1 . 1 2 2 4 . 1 2 3 4 1 0 0 . . 0 4 8 2 . 0 5 . 0 3 2 8 . 0 1 5 6 . 0 6 l 4 . 0 0 0 8 . 0 0 0 3 • . 0 0 1 6 . 1 0 .o64l . 0 3 1 2 . 0 9 8 7 . 0 0 3 2 . 0 0 1 5 . 0 0 5 7 . 1 5 . 0 9 2 8 . 0 4 6 8 . 1 2 1 3 . 0 0 7 2 . 0 0 3 5 • . 0 1 1 2 20 . 1 1 8 4 . 0 6 2 3 . 1 3 5 0 . 0 1 2 4 . 0 0 6 2 . 0 1 7 7 . 2 5 . 1 4 0 3 . 0 7 7 9 . 1 4 3 3 . 0 1 8 9 . 0 0 9 7 . 0 2 4 7 . 3 0 . 1 5 8 4 . 0 9 3 4 . 1 4 8 3 . 0 2 6 4 . 0 1 4 0 . 0 3 2 0 . 3 5 . 1 7 2 6 . 1 0 8 9 . 1 5 1 2 . 0 3 4 7 . 0 1 9 0 . 0 3 9 5 . 4 0 . 1 8 2 8 . 1 2 4 2 . 1 5 2 9 . 0 4 3 6 . 0 2 4 9 . 0 4 7 1 . 4 5 . 1 8 8 9 . 1 3 9 3 . 1 5 3 8 . 0 5 2 9 . 0 3 1 5 . 0 5 4 7 . 5 0 . 1 9 1 0 . 1 5 4 1 . 1 5 4 1 . . 0 6 2 5 . 0 3 8 8 . 0 6 2 5 . 5 5 . 1 8 8 9 . 1 6 8 3 . 1 5 3 8 • . 0 7 2 0 . 0 4 6 9 . 0 7 0 2 . 60 . 1 8 2 8 . 1 8 1 7 . 1 5 2 9 . 0 8 1 3 . 0 5 5 6 . 0 7 7 8 . 6 5 . 1 7 2 6 . 1 9 3 6 . 1 5 1 2 . 0 9 0 2 . 0 6 5 0 . 0 8 5 4 . 7 0 . 1 5 8 4 . 2 0 3 1 . 1 4 8 3 . 0 9 8 5 . 0 7 5 0 . 0 9 2 9 . 7 5 . 1 4 0 3 . 2 0 8 7 . 1 4 3 3 . 1060 . 0 8 5 3 . 1002 . . 8 0 . 1 1 8 4 . 2 0 7 7 . 1 3 5 0 . 1 1 2 5 . 0 9 5 7 • . 1 0 7 2 . 8 5 . 0 9 2 8 . 1 9 5 8 . 1 2 1 3 . 1 1 7 7 . 1 0 5 9 . 1 1 3 7 . 9 0 . 0 6 4 1 . 1 6 6 2 . 0 9 8 7 . 1 2 1 7 . 1 1 5 0 . 1 1 9 2 . 9 5 . 0 3 2 8 . 1 0 7 3 . 0 6 l 4 . 1 2 4 1 . 1220 . 1 2 3 3 66 ' TABLE 2 . b( L 1 ) a H L 2 -E I b . 0 5 . 1 0 . 1 5 . 2 0 . 2 5 . 3 0 . 3 5 . 4 0 • . 4 5 . 5 0 1 . 0 5 6 . 1 0 9 . 1 6 3 . 2 1 6 . 2 6 9 . 3 2 1 . 3 7 2 . 4 2 3 . 4 7 4 . 5 2 4 2 . 0 6 0 . 1 1 8 . 1 7 5 . 2 3 1 . 2 8 6 . 3 4 0 . 3 9 3 . 4 4 5 . 4 9 6 .546 3 . . 0 6 4 . 1 2 7 . 1 8 7 .246 . 3 0 3 . 3 5 8 . 4 1 2 .465 .516 .566 5 . 0 7 3 . 1 4 3 . 2 1 0 . 2 7 3 . 3 3 4 . 3 9 2 .448 . 5 0 1 . 5 5 2 . 6 0 1 7 . 0 8 2 . 1 5 9 . 2 3 2 . 2 9 9 . 3 6 3 . 4 2 3 . 4 7 9 . 5 3 2 . 5 8 3 . 6 3 1 10 . 0 9 6 . 1 8 3 . 2 6 2 . 3 3 4 . 4 0 1 . 4 6 3 . 5 2 0 . 5 7 3 . 6 2 2 . 6 6 8 20 . 1 3 7 . 2 5 1 .348 . 4 3 0 . 5 0 1 . 5 6 4 . 6 1 9 . 6 6 8 . 7 1 2 . 7 5 1 30 . 1 7 5 . 3 0 9 . 4 1 5 . 5 0 1 . 5 7 3 . 6 3 3 .684 . 7 2 8 . 7 6 7 ' . 8 0 1 50 . 2 4 1 . 4 o i . 5 1 5 . 6 0 1 . 667 - . 7 2 0 . 7 6 4 . . 8 0 0 . 8 3 1 . 8 5 7 70 . 2 9 7 . 4 7 1 . 5 8 6 . . 6 6 7 . 7 2 7 . 7 7 4 . 8 1 1 . 8 4 2 . 8 6 7 ' . 8 8 8 100 . 3 6 7 . 5 5 0 . 6 6 0 . 7 3 3 . 7 8 5 . 8 2 4 . 8 5 4 . 8 7 9 . 8 9 9 . 9 1 6 TABLE 3 . _ b( A 2 ) s H L 2 -E I b . 0 5 . 1 0 . 1 5 . 2 0 . 2 5 . 3 0 . 3 5 . 4 0 . 4 5 . 5 0 1 . 0 0 1 . 0 3 4 . 0 6 8 . 1 0 2 . 1 3 5 . 1 6 8 . 2 0 0 . 2 3 3 . 2 6 5 . 2 9 6 . 3 2 7 . 6 2 6 2 . 0 3 7 . 0 7 4 . 1 0 9 . 1 4 4 . 1 7 9 . 2 1 3 . 2 4 6 . 2 7 8 . 3 1 0 . 3 4 2 . 6 2 6 3 . 0 4 0 . 0 7 9 . 1 1 7 . 1 5 4 . 1 8 9 .224 . 2 5 8 . 2 9 1 . 3 2 3 . 3 5 4 . 6 2 7 5 • . 0 4 6 . 0 9 0 . 1 3 1 . 1 7 1 . 2 0 9 . 2 4 6 . 2 8 0 . 3 1 4 .346 . 3 7 7 . 6 2 9 7 . 0 5 2 . 100 . 1 4 5 . 1 8 7 ' . 2 2 7 . 2 6 5 . 3 0 0 . 3 3 4 . 3 6 6 . 3 9 6 . 6 3 0 10 . 0 6 0 . 114 , . 1 6 4 . 2 1 0 . 2 5 1 . 2 9 0 . 3 2 6 . 3 6 0 . 3 9 1 . 4 2 0 . 6 3 2 20 . 0 8 6 . 1 5 7 . 2 1 8 . 2 7 0 . 3 1 5 . 3 5 5 . 3 9 0 . 4 2 1 . 4 4 9 . 4 7 4 . 6 3 7 30 . 1 0 9 . 1 9 4 . 2 6 1 . 3 1 5 . 3 6 1 . 3 9 9 . 4 3 2 . 4 6 1 .486 . 5 0 8 . 6 4 2 50 . 1 5 1 . 2 5 2 . 3 2 5 . 3 8 0 . 4 2 3 . 4 5 7 .486 . 5 1 0 . 5 3 1 .548 . 6 5 0 70 . 1 8 7 . 2 9 7 . 3 7 0 . 4 2 3 . 4 6 3 . 4 9 4 . 5 1 9 . 5 4 0 . 5 5 7 . 5 7 2 . 6 5 6 100 . 2 3 1 .348 . 4 1 9 . 4 6 7 . 5 0 2 . 5 2 9 . 5 5 1 . 5 6 8 . 5 8 3 . 5 9 5 .664 TABLE 4. K = I}- TOWER MOMENTS HL EI a 3 5 7 10 20 30 50 70 100 .0 .1 .2 3 .5 any .2 .3 .4 .5 .6 .2 .3 .4 .5 .6 .2 .3 .4 .5 .6 .2 .3 .4 .5 .6 .984 .967 .973 .977 .979 .981 .961 .968 .973-.977 .980 .958 .966 .971 .975 .979 .960 .966 .970 .974 .978 .937 .948 .955 .960 .964 .925 .939 .948 .955 .961 .921 .935 .945 .953 .960 .924 .935 ..944 .951 .957 .953 .910 .925 .934 .941 .947 .893 .912 .925 .935 .944 .888 .907 .921 . 932 .941 .892 .907 .920 .930 .938 .926 .860 .882 .897 .908 .916 .837 .864 .884 .899 .911 .832 .858 .878 .894 .908 .838 .860 .877 .891 .904 .900 .817 .845 .864 .878 .790 .824 .848 .867 .883 .784 .816 .841 .862 .879 .793 .819 .840' .858 .873 ..865 .762 .796 .820 .838 .852 .732 .772 .802 .825 .845 .727 .765 .794 .819 .840 .739 .769 .794 .815 .833 .773 .632 .678 .712 .737 .759 .602 .652 .691 .723 .750 .600 .646 .684 .716 .743 .617 .653 .683 .709 .731 .705 .550 .601 .638 .668 .693 .524 .576 .618 .653 .684 .524 ..572 .611 .646 .676 .542 .579 .610 .637 .660 .610 .449 .502 .543 .576 .604 .430 .482 .525 .562 .594 .432 .479 .519 .553 .584 .451 .486 .516 .542 .565 .546 .389 .442 .482 .516 .546 .374 .424 .466 .502 .534 .377 .421 .459 .493 .522 .395 .427 .456 .480 .503 .480 .332 .382 .422 .456 .485 .321 .367 .406 .441 .472 .324 .365 .400 .431 .459 .340 .370 .396 .419 .439 APPENDIX 3 NUMERICAL EXAMPLES Example 1. Single Span D > I25* r .4 k/ft . f\ k / f t f \ M r > t\ < 2 5 0 ' J e 2 5 Q ' .1 „ . 1000' Given: Length of span L = 1,000 ft.. Sag of cable f = 100 ft.. EI = 1..5 ( 1 0 8 ) K ft. 2/girder AE of cable =7-0 (lO5) K *Le = 1,082 ft. *Lt.= 1,054 ft. Dead load w = 1.0 K/ft. Live load p = .4 K/ft. distributed where shown + 25 K at quarter point as shown et = 3.25 ( 1 0 ~ 4 ) Support displacement hg - h^ = -.5 ft. Formulae given in References (l) and (6) 69 Step 1. Compute and design constants. HD wL2 1.0 (IOOO)2 1250 K 8f 8 (100) L 2 ( 1 0 0 0 ) 2 .OO667 EI 1.5 ( 1 0 8 ) L e 1082 .0015 AE 7 ( 1 0 5 ) f 2L ( 1 0 0 ) 2 (IOOO) .0667 EI 1.5 (IO8) Step 2. Compute H-^ 1 . Here, it is necessary to estimate H in order to select an influence line. It is sufficiently accurate to use H = HD. Then HL2 = 1250 (.00667) = 8.3' EI H^ ' is found from the influence lines plotted in Figure 10. The influence line ordinate at the location of the point load is .139 L . The area under the curve from .25 to .50 is (.0625 - .Ol85)L . Therefore f HL' 25 (.139) (1000) .4 (.0625 - .0185) ( 1 0 0 0 ) 2 = H 100 100 = 35 + 176 = 211 K Step 3. Compute A '•• HT ' L€ AE ='- .5 - 3.25 ( 1 0 - 4 ) (1054) - 211 (.0015) = -1.17 ft. A ' = hB * hA " f t L t " "L". ^e 70 Step 4. Compute SH. Here, another estimate of H must be made in order to deter-mine . A - ^ . It is sufficiently accurate to estimate H = HD + HL1. Then HL2 = (1250 + 211) .OO667 = 9.75 EI H T 2 Figure 13 is used to determine A-, for = 9.75 and it is • 1 EI found that ^ = .277. Then 8H can be found from SH A< -1.17 -59K f 2 L A . } L E .277'(.0667) + .0015 " EI AE Step 5. Compute H. H = H D + HL' + 8H = 1250 + 211 - 59 = 1402 K Step 6. Compare the value of H' computed in step 5 with the value estimated in step 4, and repeat steps 4 and 5 until they are the same. HL2 = 1402 (.00667) =9.37 EI From'Figure 13, i ^ = .284 8H -1.17 -58 K .284 (.0667) + .0015 H = 1403 K In the case of a single span, one repetition of steps 4 and 5 should give a sufficiently accurate value of H. Example 2. Three-Span Bridge with Hinged Supports Q 25 k fA k/ft f\ k/ft i 1 750' Given: Main span length L = 1000 ft. Main span sag f = 100 ft. Side span length L„ = 500 ft. EI main span = 5.0 (lO?) K ft. 2/girder EI side span = 2 . 5 (10?) K ft. 2/girder AE of cable = 7 . 5 (lO5) K Side span rise = 112.1 ft. L e = 2 l 8 l ft. L t = 2116 ft. Dead load 1 K/ft. Live load .4 K/ft. on main span as shown + 25 K on side span where shown et = 3.25 (IO-4) Support displacement H B - h A = 0 72 Step 1. Compute HD and design constants Prom equation ( 3 9 ) HD 1 . 0 ( 1 0 0 0 ) 2 . 1250 K 8 ( 1 0 0 ) L 2 ( 1 0 0 0 ) 2 . 0 2 0 EI 5 . 0 -(10?) a 500 . 5 1000 b 500 2 5 . 0 (io?) . 5 1000 2 . 5 ( 1 0 7 ) f 2L ( 1 0 0 ) 2 (1000) . . 2 0 EI 5 (lO7) Step 2 . Compute H-^' Estimate H = HD = 1250 K HL2 = 1250 ( . 0 2 0 ) = 2 5 . 0 EI Figure 14(a) shows values of b( A i ) s plotted against Mil— EI Al for selected values of b. For HL> = 2 5 . 0 and b = . 5 , the , — N EI ordinate '-^^s is . 7 8 0 . Figure 14(b) shows values of "Al the multipliers X and Xs plotted as abscissae against the ordinate 10 ( A j) s for selected values of a. For a =• . 5 - — . • "A~l and b ( A i ) s = . 7 8 0 , the multipliers are: "Al x = . 8 3 6 x a = . 0 8 2 The main span influence line area from . 0 0 to . 7 5 is found 2 from Figure 10 to be .IO63 — X. Therefore the contribu-f tio'n to H-j^ 1 from the main span is. HL< .1063 ( 1 0 0 0 ) 2 (.836) (.4) 356 K 100 The influence line ordinate for the point load on the side span is . 1 9 4 1 XG. Therefore, the contribution to HL' from the side span is HL» .194 (1000) (.082) (25 K) 4 K 100 The total HL' = 356 + 4 = 360 K. Step 3. Compute A '. From equation (71) A' = 0 - 3.25 (IO-4) (2116) - 360 (.0029) = -1.72 ft. Step 4. Compute 8H. Estimate H = H D + HL< = 1250 + 360 = l 6 l 0 K HL2 = 1610 (.020) =32.2 EI 2 2 Figure 13 shows "A" plotted against . For H L = 32.2, 1 EI EI h±= .126 From equation (75) '8H -1.72 -52 K .126 (.20) .0029 .832 Step 5. Compute H. H = 1250 + 360 - 52 = 1558 K Step 6. Compare the value of H computed in step 5 with the value estimated in step 4. Repeat steps 4 and 5 until convergence. HL2 = 1558 (.020) = 31.2 EI 74 From Figure 13, .A ^ = .130 6H -1.72 -50 K .130 (.20) + .0029 .833 H = 1250 + 360 - 50 = 1560 K Step 7. Compare the value of H computed at the end of step 6 with the value estimated in step 2. Repeat steps 2 and 5 to convergence. From Figure 14 X = .833 x s = .083 H < .1063 ( 1 0 0 0 ) 2 (.833) (.4) .194 (1000) (.083) (25) L = . +  100 100 = 354 + 4 = 358 K ' H = 1250 + 358 - 50 = 1 5 5 8 K Step 7 will.seldom produce any significant improvement in the accuracy of H. For most suspension bridges a is usually less than .5 and b is usually less than .5. There-fore, X0 is small and not sensitive to changes in H. Since X is equal to 1-2 X , it also is not sensitive to changes • in X. E x a m p l e 3. C o n t i n u o u s S u s p e n s i o n B r i d g e 100' 4 0 0 ' 250' • 750' > < G i v e n : M a i n s p a n l e n g t h L = 1000 f t . M a i n s p a n s a g f = 100 f t . S i d e s p a n l e n g t h L a = 500 f t . M a i n s p a n EI = 1.5 ( l O 8 ) K f t . 2 S i d e s p a n EI = 7.5 (io?) K f t . 2 AE o f c a b l e = 7 . 5 (105) K S i d e s p a n r i s e - 112.1 f t . L e = 2 l 8 l f t . L t = 2116 f t . D e a d l o a d =1.0 K / f t . L i v e l o a d = .4 K / f t . d i s t r i b u t e d a s s h o w n + 25 K a t m a i n s p a n q u a r t e r p o i n t et = 3.25 (io - 4) S u p p o r t d i s p l a c e m e n t h g - h ^ = 0 7 6 Step 1 . Compute and d e s i g n c o n s t a n t s . H D ( 1 0 0 0 ) 2 ( 1 . 0 ) 1250 K 8 ( 1 0 0 ) L 2 ( 1 0 0 0 ) 2 . 0 0 6 6 7 E I 1 . 5 ( I O 8 ) L 2 l 8 l . 0 0 2 9 e _ _ AE 7 . 5 ( I O 5 ) f 2 L ( 1 0 0 ) 2 ( 1 0 0 0 ) . 0 6 6 7 E I ' 1 . 5 ( I O 8 ) a 500 . 5 1000 b 500 2 1 . 5 ( l O 8 ) . 5 1000 7 . 5 ( i o 7 ) Me .1 + ( . 5 ) ( . 5 ) . 5 0 0 f 1 . 5 + . 5 / . 5 Step 2 . Compute H ^ ' E s t i m a t e H L 2 = 1250 (.OO667) = 8 . 3 3 E I T J T 2 F i g u r e 16 shows K p l o t t e d a g a i n s t f o r s e l e c t e d v a l u e s E I o f a and b . . F o r a = . 5 and b = . 5 ( F i g u r e 1 6 ( d ) ) and H L = 8 . 3 3 , K i s f ound t o be . 8 3 6 . Then f r o m e q u a t i o n E I ( 1 0 6 ) M = . 8 3 6 ( . 5 0 0 ) = . 4 1 8 From F i g u r e 1 4 , b ( A i ) s = . 6 2 3 From F i g u r e " 1 5 , A 2 = - . 6 3 0 "Al b ( ~ ~ A 2 ) s = - . 4 0 5 " A l 77 By equation (105) T = 1 + 2 (.5 3) (.623) + .418 2 (-.630) + 2 (.5) (-.405) = 1 + 2 (.078) - .526 - 2 (.085) T = .460 The 'multipliers X and Y for the main span influence line ordinates are found from equations (108) and (109) . X 1 2.17 .460 Y .526 -1.14 .460 ~ The multipliers Xg and Yg for the side span influence line ordinates are found from equations (ill) and (112) Xs .078 .170 .460 ~ Ys .085 -.185 .460 ~ The main span influence line is made up by superimposing curves from Figure 10 and Figure 12 in accordance with equation (107). The contribution to H-^' from the main span is HL< 2.17 (1000) .4 (1000) (.106) + .139 (25) 100 1.14 (1000) .4 (1000) (.106) + .142 (25) 100 HL< = 997 - .524 = 473 K The side span influence line is made up by superimposing curves from Figure 10 and Figure 11 in accordance with equation (llO). The contribution to H-^' from the side span is HL' .170 ( 1 0 0 0 ) 2 .4 (.125 - .019) 100 • .185 ( 1 0 0 0 ) 2 .4 (.125 - .014) 100 HL' = 77 - 82 = -5 K The total value of H^ ' from main and side spans is HL< = 473 - 5 = 468 K Step 3. Compute A ' . Prom equation ( 7 l ) A ' = 0 - 3.25 (IO-4) (2116) - 468 (.0029) = -2.03 ft. Step 4. Compute 8H. Estimate H = 1250 + 468 = 1718 K HL2 = 1718 (.00667) = 11.-4 EI From Figure 13, "/T^ = .252 By equation (104) 8H -2.03 -191 K .252 (.460) (.0667) + .0029' Step 5. Compute H. H = 1250 + 468 - 191.= 1527 K Step 6. Compare the value of H computed in step 5 with the value estimated in step 4 and repeat steps 4 and 5 until convergence. HL2 = 1527 (.00667) = 1 0 . 1 EI From Figure 13, = .270 8H -2.03 -181 K .270 (.460) (.0667) + .0029 79 H = 1250 + 468 - 181 = 1537 K Step 7. Compare the value of H computed at the end of step 6 with the value estimated at the beginning of step 2 and repeat steps 2 to 6 until convergence. HL2 = 1537 (.OO667) =10.2 EI From Figure 16(d), K = . 8 l 2 M = .812 (.500) = ,406 From Figure 14, b( ^ l^s = .636 From Figure 15 , - A 2 = -.632 b("A" 2) s. = -.420 T = 1 + 2 ( . 5 ) 3 (.636) +• .403 2 (-.632) + 2 (..5) (-.420) = 1 + 2 ,(.080) - .509 - 2 .(.O85) T = .481 X =: 1 2.08 .481 ~ Y .509 -1.06 .481 ~ Xs •• .080 .166 .481 ~ Yc - .085 -.177 o .481 ~ H 1 2.08 (1000) .4 (1000) (.106) + .139 (25) L 100 - 1.06 (1000) .4 (1000) (.106) + .142 (25) 100 8o . 1 6 6 ( 1 0 0 0 ) 2 ( . 4 ) ( . 1 1 1 ) + 100 . 1 7 7 ( 1 0 0 0 ) 2 ( . 4 ) ( . 1 1 1 ) 100 = 9 4 5 - 487 + 7 4 - 69 = 4 7 3 K A' = 0 - 3 . 2 5 (IO-4) ( 2 1 1 6 ) - 4 7 3 ( . 0 0 2 9 ) = - 2 . 0 5 ft. From Figure 13 "A^ = . 2 7 0 8H - 2 . 0 5 . 2 7 0 ( . 4 8 1 ) ( . 0 6 6 7 ) + . 0 0 2 9 = - 1 7 8 K H = 1250 + 4 7 3 - 178 = 1 5 4 5 K Here, the improvement in accuracy from 1537-K to 1 5 4 5 K represents a relatively large improvement compared with what might usually be expected. For shorter or more rigid side spans, the Improvement will be less significant and step 1 might reasonably be omitted In many cases. 81 BIBLIOGRAPHY 1. Steinman, D. B., A Practical Treatise on Suspension Bridges, .2nd Ed., London, Wiley, 1929. 2. Pugsley, Sir Alfred Grenville, The Theory of Suspension  Bridges, London, Edward Arnold, 1957. 3. Johnson, Bryan and Tourniere, Theory and Practice of Modern  Framed Structures, 10th Ed., New York, Wiley, 1928. 4. Priester, G. C., "Application of Trigonometric Series to Cable Stress Analysis in Suspension Bridges", Engineering  Research Bulletin No. 12, Ann Arbor, Michigan, University of Michigan, 1929. 5. Timoshenko, S., "Theory of Suspension Bridges", Journal of  the Franklin Institute, Vol. 235 (March 1943). 6. Timoshenko, S., "The Stiffness of Suspension Bridges", Transactions, A.S.C.E., Vol. 95 (1930). 7. Steinman, B. D., "A Generalized Deflection Theory for Sus-pension Bridges", Transactions, A.S.C.E., Vol. 100, (1935). 8. Shortridge-Hardesty and Wessman, H. E., "Preliminary design of Suspension Bridges", Transactions, A.S.C.E., Vol. 101 (1936). 9. Westergaard, H. M., "On the Method of Complementary Energy and its Applications to Structures Stressed Beyond the Proportional Limit, "to Buckling and Vibrations, and to Suspension Bridges'.', Transactions, A.S.C.E., Vol. 107 (1942). 82 10. Tsien, Ling-Hi, "A Simplified Method of Analysing Suspension Bridges", Transactions, A.S.C.E., Vol. 114 (1949). 11. Gavarini,.C., "Considerations on Suspension Bridges", Acier-Stahl-Steel, Vol. 26, No. 2 (March 1 9 6 l ) , Brussels, Belgium. 12. Szidarovszky, J., "Corrected Deflection Theory of Suspension Bridges", Proceedings, A.S.C.E., Vol. 86, No. st 11 (Nov. I 9 6 0 ) . 13. Szidarovszky, J., "Practical Solution for Stiffened Sus-pension Bridges of Variable Inertia Moment and its Applica-tion to Influence Line Analysis", Acta Technica, Vol. 19, No. 3-4, Budapest (1958). 14. Heilug, R., "Eine Bernerkung zur Haengebruechentheorie", Der Stahlbau, Vol. 26, No. 2, Berlin (Feb. 1957). 15. Muller-Breslau, "Theorie der durcheinen Balken versteiften Kette", Zeitschrift der Arch un Ing, Vereins zu Hannover, 1881. 16. Rode, H. H.,, "New Deflection Theory", Kgl. Norske Viden skabers Selskabs Skrifter, Oslo, 1930. 17. Atkinson, R. J. and Southwell, R. V., "On the Problem of Stiffened Suspension Bridges and its Treatment by Relaxation Methods", Journal, Inst. Civ. Engrs., Longon, 1939. 18. Crosthwaite, C. D., "The Corrected Theory of the Stiffened Suspension Bridge", Journal, Inst. Civ. Engrs., London, 1947. 19. Crosthwaite, C. D., "Shear Deflections", Publications,. • I.A.B.S., Vol. 12, Zurich (1952). 20. Selberg, A., "Suspension Bridges", Publications, I.A.B.S., Vol. 8, Zurich (1947). 

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