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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 MB I A

S e p t e m b e r , 1963

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In presenting this thes is in pa rt ia l fulfi lment of

the requirements for an advanced degree at the University of

British Columbia, I agree that the Library s ha l l make i t fr eel y

av ai la bl e for reference and study. I fur ther agree that pe r-

mission for extensive copying of this thes is for sch ola rl y

purposes may be granted by the Head of my Department or by

hi s repres entativeso It i s understood that copying, or p u b l i

cation of this thes is for financial gain shall not be allowed

without my wr it ten p ermis sion .

The University of British Columbia-,

Vancouver 8 , Canada.

Department

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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 o f t he 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

set of t a b l e s a n d c u r v e s i s i n c l u d e d f o r u se i n t he a p p l i c a t i o n

o f th e me tho d. 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 t h e 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 m et ho d i s d i s c u s s e d a nd t h e

u se 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 t he 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 a nd v a r i o u s r e f i n e m e n t s i n t h e t h e o r y a r e d i s c u s s e d .

A c o m p u t e r 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 se a r ch and f o r t he purpo se o f t e s t i n g

t h e m a n u a l m et ho d p r o p o s e d . A d e s c r i p t i o n o f t h e 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 .

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v i i

.ACKNOWLEDGEMENTS

The author i si n d e b t e d t o Dr. R. P. Hooley f o rthe

a s s i s t a n c e , guidance andencouragement g i v e n d u r i n g the r e s e a r c h

and i n thep 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 , theauthori s

g r a t e f u l t o theN a t i o n a l Research C o u n c i l ofCanada f o rmakin g

money a v a i l a b l e f o ra r e se a rc h a s s i s t a n t s h i p ,and t o th e B r i t i s h

Columbia E l e c t r i c Company f o rthedonation of$500i n thefo rm

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

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i i i

TABLE OF CONTENTS

Page

CHAPTER 1. INTRODUCTION 1

CHAPTER 2. THEORY AND REFINEMENTS 5

G e n e r a l 5C a b l e E q u a t i o n 8G i r d e r E q u a t i o n 12

S o l u t i o n o f E q u a t i o n s ( l l ) and (35) 19E f f e c t o f R e f i n e m e n t s i n Theory on Ac cu ra cy 21

CHAPTER 3. COMPUTER PROGRAM . 25

S o l u t i o n of t he G i r d e r E q u a t i o n 27I n t e g r a t i o n o f t he Cable Eq ua t i on 32Program Linkage 33I n p u t Data f o r the Program 3^F i n a l N o t e s on the Computer P rogram 36

CHAPTER 4. DETERMINATION OF H 37

G e n e r a l 37S 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 in g Cases 38S i n g l e S p a n 39T 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 44T 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 45V a r i a b l e E I 50

CHAPTER 5. CONCLUSIONS 52

APPENDIX 1 . BLOCK DIAGRAM AND FORTRAN LI ST IN G FOR

COMPUTER PROGRAM 55

APPENDIX 2 . TABLES OF CONSTANTS 60

APPENDIX 3. NUMERICAL EXAMPLES OF CALCULATION OF H 68

BIBLIOGRAPHY 8 l

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i v

TABLE OF SYMBOLS

Geometry

L = Le ng th of spa n

B = D i f f e r e n c e i n e l e v a t i o n o f 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 m e a su r ed 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 = Incr em en t i n x

dy = Increm ent i n y

d s = . 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 d x a n d d y

LT = I f 1 -1 d x f o r a l l spans

J oWj

L e = I J L ^ | j3 d x f o r a l l spans

D e f l e c t i o n sv = 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 a nd 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 o f 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 o f 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 o f r i g h t c a b l e s u p p o rt

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 es e f f e c t . o f t empera tu re and s t r e s s e l o n g a t i o n o f

c a b l e )

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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 d ea d l o a d o f 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 o n 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 s u s pe n d er 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 e nd o f s p an

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 e nd o f s p a n

H = T o t a l h o r i z o n t a l c om po ne nt 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 c om p on en t o f d e ad 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 c om po ne nt o f c a b l e t e n s i o n d ue t o l i v e l o a d ,

t empera tu re change and su pp or t d i sp la ce me nt

H L ' = H o r i z o n t a l c om p on en t o f c a b l e t e n s i o n d ue t o l i w e l o a d

on e q u i v a l e n t b r id g e w i t h i n e x t e n s i b l e c a b l e a n d i m m o v a b l e

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 a n d

support movement

B e n d i n g Moments

M'i = Be nd ing 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 end in g moment i n g i r d e r a t r i g h t s up po 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 ca bl e

E l a s t i c a n d 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 = Yo un 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 shea r d i s t o r t i o n f o r g i r d e r o r t ru s s

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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 webw

G = Sh ea r modulu s

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 rom 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 Program

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 eo ry s o l u t i o n

= 0 f o r E l a s t i c T he or y 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 c h an g e 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 de l e t e e f f e c t o f ca b l e s l op e change

M i s c e l l a n e o u s

V E IE_E I

a = R a t i o o f s i de span l e ng th ' L g to main span l e n g t h L

b = fL sf E I7 E I

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S I M P L I F I E D CALCULATION OF CABLE

TENSION I N SUSPENSION BRIDGES

. CHAPTER1

INTRODUCTION

T h i s t h e s i s adds a few new words, and pe rhaps a few

new tho ug hts to an ar ea of s t ud 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 a mo un t o f s t u d y a nd 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 so mew hat d i f f e r e n t

f r o m th e u s u a l ' p r o b l e m s e n c o u nt e r e d b y t he 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 th e

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 wo rk has be en 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 to ov e r

come t he d i f f i c u l t i e s i n a n a l y s i n g a nd 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 .

T he p r o b l e m i n 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 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 a nd 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 a s to mi n i mi ze the bend ing s t r e s s e s

i n t he s t i f f e n i n g g i r d e r . D o u b l i n g t h e 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 t h e b e n d i n g m o m en ts .

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 . T h at 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 b et we en l o a d

1

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2

a n d 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

m et ho ds o f a n a l y s i s dependent on s u p e r p o s i t i o n a re n o t a p p l i c

a b l e i n t he 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 od i f i e d s u p e r p o s i t i o n me thod can be adap t ed to the

s o l u t i o n o f 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 o f s u s p e n s i o n b r i d g e s h as b ee n i n s p i r e d

by t w o . o b j e c t i v e s a n d a t l e a s t tw o m a i n t h e o r i e s h a v e b e e n

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 h as b e en t h e d e v e l o p m e n t

o f a n e x a c t t h e o r y o f a n a l y s i s . A s i n m os t e n g i n e e r i n g p r o b

l e m s , s o i n t h e 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 a nd would be

ex t r em e l y cumbersome to u se f o r de s i gn pu rp os es . Howeve r ,

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 c a n be o b t a i n e d b y t h e u s e o f t h e

D e f l e c t i o n T h e o r y , w h i c h t akes a c c o u n t o f t h e 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 h as

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 e d u ce t he l a b o r r e q u i r e d f o r a n a l y s i s a nd d e s i g n . A r e s u l t

h as be en th e E l a s t i c T h e o r y , w h i c h ig no re s the changes i n ge o- '

m e t r y 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 u n d er

l i v e l oa d and t empera tu re c h a n g e s . T hu s' , t h e 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 b et we en l o a d a nd s t r e s s and the u s u a l

me thods o f s u p e r p o s i t i o n can be us ed . As migh t be ex pe c t ed ,

t he 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 c an va ry w i d e l y

de pen di ng on the f l e x i b i l i t y o f t h e b r i d g e .

. Ch ap te r 2 i s devo te d to a deve lopm ent of the D e f l e c - .

t i o n The ory or forms of 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 of the many experts

i n t h e 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

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3

r e f i ne me n t s i n the the o r y may be i n c l u d e d to improve the a cc u

r a c y o f t he c a l c u l a t e d r e s u l t s an d thus t he D e f l e c t i o n Theory

eq ua t i on s may take d i f f e r e n t f orms depen d ing on the a cc u r ac y

d e s i r e d . Some o f t hese r e f i n e m e n t s a r e d i s c u s s e d a nd t h e

e f f e c t on the eq ua t i on s i s shown . The E l a s t i c T he or 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 t he 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 t he e f f e c t s on a c c u r a c y

w h i c h migh t be ex pec ted as a r e s u l t o f i n c l u s i o n o r n eg l e c t o f

some of the r e f i n e m e n t s . . No new t he or y i s to be fou nd i n

C h a p t e r 2 bu t the deve lopm en t ha s been i n c l u d e d he re t o p r o v i d e

a fr amework o f r e f e r e nc 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 ho u l d be no t ed t h a t t h r o u g h o u t t h i s w or k 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 a nd 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 here 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 lopm ent of the th eo ry tha t

s o l u t i o n o f 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 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 a d i f f e r e n t i a l e q u a t i o n and a n i n t e g r a l e q u a t i o n .

I n t he m ore g e n e r a l a nd m ore 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 m et ho ds f o r t h e s o l u t i o n o f e a c h o f these

e q u a t i o n s . T he s i m u l t a n e o u s s o l u t i o n i s f o u n d b y a c u t a n d

t r y me thod . Hence , s o l u t i o n o f a n u m er i ca l example can become

a n e x t r e m e l y l e n g t h y a nd t e d i o u s p r o c e d u r e b y h an d 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 ex i s t en ce o f compu te r s 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 b y h a n d.

The p r o b l e m 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 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 comp u te r . Ch ap t e r 3 d e s c r i b e s a p r o g r a m which

was w r i t t e n f o r t h e I B M1620 d i g i t a l com puter In or de r to

i n v e s t i g a t e s u s p en 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 tha t

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4

t he methods employed In the program are w e l l s u i t e d f o r c o m p u t er

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 rog ram

has been i n c l u d e d i n the hopes t h a t i t may se rv e as a gu id e i n

t h e p r e p a r a t i o n o f 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 key to 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 ex ac t me thods , c a b l e t e n s i o n i s f ound by

a cu t and t r y method. Ch ap te r 4 d e s c r i b e s a m e th o d, b e l i e v e d

to be new, whe reby H, the h o r i z o n t a l componen t o f ca b l e t e n s i o n

can b e f o u n d 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 u po n w h i c h the

method depends are shown and the method i s d ev el o pe d. Use of

the method r e q u i r e s the use of t a b l e s o r c u rv 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 . T he se 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

e xa m pl e s i l l u s t r a t i n g t h e a p p l i c a t i o n o f t h e m e th o d. The m et ho d

e m pl o ys 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 va lu e of H i s known. S i nc e H i s the va lu e

to be de te rm in ed and i s t he re fo re 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 of H i s

i m p r o v e d 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 a n

a c c u r a t e va l ue of H . The method may be a p p l i e d to su sp en s i on

b r i d g e s e i t h e r w i t h c o n t i nu 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 su pp or t s .

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5

CHAPTER2

THEORY ANDREFINEMENTS

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 nc e rn e d w i t h t he 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 k no wn r i g i d i t y ,

s us pe nd ed b y v e r t i c a l suspenders f r om a p e r f e c t l y c a b l e , which

i s anch ored a t tower tops o r a n c h o r a g e s . I n t h e a n a l y s i s , t h e

f o l l o w i n g s i m p l i f y i n g assumptions are made:

1. The suspenders a re I n e x t e n s i b l e .

2. The suspenders a r e so c l o se t o g e t h e r t h a t they

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. T he d ea d l o a d o f t h e b r i d g e i s d i s t r i b u t e d a l o n g

t h e g i r d e r s .

4 . Th e 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 un d er t he a c t i o n

of dead l o ad a l o n e , and c a r r i e s no be nd in g moment.

5. The dead l o a d i s co ns ta n t f o r eac h sp an , and hence

the cab le i s i n i t i a l l y p a r a b o l i c .

The above a s su mpt ions a r e u s u a l f o r the so - c a l l e d

" D e f l e c t i o n Th eor y" o r more exa c t th eo ry o f su sp en s i on b r id ge

a n a l y s i s . Les s ex ac t forms of the D e f l e c t i o n The ory i n common

use usua l ly make the f o l l o w i n g a d d i t i o n a l a s s um 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 t h e c a b l e a r e v e r y

s m a l l compared w i t h t he v e r t i c a l d e f l e c t i o n s , and

can b e n e g l e c t e d .

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6

7 . Deflections of the cable are very smallcompared

with cable ordinates, and theireffecton cable

slope can be neglected incalculationof cable

extension.8. Sheardeflections in the girders are very small

comparedwithbendingdeflectionand can be neglec

ted.

Assumptions 6, 1and 8may be excluded with little

difficultyin the derivation, and may evenbe excluded in an

analysisbydigitalcomputer. Therefore, theeffectsofhorizontaldeflections,cable slopechange,and shear deflectionare

includedhereand discussedbriefly. Itis'notto bethought

thattheirinclusion resultsin acompletetheory, butperhaps

these aresomeof themoreimportant refinements whichcan be

made. Others*havediscussed theeffectof theaboverefine

ments,-and in additionhaveintroduced, or atleastmentioned,other refinements suchastowerhorizontal force,towershorten

ingcable lock atmidspan,effectof loadsbetweenhangers,

temperaturedifferentialsbetweengirder flanges,finitehanger

spacing,weightof cable andhangers,variationof horizontal

componentof cable tension withhangerinclination,and soforth.

The DeflectionTheoryof suspension bridge analysisresultsin a non-linearrelationshipbetweenforces and deflec

tionsandhencetheprincipleof superposition andmethods

dependenton superposition are not applicable in the usual

manner. In order tosimplifytheforce-deflection relationship

"Into alinearone, it is necessary:tomakea furthersimplifying

* Reference (12)

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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

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Figure 2.

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cableisconnectedto thegirderbyverticalsuspendersand

carriesthedistributed loadq. Thecableis intension,the

horizontalcomponent ofwhichis constantand isequalto H.

Atthesupports,theverticalcomponentsof thecable tensionareyA' andVg'. Undertheactionofliveloadandtemperature

changes,thecableandgirderdeflectfromthepositionsshown

in solid linesto thepositions indicatedbydashed lines. The

cable supports deflecthorizontallythedistancesh^ and hg.

The original cable position is givenbyco-ordinatesx and y

measuredhorizontallyfromA andverticallyfromthechord joiningtheundeflected cable supportsat A and B. ApointP on

the cable deflectsfromitsinitialpositionto apointP'

horizontallyadistanceh andverticallyadistancev. Apoint

Q,on thegirder deflectsfromItsinitialpositionvertically

belowP to aposition Q,'verticallyadistancev.

CableEquation

Figure 2shows anelemental lengthof thecableat

pointP. Its undeflected position isshown as asolid line,

while its deflected position isshown as adashed line. The

lengthof theelementin theundeflected position is givenby

(ds)2

(dx)2

(dy Bdx\2

~ + \ -I J - ( 1 >

Undertheactionofliveloadsthecable deflectsas shown and

the lengthof thesameelementofcablein thedeflected posi

tion is givenby

(ds+Sds)2 (dx + dh)2 /dy Bdx dv\2

+ + . . . (2)

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ds S'dsfl 1 6ds- + -

dx dx \ 2dx

1 1dh\ dv /dy

2dx/ dx Vdx

B 1dv- + ] . . . ( 3 )

L 2dx.

Subtracting (l)from (2)andrearrangingterms, it is foundthat

dh

dx

Since ds and ha r e

both extremelysmallcomparedwithunity,dx dxtheymay bedropped from the terms 1+ i ^ s and 1 + 1 ^ 1 .

2 dx 2dxThe termi isgenerallysmallcomparedwith - over

2dx dxlimostof thespan,but may besignificant,especiallyinvery

flat cables. Expression ( 3 )then reducesto

ds6ds dh dv /dy B 1dv'

dx dx dx dx \dx L 2dx>Theextensionof thecable8ds ascausedbytemperature

expansionandstressisgivenby

8ds 6 t d s ds

dx dx AE dx

1

2

(5)

where: 6 = coefficientofthermal expansion

t = temperaturerise

.H- = changeinhorizontalcomponentofcabletension

duetoapplicationoflive load,temperature

changes, supportmovement, etc.

A = cross-sectionalareaofcableE = Young'sModulusforcablematerial

Again,sincefUlis extremelysmallcomparedwithunity,i tmayd X / \2

bedeletedfrom- the term ( 1+

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10

or,since'

ds 1 1/dy B>

dx 2\dx L,

2

(7)

then6ds etds

+HLds ds dv/dy B 1dv>

dx dxVdx L 2dxy.(8)

dx dx AE dx

A combination of equation (4)representing cablegeometryand

equation (8)representingHooke'sLawgives the cable equation

as*~ 2

dh et /ds\ 2

HT fas\ 3

+

dx dxy AEVdxJ

H Lds)AE Vdxi

dv/dy B 1dvA

dx Idx L 2dx,

(9)

Theabovecable equationmay besimplified significantlyif it

isobserved that theterm [OS-- + In expression (5)is\dx L dx/

normally lessthan .2,and LY_is generally smallcomparedwith

dxdy-IL Hence,dv_--j_-sn o ^ v e r y significantin the totalexpres-dx L dxsionand can reasonablybeneglected. Since has already

dx

beenneglectedcomparedwith unity in thesameexpression,this

amounts,to neglect of theeffectofdeflectionsoncable slope

and expression (5)becomes6ds etds HT 'dsV

dx- + - Hdx AE \dx/

(5a)

When(5a)iscombinedwith ( 4 ) , thesimplifiedcable equation

is

dh et /ds\2 HL /ds\ 3 dv(dy B I d+

dx idxy AE\dx> dx\dx

v'

L 2dx,(9a)

Itcanbeseenthat neglectofthechangeof cable slope is

reflectedin expression (9)byneglect of theterm (4JAE \dxI

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11

comparedwithunity. _Jis usuallyof theorder.001 and AE d x

Isnormallynotmuchlargerthan1, so atermoforder.001 has

beenneglected comparedwith1. Onthis basis,Timoshenko

arguesthatit is

negligible. However,it is notdifficultto

see thatagivenpercentageerrorin onetermofexpression(9)

couldbemagnifiedbysubtracting thatterm fromanotherof

similarmagnitude togivealargerpercentageerrorin^Jl.

Expression (9)can befurther simplified if is2.Mx

neglected comparedwith -5.inexpression (4). Then thedx L

cableequationbecomes

dh et /ds\2 HT Ids\ 3 dv /dy B\ = + - - ( 9 b )dx \dx/ AE \dx/ dx \dx L /

This final expression givesalinear relationshipbetweenhori

zontalandverticaldeflections.

Itshouldbenotedthatthe abovelinear relationship

betweenhorizontalandverticaldeflectionsdoes notimplythatthe structureislinear. Thecableequationhas beenreduced

toalinearequation,but anon-linear relationshipcan and does

s t i l l existbetweenstressesandapplied loads.

Ifthecableequationisintegratedoverthe span

lengthand thehorizontaldisplacementsof thesupportsare in

sertedasconstantsofintegration,thefollowing expressionresults:

h B - hAr LGt /ds\2dx

Jo

r L

o

H L ds

AE Vdx,

dx(10)

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12

or,ifL|ds_^2dx i s denotedby L, and L fd_^ .^ 3

d x isdenoted

by L.Q, then

hB "hA = t L t + %

Le

AE

_ o \ y

CL HL /dsv 2

AE \dx',

dv /dy B 1dv

dx \dx L 2 dx

dx( l i ) }

Ifthechangein cable slope can be neglected, then

- hn GtL

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Figure 4.

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13

the cableand thegirder. Figure3 shows afree-bodydiagram

ofthegirderundertheactionofapplied loads. Thereaction

RAcan befoundfrom

RA MB- MA+

(p+ w -q) (L- a) da... (12)

Then,thebendingmomentin thegirderat x,denotedby M Is

givenby

L(p+w-q)(L-a)da fx(p+w-q)(x-a)da... (13)M MA (MB-MA)x x+ +

Lo

For simplicitydefineaquantityM'equalto thebendingmomentproducedbya l l loadsexcept thoseappliedby thecable. This

isgivenby

rL(p+w)(L-a)da r x

o

(p+w)(x-a)da... (14)

... (15)

M' MA (MB-MA)x x= + +

L

Then

M M

1

x ^

L

q(L-a)da T

x

q(x-a)da-Jo L Jo

Now,i t isnecessarytoconsiderthestaticequilibrium

ofthecableundertheapplied loads. Figure4 shows the

forces actingon thedeflected cable, indicatedby adashed line.

The cable tensionsat thesupportsareresolved intocomponents

inthedirectionof thechordjoiningthedeflected pointsofsupportof thecable,andverticalcomponentsV^ andVg. The

verticalcomponentV is givenby

VA rL q( L + h B-(a+h))da

L+ h B- hA.(16)

It willbenotedthattheforcesin thedirectionof the closing

chordhavehorizontalcomponentsequalto H, thehorizontalcomponentofcable tension.' Then,fortheequilibriumof the

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14

cable

H(y-8]_+52)=VA(x-hA)-xq(x -(a+h))da ..(17)

.(18)

wherey- S]_ +S 2istheverticaldistancefrom thedeflected

positionof theclosingchordto thedeflectedcable. Itisclearfrom geometry that

8 1 B|hA (hB-hA)x>

LV L VReferencetoFigure2willshowthat

82 v h /dy B'

Idx LThe termVA(x-hA)inequation(17).now becomes

..(19)

VA(x-hfl) xA'Lq(L-a)da r L

o hB" hA+

q(hg-h)da

o hA

VA.hAA (20)

Whenit is observedthatthehorizontal deflectionsarevery

smallcomparedwiththedimensionsL, a and x,itcan beseen

thatthetermsx f (hB~h)daa n ( ^ v flhaareverysmallcomparedJo L+ho-hfl R R

withx I^ QA-k-a)da.and can beapproximated withnegligiblej o L+hB^A

errorin thetotalterm VA(x-hA)asfollows:

x ^Lq(hB-h) da x

o^ B ^ A

VAhA=hAH dy>

VdXy

Lq(hB-h)da

A

Then,thetermx. fLq(L~a) d aofequation (20)Jo L+hB'hA

neglecting a l lbut thefirsttwotermstogive

...(21)

...(22)

canbeexpanded,

x ^ q(L-a)da x

oL + hB" hA

Lq(L-a)da x(hA-hfi) fL q(L-a)da

+ ...(23)L L

Notethatthesecond termon therighthandsideofexpression

(23)ismuchsmallerthanthefirst term. Itisthereforeper-

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15

mlssibletoapproximateitasfollows

x(hA-h B)rL q(L-a)da x(hA-h B)H/dyi

dx i...(24)

o ~ \ ~ vA

Ifsubstitutionsfromexpressions ( 2 1 ) , ( 2 2 ) , (23)and(24)aremadein expression ( 2 0 ) , it isfoundthat

VA(x~hA) xfLq(L-a)da x(hffi-hB)H/dy

L \dx, A

xrL

+

q(hB-h)da hAH /dy^. . (25)

L \dxJA

If substitutionsfromexpressions ( l 8 ) , (19)and(25)aremade

inexpression (17)andtheresult iscombinedwith expression

(15) j the following expression results:

M W H y v rx+ hqda

hAH/dy) x+ H /dy\ hAr h B r

L

+ q(hB-h)da ...(26)

.dxj A \dx/AL Jo L

Sincetermsinvolving the horizontal deflections are small, it

ispermissibleto make someapproximationsintheseterms.

Specifically, it is permissibletoapproximatethesuspender

forcesq by

-H-dydx'

...(27)

Iftheaboveis substituted in(26)the expression for

bendingmomentin the girderbecomes

M M-< H y v+

B

L

hA+

(hB-hA)x h/dy B\ rxhd y da

dx L o dx2

h+A x

L

'dy\ (hA-h B)

\dx/A

d^y(hB-h)da

odx2~

.(28)

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16

Itcanbeshownthat,Ifallhorizontal displacements

are increasedby aconstantamounth*,thebendingmomentinthe

girderata llpointswillbeunchanged. Hence,ifh 0is set

equalto-h ,h may bereplacedbyzeroinexpression ( 2 8 ) ,hgmay bereplacedby hg - hA, and h may bereplacedby h - hA.

Expression (28)may bedifferentiatedtwicetogive

d2M -p -w - H

dx'

d2y d2v d2h /dy B\ dh d2y

dx2 dx2 dx2\dx Lj dx dx2...(29)

Ifhorizontaldeflectionsareneglectedin thegirder

equation, expressions (28)and(29)reducetoM-M-'- H(y +v) ... (28a)

d2M -p -w -H /d2y d2v\

T = \~?+ ~?~\ " ( 2 9 a )

dx2 \dx2 dx2/

If verticaldeflectionsareneglectedin thegirder

equation, expressions (28)

and(29)

reducetotheElasticTheoryexpressions

M= M"'1- Hy ... (28b)

d2M -p -w-Hd2y ~ = ... (29b)dx dx'-

Expression (13)canbedifferentiatedtwicetogive

q- p - w...(30)

d2M

dx'Thenequations (29b)and(30)canbecombinedtogive expression

( 2 7 ) ,anapproximaterelationshipwhichwasusedearlier.

Fromelementary strengthofmaterials,thebasic

differentialequationrelating deflectiontobendingmoment and

shear inagirderis

EId2v -M + EImd2M...(31)

dx' dx

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17

where: E

I

m

Young'sModulus forthegirder material

moment ofinertiaofthegirder

coefficientofshear distortion

m =m =

,.1 foragirder

ADEsin0cos 0foraplane truss

Aw =

G=

AT, =D

cross-sectionalareaofgirderweb

shearmodulus

cross-sectionalareaofthediagonalmember (s)

anglemeasuredfromverticaltodiagonalmember(s)

Thedeflectionsdue tosheararesmallcomparedwiththedeflectionsduetobending. Therefore,negligibleerror

2

resultsif%^isrepresentedbytheapproximateexpression (29a),dx^

Ifexpressions (28)and(29a)aresubstitutedinexpression (30),

'andtheresultingequationisdifferentiatedtwice,thefollowing

fourth-orderdifferentialequationisfound:

(1+ Hm) Eld^v 2d(El)d 3 v d2

(El)d2

vdx4

+ +dx dx3 dx^ dx...(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)c an be s o l v e d to g i v e

o

v 2 M 2Ir V 2

E I

where

CL - C Le - e

(89)

.(90)

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47

Vr x eC x

- e_ C x

L " e C L - e ~ C L( C L )2

Then the end s lo p es can be fo un d fr om

...(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 Le - e

e C L + e - C L

~ C L ^CL

e - e

From s ym m et ry o f t he g i r d e r , i t i s c l e a r tha 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 n o a p p l i e d

l o a d moment, the b e n d in g moments a t the tow er s ar e e q u a l . The

unknown moment M can be fou nd by e q u a l i z i n g end s l op es a t the

towe rs and i s g i v e n by

ab / dv-

M H l f ,dx / Ls

dv^

dx+

b / dv...(98)

2

a VdxjLs

I t ca n be shown t h a t i n t he case o f a n e x t r e m e l y s t i f f g i r d e r

w h e r e t h e 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) ca n be re du ce d

t o

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48

H

lf 1 + ab:

3 b +

2 a

... (99)

I n t h e 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 c om pu te H , i t

w i l l be n e ce s sa ry to compute M a number of t i m e s , and i t w i l l be

shown t h a t i t i s adv ant age ous to compute M e w h i c h i s i n de pe nde 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 for each new t r i a l v a l ue 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 va lu es 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 p a r a m e t e 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 f o u n d f rom t h e t a b l e s o r c u r v e s , M i s - f o u n d from

M = KJVL ...(100)

From t h e s o l u t i o n s t o e q u a t i o n s (86) and (87), i t i s f o u n d that

A 2 Mg f ILX 2

E I

A3 M3

. .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)

,CLe

-CL(103)

The t o t a l va l u e o f t he s u pp o r t m ove men t c o r r e s po nd i n g

t o t h e 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 span s i s

g i v e n by

A t H-^f L T A2.

E I. . . (10-4)

where

T 1 2 a3b ( A 1 ) , M

+

A

2 A 2 2 a b ( A 2 ) .

+ :A1

...(105)

t

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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 t he same as t hose use d f o r the case of a

t h r e e - s p a n b r i d g e w i t h h i ng ed su pp or t s and a re found i n A p p e n

d i x 2 a nd F i g u r e 14 ( a ) . V a l u e s , o f A2 and b ^ A2^ s a r e p l o t t e d

,2A A l

a g a i n s t (C L) i n F i g u r e 15 f o r s e l e c t e d va l u e s o f b a nd a r e

a l s o i n c l u d e d i n A p p e n d i x2.

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 f ou nd 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 t he same i n f l u e n c e l i n e u s ed f o r h i n g e d g i r d e r s . The

s ec o nd 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 t he case o f t he

main s pa n t he I n f l ue nc e l i n e , i s g i v e n byL

f

X v1 + Y v

2+ v

3

A i A2 +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 sh ow n p l o t t e d i n F i g u r e 12 a nd t a b u l a t e d

A ~ 2 + A ~ 3

va l ue s a r e f ound i n A pp e nd i x2.

F o r t h e l e f t s i d e s p a n , th e 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)

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50

where

. . . ( i l l )T

Y Mab(A o ) s ,S = ^ ... (112)

F i g u r e 11 shows cur ve s of v 2 and A p p e n d i x 2 h as 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 t h e r i g h t

s ide span i s o f cour se s i m i l a r t o t h a t f o r t he l e f t s i de s pa n

b u t o p p o s i t e h a n d .

The nu m e r i c a l e xa m pl e 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 o f H c o n v e r g e s r a p i d l y a nd

us e o f t he t a b l e s o r c u r ve s m akes t he 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 c an be se en t h a t t h e s o l u t i o n s t o e q u a t i o n s (58),

(86) and (87) a r e g i v e n f o r t h e s p e c i a l c a se i n w h i c h t h e g i r d e r

r i g i d i t y E I w i t h i n t h e s pa n i s c o ns t a n t . T h e r e f o r e , u s e o f t he

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 ss um pt io n of

c o n s t a n t E I w i t h i n each sp an .

I t i s p o s s i b l e t o s o l v e t h e 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 he 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

a n d t a b u l a t e s i m i l a r d a t a f o r u se i n a n a l y s i s . A s u i t a b l e

a pp r oa c h . m i gh t be t o de t e r m i ne a t y p i c a l m ode o f v a r i a t i o n o f

g i r d e r s t i f f n e s s s uc h t h a t m os t 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

ha ve ,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 co ns t a n t s mig h t t he n be

d e t e r m i n ed b y i n t e r p o l a t i o n b et we en th e 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 t h e a s s um pt i on o f c o ns t a n t

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51

g i r d e r r i g i d i t y i s a r e as on ab le 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 um er i c a l examples were s o lv e d u s in g the

c om put e r p r og r a m de s c r i be d i n t he 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 an a l ys ed as a co n t in uo us g i r d e r and w i t h h i nge s

a t t h e s u p p o r t s . The m a in e p a n g i r d e r s t i f f n e s s v a r i e d f rom a

minimum o f .5 t im es the mids pan s t i f f n e s s a t the towers to 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 . t he q u a r t e r p o i n t s .

The same b r i d ge was a n a l y s e d a s s um i ng a c on s t a n t g i r d e r r i g i d i t y

e q ua l t o t he a ve r a ge va l u e . The r e s u l t s o f t he s t u dy i n d i c a t e d

t h a t a r e as on ab ly ac cu ra t e va lu e o f H can be de t e rm ine d by

a s s u m i ng a n a v e r a g e v a l u e f o r t h e g i r d e r s t i f f n e s s . E r r o r s i n

H-^ enc oun te r ed were l e s s t han 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 h an d 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 to the av er ag e v al u e be assumed f o r each span

i n o rd e r t o de t e rmi ne the va lu e f o r H .

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TO FOLLOW PAGE 51

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Figure 13

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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 of

s u s p e n s i o n b r i d g e s b y h an d c a l c u l a t i o n s w i l l ever be a s imple

p r o c e d u r e o r one t h a t t he s t r u c t u r a l e n g i ne e r w i l l a pp r oa c h

e n t h u s i a s t i c a l l y . However , i t has been fo un 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 ev e n f o r p r e l i m i n a r y d e s i g n

i n many ca se s . Th er e f o re , t he need f o r D e f l e c t i o n Theory

s o l u t i o n s does e x i s t . I t would be ex pe d i en t t o t u r n the t a sk

over to a computer and a v o i d a l l hand c a l c u l a t i o n s , b ut 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 th e e a r l y s t ages o f

a d e s i g n , t h e r e w i l l a lw ays be a n e c e s s i t y fo r some hand c a l c u l a

t i o n s , and i t i s hoped tha t these w i l l be made somewhat e a s i e r

w i t h t he methods p r es en te d he re .

I t was ap pa re nt i n the c ha pt er on Th eo ry and R e f i n e

ments t h a t t h e s i m p l e r D e f l e c t i o n T h e o r y r e p r e s e n t e d b y e q u a t i o n s

(43) a nd ( 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 w h et h er t he 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 h an d c a l c u l a t i o n s , a nd i t i s r ec om me nd ed

t h a t t h e more r e f i n e d ve r s i o ns o f t he D e f l e c t i o n Theory be

r e s e r v e d f o r c om pu te r 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 u se i n h an d 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 f o u n d 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 nc e ( l )

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53

Once t he 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 s upe r i m p os e 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

a s g i v e n i n S t e i n m a n ' s t e x t .

Eq ua t i on s (43) and ( l i b ) fo r m the b a s i c t h eo ry f o r t he

m et hod p r e s e n t e d o f de t e r m i n i n g t he t o t a l va l ue o f c a b l e t e n s i o n .

No a p p r ox i m a t i o n no t i n he r e n t i n t he a bove e qu a t i o ns i s made f o r

t he m et hod g i v e n a nd he nce t he va l u e o f 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 m eth od h as 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 a n be s e e n i n t he s a m pl e 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 t h e m et ho d i s e x t r e m e l y e a s y 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 dge w i t h g i r d e r s h i n g e d a t t h e 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 but 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 a n y 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 t o w a r d a n 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 mu st be a n a c c u r a t e , s i m p l e

m et ho d o f d e t e r m i n i n g t h e c a b l e t e n s i o n . I t i s b e l i e v e d tha t

t he m et hod de v e l o pe d i n C ha p t e r 3 a nd i l l u s t r a t e d i n s am p le

c a l c u l a t i o n s i n A p p e n di x 3 meets t h e o b j e c t i v e s o f a c c u r a c y a nd

s i m p l i c i t y . Th er e f or e , the method sh ou ld be u s e f u l a s p a r t o f

a t o t a l m e t h o d o f a n a l y s i s .

One ap pr oa ch to a s i m p l i f i e d me thod of a n a l y s i s migh 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 .

wha tever methods a re used to comp le t e t he 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 me th od s o f 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 t he 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 a s t h e t o t a l . v a l u e o f t h e c a b l e

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tensionisknown andappliedin theequationsforthepartial

loadings,thebendingmoments anddeflectionsforthepartial

loadingcasesmay besuperimposedtogivethetotalvalues.The

key, then,is thedeterminationof thetotalcable tension,and

a simple, accuratemethodofdeterminingthecable tensionhas

beenpresentedInthiswork.

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55

APPENDIX 1

COMPUTER PROGRAM

C KEN RICHMOND CIVIL ENGINEERING THESISC PROGRAM TO ANALYSE SUSPENSION BRIDGES AND COMPUTE MAGNIFICATIONC FACTORS.CC MAIN LINE PROGRAMC

DIMENSION El(53) , P(53) , A B ( 1 3 , 5 3 ) ,A(53,5)DIMENSION E(53) , B (53) , BM(2,53)

43 READ 1 , KODE , C , R , S1 FORMAT ( 12 / ( E l 4 .7))

READ 2 , F , SIDE , RISE , T , SL IP2 FORMAT ( F 6.4 / F6.4 / F 6 . 4 / ( E 1 4 . 7 ) )

PRINT 3 , KODE , R , S3 FORMAT(//6H KODE= 14 ,3H R= E1 4. 7 ,3H S= E1 4. 7 / )

PRINT 4 , F , SIDE , RISE4 FORMAT ( 3H F= F7.4 , 6H SIDE= F1 1. 4 , 6H RISE= F l 1 . 4 / )

PRINT 5 , T , SLIP5 FORMAT ( 3H T= El7.7 , 6H SLIP= El 7.7 )

EIO =1.0 / (8.0 * F * C )IA= 17 .0 - SIDE * 20.0ID= 37 + 17 - IAIF ( KODE - 10 ) 8 , 8 , 6

6 KODE = KODE - 10DO 7 I = IA , ID , 1

7 E l ( I ) = EIOGO TO 11

8 DO9 I = IA , ID, 1READ 10 , El(I )

10 FORMAT ( F7 .4)9 El(I ) = EIO * El(I )

11 PRINT 1212 FORMAT(/23H I P El / )

DO 13 I = IA,ID,1READ 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 = 0.0SF = 0. 0GO TO ( 1 6 , 1 7 , 1 8 , 1 9 , 1 6 , 1 7 , 1 8 , 1 9 ), KODE

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56

16 DF = 1.017 SF = 1.0

GO TO 1918 DF =1.019 N2 = 1

GO TO 20020 D = 0 .0HD = 0.125 / FHL = 0.1DO 36 N = 1,2K = 0

21 HT = HL + HDIF (KODE - k ) 22,2 2 ,23

22 Hk = 2I I = IAIE = IDGO TO 400

23 Hk = 1N2 = 3GO TO 200

2k ERROR = 0. 0N2 = 2GO TO 200

25 ERROR = ERROR + SLIPPRINT 26 , HL , ERROR

26 FORMAT ( A H HL= E1 7. 7 , 7H ERROR= El 7.7 )IF (K- I ) 27 ,28 ,27

27 HL1 = HLERR = ERRORHL = HL + .1 * HDK = 1GO TO 21

28 IF(AB S(ER R0R) -1.0E -5) 3 0,3 0,2 929 DELH = ERROR *(HL - HL1)/(ERROR - ERR)

ERR = ERRORHL1 = HLHL = HL - DELHGO TO 21

30 E( IA -1 ) = -E(IA+1)DO 31 l=IA,ID,1

B M(N ,I ) = ( 400 . 0 * (2 .0 *E ( I ) -E ( I -1 ) -E ( I +1 ) ) ) * (1 . 0+D *H T * S M)31 BM(N,I) = (B M (N , I ) -S M* (P ( I )+1 .0+H T *D 2Y ) ) *E I ( I )BM(N,IA) = 0.0BM(N,ID) = 0.0IF ( KODE -k ) 33 ,33 ,32

32 BM(N,17) = 0.0BM(N ,37) = 0.0

33 SUM = 0Q = ID - IA + 1DO 3k I = IA,ID,1

3k SUM = SUM + E l ( I )

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57

3536

41

39

38

37

42

CC

c100

AVG = SUM / Q

CDL = HD / AVG

CLL = HL / AVG

CTOT = HT /AV G

PRINT 35 , CDL , CLL , CTOT

FORMAT (/ 5H CDL= El 7. 7 , 5H CLL= E l 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,1IF ( 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 ( I A - I ) = EI(IA+1)EI(ID+1) = E l ( ID -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 * XDY = 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,IAB(11,1

AB(12,1

AB(13,IGO TO

* (AL - 2.0 * X ) - BL / AL

+ DY * DY )

= 1.6E5 * E l ( I ) - 8 .0 E3 * DEI

= - 6 . 4 E 5 * E l ( l ) + 1.6E4 * DEI + 400.0 * D2EI

= SM*AB(3 ,D + 400.0* ( -1 .0-D F*DY*DY) +20.0*D F*DY*D2Y

= 9.6E 5 * El ( 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 * El(I ) + 8.0E3 * DEI

= ( P ( l ) + 1 . 0 ) * (1 .0 - SM * D2EI)= D2Y * (1 .0 - SM * D2EI)

= -D F* T* D2 Y* ( 3.0* DY*D Y + 1.0 )

= -DF*RAE*D2Y*DS*( 4 .0 *D Y* DY + 1.0 )

201,202,203) ,N1

SUBROUTINE 2

II = IA

IE = 17

AL = SIDE

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BL = RISE

N1 = 1GO TO 204

201 I I = 17IE = 37

AL = 1.0BL = 0.0N1 = 2GO TO 204

202 I I = 37IE = IDAL = SIDEBL = -RISEN1 = 3

204 GO TO (1 00 ,3 00 ,4 00 ), N2

203 GO TO (20 , 25 , 24 ), N2C

C SUBROUTINE 3C

300 DO 301 I = I I , IE , 1X = I - I I

X = .05 * X

DY = 4 .0 * F * (AL - 2 .0 * X ) - BL / ALDS = 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*DE301 B(l)= D*B(I)+RAE*HL*DS**3+T*DS*DS-DY*DE

IS = IE - 2DO 306 I = I I , IS , 2

306 ERROR = ERROR+(.05/3.0) * ( B ( l ) + 4. 0 * B(l+1) + B(l+2))GO TO ( 20 1 , 202 , 2 0 3 ) , N1

C

C SUBROUTINE 4C

400 DO 401 I = I I , IE , 1A ( l , 1 ) = AB (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 * A B ( 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 +1B(IM) = - ( P ( l l + 1 ) + 1.0 + HT *D2 Y) * ( SM / (1 .0 + HT * SM

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TO FOLLOW PAGE 59

MAIN LINE PROGRAM

\READ: KODE,

C,R, S, F,

SIDE,RISE,T,SLIP

1

READ: KODE,

C,R, S, F,

SIDE,RISE,T,SLIP

PRINT:

K(j)DE, R,S

F, SIDE, RISE

T, SLIP

COMPUTE

EIO, IA, ID

K10 KODE

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TO FOLLOW PAGE

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TO F O L L O W PAGE

) INITIALIZE

ERROR=0I

INITIALIZE

ERROR=0

, \

ERRORl < I O 5 L

COMPUTE

BM (N,IA)

TO BM (N,ID)

N2 = 2

,200)

25

ER RO R =

ERROR +

SLIP

PRINT:

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

INTERPOLATE

NEW ESTIMATE

H,

COMPUTE

A V E R A G E E I

CDL.CLL.CTOT

PRINT

CDL,CL L,

CT(|)T

N= 1,2-

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TO FOLLOW PAGE 59

= 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) ,PHI,

E(I)

COMPUTE

SM, D2Y,

RAE

I = I I , IE

COMPUTE

AB ( 1,1) TO

AB ( 13,1)

S H E E T 4 OF 6

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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

SPAN

SUBROUTINE 3

Integrate cable equation

by Simpson's Rule

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TO FOLLOW PAGE 59

SUBROUTINE 4

Compute deflections

at all points I I - l

to IE + I and store

in E(I)

I = II , IE

COMPUTE

A (1,1) TO A(T,5)

a B( I )

COMPUTE

BOUNDARY

CONDITIONS

I= I I-I,I E - I

REDUC

A ( I+1

aA(1+

TO Z

:E

,2)

2,1 )ERO

A( I E+ 1,2)

TO ZERO

COMPUTE

E ( I E +1),

E ( I E )

S H E E T 6 OF 6

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6o

APPENDIX2

TABLES

F-

P.

m

TABLE1,

v 2 + v 3

T 2 + ~A~

INPLUENCECURVES

A-

A

rL -V]_dx

Am

L

rx v 2dx

'o X Q Lrx ^

V o + V-:

A 2 + A 3

dx

L

HL2

EIX

LPl Ai As Am

1. ."4843 . 0 5 . 0311 . 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 .0681 . 0 0 3 0 . 0 0 2 4 . 0 0 3 5. 1 5 . 0 8 9 9 . 0 7 2 2 .0961 . 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 .1391 . 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 9.0211

.0271. 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 .16 65 . 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 .18 48 . 0 5 2 7 . 0 4 4 9 .053150 ' . 1 9 5 2 . 1 8 6 7 . 1 8 6 7 . 0 6 2 5 - .0541 . 0 6 2 5

. 5 5 .19 28 . 1 9 1 5 . 1 8 4 8 . 0 7 2 2 . 0 6 3 5 . 0 7 1 8

.60 . 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 .17 44 . 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 .1391 . . 1658 . 1 4 0 7 . 1 0 6 5 . 1 0 0 6 . 1 0 5 3

. 8 0 . 1 1 6 0 . 1461 . 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 . .1181 .1151 . 1 1 7 3

. 9 0 . 0 6 1 3 . 0 8 7 5 .0681 . 1 2 1 9 . 1 2 0 3 . 1 2 1 4

. 9 5 .0311 . 0 4 7 6 . 0 3 6 0 . 1 2 4 2 . 1 2 3 7 .124 0

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6 1

TABLE 1 (CONTINUED)

H L 2 xEI A l L F l p

sPm

A lA

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 .14 09 . 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 .19 24 . 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 .70 . 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 .10 02 . 1 0 5 2

. 8 0 . 1 1 6 1 . 1 4 8 2 .12 07 . 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 .121 4

95 . 0 3 1 1 . 0 4 8 8 . 0 3 6 5 .12 42. 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

. 0722 . 0 6 2 4. 0 7 1 7.60 . 1 8 5 8 . 1 9 2 6 .17 82 . 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 .14 10 . 1 0 6 5 . 0 9 9 8 . 1 0 5 1

. 8 0 . 1 1 6 1 .15 01 .12 11 . 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 .11 71

. 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 .12 42 . 1 2 3 7 .1240

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6 3

'TABLE 1 (CONTINUED)

HL 2

EI A lX

Lp

lx s F

mA

lAm

1 0 . . . 2 6 5 2 . 0 5. 10

. 1 5

. 2 0

. 2 5

. 3 0

. 3 5

. 4 0

. 4 5

. 5 0

. 55. 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

.11 64

. 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

.18 89

. 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

.10 64

. 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

. 20

. 2 5

. 30

. 3 5

. 4 0

. 4 5' .50 . 55

. 60

. 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

.16 25

. 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

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64

TABLE 1 (CONTINUED)

HL 2

EIX

V;LF

lp

sPm

A

lAs

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-.707 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

.19 10

. 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

.19 43

. 1 8 2 4

. 1 6 0 8

.12 61

. 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

.14 31

. 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

.30

. 3 5

.40

. 4 5

. 5 0

. 5 5. 6 0

. 65. 7 0. 7 5. 8 0

. 8 5 .90

. 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

. 2020

. 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

.16 40

. 1 6 3 3

.16 13

. 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

.12 26

. 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 0.12 00. 1 2 3 6

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6 5

TABLE 1 (CONTINUED)

HL 2 A-, X P P An A AEI 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 .11 72 . 0 0 7 1 . . . 0036 . 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

.30 . 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 . 55 . 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 . 65 . 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 .10 61 . 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 .10 81

. . 85 . 0 9 2 3 . 1 8 5 5 . 1 1 7 2 . 1 1 7 8 . 1 0 7 4 . 1 1 4 4.90 . 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

2 0 . 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 .106 0 . 0 8 5 3 .10 02. .80 . 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 .12 41 .122 0 . 1 2 3 3

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66

' TABLE 2 .b( L 1) a

H L 2 -E I

bH L 2 -E I . 0 5 .1 0 .1 5 .20 .2 5 .30 . 3 5 .40 . 4 5 .50

1 .056 . 1 0 9 . 163 . 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 . .064 . 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 .767 ' . 8 0 1

50 . 2 4 1 . 4 o i . 5 1 5 .6 0 1 . 6 6 7 - .7 2 0 .7 6 4 . . 8 0 0 .8 3 1 . 8 5 7

7 0 . 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 L2

-E I

bH L

2

-E I . 0 5 .1 0 .1 5 .20 .2 5 .30 .3 5 .40 .4 5 .50 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 .046 . 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 .060 .1 1 4 , . 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 .086 . 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

7 0 . 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

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TABLE4. K = I}- TOWER MOMENTS

HLEI

a3 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

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APPENDIX3

NUMERICAL EXAMPLES

Example1. SingleSpan

D

>I25*r.4 k/ft.f\ k/ft

>

f \ M

.4 k/ft.

f\ k/ftr> t\

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69

Step1. Compute anddesign constants.

HD wL2 1.0(IOOO)2 1250K

8f 8 (100)

L2

(1000)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)

Step2. Compute H-1

.Here,it is necessarytoestimateH inordertoselectan

influence line. It issufficientlyaccurateto use

H = HD. Then

HL2 =1250(.00667)= 8.3'

EI

H ' isfound fromtheinfluencelinesplottedinFigure10.The influence lineordinateat thelocationof thepoint

loadis.139L. Theareaunderthecurve from.25to.50

is (.0625 -.Ol85)L. Thereforef

HL' 25(.139) (1000) .4 (.0625 -.0185) (1000)2

= H100 100

= 35 + 176 = 211K

Step3. Compute A '

HT'L

AE

='- .5 - 3.25( 1 0 - 4 ) (1054) - 211(.0015)

= -1.17ft .

A'=hB *hA " ft L t""L".e

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70

Step4. ComputeSH.

Here,another estimateof H must bemadeinordertodeter

mine . A - ^ . It issufficientlyaccuratetoestimate

H = HD+ HL1. Then

HL2 = (1250 + 211) .OO667 = 9.75

EIH T 2

Figure13isusedtodetermine A-,for =9.75and it is1 EI

found that^ =.277. Then 8H can befoundfrom

SH A

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Example2. Three-Span Bridge with Hinged Supports

Q

25k

fA k/ftf\ k/ft

if\ k/ft

1750'

Given:

MainspanlengthL =1000ft.

Mainspan sag f =100ft .

SidespanlengthL =500ft.

EImainspan =5.0(lO?) Kft. 2/girder

EI sidespan=2.5(10?)Kft. 2/girder

AE ofcable=7.5(lO5) K

Sidespanrise=112.1ft.

L e = 2 l 8 l ft.

L t=2116ft.

Deadload1K/ft.

Live load.4K/ft. onmainspan asshown

+ 25K onsidespanwhereshown

et =3.25(IO-4)

SupportdisplacementH B- hA=0

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72

Step 1. ComputeHDanddesign constants

Promequation (39)

HD 1 . 0 ( 1 0 0 0 ) 2 . 1250 K

8 (100)

L2 (1 0 0 0 ) 2 .0 20

EI 5 . 0 -(10?)

a 50 0 . 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 . ComputeH-'

EstimateH = HD= 1250K

HL2 = 1250 ( . 0 20 ) = 2 5 . 0

EI

Figure14(a)showsvaluesofb(A i ) splotted against MilEI

A lforselected valuesof b. For HL> = 2 5 . 0and b = . 5 ,the

, N EIordinate '-^^sis . 7 80 . Figure14(b)showsvaluesof

"AlthemultipliersX and Xs plottedasabscissae againstthe

ordinate

10(A

j)sforselected valuesof a. For a =

. 5- . "A~l

and b ( A i ) s = . 7 80 , themultipliersare:

"Alx = .836

x a = .0 82

The mainspaninfluencelineareafrom . 00to . 7 5isfound2

fromFigure 10to be.IO63 X. Thereforethecontribu-

ftio'nto H-j 1fromthe mainspanis.

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HL

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74

FromFigure 13, .A ^ =.130

6H -1.72 -50K

.130 (.20) + .0029

.833

H =1250 + 360 - 50 = 1560K

Step7. Comparethe value ofHcomputedat the end of step6

with the value estimated in step2. Repeatsteps2and5

toconvergence.

FromFigure14

X = .833

x s = .083

H

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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' 400 ' 250'

750'>