1963_a7 r35 s4
TRANSCRIPT
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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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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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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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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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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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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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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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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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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 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'>