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