Analytic Tire Model Phase I: The Statically Loaded

4 Toroidal Membrane -

by John T. Tielking

July 1975

Highway Safety Research lnstitute/University of Michigan

Tuhnical Ropoct Docurntotion Page

Roprodvction of coc.plo1.d pogo w h r i r e d

I. Raport No.

UM-HSRI-PF-75-7 1. k v e r n m m t Access,m No. 3. R e c ~ p ~ m t ' r Cotoiog No. 1

4. Ttt le and Subtitle

ANALYTIC TIRE MODEL. PHASE I: THE STATICALLY LOADED TOROIDAL MEMBRANE.

7. A d ~ o r ' S)

J .T . T i e l k i n g 9. Perform~ng Orgm~xot ion Nome m d Address

Highway S a f e t y Research I n s t i t u t e U n i v e r s i t y of Michigan Huron Parkway and Baxter Road

. Ann Arbor, Michigan 48105 12. Sponsoring Agency N m e md Address

Motor Vehicle Manufacturers Assoc ia t ion 320 New Center Bui lding D e t r o i t , Michigan 48202

IS . Suppl.mmtOry Motes

-. - 5. Report Oats

~~l~ 1975 6. Pufonning Orgonizotion code

8 . Per fom~ng Orgonizotion Report No.

UM-HSRI-PF-75-7 10. Worh U n ~ t No. (TRAIS)

360934 11. Controct OI Grant NO.

13. Type of Report ond per tod Covered

F i n a l Report J u l y 30 , 1974 - June 30, 1975

14. S~onsoring Agency code

T i r e T r a c t i o n C h a r a c t e r i s t i c s A f f e c t i n g Vehicle Performance: I n t e r i m Document 10

16. Abstroct This document p r e s e n t s t h e f i n a l r e p o r t on t h e development of a

p re l iminary a n a l y t i c t i r e model. The p re l iminary model i s a n i s o t r o p i c t o r o i d a l membrane w i t h i n n e r edges bonded t o r i g i d bead r i n g s . The t o r o i d i s i n f l a t e d and d e f l e c t e d a g a i n s t a f r i c t i o n l e s s f l a t s u r f a c e . S o l u t i o n s a r e ob ta ined f o r (a ) t h e deformed shape and c o n t a c t boundary; (b) s t r a i n and s t r e s s d i s t r i b u t i o n s , i n c l u d i n g p r i n c i p a l s t r e s s e s ; ( c ) normal (hoop and r a d i a l ) and shear s t r e s s d i s t r i b u t i o n s on t h e bead r i n g ; and (d) t i r e load a s a f u n c t i o n of i n f l a t i o n p r e s s u r e and loaded r a d i u s . D i g i t a l computer programs a r e inc luded .

17. Key Words

T i r e Model Energy Method Toro ida l Membrane Contact S t r a i n s Bead S t r e s s e s

10. Dis t r ibu t ia S to tmmt

UNLIMITED

19. kcvrity Classif. (of this r-9)

UNCLASSIFIED

a. k c u r i t y Clossif. (of this pog.1

UNCLASSIFIED

21-No.ofPoges

2 1 6 22. Price

ANALYTIC TIRE MODEL

Phase I THE STAT1 CALLY LOADED TOROIDAL MEMBRANE

John T . T i e l k i n g

P r o j e c t 360934

T i r e T r a c t i o n C h a r a c t e r i s t i c s A f f e c t i n g V e h i c l e Pe r fo rmance

I n t e r i m Document 10

J u l y 1975

Sponso red by

The Motor V e h i c l e M a n u f a c t u ~ e r s A s s o c i a t i o n

'I'he a n a l y t i c t i r e model development r e p o r t e d i n

t h i s document c o n c l u d e s t h e i n i t i a l phase o f a m u l t i -

y e a r r e s e a r c h p l a n f o r a t t a i n i n g a comprehens ive a n a l y t i c a l

d e s c r i p t i o n o f t h e s t r u c t u r e of t h e pneuma t i c t i r e . The

p r i m a r y o b j e c t i v e of t h e f u l l y d e v e l o p e d a n a l y t i c t i r e

model i s t h e c a p a b i l i t y f o r c a l c u l a t i n g t h e i n f l u e n c e o f

t i r e d e s i g n v a r i a b l e s on t i r e t r a c t i o n p e r f o r m a n c e .

An i s o t r o p i c t o r o i d a l membrane i s t h e i n i t i a l

a n a l y t i c t i r e model d e s c r i b e d h e r e i n . Th is m o d e l , f o r a

s t a t i c a l l y l o a d e d t i r e i n c o n t a c t w i t h a f r i c t i o n l e s s

f l a t s u r f a c e p r o v i d e s t h e c a p a b i l i t y f o r c a l c u l a t i n g

(1 ) t h e c o n t a c t bounda ry ; ( 2 ) t h e s t r e s s d i s t r i b u t i o n i n

t h e t o r o i d a l s t r u c t u r e ; and ( 3 ) t h e s h e a r and normal s t r e s s e s

t a k e n by t h e b e a d . These v a r i a b l e s may be c a l c u l a t e d a s

f u n c t i o n s o f ( 1 ) un loaded r a d i i ; ( 2 ) i n t e r n a l p r e s s u r e ;

and ( 3 ) l o a d e d r a d i u s .

The t h e o r e t i c a l development o f t h e membrane t i r e

model and t h e computer programs n e c e s s a r y f o r i t s

o p e r a t i o n a r e documented i n t h i s r e p o r t , a l o n g w i t h a

computer s u r v e y o f i t s a n a l y t i c c a p a b i l i t i e s . Th i s work

was conduc t ed a s p a r t o f an ongoing t i r e r e s e a r c h p r o j e c t

e n t i t l e d " T i r e T r a c t i o n C h a r a c t e r i s t i c s A f f e c t i n g V e h i c l e

P e r f o r m a n c e , " s p o n s o r e d by t h e Motor V e h i c l e M a n u f a c t u r e r s

A s s o c i a t i o n .

TABLE O F CONTENTS

. . . . . . . . . . . . . . . . . 1 . INTRODUCTION 1

1.1 HSRI I s Resea r ch P l a n . . . . . . . . . . 4

1 . 2 The Scope o f t h e P r e s e n t A n a l y s i s . . . . 8

2 . GENERAL DEFORMATIONS OF MEMBRANE STRUCTURES . . . . . . . . . . . . . . . . . . 1 3

2 . 1 Deformat ion A n a l y s i s . . . . . . . . . . 1 4

2 . 2 M a t e r i a l D e s c r i p t i o n . . . . . . 1 8

. . . . . . . . . . . . . 2 . 3 E x t e r n a l Loads 23

2 .4 S t r e s s A n a l y s i s . . . . . . . . . . . . . 25

. . . . . . . . . 2 . 4 . 1 P h y s i c a l s t r e s s 2 7 2 . 4 . 2 S t r e s s t r a j e c t o r i e s . . . . . . . 30 2 . 4 . 3 P r i n c i p a l s t r e s s e s . . . 33

2 . 5 The V i r t u a l Work S o l u t i o n P r o c e d u r e . . . 36

3 . THE TOROIDAL MEMBRANE . . . . . . . . . . . . 41

3 . 1 F o r m u l a t i o n . . . . . . . . . 41

3 . 2 Approximate S o l u t i o n . . . . . . . 47

3 . 3 C o n t a c t C o n s t r a i n t . . . . 48

4 . C A L C U L A T E D RESULTS . . . . . . . . . . . . . . 51

4 . 1 Double C o n t a c t Load . . . . . . . . 53

4 . 2 A x l e - A p p l i e d Load . . . . 60

4 . 3 D i s c u s s i o n . . . . . . . . . . . . . . . 70

4 . 3 . 1 Double c o n t a c t v s . . . . . . . a x l e - a p p l i e d l o a d i n g 70

. . . . . 4 . 3 . 2 C o n t a c t s t r a i n phenomena 7 2

. . . . . 4 . 3 . 3 Numer ica l c o n s i d e r a t i o n s 75 . . . . 4 . 3 . 4 E x p e r i m e n t a l c o n f i r m a t i o n 82

. . . . . . . . . . . . . . . . . 5 . CONCLUSIONS 89

. . . . . . . . . 5 . 1 T h e o r e t i c a l Founda t i ons 89

5 . 2 R o l l i n g C o n t a c t A n a l y s i s . . . . . . . 9 2

ACKNOWLEDGEMENTS . . . . . . . . . . . . . . 9 5

REFERENCES. . . . . . . . . . . . . . . . . . 9 7

' I 'ABLE OF C0N'I'T:NTS ( C o n t . )

S Y M B O L S A N D T E R M I N O L O G Y . . . . . . . . . . . 1 0 1

A P P E N D I X B - I T E R A T I V E S O L U T I O N T E C H N I Q U E S . , . , 1 0 9

- . . . . . . . . . . A P P E N D I X C C O M P U T E R PROGRAMS 1 2 1

. . . . . . . . . . . . C.l P r i m a r y P r o g r a m s . 1 2 7

C . 1 . 1 P A T C H C . 1 . 2 B N D C.1.3 P T C H C V C . 1.4 T E N S O R

. . . . . . . . . . . C . 2 Secondary P r o g r a m s . 1 9 5

C . 2 . 1 P l o t t i n g p r o g r a m s C . 2 . 2 F i l e r e f e r e n c e

. . . . . . . . . . . . . . C . 3 P r o g r a m S i z e s 2 1 5

1 . 0 INTRODUCTIOK

A s u b s t a n t i a l e f f o r t ha s been d i r e c t e d toward t h e

s i m u l a t i o n o f v e h i c l e dynamic r e s p o n s e t o v a r i o u s d r i v e r

c o n t r o l i n p u t s . C o n s i d e r a b l e s u c c e s s ha s been a c h i e v e d

t h r o u g h v e h i c l e model ing by s i m p l e p h y s i c a l ana logy s u c h

a s a c o l l e c t i o n o f r i g i d m a s s e s , s p r i n g s , and dampers , The

weakes t l i n k i n t h e s i m u l a t i o n o f v e h i c l e dynamic r e s p o n s e

i s now r e c o g n i z e d t o be t h e s i m u l a t i o n o f t i r e t r a c t i o n

f o r c e and moment g e n e r a t i o n i n r e s p o n s e t o v a r i o u s o p e r a t i n g

c o n d i t i o n s such a s b r a k i n g and c o r n e r i n g . There h a s been

no p a r a l l e l c o n c e n t r a t e d e f f o r t t o deve lop a t i r e model

o f s o p h i s t i c a t i o n e q u a l t o t h a t o f c u r r e n t v e h i c l e mode l s .

S p o r a d i c e f f o r t s i n t h e p a s t have r e s u l t e d i n v e r y c r u d e

r e p r e s e n t a t i o n s . Only r e c e n t l y have models been deve loped

which s i m u l a t e t i r e :-esponse t o combined b r a k i n g and

c o r n e r i n g i n p u t s . The p r i m a r y o b j e c t i v e o f t h e HSRI t i r e

t r a c t i o n r e s e a r c h program i s t h e development o f m a t h e m a t i c a l

t i r e models which a r e c a p a b l e o f p r e d i c t i n g t h e s i g n i f i c a n t

t r a c t i o n * pe r fo rmance c h a r a c t e r i s t i c s o f a t i r e from b a s i c

d e s i g n and o p e r a t i n g v a r i a b l e s .

Th i s document r e p o r t s on t h e i n i t i a l p h a s e o f t h e

development o f an a n a l y t i c a l * * model wh ich , i n t h e

*The HSRI d e f i n i t i o n o f t i r e t r a c t i o n i n c l u d e s b o t h l o n g i - t u d i n a l and l a t e r a l t i r e s h e a r f o r c e s a s w e l l a s t h e a l i g n i n g moment g e n e r a t e d a t t h e t i r e - r o a d i n t e r f a c e .

**The t e rm " a n a l y t i c a l " i s a p p l i e d t o t h o s e t i r e models which a t t e m p t t o r e p r e s e n t t h e pneuma t i c t i r e s t r u c t u r e i n a comprehens ive and r e a l i s t i c manner . The t e r m " s e m i - e m p i r i c a l " i s a p p l i e d t o t i r e models which a r e deve loped f o r t h e s o l e p u r p o s e o f r e p r o d u c i n g measured t r a c t i o n d a t a and have o n l y a s l i g h t r e l a t i o n s h i p t o a c t u a l t i r e s t r u c t u r e .

1

c o n t e x t o f a t r a c t i o n r e s e a r c h program, w i l l p r o v i d e t h e

c a p a b i l i t y t o p r e d i c t ( 1 ) t h e a r e a , s h a p e , and l o c a t i o n

o f t h e t i r c - r o a d c o n t a c t r c g i o n r c l a t i v c t o t h e wheel

h u b ; and ( 2 ) t h e d i s t r i b u t i o n o f normal and s h e a r

p r e s s u r e s w i t h i n t h e c o n t a c t r e g i o n a s a f u n c t i o n o f t h e

a p p l i e d f o r c e s a t t h e wheel h u b , t h e d e s i g n o f t h e t i r e ,

and t h e f r i c t i o n a l c h a r a c t e r i s t i c s o f t h e t i r e - r o a d

i n t e r f a c e .

P r i o r t o d e v e l o p i n g an a n a l y t i c model o f t h e r o l l i n g

and s l i p p i n g pneuma t i c t i r e , i t i s f i r s t n e c e s s a r y t o

be a b l e t o c a l c u l a t e t h e d e f o r m a t i o n and s t r e s s of t h e

n o n - r o l l i n g t i r e when l o a d e d a g a i n s t a r i g i d s u r f a c e a t

a s p e c i f i e d p r e s s u r i z a t i o n . Al though t h i s s t r u c t u r a l

a n a l y s i s p rob lem i s e a s y t o d e f i n e , i t i s d i f f i c u l t t o

s o l v e . A much s i m p l e r problem i s t h a t of c a l c u l a t i n g t h e

d e f o r m a t i o n and s t r e s s i n an i n f l a t e d , b u t o t h e r w i s e u n l o a d e d ,

t i r e . I t s h o u l d be n o t e d t h a t t h i s s i m p l e r p rob lem h a s

a l r e a d y been s o l v e d by two d i f f e r e n t a n a l y t i c a l methods

[ I , 2 1 , w i t h t h e r e s u l t s v e r i f i e d by l a b o r a t o r y measu re -

m e n t s . (A t h i r d method, v i z . , f i n i t e e l emen t a n a l y s i s ,

h a s a l s o been used---with l i m i t e d s u c c e s s [ 3 , 4 1 .)

The p r i m a r y c o m p l i c a t i o n s i n t r o d u c e d by t h e l o a d i n g

o f a t i r e a g a i n s t a r i g i d s u r f a c e a r e t h a t n e i t h e r t h e

c o n t a c t p r e s s u r e d i s t r i b u t i o n n o r t h e c o n t a c t boundary

a r e known - a p r i o r i . I n s t e a d o f t h e r o t a t i o n a l symmetry

p o s s e s s e d by t h e t i r e s u b j e c t e d o n l y t o p r e s s u r i z a t i o n ,

t h e deformat ion and s t r e s s in a s t a t i c a l l y l oaded t i r e

i s asymmetr ic . No twi th s t and ing t h e d i f f i c u l t i e s i n v o l v e d ,

C la rk and h i s a s s o c i a t e s sough t an a n a l y t i c a l d e t e r m i n a -

t i o n o f t h i s asymmetr ic s t r e s s and de fo rma t ion ( i n c l u d i n g

t h e normal p r e s s u r e d i s t r i b u t i o n w i t h i n t h e a r e a o f

c o n t a c t ) d u r i n g t h e c o u r s e o f an e x t e n s i v e r e s e a r c h p r o j e c t

conduc ted under t h e a u s p i c e s of f i v e major t i r e companies

[ S ] . S i n c e t h e p o t e n t i a l energy m i n i m i z a t i o n method had

been q u i t e s u c c e s s f u l l y employed t o a n a l y z e t h e symmetr ic

de fo rma t ion and s t r e s s i n t h e p r e s s u r i z e d t i r e [ I ] , C l a rk

and h i s a s s o c i a t e s a p p l i e d t h i s method t o t h e t i r e t h a t was

s u b j e c t e d t o c o n t a c t f l a t t e n i n g . The a t t e m p t f a i l e d , one

of t h e more i m p o r t a n t r e a s o n s b e i n g t h a t t h e computing

t e chno logy p r e v a i l i n g a t t h a t t ime (1963) was unequa l t o

t h e t a s k .

A f t e r n e a r l y a decade of i n a c t i v i t y i n t h e o r e t i c a l

a n a l y s i s o f t i r e c o n t a c t , t h e F l i g h t Dynamics Labo ra to ry

o f t h e U.S. Air Force ha s r e c e n t l y s u p p o r t e d a p r o j e c t

whose o b j e c t i v e i s t h e d e t e r m i n a t i o n o f t h e s t r e s s e s

deve loped w i t h i n t h e l oaded and r o l l i n g a i r c r a f t t i r e .

The f i n i t e e lement method o f a n a l y s i s was employed and ,

as o f t h i s p o i n t i n t i m e , i t appea r s t h a t some d i f f i c u l t y

i s b e i n g e n c o u n t e r e d i n o b t a i n i n g a s o l u t i o n o f t h e " s t a n d i n g

t i r e problem" [ 6 ] . This d i f f i c u l t y d e r i v e s from t h e

immensely c o m p l i c a t e d computer code needed t o implement

t h e f i n i t e e l emen t method f o r t h e s t r e s s a n a l y s i s o f a

m u l t i - p l y t i r e ; c u r r e n t l y a v a i l a b l e h i g h - s p e e d computers

a r e unequa l t o t h e t a s k o f e x e c u t i n g t h e r e s u l t i n g

computer p rogram.

A l e s s a m b i t i o u s a n a l y s i s o f t h e s t a n d i n g t i r e p rob l em

h a s r e c e n t l y been p u b l i s h e d by D e E s k i n a z i , e t a l . [ 7 ] ,

i n which t h e t i r e i s modeled a s an i s o t r o p i c t o r o i d a l

membrane. The f i n i t e e l emen t method was employed t o

app rox ima te t h e membrane s t r u c t u r e b u t t h e number o f

e l e m e n t s was i n s u f f i c i e n t t o d e t e r m i n e t h e shape o f t h e

c o n t a c t r e g i o n o r t o p r o v i d e a d e t a i l e d a n a l y s i s o f t h e

s t r e s s e s w i t h i n t h e c o n t a c t r e g i o n . However, t h e c a l -

cu l a t e d s e c t i o n w i d t h e x p a n s i o n , due t o c o n t a c t d e f o r m a t i o n ,

a g r e e s w i t h l a b o r a t o r y measurement . Th i s f i n d i n g i n d i c a t e s

t h a t t h e f i n i t e e l emen t method i s a p o s s i b l e a l t e r n a t i v e

p r o c e d u r e f o r c o n s t r u c t i n g an a n a l y t i c t i r e mode l .

1.1 H S R I ' S RESEARCH PLAN

C o n s i d e r a t i o n o f t h e s t a t e o f t h e a r t , a s o u t l i n e d

above , l e a d s t o t h e c o n c l u s i o n t h a t t h e ene rgy m i n i m i z a t i o n

method o f f e r s s u b s t a n t i a l a d v a n t a g e s i n c a l c u l a t i n g t h e

d e f o r m a t i o n and s t r e s s o f t h e l o a d e d s t a n d i n g t i r e . F u r t h e r ,

i t i s b e l i e v e d t h a t t h e method i s a p p l i c a b l e t o f o l l o w - o n

a n a l y s e s o f r o l l i n g - c o n t a c t phenomena. A c c o r d i n g l y , HSRI

h a s p roposed a m u l t i - y e a r r e s e a r c h e f f o r t t h a t u t i l i z e s

t h e n u m e r i c a l d e s c r i p t i o n o f t h e symmet r i c i n f l a t e d t i r e ,

4

d e r i v i n g from t h e symmetric de fo rma t ion a n a l y s e s [ I , 21 ,

as t h e p o i n t o f d e p a r t u r e f o r developing t h e a n a l y t i c a l

p rocedures a p p l i c a b l e t o t h e asymmetric de fo rma t ion c a s e .

I t i s e n v i s i o n e d t h a t t h e t o t a l e f f o r t o f g e n e r a t i n g an

a n a l y t i c a l model o f t i r e t r a c t i o n on a r i g i d , t e x t u r e l e s s

s u r f a c e would r e q u i r e , a t minimum, f o u r y e a r s . Three

a n a l y t i c a l t a s k s ( s e e Table 1 ) would be performed i n

s e r i e s , each b u i l d i n g on t h e accomplishments o f t h e

p reced ing a c t i v i t i e s .

TABLE 1

TI RE ANALYSIS SCHEDULE

Task Ana lys i s Time ( y e a r s )

S tand ing T i r e a g a i n s t 2 a r i g i d f r i c t i o n a l s u r f a c e

Steady n o n - s l i p r o l l i n g 1

T r a c t i o n f o r c e g e n e r a t i o n

The s t a n d i n g t i r e c o n t a c t a n a l y s i s (Task 1) was

conceived as i n c l u d i n g t h e fo l lowing s e q u e n t i a l a c t i v i t i e s :

(1) t h e ax isymmetr ic t i r e i n f l a t i o n a n a l y s i s p u b l i s h e d i n

Reference [ I ] would be r e f o r m u l a t e d w i t h complete geomet r i c

n o n l i n e a r i t y , ( 2 ) t h e energy min imiza t ion p r o c e d u r e , as

a p p l i e d t o n o n l i n e a r axisymmetr ic problems i n Reference [ 8 ] ,

would b e u sed t o f i n d t h e deformed c o n f i g u r a t i o n * , (3)

t h e s t a n d i n g t i r e c o n t a c t a n a l y s i s would b e added t o t h e

p r e c e d i n g a n a l y s i s w h e r e i n t h e s o l u t i o n f o r t h e a r b i t r a r y

c o n t a c t boundary i s s o u g h t by t h e " s l a c k v a r i a b l e "

t e c h n i q u e , a s r e c e n t l y a p p l i e d [9] t o an i n f l a t e d membrane

c o n t a c t p rob lem. T h i s l a t t e r t e c h n i q u e , which d e r i v e s

f rom o p t i m i z a t i o n t h e o r y , h a s many advan t ages f o r t h i s

t y p e o f p rob l em, t h e main advan t age b e i n g t h a t a c o m p l e t e l y

a r b i t r a r y c o n t a c t boundary can be found w i t h o u t r e s o r t i n g

t o t h e e x t r e m e l y e x p e n s i v e g r i d s e a r c h i n g t e c h n i q u e s

r e q u i r e d when f i n i t e e l emen t methods a r e employed.

Fo l lowing t h e comple t i on o f t h i s f i r s t t a sk-v iz , t h e

s t a n d i n g c o n t a c t ana ly s i s - a " r o l l i n g c o n t a c t a n a l y s i s " would

be i n o r d e r . Logic and p rudence s u g g e s t t h a t t h i s a n a l y s i s

b e conduc t ed i n two s t a g e s . F i r s t , t h e f r e e - r o l l i n g t i r e

s h o u l d b e a n a l y z e d (Task 2 ) . Given s u c c e s s f u l c o m p l e t i o n

o f t h i s a n a l y s i s , t h e r o l l i n g , s l i p p i n g t i r e c o u l d b e t r e a t e d

(Task 3 ) . Assuming a l e v e l o f e f f o r t comparab le t o t h a t

p r o p o s e d f o r t h e " s t a n d i n g c o n t a c t a n a l y s i s " i t h a s been

e s t i m a t e d t h a t a p p r o x i m a t e l y one y e a r would b e r e q u i r e d t o

conduc t e ach s t a g e o f t h e p r o p o s e d " r o l l i n g c o n t a c t a n a l y s i s . "

- -

"Bending a s w e l l a s membrane ene rgy t e rms w i l l b e r e t a i n e d i n t h i s s o l u t i o n f o r t h e i n f l a t e d shape o f an o r t h o t r o p i c t i r e c a r c a s s . The n o n l i n e a r a n a l y s i s p r o c e d u r e p u b l i s h e d i n [ 8 ] e l i m i n a t e s t h e need f o r o b t a i n i n g a s o l u t i o n by p r e s s u r e i n c r e m e n t a t i o n o f a weakly n o n l i n e a r f o r m u l a t i o n - t h e v e r y t i m e consuming s o l u t i o n p r o c e d u r e u s e d i n [ l , 21 and a l s o i n f i n i t e e l emen t s o l u t i o n s o f t h i s p rob l em.

The e x t e n s i o n of t h e r o l l i n g c o n t a c t a n a l y s i s t o

s t r e s s i n a r o l l i n g and s l i p p i n g t i r e i s l a r g e l y a problem

o f p r o v i d i n g a r e a l i s t i c d e s c r i p t i o n of t h e f r i c t i o n a l

c h a r a c t e r i s t i c s o f t i r e - r o a d c o n t a c t . The n o n l i n e a r

deformat ion and s t r e s s a n a l y s i s of an a r b i t r a r i l y loaded

t i r e c a r c a s s w i l l y i e l d t h e d i s t r i b u t e d s h e a r f o r c e s (due

t o s u r f a c e f r i c t i o n ) as w e l l as t h e normal p r e s s u r e s i n

t h e a n a l y s i s of t h e s l i p p i n g t i r e , t hus r e q u i r i n g t h a t t h e

a n a l y s i s p r o v i d e on ly f o r an a p p r e c i a b l e zone of s l i d i n g

f r i c t i o n a l c o n t a c t . I t i s b e l i e v e d t h a t even a c rude

approximation of s l i d i n g f r i c t i o n w i l l s u f f i c e t o p r o v i d e

a v a l u a b l e e x p l a n a t i o n o f t h e manner i n which t i r e t r a c t i o n

i s i n f l u e n c e d by t h e v a r i a t i o n o f c o n t a c t a r e a and normal

p r e s s u r e a s produced by t i r e o p e r a t i n g v a r i a b l e s such as

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

1 . 2 THE SCOPE OF THE PRESENT ANALYSIS

This document p r e s e n t s t h e f i n d i n g s o b t a i n e d i n a

r e s e a r c h program des igned t o ach ieve some of t h e a n a l y t i c a l

g o a l s of Task 1, a s d i s c u s s e d i n S e c t i o n 1.1. This r e s e a r c h

e f f o r t , which has been termed "Phase I , " has produced

an a n a l y t i c t i r e model t o c a l c u l a t e t h e deformat ion and

s t r e s s i n an i s o t r o p i c t o r o i d a l membrane which i s i n f l a t e d

and loaded a g a i n s t a f r i c t i o n l e s s f l a t s u r f a c e . The

r e s u l t i n g t i r e model p r o v i d e s an a n a l y t i c a l c a p a b i l i t y f o r

c a l c u l a t i n g (1) t h e c o n t a c t boundary , ( 2 ) t h e s t r e s s

d i s t r i b u t i o n i n t h e t o r o i d a l s t r u c t u r e , and (3) t h e s h e a r

and normal s t r e s s e s a t t h e bead . These r e s u l t s may be

c a l c u l a t e d a s f u n c t i o n s of (1) un loaded r a d i i , (2 )

i n t e r n a l p r e s s u r e , and (3 ) t h e l oaded r a d i u s . S i n c e t h e

t i r e i s assumed t o be a membrane s t r u c t u r e ( f l e x u r a l

r i g i d i t y i s i g n o r e d ) , t h e i n t e r f a c i a l p r e s s u r e i n t h e

c o n t a c t r eg ion i s un i form and e q u a l t o t h e i n f l a t i o n

p r e s s u r e .

The p r e s e n t model i s g e o m e t r i c a l l y i d e n t i c a l t o t h e

model adopted by DeEskinaz i , e t a l . [ 7 ] . However, t h e

a n a l y t i c a l t r e a t m e n t employed i n t h i s s t u d y d i f f e r s i n a

number o f i m p o r t a n t r e s p e c t s :

1. The g e o m e t r i c a l n o n l i n e a r i t y i s e x a c t i n t h e p r e s e n t

s t r a i n f o r m u l a t i o n , which i s based on t h e d i f f e r e n c e

o f t h e m e t r i c t e n s o r s o f t h e deformed and undeformed

membrane. Geometr ica l n o n l i n e a r i t y i s approximated

i n [ 7 ] by r e t a i n i n g t h e s e c o n d - o r d e r terms i n t h e

s t r a i n - d i s p l a c e m e n t e q u a t i o n s .

2 . The membrane m a t e r i a l (neo-hookean) o f t h e p r e s e n t

t i r e model i s n o n l i n e a r a s de fo rma t ion i s p e r m i t t e d

on ly w i t h i n t h e c o n s t r a i n t o f i n c o m p r e s s i b i l i t y

( c o n s t a n t volume d e f o r m a t i o n ) . I n r e f e r e n c e 1 7 1 ,

t h e membrane m a t e r i a l i s assumed t o obey a l i n e a r

s t r e s s - s t r a i n r e l a t i o n and i n c o m p r e s s i b i l i t y i s

approximated by s e t t i n g t h e Po i s son r a t i o equa l t o

- 3 5 , which i s t h e average v a l u e co r re spond ing t o

c o n s t a n t volume deformat ion i n t h e s t r a i n range

e x p e c t e d .

3 . The geometry of t h e t o r o i d a l membrane, as modeled i n

t h i s s t u d y , i s d e s c r i b e d by a m e t r i c t e n s o r which i s

a cont inuous f u n c t i o n o f c u r v i l i n e a r c o o r d i n a t e s on

t h e t o r o i d a l midsur face . I n r e f e r e n c e [ 7 ] , t h e

t o r o i d a l midsur face i s approximated by a f i n i t e

number of f l a t t r i a n g u l a r e lements o f v a r i o u s s i z e s .

The above d i f f e r e n c e s ( p a r t i c u l a r l y , # 2 ) p r e c l u d e making a

q u a n t i t a t i v e comparison of c a l c u l a t e d r e s u l t s . However,

q u a l i t a t i v e comparisons s h o u l d , and do , show s i m i l a r t r e n d s .

A g e n e r a l t h e o r y f o r f i n i t e deformat ion of membrane

s t r u c t u r e s i s p r e s e n t e d i n Chapter 2 . The i n i t i a l

geometry o f t h e membrane and t h e e x t e r n a l l o a d s c a r r i e d by

t h e deformed c o n f i g u r a t i o n a r e a r b i t r a r y . A p r o c e d u r e ,

based on t h e p r i n c i p l e o f v i r t u a l work, i s p r e s e n t e d f o r

o b t a i n i n g s o l u t i o n f u n c t i o n s d e s c r i b i n g t h e deformed

c o n f i g u r a t i o n .

In Chapter 3 , t h e g e n e r a l t h e o r y and s o l u t i o n procedure

a r e a p p l i e d t o a p a r t i c u l a r s t r u c t u r e , namely, an open

t o r o i d a l membrane w i t h edges bonded t o r i g i d r i n g s ( r e p r e -

s e n t i n g t h e beads of a t i r e ) . The t o r o i d i s i n f l a t e d and

b r o u g h t i n t o s t a t i c c o n t a c t w i t h a f r i c t i o n a l e s s f l a t

s u r f a c e p e r p e n d i c u l a r t o t h e wheel p l a n e . The " o p e r a t i n g

v a r i a b l e s " a r e i n f l a t i o n p r e s s u r e and loaded r a d i u s .

Chap te r 4 p r e s e n t s r e s u l t s c a l c u l a t e d w i t h t h e

a n a l y t i c t i r e model d e r i v e d i n Chapter 3 . Two t y p e s o f

c o n t a c t l o a d i n g a r e a p p l i e d t o t h e " t i r e . " I t i s found

t h a t t h e s t r u c t u r a l s t r a i n d i s t r i b u t i o n f o r t h e s i n g l e

c o n t a c t , a x l e - a p p l i e d , l o a d i s c o n s i d e r a b l y d i f f e r e n t

from t h e s t r u c t u r a l s t r a i n d i s t r i b u t i o n produced by l o a d i n g

t h e t i r e i n p a r a l l e l c o n t a c t r e g i o n s . The c a l c u l a t e d

r e s u l t s i n c l u d e t h e i n f l u e n c e o f i n f l a t i o n p r e s s u r e and

l o a d e d r a d i u s on t h e v e r t i c a l l o a d , s t r a i n d i s t r i b u t i o n i n

t h e d e f l e c t e d t o r o i d , p r i n c i p a l s t r e s s e s , and s t r e s s

d i s t r i b u t i o n s on t h e bead r i n g .

Readers who a r e main ly i n t e r e s t e d i n t h e u t i l i t y o f

t h e a n a l y t i c t i r e model r e p o r t e d i n t h i s document may

s k i p Chapters 2 and 3 , which r e p o r t on t h e development

o f t h e t h e o r y n e c e s s a r y f o r c o n s t r u c t i o n o f t h e model ,

and p r o c e e d d i r e c t l y t o Chap te r 4 .

Chapter 5 p r e s e n t s c o n c l u s i o n s on t h e adequacy o f

t h e t h e o r e t i c a l f o u n d a t i o n ( S e c t . 5 . 1 ) u n d e r l y i n g t h e

p r e s e n t a n a l y t i c t i r e model and a p r o g n o s i s on t h e e x t e n s i o n

of t h i s model t o i n c l u d e t h e a n a l y s i s o f r o l l i n g c o n t a c t

phenomena ( S e c t . 5 .2 ) .

Appendices A and B r e p o r t on a n c i l l a r y i n v e s t i g a t i o n s

conduc t ed d u r i n g t h e c o u r s e o f t h e model deve lopment .

Appendix C p r e s e n t s l i s t i n g s and documen ta t i on o f t h e FORTRAN

I V computer programs w r i t t e n t o o b t a i n t h e c a l c u l a t e d

r e s u l t s . A copy o f t h e s o u r c e programs may b e o b t a i n e d by

s e n d i n g a t a p e t o t h e a u t h o r o f t h i s r e p o r t .

2 . 0 GENERAL DEFORMATIONS OF MEMBRANE STRUCTURES

Al though t h e r e ha s been c o n s i d e r a b l e p r o g r e s s i n t h e

development o f s o - c a l l e d "phenomenological" t h e o r i e s o f

r u b b e r e l a s t i c i t y [ l o ] , d e f o r m a t i o n and s t r e s s s o l u t i o n s

have been found o n l y f o r c a s e s o f h i g h l y i d e a l i z e d l o a d s

a p p l i e d t o t h e s i m p l e s t s t r u c t u r e s o f r u b b e r - l i k e m a t e r i a l s .

The d i f f i c u l t y i n o b t a i n i n g s o l u t i o n s f o r more complex

s t r u c t u r e s i s due t o t h e ex t reme n o n l i n e a r i t y o f t h e

e q u a t i o n s gove rn ing f i n i t e d e f o r m a t i o n o f a n o n l i n e a r

m a t e r i a l .

Th is c h a p t e r p r e s e n t s a v i r t u a l work f o r m u l a t i o n o f

t h e g e n e r a l t h e o r y f o r l a r g e e l a s t i c d e f o r m a t i o n o f a

membrane s t r u c t u r e . The i n i t i a l c o n f i g u r a t i o n o f t h e

membrane i s d e s c r i b e d by t h e t e n s o r m e t r i c o f t h e unde-

formed m i d s u r f a c e . The p rob l em, e s s e n t i a l l y , i s posed

a s one o f f i n d i n g t h e t e n s o r m e t r i c o f t h e deformed mid-

s u r f a c e . The membrane m a t e r i a l i s r e q u i r e d t o be h y p e r -

e l a s t i c i n t h e s e n s e t h a t t h e c o n s t i t u t i v e p r o p e r t i e s a r e

c o n t a i n e d i n a p o t e n t i a l f u n c t i o n a l known a s t h e s t r a i n

ene rgy d e n s i t y (phenomenolog ica l t h e o r y ) . The e x t e r n a l

l o a d s c a r r i e d by t h e deformed membrane s t r u c t u r e a r e

a r b i t r a r y , e x c e p t f o r t h e r equ i r emen t t h a t t h e y can be d e r i v e d

from a p o t e n t i a l f u n c t i o n a l whose f i r s t v a r i a t i o n i s t h e

v i r t u a l work. These r e q u i r e m e n t s a r e met by most membrane

s t r u c t u r e s found i n e n g i n e e r i n g a p p l i c a t i o n s .

The r e s e a r c h r e p o r t e d i n t h i s document ex tends t h e

f o r m u l a t i o n and s o l u t i o n p rocedure g iven by T i e l k i n g and

Feng [ 8 ] f o r n o n l i n e a r ax isymmetr ic membrane problems t o

problems i n v o l v i n g g e n e r a l f i n i t e deformat ion of an

a r b i t r a r y membrane s t r u c t u r e .

2 . 1 DEFORMATION ANALYSIS

A m a t e r i a l p o i n t Po i n t h e undeformed membrane i s

l o c a t e d by c a r t e s i a n c o o r d i n a t e s X I , X 2 , X 3 which a r e

T -.- r e f e r r e d t o t h e u n i t b a s e v e c t o r s T I , 1 2 , l3 shown i n

F igure 1. A f t e r de fo rma t ion , t h i s m a t e r i a l p o i n t i s a t

p o s i t i o n P which i s l o c a t e d by t h e c o o r d i n a t e s Y1, Y 2 ,

Y3 ( r e f e r r e d t o t h e same b a s e v e c t o r s as t h e c o o r d i n a t e s

of P o ) The a n a l y s i s of a r b i t r a r y s t r u c t u r e s i s s i m p l i f i e d

by t r e a t i n g t h e c a r t e s i a n p o i n t c o o r d i n a t e s as p a r a m e t r i c

f u n c t i o n s of c u r v i l i n e a r c o o r d i n a t e v a r i a b l e s e l , B 2 , e 3 .

Here X r a r e known f u n c t i o n s d e s c r i b i n g t h e undeformed

membrane s t r u c t u r e and Y r a r e unknown f u n c t i o n s d e s c r i b i n g

t h e deformed s t r u c t u r e . The f o r m u l a t i o n and s o l u t i o n

procedure a r e focused on o b t a i n i n g a c c u r a t e approximat i o n s

o f Y r . I n d i c i a 1 n o t a t i o n i s e x t e n s i v e l y used , w i t h L a t i n

i n d i c e s t a k i n g t h e va lues 1 , 2 , 3 . When c o n v e n i e n t , Greek

i n d i c e s t a k i n g on on ly t h e v a l u e s 1 and 2 w i l l be used .

1 4

F i g u r e 1. Base v e c t o r s and c u r v i l i n e a r c o o r d i n a t e s b e f o r e and a f t e r d e f o r m a t i o n . P o , i n t h e undeformed

m a t e r i a l , moves t o P i n t h e deformed m a t e r i a l .

The de fo rma t ion a n a l y s i s i s based on an o r t h o g o n a l

c u r v i l i n e a r c o o r d i n a t e sys tem which i s p o s i t i o n e d i n t h e

undeformed membrane s t r u c t u r e s o t h a t e l and O 2 a r e on

t h e midsu r face and O 3 i s s t r a i g h t and p e r p e n d i c u l a r t o

t h e midsur face . These c o o r d i n a t e s , O r , a r e c o n s i d e r e d t o

be e t c h e d i n t o t h e undeformed membrane and a r e convected

w i t h t h e de fo rma t ion [ l l ] .

Equat ions (1) a r e g iven a s p e c i f i c f u n c t i o n a l form by

t h e (Love-Kirchoff ) h y p o t h e s i s t h a t , a f t e r de fo rma t ion , t h e

c o o r d i n a t e l i n e O 3 i s s t r a i g h t and p e r p e n d i c u l a r t o t h e

deformed m i d s u r f a c e .

I n Equat ions and y r l o c a t e co r re spond ing p o i n t s on

t h e undeformed and deformed m i d s u r f a c e s , r e s p e c t i v e l y ; m r and n r a r e c a r t e s i a n components of u n i t v e c t o r s (ii and ii

i n F i g . 1) normal t o t h e undeformed and deformed m i d s u r f a c e s ,

r e s p e c t i v e l y ; and h3(01, e 2 , e3) i s t h e e x t e n s i o n r a t i o o f

t h e t r a n s v e r s e d imens ion , 0 3 .

Geometr ical d e s c r i p t i o n s o f t h e membrane s t r u c t u r e

b e f o r e and a f t e r deformat ion a r e d e r i v e d from t h e b a s e

v e c t o r s ( i s and Cs i n F ig . 1) o f t h e c u r v i l i n e a r c o o r d i n a t e s

a t t h e midsur face o f t h e undeformed and deformed membrane.

( un de f o rme d) (3 )

(deformed)

Measures o f s t r u c t u r a l geometry, b e f o r e and a f t e r d e f o r -

ma t ion , a r e con ta ined i n t h e m e t r i c t e n s o r s w i t h c o v a r i a n t

conlponents g and G i d c r i v e d from t h e hasc v e c t o r s (3 ) i j

and ( 4 ) .

I n Equat ions ( 5 ) and ( 6 ) , and e lsewhere i n t h i s r e p o r t ,

summation i s impl i ed by t h e r e p e a t e d i n d e x ; e . g . , Equat ion

(5) expands t o

General deformat ions o f t h e s t r u c t u r e w i l l be measured

by t h e d i f f e r e n c e of t h e two m e t r i c t e n s o r s whose components

a r e g iven by Equat ions (5) and ( 6 ) .

Equat ion ( 7 ) d e f i n e s t h e c o v a r i a n t components of t h e

Green-Sain t Venant s t r a i n t e n s o r . I t i s assumed t h a t

t h e deformat ions measured by t h e s t r a i n t e n s o r ( 7 ) a r e

uniform i n t h e e j - d i r e c t i o n . This assumpt ion , which i s

r easonab le f o r membranes, i m p l i e s t h a t t h e m e t r i c t e n s o r s

( 5 ) and ( 6 ) a r e de termined by t h e g r a d i e n t s x and r ,a

y r , a of t h e midsurface p o s i t i o n f u n c t i o n s and t h e e x t e n -

s i o n r a t i o , l 3 9 which i s now independent of 0 3 . The r

n o t a t i o n s x - - - and y = a r e used t o i n d i c a t e r , a a e a r , a a e a

g r a d i e n t s .

The m a t r i x (8) i s d i a g o n a l because t h e c u r v i l i n e a r c o o r -

d i n a t e s , O r , a r e assumed t o be o r t h o g o n a l b e f o r e

de fo rma t ion . The ze ro e lements of t h e m a t r i x ( 9 ) d e r i v e

from t h e Love-Kirchoff h y p o t h e s i s . The assumption t h a t

t h e deformat ion i s uniform i n t h e 0 3 - d i r e c t i o n a s s u r e s

t h a t t h e deformed c u r v i l i n e a r c o o r d i n a t e s O a w i l l l i e on

t h e midsur face o f t h e deformed membrane s t r u c t u r e .

2 . 2 MATERIAL DESCRIPTION

The membrane m a t e r i a l i s assumed t o be h y p e r e l a s t i c

s o t h a t i t s c o n s t i t u t i v e p r o p e r t i e s can be d e s c r i b e d by

a s t r a i n e n e r g y , o r s t o r e d e n e r g y , d e n s i t y f u n c t i o n . The

s t r a i n energy d e n s i t y f u n c t i o n i s commonly used f o r t h e

Formulat ion of a phenomenological t h e o r y o f rubber

e l a s t i c i t y ; i t r e p r e s e n t s t h e d e n s i t y of t h e change i n

Helmholtz f r e e energy of t h e m a t e r i a l upon deformat ion

which i s i s o t h e r m a l and r e v e r s i b l e . As w i l l be s e e n , t h e

s t r a i n energy d e n s i t y f u n c t i o n i s d i r e c t l y employed i n t h e

s o l u t i o n p rocedure f o r c a l c u l a t i n g g e n e r a l d e f o r m a t i o n ,

s t r a i n , and s t r e s s i n an a r b i t r a r i l y loaded membrane

s t r u c t u r e .

For i s o t r o p i c m a t e r i a l s , i t can be shown t h a t t h e s t r a i n

energy d e n s i t y i s a f u n c t i o n o f t h e t h r e e i n v a r i a n t s of

t h e s t r a i n t e n s o r .

where W denotes t h e energy d e n s i t y p e r u n i t volume o f

u n s t r a i n e d (undeformed) m a t e r i a l and I r a r e t h e s t r a i n

i i n v a r i a n t s c a l c u l a t e d from t h e mixed components, y j , of

t h e Green-Sa in t Venant s t r a i n t e n s o r .

i j where: g and ~~j a r e t h e c o n t r a v a r i a n t components o f

t h e m e t r i c t e n s o r s whose c o v a r i a n t components a r e g iven by

( 8 ) and (9) ; g and G a r e t h e d e t e r m i n a n t s o f t h e c o v a r i a n t

m a t r i x e x p r e s s i o n s ( 8 ) and ( 9 ) .

The t h e o r y d e s c r i b e d i n t h i s r e p o r t w i l l be r e s t r i c t e d

t o incompress ib le i s o t r o p i c membrane m a t e r i a l . The assump-

t i o n of i n c o m p r e s s i b i l i t y p e r m i t s an e x p r e s s i o n f o r A t o 3

be d e r i v e d by e q u a t i n g undeformed and deformed membrane

volume e l emen t s . The volume e lements a r e computed from

t h e b a s e v e c t o r s ga and ca a t a p o i n t on t h e midsur face

where t h e undeformed t h i c k n e s s i s h .

where a = fi w i t h r r

and b = w i t h

S i n c e t h e volume e l emen t e q u a t i o n may a l s o b e w r i t t e n a s

i t f o l l o w s t h a t , f o r an i n c o m p r e s s i b l e m a t e r i a l , t h e

d e t e r m i n a n t s g and G a r e e q u a l and t h e t h i r d s t r a i n

i n v a r i a n t i n E q u a t i o n s (11) i s e q u a l t o u n i t y .

I 3 = 1 ( i n c o r q p r e s s i b l e m a t e r i a l ) (15)

The s t o r e d - e n e r g y f u n c t i o n p roposed by Mooney [12]

i s an example o f a phenomenolog ica l e l a s t i c i t y t h e o r y f o r

an i n c o m p r e s s i b l e i s o t r o p i c m a t e r i a l . C o n s t i t u t i v e p r o -

p e r t i e s o f t h e s o - c a l l e d "Mooney m a t e r i a l " a r e d e s c r i b e d

by t h e ene rgy d e n s i t y

where C1 and C a r e m a t e r i a l c o n s t a n t s w i t h t h e dimension 2

of s t r e s s and a = C2/C1 The i n v a r i a n t s I1 and I 2 a r e

computed by i n t r o d u c i n g t h e m e t r i c t e n s o r s ( 8 ) and (9 ) i n t o

t h e s t r a i n i n v a r i a n t e q u a t i o n s (11) .

Equa t ions (17) and (18) may be combined t o o b t a i n a compact

e x p r e s s i o n f o r 1 2 .

The s t r a i n i n v a r i a n t s (17) and (18 o r 1 9 ) a r e e v a l u a t e d w i t h

t h e e x t e n s i o n r a t i o h 3 , which i s c a l c u l a t e d from ( 1 2 , 1 3 ,

S i n c e t h e g r a d i e n t s x a r e known f u n c t i o n s , Equa t ions r , a

(17-20) show t h e s t r a i n ene rgy d e n s i t y (16) t o depend o n l y

on t h e g r a d i e n t s o f t h e f u n c t i o n s y r ( 0 1 , 0 2 ) , which d e s c r i b e

t h e c o n f i g u r a t i o n o f t h e deformed membrane m i d s u r f a c e .

The e x t e r n a l l oads c a r r i e d by t h e deformed membrane

a r e h e r e s u b j e c t t o t h e requi rement t h a t t hey can be d e r i v e d

from a s c a l a r work f u n c t i o n a l * whose f i r s t v a r i a t i o n i s t h e

v i r t u a l work, 6V. I t may be d i f f i c u l t , however, t o

e x p l i c i t l y w r i t e such a f u n c t i o n a l if t h e e x t e r n a l l o a d s a r e

compl i ca ted . The v i r t u a l work s o l u t i o n p r o c e d u r e , d e s c r i b e d

i n S e c t i o n 2 . 5 , r e q u i r e s only t h e e x p r e s s i o n of t h e v i r t u a l

work; t h e work f u n c t i o n a l i t s e l f need n o t be w r i t t e n . The

v i r t u a l work done by a s p e c i f i c s e t o f e x t e r n a l l o a d s a c t i n g

on t h e deformed membrane w i l l i n t r o d u c e t h e s p e c i f i e d l o a d s

i n t o t h e f o r m u l a t i o n of t h e g e n e r a l deformat ion t h e o r y .

The g e n e r a l membrane deformat ion t h e o r y d e s c r i b e d i n

t h i s c h a p t e r was developed as a p r e l i m i n a r y t h e o r y n e c e s s a r y

f o r t h e c a l c u l a t i o n o f t h e deformat ion and s t r e s s i n a

pneumatic t i r e s u b j e c t e d t o d i s t r i b u t e d s u r f a c e l o a d s :

i n t e r n a l p r e s s u r e and d i s t r i b u t e d s h e a r f o r c e s produced by

t i r e - r o a d f r i c t i o n . For t h i s o b j e c t i v e , t h e deformed mem-

b rane s t r u c t u r e i s assumed t o be i n e q u i l i b r i u m w i t h t h e

f o l l o w i n g d i s t r i b u t e d loads : t a n g e n t i a l p r e s s u r e s , p a ,

a c t i n g i n t h e 8 , - d i r e c t i o n s on t h e deformed midsur face

and an i n f l a t i o n p r e s s u r e , p , a c t i n g normal t o t h e deformed

midsur face ( 0 3 - d i r e c t i o n ) . These e x t e r n a l l o a d s have t h e

dimension of f o r c e p e r u n i t of deformed midsurface a r e a .

*Lanczos [13] a p p l i e s t h e term "monogenic" t o t h i s ca t egory o f f o r c e s ( l o a d s ) . Forces-such as s l i d i n g f r i c t ion - -+h i ch cannot be d e r i v e d from a s c a l a r f u n c t i o n a l a r e termed "po lygen ic . "

The e x t e r n a l l o a d v e c t o r T i s g iven by

and t h e r e s u l t a n t f o r c e d i f f e r e n t i a l , a t a p o i n t on t h e

deformed m i d s u r f a c e , i s

where

i s t h e a r e a e lement a t t h e p o i n t on t h e deformed m i d s u r f a c e .

The d i f f e r e n t i a l of t h e v i r t u a l work a t t h i s p o i n t i s

where

i s t h e v e c t o r o f v i r t u a l d i sp lacemen t o f t h e p o i n t on t h e

deformed m i d s u r f a c e . The f o r c e d i f f e r e n t i a l , dE, does n o t

vary dur ing t h e v i r t u a l d i sp lacemen t .

The t o t a l v i r t u a l work i s o b t a i n e d by s u b s t i t u t i n g

Equat ion (23) i n t o Equat ion ( 2 5 ) and i n t e g r a t i n g over t h e

a r e a o f t h e deformed m i d s u r f a c e .

The s u b s t i t u t i o n o f E q u a t i o n s ( 2 2 ) , ( 2 4 ) , and (26) i n t o

E q u a t i o n (27) r e s u l t s i n t h e f o l l o w i n g e x p r e s s i o n f o r t h e

v i r t u a l work

which i s s u i t a b l e f o r u s e w i t h t h e v i r t u a l work s o l u t i o n

p r o c e d u r e d e s c r i b e d i n S e c t i o n 2 $ 5 . I t s h o u l d b e n o t e d

t h a t t h e t a n g e n t i a l p r e s s u r e s , p a , a r e h e r e assumed t o b e

monogenic* ( a s w e l l a s t h e normal p r e s s u r e , p ) and known

a p r i o r i . -

2 . 4 STRESS ANALYSIS

The a s sumpt ion t h a t t h e d e f o r m a t i o n ( s t r a i n ) i s u n i f o r m

a c r o s s t h e membrane t h i c k n e s s ( 0 3 - d i r e c t i o n ) i m p l i e s t h a t

t h e s t r a i n e n e r g y d e n s i t y , W , w i l l a l s o b e u n i f o r m a c r o s s

t h e t h i c k n e s s . The s t r e s s t e n s o r which i s d e r i v e d from t h e

s t r a i n ene rgy d e n s i t y w i l l t h e n b e u n i f o r m i n t h e 0 3 - d i r e c t i o n

and i t i s c o n v e n i e n t t o a n a l y z e t h e s t r e s s i n t e rms o f

s t r e s s r e s u l t a n t s , which a r e f o r c e s p e r u n i t m i d s u r f a c e l e n g t h

(undeformed o r deformed l e n g t h ) i n t h e and B 2 d i r e c t i o n s .

" t a t i c f r i c t i o n f o r c e s a r e monogenic. An e s t i m a t e o f t h e s t a t i c f r i c t i o n f o r c e s i n t h e c o n t a c t r e g i o n o f a s t a n d i n g t i r e may be b a s e d on knowledge o f t h e i n t e r f a c i a l s t r a i n i n t h e f r i c t i o n l e s s c o n t a c t r e g i o n .

The c o n t r a v a r i a n t components, T ' ~ , o f t h e Lagrange

s t r e s s r e s u l t a n t t e n s o r a r e d e r i v e d d i r e c t l y from t h e s t r a i n

energy d e n s i t y (21) .

The components T"' r e c o r d t h e p r o d u c t o f t h e components o f

t h e deformed base v e c t o r s , GB = - w i t h t h e f o r c e p e r Yr,g1r,

u n i t l e n g t h of t h e undeformed midsur face l i n e s p e r p e n d i c u l a r

t o O B ; t h e r e s u l t a n t f o r c e v e c t o r components a r e r e f e r r e d t o

t h e c a r t e s i a n u n i t base v e c t o r s Tr . As w i l l be seen i n t h e

next s e c t i o n , t h e Lagrange s t r e s s t e n s o r f a c i l i t a t e s w r i t i n g

t h e e x p r e s s i o n s n e c e s s a r y f o r t h e v i r t u a l work s o l u t i o n

p rocedure . However, i t i s n o t a conven ien t t e n s o r f o r

c a l c u l a t i n g t h e p h y s i c a l s t r e s s i n t h e deformed membrane

s t r u c t u r e .

The s t r e s s r e s u l t a n t components which a r e r e f e r r e d t o

t h e b a s e v e c t o r s C a r e c o n t a i n e d i n t h e Ki rchof f s u r f a c e 13

s t r e s s t e n s o r T", which may be c a l c u l a t e d from t h e Lagrange

t e n s o r , T a r , by s o l u t i o n of t h e t r a n s f o r m a t i o n e q u a t i o n .

The s o l u t i o n f o r i n d i c e s r = a i s

a where b3 = 131 = i s t h e J a c o b i a n de te rminan t

which appeared i n Equat ions ( 1 4 ) .

The t e n s o r components TaB r e c o r d f o r c e p e r u n i t l e n g t h

of undeformed m i d s u r f a c e , w i t h t h e f o r c e v e c t o r components

r e f e r r e d t o t h e base v e c t o r s on t h e deformed m i d s u r f a c e . B The Ki rchof f t e n s o r i s symmetr ic .

2 . 4 . 1 PHYSICAL STRESS. The p h y s i c a l t r u e s t r e s s

components , S a ~

, a r e computed from t h e c o n t r a v a r i a n t

K i rchof f t e n s o r components, T U B , accord ing t o

where G B B and G~~ a r e

[no sum on a and B )

c o v a r i a n t and cont r a v a r i a n t d i agona l

e lements o f t h e m e t r i c t e n s o r o f t h e deformed midsur face .

The components S g ive t h e t r u e s t r e s s ( a s f o r c e p e r u n i t aa

l e n g t h o f deformed midsurface) i n t h e O 1 and d i r e c t i o n s

on t h e deformed midsur face .

The SUB a r e t h e components of t h e s t r e s s r e s u l t a n t -

v e c t o r , La, r e f e r r e d t o t h e o b l i q u e v e c t o r s C 8 '

The fa g i v e t h e f o r c e p e r u n i t o f deformed l e n g t h which

a c t s a long t h e l i n e s d e f i n e d by B U = c o n s t a n t . The s t r e s s

r e s u l t a n t v e c t o r i s shown i n F igure 2 , where c1 i s normal

t o t h e l i n e e l = c o n s t a n t .

F igure 2 . S t r e s s r e s u l t a n t v e c t o r a c t i n g i n t h e deformed

midsur face ( a long a l i n e d e f i n e d by e l = c o n s t a n t ) .

The Sag a r e n o t t e n s o r components because i s n o t a C1

f u l l y i n v a r i a n t v e c t o r .

From t h e components of t h e m e t r i c t e n s o r o f t h e deformed

m i d s u r f a c e o f an i n c o m p r e s s i b l e m a t e r i a l ,

i t i s s e e n t h a t t h e p h y s i c a l s t r e s s components a r e symmet r ic

and g i v e n by

where a = A3b, from Equa t ion ( 1 2 ) .

2 . 4 . 2 STRESS TRAJECTORIES. The s t r e s s t r a j e c t o r i e s

f o l l o w t h e p r i n c i p a l axes o f t h e s t r e s s t e n s o r and i n d i c a t e

t h e d i r e c t i o n s o f t h e p r i n c i p a l s t r e s s e s . The s t r e s s

t r a j e c t o r y p l o t g i v e s a v i s u a l p i c t u r e o f how an asymmetr ic

l o a d i s c a r r i e d by a deformed s t r u c t u r e .

Le t

denote t h e a r b i t r a r y l i n e e lement p a s s i n g through an

a r b i t r a r y p o i n t i n t h e deformed m i d s u r f a c e . The l i n e

e lement i s o r i e n t e d i n t h e midsu r face by t h e u n i t normal -

v e c t o r , n , which has t h e c o v a r i a n t components na r e f e r r e d

t o t h e deformed base v e c t o r s ? a s shown i n F igu re 3 . The -

s t r e s s r e s u l t a n t v e c t o r , 1, on t h e l i n e eleitlent k t t h i s p o i n t i s

- Figure 3 . S t r e s s r e s u l t a n t v e c t o r , 1, a c t i n g on an

a r b i t r a r y l i n e e lement i n t h e deformed m i d s u r f a c e .

where z a B i s o b t a i n e d from t h e K i r c h o f f s t r e s s r e s u l t a n t

aB t e n s o r , T .

p B = h3TaB

The d i f f e r e n t i a l f o r c e v e c t o r , dF , i s

d ~ = T ds = zaBna cB ds

When t h e a r b i t r a r y l i n e e l e m e n t , d s , i s p e r p e n d i c u l a r

t o a p r i n c i p a l d i r e c t i o n , t h e f o r c e v e c t o r , dF, i s c o l i n e a r

w i t h ii and t h e r e i s no t a n g e n t i a l component o f f o r c e on

t h e l i n e e l e m e n t . I n t h i s o r i e n t a t i o n , t h e f o r c e

d i f f e r e n t i a l i s d i r e c t l y p r o p o r t i o n a l t o t h e l i n e e l emen t

v e c t o r .

where k i s t h e c o n s t a n t o f p r o p o r t i o n a l i t y . E q u a t i o n s ( 3 6 )

and (39) a r e s u b s t i t u t e d i n t o E q u a t i o n (40).

The c o n t r a v a r i a n t components , n B , of t h e u n i t normal a r e

o b t a i n e d from t h e cova r i a n t components , n u , by u s e o f t h e

m e t r i c t c n s o r ( ;Bu ( = G u B ) ,

The t r a n s f o r m a t i o n (42) and E q u a t i o n ( 3 8 ) a r e s u b s t i t u t e d

i n t o E q u a t i o n ( 4 1 ) . The r e s u l t i n g e q u a t i o n i s r e a r r a n g e d

i n t h e form o f a g e n e r a l i z e d e i g e n v a l u e p rob l em, w i t h na

d e f i n i n g an e i g e n v e c t o r which i s o r i e n t e d i n a p r i n c i p a l

d i r e c t i o n .

N o n t r i v i a l s o l u t i o n s f o r nu e x i s t when

S i n c e T ~ ' and G u B a r e r e a l symmet r ic s e c o n d - o r d e r m a t r i c e s ,

t h e r e w i l l b e two r e a l r o o t s , k Y ' of Equa t ion (44) e v a l u a t e d

a t e ach p o i n t on t h e deformed m i d s u r f a c e . A p r i n c i p a l

d i r e c t i o n , d e f i n e d by nLy) , i s found f o r e ach r o o t . S i n c e

t h e two p r i n c i p a l d i r e c t i o n s , and t h u s t h e s t r e s s t r a j e c -

t o r i e s , a r e o r t h o g o n a l , o n l y one r o o t , k , and c o r r e s p o n d i n g

p r i n c i p a l d i r e c t i o n , n u , need b e c a l c u l a t e d .

n LO TI- V

CZ1 n

k h a c

CZ1 a u

I 1

k d

Let 0 d e n o t e a s e t o f o r t h o g ~ i n a l c u r v i l i n e a r ct

c o o r d i n a t e s whi ch c o i n c i d c w i t h t h c s t r e s s t r a j e c t o r i e s .

Covar ian t and c o n t r a v a r i a n t base v e c t o r s , R and A', a r e a

a s s o c i a t e d w i t h t h e Oa c o o r d i n a t e s and a r e r e l a t e d accord ing

t o

I n t h e Oa c o o r d i n a t e sys tem, Equat ion (43) reduces

where nu = 1 . The s u p e r s c r i p t c i n d i c a t e s t h a t t h e

s t r e s s t e n s o r components a r e r e f e r r e d t o t h e Oa c o o r d i n a t e s .

S ince H~' = 0 i f a f B , i t fo l lows from Equat ion ( 4 7 ) t h a t

The e i g e n v a l u e s , k Y '

may now be immediately d e r i v e d from

Equat ion ( 4 7 ) .

The e lements o f t h e d i a g o n a l i z e d Ki rchof f s t r e s s t e n s o r

a r e t h u s computed accord ing t o

wherc k and HYY a r e o b t a i n e d w i t h t h e s o l u t i o n o f t h e Y

c YY e i g e n v a l u e problem d e f i n e d by Equat ion ( 4 3 ) . The T

a r e t h e c o n t r a v a r i a n t e lements o f t h e c a n o n i c a l form o f t h e

Ki rchof f s t r e s s t e n s o r .

The s t r e s s v e c t o r i n a p r i n c i p a l d i r e c t i o n , iy, i s

where n = 1 = . The p h y s i c a l s t r e s s com- Y

ponents , SY ' a r e d e f i n e d s o t h a t t h e p r i n c i p a l d i r e c t i o n

s t r e s s v e c t o r may be r e f e r r e d t o u n i t v e c t o r s .

where

With t h e s t r e s s t e n s o r components c a l c u l a t e d according t o

Equat ion ( 4 9 ) , Equat ion ( 5 2 ) reduces t o

The p h y s i c a l v a l u e s , SY

, of t h e p r i n c i p a l s t r e s s e s ( t r u e

s t r e s s ) a r e found t o be e x a c t l y equa l t o t h e e i g e n v a l u e s ,

k Y , d e r i v e d from Equa t ion ( 4 3 ) . Example p l o t s o f t h e

p r i n c i p a l s t r e s s e s a r e shown in 1:igurc 14 o f Chap te r 4 .

2 . 5 THE VIRTUAL WORK SOLUTION P R O C E D U R E

The s o l u t i o n o f s p e c i f i c problems i n v o l v i n g g e n e r a l

d e f o r m a t i o n s o f membrane s t r u c t u r e s may be o b t a i n e d by

d i r e c t a p p l i c a t i o n o f t h e P r i n c i p l e o f V i r t u a l Work [ 1 3 ] .

The v i r t u a l work p r i n c i p l e i s p a r t i c u l a r l y v a l u a b l e f o r

problems i n which t h e deformed membrane s t r u c t u r e i s con-

s t r a i n e d by c o n t a c t o v e r a f i n i t e s u r f a c e a r e a . I f t h e

c o n t a c t i s f r i c t i o n a l , i t may b e d i f f i c u l t , i f n o t

i m p o s s i b l e ( a s i n t h e c a s e o f s l i d i n g f r i c t i o n ) , t o w r i t e

a work f u n c t i o n a l from which t h e v i r t u a l work, 6V, can be

d e r i v e d . The absence o f a work f u n c t i o n a l p r e c l u d e s a p p l i -

c a t i o n o f t h e Minimum P o t e n t i a l Energy P r i n c i p l e , employed

f o r ax i symmet r i c problems by T i e l k i n g and Feng [ 8 ] . For

c o n s e r v a t i v e f o r c e s , t h e v i r t u a l work s o l u t i o n p r o c e d u r e

p r e s e n t e d i n t h i s s e c t i o n i s e q u i v a l e n t t o t h e R i t z Method

f o r o b t a i n i n g a minimum ene rgy s o l u t i o n .

For s t a t i c problems i n f i n i t e e l a s t i c i t y , t h e v i r t u a l

work p r i n c i p l e r educes t o t h e p o s t u l a t e t h a t , i n t h e

deformed c o n f i g u r a t i o n , t h e v i r t u a l work o f e x t e r n a l l o a d s

i s e q u a l t o t h e f i r s t v a r i a t i o n o f t h e s t r a i n ene rgy s t o r e d

i n t h e deformed s t r u c t u r e .

where b U i s t h e v a r i a t i o n o f t h e t o t a l s t r a i n energy

which i s d e r i v e d i n t h e f o l l o w i n g way.

The t o t a l s t r a i n ene rgy , U , i s o b t a i n e d by i n t e g r a t i n g

t h e s t o r e d - e n e r g y d e n s i t y (21) ove r t h e undeformed volume, v ,

o f t h e membrane m a t e r i a l .

where dv = h igl x g2 1 de1de2 = h a d 0 l d 8 ~ i s an element of

t h e undeformed volume. The v a r i a t i o n o f t h e s t r a i n energy

With t h e Lagrange s t r e s s r e s u l t a n t s , TO"*, d e f i n e d by

Equat ion ( 2 9 ) ,

Equa t ions ( 2 8 ) and ( 5 7 ) a r e now s u b s t i t u t e d i n t o t h e

v i r t u a l work p o s t u l a t e ( 5 4 ) . The p o s t u l a t e i s r e a r r a n g e d

t o o b t a i n t h e fundamental e q u a t i o n o f t h e v i r t u a l work

s o l u t i o n p rocedure .

The l e f t s i d e of Equat ion ( 5 8 ) i s t h e f i r s t v a r i a t i o n o f

t h e s o - c a l l e d " p o t e n t i a l energy" f u n c t i o n a l which i s made

s t a t i o n a r y by t h e s o l u t i o n f u n c t i o n s y ( e 1 , e 2 ) . The f i r s t r v a r i a t i o n o f t h e p o t e n t i a l energy has been d e r i v e d h e r e by

a p p l i c a t i o n o f t h e v i r t u a l work p r i n c i p l e ; t h e p o t e n t i a l

energy f u n c t i o n a l i t s e l f may n o t e x i s t f o r c e r t a i n p rob lems ,

and may be d i f f i c u l t t o w r i t e f o r o t h e r problems; i t i s n o t

needed f o r t h e v i r t u a l work s o l u t i o n p r o c e d u r e .

I t i s n o t e d t h a t t h e l e f t s i d e o f Equa t ion (58) i s a

v a r i a t i o n a l f u n c t i o n a l , 6 i l , whose independen t f u n c t i o n s

d e s c r i b e t h e deformed midsu r face c o n f i g u r a t i o n , g r a d i e n t s ,

and v i r t u a l work v a r i a t i o n s .

The s o l u t i o n f u n c t i o n s , y r , which s a t i s f y Equa t ion ( 5 9 )

w i l l be approximated by double f i n i t e s e r i e s o f g e o m e t r i c a l l y

a d m i s s i b l e ( s a t i s f y geomet r i c boundary c o n d i t i o n s ) c o o r d i n a t e

f u n c t i o n s Y i j ( e l ,BZ) w i t h unknown c o e f f i c i e n t s c f j . The

g r a d i e n t s and v a r i a t i o n s a r e s i m i i a r l y approximated ; t h e

v a r i a t i o n s e r i e s have t h e c o e f f i c i e n t v a r i a t i o n s 6cr i j '

The second and f o u r t h o f Equat ions (60) a r e s u b s t i t u t e d f o r

t h e v a r i a t i o n s , 6 y r and 6yr , a , i n Equat ion ( 5 8 ) . The

r c o e f f i c i e n t v a r i a t i o n s , 6 C i j , a r e a r b i t r a r y c o n s t a n t s and

may be f a c t o r e d ou t o f t h e i n t e g r a t i o n .

S ince t h e 6 c r j a r e independent v a r i a t i o n s , t h e i n t e g r a l i n

Equat ion (61) must be ze ro f o r each i and j index v a l u e .

'The f i r s t and t h i r d o f E q u a t i o n s (60) a r e u sed t o a p p r o x i -

i n I i qua t i ons ( 6 2 ) s ; l t c t l lc f a c t o r s ( T ~ ~ , h , y r ,L t , (; ,,,,, i j and y i J w i t h unknown by summations o f known f u n c t i o n s , y r

r ,a

c o e f f i c i e n t s , cTj . The i n t e g r a t i o n o f e ach o f E q u a t i o n s

( 6 2 ) may now b e pe r fo rmed n u m e r i c a l l y . * E q u a t i o n s ( 6 2 ) a r e

t h e n r e d u c e d t o 3 x M x N n o n l i n e a r a l g e b r a i c e q u a t i o n s .

The s o l u t i o n o f E q u a t i o n s (63) d e t e r m i n e s t h e c o e f f i c i e n t s

i n acco rdance w i t h t h e v i r t u a l work p o s t u l a t e ( 5 4 ) . The ' i j

s o l u t i o n i s r e a d i l y a c h i e v e d by e x i s t i n g r o o t f i n d i n g

t e c h n i q u e s , s u c h a s t h e Newton-Raphson i t e r a t i o n p r o c e d u r e

d e s c r i b e d i n Appendix B .

*S impson t s r u l e i s u s e d f o r t h e a n a l y t i c t i r e model d e r i v e d i n Chap te r 3 .

3 . 0 'I'till TOROIDAL MEMBRANE

The g e n e r a l d e f o r m a t i o n t h e o r y and s o l u t i o n p rocedure

d e s c r i b e d i n t h e p r e c e d i n g c h a p t e r w i l l be a p p l i e d t o

o b t a i n a s o l u t i o n f o r t h e de fo rma t ion and s t r e s s i n an

i n f l a t e d t o r o i d a l membrane, w i t h t h e i n f l a t e d t o r o i d

c o n s t r a i n e d by c o n t a c t w i t h a f r i c t i o n l e s s f l a t s u r f a c e .

3 . 1 FORMULAT ION

The geometry and c o o r d i n a t e s of t h e undeformed t o r o i d a l

midsu r face a r e shown i n F igu re 4 . The m e r i d i a n i s c i r c u l a r ,

w i t h r a d i u s R1, and t e r m i n a t e s on bead r i n g s l o c a t e d a t

B1 = + " (measured from t h e c e n t e r o f t h e m e r i d i a n ) . The

mer id i an i s c e n t e r e d a t t h e t o r o i d r a d i u s , R t .

P o i n t s on t h e undeformed midsu r face a r e l o c a t e d by

c u r v i l i n e a r c o o r d i n a t e s e l and O 2 which a r e a n g l e s measured

i n t h e m e r i d i o n a l and c i r c u m f e r e n t i a l d i r e c t i o n s , r e s p e c t i v e l y .

The same p o i n t s a r e a l s o l o c a t e d by c a r t e s i a n c o o r d i n a t e s x r which w i l l b e w r i t t e n as p a r a m e t r i c f u n c t i o n s o f O n . With

B a o r i g i n a t i n g a t t h e x 3 - a x i s , a s shown i n F i g u r e 4 , t h e

c a r t e s i a n c o o r d i n a t e s of p o i n t Po a r e

xl = Ro(B1) s i n O 2

x 2 = -R1 s i n 0 1

F i g u r e 4 . Geometry o f t h e undeformed t o r o i d a l m i d s u r f a c e .

where Ro(B1) = Rt + R1 cos 0 i s t h e r a d i u s o f t h e c i r - 1

cumference , o r l a t i t u d e , c i r c l e l o c a t e d b y t h e c o o r d i n a t e

, The m e r i d i a n r a d i u s , R1, and t h e t o r o i d r a d i u s , R t '

a r e c o n s t a n t s i n t h i s example.

With t h e undeformed midsu r face d e f i n e d by Equa t ions

( 6 4 ) , t h e m e t r i c t e n s o r o f t h e undeformed m i d s u r f a c e ,

Equa t ion ( 8 ) , ha s c o v a r i a n t components

where g i s t h e d e t e r m i n a n t o f ( 6 5 ) .

I t i s assumed t h a t t h e membrane m a t e r i a l i s d e s c r i b e d

by t h e ~ e o - h o o k e a n form o f t h e s t r a i n energy d e n s i t y f o r

an i s o t r o p i c m a t e r i a l ,

where I1 (g iven i n g e n e r a l b y Equa t ion ( 1 7 ) ) f o r t h e t o r o i d

i s

The e x t e n s i o n r a t i o A 3 i s g iven by Equa t ion (20) which was

d e r i v e d on t h e a d d i t i o n a l assumpt ion o f an i n c o m p r e s s i b l e

m a t e r i a l . For t h e t o r o i d ,

The s t r a i n energy d e n s i t y , g iven by Equa t ion (67) w i t h

Equa t ions ( 6 8 - 6 9 ) , i s now s e e n as an e x p l i c i t f u n c t i o n o f

t h e g r a d i e n t s , y r , a , of t h e f u n c t i o n s which d e s c r i b e t h e

c o n f i g u r a t i o n of t h e deformed m i d s u r f a c e .

S i n c e t h e i n f l a t e d t o r o i d i s assumed t o be i n f r i c t i o n -

l e s s c o n t a c t w i t h t h e c o n s t r a i n t s u r f a c e , t h e t a n g e n t i a l

s h e a r f o r c e s , p a , a r e z e r o and l o a d i n g i s by normal p r e s s u r e ,

p , o n l y . The v i r t u a l work Equa t ion (58) r educes t o

where t h e Lagrange s t r e s s r e s u l t a n t s , T ~ ~ , a r e o b t a i n e d

from t h e s t r a i n energy d e n s i t y ( 6 7 ) ,

and br a r e g iven by Equa t ions (14) .

The i n f l a t i o n o f t h e t o r o i d a l membrane w i l l be con-

s t r a i n e d by a f l a t s u r f a c e which i s p a r a l l e l t o t h e x1-x2

p l a n e and l o c a t e d a t Ra on t h e p o s i t i v e x j - a x i s a s shown i n

F i g u r e 5 , Although t h e r e s u l t i n g de fo rma t ion i s n o t symmetr ic

about an a x i s , t h e r e w i l l b e symmetry about t h e x l - x 3 p l a n e

( e q u a t o r i a l p l a n e ) and about t h e x 2 - x 3 p l a n e (mer id i an p l a n e

a t e 2 = 0). I n t h e i n t e r e s t o f o b t a i n i n g more r a p i d l y con-

v e r g i n g s o l u t i o n s , i t w i l l be f u r t h e r assumed t h a t t h e x l - x 2

p l a n e i s a l s o a p l a n e o f symmetry. With t h r e e o r t h o g o n a l

p l a n e s o f symmetry ( i n t e r s e c t i n g a long t h e x r - a x i s ) , i n t e g r a -

t i o n o f t h e v i r t u a l work e q u a t i o n ( 7 0 ) need o n l y be per formed

o v e r an o c t a n t o f t h e undeformed t o r o i d a l m i d s u r f a c e .

Constraint Surface

/

RI

/ il

Axis of Revolution

F i g u r e 5 , Locat ion o f c o n s t r a i n t s u r f a c e c o n t a c t i n g t h e i n f l a t e d t o r o i d .

With t h e c o o r d i n a t e s e n a s d e f i n e d i n F i g u r e 4 , t h e

s o l u t i o n f u n c t i o n s , y r , a r e e x p e c t e d t o e x h i b i t t h e f o l l o w i n g

symmetr ies :

~ ~ ( 0 ~ , 0 ~ ) : symmet r i c i n e l ; a n t i s y m m e t r i c i n O 2

Y ~ ( € J ~ , ~ ~ ) : a n t i s y m m e t r i c i n e l ; s ymmet r i c i n e 2

0 ) : symmet r i c i n 0 and e 2 ~ 3 ( ~ 1 , 2 1

The s o l u t i o n f u n c t i o n s s h o u l d a l s o b e c a p a b l e o f

d e s c r i b i n g axisymmet r i c deformat ion-should c o n t a c t n o t be

made dur i -ng t h e i n f l a t i o n .

The s o l u t i o n f u n c t i o n s , y r , and d e r i v a t i v e s , Y * , ~ ,

must s a t i s f y t h e f o l l o w i n g g e o m e t r i c boundary c o n d i t i o n s .

S i n c e t h e c o n f i g u r a t i o n w i l l n o t change a t t h e bonded edge

( e l = m b ) , t h e s o l u t i o n f u n c t i o n s must a l s o s a t i s f y

3.2 APPROXIMATE SOLUTION

The p r e c e d i n g symmetry and boundary c o n d i t i o n s a r e met

by t h e f o l l o w i n g f i n i t e s e r i e s a p p r o x i m a t i o n s .

where

i~ e e ) = cos - YI ( 1' 2 ) e l s i n ( 2 j - 1 ) e 2

i j 'IT y 2 (0 0 ) = s i n i - e l c o s ( 2 j - 2 ) e 2 1' 2 b

The d e r i v a t i v e s o f E q u a t i o n s (74) a r e u s e d f o r c a l c u l a t i n g

t h e s t r a i n i n v a r i a n t I1 a c c o r d i n g t o E q u a t i o n ( 6 8 ) and f o r

c a l c u l a t i n g b r a c c o r d i n g t o E q u a t i o n s ( 1 4 ) . The Lagrange

s t r e s s r e s u l t a n t s , T ~ ~ , a r e t h e n o b t a i n e d from E q u a t i o n (71) .

E q u a t i o n s ( 7 0 ) , o f t h e v i r t u a l work s o l u t i o n p r o c e d u r e ,

a r e t h u s r e d u c e d t o i n t e g r a l s o f known f u n c t i o n s , , which

a r e d e f i n e d by E q u a t i o n s ( 7 5 ) .

where y i j a r e o b t a i n e d by d i f f e r e n t i a t i n g Equat ions ( 7 5 ) . r , a

The m u l t i p l e i n t e g r a t i o n i s performed ove r t h e f i r s t o c t a n t

of t h e t o r o i d (0 - < 2 O b , 0 - < O 2 - < n / 2 ) by c o n v e n t i o n a l

Simpson's r u l e . Equat ions ( 7 6 ) a r e thus reduced t o 3 x M x N

n o n l i n e a r a l g e b r a i c e q u a t i o n s .

The s o l u t i o n of Equat ions ( 7 7 ) i s e a s i l y o b t a i n e d by t h e

Newton-Raphson i t e r a t i o n p rocedure d e s c r i b e d i n Appendix B .

3 . 3 CONTACT CONSTRAINT

The assumption o f f r i c t i o n l e s s c o n t a c t w i t h a r i g i d f l a t

s u r f a c e , p o s i t i o n e d p a r a l l e l t o t h e y l -yZ p l a n e as shown i n

F igure 5 , i m p l i e s a geomet r i c c o n s t r a i n t on ly on t h e s o l u t i o n

f u n c t i o n y3 . Within t h e c o n t a c t r e g i o n , whose boundary i s n o t

known - a p r i o r i , t h e deformed midsur face i s f l a t and

~ ~ ( e ~ , e ~ ) = RL ( c o n t a c t r e g i o n ) ( 7 8 )

O u t s i d e t h e c o n t a c t r e g i o n , y3 s a t i s f i e s t h e i n e q u a l i t y

c o n d i t i o n

( f r e e r e g i o n )

4 8

The g e o m e t r i c c o n s t r a i n t i s i n t r o d u c e d i n t o t h e

f o r m u l a t i o n by means o f a new f u n c t i o n , z(B1 , e 2 ) , d e f i n e d

b y t h e f o l l o w i n g e q u a t i o n which i s v a l i d o v e r t h e e n t i r e

r e g i o n o f i n t e g r a t i o n .

E q u a t i o n (80) a l l o w s z t o r e p l a c e y 3 i n t h e components o f

t h e v i r t u a l work e q u a t i o n ( 7 0 ) . The f u n c t i o n z , which

changed t h e i n e q u a l i t y c o n s t r a i n t c o n d i t i o n i n t o an e q u a l i t y

c o n s t r a i n t c o n d i t i o n , i s c a l l e d a " s l a c k v a r i a b l e , " and h a s

been u s e d p r e v i o u s l y i n o p t i m i z a t i o n prob lems when an

i n e q u a l i t y c o n s t r a i n t i s p r e s e n t [14] . The c o n t a c t boundary

i s t r a c e d by t h e l o c u s o f c o o r d i n a t e s ( e l , 0 2 ) which s u r r o u n d

t h e r e g i o n where z(B 0 ) v a n i s h e s ( o r becomes n e g l i g i b l y 1' 2

smal l -s ince o n l y an app rox ima te s o l u t i o n w i l l b e o b t a i n e d ) .

The s l a c k v a r i a b l e z w i l l e x h i b i t t h e same symmetry a s

y and must s a t i s f y t h e f o l l o w i n g g e o m e t r i c boundary 3

c o n d i t i o n s .

I t i s n o t e d t h a t z (0,O) = 0 i s n o t a boundary c o n d i t i o n

s i n c e i t w i l l n o t be known a p r i o r i i f t h e p r e s s u r e , p , i s -

s u f f i c i e n t t o p u t t h e i n f l a t e d membrane i n c o n t a c t . The

f o r m u l a t i o n w i t h s l a c k v a r i a b l e z must a l s o admi t an a x i -

symmet r ic s o l u t i o n .

The p r e c e d i n g boundary and symmetry c o n d i t i o n s a r e

met by

where Y:J ( e l , e 2 ) i s d e f i n e d by t h e t h i r d o f E q u a t i o n s ( 7 5 ) .

The c o n s t a n t s c i j , o b t a i n e d by t h e v i r t u a l work s o l u t i o n

p r o c e d u r e , now de t e rmine an app rox ima t ion o f t h e s l a c k

v a r i a b l e z ( i n s t e a d o f y 3 ) The s o l u t i o n f u n c t i o n ~ ~ ( 8 ~ , 8 ~ )

i s o b t a i n e d from ( 8 0 ) .

w i t h d e r i v a t i v e s

where z and z a r e o b t a i n e d from t h e s e r i e s app rox ima t ion , a

( 8 2 ) .

4 . 0 C A L C I J L A T E D RESU1,TS

The a n a l y s i s p r e s e n t e d i n Chap te r 3 h a s been a p p l i e d

t o de t e rmine t h e de fo rma t ion and s t r e s s e s t h a t r e s u l t

when a c i r c u l a r t o r o i d a l membrane o f i n c o m p r e s s i b l e neo -

hookean m a t e r i a l i s e x t e r n a l l y l o a d e d . The undeformed

m i d s u r f a c e geometry i s d e f i n e d by t h e m e t r i c t e n s o r g i j , Equa t ion ( 6 5 ) , and t h e m a t e r i a l p r o p e r t i e s a r e d e f i n e d by

t h e ene rgy d e n s i t y (67) w i t h Equa t ions ( 6 8 - 6 9 ) . The c a l -

c u l a t e d r e s u l t s a r e p r e s e n t e d h e r e i n i n d i m e n s i o n l e s s form-

l e n g t h s a r e d i m e n s i o n l e s s w i t h r e s p e c t t o t h e undeformed

m e r i d i a n r a d i u s R1, s t r e s s r e s u l t a n t s a r e d i m e n s i o n l e s s w i t h

r e s p e c t t o t h e p r o d u c t Clh where C1 i s t h e neo-hookean

m a t e r i a l c o n s t a n t and h i s t h e undeformed membrane t h i c k n e s s .

The c a l c u l a t i o n s have been made f o r two t o r o i d r a d i i ,

R t = 2R1 and R t = 3 . 5 R 1 The s m a l l e r r a d i u s (2R1) was

s e l e c t e d because t o r o i d t opo logy* i s more pronounced when

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

The l a r g e r r a d i u s (3.5R1) i s a p p r o x i m a t e l y t h a t o f a

p a s s e n g e r c a r t i r e . With b o t h t o r o i d s , t h e m e r i d i a n i s

bonded t o r i g i d r i n g s l o c a t e d a t e l = gb = + 135 d e g r e e s ,

Two t y p e s of c o n t a c t l o a d i n g , a s i l l u s t r a t e d i n F i g u r e

6 ( a , b ) , were a p p l i e d t o t h e i n f l a t e d t o r o i d a l membrane. The

doub le c o n t a c t l o a d i s commonly a p p l i e d t o s i m p l e models

[16-171 o f t h e s t a t i c a l l y - l o a d e d t i r e and was a p p l i e d h e r e

*The p e c u l i a r t opo logy o f t h e t o r o i d a l s t r u c t u r e makes u n u s u a l der.;ands on c l a s s i c a l methods of a n a l y s i s ; some a n a l y t i c a l t r e a t m e n t s s u c c e s s f u l w i t h s i m p l e r s t r u c t u r e s f a i l e n t i r e l y when a p p l i e d t o t h e t o r o i d . The d i f f i c u l t y w i t h l i n e a r membrane t h e o r y i s d i s c u s s e d i n Re fe r ence [ I51

5 1

P l a n e o f - - - - -

Symmetry

( a ) Double C o n t a c t Load ( b ) A x l e - A p p l i e d Load

F i g u r e 6 [ a , b ) . Two t y p e s o f t i r e c o n t a c t l o a d i n g a p p l i e d t o t h e t o r o i d a l membrane t i r e mode l .

i n o r d e r t o o b t a i n a t h i r d p l a n e o f symmetry, p e r p e n d i c u l a r

t o t h e whee l p l a n e and t o t h e v e r t i c a l p l a n e which c o n t a i n s

t h e a x l e . The a d d i t i o n a l p l a n e o f symmetry i s a s s o c i a t e d

w i t h a deformed c o n f i g u r a t i o n which i s e a s i e r t o app rox ima te

( w i t h s e r i e s o f p e r i o d i c f u n c t i o n s ) t h a n i s t h e deformed

c o n f i g u r a t i o n p roduced by t h e a x l e - a p p l i e d l o a d shown i n

F i g u r e 6 ( b ) . S a t i s f a c t o r y s o l u t i o n s have now been o b t a i n e d

f o r b o t h t y p e s o f l o a d i n g and i t i s found t h a t

t h e d i s t r i b u t i o n o f s t r u c t u r a l s t r e s s i s s i g n i f i c a n t l y

d i f f e r e n t f o r t h e doub le c o n t a c t l o a d v i s 'a v i s t h e a x l e -

a p p l i e d l o a d .

For d e t e r m i n a t i o n o f t h e c o n t a c t boundary p o i n t s , t h e

i n f l a t e d t o r o i d i s c o n s i d e r e d t o b e i n c o n t a c t o v e r t h e

r e g i o n s o f and O 2 f o r which z 2 < ,0225 R1; e , g . ,

Y3 3 .0775 K1 i s s u f f i c i e n t l y c l o s e t o Re = 3.1000 R1 t o

be c o n s i d e r e d a s a c o o r d i n a t e o f a p o i n t i n t h e c o n t a c t

r e g i o n .

The c o n t a c t boundary c u r v e , f o r a l l d e f l e c t e d t o r o i d

c a l c u l a t i o n s p r e s e n t e d i n t h i s r e p o r t , i s c l o s e l y a p p r o x i -

mated by an e l l i p s e whose ma jo r and minor a x e s , 2a and 2 b ,

a r e t h e l e n g t h and w i d t h of t h e c o n t a c t bounda ry . The

c o n t a c t a r e a i s t h e r e f o r e approx imated by t h e e l l i p t i c a l

a r e a , ~ a b , and t h e t i r e l o a d , F , r e q u i r e d t o p roduce a

s p e c i f i e d l o a d e d r a d i u s * , R e , i s

where p i s t h e i n f l a t i o n p r e s s u r e . E q u a t i o n ( 8 5 ) was u sed

t o c a l c u l a t e t h e t i r e l o a d f o r t h e l o a d - d e f l e c t i o n d a t a

p r e s e n t e d i n F i g u r e 1 5 .

4 . 1 DOUBLE CONTACT LOAD

T h i s s e c t i o n p r e s e n t s r e s u l t s c a l c u l a t e d f o r t h e doub le

c o n t a c t t y p e l o a d i l l u s t r a t e d i n F i g u r e 6 ( a ) . I t i s o f

i n t e r e s t t o compare t h e s e r e s u l t s w i t h c o r r e s p o n d i n g s o l u t i o n s

f o r t h e a x l e - a p p l i e d t y p e l o a d p r e s e n t e d below i n

S e c t i o n 4 . 2 . S i g n i f i c a n t d i f f e r e n c e s i n t h e s o l u t i o n s a r e

d i s c u s s e d i n S e c t i o n 4 . 3 . 1 .

* I t s h o u l d b e n o t e d t h a t t h e d e f l e c t e d t o r o i d a n a l y s i s i s f o r m u l a t e d (Chap te r 3 ) w i t h a p r e s c r i b e d l o a d e d r a d i u s , Rk. The l o a d r e q u i r e d t o p roduce t h i s r a d i u s i s n o t known u n t i l t h e c o n t a c t boundary i s d e t e r m i n e d -

5 3

F i g u r e 7 shows t h e deformed m e r i d i a n (11) which p a s s e s

t h r o u g h t h e c e n t e r o f t h e c o n t a c t r e g i o n . The c a l c u l a t e d

c o n t a c t boundary (111) i s supe r imposed , i n t h e same s c a l e ,

on t h i s g r a p h . The c o n t a c t boundary i s found t o b e

c l o s e l y r e p r o d u c e d by a m a t h e m a t i c a l e l l i p s e whose ma jo r

and minor a x e s a r e t h e l e n g t h and w i d t h o f t h e c o n t a c t

r e g i o n . The f l a t n e s s o f t h e t r i g o n o m e t r i c s e r i e s s o l u t i o n

f o r t h e deformed c o n f i g u r a t i o n i n t h e c o n t a c t r e g i o n i s s e e n

by examin ing t h e y3/R1 column i n Tab l e 2 which l i s t s a few

c a l c u l a t e d d a t a p o i n t s f o r t h e crown r e g i o n p l o t t e d i n

F i g u r e 7 ,

The m e r i d i o n a l (A1) and c i r c u m f e r e n t i a l (AZ) e x t e n s i o n

r a t i o s a r e p l o t t e d i n F i g u r e 7 a s f u n c t i o n s o f t h e r a d i u s

a n g l e , e l , o f t h e undeformed m e r i d i a n ( I ) . These e x t e n s i o n

r a t i o s have n o t been p r e v i o u s l y computed o r measured f o r

a t o r o i d a l membrane. I t i s s e e n t h a t t h e c i r c u m f e r e n t i a l

s t r a i n (AZ) i s r educed a s t h e c o o r d i n a t e 0 i s f o l l o w e d 1

from t h e c o n t a c t boundary t o t h e c e n t e r o f t h e c o n t a c t

r e g i o n . The m e r i d i o n a l s t r a i n (Al) i s found , q u i t e

u n e x p e c t e d l y , t o i n c r e a s e a s i s f o l l o w e d from t h e c o n t a c t

boundary t o t h e c o n t a c t c e n t e r . However, t h e o v e r a l l

m e r i d i o n a l s t r a i n i s r e l i e v e d by t h e c o n t a c t l o a d i n g . (The

h l cu rve f o r f r e e i n f l a t i o n a t t h e same p r e s s u r e i s i n c l u d e d

i n F i g u r e 7 f o r compar i son p u r p o s e s . )

F i g u r e 7 . Deformation and s t r a i n s o l u t i o n f o r i n f l a t i o n and c o n t a c t c o n s t r a i n t . I , undeformed mer id ian a t con- t a c t c e n t e r ; 11, deformed m e r i d i a n ; 111, p l a n view of c a l c u l a t e d c o n t a c t boundary ; IV, m e r i d i o n a l (A1) and c i r c u m f e r e n t i a l (X2) s t r a i n v s . p o s i t i o n ( 0 ) on t h e undeformed m e r i d i a n . ( R t = 2R1, Rp = 3.1h1, p = 1 . 4 )

T A B L E 2

C A L C U L A T E D CROWN REGlON DATA POINTS PLOTTED IN FIGURE 7

The o p p o s i t e s t r a i n b e h a v i o r i n t h e c o n t a c t r e g i o n i s

n o t e d i n F i g u r e 8 which shows h l and A 2 a s f u n c t i o n s o f t h e

r a d i u s a n g l e , 0 2 , o f t h e undeformed e q u a t o r . H e r e , h l

e x h i b i t s t h e e x p e c t e d r e d u c t i o n a s t h e e q u a t o r i s f o l l o w e d

from t h e c o n t a c t boundary t o t h e c e n t e r o f t h e c o n t a c t

r e g i o n w h i l e h 2 i s i n c r e a s i n g , a l b e i t t o a v a l u e w e l l below

t h e f r e e i n f l a t i o n l e v e l . The h1 and h 2 l e v e l s f o r f r e e

i n f l a t i o n a t t h e same p r e s s u r e a r e i n c l u d e d i n F i g u r e 8 f o r

compar i son p u r p o s e s .

The c o n v e c t e d c o o r d i n a t e e x t e n s i o n r a t i o s h and A 2 , 1

p l o t t e d i n F i g u r e s 7 and 8 , a r e c a l c u l a t e d by

Crown

F i g u r e 8 . Deformat ion and s t r a i n s o l u t i o n i n t h e e q u a t o r i a l p l a n e . I , undeformed e q u a t o r ; 1 1 , deformed e q u a t o r ; 1 1 1 , m e r i d i o n a l (XI) and c i r c u m f e r e n t i a l (12) s t r a i n v s . p o s i t i o n (82) on t h e undeformed e q u a t o r . (Same i n p u t d a t a a s f o r F i g . 7 . )

5 7

These e x t e n s i o n r a t i o s g i v e a comple te d e s c r i p t i o n o f t h e

s t r a i n f i e l d a t m i d s u r f a c e p o i n t s which l i e i n one o f t h e

p l a n e s o f d e f o r m a t i o n symmetry. These p o i n t s a r e l o c a t e d

on ly on t h e e q u a t o r ( 0 ,B2) and t h e m e r i d i a n s ( e l ,0),

( e l , ~ / 2 ) , (01 , T ) , (B1, 3n/2) . S h e a r s t r a i n i s p r e s e n t

a t a l l o t h e r p o i n t s a n d , f o r t h e s e , t h e components ( 7 ) o f

t h e G r e e n - S a i n t Venant s t r a i n t e n s o r must b e c a l c u l a t e d f o r

a comple te s t r a i n d e s c r i p t i o n .

F i g u r e 9 shows t h e p h y s i c a l s t r e s s components , S c i ~ '

c a l c u l a t e d a c c o r d i n g t o Equa t ion d i s t r i b u t e d o v e r one -

q u a r t e r o f t h e bead r i n g (B1 = 135 deg) . I t i s c l e a r l y s e e n

how t h e c o n t a c t l o a d i n g r e l i e v e s t h e un i fo rm i n f l a t i o n

p r e s t r e s s on t h e bead r i n g . I t i s n o t e d t h a t , above t h e

c e n t e r o f c o n t a c t , t h e r a d i a l s t r e s s r e l i e f i s abou t f i v e

t i m e s t h e v a l u e o f t h e r e l i e f i n t h e hoop s t r e s s .

The r e s u l t s shown i n F i g u r e s 7 - 9 were o b t a i n e d

f o r t h e doub le c o n t a c t t y p e o f l o a d i n g . Al though deforma-

t i o n and s t r e s s i n t h e c o n t a c t r e g i o n a r e n e g l i g i b l y

i n f l u e n c e d by t h e manner i n which t h e v e r t i c a l l o a d i s

a p p l i e d , t h e s i d e w a l l and bead s t r e s s d i s t r i b u t i o n s a r e

found t o be s i g n i f i c a n t l y i n f l u e n c e d by t h e manner o f v e r t i c a l

l o a d a p p l i c a t i o n . This i s c l e a r l y s e e n by compar i son o f t h e

p r e c e d i n g s o l u t i o n s w i t h s o l u t i o n s g i v e n i n t h e n e x t s e c t i o n .

Rad ia I S t ress

.70 - Inflation OhIv

.69 .. HOOP S 2 2 - Cih Stress .68 ...

.67 . I AB2 \

Shear Stress

Figure 9 . Bead s t r e s s d i s t r i b u t i o n s . (Same i n p u t d a t a a s f o r F i g . 7 . )

4 . 2 AXLE-APPLIED LOAD

The r e s p o n s e o f t h e t o r o i d a l membrane t o t h e a x l e -

a p p l i e d t y p e o f l o a d i l l u s t r a t e d i n F i g u r e 6 ( b ) i s o b t a i n e d

by making a s l i g h t change i n t h e component o f t h e doub le

c o n t a c t c o o r d i n a t e f u n c t i o n s g i v e n by E q u a t i o n s ( 7 5 ) . The

changes made a r e i n d i c a t e d i n T a b l e 3 which l i s t s t h e 8 2

components u s e d f o r b o t h l o a d i n g mechanisms. These

-- - -

TABLE 3

CIRCUMFERENTIAL COMPONEXTS OF THE COORDINATE FUNCTIONS

V e r t i c a l Load Mechanism P o i n t

C o o r d i n a t e Double C o n t a c t Ax le -App l i ed

cos ( 2 . - 1 ) O 2 I

s i n j 8 2

cos ( j - 1 ) 0 ,

cos j 0 ,

changes s i m p l y remove t h e h o r i z o n t a l p l a n e o f symmetry

and i m p l i c i t l y r e q u i r e t h e v e r t i c a l l o a d t o b e a p p l i e d t o

t h e b e a d r i n g s .

With t h e a x l e - a p p l i e d t y p e o f l o a d , t h e c o n t a c t r e g i o n

r e s p o n s e o f t h e a n a l y t i c t i r e model w i l l more c l o s e l y

r e p r e s e n t t h e r e s p o n s e o f a r e a l t i r e . The r e s p o n s e o f t h e

t i r e model t o t h e a x l e - a p p l i e d l o a d h a s been e x t e n s i v e l y

i n v e s t i g a t e d . S e v e r a l p l o t t i n g programs were w r i t t e n t o

p roduce computer-drawn p l o t s of t h e o u t p u t o f g r e a t e s t

i n t e r e s t . Many o f t h e c a l c u l a t e d r e s u l t s p r e s e n t e d i n

t h i s s e c t i o n were p l o t t e d by t h e computer .

F i g u r e s 10-14 p r e s e n t d e f o r m a t i o n , s t r a i n , and s t r e s s

s o l u t i o n s f o r an example t o r o i d a l membrane o f r a d i u s

Rt = 3 .5 R1. The r e s u l t s shown i n t h e s e f i g u r e s were

o b t a i n e d f o r d i m e n s i o n l e s s p r e s s u r e p = 1 , 7 and l o a d e d r a d i u s

RQ = 4 .57 R1. I n f r e e i n f l a t i o n a t p = 1 . 7 , t h e e q u a t o r o f

t h i s t o r o i d expands t o a r a d i u s o f y 3 = 5 . 3 3 R1. The

s o l u t i o n shown i n F i g u r e s 10 -14 t h u s c o r r e s p o n d s t o a v e r t i -

c a l d e f l e c t i o n 6 = - 7 6 R1; f o r a t o r o i d o f m e r i d i o n a l r a d i u s

R1 = 3 i n . ( t h e r a d i u s o f a s t a n d a r d au tomob i l e t i r e i n n e r

t u b e ) , t h e v e r t i c a l d e f l e c t i o n would b e 6 = 2 . 2 8 i n .

The s i n g l e c o n t a c t s t r a i n s o l u t i o n s i n t h e m e r i d i o n a l

and e q u a t o r i a l p l a n e s ( F i g . 1 1 ) e x h i b i t t h e p e c u l i a r b e h a v i o r

n o t e d i n t h e doub le c o n t a c t c a s e ( c f . F i g s , 7 - 8 ) , v i z , t h e

m e r i d i o n a l s t r a i n (A ) i n c r e a s e s w h i l e moving i n t h e m e r i - 1

d i o n a l p l a n e ( I ) from c o n t a c t boundary t o c o n t a c t c e n t e r

b u t d e c r e a s e s w h i l e moving i n t h e e q u a t o r i a l p l a n e (11) f rom

c o n t a c t boundary t o c o n t a c t c e n t e r . J u s t t h e o p p o s i t e

b e h a v i o r i s n o t e d i n t h e c i r c u m f e r e n t i a l s t r a i n ( A 2 ) . Thi s

b e h a v i o r i s d i s c u s s e d i n d e t a i l i n S e c t i o n 4 . 3 , 2 ,

The p r i n c i p a l s t r e s s d i r e c t i o n s i n d i c a t e d on t h e

deformed s i d e w a l l ( F i g . 13) a r e t h e p r i n c i p a l axes o f t h e

s t r e s s t e n s o r , and a r e c a l c u l a t e d a c c o r d i n g t o t h e t h e o r y

g i v e n i n S e c t i o n 2 . 4 . 2 . Al though s i d e w a l l s h e a r s t r a i n s i n

t h e v i c i n i t y of t h e c o n t a c t r e g i o n do n o t appear t o exceed

8 degrees (change i n a r i g h t ang le ) , t h e s e s t r a i n s can cause

p r i n c i p a l a x i s r o t a t i o n s o f up t o 4 5 d e g r e e s , as i s e v i d e n t

i n F igure 1 3 . S ince t h e t o r o i d m a t e r i a l i s i s o t r o p i c , t h e

p r i n c i p a l s t r e s s d i r e c t i o n s a r e c o i n c i d e n t wi th t h e p r i n c i p a l

s t r a i n d i r e c t i o n s . I n a c o r d - r e i n f o r c e d t i r e c a r c a s s , t h e

p r i n c i p a l s t r a i n and p r i n c i p a l s t r e s s d i r e c t i o n s may n o t c o i n c i d e .

The r e v e r s a l b e h a v i o r o f t h e maximum p r i n c i p a l s t r e s s e s

( F i g . 14) s e e n n e a r t h e c o n t a c t r e g i o n , v i z , t h e s t r e s s a t

= 0 deg i s g r e a t e r t h a n t h e s t r e s s a t e l = 45 o r 60 deg ,

i s d i f f i c u l t t o e x p l a i n b u t fo l lows t h e t r e n d of il s t r a i n

s o l u t i o n shown i n F igure 11 ( I ) . I t i s found t h a t t h e

maximum s t r e s s i s approximate ly r a d i a l and appears a t t h e

bead r i n g .

F igure 1 5 employs a c a r p e t p l o t t o p r e s e n t t h e depen-

dence o f v e r t i c a l l o a d on i n f l a t i o n p r e s s u r e and loaded

r a d i u s . Very l i n e a r l o a d - d e f l e c t i o n b e h a v i o r ( a t a f i x e d

p r e s s u r e ) i s found, w h i l e t h e l o a d - p r e s s u r e b e h a v i o r ( a t a

f i x e d loaded r a d i u s ) i s d i s t i n c t l y n o n l i n e a r . L i n e a r l o a d -

d e f l e c t i o n b e h a v i o r i s a c h a r a c t e r i s t i c e x h i b i t e d by most

t i r e s . S i n c e i t i s known t h a t when i n f l a t i o n p r e s s u r e and

t i r e l o a d a r e s i m u l t a n e o u s l y v a r i e d s o as t o m a i n t a i n

c o n s t a n t t i r e d e f l e c t i o n , t h e c o n t a c t a r e a remains e f f e c -

t i v e l y c o n s t a n t , i t appears t h a t t h e n o n l i n e a r l o a d - p r e s s u r e

r e l a t i o n s h i p e x h i b i t e d by t h e a n a l y t i c t i r e model ( F i g . 15)

i s l e s s r e a l i s t i c . The i n t r o d u c t i o n o f bending s t i f f n e s s may

s t r a i g h t e n o u t t h e p o s i t i v e s l o p e p o r t i o n o f t h e c a r p e t

p l o t shown i n F igure 1 5 .

6 2

I F i g u r e 1 0 . Deformat ion i I s o l u t i o n s f o r an a x l e - a p p l i e d I l o a d . Above: i n f l a t e d 1

1 m e r i d i a n p a s s i n g t h r o u g h i I c o n t a c t c e n t e r . L e f t :

c o n t a c t bounda ry . ( R t = 3.5R1, Ri = 4.57R1,

i p = 1 . 7 )

0 c,r, 1- I n f l a t i o n O n l y , , , _ ,,----- -.- --- I- I ---.-.I r - - - - - - - -

* - .------ -- .

R a d i a l S t r e s s

I n f l a t i o n Only _ __ __ _ - &_ - - -- .I I --. -- - I - - . - I -... -1 '8cl I /

* ' I5 t /' Hoop I" S t r e s s

S h e a r S t r e s s

F i g u r e 1 2 . Bead s t r e s s d i s t r i b u t i o n s . (Same d a t a a s f o r F i g . 1 0 . )

6 5

)J3 6.

\ P r i n c i p a l S t r e s s

- 7 D i r e c t i o n 1 3(

Figure 1 3 . P r i n c i p a l s t r e s s d i r e c t i o n s a t p o i n t s on t h e deformed s i d e w a l l . (Same d a t a a s f o r F i g . 1 0 . )

, \ l c r i d i o n a l C o o r d i n a t e ,

F i g u r e 1 6 . A f a m i l y o f c o n t a c t b o u n d a r i e s o b t a i n e d by i n c r e a s i n g t h e i n f l a t i o n p r e s s u r e w h i l e k e e p i n g t h e l o a d e d r a d i u s f i x e d . ( A x l e - a p p l i e d l o a d , Rt = 3 .5R1) R k = 4 . 5 0 1 , p v a r y i n g . )

4 . 3 DISCUSSION

4 . 3 . 1 D O U B L E CONTACT VS . AXLE-APPLIED LOADING. The

r e s u l t s o b t a i n e d f o r t h e two d i f f e r e n t methods o f t i r e model

l o a d i n g i n d i c a t e t h a t t h e i n t e r n a l s t r a i n s and s t r e s s e s

a s s o c i a t e d w i t h c o n t a c t phenomena a r e s i g n i f i c a n t l y depen-

d e n t on how t h e c o n t a c t l o a d i s a p p l i e d . I t was found

t h a t t h e s t r a i n d i s t r i b u t i o n ( o v e r t h e e n t i r e t o r o i d a l

s t r u c t u r e ) which i s g e n e r a t e d when t h e t o r o i d i s compressed

between oppos ing c o n t a c t l o a d s i s s i g n i f i c a n t l y d i f f e r e n t

from t h e s t r a i n d i s t r i b u t i o n g e n e r a t e d when a c o n t a c t l o a d

i s p roduced i n r e a c t i o n t o a d i s t r i b u t i o n o f e x t e r n a l s t r e s s

a p p l i e d t o t h e bead r ings- the a x l e - a p p l i e d l o a d .

F i g u r e 1 7 ( a , b ) p r e s e n t s a comparison o f s t r a i n s ,

d i s t r i b u t e d around t h e m e r i d i a n which p a s s e s t h rough t h e

c e n t e r o f t h e c o n t a c t r e g i o n , o b t a i n e d ( a ) f o r a doub le

c o n t a c t l o a d , F1, and (b ) an a x l e - a p p l i e d l o a d , F 2 . The l o a d s ,

F1 and F 2 , a r e d i f f e r e n t b u t a d j u s t e d t o p roduce t h e same

l o a d e d r a d i u s , Rk. The il s t r a i n d i s t r i b u t i o n produced by

a double c o n t a c t l o a d always e x h i b i t s two minima ( a p p r o x i -

ma te ly i n t h e l o c a t i o n s shown i n F i g u r e 1 7 ( a ) ) , r e g a r d l e s s

o f i n f l a t i o n p r e s s u r e o r l o a d e d r a d i u s . The c o r r e s p o n d i n g

i1 s t r a i n d i s t r i b u t i o n produced by an a x l e - a p p l i e d l o a d a lways

e x h i b i t s a s i n g l e minimum, r e g a r d l e s s o f i n f l a t i o n p r e s s u r e

o r l o a d e d r a d i u s . The c h a r a c t e r o f t h e A 2 s t r a i n d i s t r i -

b u t i o n s a r e u n a f f e c t e d ( b o t h show a s i n g l e maximum) by t h e

h l = M e r i d i o n a l E x t e n s i o n R a t i o

A 2 = C i r c u m f e r e n t i a l E x t e n s i o n R a t i o

A 3 = Thicknes s E x t e n s i o n R a t i o

h l h Z h g = 1

Crown Bead Crown Bead

F i g u r e 1 7 . S t r a i n d i s t r i b u t i o n s f o r two d i f f e r e n t t i r e l o a d s , F1 and F 2 , which p roduce t h e same l o a d e d r a d i u s , R k .

(p = 1 . 4 , RE = 3 . 1 , R t = 2.0)

t y p e o f c o n t a c t l o a d i n g . S i n c e t h e membrane m a t e r i a l i s

m a t h e m a t i c a l l y i n c o m p r e s s i b l e , t h e p r o d u c t s , h l h 2 h 3 , o f

t h e s t r a i n s i n p l a n e s o f symmetry ( a s i n F i g . 1 7 ) must

e q u a l u n i t y . Th is e x p l a i n s t h e A 3 s t r a i n d i s t r i b u t i o n

e x h i b i t s a b e h a v i o r t h a t i s i n v e r s e t o t h e b e h a v i o r o f t h e

h l d i s t r i b u t i o n .

4 . 3 . 2 CONTACT STRAIN PHENOMENA. The i n f l a t e d t o r o i d a l

s h e l l i s an e x c e l l e n t example o f a p r e s t r e s s e d s t r u c t u r e

which w i l l s u p p o r t e x t e r n a l l o a d s by s t r e s s r e l i e f i n t h e

v i c i n i t y o f t h e l o a d s . The i n f l a t e d pneuma t i c t i r e s u p p o r t s

an e x t e r n a l a x l e - a p p l i e d l o a d ( d i s t r i b u t e d on t h e bead r i n g s ) ,

and t h e r e a c t i n g c o n t a c t p r e s s u r e d i s t r i b u t i o n , by r e l i e f o f

t h e i n f l a t i o n p r e s t r e s s * i n t h e v i c i n i t y o f t h e s e l o a d s . I t

i s t h u s e x p e c t e d t h a t t h e s t r a i n d i s t r i b u t i o n i n a t i r e

(and i n an a n a l y t i c model) w i . 1 1 be r e l i e v e d i n t h e c o n t a c t

r eg ion - -w i th t h e s t r a i n lower a t t h e c e n t e r o f c o n t a c t t han

on t h e c o n t a c t bounda ry . The c o n t a c t s t r a i n d i s t r i b u t i o n s

found i n t h e a n a l y t i c t i r e model show t h a t t h e e x p e c t e d

s t r a i n r e l i e f does t a k e p l a c e - a l b e i t i n a somewhat unexpec t ed

manner .

F i g u r e 11 compares t h e h l and A 2 s t r a i n d i s t r i b u t i o n s

found a l o n g t h e deformed e q u a t o r w i t h t h e same s t r a i n s

found a l o n g t h e deformed m e r i d i a n p a s s i n g t h r o u g h t h e c e n t e r

*The r e l i e f o f t h e i n f l a t i o n p r e s t r e s s i n t h e v i c i n i t y o f t h e c o n t a c t r e g i o n r e s u l t s i n a nonuni form s t r e s s d i s t r i b u t i o n on t h e bead r i n g - - w i t h a r e s u l t a n t f o r c e which b a l a n c e s t h e t i r e l o a d .

o f t h e c o n t a c t r e g i o n . S i n c e t h e e q u a t o r and t h i s m e r i d i a n

l i e i n p l a n e s o f symmetry, t h e r e i s no s h e a r s t r a i n and

t h e s t r a i n f i e l d i s comple t e ly de t e rmined* by t h e p r i n c i p a l

e x t e n s i o n r a t i o s X l and A 2 . Upon examina t i on o f t h e s t r a i n

b e h a v i o r i n t h e c o n t a c t r e g i o n , i t i s n o t e d t h a t : i n t h e

e q u a t o r i a l p l a n e ( F i g . 1 1 ( I I ) ) , A1 d e c r e a s e s from t h e

c o n t a c t boundary t o a minimum a t t h e c e n t e r o f c o n t a c t ; b u t

i n t h e m e r i d i o n a l p l a n e ( F i g . l l ( 1 ) ) h l i n c r e a s e s from t h e

c o n t a c t boundary t o a l o c a l maximum a t t h e c e n t e r o f

c o n t a c t . The o p p o s i t e b e h a v i o r i s n o t e d upon examining t h e

X s t r a i n d i s t r i b u t i o n i n t h e same two p l a n e s . 2

To o u r knowledge, t h e c o n t a c t s t r a i n phenomena p o i n t e d

o u t i n t h e p r e v i o u s p a r a g r a p h has n o t been p r e v i o u s l y

computed n o r o b s e r v e d e x p e r i m e n t a l l y . We a r e c o n f i d e n t t h a t

i t does e x i s t i n t h e a n a l y t i c t i r e model d e f i n e d i n t h i s

r e p o r t , i . e . , i t i s n o t a consequence o f t h e n u m e r i c a l

t e c h n i q u e s employed t o o b t a i n t h e s o l u t i o n ( c f . S e c t . 4.3.3-

Numerical C o n s i d e r a t i o n s ) . I t i s c o n j e c t u r e d , a t t h i s t i m e ,

t h a t t h e c o n t a c t s t r a i n phenomena s e e n i n F i g u r e 11 a r e

s imp ly a consequence o f f l a t t e n i n g a p r e s t r e s s e d s t r u c t u r e

o f nonuni form c u r v a t u r e .

* In p l a n e s o f symmetry, A 3 i s o b t a i n e d from t h e incom- p r e s s i b i l i t y c o n d i t i o n : h l A 2 A 3 = 1.

Feng and Yang [ 1 8 ] have c a l c u l a t e d m e r i d i o n a l s t r a i n

d i s t r i b u t i o n s i n an i n f l a t e d s p h e r i c a l membrane, which i s

f l a t t e n e d a t a n t i p o d e s , t h a t a r e very s i m i l a r t o h l i n

F i g u r e 11 (11) and h 2 i n F igu re 11 ( I ) . The f l a t t e n i n g of

an i n f l a t e d t o r o i d a l membrane, however, does n o t r e s u l t i n

ax i symmet r i c de fo rma t ion because t h e i n f l a t e d c u r v a t u r e

i s nonuni form, i . e . , t h e c u r v a t u r e v a r i e s w i t h l o c a t i o n on

t h e s u r f a c e . i s t h e r e f o r e e x p e c t e d t h a t

s t r a i n and s t r e s s r e l i e f w i l l o c c u r i n a manner which

depends on t h e l o c a l c u r v a t u r e ( p r i o r t o f l a t t e n i n g )

t h e

w e l l as l o c a t i o n on t h e t o r o i d a l s u r f a c e . I t may t h e n be

c o n j e c t u r e d t h a t t h e s t r a i n r e l i e f o c c u r s i n an unba lanced

manner which r e s u l t s i n one r e l i e v e d s t r a i n a c t u a l l y showing

a h i g h e r v a l u e a t t h e c e n t e r o f c o n t a c t t h a n on t h e c o n t a c t

boundary , w h i l e t h e o t h e r r e l i e v e d s t r a i n a minimum

t h e c e n t e r o f c o n t a c t . The f a c t t h a t h l and h 2 r e v e r s e

t h e s e r o l e s when t h i s b e h a v i o r i s viewed i n p l a n e s o f

p r i n c i p a l c u r v a t u r e * l e n d s s u p p o r t t o t h e h y p o t h e s i s t h a t

t h e c u r v a t u r e o f t h e p r e s t r e s s e d t o r o i d has a s i g n i f i c a n t

i n f l u e n c e on c o n t a c t s t r a i n phenomena.

*The p l a n e s o f symmtry c o n t a i n t h e d i r e c t i o n s o f t h e p r i n c i - p a l c u r v a t u r e s ( t h e d i r e c t i o n s o f maximum and minimum c u r v a t u r e ) a s w e l l a s t h e d i r e c t i o n s o f p r i n c i p a l s t r a i n and s t r e s s .

4 . 3 . 3 N U M I i R I CAI, CONS1 1)liRA'l'TONS. 'I'hc a n a l y t i c

t i r e model s o l u t i o n s p r e s e n t e d i n t h i s r e p o r t a r e , o f

n e c e s s i t y , a p p r o x i m a t e . Even f o r a c i r c u l a r t o r o i d o f neo -

hookean m a t e r i a l ( t h e p r e s e n t a n a l y t i c t i r e model) t h e

v i r t u a l work e q u a t i o n (70) i s h i g h l y n o n l i n e a r . F u n c t i o n s

yr(B1,BZ) which e x a c t l y s a t i s f y t h e v i r t u a l work e q u a t i o n

w i l l p r o b a b l y n e v e r b e found ; t h e y n e e d n o t b e found i f

s a t i s f a c t o r y methods of a p p r o x i m a t i n g t h e s e f u n c t i o n s a r e

u t i l i z e d .

The v i r t u a l work s o l u t i o n p r o c e d u r e , d e s c r i b e d i n

S e c t i o n 2 . 5 , i s a c o n v e n i e n t method o f o b t a i n i n g app rox ima te

s o l u t i o n s . As w i t h most " d i r e c t " methods* i n v o l v i n g

v a r i a t i o n a l c a l c u l u s , t h e r e does n o t e x i s t an a n a l y t i c a l

t e c h n i q u e f o r e s t i m a t i n g t h e a c c u r a c y o f t h e app rox ima te

s o l u t i o n o b t a i n e d . I t i s , o f c o u r s e , p o s s i b l e t o s u b s t i t u t e

t h e app rox ima te s o l u t i o n back i n t o t h e v i r t u a l work

e q u a t i o n and c a l c u l a t e an e r r o r v a l u e . Th is i s se ldom

done , because t h e r e i s , a t t h i s t i m e , no way t o r e l a t e t h e

e r r o r i n s a t i s f y i n g t h e v i r t u a l work e q u a t i o n w i t h t h e e r r o r

between t h e app rox ima te s o l u t i o n and t h e e x a c t s o l u t i o n

(which remains unknown) .

E f f e c t o f Number o f Terms. A n o t u n r e a s o n a b l e method

o f e s t i m a t i n g t h e conve rgence o f an app rox ima te s e r i e s

s o l u t i o n i s s i m p l y t o run o t h e r s o l u t i o n s ( f o r t h e same

--

*The t e r m i n o l o g y " d i r e c t " i s a p p l i e d t o s o l u t i o n p r o c e d u r e s which u t i l i z e f u n c t i o n s t h a t s a t i s f y , a p r i o r i , t h e symmetry and boundary c o n d i t i o n s imposed by a p a r t i c u l a r p rob lem. A l u c i d d i s c u s s i o n o f d i r e c t methods i s g i v e n i n Chap te r 7 o f Re fe r ence [ 1 9 ] .

75

i n p u t d a t a ) w i t h f ewer and g r e a t e r numbers o f t e r m s . A

g r a p h i c a l e s t i m a t e o f t h e minimum number o f t e r m s which

a r e r e q u i r e d t o o b t a i n a conve rgen t s o l u t i o n i s o b t a i n e d

by supe r impos ing p l o t s o f t h e s o l u t i o n s f o r i n c r e a s i n g

numbers o f t e r m s .

For s o l u t i o n s t o e l a s t i c i t y p rob l ems , i t i s pe rhaps

b e t t e r t o e s t i m a t e s o l u t i o n convergence by l o o k i n g a t

s t r a i n p l o t s * r a t h e r t h a n p l o t s o f t h e deformed c o n f i g u r a t i o n

f u n c t i o n s , y r ( e l , e 2 ) . F i g u r e 18 shows t h e approx imate

A1 and h 2 s t r a i n s o l u t i o n s o b t a i n e d by t a k i n g M=N=2,3, and

4 t e rms** i n each o f t h e s e r i e s f o r ~ ~ ( 0 ~ , 8 ~ ) . Al though

c o n s i d e r a b l e change i s s e e n i n comparing t h e 2 - t e r m s o l u t i o n

w i t h t h e 3 - t e r m s o l u t i o n , t h e e f f e c t o f go ing t o a 4 - t e r m

s o l u t i o n i s n o t n e a r l y s o p ronounced . I t i s b e l i e v e d t h a t

t h e a c t u a l s t r a i n s o l u t i o n i s c l o s e l y app rox ima ted by t h e

s o l u t i o n o b t a i n e d f o r M = N = 4 . The f u n c t i o n s y,(B1,B2), a s

approx imated by 3- and 4 - t e r m s o l u t i o n s , d i f f e r i n t h e 3rd

dec imal p l a c e ; t h e i r c u r v e s would b e i n d i s t i n g u i s h a b l e on

a s t a n d a r d s i z e ( 8 1 / 2 x 11) p l o t .

* The s t r a i n s a r e c a l c u l a t e d from t h e g r a d i e n t s , ~ , , ~ ( 6 ~ , 8 ~ ) ;

t h e s e a r e more s e n s i t i v e t o t h e number o f t e rms i n an approx imate s o l u t i o n t h a n a r e t h e f u n c t i o n s , y r , t h e m s e l v e s .

**Since each s e r i e s h a s a doub le summation, t h e s e r i e s s o l u t i o n s p l o t t e d i n F i g . 1 8 a c t u a l l y have 4 , 9 , and 16 t e r m s , r e s p e c t i v e l y . A 3 - t e r m s o l u t i o n i m p l i e s a s e r i e s o f 9 t e r m s .

Comparison w i t h Another S o l u t i o n . A s a t i s f a c t o r y way

o f e s t i m a t i - n g t h e a c c u r a c y o f an app rox ima te s o l u t i o n ( i n

t h e absence o f t h e e x a c t s o l u t i o n o r e x p e r i m e n t a l c o n f i r m a -

t i o n ) i s t o compare i t w i t h a n o t h e r s o l u t i o n ( o f t h e same

prob lem) which i s o b t a i n e d by a wel l -known and s u b s t a n t i a l l y

d i f f e r e n t t e c h n i q u e . U n f o r t u n a t e l y , t h e o n l y p u b l i s h e d

s o l u t i o n o f t h e t o r o i d a l membrane c o n t a c t p rob lem 171 i s

n o t s u i t a b l e * f o r compar i son w i t h t h e s o l u t i o n o f t h e

t o r o i d a l membrane c o n t a c t p rob lem deve loped i n t h i s r e p o r t .

Hsu [ 2 0 ] h a s o b t a i n e d ax i symmet r i c i n f l a t i o n s o l u t i o n s

f o r a t o r o i d a l membrane t h a t i s n e a r l y i d e n t i c a l t o t h e

a n a l y t i c t i r e model r e p o r t e d h e r e i n ; t h e o n l y d i f f e r e n c e i s

t h a t H s u l s s o l u t i o n was o b t a i n e d by Runge-Kut ta i n t e g r a t i o n

o f t h e n o n l i n e a r f o r c e e q u i l i b r i u m e q u a t i o n s .

I n o r d e r t o v e r i f y t h e v i r t u a l work s o l u t i o n p r o c e d u r e ,

a s e p a r a t e program was w r i t t e n t o o b t a i n ax i symmet r i c i n f l a -

t i o n s o l u t i o n s f o r t h e same t o r o i d t h a t was a n a l y d by Hsu.

The r e s u l t i n g , ene rgy m i n i m i z i n g , s t r a i n s o l u t i o n s a r e com-

p a r e d i n F i g u r e 19 w i t h t h e s t r a i n s o l u t i o n s o b t a i n e d by

Runge-Kut ta i n t e g r a t i o n . The good compar i son , s e e n i n

F i g u r e 1 9 , i s p a r t i c u l a r l y s i g n i f i c a n t when one r e c a l l s t h a t

i t i s t h e deformed c o n f i g u r a t i o n f u n c t i o n s t h a t a r e a p p r o x i -

mated by t h e v i r t u a l work s o l l l t i o n p r o c e d u r e and t h e s t r a i n s ,

*The d e f l e c t e d t o r o i d s o l u t i o n o b t a i n e d i n Re f . 1 7 1 c a n n o t b e u s e d f o r compar i son p u r p o s e s b e c a u s e o f ( a ) t h e manner i n which t h e m a t e r i a l p r o p e r t i e s a r e f o r m u l a t e d , (b ) t h e a p p r o x i - ma t ion employed i n t h e n o n l i n e a r s t r a i n - d i s p l a c e m e n t e q u a t i o n s , and (c) an i n s u f f i c i e n t number o f f i n i t e e l e m e n t s employed.

Extens ion Ratio ( A )

(3) -- Integration of Equilibrium Equations

(b) x \c r# y Potential Energy Minimization

c R ~ W N Meridian Angle (el), deg

Figure 19. Comparison of meridional ( A ) and circumferential ( A 2 ) strains 1 obtained by (a) integration o f the force equilibrium equations, and (b) minimization of the potential energy functional with a 6-term trigonometric series.

il and h a r e c a l c u l a t e d from t h e g r a d i e n t s o f t h e s e 2 ' approx imated f u n c t i o n s .

O t h e r C o n s i d e r a t i o n s . Some c o n s i d e r a t i o n s h o u l d b e

g i v e n t o o t h e r a p p r o x i m a t i o n s and a s sumpt ions made i n t h e

imp lemen ta t i on o f t h e v i r t u a l work s o l u t i o n p r o c e d u r e . The

a p p r o x i m a t i o n s and a s sumpt ions l i s t e d below have n o t been

e x t e n s i v e l y i n v e s t i g a t e d f o r t h e p r e s e n t mode l ; t h e y a r e

men t ioned h e r e t o comple te t h e n u m e r i c a l a n a l y s i s d i s c u s s i o n .

1 . I n t e g r a t i o n

Numerical e v a l u a t i o n o f t h e doub le i n t e g r a l v i r t u a l

work e q u a t i o n ( 5 8 ) i s c u r r e n t l y done by c o n v e n t i o n a l

S impson ' s Ru le . C o n s i d e r a t i o n s h o u l d b e g i v e n t o t h e number

o f i n t e g r a t i o n s t a t i o n s t a k e n and t h e f e a s i b i l i t y o f u s i n g

unequa l i n t e r v a l l eng ths - - -w i th s h o r t e r i n t e r v a l s t a k e n i n

t h e v i c i n i t y o f t h e c o n t a c t r e g i o n . O t h e r i n t e g r a t i o n

p r o c e d u r e s c o u l d be c o n s i d e r e d .

2 . O p t i m i z a t i o n

The Newton-Raphson i t e r a t i o n p r o c e d u r e , employed f o r

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 a l g e b r a i c e q u a t i o n s (63),

c o u l d p o s s i b l y be r e p l a c e d w i t h some th ing more s u i t e d t o

l o n g and c o m p l i c a t e d e q u a t i o n s (which w i l l a r i s e when

o r t h o t r o p i c m a t e r i a l i s c o n s i d e r e d ) . An a l t e r n a t e t e c h n i q u e ,

t h e F l e t c h e r - Powel l method, h a s undergone p r e l i m i n a r y

i n v 5 s t i g a t i o n and i s d i s c u s s e d i n Appendix B .

3 . C o o r d i n a t e Func t i ons

I t i s t a c i t l y assumed, by t a k i n g p r o d u c t s o f t r i g o -

n o m e t r i c f u n c t i o n s a s t h e c o o r d i n a t e f u n c t i o n s ( c f .

E q . ( 7 5 ) ) , t h a t t h e e x a c t s o l u t i o n f o r a d e f l e c t e d t o r o i d a l

membrane can b e e x p r e s s e d i n t e rms of s e p a r a b l e f u n c t i o n s . "

Al though t h i s a s sumpt ion i s s u c c e s s f u l l y employed f o r t h e

s o l u t i o n o f many l i n e a r p a r t i a l d i f f e r e n t i a l e q u a t i o n s ,

t h e r e i s no a p r i o r i knowledge t h a t i t i s v a l i d f o r t h e

n o n l i n e a r p a r t i a l d i f f e r e n t i a l e q u a t i o n s which a r e i m p l i c i t l y

s o l v e d by t h e v i r t u a l work s o l u t i o n p r o c e d u r e . Some

c o n s i d e r a t i o n s h o u l d be g i v e n t o t h e f e a s i b i l i t y o f employ-

i n g o t h e r g e o m e t r i c a l l y a d m i s s i b l e c o o r d i n a t e f u n c t i o n s

( i n c l u d i n g t h e n o n s e p a r a b l e ) which may be c l o s e r t o t h e

e x a c t s o l u t i o n t h a n a r e t h e t r i g o n o m e t r i c c o o r d i n a t e

f u n c t i o n s c u r r e n t l y u s e d .

The p r e c e d i n g c o n s i d e r a t i o n s a r e r e g a r d e d a s somewhat

academic a t t h i s t i m e . They may be investigated f o r t h e

pu rpose o f " f i n e t u n i n g " t h e s o l u t i o n . I t h a s been t h e

e x p e r i e n c e o f t h e a u t h o r t h a t S impson ' s Rule i n t e g r a t i o n

w i t h t r i g o n o m e t r i c c o o r d i n a t e f u n c t i o n s w i l l e f f i c i e n t l y

p roduce s a t i s f a c t o r y s o l u t i o n s t o p rob lems such a s t h e

d e f l e c t e d t o r o i d a l membrane a n a l y z e d i n t h i s document.

- -

* e . g . , s in (O1 + 0 ) i s a s e p a r a b l e f u n c t i o n o f e l and B 2 ; 2 s i n ( e l x e 2 ) i s n o t s e p a r a b l e .

4 . 3 . 4 1;XPLKIMENTAL CONFIRMATION. A comprehensive

e x p e r i m e n t a l program aimed a t measur ing s t r a i n s a s w e l l a s

t h e deformed c o n f i g u r a t i o n o f a p r e s s u r i z e d and d e f l e c t e d

t o r o i d a l membrane h a s y e t t o b e u n d e r t a k e n . Thus, t h e

c a l c u l a t e d r e s u l t s r e p o r t e d i n t h i s document a r e m o s t l y

unconfirmed by expe r imen t . Some c o n f i r m a t i o n can be drawn

from t h e s m a l l amounts o f d a t a , on an i n n e r t u b e and a c t u a l

t i r e s , which have been p u b l i s h e d .

I n n e r Tube Expe r imen t s . Reference [ 7 ] appea r s t o b e

t h e o n l y l i t e r a t u r e s o u r c e c o n t a i n i n g e x p e r i m e n t a l d a t a on

a p r e s s u r i z e d and d e f l e c t e d t o r o i d a l membrane; t h e e x p e r i -

ments were per formed w i t h a s t a n d a r d b u t y l i n n e r t u b e .

A 1 though t h e deformed c o n f i g u r a t i o n measurements r e p o r t e d

i n [ 7 ] were made a t v e r t i c a l d e f l e c t i o n s which exceed t h e

maximum v e r t i c a l d e f l e c t i o n p o s s i b l e w i t h t h e p r e s e n t

a n a l y t i c t i r e model*, t h e r e a r e two a s p e c t s o f t h e measured

d a t a which p r o v i d e e x p e r i m e n t a l c o n f i r m a t i o n o f t h e c a l c u l a t e d

d a t a o b t a i n e d from t h e a n a l y t i c t i r e model.

1. E l l i p t i c a l Contac t Boundar ies

The b o u n d a r i e s of c o n t a c t r e g i o n s o b t a i n e d by d e f l e c t i n g

t h e i n n e r t u b e a g a i n s t a smooth ca rdboa rd s u r f a c e were found

*The a n a l y t i c t i r e model i s f o r m u l a t e d f o r u n r e s t r i c t e d de fo rma t ion ; however , t h e c o n t a c t c o n s t r a i n t , a s c u r r e n t l y i n t r o d u c e d v i a t h e s l a c k v a r i a b l e z , must b e p o s i t i o n e d a t a l o a d e d r a d i u s which exceeds t h e r a d i u s o f t h e u n i n f l a t e d e q u a t o r .

t o b e n e a r l y p e r f e c t e l l i p s e s . Th i s i s p a r t i c u l a r l y s i g n i -

f i c a n t i n view o f t h e f a c t t h a t d e f l e c t i o n s up t o o n e - h a l f

o f t h e m e r i d i o n a l r a d i u s ( R ) were imposed on t h e i n n e r 1

t u b e . The much s m a l l e r d e f l e c t i o n s imposed on t h e a n a l y t i c

t i r e model a l s o p roduced c o n t a c t r e g i o n s t h a t were bounded

by n e a r l y p e r f e c t e l l i p s e s .

2 . L i n e a r L o a d - D e f l e c t i o n Behav io r

P o i n t s on t h e l o a d - d e f l e c t i o n c u r v e f o r t h e i n n e r t u b e

a r e p l o t t e d i n F i g u r e 2 0 from d a t a g i v e n i n Tab l e 5 o f [ 7 ] .

Although o n l y t h r e e p o i n t s were measu red , t h e y do i n d i c a t e

'C -- -- V e r t i c a l

, i I I I

Load ( l b )

,- - - L - r i.l : 5

I 2.0

D e f l e c t i o n ( i n )

F i g u r e 2 0 . L o a d - d e f l e c t i o n d a t a f o r a b u t y l i n n e r t u b e . From Tab le 5 o f Re f . [ 7 ] . ( p r e s s u r e = - 4 9 p s i ) .

e s s e n t i a l l y l i n e a r l o a d - d e f l e c t i o n behav io r f o r t h e i n n e r

tube ( a s found a l s o wi th pneumatic t i r e s ) and conf i rm t h e

l i n e a r i t y of t h e l o a d - d e f l e c t i o n behav io r c a l c u l a t e d f o r

t h e a n a l y t i c t i r e model ( c f . F i g . 15) a t s m a l l e r v e r t i c a l

d e f l e c t i o n s .

Cord Load Measurements. The measurement o f cord l o a d

by s t r a i n gage t r a n s d u c e r s i s now common p r a c t i c e . A review

o f t h e t echn iques employed i s g iven by Wal t e r [ 2 1 ] . Typ ica l

cord load c y c l e s f o r a s lowly r o l l i n g b i a s - p l y t i r e a r e

shown i n F igure 2 1 , t aken from Reference [ 2 2 ] . The crown

Figure 2 1 . Typica l co rd l o a d c y c l e s f o r a b i a s - p l y t i r e . From [ 2 2 ] .

reg ion c y c l e i n F igure 2 1 i s n e a r l y symmetric about t h e

c o n t a c t c e n t e r . A symmetric c o r d l o a d c y c l e i s expec ted

because t h e crown reg ion l i e s i n t h e e q u a t o r i a l p l a n e of

symmetry and i s t h e r e f o r e f r e e of s h e a r s t r e s s . The n e a r l y

an t i symmet r i c b e h a v i o r of t h e s i d e w a l l co rd l o a d c y c l e ,

s een i n F igure 21 , i n d i c a t e s t h a t c o n s i d e r a b l e r o t a t i o n o f

t h e p r i n c i p a l s t r e s s axes i n t h e s i d e w a l l t a k e s p l a c e d u r i n g

passage over t h e c o n t a c t r e g i o n . The r o t a t i o n o f t h e

p r i n c i p a l axes i s a consequence of t h e s h e a r s t r e s s t h a t i s

induced when t h e t i r e r o l l s through t h e c o n t a c t r e g i o n o r

i s s t a t i c a l l y loaded . The p r i n c i p a l d i r e c t i o n s c a l c u l a t e d

i n t h e deformed s i d e w a l l o f t h e s t a t i c a l l y loaded a n a l y t i c

t i r e model ( s e e F ig . 13) cor respond t o p r i n c i p a l a x i s

r o t a t i o n s of up t o 4 5 d e g r e e s . S t r e s s c a l c u l a t e d i n t h e

d i r e c t i o n o f an imaginary cord embedded i n t h e s i d e w a l l o f

t h e a n a l y t i c t i r e model w i l l e x h i b i t t h e asymmetric b e h a v i o r

seen i n F igure 2 1 .

The r eason f o r t h e asymmetric c o r d l o a d b e h a v i o r i s

c l a r i f i e d by examining t h e p r i n c i p a l axes i n d i c a t e d a t A

and B i n F igure 2 2 where t h e maximum p r i n c i p a l s t r e s s i s i n

t h e d i r e c t i o n of t h e s m a l l a r rows . P r i o r t o c o n t a c t l o a d i n g ,

t h e cord l o a d (due t o i n f l a t i o n ) i s i d e n t i c a l a t a l l p o i n t s

of t h e s i d e w a l l c i r c u m f e r e n c e . A f t e r c o n t a c t l o a d i n g , t h e

maximum p r i n c i p a l s t r e s s a t A i s o r i e n t e d n e a r l y i n t h e cord

d i r e c t i o n and i n c r e a s e s t h e co rd l o a d w h i l e a t B t h e maximum

p r i n c i p a l s t r e s s has r o t a t e d away from t h e cord d i r e c t i o n ,

C c o n t a c t 4 F i g u r e 2 2 . P r i n c i p a l d i r e c t i o n s a t p o i n t s o f t h e deformed

s i d e w a l l which a r e e q u i d i s t a n t from t h e c o n t a c t c e n t e r .

t h e r e b y r e d u c i n g t h e c o r d l o a d ( t h e l o a d i s t r a n s f e r r e d t o

c o r d s i n an a d j a c e n t p l y ) . As t h e t i r e i s s t a t i c a l l y

d e f l e c t e d , o r r o l l e d s l o w l y i n t o t h e c o n t a c t r e g i o n , t h e

r o t a t i o n o f t h e p r i n c i p a l s t r e s s a x e s (which can b e c a l c u l a t e d

w i t h t h e a n a l y t i c t i r e model) i s r e s p o n s i b l e f o r asymmetr ic

c o r d l o a d c y c l e s i n a l l p o i n t s o f t h e m e r i d i a n e x c e p t a t t h e

crown.

C la rk and blcIvor [ 2 3 ] have e x p l a i n e d asymmetr ic c o r d

l o a d phenomena by c a l c u l a t i n g t h e s h e a r s t r e s s i n an a n n u l a r

membrane model o f t h e d e f l e c t e d t i r e s i d e w a l l .

Bead P r e s s u r e Measurement . P r i o r t o v e r t i c a l

d e f l e c t i o n , t h e c o n t a c t p r e s s u r e between t h e bead o f a t i r e

a n d t h e 11ead s c a t on t h e mounting ri~rl i s s y n l ~ n c t r i c abou t

t h e a x l e . When t h e t i r e i s d e f l e c t e d a g a i n s t t h e pavement ,

t h e bead c o n t a c t p r e s s u r e i s modula ted by t h e n o n a x i -

symmet r i c d i s t r i b u t i o n o f t i r e s t r e s s a t t h e b e a d r i n g .

Normal bead c o n t a c t p r e s s u r e measurements have r e c e n t l y been

p u b l i s h e d by W a l t e r [ 2 4 ] ; t h e s e measurements p r o v i d e an

i n d i c a t i o n o f t h e r a d i a l s t r e s s d i s t r i b u t i o n which s h o u l d

b e t r a n s m i t t e d t o t h e bead r i n g by t h e a n a l y t i c t i r e model .

F i g u r e 23 r ep roduces t h e bead s e a t c o n t a c t p r e s s u r e

d i s t r i b u t i o n s found f o r a r a d i a l - p l y t i r e i n [ 2 4 ] . Al though

bead p r e s s u r e i s h i g h l y dependent on rim l o c a t i o n , t h e

p r e s s u r e d i s t r i b u t i o n measured a t l o c a t i o n s 2 and 3 a r e

s i m i l a r t o t h e d i s t r i b u t i o n o f r a d i a l s t r e s s c a l c u l a t e d a t

t h e bead r i n g o f t h e a n a l y t i c t i r e model . The s i m i l a r i t y

may be due t o t h e f a c t t h a t t h e measured t i r e was o f r a d i a l

c a r c a s s c o n s t r u c t i o n , w i t h t h e t i r e l o a d b e i n g t r a n s m i t t e d

p r i m a r i l y a l o n g r a d i a l c o r d s . Re fe r ence [24] does n o t

p r e s e n t d a t a f o r o t h e r c a r c a s s c o n s t r u c t i o n s .

I n view o f t h e good compar i son o f t h e r a d i a l s t r e s s

d i s t r i b u t i o n w i t h t h e bead p r e s s u r e measurement i n F i g u r e

2 3 , i t i s b e l i e v e d t h a t t h e c a l c u l a t e d s h e a r s t r e s s d i s t r i -

b u t i o n shown i n F i g u r e 1 2 i s r e a l i s t i c , a t l e a s t f o r a

r a d i a l - p l y t i r e . No d a t a h a s been p u b l i s h e d on t h e d i s t r i b u t i o n

o f s h e a r p r e s s u r e i n t h e t i r e - r i m i n t e r f a c e .

J m (D t-t 3 (D t-t G V,

e 04 r.

co 3 t-t r

m n 3 hJ m P P U'C * H-, r. 0 n Y

t-t P- Y (D

n n - F a rt P

11 n 11 C

P P w m

m . t-t - m o F CL e u b

(D w @J

P CL

II V, rt

+ Y CD

vl V, 0 V, P ';d

F Y

B E A D C O N T A C T PRESSURE- PS I

5 . 0 CONCLUSIONS

I t i s e v i d e n t , from t h e c a l c u l a t e d r e s u l t s (Chap. 4 . 0 ) ,

t h a t t h e i s o t r o p i c t o r o i d a l membrane s t r u c t u r e i s a

p r e l i m i n a r y model f o r t h e de fo rma t ion and s t r e s s a n a l y s i s

o f t h e s t a t i c a l l y l oaded t i r e . A t e c h n i q u e f o r o b t a i n i n g

f i n i t e de fo rma t ion and s t r e s s s o l u t i o n s ha s been deve loped

and demons t r a t ed t o be v i a b l e ; t h e r e s u l t i n g computer

program i s e f f i c i e n t and s o l u t i o n s can b e o b t a i n e d by u s e

o f a medium-size computer .

Al though t h e mechanics o f how t h e pneumat ic t i r e

c a r r i e s t h e v e h i c l e l o a d have l o n g been u n d e r s t o o d i n b r o a d

t e r m s , t h e d e t a i l s ( e . g . , t h e p r i n c i p a l s t r e s s d i r e c t i o n s )

o f how t h e l o a d i s d i s t r i b u t e d i n t h e t i r e s t r u c t u r e a r e n o t

w e l l u n d e r s t o o d . The s t u d y o f s t r a i n and s t r e s s d i s t r i b u t i o n s

i n an i d e a l i z e d model o f t h e pneumat ic t i r e , such a s t h e

a n a l y t i c t i r e model d e s c r i b e d i n t h i s r e p o r t , w i l l promote an

i n c r e a s e d u n d e r s t a n d i n g of t h e s t a t i c l o a d - c a r r y i n g mechanics

o f an a c t u a l t i r e .

5 . 1 THEORETICAL FOUNDATIONS

The v i r t u a l work f o r m u l a t i o n o f t h e g e n e r a l membrane

d e f o r m a t i o n t h e o r y , p r e s e n t e d i n Chap te r 2 , i s t h e " o t h e r

s i d e o f t h e co in" i n t h e ma thema t i ca l d e s c r i p t i o n o f a

p h y s i c a l sys tem. The f o r c e e q u i l i b r i u m e q u a t i o n s f o r f i n i t e

d e f o r m a t i o n s , and t h e boundary c o n d i t i o n s , may be d e r i v e d

d i r e c t l y from t h e v i r t u a l work e q u a t i o n ( 5 8 ) . Approximate

8 9

s o l u t i o n s o f t h e v i r t u a l work e q u a t i o n a r e a l s o approx imate

s o l u t i o n s of t h e f o r c e e q u i l i b r i u m e q u a t i o n s .

TWO s i g n i f i c a n t advan t ages* o f t h e v i r t u a l work f o r m u l a t i o n

a r e :

1. The phenomenological t h e o r y o f m a t e r i a l

p r o p e r t i e s i s d i r e c t l y u t i l i z e d a s t h e s t r a i n

energy d e n s i t y whose f i r s t v a r i a t i o n i s s e t

e q u a l t o t h e v i r t u a l work.

2 . There i s no need t o e x p l i c i t l y d e r i v e a p o t e n t i a l

f u n c t i o n a l whose f i r s t v a r i a t i o n i s t h e v i r t u a l

work o f e x t e r n a l l o a d s .

The second advan t age , l i s t e d above , makes t h e v i r t u a l work

p r i n c i p l e f a r more a t t r a c t i v e t h a n t h e minimum p o t e n t i a l

energy p r i n c i p l e , which does r e q u i r e an e x t e r n a l l o a d

f u n c t i o n a l . For c o n s e r v a t i v e sys tems such a s t h e p r e s e n t

a n a l y t i c t i r e model , t h e v i r t u a l work and minimum p o t e n t i a l

energy p r i n c i p l e s a r e equ iva len t - the minimum p o t e n t i a l

energy p r i n c i p l e b e i n g d e r i v e d from t h e p r i n c i p a l o f v i r t u a l

work.

An a p p l i c a t i o n s advantage i s t h e r e l a t i v e (compared t o

working w i t h t h e e q u i l i b r i u m e q u a t i o n s ) e a s e i n d e t e r m i n i n g

approx imate f u n c t i o n s which d e s c r i b e t h e deformed c o n f i g u r a -

t i o n . The u t i l i z a t i o n o f a s e r i e s o f g e o m e t r i c a l l y

*Advantage 1 i s i n comparison t o a t t a c k i n g t h e problem w i t h f o r c e e q u i l i b r i u m e q u a t i o n s . Advantage 2 i s i n comparison w i t h t h e e a s e o f f o r m u l a t i n g a p o t e n t i a l ene rgy f u n c t i o n a l .

a d m i s s i b l e c o n f i g u r a t i o n f u n c t i o n s w i t h c o n s t a n t c o e f f i c i e n t s ,

which was employed by R i t z [ Z S ] f o r min imiz ing a p o t e n t i a l

energy f u n c t i o n a l , r e a d i l y c o n v e r t s t h e f u n c t i o n a l e q u a t i o n

( 5 8 ) o f t h e v i r t u a l work p r i n c i p l e t o a 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 a l g e b r a i c e q u a t i o n s ( 6 3 ) . The s o l u t i o n i s t h e n

o b t a i n e d by c o n v e n t i o n a l o p t i m i z a t i o n methods , such a s t h e

Newton-Raphson method employed h e r e i n .

The t h e o r e t i c a l f o u n d a t i o n s which have been deve loped

f o r t h e p r e s e n t a n a l y s i s can now b e e x t e n d e d t o account f o r o t h e r

i m p o r t a n t e f f e c t s , e . g . , t h e i n f l u e n c e of c o r d r e in fo rcemen t and

bending s t i f f n e s s , on t h e s t r u c t u r a l b e h a v i o r o f t h e

s t a t i c a l l y - l o a d e d t i r e . The t e c h n i q u e s u s e d t o i n c l u d e

c o r d r e i n f o r c e m e n t and bending s t i f f n e s s i n ax i symmet r i c

a n a l y s e s [ I , 2 1 o f t h e t i r e l oaded o n l y by i n f l a t i o n

p r e s s u r e can be u t i l i z e d f o r e x t e n d i n g t h e a n a l y t i c c a p a b i l i t y

o f t h e p r e s e n t t i r e model . A ma jo r b e n e f i t , t o b e d e r i v e d

from i n t r o d u c i n g c o r d r e i n f o r c e m e n t s , w i l l be t h e a b i l i t y

t o c a l c u l a t e t h e i n f l u e n c e o f c o r d p a t h on t h e d i r e c t i o n o f

p r i n c i p a l s t r e s s i n t h e v i c i n i t y o f t h e c o n t a c t r e g i o n . I n

s t u d y i n g t h e a n a l y t i c t i r e model s o l u t i o n s p r e s e n t e d i n

t h i s r e p o r t , i t w i l l be n o t e d t h a t t h e model h a s much more

f l e x i b i l i t y * t h a n i s e x h i b i t e d d u r i n g t h e q u a s i - s t a t i c l o a d i n g

o f a r e a l t i r e . The i n t r o d u c t i o n o f bending s t i f f n e s s w i l l

p r o v i d e t h e n e c e s s a r y c o n t r o l o f t h e t i r e model f l e x i b i l i t y .

*T!-lere i s an e x c e s s i v e amount of v e r t i c a l t r a n s l a t i o n when t h e i n f l a t e d t o r o i d a l membrane i s b r o u g h t i n t o c o n t a c t w i t h a r i g i d s u r f a c e . C o n s i d e r a b l e v e r t i c a l d e f l e c t i o n i s needed t o p roduce even a s m a l l c o n t a c t r e g i o n .

The i n t r o d u c t i o n o f t h e c o n t a c t c o n s t r a i n t b y means

o f t h e " s l a c k varia-ihle" i s a d e f i n i t c compu ta t i ona l con-

v e n i e n c e . However, a p e n a l t y i s p a i d by a c c e p t i n g t h e

r e s t r i c t i o n t h a t t h e l oaded r a d i u s canno t b e l e s s t h a n t h e

r a d i u s o f t h e u n i n f l a t e d e q u a t o r . At tempts have been made

t o remove t h i s c u r r e n t r e s t r i c t i o n on t h e l o a d e d r a d i u s b u t i t

appea r s t h a t t o do s o w i l l r e q u i r e t h e i n t r o d u c t i o n o f one

more dependent v a r i a b l e , such a s a Lagrange m u l t i p l i e r .

5 . 2 ROLLING CONTACT ANALYSIS

C l e a r l y , s i g n i f i c a n t e x t e n s i o n s Q f t h i s model a r e

r e q u i r e d t o y i e l d t h e d e f o r m a t i o n and s t r e s s of t h e l o a d e d ,

r o l l i n g t i r e . The a d d i t i o n a l e f f e c t s o f p r imary

impor t ance i n r o l l i n g t i r e per formance a r e ( a ) c e n t r i f u g a l

f o r c e s , ( b ) r h e o l o g i c a l f o r c e s , and ( c ) t a n g e n t i a l f o r c e s

p roduced by s l i d i n g f r i c t i o n . The e f f e c t o f c e n t r i f u g a l

f o r c e s on t h e ax i symmetr ic shape o f an i n f l a t e d c o r d r e i n f o r c e d

membrane model o f a t i r e h a s been s t u d i e d by W a l t e r [ 2 6 ] and

by Zorowski [ 2 7 ] . C e n t r i f u g a l f o r c e s a r e i n c l u d e d i n t h e

r o l l i n g c o n t a c t a n a l y s i s deve loped i n Re fe r ence [ 6 ] , b u t

d e f o r m a t i o n s and s t r e s s e s have n o t been c a l c u l a t e d a t v a r i o u s

r o l l i n g s p e e d s . The continuum a s p e c t s o f t h e c e n t r i f u g a l

f o r c e f o r m u l a t i o n i n [ 6 ] c o u l d e a s i l y be i n c o r p o r a t e d

i n t o t h e p r e s e n t a n a l y t i c t i r e model . The i n t r o d u c t i o n

o f r h e o l o g i c a l ( t ime -dependen t d e f o r m a t i o n ) f o r c e s i n t o

*A 4 - y e a r t i m e t a b l e f o r c o n t i n u e d development of t h e a n a l y t i c t i r e model i s g i v e n i n Table 1 (Chap. 1 ) of t h i s r e p o r t .

9 2

t h e p r e s e n t a n a l y s i s may be p o s s i b l e by means o f t h e d i s s i p a t i o n

p o t e n t i a l d i s c u s s e d by Bychawski [ 2 8 ] . Th is e x t e n s i o n o f

t he a n a l y t i c t i r e model would e n a b l e t h e r o l l i n g r e s i s t a n c e

o f a t i r e t o be c a l c u l a t e d a s a f u n c t i o n o f t i r e d e s i g n and

o p e r a t i n g v a r i a b l e s . I t i s a l s o p o s s i b l e t h a t t h e e f f e c t

o f s l i d i n g f r i c t i o n f o r c e s on t i r e de fo rma t ion and s t r e s s

cou ld be a n a l y z e d by means o f a d i s s i p a t i o n p o t e n t i a l . The

v i r t u a l work f o r m u l a t i o n i s n o t l i m i t e d , however , t o mono-

g e n i c ( d e r i v a b l e from a p o t e n t i a l ) f o r c e s and t h e r e may b e

more advantageous ways o f i n c l u d i n g s l i d i n g f r i c t i o n i n t o

t h e p r e s e n t a n a l y t i c f o r m u l a t i o n .

ACKNOWLEDGEMENTS

The a u t h o r i s i n d e b t e d t o Marie S h i h f o r t h e d e s i g n

o f an e f f i c i e n t computer program (PATCH) t o f i n d c o e f f i -

c i e n t s f o r t h e app rox ima te s e r i e s s o l u t i o n s , and t o Gar ry

H o l s t e i n f o r c o n t i n u e d development o f PATCH and t h e w r i t i n g

o f an e x t e n s i v e package o f p e r i p h e r a l programs t o d i s p l a y

t h e deformed c o n f i g u r a t i o n and a n a l y z e t h e r e s u l t i n g s t r a i n

and s t r e s s t e n s o r s . S p e c i a l t hanks a r e due t o J o e Dunne

f o r s e e i n g t h a t t h e most e f f i c i e n t u s e was made o f t h e

hardware p r o v i d e d by HSRI Computer S e r v i c e s .

The program documenta t ion was w r i t t e n by Garry

H o l s t e i n .

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W a l t e r , J.D. , " C e n t r i f u g a l E f f e c t s i n I n f l a t e d , R o t a t i n g B i a s P l y T i r e s , " T e x t i l e Resea rch J o u r n a l , Vo l . 4 0 , p . 1 , 1970 .

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F l e t c h e r , R . and P o w e l l , M. J . D . , "A Rap id ly Convergen t Descen t Method f o r M i n i m i z a t i o n , " Computer J o u r n a l , Vo l . 6 , No. 2 , p p . 1 6 3 - 1 6 8 , 1963 .

"Program FMFP ( F l e t c h e r and Powel l U n c o n s t r a i n e d S e a r c h Technique) ," System/360 S c i e n t i f i c S u b r o u t i n e Package , V e r s i o n 111 , Programmers Manual , GH20-0205-4. IBM C o r p . , White P l a i n s , N .Y.

L a t i n i n d i c e s t a k e t h e v a l u e s 1 , 2 , 3 . Greek i n d i c e s

t a k e o n l y t h e v a l u e s 1 and 2 . C o v a r i a n t components o f

v e c t o r s and o t h e r t e n s o r s a r e i n d i c a t e d by s u b s c r i p t s .

Cont r a v a r i a n t components a r e i n d i c a t e d by s u p e r s c r i p t s .

G = d e t (G. . ) 1 I

components o f t h e c r o s s p r o d u c t El x g2

components o f t h e c r o s s p r o d u c t El x c2

m a t e r i a l c o n s t a n t s

c o e f f i c i e n t s o f t h e c o o r d i n a t e f u n c t i o n s

c o e f f i c i e n t s i n t h e s e r i e s f o r y l

c o e f f i c i e n t s i n t h e s e r i e s f o r y 2

c o e f f i c i e n t s i n t h e s e r i e s f o r y 3 ( o r z )

r e s u l t a n t f o r c e i n t h e c o n t a c t r e g i o n

b a s e v e c t o r s on t h e deformed m i d s u r f a c e

deformed m a t e r i a l m e t r i c t e n s o r

b a s e v e c t o r s on t h e undeformed m i d s u r f a c e

undeformed m a t e r i a l m e t r i c t e n s o r

b a s e v e c t o r s on t h e s t r e s s t r a j e c t o r i e s

m e t r i c t e n s o r r e f e r r e d t o t h e s t r e s s

t r a j e c t o r i e s

undeformed t h i c k n e s s

i n v a r i a n t s o f t h e s t r a i n t e n s o r

c a r t e s i a n u n i t b a s e v e c t o r s

e i g e ~ v a l u e s o f t h e K i r c h o f f s t r e s s t e n s o r

u n i t v e c t o r normal t o undeformed m i d s u r f a c e

components o f ii r e f e r r e d t o r

u n i t v e c t o r normal t o deformed m i d s u r f a c e

components o f ii r e f e r r e d t o 7 r

components of t h e e i g e n v e c t o r s ( r e f e r r e d

t o p) of t h e K i r c h o f f s t r e s s t e n s o r ,

components o f v e c t o r s t a n g e n t t o t h e s t r e s s

t r a j e c t o r i e s

i n f l a t i o n p r e s s u r e (on deformed m i d s u r f a c e )

t a n g e n t i a l p r e s s u r e s (on deformed mid-

s u r f a c e ) i n t h e convec t ed O N - d i r e c t i o n s

t o r o i d l a t i t u d e r a d i u s

t o r o i d m e r i d i a n r a d i u s

t o r o i d r a d i u s ( a x i s t o c e n t e r o f m e r i d i a n

s e c t i o n )

l o a d e d r a d i u s

p r i n c i p a l s t r e s s e s

p h y s i c-:il t rue s t ress r c s u l t a n t s

e x t e r n a l l o a d v e c t o r

K i r c h o f f s t r e s s r e s u l t a n t t e n s o r

t o t a l s t r a i n e n e r g y

v i r t u a l work

undeformed m a t e r i a l vo lume

s t r a i n ene rgy d e n s i t y

c a r t e s i a n c o o r d i n a t e s o f a p o i n t i n t h e

undeformed m a t e r i a l

c a r t e s i a n c o o r d i n a t e s o f a p o i n t on t h e

undeformed m i d s u r f a c e

c a r t e s i a n c o o r d i n a t e s o f a p o i n t i n t h e

deformed m a t e r i a l

c a r t e s i a n c o o r d i n a t e s o f a p o i n t on t h e

deformed m i d s u r f a c e

g e o m e t r i c a l l y a d m i s s i b l e c o o r d i n a t e a

f u n c t i o n s

s l a c k v a r i a b l e

s t r a i n t e n s o r

Kronecker d e l t a

c o o r d i n a t e s f o l l o w i n g t h e s t r e s s

t r a j e c t o r i e s

c u r v i l i n e a r m a t e r i a l c o o r d i n a t e s ( convec t

w i t h de fo rma t ion )

convec t ed c o o r d i n a t e e x t e n s i o n r a t i o s

t r u e s t r e s s r e s u l t a n t v e c t o r

t r u e s t r e s s r e s u l t a n t v e c t o r a l o n s

t h e l i n e O a = c o n s t a n t

E u l e r i a n s t r e s s r e s u l t a n t t e n s o r

Lagrange s t r e s s r e s u l t a n t t e n s o r

bead a n g l e

d e t e r m i n a n t o f ( )

d i f f e r e n t i a l o f ( )

v a r i a t i o n o f ( )

Summation i s i m p l i e d by r e p e a t e d i n d i c e s i n component p r o d u c t s .

APPENDIX A

I'URII DE I:O mIAT I O N

T h i s append ix d e s c r i b e s t h e r e d u c t i o n o f t h e g e n e r a l

membrane d e f o r m a t i o n t h e o r y , p r e s e n t e d i n Chap te r 2 , t o

t h e s p e c i a l c a s e where t h e e x t e r n a l l o a d s a r e such t h a t

t h e p r i n c i p l e d i r e c t i o n s , f o l l o w e d by t h e c u r v i l i n e a r

c o o r d i n a t e s e a , a r e p r e s e r v e d d u r i n g t h e d e f o r m a t i o n . Such

d e f o r m a t i o n i s t e rmed "pure" a s i t i s p roduced o n l y by

e x t e n s i o n i n t h e p r i n c i p a l d i r e c t i o n s , and i s c o m p l e t e l y

measured by t h e e x t e n s i o n r a t i o s A 1 , A 2 , and A 3 .

(A- 1 )

h i = a 2 / b 2 ( i n c o m p r e s s i b l e m a t e r i a l )

S i n c e p r i n c i p a l d i r e c t i o n s a r e , by d e f i n i t i o n , o r t h o -

g o n a l , t h e b a s e v e c t o r s 9, on t h e deformed m i d s u r f a c e , w i l l

b e o r t h o g o n a l , and

The t e n s o r m e t r i c o f t h e deformed m i d s u r f a c e , G i j g i v e n

by E q u a t i o n ( 9 ) , w i l l now b e d i a g o n a l and may b e w r i t t e n

i n t e rms o f t h e e x t e n s i o n r a t i o s A r and g r a d i e n t s o f t h e

known f u n c t i o n s x which d e s c r i b e t h e c o n f i g u r a t i o n of r ' t h e undeformed m i d s u r f a c e .

The s t r a i n i n v a r i a n t s , g i v e n by E q u a t i o n (11) , now

r e d u c e t o t h e f o l l o w i n g f o r p u r e d e f o r m a t i o n o f an

i n c o m p r e s s i b l e m a t e r i a l .

The i n c o m p r e s s i b i l i t y a s sumpt ion p e r m i t s t h e e x t e n s i o n

r a t i o h 3 t o b e e l i m i n a t e d from i n v a r i a n t s I 1 and 1 2 .

I 2 = h i Z + h - 2 2 + " A";

With t h e s t r a i n i n v a r i a n t s g i v e n by ( A - 5 ) , t h e s t r a i n

ene rgy d e n s i t y W(11,12) , may be w r i t t e n a s an e x p l i c i t

f u n c t i o n o f t h e e x t e n s i o n r a t i o s A1 and A 2 .

The form o f (A-6) w i l l , o f c o u r s e , depend on t h e

phenomenolog ica l e l a s t i c i t y t h e o r y employed.

P h y s i c a l s t r e s s r e s u l t a n t s , T1 and T 2 , which have

t h e d imens ion o f f o r c e p e r u n i t l e n g t h o f undeformed

m i d s u r f a c e , a r e d e r i v e d d i r e c t l y f rom t h e s t r a i n e n e r g y

By e q u a t i n g f o r c e s on t h e c o r r e s p o n d i n g deformed and

undeformed m i d s u r f a c e l e n g t h s ,

t h e p h y s i c a l s t r e s s r e s u l t a n t s , S1 and S 2 , o f f o r c e p e r

u n i t l e n g t h o f deformed m i d s u r f a c e a r e found t o be

'l'he p h y s i c a l s t r e s s r e s u l t a n t s computed by E q u a t i o n s

( A - 9 ) , which a p p l y i n t h e c a s e o f p u r e d e f o r m a t i o n , a r e

t h e p h y s i c a l t r u e s t r e s s r e s u l t a n t s S and SZZ, g i v e n by 11

Equa t ion (25) which i s v a l i d f o r g e n e r a l d e f o r m a t i o n o f

membrane s t r u c t u r e s .

The f r e e i n f l a t i o n o f an ax i symmet r i c t o r o i d a l

membrane p roduces a s t a t e o f p u r e d e f o r m a t i o n .

A P P E N D I X B

ITERATIVE SOLUTION TECHNIQUES

The v i r t u a l work s o l u t i o n p r o c e d u r e , i n t r o d u c e d i n

S e c t i o n 2 . 5 , e s s e n t i a l l y r e d u c e s a n o n l i n e a r e l a s t i c i t y

p rob lem t o a p rob lem o f d e t e r m i n i n g t h e s e t o f c o n s t a n t s , a!

( i = 1 , 2 , . . . , N) which min imize a n o n l i n e a r m u l t i v a r i a t e

f u n c t i o n , JI.

Thi s append ix r e p o r t s on an i n v e s t i g a t i o n o f two methods ,

t h e F l e t c h e r - P o w e l l (F-P) and t h e Newton-Raphson ( N - R) ,

which may be u s e d t o f i n d t h e min imiz ing c o n s t a n t s , a;.

The N-R method i s c u r r e n t l y employed f o r o b t a i n i n g a n a l y t i c

t i r e model s o l u t i o n s . The F-P method w i l l b e an a t t r a c t i v e

a l t e r n a t i v e a s t h e complex i ty o f t h e a n a l y t i c t i r e model

i n c r e a s e s .

B . l TAYLOR EXPANSION

The m u l t i v a r i a t e T a y l o r e x p a n s i o n o f I7 abou t a s e t o f

c o n s t a n t s a t i s

where t h e d e r i v a t i v e s a r e e v a l u a t e d a t (a:, a $ , . . . ,a;) . With a i = a; + Aa. t h e expans ion (B-2) can be r e w r i t t e n

1 ' as

II(ar + Aa.) 1 - a = C * a i

where t h e h i g h e r o r d e r t e r m s , which a r e o f o r d e r (Lai)

o r s m a l l e r , a r e grouped i n t h e remainder t e rm o (na i ) 3 . I t

i s c l e a r t h a t t h e f i r s t two t e r n s on t h e r i g h t - h a n d s i d e

o f (B-3) dominate t h e expans ion of i l ( a t + Aai) - f l ( ap ) .

S i n c e , f o r s m a l l i nc remen t s h a i , t h e h i g h e r o r d e r te rms

may be n e g l e c t e d , t h e n e c e s s a r y and s u f f i c i e n t c o n d i t i o n s

f o r a ? t o be a s e t o f c o n s t a n t s which minimize l l (a i ) a r e

f o r a l l s u f f i c i e n t l y s m a l l i n c r e m e n t s , A a i S i n c e t h e

i nc remen t s a r e a r b i t r a r y , (B-4a) i s s a t i s f i e d o n l y i f

= 0

a!, a t , . . . a$

The l e f t s i d e o f (B-4b) i s a q u a d r a t i c form, w i t h a s s o c i a t e d

m a t r i x A whose e l emen t s a r e

The i n e q u a l i t y (B-4b) i s s a t i s f i e d i f t h e q u a d r a t i c form

i s p o s i t i v e d e f i n i t e ; t h e q u a d r a t i c form i s p o s i t i v e

d e f i n i t e o n l y i f t h e a s s o c i a t e d m a t r i x , A , i s p o s i t i v e

d e f i n i t e .

The energy f o r m u l a t i o n o f a n o n l i n e a r e l a s t i c i t y

problem, i n t h e c o n t e x t o f a c o n s e r v a t i v e sys t em, a s s u r e s

t h a t t h e m a t r i x A w i l l b e p o s i t i v e d e f i n i t e . " The m i n i -

m i z a t i o n problem i s reduced t o f i n d i n g t h e s e t o f

c o n s t a n t s a! which s a t i s f y Equat ion ( B - 5 ) .

-

X A check i s made on t h e p o s i t i v e d e f i n i t e n e s s of A t o v e r i f y t h e f o r m u l a t i o n and computer programming.

W i t h t h e v i r t u a l work s o l u t i o n p rocedure (Sec . 2 . 5 ) ,

t h e r e s u l t i n g n o n l i n e a r a l g e b r a i c e q u a t i o n s (63) co r r e spond

r t o Equa t ions ( B - 5 ) and t h e s o l u t i o n c o e f f i c i e n t s , C i j ,

co r re spond t o t h e min imiz ing c o n s t a n t s , a:.

B . 2 NEWTON- RAPHSON METHOD (N-R)

The N - R method i s q u i c k l y d e r i v e d from t h e f i r s t two

te rms o f t h e Tay lo r expans ion ( B - 3 ) , which can be

r e w r i t t e n as

n ( a p + Aa.) - n ( a ? ) = 1 1 C ~ a i

i=l

The d e r i v a t i v e s i n Equat ion (B- 7 ) a r e e v a l u a t e d a t t h e

i n i t i a l e s t i m a t e s o f a!, which a r e denoted by a!. The

d i f f e r e n c e on t h e l e f t - h a n d s i d e o f (B-7) w i l l be minimized

by nonzero inc remen t s Aai which cause t h e f i r s t term on

t h e r i g h t - h a n d s i d e t o v a n i s h . The min imiz ing inc remen t s

a r e o b t a i n e d by s o l v i n g t h e N s imu l t aneous l i n e a r e q u a t i o n s

which a r i s e by s e t t i n g t h e b r a c k e t e d terms i n (B-7) t o z e r o .

0 Thc i n i t i a l e s t i m a t e s o f t h e min imiz ing c o n s t a n t s , a i ,

a r e now corrected w i t h t h e nonze ro i n c r e m e n t s o b t a i n e d

b y s o l u t i o n o f ( B - 8 ) .

S i n c e t h e e s t i m a t e d c o n s t a n t s were c o r r e c t e d i n a manner

which min imizes t h e d i f f e r e n c e l ( a i ) - n(a:) ) t h e sum o f

t h e g r a d i e n t s , e v a l u a t e d a t e ach s e t o f c o n s t a n t s , must

s a t i s f y t h e f o l l o w i n g i n e q u a l i t y .

The c o r r e c t e d c o n s t a n t s , a ' a r e now u s e d t o e v a l u a t e t h e i ' d e r i v a t i v e s i n E q u a t i o n s (B-8) which can b e s o l v e d f o r

a n o t h e r s e t o f c o r r e c t i n g i n c r e m e n t s . Each s u c c e s s i v e s e t

o f c o r r e c t e d c o n s t a n t s r e d u c e s f u r t h e r t h e sum o f t h e

g r a d i e n t s a c c o r d i n g t o i n e q u a l i t y ( B - 9 ) . The c o r r e c t i o n

p r o c e d u r e i s r e p e a t e d u n t i l t h e c o r r e c t i o n s become o f

n e g l i g i b l e s i z e . Convergence t o t h e min imiz ing c o n s t a n t s ,

a * i s r a p i d , and a s s u r e d i f t h e s t a r t i n g v a l u e s , a!, a r e i ' s e l e c t e d i n a r e g i o n where t h e m a t r i x A , whose e l e m e n t s A i j

a r e g i v e n by ( B - 6 ) , i s p o s i t i v e d e f i n i t e .

E q u a t i o n s ( B - 8) a r e t h e s i m u l t a n e o u s l i n e a r e q u a t i o n s

o f t h e Newton-Raphson method ( t h e c o n s t a n t f a c t o r , 112, i s

abso rbed i n t h e c o r r e c t i o n s ) . A n a l y t i c t i r e model

s o l u t i o n s a r e o b t a i n e d by assuming i n i t i a l e s t i m a t e s ,

0 > s L k a , o f t l ic C O I I S ~ U I I ~ S for tllc funct iol l l ; "1:q. ( o n ) ) r

which w i l l app rox ima te t h e deformed c o n f i g u r a t i o n , The

i n i t i a l v a l u e s a r e c o r r e c t e d by i n c r e m e n t s , A C ~ ~ , which

a r e o b t a i n e d by s o l u t i o n o f t h e f o l l o w i n g 3 x M x N

Newton- Raphson e q u a t i o n s .

(sum on s )

i j ~F:J where Fr and - , from E q u a t i o n s ( 6 3 ) , a r e e v a l u a t e d

ac;, 0 s a t t h e i n i t i a l v a l u e s Cke The s o l u t i o n i s improved by

t h e c o r r e c t i o n s

and t h e p r o c e d u r e i s r e p e a t e d u n t i l t h e c o r r e c t i o n s have

n e g l i g i b l e e f f e c t on t h e deformed c o n f i g u r a t i o n . Care

must b e t a k e n i n s u r e t h a t t h e s t a r t i n g v a l u e s , c:~, c o r r e s p o n d t o a r e a s o n a b l e a p p r o x i m a t i o n o f t h e deformed

c o n f i g u r a t i o n . For t h e a n a l y t i c t i r e mode l , t h e s t a r t i n g

v a l u e s which c o r r e s p o n d t h e undeformed

c o n f i g u r a t i o n ( y r = x r ) , were u s e d .

A major d i s a d v a n t a g e o f t h e N-R method , when a p p l i e d

t o t h e m i n i m i z a t i o n of c o m p l i c a t e d f u n c t i o n s , i s t h e

r equ i r emen t f o r t h e s econd d e r i v a t i v e s , A i j , t o b e p r o -

grammed e x p l i c i t l y . F u r t h e r development o f t h e a n a l y t i c

t i r e model i s a lmos t c e r t a i n t o r e s u l t i n a f u n c t i o n ~ i j

whose g r a d i e n t s (which co r r e spond t o A. . ) a r e p r o h i b i t i v e l y 1 I

l e n g t h y t o p rogram.

B . 3 FLETCHER-POWELL METHOD (F-P)

The F-P method [ 2 9 ] i s an o p t i m a l s e a r c h i n g scheme

f o r f i n d i n g t h e min imiz ing c o n s t a n t s , a!, w i t h o u t p r i o r

knowledge o f t h e s econd d e r i v a t i v e s o f t h e f u n c t i o n TI.

T h i s method approx imates t h e s econd d e r i v a t i v e m a t r i x , A i j

g i v e n by ( B - 6 ) , a s t h e s e a r c h p r o g r e s s e s . The F-P method

i s programmed as S u b r o u t i n e FMFP i n t h e IBM S c i e n t i f i c

S u b r o u t i n e Package [ 3 0 ] .

The F-P method r e q u i r e s a t l e a s t N e v a l u a t i o n s o f t h e

f u n c t i o n n and i t s g r a d i e n t s , art/aai ( i = l , 2 , . . . ,N) . T e r m i n a t i o n o c c u r s when t h e g r a d i e n t s become n e g l i g i b l e .

The convergence t e s t u s e d by FMFP i s met when

where E i s an e r r o r l i m i t s e t by t h e u s e r .

The p o t e n t i a l ene rgy f u n c t i o n a l , i t s e l f , i s r e q u i r e d

i n o r d e r t o app ly t h e F - P method t o o b t a i n s o l u t i o n s from

t h e a n a l y t i c t i r e model. The g r a d i e n t s o f t h e p o t e n t i a l

ene rgy a r e t h e f u n c t i o n s F:], whose r o o t s a r e c u r r e n t l y

found by t h e N - R method. However, t h e second d e r i v a t i v e s . .

o f t h e p o t e n t i a l e n e r g y , which c o r r e s p o n d t o t h e

r e q u i r e d f o r t h e N-R method, a r e n o t needed by t h e F-P

method.

B .4 COMPARISON

A comparison o f t h e Newton-Raphson and F l e t c h e r - P o w e l l

methods was made by u s i n g b o t h methods t o o b t a i n t h e ene rgy

min imiz ing c o e f f i c i e n t s f o r a t r i g o n o m e t r i c s e r i e s

app rox ima t ion o f t h e i n f l a t e d shape o f a bonded t o r o i d a l

membrane. Th i s problem had p r e v i o u s l y been s o l v e d by Hsu

[ 2 0 ] , who used t h e Runge-Kut ta method t o i n t e g r a t e t h e

n o n l i n e a r f o r c e e q u i l i b r i u m e q u a t i o n s d e f i n e d by t h e i n f l a t e d

t o r o i d . The s o l u t i o n s o b t a i n e d by t h e N - R and F-P methods

a r e i n good agreement w i t h H s u ' s s o l u t i o n .

The computing t imes l i s t e d i n Table B-1 , be low , a r e

f o r 12* c o e f f i c i e n t s , a f , which minimize t h e n o n l i n e a r

f u n c t i o n n ( a i , a 2 , . . . , a N ) . The t imes l i s t e d i n Tab l e B - 1

were r e q u i r e d f o r t h e s o l u t i o n t o meet t h e N - R convergence

t e s t : Aai < ,002 ( i = 1 , 2 , . . . , N ) , and t h e F-P convergence

t e s t : E = .0001 ( s e e E q . ( B - 1 2 ) ) . -- - - - -

*N=12; 6 t e rms zach f o r t h e x and y c o o r d i n a t e s o f t h e i n f l a t e d m e r i d i a n .

TABLE B - 1

PDI' 1114 5 COFIPU'I'ING TIMES EQUI R E D RY TIIE NIIWTON -RAPIISON A N D FLETCHEK-POWELL METHODS TO blINIMI ZE A

blULTIVARIATE FUNCTION

Dimens ion l e s s Computing Time ( s e c ) Shape o f

Run P r e s s u r e . P N-R F - P rI

I n t h i s p a r t i c u l a r p rob l em, t h e i n f l a t i o n p r e s s u r e h a s

a s i g n i f i c a n t i n f l u e n c e on t h e c h a r a c t e r o f t h e p o t e n t i a l

e n e r g y , 2 . As t h e p r e s s u r e i n c r e a s e s , t h e p o s i t i v e

d e f i n i t e n e s s o f becomes l e s s p ronounced; g e o m e t r i c a l l y ,

t h e f u n c t i o n becomes s h a l l o w e r . A t p ' 2 . 1 , t h e d e f o r m a t i o n

o f t h i s p a r t i c u l a r t o r o i d a l membrane becomes u n s t a b l e and

t h e e n e r g y m i n i m i z a t i o n method i s u n a b l e t o f i n d a s o l u t i o n .

I t a p p e a r s t h a t t h e N-R method i s more s e n s i t i v e t o

t h e p o s i t i v e d e f i n i t e c h a r a c t e r o f rI t h a n i s t h e F-P

method. Th i s b e h a v i o r i s p e r h a p s n o t unexpec t ed a s t h e N-R

method makes u s e o f t h e s econd d e r i v a t i v e m a t r i x A , w i t h

e l e m e n t s A i j g i v e n by ( B - 6 ) , and w i l l f a i l when A i s s i n g u l a r .

b loreover , t h e N - R method may f a i l i f A i s i n d e f i n i t e ; t h i s

i s i n d i c a t i v e o f u n s t a b l e d e f o r m a t i o n . The F - P method does

n o t make u s e o f t h e c u r v a t u r e k o f TI and a p p a r e n t l y i s a b l e

t o e f f i c i e n t l y f i n d t h e minimum o f v e r y s h a l l o w f u n c t i o n s .

*The c u r v a t u r e i s i n d i c a t e d by t h e m a t r i x A .

Table B-2 l i s t s s i x o f t h e min imiz ing c o n s t a n t s found

b y t h e two methods i n runs 1 and 2 . The c o r r e s p o n d i n g

c o n s t a n t s a l l a g r e e t o t h e 3 r d s i g n i f i c a n t f i g u r e ; most of

t h e c o n s t a n t s a g r e e t o t h e 4 t h s i g n i f i c a n t f i g u r e . The

d e f o r m a t i o n , s t r a i n , and s t r e s s s o l u t i o n s , which a r e o b t a i n e d

by e v a l u a t i n g t h e t r i g o n o m e t r i c s e r i e s w i t h t h e energy

min imiz ing c o e f f i c i e n t s , were found t o a g r e e t o f o u r

s i g n i f i c a n t f i g u r e s .

TABLE B-2

COMPARISON OF MINIblIZ ING CONSTANTS FOUND BY THE NEWTON-RAPHSON AND THE FLETCHER-POWELL METHODS

x 1 0 - I x 1 0 - I x 1 0 - I x10m2 x 1 0 - ~

Run a 1 a 2 a 3 a4 a6 Method

.2892 ,2067 - . I 3 4 3 ,05342 - ,2214 ,1147 N - R 1

,2892 ,2067 - . I 3 4 3 .05341 - .2214 . I 1 4 5 F - P

,9681 ,6474 - .2769 ,1114 - ,5149 - . 2515 N - R 2

, 9681 ,6475 - . 2769 ,1114 - ,5150 - , 2515 F - P

B.5 CONCLUSIONS

The F l e t c h e r - P o w e l l method has been i d e n t i f i e d a s a

v i a b l e a l t e r n a t i v e t o t h e Newton-Raphson method f o r f i n d i n g

min imiz ing c o n s t a n t s , a ? , o f a m u l t i v a r i a t e f u n c t i o n ,

I I (a l , a 2 , . . . , a N ) Al though t h e computing t i m e , i n t h e

comparison above , was s i g n i f i c a n t l y more f o r t h e F - P method,

t h e r e i s a d i s t i n c t advan t age w i t h t h e F - P method i n

t h a t i s does n o t r e q u i r e t h e second d e r i v a t i v e s o f t h e

f u n c t i o n b e i n g min imized . For compl i ca t ed f u n c t i o n s , which

may e a s i l y a r i s c from no re complex problcms ( such as may

i n c l u d e a n i s o t r o p i c m a t e r i a l p r o p e r t i e s o r bending

s t i f f n e s s ) , t h e F - P method may a c t u a l l y p rove t o b e more

e f f i c i e n t ; t h i s w i l l be t r u e when t h e second d e r i v a t i v e s

become e x t r e m e l y compl i ca t ed and burdensome t o program.

When t h e de fo rma t ion s o l u t i o n o f t h e problem b e i n g

s o l v e d by ene rgy m i n i m i z a t i o n i s c l o s e t o i n s t a b i l i t y , t h e

computing t i m e f o r t h e N - R method i n c r e a s e s s i g n i f i c a n t l y .

The computing t ime f o r t h e F - P method a p p e a r s ( r e l a t i v e t o

t h e N - R method) t o be l e s s s e n s i t i v e t o s o l u t i o n s t a b i l i t y .

APPIINDIX C

COMPUTER PROGRAMS

T h i s append ix c o n t a i n s a l l o f t h e t i r e model computer

programs n e c e s s a r y t o o b t a i n t h e c a l c u l a t e d r e s u l t s

p r e s e n t e d i n C h a p t e r 4 . The programs a r e w r i t t e n i n

FORTRAN IV and make e x t e n s i v e u s e o f f i l e s t o r a g e hardware

( d i s k and DEC-tape) a v a i l a b l e on t h e PDP 11 /45 computer a t

HSRI. F lowchar t 1 (on t h e n e x t page) shows t h e h i e r a r c h i c a l

o r g a n i z a t i o n o f t h e package o f n i n e programs t h a t were

w r i t t e n t o o b t a i n t h e v a r i o u s t y p e s o f t i r e model o u t p u t .

A t i r e model run b e g i n s w i t h program PATCH which r e a d s

t h e i n p u t d a t a and f i n d s t h e s o l u t i o n c o e f f i c i e n t s , a i j ,

' ij , and c i j , which s a t i s f t y t h e v i r t u a l work e q u a t i o n (77) . These c o e f f i c i e n t s , and t h e i n p u t d a t a , a r e s t o r e d i n random

a c c e s s d i s k f i l e 10 f o r s u b s e q u e n t u s e by o t h e r programs

(BKD, PTCHCV, and TENSOR) which e v a l u a t e and d i s p l a y ( i n t a b l e s

and g r a p h s ) v a r i o u s c h a r a c t e r i s t i c s o f t h e t i r e model r e s u l t s .

Program BND s c a n s t h e deformed c o n f i g u r a t i o n i n t h e v i c i n i t y

o f t h e c o n t a c t r e g i o n t o l o c a t e t h e c o n t a c t bounda ry . The

boundary p o i n t s a r e s t o r e d i n f i l e 4 (DEC-tape) f o r s u b -

s e q u e n t p l o t t i n g . The o t h e r computer-drawn p l o t s p r e s e n t e d

i n t h i s document a r e o f d a t a p roduced by PTCHCV which s t o r e s

c o o r d i n a t e s i n s e q u e n t i a l a c c e s s d i s k f i l e s 2 and 3 f o r

s u b s e q u e n t p l o t t i n g by t h e Calcomp 565 d i g i t a l p l o t t e r .

Program TENSOR e v a l u a t e s components o f t h e s t r a i n and s t r e s s

t e n s o r s , p r i n c i p a l s t r e s s e s , and p r i n c i p a l s t r e s s

d i r e c t i o n s a t s e l e c t e d p o i n t s on t h e convec t ed c o o r d i n a t e

c u r v e s , A g r e a t d e a l o f i n f o r m a t i o n i s c o n t a i n e d i n t h e

s o l u t i o n c o e f f i c i e n t s ( a i j , b i j , and c i j ) ; t h e p e r i p h e r a l

programs can b e e a s i l y augmented o r supplemented t o o b t a i n

o t h e r t i r e model o u t p u t and d i f f e r e n t forms o f d i s p l a y .

Program L O O K p r i n t s o u t t h e s o l u t i o n c o e f f i c i e n t

l i b r a r y ( F i l e 10) when a check on t h e c o n t e n t s i s d e s i r e d .

S o l u t i o n c o e f f i c i e n t s from 20 t i r e model r uns can b e s t o r e d

on F i l e 1 0 , as c u r r e n t l y d e f i n e d by PATCH.

Tab l e C-1 , be low , c o n t a i n s a l i s t o f t h e s i g n i f i c a n t

FORTRAN v a r i a b l e s t h a t a r e d e f i n e d i n t h e v a r i o u s computer

p rograms . The f o l l o w i n g s e c t i o n s p r e s e n t f l o w c h a r t s ,

l i s t i n g s , and example i n p u t - o u t p u t f o r t h e programs a p p e a r i n g

i n F lowchar t 1.

TAHL1; C - 1

SI(;NII:ICANT FOR'I'MN PROGRAM VARIABLES

S u b s c r i p t s i and j range ove r t h e harmonics of e l and

e 2 , r e s p e c t i v e l y . Other lower c a s e L a t i n l e t t e r s t a k e t h e

va lues 1, 2 , and 3 . Greek l e t t e r s t a k e on ly t h e va lues 1

and 2 .

Program Theory D e s c r i p t i o n

A ( i , j 1 unknown c o e f f i c i e n t s i n

t h e s e r i e s approximations B ( i , j )

of y l ' Y 2 ' and y 3 , C ( i , j > r e s p e c t i v e l y

r Er n c a r t e s i a n components of

a v e c t o r i n t h e d i r e c t i o n

o f maximum p r i n c i p a l s t r e s s

GAMMA

GLaB

G U a B

magnitude of c o v a r i a n t base

v e c t o r s on t h e deformed

midsurface

ang le between base v e c t o r s

c o v a r i a n t components of

deformed s u r f a c e m e t r i c

t e n s o r

c o n t r a v a r i a n t components

of deformed s u r f a c e m e t r i c

t e n s o r

p r i n c i p a l s t r e s s magnitudes

TABLE C - 1 ( C o n t . )

SUB

SRr

DLY r a

TKaB

T d r )

DTYaB ( r , s )

DY r ( a )

Y r ( i , j >

Y Y l ( r , i , j )

Y Y 2 ( r , i , j )

p h y s i c a l components o f

t r u e s t r e s s

c o n v e c t e d c o o r d i n a t e

e x t e n s i o n r a t i o s

K i r c h o f f s t r e s s t e n s o r

Lagrange s t r e s s t e n s o r

c o o r d i n a t e s of a p o i n t on

t h e deformed m i d s u r f a c e

c o o r d i n a t e f u n c t i o n s

S o l u t i o n S p e c i f i c a t i o n Data

maximum ( i , j ) i n d e x

v a l u e s f o r y l and y 2

maximum ( i , j ) i n d e x

v a l u e s f o r y 3 o r z

number o f s t a t i o n s f o r

S impson ' s r u l e i n t e g r a t i o n

i n t h e and O 2 d i r e c -

t i o n s , r e s p e c t i v e l y (must

b e odd) 125

P

PHIB

C 1 H

TABLE C - 1 (Cont . )

T i r e Model I n p u t Data

P i n f l a t i o n p r e s s u r e

Ob bead a n g l e

Clh neo-hookean membrane

s t i f f n e s s

m e r i d i o n a l r a d i u s

Rt t o r o i d r a d i u s

R~ l o a d e d r a d i u s

C . 1 PRIMARY PROGRAMS

The p r imary t i r e model programs a r e PATCH, B N D , PTCHCV,

and TENSGR. These programs d e t e r m i n e t h e s o l u t i o n c o e f f i -

c i e n t s f o r a t i r e model run (PATCH), f i n d t h e c o n t a c t

boundary (BND) , and e v a l u a t e d e f o r m a t i o n , s t r a i n , and

s t r e s s i n f o r m a t i o n (PTCHCV, TENSOR) from t h e s o l u t i o n

c o e f f i c i e n t s .

C . l . l PATCH. A t i r e model r u n b e g i n s w i t h program

PATCH which r e a d s i n p u t d a t a t h a t ( a ) d e f i n e t h e t i r e model

( r a d i i and m a t e r i a l c o n s t a n t s ) , ( b ) g i v e t h e t i r e model

o p e r a t i n g v a r i a b l e s ( p r e s s u r e and l o a d e d r a d i u s ) , and ( c )

s p e c i f y t h e p r o p e r t i e s o f t h e s o l u t i o n t o b e o b t a i n e d (number

o f t e rms and i n t e g r a t i o n s t a t i o n s ) . The program u s e s t h e

c o n v e n t i o n a l S impson ' s r u l e a l g o r i t h m t o pe r fo rm t h e doub le

i n t e g r a t i o n o f Equa t ion (76) f o r e ach s e t o f i n d i c e s ( i , j ) . The i n t e g r a l s a r e accumula ted a s t h e e l e m e n t s i n t h e

m a t r i x e q u a t i o n f o r t h e Newton-Raphson i t e r a t i v e s o l u t i o n

p r o c e d u r e ( s e e Appendix B ) . S u b r o u t i n e TRISQ, a m o d i f i e d

Cholesky decompos i t i on r o u t i n e , i s c a l l e d t o s o l v e t h e

Newton-Raphson m a t r i x e q u a t i o n f o r c o e f f i c i e n t c o r r e c t i o n s .

The c o r r e c t e d c o e f f i c i e n t s * a r e t h e n u s e d t o recompute t h e

double i n t e g r a l s o f E q u a t i o n (76) and t h e Newton-Raphson

*A t i r e model run u s u a l l y s t a r t s w i t h a l l s o l u t i o n c o e f f i - c i e n t s s e t e q u a l t o z e r o . However, p r o v i s i o n i s made f o r s t a r t i n g w i t h t h e c o e f f i c i e n t s found by a p r e v i o u s r u n .

c o r r e c t i o n p r o c e s s i s r e p e a t e d . The i t e r a t i v e s o l u t i o n

p r o c e d u r e i s t e r m i n a t e d when a l l o f t h e c o e f f i c i e n t

c o r r e c t i o n s a r c l e s s t h a n I 'T 'S(=.001) . T h e p r o g r e s s of t h c

i t e r a t i v e s o l u t i o n i s mon i to r ed by p r i n t i n g o u t t h e

c o e f f i c i e n t c o r r e c t i o n s a f t e r e ach i t e r a t i o n .

The s t r u c t u r e o f program PATCH i s diagrammed i n

F lowchar t 2 , on t h e f o l l o w i n g p a g e s . Tab l e C-2 d e s c r i b e s

t h e program c o n t r o l v a r i a b l e s t h a t a r e n e c e s s a r y f o r t h e

e x e c u t i o n o f PATCH. O t h e r i n p u t d a t a a r e d e s c r i b e d i n

Tab l e C-1 . An example run f o l l o w s t h e l i s t i n g s o f PATCH

and t h e subprograms FSUM, RHSUM, and TRISQ.

START c? / Read t i r e model and s o l u t i o n I ( s p e c i f i c a t i o n d a t a .

Zero t h e i n i t i a l

s o l u t i o n c o e f f i c i e n t s .

Compute S impson ' s Rule m u l t i p l i e r s

f o r e ach i n t e g r a t i o n s t a t i o n . -1 I

L O O P

Newton-Raphson i t e r a t i o n

I Accumulate d o u b l e i n t e g r a l s o f Eq. ( 7 6 )

a t e ach i , j a s N-R m a t r i x e l e m e n t s . RHSUM

I C a l l TRISQ t o s o l v e N-R e q u a t i o n s

I f o r s o l u t i o n c o e f f i c i e n t c o r r e c t i o n s . 1

Flowcha r t 2 . PATCH

Write c o c f f i c i e n t c o r r e c t i o n s

on d i s k f i l e 6 .

C o r r e c t t h e s o l u t i o n c o e f f i c i e n t s .

C

END LOOP 4 ( c o r r e c t i o n s 1 < E

- 1

Flowchar t 2 . PATCH ( c o n t . )

(no s o l u t i o n found)

( i o i Y

W r i t e t i r e model i n p u t d a t a

and s o l u t i o n c o e f f i c i e n t s on

d i s k f i l e 1 0 - ,

YES (convergence t o a s o l u t i o n )

( l i b r a r y s t o r a g e )

Wr i t e s o l u t i o n c o e f f i c i e n t s

on d i s k f i l e 6

(PATCH O u t p u t )

LOOP

I f o r c o o r d i n a t e c u r v e s I

LOOP

S impson ' s Rule s t a t i o n s

E v a l u a t e :

i ) Deformed c o n f i g u r a t i o n , y r

i i ) E x t e n s i o n r a t i o s , h r

on d i s k f i l e 6

END LOOPS

P r i n t c o n t e n t s

o f d i s k f i l e 6 Re tu rn t o START

F lowcha r t 2 . PATCH ( c o n c l . )

TABLE C - 2

PATCH PROGRAM CONTROL DATA

FORTRAN V a r i a b l e D e s c r i p t i o n

IRECR r e c o r d number r e f e r e n c i n g

s o l u t i o n c o e f f i c i e n t s from

a p r e v i o u s r u n s t o r e d i n

d i s k f i l e 1 0

r e c o r d number s p e c i f y i n g

l o c a t i o n i n d i s k f i l e 10

where s o l u t i o n c o e f f i c i e n t s

from t h i s r u n w i l l b e

s t o r e d

c o e f f i c i e n t i n i t i a l i z a t i o n

s w i t c h , IRS=O: i n i t i a l

c o e f f i c i e n t s a r e z e r o ,

IRS=1 : i n i t i a l c o e f f i c i e n t s

a r e r e a d from d i s k f i l e 10

( r e f e r e n c e d by IRECR)

c PROGRAM P A T C H C T H I S PROGRAM DETFRHINES COORDINATE FUNCTION C O E F F f C I E N f s WHICH C P T N I M I Z E THL POTEWTIAL E N F R G Y , F R O M THLSE COEFf I C I E N T S THE C CARTESIAN LOCUS OF THE CONVECTED COORDINATE C I J R V E S C I) THETA*ZBB, C 11) T H E T A - Z z P I C 111) TH€TA- l :@. C IS COMPUTED TOGETHER WITH INwPLANE AND T R A N S V E R S E EXTENSION c R A T I O S E v r L u a T E n A L O ~ G THESE CURVES, C C C STORAGE ALLOWED FOR A M A X I M U M OF M A s h A ~ 4 r MBaNBr4 , Mco~Ca4, C AND NS1= i3 ,NSZ=ZS C

I N T E G E R PASS 01 M E N S I O N ~ ( U ~ U ) , B ( U , ~ ) ~ ~ ~ U ~ ~ ) , R H A T ( ~ Z ~ U ) I R H S ( ~ ~ ) ~ S M ~ ~ ~ , ~ ~ ) ~

1 Y ( 3 ) r Y l ( 4 , 4 l , Y 2 ( 4 r 4 ) 1 Y ~ t ~ ~ 4 ~ ~ ? Y Y 1 ( 3 , 4 , ~ ) 1 Y V 2 ( 3 ~ U r U ) r 3 Z ( U ~ U ) ~ Z ~ ~ U , U ) ~ Z Z ( Y I ~ ) ~ D Z ( ~ I I u T 1 ( 3 ) r T i ? ( 3 J , 5 T I T L E ( Z 0 ) r 6 O Y 1 ( 2 ) r D Y Z t Z ) , b V 3 ( 2 ) t 7 D T Y 1 1 ( 3 , 3 ) r D T Y 1 2 ( 3 1 3 l ~ b T Y 2 1 ~ 3 1 3 ~ ~ D T Y 2 2 ( 3 ~ 3 I t A R M ( 1 2 2 4 ) , A R C ( 4 8 )

COFMON D T Y ~ ~ , ~ T Y t Z l D T Y Z l , b T Y Z 2 , Y Y 1 , Y Y Z ~ T 1 , T Z ~ A A , 1 ~ J , K ~ L EQUIVALENCE ( H A , H B ) , ( N A , N B ) , ( A B C ( ~ ) , A [ ~ , ~ ) I ~ ( A B C ( ~ ~ ) ~ B ~ ~ ~ ~ I I P

1 [ A B C $ 3 3 ) r C ( l r l ) ) DATA P I ~ S ~ 1 ~ 1 5 9 2 ~ / r ~ T R Y ~ ~ 5 ~ ~ E ~ S / ~ 0 0 1 /

e C F I L E h I S A Y R I N T F D OUTPllT BUFFER, C F f ~ f l a Is A RANDOH ACCESS L I B R A R Y OF M I Y I M I Z I N G C O E F f I C f E N T 8 C PREVIOUSLY CALCIJLATED BY THE PATCH P R O G R A M , C

CALL A S S I G N ( b r ' D K B t P A T C H * L S T t r 131 CALL A S S I G N ( l B r ' D K 1 t [ t l B 1 3 ] C O E F F a b A T ~ 1 C ~ 2 ? ~ I E R R ~ D E F I k E F I L E 1fl(BB,50,lJ, T N D X I

C 1 T l ~ ~ t 9 ~ c m s t a e )

C R E A D ( 1 0 ' 1 )

t , c.....IF FND OF C A R D INPUT ENCOUNTEREO T E R ~ I N A T E P R O G R A M EXECUTION', C , , , O f H E P H I S F R E A D T I T L E C A R D FOR NEXT DATA SET,

R F A D ( 5 , 1 0 @ r P N D t 9 9 ) T I T L F c C S P E C I F Y T I R E MODEL PARA METERS^ C I ) NUHRER nF T E R M S I N OOUbLE S E R I E S C COOPDINATE FUNCTION EXPANSIONS C 7 1 ) NUMBER O f S T A T I ~ N S 1 N S I M P S O N ' S RULE C NUMFRICAL I M T E G R A T I n N ALGORITHM C 111) PRESSURE, I N I T I A L GEOMETRY, AND MATERIAL C PROPERTIES OF THE TOROIDAL MEMBRANE

R F A D ( S t 1 2 @ ) Y A ~ N A , ~ C ~ N C , N S ~ , N S ~ , P I P H I B , C ~ H I R I , R ~ R L C M R t M A , NBeNA (BY EQU!VALENCE) c , C , . , , , l R S IS AN O P T I O N SWITCH DETERMINING THE METHOD OF I N I T I I L ! Z A T ! Q N

C FOR THE C O O R D I b A T f F U N C T I O N C O L F F I C T E N T S I 6 I 1 I! I R S EQUALS ZFRn THE C O E F F I C I E N T S ARE I N I T I A L I Z E D A7 ZERO: C 1 1 ) , , , f l T H k R k I S E THE C O F F F I C I E N T S USED ARE THOSE PRODUCED BY A C P R E V I O U S RUN O f P A T C H , I R E C R R E I N G A P O I N T E R T O THE R E G I O N c OF FILE 1g CONTAINING THE DESIRED S T A R T I N G COEFFICIENTS',

R E A D ( 5 , Q O P ) I ~ F C R , I R E C W , I R Q b, v , t , . . , , I N I T I A L f Z € THE COEFFIC IENTS OF COORDINATE F U N C t I O Y S ,

DO 4 Ka1,Ua

E S T A B L I S H RFFEREHCE CONSTANTS T O L O A D A R R A Y S R M l t AND ~ " 8 : c..,,.

l U R l = M I N l I U B 2 = J U R l + M 2 N 2 I l J R 3 = I U R 2 + M 3 N 3

C C

HeHAX@(MA,MC) N a H A X n ( M A , N C )

C * ? .

P H I R D e P H l O C , . , , , C O N V E R T 1 0 R A D I A N MEASURE,

P H f B t P H l B * . f l l 7 4 5 3 3 C c , '

' C O M P U T E M U L T I P L I E R FOP EACH S T A T I O N I N SIHPSON'S RULE C * . . . . C f N T E G R A T l n N ALGORITHM,

H l = P H I B / ( N S l = l ) H2=PI/(NS?*l) S I N T r H l * H 2 / 9 , P I B T P P I / P H I R C 2 s S I N T + S I N T T 4 g C Z + C ?

C F I R S T COLUMN S M ( 1 , j l = S I N T S H ( 2 , 1 ) = 7 1 1 DO 1 8 1 ~ 4 , N S 1 # 2 S l + ' ( I = l , l ) = C 2

l a S W I , J ) ~ T U S H ( N S l , I ) = S I N T

C 9EC4ND AND T H I R D COLUMNS D O 1 1 I = l r N S l 8 M ( I , 3 ) t S W ( I r i ) ? S ~ ( I r I )

t l S ~ ( 1 , 2 l = S ~ ( I , 3 ) + 9 ~ ( r r 3 ) C MIDDCF COLUMNS ( 3 T O N S r l ) , IF ANY

D O 20 J ~ 4 t N S 2 r Z Da 20 I = l r N S l S H ( I , J ~ I ) = S M ( I r 3 1

20 SM(I,J)=SM(l,ZI C LAST COLUMN

04 21 I = l r N S I 21 S w ( I , N S ? ) = S M ( I r l )

C

" C B * * * * ' S e l 1SW.B AND BEGIN NEWTON-RIPHSDN L I N E b R 1 Z A I 1 0 N I D CORRECT C COORDINATE F U N C T I O N C O E F F I C I E N T $ ,

13w=0 D O 95 P A S S = ~ , H T R V

* . H R l T E ( Q , U f l B I P A S S t , , , . , Z E R O NEWTON M A T R I X C

D O 22 I = l r I lJBF9 2 2 R M A T ( 1 ) = 0 .

00 23 I ~ l r I u 6 3 2 3 R H S ( I ) = f l .

C 25 S Z 8 0 ,

D O 6 5 N 2 = l , N S ?

0 o n 0 0 T J C ) n o 0 - I - 0 .

lA luw lU NIU * m

'u * m e rU-- - TU * a CPmm Q c9 . * Q' 4 -

U ~ ~ ~ N N N W ~ ~ U ~ < < < < M U ~ V ) W ~ ) V ) ~ O I I O ~ ? U O N O U < O O 4 u P X l O C n ~ ~ ~ 0 r3C#l(ncrcr 3 O N N I N N - - ~ ~ ~ ~ - < ~ - T ~ ~ O ~ O ~ ~ * U F U ~ X ) < N J U N ~ < 4 - + < - 2 ~ 9 O w X J C l T 3 O H - 3 7

~ n n I t n n w n n - - * ~ ~ - n n n V , Z W z I# H I X- - - U I U I V W - C C ~ I I ~ ~ ~ 2 r - v M Z W - - w - + n ~ - ~ w w - C . , ~ - ~ - - - - - ~ L , U C L . C ~ E . ~ Z . ~ ~ - W T W C ~ ~ - U - - ~ + X S X J U J U B I ~ T ~ L F ~ n i n s z c r

~ w u N 1 - L a I IU@ 8 1 ~ ~ - a 8 03Wh)TUCUrU;OmNC ~ ~ ~ D R J W I I + I U ~ U B + + - + - M C ~ C NRIr W03 !% Z I I I I + L L u Z I - w < - - C Z 3 : N N P f l I # * * *S'P. I I H X J - - ~ ~ - X J X - * + ~ I w X X - ~ I@ 11 S

C W W ~ U W U O ~ I H - ~ - W = ~ I I II n c n m w n -4 ( A - S I I u w a II t s r a x o u , - ~ z z n o - w ( T I N N - 1 1 (I m - - + 3 3 - - - - n - u ~ ~ n & ~ w c l . ~ l l ~ w ~ - 9 - 9 1 - q X l & n - - 3 r Z ' V O - 0 - 4 Z - - I8 --14 8 1 0 ( < w L L r 9 Q - # C D Z t D * f l - - x J - ; 5 1 ) ~ = 9 ~ * v ) z - m t l c L . T , C n I U ' U c r c r . + u U C ) W I U ~ L d n n + M - 0 ul ,r - y ) e c ) ~ I T ) m ~ n - H + n- - - - Y G ) r U B @ i n - - > 11 U W W - n o - - b S r 0 \ C O S O C l ) < n Z Z P -

",= v-% cncn

3 + * - ~ * w S 4 ~ - - 4 1 * S ~ ~ C t w w - x o nu3 . c 3 z c n O m o c n - - ~ ~ w Z W r U z 9 C I ~ I E O V ) ~ + - - - - U * ~ N - - X+ * w u u XI N.Z+ o m w v , r u o I X - U W ~ ~ O Y U- cow

--N.CI)uOTUly+ + - O N * W . W a B t B 3 (R+ W + - Z = B o c OX-- IU .- ~ ~ - P Z O D ~ P ~ C O J ~ ~ W ~ - N ~ J u R - n CA V)IV a3 1 0 0 w m 1 I ~ ( U P L C I ~ ~ ~ ~ ~ O . ~ L ~ U D ~ ~ ~ U U L N I n J + rr 0 P. 0 G)

L t * &h)C- - H H * W W Z - 1 b NT)U S H L *) 00 U - L C A P C n t r r i A - * ( N L L R 1 . W -4 U 3- V) Z 0 - 0 X+ ~ - u t 7 J U W f v C G - * ~ - 0 3 c c Tsl Fn w B V) P 4 NN L O 3 9 - - n O N N ~ 9 Cb Z -4 -.I + a nJ- 4 - * S U O * 7 N V) m * v 9 n- COC. <<- O m t u IU + 0. HH rua *(LC.- L \ cn 2 6 1r * m N 1 U - - W Z C) CI T?u Cn t L -- N t + Z * 3 ,N- u u M .-L rn H D r% -0

1 - hl 0 s -4 xcn u w Z N H - - 0

V, N 0 < n Z C 4 u L 4 - uu < V, m u 3

C- -a .. > -13 < Z O t N .. Q z r

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m r 0 Ca 0 m - '-'. * 0 1) u u L - --I m N I

C l = l * P I R T A l : C l * S I S l N A l t S I N ( A 1 ) C O S A l o C O S t A 1 ) DO 348 Ja1,NR S T N J M l r S I N ( ( J - l ) * S 2 ) C O S J M I a C 0 3 ( ( J ~ l l * S 2 ) Y Z ( I , J ) t S I N A l * C f l $ J M l Y Y l [ Z # 11 J ) = C l * C o S A l * C O S J M l Y Y ~ ~ ~ ~ I ~ J I S ~ ( J ~ I ) * S ~ N A ~ * S X N J H ~ Y ( ~ ) ~ Y ( ~ ) + B [ I # J I * V ~ ~ I I J l O Y ~ ( ~ ) + B Y Z ( I ) + B ( I ~ J ) * V V ~ ( Z ~ I I J ) D Y 2 ( 2 ) ~ 0 Y t ? ( Z I + B l I r J ) * Y Y Z ( t ? , 1 1 Jl

34B C O N T I N U F c , . , C , , , , , C A L C U L A T E Y 3 A k D I T S D E R I V A T I V E S F R O M THE C d N 8 T R A I N t COORDINATE;

00 28 I + l # Y C D O 28 J z l r N C Y ~ ( I ~ J ) = = ~ . * Z Z * Z ( I ~ J ) Y Y ~ ( ~ , I , J I = * ~ ~ * ( Z Z * Z ~ ( I , J ) + Z C I I J ~ * O Z ( ~ ) > Y Y ~ ( ~ ~ I ~ J ) ~ * ~ , * ( ~ Z * Z Z ( I I J ) + Z ( I I J ) * D Z ( E ) )

28 C O N T I N U E Y ( J):RL.ZZ*ZZ D Y 3 ( l ) = ~ Z t * Z Z * D Z ( l 1 D Y ~ ~ ~ ~ O - ~ * * ~ Z * D Z ~ ~ )

c , . c , . . . .DEFINE S O M E C O N S T A N T S :

C C , . , . . C O H P U T E C O H P O N E N T S OF A V E C T O R NORMAL T O THC O E P O R M E O MIDSURFACE,

81~0VZ(l)*OY3(2)-~Y3(1)*OY2(2) 8 2 = D Y 3 ~ l ~ * D Y 1 ~ 2 ) - b Y 1 ~ 1 , 1 * 0 ~ 3 ( 2 ) 83~OYl(l)*DY2~2)-0YZ(!)*DYt~Z)

c , . , C. , , , ,COHPUTE T R A Y S V E R S F E X T E N S I O N R A T I O ( S R 3 ) C AND I T S T H I R D AND F O U R T H P O W E R ,

$ R 3 ~ R B * R 1 / S Q R T ( B l * B l + 82*82 + 83*831 S R 3 C s S R 3 * * 3 SR3FzSR5**4

c, C,.,,.COHPUTE D E R I V A T I V F S OF 9R3,

C l r T R B * T R l * S R 3 C DCV11~Cl*(DV3(E)*8Z=DY2~Z)*B3) D L Y 1 2 = C \ * ~ D Y Z ( l ) * B 3 - D Y 3 ~ 1 ) * R 2 ) DLY2l:CI*CDYlI2)*R3-DY3{Z)*Bt) OLY22~Cl*(DY3(1)*R1-DYl(l)rR3) DLY3l=Ci*(DY2(2)*Si~DYi(Z~*RZl OLVfZ~C1*tOYl~l)*82-D~2(1)*191)

c , C . . , . , C A L C U b A T E L A G R A N G e S T R E S S R E S U L T A N T S ,

T E Y P = 2 , * C I H T l ( l ) = T F ~ P * ~ T R j * D Y l ( l ) + S R 3 * O L Y l l ) T 2 ( l l ~ T E M P * [ T R B * O Y 1 ( L ) + S R 3 * D L Y i Z ) T 1 ( 2 ) : T F M P * ( T R l * D Y 2 ( 1 ) + S R 3 * D L V Z t ) T 2 ( 2 ) = T F M P * ( T R B * D Y Z ~ Z J + SR3*bLY22)

1 1 ( 3 ) t T E M P * t T R i * D Y 3 Q ) t S R 3 * D L Y 3 I ) 1 2 ( 3 ) = T E H P * ( T R % * D Y 3 1 2 ) + S R f * D L Y 3 E )

c . C.,.,.IF M I N I M I Z I N G C f l E F F I C I E Y T S HAVE BEEN POUND, EVALUATE C AND P R I N T LOCUS OF C O W V E C T E D COORDJNATE CURVES C AND 4 S S O C T A T E D E X T E N S I O M Q 4 T I O S .

I P ( I S W ) 9 9 ' 3 l , l Q c, CI. . . ,CDHPUTE I N m P L A N E E X T E N S I O N R A T I O S ( 3 R 1 , S R Z )

38 S R ~ ~ S Q ~ T ~ D Y ~ ( ~ ~ * B Y ~ ( ~ ~ + D Y ~ ( ~ ) * O Y ~ C ~ ~ + I ) Y S ~ ~ ~ * D Y S ~ ~ ~ ~ / R ~ SRZ=SQRT(~Yl(2)*D~l(Z~+DY2(2)*OY2(2)+DY3~2)*OYS(~))/R0

C Y R I T E ( b r 4 5 0 ) $2,SlrY[l),Y(2)rY(3),SRl,SRZ,SR3 G O T O 64

C c, C,.,,,COMPUTE L A G R A M G E STRESS D E R I V A T I V E S ,

3 1 C l a T E M P * T Q B * T R l CZ=U,*SR3C C 3 3 Q Y 3 ( 2 ) * B Z * O Y Z ( T ) * R 3 D T Y l ~ ~ l ~ l ~ r T € ~ P * T ~ l * ( l , + C 2 * D L Y l l * T R 0 * C 3 ~ B R 3 f c T R B * ( b Y 3 ( . 2 ) * * 2 +

+ D Y 2 ( L ) * * L l ) D T Y ~ 2 ( l r l l ~ C l * ( C 2 * D L Y l 2 * C 3 + S R ~ F * ( O Y ~ ( ~ ~ * ~ Y ~ ( ~ ) I C D Y Z ( E ) * O Y ~ ! ( ~ ) ) ) D T Y l l ( ~ r Z ) ~ C l * ( C 2 * D L Y ? l * C 3 + S R 3 P * D Y Z I Z ) * O Y 1 ( 2 ) 1 D T Y 1 2 ( 1 r 2 ) = C l * ( C 2 * D L Y ; E Z * C 3 * S R J F * ( D Y 2 ( 2 ) * O Y 1 ( \ ) + 0 3 1 ) DTYll(lr3)=Cl*(C2*0LY31*Cf+ SR3F* D V 3 ( O ) * D Y l ( Z ) ) OTYlZ(lr3)~Ci*(E2*DLY32*Cf+ S R 3 F * ( f l ~ * O Y S 1 Z ) * D Y I f l ) ) ) c 3 0 o Y 2 ( 1 1 * e s - o Y 3 c 1 1 ~ ~ 2 D T Y 2 1 ( 1 , l ~ = o T Y 1 2 ( l r l ) DTY22( l , l ) a T E M P t T R B * ( l , + C 2 * D L Y i2*TRl*C3* S R ~ B I T R ~ * ~ O Y ~ ( 1)**2i

t D Y 3 ( 1 ) * * 2 ) ) DTYZi(lr2):C1*(C2*OLY2l*C5+ S R 3 P * ( 8 3 ~ b Y 2 ( 1 ) * 0 V I O ] ) D T Y 2 2 ( 1 , 2 1 r C l * ( C Z * D L Y 7 Z * C 3 + 3R3F* P Y Z t l ) * D Y l ( l ) l D T Y Z ~ ( ~ , ~ I : C ~ ~ ~ C Z * D L Y ~ I * C ~ Q S~3F*(B~+OY3(ll*DYl(2))) D T Y Z Z ( i , 3 ) ~ C l * ( C 2 * D L Y 3 Z * C 3 + Q R 3 F * t b V l ( l ) * D Y S ( l ) ) ) C s z O Y 1 ( 2 ) * 8 3 ~ D Y 3 ( i ? ) w D T Y ~ ~ ~ ~ , ~ ) : T E ~ P * T R ~ * ( ~ , + C ~ * O L Y Z I ~ S T R ~ ~ C ~ ~ S R 3 F * T R B r ( D Y l ( 2 ) r ~ L +

+ D Y 3 [ 2 ? * * 2 ) ) D T Y ! Z ( ~ I ~ ) = C ~ * ( C ~ * D ~ Y ~ Z * C ~ + SRSF* (DY I ( ~ ) * o Y l ( i ) * ~ ~ f C 2 ) * 0 ~ 3 ( 1 ) 1 ) DTY!l(ZtJlrCl*(CZ*OLY31*C3+ SR3F* b Y S t Z ) * D V Z ( Z ) ) D T Y ~ ~ ( ~ ~ ~ ) S C ~ * ( C ~ * D L Y J ~ * C ~ ~ S R 3 F * ( B 1 + D Y 3 ( 2 ) * O Y Z ( I l ) ) C 3 = 0 Y 3 ( l ) * B l ~ O Y 1 1 1 ) * B f ~ T Y Z ~ ( ~ I Z ) J D T Y I Z ( Z I Z ~ O T Y 2 2 ( 2 , 2 ) r T E M P * T Q @ * ( I r + ~ 2 * ~ ~ ~ 2 2 * ~ ~ i * ~ 3 e S R S F * T R l * t D Y 3 ( l l * * Z +

+ O Y l l l ) * * Z ) ) D T Y Z l [ 2 , 3 ) ~ C l * ( C 2 * D L Y 3 1 * C 3 + S R S F * ( B ~ W ~ Y S ( ~ ) * D Y ~ ( ~ ) ~ DTV22(2,3)zcl*(CL*DLY32*C3+ S R S F * D Y f ( l ) * D V Z I l ) ) C f + ~ Y 2 ( ? ) * B l ~ O Y l ( 2 ) 2 g Z Q T Y 1 1 ( 3 , 3 ) o T E M P * T R l * { 1 , + C 2 * O L Y 3 l * T R Q * C 3 - S R ~ T * T R B * ( D Y ~ ( E ) * ~ E +

+ D Y l ( Z ~ ~ * c ? ) ) D T Y 1 Z ( 1 ~ v 3 ) ~ C l * ~ C 2 * D I Y 3 Z * C 3 t s ~ ~ ~ * ~ D Y z ( z ) * o Y ~ ( ~ I ~ D Y ~ ~ ~ I * o Y ~ ~ ~ ~ ~ ) C 3 ~ D Y l ( L ) * B ? * D Y 2 ( l ) c e l D T Y Z 1 ( 3 r 3 ) = n T Y l E ( 3 r 5 l D T Y 2 ~ ( 3 , 3 ) = T t M P * T R B * ( l . + C t * O L Y 3 2 * T R 1 I C 3 * S R 3 F * T R l * ( D Y 1 ( 1 ) * * 2 +

+ DYZ(l)**?)l C

c, * , c,,,,,cONTTNUE E V A L U A T I O N OF SECOND P A R T I A L D E R I V A T I V E S O? P t l t E N t l A ~ C ENERGY WITH RESPECT t b COOROINATE F U N C T I O Y C O E F F I C I E N T S # c $ T O R I N G UPPER T R I A N G U L A R P O R T I O N OF T Y e SYMMETRIC M A T R I X OF C SLCQNO D E R I V A T I V E S COLUMNWISE I N RMAT, c C COMPUTE D I A G O N A L INTEGSANOS F ~ , F S , F ? C COMPUTE OFF D I A G O Y A L INTEGRANOS C 2 r P 3 r P b C

L A A o l L D l s N D P l L A B 3 N b P I + l 1080 DO 48 K z l r M A 00 40 L l l , M A I D L r l L D ~ ~ L O ~ + M I N ~ 00 39 t r l , H A DO 39 J r l r r J A C l r P * Y l ( I , J ) f F ( 1 D L l 98,38435

35 F l ~ 4 A * P S U ~ ( i r l ) R M A T ( L A A ) o R H A T ( L A A ) + S R M s F i F S a A A * P S U H ( Z , Z ) L B B r L A A + L D l R M A T ( L B ~ ) ~ R M A T ( L B B ) + S R M * F ~ L A A m L A A + 1 I F ( I m K 1 3 8 r 3 b r 9 8

36 I F ( J a L ) 3 6 r 3 7 r 9 8 3 7 I b L a 0 3 0 F Z ~ A A * P S U M ( ~ ~ Z ) ~ C ~ * ~ D Y ~ ~ Z ~ * V Y I ( 2 A L ) * o y 3 ( l ~ * Y y l ? t ~ ~ ~ # L )

R M A T ( l A @ ) + R M A T ( L A B ) * S R M * ? 2 39 L A B E L A B + I

I O = ! O + i 40 L 4 0 a L A B + I0

C L A C ~ I U B F 5 + 1 IOrB DO 118 K = l , H C DO 4 8 . L t l r N C C 7 a Y Y 2 ( f r K 1 L 1 C 5 ~ V Y l ( J , K c L ) DO 45 I = l r H A 00 45 J I l , Y A C l a P * Y t ( I r J ) P ~ P A A * F S U ~ ( ~ , ~ ) . C ~ * ~ D V Z ~ ~ ) ~ ~ C L ~ D Y Z ~ Z ~ * C ~ I R M A T ( L A C I S R M A T ( L A C I + S R M * F J F ~ ~ A A * P ~ U M ( ~ , ~ ) * P * Y E ~ ~ I J ) * ( ~ Y ~ ( ~ ) * C ~ ( . L ) ~ ~ ( ~ ) * C ~ ) L B C s L A C + Y l N I R M A T ( L B C l m R M A T ( L 8 C ) + $ R M t F h

4 5 L A C s L A C + l IOrIO+l

4 6 LAC.LAC+M2N2+10 C

L D 2 ~ 2 * H l ~ i + 1 L C C ~ I U B F S t L 0 2

00 60 Ktlr M C 00 60 L o i r N C 00 55 I = l r M C 00 55 J s l r Y C F 9 r A A * F S U M ( S r 3 ) R M A T ( L C C ] S R M A T ( L C C ) ~ S R M ~ F ~ I F ( I q K 1 55r58r98

50 IF(J.1) 5 5 r b Q r 9 8 59 L c c s L C C + l 60 L C C a L C C + L D 2

C

" c,.,,, ' ' t O N l I N U C F V l L U A l I O h OF F I R S T P A R T I A L D E R I V A T I V E S OF P O I E N I I A L C ENERGY W I T H R E S P E C T T O C O O R D I N A T E F U N C T I O N C O E F F I C I E N T S # l c S T O R I N G THE R F S U L T I N G V E C T O R IN RHS', C

LS8l D O 4 2 2 1 s t , M A 00 422 J o l r N A R H ~ ~ L $ ) : K ~ S ( L J ) + ( ~ H ~ U ~ ~ I ~ ~ P * ~ ~ * V ~ ~ I I J J ~ ~ S R M L i ~ L S + M l h l R H S [ L ~ ) ~ R ~ S ( L I ) + ( R H S U ~ ( ~ ~ * P * ~ ~ * Y ~ ~ I I J ~ ) * S ~ M

422 L!JsLS+l C

L;SsIUB2+ 1 00 423 It l r H C D O 423 J+ l r N C R H S I L ~ ~ ~ R H S ( ~ ~ ) + ( R H S U Y ~ ~ I ~ P * B ~ * V J ( I , J J ) * S R H

4 3 3 L E s L 2 + 1 C C C

69 SlzSi+Hl I F ( 1 S W ,NE, 1) GQ T O 65 1sw=;! W R I T E t b r 6 5 0 )

6 5 S Z S $ L + H Z I P ( 1 S M , N E , 2 ) G O To 7 8 ISW33 N S I = l W R I T E l ~ f i 6 5 0 ) G O T O 25

78 I F ( 1 S W ,NE, 3 ) GO T O 6 6 O E L T A t S C C N O S ( T 1 M E ) H R I T E ( 6 r l S B ) D E L T A

r ' THE S O L U T I O N IS COMPLGTE, C F o t e P

C, , , . . G O T Q B P C l N N f N G OF P R O G R A Y FOR NEW T I R E M O D E L D A T A , c a r 0 i

c C

b6 00 68 f s l r I U 8 S 6 8 RHATIIUBF9+l)tRHS(I)

MAXzIUBF9+ I U R 3 ' ' MAKE A COPY OF R M A T . c*,.,.

D O 69 I a l r M A X 69 R M ( T ) s R M A ' ~ ( ~ )

140

79 I S w = S 80 CONTINUE

IP(!SW)99,9S,8¶ C

a 1 W R I T E ~ ~ , ~ R B ) P A S S C C . t t

C,~..,SYORE O E T E R M I N e b S O L U T f O N I N F I L E COEFF,DAT, c C

b , , . C...m, CREATE R E C O R O DP T I R E M O D E L PARAMFTERS,

w R I t E ( 1 B ' I R E C w ) ~ A , Y A , M C , ~ C , N S l , N S 2 , P , P ~ I B D , C l H ~ R l , R T , R L c, + , C,.,.,CREATE R E C O R D S OF M I N l M I Z I N C C O E F F I C I F N T S PROM M A T R I C E S A , 0, C;

I R € C k e I R t C W + g H R I T E ( l B C ! R F C w ) [ ( A t I r J4, I P ~ , M A I , J ~ ~ , N A ) I R E C W t l R € C w + l W R I T E I l F e l R K C w ) ( ( B ~ ~ ~ J I , X ~ ~ , ~ B ) , J ~ ~ ~ N B ! I R E C N s I R E t w t j W R I T E ( l f l C ! R E t W ) ( ( C ( I r J ) r f ~ l , H C ) r J ~ l , N C I

C C , I

C,,,,,PRINf OUT C Q E F F Z C I E N T S , W R I T E ( b r 3 6 B S ( ( A t 1 r J ) r J ~ l r 4 ) , 1 = 1 t ~ A ) W R I T E ( 6 , 3 B 0 1 1 ( 8 ( 1 r J l , J ~ i r 4 ? , 1 = 1 , ~ B ~ W R I T E ( 6 , 3 6 8 ) t ( C ( I ~ J ) r S = i r U ! , I ~ l , M C ) W R I T E ( b r 6 5 0 9 G O T O 2 5

c , . . C a m . . . I S w = g , t?OPRECY I O N S STILL 1 0 0 L A R G E , TRY AGAIN:

95 Cf lNfXNUE C R l T E l b r 7 @ 0 ) P b S S D E L T A z S E C N D S ( T 1 W E ) w R I T E ( b r l 5 P ) D E b T A

' G O Y O B E G l N N I N G OF PROGRAM FOR NEW T I R E MODEL DATA, c;..., G O T O 1

C 98 H R I T E 1 6 , 8 P B 1

t , ,

C,,,,.HOUTE OAFA B U F F E R f D I N F I L E b T O P R I N T E R UPON C A L L I N G OF EXIT, 99 CALL sPOOL(61

CALL E X I T C C C c

I@@ F O R M A T ( 2 B A 4 1 120 F O H M A T ( ~ 1 5 / 6 F 1 8 8 Y l 130 FORMAT( l H f l , 5 5 X , ' T I R F M O D E L P A R A M E T E R S ' , /, I H , T 4 X , ' A S S O C I A T E D WITH - ~ I N I T I A L I Z A ~ ~ O ~ OF C n E F F f C X E N T M A T R I C E S A AND B,!,/,lH , f 9 X , ' ( R E F E R

2ENCED I N F I L E C O E F F ; D A T R F G I N N I N G I N R E C O R D ' , 1 2 t C ) ' ) IS@ F Q R M A T ( ~ H B ~ S S X , ' I N I T I A L C O E F F I C I E N T S ' ) 258 B B R H A T ( I H 1 2 P A 4 ) 300 P O R M A T l l H ~ ~ Z X ~ b I 5 ~ b P l 3 ~ 4 ~ 385 ~ O R M A T ( I H R , ~ § X , ~ T I R E MODEL P A R I Y E T E R S t , l r l H t I Z X t 0 M A t , ' NA;,

1 C t NC',' kSle,' N S Z C , 8 X , ' P ' , B x , ' P H I B * , O X , ' C ~ H ~ ~ B X PIC R l ' ~ B X , ' 8 R L ' )

3hQ F O R M A ? ( / / l l H ~ U ( ~ ! K ~ U F ~ ~ # R , / / , I H ) ) 370 F O R ~ A T ( l H f l t I E 1 7 , B ) PBB F O R M A T ( 1 H R , 9 9 X , ' I T E R A T I O N ' t l i t l # l ~ I S U X I 'COEFFlCltNI C O R R C C l l O N I '

1 1 u5a F D R M A T { I H , l P 1 X , 5 F i f l , 4 ~ 3 E 1 7 , 8 ) 500 F O R M A I ( ' Q W 4 I R f X D I f F I C U L T Y ~ * * l t R t * # 1 4 ) baa FORMA?(I/,lH ,S¶Xt 'CONVERGENCE W I T H I T E R A T 1 d N ' I 1 2 ~ / , / W t S 4 X , r M I N I H

I I Z I N G C b E F F J C I E N T S ' ) 650 F O R P A T ( / / I , IH , I ~ Y , ' s z ' ~ B Y , ' s ~ ' , ~ x , i ~ t i ~ ' # b ~ , i~ ( i ! ) f ~ 6 ~ t ' ~ ( 3 ) ' # 14~'

+ ' S R ! ' , 1 4 x 4 ' S R 2 ' 6 1 U X t '8R3'1 700 F F R M A T ( ' B N 0 CONVERGFNCE A F T E R ' t ! 4 t 2 X , ' I T E R A T I O N S 8 ) 75a F O R M A T I ~ H ~ , ' I H I S RUN T O O K ' r P 9 , 3 r 9 SEtONDS') 8 @ 0 F b R M A T ( ' 0 E R R O R IN LOADING OF D I A G O N A L D E N S I T I E 8 I N A R R A Y R M A T P ) 9 0 0 F @ R p A T ( 3 1 5 ) 9 5 0 . F O R H I T ( I WF, 3 B l t i ~ ~ ~ ~ ~ ~ ~ ~ € b M I N I M I Z I N G C O L F F I C I L N I 8 A N D A S S ~ C I A T E D

lTlRE MODEL P A R A n E T E R S ' r l , l H r 3 S X o 0 A R E T O BE STORED I N F I L E COB?P,D ? A T B E G I N N I b G I N R E C O R D NUMBER ' r I Z I b , ' )

C c

END

c F U N C T I O N S U A P R O G R A M f S U M C T H I S PROGRAM EVALUATES PRnOUCTS OF L A G R A N G E S T R E S S D E R I V A l f V E 3 AND C C O O R D 1 N A t P P U N C T I d N D E R I V A T I V E S FOUND I N THE I N T E G R A N O OF THE I N T E G R A L C H E P S E S E N T A Y I O N OF THE SECONQ P A R T I A L D E R I V A T I V E OP P O T E N T I A L ENERIV C W I T H R E S P E C T f O C O O R D I N A T E F U N C T I O N C O E F F l t ! C N T S , C

F U N C T I O N F S U M ( I V , J H ) D I M E N S I O N b ~ ~ l l ( 3 , 3 ~ ~ ~ ~ ~ f 2 ~ 3 1 3 1 1 0 ~ ~ ~ 1 ~ f ~ ~ ~ ~ ~ ~ ~ 2 2 ( 3 ~ 3 ) ,

+ Y Y 1 t 3 , Y t 4 ! r Y Y 2 ( 3 , 4 r 4 ) 1 T I ( 3 ) ~ T 2 t 3 ) C O M M O N D T Y i 1 , ~ T Y 1 2 r ~ T V % l , b T Y i ! Z r Y Y l , Y V A ~ T l t t l ? t A A t 1 , J b K t L

C F S U ~ ~ D T V ~ ~ ( I V ~ J ~ ) * V ~ ~ ~ I V ~ ~ I J ) * Y Y ~ I J H ~ K I L ~

+ ~ @ T Y ~ ~ ( I V , J H ) * Y ~ ~ ( I V ~ I ~ J ) * Y Y ~ ~ J H I K , L ) ? tBTYl2(!VrJw)*vYl ( I V , I I J ) * Y Y E ( J H I K I L ) + t b T Y Z 2 t l V , ' J H ) * Y Y 2 C I v r l r J ) r Y V Z ( J H , K , L )

R E T U R N C

END

c F l t N C T I O k S U B P R O G R A M RHSUM c THIS P R O G R A M F V A L U A T ~ S P R O D U C T S OF L A G R A N G E S T R E S S R E S U L T A N T ) AND C COOROlNATE F U N C T I O N Dl !R IVATIVES POUND I N THE INTEGRAND OP THE I N T E G R A L C R E P R E S E N T A T I O N O F , T H E F I R S T P A R T I A L D E R I V A T I V E OF P O T E N T I A L ENERGY c w n n R E S P E C T T O C O O R D ~ N A T F PUNCTION C O E F C I C ~ L N T S , C

F U N C T I O N R H S U M [ N X ) D I M E N S I O N O T Y ~ 1 ( 3 , 3 ) , 0 ~ ~ 1 2 ( 3 , 3 ) 1 0 T ~ Z i t 3 t 3 ) r 0 ~ ~ 2 2 ( 3 ~ 3 ) ,

t C O M M O N ~ T Y ~ ~ , D T Y ~ Z ~ O T Y ~ ~ , ~ T ~ Z ~ I Y Y I , Y Y ~ , T ~ ~ T Z , A A , ~ , J ~ K ~ L R ~ S U H ~ A A * ( T l ( ~ X ) * V Y l ( N X , t ~ J ~ + T L ( N X ) * Y Y Z ( N X , I , J ) ) R E T U R N

C END

C S U B R O U T I N E T R l S Q S U B R O U T I k F T R I S Q ( b , N F Q , K E V )

C T H I S P R O C R A W CONSTRUCTS THE S O L U T I O N O? A BIMULTANEOUS L INEAR C Q U A T I O N C S Y S T E M H A V I N G A REAL, S V M M E T R l C , C O E F F I C I E N T M A T R I X BY T H E M E T H O D OF C CHOLESKV OEtOMPOSITION W I T H A T F S I FOR P O S I T I V E D f P I N I T E N E S S , C C

D I F E h S I O N A t 1 1 K E Y 6 1 LS=l IS=@ 00 5 0 IC=l,NEO K S r 0 DO 316, I P ~ l r IC IF(fR,EQ,ZC)GO T O 35 KS:KQ+IR K D = B K R r K S m I R K C t l S D o r a I p l r I R IF(I,EQ,IRlGO T O 2 B K6rKD)I K R t K R + l K C = K C + 1

10 A ( L S ) ~ A ( L S ) ~ A ( K ~ ) * A ( K R ) * A ( K C ) 20 A ( L G ) = A I L S l / A ( K S ) se L S = L S + ~ 35 KDeCl

k C = I S D O a 0 I t l , I C IF(Z,€Q,IC)GO 7 0 45 K Q = K D + I K C o K C + I

40 A ( L S ) E A ~ L S ) ~ A ( K D ) * A ( K C I * A ( K C ) a5 I S = I S + I C

I F ~ A ( L S ~ , L ~ ; @ ; I W E Y ~ - ~ f F ( A ( L S ) , € O , B , ) G O To 99

50 LSslS+1 KDo0 D O T B 1=1, kEQ KDzKD+I KReKDwI K S P P D O bP)* I [ C = l , t IF(IC,EQ,I)GO T o 4 5 K O l t K S t I C I R s I S + I C * ! K R r K R t 1

60 A ( L S ) = A ( L S ) ~ A ( K S ~ * A ( ~ R ) * A ( K R ) 6 5 A [ L 8 ) = A ( L S ) / A ( K D ) 70 LS:LS+I

L S ~ L S ~ Z KeNEQ+ 1 L r N E Q D O 9 0 . J82rkTQ I S = L 8 + 1

K R P ~ K D * ~ 00 R B 1 t 2 , J A ( L S ) ~ A ( L S ) - A ~ K ~ I * A ( I ~ I IS=IS+i K f ? t K R + L

8 0 L a L + 1 KsK-1 LrK*l K b r K B m K

90 L S ~ C S ~ l R E T U R N

99 K E Y @ @ R E T U R N

C C

END

Example PATCH Run

Four d a t a c a r d s (A-D) c o n s t i t u t e an i n p u t module f o r

program PATCH. There may be a s many i n p u t modules a s

d e s i r e d . An e n d - o f - f i l e c a r d (EOF) s h o u l d f o l l o w t h e l a s t

i n p u t module . An example i n p u t module i s g i v e n be low ,

f o l l o w e d by t h e o u t p u t p roduced by PATCH.

Card A (20A4)

TITLE = 80 a l p h a n u m e r i c c h a r a c t e r s

Card B ( 6 1 5 )

maximum i n d e x v a l u e s

N C = 4 J

NS1 = 1 3 S i m p s o n ' s r u l e s t a t i o n s

h'S2 = 19

PHIB = 1 3 5 . 0 1

Card D ( 3 1 5 )

IRS = C

r e a d i n i t i a l c o e f f i c i e n t s b e g i n n i n g i n r e c o r d 30 o f f i l e 1 0 ( i f IRS f 0)

w r i t e s o l u t i o n c o e f f i c i e n t s b e g i n n i n g i n r e c o r d 1 7 o f f i l e 10

s t a r t run w i t h i n i t i a l c o e f f i c i e n t s s e t t o ze ro

Punched Card L i s t i n g

P A T C H ( Z S C ] M A ~ N A I M C S N C : U P E ~ ; B R L E O , ~ ~ ~ R T r f , 5 4 4 , 4 U , 13 I 9 ,

I r 000 135, 11 1 9 3: 5 41,538 30 I ? 0

The computer o u t p u t from t h i s example run i s shown on

t h e f o l l o w i n g p a g e s . The e x e c u t i o n t ime on t h e PDP 11 /45

i s p r i n t e d a t t h e end o f t h e r u n .

D E T E Q t 4 l N E D r I K i M I Z T N G C O F F F I C I I h T S A N D A S S O C I A T E D T I R E , H O O E L P A R A M E T E R S ARE T P R E S T O R E D I N F I L F COEFF,DAT B E G I N N I N G I N RECORD NUMBER 17,

T I R E H O D t l P A R A H E T C R S HA NA M C NC NSl N S ? P P H I R C l H R 1 , R T

4 4 4 4 1 5 1 9 1 , Cjm0P 1 3 5 , ~ C ~ O P , 1, B O O R 1 , @Be0 31 5808

I N I T I I L C O E F F I C I E N T S R L

9,5380 O, 002 f iPav1VE 00 0, d@On;lPPPF 0R 0 . BJPBflB0RE 670 Bt0OCIR80flBE 0 0 8

I T E R A T I O N 1 COFFFICIFNT CORRECTIONS

I T E R A T I O N 2 C O E F F I C I E N T C O R H E C T l n N S

I T E R A T I O N 4 C O E F F I C I E N T C O R R E C T l O W S

C O h V E P G E h C E W I T H I T E R A T I O N 4 M I N I M I Z I N G C O E f F I C 1 E N f S

C:. 1 . 2 Rhl) . 'I'his program r eads a s e t o f s o l u t i o n

c o e f f i c i e n t s ( a I . , . . f rom d i s k f i l e 1 0 ( s t o r e d by i j ' 1~ 11

a p r e v i o u s PATCH run ) and l o c a t e s t h e c o n t a c t boundary o f

t h e deformed t i r e model t h a t i s de t e rmined by t h e c o e f f i -

c i e n t s r e a d . The boundary i s found by s e q u e n t i a l s e a r c h i n g

a t g r i d p o i n t s w i t h i n a r e c t a n g u l a r annu lus s p e c i f i e d by

t h e program c o n t r o l d a t a . The boundary p o i n t s a r e s t o r e d

i n f i l e 4 (DEC-tape) f o r subsequen t p l o t t i n g by program

P L T R N D ( l i s t e d i n Sec . C . 2 ) .

The s t r u c t u r e of program B N D i s diagrammed i n F lowcha r t s

3 and 4 , on t h e f o l l o w i n g p a g e s . Tab le C-3 d e s c r i b e s t h e

program c o n t r o l d a t a t h a t a r e n e c e s s a r y f o r t h e e x e c u t i o n o f

B N D . An example r u n , u s i n g t h e s o l u t i o n c o e f f i c i e n t s found

i n t h e example run o f PATCH, f o l l o w s t h e l i s t i n g s o f BND

and t h e subprogram TRANS.

I

Read t i r e model and s o l u t i o n

I s p e c i f i c a t i o n d a t a

Read s o l u t i o n c o e f f i c i e n t s

from d i s k f i l e 1 0

I C a l l T U N S t o t r a n s f o r m ( 0 1 , e 2 ) I

Find t h e (el , 0 2 ) c o o r d i n a t e s

of t h e c o n t a c t boundary

I c o o r d i n a t e s t o ( y l , y 2 ) c o o r d i n a t e s I

( s e e F lowchar t 4 )

I Wri t e ( y l , ) c o o r d i n a t e s on I I DEC-tape f i l e 4 and d i s k f i l e 6 (

I P r i n t c o n t e n t s 1 1 of d i s k f i l e 6 1 Retu rn t o START)

F lowchar t 3 . BND

Flowcha r t 4 . Boundary f i n d i n g algorithm.

FOKTRtZN V a r i a b l e ( s ) D e s c r i p t i o n

EPSLN boundary d e f i n i t i o n con-

s t a n t , t h e c o n t a c t

boundary e n c l o s e s mid-

s u r f a c e p o i n t s f o r which

1 z 1 < EPSLN -

d e g r e e i n c r e m e n t s f o r

boundary s e a r c h i n g i n t h e

8 and O 2 d i r e c t i o n s 1

d e g r e e d imens ions o f t h e

0 - 0 r e c t a n g l e , c e n t e r e d 1 2 about (0,O) , which i s

e x p e c t e d t o c o n t a i n t h e

boundary

r a d i a n d imens ion o f 01-02

r e c t a n g l e which i s b y -

p a s s e d i n t h e boundary

s e a r c h

r e c o r d number r e f e r e n c i n g

s o l u t i o n c o e f f i c i e n t s

s t o r e d i n d i s k f i l e 1 0

p r i n t o p t i o n s w i t c h ;

IWT > 0: p r i n t a t e ach

i n c r e m e n t ,

INT - < 0 : p r i n t e v e r y f i f t h

i nc r emen t

c P R O G R A M BUD C T Y I S PROGRAM L O C A T E S THf: C O V A C T R O I J N D A R Y , C t

I N T P G F q I N a U T E Q U I V A L F N C E ( M A , Y B ) I ( Y 4 , N q ) O A T A ~ ~ ~ ~ , i u i s s r ~ / , ~ ~ / r / , n ~ / b /

C C,..,,FILE 4 STORES O R O F R E D P A I R S D E S C R I B I N G ONF QUADRANT O f THE C S Y M M E T R T C COYTACT AnUNDARY, A G R A P H I C R E P R E S E N T A T I O N c OF T H F R O I J N ~ ) A Q Y M A Y T H E Y Y E OBTAINED USING T H E C P L O T T I N G P R O G R A M PLYBWD, C F T l F 6 I S A P R I V T E D OlJTPUl RUFFFR, C F I L E 1 0 I S A RANDnM ACCESS L I B R A R Y OF M I N I M I Z I N G C O E F F I C I E N T S C P R E V I n U S L ' f CALCULATED BY T H F PATCH PROGRAM, C

CALL A S S J G N ( ~ ~ ' ~ K ~ ~ R N ~ , L S T ' I ~ ~ ) C A L L A S S I G N ( 1 P , ~ ~ l t [ l l ~ 1 3 1 ~ f l ~ ~ ~ ~ ~ ~ ~ ; ~ ' ~ 2 2 , 1 E R R ~ D E F I N E F I L E l f l ( R Q C S W , U p 1 N D X I C A L L A S 9 I G N ( U r o D T l : I ~ ~ Q I ~ I C I J R V ~ * D A T ~ ~ ' , ~ ~ ) R E W I N D U

c c

5 TlME=SECNDS(B,l C C,..,.IF FhP OF C A R 0 I N P U T ENCOUNTERED T E R M I N A T E P R O G R A M E X E C U T I O N , C ... OTMERNTSF R E A D T I T L E CAR0 FOR N E X T DATA S E T ,

R E A D [ I N , 4 R % , E N D = l ~ B ) T I T L E W R I T k ( O T , U S 0 1 T I T L E

C , , , , , S P E C I F Y TlHF Y O D E L PARAMETESS D E S C R I B I N G C P A T C H P R O G R A M RIJh F O R A H I C Y BOUNDARY IS D ~ S I R E U ,

R F A D ( T I J ~ ~ ~ L ? ) ~ A ~ N ~ ~ ~ C I N C , ~ S ~ ~ N S ? I P ~ P H I B ~ C ~ H , R ~ L C M53M4, NBzVA (qY EQU!VALFNCEI C C,,,,.ENTFR P A H A Y F T F W $ D F 3 C R I R I N G T H E D T S C R F T E N E S S C OF THE RCIIINPASY SEARCH A L G O R I T H M , C * * . . r n T H E T A - ! L I M I T OF 9 0 1 J h i P A R Y SEARCH IS T L l l C T r J t T A - 2 L I M I T OF S O I J h D A R Y S E A R C H I S T L Z ,

R F A D ( l N r S I Q 1 F P J L N I D S ~ ~ ~ S ~ ~ T L ~ ~ T L ? , S ~ T S ~ I S Z T $ T C C... . .ZWY I S A P R I N T n P T I O N SWITCH! C 1 ) I W T , G T . 9 I N I T I A T F S A P R I N T e D C O P Y OF EVERY L I N E C E N T E R E n 1 h T 0 DISK F I L E 4. C I 1 1 1 4 T .CE, M I Y I T I A T F S P R I N T E R OllTPUT FOR FVERY c . - F I F T H L I N E , c~....IRFc~ I S A P Q T N T E R t o T H E s ~ c ~ o v OF DISK F r L E l o CONTAINING C THE D F S I R E ~ P A T C H PROGRAM T I R E MODEL PARAYETERS C AND A S S O C I A T E D Y I Y I ~ l Z I N G C O E F F I C I E N T S ,

R E A D ( I N , S P A ) 1 R E C N r I w T I R E C N P t I R E C N + S

c . C , . . . . S E T R I E V F T I R E MODEL P A R ~ M E T E W S n E S C R I 8 f Y G PATCH C P R O G R A Y R I J d N H I C H PQODUCEn M I N I M I Z I N G COEFF!CIENTS, C I N T E G R I T Y CUEC< A V A I L E D BY V I S U A L C O M P A R I S O N W I T H

C C A R D I N P U T f I R E MODEL P A R I H F T E R L 1 S 1 , R F A ~ ( I ~ @ I R E C N I ~ A F , N A F ~ ~ C F , ~ ~ F , ~ S I F , N ~ Z F , P F , P H I R F , C ~ H F , R ~ F , R T P , R L P

C C . . . . . R E T P I E V F M I N I M I 7 1 ~ G C f l E F F l C I E Y T S GENEPATLO BY P A T C H P R O G R A M ,

w R I T E ( O T , 3 5 0 ) W Q I T E ( O T , ~ ~ ~ ) ( ( A ( I r J ) , J = l 1 4 ) 0 1 z i r M I ) H R I T E ( O T 1 5 b a ) ( t B ( I ~ J ) r J = l t 4 I a I = I , M S l W R I T E ( O T r 5 6 f l ) ( E C ( I 1 J ) r J 8 l , ~ l r I ~ l r M e )

C ~ R I T E ( O T r 5 4 5 ) d R I T E C O T r 5 5 P ) E P S L N ~ O S ~ , D S Z ~ T L I , T L ~ , S ¶ T S T I S ~ T S T

C C C a e a * . T R A N S F O R M ANGl lLhH P A H A M F T E R S FROk DFGREE T O R A D I A N MEASURE,

PHIRzPHIH*,8174533 D S l = D S 1 * . B 1 7 4 5 3 ~ DS ,2aDS2* ,4174533 T C l s ? L l * , f l 1 7 4 5 3 f T L , ? l f / 2 * , f l 1 7 4 5 3 3

c , C , . a e . E S T A B L I S H C O N S T A N T S F O R C l l R Q E N T R O N ,

P I R T s P I / P ~ I Q M T a T L l / b S I + Z , N T = T L 2 / D S 2 + 2 , M T S T = S 1 T S T / D S I + i , S l T S T : D S I * M T S T M T S T r p T S T + t

e H A X 3 * 3

C C c

sz:e, C C.. a . m B F G I N B O U N D A R Y S E h R C H , c

D O sn N T Z : ~ , ~ T C O S S 2 ~ C O S ( S ? ) I F ( S Z T S T w S 2 ) 2 b r 2 h r 2 4

2 4 S 1 ~ 3 1 7 9 T M T I r Y T Q T G O 7 0 28

2 6 S l a f l ,

M T I r 1 c

28 D O 4 0 N T j a M T I r M T c , C, ..., E V A L U A T E C O N S T R A I N T C O O R D I N A T E i,

R B ~ R T + R l * C O S ( 9 1 ) D Y 1 2 ~ R u * C ~ S S L Z r S Q R T ( R l , . * D Y l 2 ) D O 3s I i l ~ M e R E * e 9 * ( Z * I * l ) A E a R Z * P I f 3 T * S 1 C O S A E ~ t f i S Z h Z ) DO 30 J t i r Y C Z~Z+C(IrJ)*COSA2*COS(J*S2)

36 C O N T I N U E C

L A B $ m A B S ( Z ) D E L T A e Z A E S * € P S L N

c, C,...;IF M A G N I T U O e QF C O N S T R A I N T C O O R D I N A T E A T C ( T H E T A * l ~ T H F T 4 * ~ ) ~ ( 8 t r 3 2 ) IS G R E A T E R THAN C E Y S L N DEFTNF ( S l r S Z ) A3 A BOUNDARY P O I N T , C 9.e OTHERWISE CONTINUE THE SEARCH ALONG THETA*^^^^ MER!D!AN,

I u n e L t r ) 4 ~ ~ ~ 8 ~ 3 8 C, ( . C, , . . ,STORE 3 1 BOUNDAQY C O O Q O I N A T F C C d R R E S P O N b I N G TO ~ ~ ~ ( N T Z J ~ ) * O S Z ,

38 B N D ( N T Z ) = S 1 I p ( 8 1 1 5 0 , 3 9 r S f l

C. . C o m m e a L I M I T OP BOUNDARY I N S Z D I R E C T 1 0 4 C H A 3 BEkN FOUND, SND WON C O N T A I N S ALL C RELEVANT ROUNbARY PARAMETFRS,

39 M A X r N T Z Gb TO 55

40 S I ~ S I + B S I C: . , , ,BOUNDARY P O I N T ALONG T H E T 4 * g s $ 2 M E R l O 1 4 N NOT FOUND, c T F R M f N A T E P R O G R A M E X E C U T I O N ,

GO TO 90 50 S2aSE+DS2

C C C

w R I T E ( O T t 7 8 5 ) ENOFILE 4

C: , , , .ROUTE DATA RUFFERLO I N F I L E 6 T O P R I N T E R UPON C A L L I N G OF EXIT; CALL S P O O L ( 0 T I C A L L E X I T

C C

5 5 N R I T E ( U r 5 0 0 ) M A X

c, , c

C , . . , , O I S P L A Y CONTACT BOUNDARY W I T H R L S P e C T TO Y 1 r Y2 C

w R I T E ( o T r 7 2 @ l C

c . * - . . T Q A N S L A T E T Y E T A - ~ - - T H E T ~ - ? ' ? F P R E S F N T A T I O N C OF CONTACT ROUNDARY TO Y l r - Y Z C O O R D I N I T E S ,

C A L L T R A N ~ ( A , S ~ Q ~ ~ R ~ , ~ ~ ~ , M A , N ~ ~ P I B T , ~ Y O ~ V I + Y ~ , M A X ) C

J T = 5 C

00 76 Jz1 ,MAX C e a a * . S T O R E Y I m - Y Z R E P R E S E N T A T I O N OF CONTACT BOUNDARY C OF1 O I S K F t L F 4 FOR LATER USE RY P L O T T I N G PROGRAM PLTBNOI

w R I T E ( 4 , 7 T 5 ) V Z ( J ) r Y l t J ) I F ( 1 d T ) 6 2 , 6 2 1 6 7

6 2 IF(JT-J¶ 6 5 8 6 5 8 7 8 6 5 J T s J T + S 4 7 Y R I T E E Q T , 7 3 0 ) Y Z ( J ) , Y l C J ) 7 8 C O N T I N U F

C D T I M € x S E C N D S ( T I q E I k R I T E [ O T , 9 Q P ) D T I M E

c . C,,.., G O TO B E G l V N I N G CIF PROGRAM FOR NEW T I R E H ~ D E L O I T A ,

G O T O 5 C

90 Y R I T E ( O T , f f l @ ) S2 C

100 E N D F I L E 4 C e . * . q ROUTE OAT4 R U F F E R E D I"iIC€ b T O P R I N T E R UPON C A L L I N G OF EXIT.

C A L L s r o u L ( o r ) C A L L E X I T

C C C

J z @ F ~ H M A T ( ~ H Q , U ~ X , ' T T R F YODEL PARAYFTEQS (USER S P E C I F I F D l % / , l Z X , ' 1 M A ' , ' Y A ' , ' YC',' N C ' , ' N S ~ ' , ' N S Z ' r O X t C P ' , b ~ , ' pH!8' Z p R X , ' t i H ' , 8 x , * R i e , 8 % , ' R X R L ' )

3 3 0 F n R Y A T ( / / , 1H ,SUX, ' P I L L C O E F F . b A T C , / t 1~ , U S X I ' C O N T E N T S OF RECORDS 1 ' 1 12, ' THROUGH ' r 1 2 )

340 F O R M A T ( I H @ , S ~ X I ' T I R E MODEL P 4 R A Y E f E Q S f ) 350 F o R M A T ( l H f l , U 9 X , ' M I N I M T Z I N G C Q F F F I C f F N T S ' ) 48VJ F O R M A T ( Z 8 A 4 ) 450 POf?YAT( 1 H l r , ? B h U ) 5 f l8 F O R M A T ( 6 1 9 / b F l 8 . 4 ) 510 F O R M A T ( 7 F 1 0 . U ) SUB F O R M A T I ~ H r t i ? X , b I S , h ~ 1 3 . 4 ) 505 F O R M A Y ( / / r l H ,47X, fROI IUDARY SEARCH PARAYETERS' ] 5 5 0 F @ R M A T ( ~ H ,SX, * E P S l Y a C , F 7 , 3 , 3 X ~ ' D Q 1 : ' , F ~ . ~ , ~ X I ~ O S Z = ' ,F7 .313X

t r 'TI,.! a ' t F 7 . 3 r 5 X , ' T L Z ' t F 7 . 3 , 3 X , ' S I T S T = ' r F 7 p 3 , 3 X ~ ' 9 2 T S T r ' r P t 7 . 3 )

5bB F O R M A T ( / / # I H r 4 ( 2 7 X , 4 E 1 7 r B , / / r l Y ) I 780 F O H H A T ( I Y ~ , ~ C O V T A € T RnLJNDARV FOR T H F T I - 2 a * , ~ 8 ; 4 , ~ M E R I D I A N N O T f

+ O ~ I t u O ' ) 7 P 5 FC!RMAT[lHP,'t3nUUDARY 4LdNC EQUATOR (THETA-I :a , ) NOT F O U N D C ) 7 2 0 F f l R M A T ( ~ Y ~ , 3 8 Y , ' C O & t A t T B 9 U Y D 4 R Y W I T H RFSPECT T O Y2 Y 1 COORDINA

~ T E S * , / / , ~ H A , ~ ~ X , * Y ? ' , ! ~ X , ~ Y ~ ~ ) 7 3 n F n R M A T ( t H t U 8 X , 2 f l U . S ) 7 3 5 F n H M 4 T ( ? F I U , 5 ) 9na FORMAT( 'E l fH1S @ \ I N T f l O Y ' , F 9 r S , ' S E C O V n S @ )

t END 16 1

C SIJBRQIJT I N F T R A N S S l J R R C l l l T I N F T R A N S ( A , R , C C , D p D ~ 2 , ~ 4 , N A , P T 8 T , R N r ! , Y ~ , Y 2 p M A X ]

C T H 1 3 P R o G R A H T R A N Q f n R M S A L O C I J S OF P O I N T S R E P R E S E M T E D I N C C I I H V I L I N E A R C O O R D I N A T E S ( T H E T A = l , T H E T A ~ ? 1 T O A C A R T E S I A N C C n O R D I N A T E RCPRFSFhTATION I N I Y I , Y Z ) , C C

D t M F N S I o V A ( U ~ U ) , H ( U ~ ~ ~ ) ~ R N D ( ~ B ~ ~ ) I Y I ( ~ B O ) C

Sr?=@, C

D O 29 N . ? t l t M A X S l = B N D [ h 2 ) R f l r P t C C * C O S ( S l ) Y 1 V : R 0 * ! 9 I N t S Z l Y2V: -CC*STN(913

C Dn 1fl 1 ~ 1 , HA RZ=,T*(?*T-I) A l : I * P I R T * S l AZcRZ*PYHT*St S I N A l = S t N l A l l COSA,?=COS(A?)

C 0 0 16 J = l , N A SINJS?=SIN(J*SZ) C O S J M l : C O S ( ( J = l l * S Z ) Y l I J r C O S A Z * S I M J S 2 Y ~ I J = S I N A I * C O S J M ~ Y ~ V = Y ~ V + A ~ I , J ) + Y I I J Y ? V = Y Z V + H ( l r J ) * Y Z I J Y I ( N Z ) = Y i V Y ? ( N , ? ) = Y Z V

?M S Z = S Z + D S Z C

R E T l J R N C C

E N 0

F i v e d a t a carcis ( - 1 c o n s t i t u t e an i n p u t module f o r

p rogram BKD. T h e r e may b e a s many i n p u t modules a s

d e s i r e d , .An e n d - o f - f i l e (€OF) s h o u l d f o l l o w t h e l a s t i n p u t

module . Program PLTBND ( l i s t e d i n S e c . C . 2 ) w i l l s u p e r -

i m p o s e , i n one s e t o f c o o r d i n a t e s , p l o t s o f a l l o f t h e

b o u n d a r i e s f o u n d i n a B N D run w i t h m u l t i p l e i n p u t m o d u l e s .

.An example i n p u t module i s g i v e n b e l o w , f o l l o w e d by t h e

r e s u l t i n g o u t p u t .

Card A (20A4j

TITLE = 80 a l p h a n u m e r i c c h a r a c t e r s

Card B (615) c o p i e s o f c a r d s B and C o f t h e PATCH

Card C ( 6 F 1 0 . 4 j i n p u t module

Card D ( 7 F 1 0 . 4 )

EPSLX = . 1 5

DS1 = .1

DSZ = .1

'I 'L1 = 3 5 . b o u n d a r y s e a r c h c o n t r o l

r-cacl solution c o c f f i cicnts b e g i n n i n g i n r c c o r d 1 7 o f f i l c 10

p r i n t a l l b o u n d a r y p o i n t s f o u n d

Punciied - Card L i s t i n g

The compute r o u t p u t f rom t h i s e x m p l e r u n i s shown on

t h e f o l l o w i n g p a g e s . The e x e c u t i o n t i m e on t h e PDP 1 1 / 4 5

i s p r i n t e d a t t h e e n d o f t h e r u n . The p l o t p r o d u c e d by

p rogram PLTBND i s a l s o sholzn.

C O N T A C T BOUNbAqY H l T H R C 3 P E C T T O Y Z -- Y l C O O R D I N A T E S

( ; . 1 . .j 1 ' ' I ' ~ ; I I t : \ . ' I ' l l i 5 1) rog I . : ~ I I I r c \ ; 1 ~ 1 ~ ;i 5 c> t o f

. . ;olut i on c-oc't'i 'icicbnt..: [ : I . I . . . fro111 d i s k f i - l c 1 0 lj' 1 1 1

[ s t o r e d I-IY a p r e v i o u s I',ZrI'C1l r u n ) and ( 1 ) e v a l u a t e s

p o i n t s , y r , on t h e d e f o r m e d m i d s u r f a c e , a l o n g c o n v e c t e d

c o o r d i n a t e c u r v e s s e l e c t e d by t h e u s e r ; and ( 2 ) e v a l u a t e s

t h e e x t e n s i o n r a t i o s , X a l o n g t h 2 s e c u r v e s . The r'

c o o r d i n a t e s , y r , and e x t e n s i o n r a t i o s , h a r e s t o r e d i n r '

d i s k f i l e 2 ( i f a l o n g a m e r i d i a n ) o r i n d i s k f i l e 3 ( i f

a l o n g a l a t i t u d e ) f o r s u b s e q u e n t p l o t t i n g b y p r o g r a m s

PLThIRL) ( p l o t m e r i d i a n ] , PLTLAT ( p l o t , l a t i t u d e ) , a n d

PLTL12 ( p l o t s t r a i n s ) .

The s t r u c t u r e o f p r o g r a m PTCHCV i s d iagrammed i n

F l o ~ i c h a r t 5 on t h e n e x t p a g e . T a b l e C-4 d e s c r i b e s t h e

p r o g r a m c o n t r o l d a t a t h a t a r e n e c e s s a r y f o r t h e e x e c u t i o n

o f I1'I'CIICV. .An example r u n , u s i n g t h e s o l u t i o n c o e f f i c i e n t s

f o u n d i n t h e example r u n o f PATCH, f o l l o w s t h e l i s t i n g o f

PTClIC1..

I s p e c i f i i a t i o n d a t a

I t

l ~ e a d s o l u t i o n c o e f f i c i e n t s 1 from d i s k f i l e 1 0 I

1

f ~ e a d program c o n t r o l d a t a

I s e l e c t i n g convec t ed c o o r d i n a t e c u r v e s I

Convected c o o r d i n a t e c u r v e s I..- "---::A $-

i I L O O P I One d e g r e e i n t e r v a l s a l o n g t h e cu rve

I E v a l u a t e y r and A r I

i h r i t e e v a l u a t i o n s o f y and h r --- ..---- r

on disk f i l e s 2 o r 3 , and 6

1 Print c o n t e n t s

E X I T 1

F' lowchar t 5 . P T C H C V

FORTR.41G l ' a r i a b l e D e s c r i p t i o n - .-

I RECN r e c o r d number r e f e r e n c i n g

s o l u t i o n c o e f f i c i e n t s

s t o r e d i n d i s k f i l e 1 0

number o f c o o r d i n a t e

c u r v e s

p r i n t o p t i o n s w i t c h ;

IhTT > 0: p r i n t a t e ach i n c r e m e n t ,

IKT < 0 ; p r i n t e v e r y f i f t h - i nc r emen t

t a b l e s p e c i f y i n g t h e

c o o r d i n a t e c u r v e s ;

CITE ( I ,1) > 1 : e v a l u a t e -

f u n c t i o n s a l o n g t h e cu rve

CITE(I,l) < 1: e v a l u a t e

f u n c t i o n s a long t h e cu rve

CVE(I,2) , f o r I = 1 , 2 , . . . ,II 'CV,

i s g i v e n i n r a d i a n s

c PROGRAM PTCHCV C T H I S P R O G R A M I O C A T t S COYVFCTED C O O R n I N A T E C l l R V E S SELECTEO B Y THE USER C AND E V A I U A T F S k Y T F N S I n N f l B T I U S ALnNC YHFSE C U R V E S , C C

D I M F r J S I O i h l A ( ~ ~ ~ ) ~ ~ ~ ( ~ I ~ ) I C ( ~ I ~ ) I Y ~ ~ ~ , ~ Y ~ ( Z ) ~ O Y ~ ( ~ ~ , D Y ~ ( ~ ) , O Z ( ~ ] ~ + C V F ( I B t Z I , T I f L E ( 2 @ )

t Q U I V A L F N C F ( Y A , M B ) , ( N A , Y H ) DATA P 1 / 3 , 1 0 1 5 9 ? 7 /

C C . . ~ . , F I L F I S A P Q I Y T E D O U T P U T SUFFER; C F I L E S ? AND 3 S T O R E THF C A R T E S I A N LOCllS OF D E S I R E D C C C N V E C T F o C O O R D I k A T F C O R V F S AND A S S n C t A T F D F X T F N S I O N c R A r r a s , A G R A P H I C R E P H ~ S F N T A T I O N M A Y T Y E N R E O B T A I N E D C U S I N G THF P L O T T I h F P R O G R A Y S P L T L , l t l P L T L 4 T r AND P L T M R D , C F I L E 10 IS A R A N D O M A C C F S S L I R R A R Y OF H I N T H I Z T N S C O E F P I C I E N T S C P R E V I O U S L Y C A L C V b A T t D BY THF P A T C H P R O G R A M , C

C A L L A $ S I G N ( ~ , ' D K ~ ~ Z P T C ~ C V . L S T ~ , ~ ~ ) C A L L h S S I G h ( ( 2 , ' O K l t I 1 l J 1 r 3 I C I J R V 2 , D A T t l 0 , E 2 ) C A L L A S S I G Y ( 3 , ' ~ l c l r 1 1 1 3 r 3 1 C ~ ~ R V 3 , 0 A T ~ 1 C ~ E 2 ) CAL.L A S S l C ~ ( j f l r ' D K 1 t [ l l $ , ' 5 1 C O E F F I O A T ~ t C ~ 2 ? ~ I E R R j D f F I N F F I l E I f l ( 8 @ , 5 Q t l - I , TNbX)

C C

T I M E = S E C N D S f B . ) R F A D ( 5 , l U P ) T I T L E dRITE(b,lh5) Tl7 l .F

C C . . . . . I R E C N I S A P O I N T E R TO T Y E R F G I O N OF D I S K F I L E 10 C O N T 4 I Y I N G C THE O E S I H F Q P A T C H P R O C W 4 M T I R E MODEL P A R A M E T E R S C AND A S S f l C l A T E n ' I I ~ I ~ I Z I ~ G C O k F f I C I E Y T S , C , . . . , N C V D E S X G N 4 T F S THE T O T A I NUMBER O f C O ' d V E C T E D C O o R D I N A T f C C I I R V E S FOR r l H f C H F U ~ C f ~ O " . r V A L U A T I O N S A S € OFSTREO, c e . . . m I W T I S A P R I N T O P T 1 0 4 S 4 I T C w f C I I 4 T , G T , @ I N I T I A T F S A P R I N T E D COPY OF E V E R Y L I N E ENTERED C J N T O D I S K F I L E S 2 AWD 1, C I T ) I W T , L E . @ I N I T I A T E S P R f l J T F R OIJTPUT F 0 9 F V E R Y F I F T H L I N E ,

R F A D ( 5 , t a P ) I S E C N , W C V a I w T C C . . . . . S P E C I F Y T I R F Y O O E I - P A R A M ~ T E R S D F S C R T B I N C P A T C H PROGRAM C HUN F O R d M I C H PTCHCV P S n G R A Y E V A L U A T I O N S ARE D E S I R E d ,

R F A D ( S r ! R B ] M A , N A , ~ C , N C , N S ~ , ~ S Z I P ~ P H I R ~ C ~ H ~ R ~ ~ R T ~ R L H R I T E ( 6 r 1 7 5 ) w R I T E ( h t 3 0 8 ) M A ~ N ~ , ~ C , ~ C , N ~ ~ ~ ~ J S E , P , P H ~ B ~ C I H , R I , R T I R L

c I R E C N P = T R E C b ' + T w R I T E ( b , 1 7 3 1 I Q F C N , I S F C Q P

C c a m . . . H F T R I F q F T I R E MnUFL f J A M 4 " F T F R S DESCRISfYG P A T C H C P R O G R A M R U V W H I C H P Q O Q U t E D Y I N l W I Z I N G C f l l ? F F I C I E N T S , C I N T E G R I T Y C H F C K A V A I L F D B y V I S U A L C f l v P A R f S O M W I T H c C A R D ? \ P U T T I 9 1 HQOFL PAHAMETFR L I S T .

H E A D ( t Q 8 1 R k C N ) Y A f , Y 4 F , ~ C F , N C F t W S l f , N S Z F , P f , P H I R F ~ C 1 h F , R ~ F t R T F , R ~ F C C , , , , . 6 ? F t R I F V E M I t d I v I 7 I V C C O E F F I C I E Y T S G E Y E R A T E D R Y P A T C H P R O G R A H ,

L

c , . , . , L ~ A ~ rvf W I T H P A R A Y F T E R S DEFIN IYG E A C H C C O N V E C T E U c O O A O J ~ A T E CURVE O f I N T E R F S T ,

# R I T E ( b r 3Sd) 'JCV D O 8 I = l r V C V R F A D t 5 , l S B ) C v F t I , l ) r C V E ( 1 ~ 7 1 N R I T E ( 6 r 1 6 0 ) l r C V E ( I , 1 ) , C V E ( I r L )

A C r ) h T l N U € C

L

C,,,,. E J T A Y L I S H CnhSTAUTS FOR C U R R E N T HUN. P f R T : P I / P H I f l H r H A X f l ( M A n M C ) N:MAXfl(NA,NC)

I: C C

D b 196 L C = ~ I N C V r b

C m * , , , LnAO C O b l S T h & T S AND I N I T ? A L I Z E PARAMETERS APPROPRIATE T O F O R M c OF C O N V E C I E D C O f l R D l ~ A T t . C I J R V E D E S I R E D ; C 1) C V E ( L C r 1 ) e l . T e 1, I N I T I A T E S F U V C T f o N E V A L U A T I O N &LONG C THE M E R I b I f l N 4 b C U R V E ( T H E T A - ? ~ C V E ( L C D ~ ) ) , C 11) C V E ( L C I ~ ) , G F D 1, 1 ~ l f l A f E S F U V C T I O N E V I L U A T I O N ALONG C THF C IRCfJMFERENTIA I , , C U R V E ( T H E T A m l = C V E I L C , E ) ) ,

S ? I = i ? , H i = 6 , H 2 = , 0 1 7 4 5 3 3 N S 1 = 1 N S Z t l f l l I F N s 2 *RITE(3r390) N ~ ~ ~ C V ~ ( L C P ~ ) ~ C V E ( C C , ~ ~

C 3fl wRIIE(6r 3863)

IYltl

I N S a 5 SZ=S21

c C

00 6 5 N Z a l r N S ? SI=S11 STNS2:SIN(S?) cnssz=cos (s? )

C r . ,

D O 64 Y l = I , N S t C,,,,.INIT$ALILF C O f l R D I N A T E F l J N C T l O N S A N 0 T H E I R I I E R I V A T T V E S ,

S I N S 1 ~ S I N [ S I ) C 0 S S 1 ~ C 0 S ~ S 1 1 A f l : R T + H l * C O S S 1 Y l l I = R B * S I N S Z DYl(Z)=Rk7*COSSL D Y ~ ( ~ ) ~ Q ~ ~ * S I ~ S ~ * S I Y S Z Y[2)a-R!*SIVSI D Y Z I l ) = - R ! * t O S S l O V 2 1 2 1 = P , ,!Z=SQRT(RLwPYI(ZII D Z [I)=(R~+$TN2i*C~SS2)i(2,*7L) DZ(ZlrY(i)/(Ze*ZZ)

c , . . C., , , , E V A l U A T F C O O U b I Y A T t F U N C T I O Y S Y l,YZ, 4 N O C O N S T R A I N T C O O R D I N A T F i,

Or) 32C I = l r M

Z 2 : - J t C f l S A ? * S I N J S Z Z Z = Z Z + C C I , Y ) * Z ~ Z ( l ) = 0 7 ( ! ) + C t I t J J * 7 1 D Z ( ~ ) ~ D I ( E ) + C ( I , J ~ * Z Z

3 2 e C O N T I N U F C

D O 3 4 5 I = t , M B C l = I * P I R T A l : C I * S l S l N A l = S f N ( A l ) C O S A l = C b S ( A I ) Do 3 4 5 J = l , N B S I N J M l = S I ~ ( ( J ~ l ) * ~ 2 ~ C O S J M l r C O S ( ( J - l ) * S 2 ) Y 2 s S I N A I * C O S J M I u Y t P = C 1 * C d 3 4 l * C O S J ~ l Y V ~ ~ : - ( J * ~ ) * S I Y A ~ * S T ~ , ! M ~ Y ( 2 ) : Y ( E l + $ ( I , J ) * Y 2 D Y ~ ~ ~ I S ~ Y ~ ( ~ ~ * ~ ( I , J ) * Y Y ~ ? D Y Z ( 2 ) a D Y ? ( 2 ) + f l ( I , J ) * Y Y 2 2

345 C O N T I N U F c . C * a * * * C A L C U L 4 T t Y 3 4 N D I T S D k R I V A T I V E S F R O M THE C O N S T R A I N T C O O R O I N A T E ,

Y t 3 ) = R L - Z 2 * 7 2 D Y 3 l ~ l : ~ 2 , , * 2 Z * O Z ( 1 ) D Y 3 ( 2 ) : ~ Z , * Z Z * O Z ( Z )

C C . . . . . C A L C U L A T E I Y e P L A N E F X T E Y S T O V R A T I O S ( $ R l r 3 R 2 ) ,

S R ~ ~ S ~ R T ( ~ V ~ ( ~ ) * @ ~ ~ ( ~ ~ + ~ Y Z ~ I I * D V ~ ( ~ ~ + O Y ~ ~ ~ ~ * D Y ~ ~ ~ ~ ~ ~ R ! S R ~ = S Q R T ( ~ Y ~ ( ~ ) * D Y ~ ~ Z ) + ~ Y ~ ( Z ) * D Y ~ ! ~ ~ ~ + ~ Y ~ ( ~ ~ * D Y ~ ~ ~ ) ) ~ ~ ~

C C

I F [ I w T . G T , B ) G O Tq 6 2 I F ( 1 V t , k F , I N 5 1 G O T O 6 3 I N 5 : 1 N 5 + 5

6 2 d R I T E ( b r ~ f l @ ) ~ ~ , S ~ , Y ( I ) , Y ( L ) ~ Y ( ~ I , S R ~ , S R ~ c . C e * , , , D l 3 4 F I L E 2 S T o Q t S F U N C T I n N E V A L U A T I O N S A L O N G C M F R I D I O N A L C l I R V F S ( T H F T A a Z o C O N S l A W T ) , c , e . , . i 7 1 S K F I L E 3 S T O R E S F U N C T I O N E V A C U A T I O N S 4 L O N G C C I H C U H F E H F N T I A L C l J R V F S ( T H E T A ~ ~ ~ C O N S T A N T ) ,

b 3 G O T O [ l r 2 ) r T F N 1 w R I T E I Z , 4 1 U ) 32,StrY(lI,VCZ~rY(I),SR1,SSZ

Gn 7 0 3 2 N R I T E ( 3 , 4 t u ) S ~ , ~ I , Y ( ~ ) ~ Y ( Z ) , Y ( T ) I S R ~ I S R ~ ! 3 I N l ~ I N 1 + 1

C hu SI=Sl+HI 6 5 S Z = S Z + H Z

C C

1 @ 4 C f 7 N T I N U F C C C

E N D F I L k r!

E N D F I L k 3 U F L T A = S F C k D S ( f I M t )

, Y R I I E ( b r 6 B B ) D E L T A C,.:,.ROUTE D A T A HUFFFRED I q F I L E 6 1 0 P R I N T E R UPON C A L L I N G OF E X I T ,

C A L L SPOOL. ( h l CALL E X I T

c c C

14CI F O R Y A T ( ? @ A U ) 15P F f ' ) R M A T [ F 5 . d , F l p n f l ) 1 6 ~ FOHMAT(1H t ~ 8 Y t I Z r / X , F 5 , @ , F 1 2 , 5 ) 165 F O R Y A T ( l H l Z B A 4 ) 1 7 @ F O R M A T ( / / , lH , 5 4 X 1 ' F ILt C n E f F , D A T C , / r 1~ r 4 5 x 4 ' C O N T E Y T S OF R E C O R D S

1 ' 1 1 2 r ' T H R f l U G H ' r T 2 ) 175 F ~ R ~ A T ( i ~ f l c ~ 2 x , ' T I ~ € Y O o t L P A R A Y E T t R S (USER SPECIFIEOl',/,l~XI'

1 H A ' , ' N A ' , ' M C ' , ' NC'I' N S l e r @ NSZ: ,bX, ' p ' , 8 x 1 V H n I f 3 ' a t R X t ' C I t i C , R X t ' P 1 * , S X , ' R T C 4 8 X t ' R C ' )

189 F f l R M A T ( b l S / h F I M , U ) 1 8 5 F n R M A T ( I H f l , 4 C 1 Y r C T I R F YnnEl P A R A M E T E R S ' ) 29d F f l R M A T ( T I S ] 3fl8 f O R H A T ( l H r l Z Y t h I S r h F 1 3 . 4 ) J i f l F Q R ~ A T ( b I 3 ~ h F l 3 . 9 ) 336 F f l R M A f ( / / r r ~ 1 2 7 X I U F 1 7 , R , / / , l H 1 ) 3U f l F @ P H A T ( l H F , U 9 Y , ' W T N T ~ 4 I L I N G C O f F F I C I f h T S ' ) 3 5 Q F O R M A I ( / / r l H , 4 1 X , I ? t C C f l N V E C T E n C Q n R D I N A T E C U R V E S OF I N T F R F S T c , / ,

1 lH r 4 5 X 1 'Ci!RVF C V E ( I r 1 ) CvE(I,21') 3Rd F O R M A I ( I H P ? ~ X I ' S L ' ~ R X , ' S i C , h X ~ ' Y f I ) ' 4 ~ X I ' ~ ( ~ I ' , ~ X , ' Y ( ~ I ' , t 4 X o ' S R 1 '

+, l u x , ' S R Z ' ) 3 4 3 F O R M A T ( / / , 1 H ~ u S X , ' ~ U R V E C V E ~ I I 1) C V E ( I I Z ) ' I 39M F n P M A T ( l 3 , F 7 , 3 r F h e O ) U0a FORHAT(IH , ~ B X , ~ F I ~ ~ . U I ~ E I ~ ~ P I U l B F ~ ~ H P P T [ 5 ~ 1 ~ ; 4 1 ? ~ 1 7 , ~ )

F O R M A T ('?THIS R U h T O O K ' r F 9 , 3 , ' S E C O N D S ' ) c C

END

1:our d a t a car(1s (A-D), p l u s t h e c o o r d i n a t e cu rve

s p e c i f i c a t i o n c a r d s ( E l , L 2 . . . , L N C V ) , c o n s t i t u t e an

i n p u t module f o r program PTCHCV. Only one i n p u t module

i s r e a d f o r each r u n , which i s t e r m i n a t e d b y c a l l i n g EXIT.

An e n d - o f - f i l e c a r d i s n o t needed . An example i n p u t

module i s g i v e n be low, f o l l o w e d b y t h e r e s u l t i n g o u t p u t .

Card A (20A4)

T I T L E = 80 a lphanumer i c c h a r a c t e r s

Card B (315)

IRECN = 1 7 r e a d s o l u t i o n c o e f f i c i e n t s b e g i n n i n g i n r e c o r d 1 7 o f f i l e 1 0

e v a l u a t e f u n c t i o n s a l o n g 2 c o o r d i n a t e c u r v e s

IWT = 1 p r i n t e v e r y e v a l u a t i o n

Card C (615) --

Card D (6F10 .4)

c o p i e s o f c a r d s B and C o f t h e PATCH i n p u t module

Card E ( F 5 , O , F10 -0) 1

Card E, - (F5 . O , F10.0)

Card 1: d e f i n e s t h e m e r i d i a n p a s s i n g t h r o u g h t h e 1

c o n t a c t c e n t e r ; c ; l l c u l a t c d d a t a w i l l b e s t o r c d i n d i s k

f i l e 2 Card C., d e f i n e s t h e e q u a t o r ; c a l c u l a t e d d a t a u

w i l l be s t o r e d i n d i s k f i l e 3 .

Punched Card L i s t i n g

The c o m p u t e r o u t p u t f rom t h i s example r u n i s shown

on t h e f o l l o ~ ~ i n g p a g e s . The e x e c u t i o n t i m e on t h e PDP 1 1 / 4 5

i s p r i n t e d a t t h e end o f t h e r u n .

m n n v E m a Q B I I , I

W W W W Q b r - C P C O U J * o P , m 3 L n Q r n 9 m 9 9 Q B L n M & M u a r z l r .

W W W W a t - n r - m a f f i l - O l - C o . N o h a J n i C . 0 N n J - - + . d o ' M N C - 4 . . . . . . . . 6) 61 61 8 . b . I

W i r l W W a o a c r " , G l - a 2 a c & & n

C . 1 . 4 TI3NSOR. T h i s p r o g r a m r e a d s a s e t o f s o l u t i o n

c o e f f i c i e n t . : ( a I . , . . f rom d i s k f i l e 1 0 ( s t o r e d i j ' I J 1.1

17). a p s c v i o u s I'!Ir1'(:tl r u n ) and e v a l u a t e s ( 1 ) t l lc c o o r d i n r j t c s ,

y r , o f p o i n t s o n t h e d e f o r m e d m i d s u r f a c e ; ( 2 ) t h e g r a d i e n t s ,

( 3 ) t h e m a g n i t u d e s , G c l , o f t h e d e f o r m e d b a s e v e c t o r s ?'s,cx,

and t h e a n g l e , GAhDIA, b e t w e e n t h e d e f o r m e d b a s e v e c t o r s ;

( 4 ) t h e e x t e n s i o n r a t i o s , X . ( 5 ) t h e p h y s i c a l c o m p o n e n t s , r '

'a9 ' o f t r u e s t r e s s ; ( 6 ) t h e t r u e p r i n c i p a l s t r e s s e s , Ky;

a n d ( 7 ) t h e c a r t e s i a n c o m p o n e n t s , Er, o f a v e c t o r i n t h e

d i r e c t i o n o f t h e maximum p r i n c i p a l s t r e s s , K l . The above

s e v e n e v a l u a t i o n s a r e made a l o n g c o n v e c t e d c o o r d i n a t e

c u r y e s , 8 = c o n s t . and O 2 = c o n s t . , s e l e c t e d by t h e u s e r . 1

The s t r u c t u r e o f p r o g r a m TENSOR i s d iagrammed i n

F l o ~ i c h a r t 6 on t h e f o l l o w i n g p a g e s . T a b l e C-5 d e s c r i b e s

t h e p r o g r a m c o n t r o l d a t a t h a t a r e n e c e s s a r y f o r t h e

e x e c u t i o n o f TENSOR, .In e x a m p l e s u n , u s i n g t h e s o l u t i o n

c o e f f i c i e n t s f o u n d i n t h e e x a m p l e r u n o f P A T C H , f o l l o \ i s

t h e l i s t i n g o f TENSOR.

f i e n d t i r e model a n d s o l u t i o n

1 s p e c i f i c a t i o n d a t a

Read s o l u t i o n c o e f f i c i e n t s I I f rom d i s k f i l e 10 J

/ I Read p rog ram c o n t r o l d a t a

I s e l e c t i n g c o n v e c t e d c o o r d i n a t e c u r v e s I

LOOP

1 Convec ted c o o r d i n a t e c u r v e s I

L O O P

I S i m p s o n ' s r u l e s t a t i o n s a l o n g t h e c u r v e /

E N D LOOP

on d i s k f i l e 6

I: lori ,chart 6 . TENSOR

I S i m p s o n ' s r u l e s t a t i o n s a l o n g t h e c u r v e I

I E v a l u a t e Ga, Ghfl\lbL-\, ).r, and S aB 1

I W r i t e Ga, G.kbIbU, , and S a B I 1 on d i s k f i l e 6

/ END LOOP I

LOOP I S i m p s o n ' s r u l e s t a t i o n s a l o n g t h e c u r v e -I 1 / E v a l u a t e h.f and Er 1

W r i t e Ky and Lr ___)

on d i s k f i l e h

P r i n t c o n t e n t s

o f d i s k f i l e 6

\

R e t u r n t o START

F l o \ $ . c h a r t 6 . TLSSOR i c o n c l . )

FORTKAL' Lyar iab l e D e s c r i p t i o n

r e c o r d number r e f e r e n c i n g

s o l u t i o n c o e f f i c i e n t s

s t o r e d i n d i s k f i l e 10

number o f c o o r d i n a t e

c u r v e s

t a b l e s p e c i f y i n g t h e

c o o r d i n a t e c u r v e s ;

CVE ( I ,1) > 1 : e v a l u a t e - f u n c t i o n s a l o n g t h e c u r v e

e l = C E ( I , 2 ) , CITE ( I ,1) < 1 : e v a l u a t e

f u n c t i o n s a l o n g t h e c u r v e

El = CITE ( I ,2) ;

(I =1 , 7 , . . . , N C V ) , CVE ( I , 2)

i s g i v e n i n r a d i a n s

c PROGRAM TFNSOR C THIS PROGRAM FVALUATFS; C 1) THE C A R T E S I A N L O C U S OF CONVECTED COORDINATE CURVES, C 11) COVARIANT CbMPONENTS OF THE DEFORMED METRIC TENSOR,

"*' C 111) P H Y S I C A L C O u P O N E ~ 7 S OF TqUE STRESS, C I V ) P R I N C I P A L D I R E C T I O N S 4 N n P R I N C I P A L S T R E S S E S , C THE AROVE ARE E V A L U 4 T E b ON CONVECTED COOROINATE CURVES C S E L F C T E D RY THE USER, C C

INTEGER R , W REAL K l r K 2 D l M F N Q I O k A C ~ , U ) ~ ~ ( U # U ~ ~ C ( ~ ~ ~ I , Y ( 3 ) , ~ 1 ~ 4 ~ 4 ~ 1 ~ ~ ~ 4 1 4 1 , ~ 3 f 4 ~ 4 ) ~

1 v Y 1 . ~ 3 , 4 , ~ ) , V ~ 2 ~ 3 ~ ~ ~ 4 ) , 2 ( 4 # 4 ~ # z 1 ~ 4 , 4 ~ ' z ~ ( 4 , ~ I # 0 ~ ( 2 1 , 2 ~ I T L E ( ~ ~ ~ ~ ~ V ~ C Z ) ~ D Y ~ ~ Z ~ ~ D Y ~ I ~ ~ I T ~ ( ~ ) ~ T E ( ~ ~ ~ C V E ~ ~ @ ~ ~ ~

EQUIVALENCE ( Y A , M B ) , ( N I # N B ) DATA R / 5 / r w / b / , P I / 3 , 1 4 1 5 9 2 7 /

c c FILF IS A PRINTED OUTPUT B U F F ~ R ' , C F I L E 1s I S A RANDOM ACCESS L I 8 R A R Y b~ M f h l M I Z I N G C O E F F I C I E N T $ C P R E V I O U S L Y CALCULATED BY THE PATCH P R O G R A M , C

C A L L A S S I G N t b ~ * D K B ~ ~ E ~ S O R ~ L $ T g t 1 4 ) C A L L A S S I G N ~ ~ ~ , ' D K ~ ~ ~ ~ ~ ~ ~ , ~ I C O E Q P ~ ~ A ~ ~ ~ ~ ~ L ~ , I E R R ) DEFINE FILE i a ( ~ e , s e , l l , I N D X )

e C

1 T T H F ~ S E C N D S ( B . c C,,,,,IF E N D OF C A R D 1'4PUT F N C O U Y T E R E D TERMINATE PROGRAM EXECUTIONI C e r n , f i T H E R N I S E R E A D T I T L E C A R D F O R N E X T DATA SET,

READ(R, IBP,FND=SS) T I T L E C,,,,.SPECIFY T I R E M O D E L PARAMETERS O f S C R I B I N G P A T C H PROGRAM C H l l N F O R WHICH T E N S O R P R O t R A M E V A L U A T I O N S ARE B E 3 1 R E b l

R E A b ( R , l S @ ) H I I Y A # M C , N C # N S ~ I N S ~ , P , ? H I B ~ C I H I R ~ I R T , R L C MReMA, N0oNA (RY E f l U f V A L F N C E ) c . , C,,,,,IRECN IS A P O I N T E R T O THL R E G I O N OF D I S K F I L E l a C O N T A I N I N G C THE D F S I R F D PATCH PROGRAM T I R E MOOEL PARAMETER$ C AND ASSOCIATED M I V I M I Z I N G C O E F F I C I E N T S ,

R E A D ( R t 3 8 P ) I R E C N C C , , , , , i ? F T R I F V F T I R E M O D E L PAR4METERS O E Q C R I R I N G PATCH C PROGRAM R U N W H I C H PRODUCED M I N I M I Z I N G C O E F F I C I E N T S , C I N T E G R I T Y CHEC4 A V A I L F D f 3 Y V I S U A L CnMPARISON W I T H C C A R D I N P U T T I R E MOOFL PARAMETER L I S T ,

R E A b ( l s C I R E C N ) ~ A F , N A P ~ ~ C F ~ N C F ~ N S l P , ~ S 2 P ~ P C , P H I B F a C l H F , R ~ F ~ R T F ~ ~ L F c . C e a e e a R E T R I E V E ~ I N I ~ I Z I ~ G C O E F F I C I E N T S GENERATED BY PATCH PROGRAM,

I R E C M ~ I R E C N + 1 R F A b ( l a C I Q E C M ) ( ( A ( I r J l r I ~ l r H A ) , J ~ l r N I ) I R E C M s I R k f w t 1 R E A D t l a C I R E C ~ ) ((B(ItJ)rI~l,HB),J~l,NB) I R E C H I I R E C M ~ ~ R E A D ( ~ u ' I R E C P ~ ( ( C ( I , J l a I = l r M C ) , J = l , N C )

c

C w R I I E ( h ' , 2 5 @ ) T I T l F wRlTE(k r .3681 N R I T E ( w , ~ O P I ) Y A , N A , Y C , N C , Y S l , N S Z , P # P H I B o C I H t R I , R f , R L J R E C N P = I R F C Y + 3 f l R I I E f H r 9 9 0 ) I R E C N t I R E C N P k R I T E i w r 3 6 3 ) W R I T E [ w r 3 P 0 ) M A P ~ ~ A F , ~ C F , N C F # N S 1 F , N 3 2 F , P F t P ~ I R P ~ C i H F I R ~ F ~ R T F , R L F W R I T E ( w r 3 7 @ ) W R I I E ( ~ ~ ~ S ~ ~ ) ( ( A ( I ~ J ~ ~ J ~ ~ ~ Y ) I I ~ ~ ~ M A ) W R I T E ~ ~ ~ ~ ~ ~ ~ ( ( ~ ( ~ ~ J ~ I J ~ I I ~ ) ~ I ~ I I M ~ ~ ~ R I T E ( w # ~ S B J ( ( C [ I ~ J ) ~ J = ~ , ~ ) I I ~ ~ I M C )

C c . Gee*., E N T P R N U V R E R OF CURVES FOR W H I C H P U N C T I Q k E V A L U A T I O N S D E S I R E D a c. C , , * * e I F NCV IS G R E A T E R THAN Z E R O D E F I N E NEW CONVECTPO C O O R D I Y A I E C C U R V E S ALONG W H I C H F U N C T I O N E V A L U A T I O N S ARC T O R E MADE, C , , , O T H E R M I $ € L!SE P R E V I O U S L Y D P F I N F D C I J R V E S ,

R F A D ( R , J R B ) hCV I F ( N C V l 1 3 t 1 3 r 3

3 . W R I T E f W # 3 8 3 1 NCV C , , , , , E N T E R P A R A Y F T P R S D E F I N I N G D E S I R F O C U R V E S ,

Ob 9 L C = t r N C V R E A D ( R 1 3 8 5 ) C V t ( L C , d I r C V E ( L C , E I W R f T E ( W t 3 9 0 ) C V E ( L C e l ~ r C V E ( L t , Z l

s c n N r I w u E G O T O 6

1 3 d R I T E ( w r 3 5 7 ) NCV C C ~ . . , . E S T A B L I S H C O N S T A N T S F O R C U R R E N T RUN:

b P H I R = P H I B * , f l 1 7 4 5 3 3 P I B T t P I / P H I B H X ~ P ~ T P / ( N S l ~ l I n ~ = P s / t ~ S ? m l I P I D O z P I ( 4 , T P I O 4 ~ 3 , * P l D U M = M A X 0 ( M A t M C ) ~ o M A X @ ( N A , N C l R T D = ~ B B . / P I

C

c , 8 I S k = w I

C D O 7 0 Ifl*ttT I F ( I S W ) 9 1 180 1 1

9 M R I T E t W r 4 @ 0 ) G O T O 15

1 0 k R I T E ( h 4 4 5 f l ) G n 10 15

1 1 W R I T f ! ( W r U ? B I 15 9 2 g S 2 1

C - 16 D O 65 N2:11NS2X

S X = S l ? S l N S 2 = S I N ( S Z f c n S 9 2 = c o S [ 9 2 )

C C

on 64 N ~ = ~ , N S I X . ,

C,..,.INfTIALIZE C O O R D I N A I E F U N C T I O N S AND T H E I R D E R I V A T I V E S , SINSl=SINIS!) C o S S l ~ C O S ( S l ) R P I R T ~ R ~ ~ C O S S I Y ( 1 I = R B * S I N S Z D Y i ( Z ) = R 0 * C o S S 2 D Y ~ t l ) z - R ! * S I N S l * S I N S 2 YIZ)=aRl*SINSl D Y Z ( l ) = ~ R l * C O S S I b Y Z t 2 ) = 0 ~ Z Z I S Q R T ( R L W D Y i ( 2 ) 1 D Z ( ~ ) = [ R ~ * S I N S I * C O S S ~ ) / ( ~ ; W ) O Z [ Z ) e Y ( i ) / ( 2 * r Z Z )

C C c . ,

C m * e e * t V A L U b T E C O O R D I N A T E F U N C T I O N S Y i r V Z , AND C O N S T R A I N T C O O R D I N A T E 2 , 00 32fl f = l t M RR=,5*(Z*I-11 C 2 s R R * P I B T A2=C2*Sl SlNAZzSIN(A2) C O S A Z 2 C O S ( A 2 1 D O 324, J a i f b $1NJS2zSIN(J*SZ) COSJSZ:COS(J*SZ) I F ( l * M A ) 2 1 @ , 2 1 B r Z 4 f l

? I @ I F ( J w N A ) 2 2 0 , 2 2 f l , 2 4 0 2 2 0 Y l ( I , J ) t C O S A Z * S I h J S ?

Y V ~ ( ~ ~ I I J ) ~ - C Z * S I N A ~ * S I Y J S Z Y Y Z ( 1 t I t J ) = J * C Q S A 2 * C O S J S Z Y t l ) = Y ( l ) + A l I , J ) * Y l l I , J ) 0 ~ 1 ( 1 ~ ~ ~ ~ 1 ( ~ ) + ~ ( 1 ~ ~ ) r v v l ( l r ~ 1 ~ ) D Y ~ ( ~ ) = ~ ~ ~ ( Z ) + A ( I ~ J ) * Y Y ? ( I I ! , J )

2 o R I F t ! -PC) ?SF, 2 5 5 1 3 2 9 2 5 5 I F ( J m N C l 2 6 P l ? b n r T Z f l Zh f l Z(I,J)=COSA?*COSJS2

Z l ( l r J l ~ ~ C 2 r S J N A 2 * C O S J S 2 Z?lI,J)=-J*COSAE*SINJSZ ZZ~ZZ+C(IIJ)*Z(I, J ) D Z C l ) : D Z ( f ) + C ( I r J ) * Z S C I , J ) Dt(?)~DZ(?)+CtI,J)*ZZ(I,J)

320 C O Y T I N U F C

D O 3 4 6 I = I , M B C l a I * P I R T A l t C j * S l S T N A I = S T N f A l ) C O s A l = c O S ~ A l ~ D O 34fl Jtl,NB SINJMltSIN((J-1)*32) COSJHlrCOS([Jnl)*S2) Y Z ( l , J ) ~ S I N A l * C O b J ~ i Y Y l ( 2 , I , J ) = ? j * C f l S A 1 * C @ $ ~ J M 1 Y Y Z ( 2 r I r J ) a ~ ( J ~ t ) * S I N A l * S T N ~ ! M l Y ~ ~ ) ~ Y ( Z ) + B I I I J ) * V ~ ( I , J ) D ~ ~ [ ~ ) = D Y ~ ( ~ I + ~ ( I I J ) * Y Y ~ , ( ~ , T I J ) D Y ~ ( ~ ) ~ O Y ~ ( ? I + B [ I I J ) * V Y ? ( E I ! O J )

3 4 9 CONTINUF

Q a q a CALCULATE y T Ah0 I T S D E R I V A T I V E S F R O M THE C f l N 3 T R A I N T C O O R D I N A T ~ ; Y ~ ~ ) = R L Q Z Z * Z Z ~ y 3 t 1 ) I * E ~ * Z Z * O Z ( 1 ) O V 3 ( 2 ) o m 2 . * 2 Z * D 2 ( 2 )

C I F ( ISW 1 6 2 , 3 @ , 30

C 30 T R 0 t l b / l R @ * R 8 )

T R 1 z I a / ( R I * R I )

c . . ' C O M P U T E COHPOYFNTS nF 4 V E C T O R NORMAL T O THE DEFORMED M ~ O S U R P A C E , C . . . . .

B S P B ~ * ~ ~ + R 2 * 8 2 + 03*83 c . * C,. , . , C A L C U & A T E TRANSVERSE EXTENSION RATIO ( S R 3 ) AND I T 3 CUBE,

8 R 3 s R e r R l / S Q R T ( 8 8 1 3 R 3 C # S R 3 * * 5

c , C,,,,,COMPUfE D E R I V A T I V E S OP SR3,

C t r T R B * T R I * d R S C DLYll~Cl*tDY3(2)*82~DY2~2)*B3) D L Y 1 Z ~ C l * ( D Y Z ( l ) * B 3 . D Y f ( l ) * 8 2 ~ D L Y Z l r C f f i t D Y I t Z ) * 8 3 r O Y 3 ( Z ~ * ~ l ~ O L Y Z Z ~ C ~ * ( O Y ~ ~ ~ ) * ~ ~ . D Y ~ ( ~ ~ * ~ J ~ D L Y 5 1 ~ C l * ~ D Y 2 t 2 ) * 8 1 ~ D Y 1 ( 2 ~ * ~ 2 ~ D L Y ~ z ~ c ~ ~ ~ D V L ( I ~ * B Z ~ O Y ~ ~ ~ ) ~ B ~ )

c, ' c r L c u C A r r I A C A A N G C S T R E S S R E I U L T A N T ~ , C , , . ,

T C M ? s 2 , * C l H Tl(llaTEnPt(TR1*OYlti)+SR3*OLY1tI T ~ ( ~ ) ~ T P M P * ( T R B * D Y I ( ~ ~ + ~ R ~ * ~ L Y I ~ ~ ) T ~ ( ~ ) ~ T C M P ~ ( T R ~ * D Y ~ ( I ) + ~ R ~ * D L V ~ I ) t 2 ( 2 ) = T E M P * ( T R B * O Y Z ~ a ¶ + S R 3 * D L Y Z ~ ) T ~ ( ~ I ~ ~ ~ M P ~ ~ T ~ ! * D Y J ~ ~ ~ + ~ R s * o L Y ~ ~ ) T ~ ( ~ ) ~ T E M P * ~ T R ~ * D Y ~ ( ~ ) + S R ~ * O L Y ~ Z )

" C q * , . , CALCULAIE INmPLANE E X T E N S I O N R A T I O S ( 8 ~ i l S R ~ ) ; SRl8GI/R1 S R Z s G 2 l R 0

C I P ( 8 2 .GT; P I 0 4 ,AND, $ 2 ;LT, T P I D Y l GO T O SS

C L L c u L q E WIRcnorF T E N s q R s ( U S I w B s DETERMINANT); C,*,.* 5 0 T K l l * ( T I ( l l * D Y 2 ~ 2 ) a T l ( a ) " b Y 1 ( 2 ) ) / B S

T K l ~ a ~ T l ( Z ~ * O V I ( l ) T 1 C l l * O V Z ( I ) ) I B 3 t ~ ~ 2 r ( f 2 l ~ ) f i 0 ~ 1 t 1) * T Z ( l ) * O Y ? ( l ¶ ) / b S

c GO T O S T

c . ' C A L C U L A ? E ,KIRCHOFF T E N S O R S (USING s j DETERMINANT) ' , C,.,c,

5 5 T K l l ~ t T ~ ~ ~ ) * D Y 3 / 2 1 ~ T l ( f ) * b Y 2 ~ Z ) ) ~ ~ u ~ ~ ~ ~ T i ( s l r ~ Y a ~ i ~ ~ t i ~ Z ~ * D Y 3 ~ t ) ) ~ B l tK22r(T2tl)*DYZ(i)*TZ~2~*OY3(1l)/Bi

C C

57 I F C 1 3 ~ ) 6 0 , b 0 1 5 8 c < t

C. . , , ,EVALUATE C O N T R A V A R I A N T COHPONEkTS OF M E T R I C TENSOR, 58 D G ~ ? G L l ~ * G L Z Z ~ G L l 2 * C L I Z

G U l l ~ O L L Z ! D C L G U l ~ s . C L l Z / D G L C U Z Z ~ G L I I / O G L

C, , 9

C # , , , , T O O f A G d N h L I Z E THC KIRCHOPF TEN3OR C O N S t R U C T C H A R A C T E R 1 8 t l C C P Q L Y N O M I A L FOR ? S S O C I 4 T L D G E N I ? R A L I Z E D E IGCNVALUt PROBLEM,

C 2 ~ 6 U l l * G U Z 2 ~ G U i Z * G U l 2 C 1 ~ S R ~ * ( Z I * T K 1 2 ~ G U l I I ~ ~ K 1 1 * G ~ J Z Z ~ T K Z 2 t G U t 1 ) C0:~R3*SRf~~TKl1*TKZ2-TK12*TK1Z)

c. 4 *

C , , , , , D P T ~ R M I N E EIGPNVALUFS' , IBRr0 IFICZ+,0060[) 5 1 0 , 5 1 0 1 5 ~ 2

St72 I F ( C 2 ~ , 6 0 0 0 1 ) 5f lSaS10#810 505 I E R a l

GO TO 5 3 0 510 D S Q R a C i * C 1 * 4 , * C 2 * C Q

fF (DSQR)520aSZB,S~B 520 1 E R a 2 530 WRITE(W,480 ) 1 E R

$0 TO 6 1 548 K l ~ ( * C 1 + S Q R T ( D S Q R ) ) l ( Z e * C 2 )

KZS(*C~*SQRT(DSQR))I(~@*CZ)

" C , O , . * ' D E T E R M I N E E I C E N V E C T O R C D R A E S P O N D l N G T O LlGENVALUE EVILI; C O E F ~ 3 R S * T K 1 2 ~ K I * G U l ~ I F ~ C O E P + ~ 0 $ 0 0 1 ) f S 0 r 5 5 8 r 5 4 5

545 I C ( C O I ! P ~ , O B B B I ) 5 6 0 , 5 5 @ r 5 5 0 590 R N l l l e

R N ~ ~ ( T K ~ ~ ~ K ~ ~ G U I ~ ) ~ C O E F G O T O z S t 0

560 R N l m l c RNZsBa

510 TMiaGUlt*RN1tGUla*RN2 T ~ 2 ~ G U l ? * R N l t G U Z Z * R N 2 t 1 ~ 1 ~ 1 ~ 0 ~ 1 ( 1 ) + ~ ~ 2 ~ ~ ~ 1 ~ ~ ~ E 2 ~ f M l * B Y 2 ( l ) + T M 2 * D Y 2 f Z l E f o T M j * D Y 3 ( 1 ) + T ~ Z * D Y 3 ( 2 ) G O TO 61

c , C , , , , , C A L C U L A T E P M Y S I C A & COHPQNFNT9 OF TRUE S T R E S S ,

bf l s ~ ~ ~ R ~ ~ ~ R ~ * s R ~ * T K ~ ~ / s R z S l Z o R @ * R l * T K l Z S Z a ~ R f l c R 0 * S R 2 * T K 2 E / S R l

C WRIl'E(WrSB0) s ~ , s ~ ~ G ~ ~ G ~ ~ G A ~ M A # s R ! I ~ R ~ , ~ R ~ I s ~ ~ I s ~ z # s ~ ~ G O TO 64

6 1 dRITE(W,580) S 2 r S l , K l , E l r E 2 r E 3 r K Z G O T O 64

6 2 HRITE(WISRB) S ~ ~ S ~ ~ ~ ~ ~ I ~ Y ~ ~ ~ ~ Y ( ~ ~ I D Y ~ ~ ~ ~ I ~ Y ~ ~ ~ ~ I O Y ~ ~ ~ ~ ~ O Y ~ ~ ~ ~ ~ 4 D Y 2 ( 2 ) 10VP(kl

C 6 4 s i a s l + n i x 6 5 32r92+H2X

C C

70 I S W = 1 3 W + l 75 CONTINUE

C C

e o DELTA=$ECNDS ( T I M E ) HRITE(W,bBBI D E L T A

C;,.,~GO T O B E G I N N I N G OF P R O G R A M FOR NEW T I R E MODEL DATA, t o T O 1

C

' ROUTE DATA BUrFEReD I N F I L E 6 T O PRINTCR UPON C A L L I N G Or EXIT: C , . , , , 95 CALL SPOOL(W)

C A L L E X I T C C c C

1 0 0 PORMAT(Z044 ) 150 F O R M A T [ 6 1 5 1 b ~ i 0 ; 4 1 258 FOAMAT( 1 ~ 1 2 ~ ~ 4 ) 380 FORHAT(1H t l a X , b 1 5 r b F 1 3 r 4 ) 350 F O R Y A T ( / / r l H r U ( 2 7 X # 4 E i ' ? f i b r / / r l H 1 ) 357 PORMAT( / / , /H r 5 0 X r ' U S E M O S T RECENTLY b E F J N E b C 4 / r l H ,U9X,'CONVECTEO

1 C O O R D Z N A T E C U R V E $ ' r / , l H , S t X r ' ( N C V = ' , 1 2 , * l r ) 360 TORMAT( lH0 ,42x r ' T I R E MODEL PARAWETERS, (USCRISPECIP IED) ' , I , 1 ~ x 1

1 M A C , NA',' qci t i N C ~ , ' NS1.p: N S Z , 8 X r ' Pg,8X, # pnfbi 2,6X, ' G I Y ' , ~ X , ~ R 1 ' , 6 % 1 ' R T 0 r 8 X , ' R L C )

36s F O R M A T ( l ~ e , sax , ~ T I R C MOOEL P A R A M E T E R S * 1 37a FORMAt(1H0149X,'MlNIMIZ1NG COEFPIC IENTS ' ) 380 F O R M A T ( I S 1 38)- FORMAT ( / I # l H ,41X, I a , CONVEClEO COORDINATE CURVES OF I N T @ R E ~ T ~ , / /

1 , i ~ rS3X,, 'CVECIr l) C V E ( X P ~ ) # ) 365 F O R M A T ( P S ~ 0 , F 1 0 ~ 0 ? 398 FORMAT (lH r S 6 X , P S a 0 ' P l a r S ) 0 0 0 F O R M A T ( ~ H ~ , ~ X , ' s a i , 7 x , i s i ~ , 7 x , ~ r t i 1 ~ , ~ x , ~ r ~ 2 ~ ~ ~ t x r , ~ y ( i ~ ~ , q x I i o

+ ~ l ( l ) ' , $ ~ , ~ ~ ~ ~ ( l ) ' , 4 ~ , ~ ~ ~ 5 ~ i ~ ~ , 5 ~ , ~ 0 ~ l t 2 ~ ' , ~ x , ~ o ~ 2 ( 2 ~ ~ , s x , ' o ~ ~ t 2 I '

+ 7 X t 0 E S c r b X r C 4 2 ' ) 480 F O R M A T t I H , ' IER 8 ' r I 2 ) 500 F O R ~ 4 T ( 1 H , , 1 1 F 1 1 a U I 6 0 0 P O R H A T ( ' @ T H I $ RUN TOOK',F91J,' SECONDS') 9 9 0 - F O R M A T ( / / , l H . S q X a ' f I L E COEFFf iDA t ' r / , l H ,4SX, 'CONTENTS OF RECORDS

1 ' 4 12, ' THROUGH ', 1 2 ) C c

END

Ilxamnlc 'I'I'NSOK Run

F ive d a t a c a r d s ( A - I : ) , p l u s t h e c o o r d i n a t e c u r v e

s p e c i f i c a t i o n c a r d s (F1, F 2 , . . . , F N C V ) , c o n s t i t u t e an

i n p u t module f o r program TENSOR. There may be a s many

i n p u t modules a s d e s i r e d . An e n d - o f - f i l e c a r d (EOF) s h o u l d

f o l l o w t h e l a s t i n p u t module . I f NCV = 0 , t h e c o o r d i n a t e

c u r v e s s p e c i f i e d i n a p r e c e d i n g i n p u t module a r e u s e d f o r

t h e f u n c t i o n e v a l u a t i o n s . An example i n p u t module i s

g i v e n be low, f o l l o w e d by t h e r e s u l t i n g o u t p u t .

Card A ( 2 0 A 4 )

TITLE = 80 a lphanumer i c c h a r a c t e r s

Card D (IS)

IRECN = 1 7 r e a d s o l u t i o n c o e f f i c i e n t s b e g i n n i n g w i t h r e c o r d 1 7 o f f i l e 1 0

Card E ( I S )

NCV = 2 e v a l u a t e f u n c t i o n s a l o n g 2 c o o r d i n a t e c u r v e s

( I f N C V - > 1 , append c a r d s F1, F Z , . . . F ' N C V ' O t h e r w i s e ,

c a r d E i s t h e l a s t c a r d i n t h e i n p u t module . )

Card F2 ( F 5 . 0 , F1O.O)

Punched Card L i s t i n g --

TENSOR 1 , ~ ~ a 4 . 5 3 R T s J , ~ 4 u 4 4 , 13 19 ,

1.006 135; 1. 1. 17

2 1 1 1~57080 8. ,78548

The computer o u t p u t from t h i s example run i s shown on

t h e f o l l o w i n g p a g e s . The e x e c u t i o n t ime on t h e PDP 11 /45

i s p r i n t e d a t t h e end o f t h e r u n .

N N n J N 6 ) m 8 5 C t r I

W W W W n - n o d N Q V , S U V V n t - r S V I

LF r- n 02 = J y c l r r b m m - . . . . . . . . F i W ( E 5

. t I

C . 2 SECONDARY PROGRAMS

'I'lle p l o t t i n g p rog ra ln s wr.i t t c n t o g raph ca l l cu ln t cd

r e s u l t s s t o r e d i n f i l e s 2 , 3 , and 4 a r e l i s t e d i n S e c t i o n

C . 2 . 1 . The f i l e r e f e r e n c e program, L O O K , which was

w r i t t e n t o r e t r i e v e and p r i n t a s e l e c t e d r e g i o n o f t h e

s o l u t i o n c o e f f i c i e n t l i b r a r y s t o r e d i n f i l e 1 0 , i s l i s t e d

i n S e c t i o n C . 2 . 2 .

C . 2 . 1 PLOTTING PROGRAhIS. Four p l o t t i n g p rog rams ,

PLTBND, PLTMRD, PLTLAT, P L T L 1 2 , were w r i t t e n t o p roduce

s p e c i f i c g r aphs o f t h e c a l c u l a t e d r e s u l t s . The f u n c t i o n

o f e ach o f t h e s e programs i s i n d i c a t e d by comment c a r d s

a t t h e head o f t h e program l i s t i n g ; t h e program l i s t i n g s

a r e g rouped i n t h e f o l l o w i n g p a g e s .

The p l o t t i n g programs make u s e o f f o u r s u b r o u t i n e s ,

PLOTST ( s t a r t p l o t ) , PLOT ( p l o t p o i n t ) , NUMBER (draw a

number ) , and P L O T N D (end p l o t ) , which a r e p a r t o f t h e

PDP 1 1 / 4 5 p l o t s o f t w a r e l i b r a r y and a r e n o t f u r t h e r d e s c r i b e d i n

t h i s r e p o r t . The p l o t o u t p u t i s p roduced by a Calcomp 565

d i g i t a l p l o t t e r , d r i v e n by t h e PDP 11 /45 i n a mu l t i p rog ram

mode.

C PROGRAM PLTRNO C T H I S P R O G R A M GRAPHS TYE C O V T A C T RnUNDARV C F R O M DATA S T O R E D I N F I L E O E V I C E 4; C C

D I H E N S I O h D A Y [ J I , T I M ( Z ) DATA B L A N K / ? '/

C C

C A L L A S 3 I C N t 4 , ' D T l t ~ l ( 0 r ~ I C U R V ~ , O A T ~ ~ ~ , Z ~ ) C C

T ~ ~ S E C N D S t e; DO 9990 131,3

9990 D A Y ( f ) s B L A N K C A L L bATE(O4Y) C A L L T I M E ( T 1 M ) C A L L P L O T S T ( , Q 0 5 , ' I ~ ' ) C A L L f ' L O T ( Z 1 0 ~ l E e 0 , ~ 3 )

C E S T A R L I S W NEW REFERENCE P O I N T PEN UP CALL P L O T I Q , f l , * l i ? e B , * Z )

C E S T A R L I S H NEW REFERENCE P O I N T r PEN DOWN w DRAWS L I N E c r L L P L O T ( ~ , Q , I , B , . S )

c

~ o l E , S f i n E L 0x40, OYaOEL 0 x 1 0 , l D v l a @ , DXi!8.,4 D Y ~ L a ~ , u + W I o T DY2o0 , D X 0 8 0 n DYB?,'S*DEL N P s 1 3 F= 1; 2

2 B CALL P L O T t X r Y t E I X = X ~ D X 0 YzY*DYB

C D O 30 I a l r r J P CALL P ~ b l ( X 4 D X l r Y + O Y l r 3 ) CALL P L O T ( X , Y l 2 ? I F ( F ! L E , , I ) G O T O 25 I F l J ,EQ, 1 ) G O T O 2 4 f F ( F , I T q i r ) DXaeOX2L

p4 CALL N u ~ B E R ( X + D X ~ ~ Y + D ~ Z , ~ ~ ~ ~ F ~ ~ ~ ~ ~ I 25 XaX-OX

Y a Y r D Y F:FmDF

30 C O N T I N U E c

I F ( J , E Q . 1 ) CALL PLOI ( I . ~~*DEL I *~ ;B+OELI .~ ) 4 0 C O N T I N U E

C C C C

CALL P L O T ( @ ; ~ ~ , ~ ~ * D E L I ~ S ) C C C c , C , , , , , G R A P H THE C O N T A C T B O U N D A R Y

1SIGNX:-1 1 3 I G N ~ t w 1 I M l t * l I n 2 3 1

C C

D O 6 8 I t l r 4 I M l o l H l * ( * l ) f H 2 r I H Z * ( r l ) I $ I G N X ~ l S I G N X * I ~ l I S I G N Y ? ! S I G N Y f i I " l 2

45 R e A O ( 4 r 2 8 @ , F N O = S S ) M A X 48 MAXMaYAY-1

R E A D ( 4 r Z L B ) Y Z l y l X , a Y ? * X S C A L E * I S I G N X Y I Y ~ * Y S C A L E * I S I Q N Y

CALL P L b T t X # Y a 3 ) C

D O 5 8 K 2 ' r l t M A Y M READ(4 t 2 l n ) Y Z t Y l XaY2rXSCALE*1SIGNX Y ~ Y l * Y S C A l E * I $ I ~ N V CALL P L Q T ( X # Y t 2 )

5B C O N T I N U E C

G O T O 45 55 I F ( l n 4 1 5 7 t b 0 8 6 f l 5 7 REWIND 4

C C

4 0 CONTINUE C C C C

CALL P L O T ( S : B * D F L # ~ ~ , ~ S ~ ~ ~ ~ c C

2$fl F O R Y A P (IS) 218 ~ 0 ~ ~ ~ ~ 1 2 ~ i 4 . 5 )

C C

Sfl9 CALL PLOTYO I T 0 t 5 C H A R G € ~ S E C Y D S ( T ~ ) / ~ ~ V I R , I ~ , W R I T E ( I T 0 # 9 9 9 4 ) C M A R G E ~ O A V a T I M

9 9 9 4 F O R H A T ( I H @ t / / ' THE CHARGE FOR T I R E P L O T S I S ' # F 7 , 2 r a HH3' / * ' ON ' # 3 ~ 4 , 3 ~ # 2 h 4 / / )

CALL E X I T C c

END

C P R O G R A M P L T M R D C T H I S P R O G R A M GRAPHS THE U N O F F ~ R M E D AND DEVORHED Wl!RIDIAN c C I V F N BY THF E Q U A T I O N THETA-2:@, C C

DfYl! fUS1ON D A Y ( S ) , t I M ( Z ) DATA R L A N K / ' @ I

C C

CALL A S S I G N ( ~ I ' D K ~ a l l l 0 t ~ ~ C I J R V Z , D A I P li,m C C

T ~ ~ S E C ~ O S ( B ; ) D O 9999 l a 1 1 3

9990 D A Y ( f ) a B L A N K CALL b A T E ( D A V ) CALL T I M E ( T 1 H ) CALI,, P b O T S T ( a Q J 8 5 ~ ' I " J ' ) CALL P L b T t 2 . 0 1 l t ? c f l r + 3 )

C ESTABLISH NEW P E F E Q t N C E P O I N T - PEN UP CALL P L b T ( 0 , 0 , . 1 2 , 0 , . 2 )

C E S T A B L I S H NEW REFERENCE P O I N T PEN DOWN * DRAWS L I N E CALL P L O T 1 3 ; 0 1 1 a 0 , * 3 )

C C A L L P L O T [ B ' . B ~ B , ~ , Z )

c * C a , , , , E S T A B L I S H SCALING R A T I O S ,

x ~ 1 . 8 ; ~ Y $ T ? 0 . 0 F z 3 a 0 00 10, 1 t l r 5 F = F p I a C A L L P L ~ T ( X S T , Y S T + , i r f i CALL P L b T ( X S T r Y S T r 2 ) C A L L N u ~ B E R ( x S ~ I Y S T * , S ~ ~ ~ ~ U I P ~ B . # w B ) X S T S X S T W Z : ~

C 10 C a N t I N u E

b, C, . . . :CONSTRUCT Y y A X I S ,

CALL P L O T ~ 4 p 0 r 0 c 0 ~ ~ ~ ) CALL P ~ O T ( O p 0 1 9 a 5 1 2 ) X S T ~ 8 , B YSTp8, la F = S , B 00 1 5 I r l t U Pa$- I, CALL P L O T ( X S T + , ~ ~ Y S T , J )

C A L L P L d T ( X S f . , f , Y S T , Z ) CALL N U M B F R ( X S T ~ , J ~ , Y S T I ~ ! Y I F I ~ ~ , * ~ ) Y S T S Y S T - 2 . 0

15 C O N T I N U E C

r ' C , , , , ,GRAPH UNDEPORMED THETAni?oB, CURVE,

XRENt@,BhXSCALE YBENsRT*YSCALE CALL P ~ , , O T I X R E N , V B F N I + ~ ) P 1 3 3 r 14159265 R A O ~ P I / 1 0 0 , I S I G N s l b E G = 1 3 5 r * R 4 D X a R l * X S C A t E * S f N ( D E G l + X 0 E N Y : R l * Y S C A L E * C f l S ( D F G j t Y B E N CALL PLOT t X I Y 1 3 1

1 0 0 5 0 T t l t 2 7 f l r S K = I + 4 M a r l 2 M

I F ( M I 3 r 5 , q 3 M M 8 2

G O Tb 5 5 4 Y M 1 3

55 00 60 Jt1,K

D€G=136e*J IP (bEG)56 ,57r 5 7

5 6 ISIGka-1 5 7 DFGREEcABS[DEG)rRAD

X = I % I C N * X S C A L F * S I N ( D E G R E E ) + X B E N YrYSCALE* COSIDEGREE)+YREN CALL P L Q T C X I Y ~ H M )

be CONTINUE 5 8 C O N T I N U E

c, . C , . . , , G R A P H PROFILE OF C O N S T R A I Y T PLANE',,

X B t N a ~ Z , r X S C A L E VBENoffL*YSCALE CALL PLOT (XBENr YBEN, 31 XBENtABS(XBEN1 CALL PLOT (XBEN, Y B E N I ~ I

KdUNT+6OUNT+ 1 GO TO (41 1 5 f l B l r KOI INT

41 ISIGNs-1 REWIND 2 R E A D ( ? , 31fl) I D l r I b Z 1 ID3rIO4r 105, ~ D ~ , D M ~ , D M ~ , D H ~ I O M ~ , O M ~ , D M ~ R E A D ( Z 1 3 9 P ) NUMRR,CVl ,CVZ Gn T O 35

5 0 8 CALL P L O T ( 7 , 0 , l Z , f l , ~ 3 ) CALL PLOTNQ 1 T 0 t S C H A R G E = S E C N D S ( T 1 ) / 3 0 8 e , / 6 , W R I T k ( I T f l , 9 9 9 4 ) C H A R G E , O A Y , T I M

9994 F O R H I T ( l H B , / / ' THE C H A R G E FOR T I R E PLOT3 IS ' ,F7 ,2 , ' H R S ' I * ' QN ' , S A U , S X I Z A U / / )

CALL E X I T C C

END

C PROGRAM P L T L A T C T H I S PROGRAM GRAPHS CONVECTFO LATTITUbE C O O R D I N A T E C U 9 V t S C DPP1NFO BY S E T T ! N G ?HI?TA-~SCONSTANT, C D A T A I S STORED I Y F I L E D E V I C E 3 . C C

D I M L N 8 I O N DAYt3),tIM(2) D A T A B L A N K / ' i /

C C

CALL A S S I G N t 3 r "Kt 1 l l l 0 r 3 1 C U R V ~ ~ O A ~ I l i , 2 2 ) C C

T ~ P S F C N D S ( B ' , DO 9998 I s 1 1 3

999s D A Y ( l ) r R L A N K CALL D A T E t 6 4 V l CALL tfM€.(TIMj CALL PLOTST, (~B95r ' IN ' ) CALL PLOT(2 ,0 t \ za01m3)

C E S T A R L I S H NEW REFERENCE P O I N T PEN UP CALL PLOT (Ba01*1Za01*? )

C E S T A R L I S H NEW REFERENCE P O I N T * PEN DOWN a DRAWS L I N E CALL PLOT(3,0,1,0,*3)

C c , 5 t

C,,,,,E$TABLISH S C A L I N G R A T I O S , X3CALF.t.,5 Y S C A L E + a S

C C

DF=ZI ~ ! 8 s r l , / 8 ,

C CALL P L b T [ B , r 5 , 2 5 , ~ 3 )

C C C c C , , . , ' ,CONSTRUCT X W A Y I S , THEN Y - A X I S ,

DO 5 0 J s l r 2 IF(J ,EQ, 2) Gb T O 10 X ~ b . 5 ~ 3 0 , D b 1 . DYt0, On lsar DY ls*, 1 D X 2 ~ X 1 8 DXZPsB, O Y Z ~ r l F r r b , GO T O 20

10 XoBp Y t 6 r 5 DXaB,

C A L L P L O T t X c Y ~ Z ) I F ( F p L T , 0 , ) G O TO 24 1PCP ,EQ, 0.1 Go T O 25 D X 2 t O X Z P CALI. N U M B E R ( X + O X Z ~ Y + D Y 2 r , 1 4 , F a B , , * @ j x r x * D % Y o Y m D Y FzP+DF C O N T I N U E

I F ( J ,EQ, 2 ) GO T O 40 C A L L P L O T ( + 3 , 5 t * 3 r S , - 3 ) G O T O 5s CALL P L O T ( B ; ~ ~ , S ~ * ~ ) COrJTINUG

R I M J s R I M * X $ C A L E CALL P L O T ( R I M S I B , , ~ )

L , C . . , , .GRAPH P R O F I L E OF C O N S T R A I N T PLANE,

a ,

Y ~ R L * Y S C A L E * ( s l r ) C A L L PLOT(Z',,Y,I) CALL P L O T ( a 2 , n Y n Z )

c C C C! , , C , , , , , G R A P H UNDEFORHED C O Y V C C T E D COORDINATE CURVE C D E P l N E D BY T H F T 4 - l a C V 2 ,

C I R ~ R T + R l * C O S ( C V Z ) C I R S = C I R * X $ C A L E CALL P L O T ( C T R S I @ , I ~ I

C C

I T U S MHEE How 1

C D O I S I t l r 3 6 8 XaCIR*COStl*RTD)*%SCaLE Y z C I R * 3 I N ~ I * R T D ) * V S C A L E CALL P L O T ( X , Y n M M ) I F ( I T . I ) 7 0 , 7 8 n 7 5

t a I T ~ T T + S MaM* (1.1) t4MrHM+M

75 C O N T I N U F C c C c . , c. . . . . G R A P H THE DEFORMED C O N V E C T E D C O O R ~ I N A T E CURVE T H E T A ~ ~ = C V Z ' ,

ISIG@.l C

D O 1 1 8 K a l r r ? I S I G = I S I G * ( * l ) I F ( K ,NE, 2 ) G O T O 98 q E 4 0 ( 3 , 2 0 0 ) I D l n I D Z n 1 0 3 , I b 4 , IDS, I O ~ , D ~ ~ I D H ~ I D H ~ , D H ~ ~ D M S ~ D M ~ R F A D ( 3 r Z I B ) N U H R , C V l r C V Z

90 R E A 0 ( 3 , 2 2 0 ) S ? r 3 l , Y 1 , Y 2 n V 3 r S R l , S R Z X ~ V 1 * X S C A L € * l S I G Y . Y f * Y S C A L E * ( ~ l . ) CALL P L f l t ( X 1 Y t 3 )

C D O 1 0 0 X ~ i r l 8 0 R E A O ( J , E Z B ) S h ~ 3 ! , Y l ~ ~ 2 ~ Y 3 r S R l r S R Z X ~ Y I * X S C A L E * I 3 I G Y = Y 3 * Y S C A L E * ( r l , ) CALL P L b T ( X n Y , 2 )

1 Q B C O N T I N U F C

R E W I N D 3 C

1 1 0 C O N T I N U E

C c

CALL P L O T ( U , , * 6 , 2 9 # - 3 ) c

2t!@ F O R M A T ( 4 1 3 8 b ~ t 3 . 4 ) 210 F O R M A T ( ~ 3 , P Z ~ B , P 6 , 4 ) 226 P O R ~ A T ( S P I B , 4 , L E l 7 * ~ )

C 500 CALL PLOTND

ITeJ=S C H A R G E ~ ~ E C M ~ $ ( T ~ ) / ~ ~ B B , ~ 6 , N R I T E ( ~ T @ , ~ ~ ~ ~ I C H A R ~ E , D A Y , T I M

9994 F O R M A T ( i n n , / / i THE C H A R G E FOR T I R E P L O T S IS ~ , F Y , E , ~ H R S V * ' ON ' , f A U 8 3 X , E A 4 / / )

C A L L ex17 C c

END

T Y I E V A EOU

PROGRAM P L T L I 3 P R O G R A M GRAP L U A T E D ALONG C ATIONS OF THE

E H3 I N w P L A N F E X T E N S I O N R A T I O S [ S R I , S R 2 ) ONVECTFO COOROlNATE CURVES G I V E N BY F O R M T H E T A * 1 = C 6 N S T A N T I AND T H E T A ~ ~ s C O N S T A N T Z ~

D A T A 19 S T O R E D I N F I L E D E V I C E S 2 AND 1, c C

D I M E N S I O N O A Y ( 3 3 , 7 1 M ( 2 ) DATA B L A N K / ' '/

C a L

C A L L A S S I G N ~ ~ O '0~11 ! l l @ # ~ I C ~ J R V ~ ; D A T ~ l i t 22) C A L L A S S I G N ( ~ ~ ' ~ K ~ ~ ~ ~ ~ B ~ ~ ~ C U R V ~ ~ D A T I ~ ~ I ~ ~ )

e C

T l = 3 E C W D S t B r ) DO 9990 l s l t 3

9990 D A Y ( I ) = B L A N K C A L L D A T E t O A Y ) C A L L T I M E ( T I M ) ,

C A L L P L O T 3 T ( , B ~ S t ' I N C ) CALL PLOfCi?,%o12,Q1*31

C E S T A B L I S H NEW REFEQENCE P O I N T * PEN UP C A L L P L O T ( B , B , ~ ~ Z ' . Q , - Z )

C ESTABl . I$H N € w REFEQENCE P O I N T PEN DOWN * DRAWS L I N E CALL P L O T ( 3 , 0 # 2 , 0 # . 3 )

C c , , r

c,.,.,ESTABLISH S C A L I N G R A T I 0 9 , X S C A L E ~ ! ~ R Q . / S . 1 4 1 5 9 2 7 ) * ( ! , /4@a X S C L E = ~ ; / U B * Y S C A L E o l 0 .

C C

OSH1=1, /8 , OFXs45, OXIUS~*XSCLE

C C C C

DO 130 L t l t Z C C

ISW=L-l GO T O ( S r i ) r L

1 C A L L P ~ O T ( B r r 0 , r ~ f ) C c , C..,,.CONSTRUCT Y - A x I 3 ,

5 Yo51 F Y r l . 5 C A L L P L O T ( B , , Y o ? )

C D O 10 13146 CALL p L o T ( , l , Y o t )

C A L L P L O T ( B , r ~ r Z ) ,

C A L L N U H B E R ( * ~ b f l r Y , , l ~ r F Y # B a ~ 1 ) PYmFY:, 1 Y a y - l a

I R CONTINU!! C

C A L L P L O T ( 0 , r f l a r 3 ) C C . . r

C,..,,CbNSTRUCT X a A X I S , I F ( I S W I l S r 1 5 r 2 0

c IS x ~ D X t f ,

F Y ~ 1 3 5 ~ N X r 4 GO TO 2 5

28 XmDX*Ur F X 1 1 6 8 ~ N X t 5

e 24 c a L L P L O T ( X ~ B ' . ~ ? )

C O S H o O S H I D O 40 I = l r N Y I F ( N X ~ I ) 3 7 r 3 7 r 3 6

37 DSHr0, 38 C A L L P L O T t X r , l t f l

C A L L P L O T ( Y t B m t 2 ) C A L L N U H $ E R ( X * D S H ~ ~ . ~ A J ~ ~ ~ ~ F X ~ B ~ ~ ~ ~ ) % 8 X * o X

F X s F X a D F X 4 f l C O N T I N U P

C c . C , . , , . C O N S T R U C T GRAPH OF IN-PLAWE E X T E N S I O N R A T I O S ( 9 R l r S R Z ) C E V A L U A T F D ALONG CONVECTED COORDINATE CURVES c T H E T A n l r C V Z AND THETAaEaCV2,

GO T O I 4 5 ~ S B ) r L 45 Y=2

GO T O 55 5 0 No3 55 04 tSPJ J s 1 r Z

K a J - 1 R E A O ( N t Z B O ) I01 r I D Z r I b 3 r I O 4 r In51 I D ~ ~ D Y ~ ~ D M ~ ~ D M ~ J O Y ~ ~ O M S ~ O M ~ R E A D ( N , ? i @ ) NUHRlCVl ,CV2 R E A D ( N r 2 2 f l ) S ~ , S ~ # Y I # Y ~ ~ Y ~ I S R I ~ S R Z I F ( I S W ) b B # b @ r 65

6 8 XtSl*XSCALE GO T O 78

6 5 XgS2*XStA l ,E 79 I F ( K 1 7 5 r 7 5 r A 0 75 Y a ( S R l = l , ) * Y S C A L E

GO T O 8CI 8 0 V o ( S R ? ~ l . ) * Y $ C A L E 85 C A L L P L O T ( X r V r 3 )

C

NUMBSNUMB-1 DO 12fl I a l rNUYB RfAD[N,22fl) 9 2 , S l r Y ! r ~ 2 r Y 3 , S R l r S R Z I F ( I 8 W ) 9 0 r 9 8 t 9 5

90 X = $ l * X S C A L t G O T O 100

95 X ~ S 2 * X S C A l E i B 0 I F ( K 1 1 0 5 t 1 0 5 r 110 IBS Y J ( S R ~ * ~ ~ ) * Y S C A L E

GO T O 115 l l a Y ~ ( S R 2 ~ 1 , ) * Y S C A L E 115 CALL P L O T I X I Y ~ Z ? 120 CONTINUF

t RFWINO N

C c

130 C O N T I N U E C C C c

CALL PLOT(6 , r -2, r - 3 ) C C

280 FORMAT(bJ3,bFl3,4) 210 PORHAT(131F?,B~Pb,4) 220 F b R ~ A T ( S F 1 0 , 4 r 2 F f 7 , 8 1

C 5BB CALL PLOTNO

IT035 CHARGE8SECNf lS (T i ) / 3h@f l , / 6 , WRITE( I T ~ , ~ ~ ~ ~ ) C W ~ R G E , O A V , T I n

9994 FORMAT(lHCJr//' THF CHARGE FOR T I R E PLOTS 13 ' ~ F Y ~ ? , ~ HRS' / * ' ON ' 1 3 A 4 r 3x1 244//)

CALL t x f r c C

END

C . 2 . 2 FILE REFERI INCE. The s t r u c t u r e o f t h e f i l e

r e f e r e n c e program LOOK i s diagrammed i n F lowcha r t 7 .

Tab l e C-6 d e s c r i b e s t h e program c o n t r o l d a t a t h a t t e l l s

LOOK which r e g i o n o f random a c c e s s d i s k f i l e 10 i s t o b e

r e t r i e v e d and p r i n t e d .

START Read program c o n t r o l d a t a

I I Copy s p e c i f i e d r e g i o n i n d i s k f i l e 10 t o d i s k f i l e 6 1

I P r i n t c o n t e n t s I - 1 o f d i s k f i l e 6 I

Flowchar t 7 . LOOK

FORTRAN Var i ab 1 e D e s c r i p t i o n

IRECS r e c o r d number r e f e r e n c i n g

t h e f i r s t g roup* o f

s o l u t i o n c o e f f i c i e n t s t o

b e p r i n t e d

number o f g roups t o b e

p r i n t e d , i f N G R P = 1

o n l y t h e f i r s t g roup i s

p r i n t e d

* A group o f s o l u t i o n c o e f f i c i e n t s i n c l u d e s a i j , b i j , an d

c . p l u s t h e s o l u t i o n s p e c i f i c a t i o n and t i r e model i n p u t ~j ' d a t a a s s o c i a t e d w i t h t h e PATCH run which g e n e r a t e d t h e

c o e f f i c i e n t s

C PROGRAM LnOK C T H I S PROGRAM P R I N T S T H t CONTENTS OF D E S I R F D RF!GIONI I N C THE RANDOM ACCESS DISK F I L l 1fl (COEFF,DAT), c, . . C,,,,,DATA CONTAIN€.D I & COEFF,DAT I S ARRANGED I N GROUPS C O N T A I N I N G c 4 R ~ C O R D S ; THF FIRST R E C O R D OF E A C H GROUP CONTAINS A LIST OF C T I R E M O O E L PARAYETKRS, THE R E M A I W I N G RECOROS C O N T A I N THE C COORDINATE FUNCTION COEFFICIENTS ASSOCIATED WITH THE C PARAMETERS OF THE F I R S T RECORD, C C

D I M E N S I O N A t U , U ) t 8 ( 4 t 4 ) , C ( 5 , 5 ) E Q U I V A L E N C E ( M A , M B ) , ( N A t N R )

c , Ce..., F I L E h I S A P R I N T E D OUTPUT BUFFER, C F I L E 10 I S A R A N D O M ACCESS L I B R A R Y OF M I N I M I Z I N G COEFFICIENTS C PREVIOUSLY CALCULATED 8V THE P A T C H P R O G R A M , C

C A L L A S S I ~ N ~ ~ ~ ' ~ U O ~ L O O K , L S T ~ , I ~ ! CALL A S S I G N ( l R , ' O K ! t [ l l f l , 3 l C O E F F , O A T j l ' t 2 2 t I E R R ) D E F I N E F I L E 1B(BB,SB,U, I N D X )

C C

w F ? I T € ( b t 1) c , C,.,,.SPECIPY RECORD NUMBFR ( I R E C S I OF T I R E MODFL PARAMETER9 C FOR F I R S T GROUP OF INTEREST, c SPECIFY T O T A L NUMRER (NGRPI OF GROUPS T O RE PRINTED;

R E A O ( 5 , 3 ) I R E C S , NGRP C C C

00 20 IT=t,NGRP c, . , C,,,.,ZERO M A T R I C E S WHICH WILL C O N T A I N COORDINATE P U N C T I O N COEFPICICNT8,

DO 2 J=l ,U DO Z 1 3 1 1 4 A t 1 1 J ) o @ , B ( I t J l = B ,

2 C ( I r J ) a P I , C

I R E C S p a I R E C S + 3

EL.:. . R E T R I E V E T I R E MODEL PARAHFTERB. R E A D ( 1 0 ' I R E C S ) ~ A ~ N A ~ ~ C ~ N C , ~ S ~ , ~ S ~ I P , P H I B I C ~ H I R ~ ~ R T , R L

1 . . I W R 1 7 € ( 6 , 7 ) 1 R E C S p I R E C S P C,,,,.RETRIEVE C D O R n I Y A T l F U N C T I O N COEFFICIENTS,

I R E C M s l R E c S + 1 READ(1M'IRECMI ( ( A ( I ~ J ~ , I ~ l r ~ A ) , J ~ i , N A ] I R E C M e I R E C M + 1 R E A D I I B ' I R E C M I ( ( R ( I , J ) , I ~ l t M R ) r J a ~ ~ N R l I R E C M ~ 1 R E C M + i R E A ~ ( 1 0 ' I R E C ~ ) ((C(I,J),I=l,MC)rJ~lrNC)

C J R I T E ( 6 , S f l ) M A ~ N A ~ M C , N C ~ N S I , N S ~ , P ~ P H I R , C ~ Y , R ~ , R T ~ R L

WRIT^(^, 1 3 ) ( ( ~ ( ~ , ~ l r J ~ l r ~ ) r I ~ l * ) w R I T E ( 6 , 13) ( ( ~ ( 1 , J l r J ~ l , ~ ) r I + J r ~ ~ ) w R I T E ( b , 1 3 ) ( t ~ ~ f r J ~ r J ~ l r 4 I r I ~ l r M C ) I R E C S a f R € C S + 4

20 CONTINUE C C L c , * * * , R O U T E D A T A RUFFERED I N F I L E 6 10 PRINTER UPON C A L L I N G OF EXIT;

CALL SPOQL(6 ) CALL E X I T

C t

1 F O R H A T ( 1 H l , 5 0 X r 'CoNTENTS OF F I L E " D K ! ~ C O E F F , D A ? ' ~ ' ~ I 3 F O R M A T ( Z I 5 ) 7 F Q R M A T ( / / # ! H , S T X , ' R E . C O R D S ' , !Z , ' THROUGH ' B I Z )

1 3 FORMAT( / / , 1H , 4 ( 3 1 X , 4 E 1 7 , 8 , / / r l H 1 ) 5 B F O R M A T ( 1 H r 1 2 X t b I 5 r b F 1 3 ~ 4 )

c C

END

Example LOOK Run

A s i n g l e d a t a c a r d ( A ) c o n s t i t u t e s an i n p u t module

f o r program L O O K . Only one i n p u t module i s r e a d f o r e ach

r u n , which i s t e r m i n a t e d by c a l l i n g EXIT.

Card A ( 2 1 5 )

IRECS = 1 7

N G R P = 1

Punched Card L i s t i n g

r e t r i e v e s o l u t i o n c o e f f i c i e n t s b e g i n n i n g w i t h r e c o r d 1 7 o f f i l e 1 0

r e t r i e v e o n l y one group o f c o e f f i c i e n t s

The f o l l o w i n g page shows t h e p r i n t o u t o f t h e r e g i o n

o f random a c c e s s d i s k f i l e 10 t h a t was r e f e r e n c e d by t h i s

example run o f L O O K .

4

8 I Li t'l N Q 6 =I rn M IF,

8

ZF Q

4

IF, N LT' I. N

6,

C . 3 PROGRAM SIZES

The c a l c u l a t e d r e s u l t s p r e s e n t e d i n t h i s r e p o r t were

o b t a i n e d w i t h a PDP 1 1 / 4 5 w i t h 6 4 K ( b y t e s ) o f c o r e memory.

Table C-7 l i s t s t h e o b j e c t and l o a d module s i z e s o f t h e

programs d e s c r i b e d i n t h e p r e c e d i n g s e c t i o n s o f t h i s

Appendix.

TABLE C - 7

MAIN PROGRAM MODULE SIZES

Nodule S i z e ( b y t e s ) Program Name O b j e c t Load

PATCH 22678 38992

TENSOR

B N D

PTCHCV

PLTBND 2084

PLTMRD 2 560

PLTLAT 2 8 4 4

L O O K

'Ilic l o a d module i s u s u a l l y rrluctl 1; irgcr t h a n t h e

o b j e c t module because o f t h e sys t em s u b r o u t i n e s t h a t a r e

b rough t i n b y t h e l i n k e d i t o r .