1

ON THE SELF-WEIGHT SAG OF PLATE-LIKE STRUCTURES

WITH APPLICATION TO MIRROR SUBSTRATE DESIGN

by

DIPAK CHANDRA TALAPATRA

B. Tech. (Hons.), Indian Ins t i t u t e of Technology, Kharagpur, 196 3

M.E. McGill University, 1968

A THESIS SUBMITTED IN PARTIAL FULFILMENT OF

THE REQUIREMENTS FOR THE DEGREE OF

DOCTOR OF PHILOSOPHY

in the Department

of

MECHANICAL ENGINEERING

We accept t h i s thesis as conforming to the

required standard

THE UNIVERSITY OF BRITISH COLUMBIA

May, 19 72

In p r e s e n t i n g t h i s t h e s i s i n p a r t i a l f u l f i l m e n t o f the

requ i rements f o r an advanced degree a t the U n i v e r s i t y o f

B r i t i s h Co lumbia , I agree t h a t the L i b r a r y s h a l l make

i t f r e e l y a v a i l a b l e f o r r e f e r e n c e and s t u d y . I f u r t h e r

agree t h a t p e r m i s s i o n f o r e x t e n s i v e c o p y i n g o f t h i s t h e s i s

f o r s c h o l a r l y purposes may be g r a n t e d by the Head o f my

Department or by h i s r e p r e s e n t a t i v e s . I t i s unders tood

t h a t p u b l i c a t i o n , i n p a r t o r i n who le , o r the c o p y i n g o f

t h i s t h e s i s f o r f i n a n c i a l g a i n s h a l l no t be a l l owed w i t h ­

out my w r i t t e n p e r m i s s i o n .

DIPAK CHANDRA TALAPATRA

Department o f M e c h a n i c a l E n g i n e e r i n g

The U n i v e r s i t y o f B r i t i s h Columbia

Vancouver 8 , Canada

Date ^ ^ - ^ 7 , n 7 'Z-

ABSTRACT

The t h e s i s i n v e s t i g a t e s the s e l f - w e i g h t sag o f p l a t e ­

l i k e s t r u c t u r e s w i t h a p p l i c a t i o n to a s t r o n o m i c a l m i r r o r s .

The e x a c t a n a l y t i c a l s o l u t i o n f o r the s e l f - w e i g h t

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

o f v a r i o u s e lementary s o l u t i o n s due to L o v e , and the v a l i d i t y

o f the e x i s t i n g approximate p rocedures i s examined.

Gu ided by the concept o f a r c h - l i k e s t r u c t u r e s f o r

optimum d e s i g n , a f i n i t e e lement f o r m u l a t i o n f o r an a x i -

symmetr i ca l s o l i d i s deve loped from f i r s t p r i n c i p l e s i n

terms o f t r i a n g u l a r r i n g - e l e m e n t s . S o l u t i o n s a re o b t a i n e d

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

enev lope o f the d i s c . The s u p e r i o r i t y o f a r c h - t y p e d e s i g n s

over s o l i d d i s c s w i t h r e s p e c t t o d e f l e c t i o n and we ight i s

e s t a b l i s h e d and t h e i r a t t r a c t i v e p o t e n t i a l demonst ra ted .

The e x t e n s i v e e x p e r i m e n t a l programme i n v o l v i n g f r o z e n

s t r e s s p h o t o e l a s t i c i t y i n c o n j u n c t i o n w i t h immersion ana logy

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

emphasizes some o f the l i m i t a t i o n s o f t h i s a p p r o a c h . I t

e s t a b l i s h e s t h a t a success o f the method i s dependent upon

the a v a i l a b i l i t y o f a s u f f i c i e n t l y s t r e s s f r e e a r a l d i t e .

A l t h o u g h not g e n e r a l l y f o l l o w e d by s t r e s s a n a l y s t s , a d i r e c t

measurement o f f r o z e n b o d y - f o r c e induced d i s p l a c e m e n t s i s

attempted. The phenomenon of continuous polymerization of

the model material, hitherto overlooked by photoelasticians,

appears to play a decisive role i n the measurement of

minute self-weight induced deformations.

The d i r e c t use of s i l i c o n e rubber models successfully

determines the displacements i n mirrors, i n the form of

s o l i d disc and arched dome, and establishes the su p e r i o r i t y

of the l a t t e r .

T A B L E O F C O N T E N T S

C h a p t e r P a g e

1. I N T R O D U C T I O N 1

1.1 T h e S e l f - W e i g h t S a g o f T h i c k P l a t e s 1

1.2 O p t i c a l , M e c h a n i c a l a n d T h e r m a l

P r o b l e m s i n L a r g e T e l e s c o p e s 2

1.2.1 T h e O p t i c a l P r o b l e m 4

1.2.2 T h e M e c h a n i c a l a n d T h e r m a l

P r o b l e m s . 8 1.3 P u r p o s e a n d S c o p e o f t h e P r e s e n t

I n v e s t i g a t i o n 12

2. A N E X A C T A N A L Y T I C A L S O L U T I O N F O R T H E S E L F -W E I G H T D E F L E C T I O N O F C I R C U L A R P L A T E S O F C O N S T A N T T H I C K N E S S 16

2.1 P r e l i m i n a r y R e m a r k s 16

2.2 S e l f - W e i g h t D e f l e c t i o n o f a C y l i n d r i c a l P l a t e S u p p o r t e d b y U n i f o r m l y D i s t r i b u t e d S h e a r a t t h e E d g e 18

2.3 R e s u l t s a n d D i s c u s s i o n 34

3. N U M E R I C A L A N A L Y S I S OF T H I C K C I R C U L A R P L A T E S OF V A R I A B L E T H I C K N E S S AND I T S A P P L I C A T I O N T O T H E D E S I G N OF M IRROR S U B S T R A T E S 44

3.1 A B r i e f R e v i e w o f t h e N u m e r i c a l T e c h n i q u e s 44

3.1.1 F i n i t e D i f f e r e n c e M e t h o d . . . . 44

3.1.2 I n t e g r a l M e t h o d 50

3.1.3 F i n i t e E l e m e n t M e t h o d 52

V

Chapter Page

3.2 Light-Weight Mirror Substrate Design Philosophies 54

3.3 F i n i t e Element Solutions for Axi-symmetric Systems 60

3. 3 . 1 Comparative Analysis of Arched Structures and S o l i d Discs . . . 6 3

3.3.2 Deflection due to Thermal Gradient 76

4. EXPERIMENTAL TECHNIQUES FOR THE ANALYSIS OF BODY FORCE DEFLECTION 80

4.1 Review of the Experimental Studies for Mirror Substrates 80

4.2 Present Experimental Investigations . . 83

4 . 2 . 1 Frozen stress p h o t o e l a s t i c i t y coupled with the immersion analogy of g r a v i t a t i o n a l stresses 83

4.2.2 Deflection study of photo-e l a s t i c models by d i r e c t measure­ment of frozen displacement . . . 94

4 . 2 . 3 Deflection analysis of a s o l i d disc and an arched dome using cold cure s i l i c o n e rubber models 1 0 2

4.3 A p p l i c a b i l i t y of the Model Results to the Prototype Mirror Substrate Design. . I l l

5. CONCLUSIONS AND RECOMMENDATIONS FOR FUTURE STUDY 1 1 4

5.1 Conclusions 1 1 4

5.2 Recommendations for Future Study . . . . 1 1 5

v i

Chapter Page

REFERENCES 120

APPENDIX 1 - F i n i t e E lement A n a l y s i s o f A x i -symmetric Body F o r c e - L o a d e d S o l i d s 1-1

APPENDIX 2 - E v a l u a t i o n o f De format ions Due to Thermal E f f e c t s 2-1

LIST OF TABLES

T a b l e Page

3.1 F i g u r e o f M e r i t Va lues 73

LIST OF FIGURES

Figure Page

1.1 Formation of Images from Distant Stars . . . . 6

1.2 Cusp-Like Surface Formed by Oblique Rays 7

1.3 Possible Ways of Supporting a Mirror • Substrate . . . . . 9

1.4 The 200-inch Hale Telescope Mirror Support Mechanism 9

1.5 Table of Substrate Forms 15

2.1 Superposition of Several Elementary Solutions 20

2.2 Notations for C y l i n d r i c a l Plate 22

2.3 Total Body Force Considered Equivalent to an Equal Uniformly Distributed Surface Pressure, q = 2pgh 35

2.4 D i s t r i b u t i o n of Radial and Shear Stresses at the End of the Plate 35

2.5 Radial Stress Variation at an Intermediate Cross-Section 36

2.6 A x i a l Stress D i s t r i b u t i o n 37

2.7 Comparison between Middle Surface Deflections as Obtained from the Present Theory (Equation !

2.17) and Emerson's Analysis (Equation 2.22) . 38

2.8 Middle and Outer Surface Deflection P r o f i l e s . 40

2.9 A Comparison Between the Present Analysis and the Thin Plate Theory Results for Several D/H Ratios 42

3.1 Axisymmetric S o l i d 45

3.2 A x i a l z-r Plane 45

i x

F i g u r e Page

3.3 Sandwich and C e l l u l a r M i r r o r S u b s t r a t e s . . . 56

3.4 C a s t e l l a t e d Beam 58

3.5 M i c h e l 1 S t r u c t u r e 58

3.6 The M i d d l e S u r f a c e D e f l e c t i o n s as P r e ­d i c t e d by the F i n i t e E lement Procedure and the A n a l y t i c a l S o l u t i o n 61

3.7 E f f e c t o f Support C i r c l e Radius on the M i d d l e S u r f a c e D e f l e c t i o n 62

3.8 A rched S t r u c t u r e s C o n s t r u c t e d w i t h i n the Space Enve lope o f the S o l i d D i s c o f D/H =6 65

3.9 Compar ison o f the Upper S u r f a c e D e f l e c t i o n s f o r S o l i d D i s c (1) w i t h t h a t o f A r c h Models (1) and (2)

(a) L a t e r a l D e f l e c t i o n 66 (b) R a d i a l D e f l e c t i o n 67

3.10 Upper S u r f a c e D e f l e c t i o n s f o r S o l i d D i s c (2) and A rch Model (3) 68

3.11 Comparison o f the Upper S u r f a c e D e f l e c t i o n s o f A r c h Models (1) and (2) as R a t i o s o f the D e f l e c t i o n o f S o l i d D i s c (1)

(a) L a t e r a l D e f l e c t i o n 69 (b) R a d i a l D e f l e c t i o n 70

3.12 Upper S u r f a c e D e f l e c t i o n s o f A r c h Model (3) as R a t i o s o f the D e f l e c t i o n o f S o l i d D i s c (2) 71

3.13 Top S u r f a c e D e f l e c t i o n o f a C a s s e g r a i n i a n S u b s t r a t e as a F u n c t i o n o f Support C i r c l e Radius 75

3.14 E f f e c t o f Temperature G r a d i e n t T = 2z on A x i a l D e f l e c t i o n o f a T h i c k C y l i n d r i c a l P l a t e , D/H = 6 77

X

F i g u r e Page

3.15 R a d i a l D e f l e c t i o n i n a T h i c k C y l i n ­d r i c a l P l a t e Due t o Temperature G r a d i e n t T = 2z 78

4.1 Immersion Ana logy f o r G r a v i t a t i o n a l S t r e s s e s 85

4.2 Schemat ic Diagram Showing Model i n Oven, Mercury Vapour Condensat ion T e c h n i q u e , and the L o c a t i o n s o f S l i c e s Taken f o r Examina t ion 87

4.3 L i g h t R e f l e c t i o n and T r a n s m i s s i o n Between Two S l i g h t l y I n c l i n e d P a r t i a l M i r r o r s 89

4.4 P a r t i a l M i r r o r s as Employed i n P o l a r i -scope f o r F r i n g e M u l t i p l i c a t i o n 89

4.5 M u l t i p l i e d F r i n g e P a t t e r n s f o r a R a d i a l S l i c e Through a Support P o i n t 90

4.6 M u l t i p l i e d F r i n g e P a t t e r n s f o r a S e c t o r S l i c e Through Two Support P o i n t s . . . . . 91

4.7 F r i n g e Photograph f o r the R ing Suppor ted S o l i d D i s c

(a) Normal Load ing 97 (b) Reverse L o a d i n g 97

4.8 S e l f - W e i g h t D e f l e c t i o n o f the R ing Supported S o l i d D i s c 98

4.9 F r i n g e Photograph f o r the S imply Supported Beam

(a) Normal L o a d i n g 99 (b) Reverse L o a d i n g 99

4.10 S e l f - W e i g h t D e f l e c t i o n o f the S imply Suppor ted Beam 100

4.11 Fo rmat ion o f Shadow Mo i re F r i n g e s 104

x i

F i g u r e Page

4.12 Support Frames f o r Models 106

4.13 Shadow Mo i re F r i n g e , P a t t e r n s f o r the S o l i d D i s c

(a) Model i n G l y c e r i n e 108 (b) Model i n A i r 108

4.14 Shadow Moi re F r i n g e P a t t e r n s f o r the Arched Model

(a) Model i n G l y c e r i n e 109 (b) Model i n A i r 109

4.15 Comparison o f G r a v i t y - I n d u c e d Upper S u r f a c e w D e f l e c t i o n s f o r the S o l i d D i s c With That o f the Arch Model 110

4.16 V a r i a t i o n o f Top S u r f a c e a w i t h P o i s s o n ' s R a t i o . . 112

5.1 A rched S u b s t r a t e Form f o r T h r e e - P o i n t Supports 117

5.2 A rched S u b s t r a t e Form w i t h R a d i a l and C i r c u m f e r e n t i a l Grooves 118

A l . l F i n i t e E lement I d e a l i s a t i o n o f A x i -symmetr ic S o l i d

(a) A c t u a l Continuum 1~2 (b) T r i a n g u l a r E lement A p p r o x i ­

mat ion 1-2

A1.2 Nodal C o o r d i n a t e s and D isp lacement Components 1-3

ACKNOWLEDGEMENT

The author wishes to express h i s i ndebtedness t o

D r . J . P . Duncan f o r h i s v a l u a b l e gu idance and c r i t i c i s m s

throughout the s tudy and p r e p a r a t i o n o f the t h e s i s . Thanks

are a l s o due t o D r . V . J . Modi f o r h i s h e l p and c o n s t r u c t i v e

s u g g e s t i o n s i n the p r e p a r a t i o n o f the f i n a l m a n u s c r i p t .

The s u g g e s t i o n s by D r . H. Vaughan and D r . D. Pos t

a re d u l y a p p r e c i a t e d .

The au thor would a l s o l i k e to thank Mr. John Hoar ,

C h i e f T e c h n i c i a n , and h i s s t a f f f o r t h e i r h e l p i n the

e x p e r i m e n t a l work.

The p r o j e c t was suppor ted by the N a t i o n a l Research

C o u n c i l , Grant No. 67-3309.

LIST OF SYMBOLS

d i a m e t e r , 2a; modulus o f r i g i d i t y

e l a s t i c i t y m a t r i x f o r i s o t r o p i c m a t e r i a l

Young 1 s modulus

e x t e r n a l n o d a l f o r c e

n o d a l f o r c e due t o body f o r c e

n o d a l f o r c e due to t h e r m a l e f f e c t

body f o r c e i n t e n s i t y i n the z d i r e c t i o n

t h i c k n e s s , 2h

s t i f f n e s s m a t r i x

suppor t c i r c l e r a d i u s / r a d i u s o f the p l a t e

average temperature r i s e i n an e lement

d i s p l a c e m e n t f u n c t i o n

a c c e l e r a t i o n due to g r a v i t y

body f o r c e v e c t o r

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

t ime

d i s p l a c e m e n t s i n x , y , z d i r e c t i o n s

mid-p lane d i s p l a c e m e n t

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

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

c o e f f i c i e n t of thermal expansion

shear s t r a i n component i n c y l i n d r i c a l coordinates

element nodal displacement

area of the tr i a n g l e

normal s t r a i n components i n c y l i n d r i c a l coordinates

s t r a i n vector

thermal s t r a i n vector

wave length of l i g h t

Poisson's r a t i o

density

normal stress components i n c y l i n d r i c a l coordinates

stress vector

shear stress component i n c y l i n d r i c a l coordinates

stress function

V<L&i.o.a.t<L& to my bzZovzd pa.ie.ntA

1. INTRODUCTION

1.1 The S e l f - W e i g h t Sag o f T h i c k P l a t e s

T h i s i n v e s t i g a t i o n i s p r i m a r i l y concerned w i t h

d e v e l o p i n g and a p p l y i n g methods f o r p r e d i c t i n g the s e l f -

we ight sag o f t h i c k p l a t e s . The i n v e s t i g a t i o n was

prompted by s e v e r a l problems a r i s i n g i n l a r g e m i r r o r

s u b s t r a t e d e s i g n and i t i s i n t h i s a rea t h a t the methods

have been u s e d . T r a d i t i o n a l l y , an a s t r o n o m i c a l m i r r o r ,

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

h a v i n g a d iameter to t h i c k n e s s r a t i o o f 8 to 1 (smal l

a p e r t u r e ) o r 6 to 1 ( l a r g e a p e r t u r e ) . These r a t i o s a re

r u l e s o f thumb proposed by R i t c h e y [1] on the b a s i s o f

e x p e r i e n c e . The s m a l l e r r a t i o , recommended f o r l a r g e

m i r r o r s , r e f l e c t s the importance o f s e l f - w e i g h t d e f l e c t i o n ,

4 2 which i s p r o p o r t i o n a l t o (Radius) / ( T h i c k n e s s ) f o r a

m i r r o r u n i f o r m l y suppor ted a t i t s p e r i p h e r y . I t i s

apparent t h a t , w i t h such s m a l l d iameter to t h i c k n e s s r a t i o s ,

a t h e o r e t i c a l i n v e s t i g a t i o n cannot be accomp l i shed by

u s i n g c o n v e n t i o n a l t h i n p l a t e t h e o r y . I t i s n e c e s s a r y to

t r e a t the m i r r o r , as a t h i c k p l a t e , t h a t i s , an a x i s y m m e t r i c

body governed by the t h r e e - d i m e n s i o n a l f i e l d equa t ions o f

e l a s t i c i t y .

2

I n m a n y i n s t a n c e s , t h e m i r r o r m a y b e s l i g h t l y c u r v e d

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

f o r w e i g h t e c o n o m y . T h e p o i n t s e r v e s t o i n d i c a t e t h e

n e c e s s i t y o f a n a l y z i n g a x i s y m m e t r i c t h i c k p l a t e s o f

v a r i a b l e t h i c k n e s s , t h a t i s , s h a l l o w d o m e s o r a x i s y m m e t r i c

s h a l l o w a r c h e d s t r u c t u r e s .

I n t h i n p l a t e t h e o r y , a c o m m o n t e c h n i q u e f o r e x a m i n ­

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

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

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

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

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

a u t o m a t i c a l l y s u p p l i e s a s o l u t i o n f o r t h e r m a l e f f e c t s . I t

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

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

t h e r m o e l a s t i c i t y , t h e r e s u l t s a p p l y t o t h e r m a l e f f e c t s . T h e

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

i s o f p a r a m o u n t i n t e r e s t a n d a n y i n f o r m a t i o n o b t a i n a b l e f r o m

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

1 . 2 O p t i c a l , M e c h a n i c a l a n d T h e r m a l P r o b l e m s i n L a r g e T e l e s c o p e s

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

s p e c u l a r , r e f l e c t i v e s u r f a c e w h o s e g e o m e t r i c a l f o r m i s

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

3

s u b s t r a t e . I f the s u b s t r a t e s e r v e s by i t s mere e x i s t e n c e

as a " c r e a t o r " o f a s u r f a c e o r i n t e r f a c e , i t s e lements

become masses which a re d i s t r i b u t e d i n space and thus a re

s u b j e c t e d t o g r a v i t a t i o n a l a t t r a c t i o n .

; The m i r r o r s u r f a c e would change i t s shape i f the

p h y s i c a l s t a t e o f the s u b s t r a t e a l t e r s . T h i s may a r i s e

when the o r i e n t a t i o n o f the m i r r o r o r the g r a v i t a t i o n a l

f i e l d changes . A change i n geometry o f the e l a s t i c s t r u c t u r e

can a l s o be induced by v a r i a t i o n s i n a b s o l u t e t empera tu re ;

o r by the e x i s t e n c e o f temperature g r a d i e n t s . F u r t h e r m o r e ,

problems o f d i m e n s i o n a l s t a b i l i t y may a r i s e from r e l a x a t i o n

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

F o r s a t i s f a c t o r y per formance o f the m i r r o r , the

s u b s t r a t e has to be so d e s i g n e d t h a t the geomet r i c form o f

the r e f l e c t i v e s u r f a c e i s m a i n t a i n e d w i t h i n an a c c e p t a b l e

l i m i t de te rmined by d i f f r a c t i o n o p t i c s . The c u r r e n t g e n e r a l

p h i l o s o p h y o f m i r r o r suppor t i s to e q u i l i b r a t e the we ight

o f each e lement o f the m i r r o r s u b s t r a t e as d i r e c t l y as

p o s s i b l e by v a r i o u s s u i t a b l y d e s i g n e d mechanisms. Thus

the e l a s t i c de fo rmat ions o f the s u b s t r a t e , which occur as

the m i r r o r i s o r i e n t e d i n d i f f e r e n t p o s i t i o n s , are

m i n i m i z e d .

4

These complex mechanisms a re q u i t e e l a b o r a t e .

I n s t e a d o f d e s i g n i n g the s u b s t r a t e as a mass ive s t r u c t u r e

w i t h many s u p p o r t s , a l t e r n a t i v e forms o f compos i te and

b u i l t - u p s u b s t r a t e s have been des igned and c o n s t r u c t e d .

Here the aim i s t o deve lop h i g h s t i f f n e s s - t o - w e i g h t r a t i o

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

shape by an a c c e p t a b l e degree o f f l e x u r e . These l i g h t ­

we igh t m i r r o r d e s i g n s can be o b t a i n e d by chang ing the

c o n v e n t i o n a l g e o m e t r i c a l forms o f the s u b s t r a t e through

r e d i s t r i b u t i o n o f m a t e r i a l . As an a l t e r n a t i v e to

u s i n g many c l o s e l y spaced s u p p o r t mechanisms, ment ioned

above, l i g h t - w e i g h t s u b s t r a t e s may be suppor ted d i r e c t l y

a t t h r e e p o i n t s o r a t s i x i n t e r n a l p o i n t s on a l i m i t e d

system o f b a l a n c e d l e v e r s p i v o t e d a t t h r e e p r imary p o i n t s

on the frame o f r e f e r e n c e .

1.2.1 The O p t i c a l Problem

The g e o m e t r i c a l form o f the m i r r o r s u r f a c e must be

m a i n t a i n e d so t h a t the images formed are d i f f r a c t i o n - l i m i t e d .

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

t i o n - l i m i t e d i f i t would m a i n t a i n the g e o m e t r i c a l form

a c c u r a t e to a s m a l l f r a c t i o n o f the wave l e n g t h o f l i g h t

b e i n g r e c e i v e d .

5

Rays coming from d i s t a n t s t a r s form images as

shown i n F i g u r e 1.1. Due to d i f f r a c t i o n e f f e c t s the images,

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

A i r y ' s d i s c s surrounded by a l t e r n a t e dark and b r i g h t r i n g s .

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

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

a lower l i m i t t o the ang le of s e p a r a t i o n between two o b j e c t s ,

1 22X

a = _ i _ — f where D = d iameter of the a p e r t u r e and X = wave

l e n g t h o f the l i g h t . T h i s e x p r e s s i o n i s determined from

R a y l e i g h ' s c r i t e r i o n which s t a t e s t h a t two images may

be r e s o l v e d when the c e n t r a l maximum o f one f a l l s ove r the

f i r s t minimum of the o t h e r . As i n d i c a t e d by the above

f o r m u l a t h i s l i m i t f o r a depends on D. In o r d e r t o make a

s m a l l , D has t o be l a r g e . Any i m p e r f e c t i o n o f the r e f l e c t ­

i n g s u r f a c e degrades t h i s l i m i t which i s r e a l l y f i x e d by

X s i n c e we cannot c o n t r o l the wave l e n g t h o f l i g h t .

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

c a l l e d the " f o c u s " o n l y f o r p r i n c i p a l r a y s . Ob l i que r a y s

focus on a c u s p - l i k e s u r f a c e (F igure 1.2). Hence, r e a l

m i r r o r s u r f a c e s are not always formed as t r u e p a r a b o l o i d s .

They are c o r r e c t e d to e f f e c t some g e o m e t r i c a l c o n t r o l o f

f o c u s s i n g i n a f o c a l p l a n e .

M i r r o r

D

| F i g u r e 1.1 Format ion o f images from d i s t a n t s t a r s

i

7

C u s p - L i k e Sur face

F o c a l P lane

— — —— P r i n c i p a l Rays

•Obl ique Rays

Figure 1.2 C u s p - l i k e surface formed by oblique rays

8

1.2.2 The M e c h a n i c a l and Thermal Problems

To m a i n t a i n a s p e c i f i e d s u r f a c e fo rm,an u n d e r l y i n g

s u b s t r a t e , on which the s u r f a c e i s formed, i s r e q u i r e d .

D u r i n g o b s e r v a t i o n s , a m i r r o r may be r o t a t e d i n t o

d i f f e r e n t a t t i t u d e s i n the g r a v i t a t i o n a l f i e l d . I f the

m i r r o r s u r f a c e i s des igned t o be a c c u r a t e when s u p p o r t e d i n

a h o r i z o n t a l p o s i t i o n , the s u r f a c e shape may change as the

m i r r o r i s moved i n t o d i f f e r e n t a t t i t u d e s . Hence a ' r i g i d '

s u b s t r a t e must be s t i f f enough t o ensure t h a t a t t e n d a n t

e l a s t i c d i s t o r t i o n s are l e s s than a c c e p t a b l e amounts.

There are a t l e a s t two p o s s i b l e ways o f s u p p o r t i n g

a m i r r o r s u b s t r a t e :

(a) S i n c e the pr imary o b j e c t i n m i r r o r d e s i g n i s

t o c r e a t e a s u r f a c e on the boundary o f a s u b ­

s t r a t e and i t s atmosphere , a t h i n p l a t e

suppor ted at many c l o s e l y spaced p o i n t s

(F igure 1.3a), l o c a t e d d i r e c t l y under the

l i n e s o f a c t i o n o f the body f o r c e s , would s e r v e

the p u r p o s e . In t h i s case the m i r r o r can be

made as t h i n as p o s s i b l e as i t mere ly p r o v i d e s

a workable s u r f a c e . The m a t e r i a l i n the sub­

s t r a t e i s then suppor ted d i r e c t l y , and i n t e r n a l

t r a n s m i s s i o n i s c o n f i n e d to d i r e c t compress ion

through the t h i c k n e s s o f the p l a t e .

/7 7 77 7 777777J7 Foundat ion

a . T h i n P l a t e

/ / / / / / / / / / Foundat ion

b. T h i c k P l a t e

F i g u r e 1.3 P o s s i b l e ways of s u p p o r t i n g

a m i r r o r s u b s t r a t e F i g u r e 1 . 4 The 2 0 0 - i n c h _H_ale t e l e s c o p e

m i r r o r suppor t :" mechanism

10

(b) By u s i n g a s t i f f s u b s t r a t e w i t h w i d e l y spaced

s u p p o r t s (F igure 1 . 3 b ) .

F i g u r e 1.4 [2] shows a t y p i c a l c o n t r o l mechanism used

by the t h i r t y - s i x suppor ts o f the 2 0 0 - i n c h Hale T e l e s c o p e

a t Mt . Pa lomar . These mechanisms are i n s t a l l e d i n c a v i t i e s

i n the m i r r o r b l a n k . As the m i r r o r i s moved i n t o d i f f e r e n t

a t t i t u d e s the we ights W and the l e v e r arms shown i n the

f i g u r e are so a r ranged t h a t a f o r c e i s e x e r t e d on band B

which j u s t b a l a n c e s the component o f g r a v i t y normal to the

d i r e c t i o n o f the o p t i c a x i s , on the s e c t i o n o f the m i r r o r

a s s i g n e d to t h i s s u p p o r t . S i m i l a r l y , a f o r c e i s e x e r t e d on

the band S, which b a l a n c e s the component, p a r a l l e l t o the

o p t i c a x i s , o f the p u l l o f the g r a v i t y on the same s e c t i o n o f

the m i r r o r . The m i r r o r i s t h e r e f o r e f l o a t i n g on the suppor t

system as i t i s moved i n t o d i f f e r e n t a t t i t u d e s .

Thermal g r a d i e n t s i n a m i r r o r s u b s t r a t e can s e r i o u s l y

deform the o p t i c a l s u r f a c e . E l a b o r a t e measures are taken

to c o u n t e r a c t t h i s e f f e c t by p r o v i d i n g l a r g e t e l e s c o p e s w i t h

e n c l o s u r e s , n o r m a l l y i n the form o f h e m i s p h e r i c a l domes.

I n s i d e and o u t s i d e temperatures are e q u a l i z e d on an a n t i c i ­

pa ted l o n g - t e r m b a s i s t o s u i t the t ime o f o b s e r v a t i o n a t

n i g h t . F u r t h e r m o r e , when m a t e r i a l s are chosen f o r m i r r o r

subs t ra tes* two impor tant p h y s i c a l p r o p e r t i e s a re taken i n t o

account ; the therma l c o e f f i c i e n t o f expans ion and c o n d u c t i v i t y

o f the m a t e r i a l .

11 E s s e n t i a l l y a m a t e r i a l w i t h a low c o e f f i c i e n t o f

expans ion i s sought s i n c e i t s ' o p t i c a l f i g u r e ' would be

s t a b l e under normal ambient temperature f l u c t u a t i o n s and i t

i s l e s s l i k e l y t o be s u s c e p t i b l e t o the rma l e f f e c t s . On

the o t h e r h a n d , h i g h t h e r m a l c o n d u c t i v i t y i s n e c e s s a r y , s i n c e

t h i s would enab le the m i r r o r t o r e a c h the rma l e q u i l i b r i u m

q u i c k l y , e s p e c i a l l y d u r i n g g r i n d i n g when i t i s n e c e s s a r y t o

d i s s i p a t e the hea t genera ted by f r i c t i o n . B e s i d e s e l e c t i n g

the m a t e r i a l f o r m i r r o r s u b s t r a t e s w i t h p r o p e r v a l u e s o f the

therma l c o e f f i c i e n t o f expans ion and t h e r m a l c o n d u c t i v i t y ,

i t s mounts must a l s o be s u i t a b l y d e s i g n e d so t h a t the therma l

d i s t o r t i o n i s m i n i m i z e d .

A l though s c a r c e , a r e f e r e n c e s h o u l d be made t o the

r e l e v a n t i n v e s t i g a t i o n s i n the f i e l d . D e f l e c t i o n a n a l y s i s o f

m i r r o r s u b s t r a t e was i n i t i a t e d by Couder (1931) [3]. He

i n v e s t i g a t e d f l a t c y l i n d r i c a l m i r r o r s by t a k i n g account o f

bend ing s t r e s s e s o n l y , the shear s t r e s s e s b e i n g n e g l e c t e d .

I t was conc luded from o b s e r v a t i o n s and an approximate a n a l y s i

t h a t the f l e x u r e o f v e r t i c a l l y mounted m i r r o r s would not be

d e t e c t a b l e up t o a d iameter o f about 120 i n c h e s . Schwesinger

(1954) [4] a n a l y z e d a v e r t i c a l l y mounted m i r r o r and took

e x c e p t i o n t o the c o n c l u s i o n o f Couder . More r e c e n t l y ,

M a l v i c k and Pearson (1968) [5] conducted d e f l e c t i o n

a n a l y s i s of C a s s e g r a i n i a n - t y p e m i r r o r s u b s t r a t e s w i t h

a s p h e r i c a l l y d i s h e d f r o n t s u r f a c e . They used the

n u m e r i c a l method o f 'dynamic r e l a x a t i o n ' t o s tudy

m i r r o r de fo rmat ions a t d i f f e r e n t a t t i t u d e s i n c o - o r d i n a t e

12

space. Kenny ( 1 9 6 8 ) [ 6 ] undertook elementary experimental

inves t i g a t i o n for c y l i n d r i c a l , arched and build-up arched

substrates and found the d e f l e c t i o n of the arched model

to be smaller than that of the equivalent r i g h t c y l i n d r i c a l

blank.

1.3 Purpose and Scope of the Present Investigation

The long established use of r i g h t c y l i n d r i c a l disc

as substrate for small aperture has led, by extension, to

the use of such s o l i d substrates even for large aperture

elements. For instance the ground-based r e f l e c t i o n

Telescope of K i t t Peak i s 1 5 6 inches i n diameter and 26

inches thick. This i s characterized by a t r a d i t i o n a l

aperture to thickness r a t i o of 6 to 1. A disc such as t h i s

s t i l l has to be supported at many discrete points by care­

f u l l y calculated and controlled forces, arranged to balance

the g r a v i t a t i o n a l forces which tend to deform the substrate

as the mirror i s manoeuvred to occupy desired p o s i t i o n s .

Thus i t i s apparent that with ground based mirrors, 4 2

as the size increases, the a /H r e l a t i o n for constant

s t i f f n e s s leads to excessive substrate weight with attendant

problems of the support design. With space-borne telescopes

the reduction of substrate weight becomes even more

important. Questions such as t o t a l mass to be accelerated

i n space and changes i n the substrate shape as the telescope

13

i s taken from the one-g f i e l d to zero-g f i e l d have to be

considered. One promising solution to the two major problems

mentioned above i s to design substrates having the highest

possible stiffness-to-weight r a t i o .

The investigation reported here considers the s e l f -

weight sag of plates i n three fundamental ways. F i r s t l y , a

th e o r e t i c a l investigation has been conducted for thick >

plates of constant thickness. This method makes use of

some re s u l t s due to Love [ 7 ] for axisymmetric bodies.

Using his r e s u l t s , an exact a n a l y t i c a l solution has been

formulated for the self-weight sag of a thick plate supported

by uniform shear forces along the periphery. Although t h i s

exact solution i s primarily of academic i n t e r e s t i t provides

an excellent check for the subsequent in v e s t i g a t i o n . Next,

the attention i s focused on a numerical solution i n con­

junction with the method of f i n i t e elements. Governing

equations have been formulated to analyze body force ;

deflections i n any axisymmetric three-dimensional structure.

The procedure i s i d e a l l y suited to examine the ef f e c t s of

arching and support.

Thirdly, some experimental methods of examining

body force deflections have been considered. The frozen

stress technique coupled with the immersion analogy for

gr a v i t a t i o n a l stress e f f e c t s i s used on epoxy-resin models.

A well known d i f f i c u l t y i n using models i s the low stress

l e v e l s induced by g r a v i t y . T h i s low s e n s i t i v i t y may be

overcome by immersing the model i n a h i g h d e n s i t y l i q u i d ,

i n t h i s case mercury . F r i n g e i n t e r p r e t a t i o n and i d e n t i f i ­

c a t i o n i s f u r t h e r improved by the t e c h n i q u e o f f r i n g e

m u l t i p l i c a t i o n . U n f o r t u n a t e l y , the e f f e c t o f m o t t l e ,

i n t r o d u c e d d u r i n g c a s t i n g , makes a c c u r a t e i n t e r p r e t a t i o n

o f the s t r e s s p a t t e r n s r a t h e r d i f f i c u l t . T h i s l e d to the

d i r e c t measurement o f f r o z e n d i s p l a c e m e n t s . The magnitude

o f the d i s p l a c e m e n t s a re i d e a l f o r o b l i q u e i n t e r f e r o m e t r i c

measurement, an o b l i q u e i n c i d e n c e i n t e r f e r o m e t e r b e i n g

a v a i l a b l e . F i n a l l y exper iments have been conducted on

moulded s i l i c o n e rubber mode ls . The r e l a t i v e l y low e l a s t i c

modulus and f a i r l y h i g h d e n s i t y , makes t h i s an i d e a l

m a t e r i a l f o r s e l f - w e i g h t d i s t o r t i o n s t u d i e s . D e f l e c t i o n s

have been measured u s i n g a M o i r e f r i n g e t e c h n i q u e a p p l i e d

o n l y r e c e n t l y to the n o n - s p e c u l a r s u r f a c e s .

In b r i e f , the o b j e c t i v e o f the p r e s e n t p r o j e c t i s

to i n v e s t i g a t e g e o m e t r i c a l forms o f s u b s t r a t e which would

y i e l d low we ight and h i g h s t i f f n e s s . The c o n f i g u r a t i o n s

s t u d i e d are i n d i c a t e d i n F i g u r e 1 .5 .

Cfojective 1 Objective 2 Objective 3

Analytical solution of constant thickness plates for various D/H ratios to show importance of shear deflection as D/H i s lowered.

F inite element solution of symmetrical arch type structures to show their superiority over constant thickness substrates.

Experimental invest iga­t ion of. models i n the forms of so l id d isc and arched done to show sup­e r i o r i t y of the l a t te r .

i

c

) J 1

f

F i g u r e 1.5 Tab le o f s u b s t r a t e forms

2. AN EXACT ANALYTICAL SOLUTION FOR THE SELF-WEIGHT DEFLECTION OF CIRCULAR PLATES OF CONSTANT THICKNESS

2.1 P r e l i m i n a r y Remarks

There are s e v e r a l t h e o r e t i c a l formulae which a re i n

c u r r e n t use f o r p r e d i c t i n g the s e l f - w e i g h t d e f l e c t i o n o f

a s o l i d , r i g h t c i r c u l a r d i s c .

The p l a t e s shown i n column 1 o f F i g u r e 1.5 may be

a n a l y z e d a p p r o x i m a t e l y by L a g r a n g i a n Theory which makes

assumpt ions s i m i l a r t o those made i n the one d i m e n s i o n a l

B e r n o u l l i - E u l e r Theory (plane s e c t i o n s remain p lane and

s t r e s s e s i n the d i r e c t i o n normal to the mid -p lane are

n e g l i g i b l e ) . The r e s u l t s are but s l i g h t l y i n e r r o r i f the

r a t i o H/D, where H i s the t h i c k n e s s and D the d i a m e t e r , i s

s m a l l (<_ — ) .

Fo r r e l a t i v e l y l a r g e r a t i o s , the assumpt ions o f the

L a g r a n g i a n Theory break down. The f i b r e s t r e s s i s no l o n g e r

l i n e a r l y d i s t r i b u t e d a c r o s s the s e c t i o n . The d e p a r t u r e from

l i n e a r i t y i s due t o the combined e f f e c t s o f s u r f a c e p r e s s u r e

and a s s o c i a t e d shear and l e a d s to a s i g n i f i c a n t d i f f e r e n c e

between the L a g r a n g i a n and t r u e s o l u t i o n s . The d i f f e r e n c e ,

f r e q u e n t l y c a l l e d "the d e f l e c t i o n due to s h e a r , " i s c a l c u l a t e d

by the s e m i - r a t i o n a l R a n k i n e - G r a s h o f f method and added t o

17

t h e L a n g r a n g i a n r e s u l t . T h i s l e a d s t o a n i m p r o v e m e n t i n

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

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

m e m b e r s .

I n r e v i e w i n g , b r i e f l y , s o m e e x i s t i n g s o l u t i o n s f o r

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

W i l l i a m s [ 8 ] s h o u l d b e m e n t i o n e d . W i l l i a m s d e t e r m i n e d

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

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

r e a c t i o n s a l o n g a s i n g l e c o n c e n t r i c c i r c l e . T h e s o l u t i o n h a s

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

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

i t s e l f . T h e a p p r o a c h , b a s e d o n t h e w o r k o f B a s s a l i [9 ] , i s

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

t i o n b y a c o m p u t e r . T h e t h e o r y s t i l l e x c l u d e s t h e

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

d e t e r m i n e d b y t h r e e d i m e n s i o n a l a n a l y t i c a l o r n u m e r i c a l

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

a k i n t o R a k i n e - G r a s h o f f • s a p p r o a c h .

A s i m p l e a p p r o x i m a t e m e t h o d o f d e t e r m i n i n g d e f o r m a t i o n s

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

b e e n d e v e l o p e d b y V a u g h a n [10]. I n t h i s m e t h o d t h e c l a s s i c a l

s o l u t i o n o f M i c h e l l [11], f o r a c l a m p e d p l a t e u n d e r a s i n g l e

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

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

e d g e . T h e b o u n d a r y m o m e n t s a n d s h e a r s a r e r e l e a s e d b y

18

s u p e r p o s i t i o n o f edge l o a d i n g s o l u t i o n s based on the use

o f a s i n g l e s i n e wave as an a p p r o x i m a t i o n to a s e r i e s .

E i t h e r o f the above a n a l y s e s , i f used t o s o l v e the

s e l f - w e i g h t d e f l e c t i o n prob lem, would r e q u i r e the d i s t r i b u t e d

body f o r c e t o be approx imated by the s u b s t i t u t i o n o f a

u n i f o r m l y d i s t r i b u t e d e x t e r n a l - s u r f a c e l o a d .

2 . 2 S e l f - W e i g h t D e f l e c t i o n o f a C y l i n d r i c a l P l a t e Supported by U n i f o r m l y D i s t r i b u t e d Shear a t the Edge

As noted e a r l i e r , the e x i s t i n g s o l u t i o n s f o r s e l f -

we ight loaded p l a t e s suppor ted a t a system o f d i s c r e t e p o i n t s

a r e , i n g e n e r a l , approx imate . An a c c u r a t e d e t e r m i n a t i o n

o f the s e l f - w e i g h t i nduced d i s p l a c e m e n t s may be

o b t a i n e d c o n s i d e r i n g s e v e r a l ax i symmetr i c systems

u s i n g methods o f Love [ 7 ] .

Assuming i s o t r o p i c e l a s t i c m a t e r i a l w i t h c o n s t a n t s

E and v , s m a l l de fo rmat ions u , v , w i n the d i r e c t i o n s

x, y , z and a body f o r c e pg per u n i t volume, an a n a l y t i c a l

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

a c i r c u l a r p l a t e f o r e q u i p o l l e n t boundary c o n d i t i o n s .

T h i s means t h a t the s t r e s s r e s u l t a n t a t the boundary i s

e x a c t l y e q u a l t o the e x t e r n a l l y a p p l i e d t r a c t i o n s though

the l o c a l i z e d s t r e s s e s may vary g r e a t l y .

The s o l u t i o n o f the above ment ioned prob lem as r e p r e ­

sented i n F i g u r e 2.1 must s a t i s f y the f o l l o w i n g boundary

c o n d i t i o n s :

( < V z =.±h = °' ( T r z ) z = ±h = ° ( 2 ' 1 ^

f\(a ) dz = 0 -h r r = a

/ - h ( o r > r = a z d z = 0 > (2.2)

/_ n(T r z) r _ a2iTadz = weight o f the p l a t e .

A l l the c o n d i t i o n s o f (2.1) and (2.2) can be s a t i s f i e d

by s u p e r p o s i t i o n o f s e v e r a l e lementary s o l u t i o n s and the

e x p r e s s i o n s f o r d i s p l a c e m e n t s can be o b t a i n e d f o l l o w i n g

L o v e ' s t rea tment o f modera te ly t h i c k p l a t e s [ 7 ] .

F i r s t l e t us c o n s i d e r the p l a t e h e l d by a u n i f o r m l y

d i s t r i b u t e d t e n s i o n , F i g u r e 2.1a. I f the s o l u t i o n f o r a

p l a t e (without body f o r c e ) s u b j e c t e d to u n i f o r m compress ion

-2 pgh (F igure 2.1b) over the upper f ace i s super imposed ,

t t I. t

z(w) /2 P gh

t t t t f

2 0

n f I r(u)

a) + ^--2pgh

* * * * * * * * *

+ b)

Figure 2.1 S u p e r p o s i t i o n of s e v e r a l elementary s o l u t i o n s

21

the f a c e z = h w i l l be f r e e o f normal s t r e s s and the r e s u l t ­

i n g d i s p l a c e m e n t s w i l l be e n t i r e l y due to the body f o r c e .

But due t o the na tu re o f the second s o l u t i o n , t h e r e w i l l be

a moment and a r a d i a l f o r c e p r e s e n t a t the edge o f the p l a t e .

To r e l i e v e the edge o f the moment and the r a d i a l f o r c e ,

two o t h e r s o l u t i o n s need to be added, as shown i n F i g u r e 2 . 1 c , d .

T h i s l e a d s to the d e s i r e d s o l u t i o n o f s e l f - w e i g h t d e f l e c t i o n

o f a p l a t e suppor ted by u n i f o r m s h e a r i n g f o r c e s a l o n g the

edges .

The n o t a t i o n s f o r the p l a t e , used i n the p r e s e n t

i n v e s t i g a t i o n are shown i n F i g u r e 2 . 2 . Body f o r c e d i s p l a c e ­

ments f o r a p l a t e h e l d by u n i f o r m l y d i s t r i b u t e d t e n s i o n on

i t s upper f ace are

u = - vpg(z+h)x/E

v = - vpg(z+h)y/E (2.3)

w = |̂(- ( z 2 + v x 2 + v y 2 + 2hz)

and i n absence o f a body f o r c e i t s d i s p l a c e m e n t s when s u b ­

j e c t e d to u n i f o r m compress ion - 2pgh over i t s upper f a c e ,

are

u = _ ( l + v ) ( 2 p g h ) x [ ( 2 _ v ) 2 3 _ 3 h 2 z _ 2 h 3 3

8 E h J q

v = _ ( l + v ) ( 2 p g h ) y [ ( 2 _ v ) z 3 _ ^2^ . 2 h 3 3 ( 1 _ v ) z ( x 2 + y 2 ) }

8 E h J *

z(w)

r

F i g u r e 2 . 2 N o t a t i o n s f o r c y l i n d r i c a l p l a t e

23

= < 1 + v ) < 2 ™ h ) [ ( l + v ) z 4 - 6 h V - 8 h 3 z + 3 ( h 2 - v z 2 ) ( x 2

+ y 2 ) 1 6 E h J

- | (1-v) ( x 2 + y 2 ) 2 ] (2.4)

S u p e r i m p o s i t i o n o f the above two systems g i v e s ,

u = _ (1+v)(2pgh)x [ ( 2 _ v ) z 3 _ 3 h 2 2 _ 2 h 3 _ 3 ( 1 _ v )

8 E h J

z ( x 2 + y 2 } ] _ v p g ( Z + h ) x

E

v = _ (1+v) ( 2 p g h ) y [ ( 2 _ v ) z 3 „ 3 ^ _ 2 h 3 3 ( 1 _ v ) z ( x 2 + y 2 } j 8 E h J

vpg(z+h)y

E

= £SL [ z 2 + 2 h z + v ( x 2 + y 2 n + d+v) (2pgh) r ( 1 + v ) z<L 6 h2 z2 2 E 1 6 E h J

- 8 h 3 z + 3 ( h 2 - v z 2 ) ( x 2 + y 2 ) - | (1-v) ( x 2 + y 2 ) 2 ] (2.5)

w i t h the r e s u l t a n t moments

24

G l " D ( K 1 + V K 2 ) + 24 + 23v + 3v'

30(1-v) pgh"

G 2 = - DfKj+vl^) + 24 + 23v + 3v'

30(1-v) pgh" (2.6)

1^ = D ( l - V ) T

2 3 2 where, D = Eh /(1 -v ) = Modulus o f r i g i d i t y o f the p l a t e

and

2 2 2 3 w 3 w 3 w

K _ o K _ o T - ° x 3x^ ^ 3 y z 3x3y

w = d e f l e c t i o n o f midd le p lane a t z = 0. o ^

I t s h o u l d be noted here t h a t the u n d e r l i n e d p a r t s

o f the e x p r e s s i o n (2.6) r e p r e s e n t moments i n the pure bend ing

o f p l a t e s . The square b r a c k e t e d term i s due t o the more

a c c u r a t e a n a l y s i s .

From ( 2.5) , by p u t t i n g z = 0 ,

2 pgv , 2. 2. pgh , 2. 2 S r 2. 2 8h , n s

W o = 2E ( X + y ) ~ I2D ( X + Y } [ X + Y ' T^T ] ( 2 * 7 )

D i f f e r e n t i a t i o n o f (2.7) y i e l d s v a l u e s o f K^, and T which

when s u b s t i t u t e d i n (2.6) g i v e s moments a t r = a :

G, = G„ = ( 3 + v ) ( 2 p g h ) a 2

+ ( 2 p g h ) h 2 ( 2 . 8 ) 16 20

H, = 0

The r e s u l t a n t f o r c e a t the edge due t o the s u p e r i m -

p o s i t i o n o f the two systems (2.3) and (2.4) i s

T = i (2pgh) • h (2.9)

Now, i n o r d e r t h a t s o l u t i o n (2.5) be f r e e o f edge

moment (2.8) and r a d i a l f o r c e ( 2 . 9 ) , d e f l e c t i o n due to a

moment o p p o s i t e to (2.8) and due t o a r a d i a l f o r c e o p p o s i t e

to (2.9) must be super imposed on ( 2 . 5 ) .

The e x p r e s s i o n s f o r d e f l e c t i o n due t o u n i f o r m t e n s i o n

a l o n g the edge are g i v e n b y ,

1-v Q

u = ~~2E~ o X

v = 9 o y (2.10)

w = -•

v6 z o

E

In the p r e s e n t case 6 Q = - ^ (2pgh)• S u b s t i t u t i n g

t h i s v a l u e o f 6Q i n (2.10) one o b t a i n s ,

26

u = - V2 E (pgh) • x

v = - (pgh) • y (2.11)

w = g- (pgh) • z

For determining d e f l e c t i o n due to a moment opposite

to that of (2.8), we choose a stress function x ^ of the form

X X = J 3(x 2+y 2) + Y , (2.12)

where 3 , Y = constants, giving

r 2 h 3 ^ c - 2 h 3 9 2 ) ( l

1 J 3y 3x^

Substituting the values of the derivatives of x ^ from (2.12)

i n the above expression,

G 1 = G 2 = | h 3 3 (2.13)

This must be negative of the value obtained i n (2.8). Hence,

27

3 = - ^ t f — (2pgh) a 2 + (2pgh)h 2 ]

with [ 7 ]

v = i (JyZ - i | i z (2.14)

Taking the reference for d e f l e c t i o n values at 0

(r, z, w = 0) ,

( w )r=0 = " §E (0) + ^ ( i 3 • 0 + y ) z=0

Y = 0

and X l = " -K I — (2pgh) a 2 + 3^.(2pgh) h 2 ] (x 2+y 2) (2.15) 1 4h 16 20

Now subs t i t u t i n g the values of and i t s derivatives i n (2.14)

the deflections due to a moment opposite to (2.8) are

obtained as:

28

u = ffL [ (2pgh) a 2 + £2- (2pgh) h 2 ] (v-1) 2Eh,:J ± 0 ^ u

3y z v = , [ 3+̂ . ( 2 pgh) a 2 + f ^ - (2pgh) h 2 ] (v-1)

2Eh J ± Q z v •

(2.16)

2,^2, - 2 w = A , [ ̂ (2pgh) a 2 + i ^ . (2pgh) h 2 ] (3x2+3y2+6vz 4Eh J X b ^ u

2 2 -3x v-3y v ]

The superimposition of (2.11) and (2.16) on (2.5) leads

to the desired solution for d e f l e c t i o n of a plate due to

i t s own weight when supported by uniformly d i s t r i b u t e d shear

along i t s edges. In c y l i n d r i c a l co-ordinates the components

of d e f l e c t i o n can be expressed as:

w = P | [ z2+2hz+vr 2 ] + 1 + v , (2pgh)[ (l+v)z 4 - 6 h 2 z 2 - 8h 3z

2 E 16Eh J

^ , 2 2S 2 3 ,, 4 , . v , . . „ . 3r 2+6vz 2-3r 2v + 3(h -vz ) r - ^ ( l - v ) r ] + =• (pgh) z + =

8 E 4Eh 3

[ ^ (2pgh) a 2 + (2pgh) h 2 ] (2.17)

1+v t n , 4 r ,„ v 3 ^2 „,_3 3 ,., ..,3 u=v = -

8Eh j (2pgh)[(2-v)z r - 3h zr - 2h r - j ( l - v ) z r ]

29

_ vpg(z-rh) r _ U - W ( p g h ) r + 3rz jS+v ( 2 p g h ) a 2 + 3^v

E 2E 2Eh 16 20

(2pgh)h 2] (v-1) .

It should be noted here that Duncan [12], i n 1958,

derived a s i m i l a r expression for w i n a simply supported

plate subjected to uniform pressure q over i t s upper face.

This was achieved by s t a r t i n g with the Legendre polynominal

solution of the basic equations of e l a s t i c i t y for an a x i -

symmetrical case as adopted by Timoshenko [13].

At t h i s stage i t would be l o g i c a l to v e r i f y i f the

above rel a t i o n s s a t i s f y the f i e l d equations and the required

boundary conditions.

The f i e l d equations for displacements without body

forces were obtained by A l l e n [14]. Manipulating stress

equilibrium and stress displacement r e l a t i o n s , including body

forces due to gravity, the modified f i e l d equations for

axisymmetric systems can be shown to be,

8 2u , 3u u l-2v 9 2u , 92w (1-v) { + ±- , } + 2 + 7 = 0

9r dr r 2 az * 3r9z

32w l-2v 92w , 8w , 9 2u , , « (1-v) + { + ~ } + - + £ i °R - pg

< 1 + v ) < 1 - 2 v ) = 0 ( 2 . 1 8 )

1

30

The v a l u e s o f u and w and t h e i r d e r i v a t i v e s from (2.17)

i d e n t i c a l l y s a t i s f y e q u a t i o n s ( 2 . 1 8 ) .

Boundary c o n d i t i o n s t o be s a t i s f i e d a r e :

i ) fh, (a ) dz = 0; ' -h r r=a

i i ) f^h ( ^ r ) r = = a

z d z = 0;

h i i i ) / , (T ) dz • 2ira = weight o f the p l a t e , -h r z r=a

From (2.17) the s t r a i n components f o r an ax i symmet r i c

system are o b t a i n e d a s :

E = |£ = £3. (z+h) + 1 + v . , (2pgh) [ 4 ( l + v ) z 3 - 1 2 h 2 z - 8h :

z d z * 16Eh

- e v z r 2 ] + i ( ^ h ) + i r ! ( 2 p ^ h ) * 2 + f i r < 2 ^ h ^ E Eh

e = | H = - i±}L. ( 2 pgh) t ( 2 - v ) z 3 - 3h 2 z - 2 h 3 - I r 3 r 8 E h 3 4

, 2, vpg (z+h) 1-v t n „ u \ . 3z r ( l - v ) z r ] - j g - (pgh) + — ^ I ^ -3+v

(2pgh)a 2 + (2pgh) h 2 ] (v-1)

31

u ~ 1 + V j (2pgh) [(2-v)z 3 - 3h 2z - 2h 3 - | ( l - v ) z r 2 ] r 8Eh

vpg(z+h) 1-v , .» , 3z r3+v / 0 . . (pgh) + [y^— • (2pgh) a E 2E 2Eh J • L O

+ i z v . (2pgh) h 2] (v-1) 20

+ S t = 3 ( l - r v ) ( 2 P g h ) (h 2r - z 2 r ) (2.19) au _ 3 z 1 .ar - 4 E h 3

For axisymmetric systems, the s t r e s s - s t r a i n r e l a t i o n s

= e + Jv„ (e + e„ + e ) r r l-2v v r 8 z'

e = e e + "I=3v ( e r + e e + ez> ( 2 ' 2 0 )

z z l-2v r 8 z

32

1+v rz 2 Yrz

Substituting the values of the s t r a i n components from (2.19)

i n (2.20) and simplifying one obtains,

o = r " (1+v) (1 ±=±jy (2pgh) [4z 3(l+v)-6h 2z-4h 3-6vzr :

- 2z 3(2-v) + | ( l - v ) z r 2 ] - 2z 3(2-v)+6h 2z+4h 3+|(l+v)zr 2 }

+ ^ P S h - v p g ( z + h ) - £$*L + { 3+*. (2pgh)a 2+ | ^ (2pgh)h 2} (1-v)

_ 2 3y z h 3 ( l - v )

20

3z

" (l+v)(I-2v) f 77 T T < 2"*> { lh 1 4 Z3 ( l + v ) - 6 h 2

Z - 4 h 3 - 6 v z r :

|_ I o n

- 2z 3(2-v)+ | ( l - v ) z r 2 ]- 2z 3(2-v)+6h 2z+4h 3+|(l-v)zr 2 }

+ ii_ea*L - vpg(z-rh)- £|1 + . (2pgh)a 2+ 3Z^(2pgh)h 2 } (1-v) 20

33

•f 3v z 3z -,

h J ( l - v ) h J

a = z

1-v

(1+v)( l -2v) i — (2pgh) { [ - 2 z 3 ( 2 - v ) + 6 h 2 z + 4 h 3 ] + 2 z 3

8 h J X " V

+ 2 v z 3 - 6 h 2 z - 4 h 3 } + pg(z+h) - 2 v 2 pg(z+h) J

1-v J

x r z - I pgr { l - ( £ ) 2 } (2.21)

The r o u t i n e a l g e b r a shows t h a t (a ) and (x ) ^ r r=a r z r=a

s a t i s f y the boundary c o n d i t i o n s ment ioned b e f o r e .

T h i s a n a l y s i s f o r the s e l f - w e i g h t d e f l e c t i o n

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

shear a t the edge , i s v a l u a b l e due to the f a c t t h a t a s t r o n o m i c a l

m i r r o r s u b s t r a t e s are s t i l l des igned i n the form of s o l i d

d i s c s f o r medium s i z e a p e r t u r e s .

In t h i s c o n t e x t o f s e l f - w e i g h t d e f l e c t i o n o f

c i r c u l a r p l a t e s , i t i s o f i n t e r e s t t o compare the mid -p lane

d e f l e c t i o n as o b t a i n e d by Emerson [ 1 5 ] ,

w = pg (2h) 16E

2 w 2 2, -5+v 3 (1-v ) (a - r ) (, •1+v

2 2 X , • a - r ) + | | (3+v) ( a 2 - r 2 )

(2 .22)

34

The above expression assumes that the t o t a l body force of

the plate i s equivalent to an equal uniformly d i s t r i b u t e d

surface pressure q = 2pgh (Figure 2.3).

2 • 3 Results and Discussion

For the plate supported by uniformly d i s t r i b u t e d

shear, the d i s t r i b u t i o n of r a d i a l and shear stress at the

end of the plate i s shown i n Figure 2.4. The v a r i a t i o n of

r a d i a l stress at an intermediate cross-section i s indicated

in Figure 2.5. It i s clear that the net r a d i a l force at

any cross-section i s zero but only the edge of the plate i s

free of moment. Furthermore, the d i s t r i b u t i o n of shear i s

parabolic.

The v a r i a t i o n of a x i a l stress a as obtained from z

the present investigation, equation (2.21), i s shown in

Figure 2.6a with the d i s t r i b u t i o n of for the boundary

and loading conditions of Figure 2.3 shown i n Figure 2.6b

t13]• It i s i n t e r e s t i n g to note that the expression (2.22),

being approximate (distributed body force replaced by a

uniform loading), v i o l a t e s the boundary condition of zero

normal stress over the top face of the plate (Figure 2.6b)

whereas the d i s t r i b u t i o n of a obtained from the present

analysis s a t i s f i e s the boundary conditions of zero normal

stress over the top and bottom faces of the plate exactly

(Figure 2.6a). A comparison of the middle plane deflections

as obtained from (2.17) and (2.22) i s shown in Figure 2.7.

Here the numerical values of physical parameters correspond

35

Uniform Load 2pgh

F i g u r e 2 .3 T o t a l body f o r c e c o n s i d e r e d e q u i v a l e n t t o an e q u a l u n i f o r m l y d i s t r i b u t e d s u r f a c e p r e s s u r e ,

a pgh

B

a) R a d i a l S t r e s s b) Shear St r e s s

F i g u r e 2« 4 D i s t r i b u t i o n o f r a d i a l and shear s t r e s s e s a t the end o f the p l a t e

37

F i g u r e 2 .6 A x i a l s t r e s s d i s t r i b u t i o n

Middle Surface

•dr 2f - B a — n H 6

Present Theory

Emerson

- 2 5 0

2 0 0

150

CO

100

5 0

o o o o o

TO

Comparison between the middle s u r f a c e d e f l e c t i o n s as o b t a i n e d from the p r e s e n t t h e o r y (equat ion 2.17) and Emerson 's a n a l y s i s (equat ion 2.22) :

E = 10.5 x 10 ; v = 0 .17 ; p = 0.0795

3 9

to that for fused s i l i c a , a conventional substrate material. As evident from equation (2.21), the d i s t r i b u t i o n of

r a d i a l stress i s non-linear. This may be attributed to the

e f f e c t of shearing stresses and normal pressures on the planes

p a r a l l e l to the surface of the plate. However, the e f f e c t

of non-linearity i n the case considered appears to be rather

n e g l i g i b l e (Figure 2.5). The r a d i a l stresses at the edge

are not zero (Figure 2.4a) but the resultant of these stresses

and t h e i r moments indeed vanish. Hence, on the basis of

Saint Venant's P r i n c i p l e , we can say that the removal of

these stresses does not a f f e c t the stress d i s t r i b u t i o n i n

the plate s i g n i f i c a n t l y at some distance away from the edge.

The expression for l a t e r a l d e f l e c t i o n w, equation

(2.17), shows i t to be a function of r and z thus suggesting

d i f f e r e n t shapes for top and bottom surfaces compared to the

mid-plane. This difference increases as the diameter to

thickness r a t i o decreases. Hence for thick c y l i n d r i c a l mirror

substrates, prediction of top surface d e f l e c t i o n from the

mid-plane analysis (as i s done i n the approximate theories)

i s not f u l l y j u s t i f i e d . Thus for accurate determination of

the top surface d e f l e c t i o n the present analysis should prove

useful. Figure 2.8 compares the deflections of the middle

and outer surfaces.

I t should be noted here that, as the outer surfaces

and middle plane (Figure 2.8) assume d i f f e r e n t shapes when

flexed, t h e i r r a d i i of curvature would not d i f f e r merely

by t h e i r normal distance apart; t h i s i n t e r v a l w i l l not be a

constant function of r. Thus the present analysis accurately

41

shows the f i n e r d e t a i l s of e l a s t i c deformation patterns,

which may be relevant i n c a l c u l a t i n g the curvatures of outer

faces. The curvature of the flexed dis c follows r e a d i l y

from (2.17).

In order to investigate the influence of shear, the

results based on present analysis are compared with the t h i n

plate theory i n Figure 2.9. The discrepancy i n d e f l e c t i o n

values increases rapidly as the diameter/thickness r a t i o i s

decreased (3.4% for D/H=10, 21.7% for D/H=4). Furthermore,

the d e f l e c t i o n diminishes quite rapidly as the D/H i s

decreased. That explains massive character of mirror substrates

where r i g i d i t y i s b u i l t up more rapidly than the associated

body forces.

It should be noted that a continuous support by

way of the p a r a b o l i c a l l y d i s t r i b u t e d shear i s not practicable.

A l o g i c a l s i m p l i f i c a t i o n i s to concentrate shear reactions

at three equispaced boundary points or at s i x i n t e r n a l

points, as mentioned e a r l i e r i n Chapter 1. These would induce

more s t r a i n compared to a continuous support because of

additional i n t e r n a l transmission of body forces to reactive

points. However, when a mirror substrate i s supported on

three equispaced pads, the problem becomes s t r i c t l y three

dimensional and one has to resort to numerical methods

aided by a d i g i t a l computer or experimental studies of

models.

r/a A compar ison between the p r e s e n t a n a l y s i s and the t h i n p l a t e t h e o r y r e s u l t s f o r s e v e r a l D/H r a t i o s :

E = 10.5 x 10°; v = 0.17; p = 0.0795

F i g u r e 2.9

43

As there i s a s i m i l a r i t y i n the nature of the body

force and thermally induced e f f e c t s i n e l a s t i c s o l i d s , i t

i s appropriate, i n t h i s context, to comment on the thermally

induced displacements.

When the temperature i n a small portion of the body

i s increased by T, d i l a t a t i o n proportional to T can be

produced without a corresponding change i n pressure. This

implies extension of a l l l i n e a r elements by aT, where a i s

the c o e f f i c i e n t of thermal expansion. In addition, i f

forces are applied to the body,the corresponding s t r a i n

obtained from s t r e s s - s t r a i n r e l a t i o n would augment the

thermal e f f e c t . Thus i t follows that the displacement due

to the thermal e f f e c t i s i d e n t i c a l to that produced by a 06 £

body force expressed as the gradient of - Y-2\> T / an& cxE!

normal surface pressure Y-2'V T l ^ ]. These would be

i n addition to the actual body force and surface t r a c t i o n

present i n the system.

Thus, the a n a l y t i c a l expressions derived for the

body force case may be used to determine thermally induced

displacements. On the other hand, thermally induced d i s ­

placements can be determined quite r e a d i l y using f i n i t e

element procedures as explained i n the following chapter.

3. NUMERICAL ANALYSIS OF THICK CIRCULAR PLATES OF VARIABLE THICKNESS AND ITS APPLICATION TO THE DESIGN OF MIRROR SUBSTRATES

An ax i symmetr i c e l a s t i c system may be d e s c r i b e d by

c y l i n d r i c a l c o o r d i n a t e s as shown i n F i g u r e 3 . 1 . Due to

symmetry i n geometry and l o a d i n g about the v e r t i c a l a x i s z ,

the system deforms o n l y i n r a d i a l and v e r t i c a l d i r e c t i o n s .

A l s o s t r e s s e s and s t r a i n s do not v a r y i n the t a n g e n t i a l

d i r e c t i o n . Thus , from a mathemat i ca l p o i n t o f v i ew , the

problem i s o n l y two d i m e n s i o n a l and i t i s s u f f i c i e n t to

a n a l y z e any a x i a l z - r p lane as shown i n F i g u r e 3 . 2 . The s t r u c ­

t u r a l forms shown i n column 2 o f F i g u r e 1.5 r e p r e s e n t

ax i symmetr i c sys tems . S e v e r a l n u m e r i c a l t e c h n i q u e s are

a v a i l a b l e which may be used f o r s o l v i n g problems o f the

p r e s e n t k i n d . I t would be u s e f u l t o comment on these p r o ­

cedures b e f o r e t r e a t i n g the prob lem i n hand.

3.1 A B r i e f Review o f the Numer i ca l Techn iques

3 . 1 . 1 F i n i t e D i f f e r e n c e Method

T h i s i s p r o b a b l y the most w i d e l y used n u m e r i c a l

t e c h n i q u e f o r s o l v i n g a v a r i e t y o f f i e l d p rob lems , i n c l u d i n g

those i n e l a s t i c i t y . The m a j o r i t y o f the problems i n s t r e s s

a n a l y s i s , when t a c k l e d by a t h e o r e t i c a l approach , reduce to

the s o l u t i o n o f one o r more d i f f e r e n t i a l e q u a t i o n s w i t h

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

45

F i g u r e 3.1 Ax i symmetr i c s o l i d F i g u r e 3.2 A x i a l z - r p lane

46

In the f i n i t e d i f f e r e n c e method, the g o v e r n i n g

e q u a t i o n s o f e l a s t i c i t y and boundary c o n d i t i o n s are r e p l a c e d

by a f i n i t e number o f unknown v a l u e s o f the dependant

v a r i a b l e s a t a number o f d i s c r e t e nodes , formed by a g r i d -

work, w i t h i n and over the boundary o f the domain.

The prob lem can be f o r m u l a t e d i n terms o f s t r e s s

f u n c t i o n s [16] o r d i s p l a c e m e n t components [ 1 4 ] , the c h o i c e b e i n g

l a r g e l y governed by the s p e c i f i c a t i o n o f the boundary c o n d i t i o n s

i n terms o f s t r e s s , d i s p l a c e m e n t o r b o t h .

In the fo rmer , the v a l u e s o f s t r e s s f u n c t i o n s o b t a i n e d

f o r the nodes g i v e o n l y an i n t e r m e d i a t e s o l u t i o n to a

p a i r o f g o v e r n i n g e q u a t i o n s . A combinat ion o f the d e r i v a t i v e s

o f the s t r e s s f u n c t i o n s y i e l d the r e q u i r e d s t r e s s components.

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

components o f d i s p l a c e m e n t s a re o b t a i n e d d i r e c t l y . A

s u i t a b l e combinat ion o f the d e r i v a t i v e s o f d i s p l a c e m e n t s g i v e

the s t r e s s components.

Once the g e n e r a l f o r m u l a t i o n o f the problem i s

e s t a b l i s h e d , the e l a s t i c r e g i o n to be s t u d i e d i s

s y s t e m a t i c a l l y d i v i d e d up by a system of c o o r d i n a t e p l a n e s

i n t e r s e c t i n g a l o n g l i n e s and at nodes . Now g o v e r n i n g e q u a t i o n s

i n terms o f n o d a l s t r e s s f u n c t i o n v a l u e s have t o be s a t i s f i e d .

These f i n i t e d i f f e r e n c e e q u a t i o n s r e p r e s e n t a s e t o f l i n e a r

a l g e b r a i c r e l a t i o n s to be s o l v e d f o r unknown s t r e s s f u n c t i o n s ,

i n g e n e r a l , by a d i g i t a l computer .

47

A s o l u t i o n o b t a i n e d by the f i n i t e d i f f e r e n c e method

i s always i n e r r o r , however s m a l l . These e r r o r s may be

reduced i n two ways [17 J :

i ) by employ ing a v e r y f i n e mesh s i z e so t h a t the

e r r o r s due to n e g l e c t e d h i g h e r o r d e r terms i n

T a y l o r ' s s e r i e s expans ions are n e g l i g i b l e ;

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

i n f i n i t e d i f f e r e n c e formulae based on a

coa rse mesh.

Both the p rocedures have d i s a d v a n t a g e s . The f i r s t

a l t e r n a t i v e may appear to be t h e o r e t i c a l l y a t t r a c t i v e as

e r r o r s d e c r e a s e i n d e f i n i t e l y w i t h a mesh s i z e . However, w i t h

a f i n e mesh, the number o f e q u a t i o n s becomes v e r y l a r g e and

the e f f o r t s i n v o l v e d i n o b t a i n i n g a s o l u t i o n ge ts p r o h i b i t i v e .

F u r t h e r m o r e , round o f f e r r o r s a l s o i n c r e a s e . The a l t e r n a t i v e

would be t o use more n o d a l p o i n t s i n the g o v e r n i n g e q u a t i o n s .

T h i s l e a d s to e x t e n s i v e m a n i p u l a t i o n s t o e l i m i n a t e the

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

d i s p l a c e m e n t f u n c t i o n s a t nodes o u t s i d e the boundary o f the

p rob lem. The f i c t i c i o u s q u a n t i t i e s are e l i m i n a t e d by u s i n g

the s p e c i f i e d boundary c o n d i t i o n s . F u r t h e r , due t o the

i n c l u s i o n o f remote nodes , the l o c a l a c c u r a c y might be

a f f e c t e d .

A comment c o n c e r n i n g a r e l a t i v e l y new n u m e r i c a l

p r o c e d u r e , deve loped by Day [18] and O t t e r e t a l . [19]

48

w o u l d b e a p p r o p r i a t e h e r e . R e f e r r e d t o a s D y n a m i c R e l a x ­

a t i o n , i t r e p r e s e n t s a n i t e r a t i v e m e t h o d f o r u s e w i t h a

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

s t r e s s - s t r a i n a n d e q u i l i b r i u m e q u a t i o n s o f e l a s t i c i t y . H o w e v e r ,

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

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

F o r a x i s y m m e t r i c b o d i e s w i t h b o d y f o r c e s d u e t o g r a v i t y ,

t h e y c a n b e e x p r e s s e d a s :

r . r z . r 6 » , + + = p u + c u

3r 3z r

3T 3a T + — z + _ r z + p = p w + c w

3r 3z r Z .

w h e r e = b o d y f o r c e i n t e n s i t y i n t h e z d i r e c t i o n

p = m a s s d e n s i t y

c = d a m p i n g f a c t o r

u , v i , w , w = v e l o c i t y a n d a c c e l e r a t i o n c o m p o n e n t s a l o n g

t h e r a n d z d i r e c t i o n s , r e s p e c t i v e l y .

T h e s y s t e m i s a n a l y z e d c o n s i d e r i n g :

i ) t h e m o t i o n o f a n e l e m e n t d u e t o i n t e r n a l s t r e s s e s

a n d i m p o s e d b o d y f o r c e s ;

i i ) t h e e l a s t i c r e l a t i o n b e t w e e n t h e s t r e s s e s a n d

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

49

The object of the c a l c u l a t i o n i s not, however, to study

the motion but to determine s t a t i c stresses and displacements

of a structure. This i s achieved by an i t e r a t i v e procedure.

Thus the method i s e s s e n t i a l l y a step-by-step integration

of damped v i b r a t i o n , using viscous damping,to ensure a t t a i n ­

ment of steady state solution.

A t y p i c a l i t e r a t i o n s t a r t s at t = 0, with a r b i t r a r i l y

chosen i n i t i a l conditions except at the boundary where d i s ­

placements and/or stresses are s p e c i f i e d . I n i t i a l l y , d i s ­

placements and v e l o c i t i e s may be chosen to be zero except at

the boundary. Substitution of chosen displacements i n the

stress displacement equations, gives stresses. The stresses

are then introduced i n the equilibrium equations to obtain

new v e l o c i t i e s . The new displacements at time t^= t + At

are obtained from the velocity-displacement r e l a t i o n ,

u^= u + uAt.

The next i t e r a t i v e step begins with the c a l c u l a t i o n

of stresses from the stress-displacement r e l a t i o n . The new

stresses are then substituted i n the equilibrium equations.

The process continues u n t i l v e l o c i t i e s and accelerations

diminish and the r i g h t hand side of the equilibrium equation

approximates to zero, thus s a t i s f y i n g the condition of

s t a t i c equilibrium.

Dynamic relaxation seems to be a very powerful

numerical technique, however, the reference system used

should be such that the coordinate surfaces conform to the

50

s u r f a c e s o f the body. Fo r curved b o u n d a r i e s c u r v i l i n e a r

c o o r d i n a t e s have to be u s e d . Mixed boundary c o n d i t i o n s

i n v o l v i n g s t r e s s e s and d i s p l a c e m e n t s do not pose any prob lem.

The main d i s a d v a n t a g e o f the method i s the d e r i v a t i o n

o f the t ime i n t e r v a l and the damping f a c t o r . S u i t a b l e v a l u e s

have t o be determined by t r i a l and e r r o r .

As can be e x p e c t e d , the a c c u r a c y o f the method

depends on the element s i z e u s e d . The r e s u l t s o b t a i n e d by

dynamic r e l a x a t i o n compare f a i r l y w e l l w i t h those due t o

o t h e r methods [18, 1 9 ] .

3 . 1 . 2 I n t e g r a l Method

I n t e g r a l methods a re f u n d a m e n t a l l y d i f f e r e n t f rom

the f i n i t e d i f f e r e n c e and f i n i t e e lement approaches . Here

a number o f f i c t i c i o u s c o n c e n t r a t e d o r l o c a l l y d i s t r i b u t e d

l o a d s , o f a r b i t r a r y v a l u e s , are a p p l i e d a l o n g the boundary

o f the model . T h e i r t o t a l e f f e c t a t an i n t e r i o r p o i n t i s

expressed as an a n a l y t i c a l summation o f the e f f e c t s o f the

i n d i v i d u a l l o a d s [ 2 0 ] .

In p lane p rob lems , f o r i n s t a n c e , the f o r c e s a p p l i e d

a t the boundary can be c o n c e n t r a t e d l oads g i v i n g r i s e to a

r a d i a l s t r e s s [21] ( a = c-°. s 9 ) w i t h i n the f i e l d . A t r TT r

the boundary , the i n t e r i o r s t r e s s e s must e q u i l i b r a t e the

e x t e r i o r s t r e s s and from t h i s c o n d i t i o n the d i s t r i b u t i o n

o f the c o n c e n t r a t e d f i c t i c i o u s l o a d s a l l over the boundary

are d e t e r m i n e d .

51

F o r ax i symmetr i c s o l i d s B o u s s i n e s q ' s s o l u t i o n o f a

f o r c e a p p l i e d n o r m a l l y t o the boundary and g i v i n g r i s e t o

s t r e s s e s a , a f l , a , x may be u s e d . The i n t e g r a t e d e f f e c t 2T t) Z JL Z

o f these components o f s t r e s s e s a t i n t e r i o r p o i n t s f o r

e v e r y f i c t i t i o u s l o a d a p p l i e d on the boundary has t o be

o b t a i n e d . A t the boundary , the sum o f i n t e r i o r s t r e s s e s

due t o a l l s u r f a c e l o a d s has to be equated t o the e x t e r n a l l y

a p p l i e d boundary s t r e s s .

I n t e g r a l methods i n v o l v e fewer unknowns than f i n i t e

d i f f e r e n c e o r f i n i t e e lements t e c h n i q u e s . They a l s o

r e s u l t i n system of e q u a t i o n s w i t h f u l l m a t r i c e s (meaning i

t h e r e are not many z e r o terms i n the c o e f f i c i e n t mat r ix )

whereas d i f f e r e n c e methods i n v o l v e spa rse m a t r i c e s o f

h i g h e r o r d e r .

U s u a l l y the i n t e g r a l p rocedures g i v e more a c c u r a t e

r e s u l t s than the f i n i t e d i f f e r e n c e method, as the s t r e s s e s

are o b t a i n e d d i r e c t l y from the a n a l y t i c a l e x p r e s s i o n f o r the

summation o f the s e p a r a t e e f f e c t s o f the i n d i v i d u a l boundary

l o a d s . On the o t h e r hand , i n the f i n i t e d i f f e r e n c e method

the s t r e s s components depend on approximate d e r i v a t i v e s o f

the d i s c r e t e s t r e s s f u n c t i o n s .

52

3 .1 .3 F i n i t e Element Method

The f i n i t e e lement method i s e s s e n t i a l l y a g e n e r a l ­

i z a t i o n o f the t h e o r y o f s t r u c t u r a l a n a l y s i s . I t makes

p o s s i b l e the a n a l y s i s o f two-and t h r e e - d i m e n s i o n a l e l a s t i c

c o n t i n u a by the techn ique s i m i l a r t o t h a t used i n the a n a l y s i s

o f o r d i n a r y framed s t r u c t u r e s .

The method was i n t r o d u c e d o r i g i n a l l y by Turner e t

a l . [22] f o r the s o l u t i o n o f complex s t r u c t u r a l problems

encountered i n the a i r c r a f t i n d u s t r y .

The impor tan t concept i n t r o d u c e d by the f i n i t e e lement

method i s the use o f two- o r t h r e e - d i m e n s i o n a l s t r u c t u r a l

e lements t o r e p r e s e n t an e l a s t i c cont inuum. The r e a l

cont inuum i s assumed to be d i v i d e d i n t o a f i n i t e number o f

e lements i n t e r c o n n e c t e d a t a f i n i t e number o f nodes . An

approx imat ion which i s employed i n f i n i t e e lement t e c h n i q u e s

i s o f a p h y s i c a l n a t u r e ; a m o d i f i e d s t r u c t u r a l system i s

s u b s t i t u t e d f o r the a c t u a l cont inuum. T h i s d i s t i n g u i s h e s

the f i n i t e e lement t e c h n i q u e from f i n i t e d i f f e r e n c e method,

where , as seen b e f o r e , the e x a c t e q u a t i o n s o f the a c t u a l

p h y s i c a l system are s o l v e d by approximate mathemat i ca l

p r o c e d u r e s .

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

an e l a s t i c cont inuum by an assembly o f s t r u c t u r a l e lements

i s not new. In f a c t , the development o f the f i n i t e e lement

concept has stemmed from an e f f o r t t o improve on the

53

Hrenn iko f f -McHenry [23,24] ' l a t t i c e a n a l o g y 1 f o r r e p r e s e n t i n g

p lane s t r e s s systems [ 2 5 ] . Important s teps i n v o l v e d i n the

a n a l y s i s may be summarized as below [ 2 6 , 2 7 ] :

i ) S t r u c t u r a l I d e a l i z a t i o n - D i v i s i o n o f the e l a s t i c

cont inuum, t o be a n a l y z e d , . i n t o an a p p r o p r i a t e l y

shaped f i n i t e e l ements ,

i i ) E lement A n a l y s i s - E v a l u a t i o n o f the s t i f f n e s s

m a t r i x f o r every element by r e l a t i n g the f o r c e s

deve loped at the e lement n o d a l p o i n t s t o the

c o r r e s p o n d i n g element d i s p l a c e m e n t s ,

i i i ) Assembly A n a l y s i s - E v o l u t i o n o f the assembly

s t i f f n e s s m a t r i x f o r the complete s t r u c t u r e by

the s u p e r p o s i t i o n o f a p p r o p r i a t e e lement s t i f f ­

nesses and d e t e r m i n a t i o n o f unknown n o d a l

d i s p l a c e m e n t s by the s o l u t i o n o f the e q u i l i b r i u m

e q u a t i o n s .

Fo r the p r e s e n t p rob lem, the f i n i t e e lement approach

was c o n s i d e r e d t o be w e l l s u i t e d as the curved b o u n d a r i e s

o f the a rched geometr ies can be r e a d i l y i d e a l i z e d by the use

o f t r i a n g u l a r e l e m e n t s . The r e q u i r e d d e f l e c t i o n can be

o b t a i n e d d i r e c t l y and the mixed boundary c o n d i t i o n s pose no

prob lem. F u r t h e r m o r e , therma l e f f e c t s can be t r e a t e d u s i n g

the normal p r o c e d u r e .

54

3.2 L i g h t - W e i g h t M i r r o r S u b s t r a t e Des ign P h i l o s o p h i e s

The s e l f - w e i g h t problem a s s o c i a t e d w i t h l a r g e m i r r o r s

has l ong been r e c o g n i z e d . The s o l i d d i s c i s not an optimum

s t r u c t u r e s i n c e m a t e r i a l near the midd le p l a n e , when i t s a x i s

c o i n c i d e s w i t h the d i r e c t i o n o f g r a v i t y , i s no t f u l l y s t r e s s e d .

Hence, i t i s a t t r a c t i v e t o d e v e l o p , f o r both ground and space

a p p l i c a t i o n s , s t r u c t u r a l forms i n which such u n d e r s t r e s s e d

m a t e r i a l has been r e d i s t r i b u t e d or removed t o more e f f e c t i v e

l o c a t i o n s . S e v e r a l p rocedures f o r a c h i e v i n g t h i s are a p p a r e n t .

The most common would be the use o f sandwiched and r i b b e d s t r u c ­

t u r e s , web and f l a n g e c o n s t r u c t i o n , a r c h c o n c e p t , e t c . [ 6 , 2 8 , 2 9 ] .

L i g h t we ight m i r r o r s have been s u c c e s s f u l l y manufac­

t u r e d f o r s a t e l l i t e and ground a p p l i c a t i o n s by the a d o p t i o n

of sandwich c o n s t r u c t i o n t e c h n i q u e . A sandwiched m i r r o r

c o n s i s t s o f two p l a t e s , s e p a r a t e d and f i x e d i n a r i g i d r e l a t i o n

to one another by s p a c e r s , one o f the e x t e r n a l s u r f a c e s

a c t i n g as m i r r o r . The arrangement o f f e r s a number o f advantages .

F i r s t l y , i t a l l o w s f o r an i n t e r n a l c i r c u l a t i o n o f a i r o r some

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

t o the rma l e q u i l i b r i u m more r a p i d l y . S e c o n d l y , a w e l l des igned

sandwich can be made as r i g i d as a s o l i d m i r r o r and t h i r d l y ,

mounting i s f a c i l i t a t e d . These i d e a s have been embodied i n

two p a t e n t s , one by R i t c h e y i n 1924 and the o t h e r by Parsons

5 5

and Rands i n 1929. F i g u r e 3.3 shows P a r s o n ' s sandwich

m i r r o r and R i t c h e y ' s c e l l u l a r c o n s t r u c t i o n [ 2 9 ] ,

The use o f r i b b e d type s t r u c t u r e was suggested by

L o r d Ross [29] as e a r l y as i n the mid-19th c e n t u r y . The

200 i n c h m i r r o r s u b s t r a t e o f the Ha le T e l e s c o p e was made

by p o u r i n g pyrex i n t o a mould i n which 114 f i r e b r i c k c o r e s

had been b o l t e d . The f i n a l p a t t e r n had the appearance

o f a honeycomb. The scheme r e s u l t e d i n a m i r r o r whose

we ight was a p p r o x i m a t e l y h a l f t h a t o f a s o l i d b l ank o f the

same s i z e when the cores were removed.

A s tudy o f the c e l l u l a r forms o f m i r r o r s u b s t r a t e s

employed to date shows t h a t v e r y few d e s i g n e r s have made use

o f the obv ious and l o g i c a l account o f c e r t a i n fundamental

theorems t o u c h i n g optimum s t r u c t u r a l d e s i g n as e n u n c i a t e d

i n 1869 by James C l a r k Maxwel l and extended i n 1904 by

A . G . M . M i c h e l l . M a x w e l l ' s theorem, which i s demonstrated i n

a condensed a r t i c l e by B a r n e t t [ 3 0 ] , shows t h a t one c l a s s o f

optimum s t r u c t u r e i s t h a t i n which a l l the m a t e r i a l employed

i s s t r e s s e d , to an a c c e p t a b l e l i m i t , i n the same sense ;

e i t h e r compress ion o r t e n s i o n . Membranes, a rches and s h e l l s

have t h i s g e n e r a l c h a r a c t e r i s t i c . M i c h e l l ' s c o r o l l a r y ,

a l s o o u t l i n e d i n the a r t i c l e by B a r n e t t , r e c o g n i z e s t h a t

s t r u c t u r a l c o n f i g u r a t i o n s cannot always be a r ranged to have

a l l the members i n compress ion o r t e n s i o n . However, M i c h e l l

demonstrated t h a t f o r a s t r u c t u r e where members i n bo th

compress ion and t e n s i o n are p r e s e n t , they must be o r t h o g o n a l

H o l e s b . R i t c h e y ' s C e l l u l a r M i r r o r

F i g u r e 3.3 Sandwich and c e l l u l a r m i r r o r substrates

57

a t a l l p o i n t s o f i n t e r s e c t i o n and s t r e s s e d to e q u a l l i m i t i n g

v a l u e s f o r the s t r u c t u r e t o be optimum. Though deve loped

and proved i n two d i m e n s i o n s , these theorems are v a l u a b l e

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

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

The m a t e r i a l employed i n c o n v e n t i o n a l beams and

p l a t e s i s not i d e a l l y d i s t r i b u t e d i n accordance w i t h the

above theorems. One wel l -known method o f r e d i s t r i b u t i o n i s

the a d o p t i o n o f Web and F lange (I-beam) c o n c e p t . T h i s can

be f u r t h e r improved by c a s t e l l a t i o n [31] o f an I-beam Web

(F igure 3 . 4 ) . The Web i s cut a l o n g a l i n e r e s e m b l i n g an

Acme t h r e a d . The r e s u l t i n g h a l v e s are then r e l o c a t e d and

welded t o g e t h e r . The l o g i c a l e x t e n s i o n o f t h i s i d e a i n t o

t h r e e d imens ions l eads t o the concepts o f l a t t i c e s and

g e o d e s i c frame works .

The domain between the f o u n d a t i o n and m i r r o r s u b s t r a t e

e lements (F igure 3.5) can be spanned by a s t r u c t u r e d e s i g n e d

a c c o r d i n g t o M i c h e l l ' s theorem. M i c h e l l ' s s t r u c t u r e c o u l d

be d e f i n e d i n the domain, where a r b i t r a r y r e a c t i o n l o c a t i o n s

and d i r e c t i o n s a re s p e c i f i e d and the body f o r c e s e x e r t e d by

the s u b s t r a t e e lements are a p p l i e d on or i n the domain. In

such a case the body f o r c e s o f the s u b s t r a t e and the d i s t r i ­

buted mass o f the M i t c h e l l ' s s t r u c t u r e i t s e l f are t r a n s m i t t e d

to the r i g i d f o u n d a t i o n through d i r e c t t e n s i o n o r compress ion

o f the s t r u c t u r a l members. F i g u r e 3.5 i l l u s t r a t e s t h i s

n o t i o n . Some such forms have been d i s c u s s e d i n d e t a i l by

Johnson [32] .

F i g u r e 3 . 4 C a s t e l l a t e d beam

F i g u r e 3 . 5 M i c h e l l s t r u c t u r e

59

The alternate approach to optimization follows from

Maxwell's o r i g i n a l theorem. By arranging the material i n

arch-like configurations, an optimum design i s l i k e l y to

r e s u l t , i n which the material tends to be stressed i n one

sense. Kenny [33] has demonstrated, by comparing deflections

of a disc to those of spherical s h e l l s , which may be con­

structed by removing materials from the d i s c , that a s h e l l

or an arch type structure i s much superior to a constant

thickness plate.

In the present investigation the second l i n e of

thought i s pursued to optimize c i r c u l a r substrates having

superior stiffness-to-weight r a t i o s . I t was thought that

arch-type substrates would be less i n t r i c a t e than ribbed

sandwiches or spac i a l l a t t i c e s which r e f l e c t the philosophy

behind Michell's c o r o l l a r y to Maxwell's theorem.

F i n a l l y i t must be stressed that, the optimum

configuration changes with each change i n the attitude of

a mirror. Thus, there can be no unique optimum configuration

for a substrate which changes attitude i n a g r a v i t a t i o n a l

f i e l d . However, i t i s customary to design a substrate with

the axis of the mirror v e r t i c a l and to assume the demonstra-

table fact that deformation due to gravity i n p a r t i c u l a r

w i l l be less severe and not c r i t i c a l when the axis i s

horizon t a l .

Hence, as with other investigations, design studies

to follow are concerned with substrates having extreme

dimensions i n a plane normal to the v e r t i c a l .

60

3.3' F i n i t e E l e m e n t S o l u t i o n s f o r A x i s y m m e t r i c Systems

S e v e r a l a x i s y m m e t r i c b o d i e s s u b j e c t e d t o a x i s y m m e t r i c

l o a d i n g were a n a l y z e d u s i n g t h e f i n i t e e l e m e n t a p p r o a c h .

D e t a i l s o f t h e m a t h e m a t i c a l t r e a t m e n t i s g i v e n i n

A p p e n d i x 1. The e l e m e n t s t i f f n e s s m a t r i c e s were l i n k e d

w i t h t h e e x i s t i n g , two d i m e n s i o n a l f i n i t e e l e m e n t computer

programme. T h i s was p o s s i b l e s i n c e t h e d e f o r m a t i o n p r o b l e m

f o r an a x i s y m m e t r i c s y s t e m i s one o f two d i m e n s i o n a l d i s ­

p l a c e m e n t and t h r e e d i m e n s i o n a l s t r e s s .

I n o r d e r t o c h e c k t h e a c c u r a c y o f t h e f i n i t e e l e m e n t

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

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

p a r a b o l i c s h e a r , was computed and compared w i t h t h e

a n a l y t i c a l s o l u t i o n ( F i g u r e 3 . 6 ) . A c l o s e a g r e e m e n t b e ­

tween t h e two s o l u t i o n s i s a p p a r e n t .

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

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

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

r a d i a l l o c a t i o n s . The a i m was t o e s t a b l i s h t h e optimum

l o c a t i o n f o r minimum d e f l e c t i o n . The r e s u l t s showed

( F i g u r e 3.7) t h e minimum d e f l e c t i o n t o o c c u r when t h e s u p p o r t

c i r c l e r a d i u s i s 0.66 o f t h e r a d i u s o f t h e p l a t e .

T h i s i s i n c l o s e a greement w i t h Emerson's [15] s t u d y

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

61

gure 3.6 The middle s u r f a c e d e f l e c t i o n as p r e d i c t e d by the f i n i t e e lement p rocedure and the a n a l y t i c a l s o l u t i o n : D

E = 10.5 x 1 0 6 ; v = 0 .17 ; p = 0.0795 ; g = 6

63

the t h r e e p o i n t s s h o u l d be l o c a t e d a t 0 .7a f o r the minimum bend­

i n g d e f l e c t i o n . S t u d i e s conducted by Kenny and W i l l i a m s

[ 8 ] f and Vaughan [io] a l s o l e d t o the same c o n c l u s i o n .

With t h i s background i t was d e c i d e d t o p roceed w i t h the a rched

s u b s t r a t e a n a l y s i s .

3 . 3 . 1 Comparat ive A n a l y s i s o f Arched S t r u c t u r e s and S o l i d D i s c s

I t i s impor tan t to r e c o g n i z e t h a t i n d e s i g n i n g a

l i g h t we ight m i r r o r s u b s t r a t e , a r e l a t i v e l y h i g h e r s t i f f n e s s -

t o - w e i g h t r a t i o may be o b t a i n e d by removing m a t e r i a l s

t o form an a r c h from a deep s o l i d d i s c (D/H = 3) r a t h e r

than from a moderate ly t h i c k d i s c (D/H = 6 ) . But the space

o c c u p i e d by the former w i l l be l a r g e r than the l a t t e r . T h i s

i n t u r n w i l l need a l a r g e r h o u s i n g and s u p p o r t i n g s t r u c t u r e

f o r the m i r r o r . So, i n the s u b s t r a t e d e s i g n , the q u e s t i o n

o f space l i m i t a t i o n has to be borne i n mind . The s u b s t r a t e

forms a n a l y z e d i n the p r e s e n t s tudy are l i m i t e d w i t h i n the

space enve lope o f the c o n v e n t i o n a l s o l i d d i s c h a v i n g d iameter

to t h i c k n e s s r a t i o of 6. F u r t h e r m o r e , a s l i g h t c u r v a t u r e

o f the r e f l e c t i n g s u r f a c e i s n e g l e c t e d . As the o b j e c t i v e

i s t o compare d e f l e c t i o n - t o - w e i g h t r a t i o s r e s u l t i n g from

changes i n the geometry o f the m i r r o r s u b s t r a t e , the e r r o r

i n t r o d u c e d by t h i s approx imat ion i s l i k e l y to be i n s i g n i f i ­

c a n t .

6 4

Figure 3.8 shows the plate previously considered

with certain material removed so that the substrate proper

remains but an arch transmits i t s body weight to the outer

c i r c u l a r boundary. The structures were analyzed by the

f i n i t e element method, which c l e a r l y demonstrated the e f f e c t

of attendant r e d i s t r i b u t i o n of substrate material on the

f l e x u r a l d e f l e c t i o n patterns. The deflections of the

upper surface of these substrates are shown i n Figures 3.9 and

3.10. For comparison, the d e f l e c t i o n of the related s o l i d

d i s c i s also included. Ratios of the deflections of the

arch models to those of the s o l i d d i s c are plotted i n

Figures 3.11 and 3.12 together with the average d e f l e c t i o n

r a t i o s .

In designing mirror substrate one i s interested i n

achieving a low weight with high s t i f f n e s s . Thus i n determin­

ing the merit of arched substrates compared to s o l i d discs,

a parameter involving both weight and d e f l e c t i o n i s desired.

For t h i s purpose a function c a l l e d 'figure of merit' has

been defined as the r a t i o of the product of weight and

de f l e c t i o n for the s o l i d disc to that for the arched sub­

s t r a t e . Figure of merit values, based on the maximum

def l e c t i o n of the upper surface, for d i f f e r e n t arch models

are shown i n Table 3.1.

65

F i g u r e 3.8 Arched s t r u c t u r e s c o n s t r u c t e d w i t h i n the enve lope o f the s o l i d d i s c o f D/H = 6

F i g u r e 3.9 Comparison o f the upper s u r f a c e d e f l e c t i o n s f o r s o l i d d i s c (1) w i t h t h a t o f a r c h models (1) and ( 2 ) : (a) l a t e r a l d e f l e c t i o n

E = 10.5 x 1 0 6 ; V = 0 . 1 7 ; p = 0.0795

67

F i g u r e

gure 3.10 Upper surface deflections for s o l i d disc (2) and arch model (3):

E = 10.5 x 106; v = 0.17; p = 0.0795

69

Arch Model (2)

1-5

O

(0

F i g u r e 3.11 Comparison o f the upper s u r f a c e d e f l e c t i o n s of a r c h models (1) and (2) as r a t i o s o f the d e f l e c t i o n o f s o l i d d i s c (1) :. (a) l a t e r a l d e f l e c t i o n

E = 10.5 x 1 0 6 ; v = 0 . 1 7 ; p = 0.0795

70

Sol id Disc (1)

Arch Model (2)

Average Deflection 'Arch Model (2) /

" 1 Arch Model (1)

Average Deflection Arch Model (1)

10 o

C 075-j?

o

Q

0 5 -

O

o 3 »

Q

•a

"5 (0

u <

O • warn

OS

r/a 02 5 0 5 0 7 5 10

F i g u r e 3 . H Comparison o f the upper s u r f a c e d e f l e c t i o n s o f a r ch models (1) and (2) as r a t i o s o f the d e f l e c t i o n o f s o l i d d i s c (1).: (b) r a d i a l d e f l e c t i o n

E = 10.5 x 1 0 6 ; v = 0 . 1 7 ; p = 0.0795

71

r/a Figure 3.12 Upper surface deflections of arch model (3)

as r a t i o s of the deflections of s o l i d d i s c (2): ,

E = 10.5 x 10 ; v = 0.17; p = 0.0795

72

A compar ison o f d e f l e c t i o n p a t t e r n s c l e a r l y

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

we ight m i r r o r s u b s t r a t e s . For r i n g suppor t r e a c t i o n s a t

S = 1, the maximum l a t e r a l d e f l e c t i o n w f o r the s o l i d d i s c

i s about two t imes h i g h e r than t h a t o f a r c h models (1) and

( 2 ) . On the o t h e r hand the we ight o f the s o l i d d i s c i s

35.9% and 60.2% h i g h e r than t h a t o f a rch models (1) and ( 2 ) ,

r e s p e c t i v e l y . In terms o f average d e f l e c t i o n r a t i o , the

a r c h models (1) and (2) o f f e r 29.7% and 17.6% r e d u c t i o n i n

w d e f l e c t i o n and the f i g u r e o f m e r i t v a l u e s are 2.84 and

4.6 8 , r e s p e c t i v e l y . For r i n g suppor t r e a c t i o n s a t S = 0 . 5 8 ,

the maximum w d e f l e c t i o n f o r the s o l i d d i s c i s about 2.8

t imes h i g h e r than t h a t of a rch model (3) whereas the we ight

o f the s o l i d d i s c i s 42.3% h i g h e r than t h a t o f a r c h model ( 3 ) .

C o n s i d e r i n g the average d e f l e c t i o n r a t i o , a r ch model (3) o f f e r s

50.39% r e d u c t i o n i n the w d e f l e c t i o n and the f i g u r e o f m e r i t

va lue i s 4 . 8 7 .

From above i t appears t h a t among the d i f f e r e n t

geometr ies c o n s i d e r e d , a r ch model (3) i s the most

advantageous .

I t s h o u l d be noted here t h a t f o r a s t r o n o m i c a l p u r ­

poses the changes i n s l o p e and c u r v a t u r e o f the r e f l e c t i n g

s u r f a c e o f the m i r r o r are o f the main i n t e r e s t . These parameters

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

r e p r e s e n t a t i v e o f the t r u e s i t u a t i o n .

F i g u r e 3.9 suggests t h a t the overhang ing type sub-

s t r a t e , a rch model ( 2 ) , g i v e s r i s e to r e v e r s e c u r v a t u r e .

O b v i o u s l y t h i s has t o be a v o i d e d , p o s s i b l y by p r o v i d i n g a

s u i t a b l e "prop" a t the edge.

TABLE 3.1

FIGURE OF MERIT VALUES

(Maximum w x weight) S o l i d d i s c (2)

(Maximum w x weight) Arch model (3)

(Maximum w x weight) S o l i d d i s c (1)

(Maximum w x weight) A rch model (2)

(Maximum w x weight) S o l i d d i s c (1)

(Maximum w x weight) A rch model (1)

= 4.87

= 4.68

= 2.84

A f i n i t e e lement a n a l y s i s o f a r c h model (2) was done

by p r e s c r i b i n g d i s p l a c e m e n t s a t nodes b and c w i t h r e f e r e n c e

t o the node a . T h i s s i m u l a t e s the presence o f a "prop"

a t the edge. The top s u r f a c e w d i s p l a c e m e n t f o r the a rch

model (2) w i t h a prop at the edge i s shown i n F i g u r e 3 . 9 a .

T h i s i s f o r p r e s c r i b e d d i s p l a c e m e n t s o f 0.0000 88 and

74

0.000112 i n . a t the nodes b and c , r e s p e c t i v e l y . The d i s ­

p lacement o f c r e l a t i v e t o a i s 0.000710 i n . when t h e r e i s

no "prop" (arch model ( 2 ) ) , and 0.000011 i n . when t h e r e i s

no " h o l e , " (arch model ( 1 ) ) . Thus the a n a l y s i s w i t h

s p e c i f i e d d i s p l a c e m e n t o f 0.000112 i n . r e p r e s e n t s a

s i t u a t i o n between a rch models (1) and ( 2 ) . The a n a l y s i s

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

a r c h model (2) can be a v o i d e d by p r o v i d i n g a "prop" a t the

edge. The exac t na tu re o f the d i s p l a c e m e n t s w i l l be governed

by the s i z e o f the " p r o p " .

The geometr ies s t u d i e d here were s e l e c t e d t o

demonstrate the p r i n c i p l e o f a r ch a c t i o n r a t h e r than e s t a b ­

l i s h i t s optimum shape. The s tudy was i n i t i a t e d w i t h a

s imp le s e m i - c i r c u l a r a r c h , so t h a t d u r i n g e x p e r i m e n t a l

i n v e s t i g a t i o n the model can be e a s i l y machined.

Arch model (3) i s the b e s t among the c o n f i g u r a ­

t i o n s s t u d i e d . T h i s i s because the suppor t r e a c t i o n s are

p r o v i d e d c l o s e to the c e n t r e o f g r a v i t y . A l s o , the f l e x u r e

o f the overhang i s reduced by removing i t s m a t e r i a l i n an

a r c h - l i k e manner.

A comment c o n c e r n i n g the C a s s e g r a i n i a n s u b s t r a t e ,

which was a n a l y z e d by the f i n i t e e lement method would be

a p p r o p r i a t e . I t s top s u r f a c e d e f l e c t i o n f o r d i f f e r e n t suppor t

c i r c l e r a d i i i s shown i n F i g u r e 3 .13 . I t i s e v i d e n t t h a t ,

f o r the geometr ies s t u d i e d , the minimum d e f l e c t i o n i s o b t a i n e d

f o r a v a l u e o f S l y i n g between 0.75 and 0 . 6 6 .

Figure 3.13 Top surface d e f l e c t i o n of a Cassegrainian sub­strate as a function of support c i r c l e radius:

E = 1 0 . 5 x 1 0 - 6 ; v = 0 . 1 7 ; p = 0 . 0 7 9 5

3.3.2 D e f l e c t i o n Due t o T h e r m a l G r a d i e n t

The d e f o r m a t i o n o f a m i r r o r s u r f a c e due t o t h e r m a l

e f f e c t s ( c h a n g e s i n a m b i e n t t e m p e r a t u r e o r p r e s e n c e o f

t h e r m a l g r a d i e n t s ) i s o f u t m o s t i m p o r t a n c e as i t may e x c e e d

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

I t i s p o s s i b l e t o t r e a t t h e r m a l e f f e c t s b y t h e same

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

l o a d e d a x i s y m m e t r i c s o l i d s . T h e r m a l e f f e c t s c a n be i n c l u d e d

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

s t r a i n s i n e a c h e l e m e n t w h i l e w r i t i n g t h e s t r e s s - s t r a i n

r e l a t i o n s i n v o l v e d . T h i s l e a d s t o an a d d i t i o n a l t e r m i n

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

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

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

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

i n A p p e n d i x 2.

The f i n i t e e l e m e n t programme d e v e l o p e d f o r t h e d i s ­

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

was m o d i f i e d t o a c c o u n t f o r any t h e r m a l e f f e c t . The

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

by u s i n g t h e r e l a t i o n s h i p ( A 2 . 4 ) .

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

p l a t e s u b j e c t e d t o l i n e a r t e m p e r a t u r e g r a d i e n t T ( z ) = Kz ,

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

i n F i g u r e s 3.14 and 3.15 N o t e t h a t t h e t o p a n d b o t t o m

100

Figure 3.14 E f f e c t of temperature gradient T = 2z on a x i a l d e f l e c t i o n of a thick c y l i n d r i c a l plate; D/H =6, a = 0.2777 xl0~6 in/in°F

78

F i g u r e 3.15 R a d i a l d e f l e c t i o n i n a t h i c k c y l i n d r i c a l p l a t e due t o a temperature g r a d i e n t T = 2 z ;

a = 0.2777 x I O - 6 in/in°F

s u r f a c e s o f the p l a t e assume d i f f e r e n t shapes . T h i s i s due

to the f a c t t h a t , i n the example , the top s u r f a c e undergoes

a temperature change o f 0°F as a g a i n s t the bottom s u r f a c e

temperature change o f 40°F. Fo r a p lane d i s c , a n a b s o l u t e

temperature change w i l l no t induce any v a r i a t i o n i n the

c u r v a t u r e , but i f s u r f a c e s are c u r v e d , a b s o l u t e temperature

change may modi fy the c u r v a t u r e .

I t s h o u l d be noted here t h a t the b o d y - f o r c e - i n d u c e d

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

c o n s t a n t , but t h a t due t o the rma l e f f e c t s w i l l depend on the

magnitude o f the temperature change. A compar ison o f the

l a t e r a l d e f l e c t i o n s ( F i g u r e s 3 . 6 , 3 . 1 4 ) shows t h a t a the rma l

g r a d i e n t o f 2°F/inch can induce d i s p l a c e m e n t s more than f o u r

t imes those due to g r a v i t y . The importance o f e q u a l i z i n g

the temperature w i t h i n a t e l e s c o p e dome-enc losure i s thus

q u i t e a p p a r e n t .

4 . EXPERIMENTAL TECHNIQUES FOR THE ANALYSIS OF BODY' FORCE DEFLECTION

The p r e s e n t p r o j e c t i s e s s e n t i a l l y t h e o r e t i c a l i n

c h a r a c t e r . However, i t was thought a p p r o p r i a t e t o undertake

e x p e r i m e n t a l s t u d i e s to s u b s t a n t i a t e the t h e o r e t i c a l

p r e d i c t i o n s o f r e d u c t i o n i n the weight t o s t i f f n e s s r a t i o

f o r m i r r o r s u b s t r a t e s by removing m a t e r i a l s i n a r c h - l i k e

manner.

E x p e r i m e n t a l i n v e s t i g a t i o n o f body f o r c e induced

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

G r a v i t y induced d i s p l a c e m e n t s a r e , i n g e n e r a l , s m a l l and

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

mode l . Thus the e x p e r i m e n t a l t e c h n i q u e s used are r e q u i r e d to

have s u f f i c i e n t r e s o l u t i o n to i d e n t i f y low o r d e r s t r e s s e s

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

4 .1 Review o f the E x p e r i m e n t a l S t u d i e s f o r M i r r o r S u b s t r a t e

S e v e r a l i n v e s t i g a t o r s have conducted e x p e r i m e n t a l

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

Emerson [15] determined t r u e c o n t o u r s ( i . e . , c o n t o u r s o f

u n i f o r m l y suppor ted p l a t e s i n absence o f bending) and the

bending d e f l e c t i o n cu rves f o r 10 5/8 i n c h d iameter o p t i c a l

81

f l a t s o f f used q u a r t z . Based on the r e l a t i o n s h i p t h a t the

bend ing d e f l e c t i o n v a r i e s i n v e r s e l y as the square o f the t h i c k ­

n e s s , the t r u e contours were o b t a i n e d by u s i n g t h r e e o p t i c a l

f l a t s o f l i k e m a t e r i a l , p r o p e r t i e s and d i a m e t e r . The p l a t e s

were t e s t e d i n p a i r s , a r ranged p a r a l l e l t o each o t h e r and

r e s t i n g , a t the v e r t i c e s o f an e q u i l a t e r a l t r i a n g l e , ove r

two s e t s o f suppor t s l o c a t e d one over the o t h e r . The

a l g e b r a i c sum o f con t our s o f the s u r f a c e s was determined

a long a d i a m e t r i c l i n e , p a r a l l e l t o two o f the s u p p o r t s ,

u s i n g a P u l f r i c h v i e w i n g i ns t rument [ 3 5 ] . The t e c h n i q u e

g i v e s q u i t e a c c u r a t e r e s u l t s but i s r e s t r i c t e d t o m i r r o r

s u b s t r a t e s i n the form of a s o l i d d i s c .

In t h i s l i n e o f s t u d i e s the s t e r e o s c o p i c approach due

to Gates [36]shou ld be ment ioned . The method r e l i e s on

the use of two i n t e r f e r o g r a m s i n which the ang le between

the i n t e r f e r i n g s u r f a c e s i s o f o p p o s i t e s i g n , bu t the

s p a c i n g and p o s i t i o n o f the f r i n g e s a r e , as f a r as the

i r r e g u l a r i t i e s o f the s u r f a c e s w i l l a l l o w , the same. The

r e l a t i v e d i s p l a c e m e n t s o f the bands a t c o r r e s p o n d i n g p o i n t s

o f each o f the p a i r o f i n t e r f e r o g r a m s are then measured

w i t h a double m i c r o s c o p e . The t e c h n i q u e a l l o w s measurement

o f d i s p l a c e m e n t s w i t h a s e n s i t i v i t y b e t t e r than 0.00 2 wave­

l e n g t h . An i m p r e s s i o n o f the r e l i e f o f the s u r f a c e may a l s o

be o b t a i n e d by v i e w i n g the i n t e r f e r o g r a m s i n a s u i t a b l e

s t e r e o s c o p e .

Dew [37, 38] d e s c r i b e d a p r e c i s e method o f d e t e r m i n i n g

g r a v i t a t i o n a l sag and u n d e f l e c t e d f i g u r e o f o p t i c a l f l a t s

82

u s i n g a F i z e a u i n t e r f e r o m e t e r . The u n d e f l e c t e d c o n f i g u r a t i o n

o f a f l a t was determined by t a k i n g the mean o f the 1 f a c e up

f i g u r e 1 and ' f a c e down f i g u r e . 1 The g r a v i t a t i o n a l sag

was o b t a i n e d by measur ing f i r s t l y the f i g u r e o f the f l a t i n

the manner under i n v e s t i g a t i o n and second ly the u n d e f l e c t e d

f i g u r e by the method d e s c r i b e d above. The t e c h n i q u e g i v e s

q u i t e a c c u r a t e r e s u l t s but i s r e s t r i c t e d to s u b s t r a t e s i n

the form o f s o l i d d i s c s o n l y .

Unwin [ 3 9 ] has e s t a b l i s h e d a t e c h n i q u e o f measur ing

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

models by u s i n g Shadow Mo i re t e c h n i q u e . The a c c e n t u a t i o n o f

g r a v i t a t i o n a l d e f o r m a t i o n s was o b t a i n e d by immersing

p o l y u r e t h a n e foam models i n w a t e r . T h i s r e s u l t e d i n a

s u f f i c i e n t number of w e l l - d e f i n e d f r i n g e s to make a n a l y s i s

p o s s i b l e .

The i n v e s t i g a t i o n by Kenny [ 6 ] though e x p l o r a t o r y

i n na tu re shou ld a l s o be ment ioned . He conducted measure­

ments o f f r o z e n d i s p l a c e m e n t s on p h o t o e l a s t i c models o f

m i r r o r s u b s t r a t e s . For t h i n p l a t e s , the t h e o r e t i c a l and

e x p e r i m e n t a l r e s u l t s showed good agreement. However, the

t h i c k p l a t e r e s u l t s e x h i b i t e d c o n s i d e r a b l e d i s c r e p a n c y .

A l s o , the t e s t s d a t a f o r two s i m i l a r t h i c k p l a t e s showed

c o n s i d e r a b l e v a r i a t i o n . The d i s c r e p a n c i e s may be a t t r i b u t e d

to u n r e l i a b l e e x p e r i m e n t a l t e c h n i q u e s and , p o s s i b l y , t o

d i f f e r e n t case h i s t o r i e s o f the mode ls .

83

4.2 Present Experimental Investigations

4 . 2 . 1 Frozen stress p h o t o e l a s t i c i t y coupled with the immersion analogy for g r a v i t a t i o n a l stresses

The frozen stress technique of ph o t o e l a s t i c i t y r e l i e s

on the a b i l i t y of epoxy resins to re t a i n at room temperature

a r e l a t i v e l y large e l a s t i c s t r a i n system induced at a higher

temperature when a suitable temperature cycle i s used

[ 4 0 - 4 2 ] . The technique of "locking" deformations due to

an applied load can be explained by means of the bi-phase

theory. The materials consist of long chain hydrocarbon

molecules, some of which are well bonded together by primary

bonds, whereas a large mass of them are less s o l i d l y held

through shorter secondary bonds. The properties of the

primary bonds do not change appreciably with temperature

whereas the secondary bonds become viscous as the temperature

increases. When a load i s applied at a high temperature the

viscous component carr i e s a very small portion of the loading

(corresponding to i t s low modulus of e l a s t i c i t y ) , the main

portion of the load being car r i e d by the primary bonds. If

the temperature i s lowered gradually to room temperature

while the load i s being maintained, the secondary bonds

become r i g i d again and "locks" the deformation of the primary

bonds. On removal of the load at room temperature the

primary bonds relax to a very small degree, but the main

portion of the deformation i s not recovered. As the "locking"

84

t a k e s p l a c e on a m i c r o s c o p i c s c a l e , the d e f o r m a t i o n and the

accompanying b i r e f r i n g e n c e i s m a i n t a i n e d i n a s l i c e o b t a i n e d

by c a r e f u l sawing o f the mode l , a v o i d i n g the g e n e r a t i o n o f

a p p r e c i a b l e h e a t .

The s l i c e thus o b t a i n e d from a t h r e e d i m e n s i o n a l model

can then be examined under the p o l a r i s c o p e l i k e a two

d i m e n s i o n a l mode l .

As ment ioned e a r l i e r , t h e main d i f f i c u l t y i n the

e x p e r i m e n t a l s tudy of body f o r c e problems i s the low s t r e s s

l e v e l induced i n models much s m a l l e r i n s i z e than the p r o t o ­

t y p e s . T h i s low s e n s i t i v i t y can be overcome by immersing

the model i n a l i q u i d o f h i g h d e n s i t y . The ana logy was

f i r s t shown by B i o t [ 4 3 ] f o r two d i m e n s i o n a l b o d i e s and by

S e r a f i m and Da C o s t a [ 4 4 ] f o r the t h r e e d i m e n s i o n a l c a s e . I t was

deve loped w i t h the assumpt ion t h a t the r a t i o o f the d e n s i t y

of the model t o the d e n s i t y o f the f l u i d was s m a l l enough

so t h a t the s t r e s s e s due to the former are n e g l i g i b l e [ 4 2 ] .

But f o r a c c u r a t e a n a l y s i s the weight o f the model s h o u l d

be taken i n t o a c c o u n t . T h i s was accomp l i shed by Parks e t a l .

[ 4 5 ] . The schemat ic p r o o f o f the immersion ana logy i s shown

i n F i g . 4 . 1 . I t i s c l e a r t h a t the advantage o f immersing

the model i n a heavy l i q u i d i s to i n c r e a s e the response o f

the model by ( K - l ) , where K i s the r a t i o o f the d e n s i t y o f

the l i q u i d i n which the model i s immersed to the d e n s i t y o f

the model m a t e r i a l .

F R E E B O D Y D I A G R A M

LOADING CONDITIONS

0. Boundary Condition*

b. Body Force*

S T R E S S S Y S T E M

L iquid Surface - j

r „ = o

R = reocl ions

b.

f \ r „ = o

R = reocl ions

b.

Liquid Sur face 0. < V - k y y

V O

b.

Y = k /

c t ^ - k / y

•M v » = 0

B.

i< . >\

•u ~-\

i

0. < V - k y y

V O

b.

Y = k /

c t ^ - k / y

•M v » = 0

C.

0. On=0

R = reaction*

b.

Y = - ( k - i ) r Is/)3 r*

D. "K^T"?' "i

a- o- n=o

- J — p = react ions k - l

b.

A. B o d y of density y submerged in a liquid of greater density k y .

B. B o d y , of density k y submerged in a liquid of It* own d e n s i t y .

C. Di f ference of above two , multiple gravi ty .

D. Gravity

4 1 Immersion analogy for g r a v i t a t i o n a l stresses [45]

86

In o r d e r t o o b t a i n the maximum a c c e n t u a t i o n o f

g r a v i t a t i o n a l s t r e s s e s mercury was used as the immersion

l i q u i d . In the s t r e s s f r e e z i n g p rocedure the model immersed

i n mercury has to b e . r a i s e d t o a temperature o f the o r d e r

o f 140°C. T h i s would g i v e r i s e t o some mercury v a p o u r , as

the b o i l i n g p o i n t o f mercury i s 180°C. As mercury vapour

i s r e a d i l y absorbed v i a r e s p i r a t o r y t r a c t and has adverse

e f f e c t s on the human body, arrangements were made i n d e s i g n ­

i n g immersion tank t o p i p e out the vapour and condense i t

i n a t e s t tube h e l d i n l i q u i d n i t r o g e n ( F i g . 4 .2).

The t h r e e - p o i n t suppor ted s o l i d d i s c m i r r o r s u b s t r a t e

and the c a l i b r a t i o n d i s c were loaded i n an oven . The

temperature was r a i s e d a t the r a t e o f 1°C per hour t o

140°C. A f t e r 48 hours a t 140°C, the temperature was lowered

at the r a t e o f 1°C per pour t o room t e m p e r a t u r e . A t the end

o f the l o a d i n g c y c l e the model was removed from the oven and

the d e s i r e d s l i c e s were cu t f o r e x a m i n a t i o n , u s i n g a

diamond impregnated s l i t t i n g wheel w i t h cop ious amount o f

l i q u i d c o o l a n t .

Though a c c e n t u a t i o n o f g r a v i t y i nduced s t r e s s e s was

a c h i e v e d by immersion o f the model i n mercury , i n o r d e r t o

improve the a c c u r a c y and t o i d e n t i f y low o r d e r f r a c t i o n a l

f r i n g e s i t was n e c e s s a r y t o use f r i n g e m u l t i p l i c a t i o n . The

n o v e l f r i n g e m u l t i p l i c a t i o n t e c h n i q u e due t o P o s t [46, 47]

was used i n the e v a l u a t i o n o f p h o t o e l a s t i c r e s u l t s .

8 7

Figure 4 . 2 Schematic diagram showing model i n oven, mercury vapour condensation technique and the location of s l i c e s taken for examination

88

F r i n g e m u l t i p l i c a t i o n i s a whole f i e l d compensat ion

t e c h n i q u e i n which the model i s put between two p a r t i a l l y

r e f l e c t i n g m i r r o r s . One o f the m i r r o r s i s s l i g h t l y i n c l i n e d .

Due to the presence o f p a r t i a l m i r r o r s , the l i g h t rays

t r a v e l back and f o r t h as shown i n F i g . 4.3. I t i s c l e a r

from the f i g u r e t h a t each ray emerges from the m i r r o r i n a

d i r e c t i o n depending on the number o f t imes the r a y has

t r a v e r s e d the mode l . The i n c l i n a t i o n s shown i n F i g . 4.3

are g r e a t l y e x a g g e r a t e d . The ray does not pass through an

unique p o i n t each t ime i t t r a v e r s e s the model . The l e n g t h

o f the l i n e over which the p h o t o e l a s t i c e f f e c t i s averaged

depends on the d i s t a n c e between the m i r r o r s and the ang le

o f i n c l i n a t i o n o f the m i r r o r . As d i f f e r e n t rays are i n c l i n e d

at d i f f e r e n t ang les w i t h r e s p e c t t o the a x i s o f the p o l a r -

i s c o p e , any one o f the r a y s can be i s o l a t e d and o b s e r v e d ,

as shown i n F i g . 4.4, w i t h o u t i n t e r f e r e n c e from o t h e r r a y s .

I f the ray which has passed through the model t h r e e t imes

i s o b s e r v e d , t h r e e t imes m u l t i p l i c a t i o n i s o b t a i n e d .

The f r i n g e m u l t i p l i c a t i o n photographs o b t a i n e d f o r a

r a d i a l s l i c e through a suppor t p o i n t and a s e c t o r s l i c e

c o n t a i n i n g two suppor t p o i n t s , o f the t h r e e - p o i n t suppor ted *

model , are shown i n F i g s . 4 . 5 - 4 . 6 .

An examina t ion o f the f r i n g e photographs i n d i c a t e s

p resence o f severe m o t t l e i n the m a t e r i a l . P h o t o e l a s t i c *

These photographs were taken a t Dr . D. P o s t ' s l a b o r a t o r y . A v e r i l l P a r k , N.Y.

M o d e l

P a r t i a l M i r r o r

P a r t i a l M i r r o r

Figure 4.3 Light r e f l e c t i o n and transmission between two _._ s l i g h t l y i n c l i n e d p a r t i a l mirrors

p Q PM PM Q A

L i g h t S o u r c e F - F i e l d L e n s e s P - P o l a r i z e r PM - P a r t i a l M i r r o r s

Q u a r t e r - W a v e P l a t e s . M - M o d e l A P - A p e r t u r e A - A n a l y z e r

Figure 4.4 P a r t i a l mirrors as employed i n a polariscope for fringe m u l t i p l i c a t i o n

9 0

Figure 4 . 5 M u l t i p l i e d fringe patterns for a r a d i a l s l i c e through a support point

F i g u r e 4.6 M u l t i p l i e d f r i n g e p a t t e r n s through two suppor t p o i n t s

92

m o t t l e i s a random b i r e f r i n g e n c e caused by l o c a l i z e d r a p i d

p o l y m e r i s a t i o n due t o p resence o f "hot s p o t s " when the

m a t e r i a l approaches g e l a t i o n [ 4 8 ] . S i n c e m o t t l e i s

ex t raneous b i r e f r i n g e n c e , i t can be thought of as n o i s e

superposed on p h o t o e l a s t i c s i g n a l [ 4 9 ] . A l t h o u g h the

v i s i b i l i t y o f m o t t l e i n c r e a s e s w i t h m u l t i p l i c a t i o n , the

s i g n a l t o n o i s e r a t i o i s the same f o r the m u l t i p l i e d p a t t e r n

as i t i s f o r the o r d i n a r y i s o c h r o m a t i c v iew.

The m u l t i p l i e d f r i n g e p a t t e r n s ( F i g . 4.5) f o r the

r a d i a l s l i c e show c l e a r f r i n g e s up to 17X. These f r i n g e s

r e p r e s e n t the magnitude o f the d i f f e r e n c e o f the secondary

p r i n c i p a l s t r e s s e s ,

V - °2 = U c -a ) 2 + 4 x 2 ' r z r z

In t h i s case the dominant s t r e s s component i s a , a and

x r z components b e i n g v e r y s m a l l . So the magnitude o f the

imposed a^ 1 ~.°2% ' ^ u e t o kody f o r c e l o a d i n g , i s q u i t e

h i g h compared t o the r e s i d u a l b i r e f r i n g e n c e caused by m o t t l e .

The f r i n g e p a t t e r n s f o r the r a d i a l s l i c e can be i n t e r p r e t e d

f a i r l y a c c u r a t e l y by d i s r e g a r d i n g the l o c a l i z e d e f f e c t s due

t o m o t t l e and drawing a smooth curve through the i s o c h r o m a t i c

f r i n g e s .

On the o t h e r hand, the m u l t i p l i e d f r i n g e p a t t e r n s

f o r the s e c t o r s l i c e ( F i g . 4.6) a re s u b s t a n t i a l l y d i s t u r b e d

by the r e s i d u a l b i r e f r i n g e n c e due t o m o t t l e . The d i f f e r e n c e

93

i n the secondary p r i n c i p a l s t r e s s e s ,

1 3 ( a r - a 0 ) + 4 T r Q

i s expec ted t o be q u i t e s m a l l as magnitudes o f the a and

oa s t r e s s e s a re a p p r o x i m a t e l y o f the same o r d e r and T Q i s u r t)

q u i t e s m a l l . The n o i s e due to r e s i d u a l b i r e f r i n g e n c e thus

becomes v e r y prominent and outweighs p h o t o e l a s t i c s i g n a l

making i t i m p o s s i b l e to a n a l y s e the f r i n g e s . S i m i l a r

phenomenon was observed i n the f r i n g e p a t t e r n s f o r the s u r f a c e

s u b - s l i c e s taken from the r a d i a l s l i c e . However, f o r the

i n t e g r a t i o n o f the s t r e s s - s t r a i n r e l a t i o n s , t o determine

d i s p l a c e m e n t s , s t r e s s components have t o be e v a l u a t e d

a c c u r a t e l y , bo th from the r a d i a l s l i c e and i t s s u b - s l i c e s .

I t s h o u l d be ment ioned here t h a t Parks e t a l . [45]

i n v e s t i g a t e d body fo rce induced s t r e s s e s i n a t h i c k - w a l l ho l l ow

c y l i n d e r u s i n g immersion a n a l o g y . However, they o n l y d e t e r m i n ­

ed the t a n g e n t i a l s t r e s s e s a l o n g the i n s i d e boundary o f a

r a d i a l s l i c e .

T h u s , a l though the a c c e n t u a t i o n o f g r a v i t y s t r e s s e s

by immers ion and i d e n t i f i c a t i o n o f low o r d e r f r a c t i o n a l

f r i n g e s by f r i n g e m u l t i p l i c a t i o n appears to be q u i t e

p r o m i s i n g , the method would demand h i g h q u a l i t y c a s t i n g ,

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

In making c a s t i n g s f o r the p r e s e n t i n v e s t i g a t i o n the

p rocedures l a i d down by Leven [50] and Kenny [51] were

94

f o l l o w e d v e r y c a r e f u l l y . As noted by s e v e r a l i n v e s t i g a t o r s

i n the f i e l d o f p h o t o e l a s t i c i t y , the amount o f m o t t l e p r e ­

sent i n c a s t i n g s i s not p r e d i c t a b l e . I d e n t i c a l c a s t i n g s

produced by i d e n t i c a l methods but a t d i f f e r e n t t imes and w i t h

d i f f e r e n t ba tches o f r e s i n s have e x h i b i t e d v a r y i n g degrees

o f m o t t l e . Hence, u n t i l a r e l i a b l e t e c h n i q u e f o r p r o d u c i n g

h i g h q u a l i t y c a s t i n g s i s e s t a b l i s h e d , i t appears t h a t the

method can not be used f o r the d e t e r m i n a t i o n o f g r a v i t y

induced d i s p l a c e m e n t s .

4.2.2 D e f l e c t i o n study o f p h o t o e l a s t i c models by d i r e c t measurement o f f r o z e n d i s p l a c e m e n t

The p h o t o e l a s t i c method i s u s u a l l y a p p l i e d to a

s i t u a t i o n where a p p r e c i a b l y l a r g e s t r a i n s are induced i n a

model p e r m i t t i n g a c c u r a t e d e t e r m i n a t i o n o f the s t r e s s e s .

D i sp lacement measurement from p h o t o e l a s t i c models i s not

v e r y common.

As p o i n t e d out i n s e c t i o n 4 . 1 , Kenny [ 6 ] has conducted

some p r e l i m i n a r y measurements o f f r o z e n d i s p l a c e m e n t s .

Though, he d i d not get j u s t i f i a b l e r e s u l t s from f r o z e n d i s ­

p lacement measurements o f the t h i c k p l a t e s , i t was d e c i d e d

to undertake e x p e r i m e n t a l d e t e r m i n a t i o n o f f r o z e n d i s p l a c e ­

ments on models i n the form of s o l i d d i s c and

arched dome. T h i s was based on the c o n v i c t i o n t h a t i f

exper iments are per formed w i t h p roper c a r e the t e c h n i q u e

might produce r e s u l t s comparable w i t h the t h e o r y .

95

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

• t r a n s i t i o n t e m p e r a t u r e 1 o f a s o l i d epoxy d i s c , 9 i n c h i n

d iameter and 1.5 i n c h i n t h i c k n e s s , and a r i n g suppor t a t

p e r i p h e r y , showed d i s p l a c e m e n t s t o be o f the o r d e r o f 10~ 4

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

i n c i d e n c e i n t e r f e r o m e t r i c measurement [ 5 2 ] . Hence i t was

d e c i d e d to use the o b l i q u e i n c i d e n c e i n t e r f e r o m e t e r , r e a d i l y

a v a i l a b l e i n the department , f o r the p r e s e n t i n v e s t i g a t i o n .

The development and d e s i g n o f t h i s apparatus i s d e s c r i b e d

by B a j a j [ 5 3 ] .

The procedure f o l l o w e d d u r i n g the measurement o f

the body f o r c e induced d i s p l a c e m e n t s i n p h o t o e l a s t i c

models may be b r i e f l y summarized as f o l l o w s :

i ) The models were s u b j e c t e d to an a n n e a l i n g c y c l e

t o remove the mach in ing s t r e s s e s .

i i ) The s u r f a c e , on which the d e f l e c t i o n measurements

were made, was then ground-wi th an adequate supp ly

o f a l i q u i d c o o l a n t and the i n i t i a l d i s p l a c e m e n t

p a t t e r n was r e c o r d e d by u s i n g the i n t e r f e r o m e t e r ,

i i i ) Now the model was put i n the oven w i t h p roper

suppor t c o n d i t i o n s and s u b j e c t e d to a s u i t a b l e

temperature c y c l e , thus f r e e z i n g the body f o r c e

induced d i s p l a c e m e n t s a t the t r a n s i t i o n tempera ture .

i v ) The f i n a l d i s p l a c e m e n t p a t t e r n was then r e c o r d e d

by p l a c i n g the model aga in i n the o b l i q u e

i n c i d e n c e i n t e r f e r o m e t e r .

96

v) The body f o r c e induced d i s p l a c e m e n t was

deduced by s u b t r a c t i n g the i n i t i a l d i s p l a c e m e n t

from the f i n a l d i s p l a c e m e n t .

F o l l o w i n g t h i s p r o c e d u r e , the f r i n g e photographs f o r

the 'normal* and ' r e v e r s e ' l o a d i n g s were o b t a i n e d f o r the

d i s c w i t h r i n g suppor t a t the p e r i p h e r y ( F i g . 4 . 7 ) . C o r r e s ­

ponding d e f l e c t i o n s are p l o t t e d i n F i g . 4 . 8 , The i n t e r f e r o ­

grams r e p r e s e n t an a r e a i n the form o f an e l l i p s e , one i n c h

minor d iameter and n ine i n c h (d iameter o f the d i s c ) major

d i a m e t e r , compressed o p t i c a l l y i n t o a c i r c u l a r l y observed

f i e l d . I t i s apparent f rom F i g . 4.8 t h a t the d e f l e c t i o n

o b t a i n e d from the normal l o a d i n g show good agreement w i t h the

t h e o r y . On the o t h e r hand the non-symmetr ic d e f l e c t i o n

p a t t e r n o b t a i n e d from the r e v e r s e l o a d i n g shows c o n s i d e r a b l e

d i s c r e p a n c y . T h e o r e t i c a l l y , the normal and r e v e r s e l o a d i n g s

s h o u l d produce the same d e f l e c t i o n p a t t e r n s . T h i s r a i s e s

a doubt as t o the a p p l i c a b i l i t y o f the f r o z e n d i s p l a c e m e n t

t e c h n i q u e i n measur ing s e l f - w e i g h t d e f o r m a t i o n s .

Hence, i n s t e a d o f p u r s u i n g f u r t h e r i n v e s t i g a t i o n s

o f m i r r o r mode ls , i t was d e c i d e d t o conduct t e s t on a

s imp le beam t o g a i n more i n s i g h t i n t o t h i s a s p e c t o f the

p rob lem. The f r i n g e photographs f o r the s i m p l y suppor ted

beam are shown i n F i g u r e 4.9 and the d i s p l a c e m e n t s o b t a i n e d

are r e p r e s e n t e d i n F i g . 4 .10 . I t i s apparent t h a t d i s ­

p lacements a s s o c i a t e d w i t h normal and r e v e r s e l o a d i n g s are

q u i t e d i f f e r e n t from each o t h e r and show a g r e a t d i s c r e p a n c y

F i g u r e 4 . 7 F r i n g e photograph f o r the r i n g suppor ted s o l i d d i s c : (a) normal l o a d i n g

F i g u r e 4 . 7 F r i n g e photograph f o r the r i n g suppor ted s o l i d d i s c : (b) r e v e r s e l o a d i n g

F i g u r e 4 . 9 F r i n g e photograph f o r t h e s i m p l y s u p p o r t e d beam: (a) normal l o a d i n g

F i g u r e 4 . 9 F r i n g e photograph f o r t h e s i m p l y s u p p o r t e d beam: (b) r e v e r s e l o a d i n g

101 w i t h the t h e o r e t i c a l d i s p l a c e m e n t p a t t e r n . I t seems every

t ime the model undergoes a temperature c y c l e , some changes

take p l a c e i n the p h y s i c a l p r o p e r t i e s o f the epoxy

m a t e r i a l s .

The e x p e r i m e n t a l i n f o r m a t i o n a v a i l a b l e so f a r , though

not d i r e c t l y u s e f u l i n a c h i e v i n g the f i n a l o b j e c t i v e , p r o v i d e s

u s e f u l i n s i g h t i n t o the l i m i t a t i o n s o f epoxy models i n the

f r o z e n d i s p l a c e m e n t t e c h n i q u e . Some o f the impor tant

o b s e r v a t i o n s are l i s t e d below:

i ) I t seems t h a t every t ime the model undergoes

a therma l c y c l e the m a t e r i a l p r o p e r t y o f the

epoxy changes . T h i s c o u l d be due t o cont inuous

p o l y m e r i z a t i o n o f the model m a t e r i a l . I t seems

the p o l y m e r i z a t i o n i s never e n t i r e l y complete

though p h o t o e l a s t i c i a n s b e l i e v e t h a t by the

c u r i n g p r o c e s s the model m a t e r i a l i s r e n d e r e d

chemi c a l l y i n e r t .

i i ) As the exper iments were per formed i n the presence

o f a i r , the oxygen can r e a c t w i t h the polymer

groups and change the c h a r a c t e r i s t i c s o f the

model m a t e r i a l near the s u r f a c e . May be by

c o n d u c t i n g exper iments i n an i n e r t atmosphere

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

i i i ) F r oz en s t r e s s p h o t o e l a s t i c i t y i s a w e l l e s t a b ­

l i s h e d t e c h n i q u e . O p p e l , a p i o n e e r i n the f i e l d

o f p h o t o e l a s t i c i t y , has made d i s p l a c e m e n t

measurements on s l i c e s o b t a i n e d from t h r e e -

d i m e n s i o n a l p h o t o e l a s t i c mode ls . These were

f o r e x t e r n a l l y a p p l i e d loads where the d i s ­

p lacements produced by the l o a d i n g were f a r i n

excess o f d i s t u r b a n c e s due t o the c o n t i n u o u s

p o l y m e r i z a t i o n . However, f o r s e l f - w e i g h t loaded

systems the minute d i s p l a c e m e n t s are e a s i l y

out -weighed by these d i s t u r b a n c e s and/or r e a c t i o n s

o f the polymer groups w i t h the oxygen o f the

s u r r o u n d i n g atmosphere.

On the b a s i s o f Kenny 's exper iments w i t h t h i c k p l a t e s

and a s s o c i a t e d d i s c r e p a n c y w i t h the t h e o r y as ment ioned

b e f o r e (exper iments w i t h t h i n p l a t e s s u b s t a n t i a t e d the

t h e o r y ) , i t appears t h a t p o l y m e r i z a t i o n a l s o depends on

s u r f a c e to volume r a t i o .

4 .2 .3 D e f l e c t i o n a n a l y s i s o f a s o l i d d i s c and an arched dome u s i n g c o l d cure s i l i c o n e rubber models

Due t o the d i f f i c u l t i e s p r e s e n t e d by the f r o z e n s t r e s s

and d i s p l a c e m e n t p h o t o - e l a s t i c approaches , i t was d e c i d e d

t o conduct exper iments w i t h c o l d cure s i l i c o n e rubber mode ls .The

o b j e c t here was to s tudy the g ross e f f e c t o f geometry

on body f o r c e induced d i s p l a c e m e n t s .

The low Young's modulus o f the c o l d cure s i l i c o n e

rubber enab les l a r g e s e l f - w e i g h t d e f l e c t i o n s i n the

103 models. The measurement of the self-weight d e f l e c t i o n

requires a technique which does not involve a physical

contact with the model. The Shadow Moire technique i s ,

thus, quite suited for the purpose.

The technique, o r i g i n a l l y proposed by Weller et a l .

[54] and developed by Theocaris [55] may be used to obtain

contour maps of non-specular surfaces of s i g n i f i c a n t a r b i t r a r y

curvature. It has been successfully applied i n studying

the geometry of the b i o l o g i c a l surfaces [56].

Here a g r i d of alternate transparent and opaque l i n e s

i s placed i n a close proximity of the object to be tested.

Collimated l i g h t incident at an angle i casts shadows of

the opaque l i n e s onto the model surface. When the grating

and i t s shadow are viewed together at an angle, o, fringes

i n d i c a t i v e of variable i n t e n s i t y i n the plane of i l l u m i n a t i o n

and observation reveal points of known r e l a t i v e l e v e l as

shown i n F i g . 4.11. I t should be pointed out that the

permissible gap between the grating and the surface i s

d i r e c t l y proportional to the p i t c h . This i s due to

d i f f r a c t i o n around the bars which degrade r e c t i l i n e a r pro­

jec t i o n of shadows at a s u f f i c i e n t distance from the grating.

To minimize th i s e f f e c t , the grating with three l e v e l l i n g

screws was mounted d i r e c t l y on the top of the model. While

taking fringe photographs the l e v e l l i n g screws were slowly

adjusted i n order to obtain fringe patterns as symmetrical

as possible.

104

& Cosec a

Figure 4.11 Formation of Shadow Moire fringes

105

In the r e l a t i o n f o r contour i n t e r v a l ' h ' (F igure

4.11), a p r o v i s i o n f o r a l t e r i n g the s e n s i t i v i t y can be s e e n .

Hence, i n s t e a d o f o b s e r v i n g the f r i n g e s n o r m a l l y , they were

photographed at an angle 0 = 45°. The g r a t i n g used had a

p i t c h o f 200 l i n e s per i n c h and was a r ranged a t i t s normal

p i t c h g r a t i n g .

The models o f s o l i d d i s c and arched m i r r o r s u b s t r a t e s

were p r e p a r e d by p o u r i n g a mix ture o f l i q u i d s i l i c o n e rubber

and a c a t a l y s t i n c a r e f u l l y p r e p a r e d moulds , as suggested

i n the Dow C o r n i n g b u l l e t i n 08-417 [57].

The Young's modulus o f s i l i c o n e rubber was determined

by a p p l y i n g a f i n i t e compress ion on a c y l i n d e r o f s i l i c o n e

rubber [58]. I t was found t o be 160 p s i .

The h i g h l y f l e x i b l e c h a r a c t e r o f the s i l i c o n e rubber

m a t e r i a l p r e s e n t e d some d i f f i c u l t y i n r e a l i z i n g p roper

boundary c o n d i t i o n s . However, by u s i n g the s u p p o r t i n g

t e c h n i q u e as i n d i c a t e d i n F i g u r e 4.12, i t was p o s s i b l e t o

approximate the boundary c o n d i t i o n s used i n the a n a l y s i s

( s e c t i o n 3.3.1) .

In o r d e r t o o b t a i n a p r e c i s e l y f l a t s u r f a c e o f the

mode l , a mould was c a r e f u l l y c o n s t r u c t e d w i t h the accuracy

o f 0.001 i n c h . However, the c u r i n g p rocess r e s u l t e d i n

some s h r i n k a g e o f the model thus l e a d i n g t o an i n i t i a l

c u r v a t u r e o f the s u r f a c e .

In o r d e r t o e l i m i n a t e the u n c e r t a i n t y c o n c e r n i n g

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

Arch Model

Figure 4.12 Support frames for models

107 were measured w i t h r e s p e c t t o t h a t i n i t i a l deformed s t a t e .

As shown by Duncan [59] , the g r a v i t y - i n d u c e d d i s ­

p lacements i n a s o l i d can be e l i m i n a t e d p a r t i a l l y by immersing

i t i n a l i q u i d o f the same d e n s i t y as the s o l i d and f u l l y

i f the s o l i d i s i n c o m p r e s s i b l e (v = 0.5). Fo r s i l i c o n e

rubber v b e i n g 0.5, i t appears t o be an i d e a l m a t e r i a l f o r

t h i s p u r p o s e . The l i q u i d s e l e c t e d was g l y c e r i n e whose

s p e c i f i c g r a v i t y i s 1.2 compared t o 1.18 f o r the model

m a t e r i a l .

The models w h i l e i n t h e i r s u p p o r t frames were put i n

a t ank . G l y c e r i n e was then s l o w l y p o u r e d , a v o i d i n g any

t r a p p e d a i r b u b b l e s , u n t i l i t s l e v e l came c l o s e to the top

s u r f a c e o f the model .

Now the Shadow Moi re f r i n g e p a t t e r n was photographed .

N e x t , the g l y c e r i n e was c a r e f u l l y d r a i n e d and the f r i n g e

photograph was taken a g a i n . These photographs are shown,

r e s p e c t i v e l y , i n F i g . 4.13 a ,b f o r the s o l i d d i s c and i n

F i g . 4.14 a ,b f o r the a rched s u b s t r a t e . As they were taken

at an ang le 0 = 4 5 ° the s u r f a c e s appear as e l l i p s e s .

By s u b t r a c t i n g the i n i t i a l d i s p l a c e m e n t s o f the

models w h i l e i n g l y c e r i n e f rom the f i n a l d i s p l a c e m e n t s i n

a i r , the b o d y - f o r c e induced d i s p l a c e m e n t s were o b t a i n e d .

They are shown i n F i g . 4.15. Note t h a t the maximum w f o r

the s o l i d d i s c i s 1.34 t imes t h a t f o r the a r c h model , On

the o t h e r hand, the we ight o f the s o l i d d i s c i s 35.9 %

h i g h e r than t h a t o f the a r c h mode l . Hence, the f i g u r e o f

1 0 8

109

110

Figure 4.15 Comparison of gravity-induced upper deflections for the s o l i d d i s c with the arch model

surface w that of

I l l

merit, based on the maximum d e f l e c t i o n , i s 2.09, thus

c l e a r l y showing the supe r i o r i t y of the arched substrate.

4.3 A p p l i c a b i l i t y of the Model Results to the Prototype Mirror Substrate Design

In applying model r e s u l t s to the prototype, i t i s

important to account for the d i f f e r i n g values of Poisson's

r a t i o . S i l i c o n e rubber has v = 0.5, whereas the prototype

mirror substrate materials have much lower values of v .

Some of the l i k e l y materials for mirror substrate, such as

fused s i l i c a and beryllium, have Poisson's r a t i o of 0.17

and 0.025, respectively. This thus raises a question re­

garding the a p p l i c a b i l i t y of the model te s t r e s u l t s .

Clutterbuck [60] from experimental and a n a l y t i c a l

studies found the peak stresses for v = 0.5 to be 10% higher

than those for v = 0.3. This was substantiated by Kenny's

[61] analysis which showed experimental investigations with

models of v = 0.5, to predict peak stresses 15% higher for

the beryllium prototype.

In the present case the e f f e c t of v on stress

d i s t r i b u t i o n and displacement i s evaluated using the

a n a l y t i c a l expressions (2.17) and (2.21). The d i s t r i b u t i o n

of r a d i a l stress a r i s shown i n F i g . 4.16. This analysis

shows that with increasing value of v , the peak stresses

also increase whereas the maximum displacement decreases.

Furthermore> i t shows that the maximum displacement predicted

by m o d e l s o f v = 0.49 w o u l d be 28.8% l o w e r f o r t h e f u s e d

s i l i c a ( v = 0.17) p r o t o t y p e . F o r t u n a t e l y , t h i s d o e s n o t

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

on g e o m e t r y .

5. CONCLUSIONS AND RECOMMENDATIONS FOR FUTURE STUDY

5.1 C o n c l u s i o n s

Important f e a t u r e s o f the i n v e s t i g a t i o n and c o n c l u ­

s i o n s based on i t may be summarized as f o l l o w s :

i ) The s e l f - w e i g h t prob lem of the t r a d i t i o n a l c i r c u l a r

d i s c m i r r o r s u b s t r a t e has been i n v e s t i g a t e d and an

a n a l y t i c a l s o l u t i o n , not p r e v i o u s l y noted i n

the l i t e r a t u r e , has been o b t a i n e d .

i i ) The f i n i t e e lement s o l u t i o n , i n terms o f t r i a n g u l a r

r i n g - e l e m e n t s , c l e a r l y demonstrates the s u p e r i o r i t y

o f a r c h - t y p e s t r u c t u r e s over s o l i d d i s c s . T h i s

would make a r c h d e s i g n s p a r t i c u l a r l y s u i t e d i n

the c o n s t r u c t i o n o f an o r b i t i n g t e l e s c o p e where

weight would be one o f the major c o n s i d e r a t i o n s ,

i i i ) The m o d i f i e d f i n i t e e lement programme, capab le

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

shows t h a t the presence o f a s m a l l temperature

g r a d i e n t can produce a p p r e c i a b l e d e f o r m a t i o n s i n

the m i r r o r s u b s t r a t e .

i v ) The f r o z e n s t r e s s p h o t o e l a s t i c i t y exper iment c l e a r l y

i n d i c a t e s i t s l i m i t a t i o n s i n e v a l u a t i n g the body f o r c e

115

induced d i s p l a c e m e n t s . T h i s i s p r i m a r i l y due t o

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

f r e e model m a t e r i a l . F r i n g e m u l t i p l i c a t i o n up t o

17X was s u c c e s s f u l l y o b t a i n e d d u r i n g the p r e s e n t

e x p e r i m e n t s .

v) The f r o z e n d i s p l a c e m e n t s t u d i e s p r o v i d e u s e f u l

i n s i g h t i n t o the l i m i t a t i o n s o f epoxy models i n

measur ing body f o r c e induced d i s p l a c e m e n t s . T h i s

i s due to the phenomenon o f con t inuous p o l y m e r i ­

z a t i o n o f epoxy r e s i n s ,

v i ) The i n v e s t i g a t i o n on c o l d cure s i l i c o n e

models c l e a r l y e s t a b l i s h e s t h e i r

u s e f u l n e s s i n s t u d y i n g s e l f - w e i g h t d e f o r m a t i o n s .

These exper iments show the s u p e r i o r i t y o f an arched

d e s i g n compared t o the d i s c c o n f i g u r a t i o n ,

v i i ) A d i f f e r e n c e i n the P o i s s o n ' s r a t i o would n a t u r a l l y

l e a d to a d i s c r e p a n c y between the model and p r o t o ­

type s t r e s s d i s t r i b u t i o n s . However, i t i s impor tan t

to r e c o g n i z e t h a t t h i s i n no way p r e c l u d e s the

use o f models i n e s t a b l i s h i n g the s u p e r i o r i t y o f

a p a r t i c u l a r geomet r i c c o n f i g u r a t i o n .

5.2 Recommendations f o r F u t u r e Study

The i n v e s t i g a t i o n p r e s e n t e d here aims at demons t ra t ing

the f a v o u r a b l e i n f l u e n c e o f the a r c h as a p p l i e d t o a m i r r o r

s u b s t r a t e d e s i g n . A s e a r c h f o r the optimum a r c h r e p r e s e n t s

116

a l o g i c a l e x t e n s i o n t o the s t u d y .

Fo r a t h r e e - p o i n t suppor t sys tem, the model can be

f u r t h e r m o d i f i e d by removing m a t e r i a l i n the form o f an

a r c h i n the t a n g e n t i a l d i r e c t i o n between suppor t p o i n t s

( F i g . 5 . 1 ) . T h i s would encourage t a n g e n t i a l t r a n s m i s s i o n

o f body f o r c e s t o r e a c t i v e p o i n t s th rough a r c h i n g r a t h e r

than f l e x u r e .

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

l e a d s t o numerous v a r i a t i o n s i n d e s i g n s . Fo r example , a

d e s i g n shown i n F i g . 5.2 appears to be p a r t i c u l a r l y p r o m i s i n g

and needs to be e x p l o r e d f u r t h e r . Here each l i t t l e b l o c k

o f m a t e r i a l i s suppor ted by the a r c h be low. The r a d i a l and

c i r c u m f e r e n t i a l c u t s r e l e a s e t r a n s m i s s i o n o f bend ing i n the

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

o f image due to d i f f r a c t i o n around the edges o f the b l o c k s .

However, a m i r r o r i n the form o f a p l a t e p l a c e d at the top

o f the a rched s u b s t r a t e may c o r r e c t t h i s (do t ted l i n e s i n

F i g . 5 . 2 ) .

The arched forms o f s u b s t r a t e s may p r e s e n t some

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

m i r r o r . There i s a l s o a p o s s i b i l i t y o f g e n e r a l ' q u i l t i n g '

o f the a p e r t u r e due t o t h e r m a l s t r a i n i n g o f s u b s t r a t e s

w i t h v a r i a b l e t h i c k n e s s . These problems need d e t a i l e d

i n v e s t i g a t i o n s .

The p r e s e n t e x p e r i m e n t a l i n v e s t i g a t i o n s show the

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

117

Figure 5.1 Arched substrate form for three-point supports

118

Figure 5.2 Arched substrate form with r a d i a l and circum­f e r e n t i a l grooves

119

o f s e l f - w e i g h t d e f l e c t i o n s . Techn iques t o produce c a s t i n g s

w i t h n e g l i g i b l e m o t t l e may be e x p l o r e d . F r o z e n d i s p l a c e m e n t

measurements s h o u l d be conducted i n an i n e r t atmosphere to

e l i m i n a t e r e a c t i o n o f the polymer groups w i t h the a tmospher i c

oxygen .

For a g e n e r a l t h r e e - d i m e n s i o n a l s u b s t r a t e system a

f i n i t e e lement t e c h n i q u e may prove t o be i d e a l . But computer

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

d i f f i c u l t i e s . However, the use o f i s o p a r a m e t r i c f i n i t e

e lements would h e l p s t r e a m l i n e the p r o c e d u r e . In any case

t h i s g e n e r a l prob lem needs f u r t h e r a t t e n t i o n .

REFERENCES

1. R i t c h e y , G.W. "On Modern R e f l e c t i n g T e l e s c o p e s , " Smi thson ian C o n t r i b u t i o n s to Knowledge, V o l . 34, 1904, pp . l O 1 1 ! ^

2. B a u s t i a n , W.W., "Genera l P h i l o s o p h y o f M i r r o r Support Sys tems ," P r o c e d i n g s o f the O p t i c a l T e l e s c o p e Technology Workshop, M a r s h a l l Space F l i g h t C e n t e r , NASA, 1970, pp . 381-387.

3. Couder , A . , B u l l . A s t r o n . , V o l . 7, 1931, p. 14.

4. Schwes inger , G . , " O p t i c a l E f f e c t o f F l e x u r e i n V e r t i c ­a l l y Mounted P r e c i s i o n M i r r o r s , " J . Op t . Soc . o f A m e r i c a , V o l . 44, 1954, p p . 417-424.

5. M a l v i c k , A . J . , and P e a r s o n , E . T . , " T h e o r e t i c a l E l a s t i c De fo rmat ion o f 4-M D iameter O p t i c a l M i r r o r U s i n g Dynamic R e l a x a t i o n , " A p p l i e d O p t i c s , V o l . 7, No. 6, June 1968, pp . 1207-1212.

6. Kenny, B . , "The D e f l e c t i o n o f O p t i c a l S t r u c t u r e s , " ER-9538, 1968, P e r k i n - E l m e r , C o n n e c t i c u t , U . S . A .

7. L o v e , A . E .H . , A T r e a t i s e on the M a t h e m a t i c a l Theory o f E l a s t i c i t y , 4th e d . , Dover P u b l i c a t i o n s , New Y o r k , T9"44, pp . 455-487, 108-109 .

8. W i l l i a m s , R., " D e f l e c t i o n S u r f a c e o f a C i r c u l a r P l a t e w i t h M u l t i p o i n t S u p p o r t , " 9136, 1968, P e r k i n - E l m e r , C o n n e c t i c u t , U . S . A .

9 . B a s s a l i , W.A . , "T ransverse F l e x u r e o f T h i n E l a s t i c P l a t e s Supported a t S e v e r a l P o i n t s , " P r o c . Cambridge P h i l . S o c , V o l . 53, 1957, p. 728.

10. Vaughan, H . , " D e f l e c t i o n o f U n i f o r m l y Loaded C i r c u l a r P l a t e s Upon E q u i s p a c e d P o i n t S u p p o r t s , " J . of_ S t r a i n A n a l y s i s , V o l . 5, No. 2, 1970, pp . 115-120.

11 . M i c h e l l , J . H . , "The F l e x u r e o f C i r c u l a r P l a t e s , " P r o c . London Math . S o c . , V o l . 34, 1902, p. 223.

12. Duncan, J . P . , " F l e x u r a l R i g i d i t y o f M u l t i p l y Connected P l a t e s , " V o l . 4, 19 58, U n p u b l i s h e d Memoir.

121

13. T imoshenko, S . , and G o o d i e r , J . N . , Theory o f E l a s t i c i t y , 2nd e d . , M c G r a w - H i l l , New York , 1951, pp . 3T9-352, 421-423.

14. A l l e n , D .N. d e G . , C h i t t y , L . , P i p p a r d , A . J . S . and Savern , R . T . S . , "The E x p e r i m e n t a l and M a t h e m a t i c a l A n a l y s i s o f A r c h Dams w i t h S p e c i a l Re fe rence to b o k a n , " P r o c . I n s t . C i v i l E n g i n e e r s , V o l . 5, P t . 1, 1956, pp . 198-258.

15. Emerson, E . B . , " D e t e r m i n a t i o n o f P laneness and Bending o f O p t i c a l F l a t s , " J n l . Res . Na t . B u r . S t a n d a r d s , V o l . 49 , No. 4, Oc tober 1952, pp . 241-24 7.

16 . S o u t h w e l l , R . V . , "Some P r a c t i c a l l y Important S t r e s s Systems i n S o l i d s o f R e v o l u t i o n , " P r o c . Roya l Soc . V o l . 180-A, J u l y 1942, pp . 367-396.

17. A l l e n , D.N. d e G . , and W i n d l e , D.W., "The F i n i t e D i f f e r e n c e A p p r o a c h , " S t r e s s A n a l y s i s , e d i t e d by Z i e n k i e w i c z , O . C . , and H o l i s t e r , G . S . , John W i l e y and Sons, London, 19 65, pp . 3-19.

18. Day, A . S . , "An I n t r o d u c t i o n to Dynamic R e l a x a t i o n , " The E n g i n e e r , V o l . 219, January 1965, pp . 218-221.

19 . O t t e r , J . R . H . , C a s s e l , A . C . C . , and Hobbs, R . E . , "Dynamic R e l a x a t i o n , " P r o c . I n s t . C i v i l E n g i n e e r s , V o l . 35, 1966, pp . 633-656.

20. A p p l , F . J . , and K o e r n e r , D .R . , "Numer ica l A n a l y s i s o f P lane E l a s t i c i t y P r o b l e m s , " P r o c . ASCE, J . Eng . Mech. D i v . , June 1968, pp . 743-751.

21. Massonnet , C . E . , "Numer ica l Use o f I n t e g r a l P r o c e d u r e s , " S t r e s s A n a l y s i s , e d i t e d by Z i e n k i e w i c z , O . C . , and H o l i s t e r , G . S . , John Wi ley and Sons, London, 1965, pp . 198-235.

22. T u r n e r , M . J . , C l o u g h , R.W., M a r t i n , H . C . , and Topp, L . J . , " S t i f f n e s s and D e f l e c t i o n A n a l y s i s o f Complex S t r u c t u r e s , " J . A e r o . S c i . , V o l . 23, 19 56, pp . 805-23.

23. H r e n n i k o f f , A . , " S o l u t i o n o f Problems i n E l a s t i c i t y by the Framework Method ," J . A p p l . M e c h . , V o l . 8, 1941, pp . A169-A175.

24. McHenry, D . , "A L a t t i c e Ana logy f o r the S o l u t i o n o f P lane S t r e s s P r o b l e m s , " J . I n s t . C i v i l E n g r s . , V o l . 21, No. 2, 1941, pp . 59-82

122

25. C l o u g h , R.W., "The F i n i t e E lement Method i n S t r u c t u r a l M e c h a n i c s , " S t r e s s A n a l y s i s , e d i t e d by Z i e n k i e w i c z , O . C . , -and H o l i s t e r , G . S . , John Wi ley and Sons , London, 1965, pp . 85-119.

26. Z i e n k i e w i c z , O . C . , and Cheung, Y . K . , The F i n i t e E lement Method i n S t r u c t u r a l and Continuum M e c h a n i c s , R e v i s e d 1 s t i m p r e s s i o n , M c G r a w - H i l l , London, 1968, p p . 46 -61 , 15-16 .

27. C l o u g h , R.W., and R a s h i d , Y . , " F i n i t e E lement A n a l y s i s o f Ax i symmet r i c S o l i d s , " P r o c . ASCE, Eng . Mech. D i v . , V o l . 91, 1965, pp . 71-85 .

28. Simmons, (5.A., "The Des ign o f L i g h t w e i g h t C e r - V i t M i r r o r B l a n k s , " P r o c e d i n g s o f the O p t i c a l T e l e s c o p e Techno logy Workshop, M a r s h a l l Space F l x g h t C e n t e r , NASA, 1970, pp . 219-239.

29. A u g u s t y n , W., " L i g h t Weight O p t i c a l E l e m e n t s - S t a t e o f the A r t 1960," 5688, 1960, P e r k i n - E l m e r , C o n n e c t i c u t , U . S . A .

30. B a r n e t t , R . L . , "Survey o f Optimum S t r u c t u r a l D e s i g n , " P r o c . SESA, V o l . 24, December 1966, pp . 19A-26A.

31. Duncan, J . P . , " S e l f - W e i g h t Loaded S t r u c t u r e s i n the Context o f L i g h t w e i g h t M i r r o r A p p l i c a t i o n s , " P r o ­ ced ings o f the O p t i c a l T e l e s c o p e Techno logy Workshop, M a r s h a l l Space F l i g h t C e n t e r , NASA, 1970, pp . 257-280.

32. Johnson,• E .W. , "Optimum M i c h e l l F rames , " Ph .D . T h e s i s , The U n i v e r s i t y o f B r i t i s h Co lumb ia , September 1970.

33. Kenny, B . , "A Compar ison o f the S e l f - W e i g h t D e f l e c t i o n s o f S p h e r i c a l S h e l l s and T h i c k C i r c u l a r P l a t e s o f Cons tant T h i c k n e s s , " 9353, 1967, P e r k i n - E l m e r , C o n n e c t i c u t , U . S . A .

34. Ho land , I. , and B e l l , K. , F i n i t e E lement Methods i n S t r e s s A n a l y s i s , T a p i r , Norway, 1969, pp . 1 0 - 1 1 .

35. P u l f r i c h , C , Z. I n s t r . , V o l . 18, 1898, p. 261.

36. G a t e s , J . W . , "The E v a l u a t i o n o f I n t e r f e r o g r a m s by D i sp lacement and S t e r e o s c o p i c Methods , 1 1 B r i t . J . A p p l . P h y s . , V o l . 5, A p r i l 1954, p p . ' 1 3 3 - 1 3 5 .

37. Dew, G . D . , "The Measurement o f O p t i c a l F l a t n e s s , " J . S c i . I n s t r u m . , V o l . 43, J u l y 1966, p p . 409-415.

38. Dew, G . D . , "Systems o f Minimum D e f l e c t i o n Supports f o r O p t i c a l F l a t s , " J . S c i . Ins t rum. V o l . 43, November 1966, pp . 809-811.

123

39. Unwin, W.B., "Simulation of Gravity Induced Defor­mation by Immersion/* 1968, Mech. Eng. Dept., The University of B r i t i s h Columbia, Vancouver, Canada.

40. Frocht, M.M., Ph o t o e l a s t i c i t y , Vols. I and I I , John Wiley and Sons, New York, 1966.

41. Dally, W.J., and Riley, W.F., Experimental Stress Analysis, McGraw-Hill, New York, 1965.

42. Hetenyi, M., Handbook of Experimental Stress Analysis, John Wiley and" Sons, New York, 1950, p. 76 8.

43. Biot, M.A., "Distributed Gravity and Temperature Loading i n Two-Dimensional E l a s t i c i t y Replaced by Boundary Pressures and Dislocations," Trans. ASME, J. Appl. Mech., Vol. 57, 1935, pp. A41-A45.

44. Serafim, J.L.,and Da Costa, J.P., "Methods and Materials for the Study of the Weight Stresses i n Dams by Means of Models," International Colloquium o f Models of Structures, June 19 59, Madrid, pp. 4~9-56.

45. Parks, V.J., D u r e l l i , A.J.,and Ferrer, L,, "Gravi­t a t i o n a l Stresses Determined Using Immersion Techniques," Trans. ASME, J . Appl. Mech., September 1967, pp. 583-590.

46. Post, D., "Isochromatic Fringe Sharpening and M u l t i p l i c a t i o n i n Ph o t o e l a s t i c i t y , " Proc. SESA, Vol. 12, No. 2, 1955, pp. 143-156.

47. Post, D., "Fringe M u l t i p l i c a t i o n i n Three-Dimensional Ph o t o e l a s t i c i t y , " J. Strain Analysis, Vol. 1, No. 5, 1965, pp. 380-388.

48. Cook, R.D., "On the Type of Photoelastic Mottle i n an Epoxy Resin," Proc. SESA, Vol. 21, No. 1, May 1954, pp. 151-152.

49. Post, D., "Photoelastic-fringe M u l t i p l i c a t i o n - F o r Tenfold Increase i n S e n s i t i v i t y , " Experimental Mechanics, Vol. 10, No. 8, August 1970 , pp. 3~0~5-312.

50. Leven, M.M., "Epoxy Resins for Photoelastic Use," Pho t o e l a s t i c i t y , edited by Frocht, M.M., Pergamon Press, New York, 1963, pp. 145-165.

51. Kenny, B., "Stress Distributions i n Thick Axi­symmetric Engineering Components," Ph.D. Thesis, S h e f f i e l d University, 1966

124

52. Duncan, J.P., "Topographical Survey of Curved Aero­dynamic Surfaces," 1970, F i n a l Report to D.R.B., Mech..Eng. Dept., The University of B r i t i s h Columbia, Vancouver, Canada.

53. Bajaj, V.K.,"Design and Development of an Oblique Incidence Interferometer," M.A. Sc. Thesis, The University of B r i t i s h Columbia, 1971.

54. Weller, R., and Shephard, B.M., "Displacement Measure­ments by Mechanical Interferometry," Proc. SESA, Vol. 6, No. 1, 1948, pp. 35-38.

55. Theocaris, P.S., Moire Fringes i n Strain Analysis, Pergamon Press, New York", 1969.

56. Duncan, J.P., Crofton, J.P., Sikka, S., and Talapatra, D., "A Technique for the Topographical Survey of B i o l o g i c a l Surfaces," Med, and B i o l . Engng., Vol. 8, 1970, pp. 425-426. "

57. Dow Corning B u l l e t i n : 08-417, 1969, Dow Corning Silico n e s Inter-America Ltd.

58. Vaughan, H., "The F i n i t e Compression of E l a s t i c S o l i d Cylinders i n the Presence of Gravity," Proc. Roy. Soc. Lond., Vol. A321, 1971, pp. 381-396.

59. Duncan, J.P., "The Elimination of Gravity-Induced Displacements i n An E l a s t i c S o l i d by Immersion i n An Ideal F l u i d , " Unpublished typescript, 1967, Mech. Eng. Dept., The University of B r i t i s h Columbia, Vancouver, Canada.

60. Clutterbuck, M., "The Dependence of Stress D i s t r i b u ­tions on E l a s t i c Constants," B r i . J n l . of Appl. Physics, Vol. 9, August 1958, pp. 323-329.

61. Kenny, B., "The E f f e c t of Poisson's Ratio on Stress D i s t r i b u t i o n s , " The Engineer, Vol. 218, October 1964, pp. 706-712T"

APPENDIX 1

FINITE ELEMENT ANALYSIS OF AXISYMMETRIC BODY-FORCE

LOADED SOLIDS

A n a l y s i s

For the purpose o f a n a l y s i s the cont inuum i s d i v i d e d

i n t o a f i n i t e number o f e lements by imag inary s u r f a c e s which

i n t e r s e c t r a d i a l s e c t i o n s i n a ne t o f l i n e s (F igure A l . l )

[ 3 4 ] . These e lements are assumed to be i n t e r c o n n e c t e d a t

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

The s t i f f n e s s c h a r a c t e r i s t i c o f an e lement i s d e ­

te rmined by assuming s u i t a b l e d i s p l a c e m e n t f u n c t i o n s . F o r a

t r i a n g u l a r ax i symmetr i c e lement we d e f i n e the s i x d i s p l a c e ­

ment components as (F igure A 1 . 2 ) ,

{<5}E

Wj

w

u. 3

w m

u m

( A l . l )

Figure A l . l

1-3

F i g u r e A1.2 Nodal c o - o r d i n a t e s and d i s p l a c e m e n t components

The functions should be so selected as to s a t i s f y the

compatibility conditions. This i s achieved by assuming

displacements that vary l i n e a r l y i n each d i r e c t i o n . The

edges of the elements would then displace as s t r a i g h t l i n e s ,

and no gaps can develop between them so long as nodal

continuity i s maintained [ 3 3 ] .

Let the displacement functions be represented by

l i n e a r polynominals as,

{A}

a 1

a 2

{T} a 3 V (A1.2)

a 4

a 5

a 6

The constants a's are determined by i n s e r t i n g the nodal co­

ordinates of the element (Figure A 1 . 2 ) ,

1-5

{6}' tc] {A}

w. 1

u.

w . 3

w m

u m

0 . 0 0 1 r i z i

1 r± z i 0 0 0

0 0 0 1 r. z. 3 D

1 r. z . 0 0 0

0 0 0 1 r z m m

1 r z 0 0 0 m m

a.

a.

a.

a .

a .

or, {A} = [ c ] " 1 {6} e (A1.3)

Substituting for {A} i n (A1.2)

{f} = {*} = {T} [ c ] " 1 {6} e = [N] {<5}e (A1.4)

The s t r a i n vector for an axisymmetric system i s

given by [19],

1-6

r z

3w Sz

9u dr

u r

9w , 3u Sr + Tz

D i f f e r e n t i a t i n g ( A 1 . 2 ) ,

{Q}

0 0 0 0 0 1

0 1 0 0 0 0

- 1 - 0 0 0 r r

0 0 1 0 1 0

{A}

a.

a.

a.

a.

ex.

and s u b s t i t u t i n g the va lue o f {A} from ( A l . 3 ) g i v e s ,

{£} = {Q> [ c ] " X { 6 } e = [B] {6Y (A1.5)

Thus s t r e s s e x p r e s s i o n becomes^

1-7

{a} [D] { e } (A1.6)

r z

w h e r e [D] i s t h e e l a s t i c i t y m a t r i x w h i c h f o r i s o t r o p i c m a t e r ­

i a l s i s g i v e n b y ,

[D] = E ( l - v )

(1+v ) ( l-2v)

1-v

v - v

v 1 -v

1

V

1 -v

0

V 1 -v

V 1 -v

1-2V 2(1 - v )

C o n s i d e r now t h e n o d a l f o r c e s

{F}

m./

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

l o a d s d u e t o b o d y f o r c e s {p} a s s o c i a t e d w i t h t h e e l e m e n t .

1-8

From v i r t u a l work considerations [32],

{F} = { [ B ] T {0} dv - I [N] T {p} dv

where integration extends over the volume of the element.

Substituting for {a} and {B} leads t o ,

{F} = ( { [ B ] T [D][B] dv ) { 6 } E -e [N] T {p} dv (A1.7)

with the s t i f f n e s s matrix of the element as,

[K] = / [ B ] T [D][B] dv (A1.8)

and nodal forces given by , ,

{F} = - J [N] T {p} dv (A1.9) P J

By superposing i n d i v i d u a l element properties, the assemblage

[K] and {F} . for the whole system can be obtained. P

1-9

The e q u i l i b r i u m e q u a t i o n s can now be s o l v e d u s i n g the f o l l o w ­

i n g r e l a t i o n s h i p f o r the whole sys tem:

{F> = [K] { 6 } e - {F} ( A L I O ) P

E v a l u a t i o n o f Element S t i f f n e s s M a t r i x and Body F o r c e s

S u b s t i t u t i n g f o r c o n s t a n t s a from (A1.3) i n t o (A l . 2 )

g i v e s the d i s p l a c e m e n t f u n c t i o n s a s :

u ° JK t u i c l + U j C 2 + u m c 3 + ( u i C 4 + u j c 5 + u m C 6 ) r + ( u i c 7

+ u j c 8 + W 2 ]

W = TK I w i C l + W j C 2 + w m C 3 + ( w i C 4 + w j C 5 + W m c 6 ) r + ( w i C 7

+ W j C g + W ^ g ) 2 ] ( A l . l l )

where, A = area of the t r i a n g l e

c, = r . z_ - z . r , c n = z . r - r . z , c~ = r . z 1 3m 3m' 2 l m i m ' 3 1

c . = z . - z , 4 3 m ' c , = z - z . . 5 m i ' c , = z . 6 1

c . = r - r . , 7 m 3 ' c Q = r. - r .

8 1 m ' C 9 = r j

The s t r a i n components are now r e a d i l y o b t a i n e d a s :

t \ e

z

Y r z

1_ 2A

c ? 0 c 8 0 c 9 0

0 c„ 0 c_ 0 c , 4 D 6

0 c 1 Q 0 c x l 0 c 1 2

C 4 C 7 C5 °8 C6 °9

w. 1

u. 1

w . 3

u.

w^ m

u m

where,

C 1 0 = C l / r + c 4 + c 7 z / r

°11 = c *>/ r + °* + ° Q z / ' r 5 ' "8

C 1 2 ~ C 3 / r + °6 + °9 z ^ r

Note t h a t the terms i n s i d e the square b r a c k e t i n (Al

i s the [B] m a t r i x . Now the s t i f f n e s s m a t r i x f o r an

1-11

axisymmetric element, computed from (A1.8) by taking the

volume i n t e g r a l over the whole rin g of material, i s

given by:

[K] •= 2TT / [ B ] T [ D ] t B ] r d r d z

where,

and,

[B] [D] [B] = 1_ 4A

k l l k12 k13 k14 k15 k16 k22 k23 k24 k25 k26

k33 k34 k35 k36 l 5ymme : r i c k44 k45 k46

k55 k56 k66

'11 C7 C7 + C 4 C 4 d 3

'12 c 7 c 4 d 3 + c 7 c 4 d 2 + c 7 d 2 c 1 0

' 1 3 C7 C8 + c 4 C 5 d 3

'14 c,- c—d „ + c,,c»d» + c.c «d„ 5 7 2 11 7 2 4 8 3

'15 C9 C7 + C 6 C 4 d 3

'16 c 6 C 7 d 2 + c7 C12 d2 + C 4 C 9 d 3

'22 c 4 c 4 + 2 c 4 c 1 0 d 2 + c 1 0 c 1 0 + c 7 c 7 d 3

k23 = c 8 C 4 d 2 + c8 C10 d2 + C 5 C 7 d 3

k24 = C5 C4 + c5 C10 d2 + c l l c 4 d 2 + C10 C11 + c 7 C 8 d 3

k25 = C 4 C 9 d 2 + C9 C10 d2 + c 6 C 7 d 3

k26 = C4 C6 + c6 c10 d2 + c4 C12 d2 + C10 C12 + c 7 C 9 d 3

k33 = C8 C8 + C 5 C 5 d 3

k34 = c 5 C 8 d 2 + c 8 C l l d 2 + C 5 C 8 d 3

k35 = C9 C8 + C 6 C 5 d 3

k36 = C6 C8 d2 + c8 C12 d2 + c 9 C 5 d 3

k44 = C5 C5 + 2 c 5 C l l d 2 + C11 C11 + C 8 C 8 d 3

k45 = C5 C9 d2 + C 9 C l l d 2 + C 6 C 8 d 3

k46 = C5 C6 + C 6 C l l d 2 + C5 C12 d2 + C11 C12 + C 8 C 9 d 3

k55 ~ C 9 C 9 + C 6 C 6 d 3

k56 *" C 6 C 9 d 2 + C12 C9 d2 + C 6 C 9 d 3

k66 = C6 C6 + 2 c 6 C 1 2 d 2 + C12 C12 + C 9 C 9 d 3

E ( l - V ) v

and d, = — — ~ , d 9 = , 1 (1-v) (l-2v) ^ 1-v

. . 1.-.2V ' ' d

3 2(1-v)

In evaluating the element s t i f f n e s s matrix,

term by term exact integration was performed.

1-13

Coming t o the body f o r c e s ,

{ F } p = - / [ N ] T {p} dv (A1.9)

Fo r body f o r c e s due t o g r a v i t y o n l y , {p} = { P Q }

I f the body f o r c e s are c o n s t a n t i t can be shown t h a t [ 3 2 ] ,

{F\} - {P } = {F M} = - 2TT r A/3 , where

p 3 P

( r . + r . + r ) r = 1 3 m (A1.13)

A l though the above r e l a t i o n i s not e x a c t , i t i s found to

g i v e q u i t e a c c u r a t e r e s u l t s .

Computer Programme

The s u b r o u t i n e f o r computing element s t i f f n e s s

m a t r i x reads e lement p r o p e r t i e s and n o d a l c o o r d i n a t e s as

i n p u t d a t a and genera tes the m a t r i x f o r each e lement which

i s then s t o r e d i n a tape by the main programme. The

programme then c a l l s the s u b r o u t i n e which forms the assemblage

s t i f f n e s s m a t r i x by s u p e r p o s i n g i n d i v i d u a l e lement e f f e c t s .

1-14

The element nodal forces due to a body force i s then calcu­

lated for each element by using equation (A1.13). Super­

position of r e s u l t s leads to the assemblage nodal forces

due to the body force. The external nodal forces (reactions)

are then added to nodal body forces to obtain the s t r u c t u r a l

load matrix. The load displacements equations for the whole

structure i s then solved to give the nodal displacements

by using the 'banded1 property of the matrix.

APPENDIX 2

EVALUATION OF DEFORMATIONS DUE TO THERMAL EFFECTS

The s t r a i n v e c t o r f o r an i s o t r o p i c m a t e r i a l due to an

average temperature change T i n the e lement i s g i v e n b y :

aT

{ E T } =

aT

aT

( A 2 . 1 )

0

where a = c o e f f i c i e n t o f the rma l e x p a n s i o n .

Fo r an e l a s t i c m a t e r i a l , the s t r e s s e s a t any p o i n t

w i t h i n the e lement are e x p r e s s e d i n terms o f the c o r r e s p o n d i n g

s t r a i n s i n c l u d i n g therma l e f f e c t s by the e l a s t i c s t r e s s -

s t r a i n r e l a t i o n ,

{a} =

r z

= [D] ({e> - U T } ) ( A 2 . 2 )

where [D] i s the e l a s t i c i t y m a t r i x d e f i n e d i n Appendix - 1.

2-2

With therma l e f f e c t s , the f o r c e d i s p l a c e m e n t r e l a t i o n

(Al.. 7) o f Appendix 1 m o d i f i e s to [ 3 2 ] ,

{F} = (/ [ B ] T [D][B] dv) { 6 } e - / [ N ] T { p } dv

- / [ B ] T [ D ] {e T> dv (A2.3)

where, - J [B] [D] {e T } dv r e p r e s e n t s the n o d a l f o r c e due

to t h e r m a l e f f e c t s . I n t e g r a t i n g over the r i n g e lement ,

{ F } T = - 2TT / [ B ] T [ D ] ' { e T ) r dr dz

N o t i n g t h a t {e T > i s a c o n s t a n t ,

{ F } T = - 2TT[B] T [D] {e T> r A

— T

S u b s t i t u t i n g f o r [B] [D] and m u l t i p l y i n g by the v e c t o r

{e } g i v e s ,

2-3

{ F } T = -d^TTr aT

c 7 + 2 c ? d 2

2 d 2 c 4 + 2 d 2 c 1 0 + c 4 + c 1 Q

c 8 + 2 d 2 C 8

2 5 2 11 5 11

c 9 + 2 d 2 c 9

2 c 6 d 2 + 2 d 2 c 1 2 + c 6 + c 1 2

( A 2 . 4 )

where d^ ••• d 3 , c j_ *'* c 9 ' A a re the same as d e f i n e d

i n Appendix 1 and ,

C 1 0 = C ± / * + °7 *^/r + C 4 ' c l l = c 2 / r + c 8 Z / / r + °5

5 1 2 = C 3 / F + C " Z 9 - + C 6 r

The components o f the n o d a l f o r c e f o r each element can

now be c a l c u l a t e d . Once t h i s i s a c h i e v e d , the element-

therma l n o d a l f o r c e s can be superposed t o ' o b t a i n the assemblage

n o d a l f o r c e s f o r the whole s t r u c t u r e .

The computer programme deve loped can now e a s i l y

2 - 4

be modified to incorporate these thermal nodal forces while

forming the s t r u c t u r a l load matrix. The solution of the f i n a l

load displacement equations to obtain the nodal displacements

follows'the same procedure as that used for the body' force.