UBC Theses and Dissertations

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UBC Theses and Dissertations

On the self-weight sag of plate-like structures with application to mirror substrate design Talapatra, Dipak Chandra 1972

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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 measure-ment 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 iga-t ion of. models i n the forms of so l id d isc and arched done to show sup-e 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 = - V 2 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 | [ z 2+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) [4z3(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 Z 3 ( 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 v2 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 sub-strate 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 \ Liquid Sur face 0. < V - k y y V O b. Y = k / c t ^ - k / y •M v » = 0 B. i< . >\ •u ~-\ i 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 circum-f 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 . 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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 . 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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,, "Gravi-t 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 Axi-symmetric 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 Aero-dynamic 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 Measure-ments 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. 

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