UBC Theses and Dissertations

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UBC Theses and Dissertations

Optimum Michell frames Johnson, Eric William 1970

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O P T I M U M M I C H E L L F R A M E S BY ERIC WILLIAM JOHNSON B.Sc. (Eng), London, 1950 M.Sc. U n i v e r s i t y o f A l b e r t a , 1965 A THESIS SUBMITTED IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY i n the Department of MECHANICAL ENGINEERING We accept t h i s t h e s i s as conforming to the r e q u i r e d standard THE UNIVERSITY OF BRITISH COLUMBIA June, 1970 In presenting t h i s thesis i n p a r t i a l f u l f i l l m e n t of the requirements f o r an advanced degree at the University of B r i t i s h Columbia, I agree that the Libr 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 for reference and study. I further agree that permission f o r extensive copying of t h i s thesis f o r s c h o l a r l y purposes may be granted by the Head of my Department or by his representatives. I t i s understood that p u b l i c a t i o n , i n part or i n whole, or the copying of t h i s thesis f o r f i n a n c i a l gain s h a l l not be allowed without my written permission. ERIC WILLIAM JOHNSON Department of Mechanical Engineering The U n i v e r s i t y of B r i t i s h Columbia Vancouver 8 , Canada Date i i ABSTRACT The theory underlying the design of minimum volume space frameworks i s reviewed. Fundamental theorems and r e l a t i o n s proposed and developed by Maxwell, M i c h e l l , Cox, Chan, Hemp, Johnson and Barnett are discussed and examined from a p r a c t i c a l viewpoint. Two dimensional frames, which are close geometric approximations to the t h e o r e t i c a l concepts of M i c h e l l , are defined and formulas established for t h e i r design and complete s o l u t i o n . Computer programmes are established for the analysis of pin jointed trusses having a wide range of parameters. The e f f e c t of changing parameters on the s t r u c t u r a l properties i s discussed. Rigid frames are analyzed by the use of STRUDL, [STRUctural Design Language], a multi-purpose computer programme for the c a l c u l a t i o n of forces and displacements i n r i g i d structures. The e f f e c t s of b i a x i a l stress i n the j o i n t s and other deviations from the t h e o r e t i c a l concept are examined. Comparison i s made with other s t r u c t u r a l designs to e s t a b l i s h the s u p e r i o r i t y and economy of the M i c h e l l design. B i r e f r i n g e n t models are made, using a numerically c o n t r o l l e d m i l l i n g machine, and tested under load i n a polariscope, to confirm the predicted stress l e v e l s i n the members and stress concentrations i n the j o i n t s . Examples are given of p r a c t i c a l a p p l i c a t i o n of Mi c h e l l space frames. The design of a high tension trans-mission tower and of a lightweight astronomical mirror support are considered. A l t e r n a t i v e solutions to both problems are suggested to provide a basis for more det a i l e d design. i v TABLE OF CONTENTS C h a p t e r Page 1 SOME THEOREMS OF OPTIMUM STRUCTURAL DESIGN 1 M a x w e l l ' s Theorem 1 M i c h e l l ' s Theorem 4 C r i t e r i a f o r D e s i g n o f Optimum S t r u c t u r e . . . 8 Volume o f S y m m e t r i c a l S t r u c t u r e s . . . . 9 .Chan's C a l c u l a t i o n o f Optimum S t r u c t u r e Volume 12 Optimum Volume f o r C a n t i l e v e r w i t h End P o i n t L o a d 19 A n a l o g y w i t h S l i p l i n e F i e l d s i n P l a n e P l a s t i c F l o w : 22 S t i f f n e s s o f an Optimum S t r u c t u r e . . . . 24 2 GEOMETRY OF APPROXIMATE STRUCTURES 26 S u p p o r t f a n s 30 M a i n S t r u c t u r e 31 N o t a t i o n 32 Geometry o f S t r u c t u r e 33 L e n g t h s o f Members 35 J o i n t C o o r d i n a t e s 35 D e f l e c t i o n 36 v C h a p t e r Page 3 FORCES IN A STATICALLY DETERMINATE MICHELL STRUCTURE 3 8 F o r c e s Due t o W e i g h t o f Members . . . . 39 F o r c e s i n t h e Members 40 C o n v e n t i o n f o r L o a d A n g l e s a t a T y p i c a l J o i n t 41 S i z e o f Members 47 R e a c t i o n s a t S u p p o r t s 47 4 ANALYSIS OF SOME PIN JOINTED MICHELL CANTILEVERS . 50 S e l e c t i o n o f D e s i g n P a r a m e t e r s 50 S y m m e t r i c a l C a n t i l e v e r s , S i n g l e L o a d . . 54 F a n A n g l e . 56 F i b r e A n g l e 57 Volume Index 57 E f f e c t o f V a r i a t i o n o f Span 66 S y m m e t r i c a l C a n t i l e v e r s , S i n g l e T i l t e d L o a d 6 8 S y m m e t r i c a l C a n t i l e v e r s , M u l t i p l e L o a d s 73 T i l t e d S y m m e t r i c a l C a n t i l e v e r s , 7 4 Skew C a n t i l e v e r s , . 76 5 THE DESIGN OF JOINTS AND OTHER MODIFICATIONS 80 P i n n e d J o i n t s 81 R i g i d J o i n t s 89 v i C h a p t e r Page B i a x i a l S t r e s s Systems i n J o i n t A r e a s 9 8 D e f l e c t i o n o f J o i n t s S u b j e c t e d t o B i a x i a l S t r e s s . ' 102 S e m i - R i g i d J o i n t s . . . . 105 E l a s t i c B u c k l i n g o f Members and Frames 107 S e l e c t i o n o f J o i n t Type and D e s i g n . . . 109 6 COMPARABLE STRUCTURES 113 I n f i n i t e F i b r e A r r a y 115 F i v e and T h r e e F i b r e M i c h e l l C a n t i -l e v e r s . . . 116 Warren T r u s s 116 Two : Bar C a n t i l e v e r 117 C a n t i l e v e r o f P a r a b o l i c S e c t i o n I l l T r i a n g u l a r P l a t e . . . . 118 C a n t i l e v e r s o f U n i f o r m C r o s s S e c t i o n 119 7 MANUFACTURE AND TESTING OF MODELS 121 F i v e F i b r e P i n J o i n t e d C a n t i l e v e r . . . . 122 S o l i d C a n t i l e v e r 126 C a n t i l e v e r W i t h L i g h t e n i n g H o l e s . . . . 127 F i v e F i b r e R i g i d C a n t i l e v e r 129 T h r e e F i b r e R i g i d C a n t i l e v e r s 133 C a l i b r a t i o n o f F r i n g e s i n Terms o f S t r e s s L e v e l 142 v i i Chapter Page 8 TOWER FOR HIGH TENSION TRANSMISSION LINE 145 S p e c i f i c a t i o n f o r Tower Design 146 Three F i b r e C a n t i l e v e r Design 148 Four F i b r e C a n t i l e v e r Design 151 9 DESIGN OF A MIRROR SUBSTRATE 15 3 Geometric Design of C a n t i l e v e r Ribs . . . 156 Symmetrical C a n t i l e v e r Rib 157 Skew C a n t i l e v e r Rib 157 Loads on Ribs and Member S i z e s 159 M u l t i - P o i n t Suspension on S i x Ribs . . . 165 Appendix (Second Volume) A NOTATION FOR STRUCTURES 1 B EQUATIONS GOVERNING GEOMETRY OF STRUCTURE 6 C COMPUTER PROGRAMMES FOR STRUCTURE DESIGN . . 17 D FORCE SYSTEM IN MICHELL CANTILEVERS . . . . 25 E DATA FOR SELECTED MICHELL CANTILEVERS . . . 42 F BIAXIAL STRESS IN JOINTS 75 G DETAILS OF COMPARABLE STRUCTURES 83 H MANUFACTURE OF PHOTOELASTIC MODELS 9 7 J DEFLECTION OF UNIFORMLY LOADED PLATES . . . 118 v i i i L I S T OF FIGURES F i g u r e Page 1.1 F o r c e s a c t i n g on a t y p i c a l s t r u c t u r e . . . . 9 1.2 C u r v i l i n e a r c o o r d i n a t e s y s t e m 12 1.3 L a y o u t o f f i b r e n e t w o r k 16 2.1 P r a c t i c a l a p p r o x i m a t i o n t o a t h e o r e t i c a l optimum s t r u c t u r e 27 2.2 T y p i c a l M i c h e l l c a n t i l e v e r 29 3.1 D e t a i l s o f a t y p i c a l member 39 3.2 C o n v e n t i o n f o r l o a d a n g l e s a t a t y p i c a l j o i n t 41 3.3 F o r c e s a c t i n g a t j o i n t J „ A T . . . 41 3.4 F o r c e s a c t i n g a t t y p i c a l j o i n t , J ^ . . . . 43 3.5 F o r c e s a c t i n g a t t y p i c a l 'A' f a n j o i n t , J a , l . . 44 3.6 F o r c e s a c t i n g a t t y p i c a l 'B' f a n j o i n t , J , 45 l , b 3.7 F o r c e s a c t i n g a t j o i n t 46 3.8 E x t e r n a l e q u i l i b r i u m o f a M i c h e l l c a n t i l e v e r 4 8 4.1 S y m m e t r i c a l c a n t i l e v e r w i t h p o i n t l o a d . . . 55 4.2 V a r i a t i o n o f f a n a n g l e w i t h number o f f i b r e s 58 4.3 V a r i a t i o n o f f i b r e a n g l e w i t h number o f f i b r e s 59 4.4 V a r i a t i o n o f volume i n d e x w i t h number o f f i b r e s 60 4.4a Volume i n d e x f o r s y m m e t r i c a l c a n t i l e v e r s , 5 " 1 6 1 XX Figure Page 4.4b Volume index f o r symmetrical c a n t i l e v e r s , § = 2 62 4.4c Volume index f o r symmetrical c a n t i l e v e r s , £ = 4 6 3 4.4d Volume index f o r symmetrical c a n t i l e v e r s , 5 " 5 • • • • 6 4 4.4e Volume index f o r symmetrical c a n t i l e v e r s , 4.5 V a r i a t i o n of s t r u c t u r e volume w i t h span . . 67 4.6 Symmetrical M i c h e l l c a n t i l e v e r w i t h t i l t e d load 6 8 4.7 V a r i a t i o n of support r e a c t i o n s w i t h angle of t i l t 70 4.8 V a r i a t i o n of s t r u c t u r e volume wit h angle of t i l t 71 4.9 C a n t i l e v e r w i t h loaded j o i n t s 73 4.10 T i l t e d symmetrical M i c h e l l c a n t i l e v e r . . . 75 4.11 T y p i c a l skew M i c h e l l c a n t i l e v e r 77 4.12 V a r i a t i o n of support r e a c t i o n s w i t h degree of skew ' 77 4.13 V a r i a t i o n of f i b r e angle and rad i u s r a t i o w i t h degree of skew . . . 79 5.1 P o s s i b l e end design f o r t y p i c a l p i n - j o i n t e d member 81 5.2a Volumes and j o i n t allowance f o r c a n t i l e v e r s ]j = 1,2 . 85 5.2b Volumes and j o i n t allowance f o r c a n t i l e v e r s 5 - 4 8 6 F i g u r e x Page 5.2c Volumes and j o i n t allowance for cantilevers 5.2d Volumes and j o i n t a l l o w a n c e f o r c a n t i l e v e r s 5.3 T y p i c a l j o i n t i n a r i g i d M i c h e l l framework. . 99 5.4 Mohr's c i r c l e f o r u n i a x i a l and b i a x i a l s t r e s s 100 5.5 D e f l e c t i o n o f M i c h e l l c a n t i l e v e r s due t o u n i a x i a l and b i a x i a l s t r e s s 104 5.6 S e m i - r i g i d j o i n t s i n a M i c h e l l framework . . 106 6.1 C o m p a r a t i v e s t r u c t u r e s 114 7.1 F i v e f i b r e p i n - j o i n t e d M i c h e l l . c a n t i l e v e r . . 123 7.2A S t r e s s p a t t e r n s i n p i n - j o i n t e d M i c h e l l c a n t i l e v e r s u b j e c t e d t o 110 l b l o a d . I n n e r end 123 7.2B S t r e s s p a t t e r n s i n p i n - j o i n t e d M i c h e l l c a n t i l e v e r s u b j e c t e d t o 110 l b l o a d . O u t e r end 125 7.3 ' S o l i d ' c a n t i l e v e r - l o a d 25 l b 126 7.4 C a n t i l e v e r w i t h l i g h t e n i n g h o l e s - l o a d 20 l b 127 7.5 F i v e f i b r e r i g i d M i c h e l l c a n t i l e v e r 130 7.6a F i v e f i b r e M i c h e l l c a n t i l e v e r - J - _ w i t h 27 l b l o a d 7 131 7.6b S t r e s s e s a t j o i n t 132 2 7.7 T h r e e f i b r e M i c h e l l c a n t i l e v e r w i t h r a d i u s f i l l e t s - 25 l b l o a d 134 7.8 J o i n t d e t a i l s i n t h r e e f i b r e M i c h e l l c a n t i l e v e r 135 •7.9 T h r e e f i b r e M i c h e l l c a n t i l e v e r s - 20 l b l o a d . 136 xi F i g u r e Page 7.10 B u c k l e d t h r e e f i b r e M i c h e l l c a n t i l e v e r . . . 138 7.11 T h r e e f i b r e M i c h e l l c a n t i l e v e r w i t h V^" r a d i u s f i l l e t s 140 1 6 7.12 D e t a i l o f s t r e s s p a t t e r n a r o u n d j o i n t J 3 2 1 4 1 7.13 T e n s i l e t e s t s p e c i m e n f o r c a l i b r a t i o n . . . 143 7.14 F r i n g e o r d e r s i n a l o a d e d s p e c i m e n 144 8.1 C l e a r a n c e f o r h i g h t e n s i o n w i r e s 147 8.2 P o s s i b l e d e s i g n f o r a t r a n s m i s s i o n t o wer -I 149 8.3 P o s s i b l e d e s i g n f o r a t r a n s m i s s i o n t o wer -I I 151 9.1 T e n t a t i v e d e s i g n f o r l a r g e m i r r o r s u b s t r a t e 155 9.2 T h r e e f i b r e s y m m e t r i c a l M i c h e l l c a n t i l e v e r r i b .' 158 9.3 T h r e e f i b r e skew M i c h e l l c a n t i l e v e r r i b . . 160 F i g u r e s (Second Volume) A l N o t a t i o n f o r M i c h e l l s t r u c t u r e s 2 A2 Members i n a t y p i c a l p a n e l 3 B l Members i n f a n 6 B2 T y p i c a l q u a d r i l a t e r a l p a n e l 7 B3 J o i n t s i n t y p i c a l M i c h e l l c a n t i l e v e r . . . . 9 B 4 F i v e f i b r e s y m m e t r i c a l c a n t i l e v e r 12 B5 D e f l e c t i o n o f a M i c h e l l f r a m e 15 D l F o r c e s a c t i n g on a M i c h e l l c a n t i l e v e r . . . 28 x i i F i g u r e Page D2 F o r c e s a t j o i n t J.,„ 29 J NN D3 F o r c e s a t a t y p i c a l i n n e r j o i n t J ^ . . . 32 D4 F o r c e s a t a t y p i c a l 'A' f a n j o i n t , J _ 34 a , l D5 F o r c e s a t a t y p i c a l ' B ' f a n j o i n t , J , , 36 l , b D6 F o r c e s a t j o i n t 37 D7 F r e e b o d y d i a g r a m s f o r f o r c e s a t s u p p o r t s 39 F l B i a x i a l s t r e s s e s a t a t y p i c a l j o i n t . . . 76 F2 A p p r o x i m a t i o n t o a b i a x i a l l y s t r e s s e d j o i n t 77 F3 E x t e n s i o n o f a t y p i c a l member 80 G l Warren t r u s s 84 G2 W i l i o t .diagram f o r d e f l e c t i o n o f a Warren t r u s s 86 G3 Two b a r c a n t i l e v e r 87 G4 C a n t i l e v e r o f p a r a b o l i c s e c t i o n 89 G'5 T r i a n g u l a r p l a t e c a n t i l e v e r 91 G6 I-beam c a n t i l e v e r 94 HI The e f f e c t o f r o t a t i o n 101 H2 Geometry o f c u t t e r o f f s e t s 102 H3 O r d e r and d i r e c t i o n o f c u t s f o r m a c h i n i n g a t y p i c a l M i c h e l l c a n t i l e v e r . . 105 J l L o a d e d d i s c and s u p p o r t s . . . . . . . . . . x i i i ACKNOWLEDGEMENTS The author wishes to express h i s g r a t i t u d e to Dr. J.P. Duncan f o r h i s i n v a l u a b l e advice, guidance and encouragement throughout a l l phases of t h i s i n v e s t i g a t i o n . The a s s i s t a n c e rendered by Professor W.O. Richmond, Dr. CR. H a z e l l and Dr. H. Vaughan i n the s o l u t i o n of problems a r i s i n g from the t h e o r e t i c a l a n a l y s i s i s much appreciated, as are the many suggestions toward the s i m p l i f i c a t i o n of the computer programmes made by Mr. S. Sikka. Mr. P. Hurren and Mr. J . Hoar, Chief Technicians, and t h e i r s t a f f , were of great p r a c t i c a l help w i t h the manufacture and t e s t i n g of models. The research was supported by grants from the Na t i o n a l Research C o u n c i l , f o r which thanks are due. Computing f a c i l i t i e s were provided by the U n i v e r s i t y Comput-ing Centre. S p e c i a l a p p r e c i a t i o n i s expressed to Mrs. M. E l l i s who converted my rough notes i n t o the f i n a l typed copy. x i v SYMBOLS A member, and l e n g t h o f member, i n 'A 1 f i b r e a s u f f i x t o i d e n t i f y j o i n t B member, and l e n g t h o f member, i n 'B' f i b r e b s u f f i x t o i d e n t i f y j o i n t C c o n s t a n t c c o n s t a n t a b b r e v i a t i o n f o r c o s 6 i n A p p e n d i x B D s u p p o r t s p a c i n g E s t r a i n e n e r g y , Young's m o d u l u s e d i l a t a t i o n FA, FB f o r c e s i n members A and B f a b b r e v i a t i o n f o r ( 2 r s i n 6 / 2 ) i n A p p e n d i x B h p o s i t i o n v e c t o r I s e c o n d moment o f a r e a o f a c r o s s - s e c t i o n J j o i n t i n M i c h e l l s t r u c t u r e K a p p r e v i a t i o n f o r l a r g e t e r m r e p e a t e d i n s e v e r a l e q u a t i o n s i n A p p e n d i x B L s p a n o f s t r u c t u r e 1 l e n g t h o f members ( M a x w e l l ' s t h e o r e m ) M b e n d i n g moment N number o f f i b r e s i n a M i c h e l l s t r u c t u r e P f o r c e p p i n . d i a m e t e r i n n o n - r i g i d j o i n t s r fan- r a d i u s s a b b r e v i a t i o n f o r s i n 0 i n A p p e n d i x B sA, sB c r o s s - s e c t i o n a l a r e a o f members A and B XV t t h i c k n e s s o f members U d i r e c t i o n v e c t o r , c u r v i l i n e a r c o o r d i n a t e V volume V d i r e c t i o n v e c t o r , c u r v i l i n e a r c o o r d i n a t e W work WA, WB w i d t h o f members WA, WB OJ d e f l e c t i o n x, y, z C a r t e s i a n c o o r d i n a t e s V r i s e o f skew s t r u c t u r e a, 3 c u r v i l i n e a r c o o r d i n a t e s 6 f a n a n g l e e l i n e a r s t r a i n n a r b i t a r y f r a c t i o n ( M i c h e l l ' s theorem) 9 a n g l e between f i b r e s v P o i s s o n ' s r a t i o p r a d i u s o f c u r v a t u r e a n o r m a l s t r e s s T s h e a r s t r e s s ty an a n g l e ty a n g l e between member and C a r t e s i a n a x e s xyi. GLOSSARY Certain words are used i n t h i s work i n a s p e c i f i c sense. These are l i s t e d below together with a few sp e c i a l i z e d terms. A reference i s given to t h e i r more formal introduction i n the body of the paper. Fan Fibre Frame Members ra d i a t i n g from a support point and not forming part of the q u a d r i l a t e r a l network of the main structure, (page 30) A continuous sequence of members joined end to end. These normally originate at a support point and are continuous to the outer boundary. In Figure 2.2, AF, FK and KM constitute one of three f i b r e s r a d i a t i n g from A. (page 27) Any structure, normally but not necessarily two dimensional, made from s t r a i g h t members joi n i n g only at t h e i r ends. The j o i n t s are r i g i d and can transfer bending moments between members. STRUDL (STRuctural Design Language) - A computer routine developed at Massachusetts I n s t i t u t e of Technology for the solution of indetermin-ate structures. (page 89 ) Truss Any structure, normally but not necessarily two dimensional, made from s t r a i g h t members joi n i n g only at t h e i r ends. The j o i n t s can rotate f r e e l y . Bending moments can not be transformed between members. Volume Index Va ( ) This i s the dimensionless quantity which appears as the l e f t hand side of equation 1.35. I t i s used i n the presentation of r e s u l t s to give more general a p p l i c a b i l i t y , (page 57) , The remaining terms, when used as adjectives, have the following meaning:-M i c h e l l The structure (frame or truss) so designated has a geometrical layout approximating to. the t h e o r e t i c a l concepts enunciated by M i c h e l l , as defined i n Chapter 2. (page 2 8) Optimum Symmetric The volume of material i n the structure so i d e n t i f i e d i s to be a minimum. (In other works optimum may be used to mean minimum cost, assembly time or other c r i t e r i a ) . A c a n t i l e v e r i s said to be symmetric i f i t s outer end, L ( J N N ) l i e s on the x axis, which i s perpendicular to the plane of the supports x v i i i and equally spaced between them. Such structures are symmetric about the x axis, (page 30) Skew A skew ca n t i l e v e r i s not symmetric, Figure 2 .2 i l l u s t r a t e s a skew c a n t i l e v e r , which may be regarded as the general case. The symmetric c a n t i l e v e r i s a sp e c i a l type of structure with zero skew. xix INTRODUCTION OPTIMISATION OF STRUCTURES From e a r l i e s t times, man has b u i l t structures of a l l types, for shelter and f o r use i n a g r i c u l t u r e , transport and industry. More recently, economic and t e c h n i c a l pressures have demanded a more r a t i o n a l analysis, rather than the empirical or semi-rational methods which long governed s t r u c t u r a l design. In general terms, structures are designed to support loads acting at designated points i n space and to transfer t h e i r e f f e c t to selected foundation points. In addition to FOUNDATION fc.nfc may be specif^ precise or QS a boondon Figure 1 The General Structure XX these s p e c i f i e d external loads, the structure has also to support i t s own weight and forces such as wind or snow loads applied to i t s surface. Walls, abutments, dams, f i l l s and s i m i l a r structures are i n t r i n s i c a l l y s o l i d and t h e i r shape i s usually well defined by t h e i r purpose. The majority of structures, how-ever, are not s o l i d and may be considered as an assembly of i n d i v i d u a l members so arranged as to r e s i s t the applied forces without f a i l u r e . In most cases, the geometric design of the structure i s not s p e c i f i e d uniquely. Normally the requirement, that c e r t a i n points i n space be connected to support the applied loads can be achieved by any one of an i n f i n i t e range of member configurations. The f i r s t stage of design thus consists of the s e l e c t i o n of a s p e c i f i c geometric layout. Each con-s t i t u e n t member may then be designed to conform with such well established bases as resistance to f a i l u r e or to buckling, as gauged by e l a s t i c or p l a s t i c theories of material behaviour. I t i s at t h i s stage that any of many such design c r i t e r i a , p a r t i c u l a r l y upper bounds on stress or d e f l e c t i o n or lower bounds on natural frequency, may dominate the d e t a i l design. The optimum design i s usually taken to be that which minimises the long term cost of construction and i n s t a l l a t i o n and, frequently, the o v e r a l l l i f e cost, including maintenance and r e p a i r . These costs are complex functions of the amount of material, labour and overhead expenses invested i n design, manufacture and maintenance. Most published work on optimum s t r u c t u r a l design i s concerned with the c r i t e r i o n of minimum volume or weight, which are often indices of cost. In aeronautical, and p a r t i c u l a r l y i n space applications, the requirement of minimum weight often dominates design for tech n i c a l reasons. Here, the cost of f u e l required to accelerate unnecessary weight, and the simultaneous reduction of pay load may overshadow a l l other considerations, even cost of manufacture. In the above discussion i t was t a c i t l y assumed that each structure was sustaining a s i n g l e , constant load system. I t i s more common i n practice for a structure to be subjected at various times to d i f f e r e n t s p e c i f i e d loadings. In such a case, there may be at le a s t two approaches to the preferred design:-a) Maximum Stress Level Design A chosen geometric layout i s solved for i n d i v i d u a l member properties, while s a t i s f y i n g the requirements of force equilibrium, displacement continuity, boundary conditions and permissible working s t r e s s . This i s done for each load condition to which the structure i s to be subjected, y i e l d -ing more equilibrium equations than unknowns. These solutions may then be examined member by member to sele c t the f i n a l design. x x i i This process i s repeated for d i f f e r e n t geometric layouts to determine the optimum minimum volume design of a l l those considered. This procedure i s perhaps of most value for cases where no single loading i s c l e a r l y dominant. b) Uniformly Stressed Dominant Load Design This method i s of p a r t i c u l a r value when a given load-ing condition predominates. A geometric layout i s selected, e i t h e r following previous designs or as a new conception. The member forces and s t r u c t u r a l displacements are then determined by a s t a t i c a l l y determinate analysis for the domin-ant loading. Member properties may then be selected to conform with such design c r i t e r i a as uniform s t r e s s , resistance to buckling, etc. Once member sizes are determined, a s t a t i c a l l y indeterminate analysis ( i f the j o i n t s are r i g i d ) w i l l indicate the magnitude of l o c a l stress concentrations caused by bending and shear forces i n the frame. The structure i s then analysed for each of the other loads which i t i s required to carry. Certain members may be subjected to greater loads under these conditions than by the dominant loading, and t h e i r cross sections w i l l require modification. A f t e r member loadings are rechecked, the procedure i s i t e r a t e d to arrive at a f i n a l design. Other geometric layouts may then be examined i n the same way u n t i l a minimum volume solut i o n i s obtained. X X I A large astronomical mirror i s an example of a dominant loading condition. O p t i c a l designers are f a m i l i a r with the fa c t that such a mirror, when h o r i z o n t a l , i s subjected to a more c r i t i c a l loading than at any other angle. A 1967 thesis by Soosar [21] surveys approaches of t h i s kind to the problem of optimisation with p a r t i c u l a r reference to structures of predetermined form, and exemplifies the method. HISTORICAL BACKGROUND TO OPTIMISATION The p r i n c i p l e s which govern the design of optimum minimum volume structures were enunciated i n a theorem by J.C. Maxwell i n 1890 [1], for simple cases i n which u n i a x i a l l y stressed s t r u c t u r a l members are either a l l i n tension or a l l i n compression. This theorem was then extended by A.G.M. Mic h e l l i n 1904 [2] to frameworks containing both t e n s i l e and compressive members, and subjected to a single load condition. He showed that such frameworks must consist of an i n f i n i t e number of members aligned along a network of mutually perpendicular curved l i n e s springing from the foundation and spanning the domain to the loads. Such a t h e o r e t i c a l structure cannot be manufactured exactly but can be approximated by a series of chords of the curves. The t h e o r e t i c a l model provides a standard to which these xxiv r e a l structures may be compared. Im p l i c i t i n Michell's proof, although not d i r e c t l y stated by him, i s the f a c t that t h i s minimum volume structure i s also more s t i f f than any other structure contained i n the same s p a c i a l domain and subjected to the same loads. Michell*s work seems to have been overlooked for nearly f i f t y years u n t i l H.L. Cox [3], [4], i n 1958 i l l u s t r a t e d i t s a p p l i c a t i o n to some simple design problems. A considerable portion of Cox's l a t e r book [16] was devoted to an analysis of Michell's theorem and i t s p r a c t i c a l a p p l i c a t i o n i n aero-n a u t i c a l and space design. In 1960, A.S.L. Chan [6] discussed c a n t i l e v e r s conforming to the M i c h e l l c r i t e r i a and derived an expression for t h e i r minimum volume. Ghista, [19], [20], has examined c r i t i c a l l y the l i m i t s of a p p l i c a b i l i t y of M i c h e l l structures, and described t h e i r use for some simple loadings. In 1966, R.L. Barnett [9] surveyed optimum s t r u c t u r a l design, describing the works mentioned above, as a sequence of investigations using e l a s t i c design to achieve minimum weight. He also discussed alternate l i n e s of develop-ment using as c r i t e r i a p l a s t i c strength design, f a i l u r e by simultaneous buckling under a l l possible modes and minimum d e f l e c t i o n . He demonstrated that the M i c h e l l structure i s also optimum for t h i s l a s t condition. In 1969 Hegemier and Prager [17] demonstrated by d i f f e r e n t methods Maxwell's and Michell's theorems and proved X X V that the M i c h e l l structure i s also optimum (has minimum volume) for loading with stationary creep and for a given natural frequency. In 1969, Sheu and Prager [18] reviewed developments i n optimum s t r u c t u r a l design between 1962 and 1968. The above paragraphs have reviewed the development of Maxwell's and Michell's theorems and t h e i r a p p l i c a t i o n to the s e l e c t i o n of optimum minimum volume solutions to various s t r u c t u r a l loading problems. In recent years a p a r a l l e l development of the theory has been used to solve problems i n p l a s t i c i t y . In 1960, W.S. Hemp [5] reexamined Michell's work and, for the f i r s t time, demonstrated i t s analogy with s l i p -l i n e f i e l d s i n plane p l a s t i c flow. In material loaded so that i t deforms p l a s t i c a l l y , shear occurs along orthogonal, curved s l i p l i n e s . These form the boundaries of small blocks of the material whose size tends to zero as a continuum i s approached. Along these l i n e s there i s r e l a t i v e movement of the blocks and no l i n e a r s t r a i n . Thus, for a given configuration of boundary t r a c t i o n s , there i s an i n f i n i t e network of s l i p l i n e s . This i s exactly analogous to Michell's i n f i n i t e s t r u c t u r a l network, whose orthogonal c u r v i l i n e a r l i n e s of constant l i n e a r s t r a i n connect r i g i d foundations and loads i n space. In both cases these t h e o r e t i c a l metworks are approximated by a l i m i t e d number of s t r a i g h t l i n e s which are chords to the c u r v i l i n e a r coordinates. xxv i Figure 2 Elementary S l i p Line F i e l d f or Extrusion Through a Die (after Johnson [23]) Consider for example, p l a s t i c flow of material i n a die with p e r f e c t l y rough, plane walls. There w i l l be no r e l a t i v e motion between the walls and the adjacent material, or along l i n e s at r i g h t angles to the walls. These are the p r i n c i p a l d i r e c t i o n s i n the material being deformed. The mutually perpendicular s l i p l i n e s — d i r e c t i o n s of r e l a t i v e movement between elements of the m a t e r i a l — w i l l be i n c l i n e d at 45° to these p r i n c i p a l d i r e c t i o n s and thus to the die surface and w i l l spread out i n an i n f i n i t e network compatible with other boundary and loading conditions. This corresponds exactly with the layout of the members forming a M i c h e l l i n f i n i t e f i b r e structure. These are i n c l i n e d at 45° to the surface of any r i g i d foundation x x v i i subjected to a d i s t r i b u t e d load. They form an orthogonal c u r v i l i n e a r network which spans the domain and extends out-ward to support the applied loads. The outer f i b r e s define the boundaries of the s t r u c t u r a l domain. In studies of p l a s t i c i t y the t h e o r e t i c a l s l i p l i n e f i e l d s are approximated by graphical solutions, using a coarse g r i d of s t r a i g h t l i n e s l a i d out according to rule i n a regular manner. These layouts are exactly analogous to those of r e a l structures approximating the t h e o r e t i c a l , minimum volume, i n f i n i t e f i b r e arrays for associated load-domain- foundation arrangements. This.approximation w i l l be further explained i n this" t h e s i s . In 1961 W. Johnson [8] extended t h i s analogy. He demonstrated that i n addition to the correspondence between s t r u c t u r a l layout and s l i p l i n e s i n the t h e o r e t i c a l case, the v e l o c i t y diagram of the s l i p l i n e f i e l d i s i d e n t i c a l i n shape with the force diagram of the structure i n the approximate solutions. Bow's notation may be used i n both cases when using graphical solutions. This correspondence may be used to obtain upper bound estimates of the volume of optimum frames from known s l i p l i n e f i e l d s or vice versa. In a book by Johnson, Sowerby and Haddow [15] t h i s analogy i s restated and i l l u s t r a t e d with examples of s l i p l i n e f i e l d s generated by extrusion through rough dies of various shapes. In p a r t i c u l a r , extrusion through a rough 45° wedge shaped die produces a s l i p l i n e f i e l d i d e n t i c a l x x v i i i with the base fans of the cantilevers described l a t e r i n the previous work. The main structure of these cantilevers i s suggested by the s l i p l i n e f i e l d s for compression of a slab between p a r a l l e l plane rough plates. In 1969 a paper by W. Johnson [23] described the use of these analogies for the solution of plate-bending and other problems. V e l o c i t y diagrams from s l i p l i n e f i e l d s are used to describe kinematic modes of deformation to obtain upper bounds for the collapse of transversely loaded pla t e s . Outline Plan of Study The present work concentrates on cantilevers as a basic element i n simple and compounded s t r u c t u r a l forms. For example, as Chan demonstrated [6] four ca n t i l e v e r s may be combined to form a beam capable of r e s i s t i n g pure bending. The major parameters—span, number of members, p o s i t i o n and type of l o a d i n g — a r e varied to determine t h e i r r e l a t i v e e f f e c t s . Formula are derived from elementary theory and computer programmes described which may be used for the so l u t i o n of c a n t i l e v e r s of any desired s i z e . The r e a l structures discussed are approximations to the corresponding t h e o r e t i c a l optimum frames, and t h e i r design introduces secondary problems not considered i n the i d e a l case. The study described i n t h i s thesis could be considered as an exploration of the degradations of the i d e a l structure for a given domain and loading system. These are i n e v i t a b l y incurred as t h i s i d e a l i s modified to s u i t f e a s i b l e methods of design, construction and use. In p a r t i c u l a r , j o i n t design raises several problems. If uniform u n i a x i a l stress d i s t r i b u t i o n i s to be s a t i s f i e d , the j o i n t s should be pinned, requiring extra material and ca r e f u l p r o f i l i n g to achieve a smooth flow of str e s s . Rigid j o i n t s introduce areas of b i a x i a l stress which modifies the s t r u c t u r a l d e f l e c t i o n . A d d i t i o n a l material i s required i f the j o i n t area i s increased to maintain a uniform stress l e v e l . In eithe r case, the r i g i d j o i n t induces secondary bending and shear stresses. S t r u c t u r a l members are often assumed to be weightless f o r elementary analysis. I f considered, the s e l f weight forces are introduced as concentrated loads at the end j o i n t s . This s i m p l i f i c a t i o n ignores the bending and shear stresses generated by the s e l f weight forces. These secondary e f f e c t s were examined using computer programmes (STRUDL) developed at Massachusetts I n s t i t u t e of Technology [ 2 2 ] . Individual members subjected to compressive loading may f a i l by buckling. The structures considered i n t h i s work were examined for t h i s p o s s i b i l i t y . The Euler c r i t i c a l load for each member, assumed to have free ends, was taken as a lower bound c r i t e r i o n . In most cases where f a i l u r e by buckling was possible, redesign of the cross section or a s l i g h t increase i n the amount of material used provided a s a t i s f a c t o r y s o l u t i o n . Buckling of the whole structure, i f made as a t h i n plate, may be prevented by the use of two p a r a l l e l systems, suitably cross connected, i n exactly the same manner as more conventional trusses and frames are designed. A series of models of t y p i c a l M i c h e l l structures were tested p h o t o e l a s t i c a l l y . The fringe patterns observed corresponded c l o s e l y with the stress d i s t r i b u t i o n predicted by theory. The stress l e v e l was found to be sensibly uniform along the greater part of the length of each member and between members. Local e l a s t i c buckling of the t h i n models could account for the observed discrepancies. J o i n t areas subjected to b i a x i a l stress exhibited stress concentrations of the same type as predicted by the t h e o r e t i c a l analysis. The i n v e s t i g a t i o n demonstrates that p r a c t i c a l forms of r i g i d jointed M i c h e l l structures have volumes which approach those of the corresponding t h e o r e t i c a l optima solutions for a s p e c i f i e d load-domain-foundation condition. I t i s probable that they w i l l provide the c l o s e s t p r a c t i c a l s o l u t i o n , since t h e i r basic geometric layout i s already known to be optimum. Since the types of degradation referred to above are associated with j o i n t s which occur i n a l l designs, the conversion of any geometric layout to a p r a c t i c a l design involves some increase i n volume from that calculated xxx i by an elementary analysis. At the l e a s t , the i d e a l M i c h e l l solution i s l i k e l y to be an excellent f i r s t approximation and provides a s t a r t i n g point for further computer analysis, as described for example i n the l a s t section of the review by Barnett [9]. Minimum volume structures have a wide ap p l i c a t i o n to many f i e l d s of design. They show to best advantage where minimum weight i s of paramount importance. They should be considered for use i n a i r c r a f t and space ve h i c l e s , where t h e i r shape permits, the M i c h e l l layout being inherently deeper i n proportion to span than most other truss designs. In t h i s connection lightening holes have been used i n airframe and ship r i b s for many years i n an empirical manner. Mi c h e l l layouts could provide guidelines for t h e i r more e f f e c t i v e design by "channeling" the s t r u c t u r a l s t r a i n into mutually perpendicular d i r e c t i o n s . In more s t a t i c applications, economy i n material can show savings i n i n s t a l l a t i o n i n remote areas, where transport costs can be s i g n i f i c a n t . A possible design for a high tension transmission tower i s included as an i l l u s t r a t i o n of t h i s a p p l i c a t i o n . These f a m i l i a r structures are designed to support known loadings at fi x e d points i n space while providing su i t a b l e clearance for e l e c t r i c a l i n s u l a t i o n . I t i s increasingly common for such towers to be s i t e d i n remote areas, often being placed on s i t e by h e l i c o p t e r . Minimum weight i s obviously of advantage. x x x i i A m o r e s p e c i a l i s e d a p p l i c a t i o n o f t h e M i c h e l l f r a m e w o r k c o n c e r n s s u p p o r t s y s t e m s f o r l a r g e m i r r o r s . T h e s u b s t r a t e s e r v e s o n l y t o s u p p o r t t h e r e f l e c t i n g s u r f a c e a n d t o m a i n t a i n i t s s h a p e a s t h e m i r r o r a s s u m e s v a r i o u s a t t i t u d e s . T h e d e s i g n c r i t e r i o n i s m i n i m u m d e f l e c t i o n o f t h e m i r r o r s u r f a c e a n d t h e s e l f w e i g h t f o r c e s f o r m t h e l a r g e r p a r t o f t h e s t r u c t u r a l l o a d i n n o r m a l u s e . T h e M i c h e l l l a y o u t i s p a r t i c u l a r l y s u i t a b l e f o r d i s t r i b u t i n g t h e p r i m a r y r e a c t i o n s o v e r t h e s u b s t r a t e , s i n c e i t i s o p t i m u m f o r a g i v e n d e f l e c t i o n a n d t h i s i n t u r n r e d u c e s t h e s e l f w e i g h t l o a d i n g s c a u s i n g t h e d e f o r m a t i o n . T h e t o l e r a b l e d e f l e c t i o n s o f t h e m i r r o r s u r f a c e — l i m i t e d b y o p t i c a l r e q u i r e m e n t s — i m p l y a g e n e r a l l y l o w l e v e l o f s t r e s s i n t h e s t r u c t u r e , w h i c h h a s a d e q u a t e r e s e r v e t o r e s i s t t h e d i f f e r e n t l o a d i n g s i n c u r r e d d u r i n g m a n u f a c t u r e a n d t r a n s p o r t t o s i t e — t e r r e s t r i a l o r i n s p a c e . T h e p r e s e n t w o r k p r o v i d e s t h e o r e t i c a l a n d e x p e r i m e n t a l e v i d e n c e o f t h e u n i f o r m d i s t r i b u t i o n o f s t r e s s i n M i c h e l l f r a m e w o r k s . T h e s e h a v e s i g n i f i c a n t l y l e s s v o l u m e t h a n m o r e f a m i l i a r s t r u c t u r a l f o r m s s u b j e c t e d t o t h e same l o a d i n g a n d maximum s t r e s s . W h i l e o f u n f a m i l i a r s h a p e , t h e f a b r i c a t i o n o f M i c h e l l s t r u c t u r e s i s s t r a i g h t f o r w a r d a n d g e n e r a l l y s i m i l a r t o t h a t o f t h e i r c o m p e t i t o r s . T h e i r u s e i n many f i e l d s w i l l y i e l d s a v i n g s i n m a t e r i a l a n d , f o r d y n a m i c a p p l i c a t i o n s , c o n t i n u e d e c o n o m i c s i n o p e r a t i o n . 1 CHAPTER 1 SOME THEOREMS OF OPTIMUM STRUCTURAL DESIGN MAXWELL'S THEOREM In 1890 C l e r k M a x w e l l p u b l i s h e d a theorem [1] which s p e c i f i e d the t o t a l volume o f a framework s u b j e c t e d t o f o r c e s imposed by an e x t e r n a l f o r c e system. T h i s paper was an e a r l y attempt t o d e f i n e optimum s t r u c t u r e s . The p r o o f i s s h o r t and i s r e p r o d u c e d here w i t h minor changes f o r c o n s i s t e n c y i n symbols. A framework i s composed of members c a r r y i n g e i t h e r c o mpressive o r t e n s i l e f o r c e s . I t i s i n e q u i l i b r u m under the a c t i o n of a system of f o r c e s P , a c t i n g a t p o i n t s i n space, whose p o s i t i o n v e c t o r s are h"N, r e f e r r e d t o a conven-i e n t o r i g i n , 0. L e t the framework be s u b j e c t e d t o an i n f i n i t e s i m a l v i r t u a l d i s p l a c e m e n t , c o n s i s t i n g o f a u n i f o r m d i l a t i o n o f space of magnitude 3e. T h i s i s so a p p l i e d t h a t the c o -o r d i n a t e o r i g i n , 0, i s u n d i s t u r b e d . E v e r y l i n e a r element o f space i s thus extended by a s t r a i n e. Then -work done by a p p l i e d f o r c e s = . . . . (1.1) , t S t r a i n e n e r g y s t o r e d b y f r a m e = Q w h e r e F r e p r e s e n t s t h e f o r c e i n a member a n d I i t s l e n g t h . T h e s u f f i c e s t a n d c i n d i c a t e r e s p e c t i v e l y t h e * t e n s i l e a n d c o m p r e s s i v e m e m b e r s . B y t h e p r i n c i p l e o f V i r t u a l W o r k , t h e e n e r g i e s e x p r e s s e d b y t h e a b o v e e q u a t i o n s a r e e q u a l . T h u s (1.2) t = l C s > M a i S u p p o s e t h a t t h e c r o s s - s e c t i o n o f e a c h member i s s o p r o p o r t i o n e d t h a t t h e s t r e s s i n a l l t e n s i l e m e m b e r s i s e v e r y w h e r e a ^ t a n d t h a t i n t h e c o m p r e s s i v e m e m b e r s i s e v e r y w h e r e a . T h e n • % • (1.3) C v c w h e r e s r e p r e s e n t s t h e c r o s s - s e c t i o n a l a r e a o f a member a n d V t , V c a r e t h e t o t a l v o l u m e s o f t h e t e n s i l e a n d c o m p r e s -s i v e m e m b e r s r e s p e c t i v e l y . T h e t o t a l v o l u m e o f t h e s t r u c t u r e V , i s t h u s F^ _ a n d F c r e p r e s e n t t h e magnitudes o f t h e s e f o r c e s , t h e d i f f e r e n c e i n t h e i r e f f e c t b e i n g a l l o w e d f o r b y t h e s i g n o f t h e c o m p r e s s i v e t e r m , f o l l o w i n g t h e u s u a l c o n v e n t i o n t h a t t e n s i l e f o r c e s a r e p o s i t i v e . V=V t +V c 3 . . . . (1.4) I f equations 1.3 are s u b s t i t u t e d i n t o 1.2 (1.5) The c o n s t a n t C i s a f u n c t i o n of the l o a d i n g imposed on the s t r u c t u r e and i s independent of the way i n which the framework i s c o n s t r u c t e d . Equations 1.3 and 1.5 may be s u b s t i t u t e d i n t o 1.4 to y i e l d I t should be noted t h a t the second term i n equations 1.6 and 1.7 i s independent of the d e s i g n o f the framework. C i s a f u n c t i o n of the e x t e r n a l l o a d i n g w h ile o*. and a are 3 t c a r b i t r a r y uniform s t r e s s e s . A minimum volume s t r u c t u r e c o n d i t i o n s may thus be deduced from equation 1.6 by c o n s i d e r i n g the minimum va l u e (1.6) (1.7) where of the f i r s t term alone. Maxwell c o n s i d e r e d the simple s o l u t i o n to t h i s o b tained by p l a c i n g e i t h e r V or V equal to zero. Such s t r u c t u r e s would c o n t a i n o n l y t e n s i o n members or o n l y compression members. Examples of these are a) t i e s or s t r u t s (and ropes) s u b j e c t e d t o equal and o p p o s i t e a x i a l f o r c e s , b) t r i a n g u l a r or t e t r a h e d a l frames w i t h f o r c e s a p p l i e d a t t h e i r v e r t i c e s and a c t i n g along l i n e s which i n t e r s e c t w i t h i n the frame, c) c a t e n a r i e s and arches. Such s t r u c t u r e s may be c o n s i d e r e d as t r i v i a l f o r the presen t i n v e s t i g a t i o n which i s concerned with a minimum volume s t r u c t u r e c o n t a i n i n g both t e n s i l e and compressive members. For such 'mixed' frames, Maxwell!s theorem had no d i r e c t use, and i t r e c e i v e d l i t t l e a t t e n t i o n f o r some year MICHELL'S THEOREM Maxwell's theorem, as expressed by equation 1.5, was g e n e r a l i z e d by A.G.M. M i c h e l l i n 1904 [2] and extended to cover 'mixed' frameworks. Again the proof i s s h o r t and i s repeated below i n m o d i f i e d form Consider again the same g e n e r a l i z e d f o r c e system as i n the Maxwell theorem, but a p p l i e d i n t u r n to a s e r i e s of d i f f e r e n t frames occupying a c e r t a i n space (which may be bounded or i n f i n i t e ) . 5 Let t h i s space, and the truss enclosed, undergo an ar b i t r a r y deformation such that no l i n e a r element of the space suffers an extension or contraction numerically greater than T){SZ) , where ( 6£ ) i s the length of the element and n i s an a r b i t r a r y small f r a c t i o n . [Note that t h i s i s an ar b i t r a r y deformation and need not be uniform as required by Maxwell's theorem.] In a t y p i c a l truss, A, containing N members, a member of length w i l l undergo a small change of length If the force acting on t h i s member i s ( a x i a l l y ) , then the increase i n s t r a i n energy stored i n the framework during the deformation i s c l e a r l y The work done by the external forces during t h i s deformation i s S^. I t i s independent of the shape of the framework and equals the change i n s t r a i n energy stored. Remembering that by d e f i n i t i o n a l l | e, | <_ n, then . • ( 1 . 9 ) 6 Suppose, however, there e x i s t s a truss, M, i n which a l l members are subjected to a uniform s t r a i n equal to n. Then, i n exactly the same way as i n 1 . 9 . . . -. ( 1 . 1 0 ) a 9 e a ( 1 « 1 1 ) Equation 1 . 7 demonstrates that i f the volume of a structure i s to be a minimum, the term n [ Z ( F l )J n=l n n must be a minimum, since a l l other variables i n t h i s equation are independent of the framework. Thus Equation 1 - 1 0 i ndicates that of a l l the possible frames that may be used, that having uniform s t r a i n i n a l l members w i l l have the minimum volume. This i s equivalent to specifying uniform stress i n a l l members for frameworks manufactured from i s o t r o p i c materials and subjected to small deformations. Unless s p e c i f i c a l l y mentioned, i t w i l l be assumed that the stresses in t e n s i l e and compressive members are numerically equal . . . . ( 1 . 1 2 ) The preceding proof s p e c i f i e d no l i m i t on the size of the space to be occupied by the frameworks under comparison. 7 I f t h e s p a c e b e c o n s i d e r e d a s i n f i n i t e , f r a m e s s a t i s f y i n g t h e c o n d i t i o n o f u n i f o r m s t r a i n w i l l h a v e a v o l u m e l e s s t h a n a n y f r a m e h a v i n g u n e q u a l s t r a i n s i n i t s m e m b e r s . I f t h e s p a c e i s b o u n d e d , t h e n f r a m e s w i t h u n i f o r m s t r a i n w i l l h a v e v o l u m e s l e s s t h a n a l l o t h e r f r a m e s l y i n g w i t h i n t h e . s p e c i f i e d b o u n d a r y . T h e r e may, h o w e v e r , b e a f r a m e o f s m a l l e r v o l u m e e x t e n d i n g b e y o n d t h e b o u n d s . I n g e n e r a l t h e m e m b e r s o f t h e o p t i m u m s t r u c t u r e , M, m u s t l i e a l o n g l i n e s o f p r i n c i p a l s t r a i n i n t h e v i r t u a l d e -f o r m a t i o n c o n s i d e r e d i n t h e p r o o f . I f t h i s w e r e n o t s o , a d i r e c t i o n c o u l d b e f o u n d a t a p o i n t i n a member f o r w h i c h t h e d i r e c t s t r a i n h a d a m a g n i t u d e l a r g e r t h a n n, w h i c h i s c o n t r a r y t o t h e s p e c i f i e d c o n d i t i o n s . I f t h e ( e q u a l ) p r i n c i p a l s t r a i n s a r e o f t h e same s i g n a t a p o i n t , t h e s t r a i n i s t h e same i n a l l d i r e c t i o n s . I f t h e y a r e o f o p p o s i t e s i g n s , t h e n t h e s t r a i n i n a d i r e c t i o n a t a n a n g l e 0 t o o n e o f t h e m , E g , i s g i v e n b y €9 = £cOS(29) < £ T h u s e i s n o t e x c e e d e d i n e i t h e r c a s e . 8 C R I T E R I A FOR D E S I G N OF OPTIMUM ( m i n i m u m v o l u m e ) S T R U C T U R E S T h e m e m b e r s o f a n o p t i m u m m i n i m u m v a l u e f r a m e w o r k m u s t l i e a l o n g d i r e c t i o n s o f p r i n c i p a l s t r e s s . T h e y m u s t f o l l o w t w o s e t s o f o r t h o g o n a l c u r v e s . S u c h s e t s may b e d i v i d e d i n t o t w o g e n e r a l c l a s s e s -a) s y s t e m s o f t a n g e n t s a n d i n v o l u t e s d e r i v e d f r o m a n y c u r v e ( e v o l u t e ) , b) o r t h o g o n a l s y s t e m s o f e q u i a n g u l a r s p i r a l s . S e t s o f c o n c e n t r i c c i r c l e s a n d t h e i r r a d i i , a n d r e c t a n g u l a r n e t w o r k s o f s t r a i g h t l i n e s , a r e s p e c i a l c a s e s o f c l a s s b . T h i s i m p l i e s t h a t t h e o p t i m u m f r a m e w o r k c o n t a i n s a n i n f i n i t e n u m b e r o f m e m b e r s , i f o r t h o g o n a l c u r v e s a r e t o b e f o l l o w e d . A f r a m e w o r k may b e c o n s t r u c t e d f r o m p a r t s o f d i f f e r -e n t s y s t e m s , p r o v i d e d t h a t d i s p l a c e m e n t s a r e c o m p a t i b l e a l o n g t h e i n t e r f a c e s . P o i n t f o r c e s may b e s u p p o r t e d b y s p e c i a l m e m b e r s , w h i c h d i s t r i b u t e t h e l o a d i n t o t h e n e t w o r k , o r b y a r r a n g i n g t h a t t h e i r p o i n t s o f a c t i o n a r e s i n g u l a r p o i n t s , w h e r e i n f i n i t e n u m b e r s o f f i b r e s c o n v e r g e . B y u s i n g d i f f e r e n t s e t s o f o r t h o g o n a l c u r v e s , s e v e r a l f r a m e s may b e s p e c i f i e d t o s a t i s f y g i v e n b o u n d a r y a n d l o a d i n g c o n d i t i o n s . I n s u c h c a s e s , f r o m e q u a t i o n s 1.7 a n d 1 . 1 0 , a l l w o u l d h a v e t h e same m i n i m u m v o l u m e f o r a g i v e n u n i f o r m s t r e s s l e v e l , a . 9 Special Case for Volume of Structure Consider any two dimensional structure as indicated in Figure 1.1 The structure i s assumed to be supported at two fixed points A and B, which are symmetrically placed about the x axis. Figure 1.1 Forces Acting on a Typical Structure A t y p i c a l load, P N, i s shown, which i s acting at L N , any point on the x axis Then RA a A X L + Ay J ( 1 1 3 ) a Bjc L + By j Total load 10 where the s u f f i c e s x and y i n d i c a t e components of these f o r c e s p a r a l l e l to the x and y axes. By elementary s t a t i c s -These equations y i e l d , by s u b s t i t u t i o n ™ L D 2 J . . . . (1.14) . . . . (1.15) (1.16) . . (1.17) No c o n c l u s i o n s can be drawn about the magnitudes of A and B without knowledge of the s t r u c t u r a l arrangement. y y However, c o n s i d e r the s p e c i a l case i n which a l l loads are a p p l i e d p e r p e n d i c u l a r to the x a x i s , so t h a t (P^) i s everywhere zero. Then a p p l y i n g equation 1.5 -11 . . . . (1.18) F o r a n y s t r u c t u r e i n w h i c h A =B ; ( t h a t i s , t h e y y v e r t i c a l r e a c t i o n s a t t h e s u p p o r t s a r e e q u a l , ) t h e r i g h t h a n d s i d e o f t h i s e q u a t i o n b e c o m e s z e r o . T h u s f r o m e q u a t i o n 1.5, t h e c o n s t a n t C i s z e r o a n d s i n c e f r o m e q u a t i o n 1.12, i t f o l l o w s t h a t . . . . (1.19) vc=vt T h e t o t a l v o l u m e o f t h e t e n s i l e m e m b e r s i s t h u s t h e same a s t h e t o t a l v o l u m e o f t h e c o m p r e s s i v e m e m b e r s i n f r a m e -w o r k s s a t i s f y i n g t h e s e s p e c i a l c o n d i t i o n s . F o r some f o r t y y e a r s t h e M i c h e l l t h e o r y l a y d o r m a n t u n t i l r e v i e w e d b y C o x [3], [4], Hemp [5] a n d o t h e r s who s h o w e d t h a t t h e s e g e o m e t r i c a l c o n d i t i o n s f o r o p t i m u m s t r u c t u r a l d e s i g n w e r e a n a l a g o u s w i t h t h o s e e x p r e s s e d b y H e n c k y ' s t h e o r e m f o r s l i p l i n e s i n p l a n e p l a s t i c f l o w . Many w o r k s h a v e b e e n p u b l i s h e d i n t h i s f i e l d a n d a n e x t e n s i v e l i s t o f r e f e r e n c e s i s i n c l u d e d i n t h e p a p e r b y R . L . B a r n e t t [9]. T h e s l i p - l i n e l a y o u t s t h e r e d i s c u s s e d i n t u r n c a n p r o v i d e i n s p i r a t i o n f o r t h e d e s i g n o f o p t i m u m s t r u c t u r e s . 12 CHAN'S CALCULATION OF THE VOLUME OF AN OPTIMUM STRUCTURE A paper by Chan [6] i n 1960 d i s c u s s e d the a p p l i -c a t i o n of M i c h e l l ' s theorem to two dimensional s t r u c t u r e s and i n p a r t i c u l a r to c a n t i l e v e r s . He d e r i v e d an e x p r e s s i o n f o r the volume of an optimum minimum volume s t r u c t u r e . His treatment i s lengthy and r i g o r o u s ; a summary of h i s paper f o l l o w s , s i n c e i t forms a t h e o r e t i c a l b a s i s to the presen t work. I t f o l l o w s from M i c h e l l ' s proof t h a t the members of an optimum s t r u c t u r e must l i e along a r r a y s of mutually p e r p e n d i c u l a r l i n e s . These may be used to d e f i n e a r i g h t handed c u r v i l i n e a r system as shown i n F i g u r e 1.2. F i g u r e 1.2 C u r v i l i n e a r Coordinate System 13 Point D has c u r v i l i n e a r coordinates a, 3. The tangent directions to the a and 3 curves at D are U and V respectively, U making a po s i t i v e angle cf) with the x axis. may be represented i n terms of the c u r v i l i n e a r coordinates da and d3 The d i r e c t s t r a i n at D i n the a d i r e c t i o n w i l l be assumed to be +e and that i n the 3 d i r e c t i o n to be -e. The shear s t r a i n i s zero since these are p r i n c i p a l d i r e c t i o n s . The rotation at D due to these strains w i l l be denoted by c o . The re l a t i o n s between the displacement components and strains and rotations, i n c u r v i l i n e a r coordinates are stated i n Love [ 7 ] . These y i e l d The inset indicates a small l i n e element DD1 which [Chan 1] (1.21) 1 cHJ + V 3 A A Set A B 3p> [Chan 2] 1 ay . JJ aB B d p AB53 [Chan 3] (1.22) o [Chan 4] [Chan 5] Also, by considering the angle cj) i n Figure 1.2 14 M « - ± 2A _ , [Chan 6] . . (1.23) M « 1 IB Equations 1.22 and.1.23 may be used to f i n d the d e r i v a t i v e s of u and v -^ J a A £ + V ¥ dU ^ - B O O + V ^ [Chan 7] doc ? bp 5J3 (1.24) Aco -u^ ; v / a v , - B e - u ^ [ c h a n 8 ] u and v may be e l i m i n a t e d between equations 1.24, jL^GO-2£^ = JL (cO+2£^== O [Chan 9] . . (1.25) and, f i n a l l y , e l i m i n a t i o n of OJ g i v e s [Chan 10] . (1.26) This i s the c o m p a t i b i l i t y equation f o r the system of c u r v i l i n e a r c o o r d i n a t e s and f o r the l a y o u t of the members of the optimum s t r u c t u r e . For s o l u t i o n of t h i s fundamental equation three cases may be co n s i d e r e d -„ v 3(f) 9<f> a ) = W = z e r o 15 Here a and 3 are both constant and the c o o r d i n a t e system becomes C a r t e s i a n . A l l s t r u c t u r a l members are s t r a i g h t i n two p e r p e n d i c u l a r a r r a y s . b) E i t h e r ~ or •—- i s zero. d a d p In t h i s case e i t h e r a or 3 i s a constant and one s e t of c o o r d i n a t e s w i l l be s t r a i g h t . They w i l l i n g e n e r a l form an envelope to an 'evolute' whose shape w i l l depend on the boundary c o n d i t i o n s . The o t h e r s e t of l i n e s w i l l be curved and w i l l form ' i n v o l u t e s ' to t h a t e v o l u t e . c) N e i t h e r ~ - nor — are z e r o . d a d p T h i s i s the more g e n e r a l case and the one t h a t w i l l be c o n s i d e r e d (the other cases a r i s i n g from t h i s as s p e c i a l v a l u e s ) . Both s e t s of l i n e s are c o n t i n u o u s l y curved and i n f l e c t i o n s i n the r e g i o n being considered may be avoided by 3d) 3d) s p e c i f y i n g t h a t there are no zeros of TA- or i n s i d e the c 9a 9 3 boundaries. For s o l u t i o n of the equations, some boundary con-d i t i o n s must be assumed, and Chan co n s i d e r e d the l a y o u t shown i n F i g u r e 1.3. From f i x e d p o i n t s , A and B, a r c s of r a d i u s r are drawn, meeting a t C such t h a t ACB = 90°. C i s the o r i g i n 16 P Figure 1.3 Layout of Fibre Network of the c u r v i l i n e a r coordinate system a, 3 , the two c i r c u l a r arcs being the respective base l i n e s . A t y p i c a l point, P i s defined by the i n t e r s e c t i o n of the coordinate l i n e s FP and GP, whose angular values are defined by the angle between the tangent at F or G and the appropriate tangent at C. The network i s bounded by the outer f i b r e s ADL and BEL. If the arcs centred on A and B are of equal radius, as drawn, the network i s symmetrical about CL. If they 17 d i f f e r , symmetry i s l o s t but the same general theory i s a p p l i c a b l e . Many d i f f e r e n t boundary c o n d i t i o n s could be s p e c i f i e d , each l e a d i n g to somewhat d i f f e r e n t end equations. This arrangement i s p a r t i c u l a r l y convenient since i t represents a simple c a n t i l e v e r . 'A' and 'B' are s i n g u l a r p o i n t s , through which pass an i n f i n i t e number of f i b r e s . They can thus r e s i s t concentrated loads and represent the support p o i n t s i n a r e a l s t r u c t u r e . A s o l u t i o n of the c o m p a t i b i l i t y equation 1.11 i s 0 SB jt oC i ^ , and, i n p a r t i c u l a r 0 ss, — oC + p . . . . (1.27) which conforms w i t h the conventions e s t a b l i s h e d by Figure 1.3. Chan then shows that the r a d i i of curvature, p and p Q, of the coordinate l i n e s at a p o i n t P(a, B) are given P by (1.28) where Ig and T^ are Bessel f u n c t i o n s of the zero and f i r s t order r e s p e c t i v e l y . 18 I f now t h e s t r u c t u r e s h o w n i n F i g u r e 1.3 made f r o m a n i s o t r o p i c m a t e r i a l , i s l o a d e d i n some m a n n e r s u c h t h a t t h e s t r e s s e s e v e r y w h e r e a r e n u m e r i c a l l y e q u a l , t h o s e i n ' 3 ' f i b r e s b e i n g o f o p p o s i t e s i g n f r o m t h o s e i n 'a' f i b r e s , t h e s t r a i n s w i l l s i m i l a r l y b e n u m e r i c a l l y e q u a l . L e t A a n d B b e f i x e d p o i n t s i n s p a c e , n o t m o v i n g d u r i n g a p p l i c a t i o n o f t h e l o a d . T h e n e q u a t i o n s 1.28 may b e u s e d t o d e r i v e t h e d e f l e c -t i o n o f p o i n t P i n t h e TJ a n d V d i r e c t i o n s . T h u s -P - P ( C h a n e q u a t i o n 63) T h e t h e o r e t i c a l s t r u c t u r e b e i n g c o n s i d e r e d c o n f o r m s t o t h e M a x w e l l a n d M i c h e l l c r i t e r i a a n d t h u s i s o p t i m u m . I t s v o l u m e i s t h u s l e s s t h a n a n y s t r u c t u r e c a r r y i n g t h e same l o a d s w i t h i n t h e same b o u n d a r i e s , b u t n o t c o n f o r m i n g t o t h e s e c r i t e r i a . T h e a c t u a l o p t i m u m v o l u m e , w h i c h c o u l d t h e n b e u s e d t o e v a l u a t e m o r e p r a c t i c a l s t r u c t u r e s , c a n now b e d e t e r m i n e d . C o n s i d e r t h e o p t i m u m s t r u c t u r e , s u b j e c t e d t o e x t e r n a l f o r c e s P^, a c t i n g a t p o i n t s J \ , w h i c h s u f f e r s m a l l d i s p l a c e -m e n t s co. a s a r e s u l t o f d e f o r m a t i o n . T h e n t h e t o t a l w o r k l d o n e b y t h e e x t e r n a l f o r c e s i s 19 71. _ §W = Y ft -^A' . . . . (1.30) A t y p i c a l member of the s t r u c t u r e , having a l e n g t h £ A and a c r o s s - s e c t i o n a l area S^, w i l l undergo a l i n e a r s t r a i n ±E and be su b j e c t e d to a uniform normal s t r e s s ±a. The s t r a i n energy s t o r e d i s thus — rM . . . . (1.31) AaT ^ where V^ . i s the volume of the optimum s t r u c t u r e . By the p r i n c i p l e of v i r t u a l work equations 1.30 and 1.31 are equal -Vm ss JL. y P.CO- . . . . (1.32) OPTIMUM VOLUME FOR CANTILEVER WITH END POINT LOAD F i n a l l y , c o n s i d e r i n p a r t i c u l a r a s t r u c t u r e as shown i n F i g u r e 1.3, s u b j e c t e d to a s i n g l e p o i n t l o a d P T, a t L, d i r e c t e d normal t o the l i n e CL. The c u r v i l i n e a r c o o r d i n a t e s of P are equal so a = 8 = <$. The displacements d e s c r i b e d by equations 1.29 are i n d i r e c t i o n s p a r a l l e l to the tangents to the a and 3 l i n e s i n t e r s e c t i n g a t L, t h a t i s a t ±45° to CL. Thus the v i r t u a l displacement, coT , of L, i n the d i r e c t i o n of the l o a d i s = ] Z E f J l ^ S ) ! ^ * 2SI,(2gj| 20 (1.33) Substituting into 1.32 VM W =^rj||+2S)l s(2S) + 2Sl1(2X) . . . . (1.34) [Chan 66] In dimensionless terms = JI [(l+2S)i;(2S)i-2a )^] • • • (1.35) It should be emphasized that the optimum structure considered above i s i n fac t purely t h e o r e t i c a l . I t i s an ideali z e d concept for mathematical manipulation, but i t s p r a c t i c a l design raises many problems, not lea s t being the join t s between the two families of f i b r e s . I t should be noted that Chan's solution expresses the volume i n terms of fan angle, 6. The span of the structure does not enter his equations. Indeed he notes on page 2 3 that the span as quoted i n his r e s u l t s i s measured d i r e c t l y from layout drawings, which are of the corresponding Mich e l l cantilevers having small -values of N. This introduces a small error i n evaluating the span of the i n f i n i t e f i b r e structure. 21 C o n v e r s e l y , i n the f o l l o w i n g work, the span i s f i r s t s p e c i f i e d as being of more p r a c t i c a l use to the d e s i g n e r . Methods are d e s c r i b e d f o r c a l c u l a t i o n o f the f a n angle of the a p p r o p r i a t e M i c h e l l s t r u c t u r e f o r a f i n i t e value of N. A r b i t r a r i l y the v a l u e of 6 f o r N = 20 i s used as an lower bound estimate o f the a c t u a l value of 6 i n an i n f i n i t e f i b r e s t r u c t u r e . The ' e r r o r ' i n both approaches i s s i m i l a r and s m a l l , amounting to perhaps 0.1% of the t r u e v a l u e . I t i s of no p r a c t i c a l consequence s i n c e the optimum s t r u c t u r a l volume i s used o n l y as a standard by which ot h e r designs may be e v a l u a t e d . To quote Chan " . . . . the knowledge of the volume of an optimum s t r u c t u r e , when i t can be found, i s most valuable'. I t r e p r e s e n t s the u l t i m a t e s t r u c t u r a l ; e f f i c i e n c y . I f . . . another form of c o n s t r u c t i o n i s sub-s t i t u t e d , the p e n a l t y i n v o l v e d i s r e a d i l y c a l c u l a t e d by comparing the weight of the proposed c o n s t r u c t i o n with the optimum s t r u c t u r e . " 22 ANALOGY WITH SLIP LINE FIELDS IN PLANE PLASTIC FLOW Equations 1.2 5 JL. + 2£jzS) = o dp v express a g e o m e t r i c a l r e s t r i c t i o n on the form o f the c u r v i l i n e a r c o o r d i n a t e s , a and 8. They are i d e n t i c a l i n form w i t h those expressed i n Hencky's Theorem f o r s l i p l i n e s i n plane p l a s t i c flow. For example, i n "The Mathe-m a t i c a l Theory of P l a s t i c i t y , " by R. H i l l (Oxford, Clarendon P r e s s , 1950) on page 135, these are quoted as . H i l l eqn. 12 '3 x and y are C a r t e s i a n axes, p i s the mean compressive s t r e s s and k i s the y i e l d s t r e s s i n shear o f the m a t e r i a l being c o n s i d e r e d . cj) has the same meaning i n both s e t s of equations. 23 M a t e r i a l which i s loaded so t h a t the s t r e s s e s are g r e a t e r than the y i e l d s t r e s s no l o n g e r behave e l a s t i c a l l y and are s a i d to be p l a s t i c . The m a t e r i a l shears along m u t u a l l y p e r p e n d i c u l a r l i n e s which are c a l l e d s l i p l i n e s . An element of m a t e r i a l e n c l o s e d between s l i p l i n e s under-goes no expansion or c o n t r a c t i o n along those l i n e s . In t h i s analogy, the c u r v i l i n e a r c o o r d i n a t e s ( s l i p l i n e s ) r e p r e s e n t l i n e s of c o n s t a n t s l i p v e l o c i t y . C o n s i d e r p l a s t i c m a t e r i a l moving p a s t a r i g i d , p e r f e c t l y rough s u r f a c e . The m a t e r i a l a c t u a l l y i n c o n t a c t w i t h the rough s u r f a c e w i l l be s t a t i o n a r y . The mutually p e r p e n d i c u l a r s l i p - l i n e s — d i r e c t i o n s of r e l a t i v e motion between elements of the m a t e r i a l — w i l l be i n c l i n e d a t 45° t o the rough s u r f a c e . In p r a c t i c e the t h e o r e t i c a l s l i p l i n e f i e l d s are approximated by g r a p h i c a l s o l u t i o n s which correspond t o those used i n d e v e l o p i n g the M i c h e l l s t r u c t u r e s d e s c r i b e d i n Chapter 2. The book by H i l l mentioned above, and a paper by W. Johnson [ 8 ] , c o n t a i n many r e f e r e n c e s t o the l a r g e volume of work reco r d e d i n t h i s f i e l d . 24 Since the analogy between s l i p l i n e f i e l d s and optimum structure layout i s exact, these solutions provide possible layouts for Mi c h e l l structures which could be used to support some system of loading, as determined by boundary conditions. STIFFNESS OF AN OPTIMUM STRUCTURE One f i n a l i n v e s t i g a t i o n should be noted. Barnett [9] i n 1966 reviewed the Maxwell and M i c h e l l theories a l -ready described. He demonstrated that the optimum structure of minimum volume i s also the s t i f f e s t structure occupying the same region of space and subjected to the same system of loads. This may be shown by modification of equation 1.32. Consider an optimum structure subjected to a single concen-trated load P. Then from equation 1.32 . . . . (1.36) where co i s the displacement of the point of ap p l i c a t i o n of P. Since i t has already been shown that i s less than the volume of any other structure, i t follows that the d e f l e c -t i o n o> i s less than that for any other structure having the same maximum stress and s t r a i n and subjected to the same load, P. If the structure i s subjected to several loads at once, each may be considered separately and the volume of the optimum structure required to support i t alone may be calculated from equation 1.34. For each of these the same conclusion may be drawn that the d e f l e c t i o n i s less than for a l l other comparable structures. These loads may then be combined and the volume of the optimum structure then required w i l l be equal to the sum of the volumes of the optimum structures for each i n d i v i d u a l load. This follows from the p r i n c i p l e of super-position since i t has been s p e c i f i e d that the deflections are small and the e l a s t i c l i m i t not exceeded. It now remains to consider some more p r a c t i c a l structures which are approximations to the optimum frames considered above. The r e s u l t i n g compromise with the theor-e t i c a l design w i l l involve an increase i n volume since the structures w i l l no longer be optimum. 26 CHAPTER 2 GEOMETRY OF APPROXIMATE STRUCTURES The optimum o r minimum volume s t r u c t u r e d i s c u s s e d i n the p r e v i o u s chapter i s t h e o r e t i c a l r a t h e r than p r a c t i c a l i n n a t u r e . I t s d e s i g n may be approximated i n r e a l s t r u c t u r e s whose volume w i l l be somewhat l a r g e r than i n the i d e a l case Modern manufacturing t e c h n i q u e s , the i n c r e a s e d use of expen-s i v e m a t e r i a l s and the c o n t i n u i n g p r e s s u r e to reduce component weights a l l h e l p to make the use af eyer more complex shapes economical. The M i c h e l l concept d i s c u s s e d i n the p r e v i o u s c h a p t e r d i d not c o n s i d e r j o i n t s a t a l l , s i n c e the members of the two s e t s of f i b r e s c r o s s a t r i g h t angles and t h e r e i s thus no t r a n s f e r between them. Any p r a c t i c a l s t r u c t u r e based on the M i c h e l l c r i t e r i a must have j o i n t s which w i l l i n t r o d u c e a f u r t h e r d i s c r e p a n c y from the i d e a l case. The g e n e r a l concept, i l l u s t r a t e d i n F i g u r e 1.3, r e a d i l y suggests a p r a c t i c a l approximation as i l l u s t r a t e d i n F i g u r e 2.1. The dashed l i n e s i n d i c a t e c u r v i l i n e a r c o o r d i n a t e s a t an a r b i -t r a r i l y chosen e q u i a n g u l a r s p a c i n g , 8. The tangents t o these curves are drawn a t the p o i n t s o f i n t e r s e c t i o n . I t w i l l be seen t h a t s u c c e s s i v e tangents along a c o o r d i n a t e are i n c l i n e d a t an angle 8 to each o t h e r . Each c u r v i l i n e a r c o o r d i n a t e l i n e can be approximated by a s e r i e s of s t r a i g h t l i n e s drawn between these p o i n t s of 27 in t e r s e c t i o n . For example PQ, QS, approximate the curve 3. Each of these chords represents, i n the approximate structure, one of a limited number of concentrated members, which replace the bundle of f i b r e s lying between the coordinates 3 ± j • MNRSQP represents a small section of such an approx-imate structure. The members form two four-sided panels, i n each of which two opposite i n t e r n a l angles are 9 0 ° while the others are (90 + 8 ) and (90 - 8 ) . It i s thus immediately obvious that such a structure does not s a t i s f y the Michell c r i t e r i a , since the members are no longer orthogonal. Its v olume must thus be larger than that of the optimum structure. I t could reasonably be expected that the difference i n volume w i l l be small since the framework follows the general layout Figure 2.1 P r a c t i c a l Approximation to a Theoretical Optimum Structure 28 of the optimum s t r u c t u r e , d i f f e r i n g only i n the l o c a l d i s -t r i b u t i o n of m a t e r i a l . I t i s proposed to r e f e r to such approximate s t r u c t u r e s as MICHELL s t r u c t u r e s , to mark h i s a s s o c i a t i o n w i t h t h e i r u n d e r l y i n g theory. The exact arrangement of a M i c h e l l s t r u c t u r e w i l l depend on the span, l o a d i n g , number of members, type of frame-work and on the g e o m e t r i c a l c o n d i t i o n s imposed by the r e q u i r e -ments of c o m p a t i b i l i t y . T h i s l a s t f a c t o r i s p a r t i c u l a r l y important s i n c e i t determines the boundary and support con-d i t i o n s f o r the system. For example i n F i g u r e 1.3, the l i n e ACFE c o u l d r e p r e s e n t a continuous e l a s t i c s u p p o r t — o r a d i s t r i b u t e d l o a d — a n d thus be used as a boundary to the s t r u c t u r e . In t h i s case the members r a d i a t i n g from B would be excluded. As has a l r e a d y been s t a t e d A and B i n t h i s , drawing are s i n g u l a r p o i n t s where i n f i n i t e numbers of f i b r e s converge, i n the optimum s t r u c t u r e . They r e p r e s e n t the p o i n t s of a p p l i c a t i o n of concentrated support f o r c e s . -t The continuous boundary concept i s perhaps more use-f u l i n the p l a s t i c flow analogy than f o r s t r u c t u r a l a n a l y s i s . In the study of s l i p l i n e s , . A C F E i n F i g u r e 1.3 would r e p r e s e n t a f i x e d s u r f a c e such as a d i e f a c e , past which the p l a s t i c m a t e r i a l i s being extruded. In the prese n t work, c a n t i l e v e r s i n p a r t i c u l a r w i l l be s t u d i e d . T h i s w i l l not g r e a t l y l i m i t the v a l u e of the r e s u l t s o b t a i n e d , s i n c e many of the equations d e r i v e d are a p p l i c a b l e to oth e r types of M i c h e l l s t r u c t u r e . The g e n e r a l arrangement o f the s t r u c t u r e s t o be c o n s i d e r e d i s shown i n F i g u r e 2.2. F i g u r e 2.2 T y p i c a l M i c h e l l C a n t i l e v e r The c a n t i l e v e r i s supported a t two f i x e d p o i n t s , A and B, and may be s u b j e c t e d to any l o a d i n g s a t one or more j o i n t s between members. These j o i n t s are i n i t i a l l y assumed t o be pinned so t h a t a l l members are s u b j e c t e d t o a x i a l f o r c e s o n l y . 30 T h e c a n t i l e v e r may b e d i v i d e d i n t o t h r e e d i s t i n c t s e g m e n t s -a) a f a n o f e q u a l l e n g t h m e m b e r s , r , r a d i a t i n g f r o m A , b) a s i m i l a r f a n o f member l e n g t h , r , r a d i a t i n g f r o m B, c ) a n e t w o r k f o r m i n g q u a d r i l a t e r a l p a n e l s e x t e n d i n g f r o m t h e f a n s t o c o m p l e t e t h e s p a n o f t h e s t r u c t u r e . S u p p o r t F a n s T h e s e a r r a y s s e r v e t o d i s t r i b u t e t h e s u p p o r t r e a c t i o n s i n t o t h e s t r u c t u r e . T h e m e m b e r s AC a n d C B a r e s o p r o p o r t i o n e d t h a t t h e y a r e p e r p e n d i c u l a r a t C. I f t h e y a r e o f e q u a l l e n g t h , r , t h e c a n t i l e v e r i s symmetric a b o u t t h e x a x i s ; i f u n e q u a l , a s i n F i g u r e 2 . 2 , t h e c a n t i l e v e r i s skew a n d L w i l l n o t l i e o n t h e x a x i s . T h e r e m a i n i n g m e m b e r s r a d i a t i n g f r o m A a n d B a r e t h e same l e n g t h a s AC a n d C B . T h e a n g l e b e t w e e n s u c c e s s i v e m e m b e r s , 9, i s c o n s t a n t . -. N o t e t h a t t h e l i n e s C F D , C G E , a p p r o x i m a t e t h e b a s e l i n e s o f t h e c u r v i l i n e a r c o o r d i n a t e s s h o w n i n F i g u r e 1.3.-T h e m e m b e r s w i l l b e i n t e n s i o n o r c o m p r e s s i o n ; c l e a r l y n o c o n c e n t r i c m e m b e r s a r e n e e d e d i n t h e s e z o n e s s i n c e t h e r e a r e n o l o a d s t o b e r e s i s t e d c i r c u m f e r e n t i a l l y . * I t may b e n o t e d t h a t a l t e r n a t i v e s u p p o r t s y s t e m s may b e u s e d t o r e p l a c e t h e s e s e g m e n t s o f t h e s t r u c t u r e . I n p a r t i c u l a r a f i x e d c o n t i n u o u s s u p p o r t c o u l d b e p r o v i d e d a l o n g t h e a r c s CFD a n d C G E . I n s u c h c a s e s t h e s e f i v e p o i n t s w o u l d b e f i x e d a n d s u f f e r n o d e f l e c t i o n u n d e r l o a d . T h i s w o u l d m o d i f y t h e d e f o r m a t i o n o f t h e s t r u c t u r e b u t w o u l d n o t a f f e c t i t s g e o m e t r y . 31 Main Structure As i l l u s t r a t e d i n Figures 2.1 and 2.2, the remainder of the structure i s composed of q u a d r i l a t e r a l frames, approx-imating the c u r v i l i n e a r network. A l l these frames are s i m i l a r , having the same arrangement of angles. The angle between members i n the fans, 9, determines the angular r e l a t i o n between a l l other members i n the frame-work. It w i l l be noted that each chain of members, t y p i c a l l y AF, FK, KM or BG, GK, KH, turn st e a d i l y i n one d i r e c t i o n , always toward the centre l i n e . This s a t i s f i e s the requirement stated in Chapter 1 that there should be no i n f l e c t i o n s i n the a or 3 l i n e s . The angle between successive members i n a chain i s always 6, except at the fan boundaries where i t i s 9/2. This arises from the geometrical requirements s p e c i f i e d i n Figure 2.1. Loads may be applied to the Michel l cantilever at any or a l l of the j o i n t s . As drawn, with equal spacing of the members (9 constant), the coordinates of the j o i n t points, D to M, depend on r , r and 0 and are fixed once these variables are s p e c i f i e d . If a Michell structure i s required to support forces at random points, a structure would have to be suitably designed with unequal spacing between members. In such a case the panels would not a l l be similar as i n the present work. This case has not been pursued as each arrangement of loads would require a unique solution. The general argu-ments developed below would, however, apply and a s i m i l a r procedure c o u l d be f o l l o w e d to s o l v e such s t r u c t u r e s . N o t a t i o n f o r S t r u c t u r e s For a l l but the s m a l l e s t of s t r u c t u r e s , a r b i t r a r y l a b e l l i n g of the j o i n t s , as i n F i g u r e 2.2, i s i m p r a c t i c a l . An o r g a n i z e d system f o r l a b e l l i n g each j o i n t , member and com-ponent p r o p e r t y was t h e r e f o r e d e v i s e d and i s d e t a i l e d i n Appendix A. B r i e f l y , i t c o n s i s t s of i d e n t i f y i n g j o i n t s by two s u f f i c e s a and b, the f i r s t r e l a t i n g t o i t s p o s i t i o n along the cha i n of members r a d i a t i n g from A, and the second to i t s p o s i t i o n along the 'B' c h a i n . These are used i n connection with the f o l l o w i n g symbols -Aa k ~ l e n g t h o f member i n 'A' c h a i n , B ^ l e n g t h of member i n 'B' c h a i n , FA , ,FB , f o r c e i n member A , , B , , ab' ab ab' ab' J , j o i n t , ab J sA,sB c r o s s - s e c t i o n a l area o f member, WA,WB width of member. Members, or t h e i r p r o p e r t i e s , are i d e n t i f i e d by the co o r d i n a t e s of the j o i n t a t i t s i n n e r end, t h a t n e a r e s t the support p o i n t s as measured along the c h a i n o f members. 32a G r a p h i c a l C o n s t r u c t i o n of S t r u c t u r e G r a p h i c a l s o l u t i o n o f a s t r u c t u r e of known span and support s p a c i n g , t o determine the f i b r e angle 6 and member l e n g t h s , i s not recommended. I t w i l l r e q u i r e the c o n s t r u c t i o n of s e v e r a l diagrams each c h a r a c t e r i s e d by one of s e v e r a l t r i a l v a l u e s o f 8. T h i s i s con t i n u e d u n t i l the va l u e of 8 g i v i n g the s p e c i f i e d span i s determined. Accuracy i s r e l a t i v e l y low s i n c e g r a p h i c a l e r r o r s are m a g n i f i e d by the • r e p e t i t i v e c o n s t r u c t i o n . For i l l u s t r a t i v e purposes or f o r o b t a i n i n g approx-imate v a l u e s of parameters, the procedure d e s c r i b e d below should be f o l l o w e d . The pr o c e s s i s i l l u s t r a t e d i n F i g u r e 2,2. 1. Locate A, B and L, t o s c a l e ( l a r g e r than a c t u a l s i z e , i f p o s s i b l e ) 2. C o n s t r u c t the r i g h t angle ACB w i t h AC = CB f o r a symmetric c a n t i l e v e r . For an unsymmetric c a n t i l e v e r , these l e n g t h s are unequal. 3. With c e n t r e s A and B draw c i r c u l a r a r c s of r a d i u s AC and BC r e s p e c t i v e l y . 4. Lay out the angles 8 around these a r c s (N-l) times t o produce the base f a n s , ADFC and BCGE. 5. Extend each r a d i a l l i n e so c o n s t r u c t e d outward a t an angle t o i t s p r e v i o u s d i r e c t i o n , t u r n i n g 32b always i n the same d i r e c t i o n . The f i r s t extension g i s at to the r a d i a l l i n e s , while a l l further extensions turn through 8. The d i r e c t i o n of turning i s always toward the centre l i n e so that l i n e s r a d i a t i n g from A turn 'downwards' and those from B turn upwards, 8 ° For example AC i s produced at ^ 'clockwise' to form the l i n e CG, i n t e r s e c t i n g r a d i a l l i n e BG at G. GE i s then drawn to 8° to CG to i n t e r s e c t BE at E. (The points C,F,D, and C,G,E are already known since they l i e on the circumference of the c i r c l e s drawn i n step 3.) S i m i l a r l y AF i s extended outward 'clockwise' 8° at TJ- to form FK while BG i s extended outward 'counterclockwise' to form GK. This completes the f i r s t q u a d r i l a t e r a l panel CFKG. This routine i s followed systematically to complete the structure, the l a s t two extensions i n t e r s e c t i n g at L. I t w i l l be seen that there are sequences of perpendicular l i n e s , such as CG, GK, KM and ML. Another set i s CF, FK, KH and HL. This forms an a l t e r n a t i v e method of construction, once the f i r s t extensions have been made. C a l c u l a t i o n of Geometry of Structures A representative M i c h e l l structure i s shown i n Figure 2.2. As can be seen the construction i s r e p e t i t i v e , the major part of the framework consisting of four sided panels a l l of the same angular shape. From t h i s diagram the nine variables may be s p e c i f i e d . These are l i s t e d below, together with the symbols which represent them i n the computer programmes used f o r s o l u t i o n of the many structures examined. L (XSPAN) Y (YSPAN) D (D) r A (RADA) r B (RADB) 0 (THETA) N (N) span of structure; distance along x-axis, from plane of supports to plane of outer-most point of structure, r i s e of structure; distance from x-axis to outermost point of structure, support spacing; length AB, "upper fan radius'; length of members A (a,o) r a d i a t i n g from support A, 'lower fan radius'; length of members B, 3 (o,b) r a d i a t i n g from support B, 'f i b r e angle 1; angle between consecutive r a d i a l members i n fans. 9 i s taken constant i n t h i s work, for any one structure, 'fan angle 1; angle included between outer members of fans, number of chains of members ra d i a t i n g from each support point, 34 6 (BETAA) angle ABC. In a symmetrical structure these reduce to s i x , since then - 45° . . . . (2.1) Y =. ZERO The six remaining variables are not a l l independent and three more may be eliminated by the following equations S =* (N-I)9 . . . . (2.2) tanfi) = A There are thus six independent parameters that must be s p e c i f i e d for a skew Mi c h e l l c a n t i l e v e r , or three for the symmetrical structure. In p r a c t i c e , i t i s usual to design a structure with a known span and support spacing, and with a known number of components, so that normally N, L, Y and D would be s p e c i f i e d . The remaining two equations to complete the solu t i o n could t h e o r e t i c a l l y be written by consideration of the geometry of the structure, by deriving expressions for L and Y i n terms of r,. , r D and 0. In practice t h i s becomes highly complex as N increases. This i s discussed i n Appendix B and examples are given for N as large as 5. The equation for the span of an N f i b r e d structure involves both s i n 0 35 and cos 8 to the power (N + 2) . Chan states that a general equation for the span could not be found and the present work substantiates his remark. For a s p e c i f i c value of N, as d i s t i n c t from the general case, an i t e r a t i v e method was found to be p r a c t i c a l and lent i t s e l f to computer solution. Values of 0 and (—) were assumed rB and used to calculate values of L and Y. These values were compared with the required values and 8 and ( r a / r ) then, changed i n a systematic way so that L and Y may be recalculated. This procedure was repeated u n t i l the error became less than a s p e c i f i e d value. The computer programme to carry out t h i s search i s detailed i n Appendix C. It i s quite rapi d l y convergent, and calculates parameters with an error of less than 0.01% i n a few cycles. The same programme, with variants i s used to calculate many other items of information about the structure being investigated. Lengths of Members The lengths of a l l the members of the framework may be calculated from f i r s t p r i n c i p l e s of geometry from Figure 2.2. The actual equations are stated in Appendix B. Joint Coordinates As for the member lengths, j o i n t coordinates may be derived d i r e c t l y from Figure 2.2. The actual equations are 3 6 stated i n Appendix B. Deflection of Structures Somewhat paradoxically, the d e f l e c t i o n of the struc-ture depends e n t i r e l y upon i t s geometry and i s independent of the loading. This follows from the design condition that the stress and s t r a i n are uniform throughout the structure. Each member i s proportioned to s a t i s f y t h i s condition. Thus once the s t r a i n i s s p e c i f i e d , the change of length of each member i s known and thus the d e f l e c t i o n can be ca l c u l a t e d . A W i l i o t diagram was used to determine the d e f l e c t i o n s , the diagram being solved t r i g n o m e t r i c a l l y to maintain accuracy. It has been assumed that the support points A and B are fix e d i n space and that the s t r u c t u r a l d e f l e c t i o n s are e n t i r e l y due to deformation of members. The equations are stated i n Appendix B. It should be noted that the above statement (that d e f l e c t i o n i s independent of loading) i s true only i f a l l members are under stress. If a structure were loaded at an inner j o i n t , some members would then be unstressed and unde-formed, i f self-weight were neglected. In such case the s t r a i n i s not uniform i n a l l members and the W i l i o t diagram i s modified. This point i s discussed i n a section of Chapter 4. 37 R i g i d i t y of Structures The structure shown i n Figure 2.2, i f assumed to be p i n - j o i n t e d , acts l i k e a mechanism. If j o i n t B, say, were replaced by a r o l l e r , free to move i n any s p e c i f i e d d i r e c t i o n , the structure would collapse by r o t a t i o n about A u n t i l an equilibrium p o s i t i o n i s reached. I t i s therefore e s s e n t i a l that both support points are fixed to maintain the i n t e g r i t y of the structure and i t s s o l u t i o n . If the j o i n t s were r i g i d , the structure can maintain a fixed configuration independent of the support points. However, t h i s r e l i e s on the r i g i d i t y of the material around the j o i n t s and any applied load would r a i s e high l o c a l stresses i n these areas, i f A and B were not both f i x e d . I t thus follows that the fixed support points should be considered as an e s s e n t i a l feature of the structure. From the conditions s p e c i f i e d above and i n Appendix B, the geometry of any M i c h e l l framework may be completely determined, once the basic parameters such as span, number of f i b r e s , arrangement of supports and so on are known. 38 CHAPTER 3 FORCES IN A STATICALLY DETERMINATE MICHELL STRUCTURE We have seen i n the p r e v i o u s chapter t h a t the geometry of a M i c h e l l s t r u c t u r e i s s p e c i f i e d by the r e q u i r e -ments l a i d down by Maxwell, M i c h e l l and Chan. These are s u f f i c i e n t to d e f i n e such a s t r u c t u r e u n i q u e l y , once i t s span, support spacing and number of f i b r e s i s s e l e c t e d to s a t i s f y a g i v e n d e s i g n . I t now remains t o d e s c r i b e the other c a l c u l a t i o n s r e q u i r e d f o r a complete s o l u t i o n o f a M i c h e l l s t r u c t u r e which i s assumed to be p i n - j o i n t e d and thus s t a t i c a l l y d eterminate. As a f i r s t approximation, the weight of each member, i f t h i s i s t o be c o n s i d e r e d , i s assumed to be concentrated e q u a l l y a t each end of the member. Thus a l l f o r c e s are a p p l i e d a t the j o i n t s and the members are assumed to be i n pure t e n s i o n or compression, w i t h no a p p l i e d bending moments. I t w i l l f u r t h e r be assumed t h a t the t h i c k n e s s o f D a l l members, p e r p e n d i c u l a r to the plane o f the s t r u c t u r e , i s c onstant, t h a t the m a t e r i a l used i s i s o t r o p i c and the a x i a l t e n s i l e or compressive s t r e s s i s everywhere n u m e r i c a l l y c o n s t a n t , d i f f e r i n g o n l y i n s i g n between members. FORCES DUE TO THE WEIGHT OF MEMBERS 39 For the purpose of the following calculations a l l dimensions w i l l be assumed to be i n inches, and forces i n pounds. The choice of these units i s somewhat a r b i t r a r y and serves merely to define p r e c i s e l y c e r t a i n constants i n -volved i n the ca l c u l a t i o n s . Figure 3.1 shows a t y p i c a l s t r u c t u r a l member made from a material of s p e c i f i c gravity p Weight of member = (Aab)fy^AQb)"tp . (3.1) Half of thi s weight i s assumed to be concentrated at each end j o i n t , J , and J , , s . The a x i a l force on the J ' ab a, (b+1) member i s F A a k a n c^ the uniform stress i s everywhere a PAab Figure 3.1 Details of a Typical Member The e f f e c t s of thi s assumption are considered i n the discussion of r i g i d j o i n t s i n Chapter 5. Thus (WAa^ttf'** FAob . . . . (3.2) Substituting t h i s into 3.1 to eliminate the term t ^ a b ^ i t follows that the weight concentrated at each j o i n t to replace the s e l f weight of t h i s member, i s given by o e • o ( 3 o 3 } where W i s a constant equal to Forces i n the Members For the purposes of the following c a l c u l a t i o n s i t w i l l be assumed that there i s a load applied at every j o i n t , LOAD(ab), and that i t acts at an angle T(ab) to the y axis. Further, for complete generality, the whole structure i s assumed to be t i l t e d so that the s e l f weight forces act at an angle + to the y axis. These d i r e c t i o n s are shown i n Figure 3.2. The equations derived below are thus com-p l e t e l y general and may be applied to any loading system. For members considered to be weightless, W i s placed equal to zero as can Load (a,b) i f no external force i s applied at a s p e c i f i c j o i n t . Analysis of the forces acting i n the s t r u c t u r a l members commences conveniently at the outer end, L, and Aa,b-. gure 3.2 Convention f o r Load Angles a t a T y p i J o i n t L O A D gure 3.3 Forces A c t i n g a t J o i n t J. 42 proceeds systematically inward to the support points. Support reactions w i l l be considered i n a succeeding section. This analysis i s the f a m i l i a r method of j o i n t s . A force diagram for the complete structure could have been drawn and a general formula could have been written applicable to a l l j o i n t s . For compilation of the computer programme i t was found convenient to consider f i v e d i f f e r e n t types of j o i n t . The d e r i v a t i o n of the following equations are given i n Appendix D. * a) The Outer J o i n t , J„„ NN From Figure 3.3, which shows the forces acting at t h i s j o i n t -9 L*^W*j-*»D+w(<v* sKw t))] M / W ' C o s Q - W [ W . S i V i ^ + f - J)+B^Mcos f£-f- f>] (3.4) * In a l l these cases i t i s assumed that the forces i n the 'A' members are t e n s i l e (+ve) and that those i n the 'B' members are compressive (-ve). The sign of the equation, a f t e r the numerical values are inserted, w i l l of course determine the truth of t h i s assumption. 4 3 b) Forces a t a T y p i c a l Inner J o i n t , J ab . ( 3 . 5 ) F i g u r e 3 . 4 shows the f o r c e s a c t i n g on a t y p i c a l j o i n t Ja k « T n e diagram a l s o a p p l i e s t o j o i n t s on the out e r f i b r e s , by p l a c i n g the a p p r o p r i a t e f o r c e s equal to zero, ^ L©AD0fc» F i g u r e 3 . 4 Forces a t t y p i c a l j o i n t , J ab FA^ b [ l + w(AQbcos(i|i+t-0)-BQ.,.b s'm(f+t-^)] + FBob [sinO* w(Bobcos(ipt-0) + BQ.,,t> cos(lp+fj) (cos0 -WJA Q ( ^ ,cos(^ -0 ) + B Q , , ^a (^^ j ] ) . FA ok [sm9 +w(A ab Sin(H>H)+Aa,b-»sin(^H-6)] ( 3 . 6 ) 44 c) Forces a t a J o i n t on the 'A' Fan (except J, ) a.o F i g u r e 3 . 5 Forces a t a T y p i c a l 'A' Fan J o i n t , J a F A a ) l [' "•W(AAi, c o s (v f -£ ) - B e - , , , Sinl>*t-f))" (cosf ^w[A a j«5 c o S(^t-f)+BQ.,,sia (4>+-f)]) p A Q ( , *^(A a pSift^-gJ+Aa.,sm(q*t)] + FB ,^ [cos w(Aa>0cos(^|)- BQ t, sin (<i>*4)] (cos | - W[Aaoco5(s>+t-|)* B a . u s / n (3 .7 ) 45 d) Forces a t a J o i n t on the 1B' Fan (except J 1 ) J=Ai.b P B o . b F i g u r e 3 . 6 Forces a t a T y p i c a l 'B' Fan J o i n t J,^ FA,b |o«|+w|A, 1bCoa(pi-t-|)- BabSia(V+HB] FA,,b.r kB,.b | i n | + w|Babcos(Y+t) + B,.bCos(s^-f ffl (cos f -W[A,,b., cos (^i-|)+80.b3in(^H)] ) PA,.b i^afl+W[Ai.b-.sin(^ -e)+Ai,bsinCM>-i-t)]] (cos | - W [\ t l H cos(^H-|)+ B ob s/n(H»+t3) ( 3 . 8 ) 46 WT. Figure 3.7 Forces at Joint F A , 0 -FA I ( I [cosf+WlA,,, sm(B-4)- Bc , ,cos ( f -B - f ]j + F B M [sin§+ W[B M s'in(j3-4)+'Bc.s'a(p-f-f); FB..= O.I ( ( -w [A l i 0 s in (H + £>o,.cos(/3-f)] ) " ^ . . . [ s i n f+W|A , & cos ( f M)+AMcos(H)]] + FB ,Jos | - .w[A I , 0 Siri(B-^ . (3.9) (|-W[A,,0S1K(BH9 + B O A cos(£-f)] ) 47 S i z e o f Members The l e n g t h of a t y p i c a l member, A ^, has a l r e a d y been determined by g e o m e t r i c a l c o n d i t i o n s as d e t a i l e d i n Cha p t e r . 2 , and the f o r c e s are known from equations 3.4 - 3.9. The width, WAab, may now be c a l c u l a t e d u s i n g the r e -quirement t h a t the a x i a l s t r e s s i s uniform. Thus WAak = FAQ,b .... ( 3 . 1 0 ) °-b I F The second moment of area of the s e c t i o n , weight of the member and other p r o p e r t i e s d e s i r e d may now be c a l c u l a t e d from standard e q u a t i o n s . Reactions a t Supports A p i n - j o i n t e d M i c h e l l s t r u c t u r e , as shown i n F i g u r e 2.8 and elsewhere, i s not of i t s e l f r i g i d and i s i n f a c t a mechanism. I t i s made"rigid by the use of two f i x e d support p o i n t s o r , i t s e q u i v a l e n t , two p e r p e n d i c u l a r support r e a c t i o n s at each support p o i n t . -Th i s i s i n d i c a t e d i n F i g u r e 3.8. T h i s i s i m p l i c i t i n the requirement s t a t e d i n Chapter 2 t h a t the angle ACB—between the inn e r fan members — i s a r i g h t angle, which was a fundamental p o i n t i n the l a y o u t o f the s t r u c t u r e . T h i s immediately r e q u i r e s AB t o be a f i x e d d i s t a n c e and thus both support p o i n t s must be f i x e d . ,. In p r a c t i c e , when r e a l s t r u c t u r e s are c o n s i d e r e d , there w i l l i n e v i t a b l y be some displacement of these p o i n t s For s m a l l relative displacements, where the change i n ACB i s i n f i n i t e s i m a l , the geometry of the system i s maintained, an assumption which i s i m p l i c i t i n a l l the above c a l c u l a t i o n s . F i g u r e 3.8 E x t e r n a l E q u i l i b r i u m of a M i c h e l l C a n t i l e v e r A t y p i c a l l o a d i s shown i n F i g u r e 3.8, tog e t h e r w i t h p e r p e n d i c u l a r components of the support r e a c t i o n s . The magni tude of these components may be determined by r e s o l u t i o n of the f o r c e s i n the members approaching the support p o i n t s . Thus -A« =|. rFA<,0cos[p-(a-l)^ + w(Aa.0FA^)sin(+)j A v -1 [F/Vo sia §-(a-1) 8] + w(Aa,0FAaj0)6os(rj] B H - J, [FB0.b sin |p+(b-i)0] + w(Bo i FBob)sia(tJ| B v =£(fB.. b cos[^+|»-!)9]+ W(B o bFBjcos(t)' the p o s i t i v e d i r e c t i o n of each reaction component being shown i n the above drawing. 50 CHAPTER 4 ANALYSIS OF SOME PIN-JOINTED MICHELL CANTILEVERS - S u b j e c t e d t o C o n c e n t r a t e d L o a d i n g s -I n t h e p r e c e d i n g c h a p t e r s t h e g e o m e t r y o f M i c h e l l s t r u c t u r e s i n g e n e r a l , and o f c a n t i l e v e r s i n p a r t i c u l a r , h as been d i s c u s s e d and e q u a t i o n s d e r i v e d c o n n e c t i n g t h e v a r i o u s p a r a m e t e r s . The f o r c e s e x e r t e d on e a c h member by t h e a p p l i e d l o a d i n g may t h e n be c a l c u l a t e d t h e e q u a t i o n s l a i d o u t i n : C h a p t e r 3. T h i s i n t u r n e n a b l e s t h e c r o s s - s e c t i o n a l a r e a ; o f e a c h member t o be c a l c u l a t e d s i n c e a u n i f o r m s t r e s s a t a l l p o i n t s has been s p e c i f i e d . A v a r i e t y o f c a n t i l e v e r s , o f v a r i o u s s p a n , r i s e and f i b r e number, were a n a l y z e d w h i l e s u b j e c t e d t o d i f f e r e n t , t y p e s o f l o a d i n g t o o b t a i n some i n s i g h t i n t o t h e way i n w h i c h t h e s t r u c t u r a l volume v a r i e d . The s i g n i f i c a n t r e s u l t s f r o m t h i s a n a l y s i s a r e d i s c u s s e d i n t h e f o l l o w i n g s e c t i o n s w h i l e t h e ' d a t a i s t a b u l a t e d i n A p p e n d i x 1 E ' . S e l e c t i o n o f D e s i g n P a r a m e t e r s The d e s i g n p a r a m e t e r s c o u l d be v a r i e d o v e r v e r y w ide l i m i t s and i t was t h e r e f o r e n e c e s s a r y t o make some a r b i t r a r y d e c i s i o n s t o r e d u c e t h e volume o f d a t a t h a t c o u l d r a p i d l y be accumulated by use of computer s o l u t i o n s . I t was t h e r e f o r e decided to base the m a j o r i t y of the s t r u c t u r e s on the de s i g n parameters of the p h o t o e l a s t i c models t h a t were to be made from b i r e f r i n g e n t m a t e r i a l and t e s t e d to c o n f i r m these c a l c u l a t i o n s . These models were to be c u t from sheet one q u a r t e r i n c h t h i c k so the t h i c k n e s s of a l l members, t , was taken as j " . T h i s i n t u r n enabled the width of a l l members, which were to be of r e c t a n g u l a r c r o s s - s e c t i o n , t o be c a l c u l a t e d from equation 3.10. The b i r e f r i n g e n t m a t e r i a l used was CR39 ('Columbia Resin') p l a s t i c produced by the Homalite Corpor-a t i o n . The manufacturers s t a t e t h a t the modulus of e l a s -t i c i t y o f CR39 i s 300,000 p s i . and i t s s p e c i f i c g r a v i t y i s 1.31. The s t r a i n was a r b i t r a r i l y chosen as 0.001. The f o l l o w i n g values were thus e s t a b l i s h e d f o r a l l models:-Modulus of E l a s t i c i t y E 300,000 p . s . i . S t r a i n z 0.001 Uniform a x i a l s t r e s s a 300 p . s . i . T hickness of a l l members t 0.25' inches S p e c i f i c G r a v i t y p 1.31 S e l f weight constant W = 0.0001577 ( i n c h e s - 1 ) r-- 1-31 x 62.4 , L~ 1728 x 300 J The use of these s p e c i f i c v a l u e s does not g r e a t l y a f f e c t the g e n e r a l i t y of the r e s u l t s o b t a i n e d . Wherever convenient, the r e s u l t s are c a l c u l a t e d or d i s p l a y e d i n terms 52 of dimensionless parameters and are thus v a l i d for any combin-ation of variables having that r a t i o . The s t r a i n was a r b i t r a r i l y chosen to conform with the basic assumption that the deflections are small compared with the dimensions of the structure. The v a r i a t i o n of d e f l e c t i o n with s t r a i n was considered for one structure and r e s u l t s are shown in Table 11 i n Appendix E. I t w i l l be seen that the d e f l e c t i o n varies d i r e c t l y with the s t r a i n . Once the s t r a i n i s selected, the uniform stress i s determined by the value of E from the f a m i l i a r equation stress s t r a i n As E increases, so w i l l the stress increase, for constant e. This w i l l not a f f e c t the d e f l e c t i o n , which de-pends only on e, but w i l l a f f e c t the cross-sectional areas of each member from equation 3.10 and thus w i l l a f f e c t the volume and weight of the structure. This i s also affected by the s p e c i f i c gravity, p , and may conveniently be con-, sidered i n the equation for W „ . 624 £ P nze E E The r a t i o — i s a property of the material used for manufacture and does not a f f e c t the optimization of struc-53 t u r a l design to perform a sp e c i f i e d function. For any given material the Maxwell/Michell conditions specify the optimum structure. Some t y p i c a l values are:-Material E Density Ratio Percent x lO-^dynes/cm^ gm/cm3 of , Be Beryllium 3.1 1.848 1.677 100.0 S i l i c o n 1.31 2.27 0.577 34.4 Pyrex 0.65 2.23 0.291 17 .38 Titanuim 1.15 4.51 0.255 15.20 Aluminum 0.62 2.699 0.230 . 13.69 Steel 1.8 7.9 0 .228 13.58 CR39* 0.018 1.31 0.014 1.37 CR39 i s used for model manufacture and i s not a constructional material. The e f f e c t of the thickness, p a r t i c u l a r l y as regards buckling of the compression members, i s also discussed l a t e r . Before considering the res u l t s of these c a l c u l a t i o n s , i t should be emphasized that these structures are s t i l l t h e o r e t i c a l i n nature. They are assumed to be pin-jointed but no provision has been made for the material i n the j o i n t s , • each member extending only for the t h e o r e t i c a l distance between j o i n t points. The e f f e c t of the extra material in j o i n t s and other approximations required to manufacture these structures i s described in Chapter 5. It should also be remembered, when considering the curves by which many of the re s u l t s are displayed, that .these 54 have no p h y s i c a l meaning except a t i n t e g r a l v a l u e s of N. A f r a c t i o n a l number of f i b r e s has no meaning. Accuracy of R e s u l t s Most of the c a l c u l a t i o n s , such as f i b r e angle, member le n g t h s , f o r c e s , e t c . were c a r r i e d out i t e r a t i v e l y by s u c c e s s i v e approximation, or by a step-by-step process a c r o s s the s t r u c t u r e . In e i t h e r case s l i g h t e r r o r s i n b a s i c p roper-t i e s are i n e v i t a b l y accumulated i n the procedure. While e i g h t s i g n i f i c a n t p l a c e s were used i n most c a l c u l a t i o n s , i t i s f e l t t h a t the f i n a l v a l u e s can s a f e l y be quoted to f o u r s i g n i f i c a n t p l a c e s and t h i s has been done i n the t a b u l a t e d r e s u l t s . The s t r u c t u r e s i n v e s t i g a t e d may c o n v e n i e n t l y be d i v i d e d i n t o the f o l l o w i n g groups f o r c o n s i d e r a t i o n . 1) Symmetrical, S i n g l e Load a t J N N P e r p e n d i c u l a r to A x i s A t y p i c a l s t r u c t u r e of t h i s type i s shown i n F i g u r e 4.1. I t i s assumed to be supported i n a v e r t i c a l plane w i t h support p o i n t s A and B i n a v e r t i c a l l i n e . The f a n r a d i i are e q u a l . A l o a d of one hundred pounds i s a p p l i e d a t the o u t e r j o i n t , L ( J N N ) i n a v e r t i c a l l y downward d i r e c t i o n . The number of f i b r e s , N, was v a r i e d from 2, the minimum p o s s i b l e , to 20 with one case having N = 100. 55 IOO lb. Figure 4.1 Symmetrical C a n t i l e v e r With P o i n t Load The major set of s t r u c t u r e s had a constant span, L, equal to 10", w h i l e the support spacing, D, was v a r i e d from 1" to 10" g i v i n g an r a t i o range of 1 to 10. An a d d i t i o n a l set of 5 f i b r e s t r u c t u r e s was examined having a constant — r a t i o of 4. Here the span, L, v a r i e d from 60 to 4 inches, D correspondingly v a r y i n g from 15 to 1 i n c h . The r e s u l t s obtained from these c a l c u l a t i o n s are p l o t t e d as curves and the f o l l o w i n g conclusions may be drawn -a) Fan Angle, & (see F i g u r e 4.2) The fan angle depends l a r g e l y on the — r a t i o and r a p i d l y approaches the value f o r N °° . Even a t l a r g e v a l u e s of the r a t i o |j , the curve f o r 6 becomes almost h o r i z o n t a l f o r N l a r g e r than 5. T h i s i n d i c a t e s t h a t i n most cases there i s l i t t l e t o be gained by u s i n g l a r g e r values of N i n p r a c t i c a l d e s i g n . Chan, i n h i s a n a l y s i s reviewed i n Chapter 1, took convenient v a l u e s of fan angle, 6, and c a l c u l a t e d the volume of the r e s u l t i n g c a n t i l e v e r [eq. 1.34 or Chan 66]. The span of the i n f i n i t e f i b r e s t r u c t u r e was not c a l c u l a t e d but measured from a s c a l e drawing of the c o r r e s p o n d i n g M i c h e l l c a n t i l e v e r . In the p r e s e n t work, the span has been s p e c i f i e d as t h i s i s a b a s i c d e s i g n parameter. The fan angle o f the M i c h e l l c a n t i l e v e r having t h i s span and the d e s i r e d number of f i b r e s was then c a l c u l a t e d . The fan angle f o r N = 20 was then used i n equation 1.34 to o b t a i n an estimate of the volume of the t h e o r e t i c a l i n f i n i t e f i b r e s t r u c t u r e . The value of 6 so used i s s l i g h t l y l e s s than the a c t u a l v a l u e f o r an i n f i n i t e f i b r e s o l u t i o n , and thus the t h e o r e t i c a l optimum volume i s a l s o a lower bound on the a c t u a l v a l u e . As d i s c u s s e d i n Appendix B, no g e n e r a l equation was d i s c o v e r e d to connect span and fan angle f o r the i n f i n i t e case. The dimensionless q u a n t i t y [-—• ] d e r i v e d on the L r l e f t s i d e of equation 1.35 i s used i n l a t e r curves to normal-i z e the data. 57 b) Fibre Angle, 6 (see Figure 4.3) The f i b r e angle, G, diminishes continuously as the number of f i b r e s increases 1 I t eventually tends to zero as N becomes very large, for a l l values of Like <5, i t i s independent of str u c t u r a l dimensions and depends only on N and — . c) Volume Index (see Figures 4.4 to 4.4e) The dimensionless parameter, ~ — , c a l l e d for conven-r ' P r ' J_I ience the Volume Index, i s plotted against N for a range of — r a t i o s . Figure 4.4 shows a l l the curves on a uniform small scale while each curve i s plotted to best advantage on d i f f e r i n g scales i n Figures 4.4a to 4.4e. In these l a t t e r cases, two curves are plotted showing the e f f e c t of the weight of the structure i t s e l f . The theory outlined i n Chapter 1 predicts an optimum minimum volume which i s shown on a l l these curves. It w i l l be seen that the "no s e l f weight" curve i s asymptopic to th i s value and indeed c l o s e l y approximates i t for N as low as 10. The "with s e l f weight" curve i s i n each case somewhat higher but also could be considered as a good approximation to the ultimate value for N larger than ten. Even for N = 5, the Volume Index i s within 5% of the th e o r e t i c a l minimum for a l l plotted values of ~ while the values for N = 3 are within 10% (5% for ~ less than 5). This further j u s t i f i e s the e a r l i e r statement that p r a c t i c a l structures need contain no more than f i v e f i b r e s per fan and yet s t i l l be excellent approximations to the t h e o r e t i c a l optimum. F i g u r e 4.2 V a r i a t i o n of Fan Angle w i t h Number of F i b r e s i n Symmetrical M i c h e l l C a n t i l e v e r s F i g u r e 4.3 V a r i a t i o n of F i b r e Angle w i t h Number of F i b r e s i n Symmetrical M i c h e l l C a n t i l e v e r s 60 ISO i>o -40 zo + + 110 3«» 4 • 6 ' 8 NUMBER OF FIBRES N TKeoK.tical Optimum. - 99'77 27-97 9-91 3 43 IO Figure 4 . 4 V a r i a t i o n of Volume Index with Number of Fibres i n Symmetrical Michell Cantilevers 346 V O L U M E INDEX. 3-45 6 $ 3 - H 343 Self W ^ h t . T h e o r e t i c a l M l n i m o m 3*433 H * * i 1 1 4- 6 8 NUMBER, OF FIBRES - N. 'O Figure 4.4a Volume Index for Symmetrical Mich e l l Cant Having L , D t O ' O Z V O L U M E I N D E X 9S>4 62 Ji>eofeticol Minimum 9'9o6 + + 4 6 ' 8 NUMBER OF FIBRES* N JL _4 J F i g u r e 4.4b Volume Index f o r Symmetrical M i c h e l l C a n t i l e v e r s Having L _ ~ D ~ ^ 29-oh V O L U M E I N D E X 28-5 2 & O h 275 63 WirK Self We^ht. No Self Weigkt.-Theoretico.1 Minimum. £7 862 H i »-4 6 8 NUMBER OF FIBR.ES N. -f ± io Figure 4.4c Volume Index for Symmetrical Mich e l l Cantilevers Having L _ . D 64 F i g u r e 4.4d Volume Index f o r Symmetrical M i c h e l l C a n t i l e v e r s Having L _ _ D 65 180.: V O L U M E I N D E X 5 $ ' I20 • With Self W e i 3 K t . No Self We^brT^^1^^^ Theoretical Minimum 98-40Z H i : H 4 - 6 8 NuMBEfe O F FiBRES N. 4 ± I O Figure 4.4e Volume Index for Symmetrical Michell Cantilevers Having L _ ,_ D Another factor j u s t i f y i n g t h i s concerns the t o t a l 2 number of members. An N-fibred framework contains N members. Manufacturing costs would tend to be proportional to the number of components and thus considerations of manufacturing economy indicate that N should be kept to a minimum. These arguments are reinforced when the j o i n t s between members are considered i n Chapter 5. EFFECT OF VARIATION OF SPAN The volumes of the set of structures having equal but varying span, are plotted i n Figure 4.5. This shows that, for the case i n which the e f f e c t s of the weight of the structure i t s e l f are ignored, the volume i s d i r e c t l y proportional to the span. This arises since the structures are a l l of i d e n t i c a l geometric shape, d i f f e r i n g only i n the scale. The force i n each member i s constant since t h i s depends only on the geometry, the stress being everywhere uniform. The cross-sectional areas are thus constant and t t o t a l volume i s proportional to the span. From the r e s u l t s Volume = 1.670 (span) However, i f the e f f e c t s of the weight of the struc-ture are included, the volume i s not d i r e c t l y proportional to the span but increases more r a p i d l y . This i s indicated by the percentage increase curve which represents -Percentage Incwase of Self Weight Vofume a s cpmpafed to Volume i f Self Vie\$hf ne^lecfeb. 67 erceatage irxcrease of Structural Volume iac-i u d m q e f f e c t s ^ self-'ooei^T?!- c o m p a r e d t& M O i J m e *|G sei£ aittoM" is neoiecred. 20 40 S P A N inches. Figure 4.5 68 (self weight volume - volume with s e l f weight ignored) x (volume with s e l f weight ignored) and the curve c l e a r l y indicates how t h i s s t e a d i l y increases. This r e f l e c t s the well known f a c t that, as bridge spans increase, the stresses i n the span are more and more due to the weight of the structure rather than to the applied load. This sets an ultimate l i m i t to the span> whatever i t s design. 2) Symmetrical Cantilever With T i l t e d End Load In the previous section we have seen that a f i v e f i b r e Michel c a n t i l e v e r provides a very s a t i s f a c t o r y approximation to the t h e o r e t i c a l optimum structure. This p a r t i c u l a r struc-ture w i l l now be used to examine the e f f e c t of applying the * , 0 " . ^ Figure 4.6 Symmetrical M i c h e l l Cantilever with T i l t e d Load l o a d a t angles o t h e r than p e r p e n d i c u l a r t o the a x i s . S p e c i f i c a l l y , a c a n t i l e v e r w i t h the f o l l o w i n g p r o p e r -t i e s was used:-Span L = 1 0 " Support s p a c i n g D = 2.5" Number o f f i b r e s N = 5 F i b r e angle 6 = 18.41° = 4 L D T h i s s t r u c t u r e was supported as shown i n F i g u r e 4.6 and s u b j e c t e d t o a one hundred pound l o a d a t L, j o i n t J N N » The d i r e c t i o n o f t h i s l o a d was v a r i e d between the l i m i t s - 4 5 ° < T < +9o° The r e a c t i o n s c o r r e s p o n d i n g t o these loads are p l o t t e d i n F i g u r e 4.7, and are perhaps of f a i r l y obvious n a t u r e . The sum of the v e r t i c a l components, A^ . and B^, must equal the v e r t i c a l component of the l o a d , (P^^ cos x ) , w h i l e the horizon-t a l r e a c t i o n s , A H and B H, r e s i s t the h o r i z o n t a l t h r u s t o f the l o a d , ( P55 s i n T ) / p l u s the moment r e q u i r e d t o counterbalance t h a t generated by the l o a d , P cos T x L. As x i n c r e a s e s the l o a d i s c a r r i e d more and more by the outer 'B' f i b r e and the c o r r e s p o n d i n g o u t e r 'A' f i b r e l o a d i s s t e a d i l y reduced. At x = 54.2°, the l o a d i s a l i g n e d w i t h member B ^ and the f o r c e i n A ^ drops t o z e r o . I f x i s f u r t h e r i n c r e a s e d the l o a d on ou t e r A members (A^g) becomes 70 — i 1 1 1 1 1 1 1 1 1 « 1 J 1 -40 -20 o 20 4o 60 so Tilt of L o a d "X. (decrees.) F i g u r e 4.7 Reactions a t Supports of a Symmetrical M i c h e l l C a n t i l e v e r w i t h T i l t e d Load a t Outer End 71 Ttlt of Load- c/ejrees F i g u r e 4.8 Volumes of Symmetrical M i c h e l l C a n t i l e v e r s Loaded a t V a r i o u s Angles compressive. Ultimately, when T becomes +90°, only the members i n the two outer f i b r e s (A^^ and B a^) are i n compres-sion, while a l l inner members are now i n tension. The inner f i b r e s now act to r e s t r a i n the outer f i b r e s from becoming more curved. Figure 4.8 indicates how the volume varies with d i r e c t i o n of loading. As might be expected the structure has a maximum volume when x = zero (pure cantilever) and declines s t e a d i l y to a minimum value when T = ±90° (pure column). The t r a d i t i o n a l method of designing a mirror that i s to be t i l t e d e n t a i l s a d e t a i l e d analysis with the mirror plane h o r i z o n t a l . When t i l t e d the mirror d e f l e c t i o n s do not exceed those encountered i n the horizontal p o s i t i o n . Figure 4.8 corroborates t h i s , i n that the maximum volume i s found at T = zero, which corresponds to the structure i n a horizontal plane. I t should be emphasised that two conditions may be i d e n t i f i e d when considering t i l t e d loads. In one case a given structure may be designed to be optimum for a given loading, say at x = zero. I t may however be subjected to loads at other values of x. Certain members may become more heavily stressed than when the structure i s h o r i z o n t a l . In the other case an optimum structure may be derived to support a given t i l t e d load. The structures considered i n t h i s section are optimum for a l l values of x such that the d i r e c t i o n of the load vector l i e s within the sector QLS (Figure 4.6) Hegemier and Prager [17] quote the l i m i t i n g value of x as -45°_< x <_ 45° which i s true for the 72a t h e o r e t i c a l structure. In the M i c h e l l t r u s s , the permissible 8 8 angle i s somewhat greater [-(45+-j) <T<+ (45+-J) ] , When the load vector f a l l s outside t h i s sector both members meeting at L carry forces of the same sign and the structure becomes, i n part, of the Maxwell type. Since the Maxwell theorem provides no geometric condition of layout, optimisation i s no longer c e r t a i n . 73 3) Symmetrical C a n t i l e v e r w i t h P o i n t Load a t Each J o i n t The 'standard' f i v e f i b r e c a n t i l e v e r examined under t h i s s e r i e s of loads i s shown i n F i g u r e 4.9. The dimensions of the s t r u c t u r e are i d e n t i c a l w i t h those i n the pr e v i o u s case, but now the one hundred pound loa d i s a p p l i e d i n sequence a t each of the twenty f i v e main s t r u c t u r a l j o i n t s . Members beyond the loa d p o i n t are un s t r e s s e d and, s t r i c t l y speaking, redundant. However, the t a b l e of f o r c e s i n each member ( d e t a i l e d i n Appendix 'E') i s intended f o r . -s» I D : > • F i g u r e 4.9 C a n t i l e v e r w i t h Loaded J o i n t s use i n d e a l i n g with combinations of l o a d s . I f s e v e r a l loads are a p p l i e d to the s t r u c t u r e s i m u l t a n e o u s l y the f o r c e i n any member may be obtained by adding the e f f e c t s due to each i n d i v i d u a l l o a d . The widths of members and other p r o p e r t i e s may then be obtained by simple c a l c u l a t i o n s i n c e the s t r e s s i s uniform. 74 The f o l l o w i n g t a b l e l i s t s the d e f l e c t i o n f o r a l o a d p l a c e d a t each j o i n t i n t u r n . The d e f l e c t i o n i s g i v e n f o r t h a t j o i n t and f o r JKC-. Load A t D e f l e c t i o n At D e f l e c t i o n Volume J o i n t That J o i n t A t J 55 Cu.In. J l , l o, 0.0025 0 , 0.00294 0.8335 J 2 , 1 ' J 1 , 2 0.0003, 0.0034 -0.0002 , 0.0060 1.1326 J 3 , 1 ' J 1 , 3 0.0012, 0.0043 -0.0003 , 0.0086 1.4238 J 4 , 1 ' J 1 , 4 -0.0009 , 0.0049 -0.0003 , 0.0083 1.6251 J 5 , 1 ' J 1 , 5 0.0012, 0.0059 -0.0004 , 0.0105 1.9737 J 2 , 2 0, 0.0051 0 , 0.0104 1.7080 J 3 , 2 ' J 2 , 3 0.0013, 0.0071 0 , 0.0142 2.3823 J 4 , 2 ' J 2 , 4 0.0040, 0.0090 0.0002 , 0.0175 2.9868 J 5 , 2 ' J 2 , 5 0.0009, 0.0100 -0.0010 , 0.0159 3.3446 J 3 , 3 0, 0.0110 0 , 0.0193 3.6795 J 4 , 3 ' J 3 , 4 0.0036, 0.0154 0.0003 , 0.0238 5.1224 J 5 , 3 ' J 3 , 5 0.0102, 0.0191 0.0057 , 0.0313 6.3633 J 4 , 4 0, 0.0237 0 , 0.0296 7.8988 J 5 , 4 ' J 4 , 5 0.0086, 0.0327 0.0074 , 0.0399 10.9196 J 5 , 5 0, 0.0501 0 , 0.0501 16.7338 4) T i l t e d Symmetrical C a n t i l e v e r s The same f i v e f i b r e c a n t i l e v e r as used i n two p r e v i o u s s e c t i o n s i s shown t i l t e d i n the v e r t i c a l plane i n F i g u r e 4.10. I t i s mounted wi t h i t s x a x i s a t an angle + t o the h o r i z o n t a l so t h a t the g r a v i t y f o r c e s due to s e l f weight are no lo n g e r p a r a l l e l to AB. 75 F i g u r e 4.10 T i l t e d Symmetrical C a n t i l e v e r A one hundred pound lo a d i s a p p l i e d a t L, p a r a l l e l t o the v e r t i c a l so t h a t T = + . The system was analyzed f o r the range o ^ t-1 < 90° and the r e s u l t s are t a b u l a t e d i n Appendix E. In a d d i t -i o n , the,analyses was repeated f o r a three f i b r e c a n t i l e v e r having the same span. As might be expected the forces and structure volumes are very s i m i l a r to those obtained i n Section 2, since the only d i f f e r e n c e i n the two cases l i e s i n the d i r e c t i o n of the s e l f weight forces, The same q u a l i f i c a t i o n regarding optimisation i s also applicable. The reactions d i f f e r considerably however due to the changes i n geometry. In the f i r s t case the load may be divided into h o r i z o n t a l and v e r t i c a l components which are r e f l e c t e d i n the support reactions. In the present case, the load and s e l f weight forces are always v e r t i c a l while the h o r i z o n t a l reactions generate a moment to r e s i s t the turning moment caused by the load. 5) Skew Cantilevers We now turn to skew cant i l e v e r s which are unsyrametrical. The fan r a d i i are unequal and the end point L does not l i e on the axis. The general arrangement of such structures i s shown i n Figure 4.11. The point C must l i e on the circumference of a c i r c l e whose diameter i s AB, ACB being a r i g h t angle. Convenient variable parameters for the s o l u t i o n of such structures are the f i b r e angle 6 and the r a t i o of the fan r a d i i — (or RADRAT i n the computer programme). The rA support spacing D and the values of L and Y i n the equations, are assumed to be s p e c i f i e d . A programme for computer solut i o n of these structures i s given i n Appendix C. A series of these skew ca n t i l e v e r s was investigated, a l l with L = 10.0", but having variable values of Y from F i g u r e 4.11 T y p i c a l Skew M i c h e l l C a n t i l e v e r Rise of stVuctuf-e. F i g u r e 4.12 V e r t i c a l Reactions a t Supports o f M i c h e l l C a n t i l e v e r s of V a r i a b l e Skew 78 zero to 2.5 inches i n steps of 0.25 i n c h . The data from these s t r u c t u r e s i s g i v e n i n Appendix E. The r e a c t i o n s are p l o t t e d i n F i g u r e 4.12 and i t i s i n t e r e s t i n g t h a t the v e r t i c a l components change i n a s t r i c t l y l i n e a l manner. The h o r i z o n t a l components remain substan-t i a l l y c o nstant, s i n c e the torque a p p l i e d to the s t r u c t u r e by the e x t e r n a l l o a d does not change g r e a t l y . F i g u r e 4.13 i n d i c a t e s the way i n which the f i b r e angle, r B 6 , and the r a t i o — , change as the skew of the s t r u c t u r e rA i n c r e a s e s . These curves are almost l i n e a r f o r l a r g e skews, and c o u l d be used to determine the s t r u c t u r e v a r i a b l e s , or a t l e a s t a c l o s e approximation to the a c t u a l v a l u e s . The volume of t h i s range of s t r u c t u r e s does not -change g r e a t l y , i n c r e a s i n g by under 3% over the range. The s t r u c t u r e s i n v e s t i g a t e d i n t h i s chapter p r o v i d e a b a s i s f o r s e l e c t i o n of s t r u c t u r e s to meet a v a r i e t y of d e s i g n s i t u a t i o n s . 7? F i g u r e 4.13 V a r i a t i o n s of F i b r e Angle and Radius R a t i o i n Some Skew M i c h e l l C a n t i l e v e r s 80 CHAPTER 5 THE DESIGN OF JOINTS AND OTHER MODIFICATIONS The range of s t r u c t u r e s c o n s i d e r e d and analyzed i n the p r e c e d i n g chapters are approximations to the optimum minimum volume s t r u c t u r e s c o n s i d e r e d by Maxwell and M i c h e l l . T h e i r d e s i g n , however, s t i l l r e q u i r e s f u r t h e r m o d i f i c a t i o n and c o n s i d e r a t i o n b e f o r e manufacture c o u l d be undertaken, p a r t i c u l a r l y i f the minimum p r a c t i c a l volume i s to be achieved. The equations d e r i v e d i n Chapter 2 and 3 may be used to c a l c u l a t e the member s i z e s f o r a s t r u c t u r e designed t o . support a s p e c i f i e d system of loads i n a g i v e n space, on the assumption t h a t a l l the j o i n t s are pinned. However, no allowance i s made f o r the m a t e r i a l neces-sary to make the a c t u a l j o i n t s , the members being assumed to have a l e n g t h equal to the d i s t a n c e between j o i n t p o i n t s and having a constant c r o s s - s e c t i o n . In f a c t , however, the a x i a l f o r c e a c t i n g along a,;-member must be t r a n s f e r r e d to i t s neighbours, p r i m a r i l y t o those l y i n g along the same f i b r e but a l s o t o some degree to those approximately p e r p e n d i c u l a r to i t . Some e x t r a m a t e r i a l must t h e r e f o r e be i n c o r p o r a t e d i n the j o i n t s to permit t h i s t r a n s f e r of f o r c e between ; members, p a r t i c u l a r l y i f the uniform s t r e s s s p e c i f i e d through-81 out the t h e o r e t i c a l s t r u c t u r e i s nowhere to be exceeded. There are s e v e r a l p r a c t i c a l approaches t o the s o l u t i o n of t h i s problem, a l l of which i n v o l v e e i t h e r a d d i t i o n a l volume or l o c a l i n c r e a s e of s t r e s s l e v e l . Some of these w i l l now be c o n s i d e r e d . The l a s t s e c t i o n o f t h i s chapter d i s c u s s e s the s i g n i f i c a n c e of these approaches i n terms of the f u n c t i o n o f the s t r u c t u r e . 1) Pinned J o i n t s Each member of the framework c o u l d be made s e p a r a t e l y w i t h ends extended beyond the dimensions s p e c i f i e d by geo- : m e t r i c a l c o n s i d e r a t i o n s , to accommodate a 'smooth' p i n p l a c e d a t the j o i n t p o i n t , w i t h i t s a x i s p e r p e n d i c u l a r to the plane of the s t r u c t u r e . • A p o s s i b l e d e s i g n i s i n d i c a t e d i n F i g u r e 5.1. F i g u r e 5.1 P o s s i b l e End Design for- T y p i c a l Member 82 For the purpose of t h i s d i s c u s s i o n i t may be assumed t h a t the members are of constant t h i c k n e s s t . A t y p i c a l member A ^ i s t h e o r e t i c a l l y o f uniform r e c t a n g u l a r s e c t i o n and would terminate a t MN as shown d o t t e d i n the diagram. A j o i n t b e a r i n g may be formed by widening the end as shown to a c i r c u l a r shape, having a diameter ( P + W A a b ) . At the c r i t i c a l plane QK, the c r o s s - s e c t i o n of the-member i s (WA ^ ) x ( t ) as i n the s t r a i g h t p o r t i o n and the s t r e s s w i l l remain f a i r l y uniform. The a x i a l f o r c e w i l l be t r a n s -f e r r e d from the member to the smooth p i n of diameter p, to the l e f t or r i g h t of QK, depending as the member i s i n t e n s i o n or compression. A f i l l e t a t F would ease the change of s e c t i o n and minimize any s t r e s s c o n c e n t r a t i o n . The e x t r a volume of m a t e r i a l per end i s g i v e n approx-imately by the equation Excess » t|^ p->WAob(4rt -^G^WAat>[2ir-4]| . . . . (5.D The p i n i s p l a c e d i n shear across a d i a m e t r a l plane and, from elementary theory, the shear s t r e s s , x, exerted on i t , i s gi v e n by (5.2) 83 The 2 i n the denominator would be i n c l u d e d i f the members were d u p l i c a t e d . As drawn i n F i g u r e 5.1, with s i n g l e members, the p i n would be s u b j e c t e d to a t w i s t i n g moment approximately about the a x i s QK. T h i s may be e l i m i n a t e d by d i v i d i n g A or A , i n t o two equal members each o f t h i c k n e s s a , D — l 3 D ~ and p l a c i n g them symmetrically on e i t h e r s i d e of the o t h e r bar. The two equations above may be used to determine the p i n diameter such t h a t uniform s t r e s s i s s u b s t a n t i a l l y ob-t a i n e d throughout the j o i n t . : The s i g n i f i c a n c e of t h i s i n c r e a s e i n volume was e v a l u a t e d by use of the s e t o f s t r u c t u r e s e a r l i e r examined i n Chapter 4, S e c t i o n 1. The p i n diameter, p, was a r b i t r a r i l y s e l e c t e d as ~", to correspond w i t h t h a t used i n the models a c t u a l l y t e s t e d . S e l e c t i o n of a l a r g e r v a l u e of p would i n c r e a s e the volume s t i l l f u r t h e r . The amount of e x t r a volume i s g i v e n i n Table 5.1 and the v a l u e s are p l o t t e d on F i g u r e s 5.2A-5.2D. There are sub-s t a n t i a l l y F i g u r e s 4.4A-4E repeated w i t h the a d d i t i o n of an upper curve i n d i c a t i n g the t o t a l volume, i n c l u d i n g j o i n t allowance. When j o i n t allowance i s c o n s i d e r e d , the volume no longer decreases c o n t i n u a l l y as N i n c r e a s e s but reaches a minimum and then r i s e s r a p i d l y , tending toward i n f i n i t y as N becomes very l a r g e . I f s t r u c t u r e s are to be made with p i n j o i n t s , there TABLE 5.1 JOINT ALLOWANCES FOR PINNED SYMMETRICAL CANTILEVERS (See Chapter 4, S e c t i o n 1 f o r d e t a i l s of s t r u c t u r e s ) L D N Net Volume J o i n t Allowance Gross Volume % Increase 1 2 8.1807 1.7716 9.9523 2l;66 3 8.1243 2.4297 10.5540 29 .91 4 8.1141 3.1987 11.3128 39.42 5 8.1104 4.0333 12.1437 49.73 7 8.1079 5.8670 13.9749 72.36 10 8.1068 9.0003 17.1071 111.02 2 2 . 12.5873 4.0605 16.6477 32 .26 3 11.9107 4.4663 16.3771 37 .50 4 11.7966 5.2985 17 .0952 44.92 5 11.7574 6.2789 18.0363 53.40 7 11.7295 8 .4894 20.2189 72.38 10 11.7173 12.2614 23.9787 104 .64 4 2 20.6597 12.8822 33.5419 62 .35 3 17.3714 11.3477 28.7191 65 .32 4 16.8940 11.7635 28.6575 69 .63 5 16.7338 12.7282 29 .4620 76 .06 7 16.6213 15.3265 31.9478 92.21 10 16.5721 20.0440 36 .6161 120.95 5 2 24.4237 19 .5259 43.9496 79 .95 3 19.4195 16.3157 35.7352 84.02 4 18.7345 16.2361 34 .9706 86.66 5 18.5071 17.0337 35.5409 92 .04 7 18.3480 19.6439 37.9919 107 .06 10 18.27 87 24.6828 42.9615 135.04 10 2 42.3208 75.9381 118.2589 179 .43 3 26.6088 57.7703 84 .3792 217.11 4 22.8859 52.2136 77.0995 209 .81 5 24 .3336 50.5002 74 . 8339 207.53 7 23.9534 51.3051 75.2585 214.19 10 23.7877 56.4998 80.2876 237.52 85 NUMBER, OF FIBRES N Figure 5.2a E f f e c t of Jo i n t Allowance on Volume of Symmetrical Michell Cantilevers So 5o > VC T J 8 3o 86 With \oint aUou3qr>ce . D ' J i i i 1 1— NUMBER OF FIBRES - N 20 F i g u r e 5.2b E f f e c t of J o i n t Allowance on Volume of Symmetrical M i c h e l l C a n t i l e v e r s - L _ . D V 87 NUMBER OP F<3«£S N F i g u r e 5.2c E f f e c t of J o i n t Allowance on Volume of Symmetrical M i c h e l l C a n t i l e v e r s - T - = 5 D 88. $ooT NUMBER or FIBRES N. F i g u r e . 5 . 2 d E f f e c t o f J o i n t A l l o w a n c e on Volume o f S y m m e t r i c a l M i c h e l l C a n t i l e v e r s -• £ - 1 0 • . 89 i s an optimum number of f i b r e s for minimum practical volume. This number seems to be between 3 and 5 for the range of ^  considered. This conclusion i s reinforced by the p r a c t i c a l consideration that the number of components should be kept to a minimum to reduce manufacturing costs. The c a l c u l a t i o n s on which the preceding curves are based r e l a t e to a series of cantilevers having a span of 10 inches. This may seem small to those interested i n s t r u c t u r a l design, but t h i s order of siz e i s of i n t e r e s t to o p t i c a l engineers. If the span i s increased, the geometric layout and member forces remain unchanged for a given r a t i o . I f the maximum deflection i s also to be kept constant, the permissible stress must be reduced i n proportion to the change of span. This w i l l increase member cross sections correspondingly and the proportion of t o t a l weight attributed to the j o i n t s remains constant. However, i f the maximum stress i s unchanged, so that the d e f l e c t i o n i s increased p r o p o r t i o n a l l y , member cross s e c t i o n a l areas are unchanged, although t h e i r lengths are increased. The proportion of material i n the jo i n t s i s thus reduced. The e f f e c t of t h i s reduction i s indicated i n Table 5.2, which i s based on data extracted from Table 5.1. Consider a cantilever of one hundred inch span, with |j equal to four. I t c a r r i e s a load of one hundred pounds and a maximum stress of three hundred p . s . i . The Net 89a volume figures from Table 5.1 are m u l t i p l i e d by ten but the J o i n t Allowances are unchanged. TABLE 5.2 JOINT ALLOWANCES FOR PINNED SYMMETRICAL CANTILEVERS L=100", £=4, Load=100 l b , 6=300 p . s . i . N Net Volume J o i n t Allowance Gross Volume % Increase 2 206.6 12.9 220.5 6.7 3 173.7 11.3 185.0 6.5 4 168.9 11.8 180.7 7.0 5 167.3 12.7 180.0 7.7 7 166.2 15.3 181.5 9.2 10 165.7 20.0 185.7 12.1 The 5 f i b r e structure now has the minimum volume, whereas with a ten inch span 3 f i b r e s was optimum. This r e s u l t indicates that each combination of parameters should be evaluated separately to determine the optimum case. The general conclusion that 5 f i b r e s represents a p r a c t i c a l upper l i m i t i s confirmed. I t i s u n l i k e l y that structures would a c t u a l l y be made i n t h i s manner. The pinned j o i n t s serve l i t t l e purpose and introduce play, causing extra unwanted d e f l e c t i o n , and the more complex design would be expensive i n labour for manufacture and assembly. 89b The concept remains of considerable value as a basic method of analysis, y i e l d i n g quite accurate r e s u l t s . 2) Rigid Joints The complete M i c h e l l framework could be cut from a single sheet of material, or, i n large s i z e s , made from sub-assemblies welded together. There i s now no problem of j o i n t design, since they are s o l i d , but the analysis i s much more complex, since the framework now i s s t a t i c a l l y indeterminate. In such case, the a x i a l forces i n the members already considered give r i s e to the primary stresses, while secondary stresses a r i s e from the bending moments and shear stresses induced at the r i g i d j o i n t s . The STRUDL programme (STRUctural Design Language) which has been compiled as part of ICES (Integrated C i v i l Engineering System) for the general solution of complex s t r u c t u r e s was used to examine the standard f i v e f i b r e n e t -work a l r e a d y c o n s i d e r e d as a pinned s t r u c t u r e . The c a n t i l e v e r had the f o l l o w i n g p r o p e r t i e s -Span L = 10" Loads a p p l i e d a t L, p a r a l l e l to AB Support Spacing D = 2.5" 1 Member t h i c k n e s s t = 0.25" Number of F i b r e s N = 5 Member widths, as i n t a b l e s F i b r e Angle 9 = 18.41° 10, Appendix E The j o i n t s are r i g i d and the s t r u c t u r e may be v i s u a l i z e d as having been c u t from a f l a t p l a t e one qu a r t e r i n c h t h i c k . > The member widths are such t h a t i n the comparable p i n - j o i n t e d s t r u c t u r e when loaded w i t h one hundred pounds a t L ( J ^ ^^) a uniform s t r e s s o f 300 p . s . i . e x i s t s a t a l l p o i n t s i n a l l members. To t h i s f i x e d s t r u c t u r e — w i t h s p e c i f i e d member s i z e s --a s e r i e s of loads were a p p l i e d s e p a r a t e l y a t L, p a r a l l e l t o AB. These loads v a r i e d from 10000 pounds t o 0.01 pound and i n each case the (constant) weight of the s t r u c t u r e was i n c l u d e d i n the f o r c e a n a l y s i s . T h i s wide range of loads r e p r e s e n t s the extreme s t a t e s of l o a d i n g to be c o n s i d e r e d . With the high l o a d s , the e f f e c t s o f the s e l f weight f o r c e s ' are n e g l i g i b l e w h ile a t the lowest l o a d s , the s e l f weight f o r c e s are of equal importance w i t h the a p p l i e d f o r c e . I t must be remembered i n t h i s a n a l y s i s t h a t the : s t r u c t u r e i s of f i x e d dimension. Thus the s t r e s s i s not, constant but v a r i e s from l o a d i n g to l o a d i n g . W i t h i n l i m i t s , however, the s t r e s s should be rea s o n a b l y c o n s t a n t through-out the s t r u c t u r e f o r any one l o a d i n g . The major data d e r i v e d from t h i s a n a l y s i s are r e -corded i n Table 5.1. Since the s t r u c t u r e i s symmetrical, data i s reproduced f o r the upper h a l f of the s t r u c t u r e o n l y . Each member i s p a i r e d w i t h i t s corresponding p a r t n e r i n the lower s e c t i o n [A ^  and ] --the o n l y d i f f e r e n c e being t h a t the 'B' members are i n compression w h i l e the 'A' members are i n ' t e n s i o n ' . S i m i l a r l y J ^ and correspond, having i d e n t i c a l d e f l e c t i o n s i n the y d i r e c t i o n and equal d e f l e c t i o n s of . op p o s i t e s i g n i n the x d i r e c t i o n . The t a b l e may be d i v i d e d i n t o t e n s e c t i o n s f o r examin-a t i o n . The f i r s t three columns name the v a r i o u s members and show t h e i r width and c r o s s - s e c t i o n a l a r e a . The next column shows the f o r c e s and s t r e s s e s f o r the pinned s t r u c t u r e p r e v i o u s l y examined. The next seven s e c t i o n s show, f o r a s e r i e s of loads ranging from 10000 l b s . to 0.01 l b s . , the f o l l o w i n g d a t a : -a) A x i a l Force i n Each Member, FORCE b) S t r e s s due to t h a t f o r c e = . The nominal con-AREA s t a n t v a l u e i s shown a t the top of the column, c) SHEAR FORCE. T h i s i s the mean of the shears a t each end of member, d) BENDING MOMENTS. The f i r s t column shows the i n n e r moment, t h a t n e a r e s t t o support A, while the second column i s t h a t a t the outer end, TABLE 5.3 92 Member Width Cross Area P i n J o i n t 1 R i g i d S t r u c t u r e 100 l b l o a d - 10000 l b Load Force l b s | Force S t r e s s Shear A10 B 0 1 0.871 0.218 65.29 6989 32120 656 .4 A l l B l l 0.759 0.190 56.90 5862 30900 473.6 A12 B 2 1 0.536 0.134 40.22 2817 21000 426 .3 A 1 3 B 3 1 0.358 0.089 26 .83 1530 17110 192.0 A14 B 4 1 0.217 0.054 16.26 7023 12960 72 .05 A 2 0 B02 1.560 0.390 117.0 11400 29240 1275. A 2 1 B12 1.371 0 .343 102.8 10100 29470 28.76 A22 B22 0.988 0.247 74.10 6841 27770 569 .8 A 2 3 B32 0.675 0.169 50.62 4641 27510 238.4 A24 B42 0.422 0.106 31.67 2897 27430 20.41 A 3 0 B 0 3 1.229 0.307 92.15 8405 27360 53.03 A 3 1 B 1 3 1.100 0.275 82.48 7577 27560 12 .04 A32 B 2 3 0.830 0.208 62.26 6071 29260 130 .9 A 3 3 B 3 3 0.598 0.150 44.88 4454 29770 75.73 A34 B 4 3 0.401 0.100 30 .06 3006 30000 19 .86 A40 B04 0.948 0.237 71.12 6474 27300 241.1 A 4 1 B14 0.868 0.217 65.07 6083 28050 43.09 A42 B24 0.690 0.172 51.73 5089 29520 18.43 A 4 3 B34 0.528 0.132 39 .57 3942 29890 18 .05 A44 B44 0.380 0.095 28.52 2858 30050 3.37 A 5 0 B05 1.353 0.338 101.5 9794 28960 1048. A 5 1 B 1 5 1.336 0 .334 100.2 9763 29240 170.1 A52 A 2 5 1.267 0.317 95.05 9338 29480 25.96 A 5 3 B35 1.202 0.301 90.17 8871 29510 31.45 A54 B45 1.140 0.285 85.53 8430 29570 85.52 T a b l e 5.3 (continued) 93 Bending Moment an a (%) R i g i d S t r u c t u r e - 1000 l b Load an Force S t r e s s Shear B.M. 5— (%) Force 916.9 90 .4 699 .2 3213 65.65 91.71 90 .4 70 .14 511.3 69 .0 586.4 3091 47 .38 51.15 69 .0 58.82 250.5 99 .5 281.8 2101 42.66 25.06 99 .6 28 .26 92.15 101 .0 153.0 1712 19 .22 9 .22 101 .0 15.34 29.04 114 .4 70.24 1296 7.21 2.91 114 .7 7 .03 2015. 67 .9 1141. 2925 127.5 201.6 67 .9 114.5 763.3 33 .1 1010. 2948 2.88 76.34 33 .1 101.3 763.3 67 .8 684.3 2771 57.0 76.34 67 .7 68.62 271.2 51 .9 464 .2 2752 23 .83 27.12 51 .9 46 .54 47.57 23 .3 289.7 2744 2 .03 4.75 23 .3 29 .03 113.3 6 .59 840.7 2737 5.31 11.35 6 .6 84.29 286.5 20 .6 758.0 2757 1.21 28 .66 20 .6 75 .99 277.8 33 .1 607.3 2927 13.10 27 .79 33 .1 60 .87 122 .1 27 .5 445.5 2978 7.57 12 .21 27 .5 44 .65 33.35 16 .6 300.6 3000 1.98 3.33 16 .6 30 .12 469 .8 45 .9 647 .5 2731 24 .13 47 .00 45 .9 64 .88 147.4 16 .8 608.5 2805 4.31 14 .74 16 .8 60.97 94.11 16 .1 501.1 2953 1.85 9.42 16 .1 51.01 49.48 14 .3 394 .3 2989 1.80 4.95 14 .3 39 .50 14.93 8 .24 285.9 3006 0.33 1.49 8 .2 28.63 2349. 106 .3 979 .5 2896 104.9 235 .0 106 .4 98 .04 524.9 24 .2 976.4 2924 17 .0 52 .49 24 .2 97 .74 294.1 17 .5 933.9 2948 2.59 34 .58 17 .5 93 .46 347 .5 19 .6 887.2 2951 3.15 34.76 19 .6 88.78 274.5 17 .1 843.1 2957 8.55 27 .44 17 .1 84 .34 T a b l e 5.3 (continued) 94 R i g i d S t r u c t u r e S t r e s s - 100 l b Load Shear B.M. a t J c c 55 am/ a R i g i d Force S t r u c t u r e - 10 S t r e s s Shear l b Load a t J 5 5 B.M. am/a 322.3 6.58 9 .20 90.4 7.24 33.3 0.68 0.95 96.0 310.1 4 .76 5.13 69 .0 6 .06 32.0 0.50 0.53 69 .3 210.7 4.29 2.52 99.7 2 .91 21.7 0.45 0 .26 101. 171.6 1.94 0.93 101.3 1.57 17.6 0.21 0.10 104. 129 .7 0.73 0.29 115.4 0.71 13.1 0.08 0 .03 124. 293.4 12.78 20.21 67.9 11.79 30.2 1.30 2 .07 67.6 295.7 0.29 7.65 33.1 10.42 30.4 0.04 0.78 32.8 277.8 5.71 7.65 67.7 7.05 28.5 0.58 0.78 67.3 275.9 2.38 2.71 51.8 4.77 28.3 0.24 0.27 50.9 274 .9 0.19 0.47 23.0 2.96 28.1 0.01 0 .04 19.9 274.4 0.54 1.15 6.6 8.65 28.2 0.07 0.12 7.0 276 .4 0.12 2.87 20.6 7.79 28.3 0.01 0 .29 20 .2 293.4 1.31 2.79 33.1 6.24 30.1 0.14 0.29 33.0 298.5 0.76 1.22 27.5 4 .57 30 .6 0.08 0.12 27 .2 300.6 0.19 0.33 16.4 3.07 30.7 0.02 0.03 14.6 273.6 2.42 4.72 46.0 6.62 27.9 0.25 0 .49 46.6 281.1 0.44 1.48 16.8 6 .22 28.7 0.05 0 .15 16.7 295.9 0.19 0.95 16.2 5.21 30 .2 0.02 0.10 16 .9 299 .15 0.18 0.50 14.3 4.03 30.5 0.02 0 .05 14.9 301.1 0.03 0.15 8.2 2.91 30.6 0.0 0.01 8.0 289 .9 10.52 23.55 106.5 9.89 29 .3 1.08 2.41 108. 292.7 1.70 5.25 24 .1 9.87 29.6 0.17 0 .52 23.9 295.0 0.26 3.46 17.5 9 .43 29.8 0.02 0 .35 17.7 295.3 0.32 3.49 19.6 8.94 29.7 0.38 0 .36 20.0 295.8 0.85 2.74 17.1 8.43 29.7 0.08 0.27 16 .8 Table 5.3 (continued) 95 Rigid Structure - 1 lb Load at J 5 5 Rigid Structure-0.1 lb Load at J 5 5 Force Stress Shear B.M. am/ a Force Stress Shear B.M. am/a . 0.95 4.34 0.09 0.12 87.6 0.32 1.45 0.03 0.04 82.2 0.79 4.15 0.07 0.07 71.3 0.26 1.37 0.03 0.02 76 .6 0.38 2.81 0.07 0 .04 110. 0.12 0.92 0.03 0 .01 129 . 0.20 2.20 0.04 0.01 126. 0.06 0.66 0.02 0.01 184. 0,08 1.42 0.02 0.01 201. 0.01 0 .26 0 .01 0.0 598. 1.52 3.90 0.15 0.26 65.8 0.49 1.27 0.04 0.08 61.3 1.34 3.90 0.01 0.09 30.5 0.43 1.25 0.01 0.02 25.0 0.89 3.61 0.07 0.09 63.4 0.28 1.12 0.02 0.02 53.8 0.59 3.52 0.02 0.03 43.9 0.18 1.04 0.0 0.0 24.8 0.35 3.36 0.01 0.01 41.3 0.09 0.89 0.01 0.01 119. 1.09 3.54 0.02 0.02 10.2 0.33 1.08 0.01 0 .01 18.7 0.97 3.54 0.01 0.03 17.5 0 .29 1.06 0.01 0.01 10.2 0.77 3.72 0.02 0.04 32.9 0.23 1.09 0.01 0.01 32.6 0.56 3.76 0.01 0.01 24 .9 0 .16 1.08 0.0 0.0 19.2 0.37 3.68 0.0 U . 0 18.7 0.10 0.98 0.0 0.0 67.2 0.80 3.36 0.04 0.07 51.8 0.21 0.90 0 .02 0 .02 67 .8 0.75 3.45 0.01 0.02 16.7 0.20 0.92 0.01 0.0 16 .6 0.63 3.64 0.01 0.02 23.0 0 .17 0.98 0.01 0.01 42 .1 0.48 3.64 0.0 0.01 19 .7 0.12 0.95 0.0 0.0 35 .5 0.34 3.54 0.0 0.0 15 .0 0.08 0.83 0.0 0.0 51.9 1.08 3.20 0.14 0.29 120. 0.20 0 .59 0 .04 0.08 180. 1.08 3.24 0.03 0.05 21.6 0 .20 0.61 0.01 0.0 10 .6 1.02 3.22 0.01 0.04 18.8 0.18 0.57 0 .01 0.01 30.4 0.95 3.18 0.02 0.05 23.7 0.16 0 .52 0.02 0 .01 44.8 0.88 3.09 0 .02 0.02 14.3 0.12 0.43 0.02 0.0 3.0 Table 5. 3 (continued) 96 Rigid Structure - 0.01 Load Self Weight Forces Only Force Stress Shear B.M. am/ a Force Stress Shear B.M. am/a - 0.253 1.163 0.020 0. 029 80.0 0.246 1.131 0.020 0.029 79.8 0.207 1.088 0 .025 0.020 77.8 0.201 1.057 0.024 0.020 78.1 0.098 0.734 0.028 0.012 138. 0.096 0.713 0.028 0 .012 138. 0.046 0.509 0.017 0.006 210. 0.044 0.492 0 .016 0.006 214. 0.008 0.140 0.009 0.003 985. 0.007 0.126 0 .009 0.003 1094 . 0.391 1.002 0.027 0.061 59.5 0.380 0 .973 0.026 0.059 59 .3 0.337 0.983 0.007 0.018 22.9 0.327 0.953 0.007 0.017 22.5 0.214 0.866 0.010 0.018 50.0 0.207 0.839 0 .010 0.017 49.2 0.134 0.792 0.002 0.004 28.6 0 .129 0.764 0.002 0.004 29 .6 0.068 0.644 0.014 0.008 159. 0.065 0.617 0.014 0 .008 166 . 0.257 0.835 0.014 0.013 24 .6 0.248 0.808 0.014 0 .013 25.4 0.222 0.807 0.006 0.003 68.8 0.214 0.780 0.006 0 .003 6.4 0.172 0.828 0.005 0.008 32 .4 0.166 0.798 0 .005 0.007 32 .3 0.122 0.813 0.003 0.002 16.5 0 .117 0.785 0.003 0.002 15 .4 0.071 0.708 0.005 0.004 92.8 0.068 0.678 0.005 0 .004 97.0 0.156 0 .656 0.013 0.019 75.7 0.149 0 .629 0.013 0.018 77.2 0.146 0.672 0.007 0.004 16.6 0 .140 0 .644 0 .007 0 .003 16 .8 0.124 0.717 0.006 0.007 51.4 0.119 0.687 0.006 0.007 52.8 0.089 0.677 0.005 0.003 43.3 0.085 0.647 0.005 0.003 45.3 0.053 0.561 0.003 0.003 73.9 0 .051 0.531 0.003 0.003 78.1 0.112 0.331 0.036 0.060 237. 0 .102 0.302 0.034 0 .057 249. 0.116 0.347 0 .011 0 . 001 5.0 0.106 0.318 0.011 0 .001 4.2 0.097 0.305 0.015 0.009 43.6 0.087 0 .276 0.276 0 .015 46.6 0.076 0.254 0.017 0.011 71.2 0.067 0 .224 0 .017 0.011 77.8 0.047 0.165 0.018 0 .003 35.8 0.039 0 .135 0 .018 0 .004 47.8 97 e) Maximum Bending S t r e s s , expressed as a percentage of the a x i a l s t r e s s . My a g = — , M being larger of the two I bending moments, WIDTH y = 2 . t x (WIDTH) 3  1 " 12 The f i n a l column a t the extreme r i g h t hand rep e a t s t h i s i n f o r m a t i o n f o r the s e l f weight f o r c e s a c t i n g alone, without e x t e r n a l l o a d . In comparing the v a r i o u s f i g u r e s , i t i s noteworthy t h a t the bending s t r e s s e s i n many members are comparable- with the a x i a l s t r e s s e s . Since these are p a r a l l e l and can be added d i r e c t l y , t h i s i m p l i e s t h a t the s t r e s s d i s t r i b u t i o n i s f a r from uniform, being as l a r g e as 2a or more a t one outer f i b r e and d e c l i n i n g u n i f o r m l y a c r o s s the s e c t i o n . The percentage r a t i o f o r a giv e n member changes but l i t t l e f o r some members wh i l e i n others the s e l f weight e f f e c t s are such t h a t these predominate a t the lower loads and g i v e r i s e t o some very high percentages. While the s t r e s s i n the pinned s t r u c t u r e i s con s t a n t f o r a l l members, i t i s by no means constant when the a x i a l f o r c e s alone are co n s i d e r e d i n the r i g i d cases. The va l u e s at the outer end of the s t r u c t u r e are reasonably constant but d e c l i n e c o n t i n u o u s l y along the N ^ f i b r e t o l e s s than 50% of the nominal v a l u e (to almost 10% i n the s e l f weight c a s e ) . 98 The f o l l o w i n g t a b l e summarizes the d e f l e c t i o n s f o r the end j o i n t J(r-LOAD DEFLECTION AT JOINT J(5.5) STRUDL ' TRUSS' STRUDL 'FRAME 1 * X Y Y 0 rads 10000 l b . -0 .010633 -4.696458 -4 .828454 -0 .592301 S e l f Weight -0 .000000 -0.000068 -0 .000095 -0 .000006 10000 + SW -0 .010634 -4 .696526 -4 .828458 -0 .592306 1000 + SW -0 .001064 -0.469714 -0 .482940 -0 .059236 100 + SW -0 .000107 -0.047033 -0 . 048380 -0 .005929 10 + SW -0 .000011 -0.004765 -0 .004923 -0 .000598 1.0 + SW -0 .0000013 -0.000538 -0 .000578 -0 .000065 0.1 + SW -0 . 0000001 -0.000115 -0 .000143 -0 .0000115 0.01 + sw -0 .0000003 -0.000073 -0 .000100 -o .000006 * A l l 'X' d e f l e c t i o n s f o r Frame are e f f e c t i v e l y Zero. BIAXIAL STRESS SYSTEMS IN JOINT AREAS F i g u r e 5.3 i n d i c a t e s the arrangement of a t y p i c a l j o i n t i n a M i c h e l l framework. The t h e o r e t i c a l shape of the members i s shown d o t t e d and i t w i l l be seen t h a t there i s c o n s i d e r a b l e o v e r l a p i n the j o i n t a rea, QRVW. Even i f secondary s t r e s s e s a r i s i n g from the r i g i d i t y of the j o i n t are n e g l e c t e d , i t i s obvious t h a t i n s i d e the area Figure 5.3 Typical Joint i n a Rigid Michell Framework 100 QRVW t h e m a t e r i a l i s s u b j e c t e d t o b i a x i a l s t r e s s , whereas t h e main p a r t o f e a c h member c a r r i e s o n l y u n i a x i a l s t r e s s , ±a. T h e s e two s t r e s s s y s t e m s may be compared u s i n g Mohr's S t r e s s C i r c l e , as shown i n F i g u r e 5.4. Members a) UNIAXIAL STRESS Tensile Klormotl Sh-cas b) BIAXIAL STRESS F i g u r e 5.4 Mohrs C i r c l e f o r U n i a x i a l and B i a x i a l S t r e s s I n F i g u r e 5.4a, two c i r c l e s a r e shown, one r e l a t i n g t o members i n t e n s i o n and t h e o t h e r s t o t h o s e i n c o m p r e s s i o n . The u n i f o r m u n i a x i a l s t r e s s i s ±o and t h e c i r c l e has a 101 r a d i u s of The maximum shear s t r e s s i s thus equal t o The b i a x i a l s t a t e i s i n d i c a t e d i n F i g u r e 5.4b, the s t r e s s e s i n the two s e t s being of equal but o p p o s i t e i n t e n s i t y , ( ± a ) . Mohr's c i r c l e now has a r a d i u s \a\ and the maximum shear s t r e s s i s twice the v a l u e f o r a u n i a x i a l system. Thus on t h i s simple a n a l y s i s , i f the maximum s t r e s s a t any p o i n t i s not to exceed a, the j o i n t volume must be doubled. T h i s may most e a s i l y be done by d o u b l i n g the t h i c k -ness l o c a l l y . I f F i g u r e 5.3b i s examined, i t w i l l be seen t h a t t h e r e i s c o n s i d e r a b l e o v e r l a p o f the t h e o r e t i c a l members i n s i d e the j o i n t area QRWV. The ends of the two A members, A , .. and A , , abut, a , £)—J. aJD t h e i r s m a l l o v e r l a p a t the bottom being equal to the t r i a n g u l a r gap above. To a f i r s t approximation the volume of the end p o r t i o n s of these two members equals the volume of QRWV. S i m i l a r l y the volume of the ends o f B . , and B , approximate! a J., io ao equal the volume of the j o i n t a r e a . Thus the t o t a l volume c a l c u l a t e d from the t h e o r e t i c a l dimensions o f the members i s about twice the volume of the j o i n t and thus equals the amount of m a t e r i a l necessary t o prov i d e uniform s t r e s s a t a l l p o i n t s . T h i s volume has a l r e a d y * I f the s t r e s s e s i n the two members are equal i n s i g n , Mohr's c i r c l e w i l l reduce to a p o i n t and the s t r e s s on a l l planes w i l l be ±cr. T h i s w i l l occur' f o r c e r t a i n l o a d i n g c o n d i -t i o n s as shown i n Table 5, Appendix E. 102 been c a l c u l a t e d and i s shown i n Chapter 4 — a n d the a p p r o p r i a t e t a b l e s i n Appendix E — a s the s t r u c t u r a l volume. I f a frame-work were cu t from a s i n g l e sheet of m a t e r i a l , or assembled from components of uniform t h i c k n e s s , a patch p l a t e of equal t h i c k n e s s would have to be added over each j o i n t t o s a t i s f y the c o n d i t i o n of uniform s t r e s s and s i m u l t a n e o u s l y i n c r e a s e the t o t a l volume to t h a t e a r l i e r c a l c u l a t e d . E x t r a volume would a l s o have to be added to p r o v i d e f i l l e t s along QR, RW, WV and VQ to smooth the change i n c r o s s -s e c t i o n and reduce s t r e s s c o n c e n t r a t i o n . In a d d i t i o n the square corners a t Q, R, W and V should be rounded f o r the same reason, which would again i n c r e a s e the volume. The percentage i n c r e a s e i n volume n e c e s s i t a t e d by . these f i l l e t s w i l l vary w i t h the d e s i g n s p e c i f i c a t i o n s , s i n c e i t i s a f u n c t i o n of the number of j o i n t s . I t s e f f e c t i s . c e r t a i n l y s m a l l f o r N l e s s than 5 which f o r p r a c t i c a l use probably r e p r e s e n t s an upper l i m i t on t h i s v a r i a b l e . The preceding paragraphs d i s c u s s i n g the volume con-s i d e r e d o n l y the e f f e c t of the primary a x i a l s t r e s s e s . The r e a l s i t u a t i o n i s complicated by the presence -of bending and shear s t r e s s e s . T h i s i s d i s c u s s e d f u r t h e r i n Appendix F. DEFLECTION OF JOINTS SUBJECTED TO BIAXIAL STRESS The s t r u c t u r a l d e f l e c t i o n s c o n s i d e r e d i n Chapter 4 were c a l c u l a t e d on the assumption t h a t the members were i n u n i a x i a l s t r e s s throughout t h e i r l e n g t h . When the j o i n t s 103 are s u b j e c t e d t o b i a x i a l s t r e s s , the e x t e n s i o n o f each member w i l l be somewhat i n c r e a s e d due to the e f f e c t of Poisson's r a t i o , assuming t h a t the f o r c e s i n each s e t of members are of d i f f e r e n t s i g n . Equations s p e c i f y i n g the amount of t h i s i n c r e a s e are d e r i v e d i n Appendix F, and the m o d i f i e d d e f l e c -t i o n s were c a l c u l a t e d f o r the range of s t r u c t u r e s d e s c r i b e d i n Chapter 4, S e c t i o n 1. The r e s u l t s o f t h i s c a l c u l a t i o n a r e ' t a b u l a t e d i n Appendix F and are shown g r a p h i c a l l y i n F i g u r e 5.5. As might be expected, the e f f e c t of b i a x i a l s t r e s s on s t r u c t u r e d e f l e c t i o n becomes more s i g n i f i c a n t as the number of members i n c r e a s e s . T h i s f o l l o w s f o r two reasons:-2 a) The number of j o i n t s i s N , and t h i s i s a j o i n t i e f f e c t . b) As N i n c r e a s e s , the l e n g t h of each member decreases and thus the p r o p o r t i o n of the l e n g t h i n the j o i n t area i n c r e a s e s . T h i s i l l u s t r a t e s again the p r e d i c t i o n s of the M i c h e l l c r i t e r i a enunciated i n Chapter 1. Here s t r u c t u r e s of l i k e g e o m e t r i c a l shape are being compared, but with d i f f e r i n g s t r e s s systems. I t can be seen t h a t the s t i f f n e s s o f the frameworks having b i a x i a l s t r e s s a t the j o i n t s i s l e s s than t h a t of the systems having uniform s t r e s s e s a t a l l p o i n t s . F i g u r e 5.5 D e f l e c t i o n o f Some M i c h e l l C a n t i l e v e r s Due t o U n i a x i a l and B i a x i a l S t r e s s e s 105 3) S e m i - r i g i d J o i n t s The s t r u c t u r a l members i n a M i c h e l l framework form two sets of members, corresponding to the orthogonal c u r v i -l i n e a r coordinates of the t h e o r e t i c a l case. As one moves along a set of these members, approximating a c u r v i l i n e a r c o o r d i n a t e , the for c e increases i n each member, toward the support p o i n t s . For any given member, the l a r g e r p o r t i o n of the force a c t i n g along i t i s t r a n s f e r r e d to i t s neighbours i n the same s e t , while only a small p r o p o r t i o n i s interchanged w i t h the adjacent members of the sets c r o s s i n g t h i s f i b r e at each j o i n t . r Advantage of t h i s f a c t could be taken when assembling a framework. The set of members forming one f i b r e extending from a support p o i n t could be f a b r i c a t e d as a s o l i d unit> e i t h e r by c u t t i n g from a s i n g l e sheet, or by welding of sub-assemblies. The two 'fans' of f i b r e s would then be assembled i n p a r a l l e l planes, as i n d i c a t e d i n Figure 5.6. The connection between the f i b r e s at the j o i n t p o i n t s could be made by gl u e i n g the f l a t surfaces together, or by use of a small diameter p i n or r i v e t placed at the geometrical j o i n t p o i n t . The connection acts merely to t r a n s f e r the forces between s e t s , and i s not r e q u i r e d to c a r r y the load as i n the case of the pinned j o i n t . In p r a c t i c e , to e l i m i n a t e t w i s t i n g i n the plane of the drawing, one set of members should be d u p l i c a t e d w i t h 106 F i g u r e 5.6 S e m i - r i g i d J o i n t s i n a M i c h e l l Framework 107 hal f the thickness i n each, and placed on either side of the other set. As a close approximation, no extra material i s needed above that t h e o r e t i c a l l y required and previously calculated. If pinned or r i v e t t e d j o i n t s are employed, some increase i n width i n the j o i n t area would be required to maintain a constant stress l e v e l . Since the transfer forces are a f r a c -t i o n of the t o t a l force, the diameter of the pin or r i v e t could be kept small and a l o c a l increase i n stress l e v e l could perhaps be tole r a t e d . The 'glued' j o i n t would be preferable, requiring less machining and providing greater r i g i d i t y . I t i s a compromise between the two solutions described e a r l i e r and should provide an e n t i r e l y s a t i s f a c t o r y design s o l u t i o n . ELASTIC BUCKLING OF MEMBERS AND FRAMES The members comprising the frameworks considered i n Chapter 4 are expressly designed for minimum volume, Members i n compression are therefore l i a b l e to f a i l by e l a s t i c buckling. A routine check was therefore made on each member i n EI TT 2 t h i s regard. As a c r i t e r i o n the Euler c r i t i c a l load, — ^ — was calculated for each member. It was assumed that the member was unconstrained at i t s ends and free to buckle i n ei t h e r d i r e c t i o n , depending 108 whether i t s width or t h i c k n e s s was the l e s s e r . R i g i d j o i n t s at the end would i n c r e a s e the r e s i s t a n c e of the member to b u c k l i n g so the c a l c u l a t e d v a l u e r e p r e s e n t e d a lower bound to i t s c r i t i c a l l o a d . T h i s c r i t i c a l l o a d was compared, as a percentage, w i t h the a c t u a l a x i a l f o r c e imposed. In most cases the r a t i o was c o n s i d e r a b l y s m a l l e r than 100%, i m p l y i n g t h a t the s t r u c t u r e c o u l d r e s i s t b u c k l i n g f o r the d e s i g n l o a d . B u c k l i n g was c o n f i n e d to two types o f case -a) where members were ve r y l o n g , u s u a l l y A N  a n d BN-1,N ' b) i s o l a t e d examples of members c a r r y i n g v e r y s m a l l l o a d s . Here the c r o s s - s e c t i o n s were v e r y f i n e i n p r o p o r t i o n t o t h e i r l e n g t h . Redesign of these c r i t i c a l members would e l i m i n a t e t h i s problem. Depending on the method of c o n s t r u c t i o n , r e d i s t r i b u t i o n of the m a t e r i a l from a r e c t a n g u l a r c r o s s - s e c t i o n t o a t u b u l a r shape, or the a d d i t i o n o f s t r e n g t h e n i n g r i b s , would be p r a c t i c a l s o l u t i o n s . The l a t t e r case would i n v o l v e an i n c r e a s e of s t r u c t u r a l volume. Many o f the s t r u c t u r e s i n v e s t i g a t e d were ve r y t h i n i n p r o p o r t i o n t o t h e i r t h i c k n e s s . Spans of 10" were common wh i l e the t h i c k n e s s was i n v a r i a b l y , g i v i n g a r a t i o of 40 to 1. T h i s c o u l d g i v e r i s e t o b u c k l i n g o f the whole frame, p a r t i c -u l a r l y i f the l o a d were not a p p l i e d e x a c t l y i n i t s p l a n e . 109 The tests i s o l a t e d a c a n t i l e v e r , but i n practice the framework would normally form part of a larger structure with s i m i l a r frameworks located i n p a r a l l e l or other planes. These frames would be interconnected and would provide mutual support against buckling, as i s common pra c t i c e i n the design of bridge trusses, cranes, towers and so on. Buckling was thus f e l t to be a minor problem i n the use of M i c h e l l frameworks, providing no more d i f f i c u l t y than i s at present encountered i n other designs. I t s incidence may occur at somewhat lower stress l e v e l s since a M i c h e l l framework i s by i t s nature, of minimum volume as compared to others working i n the same space and subjected to the same loading. SELECTION OF JOINT TYPE AND DESIGN The preceding sections of t h i s chapter describe various ways i n which the j o i n t s of a M i c h e l l structure may be designed and analyzed. The actual design selected i n a given case may well be l a r g e l y determined by the functional requirements of the structure. In many cases a structure i s designed to contain a minimum volume of material. In such cases a permissible or working stress i s s p e c i f i e d which i s as high as i s considered safe for that a p p l i c a t i o n , being determined by the l i m i t 1 1 0 of p r o p o r t i o n a l i t y of the material, a maximum set by con-side r a t i o n of fatigue or by reference to a standard code of p r a c t i c e . Since t h i s stress i s the maximum permissible anywhere i n the structure, t h i s w i l l determine the value of the b i a x i a l stress shown i n Figure 5.4b, which i s double the unaxial stress acting along the greater part of each member. Two solutions are available to the designer. He may e l e c t to dimension the j o i n t areas i n conformity with the b i a x i a l stress, and reduce the cross s e c t i o n a l area of the members so that the u n i a x i a l stress i s ra i s e d to the same l e v e l . A l t e r n a t i v e l y , the member cross sections may f i r s t be selected and extra material added i n the area of the j o i n t s so that the stress l e v e l i s maintained sensibly constant. The two approaches y i e l d the same solution and i t i s a matter of convenience which i s followed. However, i n other design s i t u a t i o n s , the s t i f f n e s s of the structure i s more important than absolute economy of material (see for example, the mirror substrates discussed i n Chapter 9 ) . I t has been shown i n Chapter 1 that a M i c h e l l structure has the l e a s t d e f l e c t i o n as well as the lea s t volume of any structure supporting the same loads i n the same space. In a given case, once the geometry of the system i s established, the d e f l e c t i o n of the structure i s d i r e c t l y r e l a t e d to the (uniform) s t r a i n , as explained i n I l l Chapter 2. The s t r e s s throughout the s t r u c t u r e i s thus f i x e d , assuming i t to be c o n s t r u c t e d from one m a t e r i a l , and may w e l l be a t a l e v e l c o n s i d e r a b l y below t h a t used i n a minimum volume d e s i g n . In t h i s case the i n c r e a s e d s t r e s s l e v e l i n the j o i n t s may w e l l be of l i t t l e s i g n i f i c a n c e and can be t o l e r -ated without the a d d i t i o n o f e x t r a m a t e r i a l i n these areas. To summarize these a l t e r n a t i v e s , i t i s convenient to r e f e r to the s t r u c t u r a l volumes l i s t e d i n Table 5.1. The volumes shown i n the column headed 'Net Volume 1 r e l a t e e q u a l l y to ' t h e o r e t i c a l ' p i n - j o i n t e d s t r u c t u r e s i n which no allowance has been made f o r j o i n t d e s i g n and t o r i g i d j o i n t e d s t r u c t u r e s i n which e x t r a m a t e r i a l has been added t o the j o i n t s to ma i n t a i n a uniform s t r e s s l e v e l throughout the s t r u c t u r e . The 'gross volume' f i g u r e s apply to p i n - j o i n t e d s t r u c t u r e s w i t h j o i n t s designed to ma i n t a i n a uniform s t r e s s l e v e l . I f h i g h e r s t r e s s l e v e l s may be t o l e r a t e d i n the j o i n t areas i n a ' d e f l e c t i o n l i m i t e d ' d e s i g n the s t r u c t u r e c o u l d be made from a sheet of uniform t h i c k n e s s . The s t r u c t u r e volume w i l l be less than the 'Net Volume' shown i n t h i s t a b l e . The e f f e c t of t h i s r e d u c t i o n i s i n d i c a t e d i n the f o l l o w i n g t a b l e which r e l a t e s t o c a n t i l e v e r s having the f o l l o w i n g dimensions:-L - 10 inches 5" ~ 4 D - 2.5 inches Sheet t h i c k n e s s - 0.25 inches Number of F i b r e s Net Volume Overlap Reduced Volume Percentage Reduction 2 20.66 3.99 16.67 19.3 3 17.37 4.48 12.89 25.8 4 16.89 4.78 12.11 28.3 5 16 .73 4.95 11.78 29 .6 7 16.62 5.17 11.45 31.1 10 16 .57 5.37 11.20 32.4 I t w i l l be noted t h a t the o v e r l a p volume i n c r e a s e s as the number of j o i n t s i n c r e a s e s . The reduced volume of these s t r u c t u r e s i s l e s s than the t h e o r e t i c a l minimum volume f o r the i n f i n i t e f i b r e s t r u c t u r e , but t h i s i s o n l y achieved because of the l o c a l excesses i n s t r e s s i n t e n s i t y . T h i s a d d i t i o n a l saving of m a t e r i a l has not been co n s i d e r e d elsewhere i n t h i s paper, s i n c e i t o n l y a p p l i e s to designs i n which the d e f l e c t i o n i s l i m i t e d t o s p e c i f i e d -v a l u e s . Where i t can be a p p l i e d , t h i s i n c r e a s e s the advantage of M i c h e l l s t r u c t u r e s over other d e s i g n s . I t i s assumed i n the above d i s c u s s i o n t h a t s t r e s s c o n c e n t r a t i o n s i n t r o d u c e d by s e c t i o n changes may be ign o r e d . These occur i n a l l types o f d e s i g n and are not unique to M i c h e l l s t r u c t u r e s . I f such c o n c e n t r a t i o n s become s i g n i f i -cant i n a s p e c i f i c case, they c o u l d presumably be reduced or e l i m i n a t e d by l o c a l m o d i f i c a t i o n o f s e c t i o n p r o f i l e . T h i s u s u a l l y i n v o l v e s o n l y a s l i g h t i n c r e a s e i n volume. 113 CHAPTER 6 COMPARABLE STRUCTURES In the p r e v i o u s c h a p t e r s , the t h e o r e t i c a l s u p e r i o r i t y of the M i c h e l l s t r u c t u r e has been e s t a b l i s h e d . I t has been shown to be a c l o s e approximation to the t h e o r e t i c a l optimum minimum volume s t r u c t u r e occupying the same space and c a r r y -i n g the same l o a d . A s e r i e s o f s t r u c t u r e s of v a r i o u s d e s igns w i l l now be examined to e s t a b l i s h a more q u a n t i t a t i v e b a s i s f o r t h i s a s s e r t i o n . An i n f i n i t e range of de s i g n parameters e x i s t so any s e l e c t i o n of s p e c i f i c v a l u e s can serve o n l y as a demonstration of these near optimum p r o p e r t i e s . , A simple c a n t i l e v e r was s e l e c t e d s i n c e M i c h e l l s t r u c t u r e s of t h i s type have a l r e a d y been s t u d i e d i n t e n s i v e l y . The a r b i t r a r y f i x e d parameters of the l o a d i n g system are i l l u s t r a t e d i n F i g u r e 6.1. A l o a d o f 100 pounds i s exer t e d a t r i g h t angles to the x a x i s , a t a d i s t a n c e of ten inches from the o r i g i n . The s t r u c t u r e i s ' b u i l t - i n ' t o a r i g i d support along the y a x i s , the support area being symmetrical about the a x i s , and having a maximum width of 2.5 i n c h e s . Where a p p r o p r i a t e , f i x e d support p o i n t s A and B are p o s i -t i o n e d a t the out e r edges of t h i s a r ea. 114 Figure 6.1 Comparative Structures A l l structures are assumed to be made from the same material, for which:-Modulus of E l a s t i c i t y E = 300,000 p . s . i . Maximum Stress a = 300 p . s . i . The space occupied by the structure i s otherwise un-r e s t r i c t e d . A l l structures are compared, as regards volume, with that of the t h e o r e t i c a l optimum framework, having an i n f i n i t e number of f i b r e s . The d e f l e c t i o n of the comparison structures i s s p e c i f i e d where possible to be i d e n t i c a l with 1 X 5 that of the f i v e f i b r e M i c h e l l structure (0.050 " ) . In some cases the s p e c i f i e d d e f l e c t i o n and stress are incompatible. Table 6.1 compares the r e s u l t s obtained from these c a l c u l a t i o n s . The various structures are described b r i e f l y below, while f u l l e r d e t a i l s and the associated c a l c u l a t i o n s may be found i n Appendix G. STRUCTURE AND TYPE VOLUME cu.in. % A I n f i n i t e Fibre Array 16.4826 100.0 B Five Fibre M i c h e l l 16.7338 101.52 C Three Fibre M i c h e l l 17.3714 105.39 D Warren Truss 22.8191 138.44 E 2 bar Cantilever 43.9233 266.48 F • Parabolic Section 50.1000 303.96 G Triangular Plate 50.1000 303.96 H C y l i n d r i c a l Cantilever 52.7911 320 .28 J I-beam Cantilever 86 .9854 527 .74 K Rectangular section 150.3000 911.87 DESCRIPTION OF COMPARATIVE STRUCTURES A. I n f i n i t e F i b r e A r r a y T h i s i s the t h e o r e t i c a l optimum s t r u c t u r e a g a i n s t which a l l the o t h e r s are compared. I t has an ~ equal to 4.0, an i n c l u d e d f i b r e angle of 74.26°, and i t s volume, from e q u a t i o n 1.19, i s l i s t e d i n Ta b l e 3 o f Appendix E as 16.4826 c u . i n . 116 B and C. 5 and 3 F i b r e M i c h e l l C a n t i l e v e r s These s t r u c t u r e s were d e s c r i b e d f u l l y i n Chapter 4, S e c t i o n 1, and t h e i r p r o p e r t i e s are o u t l i n e d i n Table 1, Appendix E. R i g i d M i c h e l l frameworks c o u l d be made, having these volumes c l o s e to the optimum v a l u e , as d e s c r i b e d i n Chapter 5, S e c t i o n 2. , The d e f l e c t i o n of the f i v e f i b r e frame 0.0501", i s taken as standard f o r the f o l l o w i n g s t r u c t u r e s . D. Warren Truss I t seems l o g i c a l t h a t the form o f s t r u c t u r e most l i k e l y to compete e f f e c t i v e l y w i t h a M i c h e l l frame, would have a s i m i l a r arrangement of members. There are. many designs of t r u s s i n r e g u l a r use, and the s e l e c t i o n of one de s i g n f o r comparison was l a r g e l y a r b i -t r a r y . The t r u s s s e l e c t e d c o n s i s t s of a number o f e q u i l a t e r a l t r i a n g u l a r p a n e ls, the d e t a i l s o f which are gi v e n i n Appendix G. F o r t i t u o u s l y , i f s i x panels are used to span ten in c h e s , the v e r t i c a l h e i g h t of the t r u s s i s 2.4744" which c l o s e l y approximates the s p e c i f i e d spacing of the support p o i n t s , 2.5 in c h e s . The volume of the t r u s s i s 22.8191 c u . i n . and i t s downward d e f l e c t i o n i s 0.06 39 i n . 117 E. Two Bar C a n t i l e v e r This i s the simplest multi-member s t r u c t u r e t h a t can be used to c a r r y the s p e c i f i e d l o a d i n g , and could be regarded as a p r e l i m i n a r y step to the true M i c h e l l framework. I t i s an optimum Maxwell frame, since the s t r e s s i s everywhere uniform, but does not s a t i s f y the M i c h e l l c r i t e r i a . . The two bars are supported at A and B and jo i n e d at L, Figure 6.1. Their c r o s s - s e c t i o n a l area i s f i x e d , the shape may be of any convenient p a t t e r n provided b u c k l i n g i s con-sidered Length of members - 10.0778" C r o s s - s e c t i o n a l area - 2.179 s q . i n . (1.4760 in,square or 1.666 in.diameter) Volume - 43.9233 c u . i n . Note t h a t the d e f l e c t i o n i s a f u n c t i o n of the geometry and thus l i m i t s the s t r e s s l e v e l to 184.98 p . s . i . ^ l e s s than the s p e c i f i e d 300 p . s . i . F. C a n t i l e v e r of P a r a b o l i c Section This i s a c a n t i l e v e r of constant width but of va r y i n g depth, so proportioned t h a t the s t r e s s i n the outer f i b r e s i s everywhere 300 p . s . i . The depth i s zero at L and a maximum at the plane AB. I t s major dimensions are:-118 Width (constant) - 2.824 i n . Root Depth - 2.661 i n . Volume - 50.1000 cu.in. G. Triangular Plate This i s an a l t e r n a t i v e to design F which aims at the same r e s u l t — t o vary the cross-sectional proportions along the c a n t i l e v e r so that the stress i n the outer f i b r e i s maintained constant. In t h i s case the depth of the ca n t i l e v e r i s maintained constant while the width increases uniformly from L toward the root. The major dimensions are:-Depth (constant) - 1.9960 i n . Width at root (max.) - 5.0200 i n . Volume - 50.1000 cu.in. It w i l l be noted that the volumes of structures F and G are i d e n t i c a l . This arises since they are both designed to s a t i s f y the same c r i t e r i a of uniform stress i n the outer f i b r e . 119 H.J.K. CANTILEVERS OF UNIFORM CROSS-SECTION The l a s t t hree s t r u c t u r e s to be c o n s i d e r e d have uniform c r o s s - s e c t i o n s and are the l e a s t e f f i c i e n t , s i n c e much of t h e i r m a t e r i a l i s s t r e s s e d below the l e v e l a t t a i n e d by a l l the m a t e r i a l i n a M i c h e l l framework. Three simple cases are c o n s i d e r e d i n t h i s c a t e g o r y : -a) a c i r c u l a r c r o s s - s e c t i o n of diameter d As can be seen i n Appendix G, a c i r c u l a r c r o s s - s e c t i o n cannot s i m u l t a n e o u s l y meet the s p e c i f i e d requirements of d e f l e c t i o n and s t r e s s . I f the d e f l e c t i o n i s maintained, the maximum s t r e s s i s 584.5 p . s . i . and the volume i s 52.7911 c u . i n . , which is- quoted i n the t a b l e . I f the maximum s t r e s s i s maintained a t 300 p . s . i . , the volume i s reduced to 13.907 c u . i n . , but the : d e f l e c t i o n becomes 0.722". b) an I beam T h i s i s more amenable to c a l c u l a t i o n than the c y l i n d e r , s i n c e depth and width may be v a r i e d independently. A v a r i e t y of s e c t i o n s may be c o n s i d e r e d ; i f the t h i c k n e s s of the web and f l a n g e s i s taken as 0.25" to conform w i t h the M i c h e l l frameworks, the f o l l o w i n g dimensions are o b t a i n e d : -Width of f l a n g e s - 16.982 i n . Depth of I-beam - 1.33067 i n . Volume - 86.9854 i n 3 . 120 c) a r e c t a n g u l a r beam A unique s o l u t i o n may be obt a i n e d by u s i n g a beam 1.33067 inches deep'by 11.295 inches wide. T h i s beam has a volume of 150.3 c u . i n . These l a s t cases w e l l i l l u s t r a t e the b a s i c s u p e r i o r -i t y o f a M i c h e l l framework. 121 CHAPTER 7 MANUFACTURE AND TESTING OF MODELS The p r e v i o u s chapters have d e s c r i b e d and d i s c u s s e d a range of M i c h e l l frameworks, p r o v i d i n g c l o s e approximations to the t h e o r e t i c a l optimum minimum volume s t r u c t u r e . These designs were themselves t h e o r e t i c a l , the v a l u e s quoted being o btained e n t i r e l y from mathematical computation. Some models were t h e r e f o r e made i n conformity w i t h the d e s i g n requirements e s t a b l i s h e d i n the p r e v i o u s c h a p t e r s . These were made from a b i r e f r i n g e n t t r a n s p a r e n t p l a s t i c , Columbia Resin CR39, and were then examined under l o a d i n a p o l a r i s c o p e t o determine the i n t e r n a l s t r e s s d i s t r i b u t i o n . The models were c u t from ~" t h i c k p o l i s h e d sheet s u p p l i e d by the manufacturers, the Homalite Corporation,,and were edge m i l l e d a t low r a t e s of feed and with ample c o o l a n t , to minimize edge e f f e c t s due to h e a t i n g . T h i s work was performed on a n u m e r i c a l l y c o n t r o l l e d m i l l i n g machine and d e t a i l s o f the programming are g i v e n i n Appendix H. Time edge e f f e c t s i n the p l a s t i c were minimized by examining the model as soon as p o s s i b l e a f t e r c u t t i n g . Dead weight l o a d i n g was used and the f r i n g e p a t t e r n s were recorded p h o t o g r a p h i c a l l y . The p o l a r i s c o p e used a d i f f u s e white l i g h t source ( f l o u r e s c e n t tubes) and employed twelve i n c h diameter p o l a r -i z e r , a n a l y z e r and q u a r t e r wave p l a t e s . 122 The v a r i o u s models t e s t e d willnow be d e s c r i b e d . 5 F i b r e P i n - j o i n t e d C a n t i l e v e r T h i s s t r u c t u r e i s i l l u s t r a t e d i n F i g u r e 7.1, and has the f o l l o w i n g major dimensions:-Number of f i b r e s N - 5 Span L - 20.0 i n . Support spacing D - 5.0 i n . L/D = 4 Members made from sheet j 1 ' t h i c k P i n j o i n t s - p i n diameter . T h i s s t r u c t u r e was made with twice the span of the other models to pro v i d e e x t r a space f o r assembly. The members were ground to width and made i n se t s of f o u r , two p a r a l l e l quarter- i n c h t h i c k p l a t e s being used to form each member. The j o i n t h o l e s were d r i l l e d and then reamed t o a t i g h t : c l e a r a n c e f i t to minimize p l a y . The j o i n t p i n s were made from lengths o f j " d i a . s t e e l d r i l l r o d . The members were d u p l i c a t e d t o e l i m i n a t e t w i s t i n g a t the j o i n t s - - t h e t o t a l member t h i c k n e s s thus being h a l f an i n c h . A t y p i c a l j o i n t thus had e i g h t members meeting a t i t and assembly became q u i t e complex. The fan members were r e p l a c e d by s o l i d aluminum p l a t e s to ease a s s e m b l y — t h e s t r e s s e s i n t h i s area would have been very d i f f i c u l t to measure s i n c e the v a r i o u s members overlapped c o n s i d e r a b l y . 123 F i g u r e 7.2A S t r e s s p a t t e r n s i n P i n n e d M i c h e l l C a n t i l e v e r C a r r y i n g 110 l b . I n n e r End 124 MEMBER SIZES IN 5 FIBRE PINNED CANTILEVER Member Width Member Width A42' B24 0.815 A21' B12 1.620 A 5 2 ' B 2 5 1.502 A23' B32 0.798 A 3 1 ' B 1 3 1.300 A 3 3 ' B 3 3 0.707 A41' B14 1.026 A43' B34 0.624 A51' B15 1.583 A 5 3 ' B 3 5 1.425 A24' B42 0.500 A 3 4 ' B 4 3 0.474 A22' B22 1.168 A44' B44 0.450 A 3 2 ' B 2 3 0.9 81 A54' B45 1.352 A l l members are i n d u p l i c a t e , each one q u a r t e r i n c h t h i c k . Members not l i s t e d i n c l u d e d i n s o l i d metal f a n s . F r i n g e p a t t e r n s seen i n the model under l o a d are shown i n F i g u r e 7.2. The model was too l a r g e to be seen completely i n one p o s i t i o n of the p o l a r i s c o p e whose components had a viewing diameter of 12 i n c h e s . I f the members forming each f i b r e , or c h a i n of l i n k s , are examined i t w i l l be seen t h a t the c o l o u r remains f a i r l y uniform along them although i t d i f f e r s between f i b r e s . T h i s i n d i c a t e s t h a t the s t r e s s l e v e l remains f a i r l y c o n s t a n t along each f i b r e . N u m e r i c a l l y , the d i f f e r e n c e i n s t r e s s l e v e l s i s 125 I F i g u r e 7.2B S t r e s s p a t t e r n s i n Pinned M i c h e l l C a n t i l e v e r C a r r y i n g 110 l b . l o a d . Outer end not l a r g e s i n c e the b l u e - p u r p l e seen i n some of the i n n e r members corresponds t o the f i r s t f r i n g e w h i l e the ye l l o w prominent i n other members r e p r e s e n t s about 85% of t h i s s t r e s s l e v e l [ approximately 155 p . s . i . a g a i n s t 180 p . s . i . ] . T h i s may be due t o comparatively s m a l l e r r o r s i n alignment, the load not being p e r p e n d i c u l a r to the c a n t i l e v e r a x i s . The s t r e s s c o n c e n t r a t i o n s around the p i n s are promin-ent. As has a l r e a d y been s t a t e d t h i s i s an i m p r a c t i c a l way to b u i l d a r e a l s t r u c t u r e . The purpose of t h i s model was l a r g e l y to demonstrate t h a t , w i t h i n l i m i t s , the s t r e s s was uniform along each f i b r e and g e n e r a l l y conforms wi t h the 126 p r e d i c t e d l e v e l s . At 110 l b s . the l o a d i n g used i n the f i g u r e s , the s t r e s s should have been u n i f o r m l y 170 p . s . i . S o l i d C a n t i l e v e r A s o l i d sheet of i " p l a s t i c was cut to the e x t e r i o r shape of the three f i b r e c a n t i l e v e r d e s c r i b e d below. I t s span was 10.0", support spacing 2.5", =r = 4. S t r e s s p a t t e r n s seen when subjected to a twenty f i v e pound l o a d are seen i n F i g u r e 7.3. F i g u r e 7.3 ' S o l i d ' C a n t i l e v e r - Load 25 l b . The bending moment exer t e d by the load on the c a n t i -l e v e r i n c r e a s e s l i n e a r l y from l e f t t o r i g h t . As the s e c t i o n 127 broadens the second moment of area i n c r e a s e s r a p i d l y (to a maximum a t the widest p o i n t ) . The maximum s t r e s s w i l l thus d e c l i n e s t e a d i l y from the r e g i o n of the l o a d to the widest p o i n t and then i n c r e a s e again toward the concentrated r e a c t i o n s which are j u s t o u t s i d e the p i c t u r e a r e a . The b l a c k c e n t r a l area i n d i c a t e s a r e g i o n o f low s t r e s s . The s t r e s s d i s t r i b u t i o n i s f a r from uniform. C a n t i l e v e r with L i g h t e n i n g Holes T h i s c a n t i l e v e r i s shown i n F i g u r e 7.4 having a span of 10 inches and an -^ r a t i o o f 4. F i g u r e 7.4 C a n t i l e v e r With L i g h t e n i n g Holes i n Arrangement Approximating to a M i c h e l l C a n t i l e v e r - Load 20 l b . 128 The e x t e r i o r shape i s s i m i l a r to t h a t of the f i v e f i b r e c a n t i l e v e r described below. The spaces between the members were approximated by a p a t t e r n of d r i l l e d holes of v a r i o u s s i z e s . This may be regarded as a crude approximation to a M i c h e l l framework and as an example of a common method of weight r e d u c t i o n of s t r u c t u r e s . , I f the spaces between the holes are considered as e q u i v a l e n t to the f i b r e s i n a M i c h e l l c a n t i l e v e r , Figure 7.4 i n d i c a t e s c l e a r l y t h a t the s t r e s s e s are f a i r l y uniform along the ' f i b r e s ' i n compression r a d i a t i n g from the lower support p o i n t . The colour of the t e n s i l e f i b r e s i s a l t o g e t h e r darker and i n d i c a t e s a somewhat lower s t r e s s l e v e l . This discrepancy i s b e l i e v e d to be due to the f a c t t h a t the s t r e s s l e v e l achieved i n the body of the c a n t i l e v e r was i n f a c t much below that which i t could have supported s a f e l y . This i s s i m i l a r to the case of the s o l i d c a n t i l e v e r where s t r e s s l e v e l was s i m i l a r l y low. U n f o r t u n a t e l y , the support holes were placed r e l a t i v e l y c l o s e to the edge of the' s t r u c t u r e and caused a s t r e s s c o n c e n t r a t i o n , the outer edge of which can be seen at the upper r i g h t of Figure 7.4. The model f r a c t u r e d i n t h i s area when a load of 25 l b . was a p p l i e d and no f u r t h e r r e s u l t s could be obtained. The s i g n i f i c a n c e of t h i s specimen l i e s i n the compar-is o n w i t h F i g u r e 7.3. Although a s u b s t a n t i a l amount of the t o t a l m a t e r i a l has been removed the general s t r e s s l e v e l has h a r d l y been changed. This i n d i c a t e s a more e f f i c i e n t use of 129 the m a t e r i a l as might be p r e d i c t e d from the M i c h e l l c r i t e r i a . F i v e F i b r e R i g i d C a n t i l e v e r T h i s model r e p r e s e n t s a r i g i d M i c h e l l framework, having f i v e f i b r e s and a span of 10" w i t h an ~ r a t i o of 4. I t i s s i m i l a r to one of the s t r u c t u r e s d e s c r i b e d i n Chapter 4, S e c t i o n 1 except t h a t the member widths have been changed. I t was somewhat a r b i t r a r i l y d e c i d e d t h a t the width of the narrowest member should be 0.100". I t was f e l t t h a t t h i n n e r members than t h i s would d i s t o r t d u r i n g c u t t i n g and would be g e n e r a l l y f r a g i l e . On the other hand, the use of f i n e s e c t i o n s would minimize the e f f e c t of the j o i n t s t i f f -ness . The dimensions of the model are l i s t e d i n Appendix H and i t s g e n e r a l p r o p o r t i o n s may be seen i n F i g u r e 7.5a, where i t i s s u b j e c t e d to a 15 l b . l o a d . The uniform d i s t r i b u t i o n of s t r e s s through a l l the members i s e v i d e n t i n t h i s p i c t u r e , as are the h i g h e r b i a x i a l s t r e s s l e v e l s i n each of the j o i n t s . F i g u r e 7.5b shows t h a t the d i s t r i b u t i o n a t a l o a d of 27 l b . i s s t i l l f a i r l y uniform, o n l y a few of the lower (compression) members having a somewhat lower l e v e l of s t r e s s than the remainder. I a) 15 l b . l o a d I b) 27 l b . loa d F i g u r e 7.5 F i v e F i b r e R i g i d M i c h e l l C a n t i l e v e r T h i s f i g u r e a l s o shows i n many of the i n n e r members the e f f e c t of the secondary bending moments. As has been s t a t e d i n Chapter 5, the e f f e c t of these bending moments i s 131 to i n c r e a s e the s t r e s s a t one s i d e o f the member and decrease i t at the ot h e r . F i g u r e 7.6a J o i n t J 3 3 With 27 l b . L o a d — F i v e F i b r e M i c h e l l C a n t i l e v e r T h i s i s w e l l shown i n F i g u r e 7.6 which i s an enlar g e d view of j o i n t ^23' A ^ f ° u r members e x h i b i t a g r a d a t i o n of co l o u r across the j o i n t . The A members are i n t e n s i o n and the B members are i n compression and the bending s t r e s s e s are i n such d i r e c t i o n s as to r o t a t e the members toward the h o r i z o n t a l a x i s . 132 T h i s i s shown d i a g r a m a t i c a l l y i n F i g u r e 7.6b. When the a x i a l and bending s t r e s s e s are added i t w i l l be seen t h a t the s t r e s s l e v e l w i l l be higher a t p o i n t s A, B. C and D than at the corresponding p o i n t s E, F, G and H. F i g u r e 7.6b S t r e s s e s a t J o i n t J ^ Th i s i s s u b s t a n t i a t e d by the p h o t o e l a s t i c f r i n g e c o l o u r s which reach hi g h e r l e v e l s a t the former s e t of p o i n t s . The g e n e r a l s t r e s s l e v e l i n a l l f o u r members, remote from the j o i n t i s : approximately 350 p . s . i . , i n d i c a t e d by the pur p l e f r i n g e . A t p o i n t s A, B and D a f r i n g e l e v e l of about 1.5 f r i n g e s (c. 500 p . s . i . ) i s a t t a i n e d , w hile a t B, E, F and G, the s t r e s s l e v e l i s reduced to the yel l o w which corresponds t o about 0.8 f r i n g e s or a l i t t l e l e s s than 300, 133 p . s . i . The c l e a r l y defined f i r s t fringe e n c i r c l e s the j o i n t area and offshoots from i t can be seen at A, B and D stretching some distance away along the member. At the centre of the j o i n t the pink nearing the second fringe can just be seen. The stress l e v e l here i s about 1.6 fringes (say 600 p . s . i . ) . These figures agree within about 10% with the stresses predicted by the STRUDL calc u l a t i o n s shown on Table 5.1 on page 94 when corrected for the difference i n loads. The higher figures from the photoelastic analysis may represent a difference between batches of the stress o p t i c a l c o e f f i c i e n t for CR39. This was assumed to be 90 p . s . i . / f r i n g e / i n c h as quoted by manufacturers, but some tolerance i s permitted. The j o i n t i s shown about three and one half times f u l l size i n t h i s p i c t u r e , which shows many small chips along the edges. These seem to be unavoidable when work-ing with t h i s b r i t t l e material. I t i s also evident that an error has been made i n cutting the side of member B23. The cutter has been advanced a l i t t l e too far and has notched the side of member A32. The e f f e c t of t h i s seems small. Three F i b r e R i g i d C a n t i l e v e r s Two models were made of the t h r e e f i b r e c a n t i l e v e r d e s c r i b e d i n S e c t i o n 1 of Chapter 4. Each had a span of ten i n c h e s and an -^ r a t i o of 4, the member widths b e i n g m o d i f i e d as f o r the f i v e f i b r e c a n t i l e v e r . The f i r s t model was cut w i t h ^" d i a . c u t t e r t o produce f i l l e t s at the j o i n t s o f i ^ - " r a d i u s . . The second model was produced 3 3 w i t h a d i a . c u t t e r so t h a t the f i l l e t r a d i u s was -g-" or s i x times t h a t i n the f i r s t model. These models were loaded s i m i l a r l y and examined t o determine the e f f e c t of f i l l e t r a d i u s on s t r e s s l e v e l i n the j o i n t a r ea. a) Model With g-" Radius F i l l e t Figure 7 . 7 shows t h i s model subjected to a load of 25 l b s . The s t r e s s d i s t r i b u t i o n i n the members i s q u i t e uniform at approximately the f i r s t f r i n g e (350 p . s . i . ) . Figure 7 . 7 25 l b . Load on Three F i b r e M i c h e l l C a n t i l e v e r With 3 / 8 " Radius F i l l e t s Stresses i n the v i c i n i t y of t y p i c a l j o i n t s may be examined i n Figure 7 . 8 which shows j o i n t s an& J32 a b ° u t twice f u l l s i z e . The purple f i r s t f r i n g e may be seen i n both j o i n t areas and the general s t r e s s p a t t e r n i s s i m i l a r to th a t seen i n the f i v e f i b r e case. The maximum s t r e s s l e v e l i n the middle of J00 i s approximately equal to that i n J o i n t J.,-. i n a) J o i n t J_„ — 27 l b . b) J o i n t J 2 2 — 27 l b . F i g u r e 7.8 J o i n t D e t a i l i n Three F i b r e M i c h e l l C a n t i l e v e r With 3/8" Radius F i l l e t s a) Model With Radius F i l l e t I b) Model With =-r" Radius F i l l e t 16 Three F i b r e M i c h e l l C a n t i l e v e r s - 20 l b . Figure 7.6a. At both j o i n t s J ^  an<-* J22 a n a r e a °^ zero stress i s seen i n the centre of the large radius acute angled f i l l e t s , suggesting that the extra material does not have much e f f e c t i n t r a n s f e r r i n g the forces between members. In j o i n t J 3 2 the obtuse comer on the outer surface theoret-i c a l l y w i l l be subjected to zero s t r e s s . The colour fringes indicate the stress l e v e l i s decreasing r a p i d l y toward the point but the resolution i s i n s u f f i c i e n t to see the black zero f r i n g e . Figure 7.9 indicates the e f f e c t of f i l l e t radius at j o i n t ^22' ^ n e w-rl-'-'':e areas seen on the model with the larger f i l l e t radius are subjected to low s t r e s s . As a f i r s t approximation, i f these areas were removed the shape of the model having the smaller f i l l e t radius would be obtained. The large f i l l e t seems thus to have i n t r o -duced more material which serves l i t t l e or no useful purpose. The maximum stress i n the j o i n t area i s somewhat reduced i n the large f i l l e t radius model. b) Model With ~ " Radius F i l l e t s 16 Two models were made of t h i s structure. The f i r s t was poorly cut and had many chips and an i r r e g u l a r f i n i s h i n the area of the j o i n t s . b) Buckled Shape Figu r e 7.10 Three F i b r e M i c h e l l C a n t i l e v e r - 25 l b . Load 139 I t w a s , h o w e v e r , l o a d e d a n d t e s t e d b u t f a i l e d b y b u c k l i n g a t 25 l b s . T h e s t r e s s p a t t e r n a t t h i s p o i n t , a n d i t s b u c k l e d s h a p e , a r e s h o w n i n F i g u r e 7.10. C o m p l e t e c o l l a p s e was o n l y a v e r t e d b y t h e s u p p o r t p r o -v i d e d b y t h e p o l a r i s c o p e e l e m e n t s . T h e b u c k l i n g was p r o b a b l y c a u s e d b y t w i s t i n g d u e t o t h e m o d e l n o t b e i n g s e t i n i t i a l l y i n a v e r t i c a l p l a n e . T h e l o a d was a p p l i e d b y a d d i n g l e a d s h o t t o a b u c k e t s u s p e n d e d f r o m t h e m o d e l . T h i s c o u l d h a v e s w u n g s i d e w a y s t o i n i t i a t e c o l l a p s e . D i s t o r t i o n o f i n d i v i d u a l m e m b e r s c a n c l e a r l y b e s e e n i n F i g u r e 7 . 1 0 a t o g e t h e r w i t h t h e v e r y h i g h b e n d i n g s t r e s s e s w h i c h a r e c l e a r l y v i s i b l e i n some a r e a s o f t h e m o d e l . I n o t h e r p a r t s t h e s t r e s s e s w e r e p r o b a b l y s o h i g h t h a t t h e f r i n g e s p r o d u c e d b y a n o n - m o n o c h r o m a t i c s o u r c e b l u r r e d i n t o o n e a n o t h e r . F i g u r e 7 . 1 0 b i n d i c a t e s t h e d e g r e e o f t w i s t i n g , s e e n b y l o o k -i n g a l o n g t h e p l a n e o f t h e m o d e l . I t i s r e m a r k a b l e t h a t t h e m o d e l s u f f e r e d n o p e r m a n e n t d a m a g e . I t r e t u r n e d t o i t s o r i g i n a l s h a p e w h e n u n l o a d e d a n d l i t t l e r e s i d u a l s t r e s s c o u l d b e o b s e r v e d . T h e s e c o n d m o d e l i s s e e n u n d e r l o a d i n F i g u r e s 7.11 a n d 7 . 1 2 . I n t h e f i r s t f i g u r e , s t r e s s . p a t t e r n s f o r l o a d s o f 15 l b . a n d 25 l b . may b e c o m p a r e d . T h e g e n e r a l s i m i l a r i t y w i t h t h e f i v e f i b r e c a s e a s s h o w n i n F i g u r e 7.5 I s o b v i o u s . T h e p a t t e r n a r o u n d t h e j o i n t s i s q u i t e d i f f e r e n t f r o m t h e t h r e e f i b r e m o d e l w i t h l a r g e r a d i u s f i l l e t s , s h o w n i n F i g u r e 7 . 7 . I 140 142 F i g u r e 7.12 shows one j o i n t , J^2' """n d e t a i l . T ) l e p a t t e r n i s s i m i l a r i n each p i c t u r e d i f f e r i n g mainly i n c o l o u r , corresponding to the d i f f e r e n c e i n s t r e s s l e v e l . The r e d u c t i o n i n s t r e s s l e v e l a t the outer obtuse angle can be seen c l e a r l y although once again the zero b l a c k f r i n g e i s not r e s o l v e d . The v a r i a t i o n i n s t r e s s l e v e l across each member due to bending s t r e s s e s i s a l s o c l e a r l y e v i d e n t . The bend-ing moments on A^^ and a r e such as to reduce the angle between them, thus re d u c i n g the s t r e s s on the lower s i d e of A ^ and the l e f t s i d e of T h i s model was not t e s t e d f o r s e v e r a l days a f t e r manufacture and e x h i b i t s c l e a r time-edge e f f e c t s . CALIBRATION OF FRINGES IN TERMS OF STRESS LEVEL Values of s t r e s s i n members and j o i n t s have been s t a t e d i n the preceding work by r e f e r e n c e to the a s s o c i a t e d f r i n g e o r d e r s . These were determined by c a l i b r a t i o n of tapered specimens s u b j e c t e d to known dead weight l o a d s . The dimensions of these c a l i b r a t i o n p i e c e s are shown i n F i g u r e 7.13. These specimens were cut from the same q u a r t e r i n c h t h i c k sheet of CR39 p l a s t i c as were the model c a n t i l e v e r s and were handled and s t o r e d i n the same manner as the a c t u a l 143 F i g u r e 7.13 T e n s i l e T e s t S p e c i m e n s f o r C a l i b r a t i o n o f B i r e f r i n g e n t M a t e r i a l m o d e l s t e s t e d . T h e m o d e l s w e r e h u n g v e r t i c a l l y i n t h e p o l a r -i s c o p e a n d l o a d e d t o o b t a i n t h e d e s i r e d f r i n g e p a t t e r n s . T h e s t r e s s i n t h e s p e c i m e n a t a n y p l a n e c o u l d t h e n b e d e d u c e d b y s i m p l e c a l c u l a t i o n . F i g u r e 7.14 s h o w s a s a m p l e s p e c i m e n u n d e r l o a d a n d t h e a s s o c i a t e d s t r e s s l e v e l s E r r o r i n e v a l u a t i n g s t r e s s i n t e n s i t i e s i s l e a s t l i k e l y i f t h e m o d e l u n d e r t e s t a n d t h e c a l i b r a t i o n s p e c i m e n c a n b e v i e w e d s i m u l t a n e o u s l y , s o t h a t c o l o u r s may b e c o m p a r e d d i r e c t l y . I n t h e p r e s e n t i n s t a n c e t h i s was i m p o s s i b l e d u e t o t h e p h y s i c a l l i m i t a t i o n s i n t h e a p p a r a t u s . T h e f r i n g e 144 I Figure 7.14 Fringe Orders i n a Loaded Specimen patterns were recorded on Kodachrome I I under constant l i g h t -ing c o n d i t i o n s . The r e s u l t i n g transparencies were then compared. The colour p r i n t s i n t h i s paper d i f f e r to some degree i n the q u a l i t y of t h e i r c olour reproduction so t h a t f r i n g e colours may vary s l i g h t l y . The statements i n the paper are based on comparison of the o r i g i n a l t r a n s p a r e n c i e s . The p h o t o e l a s t i c i n v e s t i g a t i o n of p i n - j o i n t e d and r i g i d s t r u c t u r e s has y i e l d e d r e s u l t s i n s u b s t a n t i a l agreement with those p r e d i c t e d i n Chapters 4 and 5. 145 CHAPTER 8 TOWER FOR HIGH TENSION TRANSMISSION LINE An I l l u s t r a t i o n of a P r a c t i c a l Use of the P h i l o s o p h y of the Theory The work d e s c r i b e d i n the p r e c e d i n g c h a p t e r s i n d i c a t e s t h a t M i c h e l l frames c o u l d be a p p l i e d as s o l u t i o n s f o r many s t r u c t u r a l problems. T h e i r assembly from standard s t e e l s e c t i o n s by b o l t -i n g , r e v e t t i n g or welding should r a i s e few problems s i n c e the c o n s t r u c t i o n a l techniques would be no d i f f e r e n t from those c u r r e n t l y used. Only the s t r u c t u r a l shapes are d i f f e r e n t from those p r e s e n t l y employed„ A t y p i c a l example of a k i n d of s t r u c t u r e , f o r which M i c h e l l frames would be e s p e c i a l l y s u i t a b l e , i s the t r a n s -m i s s i o n tower employed on h i g h t e n s i o n power l i n e s . The l o a d i n g on such s t r u c t u r e s i s c l o s e l y s p e c i f i e d f o r d e s i g n purposes, t o i n c l u d e the weight of the w i r e , i c e and wind lo a d s and s t r a i n i n g l o a d s , combined i n the l e a s t f a v o u r a b l e way. The tower e x i s t s t o support t h i s g e o m e t r i c a l c o n f i g u r -a t i o n determined by e l e c t r i c a l requirements. An optimum s t r u c t u r e c o u l d thus be determined c o n t a i n i n g l e s s m a t e r i a l than any other d e s i g n . T h i s l a s t f a c t o r i s of c o n s i d e r a b l e p r a c t i c a l impor-tance, s i n c e t r a n s m i s s i o n l i n e s are f r e q u e n t l y c o n s t r u c t e d through mountains and f o r e s t s t o t r a n s m i t e l e c t r i c a l energy 146 from h y d r o e l e c t r i c power p l a n t s i n remote areas t o the urban c e n t r e s of consumption. The m a t e r i a l f o r the towers f r e -q u e n t l y has t o be moved a c r o s s rugged country t o c o m p a r a t i v e l y i n a c c e s s i b l e tower s i t e s . The d i f f i c u l t y of ground access i n c r e a s i n g l y encourages the use of h e l i c o p t e r s f o r t h i s work and l i g h t weight i s o b v i o u s l y an advantage. The 500 K.V. t r a n s m i s s i o n l i n e r e c e n t l y completed i n B r i t i s h Columbia t o b r i n g power from the Peace R i v e r p r o j e c t to Vancouver was s e l e c t e d as r e p r e s e n t a t i v e of c u r r e n t p r a c t i c e i n t h i s f i e l d . Through the c o u r t e s y o f I n t e r n a t i o n a l Power and E n g i n e e r i n g C o n s u l t a n t s , L t d . minimum s p e c i f i c a t i o n s were obtained of a t y p i c a l tower i n t h i s i n s t a l l a t i o n . The i n f o r m a t i o n shown i n F i g u r e 8.1 i s based on t h e i r drawing 50011-To8-B229 and i n d i c a t e s the g e o m e t r i c a l r e s t r i c t i o n s to which any s t r u c t u r e must conform. Each conductor c o n s i s t s of four c a b l e s p l a c e d a t the corn e r s of a square of e i g h t e e n i n c h s i d e . T h i s bundle i s suspended by an i n s u l a t o r 13'2" lon g , which may swing 31° e i t h e r s i d e of v e r t i c a l due to wind, unbalanced i c e l o a d , e t c . No p a r t of the tower may come c l o s e r t o the conductor bundle than 10 '2" . The above requirements n e c e s s i t a t e a minimum h o r i z o n t a l spacing of the conductors of 35'6". In p r a c t i c e these are r a r e l y l e s s than 40' a p a r t to p r o v i d e space a t A and B f o r the support g i r d e r s . Td|a of Insulator's. N \ I / Extreme- S u i n g o f conductor. Tb|> of insulator Typically T '^above nominal ground level. Insulation Clearance 102 "from nearest uuih-e ir\ c o n d u c t o r * bundl-e.' r 8 i Conductor- B u n d l e Con ^ i g u K a f r o n . F i g u r e 8.1 Clearance f o r High Tension Wires 14 8 F i g u r e s 8.2 and 8.3 i l l u s t r a t e s two p o s s i b l e designs which s a t i s f y these d e s i g n c r i t e r i a . A s u p e r f i c i a l examin-a t i o n of F i g u r e 8.1 i n d i c a t e s t h a t the c r i t i c a l area f o r the g e o m e t r i c a l d e s i g n occurs a t A and B, a t the spaces between the c l e a r a n c e areas around the conductors. C l e a r l y , i f a M i c h e l l s t r u c t u r e i s to be used f o r the tower, the c e n t r a l conductor and i t s c l e a r a n c e must be accommodated i n s i d e the outer panel of the frame. A range of s t r u c t u r e s may be s p e c i f i e d having a s u f f i c i e n t l y l a r g e o u t e r p a n e l to permit t h i s arrangement. When the d e s i g n of the o u t r i g g e r s s u p p o r t i n g the outer conductors i s c o n s i d e r e d , i t becomes e v i d e n t t h a t i n a d d i t i o n the j o i n t s J„ „ , and J.. . must be p l a c e d as c l o s e as J N,N-1 N-1,N c p o s s i b l e t o the narrowest p a r t o f the gaps at A and B, F i g u r e 8.1. I f these j o i n t s are too low the d e s i g n w i l l be g r e a t l y r e s t r i c t e d or , e q u a l l y undesirably", the conductor spacing must be i n c r e a s e d , thus i n c u r r i n g an i n e v i t a b l e i n c r e a s e i n volume. In F i g u r e 8.2, the tower i s drawn as a standard t h r e e f i b r e c a n t i l e v e r as d e s c r i b e d i n S e c t i o n 1 of Chapter 4. I t s h e i g h t i s 100 f e e t and i t s base width 50 f e e t , (jj = 2) I t w i l l be seen t h a t the c l e a r a n c e area around the c e n t r a l con-d u c t o r j u s t f i t s i n the outer p a n e l . A more d e t a i l e d examin-a t i o n of the d e s i g n might i n v o l v e s l i g h t changes i n t h i s geometry to i n c r e a s e the c l e a r a n c e s . The o u t r i g g e r s are i d e n t i c a l skew c a n t i l e v e r s as d e s c r i b e d i n S e c t i o n 5 o f Chapter 4. For these 9 = 28°, the 149 Figure 8.2 Possible Design for a Transmission Tower U t i l i z i n g M ichell Framework 150 r a t i o o f t h e i r fan r a d i i i s 1.526, the span i s 42'6" and the f a l l i s 17', as i s the base l e n g t h . To complete the s t r u c t u r e , s t r u t s are added between and J^^ of t n e tower and J ^ i - n each o u t r i g g e r . F i g u r e 8.2 shows the tower as seen l o o k i n g along the c a b l e s . Looking s i d e on a t the c a b l e s , the tower c o u l d be b u i l t of p a r a l l e l members or of two plane M i c h e l l frames i n c l i n e d to converge a t the top to form an 'A' frame. I f needed t o s u s t a i n f o r c e s along the c a b l e s — a s an anchor tower — t h e s i d e view c o u l d a l s o be a M i c h e l l framework of i d e n t i c a l l a y o u t to t h a t i n F i g u r e 8.2, supported on fou r foundations i n the form of a square of 50 f e e t side.. In these l a t t e r cases F i g u r e 8.2 does not show the , tru e lengths of the members which would be i n c l i n e d t o the • plane of the drawing. . The h o r i z o n t a l frame formed by the two o u t r i g g e r s when seen i n p l a n , c o u l d a l s o be assembled from p a r a l l e l members making a box s e c t i o n g i r d e r , or c o u l d again be l a i d . o u t t o conform to a M i c h e l l framework. F i g u r e 8.3 i l l u s t r a t e s a tower c o n s t r u c t e d from a f o u r f i b r e M i c h e l l c a n t i l e v e r having an — r a t i o of 4. Again the t o t a l h e i g h t i s 100 f e e t and the base"width 25 f e e t . Here the cable c e n t r e l i n e B e i g h t y f e e t above nominal ground l e v e l , although t h i s c o u l d be v a r i e d by minor changes i n the tower-geometry. , 151 F i g u r e 8.3 P o s s i b l e Design f o r a Trans m i s s i o n Tower M i c h e l Framework 152 The outriggers are formed from three t r i a n g u l a r frames springing from and B^^ res p e c t i v e l y . These are not l a i d out i n conformity with the M i c h e l l c r i t e r i a for an optimum structure and serve to i l l u s t r a t e one of many ways i n which the arrangement could be made. Again, t h i s view indicates the tower p r o f i l e as seen looking along the conductors. The shape seen normal to the conductors can be varied i n the same way as described i n the f i r s t case. S t r u c t u r a l volumes have not been calculated for these two geometrical designs since no figure i s r e a d i l y a v a i l a b l e for the standard tower against which i t i s to be compared. A calculated volume may also have l i t t l e r e a l i t y unless i t i s prepared by a designer f a m i l i a r with the techniques of t h i s s p e c i a l i z e d f i e l d . Members may unwittingly be under-designed i n ignorance of worst loading combinations which may be incurred, perhaps during construction rather than i n service. The q u a l i t a t i v e designs i l l u s t r a t e d are elementary examples which are representative of the ap p l i c a t i o n of Maxwell-M i c h e l l structures to a t y p i c a l problem. I t i s not suggested that a complete design for a tower, which would be required to sustain l a t e r a l and other loads, has been completed and presented. 153 CHAPTER 9 DESIGN OF A MIRROR SUBSTRATE A var i e t y of designs for structures u t i l i z i n g M i c h e l l frameworks have been discussed i n previous chapters. These have a l l had a common purpose—to support an external force system and transfer i t to fixed foundations. The f i n a l design to be considered concerns the sub-strate for a large astronomical mirror. The t r a d i t i o n a l design for a mirror u t i l i z e s a c i r c u l a r (or rectangular) blank of glass, whose thickness i s between one si x t h and one eighth of i t s diameter. These proportions are empirical and have been found adequate to r e s i s t the forces to which the mirror i s subject. These mainly occur during manufacture when the surface i s being polished to shape. Once i n s i t u the mirror i s subjected p r i m a r i l y to s e l f weight forces which change d i r e c t i o n as the mirror i s t i l t e d to observe and follow a s p e c i f i e d s t a r . Thermal stresses are also imposed as the ambient temperature changes. These changes are normally f a i r l y slow and are minimized by surrounding structures. I t i s obviously desirable that the mirror substrate should be as l i g h t and yet as r i g i d as possible so as to minimize these d e f l e c t i o n s . M i c h e l l struc-154 t u r e s are thus p a r t i c u l a r l y a p p l i c a b l e to t h i s problem as i t has been demonstrated t h a t they c l o s e l y approach the optimum minimum volume s t r u c t u r e which i s s i m u l t a n e o u s l y the most r i g i d , as shown i n Chapter 1. The r e f l e c t i n g s u r f a c e of the m i r r o r i s formed t o some s p e c i f i e d curve w i t h an accuracy which i s o f t e n r e q u i r e d _ g to be b e t t e r than one t w e n t i e t h of a wavelength (10 inches approximately) a c r o s s the s u r f a c e which may have a diameter of as much as 150 i n c h e s . I t may be p o s s i b l e to achieve t h i s accuracy f o r a m i r r o r maintained i n a f i x e d a t t i t u d e . When t i l t e d , however, the m i r r o r w i l l sag away from i t s worked shape due to the changing d i r e c t i o n of g r a v i t y . In l a r g e m i r r o r s these changes are compensated by a p p l y i n g l o c a l c o r r e c t i n g f o r c e s . T h i s i s the f i e l d of 'Active Optics*! or 'Deformable O p t i c s ' . A somewhat d i f f e r e n t problem e x i s t s i n a mirror, d e s t i n e d f o r use i n an o r b i t i n g a s t r o n o m i c a l o b s e r v a t o r y s a t e l l i t e . In s e r v i c e , the m i r r o r w i l l be i n a w e i g h t l e s s s t a t e and w i l l not be s u b j e c t e d to changes i n s e l f weight f o r c e . However, i t s t i l l has to be shaped on e a r t h under the i n f l u e n c e of g r a v i t y . Again the s t r u c t u r e should have a minimum d e f l e c t i o n to reduce the i n e v i t a b l e c o r r e c t i o n s as much as p o s s i b l e . The m i r r o r c o n s i d e r e d here as an example has a . diameter of 48 inches and, i f made from a s o l i d b l o c k , would have a t h i c k n e s s of 8 i n c h e s . The a l t e r n a t i v e d e s i g n u t i l i z -i n g M i c h e l l s t r u c t u r e s f o r support i s i n d i c a t e d i n F i g u r e 9.1. 155 F i g u r e 9.1 T e n t a t i v e Design f o r a Large M i r r o r S u b s t r a t e The support of the m i r r o r by a d i s t r i b u t i v e s t r u c t u r e p e r m i t s the s o l i d m i r r o r blank t o be reduced to a t h i c k n e s s t s u i t i n g the requirements of o p t i c a l working and of d i f f r a c t i o n l i m i t e d o p t i c s . The d i s t r i b u t i v e s t r u c t u r a l support takes the form of M i c h e l l c a n t i l e v e r s as r a d i a l r i b s . The t h i c k n e s s t i s s e l e c t e d such t h a t the d i f f e r e n c e between the planes of the h i g h e s t and lowest p o i n t s of the d i s c i s equal t o t h a t of the edge supported s o l i d d i s c . In a r e c e n t paper, H, Vaughan [10] has i n v e s t i g a t e d a u n i f o r m d i s c of r a d i u s a supported a t d i s c r e t e p o i n t s around a c i r c l e o f r a d i u s b. He showed t h a t f o r — = , an 156 optimum f l a t n e s s occurs i f t h r e e supports are used, spaced 120° a p a r t . The deformed shape i s complex but the d e v i a t i o n s from the nominal d e s i g n s u r f a c e are a t a minimum f o r t h i s c o n f i g u r a t i o n . U s i n g h i s r e s u l t s , t h r e e r a d i a l c a n t i l e v e r s w i l l be used t o support the d i s c a t a r a d i u s of s i x t e e n i n c h e s [— = rr] . a 3 To o b t a i n a more r e a l i s t i c d e s i g n , the c a l c u l a t i o n s i n t h i s s e c t i o n w i l l be made f o r fuse d s i l i c a , a t y p i c a l m i r r o r m a t e r i a l . The p h y s i c a l c o n s t a n t s are thus -•E = 10.5 x 10 6 p . s . i . pg = 0.0795 l b . per cu. i n . s p e c i f i c g r a v i t y = 2.20 v = 0.17 For the d e s i g n of support r i b s , e was taken as 0.0001 so t h a t the d e s i g n s t r e s s was 1050 p . s . i . I t may be noted i n p a s s i n g t h a t the s e l e c t i o n of a s p e c i f i c s t r a i n was q u i t e a r b i t r a r y and has o n l y a minor e f f e c t on the volume of the s t r u c t u r e , s i n c e the g r e a t bulk of m a t e r i a l i s i n the m i r r o r s u r f a c e . The p e r m i s s i b l e s t r e s s i n the r i b s d i r e c t l y a f f e c t s the r i b d e f l e c t i o n but the r e s u l t a n t motion of the m i r r o r s u r f a c e i s a r i g i d body t r a n s l a t i o n , which may e a s i l y be compensated by a f o c u s s i n g adjustment. The s t r e s s was kept low to minimise the d e f l e c t i o n s and i n def e r e n c e t o the low r e s i s t a n c e of g l a s s t o t e n s i o n . 157 Determination of Thickness t Solutions for the c e n t r a l d e f l e c t i o n of a uniformly loaded pla t e , edge supported, may be found i n most standard texts on e l a s t i c bending of plates. For example, i n "Theory of Plates and Shells" by Timoshenko and Woinowsky, Krieger (McGraw-Hill, 1959), equation 68 on page 57 may be used:-# | _ (5+v) pgta 4  W ~ 64 (l+v)D = 0.0000316 i n . for the given d i s c . A paper by R. Williams [12] describes an exact so l u t i o n for the deformation of a c i r c u l a r d i s c of radius a, supported at n equally spaced points around a c i r c l e of radius b. The formula i s given as 4 R u = pga P [Williams 3.1] (9.1) where B i s a factor depending on the support conditions. [This equation i s i d e n t i c a l with the solution given by Yu and Pan (I.J.M.S., 8_, 1966, page 336), for the same conditions, I t has been normalized and rearranged so that i t can be expressed i n the above form]. T h e f a c t o r 3 i s a f u n c t i o n o f t h e r a d i u s r a t i o — , a ' t h e n u m b e r o f s u p p o r t s n , a n d t h e r a d i u s a t w h i c h t h e d e -f l e c t i o n i s r e q u i r e d , r , e x p r e s s e d a s t h e r a t i o — . T a b l e s a a r e g i v e n i n h i s p a p e r f o r v a r i o u s c o m b i n a t i o n s o f t h e s e f a c t o r s . U s i n g t h e s e t a b l e s , t h e v a l u e o f b f o r w h i c h t h e c e n t r a l a n d e d g e d e f l e c t i o n s (—=0, —=1.0) a r e e q u a l may b e d e t e r m i n e d b y i n t e r p o l a t i o n . T h i s g i v e s -=0.68 w h i c h a g r e e a w e l l w i t h V a u g h a n ' s f i g u r e o f 0.667, t h e d i f f e r e n c e a r i s i n g f r o m t h e i n t e r p o l a t i o n . T h e v a l u e o f 3, a n d t h e v a l u e o f oo ( 0 . 0 0 0 0 3 1 6 ) may b e s u b s t i t u t e d i n t o e q u a t i o n 9.1 t o d e t e r m i n e t h e new d i s c t h i c k n e s s . T h i s i s 4.12 i n c h e s . G E O M E T R I C D E S I G N OF C A N T I L E V E R R I B S A p r e l i m i n a r y i n v e s t i g a t i o n , u s i n g s c a l e d r a w i n g s , r e v e a l e d t h a t i t was i m p o s s i b l e t o c o n s t r u c t a M i c h e l l t y p e c a n t i l e v e r , s u p p o r t e d f r o m a c e n t r e p i l l a r a n d e x t e n d i n g o u t w a r d f o r s i x t e e n i n c h e s , w i t h i n a t o t a l d e p t h o f s i x i n c h e s . A t t h e p l a n e s o f t h e r i b s , t h e t o t a l d e p t h o f t h e s t r u c t u r e m u s t b e g r e a t e r t h a n t h e d e p t h o f t h e s o l i d d i s c . T h e s p a c e s b e t w e e n t h e r i b s c o u l d b e f i l l e d w i t h o t h e r a p p a r a t u s i n a p p l i c a t i o n s w h e r e v o l u m e i s s t r i c t l y l i m i t e d . T h i s w o u l d d e p e n d o n t h e r e q u i r e m e n t s f o r s p a c e i n w h i c h t o t i l t t h e m i r r o r s o t h a t i t may b e a l i g n e d i n s p e c i f i c d i r e c t i o n s . Two c a n t i l e v e r s , each having three f i b r e s , were selected for design analysis and are described below. Minor changes could be made to t h e i r parameters without s i g n i f i c a n t change i n the v a l i d i t y of the r e s u l t s . a) Figure 9.2 - Symmetrical Cantilever This i s a 'standard* c a n t i l e v e r design as discussed i n Chapter 4. I t s ~ r a t i o i s 2 .5. The support points A^ i s on the mirror axis and the other, at a radius of 2.4". D could be supported on a c o l l a r on the ce n t r a l pin. The t o t a l depth of d i s c and r i b i s about 12.4", although t h i s can be varied s l i g h t l y . A minimum clearance must be maintained at C to prevent contact between r i b and d i s c . The r i b could be rotated about C. Clockwise r o t a t i o n would reduce the o v e r a l l depth but increase the length of the s t r u t necessary at L to support the d i s c . Conversely counterclockwise r o t a t i o n of the r i b reduces the length of the s t r u t at L but increases the over-a l l depth. I f large rotations are made from the p o s i t i o n shown, the r i b parameters would require changing to maintain L at a radius of sixteen inches. A r i b could be designed to f i t within the six inch space. In t h i s case one support would be on the c e n t r a l axis but the other would have to be at approximately fourteen inches radius, almost below the mirror surface support point, Such a s o l u t i o n would merely transmit the problem of support to the main structure of the apparatus, and i s not a p r a c t i c a l s o l u t i o n . F i g u r e 9.2 Three F i b r e Symmetrical M i c h e l l C a n t i l e v e r f o r M i r r o r S u b s t r a t e 161 b) Figure 9.3 - Skew Mi c h e l l Cantilever The second design u t i l i z e s a skew cant i l e v e r with both support points on the c e n t r a l axis. Clearance at C i s again a l i m i t i n g factor i n the design. L may be placed closer to the disc than i s shown by moving the support points A and B downward and increasing the depth of the r i b . As drawn the span i s 16", support spacing 6.67", L 12 ) and the r i s e i s 2.33". The f i b r e angle i s 28°. The o v e r a l l depth of about eleven inches i s somewhat less than that of the symmetrical ca n t i l e v e r but the s t r u t at L i s somewhat longer. LOADS ON CANTILEVERS AND MEMBER SIZES The computer programmes described i n Appendix C were used to determine the member lengths and sizes required to maintain a uniform stress (of 1050 p.s.i.) as the mirror was moved from the horizontal to the v e r t i c a l i n f i v e degree steps. The mirror d i s c weighs approximately 593 pounds so that each r i b B subjected to a load of 197.6 pounds. Although the maximum t o t a l volume was required when the mirror was h o r i z o n t a l , i n d i v i d u a l members c a r r i e d larger loads and thus had greater volume at other angular pos i t i o n s . The maximum cross sec t i o n a l area of each member was thus selected to form a composite structure. In t h i s , the maximum stress would not exceed 1050 p . s . i . at any angle but would at some angle reach t h i s v a l u e i n each member. I t i s not an optimum s t r u c t u r e f o r any angle because of t h i s m o d i f i -c a t i o n . The composite s t r u c t u r e s thus o b t a i n e d were then a n a l y s e d u s i n g a ' S t r u d l * programme to determine the secondary s t r e s s . The r e s u l t s are t a b u l a t e d below:-a) Symmetrical M i c h e l l C a n t i l e v e r ( F i g u r e 9.2) PIN JOINTED STRUCTURE Member Length Ins. Area I n 2 Maximum Forc e A 10 L l l '12 20 '21 L22 l30 k31 ^32 01 11 21 02 12 22 03 13 23 4. 526 2. 187 2. 187 4. 526 3. 638 4. 409 4. 526 5. 281 7. 333 4. 526 2. 187 2. 187 4. 526 3. 638 4. 409 4. 526 5. 281 7. 333 0.109 114.5 0.093 97.95 0.060 63.07 0.185 194.30 0.162 170.7 0.113 118. 8 0.199 208.9 0.193 202.5 0.170 178.9 0.097 101.7 0.077 80.40 0.048 50.47 0.169 177.6 0.136 143.1 0.091 95.03 0.249 261.0 0.241 253.3 0.213 223.7 Volume - 12.25 c u . i n . F i g u r e 9.3 Three F i b r e Skew M i c h e l l C a n t i l e v e r f o r M i r r o r S u b s t r a t e 164 The STRUDL a n a l y s i s y i e l d e d the f o l l o w i n g a d d i t i o n a l d a t a : -i ) M i r r o r H o r i z o n t a l Reaction at A, H o r i z o n t a l -397.6 l b , V e r t i c a l -52.0 l b . Reaction at B, H o r i z o n t a l +397.6 l b , V e r t i c a l +249.6 l b . D e f l e c t i o n a t m i r r o r support p o i n t , L H o r i z o n t a l 0.019", V e r t i c a l 0.053". i i ) M i r r o r V e r t i c a l Reaction at A, H o r i z o n t a l +90.7 l b , V e r t i c a l +249.0 l b . Reaction at B, H o r i z o n t a l -90.7 l b , V e r t i c a l -51.4 l b . D e f l e c t i o n at m i r r o r support p o i n t , L. H o r i z o n t a l -0.19", V e r t i c a l 0.022". I t i s i n t e r e s t i n g to note t h a t the load i s almost e x a c t l y t r a n s f e r r e d between the support p o i n t s as the m i r r o r i s t i l t e d through 90°. The volume of t h i s skew s t r u c t u r e i s only s l i g h t l y l a r g e r than t h a t of the symmetrical c a n t i l e v e r . The s l i g h t e x t r a volume would be acceptable i f i t was d e s i r e d t o support the s t r u c t u r e e n t i r e l y from a c e n t r a l p i l l a r . The STRUDL a n a l y s i s y i e l d e d the f o l l o w i n g a d d i t i o n a l data. 165 b) Skew M i c h e l l C a n t i l e v e r (Figure 9.3) PIN JOINTED STRUCTURE Member Length Area Maximum Ins. I n 2 F o r c e l b s A10 3.909 0.109 114.6 A l l 2.581 0.092 96.53 A12 2.581 0.059 61.95 A20 3.909 0.187 196.7 A21 3.893 0.161 169.2 . A22 4.728 0.111 116.8 A30 3.909 0.188 197.8 A31 5.374 0.183 192.1 A32 7.512 0.162 170.2 B o i 5.401 0.104 109.3 B l l 1.868 0.085 88.9 B21 1.868 0.045 47.29 B02 5.401 0.181 189.6 B12 3.462 0.150 157.4 B22 4.149 0.085 89.18 B0 3 5.401 0.247 259.3 B13 5.261 0.240 251.7 B23 7.162 0.212 222.9 Volume - 12.40 c u . i n . i ) M i r r o r H o r i z o n t a l R e a c t i o n a t A, H o r i z o n t a l -443.8 l b , V e r t i c a l 43.3 l b . Rea c t i o n a t B, H o r i z o n t a l 443.8 l b , V e r t i c a l 154.3 l b . D e f l e c t i o n a t m i r r o r support p o i n t L. H o r i z o n t a l 0.009", V e r t i c a l 0.059". 166 i i ) M i r r o r V e r t i c a l R e a c t i o n a t A, H o r i z o n t a l +37.3 l b , V e r t i c a l 163.2 l b . Rea c t i o n at B, H o r i z o n t a l -37.3 l b , V e r t i c a l 34.4 l b . D e f l e c t i o n a t m i r r o r support p o i n t L. H o r i z o n t a l -0.009", V e r t i c a l 0.016". The above v a l u e s are s i m i l a r t o those r e p o r t e d f o r the symmetrical c a n t i l e v e r . The sag of the s t r u c t u r e i s l a r g e r when the m i r r o r i s h o r i z o n t a l and s m a l l e r when v e r t i c a l than f o r case a. The volume of e i t h e r of these examples i s 6970 c u . i n . , i n c l u d e d i s c , r i b s and c e n t r a l p i l l a r . T h i s corresponds t o a 4 8.1% s a v i n g o f volume as compared w i t h the t r a d i t i o n a l s o l i d d i s c d e s i g n . Much of t h i s s a v i n g a r i s e s from the use of an a r r a y of r i b s t o support the o p t i c a l s u r f a c e , i n p l a c e o f a c o n s i d e r a b l y t h i c k e r s o l i d d i s c . Any arrangement of r i b s w i l l show some s a v i n g . The s o l i d d i s c i s used t o e v a l u a t e volume sav i n g s f o r s p e c i f i c d e s i g n s . I t should be emphasised t h a t these are i l l u s t r a t i v e examples of the way i n which s t r u c t u r a l volume may be reduced by use of r i b s . The s o l u t i o n w i t h a b s o l u t e minimum volume r e q u i r e s f u r t h e r i n v e s t i g a t i o n . A paper by Duncan [24] i l l u s t r a t e s o t h e r r i b arrangements u s i n g M i c h e l l frameworks, and d i s c u s s e s p o s s i b l e designs f o r minimum volume s o l u t i o n s . V 1 6 7 A l t e r n a t i v e Support Arrangement for Mirror Surface Two designs have been described for the support of a c i r c u l a r mirror surface, using three r a d i a l r i b s supporting the disc at two thir d s i t s radius, the p o s i t i o n for minimum t o t a l d e f l e c t i o n for a three point suspension. Another possible arrangement consists of support-ing the d i s c at i t s centre, and also at points on the circumference of a c i r c l e whose radius w i l l be somewhat larger than that used above (b = 0.667a). The deformed shape of the disc i s indicated i n Figure 9.4. to, "E-btt ( ^ o f disc. Figure 9.4 Deformed Shape of Disc With t h i s arrangement of supports a thinner d i s c could be used while maintaining the maximum d e f l e c t i o n , ojedge' constant. For purpose of the present c a l c u l a t i o n i t w i l l be assumed that the disc surface at C w i l l be i n the 1 6 8 plane of the outer supports, S, which i s taken as fixed for the purposes of determining d e f l e c t i o n . Any deformation of the supports, S, may be regarded as a r i g i d body motion of the system and may e a s i l y be corrected. The design procedure i s somewhat complex and w i l l be described i n a series of steps. The derivation of equations used, where not extracted from standard works, i s to be found i n Appendix J . 1 . D e f l ection of a Disc with Central and Ring Supports The d e f l e c t i o n of a uniformly loaded d i s c supported at the centre and continuously around a r i n g of radius b may be determined by the superposition of two loading systems, as shown i n Figure 9.5. The two loads are re s p e c t i v e l y : -(a) a ce n t r a l point load on a l i g h t d i s c of radius a, supported at radius b (b) a uniformly loaded disc supported at radius b. 169 A 4 i 4 I i I I 4 4 1 A i 4 1 1 I 1 t S 1 1 1 1 1 4 4 4 4 4 4 4 4 1 t T t - — - • Q . — * - l (a) F i g u r e 9.5 S u p e r i m p o s e d L o a d s on a C i r c u l a r D i s c 3. 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 w i t h C e n t r a l P o i n t L o a d S o l u t i o n s o f t h i s l o a d i n g s y s t e m may be f o u n d i n most s t a n d a r d t e s t s on e l a s t i c b e n d i n g o f p l a t e s . F o r example i n " T h e o r y o f P l a t e s and S h e l l s " by Tomoshenko and W o i n o w s k y - K r i e g e r ( M c G r a w - H i l l , 1959) e q u a t i o n 89 i n A r t i c l e 19 (page 68) s t a t e s t h e d e f l e c t i o n a t any r a d i u s r as /6TTD a 9'Z In the' p r e s e n t case, t h i s equation a p p l i e s o n l y to the in n e r zone of the d i s c , f o r 0 < r < b. Thus 60,-, - "We J6TTD 9-3 The ne g a t i v e s i g n i s a p p l i e d s i n c e the c e n t r a l d e f l e c t i o n i s upwards. W i s the c e n t r a l l o a d c D = E t 3 1 2 ( 1 - v 2 ) From the above equation slope = df- 4TTD 9.4 In p a r t i c u l a r , a t the supports, r = b and slope = 9 5 In the outer s e c t i o n b <. r £ a , there i s no l o a d and the s u r f a c e i s plane. The slope i s thus c o n s t a n t , and the d e f l e c t i o n i s g i v e n by W c b(r-b) 47TTD 0+^ ) 9-4 171 4. D e f l e c t i o n o f U n i f o r m l y L o a d e d D i s c w i t h R i n g S u p p o r t T h i s l o a d i n g i s n o t s o l v e d i n T i m o s h e n k o [13] b u t t h e methods d e s c r i b e d i n C h a p t e r 3 o f t h a t work may be u s e d t o o b t a i n a s o l u t i o n . The d e r i v a t i o n i s d e s c r i b e d i n A p p e n d i x J , t h e r e s u l t s b e i n g s t a t e d b e l o w . As i n t h e above c a s e , two zones must be c o n s i d e r e d , i n s i d e and o u t -s i d e t h e s u p p o r t r i n g . F o r t h e i n n e r zone, 0 < r < b CO = 2 1 97 F o r t h e o u t e r zone, b < r < a 9* 5. D e f l e c t i o n o f D i s c w i t h Combined L o a d s » A c o n v e n i e n t c o n d i t i o n f o r t h e d e f l e c t i o n u n d e r t h e combined l o a d i n g r e q u i r e s t h e c e n t r a l d e f l e c t i o n t o be z e r o , so t h a t a l l s u p p o r t p o i n t s l i e i n t h e same p l a n e . T h i s i s i l l u s t r a t e d i n F i g u r e 9.4. I n t h i s c a s e t h e sum o f e q u a t i o n s 9.3 and 9.7 i s z e r o f o r t h e v a l u e r = z e r o . T h a t i s : -172 99 T h i s y i e l d s a v a l u e f o r t h e c e n t r a l s u p p o r t r e a c t i o n , W • 3P) - 4(itP) /^|)| H. b \ 3 sA 9. , 0 S i n c e t h e t o t a l w e i g h t o f t h e d i s c , W, i s g i v e n by t h e t o t a l r e a c t i o n on t h e r i n g s u p p o r t i s t h u s 9-H The e q u a t i o n s f o r t h e d e f l e c t i o n f o r t h e two z o n e s o f t h e p l a t e , u n d e r t h e combined l o a d i n g , a r e t h u s : -I n n e r zone 0 < r < b 0 D Z = temp O u t e r zone b < r < a 0 M P H 9'*3 -t- Wc k (T -b ' ^ 173 5. Determination of Support Radius, b For a given edge d e f l e c t i o n , below the plane of the supports, the t o t a l v a r i a t i o n , between the highest and lowest p o i n t s of the s u r f a c e , w i l l be a minimum i f no p o i n t r i s e s above the support plane. For t h i s to be s a t i s f i e d the slope at the support p o i n t s , s,must be zero ( i t w i l l a l s o be zero at C by symmetry). From equation 9.13, f o r the outer zone slope = dr 4-tt«+^> + 4 oV^v) - 2bVQ 9.14 At the supports r = b and the slope i s zero S u b s t i t u t i n g , v = 0.17, t h i s may be sol v e d to y i e l d 9*^ which has a root at a 6. Determination of Disc Thickness t The above c a l c u l a t i o n f o r support r a d i u s was i n d e -pendent of d i s c t h i c k n e s s , s i n c e t appeared o n l y i n the 'constant' term. E q u a t i o n 9.13 may thus now be s o l v e d f o r t u s i n g e q u a t i o n 9.16 to s u b s t i t u t e f o r b and the v a l u e of d e f l e c t i o n , 31.6 x 10 ^ i n c h e s , c a l c u l a t e d i n ste p 1. P u t t i n g r = a i n eq u a t i o n 9.13 S u b s t i t u t i n g a = 24, ^ = 0.7728, v = 0.17 t h i s e q u a t i o n g i v e s or t - 3'777 mc/ie& 9/7 The above c a l c u l a t i o n s have been based on the use of a continuous r i n g support, although i n f a c t s i x p o i n t supports around the c i r c l e w i l l be used. Timoshenko [13] on page 67 ( A r t i c l e 18) notes t h a t the central d e f l e c t i o n i s the same i f a g i v e n l o a d i s a p p l i e d a t one or many p o i n t around a c i r c l e of r a d i u s b. The c a l c u l a t i o n of W and the c s u p e r p o s i t i o n d e s c r i b e d i n F i g u r e 9.5 i s thus e x a c t . The d e f l e c t i o n s c a l c u l a t e d i n S e c t i o n 1 f o r a d i s c e i g h t inches t h i c k agree c l o s e l y f o r a continuous support and f o r a r a d i a 175 l i n e through a support. The edge d e f l e c t i o n on ot h e r r a d i a l l i n e s w i l l be g r e a t e r than the uniform d e f l e c t i o n here determined, but i t i s f e l t t h a t t h i s c omparatively simple c a l c u l a t i o n p r o v i d e s an a c c e p t a b l e b a s i s f o r d e s i g n . 7. Design of Support Ribs From equation 9.17 the volume and weight of the d i s c may now be c a l c u l a t e d 2 Volume = ua t = 6835 c u b i c inches . . . . 9.18 2 Weight = (ira t ) p g = 54 3.4 pounds . . . . 9.19 From equation 9.10 » 21-Q founds 9'2o (load per r i b = 13.5 lb.) T o t a l load c a r r i e d by s i x r i n g supports i s 543.4 - 81.0 = 462.4 pounds. Thus the load per support i s 77.1 pounds . . . . 9.21 The d e s i g n parameters c a l c u l a t e d above are i n d i c a t e d i n F i g u r e 9.6. The d i s c i s supported a t . t h e c e n t r e and a t s i x p o i n t s e q u a l l y spaced around a c i r c l e of r a d i u s 18.55 i n c h e s . The support i s pr o v i d e d by s i x r a d i a l r i b s each c a r r y i n g a load of 13.5 l b . a t the c e n t r e and 77.1 l b . a t the outer end. 1 7 6 Figure 9.6 Support Ribs f o r M i r r o r Disc The r i b ' c o u l d conveniently c o n s i s t of three M i c h e l l c a n t i l e v e r s , each having three f i b r e s , s p r i n g i n g from support p o i n t s at A and B, at a radius of 9.275 inches, h a l f that of the outer d i s c supports. The three c a n t i l e v e r s would have the f o l l o w i n g f u n c t i o n s : -(a) Outer load c a n t i l e v e r - This i s a skew c a n t i l e v e r r i s i n g :from A and B to the outer support p o i n t S. I t c a r r i e s a v e r t i c a l load of 77.1 pounds. (b) Inner load c a n t i l e v e r - This i s an i d e n t i c a l skew c a n t i l e v e r , as regards geometry, to (a). I t a l s o springs from A and B and r i s e s to the c e n t r a l support C, c a r r y i n g a load of 13.5 pounds. The proportions 177 of i t s members must be ad j u s t e d so t h a t the d e f l e c t i o n at C equals t h a t a t S, to s a t i s f y the de s i g n c o n d i t i o n s p e c i f i e d i n e q u a t i o n 9.9. T h i s may c o n v e n i e n t l y be achieved by making these two c a n t i l e v e r s i d e n t i c a l g e o m e t r i c a l l y . The member c r o s s s e c t i o n s w i l l d i f f e r but t h e i r l e n g t h s and j o i n t c o o r d i n a t e s w i l l be i d e n t i c a l (c) Lower moment c a n t i l e v e r - The f o r c e s a c t i o n on the r i b c a n t i l e v e r s are i n d i c a t e d i n F i g u r e 9.7. the h o r i z o n t a l plane w i l l i n t r o d u c e a h o r i z o n t a l f o r c e a t S. T h i s may be e l i m i n a t e d by a f l e x i b l e c o n n e c t i o n a t t h i s p o i n t , or by use of d i f f e r e n t maximum s t r e s s e s i n d i s c and r i b . These may be s e l e c t e d t o e q u a l i s e the deformations a t S. D i f f e r e n t i a l d e f o r m a t i o n of the d i s c and r i b i n 13-5 li JS-6 ii 7 7 / Ik. C , r 147-5* lb Reoct/ on. 9o-6 /b . F i g u r e 9.7 Force Diagram f o r a t y p i c a l r i b The two loads at S and C c r e a t e a moment which may be opposed by h o r i z o n t a l f o r c e s at C and M, t h a t a t M being supported by a t h i r d r i b s p r i n g i n g from AB. T h i s r i b a l s o w i l l be i d e n t i c a l g e o m e t r i c a l l y w i t h the o t h e r s . The f o r c e s a t C and M w i l l be generated by i n t e r a c t i o n w i t h the oth e r r i b s . The s e l e c t i o n of o v e r a l l dimensions o f these r i b s i s somewhat a r b i t r a r y and c o u l d be determined l a r g e l y by the space a v a i l a b l e f o r the s t r u c t u r e . As we have seen e a r l i e r , the volume of a s t r u c t u r e , t o c a r r y a g i v e n l o a d system, decreases as the space a v a i l a b l e f o r i t s c o n s t r u c t i o n increases. I f the f o l l o w i n g v a l u e s are s e l e c t e d : -SPAN (X SPAN) 9.275" RISE (Y SPAN) 2.0" SUPPORT SPACING (D) 3.0" then may be e a s i l y c a l c u l a t e d and i s found t o be F = 147.47 pounds 9.22 m To complete the s t r u c t u r e a ' r i g i d ' member AB has to be added t o support 'A'. The a x i a l f o r c e on t h i s member w i l l be 28.4 pounds ( t e n s i l e ) and the t r a n s v e r s e f o r c e 258.3 pounds (as determined from the r e a c t i o n s c a l c u l a t e d i n the next s e c t i o n ) . I f t h i s member i s made one i n c h square, i t s d e f l e c t i o n s under these loads w i l l be s m a l l (8.1 x 10 ^ inches a x i a l l y ) . I t c o u l d be c o n s i d e r e d as r i g i d so f a r as c a n t i l e v e r d e s i g n i s concerned. 179 8. C a l c u l a t i o n o f S i z e o f C a n t i l e v e r Members (a) Simple p i n - j o i n t e d d e s i g n - The computer pro-grammes p r e v i o u s l y d e s c r i b e d were used to determine the geometry of the c a n t i l e v e r s and the s i z e s o f each member, assuming the j o i n t s to be pinned. The p h y s i c a l c o n s t a n t s f o r fused s i l i c a were used r a t h e r than those f o r CR39 as i n the e a r l i e r c h a p t e r s . The volume of each r i b i s then composed as f o l l o w s : -Volume o f outer c a n t i l e v e r to A 3.177 Volume of i n n e r c a n t i l e v e r t o C 6.202 Volume of lower c a n t i l e v e r to M 2.589 Member AB 3.000 T o t a l volume 14.96 8 c u . i n . The t o t a l s t r u c t u r e i s thus the volume o f the d i s c (6835) p l u s t h a t o f the r i b s ( 6 x 14.97 ) or a grand t o t a l of 6925 c u b i c i n c h e s . S i n c e the d i s c d e s c r i b e d i n S e c t i o n 1 of t h i s c h a pter has a volume of 14,47 5 c u b i c inches t h i s r e p r e s e n t s a s a v i n g o f 52.2%. A n a l y s i s o f the S t r u d l s o l u t i o n shows t h a t the r e a c t i o n a t B i s a v e r t i c a l f o r c e o f 90.6 pounds (no h o r i -z o n t a l f o r c e or moment), t h a t the v e r t i c a l d e f l e c t i o n o f A i s 0.000001 inches and i t s h o r i z o n t a l d e f l e c t i o n i s 0.00003 180 inches. In general the r e s u l t s of the pinned-joint and r i g i d analysis agree well, as was discussed i n the e a r l i e r part of t h i s work. 1 CONCLUSIONS The use of M i c h e l l frameworks, t o support a g i v e n a r r a y of loads i n space and t r a n s f e r them t o fo u n d a t i o n s , r e q u i r e s l e s s m a t e r i a l than any other type of s t r u c t u r e o ccupying the same space, and s u b j e c t e d t o the same maximum s t r e s s . The s a v i n g i n volume can range from 30% or more f o r open t r u s s e s t o s e v e r a l hundred per cent f o r beams of s o l i d s e c t i o n . A M i c h e l l framework w i l l a l s o be s t i f f e r and e x h i b i t l e s s d e f l e c t i o n under the same c o n d i t i o n s than any oth e r type o f framework s u b j e c t e d to the same maximum s t r e s s . A M i c h e l l framework exceeds the optimum minimum volume o f the t h e o r e t i c a l i d e a l s t r u c t u r e by l e s s than 10%. A f i v e f i b r e d M i c h e l l frame p r o v i d e s a c l o s e approxi mation t o the i d e a l , and i s w i t h i n f i v e per cent of optimum volume. A g r e a t e r number of f i b r e s need not be used. The r e d u c t i o n i n volume of s t r u c t u r e s made p o s s i b l e by the use of M i c h e l l frameworks, renders economic t h e i r use wherever m a t e r i a l s are expensive, t h e i r c o s t of t r a n s -p o r t i s c o n s i d e r a b l e or where the s t r u c t u r e s are s u b j e c t e d to h i g h r a t e s of a c c e l e r a t i o n . M i c h e l l s t r u c t u r e s a l s o have advantages over other d e s i g n s i n cases where the d e f l e c t i o n i s l i m i t e d . In such 182 cases the maximum s t r e s s w i l l be l e s s than t h a t encountered i n other d e s i g n s . Manufacture of M i c h e l l frameworks has been eased by the i n t r o d u c t i o n of more s o p h i s t i c a t e d machine t o o l s and the use of numerical c o n t r o l . M i c h e l l - l i k e frameworks can be formed i n p l a t e s and o t h e r s o l i d members by the judicuous placement of l i g h t e n i n g h o l e s . The use of such holes has long been a f e a t u r e of marine and a e r o n a u t i c a l d e s i g n , and t h e i r use co u l d w e l l be extended. The c r i t e r i a d e s c r i b e d i n t h i s work pr o v i d e s a t h e o r e t i c a l b a s i s f o r the l a y o u t of such c u t -outs . 183 REFERENCES 1. Maxwell, J.C. S c i e n t i f i c Papers I I , O.U.P. 1890, pp. 175-177. 2. M i c h e l l , A.G.M. "The L i m i t s of Economy of M a t e r i a l i n F rame-Structures." P h i l . Mag. S e r i e s 6, 8(47) 589-597, London, November 1904. 3. Cox, H.L. "The Theory of Design," A e r o n a u t i c a l Research C o u n c i l , 19.791, January 1958. 4. Cox, H.L. " S t r u c t u r e s of Minimum Weight. The b a s i c theory of d e s i g n a p p l i e d t o the beam under pure bending." A e r o n a u t i c a l Research C o u n c i l , 19.785, January 1958. 5. Hemp, W.S. "Theory of S t r u c t u r a l Design." AGARD Report 214, October 1958. 6. Chan, A.S.L. "The Design of M i c h e l l Optimum S t r u c t u r e s , " C o l l e g e of A e r o n a u t i c s , C r a n f o r d , Report No. 142, December 1960. 7. Love, A.E.H. "A T r e a t i s e on the Mathematical Theory of E l a s t i c i t y . " C.U.P. 1927. 8. Johnson, W. "An analogy between upper bound s o l u t i o n s f o r p l a n e - s t r a i n metal working and minimum weight two d i m e n s i o n a l frames." I n t e r n a t i o n a l J o u r n a l of M e c h a n i c a l S c i e n c e , 1961, V o l . 3, pp. 239-246. 9. B a r n e t t , R.L. "Survey of Optimum S t r u c t u r a l Design." E x p e r i m e n t a l Mechanics, December 1966, p. 19A. 10. Vaughan, Henry. " D e f l e c t i o n of 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 Equispaced P o i n t Supports." J o u r n a l of S t r a i n A n a l y s i s , A p r i l 1970. 11. H i l l , R. "The Mathematical Theory of P l a s t i c i t y , " O x ford, Clarendon P r e s s , 1950. 12. W i l l i a m s , R. " D e f l e c t i o n S u r f a c e of a C i r c u l a r P l a t e w i t h M u l t i p o i n t Support," E n g i n e e r i n g Report 9136, January 1968, Perkin-Elmer. 13. Timoshenko, S. and Woinowsky, K r i e g e r . "Theory of P l a t e s and S h e l l s , " McGraw-Hill, 1959. 184 14. Johnson, W., Mellor P.B., " P l a s t i c i t y for Mechanical Engineers," Van Nostrand, 1962. 15. Johnson, W., Sowerby, R. , Haddow, J.B., "Plane-s t r a i n S l i p - l i n e F i e l d s . " Arnold, 1970. 16. Cox, H.L., "Design of Structures of Least Weight." Pergamon, 1965. 17. Hegemier, G.A. and Prager, W., "On M i c h e l l Trusses," International Journal of Mechanical Science, 1969, Vol. 11, pp. 209-215. 18. Sheu, C.Y. and Prager, W., "Recent Developments i n Optimal S t r u c t u r a l Design," Applied Mechanics Reviews, No. 21, Nol. 10, October 1968, pp. 985-992. 19. Ghista, D.N., "Optimisation of Structures with respect to weight," Dept. of C i v i l Eng., Standord, January 1965. 20. Ghista, D.N. "Fully-stressed design for a l t e r n a t i v e loads." ASCE Journal of the S t r u c t u r a l D i v i s i o n , V92, No. ST5, October 1966. 21. Soosaar, K., "Optimisation of topology and geometry of s t r u c t u r a l frames," Dept. of C i v i l Eng., Massachusetts I n s t i t u t e of Technology, May 19 67. 22. ICES STRUDL-11, Engineering Users Manual i n two volumes, Department of C i v i l Engineering, School of Engineering, Massachusetts I n s t i t u t e of Technology. 23. Johnson, W. "Upper bounds to the Load for the Transverse Bending of F l a t Rigid P e r f e c t l y P l a s t i c Plates," Int. J. Mech. S c i . 1969, Vol. 11, pp. 913-938. 24. Duncan, J.P.,"An Optimal concept for the support of Lightweight Mirror Substrate," Departmental Report, Mechanical Engineering, University of B r i t i s h Columbia. O P T I M U M M I C H E L L F R A M E S BY ERIC WILLIAM JOHNSON B.Sc. (Eng), London, 1950 M.Sc. U n i v e r s i t y of A l b e r t a , 1965 VOLUME 11 TABLE OF CONTENTS FOR APPENDIX A p p e n d i x Page A NOTATION FOR STRUCTURES 1 B EQUATIONS GOVERNING GEOMETRY OF STRUCTURE 6 C COMPUTER PROGRAMMES FOR STRUCTURE DESIGN 17 D FORCE SYSTEM IN MICHELL CANTILEVERS . . . . 25 E DATA FOR SELECTED MICHELL CANTILEVERS . . . 42 F BIAXIAL STRESS IN JOINTS . . 75 G DETAILS OF COMPARABLE STRUCTURES . . . . . 83 H MANUFACTURE OF PHOTOELASTIC MODELS . . . . 97 J DEFLECTION OF UNIFORMLY LOADED PLATES . . . 118 LIST OF FIGURES OF APPENDIX F i g u r e Page A l N o t a t i o n f o r M i c h e l l s t r u c t u r e s 2 A2 Members i n a t y p i c a l p a n e l 3 B l Members i n f a n 6 B2 T y p i c a l q u a d r i l a t e r a l p a n e l 7 B3 J o i n t s i n t y p i c a l M i c h e l l c a n t i l e v e r . . . 9 B4 F i v e f i b r e s y m m e t r i c a l c a n t i l e v e r 12 B5 D e f l e c t i o n of a M i c h e l l frame 15 D l F o r c e s a c t i n g on a M i c h e l l c a n t i l e v e r . . . 28 D2 F o r c e s a t j o i n t J„„ 29 J NN D3 F o r c e s a t a t y p i c a l i n n e r j o i n t J ^ . . . . 32 D4 F o r c e s a t a t y p i c a l 'A1 f a n j o i n t , J , . . 34 a, ± D5 F o r c e s a t a t y p i c a l 'B' f a n j o i n t , J ^ ^ . . 36 D6 F o r c e s a t j o i n t J ^ 37 D7 Freebody diagrams f o r f o r c e s a t s u p p o r t s . . 39 F l B i a x i a l s t r e s s e s a t a t y p i c a l j o i n t . . . . 76 F2 A p p r o x i m a t i o n t o a b i a x i a l l y s t r e s s e d j o i n t 77 F3 E x t e n s i o n o f a t y p i c a l member 80 G l Warren t r u s s 84 G2 W i l i o t diagram f o r d e f l e c t i o n o f a Warren t r u s s . . . 86 G3 Two bar c a n t i l e v e r 87 Figure Page G4 Cantilever of parabolic section 89 G5 Triangular plate cantilever 91 G6 I-beam cantilever 94 HI The e f f e c t of rotation 101 H2 Geometry of cutter o f f s e t s . . . . . . . 102 H3 Order and d i r e c t i o n of cuts for machining a t y p i c a l Michell cantilever 105 J l Loaded disc and supports . 1 APPENDIX A NOTATION FOR STRUCTURES A j o i n t could be i d e n t i f i e d by i t s c u r v i l i n e a r co-ordinates, a and 3. This w i l l however prove cumbersome since the angles w i l l , i n general not be i n t e g r a l values. A more convenient notation i s used i n t h i s work and i s i l l u s t r a t e d i n Figures A i and A2. The support points are i d e n t i f i e d by the l e t t e r s A and B and from these radiate chains of members. These are numbered consecutively, commencing with the base curves ACE and BCD. In the general case 'a' i s used to i d e n t i f y points on the 'A' chains and 'b' for those in,: the * 1B' chains. Joints Joints are i d e n t i f i e d by the numbers of the l i n e s intersecting at that j o i n t , written as subscripts to the c a p i t a l l e t t e r ' J ' . The general j o i n t i s thus written 'J ^ '. * It should be noted that the c u r v i l i n e a r coordinates of the jo i n t s can be obtained from these numbers. If 8 i s the coordinate change between members, (the angle between members), then the c u r v i l i n e a r coordinate of a l l points on the chain a i s given by Si m i l a r l y (a-1) 6 (b-l) 6 } A l F i g u r e A l N o t a t i o n f o r M i c h e l l S t r u c t u r e s 3 F i g u r e A2 Members i n a T y p i c a l Panel Most j o i n t s are i d e n t i f i e d on F i g u r e A l . In a symmetrical s t r u c t u r e , there are p a i r s of 'mirror image 1 j o i n t s . Such j o i n t s may be i d e n t i f i e d by exchanging the s u b s c r i p t s . Thus, as an example, J 3 7 and are such a p a i r . J o i n t s with equal s u b s c r i p t s l i e on the centre l i n e CL i n a symmetrical s t r u c t u r e . Members A member i s i d e n t i f i e d by r e f e r e n c e t o the j o i n t a t i t s inward e n d — t h a t n e a r e s t the support p o i n t . Members l y i n g along the 'a' c h a i n are i d e n t i f i e d as A.^ w h i l e the mem-bers along the 'b' c h a i n are i d e n t i f i e d as B a k * These symbols a l s o r e f e r to t h e i r l e n g t h s . Widths of Members - WA . — ab The width i s i n d i c a t e d by adding W t o the i d e n t i f i -c a t i o n symbol f o r the member. C r o s s - s e c t i o n a l Area of Members - sA , ab The c r o s s - s e c t i o n a l area i s i n d i c a t e d by the symbol sA , . ab In most cases the members are assumed to be of uniform t h i c k n e s s t . Then sA , = t (WA , ) . . . . A2 ab ab Force A c t i n g on a Member - F A a j 3 Each member i s s u b j e c t e d t o an a x i a l f o r c e , e i t h e r t e n s i l e or compressive, a c t i n g along the member. T h i s f o r c e i s i n d i c a t e d by the symbol FA^. The s i g n of the f o r c e i n d i c a t e s i t s c h a r a c t e r , t e n s i l e f o r c e s being p o s i t i v e and compressive f o r c e s n e g a t i v e , f o l l o w i n g the normal convention. The members of a t y p i c a l panel are shown i n F i g u r e A2. The q u a n t i t i e s r e l a t i n g to these members are list e d : , below. Normally, the f o r c e s i n the 'A' members are t e n s i l e and those i n 'B' members are compressive. The s i g n w i l l . however be r e t a i n e d t o minimize c o n f u s i o n , and i s necessary f o r c l a r i t y when c o n s i d e r i n g o b l i q u e l o a d i n g s . 5 Member QR RS QT TS Name A a b Ba,b+1 a ,b A a + l , b Length A a b Ba,b+1 a ,b A a + l , b Width WA , ab a ,b+l WB , a,b WA ^, , a+1 ,b Area ( c r o s s -s e c t i o n ) s A a b s B a , b + l sB , a ,b sA ., , a+1 ,b Force ( + ) F A a b (-) FB a ,b+l ( - ) F B a , b ( + > F A a + l , b Members i n Fans The members i n the fans are s i m i l a r l y d e s i g n a t e d as those d e s c r i b e d above. For a l l members r a d i a t i n g from A, b = zero while from B, a = zero. Thus t y p i c a l members i n each fan are A and B , . The same p r e f i x e s are used f o r a,o o,b c widths, areas and f o r c e s as f o r the other members. 6 APPENDIX B EQUATIONS GOVERNING GEOMETRY OF STRUCTURE LENGTHS OF MEMBERS The le n g t h s of the members of the s t r u c t u r e may be d e r i v e d as f o l l o w s : -a) Fans F i g u r e B l Members i n Fan As e a r l i e r s t a t e d Bo.b » B l From F i g u r e B l A,. b - sinf? 8 Ba,, = 2rA sirx(|) B2 7 b) Remainder of S t r u c t u r e F i g u r e B2 shows a t y p i c a l q u a d r i l a t e r a l panel of the s t r u c t u r e , t y p i c a l of a l l panels o u t s i d e the base fans, The g e n e r a l t i t l e s of the j o i n t s and s i d e s are i n d i c a t e d . F i g u r e B2 T y p i c a l Q u a d r i l a t e r a l Panel to QR. C o n s t r u c t TW p e r p e n d i c u l a r to RS and QX p e r p e n d i c u l a r Then from the geometry of the f i g u r e A a M . b ~ Aq,t> + Ba.b siaB cos Q Ba.b*i = Aa.b SitxQ + Bo.t. cos & B3 8 Once the lengths of the c i r c u m f e r e n t i a l f an members, A ( l b) a n d B ( a 1 ) ' a r e known equations B3 may be a p p l i e d t o each panel i n t u r n s t a r t i n g from C and working outward i n a syst e m a t i c manner. J o i n t C o o r d i n a t e s A t y p i c a l M i c h e l l framework i s shown i n F i g u r e B3. The c o o r d i n a t e s of each j o i n t may be found by working along each f i b r e i n sequence from the support. E i t h e r s e t may be used f o r t h i s purpose, but the s i g n convention used f o r the angles of both s e t s i s i n d i c a t e d s i n c e these are a l s o needed f o r determining d e f l e c t i o n s . The formulae f o r c a l c u l a t i n g ^ and ^ B a b a r e a s f o l l o w s ; d i f f e r e n t formulas being r e q u i r e d f o r the f a n members. 'A' Members fan members o n l y . . B4 o t h e r members fan members o n l y . . B5 other members TAa.o = [ (a - l)8 - p j VAa.b =[(Q-b)&-(P+|)] 'B' Members • V B a . b = [ (b-a )0*(p-f ) ] Figure B3 Joints i n Typical Michell Cantilever J 10 Once these angles are determined the coordinates of any j o i n t follow. X J Q . B = Q + baO B6 1 Aa.b sia(^ A0.b) b a O ' SPAN AND RISE OF STRUCTURE Equations B6 may be used to express the span, L, and r i s e , Y, i n terms of member lengths and angles i|>A and hence ultimately by equations Bl to B4 i n terms of 8, r,. and r_.. A B Figure B4 shows a f i v e f i b r e symmetrical structure. The lengths of each member are tabulated below. To save space s in 8 and cos 8 are written respectively as s and c while (2r si n 8/2) i s written as f. From these tabulated values, expressions may be• derived for the span, L, i n terms of the member lengths, which i n turn reduce to the fan radius r and the f i b r e angle 8. The angle ACO i s 45° by d e f i n i t i o n while the angle n between the axis and each successive A member i s given by n = (45-8/2). Such expressions may be written for various values of N by considering K, M, N and L i n succession as the outer point of the span. 11 Member A i.o 4 o As.c Length /(L+s) - f c / ( s + 2sz+ s 3 + sc + c) 4 - c 3 J ( s + s 2 + s ' c + c*) + c 3 f () + 5 + S C - r S C J ) C 3 i ( l + s+ sc+sc z+sc 3) - j-c 4 J (S + S* + S*C * SZC*+ C 3 ) -r C 4 j ( l + S + S i + 2s 3 + 3 4 + SC->2s*c) - r C 4 J- ^2s +3s*+ s3+ sc •+• s 3c + c ) c 4 f(s+ 3s2+ 3s3-!- s4+ sc + 2s2c + s?c\ s*c+ c 3 ) -r c 5 } (2s + 4s*> 2s3•+ sc + 2s3c + s c 3 * 53c*+ c) -r c 5 / ( l + 2s + 2s*+ 3 s 3 + 3s4+ s c •+ 3sc+ 2s 3c) -r c 5 J^2s+5s*> 5s3+4s4+ 35£+64+ sc + 3sc+ s V + 3s*c + 3 s ^ + c 3 ) - i - c 6 j£l+2a + 3sa+ 6s3+ 6 s 4 2 s * + sc + 4s*c + + sc^sV+s^e) A 5 4 ^ ( 2 s * 7 s z * IOs3 + 9s4+7ss+ 4s* + s 7+ sc + 4 s c + s c V ^ ' c ] BI.I "fe A^.i A 3 . 1 A 3 . z B3.2 A4.I As.. B4.3 A5.2 B4.4 A 5.3 igure B4 Five Fibre Symmetrical Cantilever N»2. fsU5 L ~ t -cos45 +cos (45-|)|I a (s + 2s 2 +s a +sc + s c l H - c + c ? ) ] s tcos> 45 + cos |2L (2s + S 2 + Sc + c + i)J N- + L = > c o s 4 5 + cos(45-| j = t COS 45 + cos (45-| ) — 11 + 2s + 2sz + 3s3-*- 3s*+ s5+ sc+ sc' U 3sc+2&\?£c* sc3+ s\z+ sc<+ c3+ , £ [2+2s+2s*+ s3-»- 2 s 4 - s s - ( - 2 s c •+ 2 , s l c + s 3 c + c N = 5 L = t cos 45 + cos(4S-|) Iks f 7&S /o^+9^+75:>+4s^s7+sc + 2sc2 *2s\\ 3s*c -*3sV+ sc5* 2sV»+ sfc*+ s V + sc4*/ + c^ -* c \ c^ -*-- "f C O S 45 + cas(+S-QJ | 7 I + 4&i + 4sc -c 4sh -»• + 2s»s<^  Over th i s limited range of N no coherent pattern i s v i s i b l e i n these equations, either as derived ( f i r s t line) or 2 2 after reduction by use of the equation sin 8 + cos 6 = 1 . It may well be that a more exhaustive analysis of the geometry w i l l y i e l d a general equation but the very nature of the con-struction seems to make t h i s rather u n l i k e l y . 14 D e f l e c t i o n o f the S t r u c t u r e The change i n l e n g t h o f each member i s known once the geometry of a framework has been determined. Since uniform s t r a i n i n a l l members i s a des i g n requirement, the change i n l e n g t h o f a member i s simply Ae = ± £A Q b • ..... sa the s i g n being the same as t h a t of the f o r c e FA ^ a c t i n g on the same member. The d e f l e c t i o n may thus be determined by t r i g n o m e t r i c a l s o l u t i o n o f a W i l i o t diagram. Consider a p o r t i o n o f a M i c h e l l framework as shown i n F i g u r e B5a. For convenience, assume t h a t member A, , i s i n t e n s i o n and member B, , i s i n (a,b-l) (a-l,b) compression. As a r e s u l t of these f o r c e s the members w i l l s u f f e r changes i n l e n g t h r e p r e s e n t e d by PQ and PS r e s p e c t i v e l y and the j o i n t w i l l move to a new p o s i t i o n J 1 r e l a t i v e t o the j o i n t s J , , , 4 a n d J , , The d e f l e c t i o n s of these j o i n t s J (a,b-l) (a-l,b) J w i l l cause a f u r t h e r movement of j o i n t J ^. A p o r t i o n of. the corresponding W i l i o t diagram i s shown as F i g u r e B5b. T h i s i s a v e c t o r diagram showing d e f l e c t i o n o f each j o i n t r e l a t i v e to a f i x e d o r i g i n (the support p o i n t s A and B). The c o o r d i n a t e s o f t h i s p o i n t on t h i s diagram r e p r e s e n t s the d e f l e c t i o n of * the corresponding p o i n t i n the framework. In these diagrams iJ;A and ibB are both shown p o s i t i v e as d e f i n e d i n F i g u r e B3. I t may be shown t h a t the equations s t a t e d below apply e q u a l l y f o r cases where these are n e g a t i v e . F i g u r e B5 D e f l e c t i o n o f a M i c h e l l F r a m e B9 From the f i g u r e : -Xab = *a.b-.+ £Aa,b-. cos^ Aa.b-, + QR sinS'Aa.b-! s Xo...b-^Ba./.fc>s/Viiy60./.6+ 5 R cosVBo-i.6 Yab= Ya.b-/ - HAo.b., S/Vl^ Ap.bw + QR^StPAo.b., = Yo.,J, + eBa.,.6 «« *B^ .6+ SR sin 4>B«W.A In these equations a l l terms are known except the lengths QR and SR which may be e l i m i n a t e d by c r o s s - s u b s t i t u t i o n . The f o l l o w i n g r e s u l t s are thus o b t a i n e d : -XQ|>= Xo.b-, + f A Q . b - , C o s q J A o i > - , - K eih. q>A Q . b - , . . . . BIO Y q b = Y q .b,, -EAa.b., siatfVWb-, - K «»» 4>AQ.b-i where K r e p r e s e n t s the f o l l o w i n g : -K - (Xq-i.t> -XQ.b-i)sinq>e>q.(i> - EBq-i.i, + (Tra.b-i~YQ.(.b)ooslrBo./.t « » • (H»Aa.b-i + SV&a-i.b) B l l These equations are a p p l i e d s e q u e n t i a l l y t o each j o i n t i n t u r n , working outward from C ( J ] _ j ) - * f a n Y members are u n s t r e s s e d , the same equations may be used w i t h e pl a c e d equal to zero. 17 APPENDIX C COMPUTER PROGRAMMES FOR 'DESIGN OF MICHELL STRUCTURES The equations d e s c r i b e d i n Chapters 2 and 3 and Appendices A and B were d e r i v e d from study of g e n e r a l i z e d s t r u c t u r e s . The data f o r s i x r e p r e s e n t a t i v e s t r u c t u r e s was compiled with the a i d of a desk c a l c u l a t o r . These r e s u l t s were then checked by l a r g e s c a l e g r a p h i c a l s o l u t i o n s wherever p o s s i b l e . I t was e v i d e n t t h a t these manual methods would be i m p r a c t i c a l f o r the s o l u t i o n of the wide range of s t r u c t u r e s analyzed i n Chapter 4. Computer programmes were thus designed to produce the r e q u i r e d data, the e a r l i e r c a l c u l a -t i o n s being used to check these programmes. The r o u t i n e s were not w r i t t e n as a s i n g l e u n i t . Rather each s e c t i o n was proved b e f o r e proceeding to the c a l c u l a t i o n of succeeding q u a n t i t i e s . The f o l l o w i n g programmes are f a i r l y s t r a i g h t f o r w a r d , s i n c e many s u b t i t l e s have been i n c l u d e d to e x p l a i n the purpose of each segment. Some notes are appended to a m p l i f y these comments. 18 PROGRAMME 1 - SYMMETRICAL CANTILEVERS As e x p l a i n e d i n Chapter 2, a symmetrical M i c h e l l framework i s completely s p e c i f i e d once the span, L, support s p a c i n g , D, and number of f i b r e s , N, have been s t a t e d . The f i b r e angle, 9 , may be u n i q u e l y determined from these parameters and member f o r c e s , widths and a l l other v a r i a b l e s r e q u i r e d f o r study may then e a s i l y be c a l c u l a t e d . The programme which ac h i e v e s t h i s r e s u l t may be c o n s i d e r e d i n the f o l l o w i n g s e c t i o n s : -a) Steps 1-32 There are of an a d m i n i s t r a t i v e nature nature s p e c i f y i n g the s i z e of memory areas s e t a s i d e f o r c a l c u l a t i o n and the manner i n which the data i s p r i n t e d . b) 33-87 The c a l c u l a t i o n s f o r each new s t r u c t u r e commence a t st e p 33. Here the memory areas are r e s e t t o zero and N, L and D are s p e c i f i e d , Member t h i c k n e s s , T, s t r e s s , o, and s t r a i n , e, are a l l s t a t e d here as i s the s p e c i f i c g r a v i t y of the m a t e r i a l . I f the s t r u c t u r e i s assumed w e i g h t l e s s , SPGTY i s s e t equal t o zero. The le n g t h s of the f a n members, A, » and B, , > 3 (a,o) (o,b) are c a l c u l a t e d , s i n c e these are independent of c) 88-120 T h i s i s the 'heart' of the programme as i t i s here t h a t 8 i s c a l c u l a t e d . A v a l u e of 6 i s assumed and the co r r e s p o n d i n g span, TL, c a l c u -19 l a t e d (step 101 ). TL i s then compared wi t h L. I f s m a l l e r than L, 8 i s i n c r e a s e d by 6 8 (DELTHE) and the programme r e t u r n e d t o step 72 . I f TL i s l a r g e r than L, 8 i s too l a r g e so i t i s reduced and DELTHE i s d i v i d e d by TEN be f o r e r e t u r n i n g t o step 72 . In t h i s way the ex a c t v a l u e o f 8 i s approached from below. T h i s i t e r a t i v e p rocess c o n t i n u e s u n t i l the ERROR or L — T L — ^ — i s l e s s than 0.0001%,or f o r t y c y c l e s have been r e p e a t e d . As p a r t of t h i s l o o p , the l e n g t h s o f a l l members are c a l c u l a t e d . The l a s t s teps o f t h i s s e c t i o n p r i n t these l e n g t h s . d) Steps 121-143 The angles T|»A b (PSIA) and iJ;B a b (PSIB) as d e f i n e d i n F i g u r e B3 are c a l c u l a t e d . T h i s i s an e s s e n t i a l p r e l i m i n a r y t o the c a l c u l a t i o n o f the f o r c e s . e) 144-209 The f o r c e s i n a l l members are c a l c u l a t e d from the somewhat unwieldy equations d e s c r i b e d i n Appendix D. The loads are f i r s t read i n ; t h i s programme p r o v i d i n g f o r a s i n g l e l o a d a t J N N i n any d i r e c t i o n . A simple m o d i f i c a t i o n o f steps 167-169 permits the i n t r o d u c t i o n o f mul-t i p l e l o a d i n g s , o r loads a t oth e r j o i n t s , as may be d e s i r e d . f) 210-257 J o i n t c o o r d i n a t e s , XJ , , . and Y J , , w and (ab) (ab) j o i n t d e f l e c t i o n s , X, , s and Y, , w can now be J ' (ab) (ab)' c a l c u l a t e d . Although the d e f l e c t i o n i s i n d e -pendent of the magnitude of the f o r c e a c t i n g on a member, the nature of the f o r c e — t e n s i l e or c o m p r e s s i v e — m u s t be known t o determine whether the member el o n g a t e s o r c o n t r a c t s . Steps 215-222 check t h i s and a s s i g n the c o r r e c t s i g n to the change i n l e n g t h and thus t o the d e f l e c t i o n . g) Steps 258-End The remainder of the programme c a l c u l a t e s member widths, c r o s s - s e c t i o n a l areas and oth e r p r o p e r t i e s and the support r e a c t i o n s . The f i n a l segment c a l c u l a t e s the volume of the t e n s i l e and of the compressive members and of t h e i r sum, a f t e r which the t o t a l s t r u c t u r a l weight i s * found. The specimen programme i n c l u d e s a p r i n t o u t o f data f o r a simple c a n t i l e v e r f o r which N = 2. PROGRAMME 2 - SKEW MICHELL CANTILEVER In a skew s t r u c t u r e the span L i s r e p l a c e d by two parameters, the x and y c o o r d i n a t e s of the o u t e r end J N N - These are c a l l e d XSPAN and YSPAN i n t h i s programme which c o n s i d e r s two v a r i a b l e s , 0, and the r a t i o of the f a n r a d i i , c a l l e d RADRAT. S e v e r a l methods were t r i e d f o r the c a l c u l a t i o n of THETA and RADRAT w i t h v a r y i n g s u c c e s s . Both XSPAN and YSPAN are unknown f u n c t i o n s of THETA and RADRAT, and changes i n e i t h e r v a r i a b l e c o u l d a f f e c t both l e n g t h s . The i t e r a t i v e procedures f i r s t used f a i l e d to converge on the c o r r e c t v a l u e s and u s u a l l y o s c i l l a t e d about them, r e p e a t i n g the same c a l c u l a t i o n s u n t i l terminated by a s a f e t y counter. T r i a l and e r r o r g r a p h i c a l s o l u t i o n s of some t y p i c a l examples i n d i c a t e d t h a t s m a l l changes i n THETA p r i m a r i l y a f f e c t e d XSPAN while YSPAN was c o r r e s p o n d i n g l y s e n s i t i v e t o RADRAT. I t was t h e r e f o r e d e c i d e d t h a t the c o r r e c t i o n s t o be made to the estimated v a l u e s of THETA and RADRAT should be r e l a t e d t o the corresponding e r r o r s i n XSPAN and YSPAN. An e r r o r term f o r each of these lengths was thus c a l c u l a t e d ERRX» XSPAN - TLX XSPAM . . . . c i ERRY* YSPAN-TLY Y S P A K where TLX and TLY are the d e r i v e d v a l u e s of the span and r i s e of the s t r u c t u r e . Normally ERRX and ERRY w i l l have v a l u e s i n the range " l - O ^ E1RRX < + l ' 0 . . . . C2 a minus s i g n i n d i c a t i n g that, the estimate i s too l a r g e . T h i s e r r o r term was then used to c a l c u l a t e the c o r r e c t i o n to THETA or RADRAT by use of the t y p i c a l statement 22 THETA * THETA -> (ERRX * DELTHE) • • • • 0 3 DELTHE i s s p e c i f i e d i n the programme preface as a convenient small angle ( t y p i c a l l y 1.0 degree). S i m i l a r pro-cedure a p p l i e s to RADRAT. Provided t h a t THETA and RADRAT are c l o s e to the true v a l u e , t h i s r o u t i n e converges q u i t e r a p i d l y and automatic-a l l y c o r r e c t s f o r over and under estimates, since the c o r r e c -t i o n changes s i g n w i t h the e r r o r term. However, i f the assumed values of THETA and RADRAT . are remote from the true v a l u e , the e r r o r term can be la r g e (greater than ±1.0) and the c o r r e c t i o n i s correspondingly coarse. This may be so la r g e t h a t the new value of THETA or RADRAT i s even f u r t h e r from the 'root' but on the other s i d e of i t . I f t h i s occurs the e r r o r f u n c t i o n w i l l be a l t e r n a t e l y p o s i t i v e and negative and of ever i n c r e a s i n g s i z e . The procedure i s divergent and no s o l u t i o n i s obtained. The programme was thus modified to check t h i s , by •' comparing the magnitude of the e r r o r f u n c t i o n s i n successive c y c l e s . I f e i t h e r i n c r e a s e , i n d i c a t i n g the s t a r t of a divergent s o l u t i o n , the next c o r r e c t i o n i s halved. This has been found to be most e f f e c t i v e although i t i s p o s s i b l e that the use of a very poor p r e l i m i n a r y estimate of THETA and RADRAT c o u l d generate e r r o r f u n c t i o n s so l a r g e t h a t they would s t i l l d i v e r g e . Use of F i g u r e 4.13 should a s s i s t i n t h i s r e g a r d . The programme i s otherwise s i m i l a r to Programme 1 and may be s i m i l a r l y d i v i d e d , as f o l l o w s : a 1-33 A d m i n i s t r a t i v e and Format. b 34-66 S p e c i f i c a t i o n of N, XSPAN, YSPAN, D, T, SIGMA, STRAIN and SPGTY„ c 67-149 I t e r a t i v e r o u t i n e f o r THETA and RADRAT d e s c r i b e d above. d 150-191 PSIA(ab) a n d P S I B ( a b ) . e 192-234 Fo r c e s i n members. f 235-283 J o i n t c o o r d i n a t e s and d e f l e c t i o n s . g 284-END Member p r o p e r t i e s , r e a c t i o n s and volumes. Minor V a r i a n t s S p e c i a l i z e d d a t a was produced by minor m o d i f i c a t i o n s t o these programmes, which may now be d e s c r i b e d : -W e i g h t l e s s S t r u c t u r e s W, a t the b e g i n n i n g o f s e c t i o n e, was p l a c e d equal to z e r o . V a r i a b l e S t r a i n The r e s u l t s i n Tab l e 11, Appendix E were o b t a i n e d by e n t e r i n g the d e s i r e d v a l u e o f s t r a i n as an i n p u t and c a l c u l a t -i n g from i t the c o r r e s p o n d i n g v a l u e of s t r e s s . 24 D i r e c t i o n o f S e l f Weight Forces The e quations f o r the f o r c e s i n members, as d e r i v e d i n Appendix D, a l l o w f o r a v a r i a b l e d i r e c t i o n of g r a v i t a t i o n a l a t t r a c t i o n , by the i n t r o d u c t i o n o f a v a r i a b l e a n g l e , + , (THGRAV) . The steps i n s e c t i o n e were r e p l a c e d by a s e t i n c o r p o r a t i n g t h i s v a l u e . PROGRAMME 4 - RIGID JOINT STRUCTURE As d e s c r i b e d i n Chapter 5, d a t a f o r r i g i d s t r u c t u r e s was o b t a i n e d by use of the STRUDL programme (STRUctural Design Language) which i s a p a r t o f ICES ( I n t e g r a t e d C i v i l E n g i n e e r i n g System), a v e r y complex computer r o u t i n e capable of s o l v i n g many types o f s t r u c t u r e . U n l i k e the r o u t i n e d e s c r i b e d above which c a l c u l a t e member parameters, STRUDL r e q u i r e s as i n p u t the j o i n t c o o r d i n a t e s o f the s t r u c t u r e t o g e t h e r w i t h the c r o s s - s e c t i o n a l area and second moment of area o f each member. I t thus a n a l y s e s a known s t r u c t u r e but i t does t h i s comprehensively by m a t r i x i n v e r s i o n , d e t e r m i n i n g shear f o r c e s , bending moments and r o t a t i o n s o f the members a t each r i g i d j o i n t . F u l l d e t a i l s of STRUDL may be found i n the handbooks p u b l i s h e d by MIT, where the system was developed. F O R T R A N IV G C O M P I L E R M A I N 0 4 - 2 9 - 7 0 1 9 : 2 8 : 5 3 P A G E 0 0 0 1 C PROGRAMME FOR THE S O L U T I O N OF S Y M M E T R I C A L M I C H E L L C A N T I L E V E R S . C I N C L U D I N G THE E F F E C T OF S E L F WEIGHT F O R C E S . 0 0 0 1 O I M E N S I C N A < 2 5 , 2 5 ) , E ( 2 5 1 2 5 ) , X ( 2 5 , 2 5 ) , Y ! 2 5 , 2 5 ) , P S I A ( 2 5 , 2 5 > , 1 P S I B ( 2 5 , 2 5 ) , F A 1 2 5 , 2 5 ) , F B ! 2 5 , 2 5 ) , T I L T 1 2 5 , 2 5 ) , S U L C L A I 2 5 , 2 5 ) ,  2 L O A D ! 2 5 , 2 5 ) , X J ( 2 5 , 2 5 ) , Y J < 2 5 , 2 5 ) , W A ( 2 5 , 2 5 ) , W B < 2 5 , 2 5 1 , A R E A A ( 2 5 , 2 5 3 ) , A R E A B I 2 5 , 2 5 ) , M I Z A ! 2 5 , 2 5 ) , M I Z B I 2 5 , 2 5 I , E U L C L B ( 2 5 , 2 5 ) , 4 H T I N A 1 2 5 , 2 5 ) , W T I N B ! 2 5 , 2 5 ) , E A ( 2 5 , 2 5 ) , E B ( 2 5 , 2 5 ) 0 0 0 2 R E A L L . L O A O , I T E H L , I T E M R , M I Z A . M I Z B 0 C C 3 1 F O R M A T , 1 3 ) 0 0 0 4 3 F O R M A T I F 1 5 . 6 ) 0 0 0 5 0 0 0 6 0 0 C 7 0 0 0 8 0 0 1 1 9 11 F O R M A T ( 1 H 0 , 4 H N = , I 3 , 5 X , 4 H L = , F 1 0 . 6 , 5 X , 4 H 0 = . F 1 0 . 6 . 5 X , 1 6 H L / D = , F 1 0 . 6 , 5 X , 1 2 H T H I C K N E S S = , F 1 0 . 6 , 5 X , 1 0 H S P . G T Y . = , F 6 . 3 ) _ F O R M A T ( !>IO,_8Hj;HETA_ , F 2 0 . 6 , 1 5 X , 8 H 0 E L T A = , F 2 0 . 6 , 1 5 X , 8 H E R R 0 R = , 1 F 2 0 . 6 , / ) F O R M A T ! 1 H C 5 0 X , " L E N G T H S OF M E M B E R S ' / I FORMAT I 2 5 X , 2 H A t ,I 2 , I H , , I 2 • 4 H ) = , F 1 5 . 6 , 2 0 X , 2 H B ( , 1 2 , 1 H , , 1 2 , 4 H ) = , I F 1 5 . 6 ) 0 0 0 9 13 F O R M A T , 1 H C 2 5 X , ' J O I N T C O O R D I N A T E S ' , 4 0 X , " J O I N T C E F L ECT I O N S « / ) 0 0 1 0 15 F 0 R M A T ( 6 X , 3 H X J ( , 1 2 , I H , , 1 2 , 4 H ) = , F 1 2 . 6 , 8 X , 3 H Y J ( , I 2 , 1 H , , 1 2 , 4 H ) = , 17 1 F 1 2 . 6 , 9 X , 2 H X < , 1 2 , I H , , 1 2 , 4 H ) = , F l 2 . 6 , 8 X , 2 H Y ( 1 2 , 1 H , , 1 2 , 4 H ) = 2 F 1 2 . 6 ) F 0 R M A T I 2 F 1 5 . 6 ) 0 0 1 2 0 C 1 3 0 0 1 4 1 9 F 0 R M A T ( 1 H 0 , 2 0 X , 5 H L 0 A D ( , I 2 , 1 H , , I 2 , 4 H ) = , F 1 5 . 6 , 2 0 X , 5 H T I L T ( , 1 2 , I H , , 1 1 2 , 4 H ) = . F 1 5 . 6 , / ) 21 FORMAT ( I H 0 , 6 X , ' M E M B E R ' , 7X , ' F O R C E ' , 7_X , • H I D I H ' . 7X , ' ARE A ' , 3 X , 23 1 * SECOND MOMENT* , 8 X , ' M E M B E R ' , 7 X , " F O R C E ' . 7 X , ' W 1 D T H ' , 7 X , ' A R E A * , 3 X , 2 ' S E C O N D M O M E N T ' / ) F O R M A T ( 1 H 0 , 3 X , 3 H A 1 , 12 , 1 H , , 1 2 , 1 H ) , I X , 4 F 1 2 . 6 , 9 X , 3 H B ( , 1 2 , I H , , 1 2 , 1 I H ) , 1 X , 4 F 1 2 . 6 ) 0 0 1 5 2 5 F O R M A T ( 1 H 0 , 5 X , ' V O L U M E T E N S I L E MEMBERS= • , F 9 . 6 , 5 X , 1 * VOLUME C O M P R E S S I V E M E M B E R S 3 ' , F 9 . 6 , 5 X , ' T O T A L VOLUME= • , F 9 . 6 ) 0 0 1 6 2 7 F 0 R M A T I F 1 5 . 6 ) 0 0 1 7 2 9 F 0 R M A T ( F 1 5 . 6 ) 0 0 1 8 31 F O R M A T { 1 H 0 . 2 4 H M G D U L U S OF E L A S T I C I T Y = . F 1 0 . 1 , 1 X . 6 H P . S . 1 5 X , 1 1 7 H U N I F 0 R M S T R E S S = , F 1 0 . 6 , 1 X , 6 H P . S . I . , 5 X , 2 1 7 H U N I F G R M S T R A I N = , F 1 0 . 6 , IX , 12H INC HE S / 1 N C H . ) 0 0 1 9 3 3 . . F O R M A T ( 2 X , ' M E M B E R ' , 6 X . ' F O R C E ' , 5 X , ' W I O T H ' , 4 X , ' A R E A ' , 5 X , ' I ' , 5 X , 1 ' W T / I N ' , 3 X , ' R A T I O • , 5 X , ' M E M B E R • , 6 X , ' F O R C E • , 5 X , • W l D T H ' , 4 X , • A R E A ' , 2 5 X , ' I » , 5 X , ' W T / I N « , 3 X , ' R A T I O ' , / / ) 0 0 2 0 35 F O R M A T ! 1 H 0 , 5 0 X , ' W E I G H T = ' . F 1 5 . 6 ) 0 0 2 1 0 0 2 2 0 0 2 3 0 0 2 4 0 0 2 5 0 0 2 6 0 0 2 7 3 7 3 9 F O R M A T ( F 1 5 . 6 ) F O R M A T ! I H 0 . 4 0 X , ' T H E F O L L O W I N G R E S U L T S ARE C A L C U L A T E D • , / , 4 I X , 1 I F O R . A S T R U C T U R E MADE FROM C R 3 9 P L A S T I C ' , / , 4 3 X , 2 'RAVING A SPECIFIC GRAVITY OF i . 3 i ' , 7 , 4 2 x , 3 ' T H E E F F F C T S OF S E L F WEIGHT F O R C E S • , / , 4 3 X , 4 ' A R E I N C L U O E D I N T H E S E R E S U L T S . ' , / / )  " 41 F O R M A T ! 1 H 0 . 4 5 X , ' R E A C T I O N S AT S U P P O R T P O I N T S ' ) 4 3 F O R M A T l 1 H 0 . 2 5 X , ' A T A , H O R I Z O N T A L COMPONENT = « , F 1 2 . 6 , 1 0 X , i_ ' V E R T I C A L COMPONENT = ' , _ F 1 2 . 6 I _ _ _ . 4 5 F O R M A T l IH0.25X,'AT B , H O R I Z O N T AL COMPONENT = • , F 1 2 . 6 , 1 0 X , " 1 ' V E R T I C A L COMPONENT = « , F 1 2 . 6 ) 4 9 F O R M A T ( 2 H A ( , 1 2 ,IH, , 12,2H» , F 1 1 . 6 , F 9 . 6 , 3 F 8 . 5 , F 8 . 4 , 3 X ,  51 . 2 H B I , I 2 , 1 H , , I 2 , 2 H ) , F 1 1 . 6 , F 9 . 6 , 3 F 8 . 5 , F 8 . 4 ) F O R M A T ! 1 H C 5 9 X , ' N O T E ' , / , 4 5 X , ' T H E TERM R A T I O I N THE F O L L O W I N G ' , 1 / , 4 3 X , * T A B L E I N D I C A T E S T H E P E R C E N T A G E RAT 1 0 • , / , 4 5 X , F O R T R A N IV G C O M P I L E R M A I N 0 4 - 2 9 - 7 0 1 9 : 2 8 : 5 3 P A G E 0 0 0 2 0 C 2 8 2 ' C F T H E F O R C E IN E A C H 3 ' E U L E R C R I T I C A L LOAD 4 • < E * I * l P l » * ( P I ) ) / ( L * L > ' ) 53 F O R M A T ! 1 H 0 , / / )  MEMBER TO I T S ' , / , 4 5 X , FOR B U C K L I N G ' , / , 5 2 X , 0 0 2 9 0 0 3 0 _0.O3.l-. 0 0 3 2 0 0 3 3 0 0 3 4 P I = 3 . 1 4 1 5 9 2 6 5 P I 4 = 0 . 2 5 * P I R00.T=S.QK.T..(.2...aL 9 9 0 C O N T I N U E 9 9 9 C O N T I N U E K 1 = 2  0 0 3 5 0 0 3 6 0 0 3 7 0 0 3 8 0 0 3 9 L l = 2 K = l A L L T H E A R R A Y S ARE Z E R O E D OUT F I R S T . DO 10 1 = 1 , 2 5 DO 10 J = l , 2 5 A ( I , J ) = 0 . 0  0 0 4 0 0 0 4 1 -J0_04.2_ 0 0 4 3 0 0 4 4 0 0 4 5 10 B ( I , J ) = C . O XI I , J ) = 0 . 0 _y.u.t..J.i=o_..o. P S I A I I , J ) = 0 . 0 P S I B I I , J J = 0 . 0 C O N T I N U E 0 0 4 6 J I O A J L 0 0 4 8 0 0 4 9 N I S NUMBER OF F I B R E S R E A D ( 5 , 1 ) N N.1.= N± .1 N2=N+2 NN=N-1 L I S L E N G T H OF S P A N . 0 0 5 0 -0 .0 .5JL 0 0 5 2 0 0 5 3 R E A D ( 5 , 3 ) L D IS S U P P O R T S P A C I N G R E A D J . 5 , 3 ) . D . _ . . . R A f 10 = L / D T I S T H I C K N E S S OF A L L MEMBERS I N I N C H E S . T = 0 . 2 5 0 0 5 4 _a.05.5_ 0 0 5 6 0 0 5 7 C ELMOD IS MODULUS OF E L A S T I C I T Y . E L M 0 D = 3 0 0 0 0 0 . C SXR_AJN_=__0_..OCJ . S I G M A = 3 0 0 . 0 W R I T E ( 6 , 3 1 1 EL M O D , S I G M A , S T R A I N C T H E S P E C I F I C G R A V I T Y ( S P G T Y 1 OF C R 3 9 P L A S T I C I S 1 . 3 1 0 0 5 8 0 0 5 9 _Q.0A0_ 0 0 6 1 0 C 6 2 S P G T Y = 1 . 3 1 W R I T E ( 6 , 5 ) N , L , 0 , R A T I 0 , T , S P G T Y C = I S P G T Y « 6 2 . 4 ) / 1 7 2 8 . 0 R A O I A L I S L E N G T H OF A L L MEMBERS R A O I A T ING FROM A AND B R A D I A L = D / S O R T ( 2 . 0 ) DO 20 1 = 2 , N l 0 0 6 3 0 0 6 4 _0..0.63_ 0 0 6 6 0 0 6 7 0 0 6 8 1 1 = 1 - 1 J = l _ J . J _ L J _ ! . . 2 0 A ( I , J ) = R A D I A L 8 ( J , I l = R A D I A L Cf lNT I N U E  0 0 6 9 C E P S I S P E R C E N T ERROR E P S = 0 . 0 0 0 0 0 1 _C C A L C U L A T I O N CF F A N A N G L E F O R T R A N IV G C O M P I L E R M A I N 0 4 - 2 9 - 7 0 1 9 : 2 8 = 5 3 P A G E 0 0 0 3 0 0 7 0 0 0 7 1 C F A N A N G L E , T H E T A , I S G E N E R A T E D BY AN I T E R A T I V E P R O C E S S , S T A R T I N G WITH C THE ' A P P R O X I M A T I O N ' THETA= ZERO T H E T A = o . O D E L T A = 1 0 . 0 0 0 7 2 0 0 7 3 0 0 7 4 0 0 7 5 0 0 7 6 0 0 7 7 9 0 GO TO 100 C O N T I N U E K = K+1 I F I K . G T . 4 0 ) T R = T H E T A * p i > 1 8 0 . 0 T R 2 = 0 . 5 * T R F A C T = P I 4 - T R 2 0 0 7 8 0 0 7 9 0 0 8 0 0 0 8 1 0 0 8 2 0 0 8 3 3C F A C T l = P I 4 + T R 2 A ( 2 , 2 ) = D * R 0 0 T * S I N ( T R 2 ) DO 3 0 I = 2 , N B ( I , 2 ) = A < 2 , 2 ) A < 2 , I I = A ( 2 , 2 ) C O N T I N U E 0 0 8 4 0 0 8 5 0 0 8 6 0 0 8 7 0 0 8 8 0 Q 8 9 0 0 9 0 0 0 9 1 0 0 9 2 _ 0 0 9 3 0 0 9 4 0 0 9 5 0 0 9 6 _0097_ 0 0 9 8 0 0 9 9 0 1 0 0 DO 4 0 J = 2 , N J 1 = J + 1 J 2 P _ 5 0 I = J 1 _ , N L 1 1 = 1 - 1 A I I , J ) = ( A ( I 1 , J ) + B ( I I , J ) * S I N ( T R ) ) / C O S ( T R • B ( J , I ) - A( I , J I B ( I I , J 1 l = ( A < I I , J » * S I N ( T R ) + B ( I I , J ) ) / C 0 S ( T R > A ( J l , I I )=B< I I , J 1 ) _50 C O N T I N U E _ A ( J l , j l )=B( J 1 , J I I 4 0 C O N T I N U E C TL I S C A L C U L A T E D L E N G T H OF M A I N PART OF C A N T I L E V E R 7 0 I T E M L = R A D I A L * C 0 S ( P I 4 - ( T R * F L 0 A T ( N N ) ) I I T E M R = 0 . 0 D 0 _ 7 0 J = 2 , N IT EMR= ITEMR + A ( N l , J ) * C O S ( F A C T 1 + ( T R * F L O A T ( J - N 1 ) » ) C O N T I N U E T L = 0 . 0 0 1 0 1 0 1 0 2 __0_103_ 0 1 0 4 0 1 0 ? 0 1 0 6 T L = T L + I T E M L + I T E M R G A P = L - T L E R R O R = G A P / L I F ( A 8 S ( E R R O R ) . L T . E P S ) GG TG 1 0 0 IF ( G A P . G T . 0 . 0 ) GO TO 8 0 T H E T A = T H E T A - D E L T A 0 1 0 7 0 1 0 8 0 1 0 9 0 1 1 0 0 111 0 1 1 2 8 0 D E L T A = 0 . 1 * 0 E L T A I F ( D E L T A . L T . C . 0 C 0 C 1 ) GO TO 1 0 0 T H E T A = T H E T A + D E L T A GO TO 9 C T H E T A = T H E T A + D E L T A GG TO 9 0 0 1 1 3 0 1 1 4 _0.U.5_ 0 1 1 6 0 1 1 7 0 1 1 8 1 0 0 W R I T E I 6 , 7 ) T H E T A . D E L T A . E R R O R W R I T E ( 6 , 3 9 1 WRI TE (6 , 9 ) _ . . 0 0 130 1 = 2 , N l 1 1 = 1 - 1 DO 130 J = 1 , N  0 1 1 9 0 1 2 0 0 1 2 1 1 3 0 J J = J - 1 W R I T E ( f > , l l ) l i , J J , A ( I , J ) , J J , I I , B ( J , I ) C O N T I N U E F O R T R A N IV G C O M P I L E R M A I N 0 4 - 2 9 - 7 0 1 9 : 2 8 : 5 3 PAGE 0 0 0 4 C C A L C U L A T E THE I N C L I N A T I O N S OF THE C V A R I O U S M E M B E R S P S F A ( I , J ) AND P S I B < I , J ) 0 1 2 2 B E T A A = P 1 4 0 1 2 3 BE TR 2=FAC T1 0 1 2 4 BETR2M= FACT 0 1 2 5 DO 1 8 0 J = 1 , N 0_12_6 J l = J + l 0 1 2 7 J J = J - 1 0 1 2 8 0 0 1 8 0 1 = 1 , N 0 1 2 9 I 1 = l - I 0 1 3 0 11=1+1 0 1 3 1 IF I J . E Q . l ) GO TO 1 5 0 . 0 1 3 2 P S I A ( I 1 , J ) = ( < T R * F L O A T ( 1 1 - J ) J - B E T R 2 ) 0 1 3 3 I F ! I . G T . l )G0 TO 1 4 0 0 1 3 4 P S I B ( I , J l ) = < < T R * F L O A T ( J l - 2 ) ) + B E T A A ) 0 1 3 5 GO TO 1 7 0 0 1 3 6 1 4 0 P S I B ( I , J l ) = ( I TR * F L O A T ( J l - I ) ) + B FT R2 M ) 0 1 3 7 GO TO 1 7 0 __0.13.8 . 1 5 0 P S I A ( I l , J I = H T R * F L 0 A T ( I l - 2 ) ) - B E T A A I _ . 0 1 3 9 I F ( I . G T . l ) G O TO 1 6 0 0 1 4 0 PS I B ( I , J 1 ) = ( I T R * F L O A T ( J l - 2 ) J + B E T A A ) 0 1 4 1 GO TO 1 7 0 0 1 4 2 1 6C P S I B I I , J l l = ( ( T R * F L O A T I J l - I ) ) + B E T R 2 M ) 0 1 4 3 1 7 0 C O N T I N U E _D.14_4 . ..1 8.0 CO.NT.I N.UE _.. . . . . . 0 1 4 5 WRITE ( 6 , 5 1 ) 0 1 4 6 DO 190 I = 1 , 2 5 0 1 4 7 0 0 190 J = 1 , 2 5 0 1 4 8 L O A D ( I , J ) = 0 . 0 0 1 4 9 T I LT ( I , J > =0 . 0 .___0_1.5-C F A ( I , J ) = 0 . 0 0 1 5 1 F B I I , J 1 = 0 . 0 0 1 5 2 X J ( I , J J = 0 . 0 0 1 5 3 Y J ( I , J ) = 0 . 0 0 1 5 4 WA( I , J ) = 0 . 0 0 1 5 5 WB( I , J ) = 0 . 0 0.15.6 . .. A R.E A A ( I , J ) = . .0 . .C . . .. 0 1 5 7 A R E A B ( I . J ) = 0 . 0 C 1 5 8 M I Z A ( I , J ) = 0 . 0 0 1 5 9 M I Z B I I , J ) = 0 . 0 0 1 6 0 W T I N A ( I , J ) = 0 . 0 0 1 6 1 W T I N B I I , J ) = 0 . 0 0 1 6 2 EUL.CL A( I , J . ) = C . O 0 1 6 3 E U L C L B I I , J ) = 0 . 0 0 1 6 4 E A ( I , J ) = 0 . 0 0 1 6 5 E B ( I , J ) = 0 . 0 0 1 6 6 1 9 0 C O N T I N U E C THE F O L L O W I N G S E C T I O N C A L C U L A T E S THE F O R C E S IN T H E M E M B E R S , C . A L L O W I N G FOR T H E E F F E C T S OF S E L F W E I G H T . . ...... C F I R S T S P E C I F Y A P P L I E D L O A D S , L O A D ( I , J ) , AND THE A N G L E , T I L T ( I , J ) , c THAT E A C H LOAD M A K E S WITH THE V E R T I C A L . 0 1 6 7 R E A 0 ( 5 , 1 7 ) L G A C ( N 1 , N 1 ) , T I L T ( N 1 , N 1 ) 0 1 6 8 W R I T E ( 6 , 1 9 ) N , N , L O A O ( N l , M ) , N , N , T I L T ( N 1 , N 1 ) 0 1 6 9 T I L T < N 1 , N 1 ) = ( T I L T < N 1 , N 1 ) * P I ) / 1 8 0 . 0 c W IS 7 F R 0 I N T H I S C A S E WHERE THE MEMBERS ARE A S S U M E D TO BE FORTRAN IV G COMPILER MAIN 04-29-70 19:28:53 PAGE 0005 C WEIGHTLESS. 017C W = 0.0 0171 D0WN1 = C0SITR)-(W*ICOS(FACT)) * ( A ( N 1, N)+B I N, N1))) 0172 FA(Nl»N 1 = <LCAO(N1,NI)*(COS(FACT+TILT(N1,N1 ) )+W*(<8(N,N1)* 1 SINITILTINl,N1)))I)J/DCWN1 0173 FB(N,N1I = - ( L O A 0 ( N l , N l ) * C O S ( F A C T - T I L T ( N l , N i n - W * ( A ( N l , N I * S I N 1 ( T I L T I N l , N l ) l ) ) /0GWN1 0174 DO 210 1=1,NN 0175 Nl I=N + l - I 0176 N2I=N+2"I 0177 DO 210 J=1,NN 0178 NJ=N-J 0179 N1J=N+1-J 018C N2J=N+2"J 0181 I F ( J . G T . l ) GO TO 200 0182 I F ( I . E Q . 1 ) GO TG 210 0183 200 PSI = ( I 0.5*PI J-t-PSI A( N2I ,N1J) ) 0184 PSIT=PSI-TR 0185 D0WN2=(C0SITR)-W*lAIN2I,N1J)*COS(PS IT) + BIN1 I,N2J)*SIN(PS I 1)) 0186 FAIN2I,N1J) = ((FA(N2I,N2J)*(1.0+1W*(A(N2I,N2J)*COS(PS I T ) - B ( N l I , N 2 J 1 )*SIN(PSIT ) ) ) ))-(FB(N2I,N2J)*(SIN(TR)+(W * (BIN2I,N2J)*COS<PSIT)+ 2 B ( N 1 I , N 2 J ) * C 0 S ( P S I ) ) 1 1) + (LGAD(N2I,N2J)*(CCSIPS IT+T ILT(N21,N2J ) J 3 +(W*B(N1I,N2J)*SIN(TILT(N2I,N2JI)))))/D0WN2 0187 FBI N i l , N 2 J ) =-((FA(N2 I ,N2J)* IS INITR >+(W*(A(N2I,N2J)*SIN(PS I) + 1 A I N 2 I , N 1 J ) * S I N ( P S I T ) ) ) ) )-(FBIN2I,N2J)*<1.0"(W *(A(N2I , N1J)*CCS( 2 PSI1-BIN2I,N2J)*S IN(PS I ) ) ) 1 j + ILOAC1N2 I» N 2 J ) * ( S I N ( P S I + T I L T I N 2 I , 3 N 2 J ) ) - ( W * A ( N 2 I , N 1 J 1 * S I N ( T IL T ( N 2 I , N 2 J I > ) ) > )/00WN2 0188 210 CONTINUE 0189 DO 220 1=1,NN 0190 Nl I = N*1-I C191 N2I=N+2 -I 0192 PSI = (0.5*PI«-PSIA(N2l ,1) ) 0193 PS IT=(PS I-TR2 ) 0194 00WN3=(CGS(TR2)-(W*(A(N2 I,1)*COSI PS IT)+B(N1 1,2 )*S IN(PS I)) ) ) 0195 F A I N 2 I , ! ) = ( I F A ( N 2 I , 2 ) * ( 1 . 0+( W*( A ( N2 I , 2 ) *C 0 S ( P S I T ) - B ( Nl I ,2 >* S I N ( 1 P S l T ) M ) ) - ( F B ( N 2 I , 2 ) * ( S I N ( T R ) + (W*(B(N2I,2)*C0S(PSIT) + 2 B ( N l l , 2 ) * C O S ( P S I + T R 2 ) ) ) ) ) * < L O A D ( N 2 I , 2 ) * ( C O S ( P S IT+ T I L T I N 2 I , 2 ) ) 3 +<W*B(N1I,2)*SINITILTIN2I,2)))> ) ) /DOWN 3 0196 F e i N l I , 2 > = - l ( F A ( N 2 I , 2 ) * ( S I N ( T R 2 ) * ( W * ( A ( N 2 I , 2 ) * S I N ( P S I ) + A ( N 2 I , l ) * 1 SIN(PSIT)))))-(FB(N2I,2)*(C0S(TR2 ) - ( W*(A(N21,1)*C0S(PSI+TR2 )-2 B ( N 2 I , 2 ) * S I N ( P S I ) ) ) ) ) + ( L 0 A D ( N 2 I , 2 ) * ( S I N ( P S I + T I L T ( N 2 I , 2 ) ) - I W * 3 A(N2I,1)*SIN(TILT(N2I,2)))))J/DOWN3 0197 220 CONTINUE 0198 DO 230 J=1,NN 0199 N1J=N+1-J 0200 N2J=N+2-J 0201 PSI = (0.5*PI + P S I A ( 2 , M J H 0202 PS IT=(PSI-TR2) 02C3 D0WN4 = (CCS(TR2)-(W*(A(2,N1J)*C0S(PSIT)+B11,N2J)*SIN(PSI> 1 ) ) 0204 FA(2,N1J )=((FA(2.N2J)*{COS(TR2)•(W*IA(2,N2J)*C0S<PSIT>-B(1,N2J)* 1 SINI P S I - T R ) ) ) ) ) - ( F B ( 2 , N 2 J ) * ( S I N I T P 2 t+(W*(B(1,N2J)*CCS(PSI) + 2 B(2,N2J>*COSIPSIT)) )> ) + ( LOAD! 2 ,N2 J) * ( COS ( PS IT+T IL T I 2, N2J ) ) + 3 (W*B(1.N2J)*SIN(TILT(2,N2J)I)))>/D0WN4 0205 FBI 1,N2J)=-((FA(2,N2JI*(SINITR)+1W* I A I 2,N2J)*SIN(PS I )+A(2.NIJI* 1 S I N ( P S I - T R ) ) ) ) ) - ( F B ( 2 , N 2 J ) * ( 1 . 0 + ( V . * ( B ( 2 , N 2 J ) * S I N ( P S I ) - A ( 2 , N 1 J ) * F O R T R A N IV G C O M P I L E R M A I N 0 4 - 2 9 - 7 0 1 9 : 2 8 : 5 3 PAGE 0 0 0 6 2 C O S I P S I ) ) ) ) ) + ( L O A D ( 2 , N 2 J ) * i S I N ( P S I + T I L T ( 2 , N 2 J ) J - ( W * A I 2 , N 1 J » * 3 S I N ( T I L T I 7 . N 2 J ) ) ) > > I / D 0 W N 4 0 2 0 6 2 3 0 C O N T I N U E 0 2 0 7 DOWN 5= ! 1 . 0 - t W * ( A ( 2 , l ) * S l N ( B E T A A ) » B ! l , 2 ) * C 0 S ! B E T A A ) ) ) )  0 2 0 8 F A ( 2 , 1 ) = ( ( F A ( 2 , 2 ) * ( C C S < T R 2 ) + ( W * ( A < 2 , 2 ) * S I N ( B E T A A J - B ( 1 , 2 ) * C O S < B E T A I A + T R 2 ) I > > > - ( F B I 2 , 2 ) * ( S I N ! T R 2 ) + ( W * ! B < 1 , 2 ) * S I N l B E T A A - T R 2 I + B 1 2 « 2 ) * ...2_....SJ N ( B E T A A ).) ) ).) r.( L 0 A OJ 2_,. 2 ) * (S IN (T I LT ( 2 , 2 ) - B E T A A 1 - ( W*B ( 1 , 2 ) * S INI 3 T I L T 1 2 , 2 ) ) ) ) ) ) / D 0 W N 5 0 2 0 9 F B I l , 2 ) = - ( < F A < 2 , 2 ) * ( S I N ! T R 2 > + t W * ! A l 2 , 2 ) * C 0 SI BE TAA I + A { 2 , 1) * C O S ( 1 B E T A A + T R 2 ) ) ) ) ) - 1 F B ( 2 , 2 ) * I COS 1 T R 2 ) - ! W » ( A ! 2 . 1 ) * S I N ( B E T A A - T R 2 ) -2 fit 2 , 2 ) * C O S ( B E T A A ) ) ) ) ) + 1 L O A D I 2 , 2 ) * ( C O S ( T I L T ( 2 , 2 ) - B E T A A ) - ( W * A ( 2 , 1 3 ) * S I N ( T I L T ( 2 , 2 ) ) ) ) ) I / D G W N 5 .C CAL.CUJ. AT 1 ON._Q.F_ P J . N _ J 01 NT C O O R D I N A T E S , X J I I , J ) AND Y J ( I , J ) , AND J O I N T C D E F L E C T I O N S , X ( I , J ) AND Y ( I , J ) . 0 2 1 C DO 3 0 0 I = 1 , N 1 0 2 1 1 DO 3 0 0 J = 1 , N 1  0 2 1 2 I F ( J . G T . l ) GO TO 2 6 0 0 2 1 3 I F ( I . E Q . D GO TO 3 0 0 _ J 1 2 1 4 2 6(3 .C-QNJJ NU E_ 0 2 1 5 I F | F A ( I , J ) . L T . 0 . 0 ) GO TC 2 7 0 0 2 1 6 E A I I . J ) = S T R A I N * A ( I , J ) 0 2 1 7 GO TO 2 8 0  0 2 1 8 2 7 0 E A I I . J ) = - I S T R A I N * A ( I , J ) ) 0 2 1 9 2 8 0 I F I F B I I , J ) . G T . O . O ) GO TO 2 9 0 _ Q 2 2 J D E ^ J _ , J L L _ J i ^ R A I J i * E j ( J . , A ) 0 2 2 1 GO TO 3 0 0 0 2 2 2 2 9 0 E B I I . J ) = - I S T R A I N * B I I , J ) ) 0 2 2 3 3 0 0 C O N T I N U E  0 2 2 4 DO 3 1 0 J = 1 , N 0 2 2 5 J l = J + 1 0 7 7 6 D.0_3 .1 .0_ I_^_ l . i .K '. 0 2 2 7 I I = I + l 0 2 2 8 F A C T O R = l ( X C I , J 1 ) - X I 1 1 , J ) ) * S I N I P S I B ( I , J 1 1 ) + 1 Y ( I 1 , J ) - Y I I , J 1 ) ) * C C S < 1 P S I B I I , J l ) ) - E B I I , J l l - E A l I I , J ) * S I N t P S I B I I , J l l + P S I A ( I 1 , J ) ) ) / C O S ( 2 P S I A I 1 1 , J ) + P S I B ( I , J 1 ) 1 0 2 2 9 XI I 1 , J 1 ) = ( X I I 1 , J ) + E A ( I 1 , J ) * C 0 S I P S 1 A < 1 1 , J ) ) - F A C T O R * S I N ( P S I A ( 11 , J ) ) ) _ 0 . 2 3 . 0 _ Y l I 1 , J l ) = ( Y ( I 1 , J ) - E A 1 I.1.,.JJ * S l N( P S J A ( JJ . . , J ) ) -F A C T C R * C O S ( P S I A I 1 1 . i_JJLU_ 0 2 3 1 3 1 0 C O N T I N U E C C O O R D I N A T E S OF THE P I N J O I N T S . 0 2 3 2 W R I T E ! 6 , 1 3 1 0 2 3 3 DO 3 2 0 1 = 1 , N 0 2 3 4 11 =1+1 0 2 3 5 X J I 1 1 , 1 ) = 0 . 0 0 2 3 6 X J 1 1 . I 1 ) = 0 . 0 0 2 3 7 Y J ! I l , l ) = D / 2 . 0 0 2 3 8 Y J ( 1 , 1 1 ) = - D / 2 . C 0 2 3 9 3 2 0 C O N T I N U E 0 2 4 0 DO 3 3 0 1 = 1 , N 0 2 4 1 11=1+1 0 2 4 2 X J ! 1 1 , 2 ) = A ! 1 1 , 1 ) * C O S ( P S I A ! 1 1 , 1 ) ) 0 2 4 3 Y J ! U , 2 ) = Y J I I 1 , 1 ) + ! A ( I 1 , 1 ) * S I N ! P S I A ( I 1 ,1 ) ) ) 0 2 4 4 3 3 0 C O N T I N U E 0 2 4 5 DO 3 4 0 1 = 1 , N l 0 2 4 6 DO 3 4 0 J = 2 , N 024.7 J 1 = J + 1 F O R T R A N IV G C O M P I L E R M A I N 0 4 - 2 9 - 7 C 1 9 : 2 8 : 5 3 P A G E 0 0 0 7 0 2 4 8 X J I I , J l ) = XJ< I , J ) + A( I , J J f C O S I P S I A l I , J ) ) 0 2 4 9 Y J I I , J l ) = Y J 1 I , J ) + A ( I , J ) * S I N ( P S I A 1 I , J ) ) 0 2 5 0 3 4 0 C O N T I N U E 0 2 5 1 DO 3 6 0 1 = 1 , N l 0 2 5 2 1 1 = 1 - 1 0 2 5 3 DO 3 6 0 J = 1 , N 1 0 2 5 4 J J = J - 1 0 2 5 5 I F I J . G T . l ) GO TO 3 5 0 0 2 5 6 IF 1 I . E Q . 1 ) GO TO 3 6 0 0 2 5 7 3 50 W R I T E ( 6 , 1 5 ) I I , J J , X J I I , J ) , 1 1 , J J , Y J I I , J ) , 1 1 , J J , X ( I , J ) , I I , J J , Y ( I , J ) 0 2 5 8 3 6 0 C O N T I N U E C C O M P I L E P R O P E R T I E S OF EACH M E M B E R . C F O R C E S AND W E I G H T S IN P O U N D S , WIDTHS I N I N C H E S , C R O S S S E C T I O N A L A R E A S C IN S Q U A R E I N C H E S AND MOMENTS OF I N E R T I A I N I I N C H E S ) * * 4 0 2 5 9 0 0 4 0 0 1 = 1 , N ' 0 2 6 0 11 = I + l 0 2 6 1 1 1 = 1 - 1 0 2 6 ? 0 0 4 0 0 J = l , N 0 2 6 3 J 1 = J + 1 0 2 6 4 J J = J - 1 0 2 6 5 W A I I l . J ) = I A B S I F A I U , J ) 1 ) / ( T * S I G M A ) 0 2 6 6 W B I I . J l ) = I A B S I F B I I , J l ) ) ) / 1 T * S I G M A ) C 2 6 7 A R E A A I 1 1 , J ) = ( T * W A I 1 1 , J ) ) 0 2 6 8 A R E A B I I , J 1 ) = ( T * W B < I , J 1 ) ) 0 2 6 9 M I Z A I U . J ) = I I W A I I 1 , J ) * * 3 ) * T 1 / 1 2 . 0 0 2 7 G M I Z B l i . J l ) = ( ( W e ( I , J l J * * 3 ) * T i / 1 2 . 0 0 2 7 1 W T I N A I U , J ) = I A R E A A I I 1 , J ) * S P G T Y * 6 2 . 4 ) / 1 7 2 8 . 0 0 2 7 2 W T I N e i I , J 1 ) = ( A R E A B I I , J l ) * S P G T Y * 6 2 . 4 ) / 1 7 2 8 . 0 0 2 7 3 I F ! W A I I 1 , J ) . G T . C . 2 5 ) GO TO 3 7 0 0 2 7 4 E U L C L A I 1 1 , J ) = ( A B S ( F A ( I 1 , J ) ) * A I 1 1 , J ),*A I 1 1 , J I * 1 0 0 . 0 ) / 1 I F L M 0 D * M I Z A I 1 1 , J ) * P I * P I ) 0 2 7 5 GO TO 3 80 0 2 7 6 3 7 0 E U L C L A ( U , J ) = I A 8 S ( F A ( I 1 , J ) ) * A I I 1 . J ) * A ( I I . J I * 7 6 8 0 0 . 0 ) / 1 E L M O D * 1 WAI I 1 , J ) * P I * P I ) 0 2 7 7 3 8 0 I F I W 8 I I , J 1 ) . G T . 0 . 2 5 ) GU TO 3 9 0 0 2 7 8 E U L C L H I I , J l ) = ( A E S ( F B I I , J l ) ) * B ( I , J l ) * B < I , J l ) * 1 0 0 . 0 ) / 1 ( E L M O O * M I Z B ( I , J 1 ) * P I * P I ) 0 2 7 9 GO TO 4 0 0 0 2 8 C 3 9 0 E U L C L B ( I , J l ) = I A B S I F B ( I , J l ) ) * B I I , J 1 ) * B < I , J 1 1 * 7 6 8 0 0 . 0 ) / ( E L M O D 1 * W B ( I , J 1 ) * P l * P I ) 0 2 8 1 4 0 0 C O N T I N U E C R E A C T I O N S AT S U P P O R T S . C THE H O R I Z O N T A L COMPONENTS OF THE R E A C T I O N S AT A AND B ARE C M E A S U R E D P O S I T I V E A C T I N G TO T H E R I G H T , AND T H E V E R T I C A L C COMPONENTS ARE M E A S U R E D P O S I T I V E A C T I N G U P W A R D . 0 2 8 2 F A H = 0 . 0 0 2 8 3 F A V = 0 . 0 0 2 8 4 DO 4 1 0 1 = 2 , N l 0 2 8 5 FAH = F A H - I F A I I , 1 ) * C O S I PS I A ( 1 , 1 ) ) ) 0 2 8 6 F A V = F A V + ( F A ( I , l ) * S I N ( B E T A A - ( T R * F L 0 A T ( I - 2 ) ) ) ) + I A B S ( F A ( I , l ) ) * 1 A ( I , l ) * W ) 0 2 8 7 4 1 0 C O N T I N U E 0 2 8 8 F B H = 0 . 0 0 2 8 9 F B V = 0 . 0 0 2 9 0 DO 4 2 0 J = 2 , N 1 F O R T R A N IV G C O M P I L E R M A I N 0 4 - 2 9 - 7 C 1 9 : 2 8 : 5 3 P A G E 0 0 0 8 0 2 9 1 F B H = F B H - ( F B ( 1 , J ) * S I N { P S I B < 1 , J ) ) ) 0 2 S 2 F B V = F B V - ( F B ( 1 , J ) * C 0 S ( B E T A A + ( T P * F L O A T ( J - 2 ) ) ) > + ( A B S ( F B ( l , J ) ) * 1 B < l , J ) * W ) 0 2 9 3 4 2 0 C O N T I N U E  0 2 9 4 W R I T E I 6 , 4 1 ) 0 2 9 5 W R I T E < 6 , 4 3 ) F A H , F A V _0.2.9.6 WR I . T E ( 6 , 4 5 ) f B H . F 3 V 0 2 9 7 W R I T E ( 6 , 3 3 ) 0 2 9 8 0 0 4 5 0 1 = 1 , N 0 2 9 9 I I = I - l  0 3 0 0 I I = I + l 0 3 0 1 DO 4 5 0 J = 1 , N _0.3.0.2 .J-l__=_J±J. . 0 3 0 3 J J = J - l 0 3 0 4 W R I T E ( 6 , 4 9 ) I , J J , F A ( I I , J ) , W A ( I I , J ) , A R E A A ( I I , J ) , M I Z A ( I I , J ) , 1 W T I N A ( I 1 , J ) . E U L C L A l I l , J ) , I I , J , F B ( I , J l ) , W B ( I , J l ) , A R E A B l I , J l 1 , 2 M I Z B ( I , J l ) , W T I N B I I , J l ) , E U L C L B I I , J 1 ) 0 3 0 5 4 5 0 C O N T I N U E C F I N A L S T E P IS TO D E T E R M I N E VOLUME AND WEIGHT OF S T R U C T U R E . 0 30 6 VOLUME = 0 . 0 0 3 0 7 VOL TEN = C O 0 3 0 8 VOLCOM = 0 . 0 0 3 0 9 W E I G H T = 0 . 0 0 3 1 0 DO 4 8 0 I = 1 , N JQ.3.L! DO 4 3 0 J = 1 , N 0 3 1 2 11=1+1 0 3 1 3 J 1 = J + 1 0 3 1 4 IF ( FA ( 11 , J ) . L T . 0 . 0 I GO TO 4 7 0 0 3 1 5 V O L T E N = V O L T E N + ( A ( 11 , J ) * A R E A A ( 1 1 , J ) ) 0 3 1 6 I F ( F B ( I . J D . L T . O . O ) GO TO 4 7 0 0 3 1 7 4 6 0 V O L T EN = V O L T E N + I B I I , J 1 ) * A R E A B < I , j l )) 0 3 1 8 GO TO 4 8 0 0 3 1 9 4 7 0 I F ( F 8 ( I , J 1 ) . L T . C C ) GO TO 4 8 0 0 3 2 0 V O L T E N = V O L T E N + ( B ( I , J 1 ) * A R E A B ( I , J 1 ) ) 0 3 21 4 8 G C O N T I N U E 0 3 2 2 DO 5 0 0 I = 1 , N .03.23 DO 5 0 0 J = 1 , N 0 3 2 4 11 = I + l 0 3 2 5 J l = J + 1 0 3 2 6 IF ( F A I I l . J I . G T . O . O ) GO TO 4 9 0 0 3 2 7 VOLCOM = V O L C O M + I A ( I l , J ) * A R E A A l 1 1 , J I 1 0 3 2 8 I F ( F B ( I , J 1 ) . G T . 0 . 0 ) GO TO 4 9 0 0 3 2 9 VOLCOM = V O L C C M + ( B ( I , J l ) * A R E A B ( I , J l ) ) 0 3 3 0 GO TO 5 0 0 0 3 3 1 4 9 0 I F ( F B ( I, J l ) . G T . 0 . 0 ) GO TO 5 0 0 0 3 3 2 VOLCOM = V O L C C M + ( B ( I , J 1 ) * A R E A B ( I » J l 1 ) 0 3 3 3 5 0 0 CONT INUE 0 3 3 4 VOLUME = VOLUME + V C L T E N + VGLCOM 0.3.35. WFIGHT = WEIGHT + ( V O L U M E * C ) 0 3 3 6 W R I T E ( 6 , 2 5 ) V C L T E N , V O L C O M , V O L U M E 0 3 3 7 WRITE ( 6 , 3 5 ) WEIGHT 0 3 3 8 W R I T E ( 6 , 5 3 ) 0 3 3 9 GO TO 9 9 0 C 3 4 C 9 0 0 C O N T I N U E 034.1 END E X E C U T I O N T E R M [ N A T E D » R U N - L O A D » 5 = * S 0 U R C E « 6 = » S I N K « FX 6 C U T I O N B E G I N S _MODULUS-.0F.. E L A S T I C I T Y = 3 0 0 0 0 0 . 0 P . S . I . . .UNIFORM _ST.ReSS.j_.300.000.000 _P..S...I .... N = 2 L = 1 0 . 0 0 0 0 0 0 0 = 2 . 5 0 0 0 0 0 L / 0 = 4 . 0 0 0 0 0 0 T H I C K N E S S UN1F.0RM. STRAIN...:. 0,.00100.0_I.NCHES/.I.NCH. 0 . 2 5 0 0 0 0 S P . G T Y . = 1.310 T H E T A = 6 4 . 9 4 2 3 8 3 D E L T A 0 . 0 0 0 0 1 0 THE FOLLOW ING R E S U L T S ARE C A L C U L A T E D FOR A STRUCTURE MADE FROM CR39 P L A S T I C HAVING A S P E C I F I C GRAVITY OF 1.31 ~TJiE E F F E C T S OF S E L F WEIGHT ' F O R C E S ARE INCLUDFD IN T H E S E R E S U L T S . LENGTHS OF MEMBERS A ( 1, 01 = A ( 2, A ( 2, 1 . 7 6 7 7 6 7 1 . 8 9 8 1 4 1 1 . 7 6 7 7 6 7 8 . 5 4 1 6 4 6 1. II =•_ 0, 21 = 1 , 2 1 = 1 . 7 6 7 7 6 7 l..8?81_41_ 1 . 7 6 7 7 6 7 8 . 5 4 1 6 4 6 NOTE THE TERM R A T I O IN THE FOLLOWING T A B L E I N D I C A T E S THE P E R C E N T A G E R A T I O _ OF THE FORCE IN EACH MEMBER TO I T S EULER C R I T I C A L LOAD FOR B U C K L I N G ( E » H I P I 1*1 P I I ) / I L « L I  LOAD I 2, 21 = 1 0 0 . 0 0 0 0 0 0 T I L T ! 2. 21 J O I N T COORDINATES J O I N T D E F L E C T I O N S XJ ( c. 1 ) o . c Y J I 0, 11 = ! . 2 5 0 0 0 0 XI 0, 1) = Y( 0. 11 0.0 X J ( 0. 21 = 0.0 Y J ( 0. 21 = - 1 . 2 5 0 0 0 0 XI 0. 2) = 0.0 Y( 0. 21 = 0.0 XJ( 1, 01 - 0. 0 Y J I 1, 0 ) = 1 . 2 5 0 0 0 0 XI 1. 01 = . 0.0 Y( 1, 01 0.0 X J I 1. 1) = 1 . 2 5 0 0 0 C Y J I 1. 1) = 0 . 0 0 0 0 0 1 XI 1. 1) = 0 . 0 0 0 0 0 0 Y( 1. 1) 0 . 0 0 2 5 0 0 XJ{ 1. 2) = 1 . 6 6 1 7 6 5 Y J I 1, 2) = -1 . 8 5 2 9 4 0 XI 1, 21 - 0 . 0 0 3 7 9 9 Y( 1. 21 - 0 . 0 0 5 2 8 9 XJI 2. 0) 0. C Y J I 2, 0) = 1 . 2 5 0 0 0 0 XI 2. 01 0.0 Y l 2, 0) = 0.0 X J I 2, 11 1 . 6 6 1 7 6 4 Y J I 2. 11 = 1 . 8 5 2 9 4 2 XI 2 . 1) - 0 . 0 0 3 7 9 9 Y t 2, 1) = 0 . 0 0 5 2 8 9 X J I 2. 21 1 0 . 0 0 0 0 1 0 Y J I 2, 2) 0 . 0 0 0 0 0 8 XI 2. 21 0 . 0 0 0 0 0 0 Y( 2, 2) 0 . 0 6 1 7 6 1 R E A C T I O N S AT SUPPORT POINTS AT A. HORIZONTAL COMPONENT - - 4 0 0 . 0 0 0 4 8 8  V E R T I C A L COMPONENT 4 9 . 9 9 9 9 5 4 AT B. H O R I Z O N T A L COMPONENT « 4 0 0 . 0 0 0 2 4 4 V E R T I C A L COMPONENT -FORCE . .WIDTH AREA I WT/IN RATIO.. ...MEMBER FORCE HJO.T.H 5 0 . 0 0 0 2 1 4 .AREA I H.T./_IN_ A ( 1. 01 2 0 2 . 4 9 0 0 8 2 2 . 6 9 9 8 6 7 0 . 6 7 4 9 7 0 . 4 1 0 0 0 0 . 0 3 1 9 3 6.0793 B l 0. II - 2 0 2 . 4 9 0 0 8 2 2 . 6 9 9 8 6 7 0 . 6 7 4 9 7 - 0 . 4 1 0 0 0 0 . 0 3 1 9 3 6 . 0 7 9 3 A ( 1, 11 1 4 6 . 6 7 4 e 6 6 1 . 9 5 5 6 6 5 0 . 4 8 8 9 2 0 . 1 5 5 8 3 0 . 0 2 3 1 3 7 . 0 0 9 0 B( 0, 21 - 2 7 3 . 2 0 0 6 8 4 3 . 6 4 2 6 7 5 0 . 9 1 0 6 7 1 . 0 0 6 9 8 0 . 0 4 3 0 6 6 . 0 7 9 3 A ( 2, 01 2 7 3 . 2 0 0 6 8 4 3 . 6 4 2 6 7 5 0 . 9 1 0 6 7 1 . 0 0 6 9 8 0 . 0 4 3 0 8 6 . 0 7 9 3 B( 1, 1) - 1 4 6 . 6 7 4 8 6 6 1 . 9 5 5 6 6 5 0 . 4 8 8 9 2 0 . 1 5 5 8 3 0 . 0 2 3 1 3 7 . 0 0 9 0 A ( 2, 11 2 3 0 . 4 8 8 9 6 8 3 . 0 7 3 1 8 6 0 . 7 6 8 3 0 0 . 6 0 4 6 8 0 . 0 3 6 3 4 1 4 1 . 9 3 3 5 B l 1, 21 - 2 3 0 . 4 8 8 9 6 8 3 . 0 7 3 1 8 6 0 . 7 6 8 3 0 0 . 6 0 4 6 8 0 . 0 3 6 3 4 1 4 1 . 9 3 3 5 VOLUMF TENS I L F MEMBERS^ 10.2.93581 VOLUME C O M P R E S S I V E MEMBERS= 1 0 . 2 9 3 5 8 1 TOTAL VOLUME= 2 0 . 5 8 7 1 5 8 WEIGH T = 0 . 9 7 3 8 8 6 «» F A T A L FCRTRAN ERROR: * E N D - O F - F I L E ENCOUNTERED ON READ OPERATION ERROR OCCUREO ON UNIT 5 ERROR RETURN F O R T R A N IV G C O M P I L E R M A I N 0 4 - 2 9 - 7 0 1 9 : 3 _ ! 3 9 P A G E 0 0 0 1 C PROGRAMME FOR S O L U T I O N OF M I C H E L L C A N T I L E V E R S W I T H D I F F E R I N G F A N C R A D I I . LOWER R A D I U S , R A D B , I S A S S U M E D TO B E L A R G E R THAN R A D A • 0 0 0 1 D I M E N S l O N A ! 2 5 , 2 5 ) , B ( 2 5 , 2 5 ) , X ! 2 5 , 2 5 ) , Y ( 2 5 , 2 5 ) , P S I A ( 2 5 , 2 5 ) , 1 P S I B I 7 5 , 2 5 ) , F A ! 2 5 , 2 5 ) , F B ( 2 5 , 2 5 ) , T I L T ( 2 5 , 2 5 ) , E U L C L A 1 2 5 , 2 5 ) ,  2 L O A O ( 2 5 , 2 5 1 , X J ( 2 5 , 2 5 ) , Y J ( 2 5 , 2 5 ) , W A { 2 5 , 2 5 ) , W B ( 2 5 , 2 5 ) , A R E A A ( 2 5 , 2 5 3 ) , A R E A B ( 2 5 , 2 5 ) , M I Z A ( 2 5 , 2 5 ) , M I Z B ( 2 5 , 2 5 ) , E U L C L B ( 2 5 , 2 5 ) , 4 W T I N A ( 2 5 , 2 5 ) , W T I N B l 2 5 , 2 5)_, E A ( 2 5 , 2 51 , E B ( 2 5 , 2 5 ) , , :  5 E R X ( 8 0 | , E R Y ( 8 0 > 0 0 0 2 R E A L L O A D , I T F M L . I T E M R , I T E M Y 1 , I T E M Y 2 , M I Z A , M I Z 8 0 0 0 3 1 F O R M A T ( 1 3 ) 0 0 C 4 3 F 0 R M A T ( F 1 5 . 6 ) 0 0 0 5 5 F O R M A T ! 1 H 0 , 1 5 X , 4 H N = , I 3 , 1 0 X , 4 H D = , F 1 0 . 6 , 1 0 X , 8 H X S P A N = , 1 F 1 0 . 6 , 1 0 X , 8 H Y S P A N = , F 1 0 . 6 , / / ) ~ 0 0 0 6 7 F O R M A T J I H O , • T H F T A = ' , F 8 . 5 , 4 X , ' T L X =. • , F 8 . 5 , 3 X , • T L Y = ' , F 8 . 5 , 1 2 X , ' R A D R A T = ' , F 8 . 5 , 3 X , ' R A D A = ' , F 8 . 5 , 3 X , • R A D B = ' , F 8 . 5 , 2 4X , ' B E T AC = • , F 8 . 5 , / / ) 0 0 0 7 9 F O R M A T ! 1 H C . 5 0 X , ' L E N G T H S OF M E M B E R S ' / ) 0 0 0 8 11 F O R M A T ! 2 5 X , 2 H A < , 1 2 , 1 H , , 1 2 , 4 H ) = , F 1 5 . 6 , 2 0 X , 2 H B ( , I 2 , I H , , 1 2 , 4 H ) = , "~ocoT~ I 3 1 F 1 5 . 6 ) F O R M A T ! 1 H 0 . 2 5 X , • J C I NT C O O R D I N A T E S ' , 4 0 X , " J 0 I N ' T C E F L ECT IONS ' / ) 0 0 1 0 15 F 0 R M A T ( 6 X , 3 H X J ( , 1 2 , 1 H , , I 2 , 4 H ) = , F 1 2 . 6 , 8 X , 3 H Y J ( , I 2 , I H , , 1 2 , 4 H ) = , 1 F 1 2 . 6 . 9 X . 2 H X ! , 1 2 , I H , , 1 2 , 4 H ) = , F 1 2 . 6 , 8 X , 2 H Y ( 1 2 , I H , , 1 2 , 4 H ) = , 2 F 1 2 . 6 ) 0 0 1 1 17 F 0 R M A T I 2 F 1 5 . 6 ) 0 0 1 2 1 9 F O R M A T ! 1 H 0 . 2 0 X , 5 H L C A D I , 12 , 1 H , , . I 2 , 4 H ) = , F 15 . 6 , 20 X , 5HT I L T I , I 2 , I H , , I 1 2 , 4 H ) = , F 1 5 . 6 ) 0 C 1 3 2 1 F O R M A T ( 1 H O , 6 X , • M E M B E R ' , 7 X , ' F O R C E ' , 7 X , ' W I D T H • , 7 X , • A R E A 1 , 3 X , 1 ' S E C O N D MOMENT ' , 8 X , ' M E M B E R ' , 7 X , ' F O R C E ' , 7 X , • W I D T H • , 7 X , ' A R E A ' , 3 X , 2 ' S E C O N D M O M E N T ' / ) 0 0 1 4 2 3 F 0 R M A T U H 0 , 3 X , 3 H A ( , I 2 , I H , , I 2 , 1 H ) , 1 X , 4 F 12 . 6 , 9 X , 3HB ( , I 2 , 1 H , , I 2 , * 0 0 1 5 2 5 " I 1 H ) , 1 X , 4 F 1 2 . 6 ) FORMAT 1 I H O , 5 X » ' V O L U M E T E N S I L E MEMBERS= ' , F 9 . 6 , 5 X , 1 ' V O L U M E C O M P R E S S I V E MEMBERS= • , F 9 . 6 , 5 X , • T O T A L VOLUME= ' , F 9 . 6 ) 0 0 1 6 2 7 F 0 R M A T I F 1 5 . 6 ) 0 0 1 7 2 9 F O R M A T ( F 1 5 . 6 ) 0 C 1 8 31 F O R M A T ! I H O , 2 4 H M O D U L U S OF E L A S T I C I T Y = , F 1 0 . 1 , 1 X , 6 H P . S • I . , 5 X , 1 17HUN I FORM S T R E S S = , F 1 0 . 6 , 1 X , 6 H P . S . I . , 5 X , 2 1 7 H U N I F 0 R M S T R A I N = , F 1 0 . 6 , I X , 1 2 H INC H E S / I N C H . ) 0 0 1 9 33 F O R M A T ! I H O , 5 X , ' M E M B E R ' , 4 X , ' F O R C E ' , 3 X , ' W I D T H ' , 4 X , ' A R E A ' , 5 X , 1 • I ' , 5 X , ' W T / I N . ' , 3 X , ' R A T I O ' , 5 X , ' M E M B E R ' , 4 X , « F O R C E ' , 4 X 2 • W I D T H ' , 4 X , • A R E A ' , 5 X , ' I ' , 5 X , ' W T / I N . • , 3 X , ' R A T 10 ' ) 0 0 2 0 3 5 F O R M A T ! I H O , 3 X . 2 H A ! , I 2 , 1 H , , I 2 , 1 H ) , 1 X , F 9 . 4 , 5 F 8 . 4 , 5 X , 2 H B 1 , I 2 , 1 H , , 0 C 2 1 3 7 1 1 2 , I H ) , I X , F 9 . 4 , 5 F 8 . 4 ) F O R M A T ( F l 5 . 6 ) 0 0 2 2 39 F O R M A T ! I H O , 4 0 X , ' T H E F O L L O W I N G R E S U L T S ARE C A L C U L A T E D ' , / , 4 1 X , 1 ' F O R A S T R U C T U R E MADE FROM C R 3 9 P L A S T I C • , / , 4 3 X , ? ' H A V I N G A S P E C I F I C G R A V I T Y OF 1 . 3 1 ' , / , 4 2 X , 3 " T H E E F F F C T S OF S E L F WEIGHT F O R C E S ' , / , 4 3 X , 0 02 3 " " 41 4 ' A R E I N C L U D E D IN T H E S E R E S U L T S . ' ) F O R M A T ! 1 H C 4 5 X , ' R E A C T I CNS AT S U P P O R T P O I N T S ' ) 0 0 2 4 4 3 FORMAT I I H O , 2 5 X , ' A T A , H O R I Z O N T A L COMPONENT = ' , F 1 2 . 6 , 1 0 X , 1 ' V E R T I C A L COMPONENT = ' , F 1 2 . 6 ) 0 0 2 5 4 5 F O R M A T ! I H O , 2 5 X , ' A T B , H O R I Z O N T A L COMPONENT = ' , F 1 2 . 6 , 1 0 X , 1 ' V E R T I C A L COMPONENT = « , F 1 2 . 6 ) _ Q C_26 4 7 F O R M A T ! I H O , 5 9 X , ' N O T E ' . / , 4 5 X , « T H E TERM R A T I O IN THE F O L L O W I N G ' , F O R T R A N IV G C O M P I L E R M A I N C 4 - 2 9 - 7 C 1 9 : 3 6 : 3 9 PAGE 0 0 0 2 1 / , 4 3 X , ' T A B L E I N D I C A T E S THE P E R C E N T A G E RAT 1 0 • , / , 4 5 X , 2 ' O F THE F O R C E IN E A C H MEMBER TO I T S ' , / , 4 5 X , 3 ' E U L E R C R I T I C A L L O A D FOR B U C K L I N G ' , / t 5 2 X , 4 ' ( E * I * ( P I ) * ! P I ) ) / ( L * L ) ' , / )  0 0 2 7 4 9 F O R M A T ! 1 H 0 , 4 2 X , ' E I G H T Y C Y C L E S OF C A L C U L A T I O N OF T H E T A • , / , 4 3 X , 1 • A N D RADRAT C O M P L E T E D . ERROR REMA I N S • , / , 4 5 X , ' ~ 0 0 2 8 51 ... 2 ' G R E A T E R THAN S P E C I F I E D M A X I M U M ' , / ) F t ) R M A T ( l H 0 , 5 0 X , «WEIGHT= ' . F 1 5 . 6 ) 0 0 2 9 53 F O R M A T ! I H O , / / ) 0 0 3 0 5 5 F O R M A T ( 2 H A ( , I 2 , I H , , I 2 . 2 H ) , F 1 1 . 6 , F 9 . 6 , 3 F 8 . 5 , F 8 . 4 , 3 X , I 2 H B ( , 1 2 , 1 H , , 1 2 , 2 H ) , F 1 1 . 6 , F 9 . 6 , 3 F 8 . 5 , F 8 . 4 ) 0 0 3 1 6 1 F O R M A T ! I H O , 1 5 X , • T H E T A ' , 9 X , ' D E L T H E ' , 9 X , ' R A D R A T • , 9 X , • D E L R A T • , 1 1 0 X , ' G A P X * , 1 1 X , ' G A P Y ' , / ) 0 0 3 2 63 F O R M A T ! 1 0 X , 6 F 1 5 . 8 ) 0 0 3 3 P1= 3 . 14 1 5 9 2 6 5 0 0 3 4 P I 4 = 0 . 2 5 * P I 0 0 3 5 R 0 0 T = S Q R T < 2 . C J 0 0 3 6 9 9 0 CONT INUE 0 0 3 7 9 9 9 C O N T I N U E 0 0 3 8 K= 1 C A L L THE A R R A Y S ARE Z E R O E D OUT F I R S T 0 C 3 9 DO 10 1 = 1 , 2 5 0 0 4 0 0 0 10 J = 1 , 2 5 0 0 4 1 A ! I , J 1 = 0 . 0 0 0 4 2 B( I , J ) = 0 • 0 0 0 4 3 X ! I , J ) = 0 . 0 0 0 4 4 Y ( I , J ) = 0 . 0 0 0 4 5 P S I A U , J ) = 0 . 0 0 0 4 6 P S I B 1 1 , J ) = 0 . 0 0 0 4 7 10 C O N T I N U E 0 0 4 8 DO 16 I = 1 , 8 0 0 0 4 9 ER X ( I ) = 0 . 0 0 C 5 0 E R . Y U ) = 0 . 0 0 0 5 1 16 C O N T I N U E C N IS NUMBER OF F I B R E S 0 0 5 2 R E A D ( 5 , 1 ) N 0 0 5 3 N1=N+1 0 0 5 4 N2=N+2 0 0 5 5 N N = N - 1 C X S P A N . Y S P A N , ARE C O O R D I N A T E S OF THE OUTER END OF THE S P A N . 0 0 5 6 R E A D ( 5 , 3 ) X S P A N 0 0 5 7 R E A D ! 5 , 3 ) Y S P A N C D IS S U P P O R T S P A C I N G 0 0 5 8 R E A D ! 5 , 3 J D 0 0 5 9 W R I T E ! 6 , 5 I N , D , X S P A N , Y S P A N C T I S T H I C K N E S S OF A L L M E M B E R S IN I N C H E S . 0 0 6 0 T = 0 . 2 5 C EL MOD I S MODULUS OF E L A S T I C I T Y . 0 0 6 1 E L M O D = 3 0 0 0 0 0 . 0 0 0 6 2 S I G M A = 3 0 0 . 0 0 0 6 3 S T R A I N = 0 . 0 0 1 0 0 6 4 W R I T E ( 6 , 3 1 I E L M O C , S I G M A , S T R A I N c THE S P E C I F I C G R A V I T Y ( S P G T Y ) OF C R 3 9 P L A S T I C I S 1 . 3 1 0 0 6 5 S P G T Y = 1 . 3 1 0.06.6 C = ( S P G T Y * 6 2 . 4 > / 1 7 2 8 . 0 FORTRAN IV G COMPILER MAIN 04-2~9-7CT ' 1 9 : 3 6 : 3 9 PAGE 0003 C CALCULATION OF FAN PARAMETERS. RADR IS RADIUS OF FAN CENTRED ON A,AND C RADB IS RADIUS OF FAN CENTRED ON 8 . RADB IS GREATER THAN RADA. C THETA IS FAN ANGLE AND IS GENERATED BY AN ITERATIVE PROCESS.STARTING C WITH AN APPROXIMATE VALUE CF THETA.  C DELRAT AND DELTHE ARE SPECIF IED INCREMENTS FOR RADRAT AND THETA. C RADRAT EQUALS RADB DIVIDED BY RADA 0067 RADRAT =1 .0 0068 DELRAT = 0 . 1 0069 THETA = 1 6 . 0 0070 DEL THE = 1 . 0 0071 ERR = 0 .01 0C72 EPS=0.00001 0073 19. CONTINUE 0074 KK = K -1 0075 I F 1 K . G T . 8 0 ) GC TO 120 0076 0 E N O M = S « R T ( 1.0+(R ADRA T*RADRA T1) 0077 RAC A = D/DENOM 0 078 RADB=(D*RADRAT)/DENOM 0079 TR=THETA*PI/180 .0 0C80 TR2=0.5*TR 0081 RADCON=(180.0/PI ) 0082 BETAA=ATAN(1.0/RADRAT1 0083 BETAD=(180.0*BETAA)/PI 0084 BETR2=(BETAA+TR2) 0085 BETP2M=I BET 4A-T R2) 0086 00 20 I=2,N1 0087 A l l , 1) = R A D A 0088 B( I , I l = RADB 0C89 20 CONTINUE 0090 A<2,2> = 2.0*RA03*SIN<TR2( 0091 B ( 2 , 2 ) = 2.0*RADA*SIN<TR2> 0092 DO 30 I=3,N 0093 A ( 2 , I ) = A ( 2 , 2 ) 0094 3 ( 1 , 2 1=6 ( 2 , 2 ) 0095 30 CONTINUE 0096 DO 40 J = ? , N .00.97 J l = J + l 0098 00 40 1=3 ,N1 0099 11=1-1 0100 A( I , J l = ( A ( I I , J ) + B( I I , J )*S IN(TR) )/COS(TR ) 0101 B( [I , J 1 ) = (A( I I , J ) * S I N ( T R ) + B ( I I , J ) ) / C O S ! T R ) 0102 40 CONT INUF C C TLX IS CALCULATED LENGTH OF. CANTILEVER. TL Y IS CALCULATED RISE OF CANTILEVER. 0103 ITEML=RADA*COS( (TR*FLOAT(NN) l -BETAA) 0 104 I T F M Y 1=(0 . 5 * D ) * ( R A D A * S I N ( ( T R * F L O A T ( N N ) l - B E T A A ) ) 0105 TLX=ITEML 0106 TLY=ITEMY1 0107 DO 50 J=2,N oios TLX =TLX • A(N 1 , J ) * ( C O S ( ( T R * F L 0 A T ( N 1 - J ) ) - B ETR 2) ) 0109 TL Y =TLY + A ( N 1t J ) * S I N ( ( T R * F L O A T ( N l - J ) 1 - B E T R 2 ) 0110 50 CONTINUE 0111 GAPX=XSPAN-TLX 0112 ERR X=GAPX/XSPAN 0113 GAPY _= YSPAN - . TLY _ . F O R T R A N IV G C O M P I L E R M A I N 0 4 - 2 9 - 7 0 1 9 : 3 6 : 3 9 P A G E 0 0 0 4 0 1 1 4 ERRY = G A P Y / Y S P A N 0 1 1 5 E R X ( K ) = ERRX 0 1 1 6 E R Y ( K ) = E R R Y 0 1 1 7 I F ( A B S ( E R R X ) . L T . E P . . A N D . A B S ( E R R Y ) . L T . E P S ) GO TO 120  0 1 1 8 IF ( A B S t E R X ( K ) ) . GT . ABS (EP.X ( KK ) ) ) GO TO 6 0 0 1 1 9 I F ( A B S ( E R Y ( K ) ) . G T . A 6 S I F R Y ( K K ) ) ) G 0 TG 80 _ 0 J . 2 J 3 _ _ I H E T A = . T H E T A + ( D E L T H E * E R R X . ) . . 0 1 2 1 R A D R A T = R A D R A T + < D E L R A T * E R R Y ) 0 1 2 2 K = K+1 0 1 2 3 GO TO 9 9  0 1 2 4 6 0 THETA = T H E T A + I O E L T H E * 0 . 5 * E R R X ) 0 1 2 5 I F ( A B S ( E R Y ( K I ) . G T . A B S ( E R Y I K M ) ) GO TO 7 0 .0.126 RADRAT = R A D R A T + ( O E L R A T * E R R Y ) 0 1 2 7 K = K+1 0 1 2 8 GO TO 9 9 0 1 2 9 7 0 R A D R A T = RADRAT+ I 0 . 5 * D E L R A T * E R R Y ) 0 1 3 0 K = K + 1 0 1 3 ] GO TO 9 9 0.1.3.2 8 0 R A D R A T = R A D R A T + ( 0 . 5 * D E L R A T * E R R Y ) 0 1 3 3 I F ( A B S ( E R X ( K ) I . G T . A B S I E R X ( K K ) ) ) GO TO 100 0 1 3 4 T H E T A = T H E T A + < D E L T H E * E R R X ) 0 1 3 5 K = K + 1 0 1 3 6 GO TO 9 9 0 1 3 7 1 0 0 T H E T A = T H E T A + ( G . 5 * D E L T H E * E R R X ) 0 1 3 8 K = K + 1 0 1 3 9 GO TO 9 9 0 1 4 0 120 WRITE ( 6 , 7 ) T H E T A , T L X , T L Y , R A D R A T , R A O A , R A C B , B E T A C 0 1 4 1 WRITE 1 6 , 5 3 ) 0 1 4 2 WRITE ( 6 , 3 9 ) 0 1 4 3 W R I T E ( 6 , 9 I - 0 1 4 4 0 0 130 1 = 2 , N l 0 1 4 5 1 1 = 1 - 1 0 1 4 6 DO 130 J = 1 , N 0 1 4 7 J J = J - l 0 1 4 8 W R I T E < 6 , 1 1 ) I I , J J , A ( I , J ) , J J , I I , B ( J , I ) 0 1 4 9 1 3 0 C O N T I N U E C C A L C U L A T E THE I N C L I N A T I O N S OF T H E C V A R I O U S M E M B E R S P S I A ( I , J ) AND P S I B ( I , J ) 0 1 5 0 DO 180 J = 1 , N 0 1 5 1 J 1 = J + 1 0 1 5 2 J J = J - 1 0 1 5 3 DO 1 8 0 1 = 1 , N 0 1 5 4 1 1 = 1 - 1 0 1 5 5 11=1+1 0 1 5 6 IF ( J . E O . l ) GO TO 150 0 1 5 7 P S I A d l , J ) = ( ( T R * F L O A T ( 11 - J ) ) - B E T R2 ) 0 1 5 8 I F ( I . G T . l )G0 TO 1 4 0 0 1 5 9 PS I B ( I , J l ) = ( ( T R * F L O A T ( J 1 - 2 ) ) + B E T A A ) 0.160 GO TO 1 7 0 0 1 6 1 1 4 0 P S I B ( I , J 1 ) = ( I T R * F L 0 A T ( J 1 - I D + B E T R 2 M ) 0 1 6 2 GD TO 1 7 0 0 1 6 3 1 5 0 P S I A ( I l , J ) = ( ( T R * F L O A T ( 1 1 - 2 ) l - B E T A A ) 0 1 6 4 I F ( I . G T . l )G0 TO 1 6 0 0 1 6 5 P S l B ( I , J l ) = ( ( T R * F L O A T ( J l - 2 1 l + E E T A A l . .0.1.66 GO TO 1 7 0 F O R T R A N I V G C O M P I L E R M A I N 0 4 - 2 9 - 7 0 1 9 : 3 6 : 3 9 P A G E 0 0 0 5 0 1 6 7 1 6 0 P S I B ( I , J 1 ) = ( ( T R * F L 0 A T ( J 1 - I ) ) + B E T R 2 M ) 0 1 6 8 1 7 0 C O N T I N U E 0 1 6 9 1 8 0 C O N T I N U E 0 1 7 0 W R I T E ( 6 , 4 7 ) 0 1 7 1 DO 190 I = 1 , 2 5 0 1 7 2 DO 190 J = 1 , 2 5 0 1 7 3 L O A D I I , J ) = 0 . 0 0 1 7 4 T I L T H , J ) = 0 . 0 0 1 7 5 F A ( I , J I = 0 . 0 0 1 7 6 F B I I , J 1 = 0 . 0 0 1 7 7 X J I I , J ) = 0 . 0 0 1 7 8 Y J I I , J ) = 0 . 0 0 1 7 9 WA( I , J ) = 0 . 0 0 1 8 0 WBI I , J ) = 0 . 0 0 1 8 1 A R E A A I I , J ) = O . C 0 1 8 2 AR F AB1 I , J ) = 0 . 0 0 1 8 3 M I Z A I I . J ) = 0 . 0 0 1 8 4 M I Z B I I, J 1 = C O 0 1 8 5 W T I N A I I , J 1 = 0 . 0 0 1 8 6 W T I N B I I , J 1 = 0 . 0 0 1 8 7 E U L C L A I I , J 1=0 . 0 0 1 8 8 E U L C L B I I , J ) = 0 . 0 0 189 E A I I , J ) = 0 . 0 0 1 9 0 . E B I I, J ) = 0 . 0 0 1 9 1 1 9 0 C O N T I N U E C THE F O L L O W I N G S E C T I O N C A L C U L A T E S THE F O R C E S IN THE M E M B E R S , c A L L O W I N G FOR T H E E F F F C T S OF S E L F W E I G H T . c F I R S T S P E C I F Y A P P L I E D L O A D S , L O A D I I . J ) , AND THE A N G L E , T I L T I I , J l , c THAT EACH LOAD M A K E S WITH THE V E R T I C A L . ' 0 1 9 2 R C A D ( 5 , 1 7 ) L C A n i N l , N l l , T I L T | N l , N l ) 0 1 9 3 W P I T E I 6 , 1 9 ) N , N , L O A D ! N 1 , N 1 ) , N , N , T I L T ( N 1 , N 1 ) 0 1 9 4 T I L T I N 1 , N 1 ) = I T I L T I N 1 , N 1 ) * P I ) / 1 8 0 . 0 c MEMBERS ARE A S S U M E D W E I G H T L E S S FCR T H I S C A L C U L A T I O N S IS S E T EQUAL c TO Z E R O . 0 1 9 5 W = 0 . 0 0 1 9 6 D 0 W N 1 = C 0 S ( T R 1 - W * I A l N l , N ) * S I N 1 B E T A A + T R 2 ) + D ( N , N l ) * C O S ( B E T A A - T R 2 1 ) . 0 1 9 7 . . . . _ . F A ( N l , N ) = I L O A D I N 1 , N 1 1 * ( S I N ( B ETA A+TR 2~ TI L TI N 1 , N 1 ) ) «•{ W*B { N , N l ) * I S I M T I L T I M , N 1 ) ) ) 1 ) / D 0 W N 1 0 1 9 8 F B I N , N 1 ) = - ( L 0 A D ( N 1 , N 1 ) * ( C 0 S I B E T A A - T R 2 - T I L T ( N 1 , M ) ) -1 W*A ( N l , N) * 1 S I N I T I L T I N 1 . N 1 ) 1 ) I J / D O W N l 0 1 9 9 DO 2 1 0 1 = 1 , N N 0 2 0 0 N l I = M + 1 - I 0 2 0 1 N 2 I = M + 2 " I 0 2 0 2 DO 21C J = 1 , N N 0 2 0 3 N J = N - J 0 2 0 4 N l J = N+1 - J 0 2 0 5 N 2 J = N + 2 - J 0 2 0 6 IF I J . G T . 1 ) GO TC 2 0 0 0 2 0 7 I F I I . F 0 . 1 ) GC TC 2 1 0 0 2 0 8 2 0 0 PS 1 = 1 I 0 . 5 * P I )+P SI A l N2 I , N U ) 1 0 2 0 9 PS 1T = PS l - T R 0 2 1 0 D O W N 2 = I C O S I T R ) - W * I A I N ? I , N l J I * C O S ( P S I T ) + B ( N 1 I , N 2 J ) * S I N ( P S I 1 1 1 0 2 1 1 F A ( N 2 I , N ) . J ) = ( ( F A ( N 2 I , N 2 J ) * < 1 . C - M W * ( A ( N 2 I , N 2 J ) * C 0 S I P S I T ) + B I N 1 I , N 2 J 1 I * S I N I P S I T | | ) ) ) - I F H ( N 2 I , N2 J >* ( S IN ( T R ) + ( W* ( B < N2 I, N2 J ) * C O S ( P S IT ) • 2 B ( N 1 1 , N 2 J ) * C O S ( P S I I I ) ) ) +1LOAD 1 N 2 1 , N 2 J I * I COS I P S I T + T I L T < N 2 I , N 2 J J ) FORTRAN IV G COMPILER MAIN 0 4 - 2 9 - 7 0 1 9 : 3 6 : 3 9 PAGE 0006 0212 0213 _0.2.1A_ 0215 0216 0217 3 +{W*B(N1I,N2J)*SIN{ T I L T I N 2 I , N 2 J ) ) ) ) ) ) / D 0 W N 2 FBI N i l , N 2 J ) = - ( ( F A I N 2 I , N 2 J ) * l S I N ( T R ) + ( W * < A ( N 2 I , N 2 J )*SIN(PS1 ) + 1 A ( N 2 I , N 1 J ) * S I N ( P S I T ) ) ) ) ) - ( F B ( N 2 I , N 2 J ) * ( 1 . 0 - ( W * < A ( N 2 I , N 1 J ) * C C S ( 2 P S I ) - B ( N 2 I , N 2 J i » S I N ( P S I ) ) ) ) ) » I L O A C 1 N 2 I , N 2 J ) * t S I N ( P S I+TILT1 N 2 1 , 3 N 2 J ) ) - ( W * A ( N 2 I . N 1 J ) * S I N < T I L T I N 2 I , N 2 J > ) ) ) ) ) / D C W N 2 210 CONTINUE _ DO. 220 1=1,.N.N . _ . . N l I = N +1- I N2I=N+2-I P S I = ( 0 . 5 * P I + P S I A ( N 2 I , 1 ) )  0218 0219 _022Q_ 0221 0222 0223 220 PS I T = ( P S I - T R 2 ) D0V.N3 = (CCS(TR2) - (W*(A(N2 I »1 )*COS< PS IT )+B (N11, 2)*S IN(PS I ) ) ) > F A ( N 2 I . 1) = ( (FA(N2I ,2 )* (1 .0+(W*(A(N2I ,2 )*C0S(PSIT ) -B(N1L ,2 )*S I_NL_ 1 PSIT ) ) ) ) ) - ( F B ( N 2 I , 2 ) # ( S I N ( T R ) + ( W * ( B ( N 2 I » 2 ) * C 0 S I P S I T ) + 2 BIN II » 2 ) * C 0 S (PSI + T R 2 ) ) ) ) ) + ( L O A D ( N 2 1 , 2 ) * ( C C S ( P S IT+T I L T I N 2 1 , 2 ) ) 3 + ( W « B ( N 1 I . 2 . * S I N ( T I L T ( N 2 I , 2 . ) ) . ) . /DOWN3  F B ( N 1 I , 2 ) = - ( ( F A ( N 2 I , 2 I * ( S I N I T R 2 ) + ( W * ( A ( N 2 I , 2 ) * S I N ( P S I ) + A ( N 2 I , 1 ) * 1 S I N ( P S I T ) ) ) ) ) - ( F 8 ( N 2 I » 2 ) * ( C O S ( T R 2 ) - ( W * ( A ( N 2 I , l ) * C O S ( P S I + T R 2 > -2 BIN2I , 2 ) * S I N ( P S I ) ) ) )1 + IL0ACIN2I ,2 )*(S IN(PS I+T ILTIN2 I ,2)Jr<W* 3 A I N 2 I , 1 ) * S I N I T I L T I N 2 I , 2 ) ) ) ) ) ) / 0 0 W N 3 CONTINUE DO 230 J=1,NN 0224 0225 0?2.6_ 0227 0228 0229 Nl J=N + 1 - J N2J=N+2-J P S I = ( 0 . 5 * P I + P S I A ( 2 » N 1 J ) I PS I T = ( P S 1 - T P 2 ) D0WN4=(CCS(TR2) - IW*I A 1 2 , N l J ) * COS(PS IT )+B|1,N?J )*S IN I PS I > ) > . F A ( 2 , N l J ) = t ( F A ( 2 , N 2 J ) * ( C 0 S < T R 2 ) » ( W * l A ( 2 , N l J ) * C 0 S ( P S I T ) - B t l , N 2 J ) * 1 S I N | P S I - T R ) ) ) ) ) - I F B ( 2 , N 2 J ) * I S I N I T R 2 ) + ( W * ( B ( 1 , N 2 J . * C 0 S ( P S I ) + 2 B ( 2 , N 2 J ) * C O S ( P S IT ) ) ) )) + ( L O A D ( 2 , N 2 J ) * ( C 0 S ( P S I T + T I L T ( 2 , N 2 J ) ) + 3 . (W*B( I ,N2J )*SIN(T I L T ( 2 . N 2 J ) ) ) ) ) I/C0WN4 _ F B I 1 , N 2 J ) = - ( ( F A ( 2 , N 2 J ) * ( S I M T P ) + ( W * ( A ( 2 , N 2 J ) * S INI (PS I I + A ( 2 , N 1 J )* 1 S I N ( P S I - T R ) ) ) ) ) - ( F B ( 2 , N 2 J ) * ( 1 . 0 + < W * ( B ( 2 , N 2 J ) * S I N ( P S I ) - A ( 2 , N U ) * 2 C O S ( P S I ) ) ) ) ) + ( L 0 A 0 ( 2 , N 2 J ) * ( S I N ( P S I + T I L T ( 2 , N 2 J ) ) - ( W*A ( 2 . N 1 J ) * 0230 3 S I N I T I L T I 2 . N 2 J > ) ) ))I/D0WN4 0231 230 CONTINUE ,0232 DOWN 5= ( 1 . 0 - ( W A ( A ( 2 , l ) * S I N ( B E T A A ) + B ( l t 2 ) * C OS (BETAA )_)))_ _ _ 0233 F A ( 2 , 1 ) = ( ( F A ( 2 » 2 ) * ( C 0 S ( T R 2 ) + ( W * ( A ( 2 , 2 ) * S I N ( B E T A A ) - B ( 1 , 2 )*COSIBETA 1 A + T R 2 ) ) ) ) ) - ( F B ( 2 , 2 ) * ( S I N ( T R 2 ) + <W*(EI 1 , 2 ) * S IN (BETAA -TR2) + E ( 2 , 2 ) * 2 S IN(BETAA) ) ) ) ) - ( L O A D ( 2 , 2 ) * ( S T N ( T 1 L T ( 2 , 2 ) - B E T A A ) - ( W * B ( 1 , 2 ) * S I N ( 0234 3 T I L T ( 2 , 2 ) ) ) ) ) 1 / D 0 W N 5 FBI 1 , 2 ) = - ( ( FA (2 ,2 )* (S IN (TR2)+ (W*(A(2 ,2 )<=C0S(BETAA) + A ( 2 , U * C 0 S ( _ l _ EETAA+TR2 ) ) ) ) ) - ( F B ( 2 , 2 ) * ( C O S ( T R 2 ) - ( W * ( A ( 2 , 1 ) * S I N I B E J A A - T R 2 ) -2 B ( 2 , 2 ) * C 0 S ( B E T A A ) ) ) ) I + ( L O A D ( 2 , 2 ) * ( C O S ( T I L T < 2 , 2 ) - B E T AA)-(W*A112,1 3 )*SINIT I L T ( 2 , 2 ) )) ))J/00V.N5 CALCULATION CF PIN JOINT COORD INATES ,XJ ( I , J ) AND YJ < I , J ) , AND JOINT C DEFLECTIONS , XI I , J ) AND Y ( I , J ) . 0235 DO 300 I = 1 ,N 1 _J) 23.6 _ . . . DO 300 J = 1 ,N1 0237 IF ( J . G T . 1 ) GO TO 260 023 8 IF( I . E Q . 1 ) GC TO 300 0239 260 CONTINUE 0240 0241 .0 242 I F ( F A ( I , J l . L T . O . O ) GO TO 270 EA( I , J ) = STPAIN*A< I , J ) G.O TO 2 80 F O R T R A N IV G C O M P I L E R M A I N 0 4 - 2 9 - 7 0 1 9 : 3 6 : 3 9 P A G E 0 0 0 7 0 2 4 3 0 2 4 4 0 2 4 5 0 2 4 6 2 7 0 E A ( I , J ) = - ( S T R A I N * A ( I , J ) ) 2 8 0 I F I F B I I , J l . G T . O . O ) GO TO 2 9 0 E B ( I , J ) = S T R A I N * 8 ( I , J ) GO TO 3 0 0 0 2 4 7 0 2 4 8 0 2 4 9 0 2 5 0 0 2 5 1 0 2 5 2 2 9 0 3 0 0 E B ( I , J I C O N T I N U E 0 0 3 1 0 J = 1 , N J l = J+1 DO 3 1 0 I = 1 , N I I = I + l ( S T R A I N * B ( I , J ) ) 0 2 5 3 0 2 5 4 0 2 5 5 0 2 5 6 3 1 0 F A C T O R = ( ( X ( I , J 1 ) - X ( I 1 , J ) ) * S I N ( P S I B ( I , J I ) I + ( Y ( I 1 , J ) - Y ( I , J l ) ) * C O S t 1 P S I B I I , J l ) ) - E B ( I , J l > - E A ( 1 1 , J ) * S I N ( P S I B I I , J l ) + P S I A ( I I , J ) ) ) / C O S ( 2 P S I A I I I , J 1 + PS I B ( I , J 11 I - _ _ XI I I , J 1 l = ( X ( I I , J ) + E A ( I I , J ) * C O S ( P S I A ( 1 1 , J ) 1 - F A C T O R * S I N ( P S I A ( 1 1 , J 1 1 ) Y ( U , J I ) = ( Y ( U , J I - E A ( I 1 , J ) * S I N ( P S I A ( I I , J ) ) - F A C T 0 R * C O S ( P S I A( I 1 , J ) ) 1 C O N T I N U E 0 2 5 7 0 2 5 8 0 2 5 9 0 2 6 0 0 2 6 1 C O O R D I N A T E S OF T E E P I N J O I N T S . W R I T E ( 6 , 1 3 1 DO 3 2 0 1 = 1 , N i i = i + i x j < i i , i i = o . o X J ( 1 , 1 1 1 = 0 . 0 0 2 6 2 0 2 6 3 0 2 6 4 0 2 6 5 0 2 6 6 0 2 6 7 Y J ( II, 1 1 = 0 / 2 . 0 Y J l 1 , II1= - D / 2 . 0 3 2 0 C O N T I N U E DO 3 3 0 1 = 1 , N t 1 = 1 + 1 X J ( II,2 ) = A( I 1 , 1 ) * C O S ( P S I A ( I I , 1 ) 1 0 2 6 8 0 2 6 9 JO 2 7 0 0 2 7 1 0 2 7 2 0 2 7 3 0 2 7 4 0 2 7 5 _ 0 2 7 6 0 2 7 7 0 2 7 8 0 2 7 9 Y J ( I 1 , 2 ) = Y J ( I 1 , 1 1 + ( A ( I 1 , 1 1 * S I N ( P S I A ( I 1 , 1 > » ) 3 3 0 C O N T I N U E ' DO 3 4 0 I = 1 , N 1 0 0 3 4 0 J = 2 , N J 1 = J + 1 X J ( I , J 1 ) = X J ( I , J ) + A I I , J ) * C 0 S ( P S I A ( I , J ) 1 Y J ( I , J l ) = Y J ( I , J 1 + A ( 1 , J ) * S I N ( P S I A ( I , J I 1 3 4 0 C O N T I N U E DO 36C 1 = 1 , N l I I = 1-1 DO 3 6 0 J = 1 , N 1 J J = J - 1 0 2 8 0 0 2 8 1 0 2 8 2 0 28 3 0 2 8 4 0 2 8 5 _ 0 2 8 6 0 2 8 7 0 2 8 8 I F ( J . G T . l ) GO TO 3 5 0 IF ( I . EQ . 1 1 GO TO 3 6 0 3 5 0 W R I T E ( 6 ,1 5 ) LL >JAJXJl I , J l , I I , J J , Y J ( I , J l , I I , J J , X ( I , J l , I I , J J , Y ( I , J 1 3 6 0 C O N T I N U F C C O M P I L E P R O P E R T I E S OF EACH M E M B E R . C F O R C E S AND W E I G H T S IN P O U N D S , WIDTHS IN I N C H E S , C R O S S S E C T I O N A L A R E A S C IN SQUARE I N C H E S AND MOMENTS OF I N E R T I A IN ( I N C H E S 1 * * 4 DO 4 0 0 1 = 1 , N I I = I + l I I = I - l DO 4 0 0 J = 1 , N J 1 = J + 1 0 2 8 9 0 2 9 0 0 2 9 1 _ J J = J - 1 WA( I 1 , J ! = WB( I , J 1 1 = ( A B S ( F A ( I 1 , J 1 1 ) / ( T * S I G M A ) ( A B S ( F B I I , J l 111 / ( T * S I G M A ) F O R T R A N IV G C O M P I L E R M A I N 0 4 - 2 9 - 7 0 1 9 : 3 6 : 3 9 P A G E 0 0 0 8 0 2 9 2 0 2 9 3 0 2 9 4 0 2 9 5 A R E A A I [ 1 , J ) = ( T * W A ( 11 , J ) ) AR E AB( I , J 1 > = ( T * W B ( I i J l l l M I Z A I 1 1 , J ) = ( ( W A ( 11 , J > * * 3 ) * T ) / 1 2 . 0 M I Z 8 ( I , J 1 ) = (< WB ( I » J 1 ) * # 3 ) * T ) / 1 2 . 0 0 2 9 6 0 2 9 7 _Q.29.8. 0 2 9 9 0 3 0 0 WT I N A ( I 1 , J ) = ( A R E A A I I I , J > * S P G T Y * 6 2 . 4 ) / 1 7 2 8 . 0 W T 1 N B I I , J 1 ) = ( A R E A B ( I , J l ) * S P G T Y * 6 2 . 4 ) / 1 7 2 8 . 0 I F ( W A ( 1 1 , J ) . G T . 0 . 2 5 ) GO TO 3 7 0 E U L C L A ( I l , J ) = ( A B S ( F A ( l l , J ) ) * A t I l , J ) * A ( l l , J ) * 1 1 0 . 0 ) / I I E L M O D * M I Z A < I 1 , J ) * P I * P I ) GO TO 3 8 0 0 3 0 1 . . .0302 . 3 7 0 E U L C L A ( I 1 , J ) = ( A B S ( F A ( U , J ) ) * A ( I 1 , J ) * A < 1 1 , J 1 * 7 6 8 0 0 . 0 ) / ( E L M O D * 1 WAI I 1 , J ) * P I *P I ) 3 8 0 I F ( W B ( I , J 1 1 . G T . 0 . 2 5 ) GO TO 390 0 3 0 3 0 3 0 4 E U L C L B I I , J 1 ) = ( A B S I F B ( I , J 1 ) ) * B ( I , J 1 ) * B ( I , J 1 ) # 1 0 0 . 0 ) / 1 < E L M O D * M I Z B < I , J 1 ) * P I * P I ) GO TO 4 0 0 0 3 0 5 _ 0 3 . 0 . - _ _ 3 9 0 E U L C L B I I , J 1 ) = ( A B S ( F 8 ( I , J 1 ) ) * B < I , J 1 ) * B ( I , J 1 1 * 7 6 8 0 0 . 0 ) / < E L M O O 1 * W B ( I , J 1 ) * P I * P I ) 4 0 0 C O N T I N U E C R E A C T I O N S AT S U P P O R T S . C THE H O R I Z O N T A L COMPONENTS OF THE R E A C T I O N S AT A AND B ARE C M E A S U R E D P O S I T I V E A C T I N G TC THE R I G H T , AND T H E V E R T I C A L 0 3 0 7 0.3.0 fi C COMPONENTS ARE M E A S U R E D P O S I T I V E A C T I N G U P W A R D . F A H = 0 . 0 F A V = 0 . 0 0 3 0 9 0 3 1 0 0 3 1 1 DO 4 1 0 I = 2 , N 1 F A H = F A H - < F A ( I , l ) * C O S ( P S I A U , l ) ) ) F A V = F A V + ( F A { I , 1 ) * S I N ( B E T A A - ( T R * F L O A T ( 1 - 2 1 ) ) ) + ( A B S ( F A ( 1 , 1 ) ) * 0 3 1 2 _0.3.13 1 A ( I , 1 ) * W ) 4 1 0 C O N T I N U E F B H = 0 . 0 0 3 1 4 0 3 1 5 0 3 1 6 . F B V = 0 . 0 DO 4 2 0 J = 2 , N 1 F B V = F B V - ( F B ( 1 , J ) * C O S ( B E T A A * ( T R * F L O A T ( J - 2 ) ) ) ) + ( A B S ( F B ( l , J > > * 0 3 1 7 0.3.1.8 1 B ( 1 , J ) * W ) F B H = F B H - ( F B ( 1 , J ) * S I N ( P S I B ( 1 , J ) ) 1 4 2 0 C O N T I N U E 0 3 1 9 0 3 2 0 0 3 2 1 W R I T E ( 6 , 4 1 ) W R I T E ( 6 , 4 3 ) F A H , F A V W R I T E ( 6 , 4 5 ) F B H , F B V 0 3 2 2 0 3 2 3 0.32.4 WRITE t 6 , 3 3 ) DO 4 5 0 1= 1 , N I I = I - l 0 3 2 5 0 3 2 6 0 3 2 7 11 = I + l DC 4 5 0 J = 1 , N J l = J+1 0 3 2 8 0 3 2 9 J J = J - l W R I T E ( 6 , 5 5 ) I , J J , F A ( I l . J ) , W A ( I I , J ) , A R E A A I I 1 , J ) , M I Z A l I 1 , J ) , 1 W T I N A ( I I . J ) , E U L C L A ( I 1 , J ) , I I , J , F B ( I , J 1 ) , W B I I , J 1 ) , A R E A B ( I , J 1 ) , 0 3 3 0 2 M I Z B ( I , J 1 ) , W T I N B ( I , J 1 1 , E U L C L B ( I , J 1 ) 4 5 0 C O N T I N U E C F I N A L S T E P I S TO D E T E R M I N E VOLUME AND WEIGHT OF S T R U C T U R E . 0 3 3 1 0 3 3 2 0.3.3.3 VOLUME = 0 . 0 VOL TEN = 0 . 0 VOLCOM = 0 . 0 F O R T R A N I V G C O M P I L E R M A I N 0 4 - 2 9 - 7 0 1 9 : 3 6 : 3 9 P A G E 0 C 0 9 0 3 3 4 W E I G H T = 0 . 0 0 3 3 5 DO 4 8 0 I =1,N 0 3 3 6 DO 4 8 0 J = 1 , N 0 3 3 7 1 1 = 1 + 1 0 3 3 8 J 1 = J + 1 0 3 3 9 I F I F A ( 11 , J l . L T . O . O ) GO TO 4 7 0 0 3 4 0 VOL TEN = V O L T E N + I A I 1 1 , J ) * A R E A A 1 1 1 , J ) > 0 3 4 1 I F ( F B ( I , J 1 ) . L T . 0 . 0 ) GO TO 4 7 0 0 34 2 4 6 0 VOL TEN = V O L T E N + t fit I , J l ) * A R E A B ( I , J 1 ) 1 0 3 4 3 GO TO 4 8 0 0 3 4 4 4 7C I F < F B ( I , J 1 ) . L T . O . O ) GC TO 4 8 0 0 3 4 5 VOL TEN = VOL TEN + ( B ( I , J 1 ) * A R E A B ( I , J l ) ) 0 3 4 6 4 8 0 C O N T I N U E 0 34 7 0 0 5 0 0 I = 1 ,N 0 3 4 8 DO 5 0 0 J = 1 , N 0 3 4 9 11 = I + l 0 3 5 0 J l = J+1 0 3 5 1 IF I F A ( 1 1 , J l . G T . 0 . 0 1 GO TO 4 9 C 0 3 5 2 V O L C O M = V O L C C M + l A( I i , J ) * A R E A A ( H , J I | 0 3 5 3 I F ( F B ( I , J 1 1 . G T . 0 . 0 1 GO TO 4 9 0 0 3 5 4 VOLCOM = VOLCOM + ( B < I , J 1 ) * A R E A B ( I , J 1) ) 0 3 5 5 GO TO .500 0 3 5 6 4 9 0 IF ( F B I I , J 1 ) . G T . 0 . 0 ) GO TO 5 0 0 03.5 7 VOLCOM = V O L C O M + ( B l I , J 1 ) * A R E A B ( I . J l ) ) 0 3 5 8 5.C0 .. C O N T I N U E 0 3 5 9 V O L U M E = VOLUME + VOL T E N + VOLCOM 0 3 6 0 W E I G H T = W E I G H T + I V O L U M E * C ) 0 3 6 1 W R I T E ( 6 , 2 5) V O L T E N , V O L C C M , V O L U M E 0 3 6 2 W R I T E ( 6 , 5 1 ) WEIGHT 0 3 6 3 W R I T E ( 6 , 5 3 ) 0 3 6 4 0 3 6 5 " " 9 0 0 GO TO 9 9 0 CONT INUE 0 3 6 6 END T C T A L MEMORY R E Q U I R E M E N T S 0 1 3 2 0 0 B Y T E S E X E C U T I O N TERMINATED SRUN -LOACH 5 = « S O U R C E « 6=»SINK,»  E X E C U T I O N B E G I N S N = 3 0 = 2.5OOOO0 XSPAN = 1 0 . 0 0 0 0 0 0 . .. . Y S P A N „ _ 1.000000. MODULUS OF E L A S T I C I T Y = 3 0 0 0 0 0 . 0 P . S . I . UNIFORM S T R E S S - 3 0 0 . 0 0 0 0 0 0 P . S . I . UNIFORM S T R A I N = 0 . 0 0 1 0 0 0 I N C H E S / I N C H . T H E T A - 3 5 . 8 3 9 * 9 TLX = 9 . 8 3 8 6 1 T L Y = 1 . 0 0 3 6 8 R AOR AT = 1. 2 5 6 7 1 RAO A - .1.5.5636..... RADB..?....1.. 9 5 6 4 7 BET.»0.__3.8...S.02O3. THE FOLLOWING R E S U L T S ARE C A L C U L A T E D FOR A STRUCTURE MADE FROM C R 3 9 P L A S T I C HAVING A S P E C I F I C GRAVITY OF 1.31 THE E F F E C T S OF S E L F WEIGHT FORCES ARE' INCLUDED IN T H E S E R E S U L T S . .... LENGTHS OF MEMBERS A( 1 . 01 = 1 . 5 5 6 3 5 6 81 0. 11 = 1 . 9 5 6 4 6 6 AI 1 . 11 _ 1 . 2 0 3 4 2 4 B l 1, 11 0 . 9 5 7 3 1 6 A l 1. 21 = 1 . 2 0 3 4 2 4 B< 2 . 1 I = 0 . 9 5 7 3 1 6 A l 2 . 01 = 1 . 5 5 6 3 5 6 B l Or 21 = 1 . 9 5 6 4 6 6 A l 2. 1) = 2. 1 7 5 2 2 6 B l 1. 2 ) 2 . 0 4 9 3 5 0 A l 2 , 21 = 2 . 9 6 3 5 0 3 B l 2 , 21 = 2 . 7 5 0 8 3 6 A( 3, 01 1 . 5 5 6 3 5 6 61 0. 31 = 1 . 9 5 6 4 6 6 A( 3, 1) = 3 . 3 7 3 7 6 1 B( 1 . 31 3 . 3 9 6 1 6 7 A l 3, 21 5 . 6 4 0 5 9 0 B l 2, 31 ? . . . . 5 . 5 3 1 8 1 8 NOTE THE TERM R A T I O I N THE FOLLOWING T A B L E I N D I C A T E S THE P E R C E N T A G E RATIO OF THF FORCE IN EACH MEMBER TO ITS EULER C R I T I C A L LOAD FOR B U C K L I N G IE» I « | P I > * ( P I I I / I L * L ) L O A C I 3, 31 = 1 0 0 . 0 0 0 0 0 0 T I L T ! 3> 3) = 0.0 J O I N T COORDINATES J O I N T D E F L E C T I O N S . X J l 0. 11 - Y J I 0, 11 - 1 . 2 5 0 0 0 0 XI 0. 11 = 0.0 Y( 0, 11 = 0.0 X J ( Or 21 - 0.0 Y J I 0 . 2) -1 . 2 5 0 0 0 0 XI Ot 21 = 0.0 Y( 0. 21 = 0.0 XJ 1 C, 31 z 0.0 Y J I 0. 3) = - 1 . 2 5 0 0 0 0 XI 0 . 31 = 0 .0 YC 0. 31 0.0 X J ( 1, 0) = 0.0 Y J 1 1. 0) = 1 . 2 5 0 0 0 0 XI 1. 01 = 0.0 Y l 1. 0 ) = 0.0 X J ( 1, 11 = I . 2 1 7 9 8 2 Y J I I . 1 ) 0 . 2 8 1 1 0 2 X 1 1, 1) = 0 . 0 0 0 0 0 0 Y l 1, 1) - 0 . 0 0 2 5 0 0 X J I 1, 21 = 1 . 8 8 3 7 0 6 Y J I 1. 21 • - 0 . 7 2 1 4 1 3 XI 1. 2) = - 0 . 0 0 0 7 8 0 YI 1. 21 0 . 0 0 4 4 6 2 X J l 1, 31 1 . 8 3 6 7 3 2 Y J I 1 > 31 -1 . 9 2 3 9 1 9 XI 1, 3) = - 0 - 0 0 4 1 1 5 Y( 1, 31 = 0 . 0 0 5 5 3 6 X J I 2 i 01 0.0 N < 2 , 01 = 1 . 2 5 0 0 0 0 XI 2, 0 1 = 0.0 Y l 2 t 01 = 0.0 X J I 2 • 1 1 = 1 . 5 5 4 6 5 5 Y J I 2, 1) = 1 . 1 7 7 2 6 4 XI 2 . 1) = 0 . 0 0 1 3 6 9 Y l 2, 1) = 0 . 0 0 4 0 3 7 XJI 2 < 21 = 3 . 5 9 0 9 2 5 YJ< 2, 21 = 0 . 4 1 2 2 7 2 X 1 2 . 2 ) = 0 . 0 0 1 0 8 6 Yl 2. 2) 0 . 0 1 0 9 7 6 X J I 2 . 31 = 5 . 2 3 0 3 1 2 Y J I 2. 31 = - 2 . 0 5 6 4 8 3 XI 2 . 3 ) = - 0 . 0 0 8 1 0 4 Y ( 2. 31 = 0 . 0 2 0 6 3 6 X J I 3 • 01 - 0.0 Y J I 3, 01 = 1 . 2 5 0 0 0 0 XI 3 , 0) = 0.0 Y l 3, 01 = 0(iO X J I 3 . 11 - 1 . 3 0 3 1 2 6 Y J I . 3 , 1 1 = 2 . 1 0 0 9 4 5 X I 3 . 1) = 0 . 0 0 4 5 7 3 Y l 3, 1) 0 . 0 0 4 1 5 7 X J I 3 . 21 = 4 . 5 5 8 3 5 2 Y J I 3. 2) = 2 . 9 8 7 3 8 0 XI 3 . 2) = 0 . 0 1 1 8 2 6 Y l 3, 2) = 0 . 0 1 7 9 5 0 X J I 3 . 31 9 . 8 3 8 6 1 4 Y J I 3, 31 = i . 0 0 3 6 7 5 XI 3 . 3) = 0 . 0 0 5 4 2 9 VI 3, 31 0 . 0 5 1 0 1 6 R E A C T I O N S A l SUPPORT PC I NTS AT A, , HORIZONTAL COMPONENT = - 3 9 3 . 5 4 4 4 3 4 V E R T I C A L COMPONENT = 9 . 8 5 2 9 0 5 AT R i , H O R I Z O N T A L COMPONENT = 3 9 3 . 5 4 4 4 3 4 V E R T I C A L COMPONENT 9 0 . 1 4 7 1 5 6 MEMBER FORCE WIDTH AREA I WT/IN. RATIO MEMBER FORCE WIDTH AREA . I WT/IN. . R A T I O A l l i r 01 1 1 8 . 5 5 8 0 6 0 1 . 5 8 0 7 7 3 0 . 3 9 5 1 9 0 . 0 8 2 2 9 0 . 0 1 8 6 9 4.7122 B( 0, 1 1 - 1 1 6 . 2 7 1 8 2 0 1 . 5 5 0 2 9 0 0 . 3 8 7 5 7 0 . 0 7 7 6 2 0 . 0 1 8 3 3 7 . 4 4 6 4 A l 1, , 1 1 9 5 . 0 2 9 4 B 0 1 . 2 6 7 C 5 9 0 . 3 1 6 7 6 0 . 0 4 2 3 8 0 . 0 1 4 9 8 2 . 8 1 7 3 B l 0. 21 - 187 .655045 2 . 5 0 2 0 6 7 0 . 6 2 5 5 2 0 . 3 2 6 3 3 0 . 0 2 9 5 9 7 . 4 4 6 4 A l 1 i 21 4 6 . 0 2 2 3 6 9 0 . 6 1 3 6 3 2 0 . 1 5 3 4 1 0 . 0 0 4 8 1 O.O0726 2 . 8 1 7 3 B l 0. 3) - 1 4 9 . 6 4 1 7 6 9 1 . 9 9 5 2 2 3 0.49881 0 . 1 6 5 4 8 0 . 0 2 3 6 0 7 . 4 4 6 4 A( 2i . 0) 1 8 9 . 4 7 0 3 2 2 2 . 5 2 6 2 7 1 C. 6 3 1 5 7 C . 3 3 5 8 9 0 . 0 2 9 8 8 ,4.7122 81 1. 11 - 9 1 . 4 7 9 3 0 9 1 . 2 1 9 7 2 4 0 . 3 0 4 9 3 0.0378O 0. 0 1 4 4 2 1.7826 A( 2 . 1 1 1 5 6 . 3 1 6 3 6 0 2 . 0 8 4 2 1 8 0 . 5 2 1 0 5 0 . 1 8 8 6 2 0 . 0 2 4 6 5 9.2047 B l 1 . 2) -1 5 1 . 6 2 3 4 1 3 2 . 0 2 1 6 4 5 0 . 5 0 5 4 1 0 . 1 7 2 1 4 0 . 0 2 3 9 1 8 . 1 7 0 2 A l 2 . 21 8 3 . 3 3 8 5 1 6 1 . 1 1 1 1 7 9 0 . 2 7 7 7 9 0 . 0 2 8 5 8 0 . 0 1 3 1 4 1 7 . 0 8 4 9 B( 1. 31 - 1 4 2 . 3 8 8 9 1 6 1 . 8 9 8 5 1 9 0 . 4 7 4 6 3 0 . 1 4 2 5 6 0. 0 2 2 4 5 2 2 . 4 3 7 8 A l 3, , 01 1 3 3 . 1 6 6 4 2 8 1 . 7 7 5 5 5 2 0 . 4 4 3 8 9 0 . 1 1 6 6 2 0 . 0 2 1 0 0 4 . 7 1 2 2 81 2. 11 - 4 0 . 9 5 5 3 8 3 0 . 5 4 6 0 7 2 0 . 1 3 6 5 2 0 . 0 0 3 3 9 0 . 0 0 6 4 6 1.7828 A( 3 . 1) 1 2 6 . 7 1 2 0 9 7 1 . 6 8 9 4 9 4 0 . 4 2 2 3 7 0 . 1 0 0 4 7 0 . 0 1 9 9 8 2 2 . 1 4 2 7 B l 2 . 2) - 7 4 . 1 6 3 0 8 6 0 . 9 8 8 8 4 1 0 . 2 4 7 2 1 0 . 0 2 0 1 4 0 . 0 1 1 6 9 1 4 . 7 2 0 8 A( 3 i 21 102. 7 4 1 3 9 4 1 . 3 6 9 8 8 4 0 . 3 4 2 4 7 0 . 0 5 3 5 6 0 . 0 1 6 2 0 6 1 . 8 9 4 3 B l 2. 3) - 1 1 5 . 4 5 2 5 6 0 1 . 5 3 9 3 6 7 0 . 3 8 4 8 4 0 . 0 7 5 9 9 0. 0 1 8 2 1 5 9 . 5 3 0 2 VOLUME T E N S I L E MEMBFP.S = 8 . 1 6 8 0 5 1 VOLUME C O M P R E S S I V E MEMBERS- 8 . 8 3 7 1 6 9 T O T A L VOLUME= 1 7 . 0 0 5 2 1 9 «» F A T A L FORTRAN ERROR: • E N D - O E - F I L E ENCOUNTERED ON REAP OPERATION ERROR CCCURED ON UNIT 5 ERROR RETURN t S I G N O F F 25 APPENDIX D FORCE SYSTEM IN TYPICAL MICHELL CANTILEVER The s o l u t i o n of the f o r c e s a c t i n g i n the v a r i o u s members of a M i c h e l l c a n t i l i v e r i s q u i t e s t r a i g h t f o r w a r d , f o l l o w i n g the standard techniques f o r a n a l y s i s of any s t a t i c a l l y determinate s t r u c t u r e . Each j o i n t i s i s o l a t e d and c o n s i d e r e d s e p a r a t e l y . A f r e e body diagram i s drawn f o r each j o i n t and the a p p r o p r i a t e f o r c e polygon c o n s t r u c t e d ; t h i s i s then s o l v e d , t r i g n o m e t r i c a l l y . I t i s most convenient t o commence a n a l y s i s a t j o i n t L, ( J N N ) where o n l y two members A^ N_-^ and BN_-^ N , are i n v o l v e d . Once the f o r c e s i n these members have been c a l c u -l a t e d the next i n n e r j o i n t s J .... ... and J ,„ , > may be J . (N,N-1) (N-1,N) 2 s o l v e d s i n c e o n l y .two unknown f o r c e s e x i s t a t these p o i n t s . T h i s procedure i s f o l l o w e d s y s t e m a t i c a l l y , working toward the support p o i n t s which are the l a s t t o be d e a l t w i t h . . > S o l u t i o n o f these p o i n t s y i e l d s the support r e a c t i o n s . Because o f the r e g u l a r way i n which the framework i s c o n s t r u c t e d , the freebody diagrams and f o r c e polygons are s i m i l a r i n shape f o r ' f a m i l i e s ' of j o i n t s and s i x cases o n l y need to be c o n s i d e r e d . 26 There a r e : -a) J o i n t NN. The o n l y two member j o i n t i n the s t r u c t u r e and the f i r s t to be ana l y z e d . b) A l l other j o i n t s NOT on the base f a n s . For i n n e r j o i n t s , f o u r members meet a t each p o i n t . The f o r c e s i n two of these must thus be known be f o r e the j o i n t can be s o l v e d . For j o i n t s on the outer f i b r e s , or J _ N r the same equations may be used, p l a c i n g the v a l u e s f o r the m i s s i n g outer member at zero. c) J o i n t s on the 'A' base f a n , EXCEPT J - ^ l * Four members meet a t these j o i n t s but the angles between them d i f f e r from those i n case (b) and thus separate s o l u t i o n s are r e q u i r e d . The ou t e r j o i n t , J ^ ^, has o n l y three members. d) J o i n t s on the 'B' base f a n , EXCEPT J-j^-These j o i n t s are s i m i l a r t o those on the 'A' : fa n , but of o p p o s i t e hand. They can be s o l v e d by c a r e f u l use of the 'C equations but e r r o r i s l e s s l i k e l y i f t h e i r own equations are used. In symmetrical frameworks, with symmetrical l o a d i n g s , t h i s i s unnecessary. < e) J o i n t J - ^ T h i s j o i n t i s unique i n i t s geometry and r e q u i r e s i t s own s p e c i a l s o l u t i o n . 27 f) Support P o i n t s A & B. These are the l a s t p o i n t s to be s o l v e d . Many (N) members converge a t these j o i n t s . In the f o l l o w i n g s o l u t i o n s , g e n e r a l i t y has been maintained by the i n c l u s i o n of the f o l l o w i n g v a r i a b l e s (sec-ond form i s t h a t used i n computer s o l u t i o n s ) . These are i n -d i c a t e d on F i g u r e D l : -3 - or BETAA angle ABC T , , . o r TILT , u . (a,b) (a,b) For a symmetrical s t r u c t u r e 6 = 45° r A ; : More g e n e r a l l y tan g = — - (2.2). r B Angle between d i r e c t i o n o f a p p l i e d l o a d , P, , N or LOAD, , s , and (a,b) (a,b) p e r p e n d i c u l a r t o span a x i s (y a x i s ) W or W A con s t a n t r e l a t i n g the s p e c i f i c g r a v i t y o f the s t r u c t u r a l m a t e r i a l , the uniform s t r e s s and the system of u n i t s used. For lengths i n inches, f o r c e s i n pounds S p e c i f i c G r a v i t y 62.4 Uniform S t r e s s 2 x 1728 + or THGRAV The angle between the d i r e c t i o n o f g r a v i t y and the p e r p e n d i c u l a r t o the span a x i s (y a x i s ) . 28 F i g u r e D l Forces A c t i n g on T y p i c a l M i c h e l l C a n t i l e v e r The f r e e body diagrams and f o r c e polygons f o r each type of j o i n t are now c o n s i d e r e d i n sequence:-F i g u r e D2 Forces a t J N N The f o r c e s a c t i n g a t j o i n t J N N are shown i n F i g u r e D2, together w i t h a sketch of the f o r c e polygon. The f o r c e i n A_, ., , i s assumed to be t e n s i l e w h i l e t h a t i n B„ , i s N,N-1 N-1,N assumed to be compressive. The t r u t h of these assumptions i s determined by the s i g n of equations, when the a c t u a l numerical v a l u e s are used f o r computation. 30 SELF WEIGHT FORCES The s e l f weight f o r c e , (SWT), i s taken as h a l f the weight o f each of the members meeting a t the j o i n t . Thus f o r a g i v e n member A ^ i t s weight i s g i v e n by Weight of A 0b = Aob*(WA 0. b*l)x - Aab * fFAob) x * > » 6 2 - 4 = 2 W [ A a b F A o b ] D l Thus f o r the j o i n t NN the s e l f weight f o r c e i s g i v e n by the equation (SWT) =: WJAN.^FAH.H.,-»- BH...MFBH.,.M] D2 SOLUTION OF FORCE POLYGON From F i g u r e D2 the f o l l o w i n g equations may be d e r i v e d : T V - T U + U V FB MH . N 5 ia ( :3+§ )^ ( S ^ . . D3 L U * L R + f c U = M P * Q V 31 Combining D3 and D4 FA sin fr+f-t) +(SWT) C O S 0 FEW*- F>N COS ( H - T ) * ( s w t ^ c ° 3 ( f t - | -T) cos 0 Equation D2 may then be used to e l i m i n a t e (SWT) from these equations to o b t a i n the f i n a l r e s u l t : -D5 FBM_,.M— RON N.M-l P„„ [cos^-f-t) -W[An.h-, sintt - Iff] (cos0 - W[AN.N-» SI n (p+f E V . . N cos (p-f-fj]) D6 32 b) T y p i c a l I n n e r J o i n t (y+t-0) a) F r e e b o d y D i a g r a m l ^ t f - ^ Swr-z b) F o r c e P o l y g o n F i g u r e D3 F o r c e s a t a T y p i c a l I n n e r J o i n t F i g u r e D3 i l l u s t r a t e s t h e f o r c e s a t a t y p i c a l j o i n t i n t h e main body o f t h e s t r u c t u r e . F o r edge j o i n t s FB.^ i s z e r o s i n c e t h i s member i s o m i t t e d w h i l e f o r j o i n t s Nb J J s i m i l a r l y FA i s z e r o . FA . and FB , a r e known f o r c e s aN 1 aN ab ab w h i l e FA , , and FB , , a r e t o be d e t e r m i n e d . a , b - I a - i , b The a n g l e shown as lj. i n t h e f r e e body d i a g r a m may be d e t e r m i n e d f r o m t h e e q u a t i o n H>= 90 + [(a-b)e- 1/3+f)] D7 which i s derived d i r e c t l y from equation B4. The solution of the force polygon follows a sim i l a r pattern to that described i n section A. (SWT) = W AobFbb+Aa.b-.FAQ.b-<-* B Q . b F B c b + B a - , . b F B A.,.J] D8 and thus FAa.t> |[+w|Aoi>coS^ f-9)-Bo-,.i> sin(?+f-9)]] ± B.bCcoa+t-9) + W[BQ.,, b s i a ^ f j l j cos FAo.b |>ne +W[AQb sin(H>+t) + AQ.b-.sin(4M-0)]J +FBq.bC »~ wtN»>-' o»foH} - Bob siH t)J] cos 0-w(A0 .bH cos(y+t-Q)+ Ba.,.bsin(^+^jj D9 c) T y p i c a l 'A' Fan J o i n t The f o r c e s a t a t y p i c a l 'A' f a n j o i n t are shown i n F i g u r e D4. J o i n t 11 i s s p e c i f i c a l l y excluded w h i l e j o i n t J N ^ may be c o n s i d e r e d by o m i t t i n g f o r c e FB^ ^. I n . t h i s case, the angle i|> may be obtained from equation B4 by adding 90°:- y = 90 + ( ( a-l ) 0 - p ) . . . . . . D10 F i g u r e D4 Forces a t a T y p i c a l 'A' Fan J o i n t 35 The s e l f weight force i s now given by:-(SWT)=w[Ao.*FAa,^^ • • • 0 1 1 Equations for the unknown forces, FA^ q and F B ( a _ T _ 2.) ' may be derived from the force polygon and equation D l l used to eliminate (SWT). The yi e l d s eventually -PA a , 0 = F A a J +FB... J + W [ A Q , C O S ^ T - ! ) - BO-... sin^ +t-IJ si'n6+W[BQ.,cos(4'+t-f)-+ Bo-,., cosfy+t+i)~ "cos(y »t-t)+w [BQ-,., s>n (y-*JD R\a..{staf • w[Aa.oS»'n ty+f -1) + Ao.. sin (H>+•*)]] +FBa..[cosf-W[Aa.o oos(<M+f) - Bo, Sin (4>+i)JJ . •R;. ,[sin(y^-W^q .o sin (£- jjflj cos I - W[A0O COS (H» +Bq-,., sin *• f)| D12 36 d) T y p i c a l 'B' F a n J o i n t a) F r e e b o d y D i a g r a m b) F o r c e P o l y g o n F i g u r e D5 F o r c e s a t a t y p i c a l 'B' J o i n t T h e f o r c e s a t a < B' f a n j o i n t , e x c l u d i n g j o i n t J - ^ / a r e s h o w n i n F i g u r e D5, a n d may b e c o m p a r e d w i t h t h o s e s h o w n i n F i g u r e D4. F o r t h i s c a s e : -(SWT) * W [ A , t r A , b M , b . , F A ) . b - l + B i . < > F B l . b i - B o . b F B 0 . J ] D13 37 FA l.fcrl — FAi .«> + F B , . b cos | + W[A ,.bcos (jM- f) - Bab s.'n (4>+*-0)] cos^t-|) + w[Bob Sin (Y-*)]] cosf - w[A,.t>-, cos (M-f)+Bobs*n(y+t)] D13 FB A B FAi.b C^9+W[Ai.bsin(4»+^)+Ai.b-» si a (y+ 4-0)]] •FB,.b[ I -W[A,.M«>sC^)-*B»bsin(V+^)J] V P,.b CsmC t^) - W[A..b-, Sin (T-4)]] cos f - W JA,.b-j (q>+*-|) t Bob sin ^>+*)] D14 e) Joint J 11 a) Freebody Diagram b) Force Polygon Figure D 6 Forces at Joint J The f o r c e s a c t i n g a t C, J o i n t are i n d i c a t e d i n F i g u r e D6. From the diagrams (SWT) = vs/^ l lFAn-»-A l0FA,0-»-B0lFB01 + B M F B j . . . . D I S S o l u t i o n of the f o r c e polygon y i e l d s the f o l l o w i n g r e s u l t s f o r the unknown f o r c e s FA. and F B n n . F B 0 , -FA„ +FB., d3i cosf - W|A„sin^©+ So , cos (f-MX s»n|- w|BIsin('HD+ B0.,sin(f-13+|J sin Ct-p) - W JBo.,-smCf-.4j]] i + WJA,.0 SU \ ( * -e) - Bo.» FA,, +FBM L+R, _sin|+W cos|j!-W A,.o cos(P+|->0 + A„«>s^-^) A,.o s i a ( ) 3 - f - 8 i , co*(pHj [A „ O Sin Bo., P)] D16 f) Support P o i n t s F i g u r e D7 Freebody Diagrams f o r Forces a t Supports No e x t e r n a l a p p l i e d f o r c e , P, i s shown i n the f r e e -body diagrams i n F i g u r e D7 s i n c e such f o r c e would have no e f f e c t on the s t r u c t u r e . The s e l f weight f o r c e i s however i n c l u d e d i n the c a l c u l a t i o n o f the r e a c t i o n components i n order t o pro v i d e an o v e r a l l check on the accuracy o f compu-t a t i o n . The e x t e r n a l f o r c e s a c t i n g on the e n t i r e s t r u c t u r e should be i n e q u i l i b r i u m and t h i s p r o v i d e s a t e s t f o r any e r r o r s i n the complex c h a i n o f c a l c u l a t i o n c a r r i e d out acro s s the s t r u c t u r e . C o n s i d e r i n g the support p o i n t A, the s e l f weight, f o r c e i s g i v e n by:-40 S W T = w £(A*.FAO.^ Qal D17 The v a r i o u s f o r c e s a c t i n g a t A may be r e s o l v e d h o r i z o n t a l l y and v e r t i c a l l y t o y i e l d the r e a c t i o n components. Thus:-A. - I [* D18 S i m i l a r equations may be d e r i v e d f o r the components a t support p o i n t B:-I t may perhaps f i n a l l y be p o i n t e d out t h a t the f o r c e system depends p r i m a r i l y on the shape of the s t r u c t u r e , r a t h e r than i t s span. I f s e l f - w e i g h t f o r c e s are n e g l e c t e d , the f o r c e s i n a l l members are completely independent of span s i z e and depend onl y on the angular shape of the s t r u c t u r e . The introduction of s e l f weight forces modifies t h i s since the weight of a structure increases with span. This e f f e c t may or may not be important depending on the r e l a t i v e magnitude of the applied external loadings and the weight of the structure supporting them. 42 APPENDIX E DATA ON SELECTED MICHELL CANTILEVERS A range of Miche l l cantilevers were analyzed while subjected to a variety of loads applied at the pinned j o i n t s . The choice of parameters and the conclusions reached from a study of th i s data are discussed i n Chapter 4. As explained therein, a l l structures were assumed to be cut from sheet CR39 one quarter inch thick. Thus three para-meters are common to a l l structures:-Youngs Modulus E = 300,000 p . s . i . S p e c i f i c Gravity p - 1.31 Thickness t = 0.25 inches. For a l l but one series of cal c u l a t i o n s , the s t r a i n was taken as 0.001, which therefore fixed the uniform stress as 300 p . s . i . The unit load applied i n a l l cases was one hundred pounds. This was an ar b i t r a r y selection but permits of rapid conversion of r e s u l t s to correspond with any other loading desired. The data i s presented as a series of tables i n the same sequence as the sections of Chapter 4. The tables present a summary of the information a c t u a l l y calculated • 43 s i n c e the sheer bulk of output prevents i t s i n c l u s i o n i n t h i s work. For example f o r each s t r u c t u r e i n v e s t i g a t e d , each member was c o n s i d e r e d s e p a r a t e l y and the f o l l o w i n g items c a l c u -l a t e d f o r i t : -member l e n g t h , c o o r d i n a t e s of i t s ends (before loading) d e f l e c t i o n s of i t s ends c r o s s - s e c t i o n a l area width second moment of area ( l e s s e r of two values) r a t i o of f o r c e i n member to E u l e r c r i t i c a l b u c k l i n g l o a d weight per i n c h L i t t l e of t h i s i n f o r m a t i o n , a p a r t from d e f l e c t i o n s , i s r e p r e s e n t e d here s i n c e i t can a l l be c a l c u l a t e d q u i t e simply when r e q u i r e d . A p r i n t o u t of a l l the data may be obtained from the author on r e q u e s t . TABLE 1 - Symmetrical C a n t i l e v e r s - End Load T h i s t a b l e r e l a t e s to F i g u r e 4.1 and l i s t s data f o r s t r u c t u r e s having the f o l l o w i n g p r o p e r t i e s : -SPAN L = 10" Support Spacing D = 1.0", 2.0" 2.5", 5.0", 10.0" Number of F i b r e s N = 2, 3, 4, 5, 7, 10, 20 In each case r e a c t i o n s , d e f l e c t i o n s , volumes and weights are shown wit h and without the e f f e c t of s e l f weight. the 'with s e l f weight' figure being given f i r s t i n each case, i n upright p r i n t . The 'weightless' values follow i n i t a l i c s . TABLE 2 - 5 Fibre Symmetrical Cantilever - End Point Load This table also relates to Figure 4.1 and concerns a second series of cantilevers of varying span but constant ~ r a t i o . A l l had f i v e f i b r e s but the other properties varied as follows:-60", 40", 20", 10", 5", 4". 15.0", 10.0", 5.0", 2.5", 1.25", 1.0". 4 } i n a l l cases. 5' TABLE 3 - Theoretical Optimum Structure These structures are the t h e o r e t i c a l optimum or minimum volume structures corresponding to each set of Michell frameworks considered i n Tables 1 and 2. The quan-t i t i e s quoted i n t h i s table are calculated from equation 1.20 using values derived from Figure 4.2. These values are plotted as the ultimate minimum volumes i n figures 4.4 to 4.4e. L -D -L D N = TABLE 1 DATA FOR SYMMETRICAL MICHELL CANTILEVERS WITH 1 0 0 POUND LOAD AT J , > t = 0 . 2 5 , a = 3 0 0 p . s . i . , E = 3 0 0 , 0 0 0 p . s . i . Note - Figures i n i t a l i c s r e f e r to 'weightless' s t r u c t u r e s . L L D N 9 Radius Deflec- Reactions Volume Weight S p e c i f i c D ion J ^ N A B Volume 1 1 0 .0 1 0 . 0 2 2 2 2 . 6 1 9 9 7 . 0 7 1 1 0 . 0 2 4 5 - 1 0 0 . 1 8 5 9 5 0 . 1 9 3 4 -100.0000 49. 9999 1 0 0 . 1 8 5 9 5 0 . 1 9 3 5 99.9999 50.0000 8 . 1 8 0 7 8.1667 0 . 3 8 7 0 0.3863 3 . 4 7 0 8 3. 4648 3 1 1 . 3 5 3 2 0 . 0 2 4 3 - 1 0 0 . 1 8 5 0 5 0 . 1 9 2 1 1 0 0 . 1 8 5 0 5 1 . 1 9 2 1 8 . 1 2 4 3 0 . 3 8 4 3 3 . 4 4 6 8 3 — -100.0003 49.9999 100.002 50.000 8.1106 0.3837 3.4410 4 7 . 5 7 4 1 0 . 0 2 4 3 - 1 0 0 . 1 8 5 1 5 0 . 1 9 1 8 1 0 0 . 1 8 5 1 5 0 . 1 9 1 9 8 . 1 1 4 1 0 . 3 8 3 8 3 . 4 4 2 5 4 -100.0006 49.9999 100. 0005 50. 0000 8.1004 0.3832 3.4367 5 5 . 6 8 1 9 0 . 0 2 4 3 - 1 0 0 . 1 8 4 5 5 0 . 1 9 1 7 1 0 0 . 1 8 4 5 5 0 . 1 9 1 8 8 . 1 1 0 4 0.3 8 3 7 3 . 4 4 0 9 5 — -100.0001 49.9999 100.0000 50.0000 8. 0968 0.3830 3.4352 7 3 . 7 8 8 6 0 . 0 2 4 3 - 1 0 0 . 1 8 5 0 5 0 . 1 9 1 6 1 0 0 . 1 8 5 0 5 0 . 1 9 1 7 8 . 1 0 7 9 0 . 3 8 3 6 3 . 4 3 9 9 7 — -100. 0006 49.9999 100. 0006 50.0000 8. 0943 0.38 29 3.4341 1 0 2 . 5 2 5 9 0 . 0 2 4 3 - 1 0 0 . 1 8 4 9 5 0 . 1 9 1 4 1 0 0 . 1 8 4 9 5 0 . 1 9 1 5 8 . 1 0 6 8 0 . 3 8 3 5 3 . 4 3 9 4 1 0 -100.0007 49.9998 100.0006 49.9999 8.0931 0.3829 3.4336 2 0 1 . 1 9 6 5 . - 0 . 0 2 4 3 -100..1847 5 0 . 1 9 1 0 , 100.. 1 8 4 6 50 . 1 9 1 0 8 . 1 0 5 9 . . 0 . 3 8 3 5 3 . 4 3 9 0 2 0 ' -100.000 6 49.9996 100.0006 49.9996 ' 8.0923 ' 0.3828 3.4333 (Table 1 - continued) L D L D N 9 Radius D e f l e c t i o n JNN Reactions Volume Weight S p e c i f i c Volume A B 2 10.0 5.0 2 46.3972 3.5355 0.0377 -200.5097 200.5097 12 .5873 0 .5954 10.6808 50.2976 50.2977 2 - -199.9992 199.9992 12.5594 0.5941 10. 6571 49.9999 50.0000 3 23.9915 0.0357 -200.4681 200 .4680 11.9107 0.5634 10.1067 50.2815 50.2817 3 — -200.0001 200.0000 11.8880 0.5624 10.0874 49. 9998 50.0000 4 16.0919 0.0353 -200.4631 200.4631 11.7966 0 .5580 10.0098 50.2787 50.2788 4 - -200.0010 200.0020 11.7 746 0.5570 9.9912 49.9998 49.9999 5 12.0946 0.0352 -200.4613 200.4612 11.7574 0.5562 9.9766 50.2776 50.2778 5 - -200. 0011 49.9998 11. 7356 0.5552 9.9581 49.9997 49.9998 7 8.0753 0.0351 -200.4587 200.4587 11.7295 0.5549 9 .9529 50.2770 50.2771 7 - -199.9998 199.9998 11 .707 8 0.5538 9.9345 49.9996 49.9997 10 5.3871 0.0351 -200.4603 200 .4603 11.7173 0.5543 9 .9426 50.2766 50.2767 10 — -200.0022 200.0021 11. 6958 0.5533 9.9243 49.9997 49.9998 20 2.5528 0.0351 -200.4588 200 .4588 11.7095 0 .5539 9.9359 50.2750 50.2760 - 20 — -200.0014 200.0014 11.6880 0.5529 9.9177 4 10.0 2.5 2 64.9424 1.7678 0.0618 -401.5833 401.5835 20 .6597 0 .9773 35.0600 50.4886 50 .4888 2 — -400.0005 400.0002 20.5872 0. 9739 34.9370 50.0000 50.0002 3 35.8997 0.0520 -401.1079 401.1079 17.3714 0.8218 29.4797 50.4107 50 .4110 3 — -399.9812 399.9812 17.3332 0.8200 29.4149 49.9999 50.0002 (Table 1 - continued) L D L D N 6 Radius D e f l e c t i o n JNN Reactions Volume Weight S p e c i f i c Volume A B 4 24 .3895 0.0506 -401.0779 401.0779 16 .8940 0.7992 28.6695 50.3993 50.3996 o 4 - -399.9990 399.9990 16.8592 0 . 7975 28.6105 . 49.9998 50.0001 5 18 .4137 0.0501 -401.0662 401.0662 16 .7338 0.7916 28.3977 50.3952 50.3955 5 - -400.0022 400.0024 16.7000 0. 7900 28. 3403 49.9997 50.0000 7 12 .3342 0.0498 -401.0549 401.0552 16 .6213 0.7863 28.2068 50.3922 50.3925 7 - -400.0017 400.0015 16.5882 0. 7 847 28.1506 49.9995 49.9997 10 8 .2402 0.0496 -401.0552 401.0552 16 .5721 0.7840, 28.1233 50.3906 50.3909 10 - -400.0063 400.0063 16.5394 0. 7 824 28.0678 49.9993 49.9996 20 3 .9084 0.0495 -401.0444 401.0444 16.5410 0.7825 28.0705 50.3891 50.3894 20 0 -399.9990 399.9990 16.5084 0. 7809 28.0152 49.9989 49.9992 5 10.0 2.0 2 69 .3904 1.4142 0.0730 -502.3442 502 .3442 24 .4237 1.1554 51.8110 50.5774 50.5777 2 - -499.9978 499.9978 24.3203 1.1505 51.5916 49.9999 50. 0002 3 39 .4075 0.0581 -501.4792 501.4792 19 .4195 0.9186 41.1954 50.4591 50 .4595 3 - -500.0000 500.0002 19.3748 0.9165 41.1006 49.9998 50.0002 4 26 .9230 0.0561 -501.3994 501.3994 18.7345 0 .8862 39.7423 50.4427 50 .4432 4 - -500.0032 500.0034 18. 6948 0.8844 39. 6580 49.9997 50.0002 (Table 1 - continued) L D 1 D N 6 Radius D e f l e c t i o r JNN Reactions Volume Weight S p e c i f i c Volume A B 5 20.3674 0.0554 -501.3752 501.3752 18 .5071 0.8755 39 .2599 50.4372 50 .4375 5 - -500.0042 500.0042 18.4689 0. 8737 39.1788 49.9997 50.0000 7 13.6627 0.0549 -501.3533 501.3535 18 .3480 0 . 8680 38 .9224 50 .4329 50.4333 7 - -500.0000 499.9998 18.3108 0. 8662 38. 8434 49. 9994 49.9998 10 9.1336 0.0547 -501.3533 501.3535 18 .2787 0.8647 38.7754 50.4308 50 .4311 10 - -500.007 3 500.0071 18.2419 0. 8629 38.6973 49.9941 49.9995 20 4.3338 0.0546 -501.3398 501.3398 18 .2353 0.8626 38.6833 50.4282 50 .4286 20 - -500.0000 500.0000 18.1987 . 0.8609 38.6056 49.9983 49.9987 2 79.1194 0.1259 -1008.5251 1008 .5256 42 .3208 2.0020 179.5537 51.0006 51.0011 2 - - 999.9873 999.9873 41 . 9809 1.9859 17 8.1116 49.9993 50.0000 3 49.1555 0.0796 -1003.3850 1003.3850 26.6088 1.2587 112.8927 50 .6248 50.6291 3 - -1000.0005 1000.0005 26.5373 1.2554 112.5893 49.9993 49.9998 4 34.2914 0.0745 -1003.0139 L003.0139 24.8859 1.1772 105.5829 50.5876 50.5883 4 - -1000.0012 1000.0015 24. 8283 1.1745 105.3386 49.9994 50.0001 10 10.0 1.0 5 26.1422 0.7071 0.0728 -1002.9141 L002 .9143 24.3336 1.1511 103.2397 50 .5739 50 .6748 5 . - -1000.0001 1000.0061 24. 27 99 1.1486 103.0119 49. 9990 49.9998 -(Table 1 - continued) L D L D N 0 Radius D e f l e c t i o n Reactions Volume Weight S p e c i f i c Volume JNN A B 7 17.6353 0.0717 -1002.8503 50.5648 1002 .8501 50.5657 23.9534 1.1331 101.626 6 7 -1000.0090 49.9991 1000.0093 49.9997 23.9021 1.1307 101.4090 10 11.8193 0.0712 -1002.8147 50.5599 1002.8147 50.5606 23.7877 1.1253 100.9236 10 -1000.0032 49. 9984 1000.0034 49.9991 23.7374 1.1229 100.7102 20 5.6171 0.0709 -1002.8008 50.5537 1002 .8008 50.5544 23 .6857 1.1205 100.4909 20 --1000. 0105 49.9966 1000.0107 49.9973 23.6361 1.1181 100. 2804 TABLE 2 Data For Symmetrical F i v e F i b r e M i c h e l l C a n t i l e v e r s of V a r i a b l e Span and F i x e d Support Load at J,-j. - 100 l b . t = 0.25, a = 300 p . s . i . , E = 300 ,000 p . s . i . L D L D N 8 Radius D e f l e c t i o n JNN Reactions Volume weight S p e c i f i c Volume A B Ins. Ins. Ins. Ins. l b s . l b s . c u . i n . l b 60.0 15.0 10.6066 1.3006 -406 .4343 406 .4341 101.4232 4 .7979 28.6868 52.3984 52.3987 -400. 0.022 400.0024 100. 2003 4.7400 28.3409 49.999? 50.0000 40.0 10 .0 7.0711 0.2004 -402.2778 402 .2776 67 .3423 3.1857 28.5708 51.5923 51.5927 -400. 0022 40.0 .0024 66.8002 3.1600 28.3408 49.9997 50.0000 4 20.0 5.0 5 18.414 3.5355 0.1002 -402.1335 402.1335 33.5352 1.5864 28.4558 50.7927 50 .7930 -400. 0022 400.0024 33.4000 1.5800 28.3411 49.9997 50.0000 10.0 2.5 1.7678 0.0501 -401.0662 401.0662 16 .7338 0.7916 28.3977 50.3952 50 .3955 -400.0022 400.0024 16. 7000 0.7900 28.3403 49.9997 50.0000 5.0 1.25 0.8839 0.0251 -400.5337 400 .5337 8.3585 0.3954 28 .3692 50.1971 50 .1974 -400.0022 400.0024 8.3500 0.3950 28.3403 49.9997 50.0000 4.0 1.0 0.7071 0.0200 -400.4272 400 .4275 6 .6854 0.3163 28 .3640 50.1576 50.1579 -400.0022 400.0024 6.6800 0.3160 28.3411 49.9997 50.0000 Ln o 51 T A B L E 3 • D a t a f o r C a n t i l e v e r s w i t h I n f i n i t e F i b r e N e t w o r k s b u t  W i t h F a n A n g l e s E q u a l t o t h e T w e n t y F i b r e M i c h e l l  S t r u c t u r e s D e t a i l e d i n P r e v i o u s T a b l e L D 9 o 6 o P L r 1 1.1965 22.7335 8.0927 3.43342 2 2.5528 48.5032 11.6835 9.91390 4 3.9084 74.2596 16 .4826 27.9714 5 4.3338 82.3422 18.1571 38.517 4 10 5.6171 106.725 23.5147 99.7653 . F o r a l l s t r u c t u r e s , N = 20, L = 10 .0", t = 0.25, a = 300 p . s . i . L o a d ( P T ) = 100 l b . D a t a f o r S e r i e s o f C a n t i l e v e r s W i t h I n f i n i t e F i b r e  N e t w o r k s E q u i v a l e n t t o F i v e F i b r e M i c h e l l  C a n t i l e v e r s W i t h V a r i a b l e S p a n  D e t a i l e d P r e v i o u s l y L L D e 6 m ? L r r V m 60 4 18.414° 73.656 27.8624 10.6066 98.5084 40 4 18 .414° 73.656 27.8624 7.0711 65.6726 20 4 18.414° 73.656 27.8624 3.5355 32.8358 10 4 18.414° 73.656 27 .8624 1.7678 16.4183 5 4 18.414° 73.656 27.8624 0.8839 8.2092 4 4 18.414° 73.656 27.8624 0.7071 6 .5672 N o t e : V = 1.6418 L i n t h i s c a s e . 52 TABLES 4 and 5 - Symmetrical C a n t i l e v e r With T i l t e d End Load The s t r u c t u r e and l o a d i n g are shown i n F i g u r e 4.6. The framework has the f o l l o w i n g measurements:-Span L =10.0". Support Spacing D =2.5". fr N = 5 . Load 100 l b at L (Jg 5) u n c l i n e d a t an angle T to the y a x i s . T v a r i e s from -45° to +90° i n steps o f 5°. Table 4 summarizes the r e a c t i o n s , volume, weight and d e f l e c t i o n o f L f o r t h i s range of l o a d i n g s , w hile Table 5 i l l u s t r a t e s the mass of data t h a t i s d e r i v e d i n q u i t e simple i n v e s t i g a t i o n s . T h i s t a b l e l i s t s f o r each member, the a x i a l f o r c e f o r each angular p o s i t i o n of the l o a d . Note t h a t a l - -though the s t r u c t u r e i s symmetrical the l o a d i n g i s not and thus the f o r c e i n member A ^ i s not equal n u m e r i c a l l y t o t h a t i n B, . ba These f o r c e s are equal f o r o n l y two l o a d i n g s i n t h i s t a b l e , when the t i l t i s zero and 90°, s i n c e these are symmet-r i c a l p o s i t i o n s . TABLE 4 SYMMETRICAL MICHELL CANTILEVERS WITH SKEW END LOADING N = 5, L = 10 . 0 " , 2.5", E = 300,000 p . s . i . , a = 300 p . s . i . , £ = 0.001, S p g t y 1.31. D e f l e c t i o n 0.00549", L o a d 100 p o u n d s a t v a r i o u s p o i n t s D i r e c t i o n of l o a d A . H o r . A . V e r t . j R e a c t i o n s B.Hor. V o l u m e W e i g h t D e f l e c t i o n s -45 -318.916 3.695 248.190 67.651 11.8308 0.5597 * 40 -339.341 9.579 275.049 67.717 12.8172 0 .6063 35 -357.184 15.389 299.813 67.267 13.7060 0 .6484 30 -372.308 21.082 322.297 66.306 14 .4906 0 .6855 25 -384.599 26.614 342.328 64.839 15.1649 0.7174 20 -393.962 31.945 359.753 62.880 15.7237 0.7438 15 -400.328 37.032 374.441 60.441 16.1629 0.7646 10 -403.647 41.837 386.278 57 .543 16 .4791 0.7796 - 5 -403.893 46.323 395.176 54 .207 16 .6699 0 .7886 0 -401.066 50.395 401.066 50.396 16.7338 0.7916 + 5 -395.187 54.208 403.904 46 .325 16 .6703 0.7886 0 .0 , 0.0501 10 -386.300 57.545 403.668 41.840 16.4800 0.7796 15 -374.473 60.445 400.360 37.036 16.1643 0.7647 20 -359.795 62.885 394.005 31.950 15.7255 0.7439 25 -342.380 64.846 384.651 26 .621 15.1671 0.7175 30 -322.359 66.313 372.370 21.090 14.4932 0.6856 35 -299.885 67.276 357.255 15.398 13 .7090 0 .6485 40 -275.128 67.727 339.421 9 .589 12 .8205 0.6065 45 -248.278 67.662 319.003 3.707 11.8344 0.5598 50 -219.533 67.083 296.158 - 2.204 10.7583 0.5089 55 -189.127 65.993 271.059 - 8.097 9.7324 0 .4604 -0 .0170 , 0.0265 60 -157.277 64.407 243.898 -13.929 9.3453 0.4421 -0 .0170 , 0.0265 65 -124.229 62.331 214.880 -19.655 8.8872 0.4204 -0 .0170 , 0.0265 70 - 90.237 59.780 184.226 -25.231 8.4026 0 .3975 -0 .0186 , 0.0225 75 - 55.557 56.774 152.171 -30.615 7.9381 0.3755 --0 .0186 , 0.0225 80 - 20.455 53.336 118.957 -35.767 7 .4691 0.3533 -0 .0193 , 0.0193 85 14.803 49.493 84.838 -40V645 "7 .0708 0.3345 -0 ; o 2 o i , 0.0132 90 49.948 45.274 50.073 • -45.205. 6.9496 . -0-.3286 -0 .0208 , 0.0 U ) TABLE 5 Forces i n Members of a F i v e F i b r e Symmetrical M i c h e l l C a n t i l e v e r s Subjected to a One Hundred Pound Load a t V a r i o u s Angles T i l t Volume Forces' In 'A' Members A 5 4 A 5 3 a 52 A 5 1 A 5 0 A 4 4 A 4 3 A 4 2 A 4 1 -45 11.831 104 110 116 122 123 5.63 17.5 30.6 45.1 -40 12.817 105 111 117 123 125 8.63 20.8 34 .2 49.0 -35 13.706 105 111 117 123 125 11.6 23.9 37.5 52 .6 -30 14.491 105 111 117' 123 124 14.4 26.9 40.6 55.7 -25 15.165 104 110 115 121 123 17.2 29.6 43.3 58.4 -20 15.724 101 107 113 119 120 19.8 32.1 45.7 60.7 -15 16.163 98.6 104 110 115 117 22.2 34 .4 47 .8 62.5 -10 16.479 94.9 100 106 111 113 24.5 36.4 49 .5 63.9 - 5 16 .670 90.6 95.5 101 106 107 26.6 38.1 50.8 64 .7 Zero 16.734 85.5 90.2 95.0 100 101 28.5 39.6 51.7 65.1 + 5 16.670 79.8 84.2 88.7 93.5 94.7 30.2 40.7 52 .3 64 .9 10 16.480 73.5 77.5 81.7 86.1 87.2 31.7 41.5 52 ;4 64.3 15 16.164 66.7 70.3 74.1 78.1 79.1 32.9 42.1 52.1 63.2 20 15.726 59.3 62.5 65.9 69.4 70.3 33.8 42.3 51.5 61.6 25 15.167 51.5 54.3 57.2 60.3 61.1 34.5 42.1 50.4 59.5 30 14.493 43.2 45.6 48.1 50.7 51.3 35.0 41.7 49.0 57.0 35 13.709 34.7 36 .6 38.6 40.6 41.2 35.2 40.9 47.2 54 .0 40 12.821 25.9 27.3 28.8 30.3 30.7 35.1 39 .8 45^0 50.7 45 11.834 16.9 17.8 18.8 19.8 20.0 34.7 38.5 42.5 46.9 50 10 .758 7.76 8.18 8.62 9 .08 9.20 34.1 36 .8 39 .7 42 .8 55 9 .732 -1.44 -1.52 -1.60 -1.68 -1.71 33.2 34 .8 36 .6 38.4 60 9.345 -10.6 -11.2 -11.8 -12.6 -12.6 32.1 32 .6 33.2 33.6 65 8.887 -19 .7 -20.8 -21.9 -23.1 -23.4 30.7 30.2 29.5 28 .7 70 8.403 -28.7 -30 .2 -31.9 -33.6 -34 .0 29 .1 27 .5 25.6 23 .4 75 7.938 -37.4 -39.4 -41.6 -43.8 -44.4 27 .2 24 .6 21.5 18.1 80 7.469 -45.9 -48.3 -51.0 -53 .7 -54 .4 25.2 21.5 17 .3 12.6 85 7 .071 -54 .0 -56 .9 -60 .0 -63.2 -64 .0 23.0 18 .2 12.9 6 .94 90 6 .946 -61.6 -65.0 -68.5 -72.2 -73.1 20.6 14.8 8.42 1.27 55 Table 5 (continued) A40 A34 A33 A32 A31 A30 A24 A23 A22 A21 A20 52.0 5.93 19 .1 34.9 53.7 63.1 6 .25 20.8 39 .6 63.4 75 .7. 56.1 9.10 22.9 39 .4 59.0 68.7 9.58 25.1 45.1 70.3 83.3 59.7 12 .2 26.5 43.6 63.8 73.8 12.8 29.3 50.3 76.7 90.2 62.8 15.2 29.9 47.5 6 8.1 78.3 16.0 33.2 55.1 82.5 96 .4 65.5 18.1 33.1 51.0 72.0 82.3 19.1 36.9 59 .5 87.6 102 67.7 20.9 36.0 54.1 75.3 85.6 22.0 40.3 63.4 92.1 107 69.3 23.4 38.7 56.8 78.0 88.3 24.7 43.4 66.9 95.9 110 70.5 25.8 41.1 59.1 80.1 90.2 27 .2 46 .2 69 .8 99 .0 114 71.1 28.1 43.1 60.9 81.6 91.5 29 .6 48.6 72.2 101 116 71.1 30.1 44.9 62 .3 82.5 92.2 31.7 50.6 74.1 103 117 70.6 31.8 46.3 63.1 82.7 92.1 33.5 52.3 75.4 104 117 69.6 33.4 47.3 63.6 82.4 91.3 34.2 53.5 76.1 104 117 68.1 34.6 48.0 63.5 81.4 89.8 36.5 54 .4 76 .3 103 116 66.0 35.7 48.3 62.9 79.7 87 .6 37.6 54 . 8 75.8 101 113 63.4 36.4 48.2 61.9 77 .5 84.8 38.4 54 .9 74 .8 98 .7 110 60.4 36.9 47.8 60.4 74.7 81.3 38.8 54.5 73.2 95.6 106 56.8 37.1 47 .0 58.4 71.3 77.2 39 .0 53.7 71.1 91.7 102 52.9 37.0 45.9 56.0 67.4 72.5 38.9 52 .5 68.4 87.2 96.0 48.5 36.6 44 .4 53.2 62.9 67.2 38.5 50.8 65.2 81.9 89 .8 43.8 35.9 42.6 49 .9 58 .0 61.5 37 .9 48.8 61.5 76.1 82 .8 38.8 35.0 40.4 46.3 52.7 55.3 36.9 46.5 57 .4 69 .7 75 .2 33.4 33.8 38 .0 42 .3 46.9 48.6 35.6 43.7 52 .8 62.7 67.1 27.8 32.3 35.2 38.0 40.8 41.6 34 .1 40.7 47.7 55 .3 58 .4 22.0 30.6 32.2 33.4 34 .4 34 .3 32 . 3 37 .3 42.4 47 .4 49 .3 16.0 28.7 28.9 28.6 27.7 26.7 30.3 33.7 36.7 39 .2 39.9 9.92 26.6 25.4 23.5 20.8 18.9 28.0 29.7 30 .7 30.7 30 .1 3.74 24.2 21.8 18.3 13.7 11.0 25.5 25 .6 24.5 22 .0 20 .1 -2.46 21.7 17.9 12 .9 6.54 2.94 22.8 21.3 18.1 13.1 9.91 56 Table 5 (continued) A14 A13 A12 A l l A10 •. B45 B35 B25 B 1 5 B05 B44 3.21 10.8 21.0 34.3 41.3 -16 .9 -17.8 -18.7 -19 .8 -20.0 -34.7 4.92 13.1 24.1 38.2 45.6 -25.9 -27.3 -28.8 -30.3 -30.7 -35.1 6.60 15.4 26.9 41.8 49.5 -34.7 -36 .6 -38.6 -40.6 -41.2 -35.2 8.22 17.5 29.6 45.1 53.1 -43.2 -45.6 -48.0 -50.6 -51.3 -35.0 9.78 19.4 32.0 48.0 56.3 -51.5 -54.2 -57.2 -60.3 -61.0 -34.5 11.3 21.3 34.2 50.6 59.0 -59.3 -62.9 -65.9 -69.4 -70.3 -33.8 12.7 22.9 36.1 52.8 61.3 -66 .7 -70.3 -74.1 -78.1 -79 .1 -32 .9 14.0 24.4 37.8 54.6 63.1 -73.5 -77.5 -81.7 -86 .1 -87 .2 -31.7 15.2 25.7 39.2 56 .0 64.4 -79.8 -84.2 -88.7 -93.5 -94 .7 -30.2 16.3 26.8 40.2 56.9 65.3 -85.5 -90.2 -95.0 -100 -101 -28.5 17.2 27.7 41.0 57.4 65.6 -90.6 -95.5 -101 -106 -107 -26 .6 18.1 28.4 41.4 57.5 65.5 -94.9 -100 -106 -111 -113 -24 .5 18.7 28.9 41.6 57.1 64.9 -98.6 -104 -110 -115 -117 -22.2 19 .3 29 .2 41.4 56.3 63.7 -101 -107 -113 -119 -120 -19 .8 19.7 29.2 40.8 55.1 62.1 -104 -109 -115 -121 -123 -17 .2 19 .9 29 .0 40.0 53.5 60 . 0 -105 -111 -117 -123 -124 -14.4 20.0 28.6 38.9 51.4 57.4 -105 -111 -117 -123 -125 -11.6 20.0 28.0 37 .5 49 .0 54.5 -105 -111 -117 -123 -125 -8.64 19.8 27.1 35 . 8 46.1 51.1 -104 -110 -116 -122 -123 -5.63 19 .4 26 .1 33.9 43.0 47.3 -102 -108 -114 -120 -121 -2.59 18.9 24.8 31.6 39.5 43.1 -100 -105 -111 -117 -118 + 0.48 18.3 23.4 29.2 35.7 38.6 -96.2 -101 -107 -113 -114 + 3.54 17.5 21.8 26.5 31.6 33.9 -92.0 -97 .0 -102 -108 -109 + 6 .58 16.6 20 . 0 23.6 27.3 28.8 -87.2 -91.9 -96 .9 -102 -103 +9 .56 15.5 18.1 20.5 22.8 23.6 -81.7 -86.1 -90.8 -95.7 -96.9 +12 .5 14.4 16.0 17.3 18.1 18.1 -75.6 -79.7 -84 .0 -88.5 -89.7 +15 .3 13.1 13.8 13.9 13.3 12.6 -68.9 -72.6 -76.6 -80.7 -81.7 +18.0 11.7 11.5 10.5 8.35 6 .91 -61.7 -65.0 -68.5 -72.2 -73.2 +20 .6 57 Table 5 (continued) B34 B24 B14 B04 B 4 3 _ B33 B23 B13 B03 B42 B32 -38.5 -42.5 -46.9 -48.5 -36.6 -44.4 -53.1 -62.9 -67.2 -38.5 -50.8 -39 . 8 -45.0 -50.7 -52.9 -37.0 -45.9 -56.0 -67.4 -72.5 -38.9 -52 .4 -40.9 -47.2 -54.0 -56.8 -37.0 -47.0 -58.4 -71.3 -77 .2 -39 .0 -53.7 -41.7 -49.0 -57.0 -60.4 -36.9 -47.8 -60.4 -74.7 -81.3 -38.8 -54 .5 -42.1 -50.4 -59.5 -63.4 -36.4 -48.2 -61.9 -77.5 -84.7 -38.4 -54.9 -42.3 -51.5 -61.6 -66.0 -35.7 -48.3 -62.9 -79 .7 -87 .6 -37 .6 -54 .8 -42.1 -52.1 -63.2 -68.1 -36.6 -48.0 -63.5 -81.4 -89.8 -36 .5 -54.4 -41.5 -52.4 -64.3 -69.6 -33.4 -47.3 -63.6 -82.4 -91.3 -35.2 -53.5 -40.7 -52.3 -64.9 -70.6 -31.8 -46.3 -63.1 -82.7 -92.1 -33.5 -52 .3 -39.6 -51.7 -65.1 -71.1 -30.1 -44 .9 -62.3 -82.5 -92.2 -31.7 -50.6 -38.1 -50.8 -64 .7 -71.1 -28.1 -43.1 -60.9 -81.6 -91.5 -29 .6 -48.6 -36.4 -49.5 -63.9 -70.5 -25.8 -41.1 -59.1 -80.1 -90.2 -27.2 -46 .2 -34.4 -47.8 -62.5 -69.3 -23.4 -38.7 -56 .8 -78.0 -88.3 -24.7 -43.4 -32.1 -45.7 -60.7 -67.7-20.8 -36.0 -54.1 -75.3 -85.6 -22.0 -40.3 -29 . 6 -43.3 -58 .4 -65.5 -18.1 -33.1 -51.0 -72.0 -82.3 -19 .0 -36 .9 -26.9 -40.6 -55.7 -62.8 -15.2 -29 .9 -47 .5 -68.2 -78.4 -16.0 -33.2 -23.9 -37.6 -52.6 -59.7 -12.2 -26.5 -43.6 -63.8 -73.8 -12.9 -29.3 -20.8 -34.2 -49.0 -56.1 -9 .10 -22.9 -39.4 -59.0 -68.7 -9.60 -25.1 -17.5 -30.6 -45.1 -52.0 -5.94 -19.1 -34.9 -53.7 -63.1 -6.26 -20.8 -14.1 -26 .8 -40.9 -47.6 -2.73 -15.1 -30.1 -48.0 -57.0 -2.87 -16 .3 -10.6 -22 . 8 -36 . 3 -42 . 8 + 0.51 -11.1 -25.1 -42.0 -50 .5 + 0.53 -11.6 -6.95 -18.6 -31.5 -37.7 + 3.73 -6.94 -20.0 -35.6 -43.6 + 3.93 -6 .90 -3.30 -14.2 -26.4 -32.3 +6.93 -2 .75 -14.6 -29.0 -36 .3 + 7 .30 -2.13 + 0.39 -9.80 -21.1 -26.7 + 10.1 +1.47 -9.17 -22.1 -28.8 +10.6 +2.66 + 4.07 -5.27 -15.6 -20.8 +13.1 +5.67 -3.65 -13.1 -21.0 +13.9 + 7.43 + 7.72 -0.71 -10.1 -14.8 +16 .1 + 9.83 + 1.89 -7.91 -13.1 +17.0 + 12 .1 + 11.3 + 3.86 -4.43 -8.63 +19 .0 +13.9 + 7.43 -0.70 -5.13 +20.0 + 16 .8 + 14.8 + 8.40 + 1.25 -2.49 +21.7 + 17.9 +12.9 + 6 .51 +2.91 +22.8 +21.3 Table 5 (continued) B22 B12 B02 B41 B31 B21 B l l B01 -65.2 -81.9 -89.7 -19.8 -27.1 -35.8 -46.1 -51.0 -68.4 -87.1 -96.0 -20.0 -28.0 -37 .5 -48.9 -54.4 -71.1 -91.7 -102 -20.0 -28.6 -38.9 -51.4 -57 .4 -73.2 -95.5 -106 -19 .9 -29 .0 -40.1 -53.4 -60.0 -74.8 -98.7 -110 -19.7 -29 .2 -40.9 -55.1 -62.1 -75.8 -101 -113 -19.3 -29.1 -41.4 -56 .3 -63.7 -76.3 -103 -116 -18.7 -28.9 -41.6 -57 .1 -64.9 -76.1 -103 -117 -18.0 -28.4 -41.4 -57.5 -65.5 -75.4 -104 -117 -17.2 -27 .7 -41.0 -57 .4 -65.6 -74.1 -103 -117 -16.3 -26.8 -40.2 -56 .9 -65.3 -72.2 -101 -116 -15.2 -25.7 -39.2 -56 .0 -64.4 -69.8 -99.0 -114 -14.0 -24.4 -37 .8 -54.6 -63.1 -66.9 -95.9 -110 -12.7 -22.9 -36.1 -52.8 -61.3 -63.4 -92.1 -107 -11.3 -21.3 -34 .2 -50.6 -59 .0 -59.5 -87.6 -102 -9.79 -19 .4 -32.0 -48.1 -56.3 -55.1 -82.5 -96.4 -8.22 -17 .5 -29.6 -45.1 -53.1 -50.3 -76.7 -90.2 -6.60 -15.4 -26 .9 -41.8 -49 .5 -45.1 -70.4 -83.3 -4.92 -13.1 -24.1 -38.2 -45.6 -39.6 -63.5 -75.8 -3.21 -10.8 -21.0 -34.3 -41.3 -33.8 -56.1 -67.7 -1.48 -8.42 -17.8 -30.2 -36.7 -27.7 -48.3 -59 .0 +0.27 -5.95 -14.5 -25.8 -31.8 -21.4 -40.1 -50.0 + 2 .02 -3.44 -11.1 -21.2 -26.7 -14.9 -31.6 -40.5 + 3.75 -0.90 -7.52 -16 .5 -21.3 -8.33 -22 .9 -30.7 +5.45 + 1.65 -3.93 -11.6 -15.9 -1.69 -14.0 -20.8 +7 .11 + 4.18 -0.31 -6 .68 -10.2 + 4.96 -5.0 -10.6 + 8.72 +6.68 + 3.31 -1.67 -4.56 + 11.6 + 4 .04 -0.37 + 10 .3 + 9 .13 + 6 .91 + 3.34 + 1.17 +18.1 + 13.0 +9.87 +11.7 +11.5 + 10.5 + 8.33 + 6.89 59 TABLE 6 - Load a t Each J o i n t o f a Symmetrical C a n t i l e v e r The p r e v i o u s t a b l e s have d e a l t w i t h c o n t i l e v e r s sub-j e c t e d t o an end l o a d . Table 6 i t e m i z e s the f o r c e i n each member of a f i v e fik>re c a n t i l e v e r , s u b j e c t e d to a one hundred pound l o a d p l a c e d i n s u c c e s s i o n a t each j o i n t . In a d d i t i o n the d e f l e c t i o n s o f t h a t j o i n t and of the out e r end of the : s t r u c t u r e ( J 5 5 ) are recorded. I t w i l l be noted t h a t many members are unloaded i n . each of these s t r u c t u r e s , but because of t h e i r rigid-body, t r a n s l a t i o n s and r o t a t i o n s , they c o n t r i b u t e to the d e f l e c t i o n of the remainder of t h i s s t r u c t u r e . Forces and d e f l e c t i o n s f o r m u l t i p l e l o a d i n g s may be c a l c u l a t e d from t h i s t a b l e . For a g i v e n member, the t o t a l f o r c e a c t i n g along i t w i l l be the sum o f the i n d i v i d u a l f o r c e s caused by each l o a d c o n s i d e r e d s e p a r a t e l y . The c r o s s s e c t i o n a l area and oth e r data may then be c a l c u l a t e d from the known. > uniform s t r e s s . The d e f l e c t i o n of the s t r u c t u r e under m u l t i p l e l o a d i n g i s not determined by a d d i t i o n . I t w i l l be equal t o the d e f l e c -t i o n recorded f o r the l o a d most remote from the support p o i n t s , p r o v i d e d no other l o a d l i e s o u t s i d e the area e n c l o s e d by the members l e a d i n g to t h a t most remote l o a d . For example, suppose there are s e v e r a l loads a p p l i e d to the s t r u c t u r e shown i n F i g u r e 4.9, the most remote from the supports being a t J ^ c - I f a l l the other loads are at; j o i n t s TABLE 6 Load a t J o i n t Volume l54 A 52 A 51 A 50 A 44 43 42 5.1 1.9737 5.2 3.3446 5.3 6.3633 5.4 10.920 5.5 16.734 4.1 1.6251 4.2 2.9868 4.3 5.1224 4.4 7.8988 4.5 10.920 3.1 1.4238 3.2 2 .3823 3.3 3.6795 3.4 5.1224 3.5 6.3633 2.1 1.1326 2.2 1.7080 2.3 2.3823 2.4 2.9868 2.5 3.3446 1.1 0.8335 1.2 1.1326 1.3 , 1.4238 1.4 1.6251 1.5 1.9737 85.53 61.66 90.17 31.49 65.00 95.05 -1.93 33.19 68'.50 100 .2 -33.73 -1.93 33.62 69 .39 101.5 28.52 33.53 39.57 100.6 85.51 106.1 35 .11 42.19 51.73 61.65 90 .14 111.8 o Table 6 (continued) A40 A A34 A A33 A32 A31 A30 A24 A23 A22 A21 ————— A20 28.45 28.45 28.46 33.43 34 .89 36.97 36.78 40.70 42.75 37.02 46.60 51.04 - 39 .02 53.23 60.18 55.92 35.35 48.21 63.10 70.17 37 .26 54.73 75.99 86.52 71.12 30.06 44.88 62.26 82.48 92.15 31.67 50.62 74.10 102.8 117.0 -1.829 31.91 31.92 31.90 35.10 37.19 37.00 40.95 65.83 33.51 42.17 46.10 35.33 48.17 5 4.37 96.26 28.50 39.53 51.67 57 .29 30.04 44 .83 62.19 70.63 119.4 20.56 32.84 46.40 61.32 68.25 21.66 36 .89 54 .95 76.15 86.46 30.26 32.10 61.65 62.46 33.50 37.15 85.50 90.13 91.31 28.48 39.51 44.71 100 .6 106.1 111.8 113.3 20 .54 32.82 46.36 52 .78 105.4 111.1 117.1 123.5 125.1 10.49 22 .76 36 .32 51.28 58.37 59.26 85.50 86.63 100.6 106.0 107 .4 105 .4 111.1 117 .1 118.6 99.39 104 .8 110.4 116 .4 118.0 Table 6 (continued) * 1 4 A A13 A12 A l l A10 B45 B35 B25 B15 B05 B44 16.26 19.13 26.83 20.04 29.09 40.22 18.89 28.37 41.45 56.91 14.24 21.33 32.52 47.66 65.29 -85.53 -90.17 -95.05 -100.2 -101.5 -100.6 -28.52 11.12 15.42 19 .51 18.15 23.81 29.76 19.00 25.67 33.91 42.09 . 15.97 21.45 29.38 38.91 48.19 -61.66 -65.00 -68.50 -69.39 5.84 10.55 11.96 14.63 17.40 19.96 17.21 21.04 25.22 28.17 16.06 19.46 24.15 29.00 32.35 -31.49 -33.19 -33.62 -0. 31 5.11 5.06 10.01 10.81 10.73 13.88 15.45 16.82 16.70 14.51 15.92 18.00 19.64 19.51 +1.90 +1.93 88.90 99.70 88.91 100.3 99.71 88.92 90.60 100.3 99.72 88.93 70.73 89.45 99.02 98.45 87.80 +33.73 CTl Table 6 (continued) 1 Bs4 B24 B14 B04 B43 B33 B23 B13 B03 B42 B32 -99.39 -104.8 -105.4 ,| -111.1 -117.1 -123.5 -125.1 -10.49 -22.76 -106.1 -111.8 -117.9 -119.4 -20.56 -32.84 -46 .40 -61.32 -68 .25 -21.66 -36.89 -39.57 -51.73 -65.07 -71.12 -30.06 . -44.88 -62.26 -82.48 -92.15 -31.67 -50.62 -105.4 -100.6 -106.1 -111.8 -113.3 -20.54 -85.51 -90.14 -95.02 -96.26 -28.50 -39.53 -5167 -57.29 -30.04 -33.53 -42.19 -51.69 -55.92 -35.35 -48.21 -63.10 -70.17 -37.26 -85.50 -90.13 -91.31 -61.65 -64.99 -65.83 -33.51 -42.17 -46 .10 -35.11 -40.50 -42.75 -37.02 -46.60 -51.04 -61.65 -62.46 -31.49 -31.90 -35.10 -37.19 -33.10 -33.43 -34.89 -36 .97 -30 .26 +1.83 -31.91 -28.45 -28.45 (Ti C J Table 6 (continued) B22 B12 , ... B02 B41 B31 B21, B l l B01 Load at Joint' , -88.90 -88.91 -88.92 -88.93 -87.80 5.1 -110.4 -116.4 -118.0 +0.31 -5.06 -10.73 -16.70 -19.51 5.2 -36.32 -51.28 -58.37 -5.39 -11.96 ! -19 .53 128.17 -32.35 5.3 -54.95 ' -76.15 -86.46 -11.12 -19.51 -29 .76 -42.09 -48.19 5.4 -74.10 -102.8 -117.0 -16.26 -26.83 -40 .22 -56.91 -65.29 5.5 -99.70 -99.71 -99.72 -98.45 4.1 -111.1 -117.1 -118.6 -5.11 -10.81 -16.82 -19.64 4.2 -32.82 -46.36 -52.78 -10.55 -17 .40 -25.22 -29 .00 4.3 -44.83 -62.19 -70.63 -15.42 -23.81 -33.91 -33.91 4.4 -54.73 -75.99 -86.52 -19.13 -29.09 -41.45 -47.66 4.5 -100.3 -100.3 -99.02 3.1 -100.6 -106.0 -107.4 -10.01 -15.45 -18.00 3.2 -28.48 -39.51 -44.71 -14.63 -21.04 -24.14 3.3 -35.33 -48.17 -54.37 -18.15 -25.67 -29.38 3.4 -39.02 -53.23 -60.18 -20 .04 -28.37 -32.52 3.5 -90.60 -89.45 2.1 -85.50 -86.63 -13.88 -15.92 2.2 -33.50 -37.15 -17.21 -19.46 2.3 -37.00 -40.95 -19.00 -21.45 2.4 -36.78 -40.70 -18.89 -21.33 2.5 -70.73 1.1 -59.26 -14.51 1.2 -32.10 -16.06 1.3 -31.92 -15.97 1.4 -28.46 -14.24 1.5 •C-65 Table 6 (continued) Deflection at Loaded Joint Deflection at L i J r - r 0.001 0.006 0.0 0.011 0.001 0.010 -0.001 0.016 0.010 0.019 0.006 0.031 0.009 0.033 0.007 0.040 0.0 0.050 0.0 0.050 -0.001 0.005 0.0 0.008 0.004 0.009 0.0 0.018 0.004 0.015 0.0 0.024 0.0 0.024 0.0 0.030 -0.009 0.033 -0.007 0.040 0.001 0.004 0.0 0.009 0.001 0. 007 0.0 0.014 0.0 0.011 0.0 0.019 -0.004 0.015 0.0 0.024 -0.010 0.019 -0.006 0.031 0.0 0.003 0.0 0.006 0.0 0.005 0.0 0.010 -0.001 0.007 0.0 0.014 -0.004 0.009 0.0 0.018 -0.001 0.010 0.001 0.016 0.0 0.003 0.0 0.003 0.0 0.003 0.0 0.006 -0.001 0.004 0.0 0.009 0. 001 0. 005 0 . 0 0.008 -0.001 0.006 0.0 0.011 o J , such t h a t a < 3 and b < 5, the s t r u c t u r e d e f l e c t i o n w i l l ab — — ' be e x a c t l y the same as i f o n l y J - r - were loaded. If,however, one l o a d d i d not s a t i s f y these c o n d i t i o n s but was p l a c e d , say at J _ 0 , t h i s would modify the s t r u c t u r a l deformation. Although members A--, A- 3, A_ 4, A ^ , A ^ , A ^ , B33, B 3 4 , B ^ , , B j -and B ^ - are a l l u n s t r e s s e d , the d e f l e c t i o n of j o i n t J r . 2 due. to the loa d there would modify s l i g h t l y the d e f l e c t i o n s o f a l l the j o i n t s i n t h i s a r ea. In such cases the d e f l e c t i o n s would have to be c a l c u l a t e d to s a t i s f y the imposed l o a d i n g s . T h i s behaviour i s due to the requirement t h a t s t r e s s e s are uniform. Member c r o s s - s e c t i o n s are a d j u s t e d t o ma i n t a i n the s t r e s s , and thus the s t r a i n , c o n s t a n t . ; TABLE 7 - T i l t e d Three F i b r e Symmetrical C a n t i l e v e r T h i s t a b l e c o n t a i n s i n f o r m a t i o n r e g a r d i n g a three f i b r e c a n t i l e v e r p l a c e d i n the v e r t i c a l plane but t i l t e d a t v a r i o u s angles as shown i n F i g u r e 4.10. The c a n t i l e v e r dimensions are g e n e r a l l y s i m i l a r t o the f i v e f i b r e framework p r e v i o u s l y c o n s i d e r e d and are as f o l l o w s : -Span L Support Spacing D L D Number of F i b r e s N = 10" = 2.5" = 4 = 3. TABLE 7 3 FIBRE SYMMETRICAL MICHELL CANTILEVER WITH TILTED LOAD T i l t Angle Reaction at A Reaction at B Volume Weight D e f l e c t i o n s A H A v BH B v X y -45 -221.121 230.065 221.121 -129.484 12.2778 0 .5808 —1 -40 -252.364 227.373 252.365 -126.744 13.3023 0.6293 -35 -282.663 219.298 282.663 -118.625 14 .2256 0.6730 -30 -311.095 206.084 311.095 -105.372 15 .0407 0.7115 -25 -336.797 188.131 336.797 - 87.387 15.7414 0 .7447 -20 -358.986 165.986 358.986 - 65.214 16.3221 0.7721 -15 -376.986 140.322 376.986 - 39.528 16.7785 0 .7937 -10 -390.249 111.919 390.249 - 11.110 17.1070 0 .8093 - 5 -398.372 81.642 398.372 19.177 17.3051 0.8186 Zero -401.108 50.411 401.108 50.411 17.3714 0.8218 -+ 5 -398.372 19.177 398.372 81.642 17.3051 0 .8186 10 -390.249 - 11.110 390.249 111.920 17.1070 0 .8093 0 0.0520 15 -376.986 - 39.529 376.986 140 .323 16 .7785 0.7937 20 -358.985 - 65.215 358.985 165 .987 16 .3221 0 .7721 25 -336.797 - 87.387 336 .797 188.132 15.7414 0 .7447 30 -311.095 -105.372 311.095 206.084 15.0407 0.7115 35 -282.663 -118.625 282.662 219.298 14.2256 0.6730 40 -252.364 -126.744 252.364 227 .373 13.3023 0.6293 45 -221.121 -129.484 221.121 230.065 12 .2778 0.5808 50 -189.883 -126.764 189.883 227.292 11.1600 0 .5279 55 -159.599 -118.670 159.599 219 .141 9.9575 0.4710 60 -131.189 -105.451 131.189 205.862 8.6795 0.4106 65 -105.515 - 87.510 105.515 187.858 7 .7690 0.3675 70 - 83.354 - 65.394 83.354 165.680 7 .4223 0.3511 '75 - 65.378 •'- 39.7 80 ' 65.378 140.002 7.0193- 0.3320 -0'. 0165 0.019 7 80 - 52.130 - 11.449 52.130 111.605 6.5630 0 .3105 85 - 44.010 18.734 44.010 81.354 6.2115 0 .2938 -0. 0177 0.0110 90 - 41.259 49.854 41.259 50.172 6.1264 0.2898 -0. 0184 0 TABLE 8 5 FIBRE SYMMETRICAL MICHELL CANTILEVER WITH TILTED LOAD t i l t Angle Reaction at A Reaction a t B Volume Weight AH *V BH B v -45 -223.088 228.083 223.088 -127.524 11.8283 0 .5595 -40 -253.984 225.417 253.984 -124.811 12.8151 0.6062 -35 -283.945 217.426 283.945 -116.779 13.7043 0 .6483 -30 -312.061 204.355 312.061 -103.670 14.4894 0.6854 -25 -337.475 186.598 337.475 - 85.881 15.1641 0 .7173 -20 -359 .416 164.696 359.416 - 63.952 15.7234 0 .7438 -15 -377.214 139.314 377.214 - 38.550 16.1629 0 .7646 -10 -390.329 111.224 390 .329 - 10.445 16.4792 0.7796 - 5 -398.361 81.281 398.361 + 19.507 16.6700 0 .7886 Zero -401.066 50.395 401.066 50.396 16 .7338 0.7916 + 5 -398.361 19.507 398.361 81.281 16.6700 0 .7886 10 -390.329 - 10.466 390.329 111.224 16.4792 0.7796 15 -377.214 - 38.550 377.214 139.314 16.1629 0.7646 20 -359.416 - 63.953 359.416 164.696 15.7234 0.7438 25 -337.475 - 85.882 337 .475 186.598 15 .1641 0 .7173 30 -312.061 -103.670 312.060 204.355 14.4893 0 .6854 35 -283.945 -116.779 283.945 217.427 13.7043 0 .6483 40 -253.984 -124.811 253.984 225.417 12.8151 0.6062 45 -223.088 -127.524 223.088 228.083 11.8283 0.5595 50 -192.197 -124.838 192.197 225.346 10.7517 0.5086 55 -162.248 -116.837 162.248 217.291 9 .7276 0 .4602 60 -134.152 -103.764 134.152 204.162 9.3400 0.4418 65 -108.759 - 86.021 108.759 186.362 8.8814 0 .4201 70 - 86.842 - 64.152 86.842 164.432 8.3972 0 .3972 75 - 69.062 - 38.822 69.062 139 .039 7.9326 0 .3753 80 -55.957 - 10.805 55.957 110.957 7.4638 0 .3531 85 -47.923 + 19.047 47.923 81" .'040 7.0656 0 .3342 90 -45.200 + 49.826 45.200 50.201 6.9404 0 .3283 D e f l e c t i o n s x 0.0170 0.0186 •0.0193 •0.0201 •0.0208 0.0501 0.0265 0.0225 0.0193 0 .0131 0 CO TABLE 8 - T i l t e d F i v e F i b r e Symmetrical C a n t i l e v e r s 69 T h i s data i n t h i s t a b l e i s s i m i l a r to t h a t . i n Table 7, the s t r u c t u r e s d i f f e r i n g o n l y i n the number of f i b r e s and thus i n t h e i r angular l a y o u t . I t w i l l be seen t h a t the f i g u r e s i n t h i s t a b l e are c l o s e l y s i m i l a r to those i n Table 4. The l o a d i n g s i t u a t i o n s are v i r t u a l l y i d e n t i c a l a p a r t from the d i r e c -t i o n of the s e l f weight f o r c e s . TABLE 9 - Skew M i c h e l l C a n t i l e v e r s A s e r i e s of f i v e f i b r e M i c h e l l c a n t i l e v e r s were i n v e s t i g a t e d a l l of equal span (L = 10") but of i n c r e a s i n g r i s e . The r i s e i n c r e a s e d from zero to 2.5" i n ten equal steps and the key data f o r t h i s s e r i e s o f s t r u c t u r e s i s l i s t e d i n Table 9. TABLE 10 - F i v e F i b r e Symmetrical M i c h e l l C a n t i l e v e r s I t seems a p p r o p r i a t e to conclude t h i s appendix w i t h a complete s e t of data f o r one of the s t r u c t u r e s i n v e s t i g a t e d . The framework chosen i s t h a t having f i v e f i b r e s and a span of 10", loaded w i t h one hundred pounds p e r p e n d i c u l a r t o the c a n t i l e v e r a x i s . Being symmetrical, f i g u r e s are g i v e n here-f o r h a l f the members. Table 10A l i s t s member lengths and j o i n t c o o r d i n a t e s , 10B c o n t a i n s d e t a i l s of f o r c e s and d e f l e c t i o n s w h ile the member widths and r a t i o of a x i a l f o r c e t o E u l e r c r i t i c a l b u c k l i n g load are shown i n Table 10C. TABLE 9 SKEW MICHELL CANTILEVERS X Span In. Y Span Theta Radrat. D e f l e c t i o n Volume V e r t i c a l A l b s . R eactions B l b s . 10 .0 0.250 18.420 1.058 0.0501 16 .742 40 .350 60.459 10.0 0.500 18.439 1.120 0.0502 16.757 30.319 70 .489 10.0 0.750 18.470 1.185 0.0502 16.781 20.288 80.520 10 .0 1.000 18.513 1.255 0.0503 16.816 10 .257 90 .551 10 .0 1.250 18.569 1.329 0 .0505 16 .860 0 .226 100.582 10.0 1.500 18.636 1.409 0 .0506 16 .913 - 9.805 110.613 10.0 1.750 18.715 1.494 0.0508 16 .976 -19 .836 120.646 10.0 2.000 18.806 1.587 0.0510 17 .047 -29 .878 130.689 10.0 2.251 18.908 1.687 0.0513 17.128 -39.921 140.734 10.0 2.501 19.020 1.796 0.0515 17.217 -49 .958 150.774 The h o r i z o n t a l r e a c t i o n s at A and B are +401 pounds i n a l l cases. For a l l s t r u c t u r e s N = 5 , D = 2 . 5 " , t = 0.25", E = 300,000 p . s . i . a = 300 p . s . i . , e = 0.001, Load 100 l b . at J,.,.. TABLE 10A TABLE OF MEMBER LENGTHS AND JOINT COORDINATES F i v e F i b r e M i c h e l l C a n t i l e v e r b = 0 1 2 3 4 5 a A 1.7678 — 1 B 0.0 1.25 ' 0.5657 1.25 A 1.7678 0.7845 -2 B 0.0 1.25 0.5657 1.5808 0.4588 0.8574 2.2172 0.0 A 1.7678 1.0152 1.1891 -3 . B 0.0 1.25 0.5657 1.7498 0.9987 0.9342 2.7187 0.6955 1.3805 3.6832 0.0 A 1.7678 1.2583 1.5643 1.9146 -4 B .- 0.5657 1.0151 1.5907 2.3140 0.0 1.7396 2.9977 4.4906 6.0436 1.25 1.5643 1.5870 1.1198 0.0 A 1.7678 1.5146 1.9867 2.5475 3.2092 5 A 0.0 1.5513 2.9794 4.9657 7.3969 10.000 B 1.25' 2.0977 2.6020 2.6379 1.8769 0.0 72 TABLE 10B FORCES IN MEMBERS AND JOINT DEFLECTIONS F i v e F i b r e Symmetrical M i c h e l l C a n t i l e v e r b = ->- 0 1 2 3 4 5 FA FB 1 to 65.293 0.0 0.0 56.905 -56.905 0.0 0.0025 FA FB 2 CO 117.020 0.0 0.0 102.793 -40.219 0.0012 0.0016 74.104 -74.104 0.0 0.0013 FA FB 3 CO 92.151 0.0 0.0 82.483 -26.827 0.0016 0.0011 62.257 -50.624 0.0006 0.0015 44.876 -44.876 0.0 0.0020 FA FB 4 CO 71.122 0.0 0.0 65.070 -16.261 0.0019 0.0005 51.728 -31.673 0.0013 0.0014 39 .572 -30.059 0.0009 0 .0024 28.519 -28.519 0.00 0.0033 FA FB 5 CO 101.475 0.0 0.0 100.177 0.0020 -0.0001 95.048 0.0022 0.0010 90 .171 0.0020 0.0023 85.528 . 0.0015 0.0039 0.0 0 .0055 For a s t r u c t u r e i n which -x = 10.0", D = 2.5", | = 4 . Load 100 pound a t J , , ; t = 0.25". TABLE IOC MEMBER WIDTHS AND CRITICAL BUCKLING LOAD RATIOS FOR 5 FIBRE SYMMETRICAL MICHELL CANTILEVER b = ->-a 4, 0 1 2 3 4 5 WA 0.8706 0.7587 WB 1 Q. - 0.7587 0.501 0.068 0.068 WA 1.5603 1.3706 0.9880 WB 2 g. *o 0.156 0.5362 0 040 0.135 0.9880 0.092 0.092 WA 1.2287 1. 0998 0.8301 0.5984 WB 3 O, 0.252 0.3577 0.104 0.6750 0.250 0.5984 0.647 - 0.304 0 .233 0.647 WA 0.9483 0.8676 0.6897 0.5276 0.3803 WB 4 0, "0 0.423 0.2168 0.256 0.4223 0.625 0.4008 1.601 0.3803 4.502 - 0.828 0.703 1.915 4.502 WA 1.3530 1.3357 1.2673 1.2023 1.1404 WB 5 - - - -0.208 0.156 0.299 0.546 0.963 - - - - -TABLE 11 - V a r i a t i o n o f D e f l e c t i o n With S t r a i n 74 A f i n a l t e s t was made t o determine the e f f e c t o f v a r y i n g the s t r a i n i n the s t r u c t u r e . The standard f i v e f i b r e c a n t i l e v e r w i t h a ten i n c h span was examined wi t h a range o f s t r a i n s , as d e t a i l e d i n Table 11. Uniform Uniform D e f l e c t i o n Volume ! S t r a i n S t r e s s ( J 5 5 ) 0.00001 3.0 0.000501 2053.399 0.00005 15.0 0.002505 347.830 0.0001 30.0 0.005010 170.415 0.0002 60.0 0.010020 84 .348 0.0004 120.0 0.020040 41.961. 0.0006 180.0 0.030060 27.927 0.0008 240.0 0.040080 20 .928 0.0010 300.0 0.050100 16.734 0.0020 600.0 0.100200 8.358 0.0040 1200.0 0 .200400 4.177 0.0060 1800.0 0.300600 2 .784 0.0080 2400.0 0.400800 2 .088 0.0100 3000.0 0.501000 1.670 T h i s t a b l e demonstrates t h a t the d e f l e c t i o n i s d i r e c t l y p r o p o r t i o n a l to the s t r a i n , but t h a t the volume i s o n l y approx-im a t e l y i n v e r s e l y p r o p o r t i o n a l to the s t r a i n . The d i s c r e p a n c y i s due to the e f f e c t s o f the s e l f weight f o r c e s . I f these had been n e g l e c t e d the correspondence would have been complete. 75 APPENDIX F BIAXIAL STRESS IN JOINTS The s t a t e o f s t r e s s i n the v i c i n i t y of j o i n t s i n a r i g i d Maxwell framework has been d i s c u s s e d i n a s i m p l i f i e d manner i n Chapter 5. The e f f e c t s of the secondary s t r e s s e s remains to be c o n s i d e r e d . The s t r e s s e s a c t i n g i n the v i c i n i t y of a r i g i d j o i n t are i n d i c a t e d i n F i g u r e F l . The s t r e s s e s shown i n t h i s diagram are i l l u s t r a t i v e and need not n e c e s s a r i l y r e p r e s e n t i n magnitude or d i r e c t i o n the a c t u a l s t a t e of s t r e s s a t a s p e c i f i c j o i n t . F u r t h e r , f o r c l a r i t y , the s t r e s s e s are shown separated and d i s p e r s e d along the members although i t i s a t the edges of the q u a d r i l a t e r a l QRWV t h a t they are co n s i d e r e d t o a c t . Three d i s t i n c t types of s t r e s s may be c o n s i d e r e d : -(a) the u n i a x i a l t e n s i l e / c o m p r e s s i v e s t r e s s e s o f magnitude a which are assumed to a c t i n con-v e n t i o n a l d i r e c t i o n s ; (b) bending s t r e s s e s i n each member, a r i s i n g from the r i g i d i t y of the j o i n t . These v a r y l i n e a r l y a c r o s s the s e c t i o n , r e a c h i n g maximum va l u e s a t the outer f i b r e s . (c) shear s t r e s s e s i n each member. These vary i n a p a r a b o l i c manner acr o s s the s e c t i o n r e a c h i n g a maximum a t the n e u t r a l a x i s . Distribution of S h e a r Sttess across section, U L L L l U J l i . iLUilUu Distribution of Bending StfesS across section. Uniaxial Stresses a'.=>ncj members . Figure F l B i a x i a l Stresses at a Typical Joint The sum of the f o r c e s e x e r t e d on the element QRWV by these v a r i o u s s t r e s s e s must add to zero, i f the element i s i n e q u i l i b r i u m . I t i s i m p o s s i b l e to draw a r e p r e s e n t a t i v e q u a n t i t a t i v e freebody diagram f o r an element of the j o i n t area s i n c e the e f f e c t of shear and bending s t r e s s e s v a r i e s a c r o s s the s e c t i o n . F u r t h e r , as the element i s moved from the j o i n t boundary toward i t s geometric c e n t r e , J j^, the s t r e s s e s w i l l v ary i n some complex manner as they flow between the members. As an approximation to the r e a l s t a t e , the j o i n t , , c o u l d be c o n s i d e r e d as a r e c t a n g l e as shown i n F i g u r e F2., Q R + V w F i g u r e F2 Approximation to B i a x i a l l y S t r e s s e d J o i n t : , 78 The s i d e s o f t h i s r e c t a n g l e are taken as the average of the widths of the adjacent members; t h a t i s Q R =; V W = ( W B q b * W B o - . - b ^ 2 . . . . F l Q V = R W = ( W A Q b + W A o . b-i) 2 The a x i a l f o r c e along each member imposes a uniform normal s t r e s s , a, on each f a c e of t h i s r e c t a n g l e . T h i s should a c t u a l l y be c cos 8/2, assuming the r e c t a n g l e to be o r i e n t e d a t 0/2 to the a x i s of each of the f o u r members converging a t j o i n t J a b « However, cos 8/2 i s c l o s e to 1 f o r the v a l u e s : of 9 encountered i n these c a l c u l a t i o n s , and may be dropped > N - 5 . 5 ^ 4 , 0 « 18*41*, e o a | * o - 9 « 7 M « 3 , 5 - 4 , 6 ' 3 5 - 9 0 ° . = o-9S\ I f the v a l u e s i n Table 1, Chapter 5, are examined i t w i l l be seen t h a t i n g e n e r a l the shear f o r c e s a t each j o i n t are q u i t e s m a l l and the average shear s t r e s s (shear f o r c e / c r o s s - s e c t i o n a l area) i s smal l compared w i t h a. I t w i l l thus be n e g l e c t e d as a f i r s t approximation. The bending moments however, cannot be so n e g l e c t e d although t h e i r r e l a t i v e e f f e c t v a r i e s g r e a t l y a c r o s s the s t r u c t u r e . The bending s t r e s s e s are of-comparable s c a l e w i t h the normal s t r e s s i n the area of the fans but r a r e l y 79 exceed 30% of the bending s t r e s s i n the out e r p a r t s o f the s t r u c t u r e . Thus the normal s t r e s s e s on the f a c e s o f the r e c t a n -g u l a r b l o c k c o u l d be m o d i f i e d as i n d i c a t e d i n F i g u r e F2 t o read (a ± -j) . T h i s should be read i n the sense t h a t these are o u t e r l i m i t s on the va l u e of the s t r e s s f o r most j o i n t s . Even these extreme val u e s o n l y apply a t the c o r n e r s of the j o i n t a rea, the c e n t r a l s t r e s s e s being ±c. I f these v a l u e s are accepted as r e p r e s e n t i n g the-l i m i t i n g v a l u e s of s t r e s s a t the m a j o r i t y o f outer j o i n t s > then the maximum shear s t r e s s may be determined f o r the extreme combinations of v a l u e s . By drawing Mohr 1s c i r c l e f o r the v a r i o u s cases i t w i l l be found t h a t the maximum shear s t r e s s v a r i e s from y - to - j — , which i s an i n c r e a s e o f one t h i r d i n the value a l r e a d y determined i n Chapter 5 f o r the simple case of b i a x i a l s t r e s s . T h i s maximum w i l l occur l o c a l l y i n two of the out e r corners o f the j o i n t area and i n p r a c t i c e c o u l d probably be t o l e r a t e d as a l o c a l i z e d s t r e s s c o n c e n t r a t i o n . In p r a c t i c e , i f f i l l e t s were p r o v i d e d a t the j o i n t s , the s t r e s s flow around these c o r n e r s might w e l l s u f f i c e t o r e -duce these maximum value s toward the normal.value a. 80 DEFLECTION OF STRUCTURES CONTAINING BIAXIALLY STRESSED JOINTS The d e f l e c t i o n s o f the M i c h e l l c a n t i l e v e r s a l r e a d y c o n s i d e r e d i n Chapter 4 and Appendix E are c a l c u l a t e d assuming u n i a x i a l s t r e s s e s i n a l l members. The j o i n t areas are sub- • j e c t e d t o b i a x i a l s t r e s s and w i l l thus have a somewhat l a r g e r d e f l e c t i o n due t o t h i s e f f e c t . For purposes o f c a l c u l a t i o n of the magnitude of t h i s change, c o n s i d e r the s t r e s s e s i n the j o i n t area as being u n i f o r m l y ±o [the valu e s on the axes o f the members, as shown i n F i g u r e F2. A t y p i c a l member may thus be d i v i d e d i n t o three zones as i n d i c a t e d i n F i g u r e F3. Bio* ially Sl>es6ed Z o n e s . V Untax i d l y S t r e s s e d "Zone • A .Ny V v v N. v v v V db J a b • i F i g u r e F3 E x t e n s i o n o f a T y p i c a l Member The change i n length of ( J a b J & m ^ y thus be calculated as the sum of the changes of the three zones. From the figure AL, ~ (l •*• V ) (wBqb+WSo-//>) E 2. F2 Aob - (WBab-^WBo-i-b') -.(WBQ,^, + WB<W.6+/) 2 "2. Total change , AAQfe = Al,"* AL2+ALa, E F3 The change of extension due to b i a x i a l stress i s thus given by: -Chpe (mcreose) ,n ^ / w B , b + W B q . , b * W B a . b + ^ W B 0 - , J • F4 expansion o" ncib 2. E. V / A similar expression may be written for the general bar i n the other set. 82 Change i n extension of B 2 E J The defl e c t i o n s , calculated i n t h i s manner, of the range of cantilevers e a r l i e r described i n Chapter 4, Section 1, are tabulated below and compared with the deflections of the pin jointed structures. Deflections L N Uniaxial Stress D B i a x i a l Stress 1 2 0.0245 0.0254 3 0.0243 0.0253 4 0.0243 0.0253 5 0.0243 0.0253 7 0 . 0243 0.0253 10 0.0243 0.0253 2 2 0.0377 0.0415 3 0.0357 0.0393 4 0.0353 0.0389 5 0.0352 0.0388 7 0.0351 0.0387 10 0.0351 0.0387 4 2 0.0618 0.0799 3 0.0520 0.0673 4 0.0506 0.0652 5 0.0501 0.0644 7 0.0498 0.0639 10 0.0496 0.0637 5 2 0.0730 0.1029 3 0.0581 0.0827 4 0.0561 0.0793 5 0.0554 0.0781 7 0.0549 0.0772 10 0.0547 0.0768 10 2 0.1259 0.2634 3 0.0796 0.1879 4 0.0745 0.1739 5 0.0728 0.1687 7 0.0717 0.1648 10 0.0712 0.1630 83 APPENDIX G DETAILS OF COMPARABLE STRUCTURES A range of structures are described i n Chapter 6, a l l subjected to the same loading and support conditions. These i l l u s t r a t e the s u p e r i o r i t y of the M i c h e l l type of framework i n a t y p i c a l s i t u a t i o n . The M i c h e l l structures conform to the parameters s p e c i f i e d i n Chapter 4, namely:-E = 300,000 p . s . i . Maximum stress a = 300 p . s . i . Load 100 lb at 10" from plane of supports. Where appropriate, the maximum support spacing i s 2.5", and the thickness of members, normal to the plane of the structure i s 0.25". The d e f l e c t i o n of each of the following structures i s s p e c i f i e d as 0.0501", that of the f i v e f i b r e M i c h e l l c a n t i l e v e r of s i m i l a r — r a t i o wherever t h i s i s compatible with the s p e c i f i e d s t r e s s . The design of these structures i s now discussed i n d e t a i l (structures A, B, and C having already been discussed i n Chapter 4):-D. Warren Truss The general arrangement of the truss studied i s shown in Figure Gl and i s based on the s i m i l a r structure discussed by Chan i n reference 1.6. 84 irJOO Its Figure Gl Warren Truss Preliminary analysis indicated that the structure shown above could be designed with a span of 10", an i n d i v i d u a l member length of 2.8571" and an o v e r a l l depth of 2.4744" which i s close to the 2.5 inch support spacing of the Miche l l c a n t i -lever. This seemed a reasonable compromise although i t does not follow that this arbitary design i s optimum of i t s type. The forces i n the members were calculated by means of a Maxwell diagram, solved t r i g n o m t r i c a l l y and are indicated on the figure above. 8 5 The cross-sectional area of each member may be adjusted so that the stress i n each i s uniformly 3 0 0 p . s . i . , whether i n tension or compression. The volume of the structure may then be calculated and i s found to be 2 1 . 1 7 cu.in. Since the stress i n each member i s uniform the s t r a i n w i l l s i m i l a r l y be uniform and i s £ = 0 . 0 0 1 . The extension of each member i s thus known and may be calculated by use of a W i l i o t diagram as shown i n Figure G 2 . From t h i s diagram the downward d e f l e c t i o n of L i s 0 . 0 6 3 9 inches. * In passing i t may be noted t h i s well i l l u s t r a t e s the findings of Chapter 1 . I t was there predicted that the Mi c h e l l structure (or rather the optimum frame to which the Mi c h e l l structure i s a close approximation) i s l i g h t e r yet s t i f f e r than any other. If the framework i s increased i n cross-section to reduce the d e f l e c t i o n to 0 . 0 5 0 1 " , the volume becomes N I 0 . 0 6 3 9 O N N N 2 1 . 1 7 x Q 0 5 Q 1 ' O R 2 7 . 0 0 cu. i n . Changes ia member lengths ^Sxx, are all ecjaal -fc> 0-ooa24" Excepr £8Qa>hx'k osjuals Q 00/43". O ' 4- ' 8 IZ '6 SCALE - THOUSANDTHS OP I N C H E S . o| 1 » o - o i o " * f t J > A.B (ORIGIN) D O W N W A R D (4) O F L -=> gure G2 W i l i o t Diagram for Deflection of Warren Truss 87 Two Bar C a n t i l e v e r t l O O l b . a) General Arrangement b) D e f l e c t i o n of S t r u c t u r e F i g u r e G3 Two Bar C a n t i l e v e r The general arrangement of the structure i s shown i n Figure G3. From the geometry ALO = BLO = 7.1° and, by elementary s t a t i c s , the force acting along each member i s 50 ± s i n 7 = ±(403.11) l b , AL being i n tension while BL i s in compression. From the geometry of the deflected structure, and assuming that the s t r a i n i s to be the same i n both members, by Pythagoras Gl From which, since y i s known o-oso i e ^ 0 0 0 6 2 1 " x = o - 0 0 0 / 3 " G2 The s t r a i n i n AL and BL i s thus 6 = O- Q Q 6 Z 1 - + 0 - 0 0 0 6 / 7 IO-07782 G3 Uniform stress = £.£ = I84*'98 Cross-sectional area = 403 'M - 2*1792 G4 < B 4 9 8 i . e . 1.476 inches square or 1.666 inches d i a . Volume of frame =' 43.9233 cu.in. G5 89 The compression member LB could f a i l by buckling. If EITT^ the Euler c r i t i c a l load, —= , i s taken as a c r i t e r i o n , the ax i a l load for f a i l u r e i s 11,500 pound which f a r exceeds the 403 pound force i n the member. Thus buckling i s very u n l i k e l y . Cantilever With Parabolic Section Figure G4 Cantilever of Parabolic Section The general arrangement of t h i s structure i s shown in Figure G4. I t has a constant width t and a depth h, which i s a function of length, x. It i s sp e c i f i e d that the maximum bending stress, i n an outer f i b r e , a, i s to be constant at a l l p l a n e s . Consider an element of the c a n t i l e v e r , MNQP, at a d i s t a n c e x from the l o a d . A p p l y i n g the elementary equation 3 J l I i±?) ,12 / 01- G6 I f a i s to be c o n s t a n t , equation G6 i m p l i e s t h a t the shape of the c a n t i l e v e r i s p a r a b o l i c . T h i s may be used t o s o l v e the d i f f e r e n t i a l e q uation to determine s l o p e and d e f l e c t i o n : -I .2 i !2 E X y^E \ pj fzr Thus d x LO G7 G8 G9 The con s t a n t s G and may be determined s i n c e the slope and d e f l e c t i o n are zero a t x = L. Thus GO =s • G10 91 In the present case co = 0.0501 at x = zero, a = 300 p . s . i . Thus from G10, t = 2.8228" and from G6, H = 2.661" Volume = | HLt = 50.100 cu.in, G i l G12 Triangular Plate Cantilever This structure i s shown i n Figure G5. The cantilever depth, h, remains constant while the width increases uniformly from zero at the t i p , L, A IOOlb. 'igure G5 Triangular Plate Cantilever to a maximum, T, at the root AB. By t h i s means the stress i n the outer f i b r e s may be maintained constant at a l l planes. From the formula a ' R " 1 G13 since c and y are constant, the radius of curvature, R, i s / constant and thus the beam bends to a c i r c u l a r shape, I Thus . . . G14 \ \ assuming u T i s small so that u)T may be neglected. -u Li Thus from G13, -FT or I '9960 »N. • • - . c i 5 Also, from the same equation, PL or T & P I ? C0u £ -j 6 -0200 /^ . . . G16 F i n a l l y the volume = ^ T h L - 5 O I 0 0 CU.IW. . . . G17 93 G.H.J. CANTILEVERS OF UNIFORM CROSS-SECTION For a simple c a n t i l e v e r , the d e f l e c t i o n due to a point end load i s given by the f a m i l i a r equation PL 3 W L = 3EI • . . . . G18 In the present case a l l variables are known except I which then may be calculated I = 2.21778 i n . 4 . . . . G19 Further from Y = Sr- = 0.66533 . . . . G20 M or the t o t a l depth of section i s 1.33067 i n s . These equations apply, whatever the form of the cross-section. Three cases may now be considered:-a) C y l i n d r i c a l Cross-section, Diameter d From G20, the diameter i s a maximum of 1.33067 i n . TTD4 and thus I = -g-^ — = 0.154 which i s considerably less than the value s p e c i f i e d by G19. Thus a c y l i n d r i c a l c a n t i l e v e r cannot completely s a t i s f y the conditions. If G19 i s s a t i s f i e d D = 2.5926 i n . Volume = 52.7911 cu.in. . . G21 Maximum stress = 584.5 p . s . i . A l t e r n a t i v e l y i f the diameter i s f i x e d a t 1.33067 i n . t o maintain the s t r e s s as 300 p . s . i . , the volume i s reduced t o 13.9069 c u . i n . , but the d e f l e c t i o n i s g r e a t l y i n c r e a s e d to 0.7220 i n . The f i r s t case i s quoted as more t y p i c a l . b) I Beam C a n t i l e v e r ' '103 lb. F i g u r e G 6 I Beam C a n t i l e v e r The s e c t i o n s t u d i e d i s shown i n F i g u r e G16. For convenience the t h i c k n e s s of both f l a n g e s and the web, t , are taken as e q u a l , and both f l a n g e s are assumed t o have equal width, W. The second moment, cf area of the cross-section may now be calculated about the neutral axis. ± H % 2 \Z Equations G19, G20 and G22 may be combined to connect the three variables H, t and W. Any one of these may be selected a r b i t r a r i l y , and the other two calculated from these equations. Further the volume of the structure i s given by , V = ZWLt + HLt • - • • G23 Theoretically a minimum volume i s obtained as t tends to zero, but t h i s i s impractical since the flanges would not be s e l f supporting and would buckle [for t = 0.1 i n . , the flanges are over 30" wide]. A thickness of 0.25" was therefore a r b i t r a r i l y selected as being equal to that of the members of the Michell framework. In such case:-Thickness of a l l Members =0.25" Width of Flanges W = 16.982" . . . . G24 Height of Web, H = 1.329" Volume of Structure V = 86.9854 cu.in. wt" iz G22 c) Beam of R e c t a n g u l a r ^ S e c t i o n The I beam of the p r e v i o u s s e c t i o n may be converted to a r e c t a n g u l a r s e c t i o n by p l a c i n g H equal to zero, so t h a t i t becomes a l l f l a n g e , or may be s o l v e d d i r e c t l y from equations G 1 9 and G 2 0 , knowing t h a t _ b h 3 . 1 " 12 In e i t h e r case the same unique s o l u t i o n i s o b t a i n e d : -Height 1 . 3 3 0 6 7 i n . Width 1 1 . 2 9 5 1 i n . . . . . Q 2 5 Volume 1 5 0 . 3 0 0 0 i n . 3 I t may be observed t h a t these c a n t i l e v e r s o f uniform s e c t i o n use t h e i r m a t e r i a l content very i n e f f i c i e n t l y . M a t e r i a l c l o s e to the n e u t r a l a x i s does l i t t l e work a t a l l planes, w h i l e t h a t i n the out e r f i b r e o n l y approaches the s p e c i f i e d s t r e s s l e v e l near the r o o t . The I beam has l e s s volume than the r e c t a n g u l a r s e c t i o n s i n c e much of t h i s 'dead wood' i s e l i m i n a t e d . \ \ 97 APPENDIX H MANUFACTURE OF•PHOTOELASTIC MODELS The techniques for the manufacture of good qua l i t y models from b i r e f r i n g e n t p l a s t i c s for photoelastic i n v e s t i -gation are well established. Cuts should be made with low feeds using sharp tools and great care must be taken to prevent l o c a l heating with the introduction of permanent stress patterns at zero load. The models discussed i n Chapter 7 were of r e l a t i v e l y complicated shape compared to familar t e n s i l e and bending test specimens. It was decided to m i l l these from sheet CR39 p l a s t i c , thick, using a numerically controlled m i l l -ing machine. . ;-, The table movements were operated by synchronous (SLOSYN) motors which respond to e l e c t r i c a l pulses, each pulse causing the motor to rotate by a fixed amount. Smooth surfaces could thus be cut i n two perpendicular d i r e c t i o n s — a l o n g the machine table, or at r i g h t angles to i t — b y d r i v i n g the, appropriate motor. Cuts could also be made at 45° to these directions by d r i v i n g both motors simultaneously. Surfaces not aligned with these d i r e c t i o n s could only be produced i n a series of steps r e s u l t i n g i n a rough f i n i s h 98 which was undesirable. It was therefore decided to mount the model blank on a rotary table so that each member could be turned paralled with the x or y axes for cutting. This procedure was p a r t i c u l a r l y convenient for Michell cantilevers since alternate members are perpendicular. For example,;in Figure 2.2, CF, FK, KH and HL are at 90° to one another and could thus be cut i n one setting of the table. ? The procedure used to make the models described i n Chapter 7 was as follows:-a) Joint Coordinates, Member Sizes, etc. The coordinates of the j o i n t points were found from the computer printout described i n Chapter 4 and Appendix C. The member widths were also obtained from the same source but these were modified as described i n Chapter 7 to obtain suitable models. The values used for design purposes, and those j a c t u a l l y achieved on the models, are tabulated i n Tables:1 ; and 2 for the three and f i v e f i b r e M i c h e l l Cantilevers. ; Mem-bers whose widths are not shown were not modelled separately but were merged into the s o l i d areas of the fans. b) Allowance for Angularity The model blank, as has already been stated, was; mounted on a rotary table and turned to a l i g n each cut with the machine axes. In t h i s process a point i n i t i a l l y at 99 TABLE 1 THREE FIBRE MICHELL CANTILEVER Member Width (300 p . s . i . ) Nominal Width Actual Width ~ " f i l l e t | f i l l e t A10 1.5925 0.300 - -A l l 1.2640 0.238 0.240 -A12 0.5878 0.110 0.113 • -A20 2.5540 0.480 0.482 -A21 2.0846 0.392 0.393 0 . 393 A22 1.0638 0.200 0.204 0.203 A30 1.9048 0.358 0.359 -A31 1.8122 0.340 0.341 0 .343 A32 1.4674 0.276 0.277 0.278 B01 1.5925 0.300 - -B l l 1.2640 0.238 0.240 -B21 0.5878 0.110 0.114 B02 2 .5540 0.480 0.482 -B12 2.0846 0.392 0.394 0.394 B22 1.0638 0.200 0.203 0.204 B03 1.9048 0 .358 0.360 .-B13 1.8122 0.340 0.341 0.342 B23 1.4674 0.276 0.279 0.279 100 TABLE 2 FIVE FIBRE RIGID MICHELL CANTILEVER Member Design Width 300 p . s . i . Nominal Width Actual Width •A' •B' A10' B01 0.8706 1.5603 20 02 A R 30' 03 1.2287 A.„, B„.. 0.9483 40' 04 Merged together A 5 f i ' Brm 1.3530 in A l l ' B l l 0.7587 s o l i d 'fans' A12' B21 0.5362 A13' B31 0.3577 A14' B41 0.2168 A51' B15 1.3357 0.351 0 .357 0.356 A52' B25 1.2673 0.333 0.336 0.338 A53' B35 1.2023 0.316 0.319 0.319 A54' B45 1.1404 0.300 0.300 0.303 A41' B14 0.8676 0.228 0 .230 0.229 A42' B24 0.6897 0.181 0.182 0.182 A43' B34 0.5276 0.139 0.140 0.140 44 44 0.3803 0.100 0.103 0 .102 A31' B13 1.0998 0.289 0.290 0.290 A32' B23 0.8301 0.218 0.218 0 .218 A R 33' 33 0.5984 0.157 0.160 0.161 A34' B43 0.4008 0.105 0.105 0.105 A21' B12 1.3706 0 .360 0.360 0.360 A R 22' 22 0.9880 0.260 0.260 0.260 A R 23' 32 0.6750 0.177 0.177 0.177 A24' B42 0.4223 0.110 0.110 0 .108 101 coordinates x, y w i l l move to a new pos i t i o n X, Y. The j o i n t coordinates referred to i n Section A must then be re-calculated to allow for t h i s r o t a t i o n as shown i n Figure Hi. Figure HI The E f f e c t of Rotation The point P has coordinates x, y with reference to the axes x, y. If now the axes are rotated through an angle 3 to new d i r e c t i o n s , X, Y, the coordinates of P with r e f e r -ence to the new axes are X, Y. X •= O M = ON + NM sr :>c £ 3 3 ;3 + LfS/nft . . . . HI 102 Y -a, MP s NPcos/3 •» (y -xtnnp) cosy* - 3 cos^ — smjS . . . . H2 These equations enable the new coordinates of each j o i n t point to be found as the model i s rotated to d i f f e r e n t positions. c) Offsets for Cutter Radius and Member Width The coordinates of the cutter axis may now be deter-mined for the beginning and end of each cut. Allowance must be made for the widths of the members and for the cutter radius as indicated i n Figure H2. Cutter" Figure H2 Geometry of Cutter Offsets A t y p i c a l j o i n t i s aligned so that member FJ i s p a r a l l e l to the Y axis. The coordinates of J are known from equations HI and H2. The coordinates of 0 are required for the programme for numerical control of the m i l l i n g machine. Clearly the setoff i n the X d i r e c t i o n , Sx i s £x = Wk + r =. W. + D or X Q - X J + W, + * 2 . . . . H3 •~"Z The setoff i n the y d i r e c t i o n , S Y, i s given by 5y = Pl^ l - MN 2 c o 4 © a o 3 6 3 . . . H4 If the angle FJG i s acute [(90 -6) instead of (90+6) as drawn], the coordinates of 0 become Z . . . . H5 YQ * ~ 2 * fa + s»>x Q) + W, sine + 2 c o * a 104 d) Backlash - Order of Cuts The e l e c t r o n i c c o n t r o l s have p r o v i s i o n f o r e l i m i n a -t i o n (more s t r i c t l y c o r r e c t i o n ) of b a c k l a s h i n the m i l l i n g machine leadscrew. T h i s i s done by e n s u r i n g t h a t a l l p o i n t s are approached i n the p o s i t i v e d i r e c t i o n . I f the machine i s moving n e g a t i v e l y , the s t o p p i n g p o i n t i s o v e r s h o t , the machine r e v e r s e s d i r e c t i o n and approaches the d e s i r e d l o c a t i o n i n the p o s i t i v e d i r e c t i o n . Care t h e r e f o r e has to be taken to make a l l cuts i n the p o s i t i v e d i r e c t i o n s o n l y to e l i m i n a t e the notches t h a t • would otherwise be c u t i n members due to t h i s u n c o n t r o l l e d motion. F i g u r e H3 i n d i c a t e s the order i n which the cuts were made and i s t y p i c a l of the r o u t i n e s f o l l o w e d f o r the c u t t i n g of a l l the models. e) C o m p i l a t i o n of Programme The programme f o r c o n t r o l of the machine may now be w r i t t e n f o l l o w i n g a simple r o u t i n e , which v a r i e s s l i g h t l y from system to system. The programmes f o r the three s t r u c t u r e s , which are l i s t e d a t the end of t h i s appendix, c o n s i s t of a s e r i e s of commands, each of which c o n t a i n up to f o u r words or ' b i t s ' . These a r e : -105 Figure H3 Order and Direction of Cuts for Machining a Typical Michell Cantilever 106 i ) the step number. T h i s i s o n l y f o r i d e n t i f y i n g the step and p l a y s no p a r t i n the machining p r o c e s s . I t i s d i s p l a y e d on a counter on the c o n t r o l c a b i n e t , i i ) the x c o o r d i n a t e , i i i ) the y c o o r d i n a t e . These are s t a t e d i n u n i t s o f one thousandth of an i n c h . One or both o f these c o o r d i n a t e s may be l i s t e d . These govern the movement of the c u t t i n g t o o l . In f a c t , i t i s the work t h a t i s moved p a s t a s t a t i o n a r y c u t t e r , but i t i s u s u a l to c o n s i d e r the t o o l as moving. The c o n t r o l gear a l l o w s f o r t h i s f a c t , i v ) c o n t r o l i n s t r u c t i o n s known as 'miscellaneous func-t i o n s ' . These have the f o l l o w i n g e f f e c t s : -02 Rewind tape. T h i s ends a programme. f 06 T o o l change. At the end of the step i n which t h i s appears, the machine i s stopped. T h i s p r o v i d e s an o p p o r t u n i t y f o r a t o o l t o be changed or a machine s e t t i n g to be a l t e r e d . 52 T o o l Down. . 53 T o o l Up. These commands cause the t o o l to be r a i s e d or lowered t o a p r e s e t l e v e l . 55 F a s t T r a v e r s e . T h i s i s used t o move the t o o l between c u t t i n g p o s i t i o n s . NUMERICAL TAPE CONTROL PROGRAM DEPARTMENT OF MECHANICAL ENGINEERING, UNIVERSITY OF BRITISH COLUMBIA P R E P A R E D B Y ; EWJ D A T E 30 January 70 P A R T N A M E W O R K O R D E R N O . C H E C K E D B Y D A T E R E M A R K S : 5 FIBRE MICHELL CANTILEVER. 1/8" c u t t e r . ( O r i g i n 8.000" l e f t of t u r n t a b l e c e n t r e ) . S H E E T O F ' 1 6 T A P E N O . S E Q . N O . T A B O R E O B + O R "X" I N C R E M E N T T A B O R E O B + O R "Y " I N C R E M E N T • T A B O R , E O B " M" F U N C T . E . O . B . I N S T R U C T I O N S E % E 1 T + 8000 T T 55 E 2 T - 3760 T - 2713 T 0655 E R o t a t e 324.20° 3 T + 2801 T T 52 E 4 T T + 2039 E : 5 T + 226 T - 1981 T 535 E 6 T T + 1912 T • " 52 E 7 T + 1724 E 8 T - 1654 T + 264 T 535 E 9 T + 1590 T T 52 E 10 T T + 1147 E 11 T + 282 T - 1071 T 535 E 12 T T + 1002 T 52 E 13 T + 946 E 14 T - 855 T + 342 T 535 E L5 T + 765 T T • 52 E L6 T T + 548 E 17 T + 384 T - 437 T 535 E L8 T T + 334 T 52 E L9 T + 534 E 10 T - 390 T + 484 T 535 E 11 T + 287 T T 52 E 12 T T + 207 E 13 T - 4414 T - 4365 T 06535 E R o t a t e + 18.41° >4 T + 2240 T T 52 E >5 T T + 1270 E 1 E C H . E N G . 3 - 6 9 • O R M 7 2 8 NUMERICAL TAPE CONTROL PROGRAM DEPARTMENT OF MECHANICAL ENGINEERING, UNIVERSITY OF BRITISH COLUMBIA P R E P A R E D B Y • A T E P A R T N A M E W O R K O R D E R N O . C H E C K E D B Y D A T E R E M A R K S : S H E E T „ O F . Z O T A P E N O . S E Q . N O . T A B O R E O B + O R "X" I N C R E M E N T T A B O R E O B + O R "Y " I N C R E M E N T T A B O R E O B » M " F U N C T . E . O . B . I N S T R U C T I O N S 26 T + 230 T - 1210 T 535 E 27 T T + 1158 T 52 E 28 T + 1376 E 29 T - 1291 T + 306 T 535 E 30 T + 1213 T T 52 E 31 T T + 652 E 32 T + 302 T - 569 T 535 E 33 T T + 492 T 52 E -34 T + 827 E 35 T - 706 T + 414 T 535 E 36 T + 610 T T 52 E 37 T T + 148 E 38 T - 3083 T - 2782 T - 06535 E R o t a t e + 18.41° 39 T + 1672 T T 52 E 40 T T + 665 E 41 T + 236 T - 604 T 535 E 42 T T + 550 T 52 E 43 T + 1103 E 44 T - 1003 T + 352 T 535 E 45 T + 931 T T 52 E 46 T T + 207 E 47 T - 1879 T - 1649 T 06535 E R o t a t e + 18.41° 48 T + 1220 T T 52 E 49 T T + 173 E 50 T - 4928 T - 173 T 06535 E R o t a t e + 141.14° M E C H . E N G . 3 - 6 9 F O R M 7 2 8 NUMERICAL TAPE CONTROL PROGRAM 109 DEPARTMENT OF MECHANICAL ENGINEERING, UNIVERSITY OF BRITISH COLUMBIA P R E P A R E D B Y D A T E P A R T N A M E W O R K O R D E R N O . C H E C K E D B Y D A T E R E M A R K S : S H E E T O F 3 6 T A P E N O . S E Q . N O . T A B O R E O B + O R "X" I N C R E M E N T T A B O R E O B + O R "Y" I N C R E M E N T T A B O R E O B " M " F U N C T . E . O . B . I N S T R U C T I O N S 51 T + 1220 T T 52 E 52 T - 1220 T T 535 E 53 T T + 173 T 52 E 54 T + 608 T + 306 T 06535 E R o t a t e + 18.41° 55 T + 1672 T T 52 E 56 T - 1672 T T 535 E 57 T T 665 T 52 E 58 T - 236 T - 604 T 535 E 59 T T + 550 T 52 E 60 T - 1103 T T 535 E 61 T + 1103 T T 52 E 62 T - 1031 T + .352 T 535 E 63 T + 931 T T 52 E 64 T - 931 T T 535 E 65 T T + 207 T 52 E 66 T + 2417 T - 1049 T 06535 E R o t a t e + 18.41° 67 T + 2240 T T 52 E 68 T — 2240 T T 535 E 69 T T + 1270 T 52 E 70 T 230 T _ 1210 T 535 E 71 T T + 1158 T 52 E 72 T _ 1376 T T 535 E 73 T + 1376 T T 52 E 74 T _ 1298 T + 306 T 535 E 75 T + 1213 T T 52 E M E C H . E N G . 3 - 6 9 F O R M 7 2 8 NUMERICAL TAPE CONTROL PROGRAM DEPARTMENT OF MECHANICAL ENGINEERING, UNIVERSITY OF BRITISH COLUMBIA PREPARED BY • ATE PART NAME WORK ORDER NO. CHECKED BY • ATE REMARKS: 1 r SHEET , OF -4 6 TAPE NO. SEQ. NO. TAB OR EO B + OR "X" INCREMENT TAB OR EOB + OR "Y" 1N C R EM EN T TAB O R EOB »M" FUN C T. E.O. B. INSTRUCTIONS 76 T 1213 T T 535 E 77 T T + 652 T 52 E 78 T - 302 T - 569 T 535 E 79 T T + 492 T 52 E 80 T - 827 T T 535 E 81 T + 827 T T 52 E 82 T - 731 T + 414 T 535 E 83 T + 610 T T 52 E 84 T - 610 T T 535 E 85 T T + 148 T 52 E 86 T + 4226 T — 3086 T 06535 E Rotate + 18.41° 87 T + 2801 T T 52 E 88 T - 2801 T T 535 E 89 T T + 2026 T 52 E 90 T _ 226 T _ 1968 T 535 E 91 T T + 1912 T 52 E 92 T 1724 T T 535 E 93 T + 1724 T T 52 E 94 T 1660 T + 264 T 535 E 95 T + 1590 T T 52 E 96 T 1590 T T 535 E 97 T T + 1147 T 52 E 98 T - 282 T - 1071 T 535 E 99 T T + 1002 T 52 E 100 T - 946 T T 535 E M E C H . E N G . 3 - 6 9 F O R M 7 2 8 NUMERICAL TAPE CONTROL PROGRAM DEPARTMENT OF MECHANICAL ENGINEERING, UNIVERSITY OF BRITISH COLUMBIA PREPARED BY DATE PART NAME WORK ORDER NO. CHECKED BY DATE REMARKS: SHEET OF 5 6 TAPE NO. SEQ. NO. TAB OR EO B + OR "X" INCREMENT TAB OR EO B + OR "Y" INCR EM EN T TAB OR EO B "M" FUNCT. E.O. B. INSTRUCTIONS 101 T + 946 T T 52 E 102 T - 856 T + 342 T 535 E 103 T + 765 T T 52 E 104 T - 765 T T 535 E 105 T T + 548 T 52 E 106 T - 384 T - 437 T 535 E 107 T T + 334 T 52 E 108 T - 534 T T 535 E 109 T + 534 T T 52 E 110 T - 431 T + 484 T 535 E 111 T + 287 T T 52 E 112 T - 287 T T 535 E 113 T T + 207 T 52 E 114 T - 5120 T - 2077 T 06535 E R o t a t e + 324.20° 115 T + 8000 T T 55 E 116 T + 4584 T - 562 T 55 E 117 T + 978 T T 52 E 118 T T + 1124 E 119 T - 978 E 120 T - 7973 T — 3699 T 06535 E R o t a t e + 144.2° 121 T + 2689 T T 52 E 122 T - 1150 T + 409 T 06535 E R o t a t e + 18.41° 123 T + 2850 T T 52 E 124 T - 1125 T - 139 T 06535 E R o t a t e + 18.41° 125 T + 2250 T T 52 E MECH. ENG. 3-69 FORM 728 NUMERICAL TAPE CONTROL PROGRAM DEPARTMENT OF MECHANICAL ENGINEERING, UNIVERSITY OF BRITISH COLUMBIA PREPARED BY DATE PART NAME WORK ORDER NO. CHECKED BY DATE REMARKS: SHEET , OF , 6 6 TAPE NO. SE Q. NO. TAB OR EO B + OR "X" INCREMENT TAB OR EO B + OR "Y" 1N C R EM ENT TAB OR EOB " M " FUNCT. E.O. B. INSTRUCTIONS 126 T - 1225 T • - 497 T 06535 E R o t a t e + 18.41° 127 T + 1850 T T 52 E 128 T - 850 T - 370 T 06535 E R o t a t e + 9.20° 129 T + 1157 T T 52 E 130 T + 1413 T + 1922 T 06535 E R o t a t e 331.4° 131 T + 1092 T T 52 E 132 T T + 3624 E 133 T - 1067 E 134 T - 7552 T - 5546 T 06535 E R o t a t e 151.4° 135 T + 1157 T T 52 E 136 T - 850 T + 370 T 06535 E R o t a t e + 9.20° 137 T + 1850 T T 52 E 138 T - 1200 T + 497 T 06535 E R o t a t e + 18.41° 139 T + 2225 X T 52 E 140 T - 1050 T + 139 T 06535 E R o t a t e + 18.41° 141 T + 2775 T T 52 E 142 T - 1050 T - 409 T 06535 E R o t a t e + 18.41° 143 T + 2589 T T 52 E 144 T - 3389 T + 3137 T 06535 E R o t a t e + 324.20° 145 T - 8000 T T 0255 E M E C H . E N G . 3 - 6 9 F O R M 7 2 8 NUMERICAL TAPE CONTROL PROGRAM DEPARTMENT OF MECHANICAL ENGINEERING, UNIVERSITY OF BRITISH COLUMBIA PREPARED BY EWJ DATE 1 4 F e b _ 7 0 PART NAME 3 Fibre Michell Cantilevers WORK ORDER NO. CHECKED BY DATE REMARKS: 1/16" Radius F i l l e t s Origin 8000 Left of Turntable Centre. SHEET OF 1 3 TAPE NO. SEQ. NO. TAB OR EO B + OR " X " INCREMENT TAB OR EO B + OR "Y " INCR EM EN T TAB OR EO B " M " FUNCT. E.O. B. INSTRUCTIONS E % E 1 T + 8000 T T 55 E 2 T - 4060 T — 2073 T 0655 E Rotate + 332.95° 3 T + 5103 T T 52 E 4 T T + 2606 E 5 T 669 T + 342 T 535 E 6 T + 1446 T T 52 E 7 T T + 737 E 8 T - 1789 T - 3480 T 535 E 9 T T + 2225 T 52 E 10 T + 2013 E 11 T + 140 T + 636 T 535 E 12 T T + 430 T 52 E 13 T _ 3509 T _ 3739 T 06535 E Rotate + 35.90° 14 T i + 2935 T T 52 E 15 T T + 623 E 16 T + 236 T _ 564 T 535 E 17 T T + 442 T 52 E 18 T — 851 T _ 1313 T 06535 E Rotate + 17.95° 19 T + 716 T T 52 E 20 T + 540 T + 2382 T 06535 E Rotate + 324.10° , " 21 T + 716 T T 52 E 22 T - 649 T + 606 T 535 E 23 T + 699 T T 52 E 24 T — 1443 T + 2579 T 06535 E Rotate + 324.10° 25 T + 698 T T 52 E MECH. ENG. 3-69 FORM 728 I I NUMERICAL TAPE CONTROL PROGRAM DEPARTMENT OF MECHANICAL ENGINEERING, UNIVERSITY OF BRITISH COLUMBIA PREPARED BY • ATE PART NAME WORK ORDER NO. CHECKED BY DATE REMARKS: SHEET OF 2 3 TAPE. NO. r* SEQ. NO. TAB OR EOB + OR "X" INCREMENT TAB OR EOB + OR "Y" INCREMENT TAB OR EOB "M" FUNCT. E.O. B. INSTRUCTIONS 26 T 7220 T T 06535 E Rotate + 270.00° 27 T + 698 T T 52 E 28 T - 1443 T - 2579 T 06535 E Rotate + 324.10° 29 T + 699 T T 52 E 30 T - : 649 T - 606 T 535 E . 31 T + 716 T T 52 E 32 T + 540 T - 2382 T 06535 E Rotate + 324.10° 33 T + 716 T T 52 E 34 T - 615 T + 812 T 06535 E Rotate + 17.95° 35 T 2935 T T 52 E 36 T - 2935 T T 535 E 37 T T + 623 T 52 E 38 T - 236 T - 564 T 535 E 39 T T + 442 T 52 E 40 T + 2141 T - 258 T 06535 E Rotate + 35.90° 41 T + 5103 T T 52 E 42 T - 5103 T T 535 E 43 T T + 2606 T 52 E 44 T _ 326 T — 2401 T 535 E 45 T T + 2225 T 52 E 46 ' T — 2013 T T 535 E 47 T + 2013 T T 52 E 48 T - 1789 T + 518 T 535 E 49 T + 1446 T T 52 E 50 T _ 1446 T T 535 E M E C H . E N G . 3 - 6 9 F O R M 7 2 8 NUMERICAL TAPE CONTROL PROGRAM u s DEPARTMENT OF MECHANICAL ENGINEERING, UNIVERSITY OF BRITISH COLUMBIA PREP A R ED BY DATE PART NAME WORK ORDER NO. CHECKED BY DATE REMARKS: SHEET ^ ° F 3 TAPE NO. SE 0. NO. TAB • OR EO B + OR " X " INCREMENT TAB O R EO B + OR " Y " IN CREMENT TAB OR EO B " M " FUNCT. E.O. B. INSTRUCTIONS 51 T T + 737 T 52 E 52 T - 364 T - 619 T 535 E 53 T T + 430 T 52 E 54 T - 4478 T - 1423 T 06535 E R o t a t e + 332.95° INNER CU' :s ON * [ODE L C0MPLET ED 55 T + 8000 T T 55 E 56 T + 4584 T - 438 T 55 E 57 T + 854 T T 52 E 58 T T + 876 E 59 T - 800 E . 60 T - 8522 T - 2913 T 06535 E R o t a t e + 152.95° 61 T + 5290 T T 52 E 62 T - 1812 T - 307 T 06535 E R o t a t e + 35.90° 63 T + 3572 T T 52 E 64 T - 1166 T - 830 T 06535 E R o t a t e + 17.95° 65 T + 1400 T T 52 E 66 T + 1263 T + 1921 T 06535 E R o t a t e + 333.20° 67 T + 837 T T 52 E 68 T T 3382 E 69 T. - 797 E 70 T - 8103 T - 5303 T 06535 E R o t a t e + 153.20° 71 T + 1400 T T 52 E 72 T - 1216 T + 830 T 06535 E R o t a t e + 17.95° 73 T + 3772 T T 52 E 74 T - 1912 T + 307 T 06535 E R o t a t e + 35.90° 75 T + 5240 T T 52 E 76 T - 3884 T + 2475 T 06535 E 77 T - 8000 T T 0255 E R o t a t e + 332.95° MECH. ENG. 3-69 FORM 728 NUMERICAL TAPE CONTROL PROGRAM DEPARTMENT OF MECHANICAL ENGINEERING, UNIVERSITY OF BRITISH COLUMBIA P R E P A R E D BY gT^J D A T E 7 Feb. 70 P A R T NAME Three F i b r e M i c h e l l C a n t i l e v e r WORK O R D E R NO. C H E C K E D BY D A T E REMARKS: With 3/8" Radius C u t t e r . O r i g i n 8.000" l e f t of T u r n t a b l e C e n t r e . S H E E T 1 ° F 2 T A P E NO. SEQ. NO. TA B OR EOB + OR "X" I N C R E M E N T T A B OR EO B + OR "Y" 1N C R EM E N T T A B OR EOB "M" FUNCT. E.O. B. INSTRUCTIONS E -% E 1 T + 8000 T T 55 E 2 T - 3448 T - 1761 T 0655 E 3 T + 4179 T T 52 E 4 T T + 2086 E 5 T + 1593 T + 862 T 535 E 6 T + 522 T T 52 E 7 T T + 216 E 8 T - 1165 T - 2660 T 535 E 9 T T + 1302 T 52 E 10 T + 1468 E 11 T - 2524 T - 2049 T 06535 E 12 T + 2011 T T 52 E 13 T T + 123 E 14 T - 5272 T - 123 T 06535 E 15 T + 2011 T T 52 E 16 T - 2011 T T 535 E 17 T T + 123 T . 52 E 18 T + 1905 T + 120 T 06535 E 19 T + 4179 T T 52 E 20 T - 5129 T + 504 T 535 E 21 T T + 1302 T 52 E 22 T + 950 T - 1806 T 535 E 23 T T + 2086 T 52 E 24 T - 2418 T - 280 T 535 E 25 T + 1468 T T 52 E 26 T - 1165 T + 1142 T 535 E MECH. ENG. 3-69 FORM 728 NUMERICAL TAPE CONTROL PROGRAM 117 DEPARTMENT OF MECHANICAL ENGINEERING, UNIVERSITY OF BRITISH COLUMBIA PREPARED BY DATE PART NAME WORK ORDER NO. CHECKED BY DATE REMARKS: SHEET 2 0 F 2 TAPE NO. SEQ. " NO. TAB OR EOB + OR " X " INCREMENT TAB OR EO B + OR " Y " INCREMENT TAB OR EOB " M " FUNCT. E.O. B. INSTRUCTIONS -27 T + 522' T T 52 E 28 T - 522 T T 535 E ; 29 T T + 216 T 52 E 30 T - 5154 T • - 1403 T D6535 E 31 T + 8000 T T 55 E 32 T + 4904 T - 625 T 55 E 33 T + 846 T T 52 E 34 T T + 1250 E 35 T - 846 E 36 T - 8987 T - 3412 T D6535 E 37 T + 5489 T T 52 E 38 T - 1806 T - 307 T 36535 E 39 T + 3566 T T 52 E - 40 T - 1166 T - 830 T D6535 E 41 T + 1000 T T 52 E 42 T + 1447 T + 1774 T 06535 E 43 T + ; 1303 T T 52 E 44 T T + 4300 E 45 T - 1303 E - 46 T - 7447 T - 6074 T 06535 E 47 T + 1000 T T 52 E : 48 T - 966 T + 830 T 06535 E 49 T + 3522 T T 52 E 50 T - 1762 T + 307 T 06535 E • 51 T + 5289 T T 52 E -• 52 T - 4083 T f 2787 T 06535 E 53 T - 8000 T T 0255 E MECH. ENG. 3-69 FORM 728 

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