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The lateral torsional buckling of open thin-walled beams DeVall, Ronald Homer 1972

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THE LATERAL TORSIONAL BUCKLING OF OPEN THIN-WALLED BEAMS by RONALD HOMER DEVALL B . A . S c . , U n i v e r s i t y o f B r i t i s h Columbia , 1966 M . A . S c . , U n i v e r s i t y o f B r i t i s h Columbia , 1968 A THESIS SUBMITTED IN PARTIAL FULFILMENT OF THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY i n the Department o f C i v i l E n g i n e e r i n g We accept t h i s t h e s i s as conforming t o the r e q u i r e d s t a n d a r d THE UNIVERSITY OF BRITISH COLUMBIA A p r i l 1972 In p r e s e n t i n g t h i s t h e s i s in p a r t i a l f u l f i l m e n t o f the r e q u i r e m e n t s f o r an a d v a n c e d d e g r e e a t t h e U n i v e r s i t y o f B r i t i s h C o l u m b i a , I a g r e e t h a t t h e L i b r a r y s h a l l make i t f r e e l y a v a i l a b l e f o r r e f e r e n c e and s t u d y . I f u r t h e r a g r e e t h a t p e r m i s s i o n f o r e x t e n s i v e c o p y i n g o f t h i s t h e s i s f o r s c h o l a r l y p u r p o s e s may be g r a n t e d by t h e Head o f my D e p a r t m e n t o r by h i s r e p r e s e n t a t i v e s . I t i s u n d e r s t o o d t h a t c o p y i n g o r p u b l i c a t i o n o f t h i s t h e s i s f o r f i n a n c i a l g a i n s h a l l no t be a l l o w e d w i t h o u t my w r i t t e n p e r m i s s i o n . The U n i v e r s i t y o f B r i t i s h C o l u m b i a V a n c o u v e r 8 , Canada Depar tment i i THE LATERAL TORSIONAL BUCKLING OF OPEN THIN-WALLED BEAMS ABSTRACT T h i s t h e s i s i s concerned w i t h the development o f a s t i f f n e s s m a t r i x f o r the study o f a l a r g e range o f s t a b i l i t y problems f o r beams o f a r b i t r a r y , open, t h i n - w a l l e d cross s e c t i o n s . This i s done by f i r s t d e v e l o p i n g , u s i n g a c o n s i s t e n t s e t o f common e n g i n e e r i n g assumpt ions , a n o n - l i n e a r r e l a t i o n between the forces and the displacements o f the beam. These r e l a t i o n s are then s u b s t i t u t e d i n t o the beam e q u i l i b r i u m equat ions to g i v e a se t o f t h r e e d i f f e r e n t i a l equat ions o f e q u i l i b r i u m i n terms o f the d i s p l a c e m e n t s . These d i f f e r e n t i a l equat ions are s o l v e d u s i n g an i t e r a t i o n t e c h n i q u e . A member s t i f f n e s s m a t r i x i s generated when the i t e r a t e d s o l u t i o n i s used w i t h the n o n - l i n e a r d e f l e c t i o n r e l a t i o n s . The r e s u l t i n g fourteen by fourteen m a t r i x i n c l u d e s the r e g u l a r s i x forces plus a bi-moment a t each end. The m a t r i x i s t e s t e d a g a i n s t known s o l u t i o n s and agreement i s seen to be e x c e l l e n t i n a l l cases . A l l the terms necessary f o r the b u i l d i n g o f the m a t r i x are g iven i n the Appendices . TABLE OF CONTENTS TITLE PAGE ABSTRACT TABLE OF CONTENTS LIST OF FIGURES DEFINITIONS OF SYMBOLS ACKNOWLEDGEMENTS CHAPTER 1 INTRODUCTION CHAPTER 2 PRELIMINARIES FOR DEVELOPMENT OF DIFFERENTIAL EQUATIONS 1 Displacements 2 S t r a i n s 3 C o n s t i t u t i v e Laws 4 E q u i l i b r i u m Equations CHAPTER 3 ASSEMBLY OF DIFFERENTIAL EQUATIONS 1 Discussion o f Displacement and S t r a i n 2 S t r e s s - S t r a i n Equations 3 R e l a t i n g Internal Stresses to External S t r e s s Vectors 4 R e l a t i n g the Surface S t r e s s Vector to the O v e r a l l Beam Force Resultants 5 The D i f f e r e n t i a l Equations o f S t a b i l i t y I V CHAPTER 4 SOLUTION OF DIFFERENTIAL EQUATIONS CHAPTER 5 FORCE DEFLECTION EQUATION CHAPTER 6 NUMERICAL EXAMPLES CHAPTER 7 DISCUSSION 1 Small R o t a t i o n Theory 2 Secondary S t r e s s e s 3 Constants 4 Loads 5 D i f f e r e n t i a l Equat ions 6 Symmetry and Conservat iveness 7 Approximat ions f o r Small -Terms 8 P o i n t o f A c t i o n o f A x i a l Load CHAPTER 8 CONCLUSIONS BIBLIOGRAPHY APPENDIX I THE MATRIX Page 58 73 82 91 91 91 92 96 96 97 98 99 104 106 107 APPENDIX I I TERMS NEEDED IN MEMBER MATRIX 116 V LIST OF FIGURES F i g u r e Page 1 Undeformed Beam Segment i n oc , ^) ^ C o - o r d i n a t e System 5 2 C o - o r d i n a t e s f o r P o i n t s on the Cross S e c t i o n 7 3 D e f l e c t i o n due to R o t a t i o n j3 8 4 P r i n c i p a l A x i s xt, o f Element <)s o f Cross S e c t i o n 12 5 Element taken from Cross S e c t i o n 18 6 Element r e l a t i n g Ti and (T^ 20 7 Beam S e c t i o n w i t h R e s u l t a n t Forces 21 8 Deformed and Undeformed Cross S e c t i o n 23 9 C y l i n d e r under End R o t a t i o n 30 10 G e n e r a l i z e d Displacements f o r the Beam 59 11 Beam Segment and Displacement W 75 12 Three Methods o f Connect ion f o r E c c e n t r i c A x i a l Load f o r I S e c t i o n 100 v i LIST OF SYMBOLS Denotes d i f f e r e n t i a t i o n w i t h r e s p e c t to ^-Page o f D e f i n i t i o n *• Denotes d i f f e r e n t i a t i o n w i t h r e s p e c t t o « Denotes d i f f e r e n t i a t i o n w i t h r e s p e c t to s A Area o f cross s e c t i o n 44 BM,- Bending moments about a x i s 21 8 W Bi-moment i n 'X>L^ a x i s 77 r ' - Constants i n i t e r a t i o n scheme 63 —J d s D i f f e r e n t i a l l e n g t h o f s 6 c J S , d S 0 Deformed, undeformed e lemental areas 15 d Ti A c t u a l s t r e s s v e c t o r 15 E-u Green's s t r a i n t e n s o r 13 C , j Almansi 's s t r a i n t e n s o r 14 B Young's modulus 16 f\' G e n e r a l i z e d f o r c e 58 F u n c t i o n o f ~y 27 Funct ions o f M<*, Q, y i n i t e r a t i o n scheme 66 v i i Page o f D e f i n i t i o n r.' Funct ions o f f f , f~f -f l n i t e r a t i o n scheme 69 G , G ' ; X 1 X E l a s t i c constants 15 G i ( x , f , j ^ F u n c t i o n o f X->t>}- 32 l i e , I^c. P r i n c i p a l c e n t r o i d a l moment o f i n e r t i a 46 X P P o l a r moment o f i n e r t i a , l a t e r about the s h e a r c e n t r e 50 K ; Constants used to d e f i n e s e c t i o n p r o p e r t i e s a l o n g w i t h I X 6 | I f o K , ~ So <P L fc ( =• L 75 sin GO J s L Length o f s e c t i o n M Secondary moment per u n i t l e n g t h cis 16 r*1; Constants i n d i m e n s i o n l e s s d i f f e r e n t i a l e q u a t i o n s M, = - PLVN.. M» - ' ' PL£/N, MLT= PIPLVA 46 N 3 61 Page o f D e f i n i t i o n Constants f o r cross s e c t i o n p r o p e r t i e s , c o n s i s t s o f combinations o f 1 , K; M, =• E (- - ki) - ET-N, = E C i f C *Kz, - EI R N 3 ~- E(K,3 + £K, . -Kn-K,») N, = J G - C N a =• EC^K,, - k w - 2 K O where 5 X C j , C , K , 3 ^ k . o , k „ K u a r e due to membrane a c t i o n , and a l l the r e s t are due t o secondary o r p l a t e bending s t r e s s e s u 5 4 A x i a l l o a d a l o n g a x i s 2 1 Force f lows on element . d a d s 1 7 P e r p e n d i c u l a r d i s t a n c e from o r i g i n to tangent a t p o i n t s 6 i x Page o f D e f i n i t i o n 5 C o - o r d i n a t e a long m i d - t h i c k n e s s o f the cross s e c t i o n 6 S0) S I n i t i a l and f i n a l values o f S 6 S , j K i r c h o f f s t r e s s t e n s o r 14 t T h i c k n e s s o f cross s e c t i o n 6 T, T*, T 3 Elemental f o r c e r e s u l t a n t s i n 'X-^tfx* % 17 T», T*, T 3 E lemental f o r c e r e s u l t a n t s i n oc, y 21 T Torque a l o n g y 21 ^66 j - v , w Displacements i n f i x e d g l o b a l system 4 Displacement o f o r i g i n a l o n g x,, , 11 - ^ - a . ; ^ z . Displacements o f elements ds a l o n g x.x,f>. 11 -^•&>' 1 - r s Displacement o f elements d s a l o n g ^ 11 > v 1 ) |" Dimensionless values o f M~} v*, fr Jo,} A / , F i r s t s o l u t i o n f o r J£c • 64 ^ t l ) '"A. F i r s t i t e r a t i o n f o r JX,^ 66 W Displacement a l o n g y 4 V v 0 Displacement w o f p o i n t S - o 6 61 X Page o f D e f i n i t i o n W N o n - l i n e a r w F i x e d g l o b a l c o - o r d i n a t e s y s t e m , used to d e f i n e p o s i t i o n o f p o i n t s on m i d - t h i c k n e s s C o - o r d i n a t e s o f p o i n t s s= o i n ^ - j f > ^ F i x e d a x i s d e f i n e d f o r each element J s ; p a r a l l e l t o p r i n c i p a l axes o f ds ; common o r i g i n w i t h J C ^ - . i , fx. F i x e d p r i n c i p a l axes o f element <Js s . A 1, ^ V C o - o r d i n a t e s o f c e n t r o i d i n oc^^-fr D i s p l a c e d axes s i m i l a r t o :c,>f)'^' R o t a t i o n a l d isp lacement about 3/ 4>~JB G e n e r a l i z e d d isp lacements V a r i a t i o n i n £ Deformed, undeformed normal Deformed, undeformed d e n s i t y A x i s t h a t t r a n s l a t e s b u t does not r o t a t e w i t h s e c t i o n 27 4 40 11 11 46 31 4 22 58 74 62 15 15 22 x i 0 Angle between x a x i s and ds (py e Funct ions o f % , /Cr", - cpc-! Ct\ Page o f D e f i n i t i o n 46 14 i£-w 3>„ Quasi-moments o f i n e r t i a C T f j A c t u a l s t r e s s t e n s o r Force f low 17 <3^ 3 ~ (T A c t u a l s t r e s s i n ^ d i r e c t i o n 19 A L ' ^ W . J S 44 [cp'JjCtff] Geometric m a t r i x o f 0 e v a l u a t e d a t 3=0, L 63 C©?],L©il Geometric m a t r i x o f © e v a l u a t e d a t ^ = o , L 63 C - f l " ] _ O _ i n t e g r a t e d t w i c e 67 L - O - l Means J& = t^Cl s a t i s f i e s 3 - X2>6 = LsiS] 67 ACKNOWLEDGEMENTS The a u t h o r wishes to thank h i s s u p e r v i s o r , Dr . R . F . H o o l e y , f o r h i s i n v a l u a b l e a s s i s t a n c e and encouragement d u r i n g the development o f t h i s work. G r a t i t u d e i s a l s o expressed to the N a t i o n a l Research C o u n c i l o f Canada f o r f i n a n c i a l s u p p o r t i n the form o f a Research A s s i s t a n t s h i p . 1 CHAPTER 1 INTRODUCTION The l a t e r a l t o r s i o n a l b u c k l i n g o f beams has been a s u b j e c t o f i n t e r e s t f o r many y e a r s . The f i r s t formal i n v e s t i g a t i o n began i n 1899 when P r a n d t l and M i c h e l l i n d e p e n d e n t l y developed the d i f f e r e n t i a l equat ions o f l a t e r a l t o r s i o n a l b u c k l i n g o f t h i n r e c t a n g u l a r s e c t i o n s . A few y e a r s l a t e r Timoshenko developed the equat ions f o r an I beam. From t h a t t ime o n , many authors c o n t r i b u t e d to the f i e l d and expanded the scope o f the e q u a t i o n s . Much work was done by G o o d i e r , Timoshenko, B l e i c h and V l a s s o v [ 5 ] , [7]» [ 3 ] , [ 9 ] . The r e s u l t o f t h i s work has been to d e v e l o p , by v a r i o u s methods, the d i f f e r e n t i a l e q u a t i o n s o f b u c k l i n g f o r an a r b i t r a r y open cross s e c t i o n under v a r i o u s l o a d i n g s . S o l u t i o n s to these e q u a t i o n s have been found f o r s e l e c t e d end c o n d i t i o n s and loads but i t was not u n t i l r e c e n t l y t h a t G a l l a g h e r and Barsoum [ 2 ] p u b l i s h e d a m a t r i x f o r m u l a t i o n based on an assumed d i s p l a c e m e n t f i e l d and minimum p o t e n t i a l energy. Barsoum [1] f o l l o w e d up w i t h a dynamic approach u s i n g the H a m i l t o n i a n as the s t a t i o n a r y f u n c t i o n a l to determine the parameters i n the assumed displacement f i e l d , and t h i s a l l o w e d the t reatment o f n o n - c o n s e r v a t i v e l o a d s . I t i s the purpose o f t h i s t h e s i s to d e v e l o p , by d i r e c t t reatment o f the d i f f e r e n t i a l e q u a t i o n s o f e q u i l i b r i u m , a s t i f f n e s s m a t r i x to c a l c u l a t e the b u c k l i n g l o a d o f v a r i o u s t h i n w a l l e d open cross s e c t i o n e d s t r u c t u r e s . The d i f f e r e n t i a l e q u a t i o n s w i l l be developed under the a p p l i c a t i o n o f a c o n s i s -t e n t s e t o f common e n g i n e e r i n g assumpt ions . The r e s u l t i n g m a t r i x w i l l be v a l i d under t h i s range o f assumpt ions . The d i f f e r e n t i a l e q u a t i o n s w i l l be developed u s i n g e q u i l i b r i u m e q u a t i o n s , s t r a i n e q u a t i o n s and c o n s t i t u t i v e l a w s . The r e s t r i c t i o n s p l a c e d on the development will be the m a t e r i a l remains e l a s t i c , the cross s e c t i o n r e t a i n s 2 i t s o v e r a l l geometr ic shape, the r o t a t i o n s are smal l w i t h r e s p e c t t o one, and t h e i r squares smal l w i t h r e s p e c t to themselves . As the equat ions are developed the e f f e c t o f s m a l l e r o r d e r terms w i l l be s t u d i e d and i n l i g h t o f the r e s t r i c t i o n s on r o t a t i o n s , w i l l be kept o r d i s c a r d e d . The d i f f e r e n t i a l equat ions w i l l then be i t e r a t e d to o b t a i n a s o l u t i o n . T h i s s o l u t i o n w i l l be used to c o n s t r u c t a f o r c e d e f l e c t i o n r e l a t i o n . S e v e r a l examples w i l l then be t r e a t e d and compared to known s o l u t i o n s . I t w i l l be seen t h a t agreement i s very good i n a l l t e s t e d c a s e s . The advantage o f t h i s approach l i e s i n the i n c r e a s e i n the number o f problems which become t r a c t a b l e . For i n s t a n c e , a r b i t r a r y boundary c o n d i t i o n s , v a r y i n g s e c t i o n p r o p e r t i e s and a r b i t r a r y l o a d i n g s cease to be a problem and are e a s i l y t r e a t e d by the s t i f f n e s s m a t r i x approach. 3 CHAPTER 2 PRELIMINARIES FOR DEVELOPMENT OF DIFFERENTIAL EQUATIONS The development o f the d i f f e r e n t i a l equat ions f o r the s e c t i o n w i l l be done employing e q u i l i b r i u m e q u a t i o n s , e l a s t i c c o n s t i t u t i v e laws and s t r a i n -d i s p l a c e m e n t e q u a t i o n s . F i r s t the d i s p l a c e m e n t and s t r a i n - d i s p l a c e m e n t equat ions w i l l be w r i t t e n . The r e l a t i o n s h i p between s t r a i n and s t r e s s w i l l then be examined. F i n a l l y the e q u i l i b r i u m equat ions w i l l a l l be w r i t t e n . These s e p a r a t e s e t s o f equa-t i o n s w i l l a l l be assembled u s i n g s u i t a b l e e n g i n e e r i n g a p p r o x i m a t i o n s and c o n s t r a i n t s to o b t a i n the governing d i f f e r e n t i a l equat ions f o r the problem o f l a t e r a l t o r s i o n a l b u c k l i n g . These d i f f e r e n t i a l equat ions w i l l take the form o f the o v e r a l l e q u i l i b r i u m equat ions o f a d i s p l a c e d element o f beam l e n g t h w r i t t e n i n terms o f d isp lacement d e r i v a t i v e s . For i n s t a n c e , the w e l l -known E u l e r e q u a t i o n f o r column b u c k l i n g , i s o f t h i s form. The r e s t r i c t i o n s imposed by the assumed c o n s t r a i n t s w i l l be s t u d i e d t o o u t l i n e the domain o f v a l i d i t y o f the e q u a t i o n s . The a c t u a l s t e p s o f the development are as f o l l o w s : The d isp lacements w i l l be used to c a l c u l a t e the s t r a i n s . C o n s t i t u t i v e laws w i l l then be used t o o b t a i n the s t r e s s e s i n terms o f the s t r a i n s . The s t r e s s e s w i l l be i n t e -g r a t e d o v e r the c r o s s s e c t i o n t o get the f o r c e r e s u l t a n t s , which w i l l now be i n terms o f the s t r a i n s and hence the d i s p l a c e m e n t s . S u b s t i t u t i o n o f these r e s u l t a n t s i n t o the o v e r a l l e q u i l i b r i u m equat ions g ives the d e s i r e d r e s u l t . These s teps r e q u i r e a d e t a i l e d study o f the d i s p l a c e m e n t s , s t r e s s - s t r a i n o 4 and e q u i l i b r i u m e q u a t i o n s . The f i r s t group to be s t u d i e d w i l l be the d i s -placement and s t r a i n e q u a t i o n s . 1 : 1 Displacements The main displacements i n v o l v e d are d e t a i l e d i n F i g u r e 1, which shows the beam segment undeformed and i n i t s i n i t i a l p o s i t i o n . The segment has c o n s t a n t p h y s i c a l dimensions a l o n g i t s l e n g t h and c a r r i e s no l o a d between i t s ends. The axes x , ^ , ^ are the g l o b a l axes and are f i x e d a l o n g the i n i t i a l , undeformed p o s i t i o n o f the beam. The 'fr a x i s l i e s a l o n g the l e n g t h o f the beam and the and ^ axes d e f i n e the c o - o r d i n a t e s o f p o i n t s on the m i d - t h i c k n e s s o f the cross s e c t i o n . The displacements M~,v a r e i n the d i r e c -t i o n s o f oc)f r e s p e c t i v e l y and are the displacements o f the p o i n t on the c r o s s s e c t i o n which l i e s on the .^ a x i s when undeformed. The d i s p l a c e m e n t w i s the d i s p l a c e m e n t i n the fr d i r e c t i o n o f a p o i n t on the cross s e c t i o n . S i n c e the c r o s s s e c t i o n i s assumed t o r e t a i n i t s i n i t i a l shape o n l y one d i s -placement j 3 i s r e q u i r e d to d e f i n e i t s r o t a t i o n about the y a x i s . The d isp lacements M~ , i/~, w and Ji are not a l l i n d e p e n d e n t , as the a x i a l d i s p l a c e m e n t W can be r e p r e s e n t e d by yU-> v> Ji and the cross s e c t i o n p r o p e r t i e s . The next s tep i s t o develop t h i s r e l a t i o n between , Ji and W . T h i s w i l l be done s u b j e c t to the c o n d i t i o n s t h a t the c r o s s s e c t i o n r e t a i n s i t s shape and the shear s t r a i n s a t m i d - t h i c k n e s s a r e z e r o . T h i s does n o t p r e c l u d e o u t o f p lane warping of the cross s e c t i o n . A n g u l a r d isp lacements a r e assumed s m a l l w i t h r e s p e c t to one and so t h e i r products are taken to be n e g l i g i b l e compared to themselves . T h i s a l l o w s the r o t a t i o n s to be t r e a t e d as v e c t o r s . The consequences o f t h i s assumption w i l l be examined l a t e r . A c o - o r d i n a t e system d e f i n i n g the p o s i t i o n o f p o i n t s on the cross s e c t i o n i s g i v e n i n F i g u r e 2. The axes oc^^-v, w i t h the a s s o c i a t e d d i s p l a c e -5 F I G . I U N D E F O R M E D B E A M S E G M E N T I N x , y , z C O - O R D I N A T E S Y S T E M . 6 merits W ; i s the f i x e d global system p r e v i o u s l y d e f i n e d . The co-ordinate 5 i n d i c a t e s d i s t a n c e along the c e n t r e - l i n e o f the cross s e c t i o n from the o r i g i n o f 5 . This o r i g i n i s taken at any convenient f r e e edge. Figure 2a shows one p o s s i b l e o r i g i n and d i r e c t i o n o f 5 , while Figure 2b gives an a l t e r n a t e o r i g i n and shows 3 running i n the opposite d i r e c t i o n . The thickness t may vary with s . Both the systems and 3 may be used to l o c a t e points on the cross s e c t i o n . The d i s t a n c e T i s the d i s t a n c e per-p e n d i c u l a r to the tangent l i n e of the p o i n t o f i n t e r e s t to an a r b i t r a r y o r i g i n , i n t h i s case the o r i g i n of The d i s t a n c e f has an a s s o c i a t e d s i g n : p o s i t i v e i f the swept area r ds i s clockwise about the o r i g i n , negative i f a n t i c l o c k w i s e . The d i s t a n c e f may be w r i t t e n i n terms o f oct tJ, and the angle <ft and have the sign a u t o m a t i c a l l y accounted f o r i f <f> i s d e f i n e d the f o l l o w i n g way: The angle j£ i s taken to be the angle between the p o s i t i v e d i r e c t e d j c axis and the p o s i t i v e d i r e c t e d element a5 and i s measured p o s i t i v e clockwise. T h i s means 0 i s the angle between the vectors X and ds , where ds i s p a r a l l e l to the tangent at 5 . With t h i s d e f i n i t i o n o f 4> > f may be w r i t t e n as: r =• <^  cos(<fi) +• oc s i n (<f>) Using these q u a n t i t i e s , the d i s p l a c e d shape vV can be found from the f o l l o w i n g reasoning. I f the q u a n t i t y were known, where w i s a f u n c t i o n o f 5 , then W d> 5 would be: W — W o + J" 5? <J <3w where \A/0 i s the displacement o f the p o i n t 5= o. To f i n d , three 7 F I G . 2 C O - O R D I N A T E S F O R P O I N T S ON T H E C R O S S S E C T I O N . 1M-displacements 77 , ~ and -p: must be c o n s i d e r e d . F i r s t , c o n s i d e r a p lane s e c t i o n r o t a t i o n i£ . T h i s has a s l o p e a long 5 o f ~ c o s 4> S e c o n d l y , c o n s i d e r the plane s e c t i o n r o t a t i o n . T h i s h a s ' a s l o p e a l o n g 5 o f 5in (j> . The t h i r d d e f l e c t i o n i s s l i g h t l y more c o m p l i c a t e d , d 2 F i g u r e 3 shows a smal l s t r i p o f the beam d s i n w i d t h d e f l e c t e d by a v a r y i n g w i t h ^ . FIG.3 DEFLECTION DUE TO R O T A T I O N /3 . C l e a r l y , the d e f l e c t i o n at 5 due to Ji \s> J& ? and i n the d i r e c t i o n o f the tangent a t 5 . From t h i s , the s l o p e due t o a JZ d e f l e c t i o n v a r y i n g w i t h ^ i s : Because o f the l i m i t a t i o n s on the r o t a t i o n s , these t h r e e s l o p e s may be added t o o b t a i n 3W — — C\JLL c o s <p +• cUr s i n which i s based on the s h e a r s t r a i n s b e i n g zero a t m i d - t h i c k n e s s . 9 F i n a l l y , t h i s may be w r i t t e n as: d w — c o s 4> ds + / i / * ' S i n & ds + r^ tf ds where d w i s the d i f f e r e n t i a l change o f the displacement i n the ^ d i r e c t i o n at any p o i n t 5 on the centre l i n e o f the cross s e c t i o n f o r any change c l 5 i n the 5 co-ordinate. The primes denote d i f f e r e n t i a t i o n with respect to %. Equation (1) i s developed i n s e v e r a l r e f e r e n c e s . See f o r example [6] and [9]. The f i r s t two terms on the right-hand s i d e o f Equation (1) are the con-t r i b u t i o n s o f bending with plane s e c t i o n s remaining plane. The t h i r d term represents out o f plane warping. I t should be remembered t h a t Equation (1) i s v a l i d o nly i f terms such as M.AT are n e g l i g i b l e compared to M- j Af , thus a l l o w i n g J^',/v', J$ to be t r e a t e d as v e c t o r s . The d e f l e c t i o n W can be obtained by i n t e g r a t i o n of (1) with r e s p e c t to 5 . W = W 0 — ^u! \ cos 0 ds -+- /v'j sin <p ds + J l )X Js (2) where w 0 i s the displacement of the point s=o. I t may be noted here t h a t 5= o does not need to f a l l on the edge o f the cross s e c t i o n , i t may be placed anywhere on the s e c t i o n . In t h i s development, however, i t w i l l be l e f t a t any a r b i t r a r y edge. Equation (2) may be s i m p l i f i e d by using the d i f f e r e n t i a l r e l a t i o n s : if - - s m 0 d s (3) d x _ + c o s $ ds 10 from F i g u r e 2. T h i s g i v e s : u s i n g where oc,y and x., ^a are the c o - o r d i n a t e s o f the p o i n t s 5 and 5 = o r e s p e c t -i v e l y and d e f i n i n i n g OJ , = ) r Js (e) o a l l o w s E q u a t i o n (4) to be w r i t t e n W = W o + Jbu>, + yic'CoCo - D C ) - r - ^ f ^ o ^ E q u a t i o n (7) d e s c r i b e s the deformations a l o n g the y a x i s o f the c e n t r e -l i n e o f the c r o s s s e c t i o n and can be used t o get s t r a i n s and t h e r e f o r e s t r e s s e s . I t s h o u l d be remembered t h a t terms o f the o r d e r o f AJJAT and JZAT' were taken as n e g l i g i b l e compared to terms such a s y ^ ' ; / i / ' i n t h i s development i n o r d e r t o add the angles v e c t o r i a l l y . T h e r e f o r e E q u a t i o n (7) as i t s tands i s p u r e l y l i n e a r . S i n c e i t i s d e s i r e d t o t r e a t a n o n - l i n e a r p r o b l e m , terms r e p r e s e n t i n g a x i a l f o r e s h o r t e n i n g due to r o t a t i o n s s h o u l d be i n c l u d e d , but s i n c e t h i s i s a c o m p l i c a t e d p r o c e d u r e , and s i n c e these terms w i l l be shown t o be o f no consequence f o r c e r t a i n c o n d i t i o n s , they w i l l be i n t r o d u c e d , t r e a t e d and d i s c a r d e d a t a l a t e r s t a g e . In a d d i t i o n to the d e f l e c t i o n W t h e r e are some secondary deformations a c r o s s the t h i c k n e s s o f the cross s e c t i o n . They r e p r e s e n t the e f f e c t o f p l a t e bending o f the element o f c r o s s s e c t i o n ^5 . These deformat ions are due to d e f l e c t i o n s p e r p e n d i c u l a r to the d i r e c t i o n o f d 5 and produce p l a t e 1 1 bending s t r e s s e s whereas the W d e f l e c t i o n produces membrane type s t r e s s e s . For most s e c t i o n s the p l a t e bending s t r e s s e s are i g n o r e d as t h e i r o v e r a l l e f f e c t i s n e g l i g i b l e compared to the e f f e c t o f membrane s t r e s s . However, f o r s e c t i o n s w i t h c e r t a i n g e o m e t r i c a l p r o p e r t i e s , the p l a t e bending s t r e s s may be the o n l y s t r e s s a v a i l a b l e to r e s i s t a p p l i e d l o a d . F o r example, a t h i n r e c t a n g u l a r beam behaves l i k e a p l a t e i n the weak l a t e r a l d i r e c t i o n . For t h i s reason the p l a t e bending deformations w i l l be s t u d i e d and k e p t . To o b t a i n va lues f o r these deformations some new c o - o r d i n a t e systems w i l l be i n t r o d u c e d . In F igure 4, x , c ^ a n d t h e i r a s s o c i a t e d d e f l e c t i o n s Af are the f i x e d g l o b a l system. The new system cct) t^,, and the a s s o c i -ated d e f l e c t i o n s M-^AT, are d e f i n e d such t h a t they are p a r a l l e l to the p r i n c i p a l axes o f a s m a l l element d s o f the cross s e c t i o n and share the same o r i g i n as x ^ . The system x ( J ty, a l s o d e f i n e s the p o s i t i o n o f p o i n t s on the cross s e c t i o n , and M.,,/^, r e p r e s e n t d e f l e c t i o n s o f the o r i g i n i n t h i s system. The second a x i s system ocl> and the a s s o c i a t e d d e f l e c t i o n s , are d e f i n e d t o be the p r i n c i p a l axes and the d e f l e c t i o n s r e s p e c t i v e l y o f the s m a l l element d s o f the c r o s s s e c t i o n . T h e r e f o r e , X / , y/, and sc.*, tj.L are p a r a l l e l to each o t h e r but not t o oCj^, . I t i s c l e a r t h a t s i n c e each element ds o f the c r o s s s e c t i o n must be p a r a l l e l to the tangent to 5 a t 5 , then -^/ j f> and x,t) ^ can be r e l a t e d t o X , <j/ by the angle <f> , which the tangent makes w i t h the x. a x i s . The axes oc, . ty, , and D C ^ ; ^ always remain i n the p lane o f oc,1^/ and do n o t d i s p l a c e w i t h the s e c t i o n . These axes s u f f i c e to determine the p l a t e bending d e f o r m a t i o n o f i n t e r e s t , which i s M.^ . The a l g e b r a i c r e l a t i o n between the axes a r e as f o l l o w s : X , =• oc sin (f> . + y cos cj> 12 FIG.4 PRINCIPAL AXIS x 2 , y 2 OF ELEMENT ds OF CROSS SECTION . 1 3 x c o s <f> + y> sm <f> Jl sm <j> -r- AT" C O S 96 (8) - M. cos $ -r- AT' sin (fi I t i s c l e a r t h a t s i n c e the s e c t i o n r e t a i n s i t s geometric shape , the f o l l o w i n g r e l a t i o n s h o l d : Al^ — M.,-J>y*~ M- S\Y)(j> - r / v ^ C O S if -J>(-X, COb<p-r f S / KV <P ) f \ ( 9 ) = + y3 x, cos c* + irsm <j> TJ$ S\n <f> + f COS 0 ) $ - Ms - j3 ^ } sir's - AS* -Y JZCO where M>sysVs are the d isplacements i n the A a n d / / ' d i r e c t i o n s o f the p o i n t 5 . D e r i v a t i v e s o f and are e a s i l y found by d i f f e r e n t i a t i n g (9) w i t h r e s p e c t to y and remembering t h a t <fi)x,}<j' are independent o f f o r each e lementa l s e c t i o n . T h i s completes the development o f the necessary d i s p l a c e m e n t s . 1:2 S t r a i n s The s t r a i n s necessary t o c a l c u l a t e the r e q u i r e d s t r e s s e s can be found by d i f f e r e n t i a t i o n o f the displacements i n accordance w i t h the s t r a i n t e n s o r chosen. S i n c e t h i s i s a n o n - l i n e a r development, i t would be w i s e to s t a r t o u t w i t h n o n - l i n e a r s t r a i n s and make any approximat ions l a t e r . Two common s t r a i n tensors are Green's t e n s o r E\j and A l m a n s i ' s t e n s o r Mi ~ AS, -=-14 . See, f o r example, Fung, Chapter 4 [ 4 ] . Both these are f i n i t e s t r a i n tensors and d i f f e r o n l y i n the c o - o r d i n a t e system used to r e p r e s e n t them. Green 's t e n s o r i s w r i t t e n i n terms o f the undeformed body c o - o r d i n a t e s , A l m a n s i ' s i s w r i t t e n i n terms o f the deformed body c o - o r d i n a t e s . For ex-ample, the a x i a l s t r a i n w r i t t e n i n terms o f the Green's t e n s o r and u s i n g the c o - o r d i n a t e system C C , ^ , y d e f i n e d p r e v i o u s l y becomes: where the squared terms c o n t a i n i n g ^ / 3 > / t / ^ and w i n (10) can be e v a l u a t e d by d i f f e r e n t i a t i n g the r e l a t i o n s g iven i n Equat ions (7) and ( 9 ) . The t e n s o r used i n t h i s development w i l l be the A l m a n s i , as i t r e l a t e s to the a c t u a l s t r e s s a n d , f o r reasons which w i l l become apparent l a t e r , i t i s the a c t u a l s t r e s s which w i l l be d e s i r e d . The Green 's t e n s o r was shown above merely to g i v e some i d e a o f the form o f the s t r a i n o f i n t e r e s t , and because the Green's t e n s o r can be w r i t t e n i n terms o f c o - o r d i n a t e systems a l r e a d y d e f i n e d . A t t h i s p o i n t , the s t r a i n and d isp lacement equat ions are complete . The p r i n c i p a l d e f l e c t e d shape has been developed and, u s i n g equat ions s i m i l a r to ( 1 0 ) , a l l the s t r a i n s may be o b t a i n e d . The secondary p l a t e bending deform-a t i o n s have been i n t r o d u c e d and d e t a i l e d i n F i g u r e 4 . 1:3 C o n s t i t u t i v e Laws r I t i s now o f i n t e r e s t to o b t a i n the r e l a t i o n s between the s t r e s s e s i n the c r o s s s e c t i o n and the s t r a i n s . S i n c e the s t r a i n s w i l l be d e f i n e d f o r f i n i t e - d e f o r m a t i o n , the s t r e s s t e n s o r s w i l l have to be d e f i n e d t o match. To do t h i s , Chapter 16, Fung [ 4 ] , w i l l be used t o p r o v i d e a l l the necessary d e f i n i t i o n s and r e l a t i o n s . Two s t r e s s t e n s o r s , S.-j , the K i r c h o f f s t r e s s , and 07j, 15 the a c t u a l s t r e s s , are d e f i n e d as r e l a t e d to the s t r a i n tensors E>S a n d Q/- by the f o l l o w i n g r e l a t i o n s : S . -a =• A S.-j +• ZGr E .J where A,A ) G - are e l a s t i c c o n s t a n t s . These two s t r e s s tensors are r e l a t e d by the f o l l o w i n g t r a n s f o r m a t i o n : cn,- = ^ i ^ i 5^ (12) T h i s e x p r e s s i o n i s o f course more complex than necessary f o r the problem. As w i t h the s t r a i n s , i t w i l l be d i s c u s s e d and reduced i n a f o l l o w i n g s e c t i o n . In Equat ion ( 1 2 ) , the oc and OL are measured i n the same c a r t e s i a n co-o r d i n a t e s y s t e m , but JG i s i n terms o f the deformed p o s i t i o n and cx i s i n terms o f the undeformed p o s i t i o n . The d e n s i t i e s J and a r e the deformed and undeformed d e n s i t i e s r e s p e c t i v e l y . The s t r e s s tensors are connected t o the a c t u a l s t r e s s v e c t o r d T, by the f o l l o w i n g r e l a t i o n s : Ssi lA> d So — ^ aT* where and d D are the normal v e c t o r and area o f the deformed element and lA, and c\ Spare the normal v e c t o r and area o f the undeformed e lement . Now t h a t the a c t u a l s t r e s s v e c t o r s have been g iven i n terms o f the s t r a i n s and hence i n d i r e c t l y the d i s p l a c e m e n t s , the secondary s t r e s s e s and t h e i r f o r c e 16 d i s p l a c e m e n t r e l a t i o n s h i p can be d i s c u s s e d . These s t r e s s e s , due to the p l a t e bending d e f o r m a t i o n s , can be t r e a t e d i n a much more r e l a x e d way than the s t r e s s e s d e f i n e d above. I t w i l l be assumed t h a t the p l a t e bending s t r e s s e s produce a moment per u n i t l e n g t h M t h a t i s g iven by: where M*T was d e f i n e d i n F i g u r e 4 . The moment a c t i n g on the element taken from the c r o s s s e c t i o n w i l l be d e f i n e d as and w i l l l i e a l o n g the t a n -gent to 5 i n the n e g a t i v e 5 d i r e c t i o n i n the d i s p l a c e d p o s i t i o n . T h i s completes the d i s c u s s i o n o f s t r e s s - s t r a i n o r f o r c e d e f o r m a t i o n r e l a t i o n s . Of the s t r e s s e s d e f i n e d above, o n l y the a c t u a l s t r e s s wi 11 be used. T h i s w i l l be due t o c e r t a i n problems which w i l l a r i s e , n e c e s s i t a t i n g the w r i t i n g o f e q u i l i b r i u m equat ions i n a d i s p l a c e d p o s i t i o n , which r e q u i r e the use o f the a c t u a l s t r e s s . 1:4 E q u i l i b r i u m Equat ions There are s e v e r a l s e t s o f separate e q u i l i b r i u m e q u a t i o n s to be used i n the development. The main ones o f i n t e r e s t are the equat ions r e l a t i n g the f o r c e s on the face o f an element to the o v e r a l l f o r c e r e s u l t a n t s on the beam and the e q u a t i o n s g i v i n g o v e r a l l e q u i l i b r i u m o f the beam segment i n terms o f these r e s u l t a n t s . However, there i s a problem t h a t a r i s e s from the d isp lacements t h a t must be overcome b e f o r e these s e t s o f e q u i l i b r i u m equat ions can be used. R e c a l l t h a t the aim o f t h i s development i s to o b t a i n the o v e r a l l e q u i l i b r i u m equ-a t i o n s i n terms o f the d isp lacements and t h e i r d e r i v a t i v e s . T h i s w i l l be done by r e l a t i n g the f o r c e r e s u l t a n t s on the beam to the s t r e s s e s , which i n t u r n can be found from the s t r a i n s . U n f o r t u n a t e l y , the assumptions on the 17 d i s p l a c e d shape are t h a t the shear s t r a i n s are z e r o . T h u s , o n l y the normal s t r e s s e s can be found from a s t r e s s - s t r a i n law and an a l t e r n a t i v e method must be found t o f i n d the s h e a r s . T h i s can be done by w r i t i n g e q u i l i b r i u m o f a s m a l l d i s p l a c e d element o f s i z e d s by d ^ which w i l l g i v e r e l a t i o n s between s h e a r flows and normal f o r c e s . T h i s w i l l g ive the shears i n terms o f normal f o r c e s . A second s e t o f equat ions w i l l then be developed which w i l l l i n k the normal f o r c e s , and t h e r e f o r e i n d i r e c t l y , the s h e a r s , to the normal s t r e s s i n the beam. S i n c e the normal s t r e s s can be w r i t t e n i n terms o f the s t r a i n s , t h i s a l l o w s both the shear f lows and normal f o r c e s to be w r i t t e n i n terms o f the d i s p l a c e m e n t s . T h i s procedure s u c c e s s f u l l y c i r c u m -vents the problem o f d i r e c t l y w r i t i n g the shears i n terms o f the s t r a i n s , and i t w i l l be developed before the o v e r a l l e q u i l i b r i u m equat ions f o r the beam are deve loped. To develop .the e q u a t i o n s between the shear and f o r c e f l o w s , the s m a l l element o f c r o s s s e c t i o n shown i n F i g u r e 5 w i l l be used. The q u a n t i t i e s T , ~Tt_, "T 5 a r e e lemental f o r c e r e s u l t a n t flows i n the f i x e d a x i s x i i • They a c t on the cross s e c t i o n f a c e . T x may be thought o f as a normal f o r c e , ~ [ z and T 3 as s h e a r s . The q u a n t i t i e s ^T" i j t o and % are a l s o f o r c e r e s u l t a n t f l o w s , and they a c t a l o n g planes o f the element t h a t were i n i t i a l l y p a r a l l e l to the a x i s . These r e s u l t a n t s are a l s o d e f i n e d a l o n g the f i x e d e lemental axes oc£)^/i . The r e s u l t a n t i s taken as b e i n g a l o n g the tangent d s i n the deformed shape. T h i s i s the secondary r e s u l t a n t due t o p l a t e bending e f f e c t s . Any shears a s s o c i a t e d w i t h f^j a r e a u t o m a t i c a l l y taken care o f by T L and T~3 . U s i n g the s t r e s s f lows d e f i n e d i n F i g u r e 5 and w r i t i n g e q u i l i b r i u m g i v e s the f o l l o w i n g e q u a t i o n s : 18 Z,W (a) - element of interest shown in the undisplaced cross - section (b) - enlarged view of element showing it undisplaced and displaced with respect to the element pr inc ipal ax is x2,y2,z . FIG.5 E L E M E N T TAKEN FROM CROSS-SECTION . 19 i F ^ o lis. =, - i X ( 1 5 ) There are o f course t h r e e o t h e r e q u i l i b r i u m e q u a t i o n s , but they are n o t necessary f o r the development and so are not g i v e n . The t h r e e g iven equa-t i o n s r e l a t e the normal and shear f o r c e f lows on the e lement . However, they do not as y e t r e l a t e the f o r c e f lows to the i n t e r n a l s t r e s s e s and t h e r e f o r e the s t r a i n s . S i n c e i t i s d e s i r e d t o get the f lows as a f u n c t i o n o f the s t r a i n s , the f o l l o w i n g s e t o f equat ions w i l l be developed r e l a t i n g CT^ . the a c t u a l s t r e s s i n the y d i r e c t i o n and T , the f o r c e on the element i n the ^ d i r e c t i o n . The d i s t i n c t i o n between the two must be made as "77 i s a c t i n g on a skewed s u r f a c e once the element i s d e f l e c t e d , and t h e r e f o r e need not c o i n c i d e w i t h 6~ty i n v a l u e . U s i n g F i g u r e 6 , the axes X-< , ^KI'Y are the f i x e d e lementa l p r i n c i p a l a x e s . The element i s shown d i s p l a c e d w i t h a smal l c o r n e r removed. From e q u i l i b r i u m o f the c o r n e r : T, =• (Tn i - lo'ASi (18) where - i / ^ i n the l o - l / term a r i s e s from the r a t i o o f areas o f the l i t t l e c o r n e r element and G ~ i s the a c t u a l i n t e r n a l s t r e s s a c t i n g i n the d i s p l a c e d element i n the d i r e c t i o n o f the o r i g i n a l a x e s . In o t h e r w o r d s , i t i s the normal s t r e s s i n the x t ) ^ l c u t p lane o f the c o r n e r e lement . E q u a t i o n (18) i s s i m i l a r to one o f the equat ions i n the wel l -known e l a s t i c i t y e q u a t i o n 20 x 2 , y 2 plane ( b ) (a ) - element of interest shown in undisplaced and displaced position with respect to the element principal axis x 2,y*2,z.The x 2 , y 2 plane is shown cutting the displaced element near a corner. (b) - free body diagram showing forces and stresses of interest acting on the corner of the element truncated by the x 2 , y 2 plane . FIG.6 ELEMENT RELATING Tl AND C"zz . 21 which g ives the e x t e r n a l s t r e s s v e c t o r I i n terms o f the i n t e r n a l s t r e s s and the outward normal . E q u a t i o n (18) i s enough t o e s t a b l i s h the r e l a t i o n s h i p between i n t e r n a l s t r e s s and e x t e r n a l f o r c e f l o w s . E q u a t i o n ( 1 8 ) , a l o n g w i t h Equat ions ( 1 5 ) , (16) and (17) when coupled w i t h the s t r e s s - s t r a i n l a w s , a l l o w s both shear and normal f o r c e T i " , I z. and t o be w r i t t e n i n terms o f the s t r a i n s . Now t h a t the problem o f the z e r o s h e a r s t r a i n s i s overcome, the next s tep i s to d e f i n e o v e r a l l s t r e s s r e s u l t a n t s a c t i n g on the cross s e c t i o n . These are shown i n F i g u r e 7 where V are s h e a r s , r i s an a x i a l f o r c e , B M are moments a n d T i s a torque and a l l are d e f i n e d i n the d i r e c t i o n o f the fixed a x e s . They are a l l o w e d t o t r a n s l a t e but must remain p a r a l l e l to oc ) BM2 z,w x,u T P+AP T+AT BM I +ABM I V2+AV2 BM2+ABM2 F I G . 7 B E A M S E C T I O N W I T H R E S U L T A N T F O R C E S . Using equilibrium, the overall force resultants on the entire cross section may be defined in terms of Ti, -71 and 7~3 . It is first conven-ient to define the force flows 7~,, \t and ~T3 as being analogous to T| ; T Z and. 71 but parallel to ^ D C ^ respectively rather than 22 fr} DC^T • From e q u i l i b r i u m and the geometry o f F i g u r e 4 , T , = Now • T^  = )z cos <p -r- 1^  5 in IT, J S = P C Tj. = v, T, <u =' v , C ^ H - T s w - M cosf ) d s = BM, f ( - T ! + T , w f M s . n r ) J s = B (19) (20) where and * are shown i n F i g u r e 8. They are an a x i s t h a t t r a n s l a t e s w i t h the s e c t i o n but does not r o t a t e . In the undeformed p o s i t i o n o f the beam, and 3Cj ^ are c o - i n c i d e n t . 23 FIG.8 DEFORMED AND UNDEFORMED CROSS SECTION . A f i n a l s e t o f e q u i l i b r i u m equat ions can now be w r i t t e n r e l a t i n g the o v e r a l l r e s u l t a n t s BM.V.P.T . Us ing F i g u r e 7 , the e q u i l i b r i u m equa-t i o n s about the d i s p l a c e d p o s i t i o n a r e : YlF^o - p + p v A P - O P = 0 (21) £ " F X = 0 - V , - rV, + A V , = O V, ' = 0 (23) [;M}=O T + Y ^ ' - V , o (24) E r V ° BM* +• V, - P ^ ' = - o ( 2 5 ) E M x - O ' BMi - V z + P ^ / = i o (26) 24 There are now s u f f i c i e n t equat ions between the d i s p l a c e m e n t s , s t r a i n s , c o n s t i t u t i v e laws and the e q u i l i b r i u m c o n d i t i o n s t o c o m p l e t e l y d e s c r i b e the problem. The task now i s to assemble a l l these equat ions i n t o a u s e f u l s e t o f d i f f e r e n t i a l equat ions which compactly and c o m p l e t e l y d e s c r i b e the p r o b -lem. T h i s w i l l be done i n the next s e c t i o n where a number o f s i m p l i f y i n g e n g i n e e r i n g assumptions w i l l be employed and t h e i r e f f e c t s s t u d i e d . c 25 CHAPTER 3 ASSEMBLY OF DIFFERENTIAL EQUATIONS The d i f f e r e n t i a l equat ions o f b u c k l i n g can now be developed by w r i t i n g the o v e r a l l e q u i l i b r i u m equat ions i n terms o f the o v e r a l l f o r c e r e s u l t a n t s which can be r e l a t e d t o the d isp lacements through the use o f s e l e c t e d s t r e s s and s t r a i n e q u a t i o n s . T h i s g ives the o v e r a l l e q u i l i b r i u m equat ions i n terms o f d i s p l a c e m e n t s and t h e i r d e r i v a t i v e s , and these w i l l be taken as the gov-e r n i n g d i f f e r e n t i a l e q u a t i o n s . T h i s i s e a s i e s t accompl ished by w o r k i n g through the d i s p l a c e m e n t s , then the s t r a i n s , then the s t r e s s - s t r a i n e q u a t i o n s , g e t t i n g the s t r e s s i n terms o f d i s p l a c e m e n t s , then g e t t i n g the f o r c e r e s u l t a n t s from the s t r e s s e s and s u b s t i t u t i n g them i n t o the o v e r a l l gross e q u i l i b r i u m e q u a t i o n s . T h i s can be done u s i n g the ideas and equat ions p r e s e n t e d i n the p r e v i o u s c h a p t e r . U s i n g t h i s approach, the d i s p l a c e m e n t and s t r a i n equat ions w i l l be d i s -cussed f i r s t . 3:1 D i s c u s s i o n o f Displacement and S t r a i n The c o n s t r a i n t p l a c e d on the d i s p l a c e m e n t a t t h i s p o i n t i s M.\ <w\jb are smal l compared t o one. T h i s was used i n the development o f E q u a t i o n (7) f o r w . When i t comes to e v a l u a t i n g the s t r a i n s i t would be c o n v e n i e n t i f M*\ A / e t c . were s m a l l compared t o the l i n e a r s t r a i n s , as t h i s would take and remove the squared terms from i t . I f t h i s were the c a s e , a c o n d i t i o n f o r ( 1 0 ) (27) 26 (28) To get some p h y s i c a l i d e a f o r the r e s u l t , p o s t u l a t e a beam d e f l e c t i n g i n the f o l l o w i n g t y p i c a l c u r v e : T h e r e f o r e , , ^ ^ i r — T c o s ~r L L The maximum a b s o l u t e va lues become: s - - Vo l i ; 5 i n r S u b s t i t u t i n g these i n t o Equat ion (28) and t a k i n g the case o f w j =• O g i v e s : 2 / ? | » 1^ 1 (29) In o t h e r words , the w i d t h and depth o f the s e c t i o n must be l a r g e com-pared to the d e f l e c t i o n . S i m i l a r c o n c l u s i o n s can be shown f o r d e f l e c t i o n s M. and Jt. T h i s i s undoubtedly s a t i s f i e d by a g r e a t many a c t u a l c a s e s , but u n f o r t -u n a t e l y many problems o f i n t e r e s t may v i o l a t e E q u a t i o n (29). A l o n g t h i n c a n t i l e v e r f o r i n s t a n c e may d e f l e c t s e v e r a l t imes i t s depth before b u c k l i n g . I t i s to be remembered t h a t Equat ion (29) was developed under the c o n d i t i o n s 27 t h a t the second o r d e r s t r a i n terms are smal l enough to be n e g l e c t e d . There i s always the p o s s i b i l i t y t h a t the second o r d e r terms may be l a r g e compared to w ' and s t i l l have no e f f e c t . For i n s t a n c e , the l o n g t h i n c a n t i l e v e r beam under end shear may d e f l e c t s e v e r a l t imes i t s depth and y e t the s t r a i n s are l i n e a r and the usual l i n e a r e l a s t i c a n a l y s i s i s adequate to o b t a i n mom-e n t s . I t i s worth p u r s u i n g t h i s to see i f i t i s p o s s i b l e to r e l a x E q u a t i o n ( 2 9 ) . To do t h i s , E q u a t i o n (7) w i l l f i r s t o f a l l be re-examined. Equat ion (7) W =- W 0 ^B> coi 1 - M - (*•» - x " ) •«- n S ( f o - f ) (7) as i t s tands i s l i n e a r i n uc 3 -w and js . T h i s means t h a t i t i s i n c a p a b l e o f h a n d l i n g e f f e c t s such as a x i a l f o r e s h o r t e n i n g , s i n c e t h i s e f f e c t depends on n o n - l i n e a r combinat ions o f %r')J$ ' . To i n c l u d e these e f f e c t s , i t i s c o n v e n i e n t t o d e f i n e the q u a n t i t y W , which i s the a x i a l d i s p l a c e m e n t i n the 3" d i r e c t i o n o f the p o i n t 3 and i t i n c l u d e s such e f f e c t s as a x i a l f o r e -s h o r t e n i n g as w e l l as the e f f e c t t h a t c o n t i n u i t y o f the beam and boundary c o n d i t i o n s have on a x i a l f o r e s h o r t e n i n g . C l e a r l y W can be r e l a t e d to W by the f o l l o w i n g e q u a t i o n : w = w - i I ( M . [ ) - Z ] + (30) where W i s d e f i n e d by Equat ion ( 7 ) , the squared terms are the e f f e c t s o f a x i a l f o r e s h o r t e n i n g i f the r o t a t i o n s J A ! * , , ir% can be c o n s i d e r e d to take p l a c e as r i g i d body r o t a t i o n s and i s the term which r e p r e s e n t s the e f f e c t o f c o n s t r a i n t s such as beam c o n t i n u i t y o r boundary c o n d i t i o n s . For example, i f the supports a t e i t h e r end o f the beam prevent motion o f the ends , then e l o n g a t i o n takes p l a c e d u r i n g l a t e r a l d e f l e c t i o n s and f W i s t o l o o k a f t e r e f f e c t s s i m i l a r t o t h i s , as the a x i a l f o r e s h o r t e n i n g i n t h i s 28 case cannot be regarded as due to r i g i d body r o t a t i o n s a l o n e . I f the a x i a l f o r e s h o r t e n i n g can be t r e a t e d as b e i n g due e n t i r e l y to r i g i d body r o t a t i o n s o n l y , then fC^) — O . I t s h o u l d be noted t h a t Equat ion (30) has meaning even under the o r i g i n a l c o n s t r a i n t s t h a t XL'AT\ J&v' e t c . are n e g l i g i b l e compared t o UL'JV' as i n Equat ion (30) the squared terms w i l l be o f the o r d e r L x ijjj) whereas the terms i n AJU',xr' i n w a r e o f the o r d e r sCt'oc, ir'y, , and UC'AT'« M,\W' does not imply L*C&')2 « DCM.'}y<r'. Us ing the more p r e c i s e va lue o f vv f o r w i n E q u a t i o n (10) f o r the s t r a i n E j^. , and u s i n g ui6 and V & from E q u a t i o n (9) g i v e s : - w ' - i f ( r t - i ( v v ' ) ' ( 3 1 ) In o t h e r w o r d s , the l i n e a r s t r a i n w ' i s adequate i f "F ( ^ ) — O as c l e a r l y Cw') 2 « w' . T h i s w i l l o c c u r i f ft}) = O , which i s the c o n d i t i o n t h a t the a x i a l f o r e s h o r t e n i n g can be viewed s o l e l y as the r e s u l t o f r i g i d body r o t a t i o n f o r e s h o r t e n i n g . I t w i l l now be d i s c u s s e d when t h i s o c c u r s . F i r s t o f a l l , assume MJ,^- O and o n l y J$ * O . Then the squared terms i n E q u a t i o n (30) f o r v v are f u n c t i o n s o f J>' and ^ o n l y . S i n c e j 8 ' i s c o n s t a n t across the cross s e c t i o n , the a x i a l f o r e s h o r t e n i n g v a r i e s as X and ^ . T h i s cannot be accompl ished by any r i g i d body motion o f the e n t i r e element cross s e c t i o n . T h i s d i f f e r e n t i a l f o r e s h o r t e n i n g means t h a t , s i n c e a d j a c e n t f i b r e s w i l l be. t r y i n g to change d i f f e r e n t l e n g t h s , c o n t i n u i t y c o n d i t i o n s o f the m a t e r i a l w i l l cause adjacent f i b r e s t o c o n s t r a i n each o t h e r i n some way. Because o f t h i s c o n s t r a i n t , a r i g i d body movement i s not p o s s i b l e f o r each f i b r e f o r a j& d e f l e c t i o n , and t h e r e f o r e K})! O . S i n c e Hi) equals a constant i s merely a r i g i d d i s p l a c e m e n t i n the ^ 29 d i r e c t i o n , w i l l be some f u n c t i o n o f y o t h e r than a c o n s t a n t . T h i s t h a t and t h e r e f o r e the hoped f o r r e s u l t o f Equat ion means (31) does not m a t e r i a l i z e . T h e r e f o r e , w i l l be c o n s t r a i n e d t o va lues which render (ji'^c) and (jB'tf) s m a l l compared to w ' - Some p h y s i c a l i d e a f o r t h i s can be o b t a i n e d by f o l l o w i n g a development s i m i l a r to t h a t o f Equat ion ( 2 9 ) . ' The terms to be compared are ^J3 co, from w and from W and Equat ion ( 1 0 ) . W r i t i n g the d e s i r e d c o n d i t i o n , s i m i l a r to E q u a t i o n ( 2 8 ) , g i v e s : (32) Assuming ^8 ==• _J50 ,5 in -£ and p l a c i n g the l a r g e s t v a l u e s f o r JB. > .a . . _J5 i n t o E q u a t i o n (32) g i v e s : I » \J>0 X * J , )> y. *| (33) The e f f e c t o f the jS t h a t i s be ing removed from the equat ions i s eas-i e s t v i s u a l i z e d by examining a s o l i d r i g h t c i r c u l a r c y l i n d e r as shown i n F i g u r e 9 . In F i g u r e 9 a , the undeformed c y l i n d e r i s shown. In F i g u r e 9 b , the c y l i n d e r i s shown w i t h the f i b r e s i n the p o s i t i o n they would assume under an end r o t a t i o n J& i f they were u n c o n s t r a i n e d . The a x i a l f o r e -s h o r t e n i n g would be p r o p o r t i o n a l to R , and t h e r e f o r e the s u r f a c e would not remain p l a n e . F i g u r e 9c shows the f i b r e s as they a c t u a l l y appear a l o n g w i t h the r e s u l t i n g s t r e s s . T h i s f i n a l shape i s assumed because o f the c o n s t r a i n t p l a c e d on the f i b r e s to behave as a cont inuum. A l s o note the l a c k o f a x i a l 30 FIG.9 C Y L I N D E R UNDER END R O T A T I O N 31 end r e s t r a i n t i n t h i s p a r t i c u l a r example. See Timoshenko [ 8 ] , pp. 286-291. Now t h a t O has been t r e a t e d , take the case o f yO = 0> and yU j ir =t= O . Since J6}is~~ are c o n s t a n t across the c r o s s s e c t i o n , the f o r e s h o r t e n i n g o f one f i b r e by E q u a t i o n (30) i s the same as a l l the o t h e r s . T h e r e f o r e there i s no d i f f e r e n t i a l movement a x i a l l y to r e q u i r e any c o n s t r a i n t , as was the case w i t h . However, i f the beam i s r e s t r a i n e d a x i a l l y a t the boundarys, t h i s w i l l p r e v e n t the beam from s h o r t e n i n g a x i a l l y , as i t tends to do i f i t has s l o p e s anywhere. T h i s would p r o v i d e a c o n s t r a i n t on the f i b r e s as they r o t a t e and so once aga in would have a va lue i n Equat ion (30) and would appear i n the s t r a i n . But i f t h i s i s not the c a s e , and the end i s f ree t o move a x i a l l y f o r s h o r t e n i n g due to r o t -a t i o n s , then and consequent ly T h i s then a l l o w s the d e s i r e d r e s u l t E ^ — W T h i s argument was based on the Green's t e n s o r t L i j . T h i s t e n s o r i s easy t o use compared to the Almansi s t r a i n ^ t e n s o r C ; j , as the Almansi s t r a i n i s based on the c o - o r d i n a t e system d e f i n e d by the d i s p l a c e d shape. However, the Almansi s t r a i n i s r e l a t e d t o the a c t u a l s t r e s s , and s i n c e i t i s the a c t u a l s t r e s s t h a t w i l l be used l a t e r , the t e n s o r <2,j w i l l be examined. T h i s t e n s o r l a c k s the p h y s i c a l i n t e r p r e t a t i o n t h a t can be a s s o c i a t e d w i t h the Green's t e n s o r , so the Green's t e n s o r was d i s c u s s e d f i r s t t o p r o v i d e some i d e a as to why the n o n - l i n e a r s t r a i n terms f a l l o u t . The form o f the s t r a i n t e n s o r C ; j can be taken from Fung [ 4 ] , Chapter 16, where the f o l l o w i n g r e l a t i o n between the d i s p l a c e d a x i s and the f i x e d a x i s i s d e f i n e d as : 32 % 0 - ^ + w (34) where once again W i s given by E q u a t i o n (30) and J i s , are the d i s -placements a l o n g the _JJ..; -v"" d i r e c t i o n s a t p o i n t 5 . The l i n e a r form o f ^ - 5 , 0 ^ . can be found from Equat ion ( 9 ) . -Ms =• JJL ~ J3 (9) - 1 ^ 5 = ^ - J> OC T a k i n g account o f the change i n J Z i i - v ^ due to s h o r t e n i n g because o f s l o p e s g i v e s : (35) .Ms* = -J>y +• G, where G , ^ j ^ ) i s o f the o r d e r ~ 2. \ JM s m < t „ ) OC and G j x ^ , ^ ' i s o f the o r d e r ' - I C v i m a x ) f The Almansi s t r a i n 6 ^ may be w r i t t e n a s : Now, the f o l l o w i n g d i f f e r e n t i a l r e l a t i o n s can be w r i t t e n : 33 (37) a?o a c^ a?o a^ a^o a> a?o r] w — 9 vv d x a_yv ^ + 3 W d 3-a^o .a^c a^° a^ a^ a?o ^ sL£ 4* + a j i l -r- i l a -^o 3 | o a^ a^ a?0 These are a l l t h a t i s r e q u i r e d to t r a n s f e r p o r t i o n s o f E q u a t i o n (36) from 0COi y,0 , "yQ to ZXL} LjSy y . T h i s a l l o w s p a r t s o f E q u a t i o n (36) to be w r i t t e n i n terms o f p r e v i o u s l y e v a l u a t e d f u n c t i o n s . D e n o t i n g d i f f e r e n t i a t i o n w i t h r e s p e c t to J/0 as c r * * : — - O -and u s i n g E q u a t i o n ( 3 4 ) , E q u a t i o n (37) becomes: ax a # * w a oc a ^  34 9 ^ ^ By u s i n g Equat ions ( 3 5 ) , the terms ^ f i l , , i ^ l e t c . may be eva l -u a t e d : d DC ^ <3 ? £ a^ .5 ~~ - £ - c L ( ^ J . 3^ -^.C^'^'ocXs' "T » i»vi a. X ' m a t d X A l s o , <3x a U s i n g t h e s e , Equat ions (38) become: O rv»a/c m a x 2- (39) JL w J> = J ( i - w ) S i n c e p r e v i o u s l y , the r o t a t i o n s have been r e s t r i c t e d t o be a t l e a s t as s m a l l as JJAT'<£<^U ' '« i r / e t c . i t i s c l e a r t h a t those terms 35 c o n t a i n i n g three angles as a product can c e r t a i n l y be d i s c a r d e d compared to terms such as M . S J m the equat ions f o r ^ 5 ; T>S . T h e r e f o r e , Equat ions (39) reduce t o ( 4 0 ) : - ~^8M-% + v g ( / - w ) w = ( ) + d w l ^ J + w ( i - w ) - j3 ( / - vv ) Now, from Equat ions (30) and ( 7 ) : •Ir d w - t y ^ f i _ f £ ( V + j 6 ' 3 C ^ ' d ^ 9 3C ^ D C J i 3, ^ 0 (40) (41) S u b s t i t u t i o n o f Equat ions (41) and the va lues f o r ^ £ £ 3 , ir^ i n t o the J L e q u a t i o n f o r w i n E q u a t i o n (39) g i v e s : W + w ' O - W ) ' (*2) 36 When t h i s c o m p l i c a t e d e x p r e s s i o n i s p l a c e d i n E q u a t i o n (36), the non-l i n e a r terms do not f a l l out o f the s t r a i n e q u a t i o n . However, E q u a t i o n (42) has many smal l h igh o r d e r terms i n i t . By n e g l e c t i n g terms c o n t a i n i n g t h r e e angles as products compared t o terms such as u<L i r ) y A > e t c . , r e a l i z i n g t h a t J^' i s o f the o r d e r J3 / \_ and t h a t d^± = 3 oc and 8 u P i ^ c c , Equat ion (42) becomes: When p l a c e d i n E q u a t i o n (36) f o r , and u s i n g * ~ „ * i f J L \ ^ _ ' , J L N s i n c e _ ^ £ s } J 3 J . S « , ir'5 ' e ^ becomes: - z K ^ U ' ) z f)-w ) ^ c v ) 2 0 ^ r J Since w « / » t h e n : (43) But w ' = w ' ~ £(vsf -i(>u'5)* • • • ' f ' C ? ' ) Therefore = w + f Y ^ ) Once a g a i n , the c o n c l u s i o n has been reached t h a t i f then the l i n e a r a p p r o x i m a t i o n t o the s t r a i n i s adequate. R e c a l l i n g the prev ious d i s c u s s i o n o f f OO , i t s assuming a zero v a l u e can be taken to 37 2. T , imply ( j 3 ' x ) and (jl'y) are s m a l l compared to w and t h a t t h e r e i s no a x i a l r e s t r a i n t on the beam. T h i s d i s c u s s i o n has d e a l t w i t h the a x i a l s t r a i n s f ^ and o n l y . The s h e a r s t r a i n s are not necessary as an a l t e r n a t i v e method f o r f i n d i n g a measure o f the shear f o r c e was g iven i n s e c t i o n 1:4 on E q u i l i b r i u m E q u a t i o n s . 3:2 S t r e s s - S t r a i n Equat ions Now t h a t the s t r a i n - d i s p l a c e m e n t r e l a t i o n has been dec ided upon, the c o n s t i t u t i v e r e l a t i o n <Tij A S.j + 2. & from s e c t i o n 1:3 can be used to r e l a t e s t r e s s and s t r a i n a t p o i n t s . Us ing the usual beam assumption o f P o i s s o n ' s r a t i o b e i n g z e r o and the r e s u l t s o f 3:1 changes E q u a t i o n (11) to <r„ - E e „ . - E w ' - <r ( « ) T h i s g i v e s ( T i n terms o f the d isp lacement w and Young's Modulus E . . N o t i c e t h a t the a c t u a l s t r e s s (Tand the Almansi s t r a i n were used. T h i s i s because e q u i l i b r i u m equat ions were used which r e q u i r e d a c t u a l s t r e s s e s i n d i s p l a c e d shapes t o get E q u a t i o n (13) and t h a t o n l y was r e q u i r e d to do t h i s , whereas the K i r c h o f f s t r e s s has no easy p h y s i c a l r e p r e s e n t a t i o n and to w r i t e e q u i l i b r i u m u s i n g i t would be d i f f i c u l t . 3 : 3 R e l a t i n g I n t e r n a l S t r e s s e s to E x t e r n a l S t r e s s Vectors Now t h a t the s t r e s s e s have been r e l a t e d to d i s p l a c e m e n t s , the next s t e p i s t o r e l a t e the i n t e r n a l s t r e s s e s to the e x t e r n a l s t r e s s v e c t o r s T i , )», l 3 on the beam f a c e , as i t i s these v e c t o r s t h a t are needed to get the o v e r a l l beam r e s u l t a n t s from Equat ion ( 2 0 ) . T h i s w i l l r e q u i r e some m a n i p u l a t i o n , as these e q u a t i o n s are f u l l o f n o n - l i n e a r h igh o r d e r terms t h a t w i l l be d i s c a r d -ed because o f the c o n s t r a i n t s on the r o t a t i o n s . The e q u a t i o n s o f i n t e r e s t 38 from the s e c t i o n 1:4 on E q u i l i b r i u m Equat ions a r e : - TV , + L ^M' - IOJB = o ( 1 7 ) » / (18) I, =• (T3 1 t " t o V * These equat ions g i v e s u f f i c i e n t r e l a t i o n s between 17,71, T 3 > t o a n c ' the one component (Tof the s t r e s s t e n s o r t h a t can be found from the s t r a i n ten-s o r to a l l o w them a l l to be e v a l u a t e d i n terms o f the disp lacements and t h e i r d e r i v a t i v e s . R e w r i t i n g Equat ions ( 1 6 ) , (17) and (18) and i n t e g r a t i n g ( 1 5 ) : 7 ^ = c r t - l o ^ r & _ _ / (45) to ~ ) e T, cJ5 The problem now i s to reduce Equat ion (45) by removing a l l the terms ren-dered n e g l i g i b l e by the c o n s t r a i n t s on the a n g l e s . O b v i o u s l y the term t o V ^ i s i m p o r t a n t f o r T i , but i n the equat ions f o r 7 1 and T3 i t can be h o p e f u l l y shown to be s m a l l , both i n Tivi, TT-X^and %o - S u b s t i t u t i n g f o r T . i n Tz and T3 g i v e s : 71 ^ crtvz - t o ^ 1 V2 t0ja -fV (46) 39 Now, the t o terms can be w r i t t e n as 10 ( v ^ - / ) and t o (y<li 1^1 tJ^') • P r e v i o u s l y , the equat ions were c o n s t r a i n e d to be v a l i d o n l y where ACAS were n e g l i g i b l e compared to M-}v orJ3 , o r where AJL^v,Ji c o u l d be d i s c a r d e d compared to one. A p p l y i n g these c o n d i t i o n s to the t 0 terms, i t i s obvious t h a t t o v 2 and t o l / ^ i r ^ can be d i s c a r d e d . Now, lo io I I J X ( . r i - t o i r £ ) d. This i s a c o m p l i c a t e d n o n - l i n e a r e q u a t i o n . I t w i l l h o p e f u l l y be shown t h a t t o V i i s s m a l l when d i f f e r e n t i a t e d . When expanded, t 0 becomes: lo = - j (0-t)ds + i 1„V Js - [ t 0 J, (47) Comparing the l e f t - h a n d s i d e o f Equat ion (47) w i t h the l a s t term on the r i g h t - h a n d s i d e g i v e s : t lo I V Js Now, T h e r e f o r e O t t o l [ lo V" J s > I ' l o V ' i i 0^c<^J (48) I n v o k i n g t h e p r e v i o u s l y used shape - V = iro s i n L r e s u l t s i n E q u a t i o n (48) becoming ^ < S L I f "XTQ \ too > then Equat ion (48) f i n a l l y becomes: I n i l ~ — °( I 100 L , o L f o L ^ £ o . then i t w i l l be assumed t h a t i t can s a f e l y be d i s c a r d e d w i t h r e s p e c t to c< . T h i s leads to the c o n d i t i o n S <, ^ T h i s i s a reasonable c o n d i t i o n , and i t w i l l be i n f e r r e d from t h i s s i m p l e example t h a t i f . L _ L - v ^ a x < f o b a n d S < £ then I 0 l o V z d s may be d i s c a r d e d . I t s h o u l d be noted t h a t t h i s c o n s t r a i n t tends to run c o u n t e r t o the con-s t r a i n t s expressed i n Equat ion ( 2 9 ) . Treatment o f the term To IT2 i s s l i g h t l y more c o m p l i c a t e d , because as i t s tands t o would have t o be compared to l o , which i s not very meaningful i f done d i r e c t l y . T h e r e f o r e , a d i s c u s s i o n o f t o w i l l be h e l p f u l . The term t o i s caused by two e f f e c t s : shears due to the f o r c e s ^ and shears due to the torque T . However, f o r the l i n e a r c a s e , t h a t p a r t o f 1 0 due t o V i s constant and t h e r e f o r e the a s s o c i a t e d 1 ° f o r the l i n e a r case i s z e r o . T h i s means t h a t f o r the n o n - l i n e a r p a r t 10 due to V i s second o r d e r and the product * lo vz i s t h i r d and t h e r e f o r e can be d i s c a r d e d . The % 0 p o r t i o n not due to V i s due to T and t h i s has p r e v i o u s l y been taken as smal l because o f the r e s t r a i n t s on the j 3 t e r m s , and because the s e c t i o n i s being taken to a c t as a beam, not a s h a f t . T h i s a lone would i m p l y t h a t the t o d u e to T i s s m a l l and t h e r e f o r e , s i n c e *^  would be r e a s o n a b l y smooth w i t h ^ , t h a t lo i s s m a l l . However, even g r a n t i n g t h i s , i t i s p o s s i b l e t o show the unimportance 1 1 o f t 0 i T i from another argument. 41 The term (i0 can once again be broken i n t o two p a r t s : the l i n e a r and n o n - l i n e a r terms. The n o n - l i n e a r terms can be n e g l e c t e d , as they become a 1 third o r d e r terms i n 1 0 ir^ . T h i s leaves the l i n e a r terms. They can be taken as 5 o r r 5 a ' — \ f " "' r 111 \ i i lo — J0 V W 0 ( X 0 - X ) + 1T Cfo-f) + JS CoJtcJs q „ ' I The terms to be compared are L0 and Lt. i r z . I n v o k i n g the approx-i m a t i o n s used p r e v i o u s l y g i v e s : - V = ir* s i n ^ An i n e q u a l i t y between ' f ^ ^ f t X and the _ y £ 0 } i S ^ ) ^ t f < , terms can be w r i t t e n : R e w r i t i n g the comparison u s i n g the above g i v e s : s f 3 ^ O ^ << 6 / (49) 42 / . Now, W o i s equal to a c o n s t a n t i n the l i n e a r case, terms, the comparison becomes s i m i l a r t o : For the remaining o r C l e a r l y , - j X L *J'o < ; oo lT0 Tf_ Alo TT_ L L 3 C< o ^ - ) (50) w i l l be i n general n e g l i g i b l e compared t o °( . I f , Equat ion (50) becomes: TP loo L T h i s i s merely a statement o f the f a c t L o i s o f the o r d e r t o / L (51) I f l i s o f the o r d e r /v\ / \_ T h i s a l l o w s E q u a t i o n (15) to be w r i t t e n : (52) which i s the l i n e a r e q u a t i o n f o r t c . T h e r e f o r e , under the assumptions o f (45) become: and A S Q < ~ 0 f o r an assumed s i n e shape, Equat ions L = ( f t IT, - l o - CMjS) (53) 17 =• c r i - t o t/% 43 where t o i s g iven by E q u a t i o n ( 5 2 ) . The l i m i t s S ( j and ' ^ a . * ( ^°~o presented above are not d e f i n i t e l i m i t s b u t are o n l y g iven as a guide t o i n d i c a t e t h a t around these va lues the a p p r o x i m a t i o n s are becoming d o u b t f u l . Remember t h a t these were developed on the b a s i s o f a s i m p l e s i n e curve d e f l e c t i o n and the a r b i t r a r y d e c i s i o n to d i s c a r d terms o f o r d e r V 2 0 compared t o va lues i n the range o f .zero to one. A l s o , these approximat ions took no account o f d i s t r i b u t i o n over the l e n g t h o f the s e c t i o n . For i n s t a n c e , some o f the terms b e i n g compared were m o d i f i e d by s i n e c u r v e s , others by c o s i n e c u r v e s . In some c i r c u m s t a n c e s , t h i s might a l l o w a smal l term to dominate a l a r g e r term because a t a c e r t a i n p o i n t the s i n e o r c o s i n e terms are z e r o . However, s i n c e t h i s occurs a t c e r t a i n p o i n t s and the f u n c t i o n i s non-zero over l a r g e p a r t s o f the domain e l s e w h e r e , i t was f e l t t h a t comparison o f maximum values was adequate. These l i m i t s t h e r e f o r e are based on p o s s i b l e o r d e r s o f magnitude o n l y and very rough p h y s i c a l r e a s o n i n g and s h o u l d not be construed as i n v i o l a b l e . I t i s s t i l l f e l t though t h a t , d e s p i t e the roughness o f the approximat ions i n v o l v e d , the c o n c l u s i o n s drawn are s a t i s f a c t o r y . From Equat ions ( 1 9 ) , the s t r e s s v e c t o r s i n the g l o b a l f i x e d axes a r e : T = T, .~TI — T a sin 4 — T 3 cos 4> (19) "I3 — cos <f> +-T 3 s in (j> E v e r y t h i n g about Equat ions (19) i s known except f o r \ A / 0 , the a x i a l d i s p l a c e -ment o f the o r i g i n o f 5 on the cross s e c t i o n . To e v a l u a t e W 0 » e q u i l i -brium a l o n g the 3, a x i s w i l l be used. 44 R e c a l l i n g the f i r s t e q u a t i o n o f (20), which i s i T , J s = P , and u s i n g the l a s t e q u a t i o n o f (53) g i v e s : p- - Cut - i . O J * • (»> T h i s i m p l i e s from Equat ions (44) and (7) P - E L ' C t L w . , ^ ' o , 1 + y « V x . - x U v ' ^ . - ? ) ] ) j . +• FCj") where F ( j ) = . ^ ^ I n t e g r a t i n g w i t h r e s p e c t t o S then ^ and r e a r r a n g i n g , g i v e s : W. = P » - f l u ? , - ^ ' ( x . - a ) - / w , ( y . - ^ - Iffflff <- K where 5L, ^ are the c o - o r d i n a t e s o f the c e n t r o i d i n the oc} ^ c o - o r d i n a t e system, A i s the area o f the cross s e c t i o n , and o J , = ^ £ t co, J s . S u b s t i t u t i o n o f W e i n t o E q u a t i o n (7) g i v e s : T h i s has the u n f o r t u n a t e aspect o f r e - i n t r o d u c i n g the term t „ , as i t appears i n fC}). T h e r e f o r e , s i n c e E w ' and t o = I.*C<rt) ,J s =. [0S CCE W ' H ) J 5 , the F ( j ^ term a p p e a r s , i n m o d i f i e d form i n both 0~ and to . In to , i t has the form A I ( to v'x ) J 5 which has a l r e a d y been shown to be n e g l i g i b l e under the c o n s t r a i n t s o f the p r o b l e m . I t s appearance i n <T i n the second and t h i r d o f Equat ions (53) can a l s o be n e g l e c t e d as CT. appears i n a second o r d e r 45 term a l r e a d y , and so ^ F('f) becomes a t h i r d o r d e r c o n t r i b u t i o n to these e q u a t i o n s . T h e r e f o r e , f o r the purpose o f e v a l u a t i n g T * > 1 3 and consequen-t l y H y I , , w w i l l be taken as : A t t h i s p o i n t , the shears and normal f lows T. } Tt>Ts have been r e l a t e d to the d isp lacements and then d e r i v a t i v e s . T h i s has been done through the use o f s t r e s s - s t r a i n equat ions r e l a t i n g and vVand through e q u i l i b r i u m equa-t i o n s r e l a t i n g (T,T, , T i , T 3 and t o . A l l t h i s was done by n e g l e c t i n g terms i n the equat ions which were rendered n e g l i g i b l e by the c o n d i t i o n s M^'AT' can be d i s c a r d e d compared t o JJL} 4f )JS. There i s now enough i n f o r m a t i o n to de-termine the r e s u l t a n t f o r c e s on the end o f the s e c t i o n . 3:4 R e l a t i n g the S u r f a c e S t r e s s V e c t o r to the O v e r a l l Beam Force R e s u l t a n t s U s i n g Equat ions ( 1 9 ) , (20) and ( 5 3 ) , the o v e r a l l beam r e s u l t a n t can now be w r i t t e n i n terms o f CT and t 0 and t h e r e f o r e i n terms o f the d i s p l a c e m e n t s . The r e s u l t a n t s become: V , = So Tz <U = L(L sin* - T 3 C O 5<P) JS <57> S u b s t i t u t i o n o f and T 3 g i v e s : V, = - C ( s i n * ( + l . J > +-<rt^[ -M') -costp{-lo^riv'z-CM>8)')) JS (58) S u b s t i t u t i o n f o r ° 0 , 0" and r ) , a l o n g w i t h M-x^if ^ and i n t e g r a t i n g g i v e s : -y/^ -^ o^^ vxv^ ^ •yyft^ '^-iyf 1(59) where the fo l lowing equa l i t i es are defined: \0 Kw.-a^ d s A o , - A c . = o . \0 t i c , = A T t C j - ^ J s = A ? - A * = o J A - A f using the re lat ions j | = y. = - s m ^ ) = i = cos ^  )0 cos <f> )0t (w.-c^dsds = X 1 fcG«, -o,) i s | d- ) oxt(a). -C,) <U = O - Jo 3°"t ( -«o-")cU = K 6 L cos $rf )0 t(ic-x.) ds <\s ~ X L fcCx:-^ Js | o """jx t (?o -:0 J s & r S So cos 0 L t ( j - j ) Js Js = x L ^ ( j - ^ J s o ~ L x t ( f - ? ) J s S i n ^ 1 iCwv -to,} ds\b -=• o + )0 f t (u> - <v, ) J s - K, 0 sm 4 L tCx-x)Jsds = J , e sin <f> Jo t ( y Js Js = - L 7 ^ ( 3 i s - — fs" fc3 )0 — sin j> coscf> ds — K 3 ^ x sin cosip d — K< r 5 t3 =- K «-=-) 0 7^ X c o s 2 0 ds — TZ f s i n ^ c o s 0 48 S i m i l a r l y c o s 0 + "[3 sm (p ) d s (61) By u s i n g s i m i l a r o p e r a t i o n s as those used i n , becomes: For the torque T T = I (-JiCfTjix-) rT,(*.-J>}) - M W U S (63) Expanding i n terms o f t 0 ; <T, M } Ji't and ATZ g i v e s : <M " ^ V * + M l ) ~ cos0 ( t o -<rfc T £ t j')) - Mvl]Js (64) By e x p a n d i n g , c a n c e l l i n g and throwing but terms o f t h i r d o r d e r Equat ion (64) s i m p l i f i e s c o n s i d e r a b l y . I n t r o d u c t i o n o f the values f o r JA 4 and i r l then reduce i t t o : 49 + P i i ' M f ^jif^X 1^  +- Nijt'cos <P + V / 5 i n ^ ) ] J 5 (65) Introducing 1 0 | M, 0" and in tegrat ing and adding CJS fo r the torque due to plate twis t ing g ives: T * E [ (K l 3 + K,.-Kn ~ l c t 8 K Jfi "V CKm - K 4 ) ( K * + KrK. )• + C > f i ^ Y(-K^Ka-K0j8 ^ f$x r +K , U V l « c + K 7 V / / - | f ) (66) A E -J where: C 2 ( x 2 + ^ H U - u 0 c J s - K w r r 2. ~ O "~ ) 0 11 u>. - dOw, J s = " )0 t Cw. - J 5 ~ K ij="_P C - J G = S t . Venant's tors ion constant 50 Jo r \ Q \ u -x) ds 1 r 1 i(j - j ) J5 rs t3 ' ° /2 X <f 5 , n 0 C 0 5 ^ ds So ^ 7* 5 I N ' ^ = K B ) 0 X sin 0 ds J 0 "g" x2" sin <t> cos0 ds = K « , )o 6 x ^ sin (p ds - K*, L Ir f cos1 <f> ds )o % cos ^ a s L |r ^ sin# cos (j) is Summing up the s t r e s s v e c t o r s (67) i n the d i s p l a c e d p o s i t i o n r e l a t i v e to the d e f i n e d r e s u l t a n t s g i v e s f o r the moments: BM, -~ f ( l f y * j 3 x ) - l 3 W - Mcos(«->«))<)s <68> 51 R e c a l l t h a t " X = cT-1 - l 0 V ^ The term % B v [ was shown h o p e f u l l y t o have no n o t i c e a b l e e f f e c t on t h e e q u a t i o n s f o r , T 3 and t h e r e f o r e on the shears and t o r q u e s . However, i t i s o f consequence now i n d e f i n i n g the moment, as a t f i r s t g lance i t i s a term o f importance because terms o f s i m i l a r magnitude have been kept i n p r e v i o u s e q u a t i o n s . Therefore t«, ir'z w i l l be kept i n the equat ions and examined a f t e r E q u a t i o n (68) has been expanded. E q u a t i o n (68) becomes: = <rt(«j,+v8:x> ~ lovly, - M c o s (<|>-J3) - T3 W - t o i r a ' Jb OC ^ The l a s t term on the r i g h t h a n d s i d e o f (69) can be n e g l e c t e d compared t o the second term because o f the r e s t r i c t i o n on the a n g l e s . S u b s t i t u t i n g f o r T 3 g i v e s : + btn 4 T - t o + riv'z -(M^)') ~ M cos (<p->s)] Js (70) S i n c e W i s not o f the o r d e r o f JAJL) s^r but i s o f the o r d e r o f MJX> e t c . and s i n c e M and (Mjs)1 are s m a l l , Equat ion (70) becomes: r 5/ B M . ~ J o ^ t C ^ ^ x ) - to lr z ' j , i - t o w s i n t f + M cosCcp-^)) J & ( 7 1 ) Examining the expanded form o f the t 0 terms i n (71) g i v e s : 52 T h i s i s o f the form: (72) CO AL, x.0 + AT H.0 + Wo where c< £ Red I A N C L J_V, , J T are very s i m i l a r to the shears and torques p r e v i o u s l y developed. When i n t e g r a t e d , (73) w i l l be o f the form: (74) S i n c e the s e c t i o n i s open, t h i n w a l l e d and behaving as a beam r a t h e r than as a s h a f t , T w i l l be s m a l l and t h e r e f o r e oCYuc w i l l be neg-l e c t e d . T h i s means t h a t any form o f s h a f t b u c k l i n g due to pure torque has been e l i m i n a t e d . The Vw term i s s i g n i f i c a n t w i t h r e s p e c t to the o t h e r terms i n (70) and s h o u l d be k e p t . However, when Equat ion (70) i s p l a c e d i n E q u a t i o n ( 2 5 ) , the V w term appears w i t h Vi as V, ± V, w ' Compared t o V, , i t i s i n c o n s e q u e n t i a l . T h e r e f o r e i t a l s o w i l l be d i s c a r d e d a t t h i s s t a g e , even though i t s i n s i g n i f i c a n c e does not become apparent u n t i l s u b s t i t u t i o n o f the f o r c e r e s u l t a n t s i n t o the o v e r a l l e q u i l i b r i u m e q u a t i o n s . T h e r e f o r e B M , = f . * ( < r t ( , * j s * ) - M c o » f < » - > 8 > ) j a ( 7 5 ) I n t e g r a t i o n and s u b s t i t u t i o n o f p r e v i o u s l y d e f i n e d c o n s t a n t s g i v e s : 53 S i m i l a r l y , BM, = - [ / ( l U - j B y ) ^ M s.n (76) or: (CK.-K«+^J8'+ ( - $ ^ O - K 3 W +Cl^-K 7)ir" + I (77) T h i s completes the d e t e r m i n a t i o n o f the r e s u l t a n t s t o the o r d e r o f accuracy o f the r o t a t i o n s being n e g l i g i b l e w i t h r e s p e c t to one, the squares of the r o t a t i o n s be ing n e g l i g i b l e w i t h r e s p e c t to the a x i a l s t r a i n s i f the s e c t i o n i s a x i a l l y r e s t r a i n e d , ! . ^ e ' J | , \.J3>JQZ \ ^  \Z ^ J ; | l f the s e c t i o n i s not a x i a l l y r e s t r a i n e d , ^ , a n d t ^ be ing roughly V / o o a n d s < • These f o r c e r e s u l t a n t s are very c o m p l i c a t e d because o f c o u p l i n g between the unknown displacements and t h e i r d e r i v a t i o n s . T h i s i s p a r t l y due to the use o f an a r b i t r a r y o r i g i n f o r the c o - o r d i n a t e system. I t can be shown, however, t h a t the use o f a c o - o r d i n a t e system p a r a l l e l to the p r i n c i p a l axes of the beam and w i t h i t s o r i g i n a t the shear c e n t r e w i l l uncouple the equa-t i o n s i n the l i n e a r terms. T h i s p a r t i c u l a r c o - o r d i n a t e system w i l l be used "for jQyf,^ f o r the r e s t ° f the t h e s i s . For example, see V l a s s o v [ 9 ] and B l e i c h [ 3 ] . R e w r i t i n g the r e s u l t a n t s i n t h i s new a x i s system g i v e s : 54 V4= ^ N l v ' / , ^ N , V " + PAT' -N/^V BM, =- +-N,ir" + Pf - +- Px.8 (78) B M , = + N L V - Px + N . ^ A T " 4 -T N3^ s"/ + U4fi + Ns-v-V + N 6 ^ V -where N, = E C " lace'" K 7) =• ~ E I x c N 3 * E ( K l 3 + 2 K i « , - K L 7 - K , C ) j K a - r N , = J G = C N 5 = E = E I P C C M 7 =• E ( 2 K l o - K a e - K z , ' - K „ * K M ) 55 N, = E(ZK>z - cK, (79) and these are c a l c u l a t e d i n a c o - o r d i n a t e system t h a t i s p a r a l l e l to the p r i n c i p a l axes and has i t s o r i g i n at the shear c e n t r e except J x o and which are s t i l l the values about the p r i n c i p a l c e n t r o i d a l a x i s . 3:5 The D i f f e r e n t i a l Equations o f S t a b i l i t y I n t e g r a t i o n o f the o v e r a l l beam e q u i l i b r i u m equat ions ( 2 4 ) , (25) and ( 2 6 ) , and s u b s t i t u t i n g i n the moment r e s u l t a n t s from E q u a t i o n (78) g i v e s : (80) where and F are c o n s t a n t s o f i n t e g r a t i o n . These are the f i n a l equat ions g o v e r n i n g the behaviour o f the s e c t i o n . They have been o b t a i n e d by p l a c i n g the equat ions r e l a t i n g the r e s u l t a n t s to d isp lacements i n t o the o v e r a l l e q u i l i b r i u m equat ions f o r the beam. N o t i c e t h a t a l l o f the n o n - l i n e a r terms are o f the form o f a f o r c e t imes a d i s p l a c e -ment. F o r i n s t a n c e , i n the f i r s t e q u a t i o n o f (80) the term - N, v"j3 = - B M ( > 8 A l s o note t h a t a l l the n o n - l i n e a r i t i e s are due to e q u i l i b r i u m b e i n g taken 56 about a d i s p l a c e d p o s i t i o n . Any n o n - l i n e a r i t i e s due to the s t r a i n equat ions are assumed to be rendered n e g l i g i b l e by the c o n s t r a i n t s on the r o t a t i o n s and a x i a l boundary c o n d i t i o n s . Equat ions s i m i l a r t o these are g iven i n B l e i c h [ 3 ] , Timoshenko [ 7 ] , Oden [6] and, i n p a r t i c u l a r , V l a s s o v [ 9 ] . How-e v e r , i n t h i s development, the domain o f v a l i d i t y o f the equat ions i s out -l i n e d , and the equat ions are more g e n e r a l , as they i n c l u d e the e f f e c t o f c o u p l i n g i n a n o n - l i n e a r manner, whereas the above r e f e r e n c e s g i v e l i n e a r d i f f e r e n t i a l e q u a t i o n s . F i n a l l y , note t h a t the constants 5 > < j ^ c ., 3E.>^ c. and IP a r e de-f i n e d by i n t e g r a l s o f the type \o "b y ( Si - oc) ds ~ j?x.j.c< where t h e x , ^ , terms are c o - o r d i n a t e s of the m i d - t h i c k n e s s a t p o i n t 5 . T h i s means t h a t , a l though the i£ have a form c l o s e to those o f the moments o f i n e r t i a , they are m i s s i n g the terms which account f o r the e f f e c t s o f the moment o f i n e r t i a o f the element d s about i t s own c e n t r o i d a l a x i s a t m i d -t h i c k n e s s . However, these e f f e c t s are accounted f o r by the c o n s t a n t s , K M K 3 due to s e c o n d a r y , o r p l a t e bending e f f e c t s . S i n c e they always appear c o u p l e d w i t h the $ i n the equat ions f o r f o r c e r e s u l t a n t s , i t was a s i m p l e m a t t e r to d e f i n e the values o f Xx^;l^.as J.xc- ^XLC* K7 e t c . as was done i n Equat ions ( 7 9 ) . However, the c o n s t a n t lp as d e f i n e d l a c k s the necessary second o r d e r e f f e c t s t o make i t the usual Xp . T h i s d e f i n i t i o n w i l l be kept t h o u g h , as the n a t u r e o f the s e c t i o n s c o n s i d e r e d renders the d i f f e r e n c e between the usual J - p and the lp d e f i n e d here to be n e g l i g i b l e . T h i s i s not always t r u e i n the case f o r the d i f f e r e n c e between l=<c and i ^ c o r 1^. and $ w t / . Con-s i d e r i n g a t h i n r e c t a n g u l a r s e c t i o n i t i s c l e a r t h a t about the weak a x i s 57 $ = O and X c o n s i s t s e n t i r e l y o f the second o r d e r , o r p l a t e bending e f f e c t s as r e p r e s e n t e d by K 2. o r K 7 . T h i s i s a l s o a p r o p e r t y o f the c o n s t a n t N 3 , as t h e r e e x i s t s e c t i o n s such as t h i n r e c t a n g l e s where K1 3 (which i s the n e g a t i v e o f the warping c o n s t a n t P ) i s z e r o , and the o n l y warping r e s t r a i n t i s found from second-a r y , o r p l a t e bending e f f e c t s , as g iven by Kit, Kp, K l g . 58 CHAPTER 4 SOLUTION OF DIFFERENTIAL EQUATIONS Now t h a t the equat ions have been d e v e l o p e d , i t remains to r e l a t e them to a m a t r i x type f o r m u l a t i o n as t h i s r e p r e s e n t a t i o n i s the main o b j e c t o f the s t u d y . The d e s i r e d form i s : (81) The £ are r e l a t e d to the d e f l e c t i o n s ^cc} jQ e t c . by the f o l l o w i n g equat ions and t o the beam s e c t i o n by F i g u r e 10. S i - W0 $b = - i r ' SI, =• J> S 3 - - V SE = MJ S13 = (82) is = v s,0 e v a l u a t e d a t r e s p e c t i v e ends. N o t i c e the presence o f J$ as a boundary c o n d i t i o n a t each end, making seven c o n d i t i o n s per end t o be s a t i s f i e d . T h e i r presence i s necessary t o account f o r w a r p i n g at the ends. Absence o f these terms would c o n s t r a i n p lane s e c t i o n s t o remain p lane a t the j o i n t s , o b v i o u s l y not the most general c o n d i t i o n . E q u a t i o n s (82) a l l o w the s o l u t i o n o f the d i f f e r e n t i a l equat ions i n terms 59 FIG.10 GENERALIZED DISPLACEMENTS FOR BEAM. 60 o f the v e c t o r o f boundary c o n d i t i o n s S. The end f o r c e s may be found by sub-s t i t u t i n g back i n t o the f o r c e r e s u l t a n t equat ions e v a l u a t e d a t the b o u n d a r i e s . T h i s w i l l l e a d to an e q u a t i o n o f the form: McColm t83) where the m a t r i x f u n c t i o n K(S) w i l l be taken as the s t i f f n e s s m a t r i x . The d i f f e r e n t i a l equat ions w i l l be s o l v e d u s i n g an i t e r a t i o n t e c h n i q u e t h a t employs the l i n e a r s o l u t i o n as a f i r s t a p p r o x i m a t i o n t o the d e f l e c t e d shape. In more general t e r m s , i f Li =F(i,i\t.', ) B . C . - S i s the e q u a t i o n to be s o l v e d , the s o l u t i o n w i l l be taken as where % t i s g iven by: L<X) = o B . C . = S and % n , n ^ / i s g i v e n by: B . C . =• O For t h i s problem, o n l y one i t e r a t i o n i s used, s o : (84) (85) (86) (87) (88) 61 To ease c a l c u l a t i o n s and g i v e some i d e a o f the s i z e and importance o f the t e r m s , the d i f f e r e n t i a l equat ions (78) are n o n - d i m e n s i o n a l i s e d by the f o l l o w i n g s u b s t i t u t i o n s . a ^ L ? ^ - LiX / t r - ' L ^ JB=^% (89) and E q u a t i o n s (78) become: # - M , V t (M.+ M j i ) ^ f A, | + B, i£ - M-»>2 ^ ( M r + M ^ ) > 8 •+ C > *• D, ( 9 0 ) where Ai, B,, C , } £), and E , are a r b i t r a r y c o n s t a n t s o f i n t e g r a t i o n and M , - - PLVN, M = - P L ^ / N , M 3 = N . / N . M< ~- PLVN, M 7 - N , L 7 N , Ma = - P L - V N 3 M, = -V»L7N. M..-N4L7N, M „ = P X L V N , M , Z = V , L7N3 M , 3 = N , / N , " M .« = N » L / N 3 M«- = N , L / N 3 M,T = P IAVAN 3 A2 = - N 4 L V N 3 on To s o l v e these e q u a t i o n s , the i t e r a t i o n p r e v i o u s l y mentioned w i l l be employed. T a k i n g f o r the l i n e a r case or, = O B . c = £ • • • « iT, ==• O (92) where g z are c o n s t a n t s o f i n t e g r a t i o n . The s o l u t i o n s a r e : Jiy — J, s i n l o X j " + k, c o s h AJ +- X, J +• 63 (93) In m a t r i x f o r m , Equat ions (93) become C , - l a . , b I J c 1 , <L j C\ 5 . , k ) ' (94) Using E q u a t i o n s ( 8 2 ) , (89) and (94) e v a l u a t e d a t the boundaries g i v e s , i n m a t r i x f o r m , r O j O I A / L O ' / L 0 Sin Wx c o s H A I l X . c o s l i X \ i\r\in\ ](L O /**< i (95) 64 C l e a r l y , c* - L*f]V <96> and t h e r e f o r e , from (94) and (96) ~ Or =- <pC,' = <pL<P,'J~V * <pC? = <DL$W,SX (97) E q u a t i o n s (97) are the l i n e a r s o l u t i o n i n terms o f the boundary d i s p l a c e -ments S . F o r the f i r s t i t e r a t i o n : (98) *M .t<?, -JUM.3J3. + +M„-^r t M i J B.C.*O where A } , B3, d , 0 3 a i d E 3 are a b i t r a r y constants o f i n t e g r a t i o n . 65 Some o f the constants M,- i n Equat ions (98) c o n t a i n the shears V,, V 2 and the a x i a l f o r c e P . A c c o r d i n g to Equat ions ( 2 1 ) , (22) and ( 2 3 ) , these f o r c e s are c o n s t a n t w i t h r e s p e c t t o ^ . T h e r e f o r e they w i l l be l e f t as i s i n the s o l u t i o n o f the d i f f e r e n t i a l e q u a t i o n s . However, d u r i n g the s o l u t i o n o f t h e . s t i f f n e s s m a t r i x , they w i l l be e v a l u a t e d u s i n g the c o n s t i t u t i v e equa-t i o n s f o r f o r c e s p r e v i o u s l y deve loped. Before E q u a t i o n (98) can be s o l v e d f o r . v l , ^ * - and , the q u a n t i t i e s JZt,Ar't,f%, roust be r e p l a c e d by the l i n e a r s o l u t i o n o f E q u a t i o n ( 9 7 ) . T h i s leads to terms o f the type M,- • U f ' S e Loft (99) S i n c e t h e i n t e g r a t i o n o f E q u a t i o n (98) i s the next s t e p , the presence o f terms such as (99) are a d i f f i c u l t y . The s e p a r a t i o n o f cp and & , the two f u n c t i o n s t h a t depend on ^ , make i n t e g r a t i o n d i f f i c u l t unless m u l t i p l i e d out i n f u l l . However, (99) can be viewed a s : where the terms i n l a r g e b r a c k e t s are s e a l a r s . T h e r e f o r e , t r a n s p o s i n g the l a r g e b r a c k e t s leaves the va lue unchanged. Doing t h i s t o the f i r s t b r a c k e t changes (99) t o : M; WT(wr)Tc«T« ( 1 0 0 ) Using t h i s r e d u c t i o n , and s u b s t i t u t i n g (97) i n t o (98) g i v e s : 66 -M,5 J ' W I l " ' <JTeC<p?l S -Mic. ©C<P?3 £ + E 3 B c ^ 0 ( I O I ) S o l v i n g (101) f o r the p a r t i c u l a r i n t e g r a l and adding the homogeneous s o l u t i o n g i v e s : = - f ? ^ + ^t J 3 + e z ? 3 <-f* J"1- + j * (102) / j t - +, J + K \ 4 f , ^ + j 4 s»nV« X f • l i * c o s U j * It} * ^ where 67 f f = M 4 w w r ' f f = M s Ce*3 C<P?f + M* •" , TC<D,'T,TCI^3 L 5 ? 3 " ; f f = -M 7 o , TC?t]Ltf?]" ,-MeC«]C©f3" ' ^ , W ^ 1 J ~M,5^ , Ti :^;]" Tc^]^ff' -M^cTi^r ' where the symbol means T i n t e g r a t e d twice w.r.t. 1" and C f l means J ; = s a t i s f i e s ^5 - \J> = L r j . The in t e g r a t e d values are given i n Appendix 2. Using a S i m i l a r procedure to t h a t used to get Equation (95) allows the boundary d e l f e c t i o n s , which are zero f o r t h i s i t e r a t i o n , t o be r e l a t e d to JZiyAPx and jdj. by the f o l l o w i n g s e t of equations. 6 8 O O O o o o o o o o o o =< >+ < (103) and n Ld>:l r AT P V — where i i j 0 i s t , d i f f e r e n t i a t e d once w . r . t . ^ and e v a l u a t e d a t J = O and T I , L i s r ( d i f f e r e n t i a t e d once and e v a l u a t e d a t % = / C o r 3.= L ^ • S i m i l a r d e f i n i t i o n s h o l d f o r J" i , -f , e t c . T h i s can be s i m p l i f i e d by r e w r i t i n g as f o l l o w s : M = f's' ^ ; J 3 * c*:ic; ( 1 0 4 ) lo} - r,c * i w > i i s'*• tin c[ 69 where f , =• L l l ) 0 - r ' and | t ^ -f * } etc. are a l l s imi lar ly defined. From Equations (104) c; - - t « v i ] " C f : y • fisO c l - -_->;]" r . j ' • f i r ' ) Writing J I ^ , -i?^ > ^ 3 \ a s t n e s u m o f t n e Particular and homo-geneous solution, and using (105) for the constants in the homogeneous solu-tion gives: i r 2 - i r r *f-r J ' ^ " . ] " ^ . ' ^ ^ ' ) <i<*> ^= *rY * if s , «if . , - . -s i ]"( f . ' j , *f . , . , *-Fi .o This completes the f i r s t i terat ion. Since only one iterat ion is being used in this study, Equations (106) and (97) may be added to give the values of JL^ Kr and jg that w i l l be used. 70 cpc<p.,]',s, + o , + i r i 3 - « p L ^ i " , ( ? , , s , ^ ; . 0 These equat ions f o r M,,^,^ c o n s i s t o f terms d e s c r i b i n g the l i n e a r behaviour o f the s t r u c t u r e p l u s terms h a n d l i n g the n o n - l i n e a r b e h a v i o u r . A t y p i c a l l i n e a r term i s (f) C$!]~' & ' and i t i s o f the form: The terms d e s c r i b i n g the n o n - l i n e a r behaviour may be l i n e a r i n S o r n o n - l i n e a r i n £ . Some t y p i c a l terms d e s c r i b i n g n o n - l i n e a r b e h a v i o u r a r e : and f_- ^ -71 The terms o f the form are q u i t e a b i t more c o m p l i c a t e d and l e n g t h y ; however, they s t i l l f o l l o w the same general p a t t e r n . Geometric jvj' ( Geometri c ) Geometric * L M , £ Geometric Geometric Geometric |V| ( G e o m e t r i c ) Geometric +M3 Geometric Geometric Geometric Su Sr IS 1 Geometric Geometric '0 •LM, {5* Geometric Geometric Geometric M z ( G e o m e t r i c ) Geometric * M, Geometric Geometric Geometric 72 A l l the geometric m a t r i c e s are f u n c t i o n s o f o} /, L and perhaps c o s h X L , & i n h A L • They r e s u l t from v a r i o u s f u n c t i o n s o f ^ b e i n g e v a l u a t e d a t e i t h e r ^ = o o r ^ = L • Note t h a t i n some o f the t e r m s , the d e f l e c t i o n s appear t w i c e . T h i s makes the equat ions n o n - l i n e a r i n S . A l s o n o t i c e t h a t the c o n s t a n t s may have shears V. , V z and a x i a l forces P i n them. The presence o f these n o n - l i n e a r forms w i l l be taken care o f d u r i n g the s o l u t i o n t e c h n i q u e , where an i t e r a t i v e procedure w i l l be used. During each i t e r a t i o n , the S !o , V la and P lo w i l l be e v a l u a t e d and r e - i t e r a t e d . 73 CHAPTER 5 FORCE DEFLECTION EQUATION The d isp lacements found i n the p r e v i o u s c h a p t e r can now be p l a c e d i n the f o r c e r e s u l t a n t equat ions to g ive a r e l a t i o n s h i p between the f o r c e s and the boundary d isp lacement . I f the r e l a t i o n s h i p i s e v a l u a t e d a t the b o u n d a r i e s , i t g ives the r e l a t i o n s between the f o r c e s a t the boundaries and the boundary d i s p l a c e m e n t s . As p r e v i o u s l y mentioned, t h i s can be w r i t t e n a s : I X IZ ' a s there are o n l y s i x e q u i l i b r i u m equat ions a t each end. However, because the element i s not a l i n e element but has a f i n i t e c r o s s - s e c t i o n , i t i s p o s s i b l e to have a s t r e s s f i e l d e x i s t i n g t h a t has no r e s u l t a n t but s t i l l has a gross o v e r a l l e f f e c t . As i t happens, t h i s i s the case h e r e . T h i s system o f s t r e s s e s i s c a l l e d the bi-moment. See, f o r example, [ 6 ] , [ 9 ] . I t i s c l o s e l y a s s o c i a t e d w i t h warping and the r a t e o f change o f t o r s i o n a l angu-l a r d i s p l a c e m e n t . U n f o r t u n a t e l y , s i n c e the bi-moment has no r e s u l t a n t , i t f a i l e d t o show up i n the o v e r a l l e q u i l i b r i u m e q u a t i o n s , but t h i s very p r o p e r t y a l l o w s i t to be i n t r o d u c e d now w i t h o u t any e f f e c t on the p r e v i o u s development. S i n c e p r e v i o u s l y i n t e g r a t i o n s o f s t r e s s across the c r o s s - s e c t i o n were p e r -formed i n d e v e l o p i n g the n o n - l i n e a r e q u a t i o n s , the e f f e c t o f the bi-moment i s a l r e a d y c o n t a i n e d i n the d i f f e r e n t i a l e q u a t i o n s . I t o n l y remains to get some measure o f i t s va lue t o g i v e 14 f o r c e equat ions t o correspond w i t h the 14 de-f l e c t i o n s a t the boundary. A l t h o u g h the bi-moment has no r e s u l t a n t , i t i s s t i l l capable o f doing work under c e r t a i n d i s p l a c e m e n t s . Because o f t h i s , the concept o f g e n e r a l i z e d f o r c e s and d isp lacements w i l l be used to develop the f , £ r e l a t i o n r a t h e r (83) I t i s to be noted at th is point that S is a / x M vector and f i is 74 than w r i t i n g o u t the r e s u l t a n t s as f u n c t i o n s o f the boundary d isp lacements d i r e c t l y . T h i s change, w h i l e s i g n i f i c a n t , i s one o f technique and development o n l y . I t i s i n t r o d u c e d o n l y to g a i n the measure o f a s t r e s s f i e l d which has no r e s u l t a n t . I t does not change the form o f Equat ion ( 8 3 ) . G e n e r a l i z e d forces and d isp lacements are d e f i n e d such t h a t the work done by the g e n e r a l i z e d f o r c e through the g e n e r a l i z e d d i s p l a c e m e n t i s e q u i v a l e n t to the work o f the a c t u a l s t r e s s e s through the a c t u a l d i s p l a c e m e n t s . For t h i s c a s e , the g e n e r a l i z e d displacements w i l l be taken as the S a l r e a d y d e f i n e d , and the g e n e r a l i z e d f o r c e s w i l l be taken as a c t i n g a t the d i s p l a c e -f.< + f,*k <• W , + t W , - { , 3 ^ . 3 +J>£,f (los) where - ^ W , ^ j l , * , **AJ~s and ^J^i are v i r t u a l d isp lacements about a d i s p l a c e d p o s i t i o n as Ti , Tg., ~T3 and M are a c t i n g a t the d i s p l a c e d p o s i t i o n . In o r d e r to r e l a t e *Jls, &ATS to A ^ - > ^/tr, ^_^S , which are the v i r t u a l d isp lacements o f the o r i g i n , the c o - o r d i n a t e s o f F igure-8 w i l l be used. A l s o necessary w i l l be the q u a n t i t i e s g i v e n i n F i g u r e 11, which shows a s i d e view o f the d e f l e c t e d shape W viewed a l o n g the x a x i s : ments S . The virtual work of generalized forces at 3. •= L is: where & i s a v a r i a t i o n i n £ . The v i r t u a l work o f the a c t u a l e x t e r n a l s t r e s s e s i s : (109) 75 rMjndeformed xy plane FIG.II BEAM SEGMENT AND DISPLACEMENT W I t i s c l e a r t h a t A iK' can be w r i t t e n as: where AV - % comes from the q u a n t i t i e s d e f i n e d i n F i g u r e 8 and A t ^ W comes from q u a n t i t i e s i n F i g u r e 11. By u s i n g the same r e a s o n i n g on AJJl<, , and u s i n g the r e l a t i o n s 1 = OC - J3tf and yj =- <y + J*> oc. g iven i n F i g u r e 8, AJl* and ->v^ can be w r i t t e n a s : ( n o ) = # s i n (0-_~0 + A AT'cos (<P-j&) - AJ$'(~--x.cos(P + y5in(l>) where AMJ, . comes from m a n i p u l a t i n g Equat ion (9). • 76 By analogy t o Equat ion (2), / i W may be w r i t t e n as : S u b s t i t u t i n g Equat ions (110) and (111) i n t o Equat ions (109) and u s i n g cos(<P-->8) « c o s 0 cos73 + 5ir.0 s i n j j = costf) +jQ s m <p Sin Gp-js) - Sin<jP c o s j 3 - cos<p sirv8 *S sm(p - J 3 cos<^ i n s e l e c t e d i n s t a n c e s and a l s o u s i n g Equat ions (82), g ives f o r the v i r t u a l work: + 4 $ 0 i C K^+JJX) + Tt {-fo-jxS) ~T 3 w - M c o s ( < ? - ^ 0 Js f 5 * T z ds + 4 L L T 3 Js + A o",3 { (s~\zh*jJ) + X, U - j J s ' (112) U s i n g the p r e v i o u s d e f i n i t i o n s o f V , M e t c . g iven i n the s e c t i o n on e q u i l i b r i u m e q u a t i o n s , E q u a t i o n (112) becomes: V .W. = AQZP + A & W [ B W + P w . l + ^ L B M A + P C x . - y . j B ) ] A J 6 [ B M , - P ( V ^ ^ ] + *^ V, + 4f$-V2 ^ l u T (113) (114) 77 where P BM>V e t c - a r e e v a l u a t e d a t 3 = L and where B W i s the new t e r m , the bi-moment, and i s g iven by: J. L T , ( w . - u T . ) - M(-XCOS<P +(f s i n * ) + W(T3 t - c - ^ ) -\C^t^xS)i d s The l a s t term w i l l be o f the form T w . S i n c e T was p r e v i o u s l y c o n s t r -a i n e d t o be s m a l l , and s i n c e w i s s m a l l , t h i s term w i l l be d i s c a r d e d . The bi-moment i s then g iven by. E q u a t i n g c o e f f i c i e n t s between Equat ions (113) and (108) g i v e s : f, = V, f,o = Bh, + PCxo-^P f.3 =• T (115) L = BW + Pcj, E v a l u a t i n g (114) i n terms o f prev ious constants g i v e s : (116) T h i s w i l l be the seventh " f o r c e " on the s e c t i o n and i s a measure o f the s t r e s s n e c e s s a r y to m a i n t a i n out o f p lane w a r p i n g . A c o r r e s p o n d i n g d e v e l o p -ment was done a t %~0 . g i v i n g f o u r t e e n measures o f f o r c e , c o r r e s p o n d i n g to f o u r t e e n boundary d e f l e c t i o n s . This a l l o w s the c o n n e c t i v i t y m a t r i x K to be square and to be the most general d e s c r i p t i o n o f the problem. A l l o f the f o r c e s but P have been found i n terms o f d isp lacements which can be w r i t t e n as f u n c t i o n s o f £ . However, P and W 0 a r e r e l a t e d as f o l l o w s : E A E E E 78 By d e f i n i t i o n w » = (5> $ = o and w „ =• SL G> j = L . Us ing t h i s f a c t , and Equat ions ( 8 2 ) , ^ can be s o l v e d f o r and r e p l a c e d and then P can be found. The r e s u l t o f these m a n i p u l a t i o n s y i e l d s : I t i s now p o s s i b l e to w r i t e out the f o r c e d e f l e c t i o n r e l a t i o n s h i p by u t i l i z i n g Equat ions (117) , (115) , ( 1 0 7 ) , ( 8 2 ) , (78) and ( 8 9 ) . * ^N1{(i[^ ]^ cr4:j,-4»:L^]"U'r+f:r]) -p(i-r,)-S.P(i-i') i-N,i (~<m'Jt ^D J -£ i'l) - X P + £ . P z -s. N, [ ii fr fr s3- *L teifo f; \ j; - £8 P + f n Pf + P(x-x.) -f„ P(f-j„) 6*1 i—• 1 J i_ o o ~ ~ to O a - O +• ~ - l - > ° o to I I ' • o V- »_ O O o o o I—I © I + <5si >" ui - V* o o «4 W O o po Co r - i -? 7 ' ' i . o ^ + r _ o n o I w a 0 J—> 4-? >* -+_» A* »--o 1 H 1 1 "Z *. f O __r io ^ ^ /—' r - i - + o o i&l - w ( — 1 ( CO o o ' i — i + w <=vC| - + - » 1 f J o o j 0 + -f- M -t-5j + + ~ N / "o * -f-« 0 <»—o | <V~) w 1 r o CD : r o • S i C 1 1 <vo + ~ V -Q O OO « 0 + ~ «o Hi I X 0 •+ CJ -o o - l I + oo r - i -r 1 — 1 1 °o_~ + + ~ to I r w -A* — - O ( 1 r - i -r ° o -f-r- ' ^ o t a—> + p o o I C O r- i -i r o o r G O ST? r ' I—1 o n + OO r ' Q O + ~ + -+-> /-» — N — _o a—i UJ ^1 i eo ~ + o o r N ° 0 81 + PtOi (118) E q u a t i o n s (118) are Equat ions (83) expanded where (83) i s The m a t r i x i s g iven i n Appendix I . The terms , , c e t c . a r e e v a l u a t e d i n the l a s t p a r t o f Appendix I I . 82 CHAPTER 6 NUMERICAL EXAMPLES S i n c e Y\(§) i s n o n - l i n e a r i n & as w e l l as c o n t a i n i n g unknown f o r c e s V i , V 4 and P , i t i s d i f f i c u l t t o s o l v e the equat ions L K ( S } ] £ = -(-i n a d i r e c t manner. To overcome t h i s problem the m a t r i x was d i v i d e d i n t o two b a s i c s u b - m a t r i c e s , one c o n t a i n i n g the l i n e a r s t r u c t u r a l b e h a v i o u r , one c o n t a i n i n g the n o n - l i n e a r s t r u c t u r a l b e h a v i o u r . A l o a d increment procedure was then employed to s o l v e the e q u a t i o n s . F i r s t , the complete l i n e a r s t r u c t u r e m a t r i x was generated and a found f o r a p a r t i c u l a r l o a d l e v e l . T h i s l o a d l e v e l was then r e - s o l v e d by c o n s t r u c t i n g and adding the n o n - l i n e a r p o r t i o n o f to the l i n e a r and s o l v i n g a g a i n . The n o n - l i n e a r p o r t i o n was c o n s t r u c t e d u s i n g the j u s t -c a l c u l a t e d l i n e a r d e f l e c t i o n s as w e l l as the f o r c e s V i , V , and r o b t a i n e d by m u l t i p l y i n g out the complete f o r c e d e f l e c t i o n r e l a t i o n f o r the member. T h i s procedure was r e p e a t e d , each time c o n s t r u c t i n g the non-l i n e a r p o r t i o n o f the m a t r i x from the l a s t p r e v i o u s - c a l c u l a t e d d e f l e c t i o n s and f o r c e s . T h i s i t e r a t i o n a t the g iven l o a d l e v e l was t e r m i n a t e d when S became unchanged by any f u r t h e r i t e r a t i o n . T h i s i s a secant m a t r i x approach. The l o a d l e v e l was then i n c r e a s e d and r e - i t e r a t e d , and so o n . When a z e r o determinant f o r was c a l c u l a t e d , the s t r u c t u r e was taken as b u c k l e d . Determinant p l o t s were used to determine the l o a d a t which t h i s o c c u r r e d . S e v e r a l types o f beam were s t u d i e d , and the r e s u l t s are p r e s e n t e d i n t h e f o l l o w i n g s e c t i o n . The t h e o r e t i c a l r e s u l t s f o r the channel s e c t i o n s t u d i e d were taken from V l a s s o v [9], w h i l e a l l the o t h e r s were taken from Timoshenko [7]. In the f o l l o w i n g s e c t i o n , c r i t i c a l loads from Timoshenko a r e s u b s c r i p t e d w i t h a T , from V l a s s o v w i t h a V , and the r e s u l t s o f the 83 program are s u b s c r i p t e d w i t h a p . There were f o u r c r o s s - s e c t i o n a l types a n a l y s e d : a t h i n r e c t a n g l e , a c r u c i f o r m s e c t i o n , a wide f lange and a c h a n n e l . T h i n R e c t a n g u l a r S e c t i o n I2 t = b L = l 2 0 t !!_ E/G = 3 3 elements were used i n a l l cases 1. A C a n t i l e v e r Column I1 m m . P T = + If1 El I2 PP = 100 2. PT 2. C a n t i l e v e r Beam Under End S h e a r , Warping R e s t r a i n e d A t W a l l VP PT - 4. 013 VEIGJ/L2 P P = 1.01 PT The d i f f e r e n c e between P P and P T a r i s e s because the program assumed secondary warping r e s t r a i n t a t the w a l l , whereas the Timoshenko s o l u -t i o n i g n o r e d the warping e f f e c t s e n t i r e l y . 84 3. C a n t i l e v e r Beam Under End S h e a r , Warping A l l o w e d A t Wal l 1 I R- = 4.0/3VEIGJ il PP » 1.01 p T The program assumed no warping r e s t r a i n at the w a l l , but kept the warp-i n g terms i n t e r n a l l y i n the beam. As b e f o r e , the Timoshenko s o l u t i o n i g n o r e d the warping e f f e c t s e n t i r e l y . C r u c i f o r m S e c t i o n L = I 2 0 t B/G = a.e T ] I2t*b I2t-3 elements used. 4. T o r s i o n a l B u c k l i n g Under Pure A x i a l L o a d , S imply Supported In Both Planes 1.05 ' Pr . 977 Pr 85 In the program, the ends were r e s t r a i n e d from t o r s i o n a l r o t a t i o n but f r e e t o warp. The program took account o f i n t e r n a l warping c o n s t r a i n t s . The p r o g r a m ' c r i t i c a l l o a d was compared to two c l a s s i c a l s o l u t i o n s : which i g n o r e s a l l warping r e s t r a i n t , and PrpL=- ( ^ 5 6 - j / ) JT_ Pr 4L 2 which i s developed from p l a t e t h e o r y , and consequent ly i n c l u d e s warping as w e l l as o t h e r e f f e c t s . The Wide Flange Cross S e c t i o n L=960t 1 34t E/& = Z . l l L2- JG/EF = 7* i s found from t a b l e s a g a i n s t L 1 J G - / E P f o r v a r i o u s boundary c o n d i t i o n s and l o a d s . S i x elements were used i n a l l c a s e s , e x c e p t where n o t e d . 86 5. C a n t i l e v e r Beam Under End S h e a r , Warping R e s t r a i n e d a t F i x e d End But U n r e s t r a i n e d A t Free End 1 P T = 7 . 7 6 1 / E X J G / L2 Pp = /. ooe Pr 6. Beam Simply Supported In Both P l a n e s , T o r s i o n a l R o t a t i o n R e s t r a i n e d A t Ends , Loaded A t C e n t r e , Warping U n r e s t r a i n e d A t Ends P T = 3l.i I / E I ^ T / L * P P =1.0 0 4 P T 7. Same As Example (6), But Only Two Elements Used PP * /. O C ? 0 Pr 8. Same As Example (6), But Beam F i x e d At Ends In Weak L a t e r a l D i r e c t i o n O n l y , With Warping R e s t r a i n e d A t Ends Pr ~- 6 6 . 8 1 / E I G J I \1 P P = . ?9S P r 9 . Same As Example ( 6 ) , But With A U n i f o r m l y D i s t r i b u t e d Load 1 A t The Shear Centre ( l L ) T - 53 VEXCTJ7L2 The Channel Cross S e c t i o n 20t - 2 0 t — H L = 480t 6 elements were used i n a l l cases . For c r i t i c a l l o a d s , see V l a s s o v [ 9 ] 10. Channel Beam Simply Supported In Both P l a n e s , Each E n d , T o r s i o n a l R o t a t i o n s R e s t r a i n e d A t Ends , Warping U n r e s t r a i n e d , U n i f o r m l y D i s t r i b u t e d Load 1 A t Shear Centre y v V v y 88 11. Same As Example ( 1 0 ) , But Load In Opposi te D i r e c t i o n % = I.OI 1 V I t i s i n t e r e s t i n g to note t h a t ^ p f o r case 10 i s 6.34 times t h a t f o r case 11. 12. Same As Example ( 1 0 ) , But Loaded With End Moments M Rather Than A U n i f o r m l y D i s t r i b u t e d Load M a i> n MP = 1.007 M , 13. Same As Example ( 1 2 ) , But Moment M Reversed M a M n M p = l.ooe Mv I t i s i n t e r e s t i n g to note again t h a t MP f o r case 12 i s 19.3 t imes t h a t f o r case 13. These examples were run on an IBM-360-67 machine u s i n g MTS. The channel s e c t i o n shown i n example (10) was a n a l y s e d u s i n g s i x l o a d increments w i t h t h r e e i t e r a t i o n s per l o a d i n c r e m e n t . The t o t a l t ime taken was t w e n t y - e i g h t seconds , o f w h i c h seventeen seconds were CPU t i m e . As can be seen from the above examples, agreement i s e x c e l l e n t i n a l l 89 c a s e s . Where there i s some d i s c r e p a n c y i t i s i n cases where the program c o n t a i n s secondary warping and the c l a s s i c a l s o l u t i o n s do n o t . In these i n s t a n c e s , the program s h o u l d g ive h i g h e r r e s u l t s , which i t does. Good accuracy i s a l s o o b t a i n e d even when a smal l number o f elements i s used, as f o r example s t r u c t u r e (7) which w i t h two elements g ives very good agreement compared to the c l a s s i c a l r e s u l t . A l l o f the above t e s t s are b i f u r c a t i o n type b u c k l i n g . There i s another type o f b u c k l i n g c a l l e d a m p l i f i c a t i o n b u c k l i n g and t h i s occurs when the loads tend to cause a d i s p l a c e m e n t i n the d i r e c t i o n the s e c t i o n wishes to b u c k l e . T h i s type o f b u c k l i n g i s c h a r a c t e r i z e d by one o r more o f the d e f l e c t i o n s becoming unbounded. U n f o r t u n a t e l y , there e x i s t few c l a s s i c a l s o l u t i o n s f o r a m p l i f i c a t i o n b u c k l i n g and as a r e s u l t no compar-a t i v e t e s t s were made. However, a t o t a l l y unsymmetric shape was s t u d i e d f o r a m p l i f i c a t i o n b u c k l i n g . The s e c t i o n was analyzed as a c a n t i l e v e r under end l o a d , w i t h the l o a d b e i n g a p p l i e d through the shear c e n t r e , and p a r a l l e l to o r a t a s m a l l a n g l e ^ t o the s t r o n g p r i n c i p a l p l a n e . T h i s had the e f f e c t o f c a u s i n g d isp lacements i n the l a t e r a l and t o r s i o n a l modes, thus making the problem one o f a m p l i f i c a t i o n , not b i f u r c a t i o n . The c r o s s s e c t i o n p r o p e r -t i e s and a p l o t o f c r i t i c a l l o a d versus T i s shown below. -Co = - 0 . 3 2 6 in , y 0 = o.f*/ /'n, A = 0 . 0 3 2 6 i n x =- - o . £67 i n > ^ = 0.21© i n , co,= - o. 12 is in I x * O. 483 x lo~ £ i n 4 ® t = .O I6 ' J j , = O. 786 x i o " 3 i n 4 I p - 0.951 x i o " 2 i n 4 K i 3 =-r=- -0 .101 - l o " 4 i n 6 , J = o . i 8 x s. 4 IO ir\ N7/E - o.7?3 x ; o ~ 4 in , N a / E = 0 . 8 l * N , JE - - 0 . Z I 8 * i o ~ * -3 • s lo i n -1.1° - 0 . 6 ° 0 ° 0.6° 1.1° 1.6° \|/ For t h i s p a r t i c u l a r example, the values o f E , G and |_ are r- 6 • L - IO x l o p s i . G = 3. 7 6 x i o 6 p s i . L - 18.5 in. There a r e s e v e r a l p o i n t s o f i n t e r e s t about the curve shown above. One i s the unsymmetry o f the c r i t i c a l l o a d about the o r i g i n and the o t h e r i s the s e n s i t i v i t y o f the c r i t i c a l l o a d to the angle . The dependency o f the c r i t i c a l l o a d on the angle V i s not unexpected, as the channel s e c t i o n shown p r e v i o u s l y has c r i t i c a l loads t h a t depend upon which a x i s and d i r e c t i o n i s l o a d e d . The example shown above however i s very s e n s i t i v e to V , and t h i s may be due to i t s extreme f l e x i b i l i t y i n t o r s i o n . In general the s o l u t i o n o f E K ( S ) 1 & ~ f f o r an a m p l i f i c a t i o n problem r e q u i r e s more i t e r a t i o n s to converge to a 8 v e c t o r than a b i f u r c a t i o n type problem. T h i s may cause d i f f i c u l t i e s i f the s e c t i o n i s very f l e x i b l e and i s loaded near c r i t i c a l , as a l a r g e number o f i t e r a t i o n s may be r e q u i r e d to a r r i v e a t a d e f l e c t e d shape. In some c a s e s , p a r t i c u l a r l y when one d e f l e c t i o n i n the d i r e c t i o n o f b u c k l i n g becomes l a r g e , a s tudy o f both d e f l e c t i o n s and determinants i s r e q u i r e d t o determine a c r i t i c a l l o a d . However, t h i s i s n o t always the c a s e , as t h e r e are many w e l l behaved a m p l i f i c a t i o n problems. 91 CHAPTER 7 DISCUSSION There are s e v e r a l p o i n t s o f d e t a i l t h a t s h o u l d be d i s c u s s e d , but were n o t mentioned i n the main body o f the t h e s i s , as i t was f e l t t h a t i n t r o -d u c t i o n a t a p r e v i o u s stage would be a d i g r e s s i o n from the main purpose o f the t h e s i s , t h a t i s , the assembly o f a s t i f f n e s s m a t r i x . 7:1 Smal l R o t a t i o n Theory I f the assumptions t h a t were a p p l i e d i n t h i s t h e s i s o f JMAT^CAT e t c . were i g n o r e d , then the r o t a t i o n s c o u l d not be t r e a t e d as v e c t o r s . T h i s would have the r e s u l t o f i n t r o d u c i n g terms such as MJAT'OC o r jjJjiy, i n t o the e q u a t i o n f o r a x i a l d isp lacement W . These terms are n o n - l i n e a r i n the r o t a t i o n s and the way they appear i n the e q u a t i o n f o r W depends upon which o r d e r the r o t a t i o n s are t a k e n . T h i s o r d e r dependancy c o u l d be removed by changing the c o - o r d i n a t e system to one o f E u l e r a n g l e s , but E u l e r angles are not as s t r a i g h t f o r w a r d t o use as the system chosen. The end r e s u l t would be t h a t no matter how the n o n - l i n e a r terms M-'j3^ e t c . i n w were t r e a t e d , they would c o m p l i c a t e W and immensely c o m p l i c a t e the s t r a i n - d e f l e c t i o n and s t r e s s - s t r a i n e q u a t i o n s . I t i s f o r t h i s reason t h a t the c o n s t r a i n t s on the angles were m a i n t a i n e d , as i t g r e a t l y reduces the c o m p l e x i t y o f the r e s u l t i n g equat ions w h i l e s t i l l a l l o w i n g a reasonably l a r g e f i e l d o f v a l i d i t y . 7:2 Secondary S t r e s s e s The s e c o n d a r y , o r p l a t e - b e n d i n g s t r e s s e s need o n l y be c o n s i d e r e d f o r s e c t i o n s o f two c e r t a i n t y p e s . The f i r s t type i s t y p i f i e d by the t h i n r e c t -a n g u l a r s e c t i o n , as i t r e q u i r e s the p l a t e - b e n d i n g s t r e s s e s i n o r d e r t o have any s t i f f n e s s a t a l l i n the weak p l a n e . The second type o f c r o s s - s e c t i o n 92 i s t y p i f i e d by the t h i n c r u c i f o r m o r angle s e c t i o n , where any warping r e s i s -tance must come from the secondary s t r e s s e s . In t h i s c a s e , n e g l e c t o f the secondary s t r e s s e s does not cause a zero t o r s i o n a l s t i f f n e s s , ' as p l a t e t o r s i o n a l r e s i s t a n c e would s t i l l be i n e f f e c t . However, the mathematical s o l u t i o n technique uses the q u a n t i t y A which i s the r a t i o o f p l a t e t o r s i o n s t i f f n e s s to warping t o r s i o n a l s t i f f n e s s . I f the warping i s taken as z e r o , A becomes i n f i n i t e and the equat ions break down. S e c t i o n s such as c h a n n e l s , wide f l a n g e s , e t c . which have s u b s t a n t i a l membrane s t i f f n e s s c o n t r i b u t i o n s from a l l p o s s i b l e modes need not c o n s i d e r the secondary s t r e s s e s . I t i s t h e r e f o r e recommended f o r s e c t i o n s o f t h i s type t h a t the constants K 2 , K 4 , , K?, K i & , KIT, Kia, K it , K i o , K n , K z i , Kzv be ommitted i n o r d e r to ease c a l c u l a t i o n s . 7:3 Constants In the e v a l u a t i o n o f the constants o f Equat ions (78) t h e r e are s e v e r a l p o i n t s to be n o t e d . F i r s t , placement o f the o r i g i n o f 5 a t any e x t r e m i t y g ives a non-zero c J , c o n s t a n t . I t i s p o s s i b l e to p l a c e the o r i g i n o f 5 a t a p o i n t where CJ, w i l l equal z e r o . Doing t h i s a l l o w s K,5 , or P , the warping cons-t a n t , to be w r i t t e n as K 1 3 = ^ o->z J S r a t h e r than - Uj 1 ) Ob as done i n t h i s development. They both have the same numerica l va lue i f t h e i r o r i g i n i n the x.>f p lane i s common. T h i s means t h a t t a b l e s l i s t i n g P (° r K . 3 ) m a y b e u s e d w i t h o u t having to c o n s i d e r the p o s i t i o n o f 5 = o as l o n g as the o r i g i n o f a c , ^ i s the shear c e n t r e . The placement o f 5 a t an e x t r e m i t y i n t h i s development causes some problems w i t h a x i a l l o a d s . S i n c e they are assumed a p p l i e d a t v v o =• £ , o r , which i s taken to be a t 5 = o , any a x i a l l o a d a u t o m a t i c a l l y a p p l i e s moments and bi-moments due to i t s e c c e n t r i c i t y from the shear c e n t r e T h e r e f o r e , i f an a x i a l l o a d i s to be a p p l i e d a t any o t h e r p o i n t on the c r o s s s e c t i o n , compensating moments and bi-moments must be a p p l i e d to p l a c e the a x i a l r e s u l t a n t i n the r e q u i r e d p o s i t i o n . S e c o n d l y , as was mentioned p r e v i o u s l y , the c a l c u l a t i o n o f i n v o l v e s a s i g n c o n v e n t i o n . I t i s c o n s i d e r e d p o s i t i v e i f the swept area rds i s c l o c k w i s e , n e g a t i v e i f c o u n t e r c l o c k w i s e . I f r = X 5 i n 0 + < f c o s 0 i s i n t e g r a t e d i n s t e a d , the s igns are a u t o m a t i c a l l y accounted f o r . D e f i n i n g Y i n terms o f x , y and (j> a l s o a l l o w s e a s i e r e v a l u a t i o n o f r i n terms o f c r o s s - s e c t i o n p r o p e r t i e s . T h i r d l y , the s e c t i o n constants i n Equat ions (79) were developed f o r a c r o s s - s e c t i o n which does not b r a n c h . In branched s y s t e m s , a c l o s e r look must be taken a t the e q u i l i b r i u m e q u a t i o n : T h i s was w r i t t e n f o r the element shown i n F i g u r e 5 where in 1 0 ac ted o n l y on two edges. At a b r a n c h , there are t h r e e o r more edges and a s e t o f Zo3^ * l o , > 1o4 , t o 3 . e t c . a c t i n g one to an edge. These % 0 ^ and t h e i r d e r i v a t i v e s must be i n e q u i l i b r i u m w i t h 77 and Ti . S i n c e E q u a t i o n (15) was not developed f o r t h i s s i t u a t i o n an e x t e n s i o n i s necessary f o r genera l branched systems. T h i s can be done by u s i n g a s e r i e s o f equa-t i o n s s i m i l a r to E q u a t i o n (15), Ui = iof + CT' JS where 1 0 i i s the shear f low between branches and i s the shear f low j u s t p a s t the branch a t 5 B . In t h i s e q u a t i o n (to; i s unknown. When an 94 e q u a t i o n such as t h i s has been w r i t t e n f o r each element o f s between b r a n c h e s , they may be assembled and the unknown to,- e v a l u a t e d u s i n g the f o l l o w i n g c o n -d i t i o n s : (a) The \oi must be i n e q u i l i b r i u m at each branch p o i n t . (b) The t « ; must be zero a t f r e e edges. These c o n d i t i o n s then determine what va lue t 0 ; w i l l be a t any p o i n t 5 . H a v i n g to use t h i s approach has i t s e f f e c t on the c o n s t a n t s , as they were i n t e g r a t e d over an unbranched system. T h i s can be s e e n , f o r example, i n some o f the terms f o r the torque T . One o f the c o n t r i b u t i n g terms i s Expanding t o g ives terms o f the type \0 X I t (w.-sO i s J s = K . j amongst o t h e r s . The p o r t i o n C t(u,-a,) As r e p r e s e n t s p a r t o f as determined by t 0 = \c T ' d s , an e q u i l i b r i u m e q u a t i o n . T h i s no l o n g e r holds f o r a branched system. The o t h e r two i n t e g r a l s i n v o l v e d i n K l 3 , and are geometric i n t e g r a l s and are u n a f f e c t e d by b r a n c h e s . T h i s means t h a t constants a r i s i n g from c o n s i d e r a t i o n o f the shear f lows t © w i l l have to be c a l c u l a t e d u s i n g t c ; = +• ) T , d s and a s t e p w i s e i n t e g r a t i o n procedure . T h i s means t h a t = ~ ) 0 t do, -w7.)<o, i s ^ ~~ )0 {. (w, - d 9 may not be v a l i d f o r a l l s e c t i o n s , as i t i s based on an unbranched system. However, f o r a two branched s y s t e m , the f o l l o w i n g c o n s t a n t s , l ^ o , l a c e » K 13 » which are a r e s u l t o f t o t e r m s , can be shown to take the 95 same form as an unbranched system. The i n c l u s i o n o f and Iccc i n the l i s t o f constants necessary to be i n v e s t i g a t e d a r i s e s from the f a c t t h a t t h e r e are two types o f i n t e g r a l s d e f i n i n g and Jacc • The f i r s t type i s due to the s t r e s s e s 0~" and takes the form Jo (f-jOft = Ixc. This i n t e g r a l i s u n a f f e c t e d by b r a n c h i n g i n the c r o s s - s e c t i o n , as i t i s based on <T , n o t \ 0 . The second type i s due to %o terms and takes the form So S i n $ f 0 H f - 7 ) d s c J s = - J O L C . f o r an unbranched s e c t i o n . S i n c e the i n t e r n a l i n t e g r a l i s a measure o f % 0 , i t must be m o d i f i e d t o take account o f e q u i l i b r i u m at branches . I t i s t h i s < | x c t h a t was i n v e s t i g a t e d f o r branched systems. Of c o u r s e , from know-ledge o f l i n e a r beam b e h a v i o u r , i t i s known t h a t the c o n s t a n t g i v i n g shear values i s l&x.c , so i t would be expected t h a t no matter how branched the system i s , o r what c o - o r d i n a t e system i s u s e d , the va lue t h a t s h o u l d a r i s e from the i n t e g r a l i s j £ = c t . However, t h i s i s not immediate ly c l e a r from the i n t e g r a l i , s i n (j> L Kjf-f) Js cU i f i t i s w r i t t e n i n the form necessary to handle branched systems. The i m p l i -c a t i o n o f t h i s d i s c u s s i o n seems t o be t h a t the c o - o r d i n a t e system S i s not a p a r t i c u l a r l y good one f o r branched s y s t e m s , and o n l y leads t o c o m p l i c a t i o n s i n the c a l c u l a t i o n s which i n the end g ive an expected r e s u l t . I t may be t h a t a more j u d i c i o u s c h o i c e o f c o - o r d i n a t e system i s p o s s i b l e f o r the c a l c u l a t i o n o f K,3 and the due to lo , once the nature o f these c o n s t a n t s i s understood from the development and r o l e 1 0 plays i n c a l c u l a t i n g them. I t might then be p o s s i b l e t o see t h a t i n g e n e r a l , no m a t t e r how many b r a n c h e s , the c o n s t a n t s have the same form as the un-branched s y s t e m , as i s most l i k e l y the case . F o u r t h l y , the I J C C , ! ^ values s h o u l d be c a l c u l a t e d about the c e n t r o i -dal a x i s , a l l the o t h e r s about the a x i s through the shear c e n t r e . F i n a l l y , i n some cases a general formula f o r some o f the c o n s t a n t s s h o u l d be d e r i v e d f i r s t , as u s i n g numbers d i r e c t l y may r e s u l t i n accuracy problems. 7:4 Loads The l a t e r a l loads are assumed to a c t at the shear c e n t r e . The a x i a l loads a r e taken t o a c t a t the p o i n t on the c r o s s - s e c t i o n S = 0 . I f the l a t e r a l loads a r e a p p l i e d a t any o t h e r p o s i t i o n i n the c r o s s - s e c t i o n , f o r example the top f l a n g e o f an X beam, e x t r a terms w i l l have to be i n t r o d u c e d to account f o r t h i s e f f e c t d u r i n g d e f o r m a t i o n . T h i s i s n o t d i f f i c u l t t o do , but has not been done i n t h i s development. 7:5 D i f f e r e n t i a l Equat ions I f the d i f f e r e n t i a l equat ions (78) are compared t o those o f Oden, Timoshenko, V l a s s o v , e t c . , i t w i l l be seen t h a t Equat ions (78) are much more g e n e r a l , as they do not r e q u i r e the use o f l i n e a r values o f V , M e t c . i n t h e i r e v a l u a t i o n . They are n o n - l i n e a r d i f f e r e n t i a l equat ions and they i n -c lude the p o s s i b i l i t y t h a t -VjM e t c . may change due to deformed geometry. T h i s may be o f i m p o r t a n c e , f o r i n s t a n c e , w i t h the presence o f a x i a l f o r c e s near c r i t i c a l i n one p l a n e , as they may magnify the moments t e n d i n g to cause l a t e r a l b u c k l i n g . T h i s p o s s i b i l i t y i s i g n o r e d i n the usual l i n e a r d i f f e r e n -t i a l e q u a t i o n s o f b u c k l i n g . 97 There are s e v e r a l methods o f s o l v i n g the d i f f e r e n t i a l e q u a t i o n u s i n g approximate methods. A G a l e r k i n method might be u s e d , o r any one o f s e v e r a l v a r i a t i o n s o f an i t e r a t i o n t e c h n i q u e . The i t e r a t i o n t e c h n i q u e s d i f f e r as t o what i s t r e a t e d as the l e f t h a n d s i d e o f the e q u a t i o n and what i s t r e a t e d as the r i g h t h a n d s i d e . In t h i s deve lopment , the l e f t h a n d s i d e was taken t o be the l i n e a r s t r u c t u r e e q u a t i o n s , as i t was f e l t t h a t t h i s a c h i e v e d the most even handed t r e a t e d o f the n o n - l i n e a r s t r u c t u r e terms and a l s o k e p t the e q u a t i o n s s i m p l e . . An a l t e r n a t i v e method i s to p l a c e some o f the n o n - l i n e a r s t r u c t u r e terms (which may be l i n e a r i n the d i f f e r e n t i a l e q u a t i o n ) on the l e f t h a n d s i d e , b u t t h i s has the e f f e c t o f e m p h a s i z i n g some n o n - l i n e a r s t r u c t -ure terms compared t o the o t h e r s , and c o m p l i c a t e s the s o l u t i o n c a l c u l a t i o n s . U n f o r t u n a t e l y , the t e c h n i q u e used i n t h i s t h e s i s does n o t t a k e f u l l advantage o f the n o n - l i n e a r i t y o f the e q u a t i o n s , as u s i n g one i t e r a t i o n i s e q u i v a l e n t to u s i n g l i n e a r f o r c e s i n the d i f f e r e n t i a l e q u a t i o n s o l u t i o n . E i t h e r two i t e r a t i o n s would be r e q u i r e d , o r one o f the a l t e r a n t i v e forms o f i t e r a t i o n u s e d , t o a d e q u a t e l y cover the n o n - l i n e a r i n t e r a c t i o n s i n the equa-t i o n s . F i n a l l y , s i n c e the i t e r a t i o n t e c h n i q u e o n l y approximates the c o r r e c t s o l u t i o n o f the e q u i l i b r i u m e q u a t i o n s , t h e r e i s no reason f o r the r e s u l t i n g d i s p l a c e m e n t s to g i v e e q u i l i b r i u m f o r c e s . Of c o u r s e , the f o r c e s a t a j o i n t w i l l be i n e q u i l i b r i u m , but an i n d i v i d u a l member may n o t . b e . 7:6 Symmetry and C o n s e r v a t i v e n e s s The moments, when d e f i n e d as m a i n t a i n i n g t h e i r l i n e o f a c t i o n , a r e non-c o n s e r v a t i v e when a c t i n g on f r e e edge boundary c o n d i t i o n s . T h i s w i l l p r o d -uce a non-symmetr ic member m a t r i x , [ 1 0 ] , as each member i s d e v e l o p e d under f r e e boundary c o n d i t i o n s . However, when the i n d i v i d u a l m a t r i c e s are added 98 up i n t o a s t r u c t u r e m a t r i x , i t i s the boundary c o n d i t i o n s on the s t r u c t u r e which determine the n o n - c o n s e r v a t i v e n e s s , and i n most cases they render the system c o n s e r v a t i v e and the m a t r i x symmetr ic . An example o f an e x c e p t i o n i s a c a n t i l e v e r under a p p l i e d end moment, w i t h the moment d e f i n e d as i n t h i s t h e s i s as m a i n t a i n i n g i t s d i r e c t i o n o f a c t i o n . I f any n o n - c o n s e r v a t i v e problems are e n v i s a g e d , the l o a d v e c t o r may have to be m o d i f i e d to become a f u n c t i o n o f the d i s p l a c e m e n t s . Of c o u r s e , i n d i -r e c t l y the s t r u c t u r e m a t r i x i s m o d i f i e d by t a k i n g the p a r t s o f the l o a d v e c t o r t h a t become f u n c t i o n s o f S and t r a n s f e r r i n g them to the o t h e r s i d e o f the e q u a t i o n where they can be p l a c e d i n the s t r u c t u r e m a t r i x . However, i f the l o a d i n g i s n o n - c o n s e r v a t i v e , the approach used here i s i n genera l inadequate as a dynamic approach i s best f o r the most general s o l u t i o n s [ 1 0 ] . I t i s i n t e r e s t i n g though t h a t no mention o f c o n s e r v a t i v e f o r c e s was necessary to develop the equat ions i n t h i s work. The o n l y l i m i t a t i o n s on these e q u a t i o n s , a s i d e from the r e s t r i c t i o n s on r o t a t i o n s and t o r q u e s , i s the assumption o f a s t a t i c , e l a s t i c s o l u t i o n . 7:7 A p p r o x i m a t i o n s f o r Small Terms I n some o f the approximat ions used to n e g l e c t t e r m s , the f a c t t h a t the l e n g t h o f the beam was l a r g e compared to c e r t a i n terms was u t i l i z e d . How-e v e r , the q u e s t i o n o f the v a l i d i t y o f these approximat ions f o r the elements a r i s e s , s i n c e the element l e n g t h may be q u i t e s h o r t . T h i s i s n o t a r e a l p r o b l e m , however, as the elements need o n l y be a b l e to d u p l i c a t e the a c t u a l s t r u c t u r e . I f , f o r example, the e f f e c t s were i n c l u d e d i n the element because the approximat ions about L were i n v a l i d , i t would make no d i f f e r e n c e t o the a n a l y s i s o f the l a r g e o v e r a l l s t r u c t u r e as the new terms would a l l f a l l o u t , because they are i n s i g n i f i c a n t i n a f f e c t i n g the b e h a v i o u r o f the 99 o f the l a r g e s t r u c t u r e . For i n s t a n c e , shear d e f l e c t i o n behaviour i s not u s u a l l y i n c l u d e d i n o r d i n a r y l i n e a r beam element s t i f f n e s s m a t r i c e s unless s h o r t s e c t i o n s o f beam were going to be s t u d i e d . I f shear behaviour was p l a c e d i n these e l e -ments and a l o n g t h i n beam a n a l y s e d , i t would be seen t h a t these terms con-t r i b u t e n o t h i n g to the a n a l y s i s , as f o r a l o n g t h i n beam, shear d e f l e c t i o n e f f e c t s are t r i v i a l . I t can t h e r e f o r e be concluded t h a t the elements need n o t be complete i n t h e m s e l v e s , but only must be ab le t o d u p l i c a t e the r e q u i r e d behaviour o f the a c t u a l p h y s i c a l problem. 7:8 P o i n t o f A c t i o n o f A x i a l Load In t h i s development, the a x i a l l o a d i s taken to a c t a t the f r e e edge o f the c r o s s - s e c t i o n where S~o, whereas the shear f o r c e s and moments are taken t o a c t through and about an a x i s system through the shear c e n t r e . T h i s means t h e r e are two p o i n t s o f i n t e r e s t a t each end o f the s e c t i o n , r a t h e r than the usual one p o i n t . Whi le t h i s i s d i f f e r e n t t o the usual beam a n a l y s i s where o n l y one p o i n t serves to d e f i n e a l l the d e f l e c t i o n s , i t s h o u l d not be viewed w i t h a l a r m . There are many i n s t a n c e s where two o r more p o i n t s o f r e f e r e n c e have an advantage over the usual one r e f e r e n c e p o i n t . For example, a doubly symmetric wide f l a n g e can be r e p r e s e n t e d two ways. The f i r s t and most usual method i s t o c o n c e n t r a t e a l l the f o r c e s and d e f l e c t i o n s a t the c e n t r o i d . The second method i s to use two r e f e r e n c e p o i n t s , one a t the i n t e r s e c t i o n o f the top f l a n g e and the web, one a t the i n t e r s e c t i o n o f the lower f lange and the web. Using these two p o i n t s o f r e f e r e n c e , a l l the p e r t i n e n t d e f l e c t i o n s o f the s e c t i o n can be d e s c r i b e d . However, an i m p o r t a n t advantage has been g a i n e d . I t i s now p o s s i b l e to 100 s p e c i f y s e p a r a t e boundary c o n d i t i o n s f o r each f l a n g e , which i s not p o s s i b l e us ing the c e n t r o i d as the r e f e r e n c e p o i n t . I t can t h e r e f o r e be s t a t e d t h a t the use o f two r e f e r e n c e p o i n t s g r e a t l y e n l a r g e s the type o f boundary con-d i t i o n s which may be e a s i l y h a n d l e d . T h i s d i s c u s s i o n was i n t r o d u c e d o n l y t o i n d i c a t e t h a t w h i l e two r e f e r e n c e p o i n t s f o r a beam may not be common, they may sometimes be e a s i l y i n t r o d u c e d w i t h s i g n i f i c a n t advantage. In t h i s t h e s i s there are reasons o t h e r than i n c r e a s e d u s e f u l n e s s f o r u s i n g two r e f e r e n c e p o i n t s . Some p h y s i c a l f e e l f o r these reasons can be gained from the f o l l o w i n g example. P o s t u l a t e a wide f lange beam w i t h an a x i a l l o a d a c t i n g a t a p o i n t A , which i s l o c a t e d a l o n g a l i n e p e r p e n d i c u l a r t o the web and p a s s i n g through the c e n t r o i d and shear c e n t r 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 12. FIG. 12 THREE METHODS OF CONNECTION FOR ECCENTRIC AXIAL LOAD FOR I SECTION. 101 Three d i f f e r e n t types o f c o n n e c t i o n are i l l u s t r a t e d i n F i g u r e 12. In F i g u r e 12a, the l o a d i s passed t o the s e c t i o n from A by a r i g i d arm connec-ted to the web. F igure 12b and 12c show r i g i d connect ions to the upper and lower f l a n g e s r e s p e c t i v e l y . Now, f o r t r a n s l a t i o n s and r o t a t i o n s where p lane s e c t i o n s remain p l a n e , the work done by the a x i a l f o r c e a t A i s the same f o r the t h r e e c o n n e c t i o n s shown. However, f o r a pure warping d e f o r m a t i o n , which i s c h a r a c t e r i z e d by the f l a n g e s r o t a t i n g about the ^. a x i s but i n o p p o s i t e d i r e c t i o n s , the work done by an a x i a l l o a d a t A i s d i f f e r e n t f o r a l l t h r e e c a s e s . In F i g u r e 12a, the work i s z e r o , and i n F i g u r e 12b, the work i s non-zero and o f o p p o s i t e s i g n t o the work o f F i g u r e 12c. R e c a l l i n g t h a t the work done by the a p p l i e d loads d u r i n g a warping d i s -placement i s d e f i n e d as the bi-moment, i t can be c l e a r l y seen t h a t the a x i a l f o r c e a t A e x e r t s t h r e e d i f f e r e n t bi-moments on the s e c t i o n i n F i g u r e 12, depending on how i t i s connected . T h i s i m p l i e s t h a t i t i s n o t enough t o s p e c i f y the p o s i t i o n o f the a x i a l l o a d ; i t s method o f c o n n e c t i o n to the c r o s s - s e c t i o n must a l s o be s p e c i f i e d . T h i s means t h a t p l a c i n g the a x i a l p o i n t o f r e f e r e n c e a t the shear c e n t r e would be a lmost meaningless because any e c c e n t r i c loads would be r e f e r e n c e d t o the method and p o i n t o f c o n n e c t i o n , n o t to the s h e a r c e n t r e . For the general c a s e , where the shear c e n t r e may n o t even be i n the c r o s s - s e c t i o n , i t becomes t o t a l l y meaningless to s p e c i f y the s h e a r c e n t r e as the p o i n t o f r e f e r e n c e f o r a x i a l l o a d and d e f l e c t i o n , as the f i r s t t h i n g t h a t must be done i s t o s p e c i f y some k i n d o f c o n n e c t i o n from the a x i a l l o a d to the beam, which i m p l i e s c o r r e c t i v e bi-moments. I t i s f o r these reasons t h a t the p o i n t o f r e f e r e n c e f o r a x i a l terms was taken t o be the a r b i t r a r y p o i n t S = 0 . This a l s o g ives some computat ional advantages . I f the a x i a l l o a d i s not a t 5 = 0 , then moments must be a p p l i e d t o 102 c o r r e c t the r e s u l t a n t , and these moments are c a l c u l a t e d by the e c c e n t r i c i t y o f the a x i a l l o a d from 5 = 0 , not from the shear c e n t r e . T h i s i s because the m a t r i x takes i n t o account and expects t h a t the a x i a l l o a d i s a t the p o i n t S = o • These c o r r e c t i v e moments may be f u n c t i o n s o f 8^ , the t o r s i o n a l r o t a t i o n , s i n c e the 8^ r o t a t i o n s a f f e c t the e c c e n t r i c i t y o f the a x i a l l o a d about S = o , unless the j o i n t a t which the a x i a l l o a d i s a p p l i e d i s r e s t r a i n e d t o r s i o n a l l y . The c a l c u l a t i o n o f the c o r r e c t i v e bi-moment f o r an e c c e n t r i c a x i a l l o a d cannot be done u s i n g e q u i v a l e n t f o r c e r e s u l t a n t s , as the bi-moment has no r e s u l t a n t . I n s t e a d , the c o r r e c t i v e va lue must be found u s i n g energy . For i n s t a n c e , the e x t r a work o f warping Q. due t o the e c c e n t r i c l o a d may be found by Q =• P w v where W ^ ' i s the a x i a l d isp lacement due to w a r p i n g o n l y o f the p o i n t o f a c t i o n o f P w i t h r e s p e c t to the a x i a l d isp lacement o f S = 0 . The term W e ' i s found from Equat ion (7) when a l l d e f l e c t i o n s o t h e r than js' are z e r o . T h i s g i v e s C l e a r l y , the c o r r e c t i v e bi-moment i s Pco, Once a g a i n , s i n c e co, i s o n l y meaningful f o r p o i n t s on the c r o s s -s e c t i o n , i t i l l u s t r a t e s t h a t P must i n some way be connected to some p o i n t on the c r o s s - s e c t i o n . The v a l u e o f co, i n Pu>, i s c a l c u l a t e d by i n t e g r a t i n g , to the p o i n t 5 a t which P i s connected t o the c r o s s - s e c t i o n . The c o n n e c t i o n i t s e l f may 103 cause P to be above o r below the p o i n t o f c o n n e c t i o n to the c r o s s - s e c t i o n . I f t h i s i s the c a s e , then e x t r a terms a c c o u n t i n g f o r t h i s w i l l have to be added t o P i^ . These t e r m s ~ w i l l depend on the type o f c o n n e c t i o n used. I f P i s e c c e n t r i c but on the c r o s s - s e c t i o n , then P*-0' i s the o n l y c o r r e c -t i v e term needed. 104 CHAPTER 8 CONCLUSIONS In t h i s t h e s i s , a s e t o f n o n - l i n e a r f o r c e d e f l e c t i o n r e l a t i o n s were dev-e loped u s i n g a s e t o f common e n g i n e e r i n g assumptions c o n s i s t e n t l y a p p l i e d . A d i s c u s s i o n o f the s i z e o f the terms t h a t were d i s c a r d e d on the b a s i s o f these assumptions i s g i v e n . These f o r c e - d e f l e c t i o n e q u a t i o n s , when p l a c e d i n the o v e r a l l beam equat ions o f e q u i l i b r i u m , y i e l d a s e t o f d i f f e r e n t i a l equat ions o f e q u i l i b r i u m i n terms o f the d i s p l a c e m e n t s . These equat ions are s i m i l a r i n form to those developed i n Timoshenko and Gere [7], B l e i c h [3], Oden [6] and V l a s s o v [9], except t h a t the equat ions h e r e i n are coupled non-l i n e a r e q u a t i o n s , and so more g e n e r a l . In t h i s s e n s e , the equat ions are new. The s o l u t i o n o f the d i f f e r e n t i a l equat ions was a t t a i n e d by employing an i t e r a t i o n , o r s u c c e s s i v e a p p r o x i m a t i o n method. T h i s produces a s o l u t i o n to the d i f f e r e n t i a l e q u a t i o n s , i n terms o f the boundary c o n d i t i o n s , which must be p l a c e d back i n t o the n o n - l i n e a r f o r c e - d e f l e c t i o n r e l a t i o n s and e v a l u a t e d a t the boundaries to y i e l d a s t i f f n e s s m a t r i x . I n t h i s s e n s e , the m a t r i x i s new, as i t i s the s o l u t i o n o f a more gene-r a l s e t o f n o n - l i n e a r d i f f e r e n t i a l equat ions than usual and has been developed u s i n g an a l t e r n a t i v e and q u i t e d i f f e r e n t method than the more common and w e l l developed energy methods. T h i s s t i f f n e s s m a t r i x i s n o n - l i n e a r i n the S terms and i s d i f f i c u l t t o s o l v e d i r e c t l y . A secant m a t r i x approach i n v o l v i n g s e v e r a l i t e r a t i o n s was used, a l though there are o t h e r a l t e r n a t i v e methods o f s o l v i n g n o n - l i n e a r equ-a t i o n s which c o u l d be used. The m a t r i x was used to a n a l y s e s e v e r a l c l a s s i c a l problems and the agreement was good i n a l l c a s e s . Once the m a t r i x has been d e r i v e d f o r a member, i t i s p o s s i b l e to apply v a r i o u s t r a n s f o r m a t i o n s which a l l o w the member m a t r i x to handle e c c e n t r i c 105 connect ions and e c c e n t r i c l o a d s . This i n t u r n a l l o w s the b u i l d i n g o f a r b i t -r a r y space s t r u c t u r e s w i t h unusual j o i n t and load , c o n d i t i o n s . T h i s s t e p , however, was not taken i n t h i s t h e s i s . There are s e v e r a l areas of i n v e s t i g a t i o n t h a t are open f o r f u t u r e work. F i r s t , a l t e r n a t i v e s o l u t i o n s o f the d i f f e r e n t i a l e q u a t i o n may be l o o k e d a t . The G a l e r k i n method c o u l d be u s e d , o r s e v e r a l a l t e r n a t i v e i t e r a t i o n t e c h n i q u e s c o u l d be employed. S e c o n d l y , the d i f f e r e n t i a l equat ions c o u l d be extended t o i n c l u d e the e f f e c t o f s h a f t b u c k l i n g , which was n e g l e c t e d i n t h i s t h e s i s . T h i r d l y , a b e t t e r technique than the s e c a n t m a t r i x approach f o r the s o l -u t i o n o f might be employed. F i n a l l y , the r e l a t i v e e f f e c t s o f the v a r i o u s p h y s i c a l c o n s t a n t s on the c r i t i c a l l o a d might be s t u d i e d . 106 BIBLIOGRAPHY [1] Barsoum, R . S . , " F i n i t e Element Method A p p l i e d to the Problem o f S t a b i l -i t y o f a Non-Conservat ive S y s t e m " , International Journal for Numerical Methods in Engineering, V o l . 3 , pp. 6 3 - 9 7 , (1971). [2] Barsoum, R.S . and G a l l a g h e r , R . H . , " F i n i t e Element A n a l y s i s o f T o r s i o n a l and T o r s i o n a l - F l e x u r a l S t a b i l i t y P r o b l e m s " , International Journal for Numerical Methods in Engineering, V o l . 2 , pp. 335-352, (1970) . [3] B l e i c h , F . , Buckling Strength of Metal Structures, McGraw H i l l , ( 1952) . [4] Fung, Y . C . , Foundations of Solid Mechanics , P r e n t i c e - H a l l . [5] G o o d i e r , J . N . , " F l e x u r a l - T o r s i o n a l B u c k l i n g o f Bars o f Open Cross S e c t i o n " , Bulletin No. 2 8 , C o r n e l l U n i v e r s i t y E n g i n e e r i n g Experiment S t a t i o n , January 1942. [ 6 ] Oden, J . T . , Mechanics of Elastic Structures, M c G r a w - H i l l , ( 1967) . [7] Timoshenko, S . P . and Gere , J . M . , Theory of Elastic Stability, McGraw-H i l l , (1961) . [8] Timoshenko, S . P . , Strength of Material, V o l . I I , D . Van N o s t r a n d , ( 1956) . [ 9 ] V l a s s o v , V . Z . , Thin Walled Elastic Beams, P u b l i s h e d f o r the N a t i o n a l S c i e n c e F o u n d a t i o n , Washington DC, and the Department o f Commerce, USA, by the I s r a e l Program f o r S c i e n t i f i c T r a n s l a t i o n s , (1961) . [10] Z i e g l e r , H . , "On the Concept o f E l a s t i c S t a b i l i t y " , Advn. Appl. Mech. , V o l . 4 , pp. 351-403, (1956) . 107 APPENDIX I THE MATRIX S i n c e E q u a t i o n s (118) are n o n - l i n e a r i n S , t h e r e are s e v e r a l ways o f removing a S v e c t o r to form the e q u a t i o n C K CS)l G - -f . F o r i n s t a n c e , the term NX^'AT m a y be grouped as The former grouping i s o f the form M AT', and i s the form t h a t would have been o b t a i n e d i f the equat ions were l i n e a r d i f f e r e n t i a l equat ions i n -v o l v i n g l i n e a r va lues of M , V , P i n the c o - e f f i c i e n t s . T h i s method o f grouping w i l l be used, and t h i s w i l l be used as the r u l e t o f a c t o r out the £ . F o r example, i n N>t V > the S a s s o c i a t e d w i t h v ' w i l l be taken o u t s i d e as £ . Once t h i s has been d e c i d e d , i t i s next convenient to r e p r e s e n t the m a t r i x as a sum o f f o u r s u b - m a t r i c e s K = K . + K z + K J + K 4 The l i n e a r s t r u c t u r e terms are K i and K t , whereas K3 and K ^ a r e the n o n - l i n e a r t e r m s . Expansion of Equat ions (124) under these c o n d i t i o n s y i e l d s the f o l l o w i n g : 108 13 \ 13 ll»s> 13 i 13 u 13 «8 13 i i i -Jim •13 13 13" i li 13 13 i m 13 i II X 13 + -Jjiu M 13 IW HH i n« i i H H i »« i II3^. »« \ I 1 O Symmetri \ 1 • v . 0 I las. II IK II 1W II < 7 8 9 /o N 2 i ( £ L 0 f ] " ' ) // /Z /3 /<f ^N 3 p (e 0 CP?] - ' ) •N^Ce" L*?]~') - N , l (elL*?]") K 3 -3 4 5 6 3 - N , p [ f .To " 4 -N.if?.r. - ['iff.'] 5 6 7 a 7 6 9 /o // /2 // -e .L «n" ' f ; ] / £ /3 A f II IZ /3 /4 3 -N-UC -5 + N,^[K,L -S 13 14 7 N, [, [ f i r . -Q -N.{[C - * . L * I ] " f l ] 9 - N, {. [ 10 // A3 / 4 ^ COM t 7 a 9 / o // - N 3 J > ; 1 i >o /z f •* • - e.L<pir'+n 13 »- • « f T l , L -©"LJ iT ' f? ] /4 V ( 3 , 4 ) = + P 3 = r o w j 4 - c o l u m n - P ( 7 , 8 ) -(%J0) = + P (u,4) = - P x + N 4 C * . c * f ] V 4 ^ j ^ f : . f - ^ [ * j ] " [ f f i K r , j u,B) - *Pi - N , [ [JL f *j • j*-?. WIT' uv f !] ] 114 pf • N, t ft L»:i"y * C J C f - il [*;]" D ,"f'--fir3]] (»,») - -N, [. k M Y * o + C r'-SlttrtttV #']] The a d d i t i o n o f K.> K*., K 3 and K 4 g i v e s the K m a t r i x i n If] = LK«)\{I) There are a few r e l a t i o n s which ease the c a l c u l a t i o n s o f K . For i n s t a n c e - / [<?,']" = loll A l s o , where CT] = [ T ] - / Many o f the terms are r e p e t i t i v e , so t h a t one c a l c u l a t i o n s u f f i c e s to handle s e v e r a l terms. T h i s s t i l l leaves a f o r m i d a b l e amount o f work i n assembl ing K . Appendix I I , however, can be used t o ease the c a l c u l a t i o n o f the i n d i v i d u a l terms. 116 APPENDIX I I TERMS NEEDED IN MEMBER MATRIX T h i s appendix c o n t a i n s the p a r t i c u l a r s o l u t i o n s to the f i r s t i t e r a t i o n o f the d i f f e r e n t i a l e q u a t i o n s , as w e l l as the e v a l u a t e d terms necessary t o c o n s t r u c t the m a t r i x . The p a r t i c u l a r s o l u t i o n f o l l o w s . N3 [ © ] =• L >> sink \1 } K o s W ^ , , A 3? " I ^ c o s h A J ; % } ^ O o o - 3 V \ f l 3 _ 2.1 \ ( ?* \ o o o o L 0 J - L I * y A* J , I J** A J A 1 , A1 J ) 6A s i n U j c o s h x i - ^ } A vzjf sink A| O o o o o o 118 L$Ts>] -6 A ( i Vco^yz -X 2> < n U j ) % a i l l\tnUxi -1,1 c o s ^ j ) , ^('TTV ) , O 4** ' 4X O > O ) Q . i O 0 ) O ) O ) o The r e m a i n i n g terms are those necessary to bu i ld the matrix K o o o L_ o o -1 O L L L L - 3 - L -/ 0 o o o L 0 0 / o L L L L 3 / o 0 / 0 / \ 0 ' / L o smh A c o s h A / / (x^coshA ( i ) s m k A 0 120 0. = ( 6 , ° > °> ° ) 0 u =- I G , 0 , 0 , 0 ) 0 • <*• © 0 - ( A 3, 0,0 , 0 ) © 0 - ( 0 , A ' , o, 0 ) C A c o s l o A , A s m h A , 0 j 0 1 • * © L = ( A s i n l i A , A c o s i ^ A , 0 , 0 1 r AT 1 1 , 0 =• M , C o , 0, i ,o)[cp;] -' 0 0 f ' /,£> = M . ( o, 0, o , I O C Q ! ] " ' toff 1 l ) U = M, ( 3, a, i, 0 ) [ c p ; ] " ' = M . ( i , i, 0 t © , ' ] " ' o a o f • 0 P -cl T I . O = Co, 0, o, 121 C-- M* (3, i, )>o) c*;r' f - M i (A,O, I,O) C - 0 ? 3 * M3 Ls'fCc*?]") 0 , 0 , * , o o 0 0 0 7 0 , 0 , o C<p?J' C o = M z (o, I, o, i ) [« f] i - f i 3 [ r ; T f c ? ] - ' ) T 0 , O , G , o o , 1 i ° > o , ° > ° > o O + M 3 L 5 X ] T ( W ? ] " ) T Z\co$\\ A ^ A s m h A t I f o o , o , o o o o 122 W 9 = CsmKX, cosk A, I, l) L$U £ 51 nil A j I c o s h X , 2 , 2 0 O , O , 0 o 0 0 ) ) + M * C s , ] T C u : f ) T 0 , 6 > 0 -6 O > 0 0 , 0 0 , 0 , 0 'do- (o , 1,0,0 C * ? J 0 , 0 , 0 , 0 0 , * > 2 . 0 , O > O O ) O , 0 > O c o s £X c o s h A 7 2A 5jnh X O O sinli X , c o s l a A , / , /) ca?] 6 s i n h A ; 6 c o s h X ? 6 £ s j n h A 2 c o s h X } Z. O O 124 f ; > o ,o 0 , o o -'VA2 o o o o 0 o o > 0 1 ° > 0 o 1 • - M e Co, o > O ) [ < P /A'.O) [*:]"' M7[j'JT(w;r)' o D i f ] 0 O o . ° , ° o > o , 0 , o o 1 — i -M , L r t T ( w : r ' ) 0 o o , 'o , o . ° o o 125 f 3 = - M 8 C - % o) !>?] -/ M 7[c] Tfa :r) T o o o o o , o , o o, O ) [ Q \ ] - M 1 0 L r ] T ( w f r ' ) T -%* , - z4/t , o , o o , o , o o , 0,0,0 0 , o , 0 , 0 -'A4, O)[CP;]"' -M t oLs l] T(L^rO T o , , o C<j>;]~ o 0 , o , o , o too C = - M M (~6/A\ O.;O,O)LQ1Y' + nz(-6/x>-*A\ol0)m -1 0 o I - 'VA1 o / o > o 0 > > o o o o o o • M u C - f r "VA1, ~/A\ O)C*:] M,„ [f f fWrD' - 5 * _ 2©5 o o 6 . 0 -VA* o > > O o o > > o o - - M l 6 ( * V * , 0 , O , 0 M 1 5 [ J ' ] T ( U ? ] " ) T , ° o , o , 0 0 , ° M„ [f 'J T ([«,-]") T 0 o J 3A o > ° > o o t . 0 . o -n,r[f]T(cj;r')T o • -<4 > O , o o > O o > 0 > O , o , o 128 = - M i * ( o , 1 , 0 , 0 ) L®0 - M , 3 [ J ! ] T ( [ 5 ? ] " ) T M„[r]T(wni' A / 3 , .0 o o o 0 . 0 . 0 A o o o o o , , o £ , O , O 0 , 0 , 0 o . 0 . 0 M.fCs'fCc*;]"')' , O , 0 0 0 , 0 0 , ° 0 > , 0 0 , 0 , 0 129 n -ie-O O O J c -1$ c o O co I 11 o o o O 0 c + +• s~C sr _c c o 3i u ro -/-si o O o o O u o o O u <^ c in o + o o o 0 o o o o 0 n -0 0 0 0 o o o o 0 o o o I © In 130 O o E + O O -< O O c o o O 0 c o o c VA N/ - ) - s i - t ^ o u -< —sr c m O o + c 5? «4 i . o u c O O -< c l/"> , | < T ) t < ( v f t O 0 O O o o l l5 -c C T' O O - I ' ^9 O O O O S 3 I o O u •i O C 7«3 0 o 0 o o 0 o o o o 0 / -< _sr c c v n v/> *-< + + -< c •Si o u _c VRT o o 4-4/1 o o I _sr O o •N/ vn O + _s: v n O 0 o o «5 |-I ll i J Ir. r 131 The f i , f x, l A e t c . are ^ x - f m a t r i c e s . S i n c e each row o f these m a t r i c e s may be a very c o m p l i c a t e d e x p r e s s i o n , i t w i l l be convenient to break them up. In the f o l l o w i n g pages the e t c . w i l l be w r i t t e n i n the form e t c . V,(3) T.C3) f: (1) ?\ (*) where i n d i c a t e s the I row o f the T m a t r i x . f ' O ) = L M . C o , 0 , 0 , 0 ) Co . ' ] {[(i) = - M. ( 0 , 0 , 0 , 0 ) [cp!]" • f k i ) - L M , C o , '/A\ 0 , 0 ^ * ! ] L M , [ ^ ] r ( L c D f f ) T A 3 , O , 0 O > O O > O > O 0 J O > O > O O t M 3Lr] Ta<p?n T 0 0 0 > » YA 0 0 0 > ) > 0 0 0 0 0 0 0 0 133 r | ( T " — j , l ( T " ~ J JI J O , O o , o , o > o I 0 , 0 - I -M3[ffCc*;r)T 2cosK A /A , £ 5)nhA /A , / , 2. O O O O o , O o , o _ 1 f ! M - - M , ('A, 'A, '/*, l ) L ? i ] ' -F* CO - iNsCo, o,6)W\ L i i J f f C w i r ' f _ l i A' > 0 , o , o 0 , ° > o o » o i o , o 0 , o o o , o , Mjs'rcw.-rf V A , o i o > o o , o , o o , o » o [ C P ? ] " f * W - L M f C s m h A / A 4 , cosh A / A 1 , 'A, '/OK A N A ~ A 1 J , A V £ Sin 1 K A / A 1 COSVI X 2 an ~A * ) . 2. , / O O 2 c o s h \ / A 2 , ;/s , / o , o , O o , o , o 136 f ' O ) = - Ma ( 0 , 0 , 0 , o)[tf?]' + M,(o,o 0, M7[s']T(c*:r,)r o *> o o o t o o o o o o o o > > o ) o o o > » can" C o , -Vv 0 , -"fx o o C<i>f3 o ) * o > o o o > o o o 1 o > o > _ I 137 -M 7 h ' ] T f c : r , ) T ~ A") f lz6*3A4 xl) f" o o O O z x 1 , 0 0 O o o - I Mf((i1),f-r?),-'/A',-W[*,«] O O > > o o VA* 0 o o o , o f lo) = .-ri*. ( o . o . o . o ) L«:]"' + Hu ( o , o , o , . o ) C f l P ! ] ' -M10[f l]T(L^fr)T o o o > o > o , o , o 139 - i 6 ("JA1- " h ) o o ~7; o o - V A * o o o o o o 6 4 (-'AO O , 0 6. A 1 O -2A\o o , o 0 o > 140 -1 M,3Ls3JT([*niT o > o o » ° o > o > o o o o J o o 1 o o o o r_<P?3 -i M,4 [ r i T ( c ^ f ) T o o 1 o o 1 o 0 t 0 1 o ) 0 o > , 0 o o o > 0 » i o _ I 0 o o o > 1 o o o o > o 0 0 o > > 1 0 1 0 0 1 142 i<2> V* -C O o o o c -I o j in O o ^ 4 •VI O O o TV " l o o o o o o 0 o o I <0 -r< I c vn O O 0 ° 0 o o -< VP o o l o o -< _E 0 o Jk -< 1 <^ c* *> •< CO C c —~» vO ^,—. — - -—-o o c <n -< o o i - < VO o o O u in O O o o •4 — II i Vo i O o o Vi o u c 7i o a _ , o 4 -I 'M 0 0 O 0 o o 0 O ' f. »» ft N » I ^ o o o o I —c O O 3 o O Q O o 0 o + C c + o o o o o c in + «n 0 -3 O O 8 ^ i " 0 O - i ; -i-< 3> O O c .rt I ^ I <Nf o o c I ? o o 3 L 1/1 - I . o o o o o - I -l-< c O u c 3? o -< O O cn ro 144 A s tudy o f the terms i n the appendices r e v e a l s t h a t t h e r e are many r e p -e t i t i v e m a t r i c e s i n v o l v e d i n the c a l c u l a t i o n s . In many c a s e s , o n l y the i n s e r t i o n o f a d i f f e r e n t constant and 6 makes one l a r g e group o f m a t r i c e s d i f f e r e n t from a n o t h e r . T h i s can be used t o advantage i n c o n s t r u c t i o n o f the member m a t r i x . F i r s t o f a l l , c o n s t r u c t s u b - r o u t i n e s which can handle the m a t r i x o p e r -a t i o n s which a r e c o n s t a n t l y r e p e a t i n g , such as m a t r i x X m a t r i x m a t r i x x v e c t o r v e c t o r x m a t r i x e t c . U s i n g these s u b - r o u t i n e s , those m a t r i x products which a r e c o n s t a n t t h r o u g h o u t the s o l u t i o n can be e v a l u a t e d and s t o r e d . Once i n t o the i t e r a t i o n t e c h n i q u e , the s u b - r o u t i n e s can be r e - u s e d to e v a l u a t e the r e q u i r e d terms u s i n g the d e f l e c t i o n s S and the s t o r e d c o n s t a n t m a t r i x . T h i s t e c h n i q u e a l l o w s the computer to do a l l the m u l t i p l i c a t i o n s o f the m a t r i c e s i n v o l v e d . 

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