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Buckling of thin plates using the framework method Sen, Rajan 1970

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BUCKLING OF THIN PLATES USING THE FRAMEWORK METHOD by RAJAN SEN B. Tech (Hons.) Indian I n s t i t u t e of Technology Kharagpur, I n d i a , June 1968 A THESIS SUBMITTED IN-PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF ' MASTER OF APPLIED. SCIENCE i n the department of CIVIL.ENGINEERING We accept t h i s t h e s i s as conforming to the requ i r e d standard THE UNIVERSITY OF BRITISH COLUMBIA September 1970 In presenting t h i s t h e s i s i n p a r t i a l f u l f i l l m e n t of the requirements f o r an advanced degree at the U n i v e r s i t y of B r i t i s h Columbia, I agree that the L i b r a r y s h a l l make i t f r e e l y a v a i l a b l e f o r reference and study. I f u r t h e r agree that permission f o r extensive copying of t h i s t h e s i s f o r s c h o l a r l y purposes may be granted by the Head of my Depart-ment or by h i s r e p r e s e n t a t i v e s . I t i s understood that copying or p u b l i c a t i o n of t h i s t h e s i s f o r f i n a n c i a l g a in s h a l l not be allowed without my w r i t t e n permission. Department of C i v i l Engineering The U n i v e r s i t y of B r i t i s h Columbia, Vancouver 8, Canada ABSTRACT F i n i t e element method i n v o l v i n g r ectangular bar c e l l s capable of i m i t a t i n g e l a s t i c a c t i o n i n plane s t r e s s and f l e x u r e of p l a t e s w i t h any value of the Poisson's r a t i o , i s extended to i n v e s t i g a t i o n of s t a b i l i t y of rec t a n g u l a r p l a t e s . This r e q u i r e s f o r m u l a t i o n of the s t a b i l i t y m a t r i x used f o r s o l u t i o n of the eigenvalue problem, which gives the magnitude of the - c r i t i c a l l o a d . F o u r . d i f f e r e n t examples.are solved and the r e s u l t s , compared w i t h the exact values and the a v a i l a b l e no bar s o l u t i o n s , are found to be good. A b r i e f study i s al s o made of the e f f e c t of negative e x t e n s i o n a l and f l e x u r a l s t i f f n e s s e s of the members of the c e l l and suggestion i s made on s e l e c t i o n of the d e s i r a b l e range f o r the values of the aspects r a t i o of the c e l l as rela t e d , to the values of y. TABLE OF CONTENTS CHAPTER Page I. INTRODUCTION • 1 I I . DERIVATION. OF PLANE STRESS. AND PLATE. BENDING MATRICES .. 3 C e l l P r o p e r t i e s 3 C e l l Requirements ............... 4 Plane Stress S t i f f n e s s M a t r i x .............. . .13 • Flexure C e l l ...... • . -19 C e l l Requirements f o r Flexure .............. • 20 P l a t e Bending S t i f f n e s s M a t r i x . 33 I I I . BASIC THEORY OF. INSTABILITY ' " . 35 IV. DERIVATION OF STABILITY. MATRIX • 38 Length Change of Bars . 39 C a l c u l a t i o n , of Bar Forces 56 S t a b i l i t y M a t r i x • 58 V. EXAMPLES ..... ..- ' '. ••- 68 Example 1 . . . 68 Example 2 ................ 72 Example 3 • 74 Example 4 . ... 75 E f f e c t of Negative S t i f f n e s s e s 77 VI.' CONCLUSION ' -•• 85 BIBLIOGRAPHY ' . • 87 i i LIST OF TABLES Page I. FIRST TWO COLUMNS OF.PLANE STRESS STIFFNESS MATRIX 18 I I . DISTRIBUTION FACTORS FOR FLEXURE CELL 31 I I I . ANALOGY BETWEEN THE TERMS AND LOCATIONS IN Xrr AND 56 X OF THREE OTHER CORNER BARS' 53 IV.' ELONGATIONS OF PRIMARY BARS .• ... 57 V. ELEMENTS OF STABILITY MATRIX ' ' ' - •• 62 VI. UNIFORM COMPRESSION IN. X DIRECTION ....... 71 V I I . UNIFORM COMPRESSION IN X AND Y.DIRECTIONS AND UNIFORMLY .. \ VARYING LOAD IN Y DIRECTION . ... 73-V I I I . PURE SHEAR .. . ' ' 7.6 IX. EFFECT OF HIGH y k 2 FOR UNIFORM COMPRESSION IN X DIRECTION (SIMPLY SUPPORTED) .... " 80 X. EFFECT OF HIGH.yk 2 FOR.UNIFORM COMPRESSION I N X DIRECTION (CLAMPED) . 81 XI. EFFECT OF HIGH.yk 2 FOR COMPRESSION IN TWO DIRECTIONS ... 82 X I I . EFFECT OF HIGH y k 2 FOR PURE SHEAR UNDER DIFFERING EDGE CONDITIONS • 83 X I I I . EFFECT OF HIGH uk 2 FOR PURE SHEAR EDGES SIMPLY SUPPORTED • . ... . .. 84 i i i LIST OF SYMBOLS A,,A', A" c r o s s - s e c t i o n a l areas of short side-bars of c e l l s , t o t a l , primary, secondary. A , A|, Al^ c r o s s - s e c t i o n a l areas of longer s i d e bars of c e l l s , ' t o t a l , primary, secondary. A^, A^ c r o s s - s e c t i o n a l areas of diagonal and corner bars. D e l a s t i c f l e x u r a l constant of p l a t e . E. modulus of e l a s t i c i t y . F, - Fj., '¥ a d d i t i o n a l s t r e s s e s i n members of secondary bar system. [ k ] , [K] s t i f f n e s s matrices of c e l l and of model. — — m [k ] , [K ] s t a b i l i t y matrices of c e l l and of model, -s —s m P plane s t r e s s loads. S w i t h and without s u b s c r i p t s , bar s t r e s s e s , d i r e c t . T increment of t o t a l energy of model on b u c k l i n g . U increment of energy of deformation on b u c k l i n g . V increment of p o t e n t i a l energy on b u c k l i n g . X,Y components of f o r c e s . X,Y w i t h subscripts—members of plane s t r e s s s t i f f n e s s matrix. a s m a l l e r dimension of c e l l . b dimension of p l a t e along y a x i s e, e^ displacement of po i n t of a p p l i c a t i o n of load. f c r i t i c a l i n t e n s i t y of load. k aspect r a t i o of c e l l . £ length i v m w i t h subscripts—-elements of f l e x u r a l s t i f f n e s s matrix. t thickness of p l a t e . w d e f l e c t i o n of p l a t e and of bar. w w i t h s u b s c r i p t s — n o d e and j o i n t displacements. x, y r e c t a n g u l a r coordinaty. z w i t h s u b s c r i p t s — e l e m e n t s of f l e x u r a l s t i f f n e s s matrix. a angle of diagonal bar. 8 parameter d e f i n i n g c r i t i c a l load. Y elements of s t a b i l i t y matrix. A w i t h s u b s c r i p t s — s h o r t e n i n g of p r o j e c t i o n of bar length caused by f l e x u r e 6," 8 ,6. movements of the nodes. — m n a c r i t i c a l normal u n i t s t r e s s , cr T c r i t i c a l u n i t shear s t r e s s , cr 6 , 6 w i t h and without s u b s c r i p t s — n o d a l r o t a t i o n s . Vi Poisson's R a t i o . V ACKNOWLEDGEMENT I am indebted to my s u p e r v i s o r , Dr. Hrennikoff f o r h i s time, patience and i n v a l u a b l e a s s i s t a n c e at every stage of t h i s work. I t ' s a pleasure a l s o to.acknowledge the many h e l p f u l suggestions given by Mr. C. I . Mathew. F i n a n c i a l a s s i s t a n c e i n the form of a research a s s i s t a n t s h i p and f r e e use of U.B.C.'s e x c e l l e n t computing f a c i l i t i e s i s g r a t e f u l l y acknowledged. CHAPTER I INTRODUCTION Any rig o r o u s mathematical s o l u t i o n of a s t a b i l i t y problem must s a t i s f y simultaneously the e q u i l i b r i u m , c o n s t i t u t i v e r e l a t i o n s and the s t r a i n c o m p a t i b i l i t y . Unfortunately these requirements can be met f u l l y only i n a few b u c k l i n g problems and t h i s has l e d to the use of numeri-c a l methods which y i e l d only approximate s o l u t i o n s . Of these, the v a r i a t i o n a l or energy methods and the f i n i t e d i f f e r e n c e approach have been the most widely used. . Lately,.however, a new approximate method c a l l e d the f i n i t e element method has been proposed f o r the i n v e s t i g a t i o n of s t a b i l i t y . This method was.introduced i n .1941 by Hrennikoff ( 1 ) * f o r deter-mination of s t r e s s e s and displacements i n p l a t e s subjected to plane s t r e s s and f l e x u r e . The method c o n s i s t s of r e p l a c i n g the s t r u c t u r e under c o n s i d e r a t i o n by an assembly of u n i t s or c e l l s composed of bars and conforming almost e x a c t l y to the o u t l i n e of the s t r u c t u r e . The c e l l s have a d e f i n i t e repeating p a t t e r n and are j o i n e d to each other at the corners. The bars of the c e l l s are endowed w i t h e l a s t i c p r o p e r t i e s which are determined from the co n d i t i o n s of e q u a l i t y of d e f o r m a b i l i t y of the s t r u c t u r e and i t s model under any a r b i t r a r y uniform s t r e s s condi-t i o n . The use of the framework method req u i r e s the knowledge of the Numbers s i g n i f y the refe r e n c e s , l i s t e d i n B i b l i o g r a p h y . r e l a t i o n s h i p between the corner f o r c e s and the corner displacements. The corner forces h o l d i n g the c e l l i n e q u i l i b r i u m , when one of i t s c o r n e r s ; i s given a u n i t movement along one of the coordinate .axes, are termed "the d i s t r i b u t i o n f a c t o r s . " The assembly of the d i s t r i -b u t i o n f a c t o r s corresponding to a l l u n i t movements of a l l j o i n t s i s arranged i n the form of a matrix and i s c a l l e d the s t i f f n e s s m a t r i x of the c e l l . C e l l s not i n v o l v i n g bars have also been proposed (2). U n l i k e the framework c e l l s , the "no bar" c e l l s are a mathematical a b s t r a c t i o n but they s t i l l a l l o w formation of d i s t r i b u t i o n f a c t o r s and generation of the s t i f f n e s s matrix l i k e the bar c e l l s . The model obtained by r e p l a c i n g the p l a t e w i t h bar or no bar c e l l s must be solved for. the movement of the nodes and t h i s i n v o l v e s numerous l i n e a r simultaneous equations. With the advent of high speed d i g i t a l computers t h i s method has become p r a c t i c a l and has r a p i d l y gained favor f o r s o l v i n g plane s t r e s s (3) and p l a t e bending (4) problems. Recently (5) t h i s method has been extended a l s o to the s t a b i l i t y problems. For t h i s purpose development of another m a t r i x , the s t a b i l i t y m a t r i x i s r e q u i r e d , as w i l l be discussed i n d e t a i l i n the l a t e r chapters. CHAPTER I I DERIVATION OF PLANE STRESS -AND FLEXURE STIFFNESS MATRICES C e l l P r o p e r t i e s The bar model used i n s t a b i l i t y problems must i m i t a t e the a c t i o n of the p l a t e both i n c o n d i t i o n s of plane s t r e s s w h i l e the load i s s t i l l below the c r i t i c a l i n t e n s i t y , and i n f l e x u r e as the s t r u c t u r e becomes unstable and begins to buckle. The model i s formed of equal rectangular c e l l s interconnected at the corners of the r e c t a n g l e s . D i f f e r e n t r e c -tangular c e l l s are p o s s i b l e , but the type wbich w i l l be used i n t h i s work i s p a r t i c u l a r l y convenient f o r the i n s t a b i l i t y s t u d i e s . The plane s t r e s s and f l e x u r a l p r o p e r t i e s of the c e l l may be described and deter-mined independently ,of each other. Plane Stress C e l l A' A" ka The c e l l ( f i g . 2.1) of the dimensions a and ka i s made up of two systems of bar s , the primary and the secondary. The primary system c o n s i s t s of three kinds of bars: the s i d e bars of the -lengths a and ka and the cross s e c t i o n a l areas A' and A| r e s p e c t i v e l y and of the diagonals of the areas A^. The primary system i s q u i t e s a t i s f a c t o r y to i m i t a t e F i g . 2.1 the d e f o r m a b i l i t y of the p l a t e w i t h one p a r t i c u l a r value of Poisson's r a t i o . The secondary system i s added on to make the model s u i t a b l e f o r any value.of Poisson's r a t i o . This system.is composed of subdivided s i d e bars of the areas A" and A l 1 , two on each edge, j o i n e d at the. corners to the main system and at the mid-edges to each other and to the corner bars of the areas A^. These areas are determined w i t h the a s s i s t a n c e of some'assumptions, as explained l a t e r , from the c o n d i t i o n s of equal d e f o r m a b i l i t y , as the c e l l and the p l a t e are subjected to an a r b i t r a r y uniform s t r e s s . C e l l requirements: ' A general c o n d i t i o n of.an a r b i t r a r y uniform s t r e s s i n a p l a t e may be achieved by a combination of the f o l l o w i n g s t r a i n c o n d i t i o n s : ex ; ey = yxy = 0 ey ; ex = yxy = 0 (2.1) Y X Y I ex = . cy = 0 CONDITION 1 ( F i g . 2.2) Uniform s t r e s s e s p and up are a p p l i e d to the p l a t e i n the x and y d i r e c t i o n s r e s p e c t i v e l y . The piece of p l a t e of the s i z e ka by a i s elongated i n x d i r e c t i o n an amount 6 = -JL- C 1 -^ ) k a (2.2) tE . \ / w h i l e remaining unchanged i n length i n y d i r e c t i o n . The c e l l of the same s i z e as the p l a t e , must i m i t a t e i t s deformation w h i l e acted upon by t h e c o r n e r f o r c e s s t a t i c a l l y e q u i v a l e n t t o t h e ones i n t h e p l a t e . t p # / i n pa : pa 2 2 PLATE F i g . 2.2 CELL pa uplca These c o r n e r f o r c e s e v i d e n t l y a r e y - and — | — r e s p e c t i v e l y i n x and y d i r e c t i o n s . The c o r n e r b a r s o f t h e s e c o n d a r y system do n o t work: the l o a d i n g i s s y m m e t r i c a l and i f one o f the c o r n e r b a r s , f o r example 7-8 i s • f o u n d t o be under t e n s i o n , so s h o u l d a l s o be t h e b a r 6-7. S i n c e however t h e r e i s no member a t the mid l e n g t h o f the edge 4-3 t o r e s i s t the t r a n s v e r s e component o f the s t r e s s e s i n t h e c o r n e r b a r s t h e s e b a r s must be u n s t r e s s e d . The s u b d i v i d e d s i d e b a r s 4-2 and 1-3 s i m p l y j o i n the p r i m a r y b a r s i n c a r r y i n g t h e a p p r o p r i a t e s t r e s s e s i n s p i t e o f s l i g h t t r a n s v e r s e d i s p l a c e m e n t s o c c u r r i n g a t • t h e m i d - s i d e s o f the c e l l s . T h e i r combined a r e a i s A = A^ .+ A 1 1 The b a r s 1-2 and 4-3, b o t h i n t h e p r i m a r y and t h e s e c o n d a r y systems a r e undeformed, and f o r t h i s r e a s o n t h e i r s t r e s s X i s z e r o . The e l o n g a t i o n • of the diagonal bar ( F i g . 2.2) i s .= 6 Cosct 2 2 = P (1-U )k a tE ( k 2 + l ) 1 / 2 (2.3) The-free body diagram f o r corner 1 i s shown i n F i g . 2.3. From i t s e q u i l i b r i u m , i n Y d i r e c t i o n -r x=o X 2 X l \ t 1 F i g . 2.3 X£ S i n a upka (2.4) The general r e l a t i o n between the s t r e s s e s X and the elongation of an . e l a s t i c bar of the area A and the length i s X = AEA (2.5) On s u b s t i t u t i n g i n t o Eqn(2.5) the expression f o r X^ and 6^  from the equation (2.4) and (2.3) the expression f o r the cross s e c t i o n area of the diagonal bar i s found u a t ( k 2 + l ) 3 / 2 ?k 2 (2.6) E q u i l i b r i u m of the corner i n X d i r e c t i o n gives pa X„ Cos a X l " 2~ " 2 (2.7) On r e p l a c i n g X^ by i t s expression from (2.4) and X^ by the equation (2.5), w i t h s u b s t i t u t i o n f o r the q u a n t i t y A of the elongation 6 (Eqn 2.2) of the 7. bar, the cross s e c t i o n a l area of the l a t t e r i s found (1-yk ) at 2 ( l - y 2 ) (2.8) CONDITION 2 ey, ex = yxy = 0. By a s i m i l a r analysis the t o t a l area of the side bar 1-2 i s found (k -y) at 2k ( l - y 2 ) (2.9) where A = A' + A" (2.10) CONDITION 3- yxy, ex e y = 0 The framework c e l l (Fig. 2.4).is subjected to the stress condition equivalent to uniform shear i n the plate xxy = xyx = p ^ / i n . The primary side bars are unstressed andso they have been omitted from F i g . 2.4. The diagonal bars.are evidently stressed as well as the corner bars and the subdivided side bars. p#/±n pa 2 p#/in p#/in pa 2 p#/in PLATE F i g . 2.4 CELL The f r e e body diagram f o r forces a c t i n g i n d i f f e r e n t bars at the corner 3 i s shown i n F i g . 2.5. The r e s u l t a n t of the forces and at the corner 3 i s R p a ( k2 + l ) 1 / 2 2 (2.11) E v i d e n t l y i t acts i n the d i r e c t i o n of the diagonal. The u n i t shear s t r a i n i n the p l a t e i s Y 2p (1+y) tE (2.12) F i g . 2.5 So the h o r i z o n t a l displacement <5, of the corner 3 equal to the shear displacement i n the p l a t e i s ' . _ 2p (1+y) ka tE The corresponding change i n length of the diagonal bar i s (2.13) <5 S i n a 2p (1+y) ka tE . ( k 2 + 1 ) l / 2 (2.14) T h i s . i s an e l o n g a t i o n i n the bar 2-3 and a shortening i n the bar 1-4. Using Eqn 2.5 and s u b s t i t u t i n g f o r A, A, and £ t h e i r appropriate values i t i s found that the s t r e s s i n the diagonal bar i s ya ( k 2 + l ) 1 / 2 l - y (2.15) The difference R^, of the corner force R and must be c a r r i e d by the subdivided bars (3-7) and (3-6) Then _ a ( k 2 + l ) 1 / 2 ( l - 3 y )  R l _ 2(l-y) ( 2 , 1 6 ) The stresses F and F^ i n the bars 3-7 and 3-6 are F = R i S i ™ = f c l ^ f < 2 - 1 7 > and F = R. Cosa = k*^" 3!^ (2.18) 1 1 2(l-y) It may be observed that at a l l four corners the forces R^, F and F^ are of the same magnitude but d i f f e r e n t i n sign.• Now consider one of the intermediate j o i n t s , .such as 7. Let F^ be the stress i n each of the corner bars. For equilibrium we have (Fig. 2.6) 7 2F 3 Sin ct = 2F This gives / \ F ' F Y a ( l - 3 y ) ( k 2 + l ) 1 / 2 3 3 3 2(l-y) F i g . 2.6 Note, that each corner bar c a r r i e s a stress opposite i n sign to the stresses i n edge bars of the same corner. Deformation of each corner t r i a n g l e , such as 6-3-7 (Fig. 2.7) may be v i s u a l i z e d separately from the other corner t r i a n g l e s . Let the sides 3-7 and 3-6 elongate under t h e i r respective stresses F and F^ by the amounts 7-7' = — da and 6-6' = -|- d(ka). The corner bar 7-6 i s subjected to compression F^, and so the points 7' and 6' move 10. t r a n s v e r s e l y through the distances dx and dy r e s p e c t i v e l y into' the p o s i t i o n s 7" and 6" as the corner bar shortens by the amount (7-6) - (7" - 6") = dz The s t r e s s e s i n the members of the adjacent corner t r i a n g l e 7-4-8 aire equal and opposite to the ones i n the t r i a n g l e 7-3-6 and so t h e i r corners 7 and 8 move s i m i l a r l y to the corresponding corners i n the p r e v i o u s l y discussed t r i a n g l e . The behaviour of the two remaining I t r i a n g l e s i n F i g . 2.7 i s I ' s i m i l a r to the ones j u s t • considered. ! P r o j e c t i n g the displacements j i 7-7' , 71-7",- 6-6' and 6'-6" on the l i n e 7-6 i t i s found i | that d& = dxCosa + dySina - -|-d(ka)Cosc 1 1, n. - -^daSina F i g . 2.7 From t h i s (k 2+l) 1 / 2d£ = kdx + dy - K[ ] - -|da (2.20) I t may be observed i n F i g . 2.7 that the deformed p o r t i o n s of the two perpendicular sides of a l l corner t r i a n g l e s are r e s p e c t i v e l y p a r a l l e l , to each other. By moving the t r i a n g l e 8"'-4-7'" to the r i g h t a d i s -tance 2dy, the t r i a n g l e 5 " - l - 6 ' " i n the negative d i r e c t i o n of the x a x i s a distance 2dx, and the t r i a n g l e 7"-3-6" the distances 2dx and 2dy along the y and minus x axes, the deformed c e l l becomes the parallelogram 2abc (Fig. 2.8). The deviations of•the corner angles of t h i s parallelogram from 90° represent the unit shear s t r a i n 2(l+u)p • —EE ,2dy ka 2dx F i g . 2.8 Expressing yxy from F i g . 2.8 and d i v i d i n g by two (1+u) P tE. dy dx ka a (2.21) Mu l t i p l y i n g t h i s equation by ka and subtracting from i t Eqn (2.20) we, f i n d ( k 2 + l ) d£ + | d ( k a ) + ^ = ^ I P k a (2.22) The length changes of the three bars d£, yd(ka) and yda are expressed through t h e i r stresses F^, and F and the cross s e c t i o n a l areas A^, A^ and A" s t i l l unknown. 12. This, gives 2 3/2 3 2 ( k > D • a , k_a _ a _ = 4 k ( l - y Z ) A 3 + •A1" A" . t ( l - 3 y ) U ' Z J ; This equation represents the b a s i c r e l a t i o n s e r v i n g f o r determination of the three bar areas. I t i s , . o f course, i n s u f f i c i e n t f o r f i n d i n g these three values'and s i n c e no a d d i t i o n a l equations are a v a i l a b l e , i t w i l l be assumed f o r s i m p l i c i t y that AJ = kA" and A 3 = ( k 2 + l ) 1 / 2 A " (2.24) S u b s t i t u t i n g Eqn (2.24) i n t o Eqn (2.23) we o b t a i n A" = * t ( l - 3 y ) ( k 2 + l ) ( 2 . 2 5 ) 2 k ( l - V ) = a t ( l - 3 y ) ( k 2 + l ) ( 2 i 2 6 ) 2 ( l - y Z ) and A = a t ( l - 3 y ) ( k 2 + l ) 3 / 2 ^ 2 k ( l - y Z ) The primary system bars A| and A' are found by s u b t r a c t i n g the expressions ' (2.25) and (2.26) from the combined s i d e bar areas Eqn (2.8) and (2.9) ' , „, at(2yk 2+3y-k 2) ( 2 > 2 g ) 2 ( l-y Z) A' = ^ ( 3 y k 2 + 2 y - l ) ( 2 < 2 g ) 2 ( l V ) 13. Plane Stress S t i f f n e s s M a t r i x With the bar areas known, ..the d e r i v a t i o n of expressions f o r the d i s t r i b u t i o n f a c t o r s i s a matter of conventional s t r u c t u r a l a n a l y s i s . However, the e a s i e s t way of o b t a i n i n g them i s by the use of an a u x i l i a r y c o n d i t i o n i n combination w i t h the three main con d i t i o n s discussed e a r l i e r . AUXILIARY CONDITION 4 The - c e l l i s subjected to a s t a t e of deformation i n which the four corners.are moved i n X. d i r e c t i o n i n a f l e x u r e - l i k e manner through the distances 6 as shown i n F i g . 2.9. The members 1-3 and 2-4 are thus s t r e s s e d , the f i r s t _ i n tension and the second i n compression, w i t h the st r e s s e s A 1E26 X l = Ta~ (I-uk )Et5 k ( l - y 2 ) (2.30) F i g . 2.9 The symmetry of the loa d i n g c o n d i t i o n leaves the corner members unstressed. E q u a l l y unstressed are the remaining two s i d e members and the diagonals of the c e l l s i n c e they undergo no changes i n length. Thus the c e l l i s h e l d i n e q u i l i b r i u m by the corner f o r c e s X equal to X^ as shown i n F i g . 2.9, with no Y forces c o n t r i -b u t i n g to the e q u i l i b r i u m . 14. ACTION 1 X 41 Y 41 21" 31 A c t i o n 1 2 II "31 -»-Y 11 X • t A = i 21 X. 11 F i g . 2.10 The c o n d i t i o n of the corner 1 of the c e l l being moved a distance X i n the x . d i r e c t i o n , w h i l e the other three corners remain i n t h e i r o r i g i n a l p o s i t i o n s may be obtained by combination of the three e l e -mentary c o n d i t i o n s 1, 4 and 3 as i s shown i n F i g . 2.10. Then the d i s t r i b u t i o n f a c t o r s corresponding to the displacement A^ may be found by adding up the known corner forces i n the three component c o n d i t i o n s . We o b t a i n X tEA 1 4 k ( l - y 2 ) ytEA • 4 ( l - y 2 ) X, ktEA 8(1+y) tEA 8(1+y) X, (1-yk )EtA 4 k ( l - y 2 ) (2.31) 15. I t i s i n t e r e s t i n g to observe that these d i s t r i b u t i o n f a c t o r s are a f f e c t e d by the areas of the bars only to the extent of c o n t r i b u t i o n by the c o n d i t i o n 4, sin c e the corner forces i n the c o n d i t i o n s 1 and 3 are independent of the bar areas. The d i s t r i b u t i o n f a c t o r s f o r t h i s a c t i o n are given i n Table 1. Each term i s - o b t a i n e d as x l l X l + X4 + X 3 X 2 1 - = X l X 3 - X 4 X 3 1 = X 3 - X l - X4 X 4 1 = X4 - X l - X 3 Y l l = Y l + . Y 3 Y 2 1 = Y 3 - Y l Y 3 1 = Y l - Y 3 Y 4 1 " Y l - Y 3 The f i r s t s u b s c r i p t , . i n the symbols used here f o r the d i s t r i b u t i o n f a c -t o r s , describes the number of the j o i n t and the second the number of the a c t i o n , the v e r t i c a l displacement of j o i n t 1 being a r b i t r a r i l y c a l l e d A c t i o n 1 and the h o r i z o n t a l displacement of the same j o i n t , A c t i o n 2. The d i s t r i b u t i o n f a c t o r s corresponding to the displacements A ^ of the corner 1 may be derived d i r e c t l y by a procedure s i m i l a r to the one j u s t employed. However i f i s more convenient to f i n d them by simple X manipulations w i t h the A d i s t r i b u t i o n f a c t o r s as i n d i c a t e d i n F i g . 2.11. 16. X 41 31 X * Y X 11 41 (a) (b) F i g . 2.11 The c e l l i n F i g . 2.11(a) w i t h - a l l i t s corner forces-.and .•displacements, i s r o t a t e d 180° about the y a x i s to the p o s i t i o n shown i n (b). This f i g u r e i s then turned through 90° about the z a x i s i n t o the p o s i t i o n (c) and i s compared w i t h Fig.(d) representing the c e l l w i t h the d i s p l a c e -y ment A£.• The corner forces corresponding the Fig.(d) i . e . , the required d i s t r i b u t i o n f a c t o r s , may be e v i d e n t l y copied from F i g . ( c ) making allowance f o r the d i f f e r e n t placement of the c e l l . The r a t i o k of the two s i d e s of the re c t a n g l e i n F i g . ( c ) becomes — i n F i g . ( d ) , and the dimension a i n the former f i g u r e becomes ka i n the l a t t e r . With these X s u b s t i t u t i o n s the d i s t r i b u t i o n f a c t o r s f o r A^ are t r a n s f e r r e d i n t o y those f o r A^. The d i s t r i b u t i o n f a c t o r s f o r A c t i o n 1 and A c t i o n 2 given i n Table 1 comprise the f i r s t two columns of the 8 x 8 s t i f f n e s s matrix of the c e l l . The terms of the other columns of the matrix are equal or equal and opposite i n s i g n to some terms i n the f i r s t two columns and may be obtained from the f i r s t two columns by appropriate r e v e r s a l s about some coordinate planes. Thus i n order to f i n d A^ = 1, i . e . 17. column 3 of the m a t r i x , the c e l l i n F i g . 2.11(a) with a l l i t s corner forces and displacements i s r o t a t e d 180° about the x a x i s to the p o s i t i o n shown i n F i g . 2.12(a). The corner forces corresponding to Fig..2.12(b), i . e . the r e q u i r e d d i s t r i b u t i o n f a c t o r s , may be copied from F i g . 2.12(a). F i g . 2.12 The complete equation {P} [K] {A} i s • X l l X12 X 2 1 X22 X 3 1 _ X 3 2 X 41 X42 Y l l Y12 " Y21 Y 22 "Y31 Y 32 Y 4 1 Y42 X 2 1 X22 X l l -x 1 2 X 4 1 ~ X42 X 3 1 X32 *l Y 2 1 Y 22 - Y ' l l . Y12' " Y41 ' Y42 Y 3 1 Y32 P3 X 3 1 X32 X 4 1 - " X42 X l l " X12 X 2 1 X22 X *S Y 3 1 Y 32 " Y41 Y42 - Y l l Y12 Y 2 1 Y 22 F4 X 4 1 X42 X 3 1 " X32 X 2 1 " X22 X l l X12 PI Y42 Y42 " Y31 . Y 32 " Y21 Y 22 Y l l Y12 18. TABLE I FIRST TWO COLUMNS OF PLANE STRESS STIFFNESS MATRIX A c t i o n 1 A* = 1 A c t i o n 2 X l l " 4+k 2 (l-3y) Et X12 = Et 8k (1-y 2) 8 ( l - y ) X 2 1 • = ( l - 3 y ) k Et X22 = (l-3y) Et 8 ( l - y 2 ) 8 ( l - y 2 ) X 3 1 = -4•+-k2 (1+y) Et X32 = (l-3y) Et • 8 k ( l - y 2 ) 8 ( l - y 2 ) X 4 1 = k Et X42 = . -Et , 8 ( l - y ) 8(1-y) Et . Y12 = 1 - 3y + 4k 2 Et Y l l = 8 ( l - y ) 8k(1-y 2) l-3y . Et Y 22 1 + y - 4k 2 Et Y 2 1 = 8(1-y 2) 8 k ( l - y 2 ) Y 3 1 " (l-3y) Et Y32 = - ( l - 3 y ) Et 8(1-y 2) 8 k ( l - y 2 ) Et Y42 = Et Y 4 1 = 8 ( l - y ) 8k(1-y) 19. FLEXURE CELL EI,c' The r e c t a n g u l a r f l e x u r a l bar c e l l ( F i g . 2.13) which may be used i n con-j u n c t i o n w i t h the described plane s t r e s s c e l l . c o n s i s t s of the si d e mem-bers and the diagonal members, endowed wi t h f l e x u r a l s t i f f n e s s e s E I , EI and EI ^ f o r bending out of the plane of the F i g . 2.13 c e l l and zero s t i f f n e s s e s f o r bending i n i t s plane. Since the bar c e l l of j the k i n d described here was found capable of i m i t a t i n g f l e x u r a l a c t i o n of a p l a t e only w i t h one p a r t i c u l a r value of Poisson's r a t i o , i t was found necessary f o r g e n e r a l i t y to endow the s i d e members of the c e l l w i t h t o r s i o n a l s t i f f n e s s e s , C and C^ i n a d d i t i o n to t h e i r f l e x u r a l s t i f f n e s s e s . T o r s i o n a l s t i f f n e s s i s not re q u i r e d i n the diagonal members. The f l e x u r a l c e l l , must of course be c o n s i s t e n t i n i t s geometry w i t h the plane s t r e s s c e l l and f o r t h i s reason i s must possess the corner members of the l a t t e r . Since however f l e x u r a l a c t i o n does not r e q u i r e these members, they are simply assumed to be of the s t i f f n e s s zero, a l -though they w i l l be f l e x u r a l l y deformed as explained l a t e r , they w i l l not c o n t r i b u t e to the f l e x u r a l e q u i l i b r i u m of the j o i n t s on t h e i r ends. REQUIREMENTS OF THE CELL 20. (6) The bar s t i f f n e s s e s are determined i n such a way that the c e l l deforms i n bending i n e x a c t l y the same manner as the p l a t e i n con d i t i o n s of any a r b i t r a r y uniform f l e x u r e . The e a s i e s t way to comply with t h i s necessary c o n d i t i o n . i s to make the c e l l behave as the p l a t e i n the 'following three s t a t e s : (1) Constant curvature — i n the x d i r e c t i o n , no curvature i n the y d i r e c t i o n , no t o r s i o n i n planes x and y, i . e . Mxy = 0 . (2) Constant curvature ^ i n the y d i r e c t i o n , no curvature i n the x d i r e c t i o n , ' Mxy = 0. (3) Uniform t o r s i o n a l c o n d i t i o n i n x and y planes w i t h no f l e x u r e i n x and y d i r e c t i o n s . CONDITION .1 Bending i n X . d i r e c t i o n ( F i g . 2.14) Uniform moments .m f o r bending i n x d i r e c t i o n and ^ my f o r bending i n y d i r e c t i o n are a p p l i e d to the p l a t e . The p l a t e bends to the curvature 1 . r£>. — = ~— where $ i s the d e f l e c t i o n angle on the length ka. The c e l l of the same s i z e as the p l a t e must.imitate i t s deformation w h i l e acted upon by the corner moments s t a t i c a l l y equivalent to the ones i n the p l a t e . These are Mx = ukam • 2 y and My = y anr (2.32) By the w e l l known r e l a t i o n D D$ m — = -— y r ka and so the Eqns (2.32) become Mx = ^y®$ and M y = i l T * ( 2 ' 3 3 ) E t 3 where D = 7r~T~ (2.34) 1 2 ( l - y Z ) the q u a n t i t y t being the thickness of the p l a t e . The bending moments i n the members of the c e l l are expressed i n terms of t h e i r s t i f f n e s s e s and angles of d e f l e c t i o n by the r e l a t i o n Since the members 1-2 and 3-4 remain s t r a i g h t t h e i r moments are zero M = 0 The bending moments i n the si d e members 1-3 and 2-4 are M, EI $ ka (2.36) In order to f i n d the moment i n the diagonal i t i s necessary to deter-mine i t s angle of f l e x u r a l deformation. The v e c t o r <J> i n F i g . 2.15 L ! represents the angle of r o t a t i o n of the end 1 of the diagonal 1-4 i n F i g . 2.14 i n r e l a t i o n to the end 4. The components $Cosct and $Sina of t h i s v e c t o r are r e s p e c t i v e l y the angle of f l e x u r e and t w i s t developed on the length of the diagonal. The bending Cosa $Sina j moment i n the diagonal then i s •Fig. 2.15 E I 2 $ Cosa • a ( k 2 + l ) 1 / 2 (2.37) The diagonal o f f e r s no t o r s i o n a l r e s i s t a n c e to i t s t w i s t . E q u i l i b r i u m of any j o i n t i n the c e l l such as 3 ( F i g . 2.16) gives Sina = Mx Cosa + M^ = • My Mx M=0 »-My S u b s t i t u t i o n of appropriate values f o r M^, M 2 >. Mx and My gives T = ( l ~ y k 2 ) a t 3 1 2 24(1-y ) ,,2 ,,3/2 3 y(k +1) at 2 4 k ( l - y 2 ) (2.38) (2.39) F i g . 2.116 CONDITION, 2 Bending i n y d i r e c t i o n By a s i m i l a r a n a l y s i s 2 3 (k -M)at-2 4 k ( l - y 2 ) (2.40) and expression (2.39) CONDITION 3 The p l a t e i s subjected to.uniform t o r s i o n a l moments m i n planes x and y. Since no bending moments e x i s t i n the planes x and y, the cr o s s - s e c t i o n s of the p l a t e p a r a l l e l to these dimensions remain s t r a i g h t and f o r t h i s reason the si d e bars i n the c e l l develop no bending moments. Leaving the x and y c r o s s - s e c t i o n s , passing through the centre 0, i n undisplaced p o s i t i o n the corners of the p l a t e d e f l e c t the amount 6 as shown i n F i g . 2.17.- The equivalent, moments at the corners of the c e l l are , Mx T T m a 2 xy My x m ka I xy (2.40a) 24. The angles of t o r s i o n a l r o t a t i o n of the s t r a i g h t edges of the p l a t e i n r e l a t i o n to the l i n e s p a r a l l e l to them and passing through the centre 0 are and $ 2 6 — f o r the edges 1-3 and 2-4 2 6 —- f o r the edges 1-2 and 3-4 3. (2.41) mxy/in 4. 3 * \ mxy/in -=b~ • 0 «mxy/ I « ± n r 2 1 i L * * fc i mxy/in h 2 $x 6 3 My «- My, My -*,T •«• My F i g . 2.17 These q u a n t i t i e s represent the angles of t w i s t of the s i d e members on cl llCcl the lengths y and — r e s p e c t i v e l y . Then by the f a m i l i a r r e l a t i o n s between the torque, t o r s i o n a l s t i f f n e s s and the angle of t w i s t T = 46 C k a 2 46 .C, and (2.42) ka The diagonal members develop f l e x u r a l moments which may be expressed through t h e i r angular deformations. As may be observed i n F i g . 2.18, 25. the corner 2 (as any other corner of the c e l l ) r o t a t e s i n r e l a t i o n 26 to the centre 0 through the angles i n d i c a t e d by the vec t o r s ^ and 26 . — T / • ) | The f l e x u r a l angle of r o t a t i o n of the diagonal i s — Cosct -I S i n a ka a 26 ka 46 a ( k 2 + l ) 1 / 2 (2.43) Then the bending moment of the diagonal F i g . 2.18 i s EI $2 ,.'2. ..1/2 a(k +1) 8E I 2 6 2 2 a (k+1) (2.44) The equation r e l a t i n g the t o r s i o n a l moment and the d e f l e c t i o n i n the p l a t e i s m xy D(l-y) 32W 3x3y Since 82W i s the increment of the slope 9x3y 9x 3y of the deformed s e c t i o n i n y d i r e c t i o n per u n i t length i n x d i r e c t i o n . 32W 3x3y ka v a a ' 46 ka^ (2.45) Then m xy D(l-p) 46 k a 2 26. Equilibrium of any j o i n t such as 3 (Fig. 2.19) gives M2 Sina + T± =• Mx M2 Cosa + T = My Fig. 2 19 i Substitution into these equations of i " the expressions f o r Mx and My (Eqn 2.40a) with mxy replaced from Eqn 2.45, and of the expressions (2.44) and (2.42) for M2, T and re s p e c t i v e l y r e s u l t s i n (l-3y)K Eat 24( l - y 2 ) (l-3y) E a t 3 2 4 ( l - y 2 ) 3 • (2.46) (2.47) Derivation of S t i f f n e s s Matrix A f l e x u r a l cell.possesses three degrees of freedom at each node: the displacement perpendicular to the plane of the c e l l and the r o t a -tions about the x and y axes. D i s t r i b u t i o n Factors Action 3 In t h i s action node 1 i s moved upwards without r o t a t i o n through z a distance A^, and the other nodes and the ends of the members meeting there are restrained from any movement. The corner forces and moments holding the c e l l i n the displaced p o s i t i o n are made up of the forces and bending moments on the ends of the bars 1-2, 1-3 and 1-4 meeting 27. at the corner i n question. These members carry no t o r s i o n . The remaining three members of the c e l l are unstressed. Z23 Z13 F i g . 2.20 The equal- bending moments M on the end of the'member 1-2 ( F i g . 2.20) are r e l a t e d to the end d e f l e c t i o n as f o l l o w s M =. —-— 1 a S u b s t i t u t i n g f o r I from Eqn (2.40). M = <kVf3. *1 (2.48) 4 k ( l - y )a The transverse r e a c t i o n s on.the ends of the member are (k 2-y) E t 3 A Z 2-2 2k ( l - y )a (2.49) 28. S i m i l a r l y f o r the member. 1-3 and ..the diagonal member 6EI A* (1-yk 2)' E t 3 M l = " A v 2 n 2 . ( 2 ' 5 0 ) k a 4k (1-y )a (1-yk 2) E t 3 A* F, = ir—Y- (2-51) 2k J(1-y )a 6EI A^. y ( k 2 + l ) 1 / 2 E t 3 A* M = •  1 1 = - ±— (2=52) a (k+1) 4(1-y )ka F y E t 3 A* 2 2 2 2 k ( l - y / ) a The bending moment on the ends of the diagonal member 1-4 i s resolved i n t o the components p a r a l l e l to the coordinate axes, and these provide c o n t r i b u t i o n s to the d i s t r i b u t i o n f a c t o r s at the nodes 1 and 4. In w r i t i n g down the expressions f o r the d i s t r i b u t i o n f a c -t o r s c l o s e a t t e n t i o n should be pa i d to the signs of the forces and moments. The corner forces are designated by the symbol 2 because they act i n the d i r e c t i o n of t h i s a x i s , p o s i t i v e i f d i r e c t e d upward. The x v corner moments are given the symbol m or m depending on the a x i s about which they tend to r o t a t e the node. Two s u b s c r i p t s are used i n each d i s t r i b u t i o n f a c t o r , the f i r s t i n d i c a t i n g the number of the corner where the f u n c t i o n i s a p p l i e d and the second, the d i g i t 3,. d e s c r i b i n g a r b i t r a r i l y the transverse movement of the node 1 as the a c t i o n 3. Rotations of the j o i n t 1 about the axes x and y, the subject of the 29. subsequent i n v e s t i g a t i o n w i l l be designated as the a c t i o n s 4 and 5 r e s p e c t i v e l y . A l l d i s t r i b u t i o n f a c t o r s are assembled i n Table 2. In t h i s t a b l e 3 Et" 12a(l-y •) (2.54) Ro t a t i o n Factors due'to 6 1 x The node 1 i s r o t a t e d through a p o s i t i v e angle 6^  about the x ax i s w h i l e the other nodes remained f i x e d . The p o s i t i v e r o t a t i o n i s 1 assumed clockwise l o o k i n g i n p o s i t i v e x m' 34 M m' 14 d i r e c t i o n along the a x i s of r o t a t i o n . The members 1-2, 1-4 are subjected to f l e x u r e and 1-3 to t o r s i o n . The three --.!- other members remain unstressed. The r e l a t i o n between the angle of r o t a t i o n * X 8^ and the bending moment on the end 1 of the member 1-2 i s 4Eie: M F i g . 2.21 30. S u b s t i t u t i n g I from Eqn (2.40) (k -u) E t 3 ex M = : (2.55) 6 k ( l - y Z ) On the other end of t h i s same member, the bending moment i s only h a l f as great. The rea c t i o n s on the ends of the members are (k 2-y) E t 3 9* " - L- (2.56) 4 ( l - y 2 ) k a 9 i The angle of f l e x u r e at the end 1 of the diagonal member 1-4 i s — \J2 and the bending moment on the same end 4EI 6* y(k2+l)1/2Et3e^ M = f - i - = = ^ (2.57) 1 a(k+1) 6k(l-y ) The r e a c t i o n s on the ends of the member are yEt 3 e x F ? = 5 (2.58) 4(1-y )ka The torque T^ i n the member 1-3 using Eqn (2.47) f o r i t s t o r s i o n a l s t i f f n e s s i s c.e" ( l - 3 y ) E t 3 ex T = = H 1 (2-59> 1 R a 2 4 ( l - y Z ) k The r o t a t i o n d i s t r i b u t i o n f a c t o r s derived here, by adding the necessary member r e a c t i o n and moment components at d i f f e r e n t corners, are i n c o r -porated i n Table 2 and so are the f a c t o r s of the a c t i o n 5 ( r o t a t i o n 0^) determined i n a s i m i l a r manner. TABLE I I DISTRIBUTION FACTORS FOR FLEXURE CELL 12a(l-y A c t i o n 3 A Z = 1 A c t i o n 4 e* = 1 A c t i o n 5 6^  = 1 , Z13 = 7 '«- K + k 214 = -3kL Z15 = 3L k 2 Z23 = Z24 = 3(k- £ )L Z25 = 0 Z33 = - ! ( V - 1 »• k z34 = 0 Z35 = - 3 ( ^ -V - y)L z 4 3 = 6y " *k z44 = 3yL k Z45 = -3yL' X m 1 3 = - 3kL X m14 = 2a(k+. )L 4k X m 1 5 = -2ayL X m 2 3 = - 3(k- £ )L X m 2 5 = 0 X m 3 3 = 0 X m24 = X m35 = X m45 = 0 -ayL X ° 4 3 * 3yL k X m34 = 32. rJ -m15 -2 a ( ^ + | [ l - 3 y ] ) L rJ -m 1 3 -3L k 2 X m44 = a f L k y "23 " 0 y m25 = - y a k ( l - 3 y ) L -y)L k y m l 4 = -2ayL y m24 = 0 m35 = a(~ -uk)L y m 4 3 = 3PL m34 = 0 rJ -m44 --aPL rJ -m 4 5 -aUkL 33. S t i f f n e s s M a t r i x The contents of Table 2 represent the f i r s t 3 columns of the 12 x 12 s t i f f n e s s m a t r i x [K] r e l a t i n g the nodal forces and moments of the c e l l w i t h the modal movements. The other 9 columns.of the matrix corresponding to nodal f o r c e s , brought, about by u n i t displacements of the corners 2, 3 and 4 may. be found i n a s i m i l a r manner, but i t i s much e a s i e r to determine them from the consi d e r a t i o n s of symmetry. This i s i l l u s t r a t e d on the example of the d i s t r i b u t i o n f a c t o r s of the X f i f t h column of the matrix produced by the u n i t r o t a t i o n = 1 of the node 2. Symmetrical r e v e r s a l about the x a x i s of F i g . 2.21' i n c l u d i n g a l l deformations and s t r e s s e s i n v o l v e d i n i t , i s presented i n F i g . 2.22(a). The p i c t u r e i s s e l f explanatory. Since the d i r e c t i o n of 6 r o t a t i o n of the j o i n t 2 i n F i g . 2.22,(a) i s negative i t i s f u r t h e r reversed i n F i g . 2.22(b). The corner forces and moments i n (b) are thus r e l o c a t e d , and sometimes reversed i n s i g n d i s t r i b u t i o n f a c t o r s of F i g . 2.2'1. ; : F i g . 2.22 34. When taken i n proper sequence they represent the f i f t h column of the s t i f f n e s s matrix. The complete- equation {P_} = [K] {A} i s ] rH < X rH CD >S rH CD N CN <3 X CM CD >s CM CD Csl CO <3 X co CD J » CO CD Csl s t < X s t CD >s st CD X LO S t N LO x s t 6 LO s t B LO CO Csl 1 LO X co B LO >-> CO B LO CM Csl 1 LO X CM s LO rS CM 6 LO •H N LO X rH S LO >S rH B S t -at-es] 1 St X s t 6 s t B s t CO N 1 s t X co s t >> CO B s t CM N 1 s t X CN B s t >, CM B s t rH Csl 1 s t X rH B s t >SrH B . co s t Is] CO X s t B 1 CO >•> s t 'f CO CO N CO X CO 6 CO s t B 1 CO CM N CO X CN B 1 CO CN B I CO rH Csl CO X rH B 1 CO rH B 1 LO CO Csl 1 LO X co B 1 LO J>> CO B LO CO 1 LO~ X s t B 1 LO rs. s t B LO rH N 1 LO X rH B • 1 LO >S rH B LO CN ts] LO X CN B 1 LO >-> CM B s t ro N s t X co B s t >, CO B 1 s t s t N s t X <f E s t >^  s t B s t rH Csl s t X rH S s t >s rH B 1 s t CM N s t X CM s s t B CO CO Csl CO X co 6 ' CO >s CO B 1 CO sr N co X s t 6 CO r~» s t B 1 CO iH Csl CO X rH B CO >S rH B 1 CO CN N CO X CM B CO >s CN B 1 LO CN N c LO X CM B 1 LO CM B LO rH N LO X rH S 1 LO >s rH B LO s t N LO X s t S 1 LO rS S t B LO CO Csl LO X CO B 1 LO P-.C0 B s t CN N 1 s t X CM S s t > i CN B 1 s t rH Csl s t X rH S s t rH B 1 s t s t N 1 s t X s t B s t s t -B s t CO Csl 1 <r X CO B s t CO 6 1 CO CN N CO X CM B 1 CO >%CM B CO rH Csl CO X -H B 1 CO rH B CO -vl-Cs] CO x <*• B 1 CO >> s t B CO CO Csl CO X co B 1 CO rs CO B LO rH N LO X rH B LO rS rH B LO CM Cs] LO X CN B LO >,.CM B LO CO Csl LO X co B LO >s CO LO s t N LO X s t B LO >s s t 6 s t rH Csl s t X rH B s t >, rH B s t CM N s t X CN s S i -rs CM B s t CO Csl s t X co B s t CO B s t s t N s t X s t B s t >s s t fi CO rH Csl co X rH B • CO >1, rH B CO CN Csl CO X CN B CO >, CM B CO CO N CO X co B CO co B CO s t Csl co X s r B CO rS s t B N ri X r H >, rH Csl CN X C N J>, CN CslCO X CO >-, CO C s l s t X s t >s s t It may be seen that the matrix [K] i s symmetrical about i t s diagonal, which i s i n agreement with the B e t t i ' s r e c i p r o c a l p r i n c i p a l theorem. CHAPTER I I I BASIC THEORY.OF INSTABILITY In a mathematical sense, s t a b i l i t y i m p l i e s a c o n f i g u r a t i o n where i n f i n i t e s i m a l disturbances w i l l cause only i n f i n i t e s i m a l departures from the given e q u i l i b r i u m c o n f i g u r a t i o n . The c r i t e r i o n of s t a b i l i t y of conservative holonomic systems can be formulated as fol l o w s ( 7 ) . "A conservative holonomic system i s i n a c o n f i g u r a t i o n of s t a b l e e q u i l i b r i u m i f and only i f , the value of the p o t e n t i a l energy i s a r e l a t i v e minimum." In the system to be i n v e s t i g a t e d here, i t i s assumed that the bar model i s subjected to a conservative set of inplane loads P which give r i s e to a set of a x i a l forces S i n the bars of the model. The loads P are increased p r o p o r t i o n a l l y by a common m u l t i p l i e r f to a c r i t i c a l value f- where i n s t a b i l i t y of the cr system occurs. I t i s assumed that the bar forces do not change during the bu c k l i n g deformation, which i s i n agreement w i t h the theory of p l a t e i n s t a b i l i t y . Thus i f T represents the change i n p o t e n t i a l energy during the b u c k l i n g deformation we can w r i t e T = U + V where U i s the s t r a i n energy due to f l e x u r e caused by the b u c k l i n g deformation and V i s the p o t e n t i a l energy of the e x t e r n a l loads measured from the unbuckled p o s i t i o n . 36. For s t r u c t u r a l systems made up of l i n e a r e l a s t i c bars i t i s shown (10) that the change i n p o t e n t i a l energy T i s • a q uadratic x y f u n c t i o n of the displacements W, 0 , 0 • that describe the buckled deformation. Since the f i r s t v a r i a t i o n . o f T must vanish to s a t i s f y e q u i l i b r i u m , a s u f f i c i e n t c o n d i t i o n that T be a r e l a t i v e minimum i s T >_ 0 f o r a l l p o s s i b l e b u c k l i n g deformation c o n f i g u r a t i o n s . A c r i t e r i o n f o r determining f can then be that T = 0 f o r some con-6 c r f i g u r a t i o n . This i s the f a m i l i a r Timoshenko (8) c r i t e r i o n f o r s t a b i l i t y of e l a s t i c systems. Let U = . -| 6 1 K 5. 1 T V = - & fK 6 2 — —s — where _6 represents c o l l e c t i v e l y the f l e x u r a l nodal displacements. K. i s the f l e x u r a l s t i f f n e s s matrix of the model. K i s a new m a t r i x — t h e s t a b i l i t y matrix of the model. — s ' Then T = y _6T [ K - fTCg ] _6 = 0 As T = 0 f o r _6 =j= 0 the matrix of the quadratic form [ K - f K g ] i s p o s i t i v e s e m i - s e f i n i t e ; t h e r e f o r e the c r i t i c a l load i s obtained as the lowest root of the determinantal equation I K - fK I = 0 (3.1) t q i 3 7 . To c a l c u l a t e K we note that s . V = - f P T e or -£ f P. e. — — 1 1 where e or e. represent i n plane displacements of the po i n t s of a p p l i c a t i o n of the loads P. i n the d i r e c t i o n of P., and are fu n c t i o n s c c 1 l of the b u c k l i n g displacements &. A convenient method of c a l c u l a t i n g V i s - to use a v i r t u a l work p r i n c i p l e , which s t a t e s that P 1 e = _S1 X where J? and _S are the plane s t r e s s e q u i l i b r i u m system and e and _X are a compatible displacement system caused by the b u c k l i n g displacements. I f -S i s taken as p o s i t i v e f o r compression, X represents the inplane shortening of each bar due to the out of plane b u c k l i n g deformation 6 Thus ' CHAPTER IV DERIVATION OF. STABILITY MATRIX As f o l l o w s from Eqn (3.2) d e r i v a t i o n of the s t a b i l i t y matrix of a c e l l , i n v o l v e s e v a l u a t i o n of the q u a n t i t i e s making up the sum £( S_^  ) extended to a l l bars of the plane s t r e s s c e l l , the primary as w e l l as the secondary. The bar s t r e s s e s S corresponding to the load of u n i t i n t e n s i t y are found by the plane s t r e s s a n a l y s i s , the compression being considered p o s i t i v e and the tension negative. Each of the q u a n t i t i e s A represents the d i f f e r e n c e i n length between the bar and i t s p r o j e c t i o n i n the plane of the model as the corners of the c e l l undergo t h e i r f l e x u r a l movements. Only some of these a f f e c t each A. Formation of the s t a b i l i t y m atrix of a c e l l thus reduces to the d e r i v a t i o n of expressions f o r A of d i f f e r e n t bars and these r e q u i r e the equations f o r the transverse d e f l e c t i o n s z along the length of the bars r e s u l t i n g from the f l e x u r a l nodal movements. With z known, A i s found as the f o l l o w i n g i n t e g r a l , extended over the length of the bar I . D e r i v a t i o n of expressions f o r z and A may be i l l u s t r a t e d on the example of the bar 1-2, Once these have been found, s i m i l a r expressions f o r A of the three other s i d e bars and of the diagonal bars may be w r i t t e n down by analogy. The expression z f o r the through s i d e bars i s u t i l i z e d a l s o f o r A i n the subdivided s i d e bars when these are 39. e f f e c t i v e . Some a d d i t i o n a l explanation w i l l be given f u r t h e r i n con-n e c t i o n w i t h the corner bars. LENGTH CHANGE A IN BARS (1-2) AND (4-3) The q u a n t i t y ^ s a f f e c t e d only by the corner movements X X W^ , W^ , 6^  and of the nodes 1 and 2. The f l e x u r a l c o n d i t i o n s i n these bars corresponding to these movements, are presented i n F i g s . "^4.1(a) to (d) and the d e f l e c t i o n s z may be found by making use of the moment area r e l a t i o n . Thus from F i g . 4.1(a) ( - — \ 1 _ M _ . 2 _ _M_ _^ V_ . x Zl 2* a W T X ' 3 X a . EI X ' 2 2 E I 2 M 2 Mx , &_ _ x. . ~ aEI 1 2 3 ; Since M 6 W 1 M _ 1 EI ~ 2 a 2 3 z i = w i ( T " " 2 3 } ( 4 - 2 ) a a The d e f l e c t i o n s z„, z„ and z, i n the three other cases are found 2 3 4 s i m i l a r l y as f o l l o w s ? 3 2 z 2 = W2 ( ±f- - If- ) (4.3) a a z 3 = ex (4"#) (4-4) a z 4 - 6 X ( x - ^ + 4) (4.5) a w. 2 12EI, rt x . M= 12EI 6 E I W 2 M EI M EI _ M -EI diagram (a) M I EI M EI diagram (b) M EI 4EI x M = 4EI x 3M 2a I x v M 1 3M 2a F i g . 4.1 41. The d e f l e c t i o n corresponding to the combined a c t i o n of the four nodal movements i s determined by s u p e r p o s i t i o n 2 3 3 2 3 2 z = W i < 3 | _ _ 2 x _ ) ( 3 x _ } + 0 x a a a a a 9 2 3 . a I t s d e r i v a t i v e 2 2 2 dz • 6x 6x , , T 7 , 6x 6x N , x . 3x 2x \ dx" = W l ( ~2 ~ ~T } + W 2 ( ~~T " T } + 9 1 ( T ~ IT ) a.. a a a a + ex ( i - ^  + ) ( 4 . 7 ) This expression i s squared and su b s t i t u t e d , i n t o the i n t e g r a l f o r A-^, Eqn (4.1) which gives w 2 W J WW X 1 9 = 0.6 — b 0.6 1- Y"- ( 6 X ) a + y-- ( 0 X ) a - 1.2 — = -l z a a 15 1 . 15 2 a • -o.i wx e x - o.i wx e x + o.i w2 e x + o.i w2 e x - ^ ex ex a (4.8) The values of \ i n the subdivided bars (2-5) and (5-1) are found by dz i n t e g r a t i n g the squares of ( T T ) i n Eqn (4.1) between the l i m i t s ; ' . 0 d x and ^ and again y and i r e s p e c t i v e l y . Thus (4.9) 2 2 W W WW X n o 1 . n o 2 17 , x ,2 47 . . x .2 , 1 2 25 = °- 3 ~T + ° ' 3 T T + 960 ( 61 } a + 960 ( °2 } a " °' 6 — 23 I T x , 7 x . 23 T T x 7 T T „x 1 x • x 160 W l 91 + 160 W l 62 + 160 W 2 61 - 160 W 2 62 " 60 91 92 3 42. and 2 2 W W W W " A 1 5 - 0 . 3 ^ + 0 . 3 ^ + f ^ ( e X ) 2 a + ^ ( e^) 2a-0.6-P + i i o wi 8 i - S o w i e2 - m w 2 e i + i io w2 e2 - i o 61 6 2 f (4.10) A l l these expressions f o r A are.composed of ten quadratic terms, i n v o l v i n g X X the squares of the displacements W^ , W^ , 6^, Q^' a n c* a ^ t h e i r p o s s i b l e products. Expressions (4.8), (4.9), and (4.10) may e v i d e n t l y be used f o r the A values of the through bar (4-3) and i t s s u b d i v i s i o n s (4-7) and (7-3) by simply r e p l a c i n g the i n d i c e s of the nodal displacements 2, 1 and 5 by 4, 3 and 7 r e s p e c t i v e l y (see page 43 ). LENGTH CHANGE A IN SIDE BARS (1-3) AND ;(2 V4) Only minor m o d i f i c a t i o n s are needed f o r extension of the same formulae to the two other s i d e bars F i g . (4.2). F i g . 4.2 43. In order to preserve complete s i m i l a r i t y w i t h F i g s . 4.1,- the y y p o s i t i v e angle changes 9^ and 6^ must appear above or below the plane X X of the c e l l .in the same way as 9^ and 0^ appear i n the previous f i g u r e . With t h i s p r o v i s i o n the re q u i r e d expression f o r the A values i n the bars 1-6-3 and 2-8-4 may be copied from the corresponding expressions f o r the bars 2-5-1 and 4-7-3 r e s p e c t i v e l y w i t h the f o l l o w i n g s u b s t i t u t i o n i n t h e i r formulae: the bar le n g t h a i s replaced everywhere by ka and the i n d i c e s 2, 5, 1 are replaced by 1, 6, 3 and 4-7-3 by 2-8-4 r e s -p e c t i v e l y w i t h concurrent s u b s t i t u t i o n of the angles 9^ f o r the angles 9 X. " This g i v e s : For Side Bars . 2 2 W W WW *13 = ° - 6 k ^ + ° - 6 k i - + l T ^ i ) 2 k a + i T ( e I ) 2 k a - 1 - 2 ^ a 1 -0.1 W3 0^ - 0.1 W3 6^ +, 0.1 W 9^ + 0.1 Wx dY3 - i - j 6yx QY3 ka (4.11) W 2' W 2 9 1 9 W o W / A.. = 0.6 + 0.6 - 2 - + 0X- ) a + ±r ( 0 X ) a - 1.2 34 a a 15 3 15 4 a -0.1 w„ 9 X - 0.1 w0 ex + 0.1 w. 9 X + o.'i w. 9 X - 4r eo ex a 3 3 3 4 4 3 4 4 30 3 4 (4.12). 2 2 W W WW X24 = °- 6 k a - ; + ° - 6 k f + I5 ( 6l ) 2 k a + l 5 ( Ql ) 2 k a " ^ I c i r -0.1 W 4«8j - .0.1 W4 ey +-0.1 W2 9^ + 0.1 W2 B\ - ^ 6* By ka (4.13). 44. Subdivided Side Bars 2 2 W W W W x i 6 = 0 - 3 k a - + 0 - 3 k f + i o < e i > 2 k a + ?oo < 9 3 ) 2 k a - °-6 ibr ToO W l S l + l lo W l 6 3 - l i o W 3 9 I " llo W 3 S 3 " lo 6 1 6 3 k a (4.14) l36 W W = °' 3 k f + ° - 3 i c f + Ho ( 9 1 >'ka + Ub" ( 6 3 ^ k a " °" 6 ka 7 w, ey - w„ ey + ^ w„ ey - ^ ey ey ka 160 1 1 160 1 3 160 3 1 160 3 2 60 1 3 37 2 2 W W ° - 3 - l T + 0 - 3 ^ + f io ( e 3 ) 2 a + M o ( 94 ) 2 a - 0.6 (4.15) W.W, J 4 47 i 7 T' x 23 ax 7 Dx 23 T T Qx 1 ' x Dx + 160 W 3 °3 " 160 W 3 94 " 160 W 4 6 3 + 160 W4 94 " 60 9 3 94 a (4.16) I T O / -7 „ W W • °-3 4-+ °-3 -r + &••< e 3 >2*+ Ho < < >2* -6 X 6 X a 23 x 7 x 23 x 7 T Qx 3 4 160 W 3 °3. + 160 W 3 64 + 160 W 4 6 3 " 160 W 4 64 " ~ 6 0 48 (4.17) 2 2 • °-3 + °-3 * r + i o > k a + Ho ( e I ) k a - °-6 - i r + Ho W 2 °I - llo W 2 e I " Ho W4 92 + i o W4 94 " to 92 9 I k a (4.18) 45. W„ W, 28 0.3 ka + 0.3 ka + AZ_ ( ey ) 2 k a +1Z_ 960 2 ; k a + 960 ) ka - 0.6 W 2 W 4 ka 7 „ eY , 23_ 160 W 2 G2 + 160 w„ ey + 7 2 4 160 W4 62 " flo W 4 6I oO 62 GI k a (4.19) •LENGTH CHANGE X IN DIAGONAL BARS 4-1 AND 2-3 F l e x u r a l r o t a t i o n s of the ends of t h i s diagonal 0^ and 0^ ( F i g . 4.3) are made up of the nodal r o t a t i o n s about both coordinate axes. Choosing the p o s i t i v e d i r e c t i o n s of these r o t a t i o n s as i n d i -cated i n the f i g u r e V x 3^ Cosct - 0^ Sma = (k 2 + l ) 1 / 2 ( K 0 y - 0 X) (4.20) F i g . 4.3 V x T. Cosa - 0, Sina 4 4 = (k 2 + l ) 1 / 2 ( K 0 y - 0 X) (4.21) 4 4 The p o s i t i v e d i r e c t i o n s of 0^ and 0^ are so chosen that the nodes 1 and 4 i n F i g . 4.3 correspond r e s p e c t i v e l y to the nodes 2 and 1 i n F i g . 4.1. Now the value of A^ 4 may be found from Eqn (4.8;) by using the 2 1/2 length of the diagonal a(k +1) i n place of a and r e p l a c i n g by W^ , Wx by W4, 0 X by Q± (Eqn. 4.20). and 0 X by 6 4 (Eqn. 4.21). 46. Thus 2 .. 0.6. T T 2 0.6 2 a /„Vs2 Cos a 2a .x y X. . = W, S i m + W. Sina + — (6,) — 77 6, 6, Cosa 14 a 4 a 1 15 4 Sina 15 4 4 2 , a /^x\2 „. , a ,„yN2 Cos a 2a „x v „ + — (6.) Sina + — ( S O TT: e i 6 i C o s a 15 4 15 1 Sina 15 1 1 2 W 1 W 4 + 4V ( 9 X ) Sina - 1.2 -^-^ Sina - 0.1 W,0Y Cosa + 0.1 W,GX Sina 15 1 a 4 4 4 4 - 0.1 W..6y Cosa + 0.1 W.0X Sina + 0.1 W.e7 Cosa - 0.1 W, 0 X Sina 4 1 4 1 1 4 1 4 2 + 0.1 W.O7 Cosa - 0.1 W.0X Sina - -f_- 0 Y 6 Y a 1 1 1 1 30 1 4 Sina + In 9 X eY c°sa + #n" 9 Y c°sa - IT 6 X 9 X Sina. (4.22) 30 1 4 30 4 1 30 1 4 The same expression may be used f o r the diagonal 2-3 r e p l a c i n g i n i t the i n d i c e s 1 and 4 by 2 and 3 r e s p e c t i v e l y A 0 0 = W02 Sina + — W„ 2 Sina + 4T ( 6 X ) 2 Sina + •2-f- 0 X 6 y Cosa 23 a 3 a 2 15 3 15 3 3 + 15 «l>* f l ^ + 15 «2> 2 S 1"« + f t 92 'I C«« + 15 W f t ^ 1.2 „ „ „. „ , „ „x W„W„ Sina - 0.1 W„0 X Sina - 0.1 W^8y Cosa - 0.1 W„0 X Sina a . 2 3 ' 3 3 3 2 3 2 - 0 . 1 W 9 y Cosa +0.1 W 20 X Sina +0.1 W 20 y Cosa +0.1 W 2 © X Sina +0.1 W 20 y Cosa " fo e2 e3 S i n a " % °2 63 C o S a " 30 °3 °2 C ° S a 47. LENGTH CHANGE A IN CORNER BARS The corner bar 5-6, s i t u a t e d near the node 1, i s chosen f o r i l l u s t r a t i o n of the method of determination of the length changes of the corner bars. As was pointed out e a r l i e r the corner bars are assumed to possess n e g l i g i b l e f l e x u r a l s t i f f n e s s approaching zero i n the l i m i t ; i n other words, they d e f l e c t f l e x u r a l l y i n accordance w i t h the beam theory without a f f e c t i n g the e q u i l i b r i u m of the j o i n t s on t h e i r ends, where they simply comply w i t h the d e f l e c t i o n s and slopes of the si d e members of the c e l l . I n s p e c t i o n of F i g . 4.4 shows that f l e x u r e of bar 5-6 i s a f f e c t e d by a l l three corner movements of the corners 1, 2 and 3 while being un-in f l u e n c e d by the movements of the di a g o n a l l y opposite corner 4. The d e f l e c t i o n s of the ends 5 and 6 are found from the corresponding d e f l e c -F i g . 4.4 t i o n equations of the s i d e b a r s , and • dz the end slopes from the slopes , making allowance f o r the angular: connection of the si d e and the corner bars. F i g . 4.5(a) to ( i ) present the d e f l e c t e d shapes of the bar 5-6 i n the 9 d e f l e c t i o n f i e l d s causing them. Here i s how some of the end values i n these figures-, have been c a l c u l a t e d : In F i g . 4.5(a), the end slope i s found by s u b s t i t u t i n g x = y i n the dz W., term f o r -7— (Eqn 4.7) and m u l t i p l y i n g the r e s u l t by Sihct i n order to 1 dx 48. change the slope i n the bar 1-2 into the end slope of the bar 5-6. 3W This makes 6 r = — Sina. The slope 8,. i n F i g . 4.5(a) happens to-be 5 /a b equal to the slope 6 . Deflections of the j o i n t s 5 and 6 i n the side bars are equal, and.so the r e l a t i v e d e f l e c t i o n of these points i s zero. Thus' the equation for.the d e f l e c t i o n z of the bar 5-6 may be obtained from Eqn (4.6) for the bar 1-2,.using i n i t W = W. = 0, 3W, 3W X 1 X 1 - ~2~T~ Sina and 8^ - = 7 Sina and replacing a by the length 2Sina of the bar 5-6. 3W > 5 = V 2a" Sina 2 Sina Sina Sina W, W = — 6 2 (a) (b) (c) Z * e iV= ^ Sina 5 4 V S " 3 5 2 Sina'"1, >1 z(ex) el V f f « p 6= Y~ Sina '5 4 Sina 2 Sina z ( 6 X ) V 4 Cosa ! 2 Sina j (d) (e) (f) z < 6 l > 8 6 = ^ C o s a Z((V 6 6 = 4^ C o s a (g) (h) (i) Fig. 4.5 49. As a f u r t h e r i l l u s t r a t i o n consider the deformation of the same bar i n the f i e l d 6^. From Eqns (4.4) and (4.7), w i t h f u r t h e r m u l t i p l i c a t i o n of the l a t t e r by S i n a , the downward d e f l e c t i o n and slope of the end 5 of the bar are found as shown i n F i g . 4.7(e). As the node 1 ro t a t e s through the angle 8^, the mid poi n t 6 of the bar 1-3 under-1 x goes a t o r s i o n a l r o t a t i o n y 0-^ , which becomes the f l e x u r a l r o t a t i o n 1 x = — 6. Sina of the end 6 of ,the corner bar. The d e f l e c t i o n o i l , equation of the bar 5-6 i s thus combined w i t h proper s u b s t i t u t i o n s of the equations (4.3), (4.4) and (4.5). The d e f l e c t i o n curves c o r r e s -ponding to the seven other f i e l d s i n F i g . 4.7 are derived i n a s i m i l a r way. The complete set of these d e f l e c t i o n s z, described by the sug-g e s t i v e symbols z(W^), z(8^) e t c . i s 3W 1 3W 1 2 2 z(W^) = -x— x Sina ^r~-x S i n a 1 za z a 3 z(W_) = W. [ - ~ x Sina + ^2L_ S i n 3 a ] z I la 3 a 2 3 z(W 3) = W3 [ ^ - Sin a - ±|- S i n a ] a a 2 3 z ( 6 X ) = 6 X [ - f Sina + S i n 2 a - ^ S i n 3 a ] 1 1 4 2a 2 a 2 3 (^ X \ « X r X _ , X — , Z. X _ , 0 -l 6„) = • 9„ L - T Sina - -rr- S i n a H r ' S i n a J z z 4 za z a 2 3 ( „ X \ _ x r x ,2 2x _, 3 -i 6„) = 6_ [ S i n a •+ —rr- S i n a ] 3 3 a z a 50. 3 z ( 6 y ) = K e i [ f s i n c " ^2 S i n a ] a 2 3 z ( 6 y ) = K6 y [ f Sina - — S i n 2 a + - ^ f - S i n 3 a ] Z z z B. z 2 2 3 3 z ( 6 y ) = K0 y [ - — S i n a + \ Sin a ] 3 j a z a dz Their d e r i v a t i v e s — are dx dz(W x) 3 2 x = W. ( -r— Sina - 6 S i n a —- ) dx 1 2a 2 a dz(W 2) 3 6 x 2 3 —z = W_ ( - T T— Sina H — S i n a ) dx 2 za i a dz.(W-) • _ , 2 3 T T . bx c. 2 6x . 3 . ; = w„ ( —- Sin-a Sm a ) dx 3 z 3 a a d z ( 9 l ) x 1 3x 2 . 3x 2 3 : = 6, ( - -r Sina -I S i n a - _ S i n a ) dx 1 4 a 2 a • d z ( e 2 } x 1 x 2 3x 2 3 : = 0„ ( - 7- Sina • S i n a + —r- S i n a ) dx 2 4 a 2 a d z ( 0 o ) „ 0 , 2 3 -x , 2x _. 2 6x c. 3 • v ^ = 6_. ( - — Sm a H =— Sin- a ) dx 3 a z a d z ( 6 y ) dx" 1 3 ( y Sina - 3 S i n a ^ ) d z ( 8 2 ^ ' v 1 4x. 2 6 x 2 3 J ' = ' K07, ( 4 Sina - — S i n a + Sin a ) dx z z a z a 51. dz(QY) 2 • J = KBy ( - — S i n a + S i n a ) dx 3 a 2 S u b s t i t u t i n g the square of the sum of these expressions i n t o the i n t e g r a l Eqn (4.1) y i e l d s the f o l l o w i n g expression f o r 2 2 2 3W 3W 3W X56 " S I m 1 16T + l i t + TOT +- 3§0 < e i ) 2 a + H f < 6 2 ) 2 + B o <63> * + ^ K2 ( e p 2 , + 4 (6^) 2a +-||^ Ce*) 2. - ^ 16a 1 3 32 1 1 32 16 32 1 K°1 16 + — - WW - l ^ 3 — w e x + -2-^ — W 9X + — W 8X 32 80a 2 3 160 2 1 160 2 2 80 2 3 " lo W 2 K S 1 " h ™2Ql + lo ^ 3 + h V l " lo W 3 9 2 w e x . - ^ r - T f o - V l - l o ^ - T i ^ - l o 9 ! 0X9 Xa + + 3§0 K e i 9 I a " ifo K 9 1 6 2 a " Mo K 9 i e 3 a !-+ llo 6 2 e 3 a - i o K 9 2 e I a - m K 6 2 9 2 a + Ub" K 9 2 S 3 a Too K 6 3 9 I a " W K 9 3 9 2 a " 4I0 K 6 3 6 3 a + ^ l 9 ^ a 52. A VALUES IN CORNER MEMBERS 5-8, 7-6 AND 7-8 The e a s i e s t way to determine the ex p r e s s i o n s ' f o r the changes i n length i n the three other corner members i s by making use of the general symmetry of the c e l l . The c e l l i s r o t a t e d through 180° about one of the coordinate axes x, y o.rclz so that the corner member i n question w i l l take the p o s i t i o n occupied i n F i g . 4.4 by the member 5-6 This reverses the p o s i t i v e d i r e c t i o n s of the two other coordinate axes and w i t h them the signs of two of the three vectors W, 9 and 6 . At 1 the same time the numbers of the corresponding corners and the mid edg po i n t s change a l s o . This i s i l l u s t r a t e d i n Table-3. Making the chang shown i n t h i s t a b l e the A f o r the remaining corner bars are obtained from Eqn (4.24) . 2 2 2 3W W W A58 = S i n a [ i i - + 0-3-i-+0-3jt- + ^o ( e 2 ) 2 a + i o ( e i ) 2 a + llo ( 6 4 ) 2 a + 3 f o K ' < 62> 2 a + l i K ' < 6l> 2 a  + Mo k 2 ( 6 I ) 2 a " i l l W 1 W 2 " l i W 2 W 4 + 32 W 2 6 2 + 32 W 2 [ + ••16 W 2 G 4 + 32 k W 2 6 2 + i k W 2 9 I + kl k W 2 G 4 " l i W + l i W l 6 2 - i o W 1 6 1 - lo Wl°4 " h k W l 9 2 " fo V l + f w i e I - h w 4 e 2 + lo w 4 e i - i W 4 9 4 + i f v ! TABLE I I I 54. lo k w 4 e i - i o ™sl - h e i e 2 a + h QK a x y 3lea x y 131c x y 1 x x 320 "2 G2 a + 160 e 2 6 l + 320 6 2 6 4 3 + 120 6 l \ 3k + I | r K 6 X 0 y a + K -x 0y a _ 29 K Q x e y & + |K x y a 320 w l " 2 480 1 1 960 " w l w 4 160 "4 2 + 240 V l a- 480 9 4 e 4 3 + 80 ei°2 a 40 °2 64 3 + K Q y c y 120 v l v 4 6 . a ] (4.25) , 3 • 2 0.3 2 0.3 n 2 9 ,_x.2 , 17 , x. 16 3 a 4 a 1 320 3 960 4 • 1 /0x.2 , 9k 2 ,v.2 , K 2 .v.,2 , 17k 2 / ny N2 + 120" ( V 3 + 320 ( e 3 } a + 120 (94> a + 960~ ( V a 3 3 3 x 1 v ' 1 ' x 16a 3 4 16a 1 3 32 3 3 32 3 4 16 3 1 32 W 3 K 9 3 - 16 W 3 k 9 I " 32 W 3 k 9 l " f a " W1 W4 " So V + 23. T T „x ., 3 x , 7 w, e, + -^ r- w, e, +  w k 0 y + k w y _ 9k w, 160 4 4 80 4 1 40 4 3 40 4 4 80 4 1 + Z _ w e x _ J L _ w e x + i _ w e x _ i 3 _ k w + 3k y 40 1 3 80 1 4 40 1 1 160 1°3 80 1 4 , 23 T_ 7 _y 1 nx.x-. 1 n x„x 3k „xny + 160 "Vl " 40 e 3 V + 80 63 61 a " 320 63 63 a + i o 9 3 e ^ + i l e 3 6 ^ + l 2 o e ^ ^ + 3 i • ^ 55. + JL- R x f iy a _ 29k x y _ J _ V f lx f ly-- K a _ -x y + 480 64 64 3 960 6 4 6 1 a + 160 k e i 9 3 a + 240 61 94 2 2 2 + K a _ x y K_ y y K_ y y K _ fiYfiy- i + 480 9 1 9 1 + 80 e3 94 3 40 9 1 6 3 3 + 120 6 4 9 1 & ] (4.26) c- r 3 „ 2 j 0.3 „ 2 , 0.3 „ 2 x 9 / Q x v 2 „ 17 , Q X v 2 A78 = S l n a [ 1 6 l W 4 + - 7 " W 3 + — W 2 + 320 ( V a + 960 ( V a + 3 j , ( 0 X ) 2 a + ^ - —— w w - —^— ww + — w ex + — w ex + — w 6 X 16a 3 4 16a 4 2 32 4 4 32 4 3 16 4 2 32 w4o4 16 4«3 32 ™462 80a 2 3 160 3°4 " Ho V 3 " 80 » 3 6 2 + f + lo V's " £ V? _ 7_ w x 9_ x _ ] ^ 0 x _ 13K y 3K y 40 2°4 80 2 3 40 2 2 160 2°4 80 2 3 + ^  w ey -160 2 2 1_ 40 }X A X  33 94 a + 80 32 94 a + 3K 320 3K 160' }x f i y  34°3 a -13K x y 320 94 62 a + 120 33 92 a -13K 320 33 94 480, 33°3 a + 2.9 K 960 a -3k 160 32°4 240 9 2 9 3 K 480 3 2.6 2 a + K_ 80 34 63 a -K 40 }ys y  32 64 a + K 120 32 93 (4.27) 5 6 . C a l c u l a t i o n of Bar Forces I t was shown i n Chapter I I that the s t a t e of plane s t r e s s i n a c e l l corresponding to displacement of one or more corners may be com-bined of s e v e r a l s t r a i n c o n d i t i o n s , o r i e n t e d i n the d i r e c t i o n s of the x and y axes, i n c l u d i n g the uniform normal s t r a i n s , the f l e x u r e l i k e s t r a i n s and the shear s t r a i n s . Of these i n the f i r s t two types of con-d i t i o n s the corner bars remain i n a c t i v e and the s i d e bars of the secondary system j o i n the through bars as i f they were p a r t s of them. Only the shear s t r a i n c o n d i t i o n s produce s t r e s s e s F^ i n the corner bars, and equal and opposite i n s i g n s t r e s s e s F and F^ i n the h a l f lengths of a l l s i d e bars. Thus the mean s t r e s s e s i n the two halves.of each combined s i d e bar are independent of the s t r e s s e s F and F , and may be determined from the change i n length of the t o t a l member. Under plane s t r e s s a c t i o n the model.is assumed to be completely j f r e e at the edges. F i g . 4 . 6 represents a t y p i c a l c e l l of the model i n plane . | s t r e s s a c t i o n analysed f o r displacements, , under a load of u n i t i n t e n s i t y . The I displacements U and V of i t s four cor-ners and the corner forces X and Y are i • a l l known. With the corner displacements a v a i l a b l e , the elongations of the side i and the diagonal bars are as s t a t e d i n . Table 4 , and the mean s t r e s s e s i n the y»v x-V-U F i g . 4 . 6 s i d e bars S ^ J ^34' ^13' ^ 2 4 A N D T * 1 6 s t r e s s e s i n the diagonals S^, S^^ are e a s i l y found knowing the cross s e c t i o n a l areas of the bars (Eqns. 2.8, 2.7, 2.5). TABLE IV ELONGATION OF PRIMARY BARS Bar Elongation 1-2 V l " V2 1-3 u 1 - u 3 3-4 V3 " V4 2-4 U 2 " U 4 1-4 Sina [ k(U-U.) + V -V. ] 1 4 1 4 2-3 Sina [ k(U 2-U 3) + V 3~V 2 ] Should the s t r e s s c o n d i t i o n i n the c e l l i n c l u d e also some shear a c t i o n , a d d i t i o n a l s t r e s s e s F i n the halves of the si d e bars 1-2, and 3-4, F^ i n the halves of the two other s i d e b a r s , and F 2 i n the corner bars are a l s o present. The numerical values of these s t r e s s e s stand i n r e l a t i o n F F l ~ = -TT^- = Fo (4.28) Sina Cosa 2 In a l l four corners the s t r e s s e s i n the corner bars are opposite i n si g n to the s t r e s s e s i n the adjacent halves of the side bars. The signs of a l l these s t r e s s e s are r e s p e c t i v e l y the same i n the d i a g o n a l l y opposite corners l i k e 2 and 3 and r e s p e c t i v e l y opposite to each other at the corners adjacent the same s i d e of the c e l l l i k e 1 and 2. 58. 12 Fig,. 4.7 | I t i s s u f f i c i e n t to determine e i t h e r j s t r e s s e s F or F^ from e q u i l i b r i u m of one of the nodes,like node 1 i n F i g . ! 4.7. With the st r e s s e s S,„ and S„ 12 14 known F = Y 1 - S 1 2 - S 4 Sina (4.29) .. s t r e s s e s F^ and F^ may be found'by the r e l a t i o n (4.28). E q u i l i b r i u m of the other corners must produce the same numerical values of F, F^ and • S t a b i l i t y M a t r i x of a C e l l With the e x t e n s i o n a l bar st r e s s e s known and the f l e x u r a l changes i n the bar lengths X expressed i n terms of the nodal displacements W, 0 and 6 y the elements of the s t a b i l i t y matrix are found by Eqn. (3.2). The bar s t r e s s e s S i n Eqn. (3.2) are p o s i t i v e f o r compression and negative f o r t e n s i o n . The c o n t r i b u t i o n of each h a l f side member such as 1-5 ( F i g . 4.9) may be taken i n two p a r t s : the f u l l s i d e member 1-2 wit h the mean s t r e s s S^ 2 a n < i the a d d i t i o n a l h a l f s i d e member 1-5 w i t h the s t r e s s F. This treatment i s more convenient than c o n s i d e r a t i o n of each h a l f member under the a c t i o n of . i t s : s t r e s s S + For,S-F,.as the case may be. Since the symbols F s i g n i f y s t r e s s e s of d i f f e r e n t signs i n d i f -f e r e n t members, i t i s necessary f o r the g e n e r a l i t y of the s t i f f n e s s m a t r i x to make a d e f i n i t e assumption at t h i s stage as to the d i s p o s i t i o n 59. of signs of the st r e s s e s F i n the c e l l . This i s done i n F i g . 4.9. Bars assumed to be under compression ( p o s i t i v e l y s t r e s s e d ) . Bars assumed under tension. F i g . 4.9 Here are the examples o f • c a l c u l a t i o n of the terms bf the s t a b i l i t y matrix. YW]_W1 The only expressions.for' X (among the Eqns. 4.8 to 4.19, 4;22 - 4.27) 2 which contain W^. are X^2 X^, X^, X^, X^^, X^, X^, > 5 6> ^ 7 6 a n d ^ 5 8 < The product formed according to Eqn. 3.2, w i t h the expressions f o r X being given by Eqns. 4.8, 4.11, 4.22, 4.10, 4.9, 4.14, 4.15, 4.24, 4.26 and 4.25 i s 2 2 2 2 2 (0.6 S 7„ — +0.6 S, „ — + 0.6 , — Sina - 0.3 F, 12 a 2 13 a 2 14 a 2 1 ka V - V V 3 2 W l - 0.3 F + 0.3. F + 0.3 F.- r^— + -rr~ W. F_ Sina - 0.3 F. — - Sina a a 1 ka 16a 1 2 2 a 0.3 F 2 a Si n a ) 60. Using r e l a t i o n (4.28) and r e w r i t i n g t h i s equation i n the quadratic form,obtain Y W . W . = — ( S . _ + - J ^ + S . . Sina ) - 0.825 -1 1 a • 12• k 14 a The only terms co n t a i n i n g the product 0 O B~ are X„„, X c, and X^ D given 3 z z3 j o Jo by equations 4.23, 4.24, and 4.27. Forming the product ESX o b t a i n 2 ( S 2 3 ( - | T Cosa) 9 X6 y + F 2 ( - | | Q ) Sina + F 2 ( ka) Sina ) Again using r e l a t i o n 4.28 and r e w r i t i n g i n the quadratic form ob t a i n ^ e 3 6 2 " " 30 C o S a S23 + 1§2 F k a A l l terms of the s t a b i l i t y m a t r i x are assembled i n Table 5. yw1w1 Y6xex ^ 1 ' ^ 9 1 9 1 Yw2w1 YW 2 6 X yw2ey YW2W2 Symmetric Yexex Yexey Y6^W2 Ye2ex > 92 91 Y6Y W 2 Yw.3w1 YW 3 6 X Tw3ey YW3W2 Yw3ex yw3ey YW3W3 „x x Y°39l ^ 1 . Y 0 ^ 2 ~x„x r e 3 e 2 Ye>3 Y B X 0 X YOyex yeyef Y6YW:2 y*y2 Y6 Y W 3 YOyex Y W 4 ' W 1 Yw4ey yw4w2 ^4 92 YW4ef YW4W_3 YW 4 6 X YW4ey Y W 4 W 4 yefi2 ^ 2 Y6|W:3 yefe* Y e4°4 Y O ^ Yeje£ Yeye| Y 9 ^ % ^ 2 y ^ l Y6 Y W 3 YByex Y 6 I e 3 Y 6 I W 4 (4.30) 62. TABLE V ELEMENTS OF STABILITY MATRIX yWW + 1.2 a • s + 1 - 2 b12 + ka S_. + — .13 a ^1 4 Sina - 0.825 F a = - 13F 80 " °- 1 S14 Sina - 0 .1 s12 Y © ^ = • + o.i (s1 3 + s 1 4 Cosa)* F Y W 2 W 1 = - 1.2 a S12 4 3 = - 0.1 s + ^ b12 40 i ^ 2 W l = + 19k 80 F ka S24 S13 Yw3w1 = - 1.2 ka S13 f - i ^ F 80 2 L S12 a 1 J r i = - 3kF 40 + o.i s 1 3 -= + 33F 40a a b14 Sina = + 3F 20 " °- 1 S14 Sina - 3kF 20 + o.i s u Cosa Y6 xe x - 7Fa 120 + 2a 2a + 15 b12 + 15 b14 Sina YW 26 X YW36X. Yeyey YW2ey ^ 1 Yeyey Yw3ey , Fka 2a _ _ + 192 " 15 S i 4 C o s a + § +0.1.:'- S 1 2 — S 30 12 _ 19Fka 320 19F 80 0 19Fka 320 f + 0.1 S 1 4 Sina aF a c 60 " 30 S14 S i m , 5Fka a + 192" + 30 C ° S a S14 .2 + IT KA s i 3 + ^rfSina 19kF 80 19 Fka 320 0 3kF 40 - 0.1 S 13 ye3ey_ = - 19Fka 320 = - ka „ 30 13 = + 3kF 20 0 , 1 S14 C ° S a = + 5Fka , a c r 192 + 30 S14 C ° S a = - F k2 a • a 2 60 " 30 k S i m S14 YW2W2 + 1.2 1.2 a "12 ' ka S„, + — S„„ Sina + 0.825 -24 a 23 a = - .Jf + 0.1 S 1 2 + 0.1 S 2 3 S i n a Y6^W 2 = - ~ ^ + 0.1 S.0/ + 0.1 S._ Cosa 80 24 23 Y W 3 W 2 = - 33F 1.2 „ 40a " a S 2 3 . S l n a = + f + 0 . 1 S 2 3 Sina Y 6 3 W 2 = + 3kF ^ + 0.1 S 2 3 Cosa Yw4w2 - " l l 2 s ka 24 = - 19F. 80 + 3 F k + 0 1 S 40 + U , i b24 Y 6 2 e 2 = + 15 S12 + 15 S23 S i n a + 120 F a ~y~x . 2a „ „ . , F k a y 8 2 6 2 = 15 23 a 192 ^ W 3 9 2 = " f - 0.1 S 2 3 S i n a y 6 3 ° 2 = r 30 S23 S i M + ft ^ 3 6 2 " " % S 2 3 C ° S a + i i i F k a >V2 = „x„x Y e 4 e 2 v x _ 19Fka  Y 6 4 e 2 " " 320 Y 9 2 9 2 - + I T S 2 4 + i f k ' a S ± n a S 2 3 + ifb" F k ' a YW 39 y = -.0.1 Cosa S 2 3 Y 9 3 9 2 = " fo S23 G ° S a + i l l E k a Y 9 y e y - - ^ ^ S i n a ^ ^ 4 9 2 - + l o k F - ° - 1 S 2 4 x y _ _ 19 Fka Y 9 4 9 2 " 320 y 9 4 9 2 " 30 S 2 4 y W - W . = + — S„. + f ^ 2 - S . . + — S_. S i n a + 0.825 -3 3 a 34 ka 13 a 23 a 66. Ye^3 = 13F - 0.1 S 3 4 - 0.1 S 2 3 Sina + — Ye yw 3 - 0.1 S 3 - 0.1 S 2 3 Cosa + ||£ k yW4W3 1.2 _ a b34 Te> 3 = 3F - ° - 1 S 3 4 - 4 0 = 19Fk 80 „x x 3 3 = + 120 F a + 15 S34 + 15 S23 S x ™ = ^ 2a _ _ , Fka + 1 5 S 2 3 Cosa + 1 9 2 = + S34 " f ' Y e 4 e 3 = - — S 30: 34 = 19 Fka 320 , 2ka „ , 2 a . , 2 „ „. , 7 F k 2 a  + 15 S13 + 15 k S23 S i m + 120 = 19Fk 80 = 19 Fka 320. 0 | Y W 4 W 4 = — S 0 . + r ^ 2 - S 0 . + — S. . Sina - 0.825 -a 34 ka 24 a - 14 a 67. I . Y e j w 4 = o.i s34 + 0. 1 S,, Sina 14 13F 80 Yeyw4 = - o-1 s24 - o. 1 S., Cosa 14 13kF 80 < e 4 -2a q 2a 15 -34 15 S., Sina -14 7Fa 120 y°K - - -r-=- S. . Cosa 15 14 Fka 192 15 S24 k 3 + 2 15 S14 S i n a " 120 F k a I Note: A l l bar stresses,, S p o s i t i v e i n compression. For s i g n of F s t r e s s e s see F i g . 4.9. CHAPTER V EXAMPLES The f i n i t e element method presented here has been a p p l i e d to s e v e r a l problems f o r which the exact s o l u t i o n s are a v a i l a b l e (8) and which have a l s o been solved by the f i n i t e element method employing the no bar c e l l s (9). In bar c e l l s both the s t a b i l i t y and the s t i f f n e s s matrices are a f f e c t e d by y, while, i n the no bar c e l l s the s t a b i l i t y matrix does not depend on y. In order to study the e f f e c t of va r y i n g y s e v e r a l widely divergent values of y have been used f o r the bar cel l ' s . In the course of the s o l u t i o n of these examples a n o t i c e a b l e r e d u c t i o n i n p r e c i s i o n was observed i n some cases. This was r e l a t e d to the negative and zero values of the e x t e n s i o n a l and f l e x u r a l d i f -ferences of some of the bars i n the c e l l s used i n the a n a l y s i s . . This p o i n t was made the subject of a s p e c i a l i n v e s t i g a t i o n reported at the end of t h i s s e c t i o n . Example 1 A re c t a n g u l a r p l a t e of the s i z e b x I and the thickness t i s com-pressed i n the d i r e c t i o n 'I by the s t r e s s a per u n i t area and i s not st r e s s e d i n the other d i r e c t i o n . The p l a t e i s simply supported on a l l edges as f a r as the f l e x u r a l deformation i s concerned. The exact 69. c r i t i c a l stress i s given by the expression err D a = — ( 5 . 1 ) c r b t i n which D i s found by Eqn (2.34). The comparison of t h e , c r i t i c a l values i s made on the basis of the c o e f f i c i e n t 8, which i n th i s formula i s independent of y. Three values of the Poisson's r a t i o y are used and the aspect i r a t i o 7- i s taken as 1 and 2.5. In a l l but one case, the bar model i s b . formed by (3 x 3), (4 x 4) and (6 x 6) framework, each c e l l of which, a by ka i n s i z e , i s geometrically s i m i l a r . t o the plate. In one case ( 3 x 6 ) , (4 x 8) and (6 x 12) models were employed. Bar Forces (Fig. 5.1) In view of.the symmetry of loading the corner bars are i n a c t i v e and the secondary side bars simply add t h e i r areas to the primary bars. With a=l the corner loads c a r r i e d by each c e l l are -j at and the changes i n the length and width of the c e l l are: and 62 = ( 5 , 3 ) r e s p e c t i v e l y since the edges of the plates are free to move unrestrained i n plane stress'. The shortening of the diagonal i s 6 = . & Cos a - 6 2 S i n a , - (5.4) E ( k 2 + l ) 1 / 2 a t 2 a t 2 '24 A 4 -\ - -\ / V"* / s \ / / 2 3 \ / S 1 2 1 13 1 a t a t 2 F i g . 5 . 1 2 S12. = 5 1 3 " 5 1 4 = 34 24 23 70. U s i n g t h e Eqn (2.5) and t h e e x p r e s s i o n s ( 2 . 6 ) , (2.8) and (2.9) f o r t h e c r o s s s e c t i o n a l a r e a s of the b a r s , t h e b a r s t r e s s e s caused by the l o a d o f u n i t i n t e n s i t y i . e . t h e one c o r r e s p o n d i n g t o a = 1 a r e 1 y a ( k -y) 2 k ( l - y 2 ) a ( l - y k 2 ) 2 ( l - y 2 ) 2 2 1/? y a ( k Z - y ) ( k Z + l ) J - / ^ 2 2 k ( l - y 2 ) + + (5.5) I n t h e s e e x p r e s s i o n s t h e c o m p r e s s i o n s a r e c o n s i d e r e d p o s i t i v e and t h e t e n s i o n s n e g a t i v e i n agreement w i t h the b a s i c e q u a t i o n ( 3 . 2 ) . The c r i t i c a l b u c k l i n g s t r e s s a of Eqn (5.1) i s t h e l o w e s t e i g e n v a l u e f o f the Eqn (3.1) and i t i s f o u n d by a s t a n d a r d computer p r o c e d u r e . The r e s u l t s o f b o t h b a r and no b a r s o l u t i o n s a r e p r e s e n t e d i n T a b l e 6 i n t h e form of p e r c e n t e r r o r s i n c o m p a r i s o n w i t h t h e e x a c t v a l u e s , w i t h t h e minus s i g n i n d i c a t i n g the a p p r o x i m a t e v a l u e b e i n g s m a l l e r t h a n t h e e x a c t . The same t a b l e a l s o g i v e s , t h e r e s u l t s o f b o t h b a r and no b a r c e l l s f o r a s q u a r e p l a t e w i t h i t s edges f i x e d a g a i n s t f l e x u r e and f o r a r e c t a n g u l a r p l a t e ( — = 0.6 ) whose non l o a d e d edges a r e f i x e d f o r f l e x u r e w h i l e t h e l o a d e d edges a r e s i m p l y s u p p o r t e d . 71. TABLE VI CRITICAL BUCKLING STRESSES IN PLATES PERCENT ERRORS OF FINITE ELEMENT SOLUTIONS WITH BAR AND NO BAR SOLUTIONS UNIFORM COMPRESSION IN X DIRECTION P r o b l e m Exact Soln. 0-(Eqn 5.1) Poisson's Ratio ( M ) Bar or No Bar Percent E r r o r Model Mesh 3 x 3 4 x 4 6 x 6 X. b M i t t Square P l a t e X 4.0 4.0 4.0 4.0 .3 .1 ,45 No Bar Bar Bar Bar -8.88 -10.0 -9.5 -10.25 -5.75 -5.75 -5.50 -6.50 -2.80 -2.75 -1.75 -3.0 4 X k i b X * * •-t ft t 11 tt Square P l a t e P 10.07 10.07 10.07 10.07 .3 ,1 .45 No Bar Bar Bar Bar -13.0 -13.1) -13.1 -7.80 -8.60 -8.60 -9.0 -4.56 -5.17 -4.37 -5.17 1 * ^ f t i f t tt Aspect = .6 -r-y 7.05 7.05 7.05 7.05 ,3 ,1 .45 No Bar Bar Bar Bar -15.6 -16.6 -23.0. -7.24 -10.8 -11.0 -3.74 -5.25 -6.0 b «-t t t t 11 Aspect = 2.5 r y 4.133 4.133 4.133 4.133 3 x 6 K=1.25 4 x 8 k=1.25 6 x 12 K=1.25 .3 ,1 .45 No Bar Bar Bar Bar -5.0 -11.2 -£1.55 -7.10 -4.05 -8.15 -1.97 -3.73 -2.82 -5.10 -1.80 Example 2 The only d i f f e r e n c e of t h i s example-from example 1 i s that the p l a t e i s subjected to equal s t r e s s e s a i n the d i r e c t i o n of both axes. The edges of the p l a t e are again simply supported i n f l e x u r e . The same Eqn (5.1) may be used to describe the b u c k l i n g c o n d i t i o n i n the p l a t e employing d i f f e r e n t appropriate t h e o r e t i c a l values of the co-e f f i c i e n t 3. The bar s t r e s s e s may be found as the sums of the s t r e s s e s produced by compression i n X d i r e c t i o n Eqn (5.5) and the s t r e s s e s caused by the compression i n y d i r e c t i o n . The l a t t e r may be obtained from the former by r e p l a c i n g K by and a by ka w i t h the = s t r e s s e s becoming S^^ = S^^ str e s s e s and v i c e versa. Following t h i s procedure the t o t a l s t r e s s e s are found Q = c = a ( k 2 - y ) 12 34 2k(l+y) _ = ' = , a(1-yk 2) 13 24 2(1+y) ya(k +1) 14 23 2k(l+y) F = 0 (5.6) The r e s u l t s of the s o l u t i o n are assembled i n Table 7. Also given i n Table 7 are the r e s u l t s f o r a p l a t e subjected to uniform compression s t r e s s i n the d i r e c t i o n of both axes, but with the edges of the p l a t e f i x e d against f l e x u r e . 73. TABLE VII CRITICAL-BUCKLING STRESSES IN PLATES PERCENT ERRORS OF FINITE ELEMENT SOLUTIONS WITH BAR AND NO BAR CELLS UNIFORM COMPRESSION IN TWO DIRECTIONS' AND UNIFORMLY VARYING LOAD IN X DIRECTION P r o b l e m Exact Value of g (Eqn5.1) Poisson's Ratio (y) Bar or No Bar Percent E r r o r •"' Model.- Mesh'.* •;• 3 x 3 4 x 4 | 6 x 6 H U H 1 t t t T t Square P l a t e r 2.0 2.0 2.0 2.0 .3 .1 ,45 No Bar Bar Bar Bar -12.95 -9.3 -8.96 -10.2 -5.5 -5.85 -5.2 -6.5 -2.75 -2.66 -2.32 -3.04 -* > - X 1 t t f T Square P l a t e X 5.315 5.315 5.315 5.315 ,3 .1 .45 No Bar Bar Bar Bar -16.0 -12.9 -11.4 -6.38 -8.8 -8.7 -8.76 -4.46 -4.9 -4.57 -5.26 Aspect = .6 x 9.7 9.7 9.7 9.7 .3 .1 ,45 No Bar Bar Bar Bar -7.16 -8.45 -14.3 -6.34 -5.36 -6.43 -77.1--2.84 -3.62 -4.34 -62.1 Simply supported y>>v>v Clamped i n f l e x u r e 74. Example 3 The simply supported plate of the dimension (£ x b) and the thickness t (Fig. 5.2), i s subjected to uniform shear, stress x along the edges. - " T ' • • ~ " ' _ ^ The exact c r i t i c a l stress i s given by the Eqn (5.1) as _ b *- t ; Tc'r •TT D b 2 t (5.7) F i g . 5.2 The model i s again made up of the c e l l s a xnkakgeometrically s i m i l a r to the p l a t e . With x=l, the corner loads c a r r i e d by each c e l l i n the x and y directions kat at are re s p e c t i v e l y ^ and -~ and the displacement of corners 3 and 4 i n the y d i r e c t i o n i s ka tE 2 (1+y) (5.8) i This causes an elongation i n the diagonal bar (2-3) and an equal shortening i n the diagonal bar (1-4) of the amount 6^  = 6 Sina 2(1+y) tE ka ( k 2 + l ) 1 / 2 (5.9) The stresses S ^ and S^^ are found using Eqn (2.5) with the expression for area being given by Eqn (2.6) S14 " u a ( k 2 + l ) 1 / 2 1-y - s 23 (5.10) (5.11) 75. kat . kat The s t r e s s e s i n the subdivided s i d e bars are found w i t h the help of Eqn (4.29) as a(3u-l) 2(1-y) (5.12) S.. 0 = S„. = S.._ = S 0 / = 0 since t h i s 12 34 13 24 c o n d i t i o n leaves these bars unstressed. The s t r e s s F i s p o s i t i v e s i n c e i t agrees w i t h the s i g n convention adopted f o r p o s i t i v e s t r e s s i n the secondary bars ( F i g . 4.9). The c r i t i c a l b u c k l i n g s t r e s s T i s found as before and i s the cr lowest eigenvalue f of the Eqn (3.1). The r e s u l t s f o r both bar and no bar c e l l s o l u t i o n s are presented i n Table 8 i n . t h e form of percent e r r o r s i n comparison w i t h the exact values. The s t r u c t u r e s analysed are: (1) Simply supported .square p l a t e , i n f l e x u r e (2) Square p l a t e w i t h two opposite edges f i x e d and simply supported, i n f l e x u r e I (3) Simply supported ( i n f l e x u r e ) rectangular, p l a t e w i t h — = 1.25, a l l w i t h the assumed values of y: 0.1, 0.3 and 0.45. Example 4 I b <—\ L •> A rectangular p l a t e of the s i z e (b x I) simply supported against f l e x u r e , i s subjected to a uniformly v a r y i n g load along i t s longer sides as shown i n F i g . 5.4. As before the c r i t i c a l s t r e s s F i g . 5.4 i s given by Eqn (5.1) 76. TABLE V I I I CRITICAL BUCKLING STRESSES IN PLATES PERCENT ERRORS OF FINITE ELEMENT SOLUTIONS WITH BAR AND NO BAR CELLS PURE SHEAR Exact Poisson's Bar or Percent E r r o r Value Ratio No Bar Model Mesh E r o b 1 e m of g (y) 3 x 3 |4 x 4 16 x 6 (Eqn5.i: 9.34- No Bar -12.1 -10.2 -6.47 ' S S S s S I b • y / 9.34 9.34 .3 Bar T22.0 -17.95 1-11.6 -10.3 '-6.42 • -6.07 / s S - . 1 Bar 9.34 .45 Bar -32.6 -15.1 -6.73 y Square P l a t e 12.28 No Bar -10.2 -5.86 VXVYYVX / /• l s 12.28 .3 Bar -17.2 -8.1 -4.92 • < / / b ' y 12.28 -14.0 -8.34 -4.90 y .1 Bar | — y 12.28 .45 Bar -37.6 -13.7 -5.06 Square P l a t e X - 9.92 -6.00 j ' / / 77.71 — No Bar — t b / r ' 7.71 .3 Bar -23.1 -11.8 -5.95 / / 7.71 .1 Bar -18.5 -10^35 -5.52 - S sy r y x Aspect = 1.25 7.71 .45 Bar i I -34.8 -15.9 -6.3 r- f ' t <•' Simply supported xx.yy.tti Clamped i n f l e x u r e ' In order to c a l c u l a t e the bar s t r e s s e s , the model i s analysed f o r plane s t r e s s , w i t h i t s edges r e s t r a i n e d against r i g i d body movement only. The loads a c t i n g at the nodes of the model are the s t a t i c equiva-l e n t of a load of u n i t i n t e n s i t y a c t i n g on the p l a t e ( F i g . 5.5). The bar forces are obtained using the r e l a t i o n s given i n Table 4 and Eqn 4.29. With the bar forces known, the c r i t i c a l load i s obtained as before and the r e s u l t s tabulated i n Table 7. E f f e c t of Negative S t i f f n e s s e s An i n s p e c t i o n of Eqns (2.8 and (2.38) shows that f o r k >_ 1 2 A^ and I are zero i f yk = 1 2 A^ and I are negative i f yk ;»1 Eqn (2.46) and (2.47) r e v e a l a l s o that C and are zero i f y = — C and are negative i f y > - j When the area A^ of the longer s i d e bar i s zero the c e l l i s no longer r i g i d . However the disappearance of the t o r s i o n a l s t i f f n e s s e s C and does not make the s t r u c t u r e n o n - r i g i d unless 1^ vanishes a l s o . To i n v e s t i g a t e the e f f e c t of negative and zero s t i f f n e s s e s the three examples solved e a r l i e r are solved again f o r d i f f e r e n t values of 2 y and k so that a wide range of the q u a n t i t y yk i s c o v e r e d s e v e r a l values of which make A., and I negative. 78. 0.222" 0.222 0.125" 0.125 1.333 —~+ 1.333 0.75 "*—' 0.75 1.5. —*• 1;5 2.667 2.667 2.25 •*—'2.25 1.778 L-«—' 1.778 3 x 3 model 1.375 — ^ 1.375 4 x 4 model 0.056 0.333 ~—*-\ 0.667 — 1.00 1.333 — 1.667 .0.944 0.056 0.333 0.667 1.00 1.333 1.667 0.944 Loads ( i n l b s . ) a c t i n g at nodes f o r v a r i o u s models ( i n plane s t r e s s ) f o r deter-mining bar forces i n example 4. (Based on p l a t e s i z e 7.2" x 12" x 1"). Model i s r e s t r a i n e d only to prevent r i g i d body movement i . e . 3 degrees of freedom are suppressed. 6 x 6 model F i g . 5.5 79. The r e s u l t s are summarised i n Tables 9-13. As before t h e i r p r e c i s i o n i s expressed as percentage e r r o r s of the c o e f f i c i e n t g i n 2 comparison w i t h the e l a s t i c i t y s o l u t i o n . For c e r t a i n values of yk no r e s u l t was.obtained as i n d i c a t e d by the dash symbol. The examination of the tab l e s r e v e a l s : 2 (1) The r e s u l t s are g e n e r a l l y more accurate when yk < 0.75 (2) On r e d u c t i o n of the mesh s i z e the r e s u l t s tend to diverge 2 f o r yk c l o s e to and greater than u n i t y 22 (3) In most cases no results- may be obtained as yk approaches the range of 1.15 to 1.35 (4) In the same p l a t e accuracy decreases f o r a comparable mesh s i z e as k i s increased. 80. TABLE IX PERCENT ERRORS OF FINITE ELEMENT.SOLUTION FOR VARIOUS VALUES OF ASPECT RATIO AND POISSON'S RATIO P r o b l e m Exact Soln. e (Eqn5.1) Polsson's Ratio ( y ) Aspect Ratio • of C e l l Percent E r r o r Model Mesh 4 x 6 '8 x 1'2 ' ' ' / ' ' ' '* b V J s J J } > J t 1 t t r r X 4.0 4.0 4.0 4.0 ,1 .333 . .4 .45 1.5 1.5 1.5 1.5 .225 .75 .900 1.025 -5.14 -2.66 -2.30 -2.30 -2.88 -2.04 -2.00 -2.. 58 Same as Above but = 1.75 Percent E r r o r Model Mesh 4 x 4 6 x 6 4.071 4.071 4.071 4.071 .1 .333 .4 .45 1.75 1.75 1.75 1.75. .306 1.020 1.225 1.375 -10.0 -3.26 -96.0 -7.6 -4.7 Same as above but b " 2'° 4.0 4.0 4.0 4.0 .1 ,333 ,4 ,45 • 2.0 2.0 2.0 2.0 .400 1.333 1.600 1.800 -11.15 -56.0 -8.52 -16.35 Same as above but = 2.25 4.055 4.055 4.055 4.055 ,1 ,333 .4 .45 2.25 2.25 2.25 2.25 .506 1.688 2.025 2.280 -13.6 -3.86 Same as above but = 2.50 4.133 4.133 4.133 4.133 1. .333 .4' .45 2.50 2.50 2.50 2.50 .625 2.086 2.500 2.810 -20.2 -11.3 81. TABLE X PERCENT ERRORS OF FINITE.ELEMENT SOLUTION FOR VARIOUS VALUES OF ASPECT RATIO AND POISSON'S RATIO P r o b l e m Exact Soln. 8 (Eqn5.1) Poisson's Ratio (y) Aspect Ratio (k) of C e l l yk Percent E r r o r Model Mesh 4 x 4 6 x 6 4 4/ 4 r * I t b 111111 £ = 1.25 b 9.25 9.25 9.25. 9.25 11 .333 .4 • .45 1.25 1.25 1.25 1.25 .156 .521 .625 .1.703 -7.55 -2.46 -1.1 -0.05 -3.55 +0.22 +1.1 +1.56 Same as above but = 1.50 8.33 8.33 8.33 8.33 ,1 ,333 ,4 ,45 1.50 1.50 1.50 1.50 .225 .750 .900 1.025 -9.85 -3.18 -1.71 -50.5 ±3.72 • +1.06 +2.18 -28.3 Same as above but 1.75 8.11. 8.11 8.11 8.11 ,1 ,333 .4 .45 1.75 1.75 1.75 1.75 .306 1.020 1.225 1.375 -15.7 -10.8 -75.0 -4.35 +1.80 Same as above but = 2.0 7.88 7.88 7.88. 7.88 ,1 . ,333 .4 ,45 2.0 2.0 2.0 2.0 .400 1.333 1.600 1.800 -25.0 -53.0 -6.25 -32.2 82. TABLE. XI PERCENT ERRORS OF FINITE ELEMENT SOLUTION FOR VARIOUS VALUES OF ASPECT RATIO AND POISSON'S RATIO P r o b 1 e m Exact Soln. 6 (Eqn5.1) Poisson's Ratio (y) Aspect Ratio of C e l l (k) Percent E r r o r Model Mesh 4 x 4 6 x 6 4 i an t r t 7 f £ = 1.2 D 2.44 2.44 2.44 2.44 11 .".333 .4 .45 1.2 1.2 1.2 1.2 ,144 ,480 ,576 ,650 -7.2 -3,5 -2.9 -2.64 Same as above but £ - 1 . 4 2.96 2.96 2.96 2.96 1.1 .333 .4' .45 1.4 1.-4 1.4' 1.4 ,196 ,653 ,784 ,882 -6.6 -3.78 -3.38 -3.32 Same as above but - - 1.6' 3.56 3.56 3.56 3.56 ,1 ,333 14 ,45 1.6 1,6 1.6 1.6 .256 .853 1.024 1.152 -6.52 -4.14 -4.27 Same as above but = 1. 4.24 4.24 4.24 4.24 .1 ,333 ,4 . ,45 1.8 1.8 1.8 1.8 .324 1.080 1.296 1.458 -6.4 -4.67 83. TABLE X I I PERCENT ERRORS OF FINITE SOLUTIONS FOR VARIOUS VALUES OF ASPECT RATIO AND POISSON'S RATIO P r o b l e m Exact Soln. 8 (Eqn5.1) Poisson's Ratio (y) Aspect Ratio of C e l l (k). Percent E r r o r Model Mesh 4 x 4 6 x 6 y y x y w = 1.5 r 11.50 11.50 11.50 11.50 •.1 . .333 .4 .45 1-5 1.5 1.5 1.5 .225 .750 .900 1.025 +2.960 +3.220 -10.62 -47.0 -1.01 +2.50 +1.40 -14.0 x Same as above but £ = 2.0 10.34 10.34 10.34 10.34 .1 .333 .4 . .45 2.0 2.0 2.0 200 .400 1.333 1.600 1.800 -9.10 -11.20 -1.45 +0.46 > > X < < •l l b < b 1.5 r 11.12 11.12 11.12 11.12 .1 .333 ,4 .45 1.5 1.5 1.5 1.5 .225 . .750 .900 1.025 -1.66 +1.68 -19.20 -63.5 -4.3 0.0 -0.707 -25.67 Same as above but ^ = 2.0 10.21 10.21 10.21 10.21 .1 .333 .4 .45 2.0 2.0 2.0 2.0 .400 1.333 1.600 1.800 -12.8 -30.6 -3.00 -1.17 84. TABLE X I I I PERCENT ERRORS. OF FINITE ELEMENT SOLUTION FOR VARIOUS VALUES OF ASPECT RATIO AND POISSON'S RATIO P r o b l e m /Ss S S S s • • / I / / * s / b * S S s s s S' = 1.4 Exact Soln. 3. (Eqn5..1) 7.3 7.3 7.3 7.3 Poisson's Ratio .1 .333 .4 .45 Aspect Ratio of C e l l (k) 1.4 1.4 1.4 1.-4 yk^ .196 .653 .784 ,882 Percent E r r o r Model., Mesh 4 x 4 6 x 6 - .158; +2.07 -0.6051 -11.0 -3.17 -0.43 -0.2681 -0.70 Same as above but = 1.6 7.0 7.0 7.0 7.0 .1 .333 .4 .45 1.6 1.6 1.6 1.6 .256 .853 1.024 1.152 -2.23 -1.63 -7.25 -4.55 -2.22 ^2.88 Same as above but F - 1.8 6. 6. 6. 6. .1 .33 .4 .45 1.8 1.8 1.8 1.8 .324 1.080 1.296 1.458 -4.60 ?7.00 -5.19 -3.88 \ CHAPTER VI CONCLUSION From an examination of the r e s u l t s the f o l l o w i n g conclusions can be drawn: , (1) W i t h i n the l i m i t s of a p p l i c a b i l i t y the framework method i s a v a l i d one f o r s o l v i n g s t a b i l i t y problems. (2) I t s comparison w i t h the no bar c e l l ' method i n p l a t e s when the r e s u l t s are a v a i l a b l e shows s i m i l a r p r e c i s i o n . (3) The f i n i t e element values of the c r i t i c a l load are, as a r u l e , lower.than the true values. The model thus behaves i n r e l a t i o n to i n s t a b i l i t y as a s t r u c t u r e l e s s r i g i d than the prototype. (4) As the mesh s i z e becomes f i n e r , keeping the same aspect r a t i o of c e l l s , the c r i t i c a l values of the f i n i t e element s o l u t i o n s advance monotonically towards the exact v a l u e s , somewhat f a s t e r than, i n the,inverse r a t i o of the l i n e a r c e l l s i z e s , perhaps c l o s e r to 3/2 power .of t h i s r a t i o . (5) . The c e l l proportions should p r e f e r a b l y be kept w i t h i n the l i m i t s of the aspect r a t i o 1 < k < 1.5. Outside these l i m i t s p r e c i s i o n i s reduced. (6) The c e l l s should be so proportioned w i t h reference to the value 2 of the Poissonss r a t i o that the quantity yk i s r e s t r i c t e d to remain under 0.75. 86. 2 With yk exceeding t h i s value and approaching the value of one, the s o l u t i o n at times tends to diverge on reduction of the mesh s i z e , or the approximate c r i t i c a l l o a d , normally smaller than 2 the exact l o a d , becomes greater than i t . As yk reaches the l e v e l of 1.15 - 1.3 the eigenvalue problem becomes i n s o l u a b l e by computer, p o s s i b l y due to d e f i c i e n c y of matrices i n the c e l l s i n v o l v e d . The subject of the e f f e c t s of high values of k and 2 yk r e q u i r e s f u r t h e r study. The necessary c e l l s t a b i l i t y matrices cover a l l kinds of loading described by the s t r e s s e s S. On the other hand the no bar c e l l a n a l y s i s r e q u i r e s a m u l t i p l i c i t y of s t a b i l i t y m a t r i c e s , only a few of which, a p p l i c a b l e to c e r t a i n simple types of c e l l l o a d i n g are given e x p l i c i t l y . The complete g e n e r a l i t y of the proposed bar c e l l s t a b i l i t y matrix makes i t su p e r i o r to i t s no-bar c e l l counterpart. BIBLIOGRAPHY Hren n i k o f f , A. " S o l u t i o n of Problems of E l a s t i c i t y by the Framework Method," J o u r n a l of Applied Mechanics, December 1941. Turner, M. J . , Clougb, R. W., M a r t i n , H. C., and Topp, L. J . " S t i f f n e s s and D e f l e c t i o n A n a l y s i s of Complex S t r u c t u r e s , " J o u r n a l of A e r o n a u t i c a l Sciences, V o l . 23, No. 9, September 1956. Clough, R. W. 'The F i n i t e element i n plane s t r e s s a n a l y s i s , ' Proc. 2nd A.S.C.E. Conf. on E l e c t r o n i c Computation, ' P i t t s b u r g h , Pa., September 1960. Melosh, R. J . "A S t i f f n e s s M a t r i x f o r the A n a l y s i s of Thin P l a t e s i n Bending," J . Aero S c i . V o l . 28, 1961. Gallagher, R. H., and Padlog, R. J . " D i s c r e t e Element Approach to S t r u c t u r a l I n s t a b i l i t y A n a l y s i s , " J o u r n a l of the American  I n s t i t u t e of Aeronautics and A s t r o n a u t i c s , V o l . 1, No. 6, June 1963. Kapur, K. K. and Hartz, B. J . " S t a b i l i t y of P l a t e s Using F i n i t e Element Method," J o u r n a l of Engg Mech.. Div. ASCE, EM2, V o l . 92, A p r i l , 1966. Hr e n n i k o f f , A. Mimeographed notes on Plane Stress and P l a t e Bending, i n course CE551, U n i v e r s i t y of B r i t i s h Columbia, Vancouver. Langhaar, H. L. "Energy Methods i n Applied Mechanics," John Wiley and Sons, Inc., 1962. Timoshenko, Gere. "Theory of E l a s t i c S t a b i l i t y , " McGraw-Hill Book Company, New York, 1961, pp. 348-389. Kapur, K. K. Ph.D. Thesis, U n i v e r s i t y of Washington, 1965. Lucas, W. M.', and Willems, N. "Matrix A n a l y s i s f o r S t r u c t u r a l Engineers," P r e n t i c e H a l l Inc., 1968. 

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