Open Collections

UBC Theses and Dissertations

UBC Theses Logo

UBC Theses and Dissertations

Plate bending analysis with isosceles trapezoidal bar cells Ha, Huy Kinh 1970

Your browser doesn't seem to have a PDF viewer, please download the PDF to view this item.

Notice for Google Chrome users:
If you are having trouble viewing or searching the PDF with Google Chrome, please download it here instead.

Item Metadata

Download

Media
831-UBC_1970_A7 H3.pdf [ 6.47MB ]
Metadata
JSON: 831-1.0050567.json
JSON-LD: 831-1.0050567-ld.json
RDF/XML (Pretty): 831-1.0050567-rdf.xml
RDF/JSON: 831-1.0050567-rdf.json
Turtle: 831-1.0050567-turtle.txt
N-Triples: 831-1.0050567-rdf-ntriples.txt
Original Record: 831-1.0050567-source.json
Full Text
831-1.0050567-fulltext.txt
Citation
831-1.0050567.ris

Full Text

PLATE BENDING ANALYSIS WITH ISOSCELES TRAPEZOIDAL BAR CELLS by HUY KINH HA B . A . S c , U n i v e r s i t y o f O t t a w a , May 1 9 6 8 A THESIS SUBMITTED IN PARTIAL FULFILMENT OF THE REQUIREMENTS FOR THE DEGREE OF MASTER OF APPLIED SCIENCE i n t h e D e p a r t m e n t o f C I V I L ENGINEERING We a c c e p t t h i s t h e s i s as c o n f o r m i n g t o t h e r e q u i r e d s t a n d a r d THE UNIVERSITY OF BRITISH COLUMBIA May, 1 9 7 0 I n p r e s e n t i n g t h i s t h e s i s i n p a r t i a l f u l f i l m e n t o f t h e r e q u i r e m e n t s f o r an a d v a n c e d d e g r e e a t t h e U n i v e r s i t y o f B r i t i s h C o l u m b i a , I a g r e e t h a t t h e L i b r a r y s h a l l make i t f r e e l y a v a i l a b l e f o r r e f e r e n c e and s t u d y . I f u r t h e r a g r e e t h a t p e r m i s s i o n f o r e x t e n s i v e c o p y i n g o f t h i s t h e s i s f o r s c h o l a r l y p u r p o s e s may be g r a n t e d by t h e Head o f my D e p a r t m e n t o r by h i s r e p r e s e n t a t i v e s . I t i s u n d e r s t o o d t h a t c o p y i n g o r p u b l i c a t i o n o f t h i s t h e s i s f o r f i n a n c i a l g a i n s h a l l n o t be a l l o w e d w i t h o u t my w r i t t e n p e r m i s s i o n . HUY KINH HA D e p a r t m e n t o f C i v i l E n g i n e e r i n g The U n i v e r s i t y o f B r i t i s h C o l u m b i a V a n c o u v e r 8, Canada Da t e MAY 1970 Ki.nh dang ma V2.dl0.atzd to my IhotkzK ABSTRACT A b a r c e l l i n t h e shape o f an i s o s c e l e s t r a p e z o i d i s d e v i s e d f o r u s e i n f l e x u r e p r o b l e m s o f c i r c u l a r p l a t e s . The c e l l members a r e endowed w i t h e l a s t i c p r o p e r t i e s s u c h t h a t i t d e f o r m s i n t h e same manner as a p i e c e o f p l a t e i n c o n d i t i o n o f a r b i t r a r y u n i f o r m b e n d i n g a b o u t any a x i s i n t h e p l a n e o f a p l a t e . The s t i f f n e s s m a t r i x o f t h e c e l l i s d e r i v e d e x p l i c i t l y . Two methods o f c o m p u t a t i o n o f s t r e s s e s a r e d e s c r i b e d , one by t h e n o d a l f o r c e s and t h e o t h e r by t h e n o d a l d i s p l a c e -ments . The v a l i d i t y o f t h e c e l l i s t e s t e d on two e x a m ples whose e x a c t s o l u t i o n s a r e known. One i n v o l v e s a s e m i -c i r c u l a r c l amped p l a t e u n d e r u n i f o r m l o a d , and t h e o t h e r a s i m p l y - s u p p o r t e d c i r c u l a r p l a t e u n d e r an e c c e n t r i c c o n c e n -t r a t e d l o a d . The r e s u l t s compared f a v o u r a b l y w i t h t h e t h e o r y o f e l a s t i c i t y s o l u t i o n s and t h e n o - b a r f i n i t e e l e m e n t s o l u t i o n s . Good t r e n d o f c o n v e r g e n c e o f s o l u t i o n s i s i n -d i c a t e d on r e d u c t i o n o f t h e e l e m e n t s i z e . i v TABLE OF CONTENTS C h a p t e r Page I INTRODUCTION 1 1.1 G e n e r a l 1 1.2 P l a t e E q u a t i o n s o f t h e T h e o r y o f E l a s t i c i t y 3 I I FLEXURAL CELLS IN THE SHAPE OF ISOSCELES TRAPEZOID AND TRIANGLE 6 2.1 T r a p e z o i d a l B a r C e l l 6 2.2 D e t e r m i n a t i o n o f Member P a r a m e t e r s . . 8 2.3 T r a p e z o i d a l and R e c t a n g u l a r B a r C e l l s 36 2.4 D i s t r i b u t i o n F a c t o r s o f t h e I s o s c e l e s T r a p e z o i d a l B a r C e l l 39 2.5 S t i f f n e s s M a t r i x o f I s o s c e l e s T r a p e z o i d a l B a r C e l l 51 2.6 T r a n s f o r m a t i o n M a t r i c e s 68 2.7 I s o s c e l e s T r a p e z o i d a l and T r i a n g u l a r No-Bar C e l l s 73 I I I BOUNDARY CONDITIONS IN THE PLATE MODEL 75 3.1 S i m p l y - S u p p o r t e d Edge 75 3.2 Clamped Edge 76 3.3 F r e e Edge 77 IV STRESS ANALYSIS OF THE PLATE 78 4.1 Method o f N o d a l F o r c e s 78 V C h a p t e r Page 4.2 F i r s t Method o f N o d a l D i s p l a c e m e n t s . . 86 4.3 Second Method o f N o d a l D i s p l a c e -ments 87 V FLEXURAL PROBLEMS OF CIRCULAR PLATES SOLVED BY THE F I N I T E ELEMENT METHOD . . . . 91 5.1 Example 1: S i m p l y - s u p p o r t e d S e mi-c i r c u l a r P l a t e Under U n i f o r m L o a d . . . 92 5.2 Example 2: Clamped C i r c u l a r P l a t e Under an E c c e n t r i c C o n c e n t r a t e d L o a d . . 109 V I CONCLUSIONS 121 REFERENCES 124 v i L I S T OF TABLES T a b l e Page I I - l P a r a m e t e r s f o r F l e x u r a l T r a p e z o i d a l B a r C e l l 37 I I - 2 D i s t r i b u t i o n F a c t o r s f o r F l e x u r a l T r a p e z o i d a l B a r C e l l 58 V - l . l T r a n s v e r s e D e f l e c t i o n 97 V-1.2 T a n g e n t i a l R o t a t i o n 98 V-1.3 R a d i a l R o t a t i o n 99 V-1.4 B e n d i n g Moment M r 100 V-1.5 B e n d i n g Moment M 101 V-1.6 T o r s i o n a l Moment 102 V-1.7 S h e a r Q 103 r -V-1.8 S h e a r Q. 104 t V-1.9 P e r c e n t a g e E r r o r s i n S t r e s s e s by N o d a l F o r c e and N o d a l D i s p l a c e m e n t Methods . . . . 105 V-1.10 P e r c e n t a g e E r r o r s i n S t r e s s e s by N o d a l F o r c e and N o d a l D i s p l a c e m e n t Methods . . . . 107 V-2.1 T r a n s v e r s e D e f l e c t i o n . 113 V-2.2 T a n g e n t i a l R o t a t i o n 114 V-2.3 R a d i a l R o t a t i o n 115 V-2.4 B e n d i n g Moment M r 116 V-2.5 B e n d i n g Moment M 117 V-2.6 T o r s i o n a l Moment 118 V-2.1 S h e a r Q 119 r V-2.8 Shear Q-, : I 2 0 ACKNOWLEDGEMENT The w r i t e r i s g r e a t l y i n d e b t e d t o p r o f e s s o r A. H r e n n i k o f f f o r h i s c o n s t a n t and v a l u a b l e g u i d a n c e a t e v e r y s t a g e o f t h i s work. The w r i t e r a l s o w i s h e s t o t h a n k D r . K.M. A g r a w a l , Mr. C . I . Mathew f o r t h e i r h e l p f u l d i s c u s s i o n s , and many o f h i s f r i e n d s who made h i s s t a y i n Canada a p l e a s a n t memory. Thanks a r e a l s o due t o t h e U.B.C. C o m p u t i n g C e n t e r f o r i t s e x c e l l e n t s e r v i c e s and f a c i l i t i e s . v i i i NOTATION x,y , z R e c t a n g u l a r c o o r d i n a t e s r,0 P o l a r c o o r d i n a t e s T,R,z' P l a t e c o o r d i n a t e s ( t a n g e n t i a l , r a d i a l and v e r t i c a l d i r e c t i o n s ) a ^ , a 2 / a ^ C e l l d i m e n s i o n s b i M e s h d i m e n s i o n s c D i s t a n c e f r o m c o n c e n t r a t e d l o a d t o p l a t e c e n t r e f A r b i t r a r y c o n s t a n t u s e d i n b a r c e l l h H e i g h t o f c e l l k,k' R a t i o s o f two p a r a l l e l b a s e s o f c e l l k^,k^ R a t i o s o f h e i g h t o v e r b a s e l e n g t h m..,m..,z.. D i s t r i b u t i o n f a c t o r s ( x , y , z system) as iD i l iD d e f i n e d i n t h e t e x t T R m..,m..,z!. D i s t r i b u t i o n f a c t o r s (R,T,z' system) 1 1 1 3 1 3 ' 2 m ,m N o d a l f o r c e s t r a n s f e r r e d f r o m t h e edges x y ^ q L o a d i n t e n s i t y , p o s i t i v e upwards r , r R a d i i o f c u r v a t u r e o f t h e m i d d l e s u r f a c e y o f a p l a t e i n xy and yz p l a n e s t T h i c k n e s s o f a p l a t e w V e r t i c a l d e f l e c t i o n , p o s i t i v e upwards C^,C^ Combined t o r s i o n a l r i g i d i t i e s o f p a r a l l e l members o f t h e b a r c e l l C 0 , C n ,C„ T o r s i o n a l r i g i d i t i e s o f t h e main members 3' l p ' 2p c , % .. c o f t h e b a r c e l l C^ g,C T o r s i o n a l r i g i d i t i e s o f t h e s e c o n d a r y members o f t h e b a r c e l l IX D F l e x u r a l r i g i d i t y o f t h e p l a t e E M o d u l u s o f e l a s t i c i t y o f m a t e r i a l i n t h e p l a t e and t h e members o f t h e b a r c e l l E I ^ , E l 2 Combined f l e x u r a l r i g i d i t i e s o f t h e p a r a l l e l members o f t h e b a r c e l l E 1 " 3,EI^ /EI„ F l e x u r a l r i g i d i t i e s o f t h e mai n members P P o f t h e b a r c e l l E I ^ g , E I 2 s F l e x u r a l r i g i d i t i e s o f t h e s e c o n d a r y members o f t h e b a r c e l l F_.' F2 S h e a r s i n t h e b a r c e l l members L R a t i o D/a^ M^,M ,M ,M B e n d i n g moments p e r u n i t l e n g t h o f s e c t i o n ^ r o f a p l a t e i n p l a n e s x , y , r , t r e s p e c t i v e l y M ,M x ' M r t ' M t r T o r s i o n a l moments p e r u n i t l e n g t h o f s e c t i o n ^ Y o f a p l a t e N.F.,N.D. N o d a l f o r c e and N o d a l d i s p l a c e m e n t methods o f s t r e s s a n a l y s i s f o r p l a t e model P M a g n i t u d e o f c o n c e n t r a t e d l o a d Q X'Q ,Q ,Q t S h e a r i n g f o r c e s p e r u n i t l e n g t h o f s e c t i o n ^ r o f p l a t e i n p l a n e s x , y , r , t r e s p e c t i v e l y R R a d i u s o f a p l a t e T / T ^ j T ^ / T ^ T o r q u e s i n members o f t h e b a r c e l l {f} V e c t o r o f e l e m e n t n o d a l f o r c e s { F } V e c t o r o f j o i n t l o a d s . i n p l a t e model [K] S t i f f n e s s m a t r i x o f a c e l l [S] S t i f f n e s s m a t r i x o f t h e whole model [T] T r a n s f o r m a t i o n m a t r i x [T.] S u b - m a t r i x i n [ T ] , c o r r e s p o n d i n g t o c o r n e r i { 6 } V e c t o r o f e l e m e n t n o d a l d i s p l a c e m e n t s {A} V e c t o r o f j o i n t d i s p l a c e m e n t s X a ^ , a 2 , a 3 F a c t o r s r e p r e s e n t i n g t h e o r e t i c a l s o l u t i o n f o r d e f l e c t i o n s and s l o p e s ^ l ' ^ 2 ' ^ 3 F a c t o r s r e p r e s e n t i n g t h e o r e t i c a l s o l u t i o n f o r b e n d i n g moments and t o r s i o n a l moments i n a p l a t e Y T / Y 2 F a c t o r s r e p r e s e n t i n g t h e o r e t i c a l s o l u t i o n f o r s h e a r f o r c e s i n a p l a t e 6 A n g l e between t h e b a s e and d i a g o n a l members o f a b a r c e l l 6e A n g l e between two r a d i a l l i n e s c o n f i n i n g an e l e m e n t A , A ^ V e r t i c a l d i s p l a c e m e n t o f a c o r n e r AR R a d i a l d i s t a n c e between two c i r c l e s c o n f i n -i n g an e l e m e n t X] R a t i o r/R 6.,8.,w. N o d a l r o t a t i o n s and d i s p l a c e m e n t o f c o r n e r i i i • i o f t h e e l e m e n t ( x , y , z system) T R 6.,8.,w! N o d a l r o t a t i o n s and d i s p l a c e m e n t o f c o r n e r 1 i i i o f t h e e l e m e n t (R,T,z' system) 9_.,0_. R o t a t i o n s a b o u t T and R a x e s o f j o i n t i T i ' Rx J y- P o i s s o n ' s r a t i o o f p l a t e m a t e r i a l 5 R a t i o c/R <$> R a t e o f t w i s t p e r u n i t l e n g t h o f p l a t e T/J , ^ ^ , ^ 2 ^ ' A n g l e s o f b e n d i n g i n members o f a b a r c e l l 1 CHAPTER I INTRODUCTION 1.1 G e n e r a l C l o s e d f o r m s o l u t i o n s f o r p l a t e b e n d i n g p r o b l e m s a r e o b t a i n a b l e o n l y i n c a s e s o f s i m p l e b o u n d a r y c o n d i t i o n s . I n most p r a c t i c a l s i t u a t i o n s where t h e g e o m e t r y , b o u n d a r y and l o a d i n g c o n d i t i o n s a r e c o m p l i c a t e d , one must r e s o r t t o a p p r o x i m a t e methods s u c h as t h e method o f f i n i t e d i f f e r e n c e s , t h e R a y l e i g h - R i t z t e c h n i q u e o r o t h e r v a r i a t i o n s o f t h e s t r a i n e n e r g y method [ 1 ] . W i t h t h e a d v e n t o f h i g h - s p e e d c o m p u t e r , t h e f i n i t e e l e m e n t f o r m u l a t i o n o f t h e p r o b l e m has become w i d e l y u s e d , and i t f o r m s t h e b a s i s f o r t h e p r e s e n t i n v e s t i g a t i o n . In 1941, H r e n n i k o f f [2] i n t r o d u c e d t h e c o n c e p t o f r e p l a c i n g t h e c o n t i n u o u s m a t e r i a l o f an e l a s t i c body by a s y s t e m o f framework e l e m e n t s . The b a r s w h i c h make up t h e e l e m e n t s a r e endowed w i t h e l a s t i c p r o p e r t i e s s u c h t h a t t h e d e f o r m a b i l i t y o f t h e e l e m e n t i s i d e n t i c a l w i t h t h a t o f a p i e c e o f t h e m a t e r i a l when p l a c e d i n an a r b i t r a r y u n i f o r m s t r e s s f i e l d . The e l e m e n t s a r e c o n n e c t e d a t t h e c o r n e r s and t o g e t h e r t h e y f o r m t h e g e o m e t r y o f t h e body. The f r a m e -2 work model i s s u b j e c t e d t o e x t e r n a l f o r c e s a p p l i e d a t t h e n o d e s . B o u n d a r y c o n d i t i o n s a r e e x p r e s s e d i n t e r m s o f c o n -s t r a i n t s o f movement o f t h e n o d e s . The r e s u l t s o b t a i n e d f r o m t h i s method, w i t h t h e same c o m p u t a t i o n a l e x p e n s e , were r e p o r t e d t o be c o n s i d e r a b l y s u p e r i o r t o t h o s e by t h e method o f f i n i t e d i f f e r e n c e s [ 3 ] . S q u a r e and r e c t a n g u l a r b a r c e l l s have b e e n d e v e l o p e d by H r e n n i k o f f f o r a n a l y s i s o f p l a n e s t r e s s and p l a t e b e n d i n g p r o b l e m s . I n t h e p l a n e s t r e s s c a s e t h e b a r s p o s s e s s e x t e n s i o n a l s t i f f n e s s and i n f l e x u r e t h e y have f l e x u r a l s t i f f n e s s t o r e s i s t o u t o f p l a n e b e n d i n g . M c C o r m i c k ' s [4] c e l l p o s s e s s e s b o t h forms o f s t i f f n e s s e s . I n 1963 P e s t e l [3] d e v i s e d c e l l s h a v i n g s p e c i a l d i f f e r e n -t i a l g e a r s w h i c h f a c i l i t a t e t h e c o n s t r u c t i o n o f i r r e g u l a r l y s h a p e d m o d e l s . The i d e a was e x t e n d e d t o t h r e e - d i m e n s i o n a l p r o b l e m s {2,5,6] where c e l l s i n t h e f o r m o f a cube o r p a r a l l e l e p i p e d were u s e d . The model made up o f b a r c e l l s r e p r e s e n t s a r e a l e l a s t i c s t r u c t u r e , w h i c h c a n be a c t u a l l y c o n s t r u c t e d . More t h a n a d e c a d e a f t e r t h e i n t r o d u c t i o n o f H r e n n i k o f f ' s c e l l s , a n o t h e r t y p e o f e l e m e n t was d e v i s e d [ 7 ] . T h i s new e l e m e n t i s a m a t h e m a t i c a l a b s t r a c t i o n s u i t a b l e f o r c a l c u l a t i o n s b u t i n c a p a b l e o f b e i n g p h y s i c a l l y r e p r o d u c e d . To d i f f e r e n -t i a t e t h i s n e w . c e l l f r o m t h e b a r c e l l , i t w i l l be c a l l e d t h e n o - b a r c e l l . 3 The p r e s e n t work i n t r o d u c e s a f l e x u r a l b a r - c e l l i n t h e f o r m o f an i s o s c e l e s t r a p e z o i d . Whose b a r s a r e endowed w i t h f l e x u r a l and t o r s i o n a l s t i f f n e s s e s . The a p p l i c a b i l i t y o f t h e c e l l i s d e m o n s t r a t e d by s o l v i n g two p r o b l e m s o f b e n d i n g o f s e m i - c i r c u l a r and c i r c u l a r p l a t e s . The r e s u l t s a r e compared w i t h t h o s e o b t a i n e d by t h e t h e o r y o f e l a s t i c i t y and by t h e n o - b a r f i n i t e e l e m e n t s . 1.2 P l a t e E q u a t i o n s o f t h e T h e o r y o f E l a s t i c i t y F o r s m a l l d e f l e c t i o n , t h e b e h a v i o u r o f a t h i n p l a t e i s g o v e r n e d by t h e f o u r t h o r d e r p a r t i a l d i f f e r e n t i a l e q u a t i o n ( i g n o r i n g t h e s h e a r d e f o r m a t i o n ) : .4 4 4 LJ* + 2 9 w • + "i-g = 3- n_-n 9x^ dx dy 9y I n w h i c h : w V e r t i c a l d e f l e c t i o n a t t h e p o i n t ( x , y ) , p o s i t i v e upwards q L o a d i n t e n s i t y , p o s i t i v e upwards 3 2 D = E t J / 1 2 ( l - p ) : F l e x u r a l r i g i d i t y o f t h e p l a t e t P l a t e t h i c k n e s s E M o d u l u s o f e l a s t i c i t y u P o i s s o n ' s r a t i o . The p o s i t i v e moments and s h e a r s p e r u n i t l e n g t h o f a p l a t e s e c t i o n a r e shown i n F i g u r e 1.1. T h e i r e x p r e s s i o n s i n t e r m s o f w a r e g i v e n b e l o w : 4 «« T~\ / 3 2w , 3 2w , ,, „ . x = ( —2 y T J . . . . (l-2a) 3x 3y M v = - D ( ^ + y ) . . . . (l-2b) Y 3 y Z 3x^ a 2 M_ = D(l-y) . . . . (l-2c) x y 3x3y 2 2 Q _ - D — ( — - + — - ) . . . . (l-3a) 3x 3x^ 3y^ Q , - D 1 ( _ | + i_w ) . . . . (l-3b) Y 3y 3x 3y In p o l a r c o o r d i n a t e s the above equations take the forms: (Figure 1-2) 9 2 , 1 3 . 1 3 2 v ,3 2w , 1 3w , 1 3 2w, ^ q A + — + — ( + — + —J- ^-) + -3 . = 0 2 2 2 2 2 2 3r r 3r r 39 3r r 3r r 36 D (1-4) 2 2 .. p. r 3 W , ,1 3w , 1 3 W\ n / • , , - , M r = -D[ 2~ + y( + — j ) ] . . . . ( l - 5 a ) 3 r r 3 r r 38 2 2 M = -D ( + - j j + y -) . . . . ( l - 5 b ) r 3 r r 30 3 r M r, = -M = D(l-y) (i - 1 . . . . (l-5c) r 3 r 3 0 r 38 2 2 3 ,3w 1 3w 1 3 w, . N Q - -D — (—- + + — — - ) . . . . ( l - 6 a ) 3 r 3 r r 3 r r 38 2 2 n - n 1 3 /3 W , 1 3W 1 3 WN ,, . Q. = -D ( ^ + — — + —y ^) . . . . ( l - 6 b ) r 38 3 r r 3 r r 38 F i g . 1 - 2 6 CHAPTER I I FLEXURAL CELLS IN THE SHAPE OF ISOSCELES TRAPEZOID AND TRIANGLE 2.1 T r a p e z o i d a l B a r C e l l The g e o m e t r y o f t h e c e l l i s d e s c r i b e d by t h e l e n g t h s o f t h e two p a r a l l e l s i d e s a ^ , a 2 and t h e h e i g h t h. ( F i g u r e 2.1) The r a t i o s k and k^ a r e d e f i n e d a s : l o n g e r b a s e a„ k = = — ± — s h o r t e r b a s e 1 h e i g h t , s h o r t e r b a s e a ^ The c e l l i s c o n n e c t e d w i t h t h e o t h e r c e l l s o n l y a t t h e f o u r c o r n e r s , and t h e l o a d s may be a p p l i e d o n l y t h e r e . The s t r u c t u r e o f t h e c e l l may be c o n s i d e r e d t o c o n s i s t o f two p a r t s : t h e p r i m a r y s t r u c t u r e and t h e s e c o n -d a r y s t r u c t u r e as shown i n F i g u r e 2.2. The members o f e a c h s t r u c t u r e c a n d e f o r m f r e e l y w i t h o u t i n t e r f e r e n c e f r o m members o f t h e o t h e r . T h e i r e l a s t i c modulus i s t h e same as t h a t o f t h e p l a t e . 4 uis ' X\S 3 + 6 E I 4 C =0 5 2 c 2 S , E I 2 S 1 Secondary Structure F i g . 2-2 8 The nodes 1, 2, 3, 4 a r e common t o b o t h s t r u c t u r e s , and t h e nodes 5, 6 b e l o n g o n l y t o t h e s e c o n d a r y s t r u c t u r e , t h e y a r e l o c a t e d a t t h e m i d p o i n t s o f t h e p a r a l l e l s i d e s o f t h e c e l l . The b a r s a r e i n t e r c o n n e c t e d a t t h e c o r n e r s o r nodes w i t h a x i a l h i n g e s , whose ax e s a r e n o r m a l t o t h e p l a n e o f t h e c e l l . The b a r s a r e endowed w i t h f l e x u r a l s t i f f n e s s e s f o r b e n d i n g o u t o f p l a n e o f t h e p l a t e and sometimes w i t h t o r s i o n a l s t i f f n e s s e s as w e l l . S t i f f n e s s e s f o r b e n d i n g i n t h e p l a n e o f t h e c e l l a r e t a k e n as z e r o . The t o r s i o n a l s t i f f n e s s i s d e n o t e d by l e t t e r C, and t h e f l e x u r a l s t i f f n e s s , t h e p r o d u c t o f t h e moment o f i n e r t i a and t h e modulus o f e l a s t i c i t y , i s d e s c r i b e d as E I . E a c h i s f o l l o w e d by one o r two s u b s c r i p t s , t h e n u m e r a l s u b -s c r i p t i n d i c a t i n g t h e b a r number and t h e l e t t e r s u b s c r i p t s p o r s t h e s t r u c t u r e t o w h i c h t h e b a r b e l o n g s . Due t o symmetry, b o t h d i a g o n a l b a r s have t h e same e l a s t i c p r o p e r t i e s . The s t i f f n e s s e s o f t h e b a r s a r e c a l l e d t h e p a r a m e t e r s . 2.2 D e t e r m i n a t i o n o f Member P a r a m e t e r s The b a r s t i f f n e s s e s must be so c h o s e n t h a t t h e c e l l bends i n e x a c t l y t h e same manner as t h e p l a t e i n c o n d i t i o n s o f a r b i t r a r y u n i f o r m f l e x u r e a b o u t any a x i s i n t h e p l a n e o f t h e p l a t e . The most c o n v e n i e n t way t o comply 9 w i t h t h i s n e c e s s a r y c o n d i t i o n i s t o make t h e c e l l b ehave as t h e p l a t e i n t h e f o l l o w i n g t h r e e s t a t e s . (a) P u r e t o r s i o n on t h e p l a n e s x and y , t h e p l a n e x ( i . e . t h e one p e r p e n d i c u l a r t o t h e a x i s x) b e i n g t h e p l a n e o f symmetry. T h i s means t h a t M = -M c x y yx and t h e r e a r e no b e n d i n g moments i n t h e p l a n e s x o r y. (b) P u r e b e n d i n g i n t h e x - d i r e c t i o n , w h i l e t h e c e l l r e m a i n s s t r a i g h t i n t h e y - d i r e c t i o n , no t o r s i o n i n e i t h e r p l a n e . (c) Same w i t h d i r e c t i o n s x and y i n t e r c h a n g e d . S i n c e s u p e r p o s i t i o n o f t h e s e t h r e e c o n d i t i o n s w i l l p r o d u c e u n i f o r m b e n d i n g i n any d i r e c t i o n i n r e l a t i o n t o t h e axes o f symmetry, c o m p l i a n c e w i t h t h e s e r e q u i r e m e n t s w i l l i n s u r e p r o p e r d e f o r m a b i l i t y o f t h e c e l l u n d e r any c o n d i t i o n o f u n i f o r m f l e x u r e . I t w i l l be shown l a t e r t h a t t h e above t h r e e c o n -d i t i o n s g i v e r i s e t o c e r t a i n number o f i n d e p e n d e n t e q u a t i o n s t o d e t e r m i n e t h e p a r a m e t e r s . Hence i f t h e number o f p a r a m e t e r s o f t h e c e l l i s l e s s t h a n t h e number o f e q u a t i o n s t h e c e l l b e h a v i o u r i s l i m i t e d t o s p e c i a l c a s e s , and i f i t i s g r e a t e r , t h e n t h e a d d i t i o n a l p a r a m e t e r s w i l l be a r b i t r a r y o r may be made d e p e n d e n t on e a c h o t h e r . I n some c a s e s , t h e s e f r e e p a r a m e t e r s c a n be u s e d a d v a n t a g e o u s l y t o i m p r o v e t h e b e h a v i o u r o f t h e c e l l . 2-2.1 C o n d i t i o n o f P u r e T o r s i o n C o n s i d e r a p i e c e o f p l a t e i n t h e f o r m o f an i s o s c e l e s t r a p e z o i d as shown i n F i g u r e 2-3. The t o r s i o n a l moment M a c t s i n p l a n e s x and y : xy ^ 7 M = D ( l - y ) . . . . (2-1) X Y dxdy and t h e b e n d i n g moments M = M = 0 . The d i s p l a c e m e n t s o f x y th e f o u r c o r n e r s i n r e l a t i o n t o p o i n t 0 w i l l now be d e t e r -m ined . . E q u a t i o n (2-1) g i v e s t h e r a t e o f change cf> o f t h e a n g l e o f t w i s t p e r u n i t l e n g t h o f x o r y a x i s _2 M a w _ xy dxdy D ( l - u ) S i n c e no b e n d i n g moment e x i s t s i n t h e p l a n e s x and y , t h e c r o s s - s e c t i o n s o f t h e p l a t e p a r a l l e l t o t h e s e d i r e c t i o n s r e m a i n s t r a i g h t . F u r t h e r m o r e t h e x and y c r o s s -s e c t i o n s t h r o u g h t h e c e n t r e 0 a r e a d j u s t e d by r i g i d body movement t o r e m a i n i n t h e o r i g i n a l p l a n e o f t h e c e l l b e f o r e d e f o r m a t i o n . ( F i g u r e 2-3) . R e a l i z i n g t h a t d> = — (1^ ) ^ ^ T dy dx i s t h e i n c r e m e n t o f t h e s l o p e o f t h e d e f o r m e d s e c t i o n i n th e x - d i r e c t i o n , p e r u n i t l e n g t h i n t h e y - d i r e c t i o n : F i g . 2 - 3 2A k kj a | 2A k a, 2A k a, 74 2A kk, a. 2A M i \ \ \ J _ \ . 2A k l 0 l 2A ka, 2A ka, F i g . 2 - 4 12 32w _ (2A + - 2Aj 1 4A then 3x3y a 2 a 2 h k k l a i kk, a? A = X cf) . . . . (2-2) Knowing the displacements of the corner of the plate, their angles of rotations about the axes x and y may be easily found. For example, the corner 1 rotates about the x-axis through the angle 2—• = k^af a n d a b o u t t h e Y - axis through the angle ~ = ~ . These rotations are shown in a2 K a l Figure 2.4. The displacements and rotations of the corners in Figure 2-3 are imposed on the corners of the bar-ce l l and this results in stresses in the bars. The corner move-ments are effected by the forces corresponding to the torsion condition of the.plate, and their equilibrium with the bar stresses furnishes the necessary equations to determine the parameters. The external nodal moments m and m acting on x y the c e l l may be found by transferring the uniform torques on the four edges of the plate to the adjacent corners. The operation is most conveniently done by replacing the sloping sides of the c e l l with zig-zag lines consisting of infinitesimal steps in x and y directions. (Figure 2-5). F i g . 2 - 5 14 Thus m x = M x y h / 2 = | ° d - y ) k i a i < * ) . . . . (2-3) At the node 3 or 4, m = M a n/2 + M i - (a„-aJ/2 y xy r xy 2 2 1 7 = M x y ( a i + a 2 ) / 4 = j D ( l - w ) ( l + k ) . . . . (2-4a) and a t the nodes 1 or 2, m = M a„/2 - M (a_-a n)/2 y xy 2' xy 2 2 1 7 = M x y ( a i + a 2 ) / 4 = jj; D ( l - y ) (1+k) a ^ . . . . (2-4b) The moments m and m are found to be n u m e r i c a l l y equal a t x y a l l nodes. In t h i s pure t o r s i o n c o n d i t i o n , the member (5-6) of the c e l l does not work because i t s end p o i n t s 5, 6 remain u n d i s p l a c e d and the member i s devoid of t o r s i o n a l s t i f f n e s s . Consequently the s t i f f n e s s e s o f the members (1-2), (3-4) of the secondary s t r u c t u r e are i n e f f e c t merely added to those of the primary s t r u c t u r e . Let Jl - 'IP + X l s J~2 + J 2 s C l IP + C, l s C2 2p + 2s (2-5 the s i m p l i f i e d s t r u c t u r e of the c e l l i n t h i s c o n d i t i o n i s shown i n Figure 2-6c. The members (1-2)and (3-4) i n Figure 2-6c remain s t r a i g h t a f t e r the deformation, hence no bending moments are developed i n them. The r a t e of t w i s t i s the same i n these members and i s equal to <j>. Their torques are given r e s p e c t i v e l y by: T2 = C 2 * and } . . . . (2-6 To c a l c u l a t e the s t r e s s e s i n the diagonal bars, the r e l a t i v e r o t a t i o n of one end of the bar to the other i s re q u i r e d . This i s found by combining v e c t o r i a l l y the slopes at the corners i n Figure 2-4. Rot a t i o n of corner 1 i n r e l a t i o n to corner 4 about the x-axis i s : 2A 2A 2 , 1. . 1 ,, L l . , ^ l a l kk^a^ l a l k 2 1 and about the y - a x i s : F i g . 2 - 7 17 2 A _ 2 A ' i + = k i a i * ka.^ k a 1 1 1 The f l e x u r a l and t o r s i o n a l components of these r o t a t i o n s i n the member (1-4) are found as the p r o j e c t i o n s of the two corner r o t a t i o n s on the d i r e c t i o n s r e s p e c t i v e l y p e r p e n d i c u l a r and p a r a l l e l to the member (Figure 2 - 7 ) . Thus the f l e x u r a l r o t a t i o n i s : ~ (1+k) a x <j> S i n 6 + k ^ 4> Cos 6 and the bending moment found by the u s u a l moment-curvature r e l a t i o n s h i p : M 3 = '•J ( 1 + k ) a i * S i n 6 + k i a i * C o s 6 J E I 3 / / A 3 The angle o f t w i s t i n the member (1-4) i s j (1+k) a^ cj) Cos 6 - k _ a ] _ $ S i n ^  and the torque i n t h i s member: T 3 = [ I ( 1 + k ) a i ( l ) C o s 6 _ ^ a - ^ S i n f i ] c 3 / a 3 F i g u r e 2-8 p r e s e n t s the v e c t o r diagrams of the e x t e r n a l moments m , m and the bar moments M and T at a l l x y corners o f the c e l l . In view of the symmetry of the c e l l , 18 t h e e q u i l i b r i u m r e l a t i o n s h i p between t h e s e f o r c e s i s i d e n t i c a l a t a l l c o r n e r s . From t h i s i t f o l l o w s t h a t = , and u s i n g t h e r e l a t i o n ( 2 - 6 ) : C 1 = C 2 . . . . (2-7) The moment e q u i l i b r i u m o f a c o r n e r a b o u t t h e y - a x i s g i v e s : m + T 0 S i n 6 - M 0 Cos 6 = 0 . y 3 3 S u b s t i t u t i o n i n t o t h i s e q u a t i o n o f t h e e x p r e s s i o n s f o r m , T^ and r e s u l t s i n : ~ D ( l - y ) ( l + k ) a 1 : ( J ) + [ | (1+k) a± 4> Cos 6 - k ^ 4> S i n 6] C 3 S i n 6/a 3 - [~ (1+k) a± cf> S i n 6 + k ^ <j> Cos 6] . E I 3 Cos 6/a 3 = 0 C a n c e l <j> and u s e 1 a 3 = | a1 I(1+k)2 + 4 k x 2 ] 2 2 -Cos 6 = ( l + k ) / [ ( l + k ) + 4 k x 2 ] 2 . . . . (2-8) 1 S i n 6 = 2 k 1 / [ ( l + k ) 2 + 4 k ^ ] 2 t h e n t h e e q u a t i o n becomes: Fig. 2-8 20 2 | - ( 2 k 1 ( l + k ) E I 3 + [ 4 k 1 3 - k 1 ( l + k ) ] C 3 / ( l + k ) } . 3/2 = D ( l - y ) [ ( 1 + k ) 2 + 4 k x 2 ] . . . . (2-9) E q u i l i b r i u m o f t h e moments a b o u t t h e x - a x i s a t one o f t h e b o t t o m c o r n e r s y i e l d s : m = T_ + M_ S i n S + T_ Cos6 x 2 3 3 P r o c e e d i n g w i t h t h i s e q u a t i o n as w i t h t h e one b e f o r e g i v e s : 3 2 or o R FIT + r~ { 2 k 1 ( l + k ) E i 3 + [(1+k) - 4 k 1 (1+k)] C 3 / 4 k 1 ) . 2 2 2 «"j / r\ [(1+k) + 4 k x + 4k ] " ' = D (1-y) . . . . (2-10) So f a r two r e l a t i o n s ( E q u a t i o n s (2-9) and (2-10)) between t h r e e p a r a m e t e r s E I 3 , C 3 and = have been o b t a i n e d . The a d d i t i o n a l e q u a t i o n s w i l l be d e r i v e d f r o m t h e two r e m a i n i n g d e f o r m a t i o n c o n d i t i o n s . 2-2.2 C o n d i t i o n o f P u r e B e n d i n g i n X - D i r e c t i o n The t r a p e z o i d a l a r e a o f t h e p l a t e i s b e n t u n i f o r m l y a b o u t t h e y - a x i s , w h i l e r e s t r a i n e d f r o m b e n d i n g a b o u t t h e x - a x i s . To a c c o m p l i s h t h i s , a u n i f o r m moment M i s a p p l i e d on t h e x - p l a n e and yM on t h e y - p l a n e . F i g u r e 2-9 shows t h e d e f o r m e d e l e m e n t . The a n g l e d e n o t e s t h e a n g l e s o f r o t a t i o n o f t h e ends o f t h e edge (3-4) i n r e l a t i o n t o i t s m i d d l e p o i n t , and ^ 2 s i m i l a r r o t a t i o n s o f t h e nodes 1 and 2. The c u r v a t u r e i n x - d i r e c -t i o n may be e x p r e s s e d i n t e r m s o f t h e s e a n g l e s as f o l l o w s : r x a± ka± From t h i s e q u a t i o n ^ = k\J;^  The moments M and M p e r u n i t l e n g t h o f t h e s e c t i o n x y f o u n d by t h e m o m e n t - c u r v a t u r e r e l a t i o n s h i p : M = D/r = 2DiK/a-, x x rr 1 y 1 1 As b e f o r e , t h e s i d e s o f t h e t r a p e z o i d a l a r e a a r e r e p l a c e d by t h e z i g - z a g l i n e s , and t h e edge moments a r e t r a n s f e r r e d t o t h e a d j a c e n t c o r n e r s i n t h e f o r m o f c o n c e n -t r a t i o n s m , m ( F i g u r e 2 - 10b). x y m x = I u ( a l + a 2 ) M x = 1 PD(l+k) ty1 m = ^ H h = D k . i . y 2 x l r l The d e f o r m e d shape o f t h e c e l l i n t h i s c a s e i s s y m m e t r i c a l a b o u t t h e y - a x i s , h e n c e t h e member (5-6) o f t h e s e c o n d a r y s t r u c t u r e i s i n a c t i v e . A g a i n , t h e s t i f f n e s s e s o f t h e p a r a l l e l members o f t h e two s t r u c t u r e s a r e lumped t o g e t h e r as i n F i g u r e 2-6. The members ( 3 - 4 ) , (1-2) o f t h e c e l l ( F i g u r e 2-10c) a r e b e n t w i t h o u t t w i s t a b o u t t h e y - a x i s , and t h e i r b e n d i n g moments a r e r e s p e c t i v e l y : M± = 2xpi E I 1 / a 1 M 2 = 2 * 2 E I 2 / a 2 From F i g u r e 2-9 i t may be s e e n t h a t t h e c o r n e r 1 r o t a t e s i n r e l a t i o n t o c o r n e r 4 t h r o u g h an a n g l e ( i ^ + i j ^ ) . The components o f t h i s r o t a t i o n p a r a l l e l and p e r p e n d i c u l a r t o t h e d i a g o n a l (1-4) g i v e r e s p e c t i v e l y t h e a n g l e o f t w i s t and t h e a n g l e o f b e n d i n g o f t h i s member as shown i n F i g u r e 2-11. The b e n d i n g moment i n t h e b a r (1-4) i s : M 3 = +\\> ) Cos 6 • E I 3 / a 3 and t h e t o r q u e : T 3 =. (* 1+* 2) S i n <5 • C 3 / a 3 The e x t e r n a l moments m , m and t h e b a r moments x' y M and T a c t i n g a t a l l c o r n e r s o f t h e c e l l a r e p r e s e n t e d i n F i g u r e 2-12. } . . . . (2-11) = 2 ^ E I 2 / a F i g . 2 - 12 The e q u a t i o n s o f e q u i l i b r i u m r e l a t i n g t h e s e moments a r e i d e n t i c a l a t a l l f o u r c o r n e r s , and by c o m p a r i s o n o f t h e t o p and b o t t o m c o r n e r s i t may be s e e n t h a t = M^. On c o n s i d e r a t i o n o f t h e e q u a t i o n s ( 2 - 1 1 ) i t f o l l o w s t h a t E I 1 = E I 2 From t h e moment e q u i l i b r i u m o f one o f t h e c o r n e r s a b o u t t h e x - a x i s : m = M_ S i n 6 - T_) Gos 6 o r ~ y D f l + k ) ^ = (^ 1+^ 2) E I 3 S i n 6 ° C o s 6 / a 3 -( ^ 1 + I J J 2 ) C 3 S i n 6 - C o s 6 / a 3 S i n c e (ij^+ij^) = (1+k ) ip^, t h e above e q u a t i o n r e d u c e s t o E l " 3 - C 3 = y D a 3 / S i n 26 by E q u a t i o n ( 2 - 8 ) : 8 y D a 1 E I ^ — = - • ~t 3/2* " * (2-*12) k ]_(l+k) [ ( 1 + k ) 2 + 4 k x 2 ] E q u i l i b r i u m o f moments i n y - d i r e c t i o n a t anyone o f t h e c o r n e r s g i v e s : m = M_ + T, S i n 6 + Cos 6 y 2 3 3 A f t e r s u b s t i t u t i o n o f a p p r o p r i a t e v a l u e s , t h e above 26 equation becomes: 3/2 2 (1+k) [EI_(1+k)2+ 4 C 3 k 1 2 ] = (Dk 1a 1 - 2EI2> [(1+k)2+ 4 ^ ] (2-13) The equations (2-12) and (2-13) combined with the equations (2-9) and (2-10) make up a system of four equations containing four parameters: EI^ = EI,, , = C 2 , C 3 and E I 3 - These equations are assembled below for the ease of reference: | - (2k 1(l+k) E l 3 +[4k 1 3- k 1(l+ k ) 2 ] C 3/(l+ k ) } 3/2 = D ( l - y ) [ (l+k) 2+ 4k x 2 ] . . (2-9) 2C F T " + i ~ { 2 k i ( 1 + k ) E I 3 + t ( l + k ) 3 - 4k 1 2(l+k) ] C 3 /4k 1 >. -3/2 I(l+k) 2+ 4k x 2 ] = D ( l - y ) . (2-10) SyDa.^  E I 3 - C 3 = 3/2 • • (2"1 2) k x (1+k) [ (1+k) Z+ 4k 1 / i] 3/2 2(1+k)[EI3(1+k)2+ 4 C 3 k 1 2 ] = (Dk 1a 1 - 2EI 2)[(1+k) 2+ 4 ^ ] . . . . (2-13) The equations (2-12) and (2-9) are used to determine and E I ^ : 2 2 1 / 2 C 3 = D ( l - 3 y ) (1+k) [ ( l + k r + 4 k 1 / ] a_/8k . . . (2-14) and ( l - 2 y ) ( l + k ) 2 + 4yk 2 1/2 EI = D — [ ( l + k ) ^ + 4 k / ] a, . (2-15) 8k (1+k) 1 1 With C 3 and E I ^ known, the e x p r e s s i o n s f o r = and E I 1 = E I 2 are found from Equations (2-10) and ( 2 - 1 3 ) : C l = C 2 = ^ (1—3y) [ 4 k ^ 2 - ( 1 + k ) 2 ] a 1 / 8 k 1 . . . . (2-16) E I X = E I 2 = D [ 4 k 1 2 - ( l - 2 y ) (1+k) 2] a 1 / 8 k 1 . . . . (2-17) 2-2.3 C o n d i t i o n of Pure Bending i n Y - D i r e c t i o n . In t h i s c o n d i t i o n , the c e l l i s bent about the x-a x i s and i s r e s t r a i n e d from bending about the y - a x i s . Uniform moment i s a p p l i e d on the y-plane and yM^ on the x-plane. R e f e r r i n g . t o F i g u r e 2-13, l e t be the angle of r o t a t i o n of the edge (1-2) i n r e l a t i o n to the edge ( 3 - 4 ) . In terms of t h i s angle, the c u r v a t u r e of the p l a t e i n y-d i r e c t i o n i s : F i g . 2 - 14 29 __ _ _____ - ___ r y k 1 a 1 • D Hence M = D * / k ; j a 1 . . . . (2-18) The c o r n e r moments m^, m^ a r e a g a i n f o u n d by t r a n s f e r r i n g t h e edge moments t o t h e a d j a c e n t nodes a f t e r r e p l a c i n g t h e s l o p i n g s i d e s by t h e z i g - z a g l i n e s . m x = My ( a 1 + a 2 ) / 4 = D ( l + k ) * / 4 k 1 } . . . . (2-19) m y = yM yh/2 = yD*/2 The b a r - c e l l i s now s u b j e c t e d t o t h e s e moments m J x and rriy, and i t s d e f o r m a t i o n , i n a g r e e m e n t w i t h t h a t o f t h e e l e m e n t o f t h e p l a t e s h o u l d be as shown i n F i g u r e 2-14. The members (1-2) and (3-4) o f t h e p r i m a r y s t r u c -t u r e a r e n o t s t r e s s e d , whereas t h e d i a g o n a l members a r e s u b j e c t e d t o b e n d i n g and t w i s t . The a n g l e * o f r o t a t i o n o f t h e c o r n e r 1 i n r e l a t i o n t o t h e c o r n e r 4 i s shown i n F i g u r e 2-15. The f l e x u r a l component o f t h i s a n g l e i s * S i n 6, and t h e t o r s i o n a l component IJJ Cos 6. The b e n d i n g moment M^ and t h e t o r q u e T^ i n t h e member (1-4) , as w e l l as i n t h e member (2-3) a r e : M 3 = E I - * S i n 6 / a 3 T 3 = C 3 * Cos 6/a 3 S u b s t i t u t i n g t h e v a l u e s o f E I ^ and f r o m t h e E q u a t i o n s (1-15) and (2-14) i n t o t h e above e x p r e s s i o n s : ( l - 2 y ) ( l + k ) 2 + 4 y k 2 M = D — S i n 6 . . . . (2-20) J 4 k 1 ( l + k ) T 3 = D ( l - 3 y ) (1+k) if; Cos 6/4k 1 . . . . (2-21) The c o r n e r moments m and m a r e b a l a n c e d by x y J s t r e s s e s i n t h e members o f t h e p r i m a r y and t h e s e c o n d a r y s t r u c t u r e s ; and t h e b e n d i n g moments and t o r q u e s i n t h e s e c o n d a r y members may be f o u n d by s u b t r a c t i o n f r o m t h e c o r n e r mements t h e s t r e s s c o n t r i b u t i o n f r o m t h e p r i m a r y s t r u c t u r e . I n v i e w o f t h e symmetry o f t h e c e l l and t h e • l o a d i n g , t h e e q u i l i b r i u m c o n d i t i o n s a t a l l f o u r c o r n e r s i n F i g u r e 2.16 a r e i d e n t i c a l . From t h i s i t f o l l o w s t h a t t h e b e n d i n g moments i n t h e b a r s ( 1 - 5 ) , ( 2 - 5 ) , (3-6) and (4-6) o f t h e s e c o n d a r y s t r u c t u r e must be e q u a l and so must be t h e t o r q u e s i n t h o s e b a r s . C o n s i d e r t h e c o r n e r 4. I t s e q u i l i b r i u m o f moments i n t h e y - d i r e c t i p n r e q u i r e s t h e b e n d i n g moment i n t h e s e c o n d a r y b a r (4-6) t o be: m - 1VL, Cos 6 + T n S i n 6 y 3 3 S u b s t i t u t i n g t h e v a l u e o f m^ f r o m E q u a t i o n (2-19) and t h o s e o f M., and T f r o m t h e E q u a t i o n s (2-20) , (2-21) , t h i s e x p r e s -F i g . 2 - 15 F i g . 2 - 1 6 T 6 T l=2T = 2T — » — T T F i g . 2 - 1 7 s i o n may be shown t o r e d u c e t o z e r o . T h i s means t h a t t h e members (1-2) and (3-4) o f t h e s e c o n d a r y s t r u c t u r e s h o u l d n o t d e v e l o p any b e n d i n g moments. T h i s r e s u l t a g r e e s w i t h t h e B e t t i ' s r e c i p r o c a l t h e o r e m as w i l l be e x p l a i n e d l a t e r . C a l l T t h e t o r t i o n a l moment i n t h e s e c o n d a r y b a r s ( 1 - 5 ) , ( 2 - 5 ) , (3-6) and ( 4 - 6 ) , assumed t o a c t i n t h e d i r e c t i o n s shown i n F i g u r e 2-16. The v a l u e o f T d i c t a t e d by t h e e q u i l i b r i u m o f t h e c o r n e r o f t h e c e l l i s : T = m - M_ S i n 6 - T_ Cos 6 x 3 3 3 ( l + k ) 2 - 4 k 2 = Dy = — 1// . . . . (2-22) 4 k 1 ( l + k ) The t o r s i o n a l moments T i n t h e members (1-2) and (3-4) o f t h e s e c o n d a r y s t r u c t u r e a c t upon t h e member (5-6) and b end i t u n i f o r m i l y t o t h e c u r v a t u r e ty'/h ( F i g u r e 2.17) where ' i s t h e a n g l e o f r o t a t i o n o f t h e node 5 i n r e l a t i o n t o t h e node 6 a b o u t t h e x - a x i s . I t i s d e s i r a b l e t o d i v i d e t h i s a n g l e 1 e q u a l l y between t h e nodes 5 and 6, as shown i n t h e f i g u r e . T h i s makes t h e a n g l e s o f t w i s t o f t h e f o u r members ( 4 - 6 ) , ( 6 - 3 ) , (2-5) and (5-1) e q u a l , and s i n c e t h e t o r q u e s T i n t h e s e members a r e a l s o e q u a l , t h e i r t o r s i o n a l r i g i d i t i e s must be d i r e c t l y p r o p o r t i o n a l t o t h e i r l e n g t h s : With the absence of the bending moments in the members (1-5) , (2-5) , (3-6) and (4-6) , i t is legitimate to assume their flexural r i g id i t i e s to be zero: I, I- 0 . ls 2S The flexural moment M and the torsional moment T in the secondary members may be written in terms of the member stiffnesses as follows: C l s(ij>-r )/2 T - _____ = c ( W ) / a a-/2 M = 2T = E I 4 i p ' / k 1 a 1 Then: (*-*•) = T a 1 / C l s * ' - 2T kia±/ElA Addition of the last two equations gives Ta, 2Tk na, a .EI, + 2k,a-C, = — I + -_- _ T( 1 4 1 1 ls) ^ C, EI . l K C n EI . ; Is 4 Is 4 Hence: E I 4 T = == — * . . . . (2-24) 4 a l C ~ + 2 k i a i ls Equating the expressions for T in Equation (2-22) and (2-24): 34 E I 4 v> E I 4 a i c7~ + 2 k i a i 3 ( 1 + k ) 2 - 4k 2 4k (1+k) (2-25) 'Is S i n c e t h i s e q u a t i o n c o n t a i n s two p a r a m e t e r s E I ^ and and s i n c e no o t h e r i n d e p e n d e n t e q u a t i o n s a r e a v a i l a b l e , one o f t h e q u a n t i t i e s i n v o l v e d may be assumed a r b i t r a r i l y . Such q u a n t i t y may be t h e r a t i o f = E I _ j / C ^ s . U s i n g t h e r a t i o f , t h e t h r e e s t i f f n e s s e s o f t h e s e c o n d a r y members become: 3 ( 1 + k ) 2 - 4k 2 E I . = Dya, — [ f + 2 k 1 ] . . . . (2-27) q X 4 k 1 ( l + k ) 1 3 ( 1 + k ) 2 - 4k 2 . 2k C = Dya, ± - [1 + — ± ] . . . . (2.28) x s 4 k 1 ( l + k ) f 3 ( 1 + k ) 2 - 4k 2 2k C = Dyka : ~ [1 + — i ] . . . . (2-29) S 4k n (1+k) f The s e c o n d a r y s y s t e m may be f u r t h e r s i m p l i f i e d by t h e a s s u m p t i o n o f a p a r t i c u l a r v a l u e o f t h e r a t i o f . I t i s c o n v e n i e n t t o assume f as i n f i n i t y , w h i l e k e e p i n g t h e t o r s i o n s t i f f n e s s e s C. and C„ f i n i t e . I n o t h e r words I s 2s E I . i s i n f i n i t e , and 35 3(1+k)2- 4k 2 C = Dya.. — — . . . . (2-28a) x s x 4k1(l+k) 3(1+k)2- 4k 2 C = Dyka ± - . . . . (2-29a) A S X 4kx(l+k) With in f in i te ly r ig id flexural cross member (5-6) the rotation ^ of the nodes of the secondary system are accommodated solely by the twist of the base bars. The c e l l so restricted s t i l l remains fu l ly effective in i t s f lexural action in spite of the member (5-6) staying always straight. It may be necessary at times to invert the c e l l so that the bigger base is located on the positive side of y-axis. The same equations for the bar stiffnesses would hold but the significance of the geometrical characteris-t ics of the c e l l would change, while the value f w i l l s t i l l remain in f in i t e . Providing the symbols in the inverted c e l l with the prime signs: „ O T 7 - - j . . : - V i _ small base _ a l _ 1 new ratio K = :—: , = = = r-big base ka^ k 36 .. . , h e i g h t h k l new r a t i o k n 1 = —. f—— = : - ;— 1 b i g b a s e ka-^ k new r e f e r e n c e b a s e , d e s c r i b e d b e f o r e as a ^ i s now a\ = k a ^ . W i t h i n t r o d u c t i o n o f t h e new s y m b o l s f o r t h e i n v e r t e d c e l l i t may be o b s e r v e d t h a t t h e q u a n t i t i e s 1^ = I - , = C_, and I - r e m a i n t h e same; and t h e s t i f f n e s s e s o f t h e new s e c o n d a r y b a s e b a r s , C' and C' become C' = C_ and I s 2s l s 2s C i . s = _ l s - t h e c o n s e q u e n c e o f t h e c e l l r e v e r s a l . T h i s i s p o s s i b l e due t o t h e a b s e n c e o f f i n t h e e q u a t i o n s (2-28a) and ( 2 - 2 9 a ) . I n summary, t h e e x p r e s s i o n s f o r t h e c e l l p a r a m e t e r s a r e s t a t e d i n T a b l e ( I I - l ) . 2-3 T r a p e z o i d a l and R e c t a n g u l a r F l e x u r a l B a r C e l l s I t i s a p p r o p r i a t e a t t h i s s t a g e t o c o r r e l a t e more c l o s e l y t h e a c t u a l number o f t h e c e l l p a r a m e t e r s w i t h t h e number o f e q u a t i o n s o f s t a t i c s a v a i l a b l e f o r t h e i r d e t e r m i n a t i o n and a t t h e same t i m e make a c o m p a r i s o n o f t h e t r a p e z o i d a l c e l l w i t h H r e n n i k o f f ' s r e c t a n g u l a r b a r c e l l . The t r a p e z o i d a l b a r c e l l i s u n s y m m e t r i c a l a b o u t t h e x - a x i s and s y m m e t r i c a l a b o u t t h e y - a x i s . I t s s t r u c t u r e 37 T a b l e ( I I - l ) P a r a m e t e r s f o r F l e x u r a l T r a p e z o i d a l B a r - C e l l Mi 4 C IS » I I S = 0 3 6 14 C=0 5 2 C 2 S , I 2 S = 0 I D a l 2 2 E I 1 = E I 2 = E I l p = E I 2 p = ~ t [ 4 k 1 z - ( l - 2 y ) (1+ k T ] Da (C. +C. ) =.C, = (C„ +C„ ) = l p I s 2 2p 2s 8k - ( l - 3 y ) [ 4 k 12 - ( 1 + k ) 2 ] D a l 2 2 2 2 1 / 2 3 8k U+k) H1-2JJ) ( l + k r + 4yk_/] [ ( l + k ) Z + 4 k / ] EI., = Da 1/2 C 3 = g — ( l - 3 y ) (1+k) [(1+kr + 4k/] E I 4 = D y a x C, = Dya. I s '2s Dyka. 3 ( 1 + k ) 2 - 4 k 1 2 4 k x (1+k) 3 ( 1 + k ) 2 - 4 k x 2 4 k x ( 1 + k ) 3 ( 1 + k ) 2 - 4 k x 2 •4k 1 (1+k) ( f + 2 k x ) 2k n (1+ — ± ) f 2k (1+ — f C l p ~ C l C l s f : f r e e p a r a m e t e r C 2 p C 2 C 2 s 38 c o n s i s t s o f two s u p e r i m p o s e d framework s y s t e m s . The p r i m a r y s y s t e m works i n a l l t h r e e u n i f o r m s t r e s s c o n d i t i o n s , w h e r e a s t h e s e c o n d a r y s y s t e m o n l y i n t h e u n i f o r m f l e x u r e a b o u t t h e x - a x i s . The d i a g o n a l b a r s i n t h e p r i m a r y s y s t e m meet t h e t o p b a s e members a t t h e same a n g l e , r e s e m b l i n g i n t h i s t h e r e c t a n g u l a r c e l l , b u t d i f f e r e n t f r o m i t i n t h e u n e q u a l l e n g t h s o f t h e two b a s e s . In e v e r y o n e o f t h e t h r e e u n i f o r m s t r e s s c o n d i t i o n s , t h e n o d a l moments and m a r e n u m e r i c a l l y e q u a l a t a l l f o u r c o r n e r s . C o n s i d e r i n g t h e u n i f o r m t o r s i o n a l c o n d i t i o n ( a ) , t h e two e q u a t i o n s o f e q u i l i b r i u m o f moments i n x and y d i r e c t i o n s may be w r i t t e n f o r t h e t o p n o d e s , and s i n c e t h e two b a s e members do n o t c a r r y any b e n d i n g moments, o n l y one d i s t i n c t e q u a t i o n EM = 0 i s a v a i l a b l e f o r t h e b o t t o m n o d e s . Thus t h e c o n d i t i o n (a) g e n e r a t e s t h r e e e q u a t i o n s f o r d e t e r m i n -a t i o n o f t h e b a r s t i f f n e s s e s , w h i l e i n H r e n n i k o f f ' s r e c t a n g -u l a r c e l l t h e r e w o u l d be o n l y two. In t h e f l e x u r a l c o n d i t i o n ( b ) , t h e b a s e members do n o t c a r r y t o r s i o n a l moments. By e q u a t i n g t h e b a r s t r e s s e s w i t h t h e e x t e r n a l n o d a l moments m , m , and r e a s o n -x y ' i n g as above, t h r e e more e q u a t i o n s a r e o b t a i n e d f r o m t h i s c o n d i t i o n . They c o r r e s p o n d t o t h e two e q u a t i o n s i n t h e r e c t a n g u l a r c e l l . One m i g h t e x p e c t t h a t t h r e e more e q u a t i o n s c o u l d a l s o be u t i l i z e d i n t h e s e c o n d f l e x u r a l c o n d i t i o n (c) f r o m t h e e q u i l i b r i u m o f t h e o u t s i d e j o i n t s ; however, one o f t h e s e , namely t h e moment e q u i l i b r i u m a b o u t t h e y - a x i s , i s s a t i s f i e d a u t o m a t i c a l l y by t h e B e t t i ' s r e c i p r o c a l t h e o r e m w r i t t e n down f o r t h e c o n d i t i o n s (b) and ( c ) , t h u s l e a v i n g o n l y two more i n d e p e n d e n t e q u a t i o n s . I n t h e r e c t a n g u l a r c e l l t h e same r e c i p r o c a l t h e o r e m i s a l s o e f f e c t i v e , and t h e number o f a v a i l a b l e e q u a t i o n s i n t h e c o n d i t i o n (c) i s one. Su m m a r i z i n g t h i s d i s c u s s i o n , t h e t o t a l number o f i n d e p e n d e n t e q u a t i o n s i s t h r e e i n t h e s q u a r e c e l l s , f o u r i n t h e r e c t a n g u l a r ones and e i g h t i n t h e t r a p e z o i d a l f o r f i n d i n g t h e n i n e p a r a m e t e r s EI^, , E I - , c n ' C 2 ' <"3' E I 4 ' C^ and The d e f i c i e n c y o f one e q u a t i o n i s made up by t h e a s s u m p t i o n o f an i n f i n i t e v a l u e f o r E I ^ ( o r t h e r a t i o f ) . 2-4 D i s t r i b u t i o n F a c t o r s o f t h e I s o s c e l e s T r a p e z o i d a l  B a r C e l l When t h e framework method was f i r s t i n t r o d u c e d by H r e n n i k o f f , c o m p u t e r s were n o t i n e x i s t e n c e and t h e s o l u t i o n o f t h e framework model was e f f e c t e d by a l e n g t h y p r o c e d u r e o f r e l a x a t i o n . The j o i n t s were moved one a f t e r t h e o t h e r i n n u m e r a b l e number o f t i m e s i n o r d e r t o e f f e c t t h e i r e q u i l i b r i u m . W i t h c o m p u t e r s a v a i l a b l e , t h e r o t a t i o n s and d i s p l a c e m e n t s o f a l l j o i n t s , and f o l l o w i n g t h a t , t h e c o r n e r 40 r e a c t i o n s between t h e c e l l s , may be f o u n d e a s i l y m a k i n g use o f t h e s t i f f n e s s m a t r i x o f t h e c e l l . A f l e x u r a l e l e m e n t p o s s e s s e s t h r e e d e g r e e s o f f r e e d o m a t e a c h node: t h e d i s p l a c e m e n t p e r p e n d i c u l a r t o t h e c e l l and t h e r o t a t i o n s a b o u t t h e x and y a x e s . C o n v e n -i e n c e o f s o l u t i o n r e q u i r e s t h e knowledge o f t h e n o d a l f o r c e s c o r r e s p o n d i n g t o u n i t d i s p l a c e m e n t s o f t h e n o d e s . T h e s e f o r c e s a r e r e f e r r e d t o as t h e d i s t r i b u t i o n f a c t o r s and t h e y compose t h e s t i f f n e s s m a t r i x o f t h e c e l l . 2-4.1 D i s t r i b u t i o n F a c t o r s Due t o An ( F i g u r e 2-18) The node 1 i s d i s p l a c e d a d i s t a n c e i n t h e p o s i t i v e d i r e c t i o n o f t h e z - a x i s w i t h o u t r o t a t i o n w h i l e t h e o t h e r nodes..and t h e ends o f t h e members m e e t i n g t h e r e a r e r e s t r a i n e d f r o m any movement. The i n t e r i o r j o i n t s a r e a l l o w e d t o d i s p l a c e f r e e l y t o t h e i r p o s i t i o n s o f e q u i l i b r i u m . The s t i f f n e s s m a t r i x e l e m e n t s r e p r e s e n t t h e c o r n e r f o r c e s and moments h o l d i n g t h e c e l l i n t h e d e f o r m e d s t a t e . T h ey a r e made up o f t h e f o r c e s on t h e ends o f t h e b a r s m e e t i n g a t t h e c o r n e r i n q u e s t i o n . R e f e r r i n g t o F i g u r e 2-18, t h e member (3-4) o f t h e p r i m a r y s t r u c t u r e r e m a i n s s t r a i g h t and u n t w i s t e d a f t e r t h e d e f o r m a t i o n , hence i t i s n o t s t r e s s e d . S i n c e t h e member (1-2) o f t h e s e c o n d a r y s t r u c t u r e i s d e v o i d o f f l e x u r a l s t i f f -n e s s e s , s h e a r and b e n d i n g moment c a n n o t be t r a n s m i t t e d a l o n g i t . T h i s l e a v e s t h e node 5 u n d i s p l a c e d and t h e whole s e c o n d a r y s t r u c t u r e i n a c t i v e i n t h i s d e f o r m a t i o n s t a t e . The b e n d i n g moment M 2 and t h e r e a c t i o n F 2 on t h e ends o f t h e p r i m a r y member (1-2) a r e r e l a t e d t o t h e end d e f l e c t i o n A^ as f o l l o w s : ( F i g u r e 2-19) 6 E I 2 2 2 A l M_ - — ~ A- = 3 D ( l - 3 y ) [4k- - ( 1 - k p ] a 2 4k^k 12EI, F 2 = 6D ( l - 3 y ) [ 4 k 12 - ( 1 + k ) 2 ] 4 k 1 k 3 a - 2 S i m i l a r l y f o r t h e d i a g o n a l member ( 1 - 4 ) : 6 E I - 3 D (l-2u) (l+k)2+ 4 y k . 2 M3 = "T7Al = ST [(l+k)2+ 4 k . 2 ] V 2 k i ( i + k ) A l 1 2 E I 3 A 1 1 2 D ( l - 2 y ) ( l + k ) 2 + 4 y k x 2 p - _ - j 5 2 A l J a 3 J a x z [ ( l + k ) ^ + 4 k - z ] k.^  (1+k) The b e n d i n g moments M 3 on t h e ends o f t h i s d i a g o n a l a r e r e s o l v e d i n t o t h e components p a r a l l e l t o t h e c o o r d i n a t e F i g . 2 - 1 9 a x e s , and t h e s e a r e combined w i t h t h e v a l u e s o f M 2 and F 2 » T h e s e c o m b i n a t i o n s w h i l e l e t t i n g A^ e q u a l u n i t y become t h e t e r m s o f t h e f i r s t column o f t h e s t i f f n e s s m a t r i x . The d i s t r i b u t i o n f a c t o r s due t o A^ t o g e t h e r w i t h t h e ones c o r r e s p o n d i n g t o t h e u n i t r o t a t i o n s o f t h e node 1 a r e a s s e m b l e d i n T a b l e ( I I - 2 ) . I n t h i s t a b l e , t h e c o r n e r f o r c e s a r e d e s i g n a t e d by t h e symbol z, and t h e c o r n e r moments by m o r nr d e p e n d i n g on t h e a x i s a b o u t w h i c h t h e y t e n d t o r o t a t e t h e node. F o r c e s and moments a r e p o s i t i v e i f t h e i r v e c t o r s p o i n t i n . t h e p o s i t i v e d i r e c t i o n s o f t h e a x e s . Two s u b s c r i p t s a r e u s e d i n e a c h d i s t r i b u t i o n f a c t o r , t h e f i r s t i n d i c a t i n g t h e c o r n e r , and t h e s e c o n d d e s c r i b i n g t h e a c t i o n -by a number. The number 1 i s a r b i t r a r i l y a s s i g n e d t o t h e t r a n s v e r s e movement o f t h e c o r n e r 1. R o t a t i o n s o f t h e c o r n e r 1 a b o u t t h e a x e s x and y, t h e s u b j e c t o f t h e s u b s e -q u e n t d e r i v a t i o n s , w i l l be d e s i g n a t e d t h e A c t i o n 2 and 3 r e s p e c t i v e l y . 2-4.2 D i s t r i b u t i o n F a c t o r Due t o 6^ = 1 - A c t i o n 2 The node 1 i s r o t a t e d i n p o s i t i v e d i r e c t i o n a b o u t t h e x - a x i s t h r o u g h an a n g l e 8^, w h i l e t h e o t h e r nodes r e m a i n f i x e d . The b e h a v i o u r o f t h e whole c e l l i n t h i s a c t i o n c a n be b e s t e x p l a i n e d by c o n s i d e r i n g t h e a c t i o n as a s u p e r -44 Fig . 2 - 2 0 p o s i t i o n o f t h e t h r e e d i s p l a c e m e n t modes shown i n F i g u r e 2-20. I n e a c h d i s p l a c e m e n t mode, s e v e r a l members a r e i n a c t i v e and so t h e y a r e shown d o t t e d . The d i s p l a c e m e n t mode A may be r e c o g n i z e d t o be t h e same as t h e d e f o r m a t i o n c o n d i t i o n c w i t h t h e b a s e r o t a t i o n s r e d u c e d t o 0^/4. The two b a s e members o f t h e p r i m a r y s y s t e m a r e n o t w o r k i n g . I n t h e d i s p l a c e m e n t mode B t h e p r i m a r y b a s e members a r e a g a i n i n a c t i v e f o r t h e same r e a s o n as i n t h e p r e v i o u s mode, and so i s t h e c o m p l e t e s e c o n d a r y s y s t e m b e c a u s e i t s b a s e members a r e i n c a p a b l e o f r e s i s t i n g s h e a r s w h i c h w o u l d be imposed on them by t h e a n t i s y m m e t r i c a l l y d e f o r m e d c r o s s member ( 5 - 6 ) . In d i s p l a c e m e n t mode C t h e b a s e members (3-4) , b o t h p r i m a r y and s e c o n d a r y , and t h e c r o s s member (5-6) r e -main i n a c t i v e f o r t h e same r e a s o n as i n t h e d i s p l a c e m e n t mode B, w h i l e b o t h members (1-2) work. The d i a g o n a l members a r e a c t i v e i n a l l d i s p l a c e m e n t modes, a l t h o u g h t h e combined e f f e c t i n t h e member (2-3) i s z e r o . The e a s i e s t way t o a n a l y z e t h e s t r e s s e s i n t h e member (1-4) i s by c o n -s i d e r i n g i t s t o t a l a c t i o n w i t h o u t s e p a r a t i n g i t i n t o t h e component c a s e s . F i g u r e 2-21 r e p r e s e n t s t h e r e s u l t a n t d e f o r m a t i o n s o f t h e members o f t h e c e l l i n t h e A c t i o n 2. The angle of flexure at the end 1 of the diagonal member (1-4) is 8^  Sin6 and the bending moment at this end i s : (Figure 2-22) 4EI-9* Sin6 (l-2y) (1+k)2+ 4yk, 2 M - . = = —- 2 D 6 X L i 2 9 1/2 *" U X )\ a 3 (1+k) [ (l+k)^+ 4 k . Z ] ± / Z 1 On the other end of the member, the bending moment is only half as great. The end shear: 6 E I 3 e £ _ S i n 6 (l-2y) ( l+k) 2+ 4yk. 2 g D x  3 a 3 (1+k) [ (1+k) 2+ 4k. 2] a l 1 The angle of twist in the diagonal (1-4) is x 9^ Cos6, and i t s torque i s : C 9* CosS (l-3y) (1+k)2 rp _ •- 1 _ D9^ 3 a 3 4k.[(l+k) 2+ 4 k 1 2 ] 1 / 2 1 Similarly, the torque in the base members (1-2) of Figure 2-21a i s : T- - CJ^~ = Sk^L [ 4 k 2 - (1+k)2] D9* 1 a2 ' 8kk n 1 1 The torsional moments T in the secondary members (Figure 2-21b) may be written direct ly from Equation (2-22) 47 Fig. 2 - 2 2 u s i n g t h e a n g l e i n p l a c e o f 3 ( 1 + k ) 2 - 4 k x 2 x 8k n (1+k) y D 6 1 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 a r e f o u n d by c o m b i n i n g t h e x, y and z components o f t h e end f o r c e s i n t h e members m e e t i n g a t t h e nodes and a s s u m i n g t h e c o r n e r r o t a t i o n 0^ t o be u n i t y . As an example t h e moment a t c o r n e r 1 a b o u t t h e x - a x i s , i . e . , t h e f a c t o r 1 S : m l 2 = T 2 + M 3 S i n 6 + T 3 C o s & + T n - v P y ( l - 2 y ) (l+k) 2+ 4 u k n 2 = { l i _ _ y _ l [ 4 k <L (i+k) 2] + \- . 4k, 8 k k x x (1+k) t (1+k)z+ 4 k / ] 1 4 k 1 t (l+k) 2+ 4 k x 2 ] 8 ^ (1+k) T h e s e d i s t r i b u t i o n f a c t o r s o c c u p y t h e s e c o n d column o f t h e s t i f f n e s s m a t r i x . They a r e a s s e m b l e d i n T a b l e ( I I - 2 ) . 2-4.3 D i s t r i b u t i o n F a c t o r s Due t o 6^ = 1 - A c t i o n 3 In t h i s a c t i o n , t h e j o i n t 1 i s r o t a t e d t h r o u g h a p o s i t i v e a n g l e .6^ a b o u t t h e y - a x i s , w h i l e a l l o t h e r nodes a r e r e s t r a i n e d f r o m movement. The s e c o n d a r y s y s t e m a g a i n does n o t work due t o i t s i n a b i l i t y t o c a r r y s h e a r s f r o m 49 th e node r o t a t i o n ( F i g u r e 2 - 2 3 ) . The d i s t r i b u t i o n f a c t o r s a r e a g a i n made up o f f o r c e s i n t h e members o f t h e p r i m a r y s t r u c t u r e . U s i n g t h e same p r o c e d u r e as b e f o r e , t h e moment M - on t h e end 1 o f member (1-4) i s : 4EI 6 Y - o n M 2 " — r ^ - t 4 k i - u - 2 ^ <1+k> i n £ - e l • • M The moment on t h e o t h e r end o f t h e same member i s , and t h e end s h e a r : F- = [ 4 k 2 - ( l - 2 y ) ( 1 + k ) 2 ] ~ - 9 Y 4k^k a^ The b e n d i n g moment M _ , s h e a r F_, and t o r s i o n a l moment T- a t t h e end 1 o f t h e d i a g o n a l member (1-4) a r e as f o l l o w s : ( F i g u r e 2-24) 4E I - Cos6 ( l - 2 y ) ( 1 + k ) 2 + 4yk 2 M = —— eY = ; - - 1 . D e Y J a 3 1 k 1 I ( l + k ) ^ + 4k ^ ] X / Z 1 ( l - 2 y ) ( l + k ) 2 + 4yk 2 F = — . 3 k [ ( l + k ) 2 + 4k- 2] 2 a l 1 _ C 3 Sina _ (1-3P) (1+k)  3 " a 3 1 " [ < l + k ) 2 + 4 k 1 2 ] 1 / 2 2 9 1 Fig. 2 - 2 4 51 The s h e a r and t o r q u e on t h e end 4 o f t h e d i a g o n a l a r e t h e same as on t h e end 1, and t h e b e n d i n g moment h a l f as g r e a t . The d i s t r i b u t i o n f a c t o r s a r e f o u n d i n t h e u s u a l way and a r e a s s e m b l e d i n t h e T a b l e ( I I - 2 ) , t h e y f o r m t h e t h i r d column o f t h e s t i f f n e s s m a t r i x . 2-5 S t i f f n e s s M a t r i x o f I s o s c e l e s T r a p e z o i d a l B a r C e l l T a b l e ( I I - 2 ) c o n t a i n s d a t a f o r t h e f i r s t t h r e e c olumns o f t h e (12x12) s t i f f n e s s m a t r i x [ K ] . The columns 4, 5, 6 c o r r e s p o n d i n g t o n o d a l f o r c e s , b r o u g h t a b o u t by u n i t d i s p l a c e m e n t s o f t h e c o r n e r 2, may be f o u n d i n a s i m i l a r way, b u t i t i s much e a s i e r t o d e t e r m i n e them f r o m t h e c o n -s i d e r a t i o n s o f symmetry. The method i s i l l u s t r a t e d on t h e example o f t h e d i s t r i b u t i o n f a c t o r s o f t h e f o u r t h column o f t h e m a t r i x p r o d u c e d by u n i t t r a n s v e r s e d i s p l a c e m e n t 4^ = 1 o f t h e node 2. The c e l l i n F i g u r e 2-25a ( d e f o r m a t i o n mode o f A c t i o n 1) i s r o t a t e d t h r o u g h 180° a b o u t t h e y - a x i s as shown i n F i g u r e 2-25b. S i n c e t h e d i s p l a c e m e n t i n F i g u r e 2-29b i s n e g a t i v e , i t i s f u r t h e r made p o s i t i v e i n F i g u r e 2-25c w i t h r e v e r s a l o f d i r e c t i o n s o f a l l c o r n e r a r r o w s . 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 e d i s p l a c e m e n t A „ F ig . 2 - 25 a r e s t a t e d i n F i g u r e 2.25d. T h e i r e x p r e s s i o n s may be w r i t t e n down by c o m p a r i s o n w i t h F i g u r e 2-25c. They o c c u p y t h e f o u r t h c o l u m n o f t h e s t i f f n e s s m a t r i x . The c o l u m n s 5 and 6 c o r r e s p o n d i n g t o u n i t r o t a t i o n s a t t h e c o r n e r 2 a r e f o u n d i n a s i m i l a r manner. The c o r n e r f o r c e s b r o u g h t a b o u t by u n i t d i s p l a c e m e n t s and r o t a t i o n s o f t h e c o r n e r 3 a r e d i f f e r e n t f r o m t h e ones p r o d u c e d by t h e A c t i o n s 1, 2, 3 and f o r t h i s r e a s o n t h e i r a c t i o n s w i l l be t e rmed l a , 2a and 3a. They may be most c o n v e n i e n t l y o b t a i n e d f r o m t h e a l r e a d y f o u n d e x p r e s s i o n s by r o t a t i n g t h e c e l l a b o u t t h e x - a x i s and m o d i f y i n g t h e r a t i o s k, k^ and l e n g t h a ^ . ( T a b l e I I - 3 ) F i g u r e 2-26 i l l u s t r a t e s t h e p r o c e d u r e o f s u c h c o n v e r s i o n o f t h e e f f e c t s o f t h e movement 0^ ( F i g u r e 2-26a) i n t o 6^ ( F i g u r e 2-26d) t h r o u g h t h e i n t e r m e d i a t e s t e p s o f r o t a t i o n a b o u t t h e a x i s x ( F i g u r e 2-26b) , c h a n g i n g t h e s i g n o f t h e c o r n e r movement ( F i g u r e 2-26c) and f i n a l l y a l l o w i n g f o r t h e i n v e r t e d shape o f t h e c e l l by r e p l a c i n g t h e 1 k l q u a n t i t i e s k by -^ , k^ by ^ — and a by ka.^. F o r example, t h e e x p r e s s i o n f o r m^ _ i s f o u n d as f o l l o w s : ™ Y 3 3 a = m j 3 ( r e p l a c e d ) = { 1 4 ^ * - ( l - 2 y ) ( k + 1 ) 2 ] + (k+1) [ ( l - 2 y ) (k+1) 2+ 4 y k 1 2 ] ( l - 3 y ) k]_ (k+1) — . - + ? ^ L a i k-L U k + i r + 4 k / ] ( k + i r + 4 k / 1 (d) F i g . 2 - 2 6 (c ) The d i s t r i b u t i o n f a c t o r s c o r r e s p o n d i n g t o t h e u n i t movements o f t h e node 4, t h e l a s t t h r e e columns o f t h e s t i f f n e s s m a t r i x , a r e s i m i l a r t o t h e ones p r o d u c e d by t h e movements o f t h e node 3 and a r e o b t a i n e d f r o m them i n t h e way t h e m a t r i x t e r m s i n t h e columns 4, 5 and 6 w<;re o b t a i n e d f r o m t h o s e i n t h e columns 1, 2 and 3. The s t i f f n e s s m a t r i x [K] o f t h e t r a p e z o i d a l b a r c e l l i s p r e s e n t e d i n E q u a t i o n s 2-30. I t i s s y m m e t r i c a l a b o u t t h e p r i n c i p a l d i a g o n a l i n agr e e m e n t w i t h t h e B e t t i ' s r e c i p r o c a l t h e o r e m . The s t i f f n e s s m a t r i x [K] i s t h e c o e f f i c i e n t o f p r o p o r t i o n a l i t y between t h e (12x1) v e c t o r s {f} and {6} o f t h e n o d a l f o r c e s and d i s p l a c e m e n t s i n a c c o r d a n c e w i t h t h e e q u a t i o n : { f } x y z = [ K ] x y z { 6 } x y z • • • • ( 2 " 3 0 a i n w h i c h t h e s u b s c r i p t s x y z i n d i c a t e t h e c o o r d i n a t e s y s t e m o f t h e n o d a l f u n c t i o n s . The t e r m s o f {f} and { 6 } a r e : ( F i g u r e 2-27) { f } x y z = [ z 1 ' m i ' m i / Z 2 ' m 2 ' m 2 ' Z 3 ' m 3 ' m 3 ' Z 4 ' m 4 ' m P { 6 } x y z = Iw^e^e^w^e^ey^^^e^w^e^eJ ] 56 F i g u r e 2-27 [K] = m m m m m in-ra m '11 x 11 Y 11 :21 x 21 y 21 :31 x 31 y . 31 :41 x 41 y 4.1 rn m m m '12 x 12 y 12 :22 x 22 y 22 :32 m m m 32 y 32 :42 x 42 y 42 m m m m '13 .x 13 y 13 :23 x 23 y 23 ;33 m m m x 33 y 33 :43 x 43 m -m m -m '21 x 21 y 21 : n x 11 y n ;41 X 41 y y 43 m -m 41 :31 x 31 Y 31 m '22 x 22 - z 23 _ m 2 2 '12 m -in m 12 y 12 :42 x 42 -mY m -m 42 ;32 x 32 Y 32 -m x 23 y 23 - z -m m 13 x 13 y - z -TO. m - z -m m 13 43 x 43 y 43 33 x 33 y 33 m m m m m m '11a x 11a y 11a ;21a _x 21a y 21a ;31a x 31a Y m m m itv J12a x 12a Y 12a :22a x 22a y 22a :32a 31a nv x 32a y 32a m' m m ITI' '13a x 13a y 13a ;23a x 23a y 23 a-;33a '21a '22a - z 23a m m 33a 33a m -m m -m m -m x 21a y 21a : l l a x 11a y l l a :41a x 41a y 41a m 22a -mY_ 22a '12a m x 12a -mY 12a '42a m x 42a 42a -m x 23a m Y _:3a - z -m m - z -m m 13a x 13a y 13a 43a x 43a y 43a m m '41a x 41a Y 41a m m '42a x 42a Y 42a . m m '43a x 43a Y 43a m -m '31a x 31a Y 31a m -m '32a x 32a 32a -m m '33a x 33a Y 33a (2-30 ) Ln T a b l e I I - 2 D i s t r i b u t i o n F a c t o r s f o r F l e x u r a l T r a p e z o i d a l B a r C e l l (a) A c t i o n 1 1 2 E f d - y ) a . m4| m 4 i f / Z 4 ' u ? y f / 3 1 m 3 l m 21 2 —1» —s*. ^ 4 \ . / 3 Z2I m 21 31 Z „ ( u p ) — « > nn.. m it ( l - 2 y ) ( l + k ) 2 + 4 y k x 2 ( l - 2 y ) ( l + k ) 2 { _ + } . _ '11 [ (1+k) 2+ 4 k 1 2 ] k 1 ( 1 + k ) '21 4 k x 2 - ( l - 2 y ) ( 1 + k ) 2 3 L 2 k 1 k 3 a l z 3 1 = 0 41 ( l - 2 y ) ( l + k ) 2 + 4 y k x 2 1 2 L [ ( 1 + k ) 2 + 4 k 1 2 ] k 1 ( 1 + k ) a l m x ( l - 2 y ) ( l + k ) 2 + 4 y k 1 2 1 1 ( 1 + k ) [ ( l + k ) 2 + 4 k x 2 ] 6L m 2 1 = ° m 3 1 = 0 x ( l - 2 y ) ( l + k ) 2 + 4 y k x 2 m = ^ 2_ . 6L 4 1 (1+k) [ ( l + k T + 4k ^] ( l - 2 y ) ( l + k ) 2 + 4yk 2 4 k , 2 - ( l - 2 y ) ( l + k ) 2 m I l = { 2 2 + 2 } ° k~ 1 1 ( l + k ) z + 4 k / 4 k z K l 4k 2 - ( l - 2 y ) ( 1 +k) 2 m" = « 2 1 2 k l 4 k z x m 3 1 ( l - 2 y ) ( l + k ) 2 + 4 y k x 2 Ift — —.  , — o , 4 1 ( l + k ) 2 + 4 k x 2 k l 60 ( l - 2 y ) ( 1 + k ) 2 + 4 y k - 2 z = ^ j- • 6L x z (1+k) [ (1+k) z+ 4 k / ] Z 2 2 = ° Z 3 2 ( l - 2 y ) (1+k) 2+ 4 y k - 2  z _ . 6 L * z (1+k) [ (1+k) z+ 4 k / ] ( l - 3 y ) ( l - 2 y ) (1+k) 2+ 4yk 2 m * = {- [ 4 k / - ( l + k ) Z ] + - - j A " ° 4 k i + x z 8 k k x x (1+k) [ (1+k) z+ 4 k / ] 1 ( l - 3 y ) ( l + k ) 3 3 ( l + k ) 2 - 4 ^ + ^ 5— + y} • a,L 4k- [ (1+k) z+ 4 k / ] 8 ^ (1+k) x m * - y _ iJ_L3__. [ 4 k 2_ ( 1 + k ) 2 ] } . z z 8 k x ( l + k ) 8 k k x 1 L x 3 ( 1 + k ) 2 - 4 k . 2 m* = - y a-L J Z 8 ^ (1+k) x ( l - 2 y ) ( l + k ) 2 + 4 u k . 2 ( l - 3 y ) ( l + k ) 3  m x = { • 1 . 2 k 4 2 (1+k) [ (1+k) 2+ 4 k x 2 ] 1 4k [ ( 1 + k ) 2 + 4 k 1 2 ] 3 ( 1 + k ) 2 - 4 k 1 2 • y} a.L 8 k 1 (1+k) (l+k)2(3-5y) + leyk^ a^L (1+k)2 + 4 k x 2 2 y y m22 = m32 (l+k) 2(3-7y) + 8 y k 1 2 ah m42 = 2 T~ ' (1+k/ + 4k^ 2 r 1 2 2 ( l " 2 y ) ( l + k ) 2 + 4 u k . 2 _T :13 = {~±2 t 4 k l d - 2 y ) ( l + k ) 2 ] + = - —} • 3L 4k 2 1 ( i + k ) 2 + 4 k 2 k z 9 o = ^-o [ 4 k n 2 - ( l - 2 y ) (1+k) 2] • L / J 4k k-* 1 Z 3 3 ~ 0 z ( l - 2 y ) ( l + k ) 2 + 4 y k x 2 3 L  4 3 = " ( l + k ) 2 + 4 k . 1 ° H ( 1 + k ) 2 ( 3 - 5 y ) + 16yk 2 a.L m. - = — __ . . 1 .. ( 1 + k ) 2 + 4 k x 2 2 m 2 3 = m 3 3 = 0 x ( 1 + k ) 2 ( 3 - 7 y ) + 8yk 2 a L m._ = —- ~ - — — « .... ( l + k ) 2 + 4 k x 2 2 y r r 2 2 1 (1+k) [ ( l - 2 y ) ( l + k ) 2 + 4 y k 2 ] m Y = {[4k. - ( l - 2 y ) ( 1 + k ) 2 ] • — - + _ _ _ X J 1 2kk n . . o o k 1 [ ( 1 + k ) 2 + 4 k . 2 ] 2. I + k. ( l - 3 y ) (1+k) + ^ — • -~— } - a-L ( l + k ) z + ' 4 k . z x m 2 3 = t 4 k i 2 - ( l - 2 y ) ( 1 + k ) 2 ] • — - _ L 4kk. m Y 3 - 0 y _ (1+k) [ ( l - 2 y ) ( l + k ) 2 + 4 y k - 2 ] k± ( l - 3 y ) (1+k) m 4 3 ~ ' ' * 2 2 o T ^ 0 a - i -2 k . [ ( l + k ) z + 4 k / ] ( l + k ) 2 + 4k 2 1 63 (d) Action la L = 12Et" (1-U )a x '21 a '11a '21a (l-2y) (1+k)2+ 4yk 1 2 [ (l+k)2+ 4k 1 2 ]k 1 ( l+k) 12L (l-2y) (l+k)2+ 4yk x 2 4 ^ * - (l-2y)(l+k) 2 { - , - : + ; } . 3 1 a '[(l+k) 2+ 4k 1 2 ]k 1 ( l+k) 8k. '41a 4 k 1 2 - (l-2y) (1+k)2 2kn 3L a n m 11a m 21a (l-2y) (1+k)2+ 4yk x 2 (1+k) [ (1+k)2+ 4k x 2 J 6L x ( l - 2 u ) ( l + k ) 2 + 4yk 2 m = - _ . 6L (1+k) [ ( l + k p + 4 k / ] m X n _ = 0 41a = 0 ' l l a y ( l - 2 y ) ( l + k ) 2 + 4 y k 1 2 3 L 2 1 3 ~ ( l + k ) 2 + 4 k x 2 ° * l m ( l - 2 y ) ( l + k ) 2 + 4 y k n 2 4 k , 2 - ( l - 2 y ) ( 1+k) 2  3 1 a (1+k) 2 + 4k±2 4 *I m 4 1 a = C 4 k l ( l - 2 y ) (1+k 2)] - 3 L (e) A c t i o n 2a z, „ = 0 12a 65 (l-2y) (1+k) 2+ 4uk^ 2 z = — . 6L z z a (1+k) [ ( l + k / + 4 k / ] (l-2u) (1+k) 2+ 4 u k 1 2 z = . 6L J z a (1+k) [ ( l + k p + 4k x] z42a x 3 (1+k) 2- 4 k x 2 ra, ~ = - y a,L X Z a 8 ^ (1+k) X (l-2y) (1+k) 2+ 4 y k / ( l - 3 y ) ( l + k ) 3 mx = { 1 . j v _ _ (1+k) [ (l+k) z+ 4 k / ] 1 4 k 1 [ ( l + k / + 4k H 3(1+k) 2- 4k, 2 _1_ } . L a 8 ^ (1+k) x (l-3y) (l-2y) ( l + k ) 2 + 4 y k / = { [4k / - ( l + k ) Z ] ' + ^ ~ • 4k n + J z a 8k 1 (1+k) [ (1+k) z+ 4k/'] 1 ( l - 3 y ) ( l + k ) 3 3 ( l + k ) 2 - 4 k / + i - } . a , L 4 k x [ ( l + k ) 2 + 4 k x 2 J 8k 1(l+k) 1 3(1+k) 2- 4k, 2 (l-3y) 0 _ mX = {— ±~ y •— [ 4 k / - (1+k/] }• a,L 4 Z a 8k (1+k) 8k X 1 my = 0 12a 22a ( 1 + k ) 2 (3-7y) + 8 y k 1 2 a-L ( 1 + k ) 2 + 4k±2 2 m Y 32a ( 1 + k ) 2 (3-5y) + 1 6 y k . 2 a^L ( 1 + k ) 2 + 4 k . 2 2 42a 0 66 ( f ) A c t i o n 3 a 93 m y '43a m 43a ?/ ZH' m 33a L = 1 2 E f (1-y ) a . rrv z23 a 33 a 13a / m I3a m 2 3 a m; 3a '13a '23a ( l - 2 y ) ( l + k ) 2 + 4 y k x 2 _ L ( 1 + k ) 2 + 4 k x 2 k 1 1 9 0 ( l - 2 y ) ( l + k ) 2 + 4uk 2 z . o a = ( j [ 4 k / - ( l - 2 y ) (1+k)^] + - - __} . 3 j a 4 1 ( 1 + k ) 2 + 4 k / k l z_~ = — [4k 2 - ( l - 2 y ) ( 1 + k ) 2 ] 4 k 1 x m l 3 m x ( l + k ) 2 ( 3 - 7 y ) + 8 y k 1 2 a^L 2 3 a ( 1 + k ) 2 + 4 k x 2 2 x ( l + k ) 2 ( 3 - 5 y ) + 1 6 y k x 2 a-jL 3 3 a ( 1 + k ) 2 + 4k ]_ 2 2 m x 0 = 0 43a = 0 13a (1+k) [ ( l - 2 y ) (1+k) 2+ 4 y k x 2 ] k ( l - 3 y ) (1+k) m 2 3 a = { — 1 ' ~ ; 2 ~~2 } ' a l L 2k [ ( l + k ) 2 + 4k 2 ] ( 1 + k ) + 4 k l m 3 3 a = { [ 4 k l 2 " d ^ y ) ( 1 + k ) 2 ] • — + 2k^ (1+k) [ ( l - 2 y ) ( 1 + k ) 2 + 4yk 2 ] + i — k x [ ( l + k ) 2 + 4 k x 2 ] ( l - 3 y ) k 1 ( 1 + k ) + } . a L ( l + k ) z + 4 k / x m 4 3 a = [ 4 k l 2 ~ ( 1 _ 2 y ) ( 1 + k ) 2 ] * L 2-6 T r a n s f o r m a t i o n M a t r i c e s I n t h e f i n i t e e l e m e n t model o f a c i r c u l a r p l a t e , t h e a d j a c e n t e l e m e n t s meet a t an a n g l e , and t h e i r x and y d i r e c t i o n s a t t h e common nodes do n o t c o i n c i d e . T h i s makes b a l a n c i n g o f f o r c e s a t t h e common nodes d i f f i c u l t and s u g g e s t s t h e r e p l a c e m e n t o f t h e x and y axe s by t h e T and R ax e s w i t h d i f f e r e n t d i r e c t i o n s a t t h e nodes a l o n g any c i r c u m f e r e n t i a l l i n e . ; I n t h e new s y s t e m o f a x e s , t h e d i r e c t i o n z and t h e d i s p l a c e m e n t w r e m a i n t h e same as b e f o r e . The T - a x i s ( F i g u r e 2-28) i s d i r e c t e d t a n g e n t i a l l y t o t h e c i r c l e drawn 6e t h r o u g h t h e node i n q u e s t i o n a t an a n g l e y^— t o t h e a x i s x, where 6e i s t h e c e n t r a l a n g l e o f t h e s e c t o r e n c l o s i n g t h e c e l l . The R - a x i s p o i n t s t o t h e c e n t r e o f t h e c i r c l e m a k i n g an a n g l e ~ w i t h y - a x i s . The n o d a l f o r c e s z, m , irr i n t h e x y z s y s t e m a r e T R r e l a t e d t o t h e f o r c e s z', m , m i n t h e new s y s t e m t h r o u g h t h e e q u a t i o n s . ( F i g u r e 2-28) A t nodes 1 and 3: z = z' x T^ fie R „ . 6e m = m Cos Y~ ~ m S i n 2— y T 0 . 6e , R_ <$e nr = m S i n ^ p" + m C o s 2~ S i m i l a r l y , a t t h e nodes 2 and 4: F i g . 2 - 2 9 70 m mJ mT C o s ^ + m R S i n |_. - m T S i n J-i + m RCos | i -W r i t i n g t h e above r e l a t i o n s i n m a t r i x n o t a t i o n : V f 2 f 4 xyz o o 12x12 L TRz (2.31) i n w h i c h f\ r e p r e s e n t s c o l l e c t i v e l y t h e n o d a l f o r c e s a t c o r n e r i , and [ T x ] 3x3 0 Cos 0 S i n 6e - S i n Cos 6e_ 2 6e (2-32) [T ] 2 3x3 0 Cos | -0 - S i n fie S i n f -Cos 6 e (2-33) In a more condensed form, the Equ a t i o n (2-31) may be w r i t t e n : { f }xyz = [ T ] 12x12 { f W • • ' ' <2-34> [T] i s c a l l e d the t r a n s f o r m a t i o n m a t r i x , d e f i n e d by Equation (2-31) . S i m i l a r l y the nodal displacement v e c t o r t <5 ) x y z m a Y be obtained from {6} , by the same t r a n s f o r m a t i o n : J.KZ { 6 } x y z = [ T ] 1 2 x l 2 { 6 } T R z < 2 " 3 5 > S u b s t i t u t i o n of Equations (2-34), (2-35) i n t o E q uation (2-30a) g i v e s : [ T ] { f W = C K ] x y Z C T ] { 6 } T R Z « then { f } T R z , = [ T ] " 1 [ K ) x y z [T ] { 6 ^ ^ , Since the t r a n s f o r m a t i o n matrix [T] i s an orth o g o n a l matrix, -1 T i . e . , [T] = [T] , the aboVe equation may be w r i t t e n : From t h i s e q u a t i o n , i t i n the new a x i s system can i s : be seen t h a t the s t i f f n e s s matrix 72 [ K ] [T] T [ K ] [T ] (2.37) TRz 1 x y z [ K ] TRz i s c a l l e d t h e t r a n s f o r m e d s t i f f n e s s m a t r i x . E q u a t i o n (2-36) becomes: {f> TRz = [ K ] TRz ' {6} TRz (2.38) M o d e l o f a f u l l c i r c u l a r p l a t e w o u l d r e q u i r e t h e use o f i s o s c e l e s t r i a n g u l a r e l e m e n t s i n t h e c e n t r a l r e g i o n . O n l y n o - b a r c e l l s a r e a v a i l a b l e t o r e p r e s e n t t h e s e t r i a n g l e s , w h i l e t h e b a r c e l l s may be u s e d o n l y f o r t h e e q u i l a t e r a l t r i a n g l e s . The s t i f f n e s s m a t r i c e s o f t h e i s o s c e l e s t r i a n g l e s , r e f e r r e d t o t h e x y z s y s t e m b o t h s t a t i c s and e n e r g y t y p e s a r e g i v e n i n R e f e r e n c e 9. T r a n s f o r m a t i o n o f t h e m a t r i c e s t o a new c o o r d i n a t e s y s t e m i s a g a i n n e c e s s a r y . A t t h e b a s e nodes 1 and 2, t h e d i r e c t i o n s must be T, R and z 1 t o match' t h e c h o s e n d i r e c t i o n s o f t h e a d j a c e n t t r a p e z o i d a l and t r i a n g u l a r c e l l s . A t t h e p l a t e c e n t r e where t h e v e r t i c e s o f a l l t r i a n g l e s c o i n c i d e , a common s e t o f d i r e c t i o n s X, Y i s a r b i t r a r i l y c h o s e n . ( F i g u r e 2 - 2 9 ) . The a n g l e * i n <; F i g u r e 2-29 me a s u r e d c l o c k w i s e f r o m t h e d i r e c t i o n y i n t h e t r i a n g l e t o t h e d i r e c t i o n Y o f t h e common s y s t e m a t t h e c e n t r e o f t h e c i r c l e . The t r a n s f o r m a t i o n m a t r i x f o r a t r i a n g u l a r e l e m e n t i s : [T.J o [T] 9x9 o [T-] [ T 3 ] The s i g n i f i c a n c e o f t h e s u b - m a t r i c e s [T^] and [ T 2 1 r e m a i n s t h e same as i n c a s e o f t h e t r a p e z o i d a l e l e m e n t ( E q u a t i o n s 2-32, 2 - 3 3 ) , and t h a t o f [ T - ] , c o r r e s p o n d i n g t o t h e v e r t e x node 3, i s : 'T 1 • 3 J 3x3 1 0 0 Cos * T 0 - S i n *„ 0 S i n ip Cos ijjr T The t r a n s f o r m e d s t i f f n e s s m a t r i x o f t h e t r i a n g u l a e l e m e n t i s a l s o g i v e n by E q u a t i o n (2-37) 2-7 I s o s c e l e s T r a p e z o i d a l and T r i a n g u l a r No-Bar C e l l s I n a p p l i c a t i o n o f t h e t h e o r y o f t r a p e z o i d a l b a r c e l l s , i t i s d e s i r a b l e t o compare t h e p r e c i s i o n o f i t s r e s u l t s w i t h t h a t o f t h e f i n i t e e l e m e n t method e m p l o y i n g t r a p e z o i d a l n o - b a r c e l l s . The g e n e r a l p r o c e d u r e o f t h i s method i s s u b s t a n t i a l l y t h e same as t h e one i n v o l v e d i n t h e b a r m o d e l s , b u t t h e s t i f f n e s s m a t r i c e s a r e d i f f e r e n t . I n f a c t , t h e r e a r e two k i n d s o f t h e n o - b a r s t i f f n e s s m a t r i c e s a v a i l a b l e , t h e e n e r g y k i n d and t h e s t a t i c s k i n d . The f i r s t s y m m e t r i c a l a b o u t t h e p r i n c i p a l d i a g o n a l and t h e s e c o n d u n s y m m e t r i c a l . F u l l d e t a i l s o f t h e d e r i v a t i o n o f t h e s e two k i n d s o f m a t r i c e s a r e g i v e n i n R e f e r e n c e 9. 75 CHAPTER I I I BOUNDARY CONDITIONS IN THE PLATE MODEL B o u n d a r y c o n d i t i o n s o f t h e p l a t e p r o t o t y p e a r e a p p r o x i m a t e d i n t h e p l a t e model by r e s t r a i n i n g c e r t a i n movements o f t h e edge n o d e s . The c o n d i t i o n o f t h e t r a n s v e r s e d i s p l a c e m e n t does n o t p r e s e n t any p r o b l e m , b u t th e r o t a t i o n c o n d i t i o n s r e q u i r e s p e c i a l c o n s i d e r a t i o n s s i n c e t h e y a r e n o t a l w a y s t h e same as i n t h e p l a t e p r o t o t y p e . 3-1 S i m p l y S u p p o r t e d Edge Suppose t h a t t h i s edge o f a p l a t e i s s i t u a t e d a l o n g t h e c i r c u m f e r e n c e r = R. The two o b v i o u s b o u n d a r y c o n d i t i o n s a r e : w = 0 and M = 0 r The s e c o n d c o n d i t i o n means t h a t t h e edge nodes a r e f r e e t o r o t a t e a b o u t t h e t a n g e n t i a l a x i s . I n t h e p l a t e p r o t o t y p e no r e s t r a i n i n g moments a r e a v a i l a b l e a l o n g t h e c i r c u l a r edge t o r e s i s t p o s s i b l e . t o r s i o n a l r o t a t i o n . T h i s w o u l d i n d i c a t e t h a t t h e t o r s i o n a l moments on t h e edge a r e z e r o . A d i f f e r e n t c o n c l u s i o n , however, seems t o f o l l o w f r o m c o n s i d e r a t i o n o f t h e d i f f e r e n t i a l e x p r e s s i o n f o r t h e edge t o r q u e M - / - I , \ /I 3 2 w 1 9w\ _„ _ 3w . M r t = D(l-vi) (- g r 3 e ' - --r -gg-) . On t h e edge i s n o t z e r o and i s n o t c o n s t a n t a l o n g t h e edge. T h i s makes 8 2w 7r—~-x ^ 0 a n d w i t h t h i s M . i s n o t z e r o . 9r30 r t I n t h e t h e o r y o f e l a s t i c i t y t h e i n c o n s i s t e n c y r e f e r r e d t o h e r e i s r e s o l v e d by t h e a s s u m p t i o n t h a t t h e edge t o r q u e i s c o n v e r t e d i n t o t h e s t a t i c a l l y e q u i v a l e n t edge r e a c t i o n s , w h i c h t h e s u p p o r t i s c a p a b l e o f r e s i s t i n g . The above d i s c u s s i o n s i n d i c a t e t h e p r o p e r t r e a t m e n t o f t h e t o r s i o n a l b o u n d a r y c o n d i t i o n o f t h e s i m p l y s u p p o r t e d p l a t e m o d e l . The edge nodes s h o u l d be c o n s i d e r e d f i x e d a g a i n s t t o r s i o n a l r o t a t i o n , and t h e r e s u l t a n t n o d a l t o r q u e s s h o u l d be c o n v e r t e d i n t o t h e s t a t i c a l l y e q u i v a l e n t t r a n s v e r s e f o r c e s a t t h e a d j a c e n t edge nodes on b o t h s i d e s o f t h e nodes c o n s i d e r e d . 3-2 Clamped Edge: : The two .obvious b o u n d a r y c o n d i t i o n s , t h e same i n t h e p l a t e p r o t o t y p e and t h e f i n i t e e l e m e n t m o d e l , a r e : w = 0 _ 3w A and —— = 0 9r The c o m b i n a t i o n o f t h e l a t t e r c o n d i t i o n w i t h t h e c o n d i t i o n = 0 makes ^ = 0 and w i t h t h i s M i n t h e p l a t e a t 77 t h e f i x e d edge i s a l s o z e r o . However, t h e a b s e n c e o f t h e two r o t a t i o n s o f t h e edge nodes i n t h e model d o e s n o t i n s u r e t h e a b s e n c e o f t h e edge t o r q u e , w h i c h i s i n f l u e n c e d by t h e movements o f t h e nodes one s t e p away f r o m t h e edge. Thus t h e a n a l y s t i s c o n f r o n t e d w i t h t h e c h o i c e o f one o f t h e two p o s s i b l e c o n d i t i o n s a t t h e e d g e s : t h e a b s e n c e o f t o r s i o n a l moment o r t h e a b s e n c e o f t h e t o r s i o n a l r o t a t i o n . The f i r s t a l t e r n a t i v e a p p e a r s t h e b e t t e r o f t h e two. E v e n a b e t t e r c a s e f o r a d m i s s i o n o f t o r s i o n a l r o t a t i o n a t t h e f i x e d ended b o u n d a r y nodes c a n be made when t h e model p o s s e s s e s s t r a i g h t b o u n d a r y and i s composed o f j r e c t a n g u l a r c e l l s . The model o f t h i s k i n d i s e q u i v a l e n t ! t o a two-span c o n t i n u o u s m odel s y m m e t r i c a l and s y m m e t r i c a l l y l o a d e d w i t h r e g a r d t o t h e edge i n q u e s t i o n and s i m p l y s u p p o r t e d a l o n g i t . The two h a l v e s o f t h i s c o n t i n u o u s model w o u l d d e f o r m i n a manner s i m i l a r t o t h e g i v e n o n e - s p a n m o d e l , and w o u l d have t h e i r nodes t o r s i o n a l l y r o t a t e d . The t r a p e -z o i d a l c e l l model o f a c i r c u l a r p l a t e w o u l d a p p r o a c h t h e s t r a i g h t edge, r e c t a n g u l a r c e l l model on i n f i n i t e r e d u c t i o n o f t h e s i z e o f t h e mesh. I 3.3 F r e e edge The edge nodes aire f r e e t o d i s p l a c e and t o r o t a t e . 78 CHAPTER IV STRESS ANALYSIS OF THE PLATE The n o d a l d i s p l a c e m e n t s and r o t a t i o n s o f t h e p l a t e a r e f o u n d by s o l v i n g t h e s y s t e m o f e q u a t i o n s o f n o d a l e q u i l i b r i u m : {F} , = [S] {A} , n x l nxn n x l Here {A} i s t h e column v e c t o r o f n unknown n o d a l d i s p l a c e m e n t s and r o t a t i o n s , {F} i s t h e c o r r e s p o n d i n g l o a d v e c t o r and [S] i s t h e s t i f f n e s s m a t r i x o f t h e e n t i r e model o b t a i n e d by a s s e m b l i n g t h e t r a n s f o r m e d s t i f f n e s s m a t r i c e s o f t h e i n d i v i d -u a l e l e m e n t s . F o l l o w i n g t h e c a l c u l a t i o n o f t h e n o d a l d i s p l a c e -ments and r o t a t i o n s , t h e p l a t e s t r e s s e s a r e f o u n d by one o f t h e two methods: t h e method o f n o d a l f o r c e c o n c e n t r a t i o n s o r t h e method o f n o d a l d i s p l a c e m e n t s . 4-1 Method o f N o d a l F o r c e s The n o d a l f o r c e s and moments w h i c h h o l d t h e c e l l i n e q u i l i b r i u m a r e f o u n d by E q u a t i o n ( 2 - 3 8 ) . The p l a t e s t r e s s e s , i . e . , t h e moments and s h e a r s i n t h e p l a t e , a r e t h e n c a l c u l a t e d by s p r e a d i n g t h e n o d a l f o r c e s so f o u n d o v e r t h e a p p r o p r i a t e t r i b u t a r y a r e a s (2, 8 ) . The method i s i l l u s t r a t e d by r e f e r e n c e t o F i g u r e 4.1 r e p r e s e n t i n g t h e f o u r e l e m e n t s s u r r o u n d i n g t h e common node 1 o f t h e m o d e l . In t h e a b s e n c e o f an e x t e r n a l l o a d a t t h e node 1, t h e , i n t e r n a l f o r c e s a c t i n g on t h e node due t o t h e f o u r e l e m e n t s a r e m u t u a l l y b a l a n c e d . S h o u l d an e x t e r n a l l o a d P be p r e s e n t t h e n o d a l f o r c e s z w i l l b a l a n c e i t . The f o l l o w i n g s t r e s s e s p e r u n i t l e n g t h i n TRz d i r e c t i o n s may be assumed t o e x i s t i n t h e p l a t e a t t h e node i n q u e s t i o n i n a b s e n c e o f t h e e x t e r n a l l o a d w i t h t h e i r p o s i t i v e d i r e c t i o n s a s shown i n F i g u r e 4.2. M r (m 4 + T m 3 ) / b 1 = - (m 2 + m l ) / b l M t = / R (m 2 + m- 4)/b 2 = - / R (m 1 + m 3 ) / b 2 M r t = ( m l + m 2 ) / b 1 = - / R (m 3 + m^)/b 1 M t r = (m 2 + m 4 ) / b 2 = ~ + m 3 ) / b 2 Q r = <* 3 + Z 4 ) / b l = - < z i + z 2 } / b l Q t = ( Z 2 + + z 3 ) / b 2 (4.1) M^^ and s h o u l d be n u m e r i c a l l y e q u a l and o p p o s i t e i n s i g n . S i n c e t h e y a r e n o t l i k e l y t o be so r e l a t e d , t h e two v a l u e s s h o u l d be a v e r a g e d . 80 I f a c o n c e n t r a t e d upward l o a d P i s p r e s e n t a t t h e node 1, i t s h o u l d be a p p o r t i o n e d between t h e f o u r c e l l s i n t o t h e p a r t s P-^ P-, P-, P^. The n o d a l f o r c e s z s a t i s f y t h e r e l a t i o n : z. + z- + z 3 + z 4 = P f w h i c h may be w r i t t e n : ( z 1 - P 1 ) + ( z 2 - P 2 ) + ( z - - P 3 ) + ( z 4 - P 4 ) = 0 Thus the. s h e a r s a r e o b t a i n e d f r o m t h e E q u a t i o n s (4-1) by s i m p l y r e p l a c i n g z^, z 2 , z 3 , z 4 by (z-^-P.^ , ( z 2 ~ P 2 ) , ( z 3 ~ P 3 ) and ( z 4 ~ P 4 ) . I n c a s e o f u n i f o r m l o a d t h e f o r c e s P l ' P 2 ' P 3 a n c ^ P 4 m a y ^ e t a k e n a s n o d a l l o a d s d e t e r m i n e d as f o l l o w s : 82 F i g u r e 4 - 3 T o t a l v e r t i c a l l o a d on the element of F i g u r e 4 - 3 > 1 2 2 V = T- fie (r_ - r 1 ) q. D i s t a n c e from c e n t r e of g r a v i t y of load to c e n t r e 0: (r. + r . r - + r . ) / ( r - + r-) From s t a t i c s ( f o r s m a l l angle fie) 2 V 1 + 2 V 2 = V 83 and 2 V n r n + 2 V _ r 0 = V r 1 1 2 2 c S o l v e f o r V 1 , V 2 : and V l = V ( r c " r l > / ( r 2 " r l } V 2 = V ( r 2 - r c ) / 2 ( r 2 - r x ) ? 1 = P 2 V 1 (due t o l o a d on c e l l 1 o r 2) P 3 = = V 2 (due t o l o a d on c e l l 3 o r 4) A d d i t i o n a l e x p l a n a t i o n i s needed w i t h r e g a r d t o t h e s t r e s s e s a t t h e edge. Some d e c i s i o n s r e g a r d i n g t h e s e s t r e s s e s must be b a s e d on judgment. 4-1.1 S i m p l y S u p p o r t e d Edge ( F i g u r e 4-4) The b e n d i n g moment on t h e edge i s z e r o . W i t h t h e j o i n t f i x e d a g a i n s t t o r s i o n a l r o t a t i o n f o r t h e r e a s o n e x p l a i n e d e a r l i e r t h e r e e x i s t s a b e n d i n g moment on t h e p l a n e p e r p e n d i c u l a r t o t h e edge. m R m R M t = \ ( 57/2 ~ 57/2 } = { m 2 R " m l V b2 84 F i g . 4 - 5 S i n c e t o r s i o n a l r o t a t i o n on t h e edge i s p r e v e n t e d R R (m^ + irt2 ) r* 0 and t h i s u n b a l a n c e d moment s h o u l d be c o n -v e r t e d i n t o t h e v e r t i c a l r e a c t i o n s a t t h e a d j a c e n t n o d e s . T h e i r m a g n i t u d e s and d i r e c t i o n s a r e shown i n F i g u r e 4 -5 . I n t h e same way t h e u n b a l a n c e d t o r s i o n a l moments a t t h e vnodes 2 and 3 c o n t r i b u t e v e r t i c a l f o r c e s a t t h e node 1. C a l l i n g t h i s c o n t r i b u t i o n ( p o s i t i v e u p w a r d s ) , t h e t o t a l s h e a r f o r c e i n t h e p l a t e a t t h e node 1 may be w r i t t e n : Q r = <_. + z 2 + z m ) / b 1 The s h e a r f o r c e on t h e r a d i a l p l a n e a t t h e node 1 may be t a k e n : Q t = (-2 ~ z ^ A 1 2 4-1.2 F i x e d Edge ( F i g u r e 4-4) The b e n d i n g moments a r e : T T on t h e edge: M = (m^ + m 2 )/b^ T On t h e r a d i a l : - 2m^ /b2 s i n c e t o r s i o n a l r o t a t i o n s a r e R R a l l o w e d a t t h e edge nodes and m^ = - rr^ • The s h e a r s a r e : Q r = (z1 + z 2 ) / a 1 Q t - ( z 2 - z . ) / b 2 The edge torque M^ t = 0 as i t should be, and Mfc may be T assumed to be zero although the nodal moment m^  is not l ike ly to be zero. 4-1.3 Free Edge (Figure 4-4) The shear Q , M and M , came out zero as they r ' r r t J should be and Q , may be taken as in the case of the fixed edge. 4-2 F i r s t Method of Nodal Displacements This method is applicable only for the no-bar f ini te element models. The shears and moments at any point of the element may be- found by taking the product of the element displacement vector with the so-called stress matrices. The derivation of these matrices for the isos-celes trapezoidal and triangular elements are given in Reference 9. It is convenient to have the stresses determined at the nodes. However, different cel ls meeting at the same node would have different stresses. Their average may be taken as representing the stress in the plate prototype. The va l id i ty of the stress values so determined is somewhat uncertain. When the ce l l s are different in size and shape i t is fe l t that the contributions of different ce l l s should be weighted in favour of the bigger ones, but the basis for 87 s u c h w e i g h t i n g i s n o t a p p a r e n t . The s u b j e c t i s d i s c u s s e d i n R e f e r e n c e 10. I n t h e p r e s e n t s t u d y , t h i s method o f s t r e s s d e t e r m -i n a t i o n i s n o t u s e d i n v i e w o f a d d i t i o n a l c o m p l i c a t i o n s i n v o l v e d i n t r a n s f o r m a t i o n o f t h e c o o r d i n a t e s y s t e m s f r o m x y z t o T R z 1 . 4-3 S e c ond Method o f N o d a l D i s p l a c e m e n t s I n t h i s method t h e d i f f e r e n t i a l e x p r e s s i o n s f o r moments and s h e a r s a r e w r i t t e n i n f i n i t e d i f f e r e n c e f o r m i n t e r m s o f t h e s l o p e s 8 , 8 a t t h e n e i g h b o u r i n g n o d e s . X K By d e f i n i t i o n s , t h e t a n g e n t i a l and r a d i a l r o t a t i o n s a r e : T r o t a t i o n -- -T dr R r o t a t i o n 6 9w dr , 9w R rdi From t h e s e : ~ = - 6 m 8r T and JQ = ~ r 0 R F i g u r e 4-6 shows a number o f c e l l s w i t h e q u a l h e i g h t s b- and c e n t r a l a n g l e 6e. A t a t y p i c a l i n t e r i o r node 1 v a r i o u s p a r t i a l d e r i v a t i v e s o f w w i t h r e s p e c t t o t h e v a r i a b l e s 8 and r may be a p p r o x i m a t e d by f i n i t e d i f f e r e n c e s as f o l l o w s [11]: 88 3 2w 8 9r 9r 2 9 w 9 9 6 2 = 7Q ( " r V = " { r i e ^ l Q R 2 ^ 2 S e 9 2 The m i x e d d e r i v a t i v e — — may be e v a l u a t e d i n two d i f f e r e n t 9r99 ways as f o l l o w s : 2 = i - (-6 ) = - (9 - 6 )/26« 9r96 9G 1 J 1 Z 2 ° r = — ( ~ r 6 R ) = - ( r 5 6 R 5 - r 4 e R 4 ) / 2 b ? 9r96 9r b Rb 4 R4 2 t h e i r a v e r a g e v a l u e s h o u l d be used, 3 2 ^ 2 ( " V = - ( e T 5 - 2 6 T l + 6 T 4 ) / b 2 2 9 3 9 2 ^ 2 = - ( r l 6 R 3 - 2 r l 6 R l + r l 6 R 2 ) / 6 e 2 fc=fe7 ^ = - ^ T 3 - 2 ^ T 2 ) ^ F i g . 4-6 ~ 2 ~ = 7-2 = ~ ( r 5 6 R 5 - 2 r l 6 R l + r 4 e R 4 ) / b 2 2 3r 80 3r The l a s t two d e r i v a t i v e s may a l s o be e v a l u a t e d d i f f e r e n t l y . T h ese o t h e r e x p r e s s i o n s a r e more c o m p l i c a t e d . Moments and s h e a r s a t t h e node 1 a r e o b t a i n e d by s u b s t i t u t i o n o f t h e above e x p r e s s i o n s i n t o E q u a t i o n s (1-5) and ( 1 - 6 ) . As an example: 2 2 _, r 3 w , ,1 3w , 1 3 w « , M r = - D [ ~ 2 + 3 _ + ~ \ ~ 2 } ] 3r r 3 6 - D ( - ^ 7 ^ - + h 1 „ + < e B3 - eR 2 > l > 2b- 1 26e F o r nodes on t h e b o u n d a r y , t h e same d i f f e r e n c e f o r m u l a e may a l s o be a p p l i e d w i t h s u i t a b l e e x t e n t i o n o f t h e p l a t e b e y o n d i t s b o u n d a r y . W i t h a s t r a i g h t edge, t h e p l a t e may be e x t e n d e d s y m m e t r i c a l l y o r a n t i s y m m e t r i c a l l y a b o u t t h e edge d e p e n d i n g w h e t h e r t h e edge i s cl a m p e d o r s i m p l y s u p p o r t e d . D i f f i c u l t y a r i s e s when t h e edge i s c i r c u l a r , b u t t h e p r o c e d u r e seems t o be j u s t i f i e d on i n f i n i t e r e d u c t i o n o f mesh s i z e . 91 CHAPTER V FLEXURAL PROBLEMS OF CIRCULAR PLATES SOLVED BY THE F I N I T E ELEMENT METHOD The v a l i d i t y o f t h e f i n i t e e l e m e n t method i s demon-s t r a t e d on two p r o b l e m s o f c i r c u l a r p l a t e s f o r w h i c h t h e e l a s t i c i t y s o l u t i o n s a r e a v a i l a b l e . The c i r c u l a r p l a t e i s s u b d i v i d e d by a number o f e q u i d i s t a n t c o n c e n t r i c c i r c l e s and r a d i a l l i n e s i n t o t h e a r e a s a p p r o x i m a t e d by t h e f i n i t e e l e m e n t s i n t h e shape o f i s o s c e l e s t r a p e z o i d s o v e r most o f t h e a r e a and i s o s c e l e s t r i a n g l e s i n t h e v i c i n i t y o f t h e c e n t r e ( F i g u r e 5 - 1 ) . No-bar c e l l s a r e u s e d f o r t h e t r i a n g l e s and b o t h t h e b a r and t h e n o - b a r c e l l s f o r t h e t r a p e z o i d s , w i t h employ-ment o f t h e e n e r g y s t i f f n e s s m a t r i c e s f o r t h e n o - b a r e l e m e n t s . Two w i d e l y d i f f e r e n t v a l u e s o f P o i s s o n ' s r a t i o 0.2 and 0.4 a r e u s e d t o t e s t t h e a c c u r a c y o f t h e method. The p r e c i s i o n o f t h e f i n i t e - e l e m e n t r e s u l t s i n d e f l e c t i o n s , s l o p e s , moments and s h e a r s i s r e p o r t e d i n t h e f o r m o f p e r c e n t a g e e r r o r s i n a c c o r d a n c e w i t h t h e f o r m u l a : 92 % E r r o r C a l c u l a t e d v a l u e | - [ E l a s t i c i t y v a l u e | x 1 0 0 j E l a s t i c i t y v a l u e | S h o u l d t h e c a l c u l a t e d and t h e e l a s t i c i t y v a l u e s happen t o be o f o p p o s i t e s i g n s , t h e p l u s s i g n i s u s e d i n t h e n u m e r a t o r o f t h e e q u a t i o n . The t r e n d o f c o n v e r g e n c e o f t h e c a l c u l a t e d r e s u l t s t o t h e c o r r e c t v a l u e s i s s t u d i e d by r e d u c i n g t h e mesh f i n e n e s s t o one h a l f o f t h e o r i g i n a l v a l u e s o f 6e and A R . 5-1 Example I - S i m p l y S u p p o r t e d S e m i - C i r c u l a r P l a t e U nder U n i f o r m L o a d p l a t e d e r i v e d by t h e t h e o r y o f e l a s t i c i t y i s [1] ( F i g u r e The e q u a t i o n f o r t h e d e f l e c t i o n s u r f a c e o f t h e 5-1.1) : 0 0 m+2 w Z D m=l,3,5 (A n + B n m m + C n ) S i n m9 m i n w h i c h r R A m m+5+y 7Tm(16-m 2) (2+m) [m+i (1+ y ) ] 93 B m • m+3+ii iTm(4+m) (4-m 2) [m+|(l+y) ] m iTm (16-m 2) (4-m 2) The s l o p e s o f t h e d e f l e c t i o n s u r f a c e and t h e s t r e s s e s a r e o b t a i n e d f r o m w by u s i n g t h e e q u a t i o n s ( 1 - 5 ) , ( 1 - 6 ) . The computer c a l c u l a t e s t h e s e e l a s t i c i t y f u n c t i o n s c o r r e c t t o t h e f o u r t h s i g n i f i c a n t d i g i t . Enough t e r m s i n t h e s e r i e s a r e t a k e n t o i n s u r e t h a t a c c u r a c y . Due t o symmetry, o n l y one h a l f o f t h e p l a t e n e e d be c o n s i d e r e d i n t h e f i n i t e - e l e m e n t s o l u t i o n . The p l a t e i s f i r s t a n a l y z e d u s i n g a mesh s i z e w i t h fie - 9° and AR = 0.125R, t h e s e d i m e n s i o n s a r e f u r t h e r h a l v e d t o fie = 4.5° and AR = 0.0625R. The p e r c e n t a g e e r r o r s i n d i s p l a c e m e n t s and s t r e s s e s a r e shown i n T a b l e s ( V - l . l t o V - 1 . 8 ) . The t r a n s v e r s e d e f l e c t i o n s c a l c u l a t e d u s i n g b o t h t y p e s o f c e l l s , t h e b a r and t h e n o - b a r , a r e h i g h l y a c c u r a t e . E v e n f o r t h e c o a r s e r mesh, t h e p e r c e n t a g e e r r o r i s u s u a l l y l e s s t h a n 1%. A t t h e p o i n t s where t h e t h e o r e t i c a l v a l u e i s s m a l l compared t o i t s maximum e l s e w h e r e , h i g h e r p e r c e n t a g e e r r o r i s o b s e r v e d and t h e d i s c r e p a n c y o f t h i s k i n d s h o u l d n o t be c o n s i d e r e d t o o s i g n i f i c a n t . (The c o r r e s p o n d i n g v a l u e s o f d i s p l a c e m e n t s and s t r e s s e s a r e marked w i t h an a s t e r i s k i n F i g . 5- I t h e t a b l e s , u s i n g a p p r o x i m a t e l y 10% o f t h e maximum v a l u e o f t h e f u n c t i o n i n t h e t a b l e s as t h e d i v i d i n g l i n e ) . The a c c u r a c y o f t h e r e s u l t s u s i n g t h e b a r model seems t o be a f f e c t e d by P o i s s o n ' s r a t i o , and i s g e n e r a l l y b e t t e r w i t h l o w e r v a l u e o f y . The b a r model y i e l d s s l i g h t l y b e t t e r r e s u l t s i n d e f l e c t i o n s t h a n t h e n o - b a r m o d e l , b u t t h e r e v e r s e i s t r u e i n t h e s l o p e s . S t r e s s e s i n t h e p l a t e s o b t a i n e d by t h e methods o f n o d a l f o r c e (N.F.) and n o d a l d i s p l a c e m e n t s (N.D.) f o r b o t h t y p e s o f model ( a n a l y z e d w i t h t h e f i n e r mesh) a r e shown i n T a b l e s (V-1.9 and V - 1 . 1 0 ) . In t h e m a j o r i t y o f c a s e s the N.F. method y i e l d s b e t t e r r e s u l t s i n b e n d i n g moments f o r b o t h t y p e s o f c e l l s . As t o t h e t o r q u e s and s h e a r s t h e N.F. method i s g e n e r a l l y b e t t e r w i t h t h e b a r model b u t worse w i t h t h e n o - b a r m o d e l , i n w h i c h some v e r y l a r g e e r r o r s i n M and Q, o f t h e o r d e r o f 100% a r e o b s e r v e d , t I n v i e w o f t h e l a r g e e r r o r s i n M ^ and Q o b s e r v e d i n c a l c u l a t i o n s c o n d u c t e d by t h e n o d a l f o r c e method t h i s method was a d j u d g e d i n a c c e p t a b l e f o r a l l c a l c u l a t i o n s w i t h t h e n o - b a r m o d e l . F o r t h i s r e a s o n t h e p r e c i s i o n d a t a f o r t h e n o - b a r c e l l s i n t h e T a b l e s (V-1.4 t o V-1.8) were o b t a i n e d by t h e N.D. method and f o r t h e b a r c e l l s by t h e N.F. method. I n s p e c t i o n o f t h e s e t a b l e s shows t h a t t h e ; b a r c e l l g e n e r a l l y y i e l d s b e t t e r r e s u l t s p a r t i c u l a r l y w i t h t h e s m a l l v a l u e o f t h e P o i s s o n ' s r a t i o . Good t r e n d i n c o n v e r g e n c e o f d i s p l a c e m e n t s and s t r e s s e s i s o b s e r v e d i n b o t h t y p e s o f model on r e d u c t i o n t h e e l e m e n t s i z e . TABLE ( V - l . l ) S e m i - C i r c u l a r P l a t e - U n i f o r m L o a d 4 T r a n s v e r s e D e f l e c t i o n w = a1 (qR /100D) y = 0.2 U = 0. 4 P e r c e n t a g e E r r o r P e r c e n t a g e E r r o r r/R e E l a s . AR/R= 0.125 AR/R= 0. 0625 E l a s . AR/R= 0.125 AR/R= 0.0625 S o l . 6e= :9° 6e =4 . 5° S o l . 6e = 9° 6e = 4.5° a±- B a r No-Bar B a r No -Bar a i B a r No-Bar B a r No-Bar 9 0.064* 6.9 -2.0 -0.20 -9 .64 0.062* 7.2 -1.8 0.70 -0.61 18 0 .121 1.4 -1.5 -0 .11 -0 .57 0.116 4.5 -1.4 0.73 -0 . 54 0.875 36 0.208 1.1 -1.2 -0.09 -0 .53 0.199 3.9 -1.1 0.65 -0.50 54 0 .266 0.97 -1.1 -0 .10 -0 .51 0.253 3.6 -0.94 0.58 -0 .49 72 0.298 0.93 -0.98 -0.11 -0 .50 0.283 3.4 -0.85 0.54 -0.48 90 0 . 309 0.91 -0.95 -0 .11 -0 .50 0.293 3.4 -0.83 0.53 -0.48 9 0.114 0 . 82 -1.4 -1.1 -1 .3 0.110 2.2 -1.2 -0.64 -1.3 18 0.218 0.61 -1.1 -1.1 -1 .3 0.210 2.2 -0.99 -0 .59 -1.3 0.75 36 0 . 382 0.40 -0.93 -1.1 -1 .3 0.366 2.1 -0.79 -0.00 -1.3 54 0.491 0.37 -0.83 -1.1 -1 .3 0.470 2.1 -0.68 -0.64 -1.3 72 0.554 0 .36 -0.78 -1.1 -1 .3 0.529 2.0 -0.63 -0.68 -1.3 90 0 .574 0 . 36 -0 .77 -1.1 -1 .3 0.549 2.0 -0.61 -0 .67 -1.3 18 0.288 -0.43 -0.78 -0 .76 -0 .73 0.279 0.16 -0 .69 -0 .58 -0.69 0.5 36 0.52 5 -0.40 -0.68 -0.75 -0 .72 0.508 0.28 -0.60 -0.56 -0.69 54 0.694 -0.37 -0.63 -0.75 -0 .72 0.670 0.35 -0 .50 -0 .56 -0.70 72 0 .794 -0.36 -0.60 -0.75 -0 .73 0.766 0.38 -0.41 -0.56 -0.71 18 0 .194 -0.68 -0 .75 -0.61 -0 .52 0.188 -0.24 -0.55 -0.48 -0.48 36 0.364 -0.67 -0.71 -0.61 -0 .52 0.354 -0.22 -0.78 -0.48 -0.48 0.25 54 0.495 -0 .67 -0.68 -0.61 -0 .52 0.480 -0.21 -0.44 -0.48 -0.48 '72 "0 ."576 -0Y67 -0Y66 -0.62 -0 .52 0.55 8 -0.20 -0.42 -0.49 -0.49 * V a l u e o f f u n c t i o n l e s s t h a n 10% o f maximum v a l u e i n t h e t a b l e . TABLE (V-1.2) S e m i - C i r c u l a r P l a t e - U n i f o r m L o a d 3 T a n g e n t i a l R o t a t i o n 0 = a_ (qR /100D) y = 0.2 l i - 0 .4 P e r c e n t a g e E r r o r P e r c e n t a g e •• - . ... ... . E r r o r r/R 0 E l a s . AR/R- 0.125 AR/R-0 .0625 E l a s . AR/R= 0.125 AR/R- 0.0625 S o l . 6e= 9° 6e=4 .5° . S o l . 6e= 9° 6e= 4.5° a2 B a r No-Bar Bar No-Bar . a2 B a r No-Bar B a r No-Bar 18 1.0 -1.5 -2.4 -1.2 -0.75 0.948 -3.0 -2.3 -1.5 -0.74 36 1.707 -2.4 -1.7 -0.94 -0.65 1.611 -3.3 -1.7 -1.1 -0.63 1.0 54 2 .168 -2.2 -1.5 -0.87 -0.61 ' 2.041 -2.8 -1.4 -1.0 -0 .59 72 2.428 -2.1 -1.4 -0.83 -0.6 2.282 -2.6 -1.3 -0 .96 -0 .57 90 2 .512 -2.0 -1.4 -0 .83 -0.6 2.360 -2.6 -1.2 -0 .95 -0 .57 18 0 .629 -3.2 -0.91 -0 .26 0.22 0.612 -4.7 -0 .84 -0.77 0 .20 36 1.161 -2.5 -0.84 -0.37 0 .16 1.130 -3.9 -0.74 -0.73 0.15 0.75 54 1.53 8 -2.4 -0 . 80 -0.40 0.10 1.620 -3.4 -0.68 -0.69 0 . 09 72 ,1.760 -2.4 -0 .77 -0.41 0.07 1.7 09 -3.3 -0 .65 -0 .67 0.06 90 1. 832 -2.4 -0.76 -0.42 0.06 1.779 -3.3 -0 .64 -0.66 0 .05 18 *0.07 2 6.2 5.0 -2.7 -1.9 -0.065 16. 7.5 -0 .04 -1.4 36 -0.054 18. 12 . 4.5 -5.0 -0.042 50. 20 . 2 .09 -4.9 0.5 5 4 *0.000 - - 28. 45. -0.024 - - -6.4 12 . 72 •*0.063 -27. -15. 3.4 6.7 *0.008 - - -2.4 4.1 90 *0.085 -21. -12. 2.5 5.2 *0.103 -32. -32. -2.0 3.4 18 -0.636 -0.25 -0.02 -0.61 0.25 -0.616 0.49 1.3 -0.40 -0 .16 36 -1.161 -0 .14 0.66 -0.60 -0.32 -1.124 0.67 1.2 -0.40 -0.22 .0.25 54 -1.531 -0 .11 0.54 -0.62 -0 . 38 -1.479 0.69 1.1 -0 .41 -0.27 72 -1.744 -0 .12 0.47 -0.64 -0.43 -1.684 0.69 1.1 -0.43 -0.31 90 -1.814 -0.13 0.45 -0.65 -0.44 -1.751 0.69 1.0 -0.43 -0.32 * V a l u e o f f u n c t i o n l e s s t h a n 10% o f maximum v a l u e i n t h e t a b l e . TABLE ( V - 1 . 3 ) Semi-Circular Plate Uniform Load Radial Rotation 9_. = a , (qR 3/100D) y = 0 .2 y = 0 .4 Percentage Error Percentage Error r/R 6 Elas. Sol . AR/R=0.125 6e=9° AR/R=0.0 6 25 1 6e=4.5° Elas. Sol. AR/R=0.125 6e=9° AR/R=0.06 25 6e=4.5° a 3 Bar No-Bar Bar No-Bar . a 3 Bar No-Bar Bar No-Bar 0.875 0 18 36 54 7 2 81 - 0 . 4 8 4 - 0 . 3 8 0 - 0 . 2 5 9 - 0 . 1 6 1 - 0 . 0 7 7 - 0 . 0 3 8 - 1 . 2 - 2 . 9 - 2 . 1 - 0 . 7 7 - 1 . 9 7 - 0 . 4 9 - 0 . 9 6 - 0 . 3 1 - 2 . 0 - 0 . 2 1 - 2 . 0 - 0 . 1 8 - 0 . 9 8 - 0 . 8 7 - 0 . 4 6 - 0 . 4 5 - 0 . 5 9 - 0 . 4 4 - 0 . 6 8 - 0 . 4 3 - 0 . 7 3 - 0 . 4 3 - 0 . 7 4 - 0 . 4 3 - 0 . 4 6 5 - 0 . 3 6 2 - 0 . 2 4 5 - 0 . 1 5 1 - 0 . 0 7 2 - 0 . 0 3 6 5.4 - 2 . 7 3 .1 - 0 . 6 8 3 .2 - 0 . 3 5 3.0 - 0 . 1 3 2.8 - 0 . 0 0 2.8 0 .03 0.54 - 0 . 8 3 0.83 - 0 . 4 3 0.57 - 0 . 4 1 0 .42 - 0 . 4 0 0 .35 - 0 . 3 8 0.34 - 0 . 3 8 0 .75 0 18 36 54 72 81 - 0 . 9 9 3 - 0 . 8 1 9 - 0 . 5 7 4 - 0 . 3 6 0 - 0 . 1 7 3 - 0 . 0 8 6 0.64 - 1 . 9 - 0 . 7 2 - 0 . 7 4 - 0 . 4 1 - 0 . 5 3 - 0 . 3 3 - 0 . 3 8 - 0 . 3 3 - 0 . 2 9 - 0 . 3 3 - 0 . 2 7 - 1 . 5 - 1 . 6 - 1 . 3 - 1 . 4 - 1 . 4 - 1 . 5 - 1 . 5 - 1 . 5 - 1 . 5 - 1 . 5 - 1 . 5 - 1 . 5 - 0 . 9 5 7 - 0 . 7 8 5 - 0 . 5 4 8 - 0 . 3 4 2 - 0 . 1 6 4 - 0 . 0 8 1 2 .0 - 1 . 7 2 .1 - 0 . 6 2.0 - 0 . 3 8 1.9 - 0 . 2 1 1.8 - 0 . 0 9 1.8 - 0 . 0 6 - 0 . 8 6 - 1 . 5 - 0 . 6 8 - 1 . 4 - 0 . 8 1 - 1 . 5 - 0 . 9 2 - 1 . 5 - 0 . 9 9 - 1 . 5 - 1 . 0 - 1 . 5 0 .5 18 36 54 72 - 1 . 9 0 8 - 1 . 7 0 4 - 1 . 2 9 7 - 0 . 8 5 3 - 0 . 6 6 - 0 . 6 4 - 0 . 4 7 - 0 . 5 1 - 0 . 4 2 - 0 . 4 4 - 0 . 3 5 - 0 . 3 8 - 0 . 8 2 - 0 . 7 1 - 0 . 7 7 - 0 . 7 4 - 0 . 7 6 - 0 . 7 7 - 0 . 7 8 - 0 . 7 8 - 1 . 6 4 6 - 1 . 2 5 0 - 0 . 8 2 0 - 0 . 4 0 3 0.36 - 0 . 4 4 0.53 - 0 . 3 0 . 0 . 6 0 - 0 . 2 2 0.62 - 0 . 1 5 - 0 . 5 4 - 0 . 6 7 - 0 . 5 3 - 0 . 7 0 - 0 . 5 5 - 0 . 7 3 - 0 . 5 7 - 0 . 7 4 0 .25 18 36 54 72 - 2 . 5 2 5 -2 . 363 - 1 . 9 4 0 - 0 . 6 9 5 - 0 . 7 2 - 0 . 7 0 - 0 . 6 9 - 0 . 6 2 - 0 . 6 8 - 0 . 5 8 - 0 . 7 0 - 0 . 5 6 - 0 . 6 1 - 0 . 5 2 - 0 . 6 2 - 0 . 5 3 - 0 . 6 4 - 0 . 5 5 - 0 . 6 5 - 0 . 5 7 - 2 . 2 9 3 - 1 . 8 8 1 - 1 . 3 1 6 - 0 . 6 7 3 - 0 . 2 1 - 0 . 4 7 - 0 . 1 7 - 0 . 3 8 - 0 . 1 5 - 0 . 3 3 - 0 . 1 3 - 0 . 3 0 - 0 . 4 8 - 0 . 4 8 - 0 . 4 8 - 0 . 4 9 - 0 . 5 0 - 0 . 5 0 - 0 . 5 0 - 0 . 5 2 *Value of function less than 1 0 % of maximum value in the table. TABLE (V-1.4) S e m i - C i r c u l a r P l a t e - U n i f o r m L o a d B e n d i n g Moment M = $- (qR 2/100) y = 0.2 y = 0.4 P e r c e n t a g e E r r o r P e r c e n t a g e E r r o r r/R 8 E l a s . S o l . AR/R-0.12 5 6e=9° AR/R-0.0625 6e-4.5° > E l a s . S o l . AR/R-0.125 6e=9° AR/R-0.0625 6e=4.5° • 3 1 B a r No-Bar Bar No-Bar •H B a r No-Ba_ • B a r No-Bar 0.375 18 36 54 72 90 1.937 2.775 3.185 3.382 3.442 1.7 -11. -0.31 -8.5 -0.40 -7.1 -0.39 -6.4 -0.39 -6.3 -0.19 -2.5 0.05 -1.8 - 0.09 -1.5 0.09 -1.3 0.09 -1.3 2 .073 2 .921 3.333 3.530 3 .589 0.17 -15. -1.2 - 8. -0.96 - 6.6 -0.84 - 6.0 -0.81 - 5.9 -0.64 -2.3 -0.14 -1.7 -0.06 -1.4 -0.03 -1.3 -0.03 -1.2 0.75 18 36 54 72 90 2.954 4.608 5.484 5.917 6 .04 6 -0.54 -5.0 -0.42 -4.1 -0.48 -3.5 -0.47 -3.1 -0.47 -3.1 -0.07 -2.7 -0.29 -2.6 -0.34 -2.6 -0.38 -2.5 -0.40 -2.5 3.199 4 .869 5.738 6 .165 6 .296 -2.0 - 4.7 -1.2 - 3.7 -0.9 - 3.2 -0.79 - 2.8 -0.76 - 2.8 -0.60 -2.4 -0.48 -2.4 -0.47 -2.4 -0.48 -2.3 -0.48 -2.3 0.5 18 36 54 72 90 3.149 5.632 7.300 .8.230 8.527 0.82 -2.3 0.22 -2.2 -0.19 -2.2 -0.37 -2.1 -0.41 -2.1 -0.23 -0.54 -0.03 -0.56 -0.16 -0.59 -0.22 -0.59 -0.23 -0.59 3.481 6 .014 7 .650 8 .545 8.829 -0.64 - 2.5 -0.60 - 2.0 -0.60 - 1.9 -0.58 - 1.8 -0.57 - 1.8 -0.20 -0.63 -0.24 -0.54 -0.26 -0.55 -0.28 -0.55 -0.28 -0.55 0.25 18 36 54 '72 90 .1.919 3.787 5.382 -6 .445 6.816 -0.59 -2.1 0.65 -2.3 0.76 -2.9 0 . 4 6 - 3 . 3 0.31 -3.2 -0.06 -0.38 0.21 -0.45 0 . 2 1 - 0 . 6 1 0.13 -0.74 0.10 -0.75 2.159 4.099 5.662 6.668 7 .015 -0.96 - 2.6 0.03 - 2.3 0.23 - 2.7 0.11- - 2.9 0.03 - 2.7 -0.16 -0.60 0.05 -0.52 -0.77 -0.62 0.05 -0.71 0.03 -0.68 * V a l u e o f f u n c t i o n l e s s t h a n 10% o f maximum v a l u e i n t h e t a b l e . TABLE (V-1.5) S e m i - C i r c u l a r P l a t e - U n i f o r m L o a d B e n d i n g Moment M = $_ (qR 2/100) y = 0.2 y = 0.4 P e r c e n t a g e E r r o r P e r c e n t a g e E r r o r r / R 0 E l a s . AR/R- 0 .125 AR/R-0.0625 E l a s . AR/R- 0 .125 AR/R= =0.0625 S o l . 6e= 9° 6e=4.5° S o l . 6e= 9° Se= =4.5° h B a r No-Bar B a r No-Bar h B a r No-Bar Bar No-Bar 18 1.828 19. -4.1 .4.7 -0.95 2 .049 17. -5.6 4.5 -1.2 36 2.660 16. -2.5 3.8 -0.74 2.939 15. -3.7 3.6 -0.94 0 .875 54 3.167 14. -2.1 3.2 -0.68 3.45 3 13. -3.0 3.2 -0 .81 72 3.448 13. -1.9 3.0 -0.65 3.731 13. -2.7 3.0 -0.76 90 3.538 13. -1.9 2.9 -0.66 3.819 12. -2.7 2.9 -0 .76 18 2 .404 6.5 -3.7 1.5 -1.3 2.830 6.3 -3.8 1.4 -1.6 36 3 .349 7.1 -1.8 1.6 -1.0 4.009 5.7 -2.2 1.3 -1.4 0.75 54 3 .874 6.9 -1.4 1.5 -0.96 4.645 5.5 -1.8 1.2 -1.3 72 4.160 6.7 -1.3 1.5 -0.92 4.980 5.3 -1.6 1.1 -1.3 90 4.251 6.6 -1.4 1.5 -0.95 5.085 5.2 -1.7 1.1 -1.3 18 2.619 0.47 -4.6 -0.21 -1.2 3.108 -0.58 -4.1 -0.3 -1.1 36 3.713 0.31 -2.1 -0.14 -0.62 4 .626 -0.51 -2.0 -0.3 -0 .59 0.5 54 4 .169 0.57 -1.6 -0.05 -0.51 5.376 -0.45 -1.6 -0.3 -0.52 7 2 4 .366 0.79 -1.4 0.02 -0.48 5.738 -0.42 -1.5 -0.3 -0.49 90 4.423 0.86 -1.6 0.04 -0.53 5.848 -0.4 -1.6 -0.3 -0.53 18 •1.715 -1.8 -7.7 0.45 -1.7 2.025 -2.3 -6.8 -0 .56 -1.5 36 2.627 -2.3 -5.6 -0.56 -1.1 3.254 -2.8 -4.9 -0.68 -1.1 0.25 54 2.932 -2.9 -4.9 -0.70 -1.1 3.861 -3.3 -4.3 -0.80 -0.98 72 2.969 ~ -3.3 -" -4 .7 -0.8 -1.1 4 .108 -3 .7 ' -4.3 -0.90 -1.0 - - • 90 2- 95 8 . -3.4 . -3.1. -0.8 -0.72 4.171 -3.8 -3.2 -0.92 -0 .77 * V a l u e o f f u n c t i o n l e s s t h a n 10% o f maximum v a l u e i n t h e t a b l e . TABLE ( V - 1 . 6 ) S e m i - C i r c u l a r P l a t e U n i f o r m L o a d T o r s i o n a l Moment M = 3-, ( q R 2 / 1 0 0 ) y = 0 . 2 y = 0 . 4 P e r c e n t a g e E r r o r P e r c e n t a g e E r r o r r/R 0 E l a s . Sol;.. A R / R = 0 . 1 2 5 . 6e= 9 ° A R / R = 0 . 0 6 2 5 6 e = 4 . 5 ° E l a s . S o l . A R / R = 0 . 1 2 5 6e = 9 ° A R / R = 0 . 0 6 2 5 6 e = 4 . 5 ° 6 3 Bar No-Bar Bar No-Bar 3 3 Bar No-Bar B a r No-Bar 0 . 8 7 5 1 8 3 6 5 4 7 2 8 1 - 2 . 6 6 3 - 1 . 8 5 0 - 1 . 1 5 4 - 0 . 5 5 4 - 0 . 2 7 4 1 . 8 - 2 . 2 2 . 6 - 1 . 5 2 . 3 - 1 . 1 2 . 0 - 1 . 0 2 . 0 - 1 . 0 0 . 3 5 - 0 . 8 6 0 . 0 1 - 0 . 7 4 - 0 . 1 7 - 0 . 6 8 - 0 . 2 6 - 0 . 6 5 - 0 . 2 8 - 0 . 6 5 - 1 . 9 1 7 - 1 . 3 2 3 - 0 . 8 2 1 - 0 . 3 9 3 - 0 . 1 9 4 3 . 5 - 2 . 2 3 . 4 - 1 . 4 2 . 8 - 1 . 0 2 . 4 - 0 . 8 0 2 . 3 - 0 . 7 5 1 . 2 - 0 . 8 6 0 . 7 3 - 0 . 7 3 0 . 4 7 - 0 . 6 6 0 . 3 4 - 0 . 6 2 0 . 3 1 - 0 . 6 1 0 . 7 5 1 8 3 6 5 4 7 2 8 1 - 2 . 9 0 8 - 2 . 1 6 5 - 1 . 3 9 6 - 0 . 6 8 0 - 0 . 3 3 8 3 . 7 - 1 . 9 1 . 3 - 1 . 6 0 . 7 5 - 1 . 4 0 . 5 1 - 1 . 2 0 . 4 6 - 1 . 2 0 . 1 7 - 1 . 4 0 . 3 2 - 1 . 4 0 . 4 5 - 1 . 5 - 0 . 5 - 1 . 5 - 0 . 5 4 - 1 . 5 - 2 . 1 1 4 - 1 . 5 6 7 - 1 . 0 0 7 - 0 . 4 8 9 - 0 . 2 4 3 4 . 4 - 1 . 8 2 . 7 - 1 . 5 2 . 0 - 1 . 2 1 . 7 - 1 . 1 1 . 6 - 1 . 0 0 . 4 5 - 1 . 4 0 . 2 1 - 1 . 4 0 . 0 2 - 1 . 4 - 0 . 0 8 - 1 . 4 - 0 . 1 1 - 1 . 4 0 . 5 1 8 3 6 5 4 7 2 8 1 - 2 . 6 0 0 - 2 . 3 3 3 - 1 . 6 9 8 - 0 . 8 8 1 - 0 . 4 4 4 0 . 3 5 - 0 . 8 0 . 6 5 - 1 . 5 0 . 5 9 - 1 . 8 0 . 5 0 - 1 . 9 0 . 4 8 - 1 . 9 - 0 . 1 9 - 0 . 3 6 - 0 . 1 6 - 0 . 5 6 - 0 . 1 7 - 0 . 6 7 - 0 . 1 9 - 0 . 7 2 - 0 . 2 0 - 0 . 7 3 - 1 . 9 0 6 - 1 . 7 1 1 - 1 . 2 4 4 - 0 . 6 4 5 - 0 . 3 2 5 1 . 2 - 0 . 6 8 1 . 3 - 1 . 4 0 . 9 2 - 1 . 6 0 . 5 6 - 1 . 7 0 . 4 6 - 1 . 7 0 . 0 8 - 0 . 3 6 0 . 0 4 - 0 . 5 4 - 0 . 0 8 - 0 . 6 5 - 0 . 1 8 - 0 . 7 0 - 0 . 2 1 - 0 . 7 1 0 . 2 5 1 8 3 6 5 4 7 2 8 1 - 1 . 5 1 5 - 1 . 6 2 7 - 1 . 3 9 5 " - 0 . 8 0 2 - 0 . - 4 1 5 1 . 4 2 . 0 1 . 2 1 . 1 1 . 3 0 . 0 6 1 . 4 - 0 / 6 3 1 . 5 - 0 . 8 1 0 . 4 5 0 . 6 6 0 . 3 4 0 . 4 2 0 . 3 2 0 . 1 3 0 / 3 0 - 0 . 0 8 0 . 3 0 - 0 . 1 4 - 1 . 1 1 5 - 1 . 2 0 1 - 1 . 0 3 1 - 0 . 5 9 3 - 0 . 3 0 6 3 . 8 1 . 7 4 . 2 0 . 7 7 4 . 5 - 0 . 2 4 4 Y 7 - 0 . 9 1 4 . 7 - 1 . 1 1 . 1 0 . 6 0 1 . 2 0 . 3 4 1 . 2 0 . 0 6 1 . 2 - 0 . 1 4 1 . 2 - 0 . 1 9 * V a l u e o f f u n c t i o n l e s s t h a n 1 0 % o f maximum v a l u e i n t h e t a b l e . TABLE (V-1.7) S e m i - C i r c u l a r P l a t e U n i f o r m L o a d S h e a r Q r = y1 (qR/100) y = 0 .2 y = 0 .4 P e r c e n t a g e E r r o r P e r c e n t a g e E r r o r r/R 6 . E l a s . S o l AR/R -0 .125 6e - 9 ° AR/R -0 .0625 6 e - 4 . 5 ° E l a s . S o l . AR/R -0 .12 5 6e=9° AR/R -0 .06 25 6e = 4 . 5 ° *1 Bar No-Bar Bar No-Bar yl Bar No-Bar B a r No-Bar 0.875 18 36 54 72 90 - 1 4 . 4 3 - 2 1 . 0 4 - 2 4 . 2 9 - 2 5 . 8 7 - 2 6 . 3 5 6.3 4.9 - 0 . 8 9 2 .9 - 0 . 7 6 2 .3 - 0 . 6 0 2 .1 - 0 . 5 4 2 .1 - 0 . 4 6 1.6 - 0 . 1 3 0.96 - 0 . 0 2 0.76 - 0 . 2 0 0.69 ; 0 .01 0.69 - 1 4 . 7 2 - 2 1 . 4 8 - 2 4 . 8 1 - 2 6 . 4 2 - 2 6 . 9 1 2 .2 5 .3 - 0 . 4 7 3 .0 - 0 . 3 4 2 .4 - 0 . 2 6 2 .2 - 0 . 2 4 2 .2 - 0 . 3 1 1.64 - 0 . 0 1 0.9 8 0 .05 0 .78 0 .06 0 .71 0.06 0 .71 0 .75 18 36 54 72 90 -7 .069 - 1 2 . 6 2 - 1 5 . 7 6 - 1 7 . 3 5 - 1 7 . 8 4 - 0 . 3 8 4 .2 - 1 . 2 3.7 - 0 . 9 6 2.9 - 0 . 8 0 2.4 - 0 . 7 4 2 .3 - 1 . 6 24 . - 1 . 2 2 0 . - 0 . 8 1 9 . - 0 . 6 1 8 . - 0 . 6 1 8 . -7 .326 - 1 3 . 0 4 - 1 6 . 2 7 - 1 7 . 9 1 - 1 8 . 4 1 2 .2 4 . 1 - 0 . 3 9 3.8 - 0 . 3 9 2 .9 - 0 . 3 1 2 .6 - 0 . 2 9 2 .5 - 2 . 2 2 1 . - 1 . 3 1 8 . - 0 . 8 1 1 7 . - 0 . 6 1 1 6 . - 0 . 5 6 1 6 . 0 .5 18 36 54 72 90 - 3 . 5 1 8 - 3 . 0 0 3 - 1 . 7 4 2 - 0 . 8 1 8 - 0 . 4 9 5 - 1 . 9 3.5 - 2 . 4 - 3 . 9 3.3 - 1 1 . 1 5 . - 2 4 . 29 . - 1 5 . - 1 . 3 1.3 - 1 . 4 - 1 . 0 - 0 . 3 4 - 3 . 6 1.3 8 .1 2 .5 - 7 . 1 - 3 . 3 0 1 - 2 . 6 1 0 - 1 . 2 3 1 - 0 . 2 4 0 * - 0 . 1 0 4 * - 1 0 . 4 .9 - 5 . 4 - 3 . 2 0.02 - 1 6 . 1 2 . - 8 5 . - 3 0 . 7 9 . - 3 . 0 1.6 - 1 . 8 - 0 . 9 7 - 1 . 5 - 5 . 1 - 6 . 3 - 2 9 . 15 .6 3 9 . 0 .25 18 36 54 7 2 9 0-10 .09 16 .07 19 .23 2 0 .6 9 2 1 . 1 1 5.2 1 1 . 1.7 4 .7 1.2 1.1 1.1 -o:.54 . 1 . 1 - 1 . 6 1.5 2 .8 0 .59 1.4 0 .35 0.5 0 .2 8 0 .2 0 .27 -0-..01 9.89 15 .69 18 .72 20 .10 20 .49 1.2 1 0 . - 1 . 1 4 .6 0 .02 1.5 1.3 0 .08 1.7 - 0 . 7 2 0 .45 2 .8 - 0 . 1 8 1.5 0 .08 0 .73 0 .38 0 .39 0 .48 0 .23 * V a l u e o f f u n c t i o n l e s s t h a n 10% o f maximum v a l u e i n t h e t a b l e . TABLE (V-1.8) S e m i - G i r c u l a r P l a t e - U n i f o r m L o a d S h e a r Q = y 2 (qR 2/100) y = 0. 2 V = 0.4 P e r c e n t a g e E r r o r P e r c e n t a g e E r r o r r/R 0 E l a s . AR/R= 0.125 AR/R= 0.0625 E l a s . AR/R=0 .125 AR/R=0 .0625 S o l . 6e= 9° 6e= 4.5° S o l . 6e=9 0 6e=4 .5° Y2 B a r No-Bar Bar No-Bar Y2 Bar No -B a r B a r No-Bar 18 6.907 2.9 -1.0 3.9 0 . 04 6.274 12. -1.53 2.8 0.95 36 3.66 4 -5.0 0.14 2.8 1.4 3.207 8.6 0.32 2.1 1.8 0.875 54 2 .027 -8.3 0.92 2.3 1.7 1,735 7.4 1.4 1.7 2.2 72 0.920 -9.9 1.3 2.1 1.9 0.777^ 6.7 1.8 1.4 2.4 81 "0.4 49 -10. 1.3 2.0 1.9 0.379 6.6 1.9 1.4 2.5 18 12 .55 3.6 -0.48 1.3 -22 . 11.92 3.0 -0.5 1.0 -23 . 36 6 . 603 3.3 -1.1 1.4 -30 6.129 3.7 -1.2 1.2 -32 . 0.75 54 3 .577 2.5 -0.38 • 1.1 -35. 3.267 3.6 -0 .28 1.2 -37 . 72 1. 601 1.8 0.08 0.84 -38. 1.448 3.4 0.28 1.1 -40 . 81 *0.780 1.6 . 0 .19 0.78 -38. 0 .70 4* 3.3 0.42 1.1 -40 . 18 2 4.03 -1.3 0.31 -0.41 0 .52 23.42 -1.0 0 .55 -0.36 0.57 36 14.59 -0.20 -0.84 -0.16 0.24 14.09 -0 .25 -0.63 -0.24 0.29 0.5 54 8.338 1.0 -1.1 0.11 0.24 7.997 0.50 -0.87 -0.11 0 .28 72 3.807 1.6 -0.78 0.22 0.34 3.636 0.82 -0.56 -0.05 0.39 81 1.862 -0.5 -0.67 0.24 0 .37 1.775 0 .87 -0.43 -0.04 0.43 18 34 .11 -2.1 -0 .018 -0.51 -0.30 33.51 11.4 -0.23 -0.34 -0.30 36 24.67 -2.8 -0.20 -0.66 -0.55 24.17 -1.5 0.19 -0.30 -0.48 0 . 25 54 15.79 -2.6 -1.57 -0.61 -1.1 15.43 -0.79 -1.2 -0.13 -0 .99 72 -7. 6 6 8' -1.7 • -2.5 • -0.47 -1.5' - 7 .480 0.30 -2.2 0.06 -1.35 81 3. 804 -1.3 -2.7 -0.41 -1.6 3.709 0 .67 -2.4 0.11 -1.44 * V a l u e o f f u n c t i o n l e s s t h a n 10% o f maximum v a l u e i n t h e t a b l e . TABLE (V-1.9) P e r c e n t a g e E r r o r s i n S t r e s s e s by N o d a l F o r c e (N.F.) and N o d a l D i s p l a c e m e n t (N.D.) Methods y = 0.2, AR/R = 0.065, 6e = 4.5° 9 = = 9° 6 = 18° 9 = 36° r/R Bar No-Bar B a r No-Bar B a r No-•Bar N.F. N.D. N.F. N.D. N.F. N.D. N.F. N.D. N.F. N.D. N .F . N.D. M r 0.875 0 .75 0 . 5 0.25 -0.34 0.62 0.48 -0.31 . -4.1 -3.4 -1.0 -0 .52 -0.09 -0 .13 -0.74 -3.2 -3.2 -2.8 -0.73 -0.49 0.19 -0.07 0.23 -0.06 -2.9 -3.3 -0.88 -0.46 0.33 0.06 -0.46 -2.4 -2.5 -2.7 -0 .55 -0 .38 0 .05 -0.30 -0.03 -0 .21 -1.76 -3.0 -0.95 -0 .56 0 .29 -0 .08 -0.28 -1.4 -1.8 -2 . 6 -0.56 -0.44 M t 0 .875 0 . 75 0.50 0.25 5.2 1.4 -0.16 -0.41 -2.8 -3.3 -2.5 -1.82 -0 .92 -0.1 0.17 0.32 -2.1 -2.8 -2.3 -2.3 4.7 1.5 -0.2 -0.45 -1.3 -1.7 -1.4 -1.3 -1.7 -0 .57 0.14 0.36 -0 .95 -1.3 -1.2 -1.7 3 .75 1.6 -0 .14 -0.56 -0 .10 -1.3 -0.83 -0.95 -2.1 -1.1 0 .10 0 .50 ' -0.74 -1.1 -CL62 -1.1 M r t 0.875 0 . 75 0.5 0.25 0.55 2.5 -0.18 0.6 0 -1.0 -1.5 -0.24 1.5 -99. -97. -99 . 101. -1.0 -1.3 -0 .18 0.73 0.35 2.3 -0.2 0.45 -1.78 -1.6 -0.53 1.2 -99 . -97 . -99 . 100. -0 . 86 -1.4 -0 .36 0.66 0.01 2.0 -0.16 0.34 -0 .59 -1.5 -0.81 0 .50 -99 . -98. -99 . -99 . -0.74 -1.4 -0.56 0.42 Q r 0.875 0 .75 0.5 0 .25 0 .14 -1.8 0.37 3.4 -0.04 27 . -1.4 -0.67 28 . 38. 13. 19 . 1.5 26. 2.1 4.1 -0.46 -1.6 -1.3 1.5 0.20 23 . -1.8 -0.10 21. 35. -12. 16 . 1.6 24. 1.28 2 . 8 -0.13 -1.2 -1.4 -0.08 -0.09 19 . -3.8 0 .59 13. 23. 100 . -6. 0.96 20. -1.0 1.4 Q t 0.875 0.75 0.5 0 .25 4.4 0.86 -0 .33 -0.4 5.2 -15. -0 .04 -0 .09 -23 0.22 12 . 14. 1.6 -16. 0.58 -0.44 3.9 1.3 -0.42 -0.5 4.8 -20. -0.21 -0.11 -46 . -10. 12 . 16 . 0 .84 -22 . 0.52 -0 .30 2.8 1.4 -0 .16 -0.66 4.8 -28 . -0.43 -0 .45 -79 . -42. 6.3 19. 1.4 -31. 0.24 -0.55 o TABLE (V-1.9) P e r c e n t a g e E r r o r s i n S t r e s s e s by N o d a l F o r c e (N.F.) and N o d a l D i s p l a c e m e n t (N.D.) Methods y = 0.2, AR/R = 0 .065, <5e = 4.5° 6 = = 9° 6 = 18° e = 36° r/R Bar No-Bar Bar No-Bar Bar No--Bar N.F. N.D. N.F. N.D. N.F. N.D. N.F. N.D. N.F . N.D. N .F . N.D . M r 0.875 0.75 0 . 5 0.2 5 -0.34 0 .62 0.48 -0.31 -4.1 -3.4 -1.0 -0.52 -0.09 -0.13 -0.74 -3.2 -3.2 -2.8 -0.73 -0.49 0 .19 -0.07 0.23 -0.06 -2.9 -3.3 -0.88 -0.46 0.33 0.06 -0.46 -2.4 -2.5 -2.7 -0 .55 -0 .38 0.05 -0.30 -0.03 -0 .21 -1.76 -3.0 -0.95 -0 .56 0.29 -0.08 -0.28 -1.4 -1.8 -2.6 -0.56 -0.44 M T 0.875 0.75 0.50 0.25 5.2 1.4 -0.16 -0 .41 -2.8 -3.3 -2.5 -1.82 -0.92 -0.1 0.17 0.32 -2.1 -2.8 -2.3 -2.3 4.7 1.5 -0.2 -0.45 -1.3 -1.7 -1.4 -1.3 -1.7 -0 .57 0.14 0 .36 -0 .95 -1.3 -1.2 -1.7 3.75 1.6 -0 .14 -0.56 -0 .10 -1.3 -0.83 -0.95 -2.1 -1.1 0 .10 0 .50 • -0.74 -1.1 -0..62 -1.1 M r t 0.875 0.75 0.5 0.25 0 .55 2.5 -0 .18 . 0.60 -1.0 -1.5 -0.24 1.5 -99 . -97 . -99 . 101. -1.0 -1.3 -0 .18 0.73 0.35 2.3 -0.2 0.45 -1.78 -1.6 -0.53 1.2 -99. -97 . -99 . 100 . -0.86 -1.4 -0.36 0.66 0.01 2.0 -0 .16 0.34 -0.59 -1.5 -0.81 0 .50 -99 . -98. -99 . -99 . -0.74 -1.4 -0.56 0.42 Q r 0.875 0.75 0.5 0.25 .' 0 .14 -1. 8 0.37 3.4 -0 .04 27 . -1.4 -0.67 28 . 38. 13 . 19 . 1.5 26. 2.1 4.1 -0.46 -1.6 -1.3 1.5 0.20 23. -1.8 -0 .10 21. 35. -12. 16 . 1.6 24. 1.28 2 . 8 -0 .13 -1.2 -1.4 -0.08 -0.09 19 . -3.8 0 .59 13 . 23 . 100 . -6 . 0 .96 20. -1.0 1.4 Q t 0 .875 0.75 0.5 0.25 4.4 0.86 -0 . 33 -0.4 5.2 -15. -0 .04 -0.09 -23 0.22 12. 14. 1.6 -16. 0.58 -0.44 3.9 1.3 -0.42 -0.5 4.8 -20 . -0.21 -0.11 -46 . -10. 12 . 16 . 0 .84 -22 . 0 .52 -0 .30 2.8 1.4 -0 .16 -0 .66 4.8 -28 . -0.43 -0.45 -79 . -42 . 6.3 19. 1.4 -31. 0.24 -0.55 o TABLE (V-1.9) (continued) Percentage Errors in Stresses by Nodal Force (N.F.) and Nodal Displacement (N.D.) Methods jj = 0.2, AR/R = 0 .065, <5e = 4.5° 0 = 54° 0 = 72° 0 = 90° r/R Bar No-Bar Bar No-Bar Bar No-Bar N.F. N.D. N.F. N.D. N.F. N.D . N.F. N.D. N.F. N.D. N.F . N.D . M r 0.875 0.75 0.50 0.25 0.09 -0 . 35 -0 .16 0.21 -1.3 -2.9 -0.98 -0.70 0.23 -0.20 -0.26 -0.87 -1.5 -2.6 -0.59 -0 .61 0.09 -0.38 -0.22 0.13 -1.1 -2 . 8 -0.97 -0.80 0.20 -0.26 -0 .27 -0.65 -1.3 -2.5 -0.59 -0.74 0 .09 -0.40 -0.23 0 .10 -1.1 -2.8 -0.97 -0.8 0.20 -0.28 -0.28 -0.60 -1.3 -2.5 -0 .60 -0.75 M t 0.875 0.75 0 .50 0.25 3.2 1.5 -0 .05 -0.69 -0.94 -1.3 -0.73 -0. 88 -2.1 -1.1 0.09 0.68 -0.88 -0.96 -0.51 -1.1 3.0 1.5 0.02 -0.78 -0.90 -1.2 -0.71 -0.88 -2.1 -1.2 0 .07 0 . 86 -0 .65 -0.92 -0.48 -1.1 2.9 1.5 0 .04 -0.81 -0.92 -1.3 -0.78 -0.51 -2.1 -1.2 0 .07 0.94 -0.66 -0.95 -0.53 • -0.72 M r t 0.875 0.75 0.50 0.25 -0.17 1.8 -0 .17 0.32 -0.55 -1.5 -0.90 0.07 -99 . -98. -99 . -99 . -0.68 -1.5 -0.67 0 .13 -0.26 1.6 -0.20 0.30 -0.53 -1.5 -0.92 -0.17 -99 . -98 . -99. -98. -0.65 -1.5 -0.72 -0.08 - -- -Q r 0.875 0.75 0.50 0.25 -0.02 -0. 82 -0.34 0.35 -0.17 18. -7.0 -0.62 9.7 19 . >100. -29 . 0.76 19 . -3.6 0.54 0.01 -0.6 1.3 0.28 -0.2 17 . -14. -1.1 8.1 16 . >100. -44. 0 .70 18 . -8 . 0 .19 0 .02 -0.6 -3.1 0.27 -0.20 17. -16. -1.4 7.6 15 . >100 . -49. 0 .69 18. -7.1 -0.01 Q t 0.875 0.75 0.50 0.25 2.3 1.1 0.11 -0.61 4.9 -33. -0.43 -0.94 -99 . -56. -4.1 19 . 1.72 -35. 0.24 -1.1 2.1 0.84 0.22 -0.47 4.9 -35 . -0.36 -1.3 110. -68. -12 . 18. 1.9 -38 . 0 .34 -1.5 - - --o TABLE (V-1.10) P e r c e n t a g e E r r o r s i n S t r e s s e s by N o d a l F o r c e (N.F.) and N o d a l D i s p l a c e m e n t (N.D.) Methods y = 0.4, AR/R = 0.065, 6e = 4.5° r/R 0 = 9 ° -^ =ff : — 8 = 18° 8 = 3 6 ° ; Bar No-Bar B a r No-Bar B a r No-Bar N.F . N.D. N.F. N.D. N.F. N.D. N.F. N.D. N.F . N.D . N.F. N.D . M r 0.875 0.75 0.50 0.25 -1.8 -0.78 -0.20 -0 .36 -4.24 -3.7 -1.4 -0.83 -0.22 -0.44 -1.0 -3.3 -3.0 -2.8 -1.0 -0.85 -0.64 -0.59 -0.20 -0.16 -2.7 -3.1 -1.1 -0.68 -0.30 -0. 08 -0.66 -2.7 -2.3 -2.4 -0.63 -0.60 -0 .14 -0.48 -0.24 0.05 -1.6 -2.8 -1.0 -0.68 0 .26 -0 .15 -0.40 -1.8 -1.7 -2.4 -0 .54 -0 .52 M t 0.875 0.75 0.50 0.25 5.3 1.5 -0.3 -0.5 -2.6 -3.3 -2 . 3 -1.6 -1.4 -0 .24 0.14 0.37 -2.3 -2.8 -2.1 -2.1 4.5 1.4 -0.31 -0.56 -1.4 -1.9 -1.3 -1.2 -2.0 -0.7 0.09 0.40 -1.2 -1.6 -1.1 -1.5 3.6 1.3 -0.30 -0.68 -1.1 -1.7 -0.90 -0 .94 -2.3 -1.1 0.02 0.48 -0.94 -1.4 -0 .59 -1.1 M r t 0.875 0.75 0.50 0.25 1.5 4.6 0.07 0.98 -1.0 -1.7 -0.43 1.4 -99. -97.. -99. 101. -1.0 -1.3 -0 .18 0.68 1.2 4.3 0.08 1.1 -0.70 -1.7 -0 .69 1.1 -99 . -97 . -99 . 100. -0.85 -1.4 -0.36 0.60 0.73 3.8 0 .04 1.2 -0.50 -1.5 -0.86 0.45 -99 . -97. -99 . -99. -0.73 -1.4 -0 .54 0.34 Q r 0.875 0.75 0.50 0. 25 -0 . 08 -1.9 -3.6 2.2 -0 .27 25. -1.9 -0.24 20 . 27. 5.7 10. 1.6 23 . 2.3 4.0 -0.32 -2.3 -3 . 0 0.44 -0.2 20. -1.9 0.01 15. 25. -11. 10 . 1.6 21. 1.6 2.8 -0 .01 -1.3 -1.8 -0 .18 -0.27 17 . -4 . -0 .08 -71. 18. -8 8. -6.0 1.7 18. -0.97 1.5 Q t 0.875 0.75 0.50 0.25 3.3 0.85 -0.30 -0.29 5.1 -14. -0.23 -0.15 -20. -0.44 9.0 10. 1.7 -16. 0.60 -0.48 2.8 1.0 -0.36 -0 .34 3.2 -20. -0.32 -0 .21 -41. -9 . 9.1 12. 0.95 -23. 0 .57 -0.30 2.1 1.2 -0 .24 -0.3 2.0 -29. -0 .49 -0.52 9.4 -31. 4.1 14. 1.0 31. 0.29 -0.48 TABLE (V-1.10) P e r c e n t a g e E r r o r s i n S t r e s s e s by N o d a l F o r c e (N.F.) and ^ \ N o d a l D i s p l a c e m e n t (N.D.) Methods ( c o n t i n u e d ) y = 0 ^ £ R / R = 0 . 0 6 5 , 6 e = 4 . 5 ° 0 = 54° e = 72° 0 = 9 0 ° r/R Bar No -Bar B a r NO- B a r Bar No- Bar . N.F. N.D. N.F. N.D. N.F. N.D. N.F. N.D. N.F . N.D . N.F. N.D. M r 0.875 0 .75 0 .50 0 .25 - 0 . 0 6 - 0 . 4 7 - 0 . 2 6 0 .08 - 1 . 2 - 2 . 7 - 1 . 0 - 0 . 7 7 0 .21 - 0 . 2 5 0.36 - 1 . 3 - 1 . 4 - 2 . 4 -0 .55 - 0 . 6 2 - 0 . 0 3 - 0 . 4 8 - 0 . 2 8 0 .05 - 1 . 0 . - 2 . 6 - 0 . 9 9 - 0 . 8 3 0 .18 -0 .30 - 0 . 3 6 - 1 . 1 - 1 . 3 - 2 . 3 - 0 . 5 4 - 0 . 7 1 - 0 . 0 3 - 0 . 4 8 - 0 . 2 8 0 .03 - 0 . 9 9 - 2 . 6 - 1 . 0 - 0 . 8 0 0 .17 - 0 . 3 2 - 0 . 3 6 - 1 . 0 - 1 . 2 - 2 . 3 - 0 . 5 5 - 0 . 6 8 M t 0.875 0 .75 0 .50 0 .2 5 3 .2 1.2 - 0 . 2 9 - 0 . 8 0 - 1 . 0 - 1 . 7 , - 0 . 8 4 - 0 . 9 0 - 2 . 3 - 1 . 2 - 0 . 0 2 0 .57 - 0 . 8 - 1 . 3 - 0 . 5 2 - 0 . 9 8 3.0 1.1 - 0 . 2 8 - 0 . 8 9 - 0 . 9 4 - 1 . 6 - 0 . 8 4 - 0 . 9 0 - 2 . 3 - 1 . 2 -0 .05 0.66 - 0 . 7 6 - 1 . 3 - 0 . 5 0 - 1 . 0 - 2 . 9 1.1 - 0 . 2 8 -0 .92 - 0 . 9 5 - 1 . 7 - 0 . 9 0 - 0 . 6 7 - 2 . 3 - 1 . 3 - 0 . 0 5 0 .70 - 0 . 7 6 - 1 . 3 - 0 . 5 3 -0 .77 M r t -0.875 0 .75 0 .50 0 .25 0 .46 3 .5 - 0 . 0 8 1.2 - 0 . 4 6 - 1 . 5 - 0 . 8 7 0 .06 -99 . -97 . - 9 9 . -99 . -0 .66 - 1 . 4 - 0 . 6 5 0.06 0 .34 3.3 0.18 1.2 - 0 . 4 5 - 1 . 4 - 0 . 8 7 - 0 . 1 5 -99 . -97 . -99 . - 9 9 . - 0 . 6 2 - 1 . 4 - 0 . 7 -0 .14 - - --Q r 0.875 0 .75 0 .50 0 .25 0 .05 -0 . 8 - 1 . 5 0 .08 - 0 . 3 1 16 . - 1 0 . - 0 . 6 3 6.8 1 3 . >100. -23 . 0 .78 17 . - 5 . 1 0 .73 0 .05 - 0 . 6 - 6 . 3 0.38 - 0 . 3 2 1 5 . - 5 0 . - 1 . 1 2 .4 1 1 . >100. - 3 4 . 0 .71 1 6 . -29 . 0 .39 0 .06 - 0 . 5 6 16 . 0 .48 -0 .31 1 5 . 8 5 . - 1 . 4 5 .3 1 1 . >100 . - 3 8 . 0 .71 16 . 3 9 . 0 .23 Q t 0.875 0 .75 0 .50 0 .25 1.7 1.2 - 0 . 1 -0 .13 1.5 - 3 5 . - 0 . 5 6 - 0 . 9 5 - 9 0 . - 4 7 . - 4 . 1 1 5 . 2 .2 - 3 7 . - 0 . 2 8 - 0 . 9 9 1.4 1.1 - 0 . 0 5 0.06 - 1 . 3 - 3 8 . - 0 . 5 5 - 1 . 3 100 - 5 7 . - 1 1 . 1 4 . 5 .6 - 4 0 . 0 .39 - 1 . 4 - - --- •-*-V-a-lue--of. f u n c t i o n l e s s . ,than-10%—of maximum .value i n t h e t a b l e . o CO 109 5-2 Example I I - Clamped C i r c u l a r P l a t e Under An E c c e n t r i c C o n c e n t r a t e d L o a d F i g u r e 5-2 shows a f u l l c i r c u l a r p l a t e w i t h f i x e d edge u n d e r t h e a c t i o n o f an upward c o n c e n t r a t e d l o a d P a t a d i s t a n c e c f r o m t h e c e n t r e . The d e f l e c t i o n i s g i v e n by t h e e q u a t i o n [ 1 ] : 2 P T ? 9 9 9 9 w = ~ ^ [ ( l - n ) ( i - r ) + (n - 2 n C cose) i o g n 2+g 2-2ng cosi l+ri £ - 2 n S Cos( i n w h i c h r , r c n = 5 and e = R S i n c e t h e p l a t e and l o a d i n g a r e s y m m e t r i c a l a b o u t t h e d i a m e t r i c a l l i n e t h r o u g h A, i t i s n e c e s s a r y t o model o n l y one h a l f o f t h e p l a t e . The l i m i t a t i o n on t h e s t o r a g e c a p a c i t y o f t h e computer n e c e s s i t a t e s t h e use o f a r a t h e r c o a r s e mesh w i t h 6e = 15° and AR = 0.2R. T h i s mesh s i z e i s t h e n r e d u c e d by one h a l f t o 6e = 7.5° and AR = 0.1R. The p e r c e n t a g e e r r o r s i n d i s p l a c e m e n t s and s t r e s s e s f o r two v a l u e s o f P o i s s o n ' s r a t i o u = 0.2, 0.4 a r e a s s e m b l e d i n , T a b l e s (V-2.1 t o V-2.8). I n t h e s e t a b l e s , t h e moments and Fig. 5- 2 I l l s h e a r s a r e c a l c u l a t e d u s i n g t h e methods o f n o d a l f o r c e s f o r t h e b a r m o d e l , and n o d a l d i s p l a c e m e n t s f o r t h e n o - b a r m o d e l . T h e s e two d i f f e r e n t methods o f s t r e s s c a l c u l a t i o n were f o u n d i n t h e p r e v i o u s example r e s p e c t i v e l y b e t t e r f o r t h e two t y p e s o f c e l l s u s e d . The a c c u r a c y o f t h e r e s u l t s by t h e b a r m odel a r e a g a i n s e e n t o be i n f l u e n c e d by t h e P o i s s o n ' s r a t i o , and a r e b e t t e r w i t h l o w e r v a l u e o f y. F o r t h e f i n e r mesh t h e e r r o r s i n d i s p l a c e m e n t s and s l o p e s by t h e b a r model a r e l e s s t h a n 1.5% and 3.5% f o r y = 0.2 and y = 0.4 r e s p e c t i v e l y . I n c o m p a r i s o n w i t h t h e s e t h e n o - b a r model g i v e s s l i g h t l y l e s s a c c u r a t e r e s u l t s w i t h y = 0.2 and more a c c u r a t e w i t h y = 0.4. G r e a t e r e r r o r i n s t r e s s e s t h a n i n d i s p l a c e m e n t s i s n o t e d , and t h i s i s p a r t i c u l a r l y t r u e f o r s t r e s s e s f o u n d by t h e N.D. method. The e x a c t s t r e s s e s a t t h e p o i n t o f a p p l i c a t i o n o f t h e c o n c e n t r a t e d l o a d a r e i n f i n i t e , w h i l e i n t h e f i n i t e - e l e m e n t s o l u t i o n s t h e y a r e n a t u r a l l y f i n i t e . S i n c e t h e d i s p l a c e m e n t s and s t r e s s e s change q u i t e r a p i d l y i n t h e v i c i n i t y o f t h e c o n c e n t r a t e d l o a d , l a r g e e r r o r s i n t h e f i n i t e e l e m e n t s o l u t i o n s i n t h e a r e a a d j o i n i n g t h e l o a d a r e i n e v i t a b l e . I n g e n e r a l , s t r e s s e s o b t a i n e d by t h e b a r model a r e b e t t e r e x c e p t when y has a l a r g e V a l u e when t h e r e v e r s e i s t r u e . 112 I n s p e c t i o n o f T a b l e s (V-2.1 t o V-2.8) shows t h a t c o n v e r g e n c e o f d e f l e c t i o n i s r a p i d i n b o t h t y p e s o f m o d e l , ans so i s t h e c o n v e r g e n c e i n s t r e s s e s , a l t h o u g h a t a some-what l o w e r l e v e l o f p r e c i s i o n . TABLE (V-2.1) Clamped Circular Plate-Concentrated Load 2 Transverse Deflection w = a, (PR /100D) U = 0.2 V = 0.4 Percentage Error — — — - —— Percentage Error r/R 9 Elas. AR/R = AR/R = Elas. AR/R AR/R Sol . 0.2 0.1 Sol. 0.2 0.1 6e = 15° 6e = 7 .5° 6e = 15° <5e = 7 . 5 ° a1 Bar No-Bar Bar No-Bar <*1 Bar No-Bar Bar No-Bar 0 0.264 13. 3.1 3.2 0.85 0 .264 31. 3.0 8.0 0 .04 30 0 .209 11. 3.9 2.8 1.0 0 .209 30. 4.0 7.5 1.1 0.8 60 0.135 7.2 3.6 1.9 0.96 0 .135 25. 3.6 6.0 1.0 90 0.091 4.1 4.2 1.1 1.1 0 .091 18. 4.2 4.4 1.2 120 0.069* 1.9 4.7 0.51 1.3 0.069* 14. 4.8 3.4 1.4 150 0.059* 0.8 5.0 0.22 1.4 0 .059* 12 . 5.0 3.0 1.4 180 0.056X 0.5 5.1 0.14 1.4 0.056 11. 5.1 2.8 1.4 0 0 .857 4.5 3.8 1.2 1.0 0 . 857 11. 4.3 2.9 1.1 30 0.692 5.1 4.3 1.2 1.1 0.692 12. 4.8 3.0 1.3 60 0.473 4.1 4.4 - 1.0 1.2 0.473 11. 4.9 2.7 1.3 0.6 90 0.335 2.9 5.0 0.71 1.3 0.335 9.0 5.5 2.2 1.5 120 0.2 61 1.9 5.4 0.48 1.5 0.261 7.3 6.0 1.8 1.7 150 0.224 1.4 5.7 '0.34 1.5 0.224 6.3 6.2 1.6 1.7 180 0 . 213 1.2 5.8 0.30 1.6 0.213 6.1 6.3 1.5 1.7 0.4 0 ( D 1.403 2.5 4.4 0.59 1.3 1.403 5.2 5.1 1.2 1.5 0 1.361 2.4 4.6 0.61 1.4 1.361 5.3 5.6 1.3 1.6 45 1.201 2.2 5.4 0.56 1.5 1.201 5.2 6.5 1.3 1.8 0.2 90 0.961 2.1 6.0 0.57 1.7 0.961 5.0 7.2 1.3 2.0 135 0.810 2.5 6.8 0.62 1.8 0.810 5.4 7.9 1.4 2.2 180 0.762 2.8 '6.9 0.65 1.9 0.762 5.7 ' 8.1 1.4 2.3 (1) Under Load * Value of function less than 10% of maximum value in the table. TABLE (V-2.2) Clamped C i r c u l a r P l a t e - C o n c e n t r a t e d L o a d T a n g e n t i a l R o t a t i o n 9 m = a„ (PR/100D) y = 0 .2 li = 0 . 4 P e r c e n t a g e E r r o r P e r c e n t a g e E r r o r r/R e E l a s . AR/R = AR/R = E l a s . AR/R z= AR/R = S o l . 0.2 0.1 S o l . 0.2 0.1 • 6e = 15° . 5e = 7 .5° 6e = 15° <5e = 7.5° a 2 B a r No -Bar B a r No-Bar a2 B a r No-Bar Bar Nc - B a r 0 2 . 397 -4.0 3.4 -0.79 0.88 2 .397 -7.1 4.1 -1.6 1.0 30 1.927 -2.7 4.4 -0.72 1.1 1.927 -4.4 5.4 -1.0 1.3 60 1.277 -1.1 4.5 -0.30 1.1 1.277 -0.7 5.3 -0 .10 1.3 0 . 8 9 0 0 . 881 -0.07 4.9 -0.02 1.3 0.881 1.1 5.8 0 .30 1.5 120 0 .673 -0.40 5.3 0.10 1.4 0.673 1.7 6.2 0.44 1.7 150 0.574 0.64 5.5 0.15 1.5 0 .574 2.0 6.4 0 .49 1.7 180 0. 545 0.74 5.5 0.16 1.5 0 .545 2.1 6.4 0.50 1.7 0 3.250 -5.4 3.4 -0.92 0.83 3.250 -8.8 4.5 -1.7 1.0 30 2.667 -3.3 3.9 -0.84 0.95 2 .667 -5.2 4.9 -1.2 1.2 60 1.97 8 -1.4 4.7 -0.33 1.2 1.978 -1.4 5.7 -0 .28 1.5 0.6 90 1.500 0.04 5.3 0.03 1.4 1.500 1.1 6.4 0 .31 1.7 120 1.214 0.77 5.6 0.22 1.5 1.214 2.2 6.7 0.58 1.8 150 1.067 1.2 5.7 • . 0.30 1.6 1.067 2.8 6.9 0 .69 1.9 180 1.022 1.4 5.8 0.32 1.6 1.022 3.0 6.9 0.73 1.9 0.4 o d ) 1.337 -17. -2.1 -4.7 -1.4 1.337 -28. - 1.8 -7.5 -1.5 0 -1.034 2.0 26 . 0.38 4.9 -1.034 8.1 32 . 1.1 4.3 45 0.133 27. -13. 7.0 5.0 0.133 40. - 11. 8.8 5.4 0.2 90 1.184 7.7 3.7 2.5 1.7 1.184 12. 5.3 3.6 2.3 135- •1.545 6.7 4-.7- 1.-7 1.5 • 1.545 10. 6.9 2.8 2.2 180 1.618 6.3 5.0 1.5 1.5 1.618 10. 7.6 2.6 2.2 (1) Under L o a d * V a l u e o f f u n c t i o n l e s s t h a n 10% o f maximum v a l u e i n t h e t a b l e . TABLE (V-2.3) Clamped C i r c u l a r P l a t e - C o n c e n t r a t e d L o a d R a d i a l R o t a t i o n 9 D = a., (PR/100D) y = 0. 2 M = 0 .4 P e r c e n t a g e E r r o r P e r c e n t a g e E r r o r r/R 9 E l a s . . AR/R — AR/R = E l a s . AR/R = AR/R = S o l . 0.2 0.1 S o l . 0.2 0.1 6e = 15° 6e = 7.5° 6e = 15° 6e = 7.5° . a3 B a r Nc -Ba r B a r No-Bar a 3 Bar No B a r B a r No-Bar 15 0.146 -0.25 2.3 1.9 -•1.1 0 .146 34 . 1.1 11. -0.43 30 0 .181 0.48 8.1 2.1 1.3 0.181 36. 8.0 11. 1.3 60 0 .103 1.6 6.5 1.6 0.87 0.103 39. 6.9 11. 0 .89 0.8 90 0 .052 -0.27 5.4 0.66 0.62 0.052 35 . 5.5 9 . 0.65 120 0.025 -2.3 6.3 -0.23 0.91 0.025 30. 6.7 7.4 1.0 150 0.015 -4.5 6.5 -0.76 1.2 0.015 26. 7.5 6.6 1.4 165 * 0.007 -5.3 6.4 -0.89 1.3 *0.007 25. 7.6 6.3 1.5 15 0 .597 1.8 0.98 0.60 0.55 0.597 8.2 0.54 2.5 0 .40 30 0 .780 5.0 4.7 1.2 1.3 0.780 12 . 5.3 3.3 1.5 60 0.571 5.2 3.4 1.5 0 .79 0.571 16. 3.9 4.0 0.90 0 . 6 90 0,321 5.2 3.4 1.4 0.74 0 .321 15. 3.7 3.9 0.88 120 0.166 4.5 4.1 1.1 1.0 0.166 14. 4.8 3.3 1.2 150 0.071 3.1 4.3 0.92 1.3 0.071 12. 5.4 3.0 1.5 165 ^0 .034 2.6 4.2 0.87 1.3 *0.034 11. 5.5 2.9 1.6 45 1.418 2.2 2.5 0.58 0.72 1.418 5.3 3.1 1.5 0.84 0.4 90 0.759 3.5 3.3 0.80 0.78 0.759 8.1 3.9 1.9 0.98 135 0.299 2.6 4.3 0.77 1.2 0 .299 7.0 5.5 1.8 1.5 45 1.606 3.6 2.0 0.81 0.87 1.606 6.1 2.5 1.4 1.0 0.2 90 1.296 1.3 3.2 0.34 0.73 1.296 4.3 4.1 1.1 0.98 135 0.623 -2.1 2.0 0.24' 0.89 0.623 1.2 '3.2 1.0 1.3 * V a l u e o f f u n c t i o n l e s s t h a n 10% o f maximum v a l u e i n t h e t a b l e . TABLE (V-2.4) Clamped C i r c u l a r P l a t e - C o n c e n t r a t e d L o a d B e n d i n g Moment M = 3-, P/10 y = 0.2 y = 0.4 P e r c e n t a g e E r r o r P e r c e n t a g e E r r o r r/R e E l a s . AR/R = AR/R E l a s . AR/R - AR/R = S o l . 0.2 0.1 S o l . 0.2 0.1 6e = 15° 6e = 7.5° 6e = 15° 6e = 7.5° Bar No -Ba r B a r No Bar *1 B a r No-Bar B a r No Bar 0 -1.560 -2.6 5.0 0.72 0.97 -1.560 -5.4 5.6 -1.6 1.1 30 -1.202 -0.25 9.8 0.03 2.1 -1.202 -0.69 11. -0 .05 2.3 1.0 60 -0.739 2.1 10 . 0 .62 2.3 -0.739 4.0 11. 1.2 2.5 90 -0.484 2.4 9.3 0.65 2.1 -0.484 4.6 10. 1.2 2.3 120 -0.360 2.1 7.9 0.5 3 1.9 -0.360 4.1 8.7 1.0 2.1 150 -0.303 1.8 6.9 0 .44 1.7 -0 .303 3.7 7.6 0.90 1.9 180 -0.286 1.8 6.6 0.42 1.7 -0.286 3.6 7.3 0.86 1.9 0 0.232 -16 . 118 . -9.8 13 . 0.432 -19 . 67 . -7.2 7.3 30 * 0.100 -27. 78 . -5.8 22 . 0.191 -13. 48 . -2.3 13. 60 -0.107 11. -21. 2.8 - 4.8 -0.060 15 . -41. 4.2 -8.9 0 . 6 90 -0.180 6.6 -4.4 1.7 - 0.72 -0.143 12. • -5.1 3.2 -0 .72 120 -0.192 6.4 0.77 1.7 0.50 -0.159 12 . 1.8 3.3 0 .87 150 -0.189 6.8 3.5 1.7 1.2 -0.158 13. 5.2 3.3 1.7 180 -0.186 7.0 4.3 1.7 1.3 -0.156 13 . 6.0 3.3 1.9 0 0.721 -22. 129 . -16 . 1.3 0.095 -25 . 98 . -12 . -0 .33 30 0.904 -7.0 62 . 2.9 2.6 1.014 -12 . 58 . 0.86 3.1 0.2 60 0 .786 -1.5 21. -1.5 - 2.4 0.816 -1.0 23. -1.2 -1.6 90 0.479 -11. 3.3 -5.3 0.84 0.516 -9.6 5.4 -4.5 1.8 120 0 .224 -31. -5.7 -7.0 6.5 0.285 -27 . -2.5 -6.1 6.0 150 * 0.075 -90. ' " -7.9 "-15Y 18. * 0.154 -56 . '-2.7 -11. 8.9 180 ^0.027 17 0. 2.0 -40. 45. ^ 0 . 1 1 2 -80. 1.7 -15 . 10 . * V a l u e o f f u n c t i o n l e s s t h a n 10% o f maximum v a l u e i n t h e t a b l e . TABLE (V-2.5) Clamped C i r c u l a r P l a t e - C o n c e n t r a t e d L o a d B e n d i n g Moment M = 3 9 (P/10) U = 0.2 u = 0.4 P e r c e n t a g e E r r o r P e r c e n t a g e E r r o r r/R 9 E l a s . S o l . AR/R = 0.2 6e = 15° AR/R = 0.1 6e = 7.5° E l a s . S o l . AR/R = 0.2 6e = 15° AR/R = 0.1 6e = 7.5° ^2 B a r No-Bar B a r No-Bar e2 B a r No-Bar B a r No-Bar 1.0 0 30 60 90 120 150 180 -0.312 -0.240 -0.148 -0.097 -0.072 -0.061 -0.057 87. 5.4 71. 10. 35. 10. 7. 8.7 -10. 7.5 -19. 6.6 -22. 6.4 52. 1.1 39. 2.2 17. 2.2 2.7 2.0 -5.1 1.8 -8.8 1.7 -9.9 1.6 -0.624 -0.481 -0.296 -0.194 -0.144 -0.121 -0.114 40. 5.9 36. 11. 25. 11. 14. 9.8 6.7 8.5 3.2 7.5 2.2 7.1 27. 1.2 22. 2.3 13. 2.5 7.3 2.3 4.1 2.1 2.6 1.9 2.1 1.9 0.6 0 30 60 90 120 150 180 1.009 0.457 0.206 0.143 0.120 0 .109 0.106 -1.7 11.7 10. 14. 9.2 8.9 4.8 8.7 1.6 7.3 0.98 6.5 1.1 7.3 0.95 2.8 2.1 3.2 2.2 2.2 0.95 2.3 0.41 2.0 0.25 1.7 0.22 1.8 1.015 0.459 0.175 0.100 * 0.075 * 0.065 *0.062 5.8 17. 19. 18. 15. 14. 10. 15. 6.4 12. 4.8 8.5 4.6 9.0 2.3 3.0 4.6 4.3 3.9 3.3 2.5 3.6 1.8 3.0 1.4 2.1 • 1.2 2.1 0.2 0 30 60 90 120 150 180 1.252 0 .706 0 . 306 0.271 0.337 0.393 0.412 -0.13 4 .6 0.93 23. -3.4 26. -0.41 6.0 6.5 4.2 11. 6.8 - 12. 8.9 1.1 1.6 -0.74 3.4 -0.77 1.9 4.1 4.0 3.3 3.3 2.4 2.6 2.1 2.7 1.350 0.865 0.457 0.360 0.370 0.392 0.401 I . 8 17. -0.47 32. -77. 26. -4.6 8.5 5.2 5.9 I I . 8.9 •12. • 11. 1.2 -1.5 -0.63 3.7 -2.2 1.1 2.0 4.2 2.7 4.8 2.1 4.0 1.9 3.9 * V a l u e o f f u n c t i o n l e s s t h a n 10% o f maximum v a l u e i n t h e t a b l e . TABLE (V-2.6) Clamped C i r c u l a r P l a t e - C o n c e n t r a t e d L o a d T o r s i o n a l Moment M = • $ (P/100) y = 0.2 y = 0.4 P e r c e n t a g e E r r o r P e r c e n t a g e E r r o r r/R e E l a s . S o l . AR/R = 0.2 . fie = 15° AAR/R = 0.1 •fie = 7.5° E l a s . S o l . AR/R = 0.2 fie = 15° AR/R = 0.1 fie = 7.5° &3 B a r No-Bar Bar No-Bar ^3 B a r No-Bar B a r No-Bar 0.8 15 30 60 90 120 150 165 1.156 1.577 • 1.144 0.617 0.310 0.210 *0.062 32. -9.1 30. -2.4 23. -0.92 14. 1.0 8.3 2.3 4.1 2.5 2.9 2.6 4.2 -3.9 4.1 -1.2 3.6 -0.80 2.3 -0.45 1.2 0.11 0.58 0.52 0.42 0.63 0 .867 1.182 0.858 0.463 0.233 0.098 0 .047 35. -9.4 30. -4.0 28. -0.94 20. 0.52 14. 2.6 9.1 3.8 7.9 4.0 7.2 -3.0 6.6 -1.1 6.0 -0.67 4.4 -0.30 3.0 0.32 2.1 0.79 1.9 0.91 0.6 15 30 60 90 12 0 150 165 2.553 3.008 2.256 1.376 0.753 0 . 330 0.160 35. -10. 19. -5.7 10. -3.9 7.0 -1.5 4.4 0.24 1.8 0.93 1.0 1.0 4.2 -0.82 4.6 -1.1 2.7 -0.93 1.8 -0.38 1.1 0.14 0.64 0.50 0.53 0.58 1.914 2.256 1.692 1.032 0.565 0 .248 0 .120 58. -9.3 23. -4.7 10. -3.4 6.0 -0.96 2.3 1.2 -0.82 2.3 -1.8 2.5 9.7 -0.61 5.4 -0.85 2.7 -0.79 1.5 -0.19 0.57 0.40 0.01 0.82 -0.13 0.93 0.4 45 90 135 2.422 2 .081 0.987 15.0 -5.1 8.2 -3.2 1.8 -2.4 4.9 -1.2 1.5 -0.84 0.56 -0.31 1.817 1.560 0.741 32. -5.5 8.2 -2.6 -4.2 -1.1 9.4 -1.2 1.2 -0.63 -1.3 0.02 0.2 45 90 135 -1.094 1.976 1.635 -14. 75. -13. 19.. -48. 12. 1.4 -20. 3.1 1.2 0.15 -2.6 -0 . 820 1.481 1.-226 -8.0 82. -33. 20. -55-. 11. 5.4 -19. 7.4 0.95 -0.49 -2.4 * V a l u e o f f u n c t i o n l e s s t h a n 10% o f maximum v a l u e i n t h e t a b l e . TABLE (V-2.7) Clamped Circular Plate-Concentrated Load Shear Q r = Y 2 (P-10R) U = 0 .2 V = 0.4 Percentage Error Percentage Error r/R e Elas. AR/R AR/R Elas. AR/R AR/R Sol. 0.2 0.1 Sol. 0 .2 0.1 6e = 15° 6e = 7.5° 6e = 15° 6e = 7 .5° Bar. Nc -Bar Bar No -Bar Y2 Bar No-Bar Bar No -Bar 0 -5.199 -2.7 8.1 -0 .62 0.65 -5 .199 0.65 8.2 -1.3 0.73 30 -3 . 363 1.1 32. -0 .15 11.8 -3.363 0.26 33. 0 .30 12. 1.0 60 -1.555 1.8 49. 0.58 20 . -1.555 4.3 49 . 1.3 20 . 90 -0 .835 1.4 44. 0 .37 17. -0.835 2.7 44. 0.62 17 . 120 -0.554 0.14 31. 0.03 12. -0.554 0 .54 31. 0.08 12. 150 -0.440 -0.71 21. -0 .19 7.9 -0.440 -0 .71 21. -0.20 7.8 180 -0.409 -0.87 18. -0 .27 6.5 -0.409 -1.0 17 . -0.29 6.4 0 -9 .721 -9.4 5.6 0 .12 -3.8 -9.721 -7.3 7.0 0.44 -4.2 30 -5.056 1.5 4.8 -0 .05 1.0 -5.056 1.1 5.8 -0 .01 1.3 60 -2.515 2.6 3.6 0 . 87 0.85 -2.515 1.2 4.5 0.53 1.1 0.6 90 -1.462 3.2 5 . 3 0 . 82 1.3 -1.462 1.8 6.2 0.40 1.6 120 -0.992 1.7 5.8 0.46 1.5 -0.992 0 .03 6.4 -0.02 1.7 150 -0 .788 0.50 5.8 0 .11 1.5 • -0.788 -1.4 5.9 -0 .36 1.5 180 -0 .730 0 .16 5.8 -0.03 1.4 -0.730 -1.7 5.7 -0.48 1.4 0 6.634 4.9 -7 6. 11. -10. 6 .634 -2.1 -•72. 15. -14 . 30 2.711 -19. 199 . -5.4 -26 . 2 .711 -19 . 213. -12 . -23 . 60 -0.502 89. 432 0.05 70. -0 .502 -28 . 455. -52 . 70. 0.2 90 -1.456 21. -2.5 4.3 5.3 -1.456 6.9 0 . 32 2.0 8.3 120 -1.633 -5.' -19 . 0.64^  •3.3 —•1.6 33 -3.4 -•14 .- 2.8 6.5 150 -1.622 -37. -3.8 -1.7 5.7 -1.622 -30. 4.2 -0.63 7.7 180 -1.605 -53. 11. -2.2 8.6 -1.605 -44. 20 . -2.1 9.9 * Value of function less than 10% of maximum value in the table. TABLE (V-2.8) Clamped Circular Plate-Concentrated Load Shear Q = y (P/10R) u = 0. 2 = 0.4 Percentage Error Percentage Error r/R 9 Elas . AR/R — AR/R _ Elas . AR/R AR/R Sol . 0.2 0.1 Sol. 0.2 0.1 fie =' 15° fie = 7 .5° fie = 15° fie •= 7 . 5 ° YI Bar No -Bar Bar No -Bar Tl Bar No -Bar Bar No-Bar 15 0.775 -79 . -30. -39 . 12. 0.775 -63. -38. -27. -•1.2 30 1.029 -65. -38. -32. 17. 1.029 -50 . -45. -21. 4.1 1.0 60 0 .674 -39 . -28. -21. 32 . 0.674 -28 . -36 . -11. 7.0 90 0 .334 -27. -4.4 -16. 44. 0.334 -15. -15. -6.7 28. 120 * 0.160 -22. 13. -14. 52 . * 0.160 -9.2 0.46 -4.8 35 . 150 * 0.065 -21. 22. -13. 57 . x 0.065 -7.3 11. -4.1 39 . 165 *0.031 -21. 24. -14. 58. *0 .031 -7.0 11. -3.9 40. 15 -0 .228 34. -9.6 4.3 -2.5 -0.228 49. -13. 2.5 - 4.1 30 -1.944 24. 15. 4.1 5.0 -1.944 36 . 18. 7.7 5.5 60 -0.642 21. 2.1 3.6 0.58 -0 .642 26 . 2.1 7.6 0 .66 0.6 90 -0.172 19 . 3.5 4.7 0.97 -0.172 25. 2.4 7.4 0 .94 120 -0.041 -4.9 16 . 3.2 5.2 -0.041 16 . 15. 3.5 5.2 150 * 0.008 -96. 22 . -29. 14 . -0.008 -35. 25. -7.6 14 . 165 "^ 0 .003 164. 20 . 25. 19. -0.003 -76. 26 . -15 . 19 . 15 -3 .242 31. 150 . 13. -53 . -3.242 27. 145. 8.0 - 61. 30 -4.462 16. 155. 7.3 20. -4.462 13. 154. 7.2 20 . 60 -3.463 -9.2 53. -10. 1.2 -3.463 -15. 53. -7.7 1.8 0.2 9 0 -2 .026 -38. -38. -2.6 -3.9 -2.026 -37. -37. 0.33 - 3.6 120 -1.072 -64. 106 7.1 -0.82 -1.072 -56 . 105. 6.5 0.29 - • • ' -150 * 0 .460 -81. 142- • '9.0 1.7 -0.460 -68. 142 . 4.4 4.1 165 *-0 .221 -85. 15 0. 8.5 2.2 -0.221 -69. 150. 2.7 5.1 * Value of function less than 10% of maximum value in the table. 121 CHAPTER V I CONCLUSION The f e a s i b i l i t y o f a p p l y i n g framework model made up o f i s o s c e l e s t r a p e z o i d c e l l s t o p r o b l e m s o f p l a t e f l e x u r e has b een d e m o n s t r a t e d on two e x a m p l e s . One i n v o l v i n g a s e m i - c i r c u l a r f i x e d - e n d e d p l a t e u n d e r u n i f o r m l o a d , and t h e o t h e r a s i m p l y s u p p o r t e d c i r c u l a r p l a t e u n d e r an e c c e n t r i c c o n c e n t r a t e d l o a d . B o t h b a r and n o - b a r c e l l s have been u s e d f o r c o m p a r i s o n . The p r e s e n t i n v e s t i g a t i o n l e a d s t o t h e f o l l o w i n g c o n c l u s i o n s . 1. The a c c u r a c y o f d i s p l a c e m e n t s and s t r e s s e s u s i n g t h e b a r c e l l s i s g e n e r a l l y g ood. A t t h e same t i m e i t i s a p p r e c i a b l y a f f e c t e d by t h e P o i s s o n ' s r a t i o , and i s b e t t e r w i t h t h e l o w e r v a l u e o f p. 2. D i s p l a c e m e n t s o b t a i n e d w i t h t h e b a r model a r e b e t t e r t h a n t h o s e w i t h t h e n o - b a r model f o r s m a l l v a l u e s o f y. F o r l a r g e y s u c h as 0.4, t h e r e v e r s e i s t r u e . 3. The method o f n o d a l f o r c e s (N.F.) y i e l d s e x c e l l e n t r e s u l t s f o r b e n d i n g moments i n b o t h t y p e s o f m o d e l . F o r t o r s i o n a l moments and s h e a r s , t h e N.F. method i s good with the bar cel ls but is less satisfactory and sometimes grossly in error with the no-bar ce l l s . The method of nodal displacements (ND) is applicable to both types of ce l l s and produces excellent results in calculation of torsional moments. Other stress results are good for both types of c e l l . The following table gives a comprehensive com-parison between the results obtained by the bar and no-bar ce l l s . The method giving better results is i n -dicated by the abbreviation NF or ND. The asterisk indicates the c e l l type which yields better accuracy. Bar Ce l l No-Bar C e l l y=0.2 y=0.4 y=0 .2 y=0.4 Deflection * * Bending Moments NF* NF* NF NF Torsional Moments NF* ND ND ND* Shears NF* NF ND ND* The geometry of the cel ls as defined by the ratios k and k^ undoubtedly exerts a great influence on the results obtained, part icularly with the bar ce l l s . Unfortunately, this limited study does not reveal any conclusive substance regarding this subject. 123 6. D e f l e c t i o n s and s l o p e s d e t e r m i n e d by b o t h t y p e s o f model c o n v e r g e r a p i d l y t o w a r d s t h e t r u e v a l u e s as t h e mesh s i z e i s r e d u c e d . S t r e s s e s when u s i n g N.F. method i n b a r model and N.D. method i n n o - b a r model a l s o show good t r e n d o f c o n v e r g e n c e . 124 REFERENCES 1. T i m o s h e n k o , S. and S. W o i n o w s k y - K r i e g e r . , " T h e o r y o f P l a t e s and S h e l l s . " M c G r a w - H i l l , New-York, 2nd e d . 1959 . 2. H r e n n i k o f f , A., " S o l u t i o n o f P r o b l e m s o f E l a s t i c i t y by t h e Framework Method," J . A p p l . Mech., A.S.M.E., Dec. 1941. 3. P e s t e l , E . , " I n v e s t i g a t i o n o f P l a t e and S h e l l M o d e l s by M a t r i c e s , " R e p o r t t o O f f i c e o f A e r o s p a c e R e s e a r c h , U.S. A i r F o r c e , B r u s s e l s , B e l g i u m , 1963. 4. McCormick C.W., " P l a n e S t r e s s A n a l y s i s , " P r o c e e d i n g s o f t h e ASCE, S t r u c t u r a l D i v i s i o n , A u g u s t 1963. 5. Y e t t r a m , A . L . and K. R o b b i n s , "Space Framework Method f o r T h r e e - D i m e n s i o n a l S o l i d s , " P r o c e e d i n g s o f t h e ASCE, E n g i n e e r i n g M e c h a n i c s D i v i s i o n , Dec. 1967. 6. G a n t a y a t , A., " R e c t a n g u l a r B a r and No-Bar F i n i t e E l e m e n t f o r T h r e e - D i m e n s i o n a l S t r e s s A n a l y s i s , " M.A. S c . T h e s i s , D e p t . o f C i v i l E n g i n e e r i n g , U n i v e r s i t y o f B r i t i s h C o l u m b i a , V a n c o u v e r , 1968. 7. T u r n e r , M.J., R.W. C l o u g h , H.C. M a r t i n and L . J . Topp, " S t i f f n e s s and D e f l e c t i o n A n a l y s i s o f Complex S t r u c t u r e s , " J o u r n a l o f t h e A e r o n a u t i c a l S c i e n c e s , V o l . 23, No. 9, S e p t . 1956. 8. H r e n n i k o f f , A., "The F i n i t e E l e m e n t Method A p p l i e d t o P l a n e S t r e s s and B e n d i n g o f P l a t e s , " L e c t u r e n o t e s . D e p t . o f C i v i l E n g i n e e r i n g , U n i v e r s i t y o f B r i t i s h C o l u m b i a . 9. A g r a w a l , K.M., " A n a l y s i s o f E l a s t i c S h e l l s o f R e v o l u -t i o n W i t h Membrane and F l e x u r e S t r e s s e s Under A r b i t r a r y L o a d i n g U s i n g T r a p e z o i d a l F i n i t e E l e m e n t s , " Ph.D. T h e s i s , D e p t . o f C i v i l E n g i n e e r i n g , U n i v e r s i t y o f B r i t i s h C o l u m b i a , V a n c o u v e r , A p r i l , 1968. 10. H r e n n i k o f f , A., " P r e c i s i o n o f F i n i t e E l e m e n t Method i n P l a n e S t r e s s , " P u b l i c a t i o n p e n d i n g . 11. S a l v a d o r y , M.G., and M.L. B a r o n , " N u m e r i c a l Methods i n E n g i n e e r i n g , " P r e n t i c e - H a l l I n t l . , 2nd'ed., 1961. 

Cite

Citation Scheme:

        

Citations by CSL (citeproc-js)

Usage Statistics

Share

Embed

Customize your widget with the following options, then copy and paste the code below into the HTML of your page to embed this item in your website.
                        
                            <div id="ubcOpenCollectionsWidgetDisplay">
                            <script id="ubcOpenCollectionsWidget"
                            src="{[{embed.src}]}"
                            data-item="{[{embed.item}]}"
                            data-collection="{[{embed.collection}]}"
                            data-metadata="{[{embed.showMetadata}]}"
                            data-width="{[{embed.width}]}"
                            data-media="{[{embed.selectedMedia}]}"
                            async >
                            </script>
                            </div>
                        
                    
IIIF logo Our image viewer uses the IIIF 2.0 standard. To load this item in other compatible viewers, use this url:
https://iiif.library.ubc.ca/presentation/dsp.831.1-0050567/manifest

Comment

Related Items