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Plane stress analysis with isosceles trapizoidal bar cells Charania, Hajimohammed Gulamhussin 1968

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PLANE STRESS ANALYSIS WITH ISOSCELES TRAPIZOIDAL BAR CELLS by HAJIMOHAMMED GULAMHUSAIN CHARANIA B.Tech. (Hons.), Indian I n s t i t u t e of Technology, Bombay, I n d i a , A p r i l 1 9 6 6 . A THESIS SUBMITTED IN PARTIAL FULFILMENT OF THE REQUIREMENTS FOR THE DEGREE OF MASTER OP APPLIED SCIENCE i n the Department of CIVIL ENGINEERING We accept t h i s t h e s i s as conforming to the required standard THE UNIVERSITY OF BRITISH COLUMBIA SEPTEMBER, 1968 In p r e s e n t i n g t h i s t h e s i s in p a r t i a l f u l f i l m e n t of the requirements fo r an advanced degree at the U n i v e r s i t y of B r i t i s h Columbia, I agree that the L i b r a r y s h a l l make i t f r e e 1 y . a v a i 1 a b 1 e fo r re ference and Study. | f u r t h e r agree that permiss ion fo r e x t e n s i v e copying of t h i s t h e s i s fo r s c h o l a r l y purposes may be granted by the Head of my Department or by h.i>s r e p r e s e n t a t i v e s . It i s understood that copying or p u b l i c a t i o n of t h i s t h e s i s fo r f i n a n c i a l gain s h a l l not be a l lowed wi thout my w r i t t e n p e r m i s s i o n . The U n i v e r s i t y of B r i t i s h Columbia Vancouver 8 , Canada Department ABSTRACT A bar c e l l i n the form of an i s o s c e l e s t r a p e z o i d s u i t a b l e f o r s o l u t i o n of two dimensional problems i s introduced i n v o l v i n g two new concepts, one that of overlapping bars and the other of a v a r i a b l e angle parameter. Based on the I d e n t i t y of corner displacements of the c e l l and a s i m i l a r l y shaped piece of p l a t e i n c o n d i t i o n s of uniform s t r e s s , expressions f o r the bar areas are derived i n terms of Poisson's r a t i o , the geometry of the c e l l and a v a r i a b l e angle parameter. The s t i f f n e s s matrix of the c e l l i s d e r i v e d f u r t h e r by combining d i f f e r e n t displacement modes. Two methods of c a l c u l a t i n g s t r e s s e s , one by the nodal displacements and the other by the nodal force concentrations, are described. Two examples i n v o l v i n g a r e c t a n g u l a r p l a t e and a p a r t i a l s ector of a c i r c u l a r p l a t e i l l u s t r a t e the a p p l i c a t i o n of the f i n i t e element method using t r a p e z o i d a l bar as w e l l as no-bar c e l l s and t h e i r r e s u l t s are compared with the exact e l a s t i c i t y s o l u t i o n . TABLE OF CONTENTS CHAPTER ONE CHAPTER TWO — I n t r o d u c t i o n - F i n i t e Element i n the Shape of an Isosceles Trapezoid 2.1 Trapezoidal Bar C e l l 2.2 C a l c u l a t i o n of Parameters 2.2.1 Comparison of Trapezoidal and Rectangular Bar C e l l s 2 . 3 R i g i d i t y of the Bar C e l l 2 .3.1 Choice of the angle Parameter 2.1+ D i s t r i b u t i o n Factors f o r the Bar C e l l 2.1j..l A c t i o n 1 2.14-.2 Action 2 2.1j..3 S t i f f n e s s Matrices of the Bar C e l l s 2 . 5 Trapezoidal F i n i t e Element 2 56 Transformation Matrices 2.7 Problems with P r e s c r i b e d Displacements as Boundary Conditions Page No, 1 k k 6 23 26 31 31 31 39 ^9 53 57 i 60 CHAPTER THREE - C a l c u l a t i o n of Stresses 63 3 . 1 Trapezoids with Small Bases i n Adjacent C e l l s A l t e r n -a t i n g Up and Down 63 3 . 1 . 1 Method of Nodal D i s p l a c e -ments 63 3 . 1 . 2 Method of Nodal Force 43 Concentrations 68 3 . 2 Trapezoids with Small Bases i n Adjacent C e l l s S i m i l a r l y Directed 75 3 . 2 . 1 Method of Nodal Force Concentrations 76 Page No, CHAPTER FOUR - Examples 79 I4..I Deep Rectangular Beam 79 I 4 . . I . I E l a s t i c i t y S o l u t i o n 79 1+..1.2 F i n i t e Element S o l u t i o n 8 l k»2 C i r c u l a r P l a t e 82 k.2.1 E l a s t i c i t y S o l u t i o n 82 L[.2,2 F i n i t e Element S o l u t i o n 101 I4..3 Discussion of the Results 101 i| . .3.1 Problem of Deep Rectangular Beam 101 lj . , 3.2 Problem of C i r c u l a r P l a t e 120 CHAPTER FIVE - Trapezoidal Bar C e l l and Melosh's Po s t u l a t e 122 CONCLUSION 127 BIBLIOGRAPHY 129 ACKNOWLEDGMENT The w r i t e r i s g r a t e f u l to Prof. A. Hrennikoff f o r the valuable guidance i n c a r r y i n g out the work presented i n t h i s d i s s e r t a t i o n . The w r i t e r expresses h i s g r a t i t u d e to Dr. K. M. Agrawal f o r many valuable suggestions and constant help i n computer programming. The w r i t e r also expresses h i s g r a t -i t u d e to Dr. Bulent Ovunc f o r valuable suggestions. Thanks are also due to the U.B.C. Computing Centre f o r f r e e use of the computer. CHAPTER 1 I n t r o d u c t i o n The a n a l y t i c s o l u t i o n of plane s t r e s s problems, when the body fo r c e s are absent or are constant, i s reduced to the s e l e c t i o n of an A i r y f u n c t i o n F which s a t i s f i e s the biharmonic equation, V ^ ( F ) = 0. and the given boundary c o n d i t i o n s . However, s e l e c t i o n of such a s t r e s s f u n c t i o n , except i n a few simple problems, i s not p o s s i b l e . j. ( i f In I9l;l, Hrennikoff introduced the method of r e -p l a c i n g the p l a t e under c o n s i d e r a t i o n with a plane framework c o n s i s t i n g of bar u n i t s or c e l l s . The c e l l s have a p a r t i c u l a r d e f i n i t e repeating arrangement of bars and are joined to each other only at t h e i r nodes. I t i s required that the e x t e r n a l o u t l i n e of the framework conform to the o u t l i n e of the given p l a t e . The bars have the same modulus of e l a s t i c i t y as the prototype and they must possess some appropriate c r o s s - s e c t i o n a l areas and, i n some cases, a l s o moments of i n e r t i a . These bar p r o p e r t i e s may be c a l l e d parameters and they are determined from the c o n d i t i o n that the framework must deform i n the same manner as the pl a t e when placed i n an a r b i t r a r y uniform s t r e s s f i e l d . The framework model i s subjected to the same e x t e r n a l loads i n the same l o c a t i o n s and to'the same boundary r e s t r a i n t s must be appli e d only at the c e l l nodes. * Numbers s i g n i f y the ref e r e n c e s , l i s t e d i n B i b l i o g r a p h y . - 2 -The arrangement and the number of bars i n the c e l l s must be such that the c e l l s are s u i t a b l e f o r any value of Poisson's r a t i o and that they form a r i g i d s t r u c t u r e . I f the number of independent parameters i n the c e l l i s greater than the number of equations d e s c r i b i n g an a r b i t r a r y uniform s t r e s s con-d i t i o n , the e x t r a parameters may be assigned a r b i t r a r i l y , but i f i t i s l e s s than the number of such equations the bar pa t t e r n must be modified to incorporate a d d i t i o n a l parameters. The pat-tern short of one parameter may be s u i t a b l e f o r one p a r t i c u l a r value of Poisson's r a t i o . Once the s t i f f n e s s p r o p e r t i e s of d i f f e r e n t bars com-posing the c e l l s are known the s t r e s s a n a l y s i s of the model may be accomplished by f i n d i n g the r e l a t i o n s h i p between the corner f o r c e s and corner displacements of each i n d i v i d u a l c e l l . I f one of the corners of the c e l l i s given a u n i t displacement i n the d i r e c t i o n of one of the co-ordinate axes, while a l l other corners are held i n t h e i r p o s i t i o n s , f orces are produced i n the d i r e c t i o n s of co-ordinate axes at the corners of the c e l l . These fo r c e s are needed to hold the c e l l In the deformed s t a t e and they are c a l l e d " d i s t r i b u t i o n f a c t o r s " o r " s t i f f n e s s c o e f f i c i e n t s " of the c e l l corresponding to t h i s p a r t i c u l a r corner displacement* The d i s t r i b u t i o n f a c t o r s corresponding to the displacements of a l l corners, i n both x and y d i r e c t i o n s , are arranged i n the form of a matrix c a l l e d the "element s t i f f n e s s m a t r i x " . The c e l l model r e p l a c i n g the p l a t e i s a h i g h l y indeterminate s t r u c t u r e and I t s s o l u t i o n i n v o l v e s a lar g e number of simultaneous equations. A computer i s used to generate the s t i f f n e s s matrix of the whole model and to solve these equations f o r the nodal displacements. M u l t i p l y i n g the element s t i f f n e s s matrix by the ; displacement vector of the same element the corner forces i n the elements are obtained. The s t r e s s e s i n the model may be found by using e i t h e r the corner displacements or the corner f o r c e s . Square and r e c t a n g u l a r bar c e l l s have been used i n 1 two dimensional s t r e s s c o n d i t i o n s by Hrennikoff and others, p McCormick used c e l l s whose bars were endowed with f l e x u r a l and e x t e n s i o n a l s t i f f n e s s e s . Bar c e l l s of various shapes, i n -9 e l u d i n g non-isosceles trapezoids were suggested by P e s t e l . These c e l l s i n v o lved s p e c i a l d i f f e r e n t i a l gears and d i d not r e -semble i n any way the c e l l s u t i l i z e d i n present work. The i d e a of two dimensional c e l l has been extended to a three dimensional c e l l i n the form of a cube and a p a r a l l e l e p i p e d by Hrennikoff^ 3 and r e c e n t l y by Yettram and Robins . Also no-bar c e l l s i n the form of r e c t a n g l e , t r i a n g l e , i s o s c e l e s t r apezoid and p a r a l l e l e -piped were used by d i f f e r e n t i n v e s t i g a t o r s . The present work introduces the bar c e l l i n the form of an i s o s c e l e s t r a p e z o i d , Experssions f o r the bar areas and the s t i f f n e s s matrix of the c e l l are derived i n v o l v i n g one v a r i a b l e parameter, which may be adjusted to the best advantage. The theory i s applied to two problems whose e l a s t i c i t y s o l u t i o n s are known. The r e s u l t s i n the form of displacements and s t r e s s e s at d i f f e r e n t values of the Poisson's r a t i o are com-pared with the e l a s t i c i t y values and the r e s u l t s u t i l i z i n g the no-bar t r a p e z o i d a l c e l l . CHAPTER 2 Trapezoid." F i n i t e Element i n the Form of an I s o s c e l e s 2 . 1 Trapezoidal Bar C e l l An element having the shape of an i s o s c e l e s trapezoid ( F i g . 2-1) i s defined g e o m e t r i c a l l y by the lengths of i t s p a r a l l e l sides AC=a and GE=ka and height h=k^a. The angle at the bigger base equals i j . The arrangement of bars i n the c e l l i s shown i n P i g . 2 - 2 . The corners A, C, E and G are the main nodes of the c e l l d e f i n i n g i t s o u t l i n e . At these nodes the c e l l i s joined to i t s neighbors and here i s where the e x t e r n a l loads are applied to the model. B, D, F and H are the secondary nodes, some of which may l i e outside of the c e l l o u t l i n e ; they have no contact with the s i m i l a r secondary nodes of the other c e l l s or the e x t e r n a l loads. The d e f o r m a b i l i t y of the c e l l i s determined on the b a s i s of the displacements of the main nodes only, while the secondary nodes are f r e e to move i n a manner re q u i r e d by the e q u i l i b r i u m . The h o r i z o n t a l bars having the cross s e c t i o n a l areas A and A^ are made double, with one part connecting td>the i n c l i n e d bars of the c e l l only at the main nodes, and the other p a r t provided with connections at the mid-points B and F as w e l l as at the main nodes. Thus Y - 5 -Fig. 2 - 2 h=k,a Area A = A + A Area, A, = aJ + A," Y p/t M i l l f i i I i i I 1 p(l -M)k|Q E t p(l+k)a S, B p(l+k)a yu.pk,a G | S F_ p(Kk)a t = thickness 4 \ p(l+k)a (a) (b) Fig. 2 -3 A = A' (through bar) + A" (mid-point connected bar) and A-j= A^ (through bar) « A^ (mid-point connected bar) Other bar areas and bar i n c l i n a t i o n s to the h o r i z o n t a l are stated i n P i g . 2 - 2 . To s a t i s f y the e q u i l i b r i u m at the main nodes i n a s t r e s s - f i e l d symmetrical about the x and y axes, the angles of i n c l i n a t i o n of the side bars ( c r o s s - s e c t i o n a l areas A^ and A^ ') are kept the same. This angle -Ob represents the v a r i a b l e parameter whose magnitude i n r e l a t i o n to the geometry of the c e l l w i l l be s p e c i f i e d l a t e r . 2 . 2 C a l c u l a t i o n of Parameters A general c o n d i t i o n of uniform s t r e s s i s achieved by combination of the f o l l o w i n g three types of uniform s t r e s s . ( 2 - 1 ) •7 Uhi!e_. - o. However, i t i s more convenient to use i n s t e a d the three equiv-a l e n t s t r a i n c o n d i t i o n s described as f o l l o w s : •9 U)hiJ£, - o. > & o. = o. ( 2 - 2 ) The parameters of the c e l l , the c r o s s - s e c t i o n a l areas of the bar members, are so sel e c t e d that the c e l l deforms e x a c t l y the same as the p l a t e i n the three stated s t r a i n c o n d i t i o n s . A framework c o n s i s t i n g of such c e l l s w i l l deform i d e n t i c a l l y with the p l a t e i n any a r b i t r a r y uniform s t r e s s f i e l d and thus w i l l s a t i s f y the bas i c requirement of an adequate c e l l model. A c e r t a i n number of parameters may be determined from the equations of d e f o r m a b i l i t y corresponding to the conditions (2-2). The remaining parameters, such as the angle of i n c l i n a t i o n at the base, may be s u i t a b l y assigned l a t e r . S t r a i n Condition 1 Here t i s the thickness of the p l a t e prototype, and p the normal s t r e s s i n the x d i r e c t i o n per u n i t length of p l a t e . E and are the e l a s t i c constants. P i g . 2-3 shows the p l a t e and the c e l l i n t h i s s t r e s s f i e l d . The stresses i n d i f f e r e n t bars are i n d i c a t e d by symbols S with proper numerical s u b s c r i p t s , as shown In P i g . 2-3 ( b ) . To conform to the defigrmation of the p l a t e the main nodes of the c e l l do not move i n the h o r i z o n t a l d i r e c t i o n . Since the s t r e s s c o n d i t i o n i s symmetrical about the v e r t i c a l a x i s the mid-points B and P must remain on t h i s a x i s . In t h i s s i t u a t i o n , because there i s no e x t e r n a l load at the p o i n t B to balance the v e r t i c a l components of S j , the e q i i l l i b r i u m of the j o i n t B demands tha t S^=0. A s i m i l a r argument appl i e d to the p o i n t P gives S^=0. E q u i l i b r i u m i n the v e r t i c a l d i r e c t i o n at the j o i n t s H and D demands that S^=S^ . There i s no change i n lengths of the top and bottom h o r i z o n t a l bars, both the ones going r i g h t through and the ones connected at B and P. This means that S = S x = 0. The nodal forces a c t i n g on the c e l l at the main nodes must conform to the s t r e s s c o n d i t i o n i n the p l a t e P i g . 2-3(a). - 8 -By assembling the p l a t e stresses by s t a t i c s to the corners, J the v e r t i c a l and h o r i z o n t a l components of the nodal forces at a l l four nodes bec&me ^ ^ ' ^ a n d ^ * l 3 ^ ' a - r e s p e c t i v e l y . Prom the h o r i z o n t a l and v e r t i c a l e q u i l i b r i u m of the bar s t r e s s e s and the e x t e r n a l forces at the node G ( F i g . 2-I4.). So E x a c t l y the same equations bold at a l l main nodes. This s i t u a t i o n i s brought about by the e q u a l i t y of the corresponding angles at which the bars approaching the four main j o i n t s . S o l v i n g eqns. ( 2 - 3 ) and (2-lj.) f o r S 2 and S=0. 1^ 4 '3 2 - 4 -4 ( 5th cj> +- Cos c£ -(ran 0<^  ( 2 - 5 ) an ( 2 - 6 ) ci G * L ( K - h ) - 2^^> -Van <ft ] 4 ( S i n 0 0 e Q -trar» ^ Elongation of the i n c l i n e d bar GC named bar 2 i s A , S i n <ty where ^ 1 i s the v e r t i c a l elongation of the c e l l . A general r e l a t i o n between the bar s t r e s s (S^) and the elongation of an e l a s t i c bar of length (L^) and c r o s s - s e c t i o n a l area (A b) i s Ab E <eb •5b - Lb Applying t h i s r e l a t i o n to bar 2 ^2. = Iz. ( 2 - 7 ) ( 2 - 8 ) - 9 -Equating (2-5) and (2-8) and s u b s t i t u t i n g L , - k , a - a , - K ' - f ^ i * - q » <t = # _ J l L i _ _ C r o s s - s e c t i o n a l area A2 of the bar 2 i s found as a - q t L ( k f 0 - h 2 l k . t a n 0 o 1 L 4 k , z 4 - ( k + 0 * ] 3 / 1 , , M z - " " K s k i ' - O - Z - ^ b ^ + Ck+Oianflt] The areas of the outer bars 5* and 5, i n c l i n e d at an angle -0 O w i l l be determined next. As stated before, stresses Scj and are equal. For s i m p l i c i t y the areas of these bars (A^ and A^) are assumed equal. This gives the changes i n length of these bars, because of the presence of the equal s t r e s s e s i n them, p r o p o r t i o n a l to t h e i r lengths and the lengths and may be expressed through the geometry of the c e l l by c o n s i d e r i n g F i g , 2-5 from which ( L s - + U ) S i n Go ~ k,OL_ (2-10) (2-11) S u b s t i t u t i n g + from (2-10) i n t o (2-11) a[2Ai ~ Ck-I){q^ 0] t h i s gives LV * (2-12) -10-- 1 1 -S u b s t i t u t i n g i n (2-10) then the r a t i o of the bar lengths, equal to the r a t i o of t h e i r e l o n g a t i o n s , i s Assuming the stresses i n the bars 5 and 5'7to be t e n s i l e the s t r e s s S6 i n the h o r i z o n t a l bar 6 must be compressive to s a t i s f y the h o r i z o n t a l e q u i l i b r i u m of the j o i n t s H and D. With the elongations 8c; and 8^ ( -of the bars 5 ' and 5 r e s p e c t i v e l y , the shortening of the bar 6 (the h o r i z o n t a l movement of each end being $<a/%) and the downward displacement of the node G i n r e l a t i o n to the node A, the W i l l i o t diagram i n v o l v i n g the j o i n t s A, H and G may be drawn as i n P i g . 2-6. By t a k i n g the v e r t i c a l p r o j e c t i o n of the displacements 1 Sir,&0 | o n e f l 3 i n e 0 ( 2 " 1 5 ) The area of the bar A^ i s one of the superfluous parameters which may be assigned at w i l l . I t i s convenient to assume t h i s bar to be i n f i n i t e l y s t i f f , i . e . A^ = . Then =0 and from (2-15) -12-<~ _ A> S i n Qp According to r e l a t i o n (2-7) ~ L * 0 * y > ( 2 - 1 6 ) Equating (2-6) and (2-16) and s u b s t i t u t i n g the values f o r A , , < * , L 5 a n d ^ f = (2-17) o There are two equations of d e f o r m a b i l i t y a v a i l a b l e f o r t h i s s t r e s s c o n d i t i o n , namely, the v e r t i c a l and h o r i z o n t a l elongations of the c e l l (A^ and zero r e s p e c t i v e l y ) agreeing with the s i m i l a r elongations of the p l a t e , and these have been f u l l y u t i l i z e d i n determination of the c r o s s - s e c t i o n a l areas and Acj. The a r b i t r a r y v a r i a b l e parometer @ Q w i l l be assigned l a t e r , S t r a i n Condition 2 Qy. ^ frO-A*? while- C-x - * ° -V The symbols f o r the bar str e s s e s used i n t h i s s t r a i n c o n d i t i o n are the same as In the previous one, although the values of these stresses i n two con d i t i o n s are d i f f e r e n t . In t h i s s t r a i n c o n d i t i o n ( P i g . 2-7) as i n the previous one, the r -13-jLtp/t I | | | A I 2E t (a) p/t A, pk,a /zp(l+k)a /j.p(l4k)o 4 j*|^^2/k pk,a «<8 A -4h- G / / - P ( l + k ) a (b) 2^ 2 /Zp(l4k)QL Fig. 2-7 Y A,= 2pk,a(!+^.) ta.) p(k+l)a AI P M r r s , " B pk,a 4 s ^ f ^ ^ C ^ 5 ^ " o 5 - U o 4 r~ p(k4|)g >4 p(k+l)a (b) Fig. 2 - 9 -11*. bars 3 and L\. are unstressed f o r the same reason, the Symmetry of the l o a d i n g about the v e r t i c a l a x i s . Likewise the bars 5>' and 3> have equal s t r e s s e s . The p l a t e s t r e s s e s (Pig.2-7(a) ) are assembled by s t a t i c s at the main nodes of the c e l l ( P i g . 2-7(b) ). The v e r t i c a l and h o r i z o n t a l components of n o d a l - f o r c e s , i d e n t i c a l at a l l four corners are -* yr and — r e s p e c t i v e l y . S s - 2-F,3. 2.-3 Ob) Since the stresses S^ and S£ at the j o i n t s A and G ( P i g s . 2-8(a) and 2-8(b) ) are equal, e v i d e n t l y S = S. (2-18) In order to agree with the deformation of the p l a t e the elongations of the top and bottom h o r i z o n t a l bars, both the u n i n t e r r u p t e d ones and the ones with the mid-length j o i n t s must be p r o p o r t i o n a l to t h e i r l e n g ths. Hence st r e s s e s and s t r a i n s i n these two bars are equal and t h i s makes t h e i r areas a l s o equal. Therefore A = A, (2-19) The t o t a l elongations of these bars are: of A^ ^= 2.^2. r a»d of A, , 5 , = 2 * 2 / K c o h e r e K ^ O - ^ ) The s t r e s s In the i n c l i n e d bar S 2 may be e a s i l y found from the s t r e s s - e l o n g a t i o n r e l a t i o n s h i p eqn. (2-7)• The length of the bar L 2 = J^.'^V and i t s elongation ( P i g . 2.7(b) ) C»6j = C J ^ O A ^ f A z E A , Cod c£> ( l + k ) Then S P = — — f=-. 1 — l^k, 2- [ 2 k , + < k + 0 + a n 0b J i 2 ' 2 0 ) The f o l l o w i n g r e l a t i o n s between the bar stresses are obtained from the h o r i z o n t a l and v e r t i c a l e q u i l i b r i a of the J o i n t G. ( P i g . 2-8(b) ). E l i m i n a t i n g from these two equations and s u b s t i t u t i n g S 2 from (2-20) S - S-i = —. — ' (2-23) 1 iGkiHc^^ -16-S u b s t i t u t i n g t h i s expression i n t o eqn. (2-7) f o r e i t h e r the top or the bottom h o r i z o n t a l bar, the area of these bars i s found as a i i C K + D L ^ U ^ - C i t k ) 3 - ] ^ ^ -Iran-Oo [4k,2- fCu-k)*]/ A = A i = tekfa-^Wo ( 2 " 2 U ) The e q u a l i t y of the areas A and A^ and the al g e b r a i c expressions f o r these areas are the c h a r a c t e r i s t i c s of the c e l l derived from the correspondence of deformations of the c e l l and the p l a t e i n the s t r a i n c o n d i t i o n 2. S t r a i n Condition - 3 The p l a t e and the c e l l i n t h i s s t r e s s f i e l d are shown i n F i g . 2-9. This s t r e s s f i e l d i s antisymmetrical about the v e r t i c a l a x i s and t h i s leaves s e v e r a l bars unstressed, Such i s the middle h o r i z o n t a l bar 6, f o r which there i s no more reason to be In tension than i n compression. H o r i z o n t a l e q u i l i b r i u m of the bar stre s s e s at the j o i n t s H and D shows that the i n c l i n e d bars 5 and on both sides of the c e l l must a l s o c a r r y zero s t r e s s e s . The h o r i z o n t a l through bars AC and GE are i n the same s i t u a t i o n because of no change i n length of these bars. This leaves the system of stressed bars as shown i n F i g . 2-10. F i g . .2-10 The h o r i z o n t a l and v e r t i c a l components of the load a p p l i e d at the main nodes of the c e l l are c a l c u l a t e d i n accordance with the s t r e s s c o n d i t i o n In the p l a t e and they are ^k | 0T and P^k+Oo- r e S p e c n v e i y t Numerically these components of nodal forces are the same at a l l corners. The bottom nodes Gaand E are assumed to be at r e s t and the top nodes A and C move a h o r i z o n t a l distance A 3 = ^ J J - k ( 0 _ ( F i g . 2-9). The area of the i n c l i n e d bars GC and A E , i . e . bars 2 i s known. Hence the str e s s e s S 2 c a r r i e d by these bars may be e a s i l y found by u s i n g the r e l a t i o n (2-7). I t i s d e s i r a b l e to su b s t r a c t the components of these stresses from the nodal f o r c e s , and thus to, determine the system of corner forces c a r r i e d e x c l u s i v e l y by the s o l i d l i n e bars i n F i g . 2-10 whose areas are s t i l l unknown. U n l i k e the case of square or rec t a n g u l a r bar c e l l s * i n t h i s case of the t r a p e z o i d a l bar c e l l w ith the chosen bar arrangement, two independent equations are a v a i l a b l e at each node. This makes i t p o s s i b l e to f i n d the st r e s s e s c a r r i e d by -18-a l l the bars of unknown areas i n Peg. 2-10 since only two such bars are present at each node. The areas of these bars must be found by equating the components of corner displacements of the c e l l and the p l a t e . With the bottom nodes G and E being f i x e d , the top nodes A and C move Pty ^ 3 - ' — h o r i z o n t a l l y and zero distances v e r t i c a l l y . These two equations of d e f o r m a b i l i t y are i n s u f f i c i e n t to f i n d the four unknown bar areas A", A^", A^ and A^, and so two a d d i t i o n a l c o n d i t i o n s of deformation w i l l be assumed somewhat a r b i t r a r i l y . According to t h i s assumption•the mid-length secondary nodes B and P move h o r i z o n t a l l y to the r i g h t a distance A 3 =• . With displacements of a l l j o i n t s , main and secondary, being now known the changes of length of a l l bars may be e a s i l y de-termined and since the bar st r e s s e s have already been found, the s t r e s s - l e n g t h change r e l a t i o n s h i p i n the bars (Eqn.2-7) r e s u l t s i n expressions f o r the required bar areas. Considering e q u i l i b r i u m of the node C ( F i g . 2-11) u T a. P i g . 2-11 I H - O . ^^±^ = 6 * + StCof + SA Co*? (2_25) I V = o 2- (2-26) -19-S u b s t i t u t i n g £ 2 and i n (2-2$) c-* _ bokC2.k,x0-jLvcn-k)z4- k t-U9b0+k)0-3/u ) l The two h a l f - l e n g t h bars of the top base and the bottom base behave s i m i l a r l y . The shortening of the bar AB i s equal to the elongation of the bar BC = . Therefore the raid p o i n t connected bars c a r r y equal s t r e s s of opposite s i g n . Applying eqn. (2-7) , A , # E sr _ A ; E ( w o Equaling (2-33) and (2-31;) ' •" 8ki^<i-/i*)[2k, + < i + k ) ^ e o ( 2 " 3 5 ) The areas A^ and A" of the remaining two bars are found i n a manner s i m i l a r to Aj^ and A{'. With the signs of the v stresses shown by the arrows, the equations of e q u i l i b r i u m at J ^ ffi?**^ ^ ^ the node E ( P i g . 2-12) give S* frk.a. j -Pig. 2-12 - q . ; ^ ^ - a S*a>*<j> 4 S 3 ^ ^ S " ( 2 . 3 6 ) 2V = o, = S z 5 ' . ^ * S3Sir>of ( 2_ 3 ? ) S u b s t i t u t i n g S 2 from (2-28) i n t o (2-37) ^ s- |?a -fe k , 2 0 -ju)- fl4*fo<k44)*a»i QbLk, 0- - ^ ><i1} ^ 'n 2<l-/.)[2.k,+0+K)tcihft] ( 2" 3 8 ) As before, the str e s s e s i n the bars GB and BE ( P i g . 2-9) are vim - 2 0 -These equations contain the unknown bar str e s s e s S^" and S^, while the s t r e s s S 2 can be e a s i l y found knowing the elongation and the c r o s s - s e c t i o n area of the bar 2 e a r r i n g i t . I n c i d e n t l y elongation of bar GC i s equal to shortening of bar «, Hence bars GC and AE c a r r y s t r e s s e s equal but opposite i n s i g n . Using eqn. ( 2 - 7 ) . A * E A 3 G o 3 < f > j 7 ( 2 - 2 7 ) S u b s t i t u t i n g A 2 , > A 3 and i-z. c: - bo c»<+0 ( k * i 4 g/*k, t on e.) [ 4 k f t ( k + o * l > / z -S u b s t i t u t i n g S 2 i n eqn. ( 2 - 2 6 ) ^ * ~ 2<l-/»)[2k,4-(l+kHanQi.3 ~ ( 2 ~ 2 9 ) The elongation of the bar PC i s the same as the shortening of the bar AP = S 4 - ^2>^o^ ~ C A^2.) C c ^ p . Both these bars are c a l l e d bar i\. and they c a r r y s t r e s s e s equal but opposite In sign S^. Using eqn. 2 - 7 . % = • . ( 2 - 3 0 ) where S4 - ) C Q 4 j3 a n d L4 = s f e j 3 Then 5 4 5 ^ p = — ( 2 - 3 D S u b s t i t u t i n g A 3 Gob® = 0 cmd S«n p - - = = = = • -i n t h i s expression and equaling i t to ( 2 - 2 9 ) the area A^ i s found X _ Q t [ 2 k f t l - / t ) - Q 4 - k ) 2 l - k t ( H k ) ( t - 3 A ) - t : Q ^ a , i r i - l - 4 k H 2 " ^ Q L . V i - ^ C 'lfLi.n±-..a i (2-32) -21-equal and opposite i n s i g n , while t h e i r changes i n length are the same namely, Applying the r e l a t i o n ( 2 - 7 ) . 2 (2-39) Then ^ <3ir>°< « Ag> E ( A 3 / z ^ C o M S i n S u b s t i t u t i n g A 3 , S i n * = ' - o>nc| Co<>><* - 7=^===== i n (2-39) and equaling i t to (2-38) S u b s t i t u t i n g S 2 and S3 i n (2-36) c ^ ^ fro [2-k^O-/u) - C M kX+ k, Q -f - k ) ()-3/u)-bqi7 Q-Q1 ^ ^M>>)[^+0+k)tQV)G/J ( 2 ' W Applying eqn. (2-7) to the base bar GP or PE t> =• p (2-1^2) S u b s t i t u t i n g the^values of A3 ana i n t o t h i s eqn. and equaling S" with i t s expression i n (2-I4.I) the area of the base bar i s found ^ 8^0-fS>[zk( 4- Ct+k)-ton«ol a" U 3 ) Comparing the equations (2-35) and (2-I4.3) i t may be observed that the areas A" and A£ are equal, as expected from the c o n s i d e r a t i o n presented e a r l i e r . A l l the area parameters derived above are stated In Table 2 -1 . -22-LU UJ u O (/) i LL UJ or 2 u HI 1-m < IT -J < Q o N UJ < < I-i UJ 1 -X 4 o 4 4 4 o CD c y .-,4 .01 CO rs 4 4 J ? o CD £ - 4 4 /~\ 4 r + 4 eg c a CO I 4 4 J* •4 — \s 4 o c 4 4 4> a 2 CO c4 I oo 4 J. 4 /—\ 4 I c8 4 4 i < i cb '7) c in 4 4 o < An 0 IS < « c a ^/ 4 4-4 d5 4 4 J: i O < - 2 3 -2 . 2 . 1 Comparison of Trapezoidal and Rectangular Bar C e l l s I t Is appropriate at t h i s stage to draw a comparison between the p r e s e n t l y considered t r a p e z o i d a l c e l l and the l . i i H r ennikoff's rectangular bar c e l l , c a l l i n g a t t e n t i o n to the ba s i c d i s t i n c t i o n between the two c e l l s . The p r a p e z o i d a l c e l l i s unsymmetrical about the h o r i z o n t a l a x i s and symmetrical about the v e r t i c a l a x i s . I t s bar system c o n s i s t s of two superimposed frame works, one of which, system 1 ( F i g . 2 - 1 3(a)) with the a d d i t i o n of the h o r i z o n t a l bars of the other system, works only i n s t r e s s f i e l d s symmetrical about the axis of symmetry, and the other, system 2 (Fig. 2 - 1 3 ( b ) ) , with the a d d i t i o n of the diagonal bars of the system 1, acts only i n f i e l d s antisymmetrical about the ax i s of symmetry. Furthermore the bars i n system 1 are s p e c i a l l y arranged so that they meet the top and bottom nodes at the same angles. No such angle arrangement Is needed with regard to the bars of the system 2 . The bar arrangement i n the Hrennikoff's r e c t a n g u l a r bar c e l l was n a t u r a l l y i d e n t i c a l at a l l four corners. In the symmetrical uniform s t r e s s c o n d i t i o n 1 of the t r a p e z o i d a l p l a t e prototype, the corner load components x and y, which the c e l l i s supposed to i m i t a t e , come out the same at a l l c o rners, and so two equations, i . e . the same number as i n the r e c t a n g u l a r c e l l , are a v a i l a b l e i n c o n d i t i o n 1 f o r determin-a t i o n of the bar areas. Because of the equal i n c l i n a t i o n s ( "0"o ) of the top and bottom side bars with respect to the h o r i z o n t a l , the stre s s e s c a r r i e d by them are equal i n the symmetrical c o n d i t i o n 2 as i n the p r e v i o u s l y considered c o n d i t i o n 1 . For t h i s reason Fig 2-16 - 2 5 -the s t r e s s e s c a r r i e d by the h o r i z o n t a l top and bottom bars are a l s o equal, and i n view of equal s t r a i n s i n the top and bottom bars, the areas of these bars come out the same. Thus c o n d i t i o n 2 provides two equations, based on the h o r i z o n t a l displacements of the top and bottom nodes from which the magnitude and e q u a l i t y of the areas of the top and bottom h o r i z -o n t a l bars are d e r i v e d . These two equations correspond to a s i n g l e equation i n the rec t a n g u l a r c e l l In which the top and bottom bars were equal i n lengths i n view of the symmetry of the c e l l . The equation of zero v e r t i c a l deformation of the t r a p e z o i d a l c e l l provides no new i n f o r m a t i o n , i t simply confirms, as In the r e c t a n g u l a r c e l l , B e t t i ' s r e c i p r o c a l theorem. Corner forces are equal at a l l corners a l s o i n the a n t i s y m m e t r i c a l s t r e s s c o n d i t i o n 3» By equating them to the components of the bar st r e s s e s two equations become a v a i l a b l e from which more unknown bar areas may be der i v e d . The two s i m i l a r equations i n the r e c t a n g u l a r c e l l were i n f a c t one and the same, because they i n v o l v e d s t r e s s components of the s i n g l e diagonal bar. Thus the parameters of the proposed t r a p e z o i d a l c e l l must comply with 6 equations of deformation against ij. i n the recta n g u l a r c e l l and 3 i n the square c e l l . The a c t u a l number of bars of d i f f e r e n t kinds i n the t r a p e z o i d a l c e l l i s t e n . Their areas are A ' , A ^ , A " , A ^ , A 2 , A 3 , A ^ , A ' £ , A £ and A ^ ( P i g . 2-2) and i n a d d i t i o n there i s an angle parameter ~&0 . In view of i n s u f f i c i e n c y of the number of equations a v a i l a b l e i t was necessary to make s u i t a b l e assumptions, aiming at the s i m p l i c i t y of r e s u l t s : -26-1. The areas of i n c l i n e d bars and A^ were made equal, 2, The h o r i z o n t a l bar connecting secondary nodes H and D was made a b s o l u t e l y r i g i d . 31 Two assumptions were made with regard to the displacements & [j.J of the mid-base nodes B and P i n the uniform shear f i e l d . These r e s u l t e d i n the e q u a l i t y of the secondary base bar areas A" and A£. These four assumptions together with the s i x equations of d e f o r m a b i l l t y led to ev a l u a t i o n of a l l ten bar areas i n terms of the basic geometry of the t r a p e z o i d , the Poisson's r a t i o ju+ and the parametric angle -0^ . The purpose of the l a t t e r i s to prevent the c e l l from becoming n o n - r i g i d f o r c e r t a i n combinations of i t s geometry and values of the Poisson's r a t i o as explained i n the next s e c t i o n . I t may be added that the t r a p e z o i d a l c e l l discussed here remains d i s t i n c t from Hrennikoff's r e c t a n g u l a r bar c e l l on red u c t i o n of the tra p e z o i d to the rectangle when k = 1. 2-3 R i g i d i t y of the Bar C e l l A planar framework made of s t r a i g h t bars endowed with tension - compression s t i f f n e s s e s i s r i g i d i f i t may be assembled, beginning with an I n i t i a l t r i a n g l e , node by node, by l o c a t i n g each new node with two new n o n - c o l l l n e a r bars extending from the already, located nodes ( P i g . 2-lk.) • The r e l a t i o n between the number of nodes (n) and the number of bars (b) i n such a framework i s b = 2n - 3 (2-Ul|.) P i g . 2-1J4. I f the s t r u c t u r e i s not formed i n the described manner i t may be non r i g i d even when b^-2n-3 . Consider what was described as the system 1 of the t r a p e z o i d a l c e l l ( P i g . 2 - 1 5 ) . Here b = 9 and n = 6 and so the eqn. (2-ljij.) i s s a t i s f i e d ; yet such a s t r u c t u r e i s n o n - r i g i d , as may be e a s i l y seen by c o n s i d e r i n g the instantaneous centres of d i f f e r e n t bars. Imagine that the nodes G and E are kept i n p l a c e . Then assuming t e n t a t i v e l y that the framework may change i t s shape, the bars GH and GC w i l l r o t a t e about t h e i r centre G and the bars ED and EA - about E. Then the instantaneous centres of the other four bars w i l l be found as f o l l o w s : of the bar HA-at^K, bar CD - at L, and the bars HD and AC r e s p e c t i v e l y at M and N. The symmetry of the s t r u c t u r e makes sm a l l movements of the l a s t four bars p e r f e c t l y c o n s i s t e n t , thus making the s t r u c t u r e u n s t a b l e . In the a c t u a l c e l l the bar system 1 i s c o e x i s t e n t with the system 2 ( F i g . 2-13(b) ) and the l a t t e r , although n o n - r i g i d by i t s e l f , i s capable of s t a b i l i z i n g the former because i t would not permit the i n c l i n e d movements of the nodes A and C, occurlng i n the course of d i s t o r s i o n of the system 1. The conclusion concerning n o n - r i g i d i t y of the bar system 1 has p r a c t i c a l s i g n i f i c a n c e . I n s p e c t i o n of the ex-it »• pressions f o r the bar areas A and A^, A^ and A^ shows that they a l l become zero when 2k,^ 0-/-0 - O + k ) * - * - k, 0 + k ) ( l - 3 J u ) -Un^ o - O. (2-^5) The zero magnitude of these bar areas s i g n i f i e s complete d i s -appearance of the bars of the system 2, making the c e l l non-r i g i d and u nstable. This Is where the Importance of the a r b i t -r a r y angle parameter manifests i t s e l f . The value of the - 2 8 -angle -0^ must be so chosen as to preclude the c o n d i t i o n ex-pressed by eqn. (2-L\5) • A c e l l devoid of the bars of system 2 would be i n a d m i s s i b l e a l s o i n view of i t s i n a b i l i t y to cope with unsymmetrical s t r e s s f i e l d s . The p o s s i b i l i t y of c r o s s - s e c t i o n areas of other bars becoming zero and the e f f e c t s of such zero areas on the r i g i d i t y of the c e l l must a l s o be i n v e s t i g a t e d . The c r o s s - s e c t i o n areas of the through base bars A ' and5 A-[ i s represented by the d i f f e r e n c e of expressions (2-21;) and ( 2 - 3 5 ) . The numerator of the f r a c t i o n r e p r e s e n t i n g the area A ' = A { i s 2k, (H-k)L4)ukf--(l+k)"] 4-tqnOo 116 k,4- O+k^fkk.V/u) -f-ikO+k)*] + 2k, C»+k)-+qn1 o^[4kr-//(i+-kr-k0-3/A)] (2-4G) The area A * obviously may become zero f o r c e r t a i n com-bi n a t i o n s of the geometric features of the c e l l , with some i n -fluence of the value of jjj . No other bars vanish simultaneously with A * and A { . The c e l l devoid of the bars A * and A{ i s shown i n P i g . 2-16. The c e l l of P i g . 2-16 i s s t i l l r i g i d , because the presence of the bars of system 2 , makes n o n - v e r t i c a l move-ment of the nodes A and C (nodes G and E being held i n t h e i r p o s i t i o n s ) allowed by the i n c l i n e d diagonal bars A E and GC, i m p o s s i b l e . The area of diagonal bars A E and GC (bars 2) may be-come zero (expression 2 - 9 ) when Tan ^ 0 = (2-li-7) The negative sign of t h i s expression places the angle 0 o i n t o the second quadrant, g i v i n g an unusual appearance to the c e l l . Since the p r a c t i c a l l i m i t s of Poisson's r a t i o are 0 and 1/ 2 , -29-Oq of eqn. (2-ij.?) l i e s w i t h i n the l i m i t s j o ' ^ ^ 3 0 ^ where o(Q - Ore taY) -—j- The two p o s s i b l e appearances of the c e l l i n v o l v i n g such an angle -0"d are presented i n P i g . 2-17. The c e l l s remain r i g i d i n s p i t e of the disappearance of the bars 2 since the t r i a n g l e ACF may only move v e r t i c a l l y and such motion i s prevented by the I n c l i n e d side bars. I t remains to consider the disappearance of the side bars and 5 , joined by the h o r i z o n t a l bar 6. This happens when (expression 2-17) j_ h - ( J # The c e l l of t h i s kind i s shown i n F i g . 2-18. I t i s obviously r i g i d f o r the same reason as the c e l l s of P i g . 2-17 are r i g i d . Thus the e f f e c t of the disappearance of s e v e r a l bars of system 1, occuring s i n g l y , i s not p a r t i c u l a r l y important and need not n e c e s s a r i l y be counteracted by an appropriate choice of the angle -Q^t u n l i k e the case of vanishing of the bars of the system 2. Simultaneous disappearance of the bars 1 , 1^ and 2 or of the bars l ' , l { and 5 ' , 5» 6 s t i l l leaves the c e l l r i g i d . The s i t u a t i o n i s however d i f f e r e n t when the bars 2, 5» 5* and 6 vanish simultaneously ( F i g . 2-19). The nodes A, C, F may e v i d e n t l y move eq u a l l y i n the v e r t i c a l d i r e c t i o n i n r e l a t i o n to the s t a t i o n a r y nodes G, B, E, - so the s t r u c t u r e i s not r i g i d . V e r t i c a l forces can not be transmitted from the nodes A and C to the nodes G and E. This happens when Thus proper choice of the v a r i a b l e parameter 0"o w i l l preclude a l l chances of the c e l l becoming u n s t a b l e . -30-Fig 2 - 2 0 - 3 1 -2 . 3 . 1 Choice of the Angle Parameter. Bar areas which may vanish on c e r t a i n combinations of the c e l l geometry and the s t i f f n e s s of the m a t e r i a l may al s o become negative. A bar of negative area i s of course p h y s i c a l l y impossible and, while e l a s t i c , i t i s p e c u l i a r i n s o f a r that i t shortens i n tension and lengthens i n compression, thus developing a negative energy of deformation. Experience has shown that c e l l s with negative bar areas often give r e s u l t s of i n f e r i o r p r e c i s i o n . These f a c t s have d e f i n i t e bearing on the choice of the angle parameter -0O. Normally i t i s convenient to make & o ~ ( P i g . 2 - 2 ) . This leads to simpler expressions f o r the area parameters and d i s t r i b u t i o n f a c t o r s . I f however t h i s choice of leads to negative areas of some bars or even, what I s worse, to zero areas of the bars of system 2 , or the bars 2 , £ and 5* of the system 1 , an appropriate -Q"0 conforming to the suggested requirements must be s e l e c t e d . 2.1|L D i s t r i b u t i o n Factors f o r the Bar C e l l 2.I4..I Ac t i o n 1. A * The displacement A j of corner 1 i n the X d i r e c t i o n , while the other corners remain i n t h e i r p o s i t i o n s , i s designated as a c t i o n 1 . The s u p e r s c r i p t i n the symbol A i n d i c a t e s the d i r e c t i o n of the displacement while the s u b s c r i p t s i g n i f i e s the corner's number. To s i m p l i f y the s t r u c t u r a l a n a l y s i s i n f i n d i n g the d i s t r i b u t i o n f a c t o r s (nodal forces) f o r a c t i o n 1, Z^i i s conceived as the combination of the three displacement modes A, B and C ( F i g . 2 - 2 0 ) . The d i s t r i b u t i o n f a c t o r s corresponding to the displacement A i o c c u p y y t h e ; f i r s t column of the element s t i f f n e s s m a t r i x . - 3 2 -Dlaplacement Mode A The corner forces f o r t h i s mode may be w r i t t e n d i r -e c t l y from the s t r a i n c o n d i t i o n 1 ( P i g . 2 - 3 ) of s e c t i o n 2 . 2 by s u b s t i t u t i n g A A f o r A j i n equation r e l a t i n g A i and I s • S u b s t i t u t i n g t h i s value of i n the expressions f o r the nodal forces i n P i g . 2 - 3(b) V _ E l Cl+k) A A v A t t t t A A V ^ ^ f O ^ (2-51) The c e l l i n the mode A acted upon by these nodal forces i s shown i n P i g . 2 - 2 1 . Displacement Mode B The displacements associated with the mode B are pre-sented i n P i g . 2 - 2 0(b) and again i n P i g . 2 - 2 2 . They are a n t i -symmetrical about the v e r t i c a l a x i s X. Note that the four main nodes do not move h o r i z o n t a l l y . I t i s obvious that the h o r i z o n t a l bar 6 F i g . 2 - 3(b) has no more reason to be i n tension than i n compression, and so i t must remain unstressed. The same a p p l i e s to the uninterrupted base bars AC and GE. From the h o r i z o n t a l e q u i l i b r i u m of the secondary nodes D and H , i t f o l l o w s that the i n c l i n e d bars 5* and 5 are a l s o i n a c t i v e . Thus the stressed bars are only the ones shown i n F i g . 2 - 2 2 . The s t r u c t u r e formed by these bars i s symmetrical about the v e r t i c a l a x i s , while the s t r e s s e s and displacements of these members are antisymmetrical about the same a x i s . This means that the mid-point nodes B and F do not move v e r t i c a l l y . Let t h e i r h o r i z o n t a l displacements be £p and as shown i n F i g . 2 - 2 2 . As stated the antisymmetry - 3 3 -Fig. 2-25 -3k-of displacements means al s o the antisymmetry of s t r e s s e s . Thus a l l s t r e s s e s i n the symmetrically s i t u a t e d bars are equal and opposite i n s i g n . This a p p l i e s to the bars AB and BC, GP and PE and the bars 2, 3 and k l o c a t e d on the l e f t and the r i g h t sides of the ax i s X. This a l s o r e s u l t s i n antisymmetric nodal f o r c e s . The s t r e s s e s i n a l l bars may be expressed through the displacements of t h e i r ends i n terms of the q u a n t i t i e s ST, <^ B and by the s t r e s s - e l o n g a t i o n r e l a t i o n ( 2 - 7 ) , i n which the c r o s s - s e c t i o n areas of a l l bars are known. The unknown q u a n t i t i e s &-f- and may be found from the h o r i z o n t a l e q u i l i b r i a of the secondary nodes B and P. The corner forces at the main nodes are then found from t h e i r e q u i l i b r i a at C and E. Complying with t h i s general procedure and r e f e r r i n g to F i g . 2 -22 . Elongation of bar GC = Shortening of bar AE (bar 2) = K " 1 A B Sin <b K ; Applying to t h i s bar eqn (2-7) kL^ S u b s t i t u t i n g A 2, Sin<£ , and L^. i n (2-52) <~ ^ (k-QC(k - H ) 4- 2-^ k,-bane0J[4k,2-+ (' + k / J * A B E t * A k k , ( i - | W r a k , - M i + k ) W , f c ] { 2 ' S 3 ) Elongation of bar PC;= Shortening of bar AF (bar k) = ( A B / K ) S I D p - o~B Co* f From eqn. (2-7) c. = A A E C C A B / k ) - * B C Q O ft] % " j (2-5U) S u b s t i t u t i n g A 4 r S.V, p ^ <C©4 j3 QY)aj L 4 i>9 (2-54) - 3 5 -g _ E i i ^ ( i - M - ( i 4 - k ) V k , ( i 4 k ) ( i - 3 ^ ) i q n 0 O H ^ 4 k f ) H 2 k , A B - k ? B ) ( j k ~ A-kk,2 (i-ju 2-) [>k, + (k-t-0-tqne 6] S i m i l a r l y , elongation of BE = Shortening of GB (bar 3) - A& Sin cK — Co/> o< Hence So - A * E. U e * - S"T Q ^ c Q 3 i_3 (2-56) S u b s t i t u t i n g A^, S i n o< 7 Co<5o< cind L 3 /  g _ E * h k , V k ) - Q ^ k ) 1 4 k , a + k X ' ^ Shortening of bars BC and FE i s b~r and o"B respect-i v e l y . The same r e l a t i o n ( 2 - 7 ) applied to these bars gives 4 w ^ o" A f E S"r i n bar BC, S x = — \y s" = E*M*k,V/Q-U+k)^- k,(n-kX<-3/U)+c<K)e-o^ ( 2 _ ^ 8 )  1 Ak^O-fA 1) j > k , - K i + k ) W e - e and i n bar FE, S = — B F i g . 2-23 E q u i l i b r i u m of the j o i n t s B and F ( F i g . 2-23) de-mands that = £>3 Op>, <^ (2-60) and s" = 5 ^ Co^ j3 (2-61) - 3 6 -S u b s t i t u t i n g and from eqns (2 -58) and (2 - 5 7 ) i n t o (2-55&) 2k, A B - k S"r = k o r ST - £ A B 2.G.. OT = ^ R (2-62) S i m i l a r l y s u b s t i t u t i n g S and S^ from eqns (2 - 5 9 ) and (2 -55) i n t o (2-61) 1 (ak,Afi - kSb ) = (2-63) As may be seen from these equations that &B = O"T The bar s t r e s s e s may now be found i n terms of A B by s u b s t i t u t i n g S T and S"B i n t o eqns. (2 - 5 5 ) , (2 - 5 7 ) , (2 - 5 8 ) and (2 - 5 9 ) T h u s s " = £-t\2ki:L<i~k)-<-\+k)3-+ k , C > 4 - k ) C t - 3 ^ ) w e d r A a A k k , 0 - / * x ) U k , +C\-hk)^on0ol (2-614.) E t j aki^Ci-ju) - U - t - k ) 2 ^ k, ( i - H O O s / Q i c m f r J A k ^ i - / U L ) t ; 2 k , 4 < i 4 k ) W e o ] (2- 6^) { 2.k ,\> - A) - C14-1< f -f k i ( i -He )C i -3jU) |r I k**4 k J \ & ^ k k l ( » - y U i ; U k l 4 - C » + k ) + p n e o ] <2"66> E t U l<, 2-(^-A)-(l4•)<) 2^4 - k t(^4 - k ) C>-3^ ) f[^44k, 2fAB ( " 4 k k , ( l - y U ^ U k , 4- C l + l O - f a r j e J ( 2 " 6 7 ) The required corner f o r c e s ( P i g . 2-22) are now found from the e q u i l i b r i u m of the j o i n t s C and E ( P i g . 2-2lj.) s l 5 5 S3 = S, = Cb) P i g . 2-21*; I y =. o. at E. ) X B - Sj. Sin 4- S 3 s i n x , Uk, 2 '('-^)-SLC^k)4-tc>o43- 0 [k,Ci4-k)-|(k,(^-t -k)]}EUB ( 2 _ 6 8 ) 2k 0-/Kx)[2.k, + < H - k ) W i O - J - 3 7 -2 > » o a h a ; Y B = s^CoodS + s 3 c ^ ^ _ s * S i m i l a r l y £ y = 0 at" C ; X g = >^ir> <p - S ^ S i n p 1 Ci-H<)-2k,V£)+IAk (gk-f-0 - k, (t+k) j} Et Ae, ( 2 _ ? 0 ) •2k C i-/i1) 12k, 4- 04-k)-tcin 9 i ] = o. a(~ C ; ^ 4 Co5 jB. - S^ Co/> ^ - S," and V ' _ - ( k - l ) E - t A & _ Y - -from ( 2 - 6 3 ) Displacement Mode C This shear mode (Pig.2 - 2 5 ) i s s i m i l a r to c o n d i t i o n 3 ( P i g . 2-9), and the nodal forces i n both these cases are given by the same expressions i n terms of the str e s s |? , except that the d i r e c t i o n s of p i n these two cases are opposite to each other. , N A Y%QL |p(l4-lt)kO-In mode C A c = - ^ r  and t h e r e f o r e K — ^ c ^  Y kci+/i) Then y _ k i E t A c c " ikci+A) ( 2 " 7 1 ) V C^k) E t A c * C = *kC.+/t> ( 2 " 7 2 ) The bar stresses need not be determined i n t h i s case. Combination. As i n d i c a t e d i n P i g . 2 - 2 0 , the nodal forces c o r r e s -A X ponding to the a c t i o n 1 with the displacement A i are found by combination of the modes A, B and C, whose corner displacements stand i n the f o l l o w i n g r e l a t i o n s to the displacements A j '• A a , l A , * , A B = A c = l A f . The r e s u l t a n t corner forces of Action 1 are stated i n P i g . 2-26 - 3 8 -X 4 1 - X A X B x c Y 4 | . = _ Y A + Y B " Y C | —> * 3 I - X A + X B * X V C v =Y +Y -Y '31 T A T B C Y A + Y B + Y C , 2 Y „ s Y A + Y B + Y c *2I = * A - X B - X c XA + X B + X c Fig. 2-26 A D/K A n = A 0 " 1 / 4 A E/K //4 // F '2 A E =A l / 4 A F A F = A'1 / 2 (b) (c) F ig -2 -27 A D / K Fig. 2 -28 - 3 9 -i n terms of the components X^, Y^_, Xg, Yg, Xg, Y g » - X ( r and YQ of the three c o n s t i t u e n t modes. The symbols given to these f o r c e s c o n s i s t of the l e t t e r s X or Y i n d i c a t i n g the d i r e c t i o n of the f o r c e , and the two number s u b s c r i p s , the f i r s t g i v i n g the number of the node at which the force i s a p p l i e d , and the second, the number of A X the a c t i o n , i . e . the number 1 i n case of Z\ displacement. Assuming A, — 1 t these corner forces become the terms of the s t i f f n e s s matrix of the c e l l (the d i s t r i b u t i o n f a c t o r s ) belonging to the f i r s t column. These d i s t r i b u t i o n f a c t o r s are given i n Table 2 - 2 . 2.1^.2 A c t i o n 2 The displacement mode corresponding to the Y - d i s p l a c e -ment of the corner 1, A , , i s designated the a c t i o n 2 . This a c t i o n again may be described as the combination of the three displacement modes D, E and P shown i n P i g . 2 - 2 7 . These corner forces corresponding to Action 2 occupy the second column of the s t i f f n e s s m a t r i x . Displacement Mode D The nodal forces f o r t h i s displacement mode are w r i t t e n down from the s t r a i n c o n d i t i o n 2 (Ftq 2.-7a) i n which A^. i s replaced by A 0 = • The st r e s s j? i s found i n terms of Au and then the nodal forces ( P i g . 2 - 2 8 ) are expressed through A*f>, Thus A D - — j 7 P = : ; r D 2.-fr E_ v k a O - ^ ) Hence X d = _ J * < « + l 0 E t A and Y D = 2-k 0-/u*> ( 2~ 7 3> E t k,At> k ( i - f ^ ) (2-71;) o CD c 4 + of -- a + cs 4 0> s: c 4 / CO _ + 00 4 4 /^ 4 _£* i i o cD LU ill CO 4 •J o 0 CD c 4 4 JZ i i i 00 -<s <3> *r <3 4* 4 i ill 4 oo / (<3 X CO Displacement Mode E. The displacement mode E presented In P i g . 2-27b I s d i f f e r e n t from the bas i c s t r a i n c o n d i t i o n s 1, 2 and 3 of se c t i o n 2 - 2 , and so determination of the corner forces i n t h i s mode, as i n the e a r l i e r considered mode B, re q u i r e s the knowledge of the bar s t r e s s e s . In view of the symmetry of the mode E about the v e r t i c a l a x i s , the i n c l i n e d bars 3 and I4. ( P i g . 2-3(b))not shown i n P i g . 2-29, are unstressed because of the absence of any loads to balance t h e i r v e r t i c a l components at the secondary nodes B and P. F i g . 2-29 shows only the bars which are stressed i n the Mode E. These i n c l u d e the subdivided h o r i z o n t a l bars In the bases AC and GE. Determination of the bar stresses f o l l o w s the pro-cedure used i n connection with the mode B, based on the s t r e s s length - change r e l a t i o n i n an e l a s t i c bar and the knowledge of i t s c r o s s - s e c t i o n area. Complying with the procedure mentioned above and r e -f e r r i n g to F i g . 2-29 Elongation of the bar GE = 2.A E and i t s s t r e s s S = A E 5 2 A e ^ ( 2 - 7 $ ) S i m i l a r l y shortening of the bar AC = - 2 A E / ^ and i t s s t r e s s S-, = A, E - ( I A E ) KCL_ ( 2 - 7 6 ) In view of the e q u a l i t y of the areas A and A^, the str e s s e s S and S-^  are equal and opposite i n s i g n . Note that both the through and the mid point connected h o r i z o n t a l bars (they are shown as s i n g l e bars i n F i g . 2-29) are s t r e s s e d , and Fig. 2 -32 -1+3-t h i s r e q u i r e s the use of the areas A and A-^ . S u b s t i t u t i n g the expressions f o r these areas S 1 « k k r c i - / u * ; t c n - e i " (2-77) Elongations of the bars AE and GC (bars 2) equal ( k - l ) A E _ C O < $ / ^ Hence u s i n g eqn. (2 - 7 ) S P = -ftz E ( k - Q * E Go<> <fr 2 k T I (2-78) S u b s t i t u t i n g A 2, Co-6 cmd L^. i n t o t h i s expression s _ EKk^l )L ( l+k)+2iak l 4;an-a - 0 ] [4k l 1 +( l -H|<) 1 ] , / i -Aa 2 ^ k k r ( l - > ^ t 2 k , 4 - C » + k ) ^ n 0 o ] ( 2 " 7 9 > In view of the i n f i n i t e s t i f f n e s s of the bar 6 and the symmetry of the displacement mode E, the ends of t h i s bar H and D may move only i n the v e r t i c a l d i r e c t i o n ( P i g . 2-29). C a l l t h i s displacement A E . I t i s required to express t h i s displacement i n terms of A E • As stated before the I n c l i n e d side bars 5 and 5>' c a r r y equal s t r e s s e s and i n view of the e q u a l i t y of t h e i r areas, t h e i r changes i n length must be pro-p o r t i o n a l to t h e i r l e n g t h s . This provides a c o n d i t i o n f o r ex-pr e s s i n g i n terms of A £ . P r o j e c t i n g the deformed lengths of the bars on t h e i r undeformed d i r e c t i o n s i n F i g . 2-29, the shortening of the bar 5* i s A E Sir>9- 0 - ^ Co* 0-0 (2 - 8 0 ) S i m i l a r l y the shortening of the bar 5 i s A= Co-i ®o 90 ^2-80 Since these changes i n length are p r o p o r t i o n a l to the bar lengths (eqn. (2-11;) ) A E S i n f l o - ( A E : C O ^ S O H . = J _ £ = 2 . k , - K k - l ) - b m 0 - o g A E C O A e- 0- A ^ sinO-o 2k , - ( k - O - t a ' J ^ c -kk-from t h i s ( 2 - 8 3 ) and the shortening of the bar 5 from eqn (2-81) i s fek,(k->)- ( k - 0 H q n & » l A E L 4-kk, Then the str e s s e s i n the bars 5 and 5 are A k k , L s -(2-81+) ( 2 - 8 5 ) S u b s t i t u t i n g A5 and s . E t [ C k + i ) 2 - 4 j u k , ^ ] ( k - i ) A E ^ k k , 6 - ^ z ) S i r)G-o[2.k ,H-Ck4 -0^^>3 ( 2 - 8 6 ) Now the nodal f o r c e s may be determined by the con-d i t i o n s of the h o r i z o n t a l and v e r t i c a l e q u i l i b r i a of the top arid bottom nodes E and C ( P i g . 2 -30) P i g . 2-30 ^b) Using the signs of the bar str e s s e s shown by arrows i n Pig.2 - 3 0 S u b s t i t u t i n g S 2 and S^ / X e = f^CkH) E t A . _ ( 2 _ 8 7 ) ando X-W--=-0_ ah E J Y E L - S 4- j> +- <o<? 0>d 0-o again s u b s t i t u t i n g S, S 2 and S^ ^ E t U K i - < ^ k ) a 4 - 2 k , i a n 0 - o U k 1 2 - ^ ( l + - k ) ] r A E . -1+5-X V = o. a\- C 3 X E = S s Sin 9-6 - S^ S i o cj> =- - X E X.-U- = o a\- C ) Ye' = - 5 , -+• S i C o i j> +S^Cob O b Y' = Et { k < I4-k)g- 4K 2"] + 2k,-U?0-o [>k C i+k) -2k,"2-]) A ( 2 - 8 9 ) Displacement Mode F Again as was done before i n case of the mode C, the nodal f o r c e s ( F i g . 2-31) i n t h i s displacement mode may be ob-tained from the s t r a i n c o n d i t i o n 3 of s e c t i o n 2-2 i n which A 3 i s replaced by A p . Thus j? ;may be found i n terms of , and knowing t h i s the nodal, forces X p and Y F may be e a s i l y determined. The bar s t r e s s e s are not reouired i n t h i s case, THUS, * F = r W - ' a K l e t ) M -therefore P = , . , , •.— * ik, a ( t + ^u) 4 0 ? ) ( 2 - 9 0 )  l F 8k,(i4f) ( 2 9 1 ) s As i n d i c a t e d i n F i g . 2-27 the combination of nodal forces i n modes D, E and F r e s u l t s i n corner forces correspond-Y i n g to Action 2 ( F i g . 2-32). Making A 4 u n i t y , these forces be-comeuthe d i s t r i b u t i o n f a c t o r s occupying the second column of the element s t i f f n e s s m a t r i x . They are assembled i n Table 2-3« The system adopted f o r t h e i r designation i s the same as i n the d i s -t r i b u t i o n f a c t o r s of the Acti o n 1, Knowing the f i r s t two columns of the s t i f f n e s s matrix of the c e l l , the columns 3 and I4., corresponding to u n i t d i s -placements of the mode 2 i n the X and Y d i r e c t i o n s r e s p e c t i v e l y , may be e a s i l y found by simply r e v e r s i n g F i g s 2-26 and 2-32 -1+6-O /-^  u o 4 cO CO I —1 o u p < II X Hi I i 4 O O 4 -t-r A oo ^ a •¥ -4L IS III CO c4 I c4 X oo ci + i 4 - X 4 oo o + r—»t 4 •J -4-> i l l - a . i CO 1 \ ^ O O >2 X UJ i CD P 4 + 4> a 4^ ^3 4-LU ? OO iU + 4 4 s: CT 4-J* V9 U -1*7-together with t h e i r displacements and nodal forces about t h e i r axix of symmetry X. In case of the reversed P i g . 2-32 the Y d i r e c t i o n of the displacement and of a l l nodal forces produced by i t must be f u r t h e r changed to the opposite. The d i s t r i b u t i o n f a c t o r s of the columns 3 and 1* obtained In t h i s manner are presented i n F i g . 2-33(h) & ( b ) . The c o n d i t i o n s corresponding to the v e r t i c a l d i s -placement ^ 3 and h o r i z o n t a l displacement ^ 3 of the corner 3 are designated the a c t i o n l a and a c t i o n 2a r e s p e c t i v e l y . Their a n a l y s i s may be conducted i n a manner s i m i l a r to actions 1 and 2 but the operation may be s i m p l i f i e d i n the f o l l o w i n g way. F i g . 2-3l*(a) plves the d i s t r i b u t i o n f a c t o r s of the a c t i o n 1 and i n the F i g . ( b ) , the F i g . (a ) i s shown symmet-r i c a l l y reversed about the h o r i z o n t a l a x i s . The d i r e c t i o n s of the displacement of the corner 1 and of a l l corner forces i n F i g . (b) are f u r t h e r changed to the opposite, as shown i n F i g . ( c ) . F i g . (d) presents the d i s t r i b u t i o n f a c t o r s of the a c t i o n l a , and these may be w r i t t e n down by comparison with F i g . ( c ) t a k i n g i n t o c o n s i d e r a t i o n the d i f f e r e n c e i n the geometries of the two f i g u r e s . In F i g . 2 -3 l|(d), the parameters k and k^ have the f o l l o w i n g meaning: k = bottom side (ka) ; k-. = height ( k i a ) top side (a) top side (2-92) In P i g . 2-3l*(c) the r a t i o s become: bottom side (a) = 1 ; height ( k i a ) = ki (2-93) top side (Ka) K top s i d e T k a ) K From t h i s i t f o l l o w s that the d i s t r i b u t i o n f a c t o r s of a c t i o n l a may be read o f f the corresponding corners i n F i g . 2-3l|.(c) with replacement of k by 1 and K x by K l . These Fig. 2-33 '41 r k | Q *4I *3I I a i i Y3I Y2I AX 21 X|| - - - - - - 7 , y , j , k2l Xn. ^4ia, ,^3ia Y \-2 T2I . I Y l l Y4|0„ I Y I — — A - ^ \ * I1 Y,, Y, 3ia ' 4 — s V ^ . i i T A?=l 1 1 X21 X11 (a) X41 Y3i Y2ia 3^1 (b) X41 x3l^ (c) *2ia l I Y-ia-Xiia' (d) Fig. 2-34 '42 I M2 k j Q Ic, a 3> "32 * Y32 Y22 > "22 ^ ^ Y „ Y "42a i - >/2 k a 1 \ Y| 2 Y 4 2 J2 '42a;>  3// Y32 Y22Q / 2 Y22 2^2 '12 KA2 3^2 I X22Q v32a A3=| Y. IV,' 32a 1^  Y'2a 12a (a) (b) (c) Fig. 2 -35 -1*9-d i s t r i b u t i o n f a c t o r s occupying the f i f t h column of the element s t i f f n e s s matrix are stated i n Table 2-1*. A s i m i l a r procedure i s followed i n P i g . 2-35 f o r d e r i v a t i o n of the d i s t r i b u t i o n f a c t o r s of the a c t i o n 2 a . They occupy the s i x t h column of the element s t i f f n e s s matrix and are stated i n Table 2 - 5 . As shown before i n connection with the t h i r d and f o u r t h columns of the s t i f f n e s s m a t r ix, once the d i s t r i b u t i o n . x y f a c t o r s corresponding to the displacements and A 3 are A * A Y known, the ones corresponding to the displacements and ^ 4 may be found by symmetrical r e v e r s a l s of the f i g u r e s 2-3l*(d) and 2 - 3 5 ( c ) . These are presented i n P i g . 2 - 3 6(a) and (b) • 2.1*.3 S t i f f n e s s Matrices of the Bar C e l l s . The s t i f f n e s s matrix K of the t r a p e z o i d a l c e l l ( P i g . 2-1) composed of the derived d i s t r i b u t i o n f a c t o r s I s pre-sented i n eqn. 2-91*. X l 2 x 2 1 -X22 X l l a x 1 2 a x 2 1 a - x 2 2 a n Y l l Y 1 2 " Y 2 1 Y 2 2 Y l l a Y 1 2 a " Y 2 l a Y 2 2 a X 2 1 X 2 2 X l l ~ x 12 X 2 1 a X 2 2 a X l l a " x 1 2 a Y 2 1 Y 2 2 " Y l l Y 1 2 Y 2 1 a Y 2 2 a - Y l l a Y 1 2 a K} = 8 x 8 x 3 1 x 3 2 \ l ' X l * 2 X 3 1 a X 3 2 a x l * l a " X l * 2 a Y 3 l Y 3 2 ~\l Y 3 l a Y 3 2 a ' Y l * l a Y l * 2 a x l * l X l * 2 x 3 1 -X32 X l * l a XL|2a x 3 1 a " X 3 2 a " Y 3 1 *32 Y l * l a Y l * 2 a " Y 3 1 a Structures modeled by t r a p e z o i d a l c e l l s may i n v (2-91*) upside-down trapezoids ( P i g . 2 - 3 7 ) . The s t i f f n e s s matrix of an upside-down c e l l I s d i f f e r e n t from K i n eqn (2-91*), T A B L a .X (X-GL = 2 | C L _ •Y ,x 3,Q_ Y; 4 » 4lQ_ 2 . - 4 D I S T R I B U T I O N F A C T O R S " A C T I O N i CL. ( A X 3 = I ) 8k, 0-/^) L2.k,4- Ci+k)-twh00] - {Ak, ( i4-k)4- lHne-oL04-k)%4(uk t ^] j r£t Sk, C • [2-k, + (14-k) +QM 90 ] Sk, C»-/u 2-)[2k,4-Ck4-i) Uf fc t4-k ) ^ -4k^<'-)u) 4 - - t a n ' - 9 b [ c i 4 - k ) g - ^-kZ-Ci-k^ + ^ k ^ O k - t - Q l ^ E t 8k, ( i - ^ ) [2k, 4- Ck+1) tcin G o ] - C i - s ^ ) E . t >3io_ o Fig 2 -37 - 5 3 -although i t i s composed of num e r i c a l l y the same d i s t r i b u t i o n f a c t o r s . The correspondence between the r e s p e c t i v e terms of the two c e l l s i s determined by symmetry. Using standard n o t a t i o n of the nodes ( P i g . 2-37) the s t i f f n e s s matrix Ku i s given by eqn. ( 2 - 9 5 ) . [KuJ = 8 x 8 X 3 1 a " X 3 2 a X i | . l a x i ; 2 a X 3 1 " X 3 2 V X ^ 2 _ Y 3 1 a Y 3 2 a \ l & Y i ; 2 a - Y 3 1 Y 3 2 V \ 2 x l j . l a '\2& X 3 1 a X32a V - v x 3 1 X 3 2 " Y i | . l a Y U 2 a Y 3 1 a Y 3 2 a Y ^ 2 Y 3 1 Y 3 2 x l l a " X 1 2 a X 2 1 a X 2 2 a x l l " X 1 2 x 2 1 X 2 2 - Y l l a Y 1 2 a Y 2 1 a Y 2 2 a - Y u Y 1 2 Y 2 1 Y 2 2 X 2 1 a - X 2 2 a ; x l l a x 1 2 a x 2 1 " X 2 2 X U x 1 2 - Y 2 l a Y 22a Y l l a Y 1 2 a " Y 2 1 Y 2 2 Y U Y 1 2 (2-95) The s t i f f n e s s matrices [.K") and [Ku] are symmetrical about the p r i n c i p a l d i a g o n a l , as may be observed by comparing the expressions of the corresponding terms. This symmetry i s 1 , 2 , 3 , U general f o r the bar c e l l s of a l l shapes. 2 . 5 Trapezoidal F i n i t e Element (No Bar C e l l s ) An a l t e r n a t i v e f i n i t e element method i n v o l v i n g no-bar c e l l s has been widely used r e c e n t l y , and so i t has been con-sidered d e s i r a b l e to compare the p r e c i s i o n of the bar and no-bar CtHs :s« E x p l i c i t expressions of the d i s t r i b u t i o n f a c t o r s f o r ii. 5 the t r a p e z o i d a l no bar c e l l have been derived . Two d i f f e r e n t kinds of these d i s t r i b u t i o n f a c t o r s are a v a i l a b l e : the S t a t i c s type and the Energy type. The S t a t i c s matrix i s unsymmetrical about the p r i n c i p a l diagonal but the Energy type i s symmetrical. Of the two types of matrices the energy type i s the one which i s widely used. I t s d i s t r i b u t i o n f a c t o r s are given i n Table 2-6. o -H u a 11 O 3 i 111 X a I" >-) CTi c UJ > UJ 6^ . i G o a o II -J 3 0 4-<5.s 0 0 13 4 4 i 4-4 pi -* I + + . 4 4 4 4 i c4» •it + I ri I v9 4 4 Of 4 \-/ r—i I A ] I 4 <4 -4. i CS 4 c4' 4-X. 4 O -a. i 4 cS X - 5 5 -4 Z 0 J UJ —I CD X /—* X t 4 t -ST ± 4 I It 4 c4> - 3 . i \9 I ^ 1 4 erf < 4 X Hi - 3 . 4 4 -X rJ - X | I si" 4 J. X j I -X 4 -4 i 4 •+ i <0_ A X A 4 4 d 4 -X A X I I oi c4 4 w X 4 X t i x 4 <4_ 4-i i—i ^ _ ^: 13 H - X —• \ i -X i 4 xi I -* 4 * 3 1-d -t-V9 T _^  - X I - 5 6 -z o \J ai < r-2^  + c i 1 (0 + 3 + - X X 15 d 1 -a. -X o o + -X" -X + -X -X 4^ + x CO >< - i t 4 JX -4 -X oo -X J* <4_ + U i >2 _X 3 -X + I + X NT r i x + •t Hi _X X -X + — / f—^ H A . X 4-c4 x _x + x rJ - a . i el -1 X i •+• ^ / CN ro X T ^ I v — ' I 4-_x XL •+ •+ x + X + \9 II X >2 - 5 7 -2.6 Transformation Matrices The element s t i f f n e s s matrices presented i n Eqns. (2-91+) and (2-95) are a p p l i c a b l e when i n a p l a t e model the c e l l s are arranged with the small bases i n the adjacent c e l l s o riented a l t e r n a t i v e l y up and down. This i s because the x and y d i r e c t i o n s i n the adjacent c e l l s of the model are c o n s i s t e n t . On the other hand i f the adjacent c e l l s are arranged with the small bases ori e n t e d the same way, as, f o r Instance, i n the model of a c i r c u l a r d i s c , the s t i f f n e s s matrices [K]and [KuJ i n Eqns. (2-9U) and ( 2 - 9 5 ) become u n s u i t a b l e f o r s o l u t i o n because the x and y d i r e c t i o n s i n the adjacent c e l l s do not agree. In models of t h i s kind i t i s e s s e n t i a l that the d i r e c t i o n s of displacements and dorces i n the s t i f f n e s s matrix of the c e l l be transformed to r a d i a l and t a n g e n t i a l , i . e . p a r a l l e l and perpendicular to the s l o p i n g sides of the t r a p e z o i d . Let R , T, be the forces and A and A the d i s p l a c e -ments i n r a d i a l and t a n g e n t i a l d i r e c t i o n s ( F i g . 2 - 3 8 ) . The r e -l a t i o n s between the forces X* Y and R , T at d i f f e r e n t nodes are as f o l l o w s : At nodes 1 and 3 R = X Co-i -0-, + Y S i n O i T = - X 5 i n ^ + Y Co-6 0-) At nodes 2 and 1* R = x co/b -e, - Y s i n e-, ) T = x s'm -e-, + Y s i n e , ) In matrix n o t a t i o n these r e l a t i o n s are w r i t t e n as: (2-96) Fig 2-39. - 5 9 -Q o o 1 O o r ^ T, -Si"r,-e, Co 4 ©, 0 O o o O o Yi Ra. 0 o o o o o v. o O o o o o o o o 0 o o x 3 T 3 . o o o 0 -Sir} 0, Co4 -0, o o o o o O 0 O Cos e-, o o o o o O S i n 9, Co* or t C H F x y } (2 - 9 9 ) and K y i f C ]"'{ prt } ( 2 - 100) The same transformation matrix C c J i s a p p l i c a b l e a l s o to the displacements. Thus we have K y ) « [ c ] " M D r t 5 ( 2 - 1 0 1 > Let T ^ x y l and" [ K ^ ] be the element s t i f f n e s s matrices i n the d i r e c t i o n s X, Y and R, T r e s p e c t i v e l y . Then S u b s t i t u t i n g { ? X J } and \ D x y) from $ 2 - 1 0 0 ) & ( 2 - 1 0 1 ) t c l " M p r t } - [ K x y l ^ r M ^ t } ^ 2 - 1 0 3 ) { p r t } - c c ' i E ^ r c r 1 ^ } ( 2- 1 (*> [ C] i s an orthogonal matrix and therefore [ c ] " 1 = [ o f Thus and ( 2 - 1 0 5 ) ( 2 - 1 0 6 ) ( 2 - 1 0 7 ) C e l l s i n the shape of i s o s c e l e s t r i a n g l e s ( F i g . 2-39) may be present at the centre of the d i s c model. Their s t i f f n e s s m a t r i x , r e f e r r e d to X, Y a x i s , i s given i n Ref. I4. & 5 , and as i n -60-case of the t r a p e z o i d a l c e l l s , these d i r e c t i o n s should be changed. At the base nodes 1 and 2 the new d i r e c t i o n s roust again be R and T matching the chosen d i r e c t i o n s of the adjacent trapezoids and t r i a n g l e s . At the vertex node 3 where t h i s c e l l j o i n s s e v e r a l other t r i a n g u l a r c e l l s common d i r e c t i o n s Y and H are e s t a b l i s h e d / The o r i e n t a t i o n of the c e l l shown i n P i g . 2-39 with regard to V i s described by the angle u) , other t r i a n g u l a r c e l l s w i l l have d i f f e r e n t angles u) . The transformation equations f o r the corner forces i n the t r i a n g u l a r elements i n matrix form are as f o l l o w s : V • ("Cos ^ S i n e t o c. 0 O V T, - S i n &i C o * 0 i 0 0 0 o Yi "Rz o o Cob e, - Sir, Q\ 0 o o o Co* ©, o o •v o o o o Co£ u) - Sin a) o o o Q S i n O Y 3 This transformation matrix f o r the t r i a n g l e may be given the symbol 2.7 Problems with Prescribed Displacements as Boundary Conditions Boundary c o n d i t i o n s of the f i n i t e element models may i n c l u d e p r e s c r i b e d displacements of some of the nodes, i n c l u d i n g zero displacements. This introduces a c e r t a i n p e c u l i a r i t y In the s o l u t i o n which w i l l now be described. Let the number of p o s s i b l e displacements of the nodes i n the model ( I . e . degrees of freedom) be d. Of t h i s number m displacements are known ( i n c l u d i n g zero) and n unknown. The simultaneous equations c o r r e l a t i n g the displacements and the f o r c e s w r i t t e n In matrix form are as f o l l o w s : W d x d l D j d x l = { F j d x l (2-109) -61-P a r t i t i o n l n g t K D l \%\ {*F0} K l l ( n x n ) j K 12(nxm) K 21(mxn) < $22(mxm) Prom t h i s : K n x l ) , p l ( n x l ) 2(mxl)» | P 2 ( m x l ) L K l l l 111} + [ K l 2 l i D 2 } = { F l } (2-110) (2-111) £ K - Q ] 1 S the part of the s t i f f n e s s matrix of the model i n v o l v i n g only the unknown displacements. JD^j-is the vector of the unknown displacements, J F ^ j i s the vector of the e x t e r n a l forces a p p l i e d at the j o i n t s i n the d i r e c t i o n s of the unknown displacements and f % 2 ^ | ^ 2 ( n x l ) } ^ s v e c t o r forces produced by the known displacement vector {D 2) at the nodes with unknown displacements while assuming the n unknown displacements as zeros. The fo r c e vector { F21 i n Eq n» (2-110) gives the r e a c t i o n s at the r e s t r a i n e d j o i n t s once the vector |D-^  J- of the unknown d i s -placements has been found, Eqn. (2-111) may be w r i t t e n as L K l l H D j = l F l } - [ K 1 2 H D 2 } (2*112) The d i f f e r e n c e of vectors on the r i g h t hand side may be c a l l e d { P } . Then [ K l l H D l } = { ? } (2-113) The vector of forces | F ) - a c t i n g i n n d i r e c t i o n s represents the d i f f e r e n c e of the a c t u a l a p p l i e d forces and the ones produced by the m known displacements while the other nodes remain f i x e d i n n d i r e c t i o n s . The computer may be programmed to f i n d what may be c a l l e d the f i c t i t i o u s forces f K121 \ D 2 to subtra c t them from the ex-t e r n a l l y a p p l i e d ones {F^} and then to solve the Eqn. (2-113) f o r the n unknown displacements. The nodal f o r c e concentration i n each element are then found by m u l t i p l y i n g together element and the vector of i t s -62-the s t i f f n e s s matrix of the nodal displacements. CHAPTER 3 C a l c u l a t i o n Of Stresses Computer s o l u t i o n of the f i n i t e element model of a p l a t e s t r u c t u r e subjected to plane s t r e s s gives the d i s p l a c e -ments of the nodes which are supposed to represent the d i s p l a c e -ments of the corresponding points i n the prototype. The f u r t h e r step i s the c a l c u l a t i o n of s t r e s s e s i n the s t r u c t u r e . Two methods are a v a i l a b l e f o r t h i s purpose, the method of nodal displacements and the method of nodal f o r c e c o n c e n t r a t i o n s . How they are applied depends on the arrangement of c e l l s i n the model: whether the c e l l s a l t e r n a t e i n "the small base up" and "the small base down" p o s i t i o n s or the small bases i n the adjacent c e l l s are d i r e c t e d the same way. 3.1 Trapezoids With Small Bases i n Adjacent C e l l s A l t e r n a t i n g Up and Down 3 . 1 . 1 Method of Nodal Displacements The commonly used method of nodal displacements a p p l i e s only to the models composed of no-bar c e l l s and i t i n v o l v e s the f o l l o w i n g : 1. Stresses at the nodes i n each c e l l are expressed i n terms of the displacements of i t s nodes, using the r e l a t i o n s employed i n the d e r i v a t i o n of the s t i f f n e s s matrix of the c e l l . 2. Stresses at the same node found i n t h i s way i n s e v e r a l c e l l s adjacent to i t , are averaged up, and these averages are taken as the stresses i n the prototype. This method becomes questionable when the s e v e r a l c e l l s c o n t r i b u t i n g to the s t r e s s have d i f f e r e n t shapes or form d i f f e r e n t angles at the node under c o n s i d e r a t i o n . One f e e l s that the con-t r i b u t i o n of the c e l l s B and D to the s t r e s s e s at the node 1 ( F i g . 3-1) should be greater than of the c e l l s A and C but a r a t i o n a l b a s i s f o r the assignment of s u i t a b l e weights to these c e l l s i s not apparent. In view of the questionable nature of the o u t l i n e d method a d i f f e r e n t although somewhat s p e c u l a t i v e approach i s suggested here e q u a l l y a p p l i c a b l e to the t r a p e z o i d a l c e l l models of both bar and no-bar types. The method i s based on the w e l l known e l a s t i c i t y formulae. EL ( ( 4- l £ ) ( 3 - 3 ) ix1 ~ ^ZTTju) "a F i g 3-2 shows the arrangement of the c e l l s . The d i s -placements of a l l nodes are known. I t i s required to f i n d the s t r e s s e s at the node 5» To f i n d the values of the d e r i v a t i v e s of displacements i n Eqns (3-1) to ( 3 - 3 ) an assumption i s made that the displacements vary p a r a b o l i c a l l y between any three con-secutive nodes. This allows one to evaluate the d e r i v a t i v e s of the displacements at the intermediate nodes i n the groups of three successive nodes and at the p o i n t s nearby, l i k e 2' and 8' c l o s e to the nodes 2 and 8 i n F i g . 3 - 2 . Considering the nodes 1, 2 and 3 along the l i n e (a) u i = A ! + A 2 * i + A 3 n 2 ^ u 2 = A1 + A 2 y 2 + A 3 y 2 2 V (3-lj.) u 3 * A l + V 3 + V3 2 J where y^, and y^ are the y coordinates of the nodes 1, 2 and 3 on the l i n e (a) and A-j_, A2 and A^ are appropriate c o e f f i c i e n t s . R e l a t i o n s (3-1+) may be w r i t t e n as Fig 3 - 3 Fig 3 - 4 - 6 6 -\Ua> = [Ya]{Aa} (3-5) the s u b s c r i p t a I n d i c a t i n g the l i n e (a) of the three nodes. Then {A&} = [ Y f t ] - l { * a } ( > 6 > Prom (3-1*)T th© displacement U 2 o f t D e p o i n t 2* with the coordinate Jfe'* s i t u a t e d d i r e c t l y above 5 , Is U 2 ' = f 1 W V 2 2 J {Aa} (3-7) c a l l i n g the row vector i n t h i s equation { ^ ' J - * and s u b s t i t u t i n g Aa from (3-6) »2 - t*2 V [ y a ] " 1 {TJ S} (3-8) A s i m i l a r procedure app l i e d to displacements leads to V L - i Y , ' } ' [ Y . T ' i V a } (3-9) In a s i m i l a r way the displacements of the po i n t 8' s i t -uated d i r e c t l y below 5 on the h o r i z o n t a l l i n e (c) ( F i g . 3-2) are found as U,' = { Y e } * [ Y c ^ ' i M (3-xo) W - ^ Y s [ Y c V'iVc} (3-11) With the displacements of the po i n t s 2 ' and 8 1 now known consider the p a r a b o l i c r e l a t i o n between the three successive displacements along the v e r t i c a l l i n e ( d ) . q = c 1 + C 2x + C3X 2 (3-12) Applying t h i s to the three consecutive points U 2 ' = C + C2x2-° 5 = c i + C 2 X 5 + C 3 X 5 2 a 8 ' = c 1 + c 2 x 8 ' + c 3 x 8 / 2 (3-13) Expressions (3-13) way be w r i t t e n i n matrix from as 1 M = [ * d l W (3-14) Then ] C } = [ X d J ' V d l (3-1$) D i f f e r e n t i a t i n g the eqn. (3-12) and applying the r e s u l t i n g expression to the node 5 -67-Using the row matrix X5 = [ 0 1 2 x 5 ] and the vector { c } = (3-16) the equation (3-16) becomes S i m i l a r l y (3-17) (3-18) F o l l o w i n g e x a c t l y the same approach s i m i l a r expressions may be w r i t t e n f o r the d e r i v a t i v e s of the displacements along the y a x i s du] and where 1 J s -(3-19') (3-20) 1 ^ XJ4 1 1 Us V . 1 "J* o v *6 Now us i n g the r e l a t i o n s ( 3 - D , 3-2) and(3-3), the stre s s e s at the node 5 are^found as Y J s -The p r e c i s i o n . o f the method should increase on reduction of the c e l l s i z e . As presented here, the method a p p l i e s only to -68-the s t r e s s e s at the i n t e r i o r nodes. I t may be extended to the edge nodes of the model by u s i n g the t e r m i n a l slopes of the p a r a b o l l i c displacement l i n e s . However the p r e c i s i o n of the edge str e s s e s so determined i s not expected to be high. 3 . 1 . 2 Method of Nodal Force Concentrations In t h i s method d i f f e r e n t procedures must be used f o r c a l c u l a t i o n of s t r e s s e s on the h o r i z o n t a l and v e r t i c a l s e c t i o n s of the prototype because of the zig-zag nature of the s e c t i o n f o l l o w i n g the s l o p i n g sides of the c e l l s . (a) Stresses on the H o r i z o n t a l Section The d e s c r i p t i o n of t h i s method i s given by Hrennikoff^". The diagram of stresses on the l i n e AA^ determined from the nodal concentrations i s assumed to have a polygonal shape ( F i g s . 3-3 & 3-U)« In t n e absence of l o c a l l y applied e x t e r n a l loads, the p l a t e s t r e s s e s , both normal and t a n g e n t i a l , are found as a r u l e , modified as stated below, by c o n v e r t i n g the nodal con c e n t r a t i o n s , given by the computer s o l u t i o n , i n t o the c o r r e s -ponding t r i a n g u l a r areas. Thus str e s s e s at an intermediate node n are = Cl+K)at ( 3 - 2 ^ } and CA 2.TV) l h " = O + IOat ( > 2 5 ) At the edge nodes 1 and 7 the s t r e s s e s are The s u b s c r i p t s h i n the expressions f o r ? i n d i c a t e that the s t r e s s belongs to the h o r i z o n t a l plane. The s i m i l a r shearing s t r e s s on the v e r t i c a l plane f , although t h e o r a t i c a l equal to , may come out somewhat d i f f e r e n t i n c a l c u l a t i o n . The diagram of shear stresses ^ so found i s s t a t i c a l l y e quivalent to the nodal c o n c e n t r a t i o n s . As to the normal stresses 6^ , they agree only with the j[Y c o n d i t i o n tmt not with the moment c o n d i t i o n , because the centroids of a l l t r i -angular areas of s t r e s s e s are d i s p l a c e d from the corresponding nodes. This makes i t necessary to apply the moment c o r r e c t i o n to a l l stresses found above. This i s done by computing the unbalanced moment and f i n d i n g the c o r r e c t i o n stresses by the usual f l e x u r e formula c o n s i d e r i n g the s e c t i o n AA^ as the c r o s s -s e c t i o n of a beam. The necessary formulae may be found i n Ref. ( 6 ) . They w i l l not be used i n the present work because the beam problem analyzed i n i t i s symmetrical about the v e r t i c a l a x i s and thus does not i n v o l v e a moment c o r r e c t i o n . A m o d i f i c a t i o n of the standard procedure may a l s o be needed with regard to the shear s t r e s s e s . I f the shear s t r e s s at the edge must be zero by the nature of the problem, the edge concentration T} may not be converted i n t o the t r i a n g u l a r area as i n P i g . 3—14-• The d i f f i c u l t y may be resolved by transforming T^ i n t o the p a r a b o l i c area with the base ka and the mid-point ordinate 'Vhia. - i l L - . ( P i g . 3 - 5 ) I 2. P i g . 3-5 (b) Stresses on V e r t i c a l Sections C a l c u l a t i o n of p l a t e s t r e s s e s on the v e r t i c a l s e c t i o n i s more complicated and l e s s c e r t a i n than on the h o r i z o n t a l s e c t i o n . -70-P i g . 3 - 6(a) shows a v e r t i c a l column of s e v e r a l t r a p -e z o i d a l c e l l s . The normal s t r e s s e s 6* and the shear stresses on h o r i z o n t a l sections at a l l nodes have already been de-termined and now the st r e s s e s <Sy and (the l a t t e r theor-a t i c a l l y equal to *h/ are to be found. One of the intermediate elements of the column, the element C i s shown separately i n F i g . 3-7 together with the str e s s e s a c t i n g on i t s four s i d e s . For convenience of a n a l y s i s the i n c l i n e d sides are replaced by v e r t i c a l and h o r i z o n t a l steps of i n f i n i t e s i m a l s i z e . I t w i l l be assumed that the known nodal f o r c e s X and Y a c t i n g on the c e l l are con t r i b u t e d i n accordance with s t a t i c s only be the stresses on the two sides of the c e l l adjacent to the node. Thus the for c e X^ may be viewed as caused by the normal stresses 6~x on the side 3-3'» stresses <fx on the h o r i -z o n t a l steps of the side 3-lj- and the shearing s t r e s s e s on the v e r t i c a l steps of the side 3—14-• Likewise the nodal force Y^ i s produced by the shearing stresses ^ on the side 3-3* and the h o r i z o n t a l steps of the side 3-lj-, together with the normal st r e s s e s on the same s l o p i n g s i d e . Since the stresses on the h o r i z o n t a l planes have already been found, the parts of the nodal forces contributed by them may be subtracted, thus l e a v i n g the remainders of X and Y c a l l e d X g and Y n which may be presumed to have been developed s o l e l y by the unknown st r e s s e s 'Ty and r e s p e c t i v e l y . I t i s necessary now to assume, f o r s i m p l i c i t y that the normal and shearing s t r e s s e s on the h o r i z o n t a l and v e r t i c a l planes along t h e 1 s l o p i n g sides of the c e l l (as w e l l as on the -71-Fig. 3 -8 - 7 3 -h o r i z o n t a l s i d e s ) vary l i n e a r l y . P o i n t i n g the nodal con-c e n t r a t i o n s i n the p o s i t i v e d i r e c t i o n s of the axes, and employ-i n g the us u a l sign convention f o r the normal and shearing stresses as i n d i c a t e d by the arrows i n P i g . 3-7* the remainders of the nodal concentrations at the nodes 3 and Ij. are found as fol l o w s : *32> - X-3 + (2. <TX3 + ^ 3 ) at Ck-Oat (3-26) ~ (2.6*4 ' + ^  ) kat 6 + (k-Oat 12- (3-27) Y _ Y 3 - " ( Z?h3 + u(-12- (3-28) Y + v (k-i)ot ' 12- (3-29) These expressions are derived f o r a t r a p e z o i d a l c e l l w i th the smaller base up. For an upside-down trapezoid D (Pig. 3 - 8 ) the corresponding equations become K ;03-- 3 0 ) u - ( UKS +- <rx^ ) <* - + 2 ^ ) ^ - ; ^ (3- - 3 D + ( + % : ) ^ - ( 2 ^ 4 + (3- -32) V u - is- (3- -33) As has been explained, the X g nodal forces are developed e x c l u s i v e l y by the shear s t r e s s e s lv , and the Y n forces by the normal stresses <S^ . For t h i s reason the mean stre s s e s and <*\j on the s l o p i n g sides of the c e l l C may be found thus: -714--J .2k, a t S i m i l a r expresses hold f o r the up-side down c e l l . In view of the assumed l i n e a r i t y of str e s s e s along a l l sides the expressions (3—3^ 4-) & (3 -35) may be assumed to represent the true s t r e s s e s along the s l o p i n g sides at the mid-heights of the c e l l s . The stepped s o l i d l i n e s i n P i g s . 3 - 6(b) & (c) with ordinates T and 6~ s a t i s f y i n g the equations (3-31)-) & (3 -35) thus represent somewhat u n r e a l i s t i c but c o n s i s t e n t with s t a t i c s s t r e s s d i s t r i b u t i o n s along the zig-zag l i n e 1 - 2 - 3 . . . . 9 . The stepped l i n e f o r W i s f u l l y c o n s i s t e n t with s t a t i c s , while the 6y l i n e i s c o n s i s t e n t w i t h J H equation of s t a t i c s but not with £M • The m i d - c e l l ordinates of these stepped l i n e s define the general trend of s t r e s s over the s e c t i o n , and so the required approximations to the true s t r e s s diagrams are found by j o i n i n g the mid c e l l ordinates by dotted l i n e s i n P i g s . 3-&(b) & ( c ) . Replacement of the stepped s t r e s s diagrams by the polygonal does not change the areas under the diagrams and thus does not v i o l a t e the £V a n <^ equations of s t a t i c s . The u n r e a l i s t i c square parts of the diagrams over the extreme^upper and lower h a l f c e l l s may be adjusted by judgment. Thus the 6^  diagrams may be extended to the edges at the slopes of the adjacent p a r t s . This may and may not a f f e c t the equation of s t a t i c s . The equation may be expected to be approximately s a t i s f i e d by usi n g the polygonal l i n e f o r <Sy s t r e s s e s . The square ends of the shear diagrams may be smoothed up with some minor increase of the s t r e s s e s i n the v i c i n i t y of the edges. The zig-zag s e c t i o n 1 -9 was considered as the l e f t side - 7 5 -boundary of the stack of c e l l s A to H, but the same s e c t i o n forms the r i g h t side boundary of the adjacent c e l l s on the l e f t . I t i s easy to see that i f the same c a l c u l a t i o n i s per-formed on the l e f t hand side c e l l s the nodal concentration r e -mainders X s and Y n, re s p o n s i b l e f o r 2-v and 6y stresses w i l l be somewhat d i f f e r e n t from the ones found above. Since there i s no reason f o r the preference of one side over the other i t appears l o g i c a l to average up the X g and Y n concentrations on the two sides of the zig-zag and only then proceed with the c a l c u l a t i o n of T v and str e s s e s as explained above. The formulae f o r X g and Y n values i n terms of X and Y, and the st r e s s e s 6? and on the r i g h t hand sides of c e l l s are d i f f e r e n t from the equations (3 -26) to (3 -33) f o r the l e f t hand sides used above, but they can be e a s i l y adapted from them. I f a normal c e l l (small side up) ( F i g . 3 - 7 ) i s rot a t e d through 180° i t becomes an upside down c e l l , i t s l e f t side be-comes the r i g h t s i d e , the d i r e c t i o n s of the nodal forces reverse, but the p o s i t i v e d i r e c t i o n s of stresses remain the same. From t h i s i t may be r e a l i z e d that the expressions f o r X_ and Y on the s n r i g h t hand side of the c e l l may be found from the corresponding expressions on the l e f t hand side of a reversed c e l l ( i . e . an upside down i n s t e a d of the r i g h t side up and vice-versa) i n which the signs of the terms c o n t a i n i n g and ^ are r e -versed, 3 . 2 Trapezoids With Small Bases i n Adjacent C e l l s S i m i l a r l y  D i r e c t e d . This kind of arrangement Is met with i n problems i n v o l v i n g p l a t e s i n the form of f u l l c i r c l e or a sector of i t . The a p p l i c a t i o n of the method of nodal displacements i n - 7 6 -t h i s case i s complicated and may not r e s u l t i n accuracy compar-able to the one obtained by the method of nodal force con-c e n t r a t i o n s . Only the l a t t e r ' J method i s described here. 3.2.1 Method of Nodal Force Concentrations The method of c a l c u l a t i o n of str e s s e s from the nodal displacements i n t h i s case resembles c l o s e l y the one explained e a r l i e r with regard to the str e s s e s i n the deep rect a n g u l a r beam on the plane p a r a l l e l to the bases of the t r a p e z o i d a l elements. In the present case the adjacent c e l l s are placed t o -gether forming a c i r c u l a r arc ( F i g . 3 - 9 ) or a narrow sector ( F i g . 3-10). The nodal forces N and T determined by the computer and a c t i n g i n the r a d i a l and t a n g e n t i a l d i r e c t i o n s are appli e d at the appropriate nodes i n F i g . 3 - 9 and 3-10. I f no e x t e r n a l loads are present at the node under con-s i d e r a t i o n , the normal and shearing stresses may be found by con-v e r t i n g the nodal concentrations i n t o the t r i a n g u l a r areas" i n the s t r e s s diagrams. Stresses at the intermediate nodes with some exercise of judgment may be assumed to be -0", beinfr the i n t e r n a l angle of the c e l l i n r a d i a n s , <5^  the normal s t r e s s i n r a d i a l d i r e c t i o n and ^ the shearing s t r e s s on the c i r c u l a r surface. S i m i l a r l y , the normal and shearing s t r e s s e s on the r a d i a l plane i n F i g . 3 - 9 are ^ " ( C T i M " (r...-r„„)t - 7 8 -In s p e c i a l cases c o r r e c t i o n s may be necessary as i n the deep beam problem. Thus i f the p l a t e i n F i g . 3-9 i s not a di s c but only a sector of i t with no t a n g e n t i a l e x t e r n a l forces a p p l i e d at the edge, the t a n g e n t i a l end concentration must be converted i n t o a p a r a b o l i c area r a t h e r than a t r i a n g l e , and the normal stresses at the edge may be modified f o r the e f f e c t of a moment, i n order to preserve s t a t i c s . L o c a l c o r r e c t i o n s of a s i m i l a r nature may be a l s o needed with regard to the stre s s e s a c t i n g i n the r a d i a l plane ( F i g . 3-10). CHAPTER 4 EXAMPLES Two plane s t r e s s problems, one i n v o l v i n g a deep r e c t -angular beam i n f l e x u r e and the other a c i r c u l a r p l a t e acted upon by two equal and opposite p o i n t loads along i t s diameter, are solved by the F i n i t e Element (F.E.) Method u s i n g both the bar c e l l s and the no-bar c e l l s . These s o l u t i o n s are compared with the known e l a s t i c i t y s o l u t i o n s , 4. 1 Deep Rectangular Beam 4.1.1 E l a s t i c i t y S o l u t i o n : A deep rectangular beam (Fig.Ij.-l) of small thickness t i s acted upon by the uniformly d i s t r i b u t e d load per u n i t length and i s supported i n a c e r t a i n s p e c i f i e d way at the ends. For s i m p l i c i t y of the r e s u l t s the end support i s e f f e c t e d by the shear s t r e s s e s d i s t r i b u t e d p a r a b o l i c a l l y and normal s t r e s s e s with zero r e s u l t a n t . The expressions f o r the displacements and the stresses are given by Timoshenko . (4*4) *. re. iz. s-<P 5 16 E t c : I -v-i s the v e r t i c a l d e f l e c t i o n at = O - O. (4-2) (4-3) °* ~ 4 t c 3 I 3 5 ^ " (4-4) *5 -3 4 t c (4-5) (4 - 6 ) - o 1 '^ Txy - 8 0 -q /unit lenght A A l X,u Fig. 4-q /unit lenght o to thiek'nes's' = t o to iri II o '(M) l l .5a UN) (0) I. Displacements of nodes along z ig -zag line (M) are prescribed. 2. v displacements along (N),= - v displacements along (0). X,u |*—a = l.25b—»| 1.4a Fig, 4 - 2 -81-In these expressions, the v e r t i c a l displacement U i s zero at the mid-heights of the ends while the h o r i z o n t a l d i s -placement i s zero along the l i n e of symmetry (y=o) of the beam, if.. 1.2 F i n i t e Element S o l u t i o n ! Since the s t r u c t u r e i s sym-m e t r i c a l i t i s d e s i r a b l e to make the c e l l model a l s o symmetrical about both axes. This means that the number of h o r i z o n t a l c e l l rows must be even and the number of c e l l s i n each row odd ( l i n e of symmetry passing through one of the element columns). Four-teen rows with 19 c e l l s i n each row were chosen with the c e l l and model geometry shown i n F i g . lj.-2. Since the s t r u c t u r e and the load are symmetrical about the v e r t i c a l a x i s i t i s s u f f i c i e n t to analyse only one h a l f of the model. The nodal concentrations produced by the applied load are computed by t r a n s f e r r i n g the load to the adjacent nodes by s t a t i c s . As to the c o n d i t i o n s at the supports i t was d e s i r a b l e to s p e c i f y them i n terms of the displacements along the end zig-zag l i n e s , computed by the equations (lj.-l) and (I4.-3) r a t h e r than i n terms of the f o r c e concentrations produced by the end r e a c t i o n s . Since the model possesses no nodes on the v e r t i c a l a x i s of symmetry, the symmetry of the problem i s described by assuming the h o r i z o n t a l displacements of the nodes l y i n g along the zig-zag l i n e s 0 and N near the middle of the beam equal and opposite i n sign at the corresponding nodes. At the same time the v e r t i c a l displacements U are equal at the corresponding nodes,along the same two zig-zag l i n e s . The shape of the c e l l used i s c h a r a c t e r i z e d by the r a t i o s k«l-4 and k, = o-8. Three values of Poisson's r a t i o 0 . 0 5 , 0 . 3 3 3 3 and 0.14.5 have been used. The v a r i a b l e angle parameters ^ have -82-been used. Tbe v a r i a b l e angle parameters 0 O have been s p e c i a l l y chosen to make the cross sections of a l l bars p o s i t i v e , because negative c r o s s - s e c t i o n s r e s u l t i n lower p r e c i s i o n as Is demon-s t r a t e d i n Chapter 5» Table L\.-l contains the values of d i f f e r e n t bar areas expressed i n terms of the c o e f f i c i e n t s ©f, i n the expression A = cK,at . These c o e f f i c i e n t vary f o r d i f f e r e n t values of \*~ and ^ 0 . A f t e r d e ciding on value of s u i t a b l e parameter "6"0 a r b i t r a r i l y from a small range of t h i s parameter f o r which a l l area parameters are p o s i t i v e f o r d i f f e r e n t values of the problem i s solved by the method of f i n i t e elements. Tables l\-2 to L\.-l6 show the values of d i f f e r e n t f u n c t i o n s c a l -culated by the e l a s t i c i t y formulae (E) (lj.-l) to (t|.-6) and the method of f i n i t e element (F.E.). The percentage e r r o r i s a l s o given i n accordance with the f o l l o w i n g expression % E r r o r = 1 Function (E) I 1 Function (F.E.)l (k-7) |Function ( E ) j I f both (E) and (F.E.) values of the f u n c t i o n happen to be of opposite signs the minus sign i n the numerator of the ex-pression f o r e r r o r i s replaced by p l u s . K*2 C i r c u l a r P l a t e ij.,2.1 E l a s t i c i t y S o l u t i o n : The problem of a t h i n c i r c u l a r p l a t e with two forces a c t i n g against each other along one of i t s diameters Fig.l|.-3 i s chosen i n view of a v a i l a b i l i t y of i t s t h e o r a t i c a l s o l u t i o n . (7) This s o l u t i o n , given by Musk'nelishvili , employs the theory of Complex V a r i a b l e . Pol a r Coordinates of a p o i n t , j* and -0", are presented by the Complex V a r i a b l e Z = r*g . The displacements U along I A n A n T A B L E . -4-1 . A i ^ a 'Fararn&itrs a s - C o e ff f c lea t o f Q i n A r € . a = . - ^ o . o t j wbe.Ke_ a -to)? s i d e - = \''25~b or>A t =- i b i e k f l e s S problem A- •e-o A , / a t Am/at ^/at A 3 / a t M'/aX A s / a t ICQ) C O S " 8 2 ' 5 ° o-\2\7 O ' l O g S (D' 5\322 O - 5 2 0 g 0 - S 5 4 & K b ) 0 « 3 3 3 3 o-34o<£ O- \ 370 C-1 '257 0 - G 0 3 O I C O - o - 4 s - )c rO.° 0-1325 0 - 4 H 9 O.Q3S5 ©-174 i . 0.7141 TABLE 4 - 2 X - DISPLACEMENT u AND % ERROR; AT DIFFERENT X AND Y p = 0 / 0 5 AS COEFFICIENT cj \ IN U = C - | J - I O B Y ELASTICITY A N D FINITE E L E M E N T \ P A R A M E T E R 80 a 8 2 . 5 ° ',-• • •{ 0 •COLUMN A B ' / C • - • 6/1 E W • FUNCTION? %ERROR FUNCTION %ERROR FUNCTION %ERROR FUNCTION 1 %ERROR FUNCTION % ERROR ELASTICITY 1.4703* 3.2744 4.7114 • ' «, . - 5.6821 : $ 6 , 1 2 3 1 1 U J BAR 1.4685 0.10. , 3'. 2699 : 0.14 4.7027 ' 0.18 5.6699 0.21 6 0 I O 8 8 0*23 I N N O - B A R 1.5056 -2.42 3.3686 -2.87 4.8257 -2.43 '5.802^ -2.12 6 „ 2 4 5 5 - 2 . 0 0 ELASTICITY 1.5376 3.3199 4.7243 5 . 6 5 4 6 . 6 o 0 5 0 5 |Q ui BAR 1.5340 0 . 2 3 3.3H7 0.25 4.7124 0 . 2 5 5.6407 0 . 2 5' 6 o 0 3 6 l 0o24 IN- N O - B A R 1.5750 -2 . 4 4 3.4097 -2.70 4 . 8 3 3 2 -2.31 5.7691 -2.02 6 0 I 6 5 0 -1 . 8 9 ELASTICITY 1.2861 3.0979 . 4.5403 5.5146 ; 5o?571 2 ui BAR 1.2852 0.07 3.0932 0.15 •' 4.5316 0.19 5 . 5 0 2 6 ': 0.22 5 . 9 4 3 3 0 . 2 3 IN- NO-BAR 1.2916 - 0 . 4 3 3.1453 -1.53 4.6177 ' -1.70 5.6074 *• -1.68 6 0 0 5 7 5 - 1 . 6 8 ELASTICITY 1.1280 2.9443 4.3900 5.3664 5 o 8 0 9 9 3 L J BAR 1.1273 0.06 "2.9399 0.15 4.3815 0 . 19 5.3547 ' A.2 2 5o7964 0o23 IN- N O - B A R 1.1147 1.18 2.9520 -0.26 4.4260 -0.82 5.4219 l l . 0 3 5 o 8 7 6 0 - 1 . 1 4 ELASTICITY 1.2252 3.'0163 . 4.4270 5.3614 f 5o7589 3a ui BAR 1.2220 0.26 3.0085. 0.26 4.4159 0 . 2 5 5.3482 0 . 2 5 ' 507452 0 . 2 4 IN- N O - B A R 1.2322 - 0 . 5 8 3.0385 • -0.74 4.4654 -0.87 5.4091' - 0 . 8 9 5 . 8 0 7 8 - 0 o 8 5 ELASTICITY 1.1252 2.9163 4.3270 5.2614 5 o 6 5 8 9 4a ui BAR • 1.1220 0.28 2.9085 0.27 4.3159 0 . 2 6 5.2482 0.25 506452 0 . 2 4 L N NO-BAR 1.1226 0.23 . 2.9116 0.16 4.3281 -0 . 0 3 5.2679 ' -0.12 5 . 6 6 5 5 - 0 . 1 8 ELASTICITY 0.9280 . 2.7443 4.1900 5 . 1 6 6 4 506099 5 ui . BAR 0.9273' 0.08 2.7399-' 0.16 4 . 1 8 1 6 • 0.20° 5.1*547 0.23 5*5964 0 . 2 4 I N N O - BA R 0.8982 3.21 2.7006 ' 1.59 4.1526 0.89. 5.1398 0 . 5 1 5o5917 0o32 ELASTICITY 0.8861 2 . 6 9 7 9 . 4.1403 5 . 1 1 4 6 'v 5 . 5 5 7 1 6 BAR 0.8852 0.1Q 2.6932 0.17 4.1316 0 . 2 1 5.1026 ' 0.23 5 . 5 4 3 3 0 . 2 5 I N NO-BAR 0.8553 3.47 2.6359 2.30 4.0682 1.74 5.0437 1.39 5 . 4 9 0 4 l o 2 0 ELASTICITY 1.0376 2.8200^ 4.2243 5.1546 6a ui BAR 1.0340 0.35 2.8117 0 . 2 9 4.2124 0.28: 5.1408 0.27 5*53(>l 0.26 u: NO-BAR 1.0170 1 . 9 9 2.7591 2 i l 6 , 4.1422 1.94 5.0646 1.75 .5 .4572 1 . 6 8 ELASTICITY 0.8701 2 . 6 7 4 4 ,4.1114 •5.0827 5*5231 .7 ui BAR 0 . 8 6 8 5 • .0.18 2.6699 0 . 1 7 . 4.1026 0.21 5,0699 0.24 ; 505088 0.26 I N N O - B A R 0 . 8 3 7 3 3.76 2 . 5 8 8 7 3.20' 3.9977 .' 2.7.6 4.9581 ' 2.48 5 o 3 9 7 5 2.27 • 1 '',! <\ DEEP RECTANGULAR BEAM WITH UNIFORMLY DISTRIBUTED. LOAD q#/unit length la 3 3a 4a 5 6 6a 7 POINT p : COLUMN B ROW 2 a=l.25b—>) ' ... k- 1.4a thickness 8 t t 1 1. TABLE 4 - 3 Y - D I S P L A C E M E N T V AND % ERROR AT DIFFERENT X AND Y jj. - 0.05 AS COEFFICIENT C j . IN v=c|yl0BY ELASTICITY AND FINITE ELEMENT / | ' PARAMETER 80 = 82.5 i R 0 W COLUMN A B •C ; • ! D E FUNCTION %ERROR FUNCTION %ERROR FUNCTION %ERROR FUNCTION % ERROR- FUNCTION % ERROR ELASTICITY 2.5679 2.2644 1.7275 A 1.0284 o„?^77 1 U J BAR 2 .5667 0.05 2.2623 0 .09 '1.7253" 0 . 1 3 1,0267 0 .16 0.2369 0 . 3 2 NO-BAR 2.5239 1.72 "2.2 372 1.20 1.7129 0.85 1.0180 1.00 0.2253 5ol9 ELASTICITY •2.0188 1.7732 • 1.3382 0.7728 0.1362 la UJ BAR 2.0184 0.02 1.7722 0 .06 1.3370 0.08 0.7721 0.09 0.1362. 0 . 0 NO - BAR 2.0538 - 1 . 7 3 1.8227 -2 .79 1.3799 - 3 . 1 1 0.8039- -4 .02 :< 0.1591 - 1 6 0 8 5 ELASTICITY 1.5474 1.3833 1.0638 0 . 6 360. 0 01U .73 2 UJ BAR 1.5470 0.25 1.3824 0.07 1.0628' : ,0.-10 0.6353 0.12 Oollj-70 0 o l 8 u.- NO-BAR 1.4961 3 .32 1.3341 . 3 .56 ; 1 . 0 ? Q 0 3 .27 0.6080 . 4.40 0oll96 18.77 ELASTICITY 0.7114 0.6439 0.4986 0 . 2 9 9 3 : 0.0691; 3 ui BAR 0.7113 0 . 0 2 0.6435 0.05 0.4982 0.08 0.2990 0.17 0.0693 0.-17 UL' NO-BAR 0.6645 6 . 5 9 0.5876 8.75 0.4504 9 .67 0.2582 1 3 . 0 1 i 0.0321 5 3 068 ELASTICITY 0 . 3 3 3 2 0 . 3 0 0 9 . 0.2306 0.1343 : 0.0237 3a BAR 0.3332 0.00 0.3007 0.04 0.2305 0.06 6.1342 0.07 ; 0.0237 0.0 U J NO -BAR 0.3485 -4 .59 0.3229 -7 .12 0.2589 -12.25 0.1682 -25.20 j 0.0618 -160.2-/ ELASTICITY -Q.3964 -0.3490 -0.2638 -0.1524 i -0.0269 4a U J BAR -0.3963 0.01 -0.3489' 0 .03 ' -0.2636 0.05 -0 .1524 0.0 , -0 .0269 0 . 0 NO-BAR -0 .3891 -1.84" -0.3375' 3 .29 -0.2428 7 .96 -0.1210 20.64 0 . 0 1 3 5 150 .0 ELASTICITY - 0 . 7 7 5 8 . -0.6933 -0.5330 -0 .3186 ;-0 .0738 5 uJ . BAR -0.7757 0.02 -0 .6929 0.05 -0.5326 0.08 -0.3183 0.10 . -0 .0737 0 .16 u_' NO-BAR -0.8039 - 3 . 6 3 -0.743'8 - 7 . 2 8 -0.5892 -10.53 -0.3720 -16.75 ' -0 .1203 -63ol3 ELASTICITY -1 .6117 -1 .4327 -1.0982 -0.6554 V ! - O 0 1 5 1 6 6 bj BAR -1 .6114 0.02 -1 .4318 0.06 - I . O 9 7 1 0.09 -0.6546 0.12 -0.1511; 0.18 NO-BAR -1.6326 - 1 . 3 0 -1.4.788 - 3 . 2 2 -1.1555 -5 .22 -0.7145 - 9 . 0 1 - 0 . 2 0 5 7 - 3 5 . 6 7 ELASTICITY - 2 . 0 8 1 9 -1 .8213 -1.3713 -0.7910 ; -0 .1393 6a uJ BAR -2.0815 0 . 0 2 -1 .8203 0.06 -1 .3702 ' 0*08 -0.7903 . 0 . 0 9 ; -0ol393 0.0 u: NO-BAR -2 .0771 0.23 -1.8020 l i 0 6 -1.3385 2 . 3 9 -0.7455 5.74 - 0 . 0 8 3 1 i;0o32 ELASTICITY -2.6324 - 2 . 3 1 3 7 -1.7619 -1.0477 • . - •' ' : -0 o2lj.20 7 uJ BAR -2.6310 0.05 -2.3116 0.09' -1.7597 0.13 -1.0460 0 : 1 6 -0.21^13 0 . 3 2 N O - B A R -2.6465 -0 .54 -2.3516 - 1 . 6 4 - 1 : 8 1 9 9 . - 3 . 2 9 -1.1154 - 6 . 4 6 -0*3091 -27.70 DEEP 'RECTANGULAR BEAM WITH UNIFORMLY DISTRIBUTED POINT p : C O L U M N B ROW 2 k-o=l.25b—>j < -86-T A B L E 4 - 4 N O R M A L S T R E S S . <TX A N D % E R R O R AT DIFFERENT X AND Y u. = 0-05. AS COEFFICIENT - C IN CTX-C^- BY ELASTICITY AN,D FINITE ELEMENT ':. (NODAL DISPLACEMENTS AND NODJ^ L FORCES : , ; PARAMETER 8Q - 82.5° . /y R Q W iCOLUMN } A B /*' L/ * ' . ! E FUNCfi ON % ERROR FUNCTION % ERROR FUNCTION % ERROR FUNCTION^ TERROR ! FUNCTION %ERROR \> 1 E L A S T I C I T Y - 0 . 9 8 5 1 * -0.9851+ - 0 . 9 8 5 1 * - - 0 . 9 8 5 1 * lj - 0 . 9 8 5 1 * UJ LL.-BAR -0.961+8 2 .10 -0 .9726 1.3Q - 0 . 9 7 6 7 0.88 - 0 . 9 8 0 1 * 0.51'! -0 .981*1 0 .13 NO-BAR -1.01|.95 - 6 . 5 0 -0.91*51 1*.09 - 0 . 9 5 6 5 2.91* - 0 . 9 6 9 2 1.61*! - 0 . 9 8 2 1 0.33 Q z: BAR - 0 .9508 3 . 5 2 - 0 . 9 5 1 0 3 . 5 0 - 0 . 9 5 8 6 2 . 7 2 - 0 . 9 6 7 2 1.85 ' i -0 .9758 0 . 9 7 NO-BAR -1 .1286 • -11+.53 -1 .1270 -11*.36 -1 .0813 - 9 . 7 2 -1.051*7 -7 .03h-1 .051*9 - 7 . 0 5 2 E L A S T I C I T Y -0 .8819 - 0 . 8 8 1 9 - 0 . 8 8 1 9 - 0 . 8 8 1 9 •| - 0 . 8 8 1 9 UJ u. uJ z BAR -0 . 8695 l.i+o -0 .8720 1 .12 -0 .871*6 0.83 -0 .8776 0.14.91 - 0 . 8 8 0 7 0.11+ NO-BAR -0 .8958 - 1 . 5 7 -0 .81*19 l*i'51* - 0 . 8 6 3 5 2 . 0 9 -0 .871*3 0 . 8 7 | i - 0 . 8 8 2 8 0 . 0 9 Q Z BAR' - O . 8 7 6 7 0 . 5 9 - 0 .8768 0 . 5 8 - O . 8 7 6 0 0.67 - 0 . 8 7 6 0 0 . 6 7 ! - 0 . 8 7 6 2 0 . 6 5 NO-BAR -0 . 9515 - 7 . 8 8 -1 .061*6 - 2 0 . 7 1 -1 .0557 - 1 9 . 7 0 -1.031*7 - 1 7 . 3 2 H -1 . 0 2 7 6 - 1 6 . 5 1 3 E L A S T I C I T Y - O . 7 0 8 5 - 0 . 7 0 8 5 - 0 . 7 0 8 5 - 0 . 7 0 8 5 - 0 . 7 0 8 5 UJ U J 2! BAR - O . 7 0 2 5 0.81* - 0 . 7 0 3 5 0 . 7 0 -0 .701*8 0 .51 - 0 . 7 0 6 2 /'• 0 .31 - 0 . 7 0 7 7 0 . 1 0 NO-BAR - 0 . 7 0 9 1 * -0.11+ -0 .6632 6.39 -0 .691*7 1 .95 - 0 . 7 0 7 2 ; ' 6 . 1 7 j ' - 0 . 7 1 3 0 - 0 . 6 1 * ci BAR -0 .7063 0.30 - 0 . 7 0 5 9 0.36 - 0 ; 7 0 5 7 0.39 -0 .7055/ - ' o . l * i j j _ o . 7 o 5 5 0.1*1 NO-BAR -0 .7613 -7.1*5 -0 .8683 - 2 2 . 5 6 - 0 .9018 - 2 7 . 2 9 -0 .901+0" -27.60l j-0 .9033 - 2 7 . 5 1 4 E L A S T I C I T Y - 0 . 5 0 0 0 - 0 . 5 0 0 0 - 0 . 5 0 0 0 - O . 5 0 0 6 • i i - 0 . 5 0 0 0 UJ U-' z BAR - 0 . 5 0 0 0 0.0 - 0 . 5 0 0 0 0.0 - 0 . 5 0 0 0 0.0 - 0 . 5 0 0 0 0.0 I - 0 . 5 0 0 0 0 . 0 NO-BAR - 0 . 1 - 9 9 0 0.20 -0.1*621* 7.52 -0.1+911+ 1 .72 -0 .5037f •% -0.71*1-0 .5066 - 1 . 3 1 Q z BAR - 0 . 5 0 0 0 0.0 - 0 . 5 0 0 0 0.0 - 0 . 5 0 0 0 0.0 - 0 . 5 0 0 0 ; i 0 .0 - 0 . 5 0 0 0 0 . 0 NO-BAR -0 .51*37 - 8 . 7 5 - 0 . 6 3 5 9 - 2 7 . 1 8 - 0 . 6 9 1 5 - 3 8 . 2 9 - 0 . 7 1 3 8 -1*2.761-0.7197 -1*3.93 5 E L A S T I C I T Y - 0 . 2 9 1 5 - 0 . 2 9 1 5 - 0 . 2 9 1 5 - 0 . 2 9 1 5 ' i -0 .2915 L U UJ LL: z: BAR - 0 . 2 9 7 5 - 2 . 0 5 - 0 . 2 9 6 5 - 1 . 7 0 - 0 . 2 9 5 2 - 1 . 2 5 - 0 . 2 9 3 8 ' - 0 . 7 6 i l - 0 . 2 9 2 3 - 0 . 2 1 * NO-BAR -0.291+9 - 1 . 1 7 - 0 . 2 7 2 2 6.61* - 0 . 2 9 2 5 - 0 . 3 3 - 0 . 2 9 9 5 - 2 . 7 3 i ] - 0 . 2 9 8 l - 2 . 2 5 ci z BAR -0 .2937 - 0 . 7 2 -0 .291*1 - 0 . 8 9 -0 .291*3 - 0 . 9 6 -0.291+1* - 1 . 0 0 -0.291*1* - 1 . 0 0 NO-BAR - 0 . 3 3 2 5 -11+.05 -0 .1*197 -1*3 .91* -0.1+883 -67.1*8 -0 .5226 - 7 9 . 2 6 1 - 0 . 5 3 2 7 - 8 2 . 7 3 6 E L A S T I C I T Y -0 . 1 1 8 1 - 0 . 1 1 8 1 - 0 . 1 1 8 1 - 0 . 1 1 8 1 f - 0 . 1 1 8 1 ui U z . BAR - O . I 3 0 5 -10.1)9 -O . I28O - 8 . 3 9 -0.1251* - 6 . 1 7 - 0 . 1 2 2 1 * - 3 . 6 5 " ° » H 9 i + - 1 . 0 8 NO-BAR -0.131*1+ - 1 3 . 8 9 - 0 . 1 2 5 7 -6.1*8 - 0 . 1 3 2 5 - 1 2 . 2 5 - O . I 3 0 3 - 1 0 . 3 5 t -O.123I* -1+.52 Q BAR _r>, i - 0 . 1 2 1 2 -1*.33 -0 .121*0 - 5 . 0 3 -0.12/li.O -5.01+1 -O .1238 -1+.87 NO-BAR - 0 . 1 7 0 6 -kit . 50 - 0 . 2 7 3 l i - 1 3 1 . 5 0 -0 .31+81 - 1 9 1 * . 85 -0 .3882 -228.79! -O.I4.O03 - 2 3 8 9 9 7 E L A S T I C I T Y -0 .011*6 -0 .011*6 -0.011*6 -0.011+6 I *. ( -0 .011*6 ui LL.' LL.' Z BAR -0 .0353 -11*1.82 -0 .0273 - 8 7 . 6 0 -0.023.3 - 5 9 . 8 1 -0 .0196 -31+. 331 - 0 . 1 5 8 9 - 8 . 9 7 NO-BAR - 0 . 0 7 1 5 - 3 9 0 . 3 1 - 0 . 0 5 5 8 - 2 8 2 . 8 1 -0.01*51 - 2 0 9 . 3 1 - 0 .0326 - 1 2 3 . 3 5 ; - 0 . 0 1 9 1 - 3 0 . 7 7 Ci Z BAR -0 .0 l| .92 - 2 3 7 . 7 1 -0 .01*90 - 2 3 6 . 2 6 -0.01*11* - I 8 3 . 6 8 - 0 . 0 3 2 8 - 1 2 5 . 1 8 -0.021*1 - '65.78 NO-BAR -0.151*8 - 0 . 2 6 5 5 - 0 . 3 3 1 9 - 0 . ?61+? —•—f • I - 0 . 3 6 5 1 D E E P RECTANGULAR BEAM WITH UNIFORMLY DISTRIBUTED LOAD . j. I • l 1 l l J i l l \^ V : < - 6 7 -• TABLE 4-5 NORMAL STRESS ., <T< AND % ERROR AT DIFFERENT X AND Y u = 0-05. AS COEFFICIENT C IN ^ : C 1 BY ELASTICITY AtyD FINITE ELEMENT (NODAL DISPLACEMENTS AND NODAL FORCES PARAMETER dQ - 82.5° y R G COLUMN ' •; A B ff E w FUNCTION % ERROR FUNCTION % ERROR FUNCTION % ERROR FUNCTION? %ERROR FUNCTION %ERROR E L A S T I C I T Y -o.59J4?o - 1 . 4 8 9 4 « 2 | l 4 8 7 . - 2 . 5 7 1 8 - 2 . 7 5 8 7 u; BAR - 0 . 6 5 5 7 -.10. - 1 . 5 3 7 1 - 3 . 2 1 - 2 . 1 8 3 6 - 1 . 6 2 - 2 . 5 9 5 0 - 0 . 9 0 - 2 . 7 7 1 3 - 0 . 4 6 1 ui z NO-BAR - 0 . 6 3 7 3 - 7 . 2 c - 1 . 5 9 8 2 - 7 . 3 3 - 2 . 2 4 4 2 - 4 . 4 5 - 2 . 6 4 7 5 - 2 . 9 4 - 2 . 8 1 9 3 - 2 . 2 0 U.' Q BAR - 0 . 5 8 8 1 0.9S - I . 4 8 1 9 . 0.5C - 2 . 1 3 8 4 0.4"/ - 2 . 5 6 0 5 Q .44 - 2 . 7 4 7 3 0 . 4 1 Z NO-BAR - 0 . 5 4 9 9 7.42 -1.7129 - 1 5 . 0 1 - 2 . 3 2 6 5 - 8 . 2 8 - 2 . 6 9 3 2 - 2 . 8 4 4 3 - 3 . 1 0 E L A S T I C I T Y - 0 . 2 7 9 4 - 0 . 8 7 6 3 - 1 . 3 1 5 8 - 1 . 5 9 7 8 - 1 . 7 2 2 5 u. BAR - 0 , 3 1 8 7 - 1 4 . 0 8 - 0 . 9 0 7 5 — 3 . 5 6 - 1 . 3 3 9 2 - 1 . 7 8 - 1 . 6 1 3 7 - 0 . 9 9 ! - 1 . 7 3 1 3 - 0 . 5 1 2 ui z NO-BAR - 0 . 3 6 9 4 - 3 2 . 2 4 - 0 . 9 3 8 3 - 7 . 0 8 - 1 . 3 8 7 5 - 5 . 4 5 - 1 . 6 6 7 7 ; - 4 . 3 7 | - 1 . 7 8 6 3 - 3 . 7 0 u: Q BAR - 0 . 2 7 4 9 1 . 6 1 - 0 . 8 7 1 2 0 . 5 9 - 1 . 3 0 9 4 0 . 4 9 - 1 . 5 9 0 4 o .47| - 1 . 7 1 4 6 . 0 . 4 6 Z NO-BAR - 0 . 3 0 3 5 - 8 . 6 2 - 1 . 0 6 7 5 - 2 1 . 8 2 - I . 5 0 5 7 - 1 4 . 4 3 - 1 . 7 6 3 8 / - i o . 3 8 i - 1 . 8 6 8 0 -80I45 E L A S T I C I T Y - 0 . 1 0 4 7 - 0 . 4 0 3 2 - 0 . 6 2 2 9 - 0 . 7 6 4 0 i - 0 . 8 2 6 3 •u. BAR • - 0 . 1 2 4 0 - 1 8 . 4 4 - 0 . 4 1 8 8 - 3 . 8 7 - 0 . 6 3 4 7 - 1 . 8 9 - 0 . 7 7 2 0 ' - 1 . 0 5 - 0 . 8 3 0 8 - 0 . 5 5 3 ui z: NO-BAR - 0 . 1 9 9 8 - 9 0 . 8 6 - 0 . 4 5 8 7 - 1 3 . 7 8 - 0 . 6 7 4 7 - 8 . 3 3 - 0 . 8 1 9 2 - 7 . 2 3 ; - O . 8 8 1 4 - 6 . 6 7 u.' ci BAR - 0 . 1 0 2 2 -0.l4.006 0 . 6 5 - 0 . 6 1 9 8 0 . 5 1 - 0 . 7 6 0 3 0.47j - 0 . 8 2 2 4 0 . 4 7 z NO-BAR - 0 . 1 5 3 0 - 4 6 . 1 4 - 0 . 5 7 8 1 - 4 3 . 3 8 - 0 . 8 0 1 0 - 2 8 . 5 9 - 0 . 9 3 9 6 - 2 2 . 9 9 ; - 0 . 9 9 6 2 - 2 0 . 5 6 E L A S T I C I T Y 0 . 0 0 0 . 0 0 0 . 0 0 . 0 . 0 0 • OoOO u: BAR 0 . 0 0 0 . 0 0 0 . 0 0 0 . 0 0 1 1 .. 0 . 0 0 4 ui NO-BAR - 0 . 0 8 1 9 - 0 . 0 4 7 2 - 0 . 0 3 6 2 ; - 0 . 0 3 8 5 f - 0 . 0 4 1 5 U.' ci BAR 0 . 0 0 0 . 0 0 . 0 0 . 0 1 1 0 . 0 NO-BAR - 0 . 0 5 6 9 - 0 . 1 4 7 5 - 0 . 1 5 7 6 - 0 . 1 6 8 6 I - 0 . 1 7 4 7 E L A S T I C I T Y • O . I O 4 7 0 . 4 0 3 2 0 . 6 2 2 9 0 . 7 6 4 0 • f 0 . 8 2 6 3 UJ BAR 0 . 1 2 4 0 - 1 8 . 4 4 0 . 4 1 8 8 - 3 . 8 7 0 . 6 3 4 7 - 1 . 8 9 0 . 7 7 2 0 - 1 . 0 5 ; 0 . 8 3 0 8 - 0 . 5 5 5 ui 2: NO-BAR 0 . 0 4 4 4 5 7 . 6 0 . 0 . 3 6 9 2 ; a . 4 3 0 . 5 9 7 7 4 . 0 5 0 . 7 3 5 4 3 . 7 4 0 . 7 9 2 7 4 . 0 7 U. Q BAR 0.1022 2.43 0 . 4 0 0 6 0 . 6 5 0 . 6 1 9 8 0 . 5 1 0 . 7 6 0 3 0 . 4 7 0 . 8 2 2 4 0 . 4 7 Z NO-BAR 0 . 0 4 8 5 5 3 . 7 0 0 . 2 8 4 8 2 9 . 3 6 0 . 4 7 9 0 2 3 . 1 0 0.5951 2 2 . 1 0 0 . 6 4 1 5 2 2 . 3 6 E L A S T I C I T Y 0 . 2 7 9 4 0 . 8 7 6 3 1.3158 1.5978 | 1.7225 u: BAR 0 . 3 1 8 7 - 1 4 . 0 8 0 . 9 0 7 5 - 3 . 5 6 1 . 3 3 9 2 - 1 . 7 8 1.6137 - 0 . 9 9 . 1.7313 - 0 . 5 1 6 ui z NO-BAR 0 . 2 5 0 4 1 0 . 3 5 0 . 8 5 6 0 2 . 3 2 1 . 2 8 9 3 2 . 0 1 1 .5646 2 . 0 8 1 . 6 8 2 7 2.31 u." Q BAR 0 . 2 7 4 9 1 . 6 1 0.8712 0 . 5 9 1 . 3 0 9 4 0 . 4 9 1 . 5 9 0 1 ; 0 . 4 7 1 . 7 1 4 6 0 . 4 6 Z NO-BAR 0 , 2 3 6 6 1 5 . 3 0 0 . 7 7 2 9 1 1 . 8 0 1.1577 1 2 . 0 2 1 . 3 9 9 7 1 2 . 4 0 1.14 999 1 2 . 9 3 E L A S T I C I T Y Oo594o 1 .4894 2 . 1 4 8 7 2 . 5 7 1 8 . 2 . 7 5 8 7 uJ BAR 0 . 6 5 5 7 - 1 0 . 3 9 1.5371 - 3 . 2 1 2.1836 - 1 . 6 2 2 . 5 9 5 0 - 0 . 9 0 2.7713 - 0 . 4 6 7 ui z NO-BAR 0 . 6 3 9 3 - 7 . 6 3 1 .4629 1 . 7 6 2 . 1 0 1 3 2 . 2 0 2 . 5 1 8 5 . 2 . 0 ? i 2 . 6 9 8 8 2 . 1 7 u.'- ci BAR 0 . 5 8 8 1 0 . 9 9 1 .4819 0 . 5 0 2 . 1 3 8 4 0 . 4 7 2 . 5 6 o £ 0 . 4 4 2.7473 0 « h l z NO-BAR 0 . 6 0 4 0 - 1 . 6 9 1.3562 8 . 9 4 1/9292 1 0 . 2 1 2 . 3 0 3 0 10.4-5 • 2 . 4 6 0 2 1 0 . 8 2 I I D E E P RECTANGULAR BEAM WITH UNIFORMLY DISTRIBUTED LOAD 1 / 1 1 1 1 . 1 1 1 1 1 . TABLE 4 - 6 SHEAR S T R E S S T x y AND % ERROR j AT DIFFERENT X AND Y u = 0.05. AS COEFFICIENT C INxxy =C^- BY ELASTICITY AND FINITE ELEMENT (NODAL.DISPLACEMENTS AND NODAL FORCES • ' • u PARAMETER & = 82.5° - . 1 5 W i ' S R Q W> COLUMN A B C 3 E FUNCTION % ERROR FUNCTION %ERROR FUNCTION % ERROR FUNCTION' %ERROR | FUNCTION %ERROR E L A S T I C I T Y 0.3660 0.2807 0.1951+ - • 0.1101 | Oo021*9 LL: BAR . 0.3561). 2.60 0 . 2 6 9 8 3.87 0.181*8 5.1*3 0.0999 9.32 ] 0.0153 3 8 . 5 9 \ ui Z NO-BAR 0.1+1+22 2^0.83 0.2686 1+.31 0.1696 13.18 0.081+1 , 23.62 | 0 . 0 0 1 2 9 5 . 2 0 u." ci BAR 0.3361+ 8.08 0.2633 6.19 0.1831+ 6.11* 0.1017 / 7.69 0.6198 2 0 . 2 7 z NO-BAR 0 . 3 9 2 7 -7.29 0.2705 3.62 0.191+1 0.67 0 . 1 2 2 5 -11.25 Oo 01*93 - 9 8 . 0 3 E L A S T I C I T Y 0.9290 0.7126 - 0.1+961 0.2796; 0 . 0 6 3 1 L L BAR 0.9128 1.75 0.6962 2.29 0.1+798 3.29 0.261+0 5.59 0.01*87 2 2 . 9 2 2 ui Z NO-BAR 0.9776 -5.23 0.7181 - O . 7 8 0.1+71+2 i*.l*l 0.2530 9.51 0 . 0 3 9 6 3 7 . 3 1 u. Ci BAR 0.9031 2.79 0.696^ 2.26 ,0. 1+859 2 .05 0.271+5. 1.81* 0.0628 0 . 5 9 \ Z NO-BAR 0.9511 -2.37 0.6965 2.25 / 0.1+81*1 2.1*2 0.281*0 -1.58 O . 0 8 3 6 - 3 2 . 3 3 E L A S T I C I T Y le2669 0.9717 / 0.6765 • 0.3813 o i o 8 6 i 5 u_' BAR 1.21*82 1.1+8 0.9522 2 . 0 1 / 0.6571 2.87 0.3628 l*.85 0.0691 19.76 3 ui z NO-BAR 1.2751 -0.65 0.9787 -0.72/ 0.661*8 1.73 0.3616 5.17 0 . 0 6 7 0 2 2 . 1 7 Q BAR 1.21+11 2.03 0.9551 1.70 0.6658 1.58 0.3758 1.1*3 0 . 0 8 5 7 0.1*5 z NO-BAR 1.2392 2.18 0.9353 3.71+ 0.61*91 1*.05 0 . 3 6 9 3 3.16 0 . 0 8 9 7 - l u l l * E L A S T I C I T Y 1.3795 1 . 0 5 8 0 . 0.7366 0.1+152 0 . 0 9 3 8 BAR 1.3600 1.1*1 1.0375 1.91+ 0.7162 2.77 0.3958 i*.68 0.0759 19.01* 4 ' ui Z NO-BAR 1.3558 1.71 1.0532 0.1+5 0.7272.. 1.27 0.1+002 3.60 0.0771 17.73 d BAR 1.3537 1.86 1.01+13 1.58 0.7257 1.1+7 0.1+096 1.31* 0.0933 0.1*1+ NO-BAR 1.3168 1+.51+ 1.001+1+ 5 . 0 7 0.7036 1+.1+8 0.3991 3.88 0.0913 2 . 5 7 E L A S T I C I T Y 1.2669 0.9717 0 . 6 7 6 5 O .3813 0 . 0 8 6 1 LL: BAR 1.21+82 1.1+8 0.9522 2 . 0 1 0.6571 2.87 0 . 3 6 2 8 l*.85 0 . 0 6 9 1 19.76 5 ui z NO-BAR 1.2196 3.73 0 . 9 5 0 1 2 . 2 2 0.6628 2.01 0.3669 3.77 O0O689 19.90 uJ Ci BAR 1.21*11 2.03 0.9551 1.70 0.6658 1.58 0 . 3 7 5 8 •1.1+3 0 . 0 8 5 7 0.1*5 Z NO-BAR 1.1862 6.37 0.9126 6 . 0 7 0^ 61*89 1+.08 0*3733 2.10 0.0898 >U*35 E L A S T I C I T Y 0.9290 0.7126 0.1*961 0.2796 0 . 0 6 3 1 U.' BAR 0.9128 1.75 0 . 6 9 6 2 2 . 2 9 0.1*798 : 3.29 0.261+0 5.59 0.01*87 2 2 . 9 2 6 ui NO-BAR 0.861+7 6.92 0.6769 1+.99 0.1*770 3.81* 0.2622 6.22 0.01*26 3 2 . 5 2 u. Q BAR 0 . 9 0 3 1 2.79 0 . 6 9 6 5 2.26 0.1*859 2 . 0 5 0.271+5 1.81+ 0 . 0 6 2 8 0 . 5 9 NO-BAR Oo81+65 8.88 0.6669 6.1+1 0.1*871 1.81 0 . 2 9 0 3 -3.83 0,0838 -32,66 E L A S T I C I T Y 0.3660 0 . 2 8 0 7 0.1951* 0.1101 0.021*9 uJ . BAR 0.3561+ 2.60 0.2698 3.87 0.181*8 5.1*3 0.0999 9.32 0.0153 3 8 . 5 9 7 ui NO-BAR 0 . 3 0 8 6 15.68 0 . 2 5 2 0 1 0 . 2 1 0 . 1758 10.03 0 . 0 9 0 3 17.98 0 . 0 0 2 9 ,: 88.11* LU- Ci BAR 0.3361+ 8.08 0.2633 6.19 0.1831* 6.11+ 0.1017 7.69 6 . 0 1 9 8 2 0 . 2 7 z NO-BAR 0.3098 15.31+ 0.2576 8.22 0.1967 -0.61+ 0 . 1257 -11+. 15 0.01*93 , - 9 8 . 1 0 t D E E P 2 3 4 . RECTANGULAR BEAM WITH UNIFORMLY DISTRIBUTED LOAD 11.5a Y . V o > CM v X,u thickness = t \e—Q=l.25b t -><ry i rxy TABj_E 4 - 7 X - D I S P L A C E M E N T U AND % E R R O R V \ AT DIFFERENT X AND Y /i. =! 0.3333 AS COEFFICIENT C IN u =C-|^  10 BY ELASTICITf ' AND FINITE ELEMENT; 4| . f . PARAMETER 60 - 100° • ' R 0 W. COLUMN, A B ' / C D 1 E $ FUNCTION %ERROR FUNCTION %ERROR FUNCTION %ERROR FUNCTION %ERROR FUNCTION % ERROR 1 ELASTICITY lolj.950 3 .3268 4 . 7840 5.7677 6 .2144 Ixj u: BAR 1 .4920 0.20 3 . 3 1 9 1 0 .23 4.7707 0.28 5 . 7 5 0 0 0 .31 6.1943 0 . 3 2 NO-BAR 1.5326 -'2.52 3 . 4 1 0 3 -2.51 4.8760 -1.92 5 . 8 5 9 3 -1.60 6 .3055 -1.47 la ELASTICITY 1 .5816 3 . 4 1 3 4 4.8.540 5 . 8 0 7 2 6.2166 ui BAR 1.5777 0 .25 3 .4030 0.31 4.8384 0.32 5 .7882 0.33 6 0 1 9 2 4 0 . 3 2 NO-BAR 1.6129 -2.00 3 .4873 -2.16 4 .9386 -1.74 5 . 8 9 3 5 -1.49 6 .2985 - 1 . 3 8 2 ELASTICITY 1 .3350 3 .2165 4 .7103 5.7175 6 .1746 ui U.-BAR 1»3337 0.09 3 .2098 0.21 4.6977 0.27 5.7003 0 .30 6 . 1550 0 .32 NO-BAR 1.3483 -1.00 3.2633 .-1.45 4.7^ 58 -1 .39 5.7907 -1.28 6 .2510 - 1 . 2 4 3 ELASTICITY 1 .1873 3.0986 4.6/144 5.6357 6.0991 ui IN-BAR 1.1863 0 .08 3 .0925 0.20 4/6024 0.26 5.6190 0.30 6.0799 0 .31 NO-BAR 1 . 1850 0.19 3 .1115 -0 .42 "4 .6464 -0 .63 5 .6798 -0.78 6 .1488 - 0 . 8 5 3a ELASTICITY 1.2960 3 .1864 4.6696 5 . 6 4 9 7 6.0662 ui IN-BAR 1.2929 0.24 3.1772 0.29 4 .6551 -0.31 5.6316 0.32 6 .0468 0 .32 NO-BAR 1 .2952 0.07 3 .1951 -0.27 4 . 6 9 2 4 -0 .50 5 .6814 -0.56 6 .1005 - 0 . 5 7 4a ELASTICITY 1.1960 3 .0864 4.56966 5 .5497 5.9662 ui I N BAR 1.1929 0.26 3.0772 0.30 4.5551 0.32 0.33 5 .9468 0 .32 NO-BAR 1 .1863 0.81 3.0706 0 .51 4.5592P 0 .23 5 .5454 0 .08 5 .9642 . 0 . 0 3 5 ELASTICITY 0 .9873 . 2.8986 4 .4144 5 . 4 3 5 7 I 5.8991 ui I N . BAR 0 .9863 0.09 2 .8925 0.21 4 . 4 0 2 4 0.27 5 . 4 1 9 0 0.31 ! 5.8799 0.33' NO-BAR 0 .9689 1.86 2.861x0 1.19 4.3810 0.76 5 .4084 n, Zr> i 5 . 8 7 6 3 0 .38 6 ELASTICITY 0 .9350 2 .8165 4.3103 5 .3175 ' 1 5 .7746 ui u. • BAR 0 .9337 0.13 2.8098 0.24 4.2977 0.29 5.3003 0.32 ! 5 .7550 0.34 NO-BAR 0.9131 2.34 2 .7630 1.90 4 . 2 4 3 8 1 .54 5.2490 1.29 i 5 . 7 0 7 6 6a ELASTICITY 1 .0814 219134 4 . 3 5 4 0 5 . 3 0 7 2 ! 5.7126 ui u: BAR 1.0775 0.37 2 .9030 0 .36 4.3384 0 .36 5.2882 0 .36 ! 506924 0 3 ^ NO-BAR 1 .0593 2 .05 2 .8512 2 .14 4.2712 1 .90 5 .2172 1.70 !.5.6205 1.61 •7 ELASTICITY 0 .8950 2.7268 4 . 1840 5.1677 ! 5 .614U ui I N BAR '0.8920 0.33 2.7191 0.28 4.1707 0.32 5 . 1 5 0 0 0 . 3 4 i 5 .5943 0.36 NO-BAR 0.8667 3.16 • 2.648k 2.88 4 . 0 7 7 4 2 .55 5 . 0 4 9 7 2.28 i 5 .4936 2 .15 D E E P R E C T A N G U L A R B E A M WITH UNIFORMLY DISTRIBUTED LOAD la 2 3 3a 4a 5 6 6a 7 POINT p : COLUMN B ROW 2 |«-a=l.25b —>j thickness B t t • <ry I xy - 9 0 -TABLE 4 - 8 Y - D I S P L A C E M E N T V AT DIFFERENT X AND Y /A = . 0.3333 qb 1 AND % ERROR AS COEFFICIENT C IN v = C I J 10 BY ELASTICITY AND FINITE ELEMENT PARAMETER 8n S 100° R n COLUMN A B C —1 D - E w FUNCTION %ERROR FUNCTION %ERROR FUNCTION %ERROR FUNCTION % ERROR FUNCTION % ERROR ELASTICITY 2 . 2 0 8 6 1 . 9 8 8 7 1.535.6 0 . 9 2 0 2 * 0 . 2 1 3 2 1 UJ BAR 2.2062 0.11 1.981*1 0 . 2 3 1.5311 0 . 2 9 0 . 9 1 7 2 0 . 3 2 j 0 .2127 0 . 2 5 LL NO-BAR 2 . 1 6 1 2 2.15 1 . 9 6 0 7 U + l 1.5211+ 0 . 9 2 0 . 9 1 1 7 ' 0 .911 0.201*1 I*.28 ELASTICITY 1 . 6 8 0 9 1.5156 1 . 1 6 0 9 0 . 6 7 5 8 I ' 0.1191+ IQ ui BAT? 1.6765 0 . 0 8 1.5125 0 . 2 1 1 .1576 0 . 2 8 0 . 6 7 3 6 0 . 3 3 Ooll90 0 . 3 5 Lu NO-BAR 1 . 7 1 7 9 - 2 . 1 9 1.5610 - 2 . 9 9 1 . 1 9 7 7 - 3 . 1 7 0 . 7 0 2 6 -3.95^ O d 3 8 7 - 1 6 . 1 1 ELASTICITY 1.2257 1 . 1 3 6 6 0 . 8 9 2 0 0 .5392 0„125i+ 2 ui BAR 1.2250 0 . 0 5 1.131+6 0 . 1 7 0 . 8 8 9 7 ^ 0 . 2 5 0 .5376 0 . 3 0 0 . 1 2 5 0 0 . 3 2 u- NO-BAR 1.1714-1 1*.20 1 . 0 8 9 1 1+.17 0 . 8 5 9 8 7 3 . 6 1 0.511+2 1+.61* 0 . 1 0 1 1 1 9 . 3 9 ELASTICITY 0.1*530 o . 1+1+57 0 . 3 6 0 7 / 0 . 2 2 1 5 0.0518 3 ui BAR 0.1*52 8 0.01* 0.1*1*50 0 . 1 5 0 . 3 5 9 8 / ' 0.2)* 0 . 2 2 0 8 0 . 3 0 1 0.0517 0 . 3 2 U' NO-BAR o. l*o6l* 1 0 . 2 9 0 . 3 9 1 5 1 2 . 1 7 0 .311+8, ' 1 2 . 7 0 0 . 1 8 3 1 17.31+ 0.0173 6 6 . 7 0 ELASTICITY 0 . 1 1 6 3 o : i 3 5 5 0 . 1 1 6 8 0 . 0 7 2 0 0 . 0 1 3 0 3a ui BAR . 0 . 1 1 6 2 0 . 0 9 0.1352 0 . 2 3 0 .1161* 0 . 3 5 0 . 0 7 1 8 o.hi 0 . 0 1 2 9 OJ43 u NO-BAR 0 . 1 3 6 0 - 1 6 . 8 6 0.1571+ - 1 6 . 1 2 0.11*37 - 2 2 . 9 9 0 . 1 0 3 8 -1*1*. 07 0.01*88 . -275.38 ELASTICITY - 0 . 5 3 7 1 -0.1*563 - 0 . 3 3 7 6 - 0 . 1 9 2 9 - 0 . 0 3 3 8 4 a ui BAR - 0 . 5 . 3 7 0 0 . 0 2 -0.1+560 0 . 0 7 - 0 . 3 3 7 2 0 . 1 2 - 0 . 1 9 2 6 0 . 1 5 ! - 0 . 0 3 3 8 0 . 0 LL NO-BAR - 0 . 5 2 6 3 2 . 0 2 -O.I4.I4.I4.2 2 . 6 6 - 0 . 3 1 7 7 5 . 9 1 - 0 . 1 6 2 9 0 . 0 0 5 2 H5 .51* ELASTICITY - 0 . 8 8 2 1 -0.771+8 - 0 . 5 8 9 8 - 0 . 3 5 0 7 -O0O8IO 5 ui , BAR - 0 . 8 8 1 9 0 . 0 2 . -0.771+2 0 . 0 9 - 0 . 5 8 8 9 0 . 1 5 - 0 . 3 5 0 0 0 . 1 9 ; - O . O 8 0 8 0 . 2 1 u," NO-BAR - 0 . 9 1 0 1 * - 3 . 2 1 - 0 . 8 2 5 3 - 6 . 5 1 -0 .61*63 - 9 . 5 8 -O.kOlil - 1 5 . 2 3 - 0 . 1 2 7 0 - 5 6 . 7 8 ELASTICITY -1.65/4-8 -1.1+657 - 1 . 1 2 1 2 -O.668I1. ' -0.151*6 6 ui BAR -1.651*2 o . o l * -1.1+637 0 . 1 3 - 1 . 1 1 8 9 ' 0 . 2 0 - 0 . 6 6 6 7 0.2I4. -0.151*2 0 . 2 6 u. NO-BAR - 1 . 6 7 6 5 - 1 . 3 1 - 1 . 5 1 3 7 - 3 . 2 7 - I . I 8 0 5 - 5 . 3 0 . - 0 . 7 2 8 8 - 9 . O i l -0 .2090- - 3 5 . 2 3 ELASTICITY - 2 . 1 0 1 7 -I.836I4.. - I . 3 8 1 7 - 0 . 7 9 6 7 -0.11*03 6a ui BAR - 2 . 1 0 0 3 0 . 0 7 - 1 . 8 3 3 3 0 . 1 7 -1 .3781* 0.21+ -0.791+5 0 . 2 8 - 0 . 1 3 9 8 0 . 3 0 L L NO-BAR - 2 . 0 9 6 1 0 . 2 7 - 1 . 8 1 8 8 0 . 9 6 - 1 . 3 5 0 7 ' 2.21* - 0 . 7 ^ 2 2 5 . 5 8 -0.081*0 1*0.10 ELASTICITY - 2 . 6 3 7 7 - 2 . 3 1 7 8 . -1 .761*8 - 1 . 0 l i 9 3 1 -0.21*21* f: _} p 7 ui B^AR - 2 . 6 3 5 3 0 . 0 9 - 2 . 3 1 3 3 0 . 1 9 - 1 . 7 6 0 2 0 . 2 6 -1.01*61* 0 : 2 8 ; -0.21*19 - - - 0 . 2 2 u' NO-BAR -2.651*2 - 0 . 6 3 - 2 . 3 5 9 7 - 1 . 8 0 - 1 , 8 2 5 9 -3.1+6 - 1 . 1 1 8 9 - 6 . 6 3 i - 0 . 3 1 0 2 - 2 7 . 9 5 DEEP RECTANGULAR BEAM WITH UNIFORMLY DISTRIBUTED LOAD, \|/ N ! / ^ ^ \l \l ^ la 2 3 3a 4a 5 6 6a 7 A ' B «S c .5a — > CVJ thickness - t POINT p : COLUMN B ROW 2 |«s—a=l.25b—>j t TABLE 4 - 9 NORMAL S T R E S S OJ AND % ERROR AT DIFFERENT X AND Y. u. = 0.3333 AS COEFFICIENT C IN o~x =c 5. BY ELASTICITY AND FINITE ELEMENT ; (NODAL DISPLACEMENTS AND NODAL FORCES PARAMETER 8Q = 100° R 0 W COLUMN A B C D 4' E FUNCTION % ERROR FUNCTION % ERROR FUNCTION % ERROR FUNCTION %ERROR •FUNCTION %ERROR ELASTICITY -0.9854 -0.9854 -0.9854 - 0 . 9 8 5 4 - 0 . 9 8 5 4 BAR -0.9684 1.73 - 0 . 9 7 3 9 1.17 -0.9772 O.83 -0.9806 0.49 - O . 9 8 3 9 0.15 1 Z NO-BAR -1.0196 - 3 . 4 7 -0.9437 4 . 2 4 - 0 . 9 5 6 5 2 .93 -0.9692 1 .65 - 0 . 9 8 2 1 0 . 3 4 Q BAR . - 0 . 9 4 2 5 4 . 3 5 - 0 . 9 4 2 9 4.31 - 0 . 9 4 9 9 3 .6C - 0 . 9 5 7 9 2.79 - 0 . 9 6 6 0 1.97 z NO-BAR - 1 . 1 4 5 6 - 1 6 . 2 6 -1.1909 -20.85 -1 . 1141 -13.06 - 1 . 0 6 2 6 -7.83 - 1 . 0 4 5 6 - 6 . 1 1 ELASTICITY - O . 8 8 1 9 -0.8819 - 0 . 8 8 1 9 - 0 . 8 8 1 9 - 0 . 8 8 1 9 UJ BAR - 0 . 8 7 4 1 0 . 8 9 - 0 . 8 7 4 0 0 . 8 9 - 0 . 8 7 5 7 0.7C -O.878O i 0 . 4 5 - 0 . 8 8 0 3 ' 0 . 18 2 z NO-BAR - 0 . 8 8 4 0 - 0 . 2 4 - 0 . 8 4 4 2 4 . 2 8 -0.8653 1.8c - 0 . 8 7 4 7 0 . 8 2 - 0 . 8 8 2 4 - 0 . 0 6 Q BAR -0.8724 1.08 -O.8706 • 1 .28 - 0 . 8 6 9 8 1 .3/ -0.8696 1 .39 - 0 . 8 6 9 6 i . r 4 o Z NO-BAR - 0 . 9 7 6 6 - 1 0 . 7 3 -I . 1 4 0 6 -29 .34 - 1 . 1 0 3 5 -25.12^ -1.0616 - 2 0 . 3 7 - 1 . 0 4 0 6 -17.99 ELASTICITY - 0 . 7 0 8 5 - 0 . 7 0 8 5 - 0 . 7 0 8 5 -0.7085 - 0 . 7 0 8 5 u. BAR -0.701*9 0 . 5 0 - 0 . 7 0 ^ 6 0.5*1 - 0 . 7 0 5 3 0 . 4 5 -0.7063 0 . 3 0 - 0 . 7 0 7 5 0.14 3 ui 2! NO-BAR -0.7063 0 . 3 0 - 0 . 6 6 8 2 \ 5 . 6 8 -0.6970 1.16 - 0 . 7 0 7 3 0.16 - 0 . 7 1 2 3 - 0 . 5 5 u; Ci BAR - 0 . 7 0 4 4 • 0 . 5 8 - 0 . 7 0 2 7 \0.82 - 0 . 7 0 2 2 0.86 -0.7021 0 . 8 9 - 0 . 7 0 2 1 0.89 Z NO-BAR - 0 . 7 8 4 1 - 1 0 . 6 7 - 0 . 9 3 6 3 -3^.17 - 0 . 9 5 3 5 -35 .12 - 0 . 9 4 2 4 - 3 3 . 0 2 - 0 . 9 3 3 5 -31.77 ELASTICITY - 0 . 5 0 0 0 -O.5000 - 0 . 5 0 0 0 - 0 . 5 0 0 0 - 0 . 5 0 0 0 UJ BAR - 0 . 5 0 0 0 0 . 0 - 0 . 5 0 0 0 0/. 0 - - 0 . 5 0 0 0 0 . 0 - 0 . 5 0 0 0 0 . 0 - 0 . 5 0 0 0 0 . 0 4 ui z NO-BAR - 0 . 5 0 2 4 - 0 . 4 8 - 0 . 4 6 8 0 6\4o -0.4935 1.31 - 0 . 5 0 3 5 - 0.69 - 0 . 5 0 5 7 -1 .15 u.' Ci BAR - 0 . 5 0 0 0 0 . 0 - 0 . 5 0 0 0 0 . 0 - 0 . 5 0 0 0 0 . 0 - 0 . 5 0 0 0 0 . 0 -o ;5ooo 0 . 0 Z NO-BAR - 0 . 5 6 0 6 -12.13 - 0.6881 -37.62 -0.7394 - 4 7 .81 -0.7584 - 5 1 . 6 8 - 0 .7628 - 5 2 . 5 1 ELASTICITY - 0 . 2 9 1 5 - 0 . 2 9 1 5 -0.2915 -0.2915 - 0 . 2 9 1 5 • BAR -0.2951 - 1 . 2 2 - 0 . 2 9 5 4 -1.31 -0.2947 - 1 . 0 s -0.2937 - 0 . 7 3 - 0 . 2 9 2 5 - 0 . 3 3 5 ui z NO-BAR - 0 . 3 0 2 8 -3 .84 -0.2762 5.24 -0.2933 - 0 . 6 C -0.2989 - 2.51 -0.2974 " - 2 . 0 0 Ui Q . BAR - 0 . 2 9 5 6 - 1 . 4 0 -0.2973 - 1 . 9 8 -0.2978 - 2 . 1 ; -0.2979 -2.16 -0.2979 -2.17 Z NO-BAR -0 .3415 -17 .15 - 0 . 4 5 4 3 - 5 5 . 8 3 - 0 . 5 3 1 1 -82.16 - 0 . 5 7 3 0 - 9 6 . 5 2 - 0 . 5 8 9 0 - 1 0 2 . 0 3 ELASTICITY - 0 . 1 1 8 1 - 0 . 1 1 8 1 -0.1181 - 0 . 1 1 8 1 - 0 . 1 1 8 1 U.' BAR - O . I 2 5 9 -6.62 -0.1260 - 6 . 6 8 - 0 . 1 2 4 3 - 5 . 2 ; - 0 . 1 2 2 0 - 3 . 3 4 -0„1197 - 1 . 3 6 6 ui z NO-BAR - p . 1 4 2 4 - 2 0.61 - 0 . 1 2 6 5 - 7 . 1 5 - O . 1 3 1 8 -11.614 -0.1294 - 9.60 -0.1229 - 4 . 0 9 Q BAR -0.1276 - 8 . 0 8 -0.1293 - 9 . 5 4 -0.1302 -10.26 - 0 . 1 3 0 7 - 1 0 . 4 0 - 0 . 1 3 0 4 - 1 0 . 4 1 Z NO-BAR -0.1701 -44.03 - 0 . 2 9 5 4 - 1 5 0 . 1 8 -0.3911 - 2 3 1 . 2 C - 0 . 4 4 9 1 < 5 8 0 . 4 0 - 0 . 4 7 4 2 -301.64 ELASTICITY - 0 . 0 1 4 6 - 0 . 0 1 4 6 - 0 . 0 1 4 6 - 0 . 0 1 4 6 - 0 . 0 1 4 6 UJ BAR - 0 . 0 3 1 6 • •116.77 -0.0261 - 7 8 . 9 8 - 0 . 0 2 2 8 -56.2C - 0 . 0 1 9 4 3 3 . 6 6 -0.0161 - 1 0 . 4 4 7 ui z NO-BAR -O .0707 • •385.29 - 0 . 0 5 5 1 - 2 7 8 . 17 -O.0449 -207.9C - 0 . 0 3 2 4 - •122.13 -0.0190 - 3 0 . 1 5 ul Q BAR - 0 . 0 5 7 5 • •294.19 - 0 . 0 5 7 1 -291 .57 - 0 . 0 5 0 1 - 2 4 3 . 3 * - 0 . 0 4 2 1 - -188.76 - 0 . 0 3 4 0 - 1 3 3 . 0 5 Z NO-BAR - 0 . 1 4 6 5 • •905.11 -0.2875 - 0 . 3 8 5 7 - 0 . 4 4 3 5 -0.1(462 D E E P RECTANGULAR BEAM WITH UNIFORMLY DISTRIBUTED LOAD i. ,1 • I • ' ,1 I 1 1 1 1 ' 1 / U n " l e " 9 , h " ' I TABLE 4 -10 NORMAL S T R E S S CTy AND % ERROR , AT DIFFERENT X ANDY LL = 0.3333 AS COEFFICIENT •' ! c I N Oy =c3- BY ELASTICITY AND FINITE ELEMENT , (NODAL DISPLACEMENTS AND NODAL FORCES >.» PARAMETER 9Q = 100° f 0 W. COLUMN A B ,] • c ' 6}- E ' V FUNCTIQjM % ERROR FUNCTION % ERROR FUNCTION % ERROR FUNCTION;: %ERROR FUNCTION %ERROR •Y-. ' ."ELASTICITY - 0 . 5 W o -1.1*891* -2.11*87 . - 2 . 5 7 1 8 ' - 2 . 7 5 8 7 u BAR - 0 . 6 6 1 5 - 1 1 . 3 7 - 1 . 5 3 7 0 - 3 . 2 0 - 2 . 1 8 0 0 - I.I46 - 2 . 5 8 9 5 - 0 . 6 9 - 2 . 7 6 5 0 - 0 . 2 3 L U z NO-BAR -0.61*93 - 9 . 3 1 - 1 . 5 9 3 3 - 6 . 9 7 - 2 . 2 3 l | 5 - 3 . 9 9 - 2 . 6 3 7 9 - 2 . 5 7 - 2 . 8 1 0 3 - 1 . 8 7 U ' ci BAR - 0 . 5 8 5 5 1.1*2 -1 .1*689 1.38 - 2 . 1 2 3 6 1 .17 -2 .51*51 1 , 0 3 - 2 . 7 3 2 9 0 . 9 3 z NO-BAR - 0 . 6 1 6 2 -3.71* - 1 . 7 6 8 0 - 1 8 . 7 1 -2.31*1*7 - 9 . 1 2 -2.691*8 -1*.78 -20SILOS - 2 . 9 7 E L A S T I C I T Y -0.2791*. - 0 . 8 7 6 3 - 1 . 3 1 5 8 - 1 . 5 9 7 8 - 1 . 7 2 2 5 U J BAR - 0 . 3 2 2 6 -15.1+8 - 0 . 9 0 9 2 - 3 . 7 5 -1.3381* - 1 . 7 1 - 1 . 6 1 1 3 -0.81+ - 1 . 7 2 8 2 ' - 0 . 3 3 2 L U z NO-BAR - 0 . 3 5 7 9 - 2 8 . 1 2 - 0 . 9 3 3 5 - 6 . 5 3 - 1 . 3 8 3 3 - 5 . 1 3 - 1 . 6 6 2 6 - U . 0 5 ; - 1 . 7 8 1 0 - 3 . 3 9 Ci BAR -0.271*9 1.60 - 0 . 8 6 9 0 O.83 -1.301*7 0.81* -1.581*1 t 0 . 8 6 - l o 7 0 7 6 O.87 z NO-BAR - 0 . 3 3 2 7 - 1 9 . 0 8 -1.11+71 - 3 0 . 9 0 - 1 . 5 5 9 1 -18.1*9 -1 .8001* - 1 2 . 6 7 - 1 . 8 9 7 2 -10.11* E L A S T I C I T Y -0.101*7 -OJ+032 - 0 . 6 2 2 9 -0.761*0 -Oo8263 U J BAR -0.1251* - 1 9 . 7 9 -0.1+197 -1+.09 -0.631*5 - 1 . 8 6 - 0 . 7 7 1 0 - 0 . 9 2 - 0 . 8 2 9 5 - 0 . 3 7 3. LU z: NO-BAR - 0 . 1 8 2 9 . -71*. 72 -0J.L$22 - 1 2 . 1 7 - o . 6 7 a i - 7 . 8 9 - 0 / 8 1 6 8 - 6 . 9 2 - 0 . 8 7 8 7 -6 .31* LL.' Ci .BAR - 0 . 1 0 1 7 2 .83 -0.3991+ 0.91* - 0 . 6 1 7 6 0 . 8 5 -0/.7571* , 0 . 8 6 - 0 . 8 1 9 1 0 . 8 $ z! NO-BAR - 0 . 1 7 0 8 - 6 3 . 1 1 - 0 . 6 5 2 2 - 6 1 . 7 8 - 0 . 8 6 7 3 -39 .21* -5.9971*'' - 3 0 . 5 5 -1.01*93 - 2 6 . 9 9 !. E L A S T I C I T Y OoOO ' 0 . 0 0 0 . 0 0 y 0 . 0 0 0 . 0 0 U ' BAR 0 . 0 0 0 . 0 0 0 . 0 0 0 . 0 0 0 . 0 0 4 ui z: NO-BAR - 0 . 0 6 7 6 -0.0I+13 -O.O3I+O - 0 . 0 3 6 9 - 0 . 0 3 9 8 LL; Q BAR 0 . 0 0 0 . 0 0 0 . 0 0 0 . 0 0 0 . 0 0 z NO-BAR - 0 . 0 6 9 1 - 0 . 2 1 1 7 - 0 . 2 2 6 5 - 0 . 2 3 7 B -0.21*30 E L A S T I C I T Y 0.101*7 0.1+032 0 . 6 2 2 9 0.761*0 0.8263 U : BAR 0.1251* - 1 9 . 7 9 0.1+197 -1*.09 0.631+5 - 1 . 8 5 0.7710 - 0 . 9 2 0 .8295 - 0 . 3 9 5 ui z NO-BAR 0.0528 1*9.58 0.3721* 7 .63 0 .5996 3 . 7 5 0 .7372 3 . 5 1 0.791*6 3.81* U J ci BAR 0.1017 2 . 8 3 0.3991+ 0.91* 0 .6176 0 . 8 5 0.751+5 • 0 . 8 6 0.8191 0 . 8 6 z NO-BAR 0.0378 63 .90 0 .2273 1*3.60 0.1+076 31*. 57 0 .5162 32J+3 0 .5596 3 2 . 2 7 E L A S T I C I T Y 0.2791* 0 .8763 1.3158 1 .5978 1.7225 u.' BAR 0.3262 -15.1*8 0.9092 - 3 . 7 5 1.3381+ • - 1 . 7 1 1.6113 -0.81+ 1.7282 - 0 . 3 3 6 ui z NO-BAR 0 .2507 1 0 . 2 5 0 . 8 5 6 0 2 . 3 1 1.2911+ 1 .86 1 .5673 1 . 9 2 1.6851* 2 . 1 6 u' Ci BAR 0.271*9 1 .60 0 .8690 O .83 1.301+7 0.81* 1.581+1 0 . 8 6 1.7076 O.87 z NO-BAR 0.2211 20 .86 0 . 7 H 9 I 8 . 7 8 1 .0766 1 8 . 1 8 1.301+7 1 8 . 3 5 1.3981 18 .83 E L A S T I C I T Y 0.591*0 1.1+891* 2.11*87 2 .5718 2.7587 ui BAR 0.6615 - 1 1 . 3 6 1.5370 - 3 . 2 0 2 . 1 8 0 0 -1.1*6 2 . 5 8 9 5 - 0 . 6 9 2 .7650 - 0 . 2 3 7 ui z NO-BAR 0 . 6 2 6 5 -5.1+7 1.1*61*2 1 . 6 9 2 . 1 0 5 1 2 . 0 2 2.^222 1.93 2.7020 2 . 0 5 u'- ci BAR 0 . 5 8 5 5 1.1+2 1.1*689 1.38 2 . 1 2 3 6 1.17 2.5k5l 1.03 '2 .7329 0 . 9 3 z NO-BAR 0 .5558 6.1+3 1 .2738 11+.1*7 1 . 8 2 2 7 15.17 2.1791* 1 5 . 2 6 2.3301* 1 5 . 5 3 D E E P RECTANGULAR BEAM WITH UNIFORMLY DISTRIBUTED LOAD j. I I ' I 1 1 1 I 1 1 .. TABLE 4-11 SHEAR STRESS T x y AND % ERROR AT DIFFERENT X AND Y u. = 0.3333 AS COEFFICIENT C INxXy=C-5- BY ELASTICITY AND FINITE ELEMENT (NODAL DISPLACEMENTS AND NODAL FORCES PARAMETER 9Q - 100° R < 0 ' W \ COLUMN A B c 0 E FUNCTION/ % ERROR FUNCTION %ERROR FUNCTION % ERROR FUNCTION;! %ERROR' FUNCTION %ERROR ^ELAST IC ITY 0,3660% 0.2807 o."i954 '• . 0 . 1 1 0 1 H 0 . 0 2 4 9 UJ BAR 0.3631 0 . 7 9 0 .2803 0 . 1 5 0 . 1 9 7 5 - 1 . 0 4 0.1140; p - 3 . 5 3 j 0 . 0 3 0 0 - 2 0 . 7 7 1 ui z NO-BAR 0 . i i272 •16.73 0 . 2 6 4 2 5 . 8 7 0 . 1 7 0 2 1 2 . 8 9 0 . 0 8 5 2 22.62 I 0.0020 92 .01 U. Q . BAR 0.3492 4 . 5 9 0.2739 2 . 4 3 0 . 1 9 3 5 0 . 9 7 0.1117 -1..36 ; 0.0292 - 1 7 . 5 9 z NO-BAR 0 .3956 - 8 . 1 0 0 .2727 2 . 8 6 0 .1945 0.247 0.1203 - 9 . 2 2 ; 0 . 0 4 4 4 - 7 8 . 3 7 E L A S T I C I T Y . - 0.9290 0.7126 0 . 4 9 6 1 0.2796 0 .0631 u.' BAR 0.9172 1.27 0 .7049 1 .07 0 . 4 9 2 1 0 . 8 0 0.2786 0 i 2 7 ! 0 . 0 6 4 9 - 2 . 9 5 2 z NO-BAR 0 .9646 - 3 . 8 3 0.7157 - 0 . 4 3 0 . 4 7 6 0 4 . 0 5 0 .2561 8.1)0 . 0 . 0 4 2 4 32 .88 UJ Ci BAR 0.9076 2 . 3 0 0.6990 1 . 9 0 0 . 4 8 7 5 1.72 0.2753 1.54 0.0629 O.41 z NO-BAR 0.9461 - 1 . 8 3 0.7075 0 . 7 1 0 . 4 9 2 1 0 . 8 1 0 .2878 - 2 . 9 4 : 0.0830 - 3 1 . 4 9 E L A S T I C I T Y 1.2669 0 .9717 0 . 6 7 6 5 0.3813 0 .0861 UJ BAR 1 .2523 1 .15 0.9596 1 .24 0 . 6 6 8 3 1 . 2 1 013770 1 . 1 3 0 . 0 8 5 2 1.0-9 3 NO-BAR 1.2723 - 0 . 4 3 0.9814 - 1 . 0 0 0 . 6 6 8 9 1.11 Oj.3662 3 . 9 4 0.0715 1 6 . 9 1 'UJ Ci BAR 1 .2453 , 1.71 0 . 9 5 6 3 1 . 5 8 0.6663 1.51 0,13760 1 . 3 9 0 .0856 0 . 6 0 z NO-BAR 1 .2432 1.86 0 .9557 1 . 6 4 0.6650 1 . 7 0 0'.3794 0 . 4 8 0.0947 - l O . Q l E L A S T I C I T Y 1 .3795 1 . 0 5 8 0 Oi7366 , 0 . 4 1 5 2 1 * Oo0938 UJ BAR 1.3640 1.12 1 . 0 4 4 6 1 . 2 7 0 . 7 2 7 1 1 . 2 9 0 . 4 0 9 7 f 1 . 3 2 0 .0919 2 . 0 0 4 z NO-BAR 1.3613 1 .32 1 . 0609 - 0 . 2 7 0 . 7 3 3 3 0 . 4 5 0 . 4 0 5 5 ' 22335. 0.0822 1 2 . 2 6 u." Ci BAR 1.3579. 1 .57 1 . 0 4 2 2 1 .49 . 0 . 7 2 5 9 1 .45 0 . 4 0 9 6 / 1 .36 0 . 0 9 3 1 0 . 6 3 Z NO-BAR 1 . 3 2 8 4 3 . 7 0 1 .0302 2 . 6 3 0.7227 ' 1 .88 0 .4115 ? 0 . 8 8 0 . 0 9 8 4 - 5 . 0 1 E L A S T I C I T Y 1.2669 0 .9717 0 . 6 7 6 5 0.3813 0 .0861 UJ BAR 1 .2523 1 .15 0 . 9 5 9 6 1 .24 O.6683 1 . 2 0 0.3770 1 . 1 3 0 . 0 8 5 2 1 . 0 9 5 ui z NO-BAR 1.2307 2 . 8 4 0 . 9 6 0 1 1 . 1 9 0 .6690 1.10 0.3715 2 . 5 6 0 . 0 7 3 4 14.73 UJ ci BAR 1 .2453 1.71 0 . 9 5 6 3 1.57 0.6663 1.51 0.3760 1 .39 0 . 0 8 5 6 0 . 6 0 z NO-BAR 1 . 2 0 2 4 5 . 0 9 0 . 9 3 7 8 3 . 4 8 0 .6660 1.56 0 . 3 8 3 7 - 0 . 6 2 0 . 0 9 5 3 i r-10.68 E L A S T I C I T Y 0.9290 0.7126 0 . 4 9 6 1 0.2796 0.0631 . BAR 0.9172 1.27 0 . 7 0 4 9 1 . 0 7 0 . 4 9 2 1 / 0 . 8 0 0.2789 0 . 2 7 0 . 0 6 5 0 - 2 . 9 5 s 6 ui z NO-BAR 0.8791 5 . 3 7 0 . 6 8 5 4 3 . 8 1 0.14809 3 . 0 5 0 . 2 6 4 8 5 . 3 0 0 . 0 4 5 2 2 8 . 4 2 u.' Q BAR 0.9076 2 .30 0 .6990 1 . 9 0 0 . 4 8 7 5 - 1.73 0,2753 1.54 0 . 0 6 2 9 0 .41 Z NO-BAR 0 . 8 6 4 0 7.00 0 .6939 4 . 0 2 0.4966 - 0 . 1 0 0 . 2 9 4 4 5 . 2 8 . 0 .0839 -32 .82 E L A S T I C I T Y 0.3660 O .2807 0 . 1954 0.1101 0 . 0 2 4 9 UJ BAR 0.3631 0 . 7 9 0.2803 0 . 1 5 0 . 1 9 7 5 - 1 . 0 4 0 .1140 - 3 . 5 3 0 . 0 3 0 0 ^ 0 . 7 7 7 ui z NO-BAR 0.3207 12 .38 0 . 2 3 4 $ 9 . 2 2 0 .1767 9.59 0.0908 1 7 . 5 5 0.0036 8 5 . 6 5 uj Ci BAR 0 .3492 4 . 5 9 0.2739 2 . 4 3 0 . 1 9 3 5 0 .97 0.1116 - 1 . 5 3 0.0292 -17 .59 z NO-BAR 0 . 3 2 0 1 12 .52 0.2633 6 . 1 8 0 .1982 - 1 . 4 1 0.12383 - 1 2 . 4 3 0.0448 -80.06 D E E P RECTANGULAR BEAM WITH UNIFORMLY DISTRIBUTED. LOAD TABLE 4-12 X - D I S P L A C E M E N T U AND % ERROR: AT DIFFERENT X AND Y fj. = Q.45 AS COEFFICIENT C : IN u=C^jlOBY ELASTICITY AND FINITE ELEMENT • | PARAMETER 60 * 100° " . . \ R n COLUMN A B C D E ; w i • FUNCTION %ERROR FUNCTION %ERROR FUNCTION %ERROR FUNCTION %ERROR FUNCTION % E R R O R ELASTICITY 1.5053 3.31*81* 1+.8139 5 . 8 0 3 0 6 . 2 5 2 1 1 Id BAR 1.5053 0 . 0 3.31*51 0 . 0 1 i * . 8 0 6 7 0 . 1 5 5 . 7 9 1 6 0 . 2 0 ' 6 . 2 3 7 1 * 0 . 2 3 U NO-BAR 1.51+1+1 - 2 . 5 8 3.1*302 -2.1+1+ 1+.9019 - 1 . 8 3 5 . 8 8 9 5 -1.1+9 v 6 . 3 3 7 6 - 1 . 3 7 ELASTICITY 1.5995 3.1+519 1*. 9.071* 5.8701. 6 . 2 7 9 3 la uj BAR 1.591+9 0 . 2 9 3.1*1+13 0 . 3 1 1+.8932 0 . 2 9 5.851*3 0 . 2 7 6.261*0 0.21* UJ' NO-BAR 1 .6291* - 1 . 8 7 3.5221+ -2.01+ 1+.9873 - 1 . 6 3 5.9511+ - 1 . 3 9 6 . 3 6 0 6 . - 1 . 2 9 ELASTICITY 1.3551 3 .2651* U.7803 5 . 8 0 1 1 6.261*2 2 ui BAR 1.351*9 0 . 0 1 3 . 2 6 2 2 0 . 1 0 1+.7731 0 . 1 5 5 . 7 8 9 9 0 . 1 9 6 , 2 5 0 2 0 . 2 2 u.- NO-BAR 1.3723 - 1 . 2 7 3 . 3 1 5 0 - 1 . 5 2 1*. 81*60 - 1 . 3 7 5 . 8 7 2 7 - 1 . 2 3 6o3378 - 1 . 1 7 ELASTICITY 1.2117 3 . 1 6 2 2 1+.7069 i5 .7 l+67 6 . 2 1 8 2 3 L U BAR 1.2120 - 0 . 0 3 3 . 1 5 9 9 0 . 0 7 l*.70O3 O . l i * 5 . 7 3 6 1 0 . 1 8 6.201*7 0 . 2 2 u- NO-BAR 1.211*8 - 0 . 2 6 3 . 1 8 0 6 - 0 . 5 8 1+. 71+23 - 0 . 7 5 5 .7921* - 0 . 7 9 6 . 2 6 8 2 - 0 , 8 0 ELASTICITY 1.3252 ' • 3.'2561* 1*.7965 ' 5 .7681* 6 . 1 9 2 7 3a ui BAR 1.3217 0 . 2 6 3.21*73 0 . 2 8 1+.7567 0 . 2 7 7 5 .751*0 0 . 2 5 . 6 . 1 7 8 5 0 . 2 3 LL; NO-BAR 1.3226 0 . 2 0 3 . 2 6 3 3 - 0 . 2 1 1+.7913 -0.1+6 5 . 8 0 0 0 - 0 . 5 5 6 . 2 2 8 l i ; . - 0 . 5 7 ELASTICITY 1.2252 3.1561* 1+.6695 5 .6681* 6 . 0 9 2 7 4 a ui BAR • 1 . 2 2 1 7 0 . 2 8 3.11*73 0 . 2 9 1+.6567 0 . 2 7 • 5 .651*0 0 . 2 6 6 . 0 7 8 5 0 . 2 3 U J NO-BAR 1.211*0 0 . 9 1 3 .1399 0 . 5 3 1+.6599 0 . 2 1 5 . 6 6 6 2 0.01+ 6.091+0 - 0 . 0 3 ELASTICITY 1 . 0 1 1 6 2 . 9 6 2 2 1+.5068 5.51*67 6 . 0 1 8 2 5 uj . BAR • 1.0120 - o .o l+ 2 . 9 5 9 9 0 . 0 8 1+.5003 0 . 1 5 5 . 5 3 6 1 0 . 1 9 6.001*7 0 . 2 2 u.' NO-BAR 0 . 9 9 9 0 1.25 2 . 9 3 5 0 0 . 9 2 1+.1+801+ 0 . 5 9 5.5251* 0 . 3 8 ; 6.0001* 0 . 3 0 ELASTICITY 0 . 9 5 5 1 2.8651* 1+.3803 5.1+011 v 5.861*2 6 ui BAR 0 . 9 5 5 0 0 . 0 1 2 . 8 6 2 2 0 . 1 1 1+.3731 0 . 1 7 5 . 3 8 9 9 0 . 2 1 5 . 8 5 0 2 0 .21* u NO-BAR 0 . 9 3 7 9 .1 .80 2 . 8 1 9 0 1 . 6 2 1+^ .3211+ 1.35 5.31+00 1.31 5 . 8 0 3 9 1 .03 ELASTICITY 1 . 0 9 9 5 2 . 9 5 1 9 . 1+. 1+071+ 5 . 3 7 0 1 5 . 7 7 9 3 6a ui BAR 1.091*8 O.I4-2 2.91*13 0 . 3 6 1+.3932 0 . 3 2 5.351+3 0 . 2 9 ; 5 .761*0 0 . 2 6 U I NO-BAR 1 .0780 1.95 2 . 8 9 2 8 2 . 0 0 1+.3298 1.76 5 . 2 8 6 5 1 .56 i . 5.691*5 U + 7 ELASTICITY 6 . 9 0 5 3 2.71*81* 1+.2139 5 . 2 0 3 0 ; 5 . 6 5 2 1 7 ui BAR 0 . 9 0 3 5 0 . 2 0 2.71*51 0 i l 2 1+.2067 0 . 1 7 5 . 1 9 1 6 0 ; 2 2 ' 5 .6371* 0 . 2 6 u.' NO-BAR 0 . 8 8 0 0 2 . 7 9 2 . 6 7 6 5 2 . 6 1 1+.1155 2 . 3 7 5 . 0 9 3 7 2 . 1 0 : 5.51*00 1 .98 ',,-.>-, : r,.^ (i-. ;'! s.v.-.'.-.,-. : ; - •>•> v ' - y ; » > . / ^ , ( , . . « , ^ ^ \ - - - ; - . . : - . - . ' • - . „ . . . , . v . , ./ la 2 3 3a 4a 5 6 6a 7 D E E P R E C T A N G U L A R B E A M WITH UNIFORMLY DISTRIBUTED B C II. 5a q #/unit length Y,v —r>7"-. CVJ 1 -f E y x,u — > •thickness 8 t POINT p COLUMN B ROW 2 TABLE 4 -13 Y - DISPLACEMENT V AND % ERROR AT DIFFERENT X ANDY fj. - '©.45 AS COEFFICIENT C ! IN v=C^IOBY ELASTICITY AND FINITE ELEMENT j PARAMETER B0 - 100° ' £, : j R , 0 , W-COLUMN A B C LT # ; E F U N C T I O N ; % E R R O R FUNCTION % E R R O R FUNCTION % E R R O R FUNCTION % E R R O R ' FUNCT ION % E R R O R 1 E L A S T I C I T Y 2.0605 1.8751 1.4565 A 0.8756 ! 0.2032 - *> r' • ; LJ U J B A R 2.0592 0.06 1.8731 0.11 1.4547 0.13 ' 0.8746 0.11 j 0 .2035 -0.17 N O - B A R 2.0127 2.32 1.8476 1.47 1.4430 0 .93 0.8682 0 . 8 5 ; 0.1953 3.85 la E L A S T I C I T Y 1.5417 ; •1.4095 1.0879 0.6359 • 0.1125 U J u. B A R 1.5410 0.05 1.4076 0.14 1.0859 0.19 = 0.6344 •O.23 ; 0.1120 0.46 N O - B A R 1.5792 - 2 . 4 3 1.4530 -3.09 1.1227 -3.20 0.6608 i-3.92 : 0.1302 -15.71 2 E L A S T I C I T Y 1.0932 1.0349 0.8213 0.4993 ; 0.1164 uj u_-' B A R 1.0931 0.01 1.0340 0.09 0.8202 0.13 0.4986 r 0.14 0.1163 0 . 0 5 N O - B A R 1.0422 4.66 0.9885 4 . 4 9 0.7902 3.77 0.4756 / 1 4 . 7 4 0.0936 19.63 3 E L A S T I C I T Y 0.3466 0 . 3 6 4 1 1 0.3038 0.1895 ^  o!o4^6 ui U J B A R 0.3466 -0.01 0.3638 0.07 0.3034 0.13 0.1892J 0.15 i 0.0446 0 . 0 N O - B A R 0 .3005 13*28 0.3109 14.60 0.2591 14.70 /0.1522 /L9.65 , 0.0112 74.86 3a E L A S T I C I T Y 0.0270 , 0.0674 0.0699 / 0.0464 £ 0.0086 ui B A R 0.0270 0.12 0.0672 0 .25 0.0697 0 . 3 4 0.0462 f 0.39 : 0 .0085 0.,68 N O - B A R 0 .0482 -78.68 0.0893 -32.50 0.0962 - 3 7 . 5 8 0.0773 -66.62 : . 0 . 0 4 3 4 4 0 6 . 2 2 4 a E L A S T I C I T Y -0.5951 -0 . 5 0 0 5 -0.3681 - 0 . 2 0 9 5 , - 0 . 0 3 6 7 ui U J B A R -0 .5951 0.0 - 0 . 5 0 0 4 0.03 -0.3678 0.06 - 0 . 2 0 9 3 0.09 ! - 0 . 0 3 6 6 0.16 N O - B A R -0.5826 2.10 - 0 . 4 8 7 8 2.55 -0.3481 5 . 4 1 -0.1799 14.14 . 0.0019 105.17 5 E L A S T I C I T Y -0.9259 - 0 . 8 0 8 4 -0.6132 -0.3638 ; - 0 . 0 8 4 0 ui U.' B A R -0.9260 0.00 - 0 . 8 0 8 2 0 . 0 3 -0.6128 0.06 -0.3635 ! 0 .08 : - 0 . 0 8 3 9 0 . 0 3 N O - B A R . - 0 . 9 5 4 0 . -3.03 -0.8583 - 6 . 1 7 -0.6693 -9.-15 - 0 . 4 1 7 0 -14.60 : - 0 . 1 2 9 7 -54.43 6 E L A S T I C I T Y -1.6726 -1 .4793 -1.1306 • -0.7637 v l - 0 . 1 5 5 8 ui U_ B A R -1 .6725 0.01 -1 .4784 0 . 0 6 -1.1296 0.10 -0.6730 0.11 -0 .1557 0 . 0 4 N O - B A R -1.6944 -1.31 -1 .5274 - 3 . 2 5 -1.1903 - 5 . 2 7 -0.7344 -9.01 \ - 0 . 2 1 0 3 - 3 5 . 0 4 6a E L A S T I C I T Y -2.1099 -1 .8426 -1.3860 -0.7990 i - 0 . 1 4 0 7 ui U J B A R -2.1092 0.03 - 1 . 8 4 0 7 0.10 -I .3840 0.15 -0.7975 0.18 - 0 . 1 4 0 1 0.37 N O - B A R - 2 . 1 0 3 2 0.31 - 1 . 8 2 4 9 0.96 - 1 . 3 5 5 2 2.22 -0.7546 ' -5 .55 . - 0 . 0 8 4 3 4 0 . 0 6 7 E L A S T I C I T Y -2.6399 -2.3195 -1.7659 - 1 . 0 4 9 9 ; - 0 . 2 4 2 5 ui Lu • B A R . . -2.6386 0 .05 -2.3175 0.09 -1.7641 0.11 - 1 . 0 4 9 0 o;o9 ! - 0 . 2 4 2 9 - 0 . 1 4 . N O - B A R -2.6574 -0.66 -2.3626 - 1 . 8 6 - 1 . 8 2 7 9 - 3 . 5 1 -1.1199 -6.67 ; - 0 . 3 1 0 4 -27 .99 B C 11. 5 0 v X,u 'thickness * t POINT p : COLUMN 8 ROW , 2 ,J«£—a=l.25b—>) t - 9 6 -TABLE 4-1-4 NORMAL S T R E S S CT X ' A N D % ERROR AT DIFFERENT X AND Y u= 0.45 AS COEFFICIENT C IN 0~x = C.-9- BY ELASTICITY AND FINITE ELEMENT (NODAL DISPLACEMENTS AND NODAL FORCES PARAMETER 8Q = 100° ' ' R ' Q W COLUMN A B C D E FUNCTION % ERROR FUNCTION % ERROR FUNCTION % ERROR FUNCTION %ERROR FUNCTION %ERROR 1 E L A S T I C I T Y -0.98514.- -0.9851+' -O.985I4 -0.9851+ : -0.9851+ ui u -z BAR -0.9719 1 .37 -0.9781+ 0.71 - 0 . 9 8 0 8 0 J |7 - 0 . 9 8 2 7 0 . 2 7 -0.98I+6 0 . 0 8 NO-BAR -1 .0105 -2.51; -0.91*1+0 1+.21 - 0 . 9 5 6 8 2.9C - 0 . 9 6 9 3 1 . 6 3 - 0 . 9 8 2 1 0.31+ Q z BAR. - 0 . 9 2 9 3 5 . 6 9 - 0 . 9 2 5 1 . . 6 . 12 - 0 . 9 3 3 8 5 . 2 3 -0.91+53 1+..07 - 0 . 9 5 7 6 2 . 8 3 NO-BAR -1.1631 -18.03 - 1 . 2 3 3 5 -25.17 - 1 . 1 3 6 7 - 1 5 . 3 5 - 1 . 0 6 9 8 - 8 . 5 6 -I.OI+30 -5.81+ 2 E L A S T I C I T Y -0.8819 - 0 . 8 8 1 9 - 0 . 8 8 1 9 - 0 . 8 8 1 9 - 0 . 8 8 1 9 ui u: z BAR - 0 . 8 7 5 7 0.71 -O .8753 0.75 - 0.8771 0.514 -O .8790 0 . 3 3 - 0 . 8 8 0 9 . 0 . 1 2 NO-BAR - 0 . 8 8 0 5 0.16 -Oo8l453 1+.15 - 0 . 8 6 5 9 1.81 -O.87I+8 ; 0 . 8 1 1 - 0 . 8 8 2 3 ' -0.01+ c i z • BAR -0.86714. 1.65 - 0 . 8 6 5 5 1.86 -0.861+1+ 1.9S -0.861+5 - 0 . 8 6 5 2 1.90 NO-BAR -0.9929 - 1 2 . 5 8 -1.1923 - 3 5 . 2 C - 1 . 1 3 7 9 -29 . 0 ; - 1 . 0 8 3 3 ; - 2 2 ; 8 3 -1.051+0 p l 9 . 5 2 3 E L A S T I C I T Y - 0 . 7 0 8 5 • - 0 . 7 0 8 5 - 0 . 7 0 8 5 - 0 . 7 0 8 5 " - 0 . 7 0 8 5 ui u' Z" BAR - 0 . 7 0 5 7 0 . 3 9 - 0 . 7 0 5 2 0.1+5 - 0 . 7 0 6 0 0 .35 - 0 . 7 0 6 9 0 . 2 2 - 0 . 7 0 7 9 0 . 0 8 NO-BAR - 0 , 7 0 5 5 0 . h 2 - 0 . 6 7 0 3 .5.36 - 0 . 6 9 7 9 1.1+S -0.7071+ 0.16 , - 0 . 7 1 2 1 - 0 . 5 1 Ci Z BAR - 0 . 7 0 2 0 0.91 - 0 . 7 0 0 2 1.17 - 0 . 7 0 0 2 1.16 - 0 . 7 0 0 2 1.16 - 0 . 7 0 0 1 1 . 18 NO-BAR - 0 . 7 9 8 2 - 1 2 . 6 7 - 0 . 9 8 3 6 -38.8I4 -0.9922 -1+0. - 0 . 9 7 3 6 -37.1+3 - 0 . 9 6 0 1 - 3 5 . 5 3 4 E L A S T I C I T Y -O..500O - 0 . 5 0 0 0 - 0 . 5 0 0 0 - 0 . 5 0 0 0 - o„5ooo ui u' U' z BAR - 0 . 5 0 0 0 0 . 0 - 0 . 5 0 0 0 0 . 0 - 0 . 5 0 0 0 0 . 0 - O . 5 0 0 0 : • 0 . 0 - 0 . 5 0 0 0 0 . 0 NO-BAR -O .5038 - 0 . 7 5 -0.l4.70l4. 5 . 9 ; -.0.1+91+2 1.16 - 0 . 5 0 3 3 - 0 . 6 7 -0.5051+ - 1 . 0 8 c i z BAR - 0 . 5 0 0 0 0 . 0 - 0 . 5 0 0 0 0 . 0 - 0 . 5 0 0 0 0 . 0 - 0 . 5 0 0 0 0 . 0 - 0 . 5 0 0 0 0 . 0 NO-BAR -0.5711 -111.. 22 -0.7261+ -1+5.26 -0.7772 -55.1+; - 0 . 7 9 5 3 - 5 9 . 0 7 -0.7992 - 5 9 . 8 5 5 E L A S T I C I T Y - 0 . 2 9 1 5 - 0 . 2 9 1 5 - 0 . 2 9 1 5 - 0 . 2 9 1 5 - 0 . 2 9 1 5 ui u U: z BAR -0.291+3 .0.96 -0.291+8 - 1 . 1 C -0.291+0 -O.85 - 0 . 2 9 3 1 0.51+ -0.2922 - 0 . 2 1 NO-BAR - 0 . 3 0 5 8 -1+.88 - 0 . 2 7 8 1 ^ 1+.61 - 0 . 2 9 3 6 - 0 . 7 2 - 0 . 2 9 8 6 -2.1+3 - 0.2971 - 1 . 8 9 c i z BAR - 0 . 2 9 8 0 - 2 . 2 1 - 0 . 2 9 9 8 - 2 . 8 3 -0.2998 - 2 . 8 : - 0 . 2 9 9 8 • -2.81+ - 0.2999 - 2 . 8 7 NO-BAR -0.31+79 - 1 9 . 2 8 -0.1+835 - 6 5 . 8 3 - 0 . 5 6 7 6 -91+. 6< - 0 . 6 1 5 3 ' - 1 1 1 . 0 5 - 0 . 6 3 5 2 - 1 1 7 . 8 7 6 E L A S T I C I T Y - 0 . 1 1 8 1 - 0 . 1 1 8 1 - 0 . 1 1 8 1 - 0 . 1 1 8 1 - 0 . 1 1 8 1 ui u! u' z BAR - O . I 2 I 4 3 - 5 . 3 1 -0.121+7 -5.62 - 0.1229 -1+.05 - 0 . 1 2 1 0 -2.1+5 - 0.1191 -O .87 NO-BAR - 0.11+59 - 2 3 . 5 5 - 0 . 1 2 7 1 - 7 . 6 5 - 0 . 1 316 -11.1+7 - 0.1291 - 9 . 3 2 - 0 . 1 2 2 7 -3*90 c i z BAR -0.1326 - 1 2 . 3 1 -0.131+5 - 1 3 . 8 7 - 0 . 1 3 5 6 -11+.87 - 0 . 1 3 5 5 -11+.77 -0.131+8 -11+.17 NO-BAR -0.172,6 -1+6.19 - 0 . 3 3 9 5 -170.62 -0.1+302 -261+.1..5 -0.5001+ -323.81+ - 0 . 5 3 3 7 k 3 5 2 . 0 1 7 E L A S T I C I T Y -0.011+6 -0.011+6 -0.011+6 -0.011+6 =0.011+6 ui u' u: z BAR - 0 . 0 2 8 1 ' - 9 3 . 0 0 -0.0216 -1+7.85 - 0.0192 -31*62 - 0 . 0 1 7 3 - 1 8 . 3 8 -OoOl51+ -5.69 NO-BAR -O.071I4. - 3 8 9 . 7 7 -0.051+6 -271+.9C -0.01+1+6 - 2 0 6 . 1 C -O .0322 -120.92 - 0 . 0 1 8 9 -29.71 c i z BAR - 0 . 0 7 0 7 -381+.78 -0.071+9 -1+13.53 -0.0662 - 3 5 3 . 8 7 -0.051+7 - 2 7 5 . 0 5 -0.01+21+ -190.91+ NO-BAR -O.150O - 9 2 8 . 0 0 -O .3159 • 5 0 6 6 . O C -0.1+352 . •2885>0C -0.5092 . 3 3 0 2 . 0 0 - 0 . 5 3 8 1 -359100 D E E P RECTANGULAR BEAM WITH UNIFORMLY DISTRIBUTED LOAD TABLE 4 H 5 NORMAL STRESS CTy .,AND % ERROR AT DIFFERENT X AND Y u. - 0.45/ AS COEFFICIENT C IN (ry : C i BY ELASTICITY AND FINITE ELEMENT (NODAL DISPLACEMENTS AND NODAL FORCES PARAMETER QQ = 100° R n COLUMN 'V A ' • F»M B C E w FUNCTION % ERROR FUNCTION % ERROR FUNCTION % ERROR FUNCTION | %ERROR FUNCTION %ERROR ELASTICITY - 0 . 5 9 4 0 - 1 . 4 8 9 4 - 2 . I 4 8 7 -2.5718;-: f> -2.7587 U J BAR - 0 . 6 5 4 0 -10.10 - 1 . 5 3 5 5 -3 .09 -2 .1816 - 1 . 5 3 -2.5928I4 -0 .82 -2 .7691 -0 .37 i ui 2 NO-BAR -0 .6529 - 9 .91 -1 .5915 -6 .86 - 2 . 2 3 1 5 -3 .86 - 2 . 6 3 5 2 - ' - 2 . 4 6 -2.8077 - 1 . 7 8 u : d BAR -0 .5690 4.21 - 1 . 4 5 2 0 2.51 -2 .1113 1.74 - 2 . 5 3 7 2 § 1 . 3 4 -2.7285 1 .09 2 NO-BAR - 0 . 6 4 8 8 - 9 . 2 2 -1 .8061 -21.26 -2 .3619 -9 .92 -2.6994 -4 .96 -2.8392 - 2 . 9 1 ELASTICITY - 0 . 2 7 9 4 -O .8763 -1 .3158 - 1 . 5 9 7 8 - 1 . 7 2 2 5 uJ BAR -0 .3198 -14 .47 -0 .9077 -3 .58 -1.3387 -1 .74 -1.6128 - 0 . 9 3 - I . 7 3 0 3 - 0 . 4 5 2 ui NO-BAR - 0 . 3 5 5 2 -27 . 15 -0 .9320 -6 .35 - 1 . 3 8 1 5 - 4 . 9 9 -1.6606 -3 .92 -1.7788 -3 .27 U J Q BAR -0.2687 3.82 -0 .8639 1.42 - 1 . 3 0 0 - 7 1.15 - V 5 p l 3 .1.04 - 1 . 7 0 5 5 0 . 9 9 2 NO-BAR - 0 . 3 5 1 2 -25 .71 -1.1992 -36.85 - 1 . 5 9 4 3 21.16 -1..8241 " JL-M- • -1 .1945 i l l . 15 ELASTICITY - 0 . 1 0 4 7 -O .4032 -0 .6229 -O /.7640 -0 .8263 \ U J BAR - 0 . 1 2 4 3 -18.73 -O .419I -3 .96 -0 .6346 -1 .87 -0 .7716 , - 1 . 0 0 - 0 . 8 3 0 4 - 0 . 4 9 i. ui 2! NO-BAR -0 .1782 -70 .24 - 0 . 4 5 0 4 -11 .72 -0 .6710 -7 .72 -0 .81587 - 6 . 7 8 -0 .8775 -6 .20 U J Ci BAR -0 .0991 5 .38 -0 .3969 1.55 - 0 ; 6 1 5 8 1.15 - 0 . 7 5 6 1 1.02 0 . 8 1 8 1 0 . 9 9 2 NO-BAR -O.J.838 -75.53 -0 .7009 -73.84 -0 .9099 -46 .07 - 1 . 0 3 3 9 - 3 5 . 3 4 -1.0822 - 3 0 . 9 9 ELASTICITY - 0 . 0 0 ' 0*00 0 . 0 0 0 . 0 0 0 . 0 0 U J BAR 0 . 0 0 0 . 0 0 0 . 0 0 0 . 0 0 0 . 0 0 4 ui 2 NO-BAR -O .O632 -0 .0397 - 0 . 0 3 3 5 -0 .0367 - 0 . 0 3 9 4 u.' c i BAR 0 . 0 0 0 . 0 0 0 . 0 0 0 . 0 0 0 . 0 2 NO-BAR -0 .0777 - 0 . 2 5 3 0 >• -0 .2707 -0 .2822 - 0 . 2 8 7 0 ELASTICITY 0.1047 0.4032 0.6229 0.7640 <-0.8263 U J BAR O . I2433 -18.73 0.4191 -3 .96 0.6346 -1 .87 0.7716 - 1 . 0 0 • 0 .8304 -0 .49 5 ui 2 NO-BAR 0.0558 46.67 0.3730 7.47 0.5995 3 .76 0.7371 3.51 0.7946 3.83 U J Ci BAR 0.0991 5.38 0.3969 1.55 0.6158 1.15 0.7561 1 .02 0 . 8 1 8 1 0 . 9 9 2 NO-BAR 0.0325 68.92 0.1919 5 2 . 4 1 0.3621 41 .87 0.4648 39.16 0.5052 -38 .85 ELASTICITY 0.2794 0.8763 1.3158 1.5978 •1.7225-.: U J .BAR 0.3198 - 1 M 7 0.9077a - 3 . 5 8 1.3387 -1 .73 1.6128 - 0 . 9 3 1.7303 - 0 . 4 5 6 ui 2 NO-BAR 0.2515 9.98 0.8556 2 .36 1.2914 1.85 1.5675 1.90 1.6856 2.14 u : c i BAR 0.2687 '3 .82 0.8639 1.42 1.3007 1.15 1.5813 1.04 1.7055 0 . 9 9 2 NO-BAR 0.2158 22 .74 0.6760 22 .85 1.0254 22 .07 1.2425 2 2 . 2 4 1.3297 2 2 . 8 1 ELASTICITY 0.5940 1.4894 2.1487 2.. 5718 2.7587 U J BAR 0.6540 -10.10 1.5355 -3 .09 2.1816 - 1 . 5 3 2.5928 - 0 . 8 2 ; 2;7690 -0 .37 7 u i 2 NO-BAR 0.6223 -4 .76 1.4651 1.63 2.1068 1.95 2.5236 1.87 2.7033 2.01 u.' Ci BAR 0.5690 4 .21 1.4520 . 2 . 5 1 2.1113 1.74 2.5372 1.35 2.7285 1.09 2 NO-BAR 0.5353 9 .88 1.2271 17.61 1.7573 18.2c 2 .1005 18 .32 2.2447 18 .63 DEEP RECTANGULAR BEAM WITH UNIFORMLY DISTRIBUTED LOAD 1 , I 1 1 i 1 1 1 1 1 \^• 11,5a _ > - 9 8 -T A B L E 4-16 S H E A R - S T R E S S X x y AND % E R R O R AT DIFFERENT X AND Y a = 0.45. AS COEFFICIENT ' C IN x x y =C 3- BY ELASTICITY AND FINITE ELEMENT (NODAL DISPLACEMENTS AND NODAL FORCES PARAMETER 6Q - 100° R n COLUMN A B C D ! E FUNCT ION % ERROR F U N C T I O N % E R R O R FUNCT ION % ERROR FUNCT ION % E R R O R F U N C T I O N % E R R O R E L A S T I C I T Y 0 . 3 6 6 0 0 . 2 8 0 7 0.1951+ 0 . 1 1 0 1 | 0.021+9 B A R 0 . 3 6 6 9 - 0 . 2 5 0 . 2 8 1 2 - 0 . 1 8 0 . 1 9 7 1 - 0 . 8 6 0 . 1 1 3 1 i 2 . 7 2 0 . 0 2 9 0 - 1 6 . 7 2 . 1 ui z N O - B A R 0.14.214.2 - 1 5 . 9 0 0.2631+ 6 . 1 7 0 . 1 7 0 3 12.81+ 0 . 0 8 5 2 22.62 0 . 0 0 1 8 9 2 . 8 8 u* C) B A R O . .3505 U.23 0 . 2 7 7 6 1 . 1 0 0 . 1 9 8 0 - 1 . 3 0 P.1153 -1+.6B O . O 3 I I -21+.09 ft. 2: N O - B A R 0.3971+ - 8 . 6 0 0 . 2 7 3 8 • 2.1+7 0.191+9 0 . 2 7 0 . 1 1 9 6 - B . 6 0 0.01+26 -71 .21+ E L A S T I C I T Y 0 . 9 2 9 0 0 . 7 1 2 6 ' 0.1+961 0 . 2 7 9 6 ; 0 . 0 6 3 1 u : B A R 0.9214.0 0.51+ 0 . 7 1 0 2 0 . 3 3 0.1+957 0 . 0 8 0.28/13 - 0 . 6 1 0 . 0 6 6 8 ' -5.71+ '? z N O - B A R 0.961+1+ - 3 . 8 0 0 . 7 1 5 9 -0.1+7 0.1+770 3 . « 5 0 . 2 5 7 1 8 . 0 5 0.01+30 3 1 . 8 2 L Ci B A R 0 . 9 0 8 7 2 . 1 8 0 . 7 0 3 0 1.31+ 0.1+921 0 . 8 1 0 . ^ 7 9 0 0 . 2 0 0 . 0 6 5 0 - 2 . 9 1 Z N O - B A R 0.91+69 - 1 . 9 3 0 . 7 1 2 7 - 0 . 0 3 0.1+959 0.01+ ^ 0 . 2 8 9 7 - 3 . 6 2 0 . 0 8 3 0 -31.1+0 E L A S T I C I T Y • 1 . 2 6 6 9 0 . 9 7 1 7 0 . 6 7 6 5 0 . 3 8 1 3 0 . 0 8 6 1 u . B A R 1 . 2 6 0 5 0 . 5 0 0 . 9 6 6 3 o .55 0 . 6 7 3 6 0.1+3 0 . 3 8 1 0 0 . 0 8 O . O 8 8 I - 2 . 3 6 3 ui 2; N 0 - 8 A R 1 . 2 7 6 6 - 0 . 7 7 0 . 9 8 3 6 - 1 . 2 3 0 . 6 7 1 3 0 . 7 7 0.3681+ 3 . 3 8 0 . 0 7 3 5 11+.67 U." c i B A R 1.21+71 1 . 5 6 0 . 9 6 0 8 1 . 1 1 0.6711+ 0 . 7 5 0 . 3 7 9 9 0 . 3 5 0 . 0 8 7 5 - 1 . 6 7 z N O - B A R 1.21+77 1.51 0.96I4.I4. 0.7I+ 0 . 6 7 1 8 0 . 6 9 0 . 3 8 3 7 -0.61+ 0 . 0 9 6 6 -12 .11+ E L A S T I C I T Y 1 . 3 7 9 5 - • 1 . 0 5 8 0 0 . 3 7 6 6 0.1+152 0 . 0 9 3 8 u ; B A R 1 . 3 7 2 7 0.1+9 1 . 0 5 1 8 0 . 5 9 O . 7 3 2 9 0 . 5 1 0.1+11+1 0 . 2 5 ; 0 . 0 9 5 2 • -1.51+ 4'' ui z N 0 - 8 A R 1 . 3 6 8 7 0 . 7 7 1.061+9 -0.61+ 0.7361+ 0 . 0 3 0.1+081 1 . 7 0 0.081+6 9 . 7 1 u ' c i B A R 1 . 3 5 9 9 1.1+2 I.0I4.69 1 . 0 5 0 . 7 3 1 2 0 . 7 3 0.1+136 0 . 3 7 0 . 0 9 5 1 -1.1+7 z N O - B A R 1 . 3 3 5 5 3 . 1 9 1.01+09 1 . 6 2 0 . 7 3 0 7 0 . 8 0 0.1+166 -0 .31+ 0 . 1 0 0 9 - 7 . 6 6 E L A S T I C I T Y 1 . 2 6 6 9 0 .9717. 0 . 6 7 6 5 0 . 3 8 1 3 0 . 0 8 6 1 UJ B A R 1 . 2 6 0 5 0.1+9 0 . 9 6 6 3 - 0 . 5 5 0 . 6 7 3 6 0.1+3 0 . 3 8 1 0 0 . 0 8 0 . 0 8 8 1 - 2 . 3 6 5' ui Z\ N O - B A R 1 . 2 3 9 6 2 . 1 5 0.961+7 0 . 7 2 0 . 6 7 1 8 0 . 6 8 0 . 3 7 3 5 2 . 0 3 0 . 0 7 5 3 1 2 . 5 3 U J c i B A R 1.21+71 1 . 5 6 0 . 9 6 0 8 a . 11 0.6711+ 0 . 7 5 0 . 3 7 9 9 0 . 3 5 0 . 0 8 7 5 - 1 . 6 7 z N O - B A R 1 . 2 1 0 8 1+.1+2 0.91+83 2.1+0 0 . 6 7 3 1 0 . 5 0 0 . 3 8 7 9 - 1 . 7 3 0 . 0 9 7 2 - 1 2 . 9 3 E L A S T I C I T Y 0 . 9 2 9 0 0 . 7 1 2 6 0.1+961 0 . 2 7 9 6 0 . 0 6 3 1 u : B A R 0.921+0 0.51+ 0 . 7 1 0 2 0 . 3 3 0.1+957 0 . 0 8 0 . 2 8 1 3 - 0 . 6 1 0 . 0 6 6 8 -5.71+ 6 u i z N O - B A R 0 . 8 8 7 9 1+.1+3 0 . 6 8 9 0 3 . 3 0 0.1+823 2 . 7 8 0.2651+ 5 . 0 6 0.01+58 27.1+6 u c i B A R 0 . 9 0 8 7 2 . 1 8 0 . 7 0 3 0 1.31+ 0.1+921 0 . 8 1 0 . 2 7 9 0 0 . 2 0 0 . 0 6 5 0 - 2 . 9 1 z N O - B A R 0 . 8 7 2 5 6 . 0 8 0 . 6 9 1 3 2 . 9 8 0 . 5 0 0 8 -0.91+ 0 . 2 9 6 2 -5.91+ 0.081+0 - 3 2 . 9 7 E L A S T I C I T Y 0 . 3 6 6 0 O . 2 8 0 7 0.1951+ 0 . 1 1 0 1 0.021+9 u i B A R 0 . 3 6 6 9 - 0 . 2 5 0 . 2 8 1 2 - 0 . 1 8 0.197 . 1 - 0 . 8 6 0 . 1 1 3 1 - 2 . 7 2 0 . 0 2 9 0 - 1 6 . 7 2 7 u i z N O - B A R 0 . 3 2 6 6 10.71+ 0 . 2 5 5 9 8 . 8 3 „: 0 . 1 7 6 7 9 . 6 0 0 . 0 9 0 5 1 7 . 8 0 0 . 0 0 3 3 860 71+. uj c i B A R 0 . 3 5 0 5 U .23 0 . 2 7 7 6 .1.10 0 . 1 9 8 0 - 1 . 3 . 0 O . U 5 3 -1+.68 0 . 0 3 1 1 -21+. 99 z N O - B A R 0 . 3 2 5 1 . 11 .18 0 . 2 6 5 8 5 . 3 1 , 0 . 1 9 8 8 - 1 . 7 5 0 . 1 2 3 2 -11.81+ o : . 01+31 - 7 3 . 3 2 D E E P RECTANGULAR BEAM WITH UNIFORMLY DISTRIBUTED LOAD <L -99-Fig 4 - 4 - 1 0 0 -the r a d i u s , and \f~ perpendicular to i t are given by the ex-pressions u -_ljR<,Q\[{A{^)-z [conj.Ofc)] -Co"j- :(^)} e i G j | ( l 4. 8, ^ Z G J 1 ^ ^ A ^ 3 ) - Z L C o n j . C ^ ] "Conj jj'/ ( i f . 9 ) (3-/u) p The s t r e s s e s are expressed as f o l l o w s : 0 7 = ( S , - S a ) A . (U-13) ci> = C-S, + s j cif_~ait.> u)he.re_ 5 , - 4 I f?eq| C <£>,)] <£ „ _P_ < _ R , I (U-18) The e v a l u a t i o n of the a c t u a l values of the displacements and s t r e s s e s i n these formulae was c a r r i e d out by the computer programmed f o r t h i s purpose by Dr. Bulent Ovunc , who was i n -* Research Associate i n Dept. of C i v i l Eng., IT.B.C. -101-t e r e s t e d i n d e p e n d e n t l y i n the same problem, lj.,2.2 F i n i t e Element S o l u t i o n : I n view of symmetry o n l y one q u a r t e r of the d i s c need be c o n s i d e r e d . The model (Pig.lj;-!*) c o n s i s t s of the t r a p e z o i d s and t r i a n g l e s c o n s t r u c t e d between the e q u i d i s t a n t r a d i i d i v i d i n g the c i r c l e i n t o s e c t o r s w i t h an i n t e r v a l a n g l e of 10°. The t r a p e z o i d s are e q u a l i n each p a r t i c u l a r zone c r e a t e d by two c o n c e n t r i c c i r c l e s and t h e y a re d i f f e r e n t i n d i f f e r e n t zones n ot o n l y i n the dimens-i o n s b u t a l s o i n the geometric r a t i o s k and . The p a r t of the d i s c n e a r the c e n t r e c o n s i s t s of e q u a l i s o s c e l e s t r i a n g l e s . The element s t i f f n e s s m a t r i x f o r thes e t r i a n g l e s i s a v a i l a b l e ^ » 5 # ^he t r a n s f o r m a t i o n m a t r i c e s f o r t r a p e z o i d s as w e l l as t r i a n g l e s are g i v e n i n s e c t i o n 2.6. The same t h r e e v a l u e s of P o i s s o n ' s r a t i o 0 . 0 5 , 0 .3333 arid 0.1|_5 used i n Problem 1 are a l s o used h e r e . The s e l e c t i o n of f o r w h i c h a l l a r e a s a re p o s i t i v e i s made d i f f i c u l t because of the d i f f e r e n t g e o m e t r i c r a t i o s k and k^ i n the elements of d i f f e r e n t z o n e s . Nine d i f f e r e n t t r a p e z o i d s and one t r i a n g l e are used i n the problem w i t h /*• =0 .05 and 0 . 3 3 3 3 , and e l e v e n and one when JUL* o.ltf. T a b l e s U-17, k-18 and I4.-I9 g i v e the g e o m e t r i c r a t i o s and the a r e a p a rameters f o r the elements i n d i f f e r e n t zones. Tables [j.-20 t o l\.-3>h show d i f f e r e n t f u n c t i o n s c a l c u l a t e d by the e l a s t i c i t y f o r m u l a e (I4.-8) t o (14.-18). and the f i n i t e element method t o g e t h e r w i t h the p e r c e n t a g e e r r o r c a l c u l a t e d i n accordance w i t h eqn. (I4.-7). 1|.3 D i s c u s s i o n of the R e s u l t s . 1+.3.1 Problem of Deep R e c t a n g u l a r Beam The r e s u l t s o b t a i n e d from the s o l u t i o n of t h i s problem are t a b u l a t e d i n Tables L\-2 to I4.-I6• Each t a b l e c o n t a i n s the - 1 0 1 -0 0 1±] —1 Q C3 9 .Li £ a J U 3 - H o 0 C5 10 oo 4 c 6 .L. a In < j - j < < -U. C < 8 ~L X CO 6 K CO 6 bo OO o 6 I") to v3 Co 8 K o CO o p K 6 CO 10 6o OO 0 ST r0 In CS 6 O o 6o In o d 6 x CO o O 6 N 6o O (0 o CO ft 6 CO 03 O. 6\ XT 8" b o 6 X \9 Oo O 5 o oo o o G O O o v3 3 ft 6 o V9 in Vs. b N . CO Co 8 v3 o C O 6 o to CO o In N v3 IT d NT O ID cO o 0 V3 K O Co oo .1) o 6 o o v9 K o Q NS" O 6. 6 6 NT o <s a CO N N 6) O a) 5) O 6 CO 6^  bo N (0 A "bo" o bo O V3. O CM CO to" O <0 •V9 O 6 O - 1 0 3 -oo "st-i l l —j o Q 2 E O 1  0 j .£ v. q O <£ Ti to-' 5 ?• o •In < < 4-> < < \9 bo oo K Co OO. O O O . 6 to DO to Co O in K o o 10 r0 O K on 6 d o O K to o o O o 5o r 1 o d Co 6 5) o it e» • K b oo O K Oo 6 O O (0 o " S T O o o v3 o f0 o v5 i t C O o-CO 6 o K 0 0 bo 6 o o G O Oo o p o o Oo o OA 6, o 6 o -5T o • S) ro Oo \9 oo •CO 8 to CM O O 6 o o o in 5 in CO o <? x9 O "xsF OO fO ID £} o 6 lo" oo Jj) 6 O Oo O "cT' b V3 to o b o O O o 6 o 6i oo 6) K o 6 In o NT O 6 . O CO oo K O b "6Y s In 6 1 o 6 IS" Co 6 Oct O o 6 ••p 6 o o v9 K o o o 6 O 6 o •In \3 O 6 r0 CO to I- CO K b f.0 9^ O b 6 • in[|. UJ —! 4 - . C u o VP c <n £ C a , v . r-3 \0 U o .5 in ^ 1 a ' o ii 1! — <T til -O in CO 0 b CO IP to ^9 Oo CO c4 ol Oo Co CO sr CO 0) Co a. N to Oo < 6 Q 0 6 6 o o o 6 6 O 1 6 }^ 6) O CO S) o o S 6) O 0 K s oo 6 O0 6 ID Oo O In §? 6 o ID O CO 6 J-J \* • Co O <^  0 D3 • T OO CX) t0 O ro to <N In 0 0 bo o) 0« OO O io o < 6 O O O d 6 o 6 — 6 >* -s. co ff> 0 o K 10 o bo 0 $\ o o a o IS 0 In Oo fO CD O <• 6 V o o 0 6 6 o •6 d 6 O -P < o 0 6 Oo 0 6 ro 0 5 to "a 6 i e 0 6 I 6 CO 6 O NT o 6 O 1 0 6 c < cs 6 o 55 d o ID 6 •o CO •<f 6 r0 in a IP o 10 c0 1°) 6 ID in o cO in 6 rO S 3 K. ST 6 o to cX) 8 ' f t (0 0 v9 O rO o <S 6o 6 r0 o bo o a *> o o c?) fO 0 O 6 o * TOT v» o Co cO o oo N rO (?) fO 00 do 6\ K m In — — — — ——. — ft) o £ ! r\ < ^ 10 vS O o to 0 K b In O O •s\ rO O fO rO O £> O o 6 6 6 o O 6 6 d 6 O In H in NT s. ro f0 \ & OS v& o o 6 o 5 c 6 6 6 0 O IT \ o D to ? in N fO fO £} i f — 6 6 O 6 6 6 6 d 6 d> — to to 6) o — TABLE 4 - 2 0 RADIAL DISPLACEMENT U .AND % ERROR] AT DIFFERENT f AND 9 a 0.05 AS COEFFICIENT C ] IN u = C |j BY ELASTICITY AND FINITE ELEMENT PARAMETER 80 - 82.5° • ; ' 1 0.85R 0.72R 0.50R 0.3IR . [ 0 . 2 3 R 0.I3R FUNCTION %ERROR FUNCTION %ERROR FUNCTION %ERROR FUNCTION fo ERROR FUNCTION %ERROR FUNCTION %ERROR 0° ELASTICITY -1 .3423 - 0 . 9 3 7 9 - 0 . 5 4 8 2 . - 0 . 3 1 4 4 - 0 . 2 2 8 6 - 0 . 1 2 7 2 U J u.-BAR - 1 . 2 4 0 0 7 .62 - 0 . 9 0 8 8 3 . 1 0 - 0 . 5 4 1 0 1 .32 - 0 . 3 1 2 5 0 . 6 1 - 0 . 2 2 7 7 0 .42 - 0 . 1 2 6 6 0 . 4 5 N O - B A R -1 .2988 •3.17 - 0 . 9 2 1 0 1 . 8 0 - 0 . 5 4 4 4 0 . 6 7 - 0 . 3 1 1 1 1 .03 - 0 . 2 2 4 9 1.63 - 0 . 1 2 3 7 2 .68 . io° ELASTICITY - 0 . 9 3 9 1 - 0 . 7 9 1 2 - 0 . 5 0 7 6 - 0 . 2 9 8 3 - 0 . 2 1 8 0 - 0 . 1 2 1 7 U J u: BAR - 1 . 0 1 5 9 - 8 . 1 8 - 0 . 7 9 7 4 - 0 . 7 8 - 0 . 5 0 7 4 • 0 . 5 9 - 0 . 2 9 7 0 0 . 4 2 [ - 0 . 2 1 7 3 0 ' . 3 1 - 0 . 1 2 1 2 0 . 3 8 2 . 5 5 NO-BAR - 1 . 0 0 6 1 - 7 . 1 3 - 0 . 8 0 7 9 -2.11 . - 0 . 5 0 7 6 0 . 0 2 - 0 . 2 9 5 9 0 . 7 9 - 0 . 2 1 4 8 1 .46 ' - 0 . 1 1 8 6 20° ELASTICITY - 0 . 5 6 4 4 - 0 . 5 3 8 5 - 0 . 4 0 5 6 - 0 . 2 5 3 6 - 0 . 1 8 7 9 - 0 . 1 0 6 0 ui U J BAR - 0 . 5 4 4 8 3 .47 - 0 . 5 5 6 5 - 3 . 3 4 - 0 . 4 0 8 9 - 0 . 8 3 - 0 . 2 5 3 7 - 0 . 0 7 - 0 .T880 - 0 . 0 1 - 0 . 1 0 5 8 . 0.18' NO-BAR - 0 . 5 3 4 5 5 .29 - 0 . 5 5 2 4 - 2 . 5 7 . - 0 . 4 0 9 9 - 1 . 0 7 - 0 . 2 5 3 1 0 . 1 9 - 0 . 1 8 6 1 0 . 9 5 - 0 . 1 0 3 7 2.17 . 30° ELASTICITY - 0 . 3 2 8 8 ' - 0 . 3 2 7 2 - 0 . 2 7 9 6 - 0 . 1 8 9 5 - 0 . 1 4 3 4 - 0 . 0 8 2 2 U J U : BAR - 0 . 3 3 4 0 -1 .58 - 0 . 3 2 6 0 0 . 3 6 - 0 . 2 8 4 3 - 1 . 6 6 - 0 . 1 9 0 8 - 0 . 7 0 - 0 . 1 4 4 0 - 0 . 4 5 c - 0 . 0 8 2 4 - 0 . 1 8 NO-BAR - 0 . 3 2 4 3 1 .35 - 0 . 3 1 9 4 2 .38 - 0 . 2 8 3 8 - 1 . 4 9 - 0 . 1 9 0 7 - 0 . 6 3 - 0 . 1 4 3 2 0 . 1 3 - 0 . 0 8 1 0 1 .44 40° ELASTICITY - 0 . 1 6 3 1 . - 0 . 1 6 6 3 • - 0 . 1 5 7 5 - 0 . 1 1 7 1 - 0 . 0 9 1 1 - 0 . 0 5 3 5 U J u: BAR - 0 . 1 6 0 9 1 .37 - 0 . 1 6 7 7 - 0 . 8 3 - 0 . 1 6 0 3 - 1 . 7 6 - 0 . 1 1 8 7 - 1 . 4 3 - 0 . 0 9 2 1 • -1.11 - 0 . 0 5 3 9 - 0 . 7 9 NO -BAR - 0 . 1 5 8 3 2.97 - 0 . 1 6 3 2 1 .84 - 0 . 1 5 9 1 - 1 . 0 1 - 0 . 1 1 9 0 - 1 . 6 7 -Q- -0921 - 1 . 1 6 - 0 . 0 5 3 4 0 . 0 6 50° ELASTICITY - 0 . 0 4 0 6 - 0 . 0 4 4 6 - 0 . 0 5 2 5 - 0 . 0 4 6 4 - 0 . 0 3 8 0 - 0 . 0 2 3 3 uJ u_-BAR - 0 . 0 4 0 6 ' 0 . 0 0 -0 .044.4 0 . 4 5 •. - 0 . 0 5 3 8 - 2 . 4 2 - 0 . 0 4 7 8 - 3 . 0 5 - 0 . . 0 3 9 0 - 2 . 6 7 , - 0 . 0 2 3 9 - 2 . 4 3 N O - B A R - 0 . 0 3 9 4 3 .16 - 0 . 0 4 2 6 4 ; 5 5 . - 0 . 0 5 2 4 0 . 1 5 - 0 . 0 4 8 3 - 4 . 0 9 -0 . '0398 - 4 . 5 9 - 0 . 0 2 4 3 - 4 . 0 9 . 6 0 ° ELASTICITY 0 .0494 0.0455 0 .0306 0 .0149 0 .0096 .0.0046 ui u: BAR 0 . 0 4 9 8 - 0 . 8 1 0 .0453 0 .37 0 .0299 2 . 3 2 0 .0140 6 .47 0 .0088 8 . 3 3 0 .0040 1 2 . 5 5 N O - B A R 0 .0493. . 0 . 2 5 0.0457 - 0 . 4 6 0 .0306 0 . 0 0 0 . 0 1 3 4 . 1 0 . 4 9 0 . 0 0 7 7 20 .24 0 .0029 3 6 . 5 0 . o 70 ELASTICITY 0 .1118" 0 .1081 4) . 0903 0 .0618 0.0470 0 .0270 U J u: BAR • 0 .1118 0 .00 0.1079 0 .14 0 .0899 0 . 5 0 . 0 .0612 0 . 9 6 0 .0465 1 .13 0 .0265 1 . 8 5 N O - B A R 0 . 1 1 0 5 1 .19 0 .1072 0 . 8 2 0 .0899 0 . 4 3 0 .0605 2 . 0 5 CO.0451 4 . 0 5 0 .0249 7 .84 80° ELASTICITY 0 .1486 O V 1 4 5 0 0.1262 0 . 0 9 1 1 0.0707 0 .0416 ui BAR 0 .1486 0 . 0 0 ' 0.1448 0 .17 0 .1258 0 . 3 1 0.0907* 0 . 4 1 0 .0704 0 . 4 5 . 0 .0412 0 . 9 9 N O - B A R 0 .1466 1 .33 0 .1435 , 1 .10 0 .1254 , 0 . 6 2 0 .0899 1 .24 0 . 0 6 8 9 2 .55 0 .0392 5 .64 90° ELASTICITY 0 .1608 0.1573 0 .1382 0 .1010 0 .0788 0 .0466 U J U . ' BAR 0 .1607 0 . 0 5 0 .1570 0 .17 0 .1378 0 . 2 7 . 0 .1007 0 . 3 0 . 0 .0786 0 .33 0 .0462 0 . 8 1 N O - B A R 0 . 1 5 8 6 1.37:. 0 .1555 1.15 - 0.1372 f , 0 . 6 8 0 . 0 9 9 9 - 1 .08 0 . 0 7 7 1 2.24 0 .0442 , 5 . 1 7 ' -TABLE .4-21 TANGENTIAL DISPLACEMENT V AND % . ERROR] AT DIFFERENT r AND 9 fj./• 0.05. 'AS • COEFFICIENT C i IN v =C ^ BY ELASTICITY AND FINITE ELEMENT . PARAMETER 90 - 82.5 V 0 . 8 5 R 0.72R i 0 . 5 0 R 0.31R 0 . 2 3 R 0 . I3R FUNCTION: %ERROR FUNCTION %ERROR FUNCTION %ERROR FUNCTION %ERROR FUNCTION % ERROR. FUNCTION %ERROR ELASTICITY 0 .0000 0.0000 .0.0000 0.0000 0.0000 0 .0000 0° L U BAR 0 .0000 • 0.0000 0 .0000 0.0000 0.0000 0.0000 . U J N O - B A R 0.0000 0.0000 0 .0000 0.0000 0.0000 0 .0000 ELASTICITY 0 .1562 0.1787 0 .1224 0.0728 . 0 . 0 5 3 4 0.029.9 10° U J BAR 0 .1895 - 2 1 . 3 2 0.1748 2.17 0.1195. . 2 .37 0.0720 1 .14 0.0530 0 .78 0.0296 0 . 9 0 U J NO-BAR 0 .2246 - 4 3 . 7 6 0.1727 3 .34 0 .1192 2 .58 0.0713 2 . 1 5 : 0 .0520 2 .53 0 .0286 4 .07 ELASTICITY 0 .1593 0.2415 . 0.2097 0.1333 0.0990 0.0559 20° ui BAR 0 .1600 - 0 . 4 5 . 0 .2515 - 4 . 1 4 0 .2083 0 .69 0 .1323 0 . 8 0 0 .0983 0 .64 0 .0555 ... 0 .74' U J NO-BAR 0 .1468 7.83 0.2627 - 8 . 7 8 0.2087 0 . 4 6 0.1312 1 .60 0 .0968 2 . 2 1 0 .0538 • 3 . 8 4 . 30° ELASTICITY 0 .1620 • 0.2463 0 .2512 0.1729 0.1307 0.0749 BAR 0.1610. 0 . 6 1 ' 0.2487 ^ 0 . 9 9 0 .2524 - 0 . 4 7 0 .1722 0 . 4 0 0 .1301 0 . 4 5 0 .0745 0 . 5 0 U.' NO-BAR 0 .1538 , 5 .09 0 . 2 4 6 1 ' 0 . 0 8 0 .2528 - 0 . 6 2 0 .1712 0 . 9 7 0.1284 1.76 0 .0722 3 . 5 0 ELASTICITY 0.1587 0 . 2 2 9 1 0 .2543 0 .1882 .. 0 .1452 0 .0845 40° ui BAR 0 .1559 '• 1.73 0.2299 - 0 . 3 4 0 .2562 - 0 . 7 6 0 .1880 0 . 0 8 ' ,0.1448 0 . 2 8 0 .0843 0 . 2 1 U J NO-BAR 0.1519 4 .25 0.2257 1.46 0 .2565 ' - 0 . 8 8 0.1874 0 . 4 2 0 .1433 1 . 3 1 0 .0819 3 . 0 8 ELASTICITY 0 .1444 0.1989 0 .2299 0.1802 0.1417 0.0838 50° ui % BAR 0.1420 1.66 0.1984 0 . 2 2 0 .2314 - 0 . 6 7 0 .1805 - 0 . 1 3 0 .1415 0 . 1 5 0 .0839 - 0 . 0 8 U J N O - B A R 0 .1392 4 3.62 0 .1950 1.97 0 .2305 - 0 . 2 7 0 .1801 0 . 0 8 . 0.1404 0 . 9 1 0.0816. 2 .65 . ELASTICITY 0 .1192 0.1586 0.1867 0.1526 0.1219 0 .0732 60° ui BAR 0 .1172 1 . 6 1 0 .1579 0 . 4 1 • 0.1877 - 0 . 5 6 0 .1530 - 0 . 2 3 0 .1219 0 . 0 0 0 .0735 - 0 . 3 6 • U J N O - B A R 0 .1153 3 . 2 5 0 .1552 2 .10 0 .1863 0 . 1 9 0.1527 - 0 . 0 8 0 .1212 0 . 6 1 0 .0715 2 .25 ELASTICITY 0 .0848 .. 0 .1103 0 ,1309 0 . 1 1 0 1 0 .0890 0 .0540 o 70 ui BAR • 0 .0835 . 1 .51 0.1097 0 . 5 2 0 .1315 - 0 . 4 7 0 .1103 - 0 . 2 8 0 .0890 0 . 0 0 0 .0543 - 0 . 5 7 U J N O - B A R 0 .0822 3 . 0 3 0.1079 2.14 0 .1303 0 . 4 9 0 .1102 - 0 . 1 1 0 .0886 0 .43 0 .0530 1.93 ELASTICITY 0 .0440 0.0566 0 .0673 0 .0575 0 .0468 0 .0286 80° ui BAR 0 .0434 1-.A6 ' 0.0563 0 . 5 6 0 .0676 - 0 . 4 1 0.0577 - 0 . 2 9 0 .0468 0 . 0 0 0 .0288 - 0 . 7 1 IN- N O - B A R 0.0427 2 .92 0.0554 2 . 1 5 0 .0669 0 . 6 5 0.0576 - 0 . 1 0 I 0 . 0 4 6 7 . 0 . 3 3 0 . 0 2 8 1 1 .73 ELASTICITY 0 .0000 0.0000 . 0 .0000 ... 0 .0000 0 .0000 0 .0000 90° ui BAR 0 .0000. 0 .0000 . 0 .0000 0 .0000 0 .0000 0 .0000 u. N O - B A R 0 .0000 0.0000 0 .0000 0 .0000 0 .0000 0 .0000 TABLE 4 - 2 2 RADIAL NORMAL STRESS V(TR AND % E R R O R ? AT D I F F E R E N T X AND 6 (j. * 0 - 0 5 AS C O E F F I C I E N T C I N =C-£- BY E L A S T I C I T Y A N D F I N I T E E L E M E N T K t • ( P A R A M E T E R 60- 8 2 , 5 V 0 . 8 5 R 0 . 7 2 R 0 . 5 0 R : 0 . 3 I R 0 . 2 3 R 0 . I 3 R F U N C T I O N • % E R R O R F U N C T I O N % E R R O R F U N C T I O N % E R R O R F U N C T I O N % E R R O R F U N C T I O N % E R R O R F U N C T I O N ' % E R R O R 0° E L A S T I C I T Y - 4 . 2 6 9 9 - 2 . 3 2 5 5 - 1 . 3 7 9 3 - 1 . 0 9 0 3 . - 1 . 0 2 6 0 . — 0 . 9 7 6 8 . ' LU B A R - 2 . 5 7 9 8 39 .58 - 1 . 8 8 2 5 1 9 . 0 5 - 1 . 2 8 5 0 6 .84 - 1 . 0 6 0 9 2 .69 ! - 1 . 0 0 9 0 1.66 - 0 . 9 7 0 0 - 0 . 9 4 5 0 0 .69 N O - B A R - 2 . 8 8 9 3 32 .33 - 1 . 9 0 8 7 1 7 . 9 2 - 1 . 2 9 4 1 6 . 1 8 - 1 . 0 5 5 6 . 3 . 1 8 - 0 . 9 9 6 8 2 . 8 5 . - 2 .56 10° E L A S T I C I T Y - 0 . 8 3 0 0 - 1 . 3 4 0 0 - 1 . 1 8 1 1 - 1 . 0 1 4 3 . '• - 0 . 9 6 9 4 - 0 . 9 3 3 2 LU IN-B A R - 1 . 6 4 1 8 - 9 7 . 8 2 - 1 . 4 2 2 0 - 6 . 1 1 - 1 . 1 4 4 9 3 . 0 6 - 0 . 9 9 5 1 1 .89 . - 0 . 9 5 7 0 1.28 - 0 . 9 3 0 0 0 .34 N O - B A R - 1 . 5 4 9 4 - 8 6 . 6 8 - 1 . 4 6 7 0 - 9 . 4 7 - 1 . 1 5 4 6 . 2 .24 - 0 . 9 9 2 1 2 . 1 9 - 0 . 9 4 7 0 2 . 3 1 - 0 . 9 0 3 0 3 . 2 5 20° E L A S T I C I T Y - 0 . 0 8 0 4 - 0 . 4 0 2 2 . - 0 . 7 6 3 8 - 0 . 8 1 5 7 - 0 . 8 ] 4 4 - 0 . 8 0 9 4 LU LU B A R - 0 . 0 0 3 0 96 . 25 - 0 . 5 8 4 8 - 4 5 . 4 0 - 0 . 8 0 5 2 - 5 . 4 2 - 0 . 8 1 7 5 - 0 . 2 1 - 0 . 8 1 2 5 0 .24 - 0 . 8 0 7 0 _. 0.'30 N O - B A R . 0.0646 180 .41 - 0 . 5 6 1 2 - 3 2 . 5 4 - 0 . 8 0 9 1 - 5 . 9 2 - 0 . 8 1 8 7 - 0 . 3 7 - 0 . 8 0 8 3 0 .75 - 0 . 7 9 2 0 0 . 9 1 30° E L A S T I C I T Y - 0 . 0 0 1 3 - 0 . 0 9 1 1 - 0 . 3 8 9 9 - 0 . 5 6 1 0 • - 0 . 5 9 8 5 - 0 . 6 2 5 4 LU IN-B A R - 0 . 0 0 1 5 . - 2 1 . 8 8 - 0 . 0 7 0 8 2 2 . 3 0 - 0 . 4 3 5 6 - 1 1 . 7 3 - 0 . 5 7 7 2 - 2 . 8 9 - 0 . 6 0 5 9 - 1 . 2 3 . - 0 . 6 2 6 0 0 . 0 9 N O - B A R 0.0062 591.29 - 0 . 0 4 0 0 5 6 . 0 8 - 0 . 4 3 5 7 - 1 1 . 7 6 - 0 . 5 8 2 2 - 3 . 7 9 - 0 . 6 0 7 9 - 1 . 5 6 - 0 . 6 3 0 0 • - 0 . 7 4 40° E L A S T I C I T Y 0.0094 - 0 . 0 0 2 6 - 0 . 1 4 7 6 - 0 . 3 1 2 9 - 0 . 3 6 5 0 - 0 . 4 0 8 5 u i I N B A R 0.0218 - 1 3 1 . 3 9 0 .0011 140.73 - 0 . 1 6 4 5 - 1 1 . 4 2 - 0 . 3 3 0 3 - 5 . 5 6 - 0 . 3 7 5 7 - 2 . 9 6 - 0 . 4 1 5 0 . - 1 . 5 9 N O - B A R 0 . 0 1 3 1 - 3 9 . 2 3 0.0094 462 .68 - 0 . 1 6 1 6 - 9 . 4 8 - 0 . 3 3 6 1 - 7 . 3 9 - 0 . 3 8 1 5 - 4 . 5 4 - 0 . 4 1 4 0 - 1 . 3 4 50° E L A S T I C I T Y 0.0102 0.0230 - 0 . 0 1 2 2 - 0 . 1 0 7 9 - 0 . 1 4 9 0 - 0 . 1 8 7 9 u i IN-B A R 0.0136' - 3 2 . 6 5 0.0317 - 3 7 . 8 5 - 0 . 0 1 2 1 1 . 0 1 - 0 . 1 1 8 3 - 9 . 5 9 - 0 . 1 5 7 5 - 5 . 7 5 - 0 . 1 9 4 0 - 3 . 2 6 N O - B A R 0.0112 - 9 . 7 3 0.0290 - 2 6 . 5 1 - 0 . 0 0 6 0 5 1 . 0 6 - 0 . 1 2 1 5 - 1 2 . 5 1 - 0 . 1 6 3 7 - 9 . 9 0 - 0 . 1 9 8 5 - 5 . 6 - 5 60° E L A S T I C I T Y 0.0096 0.0302 0 .0585 0 .0429 0 .0278 0 . 0 1 0 1 u i L N B A R 0.0120 - 2 4 . 9 3 0.0344 - 1 4 . 0 2 0 .0630 - 7 . 7 2 0 .0406 5 . 3 3 0 . 0 2 4 1 13.47 0 .0079 3 1 . 7 0 N O - B A R 0.0100 - 4 . 7 4 0.0328 - 8 . 5 3 0 .0662 - 1 3 . 2 7 0 .0406 5 . 3 3 0.0203 27.09 - 0 . 0 0 1 4 114 .00 o 70 E L A S T I C I T Y 0.0087 . 0 .0319 0 .0938 0 .1426 0.1557 0 .1654 u i I N B A R 0.0100 - 1 2 . 7 1 0.0350 - 9 , . 81 0 .0999 - 6 . 5 4 ' 0 . 1 4 6 0 - 2 . 3 9 0 .1566 - 0 . 5 4 0 .1670 - 0 . 9 7 N O - B A R 0 . 0 0 9 1 - 2 . 6 2 0.0333 - 4 . 6 2 0 .1009 - 7 . 5 6 0 .1484 - 4 . 0 6 0 .1558 - 0 . 0 5 0 .1555 6 . 0 0 80° E L A S T I C I T Y 0.0084 0 .0321 0 .1099 0 ,1985 0 .2322 0 .2638 u i IN-B A R 0.0092 - 9 . 6 6 0.0343 - 6 . 8 7 0 .1162 ' - 5 . 7 1 0 .2050 - 3 . 2 6 0 .2360 - 1 . 6 3 0.2695- 2.19 N O - B A R 0.0086 - 2 . 3 1 0 .0331 - 3 . 2 2 0 . 1 1 6 1 - 5 . 6 1 0 .2083 - 4 . 9 1 0 .2376 - 2 . 3 3 0 .2560 2.34 90° E L A S T I C I T Y 0.0083 0.0320 0 .1146 0 .2165 0 .2576 0 .2975 u i I N B A R 0.0089 '.' - 8 . 1 4 . 0 .0340 - 6 . 3 3 0 .1209 - 5 . 5 0 0 .2239 - 3 . 4 4 0 .2623 -o l .86 0 .2960 0 . 5 1 N O - B A R 0.0084 - 2 . 2 3 0 .0330 -2.91 0.1204 - 5 . 1 0 0 .2274 - 5 . 0 4 0 .2648 - 2 . 8 0 0 .2910 2 .18 CIRCULAR P L A T E WITH TWO POI N T . ; ; LOADS;^ :<;K. . TABLE A - 2 3 TANGENTIAL NORMAL STRESS AND % E R R O R AT D I F F E R E N T T AND 9 p * 0 . 0 5 AS C O E F F I C I E N T C I N ° e = C W T BY E L A S T I C I T Y A N D F IN ITE E L E M E N T ,; P A R A M E T E R 90" 8 2 . 5 ° i 0 . 8 5 R 0 . 7 2 R 0 . 5 0 R 0 . 3 1 R 0 . 2 3 R . O . I 3 R F U N C T I O N % E R R O R F U N C T I O N % E R R O R F U N C T I O N % E R R O R F U N C T I O N % E R R O R F U N C T I O N % E R R O R F U N C T I O N % E R R O R E L A S T I C I T Y 0.3183 0.3183 0 .3183 0.3183 0.3183 0.3183 0 ° ui B A R 0 . 5 5 6 1 - 7 4 . 7 0 0.2459 22.74 0 .2995 5 .92 0 .3105 2 .46 0.3129 1.69 0 .3100 " 2 . 5 1 IN-N O - B A R 0.1916 ' 3 9 . 8 1 0.3880 - 2 1 . 9 1 0 .3189 - 0 . 2 0 0.3204 - 0 . 6 7 0 .3243 - 1 . 8 8 0 .3135 1 . 5 1 E L A S T I C I T Y - 1 . 0 2 3 8 - 0 . 2 4 0 9 0 .1740 0 .2531 0.2667 0.276,0 10° ui B A R - 1 . 3 4 6 5 - 3 1 . 5 1 - 0 . 1 2 5 2 4 8 . 0 2 0 .1763 - 1 . 2 8 0 .2485 1 . 8 1 0.2624 1.59 0 .2705 2 .00 IN-N O - B A R - 1 . 3 5 6 9 - 3 2 . 5 3 - 0 . 1 8 7 6 22.14 0 .1866 - 7 . 2 3 0 .2572 - 1 . 6 2 0.2729 - 2 . 3 5 0 .2730 1.09 E L A S T I C I T Y - 0 . 6 5 2 7 , , - 0 . 5 7 8 9 - 0 . 1 1 4 5 0 .0840 0.1257 0.1562 ' 20° ui B A R - 0 . 6 0 4 3 7.42 - 0 . 6 8 6 3 - 1 8 . 5 5 - 0 . 0 9 9 6 13 .06 0.0849 - 1 . 0 6 0.1237 1.59 0 .1555 0 . 4 5 IN-N O - B A R - 0 . 5 0 5 0 22.63 - 0 . 7 4 0 0 - 2 7 . 8 3 - 0 . 0 9 4 9 1 7 . 1 5 0.0909 - 8 . 1 8 • 0 .1313 - 4 . 4 4 0 .1582 1.28 3 0 ° E L A S T I C I T Y - 0 . 3 7 9 3 ' - 0 . 5 2 9 4 - 0 . 3 4 4 7 - 0 . 1 2 9 8 > - 0 . 0 6 9 8 - 0 . 0 2 1 8 U I B A R - 0 . 3 7 5 8 . 0 .92 - 0 . 5 3 2 2 - 0 . 5 3 - 0 . 3 4 6 1 - 0 . 4 2 - 0 . 1 2 7 2 i ; 9 5 - 0 . 0 7 0 5 - 0 . 9 6 - 0 . 0 1 7 3 2 2 . 5 0 . U J N O - B A R - 0 . 3 1 6 5 16 .55 - 0 . 5 2 2 1 1.37 - 0 . 3 4 8 8 - 1 . 1 9 - 0 . 1 2 5 3 3^45 - 0 . 0 6 7 6 3 . 1 2 - 0 . 0 1 4 8 4 7 . 3 0 E L A S T I C I T Y - 0 . 2 4 8 8 - 0 . 4 2 5 0 - 0 . 4 6 4 1 - 0 . 3 3 3 4 - 0 , 2 7 9 9 - 0 . 2 3 1 5 4 0 ° ui B A R - 0 . 2 3 6 9 4 .80 - 0 . 4 2 9 3 - 1 . 0 2 - 0 . 4 7 4 9 - 2 . 3 3 - 0 . 3 3 3 1 . 0 . 0 9 - 0 . 2 8 0 7 - 0 : 2 9 - 0 . 2 2 4 5 . 3 .04 u,- N O - B A R - 0 . 2 1 4 2 13 .90 - 0 . 4 0 6 3 4 . 4 0 - 0 . 4 8 2 0 - 3 .'86 - 0 . 3 3 5 1 ' - 0 . 4 9 - 0 . 2 8 3 7 r l . 3 6 - 0 . 2 2 3 5 3 .47 5 0 ° E L A S T I C I T Y - 0 . 1 8 1 9 - 0 . 3 4 4 3 - 0 . 5 0 7 1 - 0 . 4 9 7 0 - 0 . 4 7 2 7 - 0 . 4 4 4 6 ui B A R - 0 . 1 7 4 5 4 .04 - 0 . 3 4 0 4 1.14 - 0 . 5 1 3 3 - 1 . 2 4 - 0 . 4 9 9 1 - 0 . 4 2 - 0 . 4 7 4 1 - 0 . 3 1 - 0 . 4 4 0 0 ' 1 .04 IN-N O - B A R - 0 . 1 6 1 4 11.28 - 0 . 3 2 6 2 5 .26 - 0 . 5 1 3 1 - 1 . 1 9 - 0 . 5 0 3 0 - 1 . 2 1 • z 0 . 4 8 2 8 - 2 . 1 4 - 0 . 4 4 0 0 . 1 . 0 4 -E L A S T I C I T Y - 0 . 1 4 5 1 - 0 . 2 9 0 7 - 0 . 5 1 3 2 - 0 . 6 1 3 5 - 0 . 6 2 9 2 - 0 . 6 3 5 8 6 0 ° ui B A R - 0 . 1 3 8 6 c ' 4 .53 - 0 . 2 8 6 6 1 . 4 1 J - 0 . 5 1 6 8 - 0 . 6 8 - 0 . 6 1 6 3 - 0 . 4 6 - 0 . 6 3 0 7 - 0 . 2 5 - 0 . 6 3 5 0 0 . 1 3 IN-N O - B A R - 0 . 1 3 1 2 9 .60 - 0 . 2 7 6 4 . 4 . 9 1 - 0 . 5 1 1 6 0 . 3 1 - 0 . 6 2 0 1 - 1 . 0 7 - 0 . 6 4 3 6 - 2 . 2 9 - 0 . 6 4 0 0 - 0 . 6 5 o 7 0 E L A S T I C I T Y - 0 . 1 2 4 6 , - 0 . 2 5 7 5 - 0 . 5 0 6 7 - 0 . 6 8 8 0 - 0 . 7 4 1 3 - 0 . 7 8 5 6 ui B A R - 0 . 1 1 9 2 4 .34 - 0 . 2 5 3 0 1.77 - 0 . 5 0 7 8 - 0 . 2 2 - 0 . 6 9 0 5 - 0 . 3 6 - 0 . 7 4 2 5 - 0 . 1 5 - 0 . 7 9 1 0 - 0 . 6 9 IN-N O - B A R - 0 . 1 1 4 1 8 .43 - 0 . 2 4 5 9 . 4 . 5 1 - 0 . 5 0 0 8 1 .16 - 0 . 6 9 2 4 - 0 . 6 4 - 0 . 7 5 7 7 - 2 . 1 9 - 0 . 7 9 9 0 - 1 . 7 1 E L A S T I C I T Y - 0 . 1 1 4 1 - 0 . 2 3 9 6 - 0 . 4 9 9 4 - 0 . 7 2 8 7 - 0 . 8 0 8 2 - 0 . 8 8 0 5 8 0 ° ui B A R - 0 . 1 0 9 2 4 . 3 1 - 0 . 2 3 5 2 1.84 - 0 : 4 9 9 4 0 . 0 0 - 0 . 7 3 0 7 - 0 . 2 8 - 0 . 8 0 8 7 - 0 . 0 6 - 0 . 8 9 1 0 ' - 1 . 1 9 U : N O - B A R - 0 . 1 0 5 2 7.76 - 0 . 2 2 9 5 4 . 2 1 - 0 . 4 9 1 7 1 , 5 5 - - 0 . 7 3 0 8 - 0 . 3 0 . - 0 . 8 2 4 7 - 2 . 0 4 - 0 . 9 0 1 5 - 2 . 3 9 E L A S T I C I T Y - 0 . 1 1 0 8 - 0 . 2 3 3 9 . - 0 . 4 9 6 6 - 0 . 7 4 1 4 - 0 . 8 3 0 2 - 0 . 9 1 3 0 : 9 0 ° ui B A R - 0 . 1 0 6 1 4.27 - 0 . 2 2 9 6 1.87 - 0 . 4 9 6 2 0 . 0 8 - 0 . 7 4 3 3 - 0 . 2 4 \ - 0 . 8 3 0 5 - 0 . 0 3 - 0 . 9 2 5 0 - 1 . 3 2 • u.-N O - B A R - 0 . 1 0 2 5 7 .55 - 0 . 2 2 4 3 4 . 1 1 - 0 . 4 8 8 3 1 .66 - 0 . 7 4 2 8 • - 0 . 1 8 - 0 . 8 4 6 7 - 1 . 9 8 . - 0 . 9 3 5 0 . - 2 . 4 1 . CIRCULAR PLATE - WITH TWO POINT LOADS ' T A B L E 4 - 2 4 SHEAR STRESS X r e AND % E R R O R AT D I F F E R E N T T A N D 9 LL•• 0 . 0 5 AS C O E F F I C I E N T C IN T R E = C ^ - BY E L A S T I C I T Y A N D F I N I T E E L E M E N T j P A R A M E T E R 9Q = 8 2 . 5 ° V 0 . 8 5 R 0 . 7 2 R 0 . 5 0 R 0 . 3 I R 0 . 2 3 R 0 . I 3 R F U N C T I O N % E R R O R F U N C T I O N % E R R O R F U N C T I O N % E R R O R F U N C T I O N % E R R O R F U N C T I O N % E R R O R F U N C T I O N % E R R O R E L A S T I C I T Y 0.0000 0.0000 • 0 .0000 0 .0000 0.0000 0 .0000 0 ° U J B A R 0.0000 0.0000 0 .0000 0 .0000 0.0000 0.0000 U J N O - B A R .0 .0000 0.0000 0 .0000 0 .0000 '' 0 .0000 0.0000 E L A S T I C I T Y 1.0.718 0.8843 0 .4359 0 .2840 0 .2520 0 . 2 2 8 1 10° ui B A R 1.2577 - 1 7 . 3 3 0.7754 1 2 . 3 2 0 .3995 8 .34 0 . 2 7 5 1 3 . 1 2 . 0 .2469 2 .00 0 .2250 1.36 IN- N O - B A R 1.2740 - 1 8 . 8 6 0.7779 1 2 . 0 4 0 .3909 1 0 . 3 2 0 .2700 4 . 9 7 0 . 2 4 5 1 2.74 0 .2150 5 .74 E L A S T I C I T Y 0 . 3 1 5 1 0 .6401 0 .6232 0 .4932 0.4549 0 .4236 20° ui B A R 0.2258 28.33 0 .7221 - 1 2 . 8 2 0 .6099 2 .13 0 . 4 8 3 1 2 .04 0 .4476 1.60 0 .4185 ... 1 . 2 1 U J N O - B A R 0.2070 34 .30 0 .7441 - 1 6 . 2 5 0 .6032 3 . 2 1 0 .4770 3 . 2 9 0 .4462 1 . 9 1 0 .4015 5 . 2 3 E L - A S T I C I T Y 0.1144 0.3449 0 .5872 0 . 5 9 2 1 0 . 5 7 7 1 0 .5603 3 0 ° U I B A R 0 . 1 3 5 1 . - 1 8 . 1 3 0.3279 4 . 9 1 0 .5989 - 1 . 9 9 0 .5867 0 . 9 2 0 .5705 1.13 0 .5540 1.14 IN- N O - B A R 0.1102 3 .69 0.3138 9 . 0 1 0 .5969 - 1 . 6 5 0 .5836 1 .45 0 .5724 0 . 8 0 0 .5450 . .2.74 E L A S T I C I T Y 0 . 0 5 1 5 0.1863 0 . 4 6 3 1 0 .5896 0 .6110 0 . 6 2 3 1 4 0 ° ui B A R 0.0438 14 .93 0.1903 - 2 . 1 6 0 .4722 - 1 . 9 6 0 .5882 6 . 2 4 0 . 6 0 6 1 0 .80 0.-6185 0 . 7 3 u- N O - B A R 0.0430 1 6 . 5 3 0.1754 . 5 . 8 5 0 .4749 - 2 . 5 5 0 .5899 - 0 . 0 5 0 .6124 - 0 . 2 3 O. :6000 3 . 7 1 E L A S T I C I T Y 0.0265 0.1052 0.3337 0 .5165 0.5678 0.6087 5 0 ° ui B A R 0.0272 - 2 . 6 9 0.0126 2 .47 0 . 3 3 6 1 - 0 . 7 1 0 .5160 ; 0 . 1 0 0 .5640 0 . 6 6 0 .6055 0 . 5 2 IN- N O - B A R 0.0260 1.68 0 .0968- j 7 .94 0 .3332 0 . 1 5 0 .5202 - 0 . 7 2 0 .5737 - 1 . 0 3 0 .5885 3 . 3 2 . - E L A S T I C I T Y - 0.0145 0.0608 0 .2250 0 . 4 0 4 1 0 .4672 0.5237 6 0 ° ui B A R 0.0134 7.57 0.0604 0 . 6 8 0 .2258 - 0 . 3 6 0 .4030 0 .27 0 .4639 0 .70 0 .5210 0 . 5 2 U J N O - B A R 0.0138 4 .94 0.0570 . 6 . 1 6 • 0 .2222 1.27 0 .4075 - 0 . 8 3 0 .4740 - 1 . 4 6 0 .5110 . 2 . 4 5 E L A S T I C I T Y 0.0079 , 0.0339 0 .1377 0 .2740 0 .3290 0 .3819 7 0 ° U I B A R 0.0077 . i 2 .20 0.0332 1.97 0 .1377 0 . 0 0 0 .2727 0 . 4 7 " 0 .3263 0 . 8 1 0 .3810 0 . 2 3 U J N O - B A R 0.0078 1 .15 0.0318 6 . 2 1 0 .1348 2 .07 0 .2756 - 0 . 5 8 0 . 3 3 4 4 - 1 . 6 4 0 .3750 1 . 8 1 E L A S T I C I T Y 0.0035 . 0 .0152 0 .0652 0.1377 0 .1693 0 .2009 8 0 ° ui B A R . 0.0033 ' 5 . 0 0 0.0150 1 .52 0 . 0 6 5 1 0 . 1 0 0 .1369 0'.60 0 .1677 0 . 9 0 .0.2005 - 0 . 2 0 IN-N O - B A R 0.0034 2 . 1 1 0.0144 5 .64 0 .0637 2 .34 0 .1383 - 0 . 3 8 0 . 1 7 2 1 - 1 . 6 7 0 .1972 1 .85 E L A S T I C I T Y 0.0000 0.0000 0 .0000 0 . 0 0 0 0 0 .0000 0 .0000 9 0 ° ui B A R 0.0000 0.0000 0 . 0 0 0 0 : 0 . 0 0 0 0 0 .0000 0.0000 u.- N O - B A R O'.OOOO 0.0000 0 . 0 0 0 0 0 .0000 . . . . •-• 0.0000 0: 0.000 ;• VI' • • -. • • . • < , - • . • ; • . ' • \ TABLE 4 - 2 5 RADIAL DISPLACEMENT U . AND % ERROR 1 AT DIFFERENT f AND 8 u.''• 0.3333 AS COEFFICIENT C'' p ~ • f IN u = C ^ BY ELASTICITY AND FINITE ELEMENT i PARAMETER 80" 1 0 0 ° ' A 0.85 R 0.72R ;0.50R 0.31 R : 0 . 2 3 R O . I 3 R F U N C T I O N % E R R O R F U N C T I O N %ERROR F U N C T I O N %ERROR F U N C T I O N %ERROR F U N C T I O N % E R R O R F U N C T I O N % E R R O R E L A S T I C I T Y - 1 . 4 1 9 0 - 1 . 0 0 2 8 - 0 . 5 9 3 3 - 0 . 3 4 2 3 - 0 . 2 4 9 3 - 0 . 1 3 8 9 0° UJ B A R - 1 . 3 4 7 0 5.07 - 0 . 9 7 5 6 2 .72 -0.585*6 1 .29 - 0 . 3 3 9 5 0 . 8 0 - 0 . 2 4 7 6 0 . 7 2 - 0 . 1 3 8 0 0 . 6 6 IN- N O - B A R - 1 . 4 1 1 0 . 0 . 5 5 - 0 . 9 9 0 1 1.27 - 0 . 5 9 0 4 0 . 4 8 - 0 . 3 3 9 4 0 . 8 7 - 0 . 2 4 5 7 1.46 - 0 . 1 3 5 5 2 . 4 1 E L A S T I C I T Y - 0 . 9 5 4 5 - 0 . 8 2 8 4 - 0 . 5 4 3 4 - 0 . 3 2 2 1 - 0 . 2 3 5 9 - 0 . 1 3 1 9 io° ui B A R - 1 . 0 1 5 2 - 6 . 3 6 - 0 . 8 3 8 7 - 1 . 2 4 . - 0 . 5 4 0 6 0 . 5 2 - 0 . 3 2 0 3 0 . 5 6 - 0 . 2 3 4 6 0 . 5 5 - 0 . 1 3 1 2 0 .57 IN- N O - B A R - 1 . 0 0 8 3 - 5 . 6 4 - 0 . 8 4 8 9 . - 2 . 4 8 - 0 . 5 4 4 2 - 0 . 1 4 - 0 . 3 2 0 0 0 . 6 5 : - 0 . 2 3 2 8 . 1 . 3 1 - 0 . 1 2 8 9 2 . 3 1 E L A S T I C I T Y - 0 . 5 3 1 9 - 0 . 5 3 0 3 - 0 . 4 1 8 1 - 0 . 2 6 6 1 - 0 . 1 9 8 0 . - 0 . 1 1 2 1 20° ui B A R - 0 . 5 0 4 7 5 . 1 1 - 0 . 5 4 1 4 - 2 . 1 2 - 0 . 4 2 1 8 - 0 . 8 9 - 0 . 2 6 6 3 - 0 . 0 7 • - 0 . 1 9 7 9 0 . 0 6 . - 0 . 1 1 1 8 0 . 2 6 IN- N O - B A R - 0 . 4 9 6 4 6.67 - 0 . 5 4 0 4 - 1 . 9 1 - 0 . 4 2 2 6 - 1 . 0 9 ' - 0 . 2 6 5 8 0 . 1 1 . i - 0 . 1 9 6 4 . 0 . 8 5 - 0 . 1 0 9 8 2 . 0 1 30° E L A S T I C I T Y - 0 . 2 6 6 3 - 0 . 2 8 2 1 - 0 . 2 6 3 5 - 0 . 1 8 5 7 - 0 . 1 4 1 9 - 0 ^ 0 8 2 0 Ui B A R - 0 . 2 7 3 4 . - 2 . 6 8 - 0 . 2 7 8 3 1 .38 - 0 . 2 6 7 4 - 1 . 4 8 - 0 . 1 8 7 4 - 0 . 8 9 : - 0 . 1 4 2 9 - 0 . 7 1 - 0 . 0 8 2 2 > - 0 . 2 8 N O - B A R - 0 . 2 6 0 3 2 .20 - 0 . 2 7 1 7 3 . 6 9 - 0 . 2 6 7 1 - 1 / 3 4 - 0 . 1 8 6 9 - 0 . 6 5 - 0 . 1 4 1 7 0 . 1 0 - 0 , 0 8 0 8 1.44 E L A S T I C I T Y - 0 . 0 7 7 3 - 0 . 0 9 3 1 - 0 . 1 1 3 8 - 0 . 0 9 5 0 - 0 . 0 7 5 9 - 0 . 0 4 5 5 40° ui B A R - 0 . 0 7 4 8 3 . 2 5 - 0 . 0 9 4 6 v - 1 . 5 5 - 0 . 1 1 5 3 - 1 . 2 8 - 0 . 0 9 6 9 - 1 . 9 8 - 0 . 0 7 7 5 - 2 . 0 0 - 0 . 0 4 6 1 - 1 . 3 5 u: N O - B A R - 0 . 0 7 1 1 7 .96 - 0 " v 0 8 8 6 4 . 8 1 -0 .1144' - 0 . 5 1 , - 0 . 0 9 6 7 , - 1 . 7 9 - 0 . 0 7 6 9 - 1 . 2 9 - 0 . 0 4 5 4 0 . 1 9 E L A S T I C I T Y 0 . 0 6 4 1 00.0504 0 .0149 - 0 . 0 0 6 4 - 0 . 0 0 9 1 - 0 . 0 0 7 3 50° ui B A R 0 . 0 6 3 5 0 .93 0.0507 - 0 . 5 9 0 .0144 3 . 4 6 - 0 . 0 0 7 7 - 2 0 . 5 1 - 0 . 0 1 0 4 - 1 4 . 8 3 - 0 . 0 0 8 1 - 1 0 . 2 3 IN N O - B A R 0.0659 - 2 . 8 1 0.0534 - 5 . 9 5 0 .0160 - 7 . 0 3 - 0 . 0 0 7 9 • - 2 2 . 0 7 - 0 . 0 1 0 6 - 1 6 . 1 2 - 0 . 0 0 8 1 - 1 1 . 0 1 -E L A S T I C I T Y 0 . 1 6 9 1 0 .1572 0 .1169 0 ;0704 0 .0510 0 .0280 60° ui B A R 0.1692 - 0 . 0 3 0.1567 0 . 3 3 0 .1164 . 0 . 4 1 0 .0697 0 . 1 0 • 0 .0502 1.57 0 .0274 2 .27 . IN- N O - B A R 0.1693 - 0 . 0 9 0.1769 - 0 . 4 4 0 .1176 - 0 . 6 5 0 .0694 1 .38 0.0495. 2 .98 . 0 . 0 2 6 6 5 . 0 6 o 70 E L A S T I C I T Y 0.2424 0.2317 0.1901 0.1?Q? 0 . 0 9 8 1 0 .0565 UI B A R • 0 .2419 . 0 . 2 1 0 .2311 0 . 2 2 0 .1896 0 . 2 8 0 .1288 0 . 2 6 0 .0979 0 .24 0 . 0 5 6 1 0 . 6 9 IN-N O - B A R 0 . 2 4 1 1 0 . 5 3 0 .2310 0 . 2 7 0 .1903 - 0 . 1 0 0 .1285 0 . 4 9 0 .0967 1 .38 0 .0547 3 . 1 3 E L A S T I C I T Y 0.2858 0.2758 0 . 2 3 4 2 0 .1658 0 .1280 ' 0 .0749 80° ui B A R 0.2852 0 . 2 0 0 . 2 7 5 1 0 . 2 6 0 . 2 3 3 5 0.291 n . l f iS f i o . m 0 .1282 - 0 . 1 1 0.0747 0 . 2 2 IN- N O - B A R 0.2838 0 . 7 1 . 0 .2743 < 0 . 5 4 0 .2338 0 . 1 5 0 .1653 0 . 3 0 : 0 .1268 0 . 9 3 0 .0729 2 .56 E L A S T I C I T Y 0.3002 0.2904 0 .2488 0 .1783 0 .1383 0 .0812 90° ui B A R 0.2996 . 0 . 2 3 0.2897 0 . 2 6 0 . 2 4 8 1 0 . 2 9 0 . 1 7 8 1 0 . 0 6 0 .1385 - 0 . 1 9 0 .0812 0 . 0 0 IN N O - B A R 0.2979 0 . 7 6 0.2887 0 . 6 1 0 .2483 0 . 2 1 0 .1778 0 . 2 6 0 . 1 3 7 1 0 . 8 1 0 .0793 2 . 4 2 •: CIRCULAR P L A T E - WITH TWO POINT LOADS - 1 1 f -. TABLE 4-26 TANGENTIAL ' DISPLACEMENT . V . AND % ERROR AT DIFFERENT T AND 6 fj. = 0 . 3 3 3 3 AS COEFFICIENT C IN v = C £ BY ELASTICITY AND FINITE ELEMENT.. PARAMETER 60 - 100° # 0.85 R 0 .72R 0.50R 0.31 R ' ' 0 .23R 0.I3R • B \ - FUNCTION }fo ERROR FUNCTION %ERROR FUNCTION %ERROR FUNCTION %ERROR FUNCTION %ERROR FUNCTION %ERROR 0 ° ELASTICITY 0 .0000 0.0000 0.0000 0 .0000 0 .0000 0.0000 U J U. ' BAR 0 .0000 0.0000 0 .0000 0 .0000 0.0000 0 .0000 • 0.0000 N O - B A R 0.0000 0.0000 0 .0000 0 .0000 0 .0000 10° ELASTICITY 0.2730 0.2575 0 . 1 6 2 1 0.0938 0.0683 0.0380 hj u.-BAR 0 .2951 - 8 . 0 7 0.2508 2 .59 0 .1582 2 .40 0 .0927 . 1 1 .17 0.0677 0 .85 0.0376 1.14 NO-BAR 0 . 3 4 7 1 - 2 7 . 1 2 0.2502 2 .83 0 .1588 2 . 0 1 0 .0925 ,;; 1 .42 0.0672 l r 6 5 0.0368 3 . 1 8 20° ELASTICITY 0.2876 0.3513 0 .2780 0.1717 0.1266 0 . 0 7 1 1 Li i f BAR 0 .2864 . 0 .42 • 0.3566 - 1 . 5 0 0.2754 0 . 9 2 0 .1702 0 . 8 2 0 .1258 0 .68 0 .0705 0 .93 NO-BAR 0 . 2 7 9 1 2.94 0 .3762 - 7 . 0 8 0 .2776 0 .14 0 .1700 0 . 9 6 0 .1249 • 1.39 0 .0690 2 .95 3 0 ° ELASTICITY 0 .2831 • 0.3590 0 .3332 0 .2226 0 .1672 0 .0953 U J IN-BAR 0.2843 - 0 . 4 4 0.3588 0 . 0 5 0 .3332 0 . 0 0 0 .2216 '? 0 . 4 5 0.1664 0 .48 0.0947 0 .60 NO-BAR 0.2753 2 .75 0.3604 - 0 . 3 8 0.3357 - 0 . 7 5 0 .2216 0 . 4 5 0 .1655 1.02 0 .0928 2 . 6 1 4 0 ° ELASTICITY 0 .2665 0.3327 i- 0.3374 0 .2424 0.1857 . 0 . 1 0 7 5 . •uj m. BAR 0 .2652 i 0 .48 0.3332 - 0 . 1 4 0 .3379 - 0 . 1 5 0 .2419 0 . 2 0 0 .1852 0 .30 0 .1073 . 0 .22 N O - B A R 0.2596 2.57 0.3301 0 . 7 8 0 .3406 - 0 . 9 4 0 .2423 0 . 0 2 0 .1845 0 .66 0 .1052 2.19 5 0 ° ELASTICITY 0 . 2 3 5 1 0.2873 0 .3050 0 . 2 3 2 1 . 0 .1813 0.1067 ui IN-BAR • 0 .2335 . 0 .67 '.0.2870 0 . 1 0 0 .3054 - 0 . 1 3 0 .2319 * 0 .09 0.1809 0 :20 0 .1069 - 0 . 1 7 N O - B A R 0.2297 ... , 2 . 3 1 0-.2835 1 .33 0 .3062 - 0 . 4 0 0 .2326 - 0 . 2 1 0 . 1 8 0 7 0 .36 0 .1048 1.76-6 0 ° ELASTICITY 0.1898 0.2279 . 0 .2476 0 ,1966 0 .1560 0.0932 ui IN-BAR 0 .1882 0 .84 0.2275 0 . 1 9 0 .2480 - 0 . 1 3 0 .1965 0 .07 0.1557 0.17 0.0937 - 0 . 5 3 N O - B A R .0.1857 _ 2 .15 0.2246 1 .48 0 .2476 0 . 0 0 0 .1972 - 0 . 3 0 0 . 1 5 5 8 0 .16 . 0 . 0 9 1 9 •1.37 o 7 0 ELASTICITY 0 . 1 3 3 1 , 0.1580 0 .1736 0 .1418 0.1138 0.0687 U I L N BAR • 0 .1320 0 .87 0 .1575 0 .27 0 .1738 - 0 : 1 1 0 .1416 0 . 0 8 0 .1136 0.17 0 .0693 - 0 . 8 1 N O - B A R 0.1304 2 .05 0.1555 1 .55 0 . 1 7 3 1 0 .27 0 .1422 - 0 . 2 9 0 .1138 0 .00 0 .0680 1.05 3 0 ° ELASTICITY 0 .0686 0.0808 0 .0893 0 . 0 7 4 1 0 .0599 0.0364 ui IN-BAR O .OS0O 0.89 0.0806 0.3*1 0 .0894 - 0 . 1 0 0 .0740 0 . 0 9 0 .0598 0 .18 0 .0368 - 0 . 9 9 N O - B A R 0 .0672 2 .00 0.0795 1.57 0 .0889 0 . 4 1 . 0 r 0 7 4 2 • - 0 . 2 7 0 .0599 0 .00 0 . 0 3 6 1 0 . 8 5 9 0 ° ELASTICITY 0 .0000 0.0000 0 .0000 0 .0000 0 .0000 0 .0000 ui BAR •0,0000 0.0000 0 .0000 0 .0000 0 .0000 0.0000. N O - B A R 0 .0000 0.0000 o.oooo • 0.0000 0 .0000 0 .0000 ! TABLE 4 - 2 7 RADIAL NORMAL STRESS 07-. A N D % ERRORJ AT DIFFERENT r AND Q JJL•» 0.3333 AS COEFFICIENT C j IN CTr =C £ BY ELASTICITY AND FINITE ELEMENT 1 PARAMETER 0O= 100° •'/.-.' J V 0.85R 072R 0.50 R 0.31 R } 0 . 2 3 R 0.13 R F U N C T I O N %ERROR FUNCTION %ERROR FUNCTION %ERROR FUNCTION %ERROR F U N C T I O N % E R R O R F U N C T I O N %ERROR E L A S T I C I T Y - 4 . 2 6 9 9 - 2 . 3 2 5 5 - 1 . 3 7 3 2> - 1 . O 3 0 3 - 1 . 0 2 6 0 - 0 . 9 7 6 8 0 ° B A R - 2 . 7 4 2 8 35 .76 - 1 . 8 9 6 3 1 8 . 4 6 - 1 . 2 8 1 4 7 .10 - 1 . 0 5 2 9 3 . 4 3 - 1 . 0 0 0 9 2 .45 - .0 .9660 1 . 1 1 U . ' N O - B A R - 3 . 0 4 6 3 28.66 - 1 . 9 5 0 7 1 6 . 1 2 - 1 . 3 0 7 0 - 5 .24 - 1 . 0 6 3 2 2 .49 - 1 , 0 0 3 4 2 . 2 1 - 0 . 9 5 0 0 2.74 E L A S T I C I T Y - 0 . 8 3 0 0 - 1 . 3 4 0 0 - 1 . 1 8 1 1 - 1 . 0 1 4 3 - 0 . 9 6 9 4 - 0 . 9 3 3 , 2 10° B A R - 1 . 5 2 6 4 - 8 3 . 9 0 - 1 . 4 3 1 3 - 6 . 8 1 - 1 . 1 4 4 9 3 . 0 6 - 0 . 9 9 0 4 2 .36 - 0 . 9 5 1 6 1.84 ^ 0 : 9 2 0 0 1 . 4 1 IN- N O - B A R - 1 . 4 4 7 4 - 7 4 . 3 9 - 1 . 4 6 5 9 - 9 . 3 9 - 1 , 1 6 1 6 1 .65 - 0 . 9 9 8 2 1 .59 - 0 . 9 5 2 5 • 1 . 7 4 ' - 0 . 9 1 2 0 2 .28 E L A S T I C I T Y - 0 . 0 8 0 4 - 0 . 4 0 2 2 - 0 . 7 6 3 8 - 0 . 8 1 5 7 - 0 . 8 1 4 4 - 0 . 8 0 9 4 20° ui B A R - 0 . 0 2 7 8 65.40 - 0 . 5 5 4 3 - 3 7 . 8 3 - 0 . 8 0 8 2 - 5 . 8 0 - 0 . 8 1 9 4 - 0 . 4 5 - 0 . 8 1 3 1 0 .16 - 0 . 8 0 5 0 - 0 .54 IN- N O - B A R 0.0405 150.36 - 0 . 5 3 3 4 - 3 2 . 6 3 - 0 . 8 0 6 0 - 5 . 5 3 - 0 . 8 2 1 1 - 0 . 6 5 - 0 . 8 1 1 3 0 .39 - 0 . 7 9 5 0 1 .78 3 0 ° E L A S T I C I T Y - 0 . 0 0 1 3 - 0 . 0 9 1 1 - 0 . 3 8 9 9 - 0 . 5 6 1 0 - 0 . 5 9 8 5 - 0 . 6 2 5 4 U J B A R - 0 . 0 1 3 6 . - 9 7 7 . 5 0 - 0 . 0 8 0 6 1 1 . 5 0 - 0 . 4 3 6 1 - 1 1 . 8 6 - 0 . 5 8 3 5 - 4 . 0 2 , - 0 . 6 1 1 8 - 2 . 2 2 - 0 . 6 3 0 0 - 0 . 7 4 IN- N O - B A R 0.0064 609.2 - 0 . 0 4 7 4 47 .97 - 0 . 4 2 9 4 - 1 0 . 1 3 - 0 . 5 8 0 8 -3.53 ; - 0 . 6 0 7 6 - 1 . 5 3 - 0 . 6 2 0 0 0 . 8 6 E L A S T I C I T Y 0.0094 - 0 . 0 0 2 6 - 0 . 1 4 7 6 - 0 . 3 1 2 9 - 0 . 3 6 5 0 - 0 . 4 0 8 5 4 0 ° ui B A R 0.0205 - 1 1 8 . 1 1 - 0 . 0 0 5 5 - 1 1 0 . 7 4 - 0 . 1 6 3 5 - 1 0 . 8 0 - 0 . 3 3 6 1 - 7 . 4 0 - 0 . 3 8 3 1 - 4 . 9 7 - 0 . 4 2 0 5 . - 2 . 9 6 u,- N O - B A R 0 .'0134 - 4 2 . 7 1 0 .0086 4 3 2 . 4 5 ^ -0 .1582 - 7 . 1 8 - 0 . 3 3 2 4 - 6 . 2 2 - 0 . 3 7 8 7 - 3 . 7 6 - 0 . 4 1 0 0 - 0 . 3 7 5 0 ° E L A S T I C I T Y 0.0102 0 ,0230 - .0 .0122 - 0 . 1 0 7 9 - 0 . 1 4 9 0 - 0 . 1 8 7 9 ui B A R ' 0.0146 - 4 2 . 5 8 0 .0302 - 3 1 . 6 0 - 0 . 0 1 3 1 - 6 . 9 5 - 0 . 1 2 0 5 - 1 1 . 6 6 - 0 . 1 6 2 3 - 8 . 9 6 - 0 . 1 9 8 5 - 5 . 6 5 IN- N O - B A R 0.0112 - 9 . 7 4 0 . 0 2 9 1 - 2 6 . 7 4 0 - 0 . 0 0 5 5 5 4 . 7 9 - 0 . 1 1 7 6 - 8 . 9 2 i - 0 . 1 5 9 5 - 7 . 0 5 - 0 . 1 9 5 0 - 3 . 8 0 E L A S T I C I T Y 0.0096 0.0302 0 .0585 0 .0429 0.0278 0 . 0 1 0 1 6 0 ° ui B A R 0.0130 - 3 6 . 0 1 0 .0348 . - 1 5 . 4 5 0 .0623 - 6 . 6 2 0 .0414 3 . 4 8 0.0234 1 5 . 7 2 0 .0056 4 4 . 5 0 N O - B A R 0 . 0 1 0 1 - 5 . 1 5 .0.0328 . - 8 . 5 7 0 .0663 ' - 1 3 . 4 8 0 .0436 - 1 . 5 9 0 .0247 11 .38 . 0 .0034 6 5 . 5 0 E L A S T I C I T Y 0.0087 0.0319 0 .0938 0 .1426 0.1557 .0 .1654 7 0 ° U I B A R 0.0109 - 2 3 . 0 8 0 .0362 - 1 3 . 4 2 0 .1004 - 7 . 0 3 0 .1483 - 3 . 9 7 . 0 .1593 - 2 . 3 1 0 .1687 - 2 . 0 0 IN-N O - B A R 0 . 0 0 9 1 - 3 . 1 0 0.0334 - 4 . 9 0 0 . 1 0 1 1 - 7 . 7 8 0 .1504 - 5 . 4 7 0 .1596 - 2 . 4 7 0 .1620 2 .06 E L A S T I C I T Y 0.0084 0 . 0 3 2 1 0 .1099 0 . 1 9 8 5 0 .2322 0 .2638 8 0 ° ui B A R 0.0099 - i 8 . 8 6 0 .0356 - 1 0 . 9 2 0 .1173 - 6 . 7 1 0 .2079 - 4 . 7 1 0 .2406 - 3 . 6 1 • 0.274YJ . - 3 . 8 6 I N N O - B A R 0.0086 - 2 . 7 4 0 .0332 - 3 . 5 5 0 .1164 - 5 . 8 4 0 .2097 - 5 . 6 2 0.2407 ^ 3 . : 6 5 r 0.2635 . 0 . 1 1 E L A S T I C I T Y . .0 .0083 0 .0320 0 .1146 0 . 2 1 6 5 0 .2576 0 .2975 9 0 ° ui B A R . 0.0097 - L 7 . 0 5 0.0354 - 1 0 . 4 9 0 .1222 - 6 . 6 9 0 .2270 - 4 . 8 6 0 .2674 - 3 . 8 4 0 .3100 - 4 . 2 0 I N NO-BAR . - 0 . 0 0 8 5 - 2 . 7 1 0 . 0 3 3 1 - 3 . 2 6 0 .1207 - 5 . 3 5 0 . 2 2 8 6 - 5 . 6 1 '•: / 0 .2676 - 3 . 8 8 0 .2985 , - 0 . 3 4 TABLE 4 , -28 'TANGENTIAL NORMAL STRESS <TE AND % E R R O R AT DIFFERENT r AND 9 JJ. * 0.3333 AS COEFFICIENT C IN Rt BY ELASTICITY AND FINITE ELEMENT PARAMETER 0 f t = 100* \ 0.85R 0.72 R 0.50R 0.31 R 0.23 R 0.13 R FUNCTION %ERROR FUNCTION % ERROR FUNCTION %ERROR FUNCTION %ERROR F U N C T I O N % E R R O R F U N C T I O N %ERROR .0° E L A S T I C I T Y 0.3183 0 .3183 0 .3183 0 .3183 0 .3183 0 .3183 ui IN-B A R 0.4779 - 5 0 . 1 3 0.2924 8 . 1 3 0 .3013 5 .34 0 .3107 2 .39 0 .3128 1 . 7 1 0 .3100 2 .65 N O - B A R . 0 .0886 7 2 . 1 7 0 .4219 - 3 2 . 5 4 0 . 3 2 3 1 - 1 . 5 1 ; 0 .3197 - 0 . 4 6 0 .3215 - 0 . 9 9 0 .3165 0 . 5 6 10° E L A S T I C I T Y - 1 . 0 2 3 8 - 0 . 2 4 0 9 0 .1740 ; .. • 0 . 2 5 3 1 . 0 . 2 6 6 7 0.2760 ui IN-B A R - 1 . 2 4 7 0 - 2 1 . 8 0 - 0 . 1 7 3 5 2 8 . 0 0 0 . 1 7 5 1 - 0 . 6 3 0 .2484 1 .88 0 .2623 1 .62 0 .2695 . . 2 .40 0 . 1 8 N O - B A R - 1 . 2 9 6 3 - 2 6 . 6 1 - 0 . 2 2 6 1 6 .14 0 .1866 - 7 . 2 1 0 .2558 - 1 . 0 4 0 .2698 - 1 . 1 7 0 . 2 7 6 5 ' " 20° E L A S T I C I T Y - 0 . 6 5 2 7 - 0 . 5 7 8 9 - 0 . 1 1 4 5 ' 0 .0840 > 0.1257 - 0".1562 ui B A R - 0 . 6 0 6 3 7 . 1 1 - 0 . 6 5 3 1 . - 1 2 . 8 1 - 0 . 1 0 4 6 8 . 6 9 0 .0836 0 . 4 7 0 .1233 1 .90 0.1552 0 .64 N O - B A R - 0 . 5 1 3 2 . 21.38 - 0 . 7 1 8 0 - 2 4 . 0 2 - 0 . 0 9 9 4 13 .17 0 .0880 - 4 . 7 4 0 .1275 - 1 . 4 2 0 .1610 - 3 . 0 8 3 0 ° E L A S T I C I T Y - 0 . 3 7 9 3 - 0 . 5 2 9 4 - 0 . 3 4 4 7 - 0 . 1 2 9 8 - 0 . 0 6 9 8 - 0 . 0 2 1 8 U I B A R - 0 . 3 9 7 9 . . . - 4 . 8 8 - 0 . 5 1 8 5 2 .06 - 0 . 3 4 9 5 - 1 . 4 1 - 0 . 1 2 9 9 - 0 . 0 8 - 0 . 0 7 1 6 - 2 . 6 0 - 0 . 0 1 6 9 ' 2 2 . 5 0 N O - B A R - 0 . 3 1 8 5 16.03 - 0 . 5 2 1 1 1.57 - 0 . 3 5 0 6 - 1 . 7 3 - 0 . 1 2 8 6 0 . 9 2 - 0 . 0 7 1 6 - 2 . 6 0 - 0 . 0 1 3 4 3 8 . 7 0 4 0 ° E L A S T I C I T Y - 0 . 2 4 8 8 ; -rO.4250 - 0 . 4 6 4 1 - 0 . 3 3 3 4 - 0 . 2 7 9 9 - 0 . 2 3 1 5 tii I N B A R - 0 . 2 5 1 6 - 1 . 1 2 - 0 . 4 3 0 8 - 1 . 3 6 - 0 . 4 7 1 4 - 1 . 5 6 - 0 . 3 3 5 9 - 0 . 7 6 - 0 . 2 8 2 6 - 0 . 9 5 - 0 . 2 2 4 0 3 . 2 5 N O - B A R - 0 . 2 1 4 4 13.84 - 0 . 4 0 7 0 4 . 2 3 - 0 . 4 7 9 9 - 3 . 4 1 - 0 . 3 3 7 2 - 1 . 1 4 - 0 . 2 8 7 0 - 2 . 5 1 - 0 . 2 2 2 5 3 . 9 0 5 0 ° E L A S T I C I T Y - 0 . 1 8 1 9 - 0 . 3 4 4 3 - - 0 ; 5 0 7 1 - 0 . 4 9 7 0 - 0 . 4 7 2 7 - 0 . 4 4 4 6 , ui I N B A R - 0 . 1 8 3 3 - 0 . 7 7 - 0 . 3 4 5 2 - 0 . 2 6 - 0 . 5 0 9 4 - 0 . 4 6 - 0 . 5 0 0 3 ' - 0 . 6 7 - 0 . 4 7 5 8 - 0 . 6 6 - 0 . 4 3 9 0 1.26-N O - B A R - 0 . 1 6 1 5 11.19 - 0 . 3 2 6 5 5 .17 - 0 . 5 1 1 6 - 0 . 9 0 - 0 . 5 0 3 4 - 1 . 3 0 - 0 . 4 8 4 5 - 2 . 5 0 - 0 . 4 4 1 0 0 . 8 1 -6 0 ° E L A S T I C I T Y - 0 . 1 4 5 1 - 0 . 2 9 0 7 - 0 . 5 1 3 2 - 0 . 6 1 3 5 - 0 . 6 2 9 2 - 0 . 6 3 5 8 ui U J B A R - 0 . 1 4 3 8 0 . 9 2 - 0 . 2 9 1 8 - 0 . 3 8 - 0 . 5 1 4 8 - 0 . 3 1 - 0 . 6 1 5 4 - 0 . 3 1 - 0 . 6 3 1 1 - 0 . 3 1 - 0 . 6 3 5 0 0 . 1 2 . N O - B A R - 0 . 1 3 1 2 9 .60 - 0 . 2 7 6 7 . 4 . 8 0 - 0 . 5 1 0 9 0 .44 - 0 . 6 1 9 3 - 0 . 9 4 - 0 . 6 4 3 6 - 2 . 2 9 - 0 . 6 3 8 5 - 0 . 4 3 o 7 0 E L A S T I C I T Y - 0 . 1 2 4 6 . - 0 . 2 5 7 5 - 0 . 5 0 6 7 - 0 . 6 8 8 0 - 0 . 7 4 1 3 - 0 . 7 8 5 6 ui U J B A R - 0 . 1 2 2 7 . 1 . 5 1 - 0 . 2 5 7 5 0 . 0 0 - 0 . 5 0 7 5 - 0 . 1 5 - 0 . 6 8 8 1 - 0 . 0 1 . - 0 . 7 4 1 2 0 .04 - 0 . 7 9 1 0 - 0 . 6 9 N O - B A R - 0 . 1 1 4 1 8 . 4 6 - 0 . 2 4 6 2 4."4 2 - 0 . 5 0 0 5 1 .22 - 0 . 6 9 1 1 - 0 . 4 4 - 0 . 7 5 6 4 . ' - 2 . 0 1 - 0 . 7 9 8 0 - 1 . 5 8 8 0 ° E L A S T I C I T Y - 0 . 1 1 4 1 - 0 . 2 3 9 6 - 0 . 4 9 9 4 - 0 . 7 2 8 7 - 0 . 8 0 8 2 - 0 . 8 8 0 5 ui U J B A R - 0 . 1 1 J 3 1 .90 - 0 . 2 3 9 2 0 . 1 6 - 0 . 5 0 0 1 - 0 . 1 2 - 0 . 7 2 7 6 0 . 1 4 - 0 . 8 0 6 0 0 . 2 6 - 0 . 8 9 0 0 - 1 . 0 8 N O - B A R 7.0.1052-. 7 .80 - 0 . 2 2 9 7 • 4 . 1 3 - 0 . 4 9 1 6 1 .57 - 0 . 7 2 9 4 - 0 . 1 0 - 0 . 8 2 2 6 - 1 . 7 9 - 0 . 9 0 0 0 - 1 . 1 3 9 0 ° E L A S T I C I T Y - .0 .1108 - 0 . 2 3 3 9 - 0 . 4 9 6 6 - 0 . 7 4 1 4 - 0 . 8 3 0 2 - 0 . 9 1 3 0 ' ui U J B A R - 0 . 1 0 8 6 1.98 - 0 . 2 3 3 4 ',, 0 . 2 4 - 0 . 4 9 7 1 - 0 . 1 0 - 0 . 7 4 0 0 0 . 2 0 - 0 . 8 2 7 4 0 . 3 4 - 0 . 9 2 6 0 - 1 . 4 0 NO-BAR - 0 . 1 0 2 4 7 .60 - 0 . 2 2 4 5 4 . 0 3 - 0 . 4 8 8 2 1 .68 - 0 . 7 4 1 3 0 . 0 2 - 0 . 8 4 4 3 - 1 . 7 0 - 0 . 9 3 5 5 . - 2 . 4 8 CIRCULAR PLATE WITH TWO POINT LOADS SYM thickness- t U = RADIAL DISR V TANGENTIAL DISR QUARTER OF THE CIRCULAR PLATE ' • H S ^ V ^ i f i ^ V ; '•> '•••'•'•iJk TABLE 4 - 2 9 SHEAR S T R E S S T r e A N D % E R R O R AT DIFFERENT T AND 9 p * Oi3333 AS COEFFICIENT C IN Tre "C BY ELASTICITY AND FINITE ELEMENT PARAMETER 90 = 1 0 0 ° V 0 . 8 2 R 0 7 2 R 0 . 5 0 R 0 . 3 I R 0 . 2 3 R 0 . 1 3 R *\ F U N C T I O N % E R R O R F U N C T I O N % E R R O R F U N C T I O N % E R R O R F U N C T I O N % E R R O R F U N C T I O N % E R R O R F U N C T I O N % E R R O R E L A S T I C I T Y 0.0000 0 .0000 0 .0000 0 .0000 0 .0000 0 .0000 0° L U B A R 0.0000 0 .0000 0.0000^ 0 .0000 0.0000 0 .0000 IN- N O - B A R 0.0000 0 .0000 0 .0000 0 .0000 0 .0000 0.0000 E L A S T I C I T Y 1.0718 • 0.8843 0.4359 0 .2840 0 .2520 0.2281. 10° L U B A R 1.3241 - 2 3 . 5 4 0.8174 7.57 0 .4067 6 .69 0 .2779 2 .15 0.2487 1 .30 0 .2240 1 .80 IN- N O - B A R 1.3394 - 2 4 . 9 6 0 .7843 , 1 1 . 3 1 0 .3960 9 . 1 5 0 .2726 4 . 0 1 0 . 2 4 7 2 • 1.90 0 .2210 3 . 1 0 E L A S T I C I T Y 0 . 3 1 5 1 0 . 6 4 0 1 0 .6232 0 .4932 0 .4549 0 .4236 20° ui B A R ' 0 . 1 6 1 1 48 .88 0 .6976 - 8 . 9 8 0 .6163 1 . 1 1 0 .4875 1 .16 0 .4508 0 . 9 0 0.4165_. 1 .72 IN- N O - B A R 0 . 1 7 3 1 45 .05 0 .7496 - 1 7 . 1 1 0.6077 2 .49 0 .4803 2 . 6 1 0 . 4 4 9 3 1.24 0 .4135 2 .45 30° E L A S T I C I T Y 0.1144 0.3449 0 .5872 0 . 5 9 2 1 0 . 5 7 7 1 0 .5603 L U B A R 0.1605. . - 4 0 . 3 3 0 .3102 1 0 . 0 5 0 .5957 - 1 . 4 4 0 . 5 9 0 1 0 .34 V 0 . 57 39 0 . 5 5 0 .5565 0 . 6 8 IN- N O - B A R 0.1202 - 5 . 0 9 0 .3066 1 1 . 1 1 0 .5938 - 1 . 1 3 0 . 5 8 5 1 - , 1 .19 0 . 5 7 5 1 0 . 3 5 0 .5500 1.84 E L A S T I C I T Y 0.0515 0 .1863 0-4<o2)l 0.5896 0 .6110 0 . 6 2 3 1 40° ui B A R 0.0404 21.57 0 .1965 - 5 . 4 9 0 .4635 - 0 . 0 8 0 .5882 0 . 2 4 0 .6079 .0.50 0 .6180 . 6 . 8 2 u N O - B A R 0.0400 22.38 0 .1784 4 . 2 5 0 .4724 - 2 . 0 2 0 .5892 0 . 0 6 0 .6135 - 0 . 4 0 0 .6160 1.14 50° E L A S T I C I T Y 0.0265 0 .1052 0.3337 0 .5165 0 .5678 0.6087 ui B A R 0.0305 - 1 5 . 2 3 0 .1033 1 .76 0 .3312 , 0 . 7 5 0 .5125 0 .77 0 .5630 0 ,84 0 .6045 0 . 6 9 IN- N O - B A R 0.0265 . 0 ,04 0.0964 8 .37 0 .3313 0 . 7 4 0 .5179 - 0 . 2 7 0 . 5 7 3 1 - 0 . 9 4 0 . 6 0 6 1 0 . 3 6 -E L A S T I C I T Y . 0 .0145 0.0608 0 .2250 0 . 4 0 4 1 . 0 . 4 6 7 2 0.5237 60° ui B A R 0.0129 11.24 0 .0626 - 2 . 9 1 0 .2243 0 .34 0 .3984 1 .40 6.4607 1.39 0 .5190 0 . 9 0 IN- N O - B A R 0.0134 7 .72 0 .0574 5 . 6 6 0 .2217 1.47 0 . 4 0 5 1 - 0 . 2 3 .; 0 .4725 - 1 . 1 4 0 .5200 0 . 7 0 o 70 E L A S T I C I T Y 0.0079 , 0 .0339 • 0.1377 0 .2740 • 0.3290 0.3819 ui B A R • 0 .0079 . 0 . 0 0 0.0338 0 . 2 0 0 .1372 0 . 3 9 0 . 2 6 9 1 1 .79 0 .3227 1 . 9 1 0 .3830 - 0 . 2 9 IN- N O - B A R 0.0077 1.97 0 .0317 i :6 .32 0 . 1 3 4 5 2 .26 0 .2738 • 0 . 0 8 •: 0.3329 - 1 . 1 8 0 .3780 . 1 .02 E L A S T I C I T Y ' 0.0035 0 .0152 0 .0652 0.1377 0 .1693 0 .2009 80° ui B A R 0.0032 8 .74 0 .0154 - 1 . 1 3 0^0651 0 . 0 8 0 .1350 1 .98 0 .1655 . 2 . 2 2 0 .2065 0 . 1 0 I N N O - B A R 0.0033 4 . 2 9 0 .0144 5 . 5 3 0 .0636 2 . 4 3 0 .1374 0 .27 : 0.1712 - 1 . 1 2 0 .2015 - 0 . 3 0 E L A S T I C I T Y 0.0000 o •' 0 .0000 0 "• 0 . 0 0 0 0 0 .0000 0 .0000 0 .0000 90° ui B A R 0.0000 „ • n., 0.0000 0 .0000 0 .0000 0 .0000 • 0 .0000 L N N O - B A R 0.0000 . 0 .0000 0 .0000 0 .0000 0 .0000 0 . 0 0 0 0 . C I R C U L A R P L A T E WITH TWO ' POINT LOADS v. U = RADIAL DISP. V « TANGENTIAL DISR - 'MU-TABLE 4 - 3 0 RADIAL DISPLACEMENT U AND % E R R O R AT DIFFERENT f AND Q /x * 0 . 4 5 AS COEFFICIENT C IN u = C |-f BY ELASTICITY AND' FINITE ELEMENT PARAMETER 0O= 1 1 2 . 5 ° V 0 . 8 5 R 0 . 7 2 R 0 . 5 2 R 0 . 3 7 R ] 0 . 31 R 0 . 2 2 R F U N C T I O N % E R R O R F U N C T I O N % E R R O R F U N C T I O N % E R R O R F U N C T I O N % E R R O R JFUNCTION % E R R O R F U N C T I O N % E R R O R E L A S T I C I T Y - 1 . 4 5 0 6 - 1 . 0 2 9 6 - 0 . 6 4 2 8 - 0 . 4 2 9 8 - 0 . 3 5 3 9 - 0 . 2 4 6 3 0° B A R - 1 . 4 0 2 7 3 . 3 0 - .1 .0069 2 . 2 1 - 0 . 6 3 5 6 1 . 1 1 - 0 . 4 2 6 8 0 . 6 9 - 0 . 3 5 1 7 0 .60 - 0 . 2 4 4 8 0 . 5 8 U - N O - B A R . - 1 . 4 5 6 6 - 0 . 4 1 - 1 . 0 1 8 3 1 .10 . - 0 . 6 4 0 9 0 . 2 9 - 0 . 4 2 9 1 0 . 1 5 - 0 . 3 5 3 3 0 .16 - 0 . 2 4 5 6 0 . 2 6 E L A S T I C I T Y - 0 . 9 6 0 8 - 0 . 8 4 3 8 - 0 . 5 8 3 5 - 0 . 4 0 0 7 i - 0 . 3 3 2 0 - 0 . 2 3 2 5 10° ui B A R - 1 . 0 0 8 2 - 4 . 9 3 - 0 . 8 5 5 8 - 1 . 4 2 - 0 . 5 8 1 9 0 . 2 8 - 0 . 3 9 9 1 0 . 4 0 I - 0 . 3 3 0 6 0.4,0 - 0 . 2 3 1 4 0 . 4 5 LU N O - B A R - 1 . 0 0 8 9 . - 5 . 0 0 - 0 . 8 6 5 0 - 2 ; 5 1 - 0 , 5 8 5 4 - 0 . 3 3 - 0 . 4 0 0 9 - 0 . 0 4 j - 0 . 3 3 1 9 0 .03 - 0 . 2 3 2 1 0 .17 E L A S T I C I T Y - 0 . 5 1 8 5 - 0 . 5 2 6 9 - 0 . 4 3 7 2 - 0 . 3 2 1 8 | - 0 . 2 7 1 3 - -6 .1935 20° ui B A R - 0 . 4 9 1 8 5.17 - 0 . 5 3 4 9 - 1 . 5 2 - 0 . 4 4 1 8 - 1 . 0 6 - 0 . 3 2 2 7 - 0 . 2 9 | - 0 . 2 7 1 6 -Ov.12 - 0 . 1 9 3 3 - - 0 . 0 8 U J N O - B A R . -0 .4809 7.26 - 0 . 5 3 6 2 - 1 . 7 6 ' - 0 . 4 4 2 4 - 1 . 2 0 - 0 . 3 2 3 5 - 0 . 5 1 ; - 0 . 2 7 2 2 - 0 . 3 4 - 0 . 1 9 3 7 - 0 . 1 2 E L A S T I C I T Y - 0 . 2 4 0 5 J - 0 . 2 6 3 6 - 0 . 2 6 1 2 - 0 . 2 1 2 5 i - 0 . 1 8 4 2 - 0 . 1 3 5 6 30° U J B A R - 0 . 2 4 7 1 -2: 76 - 0 . 2 5 9 2 1 , 6 5 - 0 . 2 6 4 8 - 1 . 3 5 - 0 . 2 4 1 7 - 1 . 0 4 ( - 0 . 1 8 5 7 - 0 . 8 3 - 0 . 1 3 6 3 - 0 . 5 4 U J N O - B A R - 0 . 2 3 4 8 2.39 - 0 . 2 5 2 6 4 . 1 7 - 0 , 2 6 4 5 - 1 . 2 7 - 0 . 2 1 4 7 - 1 . 0 6 1 - 0 . 1 8 5 8 - 0 . 8 7 - 0 . 1 3 6 4 - 0 . 6 3 E L A S T I C I T Y - 0 . 0 4 1 9 - 0 . 0 6 3 0 - 0 . 0 9 4 4 - 0 . 0 9 3 7 j - 0 . 0 8 5 9 - 0 . 0 6 7 3 40° ixi B A R - 0 . 0 3 9 7 5.28 . - 0 . 0 6 4 1 - 1 . 8 1 - 0 . 0 9 5 2 - 0 . 8 7 - 0 . 0 9 5 5 - 1 . 9 9 1 - 0 . 0 8 7 6 - 1 . 9 7 - 0 . 0 6 8 5 - 1 . 8 0 U J N O - B A R - 0 . 0 3 5 8 14.74 - 0 . 0 5 8 5 7 . 1 5 - 0 . 0 9 4 5 - 0 , 1 0 - 0 . 0 9 5 4 - 1 . 8 2 : - - 0 . 0 8 7 5 - 1 . 8 2 - 0 . 0 6 8 5 - 1 . 7 4 50° E L A S T I C I T Y 0.1072 0 ,0895 ; 0 . 0 4 7 1 CO.0181 : 0 .0100 0 .0023 • ui B A R 0.1067 0 .50 0 .0900 - 0 . 5 3 0 ,0469 0 . 3 1 0 .0172 5 .44 ' 0 .0088 11.77 0 . 0 0 1 1 51.10. IN- N O - B A R 0.1090 - 1 . 5 8 0 .0926 - 3 . 4 8 . 0 . 0 4 8 5 ' - 3 . 0 5 0 .0174 3 . 9 8 ; 0 . 0 0 9 0 ' 10.27 0 .0012 48.09-E L A S T I C I T Y 0.2184 0 .2032 0 . 1 5 8 1 0 .1124 0 .0933 0 .0649 60° ui B A R 0.2186 - 0 . 0 7 0 .2029 0 . 1 3 0 .1580 0 . 0 5 0 . 1 1 2 1 ' 0 . 3 2 : 0 .0927 0 .58 0 .0642 1.14 U J N O - B A R 0.2187 - 0 . 1 3 ' 0 . 2 0 4 1 - 0 . 4 4 1 0 .1593 •-0.72 0 .1125 - 0 . 0 2 , 0 .0929 0 .39 0 .0642 1.09 o 70 E L A S T I C I T Y 0.2962.. 0 .2826 0 .2375 0 .1830 i 0 .1569 0 .1143 U J B A R • 0 . 2 9 5 9 . 0 .10 0 .2824 0 .07 0 .2374 0 . 0 4 0 .1830 0 . 0 0 0 .1568 0 .04 0 .1140 0 . 2 0 U J N O - B A R 0.2952 0 .33 0 .2823 0 . 0 9 0 .2382 - 0 . 2 8 0 .1834 - 0 . 2 3 0 .1571 - 0 . 1 0 0 .1140 0 . 2 0 ' E L A S T I C I T Y 0.3423 0.3297 0 . 2 8 5 1 0 .2265 0 .1966 0 .1456 80° ui B A R 0.3420 0 . 1 0 ' 0 .3293 0 . 1 1 0 .2849 0 . 0 6 0 .2266 - 0 . 0 6 0 .1968 - 0 . 1 1 0 .1458 - 0 . 1 1 IN-N O - B A R 0.3407 0.47 0 .3287 0 . 3 0 0 .2854 - 0 . 0 9 0 .2270 - 0 . 2 6 0 . 1 9 7 1 - 0 . ' 24 0 .1458 - 0 . 1 0 E L A S T I C I T Y 0,3577 . 0 .3453 0.3010. 0 . 2 4 1 1 0 . 2 1 0 1 0 .1564 S0° ui B A R 0.3573 . 0 . 1 1 0 .3449 0 . 1 0 0 .3008 0 . 0 6 0 . 2 4 i 3 - 0 . 0 7 0 .2104. - 0 . 1 4 . 0 .1567 - 0 . 2 0 . u.- N O - B A R •0.3558 0 . 5 1 - 0 . 3 4 4 1 ^0.35 0 . 3 0 1 1 - 0 . 0 3 0 .2417 - 0 . 2 6 0 .2107 - 0 . 2 7 0 .1567 . - 0 . 1 8 CIRCULAR PLATE WITH TWO POINT LOADS -116-TABLE 4 -31 TANGENTIAL DISPLACEMENT V A N D % E R R O R AT D I F F E R E N T T AND B fj. « 0.45 AS C O E F F I C I E N T C IN v =C P A R A M E T E R BY E L A S T I C I T Y A N D ' F I N I T E E L E M E N T II2.5 V 0 . 8 5 R 0 .72R 0 .52 R : 0 .37 R 0 . 3 I R 0 .22 R *\ FUNCTION %ERROR FUNCTION % ERROR FUNCTION %ERROR .FUNCTION %ERROR FUNCTION % E R R O R F U N C T I O N % E R R O R E L A S T I C I T Y 0.0000 0 .0000 . 0 .0000 0 .0000 0.0000 0 .0000 0 ° U J B A R 0.0000 0 .0000 0 .0000 0 .0000 0.0000 0 .0000 U.' N O - B A R . 0 .0000 0 .0000 0 .0000 0 .0000 0.0000 0 .0000 E L A S T I C I T Y 0.3213 ' 0 .2900 0 .1876 0.1247 0 .1024 0 .0710 10° B A R 0.3414 - 6 . 3 2 0 . 2 8 4 1 2 .02 0 .1835 . 2 .16 0 . 1 2 3 1 1.28. 0 .1014 1 .00 0.0705' 0 .74 N O - B A R 0.3985 - 2 4 . 0 9 0 .2818 2 .92 0 .1839 1.97 0 .1232 1 . 2 1 0.1014 1 . 0 1 0 .0704 0 . 8 6 E L A S T I C I T Y 0.3404 0 .3966 0 .3180 0 . 2 2 5 1 0 .1875 0.1319 20° ui B A R 0.3379 0 . 7 5 0 .4008 - 1 . 0 5 0 .3159 0 ,67 0 .2234 0 . 7 6 0 .1862 0 . 6 8 0 .1312 - 0 . 5 6 u N O - B A R 0.3333 2 . 1 1 0 .4234 - 6 . 7 6 0 .3180 0 . 0 0 0 .2237 0 . 6 4 0 .1862 0 .67 0 .1310 0 .69 E L A S T I C I T Y 0.3329 0.4054 0 .3768 0 . 2 8 6 5 ' 0 . 2 4 3 1 0 .1745 * 3 0 ° U I B A R - n . A D 0 .403 '6 / O.AS 0 . 3 7 6 9 - 0 . 0 3 0 .2857 0 . 2 8 0.2424 0 . 3 2 0 .1739 0 . 3 2 U : N O - B A R 0.3254 2 .25 0 .4069 - 0 . 3 6 0 .3798 . - 0 . 7 9 0 .2863 0 . 0 9 0 .2425 ' 0 . 2 6 0.1737 0 . 4 5 E L A S T I C I T Y 0v3109 0.3754 0 . 3 7 8 1 0.3059 0.2647 0 .1942 4 0 ° ui B A R 0.3097 0 .39 . 0 . 3 7 5 1 0 . 0 9 0 . 3 7 8 1 0 . 0 0 0 .3059 . 0 . 0 0 0 .2645 , 0 . 0 8 0 .1940 0 .09 U J N O - B A R 0.3043 2 . 2 1 0 .3732 0 . 5 8 0 .3816 - 0 . 9 2 0 .3069 - 0 . 3 0 0 .2648 - 0 . 0 6 .0.1938 . 0 . 2 0 E L A S T I C I T Y 0.2725 6.3238 0 .3397 0 . 2 8 8 1 0 .2535 0 .1900 5 0 ° ui B A R 0.2708 0 .62 0 .3230 0 . 2 3 . 0 . 3 3 9 4 . 0 .07 0 . 2 8 8 2 : - 0 . 0 3 0.2536 - 0 . 0 4 0 .1902 - 0 . 1 0 IN- N O - B A R 0.2670 1.99 .0 .3202 . 1 .10 0.3409 - 0 . 3 5 0 .2893 - 0 . 4 2 0 .2542 - 0 . 2 7 0 .1900 - 0 . 0 2 -E L A S T I C I T Y 0.2189 0 .2565 0.2747 0.2407 0.2147 0.1637 6 0 ° ui B A R 0.2172 0 .79 0 .2558 0 . 2 6 0 .2744 0 . 0 9 0 .2408 - 0 . 0 1 0 .2148 - 0 . 0 6 0 . 1 6 4 1 - 0 . 2 3 IN- N O - B A R 0.2148 1.86 0.2534 1 .22 1 0 .2747 . 0 . 0 0 0.2417 - 0 . 4 . 1 0 .2155 - 0 . 3 5 0 . 1 6 4 1 - 0 . 2 3 E L A S T I C I T Y 0.1530 .. 0 .1776 0 . 1 9 2 1 0 .1720 0 .1548 0 .1196 o 7 0 ui B A R • 0.1517 0 .84 0 .1770 ' 0 . 3 2 0.1919 0 . 1 1 0 .1719 0 . 0 1 0 .1549 - 0 . 0 5 0 .1200. - 0 . 3 1 I N N O - B A R 0.1503 1.77 0 .1753 1 .28 0 .1917 0 . 2 1 . 0 .1725 , - 0 . 3 4 0 .1544 - 0 . 3 8 0 .1200 - 0 . 3 2 E L A S T I C I T Y 0.0787 0 .0980 0 .0987 0 .0894 r 0.0809 0 .0630 8 0 ° ui B A R 0.0780 0 .87 0 .0905 0 . 3 5 0 .0986 0 . 1 2 0 .0893 0 . 0 3 0 .0809 0 . 0 0 0 .0632 - 0 . 3 6 I N N O - B A R 0.0773 1 .72 0 .0896 1 .30 0 .0983 0 . 3 3 0 .0896 - 0 . 2 8 0 .0812 - 0 . 3 7 0 .0633 - 0 . 3 8 E L A S T I C I T Y . 0 . 0 0 0 0 0 .0000 0 .0000 0 .0000 0 .0000 0 .0000 ' 9 0 ° ui B A R 0.0000 0 .0000 ' ':'.''•.'' ' 0.0000 ' . '•;- ' ' : ,•.' ; : ' 0.0000 0 .0000 0 .0000 U J N O - B A R •0.0000 0 .0000 0 .0000 0 .0000 0 .0000 ' 0 . 0 0 0 0 CIRCULAR PLATE WITH TWO . POINT LOADS ;^:s;;x>:g ... % '•-. ' / U = RADIAL DISR V * TANGENTIAL DlSR';;5|;Blfi|{;;\?: QUARTER OF THE CIRCULAR; PLATE ; TABLE 4 - 3 2 RADIAL NORMAL STRESS 07 . AND % ERROR AT D I F F E R E N T X AND 9 fx '• 0.45 , 'AS C O E F F I C I E N T C \ , IN CTr = C.-^  BY E L A S T I C I T Y A N D F I N I T E E L E M E N T P A R A M E T E R 0 O = 112.5° \ V 0 . 8 5 R 0.72R 0.52R 0.37 R 0 .3IR 0.22 R FUNCTION %ERROR FUNCTION %ERROR FUNCTION %ERROR FUNCTION %ERROR F U N C T I O N % E R R O R F U N C T I O N % E R R O R E L A S T I C I T Y = 4 . 2 6 9 9 - 2 . 3 2 5 5 - 1 . 4 2 6 8 - 1 . 1 5 6 9 - 1 . 0 9 0 3 - 1 . 0 1 9 7 0 ° uj B A R • - 2 . 8 5 2 4 3 3 . 2 0 - 1 . 9 1 8 5 1 7 . 5 0 - 1 . 3 2 6 6 7 . 0 2 - 1 . 1 1 6 7 3 . 4 8 - 1 . 0 6 2 9 2 . 5 1 - 1 . 0 0 7 8 1 . 1 9 IN- N O - B A R . - 3 . 1 0 1 8 2 7 . 3 6 - 1 . 9 6 9 8 1 5 . 2 9 - 1 . 3 5 8 3 4 . 8 0 - 1 . 1 3 6 9 1 . 7 3 - 1 . 0 8 0 1 0 . 9 3 - 1 . 0 2 1 5 , . - 0 . 1 8 E L A S T I C I T Y - 0 . 8 3 0 0 - 1 . 3 4 0 0 - 1 . 2 0 2 8 - 1 . 0 5 7 9 - 1 . 0 1 4 3 - 0 . 9 6 4 8 10° ui B A R - 1 . 4 5 4 3 - 7 5 . 2 2 - 1 . 4 3 1 6 - 6 . 8 3 - 1 . 1 7 1 0 . 2 . 6 5 - 1 . 0 3 5 0 2 . 1 7 - 0 . 9 9 6 8 1 . 7 2 - 0 . 9 5 6 3 0 . 8 8 IN- N O - B A R - 1 . 4 1 1 9 - 7 0 . 1 1 - 1 . 4 6 4 2 - 9 . 2 6 - 1 . 1 8 9 6 1 . 0 9 - 1 . 0 5 0 0 0 . 7 4 - 1 . 0 1 0 4 0 . 3 8 - 0 . 9 6 8 3 - 0 . 3 6 E L A S T I C I T Y - 0 . 0 8 0 4 - 0 . 4 0 2 2 : - 0 . 7 4 8 8 - 0 . 8 1 1 4 - 0 . 8 1 5 7 - 0 . 8 1 3 9 20° ui B A R - 0 . 0 4 4 1 4 5 . 1 8 - 0 . 5 3 6 1 - 3 3 . 3 0 - 0 . 7 9 6 0 - 6 . 3 1 - 0 . 8 1 9 8 - 1 . 0 3 - 0 . 8 1 8 1 - 0 . 3 0 - 0 . 8 1 3 5 - 0 . 0 5 IN- N O - B A R 0 . 0 3 2 0 1 3 8 . 8 3 - 0 . 5 2 4 2 - 3 0 . 3 3 - 0 . 7 9 2 8 - 5 . 8 8 - 0 . 8 2 4 2 . - 1 . 5 7 - 0 . 8 2 4 2 - 1 . 0 4 - 0 . 8 2 0 8 - 0 . 8 4 E L A S T I C I T Y - 0 . 0 0 1 3 - 0 . 0 9 1 1 - 0 . 3 6 4 5 - 0 . 5 2 0 6 - 0 . 5 6 1 0 - 0 . 6 0 2 1 3 0 ° U I B A R - 0 . 0 1 3 1 - 9 3 3 . 6 0 - 0 . 0 8 3 0 8 . 8 0 - 0 . 4 0 4 1 - 1 0 . 8 7 - 0 . 5 4 3 6 - 4 . 4 1 - 0 . 5 7 6 1 - 2 . 7 0 - 0 . 6 0 8 3 - 1 . 0 3 IN- N O - B A R 0 . 0 0 7 1 6 6 4 . 3 9 - 0 . 0 4 7 5 ° 4 7 . 8 1 - 0 . 3 9 8 0 - 9 . 1 9 - 0 . 5 4 1 5 - 3 . 9 9 - 0 . 5 7 6 0 - 2 . 6 8 - 0 . 6 1 0 7 • - 1 . 4 3 E L A S T I C I T Y 0 . 0 0 9 4 - 0 . 0 0 2 6 - 0 . 1 2 9 6 - 0 . 2 6 5 2 - 0 . 3 1 2 9 - 0 . 3 7 0 4 4 0 ° ui B A R . 0 . 0 1 8 7 - 9 8 \ 7 4 - 0 . 0 0 6 9 - 1 6 5 . 8 0 - 0 . 1 4 0 0 - 8 . 0 0 - 0 . 2 8 2 5 - 6 . 5 6 - 0 . 3 2 7 8 - 4 . 7 5 - 0 . 3 7 8 4 - 2 . 1 5 u.- N O - B A R 0 . 0 1 3 7 - 4 5 . 6 6 0 . 0 0 7 9 4 0 4 . 8 8 - 0 . 1 3 4 5 - 3 . 7 9 - 0 . 2 7 9 1 - 5 . 2 7 - 0 . 3 2 5 2 - 3 . 9 2 - 0 . 3 7 7 4 - 1 . 9 0 E L A S T I C I T Y 0 . 0 1 0 2 0 . 0 2 3 0 - 0 . 0 0 4 8 - 0 . 0 7 5 1 • - 0 . 1 0 7 9 - 0 . 1 5 3 6 5 0 ° ui • . B A R 0 . 0 1 4 9 - 4 5 . 1 7 6 . 0 2 8 8 - 2 5 . 3 8 - 0 . 0 0 4 9 - 1 . 7 3 - 0 . 0 8 1 6 - 8 . 6 8 - 0 . 1 1 5 7 - 7 . 1 2 - 0 . 1 5 9 1 - 3 . 5 7 IN- N O - B A R 0 . 0 1 1 2 - 9 . 8 0 . 0 . 0 2 9 2 - 2 7 . 0 4 0 . 0 0 3 6 1 7 4 . 1 8 - 0 . 0 7 8 1 - 3 . 9 8 - 0 . 1 1 2 6 - 4 . 3 3 - 0 . 1 5 6 5 - 1 . 9 1 E L A S T I C I T Y 0 . 0 0 9 6 0 . 0 3 0 2 0 . 0 5 7 6 0 . 0 5 1 9 • 0 . 0 4 2 9 0 . 0 2 5 9 6 0 ° ui B A R 0 . 0 1 3 0 - 3 5 . 3 6 0 . 0 3 4 3 - 1 3 . 5 7 0 . 0 6 0 6 - 5 . 1 2 0 . 0 5 2 4 - 0 . 9 1 ' 0 . 0 4 1 9 2 . 3 8 0 . 0 2 4 2 6.32 IN" N O - B A R 0 . Q 1 0 0 - 4 . 6 7 0 . 0 3 2 7 . - 8 . 2 9 0 . 0 ' 6 4 9 - 1 2 . 6 7 0 . 0 5 6 2 - 8 . 1 2 ; 0 . 0 4 5 2 - 5 . 5 0 0 . 0 2 7 5 - 6 . 2 8 E L A S T I C I T Y 0 . 0 0 8 9 ] , 0 . 0 3 1 9 0 . 0 8 7 8 0 . 1 2 9 6 0 . 1 4 2 6 0 . 1 5 7 0 -o 7 0 U I B A R • 0 . 0 1 1 0 - 2 3 . 8 4 0 . 0 3 5 8 - 1 2 . 1 5 0 . 0 9 2 3 ' - 5 . 2 0 0 . 1 3 3 2 - 2 . 8 1 j 0 . 1 4 5 3 - 1 . 9 2 0 . 1 5 8 3 - 0 . 8 4 I N N O - B A R 0 . 0 0 9 1 - 2 . 3 6 0 . 0 3 3 3 - 4 . 4 7 0 . 0 9 3 7 - 6 . 7 8 • 0 . 1 3 6 9 - 5 . 6 6 0 . 1 4 9 1 - 4 . 5 4 0 . 1 6 2 1 . - 3 . 2 3 E L A S T I C I T Y 0 . 0 0 8 4 0 . 0 3 2 1 0 . 1 0 1 2 . 0 . 1 7 0 7 0 . 1 9 8 5 0 . 2 3 6 0 8 0 ° ui B A R 0 . 0 1 0 0 - 1 8 . 6 0 0 . 0 3 5 3 - 9 . 9 7 0 . 1 0 6 1 - 4 . 8 1 0 . 1 7 5 6 - 2 . 8 9 •>, 0 . 2 0 2 9 - 2 . 1 7 0 . 2 3 8 8 - 1 . 2 0 IN- N O - B A R 0 . 0 0 8 5 - 1 . 4 1 0 . 0 3 3 0 - 2 . 8 2 0 . 1 0 5 9 - 4 . 6 3 0 . 1 7 8 7 - 4 . 7 0 0 . 2 0 6 8 - 4 . 1 5 0 . 2 4 2 9 - 2 . 9 5 E L A S T I C I T Y 0 . 0 0 8 3 0 . 0 3 2 0 0 . 1 0 5 0 0 . 1 8 3 4 0 . 2 1 6 5 0 . 2 6 2 2 9 0 ° ui . B A R 0 . 0 0 9 7 . - 1 6 . 8 5 0 . 0 3 5 1 . - 9 . 5 4 o . l i o o - 4 . 7 5 0 . 1 8 8 7 - 2 . 8 7 0 . 2 2 1 2 - 2 . 1 8 0 . 2 6 5 5 - 1 . 2 4 NO-BAR ' 0 . 0 0 8 4 ' - 1 7 2 3 0 . 0 3 2 8 ; - 2 . 4 3 0 . 1 0 9 3 - 4 . 1 0 0 . 1 9 1 6 - 4 . 4 2 0 . 2 2 5 1 - 4 . 0 0 . 0 . 2 6 9 8 . - 2 . 8 7 CIRCULAR PLATE WITH TWO POINT LOADS, •i >,; •' •••'•/••' ' ,V;.. '•• • v. ., \ TABLE . 4 - 3 3 TANGENTIAL NORMAL STRESS CTe A N D % ERROR! .V:';-.1 AT DIFFERENT t AND 6 fx * ' 0.45 AS COEFFICIENT C j IN Oe = C-£-f BY ELASTICITY AND FINITE ELEMENT j PARAMETER 60 = 112.5* . | .' . • . ' . . " • • . • f ' . • ' . . - • V 0 . 8 5 R * 0 . 7 2 R 0 . 5 2 R ^ 0 . 3 7 R 0 - 3 I R 0 - 2 2 R . *\ F U N C T I O N ^ % E R R O R F U N C T I O N % E R R O R F U N C T I O N % E R R O R F U N C T I O N % E R R O R F U N C T I O N % E R R O R F U N C T I O N % E R R O R 0° E L A S T I C I T Y 0.3183 0 .3183 0 . 3 1 8 3 . 0 .3183 0.3183 0 .3183 L U u.-B A R '.• 0.4093 - 2 8 . 5 8 . 0 . 3 2 4 1 - 1 . 8 1 0 .3023 5 . 0 2 0 .3085 31 OS 0.31 (05 2.44 0 .3117 0.3199 2 .06 N O - B A R . 0 .0525 83 .50 0 .4373 - 3 7 . 3 9 0 . 3 2 5 1 ' - 2 . 1 2 0 .3197 - 0 . 4 4 0.3197 - 0 . 4 4 - 0 . 5 1 10° E L A S T I C I T Y - 1 . 0 2 3 8 - 0 . 2 4 0 9 0 .1583 0 .2375 0 . 2 5 3 1 0 .2679 uj u.-B A R - 1 . 1 9 9 5 - 1 7 . 1 6 - 0 . 2 0 7 6 1 3 . 8 5 0 .1612 , - 1 . 8 4 0 .2329 1 .94 0.2484 1.87 0 . 2 6 3 1 " 1 .78 N O - B A R - 1 . 2 7 5 5 - 2 4 . 5 9 - 0 . 2 5 2 7 ~ - 4 . 9 1 0 .1705 - 7 . 7 3 0 .2413 - 1 . 5 7 • 0 . 2 5 6 1 - 1 . 1 5 0 .2702 - 0 . 8 7 20° E L A S T I C I T Y - 0 . 6 5 2 7 - 0 . 5 7 8 9 - 0 . 1 4 6 9 0 .0392 0 .0840 0.1297 ui IN-B A R - 0 . 6 0 9 2 6 .66 - 0 . 6 4 5 5 - 1 1 . 5 1 - 0 . 1 4 0 3 ; . 4 . 4 2 0 .0412 - 4 . 9 2 0 .0840 0 . 0 0 0.128.7... 6.79 N O - B A R - 0 . 5 1 5 9 20.96 - 0 . 7 2 6 2 - 2 5 . 4 4 - 0 . 1 3 7 0 6 . 7 2 0 .0453 - 1 5 . 3 7 0 .0883 - 5 . 0 3 0 . 1 3 3 1 - 2 . 5 9 30° E L A S T I C I T Y - 0 . 3 7 9 3 - 0 . 5 2 9 4 - 0 ; 3 7 0 6 - 0 . 1 8 7 7 - 0 . 1 2 9 8 - 0 . 0 6 3 7 U I u-B A R - 0 . 4 0 1 1 . - 5 . 7 4 - 0 . 5 1 3 0 3 . 0 9 - 0 . 3 8 0 6 - 2 . 7 0 - 0 . 1 8 7 3 0 . 2 2 - 0 . 1 2 9 1 0 .54 - 0 . 0 6 2 0 2 .66 N O - B A R - 0 . 3 1 9 7 15.72 - 0 . 5 1 9 7 1 . 8 3 : - 0 . 3 8 5 3 - 3 . 9 6 • - 0 . 1 8 6 5 0 . 6 6 - 0 . 1 2 7 7 1.58 - 0 . 0 6 0 7 4 . 6 6 40° E L A S T I C I T Y - 0 . 2 4 8 8 - 0 . 4 2 5 0 - 0 . 4 7 3 0 - 0 . 3 7 8 1 - 0 . 3 3 3 4 - 0 . 2 7 4 0 ui IN-B A R - 0 . 2 5 5 9 - 2 . 8 7 . - 0 . 4 2 7 7 - 0 . 6 4 - 0 . 4 7 9 9 - 1 . 4 6 - 0 . 3 8 1 7 - 0 . 9 6 - 0 . 3 3 5 1 - 0 . 5 2 - 0 . 2 7 2 1 0 .69 N O - B A R - 0 . 2 1 4 4 13.82 - 0 . 4 0 6 2 4 . 4 2 - 0 . 4 9 4 9 - 4 . 6 3 - 0 . 3 8 3 7 - 1 . 4 9 . - 0 . 3 3 6 0 - 0 . 7 7 - 0 . 2 7 3 7 0 . 1 3 50° E L A S T I C I T Y - 0 . 1 8 1 9 ' - 0 . 3 4 4 3 - 0 . 5 0 0 8 - 0 . 5 1 0 7 - 0 . 4 9 7 0 - 0 . 4 6 9 6 ui IN-B A R - 0 . 1 8 5 3 - 1 . 8 7 - 0 . 3 4 4 3 0 . 0 0 - 0 . 5 0 1 5 - 0 . 1 4 - 0 . 5 1 4 6 - 0 . 7 6 - 0 . 5 0 0 1 - 0 . 6 3 - 0 . 4 6 9 2 0 .07 N O - B A R - 0 . 1 6 1 6 . 11 .14 - 0 . 3 2 4 6 5 . 7 3 - 0 . 5 0 6 7 - 1 . 1 9 - 0 . 5 1 8 6 - 1 ' . 54. - 0 . 5 0 3 1 - 1 . 2 3 - 0 . 4 7 3 3 - 0 . 7 9 60° E L A S T I C I T Y - 0 . 1 4 5 1 - 0 . 2 9 0 7 - 0 . 4 9 6 8 -' - 0 . 5 9 2 5 - 0 . 6 1 3 5 - 0 . 6 3 0 3 ui IN-B A R - 0 . 1 4 4 8 0 .26 - 0 . 2 9 1 3 - 0 . 2 4 - 0 . 4 9 6 9 - 0 . 0 2 - 0 . 5 9 4 3 - 0 . 3 0 , - 0 . 6 1 5 8 - 0 . 3 8 - 0 . 6 3 1 8 - 0 . 2 3 N O - B A R . - 0 . 1 3 1 2 9 . 6 1 - 0 . 2 7 4 7 . 5 . 4 8 - 0 . 4 9 5 0 1 0 . 3 6 - 0 . 5 9 8 5 - 1 . 0 1 . - 0 . 6 2 0 2 - 1 . 0 9 - 0 . 6 3 8 2 - 1 . 2 4 70° E L A S T I C I T Y - 0 . 1 2 4 6 , - 0 . 2 5 7 5 - 0 . 4 8 4 5 - 0 . 6 3 8 1 - 0 . 6 8 8 0 - 0 . 7 4 7 0 ui L L B A R • - 0 . 1 2 2 9 1 .39 - 0 . 2 5 7 2 0 . 1 4 - 0 . 4 8 4 0 ' 0 . 1 0 - 0 . 6 3 8 2 - 0 . 0 1 - 0 . 6 8 8 8 . -*0.11 -0 .7 .499 - 0 . 3 8 N O - B A R - 0 . 1 1 3 9 ; 8 . 5 8 - 0 . 2 4 4 1 5 . 2 2 - 0 . 4 7 8 2 . 1 . 3 0 - 0 ^ 6 4 0 4 - 0 , 3 6 " 1 - 0 . 6 9 3 5 - 0 . 7 9 - 0 . 7 5 8 1 - 1 . 4 8 80° E L A S T I C I T Y - 0 . 1 1 4 1 - 0 . 2 3 9 6 - 0 . 4 7 4 3 - 0 . 6 6 0 2 - 0 . 7 2 8 7 - 0 . 8 1 6 9 ui IN-B A R - 0 . 1 1 1 7 . 2 .08 - 0 . 2 3 8 8 0 . 3 2 - 0 . 4 7 3 9 0 . 1 0 - 0 . 6 5 9 4 0 . 1 3 - 0 . 7 2 8 3 0 . 0 5 - 0 . 8 2 0 6 - 0 . 4 5 N O - B A R -0 .10 '49 8 . 0 2 - 0 . 2 2 7 7 4 . 9 8 - 0 . 4 6 6 3 ,1.70 - 0 . 6 5 9 9 , 0 . 0 4 - 0 . 7 3 2 5 - 0 . 5 3 . - 0 . 8 2 9 9 - 1 . 5 9 90° E L A S T I C I T Y - 0 , 1 1 0 8 - 0 . 2 3 3 9 -A : . . - 0 . 4 7 0 6 - 0 . 6 6 6 7 - 0 . 7 4 1 4 - 0 . 8 4 0 1 ui I N B A R -0 ."1083 2.28 - 0 . 2 3 3 0 0 . 4 2 - 0 . 4 7 0 1 : V 0 . 1 1 - 0 . 6 6 5 6 0 . 1 8 - 0 . 7 4 0 7 0 . 1 0 - 0 . 8 4 4 0 - 0 . 4 7 N O - B A R - 0 . 1 0 2 0 • ' "7.-84 - 0 . 2 2 2 5 ^•4.90 - 0 . 4 6 2 0 V' 1 .83 ' - 0 . 6 6 5 6 \ 0 . 1 8 - 0 . 7 4 4 7 - 0 . 4 4 - 0 . 8 5 3 8 . - 1 . 6 3 . . . . . . ... vr • . I TABLE 4 - 3 4 1 SHEAR S T R E S S T r e A N D % E R R O R ; AT D I F F E R E N T X AND 9 fin ' 0 . 4 5 AS C O E F F I C I E N T C IN ' X R E = C BY E L A S T I C I T Y A N D F I N I T E E L E M E N T x1 P A R A M E T E R 90 = 112.5 V 0 . 8 5 R 0 . 7 2 R 0 . 5 2 R ; 0 . 3 7 R 0 . 3 I R 0 . 2 2 R FUNCTION %ERROR FUNCTION % ERROR FUNCTION %ERROR FUNCTION %ERROR F U N C T I O N % E R R O R F U N C T I O N %ERROR E L A S T I C I T Y 0.0000 0 . 0 0 0 0 . 0 . 0 0 0 0 , 0 .0000 0 .0000 0 .0000 0° B A R 0.0000 0.0000 0 .0000 ; 0.0000 ooooo O-OOOO IN-N O - B A R . 0 .0000. 0 .0000 0 .0000 0 .0000 0 .0000 0 .0000 E L A S T I C I T Y 1.0718 0.8843 0.4614 0 .3180 0 .2840 0.2489 10° ui . B A R 1.3463 •. - 2 5 . 6 0 0 .8396 5 . 0 5 0.4294 6 . 9 3 0 .3056 3 . 9 3 0.27i51 3 . 1 5 0.-2428 2 .43 IN-N O - B A R 1.3713 - 2 7 . 9 4 0 .7909 10 .56 0 .4215 8 . 6 4 n.inu 4 .fin 0.27141 3 .50 0 .2432 2..29 E L A S T I C I T Y 0 . 3 1 5 1 0 .6401 ;•' 0.6375 0 .5295 0.49i32 0 .4509 20° ui B A R 0.1407 55 . 35 0 .6880 - 7 . 4 8 , 0 .6334 ' 0 . 6 5 0.5194 1 . 9 1 0.4837 1.92 0 .4425 ... 1.87 U J N O - B A R . 0 . 1 5 4 5 5 0 . 9 5 0 .7580 - 1 8 . 4 2 0.6294 ' 1 .28 0.5160 • 2 .56 0.48 r 16 2.36 0 .4429 1.77 30° E L A S T I C I T Y 0.1144 0.3449 0 .5786 0 .6004 0 . 5 9 2 1 0 . 5 7 5 1 U I B A R 0.1667 - - 4 5 . 7 4 0 .3022. 1 2 . 3 8 0 .5878 - 1 . 5 9 0 .5992 0 . 2 1 0 .5880 0 .69 0 .5683 1 .20 U J N O - B A R 0.1256 - 9 . 8 0 0 .2988 1 3 . 3 6 0 . 5 8 9 9 - 1 . 9 5 0 .5965 - 0 . 6 5 0 .5864 0.97 0 .5690 1.08 E L A S T I C I T Y 0.0515 0.1863 0 .4422 0 .5629 0 .5896 0 .6128 40° ui B A R 0.0404 21.57 0 .1953 - 4 . 8 5 0 .4405 0 .37 0 .5644 - 0 . 2 7 0 .5893 0 .04 0 .6089 . 0 .64 U J N O - B A R 0.0386 25,07 0 .1778 4 . 5 3 0 .4533 - 2 . 5 1 0 .5666 - 0 . 6 4 0 .5899 - 0 . 0 4 0 .6104 0 , 4 0 50° E L A S T I C I T Y 0.0265 0.1052 0 .3113 0.4670'J 0 .5165 ] 0.5730 ui B A R 0.0318' - 2 0 . 0 6 0.1017 3 .27 0 .3079 1 .10 0.4663. 0 . 1 5 0 . 5 1 6 4 0 .02 0 . 5 7 1 1 0 .34 IN-N O - B A R 0 . 0 2 7 1 - 2 . 4 6 0 .0944 1 0 . 2 2 0.3077 1 . 1 5 0 .4706 - 0 . 7 6 0 .5197 - 0 . 6 1 0 .5739 - 0 . 1 5 -E L A S T I C I T Y 0.0145 0 .0608 0 .2066 0 .3496 • 0 . 4 0 4 1 0 . 4 7 4 1 60° ui B A R 0.0132 9 .24 0 .0622 - 2 . 2 9 0 .2052 0 . 6 6 0 .3475 0 . 6 1 0 .4029 0 .30 0 .4728 0.27 IN-N O - B A R 0.0134 7 .63 0 . 0 5 6 8 / 6 .49 0 .2028 . 1 . 8 2 0.3513 - 0 . 4 9 0 .4069 - 0 . 6 9 0 .4765 - 0 . 5 1 E L A S T I C I T Y 0.0078 , • 0.0339 0 . 1 2 5 1 0 .2296 0 .2740 • 0 .3352 70° U I B A R • 0 .0082 - 3 . 9 1 0 .0335 1 .19 0 . 1 2 4 1 0 . 7 8 0 .2278 0 . 8 2 0 .2725 0 .54 0 . 3 3 4 1 0 .34 U J N O - B A R 0.0079 ••' - 0 . 4 1 0 .0313 7 .43 0 .1215 2 .58 0 .2295 . 0 .04 0.2757 - 0 . 6 2 0 .3376 - 0 . 7 1 E L A S T I C I T Y 0.0034 0 .0152 0 .0589 0 .1133 0.1377 . - . 0 . 1 7 2 9 80° ui B A R 0.0033 6.44 0 .0153 - 0 . 5 6 0 .0586 0 . 4 3 0 .1122 0 . 9 1 0 .1368 0 .67 0 .1722 0 . 4 3 I N N O - B A R 0.0034 0 .00 0 . 0 1 4 3 , 6 .17 0 .0572 . 2.94 0 .1129 . 0 . 2 9 0 .1384 - 0 . 4 7 0 .1743 - 0 . 7 9 E L A S T I C I T Y 0.0000 0 .0000 0 . 0 0 0 0 . 0 .0000 0 .0000 0 .0000 90° ui B A R 0.0000 . . 0 .0000 0 .0000 0 .0000 ' ., •• 0.0000 0 .0000 I N N O - B A R 0.0000 0 .0000 • 0 .0000 0 .0000 0 .0000 0 .0000 . v a l u e s of one f u n c t i o n c a l c u l a t e d by d i f f e r e n t means. The e r r o r s i n X and Y d i s p l a c e m e n t s f o r a l l v a l u e s of P o i s s o n ' s r a t i o i n case of the framework model a r e c o n s i s t e n t l y l e s s than Vfot When u s i n g the No-Bar c e l l s , the e r r o r i n X - d i s p l a c e m e n t s i s about Z% and i n Y - d i s p l a c e m e n t s i t i s c o n s i d e r a b l y greater,, S t r e s s e s o b t a i n e d by b o t h , the method of n o d a l d i s -p lacements and the method of n o d a l f o r c e c o n c e n t r a t i o n s , are equal l y good i n case of the framework model. T h i s e r r o r i s about 2% to yfo. The p e r c e n t a g e e r r o r i s g r e a t e r f o r s m a l l v a l u e s of s t r e s s e s and i t improves c o n s i d e r a b l y f o r l a r g e r s t r e s s e s , When u s i n g the No-bar c e l l s , the s t r e s s e s o b t a i n e d by the method of n o d a l d i s p l a c e m e n t s are as much as \\Ofo to $0% i n e r r o r and f o r t h i s r e a s o n u n a c c e p t a b l e 0 T h i s i s because of the l a r g e r e r r o r s i n the computed d i s p l a c e m e n t s and the s e n s i t i v i t y of the method to these e r r o r s . The method of n o d a l f o r c e c o n c e n t r a t i o n s thus p r o v e s t o be more r e l i a b l e f o r c a l c u l a t i n g s t r e s s e s when u s i n g the no-bar c e l l s , l\.«.3o2 Problem of C i r c u l a r P l a t e A p a r t ! f r o m the v i c i n i t y of the c o n c e n t r a t e d l o a d s good r e s u l t s are o b t a i n e d i n c a l c u l a t i o n of d i s p l a c e m e n t s both by bar c e l l s and no-bar c e l l s . T h i s a p p l i e s t o a l l t h r e e v a l u e s of P o i s s o n ' s r a t i o u sed. However the p r e c i s i o n w i t h the b a r c e l l s i s perhaps t w i c e as good f o r , b o t h r a d i a l and t a n g e n t i a l d i s p l a c e m e n t s e x c e p t i n c l o s e p r o x i m i t y to the l o a d (no f a r t h e r away than two s t e p s ) . As t o the s t r e s s e s determined by the n o d a l f o r c e con-c e n t r a t i o n , not c o u n t i n g the point- of a p p l i c a t i o n of the l o a d , t hey are about i n e r r o r i n the case of normal s t r e s s e s i n the - 1 2 1 -r a d i a l d i r e c t i o n and about 2% i n the case of normal s t r e s s e s i n the t a n g e n t i a l d i r e c t i o n and the s h e a r i n g s t r e s s ? re- „ The bar c e l l s and the no-bar c e l l s g i v e s t r e s s e s of comparabie. p r e c i s i o n * • CHAPTER 5 T r a p e z o i d a l Bar C e l l , a n d Melosh's P o s t u l a t e I t i s s t a t e d i n s e c t i o n lj.,1.2 t h a t the r e s u l t s o b t a i n e d by u s i n g c e l l s w i t h p o s i t i v e b a r a r e a are b e t t e r than when seme are a s are n e g a t i v e . T h i s statement seems to be i n a c c o r d w i t h 8 the o b s e r v a t i o n made by Melosh w i t h r e g a r d t o c o m p a r a t i v e com-p u t a t i o n a l q u a l i t i e s of d i f f e r e n t s t i f f n e s s m a t r i c e s of f i n i t e e l e m e n t s . Melosh c o n s i d e r s no bar f i n i t e elements d e v o i d of f r e e body movements. I f an element of t h i s k i n d i s f o r example q u a d r i l a t e r a l i t s s t a t e of d e f o r m a t i o n i s d e s c r i b e d by 8 - 3 = 5 n o d a l d i s p l a c e m e n t s and i t s s t i f f n e s s m a t r i x has 5*5 s i z e . F u r t h e r m o r e the m a t r i c e s of c e l l s c o n s i d e r e d by Melosh have a l l p o s i t i v e e i g e n v a l u e s . Prom g e n e r a l c o n s i d e r a t i o n s b u t w i t h o u t a s p e c i f i c p r o o f 8 Melosh p o s t u l a t e s t h a t the m a t r i x w i t h the s m a l l e s t e i g e n v a l u e s c h a r a c t e r i z e d by the s m a l l e s t t r a c e ( t h e sum of the p r i n c i p a l d i a g o n a l terms) produces the b e s t r e s u l t . I n the f o l l o w i n g the deep beam problem i s s o l v e d a g a i n u s i n g the same c e l l and M = 0 . 3 3 3 3 , b u t u t i l i z i n g t h r e e d i f f e r e n t v a l u e s of the angl e parameter ~Q0 , some of which r e s u l t i n negat-i v e b a r a r e a s . These have been chosen i n the f o l l o w i n g way. 1. For one v a l u e of •&<, a l l b a r areas a re p o s i t i v e , a l l e i g e n v a l u e s of the m a t r i x are p o s i t i v e and t h e i r sum, the t r a c e , s m a l l . . 2. F o r a n o t h e r 0 o , one of the bar areas i s n e g a t i v e , a l l e i g e n v a l u e s a g a i n axse p o s i t i v e , and the t r a c e g r e a t e r than i n the f i r s t c a s e , 3. For a t h i r d -Q-0 , s e v e r a l b a r areas are n e g a t i v e and so - 1 2 3 -are some of the eigenvalues. These data are presented i n Table 5-1* The c e l l of Case 3 does not belong to category considered by Melosh, but the c e l l s 1 and 2 do. Of these, according to Melosh, the f i r s t should give the better p r e c i s i o n . The three c e l l s are used for c a l c u l a t i o n of displace-ments i n the problem of the deep beam analysed before at the nodes indicated i n F i g s 5>-l« ^be re s u l t s including the % error are assembled i n Table 5 - 2 , The displacements are a l l less than 2% i n error i n f i r s t two cases, although the precision of the Case 1 i s from 2 to 5 times better than that of the Case 2 0 The Case 3 i s d e f i n i t e l y worst of a l l , some functions especially those which are numerically small, being more than 10% out. These r e s u l t s , admittedly limited i n scope, confirm both the d e s i r a b i l i t y of using c e l l s with a l l positive bar areas, and employing s t i f f n e s s matrices with a l l positive eigenvalues and least trace. -1214.-a = \>15 b t = thickne T A B L E S - I A r e a J s c t r a m e . t e . i ' s , e i g e n v a l u e s a n j { i ' o c L °£ S+i-T-Tne.se> M a t r i c e s u)iih__ d J f{o. r e n t cm <j jpqrq In <L*ter -fy> • fL 6 - 3 3 3 3 A - r^ao o) -e-0 - ioo° (2) &0*l'35° (3) O o - 60° A , o-34o<& at I ' 4 c 6 3 at - 0 - 6 3 4 4 at A," O. 15.57 °X I - 8 8 6 7 *t - 0 -ito '2-2- at A ?_ O- 1 3 70 at ~6 - i S 2 . o at !• £522.1 qt A 3 O. 61 62. tft 9 » 2 . s r o 4 a t -| -2.853 at A A O- 6 0 30 at 3 - Q S " 2 3 at -)-2s7g at at 4-27-3 0 at O 633.2 at e igenvalues > l« 8 2 . 1 6 3 < o 4 85 1 • 91 6 3 2. o . 814-9 S"l G 3 0. 81 02_ 3 0 - 7 0 3 0 I - 8 0 0 I O - 6 0 98 4 O. £JI30. O ? 0 3 3 - 0 - 4 ^ . 0 0 .5" O..S3 87 - 1' I 0 6 0 t r a c t ! . - 4 - - QSOJo 1 = 7 0 3 3 126-11! ~ LL < ^ I T ^ - tD 5 — '13 sr ^ 3 o -a-___ r- \f) 1 ^ |0 vO ST O O Cj r- i _ G ?J C r« • -• ~o -0 ^£ o X < vj o I -—> J o in z UJ V-~Z-T1 <3 o — N \ X O •Ll! K X -O VP 4-> o to (A ol I ro oo si o 5") K K I Ofl ro & N T ci O in Co Co 6 V9 K <0 K K 6 C O I O 0 K O* O O o o CD Oo CO <0 o <0 rO o in Oo bo ro 6 S N o In o K ft 5} ro T J T T 6 CD CD to •"9 6 in 6 rO o ST rO ID o i I Co 10 sr 4) 6 ) o I rO 10 CX) C O 0 IB c4 6 O CO 8 v9 ID 6 i 6 6 in Oo 5) o <5> 6 N CX) Oo CO In to ii) X T Co i o rO! <o O | 6 ro Ol i Oo - j in : 6) o i o l o o o «0 CO _ i _ K in •J) rO rO oo ; 1 Oo 6 10 o O0 oi sr o o 0^ oo ro in Oo ! v£ o! ^ j Oo O | 6 OS ! 101 ro o^ !~t o 5* 9 IF ro o ! 6 NT I ^ V9 -1 (0 ro oo 00 0^ 6 oi ! rO j ol CO j oi j ^ o \ o\ 6 "4"; !ni -to oo j CO ! — O j O ; O I O K i O | •? I ^ i ro I ; ^ 1 Lf»: 9^ VO o CO In o ro i0 \3 oo C O to "oTTnT V3 | co i rO i — CH| tO 115 ro; • i .-0 : CO i ro! <H\ ~ j v5> | ro I s{~ O o 6 Co 6 fO 1? 6 o o 6 8 c3 In Do 10 6 co rO O o 6 o o o b Oo LO rO <5o Ol 6 o *o 6 <9 rO ot CO 10 fO 0^ TO rO ro 1A - 1 2 7 -C OWC LIT ST QNS The present i n v e s t i g a t i o n leads to the f o l l o w i n g c o n c l u s i o n s o 1. C e l l s i n the shape of i s o s c e l e s trapezoids are qu i t e s u i t a b l e f o r plane s t r e s s analysis» They are probably i n f e r i o r to r e c t a n g u l a r c e l l s i n problems i n v o l v i n g r e c t a n g u l a r beams, i n view of some complications a r i s i n g i n determination of stresses e i t h e r from nodal displacements or nodal force c o n c e n t r a t i o n s . On the other hand the t r a p e z o i d a l shape seems i d e a l i n a p p l i c a t i o n to p l a t e s i n the form of f u l l c i r c l e s or s e c t i o n of c i r c l e s . 2. Judging by the examples of the deep r e c t a n g u l a r beam and the c i r c u l a r d i s c , the f i n i t e element method making use of t r a p -e z o i d a l bar c e l l s i s a valid.method of s t r e s s a n a l y s i s . 3. The c e l l s used contain three novel f e a t u r e s , double bar areas, a v a r i a b l e angle parameter and extension of some bars be-yond the o u t l i n e of the c e l l . None of these has been found to exert an unfavourable e f f e c t on the v a l i d i t y of the c e l l , although the l a s t f e a t u r e gives the c e l l an unusual appearance, ij.e An admittedly l i m i t e d i n v e s t i g a t i o n confirmed the pre-v i o u s l y held s u s p i c i o n that c e l l s with negative bar areas pro-duce lower p r e c i s i o n . 5. The adju s t a b l e v a r i a b l e angle parameter has been found h i g h l y b e n e f i c i a l , a l l o w i n g one to avoid negative bar areas i n c e l l s of c e r t a i n geometries with c e r t a i n values of Poisson's r a t i o , and through that improving the p r e c i s i o n of the r e s u l t s . 6. Of the two methods f o r determination of s t r e s s e s , i n a rec t a n g u l a r beam the method.of nodal displacements has been found i n f e r i o r to the method of nodal force concentrations i n - 1 2 8 -view of i t s s e n s i t i v i t y t o even s m a l l e r r o r s , i n h e r e n t i n c a l -c u l a t e d d i s p l a c e m e n t s . 7. The method of n o d a l f o r c e c o n c e n t r a t i o n s has been found q u i t e s a t i s f a c t o r y f o r c a l c u l a t i o n of s t r e s s e s b oth i n the deep beam and i n the c i r c u l a r d i s c . 8„ On comparison w i t h the no-bar c e l l s , the r e s u l t s o b t a i n e d w i t h the bar c e l l s have proved on the whole, more p r e c i s e , b oth i n d i s p l a c e m e n t s and s t r e s s e s , , 9. L i m i t e d s t u d y i n d i c a t e d p a r t i a l c o n c u r r e n c e of the r e -s u l t s w i t h the p r i n c i p l e e n u n c i a t e d by Welosb r e l a t i n g the q u a l i t y o f the s t i f f n e s s m a t r i x of the c e l l w i t h i t s c h a r a c t e r -i s t i c r o o t s and trace,. -129-BIBLIOGRAPHY 1. Hrennikoff, A., "Solution oi* Problems of E l a s t i c i t y by the Framework Method", Journal of Applied Mechanics, ASME., December 19lj-l. 2o McCormick, C.W., "Plane Stress Analysis", Proceedings of the ASCE, Structural D i v i s i o n , August, 1963. 3. Yettram, A.L., and Robbins, K., "Space - Framework Method for Three-Dimensional Solids", Proceedings of the ASCE, Engineering Mechanics D i v i s i o n , December 1967* I;. Hrennikoff, A., Mimeograph notes on Course CE 551* at University of B r i t i s h Columbia, Vancouver. 5. Agrawal, K.M., "Analysis of E l a s t i c Shells of Revolution with Membrane and Flexure Stresses Under Arbitrary Load-ing Using Trapezoidal F i n i t e Elements", Ph.D Thesis Dept. of C i v i l Engineering, University of B r i t i s h Columbia, Vancouver, 19680 6 . Timosbenko and Goodier, "Theory of E l a s t i c i t y " , McG-raw H i l l Book Company. 7. M u s k b e l i s b v i l i , N.I., "Some Basic Problems of the Mathematical Theory of E l a s t i c i t y " , Translated from Russian by J.R.M. Radok, P. Noordboff Ltd., 1963, PP 330 - 338 0 8. Melosh, R.J'., "Structural Analysis of Solids", Proceedings of ASCE, Structural D i v i s i o n , August 1963.. 9. Pestel, E., "Investigation of Plate and Shell Models by Matrices", Report to Office of Aerospace Research, U.S.Air Force, Brussels, Belgium, 1963« 

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