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Finite element method in application to plane stress problems Mahapatra, Bijaya Chandra 1967

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FINITE ELEMENT METHOD IN APPLICATION TO PLANE STRESS PROBLEMS by BIJAYA C. MAHAPATRA B.Tech (Hons), I.I.T., Kharagpur, India, Feb., 1964 A THESIS SUBMITTED IN PARTIAL FULFILMENT OF THE REQUIREMENTS FOR THE DEGREE OF MASTER OF APPLIED SCIENCE in the Department of CIVIL ENGINEERING We accept this thesis as conforming to the required standard THE UNIVERSITY OF BRITISH COLUMBIA September, 1967. 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 l y a v a i l a b l e fo r re fe rence 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 hi.'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 . Department The U n i v e r s i t y of B r i t i s h Columbia Vancouver 8, Canada (1) ABSTRACT Framework cel ls and no-bar cel ls are used for solving plane stress problems by the Finite Element Method of Analysis and the results obtained by using the two types of cel ls are compared with the e last ic i ty solution in two specific examples. The precision of the Finite Element Method is tested at different values of the Poisson's ratio of the material. The stresses found by the Finite Element Method are calculated in two ways: from the corner forces and the corner displacements, and the quality of the results employing the methods is compared. The precision of the results obtained with the bar cel ls has been found consistently better than with the no-bar c e l l s . Comparable precision has been observed in the stresses found from the corner forces and the corner displacements. ( i i ) ACKNOWLEDGEMENT The writer is greatly indebted to Professor A. Hrennikoff for his i n -valuable guidance in the preparation of the thesis. The writer wishes also to express his gratitude to Dr. Bulent 6vun$ and Mr. K.M. Agrawal for many valuable suggestions. Thanks are also due to the U.B.C. Computing Centre for free use of the computer. ( i i i ) INDEX Page No. CHAPTER ONE — INTRODUCTION 1 CHAPTER TWO — DERIVATION OF STIFFNESS MATRICES 3 2.1 Bar Type Cells 3 2.2 Rectangular Pattern with Auxil iary Members 3 2.3 Rigidity of the Cell 9 2.4 Stiffness Coefficients for the Cell 11 2.5 No-Bar Cells 16 2.6 Stiffness Coefficients of the No-Bar Cell 18 2.7 Symmetry of Stiffness Matrices 20 2.8 Comparison of Matrices 20 2.9 Solution of Plate Model by Finite Element Method 20a CHAPTER THREE — EXAMPLES OF SOLUTION BY FINITE ELEMENT THEORY 3.1 Example 1 21 3.2 Example 2 24 3.3 Discussion 27 CHAPTER FOUR — ACCURACY OF FINITE ELEMENT METHOD AND POISSON'S RATIO 28 (iv) Index Page No. CHAPTER FIVE — DETERMINATION OF STRESSES FROM NODE DISPLACEMENTS 30 CONCLUSIONS 33 BIBLIOGRAPHY 34 Genii 15.9.67 ChapteA Om INTRODUCTION The analytic solution of plane stress problems involves biharmonic equation subject to given boundary conditions. Unfortunately, however, selection of proper stress function, satisfying both the differential equation and the boundary conditions, is rarely possible and so i t is very seldom that the problems can be solved exactly. An ingenious method of solution was, however, introduced by Prof. •1* Hrennikoff in 1941 . The method consists of replacing the plate under con-sideration with a plane framework consisting of bar units or c e l l s . The cel ls have a certain definite repreating pattern and are joined to each other at the corners. The external outline of the framework must conform to the outline of the given plate, and the framework must carry exactly the same forces and in the same locations as the plate. The bars have the same modulus of e last ic i ty as the prototype and they must possess some appropriate cross-sectional areas, determined from the condition that the framework deforms in the same manner as the plate when placed in an arbitrary uniform stress f i e l d . The use of the framework method requires the knowledge of a relat ion-ship between the corner forces and the corner displacements. If one of the corners of a cel l is given unit displacement in the direction of one of the co-ordinate axes x or y, while the other corners are held in posit.-fon, the x and y components of the corner forces required to hold the cel l in the deformed state are called the "distribution factors" or "stiffness coefficients" of the c e l l . The distribution factors corresponding to * Numbers signify the references, l isted in the Bibliography - 2 - Introduction displacements of a l l corners, both in x and y directions, are arranged in the form of a matrix called the "stiffness matrix of the c e l l " . The model obtained by replacing the plate with elast ic f in i te cel ls is highly indeterminate and i t s solution involves numerous linear simul-taneous equations. Originally, the solution was achieved in a laborious way by the relaxation method. Subsequent introduction oof digi tal com-puters drastical ly reduced the labor of computation. Cells not involving bars have been also proposed. Unlike the frame-work cel ls the no-bar cel ls cannot be constructed physically, but l ike their framework counterpart, they allow one to obtain a set of d i s t r i -bution factors which may be used in the solution of plane stress problems. It is desirable to compare both the bar cel l and the no-bar cel l solutions with the e last ic i ty solutions, where such are available. The distribution factors, for both types of cel ls are functions of the Poisson's ratio of the material. The effect of the Poisson's ratio on the precision of the solution is also examined. The plate stresses in the Finite Element solution are found in two ways: from the nodal concentrations and from the joint displacements. The quality of the two results is compared. Cliapt&i Two DERIVATION OF STIFFNESS MATRICES 2.1 BAR TYPE CELLS Different patterns of bar. type cel ls and determination of their para-meters have been discussed in ref. 1 and 2. However, i t is f e l t desirable to repeat parts of this discussion in application to rectangular cel ls in the present work. 2.2 RECTANGULAR PATTERN WITH AUXILIARY MEMBERS This pattern is applicable to any arbitrary value of Poisson's rat io . It also allows one to use different sizes of cel ls in the same problem. Consider a rectangular isotropic plate of uniform thickness ( f ig . l a ) . The corresponding framework cel l is shown in f i g . l b . t* CL - H l-s — a Fig. la Fig. lb By an arbitrary assumption, the inner rectangle has dimensions half as great as those of the outer rectangle; the diagonals pass over each other at the centre without being connected. The external forces are assumed to be applied only at the main jo ints , and not at the joints of the inner rectangle. The deformability of the framework is judged on the basis of displacements of the main joints only, while the inner rectangles are viewed simply as an internal mechanism insuring proper deformability of the framework as described by the main jo ints . - 4 - Derivation of Stiffness Matrices 2.2 RECTANGULAR PATTERN WITH AUXILIARY MEMBERS (cont'd) The cel l has five kinds of bar members: two kinds of outer bars, two of inner bars, and one of diagonal bars, assumed to possess the same area on the fu l l length. However, i t has been demonstrated^ that such a cel l can have the maximum of four independent characteristics, and so the area of the vertical bars of the inner rectangle is assigned at w i l l , the area k times the one of the inner horizontal bar, i . e . kA^. It is easy to see that under a l l conditions, the opposite sides of the inner rectangle are stressed equally. This follows from conditions of equilibrium of the four joints of the inner rectangle. Thus at jo int (8) , Fig. 2 S2^ sin<>£ = S^ CosoC and at jo int (6) 1 * S 2 sin<*i = S 2 Cosoi. Therefore S 2 = S 2 Coto£ = kS 2 ) * ) and S 0 = S0 ( 1 1 ] Fig. 2 Similarly considering the equilibrium of joints (5) and (7) one gets S,!, = which shows that the bars on the opposite sides of the inner rectangle are stressed equally under a l l circumstances. The four independent parameters A, A-|, A,,, and A 3 of the cel l must be determined from the conditions of equal deformability with the plate in an arbitrary uniform stress f i e l d . - 5 - Derivation of Stiffness Matrices 2.2 RECTANGULAR PATTERN WITH AUXILIARY MEMBERS (cont'd) A general condition of uniform stress is achieved by combination of the following three types of uniform stress states: xy <TX = (Ty = (2) It is more convenient, however, to make use of the following three strain conditions which lead to the same result as (2): xy xy x = " xy Cx = *y (3) CONDITION 1 £ x y.ty = V*xy = o Uniform stresses p^  and y U P - | are applied in x and y directions .respectively ( f ig . 3). pr — t t t t t Fig. 3 - 6 - Derivation of Stiffness Matrices 2.2 RECTANGULAR PATTERN WITH AUXILIARY MEMBERS (cont'd) Since there is no deformation in horizontal bars and '1 V . S where ka 1 (4) Transferring the edge stresses to corners by simple statics and analyzing an outside corner of the c e l l , one gets: and S1 = fcf-SA.Cos* = • * O - / * K * ) Final ly , equating (4) and (5) (5) A, CONDITION 2 ' ~ ~ . a t (6) Y ' — Y 2 2 © Fig. 4 This condition is analogous to Condition 1, carrying out the analysis in a similar manner, one gets: • out (7) - 7 - Derivation of Stiffness Matrices 2.2 RECTANGULAR PATTERN WITH AUXILIARY MEMBERS (cont'd) CONDITION 3 Pure shear condition is assumed as shown in Fig. 5. PLATE C E L L Fig. 5 From the deformability conditions (Fig. 5), i t follows that .the side members .are unstressed while the outer parts of the diagonals are equally stressed with tension in one diagonal and compression in the other. i . e. S 3= f y - f - Vi+fc1 t e n * i o n . . - a n d 5$- f^JtV'n-K*- Comp. (8) The equality of the displacements of the corners (T) and (3) of the rectangle leads to equality of deformations of the two diagonals, one in tension, the other in compression by the amount Coz>aL The antisymmetry of the loading condition demands that the stresses in the opposite sides of the inner rectangle must be equal and opposite. But, according to the statement of equation (1)., they must also be equally stressed, thus the sides of the inner rectangle, in condition 3, must remain unstressed, i .e S 0 = KS0 = 0. - 8 - Derivation of Stiffness Matrices 2.2 RECTANGULAR PATTERN WITH AUXILIARY MEMBERS (cont'd) With stresses S 2 , being zero, the diagonals are stressed with the same stress on their f u l l length and the change in length of the diagonal substituting the value of from (8) and equating equation (9) to osCoscL one gets: at To evaluate the area A 2 , apply Condition 2 (Fig. 4) again. Consider the inner rectangle: Unit strain in the horizontal bar = 5 2 /AZB (11) Unit strain in the vertical bar = s'2 / K A 2 E (12) But S 2 = KS2 from (1) Thus, from (11) and (12), the unit strains in the horizontal and vertical bars are equal. Since the load is symmetrical about both axes, the inner rectangle remains a geometrically similar rectangle after deformation. i . e . : Unit strain in side members: = Unit strain in diagonals. i > e - : , - S 2 = S 4 OR S 9 = _V. S. • (13) A2E A 3 E A 3 consider the equilibrium of joint (4): - 9 - Derivation of Stiffness Matrices 2.2 RECTANGULAR PATTERN WITH AUXILIARY MEMBERS (cont'd) S3 Cos* = - I op. S 3 = (14) Similarly , from equilibrium of Joint (8) S3 = S 4 ->- S 2' Cos °£ + S 2 SvnoL = 5^ + S2v"> -t-k2- ( 1 5 ) From (13) , . (14.),. .and (15) Cosot 4 A 3 ^ 2 Cos 4 A3 (16) The total elongation of the diagonal member 2 A 3 E where £2 = 6-A1*) £ t Substituting the value of S3 from (14) in the above equation, one gets: 5 4 = - f e r ^ - • (17) Final ly , equating (16) and (17) and substituting the value of A3 from ( 1 Q > : A - ~t 2.3 RIGIDITY OF THE CELL It is necessary to see that the cel l has enough bar members in relation to the number of joints to behave as a r ig id structure. The cel l has 14 members and 8 jo ints , not counting the points of intersection of the diagonals. In s tat ica l ly determinate r ig id structures, the numbers - 10 - Derivation of Stiffness Matrices 2.3 RIGIDITY OF THE CELL (cont'd) of bars (b) and joints (j) are related by the equation of b = 2 j -3 . Hence, in the present case, the number of bars required = 2 x 8 - 3 = 1 3 which is one less than the number of bars present. The cel l i s , there-fore, indeterminate and is obviously r ig id . The cel l may become non-rigid in special circumstances. Examination of the expressions for A and A^  shows that the main horizontal bar be-2 i comes zero when k -/ fc = o, i . e . : k = i f . Similar ly , the vertical bar vanishes when k •= '/V^ tT . Both these conditions signify that i f the ratio of the shorter side to the longer side equals V^c-* t the area of the longer bar becomes zero. The total number of bars in the cel l then becomes 12 making the structure non-rigid. A model con-sist ing of such cel ls may s t i l l perform adequately in conditions of uni-form stress but i t becomes a mechanism when the stress is no longer uniform as is the case in a l l practical problems. The conclusion is that the cel l with a side ratio of V m a y n °t be permitted. The area A 3 of the diagonal can never be zero. The areas A 2 and kA2 of the auxil iary bars become zero only when = 1/3. Then the mid-length hinges on the diagonals disappear and the cel l reduces to that of a simple rectangular pattern maintaining i t s r ig id i t y . The cel l may become structurally unstable under compressive stress, in view of the absence of e last ic restraint to rotation of the inner rectangle about an axis perpendicular to the plane of the framework. However, this does not affect the calculation of bar stresses since the model is not used for investigation of s tab i l i t y . - 11 - Derivation of Stiffness Matrices 2.3 RIGIDITY OF THE CELL (cont'd) As varies between the l imits 1/2 and o, A 2 changes a l l way from + inf in i ty through zero to a negative-value. The area A becomes negative Negative values of A, A-j or A 2 make i t impossible to construct a physical model of the cel l but this does not affect i t s appl icabi l i ty in calculations. 2.4 STIFFNESS COEFFICIENTS FOR THE CELL Calculation of distribution factors, after obtaining the areas of the bars, is a matter of ordinary structural analysis. However, the easiest way of obtaining them is by the use of an auxil iary condition in combination with the three main conditions discussed above. AUXILIARY CONDITION - 4 The cel l is subjected to a state of deformation.in which one of i t s main vertical members is lengthened and the other shortened by the amount :.2 S ( F i g . 6). x when K < ff/lC • Similar ly , the area A, becomes negative for K>V/^ JT. Fig. 6 It follows from antisymmetry of the stress condition about the vertical axis that a l l bars other than the two main vertical mem-bers remain unstressed. The stresses in the vertical sidebars, - 12; - Derivation of Stiffness Matrices 2.4 STIFFNESS COEFFICIENTS FOR THE CELL (cont'd) and corner forces equal to them, are given by the equation: 2 A i E r _ i - / * k z . / * = S ' =• ~ k a 4 7 7 ^ 7 * E t * 4 It should be noted that in the previous three conditions, the corner forces were obtained from the plate stresses and not from the bar stresses as is done in the Condition 4. Before combining the three conditions to produce unit displacement of any corner, i t is necessary to demonstrate the nomenclature for the stiffness coefficients of the c e l l . The eight directions of displacement of the cel l corners are shown in Fig. 7(a) and the eight corner forces corresponding to unit displace-ment in direction 1 are shown in Fig. 7(b). Fig. 7(a) Fig. 7(b) In general, = ^ o r c e required l n the direction i to produce a unit displacement in the direction j , in the absence of a l l other displacements. The relationship between the corner forces and the corner displace-ment for a quadrilateral cel l may then be written by the matrix equation (1 - 13 - Derivation of Stiffness Matrices 2.4 STIFFNESS COEFFICIENTS FOR THE CELL (cont'd) " k n k12 k13 k14 k15 k16 k17 k18 F 2 k21 k22 k23 k24 k25 k26 k27 k28 A a F 3 k31 k32 k33 k34 k35 k36 k37 k38 A 3 F 4 k41 k42 k43 k44 k45 k46 k47 k48 * 4 F 5 k51 k52 k53 k54 k55 k56 k57 k58 A * F 6 k61 K62 k63 k64 k65 k66 k67 k68 "Aft F 7 k71 k72 k73 k74 k75 k76 k77 k78 A 7 LF-J k81 k82 k83 k84 k85 k86 k87 k88 ( A 8 18) The eight corner forces corresponding to any one of the main corner displacements may be found directly by moving the corner in question, but they are more conveniently determined by combining the several basic conditions analyzed above. Not a l l the conditions need to be used to produce the unit displacement of a particular corner. For example, to 1 obtain a unit displacement in x direction of corner (T), (Aj=1),.only-three conditions are used (Fig. 8). The distribution factors are obtained by adding up the corner forces of individual conditions (Figs. 3, 5, and 6) with the values of "p"s replaced by corresponding S values expressed in terms of A, (Fig. 8). - 14 - Derivation of Stiffness Matrices 2.4 STIFFNESS COEFFICIENTS FOR THE CELL (cont'd) k*' (d) Fig. 8 The eight forces of figure 8(d) represent the elements of Column 1 of the stiffness matrix (k).:in equation (18). A similar set of eight forces can be obtained by giving unit displacement to corner (T) in the y direction. The distribution factors corresponding to unit displacement of corner (T) in x and y directions are summarized below in Table 1: TABLE 1 Corner (JJ Unit Displacement in y Direction Corner (JJ Unit Displacement in x Direction v l l 4 + K z c i -3/n) 8K 0-/u») v21 E C Et c12 = v22 Et - 15 - Derivation of Stiffness Matrices 2.4 STIFFNESS COEFFICIENTS FOR THE CELL (cont'd) TABLE 1 (cont'd) Corner (1) Unit Displacement in x Direction Corner (1) Unit Displacement in y Direction ' '31 E t K32 i - 3 / ^ Ml 8c*'- A * - 1 ) Et K42 0 - 3 / * ) If /*» - 4 K 2 E t '51 - 4 +K a(i»/H) h 8Kcr^uTr v52 1 - 3 / * . '61 E t v62 71 E t 72 EE 8 6-/*) '81 60 -/O '82 E t The corner forces produced by the displacements of corners (2), (3), and (4) are numerically equal to the corresponding forces resulting from displacements of the joint (T), presented in Table 1, although some of them have opposite signs. This correspondence and. the signs are determined by inspection. The stiffnex matrix expressed in terms of the coefficients of Table 1 is given in equation (19). 8x8 k l l k12 k31 k41 k51 k61 k71 k81 k21 k22 k32 k42 k52 k62 k72 k82 k31 k32 k n "k21 k71 "k81 k51 "k61 k41 k42 "k21 k22 -k 7 2 k82 "k52 k62 k51 k52 k71 "k72 k l l "k21 k31 "k41 k61 k62 "k81 k82 "k21 k22 k32 k42< k71 k72 k51 "k52 k31 k32 k l l k21 k81 k82 -k61 k62 "k41 k42 k21 k22 (19) - 16 - Derivation of Stiffness Matrices 2.4 STIFFNESS COEFFICIENTS FOR THE CELL (cont'd) Examination of the matrix (19) reveals that i t is symmetrical about the main diagonal. i.e.: This result is in agreement with Bett's rec:t(3jr<?cal theorem. 2.5 NO-BAR CELL 4 Finite Element cel ls not made up of bars have also been proposed. Their nature may be explained as follows: Imagine a piece of rectangular plate placed in a stress f i e l d , preferably a simple case such as that of the uniform stress f i e l d (Fig. 9(a)). The deformation of the plate under this stress f i e l d is also shown in the figure. Fig. 9(a) S= K-XJ K / T Iff FINITE ELEMENT CELL Fig. 9(b) The stresses acting on the edges of the plate are assembled at the cor-ners in a manner consistent with statics such as the one employed ear l ier in the framework cel ls (Fig. 3). A different method based on energy considerations may also be used for this purpose. The transfer of edge stresses to the corners transforms the piece of plate into a no-bar cel l (Fig. 9(b)) to be used in the analysis. The ce l l must satisfy several - 17 - Derivation of Stiffness Matrices 2.5 NO-BAR CELL (cont'd) of such stress conditions. Proper combinations of these would result in single corner displacements and would allow one to determine the d ist r ibu-tion factors. A plate polygon of N-sides has 2N degrees of freedom of i t s corners as they move while the cel l deforms under stress. Three of these d is -placements may be affected by r ig id body movements without imposing any stresses on the c e l l . This leaves (2N-3) independent stress conditions required for derivation of stiffness matrix, irrespective of actual shape of the f in i te element. A rectangular cel l thus demands five (2N-3=5) stress conditions for obtaining i t s stiffness matrix. Of these f i ve , three must correspond to the uniform states of stress, i . e . : the same states as used for the bar type c e l l s . The corner forces obtained for these three con-ditions are equally applicable for the present case since they were found from the plate prototype and not from the bar stresses as explained ear l ier . The two extra conditions to be used now are shearless bending in the plane of the plate with stresses acting in x and y directions. CONDITION - 4* - Bending with stresses acting in x Direction (Fig. 10(a)). No stresses of any kind are present on the.vertical sides of the element and no shear stresses on the horizontal sides. *c--f> 4 f Ko. •< — - CX >-r K Y Fig. 10(a) i i / Fig. 10(b) if x c 6 - 18 - Derivation of Stiffness Matrices 2.5 NO-BAR CELL (cont'd) The corner forces for the cel l (Fig. 10(b)) s tat ica l ly equivalent to the flexural stresses are found thus: X .a = (2 • ) x (P4 ) c 3 a ^a OR: X, CONDITION 5* - This condition is similar to Condition 4* with stresses acting in y direction (Fig. 11) •a +5 \ f - - ^ / 2.6 STIFFNESS COEFFICIENTS OF. THE NO-BAR CELL Conner (T) is made to displace a unit distance A^=l by bining the three conditions as shown in Fig. 12. com-CJOLN.D--- 1 Fig. 12 - 19 - Derivation of Stiffness Matrices 2.6 STIFFNESS COEFFICIENTS OF THE NO-BAR CELL (cont'd) The distribution factors are obtained by adding up the corresponding corner forces of the individual conditions as was done with the bar c e l l s . The set of distribution factors corresponding to unit displacement of corner (T) in direction 2 (A2=l ) may be obtained in a similar manner from the three conditions 2 , 3, and 5*. The results are summarized in Table 2 . TABLE 2 Corner (1) Unit Displacement in x Direction =^1 Corner (1) Unit Displacement in y Direction 2=1 " l l = ] Et £ t '12 80- AO '21 8 (i-M) '22 12 8 K 6 - » A ) Et '31 .UK <<-/«•») E t '32 '41 ^r- E t '42 8 0 - ^ ) '51 Et E t '52 '61 '62 (2fA*2>* _ ' ^  Et 71 E t 72 g O - A*) '81 E t '82 , i z Q~,u a ) 8*. fit AO, E t The distribution factors corresponding to unit displacements of corners (2), (3) and (4) are obtained from Table 2 in the same way as for the bar type c e l l . Equation 19 is equally applicable in,the present case. The same set of distribution factors have been obtained from the energy consideration in Ref. 4. - 20 - Derivation of Stiffness Matrices 2.7 SYMMETRY OF STIFFNESS MATRICES The matrix formed of the coefficients of Table 2 l ike the matrix described earl ier is symmetrical about the principal diagonal. Symmetry of the stiffness matrix of a ce l l made up of bars is the direct consequence of Bett i 's reciprocal theorem, since the framework ce l l is a real e last ic structure and Bett i 's theorem is applicable to a l l l inear e last ic structures. As to the no-bar c e l l s , i t s stiffness matrix is always symmetrical when i ts terms are determined from the energy considerations. With i t s terms found by the Law of the Lever, as described above the matrix is symmetrical or unsymmetrical depending upon the cel l i t s e l f being symmetrical or unsymmetrical. Thus the matrix is symmetrical for a rectangular cel l 3 and unsymmetrical for a cel l in the form of a trapizoid. 2.8 COMPARISON OF MATRICES It is interesting to note that some of the corresponding te,nms in Tables 1 and 2 are equal and i t is easy to see the reason for this peculiarity. The elements in Column 1 of Table 1 were obtained by com-bining the Condition 4 with Conditions 1 and 3 (Fig. 8). The corresponding elements in Table 2 were obtained by combining Condition 4* with the same conditions 1 and 3 (Fig. 12). Thus the contribution of the Conditions land 3 to the two sets of distribution factors is the same. Condition 4 and Condition 4* have no corner forces in the y Direction, thus making the total corner forces in the y Direction equal for both types of c e l l s . That is k ^ , k ^ , k g l , and kg-j are the same for both. On the other hand, the Condition 4 and 4* contribute different corner forces in the x Direction thus making the distribution factors in the x Direction ( k n n , , k ^ , and k 7 1 ) unequal for the two types of c e l l s . - 20a - Derivation of Stiffness Matrices 2.8 COMPARISON OF MATRICES (cont'd) A similar reasoning shows that, the distribution factors of the Column 2 in x Direction are the same in both Tables. 2.9 SOLUTION OF PLATE MODEL BY FINITE ELEMENT METHOD The plate under consideration is replaced by a model consisting of many cel ls of proper shape and size and is provided with the same boundary conditions and the loads "F" as the prototype. The solution of the model involves determination of a l l nodal displacements for which purpose the stiffness matrix (K) of the model is set up by the computer by combining the stiffness matrices (k) of the individual c e l l s . The computer solves the system of simultaneous equations: The plate stresses are subsequently calculated by dividing the nodal forces by the corresponding tributary areas equal to the product of the length or breadth of the cel l and the thickness of the plate. An alternative procedure of finding the stresses directly from the displacements is discussed in Chapter V. the c e l l s . Thus: Cha.pte/1 Jh/itd EXAMPLES OF SOLUTION BY FINITE ELEMENT THEORY Finite element solutions, making use of different matrices, pro-cedures and values of Poisson's ratio are compared on two examples for which there exist rigorus e last ic i ty solutions. 3.1 EXAMPLE 1 A thin semi-infinite plate is acted upon by a concentrated force P per unit thickness directed perpendicularly to the straight edge (Fig. 13) P/UNIT T H I C K N E S S The expressions for the displacements u and v in x and y Directions are ' 7T E. 7T£ (20) The significance of the symbols is given in Fig. 13. D is the distance from the origin; of a point on x-axis which is considered stationary. - 22 Examples of Solution by Finite Element Theory 3.1 EXAMPLE 1 (cont'd) The expressions for stress are given by: <r = - ^ • c o s 4 © (21) 6L = S;»4e In view of the symmetry of the structure and the loading system i t is suff icient to analyze one-half of the model which is assumed to consist of 12x6 cel ls of size (axa).- The immovable point F is located at the bot-tom of the model. The boundary conditions are prescribed (Fig. 14) in terms of the known displacements along the boundary BCF, determined by a rt r : I IV -6 MO. <9>*z %—H -An\$ OF S Y M M E T R Y Fig. 14 Y DISPLACEMENTS ARE APPLIED 0 - Indicates no d is -placement 1 - Indicates possible displacement in that direction. - 23 - Examples of Solution by Finite Element Theory 3.1 EXAMPLE 1 (cont'd) the e last ic i ty equation (20). No horizontal displacements are present along the axis of symmetry AF. The stresses and displacements at the nodal points along the x-axis are shown in Plate 1. The plate consists of several graphs presenting stresses and displacements along the x-axis for the values of Poisson's ratio f*-= o.o, o.2 and o.4. Each graph contains three curves: the curve of stresses or displacements determined by e last ic i ty solution and the two curves of the percentage deviations of the bar and no-bar models solutions from the e last ic i ty values. At most of the points, the errors are below 10% and usually much below this f igure, except where the function i t s e l f (stress or displacement) is very small. However, greater errors in (Tx are found at the node next to the point of application of the load P. The reason for this discrepancy is quite clear from the sign of the error and the nature of the true solution. At the point P the e last ic i ty stress (Tx is in f in i te and at the adjacent node directly below P, i t is very high retreating fast both ways to the le f t and to the right from the axes of symmetry. The nodal concentration in x Direction naturally reflects the mean stress (Jx over a certain adjacent area and th is , of course, is smaller than the true stress on the. axis of symmetry. The Finite Element Stress tTx is thus numerically smaller than the true stress. The error of this type when the e last ic i ty stress exceeds numerically the Finite Element stress is called negative. Cfy andTxy stresses on the x-axis are zero. The high percentage error has a practical significance only when the values of the function (stresses or displacements) are themselves - 24 - Examples of Solution by Finite Element Theory 3.1 EXAMPLE 1 (cont'd) large and not when they are very small. The results obtained at the joints near the applied load P can be made closer to the e last ic i ty solution by using smaller rectangular cel ls joined to the bigger cel ls below by appropriate triangular elements. However, the results w i l l be affected by the presence of the triangular cells^and the direct comparison of the precision of the f in i te element method with regard to stresses and deflections while using the rectangular bar and no-bar cel ls w i l l be obscured. In Plates 2 and 3 are presented the data pertaining to the Sections AA and BB similar to the ones of Plate 1. Here again, the results are much the same as in Plate 1, i . e . : the errors are reasonably small and con-sistently better with the bar c e l l s . 3.2 EXAMPLE 2 In this example a point load is applied within an in f in i te plate (Fig. 15). Fig. 15 The exact values of stresses and displacements are: (2 2) - 25 - Examples of Solution by Finite Element Theory 3.2 EXAMPLE 2 (cont'd) UL -C H-AO* S i n 0 • Cos 6 4 7 T E (23) The significance of the terms in these equations is the same^and as in Example 1. D is again the distance from the origin of the point on the x-axis which is considered stationary. In view of the symmetry of the structure i t is suff ic ient to analyze only one-quarter of the model which is assumed to consist of 12x6 cel ls of the size a x a. The immovable point on the x-axis is located at the bottom of the model. The boundary conditions for the model are prescribed in terms of i t s known displacements along BCF. (Fig. 14), determined from equation (23), as in Example 1, with the exception that the y-displacements (v) along the y-axis AB are also specified to be zero. The results obtained by the f in i te element theory are shown graphically in Plates 4, 5, and 6 in the same way as in the Example 1. The stresses . and displacements at nodal points along x-axis are shown in Plate 4. Because of symmetry v andTxy have zero values along this axis. The values of U. obtained by using both types of cel ls have an error less than 3%, but the results obtained with no-bar model are slightly better than the bar model. The values of (Tx obtained by using the bar model are much better than'those obtained with the no-bar model. Both models give a high percentage error at the node next to the point of applica-tion of the load P, and the reason for this discrepancy has been ex-plained in Example 1. tTy values are sometimes better with the no-bar model and sometimes with the bar model. - 26 - Examples of Solution by Finite Element Theory 3.2 EXAMPLE 2 (cont'd) The stresses and displacements along the Sections AA and BB passing through the middle of the model are presented in the Plates 5 and 6. The results obtained with the bar model are consistently better than with the no-bar model. 3.3 DISCUSSION The difference between the distribution factors of Table 1 (bar cel l ) and Table 2 (no-bar cel l ) is that the former are based on imitation of deformabilities of the prototype by the model in conditions of uniform stress, while in the lat ter , the model and the prototype are made to deform in the same manner not only under uniform stress, but also in conditions of shearless bending. For this reason, one might expect a greater accuracy of the no-bar c e l l s . However, the results of the two examples do not confirm this expectation. The plate stress distribution in practical engineering problems is never uniform. At the same time, the manner of deviation of the plate stress from uniformity is not l ike ly to be that of shearless bending. Hence, from the theoretical point of view, the results obtained with the no-bar model need not necessarily be better than with the bar model. Actually, in the two given examples, the bar model was found to be superior to.,: the other. As the size of the mesh decreases, the stresses in the v ic in i ty of the individual cel ls become more uniform and the results obtained by either type of cel ls should become more precise and in the l imi t should 3 converge to the e last ic i ty solution. Therefore, the use of either type of cel l appears quite proper in the Finite Element Method of Analysis, but the precision is expected to be better with the bar c e l l s . - 27 - Examples of Solution by Finite Element Theory 3.3 DISCUSSION (cont'd) Although the shape of the cel ls in both examples was square, i t is f e l t that the results obtained with oblong cel ls would not be much d i f -ferent as long as the ratio of the sides of the rectangles differs suf-f ic ient ly from the relation K = "J~f£> . ChtipteA faun. ACCURACY OF FINITE ELEMENT METHOD AND POISSON'S RATIO The precision of the results obtained by the Finite Element Method seems to depend on the value of the Poisson's rat io , and so i t is desirable to study more closely this dependence. In the e last ic i ty solution, the displacements are always functions of /I* , but the stresses are influenced by ft- only when the loads are applied within the area of the plate, as in example 2, and are independent of with the loads acting on the periphery of the area, as in Example 1. The distribution factors in the Finite Element Method depend on . Then one may expect that in the problems of the latter type the effect of on the stresses must somehow cancel out. The effect of ^ on the percentage error and displacements found by the Finite Element Method is demonstrated in Plates 7, 8, 9 and 10. The f i r s t two of these refer to the Example 1 and the last two refer to the Example 2. The exact values of the functions and the percentage errors are found at six nodes located on the lines 15-21 and 36-42 of Figure 17. Five values o f r a n g i n g from 0.0 to 0.4 at an interval of 0 .1, are used for the above purpose. The Plates 7 and8 show that in most cases the curves of percent error versus /IA. for stresses are l inear. In some cases the errors change from positive to negative or vice versa. The percentage errors for dis -placements " UL" do not change much with the change in / U . . Their graphs are l inear with small slopes in either direction. The displacements " v " have small absolute values at some of the joints under consideration resulting in a high percentage error by the Finite Element Method. The - 29 - Accuracy of Finite Element Method and Poisson's Ratio curves of errors of y at such points are not shown in the figure. In Example 2, the formula (22) showsr.that Mr Pi where f(r ,e) is constant for any particular point and (6"") represents any one of the stress components t^x, 0Jyt and f x y at the same point for different values of . Equation (24) shows that the stresses at any point vary l inearly with . The percentage error of stresses at joints 17, 19, and 21 (Plate 9) vary almost l inearly with the Poisson's Ratio and the amount of variation is signif icant as seen from the graphs, while for joints 38, 40, and 42 the variation is less signif icant. Like the previous example, thee errors at times increase and at others decrease with increase of the Poisson's Ratio. The plots of percentage error of displacements versus ^ are also nearly l inear in most cases, although the amount of variation from the true values is very small. No substantial difference is noticeable in this respect between the two examples in the f i r s t of which the true stresses are independent of the Poisson's Ratio, and in the second one are functions of i t . The percentage errors in both examples are comparable. Chapttn. Vivz DETERMINATION OF STRESSES FROM NODE DISPLACEMENTS In Chapter 3, the Plate stresses were calculated from the nodal force concentrations. As an alternative procedure, they may also be found from the joint displacmenits. Stresses at a point in terms of strain components are: - fr 0 *+A*Y) The strains may be evaluated in two different ways. a) Consider an inter ior jo int (5) in Fig. 16. I K ex — cu — > H ko. i f ko-i Y. 1 8 Y Fig. 16 The average strains at joint (5), in the simplest form, may be given in terms of i t s displacements of i t s adjacent jo ints , thus: 2 k. a. U6-U4 (2 0 V 8 - V 4 ) 2 a 2Kcx where u and v are the displacements in x and y directions res-pectively. The subscript shows the corresponding jo int . Substitu-tion of equation (26) in (25) leads to: L-2KO. 2 a. P/UNIT THICKNESS A 1 X A 1 NUMBERING OF JOINTS AND SECTIONS ALONG WHICH STRESSES AND DISPLACEMENTS ARE COMPARED FIG. 17 - 31 - Determination of Stresses From Node Displacements (27) b) A more elaborate expression for the strains at the node 5 making use of the four surrounding c e l l s , leads to the following expres-The no-bar model was used for comparison of stresses found by equations (27) and (28) in the Examples 1 and 2. At some nodal points, the results are better with equation (27) and at others-with equation (28). The more elaborate formula appeared to have no extra precision of results over i t s more simple counterpart. Stresses obtained from the corner forces and the nodal displacements (equation 28) were compared in accuracy for the bar and the no-bar models. The results are shown in Plates 11, 12, 13, and 14. The Plates consist of several figures each containing three curves showing the e last ic i ty solution and the percentage error in stresses employing the two methods. The curves show that for the bar model the results are s l ight ly better with the corner force, although the difference is - 32 - Determination of Stresses From Node Displacements very small. For the no-bar model the results at some points are better with the corner forces while at the other points, they are better with the nodal displacements. Thus, both the nodal force and the nodal displacement methods appear adequate for calculation of stresses, but the results obtained have not been extensive enough to allow one to form a definite conclusion as to the superiority of one method over the other. CONCLUSIONS Actual solution of examples shows that: It is quite proper to use both the bar type and the no-bar type rectangular cel ls in the Finite Element Method of Analysis. The precision of the values of stresses and displacements obtained with the bar model is better than with the no-bar model. The accuracy of the Finite Element Method depends upon the property of the material. The nature of variation of the percentage error in stresses and displacements at any point is l inear in y^ - in most cases. No noticeable difference in the percentage error in stresses for different values of is observed between the two examples, in one of which the true stresses depend on ^ and in the other do not. The plate stresses may be calculated either by employing the nodal force concentrations or the joint displacements. The results ob-tained by the two methods are comparable in precision. BIBLIOGRAPHY 1. A. Hrennikoff, "Solution of Problems of E last ic i ty by the Frame-work Method", Journal of Applied Mechanics, A.S.M.E., New York, Vol. 63, Dec. 1941. 2. A. Hrennikoff, "Framework Method and Its Technique for Solving Plane Stress Problems", Publications of International Association  for Bridge and Structural Engineering, Zurich, Switzerland, Vol. 9-1949. 3. A. Hrennikoff, Mimeographed Notes On Plane Stress, in Course CE 551. 4. R.W. Clough, "The Finite ELement Method in Plane Stress Analysis", Proceedings of A.S.C.E. 2nd Conference on Electronic Computation, Pittishburgh, Sept., 1960. 5. Timoshenko and Goodier, "Theory of E las t i c i t y " , McGraw H i l l Book Company. RESULTS A t NO DAL P 0 I NTS ALONG X — i A X R E S U L T S A L O N G S E C T I O N B B _ _ j R E S U L T S A L O N G , S E C T I O N A A RESULTS AT NODAL POINTS ON :X - AXIS f -T RESULTS ALONG SECTION BB I RESULTS ALONG SECT ION: A A IS OA G.0 to* g RESULTS ALONG S E C T I O N f B B R E S U L l T S . A L O N G S E C T I O N A A , _ L -I L T T X ... t I X ...i_L : J: ::m. j.i. - L L n . J..1 -!_].. X L .tlii± Ij.L! I. "I 1 T Hit i F F H , P . T T : L t J ± • X X X X t l L f p j—} P J X [ T X L -r i-1 I.X.L-U-ii!'|T1TiT X L X L X ; . I i ! -i-,- r |-. - i - j - i - , -(45 r r T ' -xxrX - L x: j.. - i. i i_ LJXx np: LHLXL ! - 1 -'-.14- -!.- - V-L '-J-4-4 ] - i . : l - y 4 i : ^ H : b f f l : t x XXjXXTXL :btt±|±4± ! LX L P ! i •4-h-T ±= t ' T I T X L X T " ! T m - 4 - t -- L r r x m ; i I 1 •,-(-!-,• X t X 1—i --j- :--!-•]-P t - l -n i i x x u P j XH g x. L J X L i X - i X X j p L P .: LL - H -i...L. T - i -• L L x "i i .LP -4-+-H - L P . r . . | . t X L L I X M i l l r r T l H I- L J.4—J-M - i l . i t •H T T r! -j-4-L.U -1-L.i-, i - h f l - H j - H -•• H X P L X L H I X ' --H-+-H-XX i±tM4"tUtL X B L H J X -' I LX 44 J_L.!'" i..L'.Ep_L. I ! L f.! L L X L . L M-H--H-H -A-i H- 4fH-iM m . PTE P P . LX _ I f-t-4 - - X X .L tt -t p.. L : E X A M P U E J T X - i -i± T _|J! -H-S-i i I K X X r r X XiXT . L.j.Xj_L ixriz T T " fcB-H _ L . - i L L _.x^i±Hx x J_L; T m X f f l - 3 ' . . . . . m t f m " * X H 4 L i T l t o " ' m 4 4 i l - ± l -I . L L [.I ..4-1X ri J" 1 ! J_. I P , ! XI- ! I 1 4 . J _ J L .. .HQ S t X X Q . L 4X i ! 4-: I ' l l L p _ . i ;.|_ i .LL. • 1 1 L L L i L p 1 P f—1—I r—J- -j—1 ( f t i s , j. L P rrLf-i# J.-LL ! .Li it-H-ii-H r H H '"' • t / . j L V u p - u i j r; • XI - r r r r 1 I - L L I 1 4-j-. H^  T T Q r 0 L L T T r t t E l T . t t r : j : f r t m x • i l l ! ! ! P : 1 - L :Li-. IJLT i_u.T JLLL.L I. ! 1 1 [-La_|.j. L] . 1 _LJ_P. ;..J.: 1 T -P IT - -M- : - i -1 !_.-! L • 4 r H t i - ! - U u u - u Li4£hT14 1_ L_. j _ i 1 1 ^•P±l I i 1 HJJ-htitill "r p i-l T'L£" H-H-! - -iLTT.LJ.J..j..1 i . ' ^ P L t r p t 45—-L~+-;-i i r i t E - i n r i - j - i i ~ 4 - i - r . - n t T i T j o y 4;:;H . ;T _, P I L ? : r .j L...j I L P - i T% ERROR jBjYJ" B^RlMOWL"H\j : : q . , ,_L Us. _i L P til !4 i - M ^ i - l - ! 'XLS LN4--I P i | - H H-i-Li h-!.m-H4 xtH "1LL r j ; t-.-i k—-^-]JU-ilt I.' L i J X L ' X i - L j . . - + T TJLJJ , x L U . : t L M ILrLLuX^-l- L J ... :3.n| i|h|La[^ riAT-^N*:iT»o:iNT.n X L X -i-rj-• i;-ri - ^ T T-rf-!-44 cLIGUftES. -ITtl'ftT-HXt L!N*:JT»ai .jLiA^E^^lUfJ:E^.Rr|F^U L L i i;X ;4 : JOJfNT :.:(;;2il4 rj±r fi tt LctLl-tttixt^ Hx.itttirH-i~r t : - w • 1/3 •4-1-6 1 H i - -! [ i r ! l-bl- - H - L rP _ l _ -1 •-•I r~r 4- n "14-4 r-1..1- +r|.f - i T £ H ^ : . - i:F-:X:+-i i i ! II l l | , i I T I I 1 ; , 1 I i I ! j ; . ! u T p , n 1 1 * , 1 "i4 TTJ_ _.L L - L 1 N H i . x i l i . L L | _ X L i X L l T l H--I-H-JZLLLLLL i x ^ ' 1 -x t : - - L - l . L l I Tl xrosr •i 1- III •; ^ . -d-r ; 4 ; T W .1.  _i_ Y'L . 1-1. X T .p i. _ U . L L . U 4 - - L - P ' L P 1 IL I F f i ; T P H.j...uj:iq7; P T S 3 - U 4-,4-. i - i . H4HH- -H-i-l-04 T T r r x f T " T L ' T ' ! -1-H-x x i " l L .i..L X T rr-J T - U . X - X - 4 - L L P P . I --I --i I : I - - f L -I-H j-! rr-f-r J L D : T F X L J - I - J . I L xhL-ii T T T X 3*^ . L TX1 -;-i-(-PI : | ± . f : •|;1X H4-X--H I- ! H:xiXJX . p r i ! _ L | Pi--1- -;- 1 . .txt-li IXJ-LL 1 1 1 i 1 i I 1 i ^ L X i i N r ! i_i. "PT 1 1 1 XX -LXi 11 L L -^^dpipjlp^l^Ptp. •Ell LVALUES.!. r 0 F |V AT. .@ WRE V!E|RYL"bMA-j-lL-.-H-4XL; "\: fr r i % tRROft BY; | F. !E.. METH © D L j S iLARGp' ; - H i " j t j X " . f U . _ L L 4 _ U X ifFF TxrUTf-iH 1 _i M - L L P T P " X L T X 4 Xbdi TT'l I' •.-rt x 1 T TT 4 X .1 . 1 P_L 1 - x-b i x : .1 L-f-4-fe ] 4 - { i h h H n ^ # - H ^ f - 4 l X h ^ - f - - ^ - p - i 7 1.. - i -r X i -Pi--P r[-'-F I X -X-" .• . i . .1:i-i ; 4 J j l i" h M : ± j X x x x t a . P I I I I p ! I I •Vrfprprmtt^Puz T - i - H i H l H-xFH+Ht 1 1 L. L.I "I .JXrAlNpX^ .dTiL^  • • , I I ! • 1 1 I ' . i I I : ! I \— -r •1:TT" ± . i L X I J X-x - i x H !4XX.rj3: • i -1T . . . 5 . ±1-- L L hf-T HLLXX^-I. TH-P- rH i-Ki 1 1 r r r i TTrrlT^n „ Pi P XCJ_1.L I a. 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' , ' j I '- I - l - l -i+iii-iiLJiL L-u_i_L.j_i_r ...44.:.. ~T~*"i"i~ ~T ixr4nr™~ ."iipr" r ,j i;i,p.:,Lrrr " T,V. p,.,:7|rripr"" -I — ~ —• 4-+--'-.P_L ^-4-i-J-j-iJ.. -i-i-.;—. 11; ...|.J..;_...^ _...^ .4_u.L.h!J: .o^gflijfg 7rp._r-i:;x":^~^m :::::: i . , ; P'l 9 9 • ;. • •• * — : ~ T - , ; r . ¥ m :m:.pr* J S C ' S iXL.' 1 ' _±1L *-v+-pr - S — r - H r ' 4 - ! - ' • 1 44-:t±±±± Ll I 1 -1 J. !._i_T-- H_irj_L4i Li Illilii J _ i -- 4 -J 4 J : L . .i_iij r . : i T .44^111144 _LJJ4 - { -X4- .J .J r . rL.Lj_i . . f ^ p N ^ . . . . . . . . . _ L L . . U _ } . J _ U . r .Li4_i- _ . , L - X I ^ V J . . X _ . 44444 !^44_U444J I L ^ ~ ^ . . -4-4-J_4_i. ' - - i - i - J_:- i . . -'-.. - . . 4?5- J—i—. _ L i f 1 i L i L t i ! i I i j . f I i T I I ''j':TJ.T_':_.!J.4 n.i_L..r. n~ n n i T ; - ' ' I 4 I i ! r nT4±a4d4Tril:i4 J J L I 4 4 - U . - L 4 - 4 4 4 . ; J _ J J i : Tj ; i i i i l l J:tEEBn±^ E4Tf -lIIi-.4_t_4Li-J_I4..4.. ,' J4---L4-4.-H--4^4-4-PH-hM-H-r+x L J L T G ; ± 4 : 4 ^ . . U J i i T i j n 1 i T I n r T T , i T T i X t t i n r r - j - v - i • "": L "o! i 44LI±4±bbbHd:r a4:4;4iaiH+rH± 4 4 - t ^ H - r f f 4 r H 4 R E S U L T S ALONG B B R E S U L T S A L O N G A A RESULTS ALONG SECTION BB RESULTS ALONG SECTION A A R E S U L T S ALONG S E C T I O N BB - - .RESULTS A L O N G S E C T I O N . AA? i ' - i t RESULTS ALONG SECT ION B B R E S U L T S A L O N G SECT ION A A 

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