E - \u2014 [ 7 ] (4.100) da - - da T - - . . da f 1=1 -where Q and 1? are the nodal variables for the nodes on TQ and Tj; E\u00b0 is the number of elements between the contours TQ and and S\u00b0 their element matrices. The change in the element matrices can be calculated directly as SS? 3S? 3x. = T^\" (4.101) 3a 3x. 3a 3 where the nodal coordinates x^ are thought of as functions of the crack length a. The derivatives 3x_.\/3a are then unity or zero, depending on whether or not x^ . is the x-coordinate of a node located on TQ, respectively. Alternatively, 3S^\/3a may be approximated by a simple forward f i n i t e difference scheme 3S? AS? \"v \u2014\u2022 - \u2014 -\u2022 = \"T [S? -S? ]. (4.102) 3a Aa Aa -x -\u2022\u2022 \"a+Aa \"\"\"a Here S\u00b0 are the element matrices for the elements between the contours T 0 and Ti, calculated for the i n i t i a l crack length a, and S? when x coordinate a+Aa of each of the nodes lying on TQ have been incremented by Aa. The equations (4.98) and (4.100) suggest that to calculate the stress intensity factor R^., the master f i n i t e element matrix equation need only be solved once, i.e. for the i n i t i a l crack length a. After obtaining this solution, pre- and post-multiplying the differentiated element matrices of equation (4.102) with the solution vectors for the corresponding nodal variables and then summing these over a l l the elements between TQ and Tj yields the rate of change of potential energy in the discrete sense ATr\u201e\/Aa. Alternatively, the differentiated matrices can be assembled and then M pre- and post-multiplied by the solution vectors u and t for the nodal dis-placements and stresses, respectively, for the nodes on TQ and Ti , as in 77 equation (4.100) a+Aa -so ][\u201e]. (4.103) Finally, the stress intensity factor Kj for plane strain state can be computed from the requirement that i t be internal to the body and enclose the crack tip. It can also be shrunk to a single node at the crack tip so that the sum-mation in (4.100) extends over the elements adjacent to the crack tip only A glance at Figure 3 and the procedure outlined above for determining the potential energy release rate indicate that It is an area-analogue of the path independent J-integral. (4.104) The contour TQ to be translated is thus far arbitrary except for 78 CHAPTER 5 APPLICATION OF BOUNDARY CONDITIONS So far only the homogeneous boundary conditions have been considered in the development of the theory of mixed methods. The treat-ment of homogeneous, mixed homogeneous and non-homogeneous boundary condi-tions , how these can be incorporated in the mixed f i n i t e element method, and equivalence to boundary residual concept are presented in this chapter. For i l l u s t r a t i o n purposes the plane stress linear e l a s t i c i t y problem with unit thickness i s considered. The governing differential equations are - T . . . = f. (5.1a) T.. = 2ue.. + Xe 6.. (5.lb) 13 13 kk 13 e = l\/2(u, , + u. ) (5.1c) i j i\u00bbj 3,1 where x \u201e and e are symmetric second order tensors, and i=j=l,2. Equations (5.1a) are the equilibrium equations relating stress gradients to the body forces f^, (5.1b) are the constitutive equations where X and u are Lam\u00a3s constants and (5.1c) are the kinematic equations relating strains to displacement gradients. Assuming that the equations (5.1c) are satisfied identically equations (5.1) reduce to the following set of equations - T = f ; i=j=l,2 (5.2) \u2022\u2022-J >J -1-1 \/ 2 ( U i , 3 + U 3 , i ) \" W k l \" 0 5 i-J-K-J-1.2. (5.3) Where C^j^j is the fourth order compliance tensor. For E, the Young's modulus and v, the Poisson's ratio defined as 79 _ u(3A+u), _ J (X+y) ' V 2(X+u) the equations (5.2) and (5.3), for a plane stress problem, can be written in the following form: x - x = fx xx,x xy,y - x - x = fy xy,x yy,y (5.4) (5.5) u, - - (x - vx ) = 0 x E xx yy v, - - (-vx + x ) = 0 y E xx yy 2(l+v)_ u, + v, - \u2014 x = 0. y x E xy (5.6) (5.7) (5.8) The equations (5.4) to (5.8) can now be put into matrix form 0 0 0 0 5 3x 0 0 T-3y 3y 3x 3x u 3y 0 _3y_ _3x ^ - i 0 E E 0 0 -2(l+v) v XX yy xy which takes the equivalent form of (3.35) as \"0 T*l IT -C u f X 0 0 0 0 (5.9) (5.10) or Where AA = p, u =; or = u and u 2 X =, xx yy xy v X* = - x = - i - o 3x 3y 0 3_ 3_ \"3y ~3x (5.10a) (5.10b) (5.10c) (5.10d) 80 T T T A = and f =, x y and the compliance matrix C is a symmetric positive definite matrix 1 -v 0 ,C=f -v 1 0 0 0 2(l+v) The energy product of definition 3.4.3, for unit thickness, becomes (AA,A) = [A,A] = \/ [uTT*x + xTTu - xVrjdfi. (5.10e) (5.11) This, on integration by parts of the f i r s t term on the right hand side, yields [A,A] = - \u00a3 x u-nds - $ x u-sds + \/ [ 2 T T T U - T T C T ] d f i (5.12) where T and x are stresses normal and tangential to the boundary and n nn ns J -and s are unit outward normal and tangential vectors, respectively. If the boundary conditions are homogeneous, the boundary integrals i n (5.12) can be dropped resulting in the energy product of equation (3.73). fA,A] A = \/ [2x Tu - x Cxjdfi. (5.12a) The mixed variational principle of (3.77) for plane stress linear el a s t i c i t y can now be written as F(A) = J n [2xTTu-xTCx]dft - 2 \/ f Tudfi. (5.13) T W and A= E H.. - - - A 5.1 Homogeneous Boundary Conditions The homogeneous boundary conditions for plane stress can be expressed as on S m x . . n. =0 u. = 0 l (5.14) on S u 81 where ru's are the components of unit outward normal, S_ and are the portions of the boundary S where the stresses and the displacements are prescribed to be zero, respectively. The element matrices can be generated directly from (5.13) (Appendix A) and assembled by using the procedure discussed in chapter 4. Since the stresses are incorporated into the functional F(A) as natural boundary conditions only the kinematic boundary conditions on u are to be satisfied. This can be achieved by forcing the corresponding displacement nodal variables to be zero on the boundary S^. The process is identical to forcing the homogeneous kinematic boundary conditions in the displacement method. 5.2 Homogeneous Mixed Boundary Conditions The boundary conditions for the plane stress problem of equations (5.4) to (5.8) in this case are u. = 0 on S x u T . . n . = 0 on S_ (5.15) T . , n , + au. = 0 on S\u201e xj j x M where S^, S_ and S^ are the portions of the boundary S on which displacements, stresses and mixed conditions are specified, respectively, and a is a constant. Consider the energy product of (5.11). Since the variable u and x are in the f i e l d of definition of operator A, they must satisfy a l l the boundary conditions in equations (5.15). The energy product of two elements w A and A from D, is - - A = IQ ^-T-*~ + I T - - \" d i s -integration by parts of the f i r s t two terms on the right hand side yields 82 [A,A] = - $ x u>nds + $ f u\u00bbnds - <$ x u-sds +, i.e. functions u! and x x.'. , both continuous, such that i j u! = u? on S x x u T i j n j = T i \u00b0 n ST ( 5 - 1 9 ) T ! . n . + au! = C? on S\u201e. i] ] x x M Define new variables u 2 and x 2. in the following way \u00b1 \u00b1 3 u'. X X . . * . . 13 i3 13 (5.20) or A\" = A - A' . (5.20a) Substituting equations (5.20) into (5.2), (5.3) and (5.18) yields -x'.'. . = f. + x'. . . = f.1 (5.21) 13 \u00bbJ i 13 ,3 i u1.' = 0 on S X u x^n. =0 on S_ (5.23) T'.'.n. + au*.f = 0 on S.\u201e. 13 3 i M 84 Thus the problem in A\" has homogeneous boundary conditions, and the only difference in (5.2), (5.3) and (5.21), (5.22), respectively, is the introduction of the term T \\ . . on the right hand side of (5.21) and [-1\/2 I J >J (u! ,+u! .)+C... , T ' , ] in (5.22). If these terms were known, i t would be i , j j , i l j k l k l j possible to obtain an approximate solution for A\" and convergence would be w ensured i f the coordinate functions were complete in the space and the associated energy product as given in equation (5.17). \u2014 \u2022 \u2014 T Let the f i n i t e element approximations for A\"=

be M k-l cb.u\" ; 1=1,2. k=l 1 l k (5.24) or where i j A\" T -M E I | ) . . T V . , ; i , j = l , 2 . I = 1 i J i J l T *N N\" ik 131 Then the required element matrix equation, from section 4.2, is . * M ] A = (P ,*M) here (5.27) T P = f.' 0 1 0 s\u00ab Thus a solution for A\" can be obtained such that |A\" - A\" I -> 0, as M and N \u00bb. (5.28) It i s worth noting that here the kinematic nonhomogeneous boundary conditions are conveniently incorporated through the load vector. Such a procedure cannot be achieved in the displacement approach because i t would 85 require displacements to satisfy the nonhomogeneous boundary conditions and have continuous f i r s t derivatives. However, a solution may be obtained directly for u. and T,. 1 i j without actually knowing A1. This is accomplished by assuming approximations for u. and x.. as M -k u. = X i> .u., ; i=l,2. I , , I ik k=l (5.29) N -2 T i i = 2t\u00b1 V i J i J 1 ' j ' = 1 ' 2 ' A = $ w -M -M T TN 7k -1 where $ w =

, \\b . . > -M T i T i j -N T and w_, = . -M ik i j I The sets of functions {<^ } and {ij;^ } are complete in H^ space considered with respect to the nonhomogeneous boundary conditions. Therefore as the function A\"+A' satisfies such nonhomogeneous boundary conditions, i t follows that there exist u., and x.., such that ik i j 1 |A - (A\"+A')| -> 0, as M and N + ~. (5.30) Now the function A-(A\"+A') has homogeneous forced boundary conditions; N therefore the energy product of i t with any arbitrary function 0 ^ in the space w H^ , with homogeneous forced boundary conditions, would also vanish in lieu of (5.30). Therefore IA - (A\"+A'), 0 ^ J A = 0. (5.31) Since 0^ is arbitrary, i t can be replaced by 0^ in (5.26) and equation (5.31) can be written as 86 [A - ( A , , + A ' ) , * J J ] A = 0. (5.32) From linearity of the operator A, (5.33) and substituting this into equation (5.27) yields (5.34) But or \/ T A r (2 \u2022 \u2022M> \" \/ft fV 0 1 o % l . r ( f ^ k ) ( g M . *\"?.), + i >dft. d n Therefore from (5.21) \/ft \/ft *_ d n and integrating by parts the second term on the right hand side yields \/ n ( f i ' < ) d f i - \/ft ( f_*_ - T i j ^ f j > d n +* T _ J n J * i d s -Using equations (5.19) and the fact that * i d s M (5.35) and from (5.22) \/ft <*__*__ > d n = \/ft { - 1 \/ 2 ( u i , j + u j , i > + c u k _ T k i } * q i . d n - ( 5 ' 3 6 ) Therefore = <\/, + \/ S t V _ d * + J C^ kds) - [ A ' , $ A (5.37) T M because, from (5.17) 87 \/ . T ! .cb k .dft + f au : cp kd M . J {i \/ 2 ( u ! . + u! .) - c . . . . T' }t|>?.dn. (5.38) Now equation (5.37) enables (5.34) to be written as 0 L- . 0 X f\u00b1*lda + \/sT V i d s + U V i d s T M (5.39) 0 \u2014 \u2014 \u2014 T which are the mixed Galerkin equations that govern a s o l u t i o n f o r A= w i n when the boundary conditions are nonhomogeneous. Writing out i n f u l l the equation (5.39) k V i , j d f i + j s M a V i d s = k f \u00b1 < d n + u T>J D S + \/s c i * i d s ' M T M M k=l , 2 , . . . N. M. (5.40) (5.41) The equations (5.40) and (5.41) require that the approximate s o l u t i o n f o r u^ given by (5.29) should s a t i s f y the nonhomogeneous forced boundary conditions and the coordinate functionsthe homogeneous forced boundary conditions on S . u In the f i n i t e element method i t i s not necessary to introduce N -N d i f f e r e n t approximations A corresponding to and * coordinate functions. - . -M -M N The coordinate functions of a f i n i t e element approximation associated with the degrees of freedom that do not l i e on s a t i s f y homogeneous condi-tions on S^, i . e . vanish on S^. Therefore the equations (5.40) and (5.41) can be solved by spec i f y i n g the values of u ^ of (5 .29) , the nodal degrees of freedom on S^. The remaining coordinate functions s a t i s f y homogeneous forced boundary conditions and thus only one set of coordinate functions need be introduced. 88 To show that the approximate solution A of (5.29) converges in the energy norm, rewrite (5.31) from linearity of the operator A in the form [_>\u00a7M]A = t A \" + A , , ^ J ] A . (5.42) Also from equation (5.20a) [_>_M]A = [ ^ ' ^ M ] A ' ( 5 ' 4 3 ) Subtracting (5.42) from (5.43) and from linearity of the operator A; Using property ( i i i ) of theorem 3.4.2 (Schwarz inequality) But from (5.28), |A\"-A\"|-K); as M and N^; therefore [A-A,e*J]A + 0, as M and N-*\u00bb. - - -MA N Since 0 ^ is an arbitrary function with homogeneous forced boundary conditions, i t may be set equal to A-A. Then U-A.A-A] 0, as M and N-*\u00bb, which implies that l_~_lA \u00b0> a s M a n d N~*\"- (5.44) That i s , the approximate solution to the nonhomogeneous boundary condition problem converges i n the energy norm of H^. One of the advantages that the mixed method offers li e s i n different ways of incorporating the boundary conditions. So far the natural boundary conditions have been associated with stresses, a consequence of extracting the boundary integrals from the equilibrium equations. This led to constrain-ing of forced boundary conditions on u on S^ through the nodal variables. It w i l l be demonstrated here that in fact a l l nonhomogeneous 89 boundary conditions can be incorporated through boundary integrals from both equilibrium and constitutive-kinematic equations through the Hellinger-Reissner's mixed variational principle since i t yields the same equations as the mixed Galerkin method for T^T theory (Chapter 3). The Hellinger-Reissner mixed variational principle for plane stress with unit thickness and zero body forces f and f can be written as x y I = \/_ [ T U , + T v, +T (u, +v, ) - - ^ x 2 +x2 - 2 V T x +2(1+V ) T 2 } ] dft Ju xx x yy y xy 'y 'x 2E xx yy xx yy xy - J- (up +vp\" )ds - \/ [(u-u)p +(v-v)p Ids (5.45) o 2 x y o 2 x y where p . = T . . n . . S J is the portion of the boundary S on which the stresses T . . or p. are prescribed. The part So has the displacements u. (u and v) prescribed on i t . The stress boundary conditions of (5.18a) and (5.18b) on S \u201e and S \u201e can be considered to be on part S i while S would coincide with T M u S 2 . Therefore p. = x..n. = T\u00b0 on S, i i j J i > p. = x n. = C? - au. on S_, i i ] J i i M (5.46) and the f i r s t variation of I with respect to u. and x.. gives & 1 - \/fl [ \\ j & U \u00b1 , 3 ] d n + \/fl ^ ^ ^ I J ^ . ^ ^ i j k i ^ 1 ^ 1 1 \" \" jST T\u00b0 (5.47) 6 V S \" \/sM < C!hV 6 ui \" \/Su ( V U S ) 6 ( T l j B J ) d 8 \" J S u T U n j f i u \u00b1 \" \u00b0-Now assume approximations for u_^ and i as M k u. = \u00a3 \u00abJ).u ; 1=1,2. (5.48) 1 k=l 1 l k N I and x = E * , , T ; i,j=l,2. (5.49) .^J .2=1 90 Therefore M k - N 1 6u. = Z.6uM ; fix.. = E I|>..6T.,., (5.50) 1 k=l 1 1 3 1=1 1 J i j N and fiu. . = E d> .6u., . (5.51) i, J k = 1 i . J ik Substitution of (5.48) to (5.51) into (5.47) yields 5 1 \" j x l \/ n V i , i d ^ s _ TS*ids\"\/sM ( C i \" ^ i ^ i d s - \/ s u ^ i jV i d s ] 6 u i k + qL { 1 \/ 2 ( ^ i , J + ^ , i ) - C i J k i\\i }*i j d f i -\/s u \" u \/ i j n j d s + \/ s u u _ * i j n j d 8 ] fix..q = 0. (5.52) Since 61 vanishes for arbitrary variations 6u., and fix.. , the following ik i j q ' equations are obtained: k V i , j d n + 'sM a V_ d s ~ j s ^ V i d s = J s T T?*i d s + 'sM c i * i d s ; M u J J T M k-1,2, . . . M. (5.53) q-1,2, . . . N. (5.54) The equations (5.53) and (5.54) contain 2M+3N equations for 2M+3N unknowns with a symmetric matrix of coefficients. It is interesting to note that the nonhomogeneous forced boundary conditions are applied through the displace-ment vector in (5.54) and hence need not be constrained as was done in the previous case. Except for boundary integrals over these equations are exactly the same as (5.40) and (5.41). 91 5.4 Boundary Residual Concept An alternate procedure for incorporating boundary conditions in the mixed Galerkin method is similar to the boundary residual concept presented by Finlayson and Scriven [8] and Finlayson [7] in which the domain residual together with boundary residuals are made orthogonal to the shape functions of the approximate solution. Consider the plane stress linear e l a s t i c i t y problem of equation (5.9) with nonhomogeneous boundary conditions of equations (5.18) and approximations for u_^ and i of equations (5.48) and (5.49). The substitution of u^ and x into the f i e l d equations (5.9) and boundary conditions (5.18) yields the following residuals: R e i = [-\\i,ff\u00b1] i n f i (5'55) R c i = [ 1 \/ 2 ^ , i + ~ U i A ) - % ^ l \\ l ] \u00b1 n B \u00b0-56) R u i \" \" [ V U i J o n Su ( 5 - 5 7 ) *r\u00b1= [ V r T i ] O N S T ( 5 - 5 8 ) R... = [ T . .n.+au.-C1?] on S\u201e (5.59) Mi l j ] i i M where R . and R . are the domain residuals for the equilibrium and kinematic-el c i constitutive equations, respectively; R^ , R^, and R^ are the boundary residuals on S , S m and S w, respectively, u T M If the residuals R^ from equation (5.55) along with R^ and R^ from equations (5.58) and (5.59) are made orthogonal to the shape functions for u_^ and residuals R^ from (5.56) along with residual R^ from (5.57) to the shape functions for x \u201e , the following equations result: k ^ i j f j - f i ^ i d n + l s S v T i ) d s + Un *i d s \u2022 0 ; k=l,2, . . . M. (5.60) 92 \/ft [1\/2('Z\u00b1,3+'U3,\u00b1 ) ~ W i j ^ V \" \" ^ *i_ q=l,2, N. (5.61) Assuming f.=0; and applying Gauss' theorem to (5.60) yields M k=l,2, M. (5.62) \/ft [ 1 \/\u00ab ui, j 4\u00bb J li)\"Vu ]* ,\/^s u ..*?.n.ds = L u q=l,2, N. (5.63) The equations (5.62) and (5.63) are the same as equations (5.53) and (5.54) in the previous section. Therefore in linear e l a s t i c i t y , the equations obtained by the mixed Galerkin method, the mixed variational principle and the boundary residual concept are the same. In the displacement approach, Hutton [15] showed that the equations obtained from the Galerkin Method and the boundary residual concept for approximations from wider class would be identical i f the forced boundary conditions were either homogeneous or were identically satisfied by the f i n i t e element approximations when nonhomogeneous. However, the f l e x i b i l i t y offered by the mixed methods in incorporating the boundary conditions, forced or natural, provides a wider equivalence. 93 CHAPTER 6 EIGENVALUE ANALYSIS OF THE ELEMENT MATRIX An eigenvalue-eigenvector analysis of the element matrix arising from the mixed method i s presented in this chapter. Various combinations of displacement and stress approximations over a triangular and a rectangular element are considered. Two problems are included, namely the linear elasticity plane stress and the linear part of the Navier-Stokes equations. It i s anticipated that the analysis of eigenvalues w i l l provide some insight as to choice of approximations for the dependent variables involved so that completeness is achieved. 6.1 Linear Elasticity Problem Consider the matrix equation (5.9) for the plane stress problem with zero body forces, i.e. f =f =0, J x y 0 0 3 ~3x 0 3 \"3y u 0 0 0 0 3 \"3y 3 ~3x V 0 JL 3x 0 1 \"E v_ E 0 T XX = 0 0 3 3y E 1 ~E 0 T yy 0 3 3 0 0 \" -2(l+v) 0 dy 3x E T (6.1) The variational principle for (6.1) with homogeneous boundary conditions (section 5.1) can be written as I = \/_ [x u, +T v, +T (u, +v, )-^{T2 +T2 -2VT T +2(1+V)T2 }]dn. (6.2) J Q xx 'x yy y xy y x 2E xx yy xx yy xy Since I represents strain energy, inspection of the right hand side in (6.2) suggests the following three rigid body modes which yield zero strain energy: 94 T = T T = X X yy xy T = T = T = X X yy xy T = T = T X X yy xy (i) u = constant, v = 0 ; = = 0 xx ( i i ) u = 0 , v = constant; x ^ = X t t = x^_ = 0 (6.3) ( i i i ) u = -cy , v = cx These r i g i d body modes are expected to be removed by the specified kinematic boundary conditions. Furthermore, i t is required by the functional of (6.2) that the displacements satisfy the kinematic boundary conditions while the stresses emerge as natural boundary conditions. Therefore the fi n i t e element approximations should be in compliance with these requirements, i.e. the matrix of equations (4.19) should exhibit the ri g i d body modes of (6.3) . The independently chosen approximations for the stresses and the displacements have to comply with the completeness requirement (i) of section 4.4. The mean convergence of strains from the assumed stresses to the strains derived from the assumed displacements would be assured for a fi n i t e number of degrees of freedom i f the former contains a l l the strain modes and perhaps more than the strain modes in the latter. It is assumed that the displacements possess a l l the rigid body and constant strain modes and that the stresses possess a l l the constant stress modes. It is now asserted that the violation of the completeness requirement (i) results in a hypersingular element matrix, i.e. the number of zero eigenvalues greater than the rigi d body modes expected in a problem. The eigenvectors for the extra zero eigenvalues correspond to mechanisms which are defined as the kinematic freedoms possible when the material has no elastic stiffness. This is illustrated by the following example. e * Consider one element domain fi and the approximate solutions for u,v,x ,x and x as xx yy xy 95 u = ax + by + c x 2 + dxy + ey 2 v = bx + fy + gx 2 + hxy + i y 2 x x x = 3 (6.4) x = k yy T = I. xy The polynomials f o r u and v are so chosen that the r i g i d body modes have been eliminated by s a t i s f y i n g the kinematic boundary conditions. Therefore e = u, = a + 2cx + dy xx x \u00a3 = v, = f + hx + 2iy (6.5) yy x Y = u, + v, = 2b + (d+2g)x + (2e+h)y. xy y x From the equations (6.5) the s t r a i n s derived from u and v are complete l i n e a r polynomials. Therefore the mean convergence of constant s t r a i n s from x , x and T i n (6.4) to the s t r a i n s i n (6.5) would not occur and the com-yy xy \u2022 pleteness requirement ( i ) i s v i o l a t e d . The parameters i n (6.4) are to be determined from the v a r i a t i o n a l formulation. The s u b s t i t u t i o n of the ex-pressions i n equations (6.4) into (6.2) y i e l d s I = [j(Aa+2ac+Bd) + k(Af+ah+2gi) + 1(2Ab+ad+2ag+23e+3h)] A 2E [j2+k 2-2vjk+2(l+v)I 2] (6.6) vhere A = \/ dfi, a = \/ xdft and 3 = \/ ydft. ft6 ft6 QE rhe system of equations governing the one element domain i s obtained by making 1 stationary with respect to the unknowns a,b,c,d, . . . I. This i s 96 0 0 0 0 0 0 0 0 1 A 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 2a 0 0 0 0 0 0 1 e 0 0 0 0 0 1 0 0 0 0 0 1 0 0 0 0 1 0 0 0 1 0 symmetric 0 1 0 0 0 0 a 0 28 A 0 0 2a A vA \"E E 0 -I \u00ab -2(l+v)A (6.7) J k I It is observed that the f i r s t and the third rows; second, f i f t h and the seventh rows; and sixth and the ninth rows are the same except for some multiples, while the fourth and the eighth rows are linear combinations of the f i r s t and the second rows and the second and the sixth rows, respectively, rherefore only six of the twelve equations in (6.7) are linearly independent. Hence the element matrix has a rank of 6 instead of 12 and therefore is singular. As a consequence c,d,e,g,h and i are indeterminate and these are the coefficients of the quadratic terms i n the polynomials for u and v, equations (6.4). For a matrix of the form (4.19), 0 (m+n)x(m+n) mxm T a nxm mxn - b nxn (6.8) 97 w h i c h i s r e a l a n d s y m m e t r i c , t h e e i g e n v a l u e s a r e r e a l a n d b e c a u s e o f t h e i n d e f i n i t e n e s s o f t h e m a t r i x t h e s e c a n b e e i t h e r p o s i t i v e , z e r o o r n e g a t i v e . F u r t h e r , i f t h e n x n b s u b m a t r i x i s p o s i t i v e d e f i n i t e a n d r t h e r a n k o f t h e m a t r i x S, i t i s s h o w n i n A p p e n d i x B t h a t t h e e i g e n v a l u e d i s t r i b u t i o n i s o f t h e f o l l o w i n g t y p e : ( i ) m p o s i t i v e a n d n n e g a t i v e e i g e n v a l u e s i f r=m+n; ( i i ) ( r - n ) p o s i t i v e , ( m + n - r ) z e r o a n d n n e g a t i v e e i g e n v a l u e s i f r and T = and [S]= - - - - -xx -yy -xy 0 a T z a b as i n (6.8), and [I] is the identity matrix. The qualitative description of the eigenvalues and the composition of the eigenvectors for a l l the combinations of interpolations used for the displacements and stresses over a triangular element appears in Table I and that for a rectangular element is l i s t e d i n Table II. For both triangu-lar and rectangular elements, the number of negative eigenvalues corresponded to the number of stress degrees of freedom, whereas the three zero eigen-values for th_ expected rig i d body modes are obtained only for the displace-ment stress combinations which satisfy the completeness requirement ( i ) . In the cases where more than three zero eigenvalues are obtained, the number of extra zeroes corresponded to the number of modes present i n the strains derived from the assumed displacements that were not contained in the strains from the assumed stresses. The eigenvectors are composed of the same dis-tribution as the assumed approximations for displacements and stresses i n a l l cases except for rigid body modes where u and v satisfied u,x=0, v, =0, u, +v, =0 and stresses were zero, y x It is essential for convergence in the energy sense that a mixed fin i t e element formulation conform to the completeness requirements of 1 0 1 section 4.4, and be able to represent r i g i d body modes and the constant s t r a i n s (with completeness requirement (i) s a t i s f i e d and assumed dis p l a c e -ments that possess constant s t r a i n s , i t i s implied that the corresponding assumed stresses would contain the constant s t r e s s e s ) . To include r i g i d body modes and constant s t r a i n s i n the assumed displacements i s a simple matter. However for c e r t a i n combinations of assumed displacements and stresses, i t i s not quite obvious that completeness i s achieved, e s p e c i a l l y for incomplete polynomials. For example using biquadratic displacements and b i l i n e a r stresses over a rectangular element y i e l d s four zero eigen-values. A scheme to check completeness requirement (i) and trace the terms i n the polynomials used f o r the displacements which correspond to s t r a i n s not included i n the assumed stresses i s presented here. The poly-nomials considered are the ones mentioned i n the example above. The assumed biquadratic displacements u and v are u = ai+a2X+a3y+at +x 2+a5xy+a5y 2+a7X 2y+a3xy 2 v = bi+b2X+b3y+bi +x +b5Xy+bgy +byx^y+b8xy z and the b i l i n e a r stresses are Tx x = C! + c 2 x + c 3 y + c 4xy (6.14) T = d 1 + d 2x + d 3y + d^xy (6.15) x x y = e! + e 2x + e 3y + ei+xy. The s t r a i n s derived from u and v are e = u, = a2+2aitx+a5y+2a7xy+agy2 (6.16a) XX X e y y = v, = b 3+b 5x+2b 6y+b 7x 2+2b 8xy (6.16b) Y\u201e\u201e = u, +v = (a3+b2)+(a5+2b 1 +)x+(2a 6+b 5)y+a 7x 2+2(a 8+b 7)xy+b 8y 2. (6.16c) xy y x 102 Now using the generalized Hooke's Law for the plane stress problem, the strains corresponding to the assumed stresses in (6.15) are Sex = E[Txx~V^yy-' = E [ ( cl- v dl)+(c2-vd 2)x+(c3-vd 3)y+(c l t-vdt t)xy] (6.17a) Sy = E 1\"~ V^xx + Xyy ] = f [ ( d l - v c l ) + ( d 2 - v c 2 ) x + ( d 3 - v c 3 ) y + ( d ' t - v c i + ) x y ] (6.17b) 2(l+v) r 2(l+v) r , . . . fe. i t \\ Yxy = \u2014 E Txy = \u2014 E Ui+e^egy+e^xy]. (6.17c) If the displacements were assumed to be a complete polynomial of degree p and stresses to be a complete polynomial of degree (p-1) , then completeness would be achieved. However, the polynomials considered here are not complete and comparison of equations (6.16) with (6.17) shows that certain terms in the derived strains are not contained in the corresponding strains from the assumed stresses, i.e. a 8y 2 in (6.16a), b 7x 2 (6.16b), a 7x 2 and b 8y 2 in (6.16c). But a 7 and b 8 as coefficients of the bilinear terms in the derived strains e and e match with the coefficients of the bilinear terms in e and e xx yy xx yy of (6.17), respectively; while (a 8+a 7) appearing as coefficient of the bilinear term in the derived shear strain Y matches with the coefficient xy of the bilinear term in Y of (6.17). Therefore only one of the coefficients xy a 7,a 8,b 7 and b 8 is indeterminate, hence i t leads to only one mechanism besides the three rigid body modes. This is confirmed by the results obtained for biquadratic u and v and bilinear stresses over a rectangular element, Table II. For the same combination, the mode shape for the mechanism after elimination of the rigid body modes, is illustrated in Figure 6. The static condensation by elimination of stresses was performed for a l l of the combinations of displacements and stresses appearing in Tables I and II. The condensed element matrix is found to be exactly the same as the stiffness matrix one would obtain from the displacement method using identical assumed displacements over an element except for the combinations 103 where the completeness requirement (i) i s violated which possess the same number of mechanisms. 6.2 Linear Part of the Navier-Stokes Equations The basic equations governing the two dimensional, steady, incom-pressible flow are the well-known Navier-Stokes equations p(uu, + vu, ) + p, - uV2u =0 in tt (6.18) x y x p(uv, + w, ) + p, - yV 2v = 0 in tt (6.19) x y y u, + v, = 0 in tt (6.20) x y where u,v are the x,y components of velocity, respectively, p is the f l u i d density, p, the pressure, u , the dynamic viscosity, and tt the open domain of the problem. In terms of deviatoric stresses directly, the equations (6.18) and (6.19) can be written as p(uu, + vu, ) + (p, - T - T ) = 0 in tt (6.21) x 'y r'x xx,x xy,y p(uv, + w, ) + (p, - T \u2014 T ) = 0 i n tt (6.22) x 'y y xy,x yy\u00bby where T and T are the normal deviatoric stresses in the x and y directions, xx yy respectively; and T is the shear stress. The equations relating deviatoric stresses to velocities for a Newtonian flu i d are Txx = - 3^(u'x + V + 2^ U'x ( 6 ' 2 3 ) Tyy = \" 3 ^ ' x + v \u00bb y > + 2 y V ' y ( 6 \" 2 4 ) x = p(u, + v, ). (6.25) xy y x These equations can be put into an alternate form by solving for the velocity gradients as 104 1. 1 , = \u2014 ( T - -T ) u xx 2 yy v y 2 xx yy in ft in ft (6.26) U , + V , = \u2014T y x y xy in ft. Now the equations (6.21), (6.22), (6.20) and (6.26) can be put into the matrix operator form for mixed formulation as t 9 _, 9 \\ p ^ T x ^ o f - -9_ _3_ 3x 9x 0 8_ \"3y 3 3x 3_ 3x 9_ 3y \/> 9 x 9 \\ p ( V x ^ 3 3y 0 3_ 3y 3_ 3x 3_ 3y .L. J _ 3y 3x 1 M 1 2y ~ 0 2y ~y 0 0 0 1 y X X yy xy = 0 in ft. (6.27) For steady creeping flow, a special case of incompressible, steady Newtonian flow, the nonlinear part of the matrix operator above, which makes i t non-symmetric, can be dropped. Thus the following f i r s t order, linear differen-t i a l equations result and involve a symmetric matrix d i f f e r e n t i a l operator 0 0 0 0 9_ 3_ \"3x ~9y 0 9_ _9_ 3x ~9x f 0 3y 0 0 0 - \u2014 3y _3_ _3_ \"3y 9x 0 1 0 1_ 2y 0 0 v X X = 0 i n ft; (6.28) 105 comprising six equations for six unknowns. Here, in lieu of T*T theory 1* = -T = 9-- 9- 0 - a -3x 3x 3y 1- o 3y 3y 3x T T and for u= and x=

, the analogous form'of equations xx yy xy (6.28) to the plane stress elasticity equations (5.10) is \" 0 u = 0 T -F X or AA = 0 where F = ' 0 0 0 0 \" 0 1 u 2y 0 -1_ 1 2u y 0 0 0 0 1 y (6.29) (6.29a) (6.30) which i s a positive semidefinite matrix. In tensor notation, the equations (6.28) take the form p,. - x.. . = 0; i=j=l,2; in ft i I J ,2 u. . = 0; i=l,2; i n 2 ( u i . i + U i . i ) - C n i T k i = \u00b0i i=J=k=I=l,2; in ft 2 v u i , j T U j , i ; \" \" i j k l l k l and can be subjected to some boundary conditions analogous to equations (5.18); on S (6.31a) (6.31b) (6.31c) u. = uV l l (-P<$. . + x. .)n. _ T 0 on S\u201e (6.32) (-p<5. . + x. .)n. + au. = C\u00b0 i j i j 2 i I on S. M 106 where S +Sm+S =S, the boundary of domain fi. u T M J The matrix equation (6.28) is similar to matrix equation (6.1) except for pressure p, the continuity equation as zero divergence of velocities and the deviatoric normal stresses. The mixed variational principle for these equations for homogeneous boundary conditions can be derived as I = \/_. [-p(u, + V , ) + X U, + T V, + T (U, + V , ) ; fi x y xx x yy y xy y x - ~ { T 2 + T 2 - T T + T 2 }]dfi (6.33) 2u xx yy xx yy xy which is also similar to the mixed variational principle for the linear elasticity plane stress problem in (6.2) except for the term p(u,x+v,^) and requires velocities to satisfy the kinematic boundary conditions. The variational principle of (6.33) also gives the three rigid body modes as in (6.3); (i) u = constant, v = 0; x = x X X ( i i ) u = 0, v = constant; x = x = x =0 (6.34) xx ( i i i ) u = -cy; v = cx; = x while pressure can be arbitrary, since the incompressibility leads to the divergence of u and v to be zero rather than be proportional to pressure. If the f i n i t e element approximations for the variables involved in (6.33) are chosen over an element domain fi as u = E d> .u. 1=1 (6.35a) m v = E cb. v. X = 0 yy xy X 0 yy xy X = 0 yy xy 107 _ n x = E ib. x XX . T l XXI 1=1 T = E \\b.T yy ' i = 1 i yyi (6.35c) T = E \\b ,T x y 1=1 x y Then the substitution into the functional I of (6.33) and setting i t s f i r s t variation 61 to zero for stationarity yields the following matrix equation: 0 0 e a 0 0 f 0 0 0 0 0 u-0 f , 2y-0 0 0 Ol b \" u b a V Ol 0 p U 2y-0 T -XX - i d y-0 T -yy Ol - i d y- T -xy = 0. (6.36) Here u=, v= , P= , etc., are the linear vectors of nodal degrees xxn \u00b0 of freedom. The submatrices a,b,d,e and f are obtained in the following manner: 3 i j \" \/ ne*i,x*J b i j \" \/ne+i.y^ dn 1=1,2, e 3=1,2, dVi i 5 j = l , 2 , d i i = \/ n e * i * j e j=l,2, . f i j = -la**i,y!U dQ, m; n. . n. m; 1. (6.37) 108 Note that the matrix d is symmetric and positive definite. The matrix of equation (6.36) is symmetric, therefore the eigenvalues of the matrix are real. Further, i t is indefinite and i f the rank of the matrix is (2m+I+3n), then from Appendix B i t has 2m positive and (I+3n) negative eigenvalues. The choice of polynomials for u,v,x , XX x and s t i l l has to comply with the same completeness requirements set out in the previous section. However the requirements on the pressure f i e l d are dubious because i t is not related to the strains. The obvious question that arises here i s : what are the completeness requirements on p? Further i t is not possible to ask for mean convergence of pressure to the volumetric strain as was done for the stresses in the plane stress linear elasticity problem because of the incompressibility constraint. In the f i n i t e element application to the Navier-Stokes equations (6.18) to (6.20) using the primitive dependent variables u,v and p, a similar situation was faced by Taylor and Hood [34] and Olson and Tuann [26], One of the possible variational principles for the linear part of these equations used i n the reference [26] i s J(U'V'P) \" \/ f t [ ^ ' x S ^ \u00bb y + V \u00bb \/ 1 \" p ( U ' x + V ' y ) ] d - \/ (Xu + Yv)ds (6.38) T, where (X,Y) are the specified traction on the boundary S^ _. The term P( u> x + v,^)dft appears in both variational principles, I of (6.33) and J(u,v,p) of (6.38). Therefore the requirement that the pressure interpolation should be one degree less than those for the velocity components, as found by Olson and Tuann [26], is also expected here and indeed confirmed by the numerical results. A different explanation for such a requirement is pre-sented here. 109 Since the divergence of u and v i n equation (6.20) i s zero, i t acts as a constraint equation. If the approximations used for u and v involve complete polynomials of degree m, then u,^ and v, would involve complete polynomials of degree (m-1). Therefore the unknowns corresponding to the (m-1) complete polynomial for u,^ must be r e l a t e d to the unknowns corresponding to (m-1) complete polynomial f o r v, i n order to s a t i s f y continuity i n the discr e t e sense. Assuming that U,V,T ,T and x have been chosen properly, xx yy xy r r i . e . do not y i e l d any mechanisms except for the r i g i d body modes, then the d i s c r e t i z e d continuity equations < e T f T > j \" } = 0 (6.39) should have a rank not less than m+\"'\"C2, i . e . combinations of (m+1) taken 2 at a time. However i n the f i n i t e element formulation, equation (6.36), the number of d i s c r e t i z e d continuity equations i s associated with the number of degrees of freedom for pressure thus l i m i t i n g i t to m+~'\"C2. As a consequence, the pressure should have degrees of freedom not more than C2y which implies a complete polynomial of degree not higher than (m-1). Consider the following schematic representation of the complete polynomials for u,v and p. V e l o c i t y u 1 Constant a\\ x y Linear a2 a 3 x 2 xy y 2 Quadratic a^ a$ ag x 3 x 2y xy 2 y 3 Cubic a 7 a 8 ag a 1 0 x 4 x 3y x 2 y 2 xy 3 y 4 Quartic a n a\\z al3 alh a15 etc. etc. etc. (a) 110 Thus i f the a.'s are the coefficients of the polynomial for u, the complete quadratic for u can be written as u = ai + a 2x + a 3y + a^x2 + a$y.y + agy 2. (6.40) Replacing a.'s by b^'s in scheme (a) for v; the complete quadratic for v is v = hi + b 2x + b 3y + bi+x2 + b 5xy + b 6y 2. (6.41) Similarly for pressure p; p = ci + c 2x + c 3y + c^x 2 + C5xy + C g y 2 . (6.42) The partial derivatives u,^ and v, can also be written schematically as complete polynomials, \\\u00b0 u'x \\ a i \\ \\ 1 \\ 0 Constant a 2 \\a 3 \u2022 \\ \\ x y \\ 0 Linear 2a!,. a^ \\ag 2\\ \\ ( M x z xy y ^0 Quadratic 3aj 2aQ ag \\a-io x 3 x 2y xy 2 y3^\\0 Cubic 4an3a^ 2 23I 33T_I\\3 15 etc. etc. etc. 0\/ v, b l 7 \/ 7 \/ 0\/ 1 Constsnt b 2\/ b 3 \/ \/ 0\/ x y Linear b^\/bc, 2bg \/ \/ ( O 0\/ x 2 xy y 2 Quadratic by\/bg 2bg 3bj_o 0\/^x3 x 2y xy 2 y 3 Cubic b^\/bi 2 2b^ 33bji+4bi5 etc. etc. etc. Consider only the non-zero terms to obtain u, +v, 3s J x y (u,x+v,y) 1 Constant (a 2+b 3) x y Linear (la^+b^) (a5+2bg) x 2 xy y 2 Quadratic (3a7+bg) (2ag+2bg) (sg+3bio) x 3 x 2y xy 2 y 3 Cubic (4a n+b 1 2) (3a 1 2+2b 1 3) (2a 1 3+3b 1 1 +) (a l l f+4b 1 5) etc. etc. etc. (d) I l l and p; 1 x y Pressure p Constant Linear c2 c3 x 2 xy y 2 Quadratic c^ C5 eg x 3 x 2y x y 2 y 3 Cubic C 7 CQ Cg C^Q 4 3 2 2 3 4 (e) x H x ay x z y z xy 3 y H Quartic C)i Cj2 c13 c14 c15 etc. etc. etc. The d i s c r e t i z e d continuity equations can now be obtained over the domain i n the following manner: J o e P ( u \u00bb v + v, )dfl = 0. 9c. ^ n e F V \" \u00bb x ' V V 1 ^ (6.43) Let a i j = Joe xVdft e; 1=3=0,1,2,3, 'fte then from schemes (d) and (e) the following matrix form for the d i s c r e t i z e d continuity equations i s obtained; a00 2 u 1 0 a01 3 a 2 0 2 a u <*02. . . a o o a10 2 a 0 i a 2 0 2 a n 3ao2-<*10 2a 2o \u00ab 1 1 3 a30 2 a 2 1 a 1 2 . \u2022 -aiO a20 2 a n \u00ab30 2 a 2 1 3 a 1 2 . <*01 2 a n <*02 3 a 2 i 2a 12 \u00b0<03' . -a 0 1 a x l 2 a 0 2 a 2 1 2 a 1 2 3a 03-\"20 2 a30 a 2 1 3aito 2 a 3 l a 2 2 . . . a 2 0 C30 2 a 2 1 \u00b0\"t0 2 a 3 l 3a2 2. an 2 a 2 1 a 1 2 3 a 3 l 2 a 2 2 C13- . . a n a 2 i 2 a 1 2 \u00ab31 2 a 2 2 3 a 1 3 . \u00ab0 2 2a 12 a03 3 a 2 2 2\u00ab13 ag u. . . a o z a 1 2 2ao3 a 2 2 2^13 3a 0 t t. C 3 0 2a 4 0 a31 3 a 5 0 2aiti a32- \u2022 ^ 30 aifO 2 a 3 i \u00ab50 2ai+l 3 a 3 2 . a21 2a31 a 2 2 3 a 4 1 2 a 3 2 a23- . . a 2 i a31 2 a 2 2 a41 2 a 3 2 3 a 23-a 1 2 2 a 2 2 \u00ab 1 3 3 a 3 2 2 a 2 3 \u2022 . . a 1 2 \u00bb2 2 2ai3 <*32 2 a 23 3 a 1 4 . <*03 2\u00ab13 a04 3 a 2 3 2a ih C Q 5 - \u2022 -C03 \u00ab13 2aQit a 2 3 2 a m 3a 0 5-(Nx2Q) a4 a5 a7 as a 9 b 3 b 5 b 5 b 8 bg b10 = 0. (6.44) 112 Here N= n + 2C2, where n i s the degree of complete polynomial for p; Q=in+^C2> m being the degree of polynomials used f o r v e l o c i t i e s u and v; e.g. for t h u and v cubic, m=3 and Q=6. I t can be observed that every (Q+j) column th i s either equal to or a simple multiple of the j column. Therefore the rank of the matrix of c o e f f i c i e n t s i n (6.44) i s at most Q. Further the rank of the matrix i s s t i l l Q even i f N i s greater than Q. Thus for N greater than Q, a l l the d i s c r e t i z e d continuity equations of (6.44) are not indepen-dent. Now consider the case where u,v and p are complete quadratics, i . e . m=n=2 and Q=3, N=6. The r e s u l t i n g d i s c r e t i z e d i n c o m p r e s s i b i l i t y constraints are \u00ab00 2 a 1 0 a o i a00 <*10 2a o f a 2 <*10 2 a 2 0 a n a10 a20 2on ait a01 2a u <*02 a01 a n 2 a 0 2 as \u00ab20 2<*30 a 2 i \u00ab20 \u00ab30 2 a 2 i b 3 a n 2 a 2 i a 1 2 a n a 2 i 2 a 1 2 b 5 c<02 2 a 1 2 a03 \u00ab02 a 1 2 2\u00ab0 3 b 6 0. (6.45) C l e a r l y the rank of the matrix of c o e f f i c i e n t s i n (6.45) i s 3. Therefore three of the constraints i n (6.45) are not independent. The corresponding contribution from p(u\u00bbx+v,^)dfi to the equi-l i b r i u m equations i s a00 aio \u00ab01 a 2 0 a n ao2 c l 2a io 2a 2o 2 a n 2 a 3 0 2 a 2 i 2ai2 c2 a01 a n ao2 a 2 i a 1 2 ao3 c3 aoo a10 a01 a20 a n a02 aiO a20 a n a30 a21 a 1 2 c5 2 a 0 1 2a n 2 a 0 2 2 a 2 i 2 a 1 2 2a 03 c6 = ac. (6.46) 113 where the matrix of coefficients is simply the transpose of the matrix in (6.45). This is the equivalent of e f part of the equilibrium equations in (6.36). In general, ac cannot be equal to zero, i.e. pressure cannot be in equilibrium by i t s e l f . Therefore ac = 0 (6.47) should not have any non-zero solutions. This can be possible i f the rank of matrix a is the same as the degrees of freedom c^'s, i.e. 6 in the example considered. But the rank of a is obviously 3, thus leading to indeterminate c^'s which causes a self-equilibrating pressure f i e l d and non-unique solutions. However, this situation can be avoided i f the rank of the matrix a is equal to the number of constraints. This is possible i f the pressure distribution is taken as linear for quadratic distributions in u and v. The equations (6.45) and (6.46) then reduce to arjo 2a 1 0 a 0 1 a 0 0 a 1 0 -2a0i a 1 0 2 c*20 a l l a 10 a 20 2 a l l a 0 1 2 a u a 0 2 a 0 1 a n 2 a 0 2 a 2 ak A5 b 5 0, (6.48) and a 00 \"10 a 01 2 a 1 0 2 a 2 0 2 a n a 01 a l l \u00ab02 a 00 a 1 0 a 0 1 a 10 <*20 a n 2 a o i 2aii 2ao2 ^2 C3 = ac, (6.49) 114 respectively. It can now be concluded that in general N should not be greater than Q in equation (6.44) to avoid self-equilibrating systems in pressure and is equivalent to saying that the degree of interpolating polynomial for pressure should not be larger than (m-1) where m is the degree of complete polynomials used for the velocity components u and v. The pressure on the boundary is incorporated as a natural boundary condition i n the functional of (6.33). However, inspection of equations (6.31) reveals that pressure needs to be fixed at some point in the domain fi as adatum. If the approximating polynomial in the f i n i t e element formula-tion f a i l s to comply with the requirement concluded above, then to avoid self-equilibrating systems, the pressure needs to be specified at more than one point on the boundary depending on the number of self-equilibrating modes present. At the same time the completeness requirement (i) for mean convergence of the stresses to velocities should not be overlooked. Therefore a consistent formulation of the element matrix would have only three rigid body modes, as in equations (6.34). Various combinations of interpolations for the velocities, pressure and stresses (u,v,p,x , T and T ) over a triangular and a rectangular ' xx' yy xy & & element (Figures 4 and 5, expect for addition of pressure degree of freedom at the nodes) are considered. The following eigenvalue problem is 0 a B \" \"V 0 0 \" T a 0 0 p - A 0 I -p 0 P = 0 3 T 0 -b T 0 0 I - T T (6.50) (2m+J+3n)x(2m+I+3n) (2m+l+3n)x(2m+If3n) 115 where 6 = , X lx3n T T T =-xx -yy -xy a 0 b e , a = 0 b a 2mxl f and b 3nx3n i-h o d 0 The submatrices \u00a3jb,d,e,f, and the sub-column vectors are the same as in the matrix equation (6.36) whose derivation is similar to that of plane stress problem (Appendix A), where as I. , U and I are the identity matrices. 2mx2m 1%1 3nx3n It was mentioned earlier in this section that i f the rank of this matrix is (2m+I+3n), then i t has 2m positive and (I+3n) negative eigenvalues. If there are q ri g i d body modes and r mechanisms which correspond to zero eigenvalues, and since these correspond to indeterminacies of the. u and v degrees of free-dom (2m), then only (2m-q-r) eigenvalues are positive. Similarly, i f there are p indeterminacies of pressure degrees of freedom (1), then (I+3n-p) eigenvalues are negative. The distribution of negative, zero and positive eigenvalues and the composition of eigenvectors for different combinations of interpolations for u,v,p and the stresses x's are presented in Tables III and IV for a t r i -angular and a rectangular element, respectively. It can be observed that the combinations of interpolations which do not comply with the requirements on pressure and stresses resulted i n more than three eigenvalues required for rigid body modes. The self-equilibrating modes in pressure are obtained when the pressure interpolation polynomial i s of the same degree as the polynomials for u and v and with the exception of linear u,v,p,xxx,x^^ and x over a triangular element, have the same distribution as the interpolation xy polynomial while u,v and the x's are zero. The mechanisms are obtained when u,v are quadratic and x's constant over a triangular element; u,v biqua-dratic, x's bilinear over, a rectangular element, as.in the plane stress 116 problem of the previous section. In addition self-equilibrating pressure modes result when the pressure is quadratic for the triangular element. Again, when the rigid body modes are eliminated, the mechanisms seem to possess the same distribution for u and v as the assumed interpola-tions while pressure and the stresses are zero. In recognizing the rigi d body modes of equations (6.34), the pressure was said to be arbitrary. However, the eigenvectors for zero eigen-values associated with the rigid body modes and mechanisms displayed zero pressure in the element. This can be explained by spli t t i n g the eigenvalue problem of equation (6.50) in the following manner: - XI 6 + ag + \u00a7T = 0 (6.51a) aT6 - XI p = 0 (6.51b) -p-BT6 - [b + XI ] \u2022= 0. (6.51c) Now the rigid body modes consist of u=a-cy and v=b+cy, therefore i t is T T clear that a 6 and B 6 are zero since these involve derivatives u, . v, and - - - - x y (u,^+v,x). The stresses are zero from (6.34), hence from equation (6.51a) ap_ = 0. (6.52) If the rank of a is equal to the number of constrians, i.e. the degrees of freedom in pressure, then the equation (6.52) is only true i f p=0. There-fore pressue i s zero everywhere i n the domain. Hence the equations (6.34) for rigid body modes are modified here (i) u = Constant, v = 0; p = x = x xx ( i i ) u = 0, v = constant; p = x =x =x =0 (6.53) R XX V \" ( i i i ) u = cy, v = cx; p = Txx = T Thus a consistent formulation should not have more then three zero eigen-values for the rigid modes of equations (6.53). Finally static condensation = X = 0 yy xy = X = 0 yy xy = X = 0. yy xy 117 of the matrix equation (6.36) by eliminating the stresses, for the cases where no mechanisms are present, gives exactly the same matrix equation as one would obtain from the functional of (6.38) using the same u,v and p interpolation polynomials. In fact, the results of the eigenvalue analysis for such cases are very similar to those presented by Olson and Tuann [25]. In concluding this chapter i t should be pointed out that for certain combinations of approximate displacements and stresses (which indeed comply with the completeness requirement (i) and exhibit proper eigenvalues and eigenvectors over one element domain) the assembled element matrices according to certain continuity requirements yield self-equilibrating modes over the f u l l domain. The typical example i s that of plane linear elasticity with quadratic displacements, linear stresses over a triangular element and both continuous across the interelement boundaries. In this case, the boundary integrals on the common boundaries amongst adjacent elements cancel each other. Thus the element matrices for the elements, which do not have edges coinciding with the boundary of the problem domain, can be formulated without extracting the boundary integrals from the energy product. Then the contribution to the energy product from the integrals like \/g U . T . . .dfi is zero for the degrees of freedom at the vertex nodes. fie 1 ^ \u00bb3 Thus zero rows and columns are obtained for displacement degrees of freedom at a l l internal vertices of the triangular elements when the element matrices are assembled. This then gives self-equilibrating modes over the f u l l domain. To check the existence of such modes, the element matrices for a certain for-mulation can be assembled so that there is at least one internal vertex node for the triangular elements and a corner node for the quadilateral elements, and then an eigenvalue-eigenvector analysis performed on the resulting matrix. 118 CHAPTER 7 APPLICATIONS OF THE MIXED FINITE ELEMENT METHOD The applications of the mixed f i n i t e element method to beam bending, plane linear elasticity and the problems with stress concentrations and singularities are presented in this chapter. The strain energy conver-gence rates for various formulations are predicted and compared with the numerical results obtained from the f i n i t e element analysis. The energy convergence of the mixed f i n i t e element method for plane e l a s t i c i t y with stress singularities, established in section 4.6, is also demonstrated with a numerical example. Finally the stress intensity factor K^ . for plates with symmetric edge cracks and a central crack are calculated and compared with the nearly exact values available. 7.1 Beam Problem Using the nomenclature of Figure 7, the following four f i r s t order f i e l d differential equations for simple beam theory result: - g - q - 0 (7.1) - f - V - 0 (7.2) de M dx EI 0 (7.3) \u00a3 - e = 0 (7.4) where (7.1) and (7.2) are the equilibrium equations (7.3) is the constitutive relationship (E=Young's Modulus, I=moment of inertia) and (7.4) is the con-straint equation arising from the assumption of plane sections remaining plane after deformation. If equations (7.2) and (7.4) are satisfied exactly, V and 0 in equations (7.1) and (7.3) can be eliminated and two second order aquations are obtained 119 d M n d 2v M _ 7 \" FT = \u00b0-(7.5) (7.6) dx z EI The mixed method is applied to both systems, namely the four f i r s t order equations (7.1) to (7.4) and the two second order equations (7.5) and (7.6). 7.1.A Two Second Order Equations The equations (7.5) and (7.6) can be put into the matrix form as (7.7) or where D= \"0 D2' V q M 0 AA = f (7.7a) dx* Here the matrix operator A is symmetric, i.e. (AA,A) = (A,AA), and the energy product is given by (AA, A) = \/ X 2 [M'-v+v'^-^rldx. (7.8) Integration of the f i r s t and the second term by parts yields (AA,A) = M'v|X2 + v'M|X2 - | X 2 [2M'v'+^]dx. The mixed variational principle can now be expressed as (7.9) XM = [2M'V4| r2qv]dx. (7.10) It can be observed from (7.10) that the continuity requirement has been reduced by one order compared to the Potential Energy approach with the variational form as I D = JI2 [EIv,,2-2qv]dx. (7.11) 120 This allows one to use lower order polynomials for approximating v and M. For a stationary point, the f i r s t variation of I in (7.10) i s zero 61 = -\/A2[2M'6v,+2v,6M,+^M-2q6v]dx = 0. M . hi Again integration of the f i r s t and the second term on the right hand side yields 6L, = \/ X2 2(M\"-q)6vdx+\/X2 2 (v\"-^r) 6Mdx-M' 6v I X 2 - v ' 6M jXz = 0. (7.12) M J x 1 ^ ; xj x EI 'xi 'x! This indicates that the forced boundary conditions are implied on the variables v and M which are different from v and v', one would find from the variational principle in equation (7.11) (ordinary Potential Energy Theorem). The two boundary terms in (7.9) could have also been obtained by twice integrating by parts the f i r s t term in (7.8) giving a variational principle of the form J M = [ 2 v \" M - i f - 2 q v ] d x ( 7 , 1 3 ) which involves a second derivative v, thereby requiring the same continuity requirement on v as the potential energy approach. It can be shown that the forced boundary conditions for the mixed variational principle in (7.13) are implied on v and v'. Thus the mixed method offers f l e x i b i l i t y not just in incorporating the boundary conditions as mentioned in Chapter 5, but also in continuity requirements when dealing with higher order operators. The mixed variational principle in (7.10) should be distinguished from the one in (7.13) in that the latter follows from the energy product in symmetric form as defined in equation (3.73). As i t i s advantageous to have reduced continuity requirements in f i n i t e element analysis for problems involving higher order operators, e.g. (7.7), the boundary terms have to be extracted from both the equilibrium (7.5) and the constitutive (7.6) equations. The effect of such a formulation on the convergence i s considered. In the simple beam bending theory the stress, which is the bending 121 moment M, is proportional to the curvature, the second derivative of displacement, equation (7.6). In order to obtain improved convergence in the strain energy the approximations for M and v have to comply with the requirement for mean square convergence of M to v\", so that the error in strain energy can be e s t i -mated from equation (3.71). This can be rewritten as [A-A0,A-A0] = (M-M0,M-M0) (7.14) for two second order beam equations. Here Mo=EIV\" is the exact bending moment distribution, M, the f i n i t e element approximation, and V Q , the exact solution for the deflection v. Let the approximations for v and M be (7.15) M v = E $ .v. i - l 1 1 N M = E Y.M., J - l 2 3 where $. and Y. e H^=VxM, the cross product space ($. e V, Y. e M). i J A l j The substitution of (7.15) into (3.68) gives p . _ , - a . , + , \u00a7 _ \u00ab . , and minimization with respect to M,. yields N M E (Y.,Y.)M. = Z (Y.,$'.')v., i=l,2, . . ., N. (7.16) j = l 1 J J j = 1 I J J However, the mixed variational principle in (7.10) gives the following equation at extremum instead of (7.16), i.e. N N E . (Y.,Y.)M. = - E (Y!,$!)v., i=l,2, . . ., N. (7.17) j = l 1 J J j=l 1 J J The right sides of (7.16) and (7.17) imply that (4\\ > v\") = _ ( r ,v'), (7.17a) which means that the derivative of v', which is v\", is taken as a generalized derivative. This is because v' is only required to be piecewise continuous and therefore would not possess an ordinary derivative everywhere. 122 As was done in Chapter 6, inspection of the functional I in (7.10), for g=0, reveals that the element matrix should have the following r i g i d body mode v = constant, M = 0. After trying out different polynomials for v and M, i t i s found that the inter-polations should be of the same degree in order to achieve the only r i g i d body mode as mentioned above. Such approximations also meet the requirement (i) of completeness. If the boundary conditions are homogeneous equations (7.16) and (7.17) are equivalent and the error in the energy product can be estimated from (7.14). A beam of uniform cross section with constant stiffness EI, length 1 is subjected to constant load per unit length q. The following four cases of boundary conditions are considered, Figure 8: (1) simply supported beam (S.S); v(0)=M(0)=v(J)=M(l)=0. (2) cantilever; v(0)=M(l)=0. (3) both ends clamped (fixed-fixed); v(0)=v(J)=0. (4) one end clamped and the other in a vert i c a l guide (fixed-guided); v(0)=0. (Tangent to the elastic curve at vert i c a l guide remains horizontal). Three different combinations for approximating v and M within the element are chosen (i) v-linear, M-linear; ( i i ) v-quadratic, M-quadratic; ( i i i ) v-cubic, M-cubic. The nodes per element and the number of degrees of freedom per node for the combinations ( i ) , ( i i ) and ( i i i ) are shown in Figure 9. The derivation of the element matrices i n a l l three cases is analogous to that of plane stress problem presented in Appendix A. In each case the beam is divided into 123 elements of equal length, 1^. The following quantities, where necessary, are tabulated for presentation purposes: 6 = deflection (v); 0 = rotation (v*); M = moment; V = shear (M') ; U = strain energy. The subscript M stands for middle, Q for quater point, E for end, RE for right hand end, and LE for l e f t hand end. (i) v-linear, M-linear The results are shown in Table V and the plots of quantities of interest versus the number of elements to show convergence appear in Figures 10. Linear approximations for v and M provide the continuity of v and M across the nodes and do not violate the completeness requirement ( i ) . Since the second derivative of linear approximation for v vanishes, equation (7.16) is satisfied only in a generalized sense, i.e. equation (7.17). For linear approximations within the element, the error in both v and M can be shown to be 0(1 2) and Q(lk) in I I M I I 2 , where I is the element length ( I =4-). From Figure 10(a), the relative error in strain energy converges as N~2 for the simply supported and cantilever configurations (cases 1 and 2, respectively), from below in case 1 (Table V(a)) and from above in case 2 (Table V(b)); and N - 4 from below for the fixed-fixed and fixed-guided ones (cases 3 and A, respectively) (Tables V(c) and V(d)). The expected rate of strain energy con-vergence is obtained when the moments are not forced to be zero (not as the forced homogeneous boundary condition on stress). Perhaps, for the cases when the bending moment is forced to be zero on the boundary, some lower 124 order error terms prevail. Figure 10(b) i l l u s t r a t e s the monotonic convergence of mid-point d e f l e c t i o n 6^ f o r the simply supported beam from below and t i p d e f l e c t i o n 6\u201e for the c a n t i l e v e r from above, whereas the r e l a t i v e error con-E verges as N - 2 , Figure 10(c). The bending moments at the nodes are exact for both cases 1 and 2. However, the reverse i s true for the other two cases, as can be observed from Tables V(c) and V(d). The computed de f l e c t i o n s at the nodes are exact, whereas the r e l a t i v e error i n the fixed moment con-verges as N - 2 , as i l l u s t r a t e d i n Figure 10(d). The other q u a n t i t i e s , appearing i n Tables V, seem to converge with increasing number of elements N f o r a l l four cases. ( i i ) v-quadratic, M-quadratic With v and M both quadratic, the completeness requirements f o r the energy convergence are s a t i s f i e d since the va r i a b l e s v and M are continuous across the nodes and have piecewise continuous f i r s t and second d e r i v a t i v e s . The errors i n v and M can be shown to be 0(1 3) which leads to an error of e O(l^) i n ||M|| 2. However, a quadratic approximation for M i s capable of representing the exact s o l u t i o n MQ for the constant load q along the beam length. Therefore the s t r a i n energy from the f i n i t e element s o l u t i o n i s expected to be exact. This i s confirmed by the r e s u l t s i n Tables VI for a l l four cases. Also the moments and the derived shears obtained are exact. I t i s i n t e r e s t i n g to note that the mid-deflections 6^ , i n a l l four cases, f o r an even number of elements obtained are exact and the r e l a t i v e error f o r an odd number of elements along the beam length converges as N _ t |, Figure 11; while the end computed d e f l e c t i o n 6 f or cases 2 and 4 are exact for odd RE and even number of elements. When a fa s t e r convergence i s observed for the displacement, the r e l a t i v e error i n the derived r o t a t i o n , i n a l l four cases appears to converge as N - 2 , Figure 11. Tables VI also i n d i c a t e convergence 125 of other nodal variables. ( i i i ) v-cubic, M-cubic With cubic approximations, both M and v and their f i r s t deriva-tives are continuous. Here again, the completeness requirements are satis-fied and like the previous approximations ( i i ) the strain energy is expected to be exact as well as the moments and the shears. The numerical results presented in Tables VII confirm this. The end deflections in the cases 2 and 4 are also exact. However, the relative error in the mid-deflection 6^ converges as N - ! + and in rotations as N~3 for a l l four cases; Figure 1 2 . 7.1.B Four First Order Equations The four f i r s t order beam equations (7.1) to (7.4) can be put into the matrix form as 0 D -EI V q 0 0 M 0 V 0 or AA = f where the matrix operator A is symmetric, i.e. (AA,A)=(A,AA). The energy product is given by \/ A \u00bb ,\\ f r dV dM\u201e \u201e \u201e, de,, M2,dvT7, , < A A . A ) = j1 [ - ^ - d ^ - 2 v e ^ - - + d - V ] d x . (7.18) (7.18a) (7.19) Integrating by parts the f i r s t and the fourth terms on the right hand side gives ,dv \u201edM\u201e (AA ,A) = -Vv|X2 + 9M|X2 + f [ 2 V ^ - 2 ^ 6 - 2 v e - ^ ] d x 'xj 'xi J l dx dx EI = -Vv| X 2 + 6M|X2 + [A,A] . x l (7.20) 126 Here [A,A] is the modified energy product of equation (3.73); The equation (7.20a) then leads to the mixed variational principle for the equations (7.18) with forced boundary conditions on v and M. The mixed Galerkin Method (by adding boundary residuals to the residuals of equations (7.1) and (7.3), section 5.4) and the mixed variational principle (7.21) would yield the same element matrix since the problem is linear and self-adjoint. Appendix C shows that the linear elasticity equation (5.9) yields the same energy product (7.19) as equations (7.18) when the basic assumptions of the simple beam bending theory are incorporated and the shear strain energy term, involving V 2, is neglected. Therefore the error in the energy product can be predicted from equation (3.71) provided approximations for the displace-ments (v,6) and stresses (M,V) are complete. Linear approximations are chosen for v,9,M and V. The element nodes and the nodal degrees of freedom are the same as illustrated in Figure 9, combination ( i i i ) . Further, the approximations chosen are complete and the error i n the energy product is governed by the mean square error in M, i.e. error in ||M|| 2, which is 0(1^) for the linear M distribution. Again four cases of boundary conditions (Figure 8) are considered. The results obtained from the f i n i t e element analysis are tabulated in Table VIII and convergence plots are shown in Figures 13. It is observed from Figure 13(a) that the relative error in strain energy converges as N _ l + for a l l four cases. Moments and shears at the nodes in cases 1 and 2 for both even and odd number of elements N are exact. However for cases 3 and 4, shears 127 are exact for both even and odd N but only end moments are exact for N even and they converge as N-1* for N odd in case 3. From Figure 13(b), the rela-tive error in rotation 0 converges as N - 2 in a l l cases. Deflections 6 at the free end for N even in case 2 and the guided end for N odd and even in case 4 are exact, converging as N - i +for odd N in case 2; whereas the mid-deflections 6 appear to converge to the exact value in an oscillatory manner. It has been mentioned in Chapters 3 and 5 and section 7.1.A that the mixed methods allow f l e x i b i l i t y in incorporating the boundary conditions. If the f i r s t and the second terms on the right hand side of equation (7.19) are integrated by parts, the following mixed variational principle results: X M = h [2V^+2Md|-2ve-g-2qv]dx (7.22) with forced boundary conditions on v and 0, the same as for the Potential Energy Theorem. Again linear v,0,M and V are used. Since the completeness requirements are not altered by shifting the forced boundary condition from M to 0, the convergence of strain energy is s t i l l expected to be N-t+. The strain energy for cases 1, 2 and 3 is computed and tabulated in Table IX. The relative error in strain energy versus N is then plotted in Figure 14 and in a l l cases the strain energy i s found to converge as N-t+. Next the shear strain energy term V 2 is also included and the mixed variational principle of (7.22) now includes an additional term (Appendix C) V - \/ I ^ a ^ - w o - g - ^ v ^ ^ ( 7 . 2 3 ) where v is poisson's ratio and h the height of the beam. Again the forced boundary conditions are on the variables v and 0. Using the same linear 128 approximations for v,9,M and V within an element, which s t i l l complies with the completeness requirements, cases 1, 2 and 3 are analysed and the results appear in Tables X. In the analysis, v is taken as 0.25 and h as ^ - The relative error i n strain energy converges as N _ t + i n a l l three cases as shown Figure 15(a). The shear V, in cases 1 and 3, is exact and converges as N~2 for the cantilever, case 2; the relative errors in the mid-moment for case 1, the fixed moment for case 2 and the end moments for case 3 converge as N\"-2; the mid-deflection <5W for cases 1 and 3 converges as N - 2 while the free end deflection 6 of the cantilever converges as N~^ as shown in Figures 15. However, the end rotation 8 for the cases 1 and 2 is exact for N even (Tables X) and from Figure 15(b) i t appears to converge as N-** for N odd in case 2. Despite the several different convergence rates observed for basic variables v,9,M and V for the three different formulations considered above, the relative error in the strain energy in a l l cases converges as N-1+. Further, where the boundary integrals were taken out from the equilibrium equations, i.e. forced boundary conditions on v and 0, the strain energy converges from below. Alternately i t is from above in the case where the forced boundary conditions are on the displacement v and the moment M. 7.2 Plane Linear Elas t i c i t y The stress and the displacement are chosen to be linear within a three node triangular element with five degrees of freedom per node (Figure 4) and are forced to be continuous across the interelement boundaries by equating the nodal variables at common nodes. These approximations satisfy the completeness requirements. 129 The derivation of the element matrix and the consistent load vector arising either from body forces or nonhomogeneous stress boundary conditions are given in Appendix A. The f i n i t e element thus formulated is then applied to solve the following problems. 7.2.A Plane Stress: Square Plate with Parabolically Varying End Loads A square plate with parabolically varying end loads is shown in Figure 16. Since the problem is symmetric about the x and y axes, only a quarter of the plate ABCD is considered in the f i n i t e element analysis with the forced boundary conditions u=0 on AD and v=0 on AB. Since the displacements u, v and the stresses x , x and x xx yy xy are assumed linear, the error in the stresses is 0(2 2), where 1 is the e e largest diameter within the element. Further, since such approximations satisfy the completnness requirements, and displacements and stresses are continuous across the interelement boundaries, equation (3.71) holds and error in the strain energy is expected to be the mean square error in the stresses, i.e. 0(1^), which is 0(N_1+) for a uniform grid, Figure 16. The numerical results for some of the stresses and displacements at points A,B,C, and D and the strain energy from the mixed f i n i t e element analysis for various grids are presented in Table XI. Also presented in Table XI are the results from the displacement element, obtained by Cowper, Lindberg and Olson [5], (using f u l l cubics for u and v displacements over a triangular element with six degrees of freedom (u,u ,u ,v,v ,v ) at x y x y vertices and two (u,v) at the centroid) for comparison. Since no attempt is made to satisfy the stress boundary conditions, the correct values of stresses on the boundary are obtained only in the limit of N-^. The convergence plots for the mixed element are shown in Figures 17 130 and those for the displacement element in Figure 18, [5J. The energy con-vergence rate for the former appears to be very close to N - 1 + (Figure 17(a)) as predicted. Although the error in strain energy from the displacement element is much smaller than from the mixed, owing to the fact that the former is a much more refined element, the energy convergence rate is slightly less than N-^, Figure 18, which is lower than the predicted assymp-totic rate of N - 6. The other interesting observation about the energy convergence for the mixed f i n i t e element is the convergence from below when the forced boundary conditions are on the displacement variables which has also been observed in section 7.I.B. The convergence rate for the stresses from the mixed element, Figure 17(c), appears to be close to N~2 for N larger than 8. However, peculiar kinks are observed and can be associated with the fact that certain stresses were fortuitously close to their exact values for a certain grid; e.g. N g f\u00b0 r N=6, etc. Figure 17(b) shows the convergence of displacements indicating faster convergence for u^ and (close to N-l+) than u^ and v^ (close to N \u2014 2). This is also observed for the displacement element and N greater than 6, Figure 18. In the mixed element, the kinks are observed in the convergence plots for the displacement u,, at N=6 and v at N=6 and N=10, a C which can be associated with slightly larger errors for such grids. 7.2.B Cantilever (Plane Stress) The dimensions, loading and the material properties are detailed in Figure 19(a). Two types of boundary conditions are considered at the fixed end as indicated in Figures 19(b) and 19(c); B.C.I and B.C.2. The latter is used for comparison purposes since the solutions using various f i n i t e elements for B.C.2 are available in the literature, while the former is considered to investigate the energy convergence. 131 (a) Cantilever with Boundary Conditions B.C.I In an effort to investigate the energy convergence, the exact value of strain energy is necessary. Besides such boundary conditions being easily incorporated in the f i n i t e element analysis, i t is also possible to obtain an elasticity plane stress solution under the assumption of bilinear normal stress in the x-direction and the shear stress independent of x and quadra-t i c in y. The solutions for the displacements u and v along with the strain energy for the boundary conditions B.C.I are presented in Appendix D. Since the part of the boundary between A and F, and E and F can move in either direction and i f the stresses on the l e f t face were not in-cluded in the f i n i t e element analysis through a consistent load vector, a stress-free boundary w i l l be simulated. This violates the assumptions of the elasticity solution which shall put in doubt the validity of the exact strain energy to be used in the error analysis. Therefore the stresses on the l e f t end are included in the consistent load vector. The typical mesh used in the analysis is shown in Figure 22 and the numerical results are presented in Table XII. The results indicate that the stress T at x=12 inches and y=-6 inches, and the tip deflection xx are converging to the exact values in an oscillatory manner. However, the strain energy is converging to the correct value from below. The convergence plots are shown in Figures 20. In Figure 20(a), the plot of relative error in strain energy versus N the number of elements in the beam depth, the inclusion of N=6 leads to a kink in the plot. Note the grid for N=6 does not contain the previous grids for N=2 and N=4, and excluding this the energy appears to converge as N - L f as predicted. The same behaviour is also observed in Figure 20(c) for the relative error in the stress T versus N for N>4 \u00b0 xx (close to N - 2 without N=6). This is not surprising since the error in strain energy is governed by the mean square error in the stresses. It is gratifying 132 that the energy convergence rate is about double that for stress as predicted. Figure 20(b) does not indicate any definite convergence rate for the tip deflection 6 and perhaps, more data is required to establish any trend of convergence. (b) Cantilever with Boundary Conditions B.C.2 Unfortunately the exact solution for the boundary conditions B.C.2 is not available and hence the exact strain energy is not known. The problem is solved to compare with the solutions obtained by using different displacement f i n i t e elements for various grids. These are readily available in literature, e.g. Gallagher [9]. Here the boundary AE (Figures 19) is pre-vented from moving in either direction and leads to stress singularities at corners A and E. Furthermore the shear stress and normal stress distributions at the fixed boundary are not the same as assumed in the beam theory. How-ever the results are compared with the beam theory [35] which provides an upper bound for the tip deflection from the displacement f i n i t e element. The numerical results from the mixed f i n i t e element analysis for various grids (Figure 22) are tabulated in Table XIII. Again the stress T at x=12 inches and y=-6 inches, and the tip deflection appear to converge X X in an oscillatory manner. The strain energy for N=8 is slightly higher than the strain energy 1\/2P6 obtained from the beam theory whereas in the previous examples, when the boundary integrals were extracted from the equilibrium equations, the energy converged from below. Since the exact numerical value is not known, the convergence of strain energy is rather d i f f i c u l t to establish. Table XIV shows the comparison of numerical results from the mixed f i n i t e element with those from the displacement models, e.g. constant stress triangle (C.S.T.), linear stress triangle (L.S.T.) and quadratic stress 133 triangle (Q.S.T.) elements, for the various grids shown in Figures 21. The mixed f i n i t e element used here appears to perform slightly poorer than the L.S.T., which uses quadratic approximations for the displacements and much better than the C.S.T., using linear displacements. The Figures 23(a) and 23(c) also indicate fast convergence of the tip deflection with increas-ing degrees of freedom and N, the number of elements in the beam depth, respectively, relative to the other elements. The graph of strain energy versus N shown in Figure 23(b) also shows rapid convergence. Finally the relative error in tip deflection is plotted against N for the mixed f i n i t e element and other displacement f i n i t e elements in Figure 24. It appears as in case (a) that more data is required to establish the convergence rate. However the plot .does exhibit fast convergence. It should be noted that the tip deflection from beam theory is used as exact solution in plotting these curves, and i t is in error i t s e l f . 7.2.C Stress Concentration around a Circular Hole (Plane Strain) A square plate (plane strain) with a circular hole in the middle (Figures 25) loaded by a uniform uniaxial stress to is considered. The diameter of the hole is one eighth of the plate width and the plate is of unit thickness. The plane strain state is analysed for both isotropic and orthotropic cases. It is demonstrated in Appendix A, how the element matrix for a plane stress isotropic case is modified for plane strain isotropic and orthotropic cases. The procedure is much simpler than for a displacement fi n i t e element. Because of symmetry only a quarter of the problem is con-sidered. The grid and the boundary conditions used in the fi n i t e element analysis are shown in Figure 26. This is essentially the same as used by Zienkiewiz, Cheung and Stagg [42] for constant stress triangular elements. 134 A comparison between the analytical solutions (for the isotropic case, Timoshenko and Goodier [35], and the orthotropic case, Savin [31]) for an in f i n i t e plate with a circular hole in the middle ( T along edge BC and T along edge AE) and the solutions obtained from the mixed f i n i t e element yy analysis is shown in Figure 27. A similar comparison with the solutions from the constant stress triangles [42] are shown in Figure 28. The mixed f i n i t e element solution shows excellent agreement with the analytical solution, and further the stresses are obtained directly at the nodes. The constant stress triangles also show good agreement with the exact solution, but the stresses are computed by averaging at nodes from the neighbouring elements, assuming the constant stress within the element to be the stress level at the node. Further the concentrations occuring at the boundaries are obtained by extrapo-lation. 7.3 Stress Singularities The strain energy convergence for plane stress elasticity with stress singularities, established in section 4.6.A, is demonstrated by a numerical example. The stress intensity factor K^. i s then determined from the method described in section 4.6.B for rectangular plates with symmetric edge cracks and a central crack (mode type I, Figure 1). 7.3.A Strain Energy Convergence The problem of a square plate with symmetric edge cracks (mode type I) is considered. Figure 29(a) shows the problem description and Figure 29(b) illustrates the f i n i t e element idealization of the quarter of the plate con-sidered because of the symmetry about the x and y axes. The problem is solved using mixed f i n i t e elements for various grid sizes for two cases. The stress x is kept continuous across point D (the crack tip) in the f i r s t case and 135 in the second case, an extra node is introduced along the x-axis next to the original one at D and only U , V , T and x degrees of freedom are equated at xx xy the two nodes, thus allowing x to be discontinuous across point D, the crack yy tip. (See distributions in Figure 32). The numerical results for both cases are presented in Tables XV. The strain energy is converging from below in both cases, while the peak stress T v v D a t the crack tip is about 28 percent higher when the normal stress i s discontinuous across the point D, than for the case when i t is continuous. The plots of strain energy versus the mesh size appear in Figures 30. The shape of the curves in both cases are very similar and they exhibit faster convergence than just linear as might have been expected. Figure 30(a) shows a comparison with the solutions obtained using various other elements. The present mixed element definitely shows a faster strain energy convergence than the constant stress triangles, the linear stress triangles, and the hybrid stress rectangles with cubic stress distribution within the element and quadratic displacements along the boundaries. The convergence rate is indicated by the plot of the relative error in strain x 2L 2 energy (exact U=3.228\u2014, Tong and Pian [38]) versus N, the number of elements C i t along the edge OA, Figure 31. It can be observed that the convergence rate approaches N - 2 as N gets larger, for both cases. It is clearly faster than N - 1 indicating the cancellation of the errors in the energy product of equation (4.84) due to stress singular terms. Further, a slightly larger error is observed in the case of discontinuous normal stress at the crack tip. Finally the normal stress x is plotted along the edge OA in Figures 32. In both cases, a small zone of compression is observed on the stress free edge of the crack with a peak value of about XQ (the applied stress on edge BC) close to the crack tip. 136 7.3.B Evaluation of the Stress Intensity Factor K^ . Two plane s t r a i n problems, rectangular plates one with symmetric edge cracks and the other with a ce n t r a l crack are considered. The d e t a i l s for these are shown i n Figures 33 and the f i n i t e element i d e a l i z a t i o n i n Figure 34. The layout of the mesh, used i n both problems, i s analogous to the one used by Parks [27] with the exception that the present elements are tria n g u l a r . Also indicated i n Figure 34 are the ra t i o s of the r a d i i y\u201e r0 to the crack length a (0, 0.1, 0.2 and 0.5). These are then used to calcu-l a t e the p o t e n t i a l energy release rate for a crack extension of Aa i n the f i n i t e element analysis as described i n section 4.6.B. Because of symmetry about the x and y axes, only a quarter of the problem i s considered i n each case and th i s i s shown i n Figures 33 as shaded areas along with the respective boundary conditions. Although, i t i s s u f f i c i e n t to solve each problem once for the i n i t i a l crack length a (section 4.6.B), at present the f i n i t e element analy-s i s i s performed every time when the countour TQ i s translated along with the i n t e r i o r nodes by the amount Aa=5xl0 6 a i n the x- d i r e c t i o n . The p o t e n t i a l energy release rate, i n the d i s c r e t i z e d form can be expressed as A 7 IM G I = -AT <7-55> and An., = TL, - TL, M Mr Mg 1 0 where TL, i s the p o t e n t i a l energy associated with the i n i t i a l crack and TL, MQ Mr 1 0 when the crack t i p has been m o v e d by the amount Aa. Then the crack i n t e n s i t y factor i s calculated by 137 The numerical results for the plate with symmetric edge cracks are lis t e d in Table XVI and those for the plate with a central crack in Table VII. The crack intensity factor for the former is compared with the nearly exact K^. obtained by Bowie [2] and for the latter, with K by Bowie and Neal [3]. In both cases the results obtained are in excellent agreement with the references. It is seen that the least percentage error is obtained for the contour Tn with radius Y^ =0.la, and the worst for y p =0, i.e. only R 0 R 0 the crack tip node is translated. The former is also associated with the highest potential energy release rate which varies for different sets of nodes defining TQ. When the calculated value of G^. or J-integral is in fact indepen-dent of the particular set of nodes defining TQ, the mesh may be called optimal. Thus, i t suffices, for optimal meshes, to move only the exterior node defining the crack tip, thereby altering the boundary, regardless of the particular set of interior nodes comprisng the contour TQ. Alternatively, non-optimal meshes w i l l exhibit some path dependence in the calculated values of G^.. In such cases, personal judgement and experience can help determine the best value of G^.. In Table XVIII, a comparison with stress intensity factors obtained from the energy release rate by other authors is presented. It can be seen that excellent accuracy is obtained with much fewer mixed f i n i t e elements and degrees of freedom than the corresponsing displacement models. Finally the plots of normal stress on the cracked face OA (Figures 33) are shown in Figures 35. Note that a higher peak stress is obtained at the crack tip than the peak stress Indicated in Figure 32(a), probably because of the refined mesh near the crack tip. 138 CHAPTER 8 CONCLUSIONS A detailed investigation of the theoretical foundation and practical aspects of applying mixed methods of approximate analysis for continuum and fi n i t e element analysis has been presented. Mixed methods always involve Indefinite operators and consequently a new energy product and i t s associated energy norm had to be introduced for these special operators. The fields of definition of such operators were so restricted that when obeyed, the energy product was found to be positive definite and represented twice the strain energy. These new concepts then formed the bases for establishing the energy convergence of complete approximations for displacements and stresses. It was found that the energy convergence implied the mean square convergence of such approximations to the exact values. The completeness requirements for continuum analysis were defined in two steps: (i) the mean square convergence of the strains from the approxi-mate stresses to the strains derived from the approximate displacements; (i i ) convergence of the energy norm. The alternate form of the requirement (i) is as follows: The strains from the stress approximations should possess at least all the strain modes that are present in the strains derived from the displacement approximations. It was also concluded that a violation of this requirement leads to mechanisms and this was confirmed by the eigenvalue-eigenvector analysis of an element matrix. The presence of mechanisms also indicates the breakdown of positive definiteness of the energy product, hence re-quirement (i) is the prerequisite of ( i i ) . The error in the energy product was shown to be proportional to the mean square error in the stress approximation when completeness is satisfied. This leads to much faster convergence in the 139 strain energy calculated from the mixed method than obtainable from the corresponding displacement method, i.e. the latter with identical displace-ment approximations as used in the mixed method. The foregoing concepts for the continuum were then extended to the f i n i t e element method and the corresponding completeness c r i t e r i a were established. These were found to be as follows: a) the displacement approximations should include a l l rig i d body and con-stant strain modes; b) the stress approximations should include a l l constant stress modes; c) the same as requirement (i) for the continuum, i.e. a l l the displacement strain modes should be included in the stress (strain) approximations. It was also concluded that for complete approximations the strain energy convergence cannot be any faster than for that calculated from the corres-ponding displacement model, unless the stresses are made continuous across the interelement boundaries. In the example of beam bending with four f i r s t order equations the use of linear interpolations for the four basic variables resulted in a predicted mean square error in stresses of 0(N _ l f). Therefore the predicted error in strain energy was also of 0(N_ 1 +) and this was confirmed by the numerical examples. The plane stress triangular element using linear interpolations for both displacements and stresses yielded a predicted error in stresses of 0(N - Z) , and a mean square error and strain energy error of 0(N - t +). In the numerical applications of this element, the energy convergence rate was indeed found to be 0(N - 1 +) for the plane stress square plate with parabolic end loads and nearly the same for the plane stress cantilever. In comparison the corresponding displacement element (C.S.T.) yields a convergence rate of only 0(N - 2). A faster energy convergence rate (nearly 0(N - 2)) was also 140 observed for the plane stress square plate with symmetric edge cracks for which the strain energy converges only linearly with N, even with higher order displacement or hybrid-type f i n i t e elements, Tong and Pian [38]. Excellent accuracy was also obtained for the stresses around a circular hole in the middle of a square plate subjected to uniaxial com-pression for plane strain isotropic and orthotropic cases. Finally the crack intensity factors (K^) computed for plane strain rectangular plates, one with symmetric edge cracks and the other with a central crack, yielded errors of only 1.97% and 0.89%, respectively. The matrix equations to be solved in the mixed f i n i t e element analysis are always indefinite and have zeroes on the diagonals for the displacement degrees of freedom. The method of Gaussian elimination with partial pivoting was successfully employed to solve such equations. Hence i t is concluded that the indefinite nature of the mixed method equations presents no special d i f f i c u l t i e s . In general, methods involving indefinite operators preclude obtaining upper or lower bounds on energy. In the applications of the mixed fi n i t e element method discussed herein, the energy was observed to converge in some cases from above and in some from below. However, in the examples where the variational principle was formed by extracting the boundary integrals from the equilibrium equations, the strain energy always converged from below. In the examples solved, far more accurate results were obtained by using the mixed f i n i t e element method than the corresponding displacement method with the same displacement approximations. However, the mixed method required more degrees of freedom for the same number of elements. On the other hand, the results for the stress concentration and stress singular problems were generally more accurate even using fewer elements (and total 141 number of degrees of freedom) than for the displacement models. Hence i t seems f a i r to conclude that the mixed f i n i t e element method can produce more efficient solutions for problems involving stress concentrations or singularities. 142 BIBLIOGRAPHY 1. Anderson, G.P., Ruggles, V.L., and Stibor, G., \"Use of Finite Element Computer Programs in Fracture Mechanics\", International Journal of Fracture Mechanics, Vol. 7, No. 1, 1971, pp. 63-76. 2. Bowie, O.L., \"Rectangular Tensile Sheet with Symmetric Edge Cracks\", Journal of Applied Mechanics, 31, 1964, pp. 208-212. 3. Bowie, O.L., and Neal, D.M., \"A Note on the Central Crack in a Uniformly Stressed Strip\", Engineering Fracture Mechanics, Vol. 2, 1970, pp. 181-182. 4. Cook, R.D., Concepts and Applications of Finite Element Analysis, John Wiley & Sons, Inc., New York, 1974. 5. Cowper, G.R., Lindberg, G.M., and Olson, M.D., \"A Shallow Shell Finite Element of Triangular Shape\", International Journal of Solids Structures, Vol. 4, 1970, pp. 1133-1156. 6. Dunham, R.S., and Pister, K.S., \"A Finite Element Application of the Hellinger-Reissner Variational Theorem\", Proceedings of the Second Conference on Matrix Methods in Structural Mechanics, Wright-Patterson Air Force Base, Ohio, AFFDL-TR68-150, 1968, pp. 471-487. 7. Finlayson, B.A., The Method of Weighted Residuals and Variational Principles, Academic Press, New York, 1972. 8. Finlayson, B.A., and Scriven, L.E., \"The Method of Weighted Residuals\u2014 A Review\", Applied Mechanics Review, 19, 1966, pp. 735-748. 9. Gallagher, R.H., Finite Element Analysis, Prentice-Hall, Englewood C l i f f s , New Jersey, 1975. 10. Hellan, K., \"Analysis of Elastic Plates in Flexure by a Simplified Finite Element Method\", Acta Polytechnica Scandinavica, C i v i l Engineering Series No. 46, Trondheim, 1967. 11. Hellwig, G., Differential Operators of Mathematical Physics, Addison-Wesley Publishing Company, London, (transl.) 1967. 12. Herrmann, L.R., \"A Bending Analysis of Plates\", Proceedings of the First Conference on Matrix Methods in Structural Mechanics, Wright-Patterson Air Force Base, Ohio, AFFDL-TR66-80, 1966, pp. 577-604. L3. Herrmann, L.R., \"Elasticity Equations for Incompressible and Nearly Incompressible Materials by a Variational Theorem\", American Institute of Aeronautics and Astronautics Journal, Vol. 3, No. 10, 1965, pp. 1896-1900. L4. Huebner, K.H., The Finite Element Method for Engineers, John Wiley & Sons, New York, 1975. L5. Hutton, S.G., \"Finite Element Method\u2014A Galerkin Approach\", Ph.D. Thesis, University of British Columbia, Canada, September 1971. 143 16. Hutton, S.G., and Anderson, D.L., \"Finite Element Method: A Galerkin Approach\", American Society of Civil Engineers, Engineering Mechanics Division, October, 1971, pp. 1503-1520. 17. Lorch, E.R., Spectral Theory, University Texts in the Mathematical Sciences, Oxford University Press, 1962. 18. Mikhlin, S.G., Variational Methods in Mathematical Physics, The Macmillan Company, New York, 1964. 19. Oden, J.T., Finite Elements of Nonlinear Continua, McGraw-Hill, New York, 1972. 20. Oden, J.T., \"Generalized Conjugate Functions for Mixed Finite Element Approximations of Boundary-Value Problems\", The Mathematical Foundations of the Finite Element Method\u2014with Applications to Partial Differential Eguations, A.K.. Aziz edition, Academic Press, New York, 1972, pp. 629-669. 21. Oden, J.T. , and Reddy, J.N., \"On Dual Complementary Variational Principles in Mathematical Physics\", International Journal of Engineering Science, Vol. 12, 1974, pp. 1-29. 22. Oden, J.T., \"A General Theory of Finite Elements I Topological Con-siderations\", International Journal for Numerical Methods in Engineering, Vol. 1, No. 3, 1969, pp. 205-221. 23. Oden, J.T., \"A General Theory of Finite Elements II Applications\", International Journal for Numerical Methods in Engineering, Vol. 1, No. 3, 1969, pp. 247-260. 24. Oliveira, E.R.A., \"Theoretical Foundations of the Finite Element Method\", International Journal of Solids Structures, Vol. 4, 1968, pp. 929-952. 25. Oliveira, E.R.A., \"Completeness and Convergence in the Finite Element Method\", Proceedings of the Second Conference on Matrix Methods in Structural Mechanics, Wright-Patterson Air Force Base, Ohio, AFFDL-TR68-150, 1968, pp. 1061-1090. 26. Olson, M.D., and Tuann, S.Y., \"Primitive Variables versus Stream Function Finite Element Solutions for the Navier-Stoke 1s Equations\", International Centre for Computer Aided Design\u2014Proceedings of the Second International Symposium on Finite Element Methods in Flow Problems, S. Margherita Ligure (Italy), June 14-18, 1976. Also see: Tuann, S.Y., and Olson, M.D., \"A Study of Various Finite Element Methods for the Navier Stoke's Equations\", Structural Research Series, Report No. 14, Department of C i v i l Engineering, University of British Columbia, Vancouver, Canada, May 1976. 144 27. Parks, D.M., \"A Stiffness Derivative Finite Element Technique for Determination of Crack Tip Stress Intensity Factor\", International Journal of Fracture, Vol. 10, No. 4, December 1974, pp. 487-501. 28. Pian, T.H.H., and Tong, P., \"Basis of Finite Element Methods for Solid Continua\", International Journal for Numerical Methods in Engineering, Vol. 1, 1969, pp. 3-28. 29. Rice, J.R., \"A Path Independent Integral and the Approximate Analysis of Strain Concentration by Notches and Cracks\", Journal of Applied Mechanics, Vol. 35, 1968, pp. 379-386. 30. Reddy, J.N., and Oden, J.T. , \"Convergence of Mixed Finite Element Approximations of a Class of Linear Boundary-Value Problems\", Journal of Structural Mechanics, Vol. 2, 1973, pp. 83-108. 31. Savin, G.N., Stress Concentrations around Holes, Pergamon Press,'1961. 32. Strang, G., Linear Algebra and its Applications, Academic Press, New York, 1976. 33. Strang, G., and Fix, G.F., An Analysis of the Finite Element Method, Prentice Hall, Englewood C l i f f s , New Jersey, 1973. 34. Taylor, C., and Hood, P., \"A Numerical Solution of the Navier Stoke's Equations using the Finite Element Technique\", Computer and Fluids, Vol. 1, 1973, pp. 73. 35. Timoshenko, S.P., and Goodier, J.N., Theory of Elasticity, 3rd edition, McGraw H i l l , New York, 1970. 36. Tong, P., \"New Displacement Hybrid Finite Element Model for Continua\", International Journal for Numerical Methods in Engineering, Vol. 2, 1970, pp. 73-83. 37. Tong, P., and Pian, T.H.H., \"The Convergence of Finite Element Method in Solving Linear Elastic Problems\", International Journal of Solids Structures, Vol. 3, 1967, pp. 865-879. 38. Tong, P., and Pian, T.H.H., \"On the Convergence of the Finite Element Method for Problems with Singularity\", International Journal of Solids Structures, Vol. 9, 1973, pp. 313-321. 39. Watwood, V.B., Jr., \"The Finite Element for Prediction of Crack Behaviour\", Nuclear Engineering Design II, 1969, pp. 323-332. 40. Wiinderlich, W., \"Discretisation of Structural Problems by a Generalized Variational Approach\", Proceedings of the International Association for Shell Structures, Pacific Symposium on Hydrodynamically Loaded Shells\u2014 Part I, Honolulu, Hawaii, October 10-15, 1971. 41. Zienkiewicz, O.C, The Finite Element Method in Engineering Science, McGraw H i l l , London, 1971. 145 Zienkiewicz, O.C. , Cheung, Y.K., and Stagg, K.G., \"Stresses in Aniso-tropic Media with Particular Reference to Problems of Rock Mechanics\", Journal of Strain Analysis, Vol. 1, No. 2, 1966, pp. 172-182. Balakrishnan, A.V., Applied Functional Analysis, Applications of Mathemati 3, Springer-Verlag, New York, 1976. Degrees of Sign of No. of Interpolation Freedom Eigen- Eigen- Composition of Eigenvectors j u,v x's* u,v x's Total values values (-) 3 X 's constant; u,v linear. Linear Constant 6 3 9 (0) (+) 3t 3 X X 's=u, =v, =u, +v, =0.t x y y x 's constant; u,v linear. (-) 9 X 's,u,v linear. Linear Linear 6 9 15 (0) (+) 3 3 X X 's=u, =v, =u, +v, =0. | x y y x j 1s,u,v linear. (-) 3 X 's constant; u,v quadratic. Quadratic Constant 12 3 15 (0) (+) 9** 3 X X 's=0;{u, =v, =u, +v, =0;u,v quadratic}, j x y y x 's constant; u,v quadratic. (-) 9 X 's linear; u,v quadratic Quadratic Linear 12 9 21 (0) (+) 3 9 X X s=u, =v, =u, +v, =0. x y y x 's linear; u,v quadratic. (-) 18 X 's,u,v quadratic. Quadratic Quadratic 12 18 30 (0) (+) 3 9 X X 's=u, =v, =u, +v, =0. x y y x s,u,v quadratic. *x's: A l l stresses x ,x and x have the same type of interpolation. xx yy xy **Extra zero eigenvalues are associated with mechanisms which have the same u,v distributions as the approximating polynomials. tRigid body modes (u, =0;v, =0;u, +v, =0). x y y x TABLE I: Eigenvalues and eigenvectors of element matrix for triangular elements using different combinations of interpolations for u,v and x's; linear elasticity plane stress. Interpolation Degrees of Freedom Sign of Eigen-values No. of Eigen-values Composition of Eigenvectors u ,v X ,x ,x xx yy xy u,v x's Total Bilinear Constant 8 3 11 (-) (0) (+) 3 3 x's constant; u,v bilinear. x's=0;{u, =v, =u, +v, =0;u,v bilinear.} x y y x x's constant; u,v bilinear. Bilinear Bilinear 8 12 20 (-) (0) (+) 12 3t 5 x's,u,v bilinear. x's=u, =v, =u, +v, =0.t x y y x x's,u,v bilinear. * Biquadratic; Bilinear 16 12 28 (-) (0) (+) 12 4** 12 x's bilinear; u,v biquadratic. x's=0;{u, =v, =u, +v, =0;u,v biquadratic}, x y y x x's bilinear; u,v biquadratic. I Biquadratic Biquadratic 16 24 40 (-) (0) (+) 24 3 13 x's,u,v biquadratic. x's=u, =v, =u, +v, =0. x y y x x's,u,v biquadratic. *Full quadratic in x and y plus x zy and xy . **Extra zero eigenvalues are associated with mechanisms which have the same u,v distributions as the approximating polynomials. tRigid body modes (u, =0;v, =0;u, +v, =0). \u00b0 J x y y x TABLE II; Eigenvalues and eigenvectors of element matrix for rectangular elements using different combinations of interpolations for u,v and x's; linear e l a s t i c i t y plane stress. Degrees of Sign of No. of 1 Interpolations Freedom Eigen- Eigen- Composition of Eigenvectors u,v P T'S u,v P x's Total values values (-) 10 p const.;U,V,T's linear. 2++3+0 3 p linear;u=v=x's=0. Linear Linear Linear 6 3 9 18 (0) (+) p=T s=u, =v. =u, +v, =0. x y y x p const.;U,V,T's linear. (-) 12 p,x's linear;U,V quadratic. Quadratic Linear Linear 12 3 9 24 (0) (+) 0+3*+0 9 P=T'S=U, =V, =U, +V, =0. x y y x p,x's linear;U,V quadratic. (-) 6 T'S const.; p,u,v quadratic. p quadratic;U=V=T's=0.^ f ] Quadratic Quadratic Constant 12 6 3 21 (0) (+) 3+3+6# 3 (P=T'S=U, =v, =U, +V, =0\/. 1 x 'y 'y 'x )' p=x's=0;u,v quadratic. T'S const.;p,u,v quadratic. (-) 21 p linear; U,V,T'S quadratic. Quadratic Linear Quadratic 12 3 18 33 (0) (+) 3 9 P=T'S=U, =V, =U, +V, =0. I v \u00bb x 'y 'y 'x p linear;U,V,T's quadratic. (-) 21 P,T'S,U,V quadratic, p quadratic;U=V=T's=0. Quadratic Quadratic Quadratic 12 6 18 36 (0) (+) 3+3+0 9 p=T s=u, =v, =u, +v, =0. x y y x p,TTs,u,v quadratic. tSelf-equilibrating modes in pressure. *Rigid body modes (u, =0;v, =0;u, +v, =0). \/\/Mechanisms. x y y x Note: The number of zero eigenvalues appear in the order; self-equilibrating, r i g i d body and M mechanisms respectively. Proper rigid body modes also have P=T'S=0. \u00b0\u00b0 TABLE III: Eigenvalues and eigenvectors of element matrix for triangular elements using different combinations of interpolations for u,v,p and T'S (T ,T ,T ); two-dimensional, r r xx yy xy incompressible creeping flow (linear part of the Navier-Stokes equations). r Interpolations Degrees of Freedom Sign of Eigen-values (-) (0) (+) No. of Eigen-values 15 1^+3+0 5 Composition of Eigenvectors u,v P x's u,v P x's Total B i l i n e a r B i l i n e a r B i l i n e a r 8 4 12 24 p,x's,u,v b i l i n e a r , p bilinear;x's=u=v=0. p=x s=u, =v, =u, +v, =0. x y y x p,x's,u,v b i l i n e a r . Biquadratic B i l i n e a r B i l i n e a r 16 4 12 32 (-) (0) (+) 16 0+3+1\/\/ 12 p,x's b i l i n e a r ; u , v biquadratic. p=x's=u, =v, =u, +v, =0. { x y y x } p=x's=0;u,v biquadratic. p,x's b i l i n e a r ; u , v biquadratic. Biquadratict B i l i n e a r Biquadratic 16 4 24 44 (-) (0) (+) 28 0+3*+0 13 p b i l i n e a r , x ' s , u , v biquadratic. p=x's=u, =v, =u, +v, =0. x y y x p b i l i n e a r ; x ' s , u , v biquadratic. Biquadratic Biquadratic Biquadratic 16 8 24 48 (-) (0) (+) 26 2+3+0 13 p,x's,u,v biquadratic. p biquadratic;x's=u=v=0. p = x s=u > x=v, y=u, y+v, x-0. p,x's,u,v biquadratic. t S e l f e q u i l i b r a t i n g modes i n pressure. *Rigid body modes (u, x=v, y=u, y+v, x=0). \/\/Mechanisms. Note: The number of zero eigenvalues appear i n the order: s e l f - e q u i l i b r a t i n g , r i g i d body and mechanisms, respectively. Proper r i g i d body modes also have p=x's=0. TABLE IV: Eigenvalues and eigenvectors of element matrix for rectangular elements using d i f f e r e n t combinations of inte r p o l a t i o n s for u,v,p and x's (x ,x ,x ): two-dimensional incompressible r xx yy xy creeping flow ( l i n e a r part of the Navier-Stokes equations). 150 j N \u00b0 -I o f 1026\\\u201eEI M 1029_EI hi i o v E 103UEI I I Elem. q l 3 q i q 2-f 5 I j 2 1.041667 2.083333 2.500 2.604167 1 4 1.236979 3.515625 3.750 3.743489 1 6 1.273148 3.858025 4.167 3.975909 8 1.285807 3.987630 4.375 4.058838 1 10 1.291667 4.050000 4.500 4.097500 I 12 1.294850 4.084680 4.583 4.118575 1 14 1.296769 4.105928 4.643 4.131308 I 16 1.298014 4.119873 4.688 4.139582 I 18 1.298868 4.129515 4.722 4.145260 [ 20 1.299479 4.136458 4.750 4.149333 J EXACT 1.302083 4.166667 5.000 4.166667 TABLE V(a): Simply supported beam; moments at the nodes are exact. No. of Elem. 1C 2 9 L E E I 1026 El M 1 0 e M E I ' 1 0 6 R E E I l o e ^ E i 102UEI q l 3 q l 4 q l 3 q l 3 2 \u2014 4. 687500 \u2014 1. 354167 1.770833 2 .864583 4 5. 33850 4. 492190 1. 601563 1. 276042 1.705730 2 .587891 6 3. 74228 4. 456020 1. 554784 1. 261574 1.685957 2 .538795 8 2. 88086 4. 443360 1. 531576 1. 256510 1.678060 2 .521770 10 2. 34167 4. 437500 1. 517500 1. 254167 1.674167 2 .513917 12 1. 97242 4. 434320 1. 508005 1. 252894 1.671971 2 .509659 14 1. 70372 4. 432400 1. 501154 1. 252126 1.670616 2 .507093 16 1. 49940 4. 431150 1. 495972 1. 251628 1.669723 2 .505430 18 1. 33887 4. 430300 1. 491912 1. 251286 1.669095 2 .504290 20 1. 20938 4. 429690 1. 488646 1. 251042 1.668646 2 .503474 EXACT 0. 00000 4. 427083 1. 45833 1. 250000 1.666667 2 .500000 | TABLE V(b): Cantilever; moments at the nodes are exact. TABLES V: Numerical results for two second order beam equations; v-linear, M-linear. Rotations and shears are derived from v and M, respectively. 151 No. of j Elem. 103617EI E 1036 EI M -10 ^ 104UEI j q l 2 q l 2 q 2j5 1 2 \u2014 2.60417 6.25000 6.250 6.510417 | 4 5.859 2.60417 4.68750 7.813 6.917320 6 4.822 2.60417 4.39815 8.102 6.939086 8 3.988 2.60417 4.29688 8.203 6.942750 10 3.375 2.60417 4.25000 8.250 6.943750 12 2.918 2.60417 4.22458 8.275 6.944111 14 2.566 2.60417 4.20918 8.291 6.944264 16 2.289 2.60417 4.19922 8.301 6.944338 18 2.065 2.60417 4.19239 8.308 6.944380 j 20 1.880 2.60417 4.18750 8.313 6.944401 J EXACT 0.000 2.60417 4.16667 8.333 6.944444 TABLE V(c): Beam with both ends fixed, deflections at the nodes are exact. No. of Elem. 102 S L E E I 1C 2 6 M E I M 1 0 2 6 R E E I 1 0 2 e R E E I -1 0 M L E 1 0 MRE 102UEI q l 3 q l 4 q l 4 q l 3 q l 2 q l 2 q 2 i 5 2 \u2014 2. 34375 4. 16667 \u2014 3 .12500 1.87500 1.1067708 4 3. 1901 2. 34375 4. 16667 2.0182 3 .28125 1.71875 1.1108398 6 2. 3341 2. 34375 4. 16667 1.3696 3 .31019 1.68982 1.1110575 8 1. 8305 2. 34375 4. 16667 1.3467 3 .32031 1.67969 1.1110942 10 1. 5042 2. 34375 4. 16667 0.8292 3 .32500 1.67500 1.1111042 12 1. 2756 2. 34375 4. 16667 0.6920 3 .32755 1.67245 1.1111078 14 1. 1070 2. 34375 4. 16667 0.5937 3 .32909 1.67092 1.1111093 16 0. 9776 2. 34375 4. 16667 0.5198 3 .33008 1.66992 1.1111101 18 0. 8752 2. 34375 4. 16667 0.4623 3 .33076 1.66924 1.1111104 20 0. 7922 2. 34375 4. 16667 0.4162 3 .33125 1.66875 1.1111107 j EXACT 0. 0000 2. 34375 4. 16667 0.0000 3 .33333 1.66667 1.1111111 TABLE V(d): Beam with one end fixed and the other guided; deflections at the nodes are exact. TABLES V: Numerical results for two second order beam equations; v-linear, M-linear. Rotations and shears are derived from v and M, respectively. 152 fi No. of Elem. 10*6^1 10 2 e EI h 1036 QEI 10 3UEI q l * q l 3 1 1.250000 5.000000 \u2014 4.16667 2 1.302083 4.791667 9.244792 4.16667 3 1.301440 4.506173 \u2014 4.16667 4 1.302083 4.375000 9.277344 4.16667 5 1.302000 4.306667 \u2014 .4.16667 6 1.302083 4.266975 9.276942 4.16667 7 1.302062 4.241983 \u2014 4.16667 8 1.302083 4.225260 9.277344 4.16667 15 1.302082 4.184197 \u2014 4.16667 16 1.302083 4.183129 9.277344 4.16667 1 EXACT 1.302083 4.166667 9.277344 4.16667 j TABLE VI(a): Simply supported beam. No. of 1 Elem. 102 9. L EEI 10 26 EI M 106 EI M 1 0 6 R E E I 1 0 9 R E E I 102UEI q l 3 q l 4 q l 3 q l 3 q 2 2 5 1 5.0000 4.375000 1.250000 1.25 2.000000 2.50 2 1.6667 4.427083 1.562500 1.25 1.708333 2.50 3 0.8025 4.426440 1.435185 1.25 1.679012 2.50 4 0.4688 4.427083 1.484375 1.25 1.671876 2.50 5 0.3067 4.427000 1.450000 1.25 1.669333 2.50 6 0.2161 4.427083 1.469907 1.25 1.668210 2.50 7 0.1604 4.427062 1.454082 1.25 1.667637 2.50 8 0.1237 4.427083 1.464844 1.25 1.667318 2.50 15 0.0360 4.427082 1.457407 1.25 1.666763 2.50 16 0.0317 4.427083 1.459961 1.25 1.666747 2.50 | EXACT 0.0000 4.427083 1.458333 1.25 1.666667 2.50 TABLE VI(b); Cantilever TABLES VI: Numerical r e s u l t s f o r two second order beam equations; v-quadratic, M-quadratic. Rotations and shears are derived. Moments and shears are exact i n a l l cases. 153 No. of Elem. 1O30 EI E 1036 QEI 10 39 QEI 1036 EI M lO^UEI ql 3 ql1* qi 3 ql 4 q 2 ! 5 1 8.333 \u2014 \u2014 2.083333 6.94444 2 6.250 1.432292 5.20833 2.604167 6.94444 3 3.395 \u2014 \u2014 2.597737 6.94444 4 2.083 1.464844 9.11458 2.604167 6.94444 5 1.400 \u2014 \u2014 2.603333 6.94444 6 1.003 1.464442 7.52315 2.604167 6.94444 7 0.753 \u2014 \u2014 2.603950 6.94444 8 0.586 1.464844 8.13802 2.604167 6.94444 15 0.175 \u2014 \u2014 2.604156 6.94444 16 0.155 1.464844 7.89388 2.604167 6.94444 EXACT 0.000 1.464844 7.81250 2.604167 6.94444 TABLE VI(c): Beam with both ends fixed. No. of Elem. i 0 2 e L E E i i o V 1 lO^JSI M 10 26 R EEI I O 3 G R E E I 102UEI ql 3 ql1* ql 3 ql1* ql 3 q 2 ! 5 1 5.0000 2.291670 4.16667 4.16667 33.3333 1.111111 2 1.6667 2.343750 7.29167 4.16667 04.1667 1.111111 3 0.8025 2.343107 6.01852 4.16667 01.2350 1.111111 4 0.4688 2.343750 6.51042 4.16667 00.5210 1.111111 5 0.3067 2.343667 6.16667 4.16667 00.2667 1.111111 6 0.2161 2.343750 6.36574 4.16667 00.1543 1.111111 7 0.1603 2.343728 6.20750 4.16667 00.0972 1.111111 8 0.1348 2.343750 6.27010 4.16667 00.0651 1.111111 15 0.0361 2.343749 6.24074 4.16667 00.0099 1.111111 16 0.0317 2.343750 6.26628 4.16667 00.0082 1.111111 EXACT 0.0000 2.343750 6.25000 4.16667 00.0000 1.111111 B^LE VI(d): Beam with one end fixed and the other guided. IBLES VI: Numerical results for two second order beam equations; v-quadratic, M-quadratic. Rotations and shears are derived. Moments and shears are exact in a l l cases. 154 No. of Elem. l O ^ E I 10 26 EI M 103UEI ql3 q^ q 2 ! 5 1 5.000000 \u2014 4.166667 2 4.305556 1.307870 4.166667 3 4.209877 \u2014 4.166667 4 4.185049 1.302594 4.166667 5 4.176092 \u2014 4.166667 6 4.172123 1.302189 4.166667 EXACT 4.166667 1.302083 4.166667 TABLE VII(a) : Simply supported beam. Mo. of Elem. 10 39 T JEI Lh, 1026\\,EI M 108WEI M 1 0 6 R E E I loe^Ei 102UEI J ql3 q^ ql3 ql4 ql3 q 2! 5 1 8.3333 \u2014 \u2014 1.250 1.583333 2.50 2 1.3889 4.432870 1.458333 1.250 1.652778 2.50 3 0.4321 \u2014 \u2014 1.250 1.662346 2.50 4 0.1838 4.427594 1.458333 1.250 1.664828 2.50 5 0.0943 \u2014 \u2014 1.250 1.665724 2.50 6 0.0546 4.427189 1.458333 1.250 1.666121 2.50 EXACT 0.0000 4.427083 1.458333 1.250 1.666667 2.50 TABLE VII(b): Cantilever TABLES VII: Numerical results for two second order beam equations, v-cubic, M-cubic. Moments and shears are exact in a l l cases. 155 No. of Elem. 1039T,EI E 10 36 EI M lO^UEI ql * q 2! 5 1 8.3333 \u2014 6.94444 2 1.3889 2.662037 6.94444 3 0.4321 \u2014 6.94444 4 0.1838 2.609273 6.94444 5 0.0943 \u2014 6.94444 6 0.0546 2.605228 6.94444 EXACT 0.0000 2.604167 6.94444 TABLE VII(c): Beam with both ends fixed. No. of Elem. io 3 e L EEi i o ' V 1 1029 EI M 10 26 R EEI I O 3 S R E E I 102UEI q l 3 q l * q l 3 q l * q l 3 q 2 ! 5 1 8.3333 \u2014 \u2014 4.166667 -8.3333 1.111111 2 1.3889 2.349537 6.250 4.166667 -1.3889 1.111111 3 0.4321 \u2014 \u2014 4.166667 -0.4321 1.111111 4 0.1838 2.344261 6.250 4.166667 -0.1838 1.111111 5 0.0943 \u2014 \u2014 4.166667 -0.0943 1.111111 6 0.0546 2.343856 6.250 4.166667 -0.0546 1.111111 EXACT 0.0000 2.343750 6.250 4.166667 0.0000 1.111111 CABLE VII(d): Beam with one end fixed and the other guided. CABLES VII: Numerical results for two second order beam equations, v-cubic, M-cubic. Moments and shears are exact in a l l cases. 156 No. of Elem. 10 26 EI M i o2 e E I E 10 3UEI 1 ql 1* q l 3 1 2 1.852 5.5560 4.6300 3 \u2014 4.5730 4.2337 4 1.360 4.5140 4.1956 5 \u2014 4.3380 4.1766 6 1.338 4.3210 4.1723 j 1 EXACT 1.302 4.1667 4.1667 j TABLE VI I I ( a ) : Simply supported beam; nodal moments and shears are exact. No. of Elem. 1 0 9 L E E I ^ V 1 1 0 6 R E E I 1 0 6 R E E I 102UEI q l 3 q l 4 q l 3 q l 4 q i 3 q 2 ! 5 1 1.6667 \u2014 \u2014 1.6667 1.6667 4.1667 2 0.5555 3.9350 1.2500 1.2500 1.9444 2.5463 3 0.2309 \u2014 \u2014 1.2551 1.5387 2.5324 4 0.1389 4.4840 1.5625 1.2500 1.7361 2.5029 5 0.0844 \u2014 \u2014 1.2507 1.6169 2.5043 6 0.0617 4.3470 1.4352 1.2500 1.6975 2.5006 EXACT 0.0000 4.4271 1.4583 1.2500 1.6667 2.5000 TABLE VIII(b): Cantilever; nodal moments and shears are exact. TABLES VIII: Numerical r e s u l t s f o r four f i r s t order equations, forced boundary conditions on v and M; v,0,M and V a l l l i n e a r . 157 No. of Elem. 1O20\u201eEI h, 1036 KI M 10\\ 10 ^ lO'+UEI q l 3 q l * q l 2 q l 2 q 2! 5 1 2 3 4 5 6 1.3889 0.9150 0.3472 0.3442 0.1543 4.62960 3.18290 2.57210 8.3333 4.1667 4.6296 -8.3333 -8.2305 -8.3333 -8.3333 -8.3333 -1.1574 8.4675 7.2338 7.1547 7.0016 EXACT 0.0000 2.60417 4.1667 -8.3333 6.9444 TABLE VIII(c): Beam with both ends fixed; shears are exact. No. of Elem. 102 6 L E E I 1C 2 6 M E I M 1C 2 6 R E E I 102 6RE E I \" 1 0 M L E 1 0 MRE ' 102UEI q l 3 q l * q l * q l 3 q l 2 q l 2 q 2! 5 1 4. 1666 \u2014 4. 16667 4. 1667 2.5000 2.50000 1.0466 2 5. 5555 2. 54629 4. 16667 2. 7778 3.3333 1.66667 1.1574 3 1. 3775 \u2014 4. 16667 0. 4515 3.3230 1.67695 1.1263 4 1. 3888 2. 40158 4. 16667 0. 6944 3.3333 1.66667 1.1141 0. 5108 \u2014 4. 16667 0. 1775 3.3320 1.66800 1.1132 6 0. 6172 2. 34050 4. 16667 0. 3086 3.3333 1.66667 1.1117 EXACT 0. 0000 \u2022 2. 34375 4. 16667 0. 0000 3.3333 1.66667 1.1111 [ TABLE VIII(d):' Beam with one end fixed and the other guided; shears are exact. TABLES VIII: Numerical results for four f i r s t order equations, forced boundary conditions on v and M; v,0,M and V a l l linear. 158 No. of Elem. (1) Simply Supp. 103UEI (2) Cantilever 102UEI q 2 ^ (3) Fixed-fixed lO^UEI q 2 ^ 1 2 3.472222 1.388884 __ 3 4.034536 2.194787 . \u2014 4 4.123264 2.365451 6.510467 5 4.133495 2.440016 6.521569 6 4.158093 2.467577 6.858711 8 4.163954 2.488878 6.917318 10 4.165556 2.495289 6.933333 12 4.166131 2.497662 6.939086 16 4.166497 2.499243 6.942750 j EXACT 4.166667 2.500000 6.944444 TABLE IX: Strain energy estimations for four f i r s t order beam equations with forced boundary conditions on v and 9; M,V,9 and v a l l linear. |No. of Elem. 1026 EI M 1C 2 e E E i 1 0 MM 103UEI q l 3 q i 2 q l 2 q 2 ! 5 1 2 2.43056 4. 16667 0.83333 0 .08333 6 .076389 3 \u2014 4. 16816 \u2014 0 .00953 6 .762226 4 2.17014 4. 16667 1.45833 0 .02083 6 .727431 5 \u2014 4. 16793 \u2014 0 .00397 6 .769543 6 2.13049 4. 16667 1.20370 0 .00926 6 .762251 8 2.10504 4. 16667 1.30208 0 .00521 6 .768121 10 2.10056 4. 16667 1.23333 0 .00333 6 .769722 12 2.09298 4. 16667 1.27315 0 .00231 6 .770298 16 2.08876 4. 16667 1.26302 0 00130 6 .770664 EXACT 2.08333 4. 16667 1.25000 0 00000 6 .770833 TABLE X(a): Simply supported beam, shears are exact. TABLES X: Numerical results for four f i r s t order beam equations; forced boundary conditions on v and 6; shear strain energy included; M,V,e,v a l l linear. 159 No. of Elem. 1 0 6 R E E I 1 0 9 R E E I 1 0 M L E VLE 102UEI q l * q i 3 q l 2 qi q 2 ! 5 1 0.84821 1.07143 1. 07143 1. 35714 3 .794643 2 1.43430 1.66667 3. 39744 1. 46154 3 .151709 3 1.54113 1.63422 4. 33756 1. 17524 3 .487009 4 1.55303 1.66667 4. 56439 1. 13636 3 .513652 5 1.55948 1.66195 4. 74883 1. 07071 3 .533663 6 1.56056 1.66667 4. 80288 1. 06272 3 .535970 8 1.56188 1.66667 4. 88839 1. 03571 3 .539845 10 1.56225 1.66667 4. 92835 1. 02299 3 .540917 12 1.56238 1.66667 4. 95016 1. 01601 3 .541304 16 1.56246 1.66667 4. 97192 1. 00904 3 .541552 EXACT 1.56250 1.66667 5. 00000 1. 00000 3 .541667 TABLE X(b): Cantilever. No. of Elem. 10 26 EI M 1 O 2 M M 10 2M^ 103UEI q l * q l 2 q i 2 q 2! 5 1 2 1.04167 0.00000 0.00000 2.604167 3 \u2014 \u2014 6.51466 3.242249 4 1.12847 6.25000 6.25000 3.255208 5 \u2014 \u2014 7.58171 3.289003 6 1.05024 3.70370 7.40741 3.290038 8 1.06337 4.68750 7.81250 3.295898 10 1.04500 4.50000 8.00000 3.297500 12 1.05131 4.39815 8.10185 3.298075 16 1.04709 4.29688 8.20313 3.298442 EXACT 1.04167 4.16667 8.33333 3.298611 1 TABLE X(c): Beam with both ends fixed; shears are exact. TABLES X: Numerical results for four f i r s t order beam equations; forced boundary conditions on v and 6; shear strain energy included; M,V\u00a3,v a l l linear. FINITE ELEMENT 10Etu_ 102Etu lOEtv 10Etv D -ION , | xxA ) N f )L (1-v Z)N nL ( l - V -)N0L (l-v ; -)N L Nn 1 GRID N Elem. A* Elem. Bt Elem. A Elem. B Elem. A Elem. B Elem. A Elem. B Elem. A Elem. B J l x l 2 1.054258 1.507941 -1.1435 2.1934 2.99336 1.31824 4.332647 5.085466 0.00000 1.44190 2 x 2 4 1.443990 1.519821 -0.7670 1.8684 1.45335 1.28574 4.849820 5.073595 2.01236 1.40137 3 x 3 6 1.463880 1.519773 0.6555 1.8046 1.34960 1.27936 4.974455 5.073633 1.41893 1.40559 4 x 4 8 1.498170 1.519862 1.3373 1.7896 1.28864 1.27787 5.016690 5.073544 1.58424 1.40789 5 x 5 10 1.508706 1.519900 1.5268 1.7852 1.28480 1.27742 5.036405 5.073507 1.28638 1.40880 6 x 6 12 1.512260 1.6488 1.27514 5.047228 1.49870 EXACT 1.519928 1.519928 1.7837 1.7837 1.27727 1.27727 5.073478 5.073478 1.40954 1.40954 j ION . yyA ION yyB 102N xxB 1 0 NxxC 10Et2U Nn No N 0 N 0 ( l - v 2 ) L 2 N 2 N Elem. A Elem. B Elem. A Elem. B Elem. A Elem. B Elem. A Elem. B Elem. A Elem. B 2 6.66667 8. 55810 4.04167 4.70735 6.2500 3.9928 0.0000 0.3181 2.553610 2.787981 4 9.16957 8. 59863 3.54268 4.17500 0.9848 0.4235 0.7266 -0.0005 2.746331 2.793366 6 7.75831 8. 59441 4.31117 4.11902 4.8662 0.0848 -0.4544 -0.0299 2.782543 2.793540 8 8.92269 8. 59211 4.02879 4.10971 3.0654 0.0329 -0.3742 -0.0285 2.789962 2.793562 10 8.39253 8. 59120 4.05786 4.10767 1.9439 0.0166 -0.2945 -0.0233 2.792055 2.793567 12 8.74161 4.13723 1.5472 -0.2530 2.792746 EXACT 8.59046 8. 59046 4.10670 4.10670 0.0000 0.0000 0.0000 0.0000 2.793570 2.793570 *Mixed Finite Element; displacements and stresses linear. tDisplacement Finite Element; u and v f u l l cubics [5]. TABLE XI: Numerical results for parabolically loaded plane stress problem. 1 No. of Elem. 8 in Beam Depth N Total No. of Elem. Degrees of Freedom Tip Deflec-t i o n 6 c v(L,0).(in.) T @ X X x=12\",y=-6\" k s i Longitudinal D e f l e c t i o n @ B=UB. (in.) St r a i n Energy U(k-in.) 2 32 131 0.336062 53.084 0.059717 6.731654 4 128 421 0.355121 60.501 0.064024 7.136889 6 288 871 0.355093 59.612 0.063980 7.140213 8 512 1481 0.355459 60.148 0.064038 7.145934 EXACT (ELASTICITY) 0.355333 60.000 0.064000 7.146667 TABLE XII: Numerical re s u l t s for the ca n t i l e v e r (plane stress) with boundary conditions B.C.I (Figure 19). Mixed f i n i t e element; displacements and stresses l i n e a r . No. of Elem. Total Degrees Tip Deflec- T @ X X Longitudinal S t r a i n i n Beam No. of of t i o n 5 = c x-12\",y=-6\" Def l e c t i o n @ Energy 1 Depth N Elem. Freedom v(L,0).(in.) k s i B=UB. (in.) U(k-in.) I 1 8 46 0.245120 39.107 0.042410 4.902405 I 2 32 129 0.3359427 52.694 0.059602 6.718577 4 128 415 0.355464 60.469 0.064064 7.109509 6 288 861 0.355698 59.734 0.064087 7.114006 8 512 1467 0.355952 60.125 0.064146 7.119015 J [ EXACT (BEAM THEORY) 0.355833 60.000 7.116667 J TABLE XIII: Numerical results f o r the ca n t i l e v e r (plane stress) with boundary conditions B.C.2 (Figure 19). Mixed f i n i t e element; displacements and stresses l i n e a r . 162 Element Type Mesh N Number of Degrees of Freedom Tip Deflection 6=vc (in.) x - Normal X X Stress @ x=12\" y=-6\". (ksi) A-1 2 48 0.19819 33.407 j C.S.T A-2 4 160 0.30556 51.225 A-3 8 576 0.34188 57.342 B-1 1 48 0.34872 L.S.T B-2 2 160 0.35506 59.145 B-3 4 576 0.35569 60.024 j C-1 1 68 0.35373* 58.973* I Q.S.T C-2 2 214 0.35506 59.843 I C-3 4 268 0.35580 59.993 MIXED M-1 1 46 0.24512* 39.108 DISPL. M-2 2 129 0.33594 52.694 AND M-4 4 415 0.35547 60.469 STRESSES M-6 6 861 0.35570 59.734 I LINEAR M-8 8 1467 0.35595 60.125 1 j BEAM THEORY 0.35583 60.000 1 *Average of values at y=6\" and y=-6\". TABLE XIV: Comparison amongst C.S.T., L.S.T., Q.S.T., and mixed f i n i t e element. Cantilever with boundary conditions B.C.2 (Figure 19). E U A E V A E v c E UD TxxD T _ TxxA EU* T 0 L T0 T0 2 0.4633 1.4778 1.1222 0.2440 0.7052 1.3835 0.6897 2.738912 4 0.4571 1.7771 1.1737 0.4271 1.4203 2.3751 -0.7849 3.122624 6 0.4507 1.7810 1.1771 0.4787 1.8202 3.0010 0.5045 3.166420 8 0.4739 1.8312 1.1849 0.5216 2.1230 3.4100 -0.2607 3.196404 10 0.4510 1.8186 1.1856 0.5364 2.4049 3.8175 0.0071 3.203674 12 0.4610 1.8272 1.1869 0.5546 2.6225 4.1357 0.0469 3.212178 TABLE XV(a): x continuous at the crack tip D. N E U A E V A E v c E uD T xxD TyyD TxxA EU* x 0 L T Q L x 0L x 0 L T0 T0 T0 2 0.0544 2.9301 2.8883 -0.7285 -0.1440 2.2921 -0.7936 1.704874 4 0.4568 1.7749 1.1719 0.4281 1.4257 3.0437 -0.7605 3.113852 6 0.4461 1.7709 1.1762 0.4817 1.8396 3.8743 0.4800 3.159226 8 0.4732 1.8273 1.1837 0.5252 2.1493 4.3664 -0.2376 3.191367 10 0.4996 1.8147 1.1848 0.5404 2.4361 4.8837 -0.0143 3.199594 12 0.4600 1.8241 1.1862 0.5585 2.6562 5.2798 0.0688 3.208874 TABLE XV(b): x discontinuous at the crack tip D. yy TABLES XV: Numerical results for the plane stress problem of square plate with symmetric edge cracks, Figure 29. UE O N Co *Exact value: U = 3.228 2 t 2-. T\u00b0 ng and Pian [38] T n L t 164 E7TM 106EATL, M aE Air . M K I Error % a T 2hbt T^hbt T 2hbt Aa T 0\/a 0.0 1.05942309 1.385584 0.274918 1.90402 10.98 0.1 1.05942339 1.680112 0.333358 2.09664 1.97 0.2 1.05942331 1.607250 0.318898 2.05067 4.12 J 0.5 1.05942333 1.626610 0.322740 2.06300 3.55 I n i t i a l Crack 1.05942171 EXACT K ; ref. [2] 2.13884 TABLE XVI: Stress intensity factors from the f i n i t e element analysis of the rectangular plate with symmetric edge cracks, Figure 33(a). (Mixed f i n i t e element; displacements and stresses linear). r r 0 E*M 106EATT M aE ATT\u201e M K I Error % a T 2hbt T 2hbt T 2hbt a T 0\/a 0.0 1. 04790510 1.408614 0.279486 1.91978 8.98 0.1 1. 04790543 1.730814 0.343416 2.12804 0.89 0.2 1. 04790535 1.645374 0.326464 2.07485 1.63 0.5 1. 04790537 1.664358 0.330230 2.08679 1.06 I n i t i a l Crack 1 1. 04790370 EXACT K ; ref. [3] 2.10922 TABLE XVII: Stress intensity factors from the fi n i t e element analysis of the rectangular plate with a central crack, Figure 33(b). (Mixed f i n i t e element; displacements and stresses linear). Author(s) Number of Elements Degrees of Freedom Accuracy of K Error % Type of Element Watwood [38] 470 956 2.00 Triangular and rectangular Anderson et al. [1] 1470 3000 0.14 Quadrilateral Present result 174 505 1.97 Mixed triangles.* Symm. edge cracks Present result 174 505 0.89 Mixed triangles. Central crack TABLE XVIII: Comparison of stress intensity factors obtained from energy release rate using different elements and procedures. *Plane strain mixed f i n i t e element; displacements and stresses linear. 166'-FIG. 2 ! Typical contour for evaluation of J-integral 167 FIG.3' Accommodation of crack extension Aa by advancing nodes on the path TQ . 0 x FIG.4= Node numbers and degrees of freedom for a triangular element. H u 4 , v 4 ^4 4 4 T xx,Tyy,Txy 4 ? 8 * u\u201ev, 1 xx >Lyy>t xy 7 -o--o-5 U 3 . V 3 3 3 3 Txx,Tyy,Txy \u2022 \u00b0 3 46 u 2 ,v 2 2 2 2 1 xx >l y y>lxy 0 FIG.5-- Node numbers and degrees of freedom for a rectangular element. 169 FIG.7 : Fo rces act ing on an inf in i tes imal beam element 170 v(o) = M(o)=0 vU) = MU)=0 Cased) Simply supported (S.S.) v(o)=0 MU) = 0 1===--A Case(2) Cantilever v(o) = 0 A vU)=0 \u00a7 Case(3) Both ends clamped (fixed- fixed) v(o)=0 i l \u00ab ^ Case(4) One end clamped and the other in a vertical guide (fixed - guided) Deflected elastic curve FIG.8 ! Forced boundary conditions on v and M for the beam problem. 171 V l v 2 M i M o 6 X, ( i ) v - l i n e a r , M - l i n e a r v, v 2 v 3 M| M 2 M 3 I <5 o cW 2 (ii) v - quad ra t i c , M - q u a d r a t i c 0, 0 2 M, M 2 Vl v 2 i <5 J p (iii) v - cub i c , M - c u b i c FIG.9= Degrees of f r eedom for the beam e l e m e n t - t w o s e c o n d order e q u a t i o n s . 172 I 2 4 6 8 10 20 40 60 80 100 Number of element along beam length\"N\" FIG.10(a) Two second order beam equations ; displacement linear and moment linear. FIG.IO(c):Two second order beam equations; displacement linear and moment linear. 1.75 FIG.IO(d):Two second order beam equations; displacement linear and moment linear. _ Relative error O Mid and end O ,L rotations ' 9LT 1 7 7 6 1 2 4 6 Number of elements N FIG.I2! Two second order beam equations, displacement cubic and moment cubic. Relative error in strain energy Rotation at guided end even no.of elements Fixed end rotation .odd no. of elements 2 4 6 8 1 Number of elements FIG. I3 \u2014 ' \u2014> Q CO -1 CD T3 D r T3 O CO CD Q 3 5 CO CD - \u2022 3 \u2014 Q O ro o O CD nv CO CD tre -\u00bb J > Crt CD o CO 3 ce C D O X o CO CD CL -\\ T J O CD JO CD o 3 bo CD \u2014 3 a \u2014t- Q Relative error in strain energy ~~ O i O 1 Re 3 , < la ti ve error in disp \\ c >lacement 5 \u00bb78T 185 FIG.I7(c): Stress convergence. FIG.17 ' Parabolicolly loaded plane stress problem using mixed element; displacements and stresses linear. 186 FIG. 18 : Pa ra bo Ii ca lly loaded plane stress problem using displacement element (cubic displacements ), Ref. C5]. 187 Tota l load P = 4 0 D h = l2 T o t a l l oad P= 4 0 k E = 3 0 0 0 0 ksi v = 0 . 2 5 t = l \" FIG. 19(a) Cantilever beam and load system L P a r a b o l i c a l l y v a r y i n g end shear X U _h_ 2 h_ 2 P a r a b o l i c s h e a r = \u00a3 ( # - \/ ) u ( 0 , 0 ) = v ( 0 , 0 ) = u(0,-\u00a7-) = u(0,-4) = 0 \u2022xy g l v 4 FIG. 19(b) Fixed end boundary conditions B.C.I. x u 2 2 u( 0 , y ) = v ( 0 , y ) = 0 FIG.19(c) Fixed end boundary conditions B.C. 2 . FIGS.I9: Linear elasticity cantilever problem and forced boundary conditions used . 4 6 8 10 Number of elements in beam depth FlG.20(a) FIG.20(b) FIG.20(c) FlG.20= Convergence plots for the cantilever with boundary conditions B.C.I, using mixed finite ele me nts ; stresses and displacements all linear. 189 FIG.2l(a) ! Grids used for the constant stress triangular elements FIG.2l(b) : Grids used for the linear stress tr iangles 191 ~6-MESH C -3 - 42 Q.S.T's N = 4 FIG.21 (c) : Grids used for the quadratic stress triangles 192 ^ 4N - Ele me nts MESH M-4 128 MIXED FINITE ELEMENTS Fl G.22: Ty p i c a I mesh used for mixed finite ele ment ( N = 4 ). 0 200 400 600 800 1000 1200 1400 1600 Degrees of freedom FIG.23 (a): Plot of tip deflection vs. total degrees of freedom. 193 0 I 2 3 4 5 6 7 8 9 10 N = Number of elements in beam depth FIG.23(b) ! Plot of strain energy versus number of elements in beam depth. 285 270 266.875 I 2 3 4 5 6 7 8 9 10 N = Number of elements in beam depth FIG.23(c)-Plot of tip deflection versus number of elements in beam depth. FIGS.23 : Plots for the cantilever with boundary conditions B.C. 2,using mixed finite element; stresses and displacements all linear. 194 FIG.24=Convergence of tip deflection for C.S.T., L.S.T. , Q.S.T.,and mixed finite element with stresses and displacements linear for boundary conditions B.C.2. 195 2L r = 8 unit thickness E. i.o. v . o . i . s-zjjTT) FIG.25(a) : Isotropic case. 2L unit thickness E, = 1.0, E2=3.0,z\/,=0.l,i\/2=0,G|2=0.42 FIG.25(b)' Orthotropic case. FIGS.25 : Model for infinite plate by a square plate with a circular hole at centre, isotropic and orthotropic. 196 FIG.26: Finite element grid for the square plate with a circular hole in the middle. Isotropic and orthotropic cases. 197 Exact solution infinite plate Finite element solution for FIG.27(a)- Isotropic,E= 1.0,v = O.I and G = 2(1+1\/) FIG.27(b):Orthotropic,E,= I.O,E2=3.0, \u2022v,=O.I ,v2=0.0 and G | 2=0.42 FIGS. 27 : Comparison of theoretical and finite element (mixed ) results for infinite plate with a circular hole in the middle. E x a c t s o l u t i on for i n f i n i t e p l a te o F i n i t e e l e m e n t s o l u t i o n FIG.28(a)' Isotropic FIG.28(b) = Orthotropic FIGS.28= Comparison of theoretica and constant stress finite element, Zienkiewicz [42 .results with the same material properties as used in the mixed method,* T q = -I.O. 198 2 L \u2014 t t t t i tv) 2 L L x(u) \" yy= To i i i i i i FlG.29(a) : Square plate with edge cracks under uniaxial tension , v - 0.3 . rIG.29(b) Finite element ~~ dealization of the quarter slate along with boundary :onditions ( N = 4 ). L IGS.29s Plane stress problem considered for investigation of energy convergence in the case of stress s ingu lar i t ies . FIG.30(a): Tyy continuous at the crack tip D and FIG.30(b): r y y discontinuous at the comparison with results from other crack tip D. elements. ( Tong and PianC381 ). FIGS.30 : Plots of strain energy versus the mesh size for the plane stress problem with symmetric edge eracks -Figure 29. 200 O ^\"yy stress continuous at the crack tip D. A T y y stress discontinuous at the crack tip D. FIG. 31 \u2022 Strain energy convergence for the plane stress problem square plate with symmetric edge cracks-Figure 29. 201 -2 - 1 \u20222-1 FIG.32( a)-Tyy continuous at the crack tip D. FIG.32(b): r y y discontinuous at the crack tip D . FIGS.32: Normal stress distribution along the middle of a square plate with symmetric edge cracks-Figure 29(b),(N= 12). 202 FIG.33(a): Symmetric edge cracks Fl G. 33(b): Centra I crack FIGS.33 ; Rectangular plates with cracks used for determining the crack intensity factor Kj. The quarter plate considered for the finite element analysis along with boundary conditions shown as shaded area . FIG.34 : Finite element mesh used for determining the crack intensity factor K I f u s e d for both symmetric edge and central cracks. 204 - 2 - 1 - 2 - 1 FIG. 35(a):Sy mmetric edge cracks. FIG. 35(b ): Centra I Crack. FIGS.35* Normal stress distribution along the edge OA of the rectangular plates with cracks - Figure 33. Boundary conditions' u ( L ,0 ) = v ( L,0 )= u( L,c) = u ( L,-c )= 0 FIG.36 : Boundary tractions and conditions for cantilever linear elasticity solution (Appendix D). ho o Ln 2Q6. APPENDIX A CALCULATION OF THE FINITE ELEMENT MATRIX; LINEAR DISPLACEMENTS AND STRESSES OVER A TRIANGULAR ELEMENT The element matrix equation f o r plane stress l i n e a r e l a s t i c i t y i s derived here f o r l i n e a r displacements and l i n e a r stresses i n a tr i a n g u l a r element using i s o t r o p i c material. Also demonstrated are the modifications necessary to a l t e r the element matrix f o r plane s t r a i n i s o t r o p i c and orthotropic materials. The element geometry i s shown i n Figure 4. Using area coordinates Pl>P2>P3\u00bb [43-]; the l i n e a r approximations f o r u. and T . . can be written i l as u\u00b1 = < D 1 Pz P3> T\u00b1 j =u i l u i 2 ui3 . J. W. 1=1,2. (A.l) i=j=l,2. (A. 2) i j -3 i j Note t e n s o r i a l notation i s implied. Comparing equations (A.l) and (A.2) with (4.1) gives *k = *k = p k ; k=l,2,3. Consider the mixed v a r i a t i o n a l p r i n c i p l e for homogeneous boundary conditions i n equation (5.13) F(A) = [A,A] A - 2JQ fTudft. = jn [2x TTu-x TCx]dfi - 2 \/ f Tudfi where T = < T T T > xx yy xy u = =\" T f = ; x y < T 1 1 T22 T 1 2 > ; T (A. 3) (A. 4) (A. 5) (A.6) 207 T = 9_ 9x 0 f -9_ 9y 9_ 9y 9_ 9x (A. 7) and 1 -v -v 0 1 0 Substituion of (A.l) and (A. 2) into Q y i e l d s \"a 0 \" F(A) = T 2T 0 b b a _ Ci - ~T T (A.8) T -c - vc 0 -vc c 0 0 0 2(l+v)c In (A.9) the submatrices a,b,c,d,e,x and tl are given by a. . = \/ p .p . dfi; i=j=3. e f n p.p. dsl; i=j=3. Jcl ,y J C i j = ~ I k p i p j d f i ' i = J = 3 ' d. = \/. f p.dfi; I 1 n x i e. = \/. f p.dfi; l ' U y l 1=1,2,3. 1=1,2,3. < T 1 T2 T3 T l T2 T3 T l T2 T3 >T xx xx xx yy yy yy xy xy xy T u = T - x x xx yy xy z z w \\ r i r 3 3 . 1 L r 2 2 T 2 xx yy xy - T3 xx yy xy (A.2 4 ) and the matrix E becomes R) 0 E 1 5 x 1 5 a l l 0 b n 0 0 a 2 1 0 b 2 1 0 0 a31 0 b31 0 b u a n 0 0 0 b 2 1 a 2 1 0 0 0 b31 a31 c l l f 11 o a 1 2 0 c12 f l 2 0 a13 0 c13 f l 3 0 c n 0 0 b 1 2 f l 2 c12 0 0 b13 f l 3 c13 0 8 l l b 1 2 a 1 2 0 0 \u00a712 b13 a13 0 0 813 0 0 a 2 2 0 b 2 2 0 0 a32 0 b32 0 0 b22 a 2 2 0 0 0 b32 a32 c 2 2 f 22 0 a2 3 0 c23 f23 0 Symmetric c 2 2 0 0 b23 f 23 c2 3 0 \u00a722 b23 a2 3 0 0 823 0 0 a33 0 b33 0 0 b33 a33 c33 f 33 0 c33 0 833 (A.2 5 ) where [f] = -v[c] and [g] = 2 ( l + v ) [ c ] . The corresponding entries i n the load vector p are also interchanged and the modified load vector becomes 210 {p} = T. (A.26) The equation (A.20) alters to EA = p (A.27) The boundary conditions associated with the plane elasticity problems considered in chapter 7 are of the type u. = 0 on S l u T..n. = 0 on S (A.28) 1 ] J T T . . n . = T? on Sm i j J l T when the body forces f and f are zero. Here S and S are the portions x y u x of the boundary S where the displacements and stresses are zero, respectively; and the portion on which the tractions T_^ are prescribed. The equations (A. 28) are similar to equations (5.18) i f u(? = c?=a=0 and S same as S . Thus 1 1 M T equation (A.19) is similar to the equations (5.40) and (5.41) except that the former is expressed in the matrix form. Therefore the sub-load vectors d and e in the element matrix equation arise from the boundary integral \/ T?(j>.ds wh ere the element boundary coincides with S and at present is identical to i t s generation in the displace-ment method. The procedure for deriving the element matrix E in (A.25), outlined above, is quite general for plane elasticity. The only change that needs to be introduced in switching from isotropic plane stress to isotropic or orthotropic plane strain l i e s in the compliance matrix C of (A.8). For isotropic plane strain, the compliance matrix C is given by 1 - 7 * - 0 l-\\> l-v 2 0 0 T-=- , l-v J (A.29) 211 and for orthotropic plane strain C = (1-nv2) -v1(l+vz) - v i ( l + v 2 ) \u00b1 (l-vf) 0 1 '12 where n=^?'> elastic constants E 1 } v i and G 1 2 are associated with behaviour normal to the plane of strata; and the elastic constants E 2,v 2 and G 2 1 (G 2 1 is independent here) with the plane of strata, as shown in Figure 25(b). 212 APPENDIX B EIGENVALUE DISTRIBUTION FOR A REAL SYMMETRIC MATRIX A quadratic form in n-variables, x i , X 2 , . . . x^, is an expression of the type n n m T Q = I E a..x.x. = x ax (B.l) 1-1 J - l 1 J 1 J \" \" where a is a symmetric matrix of constants. If the coefficients a., and the variables x. are restricted to real values, then Q is real. Let v be I the rank of matrix a. Now there exists a nonsingular transformation x = ty, (B.2) Strang [32], such that the coefficients t can be chosen so that Q reduces to Q = yf+yf+ \u2022 \u2022 \u2022 + y i - y i + r \u2022 \u2022 \u2022 -y2.- ( B - 3 ) Equation (B.3) is called the canonical form of the quadratic form Q. The number of positive terms in (B.3), denoted by I, is called the index of the quadratic form. It is determined uniquely by the matrix a. The types of a quadratic form are determined by the rank r and the index I, as follows: (a) Q i s positive definite i f , and only i f , I=n. (b) Q is positive semidefinite i f , and only i f , I=r 0 a u T (B.4) a -b _ V where - is a symmetric positive definite matrix, and - is a real n A n \u2022 mxn rectangular matrix. Assuming that the rank r of the real symmetric matrix 0 a l -j r i s m+n, then how many positive and negative real eigenvalues does a D this matrix possess? This necessitates determination of index I of the quadratic form (B.4). Consider the negative of the quadratic form in (B.4) T T Ql = -Q = p T - i r .. b - a V \u2022 \u2014 - a O 1 u (B.5) The (m+n)x(m+n) square matrix of equation (B.5) can be written as b l l b12 \u2022 \u2022 \u2022 b i n \" a l l - a 2 i . . \u2022 _ aml b21 b 2 2 . . . b 2 n - a 1 2 - a 2 2 . . \u2022 _am2 \u2022 \u2022 \u2022 * \u2022 \u2022 \u2022 \u2022 \u2022 bnl bn2 \u2022 . . b nn ~a l n ~ a2n \u2022 \u2022 . -a mn -an \" a12 \u2022 \u2022 # - a l n - a 2 i - a 2 2 . \u2022 -- a2n 0 . \u2022 \u2022 \u2022 \u2022 MxN _ aml - am2 \u2022 . .-a mn (B.6) The principal minors for the matrix (B.6) can be written as 214 Mi ; M 2 = b l l bi 2 ; M 3 = b l l b 1 2 B 1 3 B 2 1 b 2 2 B 2 1 B 2 2 B 2 3 B 3 1 B 3 2 B 3 3 etc. Since the submatrix b is positive definite, therefore a l l principal minors up to M r are positive. The M N + 1 principal minor as determinant of the matrix \u00bb11 '12 \u00bb21 D 2 2 3 n l Dn2 [_-an - a 1 2 b l n ~ a l l b 2 n _ a 1 2 nn 1 1 1 . -a ln (B.7) which has a zero on the diagonal, cannot be positive. This holds because B is not positive definite. Therefore M ,-, is either negative or zero. -n+1 r n+1 The latter cannot be true since i f m=l, i.e. only one degree of freedom in u; then the quadratic form is positive semidefinite which is not true. Hence M , 1 is negative. As for the principal minors M ,\u201e to M , , these n+1 \u00b0 r n+2 n+m can be either positive, zero or negative. However for rank r=m+n no two consecutive M^ 's can be zero; i f M^ and a r e z e r o then the rank of the matrix (B.6) is k or less. Further, any zero in an i n d i c i a l sequence lies between adjacent terms with opposite signs. Therefore the i n d i c i a l sequence can be arranged as (l,Mi,M2 . . . ,M ,M , .. , . . .,M , ). The index I of the quadratic form equals the n n+1 n+m number of permanences of sign in any i n d i c i a l sequence where any zero entry is given an arbitrary sign. Therefore the index I for Q, in (B.5) is n since the number of permanences in the i n d i c i a l sequence above is n; i.e. the signs of principal minors Mj to M^ are positive. Now the quadratic form Q, i n (B.5) through nonsingular transformation of the type (B.2) can 215 be expressed in the form of (B.3) as Ql = -Q = v2+v2+ . . . +v2-u2-u?- . . . -u 2; (B.8) 1 ^ n J- ^ m or Q = u2+u2+ . . . +u 2-v 2-v 2- . . . -v 2. (B.9) Hence i t can be deduced from (B.9) that the matrix (4.19) has m positive and n negative real eigenvalues. Next, consider the fi n i t e element matrix equation (6.36) for the linear part of the Navier-Stokes equations. This is expressed in a slightly different form as follows: 0 a \u00a3 u T a 0 0 P = 0 (B.10) \/ 0 \"I. T where y i s a symmetric, positive definite (nxn) matrix. A rearrangement of (B.10) yields 0 3 a u SA = 3 T \"Y 0 T = 0 T a 0 0 P-where T T T A = (m+n+J)x(m+n+l) 0 3 a T 3 Y 0 T a 0 0 (B.ll) (B.12) (B.13) and a and 3 are (mxj) and (mxn) rectangular matrices, respectively. Let the rank r of the matrix S be (m+n+l). The matrix S is symmetric and indefinite, therefore a l l i t s eigenvalues are real and i t remains to be determined as to how many of these are positive or negative. The quadratic form of the mixed variational principle associated with (B.ll) is 216 Q = 2U TBT - T T Y T + 2u Tap. ( B . 1 4 ) The f i r s t two terms on the right hand side are i d e n t i c a l to the quadratic form i n ( B . 4 ) . Further, i t i s possible to diagonalize the submatrix 0 6] B T - Y through a transformation of the type (B.2) as was done for (B..6) Let such a nonsingular transformation be given by Qn Q12 .9.2 I Q22 T L \u2014 J (B.15) and Qn Q12 2 Q21 Q22 0 0 0 1 (B.16) to be complete, where I i s an ( i x l ) i d e n t i t y matrix. The s u b s t i t u t i o n of th i s transformation i n the quadratic form Q i n (B.14) (af t e r some algebraic manipulations) y i e l d s ; T0 0 -X n T T 2 Ql1 \u00ab Ql2 0 u f p (B.17) where X and X are (mxm) and (nxn) diagonal matrices with p o s i t i v e e n t r i e s , r e s p e c t i v e l y . The f u n c t i o n a l form i n (B.14) for transformed variables then leads to the following matrix equation: SA . m 0 Q n a -A Q 1 2a T T a Qi1 a Q 1 2 0 = 0. (B.18) 217 The matrix S of the coefficients in (B.18) and S of the coefficients in (B. l l ) , both have the same number of positive and negative eigenvalues. This is because of the congruence transformation and the Law of Inertia. Therefore i t is only required to find the positive and negative eigenvalues for S. T T Let Qua=6 and Qjia=n, then the matrix S can be written as A? 0 0 A \u2122 0 symmetric 0 0 0 0 (m*n) 1 L _ -X1 0 0 -An 0 -A-5 1 1 6 12 ( S21 ( 522 5ml 6m2 oTnnni2 0 jH21H22 0 n n l r l n 2 I I '11 >21 ml \" I I 121 n l 0 ( i x l ) (B.19) The principal minors for the matrix (B.19) can be written as M i =UiI; M 2 = AT 0 , m A? 0 0 0 Af 0 etc. ,0 0 A m Since A\u2122 and A^ are a l l positive real numbers, hence a l l the principle minors up to Mm are positive. The (m+l) t h principle minor is simply 218 -X1^!^. Therefore i t is negative and the sign alternates thereafter up to (m+n)1\"*1. Beyond (m+n), the principal minors can be either positive, zero, or negative and for rank r=mfn+1, the zero entry would only appear between a positive and a negative entry in the i n d i c i a l sequence. Clearly the index I is m and m m+1 n+m n+m+1 m+n+1 hence only m eigenvalues of the matrix S are positive while (n+l) eigen-values are negative, which also holds for matrix S in (B.13). 219 APPENDIX C EQUIVALENCE OF ENERGY PRODUCTS FOR FOUR FIRST ORDER BEAM EQUATIONS AND PLANE LINEAR ELASTICITY EQUATIONS The energy products for the four f i r s t order beam equations (7.18) and the plane stress linear elasticity equations (5.19) after introducing the simple beam theory assumptions are shown to be the same when the shear energy contribution is dropped. The four f i r s t order beam equations 0 0 0 0 0 D _D -1 1_ ' E I 0 where D^J-; lead to the following energy product; ) V q e 0 ) M 0 V i_ J 0 ( C l ) (AlA,A) = ! [ _ v ^ - e | i - 2 v e + l ^ - ^ f ^ ] d x (C.2) - - - . J1 dx dx dx EI dx From equation (5.11), the energy product for plane stress linear elasticity (unit thickness) is (A2A,A) = \/ [u TT*T+T TTu-T TCx ] d n (C.3) ,d6 M2, \u201edv, for A = and where X* = -T = _9 A = 0 T* T -C - ~ 0 8_ \"3y '3y 3_ '3x -v 0 -v 0 1 0 0 2(l+ v) 220 T T and u=; 1~' When t n e basic assumptions of plane sections remain plane after bending and small deflections, following the nomenclature of Figure 7, the following quantities are obtained: 0 (C.4a) yy x = _ Itt XX I V ,h2 2, T x y = 2 i ( r - y2> (C.4b) (C.4c) and u = -y9. (C.4d) He re I is the moment of inertia about the z\u2014axis; h, the height of the beam, and M,V,0 and v are functions of x only. Since x =0: the matrix yy operators T* and T and the matrix C reduce to 3_ \"9x 9_ \"9y 3_ 3x (C4e) 5-1 1 0 0 2(l+v) (C.4f) Now the substitution of equations (C.4) into (C.3) yields the following. The f i r s t term on the right hand side of (C.3) becomes Ja u T*xdft = ji <-y6 v> 3_ _3_ 3x ~3y 0 3x My_ \" i V ,h2 2 . dydx f f2 \/ Z 2 \u00ab d M y.2flw v d V ^ h 2 2SXA A j l hi I dx I 6 V-2I d^ (4- y ) } d y d x -2 h r2 Since for unit thickness, y2dy=I; therefore integration on y yields; \"2 221 r uTT*T I 2I V4 1 . 0 0 2(l+v) f My_ ~I dydx V ,h2 2 I M f \/ l \/I ^ + g 2 ( | - 2 - y 2 ) 2 > d y d x . r2 ry2., 9.h 2 ,h2 h. 2 This, after integration on y, yields a Adding the equations (C.5), (C.6) and (C.7) gives, ( C 6 ) (C.7) \/. [u TT*x+T TTu-T TCT]dfi = f {-vf-0^-2V0+M^-^fV^+^4 2V 2]dx (C.S) tt - - - - \u2014 - \u2014 ' l dx dx dx EI dx 5EI where j1 5 E I h 2V 2dx is the contribution to the energy product from the shear stress due to flexure. This is only significant for short beams and when i t is dropped, the energy product in (C.3) becomes 222 - \/ 2 [ - v i - ^ f - ^ - E - I ^ ^ - ' \u00ab=\u2022' This is exactly the same as in equation (C.2). 223 APPENDIX D ELASTICITY SOLUTIONS FOR A CANTILEVER An elas t i c i t y solution for the cantilever with the boundary tractions and the boundary conditions shown in Figure 36 is derived. It is also shown that the strain energy computed from the normal stress T and the shear stress x distributions is actually equal to half the xx xy J n total work done by the boundary tractions in moving through the displace-ments obtained from the elasticity solution on the boundaries. The stress distributions are taken as x ( D . l a ) XX I x = 0 (D.lb) yy Txy \" - h ( c 2 ~ ? 2 ) ( D - 1 C ) where P is the total load due to the shear stress at the ends. The equations (D.l) identically satisfy the equilibrium Vl+(j)=0 for T x x = g ^ 2 > x = % and x =--\u2014j\u2014 everywhere inside the cantilever, as well as yield yy 3x z xy . 3x3y J ' 3 the same stresses on the boundary as applied tractions. The corresponding strains, by applying Hooke's law, are 3u Txx Pxy , \u201e . \u00a3 = T:\u2014 = ~Tr~ = - -^r~ (D.2a) xx 3x E EI 3v xx vPxy . N eyy = ^ = ~Y- = -ET (D'2B) Y = Iii + = I S = _ i ( c2 2) ( D > 2 c ) Yxy 3y 3x G 2IG ^ y ; ^.u.zcj E where ^ =2(l+v)' ^ e s t r a ^ - n s derived in (D.2) also satisfy identically the 224 82e 32e 8 2 Y compatibility equation ( 9 y 2 + ^J? = dxd*J) \u2022 By integration of (D.2a) and (D.2b), u and v are obtained as u = - f f ^ + f i ( y ) ; v = ^ f f ^ + f 2 ( x ) (D.3) in which the functions f\\ and f 2 are as yet unknown functions of y and x only, respectively. Substitution of u and v above into equation (D.2c) yields _ P x j dfl(y) + v P Z j + d f 2 ( x ) =__J_ 2 , 2EI dy 2EI dx 2IG ^ Y J K ' In equation (D.4) some terms are functions of x only, some are functions of y only, and one is independent of both x and y. These can be grouped as w x ) = _ P x j + df 2(x) * W 2EI + dx g ( y ) = y l l l + dfi(y) _ Pzi 2EI dy 2IG Pc 2 K = - (constant). Thus (D.3) becomes F(x) + G(y) = K. (D.5) It can be concluded from (D.5) that F(x) and G(y) must be constants. Denoting these by d and e, respectively, therefore d + e = k a n d df 2(x) = Pxj dfi(y) = _ vPyj. P y j . + a n d dx 2EI + d ' dy 2EI + 2IG 6\" Integrating these yields the functions fi(y) and f 2 ( x ) ; Px^ f 2(x) = + dx + h. 225 Substitution in the expression for u and v in (D.3) gives Px 2y vPy3 _,_ Py 3 , . The four constants d,e,g and h can now be determined from the four boundary conditions (Figure 36) u(L,0) = v(L,0) = u(L,c) = u(L,-c) = 0 (D.8) and these are found to be d - - m [ 3 + ( 4 + 5 v ) i7 ] g = 0 and h = ||1 [1 + f(4 +5v) . Finally the substitution for d,e,g and h in the equations (D.6) and (D.7) and letting E,=j- and r\\=2- gives J_i c u(5,n) = H^[^n(l-C2)-(^)^r1(l-n2)] (D.9) P T 3 T 1 ^ 1 r 2 v(c,n) = H I[i-fc:4ic- 34i^ 2-{3vc:n 2+(4+5v)(i-c:)}]. (D.IO) Therefore the tip deflection 6 at C (Figure 36) is given by PT 3 1 r - 2 6 = v(0,0) = [14j(4+5v)^2-] ( D . l l ) while the longitudinal deflection u at B, where 5=0 and n=-l, is obtained as UB \" \"V'-V = - l i T - The strain energy in the cantilever can be computed by using the assumed stresses in (D.l) and the corresponding strains in (D.2) in the usual manner. 226 U = Tr \/k f C [T e +T e +T Y ldydx 2 1 o -c xx xx yy yy xy xy After i n t e g r a t i o n , the s t r a i n energy i n the can t i l e v e r i s p2j 3 i o \u201e2 U = -ggY\" U+;p(l+v)^r]. (D.13) The work done on the boundary can be computed from the following l i n e i n t e g r a l W = f (x..n.u.)ds (D.14) where i s the portion of the boundary where the stresses are prescribed and n^ . , the unit outward normal. For the c a n t i l e v e r i n Figure 36, this i n t e g r a l takes the form W = C T x \/ \u00b0 > y > v ( \u00b0 > y ) d y + \/ ! C T x y ( L , y ) v ( L , y ) d y + jC_c T x x ( L , y ) u ( L , y ) d y = Wx + W2 + W3. (D.15) Here T x y ( \u00b0 ' y ) = \" f l ( c 2~y 2> v(0,y) [ I 4 | ( 4 + 5 v ) ^ ] . Thus the f i r s t i n t e g r a l i n the r i g h t hand side of (D.15) i s given by 2 3 2 wi = C - 3EI5\" ( c 2 _ y 2 ) ti+j<4+5 )^i2-]dy and a f t e r i n t e g r a t i o n W i = l i i ~ t i 4 i ( 4 + 5 v ) i > ] - (D-16) Next, i x ^ ( L , y ) i s the same as x (0,y) and v(L,y) i s given by , T N _ v PL 2 v(L,y) - \"J g l y ' Then the second i n t e g r a l W2 i s 227 W2 = j _ c ~ 4 ^ f l y z ( c ^ - y 2 ) d y and leads to F i n a l l y to obtain W3 p 2 P 2 j W 2 = - i o \u2014 \u2022 and u(L,y) = | * ( y 2 - c 2 ) Therefore which gives 3 J-c \u2014 6 \u2014 ET2\" y ( y c ) d y _ (2+v) P 2 c 2 L W3 - 1 5 E I . (D.18) Adding (D.16), (D.17) and (D.18) y i e l d s P T 3 i i r2 W = Wj + W2 + W3 = [14y^(l+v)^2-]. (D.19) Comparison of (D.19) with (D.13) gives W = 2U. (D.20) Therefore the s t r a i n energy computed from the stress d i s t r i b u t i o n s i n (D.l) i s exactly h a l f the work done by the boundary tractions i n going through boundary displacements as obtained from the e l a s t i c i t y solutions (D.9) and (D.10). ","attrs":{"lang":"en","ns":"http:\/\/www.w3.org\/2009\/08\/skos-reference\/skos.html#note","classmap":"oc:AnnotationContainer"},"iri":"http:\/\/www.w3.org\/2009\/08\/skos-reference\/skos.html#note","explain":"Simple Knowledge Organisation System; Notes are used to provide information relating to SKOS concepts. 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