Mixed finite element models for plate bending analysis theory

comjmters & Smcrwes Printed in the U.S.A. Vol. 19, No. 3, pp. 431-445, 0 0 4 5 -7 9 4 9 /u 1984 Pergamon zyxwvutsrqponmlkjihgfedcbaZYXWVUTSRQPONMLKJIHGFEDCBA s3 .0 0 + .I 0 Press Ltd. MIXED FINITE ELEMENT MODELS FOR PLATE BENDING ANALYSIS THEORY D. KARAMANLIDIS~ and S. N. ATLURI$ Center for the Advancement of Computational Mechanics, School of Civil Engineering, Georgia Institute of Technology, Atlanta, GA 30332, U.S.A. zyxwvutsrqponmlkjihgfedcbaZYXW Abstract-The theoretical background of mixed finite element models, in general for nonlinear problems, is briefly reexamined. In the first part of the paper, several alternative “mixed” formulations for 3-D continua undergoing large elastic deformations under the action of time dependent external loading are outlined and are examined incisively. It is concluded that mixed finite element formulations, wherein the interpolants for the stress field satisfy only a part of the domain equilibrium equations, are not only consistent from a theoretical standpoint but are also preferable from an implementation point of view. In the second part of the paper, alternative variational bases for the development of thin-plate elements are presented and discussed in detail. In light of this discussion, it is concluded that the “bad press” generated in the past concerning the practical relevance of the so-called assumed stress hybrid finite element model is not justified. Moreover, the advantages of this type of elements as compared with the “assumed displacement” or alternative mixed elements are outlined. INTRODUCTION forming thin plate elements have never been used widely in practice. This is because higher order trial functions must be used (e.g. complete 5th order pation of many researchers during the past two polynomial in the case of a triangular element) in decades. Recalling that the finite element method such elements. As the elements generalized degrees(FEM) was born in the mid-fifties as a natural of-freedom, displacements w, rotations &/ax, dw/dy extension of the two classical matrix methods of and curvatures a 2 w /a x 2 , a 2 w /M y, a 2 w /a y2 at the structural analysis, namely the displacement method vertices as well as well as displacement derivatives and the force method, it is not surprising to note that awlan at the midside nodes may be defined. Intromost of the plate bending elements proposed so far duction of second-order derivatives as nodal paramare based on these two approaches. Comprehensive eters enforces C2 continuity (“Over-compatibility”) at reviews on plate bending elements were presented by the element vertices. This is not only superfluous Gallagher [ 181, Brebbia and Connor [ IO], Holand [24], from a mathematical point of view but also makes the Miiller and Mtiller[36], and Schwarz[46] to name just application of elements of this type in analyses of a few. engineering structures (wherein, almost exclusively, The development of assumed displacement plate due to discontinuities in geometry, material properbending elements based upon the Kirchhoff theory of ties and loading, the curvatures may be disconthin plates requires that the trial functions fulfill ri zyxwvutsrqponmlkjihgfedcbaZYXWVUTSRQPONM tinuous) impossible. In a recent publication by Carapriori the requirement that at the interelement boundmanlian et al. [ 121,a triangular plate bending element aries not only the lateral displacement w but also its has been suggested using displacements and their first first derivative &~/an (n is the outward normal to the derivatives as elemental degrees of freedom. Clearly, boundary) be continuous (so-called C’ continuity). this element does not enforce C2 continuity,4 however Additionally, the basic requirement of exact repreapplication of conforming elements in practical strucsentation of rigid body modes as well as constant tural analyses still remains neither an easy nor an strain modes in each element should be met in order inexpensive task. On the basis of these experiences, it to ensure convergence to the exact solution as the was natural to ponder whether or not reliable, costelement size decreases. effective elements could be developed by relaxing the Plate bending elements fulfilling these two requirerestrictions imposed by the strict C’ continuity rements (so-called fully conforming or compatible elequirement of Kirchhoff theory. The flrst attempt ments) have been suggested by Withum[54], towards this goal, the so-called non-conforming eleFelippa[l6], Bosshard[9], Bell[7], and others during ment concept introduced in the mid sixties, appeared the sixties. In spite of their inclusion in the element to be unsuccessful. In this scheme, the C’ continuity libraries of some general purpose finite element codes is maintained only at distinct points (nodes) rather like ASAS, ASKA, MARC and PRAKSI, conthan along the entire interelement boundary. Although, in some cases, elements of this kind happened to give satisfactory results, from a mathetPost-Doctoral Fellow. matical point of view they are not acceptable (see, fRegents’ Professor of Mechanics. e.g. [13,49]) since they do not always guarantee con§Peano[40], too, has succeeded in establishing CL elements without enforcing supertmous C* continuity at vergence of the approximate solution to the exact one vertices. as the element’s size decreases. The development of simple and effective finite elements for thin-plate bending has been the preoccu- 431 D. KARAM ANLIDIS and S. N. ATLIJRI 432 zyxwvutsrqponmlkjihgfedcbaZYXWVUTSRQPONMLKJIHGFEDCBA The difficulty of imposing C’ continuity has resulted in many alternative approaches to the problem. Two such basic alternatives are as follows. Firstly, the C’ continuity requirement can be relaxed ci priori, but enforced li posteriori, by means of Lagrangian multipliers, as a natural constraint (Euler/Lagrange equation) of an extended variational principle. The same goal can be also achieved by utilizing the so-called discrete Kirchhoff or a more general penalty function theory. The basic idea behind these finite element concepts can be understood as an attempt to “feedback”, rather than to neglect (as it is the case within the nonconforming concept), the strain energy produced on the interelement boundary, due to normal slope discontinuities, to the total energy balance of the structure under consideration. Elements of this kind, the so-called hybrid displacement elements, have been suggested in the past by Yamamoto[56], Tong[SO], Harvey and Kelsey[21], and Kikuchi and Ando[33] among others. They reportedly give good results[21,33,34] but may in some cases become numerically unstable (as shown by Mang and Gallagher[35]). On the basis of discrete Kirchoff or penalty function theory, plate bending finite elements have been developed in the past by Wempner et al.[53], Dhatt[lS], Fried[l7], and others. The necessity of ful6lling the C’ continuity stems from the appearance of second-order (displacement) derivatives in the energy functional which, in turn, is due to the utilization of Kirchhoff s thin plate theory within the element’s formulation. Therefore, a second alternative to overcome the C’ continuity problem consists in either treating the plate bending problem as a special case of the 3-D elasticity problem or in employing plate theories such as Reissner’s or Mindlin’s wherein the relation between displacement derivatives and rotations is no longer imposed. Among the tirst papers wherein plate bending elements utilizing Reissner’s theory have been proposed are those due to Smith[48] and Pryor et al.[44]. Utilization of 3-D elasticity theory towards the development of C continuous elements for thick- and thin-walled plates has been reported in the pioneering paper by Ahmad et al. [l]. The major difficulties that one may expect in such a straightforward use of the 3-D theory for plate bending have been summarized in[l] as follows: (i) The fact that a thin plate or shell, in general, undergoes negligible “thickness stretch” leads to large stiffness coefficients for the corresponding displacements, which, in turn, produces an illconditioned coefficient matrix in the final matrix equation. (ii) Due to the fact that no account is taken dpriori of the plausible kinematic constraints (i.e. straight lines remain straight after the deformation, etc.) of a plate or shell, an unnecessarily large number of DOF has to be introduced to model the structure under investigation. In order to overcome these difficulties, the following remedies were proposed in [ 11:(i) the strain energy corresponding to stresses u,, (transverse normal stress) was ignored, (ii) upon enforcing the constraint that an edge of the plate or shell remain straight after deformation its six DOF can be replaced by the standard five engineering DOF (three translations and two rotations) associated with a point on the middle surface. The early attempts in utilizing this concept were promising; however, it was soon recognized that elements of this kind are useless as far as the analysis of thin structures is concerned, since they become overly stiff as their thickness-to-length ratio approaches zero (the so-called “locking phenomenon”). In Refs.[39, 571 the so-called “uniform” and “selective” reduced integration techniques have been proposed and isoparametric elements have been developed which do not lock. Even now, the use of these heuristical techniques is a rather controversial matter (see Refs. [25,26,36, 371 for elaborate discussions on the subject). The most serious drawback of this type of elements lies in the fact that they eventually possess so-called spurious kinematic modes which may or may not disappear upon assembling the individual elements to form the given structure. Thus, the quest for a simple plate bending element still goes on and is justified. A common feature of all the element types discussed so far is the use of displacements as the primal (independent) variables of the approximation process, with stresses, on the other hand, being treated as dependent variables. The process of differentiation, required in order to calculate stresses in terms of displacements, leads to an inevitable loss in accuracy-an element feature which is most undesirable from an engineering point of view (e.g. application for design purposes). Mixed methods utilizing simultaneous use of displacements and stresses (or resultants and/or couples) as independent variables enable the calculation of both the quantities with the same order of accuracy. Mixed models for plate bending problems have been suggested independently, almost simultaneously, by Giencke[ 191, Herrmann[23], and Hellan[22] in 1967. While Giencke’s method uses the principle of virtual displacements and the principle of virtual forces simultaneously, the mixed-type element due to Herrmann utilizes the Reissner energy functional. On the other hand, the idea behind Hellan’s method consists in introducing independent approximations (trial functions) for the lateral displacement w and the stress resultants M ,,, M yy, and M xy into the corresponding plate bending differential equations. More systematic approaches based on rigorous variational arguments were given first by Prager[43] and a short time later by Bufler[l 11. In particular, in those papers, extended versions of the classical variational theorems (including the one due to Reissner) were developed, such that discontinuities along the interelement boundaries in tractions as well as in displacements can be accounted for consistently. A comprehensive discussion of the fundamental ideas behind mixed finite element models for plates was given by Connor[l4]. The so-called assumed stress hybrid method can be regarded as a special case of a mixed method. Within this approach, the stress field approximation is chosen so as to satisfy the equilibrium equations d priori. The variational theorem for such a model was given by Prager[43] and Bufler[ll]. However, a plane stress finite element based on essentially the same idea was discovered heuristically by Pian in 1964[41]. Triangular plate bending hybrid stress elements were first suggested by Sevem and Taylor[47]. Since the late sixties, numer- Mixed finite element models for plate bending analysis ous mixed elements for plate bending problems have been suggested. According to the relevant literature, elements of this kind have been applied successfully in a variety of engineering problems covering linear as well as nonlinear, static as well as dynamic analyses. In spite of their demonstrated potential, elements of this type have never received their due attention in practical engineering applications. As a matter of fact, only very few of the general purpose finite element codes available in today’s commercial software environment contain mixed-type elements in lIdAu, 433 partial derivatives with respect to Cartesian coordinates xi in Q(w and time t, respectively. We use Cartesian coordinates throughout. By oii we denote the Cauchy-Euler (true) stress tensor; A& denotes the tensor of Truesdell incremental stress; ui the displacement vector; Fi and Ti the prescribed body force and surface traction, respectively; while E. and M represent point loads and masses, respectively. In updated Lagrangian formulation (see Ref.[31]) and under consideration of large deflection as well as dynamic effects, the principle due to Reissner for the Q(“‘+‘) state is expressed by pAii,Au,dV+ 1 MAiiiAui [- U~~AS~)+U~~AU~,~AU~~+AS~A~~,~]~V AS,) = c k . . . . . . . . . . . . . . . . . . . . . . . .. . . . . m zyxwvutsrqponmlkjihgfedcbaZYXWVUTSRQPONMLKJIHGFEDCBA o . . . . . ‘o. . . . . . . . . . . . . . . . . . . . a AjiAuidV 0 -c m -zLm AzAu,dS m i Lvm -c A&Aui k . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 0 +cIsC$uij+(piii-~~.Au,ldv-S*_tdu.dS}+~(~~i-~i)Aui m VNBI ................................... .............................. - /.Q - ; (uij + uj,i - u~,~u~~) d V ...................................... = stationary. 8 their libraries. It seems therefore worthwhile to inIn the above, p’ijis the Almansi strain tensor, and ui vestigate in this paper whether or not mixed finite are displacements from Q(O)to a(“?. The first term in elements do represent a reliable, cost-effective basis to eqn (1) represents the incremental complementary carry out numerical analysis of plate-like engineering strain energy density which is related to the third term structures. For this purpose, the mathematical forby means of the so-called canonical, contact or mulations of several mixed element variants will be Legendre-Euler transformation reexamined and discussed in detail. The authors’ primary goal is to provide sufficient information so AS,d, = VA + Ud, (2) that it may be possible to reassess the accuracies of where U,, = U,(AuiJ is the incremental strain energy the “bad press” received by the mixed elements (see, density. The second term is due to geometrical none.g. [20,45]). zyxwvutsrqponmlkjihgfedcbaZYXWVUTSRQPONMLKJIHGFEDCBA linearity leading to the so-called “initial stress” or VARIATIONAL BASES FOR MIXED “geometric stiffness” matrix. The fourth term is due FINITE ELEMENT MODELS to the inclusion of inertia effects, whilst the fifth term Mixed finite element models are, in general, estabrepresents the incremental work by the external aplished on the basis of the so-called Reissner variplied forces. The final two terms are “correction” ational princiRle. This is a two-field principle with terms due to the fact that in the state Q(“? neither both the displacements and stresses as independent total equilibrium nor total compatibility is exactly variables. Although this paper deals primarily with satisfied. In variational eqn (1) the only essential boundary plate bending problems, for reasons which would become apparent, we start from a functional formuconditions to be met by the primal variables Au, and lation for a general 3-D solid. We assume that the AS, are: continuum under consideration is divided into (m) (a) the displacement boundary conditions elements and denote by V,, the volume, by pN the density, by aV,, the entire surface, by S,, and S,, Aui-At&=0 on S,, the parts of the surface on which tractions and (3) displacements are respectively imposed, and by S,, the interelement boundary of the mth element in the and (b) the interelement displacement continuity conreference configuration 8”. Furthermore, A(. . _) ditions and (T) denote respectively the incremental and Aui+ - Aui- = 0 on S,,.,,,, prescribed quantities; (. . .),i and (. : .) symbolize the (4) D. KARAMANLIDIS and S. N. ATLIJRI 434 where the symbols (. . .)+ and (. . .)- refer to the positive and negative part of S,,, respectively. Taking the variation of the functional n with respect to Au, and AS,, we obtain the following Euler-Lagrange equations of the state Q@‘+‘): variational equation + AP=(R, + G%-‘RJ Equilibrium I + Ax%IAC-~Au,: ogJ + AS, + (akjAui,k),j+pi + Api = p(iii + AZ&) in I’,.,,,,. = stationary. (5) Strain-displacement compatibility 6ASq: /JO+ 2 - k (uu + uj,i- Uk,iUk,j) ‘I - ; (Au,~ + AuJ = 0 in VNm. (6) Interelement traction reciprocity Upon introducing Au=NAB (12) only displacement quantities occur. In eqn (11) the matrix K = GrI-I-‘G refers to the elemental “smalldeflection” stiffness matrix. Finally, taking the variation of the functional lIN with respect to Aii leads to the expression 6rZ, = c 6Ai={(K + K”)Ai + R, + G%-‘R,} m +6Ax%lAC-6Ax=Aij=O 6Au; (oijvj + AS,vj + CT~~AU~,~V~)+ + (a,vj + AS,vj + ukjAuj,,vj)- = 0 on (7) S,,. and upon assembling differential equations AS=P/?’ in V,, (13) to the system of linearized MAi+KAx=Aij+r. in eqn (1) the trial functions and Ax=Aq (8a,b) where Au = LAu,, Au,, Au,J*; N is the matrix of normalized trial functions (“shape functions”); A i is the elemental nodal displacement vector; AS = LAS,,, AS,,, AS,,, A&, AS,,, A2QT; P is the matrix of stress trial functions; and fl denotes the generalized stress coefficients; we obtain the discretized functional (14) A different type of mixed model can be developed upon using, instead of eqn (8b), interelementcontinuous stress field approximation AS=LAS VN,,, in (8~) where L denotes the matrix of stress shape functions and A s the elemental nodal stress vector. In this case, we obtain from eqn (1) the discretized variational equation nddS,di)=C -~dS%AS+~Aig”At m{ +Af%=GAii+Ati%,+A@& I + Ax%lAi-- AxTAd = stationary. + Ax?MA%-(9) Ax=Aij = stationary (15) In eqn (9) H refers to the (elemental) flexibility or upon assembling matrix, K” to the geometric stiffness matrix, G to the stress-displacement matrix, R< and Rc to the residual ZYI~Az,Ax)~ -~AzPAz+~Ax~“Ax force vector due equilibrium imbalance and incompatibility respectively, M to the (overall) mass matrix, +AzT~Ax+AxTre+AzTrc Ax to the displacement vector, and Aij to the external load vector. -I- Ax%lACAx=Ai-j Choosing the p’s to be independent for each individual element (i.e. a discontinuous stress field = stationary (16) approximation) leads to the statement zyxwvutsrqponmlkjihgfedcbaZYXWVUTSRQPONMLKJIHGFEDCB where AZ denotes the global stress vector. Taking the Tan, variation of llN with respect to AZ and Ax, we get or 6nN=6B aS @=(-H/l +GAi+R,)=O. (104 an, E - GAz=FAz + 6AxrK”Ax + 6AzTrAx UOb) + 6AxTrTAz + 6Ax=r, + GAz=r, +6Ax%IAi-6Ax=Aij=O (17) Solving for /I (upon assuming det H # 0) gives and finally the mixed system of differential equations /I = zyxwvutsrqponmlkjihgfedcbaZYXWVUTSRQPONMLKJIHGFEDCBA H-‘GA fi + H-‘R,. (11) MA%+K”Ax+rTAz=Aq-rr, Upon introduction of eqn (11) in eqn (9) the J’s can be eliminated on the element level, so that in the final TAx+FAz=r,. (W Wb) 435 Mixed finite element models for plate bending analysis In the past, mixed finite element models of both the types presented above have been developed and applied in structural analyses of linear and nonlinear, static and dynamic problems. The question is, of (where a, is an arbitrary vector) is applied to the course, which of these two presented alternative second and third term of eqn (1) to yield approaches should be preferred over the other? Attention should, therefore, be directed towards some specific aspects of the two alternative mixed finite element approaches, eqns @a, b) and (8a, c), respectively. Firstly, for a static or quasistatic problem, eqn (14) and (18a, b) degenerate to an algebraic system of equations with positive definite and indefinite coefficient matrices, respectively. The computational advantage of the former over the latter does not lie, however, in the positive definiteness of the coefficient matrix, as pointed out in[l7] but rather in the fact that the overall number of unknowns as well as the matrix-bandwidth is smaller, provided We obtain thereafter the modified mixed variational that the same element mesh and interpolants of the equation same order have been used. Secondly, as pointed out fldjdu, AS& = c - u,,(AS,/) by Harbord et al.[20] and Wunderlich[55], the solum U VN#l# II tion of eqns (18a, b) immediately yields also the mechanical variables (stresses) which are of basic dV - CJ@;Auk,,idukj zyxwvutsrqponmlkjihgfedc interest in the analysis and design of a structure; while for achieving the same purpose in the case of eqn (14), an element-by-element post-processing has to be (CkjAt+ + AS$V~AU~dS av,, carried out on the basis of eqn (11). It is doubtful; however, that such a direct solution of stresses from pA&Au,dV +&kfAiiiAu, zyxwvutsrqpon eqns (18a, b) can be considered to be a significant k advantage. Moreover, in the authors’ opinion, there is no mathematical basis for claiming that the aforeAThi dS - 1 dFtAui mentioned post-processing introduces a loss in accuk racy within the stress calculation[20,45]. Thirdly, it is + (correction terms) = stationary. seen from variational eqn (1) that when interelement(20a) continuous stress interpolants, eqn (SC), are introduced, in the assembled structure under investigation, To the best of the authors’ knowledge, a variational at each nodal point continuity in stresses will be principle of this kind has been formulated and impleimposed as well. This poses problems when concenmented in finite element large deflection static analtrated edge/nodal loads are present. Therefore, from yses for the first time in Ref.[2]. Comprehensive this point of view, the use of mixed element models reviews on various alternative formulations taking based on eqns (8, c) is limited mostly to academic into account material nonlinearities, too, were given situations. Evidently, this criticism applies to models in Refs. [3-51. On the basis of the discussion presented based on eqn (Sa, c) but not to those based on eqns there, it becomes obvious that implementation of eqn @a, b). In their critical commentary on the practical (20a) in finite element analyses of general-form strucrelevance of mixed finite element models, Harbord et zyxwvutsrqponmlkjihgfedcbaZYXWVUTSRQPONM tures (like shells or 3-D continua) is a rather complial. [ 181appear to have overlooked this very important cated task. This is simply because of the difficulty in fact. finding trial functions which satisfy dpriori eqn (19a). Modzjied mixed models. In variational eqn (l), the The situation changes dramatically when one introstress-field is not subject ri priori to any constraints duces only the linear part of (19a), i.e. wirhin the element. The aim of this section is to develop alternative formations wherein the stress AS,=0 in VN,,, (W field is subject to ri priori constraint conditions within an element. In such a case, the functional in eqn (1) as an ri priori constraint in eqn (1). Application of will be appropriately modified. The computational divergence theorem on the third term in eqn (1) and advantages of such modified variational formations subsequent introduction of eqn (19b) lead to the over that of eqn (1) will be pointed out. variational statement Obviously, the constraint conditions should be chosen in such a way that they are, on the one hand, m~hanically meaningf~, and, on the other hand, the IldAu, AS,) s c - u,c(AS,) m 1s V.Vt?l [ trial functions which guarantee their ci priori fulfillment can be found more or less easily. + q; Auk,,Aukj dV Thus, a reasonable first choice is the incremental part of the static eq~~b~~ equations, i.e. AS,yjd~, dS AS,, + (a/Ju,,),, + AES,= 0 VNm* (19a) avNm 1 cs 1 +s In order to introduce divergence theorem eqn (19a) into eqn (l), the +Cm 5 pAi&Au, dV + 1 MAiiiAy k VNSl 436 zyxwvutsrqponmlkjihgfedcbaZYXWVUTSRQPONMLKJIHGFEDCBA D. K ARAM AN LI DI S and S. N. ATLUR~ - (nkjAui,kj,j + ApiAu, d V +lsaNmdP,duidS) -~Afidlr, + (correction terms) = stationary. (20b) It is worth mentioning that trial functions for stresses which guarantee h priori satisfaction of eqn (19b) can be established relatively easily, either by direct integration (in the case of a beam) or by means of static-geometry analogy or by use of well-known stress functions. Apparently, Pirotin [42] was the first investigator to utilize a variational equation of this kind in finite element analyses of geometrically nonlinear beam and shell problems. It was found that, for this class of problems, finite element models on the basis of eqn (20b) are more efficient than those based on eqn (1) or on the standard assumed displacement formulation. Subsequently, Boland[8] employed variational equations of the same kind as eqns (ZOa, b) and developed what he called a “consistent” finite element model for geometrically nonlinear beam problems and “inconsistent” models for large deflection analysis of beams and shells. The ‘“consistent” model, actually, is based on a variational equation similar to eqn (20a) whilst the “inconsistent” one is related to eqn (20b), Numerical test studies carried out in Ref. [8] clearly indicate that for the solution of geometrically nonlinear problems the “inconsistent” model is not only easier implementable but also more efficient than the “consistent” one. Further applications of eqn (20b) on geometrically nonlinear beam and shell stability problems were reported in Refs. [25,28,29,30]. Boland’s models have been examined and discussed at some length in [ZS]. It was pointed out that in both finite element models some inconsistencies in implementation have been introduced. After having pointed out two alternative mo~~catio~s to the formulations of eqn (l), we now direct our attention to the following two important questions: (i) Do eqns (20a, b) represent correct variational equations from the mechanics point of view? (ii) If so, do finite element modeis based on these offer any computational or other advantages over those based on eqn (I)? We first consider the theoretical validity of eqns (20a) and (20b) as applied to the original c~~~~~u~ and not to a finite element assembly of the descritized continuum. Thus, we consider the cases wherein the summation over elements, denoted by (X,) is removed, and the ~nte~ations are carried out over V and i?V, etc. where V is the entire continuum; moreover, we interpret the constraint conditions (19a) and (19b) to be imposed in the entire continuum and nof inde~ndently in each element. ‘in such a situation, the vanishing of the variation of the appropriately modified [c, being removed in eqns (20a, b)] functionals of eqn (20a, b) with respect to AS’, leads, in both the cases, to the com~at~b~~ty condition, eqn (6). However, the variation of the modified functionals of (2Oa, b) with respect to Au, leads (upon dropping the correction terms) to: -p&i,=0 in V (ala) L@- pdii, = 0 in V. (2lb) Note that in (21a, b), V represents the entire continuum. Comparing with the correct linear mumenturn balance condition, eqn (6), it may be seen that eqns (21a, b) represent theoretically meaningless balance conditions in the entire continuum. Now we consider the effects of constraints of the type of eqn (19a) and (19b) imposed separately in each element, V,,, on the stationarity of the functionals in (20a, b), respectively, both of which are defined for a$nite element assmbly of the discretized continuum. Let the displacement field Au, in each element be assumed as: Atii = N,Aq,(r = 1 . . . n,) in VHm (22a) where ~1~is the number of nodal-displacement degrees of freedom for each element. We use the standard Galerkin weighted residual finite element approach wherein the trial functions (i.e. in A@ are the same as the test functions (i.e. in variation ~Au,). Thus, 6du, = N,Gdq, (F = 1 . . . nJ in V,,. (22b) We now consider the variation of the finite element functional of eqn (20a). + (correction terms) = 0. (23) In (23) we have assumed Ati, to be held fixed. Noting that 6dS, is subject to the constraint as in (19a) i.e. and using (22b), one may rewrite (23) as: - x dFiN,SAq, + correction terms) = 0. zyxwvutsrqponm (25) k The first term on the r.h.s. of eqn (25) [wherein Mixed finite element models for plate bending analysis 437 Now, for any finite element patch as in Fig. 1, the dt+,, = f(du,, + A+)] leads to the incremental compatibility condition. In interpreting the remainder of vanishing of the second through eight terms on the r.h.s. of (26) implies the balance, in a weighted the terms, one must note the nature of local support residual sense over the patch, between: (i) the virtual of the finite element basis functions for du, as introwork of initial stresses, due to geometric nonlinearity, duced in eqn (22a). From the nature of such finite in the elements of the patch, (ii) the virtual work of element basis functions, one sees that a virtual varithe unequilibrated incremental tractions at interation 69, at a particular node introduces a variation element boundaries within the patch, (iii) the virtual in displacements only in the patch of elements surwork of the external tractions over the patch. It is in rounding the node in question (see Fig. 1). Thus, for this sense once again that the enforcement of the the patch in question, the vanishing of the second partial constraint as in eqn (19b) in each element does through sixth terms on the r.h.s. of eqn (25) for any still lead to a physically meaningful finite element arbitrary 69, implies the balance, in a weighted model. residual sense over the patch, between: (i) the virtual We now consider the final form of the finite work of unequilibrated tractions [note that in element algebraic equation that arises through the use Zmlav,, ( ) d V, equal boundary segment occurs twice, of eqns (l), (20a), and (20b), respectively. The diswith vi+ = - vi-] at the interelement boundaries cretization process of eqns (20a, b) and (1) calls for within the patch, (ii) the virtual work of external the introduction of appropriate trial functions for A II tractions on the patch, and (iii) the virtual work of and AS as given below: inertia forces. It is this sense that the enforcement of the constraint of eqn (19a) in each element does still lead to a physically meaningful finite element model. Au=NAi and AS Likewise, the vanishing of the first variation of the functional in eqn (20b), when 6AS, are subject to the for eqn (20a) for eqns (20b, 1). constraint of eqn (19b) and when 6Aui are of the form in eqn (22b), can easily be shown to lead to: zyxwvutsrqponmlkjihgfedcbaZYXWVUTSRQPONMLKJIHGFEDCBA Considering the element patch shown in Fig. 1 only, we obtain II x 6Aq,dV+ I JVmn x 6Aq, dS + Ai-i%,Aii- AiiTGJ (A,!$vjNi,)GAq, dS zyxwvutsrqponmlkjihgfedcbaZYXWVUTSRQPONMLKJIHGFEDC -A~TA~,-~Ail*Ai+~AiT~~A~ (pAiiiNi,)GAq, d V + + /?‘H/3 -; (A TtNir) s &lN!n (Apfli,)GAq, dV i Vh +AtiTG,T~+;AP%,dIfAiTA& + Ai7hlA a = stationary (27a) + C (MAii,Ni~)GAq~ - C (AFJfi,)SAq, zyxwvutsrqponmlkjihgfedcbaZYXWVUTSRQPONMLKJIHGFEDCBA k + k correction terms = 0. Wb) = stationary + AiTG3/l + Ai%iAi= stationary Fig. 1. Element patch. AiTAij3 (27~) where, for simplicity, surface tractions, concentrated forces, and correction terms have been omitted. As for the meaning of the symbols used in eqns (27a-c), we refer the reader to Table 1. D. KARAMANLIDIS zyxwvutsrqponmlkjihgfedcbaZYXWVUTSRQPONMLK and S. N. ATLURI 438 Table 1. Discretization of the functionals in eqns (1) and (20a, b) . . . . . . . . . . . . . . . . . . 1.1. &JAc(ASij)dV iDTHE - -:eTFB J;aij Auk, jdV L("kjAui,k . . . . . . . . . . . . . . . . . .f;. + ASij),jAuidV + ASij)ujAuidS for + A$T& Eqs. + ;A;TK2Ai (1, 20b) zyxwvutsrqponmlkjihgfedcbaZYXWVUTSRQ for Eq. (20.3) BTG _. + ;AtjTICIAc + APTA;-1 for Eqs. .*...:<. (1, ZOa,b) for Eq. (1) for Eq. (1) ..*.I.:. for AP~;iAui,J~ . . . . . . . . . . . . . . . . . . . .I.:. AcT,y$ ,-&,$buid” AGT&j3 for Eqs. (1, Eq. (20~1) ZOa,b) I . . . . . . . . . . . . . . . . . . . ..I.:. for Eqs. (1, 20a,b) I Using the notation Aii, Ail,,,,, + Ai=t?,,A~ + AfTO,,,Ai+ + Aii=&,A& = LA&,=, Ai,=, A ii,=]=; = LAiiJT, A&=, Ai,=JT + AiiTB,,Aii - Ai=A& - AP;=A$,+ = stationary A%,,,+* = LA&*, AiiLT, Aii,=d=; Ai$,,+3 = /_AiiLT,A&,=, Aii,=J= and n patc,,= - (28b) m+3 1 c ifi%I/?-;AiTB,Ai-Ai%.,Aiv n=m Ai = LAG,=, AC,=, Ai&=, A&,=J; Ati = Aii, y zyxwvutsrqponmlkjihgfedcbaZYXWVUTSRQPONMLKJIHGFEDCBA =LBmT,8~+,,BI+z,BI+3_lT -~A~~~,dQ--~diTC,d~-diTC,,dQ Eqns (27a-c) can be written alternatively below (see Table 2 also): as shown -; Ati=C,,A~ +;Ai%,Ai’f - A;=S,y - AG=S,y Ai%,,AG +~AQ7D,,Aiu+A)TT,y+A~~,y + AS=fl,A; + Ati=tl,A; + A+=tlo,A~ + A~=B,,,,A~ - A ?=A$” - AiTA$W = stationary. (28c) Taking now the variation of the functional with respect to Ati we obtain, in the case of eqn (28a): e,,A~+e,A~+(-A,,-B,,+D,,+E,,)A~ +(- A;w-B;w+D;w+E,T,)Ai +(-R,+T,)y-A&++&=0 while in the case of eqn (28b), we obtain: Pa) 439 Mixed finite element models for plate bending analysis Table 2. Explanation of the symbols used in eqns (28a, b, c) zyxwvutsrqponmlkjihgfedcb I n=m and finally in the case of eqn (28c), we obtain: based, for the most part, on the classical principle of minimum potential energy. In the case of a linear elastic problem, and upon neglecting inertia effects as well as transverse shear deformations (Kirchhoff plate theory), the principle reads e,,d~+e,d~+(-B,,-C,,+D,,)d~ +(-B,T, - C,; + D,T,))d8 +(--S,+T,Jr -d&=0 (29~) which represent the equilibrium conditions at the nodal point I, in each of the three cases (under the use of eqns (20a), (20b), and (l), respectively). In spite of seemingly different nature of eqns (29a-c), it can be concluded now that: (i) upon consideration of the three alternative discretized functionals in eqns (20a, b) and (I), the corresponding Euler-Lagrange equations do contain, in all the three cases, terms of the same type; therefore, in all of these three cases, the associated finite element models should be regarded as being correct; and (ii) implementation of finite element models based on eqn (20b) is easier than those models based on eqns (20a) and (l), since fewer terms have to be developed on the element level. This is especially true in the case of large deflection problems. Finite element models for thin plates in bending Assumed displacement moa’els. As already men- tioned, the development of plate bending elements is n(w)=? ; H [(w,, + %J* - 2(1 - v)(w,, . ~,~,c- &>I dA - :..dA 1& -1 q.w.d.s cwn ] - c (Fz . w - n;i,. w,, + n;i. w,,) zyxwvutsrqpon k = Minimum (30) where D = E. h3/12. (1 - vz) denotes the plate bending stiffness; E is the Young modulus; v the Poisson ratio; h the plate thickness, ~7=p(x, v) the lateral body force; q = q(s) the distributed loading acting along the boundary C,; and pZ, aX, and n;i, external loads acting on the nodal point k. In eqn (30) the primal variable w is subject to the subsidiary conditions D. 440 KARAMANLIDIS on w-W=O;w,,-W,,=O;w,,-W,,=O C, (3 1a-c) zyxwvutsrqponmlkjihgfedcbaZYXWVUTSRQPON and S. N. ATLURI zyxwvutsrqponmlkjihgfedcbaZYXWV functional with respect to 1 and p, one obtains as natural constraints the C’ continuity conditions and w+-w-=O;w,+,-w;,=O;w,+,-w;,=O on C, (32a-c) where C, is the interelement boundary. Upon taking the variation of the functional with respect to w, eqn (30) yields (i) the plate bending equilibrium equation +p = 0 A4X.X.= + 2MV.V + MYYJY and (ii) the interelement tions on C, in A, + w, x- 3,, = 0; w,+Y- c’,,=O on C,+ w, x- 6’,, = 0; w;~ - 3,, = 0 on C,-. and On the other hand, taking the variation with respect to w leads to the expressions (a)* =(M,;v,+M,;v,)*;oL)* (33) = (M Xy v, + M ,,,, . v,) * on &4, traction reciprocity condi- which can be used in order to eliminate the Lagrangian multipliers 1 and p from eqn (36). Elements of this type are called hybrid displaceMxy . v,)’ + (M,, . v, + Mx,, . v,)- = 0 Wxx. v, + zyxwvutsrqponmlkjihgfedcbaZYXWVUTSRQPONMLKJIHGFEDCBA ment elements. These, which reportedly give good (344 zyxwvutsrqponmlkjihgfedcbaZYXWVUTSRQPONM results in some cases, can, however, become numerically unstable (see, e.g. Mang and Gallagher1351 for (M Xy v, + Myy . vJ+ + (M ,, . v, + MYY. vY)- = 0 a detailed discussion in this subject). (34b) M ixed models + (Wxw+ Mxy,J vx+ Of,,, + M,y,J v,)- = 0 (34c) where use of the constitutive equations M,, = - D . (w,, + v . w,,) (35a) Myy = - D . (w, + v . w,xx) (35b) M X,,= - D . (1 - v) . w+ (35c) has been made. The main drawbacks of the variational eqn (30), when used as a basis for the development of plate bending elements, are: (i) a priori satisfaction of the essential constraint conditions eqns (32a-c) requires the use of higher order trial functions (see Gallagher [ 181 and Scharpf [45] for comprehensive discussions) and (ii) due to the occurence of secondorder derivatives in eqns (35a-c) the bending moments are calculated with a significant loss in accuracy. In order to overcome the first difficulty, in the past, several authors (e.g. [I 1,33,2 l] suggested modified principles with relaxed interelement continuity requirements. Within a variational formulation of this kind, C? continuity conditions are satisfied d posteriori as natural constraints. In order to understand these concepts, let us consider an extended variational principle: n*(w, I?‘,a,p) = n(w) - C As the point of departure for the developedment of mixed finite element models for plate bending analysis, the Reissner variational principle - 2 . (1 + v) . (M,, . M ,, - M :,)) .w,xx -M,, .wTyy -=f,, .w,xy dA .......*.........* II -V n A, - M I, zyxwvutsrqponmlkjihgfedcbaZYXWVUTSR p-w.dA+[cOm$.w.dS} - c [Fz. w - XIX w,x + lay . w,J = staiionary (37a) can be used. The d priori constraints in eqn (37a) are eqns (31a-c) and (32a-c); therefore, as far as C’ continuity is concerned, eqn (37a) does not offer any advantages over the principle of the minimum of potential energy, eqn (30). In the following, alternative mixed formulations will be given, such that the fulfillment of C’ continuity can be either avoided or easily achieved. A first alternative can be developed by introducing eqns (32b, c) into eqn (32a) by means of Lagrangian multipliers. The resulting variational principle reads: 1 . (3,x - w,,) dS m s - 2. D . (1 - v’) + aA P . (k, - w,,JdS . (CM,, + Myy)* - 2 (1 + v) . (M,, . Myy - Mzy)) m = stationary (36) which has been obtained from eqn (30) upon introduction in it of eqns (32b, c) by means of Lagrangian multipliers. It is seen that taking the variation of the II My,, w,,,~- 2 . Mxy . w,xy dA .-. .Mx, . . . . .w,,, . . . -. . . . . . . . . . . . . . . . . . . . . . . - p.w-dA+~cOmq.w.d~) 441 Mixed finite element models for plate bending analysis -ax~w,,+d+v,y) -~(F*?v k In the event that the moment field is assumed so as to fulfill zyxwvutsrqponmlkjihgfedcbaZYXWVUTSR dpriori the traction reciprocity conditions on Cm, namely, I- + + c amwxx.“x + Mxy. “J . (WY,- +xX) J i&y.v, + M,, . v,) . (w,, - ~,Jl dS = stationary. (39a) (M,,, v, + MXY.v,)+(M ,, . v, + M x,. v,)- = 0 (38a) . v,)- = 0 (39b) (MXy. v, + kf,, . v,)’ + (M,, . v, + A4,zY Discretization of eqn (38a) calls for the introduction of Co and C’ continuous approximations for w and fi, respectively, while the traction reciprocity conditions on C,,, are met ci posteriori. Upon application of the divergence theorem to the terms underscored by. . ., an alternative formulation of eqn (38a) can be obtained: then the variation of the functional in eqn (37b) with respect to M,,, M yy, and M xy gives as ci posteriori conditions + w, x-w, x =O; w:~- w;,=O on C,. In the past, several authors have proposed elements of this type with Co continuous moment and displacement field approximations. For rather academic examples, they reportedly give good results (see Refs. [20,55]); for practical purposes, however, . (WL + &J2 zyxwvutsrqponmlkjihgfedcbaZYXWVUTSRQPONMLKJIH they are as useful as the fully-compatible assumed - 2. D . zyxwvutsrqponmlkjihgfedcbaZYXWVUTSRQPONMLKJIHGFEDCBA (1 - v*) displacement elements. A concise overview on the -2.u +v)$i4xxd4yy-M~y)) alternative mixed formulations presented so far is given in Table 3. + Wx,, + Mxyy). w,, + Wxy x + Myyg). w,y dA . . . . . . . . . . . . . . . . . . . . . . . . ... . . . . . . . . . . . zyxwvutsrqponmlkjihgfedcbaZYXWVUTSRQPONMLKJIHG In light of Table 4, it is apparent that none of the proposed models can be considered as being fully satisfactory. jj.w.dA +JcV&fi.dS} A more appealing alternative mixed approach can be obtained from eqns (38a) or (38b) upon intro-C(&G -n;i,~~,,+E;i,~3,,) zyxwvutsrqponmlkjihgfedcbaZYXWVUTSRQPONMLKJIHGFEDCBA duction of the plate bending equilibrium condition k 1I -Iis nA - . v, + WXX " xs Mx; v, + Myy. v,). Kiy dS 1 = stationary. + M x.x+x+ 2M xy _xy+ Myyay+ F = 0 v,). bc,, a.4I as an essential constraint. ational equation reads (M,,. (38b) (40) The thus-obtained vari- n (M,,, M,,, M,,, c) wherein only first-order derivatives occur. This fact has, according to Harbord et al.[19], a positive influence on the element convergence performance; therefore, eqn (38b) should be preferred over eqn (38a). Another possibility for the development of Co continuous mixed elements is described next. Applying the divergence theorem to the terms underscored by. . . . , we obtain from eqn (37a) another alternative formulation - 20 + “) . Wx, . M,, - M,,)*) s + aA[((Mxx, + Mxy,y)*v, + Wxy,.x + M,,) . “,>I. fi -(M ;~,+M ,,~v,)~~~ - (M ,, . v, + M , . v,) . d ,,I,, - 1,4 .@ dS} - c (F2 . D - n;i, . c,, + hq .3,,) k = stationary. (41) Equation (41) represents the variational basis for what one customarily calls the “assumed stress hybrid finite element method”. Within this approach, discretization of the functional calls for introduction of independent interpolants for bending moment and displacement field in the interior and on the boundary of the element, respectively. zyxwvutsrqponmlkjihgfedcba a=Pfi+ti; - jj.w.dA c (Fz . w - +[cO&w’dS} n;i,.w,, + q, . w,),) k = stationary. (37b) *=RRi Wa, b) where Q = LM ,.., M ,,,,, M ,,J’, + = Li?, - C,,JT, e den_otes an arbitrary partial solution of eqn (40), N = N(s, n), and s and n are Cartesian coordinates on the element’s boundary. Introducing eqns (42a, b) in eqn (41), and following the procedure outlined previously, we obtain, after the elimination of /3’s on the D. zyxwvutsrqponmlkjihgfedcbaZYXWVUTSRQPONMLKJIHGFEDC KARAMANLIDIS and S. N. ATLIJRI 442 Table 3. On mixed plate models based on eqns (38a) and (37b) VARIATIONAL BASIS Eq. (38a) Eq. PRIMAL VARIABLES w, I - w,x w, Y WTx = cu on w,x + w,y w= 0 wTx- 1 a* c ;.x =o + --w - w m _+ --w _- =o _+ xv w - w _- w,x - - = “.y = a M C m xx M *V +ic ‘/I x xy .v,+M Y = 0 I net, on c, 0” cr )++(Mxx.“,+?,xV.“y)-=0 *" )++(Mw.vx+l.lyy."y)-' YY Y 0 w c zyxwvutsrqponmlkjihgfedcbaZYXWVUTSRQPONMLKJIHGFEDCBA TRIAL FUNCTIONS FINAL ALGEBRAIC MATRIX EQUATION M - w. = zyxwvutsrqponmlkjihgfedcbaZYXWVUTSRQPONMLKJIHGFEDCB 0 w, - w, = 0 zyxwvutsrqponmlkjihgfedcbaZYXWVUTSRQPONMLKJIHGFED Y Y Y -ci=o w (iii) M 1 0 w+-;=O i M,,, YY' ;-;=o a F: 5 (37b) w = N; ._ g LO .. = Ku = ; _. _ Table 4. Comparison between two mixed finite elements models for thin plate analysis DISADVANTAGES ADVANTAGES *essential constraint conditions easy to fulfill l incorporation elements packages of such in conunercial possible *coefficient matrix of the final algebraic equations positive definite lsmaller number of on system level l smaller ables level on *formation matrices .large number of elemental variables requires elimination on the element's level *formation matrices of elemental rather expensive *application to nonlinear problems not efficient unknowns number of varithe element's of elemental inexpensive *application to nonlinear problems efficient *application to real engineering structures impossible in general *large number of unknowns on system level -coefficient matrix of the final algebraic equations indefinite *bandwidth large lincorporation cial packages in commerimpossible Mixed finite element models for plate bending analysis element 443 kinematic dof, respectively. Formation of H-i and the right-hand multiplication by G requires . na) multiplications, while the solution of (nl/2 + ni ; ii%i - iir(q + zyxwvutsrqponmlkjihgfedcbaZYXWVUTSRQPONMLKJIHGFEDCBA GTH-‘$ - 1) zyxwvutsrqponmlkjihgfedcbaZYXWVUTSRQPONMLKJIHG eqn (40) needs (ns3/6 + zyxwvutsrqponmlkjihgfedcbaZ nb 2. na) multiplications for the direct formation of x = H-‘G. zyxwvutsrqponmlkjihgfe level, n(ii) = ; 1 -2iiT = stationary (43) SUMMARY AND CONCLUSIONS k In the present paper, an attempt was made to give where an overview of several alternative finite element models for thin plate bending analysis. The central focus of the paper has been the so-called mixed elements, (. . .) dS zyxwvutsrqponmlkjihgfedcbaZYXWVUTSRQPONMLKJIHG _J...)dA=$I’HB+br+; wherein, in the interior of a single element separate 1 s a.4 interpolants for displacements and stresses are intro=/lTGtl+~=~ duced. The practical relevance of elements of this W a, b) type has been, in the past, a controversial matter; K = G%-‘G is the elemental stiffness matrix, and ij therefore, one of the main goals of the paper has been and 1 the load vector due to distributed and concento present a concise re-examination of the theoretical trated loading, respectively. background of mixed elements. In the tlrst part of the Since the publication of Pian’s paper[41] in the present paper dealing with 3-D elasticity, large mid-sixties, finite elements of this type have been used deflections and dynamic phenomena as well have successfully for carrying out numerical analyses of a been considered. In this light, the justification of variety of engineering mechanics problems. Thus, one critical comments on the relevance of mixed elements is not surprised to note that mixed hybrid plate made by several authors in the past can be seen to be elements have been included in a few commercial questionable. In the second part of the present paper, finite element packages. Furthermore, a recently pubwhich deals with elastic thin plates in bending, several lished comparative study[6] on a series of triangular alternative variational finite element formulations plate bending elements clearly demonstrated the suhave been presented. Their advantages and disadvanperiority of an element developed on the basis of eqn tages when compared with each other have been (41) over well-known commercial assumed displaceoutlined. It was found that so-called hybrid mixed ment elements. However, mixed hybrid plate ele- formulations provide rational bases for the develments have been in the past subject of criticisms, too. opment of effective yet easily implementable eleSome of these are noted below: ments. Extensive numerical studies carried out by (1) According to Fried[35] inclusion of distributed means of a newly developed element of this type on loads and inertia forces within finite element models well-selected examples are being presented in a combased on eqn (41) should be regarded as not straightpanion paper[32]. forward. This criticism is meaningless as can be seen from the above developments. He further claims that “ . . . there is no reason whatsoever, neither comAcknowledgements-The first author (D.K.) gratefully acknowledges a habilitation stipend provided by the German putational nor theoretical for not assuming the inteScience Foundation under grant Ka 487/4. The authors also rior stresses (strains) to be derivable from a displacement functions”. The computational reason, of acknowledge the partial support provided by the Georgia Institute of Technology. It is a pleasure to thank Ms. J. course, is: calculation of stresses and stress resultants without loss in accuracy. The theoretical reason is: in Webb for her assistance in the preparation of this typescript. a MIXED MODEL, by its very nature, stress must not be assumed to be derivable from displacements. Furthermore, imposing the equilibrium equation (41) REFRRRNCES to be satisfied dpriori has both computational as well 1. S. Ahmad, B. M. Irons and 0. C. Zienkiewicz, Analysis as theoretical advantages. of thick and thin shell structures by curved finite elements. Int. J. Num. Meth. Engng 2, 419-451 (1970). Upon realizing that neither eqn (38a) nor eqn (37b) 2. S. N. Atluri, On the hybrid stress finite element model represents an appropriate basis for the development for incremental analysis of large deflection problems. of generally applicable plate bending elements (in the Int. J. Solidr Structures 9. 1177-l 191 (1973). first computational effort to high and in the second 3. S. N. Atluri and H. Murakawa, On hybrid finite it is inconvenient to impose moment field continuity), element models in nonlinear solid mechanics. Finite eqn (41) seems to be a balanced compromise. Elements in Nonlinear Mechanics (Edited by P. G. (2) Several authors, such as Horrigmoe[25] and Bergan et al.), Vol. 1, pp. 3-41. Tapir, Norway (1977). Batoz et al. [6], developed special algorithms in order 4. S. N. Atluri, On some new general and complementary to reduce the computational effort for the inversion energy principles for the rate problem of f&rite strain, of the matrix H-‘. However, since H-‘G but not H-’ classical elasto-plasticity. J. Structural Mech. 9, 61-92 (1980). is needed, this is not necessary. Using one of the 5. S. N. Atluri, Rate complementary energy principles, standard equation solvers with H as coefficient matrix finite strain plasticity and finite elements. Proc. Svmo. and G as right-hand side vector, H-‘G can be directly on Variation& Meth: Mech. of Solicis. Pergamon he&, evaluated as follows: H (9 x ns) x G (5 x na) = (“a x na) (45) n, and ns are the numbers of stress parameters and Oxford (1981). 6. J.-L. Batoz, K.-J. Bathe and L.-W. Ho, A study of three-node triangular plate bending elements. Znf. J. Num. Meth. Engng 15, 1771-1812 (1980). 7. K. Bell, A refined triangular plate bending finite element. Znt. J. Num. Meth. Engng 1, 101-122 (1969). 444 D. zyxwvutsrqponmlkjihgfedcbaZYXWVUTSRQPONMLKJIHGFEDC KARAMANLIDIS and S. N. ATLURI 8. P. L. Boland, Large deflection analysis of thin elastic structures by the assumed stress hybrid finite element method. Thesis presented to the Massachusetts Institute of Technology at Cambridge, Massachusetts, in 1975, in partial fulfillment of the requirements for the degree of Doctor of Philosophy. 9. W. Bosshard, Ein neues, vollvertriigliches endliches Element fur Plattenbiegung. Abhandlung IYBH, Zurich, pp. 27-40 (1969). 10. C. Brebbia and J. Connor, Plate bending. Finite Element Technics in Struct~al ~ec~ics (Edited by H. Tottenham and C. Brebbia). Southampton University Press (1970). 11. H. Bufler, Die verallgemeinerten Variationsgleichungen der dtinnen Platte bei Zulassung diskontinuierlicher ~~nit~r~fte und Ve~chiebun~n. Zng~i~-Arch~ 39, 330-340 (1970). 12. C. Caramanlian, K. A. Selby and G. T. Will, A quintic conforming plate bending triangle. Znt. J. Num. Meth. Engng 12, 1109-I 130 (1978). 13. Ph. G. Ciarlet, The Finite Element Method~or EZZiptic Problems. Noah-Holland, Amsterdam (1978). 14. J. Connor, Mixed models for plates. Finite Element Techniques in Structural Mechanics (Edited by H. Tottenham and C. Brebbia). Southampton University Press (1970). 15. G. Dhatt, Numerical analysis of thin shells by curved triangular elements based on discrete Kirchhoff hypothesis”. Proc. ASCE Symp. on Appl. of Finite Element Meth. in Civil Engng, Vanderbilt University, Nashville (1969). 16. C. A. Felippa, Refined finite element analysis of linear and nonlinear two-dimensional structures. Thesis presented to the University of California at Berkeley, California, in 1966, in partial fulfillment of the requirements for the degree of Doctor of Philosophy. 17. I. Fried, Residual energy balancing technique in the generation of plate bending finite elements. Comput. Structures 4, 771-778 (1974). 18. R. H. Gallagher, Analysis of Plate and Shell Structures. Finite Element Meth. in Civil Engng, Vanderbilt University, Nashville (1969). 19. E. Giencke. Ein einfaches und genaues finites Verfahren zur Berechnung von orthotrop& Scheiben und Platten. Der Stahlbau 36, 2-268 and 303-315 (1967). 20. R. Harbord. B. Krdnlin and R. Schroder, Schalenelemente in ’ gemischier Darstellun~~eo~e, Kritik, Beispiele. Zngenieur-Archiv 47, 207-222 (1978). 21 J. W. Harvey and S. Kelsey, Triangular plate bending elements with enforced compatibility. J. AZAA 9, 1023-1026 (1971). 22. K. Hellan, Analysis of elastic plates in flexure by a simolified finite element method. Actu Polytechnica Scandinavica 46, l-29 (1967). 23. L. R. Herrmann, Finite-element bending analysis for plates. J. Engng Mech., ASCE 93, 13-26 (1967). 24. I. Holand, Membrane and plate bending elements. Proc. World Cong. on Finite Element Meth. Structural Mech., Boumemouth/Dorset (1975). 25. G. Horrigmoe, Finite element instability analysis of free-form shells. Rep. 77-2, University of Trondheim (1977). 26. ‘I. J. R. Hughes, M. Cohen and M. Haroun, Reduced and selective integration techniques in the finite element analysis of plates. Nuclear Engng Design 46, 203-222 (1978). 21. T. J. R. Hughes and T. F. Tezduyar, Finite elements based upon Mindlin plate theory with particular reference to the four-node bilinear isoparametric element. J. Appl. Mech. ASME 48, 587-596 (1981). 28. D. Karamanlidis, Finite Elementmodelle xur numerischen Berechnung des geometrisch nichtlinearen Verhaltens ebener R~mentra~erke im unter- und iiber- kritischen Bereich. Res. Rep. VDI-Z l/63, Verband Deutscher Ingenieure, Dusseldorf (1980). 29. D. Karamanlidis, A. Honecker and K. Knothe, Large deflection finite element analysis of pre- and postcritical response of thin elastic frames. Proc., European-U.S. Workshop on Nonline~ Finite Element Anal. in Structural Mech., Bochum (1980). 30. D. Karamanlidis and K. Knothe, Geometrisch nichtlineare Berechnung von ebanen Stabwerken auf der Grundlage eines gemischt-hybriden Finite-ElementeVerfahrens. Zng~~~r-Arc~v 50, 377-392 (1981). 3 1. D. Karamanlidis, Beitrag xur linearen und nichtlinearen Elastokinetik der Systeme und Kontinua (TheorieBerechnunasmodelle-Anwendungsbeispiele). Habilitation ThesisrTechnical University-of Berlin (1983). 32. D. Karamanlidis. H. Le The and S. N. Atluri. Mixed finite element models for plate bending analysis: applications. Paper submitted for publication (May 1983). 33. F. Kikuchi and Y. Ando, A new variational functional for the finite element method and its application to plate and shell problems. Nuclear Engng Design 21, 95-113 (1972). \-- --,34. F. Kikuchi and Y. Ando, Some finite element solutions for plate bending problems by simplified hybrid displacement method. Nuclear Engng Design 23, 155-178 (1972). 35. H. Mang and R. H. Gallagher, A critical assessment of the simplified hybrid displacement method. Znt. J. Num. Meth. Engng 11, 145-167 (1977). 36. G. Miiller and F. Miiller, Platten-Theorie, Berechnung, Bemessung. Rep. 76-1, University of Stuttgart (1977). 37. A. K. Noor and C. M. Andersen, Mixed models and reduced/selective integration dispiacement models for nonlinear shell analysh. Nonlinear Finite Element Analvsis of Plates and Shells. (Edited bv T. J. R. Hushes et al.), ASME-AMD 48, pp. 119-146 (1981). 38. H. Parisch, A critical survey of the 9-mode degenerated shell element with special emphasis on thin shell application and reduced integration. Comput. Meth. Appl. Mech. Engng 20, 323-350 (1979). 39. S. F. Pawsey and R. W. Clot@, Improved numerical intenration of thick shell fmite elements. Znt. J. Num. Me& Engng 3 575-586 (1971). 40. A. G. Peano. H>erarchies of conforming finite elements. Thesis presented to Washington Unive&ty at St. Louis, Missouri, in 1975, in partial fulfillment of the requirements for the degree of Doctor of P~lo~phy. 41. T. H. H. Pian, Derivation of element stiffness matrices by assumed stress distributions. J. AZAA 2, 1333-1336 (1964). 42. S. D. Pirotin, Incremental large deflection analyses of elastic structures. Thesis presented to the Massachusetts Institute of Technology at Cambridge, Massachusetts, in 1971, in partial fulfillment of the requirements for the degree of Doctor of Philosophy. 43. W. Prager, Variational principles for elastic plates with relaxed ~ntin~ty requirements. Znt. J. So&& Structures 4, 837-844 (1968). 44. Ch. W. Pryor, R. M. Barker and D. Frederick, Finite element bending analysis of Reissner plates. J. Engng Mech., ASCE 96, 967-983 (1970). ~wend~gen von Element45. D. W. Scharpf, programmen im Briicken-und Hochbau. Finite Eiemente in der Baupraxis (Edited by P. J. Paht et al.). Wilhelm Ernst, Berlin, (1978). 46. H. R. Schwarz, Methode derfiniten Elemente. TeubnerVerlag, Stuttgart (1980). 47. R. T. Sevem and P. R. Taylor, The finite element method for flexure of slabs when stress distributions are assumed. Proc. Inst. Civil Engng 34, 153-170 (1966). 48. S. M. Smith, A finite element analysis for moderately thick rectangular plates in bending. Znt. J. Mech. Sci. 10, 563-570 (1968). Mixed finite element models for plate bending analysis 49. F. Stummel, Remarks concerning the patch test for nonconforming finite elements. ZAMM 58, 124-126 (1978). 50. P. Tong, New displacement hybrid finite element models for solid continua. ht. J. Num. Meth. Engng 2,73-83 (1970). 51. P. Tong, S. T. Mau and T. H. H. Pian, Derivation of geometric stiffness and mass matrices for finite element hybrid models. Znt. J. Sol& Structures 10,919-932 (1974). 52. K. Wash& Variational Method in Elasticity and Plasticity, 3rd Edn. Pergamon Press, Oxford (1982). 53. G. A. Wempner, J. T. Gden and D. A. Kross, Finite element analysis of thin shells. J. Engng Mech. 10, 1273-1294 (1968). 445 54. D. Withum, Rerechnung von Platten nach dem Ritxschen Verfahren mit Hilfe dreieckf&miger Maschennetze. Rep. 9, Technical University of Hannover (1966). 55. W. Wunderlich, Mixed models for plates and shellsprinciples elements, examples. Hybrid and Mixed Elements. (Edited by S. N. Athni, et ol.). Wiley, Chichester (1983). 56. Y. Yamamoto, A formulation of matrix displacement method. Internal Ren. Massachusetts Institute of Technology (1966). 57. 0. C. Zienkiewicz. R. L. Tavlor and F. M. Too. Reduced integration technique*in general analysis of plates and shells. Znt. J. Num. Meth. Engng 3, 275-290 (1971).