Mixed Finite Element Models for Plate Bending Analysis

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 The development of simple and effective finite elements for thin-plate bending has been the preoccupation of many researchers during the past two decades. Recalling that the finite element method (FEM) was born in the mid-fifties as a natural extension of the two classical matrix methods of structural analysis, namely the displacement method and the force method, it is not surprising to note that most of the plate bending elements proposed so far are based on these two approaches. Comprehensive reviews on plate bending elements were presented by Gallagher [ 181, Brebbia and Connor [ IO], Holand [24], Miiller and Mtiller[36], and Schwarz[46] to name just a few. The development of assumed displacement plate bending elements based upon the Kirchhoff theory of thin plates requires that the trial functions fulfill ri priori the requirement that at the interelement boundaries not only the lateral displacement w but also its first derivative &~/an (n is the outward normal to the boundary) be continuous (so-called C’ continuity). Additionally, the basic requirement of exact representation of rigid body modes as well as constant strain modes in each element should be met in order to ensure convergence to the exact solution as the element size decreases. Plate bending elements fulfilling these two requirements (so-called fully conforming or compatible elements) have been suggested by Withum[54], Felippa[l6], Bosshard[9], Bell[7], and others during the sixties. In spite of their inclusion in the element libraries of some general purpose finite element codes like ASAS, ASKA, MARC and PRAKSI, contPost-Doctoral Fellow. fRegents’ Professor of Mechanics. §Peano[40], too, has succeeded in establishing CL elements without enforcing supertmous C* continuity at vertices. 431 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 polynomial in the case of a triangular element) in such elements. As the elements generalized degreesof-freedom, displacements w, rotations &/ax, dw/dy and curvatures a2w/ax2, a2w/My, a2w/ay2 at the vertices as well as well as displacement derivatives awlan at the midside nodes may be defined. Introduction of second-order derivatives as nodal parameters enforces C2 continuity (“Over-compatibility”) at the element vertices. This is not only superfluous from a mathematical point of view but also makes the application of elements of this type in analyses of engineering structures (wherein, almost exclusively, due to discontinuities in geometry, material properties and loading, the curvatures may be discontinuous) impossible. In a recent publication by Caramanlian et al. [ 121, a triangular plate bending element has been suggested using displacements and their first derivatives as elemental degrees of freedom. Clearly, this element does not enforce C2 continuity,4 however application of conforming elements in practical structural analyses still remains neither an easy nor an inexpensive task. On the basis of these experiences, it was natural to ponder whether or not reliable, costeffective elements could be developed by relaxing the restrictions imposed by the strict C’ continuity requirement of Kirchhoff theory. The flrst attempt towards this goal, the so-called non-conforming element concept introduced in the mid sixties, appeared to be unsuccessful. In this scheme, the C’ continuity is maintained only at distinct points (nodes) rather than along the entire interelement boundary. Although, in some cases, elements of this kind happened to give satisfactory results, from a mathematical point of view they are not acceptable (see, e.g. [13,49]) since they do not always guarantee convergence of the approximate solution to the exact one as the element’s size decreases. 432 D. KARAMANLIDIS and S. N. ATLIJRI 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 thickand 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,,, Myy, and Mxy 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, numerMixed finite element models for plate bending analysis 433 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 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 lIdAu, AS,) = c [U~~AS~)+U~~AU~,~AU~~+AS~A~~,~]~V pAii,Au,dV+ 1 MAiiiAui m . . . . . . . . . . . . . . . . . . . . . . . . . . . . . k a 0 o . . . . . ‘o. . . . . . . . . . . . . . . . . . . . -zLm AjiAuidV -c m i AzAu,dS -c A&Aui Lvm k . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .