The Mitc7 and Mitc9 Plate Bending Elements

We present in this paper two plate bending elements that have been proposed and mathematically analyzed previousIy by Bathe, Brezzi and Fortin: the MITC7 (a triangular) element and the MITC9 (a quad~late~l) element. The formation of these elements is summarized and example solutions are given that demonstrate the high accuracy of these elements. 1. lNTRODUCHON Although much research effort has been spent on the development of reliable and efficient plate and shell elements, there is still a large interest in arriving at improved elements. During the recent years we have concentrated on the development of elements based on Mix~-Inte~olat~ Tensorial Components (i.e. our MITC elements) and have proposed the 4-node MITC4 element [l, 21, the 8-node MITC8 element [3] and a complete family of new elements [4, 51. The 4-node and &node elements have been developed for general shell analysis and are available in ADINA [6], whereas the elements given in [4, 51 have so far only been proposed for plate analysis, but have excellent potential to be extended for applicability to effective general shell analysis as well. We mentioned before [2,3] that the extension of a plate element to a general (linear and nonlinear) shell element usually represents a major step and is frequently not possible. It is therefore most appropriate to concentrate directly on the development of general nonlinear shell analysis capabilities which can then also be used in the linear analysis of plates. Hence, we have emphasized in our work the development of general shell elements and proposed the MITC4 and MITCS elements. To understand the behavior of these elements and possibly to design additional elements, we conducted mathematical analyses which were, however, so far only possible for the case of the linear analysis of plates. These theoretical considerations led us to some interesting and quite general results, which are in essence based on an analogy that can be drawn between the analysis of incompressible media and the analysis of Reissner-Mindlin plate problems. The mathematical analyses also identified additional elements for the plate bending problem and since these elements are constructed much like the MITC4 and’ MITC8 elements, we can be quite confident that they can be extended to general shell analysis as well. The objective in this paper is to summarize the formation of two of the new elements for the plate bending problem and present a numerical evaluation. The elements considered are a ‘I-node triangular element, the MITC7 element, and a g-node quadrilateral element, the MITC9 element. Both these ela ments pass the patch test, they contain no spurious zero energy mode and, as demonstrated in the paper, have excellent predictive capability. 2. THE PLATE BENDING PROBLEM CONSIDERED We consider the spaces: Q= (H:(D))2 and IY = H:(Q) and a load function~~ven in L2(Q). The sequence of problems under consideration is: P,: inf 57@, e) %Ee,zvusW 2 +; IIe-Vw((:-P(f, w) where (13/2)u(9, 9) is the bending internal energy, and (i11/2) II 8 VW II : is the shear energy. 1) II ,, and ( , ) represent respectively the norm and the inner product in L%(Q). Assume now that we are given the ii&e element subspaces Q% ce and Wh c W. The corresponding discretized problem is described by In general, pti ‘locks’ for small t. A common procedure is to reduce the influence of the shear energy. We consider here the case in which the reduction is carried out in the following way: we assume that we are given a third finite element space, r,, and a linear operator R which takes values in r, . Then we use 11 R(8, VW,) 11: instead of I{ 6, VW, 11: in the shear energy. In our formulation we further assume that RVw, = Vw, for all wli E W, (1) 797 798 KLAUS-JORGEIN BAG ef al. so that the discretized problem takes its final form + ; /I R6,, VW,, I/; t’(l; y,). Setting 7=lt-2(O-Vw) and 7h=~t-2(R0~-Vwh) (2) the Euler equations of P, and Pi* are, respectively a(O,q)+(y,tl-VU=(S,U Vsto0,KoW 7 = It -‘(xl VW) (3)