C~puters & Stnetwes Vol. 47, No. 1, pp. 125-141, 1993 US*7~9~3 s&a0 + fJ.00 Rioted in Great Britain. 0 1993 Pngsmon Rcss zyxwvutsrqp Ltd zyxwvut THE PERFORMANCE OF A REFORMULATED FOUR-NODE PLATE LENDING ELEMENT IN MODERATELY THICK TO VERY THIN PLATE APPLICATIONS D. BRIASSOULIS Department of Agr~c~lt~~ En~n~~ng, Agricultural university of Athens, Greece (Received 9 December 1991) Ahstraet-This paper analyzes the behavior of a reformulated four-node Mindlin plate bending element in thin to very thin and in moderately thick element applications. The shear locking mechanism is rejected with the proposed formulation whereas machine-related locking remains the only problem potentially developed in thin element apphcations of the ~fo~~at~ element. The solution to the machine locking problem may be achieved with the implantation of a simple technique rendering the element completeiy locking-free. In moderately thick element applications, the reformulated four-node plate bending element is shown to exhibit an excellent performance in accordance with Mindlin-Reissner theory. This performance, if coupled with an analogous behavior of the element in modeling boundary layers developing in thin plate applications (a work to appear soon), renders the reformulated four-node Mindlin plate bending element one of the most reliable and robust, yet simple, Mindlin Co plate bending elements. 1. INTRODUCIION The widespread use of the finite element method over the whole range of engineering applications 11,Z] has placed this powerful analytical tool within the reach of users often lacking the necessary background knowledge and using the method in the form of ‘black box’ computer code [3]. This situation, combined with the fact that many finite elements exhibit, under certain conditions, problematic behavior not always detectable or anticipated by the inexperienced user, may produce serious errors in engineering analysis. Nevertheless, the requirement for ‘reason’ [3] in the applications of the finite element method mostly concerns those devoted to the edu- cational aspects of engineering. However, it is necessary at least, that the finite elements already available in commercial codes, or the elements under development satisfy robustness and reliability, as these two requirements are introduced and defined in [3]. Historically, applications of the classical Co struc- tural finite elements have been associated with two major deficiencies described under the terms locking and zero energy modes [ 1,2,4-61. In particular, shear and membrane locking have been the dominating inherent problems of these elements in the last 30 years. Machine locking should be distinguished from shear and membrane locking as it does not represent an inherent element probtem but rather, a purely machine-dependent phenomenon (as defined in IS]). Machine locking can, because of its nature, be allevi- ated relatively easily in contrast to the shear and membrane locking problems [ 1,2,8,9]. During the last few years, research has been di- rected toward the development of alternative element formulations aimed at eliminating the locking prob- lems (e.g., mixed or hybrid formulations [lo, 111, enhanced shear strain inte~olations 112, 131 and formulations based on the imposition of discrete Kirchhoff constraints [ 14, 151). Various techniques of reducing the order of the numerical integration, introduced in an attempt to eliminate the shear and membrane locking problems [4-61, may be described as a double-edged sword as they have been associated with the development of spurious kinematic modes in many cases of shell and plate element applications (zero energy modes or mechanisms). Recently, new mixed approximations were introduced involving specification of the transverse displacements, the rotations and the shear resultants as independent variables, whereas discrete constraints were used to express the shear resultants in terms of the displace- ment parameters [16, 171. In another development, research focusing on the analysis of the locking phenomena and the redefini- tion of the associated problems [18,19], has led to a new formulation for the Co structural elements built at the finite element assembly level. This alternative formulation was proposed in [20], aiming at the removal of the shear and membrane locking mechan- isms and at avoiding the potential development of zero energy modes, usually developed with the classical fo~ulation under reduced integration. The complete elimination of all locking problems was confirmed with extensive numerical applications [20-231. Unfortunately, it was also revealed that this formulation may yield flexible (i.e. soft elastic) I25