a simple four-node quadrilateral finite element for plates

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m __ ELI JOURNAL OF COMPUTATIONAL AND APPLIED MATHEMATICS ELSEVIER Journal of Computational and Applied Mathema tics 50 (1994) 361-3 73 A simple four-node quadrilateral finite element for plates Wilfrie d B. K&zig a,*, Jian-Wu Zhang b a nstitute or Statics Dy namics, Ruhr University, D-44780 Bochum, Germany b Depart ment of Mechanical Engineering, Jiao Tong University, Shanghai 200030, China Received 14 May 1992; revised 22 December 1992 bstract A simple and accurate four-node quadrilate ral finite element based on the Mindlin plate theory and Kirchhoff constraints is presented for general thin plate bending applications. The derivation of the element stiffness properties is straightforward, starting with a specified eight-node interpolation; usual discrete Kirchhoff ( DK) constraints are employed to constrain out the four midside nodes of the element. The present resulting DK element passes patch tests with elements of arbitrary and even highly distorted mesh types. Numerical studies of the element convergence behaviours are undertaken for various plate bending problems so far investigated. It is indicated from comparative examples that fairly good convergence characteristics have been achieved. Key words: Plate theory; Mindlin-Reissner theory with discrete Kirchhoff constraints; Finite-element method 1 I ntrod uct i on The Mindlin-Reissner theory based fi nite el ements lock or become overstiff in thin plate limit when they are used to model plate structures. The locking p henom enon arises due to incompatible deformation mode s introduced in the indepen dent interpolation for transverse and rotational displacements under the Kirchhoff shear constraint. Much attention has been focused in the past decade s on investigation of shear locking and related numerical difficulties. Particular approaches such as reduced/selective integration and kinematic mode control for alleviating these problems may be found in [1,4,9 ,15,16 ]. With regard to the inconsistency of Kirchhoff mode s there has been considerable effort in the development of improved quadrilater al finite e lements based o n the Mindlin plate theory for bending analysis of thin plates. Hughes and Tezduyar [lo] developed a quadrilatera l element using bilinear Lagrangian functions for nodal rotational displacements. Four- and nine-node quadrilateral elements were proposed in [3,8] , both applying substitutive transverse * Corresponding author. 0377-0427/94/ 07.00 0 1994 Elsevier Science B.V. All rights reserved SSDI 0377.0427(93)E0118-6

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