INTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN ENGINEERING Int. J. Numer. Meth. Engng 2001; 51:919–942 An improved three-node hybrid-mixed element for Mindlin= Reissner plates Rezak Ayad 1;*;† , Alain Rigolot 2 and Nabyl Talbi 3 1 Groupe M ecanique; Mat eriaux et Structure; EA 2617; UFR Sciences; Universit e de Reims Champagne-Ardenne; BP1039; 51687 Reims Cedex 2; France 2 Laboratoire de Mod elisation en M ecanique (LMM); UMR-CNRS No 7607; Universit e Pierre et Marie Curie (Paris VI); 4 place Jussieu; Tour 66; 75252 Paris Cedex 05; France 3 D epartement de G enie Maritime; Universit e des Sciences Technologiques d’Oran; 31000 Oran; Alg erie SUMMARY Based on the mixed shear projected (MiSP) approach [6], an enhanced bending approximation for homogeneous isotropic plates is presented. Some hard benchmark tests, as skew plate (30 ◦ ) problem for instance, have often shown poor convergence when low order elements (3- or 4-node element) are developed using linear approximations for kinematic variables. To put right this weakness, we pro- pose a high-order interpolation for rotational dofs which results in more rich bending curvatures. The mid-side rotational nodes are eliminated using a combination of local discrete kinematic and consti- tutive Mindlin hypothesises. The derived 3-node triangular element, called MiSP3+, is free of shear locking and passes all patch-tests for thick and thin plates in an arbitrary mesh. Copyright ? 2001 John Wiley & Sons, Ltd. KEY WORDS: nite element; plate bending; transversal shear; Reissner–Mindlin; shear locking; hybrid-mixed 1. INTRODUCTION 1.1. Synthesis of triangular nite elements for plate bending= shear problems During the last three decades, signicant eorts have been oriented to the formulation of sim- ple and ecient triangular and quadrilateral plate and shell nite elements based on Reissner– Mindlin theory [1; 2], which are valid for static and dynamic, linear and non-linear analysis. A synthesis of the successive developments is made (see References [3–9]). The nite ele- ments derived from the rst-order theory take into account the eects of transversal shear and require only C 0 continuity for the three independent kinematic variables: the transversal dis- placement w and the rotations of the normal vector to the plate middle surface  x ; y . These elements, which are based on the displacement formulation using the exact integration for ∗ Correspondence to: Rezak Ayad, Ecole Sup d’Ingenieurs en Emballage et Con., Esplanade Roland Garros, BP1029, 51686 Reims, France † E-mail: rezak.ayad@univ-reims.fr Received 20 March 2000 Copyright ? 2001 John Wiley & Sons, Ltd. Revised 26 September 2000