Contpurers 6 Sfn~~urcs Vol. 25,No. 6. pa. 811-824, 1987 p&cd zyxwvutsrqponmlkjihgfedcbaZYXWVUTSRQPONMLKJIHGFEDCBA in G re a tBrita in. 0045.794987 s3.00 + 0.00 Q 1987 F’qmoa Jourr&Ltd. zyxwvutsrqp A SIMPLE AND EFFICIENT SHEAR-FLEXIBLE PLATE BENDING ELEMENT REAZ A. CHAUDHURI Department of Civil Engineering, University of Utah, Salt Lake City, UT 84112, U.S.A. zyxwvutsrqponmlkjihg (Received 30 June 1986) Abtract-A shear-flexible triangular element formulation, which utilizes an assumed quadratic displace- ment potential energy approach and is numerically integrated using Gauss quadrature, is presented. The Reissner/Mindlin hypothesis of constant cross-sectional warping is directly applied to the three- dimensional elasticity theory to obtain a moderately thick-plate theory or constant shear-angle theory (CST), wherein the middle surface is no longer considered to be the reference surface and the two rotations are replaced by the two in-plane displacements as nodal variables. The resulting finite-element possesses 18 degrees of freedom (DGF). Numerical results are obtained for two different numerical integration schemes and a wide range of meshes and span-to-thickness ratios. These, when compared with avaiiabk exact, series or finite-element solutions, demonstrate accuracy and rapid convergence characteristics of the present element. This is especially true in the case of thin to very thin plates, when the present element, used in conjunction with the reduced integration scheme, outperforms its counterpart, based on discrete Kirchhoff constraint theory (DKT). NOTATION length of a square plate strain-displacement relation at the reference sur- face of the mth element elastic stiffness; i, j = 1, . . . (5 integrated (through thickness) material matrix nodal displacement vector of the mth element Young’s modulus of an isotropic material consistent load vector of the mth element stiffness matrix of the Mth element uniform pressure loading applied pressure on the top surface of the mth element thickness of a plate strain energy stored in the plate displacement components, i = 1,2,3, at a point inside the plate displacement components, i = 1,2,3, at a point on the reference surface nodal dispiacement components, i = 1,2,3 potential due to external conservative forces Cartesian coordinates, i = 1.2.3 engineering shear straincomponents, i,j = 1,2,3; i #j, at a point inside the plate engineering shearing strains, i,j = I, 2,3; i #j, at a point on the reference surface reference surface area of the m th element tensorial strain components, i,j = 1,2,3 at a point inside the plate tensorial strain components, i,j = 1,2,3 at a point on the reference surface Poisson’s ratio of an isotropic material total potential energy functional normal stress components, i,j = I, 2. i -j at a point inside the plate shear stress componmts, i.j = 1,2,3, i #j. at a point inside the plate applied distributed normal forces over the bot- tom (reference) and the top surface of the plate respectively shape function vector INTRODUCTION A simple but efficient plate bending element has been almost as elusive to modem day researchers as the philosopher’s stone to the ancients. No particular plate bending element has yet emerged as the ‘best element’ and that this is not due to lack of efforts has been amply demonstrated by a recent review paper [ 11, which has catalogued 88 (27 rectangular, 43 triangular, 17 quadrilateral and one polygonal) plate bending elements and 154publications on the subject. The majority of these elements are based on Kirchhoff hypothesis, which neglects transverse shear deform- ation altogether. However, more zyxwvutsrqponmlkjihgfedcbaZYXW recently, elements, which-while admitting the Reissner/Mindlin hy- pothesis of constant transverse shear deformation for moderately thick plates-also satisfy the Kirchhoff hypothesis in the limiting case of thin plates, by employment of one of such mechanisms as reduced integration [2,3], penalty methods [4, SJ, energy bal- ancing [a], mixed formulation [I, discrete Kirchhoff constraints [8,9] and so forth, have become popular. The aforementioned elements, without a single excep- tion, are represented by their middle surfaces, which necessitates admitting rotations as nodal variables. Representation of a plate by its middle surface, where the in-plane displacements vanish, was a necessity to early investigations [lo] in the classical plate theory (CRT), based on Kirchhoff hypothesis, seeking closed-form solutions. This is because such a repre- sentation helped reduce the number of independent variables from three (u,, i = 1,2,3) to one (i.e. u,), while increasing the order of the partial differential equation(s) from two to four. This practice was 817