On elimination of shear locking in the element‐free Galerkin method

In this study, a method for completely eliminating the presence of transverse shear locking in the application of the element-free Galerkin method (EFGM) to shear-deformable beams and plates is presented. The matching approximation fields concept of Donning and Liu has shown that shear locking effects may be prevented if the approximate rotation fields are constructed with the innate ability to match the approximate slope (first derivative of displacement) fields and is adopted. Implementation of the matching fields concept requires the computation of the second derivative of the shape functions. Thus, the shape functions for displacement fields, and therefore the moving least-squares (MLS) weight function, must be at least C1 continuous. Additionally, the MLS weight functions must be chosen such that successive derivatives of the MLS shape function have the ability to exactly reproduce the functions from which they were derived. To satisfy these requirements, the quartic spline weight function possessing C2 continuity is used in this study. To our knowledge, this work is the first attempt to address the root cause of shear locking phenomenon within the framework of the element-free Galerkin method. Several numerical examples confirm that bending analyses of thick and thin beams and plates, based on the matching approximation fields concept, do not exhibit shear locking and provide a high degree of accuracy for both displacement and stress fields. Copyright © 2001 John Wiley & Sons, Ltd.

[1]  Mark A Fleming,et al.  Meshless methods: An overview and recent developments , 1996 .

[2]  R. Cook,et al.  Concepts and Applications of Finite Element Analysis , 1974 .

[3]  Richard H. Macneal,et al.  A simple quadrilateral shell element , 1978 .

[4]  T. Belytschko,et al.  Element‐free Galerkin methods , 1994 .

[5]  J. Oden,et al.  H‐p clouds—an h‐p meshless method , 1996 .

[6]  Thomas J. R. Hughes,et al.  A LARGE DEFORMATION FORMULATION FOR SHELL ANALYSIS BY THE FINITE ELEMENT METHOD , 1981 .

[7]  Wing Kam Liu,et al.  Wavelet and multiple scale reproducing kernel methods , 1995 .

[8]  P. Lancaster,et al.  Surfaces generated by moving least squares methods , 1981 .

[9]  T. Belytschko,et al.  Analysis of thin shells by the Element-Free Galerkin method , 1996 .

[10]  T. Belytschko,et al.  A C0 triangular plate element with one‐point quadrature , 1984 .

[11]  Wing Kam Liu,et al.  MESHLESS METHODS FOR SHEAR-DEFORMABLE BEAMS AND PLATES , 1998 .

[12]  Wing Kam Liu,et al.  Reproducing kernel particle methods , 1995 .

[13]  T. Hughes.  Generalization of selective integration procedures to anisotropic and nonlinear media , 1980 .

[14]  Ted Belytschko,et al.  Analysis of thin plates by the element-free Galerkin method , 1995 .

[15]  Robert L. Taylor,et al.  Linked interpolation for Reissner‐Mindlin plate elements: Part II—A simple triangle , 1993 .

[16]  Martin Cohen,et al.  The “heterosis” finite element for plate bending , 1978 .

[17]  O. C. Zienkiewicz,et al.  Reduced integration technique in general analysis of plates and shells , 1971 .

[18]  K. Bathe,et al.  A four‐node plate bending element based on Mindlin/Reissner plate theory and a mixed interpolation , 1985 .

[19]  Thomas J. R. Hughes,et al.  A simple and efficient finite element for plate bending , 1977 .

[20]  T. Hughes,et al.  Finite Elements Based Upon Mindlin Plate Theory With Particular Reference to the Four-Node Bilinear Isoparametric Element , 1981 .

[21]  Hirohisa Noguchi,et al.  Element free analyses of shell and spatial structures , 2000 .

[22]  E. Hinton,et al.  A nine node Lagrangian Mindlin plate element with enhanced shear interpolation , 1984 .

[23]  Worsak Kanok-Nukulchai,et al.  A simple and efficient finite element for general shell analysis , 1979 .

[24]  O. C. Zienkiewicz,et al.  Linked interpolation for Reissner‐Mindlin plate elements: Part III—An alternative quadrilateral , 1994 .

[25]  Wing Kam Liu,et al.  Stress projection for membrane and shear locking in shell finite elements , 1985 .

[26]  S. Timoshenko,et al.  THEORY OF PLATES AND SHELLS , 1959 .

[27]  T. Belytschko,et al.  A new implementation of the element free Galerkin method , 1994 .

[28]  C. Duarte,et al.  hp-Clouds in Mindlin's thick plate model , 2000 .