A linear smoothed quadratic finite element for the analysis of laminated composite Reissner–Mindlin plates

Abstract It is well known that the high-order elements have significantly improved the accuracy of solutions in the traditional finite element analysis, but the performance of high-order elements is restricted by the shear-locking and distorted meshes for the plate problems. In this paper, a linear smoothed eight-node Reissner-Mindlin plate element (Q8 plate element) based on the first order shear deformation theory is developed for the static and free vibration analysis of laminated composite plates, the computation of the interior derivatives of shape function and isoparametric mapping can be removed. The strain matrices are modified with a linear smoothing technique by using the divergence theorem between the nodal shape functions and their derivatives in Taylor’s expansion. Moreover, the first order Taylor’s expansion is also employed for the construction of stiffness matrix to satisfy the linear strain distribution. Several numerical examples indicate that the novel Q8 plate element has good performance to alleviate the shear-locking phenomenon and improve the quality of the solutions with distorted meshes.

[1]  Guirong Liu,et al.  A face‐based smoothed finite element method (FS‐FEM) for 3D linear and geometrically non‐linear solid mechanics problems using 4‐node tetrahedral elements , 2009 .

[2]  Liviu Librescu,et al.  Analysis of symmetric cross-ply laminated elastic plates using a higher-order theory. II - Buckling and free vibration , 1988 .

[3]  Olivier Polit,et al.  A new eight‐node quadrilateral shear‐bending plate finite element , 1994 .

[4]  Dean Hu,et al.  A fully smoothed finite element method for analysis of axisymmetric problems , 2016 .

[5]  Hung Nguyen-Xuan,et al.  Static and free vibration analyses of composite and sandwich plates by an edge-based smoothed discrete shear gap method (ES-DSG3) using triangular elements based on layerwise theory , 2014 .

[6]  Sohichi Hirose,et al.  NURBS-based isogeometric analysis of buckling and free vibration problems for laminated composites plates with complicated cutouts using a new simple FSDT theory and level set method , 2016 .

[7]  Sohichi Hirose,et al.  A cutout isogeometric analysis for thin laminated composite plates using level sets , 2015 .

[8]  J. N. Reddy,et al.  Buckling and vibration of laminated composite plates using various plate theories , 1989 .

[9]  K. Y. Dai,et al.  A mesh-free method for static and free vibration analysis of shear deformable laminated composite plates , 2004 .

[10]  J. N. Reddy,et al.  Free vibration of antisymmetric, angle-ply laminated plates including transverse shear deformation by the finite element method , 1979 .

[11]  Xiangyang Cui,et al.  A Smoothed Finite Element Method (SFEM) for Linear and Geometrically Nonlinear Analysis of Plates and Shells , 2008 .

[12]  Nam Mai-Duy,et al.  Free vibration analysis of laminated plate/shell structures based on FSDT with a stabilized nodal-integrated quadrilateral element , 2008 .

[13]  Stéphane Bordas,et al.  Linear smoothed polygonal and polyhedral finite elements , 2017 .

[14]  K. M. Liew,et al.  SOLVING THE VIBRATION OF THICK SYMMETRIC LAMINATES BY REISSNER/MINDLIN PLATE THEORY AND THEp-RITZ METHOD , 1996 .

[15]  T. Belytschko,et al.  Consistent element‐free Galerkin method , 2014 .

[16]  Chang-Koon Choi,et al.  Coupled use of reduced integration and non‐conforming modes in quadratic Mindlin plate element , 1989 .

[17]  Stéphane Bordas,et al.  A fully smoothed XFEM for analysis of axisymmetric problems with weak discontinuities , 2017 .

[18]  Silvia Bertoluzza,et al.  A high order collocation method for the static and vibration analysis of composite plates using a first-order theory , 2009 .

[19]  Hung Nguyen-Xuan,et al.  AN EDGE-BASED SMOOTHED FINITE ELEMENT METHOD FOR ANALYSIS OF LAMINATED COMPOSITE PLATES , 2013 .

[20]  K. M. Liew,et al.  Vibration analysis of symmetrically laminated plates based on FSDT using the moving least squares differential quadrature method , 2003 .

[21]  Gui-Rong Liu,et al.  a Mesh-Free Method for Static and Free Vibration Analyses of Thin Plates of Complicated Shape , 2001 .

[22]  Hung Nguyen-Xuan,et al.  A cell‐based smoothed discrete shear gap method using triangular elements for static and free vibration analyses of Reissner–Mindlin plates , 2012 .

[23]  Gautam Dasgupta,et al.  Integration within Polygonal Finite Elements , 2003 .

[24]  Hung Nguyen-Xuan,et al.  Isogeometric analysis of laminated composite and sandwich plates using a layerwise deformation theory , 2013 .

[25]  J. N. Reddy,et al.  Analysis of laminated composite plates using a higher‐order shear deformation theory , 1985 .

[26]  J. Whitney,et al.  The Effect of Transverse Shear Deformation on the Bending of Laminated Plates , 1969 .

[27]  Trung Nguyen-Thoi,et al.  A cell-based smoothed discrete shear gap method (CS-FEM-DSG3) using layerwise theory based on the C0-HSDT for analyses of composite plates , 2014 .

[28]  William Portilho de Paiva,et al.  Modal analysis of anisotropic plates using the boundary element method , 2011 .

[29]  Guirong Liu,et al.  An edge-based smoothed finite element method (ES-FEM) for static, free and forced vibration analyses of solids , 2009 .

[30]  H. Nguyen-Xuan,et al.  A smoothed finite element method for plate analysis , 2008 .

[31]  E. Carrera.  Theories and finite elements for multilayered, anisotropic, composite plates and shells , 2002 .

[32]  Y. Stavsky,et al.  Elastic wave propagation in heterogeneous plates , 1966 .

[33]  Mo Shing Cheung,et al.  Static and vibration analysis of anisotropic composite laminates by finite strip method , 1993 .

[34]  J. Reddy,et al.  Stability and vibration of isotropic, orthotropic and laminated plates according to a higher-order shear deformation theory , 1985 .

[35]  E. Hinton,et al.  A study of quadrilateral plate bending elements with ‘reduced’ integration , 1978 .

[36]  Xu Han,et al.  Selective smoothed finite element methods for extremely large deformation of anisotropic incompressible bio‐tissues , 2014 .

[37]  T. Q. Bui,et al.  An efficient meshfree method for vibration analysis of laminated composite plates , 2011 .

[38]  Chang-Koon Choi,et al.  Improvement of quadratic finite element for mindlin plate bending , 1992 .

[39]  Gui-Rong Liu,et al.  A smoothed finite element method for analysis of anisotropic large deformation of passive rabbit ventricles in diastole , 2015, International journal for numerical methods in biomedical engineering.

[40]  K. Y. Dai,et al.  Static and free vibration analysis of laminated composite plates using the conforming radial point interpolation method , 2008 .

[41]  Stéphane Bordas,et al.  Isogeometric analysis of laminated composite and sandwich plates using a new inverse trigonometric shear deformation theory , 2014 .

[42]  Rezak Ayad,et al.  An improved four‐node hybrid‐mixed element based upon Mindlin's plate theory , 2002 .

[43]  Jiun-Shyan Chen,et al.  Non‐linear version of stabilized conforming nodal integration for Galerkin mesh‐free methods , 2002 .

[44]  Chang-Koon Choi,et al.  Quadratic NMS Mindlin‐plate‐bending element , 1999 .

[45]  Raghu Natarajan,et al.  Finite element analysis of laminated composite plates , 1979 .

[46]  Stéphane Bordas,et al.  On the performance of strain smoothing for quadratic and enriched finite element approximations (XFEM/GFEM/PUFEM) , 2011 .

[47]  M. H. Aliabadi,et al.  Fracture mechanics analysis of anisotropic plates by the boundary element method , 1993 .

[48]  Irwan Katili,et al.  A new discrete Kirchhoff-Mindlin element based on Mindlin-Reissner plate theory and assumed shear strain fields. I - An extended DKT element for thick-plate bending analysis. II - An extended DKQ element for thick-plate bending analysis , 1993 .

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

[50]  A. W. Leissa,et al.  Analysis of Heterogeneous Anisotropic Plates , 1969 .

[51]  K. Bathe,et al.  A continuum mechanics based four‐node shell element for general non‐linear analysis , 1984 .

[52]  Renato Natal Jorge,et al.  Free vibration analysis of symmetric laminated composite plates by FSDT and radial basis functions , 2005 .

[53]  Guowei Ma,et al.  A novel integration scheme for solution of consistent mass matrix in free and forced vibration analysis , 2016 .

[54]  Hung Nguyen-Xuan,et al.  Static, free vibration, and buckling analysis of laminated composite Reissner–Mindlin plates using NURBS‐based isogeometric approach , 2012 .

[55]  S Natarajan,et al.  Linear smoothed extended finite element method , 2017, 1701.03997.

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

[57]  Irwan Katili,et al.  A new discrete Kirchhoff-Mindlin element based on Mindlin–Reissner plate theory and assumed shear strain fields—part II: An extended DKQ element for thick-plate bending analysis , 1993 .

[58]  K. M. Liew,et al.  Moving least squares differential quadrature method and its application to the analysis of shear deformable plates , 2003 .

[59]  Xikui Li,et al.  Quadratically consistent one-point (QC1) quadrature for meshfree Galerkin methods , 2012 .

[60]  Warna Karunasena,et al.  Buckling and vibration analysis of laminated composite plate/shell structures via a smoothed quadrilateral flat shell element with in-plane rotations , 2011 .

[61]  E. Reissner,et al.  Bending and Stretching of Certain Types of Heterogeneous Aeolotropic Elastic Plates , 1961 .

[62]  Jiun-Shyan Chen,et al.  A stabilized conforming nodal integration for Galerkin mesh-free methods , 2001 .

[63]  K. Y. Dai,et al.  A Smoothed Finite Element Method for Mechanics Problems , 2007 .

[64]  Hung Nguyen-Xuan,et al.  Isogeometric Analysis of Laminated Composite Plates Using the Higher-Order Shear Deformation Theory , 2015 .

[65]  Chien H. Thai,et al.  Analysis of laminated composite plates using higher-order shear deformation plate theory and node-based smoothed discrete shear gap method , 2012 .

[66]  Ted Belytschko,et al.  Second‐order accurate derivatives and integration schemes for meshfree methods , 2012 .

[67]  J. Reddy.  Mechanics of laminated composite plates and shells : theory and analysis , 1996 .

[68]  Rolf Stenberg,et al.  A stable bilinear element for the Reissner-Mindlin plate model , 1993 .

[69]  Guirong Liu,et al.  A node-based smoothed finite element method (NS-FEM) for upper bound solutions to solid mechanics problems , 2009 .