Accuracy of refined finite elements for laminated plate analysis

Abstract This paper compares and evaluates various plate finite elements to analyze the static response of laminated plate when varying the thickness ratio, orthotropic ratio and the stacking sequence of the lay-out. Plate elements were created from different theoretical assumptions for the displacement distribution along the thickness direction. Classical (Kirchhoff and Reissner–Mindlin), known refined (Reddy, Pandya, and Kant), and other higher-order displacement fields were then implemented up-to fourth-order expansion. Following this the Carrera Unified Formulation was used to derive finite element matrices in terms of fundamental nuclei which consist of 3 × 3 arrays. The accuracy of a given plate element was established in terms of the error vs. thickness-to-length parameter. A significant number of finite elements for plates were implemented and compared using displacement and stress variables for various plate problems. In this paper concluding results are presented as number and type of required displacement variables vs. accuracy on a given stress/displacement parameter. These diagrams are labelled as Best Plate Theories BPT curves, which differ by changing geometry, material data and lay-out as well as loading and boundary conditions.

[1]  É. Grigolyuk,et al.  State of the art of the theory of multilayer shells , 1972 .

[2]  E. Carrera.  Historical review of Zig-Zag theories for multilayered plates and shells , 2003 .

[3]  A. Leissa.  A Review of Laminated Composite Plate Buckling , 1987 .

[4]  Erasmo Carrera,et al.  Classical and advanced multilayered plate elements based upon PVD and RMVT. Part 2: Numerical implementations , 2002 .

[5]  Erasmo Carrera,et al.  Classical and advanced multilayered plate elements based upon PVD and RMVT. Part 1: Derivation of finite element matrices , 2002 .

[6]  E. Carrera.  Developments, ideas, and evaluations based upon Reissner’s Mixed Variational Theorem in the modeling of multilayered plates and shells , 2001 .

[7]  Erasmo Carrera,et al.  Transverse Normal Stress Effects in Multilayered Plates , 1999 .

[8]  Ya. M. Grigorenko,et al.  Solution of problems and analysis of the stress—Strain state of nonuniform anisotropic shells (Survey) , 1997 .

[9]  E. Reissner.  The effect of transverse shear deformation on the bending of elastic plates , 1945 .

[10]  Ahmed K. Noor,et al.  Assessment of Shear Deformation Theories for Multilayered Composite Plates , 1989 .

[11]  V. Vasil’ev,et al.  On Refined Theories of Beams, Plates, and Shells , 1992 .

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

[13]  Tarun Kant,et al.  Finite element analysis of laminated composite plates using a higher-order displacement model , 1988 .

[14]  Erasmo Carrera,et al.  Analysis of thickness locking in classical, refined and mixed multilayered plate theories , 2008 .

[15]  Mohamed Samir Hefzy,et al.  Review of Knee Models , 1988 .

[16]  Tarun Kant,et al.  Numerical analysis of thick plates , 1982 .

[17]  Ahmed K. Noor,et al.  Computational Models for Sandwich Panels and Shells , 1996 .

[18]  G. Kirchhoff,et al.  Über das Gleichgewicht und die Bewegung einer elastischen Scheibe. , 1850 .

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

[20]  S A Ambartsumian.  Fragments of the Theory of Anisotropic Shells , 1990 .

[21]  Holm Altenbach,et al.  Theories for laminated and sandwich plates , 1998 .

[22]  Erasmo Carrera,et al.  MITC technique extended to variable kinematic multilayered plate elements , 2010 .

[23]  Mohamad S. Qatu,et al.  Vibration of Laminated Shells and Plates , 2004 .

[24]  Erasmo Carrera,et al.  Multilayered Finite Plate Element based on Reissner Mixed Variational Theorem. Part I: Theory., , 2002 .

[25]  E. Carrera.  Theories and Finite Elements for Multilayered Plates and Shells:A Unified compact formulation with numerical assessment and benchmarking , 2003 .

[26]  R. D. Mindlin,et al.  Influence of rotary inertia and shear on flexural motions of isotropic, elastic plates , 1951 .

[27]  Rakesh K. Kapania,et al.  A Review on the Analysis of Laminated Shells Virginia Polytechnic Institute and State University , 1989 .

[28]  Erasmo Carrera,et al.  Multilayered Finite Plate Element based on Reissner Mixed Variational Theorem. Part II: Numerical Analysis, , 2002 .

[29]  J. Reddy,et al.  THEORIES AND COMPUTATIONAL MODELS FOR COMPOSITE LAMINATES , 1994 .

[30]  Erasmo Carrera,et al.  Analysis of thickness locking in classical, refined and mixed theories for layered shells , 2008 .

[31]  S. A. Lur'e,et al.  Kinematic models of refined theories concerning composite beams, plates, and shells , 1996 .

[32]  Erasmo Carrera,et al.  Guidelines and Recommendations on the Use of Higher Order Finite Elements for Bending Analysis of Plates , 2011 .

[33]  A. Noor,et al.  Assessment of Computational Models for Multilayered Composite Shells , 1990 .

[34]  Erasmo Carrera,et al.  Guidelines and Recommendations to Construct Theories for Metallic and Composite Plates , 2010 .

[35]  Rakesh K. Kapania,et al.  Recent advances in analysis of laminated beams and plates. Part I - Sheareffects and buckling. , 1989 .

[36]  S. A. Ambartsumyan,et al.  Theory of anisotropic shells , 1964 .

[37]  Ya. M. Grigorenko.  Approaches to the numerical solution of linear and nonlinear problems in shell theory in classical and refined formulations , 1996 .

[38]  Tarun Kant,et al.  An unsymmetric FRC laminate C° finite element model with 12 degrees of freedom per node , 1988 .

[39]  Erasmo Carrera,et al.  Mixed layer-wise models for multilayered plates analysis , 1998 .

[40]  Erasmo Carrera,et al.  Evaluation of Layerwise Mixed Theories for Laminated Plates Analysis , 1998 .

[41]  G. Jemielita,et al.  On kinematical assumptions of refined theories of plates : a survey , 1990 .