Axiomatic/asymptotic PVD/RMVT-based shell theories for free vibrations of anisotropic shells using an advanced Ritz formulation and accurate curvature descriptions

Abstract The Hierarchical Trigonometric Ritz Formulation (HTRF) has earlier been successfully developed for plates [1] , [2] , [3] , [4] and shells [5] using the Principle of Virtual Displacements (PVD). In this paper the HTRF is significantly extended with the help of Reissner’s Mixed Variational Theorem (RMVT) so as to deal with the free vibrations of doubly-curved anisotropic laminated composite shells. The interlaminar equilibrium of the transverse normal and shear stresses is fulfilled a priori by exploiting the use of Lagrange multipliers. The transverse normal and shear stresses thus become primary variables within the formulation and are always modeled with a Layer-Wise kinematics description. Equivalent Single Layer, Zig-Zag and Layer-Wise approaches are instead efficiently used for the displacement primary variables. Appropriate expansion orders for each displacement or stress unknown are selected depending on the required accuracy and the computational cost. Axiomatic/asymptotic shell theories are then developed by virtue of a deep study on the effectiveness of each term both in the displacements and in the transverse stresses fields. Next exact and/or accurately approximated curvature descriptions are taken into account. Cylindrical, spherical and hyperbolic paraboloidal shells are investigated. The proposed advanced quasi-3D shell models are assessed by comparison with 3D elasticity solutions.

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

[2]  K. Washizu.  Variational Methods in Elasticity and Plasticity , 1982 .

[3]  Mohamad S. Qatu,et al.  Static and vibration analyses of thick deep laminated cylindrical shells using 3D and various shear deformation theories , 2012 .

[4]  E. Carrera,et al.  Radial Basis Functions collocation for the bending and free vibration analysis of laminated plates using the Reissner-Mixed Variational Theorem , 2013 .

[5]  L. Demasi.  ∞6 Mixed plate theories based on the Generalized Unified Formulation. Part V: Results , 2009 .

[6]  C. W. Bert.  Damping of Composite and Sandwich Panels: Part II , 1976 .

[7]  E. Viola,et al.  General higher-order equivalent single layer theory for free vibrations of doubly-curved laminated composite shells and panels , 2013 .

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

[9]  A. E. H. Love,et al.  The Small Free Vibrations and Deformation of a Thin Elastic Shell , 1887 .

[10]  J. N. Reddy,et al.  Exact Solutions of Moderately Thick Laminated Shells , 1984 .

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

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

[13]  Anthony N. Palazotto,et al.  Transverse shear deformation in orthotropic cylindrical pressure vessels using a higher-order shear theory , 1989 .

[14]  Francesco Tornabene,et al.  2-D GDQ solution for free vibrations of anisotropic doubly-curved shells and panels of revolution , 2011 .

[15]  Ekkehard Ramm,et al.  Hybrid stress formulation for higher-order theory of laminated shell analysis , 1993 .

[16]  Erasmo Carrera,et al.  CZ° requirements—models for the two dimensional analysis of multilayered structures , 1997 .

[17]  Mohamad S. Qatu,et al.  Vibration of doubly curved shallow shells with arbitrary boundaries , 2012 .

[18]  L. Donnell,et al.  Stability of Thin-Walled Tubes Under Torsion , 1934, Journal of Fluids Engineering.

[19]  E. Carrera,et al.  Advanced variable kinematics Ritz and Galerkin formulations for accurate buckling and vibration analysis of anisotropic laminated composite plates , 2011 .

[20]  W. Flügge.  Stresses in Shells , 1960 .

[21]  E. Carrera,et al.  Free vibration analysis of sandwich plates with anisotropic face sheets in thermal environment by using the hierarchical trigonometric Ritz formulation , 2013 .

[22]  Erasmo Carrera,et al.  Multilayered Shell Theories Accounting for Layerwise Mixed Description, Part 2: Numerical Evaluations , 1999 .

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

[24]  Wilhelm Flügge,et al.  Statik und Dynamik der Schalen , 1962 .

[25]  E. Carrera,et al.  Accurate free vibration analysis of thermo-mechanically pre/post-buckled anisotropic multilayered plates based on a refined hierarchical trigonometric Ritz formulation , 2013 .

[26]  J. Whitney,et al.  A Refined Theory for Laminated Anisotropic, Cylindrical Shells , 1974 .

[27]  E. Reissner.  On a Certain Mixed Variational Theorem and on Laminated Elastic Shell Theory , 1987 .

[28]  E. Carrera,et al.  Thermo-Mechanical Buckling Analysis of Anisotropic Multilayered Composite and Sandwich Plates by Using Refined Variable-Kinematics Theories , 2013 .

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

[30]  J. N. Reddy,et al.  General two-dimensional theory of laminated cylindrical shells , 1990 .

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

[32]  Erasmo Carrera,et al.  Analysis of laminated doubly-curved shells by a layerwise theory and radial basis functions collocation, accounting for through-the-thickness deformations , 2011 .

[33]  C. C. Chao,et al.  COMPARISON OF NATURAL FREQUENCIES OF LAMINATES BY 3-D THEORY, PART II: CURVED PANELS , 2000 .

[34]  S. T. Gulati,et al.  Effects of Anisotropy in Axisymmetric Cylindrical Shells , 1967 .

[35]  A. Love.  A treatise on the mathematical theory of elasticity , 1892 .

[36]  Frithiof I. Niordson,et al.  Theory of Thin Shells , 1969 .

[37]  S. Bertoluzza,et al.  A wavelet collocation approach for the analysis of laminated shells , 2011 .

[38]  E. Reissner.  On a certain mixed variational theorem and a proposed application , 1984 .

[39]  W. T. Koiter,et al.  The Theory of Thin Elastic Shells , 1961 .

[40]  E. Carrera,et al.  Selection of appropriate multilayered plate theories by using a genetic like algorithm , 2012 .

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

[42]  George Biddell Airy,et al.  IV. On the strains in the Interior of beams , 1863, Philosophical Transactions of the Royal Society of London.

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

[44]  Erasmo Carrera,et al.  Axiomatic/asymptotic evaluation of multilayered plate theories by using single and multi-points error criteria , 2013 .

[45]  E. Reissner.  On a mixed variational theorem and on shear deformable plate theory , 1986 .

[46]  E. Carrera,et al.  Coupled thermoelastic effect in free vibration analysis of anisotropic multilayered plates and FGM plates by using a variable-kinematics Ritz formulation , 2014 .

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

[48]  Teh-Min Hsu,et al.  A theory of laminated cylindrical shells consisting of layers of orthotropic laminae , 1970 .

[49]  C. W. Bert.  Literature Review : Damping of Composite and Sandwich Panels: Part I , 1976 .

[50]  Luciano Demasi.  Refined multilayered plate elements based on Murakami zig–zag functions , 2005 .

[51]  Jack R. Vinson,et al.  Laminated Transversely Isotropic Cylindrical Shells , 1971 .

[52]  Erasmo Carrera,et al.  Multilayered Shell Theories Accounting for Layerwise Mixed Description, Part 1: Governing Equations , 1999 .

[53]  E. Carrera.  On the use of the Murakami's zig-zag function in the modeling of layered plates and shells , 2004 .

[54]  Erasmo Carrera,et al.  A unified formulation for finite element analysis of piezoelectric adaptive plates , 2006 .

[55]  E. Carrera.  A Reissner’s Mixed Variational Theorem Applied to Vibration Analysis of Multilayered Shell , 1999 .

[56]  Erasmo Carrera,et al.  Advances in the Ritz formulation for free vibration response of doubly-curved anisotropic laminated composite shallow and deep shells , 2013 .

[57]  Free vibrations of thick, layered cylinders having finite length with various boundary conditions , 1972 .