Advances in the Ritz formulation for free vibration response of doubly-curved anisotropic laminated composite shallow and deep shells

The hierarchical trigonometric Ritz formulation (HTRF) developed in the framework of the Carrera unified formulation (CUF), for the first time, is extended to shell structures in order to cope with the free vibration response of doubly-curved anisotropic laminated composite shells. The HTRF is the outcome of the combination of advanced shell theories hierarchically generated via the CUF with the trigonometric Ritz method. It is based on so-called Ritz fundamental primary nuclei obtained by virtue of the principle of virtual displacements (PVD). The PVD is further used to derive the governing differential equations and natural boundary conditions. Donnell–Mushtari’s shallow shell-type equations are given as a particular condition. Several shell geometries accounting for thin and thick shallow cylindrical and spherical shells, deep cylindrical shells and hollow circular cylindrical shells, with cross-ply and angle-ply staking sequences are investigated. CUF-based refined shell models are assessed by comparison with the 3D elasticity solution. Convergence and accuracy of the presented formulation are examined. The effects of significant parameters such as stacking sequence, length-to-thickness ratio and radius-to-length ratio on the circular frequency parameters are discussed.

[1]  L. Demasi.  Partially Zig-Zag Advanced Higher Order Shear Deformation Theories Based on the Generalized Unified Formulation , 2012 .

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

[3]  L. Demasi.  ∞6 Mixed plate theories based on the Generalized Unified Formulation.: Part II: Layerwise theories , 2009 .

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

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

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

[7]  Jianqiao Ye,et al.  Three-dimensional vibration of laminated cylinders and cylindrical panels with symmetric or antisymmetric cross-ply lay-up , 1994 .

[8]  Luciano Demasi,et al.  ∞6 Mixed plate theories based on the Generalized Unified Formulation. Part I: Governing equations , 2009 .

[9]  Bo Liu,et al.  Exact characteristic equations for free vibrations of thin orthotropic circular cylindrical shells , 2012 .

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

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

[12]  L. Demasi.  ∞6 Mixed plate theories based on the Generalized Unified Formulation. Part III: Advanced mixed high order shear deformation theories , 2009 .

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

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

[15]  Erasmo Carrera,et al.  Elastodynamic Behavior of Relatively Thick, Symmetrically Laminated, Anisotropic Circular Cylindrical Shells , 1992 .

[16]  H. Matsunaga.  Vibration and buckling of cross-ply laminated composite circular cylindrical shells according to a global higher-order theory , 2007 .

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

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

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

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

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

[22]  Y. C. Das.  Vibrations of orthotropic cylindrical shells , 1964 .

[23]  Mohamad S. Qatu,et al.  Accurate equations for laminated composite deep thick shells , 1999 .

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

[25]  W. Soedel,et al.  SIMPLIFIED EQUATIONS AND SOLUTIONS FOR THE VIBRATION OF ORTHOTROPIC CYLINDRICAL SHELLS , 1983 .

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

[27]  Stanley B. Dong,et al.  Free Vibration of Laminated Orthotropic Cylindrical Shells , 1968 .

[28]  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 .

[29]  Erasmo Carrera,et al.  Vibration Analysis of Anisotropic Simply Supported Plates by Using Variable Kinematic and Rayleigh-Ritz Method , 2011 .

[30]  K. M. Liew,et al.  Free vibration of two-side simply-supported laminated cylindrical panels via the mesh-free kp-Ritz method , 2004 .

[31]  G. B. Warburton,et al.  The Flexural Vibrations of Thin Cylinders , 1953 .

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

[33]  K. M. Liew,et al.  A higher order theory for vibration of shear deformable cylindrical shallow shells , 1995 .

[34]  A. Leissa,et al.  Vibration of shells , 1973 .

[35]  K. Liew,et al.  A Ritz vibration analysis of doubly-curved rectangular shallow shells using a refined first-order theory , 1995 .

[36]  R N Arnold,et al.  Flexural vibrations of the walls of thin cylindrical shells having freely supported ends , 1949, Proceedings of the Royal Society of London. Series A. Mathematical and Physical Sciences.

[37]  Kostas P. Soldatos,et al.  Comparative dynamic studies for symmetric cross-ply circular cylindrical shells on the basis of a unified shear deformable shell theory , 1995 .

[38]  D. J. Johns,et al.  A Comparison of the Characteristic Equations in the Theory of Circular Cylindrical Shells , 1961 .

[39]  X. Zhao,et al.  Vibration analysis of laminated composite cylindrical panels via a meshfree approach , 2003 .

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

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

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

[43]  Joseph Callahan,et al.  A closed-form solution procedure for circular cylindrical shell vibrations , 1999 .

[44]  Arcangelo Messina,et al.  The influence of boundary conditions and transverse shear on the vibration of angle-ply laminated plates, circular cylinders and cylindrical panels , 2001 .

[45]  Mohamad S. Qatu,et al.  Recent research advances on the dynamic analysis of composite shells: 2000-2009 , 2010 .

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

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

[48]  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 .

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

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

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

[52]  Erasmo Carrera.  A class of two-dimensional theories for anisotropic multilayered plates analysis , 1995 .

[53]  Arcangelo Messina,et al.  Ritz-type dynamic analysis of cross-ply laminated circular cylinders subjected to different boundary conditions , 1999 .

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

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

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

[57]  E. Carrera.  A study of transverse normal stress effect on vibration of multilayered plates and shells , 1999 .

[58]  Luciano Demasi,et al.  ∞3 Hierarchy plate theories for thick and thin composite plates: The generalized unified formulation , 2008 .

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

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

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

[62]  R. Jorge,et al.  Static and free vibration analysis of composite shells by radial basis functions , 2006 .

[63]  L. Demasi.  ∞6 Mixed plate theories based on the Generalized Unified Formulation. Part IV: Zig-zag theories , 2009 .

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

[65]  Erasmo Carrera,et al.  Analysis of laminated shells by a sinusoidal shear deformation theory and radial basis functions collocation, accounting for through-the-thickness deformations , 2011 .

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

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

[68]  E. Reissner.  A New Derivation of the Equations for the Deformation of Elastic Shells , 1941 .

[69]  Kevin Forsberg.  Influence of Boundary Conditions on the Modal Characteristics of Thin Cylindrical Shells , 1964 .

[70]  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 .

[71]  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 .

[72]  V. V. Novozhilov,et al.  Thin shell theory , 1964 .

[73]  K. Liew,et al.  A review of meshless methods for laminated and functionally graded plates and shells , 2011 .