The local GDQ method applied to general higher-order theories of doubly-curved laminated composite shells and panels: The free vibration analysis

Abstract This paper presents a general two-dimensional approach for solving doubly-curved laminated composite shells using different kinematic expansions along the three orthogonal directions of the curvilinear shell model. The Carrera Unified Formulation (CUF) with different thickness functions along the three orthogonal curvilinear directions is applied to completely doubly-curved shells and panels, different from spherical and cylindrical shells and plates. Furthermore, the fundamental nuclei for doubly-curved structures are presented in their explicit form for the first time by the authors. These fundamental nuclei also allow to consider doubly-curved structures with variable thickness. In addition, the theoretical model includes the Murakami’s function (also known as zig-zag effect). For some problems it is useful to have an in-plane kinematic expansion which is different from the normal one. The 2D free vibration problem is numerically solved through the Local Generalized Differential Quadrature (LGDQ) method, which is an advanced version of the well-known Generalized Differential Quadrature (GDQ) method. The main advantage of the LGDQ method compared to the GDQ method is that the former can consider a large number of grid points without losing accuracy and keeping the very good stability features of GDQ method as already demonstrated in literature by the authors.

[1]  E. Viola,et al.  Static analysis of completely doubly-curved laminated shells and panels using general higher-order shear deformation theories , 2013 .

[2]  D. L. Young,et al.  Local Differential Quadrature Method for 2-D Flow and Forced-Convection Problems in Irregular Domains , 2009 .

[3]  Phillip L. Gould.  Thin Plates and Shells , 2013 .

[4]  Erasmo Carrera,et al.  Analysis of thick isotropic and cross-ply laminated plates by generalized differential quadrature method and a Unified Formulation , 2014 .

[5]  A. Ceruti,et al.  Mixed Static and Dynamic Optimization of Four-Parameter Functionally Graded Completely Doubly Curved and Degenerate Shells and Panels Using GDQ Method , 2013 .

[6]  Mohammad Mohammadi Aghdam,et al.  Non-linear bending analysis of laminated sector plates using Generalized Differential Quadrature , 2010 .

[7]  Erasmo Viola,et al.  Vibration analysis of spherical structural elements using the GDQ method , 2007, Comput. Math. Appl..

[8]  L. Donnell,et al.  Beams, plates and shells , 1976, Finite Element Analysis.

[9]  S. G. Lekhnit︠s︡kiĭ.  Theory of elasticity of an anisotropic body , 1981 .

[10]  Raymond D. Mindlin,et al.  An Introduction to the Mathematical Theory of Vibrations of Elastic Plates , 2006 .

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

[12]  Nicholas Fantuzzi,et al.  Generalized differential quadrature finite element method for cracked composite structures of arbitrary shape , 2013 .

[13]  Francesco Tornabene,et al.  Free vibrations of laminated composite doubly-curved shells and panels of revolution via the GDQ method , 2011 .

[14]  Phillip L. Gould,et al.  Analysis of Shells and Plates , 1987 .

[15]  Arthur W. Leissa,et al.  Vibration of Plates , 2021, Solid Acoustic Waves and Vibration.

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

[17]  Phillip L. Gould,et al.  Finite Element Analysis of Shells of Revolution , 1985 .

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

[19]  A. Kalnins,et al.  Thin elastic shells , 1967 .

[20]  Alfredo Liverani,et al.  FGM and laminated doubly curved shells and panels of revolution with a free-form meridian: A 2-D GDQ solution for free vibrations , 2011 .

[21]  Nasser Talebbeydokhti,et al.  Numerical modelling of the mild slope equation using localised differential quadrature method , 2012 .

[22]  I. I. Vorovich,et al.  Nonlinear Theory of Shallow Shells , 1999 .

[23]  Alfredo Liverani,et al.  General anisotropic doubly-curved shell theory: A differential quadrature solution for free vibrations of shells and panels of revolution with a free-form meridian , 2012 .

[24]  K. M. Liew,et al.  Free vibration analysis of functionally graded conical shell panels by a meshless method , 2011 .

[25]  Daniel J. Inman,et al.  2-D differential quadrature solution for vibration analysis of functionally graded conical, cylindrical shell and annular plate structures , 2009 .

[26]  Mohammad Reza Forouzan,et al.  Large deflection analysis of thermo-mechanical loaded annular FGM plates on nonlinear elastic foundation via DQM , 2010 .

[27]  Marco Amabili,et al.  Nonlinear Vibrations and Stability of Shells and Plates , 2008 .

[28]  Yaojun Ge,et al.  Generation of inflow turbulence using the local differential quadrature method , 2013 .

[29]  Nicholas Fantuzzi,et al.  Static analysis of functionally graded conical shells and panels using the generalized unconstrained third order theory coupled with the stress recovery , 2014 .

[30]  Francesco Tornabene,et al.  Modellazione e soluzione di strutture a guscio in materiale anisotropo , 2007 .

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

[32]  Moshe Eisenberger,et al.  Exact vibration analysis of variable thickness thick annular isotropic and FGM plates , 2007 .

[33]  Romesh C. Batra,et al.  Free vibration of three-layer circular cylindrical shells with functionally graded middle layer , 2010 .

[34]  Elena Ferretti,et al.  GDQFEM Numerical Simulations of Continuous Mediawith Cracks and Discontinuities , 2013 .

[35]  A. Saada.  Elasticity : theory and applications , 1993 .

[36]  Nicholas Fantuzzi,et al.  Numerical investigation of functionally graded cylindrical shells and panels using the generalized unconstrained third order theory coupled with the stress recovery , 2012 .

[37]  Mohamed Nassar,et al.  Vibration analysis of structural elements using differential quadrature method , 2012, Journal of advanced research.

[38]  Zekeriya Girgin,et al.  Buckling Analyses of Axially Functionally Graded Nonuniform Columns with Elastic Restraint Using a Localized Differential Quadrature Method , 2013 .

[39]  W. Soedel.  Vibrations of shells and plates , 1981 .

[40]  C. Bert,et al.  Differential Quadrature Method in Computational Mechanics: A Review , 1996 .

[41]  N. Rogacheva.  The Theory of Piezoelectric Shells and Plates , 1994 .

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

[43]  S. C. Dixon,et al.  Flutter, vibration, and buckling of truncated orthotropic conical shells with generalized elastic edge restraint, supplement , 1970 .

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

[45]  Alfredo Liverani,et al.  Static analysis of laminated composite curved shells and panels of revolution with a posteriori shear and normal stress recovery using generalized differential quadrature method , 2012 .

[46]  J. G. Simmonds,et al.  The Nonlinear Theory of Elastic Shells , 1998 .

[47]  Ashraf M. Zenkour,et al.  Analysis of Sandwich Plates by Generalized Differential Quadrature Method , 2013 .

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

[49]  Moshe Eisenberger,et al.  Dynamic stiffness vibration analysis of thick spherical shell segments with variable thickness , 2010 .

[50]  Erasmo Viola,et al.  Vibration Analysis of Damaged Circular Arches with Varying Cross-section , 2005 .

[51]  Zhi Zong,et al.  A localized differential quadrature (LDQ) method and its application to the 2D wave equation , 2002 .

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

[53]  Elena Ferretti,et al.  Soft Core Plane State Structures Under Static Loads UsingGDQFEM and Cell Method , 2013 .

[54]  Erasmo Viola,et al.  Free vibration analysis of functionally graded panels and shells of revolution , 2009 .

[55]  G. E. Mase,et al.  Continuum Mechanics for Engineers , 1991 .

[56]  I. S. Sokolnikoff.  Mathematical theory of elasticity , 1946 .

[57]  A. L. Goldenveizer.  THEORY OF ELASTIC THIN SHELLS , 1962 .

[58]  Chang Shu,et al.  Free vibration analysis of laminated composite cylindrical shells by DQM , 1997 .

[59]  C. R. Calladine,et al.  Theory of Shell Structures , 1983 .

[60]  Philippe G. Ciarlet,et al.  Theory of Shells , 2000 .

[61]  Erasmo Viola,et al.  2-D solution for free vibrations of parabolic shells using generalized differential quadrature method , 2008 .

[62]  D. L. Young,et al.  The localized differential quadrature method for two-dimensional stream function formulation of Navier–Stokes equations , 2011 .

[63]  Erasmo Viola,et al.  Analytical and numerical results for vibration analysis of multi-stepped and multi-damaged circular arches , 2007 .

[64]  Nicholas Fantuzzi,et al.  Radial basis function method applied to doubly-curved laminated composite shells and panels with a General Higher-order Equivalent Single Layer formulation , 2013 .

[65]  J. N. Reddy,et al.  Energy principles and variational methods in applied mechanics , 2002 .

[66]  Alessandro Marzani,et al.  Nonconservative stability problems via generalized differential quadrature method , 2008 .

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

[68]  Nicholas Fantuzzi,et al.  Mechanics of Laminated Composite Doubly-Curved Shell Structures. The Generalized Differential Quadrature Method and the Strong Formulation Finite Element Method , 2014 .

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

[70]  Zhi Zong,et al.  Advanced Differential Quadrature Methods , 2009 .

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

[72]  Erasmo Viola,et al.  FREE VIBRATIONS OF FOUR-PARAMETER FUNCTIONALLY GRADED PARABOLIC PANELS AND SHELLS OF REVOLUTION , 2009 .

[73]  C. Shu.  Differential Quadrature and Its Application in Engineering , 2000 .

[74]  J. Douglas Faires,et al.  Numerical Analysis , 1981 .

[75]  A. Ceruti,et al.  Free-Form Laminated Doubly-Curved Shells and Panels of Revolution Resting on Winkler-Pasternak Elastic Foundations: A 2-D GDQ Solution for Static and Free Vibration Analysis , 2013 .

[76]  J. N. Reddy,et al.  FGM and Laminated Doubly-Curved and Degenerate Shells Resting on Nonlinear Elastic Foundations: A GDQ Solution for Static Analysis with a Posteriori Stress and Strain Recovery , 2013 .

[77]  Snehashish Chakraverty,et al.  Vibration of Plates , 2008 .

[78]  Reza Madoliat,et al.  Static analysis of cross-ply laminated plates with integrated surface piezoelectric layers using differential quadrature , 2009 .

[79]  Francesco Tornabene.  Meccanica delle Strutture a Guscio in Materiale Composito. Il metodo Generalizzato di Quadratura Differenziale , 2012 .

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

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

[82]  Akbar Alibeigloo,et al.  Static analysis of functionally graded cylindrical shell with piezoelectric layers using differential quadrature method , 2010 .

[83]  Erasmo Viola,et al.  Free vibrations of three parameter functionally graded parabolic panels of revolution , 2009 .

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

[85]  Š. Markuš,et al.  The mechanics of vibrations of cylindrical shells , 1988 .

[86]  J. N. Reddy,et al.  Theory and analysis of elastic plates , 1999 .

[87]  Nicholas Fantuzzi,et al.  Generalized Differential Quadrature Finite Element Method for vibration analysis of arbitrarily shaped membranes , 2014 .

[88]  Alfredo Liverani,et al.  Laminated composite rectangular and annular plates: A GDQ solution for static analysis with a posteriori shear and normal stress recovery , 2012 .

[89]  J. R. Banerjee,et al.  An exact dynamic stiffness element using a higher order shear deformation theory for free vibration analysis of composite plate assemblies , 2013 .

[90]  C. Bert,et al.  The behavior of structures composed of composite materials , 1986 .

[91]  Francesco Tornabene,et al.  Free vibration analysis of functionally graded conical, cylindrical shell and annular plate structures with a four-parameter power-law distribution , 2009 .

[92]  A. Leissa,et al.  Vibrations of continuous systems , 2011 .

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

[94]  K. Y. Lam,et al.  A numerical study of wave propagation in a poroelastic medium by use of localized differential quadrature method , 2004 .

[95]  D. Chapelle,et al.  The Finite Element Analysis of Shells - Fundamentals , 2003 .

[96]  B. Sobhani Aragh,et al.  THREE-DIMENSIONAL ANALYSIS FOR THERMOELASTIC RESPONSE OF FUNCTIONALLY GRADED FIBER REINFORCED CYLINDRICAL PANEL , 2010 .

[97]  Moshe Eisenberger,et al.  Exact vibration frequencies of segmented axisymmetric shells , 2006 .

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

[99]  J. N. Reddy,et al.  Winkler–Pasternak foundation effect on the static and dynamic analyses of laminated doubly-curved and degenerate shells and panels , 2014 .

[100]  I. S. Sokolnikoff,et al.  Tensor Analysis: Theory and Applications , 1952 .

[101]  Erasmo Carrera,et al.  Static analysis of doubly-curved anisotropic shells and panels using CUF approach, differential geometry and differential quadrature method , 2014 .

[102]  Hua Li,et al.  Rotating Shell Dynamics , 2005 .

[103]  Erasmo Viola,et al.  Vibration analysis of conical shell structures using GDQ Method , 2006 .

[104]  Francesco Tornabene,et al.  Free vibrations of anisotropic doubly-curved shells and panels of revolution with a free-form meridian resting on Winkler–Pasternak elastic foundations , 2011 .

[105]  Erasmo Viola,et al.  Static analysis of functionally graded doubly-curved shells and panels of revolution , 2013 .

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

[107]  Nicholas Fantuzzi,et al.  Generalized Differential Quadrature Finite Element Method applied to Advanced Structural Mechanics , 2013 .

[108]  Jian-an Sun,et al.  Upwind local differential quadrature method for solving incompressible viscous flow , 2000 .

[109]  K. M. Liew,et al.  Differential quadrature element method: a new approach for free vibration analysis of polar Mindlin plates having discontinuities , 1999 .

[110]  E. Viola,et al.  General higher-order shear deformation theories for the free vibration analysis of completely doubly-curved laminated shells and panels , 2013 .