In this paper, the Generalized Differential Quadrature (GDQ) method is applied to study the dynamic behaviour of functionally graded (FGM) and laminated doubly-curved shells and panels of revolution with a free-form meridian. The First-order Shear Deformation Theory (FSDT) is used to analyse the above mentioned moderately thick structural elements. In order to include the effect of the initial curvature a generalization of the Reissner-Mindlin theory, proposed by Toorani and Lakis, is adopted. The governing equations of motion, written in terms of stress resultants, are expressed as functions of five kinematic parameters, by using the constitutive and kinematic relationships. The solution is given in terms of generalized displacement components of points lying on the middle surface of the shell. Simple Rational Bézier curves are used to define the meridian curve of the revolution structures. Firstly, the Differential Quadrature (DQ) rule is introduced to determine the geometric parameters of the structures with a free-form meridian. Secondly, the discretization of the system by means of the GDQ technique leads to a standard linear eigenvalue problem, where two independent variables are involved. Results are obtained taking the meridional and circumferential co-ordinates into account, without using the Fourier modal expansion methodology. Comparisons between the Reissner-Mindlin and Toorani-Lakis theory are presented. Furthermore, GDQ results are compared with those obtained by using commercial programs such as Abaqus, Ansys, Nastran, Straus and Pro/Mechanica. Very good agreement is observed. Finally, different lamination schemes are considered to expand the combination of the two functionally graded four-parameter power-law distributions adopted. The treatment is developed within the theory of linear elasticity, when materials are assumed to be isotropic and inhomogeneous through the lamina thickness direction. A two-constituent functionally graded lamina consists of ceramic and metal that are graded through the lamina thickness. A parametric study is performed to illustrate the influence of the parameters on the mechanical behaviour of shell structures considered. KEYWORDS: Free vibrations, Doubly-Curved Shells of Revolution, Rational Bézier Curves, Laminated Composite Shells, Functionally Graded Materials, First-Order Shear Deformation Theory, Generalized Differential Quadrature Method.

F. Tornabene, A. Liverani, G. Caligiana (2011). FGM and laminated doubly curved shells and panels of revolution with a free-form meridian: A 2-D GDQ solution for free vibrations. INTERNATIONAL JOURNAL OF MECHANICAL SCIENCES, 53(6), 446-470 [10.1016/j.ijmecsci.2011.03.007].

### FGM and laminated doubly curved shells and panels of revolution with a free-form meridian: A 2-D GDQ solution for free vibrations

#### Abstract

In this paper, the Generalized Differential Quadrature (GDQ) method is applied to study the dynamic behaviour of functionally graded (FGM) and laminated doubly-curved shells and panels of revolution with a free-form meridian. The First-order Shear Deformation Theory (FSDT) is used to analyse the above mentioned moderately thick structural elements. In order to include the effect of the initial curvature a generalization of the Reissner-Mindlin theory, proposed by Toorani and Lakis, is adopted. The governing equations of motion, written in terms of stress resultants, are expressed as functions of five kinematic parameters, by using the constitutive and kinematic relationships. The solution is given in terms of generalized displacement components of points lying on the middle surface of the shell. Simple Rational Bézier curves are used to define the meridian curve of the revolution structures. Firstly, the Differential Quadrature (DQ) rule is introduced to determine the geometric parameters of the structures with a free-form meridian. Secondly, the discretization of the system by means of the GDQ technique leads to a standard linear eigenvalue problem, where two independent variables are involved. Results are obtained taking the meridional and circumferential co-ordinates into account, without using the Fourier modal expansion methodology. Comparisons between the Reissner-Mindlin and Toorani-Lakis theory are presented. Furthermore, GDQ results are compared with those obtained by using commercial programs such as Abaqus, Ansys, Nastran, Straus and Pro/Mechanica. Very good agreement is observed. Finally, different lamination schemes are considered to expand the combination of the two functionally graded four-parameter power-law distributions adopted. The treatment is developed within the theory of linear elasticity, when materials are assumed to be isotropic and inhomogeneous through the lamina thickness direction. A two-constituent functionally graded lamina consists of ceramic and metal that are graded through the lamina thickness. A parametric study is performed to illustrate the influence of the parameters on the mechanical behaviour of shell structures considered. KEYWORDS: Free vibrations, Doubly-Curved Shells of Revolution, Rational Bézier Curves, Laminated Composite Shells, Functionally Graded Materials, First-Order Shear Deformation Theory, Generalized Differential Quadrature Method.
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2011
F. Tornabene, A. Liverani, G. Caligiana (2011). FGM and laminated doubly curved shells and panels of revolution with a free-form meridian: A 2-D GDQ solution for free vibrations. INTERNATIONAL JOURNAL OF MECHANICAL SCIENCES, 53(6), 446-470 [10.1016/j.ijmecsci.2011.03.007].
F. Tornabene; A. Liverani; G. Caligiana
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Utilizza questo identificativo per citare o creare un link a questo documento: `https://hdl.handle.net/11585/102914`
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