A comparison between a 3D exact shell solution and 2D numerical approaches is proposed in the present paper with a specific focus on the free vibrations of multi-layered composite plates and shells. The validation of new plate and shell structural models is often carried out imposing the cylindrical bending conditions. The reference 3D model here employed is developed using an orthogonal and mixed curvilinear coordinate system. The equilibrium equations are proposed for spherical shells, and they can easily degenerate in those for cylindrical shells, cylinders and plates. In the free vibration analysis, the analytical solution of the system can be obtained using a harmonic form for the displacements by imposing arbitrary half-wave numbers in both the in-plane directions. This harmonic formulation automatically leads to simply supported boundary conditions. The cylindrical bending can be easily studied imposing that one of the two half-wave numbers is equal zero. The numerical solutions of the problem are obtained using different 2D models. A classical 2D Finite Element (FE) model and a 2D Generalized Differential Quadrature (GDQ) method are compared with the 3D analytical reference solution. The 2D FE model uses a refined mesh based on Quad8 elements in the framework of a commercial finite element code. The 2D GDQ model solves the problem using a non-uniform Chebyshev-Gauss-Lobatto grid and they are based on refined 2D kinematic assumptions. The proposed benchmarks show that both the 2D numerical models (FE and GDQ ones) are able to provide almost all the natural frequencies of the investigated shells with a significant precision. The 3D model obtains all the vibration modes with a single run where all the edges are simply supported. However, both the 2D models can miss some modes when the simply supported boundary conditions are applied to all the edges in a no coherent way. Some of these modes are cylindrical bending modes and they can be recovered by opportunely modifying the boundary conditions in 2D numerical models. In the case of plates and cylinders, all the missing frequencies can be obtained imposing the simply supported conditions only for the edges parallel to the direction with zero half-wave number and using the free boundary conditions for the other two parallel sides. When a cylindrical shell panel is considered, the modes with zero half-wave number in the curvilinear direction cannot be obtained because the cylindrical bending conditions cannot be imposed. In the case of cylindrical bending modes, it has been observed that the 2D numerical solutions converge to the 3D exact model when the length of the simply supported sides increases in order to reduce the boundary effects corresponding to the free edges.

Cylindrical bending in composite structures by means of analytical and numerical 2D/3D shell models / Torre, R.; Brischetto, S.; Tornabene, F.. - (2019). ((Intervento presentato al convegno ICCS22 – 22nd International Conference on Composite Structures & CCCS1 - 1st Chinese Conference on Composite Structures tenutosi a Wuhan (China) nel 31st October - 3rd November, 2019.

### Cylindrical bending in composite structures by means of analytical and numerical 2D/3D shell models

#### Abstract

A comparison between a 3D exact shell solution and 2D numerical approaches is proposed in the present paper with a specific focus on the free vibrations of multi-layered composite plates and shells. The validation of new plate and shell structural models is often carried out imposing the cylindrical bending conditions. The reference 3D model here employed is developed using an orthogonal and mixed curvilinear coordinate system. The equilibrium equations are proposed for spherical shells, and they can easily degenerate in those for cylindrical shells, cylinders and plates. In the free vibration analysis, the analytical solution of the system can be obtained using a harmonic form for the displacements by imposing arbitrary half-wave numbers in both the in-plane directions. This harmonic formulation automatically leads to simply supported boundary conditions. The cylindrical bending can be easily studied imposing that one of the two half-wave numbers is equal zero. The numerical solutions of the problem are obtained using different 2D models. A classical 2D Finite Element (FE) model and a 2D Generalized Differential Quadrature (GDQ) method are compared with the 3D analytical reference solution. The 2D FE model uses a refined mesh based on Quad8 elements in the framework of a commercial finite element code. The 2D GDQ model solves the problem using a non-uniform Chebyshev-Gauss-Lobatto grid and they are based on refined 2D kinematic assumptions. The proposed benchmarks show that both the 2D numerical models (FE and GDQ ones) are able to provide almost all the natural frequencies of the investigated shells with a significant precision. The 3D model obtains all the vibration modes with a single run where all the edges are simply supported. However, both the 2D models can miss some modes when the simply supported boundary conditions are applied to all the edges in a no coherent way. Some of these modes are cylindrical bending modes and they can be recovered by opportunely modifying the boundary conditions in 2D numerical models. In the case of plates and cylinders, all the missing frequencies can be obtained imposing the simply supported conditions only for the edges parallel to the direction with zero half-wave number and using the free boundary conditions for the other two parallel sides. When a cylindrical shell panel is considered, the modes with zero half-wave number in the curvilinear direction cannot be obtained because the cylindrical bending conditions cannot be imposed. In the case of cylindrical bending modes, it has been observed that the 2D numerical solutions converge to the 3D exact model when the length of the simply supported sides increases in order to reduce the boundary effects corresponding to the free edges.
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Utilizza questo identificativo per citare o creare un link a questo documento: `http://hdl.handle.net/11583/2767792`