Shock and Vibration

Volume 2016 (2016), Article ID 2373862, 17 pages

http://dx.doi.org/10.1155/2016/2373862

## Boundary Conditions in 2D Numerical and 3D Exact Models for Cylindrical Bending Analysis of Functionally Graded Structures

^{1}DICAM Department, University of Bologna, Bologna, Italy^{2}Department of Mechanical and Aerospace Engineering, Politecnico di Torino, Torino, Italy

Received 8 August 2016; Revised 19 October 2016; Accepted 26 October 2016

Academic Editor: Yuri S. Karinski

Copyright © 2016 F. Tornabene et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

The cylindrical bending condition for structural models is very common in the literature because it allows an incisive and simple verification of the proposed plate and shell models. In the present paper, 2D numerical approaches (the Generalized Differential Quadrature (GDQ) and the finite element (FE) methods) are compared with an exact 3D shell solution in the case of free vibrations of functionally graded material (FGM) plates and shells. The first 18 vibration modes carried out through the 3D exact model are compared with the frequencies obtained via the 2D numerical models. All the 18 frequencies obtained via the 3D exact model are computed when the structures have simply supported boundary conditions for all the edges. If the same boundary conditions are used in the 2D numerical models, some modes are missed. Some of these missed modes can be obtained modifying the boundary conditions imposing free edges through the direction perpendicular to the direction of cylindrical bending. However, some modes cannot be calculated via the 2D numerical models even when the boundary conditions are modified because the cylindrical bending requirements cannot be imposed for numerical solutions in the curvilinear edges by definition. These features are investigated in the present paper for different geometries (plates, cylinders, and cylindrical shells), types of FGM law, lamination sequences, and thickness ratios.

#### 1. Introduction

The cylindrical bending conditions have a great diffusion in the open literature because they allow an immediate and simple verification of several plate and shell models. Some of these plate models which use the cylindrical bending hypotheses for the verifications are discussed in the following. Chen et al. [1] used the state-space method for the investigation of a simply supported cross-ply laminated plate embedding viscous interfaces in cylindrical bending. Chen and Lee [2, 3] also used the state-space method for the cylindrical bending analysis of simply supported angle-ply laminated plates with interfacial damage and simply supported angle-ply laminated plates subjected to a static load. An experimental three-roller cylindrical bending investigation for plates was proposed by Gandhi and Raval [4]; the comparison with analytical solutions was also conducted. Oral and Darendeliler [5] proposed a methodology for the design of plate-forming dies in cylindrical bending using optimization techniques which allow the cost reductions of die production. Considering the large deflection of a thin beam, under certain conditions, the solution of the plate problem is not unique [6]. The bending-gradient plate theory, seen as an extension of the Reissner-Mindlin plate model, was used in [7] for the cylindrical bending analysis of laminated plates. Nimbolkar and Jain [8] used the same Reissner-Mindlin plate theory for the cylindrical bending investigation of composite and elastic plates subjected to the mechanical transverse load under plain strain conditions. Sayyad and Ghugal [9] investigated laminated plates using a th order shear deformation theory under cylindrical bending requirements. Some of the most important three-dimensional exact plate solutions were proposed by Pagano [10] and Pagano and Wang [11] for the cylindrical bending analysis of multilayered composite plates. Saeedi et al. [12] developed a 2D plate layer-wise model for the analysis of delamination growth in multilayered plates subjected to cylindrical bending loadings. This method is alternative to the finite element method. Shu and Soldatos [13] developed a new stress analysis method to investigate the stress distributions in angle-ply laminated plates when subjected to cylindrical bending. Further cylindrical bending analyses for plates consider the inclusion of functionally graded material (FGM) layers; typical examples are [14, 15]. In [16–21], the cylindrical bending analysis was proposed for multilayered plates embedding piezoelectric and/or FGM layers. The cylindrical bending analysis for multilayered composite and/or piezoelectric plates proposed in [22–24] includes the application of thermal loads. The cylindrical bending analysis of shells is not so diffused in the literature. Few cases can be found, and they concern the cylindrical bending conditions imposed to cylindrical shells. Simply supported angle-ply laminated cylindrical shells in cylindrical bending were investigated in [25] using the state-space formulation for the bending and free vibration analyses. A three-dimensional piezoelectric model based on the perturbation method for the cylindrical bending electromechanical load applications to cylindrical shells was proposed in [26]. Yan et al. [27] analyzed the cylindrical bending behavior of a simply supported angle-ply laminated cylindrical shell embedding viscoelastic interfaces.

The authors proposed the free vibration analysis of simply supported FGM plates and shells in [28, 29] where the 3D exact shell model was compared with several 2D numerical models such as the classical finite element (FE) one and the classical and refined Generalized Differential Quadrature (GDQ) methods. Similar comparisons were also proposed in [30, 31] for one-layered and multilayered isotropic, composite, and sandwich plates and shells. In such analyses, the considered geometries are plates, cylinders, and cylindrical and spherical shell panels. Low and high frequencies were calculated for thick and thin simply supported plates and shells with constant radii of curvature. The comparison between the 2D numerical methods (FE and GDQ solutions) and the 3D exact solution is possible only if an exhaustive vibration mode investigation is conducted. This investigation allows understanding how to make the comparison between numerical and analytical methods. This comparison is not easy because the exact 3D solution gives infinite vibration modes for all the possible combinations of half-wave numbers () and 2D numerical models propose a finite number of vibration modes because they use a finite number of degrees of freedom in the plane and in the thickness direction. It should be underlined that the 3D exact model is able to investigate any kind of geometric ratio (from extremely thick to very thin structures); nevertheless it can model only simply supported boundary conditions and it can investigate structures wherein constant radii of curvature are present. The 2D numerical approaches are more versatile (in terms of boundary conditions and type of structure analyzed); however they can be used only for moderately thick structures. It is obvious that thick structures can be analyzed only when higher order displacement fields are set in such 2D models. Since commercial codes implement only through the thickness linear theories, they can investigate accurately only moderately thick shells. The method proposed in [28–31] to compare the 3D exact shell model and the 2D numerical models is based on the calculation of the frequencies via the 2D numerical code and their appropriate visualization of the relative vibration modes via the GDQ and/or FE method. After this visualization, the evaluation of the 3D exact frequencies is possible by means of the appropriate imposition of half-wave numbers obtained from the vibration mode study. From this procedure, it is clear how the 3D analysis could give some frequencies that cannot be calculated by the 2D numerical codes. This feature was not the aim of the papers [28–31], and for this reason these missed frequencies are investigated in the present new work for FGM structures.

This new analysis is conducted calculating the first 18 frequencies for one-layered and sandwich FGM plates and shells by means of the 3D exact model. The half-wave numbers are combined along the two directions and in the plane considering the simply supported boundary conditions for the four edges. In these first 18 frequencies there are some modes that cannot be calculated by means of the 2D numerical models because they are not solutions of the employed mathematical model with the four simply supported edges. In general, these missed modes by the 2D numerical models correspond to the cylindrical bending conditions. These conditions in the 3D exact solution mean that one of the two half-wave numbers is equal to zero through an edge direction with a transverse displacement different from zero along the perpendicular edge. These particular frequencies are not solutions of the 2D numerical models built with the four edges which are simply supported, but a different mathematical model must be defined where two edges are simply supported and the other two are free. These conditions allow the cylindrical bending, with the transverse displacement through two parallel edges different from zero, in numerical models in order to obtain the missed frequencies and modes. This condition is possible only when the zero half-wave number is imposed through a rectilinear side because the half-wave number equals zero in the curvilinear edge does not mean that all the derivatives in the that direction are zero because of the presence of the curvature. This last condition is in conflict with the definition of cylindrical bending.

The 3D exact shell model is described and validated in Section 2. It uses the equilibrium equations written using the generic curvilinear orthogonal coordinates. Such equations are solved in closed form by means of simply supported boundary conditions. The differential equations along , written in layer-wise form, are solved using the exponential matrix method (also known as transfer matrix method or the state-space approach). The 2D FE and GDQ models are proposed and opportunely validated in Section 3. The 2D FE model makes use of a very common commercial finite element code. The 2D GDQ models give results by means of an in-house academic software and they are based on classical and refined shell models. The results about the comparisons between 3D exact and 2D numerical models, with the remark about the boundary conditions, are shown Section 4. Section 5 gives the main conclusions.

#### 2. Exact Solution of a 3D Shell Model

The three-dimensional exact shell model employed for the comparisons proposed in Section 4 has been elaborated in [32–40] for the free frequency analyses of single-layered and multilayered isotropic, orthotropic, composite, FGM, and sandwich plates and shells with constant radii of curvature and for the free vibrations of single- and double-walled carbon nanotubes including van der Waals interactions. The equilibrium equations proposed in the general orthogonal curvilinear coordinate system () are valid for both plates and shells with constant radii of curvature. For the sake of clarity, the plates and shells geometries within their local coordinate systems are depicted in Figure 1. The exact form of these equations is obtained using simply supported boundary conditions, while the differential equations in are solved by means of the exponential matrix method (also known as transfer matrix method or the state-space approach).