Mathematical Problems in Engineering

Volume 2015, Article ID 124296, 13 pages

http://dx.doi.org/10.1155/2015/124296

## Free Vibration Analysis of Symmetrically Laminated Folded Plate Structures Using an Element-Free Galerkin Method

^{1}College of Civil Engineering and Architecture, Guangxi University, Nanning 530004, China^{2}Guangxi Key Laboratory of Disaster Prevention and Engineering Safety, Guangxi University, Nanning 530004, China

Received 21 July 2014; Revised 14 November 2014; Accepted 14 November 2014

Academic Editor: Kim M. Liew

Copyright © 2015 L. X. Peng. 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

An element-free Galerkin method for the solution of free vibration of symmetrically laminated folded plate structures is introduced. Employing the mature meshfree folded plate model proposed by the author, a folded laminated plate is simulated as a composite structure of symmetric laminates that lie in different planes. Based on the first-order shear deformation theory (FSDT) and the moving least-squares (MLS) approximation, the stiffness and mass matrices of the laminates are derived and supposed to obtain the stiffness and mass matrices of the entire folded laminated plate. The equation governing the free vibration behaviors of the folded laminated plate is thus established. Because of the meshfree characteristics of the proposed method, no mesh is involved to determine the stiffness and mass matrices of the laminates. Therefore, the troublesome remeshing can be avoided completely from the study of such problems as the large deformation of folded laminated plates. The calculation of several numerical examples shows that the solutions given by the proposed method are very close to those given by ANSYS, using shell elements, which proves the validity of the proposed method.

#### 1. Introduction

Because of high strength/weight ratio, easy forming, and low cost, folded plate structures have been widely used in many engineering branches, such as roofs, corrugated-cores, and cooling towers. They have much higher load carrying capacity compared to flat plates. Before the invention of fiber-reinforced material, folded plates were often made of medal or timber. The application of fiber-reinforced material to folded plate structures was a remarkable advance in engineering, which combined the advantages of fiber-reinforced material and folded plate structure directly and made the structure even lighter and stiffer.

The study of isotropic folded plates had a quite long history, and a variety of methods had emerged. In early days, researchers were short of powerful numerical tools and tried to analyze the structures with various approximations. The beam method and the theory that ignores relative joint displacement were introduced [1]. Although the methods were weak in dealing with generalized folded plate problems, they were simple and fulfilled the demand of fast and easy computation in engineering. Therefore, they are still used in some design environments, where accurate analysis is not the first concern. Researchers such as Gaafar [2], Yitzhaki [3], Yitzhaki and Reiss [4], and Whitney et al. [5] were the first to consider the relative joint displacement of the structures in their methods, which led to more precise analysis results. Goldberg and Leve [6] used the two-dimensional theory of elasticity and the two-way slab theory to analyze folded plates. In their method, both the simultaneous bending and the membrane action of a folded plate were taken into account, and the degree of freedom (DOF) of each point along the joint of the folded plate was chosen to be four (three components of translation and one rotation). Niyogi et al. [7] considered this method as the first to give an exact static solution for folded plates. Yitzhaki and Reiss [4] took the moments along the joints of folded plates as unknown and applied the slope deflection method to the analysis of the folded plates. Bar-Yoseph and Hersckovitz [8] proposed an approximated method for folded plates based on Vlasov’s theory of thin-wall beams. Their method considered a folded plate as a monolithic structure composed of longitudinal beams, which can give good results for long folded plates. Bandyopadhyay and Laad [9] compared two classical methods for folded plates and studied the suitability of these methods for the preliminary analysis of folded plate structures. Lai et al. [10] gave an equation of the middle surface of a simply supported cross V-shaped folded plate roof by using the inclined coordinate system and generalized functions, sign function and step function, and carried out a nonlinear analysis for the folded plate.

The development of computation techniques and computers has aroused research interest in numerical methods for folded plates. A number of methods, such as the finite strip methods (Cheung [11], Golley and Grice [12], Eterovic and Godoy [13]), the combined boundary element-transfer matrix method (Ohga et al. [14]), and the finite element methods (FEM) (Liu and Huang [15], Perry et al. [16], Niyogi et al. [7], and Duan and Miyamoto [17]), have been introduced to solve folded plate problems. Among these methods, the FEMs are the most successful. They are very versatile as they can deal with the problems with complicated geometry, boundary conditions, or loadings easily. However, FEM also has disadvantages. Their solution of a problem is based on the meshes that discretize the problem domain, and any dramatic change of the problem domain will lead to remeshing of the domain, which results in programming complexity, diminished accuracy, and long computation time. Regarding the disadvantage, some researchers proposed the element-free, meshfree, or meshless methods [18–22]. As alternatives to FEMs, the meshfree methods construct their approximated solution of a problem completely in terms of a set of ordered or scattered points that discretize the problem domain; that is, their solution relies on the points other than meshes. No element is required. Without the limit of meshes, the meshfree methods are more applicable than the FEMs and avoid the aforementioned difficulties caused by remeshing in the FEMs.

Bui et al. [23], Bui and Nguyen [24], and Somireddy and Rajagopal [25] have introduced the meshfree methods for vibration analysis of laminated plates. However, few studies on folded laminated plates have been found. There are only Niyongi et al. [7] and Lee et al.’s [26] work on vibration and the author’s work on bending with a meshfree method, which is also the motive for this paper.

The objective of this paper is to introduce an element-free Galerkin method based on the first-order shear deformation theory (FSDT) [27, 28] for the free vibration analysis of folded laminated plates. A symmetrical folded laminated plate is regarded as a composite structure composed of symmetric laminates. The analysis process includes (a) deriving the stiffness and mass matrices of the symmetric laminates that make up a folded plate by the element-free Galerkin method; (b) considering the laminates as super elements and superposing their stiffness and mass matrices to obtain the global stiffness and mass matrices of the folded plate. Some numerical examples are used to demonstrate the convergence and accuracy of the proposed method. The calculated results are compared with the results from the finite element analytical software ANSYS. The proposed method may be used as a potential meshfree tool for the analysis of laminated shell structures.

#### 2. Moving Least-Squares Approximation

In the moving least-squares approximation (MLS) [18], a function in a domain can be approximated by in the subdomain andwhere are the monomial basis functions, are the corresponding coefficients, is a factor that measures the domain of influence (or the support) of the nodes, and is the number of basis functions. In this paper, the quadratic basis () are used for the laminates. The unknown coefficients are obtained by the minimization of a weighted discrete normwhere or is the weight function that is associated with node , outside , is the number of nodes in that make the weight function , and are the nodal parameters. The minimization of in (2) with respect to leads to a set of linear equationswhereThe coefficients are then derived from (4):By substituting (7) into (1), the approximation is expressed in a standard form aswhere the shape function is given byFrom (6), we obtainand thus (9) can be rewritten as

#### 3. Meshless Model of a Laminate

The first step in our analysis is to obtain the stiffness and mass matrices of the laminates that make up a folded plate. The meshless model for a laminate in the local coordinate, as shown in Figure 1, is prescribed with a set of nodes. The DOF of every node is , where , , and are the nodal translations of the laminate in the -direction, -direction, and -direction, respectively. and are the rotation about the -axis and the -axis, respectively. The laminate is assumed to have layers, and the thickness of each layer is (). Therefore, the thickness of th layer is .