Research Article  Open Access
Natural Frequencies of Shear Deformable Plates by Polyharmonic Splines
Abstract
We use the thirdorder shear deformation theory and a collocation technique with polyharmonic splines to predict natural frequencies of moderately thick isotropic plates. The natural frequencies of vibration are computed for various plates and compared with some available published results. Through numerical experiments, the capability and efficiency of the present method for eigenvalue problems are demonstrated, and the numerical accuracy and convergence are thoughtfully examined.
1. Introduction
Radial basis functions (RBFs) have recently proved to be an excellent technique for interpolating data and functions. A radial basis function can be considered a spline that depends on the Euclidean distance between distinct data centers , also called nodal or collocation points. Not only RBFs are adequate to scattered data approximation and in general to interpolation theory, as proposed by Hardy [1], but also excellent for solving partial differential equations (PDEs) as first introduced by Kansa [2].
Kansa proposed an unsymmetric RBF collocation method based upon multiquadric interpolation functions, in which the shape parameter is considered to be variable across the problem domain. The distribution of the shape parameter is obtained by an optimization approach, in which the value of the shape parameter is assumed to be proportional to the curvature of the unknown solution of the original partial differential equation. In this way, it is possible to reduce the condition number of the matrix at the expense of implementing an additional iterative algorithm. In the present work, we will implement the unsymmetric global collocation method in a form that is independent of this shape parameter, based on unshifted polyharmonic splines. In some respect this can be seen as a more stable form than multiquadrics.
Structures composed of laminated materials are among the most important structures used in modern engineering and, especially, in the aerospace industry. Such lightweight structures are also being increasingly used in civil, mechanical, and transportation engineering applications. The rapid increase of the industrial use of these structures has necessitated the development of new analytical and numerical tools that are suitable for the analysis and study of the mechanical behavior of such structures. The behavior of structures composed of advanced composite materials is considerably more complicated than for isotropic ones. The strong influences of anisotropy, the transverse stresses through the thickness of a laminate, and the stress distributions at interfaces are among the most important factors that affect the general performance of such structures. The use of shear deformation theories has been the topic of intensive research, as in [7–19], among many others.
The analysis of laminated plates by finite element methods is now considerably established. The use of alternative methods such as the meshless methods based on radial basis functions is attractive due to the absence of a mesh and the ease of collocation methods. The use of radial basis function for the analysis of structures and materials has been previously studied by numerous authors [20–31]. More recently the authors have applied RBFs to the static deformations of composite beams and plates [32–35]. Recent meshless techniques were discussed in [36–42].
In this paper the use of radial basis functions to isotropic and composite plates using a thirdorder shear deformation theory is investigated. The quality of the present method in predicting free vibrations of isotropic and laminated composite plates is compared and discussed with other methods in some numerical examples.
2. The Radial Basis Function Method
2.1. The Eigenproblem
Radial basis functions (RBF) approximations are gridfree numerical schemes that can exploit accurate representations of the boundary, are easy to implement, and can be spectrally accurate [43, 44].
In this section the formulation of a global unsymmetrical collocation RBFbased method to compute eigenvalues of elliptic operators is presented.
Consider a linear elliptic partial differential operator and a bounded region in with some boundary . The eigenproblem looks for eigenvalues () and eigenvectors that satisfy where is a linear boundary operator. The eigenproblem of (1) is replaced by a finitedimensional eigenvalue problem, based on RBF approximations.
The operator is approximated by a matrix that incorporates the boundary conditions and then solve the eigenvalues and eigenvectors of this matrix by standard techniques.
2.2. Radial Basis Functions
The radial basis function approximation of a function is given by where is a finite set of distinct points (centers) in . The coefficients are calculated so that satisfies some boundary conditions. Although many other functions could be used, as illustrated by several authors [1, 2, 45], here we use unshifted polyharmonic splines in the form where the Euclidean distance is real and nonnegative.
Considering distinct interpolations, and knowing , we find by the solution of a linear system , where , and = . The RBF interpolation matrix is positive definite for some RBFs [46] but in general provides illconditioned systems. This illconditioning problem is due to full matrices produced by the collocation method and worsens with the increase of number of nodes. However it was shown by Schaback that the more illconditioned the problem the better the solution, until a flatness of the function occurs. In our numerical experiments this illconditioning did not produce any inconvenience to quality of solution.
2.3. Solution of the Eigenproblem
We follow a simple scheme for the solution of the eigenproblem (1). We consider nodes in the interior of the domain and nodes on the boundary, with .
We denote interpolation points by and . For the interior points we have that or where
For the boundary conditions we have or
Therefore we can write a finitedimensional problem as a generalized eigenvalue problem where
3. ThirdOrder Theory
The displacement field for the thirdorder shear deformation theory of Hardy [1] is obtained as where and are the inplane displacements at any point , , denote the inplane displacement of the point on the midplane, is the deflection, and and are the rotations of the normals to the midplane about the  and axes, respectively.
The straindisplacement relationships are given as
Therefore strains can be expressed as where and .
A laminate can be manufactured from orthotropic layers (or plies) of preimpregnated unidirectional fibrous composite materials. Neglecting for each layer, the stressstrain relations in the fiber local coordinate system can be expressed as where subscripts 1 and 2 are, respectively, the fiber and the normal to fiber inplane directions, 3 is the direction normal to the plate, and the reduced stiffness components, , are given by in which , and are materials properties of the lamina.
By performing adequate coordinate transformation, the stressstrain relations in the global  coordinate system can be obtained as
The thirdorder theory of Hardy [15, 16] satisfies zero transverse shear stresses on the bounding planes.
The equations of motion of the thirdorder theory are derived from the principle of virtual displacements: where is the external distributed load and with where take the symbols . The resultants denote the inplane force resultants, the moment resultants, the shear resultants and and denote the higherorder stress resultants: where take the symbols .
If needed, the firstorder shear deformation theory equations are readily obtained from the thirdorder equations, just by putting .
4. Numerical Examples
The eigenvalues presented in this problem are expressed in terms of the nondimensional frequency parameter . Unlike firstorder shear deformation theories the present approach does not incorporate shear correction factors in the constitutive matrix. The Poisson ratio, , is taken to be 0.3 throughout the present problem. Figure 1 illustrates the grid scheme for points.
In Tables 1, 2, 3, 4, 5, and 6 we compare the present results with those of Liew et al. [3]. We consider here simply supported (SSSS) and clamped (CCCC) boundary conditions in all edges, thin and thick plates and square and rectangular plates. The results obtained in Tables 1 to 6 show that the present method presents very close results to those proposed by Liew et al. [3]. However, for thin plates we need more points than in thick plates in order to get convergent and accurate frequencies. The reason for this behaviour lies in the worse conditioning of thin plates. In Liew et al. [3] thin plates also showed slower convergence.






In Tables 7 and 8 we analyse thin and thick plates for SSSS and CCCC boundaries. Present results are compared with classical theory of Leissa [4], firstorder theory of Dawe [5], and orthogonal polynomials of Liew et al. [3]. The present method shows very accurate results when compared to Liew et al. [3] and FSDT of Dawe [5], for both SSSS and CCCC boundary conditions.


5. Conclusions
Radial basis functions are increasingly popular in the analysis of science and engineering problems.
In this paper, we use the thirdorder shear deformation theory and a collocation technique with polyharmonic splines to predict natural frequencies of moderately thick isotropic plates. The natural frequencies of vibration are computed for various plates and compared with some available published results. The formulation for higherorder plates and their interpolation with polyharmonic splines has been presented.
Through numerical experiments, the capability and efficiency of the present method for eigenvalue problems are demonstrated, and the numerical accuracy and convergence are thoughtfully examined.
When compared to other RBFs, polyharmonic splines show good stability and accuracy. However, proper modelling of plates should consider both a good numerical technique and an adequate shear deformation theory.
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Copyright © 2013 A. J. M. Ferreira and A. M. Zenkour. 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.