Journal of Applied Mathematics

Volume 2013 (2013), Article ID 284208, 7 pages

http://dx.doi.org/10.1155/2013/284208

Research Article

Natural Frequencies of Shear Deformable Plates by Polyharmonic Splines

A. J. M. Ferreira1,2 and A. M. Zenkour2,3

¹Faculdade de Engenharia da Universidade do Porto, Porto, Portugal

²Department of Mathematics, Faculty of Science, King Abdulaziz University, P.O. Box 80203, Jeddah 21589, Saudi Arabia

³Department of Mathematics, Faculty of Science, Kafr El-Sheikh University, Kafr El-Sheikh 33516, Egypt

Received 10 July 2013; Accepted 16 September 2013

Academic Editor: Carla Roque

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.

Abstract.

We use the third-order 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 third-order 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 grid-free 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 RBF-based 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 finite-dimensional 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 ill-conditioned systems. This ill-conditioning 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 ill-conditioned the problem the better the solution, until a flatness of the function occurs. In our numerical experiments this ill-conditioning 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 finite-dimensional problem as a generalized eigenvalue problem

where

3. Third-Order Theory

The displacement field for the third-order shear deformation theory of Hardy [1] is obtained as

where and are the in-plane displacements at any point , , denote the in-plane 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 strain-displacement 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 stress-strain 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 in-plane 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 stress-strain relations in the global -- coordinate system can be obtained as

The third-order theory of Hardy [15, 16] satisfies zero transverse shear stresses on the bounding planes.

The equations of motion of the third-order 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 in-plane force resultants, the moment resultants, the shear resultants and and denote the higher-order stress resultants:

where take the symbols .

If needed, the first-order shear deformation theory equations are readily obtained from the third-order equations, just by putting .

4. Numerical Examples

The eigenvalues presented in this problem are expressed in terms of the nondimensional frequency parameter . Unlike first-order 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.

Figure 1: regular grid for discretization of the square laminated plate.

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.

Table 1: Convergence study of frequency parameters, for rectangular isotropic plates with SSSS boundaries, .

Mode

Liew et al. [3]

1
2.0146
2.0089
2.0087
2.0000

2
5.0630
5.0343
5.0218
5.0000

3
5.0644
5.0371
5.0255
5.0000

4
8.1648
8.0717
8.0508
7.9999

5
9.8773
9.9908
10.0138
9.9998

6
10.2520
10.2462
10.0472
9.9998

7
12.7965
12.9815
13.0339
12.9997

8
12.8308
13.1027
13.0579
12.9997

Table 2: Convergence study of frequency parameters, for rectangular isotropic plates with SSSS boundaries, .

Mode

Liew et al. [3]

1
1.7725
1.7700
1.7688
1.7659

2
3.8769
3.8717
3.8698
3.8576

3
3.8770
3.8717
3.8699
3.8576

4
5.6201
5.6063
5.6009
5.5729

5
6.5939
6.6093
6.6149
6.5809

6
6.6098
6.6127
6.6153
6.5809

7
8.0235
8.0092
8.0044
7.9470

8
8.0250
8.0099
8.0046
7.9470

Table 3: Convergence study of frequency parameters, for rectangular isotropic plates with SSSS boundaries, .

Mode

Liew et al. [3]

1
1.2528
1.1294
1.1511
1.1600

2
1.7969
1.5886
1.6694
1.6400

3
3.5791
2.4060
2.4977
2.4400

4
4.1586
2.8928
3.7214
3.5600

5
4.1586
4.5265
4.4835
4.1600

6
6.1938
5.4430
5.1762
4.6400

7
6.2026
5.7588
7.0559
5.0000

8
6.2026
6.4153
7.5293
5.4399

Table 4: Convergence study of frequency parameters, for rectangular isotropic plates with SSSS boundaries, .

Mode

Liew et al. [3]

1
1.0699
1.0722
1.0732
1.0741

2
1.4836
1.4789
1.4702
1.4768

3
2.1267
2.1114
2.1041
2.1059

4
2.9901
2.9429
2.9328
2.9145

5
3.3120
3.3224
3.3288
3.3191

6
3.6449
3.6403
3.6375
3.6306

7
4.0418
3.9265
3.9239
3.8576

8
4.1796
4.1508
4.1451
4.1281

Table 5: Convergence study of frequency parameters, for rectangular isotropic plates with CCCC boundaries, .

Mode

Liew et al. [3]

1
3.9078
3.7202
3.6892
3.6460

2
8.5057
7.7031
7.5817
7.4362

3
8.5057
7.7032
7.5818
7.4362

4
14.7680
11.9358
11.4953
10.9643

5
15.8127
13.7424
13.5536
13.3315

6
17.1194
14.0909
13.7236
13.3947

7
25.6747
18.6394
17.7457
16.7173

8
25.6747
18.6396
17.7458
16.7173

Table 6: Convergence study of frequency parameters, for rectangular isotropic plates with CCCC boundaries, .

Mode

Liew et al. [3]

1
2.5935
2.5956
2.5976
2.6803

2
4.5153
4.5295
4.5366
4.6744

3
4.5153
4.5295
4.5366
4.6744

4
6.0888
6.1042
6.1166
6.2748

5
6.9553
6.9909
7.0045
7.1481

6
7.0281
7.0730
7.0833
7.2466

7
8.2813
8.3096
8.3290
8.4803

8
8.2813
8.3096
8.3290
8.4803

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], first-order 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.

Table 7: Comparison study of frequency parameters, for square isotropic plates with SSSS boundaries.

Method of solution
Mode

(1, 1)
(1, 2)
(2, 1)
(2, 2)
(1, 3)
(3, 1)

0.01
Classical theory [4]
2.000
5.000
5.000
8.000
10.000
10.000

FSDT, Ritz method [5]
1.999
4.995
4.995
7.998
9.981
9.981

Orthogonal polynomials [3]
1.999
4.995
4.995
7.998
9.981
9.981

Present, grid
1.9695
5.1285
5.1286
7.9350
10.4481
10.4689

Present, grid
1.9950
5.0563
5.0563
8.0279
10.1396
10.1719

Present, grid
1.9997
5.0189
5.0189
8.0180
10.0261
10.0464

0.1
Classical theory [4]
2.000
5.000
5.000
8.000
10.000
10.000

3D analysis [6]
1.934
4.662
4.622
7.103
8.662
8.662

FSDT, Ritz method [5]
1.931
4.605
4.605
7.064
8.605
8.605

Orthogonal polynomials [3]
1.931
4.605
4.605
7.064
8.605
8.605

Present, grid
1.9387
4.6608
4.6608
7.1929
8.7181
8.7134

Present, grid
1.9386
4.6448
4.6448
7.1589
8.7070
8.7134

Present, grid
1.9376
4.6394
4.6394
7.1449
8.7091
8.7110

Table 8: Comparison study of frequency parameters, for square isotropic plates with CCCC boundaries, .

Mode

CPT [4]
FSDT (Ritz) [5]
Liew et al. [3]

1, 1
3.2900
3.2478
3.2461
3.646
3.297
3.297
3.292

1, 2
6.2129
6.1577
6.1648
7.436
6.290
6.290
6.276

2, 1
6.2129
6.1577
6.1648
7.436
6.290
6.290
6.276

2, 2
8.8256
8.6192
8.6169
10.964
8.837
8.842
8.792

1, 3
9.8350
10.0951
10.1473
13.333
10.376
10.376
10.356

3, 1
9.9634
10.2240
10.2712
13.395
10.465
10.461
10.455

5. Conclusions

Radial basis functions are increasingly popular in the analysis of science and engineering problems.

In this paper, we use the third-order 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 higher-order 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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