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.
Radial basis functions (RBFs) have recently proved to be an excellent technique for interpolating data and functions. A radial basis function
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 [
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 [
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.
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 [
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
The operator
The radial basis function
Considering
We follow a simple scheme for the solution of the eigenproblem (
We denote interpolation points by
For the boundary conditions we have
Therefore we can write a finite-dimensional problem as a generalized eigenvalue problem
The displacement field for the third-order shear deformation theory of Hardy [
The strain-displacement relationships are given as
Therefore strains can be expressed as
A laminate can be manufactured from orthotropic layers (or plies) of preimpregnated unidirectional fibrous composite materials. Neglecting
By performing adequate coordinate transformation, the stress-strain relations in the global
The third-order theory of Hardy [
The equations of motion of the third-order theory are derived from the principle of virtual displacements:
If needed, the first-order shear deformation theory equations are readily obtained from the third-order equations, just by putting
The eigenvalues presented in this problem are expressed in terms of the nondimensional frequency parameter
In Tables
Convergence study of frequency parameters,
Mode |
|
|
|
Liew et al. [ |
---|---|---|---|---|
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 |
Convergence study of frequency parameters,
Mode |
|
|
|
Liew et al. [ |
---|---|---|---|---|
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 |
Convergence study of frequency parameters,
Mode |
|
|
|
Liew et al. [ |
---|---|---|---|---|
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 |
Convergence study of frequency parameters,
Mode |
|
|
|
Liew et al. [ |
---|---|---|---|---|
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 |
Convergence study of frequency parameters,
Mode |
|
|
|
Liew et al. [ |
---|---|---|---|---|
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 |
Convergence study of frequency parameters,
Mode |
|
|
|
Liew et al. [ |
---|---|---|---|---|
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
Comparison study of frequency parameters,
|
Method of solution | Mode | |||||
---|---|---|---|---|---|---|---|
(1, 1) | (1, 2) | (2, 1) | (2, 2) | (1, 3) | (3, 1) | ||
0.01 | Classical theory [ |
2.000 | 5.000 | 5.000 | 8.000 | 10.000 | 10.000 |
FSDT, Ritz method [ |
1.999 | 4.995 | 4.995 | 7.998 | 9.981 | 9.981 | |
Orthogonal polynomials [ |
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 [ |
2.000 | 5.000 | 5.000 | 8.000 | 10.000 | 10.000 |
3D analysis [ |
1.934 | 4.662 | 4.622 | 7.103 | 8.662 | 8.662 | |
FSDT, Ritz method [ |
1.931 | 4.605 | 4.605 | 7.064 | 8.605 | 8.605 | |
Orthogonal polynomials [ |
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 |
Comparison study of frequency parameters,
Mode |
|
|
|
CPT [ |
FSDT (Ritz) [ |
Liew et al. [ |
|
---|---|---|---|---|---|---|---|
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 |
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.