Convergence of frequency parameters of completely free FGM square plates with different thicknesstowidth ratios
Abstract
Threedimensional (3D) vibration analysis of thick functionally graded plates and cylindrical shells with arbitrary boundary conditions is presented in this chapter. The effective material properties of functionally graded structures vary continuously in the thickness direction according to the simple powerlaw distributions in terms of volume fraction of constituents and are estimated by Voigt’s rule of mixture. By using the artificial spring boundary technique, the general boundary conditions can be obtained by setting proper spring stiffness. All displacements of the functionally graded plates and shells are expanded in the form of the linear superposition of standard 3D cosine series and several supplementary functions, which are introduced to remove potential discontinuity problems with the original displacements along the edge. The RayleighRitz procedure is used to yield the accurate solutions. The convergence, accuracy and reliability of the current formulation are verified by numerical examples and by comparing the current results with those in published literature. Furthermore, the influence of the geometrical parameters and elastic foundation on the frequencies of rectangular plates and cylindrical shells is investigated.
Keywords
 Threedimensional elasticity theory
 functional graded materials
 plate and cylindrical shell
 general boundary conditions
1. Introduction
Functionally graded materials (FGMs) are a new type of composite materials with smooth and continuous variation in material properties in desired directions. This is achieved by gradually varying the volume fraction of the constituent materials. Such materials possess various advantages over conventional composite laminates, such as smaller stress concentration, higher fracture toughness and improved residual stress distribution. Recently, the FGMs have been used to build plate and shell components in various engineering applications, especially mechanical, aerospace, marine and civil engineering. In some cases, those FGM plates and shells are frequently subjected to dynamic loads, which leads to the vibration behaviours, which may cause fatigue damage and result in severe reduction in the strength and stability of the whole structures. Therefore, the vibration analysis of the FGM plates and shells is required and it is important to provide insight into dynamic behaviours and optimal design.
It is well known that vibration problems deal with two main concepts: plate and shell theories and computational approaches. The development of plate and shell theories has been subjected to significant research interest for many years, and many plate and shell theories have been proposed and developed. The main plate and shell theories can be classified into two categories: twodimensional (2D) plate and shell theories, including classic plate and shell theory (CPT) [1–4], the firstorder shear deformation theory (FSDT) [5–16], and the higher order shear deformation theory (HSDT) [17–26], and threedimensional (3D) theory of elasticity [27–35]. However, all 2D theories are approximate because they were developed based on certain kinematic assumptions that result in relatively simple expression and derivation of solutions. Actually, 3D elasticity theory, which does not rely on any hypotheses about the distribution field of deformations and stress, not only provides realistic results but also allows for further physical insight. More attempts have been made for 3D vibration analysis of plates and shells in the recent decades. Furthermore, many analytical, semianalytical and numerical computational methods have also been developed, such as Ritz method, statespace method, differential quadrature method (DQM), Galerkin method, meshless method, finite element method (FEM) and discrete singular convolution (DSC) approach.
However, a close scrutiny of the literature in this field reveals that most investigations were carried out based on 2D plate and shell theories, and a general 3D solution for this subject seems to be limited. Moreover, the review also reveals that most of previous research efforts were restricted to vibration problems of FGM plates and shells with limited sets of classical boundary conditions. It is well recognized that there exist various possible boundary restraint cases for plates and shells in practical assembly and engineering applications. Consequently, it is necessary and of great significance to develop a unified, efficient and accurate method that is capable of universally dealing with FGM plates and shells with general boundary conditions.
In view of these apparent voids, the aim of this chapter is to develop an accurate semianalysis method that is capable of dealing with vibrations of FGM plates and shells with general boundary conditions, including classical boundaries, elastic supports and their combinations and to provide a summary of known 3D results of plates and shells with general boundary conditions, which may serve as benchmark solutions for future researches in this field.
In this chapter, 3D vibration analysis of thick functionally graded plates and cylindrical shells with arbitrary elastic restraints is presented. The effective material properties of functionally graded structures vary continuously in the thickness direction according to the simple powerlaw distributions in terms of volume fraction of constituents and are estimated by Voigt’s rule of mixture. By using the artificial spring boundary technique, the general boundary conditions can be obtained by setting proper spring stiffness. All displacements of the functionally graded plates and shells are expanded in the form of the linear superposition of standard 3D cosine series and several supplementary functions, which are introduced to remove potential discontinuity problems with the original displacements along the edge. The RayleighRitz procedure is used to yield the accurate solutions.
2. Theoretical formulations
2.1. Preliminaries
A differential element of a shell with uniform thickness
where
In engineering applications, plates and shells are the basic structural elements. For the sake of brevity, this chapter will be confined to rectangular plates and cylindrical shells. According to Fig. 2, the coordinate systems and Lame coefficients are given as follows [36]: for rectangular plates,
where
where
where
where
In this work, the general boundary conditions can be described in terms of three groups of springs (
where the superscripts
2.2. Energy functional
The energy functional of plates or shells can be expressed as follows:
where
The kinetic energy
where the over dot represents the differentiation with respect to time.
The strain energy
Substituting Eqs. (1–16) into Eq. (15) together with Lame coefficients, one can obtain the explicit expressions of strain energy for rectangular plates and cylindrical shells.
The potential energy (
where
2.3. Admissible functions
It is crucially important to construct the appropriate admissible displacement functions in the Rayleigh–Ritz method. Beam functions, orthogonal polynomials and Fourier series are often used as displacement functions of plates and shells. However, the use of beam function will lead to at least a very tedious solution process [38]. The problem with using a complete set of orthogonal polynomials is that the higherorder polynomials tend to become numerically unstable because of the computer roundoff errors [38, 39]. These numerical difficulties can be avoided by the Fourier series because the Fourier series constitute a complete set and exhibit an excellent numerical stability. However, when the displacements are expressed in terms of conventional Fourier series, discontinuities potentially exist in the original displacements and their derivatives. In this chapter, a modified Fourier series defined as the linear superposition of a 3D Fourier cosine series and some auxiliary polynomial functions is used to express the displacement components, which are given as follows [40–43]:
where
It is easy to verify that
The similar conditions exist for the
2.4. Solution procedure
Substituting Eqs. (14–16) into Eq. (13) together with the displacement functions defined in Eqs. (17–19) and performing the Rayleigh–Ritz operation, a set of linear algebraic equation against the unknown coefficients can be obtained as follows:
where
where
The frequencies can be determined by solving Eq. (25) via the eigenfunction of MATLAB program. The mode shape corresponding to each frequency can be obtained by back substituting the eigenvector to the displacement functions in Eqs. (17–19).
3. Numerical examples and discussion
In this section, several vibration results of FGM plates and cylindrical shells with general boundary conditions are presented to illustrate the accuracy and reliability of the current formulation. To simplify presentation, C, S, F and E denote the clamped, simply supported, free and elastic restraints. Three types of elastic boundary conditions designated by symbols E_{1}, E_{2} and E_{3} are considered. E_{1}type edge is considered to be elastic in normal direction; the support type E_{2} only allows elastically restrained displacement in both tangential directions; when all of three displacements along the edges are elastically restrained, the edge support is defined by E_{3}. The expressions of the different boundary conditions along the edge
Free boundary condition (F):
Clamped boundary condition (C):
Simply supported boundary condition (S):
First type of elastic restraint (E_{1}):
Second type of elastic restraint (E_{1}):
Three type of elastic restraint (E_{1}):
A simple letter is used to describe the boundary conditions of structure. For example, SFCE denotes a plate having simply supported boundary condition at
3.1. Rectangular plates
In this section, several numerical examples concerning the free vibration of FGM rectangular plates with different geometrical parameters and boundary conditions have been investigated to verify the convergence, accuracy and reliability of the present method. Some new vibration results of rectangular plates with elastic boundary conditions are given. Unless stated otherwise, the material properties for ceramic and metallic constituents of FGM plates are given as follows:
3.1.1. Convergence study
Theoretically, there are infinite terms in the modified Fourier series solution. However, the series is numerically truncated, and only finite terms are counted in actual calculations. The convergence of this method will be checked. Table 1 presents the first seven frequency parameters Ω of completely free FGM square plates. The frequency parameter Ω is defined as follows:
The geometrical parameters are given as follows:









0.1  9 × 9 × 4  2.9579  4.3853  5.4058  7.4361  7.4361  12.903  12.903 
11 × 11 × 4  2.9558  4.3851  5.4054  7.4293  7.4293  12.901  12.901  
11 × 11 × 8  2.9524  4.3802  5.4018  7.4256  7.4256  12.898  12.898  
13 × 13 × 8  2.9514  4.3802  5.4016  7.4225  7.4225  12.897  12.897  
13 × 13 × 10  2.9513  4.3800  5.4015  7.4223  7.4223  12.897  12.897  
0.2  9 × 9 × 4  2.7261  4.0298  4.9324  6.4506  6.4506  10.093  10.642 
11 × 11 × 4  2.7257  4.0297  4.9322  6.4492  6.4492  10.093  10.640  
11 × 11 × 8  2.7250  4.0287  4.9315  6.4485  6.4485  10.093  10.639  
13 × 13 × 8  2.7247  4.0286  4.9314  6.4479  6.4479  10.093  10.637  
13 × 13 × 10  2.7247  4.0286  4.9313  6.4478  6.4478  10.093  10.637  
0.5  9 × 9 × 4  2.0442  2.8571  3.4668  3.9777  3.9777  4.0369  4.3056 
11 × 11 × 4  2.0442  2.8571  3.4667  3.9776  3.9776  4.0369  4.3055  
11 × 11 × 8  2.0441  2.8568  3.4665  3.9773  3.9773  4.0368  4.3053  
13 × 13 × 8  2.0440  2.8568  3.4665  3.9772  3.9772  4.0368  4.3052  
13 × 13 × 10  2.0440  2.8568  3.4664  3.9772  3.9772  4.0368  4.3052 
As aforementioned, the boundary conditions can be easily obtained via changing the value of boundary springs. Therefore, the accuracy of the current method is strongly influenced by the values of springs’ stiffness. To determine the appropriate values of spring’s stiffness, the effects of elastic parameters on the frequencies of the FGM plate are investigated. The elastic parameter















10^{−1}  0.0167  0.0464  0.0654  0.0167  0.0654  0.0802  0.0650  0.0654  0.1117 
10^{0}  0.0419  0.1463  0.2069  0.0419  0.2069  0.2533  0.2028  0.2061  0.3514 
10^{1}  0.1285  0.4615  0.6536  0.1286  0.6513  0.7985  0.6294  0.6301  1.1044 
10^{2}  0.3984  1.4270  2.0440  0.3996  1.9750  2.4251  1.5056  1.7178  3.2905 
10^{3}  1.0903  2.8583  3.7961  1.1184  3.0710  4.2125  2.0002  2.7732  5.9372 
10^{4}  1.8471  3.1625  4.5503  2.0317  3.8426  4.3529  2.0785  3.0227  6.4327 
10^{5}  2.0625  3.2758  4.5745  2.3675  4.2459  4.3815  2.0880  3.0600  6.5168 
10^{6}  2.0904  3.2915  4.5774  2.4305  4.3436  4.3859  2.0891  3.0657  6.5299 
10^{7}  2.0934  3.2932  4.5777  2.4464  4.3740  4.3870  2.0893  3.0664  6.5316 
10^{8}  2.0937  3.2934  4.5778  2.4490  4.3790  4.3871  2.0893  3.0665  6.5318 
10^{9}  2.0937  3.2934  4.5778  2.4493  4.3796  4.3871  2.0893  3.0665  6.5319 
10^{10}  2.0937  3.2934  4.5778  2.4493  4.3796  4.3871  2.0893  3.0665  6.5319 
3.1.2. Plate with general boundary conditions
To illustrate the accuracy of the present method, the comparisons of the current results with those in the published literature are presented. Table 3 presents the first two frequency parameters of the FGM square plates with different boundary conditions. The results are compared with those presented by Huang et al. [32] using the Ritz method on the basis of 3D elasticity theory. Table 4 presents the fundamental frequency parameters of the FGM square plates with SSSS boundary conditions. Numerical vibration results for the same problems have been reported by HosseiniHashemi et al. [18] and Matsunaga [20] using HSDTs, showing that excellent agreement of the results is achieved.














Ω_{1}  3.406  3.406  0.000  0.6637  0.6657  0.347  3.400  3.421  0.618 
Ω_{2}  6.296  6.296  0.000  1.432  1.434  0.140  3.820  3.840  0.524 
Ω_{3}  6.296  6.296  0.000  2.154  2.158  0.186  5.774  5.787  0.225 
Ω_{4}  7.347  7.345  0.027  3.396  3.405  0.265  5.976  5.989  0.218 
Ω_{5}  7.347  7.345  0.027  4.347  4.348  0.023  7.609  7.657  0.631 








0.1  Ref. [18]  0.0577  0.0490  0.0443  0.0381  0.0364  0.0293 
Ref. [20]  0.0577  0.0492  0.0442  0.0381  0.0364  0.0293  
Present  0.0578  0.0491  0.0443  0.0381  0.0364  0.0294  
0.2  Ref. [18]  0.2113  0.1807  0.1631  0.1378  0.1301  0.1076 
Ref. [20]  0.2121  0.1819  0.1640  0.1383  0.1306  0.1077  
Present  0.2122  0.1816  0.1640  0.1383  0.1306  0.1080 














CSSS  0.1  5.660  5.235  4.434  4.272  12.66  11.71  10.01  9.671 
0.3  4.415  4.096  3.263  3.083  11.01  10.20  8.354  7.676  
0.5  3.391  3.156  2.435  2.265  6.866  6.464  5.023  4.591  
CCSS  0.1  6.416  5.936  5.008  4.819  17.17  15.88  13.48  13.00 
0.3  4.807  4.464  3.520  3.313  13.68  12.69  10.12  9.566  
0.5  3.582  3.336  2.549  2.366  10.54  9.807  7.548  7.037  
CCCS  0.1  7.437  6.884  5.772  5.544  17.71  16.38  13.90  13.40 
0.3  5.264  4.894  3.815  3.572  14.01  13.00  10.36  9.782  
0.5  3.797  3.540  2.681  2.479  10.75  10.00  7.679  7.153  
CFFF  0.1  0.864  0.799  0.687  0.664  0.862  0.797  0.687  0.665 
0.3  0.816  0.755  0.637  0.613  0.845  0.781  0.669  0.647  
0.5  0.746  0.690  0.568  0.543  0.821  0.759  0.645  0.622  
CCFF  0.1  1.684  1.558  1.330  1.285  4.230  3.911  3.356  3.246 
0.3  1.473  1.363  1.125  1.076  3.900  3.608  3.021  2.902  
0.5  1.253  1.160  0.932  0.885  3.479  3.220  2.627  2.506  
CCCF  0.1  5.643  5.223  4.393  4.222  7.628  7.056  6.031  5.825 
0.3  4.074  3.785  2.956  2.774  6.594  6.107  5.009  4.781  
0.5  2.944  2.738  2.080  1.932  5.503  5.102  4.053  3.831 
Several new numerical results for free vibration of FGM plates with general boundary conditions, including classical and elastic boundary conditions, are presented in Tables 5 and 6. The geometrical parameters are given as:














E_{1}E_{1}E_{1}E_{1}  0.1  4.798  4.437  3.786  3.655  12.138  11.228  9.626  9.303 
0.3  4.113  3.837  3.142  2.983  11.145  10.426  8.736  8.339  
0.5  3.444  3.236  2.522  2.352  10.019  9.439  7.495  7.035  
E_{2}E_{2}E_{2}E_{2}  0.1  1.975  2.006  2.090  2.109  6.541  6.592  6.748  6.780 
0.3  3.043  3.025  2.844  2.773  9.601  9.476  8.804  8.569  
0.5  3.083  2.979  2.487  2.343  9.597  9.238  7.680  7.238  
E_{3}E_{3}E_{3}E_{3}  0.1  1.823  1.823  1.828  1.826  5.690  5.649  5.586  5.563 
0.3  2.594  2.541  2.357  2.298  7.867  7.670  7.063  6.874  
0.5  2.724  2.634  2.234  2.123  8.308  8.014  6.799  6.471 
3.2. Cylindrical shells
This section is concerned with the free vibration of FGM cylindrical shells with different boundary conditions. The convergence, accuracy and reliability of the present method are demonstrated by numerical examples and comparisons. New numerical results for the FGM cylindrical shells with the elastic boundary conditions are also presented. Unless stated otherwise the material properties for ceramic and metallic constituents of FGM cylindrical shells are given as follows:
3.2.1. Convergence study
The convergence studies of the first two frequencies for the completely free cylindrical shells with different circumferential wave numbers
















0.1  10 × 10  675.95  775.84  72.216  93.963  202.45  236.49  383.66  422.39 
11 × 11  675.95  775.84  72.213  93.900  202.44  236.37  383.64  422.20  
12 × 12  675.95  775.82  72.211  93.898  202.43  236.36  383.61  422.19  
13 × 13  675.95  775.82  72.209  93.859  202.42  236.28  383.60  422.07  
14 × 14  675.95  775.81  72.208  93.858  202.42  236.28  383.58  422.06  
0.2  10 × 10  702.69  829.15  156.06  195.32  426.43  484.16  782.43  843.50 
11 × 11  702.69  829.14  156.06  195.27  426.42  484.05  782.41  843.32  
12 × 12  702.69  829.12  156.05  195.27  426.41  484.04  782.37  843.29  
13 × 13  702.68  829.12  156.05  195.24  426.40  483.98  782.36  843.19  
14 × 14  702.68  829.11  156.05  195.24  426.39  483.97  782.34  843.18  
0.5  10 × 10  813.89  990.82  472.71  513.27  1119.84  1157.84  1798.22  1821.60 
11 × 11  813.88  990.82  472.71  513.25  1119.82  1157.80  1798.18  1821.52  
12 × 12  813.88  990.81  472.70  513.25  1119.81  1157.78  1798.15  1821.49  
13 × 13  813.88  990.81  472.70  513.24  1119.80  1157.76  1798.13  1821.45  
14 × 14  813.88  990.81  472.70  513.24  1119.80  1157.75  1798.12  1821.43 















10^{−1}  6.0482  156.26  426.50  6.0495  156.10  426.42  6.3136  156.05  426.40 
10^{0}  19.080  158.13  427.41  19.120  156.58  426.54  19.945  156.05  426.41 
10^{1}  58.929  174.72  435.81  60.159  161.25  427.76  62.811  156.06  426.42 
10^{2}  153.57  251.42  482.58  181.20  197.82  438.59  190.65  156.13  426.59 
10^{3}  233.55  318.79  529.79  404.21  306.64  485.62  424.98  156.65  427.85 
10^{4}  250.70  330.93  538.16  506.09  361.57  523.38  510.67  157.86  430.84 
10^{5}  253.21  332.47  539.14  519.80  369.75  532.12  521.32  158.32  432.03 
10^{6}  253.55  332.66  539.26  521.22  370.71  533.44  522.45  158.38  432.19 
10^{7}  253.62  332.69  539.28  521.37  370.89  533.83  522.56  158.39  432.20 
10^{8}  253.65  332.70  539.29  521.38  370.94  533.98  522.58  158.39  432.21 
10^{9}  253.66  332.71  539.29  521.38  370.95  534.00  522.58  158.39  432.21 
10^{10}  253.66  332.71  539.29  521.38  370.95  534.00  522.58  158.39  432.21 
It is significant to investigate the effects of elastic parameters on the frequencies of the cylindrical shells. The cylindrical shells are restrained by only one kind of spring whose stiffness parameter ranges from 10^{−1} to 10^{10} at
3.2.2. Cylindrical shells with general boundary conditions
To illustrate the accuracy of the present method, the comparisons of the current results with those in published literature are presented. Table 9 presents the first three frequency parameters
Several new numerical results for free vibration of FGM cylindrical shells with general boundary conditions, including classical and elastic boundary conditions, are presented in Tables 11 and 12. The geometrical parameters are given as follows:
















CC  1  1.7860  1.7972  1.7905  2.6043  2.6222  2.6050  3.4148  3.4192  3.4245 
2  1.7452  1.7573  1.7500  3.2942  3.3114  3.2949  3.4921  3.5150  3.5012  
3  1.8867  1.8862  1.8912  3.6024  3.6320  3.6099  3.9416  3.9257  3.9447  
4  2.1966  2.2072  2.2004  3.8126  3.8228  3.8193  4.2757  4.3215  4.2783  
5  2.6385  2.6617  2.6415  4.1302  4.1327  4.1364  4.7010  4.7322  4.7031  
CF  1  0.7514  0.7546  0.7516  1.7563  1.7692  1.7568  1.8800  1.8996  1.8812 
2  0.6620  0.6713  0.6622  1.8962  1.9256  1.8980  2.1305  2.1557  2.1324  
3  0.9246  0.9301  0.9247  2.0610  2.0668  2.0630  2.5165  2.5482  2.5179  
4  1.4021  1.4282  1.4021  2.4030  2.4646  2.4049  2.9919  3.0342  2.9930  
5  1.9814  2.0228  1.9814  2.8666  2.8571  2.8684  3.5251  3.5628  3.5258  
FF  1  0.0000  0.0000  0.0000  0.0001  0.0001  0.0003  1.0710  1.0734  1.0709 
2  0.2576  0.2608  0.2576  0.3800  0.3831  0.3799  1.3533  1.3594  1.3532  
3  0.6884  0.6890  0.6884  0.9253  0.9377  0.9252  1.8689  1.8794  1.8689  
4  1.2302  1.2525  1.2302  1.5160  1.5307  1.5158  2.4754  2.4917  2.4753  
5  1.8427  1.8694  1.8426  2.1343  2.1532  2.1341  3.1169  3.1417  3.1169 














f_{1}  152.93  152.13  150.03  148.67  149.29  147.76  148.75  147.10 
f_{2}  152.93  152.13  150.03  148.67  149.29  147.76  148.75  147.10 
f_{3}  220.06  219.31  212.94  211.89  212.22  211.00  219.49  218.00 
f_{4}  220.06  219.31  212.94  211.89  212.22  211.00  219.49  218.00 
f_{5}  253.78  254.30  250.74  250.36  249.31  248.68  243.43  242.86 
f_{6}  253.78  254.30  250.74  250.36  249.31  248.68  243.43  242.86 
f_{7}  383.55  384.04  370.63  370.69  369.46  369.21  383.71  382.79 
f_{8}  383.55  384.04  370.63  370.69  369.46  369.21  383.71  382.79 
f_{9}  420.51  420.86  415.47  414.68  412.97  411.88  402.56  401.57 
f_{10}  431.45  428.75  420.39  416.91  418.46  414.66  423.57  419.16 









CC  0.1  390.33  377.23  374.78  375.39  379.94  379.62  376.24 
0.3  621.15  599.34  594.11  591.68  594.96  595.70  593.23  
0.5  701.10  686.38  681.10  674.46  668.80  666.26  663.96  
CS  0.1  378.25  365.82  363.43  363.82  367.81  367.48  364.19 
0.3  567.55  548.06  543.23  540.49  542.33  542.55  540.43  
0.5  648.20  634.72  629.77  623.22  617.07  614.23  612.09  
SS  0.1  367.68  355.76  353.44  353.68  357.26  356.85  353.71 
0.3  528.69  510.67  506.15  503.35  504.53  504.51  502.59  
0.5  608.85  596.84  592.29  585.94  579.40  576.22  574.05  
CF  0.1  153.27  149.63  148.75  148.30  148.46  147.80  146.56 
0.3  257.65  254.87  253.62  251.58  248.56  246.32  244.47  
0.5  263.06  261.04  260.07  258.44  255.66  253.17  250.86  
SF  0.1  149.92  146.38  145.51  145.06  145.17  144.51  143.30 
0.3  247.41  245.02  243.86  241.81  238.61  236.30  234.48  
0.5  248.50  246.92  246.06  244.45  241.51  238.95  236.67 









E_{1}E_{1}  0.1  368.31  356.53  354.28  354.66  358.44  358.13  355.04 
0.3  558.81  545.07  542.41  542.83  548.35  550.42  549.32  
0.5  669.03  660.42  657.10  652.86  649.27  647.64  645.98  
E_{2}E_{2}  0.1  82.564  91.860  95.778  101.94  109.61  113.78  116.41 
0.3  381.14  384.20  386.62  392.04  400.91  405.96  408.78  
0.5  539.57  539.80  540.83  544.12  551.17  556.07  559.18  
E_{3}E_{3}  0.1  80.874  89.444  93.013  98.592  105.46  109.13  111.41 
0.3  331.30  347.67  354.18  364.51  377.62  384.51  388.49  
0.5  515.44  522.81  525.80  531.29  539.88  545.19  548.45 
4. Conclusions
A new 3D exact solution for free vibration analysis of thick functionally graded plates and cylindrical shells with arbitrary boundary conditions is presented in this chapter. The effective material properties of functionally graded structures vary continuously in the thickness direction according to the simple powerlaw distributions in terms of volume fraction of constituents and are estimated by Voigt’s rule of mixture. By using the artificial spring boundary technique, the general boundary conditions can be obtained by setting proper spring stiffness. All displacements of the functionally graded plates and shells are expanded in the form of the linear superposition of standard 3D cosine series and several supplementary functions, which are introduced to remove potential discontinuity problems with the original displacements along the edge. The RayleighRitz procedure is used to yield the accurate solutions. The convergence, accuracy and reliability of this formulation are verified by numerical examples and by comparng the current results with those in published literature. The influence of the geometrical parameters and elastic foundation on the frequencies of rectangular plates and cylindrical shells is investigated.
Acknowledgments
The authors gratefully acknowledge the financial support from the National Natural Science Foundation of China (Nos. 51175098 and 51279035) and the Fundamental Research Funds for the Central Universities of China (No. HEUCFQ1401).
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