Abstract
Due to the high specific strength and stiffness, laminated composite plates, especially midplane symmetric laminated composite plates, are frequently used as structural component in aeronautical and aerospace engineering. Since obtaining analytic solutions are difficult even for simply supported midplane symmetric laminated composite plates, numerical methods have to be used to obtain approximate solutions. To evaluate various numerical methods, benchmark solutions are needed. In this article, highly accurate frequencies of simply supported angleply midplane symmetric laminated composite plates with two sets of equivalent material properties are obtained by the modified differential quadrature method and presented to serve as the benchmark solutions.
1. Introduction
Due to the high specific strength and stiffness, laminated composite plates are frequently used as structural component in aeronautical and aerospace engineering. Their static, buckling and free vibration behavior is of important to the designers and thus has been received great attentions [1, 2]. Among various types of laminations, the midplane symmetric laminates are widely used in practice. The decoupling of inplane and outofplane deformation makes the production of a flat plate as well as analysis much simpler than the general laminations.
In free vibration analysis, the laminated composite plates are usually equivalent to anisotropic plates. Analytical solutions are rarely available even for rectangular anisotropic plates with simple supported boundary conditions. Therefore, various approximate approaches [14] and numerical methods [59] have been employed for solutions.
In literature, two equivalent ways of expressing the material properties are commonly used. Take the Eglass/epoxy (E/E) material as an example, the material properties expressed in one way, called the material system I (MSI), are ${E}_{1}=$ 60.7 GPa, ${E}_{2}=$ 24.8 GPa, ${G}_{12}=$ 12.0 GPa and ${\nu}_{12}=$ 0.23 [3, 8, 10], and the material properties expressed in the other way, called the material system II (MSII), are ${E}_{1}/{E}_{2}=$ 2.45, ${G}_{12}/{E}_{2}=$ 0.48 and ${\nu}_{12}=$ 0.23 [4, 6, 7]. Researchers often do not distinguish one set of material properties from the other since they are regarded equivalent. The choice mainly depends on their personal preference. In references [4, 6, 7], the results are obtained based on the MSII, but compared with the upper bound solutions with the MSI [3]. In reference [8], the material properties of MSI are given, but the data are actually obtained with the ones with MSII. Occasionally this might cause misunderstanding to the readers, although it is not difficult to tell that MSII is actually used in their calculation by looking at the exact solutions of special orthotropic rectangular plates, since the exact solutions for the two sets of equivalent material systems are slightly different and the corresponding Ritz solutions of special orthotropic plates reported in [3] are also exact solutions [10]. The difference in solutions for the laminated composite plates with the two sets of equivalent material properties is really small and negligible from the practical point of view.
From the computational point of view, however, the small difference may be important in testing the accuracy and efficiency of new numerical methods. Very accurate benchmark solutions are required in such cases. The data reported in [3, 4] are not very accurate due to either the lower rate of convergence of the method or the extra constraints implicitly enforced in the test functions [11]. More terms in the series of the test functions are needed to obtained solutions with higher accuracy by using the Ritz method.
The primary objective of this paper is to provide highly accurate benchmark frequencies for simply supported square laminated composite plates with two sets of equivalent material properties. The modified differential quadrature method proposed by the author is used to obtain accurate solutions. The slight difference in the frequencies of the midplane symmetric laminates with two sets of equivalent material systems is clearly demonstrated.
2. Basic equations and solution procedures
2.1. Governing equation and expression of boundary condition
Denote the length, width and total thickness of the rectangular laminated composite plate by $a$, $b$, and $h$. The governing equation for the free vibration analysis of a midplane symmetric laminated composite plate is given by:
where ${\stackrel{}{D}}_{ij}$ are the effective bending and twisting stiffness [12], $w\left(x,y\right)$ is the deflection, $\rho $ and $\omega $ are the mass density and circular frequency, respectively.
The expressions of simply supported boundary conditions are:
where the expressions of bending moments ${M}_{x}$ and ${M}_{y}$ are:
2.2. Modified differential quadrature method and solution procedures
For completeness considerations, the modified differential quadrature method (modified DQM) and solution procedures are briefly introduced.
Denote ${N}_{x}$ and ${N}_{y}$the numbers of grid points in $x$ and $y$ directions, and (${x}_{i}$, ${y}_{j}$) ($i=$ 1, 2,..., ${N}_{x}$; $j=$ 1, 2,..., ${N}_{y}$) the grid points. In the modified DQM, two additional derivative degrees of freedom at end points are introduced by using the method of modification of weighting coefficient3 (MMWC3) proposed by the author [9].
For simplicity and demonstration of the method, take a onedimensional problem as an example. In the ordinary differential quadrature method, the first order derivative of the solution $w\left(x\right)$ with respect to $x$ at grid point ${x}_{i}$ is approximated as:
where ${A}_{ij}^{x}$ is called the weighting coefficient, which can be explicitly computed by:
To apply multiple boundary conditions rigorously, two additional degrees of freedom (DOFs) are introduced during formulation the weighting coefficient of the second order derivatives at two end points by using the MMWC3 [9], namely:
$=\sum _{j=1}^{{N}_{x}}{\stackrel{~}{B}}_{ij}^{x}{w}_{j}+{A}_{i1}^{x}{w}_{1}^{\text{'}}+{A}_{iN}^{x}{w}_{N}^{\text{'}}=\sum _{j=1}^{N+2}{\stackrel{~}{B}}_{ij}^{x}{\delta}_{j},\left(i=1,{N}_{x}\right),$
where ${\delta}_{j}={w}_{j}\text{}\text{(}j=\text{1, 2, ...,}{N}_{x}\text{)}$, ${\delta}_{N+1}={w}_{1}^{\text{'}}$, ${\delta}_{N+2}={w\text{'}}_{N}$, ${\stackrel{~}{B}}_{i(N+1)}^{x}={A}_{i1}^{x}$, ${\stackrel{~}{B}}_{i(N+2)}^{x}={A}_{iN}^{x}$.
At all inner points, the weighting coefficients of the second order derivative are the same as the ordinary DQM, namely:
In the modified DQM, the weighting coefficients of the third and the fourth order derivatives, denoted by ${\stackrel{~}{C}}_{ij}^{x}$, ${\stackrel{~}{D}}_{ij}^{x}$ are computed by:
The weighting coefficients of the first to fourthorder derivatives with respect to $y$ can be calculated in a similar way, simply replacing $x$ and ${N}_{x}$ in Eq. (5) to Eq. (9) by $y$ and ${N}_{y}$. Since only square plates ($a=b$) are considered, thus ${N}_{x}={N}_{y}=N$. In terms of the modified differential quadrature (DQ), the bending moments at corresponding boundary points can be expressed as:
In terms of the DQ, the governing equation at all grid points can be expressed as:
where superscripts $x$ and $y$ mean that the weighting coefficients of the corresponding derivatives are taken with respect to x and $y$, ${\stackrel{}{w}}_{ik}$ contains the deflection ${w}_{il}$ as well as the firstorder derivative with respect to $x$ or $y$ along boundary points, introduced by the method of modification of weighting coefficient3 (MMWC3), ${\stackrel{~}{w}}_{kl}$, ${\stackrel{~}{w}}_{jk}$ and ${\stackrel{~}{w}}_{ik}$ are only a part of ${\stackrel{}{w}}_{ik}$. There are $\left(N+2\right)\times \left(N+2\right)4$ degrees of freedom (DOFs) in total. From Eq. (7), it is clearly seen that ${B}_{ij}^{x}$$(i=1,N)$ are different from ${\stackrel{~}{B}}_{ij}^{x}$$(i=1,N)$, and ${B}_{lk}^{y}$$(l=1,\text{}N)$ are different from ${\stackrel{~}{B}}_{lk}^{y}$$(l=1,\text{}N)$.
The bending moment equation is placed at the position where the DOF of the firstorder derivative with respect to $x$ or $y$ at corresponding boundary point is. Enforcing the simply supported boundary conditions rigorously yields following partitioned matrix equations, namely:
where $\mathrm{\Omega}=\omega {a}^{2}\sqrt{\rho h/{D}_{0}}$ is called the frequency parameter, ${D}_{0}={E}_{1}{h}^{3}/\left[12\right(1{\nu}_{12}{\nu}_{21}\left)\right]$, ${E}_{1}$, ${v}_{12}$ and ${v}_{21}$ are the modulus of elasticity in the fiber direction, as well as the major and minor Poisson’s ratios, respectively. The vector $\left\{{w}_{\alpha}\right\}$ contains only the nonzero DOFs of the deflection at all inner grid points and its dimension is $\left(N\u20132\right)\times \left(N\u20132\right)$.
After eliminating $\left\{{w}_{\beta}\right\}$, Eq. (13) can be rewritten in the following matrix equation:
where $\left[\stackrel{}{K}\right]=[{K}_{\alpha \alpha}{K}_{\alpha \beta}{K}_{\beta \beta}^{1}{K}_{\beta \alpha}]$.
Solving Eq. (14) by a standard eigensolver yields the frequency parameters.
To achieve the fastest rate of convergence and obtain reliable and accurate solutions, following grid points are used in the modified DQM:
The exact frequency parameters ($\mathrm{\Omega}$) for especially orthotropic rectangular plates can be calculated analytically by [1, 3]:
where $m$ and $n$ are the half wave number of the vibration mode in $x$ and $y$ directions, respectively.
3. Results and discussion
Three materials of lamina, i.e., Eglass/epoxy (E/E), Boron/epoxy (B/E) and Graphite/epoxy (G/E), are considered. The material parameters directly taken from [3, 4] are listed in Table 1. For each material, two sets of equivalent material constants are given. Among the three materials, Graphite/epoxy exhibits the highest anisotropy, since ${E}_{1}/{E}_{2}$ is the largest.
Table 1Material property of two sets of equivalent material constants
Materials  Material system I (MSI) [3]  Material system II (MSII) [4]  
${E}_{1}$ (GPa)  ${E}_{2}$ (GPa)  ${G}_{12}$ (GPa)  ${\nu}_{12}$  ${E}_{1}/{E}_{2}$  ${G}_{12}/{E}_{2}$  ${\nu}_{12}$  
E/E  60.7  24.8  12.0  0.23  2.45  0.48  0.23 
B/E  209.  19.0  6.40  0.21  11.0  0.34  0.21 
G/E  138.  8.96  7.10  0.30  15.4  0.79  0.30 
Denote $\theta $ the fiber orientation angle. Four angles, i.e., $\theta =$ 0°, 15°, 30° and 45°, are considered. The relative bendingtwisting coupling coefficients ${D}_{16}/{D}_{0}$ and ${D}_{26}/{D}_{0}$, which reflect the degrees of anisotropy, are listed in Table 2.
Table 2Relative bendingtwisting coefficients of angleply (θ/θ/θ) laminated plates
$\theta $°  Eglass/epoxy (E/E)  Boron/epoxy (B/E)  Graphite/epoxy (G/E)  
${D}_{16}/{D}_{0}$  ${D}_{26}/{D}_{0}$  ${D}_{16}/{D}_{0}$  ${D}_{26}/{D}_{0}$  ${D}_{16}/{D}_{0}$  ${D}_{26}/{D}_{0}$  
0  0.000000  0.000000  0.000000  0.000000  0.000000  0.000000 
15  0.122312  0.012555  0.214391  0.012882  0.205801  0.027967 
30  0.176432  0.079666  0.297578  0.096069  0.291366  0.113533 
45  0.147858  0.147858  0.227273  0.227273  0.233768  0.233768 
It is seen that ${D}_{16}/{D}_{0}$ is the largest when$\theta =$ 30°. This perhaps is the reason why the convergence study is performed for $\theta =$ 30° in [3], since the higher the anisotropy, the lower the rate of convergence for various approximate and numerical methods. Although ${D}_{16}/{D}_{0}$ is the second largest when $\theta =$ 45°, however, ${D}_{26}/{D}_{0}$ is the largest. Thus, convergence studies are performed for both $\theta =$ 30°and $\theta =$ 45° in present investigations. Corresponding results are listed in Table 3 and Table 4, respectively. Midplane symmetric angleply square plates with all edges simply supported, denoted by SSSS, are investigated.
From Table 3 and Table 4, it is clearly seen that the rate of convergence of the DQM is high. The rate of convergence of the DQM for $\theta =$ 30° is higher than the one for $\theta =$ 45°. This indicates that the anisotropy of the (45°/–45°/45°) square plates is higher than the one of the (30°/–30°/30°) square plates for the same material and the anisotropy of the graphite/epoxy square plates with (45°/–45°/45°) is the highest.
To ensure the high accuracy of solutions, the frequency parameters of threelayer angleply ($\theta /\theta /\theta $) square plates with all edges simply supported are obtained by the modified DQM with 31×31 grid points and are presented in Tables 57. The DQ solutions contain results using two sets of equivalent material constants listed in Table 1 and are all below the upper bound solutions cited from [3]. Note that the Ritz data reported in [3] are exact only for the case of $\theta =$ 0°.
In Table 5, Table 6, and Table 7, the exact solutions for $\theta =$ 0° are recomputed by using Eq. (16) with the corresponding material constants, since the existing exact solutions are only accurate to two places of decimals. It is observed that the DQ data are exactly the same as the recomputed exact solutions. The exact solutions with MSI of materials E/E and G/E are slightly higher than the corresponding ones with MSII, and the exact solutions with MSI of material B/E are slightly lower than the corresponding ones with MSII. This trend remains the same in the DQ solutions for other fiber orientation angles. It seems that this trend is mainly caused by the difference of ${G}_{12}$, since ${G}_{12}$ in MSI of materials E/E and G/E is also slightly larger than ${G}_{12}$ in MSII and ${G}_{12}$ in MSI of material B/E is smaller than ${G}_{12}$ in MSII.
Table 3Convergence of frequency parameters for angleply (30°/–30°/30°) SSSS square plates (MSI)
Material  $N$  Mode numbers  
1  2  3  4  5  6  7  8  
E/E  11  15.8619  35.8018  42.5515  61.3169  71.6273  85.6521  93.5636  108.7378 
15  15.8621  35.8021  42.5519  61.3176  71.6287  85.6529  93.5625  108.7262  
19  15.8621  35.8021  42.5520  61.3177  71.6288  85.6529  93.5626  108.7263  
23  15.8622  35.8021  42.5520  61.3177  71.6289  85.6530  93.5626  108.7264  
27  15.8622  35.8021  42.5520  61.3177  71.6289  85.6530  93.5626  108.7264  
[3]  15.90  35.86  42.62  61.45  71.71  85.72  93.74  108.9  
B/E  11  11.9625  22.4074  35.4364  37.4339  49.2075  55.9908  70.5661  73.0071 
15  11.9648  22.4100  35.4424  37.4329  49.2104  55.9665  70.4988  72.9975  
19  11.9655  22.4109  35.4444  37.4329  49.2123  55.9665  70.4998  72.9982  
23  11.9658  22.4112  35.4453  37.4329  49.2132  55.9665  70.5002  72.9986  
27  11.9659  22.4114  35.4457  37.4329  49.2136  55.9665  70.5005  72.9988  
[3]  12.21  22.78  35.86  37.90  50.04  56.70  71.36  73.57  
G/E  11  11.6857  21.5346  35.4172  35.5276  48.6468  52.6563  69.1293  71.4666 
15  11.6894  21.5392  35.4255  35.5259  48.6519  52.6272  69.0619  71.4086  
19  11.6906  21.5407  35.4286  35.5259  48.6552  52.6272  69.0637  71.4088  
23  11.6911  21.5414  35.4299  35.5259  48.6569  52.6272  69.0645  71.4089  
27  11.6914  21.5417  35.4306  35.5259  48.6576  52.6272  69.0649  71.4090  
[3]  11.97  21.97  35.88  36.04  49.60  53.43  70.04  72.35 
Table 4Convergence of frequency parameters for angleply (45°/–45°/45°) SSSS square plates (MSI)
Material  $N$  Mode numbers  
1  2  3  4  5  6  7  8  
E/E  11  16.0871  36.8624  41.7104  61.6715  76.9472  79.8778  94.4474  108.7482 
15  16.0876  36.8626  41.7116  61.6726  76.9474  79.8804  94.4454  108.7347  
19  16.0877  36.8627  41.7120  61.6728  76.9474  79.8810  94.4456  108.7352  
23  16.0878  36.8627  41.7121  61.6728  76.9474  79.8812  94.4456  108.7354  
27  16.0878  36.8627  41.7121  61.6728  76.9474  79.8813  94.4456  108.7355  
[3]  16.14  36.93  41.81  61.85  77.04  80.00  94.68  109.0  
B/E  11  12.3054  24.1007  33.5834  39.5290  53.7288  58.3472  64.9806  76.8154 
15  12.3196  24.1000  33.6269  39.5290  53.7162  58.3201  65.0412  76.7477  
19  12.3253  24.0999  33.6443  39.5297  53.7162  58.3198  65.0666  76.7506  
23  12.3281  24.0999  33.6528  39.5301  53.7162  58.3197  65.0794  76.7522  
27  12.3298  24.0999  33.6576  39.5303  53.7162  58.3197  65.0868  76.7532  
[3]  12.71  24.51  34.44  40.23  54.44  59.40  66.38  78.00  
G/E  11  11.8647  23.2991  33.2088  37.7016  53.3682  55.2007  64.6746  75.3103 
15  11.8774  23.2987  33.2480  37.7018  53.3533  55.1709  64.7192  75.2411  
19  11.8824  23.2987  33.2641  37.7028  53.3533  55.1707  64.7400  75.2449  
23  11.8850  23.2987  33.2720  37.7033  53.3533  55.1707  64.7507  75.2470  
27  11.8865  23.2987  33.2765  37.7036  53.3533  55.1707  64.7571  75.2482  
[3]  12.31  23.72  34.14  38.45  54.10  56.31  66.20  76.23 
Table 5Frequency parameters of angleply (θ/θ/θ) SSSS square plates (E/E, N= 31)
$\theta $°  Methods  Mode numbers  
1  2  3  4  5  6  7  8  
0  DQM (I)  15.19467  33.29959  44.41877  60.77869  64.52979  90.30141  93.66415  108.5563 
Exact (I)  15.19467  33.29959  44.41877  60.77869  64.52979  90.30141  93.66415  108.5563  
DQM (II)  15.17055  33.24847  44.38711  60.68220  64.45675  90.14548  93.63063  108.4588  
Exact (II)  15.17055  33.24847  44.38711  60.68220  64.45675  90.14548  93.63063  108.4588  
15  DQM (I)  15.4150  34.0748  43.8514  60.8068  66.6413  91.3847  91.5001  108.8889 
[3]  15.43  34.09  43.87  60.85  66.67  91.40  91.56  108.9  
DQM (II)  15.3959  34.0299  43.8199  60.7327  66.5601  91.3403  91.3773  108.7845  
30  DQM (I)  15.8622  35.8021  42.5521  61.3177  71.6289  85.6530  93.5627  108.7265 
[3]  15.90  35.86  42.62  61.45  71.71  85.72  93.74  108.9  
DQM (II)  15.8534  35.7679  42.5238  61.2745  71.5463  85.5891  93.4889  108.6531  
45  DQM (I)  16.0880  36.8627  41.7122  61.6729  76.9474  79.8813  94.4456  108.7356 
[3]  16.14  36.93  41.81  61.85  77.04  80.00  94.68  109.0  
DQM (II)  16.0842  36.8321  41.6880  61.6430  76.8622  79.8129  94.3878  108.6515 
Table 6Frequency parameters of angleply (θ/θ/θ) SSSS square plates (B/E, N= 31)
$\theta $°  Methods  Mode numbers  
1  2  3  4  5  6  7  8  
0  DQM (I)  11.03935  17.36394  30.90502  40.37093  44.15742  51.12759  53.26851  69.45577 
Exact (I)  11.03935  17.36394  30.90502  40.37093  44.15742  51.12759  53.26851  69.45577  
DQM (II)  11.04440  17.37677  30.92123  40.37645  44.17759  51.14502  53.30614  69.50708  
Exact (II)  11.04440  17.37677  30.92123  40.37645  44.17759  51.14502  53.30614  69.50708  
15  DQM (I)  11.3047  19.0789  33.1642  38.7790  45.2024  51.9267  59.1244  72.3957 
[3]  11.37  19.21  33.32  38.86  45.46  52.14  59.48  72.77  
DQM (II)  11.3089  19.0890  33.1790  38.7854  45.2210  51.9463  59.1566  72.4253  
30  DQM (I)  11.9660  22.4115  35.4460  37.4329  49.2139  55.9665  70.5006  72.9989 
[3]  12.21  22.78  35.86  37.90  50.04  56.70  71.36  73.57  
DQM (II)  11.9678  22.4180  35.4527  37.4446  49.2295  55.9816  70.5315  73.0123  
45  DQM (I)  12.3308  24.0999  33.6606  39.5305  53.7162  58.3197  65.0916  76.7538 
[3]  12.71  24.51  34.44  40.23  54.44  59.40  66.38  78.00  
DQM (II)  12.3315  24.1065  33.6659  39.5399  53.7361  58.3325  65.1028  76.7794 
Table 7Frequency parameters of angleply (θ/θ/θ) SSSS square plates (G/E, N= 31)
$\theta $°  Methods  Mode  
1  2  3  4  5  6  7  8  
0  DQM (I)  11.28972  17.13178  28.69169  40.74023  45.15887  45.78291  54.08234  68.14209 
Exact (I)  11.28972  17.13178  28.69169  40.74023  45.15887  45.78291  54.08234  68.14209  
DQM (II)  11.28718  17.12536  28.68364  40.73740  45.14874  45.77484  54.06362  68.13483  
Exact (II)  11.28718  17.12536  28.68364  40.73740  45.14874  45.77484  54.06362  68.13483  
15  DQM (I)  11.3927  18.5447  31.0178  39.0659  45.6444  47.9654  58.2869  67.4789 
[3]  11.46  18.69  31.20  39.15  45.91  48.19  58.70  67.84  
DQM (II)  11.3906  18.5398  31.0109  39.0628  45.6354  47.9567  58.2711  67.4670  
30  DQM (I)  11.6916  21.5420  35.4311  35.5259  48.6582  52.6273  69.0651  71.4091 
[3]  11.97  21.97  35.88  36.04  49.60  53.43  70.04  72.35  
DQM (II)  11.6908  21.5389  35.4278  35.5204  48.6507  52.6203  69.0502  71.4019  
45  DQM (I)  11.8875  23.2987  33.2794  37.7038  53.3533  55.1707  64.7612  75.2490 
[3]  12.31  23.72  34.14  38.45  54.10  56.31  66.20  76.63  
DQM (II)  11.8873  23.2956  33.2768  37.6996  53.3432  55.1652  64.7556  75.2369 
4. Conclusions
The free vibration of midplane symmetric angleply laminated composite square plates with all edges simply supported is successfully solved by using the modified differential quadrature method (modified DQM). Three material systems are considered. The rate of convergence of the modified DQM is investigated. The results are tabulated for references.
Based on the results reported herein, one may conclude that the DQ data are highly accurate and can be served as the benchmark solutions. The difference in solutions of the midplane symmetric angleply laminated composite plates with two sets of equivalent material constants is clearly seen and thus care should be taken when highly accurate results are needed for comparisons in testing newly developed numerical methods. However, the difference is small and negligible from the practical point of view.
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About this article
The project is partially supported by the Priority Academic Program Development of Jiangsu Higher Education Institutions.