Benchmark solutions of the free vibration of simply supported laminated composite plates
Xinwei Wang^{1}
^{1}State Key Laboratory of Mechanics and Control of Mechanical Structures, Nanjing University of Aeronautics and Astronautics, No. 29 Yudao Street, Nanjing, 210016, China
^{1}Corresponding author
Mathematical Models in Engineering, Vol. 3, Issue 2, 2017, p. 98105.
https://doi.org/10.21595/mme.2017.19257
Received 28 September 2017; accepted 10 October 2017; published 31 December 2017
JVE Conferences
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.
Keywords: laminated composite plate, equivalent material properties; benchmark solution, free vibration; modified differential quadrature method.
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 1. Material 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 2. Relative bendingtwisting coefficients of angleply ($\theta /\theta /\theta $) 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 3. Convergence 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 4. Convergence 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 5. Frequency parameters of angleply ($\theta /\theta /\theta $) 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 6. Frequency parameters of angleply ($\theta /\theta /\theta $) 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 7. Frequency parameters of angleply ($\theta /\theta /\theta $) 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.
Acknowledgements
The project is partially supported by the Priority Academic Program Development of Jiangsu Higher Education Institutions.
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