Khorshidi, K., Khodadadi, M. (2016). Precision Closed-form Solution for Out-of-plane Vibration of Rectangular Plates via Trigonometric Shear Deformation Theory. Mechanics of Advanced Composite Structures, 3(1), 31-43. doi: 10.22075/macs.2016.371

Korosh Khorshidi; Mohammad Khodadadi. "Precision Closed-form Solution for Out-of-plane Vibration of Rectangular Plates via Trigonometric Shear Deformation Theory". Mechanics of Advanced Composite Structures, 3, 1, 2016, 31-43. doi: 10.22075/macs.2016.371

Khorshidi, K., Khodadadi, M. (2016). 'Precision Closed-form Solution for Out-of-plane Vibration of Rectangular Plates via Trigonometric Shear Deformation Theory', Mechanics of Advanced Composite Structures, 3(1), pp. 31-43. doi: 10.22075/macs.2016.371

Khorshidi, K., Khodadadi, M. Precision Closed-form Solution for Out-of-plane Vibration of Rectangular Plates via Trigonometric Shear Deformation Theory. Mechanics of Advanced Composite Structures, 2016; 3(1): 31-43. doi: 10.22075/macs.2016.371

Precision Closed-form Solution for Out-of-plane Vibration of Rectangular Plates via Trigonometric Shear Deformation Theory

In this study, the new refine trigonometric shear deformation plate theory is used to study the out-of-plane vibration of the rectangular isotropic plates with different boundary conditions. The novelty of the research is that the analytical precision closed-form solution is developed without any use of approximation for a combination of six different boundary conditions; specifically, two opposite edges are simply supported hard and any of the other two edges can be simply supported hard, clamped or free. The equations of motion and natural boundary conditions, using Hamilton’s principle are derived. The present analytical precision closed-form solution can be obtained with any required accuracy and can be used as benchmark. Based on a comparison with the previously published results, the accuracy of the results is shown. Finally, the effect of boundary conditions, variations of aspect ratios and thickness ratios on natural frequency parameters is shown and the relation between natural frequencies for different plates is examined and dis-cussed in detail.

Precision Closed-form Solution for Out-of-plane Vibration of Rectangular Plates via Trigonometric Shear Deformation Theory

K. Khorshidi^{*}, M. Khodadadi^{ }

Department of Mechanical Engineering, Faculty of Engineering, Arak University, Arak, Iran

Paper INFO

ABSTRACT

Paper history:

Received 2015-12-26

Revision Received 2016-02-09

Accepted 2016-02-23

In this study, the new refine trigonometric shear deformation plate theory is used to study the out-of-plane vibration of the rectangular isotropic plates with different boundary conditions. The novelty of the research is that the analytical precision closed-form solution is developed without any use of approximation for a combination of six different boundary conditions; specifically, two opposite edges are simply supported hard and any of the other two edges can be simply supported hard, clamped or free. The equations of motion and natural boundary conditions, using Hamilton’s principle are derived. The present analytical precision closed-form solution can be obtained with any required accuracy and can be used as benchmark. Based on a comparison with the previously published results, the accuracy of the results is shown. Finally, the effect of boundary conditions, variations of aspect ratios and thickness ratios on natural frequency parameters is shown and the relation between natural frequencies for different plates is examined and discussed in detail.

Moderately thick plates are important structural elements. They are widely used in various engineer-ing applications such as aircrafts, space structures, ships and submarines. In order to solve plate prob-lems, two main steps must be taken: the choice of the plate theory and the type of solution method. The most commonly used plate theories can be classified into four main categories: Classical Thin Plate Theory (CPT) (based on the hypothesis that straight lines normal to the undeformed midplane remain straight and normal to the deformed mid plane and do not undergo thickness stretching.), Leissa [1], First-order Shear Deformation Plate Theory (FSDT) (based on the assumption that straight lines normal to the undeformed midplane remain straight but not necessarily normal to the deformed midplane and in this theory the transverse shear strain distribution is assumed to be constant through the plate thickness and therefore shear correction factor is required to account for the strain energy of shear deformation), Reissener [2 ,3], Mindlin[4], and Kim and Cho [5], Third-order Shear Deformation Plate Theory (TSDT) and three-dimensional elasticity theory (3-D). According to a comprehensive survey of literature, it is found that a wide range of researches has been carried out on free vibration of the rectangular and circular plates that most of them have used CPT, FSDT, TSDT and 3-D [6]. In order to deal with moderately thick plates, the trigonometric shear deformation plate theory was introduced to take into account the transverse shear strains and rotary inertia. Five variables are used in this theory to describe the deformation: three displacements of the middle surface and two rotations. In case of flat plates (without geometric imperfections), the in-plane displacements are uncoupled from the transverse displacement and rotations. Several publications can be found, in existing literature, concerning the investigation of trigonometric shear deformation plate theory. Ferreira et al. [7] analyzed symmetric composite plates using a meshless method based on global multi quadric radial basis functions. They used the trigonometric shear deformation theory which this trigonometric theory uses trigonometric functions through the thickness direction, allowing for zero transverse shear stresses at the top and bottom surfaces of the plate. Xiang and Wang [8] considered the free vibration analysis of symmetric laminated composite plates using the trigonometric shear deformation theory. Mantari et al. [9, 10] developed a new trigonometric shear deformation theory for isotropic and composite laminated and sandwich plates. Tounsi et al. [11] presented a Refined Trigonometric Shear Deformation Theory (RTSDT) by taking into account the transverse shear deformation effects for the thermo-elastic bending analysis of the functionally graded sandwich plates. Tornabene et al. [12] developed a general formulation of a 2D higher-order equivalent single layer theory including the stretching and zig-zag effects for free vibrations of thin and thick doubly-curved laminated composite shells and panels with different curvatures. Rango et al [13] presented the formulation of an enriched macro element suitable for analyzing the free vibration response of the composite plate based on the Trigonometric Shear Deformation Theory (TSDT). Sahoo and Singh [14] proposed a new trigonometric zig-zag theory for the static analysis of the laminated composite and the sandwich plates. When the equations of motion are derived using each of plate theories, these partial differential equations must be solved through numerical methods, semi-analytical methods or exact analytical methods. The exact free vibration and buckling analysis of rectangular plates has been studied by many researchers using CPT, FSDT and TSDT. Vel and Batra [15] presented a three-dimensional exact solution for free and forced vibrations of simply supported functionally graded rectangular plates. HosseiniHashemi and Arsanjani [16] derived the dimensionless equations of motion from the Mindlin plate theory to study the transverse vibration of thick rectangular plates without further usage of any approximate method. Hosseini-Hashemi et al [17] presented an exact solution to study the buckling of in-plane loaded isotropic rectangular plates with different boundary conditions. The proposed rectangular plates have two opposite edges simply-supported, while all possible combinations of free, simply-supported and clamped boundary conditions are applied to the other two edges. Hosseini-Hashemi et al [18] investigated the structural-acoustic radiation of vibrating rectangular Mindlin plates in different combinations of classical boundary conditions. Hosseini-Hashemi et al [19] presented an analytical solution for free vibration analysis of moderately thick rectangular plates, which is composed of Functionally Graded Materials (FGMs) and is supported by either Winkler or Pasternak elastic foundations. Khorshidi [20-21] analyzed the dynamic response of the moderately thick isotropic rectangular plates using an exact closed-form procedure. Hosseini-Hashemi et al. [22] presented an exact closed-form procedure for free vibration analysis of moderately thick rectangular plates having two opposite edges simply supported (i.e. Levy-type rectangular plates) based on the Reissner–Mindlin plate theory. Liu and Xing [23] obtained an exact closed-form solution for free vibrations of orthotropic rectangular Mindlin plates using the separation of variables. Dozio [24] presented an exact solution for free vibration of rectangular cross-ply laminated plates with at least one pair of opposite edges simply supported using refined kinematic theories of variable order. Leissa [25] presented an exact solution for the six cases of vibrating thin rectangular plates having two opposite sides simply-supported and the Ritz method for the remaining 15 cases which involved the possible combinations of clamped, simply-supported, and free edge conditions. Liew et al. [26] analyzed the transverse vibration of thick rectangular plates using the Rayleigh-Ritz procedure. Liew et al. [27] present the vibration analysis of shear deformable plates, which is formulated on the basis of first-order Mindlin theory. Malik and Bert [28] presented an accurate three-dimensional elasticity solution for free vibrations of six types of plates having free lateral surfaces, two opposite sides simply supported, and two other sides having combinations of simply supported, clamped, and free boundary conditions via the differential quadrature method. Liew et al. [29] formulated three-dimensional Ritz method for the vibration analysis of homogeneous, thick, rec-tangular plates with arbitrary combinations of boundary constraints. Zhou et al. [30] presented three-dimensional vibration analysis of thick rectangular plates using Chebyshev polynomial and Ritz method.

The objective of this study is to determine the free vibration response of rectangular plates using the trigonometric shear deformation plate theory. Such equations for moderately thick plates are not available in the literature. In order to fill this apparent void, the present work is carried out by providing the exact free vibration analysis for six cases of a rectangular plate having two opposite sides simply supported. The other two edges may be given by any possible combination of free, simply-supported and clamped boundary conditions. The integrated equa-tions of motion in terms of the resultant stresses are derived from the trigonometric shear deformation plate theory for moderately thick rectangular plates. This is done by considering the transverse shear deformation and rotary inertia. The exact transverse deflection and the exact displacements along and axes are derived for the first time. The present analytical solution can be obtained with any required accuracy and can be used as benchmark. Based on a comparison with the previously published results, the accuracy of the results is shown. Finally, the effect of boundary conditions, variations of aspect ratios and thickness ratios on natural frequency parameters and the relation between natural frequencies for diffrent plates are examined and discussed in detail.

2. Governing Equations of Motion

A flat, isotropic, rectangular plate with uniform thickness , length , width , modulus of elasticity , Poisson's ratio , and density is shown in Fig. 1. The displacement components and are the in-plane displacements of middle surface in and directions respectively and wis the deflection of middle surface in direction.The two edges of the plate parallel to the direction are assumed to be simply supported while the other two edges may have any combinations of clamped, free or simply supported boundary conditions.

Based on the trigonometric shear deformation theory, the displacement field can be described as the following [10]:

(1a)

(1b)

,

(1c)

Where and are the rotations of the transverse normal about and axes, respectively and

.

(2)

In Eqs. (1a) and (1b) the sinusoidal function is assigned according to the shear stress distribution through the thickness of the plate. Using Hamilton’s principle (see appendix A), the governing differential equations of motion are as follows:

(3a)

(3b)

(3c)

Figure 1. The geometry of a recangular plate

Where the stress resultants ( , , , and ) are exhibited in appendix A.

The governing differential equations of motion in terms of displacement field ( , and ) can be rewritten as what follows:

(4a)

(4b)

(4c)

Where

(5a)

(5b)

(5c)

(5d)

(5e)

(5f)

(5f)

(5g)

(5h)

For generality and convenience, the coordinates are normalized with respect to the plate planar dimensions and the following nondimensional terms are introduced.

(6a-6g)

(7a,7b)

Where is the vibration frequency of the plate, is the frequency parameter, is the flexural rigidity, is aspect ratio, is thickness ration and

(8a)

(8b)

Substituting nondimensional terms into Eq. (4), the nondimensional governing differential equations of motion are expressed as follows:

(9a)

(9b)

(9c)

Where comma-subscript convention represents the partial differentiation with respect to the normalized coordinates and

(10a-10c)

According to trigonometric shear deformation plate theory, the boundary conditions for an edge parallel to the ( or ) are given by:

Hard simply support boundary conditions:

(11a)

Clamped boundary conditions:

(11b)

Free boundary conditions:

(11c)

Where

(11d)

Corresponding boundary conditions for the simply supported edge at both or are obtained by interchanging subscripts 1 and 2 in equations (11).

After differentiating Eqs. (9b) and (9c) with respect to and , respectively, the two obtained equations should be added together. Thus, we have the following equations:

(12)

And the Eq. (9a) can be rewritten as follows:

(13)

In order to solve Eqs. (9a)–(9c), it is necessary to obtain , first. Next, substituting Eq. (12) into Eq. (13), the potential function can be given by the following equation

(14a)

Where

(14b)

(14c)

(14d)

Considering Eqs. (12-14a) and after some mathematical manipulations, the following equation can be obtained:

(15)

Where

(16a)

(16b)

(16c)

Eq. (15) can be written as what follows:

(17)

Where , and are the roots of following equation:

(18)

Based on the superposition principle we can write the following solution to Eq. (15), as:

(19)

Where , and are potentials satisfying the differential equations:

,

(20a)

,

(20b)

,

(20c)

And the potential functions , and are defined as follows:

(21a)

(21b)

(21c)

Where

(22a)

(22b)

(22c)

(22d)

Substituting Eqs. (14a) and (19) into Eq. (9), the nondimentional rotations can be expressed as the follwing:

(23a)

(23b)

In order to find the coefficient , the following coefficients are obtained by substituting Eqs. (23a) and (23b) into Eqs. (9),:

(24a)

(24b)

and

(25a)

(25b)

Where

(26)

Substituting Eq. (23) into Eq. (9a), the following equation is obtained as:

.

(27)

The potential functions so that simultaneously satisfies Eqs. (25) and (27), and it is defined as follows:

(28)

Finally, the and are introduced as what follows:

(29a)

(29b)

(29c)

and

(30a)

(30b)

(30c)

(30d)

Using the separation of variables method, one set of solutions for Eq. (30) can be written as what follows:

(31a)

(31b)

(31c)

(31d)

Where

(32a)

(32b)

(32c)

(32d)

Note that Eq. (31) is one set of solutions for Eq. (30), The boundary conditions of plate at and are assumed simply supported, then Eq. (31) are reduced as follow:

(33a)

(33b)

(33c)

(33d)

and

.

(34)

Substituting Eqs. (33) into Eqs. (29) and substituting the results into the three appropriate boundary conditions along the edges at and (Eqs. (11)), leads to a characteristic determinant of the eight-order for each . Expanding the determinant and collecting terms yield a characteristic equation.

3. Comparison Studies

In order to validate the accuracy of the present method, a comparison has been carried out with the previously published results by Leissa [25], Liew et al. [26], Hosseini-Hashemi and Arsanjani [16], Malik and Bert [28], and Zhou et al. [30] for both thin (τ=0.001) and moderately thick (τ=0.1 and τ=0.2) isotropic square plates for all the six considered boundary conditions. The present results are shown in Tabels 1 and 2, and are compared with other well-known solutions (e.g. exact solution by Hosseini-Hashemi and Arsanjani [16], Rayleigh Ritz method by Leissa [25], Liew et al. [26] and Zhou et al. [30] and differential quadrature method by Malik and Bert [28]) and different plate theories (e.g. classical plate theory by Leissa [25], first-order shear deformation plate theory by Hosseini-Hashemi and Arsanjani [16]) and three dimensional elasticity (by Leissa et al. [25], Liew et al. [26], and Zhou et al. [30]). From the results shown in Table 1, it can be observed that there is an excellent agreement between the present results and those given by Leissa [25], Liew et al. [26], Hosseini-Hashemi and Arsanjani[16], Malik and Bert [28], and Zhou et al. [30].

4. Results and Discussion

The natural frequency parameters obtained from the exact characteristic equations presented in Section 3 have been expressed in dimensionless form where the symbols are defined in Section 2. The numerical calculations have been performed for each of the six different boundary conditions. In the numerical calculations, Poisson’s ratio v=0.3 has been used. The results are given in Table 2 for the thickness to length ratios τ=0.001, τ=0.1, and τ=0.2 over a range of a aspect ratios and . In Table 2, the results are given for the first five nondimensional natural frequency parameters of the isotropic rectangular plates. The results are presented with considerable accuracy simply because they are easily obtained for the accuracy given, and because they may be used as a benchmark. For all six cases the wave forms are, of course, sine functions in the x_{1} direction, according to their corresponding equations of transverse displacement. Furthermore, the wave forms in the x_{2 }direction are sine function exactly for the case only, whereas for the other cases the forms are only approximately sinusoidal.

4.1. Theeffect of plate aspect ratio on the natural frequency parameters

In order to study the effect of aspect ratio on the vibration behavior of the plates, consideration may now be focused on Tabels 2-4 and Figure 2. From the results presented in these tables, it is observed that the nondimensional natural frequency parameter , except for the first nondimensional natural frequency of the plates, for the rest of considered six plates increases with increasing plate aspect ratio (a/b), if the relative thickness ratio is kept constant. It seems this different behavior of plates, with respect to the rest of plates, is due to having two parallel edges free boundary conditions.Considering the results presented in Table 3 and Figure 2, one may observe that, the half wave in the direction decreases and the half waves in the direction increase with increasing plate aspect ratio (a/b), if the relative thickness ratio is kept constant. This observation indicates that, between two plates having an identical b, thickness h and boundary conditions, the one which has longer width a behaves like a beam.

Figure 2. The effects of aspect ratio on the nondimensional frequancy ( ).

Table 1. The comparison study of the natural frequency parameter ( )for , , , , and boundary conditions of square plate for different thickness ratios.

BCs

Methods

Malik [28]

0.1

19.0901

45.6193

45.6193

70.1040

85.4878

Liew et al [26]

19.0898

45.6193

45.6193

70.1038

85.4876

Zhou et al [30]

19.0898

45.6193

45.6193

70.1038

85.4876

Present study

19.0661

45.4917

45.4917

69.8213

85.0830

Hashemi [16]

0.1

22.4260

47.2245

52.3247

74.4019

86.2191

Malik [28]

22.4535

47.2761

52.4356

74.5481

86.3542

Present study

22.4047

47.1387

52.2487

74.2516

85.9542

Hashemi [16]

0.2

22.5355

40.0654

45.3350

59.3313

66.0079

Present study

22.5597

40.1049

45.4333

59.4424

66.1755

Leissa [25]

0.001

9.6314

16.1348

36.7256

38.9450

46.7381

Hashemi [16]

9.6311

16.1313

36.7161

38.9433

46.7317

Present study

9.6310

16.1314

36.7165

38.9436

46.7319

Hashemi [16]

0.2

10.6981

23.1532

32.7157

43.5740

45.3051

Malik [28]

10.7216

23.2565

32.9299

43.9289

45.6888

Present study

10.8240

23.5908

31.8004

44.5052

45.8714

Hashemi [16]

0.1

12.2606

30.4743

38.7128

55.9736

62.9527

Malik [28]

12.2623

30.5095

38.7264

56.0240

63.0725

Present study

12.2519

30.4373

38.6425

55.8560

62.8485

38

Table 2. The first five natural frequency parameters ( ) for , , , , and boundary conditions of rectangular plates with different aspect and thickness ratios.

BCs

0.001

0.5

49.3476

78.9557

128.302

167.778

197.385

0.1

45.4917

69.8213

106.765

133.770

152.821

0.2

38.2052

55.2943

95.4108

106.562

106.562

0.001

2

12.3370

19.7391

32.0760

41.9455

49.3476

0.1

12.0678

19.0661

30.3643

45.4917

45.4917

0.2

11.3729

17.4553

26.6913

33.4425

38.2052

0.001

0.5

69.3257

94.5830

140.200

206.688

208.381

0.1

59.4495

79.1242

112.462

151.451

156.336

0.2

45.4333

59.4424

81.3549

101.807

107.923

0.001

2

12.9185

21.5335

35.2111

42.2393

50.4307

0.1

12.5941

20.6199

32.8994

39.3201

46.2699

0.2

11.7847

18.5629

28.2848

33.5677

38.6280

0.001

0.5

95.2594

115.799

156.350

218.961

254.120

0.1

75.3708

90.2390

119.256

160.420

168.269

0.2

53.3295

64.1280

83.9169

107.449

109.436

0.001

2

13.6857

23.6462

38.6936

42.5863

51.6737

0.1

13.2755

22.4047

35.6292

39.5722

47.1387

0.2

12.2972

19.7810

29.9276

33.7067

39.0840

0.001

0.5

9.5076

27.3596

38.4758

64.2026

87.0925

0.1

9.3259

24.9369

35.9366

56.4028

75.9733

0.2

8.8851

21.2688

30.9574

45.1618

59.2599

0.001

2

9.7322

11.6743

17.6556

27.7016

39.1518

0.1

9.5554

11.3716

16.8869

26.1218

36.5820

0.2

9.0899

10.7008

15.5425

23.1668

31.5120

0.001

0.5

16.1141

46.6708

75.1191

95.83-3

110.659

0.1

15.5981

43.2958

66.8379

83.4127

94.3631

0.2

14.5425

37.0571

53.4665

65.3572

72.7286

0.001

2

10.2961

14.7587

23.6025

37.0899

39.4497

0.1

10.1062

14.3659

22.6312

34.8247

36.9407

0.2

9.23518

13.4236

20.4762

30.2362

31.8004

0.001

0.5

22.8153

50.7489

98.7753

99.7726

132.256

0.1

21.1679

45.4951

81.0143

83.8251

103.514

0.2

18.4830

37.4621

58.4731

64.0231

73.6141

0.001

2

10.4221

15.7439

25.7668

40.5452

40.5831

0.1

10.1999

15.1839

24.3551

37.1590

37.4081

0.2

9.66068

13.9690

21.5178

31.6148

31.9010

39

Table 3. The first six natural frequency parameters ( ) in terms of wave numbers (m,n) with different aspect and thickness ratios 0.1.

BCs

1

2

4

1

2

4

1

2

4

1

2

4

1

2

4

1

2

4

; m is the mode sequence in x direction and n is

4.2. The effect of plate thickness ratio () on the natural frequency parameters

The influence of thickness ratio on the nondimensional natural frequency parameter can also be examined for plates with specific boundary conditions by keeping the aspect ratio constant while varying the thickness ratio. From the results presented in Tables 1-3 and Figure 3, it can be easily observed that, as the thickness ratio increases from to the nondimensional natural frequency parameter decreases. Such behavior is due to the influence of the transverse shear deformation in the plates.

4.3. The effect of plate boundary conditions on the natural frequency parameters

To study the effect of boundary conditions on the nondimensional natural frequency parameter , consideration may now be focused on the values of listed in a specific column of Tables 1-3. From the results presented in these tables, it is observed that the lowest nondimensional natural frequency parameter corresponds to plates subjected to less edge constraints. As the number of supported edges increases, the values of also increase. Among all six boundary conditions listed in Tables 1-3, it can be seen that the lowest and highest values of correspond to and cases, respectively. Thus, the higher constraints at the edges increase the flexural rigidity of the plate, resulting in a higher nondimensional natural frequency parameter response.

Figure 3. The effects of thickness to length ratio on the nondimensionalfrequancy ( ).

40

4.4. Complementary results

In order to satisfy Eq. (35) (case of boundary conditions), it is necessary that

(35)

((36

Where ( ) is integer values.

Using Eq. (36), between two plates having identical thickness ratio and boundary condition, the dimensionless natural frequency given in Table 3 for , and , may be related to , and through Eq. (37).

(37)

As an example for two simply supported plates having identical thickness ratio and mode number in direction ( ), the nondimensional natural frequency parameter for and ( ) is the same as those of and , because

= =(1)(2)=(2)(1)

(38)

This is because for , the simply supported boundary condition of selected plate is duplicated at the nodal lines ( ). Similarly, for two simply supported plates having identical thickness ratio and mode number in direction ( ), the nondimensional natural frequency parameter for and ( ) is the same as those of and ,

((39

Focusing now on two simply supported plates having identical thickness ratio, the nondimensional natural frequency parameter for , and ( ) is the same as those of , and . Thus, some additional results regarding other mode numbers in and directions and aspect ratio not covered in Tables 1 and 3, can be obtained from the same table through Eq. (93).

5. Conclusions

In this study the trigonometric shear deformation plate theory is used to study the flextural vibration behavior of moderately thick rectangular with different boundary conditions.). The exact closed-form vibration equations are derived from the six cases having two opposite edges simply supported hard and any of the other two edges can be hard simply supported, clamped or free. The six cases considered are namely: , , , , and plates. The advantages of the proposed closed-form solution are the following:

1- They are capable of predicting the natural frequency parameters with high accuracy within the validity of the trigonometric shear deformation plate theory since an exact analytical solution is used.

2- They provide a closed-form vibration equation that can be easily solved numerically by researchers and engineers.

Using numerical data provided previously, the effect of different parameters including boundary conditions, aspect ratio and thickness ratio on the nondimensional natural frequency parameter is examined and discussed in detail. The obtained results show the accuracy of the trigonometric shear deformation plate theory. The nondimensional natural frequency parameter , except for the first nondimensional natural frequency of the plates, decreases with increasing plate aspect ratio. The nondimensional natural frequency parameter of the plate increases monotonically, as the thickness ratio increases. For all values of aspect ratio and thickness ratio, the nondimensional natural frequency parameter corresponding to clamped boundary conditions possesses higher values in comparison with free and simply supported boundary conditions.

Appendix A

In this section, the Hamilton’s principle is used to obtain the governing differential equation for free vibration of moderately thick isotropic rectangular plates under the hypothesis of the trigonometric shear deformation theory. The Hamilton’s principle is obtained as follows:

,

(A1)

Where is the kinetic energy of the plate and is the elastic strain energy of the plate. The kinetic energy, including rotary inertia, and the elastic strain energy are given by the following equation:

(A2)

(A3)

According to the trigonometric shear deformation theory, the following strain-displacement relations are given:

41

(A4)

(A5)

(A6)

(A7)

(A8)

Substituting Eqs. (A4-A8) into Eqs. (A2-A3), the Eq. (A1) can be rewritten as what follows:

(A9)

Where the stress resultants ( , , , and ) are defined by:

(A10)

(A11)

(A12)

According to the trigonometric shear deformation theory, the following stress-displacement relations, under the hypothesis , are given:

(A13)

41

(A14)

(A15)

(A16)

(A17)

Finally, the governing differential equations of motion are given in absence of the applied load and in terms of the stress resultants by Hamilton's principle as follows:

[2] Reissner E, On the theory of bending of elastic plates. J Math Phys, 1944; 23(3): 184–191.

[3] Reissner E, The effect of transverse shear deformation on the bending of elastic plates. ASME J Appl Mech, 1945; 12(5): 69–77.

[4] Mindlin RD, Influence of rotatory inertia and shear on flexural motions of isotropic, elastic plates. ASME J Appl Mech, 1951; 18(5): 31–38.

[5] Kim J, Cho M, Enhanced first-order shear deformation theory for laminated and sandwich plates. J Appl Mech, 2005; 72(6): 809–817.

[6] Reddy JN, Theory and analysis of elastic plates and shells. CRC Press, 2007.

[7] Ferreira AJM, Roque CMC, Jorge RMN, Electrochemical Analysis of composite plates by trigonometric shear deformation theory and multi quadrics. Comput Struct, 2005; 83(1): 2225–2237.

[8] Xiang S, Wang KM, Free vibration analysis of symmetric laminated composite plates by trigonometric shear deformation theory and inverse multi quadric RBF. Thin-Walled Struct, 2009; 47(6): 304–310.

[9] Mantari JL, Oktem AS, Guedes Soares C, A new trigonometric shear deformation theory for isotropic, laminated composite and sandwich plates. Int J Solids Struct, 2012; 49(4): 43–53.

[10] Mantari JL, Oktem AS, Guedes Soares C, A new trigonometric layerwise shear deformation theory for the finite element analysis of laminated composite and sandwich plates. Comput Struct, 2012; 94(7): 45–53.

[11] Tounsi A, Houari MSA, Benyoucef S, Bedia EAA, A refined trigonometric shear deformation theory for thermoelastic bending of functionally graded sandwich plates. Aero Sci Technol, 2013; 24(6): 209–220.

[12] Tornabene F, Viola E, Fantuzzi N, General higher-order equivalent single layer theory for free vibrations of doubly-curved laminated composite shells and panels. Compos Struct, 2013; 104(31): 94–117.

[13] Rango RF, Nallim LG, Oller S, Formulation of enriched macro elements using trigonometric shear deformation theory for free vibration analysis of symmetric laminated composite plate assemblies. Compos Struct, 2015; 119(2): 38–49.

[14] Sahoo R, Singh BN, A new trigonometric zigzag theory for static analysis of laminated composite and sandwich plates. Aero Sci Technol, 2014; 35(5): 15–28.

[15] Vel SS, Batra RC, Three-dimensional exact solution for the vibration of functionally graded rectangular plates. J Sound Vib, 2004; 272: 703–730.

[16] Hosseini-Hashemi S, Arsanjani M, Exact characteristic equations for some of classical boundary conditions of vibrating moderately thick rectangular plates. Int J Solids Struct, 2005; 42(10): 819–853.

[17] Hosseini-Hashemi S, Khorshidi K, Amabili, M, Exact solution for linear buckling of rectangular Mindlin plates. J Sound Vib, 2008; 315(3): 318–342.

[18] Hosseini-Hashemi S, Khorshidi K, Rokni Damavandi Taher H, Exact acoustical analysis of vibrating rectangular plates with two opposite edges simply supported via Mindlin plate theory. J Sound Vib, 2009; 322(3): 883–900.

[19] Hosseini-Hashemi S, Rokni Damavandi Taher H, Akhavan H, Omidi M, Free vibration of functionally graded rectangular plates using first-order shear deformation plate theory. Int J Eng Sci, 2010; 34(2): 1276–1291.

[20] Khorshidi K, Elasto-plastic response of impacted moderatly thick rectangular plates with different boundary conditions. Procedia Eng, 2011; 10(2): 1742–1747.

[21] Khorshidi K, Vibro-acoustic analysis of Mindlin rectangular plates resting on an elastic foundation. Sci Iranica, 2008; 18(1): 45–52.

[22] Hosseini-Hashemi S, Fadaee M, Atashipour SR, A new exact analytical approach for free vibration of Reissner–Mindlin Functionally graded rectangular plates. Int J Mech Sci, 2011; 53(7): 11–22.

[23] Liu, B., Xing, Y., Exact solutions for free vibrations of orthotropic rectangular Mindlin plates. Compos Struct, 2011. 93(4): 1664–1672.

[24] Dozio L, Exact vibration solutions for cross-ply laminated plates with two opposite edges simply supported using refined theories of variable order. J Sound Vib, 2014; 333(2): 2347–2359.

[25] Leissa AW, The free vibration of rectangular plates. J Sound Vib, 1973; 31(3): 257–293.

[26] Liew KM, Xiang Y, Kitipornchai S, Transverse vibration of thick rectangular plates-I. Comprehensive sets of boundary conditions. Comput Struct, 1993; 49(1): 1–29.

[27] Liew KM, Hung KC, Lim MK, Vibration of Mindlin plates using boundary characteristic orthogonal polynomials. J Sound Vib, 1995; 182(1): 77–90.

43

[28] Malik M, Bert CW, Three-dimensional elasticity solutions for free vibrations of rectangular plates by the differential quadrature method. Int J Solids Struct, 1998; 35(4): 299–318.

[29] Liew KM, Hung KC, Lim MK, A continuum three-dimensional vibration analysis of thick rectangular plates. Int J Solids Struct, 1993; 30(24): 3357–3379.

[30] Zhou D, Cheung YK, Au FTK, Lo SH, Three-dimensional vibration analysis of thick rectangular plates using Chebyshev polynomial and Ritz method. Int J Solids Struct, 2002; 196(49): 4901–4910.

[2] Reissner E, On the theory of bending of elastic plates. J Math Phys, 1944; 23(3): 184–191.

[3] Reissner E, The effect of transverse shear deformation on the bending of elastic plates. ASME J Appl Mech, 1945; 12(5): 69–77.

[4] Mindlin RD, Influence of rotatory inertia and shear on flexural motions of isotropic, elastic plates. ASME J Appl Mech, 1951; 18(5): 31–38.

[5] Kim J, Cho M, Enhanced first-order shear deformation theory for laminated and sandwich plates. J Appl Mech, 2005; 72(6): 809–817.

[6] Reddy JN, Theory and analysis of elastic plates and shells. CRC Press, 2007.

[7] Ferreira AJM, Roque CMC, Jorge RMN, Electrochemical Analysis of composite plates by trigonometric shear deformation theory and multi quadrics. Comput Struct, 2005; 83(1): 2225–2237.

[8] Xiang S, Wang KM, Free vibration analysis of symmetric laminated composite plates by trigonometric shear deformation theory and inverse multi quadric RBF. Thin-Walled Struct, 2009; 47(6): 304–310.

[9] Mantari JL, Oktem AS, Guedes Soares C, A new trigonometric shear deformation theory for isotropic, laminated composite and sandwich plates. Int J Solids Struct, 2012; 49(4): 43–53.

[10] Mantari JL, Oktem AS, Guedes Soares C, A new trigonometric layerwise shear deformation theory for the finite element analysis of laminated composite and sandwich plates. Comput Struct, 2012; 94(7): 45–53.

[11] Tounsi A, Houari MSA, Benyoucef S, Bedia EAA, A refined trigonometric shear deformation theory for thermoelastic bending of functionally graded sandwich plates. Aero Sci Technol, 2013; 24(6): 209–220.

[12] Tornabene F, Viola E, Fantuzzi N, General higher-order equivalent single layer theory for free vibrations of doubly-curved laminated composite shells and panels. Compos Struct, 2013; 104(31): 94–117.

[13] Rango RF, Nallim LG, Oller S, Formulation of enriched macro elements using trigonometric shear deformation theory for free vibration analysis of symmetric laminated composite plate assemblies. Compos Struct, 2015; 119(2): 38–49.

[14] Sahoo R, Singh BN, A new trigonometric zigzag theory for static analysis of laminated composite and sandwich plates. Aero Sci Technol, 2014; 35(5): 15–28.

[15] Vel SS, Batra RC, Three-dimensional exact solution for the vibration of functionally graded rectangular plates. J Sound Vib, 2004; 272: 703–730.

[16] Hosseini-Hashemi S, Arsanjani M, Exact characteristic equations for some of classical boundary conditions of vibrating moderately thick rectangular plates. Int J Solids Struct, 2005; 42(10): 819–853.

[17] Hosseini-Hashemi S, Khorshidi K, Amabili, M, Exact solution for linear buckling of rectangular Mindlin plates. J Sound Vib, 2008; 315(3): 318–342.

[18] Hosseini-Hashemi S, Khorshidi K, Rokni Damavandi Taher H, Exact acoustical analysis of vibrating rectangular plates with two opposite edges simply supported via Mindlin plate theory. J Sound Vib, 2009; 322(3): 883–900.

[19] Hosseini-Hashemi S, Rokni Damavandi Taher H, Akhavan H, Omidi M, Free vibration of functionally graded rectangular plates using first-order shear deformation plate theory. Int J Eng Sci, 2010; 34(2): 1276–1291.

[20] Khorshidi K, Elasto-plastic response of impacted moderatly thick rectangular plates with different boundary conditions. Procedia Eng, 2011; 10(2): 1742–1747.

[21] Khorshidi K, Vibro-acoustic analysis of Mindlin rectangular plates resting on an elastic foundation. Sci Iranica, 2008; 18(1): 45–52.

[22] Hosseini-Hashemi S, Fadaee M, Atashipour SR, A new exact analytical approach for free vibration of Reissner–Mindlin Functionally graded rectangular plates. Int J Mech Sci, 2011; 53(7): 11–22.

[23] Liu, B., Xing, Y., Exact solutions for free vibrations of orthotropic rectangular Mindlin plates. Compos Struct, 2011. 93(4): 1664–1672.

[24] Dozio L, Exact vibration solutions for cross-ply laminated plates with two opposite edges simply supported using refined theories of variable order. J Sound Vib, 2014; 333(2): 2347–2359.

[25] Leissa AW, The free vibration of rectangular plates. J Sound Vib, 1973; 31(3): 257–293.

[26] Liew KM, Xiang Y, Kitipornchai S, Transverse vibration of thick rectangular plates-I. Comprehensive sets of boundary conditions. Comput Struct, 1993; 49(1): 1–29.

[27] Liew KM, Hung KC, Lim MK, Vibration of Mindlin plates using boundary characteristic orthogonal polynomials. J Sound Vib, 1995; 182(1): 77–90.

[28] Malik M, Bert CW, Three-dimensional elasticity solutions for free vibrations of rectangular plates by the differential quadrature method. Int J Solids Struct, 1998; 35(4): 299–318.

[29] Liew KM, Hung KC, Lim MK, A continuum three-dimensional vibration analysis of thick rectangular plates. Int J Solids Struct, 1993; 30(24): 3357–3379.

[30] Zhou D, Cheung YK, Au FTK, Lo SH, Three-dimensional vibration analysis of thick rectangular plates using Chebyshev polynomial and Ritz method. Int J Solids Struct, 2002; 196(49): 4901–4910.