Document Type: Research Paper
Authors
Arak University
Abstract
Keywords

Mechanics of Advanced Composite Structures 3 (2016) 3143 

Semnan University 
Mechanics of Advanced Composite Structures journal homepage: http://macs.journals.semnan.ac.ir 
Precision Closedform Solution for Outofplane 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 20151226 Revision Received 20160209 Accepted 20160223 
In this study, the new refine trigonometric shear deformation plate theory is used to study the outofplane vibration of the rectangular isotropic plates with different boundary conditions. The novelty of the research is that the analytical precision closedform 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 closedform 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. 



Keywords: Vibration Precision closedform solution Trigonometric shear deformation theory



© 2016 Published by Semnan University Press. All rights reserved. 
Moderately thick plates are important structural elements. They are widely used in various engineering applications such as aircrafts, space structures, ships and submarines. In order to solve plate problems, 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], Firstorder 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], Thirdorder Shear Deformation Plate Theory (TSDT) and threedimensional elasticity theory (3D). 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 3D [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 inplane 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 thermoelastic bending analysis of the functionally graded sandwich plates. Tornabene et al. [12] developed a general formulation of a 2D higherorder equivalent single layer theory including the stretching and zigzag effects for free vibrations of thin and thick doublycurved 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 zigzag 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, semianalytical 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 threedimensional 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. HosseiniHashemi et al [17] presented an exact solution to study the buckling of inplane loaded isotropic rectangular plates with different boundary conditions. The proposed rectangular plates have two opposite edges simplysupported, while all possible combinations of free, simplysupported and clamped boundary conditions are applied to the other two edges. HosseiniHashemi et al [18] investigated the structuralacoustic radiation of vibrating rectangular Mindlin plates in different combinations of classical boundary conditions. HosseiniHashemi 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 [2021] analyzed the dynamic response of the moderately thick isotropic rectangular plates using an exact closedform procedure. HosseiniHashemi et al. [22] presented an exact closedform procedure for free vibration analysis of moderately thick rectangular plates having two opposite edges simply supported (i.e. Levytype rectangular plates) based on the Reissner–Mindlin plate theory. Liu and Xing [23] obtained an exact closedform 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 crossply 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 simplysupported and the Ritz method for the remaining 15 cases which involved the possible combinations of clamped, simplysupported, and free edge conditions. Liew et al. [26] analyzed the transverse vibration of thick rectangular plates using the RayleighRitz procedure. Liew et al. [27] present the vibration analysis of shear deformable plates, which is formulated on the basis of firstorder Mindlin theory. Malik and Bert [28] presented an accurate threedimensional 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 threedimensional Ritz method for the vibration analysis of homogeneous, thick, rectangular plates with arbitrary combinations of boundary constraints. Zhou et al. [30] presented threedimensional 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, simplysupported and clamped boundary conditions. The integrated equations 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.
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 inplane 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.
(6a6g) 

(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 commasubscript convention represents the partial differentiation with respect to the normalized coordinates and
(10a10c) 
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. (1214a) 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 eightorder for each . Expanding the determinant and collecting terms yield a characteristic equation.
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], HosseiniHashemi 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 wellknown solutions (e.g. exact solution by HosseiniHashemi 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], firstorder shear deformation plate theory by HosseiniHashemi 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], HosseiniHashemi and Arsanjani[16], Malik and Bert [28], and Zhou et al. [30].
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. The effect 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 24 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 


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 
95.833 
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 



; 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 13 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 13. 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 13, 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).
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 closedform 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 closedform 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 closedform 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 straindisplacement relations are given:

(A4) 

(A5) 

(A6) 

(A7) 

(A8) 
Substituting Eqs. (A4A8) into Eqs. (A2A3), 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 stressdisplacement relations, under the hypothesis , are given:
(A13) 



(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:
(A18) 

(A19) 

(A20) 
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