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

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

1. 1.      Introduction

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.

1. 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]:

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

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:

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:

Where

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

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

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

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

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:

Clamped boundary conditions:

Free boundary conditions:

Where

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:

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

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

Where

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

Where

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

Where ,  and are the roots of following equation:

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

Where ,  and  are potentials satisfying the differential equations:

And the potential functions ,  and  are defined as follows:

Where

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

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

and

Where

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

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

Finally, the  and  are introduced as what follows:

and

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

Where

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:

and

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.

1. 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].

1. 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 x₁ direction, according to their corresponding equations of transverse displacement. Furthermore, the wave forms in the x₂ 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 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.

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