Abstract

A series solution for the transverse vibration of Mindlin rectangular plates with elastic point supports around the edges is studied. The series solution for the problem is obtained using improved Fourier series method, in which the vibration displacements and the cross-sectional rotations of the midplane are represented by a double Fourier cosine series and four supplementary functions. The supplementary functions are expressed as the combination of trigonometric functions and a single cosine series expansion and are introduced to remove the potential discontinuities associated with the original admissible functions along the edges when they are viewed as periodic functions defined over the entire - plane. This series solution is approximately accurate in the sense that it explicitly satisfies, to any specified accuracy, both the governing equations and the boundary conditions. The convergence, accuracy, stability, and efficiency of the proposed method have been examined through a series of numerical examples. Some numerical examples about the nondimensional frequency and mode shapes of Mindlin rectangular plates with different point-supported edge conditions are given.

1. Introduction

Rectangular plate is of great importance in various engineering branches, such as aerospace, electronics, and mechanical, nuclear, and marine engineering. A better understanding of its dynamic characteristics is meaningful when dealing with the design of a plate-type structure. In the early stage of research about rectangular plate, the majority of research was about thin plate based on classical Kirchhoff hypothesis. However, for the shortage of consideration about transverse shear deformation and rotary inertia, vibration frequencies of thick plates are overestimated. More and more investigations about Mindlin plates began to attract attention after the Mindlin first-order plate theory was proposed by Mindlin [1]. The vibration characteristics of Mindlin plates have been well investigated by researches with classical and elastic edge support [2–5]. Few literatures focused on the vibration behavior of Mindlin plates with elastic point supports around the edges. However, the boundary conditions of the plates are not always classical and elastic around the edges in practical engineering applications. And the boundary conditions of elastic point supports around the edges do exist. So it is much of great significance to study the vibration behavior of Mindlin plates with elastic point supports around the edges.

In the field of vibration analysis of structures subject to elastic edge/point supports, Takashi and Jin [6] used the collocation method to investigate Mindlin plates with constant thickness and two opposite edges simply supported. Jiarang and Jianqiao [7] established the three-dimensional state equation for the laminated thick orthotropic plate with simply supported edges and obtained the numerical results. Ohya et al. [3] investigated the free vibration characteristics of rectangular Mindlin plates which have simultaneous elastic edge and internal supports via the superposition method. Hosseini-Hashemi et al. [8] proposed an exact closed-form procedure to solve free vibration of moderately thick rectangular plates with two opposite edges simply supported. Dozio et al. [9] developed a Ritz method using a set of trigonometric functions to obtain accurate modal properties of rectangular plates with arbitrary thickness. Maretic [10] analyzed the transverse vibration of a circular plate loaded by uniform pressure along its edge. Based on the shear deformable plate theory, Bahmyari [11] used the Element-Free Galerkin Method to analyze the free vibration of inhomogeneous moderately thick plates with point supports resting on a two-parameter elastic foundation. The results show that the vibration results obtained are in a very good agreement with the available literatures in spite of using low numbers of nodes. Bashmal et al. [12] investigated the in-plane free vibrations of an elastic and isotropic annular disk with elastic constraints at the inner and outer boundaries. And the inner and outer boundaries are applied either along the entire periphery of the disk or at a point. Esendemir [13] studied polymer-matrix composite beam of arbitrary orientation supported from two ends acted upon with a force at the midpoint by an analytical elastoplastic stress analysis. Foyouzat et al. [14] presented an analytical-numerical approach to determine the dynamic response of thin plates resting on multiple elastic point supports with time-varying stiffness. Gan et al. [15] studied the effect of intermediate elastic support on the vibration of functionally graded Euler-Bernoulli beams excited by a moving point load. Kucuk et al. [16] carried out the analytical elastic-plastic stress analysis on an aluminum metal-matrix composite beam reinforced by unidirectional steel fibers supported at the ends acted upon with a force at the midpoint. Hosseini-Hashemi et al. [17–19] used the generalized differential quadrature method to study the buckling analysis and dynamic transverse vibration characteristics. Setoodeh and Karami [20] employed a three-dimensional elasticity based layerwise finite element method (FEM) to study the static, free vibration, and buckling responses of general laminated thick composite plates. In this paper, the elastic line and point supports are successfully incorporated for thick plates. On the basis of three-dimensional elasticity theory, Xu and Zhou [21] studied the bending of simple-supported rectangular plate on point supports, line supports, and elastic foundation. Wu [22] analyzed the free vibration of arbitrary quadrilateral thick plates with internal columns and elastic edge supports by using the powerful pb-2 Ritz method and Reddy’s third-order shear deformation plate theory. Wu and Lu [23] studied the free vibration of rectangular plates with internal columns and elastic edge supports using the powerful pb-2 Ritz method. Other literatures related to this field are shown in [24–28].

From the literatures review, we can know that most researches focused on the classical plates. Only few literatures studied the vibration characteristics of Mindlin plates with point supports around the edges. And these literatures mainly used the collocation method, analytical-numerical method, and so on. To the best of the authors’ knowledge, there are no published literatures focused on vibration characteristics of Mindlin plates with point supports around the edges by the method of modified Fourier. Thus, a unified, efficient, and accurate formulation to deal with the free vibration of Mindlin plates subjected to arbitrary point-supported boundary condition is necessary and much of great significance.

In this paper, a modified Fourier solution for the free vibration of Mindlin rectangular plates with elastic point supports around the edges is proposed. The vibration displacements and the cross-sectional rotations of the midplane are represented by a double Fourier cosine series and four supplementary functions. The supplementary functions are in the form of the combination between trigonometric functions and a single cosine series expansion. By importing supplementary functions, the discontinuity of the boundary condition is overcome. The change of the boundary conditions can be easily achieved by only varying the stiffness of the boundary springs around all edges of the plates without involving any change to the solution procedure. The natural frequencies of Mindlin rectangular plates are obtained by using the Rayleigh–Ritz method. Effectiveness and stability of this method will be verified by comparing with the results of FE modeling. Some numerical examples of free vibration for Mindlin rectangular plates with different aspect ratio and thickness are conducted under different point-supported conditions.

2. Theoretical Formulations

2.1. Point-Supported Edge Conditions

The rectangular Mindlin plate with arbitrary elastic point edge supports is shown in Figure 1. The boundary conditions are presented by three kinds of restraining springs [29–34], namely, translational, rotational, and torsional springs. Springs are evenly arranged on each edge of Mindlin plate. Through changing the stiffness of springs, different boundary conditions can be achieved [35, 36]. The governing differential equations of a Mindlin plate about its free vibration are as follows:

Figure 1 

A Mindlin plate with arbitrary elastic point edge supports.

In the formulas above, transverse displacement is presented by , is the slope along direction, and is the slope along direction. means the mass density, is Poisson’s ratio, is the shear correction factor, and the thickness of the plate is . is the shear modulus, is Poisson’s ratio, and is the flexural rigidity.

Besides, the bending moment is expressed by (4) and (5), twisting moment is calculated by (6), and the transverse shearing forces in plates are expressed by (7) and (8).

There are three kinds of forces along every edge, namely, the bending moment, the twisting moment, and the shearing forces. Rotational, torsional, and translational springs along every edge are the counterparts to these three forces in this paper. The boundary conditions for an elastically restrained rectangular plate are as follows.

At ,

, , and represent the stiffness of the translation spring, rotational spring, and torsional spring, respectively, on the boundary edge ; the corresponding expression results in where is the elastic point supports numbers on edge , is the Dirac delta function, and , , and are, respectively, the stiffness of the translational, rotational, and torsional springs at the support located at . Similarly, the remaining boundary conditions on the other three edges can be expressed as follows.

At ,At ,At ,

Equations (9)–(32) represent a set of general boundary conditions by setting the spring stiffness to appropriate values. Based on (4)–(8), the boundary condition can be finally written as follows.

At ,At ,At ,At ,

According to the Mindlin plate theory, the transverse displacement of the plate middle surface and the rotations of the cross section, respectively, along the direction and the direction are utilized. Based on the traditional solution method, the admissible functions usually express a Fourier series expansion, because the Fourier functions constitute a complete set and exhibit an excellent numerical stability in the previous study [37–43]. We found the conventional Fourier series expression to have some defects which contain the convergence problem along the boundary edges except for a few simple boundary conditions, and the derivatives of a Fourier series cannot be obtained simply through term-by-term differentiation.

In this study, in order to overcome the shortcoming with the conventional Fourier series expression, the admissible functions will be expressed as a more wholesome form of Fourier series expansion:where , , , , , , , , and represent the unknown coefficients, , . The specific expressions of the auxiliary functions and are defined as

As shown in (45)–(47), the supplementary functions , , , and are used for the and direction displacement expressions. It is easy to find that

Similar conditions are present in the supplementary function in -direction, namely, and . Whereas these conditions are not necessary, the existence of these conditions is purely for the sake of simplifying follow-up mathematical expressions and derivation process.

Examining the admissible functions in (45)–(47), one will notice that, besides the standard 2D Fourier series defined over the domain , four additional 1D Fourier expansions are also included in the admissible functions expressions. In light of (52), it is not difficult to see that the fourth term (or the third single Fourier series) on the right side of (45) is equal to the normal derivative of the displacement function at edge . Thus, this term will actually inherit any potential discontinuity associated with the normal derivative at the boundary . Similarly, the other three terms are used to take care of the possible discontinuities at the remaining edges. Thus, the 2D Fourier expansion will now represent a periodic (residual displacement) function defined over the entire plane. The convergence characteristic for the Fourier series representation of such a function has been well established in mathematics. Namely, the Fourier series will converge with a speed of at least.

By substituting (45)–(47) into boundary condition (33) and expanding all the -related terms, except for those containing , into Fourier cosine series, one will have

The expressions for Fourier cosine expansion coefficient of related functions can be found in Appendix A. To establish the relationship between the Fourier coefficients in (54), the terms on the left side shall also be expressed in the form of cosine series, resulting inwhere and are Kronecker delta. Using similar procedures, the other seven boundary condition equations can be obtained. Thus, a total of twelve constraint equations can be derived as follows.

At ,At ,At ,At ,