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Abstract

This paper presents a three-dimensional formulation for the free vibrations of thick rectangular plates with general boundary conditions and resting on elastic foundations. The general boundary conditions are imposed by means of penalty function method. The displacements of the plates are expressed by a three-dimensional cosine series and some simple polynomial functions which introduced to ensure and accelerate the convergence of the series representation. All the unknown coefficients can be obtained by using the Rayleigh–Ritz method. Comparisons of the present results with those in available literature demonstrate the accuracy and reliability of the present formulation. Furthermore, parametric investigations are presented including the effects of boundary conditions, geometrical parameters, and elastic foundation.

Keywords

Three-dimensional elasticity theory thick plate general boundary restraints elastic foundations

Introduction

Plates are the basic structural elements in most engineering structures, such as ships, automobiles, and architectures. During their operation, those plates are commonly subjected to oscillating loads, which lead to the vibration. Consequently, it is of paramount importance to thoroughly understand the vibration behaviors of plates thus reduce structural vibrations through proper design. In addition, there exists a variety of possible boundary restraint cases for plates in practical assembly and engineering applications. Development of the reliable and efficient techniques for vibration analysis of plates with general boundary conditions is the subject of significant research interest of the present study.

Significant advances on the subject have been achieved during the past few decades. A larger variety of plate theories have been proposed and developed. In most cases, the two-dimensional (2D) theories such as the classic plate theory,^1–6 the first-order shear deformation theory^7–17 and the higher order deformation theory^18–24 were adopted to analyze the vibration problems of those structures. However, since there are some certain kinematic assumptions, those 2D theories only provide approximate solutions. In order to obtain realistic results, an increasing number of researchers employed three-dimensional (3D) elasticity theory to study the vibration problems of plates. Due to no hypotheses are assumed for the distribution field of the deformations and stresses, the 3D analysis not only provides a full spectrum of vibration solutions but also allows further physical insight. Malik and Bert²⁵ provided 3D solutions for vibration rectangular plates using differential quadrature method. Liew et al.²⁶ developed a 3D Ritz method for vibration of thick plates in which the effects of thickness ratios and end constraints on the frequencies and mode shapes were investigated. Zhou and co-authors^27–30 analyzed 3D vibration characteristics of plates using Ritz method combined with Chebyshev polynomial. Naginoa et al.³¹ studied 3D free vibration of rectangular plates via a B-spline Ritz method. A comparison between classical 2D finite elements and an exact 3D solution for free vibration of plates and shells was carried out by Brischetto and Torre³². Brischetto³³ proposed 3D equilibrium equations in general orthogonal curvilinear coordinates for free vibration of plates and shells. An analytical solution for free vibration of an incompressible isotropic linear elastic rectangular plate with simply supported boundary conditions was provided by Aimmanee and Batra.³⁴ It is noted that the Carrera Unified Formulation (CUF) is a powerful framework of analysis of beam, plate, and shell structures and is capable of detecting 3D effects on vibration behavior.³⁵ Ferreira et al.³⁶ combined CUF and a radial basis function collocation technique to study vibration behavior of isotropic and cross-ply laminated plates. Later, they proposed a combination of CUF and the generalized differential quadrature technique for vibration analysis of thick isotropic and cross-ply laminated plates.³⁷ Rezaei and Saidi³⁸ used CUF in conjunction with state space method to analyzed Levy-type rectangular porous-cellular plates.

A review of the scientific literature in this filed reveals that the available 3D solutions of plates are limited. In addition, most of the previous studies about vibration problems of plates are restricted to classical boundary conditions. It is well recognized that there exists a variety of possible boundary restraint cases especially elastic restraints for plates in practical assembly and engineering applications. Plates on elastic foundations have been widely adopted by many researchers to model interaction between elastic media and plates for various engineering plate problems. Consequently, it is necessary to study vibration behavior of plate with general boundary conditions and resting on elastic foundations. This paper presents a 3D formulation for the free vibrations of thick rectangular plates with general boundary conditions and resting on elastic foundations. The general boundary conditions are imposed by means of penalty function method. The displacements of the plates are expressed by a 3D cosine series and some simple polynomial functions which introduced to ensure and accelerate the convergence of the series representation.^39–43 All the unknown coefficients can be obtained by using the Rayleigh–Ritz method. Comparisons of the present results with those in available literature demonstrate the accuracy and reliability of the present formulation. Furthermore, parametric investigations are presented, including the effects of boundary conditions, geometrical parameters, and elastic foundation.

Theoretical formulations

Preliminaries

Consider a rectangular plate on elastic foundation with uniform thickness h, as depicted in Figure 1. The length and width of the plate are denoted by a and b, respectively. The geometry and dimensions are defined with respect to an orthogonal coordinate system (x, y, and z) which is located on the bottom surface. The displacement components along x, y, and z directions of a point in the plate are represented by u, v, and w, respectively. According to 3D elasticity theory, the relationships between strains and displacements are given as follows⁴⁴

\begin{array}{l} ε_{x x} = \frac{\partial u}{\partial x}; ε_{y y} = \frac{\partial v}{\partial y}; ε_{z z} = \frac{\partial w}{\partial z} \\ γ_{x y} = \frac{\partial v}{\partial x} + \frac{\partial u}{\partial y}; γ_{x z} = \frac{\partial w}{\partial x} + \frac{\partial u}{\partial z}; γ_{y z} = \frac{\partial v}{\partial z} + \frac{\partial w}{\partial y} \end{array}

(1)

Figure 1.

Geometry and coordinate system of a plate on elastic foundation.

Based on Hook’s law, the constitutive relations are given in a matrix form as follows

[\begin{matrix} σ_{x x} \\ σ_{y y} \\ σ_{z z} \\ σ_{y z} \\ σ_{x z} \\ σ_{x y} \end{matrix}] = [\begin{matrix} Q_{11} & Q_{12} & Q_{13} & 0 & 0 & 0 \\ Q_{12} & Q_{22} & Q_{23} & 0 & 0 & 0 \\ Q_{13} & Q_{23} & Q_{33} & 0 & 0 & 0 \\ 0 & 0 & 0 & Q_{44} & 0 & 0 \\ 0 & 0 & 0 & 0 & Q_{55} & 0 \\ 0 & 0 & 0 & 0 & 0 & Q_{66} \end{matrix}] [\begin{matrix} ε_{x x} \\ ε_{y y} \\ ε_{z z} \\ γ_{y z} \\ γ_{x z} \\ γ_{x y} \end{matrix}]

(2)

where the stiffness coefficients

Q_{i j}

(i, j = 1–6) are defined as follows

\begin{array}{l} Q_{11} = Q_{22} = Q_{33} = \frac{E (1 - μ)}{(1 + μ) (1 - 2 μ)} \\ Q_{12} = Q_{13} = Q_{23} = \frac{E μ}{(1 + μ) (1 - 2 μ)} \\ Q_{44} = Q_{55} = Q_{66} = \frac{E}{2 (1 + μ)} \end{array}

(3)

In this work, the bottom surface of the plate is assumed to be continuously rested on an elastic foundation represented by Winkler/Pasternak model. The boundary conditions are imposed by penalty factor of stiffness type, namely, artificial linear springs (k_u, k_v, and k_w). Therefore, the boundary conditions can be defined as follows

k_{u}^{x 0} u = σ_{x x}, k_{v}^{x 0} v = σ_{x y}, k_{w}^{x 0} w = σ_{x z}; k_{u}^{x a} u = σ_{x x}, k_{v}^{x a} v = σ_{x y}, k_{w}^{x a} w = σ_{x z}

(4)

k_{u}^{y 0} u = σ_{x y}, k_{v}^{y 0} v = σ_{y y}, k_{w}^{y 0} w = σ_{y x}; k_{u}^{y b} u = σ_{x y}, k_{v}^{y b} v = σ_{y y}, k_{w}^{y b} w = σ_{y x}

(5)

where the subscripts and superscripts indicate the directions and locations of the artificial springs.

Energy functional

The kinetic energy T of the rectangular plate can be expressed as follows

T = \frac{1}{2} \int_{V} ρ [{(\frac{d u}{d t})}^{2} + {(\frac{d v}{d t})}^{2} + {(\frac{dw}{d t})}^{2}] d V

(6)

where V is the volume and ρ is the mass density per unit volume.

The strain energy U can be written in integral form as follows

U = \frac{1}{2} \int_{V} (σ_{α α} ε_{α α} + σ_{β β} ε_{β β} + σ_{z z} ε_{z z} + σ_{β z} γ_{β z} + σ_{α z} γ_{α z} + σ_{α β} γ_{α β}) d V

(7)

The potential energy P can be into two components i.e. potential energy stored in the boundary springs (P_bs) and potential energy stored in elastic foundation (P_ef). The P_bs and P_ef are given as follows

P_{b s} = \frac{1}{2} [\int_{S_{x i}} (k_{u}^{x i} u^{2} + k_{v}^{x i} v^{2} + k_{w}^{x i} w^{2}) d S_{x i} + \int_{S_{y i}} (k_{u}^{y i} u^{2} + k_{v}^{y i} v^{2} + k_{w}^{y i} w^{2}) d S_{y i}]

(8)

P_{e f} = \frac{1}{2} \iint_{S} {K_{r} w^{2} + K_{g} {(\frac{\partial w}{\partial x})}^{2} + K_{g} {(\frac{\partial w}{\partial y})}^{2}} d S

(9)

where

S_{x i}

and

S_{y i}

denote the area of boundary surfaces and

S

denotes the interface area between plates and elastic foundation.

The energy functional of plates can be expressed as follows

Π = T - U - P

(10)

Admissible functions

In Rayleigh–Ritz method, the only constriction to admissible displacement functions is that they should satisfy the geometrical boundary conditions. In this work, the geometrical boundary conditions are imposed by the penalty function method. Consequently, the choice and construction of admissible functions become flexible. Compared with polynomials which maybe lead to instability, Fourier series exhibit an excellent numerical stability.⁴⁵ Unfortunately, there exists convergence problem corresponding to the use of Fourier series. Therefore, some auxiliary functions are introduced to eliminate this problem. The displacement components can be expressed as follows

u (x, y, z, t) = {\begin{cases} \sum_{m = 0}^{\infty} \sum_{n = 0}^{\infty} \sum_{q = 0}^{\infty} A_{m n q} \cos λ_{m} x \cos λ_{n} y \cos λ_{q} z + \\ \sum_{l = 1}^{2} \sum_{n = 0}^{\infty} \sum_{q = 0}^{\infty} a_{l n q} ξ_{l x} (x) \cos λ_{n} y \cos λ_{q} z + \\ \sum_{m = 0}^{\infty} \sum_{l = 1}^{2} \sum_{q = 0}^{\infty} {\bar{a}}_{l m q} \cos λ_{m} x ξ_{l y} (y) \cos λ_{q} z + \\ \sum_{m = 0}^{\infty} \sum_{n = 0}^{\infty} \sum_{l = 1}^{2} {\tilde{a}}_{l m n} \cos λ_{m} x \cos λ_{n} y ξ_{l z} (z) + \end{cases}} e^{j ω t}

(11)

v (x, y, z, t) = {\begin{cases} \sum_{m = 0}^{\infty} \sum_{n = 0}^{\infty} \sum_{q = 0}^{\infty} B_{m n q} \cos λ_{m} x \cos λ_{n} y \cos λ_{q} z + \\ \sum_{l = 1}^{2} \sum_{n = 0}^{\infty} \sum_{q = 0}^{\infty} b_{l n q} ξ_{l x} (x) \cos λ_{n} y \cos λ_{q} z + \\ \sum_{m = 0}^{\infty} \sum_{l = 1}^{2} \sum_{q = 0}^{\infty} {\bar{b}}_{l m q} \cos λ_{m} x ξ_{l y} (y) \cos λ_{q} z + \\ \sum_{m = 0}^{\infty} \sum_{n = 0}^{\infty} \sum_{l = 1}^{2} {\tilde{b}}_{l m n} \cos λ_{m} x \cos λ_{n} y ξ_{l z} (z) + \end{cases}} e^{j ω t}

(12)

w (x, y, z, t) = {\begin{cases} \sum_{m = 0}^{\infty} \sum_{n = 0}^{\infty} \sum_{q = 0}^{\infty} C_{m n q} \cos λ_{m} x \cos λ_{n} y \cos λ_{q} z + \\ \sum_{l = 1}^{2} \sum_{n = 0}^{\infty} \sum_{q = 0}^{\infty} c_{l n q} ξ_{l x} (x) \cos λ_{n} y \cos λ_{q} z + \\ \sum_{m = 0}^{\infty} \sum_{l = 1}^{2} \sum_{q = 0}^{\infty} {\bar{c}}_{l m q} \cos λ_{m} x ξ_{l y} (y) \cos λ_{q} z + \\ \sum_{m = 0}^{\infty} \sum_{n = 0}^{\infty} \sum_{l = 1}^{2} {\tilde{c}}_{l m n} \cos λ_{m} x \cos λ_{n} y ξ_{l z} (z) + \end{cases}} e^{j ω t}

(13)

where λ_m=mπ/a, λ_n=nπ/b, λ_q=qπ/h,

A_{m n q}

a_{l n q}

{\bar{a}}_{l m q}

{\tilde{a}}_{l m n}

B_{m n q}

b_{l n q}

{\bar{b}}_{l m q}

{\tilde{b}}_{l m n}

C_{m n q}

c_{l n q}

{\bar{c}}_{l m q}

, and

{\tilde{c}}_{l m n}

are the unknown coefficients. ω and t are the circular frequency and time variable, respectively.

ξ_{l x}

ξ_{l y}

, and

ξ_{l z}

are the closed-form auxiliary polynomials functions. According to 3D theory of elasticity, at least first-order derivatives of the auxiliary functions need exists which ensures the continuity of the displacements and their derivatives at any point on the plate. The selected auxiliary functions are given as follows

ξ_{1 x} = x {(\frac{x}{a} - 1)}^{2}, ξ_{2 x} = \frac{x^{2}}{a} (\frac{x}{a} - 1)

(14)

ξ_{1 y} = y {(\frac{y}{b} - 1)}^{2}, ξ_{2 y} = \frac{y^{2}}{b} (\frac{y}{b} - 1)

(15)

ξ_{1 z} = z {(\frac{z}{h} - 1)}^{2}, ξ_{2 z} = \frac{z^{2}}{h} (\frac{z}{h} - 1)

(16)

It is easy to verify that

\begin{matrix} ξ_{1 x} (0) = ξ_{1 x} (a) = ξ_{1 x}^{'} (a) = 0, ξ_{1 x}^{'} (0) = 1 \\ ξ_{2 x} (0) = ξ_{2 x} (a) = ξ_{2 x}^{'} (0) = 0, ξ_{2 x}^{'} (a) = 1 \end{matrix}

The similar conditions exist for the y- and z-related polynomials. The introductions of these auxiliary functions are for the purpose of removing the discontinuities of the original displacement functions and their derivatives at edges.

Solution procedure

Substituting equations (6) to (9) and (11) to (13) into equation (10), and minimizing energy functional with respect to the unknown coefficients, one can obtain a set of linear algebraic equation as follows

{K - ω^{2} M} X = 0

(17)

where K and M is the total symmetric stiffness and mass matrices for the structure. X is the column vector given as follows

X = {[X_{u}, X_{v}, X_{w}]}^{T}

where

\begin{matrix} X_{u} = {\begin{cases} A_{000}, \dots, A_{m n q}, \dots, A_{M N Q}, a_{100}, \dots, a_{l n q}, \dots, a_{2 N Q}, \\ {\bar{a}}_{100}, \dots, {\bar{a}}_{l m q}, \dots {\bar{a}}_{2 M Q}, {\tilde{a}}_{100}, \dots, {\tilde{a}}_{l m n}, \dots {\tilde{a}}_{2 M N} \end{cases}} \\ X_{v} = {\begin{cases} B_{000}, \dots, B_{m n q}, \dots, B_{M N Q}, b_{100}, \dots, b_{l n q}, \dots, b_{2 N Q}, \\ {\bar{b}}_{100}, \dots, {\bar{b}}_{l m q}, \dots {\bar{b}}_{2 M Q}, {\tilde{b}}_{100}, \dots, {\tilde{b}}_{l m n}, \dots {\tilde{b}}_{2 M N} \end{cases}} \\ X_{w} = {\begin{cases} C_{000}, \dots, C_{m n q}, \dots, C_{M N Q}, c_{100}, \dots, c_{l n q}, \dots, c_{2 N Q}, \\ {\bar{c}}_{100}, \dots, {\bar{c}}_{l m q}, \dots {\bar{c}}_{2 M Q}, {\tilde{c}}_{100}, \dots, {\tilde{c}}_{l m n}, \dots {\tilde{c}}_{2 M N} \end{cases}} \end{matrix}

All frequencies and corresponding mode shapes can be obtained by solving equation (17).

Numerical examples and discussion

In this section, a variety of numerical examples on free vibration of thick rectangular plates with general boundary conditions and resting on elastic foundation are given to validate accuracy and reliability of the present method. The typical boundary conditions, namely, clamped (C), simply-supported (S), free (F), and elastic (E) restraints, are considered. For the sake of brevity, the boundary conditions are described uniformly by letter strings. For example, the symbol CSFE represents a rectangular plate with clamped end at x = 0, simply supported edge at y = 0, free boundary condition at x = a, and elastic restraint at y = b. The frequency parameter Ω is defined as follows

Ω = ω b^{2} \sqrt{ρ h / D}

where

D = E h^{3} / 12 (1 - μ^{2})

Convergence study

Theoretically, the exact solution can be obtained for the superposition of infinite number in series expression shown in equations (11) to (13). However, in actual calculation, the infinite series must be truncated to M, N, and Q. Consequently, the convergence of the current formulation needs to be checked firstly. Table 1 presents the first seven dimensional frequency parameters Ω of a rectangular plate with different thickness-to-width ratios subjected completely free boundary conditions. The geometrical parameters used in analysis are as follows: a/b = 1, h/b = 0.2, and 0.5. It is obvious that frequency parameters monotonically converge as the truncated numbers increase. M × N × Q = 14 × 14 × 10 will be selected as the truncated numbers in the following calculations.

Table 1.

Convergence of frequency parameters of completely free square plates with different thickness-to-width ratio h/b (μ = 0.3).

h/b	M × N × Q	Ω₁	Ω₂	Ω₃	Ω₄	Ω₅	Ω₆	Ω₇
0.2	10 × 10 × 6	11.713	17.434	21.253	27.652	27.652	40.192	42.776
	11 × 11 × 6	11.712	17.434	21.253	27.651	27.651	40.192	42.776
	12 × 12 × 8	11.711	17.433	21.252	27.650	27.650	40.192	42.775
	13 × 13 × 8	11.711	17.433	21.252	27.649	27.649	40.192	42.775
	14 × 14 × 10	11.711	17.433	21.252	27.648	27.648	40.192	42.775
	15 × 15 × 10	11.710	17.433	21.252	27.648	27.648	40.192	42.775
0.5	10 × 10 × 6	8.7803	12.515	14.962	16.072	17.030	17.030	17.632
	11 × 11 × 6	8.7802	12.515	14.962	16.072	17.030	17.030	17.632
	12 × 12 × 8	8.7801	12.515	14.962	16.072	17.030	17.030	17.632
	13 × 13 × 8	8.7801	12.515	14.962	16.072	17.030	17.030	17.632
	14 × 14 × 10	8.7801	12.515	14.962	16.072	17.030	17.030	17.631
	15 × 15 × 10	8.7800	12.515	14.961	16.072	17.030	17.030	17.631

In this work, the boundary conditions are handled by artificial boundary springs. In order to determine the appropriate value of the springs’ stiffness for different boundary conditions, it is necessary to investigate the influence of springs’ stiffness on the frequencies of the plate. The variations of the first three dimensional frequency parameters with different value of springs’ stiffness are depicted in Figure 2. The geometrical parameters are as follows: a/b = 1 and h/b = 0.2. The elastic parameters Γ_i are defined as Γ_i = lg(k_i/D) (i = u, v, and w). The plate is assumed to have free boundary conditions at y = constant and be restrained by only one kind of boundary springs at x = constant. It is obvious that in certain range, the increase in the elastic parameter leads to increase in the frequency parameters, and out of the range, the frequency parameters are almost constant. Based on this, the elastic parameters are defined as follows: Γ = 9 for clamped boundary condition and Γ = 2 for elastic boundary condition.

Figure 2.

Variations of the first three frequency parameters of the plates with h/b = 0.2 with different value of spring stiffness.

Plate with general boundary conditions

In this subsection, free vibration of the plates with general boundary conditions is investigated. Apart from aforementioned classical boundary conditions, three types of elastic boundary conditions (i.e. E₁, E₂, and E₃) are also considered. The definitions of the boundary conditions and corresponding value of Γ are given in Table 2.

Table 2.

The value of Γ for different boundary conditions (x = constant).

Boundary set	Essential condition	Γ_u	Γ_v	Γ_w
F	No constraints	0	0	0
C	u = v = w = 0	9	9	9
S	v = w = 0	0	9	9
E₁	u ≠ 0, v = w = 0	2	9	9
E₂	v ≠ 0, u = w = 0	9	2	9
E₃	w ≠ 0, u = v = 0	9	9	2

Some comparisons of the present results with those in available literature are presented to demonstrate the accuracy and reliability of the present formulation. The first seven frequency parameters of rectangular simply supported plates with different length-to-width rations and thickness-to-width rations are given in Table 3. The same vibration problem has been studied by Liew et al.²⁶ using Ritz method in conjunction with 3D theory of elasticity. It can be seen that excellent agreement of the results is achieved. Table 4 presents dimensional frequency parameters Ω of rectangular plate with different boundary conditions. The geometrical parameters are as follows: h/b = 0.2; a/b = 0.5, 1, and 2. The results are compared with those given by Malik and Bert²⁵ using the differential quadrature (DQ) method. It is observed that results show a great agreement.

Table 3.

First seven frequency parameters of rectangular plates with SSSS boundary conditions (μ = 0.3).

a/b	h/b		Ω₁/π²	Ω₂/π²	Ω₃/π²	Ω₄/π²	Ω₅/π²	Ω₆/π²	Ω₇/π²
1	0.1	Liew et al.²⁶	1.9342	4.6222	4.6222	6.5234	6.5234	7.1030	8.6617
	0.1	Present	1.9350	4.6237	4.6237	6.5234	6.5234	7.1071	8.6627
	0.2	Liew et al.²⁶	1.7758	3.2617	3.2617	3.8991	3.8991	4.6127	5.6524
	0.2	Present	1.7760	3.2617	3.2617	3.8996	3.8996	4.6127	5.6538
	0.3	Liew et al.²⁶	1.5895	2.1745	2.1745	3.0752	3.2406	3.2406	4.3489
	0.3	Present	1.5896	2.1745	2.1745	3.0752	3.2409	3.2409	4.3489
	0.4	Liew et al.²⁶	1.4131	1.6309	1.6309	2.3064	2.7254	2.7254	–
	0.4	Present	1.4132	1.6309	1.6309	2.3064	2.7255	2.7255	3.2617
2	0.1	Liew et al.²⁶	1.2237	1.9342	3.0825	3.2617	3.9715	4.6222	4.6222
	0.1	Present	1.2244	1.9363	3.0852	3.2617	3.9725	4.6258	4.6264
	0.2	Liew et al.²⁶	1.1555	1.6309	1.7758	2.7197	3.2617	3.2617	3.4110
	0.2	Present	1.1557	1.6309	1.7764	2.7205	3.2617	3.2617	3.4113
	0.3	Liew et al.²⁶	1.0669	1.0872	1.5895	2.1745	2.1745	2.3443	2.4311
	0.3	Present	1.0670	1.0872	1.5898	2.1745	2.1745	2.3447	2.4311
	0.4	Liew et al.²⁶	0.8154	0.9748	1.4131	1.6309	1.6309	1.8233	2.0231
	0.4	Present	0.8154	0.9748	1.4133	1.6309	1.6309	1.8233	2.0234

Table 4.

First two frequency parameters of plates with different boundary conditions (h/b = 0.2, μ = 0.3).

Boundary conditions	a/b = 0.5		a/b = 1		a/b = 2
Boundary conditions	Malik and Bert²⁵	Present	Malik and Bert²⁵	Present	Malik and Bert²⁵	Present
SSSS	1.2809	1.2809	2.7894	2.7898	7.2605	7.2617
SSSS	1.5312	1.5313	5.1235	5.1235	10.247	10.247
SCSF	0.6404	0.6404	1.8141	1.8154	3.3770	3.3840
SCSF	1.3176	1.3178	2.5617	2.5617	7.2563	7.2627
SFSF	1.2523	1.2523	1.4326	1.4326	1.4850	1.4850
SFSF	1.2809	1.2809	2.2477	2.2480	3.9700	3.9715

Table 5 presents fundamental frequency parameters Ω of the rectangular plate with different classical and elastic boundary conditions. The geometrical parameters used are taken to be as follows: a/b = 0.5, 1, and 2; h/b = 0.1, 0.2, 0.3, and 0.4. It is obvious that the geometrical parameters and boundary conditions can alter the fundamental frequency parameters of the rectangular plate. In all cases, the increase in the length-to-width ratios leads to decrease in the frequency parameters, which means the frequency parameters are dominated by mass of the plate. The effects of thickness-to-width ratios on the frequency parameters strongly depend on the boundary conditions. For the plate with classical boundary conditions, the frequency parameters decrease as thickness-to-width ratios h/b increases, while an inverse behavior can be obtained for plate with elastic boundary conditions. 3D mode shapes for rectangular thick plate with different boundary conditions are depicted in Figures 3 to 5.

Table 5.

Foundational frequency parameters Ω of rectangular plates with different boundary conditions (μ = 0.3).

a/b	h/b	SSSF	CFFF	SCCC	SSSS	E₁E₁E₁E₁	E₂E₂E₂E₂	E₃E₃E₃E₃
0.5	0.1	32.1913	13.5805	62.7665	47.3187	0.4478	4.4453	7.5223
	0.3	10.7306	11.2270	36.2957	21.4612	2.1529	6.8197	12.0824
	0.5	6.4383	8.9038	24.1034	12.8767	3.8224	7.0547	13.7252
1	0.1	11.3834	3.4504	29.4482	22.5072	0.4478	4.4571	5.9159
	0.3	9.8946	3.2436	20.2826	17.1854	2.1543	7.1465	8.8329
	0.5	6.4383	2.9426	14.4431	12.8767	3.8305	7.6039	9.4921
2	0.1	3.9692	0.8605	22.6764	16.6699	0.1598	3.1249	4.7040
	0.3	3.7012	0.8431	15.9860	13.0548	0.7666	4.4359	6.7842
	0.5	3.3518	0.8175	11.4342	9.9465	1.3532	4.2919	7.2880

Figure 3.

First four mode shapes of rectangular plate with CFFF boundary conditions (a/b = 2, h/b = 0.5).

Figure 4.

First four mode shapes of rectangular plate with SSSF boundary conditions (a/b = 2, h/b = 0.5).

Figure 5.

First four mode shapes of rectangular plate with E₁E₁E₁E₁ boundary conditions (a/b = 2, h/b = 0.5).

Plate resting on elastic foundation

The comparison of first seven frequency parameters for free vibration of a rectangular plate on elastic foundation with free boundary conditions is present in Table 6. The non-dimensional foundation parameters k_g = 100, k_s = 10 which are defined as follows:

k_{r} = K_{r} a^{4} / D, k_{g} = K_{g} a^{2} / D

Table 6.

First seven frequency parameters of square plates on elastic foundation with free edges (k_r = 100, k_s = 10, μ = 0.3).

h/b		Ω₁/π²	Ω₂/π²	Ω₃/π²	Ω₄/π²	Ω₅/π²	Ω₆/π²	Ω₇/π²
0.01	Ye et al.¹⁶	1.0132	1.4869	1.4869	2.2626	3.1804	3.4473	4.3948
0.01	Present	1.0132	1.4877	1.4877	2.2679	3.1827	3.4498	4.4106
0.2	Ye et al.¹⁶	1.0108	1.4381	1.4381	2.0931	2.8167	3.0501	3.6573
0.2	Present	1.0108	1.4389	1.4389	2.0940	2.8200	3.0522	3.6596
0.5	Ye et al.¹⁶	0.92415	1.0643	1.0643	1.4970	1.6355	1.6898	1.7459
0.5	Present	0.92415	1.0644	1.0644	1.4970	1.6355	1.6899	1.7459

In this study, the results of the square plates are compared with the 3D solutions.¹⁶ It can be seen that excellent agreement of the results is achieved.

Several new numerical results for free vibration of rectangular plates resting on elastic foundation are presented in Tables 7 and 8. The geometry parameters are given as follows: h/b = 0.2, a/b = 1 and 2. Various boundary conditions i.e SSSS, CSCS, CFCF, E₂E₂E₂E₂, and E₃E₃E₃E₃, are considered. It is observed that for a plate with given size, some of the frequency parameters increase with increase in the foundation stiffness, while some of those do not distinctly vary with foundation stiffness. It is also found that frequencies of elastically restrained plates are not sensitive to the change of foundation stiffness.

Table 7

First three frequency parameters Ω of square plates on elastic foundation with different classical boundary conditions (h/b = 0.2, μ = 0.3).

Boundary conditions	k _r	k _s	a/b = 1			a/b = 2
Boundary conditions	k _r	k _s	Ω₁	Ω₂	Ω₃	Ω₁	Ω₂	Ω₃
SSSS	10	0	17.7867	32.1917	32.1917	11.4324	16.0959	17.5482
		10	22.2470	32.1917	32.1917	12.6346	16.0959	18.7864
		100	32.1917	32.1917	44.2349	16.0959	20.4056	27.3613
	100	0	19.9546	32.1917	32.1917	11.6612	16.0959	17.6940
		10	23.9846	32.1917	32.1917	12.8414	16.0959	18.9220
		100	32.1917	32.1917	44.9877	16.0959	20.5311	27.4508
	1000	0	32.1917	32.1917	34.3406	13.7355	16.0959	19.0870
		10	32.1917	32.1917	36.6286	14.7456	16.0959	20.2249
		100	32.1917	32.1917	45.5260	16.0959	21.7423	28.3278
CSCS	10	0	22.8985	32.1917	40.4434	12.3840	16.0959	19.9420
		10	26.6440	32.1917	45.2788	13.5218	16.0959	21.0743
		100	32.1917	46.9699	56.9579	16.0959	21.0498	29.1508
	100	0	24.6412	32.1917	41.3809	12.5956	16.0959	20.0708
		10	28.1297	32.1917	46.0794	13.7154	16.0959	21.1958
		100	32.1917	47.6800	56.9582	16.0959	21.1715	29.2351
	1000	0	32.1917	37.3981	49.4356	14.5381	16.0959	21.3132
		10	32.1917	39.5976	53.1087	15.5136	16.0959	22.3706
		100	32.1917	53.7177	56.9611	16.0959	22.3492	30.0634
CFCF	10	0	18.1212	20.4131	29.4429	5.2455	7.8518	13.1569
		10	20.7769	24.8650	29.4524	5.8989	9.7961	13.1581
		100	29.4656	35.6691	46.4614	9.7895	13.1635	19.3964
	100	0	20.3703	22.3747	29.4435	5.7499	8.1856	13.1569
		10	22.7469	26.4664	29.4540	6.3513	10.0647	13.1582
		100	29.4657	36.7568	47.2044	10.0674	13.1636	19.5298
	1000	0	29.4438	35.2324	36.0897	9.4000	10.9719	13.1574
		10	29.4504	36.5459	38.5566	9.7779	12.4287	13.1596
		100	29.4664	45.7691	52.8134	12.5049	13.1637	20.8134

Table 8.

First three frequency parameters Ω of square plates on elastic foundation with elastic boundary conditions (h/b = 0.2, μ = 0.3).

Boundary conditions	k _r	k _s	a/b = 1			a/b = 2
Boundary conditions	k _r	k _s	Ω₁	Ω₂	Ω₃	Ω₁	Ω₂	Ω₃
E₂E₂E₂E₂	0	0	6.1621	6.1621	7.7814	4.1159	6.0252	6.1958
		10	6.1622	6.1622	8.2630	4.1159	6.1958	6.3508
		100	6.1623	6.1623	8.7443	4.1160	6.1959	7.2050
	10	0	6.1621	6.1621	8.3899	4.1159	6.0760	6.1958
		10	6.1622	6.1622	8.8421	4.1159	6.1958	6.3992
		100	6.1623	6.1623	9.2968	4.1160	6.1959	7.2480
	100	0	6.1621	6.1621	10.8101	4.1159	6.1958	6.5152
		10	6.1622	6.1622	10.8103	4.1159	6.1958	6.8189
		100	6.1623	6.1623	10.8106	4.1160	6.1959	7.6242
E₃E₃E₃E₃	0	0	7.7982	8.7891	8.7891	6.0482	7.4861	7.6027
		10	8.2690	8.7892	8.7892	6.3666	7.4861	7.6027
		100	8.7450	8.7892	8.7892	7.2080	7.4861	7.6027
	10	0	8.4056	8.7891	8.7891	6.0989	7.4861	7.6027
		10	8.7892	8.7892	8.8478	6.4149	7.4861	7.6027
		100	8.7892	8.7892	9.2975	7.2510	7.4861	7.6027
	100	0	8.7891	8.7891	12.4712	6.5367	7.4861	7.6027
		10	8.7892	8.7892	12.4714	6.8337	7.4861	7.6027
		100	8.7892	8.7892	12.4716	7.4861	7.6027	7.6270

Conclusions

A 3D formulation for the free vibrations of thick rectangular plates with general boundary conditions and resting on elastic foundations is presented. The general boundary conditions are imposed by means of penalty function method. The displacements of the plates are expressed by a 3D cosine series and some simple polynomial functions which introduced to ensure and accelerate the convergence of the series representation. All the unknown coefficients can be obtained by using the Rayleigh–Ritz method. The effects of boundary conditions, geometrical parameters, and elastic foundation on the vibration behavior of the plate are also investigated. It can be found that the increase in the length-to-width ratios leads to decrease in the frequency parameters, and the effects of thickness-to-width ratios on the frequency parameters strongly depend on the boundary conditions. It is observed that the foundation stiffness have significant influence on frequencies.

Furthermore, in contrast to most existing techniques, the current method can be universally applicable to a variety of boundary conditions including all the classical cases, elastic restraints, and their combinations. The change of the boundary conditions is as easy as modifying material properties and geometry dimensions without the need of making any change to the solution procedure. In addition, the proposal solution can be readily applied to shells with more complex boundaries like point supports, non-uniform elastic restraints, mix boundaries, partial supports, and their combinations.

Footnotes

Declaration of conflicting interests

The author(s) declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.

Funding

The author(s) disclosed receipt of the following financial support for the research, authorship, and/ or publication of this article: The authors gratefully acknowledge the financial support from the National Natural Science Foundation of China (No. 51705071).

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Three-dimensional free vibration of thick plates with general end conditions and resting on elastic foundations

Abstract

Keywords

Introduction

Theoretical formulations

Preliminaries

Energy functional

Admissible functions

Solution procedure

Numerical examples and discussion

Convergence study

Plate with general boundary conditions

Plate resting on elastic foundation

Conclusions

Footnotes

Declaration of conflicting interests

Funding

References