# Free and Forced Vibration of the Moderately Thick Laminated Composite Rectangular Plate on Various Elastic Winkler and Pasternak Foundations

Authors: Dongyan Shi; Hong Zhang; Qingshan Wang; Shuai Zha
Journal: Shock and Vibration
 (2017)

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## Abstract

An improved Fourier series method (IFSM) is applied to study the free and forced vibration characteristics of the moderately thick laminated composite rectangular plates on the elastic Winkler or Pasternak foundations which have elastic uniform supports and multipoints supports. The formulation is based on the first-order shear deformation theory (FSDT) and combined with artificial virtual spring technology and the plate-foundation interaction by establishing the two-parameter foundation model. Under the framework of this paper, the displacement and rotation functions are expressed as a double Fourier cosine series and two supplementary functions which have no relations to boundary conditions. The Rayleigh-Ritz technique is applied to solve all the series expansion coefficients. The accuracy of the results obtained by the present method is validated by being compared with the results of literatures and Finite Element Method (FEM). In this paper, some results are obtained by analyzing the varying parameters, such as different boundary conditions, the number of layers and points, the spring stiffness parameters, and foundation parameters, which can provide a benchmark for the future research.


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## Body

## 1. Introduction

A variety of composite plates on the elastic foundations are widely used in civil engineering, like rigid road plate, airport pavement, building foundations, dock platform, and so on. It is of great significance for design, use, and maintenance to study the mechanical properties of composite plates on elastic foundation. Therefore, the study on the vibration characteristics of laminated plates on the elastic foundations has been one of the hot spots in recent years. According to the existing research, the elastic foundational plate models can be divided into two categories which are Winkler-type foundation and two-parameter foundation.Winkler foundation is the first proposed linear elastic theoretical model. Laura et al. [1] solved the free vibrations of a circular thin plate on Winkler foundation with varying thickness by using linear analysis and the Rayleigh-Schmidt method. Liew et al. [2] extended an approximate analysis of rectangular Mindlin plates resting on Winkler foundations based on the differential quadrature method (DQM). The plates were subject to arbitrary combination of free, simply supported, and clamped boundary conditions. Gupta et al. [3] discussed the effect of Winkler foundation on axisymmetric vibrations of polar orthotropic circular plates with variable thickness based on the classical plate theory. Xiang [4] investigated the effect of multisegment Winkler foundations on the rectangular Mindlin plates. They divided the four sides into two groups, in which the two parallel edges were one group. A set of edges were simply supported and another group was the combinations of uniform and classical boundary conditions. Younesian et al. studied strongly nonlinear generalized duffing oscillators [5] by using He’s frequency–amplitude formulation and He’s energy balance method and then proposed a closed form expression for the dynamic response of an elastic plate rested on a nonlinear elastic Winkler foundation [6]. Ansari et al. investigated the forced vibration of microbeam structures supported by nonlinear viscoelastic-type foundation [7], Kelvin–Voight foundation [8], and Winkler and Pasternak foundations [9] based on the Galerkin approach and multiple time-scales method. There are many related researches on the Winkler foundation, and we can refer to the relevant refs. [10–12]. The calculated results of the Winkler-type foundation plate model are very different from the actual results. In some special conditions, especially, the results could not meet the requirements any more by using the Winkler foundation model.With the deepening of the research, the two-parameter elastic foundation model is proposed. Two independent parameters are used to represent the compressive strength and shear strength of the soil, which can overcome the inherent defects of the Winkler foundation model and effectively eliminate the discontinuity. In the two-parameter model, the Pasternak foundation is widely used. Nedri et al. [13] studied the free vibration analysis of simply supported thick laminated rectangular composite plates resting on elastic Pasternak-type foundations based on refined hyperbolic shear deformation theory. Baltacıoğlu et al. [14] used the discrete singular convolution method to realize nonlinear static analysis of a thick laminated rectangular composite plate on nonlinear foundation by using the FSDT. Singh et al. [15] dealt with the postbuckling shear deformation of the laminated composite plates on the Pasternak-type foundation on the basis of HSDT and random system properties. It is the first time that a perturbation technique has been successfully combined with direct iterative technique by neglecting the changes in nonlinear stiffness matrix due to random variation of transverse displacements during iteration. Malekzadeh et al. [16] investigated the effect of nonideal simple supports and initial stresses on the vibration of laminated rectangular plates on Pasternak foundation based on the Lindstedt-Poincare perturbation technique. Kutlu et al. [17] derived a mixed-type finite element formulation to study the dynamic response of the Mindlin plate–arbitrarily orthotropic Pasternak foundation interaction by applying the Gâteaux differential. Tornabene et al. [18] considered the static and dynamic analyses of laminated doubly curved and degenerate shells and panels on the Winkler and Pasternak foundations by using the generalized differential quadrature (GDQ) method and FSDT. Jahromi et al. [19] studied free vibration of moderately thick rectangular plate partially resting on Pasternak foundation based on GDQ and FSDT. The boundary conditions of plate were considered as combinations of free, simple, or clamped support. Li and Zhang [20] investigated free vibration analysis of magnetoelectroelastic plate resting on a Pasternak-type foundation based on Mindlin plate theory. Khalili et al. [21] used the Lindstedt-Poincare perturbation technique to study the buckling of nonideal rectangular laminated plate on Pasternak foundation. One edge of the plate was allowed a small nonzero deflection and movement. Dehghany and Farajpour [22] dealt with exact solution for free vibration analysis of simply supported rectangular plates on Pasternak foundation on the basis of three-dimensional elasticity theory. Briscoe et al. [23] presented a solution for the buckling strength of simply supported plates on the Pasternak-type foundation under in-plane bending loads by using the minimum potential energy principle. Thai et al. [24] proposed a simple refined theory for studying bending, buckling, and vibration of thick plates resting on Pasternak foundation. The boundary conditions of rectangular plates were expressed as two opposite edges with simply supported plates and the other two edges with arbitrary boundary conditions. Idowu et al. [25] proposed a fourth-order partial differential equation to study the dynamic effects of viscous damping on isotropic rectangular plates resting on Pasternak foundation subjected to moving loads. Bahmyari and Khedmati [26] used shear deformable plate theory and Element-Free Galerkin Method to study the free vibration analysis of nonhomogeneous moderately thick plates with point supports resting on Pasternak elastic foundation. Kiani et al. [27] considered instability of simply supported sandwich plates with functionally graded material (FGM) face sheets resting on the Pasternak foundations based on FSDT. There have been a lot of literatures of the plate resting on the Pasternak-type foundation. We can also check the relevant references, such as [28, 29]. However, these literatures have great limitations for the moderately thick or thick laminated rectangular plates resting on the elastic foundation which have arbitrary and special boundary conditions. It is necessary to study the effects of boundary conditions on the free and forced vibration characteristics of plate resting on the elastic foundation.Stimulated by the restriction of the plate boundary conditions in the existing researches, the free and forced vibration characteristics of the moderately thick laminated rectangular plates are analyzed which rest on the Winkler or Pasternak foundations and have various uniform supported and multipoints supported boundary conditions. It should be pointed out that the free vibration analysis of the moderately thick laminated composite rectangular plate with nonuniform boundary conditions [30] has been done previously. However, it only discussed the free vibration analysis of the laminated plate with partial supports and multipoints supports. In this proposition, the free and forced vibration analyses of uniform and multipoints supported laminated rectangular plate resting on the Winkler and Pasternak foundations are studied in this paper. As we all know, there are many structural forms of plate in engineering applications, like stiffened plates [31–34], cracked rectangular plates [35–38], corroded plates [39–43], and so on. As the most common basic model, the rectangular plate structure is widely studied. An IFSM is extended to study the free and forced vibration of rectangular plates on the elastic Winkler-type and Pasternak-type foundation. This method is previously studied by Zhang and Li [44, 45]. According to the FSDT, the five displacement functions can be written as feasible period superposition functions. Their specific expressions are a double Fourier cosine series and two supplementary functions by ignoring the influence of boundary conditions. On the basis of the traditional Fourier series, these supplementary functions are added to eliminate the discontinuous or jumping phenomenon in the boundaries which are regarded as a periodic function and defined within the entire coordinates of laminate plate. These unknown coefficients are defined in the generalized coordinates which can be solved by Rayleigh-Ritz procedure. It is very easy to realize the change of different boundary conditions by changing the stiffness value of the five springs on the four edges. The results obtained by this method are compared with those results obtained by literatures and FEM, which show good agreement. The work mainly deals with the consequences of the practical and significant constraints such as various uniform supports, multipoints supports, and various values of the foundation parameters.


## 2. Theoretical Analysis

A combination technique is used to get the vibration characteristics of orthotropic laminated plate structure based on artificial spring technique and Rayleigh-Ritz method. This plate under uniform supports or multipoints supports is rested on elastic Winkler-type and Pasternak-type foundation.


### 2.1. Establishment of the Model

As shown in Figure1, a laminated rectangular plate model is established to analyze vibration characteristics. For the plate, the length, width, and thickness are a, b, and h. Establish a coordinate system in the plate mid-surface. In this coordinate system, the x, y, and z represent the length, width, and thickness directions of the studied plate. Symbol θ is the laying angle between the layer fiber direction and the x-axis. For the sake of brevity, it is supposed that every lamina has the same material properties and thickness. For the elastic foundation, Kw and Ks are defined as linear Winkler foundation and linear Pasternak foundation parameters. Five types of spring are used to describe the boundary conditions of laminated foundation plate, which are linear springs (ku,kv, andkw) and rotational springs (Kx, Ky), respectively. The arbitrary boundary conditions can be realized by setting the stiffness values of the five different springs [46–50]. For instance, the free boundary condition can be easily gained when the spring stiffness values on the four edges are zero. All the stiffness values are set to a large value to achieve the clamped boundary conditions in the numerical calculation.Figure 1
Rectangular plate resting on elastic foundations.


### 2.2. Relationship between Kinematics and Stress

According to the displacements and rotations of the middle surface for the established plate model, the displacements can be expressed based on FSDT [51–53].(1)Ux,y,z,t=ux,y,z,t+zϕxx,y,t,Vx,y,z,t=vx,y,z,t+zϕyx,y,t,Wx,y,z,t=wx,y,z,t,where u, v, and w represent the middle surface displacements of the plate in the x, y, and z directions and ϕx and ϕy are the rotations of transverse normal for x- and y-axes, respectively. Then t is the time variable.For rectangular laminated plates, the linear strains-displacement relations ofk’th layer can be got according to the strain–stress relationship of elasticity theory(2)εxx=εxx0+zkχxx,εyy=εyy0+zkχyy,γyz=γyz0,γxz=γxz0,γxy=γxy0+zkχxy.According to the above formulas,εxx0, εyy0, and γxy0 express the normal and shear strains in the o-xyz coordinate system. The symbols of transverse shear strains are marked as γxz and γyz which can be regarded as constant and ignore the thickness change. χxx, χyy, and χxy represent the corresponding curvature and twist changes. Besides, zk expresses the thickness variable range of k’th layer. Expressions for strain and generalized displacement relations can be established as(3)εxx0=∂u∂x,γxz0=∂w∂x+ϕx,χxx=∂ϕx∂x,εyy0=∂v∂y,γyz0=∂w∂y+ϕy,χyy=∂ϕy∂y,γxy0=∂u∂y+∂v∂x,χxy=∂ϕx∂y+∂ϕy∂x.According to the generalized Hooke’s law, the relation between stress and strain can be expressed as(4)σxxσyyτxyτyzτxz=Q11k¯Q12k¯Q16k¯00Q12k¯Q22k¯Q26k¯00Q16k¯Q26k¯Q66k¯00000Q44k¯Q45k¯000Q45k¯Q55k¯εxxεyyγxyγyzγxzin which σxx and σyy are the normal stresses and τyz, τxz, and τxy are shear stresses in the o-xyz coordinate system. Besides, the specific expression of k’th layered stiffness coefficients Qplk¯(p,l=1,2,4,5,6) [54, 55] can be written as(5)Q11k¯=Q11kcos4⁡θk+2Q12k+2Q66kcos2⁡θksin2θk+Q22ksin4⁡θk,Q12k¯=Q11k+Q22k-4Q66kcos2θksin2θk+Q12ksin4θk+cos4θk,Q22k¯=Q11ksin4⁡θk+2Q12k+2Q66kcos2⁡θksin2⁡θk+Q22kcos4⁡θk,Q16k¯=Q11k-Q12k-2Q66kcos3⁡θksin⁡θk+Q12k-Q22k+2Q66kcos⁡θksin3⁡θk,Q26k¯=Q11k-Q12k-2Q66kcos⁡θksin3⁡θk+Q12k-Q22k+2Q66kcos3⁡θksin⁡θk,Q66k¯=Q11k+Q22k-2Q12k-2Q66kcos2⁡θksin2⁡θk+Q66ksin4⁡θk+cos4⁡θk,Q44k¯=Q44kcos2⁡θk+Q55ksin2⁡θk,Q45k¯=Q55k-Q44kcos⁡θksin⁡θk,Q55k¯=Q55kcos2⁡θk+Q44ksin2⁡θk,where θ is the laying angle and Qplk represents the material coefficients of the k’th layer which can be obtained by building the relationships with longitudinal modulus E1k, the transverse modulus E2k, Poisson’s ratios μ12k and μ21k, and shear moduli G12k, G13k, and G23k.(6)Q11k=E1k1-μ12kμ21k,Q12k=μ21kQ11k,Q22k=E2k1-μ12kμ21k,Q44k=G23k,Q55k=G13k,Q66k=G12k.In order to seek simplicity, the material constants of each layer are the same, which can be expressed asEik=Ei, G12k=Gi,j, and μi,jk=μi,j(i,j=1,2,3). In addition, the relationship of μ12 and μ21 is μ12E2=μ21E1. It should be pointed out that the isotropic plate resting on the elastic foundations can be easily analyzed by letting E1=E2, G12=G13=G23=E1/(2+2μ12). By introducing the shear correction factor κ, the relation between the generalized forces and strains can be finally obtained [51, 52]. The specific expressions are shown as follows:(7)NxNyNxyMxMyMxy=A11A12A16B11B12B16A12A22A26B12B22B26A16A26A66B16B26B66B11B12B16D11D12D16B12B22B26D12D22D26B16B26B66D16D26D66εxx0εyy0γxy0χxxχyyχxy,QxQy=κA55A45A45A55γxzγyzin which the generalized forces include normal forces (Nx, Ny), shear force (Nxy), bending moments (Mx, My), twisting moment (Mxy), and transverse shear forces (Qx, Qy). In addition, Aij, Bij, and Dij are extensional, extensional-bending, and bending stiffness coefficients, whose expressions are written as(8)Aij=∑k=1NQijk¯Zk+1-Zk,Bij=12∑k=1NQijk¯Zk+12-Zk2,Dij=13∑k=1NQijk¯Zk+13-Zk3in which N donates the total number of layers. Moreover, when we study the vibration characteristics of the isotropic plate, the extensional-bending stiffness coefficients Bij are zero and Aij and Dij are independent.


### 2.3. Energy Equation

The main work of this paper is to investigate the vibration characteristics of the moderately thick laminated composite rectangular plate with uniform and multipoints supports which is rested on the elastic foundations. Rayleigh-Ritz energy method is extended to study the free and forced vibration of plate on the Winkler and Pasternak foundations. The Lagrangian energy function for the laminated plate can be written as(9)LP=TP-UP-Uf-Vsprings-Wext.T
P expresses the total kinetic energy of laminated plate whose expression is(10)Tp=12∫0a∫0bI0∂u∂t2+I0∂v∂t2+I0∂w∂t2+I2∂ϕx∂t2+I2∂ϕy∂t2+2I1∂u∂t∂ϕx∂t+2I1∂v∂t∂ϕy∂tdydx,I0=∑k=1N∫ZkZk+1ρkdz,I1=∑k=1N∫ZkZk+1ρkzdz,I2=∑k=1N∫ZkZk+1ρkz2dzin which I0, I1, and I2 are the inertia terms of the plate, and ρk is area density of the k’th layer.U
P expresses the strain energy for the moderately thick plates:(11)UP=12∫0a∫0bNxεxx0+Nyεyy0+Nxyεxy0+Mxχxx+Myχyy+Mxyχxy+Qxγxz+Qyγyzdxdy.Substituting (3), (7), and (8) into (11) can obtain the relations between strain energy and displacements in mid-surface of plate. Then, strain energy expression can be written as a superposition of three components, which are stretching energy (US), bending energy (UB), and bending–stretching coupling energy (UBS).(12a)US=12∫0a∫0bA11∂u∂x2+A22∂v∂y2+2A12∂u∂x∂v∂y+A66∂u∂y+∂v∂x2+2A16∂u∂y+∂v∂x∂u∂x+2A26∂u∂y+∂v∂x∂v∂y+κcA44∂w∂y+ϕy2+κcA55∂w∂x+ϕx2+2κcA45∂w∂x+ϕx∂w∂y+ϕydxdy,(12b)UB=12∫0a∫0bD11∂ϕx∂x2+D22∂ϕy∂y2+2D12∂ϕx∂x∂ϕy∂y+D66∂ϕx∂y+∂ϕy∂x2+2D16∂ϕx∂y+∂ϕy∂x∂ϕx∂x+2D26∂ϕx∂y+∂ϕy∂x∂ϕy∂ydxdy,(12c)UBS=∫0a∫0bB11∂u∂x∂ϕx∂x+B12∂u∂x∂ϕy∂y+∂v∂y∂ϕx∂x+B22∂v∂y∂ϕy∂y+B66∂v∂x+∂u∂y∂ϕx∂y+∂ϕy∂x+B16∂u∂x+B26∂v∂y∂ϕx∂y+∂ϕy∂x+B16∂ϕx∂x+B26∂ϕy∂y∂v∂x+∂u∂ydxdy.The strain energy due to the Winkler and Pasternak foundations is given by(13)Uf=12∫0a∫0bKWw2+KS∂w∂x2+∂w∂y2dxdy.For the elastic Winkler foundation, we just need to set the Pasternak foundation parameterKS to zero. Then, the influence of the free vibration for the plate which is on the Winkler-type elastic foundation can be studied with the change of the Winkler foundation parameterKW. In this paper, the main interests are studying the special boundary constraints and variation of Winkler and Pasternak foundations by introducing artificial virtual spring technology, which can be found in [56–58]. So the cases of uniform supported and multipoints supported boundary conditions whose plates are rested on the Winkler and Pasternak foundations will be considered here.V
s
p
r
i
n
g
s
u
n
i
f
o
r
m is the potential energy on the four edges of the plate with elastic uniform boundary conditions. It is simulated with five kinds of springs evenly distributed on four edges.(14)Vspringsuniform=12∫0b∫-h/2h/2kx0uux,y2+kx0vvx,y2+kx0wwx,y2+Kx0xϕxx,y2+Kx0yϕyx,y2x=0+kxauux,y2+kxavvx,y2+kxawwx,y2+Kxaxϕxx,y2+Kxayϕyx,y2x=adzdy+12∫0a∫-h/2h/2ky0uux,y2+ky0vvx,y2+ky0wwx,y2+Ky0xϕxx,y2+Ky0yϕyx,y2y=0+kybuux,y2+kybvvx,y2+kybwwx,y2+Kybxϕxx,y2+Kybyϕyx,y2y=bdzdx.For the cases of multipoints supports, the supported points are evenly distributed on the four edges. For intuitive understanding, Figure2 gives the distributions of 4-point, 8-point, and 16-point supported boundary conditions. They can be regarded as the discretization of uniform support. Therefore, the potential energy Vspringspoints for the plate with multipoints supports can be written as(15)Vspringspoints=12∑r=0NRkx0uuxr,yr2+kx0vvxr,yr2+kx0wwxr,yr2+Kx0xϕxxr,yr2+Kx0yϕyxr,yr2xr=0+12∑r=0NRkxauuxr,yr2+kxavvxr,yr2+kxawwxr,yr2+Kxaxϕxxr,yr2+Kxayϕyxr,yr2xr=a+12∑s=0NSky0uuxs,ys2+ky0vvxs,ys2+ky0wwxs,ys2+Ky0xϕxxs,ys2+Ky0yϕyxs,ys2ys=0+12∑s=0NSkybuuxs,ys2+kybvvxs,ys2+kybwwxs,ys2+Kybxϕxxs,ys2+Kybyϕyxs,ys2ys=b.Figure 2
Laminated plate with various multipoints supported boundary conditions.
(a)
4-point support
(b)
8-point support
(c)
16-point supportW
e
x
t expresses the work done by the external excitation force on the moderately thick plates:(16)Wext=∬Sfuu+fvv+fww+fϕxϕx+fϕyϕydxdy,where fi(i=u,v,w,ϕx,ϕy) is the external load distribution function on the plate. In this paper, the normal harmonic point force Fi on the plate is applied to study the flexural vibration behavior of the composite plate. Therefore, the load distribution function fi can be expressed as(17)fi=Fiδx-xeδy-ye,where (xe, ye) is the position of point force. δ is the 2D Dirac function.


### 2.4. Kinematics Balance Equation

The kinematics balance equations for the moderately thick plate can be obtained by Hamilton’s principle [51, 52].(18)∂Nx∂x+∂Nxy∂y=I0∂2u∂t2+I1∂2ϕx∂t2,∂Nxy∂x+∂Ny∂y=I0∂2v∂t2+I1∂2ϕy∂t2,∂Mx∂x+∂Mxy∂y-Qx=I1∂2u∂t2+I2∂2ϕx∂t2,∂Qx∂x+∂Qy∂y=I0∂2w∂t2,∂Mxy∂x+∂My∂y-Qy=I1∂2v∂t2+I2∂2ϕy∂t2.As we can see from (18), the displacement functions required second-order derivatives. The kinematics balance equations can be rewritten as a matrix form by combining (3), (7), and (18) simultaneously.(19)L11L120L14L15L12L220L24L2500L33L34L35L14L24L34L44L45L15L25L35L45L55-M1100M1400M2200M2500M3300M1400M4400M2500M55uvwϕxϕy=FuFvFwFϕxFϕy,where linear differential operators are expressed as Lij and Mij. The specific expressions are given below:(20)L11=A11∂2∂x2+2A16∂2∂x∂y+A66∂2∂y2,L12=A16∂2∂x2+A12+A66∂2∂x∂y+A26∂2∂y2,L14=B11∂2∂x2+2B16∂2∂x∂y+B66∂2∂y2,L15=B16∂2∂x2+B12+B66∂2∂x∂y+B26∂2∂y2,L22=A66∂2∂x2+2A26∂2∂x∂y+A22∂2∂y2,L24=B16∂2∂x2+B12+B66∂2∂x∂y+B26∂2∂y2,L25=B66∂2∂x2+2B26∂2∂x∂y+B22∂2∂y2,L33=-A55∂2∂x2-2A45∂2∂x∂y-A44∂2∂y2,L34=-A55∂∂x-A45∂∂y,L44=D11∂2∂x2+2D16∂2∂x∂y+D66∂2∂y2-A55,L35=-A45∂∂x-A44∂∂y,L45=D16∂2∂x2+D12+D66∂2∂x∂y+D26∂2∂y2-A45,L55=D66∂2∂x2+2D26∂2∂x∂y+D22∂2∂y2-A44,M11=M22=M33=-I0∂2∂t,M14=M25=-I1∂2∂t,M44=M55=-I2∂2∂t.


### 2.5. Displacement Expression

An IFSM is applied to express the displacements of the moderately thick laminated rectangular plate on the elastic foundations. The expressions ignore the boundary conditions and eliminate the discontinuous or jumping phenomenon in the boundaries. They can be regarded as a periodic function which is defined within the entire coordinates [59]. The specific displacement expressions of the plate are given below:(21)uvwϕxϕy=P00000P00000P00000P00000PAmnBmnCmnDmnEmn,(22)P=cos⁡λ0xcos⁡λ0y,…,cos⁡λ0xcos⁡λny,…,cos⁡λmxcos⁡λny,…,cos⁡λMxcos⁡λNy,sin⁡λ-2xcos⁡λ0y,…,sin⁡λ-2xcos⁡λny,…,sin⁡λ-2xcos⁡λNy,…,sin⁡λ-1xcos⁡λNy,cos⁡λ0xsin⁡λ-2y,cos⁡λ0xsin⁡λ-1y,…,cos⁡λmxsin⁡λ-2y,…,cos⁡λMxsin⁡λ-1y,(23a)Amn=A0,01,…,A0,n1,…,Am,n1,…,AM,N1,A-2,02,…,A-2,n2,…,A-2,N2,…,A-1,N2,A0,-23,A0,-13,…,Am,-23,…,AM,-13T,(23b)Bmn=B0,01,…,B0,n1,…,Bm,n1,…,BM,N1,B-2,02,…,B-2,n2,…,B-2,N2,…,B-1,N2,B0,-23,B0,-13,…,Bm,-23,…,BM,-13T,(23c)Cmn=C0,01,…,C0,n1,…,Cm,n1,…,CM,N1,C-2,02,…,C-2,n2,…,C-2,N2,…,C-1,N2,C0,-23,C0,-13,…,Cm,-23,…,CM,-13T,(23d)Dmn=D0,01,…,D0,n1,…,Dm,n1,…,DM,N1,D-2,02,…,D-2,n2,…,D-2,N2,…,D-1,N2,D0,-23,D0,-13,…,Dm,-23,…,DM,-13T,(23e)Emn=E0,01,…,E0,n1,…,Em,n1,…,EM,N1,E-2,02,…,E-2,n2,…,E-2,N2,…,E-1,N2,E0,-23,E0,-13,…,Em,-23,…,EM,-13T,where λm=mπ/a and λn=nπ/b. Besides, Amn, Bmn, Cmn, Dmn, and Emn are 2D Fourier coefficients vector, which are composed by Amni, Bmni, Cmni, Dmni, and Emni(i=1,2,3), respectively.There have been a lot of references about the Rayleigh-Ritz technology, such as [60–62], and Ritz like methods, such as [63–65]. In this paper, based on the Rayleigh-Ritz technology, a set of linear algebraic equations directed against Fourier unknown coefficients on displacement equations can be got by combining (18)–(23e) simultaneously. A matrix form can be obtained by transformation(24)Κ-ω2MG=F¯.When we study the free vibration of the plate, we only need to ignore the external force vectorF¯. K, M, and G are stiffness matrix, mass matrix, and unknown Fourier coefficients vector, separately. The specified expressions of K, M, and F¯ are listed in the Appendix. It should be pointed out that G is a combination of the five 2D Fourier coefficients vectors, whose form is(25)G=AmnBmnCmnDmnEmnT.Form (24), we can easily find that the circular frequency ω is the square root of eigenvalues and G is the eigenvectors. When we study the vibration response of the elastic supported plates resting on the elastic Winkler or Pasternak foundations, G can be obtained by giving a frequency value. Then, the exact displacements expressions of plate can be got by putting G into (21)–(23e). In order to explain the innovation of this paper more intuitively, a brief illustration is given in Figure 3. Compared with [30, 66], this paper has three main original ideas. Firstly, this paper focuses on the effect of elastic foundation on the plate. Secondly, the displacement expressions of the plate are improved. Thirdly, forced vibration response of plate on the elastic foundation is studied by introducing a harmonic point force. In addition, the corresponding results are also given in the next section.Figure 3
The flowchart about the solution for the laminated plate on the elastic foundation.


## 2.1. Establishment of the Model

As shown in Figure1, a laminated rectangular plate model is established to analyze vibration characteristics. For the plate, the length, width, and thickness are a, b, and h. Establish a coordinate system in the plate mid-surface. In this coordinate system, the x, y, and z represent the length, width, and thickness directions of the studied plate. Symbol θ is the laying angle between the layer fiber direction and the x-axis. For the sake of brevity, it is supposed that every lamina has the same material properties and thickness. For the elastic foundation, Kw and Ks are defined as linear Winkler foundation and linear Pasternak foundation parameters. Five types of spring are used to describe the boundary conditions of laminated foundation plate, which are linear springs (ku,kv, andkw) and rotational springs (Kx, Ky), respectively. The arbitrary boundary conditions can be realized by setting the stiffness values of the five different springs [46–50]. For instance, the free boundary condition can be easily gained when the spring stiffness values on the four edges are zero. All the stiffness values are set to a large value to achieve the clamped boundary conditions in the numerical calculation.Figure 1
Rectangular plate resting on elastic foundations.


## 2.2. Relationship between Kinematics and Stress

According to the displacements and rotations of the middle surface for the established plate model, the displacements can be expressed based on FSDT [51–53].(1)Ux,y,z,t=ux,y,z,t+zϕxx,y,t,Vx,y,z,t=vx,y,z,t+zϕyx,y,t,Wx,y,z,t=wx,y,z,t,where u, v, and w represent the middle surface displacements of the plate in the x, y, and z directions and ϕx and ϕy are the rotations of transverse normal for x- and y-axes, respectively. Then t is the time variable.For rectangular laminated plates, the linear strains-displacement relations ofk’th layer can be got according to the strain–stress relationship of elasticity theory(2)εxx=εxx0+zkχxx,εyy=εyy0+zkχyy,γyz=γyz0,γxz=γxz0,γxy=γxy0+zkχxy.According to the above formulas,εxx0, εyy0, and γxy0 express the normal and shear strains in the o-xyz coordinate system. The symbols of transverse shear strains are marked as γxz and γyz which can be regarded as constant and ignore the thickness change. χxx, χyy, and χxy represent the corresponding curvature and twist changes. Besides, zk expresses the thickness variable range of k’th layer. Expressions for strain and generalized displacement relations can be established as(3)εxx0=∂u∂x,γxz0=∂w∂x+ϕx,χxx=∂ϕx∂x,εyy0=∂v∂y,γyz0=∂w∂y+ϕy,χyy=∂ϕy∂y,γxy0=∂u∂y+∂v∂x,χxy=∂ϕx∂y+∂ϕy∂x.According to the generalized Hooke’s law, the relation between stress and strain can be expressed as(4)σxxσyyτxyτyzτxz=Q11k¯Q12k¯Q16k¯00Q12k¯Q22k¯Q26k¯00Q16k¯Q26k¯Q66k¯00000Q44k¯Q45k¯000Q45k¯Q55k¯εxxεyyγxyγyzγxzin which σxx and σyy are the normal stresses and τyz, τxz, and τxy are shear stresses in the o-xyz coordinate system. Besides, the specific expression of k’th layered stiffness coefficients Qplk¯(p,l=1,2,4,5,6) [54, 55] can be written as(5)Q11k¯=Q11kcos4⁡θk+2Q12k+2Q66kcos2⁡θksin2θk+Q22ksin4⁡θk,Q12k¯=Q11k+Q22k-4Q66kcos2θksin2θk+Q12ksin4θk+cos4θk,Q22k¯=Q11ksin4⁡θk+2Q12k+2Q66kcos2⁡θksin2⁡θk+Q22kcos4⁡θk,Q16k¯=Q11k-Q12k-2Q66kcos3⁡θksin⁡θk+Q12k-Q22k+2Q66kcos⁡θksin3⁡θk,Q26k¯=Q11k-Q12k-2Q66kcos⁡θksin3⁡θk+Q12k-Q22k+2Q66kcos3⁡θksin⁡θk,Q66k¯=Q11k+Q22k-2Q12k-2Q66kcos2⁡θksin2⁡θk+Q66ksin4⁡θk+cos4⁡θk,Q44k¯=Q44kcos2⁡θk+Q55ksin2⁡θk,Q45k¯=Q55k-Q44kcos⁡θksin⁡θk,Q55k¯=Q55kcos2⁡θk+Q44ksin2⁡θk,where θ is the laying angle and Qplk represents the material coefficients of the k’th layer which can be obtained by building the relationships with longitudinal modulus E1k, the transverse modulus E2k, Poisson’s ratios μ12k and μ21k, and shear moduli G12k, G13k, and G23k.(6)Q11k=E1k1-μ12kμ21k,Q12k=μ21kQ11k,Q22k=E2k1-μ12kμ21k,Q44k=G23k,Q55k=G13k,Q66k=G12k.In order to seek simplicity, the material constants of each layer are the same, which can be expressed asEik=Ei, G12k=Gi,j, and μi,jk=μi,j(i,j=1,2,3). In addition, the relationship of μ12 and μ21 is μ12E2=μ21E1. It should be pointed out that the isotropic plate resting on the elastic foundations can be easily analyzed by letting E1=E2, G12=G13=G23=E1/(2+2μ12). By introducing the shear correction factor κ, the relation between the generalized forces and strains can be finally obtained [51, 52]. The specific expressions are shown as follows:(7)NxNyNxyMxMyMxy=A11A12A16B11B12B16A12A22A26B12B22B26A16A26A66B16B26B66B11B12B16D11D12D16B12B22B26D12D22D26B16B26B66D16D26D66εxx0εyy0γxy0χxxχyyχxy,QxQy=κA55A45A45A55γxzγyzin which the generalized forces include normal forces (Nx, Ny), shear force (Nxy), bending moments (Mx, My), twisting moment (Mxy), and transverse shear forces (Qx, Qy). In addition, Aij, Bij, and Dij are extensional, extensional-bending, and bending stiffness coefficients, whose expressions are written as(8)Aij=∑k=1NQijk¯Zk+1-Zk,Bij=12∑k=1NQijk¯Zk+12-Zk2,Dij=13∑k=1NQijk¯Zk+13-Zk3in which N donates the total number of layers. Moreover, when we study the vibration characteristics of the isotropic plate, the extensional-bending stiffness coefficients Bij are zero and Aij and Dij are independent.


## 2.3. Energy Equation

The main work of this paper is to investigate the vibration characteristics of the moderately thick laminated composite rectangular plate with uniform and multipoints supports which is rested on the elastic foundations. Rayleigh-Ritz energy method is extended to study the free and forced vibration of plate on the Winkler and Pasternak foundations. The Lagrangian energy function for the laminated plate can be written as(9)LP=TP-UP-Uf-Vsprings-Wext.T
P expresses the total kinetic energy of laminated plate whose expression is(10)Tp=12∫0a∫0bI0∂u∂t2+I0∂v∂t2+I0∂w∂t2+I2∂ϕx∂t2+I2∂ϕy∂t2+2I1∂u∂t∂ϕx∂t+2I1∂v∂t∂ϕy∂tdydx,I0=∑k=1N∫ZkZk+1ρkdz,I1=∑k=1N∫ZkZk+1ρkzdz,I2=∑k=1N∫ZkZk+1ρkz2dzin which I0, I1, and I2 are the inertia terms of the plate, and ρk is area density of the k’th layer.U
P expresses the strain energy for the moderately thick plates:(11)UP=12∫0a∫0bNxεxx0+Nyεyy0+Nxyεxy0+Mxχxx+Myχyy+Mxyχxy+Qxγxz+Qyγyzdxdy.Substituting (3), (7), and (8) into (11) can obtain the relations between strain energy and displacements in mid-surface of plate. Then, strain energy expression can be written as a superposition of three components, which are stretching energy (US), bending energy (UB), and bending–stretching coupling energy (UBS).(12a)US=12∫0a∫0bA11∂u∂x2+A22∂v∂y2+2A12∂u∂x∂v∂y+A66∂u∂y+∂v∂x2+2A16∂u∂y+∂v∂x∂u∂x+2A26∂u∂y+∂v∂x∂v∂y+κcA44∂w∂y+ϕy2+κcA55∂w∂x+ϕx2+2κcA45∂w∂x+ϕx∂w∂y+ϕydxdy,(12b)UB=12∫0a∫0bD11∂ϕx∂x2+D22∂ϕy∂y2+2D12∂ϕx∂x∂ϕy∂y+D66∂ϕx∂y+∂ϕy∂x2+2D16∂ϕx∂y+∂ϕy∂x∂ϕx∂x+2D26∂ϕx∂y+∂ϕy∂x∂ϕy∂ydxdy,(12c)UBS=∫0a∫0bB11∂u∂x∂ϕx∂x+B12∂u∂x∂ϕy∂y+∂v∂y∂ϕx∂x+B22∂v∂y∂ϕy∂y+B66∂v∂x+∂u∂y∂ϕx∂y+∂ϕy∂x+B16∂u∂x+B26∂v∂y∂ϕx∂y+∂ϕy∂x+B16∂ϕx∂x+B26∂ϕy∂y∂v∂x+∂u∂ydxdy.The strain energy due to the Winkler and Pasternak foundations is given by(13)Uf=12∫0a∫0bKWw2+KS∂w∂x2+∂w∂y2dxdy.For the elastic Winkler foundation, we just need to set the Pasternak foundation parameterKS to zero. Then, the influence of the free vibration for the plate which is on the Winkler-type elastic foundation can be studied with the change of the Winkler foundation parameterKW. In this paper, the main interests are studying the special boundary constraints and variation of Winkler and Pasternak foundations by introducing artificial virtual spring technology, which can be found in [56–58]. So the cases of uniform supported and multipoints supported boundary conditions whose plates are rested on the Winkler and Pasternak foundations will be considered here.V
s
p
r
i
n
g
s
u
n
i
f
o
r
m is the potential energy on the four edges of the plate with elastic uniform boundary conditions. It is simulated with five kinds of springs evenly distributed on four edges.(14)Vspringsuniform=12∫0b∫-h/2h/2kx0uux,y2+kx0vvx,y2+kx0wwx,y2+Kx0xϕxx,y2+Kx0yϕyx,y2x=0+kxauux,y2+kxavvx,y2+kxawwx,y2+Kxaxϕxx,y2+Kxayϕyx,y2x=adzdy+12∫0a∫-h/2h/2ky0uux,y2+ky0vvx,y2+ky0wwx,y2+Ky0xϕxx,y2+Ky0yϕyx,y2y=0+kybuux,y2+kybvvx,y2+kybwwx,y2+Kybxϕxx,y2+Kybyϕyx,y2y=bdzdx.For the cases of multipoints supports, the supported points are evenly distributed on the four edges. For intuitive understanding, Figure2 gives the distributions of 4-point, 8-point, and 16-point supported boundary conditions. They can be regarded as the discretization of uniform support. Therefore, the potential energy Vspringspoints for the plate with multipoints supports can be written as(15)Vspringspoints=12∑r=0NRkx0uuxr,yr2+kx0vvxr,yr2+kx0wwxr,yr2+Kx0xϕxxr,yr2+Kx0yϕyxr,yr2xr=0+12∑r=0NRkxauuxr,yr2+kxavvxr,yr2+kxawwxr,yr2+Kxaxϕxxr,yr2+Kxayϕyxr,yr2xr=a+12∑s=0NSky0uuxs,ys2+ky0vvxs,ys2+ky0wwxs,ys2+Ky0xϕxxs,ys2+Ky0yϕyxs,ys2ys=0+12∑s=0NSkybuuxs,ys2+kybvvxs,ys2+kybwwxs,ys2+Kybxϕxxs,ys2+Kybyϕyxs,ys2ys=b.Figure 2
Laminated plate with various multipoints supported boundary conditions.
(a)
4-point support
(b)
8-point support
(c)
16-point supportW
e
x
t expresses the work done by the external excitation force on the moderately thick plates:(16)Wext=∬Sfuu+fvv+fww+fϕxϕx+fϕyϕydxdy,where fi(i=u,v,w,ϕx,ϕy) is the external load distribution function on the plate. In this paper, the normal harmonic point force Fi on the plate is applied to study the flexural vibration behavior of the composite plate. Therefore, the load distribution function fi can be expressed as(17)fi=Fiδx-xeδy-ye,where (xe, ye) is the position of point force. δ is the 2D Dirac function.


## 2.4. Kinematics Balance Equation

The kinematics balance equations for the moderately thick plate can be obtained by Hamilton’s principle [51, 52].(18)∂Nx∂x+∂Nxy∂y=I0∂2u∂t2+I1∂2ϕx∂t2,∂Nxy∂x+∂Ny∂y=I0∂2v∂t2+I1∂2ϕy∂t2,∂Mx∂x+∂Mxy∂y-Qx=I1∂2u∂t2+I2∂2ϕx∂t2,∂Qx∂x+∂Qy∂y=I0∂2w∂t2,∂Mxy∂x+∂My∂y-Qy=I1∂2v∂t2+I2∂2ϕy∂t2.As we can see from (18), the displacement functions required second-order derivatives. The kinematics balance equations can be rewritten as a matrix form by combining (3), (7), and (18) simultaneously.(19)L11L120L14L15L12L220L24L2500L33L34L35L14L24L34L44L45L15L25L35L45L55-M1100M1400M2200M2500M3300M1400M4400M2500M55uvwϕxϕy=FuFvFwFϕxFϕy,where linear differential operators are expressed as Lij and Mij. The specific expressions are given below:(20)L11=A11∂2∂x2+2A16∂2∂x∂y+A66∂2∂y2,L12=A16∂2∂x2+A12+A66∂2∂x∂y+A26∂2∂y2,L14=B11∂2∂x2+2B16∂2∂x∂y+B66∂2∂y2,L15=B16∂2∂x2+B12+B66∂2∂x∂y+B26∂2∂y2,L22=A66∂2∂x2+2A26∂2∂x∂y+A22∂2∂y2,L24=B16∂2∂x2+B12+B66∂2∂x∂y+B26∂2∂y2,L25=B66∂2∂x2+2B26∂2∂x∂y+B22∂2∂y2,L33=-A55∂2∂x2-2A45∂2∂x∂y-A44∂2∂y2,L34=-A55∂∂x-A45∂∂y,L44=D11∂2∂x2+2D16∂2∂x∂y+D66∂2∂y2-A55,L35=-A45∂∂x-A44∂∂y,L45=D16∂2∂x2+D12+D66∂2∂x∂y+D26∂2∂y2-A45,L55=D66∂2∂x2+2D26∂2∂x∂y+D22∂2∂y2-A44,M11=M22=M33=-I0∂2∂t,M14=M25=-I1∂2∂t,M44=M55=-I2∂2∂t.


## 2.5. Displacement Expression

An IFSM is applied to express the displacements of the moderately thick laminated rectangular plate on the elastic foundations. The expressions ignore the boundary conditions and eliminate the discontinuous or jumping phenomenon in the boundaries. They can be regarded as a periodic function which is defined within the entire coordinates [59]. The specific displacement expressions of the plate are given below:(21)uvwϕxϕy=P00000P00000P00000P00000PAmnBmnCmnDmnEmn,(22)P=cos⁡λ0xcos⁡λ0y,…,cos⁡λ0xcos⁡λny,…,cos⁡λmxcos⁡λny,…,cos⁡λMxcos⁡λNy,sin⁡λ-2xcos⁡λ0y,…,sin⁡λ-2xcos⁡λny,…,sin⁡λ-2xcos⁡λNy,…,sin⁡λ-1xcos⁡λNy,cos⁡λ0xsin⁡λ-2y,cos⁡λ0xsin⁡λ-1y,…,cos⁡λmxsin⁡λ-2y,…,cos⁡λMxsin⁡λ-1y,(23a)Amn=A0,01,…,A0,n1,…,Am,n1,…,AM,N1,A-2,02,…,A-2,n2,…,A-2,N2,…,A-1,N2,A0,-23,A0,-13,…,Am,-23,…,AM,-13T,(23b)Bmn=B0,01,…,B0,n1,…,Bm,n1,…,BM,N1,B-2,02,…,B-2,n2,…,B-2,N2,…,B-1,N2,B0,-23,B0,-13,…,Bm,-23,…,BM,-13T,(23c)Cmn=C0,01,…,C0,n1,…,Cm,n1,…,CM,N1,C-2,02,…,C-2,n2,…,C-2,N2,…,C-1,N2,C0,-23,C0,-13,…,Cm,-23,…,CM,-13T,(23d)Dmn=D0,01,…,D0,n1,…,Dm,n1,…,DM,N1,D-2,02,…,D-2,n2,…,D-2,N2,…,D-1,N2,D0,-23,D0,-13,…,Dm,-23,…,DM,-13T,(23e)Emn=E0,01,…,E0,n1,…,Em,n1,…,EM,N1,E-2,02,…,E-2,n2,…,E-2,N2,…,E-1,N2,E0,-23,E0,-13,…,Em,-23,…,EM,-13T,where λm=mπ/a and λn=nπ/b. Besides, Amn, Bmn, Cmn, Dmn, and Emn are 2D Fourier coefficients vector, which are composed by Amni, Bmni, Cmni, Dmni, and Emni(i=1,2,3), respectively.There have been a lot of references about the Rayleigh-Ritz technology, such as [60–62], and Ritz like methods, such as [63–65]. In this paper, based on the Rayleigh-Ritz technology, a set of linear algebraic equations directed against Fourier unknown coefficients on displacement equations can be got by combining (18)–(23e) simultaneously. A matrix form can be obtained by transformation(24)Κ-ω2MG=F¯.When we study the free vibration of the plate, we only need to ignore the external force vectorF¯. K, M, and G are stiffness matrix, mass matrix, and unknown Fourier coefficients vector, separately. The specified expressions of K, M, and F¯ are listed in the Appendix. It should be pointed out that G is a combination of the five 2D Fourier coefficients vectors, whose form is(25)G=AmnBmnCmnDmnEmnT.Form (24), we can easily find that the circular frequency ω is the square root of eigenvalues and G is the eigenvectors. When we study the vibration response of the elastic supported plates resting on the elastic Winkler or Pasternak foundations, G can be obtained by giving a frequency value. Then, the exact displacements expressions of plate can be got by putting G into (21)–(23e). In order to explain the innovation of this paper more intuitively, a brief illustration is given in Figure 3. Compared with [30, 66], this paper has three main original ideas. Firstly, this paper focuses on the effect of elastic foundation on the plate. Secondly, the displacement expressions of the plate are improved. Thirdly, forced vibration response of plate on the elastic foundation is studied by introducing a harmonic point force. In addition, the corresponding results are also given in the next section.Figure 3
The flowchart about the solution for the laminated plate on the elastic foundation.


## 3. Results and Discussions

In this section, some calculation cases involving symmetrically and antisymmetrically laminated plates resting on the elastic Winkler and Pasternak foundations are considered, which have various uniform or multipoints supported boundary conditions. Comparisons are made with the available results of literatures and FEM. For the sake of brevity, the letters F, S, and C are used to represent completely free, simply supported, and clamped edges. In addition, a simplified combination of letters (in Table3) is used to characterize the boundary condition of the antisymmetrically laminated plates. For example, the SCSC shows that the laminated plates have S, C, S, and C at the four edges of x0, y0, xa, and yb, respectively. Besides, material properties of the elastic lamina are supposed asMaterial I: E1/E2=10; G12=G13=0.5E2; G23=0.2E2; μ12=0.25; μ12E1=μ21E2Material II: E1/E2=40; G12=G13=0.6E2; G23=0.5E2; μ12=0.25; μ12E1=μ21E2


### 3.1. Convergence Analysis

For the displacement admissible functions constructed in (21)–(23e), the appropriate truncation value should be set in numerical evaluation. The size of the values of M and N is the direct representation of the convergence of the method. Table 1 shows the convergence of the first eight frequency parameters Ω of the angle-ply (0/90°) rectangular laminated plates on the elastic Winkler foundation. The geometrical dimensions are a/b=1, h/a=0.1, the material type isMaterial I, and the foundation parameters are kW=10 and kS=0.Table 1
Convergence of frequency parameterΩfor angle-ply (0/90°) rectangular laminate plates with free and CCCC boundary constraints (Material I).
BC
M
×
N
Mode number
1
2
3
4
5
6
7
8
FFFF
8
×
8
3.147
3.147
3.162
5.392
10.299
10.563
13.076
13.076
10
×
10
3.147
3.147
3.162
5.388
10.299
10.563
13.071
13.071
12
×
12
3.147
3.147
3.162
5.386
10.299
10.563
13.069
13.069
14
×
14
3.147
3.147
3.162
5.386
10.299
10.563
13.068
13.068
16
×
16
3.147
3.147
3.162
5.385
10.299
10.563
13.068
13.068
18
×
18
3.147
3.147
3.162
5.385
10.299
10.563
13.067
13.067
20
×
20
3.147
3.147
3.162
5.385
10.299
10.563
13.067
13.067
22
×
22
3.147
3.147
3.162
5.385
10.299
10.563
13.067
13.067
CCCC
8
×
8
13.581
24.121
24.121
31.944
38.209
38.448
44.141
44.141
10
×
10
13.580
24.119
24.119
31.942
38.205
38.444
44.137
44.137
12
×
12
13.580
24.119
24.119
31.941
38.204
38.443
44.135
44.135
14
×
14
13.580
24.119
24.119
31.940
38.203
38.443
44.134
44.134
16
×
16
13.580
24.118
24.118
31.940
38.203
38.443
44.134
44.134
18
×
18
13.580
24.118
24.118
31.940
38.203
38.442
44.133
44.133
20
×
20
13.580
24.118
24.118
31.940
38.203
38.442
44.133
44.133
22
×
22
13.580
24.118
24.118
31.940
38.203
38.442
44.133
44.133From Table1, we can find that the present method shows good convergence. For example, for the free plate on the Winkler foundation, the biggest difference for the worst case which is made is the contrast of 8 × 8 and 18 × 18 being less than 0.13%, while the biggest difference for the worst case which is made is the contrast of 14 × 14 and 22 × 22 being less than 0.018%. It is not difficult to find that the biggest difference for the worst case which is made is the contrast of 18 × 18 and 22 × 22 being zero. In order to make the results more accurate, all the truncated values of M and N are 18 in the next calculations.


### 3.2. The Uniform Supported Plate on Elastic Foundations

In this section, free and forced vibration characteristics of the thin and moderately thick plate are discussed. The accuracy of the present method is verified by being compared with the results of Shen et al. [67]. The first frequency parameters Ω of a three-layered square plate which are simply supported and rest on different elastic foundations are displayed in Table 2. From the table, we can see that the first frequency increases when the two kinds of foundation parameters and the length-thickness ratios a/h increase. In addition, this method is also applicable to the analysis of thin plates. In Table 3, the first six frequency parameters Ω of the two angle-ply (0/90°) clamped square thin plates on the Winkler foundation are given. The geometry parameters of the plate layers are a/b=1, h/b=0.002, and the material type isMaterial II. The results obtained by the presented method are in good agreement with the FEM results.Table 2
Frequency parametersΩfor angle-ply (0/90°/0) square plate with simply supported boundary constraints and different foundation parameters and length-thickness ratios (Material II).
k
W
k
S
Method
a
/
h
50
20
10
5
0
0
Shen et al. [67]
18.689
17.483
14.702
10.263
Present
18.645
17.501
14.730
10.247
Error (%)
0.239
0.101
0.193
0.154
100
0
Shen et al. [67]
21.152
20.132
17.753
14.244
Present
21.155
20.148
17.776
14.233
Error (%)
0.016
0.078
0.129
0.078
100
10
Shen et al. [67]
25.390
24.536
22.596
19.879
Present
25.393
24.549
22.614
19.870
Error (%)
0.013
0.052
0.080
0.047Table 3
Frequency parameterΩfor angle-ply (0/90°) square thin plate on the Winkler foundation with CCCC boundary condition (Material II).
k
W
Method
Mode shape
1
2
3
4
5
6
0
Present
24.138
49.879
49.879
68.085
92.529
92.703
FEM
24.281
50.210
50.210
68.589
93.418
93.503
10
Present
24.344
49.979
49.979
68.159
92.583
92.757
FEM
24.488
50.412
50.412
68.664
93.502
93.658
100
Present
26.127
50.871
50.871
68.816
93.068
93.240
FEM
26.085
51.311
51.311
69.362
93.938
94.081For laminated plate on elastic Pasternak foundation, it is significant to study the change of natural frequency under various boundary conditions. Table4 shows the first three frequency parameters Ω for the two-layered [+45°/−45°] rectangular plate. It lists the change of Ω of the plate which rests on the elastic Pasternak-type foundation. There are six kinds of boundary constraints, that is, FFFF, FSFS, FCFC, SSSS, and CCCC. Five kinds of anisotropic ratios E1/E2 are given, which are 10, 20, 30, 40, and 50. In addition, two foundation parameters are kW=100 and kS=10, respectively. An interesting phenomenon is that the increase of anisotropic ratio has little effect on the natural frequency. But the frequency parameters increase in general with the binding force of the BC increasing as exhibited in Table 4.Table 4
Frequency parametersΩfor angle-ply (+45°/−45°) rectangular plate with different BC and anisotropic degrees.
E
1
/
E
2
Mode number
Boundary condition
FFFF
FSFS
FCFC
SSSS
SCSC
CCCC
10
1
10.000
15.642
17.433
32.963
35.789
39.335
2
14.289
24.010
25.816
44.438
46.692
49.321
3
22.965
34.155
36.133
59.647
61.544
63.734
20
1
10.000
15.947
17.864
35.833
38.249
41.351
2
14.315
25.272
26.912
48.419
50.112
52.108
3
23.021
36.639
39.319
64.457
65.724
67.269
30
1
10.000
16.194
18.225
37.841
40.020
42.822
2
14.334
26.253
27.782
51.054
52.433
54.058
3
23.058
38.118
40.927
67.556
68.497
69.683
40
1
10.000
16.409
18.542
39.381
41.388
43.956
2
14.348
27.069
28.510
52.985
54.153
55.523
3
23.085
39.216
42.035
69.768
70.511
71.466
50
1
10.000
16.604
18.825
40.616
42.485
44.862
2
14.359
27.770
29.137
54.476
55.488
56.671
3
23.105
40.097
42.891
71.438
72.050
72.846The effects of the boundary spring stiffness on free vibration behaviors of the laminated plate are investigated based on parametric study, which can deepen the understanding. Five elastic boundary restraint ratios areΓβ(β = u, v, w, ϕx, and ϕy), that is, Γu=ku/D; Γv=kv/D; Γw=kw/D; Γx=Kx/D; Γy=Ky/D. D is the structural stiffness coefficient, whose expression is D=E2h3/12(1-μ122). Figure 4 exhibits the curves of lowest three frequency parameters Ω with the change of restraint parameters for the laminated composite rectangular plates resting on the elastic foundation. The unsymmetrical lamination scheme [45°/−45°/45°/−45°] of the composite plate is brought. The laminated plates are supported by a set of spring component on the four edges with stiffness values varying from 10-8D to 1012D. The geometry constants of the plate layers are a/b=2, h/b=0.2, and the material type isMaterial II. Two foundation parameters are kW=10 and kS=10, respectively. From the figures, it can be seen clearly that when the boundary spring stiffness parameter is less than 10-2D, the change of the spring stiffness values has little effect on the natural frequency of the laminated plate. The laminated plate frequency will increase quickly with the increase of the stiffness when the value changes between 10-2D and 103D. But when the value is more than 103D, the natural frequency of the plate is almost constant. In addition, we can also find that the greatest impact on the frequency is the spring parameters Γu and Γv. For Γx and Γy, they almost have no effect on the frequency. The interesting thing is that Γw has no effect on the first two frequencies, while the effect on the third frequency is greater than the elastic spring parameters of Γu and Γv.Figure 4
Frequency parametersΩ versus elastic spring stiffness parameter Γ for the plate with uniform boundary.As mentioned earlier, the isotropic plates can be easily obtained by lettingE1=E2 and G12=G13=G23=E1/(2+2μ12). Figure 5 gives the transverse vibration displacement level on z direction of isotropic plates in the frequency range of 0–800 Hz by exerting a harmonic point force on the simple supported plate. The amplitude of the point force is 1 N. It will be applied to the following cases of forced vibrations. The position of point force is at (a/10, b/10). The positions of the two observation points are at (a/2, b/2) and (7a/8, 7b/8). Material parameters of plate are E1=E2=1.85e11 N/m2, μ12=0.3, and ρ=1600 kg/m2. The geometry parameters of the plate are a/b=1, h1/b=0.01, and h2/b=0.005. The results obtained by the present method are compared with those obtained by FEM which show good agreement. It shows that this method is also applicable to the thin plate structures. As we can see from Figure 5, the thickness of the plate has a great effect on the displacement response. The vibration displacement level increases with the decrease of the plate thickness.Figure 5
Vibration displacement response of the simple support isotropic plate with varying thickness.
(a)
Displacement response at (a/2, b/2)
(b)
Displacement response at (7a/8, 7b/8)In order to further study the effect of elastic support on the transverse vibration response of the four-layered [0/90°/0/90°] rectangular plate, three elastic boundary conditions are proposed according to Figure4. They are E1=10-1D, E2=101D, and E3=102D. The geometry constants of the plate layers are a/b=2, h/b=0.2, and the material type isMaterial II. Two foundation parameters are kW=100 and kS=100, respectively. The position of point force is at (a/8, b/8). The positions of the two observation points are at (a/4, b/4) and (7a/8, 7b/8). It can be seen from Figure 6 that the values of natural frequencies increase in the frequency range of 0–1000 Hz with the increase of the boundary springs stiffness. Moreover, the vibration displacements decrease with the increase of the boundary springs stiffness.Figure 6
Vibration displacement response of the laminated foundation plate with various elastic supports.
(a)
Displacement response at (a/4, b/4)
(b)
Displacement response at (7a/8, 7b/8)


### 3.3. The Multipoints Supported Plate on Elastic Foundations

In this section, the free and forced vibrations of laminated plates on elastic foundation with various multipoints supported boundary constraints are investigated. For this plate model, the geometrical dimensions area=b=1 m, h=0.2 m, and the material type isMaterial II. In order to simplify, various constraint values of multipoints supports are set. For example, K7 expresses that the stiffness value of five kinds of spring is 107. As shown in Table 5, the first six of Ω for the plate model under consideration with 8-point support, resting on the Winkler foundations, by using the method proposed in this paper are compared with the results of numerical calculation which use the ABAQUS model. The results of the two solutions are in good agreement with K7 elastic supported boundary conditions. Besides, there is a cognitive phenomenon that the frequency parameters Ω increase gradually, especially after the fourth frequency, when the Winkler parameter kW increases.Table 5
Frequency parameterΩfor angle-ply (0/90°) square plate with 8-point elastic supported boundary constraints and different foundation parameters (Material II).
k
W
k
S
Method
Mode shape
1
2
3
4
5
6
0
0
Present
1.185
1.218
1.218
1.923
2.944
2.944
FEM
1.202
1.219
1.219
1.856
2.962
2.962
10
0
Present
1.218
1.218
1.923
3.377
4.280
4.280
FEM
1.219
1.219
1.856
3.374
4.388
4.388
100
0
Present
1.218
1.218
1.924
10.070
10.254
10.254
FEM
1.219
1.219
1.856
10.006
10.625
10.625Based on the conclusions drawn from Table5, we have proved the accuracy of the present method. So we will directly give some frequency values of the antisymmetric laminated plate with the 8-point K7 elastic supports under the elastic Pasternak foundation. Table 6 shows some of the frequency parameters of the two kinds of cross-ply square plates, that is, angle-ply [0/90°] and [0/90°/0/90°]. This plate model is the same as the model considered in Table 4. It is not hard to find that the natural frequency increases when foundation parameters increase, and the effect of the Winkler parameter kW on the plate frequency is much larger than the Pasternak foundation parameter ks. In addition, the frequency of the four-layered plate is a little bigger than the frequency of the two-layered plate. In the following research, we will give the frequency of the plate on the elastic foundation changing with the increase of the number of layers.Table 6
Frequency parameterΩfor two cross-ply square plates with 8-point elastic supported boundary constraints and different foundation parameters (Material II).
k
W
k
S
0/90°
0/90°/0/90°
2
4
6
2
4
6
0
0
1.217
1.923
2.944
1.219
1.926
3.146
10
1.218
1.924
10.531
1.219
1.926
10.648
100
1.218
1.924
22.804
1.219
1.926
23.149
10
0
1.218
3.377
4.280
1.219
3.379
4.423
10
1.218
3.385
10.982
1.219
3.385
11.094
100
1.218
3.390
22.804
1.219
3.390
23.149
100
0
1.218
10.070
10.254
1.219
10.071
10.316
10
1.218
10.073
14.414
1.219
10.073
14.501
100
1.218
10.074
22.804
1.219
10.074
23.149In this part, we discuss the effect of boundary spring stiffness on the plate vibration. Here, we set up five kinds of elastic boundary restraint parametersΓβ(β=u, v, w, ϕx, and ϕy) of multipoints supports, just like the parameters in Figure 4. Figure 7 exhibits the curves of lowest three frequency parameters Ω with the change of restraint parameters of the laminated composite rectangular plates resting on the elastic foundation. The symmetrical lamination scheme [0/90°/0/90°/0] of the composite plate is studied. This laminated plate is supported by a set of spring components on the 8 points with stiffness values varying from 10-8D to 1012D. The geometry constants of the plate layers are a/b=1, h/b=0.2. The material properties and the nondimensional foundation parameters are same as the parameters in Figure 4. From the figures, we can clearly see that the growth trend of the five parameters curves is the same as the growth trend in Figure 4. But the different place is that the effect of Γw on the third frequency is same as the elastic spring parameters Γu and Γv in Figure 7. Besides, frequencies are different. It shows that the different supports have a great influence on the frequency.Figure 7
Frequency parametersΩ versus elastic parameter Γ for the plate with 8-point supported boundary conditions.As we all know, the bearing capacity of the composite lamina is directly influenced by the fibers direction and the layer’s number. Therefore, it is of great significance to study the effect of layer’s number and angles on the vibration characteristics of laminated plates on the elastic foundations. Figure8 depicts the first three-order frequency parameters Ω of the laminated plate on the Pasternak-type foundation. The schemes of layer’s number and angles are [0/θ]n, where n stands for the half of the layer’s number and θ is the angle. In this figure, there are four kinds of angle schemes which are θ=30°, 45°, 60°, and 90°. According to Figure 7, the clamped boundary constraint is simulated by setting the values of five parameters to greater than 103D. Therefore, the elastic boundary restraint parameter Γβ(β=u, v, w, ϕx, and ϕy) is 106. The layers of the plate have equal thickness and the material type isMaterial II. The geometric parameters are a/b=2, h/b=0.2. In addition, two linear foundation parameters are kW=100 and kS=10. Initially, the frequencies begin to increase rapidly with the increase of layer’s number. Then, the frequencies tend to be stable until the number of layers exceeds 12. For the first and second frequencies, the values increase when the angles increase. However, this phenomenon does not apply to the third-order frequency.Figure 8
Frequency parametersΩ versus number of layers n for a [0/θ]n layered plate with middle-points supported boundary conditions.The laminated plates on the elastic foundation with various multipoints supports extensively exist in engineering applications, like point supported glass curtain wall, spot-welded plate, and so on. This requires us to understand the vibration characteristics of multipoint supported plate. Figure9 considers the multipoints supported plate resting on the Pasternak foundation. These points are evenly distributed on the four edges. The boundary restraint parameter Γβ(β=u, v, w, ϕx, and ϕy) is 106 and the two foundation parameters are kW=100 and kS=10. The geometrical dimensions and material properties are the same as Figure 8. Based on the conclusions drawn from Figure 8, the lowest four frequency parameters Ω of the [0/45°]6 layered plate which is rested on Pasternak-type foundation against the number of points q are depicted in Figure 9. In addition, we also give the Ω of the same plate model with the equivalent uniform support as the contrast data. The number of supported points on the four edges gradually increases from 8 to 70. When the number of supported points increases, the first four frequency parameters increase firstly. Then, they tend to be stable and are similar to the frequencies of the same plate with uniform support when the number of points exceeds 48. The biggest difference of the worst case which is made is the contrast of 44 points and uniform support one being less than 0.182%. Then the biggest difference of the worst case which is made is the contrast of 48 points and uniform support one being about 0.069%. So, the natural frequencies of these point supports are finally converged to the ones of the corresponding uniform supported plate. Therefore, the multipoints support can be used to replace the uniform boundary conditions. For example, multiple bolts are tightened to simulate the uniform clamped boundary conditions in the experiment.Figure 9
Frequency parametersΩ versus points number q for layered plate with multipoints supported boundary conditions.
(a)
The 1st and 2nd frequency parameters
(b)
The 3rd and 4th frequency parametersFigure10 gives the transverse vibration displacement level on z direction of isotropic plates in the frequency range of 0–400 Hz. The plate with simple support on the four endpoints is encouraged by a harmonic point force. The geometry parameters and material parameters of plate are same as Figure 5. The position of point force is at (a/10, b/10), and the two observation points are at (a/2, b/2) and (7a/8, 7b/8), respectively. The results obtained by FEM are given here in order to verify the accuracy of the present method. The vibration response curves at different observation points are very different. These rules are the same as those in Figure 5. It should be pointed out that the number of resonant peaks obviously increases at the observation point which is far from the position of point force.Figure 10
Vibration displacement response of the simple supported isotropic plate on endpoints with varying thickness.
(a)
Displacement response at (a/2, b/2)
(b)
Displacement response at (7a/8, 7b/8)Figure11 studies the effect of elastic foundation parameters on the vibration response of the plate with 8-point elastic support. The five kinds of combinations are kW=10 and kS=0, kW=10 and kS=10, kW=10 and kS=50, kW=50 and kS=0, and kW=100 and kS=0. The boundary condition of plate is E2E2E2E2. The geometry constants of the plate layers are a/b=1, h/b=0.2, and the material type isMaterial II. The position of point force is at (a/5, b/5). The positions of the two observation points are at (2a/5, 2b/5) and (7a/10, 7b/10). Four groups of comparison curves of vibration displacement levels in the frequency range of 0–1500 Hz are given in Figure 11. It shows the vibration response with the change of Winkler-type foundation parameters kW in (a) and (b). With the increase of the Winkler parameter kW, the natural frequency increases, the displacement response wave moves back in the frequency domain, and the wave crest is attenuated. In (c) and (d) of Figure 11, the effect of Pasternak parameter kS on the vibration response of plate is studied. As we can see, the displacement wave moves back in the frequency domain with the increase of kS. Moreover, there is an interesting finding that the peak value is largest when the foundation combination is kW=10 and kS=50.Figure 11
Vibration displacement response of the 8-point elastic supported laminated plate on the various elastic foundations.
(a)
Displacement response at (2a/5, 2b/5)
(b)
Displacement response at (7a/10, 7b/10)
(c)
Displacement response at (2a/5, 2b/5)
(d)
Displacement response at (7a/10, 7b/10)At the end of this paper, the three-dimensional view models are adopted to understand the vibration characteristics of multipoints supported laminated plates on the elastic Pasternak foundations. The geometric dimensions of these moderately thick [0/45°]6 layered plates are a=b=1 m, h=0.2 m, and the material properties are the same as the material I. Five boundary restraint parameters are Γβ=106(β=u, v, w, ϕx, and ϕy). In addition, two linear foundation parameters are kW=10 and kS=10, respectively. For comparison, mode shapes of the uniform supported plate on the Pasternak foundation and uniform supported plate without foundation are also given in Figure 12. As shown in Figure 12, the mode shapes tend to approach uniform boundary conditions when the number of points increases gradually. In addition, we can intuitively see the effect of the elastic foundation on the mode shapes of plate.Figure 12
Mode shapes of multipoints supported laminated plate on the Pasternak foundation.


## 3.1. Convergence Analysis

For the displacement admissible functions constructed in (21)–(23e), the appropriate truncation value should be set in numerical evaluation. The size of the values of M and N is the direct representation of the convergence of the method. Table 1 shows the convergence of the first eight frequency parameters Ω of the angle-ply (0/90°) rectangular laminated plates on the elastic Winkler foundation. The geometrical dimensions are a/b=1, h/a=0.1, the material type isMaterial I, and the foundation parameters are kW=10 and kS=0.Table 1
Convergence of frequency parameterΩfor angle-ply (0/90°) rectangular laminate plates with free and CCCC boundary constraints (Material I).
BC
M
×
N
Mode number
1
2
3
4
5
6
7
8
FFFF
8
×
8
3.147
3.147
3.162
5.392
10.299
10.563
13.076
13.076
10
×
10
3.147
3.147
3.162
5.388
10.299
10.563
13.071
13.071
12
×
12
3.147
3.147
3.162
5.386
10.299
10.563
13.069
13.069
14
×
14
3.147
3.147
3.162
5.386
10.299
10.563
13.068
13.068
16
×
16
3.147
3.147
3.162
5.385
10.299
10.563
13.068
13.068
18
×
18
3.147
3.147
3.162
5.385
10.299
10.563
13.067
13.067
20
×
20
3.147
3.147
3.162
5.385
10.299
10.563
13.067
13.067
22
×
22
3.147
3.147
3.162
5.385
10.299
10.563
13.067
13.067
CCCC
8
×
8
13.581
24.121
24.121
31.944
38.209
38.448
44.141
44.141
10
×
10
13.580
24.119
24.119
31.942
38.205
38.444
44.137
44.137
12
×
12
13.580
24.119
24.119
31.941
38.204
38.443
44.135
44.135
14
×
14
13.580
24.119
24.119
31.940
38.203
38.443
44.134
44.134
16
×
16
13.580
24.118
24.118
31.940
38.203
38.443
44.134
44.134
18
×
18
13.580
24.118
24.118
31.940
38.203
38.442
44.133
44.133
20
×
20
13.580
24.118
24.118
31.940
38.203
38.442
44.133
44.133
22
×
22
13.580
24.118
24.118
31.940
38.203
38.442
44.133
44.133From Table1, we can find that the present method shows good convergence. For example, for the free plate on the Winkler foundation, the biggest difference for the worst case which is made is the contrast of 8 × 8 and 18 × 18 being less than 0.13%, while the biggest difference for the worst case which is made is the contrast of 14 × 14 and 22 × 22 being less than 0.018%. It is not difficult to find that the biggest difference for the worst case which is made is the contrast of 18 × 18 and 22 × 22 being zero. In order to make the results more accurate, all the truncated values of M and N are 18 in the next calculations.


## 3.2. The Uniform Supported Plate on Elastic Foundations

In this section, free and forced vibration characteristics of the thin and moderately thick plate are discussed. The accuracy of the present method is verified by being compared with the results of Shen et al. [67]. The first frequency parameters Ω of a three-layered square plate which are simply supported and rest on different elastic foundations are displayed in Table 2. From the table, we can see that the first frequency increases when the two kinds of foundation parameters and the length-thickness ratios a/h increase. In addition, this method is also applicable to the analysis of thin plates. In Table 3, the first six frequency parameters Ω of the two angle-ply (0/90°) clamped square thin plates on the Winkler foundation are given. The geometry parameters of the plate layers are a/b=1, h/b=0.002, and the material type isMaterial II. The results obtained by the presented method are in good agreement with the FEM results.Table 2
Frequency parametersΩfor angle-ply (0/90°/0) square plate with simply supported boundary constraints and different foundation parameters and length-thickness ratios (Material II).
k
W
k
S
Method
a
/
h
50
20
10
5
0
0
Shen et al. [67]
18.689
17.483
14.702
10.263
Present
18.645
17.501
14.730
10.247
Error (%)
0.239
0.101
0.193
0.154
100
0
Shen et al. [67]
21.152
20.132
17.753
14.244
Present
21.155
20.148
17.776
14.233
Error (%)
0.016
0.078
0.129
0.078
100
10
Shen et al. [67]
25.390
24.536
22.596
19.879
Present
25.393
24.549
22.614
19.870
Error (%)
0.013
0.052
0.080
0.047Table 3
Frequency parameterΩfor angle-ply (0/90°) square thin plate on the Winkler foundation with CCCC boundary condition (Material II).
k
W
Method
Mode shape
1
2
3
4
5
6
0
Present
24.138
49.879
49.879
68.085
92.529
92.703
FEM
24.281
50.210
50.210
68.589
93.418
93.503
10
Present
24.344
49.979
49.979
68.159
92.583
92.757
FEM
24.488
50.412
50.412
68.664
93.502
93.658
100
Present
26.127
50.871
50.871
68.816
93.068
93.240
FEM
26.085
51.311
51.311
69.362
93.938
94.081For laminated plate on elastic Pasternak foundation, it is significant to study the change of natural frequency under various boundary conditions. Table4 shows the first three frequency parameters Ω for the two-layered [+45°/−45°] rectangular plate. It lists the change of Ω of the plate which rests on the elastic Pasternak-type foundation. There are six kinds of boundary constraints, that is, FFFF, FSFS, FCFC, SSSS, and CCCC. Five kinds of anisotropic ratios E1/E2 are given, which are 10, 20, 30, 40, and 50. In addition, two foundation parameters are kW=100 and kS=10, respectively. An interesting phenomenon is that the increase of anisotropic ratio has little effect on the natural frequency. But the frequency parameters increase in general with the binding force of the BC increasing as exhibited in Table 4.Table 4
Frequency parametersΩfor angle-ply (+45°/−45°) rectangular plate with different BC and anisotropic degrees.
E
1
/
E
2
Mode number
Boundary condition
FFFF
FSFS
FCFC
SSSS
SCSC
CCCC
10
1
10.000
15.642
17.433
32.963
35.789
39.335
2
14.289
24.010
25.816
44.438
46.692
49.321
3
22.965
34.155
36.133
59.647
61.544
63.734
20
1
10.000
15.947
17.864
35.833
38.249
41.351
2
14.315
25.272
26.912
48.419
50.112
52.108
3
23.021
36.639
39.319
64.457
65.724
67.269
30
1
10.000
16.194
18.225
37.841
40.020
42.822
2
14.334
26.253
27.782
51.054
52.433
54.058
3
23.058
38.118
40.927
67.556
68.497
69.683
40
1
10.000
16.409
18.542
39.381
41.388
43.956
2
14.348
27.069
28.510
52.985
54.153
55.523
3
23.085
39.216
42.035
69.768
70.511
71.466
50
1
10.000
16.604
18.825
40.616
42.485
44.862
2
14.359
27.770
29.137
54.476
55.488
56.671
3
23.105
40.097
42.891
71.438
72.050
72.846The effects of the boundary spring stiffness on free vibration behaviors of the laminated plate are investigated based on parametric study, which can deepen the understanding. Five elastic boundary restraint ratios areΓβ(β = u, v, w, ϕx, and ϕy), that is, Γu=ku/D; Γv=kv/D; Γw=kw/D; Γx=Kx/D; Γy=Ky/D. D is the structural stiffness coefficient, whose expression is D=E2h3/12(1-μ122). Figure 4 exhibits the curves of lowest three frequency parameters Ω with the change of restraint parameters for the laminated composite rectangular plates resting on the elastic foundation. The unsymmetrical lamination scheme [45°/−45°/45°/−45°] of the composite plate is brought. The laminated plates are supported by a set of spring component on the four edges with stiffness values varying from 10-8D to 1012D. The geometry constants of the plate layers are a/b=2, h/b=0.2, and the material type isMaterial II. Two foundation parameters are kW=10 and kS=10, respectively. From the figures, it can be seen clearly that when the boundary spring stiffness parameter is less than 10-2D, the change of the spring stiffness values has little effect on the natural frequency of the laminated plate. The laminated plate frequency will increase quickly with the increase of the stiffness when the value changes between 10-2D and 103D. But when the value is more than 103D, the natural frequency of the plate is almost constant. In addition, we can also find that the greatest impact on the frequency is the spring parameters Γu and Γv. For Γx and Γy, they almost have no effect on the frequency. The interesting thing is that Γw has no effect on the first two frequencies, while the effect on the third frequency is greater than the elastic spring parameters of Γu and Γv.Figure 4
Frequency parametersΩ versus elastic spring stiffness parameter Γ for the plate with uniform boundary.As mentioned earlier, the isotropic plates can be easily obtained by lettingE1=E2 and G12=G13=G23=E1/(2+2μ12). Figure 5 gives the transverse vibration displacement level on z direction of isotropic plates in the frequency range of 0–800 Hz by exerting a harmonic point force on the simple supported plate. The amplitude of the point force is 1 N. It will be applied to the following cases of forced vibrations. The position of point force is at (a/10, b/10). The positions of the two observation points are at (a/2, b/2) and (7a/8, 7b/8). Material parameters of plate are E1=E2=1.85e11 N/m2, μ12=0.3, and ρ=1600 kg/m2. The geometry parameters of the plate are a/b=1, h1/b=0.01, and h2/b=0.005. The results obtained by the present method are compared with those obtained by FEM which show good agreement. It shows that this method is also applicable to the thin plate structures. As we can see from Figure 5, the thickness of the plate has a great effect on the displacement response. The vibration displacement level increases with the decrease of the plate thickness.Figure 5
Vibration displacement response of the simple support isotropic plate with varying thickness.
(a)
Displacement response at (a/2, b/2)
(b)
Displacement response at (7a/8, 7b/8)In order to further study the effect of elastic support on the transverse vibration response of the four-layered [0/90°/0/90°] rectangular plate, three elastic boundary conditions are proposed according to Figure4. They are E1=10-1D, E2=101D, and E3=102D. The geometry constants of the plate layers are a/b=2, h/b=0.2, and the material type isMaterial II. Two foundation parameters are kW=100 and kS=100, respectively. The position of point force is at (a/8, b/8). The positions of the two observation points are at (a/4, b/4) and (7a/8, 7b/8). It can be seen from Figure 6 that the values of natural frequencies increase in the frequency range of 0–1000 Hz with the increase of the boundary springs stiffness. Moreover, the vibration displacements decrease with the increase of the boundary springs stiffness.Figure 6
Vibration displacement response of the laminated foundation plate with various elastic supports.
(a)
Displacement response at (a/4, b/4)
(b)
Displacement response at (7a/8, 7b/8)


## 3.3. The Multipoints Supported Plate on Elastic Foundations

In this section, the free and forced vibrations of laminated plates on elastic foundation with various multipoints supported boundary constraints are investigated. For this plate model, the geometrical dimensions area=b=1 m, h=0.2 m, and the material type isMaterial II. In order to simplify, various constraint values of multipoints supports are set. For example, K7 expresses that the stiffness value of five kinds of spring is 107. As shown in Table 5, the first six of Ω for the plate model under consideration with 8-point support, resting on the Winkler foundations, by using the method proposed in this paper are compared with the results of numerical calculation which use the ABAQUS model. The results of the two solutions are in good agreement with K7 elastic supported boundary conditions. Besides, there is a cognitive phenomenon that the frequency parameters Ω increase gradually, especially after the fourth frequency, when the Winkler parameter kW increases.Table 5
Frequency parameterΩfor angle-ply (0/90°) square plate with 8-point elastic supported boundary constraints and different foundation parameters (Material II).
k
W
k
S
Method
Mode shape
1
2
3
4
5
6
0
0
Present
1.185
1.218
1.218
1.923
2.944
2.944
FEM
1.202
1.219
1.219
1.856
2.962
2.962
10
0
Present
1.218
1.218
1.923
3.377
4.280
4.280
FEM
1.219
1.219
1.856
3.374
4.388
4.388
100
0
Present
1.218
1.218
1.924
10.070
10.254
10.254
FEM
1.219
1.219
1.856
10.006
10.625
10.625Based on the conclusions drawn from Table5, we have proved the accuracy of the present method. So we will directly give some frequency values of the antisymmetric laminated plate with the 8-point K7 elastic supports under the elastic Pasternak foundation. Table 6 shows some of the frequency parameters of the two kinds of cross-ply square plates, that is, angle-ply [0/90°] and [0/90°/0/90°]. This plate model is the same as the model considered in Table 4. It is not hard to find that the natural frequency increases when foundation parameters increase, and the effect of the Winkler parameter kW on the plate frequency is much larger than the Pasternak foundation parameter ks. In addition, the frequency of the four-layered plate is a little bigger than the frequency of the two-layered plate. In the following research, we will give the frequency of the plate on the elastic foundation changing with the increase of the number of layers.Table 6
Frequency parameterΩfor two cross-ply square plates with 8-point elastic supported boundary constraints and different foundation parameters (Material II).
k
W
k
S
0/90°
0/90°/0/90°
2
4
6
2
4
6
0
0
1.217
1.923
2.944
1.219
1.926
3.146
10
1.218
1.924
10.531
1.219
1.926
10.648
100
1.218
1.924
22.804
1.219
1.926
23.149
10
0
1.218
3.377
4.280
1.219
3.379
4.423
10
1.218
3.385
10.982
1.219
3.385
11.094
100
1.218
3.390
22.804
1.219
3.390
23.149
100
0
1.218
10.070
10.254
1.219
10.071
10.316
10
1.218
10.073
14.414
1.219
10.073
14.501
100
1.218
10.074
22.804
1.219
10.074
23.149In this part, we discuss the effect of boundary spring stiffness on the plate vibration. Here, we set up five kinds of elastic boundary restraint parametersΓβ(β=u, v, w, ϕx, and ϕy) of multipoints supports, just like the parameters in Figure 4. Figure 7 exhibits the curves of lowest three frequency parameters Ω with the change of restraint parameters of the laminated composite rectangular plates resting on the elastic foundation. The symmetrical lamination scheme [0/90°/0/90°/0] of the composite plate is studied. This laminated plate is supported by a set of spring components on the 8 points with stiffness values varying from 10-8D to 1012D. The geometry constants of the plate layers are a/b=1, h/b=0.2. The material properties and the nondimensional foundation parameters are same as the parameters in Figure 4. From the figures, we can clearly see that the growth trend of the five parameters curves is the same as the growth trend in Figure 4. But the different place is that the effect of Γw on the third frequency is same as the elastic spring parameters Γu and Γv in Figure 7. Besides, frequencies are different. It shows that the different supports have a great influence on the frequency.Figure 7
Frequency parametersΩ versus elastic parameter Γ for the plate with 8-point supported boundary conditions.As we all know, the bearing capacity of the composite lamina is directly influenced by the fibers direction and the layer’s number. Therefore, it is of great significance to study the effect of layer’s number and angles on the vibration characteristics of laminated plates on the elastic foundations. Figure8 depicts the first three-order frequency parameters Ω of the laminated plate on the Pasternak-type foundation. The schemes of layer’s number and angles are [0/θ]n, where n stands for the half of the layer’s number and θ is the angle. In this figure, there are four kinds of angle schemes which are θ=30°, 45°, 60°, and 90°. According to Figure 7, the clamped boundary constraint is simulated by setting the values of five parameters to greater than 103D. Therefore, the elastic boundary restraint parameter Γβ(β=u, v, w, ϕx, and ϕy) is 106. The layers of the plate have equal thickness and the material type isMaterial II. The geometric parameters are a/b=2, h/b=0.2. In addition, two linear foundation parameters are kW=100 and kS=10. Initially, the frequencies begin to increase rapidly with the increase of layer’s number. Then, the frequencies tend to be stable until the number of layers exceeds 12. For the first and second frequencies, the values increase when the angles increase. However, this phenomenon does not apply to the third-order frequency.Figure 8
Frequency parametersΩ versus number of layers n for a [0/θ]n layered plate with middle-points supported boundary conditions.The laminated plates on the elastic foundation with various multipoints supports extensively exist in engineering applications, like point supported glass curtain wall, spot-welded plate, and so on. This requires us to understand the vibration characteristics of multipoint supported plate. Figure9 considers the multipoints supported plate resting on the Pasternak foundation. These points are evenly distributed on the four edges. The boundary restraint parameter Γβ(β=u, v, w, ϕx, and ϕy) is 106 and the two foundation parameters are kW=100 and kS=10. The geometrical dimensions and material properties are the same as Figure 8. Based on the conclusions drawn from Figure 8, the lowest four frequency parameters Ω of the [0/45°]6 layered plate which is rested on Pasternak-type foundation against the number of points q are depicted in Figure 9. In addition, we also give the Ω of the same plate model with the equivalent uniform support as the contrast data. The number of supported points on the four edges gradually increases from 8 to 70. When the number of supported points increases, the first four frequency parameters increase firstly. Then, they tend to be stable and are similar to the frequencies of the same plate with uniform support when the number of points exceeds 48. The biggest difference of the worst case which is made is the contrast of 44 points and uniform support one being less than 0.182%. Then the biggest difference of the worst case which is made is the contrast of 48 points and uniform support one being about 0.069%. So, the natural frequencies of these point supports are finally converged to the ones of the corresponding uniform supported plate. Therefore, the multipoints support can be used to replace the uniform boundary conditions. For example, multiple bolts are tightened to simulate the uniform clamped boundary conditions in the experiment.Figure 9
Frequency parametersΩ versus points number q for layered plate with multipoints supported boundary conditions.
(a)
The 1st and 2nd frequency parameters
(b)
The 3rd and 4th frequency parametersFigure10 gives the transverse vibration displacement level on z direction of isotropic plates in the frequency range of 0–400 Hz. The plate with simple support on the four endpoints is encouraged by a harmonic point force. The geometry parameters and material parameters of plate are same as Figure 5. The position of point force is at (a/10, b/10), and the two observation points are at (a/2, b/2) and (7a/8, 7b/8), respectively. The results obtained by FEM are given here in order to verify the accuracy of the present method. The vibration response curves at different observation points are very different. These rules are the same as those in Figure 5. It should be pointed out that the number of resonant peaks obviously increases at the observation point which is far from the position of point force.Figure 10
Vibration displacement response of the simple supported isotropic plate on endpoints with varying thickness.
(a)
Displacement response at (a/2, b/2)
(b)
Displacement response at (7a/8, 7b/8)Figure11 studies the effect of elastic foundation parameters on the vibration response of the plate with 8-point elastic support. The five kinds of combinations are kW=10 and kS=0, kW=10 and kS=10, kW=10 and kS=50, kW=50 and kS=0, and kW=100 and kS=0. The boundary condition of plate is E2E2E2E2. The geometry constants of the plate layers are a/b=1, h/b=0.2, and the material type isMaterial II. The position of point force is at (a/5, b/5). The positions of the two observation points are at (2a/5, 2b/5) and (7a/10, 7b/10). Four groups of comparison curves of vibration displacement levels in the frequency range of 0–1500 Hz are given in Figure 11. It shows the vibration response with the change of Winkler-type foundation parameters kW in (a) and (b). With the increase of the Winkler parameter kW, the natural frequency increases, the displacement response wave moves back in the frequency domain, and the wave crest is attenuated. In (c) and (d) of Figure 11, the effect of Pasternak parameter kS on the vibration response of plate is studied. As we can see, the displacement wave moves back in the frequency domain with the increase of kS. Moreover, there is an interesting finding that the peak value is largest when the foundation combination is kW=10 and kS=50.Figure 11
Vibration displacement response of the 8-point elastic supported laminated plate on the various elastic foundations.
(a)
Displacement response at (2a/5, 2b/5)
(b)
Displacement response at (7a/10, 7b/10)
(c)
Displacement response at (2a/5, 2b/5)
(d)
Displacement response at (7a/10, 7b/10)At the end of this paper, the three-dimensional view models are adopted to understand the vibration characteristics of multipoints supported laminated plates on the elastic Pasternak foundations. The geometric dimensions of these moderately thick [0/45°]6 layered plates are a=b=1 m, h=0.2 m, and the material properties are the same as the material I. Five boundary restraint parameters are Γβ=106(β=u, v, w, ϕx, and ϕy). In addition, two linear foundation parameters are kW=10 and kS=10, respectively. For comparison, mode shapes of the uniform supported plate on the Pasternak foundation and uniform supported plate without foundation are also given in Figure 12. As shown in Figure 12, the mode shapes tend to approach uniform boundary conditions when the number of points increases gradually. In addition, we can intuitively see the effect of the elastic foundation on the mode shapes of plate.Figure 12
Mode shapes of multipoints supported laminated plate on the Pasternak foundation.


## 4. Conclusions

This paper extends an improved Fourier series solution to solve the free and forced vibration analysis of the moderately thick laminated rectangular plate on the elastic Winkler and Pasternak foundations which has various uniform and multipoints supported boundary constraints. The five displacement functions are all expressed as a series of two-dimensional Fourier series which can ignore the influence of boundary conditions. The introduction of two supplementary polynomials of the displacement functions can effectively eliminate the discontinuous or jumping phenomenon on the boundaries. The formulations are based on FSDT and the interaction of plate-foundation model and the artificial spring technique. The different boundary constraints can be easily realized by setting different elastic restraint stiffness. It can effectively simulate the boundary conditions of the plate structures in real world. The results of the present method show good agreement with the results of existing literatures and FEM. The results of free vibration for thin and moderately thick plates with various foundation parameters, different uniform boundary conditions, and multipoint supports are examined in this investigation. In addition, the lamination schemes and geometric parameters have shown a great influence on the vibration frequency of the plate. The effects of the increase of layer’s number, changes of the laying angle, and increase of the number of supported points on natural frequency of the plates are presented. The forced vibration response of plate is studied by applying an external harmonic point force. Vibration displacement response of the plate with various thicknesses, elastic boundary constraints, and elastic foundation parameters is investigated, which can provide a benchmark for the future research.



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*Source: 7820130-2017-07-11.xml*