# A New Efficient Modified First-Order Shear Model for Static Bending and Vibration Behaviors of Two-Layer Composite Plate

Authors: Vu H. Nam; Nguyen H. Nam; Pham V. Vinh; Dang N. Khoa; Do V. Thom; Phung V. Minh
Journal: Advances in Civil Engineering
 (2019)

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## Abstract

A two-layer (connected by stubs) partial composite plate is a structure with outstanding advantages which can be widely applied in many fields of engineering such as construction, transportation, and mechanical. However, studies are scarce in the past to investigate this type of structure. This paper is based on the new modified first-order shear deformation plate theory and finite element method to develop a new four-node plate element with nine degrees of freedom per node for static bending and vibration analysis of the two-layer composite plate. The numerical results are compared to published data for some special cases. The effects of some parameters such as the boundary condition, stiffness of the connector stub, height-to-width ratio, thickness-to-thickness ratio between two layers, and aspect ratio are also performed to investigate new numerical results of static bending and free vibration responses of this structure.


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## Body

## 1. Introduction

Two-layer beam and plate are among the most commonly used structures in many fields of engineering, such as construction, transportation, and mechanical, because of their advantages in comparison with one-layer structures; we can see these structures in practice such as steel-concrete composite beams and plates, layered wooden beams and plates, and wood-concrete and timber-steel floor structures. Some advantages of this type of composite structure can be considered such as easy manufacturing, taking advantage of the material properties of two components which made the structure. In fact, there are many different grafting methods to manufacture two-layer beams or plates; for example, two components of a structure are sandwiched together at the edges, using glue or connector stub to bond two parts at the contact surface. Because of these advantages, many researchers have focused on the mechanical analysis of these structures.Two-layer beam has been investigated by Foraboschi [1, 2] using an analytical method. Kryžanowski et al. [3] developed an analytical method to evaluate exact critical forces of two-layer composite columns. Grognec et al. [4] used the Timoshenko beam model for buckling analysis of two-layer composite beams with partial interaction. He and Yang [5, 6] presented the finite element method and higher order beam theory for buckling and dynamic problem of the two-layer composite beam. He et al. [7] carried out an analytical solution for free vibration and buckling of two- and three-layer composite beams based on a higher order beam theory. Hozan et al. [8] analyzed a two-layer composite planar beam with exact geometric and material nonlinearities as well as the finite slip between the layers. Wu and coworkers [9, 10] used the 2D elasticity solution for analysis of two-layer beams with the viscoelastic interlayer and arbitrarily shaped interface. Large deformation analysis of two-layer composite beams was studied [11]. Based on the higher order beam theory and analytical model, Wen et al. [12] investigated the flexural response of the two-layer composite beam. Do et al. [13] presented static bending analysis of three-layer beams made of functionally graded materials in the thermal environment based on FEM and Mindlin theory. Cas et al. [14] used an analytical method to investigate the mechanical behavior of two-layer composite beams with interlayer slips. Ke et al. [15] used experimental and numerical FE methods to study local web buckling of single-coped beam connections with the slender web.Latham et al. [16] developed a new finite element based on a new plate theory due to the Reissner plate theory for static bending of two-layer plates. Foraboschi and colleagues [17, 18] used an analytical model and experimental method for static bending of laminated glass plates. Layered plate with discontinuous connection was investigated by Foraboschi [19] using the exact mathematical model replacing discontinuous connections with a fictitious continuous medium. Foraboschi [20, 21] also detailed the mechanical behavior of three-layer plates using the exact mathematical model and elasticity solution. Wu et al. [22] applied the 3D exact solution to explore the two-layer plate bonded by a viscoelastic interlayer. Vidal et al. [23] presented a new approach to study the deflection and stress of the multilayer composite plate. Alinia et al. [24] applied the numerical nonlinear finite element method and theoretical p-Ritz energy method to analyze inelastic buckling and postbuckling behaviors of stocky plates under combined shear and in-plane bending stresses.This paper aims to use the finite element method (FEM) based on a new modified first-order shear deformation plate theory (FSDT) to analyze static bending and free vibration of the two-layer composite plate. The proposed method shows the simple formulations and computational efficiency. The accuracy and reliability of the proposed method are validated with other published results. Several numerical examples and influence of some parameters on static bending and free vibration of the two-layer composite plate are also investigated; these new results have significant impacts on the use of this structure in practice.The organization of this paper is as follows: Section2 presents the finite formulation of static bending and free vibration of the two-layer composite plate based on a new modified first-order shear deformation plate theory. New numerical results for bending and free vibration analysis of these plates are computed and discussed in Section 3. Some major conclusions are given in Section 4.


## 2. Geometry and Theoretical Formulation

### 2.1. Geometry and Assumptions

The geometry model of the problem is a two-layer plate including two isotropic plates which bonded together at the contact surface using connector stubs, as shown in Figure1. Two parts are of the same size in the x−y plane with thickness hm for the bottom layer and hn for the top layer. To avoid the repetition of similar equations for the bottom layer and top layer, we use the subscript m for the bottom layer, subscript n for the top layer, and subscript st for the connector stub.Figure 1
Geometrical notation of the two-layer plate.The assumptions of the two-layer composite plate include the following: the materials of each layer are linear, elastic, and isotropic; the displacement and rotation of the plate are small; there is no delamination phenomenon between two layers; the deflection of the stub is in the contact interface of two components; and the mass of the stub is much smaller than the mass of the plate, so we assume that it is neglected.


### 2.2. Discrete Finite Element Equations

#### 2.2.1. New Modified First-Order Shear Deformation for Plate Elements

The basic equations of the Mindlin plate theory are [25](1)Mx=D∂ψx∂x+ν∂ψy∂y,My=D∂ψx∂y+ν∂ψy∂x,Mxy=Myx=121−νD∂ψx∂y+∂ψy∂x,Qx=S∂w∂x+ψx,Qy=S∂w∂y+ψy,where(2)D=Eh3121−ν2,S=kGh,G=E21+ν,in which k = 5/6 is the shear correction.Equilibrium of moments aboutx- and y-axis and transverse force leads to(3)∂Mx∂x+∂Mxy∂y−Qx=mx,∂My∂y+∂Mxy∂x−Qy=my,∂Qx∂x+∂Qy∂y=−q.Substituting equation (1) into equation (3), we get the following:(4)DS∂2ψx∂x2+121−ν∂2ψx∂y2+121+ν∂2ψy∂x∂y−∂w∂x+ψx=0,DS∂2ψy∂y2+121−ν∂2ψy∂x2+121+ν∂2ψx∂x∂y−∂w∂y+ψy=0,Δw+∂ψx∂x+∂ψy∂y=0,where Δ=∂2/∂x2+∂2/∂y2 is the Laplace differential operator.Assume that the total deflection consists of two parts which are bending deflection and transverse shear, while angles of the plate cross-sectional slope are a result of rotation of pure bending and shear angles:(5)w=wb+ws,ψx=φx+θx,ψy=φy+θy,where φx=−∂wb/∂xandφy=−∂wb/∂y are rotations due to pure bending and θxandθy are the shear angles.Substituting equation (5) into equation (4), we get two differential equations:(6)∂∂xDSΔwb+ws=DS∂2θx∂x2+121−ν∂2θx∂y2+121+ν∂2θy∂x∂y−θx,(7)∂∂yDSΔwb+ws=DS∂2θy∂y2+121−ν∂2θy∂x2+121+ν∂2θx∂x∂y−θy,(8)Δws=−∂θx∂x+∂θy∂y.Equations (6) and (7) can be rewritten in the following forms:(9)∂Fwb,ws∂x=g1θx,θy,∂Fwb,ws∂y=g2θx,θy,and their integrals per x and y read Fwb,ws=∫g1θx,θydx=∫g2θx,θydy; according to the structure of g1andg2 in equations (6) and (7), the reasonable solution is that both g1 and g2 must be set to zero; as a consequence of that consideration, the relation between transverse shear and bending is(10)ws=−DSΔwb.The total deflection is(11)w=wb−DSΔwb.Using the new modified first-order shear deformation theory, the displacement field of the plate can be rewritten in the following form [26]:(12)ux,y,z=ux,y−zφx,vx,y,z=vx,y−zφy,wx,y,z=wbx,y+wsx,y.The strain-displacement relations may be written as follows [26]:(13)εx=∂u∂x−z∂2wb∂x2,εy=∂v∂y−z∂2wb∂y2,γxy=∂u∂y+∂v∂x−2z∂2wb∂x∂y,γxz=∂ws∂x,γyz=∂ws∂y.Equation (13) can be rewritten in the matrix form as(14)ε=ε0+zε1,where(15)ε0=∂u∂x∂v∂y∂u∂y+∂v∂x,ε1=−∂2wb∂x2∂2wb∂y22∂2wb∂x∂y,and(16)γ=∂ws∂x∂ws∂y.


#### 2.2.2. Finite Element Formulations

The plate is discretized using the quadrilateral four-node element, as shown in Figure2. Each node has nine degrees of freedom including four axial, one transversal, and four rotational displacements.Figure 2
A two-layer composite plate element.
(a)
(b)The element nodal displacement vector can be given as(17)q=q1q2q3q4T,where(18)qj=ujmvjmujnvjnwjφxjmφyjmφxjnφyjnT,j=1,4¯.The nodal displacement vector of each component is(19)qi=q1iq2iq3iq4iT,i=m,n,where(20)qji=ujivjiwjφxjiφyjiT,j=1,4¯,i=m,n.In equations (18) and (20), u,v,andw are, respectively, the x−andy-axis displacements and transversal displacement and φxandφy denote the bending rotations of the x−andy−axis, respectively.The static bending deflectionwb is assumed in a polynomial form. Then, the transverse shear ws and bending rotations φxandφy are calculated using the above equations, while the axial displacements uandv are approximated using Lagrangian shape functions.The displacementsuandv may be approximated as(21)u=∑j=14Njuj,v=∑j=14Njvj,where(22)Nj=141−ξjξ1−ηjη,j=1,4¯.The bending deflectionwb is approximated using a polynomial form as(23)wbi=Pbiai,i=m,n,where ai is a vector of polynomial coefficients and(24)Pbi=1,ξ,η,ξ2,ξη,η2,ξ3,ξ2η,ξη2,η3,ξ3η,ξη3,i=m,n,where ξ=2x−a/a,η=2y−b/b. According to equation (10), the transverse shear is given by(25)wsi=−DiSiΔwbi=Psiai,where(26)Psi=−0,0,0,2αi,0,2βi,6αiξ,2αiη,2βiξ,6βiη,6αiξη,6βiξη,i=m,n,in which αi=4Di/Sia2,β=4Di/Sib2.The total static deflection of the plate is(27)wi=wbi+wsi=Pbi+Psiai,i=m,n.The rotations of cross sections on the neutral plane are(28)φxi=−2a∂Pbi∂ξai=−2a0,1,0,2ξ,η,0,3ξ2,2ξη,η2,0,3ξ2η,η3ai=Pφxiai,i=m,n,(29)φyi=−2b∂Pbi∂ηai=−2b0,0,1,0,ξ,2η,0,ξ2,2ξη,3η2,ξ3,3ξη2ai=Pφyiai,i=m,n.Taking coordinate valuesξandη for each node into equations (27)–(29), we get(30)di=Ciai⟹ai=Ci−1di,i=m,n,where di=d1id2id3id4iTdji=wjiφxjiφyjiT,j=1,4¯,i=m,n, and matrix Ci is shown in Appendix.Taking equation (30) into account, we get(31)wbi=PbiCi−1di=Qbidi,i=m,n,(32)wsi=PsiCi−1di=Qsidi,i=m,n,(33)wi=wbi+wsi=Pbi+PsiCi−1di=PiCi−1di,i=m,n.Substituting equations (21) and (31) into equation (14), we obtain(34)εi=L0iui+ziL1idi=L0iziL1iuidi=L0iziL1iqi,where(35)L0i=L01iL02iL03iL04i,L0ji=∂Nj∂x00∂Nj∂y∂Nj∂y∂Nj∂x,j=1,4¯,i=m,n.(36)L1i=HbiCi−1,Hbi=4a2∂2Pbi∂ξ24b2∂2Pbi∂η24ab∂2Pbi∂ξ∂η,i=m,n.By substituting equation (25) into equation (16), we get(37)γi=Lsidi,i=m,n,where(38)Lsi=HsiCi−1,Hsi=2a∂Psi∂ξ2b∂Psi∂η,i=m,n.According to Hooke’s law, the relationship between stresses and strains can be given as(39)σi=Diεi,τi=Giγi,i=m,n,where(40)Di=Ei1−νi21νi0νi10001−νi2,Gi=kEi21+νi1001,i=m,n.The calculation model for the connector stub is shown in Figure3.Figure 3
Axial displacement assuming a two-layer composite plate with connector stubs.By using Lagrangian approximation for both axial displacement and rotation of two components to determine the axial deflection of the connector stub, the interfacial slips of two components are given by(41)δst=ustvst=un−hn2vn−hn2−um−hn2vm−hn2=un0−um0+hnφxn−hmφxm2vn0−vm0+hnφyn−hmφym2=Nstqe,where(42)Nst=Nst1Nst2Nst3Nst4,(43)Nstj=−Nj0Nj00−0.5hmNj00.5hnNj00−Nj0Nj00−0.5hmNj00.5hnNj,j=1,4,¯in which ust and vst are the deflections of the connector stub.The strain energy of two components and connector stub can be given by(44)∏e=∏ep+∏est=∑i=m,n12∫Ae∫hiεiTDiεi+τiTGiγidzidA+12∫Aekstust2+vst2dA=∑i=m,n12∫Ae∫hiεiTDiεi+τiTGiγidzidA+12∫AeqeTNstTkstNstqedA.The kinetic energy is(45)Te=12∑i=m,n∫Ae∫hiρiw˙2dzidA,or(46)Te=12∑i=m,n∫Ae∫hid˙iTCi−TPiTρiPiCi−Td˙idzidA.Equation (46) can be rewritten in the shorter form as follows:(47)Te=12∑i=m,n∫Aem^id˙iTCi−TPiTPiCi−Td˙idA,where m^i=∫hiρidzi,i=m,n.The work done by external distribution normal force acting on the top surface of layern is(48)Ve=∫AeqwdA=∫AeqQndndA,where q is the uniform load.The Lagrangian function of the plate element is defined by(49)Le=Te−∏e.Using Lagrange’s equations, the governing equations of motion of the plate element are given by(50)ddt∂Le∂q˙e−∂Le∂qe=∂Ve∂qe.Substituting equations (44), (46), and (48) into equation (50) considering equation (49), the equation of motion of the plate can be obtained in the matrix form as(51)Mem+Menq¨e+Kem+Ken+Kestqe=Fem+Fen,in which Mei,KeiandKest, and Fei are, respectively, the element mass matrix, the element stiffness matrix, and element force vector, which are defined as(52)Mei=ab4m^i∫−11∫−11Ci−TPiTPiCi−Tdξdη,i=m,n,(53)Kei=ab4∫−11∫−11∫hiL0iziL1iTDiL0iziL1idzidξdη+ab4∫−11∫−11∫hiLsiTGiLsidzidξdη,i=m,n,(54)Kest=ab4∫−11∫−11NstTkstNstdξdη,(55)Fen=ab4∫−11∫−11qQndξdη,Fem=0.Note that we need not any selective reduced integration or reduced integration scheme to calculate the matrices and vectors in equations (52)–(55). When kst gets a high value, displacements of the top and bottom layers at the contact surface are the same; from equation (41), the stiffness matrix of the stub in equation (54) gets the value 0: it means that the structure has no stubs and then the displacement is continuous at the interfaces between two layers. After assembling the element matrix and the element vector into the global matrix and global vector, the governing equation of motion of the plate is obtained as(56)MU¨+KU=F,where M,K,andF are, respectively, the global mass matrix, global stiffness matrix, and global force vector and U is the vector of unknown nodal displacements.


### 2.3. Static Bending and Free Vibration Problem

For the static bending problem, ignoringU¨, equation (56) becomes(57)KU=F.Solving equation (57), we get the vector of unknown nodal displacements U.For free vibration analysis, neglecting the effect of external force vectorF and assuming u=Ueiωt in equation (56), we get the following eigenvalue problem:(58)K−ω2MU=0.Solving equation (58), we get ω, which denotes the natural frequencies of the plate; for each value of ω, we have an eigenvector.


## 2.1. Geometry and Assumptions

The geometry model of the problem is a two-layer plate including two isotropic plates which bonded together at the contact surface using connector stubs, as shown in Figure1. Two parts are of the same size in the x−y plane with thickness hm for the bottom layer and hn for the top layer. To avoid the repetition of similar equations for the bottom layer and top layer, we use the subscript m for the bottom layer, subscript n for the top layer, and subscript st for the connector stub.Figure 1
Geometrical notation of the two-layer plate.The assumptions of the two-layer composite plate include the following: the materials of each layer are linear, elastic, and isotropic; the displacement and rotation of the plate are small; there is no delamination phenomenon between two layers; the deflection of the stub is in the contact interface of two components; and the mass of the stub is much smaller than the mass of the plate, so we assume that it is neglected.


## 2.2. Discrete Finite Element Equations

### 2.2.1. New Modified First-Order Shear Deformation for Plate Elements

The basic equations of the Mindlin plate theory are [25](1)Mx=D∂ψx∂x+ν∂ψy∂y,My=D∂ψx∂y+ν∂ψy∂x,Mxy=Myx=121−νD∂ψx∂y+∂ψy∂x,Qx=S∂w∂x+ψx,Qy=S∂w∂y+ψy,where(2)D=Eh3121−ν2,S=kGh,G=E21+ν,in which k = 5/6 is the shear correction.Equilibrium of moments aboutx- and y-axis and transverse force leads to(3)∂Mx∂x+∂Mxy∂y−Qx=mx,∂My∂y+∂Mxy∂x−Qy=my,∂Qx∂x+∂Qy∂y=−q.Substituting equation (1) into equation (3), we get the following:(4)DS∂2ψx∂x2+121−ν∂2ψx∂y2+121+ν∂2ψy∂x∂y−∂w∂x+ψx=0,DS∂2ψy∂y2+121−ν∂2ψy∂x2+121+ν∂2ψx∂x∂y−∂w∂y+ψy=0,Δw+∂ψx∂x+∂ψy∂y=0,where Δ=∂2/∂x2+∂2/∂y2 is the Laplace differential operator.Assume that the total deflection consists of two parts which are bending deflection and transverse shear, while angles of the plate cross-sectional slope are a result of rotation of pure bending and shear angles:(5)w=wb+ws,ψx=φx+θx,ψy=φy+θy,where φx=−∂wb/∂xandφy=−∂wb/∂y are rotations due to pure bending and θxandθy are the shear angles.Substituting equation (5) into equation (4), we get two differential equations:(6)∂∂xDSΔwb+ws=DS∂2θx∂x2+121−ν∂2θx∂y2+121+ν∂2θy∂x∂y−θx,(7)∂∂yDSΔwb+ws=DS∂2θy∂y2+121−ν∂2θy∂x2+121+ν∂2θx∂x∂y−θy,(8)Δws=−∂θx∂x+∂θy∂y.Equations (6) and (7) can be rewritten in the following forms:(9)∂Fwb,ws∂x=g1θx,θy,∂Fwb,ws∂y=g2θx,θy,and their integrals per x and y read Fwb,ws=∫g1θx,θydx=∫g2θx,θydy; according to the structure of g1andg2 in equations (6) and (7), the reasonable solution is that both g1 and g2 must be set to zero; as a consequence of that consideration, the relation between transverse shear and bending is(10)ws=−DSΔwb.The total deflection is(11)w=wb−DSΔwb.Using the new modified first-order shear deformation theory, the displacement field of the plate can be rewritten in the following form [26]:(12)ux,y,z=ux,y−zφx,vx,y,z=vx,y−zφy,wx,y,z=wbx,y+wsx,y.The strain-displacement relations may be written as follows [26]:(13)εx=∂u∂x−z∂2wb∂x2,εy=∂v∂y−z∂2wb∂y2,γxy=∂u∂y+∂v∂x−2z∂2wb∂x∂y,γxz=∂ws∂x,γyz=∂ws∂y.Equation (13) can be rewritten in the matrix form as(14)ε=ε0+zε1,where(15)ε0=∂u∂x∂v∂y∂u∂y+∂v∂x,ε1=−∂2wb∂x2∂2wb∂y22∂2wb∂x∂y,and(16)γ=∂ws∂x∂ws∂y.


### 2.2.2. Finite Element Formulations

The plate is discretized using the quadrilateral four-node element, as shown in Figure2. Each node has nine degrees of freedom including four axial, one transversal, and four rotational displacements.Figure 2
A two-layer composite plate element.
(a)
(b)The element nodal displacement vector can be given as(17)q=q1q2q3q4T,where(18)qj=ujmvjmujnvjnwjφxjmφyjmφxjnφyjnT,j=1,4¯.The nodal displacement vector of each component is(19)qi=q1iq2iq3iq4iT,i=m,n,where(20)qji=ujivjiwjφxjiφyjiT,j=1,4¯,i=m,n.In equations (18) and (20), u,v,andw are, respectively, the x−andy-axis displacements and transversal displacement and φxandφy denote the bending rotations of the x−andy−axis, respectively.The static bending deflectionwb is assumed in a polynomial form. Then, the transverse shear ws and bending rotations φxandφy are calculated using the above equations, while the axial displacements uandv are approximated using Lagrangian shape functions.The displacementsuandv may be approximated as(21)u=∑j=14Njuj,v=∑j=14Njvj,where(22)Nj=141−ξjξ1−ηjη,j=1,4¯.The bending deflectionwb is approximated using a polynomial form as(23)wbi=Pbiai,i=m,n,where ai is a vector of polynomial coefficients and(24)Pbi=1,ξ,η,ξ2,ξη,η2,ξ3,ξ2η,ξη2,η3,ξ3η,ξη3,i=m,n,where ξ=2x−a/a,η=2y−b/b. According to equation (10), the transverse shear is given by(25)wsi=−DiSiΔwbi=Psiai,where(26)Psi=−0,0,0,2αi,0,2βi,6αiξ,2αiη,2βiξ,6βiη,6αiξη,6βiξη,i=m,n,in which αi=4Di/Sia2,β=4Di/Sib2.The total static deflection of the plate is(27)wi=wbi+wsi=Pbi+Psiai,i=m,n.The rotations of cross sections on the neutral plane are(28)φxi=−2a∂Pbi∂ξai=−2a0,1,0,2ξ,η,0,3ξ2,2ξη,η2,0,3ξ2η,η3ai=Pφxiai,i=m,n,(29)φyi=−2b∂Pbi∂ηai=−2b0,0,1,0,ξ,2η,0,ξ2,2ξη,3η2,ξ3,3ξη2ai=Pφyiai,i=m,n.Taking coordinate valuesξandη for each node into equations (27)–(29), we get(30)di=Ciai⟹ai=Ci−1di,i=m,n,where di=d1id2id3id4iTdji=wjiφxjiφyjiT,j=1,4¯,i=m,n, and matrix Ci is shown in Appendix.Taking equation (30) into account, we get(31)wbi=PbiCi−1di=Qbidi,i=m,n,(32)wsi=PsiCi−1di=Qsidi,i=m,n,(33)wi=wbi+wsi=Pbi+PsiCi−1di=PiCi−1di,i=m,n.Substituting equations (21) and (31) into equation (14), we obtain(34)εi=L0iui+ziL1idi=L0iziL1iuidi=L0iziL1iqi,where(35)L0i=L01iL02iL03iL04i,L0ji=∂Nj∂x00∂Nj∂y∂Nj∂y∂Nj∂x,j=1,4¯,i=m,n.(36)L1i=HbiCi−1,Hbi=4a2∂2Pbi∂ξ24b2∂2Pbi∂η24ab∂2Pbi∂ξ∂η,i=m,n.By substituting equation (25) into equation (16), we get(37)γi=Lsidi,i=m,n,where(38)Lsi=HsiCi−1,Hsi=2a∂Psi∂ξ2b∂Psi∂η,i=m,n.According to Hooke’s law, the relationship between stresses and strains can be given as(39)σi=Diεi,τi=Giγi,i=m,n,where(40)Di=Ei1−νi21νi0νi10001−νi2,Gi=kEi21+νi1001,i=m,n.The calculation model for the connector stub is shown in Figure3.Figure 3
Axial displacement assuming a two-layer composite plate with connector stubs.By using Lagrangian approximation for both axial displacement and rotation of two components to determine the axial deflection of the connector stub, the interfacial slips of two components are given by(41)δst=ustvst=un−hn2vn−hn2−um−hn2vm−hn2=un0−um0+hnφxn−hmφxm2vn0−vm0+hnφyn−hmφym2=Nstqe,where(42)Nst=Nst1Nst2Nst3Nst4,(43)Nstj=−Nj0Nj00−0.5hmNj00.5hnNj00−Nj0Nj00−0.5hmNj00.5hnNj,j=1,4,¯in which ust and vst are the deflections of the connector stub.The strain energy of two components and connector stub can be given by(44)∏e=∏ep+∏est=∑i=m,n12∫Ae∫hiεiTDiεi+τiTGiγidzidA+12∫Aekstust2+vst2dA=∑i=m,n12∫Ae∫hiεiTDiεi+τiTGiγidzidA+12∫AeqeTNstTkstNstqedA.The kinetic energy is(45)Te=12∑i=m,n∫Ae∫hiρiw˙2dzidA,or(46)Te=12∑i=m,n∫Ae∫hid˙iTCi−TPiTρiPiCi−Td˙idzidA.Equation (46) can be rewritten in the shorter form as follows:(47)Te=12∑i=m,n∫Aem^id˙iTCi−TPiTPiCi−Td˙idA,where m^i=∫hiρidzi,i=m,n.The work done by external distribution normal force acting on the top surface of layern is(48)Ve=∫AeqwdA=∫AeqQndndA,where q is the uniform load.The Lagrangian function of the plate element is defined by(49)Le=Te−∏e.Using Lagrange’s equations, the governing equations of motion of the plate element are given by(50)ddt∂Le∂q˙e−∂Le∂qe=∂Ve∂qe.Substituting equations (44), (46), and (48) into equation (50) considering equation (49), the equation of motion of the plate can be obtained in the matrix form as(51)Mem+Menq¨e+Kem+Ken+Kestqe=Fem+Fen,in which Mei,KeiandKest, and Fei are, respectively, the element mass matrix, the element stiffness matrix, and element force vector, which are defined as(52)Mei=ab4m^i∫−11∫−11Ci−TPiTPiCi−Tdξdη,i=m,n,(53)Kei=ab4∫−11∫−11∫hiL0iziL1iTDiL0iziL1idzidξdη+ab4∫−11∫−11∫hiLsiTGiLsidzidξdη,i=m,n,(54)Kest=ab4∫−11∫−11NstTkstNstdξdη,(55)Fen=ab4∫−11∫−11qQndξdη,Fem=0.Note that we need not any selective reduced integration or reduced integration scheme to calculate the matrices and vectors in equations (52)–(55). When kst gets a high value, displacements of the top and bottom layers at the contact surface are the same; from equation (41), the stiffness matrix of the stub in equation (54) gets the value 0: it means that the structure has no stubs and then the displacement is continuous at the interfaces between two layers. After assembling the element matrix and the element vector into the global matrix and global vector, the governing equation of motion of the plate is obtained as(56)MU¨+KU=F,where M,K,andF are, respectively, the global mass matrix, global stiffness matrix, and global force vector and U is the vector of unknown nodal displacements.


## 2.2.1. New Modified First-Order Shear Deformation for Plate Elements

The basic equations of the Mindlin plate theory are [25](1)Mx=D∂ψx∂x+ν∂ψy∂y,My=D∂ψx∂y+ν∂ψy∂x,Mxy=Myx=121−νD∂ψx∂y+∂ψy∂x,Qx=S∂w∂x+ψx,Qy=S∂w∂y+ψy,where(2)D=Eh3121−ν2,S=kGh,G=E21+ν,in which k = 5/6 is the shear correction.Equilibrium of moments aboutx- and y-axis and transverse force leads to(3)∂Mx∂x+∂Mxy∂y−Qx=mx,∂My∂y+∂Mxy∂x−Qy=my,∂Qx∂x+∂Qy∂y=−q.Substituting equation (1) into equation (3), we get the following:(4)DS∂2ψx∂x2+121−ν∂2ψx∂y2+121+ν∂2ψy∂x∂y−∂w∂x+ψx=0,DS∂2ψy∂y2+121−ν∂2ψy∂x2+121+ν∂2ψx∂x∂y−∂w∂y+ψy=0,Δw+∂ψx∂x+∂ψy∂y=0,where Δ=∂2/∂x2+∂2/∂y2 is the Laplace differential operator.Assume that the total deflection consists of two parts which are bending deflection and transverse shear, while angles of the plate cross-sectional slope are a result of rotation of pure bending and shear angles:(5)w=wb+ws,ψx=φx+θx,ψy=φy+θy,where φx=−∂wb/∂xandφy=−∂wb/∂y are rotations due to pure bending and θxandθy are the shear angles.Substituting equation (5) into equation (4), we get two differential equations:(6)∂∂xDSΔwb+ws=DS∂2θx∂x2+121−ν∂2θx∂y2+121+ν∂2θy∂x∂y−θx,(7)∂∂yDSΔwb+ws=DS∂2θy∂y2+121−ν∂2θy∂x2+121+ν∂2θx∂x∂y−θy,(8)Δws=−∂θx∂x+∂θy∂y.Equations (6) and (7) can be rewritten in the following forms:(9)∂Fwb,ws∂x=g1θx,θy,∂Fwb,ws∂y=g2θx,θy,and their integrals per x and y read Fwb,ws=∫g1θx,θydx=∫g2θx,θydy; according to the structure of g1andg2 in equations (6) and (7), the reasonable solution is that both g1 and g2 must be set to zero; as a consequence of that consideration, the relation between transverse shear and bending is(10)ws=−DSΔwb.The total deflection is(11)w=wb−DSΔwb.Using the new modified first-order shear deformation theory, the displacement field of the plate can be rewritten in the following form [26]:(12)ux,y,z=ux,y−zφx,vx,y,z=vx,y−zφy,wx,y,z=wbx,y+wsx,y.The strain-displacement relations may be written as follows [26]:(13)εx=∂u∂x−z∂2wb∂x2,εy=∂v∂y−z∂2wb∂y2,γxy=∂u∂y+∂v∂x−2z∂2wb∂x∂y,γxz=∂ws∂x,γyz=∂ws∂y.Equation (13) can be rewritten in the matrix form as(14)ε=ε0+zε1,where(15)ε0=∂u∂x∂v∂y∂u∂y+∂v∂x,ε1=−∂2wb∂x2∂2wb∂y22∂2wb∂x∂y,and(16)γ=∂ws∂x∂ws∂y.


## 2.2.2. Finite Element Formulations

The plate is discretized using the quadrilateral four-node element, as shown in Figure2. Each node has nine degrees of freedom including four axial, one transversal, and four rotational displacements.Figure 2
A two-layer composite plate element.
(a)
(b)The element nodal displacement vector can be given as(17)q=q1q2q3q4T,where(18)qj=ujmvjmujnvjnwjφxjmφyjmφxjnφyjnT,j=1,4¯.The nodal displacement vector of each component is(19)qi=q1iq2iq3iq4iT,i=m,n,where(20)qji=ujivjiwjφxjiφyjiT,j=1,4¯,i=m,n.In equations (18) and (20), u,v,andw are, respectively, the x−andy-axis displacements and transversal displacement and φxandφy denote the bending rotations of the x−andy−axis, respectively.The static bending deflectionwb is assumed in a polynomial form. Then, the transverse shear ws and bending rotations φxandφy are calculated using the above equations, while the axial displacements uandv are approximated using Lagrangian shape functions.The displacementsuandv may be approximated as(21)u=∑j=14Njuj,v=∑j=14Njvj,where(22)Nj=141−ξjξ1−ηjη,j=1,4¯.The bending deflectionwb is approximated using a polynomial form as(23)wbi=Pbiai,i=m,n,where ai is a vector of polynomial coefficients and(24)Pbi=1,ξ,η,ξ2,ξη,η2,ξ3,ξ2η,ξη2,η3,ξ3η,ξη3,i=m,n,where ξ=2x−a/a,η=2y−b/b. According to equation (10), the transverse shear is given by(25)wsi=−DiSiΔwbi=Psiai,where(26)Psi=−0,0,0,2αi,0,2βi,6αiξ,2αiη,2βiξ,6βiη,6αiξη,6βiξη,i=m,n,in which αi=4Di/Sia2,β=4Di/Sib2.The total static deflection of the plate is(27)wi=wbi+wsi=Pbi+Psiai,i=m,n.The rotations of cross sections on the neutral plane are(28)φxi=−2a∂Pbi∂ξai=−2a0,1,0,2ξ,η,0,3ξ2,2ξη,η2,0,3ξ2η,η3ai=Pφxiai,i=m,n,(29)φyi=−2b∂Pbi∂ηai=−2b0,0,1,0,ξ,2η,0,ξ2,2ξη,3η2,ξ3,3ξη2ai=Pφyiai,i=m,n.Taking coordinate valuesξandη for each node into equations (27)–(29), we get(30)di=Ciai⟹ai=Ci−1di,i=m,n,where di=d1id2id3id4iTdji=wjiφxjiφyjiT,j=1,4¯,i=m,n, and matrix Ci is shown in Appendix.Taking equation (30) into account, we get(31)wbi=PbiCi−1di=Qbidi,i=m,n,(32)wsi=PsiCi−1di=Qsidi,i=m,n,(33)wi=wbi+wsi=Pbi+PsiCi−1di=PiCi−1di,i=m,n.Substituting equations (21) and (31) into equation (14), we obtain(34)εi=L0iui+ziL1idi=L0iziL1iuidi=L0iziL1iqi,where(35)L0i=L01iL02iL03iL04i,L0ji=∂Nj∂x00∂Nj∂y∂Nj∂y∂Nj∂x,j=1,4¯,i=m,n.(36)L1i=HbiCi−1,Hbi=4a2∂2Pbi∂ξ24b2∂2Pbi∂η24ab∂2Pbi∂ξ∂η,i=m,n.By substituting equation (25) into equation (16), we get(37)γi=Lsidi,i=m,n,where(38)Lsi=HsiCi−1,Hsi=2a∂Psi∂ξ2b∂Psi∂η,i=m,n.According to Hooke’s law, the relationship between stresses and strains can be given as(39)σi=Diεi,τi=Giγi,i=m,n,where(40)Di=Ei1−νi21νi0νi10001−νi2,Gi=kEi21+νi1001,i=m,n.The calculation model for the connector stub is shown in Figure3.Figure 3
Axial displacement assuming a two-layer composite plate with connector stubs.By using Lagrangian approximation for both axial displacement and rotation of two components to determine the axial deflection of the connector stub, the interfacial slips of two components are given by(41)δst=ustvst=un−hn2vn−hn2−um−hn2vm−hn2=un0−um0+hnφxn−hmφxm2vn0−vm0+hnφyn−hmφym2=Nstqe,where(42)Nst=Nst1Nst2Nst3Nst4,(43)Nstj=−Nj0Nj00−0.5hmNj00.5hnNj00−Nj0Nj00−0.5hmNj00.5hnNj,j=1,4,¯in which ust and vst are the deflections of the connector stub.The strain energy of two components and connector stub can be given by(44)∏e=∏ep+∏est=∑i=m,n12∫Ae∫hiεiTDiεi+τiTGiγidzidA+12∫Aekstust2+vst2dA=∑i=m,n12∫Ae∫hiεiTDiεi+τiTGiγidzidA+12∫AeqeTNstTkstNstqedA.The kinetic energy is(45)Te=12∑i=m,n∫Ae∫hiρiw˙2dzidA,or(46)Te=12∑i=m,n∫Ae∫hid˙iTCi−TPiTρiPiCi−Td˙idzidA.Equation (46) can be rewritten in the shorter form as follows:(47)Te=12∑i=m,n∫Aem^id˙iTCi−TPiTPiCi−Td˙idA,where m^i=∫hiρidzi,i=m,n.The work done by external distribution normal force acting on the top surface of layern is(48)Ve=∫AeqwdA=∫AeqQndndA,where q is the uniform load.The Lagrangian function of the plate element is defined by(49)Le=Te−∏e.Using Lagrange’s equations, the governing equations of motion of the plate element are given by(50)ddt∂Le∂q˙e−∂Le∂qe=∂Ve∂qe.Substituting equations (44), (46), and (48) into equation (50) considering equation (49), the equation of motion of the plate can be obtained in the matrix form as(51)Mem+Menq¨e+Kem+Ken+Kestqe=Fem+Fen,in which Mei,KeiandKest, and Fei are, respectively, the element mass matrix, the element stiffness matrix, and element force vector, which are defined as(52)Mei=ab4m^i∫−11∫−11Ci−TPiTPiCi−Tdξdη,i=m,n,(53)Kei=ab4∫−11∫−11∫hiL0iziL1iTDiL0iziL1idzidξdη+ab4∫−11∫−11∫hiLsiTGiLsidzidξdη,i=m,n,(54)Kest=ab4∫−11∫−11NstTkstNstdξdη,(55)Fen=ab4∫−11∫−11qQndξdη,Fem=0.Note that we need not any selective reduced integration or reduced integration scheme to calculate the matrices and vectors in equations (52)–(55). When kst gets a high value, displacements of the top and bottom layers at the contact surface are the same; from equation (41), the stiffness matrix of the stub in equation (54) gets the value 0: it means that the structure has no stubs and then the displacement is continuous at the interfaces between two layers. After assembling the element matrix and the element vector into the global matrix and global vector, the governing equation of motion of the plate is obtained as(56)MU¨+KU=F,where M,K,andF are, respectively, the global mass matrix, global stiffness matrix, and global force vector and U is the vector of unknown nodal displacements.


## 2.3. Static Bending and Free Vibration Problem

For the static bending problem, ignoringU¨, equation (56) becomes(57)KU=F.Solving equation (57), we get the vector of unknown nodal displacements U.For free vibration analysis, neglecting the effect of external force vectorF and assuming u=Ueiωt in equation (56), we get the following eigenvalue problem:(58)K−ω2MU=0.Solving equation (58), we get ω, which denotes the natural frequencies of the plate; for each value of ω, we have an eigenvector.


## 3. Numerical Results and Discussion

We now focus on numerically studying the static bending and free vibration responses of the two-layer composite plate. Each edge of the plate can be under the simply supported boundary condition or clamped boundary condition. For the simply supported edge,(59a)vn=vm=w=0,φyn=φym=0,atx=0,a,(59b)un=um=w=0,φxn=φxm=0,aty=0,b.For the clamped edge,(60)un=um=vn=vm=w=0,φxn=φxm=φyn=φym=0,atx=0,aandy=0,b.


### 3.1. Static Bending Analysis of Two-Layer Composite Plates

#### 3.1.1. Accuracy Studies

To verify the proposed method, we compare the deflections in two cases of materials: homogeneous and composite materials. Firstly, a square homogeneous plate under contribution loadq=−1N is investigated, and the length and the width are a=b=1m. The plate is simply supported at all edges (SSSS) or fully clamped (CCCC). In the published works [27, 28], the plate has one layer made of homogeneous material with E=10920MPa, Poisson’s ratio ν=0.3, and the thickness h. But in this work, we assume that the plate has two layers connected by stubs with two cases—case 1: hn = 0.999 h, hm = 0.001 h, and kst = 0 and case 2: hn = hm = 0.5 h and kst = 1010E. In both cases, material properties of two layers are set to be En=Em=E=10920MPaandνn=νm=ν=0.3, thicknesses of the top and bottom layer are hn and hm, respectively, and hn + hm = h. The nondimensional deflection is defined as(61)w¯=wmaxDqa4,where D is given in equation (2).Table1 shows a comparison of the nondimensional deflections of SSSS and CCCC plates obtained by this method and other approaches (FEM [28] and analytical solution [28]). We meet a good agreement among three solutions.Table 1
Comparison of nondimensional deflection of square plates witha/h = 10 and a/h = 10000.
Source
SSSS
CCCC
a
/
h
=
10
a
/
h
=
10000
a
/
h
=
10
a
/
h
=
10000
Ferreira [28]
0.004271
0.004060
0.001503
0.001264
Exact solution [28]
0.004270
0.004060
—
0.001260
Present (hn=0.999h,hm=0.001h,andkst=0)
0.004290
0.004079
0.001508
0.001272
Present (hn=hm=0.5handkst=1010E)
0.004300
0.004079
0.001523
0.001291We see that when the thickness of one layer is much higher than that of the other layer and the stiffness of the connector stub is very small, the behavior of the two-layer plate is the same as the behavior of the one-layer plate with the same thickness. In addition, when two layers have the same thickness and material, the stiffness of the connector stub gets a much higher valuekst = 1010E, and then the two-layer plate is considered as the one-layer plate with the thickness equal to the sum of thickness of the two components.In order to further confirm the accuracy of this method, we compare the nondimensional deflection of a square SSSS composite plate (a/b = 1 and the total thickness h = a/10) with two layers (0/90°) which have the same thickness h/2;the  material  properties  areE1/E2=25,G12=G13=0.5E2,G23=0.2E2,andν12=0.25. In this comparison, the plate is modeled as hn = hm = 0.5 h, kst = 1010E2, and the fiber directions of the top and bottom layers are 0° and 90°, respectively. Table 2 shows the comparison of the nondimensional deflection of this composite plate subjected to uniformly distributed load and central concentrated load. The nondimensional deflection of the plate is calculated by the following formula:(62)w¯=wmax100E2h3a4q.Table 2
Comparison of the nondimensional deflection of a square SSSS composite plate subject to uniformly distributed load and central concentrated load.
w
¯
Uniformly distributed load
Central concentrated load
Reddy [29]
Present
Reddy [29]
Present
1.6955
1.6538
4.6664
4.6421It is shown plainly that the present results have a good agreement with the results given in [29] (analytical method). According to the above two comparisons, the present results have a good agreement with other published ones.


#### 3.1.2. Numerical Results for Static Bending Analysis of Two-Layer Composite Plates

In this section, a rectangular two-layer composite plate subjected to the uniform loadq is investigated. The boundary conditions of the plate are SSSS (fully simply supported), CCCC (fully clamped), CSCS (two opposite edges are simply supported and two other edges are clamped), and CCSS (two adjacent edges are clamped and two other edges are simply supported). The sides, thickness, and material properties of the plate are set to be a=1m, h=a/100, Em=200GPa, En=10GPa, a/b=open, hn/hm=open, and kst/Em=open. The nondimensional displacement of this plate is normalized by(63)w¯=w100Emh312qa41−νm2.(1) Influence of the Length-to-Width Ratio a/b. In order for studying the effect of the aspect ratio a/b on the mechanical bending behavior of the two-layer composite plate, this ratio is set to vary from 0.5 to 4, while hn/hm=1, hn/hm=2, and the ratio of stiffness of the connector stub kst/Em=1 are considered. The computed results are shown in Table 3 and Figures 4(a) and 4(b). According to Table 3 and Figure 4, it is seen that when the aspect ratio a/b increases, the nondimensional deflection decreases. In addition, when the ratio a/b>3, the nondimensional deflections of both boundary conditions CCCC and CSCS of plates are very close to each other.Table 3
Nondimensional deflection of the plate as a function of ratioa/b with kst/Em=1.
a
/
b
h
n
/
h
m
=
1
h
n
/
h
m
=
2
SSSS
CCCC
CSCS
CCSS
SSSS
CCCC
CSCS
CCSS
0.5
6.4728
1.6252
5.4088
3.1243
12.5027
3.1584
10.4767
6.0528
0.625
5.3098
1.4770
3.8595
2.7089
10.2589
2.8726
7.4844
5.2510
0.75
4.2368
1.2639
2.6511
2.2449
8.1874
2.4598
5.1458
4.3531
0.875
3.3289
1.0313
1.8013
1.7999
6.4339
2.0083
3.4991
3.4914
1.0
2.5968
0.8135
1.2304
1.4124
5.0196
1.5849
2.3919
2.7406
1.2
1.7421
0.5350
0.6889
0.9375
3.3683
1.0431
1.3408
1.8201
1.4
1.1793
0.3464
0.4036
0.6196
2.2806
0.6760
0.7866
1.2035
1.6
0.8107
0.2259
0.2475
0.4139
1.5683
0.4413
0.4830
0.8046
1.8
0.5676
0.1501
0.1582
0.2818
1.0984
0.2936
0.3094
0.5482
2.0
0.4050
0.1020
0.1051
0.1958
0.7839
0.1999
0.2058
0.3812
2.2
0.2943
0.0711
0.0721
0.1389
0.5699
0.1395
0.1416
0.2707
2.4
0.2177
0.0507
0.0510
0.1006
0.4217
0.0997
0.1003
0.1962
2.6
0.1637
0.0370
0.0371
0.0742
0.3172
0.0729
0.0730
0.1450
2.8
0.1249
0.0276
0.0275
0.0558
0.2423
0.0544
0.0544
0.1091
3.0
0.0967
0.0209
0.0209
0.0426
0.1877
0.0415
0.0414
0.0835
3.2
0.0759
0.0162
0.0162
0.0331
0.1473
0.0321
0.0322
0.0650
3.4
0.0602
0.0127
0.0127
0.0261
0.1170
0.0253
0.0254
0.0512
3.6
0.0484
0.0101
0.0101
0.0208
0.0940
0.0202
0.0203
0.0409
3.8
0.0392
0.0082
0.0082
0.0168
0.0763
0.0164
0.0164
0.0331
4.0
0.0321
0.0067
0.0067
0.0137
0.0626
0.0134
0.0134
0.0270Figure 4
The influence of ratioa/b on the nondimensional deflection of the plate with kst/Em=1: (a) hn/hm=1; (b) hn/hm=2.
(a)
(b)(2) Influence of the Thickness-to-Thickness Ratio hn/hm. Next, the research on the variation of deflection affected by the relationship between the bottom layer thickness and the top layer thickness is now investigated. The ratio hn/hm increases from 1 to 4, kst/Em=1, and the length-to-width ratio a/b is set to be 1 and 2. The nondimensional deflection of the plate with different boundary conditions is shown in Table 4 and Figure 5. The deflection of the plate increases when increasing ratio hn/hm; this can be explained by the fact that Young’s modulus of the layer n is less than that of layer m, and the stiffness of the two-layer plate becomes smaller when the ratio hn/hm increases.Table 4
Nondimensional deflection of the plate as a function of ratiohn/hm with kst/Eb=1.
h
n
/
h
m
a
/
b
=
1
a
/
b
=
2
SSSS
CCCC
CSCS
CCSS
SSSS
CCCC
CSCS
CCSS
1.0
2.5968
0.8135
1.2304
1.4124
0.4050
0.1020
0.1051
0.1958
1.2
3.1576
0.9907
1.4977
1.7187
0.4926
0.1243
0.1281
0.2384
1.4
3.6969
1.1617
1.7556
2.0138
0.5769
0.1460
0.1503
0.2795
1.6
4.1941
1.3201
1.9940
2.2864
0.6546
0.1661
0.1710
0.3175
1.8
4.6367
1.4618
2.2070
2.5297
0.7239
0.1841
0.1896
0.3516
2.0
5.0196
1.5849
2.3919
2.7406
0.7839
0.1999
0.2058
0.3812
2.2
5.3436
1.6895
2.5488
2.9194
0.8348
0.2133
0.2196
0.4063
2.4
5.6129
1.7769
2.6798
3.0684
0.8771
0.2246
0.2312
0.4273
2.6
5.8338
1.8489
2.7875
3.1909
0.9118
0.2340
0.2408
0.4447
2.8
6.0131
1.9077
2.8752
3.2906
0.9400
0.2416
0.2487
0.4588
3.0
6.1575
1.9552
2.9461
3.3710
0.9628
0.2478
0.2551
0.4702
3.2
6.2731
1.9934
3.0030
3.4355
0.9810
0.2529
0.2603
0.4795
3.4
6.3651
2.0239
3.0485
3.4870
0.9956
0.2569
0.2644
0.4868
3.6
6.4379
2.0482
3.0847
3.5278
1.0071
0.2602
0.2678
0.4927
3.8
6.4954
2.0676
3.1133
3.5602
1.0162
0.2628
0.2704
0.4974
4.0
6.5405
2.0828
3.1359
3.5856
1.0234
0.2649
0.2726
0.5011(3) Influence of the Stiffness of the Connector Stub. Table 5 and Figure 6 show the nondimensional deflections as a function of the ratio kst/Em of a square two-layer composite plate with hn/hm=1 and hn/hm=2. The stiffness of the stub varies from 0 to 106Em; when it changes from 0 to Em, this plate is stiffer and the deflection decreases; but when the stiffness of the stub > Em, then the effect of the stub goes to zero so that the two-layer plate connected by the stub becomes a two-layer one with no stub and the displacements at the contact place are the same; it means that there is no slip between two layers at the contact place.Table 5
Nondimensional deflection of the plate as a function of ratiokst/Em with a/b=1.
k
st
/
E
m
h
n
/
h
m
=
1
h
n
/
h
m
=
2
SSSS
CCCC
CSCS
CCSS
SSSS
CCCC
CSCS
CCSS
0
3.1903
0.9951
1.5068
1.7316
9.4673
2.9526
4.4712
5.1381
10–2
2.6790
0.8708
1.3056
1.4862
5.5538
1.9698
2.8925
3.2279
10–1
2.6051
0.8214
1.2402
1.4215
5.0730
1.6363
2.4560
2.7995
100
2.5968
0.8135
1.2304
1.4124
5.0196
1.5849
2.3919
2.7406
101
2.5959
0.8126
1.2293
1.4114
5.0142
1.5792
2.3850
2.7343
102
2.5958
0.8125
1.2292
1.4113
5.0137
1.5787
2.3843
2.7336
103
2.5958
0.8125
1.2292
1.4113
5.0136
1.5786
2.3842
2.7336
104
2.5958
0.8125
1.2292
1.4113
5.0136
1.5786
2.3842
2.7336
105
2.5958
0.8125
1.2292
1.4113
5.0136
1.5786
2.3842
2.7336
106
2.5958
0.8125
1.2292
1.4113
5.0136
1.5786
2.3842
2.7336Figure 5
The effect of ratiohn/hm on the nondimensional transverse deflection of the plate with kst/Em=1: (a) a/b=1; (b) a/b=2.
(a)
(b)Figure 6
The effect of ratiokst/Em on nondimensional deflection of the plate with a/b=1: (a) hn/hm=1; (b) hn/hm=2.
(a)
(b)To show clearly the slip between the top and bottom layers at the contact place, we depicted the ratioRmn1 and Rmn2 in Figures 7 and 8, respectively, in which Rmn1 and Rmn2 are defined as(64)Rmn1=uma4,b2,hm2−una4,b2,−hn21000Emh312qa41−νm2,Rmn2=umx,b2,hm2−unx,b2,−hn21000Emh312qa41−νm2.Figure 7
The effect of ratiokst/Em on Rmn1 of the plate with hn/hm=1.Figure 8
The slip between the top and bottom layers at the contact place varying along thex-direction, with kst/Em = 0.1 and hn/hm=1.We can see again in Figure7 that when the stiffness of the stub increases, the slip between two layers decreases and tends to zero. By looking at Figure 8 (the ratio kst/Em = 0.1 is employed), it is very interesting to see that the minimum of slip between two layers at the contact place appears at the central plate with symmetric boundary conditions (CCCC, SSSS, and CSCS). The slip gets a maximum value at the simply supported edge, and this slip strongly depends on the boundary condition.(4) Influence of Boundary Condition. We further study the deflection affected by the boundary condition. A square plate with hn/hm = 1-2 and kst/Em = 0-1 is considered. Figure 9 shows the displacement at x = a/2 and along the y direction in four boundary conditions. The boundary condition has strong effects on the deflection magnitude and deformation shape of the plate. It is shown that the plate with the simple support at all edges has the highest deflection and the one with the fully clamped support has the smallest deflection. The deformation shapes of the SSSS plate, CCCC plate, and CSCS plate are symmetric, while the deformation of the CCSS plate is asymmetric.Figure 9
Nondimensional deflection of the plate along the line partx=a/2 with a/b=1 and (a) hn/hm=1, kst/Em=0, (b) hn/hm=1, kst/Em=1, (c) hn/hm=2, kst/Em=0, and (d) hn/hm=2, kst/Em=1.
(a)
(b)
(c)
(d)


### 3.2. Free Vibration Analysis

#### 3.2.1. Accuracy Studies

To confirm the accuracy of this work, we first present a comparison of frequencies of homogeneous CCCC and SSSS plates, witha = b = 1 m, h = a/10 and a/100, E = 10920 MPa, and Poisson’s ratio ν=0.3; similar to the bending analysis, hn = hm = 0.5 h and kst = 1010E are again used for this comparison. The nondimensional frequencies are defined by(65)ω¯i=ωiaρG.The results are listed in Tables6 and 7, in which Ferreira theory [28] was based on FEM and Mindlin theory [28] was based on the analytical solution. As expected, the results obtained by this work match well with those derived from other solutions.Table 6
Comparison of the nondimensional frequencies of an SSSS plate.
a
/
h
Mode
Ferreira [28]
Mindlin [28]
Present
10
1
0.9346
0.930
0.9271
2
2.2545
2.219
2.2055
3
2.2545
2.219
2.2055
4
3.4592
3.406
3.3600
5
4.3031
4.149
4.1123
6
4.3031
4.149
4.1123
100
1
0.0965
0.0963
0.0952
2
0.2430
0.2406
0.2343
3
0.2430
0.2406
0.2343
4
0.3890
0.3847
0.3680
5
0.4928
0.4807
0.4580
6
0.4928
0.4807
0.4580Table 7
Comparison of the nondimensional frequencies of a CCCC plate.
a
/
h
Mode
Ferreira [28]
Liew et al. [27]
Present
10
1
1.5955
1.5582
1.5710
2
3.0662
3.0182
2.9927
3
3.0662
3.0182
2.9927
4
4.2924
4.1711
4.1829
5
5.1232
5.1218
4.9441
6
5.1730
5.1594
4.9893
100
1
0.1750
0.1743
0.1683
2
0.3635
0.3576
0.3431
3
0.3635
0.3576
0.3432
4
0.5358
0.5240
0.4978
5
0.6634
0.6465
0.6014
6
0.6665
0.6505
0.6057Next, we compare the nondimensional frequencies of an SSSS composite plate 0/90°. The parameters of this plate are set to beE1/E2=25, G12=G13=0.5E2, G23=0.2E2, ν12=0.25, a/b=1, and a/h=10. The nondimensional frequencies of the plate are normalized by(66)ω˜i=ωib2π2ρhD22,i=1−6.The comparison is shown in Table8; note that the results in [29] are obtained from the exact solution. We can see that the present results and reference results are close to each other. From two comparisons, we conclude that the accuracy of this approach is accepted.Table 8
Comparison of the nondimensional frequenciesω˜i of a square SSSS composite plate.
Mode
E
1
/
E
2
=
10
E
1
/
E
2
=
20
Reddy [29]
Present
Reddy [29]
Present
1
1.183
1.151
0.990
1.032
2
3.174
3.209
2.719
2.966
3
3.174
3.209
2.719
2.966
4
4.733
5.057
3.959
4.101
5
6.666
6.399
5.789
5.806
6
6.666
6.399
5.789
5.806


#### 3.2.2. Numerical Results for Free Vibration Analysis of Two-Layer Composite Plates

In this section, we research the natural frequency of the two-layer composite plate utilizing the proposed method. The two-layer composite plate (lengtha, width b, thickness of the top layer hn, and thickness of the bottom layer hm) is studied herein. The nondimensional fundamental frequency is defined as(67)ω⌣=ω1⋅100⋅a21+νmρEm.(1) Influence of Length-to-Width Ratio. Table 9 and Figure 10 show the dimensionless fundamental frequencies as a function of aspect ratio a/b with hn/hm=1 and kst/Em=1 of two-layer composite plates altered by the boundary condition. It is evident that the highest and the smallest natural frequencies are found for CCCC and SSSS plates, respectively. When the aspect ratio a/b increases, the nondimensional fundamental frequency increases, and the fundamental frequencies of CCCC and CSCS plates are close when the aspect ratio a/b > 2.6; this is a similar behavior to the static bending problem.Table 9
Influence of ratioa/b on nondimensional fundamental frequency of the plate with hn/hm=1 and kst/Em=1.
a
/
b
Boundary
SSSS
CCCC
CSCS
CCSS
0.5
0.8439
1.6701
0.9329
1.2115
1.0
1.3505
2.4421
1.9700
1.8434
1.2
1.6474
3.0049
2.6189
2.2592
1.4
1.9983
3.7145
3.3955
2.7687
1.6
2.4029
4.5647
4.2971
3.3698
1.8
2.8614
5.5501
5.3220
4.0606
2.0
3.3737
6.6663
6.4691
4.8395
2.2
3.9398
7.9101
7.7374
5.7053
2.4
4.5595
9.2791
9.1262
6.6570
2.6
5.2330
10.7714
10.6349
7.6938
2.8
5.9601
12.3857
12.2627
8.8151
3.0
6.7408
14.1208
14.0093
10.0203
3.2
7.5751
15.9757
15.8739
11.3092
3.4
8.4628
17.9494
17.8561
12.6811
3.6
9.4040
20.0412
19.9553
14.1359
3.8
10.3985
22.2503
22.1709
15.6730
4.0
11.4463
24.5760
24.5023
17.2922Figure 10
The effect of ratioa/b on nondimensional fundamental frequencies of the plate with hn/hm=1 and kst/Em=1.(2) Influence of Thickness-to-Thickness Ratiohn/hm. To study the influence of thickness-to-thickness ratio hn/hm, a two-layer square composite plate with kst/Em=1 is used for this analysis. The nondimensional fundamental frequencies as a function of the ratio hn/hm are tabulated in Table 10 and shown in Figures 11 and 12. The nondimensional fundamental frequency varies nonlinearly with the increasing ratio hn/hm; the minimum value of fundamental frequency will be obtained when the ratio hn/hm is in the range of 1.8 to 2.6.Table 10
Influence of ratiohn/hm on nondimensional fundamental frequency of the square plate with a/b=1 and kst/Em=1.
h
n
/
h
m
Boundary
SSSS
CCCC
CSCS
CCSS
1.0
1.3505
2.4421
1.9700
1.8434
1.2
1.2757
2.3022
1.8582
1.7396
1.4
1.2231
2.2024
1.7789
1.6661
1.6
1.1872
2.1332
1.7241
1.6156
1.8
1.1641
2.0873
1.6880
1.5825
2.0
1.1506
2.0591
1.6662
1.5627
2.2
1.1445
2.0445
1.6552
1.5530
2.4
1.1439
2.0403
1.6526
1.5511
2.6
1.1476
2.0439
1.6562
1.5550
2.8
1.1544
2.0535
1.6646
1.5633
3.0
1.1635
2.0676
1.6765
1.5749
3.2
1.1744
2.0850
1.6911
1.5889
3.4
1.1866
2.1049
1.7077
1.6048
3.6
1.1997
2.1266
1.7256
1.6219
3.8
1.2133
2.1496
1.7446
1.6399
4.0
1.2274
2.1734
1.7642
1.6585Figure 11
The effect of ratiohn/hm on nondimensional fundamental frequencies of the plate with a/b = 1 and kst/Em=1.Figure 12
Nondimensional fundamental frequencies as a function ofhn/hm and a/b of the SSSS plate with kst/Em=1.(3) Influence of Stiffness of the Connector Stub. In this exploration, the effect of stiffness of the connector stub on the nondimensional fundamental frequency of the two-layer composite plate is studied. A square two-layer plate with a/b=1 and kst/Em varying from 0 to 106 is investigated. Table 11 and Figure 13 report the computed results of this plate with hn/hm = 1; we can see that increasing the ratio kst/Em from 0 to 1 leads to the increase in the frequencies. When the ratio kst/Em>1, the nondimensional fundamental frequency is almost unchanged.Table 11
Influence of ratiokst/Em on nondimensional fundamental frequency of the plate with a/b=1 and hn/hm=1.
k
st
/
E
m
Boundary
SSSS
CCCC
CSCS
CCSS
0
1.2206
2.2205
1.7881
1.6710
10–2
1.3292
2.3636
1.9142
1.7976
10–1
1.3482
2.4308
1.9625
1.8376
100
1.3505
2.4421
1.9700
1.8434
101
1.3507
2.4433
1.9708
1.8441
102
1.3508
2.4435
1.9709
1.8441
103
1.3508
2.4435
1.9709
1.8441
104
1.3508
2.4435
1.9709
1.8441
105
1.3508
2.4435
1.9709
1.8441
106
1.3506
2.4434
1.9710
1.8439Figure 13
The effect of ratiokst/Em on nondimensional fundamental frequencies of the plate with a/b = 1 and hn/hm=1.Figures14 and 15 plot nondimensional fundamental frequencies of the plate as a function of ratios a/b, hn/hm, and kst/Em. In addition, the first six mode shapes of plates with boundary conditions such as SSSS, CCCC, CSCS, and CCSS are shown in Figures 16–19 with a/b = 1, hn/hm = 1, and kst/Em = 10−1. It is seen that the boundary condition has an effect on mode shapes as well as on the value of frequencies.Figure 14
Nondimensional fundamental frequencies as a function ofkst/Em and hn/hm of the SSSS plate with a/b=1.Figure 15
Nondimensional fundamental frequencies as a function ofkst/Em and a/b of the SSSS plate with hn/hm=1.Figure 16
The first six mode shapes of a two-layer square SSSS composite plate. (a) Mode 1. (b) Mode 2. (c) Mode 3. (d) Mode 4. (e) Mode 5. (f) Mode 6.
(a)
(b)
(c)
(d)
(e)
(f)Figure 17
The first six mode shapes of a two-layer square CCCC composite plate. (a) Mode 1. (b) Mode 2. (c) Mode 3. (d) Mode 4. (e) Mode 5. (f) Mode 6.
(a)
(b)
(c)
(d)
(e)
(f)Figure 18
The first six mode shapes of a two-layer square CSCS composite plate. (a) Mode 1. (b) Mode 2. (c) Mode 3. (d) Mode 4. (e) Mode 5. (f) Mode 6.
(a)
(b)
(c)
(d)
(e)
(f)Figure 19
The first six mode shapes of a two-layer square CCSS composite plate. (a) Mode 1. (b) Mode 2. (c) Mode 3. (d) Mode 4. (e) Mode 5. (f) Mode 6.
(a)
(b)
(c)
(d)
(e)
(f)


## 3.1. Static Bending Analysis of Two-Layer Composite Plates

### 3.1.1. Accuracy Studies

To verify the proposed method, we compare the deflections in two cases of materials: homogeneous and composite materials. Firstly, a square homogeneous plate under contribution loadq=−1N is investigated, and the length and the width are a=b=1m. The plate is simply supported at all edges (SSSS) or fully clamped (CCCC). In the published works [27, 28], the plate has one layer made of homogeneous material with E=10920MPa, Poisson’s ratio ν=0.3, and the thickness h. But in this work, we assume that the plate has two layers connected by stubs with two cases—case 1: hn = 0.999 h, hm = 0.001 h, and kst = 0 and case 2: hn = hm = 0.5 h and kst = 1010E. In both cases, material properties of two layers are set to be En=Em=E=10920MPaandνn=νm=ν=0.3, thicknesses of the top and bottom layer are hn and hm, respectively, and hn + hm = h. The nondimensional deflection is defined as(61)w¯=wmaxDqa4,where D is given in equation (2).Table1 shows a comparison of the nondimensional deflections of SSSS and CCCC plates obtained by this method and other approaches (FEM [28] and analytical solution [28]). We meet a good agreement among three solutions.Table 1
Comparison of nondimensional deflection of square plates witha/h = 10 and a/h = 10000.
Source
SSSS
CCCC
a
/
h
=
10
a
/
h
=
10000
a
/
h
=
10
a
/
h
=
10000
Ferreira [28]
0.004271
0.004060
0.001503
0.001264
Exact solution [28]
0.004270
0.004060
—
0.001260
Present (hn=0.999h,hm=0.001h,andkst=0)
0.004290
0.004079
0.001508
0.001272
Present (hn=hm=0.5handkst=1010E)
0.004300
0.004079
0.001523
0.001291We see that when the thickness of one layer is much higher than that of the other layer and the stiffness of the connector stub is very small, the behavior of the two-layer plate is the same as the behavior of the one-layer plate with the same thickness. In addition, when two layers have the same thickness and material, the stiffness of the connector stub gets a much higher valuekst = 1010E, and then the two-layer plate is considered as the one-layer plate with the thickness equal to the sum of thickness of the two components.In order to further confirm the accuracy of this method, we compare the nondimensional deflection of a square SSSS composite plate (a/b = 1 and the total thickness h = a/10) with two layers (0/90°) which have the same thickness h/2;the  material  properties  areE1/E2=25,G12=G13=0.5E2,G23=0.2E2,andν12=0.25. In this comparison, the plate is modeled as hn = hm = 0.5 h, kst = 1010E2, and the fiber directions of the top and bottom layers are 0° and 90°, respectively. Table 2 shows the comparison of the nondimensional deflection of this composite plate subjected to uniformly distributed load and central concentrated load. The nondimensional deflection of the plate is calculated by the following formula:(62)w¯=wmax100E2h3a4q.Table 2
Comparison of the nondimensional deflection of a square SSSS composite plate subject to uniformly distributed load and central concentrated load.
w
¯
Uniformly distributed load
Central concentrated load
Reddy [29]
Present
Reddy [29]
Present
1.6955
1.6538
4.6664
4.6421It is shown plainly that the present results have a good agreement with the results given in [29] (analytical method). According to the above two comparisons, the present results have a good agreement with other published ones.


### 3.1.2. Numerical Results for Static Bending Analysis of Two-Layer Composite Plates

In this section, a rectangular two-layer composite plate subjected to the uniform loadq is investigated. The boundary conditions of the plate are SSSS (fully simply supported), CCCC (fully clamped), CSCS (two opposite edges are simply supported and two other edges are clamped), and CCSS (two adjacent edges are clamped and two other edges are simply supported). The sides, thickness, and material properties of the plate are set to be a=1m, h=a/100, Em=200GPa, En=10GPa, a/b=open, hn/hm=open, and kst/Em=open. The nondimensional displacement of this plate is normalized by(63)w¯=w100Emh312qa41−νm2.(1) Influence of the Length-to-Width Ratio a/b. In order for studying the effect of the aspect ratio a/b on the mechanical bending behavior of the two-layer composite plate, this ratio is set to vary from 0.5 to 4, while hn/hm=1, hn/hm=2, and the ratio of stiffness of the connector stub kst/Em=1 are considered. The computed results are shown in Table 3 and Figures 4(a) and 4(b). According to Table 3 and Figure 4, it is seen that when the aspect ratio a/b increases, the nondimensional deflection decreases. In addition, when the ratio a/b>3, the nondimensional deflections of both boundary conditions CCCC and CSCS of plates are very close to each other.Table 3
Nondimensional deflection of the plate as a function of ratioa/b with kst/Em=1.
a
/
b
h
n
/
h
m
=
1
h
n
/
h
m
=
2
SSSS
CCCC
CSCS
CCSS
SSSS
CCCC
CSCS
CCSS
0.5
6.4728
1.6252
5.4088
3.1243
12.5027
3.1584
10.4767
6.0528
0.625
5.3098
1.4770
3.8595
2.7089
10.2589
2.8726
7.4844
5.2510
0.75
4.2368
1.2639
2.6511
2.2449
8.1874
2.4598
5.1458
4.3531
0.875
3.3289
1.0313
1.8013
1.7999
6.4339
2.0083
3.4991
3.4914
1.0
2.5968
0.8135
1.2304
1.4124
5.0196
1.5849
2.3919
2.7406
1.2
1.7421
0.5350
0.6889
0.9375
3.3683
1.0431
1.3408
1.8201
1.4
1.1793
0.3464
0.4036
0.6196
2.2806
0.6760
0.7866
1.2035
1.6
0.8107
0.2259
0.2475
0.4139
1.5683
0.4413
0.4830
0.8046
1.8
0.5676
0.1501
0.1582
0.2818
1.0984
0.2936
0.3094
0.5482
2.0
0.4050
0.1020
0.1051
0.1958
0.7839
0.1999
0.2058
0.3812
2.2
0.2943
0.0711
0.0721
0.1389
0.5699
0.1395
0.1416
0.2707
2.4
0.2177
0.0507
0.0510
0.1006
0.4217
0.0997
0.1003
0.1962
2.6
0.1637
0.0370
0.0371
0.0742
0.3172
0.0729
0.0730
0.1450
2.8
0.1249
0.0276
0.0275
0.0558
0.2423
0.0544
0.0544
0.1091
3.0
0.0967
0.0209
0.0209
0.0426
0.1877
0.0415
0.0414
0.0835
3.2
0.0759
0.0162
0.0162
0.0331
0.1473
0.0321
0.0322
0.0650
3.4
0.0602
0.0127
0.0127
0.0261
0.1170
0.0253
0.0254
0.0512
3.6
0.0484
0.0101
0.0101
0.0208
0.0940
0.0202
0.0203
0.0409
3.8
0.0392
0.0082
0.0082
0.0168
0.0763
0.0164
0.0164
0.0331
4.0
0.0321
0.0067
0.0067
0.0137
0.0626
0.0134
0.0134
0.0270Figure 4
The influence of ratioa/b on the nondimensional deflection of the plate with kst/Em=1: (a) hn/hm=1; (b) hn/hm=2.
(a)
(b)(2) Influence of the Thickness-to-Thickness Ratio hn/hm. Next, the research on the variation of deflection affected by the relationship between the bottom layer thickness and the top layer thickness is now investigated. The ratio hn/hm increases from 1 to 4, kst/Em=1, and the length-to-width ratio a/b is set to be 1 and 2. The nondimensional deflection of the plate with different boundary conditions is shown in Table 4 and Figure 5. The deflection of the plate increases when increasing ratio hn/hm; this can be explained by the fact that Young’s modulus of the layer n is less than that of layer m, and the stiffness of the two-layer plate becomes smaller when the ratio hn/hm increases.Table 4
Nondimensional deflection of the plate as a function of ratiohn/hm with kst/Eb=1.
h
n
/
h
m
a
/
b
=
1
a
/
b
=
2
SSSS
CCCC
CSCS
CCSS
SSSS
CCCC
CSCS
CCSS
1.0
2.5968
0.8135
1.2304
1.4124
0.4050
0.1020
0.1051
0.1958
1.2
3.1576
0.9907
1.4977
1.7187
0.4926
0.1243
0.1281
0.2384
1.4
3.6969
1.1617
1.7556
2.0138
0.5769
0.1460
0.1503
0.2795
1.6
4.1941
1.3201
1.9940
2.2864
0.6546
0.1661
0.1710
0.3175
1.8
4.6367
1.4618
2.2070
2.5297
0.7239
0.1841
0.1896
0.3516
2.0
5.0196
1.5849
2.3919
2.7406
0.7839
0.1999
0.2058
0.3812
2.2
5.3436
1.6895
2.5488
2.9194
0.8348
0.2133
0.2196
0.4063
2.4
5.6129
1.7769
2.6798
3.0684
0.8771
0.2246
0.2312
0.4273
2.6
5.8338
1.8489
2.7875
3.1909
0.9118
0.2340
0.2408
0.4447
2.8
6.0131
1.9077
2.8752
3.2906
0.9400
0.2416
0.2487
0.4588
3.0
6.1575
1.9552
2.9461
3.3710
0.9628
0.2478
0.2551
0.4702
3.2
6.2731
1.9934
3.0030
3.4355
0.9810
0.2529
0.2603
0.4795
3.4
6.3651
2.0239
3.0485
3.4870
0.9956
0.2569
0.2644
0.4868
3.6
6.4379
2.0482
3.0847
3.5278
1.0071
0.2602
0.2678
0.4927
3.8
6.4954
2.0676
3.1133
3.5602
1.0162
0.2628
0.2704
0.4974
4.0
6.5405
2.0828
3.1359
3.5856
1.0234
0.2649
0.2726
0.5011(3) Influence of the Stiffness of the Connector Stub. Table 5 and Figure 6 show the nondimensional deflections as a function of the ratio kst/Em of a square two-layer composite plate with hn/hm=1 and hn/hm=2. The stiffness of the stub varies from 0 to 106Em; when it changes from 0 to Em, this plate is stiffer and the deflection decreases; but when the stiffness of the stub > Em, then the effect of the stub goes to zero so that the two-layer plate connected by the stub becomes a two-layer one with no stub and the displacements at the contact place are the same; it means that there is no slip between two layers at the contact place.Table 5
Nondimensional deflection of the plate as a function of ratiokst/Em with a/b=1.
k
st
/
E
m
h
n
/
h
m
=
1
h
n
/
h
m
=
2
SSSS
CCCC
CSCS
CCSS
SSSS
CCCC
CSCS
CCSS
0
3.1903
0.9951
1.5068
1.7316
9.4673
2.9526
4.4712
5.1381
10–2
2.6790
0.8708
1.3056
1.4862
5.5538
1.9698
2.8925
3.2279
10–1
2.6051
0.8214
1.2402
1.4215
5.0730
1.6363
2.4560
2.7995
100
2.5968
0.8135
1.2304
1.4124
5.0196
1.5849
2.3919
2.7406
101
2.5959
0.8126
1.2293
1.4114
5.0142
1.5792
2.3850
2.7343
102
2.5958
0.8125
1.2292
1.4113
5.0137
1.5787
2.3843
2.7336
103
2.5958
0.8125
1.2292
1.4113
5.0136
1.5786
2.3842
2.7336
104
2.5958
0.8125
1.2292
1.4113
5.0136
1.5786
2.3842
2.7336
105
2.5958
0.8125
1.2292
1.4113
5.0136
1.5786
2.3842
2.7336
106
2.5958
0.8125
1.2292
1.4113
5.0136
1.5786
2.3842
2.7336Figure 5
The effect of ratiohn/hm on the nondimensional transverse deflection of the plate with kst/Em=1: (a) a/b=1; (b) a/b=2.
(a)
(b)Figure 6
The effect of ratiokst/Em on nondimensional deflection of the plate with a/b=1: (a) hn/hm=1; (b) hn/hm=2.
(a)
(b)To show clearly the slip between the top and bottom layers at the contact place, we depicted the ratioRmn1 and Rmn2 in Figures 7 and 8, respectively, in which Rmn1 and Rmn2 are defined as(64)Rmn1=uma4,b2,hm2−una4,b2,−hn21000Emh312qa41−νm2,Rmn2=umx,b2,hm2−unx,b2,−hn21000Emh312qa41−νm2.Figure 7
The effect of ratiokst/Em on Rmn1 of the plate with hn/hm=1.Figure 8
The slip between the top and bottom layers at the contact place varying along thex-direction, with kst/Em = 0.1 and hn/hm=1.We can see again in Figure7 that when the stiffness of the stub increases, the slip between two layers decreases and tends to zero. By looking at Figure 8 (the ratio kst/Em = 0.1 is employed), it is very interesting to see that the minimum of slip between two layers at the contact place appears at the central plate with symmetric boundary conditions (CCCC, SSSS, and CSCS). The slip gets a maximum value at the simply supported edge, and this slip strongly depends on the boundary condition.(4) Influence of Boundary Condition. We further study the deflection affected by the boundary condition. A square plate with hn/hm = 1-2 and kst/Em = 0-1 is considered. Figure 9 shows the displacement at x = a/2 and along the y direction in four boundary conditions. The boundary condition has strong effects on the deflection magnitude and deformation shape of the plate. It is shown that the plate with the simple support at all edges has the highest deflection and the one with the fully clamped support has the smallest deflection. The deformation shapes of the SSSS plate, CCCC plate, and CSCS plate are symmetric, while the deformation of the CCSS plate is asymmetric.Figure 9
Nondimensional deflection of the plate along the line partx=a/2 with a/b=1 and (a) hn/hm=1, kst/Em=0, (b) hn/hm=1, kst/Em=1, (c) hn/hm=2, kst/Em=0, and (d) hn/hm=2, kst/Em=1.
(a)
(b)
(c)
(d)


## 3.1.1. Accuracy Studies

To verify the proposed method, we compare the deflections in two cases of materials: homogeneous and composite materials. Firstly, a square homogeneous plate under contribution loadq=−1N is investigated, and the length and the width are a=b=1m. The plate is simply supported at all edges (SSSS) or fully clamped (CCCC). In the published works [27, 28], the plate has one layer made of homogeneous material with E=10920MPa, Poisson’s ratio ν=0.3, and the thickness h. But in this work, we assume that the plate has two layers connected by stubs with two cases—case 1: hn = 0.999 h, hm = 0.001 h, and kst = 0 and case 2: hn = hm = 0.5 h and kst = 1010E. In both cases, material properties of two layers are set to be En=Em=E=10920MPaandνn=νm=ν=0.3, thicknesses of the top and bottom layer are hn and hm, respectively, and hn + hm = h. The nondimensional deflection is defined as(61)w¯=wmaxDqa4,where D is given in equation (2).Table1 shows a comparison of the nondimensional deflections of SSSS and CCCC plates obtained by this method and other approaches (FEM [28] and analytical solution [28]). We meet a good agreement among three solutions.Table 1
Comparison of nondimensional deflection of square plates witha/h = 10 and a/h = 10000.
Source
SSSS
CCCC
a
/
h
=
10
a
/
h
=
10000
a
/
h
=
10
a
/
h
=
10000
Ferreira [28]
0.004271
0.004060
0.001503
0.001264
Exact solution [28]
0.004270
0.004060
—
0.001260
Present (hn=0.999h,hm=0.001h,andkst=0)
0.004290
0.004079
0.001508
0.001272
Present (hn=hm=0.5handkst=1010E)
0.004300
0.004079
0.001523
0.001291We see that when the thickness of one layer is much higher than that of the other layer and the stiffness of the connector stub is very small, the behavior of the two-layer plate is the same as the behavior of the one-layer plate with the same thickness. In addition, when two layers have the same thickness and material, the stiffness of the connector stub gets a much higher valuekst = 1010E, and then the two-layer plate is considered as the one-layer plate with the thickness equal to the sum of thickness of the two components.In order to further confirm the accuracy of this method, we compare the nondimensional deflection of a square SSSS composite plate (a/b = 1 and the total thickness h = a/10) with two layers (0/90°) which have the same thickness h/2;the  material  properties  areE1/E2=25,G12=G13=0.5E2,G23=0.2E2,andν12=0.25. In this comparison, the plate is modeled as hn = hm = 0.5 h, kst = 1010E2, and the fiber directions of the top and bottom layers are 0° and 90°, respectively. Table 2 shows the comparison of the nondimensional deflection of this composite plate subjected to uniformly distributed load and central concentrated load. The nondimensional deflection of the plate is calculated by the following formula:(62)w¯=wmax100E2h3a4q.Table 2
Comparison of the nondimensional deflection of a square SSSS composite plate subject to uniformly distributed load and central concentrated load.
w
¯
Uniformly distributed load
Central concentrated load
Reddy [29]
Present
Reddy [29]
Present
1.6955
1.6538
4.6664
4.6421It is shown plainly that the present results have a good agreement with the results given in [29] (analytical method). According to the above two comparisons, the present results have a good agreement with other published ones.


## 3.1.2. Numerical Results for Static Bending Analysis of Two-Layer Composite Plates

In this section, a rectangular two-layer composite plate subjected to the uniform loadq is investigated. The boundary conditions of the plate are SSSS (fully simply supported), CCCC (fully clamped), CSCS (two opposite edges are simply supported and two other edges are clamped), and CCSS (two adjacent edges are clamped and two other edges are simply supported). The sides, thickness, and material properties of the plate are set to be a=1m, h=a/100, Em=200GPa, En=10GPa, a/b=open, hn/hm=open, and kst/Em=open. The nondimensional displacement of this plate is normalized by(63)w¯=w100Emh312qa41−νm2.(1) Influence of the Length-to-Width Ratio a/b. In order for studying the effect of the aspect ratio a/b on the mechanical bending behavior of the two-layer composite plate, this ratio is set to vary from 0.5 to 4, while hn/hm=1, hn/hm=2, and the ratio of stiffness of the connector stub kst/Em=1 are considered. The computed results are shown in Table 3 and Figures 4(a) and 4(b). According to Table 3 and Figure 4, it is seen that when the aspect ratio a/b increases, the nondimensional deflection decreases. In addition, when the ratio a/b>3, the nondimensional deflections of both boundary conditions CCCC and CSCS of plates are very close to each other.Table 3
Nondimensional deflection of the plate as a function of ratioa/b with kst/Em=1.
a
/
b
h
n
/
h
m
=
1
h
n
/
h
m
=
2
SSSS
CCCC
CSCS
CCSS
SSSS
CCCC
CSCS
CCSS
0.5
6.4728
1.6252
5.4088
3.1243
12.5027
3.1584
10.4767
6.0528
0.625
5.3098
1.4770
3.8595
2.7089
10.2589
2.8726
7.4844
5.2510
0.75
4.2368
1.2639
2.6511
2.2449
8.1874
2.4598
5.1458
4.3531
0.875
3.3289
1.0313
1.8013
1.7999
6.4339
2.0083
3.4991
3.4914
1.0
2.5968
0.8135
1.2304
1.4124
5.0196
1.5849
2.3919
2.7406
1.2
1.7421
0.5350
0.6889
0.9375
3.3683
1.0431
1.3408
1.8201
1.4
1.1793
0.3464
0.4036
0.6196
2.2806
0.6760
0.7866
1.2035
1.6
0.8107
0.2259
0.2475
0.4139
1.5683
0.4413
0.4830
0.8046
1.8
0.5676
0.1501
0.1582
0.2818
1.0984
0.2936
0.3094
0.5482
2.0
0.4050
0.1020
0.1051
0.1958
0.7839
0.1999
0.2058
0.3812
2.2
0.2943
0.0711
0.0721
0.1389
0.5699
0.1395
0.1416
0.2707
2.4
0.2177
0.0507
0.0510
0.1006
0.4217
0.0997
0.1003
0.1962
2.6
0.1637
0.0370
0.0371
0.0742
0.3172
0.0729
0.0730
0.1450
2.8
0.1249
0.0276
0.0275
0.0558
0.2423
0.0544
0.0544
0.1091
3.0
0.0967
0.0209
0.0209
0.0426
0.1877
0.0415
0.0414
0.0835
3.2
0.0759
0.0162
0.0162
0.0331
0.1473
0.0321
0.0322
0.0650
3.4
0.0602
0.0127
0.0127
0.0261
0.1170
0.0253
0.0254
0.0512
3.6
0.0484
0.0101
0.0101
0.0208
0.0940
0.0202
0.0203
0.0409
3.8
0.0392
0.0082
0.0082
0.0168
0.0763
0.0164
0.0164
0.0331
4.0
0.0321
0.0067
0.0067
0.0137
0.0626
0.0134
0.0134
0.0270Figure 4
The influence of ratioa/b on the nondimensional deflection of the plate with kst/Em=1: (a) hn/hm=1; (b) hn/hm=2.
(a)
(b)(2) Influence of the Thickness-to-Thickness Ratio hn/hm. Next, the research on the variation of deflection affected by the relationship between the bottom layer thickness and the top layer thickness is now investigated. The ratio hn/hm increases from 1 to 4, kst/Em=1, and the length-to-width ratio a/b is set to be 1 and 2. The nondimensional deflection of the plate with different boundary conditions is shown in Table 4 and Figure 5. The deflection of the plate increases when increasing ratio hn/hm; this can be explained by the fact that Young’s modulus of the layer n is less than that of layer m, and the stiffness of the two-layer plate becomes smaller when the ratio hn/hm increases.Table 4
Nondimensional deflection of the plate as a function of ratiohn/hm with kst/Eb=1.
h
n
/
h
m
a
/
b
=
1
a
/
b
=
2
SSSS
CCCC
CSCS
CCSS
SSSS
CCCC
CSCS
CCSS
1.0
2.5968
0.8135
1.2304
1.4124
0.4050
0.1020
0.1051
0.1958
1.2
3.1576
0.9907
1.4977
1.7187
0.4926
0.1243
0.1281
0.2384
1.4
3.6969
1.1617
1.7556
2.0138
0.5769
0.1460
0.1503
0.2795
1.6
4.1941
1.3201
1.9940
2.2864
0.6546
0.1661
0.1710
0.3175
1.8
4.6367
1.4618
2.2070
2.5297
0.7239
0.1841
0.1896
0.3516
2.0
5.0196
1.5849
2.3919
2.7406
0.7839
0.1999
0.2058
0.3812
2.2
5.3436
1.6895
2.5488
2.9194
0.8348
0.2133
0.2196
0.4063
2.4
5.6129
1.7769
2.6798
3.0684
0.8771
0.2246
0.2312
0.4273
2.6
5.8338
1.8489
2.7875
3.1909
0.9118
0.2340
0.2408
0.4447
2.8
6.0131
1.9077
2.8752
3.2906
0.9400
0.2416
0.2487
0.4588
3.0
6.1575
1.9552
2.9461
3.3710
0.9628
0.2478
0.2551
0.4702
3.2
6.2731
1.9934
3.0030
3.4355
0.9810
0.2529
0.2603
0.4795
3.4
6.3651
2.0239
3.0485
3.4870
0.9956
0.2569
0.2644
0.4868
3.6
6.4379
2.0482
3.0847
3.5278
1.0071
0.2602
0.2678
0.4927
3.8
6.4954
2.0676
3.1133
3.5602
1.0162
0.2628
0.2704
0.4974
4.0
6.5405
2.0828
3.1359
3.5856
1.0234
0.2649
0.2726
0.5011(3) Influence of the Stiffness of the Connector Stub. Table 5 and Figure 6 show the nondimensional deflections as a function of the ratio kst/Em of a square two-layer composite plate with hn/hm=1 and hn/hm=2. The stiffness of the stub varies from 0 to 106Em; when it changes from 0 to Em, this plate is stiffer and the deflection decreases; but when the stiffness of the stub > Em, then the effect of the stub goes to zero so that the two-layer plate connected by the stub becomes a two-layer one with no stub and the displacements at the contact place are the same; it means that there is no slip between two layers at the contact place.Table 5
Nondimensional deflection of the plate as a function of ratiokst/Em with a/b=1.
k
st
/
E
m
h
n
/
h
m
=
1
h
n
/
h
m
=
2
SSSS
CCCC
CSCS
CCSS
SSSS
CCCC
CSCS
CCSS
0
3.1903
0.9951
1.5068
1.7316
9.4673
2.9526
4.4712
5.1381
10–2
2.6790
0.8708
1.3056
1.4862
5.5538
1.9698
2.8925
3.2279
10–1
2.6051
0.8214
1.2402
1.4215
5.0730
1.6363
2.4560
2.7995
100
2.5968
0.8135
1.2304
1.4124
5.0196
1.5849
2.3919
2.7406
101
2.5959
0.8126
1.2293
1.4114
5.0142
1.5792
2.3850
2.7343
102
2.5958
0.8125
1.2292
1.4113
5.0137
1.5787
2.3843
2.7336
103
2.5958
0.8125
1.2292
1.4113
5.0136
1.5786
2.3842
2.7336
104
2.5958
0.8125
1.2292
1.4113
5.0136
1.5786
2.3842
2.7336
105
2.5958
0.8125
1.2292
1.4113
5.0136
1.5786
2.3842
2.7336
106
2.5958
0.8125
1.2292
1.4113
5.0136
1.5786
2.3842
2.7336Figure 5
The effect of ratiohn/hm on the nondimensional transverse deflection of the plate with kst/Em=1: (a) a/b=1; (b) a/b=2.
(a)
(b)Figure 6
The effect of ratiokst/Em on nondimensional deflection of the plate with a/b=1: (a) hn/hm=1; (b) hn/hm=2.
(a)
(b)To show clearly the slip between the top and bottom layers at the contact place, we depicted the ratioRmn1 and Rmn2 in Figures 7 and 8, respectively, in which Rmn1 and Rmn2 are defined as(64)Rmn1=uma4,b2,hm2−una4,b2,−hn21000Emh312qa41−νm2,Rmn2=umx,b2,hm2−unx,b2,−hn21000Emh312qa41−νm2.Figure 7
The effect of ratiokst/Em on Rmn1 of the plate with hn/hm=1.Figure 8
The slip between the top and bottom layers at the contact place varying along thex-direction, with kst/Em = 0.1 and hn/hm=1.We can see again in Figure7 that when the stiffness of the stub increases, the slip between two layers decreases and tends to zero. By looking at Figure 8 (the ratio kst/Em = 0.1 is employed), it is very interesting to see that the minimum of slip between two layers at the contact place appears at the central plate with symmetric boundary conditions (CCCC, SSSS, and CSCS). The slip gets a maximum value at the simply supported edge, and this slip strongly depends on the boundary condition.(4) Influence of Boundary Condition. We further study the deflection affected by the boundary condition. A square plate with hn/hm = 1-2 and kst/Em = 0-1 is considered. Figure 9 shows the displacement at x = a/2 and along the y direction in four boundary conditions. The boundary condition has strong effects on the deflection magnitude and deformation shape of the plate. It is shown that the plate with the simple support at all edges has the highest deflection and the one with the fully clamped support has the smallest deflection. The deformation shapes of the SSSS plate, CCCC plate, and CSCS plate are symmetric, while the deformation of the CCSS plate is asymmetric.Figure 9
Nondimensional deflection of the plate along the line partx=a/2 with a/b=1 and (a) hn/hm=1, kst/Em=0, (b) hn/hm=1, kst/Em=1, (c) hn/hm=2, kst/Em=0, and (d) hn/hm=2, kst/Em=1.
(a)
(b)
(c)
(d)


## 3.2. Free Vibration Analysis

### 3.2.1. Accuracy Studies

To confirm the accuracy of this work, we first present a comparison of frequencies of homogeneous CCCC and SSSS plates, witha = b = 1 m, h = a/10 and a/100, E = 10920 MPa, and Poisson’s ratio ν=0.3; similar to the bending analysis, hn = hm = 0.5 h and kst = 1010E are again used for this comparison. The nondimensional frequencies are defined by(65)ω¯i=ωiaρG.The results are listed in Tables6 and 7, in which Ferreira theory [28] was based on FEM and Mindlin theory [28] was based on the analytical solution. As expected, the results obtained by this work match well with those derived from other solutions.Table 6
Comparison of the nondimensional frequencies of an SSSS plate.
a
/
h
Mode
Ferreira [28]
Mindlin [28]
Present
10
1
0.9346
0.930
0.9271
2
2.2545
2.219
2.2055
3
2.2545
2.219
2.2055
4
3.4592
3.406
3.3600
5
4.3031
4.149
4.1123
6
4.3031
4.149
4.1123
100
1
0.0965
0.0963
0.0952
2
0.2430
0.2406
0.2343
3
0.2430
0.2406
0.2343
4
0.3890
0.3847
0.3680
5
0.4928
0.4807
0.4580
6
0.4928
0.4807
0.4580Table 7
Comparison of the nondimensional frequencies of a CCCC plate.
a
/
h
Mode
Ferreira [28]
Liew et al. [27]
Present
10
1
1.5955
1.5582
1.5710
2
3.0662
3.0182
2.9927
3
3.0662
3.0182
2.9927
4
4.2924
4.1711
4.1829
5
5.1232
5.1218
4.9441
6
5.1730
5.1594
4.9893
100
1
0.1750
0.1743
0.1683
2
0.3635
0.3576
0.3431
3
0.3635
0.3576
0.3432
4
0.5358
0.5240
0.4978
5
0.6634
0.6465
0.6014
6
0.6665
0.6505
0.6057Next, we compare the nondimensional frequencies of an SSSS composite plate 0/90°. The parameters of this plate are set to beE1/E2=25, G12=G13=0.5E2, G23=0.2E2, ν12=0.25, a/b=1, and a/h=10. The nondimensional frequencies of the plate are normalized by(66)ω˜i=ωib2π2ρhD22,i=1−6.The comparison is shown in Table8; note that the results in [29] are obtained from the exact solution. We can see that the present results and reference results are close to each other. From two comparisons, we conclude that the accuracy of this approach is accepted.Table 8
Comparison of the nondimensional frequenciesω˜i of a square SSSS composite plate.
Mode
E
1
/
E
2
=
10
E
1
/
E
2
=
20
Reddy [29]
Present
Reddy [29]
Present
1
1.183
1.151
0.990
1.032
2
3.174
3.209
2.719
2.966
3
3.174
3.209
2.719
2.966
4
4.733
5.057
3.959
4.101
5
6.666
6.399
5.789
5.806
6
6.666
6.399
5.789
5.806


### 3.2.2. Numerical Results for Free Vibration Analysis of Two-Layer Composite Plates

In this section, we research the natural frequency of the two-layer composite plate utilizing the proposed method. The two-layer composite plate (lengtha, width b, thickness of the top layer hn, and thickness of the bottom layer hm) is studied herein. The nondimensional fundamental frequency is defined as(67)ω⌣=ω1⋅100⋅a21+νmρEm.(1) Influence of Length-to-Width Ratio. Table 9 and Figure 10 show the dimensionless fundamental frequencies as a function of aspect ratio a/b with hn/hm=1 and kst/Em=1 of two-layer composite plates altered by the boundary condition. It is evident that the highest and the smallest natural frequencies are found for CCCC and SSSS plates, respectively. When the aspect ratio a/b increases, the nondimensional fundamental frequency increases, and the fundamental frequencies of CCCC and CSCS plates are close when the aspect ratio a/b > 2.6; this is a similar behavior to the static bending problem.Table 9
Influence of ratioa/b on nondimensional fundamental frequency of the plate with hn/hm=1 and kst/Em=1.
a
/
b
Boundary
SSSS
CCCC
CSCS
CCSS
0.5
0.8439
1.6701
0.9329
1.2115
1.0
1.3505
2.4421
1.9700
1.8434
1.2
1.6474
3.0049
2.6189
2.2592
1.4
1.9983
3.7145
3.3955
2.7687
1.6
2.4029
4.5647
4.2971
3.3698
1.8
2.8614
5.5501
5.3220
4.0606
2.0
3.3737
6.6663
6.4691
4.8395
2.2
3.9398
7.9101
7.7374
5.7053
2.4
4.5595
9.2791
9.1262
6.6570
2.6
5.2330
10.7714
10.6349
7.6938
2.8
5.9601
12.3857
12.2627
8.8151
3.0
6.7408
14.1208
14.0093
10.0203
3.2
7.5751
15.9757
15.8739
11.3092
3.4
8.4628
17.9494
17.8561
12.6811
3.6
9.4040
20.0412
19.9553
14.1359
3.8
10.3985
22.2503
22.1709
15.6730
4.0
11.4463
24.5760
24.5023
17.2922Figure 10
The effect of ratioa/b on nondimensional fundamental frequencies of the plate with hn/hm=1 and kst/Em=1.(2) Influence of Thickness-to-Thickness Ratiohn/hm. To study the influence of thickness-to-thickness ratio hn/hm, a two-layer square composite plate with kst/Em=1 is used for this analysis. The nondimensional fundamental frequencies as a function of the ratio hn/hm are tabulated in Table 10 and shown in Figures 11 and 12. The nondimensional fundamental frequency varies nonlinearly with the increasing ratio hn/hm; the minimum value of fundamental frequency will be obtained when the ratio hn/hm is in the range of 1.8 to 2.6.Table 10
Influence of ratiohn/hm on nondimensional fundamental frequency of the square plate with a/b=1 and kst/Em=1.
h
n
/
h
m
Boundary
SSSS
CCCC
CSCS
CCSS
1.0
1.3505
2.4421
1.9700
1.8434
1.2
1.2757
2.3022
1.8582
1.7396
1.4
1.2231
2.2024
1.7789
1.6661
1.6
1.1872
2.1332
1.7241
1.6156
1.8
1.1641
2.0873
1.6880
1.5825
2.0
1.1506
2.0591
1.6662
1.5627
2.2
1.1445
2.0445
1.6552
1.5530
2.4
1.1439
2.0403
1.6526
1.5511
2.6
1.1476
2.0439
1.6562
1.5550
2.8
1.1544
2.0535
1.6646
1.5633
3.0
1.1635
2.0676
1.6765
1.5749
3.2
1.1744
2.0850
1.6911
1.5889
3.4
1.1866
2.1049
1.7077
1.6048
3.6
1.1997
2.1266
1.7256
1.6219
3.8
1.2133
2.1496
1.7446
1.6399
4.0
1.2274
2.1734
1.7642
1.6585Figure 11
The effect of ratiohn/hm on nondimensional fundamental frequencies of the plate with a/b = 1 and kst/Em=1.Figure 12
Nondimensional fundamental frequencies as a function ofhn/hm and a/b of the SSSS plate with kst/Em=1.(3) Influence of Stiffness of the Connector Stub. In this exploration, the effect of stiffness of the connector stub on the nondimensional fundamental frequency of the two-layer composite plate is studied. A square two-layer plate with a/b=1 and kst/Em varying from 0 to 106 is investigated. Table 11 and Figure 13 report the computed results of this plate with hn/hm = 1; we can see that increasing the ratio kst/Em from 0 to 1 leads to the increase in the frequencies. When the ratio kst/Em>1, the nondimensional fundamental frequency is almost unchanged.Table 11
Influence of ratiokst/Em on nondimensional fundamental frequency of the plate with a/b=1 and hn/hm=1.
k
st
/
E
m
Boundary
SSSS
CCCC
CSCS
CCSS
0
1.2206
2.2205
1.7881
1.6710
10–2
1.3292
2.3636
1.9142
1.7976
10–1
1.3482
2.4308
1.9625
1.8376
100
1.3505
2.4421
1.9700
1.8434
101
1.3507
2.4433
1.9708
1.8441
102
1.3508
2.4435
1.9709
1.8441
103
1.3508
2.4435
1.9709
1.8441
104
1.3508
2.4435
1.9709
1.8441
105
1.3508
2.4435
1.9709
1.8441
106
1.3506
2.4434
1.9710
1.8439Figure 13
The effect of ratiokst/Em on nondimensional fundamental frequencies of the plate with a/b = 1 and hn/hm=1.Figures14 and 15 plot nondimensional fundamental frequencies of the plate as a function of ratios a/b, hn/hm, and kst/Em. In addition, the first six mode shapes of plates with boundary conditions such as SSSS, CCCC, CSCS, and CCSS are shown in Figures 16–19 with a/b = 1, hn/hm = 1, and kst/Em = 10−1. It is seen that the boundary condition has an effect on mode shapes as well as on the value of frequencies.Figure 14
Nondimensional fundamental frequencies as a function ofkst/Em and hn/hm of the SSSS plate with a/b=1.Figure 15
Nondimensional fundamental frequencies as a function ofkst/Em and a/b of the SSSS plate with hn/hm=1.Figure 16
The first six mode shapes of a two-layer square SSSS composite plate. (a) Mode 1. (b) Mode 2. (c) Mode 3. (d) Mode 4. (e) Mode 5. (f) Mode 6.
(a)
(b)
(c)
(d)
(e)
(f)Figure 17
The first six mode shapes of a two-layer square CCCC composite plate. (a) Mode 1. (b) Mode 2. (c) Mode 3. (d) Mode 4. (e) Mode 5. (f) Mode 6.
(a)
(b)
(c)
(d)
(e)
(f)Figure 18
The first six mode shapes of a two-layer square CSCS composite plate. (a) Mode 1. (b) Mode 2. (c) Mode 3. (d) Mode 4. (e) Mode 5. (f) Mode 6.
(a)
(b)
(c)
(d)
(e)
(f)Figure 19
The first six mode shapes of a two-layer square CCSS composite plate. (a) Mode 1. (b) Mode 2. (c) Mode 3. (d) Mode 4. (e) Mode 5. (f) Mode 6.
(a)
(b)
(c)
(d)
(e)
(f)


## 3.2.1. Accuracy Studies

To confirm the accuracy of this work, we first present a comparison of frequencies of homogeneous CCCC and SSSS plates, witha = b = 1 m, h = a/10 and a/100, E = 10920 MPa, and Poisson’s ratio ν=0.3; similar to the bending analysis, hn = hm = 0.5 h and kst = 1010E are again used for this comparison. The nondimensional frequencies are defined by(65)ω¯i=ωiaρG.The results are listed in Tables6 and 7, in which Ferreira theory [28] was based on FEM and Mindlin theory [28] was based on the analytical solution. As expected, the results obtained by this work match well with those derived from other solutions.Table 6
Comparison of the nondimensional frequencies of an SSSS plate.
a
/
h
Mode
Ferreira [28]
Mindlin [28]
Present
10
1
0.9346
0.930
0.9271
2
2.2545
2.219
2.2055
3
2.2545
2.219
2.2055
4
3.4592
3.406
3.3600
5
4.3031
4.149
4.1123
6
4.3031
4.149
4.1123
100
1
0.0965
0.0963
0.0952
2
0.2430
0.2406
0.2343
3
0.2430
0.2406
0.2343
4
0.3890
0.3847
0.3680
5
0.4928
0.4807
0.4580
6
0.4928
0.4807
0.4580Table 7
Comparison of the nondimensional frequencies of a CCCC plate.
a
/
h
Mode
Ferreira [28]
Liew et al. [27]
Present
10
1
1.5955
1.5582
1.5710
2
3.0662
3.0182
2.9927
3
3.0662
3.0182
2.9927
4
4.2924
4.1711
4.1829
5
5.1232
5.1218
4.9441
6
5.1730
5.1594
4.9893
100
1
0.1750
0.1743
0.1683
2
0.3635
0.3576
0.3431
3
0.3635
0.3576
0.3432
4
0.5358
0.5240
0.4978
5
0.6634
0.6465
0.6014
6
0.6665
0.6505
0.6057Next, we compare the nondimensional frequencies of an SSSS composite plate 0/90°. The parameters of this plate are set to beE1/E2=25, G12=G13=0.5E2, G23=0.2E2, ν12=0.25, a/b=1, and a/h=10. The nondimensional frequencies of the plate are normalized by(66)ω˜i=ωib2π2ρhD22,i=1−6.The comparison is shown in Table8; note that the results in [29] are obtained from the exact solution. We can see that the present results and reference results are close to each other. From two comparisons, we conclude that the accuracy of this approach is accepted.Table 8
Comparison of the nondimensional frequenciesω˜i of a square SSSS composite plate.
Mode
E
1
/
E
2
=
10
E
1
/
E
2
=
20
Reddy [29]
Present
Reddy [29]
Present
1
1.183
1.151
0.990
1.032
2
3.174
3.209
2.719
2.966
3
3.174
3.209
2.719
2.966
4
4.733
5.057
3.959
4.101
5
6.666
6.399
5.789
5.806
6
6.666
6.399
5.789
5.806


## 3.2.2. Numerical Results for Free Vibration Analysis of Two-Layer Composite Plates

In this section, we research the natural frequency of the two-layer composite plate utilizing the proposed method. The two-layer composite plate (lengtha, width b, thickness of the top layer hn, and thickness of the bottom layer hm) is studied herein. The nondimensional fundamental frequency is defined as(67)ω⌣=ω1⋅100⋅a21+νmρEm.(1) Influence of Length-to-Width Ratio. Table 9 and Figure 10 show the dimensionless fundamental frequencies as a function of aspect ratio a/b with hn/hm=1 and kst/Em=1 of two-layer composite plates altered by the boundary condition. It is evident that the highest and the smallest natural frequencies are found for CCCC and SSSS plates, respectively. When the aspect ratio a/b increases, the nondimensional fundamental frequency increases, and the fundamental frequencies of CCCC and CSCS plates are close when the aspect ratio a/b > 2.6; this is a similar behavior to the static bending problem.Table 9
Influence of ratioa/b on nondimensional fundamental frequency of the plate with hn/hm=1 and kst/Em=1.
a
/
b
Boundary
SSSS
CCCC
CSCS
CCSS
0.5
0.8439
1.6701
0.9329
1.2115
1.0
1.3505
2.4421
1.9700
1.8434
1.2
1.6474
3.0049
2.6189
2.2592
1.4
1.9983
3.7145
3.3955
2.7687
1.6
2.4029
4.5647
4.2971
3.3698
1.8
2.8614
5.5501
5.3220
4.0606
2.0
3.3737
6.6663
6.4691
4.8395
2.2
3.9398
7.9101
7.7374
5.7053
2.4
4.5595
9.2791
9.1262
6.6570
2.6
5.2330
10.7714
10.6349
7.6938
2.8
5.9601
12.3857
12.2627
8.8151
3.0
6.7408
14.1208
14.0093
10.0203
3.2
7.5751
15.9757
15.8739
11.3092
3.4
8.4628
17.9494
17.8561
12.6811
3.6
9.4040
20.0412
19.9553
14.1359
3.8
10.3985
22.2503
22.1709
15.6730
4.0
11.4463
24.5760
24.5023
17.2922Figure 10
The effect of ratioa/b on nondimensional fundamental frequencies of the plate with hn/hm=1 and kst/Em=1.(2) Influence of Thickness-to-Thickness Ratiohn/hm. To study the influence of thickness-to-thickness ratio hn/hm, a two-layer square composite plate with kst/Em=1 is used for this analysis. The nondimensional fundamental frequencies as a function of the ratio hn/hm are tabulated in Table 10 and shown in Figures 11 and 12. The nondimensional fundamental frequency varies nonlinearly with the increasing ratio hn/hm; the minimum value of fundamental frequency will be obtained when the ratio hn/hm is in the range of 1.8 to 2.6.Table 10
Influence of ratiohn/hm on nondimensional fundamental frequency of the square plate with a/b=1 and kst/Em=1.
h
n
/
h
m
Boundary
SSSS
CCCC
CSCS
CCSS
1.0
1.3505
2.4421
1.9700
1.8434
1.2
1.2757
2.3022
1.8582
1.7396
1.4
1.2231
2.2024
1.7789
1.6661
1.6
1.1872
2.1332
1.7241
1.6156
1.8
1.1641
2.0873
1.6880
1.5825
2.0
1.1506
2.0591
1.6662
1.5627
2.2
1.1445
2.0445
1.6552
1.5530
2.4
1.1439
2.0403
1.6526
1.5511
2.6
1.1476
2.0439
1.6562
1.5550
2.8
1.1544
2.0535
1.6646
1.5633
3.0
1.1635
2.0676
1.6765
1.5749
3.2
1.1744
2.0850
1.6911
1.5889
3.4
1.1866
2.1049
1.7077
1.6048
3.6
1.1997
2.1266
1.7256
1.6219
3.8
1.2133
2.1496
1.7446
1.6399
4.0
1.2274
2.1734
1.7642
1.6585Figure 11
The effect of ratiohn/hm on nondimensional fundamental frequencies of the plate with a/b = 1 and kst/Em=1.Figure 12
Nondimensional fundamental frequencies as a function ofhn/hm and a/b of the SSSS plate with kst/Em=1.(3) Influence of Stiffness of the Connector Stub. In this exploration, the effect of stiffness of the connector stub on the nondimensional fundamental frequency of the two-layer composite plate is studied. A square two-layer plate with a/b=1 and kst/Em varying from 0 to 106 is investigated. Table 11 and Figure 13 report the computed results of this plate with hn/hm = 1; we can see that increasing the ratio kst/Em from 0 to 1 leads to the increase in the frequencies. When the ratio kst/Em>1, the nondimensional fundamental frequency is almost unchanged.Table 11
Influence of ratiokst/Em on nondimensional fundamental frequency of the plate with a/b=1 and hn/hm=1.
k
st
/
E
m
Boundary
SSSS
CCCC
CSCS
CCSS
0
1.2206
2.2205
1.7881
1.6710
10–2
1.3292
2.3636
1.9142
1.7976
10–1
1.3482
2.4308
1.9625
1.8376
100
1.3505
2.4421
1.9700
1.8434
101
1.3507
2.4433
1.9708
1.8441
102
1.3508
2.4435
1.9709
1.8441
103
1.3508
2.4435
1.9709
1.8441
104
1.3508
2.4435
1.9709
1.8441
105
1.3508
2.4435
1.9709
1.8441
106
1.3506
2.4434
1.9710
1.8439Figure 13
The effect of ratiokst/Em on nondimensional fundamental frequencies of the plate with a/b = 1 and hn/hm=1.Figures14 and 15 plot nondimensional fundamental frequencies of the plate as a function of ratios a/b, hn/hm, and kst/Em. In addition, the first six mode shapes of plates with boundary conditions such as SSSS, CCCC, CSCS, and CCSS are shown in Figures 16–19 with a/b = 1, hn/hm = 1, and kst/Em = 10−1. It is seen that the boundary condition has an effect on mode shapes as well as on the value of frequencies.Figure 14
Nondimensional fundamental frequencies as a function ofkst/Em and hn/hm of the SSSS plate with a/b=1.Figure 15
Nondimensional fundamental frequencies as a function ofkst/Em and a/b of the SSSS plate with hn/hm=1.Figure 16
The first six mode shapes of a two-layer square SSSS composite plate. (a) Mode 1. (b) Mode 2. (c) Mode 3. (d) Mode 4. (e) Mode 5. (f) Mode 6.
(a)
(b)
(c)
(d)
(e)
(f)Figure 17
The first six mode shapes of a two-layer square CCCC composite plate. (a) Mode 1. (b) Mode 2. (c) Mode 3. (d) Mode 4. (e) Mode 5. (f) Mode 6.
(a)
(b)
(c)
(d)
(e)
(f)Figure 18
The first six mode shapes of a two-layer square CSCS composite plate. (a) Mode 1. (b) Mode 2. (c) Mode 3. (d) Mode 4. (e) Mode 5. (f) Mode 6.
(a)
(b)
(c)
(d)
(e)
(f)Figure 19
The first six mode shapes of a two-layer square CCSS composite plate. (a) Mode 1. (b) Mode 2. (c) Mode 3. (d) Mode 4. (e) Mode 5. (f) Mode 6.
(a)
(b)
(c)
(d)
(e)
(f)


## 4. Conclusions

In this work, new numerical results of mechanical behaviors and free vibration responses of the two-layer composite plate are explored, in which two layers are connected by stubs. We used the finite element method combined with the new modified first-order shear deformation theory, which has the following advantages:(i)
Simple formulations for theoretical representation(ii)
No need for reduced integration or selective reduced integration scheme for the proposed methodThe computed results of static bending and free vibration obtained by this approach are also compared to other solutions, showing a good agreement. We gained insight into the responses of deflections and natural frequencies. Besides, some geometrical and physical properties of this structure are also examined. Finally, from new numerical results, several conclusions may be achieved as follows:(i)
The stiffness of the stub has a strong effect on static bending deflections and natural frequencies of the two-layer composite plate whenkst/Em < 1. When kst/Em > 1, there is no slip between the top and the bottom layers at the contact place, and the static bending and free vibration responses of this plate are similar to those of the one-layer composite plate without stubs.(ii)
Likewise, the boundary condition has great effect on the slip between two layers, the minimum slip appears at the center of the plates with the symmetric boundary condition, and the slip has the greatest value at the simply supported edge.(iii)
When the ratiohn/hm increases, the static bending deflection increases, but the fundamental frequency varies as a nonlinear function of this ratio.(iv)
The higher the ratioa/b, the smaller the deflection and the higher the fundamental frequency.The new results of this work are useful for calculation, design, and testing, as well as for giving the optimal solution for the two-layer plate and shell in engineering and technologies. This study suggests some further works on buckling, dynamic response, and heat transfer problems of the two-layer composite plate using different plate theories.



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