# Free Vibration of Functionally Graded Sandwich Shallow Shells on Winkler and Pasternak Foundations with General Boundary Restraints

Authors: Xiancheng Wang; Wei Li; Jianjun Yao; Zhenshuai Wan; Yu Fu; Tang Sheng
Journal: Mathematical Problems in Engineering
 (2019)

---

## Abstract

In this investigation, an exact method based on the first-order shear deformation shallow shell theory (FSDSST) is performed for the free vibration of functionally graded sandwich shallow shells (FGSSS) on Winkler and Pasternak foundations with general boundary restraints. Vibration characteristics of the FGSSS have been obtained by the energy function represented in the orthogonal coordinates, in which the displacement and rotation components consisted of standard double Fourier cosine series and several closed-form supplementary functions are introduced to eliminate the potential jumps and boundary discontinuities. Then, the expansion coefficients are determined by using Rayleigh-Ritz method. The proposed method shows good accuracy and reliability by comprehensive investigation concerning free vibration of the FGSSS. Numerous new vibration results for FGSSS on Winkler and Pasternak foundations with various curvature types, geometrical parameters, and boundary restraints are provided, which may serve as benchmark solutions for future research. In addition, the effects of the inertia, shear deformation, and foundation coefficients on free vibration characteristic of FGSSS are illustrated.


---

## Body

## 1. Introduction

The functionally graded (FG) materials are widely used in aerospace, automobile, and civil engineering due to continuous variation of material properties along the thickness direction [1–7]. As is known to us, the FG shallow shells-structures, i.e., FG plate, FG circular cylindrical shallow shell, FG spherical shallow shell, and FG hyperbolic paraboloidal shallow shell, have attracted considerable attention for its high strength and stiffness [8–12]. It is noticed that the FG shallow shells are unavoidably suffered from dynamic loads, which can lead to fatigue wear and structural damage [13–15]. Therefore, it is essential to study the free vibration characteristics of FG shallow shell structures.Recently, extensive research efforts focused on vibration of FG shallow shells have been made by various shell theories, such as classical shallow shell theory (CSST) [16], first-order shear deformation theory (FSDT) [17], and higher-order shear deformation theory (HSDT) [18]. Furthermore, numerous calculation methods, i.e., Ritz method, finite element method (FEM), generalized differential quadrature method, and wave propagation approach, have also been developed [19–27].Most of the above researches are limited to the classical boundary restraints (clamped, free, simply supported, and shear-diaphragm supported), which requires constant modification of the solution procedure according to the variation of the boundary restraints [28–30]. However, in practical engineering applications, the boundary restraints are not always in certain classical case. There are many possible boundaries such as nonuniform boundaries, elastic edge boundaries, and point-supported boundaries [31–34]. Consequently, an efficient and accurate method for vibration analysis of FG shallow shells with general boundary restraints including classical and elastic edge boundaries is needed. On the other hand, among the FG shallow shells, the existing results of FG sandwich shallow shells (FGSSS) are too scarce for the comparative studies and engineering applications [35–37]. Talebitooti et al. [38] studied the sound transmission across FG laminated sandwich cylindrical shells by using three-dimensional elasticity theory. Hao et al. [39] investigated the nonlinear forced vibrations and natural frequency of FG doubly curved shallow shells with a rectangular base based on the third-order shear deformation theory. Sofiyev [40] presented a modified form of FSDT to solve the bucking problem of FG sandwich truncated conical shells. Trinh and Kim [41] employed the fourth-order Runge-Kutta method to obtain analytical closed-form solutions for thin FGSSS with double curvature testing on elastic bases.Therefore, this paper presents an exact method for the free vibration of FGSSS on Winkler and Pasternak foundations based on first-order shear deformation shallow shell theory (FSDSST). Regardless of boundaries, the standard double Fourier cosine series and several closed-form supplementary functions are introduced to describe the displacement and rotation components of FGSSS so as to eliminate the possible jumps and boundary discontinuities. The current results are checked by comparing with those results published in other literature. Numerous new vibration results for FGSSS with different curvature types, geometrical parameters, and boundary restraints resting on Winkler and Pasternak foundations are presented. The results show that vibration frequencies of FGSSS are strongly influenced by the boundaries. Furthermore, the effect of inertia, shear deformation, and foundation coefficients on free vibration characteristic is comprehensively investigated by comparing FSDSST with CSST.


## 2. Materials and Methods

### 2.1. Model Description and Material Properties

The basic configuration of FGSSS in rectangular planform is depicted in Figure1. Herein, the length, width, and total thickness are represented bya,b, andh, respectively. The FGSSS considered here are characterized by the middle surface, which can be obtained by [45](1)z=-x22Rx+y22Ry+xyRxywhere Rx and Ry denote the radii of curvature in the directions ofx andy, respectively. Rxy is the corresponding radius of twist. In this work, we set Rx,  Ry, and Rxy as constants. Thex andy coordinates are parallel to boundaries so that Rxy is infinity. Herein, three types of independent springs including translational, rotational, and torsional springs are used to realize the given boundaries of FGSSS by setting the stiffness of the springs (x=0,a,y=0,b) as various values, which are Kφx,  Kφy,  kφu, kφv, and kφw(φ=x0,x1,y0,y1), respectively. For instance, the free boundary can be easily generated by setting the spring stiffness values into zero, and the stiffness values are set to infinite (a larger number, 1015 N/m) to achieve a clamped boundary restraint. On the other hand, for the elastic foundations, KW and KS are denoted as linear Winkler foundation and Pasternak foundation coefficients, respectively.Figure 1
Boundary restraints, linear Winkler and Pasternak foundation for a FGSSS.In this analysis, different curvature types of FGSSS models, namely, plate (Rx=Ry=Rxy=∞), circular cylindrical shell (Rx=R,  Ry=Rxy=∞), spherical shell (Rx=Ry=R,  Rxy=∞), and hyperbolic paraboloidal shell (Rx=-Ry=R,  Rxy=∞), are shown in Figure 2.Figure 2
Types of curvatures for FGSSS on rectangular planforms: (a) plate; (b) circular cylindrical shell; (c) spherical shell; (d) hyperbolic paraboloidal shell.
(a) (b) (c) (d)Typically, FG layers of shallow shell are made from a mixture of metal and ceramic materials in different proportions. The effective material properties are assumed to vary continuously through the thickness and can be obtained:(2a)Ez=Ec-EmVcz+Em(2b)ρz=ρc-ρmVcz+ρm(2c)μz=μc-μmVcz+μmwhereE,ρ,andμ represent Young’s modulus, mass density, and Poisson ratio, respectively, and the subscriptsc andm represent the ceramic and metal phase, respectively. Vc(z) is the volume fraction of ceramic constituent. As shown in Figure 3, the representative FGSSS are considered in this work: Type 1 is composed of FG face sheets and homogeneous middle layer (Figure 3(a)); Type 2 has homogeneous face sheets and FG middle layer (Figure 3(b)). Vc(z) of FG sandwich shallow shell in the thickness direction is expressed as(3a)Type1-1:Vc=z-z1z2-z1p1z∈z1,z21z∈z2,z3z-z4z3-z4p3z∈z3,z4Type1-2:Vc=z-z1z1-z2p1z∈z1,z20z∈z2,z3z-z3z4-z3p3z∈z3,z4(3b)Type2-1:Vc=1z∈z1,z2z-z3z2-z3p2z∈z2,z30z∈z3,z4Type2-2:Vc=0z∈z1,z2z-z2z3-z2p2z∈z2,z31z∈z3,z4where the subscripts p1, p2, and p3 are the gradient index used to determine the FG materials and only take nonnegative values. Typical values for metal and ceramic used in the FG layer of shallow shells are listed in Table 1. Type 1-1 and Type 1-2 FGSSS are composed of FG face sheets and homogeneous middle layer, while Type 2-1 and Type 2-2 FGSSS are composed of homogeneous face sheets and FG middle layer. It is found that the bilayered FGSSS can be acquired assigning appropriate ratios of thickness for each layer. And the thickness of each layer from bottom to top is expressed by the combination of three numbers; for instance, “2-3-2” denotes thath1:h2:h3=2:3:2. To describe the behavior of (3a) and (3b), the volume fractions Vc(z) in the direction of thickness for the FGSSS with various gradient indexp are shown in Figure 4.Table 1
Main materials properties of the used FG layer.
Material Properties E(GPa) μ ρ(kg/m3) Metal (Al) 70 0.3 2702 Ceramic (Al2O3) 380 0.3 3800Figure 3
The material variation along the thickness of the FGSSS: (a) Type 1: FG face sheets and homogeneous middle layer; (b) Type 2: homogeneous face sheets and FG middle layer.Figure 4
Variation of volume fractionVc through the thickness in various types: (a) Type 1-1; (b) Type 1-2; (c) Type 2-1; (d) Type 2-2.
(a) (b) (c) (d)


### 2.2. Stress-Strain Relations and Stress Resultants

To describe the shell clearly,u,v, and w represent the displacements in thex,y, andz directions, andt is the time variable, respectively. The assumed displacement field for the FGSSS based on the FSDSST can be given by(4)Ux,y,z,t=ux,y,t+zϕxx,y,tVx,y,z,t=vx,y,t+zϕyx,y,tWx,y,z,t=wx,y,twhere ϕx and ϕy represent the rotations of the reference surface to they andx axes, respectively. For the FGSSS the strains can be defined as(5)εxεyγxyT=εx0+zχxεy0+zχyγxy0+zχxyT=∂u∂x∂v∂y∂u∂y+∂v∂xT+z∂ϕx∂x∂ϕy∂y∂ϕx∂y+∂ϕy∂xT(6)γxzγyzT=γxz0γyz0T=∂u∂z+∂w∂x∂v∂z+∂w∂yT=ϕx+∂w∂xϕy+∂w∂yTwhere the normal and shear strains in the directions ofx,y, andz are εx  εy and γxy.  γxz and γyz are transverse shear strains. According to generalized Hooke’s law [46], the stress-strain relations of the shallow shells can be written as(7)σxσyτyzτxzτxy=Q11zQ12z00Q16zQ12zQ22z00Q26z00Q44zQ45z000Q45zQ55z0Q16zQ26z00Q66zεxεyγyzγxzγxywhere the material elastic stiffness coefficients Qij(z) are defined in terms of the materials properties as(8)Q11z=Ez1-μz2Q44z=G23zQ12z=Ezμz1-μz2Q55z=G13zQ22z=Ez1-μz2Q66z=G12zIt should be noted thatG12(z)=G13(z)=G23(z)=E(z)/(2+2μ(z)). By carrying the integration of stresses over the plate thickness, the force and moment resultants are obtained as(9a)Nx,Ny,NxyT=∫-h/2h/2σx,σy,τxydz(9b)Mx,My,MxyT=∫-h/2h/2σx,σy,τxyzdz(9c)Qx,QyT=∫-h/2h/2τxz,τyzdzwhere Nx,  Ny,  Nxy are the normal and shear forces, Mx,  My, and Mxy are the bending and twisting moments. Qx,  Qy are the transverse shear forces. Performing the integration in (9a), (9b), and (9c) yields (10)NxNyNxy=A11A12A16A12A22A26A16A26A66∂u∂x,∂v∂y,∂u∂y+∂v∂xT+B11B12B16B12B22B26B16B26B66∂ϕx∂x,∂ϕy∂y,∂ϕx∂y+∂ϕy∂xT(11)MxMyMxy=B11B12B16B12B22B26B16B26B66∂u∂x,∂v∂y,∂u∂y+∂v∂xT+D11D12D16D12D22D26D16D26D66∂ϕx∂x,∂ϕy∂y,∂ϕx∂y+∂ϕy∂xT(12)QxQy=κA44A45A45A55∂u∂z+∂w∂x,∂v∂z+∂w∂yTwhere κ denotes the shear correction coefficient and is usually taken as 5/6. The stiffness coefficients Aij, Bij, and Dij can be obtained as(13)Aij,Bij,Dij=∑k=13∫zkzk+1Qijk1,z,z2dz


### 2.3. Energy Functions and Governing Equations

A modified Fourier version based on Rayleigh-Ritz method is performed. The Rayleigh-Ritz method is a powerful tool in the field of vibration analysis, in which the undetermined coefficients in the displacement can be obtained by minimizing the Lagrangian energy function expression or making them equal to zero [47]. Then, a series of equations can be summed up in matrix form as a standard characteristic equation, which can be easily solved to obtain the desired frequencies and modes. The Lagrangian energy function of the FGSSS can be written as(14)L=T-Us-Usp-Uf+WeThe strain energyUs for the FGSSS during vibration is given in an integral form by(15)Us=12∫0a∫0bNxεx0+Nyεy0+Nxyγxy0+Mxχx+Myχy+Mxyχxy+Qxγxz0+Qyγyz0dydxSubstituting (5), (6), and (10)-(12) into (15), it can be written in terms of displacements and rotations components as(16)Us=12∫0a∫0bA11∂u∂x2+A22∂v∂y2+56A44ϕx+∂w∂x2+56A55ϕy+∂w∂y2+A66∂u∂y+∂v∂x2+2A12∂u∂x∂v∂y+2A16∂u∂x∂u∂y+∂u∂x∂v∂x+2A26∂v∂x∂v∂y+∂u∂y∂v∂y+53A45ϕx+∂w∂xϕy+∂w∂y+2B11∂u∂x∂ϕx∂x+2B22∂v∂y∂ϕy∂y+2B66∂u∂y+∂v∂x∂ϕx∂y+∂ϕy∂x+2B12∂u∂x∂ϕy∂y+∂v∂y∂ϕx∂x+2B16∂u∂x∂ϕx∂y+∂u∂x∂ϕy∂x+∂u∂y∂ϕx∂x+∂v∂x∂ϕx∂x+2B26∂v∂y∂ϕx∂y+∂v∂y∂ϕy∂x+∂u∂y∂ϕy∂y+∂v∂x∂ϕy∂y+D11∂ϕx∂x2+D22∂ϕy∂y2+D66∂ϕx∂y+∂ϕy∂x2+2D12∂ϕx∂x∂ϕy∂y+2D16∂ϕx∂y∂ϕx∂x+∂ϕx∂x∂ϕy∂x+2D26∂ϕx∂y∂ϕy∂y+∂ϕy∂x∂ϕy∂ydydxThe kinetic energyT, external work We, and deformation strain energyUsp of the FGSSS can be seen in [48]. Additionally, the strain energy based on the Winkler and Pasternak foundations is (17)Uf=12∫0a∫0bkWw2+kS∂w∂x2+kS∂w∂y2dydxThe governing equations for the FGSSS can be written in matrix form by combining (5), (6), and (10)-(12) based on Hamilton’s principle: (18)L11L12L13L14L15L12L22L23L24L25-L13-L23L33L34L35L14L24-L34L44L45L15L25-L35L45L55-w2-I000-I100-I000-I100-I000-I100-I200-I100-I2uvwϕxϕy=-px-py-pz-mx-myThe coefficients of the linear operatorLij are given in Appendix A.


### 2.4. Admissible Displacement Functions

In this subsection, the free vibration of FGSSS on the Winkler and Pasternak foundation with general boundary restraints are considered. The displacement and rotation components of FGSSS consisted of standard double Fourier cosine series and several closed-form supplementary functions are introduced to remove the potential jumps and boundary discontinuities, which can be expressed as(19)ux,y,t=∑m=0M∑n=0NAmncos⁡λmxcos⁡λny+∑l=12∑n=0Nalnζlaxcos⁡λny+∑l=12∑m=0Mblmζlbycos⁡λmxeiwt(20)vx,y=∑m=0M∑n=0NBmncos⁡λmxcos⁡λny+∑l=12∑n=0Nclnζlaxcos⁡λny+∑l=12∑m=0Mdlmζlbycos⁡λmxeiwt(21)wx,y,t=∑m=0M∑n=0NCmncos⁡λmxcos⁡λny+∑l=12∑n=0Nelnζlaxcos⁡λny+∑l=12∑m=0Mflmζlbycos⁡λmxeiwt(22)ϕxx,y=∑m=0M∑n=0NDmncos⁡λmxcos⁡λny+∑l=12∑n=0Nglnζlaxcos⁡λny+∑l=12∑m=0Mhlmζlbycos⁡λmxeiwt(23)ϕyx,y=∑m=0M∑n=0NEmncos⁡λmxcos⁡λny+∑l=12∑n=0Nilnζlaxcos⁡λny+∑l=12∑m=0Mjlmζlbycos⁡λmxeiwtwhere λm=mπ/a, λn=nπ/b.M andN are truncation numbers with respect to variablesx andy. Amn, Bmn, Cmn, Dmn, and Emn represent the Fourier expansion coefficients of the cosine Fourier series, respectively. aln, blm, cln, dlm, eln, flm, gln, hlm, iln, and jlm are the corresponding supplement coefficients. The auxiliary polynomial functions ζla(x) and ζlb(y) are introduced to remove all the potential discontinuities associated with the first-order derivatives at the boundaries, which can be expressed as follows:(24)ζ1ax=xxa-12ζ2ax=x2axa-1ζ1by=yyb-12ζ2by=y2byb-1It can be verified that(25)ζ1a′0=1,ζ1a0=ζ1aa=ζ1a′a=0ζ2a′a=1,ζ2a0=ζ2aa=ζ2a′0=0ζ1b′0=1,ζ1b0=ζ1bb=ζ1b′b=0ζ2b′b=1,ζ2b0=ζ2bb=ζ2b′0=0Alternately, all the expansion coefficients in (19)-(23) are treated equally and independently as the orthogonal coordinates and solved from the Rayleigh-Ritz method. The vibration characteristic equation can be summed up:(26)K-ω2MG=0where the coefficient eigenvector G=Gu,Gv,Gw,Gϕx,GϕyT is the unknown expansion coefficient that appears in the series expansions and can be determined by(27)Gu=A00,A01,⋯,Am0,Am1,⋯Amn,⋯,AMN,a10,⋯,aln,⋯,a2N,b10,⋯,blm,⋯,b2MGv=B00,B01,⋯,Bm0,Bm1,⋯Bmn,⋯,BMN,c10,⋯,cln,⋯,c2N,d10,⋯,dlm,⋯,d2MGw=C00,C01,⋯,Cm0,Cm1,⋯Cmn,⋯,CMN,e10,⋯,eln,⋯,e2N,f10,⋯,flm,⋯,f2MGϕx=D00,D01,⋯,Dm0,Dm1,⋯Dmn,⋯,DMN,g10,⋯,gln,⋯,g2N,h10,⋯,hlm,⋯,h2MGϕy=E00,E01,⋯,Em0,Em1,⋯Emn,⋯,EMN,i10,⋯,iln,⋯,i2N,j10,⋯,jlm,⋯,j2MK andM represent the stiffness and mass matrix of the FGSSS, respectively. Both of them are matrices and can be expressed as [48](28)K=KuuKuvKuwKuϕxKuϕyKuvKvvKvwKvϕxKvϕyKuwKvwKwwKwϕxKwϕyKuϕxKvϕxKwϕxKϕxϕxKϕxϕyKuϕyKvϕyKwϕyKϕxϕyKϕyϕyM=Muu00Muϕx00Muu00Mvϕy00Mww00Muϕx00Mϕxϕx00Mvϕy00MϕyϕyThe stiffness matrixK and mass matrixM are given in Appendix B.


## 2.1. Model Description and Material Properties

The basic configuration of FGSSS in rectangular planform is depicted in Figure1. Herein, the length, width, and total thickness are represented bya,b, andh, respectively. The FGSSS considered here are characterized by the middle surface, which can be obtained by [45](1)z=-x22Rx+y22Ry+xyRxywhere Rx and Ry denote the radii of curvature in the directions ofx andy, respectively. Rxy is the corresponding radius of twist. In this work, we set Rx,  Ry, and Rxy as constants. Thex andy coordinates are parallel to boundaries so that Rxy is infinity. Herein, three types of independent springs including translational, rotational, and torsional springs are used to realize the given boundaries of FGSSS by setting the stiffness of the springs (x=0,a,y=0,b) as various values, which are Kφx,  Kφy,  kφu, kφv, and kφw(φ=x0,x1,y0,y1), respectively. For instance, the free boundary can be easily generated by setting the spring stiffness values into zero, and the stiffness values are set to infinite (a larger number, 1015 N/m) to achieve a clamped boundary restraint. On the other hand, for the elastic foundations, KW and KS are denoted as linear Winkler foundation and Pasternak foundation coefficients, respectively.Figure 1
Boundary restraints, linear Winkler and Pasternak foundation for a FGSSS.In this analysis, different curvature types of FGSSS models, namely, plate (Rx=Ry=Rxy=∞), circular cylindrical shell (Rx=R,  Ry=Rxy=∞), spherical shell (Rx=Ry=R,  Rxy=∞), and hyperbolic paraboloidal shell (Rx=-Ry=R,  Rxy=∞), are shown in Figure 2.Figure 2
Types of curvatures for FGSSS on rectangular planforms: (a) plate; (b) circular cylindrical shell; (c) spherical shell; (d) hyperbolic paraboloidal shell.
(a) (b) (c) (d)Typically, FG layers of shallow shell are made from a mixture of metal and ceramic materials in different proportions. The effective material properties are assumed to vary continuously through the thickness and can be obtained:(2a)Ez=Ec-EmVcz+Em(2b)ρz=ρc-ρmVcz+ρm(2c)μz=μc-μmVcz+μmwhereE,ρ,andμ represent Young’s modulus, mass density, and Poisson ratio, respectively, and the subscriptsc andm represent the ceramic and metal phase, respectively. Vc(z) is the volume fraction of ceramic constituent. As shown in Figure 3, the representative FGSSS are considered in this work: Type 1 is composed of FG face sheets and homogeneous middle layer (Figure 3(a)); Type 2 has homogeneous face sheets and FG middle layer (Figure 3(b)). Vc(z) of FG sandwich shallow shell in the thickness direction is expressed as(3a)Type1-1:Vc=z-z1z2-z1p1z∈z1,z21z∈z2,z3z-z4z3-z4p3z∈z3,z4Type1-2:Vc=z-z1z1-z2p1z∈z1,z20z∈z2,z3z-z3z4-z3p3z∈z3,z4(3b)Type2-1:Vc=1z∈z1,z2z-z3z2-z3p2z∈z2,z30z∈z3,z4Type2-2:Vc=0z∈z1,z2z-z2z3-z2p2z∈z2,z31z∈z3,z4where the subscripts p1, p2, and p3 are the gradient index used to determine the FG materials and only take nonnegative values. Typical values for metal and ceramic used in the FG layer of shallow shells are listed in Table 1. Type 1-1 and Type 1-2 FGSSS are composed of FG face sheets and homogeneous middle layer, while Type 2-1 and Type 2-2 FGSSS are composed of homogeneous face sheets and FG middle layer. It is found that the bilayered FGSSS can be acquired assigning appropriate ratios of thickness for each layer. And the thickness of each layer from bottom to top is expressed by the combination of three numbers; for instance, “2-3-2” denotes thath1:h2:h3=2:3:2. To describe the behavior of (3a) and (3b), the volume fractions Vc(z) in the direction of thickness for the FGSSS with various gradient indexp are shown in Figure 4.Table 1
Main materials properties of the used FG layer.
Material Properties E(GPa) μ ρ(kg/m3) Metal (Al) 70 0.3 2702 Ceramic (Al2O3) 380 0.3 3800Figure 3
The material variation along the thickness of the FGSSS: (a) Type 1: FG face sheets and homogeneous middle layer; (b) Type 2: homogeneous face sheets and FG middle layer.Figure 4
Variation of volume fractionVc through the thickness in various types: (a) Type 1-1; (b) Type 1-2; (c) Type 2-1; (d) Type 2-2.
(a) (b) (c) (d)


## 2.2. Stress-Strain Relations and Stress Resultants

To describe the shell clearly,u,v, and w represent the displacements in thex,y, andz directions, andt is the time variable, respectively. The assumed displacement field for the FGSSS based on the FSDSST can be given by(4)Ux,y,z,t=ux,y,t+zϕxx,y,tVx,y,z,t=vx,y,t+zϕyx,y,tWx,y,z,t=wx,y,twhere ϕx and ϕy represent the rotations of the reference surface to they andx axes, respectively. For the FGSSS the strains can be defined as(5)εxεyγxyT=εx0+zχxεy0+zχyγxy0+zχxyT=∂u∂x∂v∂y∂u∂y+∂v∂xT+z∂ϕx∂x∂ϕy∂y∂ϕx∂y+∂ϕy∂xT(6)γxzγyzT=γxz0γyz0T=∂u∂z+∂w∂x∂v∂z+∂w∂yT=ϕx+∂w∂xϕy+∂w∂yTwhere the normal and shear strains in the directions ofx,y, andz are εx  εy and γxy.  γxz and γyz are transverse shear strains. According to generalized Hooke’s law [46], the stress-strain relations of the shallow shells can be written as(7)σxσyτyzτxzτxy=Q11zQ12z00Q16zQ12zQ22z00Q26z00Q44zQ45z000Q45zQ55z0Q16zQ26z00Q66zεxεyγyzγxzγxywhere the material elastic stiffness coefficients Qij(z) are defined in terms of the materials properties as(8)Q11z=Ez1-μz2Q44z=G23zQ12z=Ezμz1-μz2Q55z=G13zQ22z=Ez1-μz2Q66z=G12zIt should be noted thatG12(z)=G13(z)=G23(z)=E(z)/(2+2μ(z)). By carrying the integration of stresses over the plate thickness, the force and moment resultants are obtained as(9a)Nx,Ny,NxyT=∫-h/2h/2σx,σy,τxydz(9b)Mx,My,MxyT=∫-h/2h/2σx,σy,τxyzdz(9c)Qx,QyT=∫-h/2h/2τxz,τyzdzwhere Nx,  Ny,  Nxy are the normal and shear forces, Mx,  My, and Mxy are the bending and twisting moments. Qx,  Qy are the transverse shear forces. Performing the integration in (9a), (9b), and (9c) yields (10)NxNyNxy=A11A12A16A12A22A26A16A26A66∂u∂x,∂v∂y,∂u∂y+∂v∂xT+B11B12B16B12B22B26B16B26B66∂ϕx∂x,∂ϕy∂y,∂ϕx∂y+∂ϕy∂xT(11)MxMyMxy=B11B12B16B12B22B26B16B26B66∂u∂x,∂v∂y,∂u∂y+∂v∂xT+D11D12D16D12D22D26D16D26D66∂ϕx∂x,∂ϕy∂y,∂ϕx∂y+∂ϕy∂xT(12)QxQy=κA44A45A45A55∂u∂z+∂w∂x,∂v∂z+∂w∂yTwhere κ denotes the shear correction coefficient and is usually taken as 5/6. The stiffness coefficients Aij, Bij, and Dij can be obtained as(13)Aij,Bij,Dij=∑k=13∫zkzk+1Qijk1,z,z2dz


## 2.3. Energy Functions and Governing Equations

A modified Fourier version based on Rayleigh-Ritz method is performed. The Rayleigh-Ritz method is a powerful tool in the field of vibration analysis, in which the undetermined coefficients in the displacement can be obtained by minimizing the Lagrangian energy function expression or making them equal to zero [47]. Then, a series of equations can be summed up in matrix form as a standard characteristic equation, which can be easily solved to obtain the desired frequencies and modes. The Lagrangian energy function of the FGSSS can be written as(14)L=T-Us-Usp-Uf+WeThe strain energyUs for the FGSSS during vibration is given in an integral form by(15)Us=12∫0a∫0bNxεx0+Nyεy0+Nxyγxy0+Mxχx+Myχy+Mxyχxy+Qxγxz0+Qyγyz0dydxSubstituting (5), (6), and (10)-(12) into (15), it can be written in terms of displacements and rotations components as(16)Us=12∫0a∫0bA11∂u∂x2+A22∂v∂y2+56A44ϕx+∂w∂x2+56A55ϕy+∂w∂y2+A66∂u∂y+∂v∂x2+2A12∂u∂x∂v∂y+2A16∂u∂x∂u∂y+∂u∂x∂v∂x+2A26∂v∂x∂v∂y+∂u∂y∂v∂y+53A45ϕx+∂w∂xϕy+∂w∂y+2B11∂u∂x∂ϕx∂x+2B22∂v∂y∂ϕy∂y+2B66∂u∂y+∂v∂x∂ϕx∂y+∂ϕy∂x+2B12∂u∂x∂ϕy∂y+∂v∂y∂ϕx∂x+2B16∂u∂x∂ϕx∂y+∂u∂x∂ϕy∂x+∂u∂y∂ϕx∂x+∂v∂x∂ϕx∂x+2B26∂v∂y∂ϕx∂y+∂v∂y∂ϕy∂x+∂u∂y∂ϕy∂y+∂v∂x∂ϕy∂y+D11∂ϕx∂x2+D22∂ϕy∂y2+D66∂ϕx∂y+∂ϕy∂x2+2D12∂ϕx∂x∂ϕy∂y+2D16∂ϕx∂y∂ϕx∂x+∂ϕx∂x∂ϕy∂x+2D26∂ϕx∂y∂ϕy∂y+∂ϕy∂x∂ϕy∂ydydxThe kinetic energyT, external work We, and deformation strain energyUsp of the FGSSS can be seen in [48]. Additionally, the strain energy based on the Winkler and Pasternak foundations is (17)Uf=12∫0a∫0bkWw2+kS∂w∂x2+kS∂w∂y2dydxThe governing equations for the FGSSS can be written in matrix form by combining (5), (6), and (10)-(12) based on Hamilton’s principle: (18)L11L12L13L14L15L12L22L23L24L25-L13-L23L33L34L35L14L24-L34L44L45L15L25-L35L45L55-w2-I000-I100-I000-I100-I000-I100-I200-I100-I2uvwϕxϕy=-px-py-pz-mx-myThe coefficients of the linear operatorLij are given in Appendix A.


## 2.4. Admissible Displacement Functions

In this subsection, the free vibration of FGSSS on the Winkler and Pasternak foundation with general boundary restraints are considered. The displacement and rotation components of FGSSS consisted of standard double Fourier cosine series and several closed-form supplementary functions are introduced to remove the potential jumps and boundary discontinuities, which can be expressed as(19)ux,y,t=∑m=0M∑n=0NAmncos⁡λmxcos⁡λny+∑l=12∑n=0Nalnζlaxcos⁡λny+∑l=12∑m=0Mblmζlbycos⁡λmxeiwt(20)vx,y=∑m=0M∑n=0NBmncos⁡λmxcos⁡λny+∑l=12∑n=0Nclnζlaxcos⁡λny+∑l=12∑m=0Mdlmζlbycos⁡λmxeiwt(21)wx,y,t=∑m=0M∑n=0NCmncos⁡λmxcos⁡λny+∑l=12∑n=0Nelnζlaxcos⁡λny+∑l=12∑m=0Mflmζlbycos⁡λmxeiwt(22)ϕxx,y=∑m=0M∑n=0NDmncos⁡λmxcos⁡λny+∑l=12∑n=0Nglnζlaxcos⁡λny+∑l=12∑m=0Mhlmζlbycos⁡λmxeiwt(23)ϕyx,y=∑m=0M∑n=0NEmncos⁡λmxcos⁡λny+∑l=12∑n=0Nilnζlaxcos⁡λny+∑l=12∑m=0Mjlmζlbycos⁡λmxeiwtwhere λm=mπ/a, λn=nπ/b.M andN are truncation numbers with respect to variablesx andy. Amn, Bmn, Cmn, Dmn, and Emn represent the Fourier expansion coefficients of the cosine Fourier series, respectively. aln, blm, cln, dlm, eln, flm, gln, hlm, iln, and jlm are the corresponding supplement coefficients. The auxiliary polynomial functions ζla(x) and ζlb(y) are introduced to remove all the potential discontinuities associated with the first-order derivatives at the boundaries, which can be expressed as follows:(24)ζ1ax=xxa-12ζ2ax=x2axa-1ζ1by=yyb-12ζ2by=y2byb-1It can be verified that(25)ζ1a′0=1,ζ1a0=ζ1aa=ζ1a′a=0ζ2a′a=1,ζ2a0=ζ2aa=ζ2a′0=0ζ1b′0=1,ζ1b0=ζ1bb=ζ1b′b=0ζ2b′b=1,ζ2b0=ζ2bb=ζ2b′0=0Alternately, all the expansion coefficients in (19)-(23) are treated equally and independently as the orthogonal coordinates and solved from the Rayleigh-Ritz method. The vibration characteristic equation can be summed up:(26)K-ω2MG=0where the coefficient eigenvector G=Gu,Gv,Gw,Gϕx,GϕyT is the unknown expansion coefficient that appears in the series expansions and can be determined by(27)Gu=A00,A01,⋯,Am0,Am1,⋯Amn,⋯,AMN,a10,⋯,aln,⋯,a2N,b10,⋯,blm,⋯,b2MGv=B00,B01,⋯,Bm0,Bm1,⋯Bmn,⋯,BMN,c10,⋯,cln,⋯,c2N,d10,⋯,dlm,⋯,d2MGw=C00,C01,⋯,Cm0,Cm1,⋯Cmn,⋯,CMN,e10,⋯,eln,⋯,e2N,f10,⋯,flm,⋯,f2MGϕx=D00,D01,⋯,Dm0,Dm1,⋯Dmn,⋯,DMN,g10,⋯,gln,⋯,g2N,h10,⋯,hlm,⋯,h2MGϕy=E00,E01,⋯,Em0,Em1,⋯Emn,⋯,EMN,i10,⋯,iln,⋯,i2N,j10,⋯,jlm,⋯,j2MK andM represent the stiffness and mass matrix of the FGSSS, respectively. Both of them are matrices and can be expressed as [48](28)K=KuuKuvKuwKuϕxKuϕyKuvKvvKvwKvϕxKvϕyKuwKvwKwwKwϕxKwϕyKuϕxKvϕxKwϕxKϕxϕxKϕxϕyKuϕyKvϕyKwϕyKϕxϕyKϕyϕyM=Muu00Muϕx00Muu00Mvϕy00Mww00Muϕx00Mϕxϕx00Mvϕy00MϕyϕyThe stiffness matrixK and mass matrixM are given in Appendix B.


## 3. Numerous Results and Discussion

In this section, a comprehensive free vibration analysis for FGSSS on Winkler and Pasternak foundations with general boundary restraints is presented. Firstly, the current results are checked by comparing with those results published in other literature. Then, the vibration behaviors for FGSSS on Winkler and Pasternak foundations with various curvature types, distribution types, geometrical parameters, and boundary restraints are studied. Finally, the influence of vibration parameters including inertia, shear deformation, and foundation coefficient on the free vibration is illustrated.


### 3.1. FGSSS with General Boundary Restraints

In the engineering applications, the letters F, C, S, and SD represent the completely free, completely clamped, simply supported, and shear-diaphragm supported boundary restraints, respectively. Besides the aforementioned classical boundary restraints, three elastic boundary restraints, denoted by E1, E2, and E3, are considered in this paper. Taking edge at x=0, for example, the seven boundaries and corresponding restraint parameters are shown in Table 2.Table 2
The corresponding elastic restraint parameters for various boundaries.
Boundary type Restraint parameters F k x 0 u = k x 0 v = k x 0 w = K x 0 x = K x 0 y = 0,  Nx=Nxy=Qx=Mx=Mxy=0 C k x 0 u = k x 0 v = k x 0 w = K x 0 x = K x 0 y = 1 0 7 D,   u=v=w=ϕx=ϕy=0 S k x 0 u = k x 0 v = k x 0 w = K x 0 y = 1 0 7 D ,    K x 0 x = 0,   u=v=w=Mx=ϕy=0 SD k x 0 v = k x 0 w = K x 0 y = 1 0 7 D ,   k x 0 u = K x 0 x = 0 E1 k x 0 w = K x 0 x = K x 0 y = 1 0 7 D ,   k x 0 u = k x 0 v = 15 D,   w=ϕx=ϕy=0,u≠0,v≠0 E2 k x 0 u = k x 0 v = K x 0 x = K x 0 y = 1 0 7 D ,  k x 0 w = 10 D,   u=v=ϕx=ϕy=0,w≠0 E3 k x 0 u = k x 0 v = k x 0 w = 1 0 7 D ,   K x 0 x = K x 0 y = 5 D,  u=v=w=0,ϕx≠0,ϕy≠0The reference bending stiffnessDm=Emh3/12(1-μm2). The accuracy and reliability of the current results are displayed in Tables 3 and 4. Table 3 compares the nondimensional frequency parameters ϖ=ωb2/hρm/Em of simply supported square FG sandwich plates with FG face sheets and homogeneous middle layer (1-0-1, 1-1-1, 1-2-1, 2-1-2, 2-2-1, and 1-8-1). The thickness-length ratios are taken to beh/a=0.01, 0.1, and 0.2. The gradient indexes are given asp=0.5, 5, and 10. Young’s modulus, mass density, and Poisson ratio are obtained in Table 1. The solutions given by Li et al. [42], by the 3D solutions and those of Hadji et al. [43], and by the RPT are provided for the comparisons. The differences are less than 1.530% and 2.015% for the worst case, respectively.Table 3
Comparison of non-dimensional frequency parametersϖ=ωb2/hρm/Em of simply supported square FG sandwich plates with homogeneous middle layer.
h/a p Theory ϖ 1-0-1 1-1-1 1-2-1 2-1-2 2-2-1 1-8-1 0.01 0.5 3D [42] 1.48244 1.56046 1.61915 1.52355 1.59031 1.76357 RPT [43] 1.48241 1.56042 1.61912 1.52353 1.59030 1.76354 Present 1.48248 1.56023 1.61913 1.52358 1.59034 1.76359 5 3D [42] 0.96563 1.06309 1.19699 0.99903 1.13020 1.56988 RPT [43] 0.96564 1.06309 1.19697 0.99903 1.13019 1.56985 Present 0.96566 1.06313 1.19697 0.99907 1.13024 1.56980 10 3D [42] 0.95042 1.01237 1.14408 0.95934 1.08065 1.54164 RPT [43] 0.95044 1.01236 1.14406 0.95937 1.08065 1.54162 Present 0.95041 1.01234 1.14408 0.95935 1.08065 1.54166  0.1 0.5 3D [42] 1.44614 1.52131 1.57668 1.48608 1.54926 1.71130 RPT [43] 1.44423 1.51921 1.57450 1.48408 1.54710 1.70901 Present 1.44485 1.51975 1.57461 1.48415 1.54750 1.70920 5 3D [42] 0.94476 1.04532 1.17567 0.98103 1.10983 1.52993 RPT [43] 0.94598 1.04465 1.17396 0.98184 1.10881 1.52792 Present 0.94552 1.04434 1.17355 0.98142 1.10874 1.52805 10 3D [42] 0.92727 0.99523 1.12466 0.94078 1.06104 1.50333 RPT [43] 0.92838 0.99550 1.12313 0.94296 1.06090 1.50138 Present 0.92836 0.99550 1.12313 0.94295 1.06080 1.50137  0.2 0.5 3D [42] 1.35358 1.42178 1.46940 1.39053 1.44535 1.58186 RPT [43] 1.34743 1.41508 1.46251 1.38410 1.43843 1.57476 Present 1.35015 1.41682 1.46368 1.38515 1.43901 1.57592 5 3D [42] 0.89086 0.99798 1.11900 0.93362 1.05607 1.42845 RPT [43] 0.89462 0.99545 1.11318 0.93594 1.05228 1.42197 Present 0.89504 0.99561 1.11285 0.93480 1.05496 1.42372 10 3D [42] 0.86833 0.94984 1.07290 0.89228 1.00949 1.40568 RPT [43] 0.87178 0.95033 1.06754 0.89918 1.00848 1.39932 Present 0.86963 0.95012 1.06950 0.89541 1.00887 1.40105Table 4
Comparisons of nondimensional frequency parametersϖ=ωb2/hρ0/E0 for Type 1-1 and FGSSS with completely free boundary restraints, respectively (a=b, h/b=0.1, and Rxy=∞).
Mode a/R=0.2 a/R=0.5 12×12 13 ×13 14×14 12×12 13×13 14×14 Jin et al. [44] Pre-sent Jin et al. [44] Pre-sent Jin etal. [44] Pre-sent Jin et al. [44] Pre-sent Jin et al. [44] Pre-sent Jin et al. [44] Pre-sent Type1-1: FG circular cylindrical shallow shell  1 0.903 0.907 0.902 0.905 0.902 0.905 0.901 0.903 0.901 0.902 0.901 0.902 2 1.352 1.361 1.352 1.359 1.352 1.358 1.405 1.411 1.405 1.411 1.405 1.410 3 1.662 1.670 1.662 1.666 1.662 1.659 1.734 1.738 1.734 1.735 1.734 1.735 4 2.279 2.285 2.278 2.281 2.278 2.281 2.272 2.275 2.271 2.276 2.271 2.276  Type1-1: FG spherical shallow shell  1 0.902 0.905 0.902 0.905 0.902 0.902 0.899 0.904 0.898 0.901 0.898 0.897 2 1.335 1.336 1.335 1.336 1.335 2.334 1.322 1.325 1.322 1.325 1.322 1.325 3 1.696 1.691 1.696 1.694 1.696 1.695 1.900 1.905 1.900 1.905 1.900 1.905 4 2.279 2.284 2.279 2.284 2.279 2.284 2.274 2.276 2.274 2.275 2.274 2.275  Type1-1: FG hyperbolic paraboloidal shallow shell  1 0.902 0.905 0.902 0.905 0.902 0.905 0.899 0.896 0.899 0.894 0.899 0.894 2 1.399 1.395 1.399 1.395 1.399 1.394 1.634 1.635 1.634 1.637 1.634 1.638 3 1.648 1.645 1.648 1.645 1.648 1.654 1.683 1.687 1.683 1.687 1.683 1.687 4 2.283 2.285 2.282 2.286 2.282 2.286 2.294 2.296 2.293 2.296 2.293 2.297  Type2-1: FG circular cylindrical shallow shell  1 0.835 0.845 0.835 0.845 0.835 0.844 0.833 0.835 0.833 0.834 0.833 0.834 2 1.249 1.245 1.249 1.245 1.249 1.245 1.302 1.306 1.302 1.308 1.302 1.308 3 1.537 1.550 1.537 1.552 1.537 1.553 1.604 1.612 1.604 1.610 1.604 1.610 4 2.105 2.110 2.105 2.110 2.104 2.108 2.104 2.107 2.104 2.107 2.104 2.109  Type2-1: FG spherical shallow shell  1 0.835 0.838 0.834 0.835 0.834 0.832 0.830 0.834 0.830 0.835 0.830 0.835 2 1.235 1.241 1.235 1.239 1.235 1.238 1.227 1.231 1.227 1.234 1.227 1.234 3 1.570 1.574 1.570 1.574 1.570 1.574 1.765 1.768 1.765 1.767 1.764 1.767 4 2.106 2.110 2.105 2.110 2.105 2.108 2.105 2.109 2.105 2.114 2.105 2.114  Type2-1: FG hyperbolic paraboloidal shallow shell  1 0.836 0.841 0.835 0.838 0.835 0.837 0.832 0.835 0.832 0.836 0.832 0.836 2 1.291 1.294 1.291 1.294 1.291 1.294 1.507 1.521 1.507 1.521 1.507 1.520 3 1.522 1.532 1.522 1.537 1.522 1.540 1.555 1.547 1.555 1.549 1.555 1.551 4 2.101 2.104 2.101 2.114 2.104 2.107 2.106 2.115 2.105 2.114 2.105 2.114Meanwhile, The frequency parametersϖ=ωb2/hρ0/E0  (ρ0=1kg/m3,  E0=1Gpa) for Type 1-1 and Type 2-1 FGSSS with completely free boundary restraints are presented in Table 4. All layers are assumed to be of equal thickness. The geometrical parameters are given asa=b andh/b=0.1. The length-radii ratios used for this analysis area/R=0.2 and 0.5, and the truncation number isM=N=12, 13, and 14. The frequency results are compared with the solutions given by Jin et al. [44]. A well agreement can be obtained.Numerous new results for fundamental frequenciesf (Hz) are presented in Table 5 for Type 1-1 FGSSS with a variety of general boundary restraints. The gradient index and geometrical parameters are given asp=5, a/b=1,  h/a=0.1,  Rx/a=2 and 5. And the thickness for each layer is set to be equal. Young’s modulus, mass density, and Poisson ratio are showed in Table 1. The boundary restraints including SSSS, CCCC, CFCF, SCSC, CSDCSD, CE1E1E1, SE2SE2, and E3E3E3E3 are considered. It is obvious that the values of fundamental frequencies corresponding to different boundaries are quite sensitive to the change of geometrical parameters. The frequencies of the FGSSS increase with the increase of radius ratio (Rx/Ry) and the decrease of radius-length ratio (Rx/a). The first four mode shapes for the plate, circular cylindrical shallow shell, and spherical shallow shell with CCCC, SSSS, and SE2SE2 boundaries are depicted in Figures 5, 6, and 7, respectively. The radius-length ratioRx/a is set as 2.Table 5
The first four frequenciesf (Hz) for Type 1-1 (1-1-1) FGSSS with various boundary restraints and radius-length ratios (p=5, a/b=1, and h/a=0.1).
R x/Ry R x/a Mode Boundary restraints SSSS CCCC CFCF SCSC CSDCSD CE1E1E1 SE2SE2 E3E3E3E3 Plate ∞ 1 66.545 79.857 72.148 58.102 75.279 39.310 74.196 21.214 2 93.851 105.51 96.792 73.858 99.082 51.903 98.154 35.150 3 138.47 140.48 123.95 97.593 123.27 71.192 102.21 45.211 4 159.68 173.65 143.54 120.97 135.36 98.115 125.34 59.633  +1 5 1 68.563 80.374 75.156 60.154 78.364 40.151 77.458 23.157 2 97.471 106.35 85.215 79.585 95.544 53.31 90.17 41.405 3 135.54 133.37 98.410 89.361 126.55 79.87 107.01 58.649 4 152.12 159.51 112.12 103.52 135.10 108.81 130.97 61.056 2 1 76.167 87.355 78.124 72.195 83.954 43.486 80.853 25.699 2 90.544 106.05 95.439 80.398 97.610 55.439 98.398 46.378 3 99.202 117.08 106.25 97.403 116.21 86.925 107.403 71.642 4 119.14 139.52 115.58 115.46 130.47 100.54 125.468 76.557  0 5 1 64.479 78.545 67.191 58.235 74.407 36.419 70.435 21.244 2 89.655 96.405 85.366 78.981 89.541 55.661 98.918 45.112 3 105.24 120.94 109.01 99.210 111.24 79.015 119.21 55.880 4 126.50 135.79 122.47 119.58 125.19 95.471 139.58 73.385 2 1 70.571 82.998 73.164 64.771 79.547 39.486 75.761 23.446 2 86.454 89.699 89.101 82.913 85.254 58.212 93.157 47.254 3 102.15 117.20 117.80 100.24 100.30 82.172 109.97 60.351 4 105.85 126.79 128.58 115.38 123.21 90.309 110.58 76.245  -1 5 1 56.152 69.385 59.526 51.402 65.265 32.863 61.626 20.137 2 79.544 89.510 69.624 77.273 84.541 59.784 76.587 37.560 3 100.58 111.34 85.060 94.783 100.24 80.479 92.129 54.783 4 126.24 130.75 115.11 109.23 135.97 92.442 127.66 57.311 2 1 65.931 79.312 68.214 60.109 75.542 38.651 70.721 22.192 2 79.654 95.702 85.824 82.562 89.032 60.082 95.104 45.595 3 98.312 123.54 103.96 95.351 112.33 79.364 107.31 53.102 4 105.29 126.50 120.54 100.33 135.61 87.115 120.25 69.548Figure 5
Mode shapes for plates with boundary restraints: (a) CCCC; (b) SSSS; (c) SE2SE2.
(a) (b) (c)Figure 6
Mode shapes for circular cylindrical shallow shells with boundary restraints: (a) CCCC; (b) SSSS; (c) SE2SE2.
(a) (b) (c)Figure 7
Mode shapes for spherical shallow shells with boundary restraints: (a) CCCC; (b) SSSS; (c) SE2SE2.
(a) (b) (c)


### 3.2. FGSSS Resting on Winkler and Pasternak Foundations

In this subsection, the free vibration of Type1-2 (1-1-1) FGSSS on Winkler and Pasternak foundations with general boundary restraints is investigated. The geometric parameters and gradient index of the models area=b=1 m,h/a=0.1,Rx/a=2,p=0.5 and 5. Winkler coefficient KW and Pasternak coefficientKS are taken to be 10 and 100. Young’s modulus, mass density, and Poisson ratio of the structures are obtained in Table 1. As known in Table 6, numerous new solutions of fundamental frequenciesf (Hz) for the FG sandwich plates, circular cylindrical, spherical, and hyperbolic paraboloidal shallow shells on Winkler and Pasternak foundations with various boundaries (SSSS, CCCC, CFCF, SCSC, CSDCSD, CE1E1E1, SE2SE2, and E3E3E3E3) are presented. These results can be used as the benchmark solutions for future research in this field. From the results, it is clear that the variation of the Winkler and Pasternak coefficients has the significant effect on frequencies of the shallow shells. In order to further explore the influence of foundation coefficients KW andKS on the frequency parameter of the FGSSS, the detailed parametric study will be illustrated in the next subsection.Table 6
Fundamental frequenciesf (Hz) for Type 1-2 (1-1-1) plate, circular cylindrical spherical and hyperbolic paraboloidal shallow shell on Winkler and Pasternak foundations with various boundary restraints.
K W K S ρ Boundary Restraints SSSS CCCC CFCF SCSC CSDCSD CE1E1E1 SE2SE2 E3E3E3E3 FG sandwich plate  10 10 0.5 69.151 99.615 89.141 94.136 96.018 43.372 77.252 33.384 5 58.754 78.703 68.155 72.151 74.712 39.165 62.801 30.728 100 0.5 77.465 113.40 96.752 102.41 109.86 52.692 84.752 36.832 5 62.938 92.796 78.234 81.182 88.751 45.648 71.867 31.659 100 10 0.5 78.746 115.76 99.268 105.15 110.21 53.701 84.736 36.685 5 62.998 95.398 81.680 85.145 91.327 46.408 71.651 30.625 100 0.5 93.545 131.45 113.04 121.96 128.63 58.310 101.71 38.547 5 73.742 112.72 80.124 96.584 106.23 52.522 78.087 33.201  FG sandwich circular cylindrical shallow shell  10 10 0.5 63.850 87.642 78.359 85.209 83.678 40.214 71.065 29.548 5 54.243 69.245 61.953 69.321 65.251 35.247 56.428 27.651 100 0.5 70.955 102.96 90.145 94.547 98.932 46.367 75.992 31.287 5 58.987 93.970 71.625 78.520 90.240 41.140 70.222 29.748 100 10 0.5 72.568 109.64 92.323 98.336 105.63 46.215 80.571 32.214 5 60.962 94.353 74.254 81.351 89.321 40.162 72.678 28.967 100 0.5 84.647 125.21 103.35 112.42 118.96 50.271 89.524 35.124 5 68.166 110.79 78.547 90.244 106.71 45.214 75.215 31.258  FG sandwich spherical shallow shell  10 10 0.5 54.156 69.759 60.297 64.392 64.253 35.157 58.739 25.172 5 46.394 57.443 52.061 54.163 52.360 32.658 49.154 22.698 100 0.5 63.582 95.245 88.570 89.571 93.575 37.687 68.215 26.215 5 48.151 81.524 70.657 76.408 77.074 34.186 53.216 24.154 100 10 0.5 65.696 102.75 85.041 92.019 98.709 39.398 74.155 26.687 5 50.168 88.178 70.992 74.326 84.171 35.187 65.397 25.154 100 0.5 74.963 116.32 95.891 104.45 110.38 44.163 82.357 29.696 5 67.490 105.88 74.126 85.247 101.49 40.352 70.644 27.215  FG sandwich hyperbolic paraboloidal shallow shell  10 10 0.5 34.248 43.435 38.423 40.317 40.457 31.971 36.215 22.198 5 30.157 38.480 34.767 36.051 36.370 28.645 32.367 20.067 100 0.5 45.955 64.016 54.031 57.309 58.036 34.196 50.397 24.548 5 39.221 52.482 45.021 46.322 49.467 30.677 42.255 22.001 100 10 0.5 48.251 69.245 60.652 63.567 63.259 35.672 53.219 25.157 5 38.195 53.522 45.682 49.294 50.245 31.015 41.488 24.695 100 0.5 50.003 82.795 62.052 68.088 78.590 32.359 58.154 28.964 5 41.326 72.278 49.992 53.355 66.048 30.248 45.570 26.175


### 3.3. Studies on Free Vibration Parameters

In this subsection, the influence of vibration parameters including inertia, shear deformation, and foundation coefficient on the free vibration is illustrated. For CSST, the inertia and shear deformation are not considered in solving the vibration characteristic of FG shallow shells. Consequently, the modified FSDSST is addressed to tackle above problems.The difference of frequency parameters Ω obtained by CSST and FSDSST for simply supported Type1-2 FGSSS with various thickness ratios and curvature types is shown in Figure 8. The gradient index and geometric parameters used arep=2,a/b=1, andRx/a=2. It is clear that the difference of frequency parameters grows with the increase of thickness ratios for all curvature types. Mode shape of (2, 2) displayed larger errors than that of (1, 1), (2, 1), and (1, 2). Hence, CSST is not suitable for the FGSSS due to the important role of inertia and shear deformation in frequency calculation of high-order mode. Furthermore, the difference of frequency parameters Ω obtained by CSST and FSDSST for simply supported FG spherical shallow shells with different thickness ratios and anisotropic degrees are plotted in Figure 9. It can be seen that the difference of frequency parameters increases with the increase of anisotropic degrees, which suggests that the effect of inertia and shear deformation is magnified when anisotropic degrees are taken to be larger values. However, due to the influence of mode shape and anisotropic degrees, the difference of frequency parameters for some FG spherical shallow shells may show a decreasing trend after reaching a certain thickness ratio. Furthermore, the variations of frequency parameters Ω versus the foundation coefficients for simply supported Type 1-2 (1-1-1) FGSSS are presented in Figure 10. It can be easily obtained that the change in frequencies parameters is very small when the Winkler and Pasternak foundation coefficients are less than 105, while when the values of Winkler and Pasternak foundation coefficients are between 105 and 109, the frequencies parameters increase rapidly, which is a sensitive range. In the case of greater than 109, the effect of Winkler and Pasternak foundation coefficients on the frequencies parameters Ω can be neglected.Figure 8
The difference of frequency parameters Ω obtained by CSST and FSDSST for simply supported FGSSS with different thickness ratiosh/a and curvature types: (a) plate; (b) circular cylindrical shallow shell; (c) spherical shallow shell; (d) hyperbolic paraboloidal shallow shell. (m andn represent the modal number in the direction ofx andy, respectively).
(a) (b) (c) (d)Figure 9
The difference of frequency parameters Ω obtained by CSST and FSDSST for simply supported FG spherical shallow shells with different thickness ratiosh/a and anisotropic degreesEm/Ec: (a) (1, 1); (b) (2, 1); (c) (1, 2); (d) (2, 2).
(a) (b) (c) (d)Figure 10
Variation of the frequency parameters versus the foundation coefficients for simply supported FGSSS with various curvature types. (a) plate; (b) circular cylindrical shallow shell; (c) spherical shallow shell; (d) hyperbolic paraboloidal shallow shell.
(a) (b) (c) (d)


## 3.1. FGSSS with General Boundary Restraints

In the engineering applications, the letters F, C, S, and SD represent the completely free, completely clamped, simply supported, and shear-diaphragm supported boundary restraints, respectively. Besides the aforementioned classical boundary restraints, three elastic boundary restraints, denoted by E1, E2, and E3, are considered in this paper. Taking edge at x=0, for example, the seven boundaries and corresponding restraint parameters are shown in Table 2.Table 2
The corresponding elastic restraint parameters for various boundaries.
Boundary type Restraint parameters F k x 0 u = k x 0 v = k x 0 w = K x 0 x = K x 0 y = 0,  Nx=Nxy=Qx=Mx=Mxy=0 C k x 0 u = k x 0 v = k x 0 w = K x 0 x = K x 0 y = 1 0 7 D,   u=v=w=ϕx=ϕy=0 S k x 0 u = k x 0 v = k x 0 w = K x 0 y = 1 0 7 D ,    K x 0 x = 0,   u=v=w=Mx=ϕy=0 SD k x 0 v = k x 0 w = K x 0 y = 1 0 7 D ,   k x 0 u = K x 0 x = 0 E1 k x 0 w = K x 0 x = K x 0 y = 1 0 7 D ,   k x 0 u = k x 0 v = 15 D,   w=ϕx=ϕy=0,u≠0,v≠0 E2 k x 0 u = k x 0 v = K x 0 x = K x 0 y = 1 0 7 D ,  k x 0 w = 10 D,   u=v=ϕx=ϕy=0,w≠0 E3 k x 0 u = k x 0 v = k x 0 w = 1 0 7 D ,   K x 0 x = K x 0 y = 5 D,  u=v=w=0,ϕx≠0,ϕy≠0The reference bending stiffnessDm=Emh3/12(1-μm2). The accuracy and reliability of the current results are displayed in Tables 3 and 4. Table 3 compares the nondimensional frequency parameters ϖ=ωb2/hρm/Em of simply supported square FG sandwich plates with FG face sheets and homogeneous middle layer (1-0-1, 1-1-1, 1-2-1, 2-1-2, 2-2-1, and 1-8-1). The thickness-length ratios are taken to beh/a=0.01, 0.1, and 0.2. The gradient indexes are given asp=0.5, 5, and 10. Young’s modulus, mass density, and Poisson ratio are obtained in Table 1. The solutions given by Li et al. [42], by the 3D solutions and those of Hadji et al. [43], and by the RPT are provided for the comparisons. The differences are less than 1.530% and 2.015% for the worst case, respectively.Table 3
Comparison of non-dimensional frequency parametersϖ=ωb2/hρm/Em of simply supported square FG sandwich plates with homogeneous middle layer.
h/a p Theory ϖ 1-0-1 1-1-1 1-2-1 2-1-2 2-2-1 1-8-1 0.01 0.5 3D [42] 1.48244 1.56046 1.61915 1.52355 1.59031 1.76357 RPT [43] 1.48241 1.56042 1.61912 1.52353 1.59030 1.76354 Present 1.48248 1.56023 1.61913 1.52358 1.59034 1.76359 5 3D [42] 0.96563 1.06309 1.19699 0.99903 1.13020 1.56988 RPT [43] 0.96564 1.06309 1.19697 0.99903 1.13019 1.56985 Present 0.96566 1.06313 1.19697 0.99907 1.13024 1.56980 10 3D [42] 0.95042 1.01237 1.14408 0.95934 1.08065 1.54164 RPT [43] 0.95044 1.01236 1.14406 0.95937 1.08065 1.54162 Present 0.95041 1.01234 1.14408 0.95935 1.08065 1.54166  0.1 0.5 3D [42] 1.44614 1.52131 1.57668 1.48608 1.54926 1.71130 RPT [43] 1.44423 1.51921 1.57450 1.48408 1.54710 1.70901 Present 1.44485 1.51975 1.57461 1.48415 1.54750 1.70920 5 3D [42] 0.94476 1.04532 1.17567 0.98103 1.10983 1.52993 RPT [43] 0.94598 1.04465 1.17396 0.98184 1.10881 1.52792 Present 0.94552 1.04434 1.17355 0.98142 1.10874 1.52805 10 3D [42] 0.92727 0.99523 1.12466 0.94078 1.06104 1.50333 RPT [43] 0.92838 0.99550 1.12313 0.94296 1.06090 1.50138 Present 0.92836 0.99550 1.12313 0.94295 1.06080 1.50137  0.2 0.5 3D [42] 1.35358 1.42178 1.46940 1.39053 1.44535 1.58186 RPT [43] 1.34743 1.41508 1.46251 1.38410 1.43843 1.57476 Present 1.35015 1.41682 1.46368 1.38515 1.43901 1.57592 5 3D [42] 0.89086 0.99798 1.11900 0.93362 1.05607 1.42845 RPT [43] 0.89462 0.99545 1.11318 0.93594 1.05228 1.42197 Present 0.89504 0.99561 1.11285 0.93480 1.05496 1.42372 10 3D [42] 0.86833 0.94984 1.07290 0.89228 1.00949 1.40568 RPT [43] 0.87178 0.95033 1.06754 0.89918 1.00848 1.39932 Present 0.86963 0.95012 1.06950 0.89541 1.00887 1.40105Table 4
Comparisons of nondimensional frequency parametersϖ=ωb2/hρ0/E0 for Type 1-1 and FGSSS with completely free boundary restraints, respectively (a=b, h/b=0.1, and Rxy=∞).
Mode a/R=0.2 a/R=0.5 12×12 13 ×13 14×14 12×12 13×13 14×14 Jin et al. [44] Pre-sent Jin et al. [44] Pre-sent Jin etal. [44] Pre-sent Jin et al. [44] Pre-sent Jin et al. [44] Pre-sent Jin et al. [44] Pre-sent Type1-1: FG circular cylindrical shallow shell  1 0.903 0.907 0.902 0.905 0.902 0.905 0.901 0.903 0.901 0.902 0.901 0.902 2 1.352 1.361 1.352 1.359 1.352 1.358 1.405 1.411 1.405 1.411 1.405 1.410 3 1.662 1.670 1.662 1.666 1.662 1.659 1.734 1.738 1.734 1.735 1.734 1.735 4 2.279 2.285 2.278 2.281 2.278 2.281 2.272 2.275 2.271 2.276 2.271 2.276  Type1-1: FG spherical shallow shell  1 0.902 0.905 0.902 0.905 0.902 0.902 0.899 0.904 0.898 0.901 0.898 0.897 2 1.335 1.336 1.335 1.336 1.335 2.334 1.322 1.325 1.322 1.325 1.322 1.325 3 1.696 1.691 1.696 1.694 1.696 1.695 1.900 1.905 1.900 1.905 1.900 1.905 4 2.279 2.284 2.279 2.284 2.279 2.284 2.274 2.276 2.274 2.275 2.274 2.275  Type1-1: FG hyperbolic paraboloidal shallow shell  1 0.902 0.905 0.902 0.905 0.902 0.905 0.899 0.896 0.899 0.894 0.899 0.894 2 1.399 1.395 1.399 1.395 1.399 1.394 1.634 1.635 1.634 1.637 1.634 1.638 3 1.648 1.645 1.648 1.645 1.648 1.654 1.683 1.687 1.683 1.687 1.683 1.687 4 2.283 2.285 2.282 2.286 2.282 2.286 2.294 2.296 2.293 2.296 2.293 2.297  Type2-1: FG circular cylindrical shallow shell  1 0.835 0.845 0.835 0.845 0.835 0.844 0.833 0.835 0.833 0.834 0.833 0.834 2 1.249 1.245 1.249 1.245 1.249 1.245 1.302 1.306 1.302 1.308 1.302 1.308 3 1.537 1.550 1.537 1.552 1.537 1.553 1.604 1.612 1.604 1.610 1.604 1.610 4 2.105 2.110 2.105 2.110 2.104 2.108 2.104 2.107 2.104 2.107 2.104 2.109  Type2-1: FG spherical shallow shell  1 0.835 0.838 0.834 0.835 0.834 0.832 0.830 0.834 0.830 0.835 0.830 0.835 2 1.235 1.241 1.235 1.239 1.235 1.238 1.227 1.231 1.227 1.234 1.227 1.234 3 1.570 1.574 1.570 1.574 1.570 1.574 1.765 1.768 1.765 1.767 1.764 1.767 4 2.106 2.110 2.105 2.110 2.105 2.108 2.105 2.109 2.105 2.114 2.105 2.114  Type2-1: FG hyperbolic paraboloidal shallow shell  1 0.836 0.841 0.835 0.838 0.835 0.837 0.832 0.835 0.832 0.836 0.832 0.836 2 1.291 1.294 1.291 1.294 1.291 1.294 1.507 1.521 1.507 1.521 1.507 1.520 3 1.522 1.532 1.522 1.537 1.522 1.540 1.555 1.547 1.555 1.549 1.555 1.551 4 2.101 2.104 2.101 2.114 2.104 2.107 2.106 2.115 2.105 2.114 2.105 2.114Meanwhile, The frequency parametersϖ=ωb2/hρ0/E0  (ρ0=1kg/m3,  E0=1Gpa) for Type 1-1 and Type 2-1 FGSSS with completely free boundary restraints are presented in Table 4. All layers are assumed to be of equal thickness. The geometrical parameters are given asa=b andh/b=0.1. The length-radii ratios used for this analysis area/R=0.2 and 0.5, and the truncation number isM=N=12, 13, and 14. The frequency results are compared with the solutions given by Jin et al. [44]. A well agreement can be obtained.Numerous new results for fundamental frequenciesf (Hz) are presented in Table 5 for Type 1-1 FGSSS with a variety of general boundary restraints. The gradient index and geometrical parameters are given asp=5, a/b=1,  h/a=0.1,  Rx/a=2 and 5. And the thickness for each layer is set to be equal. Young’s modulus, mass density, and Poisson ratio are showed in Table 1. The boundary restraints including SSSS, CCCC, CFCF, SCSC, CSDCSD, CE1E1E1, SE2SE2, and E3E3E3E3 are considered. It is obvious that the values of fundamental frequencies corresponding to different boundaries are quite sensitive to the change of geometrical parameters. The frequencies of the FGSSS increase with the increase of radius ratio (Rx/Ry) and the decrease of radius-length ratio (Rx/a). The first four mode shapes for the plate, circular cylindrical shallow shell, and spherical shallow shell with CCCC, SSSS, and SE2SE2 boundaries are depicted in Figures 5, 6, and 7, respectively. The radius-length ratioRx/a is set as 2.Table 5
The first four frequenciesf (Hz) for Type 1-1 (1-1-1) FGSSS with various boundary restraints and radius-length ratios (p=5, a/b=1, and h/a=0.1).
R x/Ry R x/a Mode Boundary restraints SSSS CCCC CFCF SCSC CSDCSD CE1E1E1 SE2SE2 E3E3E3E3 Plate ∞ 1 66.545 79.857 72.148 58.102 75.279 39.310 74.196 21.214 2 93.851 105.51 96.792 73.858 99.082 51.903 98.154 35.150 3 138.47 140.48 123.95 97.593 123.27 71.192 102.21 45.211 4 159.68 173.65 143.54 120.97 135.36 98.115 125.34 59.633  +1 5 1 68.563 80.374 75.156 60.154 78.364 40.151 77.458 23.157 2 97.471 106.35 85.215 79.585 95.544 53.31 90.17 41.405 3 135.54 133.37 98.410 89.361 126.55 79.87 107.01 58.649 4 152.12 159.51 112.12 103.52 135.10 108.81 130.97 61.056 2 1 76.167 87.355 78.124 72.195 83.954 43.486 80.853 25.699 2 90.544 106.05 95.439 80.398 97.610 55.439 98.398 46.378 3 99.202 117.08 106.25 97.403 116.21 86.925 107.403 71.642 4 119.14 139.52 115.58 115.46 130.47 100.54 125.468 76.557  0 5 1 64.479 78.545 67.191 58.235 74.407 36.419 70.435 21.244 2 89.655 96.405 85.366 78.981 89.541 55.661 98.918 45.112 3 105.24 120.94 109.01 99.210 111.24 79.015 119.21 55.880 4 126.50 135.79 122.47 119.58 125.19 95.471 139.58 73.385 2 1 70.571 82.998 73.164 64.771 79.547 39.486 75.761 23.446 2 86.454 89.699 89.101 82.913 85.254 58.212 93.157 47.254 3 102.15 117.20 117.80 100.24 100.30 82.172 109.97 60.351 4 105.85 126.79 128.58 115.38 123.21 90.309 110.58 76.245  -1 5 1 56.152 69.385 59.526 51.402 65.265 32.863 61.626 20.137 2 79.544 89.510 69.624 77.273 84.541 59.784 76.587 37.560 3 100.58 111.34 85.060 94.783 100.24 80.479 92.129 54.783 4 126.24 130.75 115.11 109.23 135.97 92.442 127.66 57.311 2 1 65.931 79.312 68.214 60.109 75.542 38.651 70.721 22.192 2 79.654 95.702 85.824 82.562 89.032 60.082 95.104 45.595 3 98.312 123.54 103.96 95.351 112.33 79.364 107.31 53.102 4 105.29 126.50 120.54 100.33 135.61 87.115 120.25 69.548Figure 5
Mode shapes for plates with boundary restraints: (a) CCCC; (b) SSSS; (c) SE2SE2.
(a) (b) (c)Figure 6
Mode shapes for circular cylindrical shallow shells with boundary restraints: (a) CCCC; (b) SSSS; (c) SE2SE2.
(a) (b) (c)Figure 7
Mode shapes for spherical shallow shells with boundary restraints: (a) CCCC; (b) SSSS; (c) SE2SE2.
(a) (b) (c)


## 3.2. FGSSS Resting on Winkler and Pasternak Foundations

In this subsection, the free vibration of Type1-2 (1-1-1) FGSSS on Winkler and Pasternak foundations with general boundary restraints is investigated. The geometric parameters and gradient index of the models area=b=1 m,h/a=0.1,Rx/a=2,p=0.5 and 5. Winkler coefficient KW and Pasternak coefficientKS are taken to be 10 and 100. Young’s modulus, mass density, and Poisson ratio of the structures are obtained in Table 1. As known in Table 6, numerous new solutions of fundamental frequenciesf (Hz) for the FG sandwich plates, circular cylindrical, spherical, and hyperbolic paraboloidal shallow shells on Winkler and Pasternak foundations with various boundaries (SSSS, CCCC, CFCF, SCSC, CSDCSD, CE1E1E1, SE2SE2, and E3E3E3E3) are presented. These results can be used as the benchmark solutions for future research in this field. From the results, it is clear that the variation of the Winkler and Pasternak coefficients has the significant effect on frequencies of the shallow shells. In order to further explore the influence of foundation coefficients KW andKS on the frequency parameter of the FGSSS, the detailed parametric study will be illustrated in the next subsection.Table 6
Fundamental frequenciesf (Hz) for Type 1-2 (1-1-1) plate, circular cylindrical spherical and hyperbolic paraboloidal shallow shell on Winkler and Pasternak foundations with various boundary restraints.
K W K S ρ Boundary Restraints SSSS CCCC CFCF SCSC CSDCSD CE1E1E1 SE2SE2 E3E3E3E3 FG sandwich plate  10 10 0.5 69.151 99.615 89.141 94.136 96.018 43.372 77.252 33.384 5 58.754 78.703 68.155 72.151 74.712 39.165 62.801 30.728 100 0.5 77.465 113.40 96.752 102.41 109.86 52.692 84.752 36.832 5 62.938 92.796 78.234 81.182 88.751 45.648 71.867 31.659 100 10 0.5 78.746 115.76 99.268 105.15 110.21 53.701 84.736 36.685 5 62.998 95.398 81.680 85.145 91.327 46.408 71.651 30.625 100 0.5 93.545 131.45 113.04 121.96 128.63 58.310 101.71 38.547 5 73.742 112.72 80.124 96.584 106.23 52.522 78.087 33.201  FG sandwich circular cylindrical shallow shell  10 10 0.5 63.850 87.642 78.359 85.209 83.678 40.214 71.065 29.548 5 54.243 69.245 61.953 69.321 65.251 35.247 56.428 27.651 100 0.5 70.955 102.96 90.145 94.547 98.932 46.367 75.992 31.287 5 58.987 93.970 71.625 78.520 90.240 41.140 70.222 29.748 100 10 0.5 72.568 109.64 92.323 98.336 105.63 46.215 80.571 32.214 5 60.962 94.353 74.254 81.351 89.321 40.162 72.678 28.967 100 0.5 84.647 125.21 103.35 112.42 118.96 50.271 89.524 35.124 5 68.166 110.79 78.547 90.244 106.71 45.214 75.215 31.258  FG sandwich spherical shallow shell  10 10 0.5 54.156 69.759 60.297 64.392 64.253 35.157 58.739 25.172 5 46.394 57.443 52.061 54.163 52.360 32.658 49.154 22.698 100 0.5 63.582 95.245 88.570 89.571 93.575 37.687 68.215 26.215 5 48.151 81.524 70.657 76.408 77.074 34.186 53.216 24.154 100 10 0.5 65.696 102.75 85.041 92.019 98.709 39.398 74.155 26.687 5 50.168 88.178 70.992 74.326 84.171 35.187 65.397 25.154 100 0.5 74.963 116.32 95.891 104.45 110.38 44.163 82.357 29.696 5 67.490 105.88 74.126 85.247 101.49 40.352 70.644 27.215  FG sandwich hyperbolic paraboloidal shallow shell  10 10 0.5 34.248 43.435 38.423 40.317 40.457 31.971 36.215 22.198 5 30.157 38.480 34.767 36.051 36.370 28.645 32.367 20.067 100 0.5 45.955 64.016 54.031 57.309 58.036 34.196 50.397 24.548 5 39.221 52.482 45.021 46.322 49.467 30.677 42.255 22.001 100 10 0.5 48.251 69.245 60.652 63.567 63.259 35.672 53.219 25.157 5 38.195 53.522 45.682 49.294 50.245 31.015 41.488 24.695 100 0.5 50.003 82.795 62.052 68.088 78.590 32.359 58.154 28.964 5 41.326 72.278 49.992 53.355 66.048 30.248 45.570 26.175


## 3.3. Studies on Free Vibration Parameters

In this subsection, the influence of vibration parameters including inertia, shear deformation, and foundation coefficient on the free vibration is illustrated. For CSST, the inertia and shear deformation are not considered in solving the vibration characteristic of FG shallow shells. Consequently, the modified FSDSST is addressed to tackle above problems.The difference of frequency parameters Ω obtained by CSST and FSDSST for simply supported Type1-2 FGSSS with various thickness ratios and curvature types is shown in Figure 8. The gradient index and geometric parameters used arep=2,a/b=1, andRx/a=2. It is clear that the difference of frequency parameters grows with the increase of thickness ratios for all curvature types. Mode shape of (2, 2) displayed larger errors than that of (1, 1), (2, 1), and (1, 2). Hence, CSST is not suitable for the FGSSS due to the important role of inertia and shear deformation in frequency calculation of high-order mode. Furthermore, the difference of frequency parameters Ω obtained by CSST and FSDSST for simply supported FG spherical shallow shells with different thickness ratios and anisotropic degrees are plotted in Figure 9. It can be seen that the difference of frequency parameters increases with the increase of anisotropic degrees, which suggests that the effect of inertia and shear deformation is magnified when anisotropic degrees are taken to be larger values. However, due to the influence of mode shape and anisotropic degrees, the difference of frequency parameters for some FG spherical shallow shells may show a decreasing trend after reaching a certain thickness ratio. Furthermore, the variations of frequency parameters Ω versus the foundation coefficients for simply supported Type 1-2 (1-1-1) FGSSS are presented in Figure 10. It can be easily obtained that the change in frequencies parameters is very small when the Winkler and Pasternak foundation coefficients are less than 105, while when the values of Winkler and Pasternak foundation coefficients are between 105 and 109, the frequencies parameters increase rapidly, which is a sensitive range. In the case of greater than 109, the effect of Winkler and Pasternak foundation coefficients on the frequencies parameters Ω can be neglected.Figure 8
The difference of frequency parameters Ω obtained by CSST and FSDSST for simply supported FGSSS with different thickness ratiosh/a and curvature types: (a) plate; (b) circular cylindrical shallow shell; (c) spherical shallow shell; (d) hyperbolic paraboloidal shallow shell. (m andn represent the modal number in the direction ofx andy, respectively).
(a) (b) (c) (d)Figure 9
The difference of frequency parameters Ω obtained by CSST and FSDSST for simply supported FG spherical shallow shells with different thickness ratiosh/a and anisotropic degreesEm/Ec: (a) (1, 1); (b) (2, 1); (c) (1, 2); (d) (2, 2).
(a) (b) (c) (d)Figure 10
Variation of the frequency parameters versus the foundation coefficients for simply supported FGSSS with various curvature types. (a) plate; (b) circular cylindrical shallow shell; (c) spherical shallow shell; (d) hyperbolic paraboloidal shallow shell.
(a) (b) (c) (d)


## 4. Conclusions

This paper presents a modified Fourier method for free vibration of FGSSS on Winkler and Pasternak foundations based on FSDSST. Vibration characteristics of the shallow shells have been obtained by the energy function represented in the orthogonal coordinates, in which the displacement and rotation components are described as a combining form of standard double Fourier cosine series and several closed-form supplementary functions in order to eliminate the potential jumps and boundary discontinuities. The present method displays better reliability and accuracy. Then, numerous new results for FGSSS on Winkler and Pasternak foundations with various curvature types, geometrical parameters, and boundary restraints are presented, which may be used for benchmark solutions in the future research.In addition, a comprehensive investigation concentrated on the free vibration characteristics of FGSSS is performed. The results show that the inertia, shear deformation, and foundation coefficients are verified to affect significantly the vibration frequencies of FGSSS. The difference of frequency parameters increases with the increase of anisotropic degrees, which proves that the effect of inertia and shear deformation is magnified when anisotropic degrees are taken to be larger values. And the change in frequencies parameters is small when the Winkler and Pasternak foundation coefficients are less than 105, while when the values of Winkler and Pasternak foundation coefficients are between 105 and 109, the frequencies parameters increase rapidly. In the case of greater than 109, the effect of foundation coefficients on the frequencies parameters can be neglected.



---

*Source: 7527148-2019-03-13.xml*