# Free Vibration Analysis of the Unified Functionally Graded Shallow Shell with General Boundary Conditions

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

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## Abstract

The free vibration analysis of the functionally graded (FG) double curved shallow shell structures with general boundary conditions is investigated by an improved Fourier series method (IFSM). The material properties of FG structures are assumed to vary continuously in the thickness direction, according to the four graded parameters of the volume distribution function. Under the current framework, the displacement and rotation functions are set to a spectral form, including a double Fourier cosine series and two supplementary functions. These supplements can effectively eliminate the discontinuity and jumping phenomena of the displacement function along the edges. The formulation is based on the first-order shear deformation theory (FSDT) and Rayleigh-Ritz technique. This method can be universally applied to the free vibration analysis of the shallow shell, because it only needs to change the relevant parameters instead of modifying the basic functions or adapting solution procedures. The proposed method shows excellent convergence and accuracy, which has been compared with the results of the existing literatures. Numerous new results for free vibration analysis of FG shallow shells with various boundary conditions, geometric parameter, material parameters, gradient parameters, and volume distribution functions are investigated, which may serve as the benchmark solution for future researches.


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## Body

## 1. Introduction

The functionally gradient material (FGM) can be got by making the material properties of the structures change continuously in the thickness direction. This can make it possible to weaken or eliminate the internal defect and improve the bearing capacity of the structure and environmental adaptability. As the basic structure, shallow shell is widely used in the fields of aeronautics, astronautics, ship engineering, petrochemical container, and civil engineering. In the application of engineering practice, the main body of structures is made up of plate, cylindrical shell, spherical shell, hyperbolic paraboloidal shell, and so on. Therefore, it is of great significance to establish a unified analytical model to study the free vibration, buckling, and postbuckling characteristics of FG double curved shallow shells.The double curved shallow shell structures have many special forms. The plate is one of the special forms by setting the curvature of two directions to infinity. As the most common form, there have been many results of the research on FG plate. Matsunaga [1] analyzed the free vibration and buckling stresses of simply supported FG plates according to a 2D higher-order deformation theory (HSDT). Nguyen et al. [2] established a FSDT plate model to study a simply supported square plate and a cylindrical bending sandwich plate clamped at both ends. Zhao et al. [3, 4] employed FSDT to investigate the free vibration and buckling analysis of FG plates on the basis of the element-free kp-Ritz method. Shen et al. [5, 6] investigated the nonlinear bending of simply supported FG plates reinforced by single-walled carbon nanotubes subjected to a transverse uniform or sinusoidal load in thermal environments based on a HSDT. Belabed et al. [7] obtained an efficient and simple higher-order shear and normal deformation theory for bending and free vibration analysis of simply supported FG plates. Neves et al. [8, 9] presented a quasi-3D sinusoidal shear deformation theory and quasi-3D HSDT for the static and free vibration and buckling analysis of FG plates by applying Carrera’s unified formulation and meshless technique. Tounsi et al. [10] presented a refined trigonometric shear deformation theory (RTSDT) for the thermoelastic bending analysis of FG sandwich plates. Benachour et al. [11] used four-variable refined plate theory for free vibrations of FG clamped plates with arbitrary gradient based on Navier technique and Ritz method. Thai and Vo [12, 13] developed a new sinusoidal shear deformation theory for bending, buckling, and vibration of simply supported FG plates. Although a lot of researches about the vibration characteristics of FG plates have been done, there are many limitations in these studies, such as the various boundary conditions, varying gradient parameters, and volume distribution functions.The FG cylindrical shell can be seen as the single curved FG shallow shell. For this kind of cylindrical shell structure, the researches on the vibration characteristics have been very mature. Sheng and Wang [14, 15] investigated thermal vibration, buckling, and dynamic stability analysis of a FG cylindrical shell based on rotary inertia, FSDT, and modal expansion method. Vel [16] presented an exact elasticity solution for the free and forced vibration of simply supported FG cylindrical shells by using either the Mori–Tanaka method or asymptotic expansion homogenization (AEH) method. Shah et al. [17] studied the vibration behaviors of FG cylindrical shells on the Winkler and Pasternak foundations. Matsunaga [18] examined free vibration and stability of FG simply supported circular cylindrical shells by applying two-dimensional HSDT. Zahedinejad et al. [19] proposed a semi-analytical 3D free vibration analysis of functionally graded curved cylindrical panels with two opposite edges simply supported and arbitrary boundary conditions at the other edges based on the differential quadrature method (DQM). Santos et al. [20] developed a semi-analytical finite element model for the analysis of FG cylindrical shells by using the 3D linear elastic theory. Sepiani et al. [21] investigated vibration and buckling analysis of two-layered functionally graded cylindrical shell, considering the effects of transverse shear and rotary inertia based on FSDT and the classical shell theory (CST). Bagherizadeh et al. [22] studied the mechanical buckling of simply supported FG cylindrical shell on the Pasternak foundations based on HSDT. Beni et al. [23] analyzed the free vibration of size-dependent shear deformable simply supported FG cylindrical shell on the basis of FSDT and modified couple stress theory. From these results, it is not difficult to find that most of the boundary conditions of the cylindrical shell are simply supported boundary conditions or fixed boundary conditions. The effects of the elastic boundary conditions on the vibration are rarely mentioned.As a kind of shallow shell structure with double curvature, some structural forms are often used in engineering practice, such as spherical shells, hyperbolic paraboloidal shell, and drum shell. It is not difficult to find that these structures can be obtained by changing the geometric parameters and the coordinates. Therefore, a unified analytical model is developed to study the vibration characteristics of varying FG shallow shell structures. Tornabene [24–26] focused on the free vibration and dynamic behavior of FG conical, cylindrical shells and annular plates with a four-parameter power-law distribution based on the 2D GDM and FSDT. Besides, Tornabene et al. [27] applied higher-order equivalent single layer theories to study free vibrations of free-form FG doubly curved shells. Neves et al. [28] applied radial basis functions collocation to examine the free vibration analysis of FG shells which included spherical as well as cylindrical shell panels with all edges clamped or simply supported based on HSDT. Pradyumna et al. [29] carried out free vibration analysis of FG curved panels using a higher-order finite element formulation. Matsunaga [30] analyzed natural frequencies and buckling stresses of FG shallow shells based on 2D HSDT. Three types of simply supported shallow shells with positive, zero, and negative Gaussian curvature were considered. Alijani et al. [31] investigated the nonlinear forced vibrations of simply supported FG doubly curved shallow shells based on Donnell’s nonlinear shallow shell theory and the Galerkin method. Su et al. [32] presented a unified solution for vibration analysis of FG cylindrical, conical shells and annular plates with general boundary conditions by applying the FSDT and Rayleigh-Ritz procedure. Qu et al. [33] described a unified formulation for free, steady-state, and transient vibration analyses of FG shells with arbitrary boundary conditions on the basis of the FSDT. In this paper, the free vibrations of functionally graded cylindrical, conical, and spherical shells with different combinations of free, shear-diaphragm, simply supported, clamped, and elastic-supported boundary conditions were discussed. Alijani et al. [34] employed a pseudo-arclength continuation and collocation scheme to analyze thermal effects on nonlinear vibrations of simply supported FG doubly curved shells based on the HSDT. From the existing literatures, we can find that there have been a lot of discussions about the free vibration or buckling of the FG cylindrical shell, spherical shell, and conical shell. However, there are some limitations in the study of hyperbolic paraboloidal shell, which is a kind of the shallow shell structure. Moreover, the research on the multigradient parameters also has some limitations.Stimulated by the restriction of the varying boundary conditions and establishment of unified model in the current literature researches, the free vibration analysis of the unified FG shallow shell with varying general boundary constraints is developed. Based on the FSDT, an improved Fourier series solution is extended, which is previously proposed by Li et al. [35, 36], to study the vibration of FG shallow shell with general boundary constraints. The five displacement functions are all written as a feasible superposition of the 2D trigonometric series expansion, ignoring the influence of boundary conditions, in spectral form, as a double Fourier cosine series and two supplementary functions. On the basis of the traditional Fourier series, these supplementary functions are added to eliminate the discontinuous or jumping phenomenon in the boundaries which are regarded as periodic functions and defined within the entire coordinates of FG shallow shells. All these unknown coefficients are defined in the generalized coordinates which can be solved by Rayleigh-Ritz procedure [37–40]. It is very easy to realize the change of different boundary conditions by changing the value of the five springs in the plate’s four edges [41–45]. This method can realize parametric study without modifying the main program. Besides, the results obtained by the present method show good convergence and accuracy by being compared with those results obtained by literatures. It should be pointed out that the varying thickness has a great effect on the natural frequencies of the plates and shells. Many references have already been devoted to the study of the free vibration of multilayered plates [46–49] and shells [50–55] by using advanced analytical models based on variable through-the-thickness kinematic. Therefore, in this paper, the effects of various thicknesses on the natural frequencies of the FG shallow shells are manifested in various aspect ratios, such as length-thickness ratios and width-thickness ratios. Besides, we also study the effect of the varying boundary constraints, four kinds of gradient parameters, two different volume distribution functions, and various geometric parameters on the free vibration of the FG shallow shells.


## 2. Theoretical Formulations

### 2.1. Description of the Model

As shown in Figure1, a FG double curved shallow shell model is established to analyze the vibration characteristics. A coordinate system in the mid-surface of the double curved shallow shell is established. In this coordinate system, x, y, and z represent the length, width, and thickness directions. Besides, a and b are signed as the length along x and y directions and h is the thickness of the z direction. Rx and Ry are the radii of the curvature along x and y directions, respectively. An artificial spring technique is used to gain the vibration characteristics of FG shallow shell structures. Five types of springs are used to simulate the boundary conditions of the model, which are linear springs (ku,kv,kw) and rotational springs (Kx,Ky), respectively. Different boundary conditions can be realized by setting different stiffness values of the springs. For instance, the fixed support can be easily gained when the spring coefficients on the four edges are infinity which is set to 1E15 in the numerical calculation.Figure 1
Geometric parameters and coordinate system of FG shallow shell with elastic boundary conditions.Figure2 shows the various studied FG shallow shells which are used to validate the accuracy and versatility of the proposed method. It should be pointed out that we can easily obtain a rectangular plate structure by setting the Rx=Ry=∞, as shown in Figure 2(a). The cylindrical shell can be gained when Rx=∞, Rβ=R, as the model in Figure 2(c). The spherical shell and hyperbolic paraboloidal shell can be obtained when Rx=Ry=R and Rx=-Ry=R as shown in Figures 2(b) and 2(d).Figure 2
FG shallow shells with various structure types: (a) plate; (b) spherical shell; (c) cylindrical shell; (d) hyperbolic paraboloidal shell.
(a)
Plate
(b)
Spherical shell
(c)
Cylindrical shell
(d)
Hyperbolic paraboloidal shell


### 2.2. Admissible Displacement Functions

In order to eliminate the discontinuous or jumping phenomenon in the boundaries of FG shallow shells with arbitrary elastic supports, we propose an IFMS to express the displacement and rotation functions [59–61] which can be regarded as periodic functions: (1a)u0x,y,t=e-jωt∑m=0+∞∑n=0+∞Amn1cos⁡λmxcos⁡λny+∑m=-2-1∑n=0+∞Amn2sin⁡λmxcos⁡λny+∑n=-2-1∑m=0+∞Amn3cos⁡λmxsin⁡λny(1b)v0x,y,t=e-jωt∑m=0+∞∑n=0+∞Bmn1cos⁡λmxcos⁡λny+∑m=-2-1∑n=0+∞Bmn2sin⁡λmxcos⁡λny+∑n=-2-1∑m=0+∞Bmn3cos⁡λmxsin⁡λny(1c)w0x,y,t=e-jωt∑m=0+∞∑n=0+∞Cmn4cos⁡λmxcos⁡λny+∑m=-2-1∑n=0+∞Cmn2sin⁡λmxcos⁡λny+∑n=-2-1∑m=0+∞Cmn3cos⁡λmxsin⁡λny(1d)φxx,y,t=e-jωt∑m=0+∞∑n=0+∞Dmn1cos⁡λmxcos⁡λny+∑m=-2-1∑n=0+∞Dmn2sin⁡λmxcos⁡λny+∑n=-2-1∑m=0+∞Dmn3cos⁡λmxsin⁡λny(1e)φyx,y,t=e-jωt∑m=0+∞∑n=0+∞Emn1cos⁡λmxcos⁡λny+∑m=-2-1∑n=0+∞Emn2sin⁡λmxcos⁡λny+∑n=-2-1∑m=0+∞Emn3cos⁡λmxsin⁡λnyin which λm=mπ/a, λn=nπ/b. The unknown 2D Fourier coefficients expansions are Amni, Bmni, Cmni, Dmni, and Emni (i=1,2,3), respectively.


### 2.3. Kinematic Relations and Stress Resultants

The displacement of previously mentioned FG shallow shell model can be expressed according to the displacements and rotations of the middle surface [62, 63], which is based on the theory of the FSDT.(2)Ux,y,t=u0x,y,t+zφxx,y,tVx,y,t=v0x,y,t+zφyx,y,tWx,y,t=w0x,y,t,where t is the time variable. Then u0, v0, and w0 represent the middle surface displacements in x, y, and z direction, and the rotations of transverse normal for x- and y-axes are denoted by φx and φy, respectively. According to the small deformation elasticity theory and linear strains-displacement relations, the relationship of strain–stress can be expressed as(3)εxx=εxx0+zχxx,εyy=εyy0+zχyy,γyz=γyz0,γxz=γxz0,γxy=γxy0+zχxy,where εxx0, εxx0, and γxy0 denote the symbols of the normal and shear strains and χxx, χyy, and χxy express curvature and twist changes. Transverse shear strains (γxz,γyz) are regarded as constant, ignoring the thickness change. These corresponding quantities can be written as(4)εxx0=∂u0∂x+w0Rx,γxz0=∂w0∂x+φx-uRx,χxx=∂φx∂x,εyy0=∂v0∂y+w0Ry,γyz0=∂w0∂y+φy-v0Ry,χyy=∂φy∂y,γxy0=∂u0∂y+∂v0∂x,χxy=∂φx∂y+∂φy∂x.Based on the generalized Hooke’s law, the relationship between stress and strain can be obtained by(5)σxxσyyτxy=Q11Q120Q12Q11000Q66εxxεyyγxy,τxzτyz=Q6600Q66γxzγyz,where the normal stresses are σxx and σyy, shear stresses are τxz, τyz, and τxy, and Qef (e,f=1,2,6) represent the material coefficients which can be solved as follows:(6)Q11=Ez1-μ2z,Q12=μzEz1-μ2z,Q66=Ez21+μz.Young’s modulus, density, and Poisson’s ratio areE(z), ρ(z), and μ(z) which can be expressed as functions in the thickness direction.(7a)Ez=EC-ESVC+ES(7b)ρz=ρC-ρSVC+ρS(7c)μz=μC-μSVC+μSin which C and S represent the two different material constituents which show smooth and continuous mechanical behavior. Here, the volume fraction (VC) will be represented by two general types of volume distribution function based on the four gradient parameters (i,j,c,p).(8a)FGMΙi/j/c/p:VC=1-i12+zh+j12+zhcp(8b)FGMΙΙi/j/c/p:VC=1-i12-zh+j12-zhcp,where i, j, and c determine the variation of the material in the thickness direction and p is the gradient power-law exponent which can only take the positive. When the value of p is infinite, the structure of FGMΙ is composed of the pure C material, but the structure of FGMΙΙ is composed of the pure S material. When the value of p is 0, the material of structure is exactly the opposite. This relationship of volume fractions should be met as VC+VS=1.The constitutive equation of force and moment contacted with strain can be expressed as(9)NxNyNxyMxMyMxy=A11A120B11B120A12A110B12B11000A6600B66B11B120D11D120B12B110D12D11000B6600D66εxx0εyy0γxy0χxxχyyχxy,QxQy=κcA6600A66γxzγyz,where the stress (Nx,Ny), shear force (Nxy), torque (Mx,My), torsional moment (Mxy), and transverse shear force (Qx,Qy) resultants can be obtained by establishing the relationship with strains. Besides, the shear correction factor is Кc which is generally set to 5/6. The extensional, extensional-bending, and bending stiffness coefficients are Aef, Bef, and Def which can be gained by the integration over the thickness:(10)Aef,Bef,Def=∫-h/2h/2Qefz1,z,z2dz.Through the derivation of the formula, we can establish the relationship between internal force and kinematics of the provided FG shallow shell model.


### 2.4. Energy Expressions

As mentioned earlier, the main work of our study is the free vibration characteristics of FG shallow shell with elastic supports. Thus, we study the FG shallow shell vibrations based on Rayleigh-Ritz energy method [64–67] which can obtain more accurate results. The Lagrangian energy function for the shallow shell can be written as(11)L=T-Ushell-Usprings.The total kinetic energyT of the FG shallow shell is written as(12)T=12∫0a∫0bI0∂u0∂t2+∂v0∂t2+∂w0∂t2+2I1∂u0∂t∂φx∂t+∂v0∂t∂φy∂t+I2∂φx∂t2+∂ψy∂t2dydx,where I0, I1, and I2 are expressed as(13)I0I1I2=∫-h/2h/2ρz1,z1,z2dz.U is the strain energy stored inside the FG shallow shell:(14)Ushell=12∫0a∫0bNxεxx0+Nyεyy0+Nxyεxy0+Mxχxx+Myχyy+Mxyχxy+Qxγxz+Qyγyzdydx.By combining (4), (9), and (14) together, we can divide the strain energy formula into three parts, which can be rewritten as Ushell=Us+Ub+Ubs. In this expression, Us, Ub, and Ubs are stretching energy, bending energy, and bending–stretching coupling energy which can be expressed as follows:(15a)Us=12∫0a∫0bA11∂u0∂x+w0Rx2+A11∂v0∂y+w0Ry2+2A12∂u0∂x+w0Rx∂v0∂y+w0Ry+A66∂u0∂y+∂v0∂x2+κcA66∂w0∂x+φx-u0Rx2+κcA66∂w0∂y+φy-v0Ry2dydx(15b)Ub=12∫0a∫0bD11∂φx∂x2+D11∂φy∂y2+2D12∂φx∂x∂φy∂y+D66∂φx∂y+∂φy∂x2dydx(15c)Ubs=∫0a∫0bB11∂u0∂x+w0Rx∂φx∂x+B12∂u0∂x+w0Rx∂φy∂y+∂v0∂y+w0Ry∂φx∂x+B11∂v0∂y+w0Ry∂φy∂y+B66∂v0∂x+∂u0∂y∂φx∂y+∂φy∂xdydx.For the arbitrary elastic-supported case, the potential energy with the five kinds of restrained springs [68–74], which is the simulation of boundary conditions, should be written as (16)Usprings=12∫0b∫-h/2h/2kx0uu0x,y2+kx0vv0x,y2+kx0ww0x,y2+Kx0xφxx,y2+Kx0yφyx,y2x=0+kxauu0x,y2+kxavv0x,y2+kxaww0x,y2+Kxaxφxx,y2+Kxayφyx,y2x=adzdy+12∫0a∫-h/2h/2ky0uu0x,y2+ky0vv0x,y2+ky0ww0x,y2+Ky0xφxx,y2+Ky0yφyx,y2y=0+kybuu0x,y2+kybvv0x,y2+kybww0x,y2+Kybxφxx,y2+Kybyφyx,y2y=bdzdx.


### 2.5. Equation Solving

Partial derivative of unknown coefficients by Lagrange equationL is zero based on the Rayleigh-Ritz technology [75–78]. Therefore, a matrix form can be obtained by substituting (1a)–(1e) into (11)–(16), which can be written as (17)Κ-ω2MG=0,where the stiffness and mass matrixes for the FGM shallow shell are denoted by K and M. The unknown Fourier coefficients can be expressed as a vector in the form of(18)G=Gu0Gv0Gw0GφxGφyT.So far, the natural frequencies and coefficientsG can be easily got by solving the eigenvalues and eigenvectors of a standard matrix in (18). Substitute G into (1a)–(1e) and get the specific displacement expressions of the FG shallow shell. Then, the corresponding mode shape can be immediately obtained.


## 2.1. Description of the Model

As shown in Figure1, a FG double curved shallow shell model is established to analyze the vibration characteristics. A coordinate system in the mid-surface of the double curved shallow shell is established. In this coordinate system, x, y, and z represent the length, width, and thickness directions. Besides, a and b are signed as the length along x and y directions and h is the thickness of the z direction. Rx and Ry are the radii of the curvature along x and y directions, respectively. An artificial spring technique is used to gain the vibration characteristics of FG shallow shell structures. Five types of springs are used to simulate the boundary conditions of the model, which are linear springs (ku,kv,kw) and rotational springs (Kx,Ky), respectively. Different boundary conditions can be realized by setting different stiffness values of the springs. For instance, the fixed support can be easily gained when the spring coefficients on the four edges are infinity which is set to 1E15 in the numerical calculation.Figure 1
Geometric parameters and coordinate system of FG shallow shell with elastic boundary conditions.Figure2 shows the various studied FG shallow shells which are used to validate the accuracy and versatility of the proposed method. It should be pointed out that we can easily obtain a rectangular plate structure by setting the Rx=Ry=∞, as shown in Figure 2(a). The cylindrical shell can be gained when Rx=∞, Rβ=R, as the model in Figure 2(c). The spherical shell and hyperbolic paraboloidal shell can be obtained when Rx=Ry=R and Rx=-Ry=R as shown in Figures 2(b) and 2(d).Figure 2
FG shallow shells with various structure types: (a) plate; (b) spherical shell; (c) cylindrical shell; (d) hyperbolic paraboloidal shell.
(a)
Plate
(b)
Spherical shell
(c)
Cylindrical shell
(d)
Hyperbolic paraboloidal shell


## 2.2. Admissible Displacement Functions

In order to eliminate the discontinuous or jumping phenomenon in the boundaries of FG shallow shells with arbitrary elastic supports, we propose an IFMS to express the displacement and rotation functions [59–61] which can be regarded as periodic functions: (1a)u0x,y,t=e-jωt∑m=0+∞∑n=0+∞Amn1cos⁡λmxcos⁡λny+∑m=-2-1∑n=0+∞Amn2sin⁡λmxcos⁡λny+∑n=-2-1∑m=0+∞Amn3cos⁡λmxsin⁡λny(1b)v0x,y,t=e-jωt∑m=0+∞∑n=0+∞Bmn1cos⁡λmxcos⁡λny+∑m=-2-1∑n=0+∞Bmn2sin⁡λmxcos⁡λny+∑n=-2-1∑m=0+∞Bmn3cos⁡λmxsin⁡λny(1c)w0x,y,t=e-jωt∑m=0+∞∑n=0+∞Cmn4cos⁡λmxcos⁡λny+∑m=-2-1∑n=0+∞Cmn2sin⁡λmxcos⁡λny+∑n=-2-1∑m=0+∞Cmn3cos⁡λmxsin⁡λny(1d)φxx,y,t=e-jωt∑m=0+∞∑n=0+∞Dmn1cos⁡λmxcos⁡λny+∑m=-2-1∑n=0+∞Dmn2sin⁡λmxcos⁡λny+∑n=-2-1∑m=0+∞Dmn3cos⁡λmxsin⁡λny(1e)φyx,y,t=e-jωt∑m=0+∞∑n=0+∞Emn1cos⁡λmxcos⁡λny+∑m=-2-1∑n=0+∞Emn2sin⁡λmxcos⁡λny+∑n=-2-1∑m=0+∞Emn3cos⁡λmxsin⁡λnyin which λm=mπ/a, λn=nπ/b. The unknown 2D Fourier coefficients expansions are Amni, Bmni, Cmni, Dmni, and Emni (i=1,2,3), respectively.


## 2.3. Kinematic Relations and Stress Resultants

The displacement of previously mentioned FG shallow shell model can be expressed according to the displacements and rotations of the middle surface [62, 63], which is based on the theory of the FSDT.(2)Ux,y,t=u0x,y,t+zφxx,y,tVx,y,t=v0x,y,t+zφyx,y,tWx,y,t=w0x,y,t,where t is the time variable. Then u0, v0, and w0 represent the middle surface displacements in x, y, and z direction, and the rotations of transverse normal for x- and y-axes are denoted by φx and φy, respectively. According to the small deformation elasticity theory and linear strains-displacement relations, the relationship of strain–stress can be expressed as(3)εxx=εxx0+zχxx,εyy=εyy0+zχyy,γyz=γyz0,γxz=γxz0,γxy=γxy0+zχxy,where εxx0, εxx0, and γxy0 denote the symbols of the normal and shear strains and χxx, χyy, and χxy express curvature and twist changes. Transverse shear strains (γxz,γyz) are regarded as constant, ignoring the thickness change. These corresponding quantities can be written as(4)εxx0=∂u0∂x+w0Rx,γxz0=∂w0∂x+φx-uRx,χxx=∂φx∂x,εyy0=∂v0∂y+w0Ry,γyz0=∂w0∂y+φy-v0Ry,χyy=∂φy∂y,γxy0=∂u0∂y+∂v0∂x,χxy=∂φx∂y+∂φy∂x.Based on the generalized Hooke’s law, the relationship between stress and strain can be obtained by(5)σxxσyyτxy=Q11Q120Q12Q11000Q66εxxεyyγxy,τxzτyz=Q6600Q66γxzγyz,where the normal stresses are σxx and σyy, shear stresses are τxz, τyz, and τxy, and Qef (e,f=1,2,6) represent the material coefficients which can be solved as follows:(6)Q11=Ez1-μ2z,Q12=μzEz1-μ2z,Q66=Ez21+μz.Young’s modulus, density, and Poisson’s ratio areE(z), ρ(z), and μ(z) which can be expressed as functions in the thickness direction.(7a)Ez=EC-ESVC+ES(7b)ρz=ρC-ρSVC+ρS(7c)μz=μC-μSVC+μSin which C and S represent the two different material constituents which show smooth and continuous mechanical behavior. Here, the volume fraction (VC) will be represented by two general types of volume distribution function based on the four gradient parameters (i,j,c,p).(8a)FGMΙi/j/c/p:VC=1-i12+zh+j12+zhcp(8b)FGMΙΙi/j/c/p:VC=1-i12-zh+j12-zhcp,where i, j, and c determine the variation of the material in the thickness direction and p is the gradient power-law exponent which can only take the positive. When the value of p is infinite, the structure of FGMΙ is composed of the pure C material, but the structure of FGMΙΙ is composed of the pure S material. When the value of p is 0, the material of structure is exactly the opposite. This relationship of volume fractions should be met as VC+VS=1.The constitutive equation of force and moment contacted with strain can be expressed as(9)NxNyNxyMxMyMxy=A11A120B11B120A12A110B12B11000A6600B66B11B120D11D120B12B110D12D11000B6600D66εxx0εyy0γxy0χxxχyyχxy,QxQy=κcA6600A66γxzγyz,where the stress (Nx,Ny), shear force (Nxy), torque (Mx,My), torsional moment (Mxy), and transverse shear force (Qx,Qy) resultants can be obtained by establishing the relationship with strains. Besides, the shear correction factor is Кc which is generally set to 5/6. The extensional, extensional-bending, and bending stiffness coefficients are Aef, Bef, and Def which can be gained by the integration over the thickness:(10)Aef,Bef,Def=∫-h/2h/2Qefz1,z,z2dz.Through the derivation of the formula, we can establish the relationship between internal force and kinematics of the provided FG shallow shell model.


## 2.4. Energy Expressions

As mentioned earlier, the main work of our study is the free vibration characteristics of FG shallow shell with elastic supports. Thus, we study the FG shallow shell vibrations based on Rayleigh-Ritz energy method [64–67] which can obtain more accurate results. The Lagrangian energy function for the shallow shell can be written as(11)L=T-Ushell-Usprings.The total kinetic energyT of the FG shallow shell is written as(12)T=12∫0a∫0bI0∂u0∂t2+∂v0∂t2+∂w0∂t2+2I1∂u0∂t∂φx∂t+∂v0∂t∂φy∂t+I2∂φx∂t2+∂ψy∂t2dydx,where I0, I1, and I2 are expressed as(13)I0I1I2=∫-h/2h/2ρz1,z1,z2dz.U is the strain energy stored inside the FG shallow shell:(14)Ushell=12∫0a∫0bNxεxx0+Nyεyy0+Nxyεxy0+Mxχxx+Myχyy+Mxyχxy+Qxγxz+Qyγyzdydx.By combining (4), (9), and (14) together, we can divide the strain energy formula into three parts, which can be rewritten as Ushell=Us+Ub+Ubs. In this expression, Us, Ub, and Ubs are stretching energy, bending energy, and bending–stretching coupling energy which can be expressed as follows:(15a)Us=12∫0a∫0bA11∂u0∂x+w0Rx2+A11∂v0∂y+w0Ry2+2A12∂u0∂x+w0Rx∂v0∂y+w0Ry+A66∂u0∂y+∂v0∂x2+κcA66∂w0∂x+φx-u0Rx2+κcA66∂w0∂y+φy-v0Ry2dydx(15b)Ub=12∫0a∫0bD11∂φx∂x2+D11∂φy∂y2+2D12∂φx∂x∂φy∂y+D66∂φx∂y+∂φy∂x2dydx(15c)Ubs=∫0a∫0bB11∂u0∂x+w0Rx∂φx∂x+B12∂u0∂x+w0Rx∂φy∂y+∂v0∂y+w0Ry∂φx∂x+B11∂v0∂y+w0Ry∂φy∂y+B66∂v0∂x+∂u0∂y∂φx∂y+∂φy∂xdydx.For the arbitrary elastic-supported case, the potential energy with the five kinds of restrained springs [68–74], which is the simulation of boundary conditions, should be written as (16)Usprings=12∫0b∫-h/2h/2kx0uu0x,y2+kx0vv0x,y2+kx0ww0x,y2+Kx0xφxx,y2+Kx0yφyx,y2x=0+kxauu0x,y2+kxavv0x,y2+kxaww0x,y2+Kxaxφxx,y2+Kxayφyx,y2x=adzdy+12∫0a∫-h/2h/2ky0uu0x,y2+ky0vv0x,y2+ky0ww0x,y2+Ky0xφxx,y2+Ky0yφyx,y2y=0+kybuu0x,y2+kybvv0x,y2+kybww0x,y2+Kybxφxx,y2+Kybyφyx,y2y=bdzdx.


## 2.5. Equation Solving

Partial derivative of unknown coefficients by Lagrange equationL is zero based on the Rayleigh-Ritz technology [75–78]. Therefore, a matrix form can be obtained by substituting (1a)–(1e) into (11)–(16), which can be written as (17)Κ-ω2MG=0,where the stiffness and mass matrixes for the FGM shallow shell are denoted by K and M. The unknown Fourier coefficients can be expressed as a vector in the form of(18)G=Gu0Gv0Gw0GφxGφyT.So far, the natural frequencies and coefficientsG can be easily got by solving the eigenvalues and eigenvectors of a standard matrix in (18). Substitute G into (1a)–(1e) and get the specific displacement expressions of the FG shallow shell. Then, the corresponding mode shape can be immediately obtained.


## 3. Numerical Results and Discussion

In this part, a number of examples for the different FG shallow shells with varying elastic boundary conditions will be discussed. In the numerical calculation, the fast convergence of this method is proved by setting different truncation value of the displacement function expressions. By comparing with the available results extracted from the existing literatures, it proves the accuracy of the proposed method and model in this paper. It should be noted that the frequency parameters used in this paper areΩ=ωa2/h/(ρc/Ec)1/2. In order to facilitate the description, this paper uses alphabetic strings to represent the boundary constraints (BC). For example, SCFE represents a kind of boundary constraints by setting the spring values of the four edges to the numbers shown in Table 1. Specifically, S is simply supported, C is clamped edge, F is free edge, and E is elastic support. Besides, the material properties involved in this paper are presented here, and we will not give special explanation anymore in the following example:Al:E=70 GPa; μ=0.3; ρ=2707 kg/m3Al2O3: E=380 GPa; μ=0.3; ρ=3800 kg/m3ZrO2: E=151 GPa; μ=0.3; ρ=3000 kg/m3.Table 1
Spring stiffness values for the various boundary conditions of FGM shallow shell.
Edges
BC
Spring stiffness values
k
u
k
v
k
w
K
x
K
y
x
=
0
/
x
=
a
F
0
0
0
0
0
C
1015
1015
1015
1015
1015
S
0
1015
1015
0
1015
E1
108
1015
1015
1015
1015
E2
108
108
1015
1015
1015
E3
108
108
108
1015
1015
y
=
0
/
y
=
b
F
0
0
0
0
0
C
1015
1015
1015
1015
1015
S
1015
0
1015
1015
0
E1
1015
108
1015
1015
1015
E2
108
108
1015
1015
1015
E3
108
108
108
1015
1015


### 3.1. Convergence Study

As previously mentioned in (1a)–(1e), we can get the convergence criterion by the truncation value of M and N in the displacement function expressions in the numerical calculation. Therefore, the values of M and N are the direct representation of the convergence for the proposed method. In Table 2, we can get the convergence of first eight frequency parameters Ω for the various shallow shells structures with SSSS boundary constraints whose functionally graded material is Al/Al2O3. Their geometrical dimensions are a/b=1, h/a=0.1, and R=1 m.Table 2
Convergence of frequency parameterΩ for Al/Al2O3 shallow shells with SSSS boundary constraints (FGMΙ(i=1/j=0/c/p=1), R=1, a/b=1, h=0.1).
Type
M
×
N
Mode number
1
2
3
4
5
6
7
8
Plate
8
×
8
4.417
10.585
10.585
16.195
16.195
16.303
19.910
19.910
10
×
10
4.417
10.585
10.585
16.195
16.195
16.303
19.908
19.908
12
×
12
4.417
10.585
10.585
16.195
16.195
16.303
19.907
19.907
14
×
14
4.417
10.585
10.585
16.195
16.195
16.303
19.906
19.906
16
×
16
4.417
10.585
10.585
16.195
16.195
16.303
19.906
19.906
18
×
18
4.417
10.585
10.585
16.195
16.195
16.303
19.906
19.906
Cylindrical shell
8
×
8
5.739
10.493
12.308
16.053
16.195
16.613
19.880
21.162
10
×
10
5.739
10.493
12.308
16.053
16.195
16.613
19.877
21.159
12
×
12
5.739
10.493
12.308
16.053
16.195
16.613
19.877
21.159
14
×
14
5.739
10.493
12.308
16.053
16.195
16.613
19.876
21.158
16
×
16
5.739
10.493
12.308
16.053
16.195
16.613
19.876
21.158
18
×
18
5.739
10.493
12.308
16.053
16.195
16.613
19.876
21.158
Spherical shell
8
×
8
8.946
13.139
13.139
16.053
16.053
18.082
21.422
21.422
10
×
10
8.946
13.139
13.139
16.053
16.053
18.082
21.419
21.419
12
×
12
8.946
13.139
13.139
16.053
16.053
18.082
21.419
21.419
14
×
14
8.946
13.139
13.139
16.053
16.053
18.082
21.418
21.418
16
×
16
8.946
13.139
13.139
16.053
16.053
18.082
21.418
21.418
18
×
18
8.946
13.139
13.139
16.053
16.053
18.082
21.418
21.418
Hyperbolic paraboloidal shell
8
×
8
3.609
10.940
11.121
15.434
16.053
16.334
20.271
20.706
10
×
10
3.609
10.940
11.121
15.434
16.053
16.334
20.269
20.703
12
×
12
3.609
10.940
11.121
15.434
16.053
16.334
20.268
20.703
14
×
14
3.609
10.940
11.121
15.434
16.053
16.334
20.268
20.702
16
×
16
3.609
10.940
11.121
15.434
16.053
16.334
20.268
20.702
18
×
18
3.609
10.940
11.121
15.434
16.053
16.334
20.268
20.702The method presented in this paper shows fast convergence which can be found in Table1. For example, the biggest difference for the worst case of the plate which is made by the contrast of 8 × 8 and 18 × 18 is less than 0.02%. It is not difficult to find that the biggest difference for the contrast of 8 × 8 and 18 × 18 of the cylindrical shell is also less than 0.02%. Besides, for the spherical shell and hyperbolic paraboloidal shell, the biggest difference between these two truncated values is 0.017% and 0.019%. Through these results, we can find that the method shows good uniform convergence for different structures which shows the accuracy of the unified model. In addition, it is necessary to point out that, in order to make the results obtained in the following numerical examples more accurate, we use the invariant truncation values of M=N=14.


### 3.2. Validation and Some New Results

In this section, the accuracy of the proposed method is verified by being compared with the literature results. In addition, this paper also gives some new results, which can provide a benchmark for the future research. The first five frequency parametersΩ for Al/Al2O3 square plate with SSSS boundary constraints and varying length-thickness ratios a/h and gradient power-law exponent parameter p are given in Table 3. The geometric parameters are a=b=1 m and the material parameters of alumina and aluminum have been given previously. The results of the present method are compared with those of Jin et al. [56] which show good agreement. From Table 3, we can easily find that the frequency parameter Ω of the square plate is decreasing gradually with the increase of a/h and p. In Table 4, the first frequency parameters Ω for Al/ZrO2 plate with various boundary constraints are given. The influence of different length-width ratio a/b, thickness-width ratio h/b, and gradient power-law exponent parameter p on the plate frequency is also discussed. The results of the free vibration under SSSS boundary conditions are compared with those of Kiani et al. [57], which show good agreement. It is worth noting that these results in Table 4 are consistent with the results obtained in Table 3.Table 3
Frequency parametersΩ for Al/Al2O3 square plate with SSSS boundary constraints, varying geometric parameters, and gradient power-law exponent p (FGMΙ(i=1/j=0/c/p)).
h
/
a
p
Method
Mode number
1
2
3
4
5
0.1
0
Present
5.769
13.76
13.76
19.48
19.48
[56]
5.779
13.81
13.81
19.48
19.48
1
Present
4.417
10.59
10.59
16.19
16.19
[56]
4.428
10.63
10.63
16.20
16.20
5
Present
3.785
9.004
9.004
12.63
12.63
[56]
3.774
8.931
8.931
12.64
12.64
0.2
0
Present
5.280
9.741
9.741
11.55
11.55
[56]
5.304
9.742
9.742
11.65
11.65
1
Present
4.076
8.086
8.086
9.005
9.005
[56]
4.100
8.089
8.089
9.108
9.108
5
Present
3.447
6.297
6.297
7.496
7.496
[56]
3.405
6.296
6.296
7.344
7.344Table 4
Frequency parametersΩ for Al/ZrO2 plate with various boundary constraints, varying geometric parameters, and gradient power-law exponent p (FGMΙΙ(i=1/j=0/c/p)).
BC
a
/
b
h
/
b
Method
p
=
0
p
=
0.5
p
=
1
p
=
2
p
=
5
p
=
10
p
=
∞
SSSS
1
0.1
Present
5.7693
5.1932
4.9504
4.7750
4.6315
4.5071
4.1352
[57]
5.7693
5.1932
4.9504
4.7750
4.6315
4.5071
4.1353
0.02
Present
5.9647
5.3631
5.1135
4.9382
4.7988
4.6710
4.2753
[57]
5.9647
5.3631
5.1135
4.9382
4.7988
4.6710
4.2753
2
0.1
Present
14.6070
13.1429
12.5296
12.0910
11.7358
11.4215
10.4698
0.02
Present
14.9198
13.4149
12.7906
12.3523
12.0040
11.6844
10.6940
CCCC
1
0.1
Present
9.8425
8.8831
8.4665
8.1474
7.8691
7.6511
7.0548
0.02
Present
10.8410
9.7492
9.2954
8.9754
8.7198
8.4872
7.7705
2
0.1
Present
27.5886
24.8797
23.7153
22.8390
22.0870
21.4798
19.7746
0.02
Present
29.6598
26.6713
25.4299
24.5558
23.8588
23.2228
21.2592
CSCS
1
0.1
Present
8.0703
7.2786
6.9375
6.6802
6.4591
6.2816
5.7845
0.02
Present
8.7301
7.8505
7.4851
7.2277
7.0225
6.8352
6.2574
2
0.1
Present
16.0659
14.4609
13.7857
13.2987
12.9004
12.5534
11.5155
0.02
Present
16.5502
14.8811
14.1886
13.7021
13.3152
12.9606
11.8626
E1E2E1E2
1
0.1
Present
1.6260
1.6528
1.6666
1.6807
1.6951
1.7018
1.7098
0.02
Present
10.8410
9.7492
9.2954
8.9754
8.7198
8.4872
7.7705
2
0.1
Present
5.6307
5.7230
5.7706
5.8192
5.8686
5.8913
5.9186
0.02
Present
29.6598
26.6713
25.4299
24.5559
23.8588
23.2228
21.2592
E2E3E2E3
1
0.1
Present
1.6249
1.6514
1.6651
1.6790
1.6931
1.6995
1.7073
0.02
Present
10.4114
9.4120
8.9917
8.6918
8.4499
8.2330
7.5680
2
0.1
Present
5.6174
5.7067
5.7522
5.7982
5.8442
5.8650
5.8896
0.02
Present
27.4294
25.0188
23.9872
23.2505
22.6568
22.1105
20.3974Cylindrical shell, as a special form of shallow shell with double curvature, is well known and widely used in engineering practice. The free vibration characteristics of FG cylindrical shells are studied based on the unified doubly curved shallow shell model established in this paper. Table5 gives the first nine frequencies of FGMΙi=1/j=0/c/p cylindrical shell with FFCF boundary constraints and varying gradient parameter of power-law exponent p. There are two kinds of functionally graded materials whose properties are Ec=168 GPa, Em=70 GPa, μc=μm=0.3, pc=5700 kg/m3, pm=2707 kg/m3. The geometric parameters of cylindrical shell are h=0.1 m, R=1 m, a=2 m, b=2π/3 m. The first frequency parameter Ω for Al/ZrO2 cylindrical shell with SSSS boundary constraints is given in Table 6. At the same time, the effects of various geometric parameters, including length-thickness ratio a/h and radius-length ratio R/a, and power-law exponent p are also discussed. The accuracy of the results is verified by being compared with the related references [26, 57, 58]. As shown in Tables 5 and 6, the frequency of the cylindrical shell is reduced gradually when the radius R, the thickness h, or the gradient power-law exponent p increase. In Table 7, we give the first-order frequency parameter Ω of free vibration of Al/Al2O3 cylindrical shells with different boundary conditions and varying geometric parameters of a/h and h/b. In the studied cylindrical shell model, the radius is R=2 m. In this table, the effects of two general types of volume distribution function (FGMΙi=1/j=0/c/p=1 and FGMΙΙi=1/j=0/c/p=1) on the frequency of cylindrical shells are also given. It should be pointed out that the frequencies of the cylindrical shell under FGMI and FGMII have some small changes when the shell has the same boundary conditions and geometric parameters.Table 5
Frequencies (Hz) ofFGMΙ(i=1/j=0/c/p) cylindrical shell with FFCF boundary constraints and varying gradient power-law exponent p (Ec=168 GPa, Em=70 GPa, μc=μm=0.3, pc=5700 kg/m3, pm=2707 kg/m3, h=0.1, R=1, a=2, b=2π/3).
p
Method
Mode number
1
2
3
4
5
6
7
8
9
0
Present
61.03
94.83
153.13
241.67
276.26
291.68
355.13
450.74
512.49
[58]
61.02
94.82
153.13
241.39
275.83
291.45
355.14
450.76
511.92
[26]
61.06
94.92
153.11
241.75
276.37
291.94
355.07
450.82
512.48
0.6
Present
59.11
92.94
148.16
234.52
266.87
287.37
342.44
442.01
495.33
[58]
59.24
93.14
148.51
234.79
267.06
287.85
343.22
443.06
495.89
[26]
59.13
93.03
148.14
234.60
266.97
287.63
342.38
442.09
495.31
1
Present
58.79
92.39
147.51
233.23
265.51
285.52
341.09
439.41
492.81
[58]
58.96
92.65
147.94
233.64
265.86
286.16
342.08
440.72
493.67
[26]
58.82
92.48
147.49
233.31
265.61
285.77
341.04
439.49
492.80
5
Present
59.93
91.45
151.59
236.35
272.08
278.81
353.89
435.11
504.54
[58]
60.09
91.69
152.02
236.75
272.42
279.34
354.89
436.34
505.35
[26]
59.96
91.54
151.57
236.42
272.19
279.06
353.83
435.18
504.56
20
Present
59.13
90.21
149.31
233.26
268.47
275.15
348.37
429.12
497.85
[58]
59.19
90.29
149.49
233.27
268.37
275.23
348.80
429.63
497.87
[26]
59.16
90.30
149.29
233.32
268.57
275.41
348.32
429.19
497.87
50
Present
58.18
89.52
146.47
229.93
263.84
274.07
340.84
425.59
489.36
[58]
58.20
89.55
146.55
229.79
263.58
273.98
341.04
425.83
489.06
[26]
58.21
89.60
146.45
230.00
263.95
274.32
340.79
425.66
489.36
100
Present
57.72
89.21
145.09
228.32
261.56
273.66
337.14
424.02
485.17
[58]
57.73
89.21
145.14
228.12
261.23
273.51
337.25
424.15
484.75
[26]
57.75
89.29
145.08
228.39
261.66
273.90
337.09
424.10
485.16
∞
Present
57.17
88.83
143.43
226.37
258.77
273.21
332.64
422.20
480.04
[58]
57.16
88.81
143.43
226.11
258.36
273.00
332.65
422.21
479.49
[26]
57.19
88.91
143.41
226.44
258.87
273.46
332.59
422.27
480.02Table 6
Frequency parametersΩ for Al/ZrO2 cylindrical shell with SSSS boundary constraints and varying geometric parameters and gradient power-law exponent p (FGMΙΙ(i=1/j=0/c/p), a/b=1).
a
/
h
R
/
a
Method
p
=
0
p
=
0.5
p
=
1
p
=
2
p
=
5
p
=
10
p
=
∞
10
2
Present
6.1667
5.5612
5.2958
5.0890
4.9112
4.7782
4.4201
[57]
6.1552
5.5522
5.2876
5.0812
4.9032
4.7700
4.4119
3
Present
5.9510
5.3592
5.1052
4.9150
4.7563
4.6288
4.2655
[57]
5.9456
5.3548
5.1012
4.9111
4.7524
4.6249
4.2616
5
Present
5.8357
5.2521
5.0046
4.8235
4.6749
4.5501
4.1829
[57]
5.8337
5.2504
5.0031
4.8220
4.6734
4.5486
4.1814
10
Present
5.7860
5.2069
4.9625
4.7856
4.6411
4.5170
4.1472
[57]
5.7855
5.2065
4.9622
4.7852
4.6407
4.5166
4.1469
20
Present
5.7735
5.1960
4.9527
4.7768
4.6333
4.5091
4.1382
[57]
5.7734
5.1959
4.9526
4.7767
4.6332
4.5090
4.1382
100
Present
5.7695
5.1931
4.9503
4.7748
4.6314
4.5070
4.1354
[57]
5.7695
5.1931
4.9503
4.7748
4.6314
4.5070
4.1354
50
2
Present
13.6959
12.5601
11.9594
11.3453
10.6948
10.3516
9.8168
[57]
13.6904
12.5575
11.9578
11.3440
10.6928
10.3488
9.8128
3
Present
10.1888
9.3097
8.8667
8.4394
8.0024
7.7542
7.3030
[57]
10.1854
9.3077
8.8651
8.4379
8.0006
7.7522
7.3006
5
Present
7.7632
7.0509
6.7181
6.4289
6.1535
5.9732
5.5644
[57]
7.7616
7.0497
6.7171
6.4280
6.1525
5.9721
5.5633
10
Present
6.4625
5.8318
5.5589
5.3505
5.1714
5.0289
4.6321
[57]
6.4620
5.8314
5.5586
5.3502
5.1711
5.0285
4.6317
20
Present
6.0930
5.4840
5.2283
5.0443
4.8945
4.7630
4.3673
[57]
6.0929
5.4839
5.2282
5.0442
4.8944
4.7629
4.3672
100
Present
5.9699
5.3680
5.1181
4.9424
4.8026
4.6747
4.2790
[57]
5.9699
5.3680
5.1181
4.9424
4.8026
4.6747
4.2790Table 7
Frequency parametersΩ for Al/Al2O3 cylindrical shell with different BC and varying geometric parameters.
Material type
a
/
b
h
/
b
BC
SSSS
SE1SE1
SCSC
CFCF
CE2CE2
CCCC
E1E2E1E2
E3FE3F
F
G
M
Ι
(
i
=
1
/
j
=
0
/
c
/
p
=
1
)
1
0.1
4.812
6.428
7.204
5.027
7.734
8.446
1.108
0.777
0.05
6.059
7.610
9.742
5.882
8.864
10.908
3.132
2.097
0.01
14.011
17.976
22.193
8.862
19.552
23.812
17.424
7.511
2
0.1
11.358
20.257
25.185
5.600
20.933
25.740
3.838
2.085
0.05
12.807
21.974
35.872
7.241
22.612
36.379
10.845
4.570
0.01
34.617
43.664
65.965
12.143
43.049
68.030
35.525
10.029
F
G
M
Ι
Ι
(
i
=
1
/
j
=
0
/
c
/
p
=
1
)
1
0.1
4.766
6.357
7.144
5.032
7.653
8.393
1.108
0.777
0.05
6.034
7.568
9.707
5.887
8.816
10.876
3.132
2.097
0.01
13.992
17.949
22.166
8.863
19.524
23.783
17.410
7.512
2
0.1
11.190
19.946
24.954
5.603
20.604
25.517
3.838
2.085
0.05
12.720
21.796
35.747
7.244
22.425
36.257
10.845
4.571
0.01
34.589
43.622
65.857
12.143
43.005
67.921
35.484
10.028The first frequency parameterΩ of Al/ZrO2 spherical shell with SSSS boundary constraints is given in Table 8. The effects of various geometric parameters, including length-thickness ratio a/h and radius-length ratio R/a, and power-law exponent p are also discussed in this table. It shows good agreement when the results obtained by the proposed method are compared with those results obtained by Kiani [57]. We can find that, with the radius increasing, the frequency of the spherical shell decreases when other parameters remain unchanged. Some new results are given in Table 9. In this table, we give the first-order frequency parameter Ω of the free vibration of Al/Al2O3 spherical shell with varying boundary conditions and geometric parameters (a/h,h/b). Geometry parameters of this spherical shell are radius R=2 m and b=1 m, respectively. Under the two general types of volume distribution function of FGMΙ and FGMΙΙ, the first-order frequency of free vibration of spherical shell is given in Table 9. The frequencies of the spherical shell under FGMI and FGMII have some small changes which has the same laws shown in Table 7. For the free vibration of this FG spherical shell, it is significant to study the change of natural frequency under different boundary constraints.Table 8
Frequency parametersΩ for Al/ZrO2 spherical shell with SSSS boundary constraints, varying geometric parameters, and gradient power-law exponent p (FGMΙΙ(i=1/j=0/c/p)).
a
/
h
R
/
a
Method
p
=
0
p
=
0.5
p
=
1
p
=
2
p
=
5
p
=
10
p
=
∞
10
2
Present
7.4580
6.7593
6.4304
6.1422
5.8730
5.7072
5.3457
[57]
7.4599
6.7612
6.4322
6.1440
5.8747
5.7088
5.4370
3
Present
6.5826
5.9454
5.6595
5.4275
5.2222
5.0792
4.7182
[57]
6.5834
5.9462
5.6603
5.4283
5.2230
5.0798
4.7186
5
Present
6.0765
5.4745
5.2142
5.0165
4.8500
4.7197
4.3554
[57]
6.0767
5.4748
5.2145
5.0168
4.8503
4.7200
4.3556
10
Present
5.8478
5.2629
5.0149
4.8335
4.6846
4.5595
4.1915
[57]
5.8479
5.2630
5.0150
4.8336
4.6847
4.5596
4.1915
20
Present
5.7891
5.2096
4.9651
4.7881
4.6435
4.5193
4.1494
[57]
5.7891
5.2096
4.9651
4.7881
4.6336
4.5194
4.1494
100
Present
5.7701
5.1935
4.9505
4.7750
4.6316
4.5073
4.1358
[57]
5.7701
5.1935
4.9505
4.7750
4.6316
4.5073
4.1358
50
2
Present
25.3976
23.3588
22.2354
21.0343
19.7303
19.0801
18.2042
[57]
25.3980
23.3591
22.2357
21.0345
19.7305
19.0803
18.2044
3
Present
17.5981
16.1645
15.3894
14.5770
13.7045
13.2582
12.6137
[57]
17.5982
16.1646
15.3895
14.5771
13.7046
13.2583
12.6138
5
Present
11.6130
10.6300
10.1229
9.6194
9.0953
8.8085
8.3238
[57]
11.6130
10.6300
10.1230
9.6194
9.0953
8.8086
8.3238
10
Present
7.7757
7.0622
6.7288
6.4393
6.1635
5.9830
5.5734
[57]
7.7758
7.0621
6.7288
6.4393
6.1635
5.9830
5.5734
20
Present
6.4655
5.8346
5.5615
5.3531
5.1738
5.0313
4.6342
[57]
6.4655
5.8345
5.5615
5.3531
5.1739
5.0313
4.6342
100
Present
5.9855
5.3827
5.1321
4.9553
4.8143
4.6859
4.2902
[57]
5.9855
5.3827
5.1321
4.9553
4.8143
4.6859
4.2902Table 9
Frequency parametersΩ for Al/Al2O3 spherical shell with different BC and varying geometric parameters.
Material type
a
/
b
h
/
b
BC
SSSS
SE1SE1
SCSC
CFCF
CE2CE2
CCCC
E1E2E1E2
E3FE3F
F
G
M
Ι
(
i
=
1
/
j
=
0
/
c
/
p
=
5
)
1
0.1
4.925
6.125
6.409
5.055
7.243
7.894
1.322
0.822
0.05
7.477
8.481
9.196
7.285
9.747
11.361
3.415
2.159
0.01
32.351
32.787
34.926
17.654
33.135
37.776
24.646
7.302
2
0.1
15.821
21.297
24.259
11.287
20.579
25.672
4.597
2.176
0.05
27.257
31.297
36.784
13.356
30.304
39.615
11.859
4.494
0.01
128.022
130.877
139.230
25.017
96.411
141.174
83.190
15.355
F
G
M
Ι
Ι
(
i
=
1
/
j
=
0
/
c
/
p
=
5
)
1
0.1
4.811
5.978
6.271
5.008
7.079
7.773
1.335
0.822
0.05
7.403
8.388
9.109
7.268
9.643
11.283
3.435
2.155
0.01
32.298
32.732
34.864
17.628
33.064
37.709
24.590
7.294
2
0.1
15.500
20.813
23.870
11.219
20.774
25.311
4.643
2.167
0.05
27.018
30.980
36.437
13.223
29.965
39.262
11.929
4.466
0.01
127.816
130.666
138.997
24.983
96.203
140.922
83.010
15.328As mentioned earlier, the hyperbolic shell is a special form which can be obtained by setting the radius of curvature asRx=-Ry=R. It should be noted here that the plate and the hyperbolic paraboloidal shell are all about the symmetrical distribution of the middle surface. So, for the two kinds of structural distribution of FGMΙ and FGMΙΙ, the frequencies for free vibration of the plate and the hyperbolic paraboloidal shell have no difference. Therefore, the following research does not distinguish under the two general types of volume distribution function. Table 10 gives the first-order frequency of the Al/ZrO2 hyperbolic paraboloidal shell with E1E2E1E2 boundary constraints. The geometry parameter is set as a=b=1 m. With the increase of gradient power-law exponent parameter p, the frequency decreases gradually. Furthermore, the increase of radius and thickness leads to the decrease of frequency. In Table 11, the first three frequency parameters Ω of Al/Al2O3 hyperbolic paraboloidal shell with different boundary constraints and varying geometric parameters (a/b,h/b) are given. In this table, the radius remains constant and is set to R=2 m. It is easy to see from the table that the boundary conditions have a great influence on the frequency for the free vibration of the FG hyperbolic paraboloidal shell. There is an interesting phenomenon that the increase of thickness-width ratio h/b has little effect on the natural frequency. But the length-width ratio a/b has a great influence on the frequency; that is, the frequency increases with the increase of a/b.Table 10
Frequency parametersΩ for Al/ZrO2 hyperbolic paraboloidal shell with E1E2E1E2 boundary constraints, varying geometric parameters, and gradient power-law exponent p (a=b=1, (i=1/j=0/c/p)).
a
/
h
R
/
a
p
=
0
p
=
0.5
p
=
1
p
=
2
p
=
5
p
=
10
p
=
∞
10
1
2.493
2.378
2.328
2.289
2.256
2.234
2.182
2
1.879
1.859
1.852
1.849
1.849
1.847
1.839
3
1.743
1.747
1.750
1.756
1.764
1.767
1.768
5
1.669
1.687
1.697
1.708
1.720
1.725
1.731
10
1.637
1.661
1.674
1.687
1.701
1.707
1.715
20
1.629
1.655
1.668
1.682
1.696
1.703
1.711
100
1.626
1.653
1.667
1.681
1.695
1.702
1.710
50
1
12.492
11.762
11.461
11.240
11.054
10.902
10.479
2
11.500
10.511
10.097
9.795
9.545
9.333
8.719
3
11.170
10.126
9.689
9.376
9.120
8.897
8.230
5
10.968
9.893
9.446
9.127
8.871
8.642
7.944
10
10.874
9.786
9.334
9.014
8.758
8.527
7.815
20
10.849
9.759
9.305
8.985
8.730
8.497
7.782
100
10.841
9.750
9.296
8.976
8.720
8.488
7.771Table 11
Frequency parametersΩ for Al/Al2O3 hyperbolic paraboloidal shell with different BC and varying geometric parameters (R=2, (i=1/j=0/c/p=1)).
a
/
b
h
/
b
Mode number
BC
SCSC
SE2SE2
CE3CE3
E1E1E1E1
E2 E2E2E2
E3 E3E3E3
E2E3E2E3
1
0.1
1
7.169
0.964
6.901
7.231
1.331
1.096
1.289
2
11.951
5.898
8.566
14.497
1.339
1.096
1.314
3
13.929
11.471
12.740
14.603
1.897
1.101
1.892
0.05
1
9.850
2.359
10.023
7.846
3.284
2.994
3.266
2
14.100
6.290
12.273
16.727
3.296
3.096
3.274
3
16.185
12.807
16.240
16.805
4.689
3.096
4.688
0.01
1
30.627
8.835
31.145
11.946
11.195
11.109
11.165
2
32.776
21.977
31.649
25.687
25.177
23.159
23.378
3
32.924
24.532
34.702
30.369
25.363
23.169
24.899
2
0.1
1
26.447
3.857
16.542
22.198
4.566
3.730
4.257
2
28.675
18.452
18.734
25.895
4.643
3.783
4.580
3
35.870
19.907
20.530
32.713
6.258
3.787
6.177
0.05
1
39.399
9.437
23.876
27.912
11.389
10.052
11.031
2
40.132
23.845
28.840
28.854
11.440
10.494
11.411
3
44.907
25.160
34.898
41.508
15.517
10.653
12.468
0.01
1
117.50
35.342
61.962
42.255
39.025
38.643
38.729
2
122.52
59.276
74.310
79.353
61.277
56.844
58.321
3
130.65
65.141
91.389
92.038
70.915
66.853
67.384


### 3.3. Parametric Study

In this section, the free vibration characteristics of FG shallow shells are studied by using the four kinds of special structures with varying boundary constraints and gradient power-law exponent parameterp. In order to compare conveniently, this section uses Al/Al2O3 as the function gradient material, and its volume distribution functions are all FGMΙ.Firstly, the effect of gradient power-law exponent parameterp on the vibration characteristics of the Al/Al2O3 shallow shells is studied. Figure 3 shows the first-order natural frequency parameters Ω for the free vibration of plate, cylindrical shell, spherical shell, and hyperbolic paraboloidal shell with varying power-law exponent p and six kinds of boundary constraints. In this case, the geometrical dimensions are a/b=1, h/b=0.05, and R=2 m. It can be seen from Figure 3 that, under different boundary conditions, the effects of the function gradient power-law exponent parameter p on the vibration characteristics of the shells cannot give a definite conclusion easily. The effects of different boundary conditions on the vibration characteristics of shell structures may be very different. For example, for the FG shallow shell with the boundary constraints of CE2CE2, SCSC, and E1E1E1E1, the first-order frequency decreases with the increase of p. In particular, the frequency decreases very quickly, when p is between 0.1 and 100. However, for the four kinds of shallow shells with varying boundary constraints of E2E2E2E2, E3E3E3E3, and SE3SE3, the frequency increases when p increases.Figure 3
Variations of the first frequency parametersΩ of the Al/Al2O3 shallow shells with different boundary constraints: BD1: CE2CE2; BD2: SCSC; BD3: E1E1E1E1; BD4: E2E2E2E2; BD5: E3E3E3E3; BD6: SE3SE3.
(a)
Plate
(b)
Cylindrical shell
(c)
Spherical shell
(d)
Hyperbolic paraboloidal shellThe effects of material parametersa, b, and c on the vibration characteristics of the Al/Al2O3 shallow shells with E3E3E3E3 are discussed. Figure 4 shows the frequency parameters Ω of the shallow shells with the varying gradient power-law exponent parameters p and four types of material gradient parameters. Four kinds of gradient parameters types are Type 1: a=1, b=c=0; Type 2: a=1, b=0.5, c=2; Type 3: a=0, b=-0.5, c=2; and Type 4: a=1, b=1, c=4. From Figure 4, it can be found that the material gradient parameters have great effect on the vibration characteristics of the FG shallow shells. The changes of frequencies for the shallow shells with Type 1, Type 2, and Type 3 are the same, with the frequency increasing quickly when p increases from zero to 5 and then decreasing with p continuing to increase. But the frequency of the shallow shells with Type 4 is quite different with them. Especially when p increases to 5, the frequency increases and then decreases slowly. Therefore, it is very important to study the vibration characteristics of FG shells with the varying material gradient parameters. This is because the change of material gradient parameters will completely change the mechanical properties of FG structure in the thickness direction. It is the reason why FGM has more advantages than traditional materials.Figure 4
Variations of the first frequency parametersΩ of the Al/Al2O3 shallow shells with E3E3E3E3 and different material types: Type 1: a=1, b=c=0; Type 2: a=1, b=0.5, c=2; Type 3: a=0, b=-0.5, c=2; Type 4: a=1, b=1, c=4.
(a)
Plate
(b)
Cylindrical shell
(c)
Spherical shell
(d)
Hyperbolic paraboloidal shellAccording to the above results, the 3D view model shapes of the Al/ZrO2 shallow shells with CCCC boundary constraints and volume distribution functions FGMΙ(i=1/b=0/c/p=1) are given in Figure 5. The geometric parameters of the plate, cylindrical shell, and spherical shell in these figures are all h=0.05 m, R=1 m, but the length-width ratio a/b is varied which has been given in Figure 5. These 3D view models are adopted to understand the free vibration characteristics of FG shallow shells.Figure 5
Mode shapes of the Al/ZrO2 shallow shells with CCCC boundary condition.


## 3.1. Convergence Study

As previously mentioned in (1a)–(1e), we can get the convergence criterion by the truncation value of M and N in the displacement function expressions in the numerical calculation. Therefore, the values of M and N are the direct representation of the convergence for the proposed method. In Table 2, we can get the convergence of first eight frequency parameters Ω for the various shallow shells structures with SSSS boundary constraints whose functionally graded material is Al/Al2O3. Their geometrical dimensions are a/b=1, h/a=0.1, and R=1 m.Table 2
Convergence of frequency parameterΩ for Al/Al2O3 shallow shells with SSSS boundary constraints (FGMΙ(i=1/j=0/c/p=1), R=1, a/b=1, h=0.1).
Type
M
×
N
Mode number
1
2
3
4
5
6
7
8
Plate
8
×
8
4.417
10.585
10.585
16.195
16.195
16.303
19.910
19.910
10
×
10
4.417
10.585
10.585
16.195
16.195
16.303
19.908
19.908
12
×
12
4.417
10.585
10.585
16.195
16.195
16.303
19.907
19.907
14
×
14
4.417
10.585
10.585
16.195
16.195
16.303
19.906
19.906
16
×
16
4.417
10.585
10.585
16.195
16.195
16.303
19.906
19.906
18
×
18
4.417
10.585
10.585
16.195
16.195
16.303
19.906
19.906
Cylindrical shell
8
×
8
5.739
10.493
12.308
16.053
16.195
16.613
19.880
21.162
10
×
10
5.739
10.493
12.308
16.053
16.195
16.613
19.877
21.159
12
×
12
5.739
10.493
12.308
16.053
16.195
16.613
19.877
21.159
14
×
14
5.739
10.493
12.308
16.053
16.195
16.613
19.876
21.158
16
×
16
5.739
10.493
12.308
16.053
16.195
16.613
19.876
21.158
18
×
18
5.739
10.493
12.308
16.053
16.195
16.613
19.876
21.158
Spherical shell
8
×
8
8.946
13.139
13.139
16.053
16.053
18.082
21.422
21.422
10
×
10
8.946
13.139
13.139
16.053
16.053
18.082
21.419
21.419
12
×
12
8.946
13.139
13.139
16.053
16.053
18.082
21.419
21.419
14
×
14
8.946
13.139
13.139
16.053
16.053
18.082
21.418
21.418
16
×
16
8.946
13.139
13.139
16.053
16.053
18.082
21.418
21.418
18
×
18
8.946
13.139
13.139
16.053
16.053
18.082
21.418
21.418
Hyperbolic paraboloidal shell
8
×
8
3.609
10.940
11.121
15.434
16.053
16.334
20.271
20.706
10
×
10
3.609
10.940
11.121
15.434
16.053
16.334
20.269
20.703
12
×
12
3.609
10.940
11.121
15.434
16.053
16.334
20.268
20.703
14
×
14
3.609
10.940
11.121
15.434
16.053
16.334
20.268
20.702
16
×
16
3.609
10.940
11.121
15.434
16.053
16.334
20.268
20.702
18
×
18
3.609
10.940
11.121
15.434
16.053
16.334
20.268
20.702The method presented in this paper shows fast convergence which can be found in Table1. For example, the biggest difference for the worst case of the plate which is made by the contrast of 8 × 8 and 18 × 18 is less than 0.02%. It is not difficult to find that the biggest difference for the contrast of 8 × 8 and 18 × 18 of the cylindrical shell is also less than 0.02%. Besides, for the spherical shell and hyperbolic paraboloidal shell, the biggest difference between these two truncated values is 0.017% and 0.019%. Through these results, we can find that the method shows good uniform convergence for different structures which shows the accuracy of the unified model. In addition, it is necessary to point out that, in order to make the results obtained in the following numerical examples more accurate, we use the invariant truncation values of M=N=14.


## 3.2. Validation and Some New Results

In this section, the accuracy of the proposed method is verified by being compared with the literature results. In addition, this paper also gives some new results, which can provide a benchmark for the future research. The first five frequency parametersΩ for Al/Al2O3 square plate with SSSS boundary constraints and varying length-thickness ratios a/h and gradient power-law exponent parameter p are given in Table 3. The geometric parameters are a=b=1 m and the material parameters of alumina and aluminum have been given previously. The results of the present method are compared with those of Jin et al. [56] which show good agreement. From Table 3, we can easily find that the frequency parameter Ω of the square plate is decreasing gradually with the increase of a/h and p. In Table 4, the first frequency parameters Ω for Al/ZrO2 plate with various boundary constraints are given. The influence of different length-width ratio a/b, thickness-width ratio h/b, and gradient power-law exponent parameter p on the plate frequency is also discussed. The results of the free vibration under SSSS boundary conditions are compared with those of Kiani et al. [57], which show good agreement. It is worth noting that these results in Table 4 are consistent with the results obtained in Table 3.Table 3
Frequency parametersΩ for Al/Al2O3 square plate with SSSS boundary constraints, varying geometric parameters, and gradient power-law exponent p (FGMΙ(i=1/j=0/c/p)).
h
/
a
p
Method
Mode number
1
2
3
4
5
0.1
0
Present
5.769
13.76
13.76
19.48
19.48
[56]
5.779
13.81
13.81
19.48
19.48
1
Present
4.417
10.59
10.59
16.19
16.19
[56]
4.428
10.63
10.63
16.20
16.20
5
Present
3.785
9.004
9.004
12.63
12.63
[56]
3.774
8.931
8.931
12.64
12.64
0.2
0
Present
5.280
9.741
9.741
11.55
11.55
[56]
5.304
9.742
9.742
11.65
11.65
1
Present
4.076
8.086
8.086
9.005
9.005
[56]
4.100
8.089
8.089
9.108
9.108
5
Present
3.447
6.297
6.297
7.496
7.496
[56]
3.405
6.296
6.296
7.344
7.344Table 4
Frequency parametersΩ for Al/ZrO2 plate with various boundary constraints, varying geometric parameters, and gradient power-law exponent p (FGMΙΙ(i=1/j=0/c/p)).
BC
a
/
b
h
/
b
Method
p
=
0
p
=
0.5
p
=
1
p
=
2
p
=
5
p
=
10
p
=
∞
SSSS
1
0.1
Present
5.7693
5.1932
4.9504
4.7750
4.6315
4.5071
4.1352
[57]
5.7693
5.1932
4.9504
4.7750
4.6315
4.5071
4.1353
0.02
Present
5.9647
5.3631
5.1135
4.9382
4.7988
4.6710
4.2753
[57]
5.9647
5.3631
5.1135
4.9382
4.7988
4.6710
4.2753
2
0.1
Present
14.6070
13.1429
12.5296
12.0910
11.7358
11.4215
10.4698
0.02
Present
14.9198
13.4149
12.7906
12.3523
12.0040
11.6844
10.6940
CCCC
1
0.1
Present
9.8425
8.8831
8.4665
8.1474
7.8691
7.6511
7.0548
0.02
Present
10.8410
9.7492
9.2954
8.9754
8.7198
8.4872
7.7705
2
0.1
Present
27.5886
24.8797
23.7153
22.8390
22.0870
21.4798
19.7746
0.02
Present
29.6598
26.6713
25.4299
24.5558
23.8588
23.2228
21.2592
CSCS
1
0.1
Present
8.0703
7.2786
6.9375
6.6802
6.4591
6.2816
5.7845
0.02
Present
8.7301
7.8505
7.4851
7.2277
7.0225
6.8352
6.2574
2
0.1
Present
16.0659
14.4609
13.7857
13.2987
12.9004
12.5534
11.5155
0.02
Present
16.5502
14.8811
14.1886
13.7021
13.3152
12.9606
11.8626
E1E2E1E2
1
0.1
Present
1.6260
1.6528
1.6666
1.6807
1.6951
1.7018
1.7098
0.02
Present
10.8410
9.7492
9.2954
8.9754
8.7198
8.4872
7.7705
2
0.1
Present
5.6307
5.7230
5.7706
5.8192
5.8686
5.8913
5.9186
0.02
Present
29.6598
26.6713
25.4299
24.5559
23.8588
23.2228
21.2592
E2E3E2E3
1
0.1
Present
1.6249
1.6514
1.6651
1.6790
1.6931
1.6995
1.7073
0.02
Present
10.4114
9.4120
8.9917
8.6918
8.4499
8.2330
7.5680
2
0.1
Present
5.6174
5.7067
5.7522
5.7982
5.8442
5.8650
5.8896
0.02
Present
27.4294
25.0188
23.9872
23.2505
22.6568
22.1105
20.3974Cylindrical shell, as a special form of shallow shell with double curvature, is well known and widely used in engineering practice. The free vibration characteristics of FG cylindrical shells are studied based on the unified doubly curved shallow shell model established in this paper. Table5 gives the first nine frequencies of FGMΙi=1/j=0/c/p cylindrical shell with FFCF boundary constraints and varying gradient parameter of power-law exponent p. There are two kinds of functionally graded materials whose properties are Ec=168 GPa, Em=70 GPa, μc=μm=0.3, pc=5700 kg/m3, pm=2707 kg/m3. The geometric parameters of cylindrical shell are h=0.1 m, R=1 m, a=2 m, b=2π/3 m. The first frequency parameter Ω for Al/ZrO2 cylindrical shell with SSSS boundary constraints is given in Table 6. At the same time, the effects of various geometric parameters, including length-thickness ratio a/h and radius-length ratio R/a, and power-law exponent p are also discussed. The accuracy of the results is verified by being compared with the related references [26, 57, 58]. As shown in Tables 5 and 6, the frequency of the cylindrical shell is reduced gradually when the radius R, the thickness h, or the gradient power-law exponent p increase. In Table 7, we give the first-order frequency parameter Ω of free vibration of Al/Al2O3 cylindrical shells with different boundary conditions and varying geometric parameters of a/h and h/b. In the studied cylindrical shell model, the radius is R=2 m. In this table, the effects of two general types of volume distribution function (FGMΙi=1/j=0/c/p=1 and FGMΙΙi=1/j=0/c/p=1) on the frequency of cylindrical shells are also given. It should be pointed out that the frequencies of the cylindrical shell under FGMI and FGMII have some small changes when the shell has the same boundary conditions and geometric parameters.Table 5
Frequencies (Hz) ofFGMΙ(i=1/j=0/c/p) cylindrical shell with FFCF boundary constraints and varying gradient power-law exponent p (Ec=168 GPa, Em=70 GPa, μc=μm=0.3, pc=5700 kg/m3, pm=2707 kg/m3, h=0.1, R=1, a=2, b=2π/3).
p
Method
Mode number
1
2
3
4
5
6
7
8
9
0
Present
61.03
94.83
153.13
241.67
276.26
291.68
355.13
450.74
512.49
[58]
61.02
94.82
153.13
241.39
275.83
291.45
355.14
450.76
511.92
[26]
61.06
94.92
153.11
241.75
276.37
291.94
355.07
450.82
512.48
0.6
Present
59.11
92.94
148.16
234.52
266.87
287.37
342.44
442.01
495.33
[58]
59.24
93.14
148.51
234.79
267.06
287.85
343.22
443.06
495.89
[26]
59.13
93.03
148.14
234.60
266.97
287.63
342.38
442.09
495.31
1
Present
58.79
92.39
147.51
233.23
265.51
285.52
341.09
439.41
492.81
[58]
58.96
92.65
147.94
233.64
265.86
286.16
342.08
440.72
493.67
[26]
58.82
92.48
147.49
233.31
265.61
285.77
341.04
439.49
492.80
5
Present
59.93
91.45
151.59
236.35
272.08
278.81
353.89
435.11
504.54
[58]
60.09
91.69
152.02
236.75
272.42
279.34
354.89
436.34
505.35
[26]
59.96
91.54
151.57
236.42
272.19
279.06
353.83
435.18
504.56
20
Present
59.13
90.21
149.31
233.26
268.47
275.15
348.37
429.12
497.85
[58]
59.19
90.29
149.49
233.27
268.37
275.23
348.80
429.63
497.87
[26]
59.16
90.30
149.29
233.32
268.57
275.41
348.32
429.19
497.87
50
Present
58.18
89.52
146.47
229.93
263.84
274.07
340.84
425.59
489.36
[58]
58.20
89.55
146.55
229.79
263.58
273.98
341.04
425.83
489.06
[26]
58.21
89.60
146.45
230.00
263.95
274.32
340.79
425.66
489.36
100
Present
57.72
89.21
145.09
228.32
261.56
273.66
337.14
424.02
485.17
[58]
57.73
89.21
145.14
228.12
261.23
273.51
337.25
424.15
484.75
[26]
57.75
89.29
145.08
228.39
261.66
273.90
337.09
424.10
485.16
∞
Present
57.17
88.83
143.43
226.37
258.77
273.21
332.64
422.20
480.04
[58]
57.16
88.81
143.43
226.11
258.36
273.00
332.65
422.21
479.49
[26]
57.19
88.91
143.41
226.44
258.87
273.46
332.59
422.27
480.02Table 6
Frequency parametersΩ for Al/ZrO2 cylindrical shell with SSSS boundary constraints and varying geometric parameters and gradient power-law exponent p (FGMΙΙ(i=1/j=0/c/p), a/b=1).
a
/
h
R
/
a
Method
p
=
0
p
=
0.5
p
=
1
p
=
2
p
=
5
p
=
10
p
=
∞
10
2
Present
6.1667
5.5612
5.2958
5.0890
4.9112
4.7782
4.4201
[57]
6.1552
5.5522
5.2876
5.0812
4.9032
4.7700
4.4119
3
Present
5.9510
5.3592
5.1052
4.9150
4.7563
4.6288
4.2655
[57]
5.9456
5.3548
5.1012
4.9111
4.7524
4.6249
4.2616
5
Present
5.8357
5.2521
5.0046
4.8235
4.6749
4.5501
4.1829
[57]
5.8337
5.2504
5.0031
4.8220
4.6734
4.5486
4.1814
10
Present
5.7860
5.2069
4.9625
4.7856
4.6411
4.5170
4.1472
[57]
5.7855
5.2065
4.9622
4.7852
4.6407
4.5166
4.1469
20
Present
5.7735
5.1960
4.9527
4.7768
4.6333
4.5091
4.1382
[57]
5.7734
5.1959
4.9526
4.7767
4.6332
4.5090
4.1382
100
Present
5.7695
5.1931
4.9503
4.7748
4.6314
4.5070
4.1354
[57]
5.7695
5.1931
4.9503
4.7748
4.6314
4.5070
4.1354
50
2
Present
13.6959
12.5601
11.9594
11.3453
10.6948
10.3516
9.8168
[57]
13.6904
12.5575
11.9578
11.3440
10.6928
10.3488
9.8128
3
Present
10.1888
9.3097
8.8667
8.4394
8.0024
7.7542
7.3030
[57]
10.1854
9.3077
8.8651
8.4379
8.0006
7.7522
7.3006
5
Present
7.7632
7.0509
6.7181
6.4289
6.1535
5.9732
5.5644
[57]
7.7616
7.0497
6.7171
6.4280
6.1525
5.9721
5.5633
10
Present
6.4625
5.8318
5.5589
5.3505
5.1714
5.0289
4.6321
[57]
6.4620
5.8314
5.5586
5.3502
5.1711
5.0285
4.6317
20
Present
6.0930
5.4840
5.2283
5.0443
4.8945
4.7630
4.3673
[57]
6.0929
5.4839
5.2282
5.0442
4.8944
4.7629
4.3672
100
Present
5.9699
5.3680
5.1181
4.9424
4.8026
4.6747
4.2790
[57]
5.9699
5.3680
5.1181
4.9424
4.8026
4.6747
4.2790Table 7
Frequency parametersΩ for Al/Al2O3 cylindrical shell with different BC and varying geometric parameters.
Material type
a
/
b
h
/
b
BC
SSSS
SE1SE1
SCSC
CFCF
CE2CE2
CCCC
E1E2E1E2
E3FE3F
F
G
M
Ι
(
i
=
1
/
j
=
0
/
c
/
p
=
1
)
1
0.1
4.812
6.428
7.204
5.027
7.734
8.446
1.108
0.777
0.05
6.059
7.610
9.742
5.882
8.864
10.908
3.132
2.097
0.01
14.011
17.976
22.193
8.862
19.552
23.812
17.424
7.511
2
0.1
11.358
20.257
25.185
5.600
20.933
25.740
3.838
2.085
0.05
12.807
21.974
35.872
7.241
22.612
36.379
10.845
4.570
0.01
34.617
43.664
65.965
12.143
43.049
68.030
35.525
10.029
F
G
M
Ι
Ι
(
i
=
1
/
j
=
0
/
c
/
p
=
1
)
1
0.1
4.766
6.357
7.144
5.032
7.653
8.393
1.108
0.777
0.05
6.034
7.568
9.707
5.887
8.816
10.876
3.132
2.097
0.01
13.992
17.949
22.166
8.863
19.524
23.783
17.410
7.512
2
0.1
11.190
19.946
24.954
5.603
20.604
25.517
3.838
2.085
0.05
12.720
21.796
35.747
7.244
22.425
36.257
10.845
4.571
0.01
34.589
43.622
65.857
12.143
43.005
67.921
35.484
10.028The first frequency parameterΩ of Al/ZrO2 spherical shell with SSSS boundary constraints is given in Table 8. The effects of various geometric parameters, including length-thickness ratio a/h and radius-length ratio R/a, and power-law exponent p are also discussed in this table. It shows good agreement when the results obtained by the proposed method are compared with those results obtained by Kiani [57]. We can find that, with the radius increasing, the frequency of the spherical shell decreases when other parameters remain unchanged. Some new results are given in Table 9. In this table, we give the first-order frequency parameter Ω of the free vibration of Al/Al2O3 spherical shell with varying boundary conditions and geometric parameters (a/h,h/b). Geometry parameters of this spherical shell are radius R=2 m and b=1 m, respectively. Under the two general types of volume distribution function of FGMΙ and FGMΙΙ, the first-order frequency of free vibration of spherical shell is given in Table 9. The frequencies of the spherical shell under FGMI and FGMII have some small changes which has the same laws shown in Table 7. For the free vibration of this FG spherical shell, it is significant to study the change of natural frequency under different boundary constraints.Table 8
Frequency parametersΩ for Al/ZrO2 spherical shell with SSSS boundary constraints, varying geometric parameters, and gradient power-law exponent p (FGMΙΙ(i=1/j=0/c/p)).
a
/
h
R
/
a
Method
p
=
0
p
=
0.5
p
=
1
p
=
2
p
=
5
p
=
10
p
=
∞
10
2
Present
7.4580
6.7593
6.4304
6.1422
5.8730
5.7072
5.3457
[57]
7.4599
6.7612
6.4322
6.1440
5.8747
5.7088
5.4370
3
Present
6.5826
5.9454
5.6595
5.4275
5.2222
5.0792
4.7182
[57]
6.5834
5.9462
5.6603
5.4283
5.2230
5.0798
4.7186
5
Present
6.0765
5.4745
5.2142
5.0165
4.8500
4.7197
4.3554
[57]
6.0767
5.4748
5.2145
5.0168
4.8503
4.7200
4.3556
10
Present
5.8478
5.2629
5.0149
4.8335
4.6846
4.5595
4.1915
[57]
5.8479
5.2630
5.0150
4.8336
4.6847
4.5596
4.1915
20
Present
5.7891
5.2096
4.9651
4.7881
4.6435
4.5193
4.1494
[57]
5.7891
5.2096
4.9651
4.7881
4.6336
4.5194
4.1494
100
Present
5.7701
5.1935
4.9505
4.7750
4.6316
4.5073
4.1358
[57]
5.7701
5.1935
4.9505
4.7750
4.6316
4.5073
4.1358
50
2
Present
25.3976
23.3588
22.2354
21.0343
19.7303
19.0801
18.2042
[57]
25.3980
23.3591
22.2357
21.0345
19.7305
19.0803
18.2044
3
Present
17.5981
16.1645
15.3894
14.5770
13.7045
13.2582
12.6137
[57]
17.5982
16.1646
15.3895
14.5771
13.7046
13.2583
12.6138
5
Present
11.6130
10.6300
10.1229
9.6194
9.0953
8.8085
8.3238
[57]
11.6130
10.6300
10.1230
9.6194
9.0953
8.8086
8.3238
10
Present
7.7757
7.0622
6.7288
6.4393
6.1635
5.9830
5.5734
[57]
7.7758
7.0621
6.7288
6.4393
6.1635
5.9830
5.5734
20
Present
6.4655
5.8346
5.5615
5.3531
5.1738
5.0313
4.6342
[57]
6.4655
5.8345
5.5615
5.3531
5.1739
5.0313
4.6342
100
Present
5.9855
5.3827
5.1321
4.9553
4.8143
4.6859
4.2902
[57]
5.9855
5.3827
5.1321
4.9553
4.8143
4.6859
4.2902Table 9
Frequency parametersΩ for Al/Al2O3 spherical shell with different BC and varying geometric parameters.
Material type
a
/
b
h
/
b
BC
SSSS
SE1SE1
SCSC
CFCF
CE2CE2
CCCC
E1E2E1E2
E3FE3F
F
G
M
Ι
(
i
=
1
/
j
=
0
/
c
/
p
=
5
)
1
0.1
4.925
6.125
6.409
5.055
7.243
7.894
1.322
0.822
0.05
7.477
8.481
9.196
7.285
9.747
11.361
3.415
2.159
0.01
32.351
32.787
34.926
17.654
33.135
37.776
24.646
7.302
2
0.1
15.821
21.297
24.259
11.287
20.579
25.672
4.597
2.176
0.05
27.257
31.297
36.784
13.356
30.304
39.615
11.859
4.494
0.01
128.022
130.877
139.230
25.017
96.411
141.174
83.190
15.355
F
G
M
Ι
Ι
(
i
=
1
/
j
=
0
/
c
/
p
=
5
)
1
0.1
4.811
5.978
6.271
5.008
7.079
7.773
1.335
0.822
0.05
7.403
8.388
9.109
7.268
9.643
11.283
3.435
2.155
0.01
32.298
32.732
34.864
17.628
33.064
37.709
24.590
7.294
2
0.1
15.500
20.813
23.870
11.219
20.774
25.311
4.643
2.167
0.05
27.018
30.980
36.437
13.223
29.965
39.262
11.929
4.466
0.01
127.816
130.666
138.997
24.983
96.203
140.922
83.010
15.328As mentioned earlier, the hyperbolic shell is a special form which can be obtained by setting the radius of curvature asRx=-Ry=R. It should be noted here that the plate and the hyperbolic paraboloidal shell are all about the symmetrical distribution of the middle surface. So, for the two kinds of structural distribution of FGMΙ and FGMΙΙ, the frequencies for free vibration of the plate and the hyperbolic paraboloidal shell have no difference. Therefore, the following research does not distinguish under the two general types of volume distribution function. Table 10 gives the first-order frequency of the Al/ZrO2 hyperbolic paraboloidal shell with E1E2E1E2 boundary constraints. The geometry parameter is set as a=b=1 m. With the increase of gradient power-law exponent parameter p, the frequency decreases gradually. Furthermore, the increase of radius and thickness leads to the decrease of frequency. In Table 11, the first three frequency parameters Ω of Al/Al2O3 hyperbolic paraboloidal shell with different boundary constraints and varying geometric parameters (a/b,h/b) are given. In this table, the radius remains constant and is set to R=2 m. It is easy to see from the table that the boundary conditions have a great influence on the frequency for the free vibration of the FG hyperbolic paraboloidal shell. There is an interesting phenomenon that the increase of thickness-width ratio h/b has little effect on the natural frequency. But the length-width ratio a/b has a great influence on the frequency; that is, the frequency increases with the increase of a/b.Table 10
Frequency parametersΩ for Al/ZrO2 hyperbolic paraboloidal shell with E1E2E1E2 boundary constraints, varying geometric parameters, and gradient power-law exponent p (a=b=1, (i=1/j=0/c/p)).
a
/
h
R
/
a
p
=
0
p
=
0.5
p
=
1
p
=
2
p
=
5
p
=
10
p
=
∞
10
1
2.493
2.378
2.328
2.289
2.256
2.234
2.182
2
1.879
1.859
1.852
1.849
1.849
1.847
1.839
3
1.743
1.747
1.750
1.756
1.764
1.767
1.768
5
1.669
1.687
1.697
1.708
1.720
1.725
1.731
10
1.637
1.661
1.674
1.687
1.701
1.707
1.715
20
1.629
1.655
1.668
1.682
1.696
1.703
1.711
100
1.626
1.653
1.667
1.681
1.695
1.702
1.710
50
1
12.492
11.762
11.461
11.240
11.054
10.902
10.479
2
11.500
10.511
10.097
9.795
9.545
9.333
8.719
3
11.170
10.126
9.689
9.376
9.120
8.897
8.230
5
10.968
9.893
9.446
9.127
8.871
8.642
7.944
10
10.874
9.786
9.334
9.014
8.758
8.527
7.815
20
10.849
9.759
9.305
8.985
8.730
8.497
7.782
100
10.841
9.750
9.296
8.976
8.720
8.488
7.771Table 11
Frequency parametersΩ for Al/Al2O3 hyperbolic paraboloidal shell with different BC and varying geometric parameters (R=2, (i=1/j=0/c/p=1)).
a
/
b
h
/
b
Mode number
BC
SCSC
SE2SE2
CE3CE3
E1E1E1E1
E2 E2E2E2
E3 E3E3E3
E2E3E2E3
1
0.1
1
7.169
0.964
6.901
7.231
1.331
1.096
1.289
2
11.951
5.898
8.566
14.497
1.339
1.096
1.314
3
13.929
11.471
12.740
14.603
1.897
1.101
1.892
0.05
1
9.850
2.359
10.023
7.846
3.284
2.994
3.266
2
14.100
6.290
12.273
16.727
3.296
3.096
3.274
3
16.185
12.807
16.240
16.805
4.689
3.096
4.688
0.01
1
30.627
8.835
31.145
11.946
11.195
11.109
11.165
2
32.776
21.977
31.649
25.687
25.177
23.159
23.378
3
32.924
24.532
34.702
30.369
25.363
23.169
24.899
2
0.1
1
26.447
3.857
16.542
22.198
4.566
3.730
4.257
2
28.675
18.452
18.734
25.895
4.643
3.783
4.580
3
35.870
19.907
20.530
32.713
6.258
3.787
6.177
0.05
1
39.399
9.437
23.876
27.912
11.389
10.052
11.031
2
40.132
23.845
28.840
28.854
11.440
10.494
11.411
3
44.907
25.160
34.898
41.508
15.517
10.653
12.468
0.01
1
117.50
35.342
61.962
42.255
39.025
38.643
38.729
2
122.52
59.276
74.310
79.353
61.277
56.844
58.321
3
130.65
65.141
91.389
92.038
70.915
66.853
67.384


## 3.3. Parametric Study

In this section, the free vibration characteristics of FG shallow shells are studied by using the four kinds of special structures with varying boundary constraints and gradient power-law exponent parameterp. In order to compare conveniently, this section uses Al/Al2O3 as the function gradient material, and its volume distribution functions are all FGMΙ.Firstly, the effect of gradient power-law exponent parameterp on the vibration characteristics of the Al/Al2O3 shallow shells is studied. Figure 3 shows the first-order natural frequency parameters Ω for the free vibration of plate, cylindrical shell, spherical shell, and hyperbolic paraboloidal shell with varying power-law exponent p and six kinds of boundary constraints. In this case, the geometrical dimensions are a/b=1, h/b=0.05, and R=2 m. It can be seen from Figure 3 that, under different boundary conditions, the effects of the function gradient power-law exponent parameter p on the vibration characteristics of the shells cannot give a definite conclusion easily. The effects of different boundary conditions on the vibration characteristics of shell structures may be very different. For example, for the FG shallow shell with the boundary constraints of CE2CE2, SCSC, and E1E1E1E1, the first-order frequency decreases with the increase of p. In particular, the frequency decreases very quickly, when p is between 0.1 and 100. However, for the four kinds of shallow shells with varying boundary constraints of E2E2E2E2, E3E3E3E3, and SE3SE3, the frequency increases when p increases.Figure 3
Variations of the first frequency parametersΩ of the Al/Al2O3 shallow shells with different boundary constraints: BD1: CE2CE2; BD2: SCSC; BD3: E1E1E1E1; BD4: E2E2E2E2; BD5: E3E3E3E3; BD6: SE3SE3.
(a)
Plate
(b)
Cylindrical shell
(c)
Spherical shell
(d)
Hyperbolic paraboloidal shellThe effects of material parametersa, b, and c on the vibration characteristics of the Al/Al2O3 shallow shells with E3E3E3E3 are discussed. Figure 4 shows the frequency parameters Ω of the shallow shells with the varying gradient power-law exponent parameters p and four types of material gradient parameters. Four kinds of gradient parameters types are Type 1: a=1, b=c=0; Type 2: a=1, b=0.5, c=2; Type 3: a=0, b=-0.5, c=2; and Type 4: a=1, b=1, c=4. From Figure 4, it can be found that the material gradient parameters have great effect on the vibration characteristics of the FG shallow shells. The changes of frequencies for the shallow shells with Type 1, Type 2, and Type 3 are the same, with the frequency increasing quickly when p increases from zero to 5 and then decreasing with p continuing to increase. But the frequency of the shallow shells with Type 4 is quite different with them. Especially when p increases to 5, the frequency increases and then decreases slowly. Therefore, it is very important to study the vibration characteristics of FG shells with the varying material gradient parameters. This is because the change of material gradient parameters will completely change the mechanical properties of FG structure in the thickness direction. It is the reason why FGM has more advantages than traditional materials.Figure 4
Variations of the first frequency parametersΩ of the Al/Al2O3 shallow shells with E3E3E3E3 and different material types: Type 1: a=1, b=c=0; Type 2: a=1, b=0.5, c=2; Type 3: a=0, b=-0.5, c=2; Type 4: a=1, b=1, c=4.
(a)
Plate
(b)
Cylindrical shell
(c)
Spherical shell
(d)
Hyperbolic paraboloidal shellAccording to the above results, the 3D view model shapes of the Al/ZrO2 shallow shells with CCCC boundary constraints and volume distribution functions FGMΙ(i=1/b=0/c/p=1) are given in Figure 5. The geometric parameters of the plate, cylindrical shell, and spherical shell in these figures are all h=0.05 m, R=1 m, but the length-width ratio a/b is varied which has been given in Figure 5. These 3D view models are adopted to understand the free vibration characteristics of FG shallow shells.Figure 5
Mode shapes of the Al/ZrO2 shallow shells with CCCC boundary condition.


## 4. Conclusions

In this paper, an improved Fourier series solution is proposed to solve the free vibration analysis of the unified FG double curvature shallow shells with general boundary constraints. The theoretical analysis is based on the FSDT and the Rayleigh-Ritz technique. The energy equation is solved by five displacement functions which are expressed as a series of 2D Fourier series. By introducing the product form of sine and cosine function, we can effectively eliminate the discontinuous or jumping phenomenon of the displacement function on the boundaries. Artificial virtual spring technique is used to simulate the boundary conditions; that is, five groups of springs are arranged on the four edges. They are linear springs(ku,kv,kw) and rotational springs (Kx,Ky), respectively. Different boundary conditions can be realized by setting different stiffness values of the springs.Through the analysis and discussion of the results, we can draw the following conclusions:(1)
The proposed method shows good convergence and the accuracy is verified by being compared with the existing literatures.(2)
The frequencies of the FG shall shells are changed a lot when the boundary conditions change. The results show that the spring stiffness is directly proportional to the frequency of the shallow shells.(3)
The geometric parameters also have a great influence on the free vibration of the FGM shallow shells. The frequency parameter of the square plate decreases gradually with the increase of length-thickness ratioa/h while the frequency of the cylindrical shell reduces gradually when the radius R and the thickness h increase.(4)
The frequencies of the shallow shells are inversely proportional to the gradient index. The volume distribution functions only affect the cylindrical shells and spherical shells and have no effect on plates and hyperbolic paraboloidal shells.It is of great value to study the free vibration characteristics for practical engineering applications. For example, the parametric studies of this paper provide the technical support and theoretical basis for the vibration reduction, noise reduction, and so on. In the following research, we aim to study the forced vibration analysis of the unified functionally graded shallow shell with general boundary conditions. And the work finished in this paper has laid a foundation for further study.



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*Source: 7025190-2017-09-11.xml*