# The Modified Fourier-Ritz Approach for the Free Vibration of Functionally Graded Cylindrical, Conical, Spherical Panels and Shells of Revolution with General Boundary Condition

Authors: Lijie Li; Haichao Li; Fuzhen Pang; Xueren Wang; Yuan Du; Shuo Li
Journal: Mathematical Problems in Engineering
 (2017)

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## Abstract

The aim of this paper is to extend the modified Fourier-Ritz approach to evaluate the free vibration of four-parameter functionally graded moderately thick cylindrical, conical, spherical panels and shells of revolution with general boundary conditions. The first-order shear deformation theory is employed to formulate the theoretical model. In the modified Fourier-Ritz approach, the admissible functions of the structure elements are expanded into the improved Fourier series which consist of two-dimensional (2D) Fourier cosine series and auxiliary functions to eliminate all the relevant discontinuities of the displacements and their derivatives at the edges regardless of boundary conditions and then solve the natural frequencies by means of the Ritz method. As one merit of this paper, the functionally graded cylindrical, conical, spherical shells are, respectively, regarded as a special functionally graded cylindrical, conical, spherical panels, and the coupling spring technology is introduced to ensure the kinematic and physical compatibility at the common meridian. The excellent accuracy and reliability of the unified computational model are compared with the results found in the literatures.


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## Body

## 1. Introduction

As the based structure elements, functionally graded cylindrical, conical, spherical panels and shells have a wide application in various engineering constructions, such as aircraft, space vehicles, and structures of military industries. Therefore, the knowledge of vibration characteristics of the functionally graded cylindrical, conical, spherical panels and shells is of particular importance for the predesign of the engineering structures.So far, a huge amount of researches focused on the vibration analysis of functionally graded cylindrical, conical, spherical panels and shells by using different methods and shell theories. Tornabene and Viola [1–3] studied the free vibration of the moderately thick four-parameter functionally graded conical, cylindrical shells and annular plates under some classical boundary conditions by means of the generalized differential quadrature method on the basis of the first-order shear deformation theory. Zhao and Liew [4, 5] presented the element-free kp-Ritz method to investigate the free vibration analysis of mechanical and thermal buckling of functionally graded conical shell panels with classical boundary conditions. Aragh and Hedayati [6] dealt with the free vibration and static response of a two-dimensional functionally graded (2D FGM) metal/ceramic open cylindrical shell with classical boundary conditions by using 2D generalized differential quadrature method. Su et al. [7–9] applied the modified Fourier series and Rayleigh–Ritz method to analyze the free vibrations of functionally graded open and closed shells including cylindrical, conical, and spherical ones with general boundary conditions based on first-order shear deformation theory. Sofiyev and Kuruoglu [10–13] presented a theoretical approach on the basis of the Galerkin method to solve vibration problems of functionally graded (FG) truncated and complete conical shells under mixed classical boundary conditions and resting on elastic foundations. Kim [14] used an analytical method to investigate the free vibration characteristics of FGM cylindrical shells partially resting on elastic foundation with an oblique edge. Jin et al. [15, 16] presented a Haar Wavelet Discretization (HWD) method-based solution approach for the free vibration analysis of functionally graded (FG) cylindrical, spherical, and parabolic shells of revolution with arbitrary boundary conditions. Hosseini-Hashemi et al. [17] used the state space approach to investigate the free vibration of Levy-type thick functionally graded (FG) circular cylindrical shell panels to identify the validity range of two common shell theories namely Donnell and Sanders theories. Iqbal et al. [18] applied the wave propagation approach to study the vibration characteristics of functionally graded material circular cylindrical shells with classical boundary conditions. Shah et al. [19] employed the wave propagation method to study the vibrations of functionally graded cylindrical shells based on the Winkler and Pasternak foundations. Loy et al. [20, 21] studied the vibration of cylindrical shells made of a functionally gradient material (FGM) composed of stainless steel and nickel by means of the Rayleigh–Ritz method under some selected boundary conditions. Najafizadeh and Isvandzibaei [22] used the Rayleigh–Ritz method and third order shear deformation shell theory to analyze the free vibrations of the thin cylindrical shells under ring supports and classical boundary conditions. Naeem et al. [23] applied the Ritz formulation to study the vibration frequencies of functionally graded circular cylindrical shells with classical boundary conditions. Taghvaeipour et al. [24] presented a new finite element formulation to study the structural analysis of functionally graded hollow cylinders. Pradyumna and Bandyopadhyay [25] employed the finite element method to investigate the free vibration analysis of functionally graded curved panels including the cylindrical and spherical panels based on the higher-order shear deformation theory. Neves et al. [26] dealt with free vibration problems of functionally graded shells with classical boundary conditions based on the higher-order shear deformation theory and the radial basis functions collocation. Qu et al. [27] described a general formulation for free, steady-state, and transient vibration analyses of functionally graded shells of revolution subjected to arbitrary boundary conditions by means of a modified variational principle in conjunction with a multisegment partitioning procedure. Santos et al. [28, 29] developed a semianalytical axisymmetric finite element model using the 3D linear elastic theory to study the free vibrations of functionally graded cylindrical shells made up of isotropic properties. Vel [30] presented an exact elasticity solution for the free and forced vibration of functionally graded cylindrical shells with classical boundary conditions.The above review indicates that there exist some literatures on the free vibration of a functionally graded cylindrical, conical, spherical panels and shells, and those numerical results are very useful for the practical project application. However, from the literatures review, we also know that the most existing numerical solutions for the title problem usually take account of a onefold computational model instead of unified computational model. For instance, Su et al. [7–9] divided the open and closed functionally graded cylindrical, conical, spherical shells to study and resort different admissible functions by means of the modified Fourier series technology. However, in practical project application, the structural forms are generally unknown. Thus, the establishment of a unified, efficient, and accurate computational model for free vibration analysis of functionally graded cylindrical, conical, spherical panels and shells with general boundary conditions is necessary and significant.In previous study, an improved Fourier series method was proposed by Li [31, 32] to determine the vibration of a single beam under arbitrary boundary conditions. The method has been subsequently exploited to determine the vibration behaviors of structures [33–48]. Recently, a new method of Fourier-Ritz approach was proposed by Jin et al. [49], based on the improved Fourier series method to analyze the free vibration of laminated functionally graded shallow shells with general boundary conditions. In contrast to most existing techniques, the modified Fourier-Ritz method can be universally applicable to a variety of boundary conditions including all the classical cases, elastic restraints, and their combinations without the need of making any changes of the solution procedure. The purpose of this paper is to extend a modified Fourier-Ritz approach to presents a unified computational model for the free vibration of four-parameter functionally graded moderately thick cylindrical, conical, spherical panels and shells of revolution with general boundary conditions on the basis of the first-order shear deformation theory. Applying the modified Fourier-Ritz approach, the admissible functions of the structure elements are expanded into the improved Fourier series which consist of two-dimensional (2D) Fourier cosine series and auxiliary functions to eliminate all the relevant discontinuities of the displacements and their derivatives at the edges regardless of boundary conditions and then solving the natural frequencies by means of the Ritz method. In order to obtain the unified computational model of the functionally graded cylindrical, conical, spherical panels and shells of revolution, as one merit of this paper, the coupling spring technology is introduced to ensure the kinematic and physical compatibility at the common meridian, if a complete shell of revolution needs considering. The excellent accuracy and reliability of the unified computational model are compared with the results found in the literatures.


## 2. Theoretical Formulations

### 2.1. Description of the Model

As shown in Figure1, the basic configuration of the problem considered here is an FG shell. An orthogonal curvilinear coordinate system composed of coordinates α, β, and z is located on the reference surface (z=0) of the shell. Rα and Rβ denote the values of the principal radii of curvature of the reference surface along α and β axes, respectively. Lα and Lβ denote the lengths in α and β directions, respectively. The thickness of shell element is represented by h. As shown in Figure 2, the geometry of panels and shells considered in this paper is a surface of revolution with a cylindrical, conical, and spherical meridian. Based on Figure 2, the coordinate systems and quantities Rα, Rβ, Lα, and Lβ are given as follows: for cylindrical meridian, α=x, β=θ, Rα=∞, Rβ=R, Lα=L, and Lβ=ϕ(0≤ϕ≤2π); for conical meridian, α=s, β=θ, Rα=∞, Rβ=s×tan⁡α0, Lα=L0, and Lβ=ϕ(0≤ϕ≤2π); for spherical meridian, α=φ, β=θ, Rα=Rβ=R, Lα=φ(φ=φ1-φ0), and Lβ=ϕ(0≤ϕ≤2π). The details of the geometric configuration of the FG panels and shells of revolution structures for verifying the accuracy and versatility of the proposed approach are shown in Figure 3.Figure 1
Geometry and notations of a FGM shell.Figure 2
Definition of shell coordinate system: (a) cylindrical shell; (b) conical shell; (c) spherical shell.
(a)
(b)
(c)Figure 3
FG panel and shell structures.


### 2.2. Kinematic Relations and Stress Resultants

According to the FSDT assumptions, the displacement components of an arbitrary point in the FGM panels and shells are expressed in terms of the displacements and rotations of the reference surface, as given as follows:(1a)Uα,β,z,t=u0α,β,t+zψαα,β,t(1b)Vα,β,z,t=v0α,β,t+zψβα,β,t(1c)Wα,β,z,t=w0α,β,t,where u0, v0, and w denote the displacements of corresponding point on reference surface in the α, β, and z directions, respectively. ψα and ψβ are the rotations of the normal to the reference surface about the β and α direction, respectively, and t is the time. Relationships between strains and displacements along the shell reference (middle) surface (z=0) are represented by the following:(2)εα0=1A∂u0∂α+v0AB∂A∂β+w0Rα,εβ0=1B∂v0∂β+u0AB∂B∂α+w0Rβ,γαβ0=BA∂∂αv0B+AB∂∂βu0Aκα=1A∂ψα∂α+ψβAB∂A∂β,κβ=1B∂ψβ∂β+ψαAB∂B∂α,καβ=BA∂∂αψβB+AB∂∂βψαAγαz0=ψα-u0Rα+1A∂w0∂α,γβz0=ψβ-v0Rβ+1B∂w0∂β,where the symbol A and B are the Lamé parameters. According to Figure 2, the Lamé parameters of the above types panels and shells can be defined as follows: for case of the cylindrical meridian, A=1 and B=R; for case of the conical meridian, A=1 and B=s×sin⁡α0; for case of the spherical meridian, A=R and B=R×sin⁡φ.The constitutive equations related the force and moment resultants to strains and curvatures of the reference surface are given in the matrix form:(3)NαNβNαβMαMβMαβ=A11A120B11B120A12A110B12B11000A6600B66B11B120D11D120B12B110D12D11000B6600D66εα0εβ0γαβ0κακβκαβQαQβ=κA6600κA66γαz0γβz0,where Nα,Nβ, and Nαβ are the in-plane force resultants, Mα,Mβ, and Mαβ are moment resultants, and Qα and Qβ are transverse shear force resultants. The shear correction factorκ is computed such that the strain energy due to transverse shear stresses in (3) equals the strain energy due to the true transverse stresses predicted by the three-dimensional elasticity theory. In application, the shear correction factor does not have to be the same in different geometric directions and depended on many factors such as the geometric and material parameters [50]. However, the aim of in this paper is to study the vibration analysis of vibration analysis of functionally graded cylindrical, conical, spherical panels and shells with general boundary conditions. Thus, in order simplify this study and based on existing literature [40, 51], the shear correction factor κ selected a generic parameter κ=5/6 in the next calculation. Aij, Bij, and Dij (i,j=1,2 and 6) are the extensional, extensional-bending coupling and bending stiffness, and they are, respectively, expressed as(4)Aij,Bij,Dij=∫-h/2h/2Qijz1,z,z2dz,where the elastic constants Qijz are functions of thickness coordinate z, which are defined as(5)Q11z=Ez1-μ2z,Q12z=μzEz1-μ2z,Q66z=Ez21+μz.Typically, the functionally graded materials are made of two or more constituent phases which have smooth and continuous mechanical behavior in one or more directions. In this paper, it is assumed that the FG panels and shells are made of a mixture consisting of ceramic and metal. Young’s modulusE(z), density ρ(z), and Poisson’s ratio μ(z) are assumed to vary continuously through the panels thickness and can be expressed as a linear combination:(6)Ez=Ec-EmVc+Emρz=ρc-ρmVc+ρmμz=μc-μmVc+μmin which the subscripts c and m represent the ceramic and metallic constituents, respectively. And the volume fraction Vc follows the two general four-parameter power-law distributions [1]:(7)FGMΙa/b/c/p:Vc=1-a12+zh+b12+zhcpFGMΙΙa/b/c/p:Vc=1-a12-zh+b12-zhcp,where p is the power-law exponent and takes only positive values, a, b, and c are the material parameters, and their values determine the volume fraction through the general four-parameter power-law distributions. In addition, for (7), the difference between the two general four-parameter power-law distributions is decided by 1/2-z/h and 1/2+z/h. In order to get more intuitive description, Figure 4 shows the variations of the volume fraction Vc for different distributions and values of the power-law exponent. Form Figure 4, we can know that the volume fraction Vc with FGMI and FGMII is symmetric with reference to z=0. The volume fraction of all the constituent materials should add up to one; that is,(8)Vc+Vm=1.Figure 4
Variations of the volume fractionVc through the shell thickness for different values of power-law exponent p: (a) FGMI(a=1/b=0/c/p); (b) FGMII(a=1/b=0/c/p); (c) FGMI(a=1/b=0.5/c=2/p); (d) FGMII(a=1/b=0.5/c=2/p); (e) FGMI(a=0/b=-0.5/c=2/p); (f) FGMII(a=0/b=-0.5/c=2/p); (g) FGMI(a=1/b=1/c=4/p); (h) FGMII(a=1/b=1/c=4/p).
(a)
(b)
(c)
(d)
(e)
(f)
(g)
(h)When the value ofp equals zero or infinity, the homogeneous isotropic material can be obtained as a special case of the functionally graded material.The strain energy (Us) of the moderately thick functionally graded panels and shells of revolution can be defined as(9)Us=12∭VNαεα0+Nβεβ0+Nαβγαβ0+Mακα+Mβκβ+Mφβκφβ+Qαγαz0+Qβγβz0dV.Substituting (2) and (3) into (9), the strain energy expression of the structure can be written in terms of middle surface displacements and rotations. For convenience, the strain energy expression is divided into three components; that is, Us=US+UB+UBS, where US, UBS, and UB indicate Stretching, Bending, and Bending-Stretching coupling energy expressions, respectively.(10)US=12∬SA111A∂u0∂α+v0AB∂A∂β+w0Rα2+A111B∂v0∂β+u0AB∂B∂α+w0Rβ2+κA66ψα-u0Rα+1A∂w0∂α2+A66BA∂∂αv0B+AB∂∂βu0A2+2A121A∂u0∂α+v0AB∂A∂β+w0Rα1B∂v0∂β+u0AB∂B∂α+w0Rβ+κA66ψβ-v0Rβ+1B∂w0∂β2UBS=12∬S2B111A∂u0∂α+v0AB∂A∂β+w0Rα1A∂ψα∂α+ψβAB∂A∂β+2B111B∂v0∂β+u0AB∂B∂α+w0Rβ1B∂ψβ∂β+ψαAB∂B∂α+2B121A∂u0∂α+v0AB∂A∂β+w0Rα1B∂ψβ∂β+ψαAB∂B∂α+2B121B∂v0∂β+u0AB∂B∂α+w0Rβ1A∂ψα∂α+ψβAB∂A∂β+2B66BA∂∂αv0B+AB∂∂βu0ABA∂∂αψβB+AB∂∂βψαAdSUB=12∬SD111A∂ψα∂α+ψβAB∂A∂β2+D111B∂ψβ∂β+ψαAB∂B∂α2+2D121A∂ψα∂α+ψβAB∂A∂β1B∂ψβ∂β+ψαAB∂B∂α+D66BA∂∂αψβB+AB∂∂βψαA2dS.The corresponding kinetic energy(T) function of the moderately thick FG panels and shells of revolution can be given as(11)T=12∬SI0∂u0∂t2+∂v0∂t2+∂w0∂t2+2I1∂u0∂t∂ψα∂t+∂v0∂t∂ψβ∂t+I2∂ψα∂t2+∂ψβ∂t2dS,where (12)I0I1I2=∫-h/2h/2ρz1,z1,z2dz.Since the main focus of this paper is to develop a unified computational model to study the vibration of moderately thick FG panels and shells of revolution with general boundary conditions, in order to satisfy the request, the artificial spring boundary technique is adopted here. In this technique, five groups of boundary restraining springs are arranged at all sides of the FG panels and shells of revolution to separately simulate the general boundary conditions. Then the equations describing general elastic supported moderately thick FG panels and shells of revolution can be written as follows:(13a)kα0uu0=Nα,kα0vv0=Nαβ,kα0ww0=Qα,Kα0αψα=Mα,Kα0βψβ=MαβAtα=0,(13b)kα1uu0=-Nα,kα1vv0=-Nαβ,kα1ww0=-Qα,Kα1αψα=-Mα,Kα1βψβ=-MαβAtα=Lα,(13c)kβ0uu0=Nβ,kβ0uv0=Nαβ,kβ0ww0=Qβ,Kβ0αψα=Mβ,Kβ0βψβ=MαβAtβ=0,(13d)kβ1uu0=-Nβ,kβ1uv0=-Nαβ,kβ1ww0=-Qβ,Kβ1αψα=-Mβ,Kβ1βψβ=-MαβAtβ=Lβ.Thus, as one of the merits of the present study, the unified treatment in dealing with the panels under general boundary conditions can be achieved by assigning the stiffness of the boundary springs with various values. For example, the free boundary condition can be readily obtained by setting the spring coefficients to zero and the clamped boundary can be obtained by assigning the springs’ stiffness to infinity. Table1 gives the corresponding spring stiffness values for the considered boundaries in which the symbol D represents the bending stiffness D=Ech3/12(1-μc2). Therefore, the potential energy Usp stored in the boundary springs is given as(14)Usp=12∫kα0uu02+kα0vv02+kα0ww02+Kα0αψα2+Kα0βψβ2α=0kα1uu02+kα1vv02+kα1ww02+Kα1αψα2+Kα1βψβ2α=LαBdβ+12∫kβ0uu02+kβ0vv02+kβ0ww02+Kβ0αψα2+Kβ0βψβ2β=0kβ1uu02+kβ1vv02+kβ1ww02+Kβ1αψα2+Kβ1βψβ2β=LβAdα.Table 1
The corresponding spring stiffness values for general boundary conditions.
Edges
BC
Essential conditions
Corresponding spring stiffness values
Γ
u
Γ
v
Γ
w
Γ
x
Γ
θ
x
=
c
o
n
s
t
a
n
t
F
N
α
=
N
α
β
=
Q
α
=
M
α
=
M
α
β
=
0
0
0
0
0
0
C
u
0
=
v
0
=
w
0
=
ψ
α
=
ψ
β
=
0
108D
108D
108D
108D
108D
S
u
0
=
v
0
=
w
0
=
M
α
=
ψ
β
=
0
108D
108D
108D
0
108D
E1
u
0
≠
0; v0=w0=ψα=ψβ=0
102D
102D
108D
108D
108D
E2
v
0
≠
0; u0=w0=ψα=ψβ=0
108D
108D
102D
108D
108D
E3
u
0
≠
0, v0≠0; w0=ψα=ψβ=0
108D
108D
108D
102D
102D
θ
=
c
o
n
s
t
a
n
t
F
N
β
=
N
α
β
=
Q
β
=
M
β
=
M
α
β
=
0
0
0
0
0
0
C
u
0
=
v
0
=
w
0
=
ψ
α
=
ψ
β
=
0
108D
108D
108D
108D
108D
S
u
0
=
v
0
=
w
0
=
ψ
α
=
M
β
=
0
108D
108D
108D
108D
0
E1
u
0
≠
0; v0=w0=ψα=ψβ=0
102D
102D
108D
108D
108D
E2
v
0
≠
0; u0=w0=ψα=ψβ=0
108D
108D
102D
108D
108D
E3
u
0
≠
0, v0≠0; w0=ψα=ψβ=0
108D
108D
108D
102D
102DAs mentioned before, we can know that the main focus and merits of this paper are to present a unified computational model of the functionally graded cylindrical, conical, spherical panels and shells of revolution. However, the previous studies are most confined to the signer geometric configuration, that is, panels and shells, the difficulty of which is that the admissible functions of the panels do not fit to the shells. As we all know, in addition to the external boundary conditions, the kinematic and physical compatibility should be satisfied at the common meridian ofβ=0 and 2π, if a complete shell of revolution needs considering [3, 53–55]. The kinematic compatibility conditions include the continuity of displacements. The physical compatibility conditions can only be the five continuous conditions for the generalized stress resultants. Thus, to consider the complete shell of revolution characterized by ϕ=2π, it is necessary to implement the kinematic and physical compatibility conditions between the two computational meridians of β=0 and 2π.The kinematic compatibility conditions are as follows:(15)u0α,0,t=u0α,2π,t,v0α,0,t=v0α,2π,t,w0α,0,t=w0α,2π,tψαα,0,t=ψαα,2π,t,ψβα,0,t=ψβα,2π,t.The physical compatibility conditions are as follows:(16)Nβα,0,t=Nβα,2π,t,Nαβα,0,t=Nαβα,2π,t,Qβα,0,t=Qβα,2π,tMβα,0,t=Mβα,2π,t,Mαβα,0,t=Mαβα,2π,t.In order to achieve the above requirements, the authors draw on the experience of two elastically coupled rectangular plates [56] to present a coupling spring technique. As one innovation point of present study, three groups of linear and two groups of coupling springs are applied to imitate the kinematic compatibility and physical compatibility conditions of functionally graded panels at the common meridian of β=0 and 2π when a complete shell of revolution is considered. Therefore, the potential energies (Ucp) stored in the five types of coupling springs can be defined as follows:(17)Ucp=12∫-h/2h/2∫0Lαkucuu0β=0-u0β=2π2+kvcvv0β=0-v0β=2π2+kwcww0β=0-w0β=2π2+Kαcαψαβ=0-ψαβ=2π2+Kβcβψββ=0-ψββ=2π2Bdαdz.It should be noted that the stiffness of corresponding springs used at the common meridian ofβ=0 and 2π of functionally graded panels is revalued to be zero automatically.


### 2.3. The Modified Fourier-Ritz Approach

The modified Fourier-Ritz approach was proposed by Jin et al. [49] to solve the free vibrations of laminated functionally graded shallow shells with general boundary conditions. The presented method consists of two main steps: firstly, the admissible functions of the structure elements are expanded into the modified Fourier series which consist of the two-dimensional (2D) Fourier cosine series and auxiliary functions to eliminate all the relevant discontinuities of the displacements and their derivatives at the edges regardless of boundary conditions; secondly, the natural frequencies of the structure elements are obtained by using the variational operation (Ritz method) in terms of the unknown the expanded coefficient. In contrast to most existing techniques, the modified Fourier-Ritz method can be universally applicable to a variety of boundary conditions including all the classical cases, elastic restraints, and their combinations without the need of making any changes to the solution procedure. Therefore, each of the referenced surface displacements or rotations of the normal of a shell can be written as an improved Fourier series as follows:(18a)u0α,β,t=∑m=0∞∑n=0∞Amnucos⁡λmαcos⁡λnβ+∑l=12χlβ∑m=0∞amlcos⁡λmα+∑l=12ζlα∑n=0∞bnlcos⁡λnβejωt(18b)v0α,β,t=∑m=0∞∑n=0∞Bmnvcos⁡λmαcos⁡λnβ+∑l=12χlβ∑m=0∞cmlcos⁡λmα+∑l=12ζlα∑n=0∞dnlcos⁡λnβejωt(18c)w0α,β,t=∑m=0∞∑n=0∞Cmnwcos⁡λmαcos⁡λnβ+∑l=12χlβ∑m=0∞emlcos⁡λmα+∑l=12ζlα∑n=0∞fnlcos⁡λnβejωt(18d)ψαα,β,t=∑m=0∞∑n=0∞Dmnαcos⁡λmαcos⁡λnβ+∑l=12χlβ∑m=0∞gmlcos⁡λmα+∑l=12ζlα∑n=0∞hnlcos⁡λnβejωt(18e)ψβα,β,t=∑m=0∞∑n=0∞Emnβcos⁡λmαcos⁡λnβ+∑l=12χlβ∑m=0∞kmlcos⁡λmα+∑l=12ζlα∑n=0∞qnlcos⁡λnβejωt,where λm=mπ/Lα, λn=nπ/Lβ, and Amnu, Bmnv, Cmnw, Dmnφ, and Emnθ are the Fourier coefficients of two-dimensional Fourier series expansions for the displacements functions, respectively. aml, bnl, cml, dnl, eml, fnl, gml, hnl, kml, and qnl are the supplemented coefficients of the auxiliary functions χlβ and ζl(α), where l=1,2. The two types of auxiliary functions χlβ and ζl(α) are selected to remove all the discontinuities potentially associated with the first-order derivatives at the boundaries. Thus, the function sets are capable of representing any free vibration motion of the shell. The two types of auxiliary functions are given as(19a)ζα=ζ1φ=Lα2πsin⁡πα2Lα+Lα2πsin⁡3πα2Lαζ2φ=-Lα2πcos⁡πα2Lα+Lα2πcos⁡3πα2Lα(19b)χβ=χ1β=Lβ2πsin⁡πβ2Lβ+Lβ2πsin⁡3πβ2Lβχ2β=-Lβ2πcos⁡πβ2Lβ+Lβ2πcos⁡3πβ2Lβ.It is easy to verify that(20a)ζ10=ζ1Lα=ζ1′Lα=0,ζ1′0=1(20b)ζ20=ζ2Lα=ζ2′0=0,ζ2′Lα=1(20c)χ10=χ1Lβ=χ1′Lβ=0,χ1′0=1(20d)χ20=χ2Lβ=χ2′0=0,χ2′Lβ=1The Lagrangian energy function (L) of the moderately thick functionally graded panels and shells of revolution can be written as(21)L=T-Us-Usp-Ucp.Substituting (9), (11), (14), and (17) into (21), then the Lagrangian expression is minimized by taking its derivatives with respect to these coefficients:(22)∂L∂ι=0,ι=Amnu,aml,bnl,Bmnv,cml,dnl,…,Emnβ,gml,hnl.Since the displacements and rotation components of the moderately thick functionally graded panels and shells of revolution are chosen asM and N to obtain the results with acceptable accuracy, a total of 5×(M+1)×(N+1)+10×(M+N+2) equations are obtained. They can be summed up in a matrix form:(23)K-ω2MH=0,where K, M, and H, respectively, represent the stiffness matrix, mass matrix, and vector of the unknown coefficients for the panel and shell. The detailed expressions for above matrices are given in Appendix. By solving (23), the frequencies (or eigenvalues) of moderately thick functionally graded panels and shells of revolution can be readily obtained and the mode shapes can be yielded by substituting the corresponding eigenvectors into series representations of displacement components.


## 2.1. Description of the Model

As shown in Figure1, the basic configuration of the problem considered here is an FG shell. An orthogonal curvilinear coordinate system composed of coordinates α, β, and z is located on the reference surface (z=0) of the shell. Rα and Rβ denote the values of the principal radii of curvature of the reference surface along α and β axes, respectively. Lα and Lβ denote the lengths in α and β directions, respectively. The thickness of shell element is represented by h. As shown in Figure 2, the geometry of panels and shells considered in this paper is a surface of revolution with a cylindrical, conical, and spherical meridian. Based on Figure 2, the coordinate systems and quantities Rα, Rβ, Lα, and Lβ are given as follows: for cylindrical meridian, α=x, β=θ, Rα=∞, Rβ=R, Lα=L, and Lβ=ϕ(0≤ϕ≤2π); for conical meridian, α=s, β=θ, Rα=∞, Rβ=s×tan⁡α0, Lα=L0, and Lβ=ϕ(0≤ϕ≤2π); for spherical meridian, α=φ, β=θ, Rα=Rβ=R, Lα=φ(φ=φ1-φ0), and Lβ=ϕ(0≤ϕ≤2π). The details of the geometric configuration of the FG panels and shells of revolution structures for verifying the accuracy and versatility of the proposed approach are shown in Figure 3.Figure 1
Geometry and notations of a FGM shell.Figure 2
Definition of shell coordinate system: (a) cylindrical shell; (b) conical shell; (c) spherical shell.
(a)
(b)
(c)Figure 3
FG panel and shell structures.


## 2.2. Kinematic Relations and Stress Resultants

According to the FSDT assumptions, the displacement components of an arbitrary point in the FGM panels and shells are expressed in terms of the displacements and rotations of the reference surface, as given as follows:(1a)Uα,β,z,t=u0α,β,t+zψαα,β,t(1b)Vα,β,z,t=v0α,β,t+zψβα,β,t(1c)Wα,β,z,t=w0α,β,t,where u0, v0, and w denote the displacements of corresponding point on reference surface in the α, β, and z directions, respectively. ψα and ψβ are the rotations of the normal to the reference surface about the β and α direction, respectively, and t is the time. Relationships between strains and displacements along the shell reference (middle) surface (z=0) are represented by the following:(2)εα0=1A∂u0∂α+v0AB∂A∂β+w0Rα,εβ0=1B∂v0∂β+u0AB∂B∂α+w0Rβ,γαβ0=BA∂∂αv0B+AB∂∂βu0Aκα=1A∂ψα∂α+ψβAB∂A∂β,κβ=1B∂ψβ∂β+ψαAB∂B∂α,καβ=BA∂∂αψβB+AB∂∂βψαAγαz0=ψα-u0Rα+1A∂w0∂α,γβz0=ψβ-v0Rβ+1B∂w0∂β,where the symbol A and B are the Lamé parameters. According to Figure 2, the Lamé parameters of the above types panels and shells can be defined as follows: for case of the cylindrical meridian, A=1 and B=R; for case of the conical meridian, A=1 and B=s×sin⁡α0; for case of the spherical meridian, A=R and B=R×sin⁡φ.The constitutive equations related the force and moment resultants to strains and curvatures of the reference surface are given in the matrix form:(3)NαNβNαβMαMβMαβ=A11A120B11B120A12A110B12B11000A6600B66B11B120D11D120B12B110D12D11000B6600D66εα0εβ0γαβ0κακβκαβQαQβ=κA6600κA66γαz0γβz0,where Nα,Nβ, and Nαβ are the in-plane force resultants, Mα,Mβ, and Mαβ are moment resultants, and Qα and Qβ are transverse shear force resultants. The shear correction factorκ is computed such that the strain energy due to transverse shear stresses in (3) equals the strain energy due to the true transverse stresses predicted by the three-dimensional elasticity theory. In application, the shear correction factor does not have to be the same in different geometric directions and depended on many factors such as the geometric and material parameters [50]. However, the aim of in this paper is to study the vibration analysis of vibration analysis of functionally graded cylindrical, conical, spherical panels and shells with general boundary conditions. Thus, in order simplify this study and based on existing literature [40, 51], the shear correction factor κ selected a generic parameter κ=5/6 in the next calculation. Aij, Bij, and Dij (i,j=1,2 and 6) are the extensional, extensional-bending coupling and bending stiffness, and they are, respectively, expressed as(4)Aij,Bij,Dij=∫-h/2h/2Qijz1,z,z2dz,where the elastic constants Qijz are functions of thickness coordinate z, which are defined as(5)Q11z=Ez1-μ2z,Q12z=μzEz1-μ2z,Q66z=Ez21+μz.Typically, the functionally graded materials are made of two or more constituent phases which have smooth and continuous mechanical behavior in one or more directions. In this paper, it is assumed that the FG panels and shells are made of a mixture consisting of ceramic and metal. Young’s modulusE(z), density ρ(z), and Poisson’s ratio μ(z) are assumed to vary continuously through the panels thickness and can be expressed as a linear combination:(6)Ez=Ec-EmVc+Emρz=ρc-ρmVc+ρmμz=μc-μmVc+μmin which the subscripts c and m represent the ceramic and metallic constituents, respectively. And the volume fraction Vc follows the two general four-parameter power-law distributions [1]:(7)FGMΙa/b/c/p:Vc=1-a12+zh+b12+zhcpFGMΙΙa/b/c/p:Vc=1-a12-zh+b12-zhcp,where p is the power-law exponent and takes only positive values, a, b, and c are the material parameters, and their values determine the volume fraction through the general four-parameter power-law distributions. In addition, for (7), the difference between the two general four-parameter power-law distributions is decided by 1/2-z/h and 1/2+z/h. In order to get more intuitive description, Figure 4 shows the variations of the volume fraction Vc for different distributions and values of the power-law exponent. Form Figure 4, we can know that the volume fraction Vc with FGMI and FGMII is symmetric with reference to z=0. The volume fraction of all the constituent materials should add up to one; that is,(8)Vc+Vm=1.Figure 4
Variations of the volume fractionVc through the shell thickness for different values of power-law exponent p: (a) FGMI(a=1/b=0/c/p); (b) FGMII(a=1/b=0/c/p); (c) FGMI(a=1/b=0.5/c=2/p); (d) FGMII(a=1/b=0.5/c=2/p); (e) FGMI(a=0/b=-0.5/c=2/p); (f) FGMII(a=0/b=-0.5/c=2/p); (g) FGMI(a=1/b=1/c=4/p); (h) FGMII(a=1/b=1/c=4/p).
(a)
(b)
(c)
(d)
(e)
(f)
(g)
(h)When the value ofp equals zero or infinity, the homogeneous isotropic material can be obtained as a special case of the functionally graded material.The strain energy (Us) of the moderately thick functionally graded panels and shells of revolution can be defined as(9)Us=12∭VNαεα0+Nβεβ0+Nαβγαβ0+Mακα+Mβκβ+Mφβκφβ+Qαγαz0+Qβγβz0dV.Substituting (2) and (3) into (9), the strain energy expression of the structure can be written in terms of middle surface displacements and rotations. For convenience, the strain energy expression is divided into three components; that is, Us=US+UB+UBS, where US, UBS, and UB indicate Stretching, Bending, and Bending-Stretching coupling energy expressions, respectively.(10)US=12∬SA111A∂u0∂α+v0AB∂A∂β+w0Rα2+A111B∂v0∂β+u0AB∂B∂α+w0Rβ2+κA66ψα-u0Rα+1A∂w0∂α2+A66BA∂∂αv0B+AB∂∂βu0A2+2A121A∂u0∂α+v0AB∂A∂β+w0Rα1B∂v0∂β+u0AB∂B∂α+w0Rβ+κA66ψβ-v0Rβ+1B∂w0∂β2UBS=12∬S2B111A∂u0∂α+v0AB∂A∂β+w0Rα1A∂ψα∂α+ψβAB∂A∂β+2B111B∂v0∂β+u0AB∂B∂α+w0Rβ1B∂ψβ∂β+ψαAB∂B∂α+2B121A∂u0∂α+v0AB∂A∂β+w0Rα1B∂ψβ∂β+ψαAB∂B∂α+2B121B∂v0∂β+u0AB∂B∂α+w0Rβ1A∂ψα∂α+ψβAB∂A∂β+2B66BA∂∂αv0B+AB∂∂βu0ABA∂∂αψβB+AB∂∂βψαAdSUB=12∬SD111A∂ψα∂α+ψβAB∂A∂β2+D111B∂ψβ∂β+ψαAB∂B∂α2+2D121A∂ψα∂α+ψβAB∂A∂β1B∂ψβ∂β+ψαAB∂B∂α+D66BA∂∂αψβB+AB∂∂βψαA2dS.The corresponding kinetic energy(T) function of the moderately thick FG panels and shells of revolution can be given as(11)T=12∬SI0∂u0∂t2+∂v0∂t2+∂w0∂t2+2I1∂u0∂t∂ψα∂t+∂v0∂t∂ψβ∂t+I2∂ψα∂t2+∂ψβ∂t2dS,where (12)I0I1I2=∫-h/2h/2ρz1,z1,z2dz.Since the main focus of this paper is to develop a unified computational model to study the vibration of moderately thick FG panels and shells of revolution with general boundary conditions, in order to satisfy the request, the artificial spring boundary technique is adopted here. In this technique, five groups of boundary restraining springs are arranged at all sides of the FG panels and shells of revolution to separately simulate the general boundary conditions. Then the equations describing general elastic supported moderately thick FG panels and shells of revolution can be written as follows:(13a)kα0uu0=Nα,kα0vv0=Nαβ,kα0ww0=Qα,Kα0αψα=Mα,Kα0βψβ=MαβAtα=0,(13b)kα1uu0=-Nα,kα1vv0=-Nαβ,kα1ww0=-Qα,Kα1αψα=-Mα,Kα1βψβ=-MαβAtα=Lα,(13c)kβ0uu0=Nβ,kβ0uv0=Nαβ,kβ0ww0=Qβ,Kβ0αψα=Mβ,Kβ0βψβ=MαβAtβ=0,(13d)kβ1uu0=-Nβ,kβ1uv0=-Nαβ,kβ1ww0=-Qβ,Kβ1αψα=-Mβ,Kβ1βψβ=-MαβAtβ=Lβ.Thus, as one of the merits of the present study, the unified treatment in dealing with the panels under general boundary conditions can be achieved by assigning the stiffness of the boundary springs with various values. For example, the free boundary condition can be readily obtained by setting the spring coefficients to zero and the clamped boundary can be obtained by assigning the springs’ stiffness to infinity. Table1 gives the corresponding spring stiffness values for the considered boundaries in which the symbol D represents the bending stiffness D=Ech3/12(1-μc2). Therefore, the potential energy Usp stored in the boundary springs is given as(14)Usp=12∫kα0uu02+kα0vv02+kα0ww02+Kα0αψα2+Kα0βψβ2α=0kα1uu02+kα1vv02+kα1ww02+Kα1αψα2+Kα1βψβ2α=LαBdβ+12∫kβ0uu02+kβ0vv02+kβ0ww02+Kβ0αψα2+Kβ0βψβ2β=0kβ1uu02+kβ1vv02+kβ1ww02+Kβ1αψα2+Kβ1βψβ2β=LβAdα.Table 1
The corresponding spring stiffness values for general boundary conditions.
Edges
BC
Essential conditions
Corresponding spring stiffness values
Γ
u
Γ
v
Γ
w
Γ
x
Γ
θ
x
=
c
o
n
s
t
a
n
t
F
N
α
=
N
α
β
=
Q
α
=
M
α
=
M
α
β
=
0
0
0
0
0
0
C
u
0
=
v
0
=
w
0
=
ψ
α
=
ψ
β
=
0
108D
108D
108D
108D
108D
S
u
0
=
v
0
=
w
0
=
M
α
=
ψ
β
=
0
108D
108D
108D
0
108D
E1
u
0
≠
0; v0=w0=ψα=ψβ=0
102D
102D
108D
108D
108D
E2
v
0
≠
0; u0=w0=ψα=ψβ=0
108D
108D
102D
108D
108D
E3
u
0
≠
0, v0≠0; w0=ψα=ψβ=0
108D
108D
108D
102D
102D
θ
=
c
o
n
s
t
a
n
t
F
N
β
=
N
α
β
=
Q
β
=
M
β
=
M
α
β
=
0
0
0
0
0
0
C
u
0
=
v
0
=
w
0
=
ψ
α
=
ψ
β
=
0
108D
108D
108D
108D
108D
S
u
0
=
v
0
=
w
0
=
ψ
α
=
M
β
=
0
108D
108D
108D
108D
0
E1
u
0
≠
0; v0=w0=ψα=ψβ=0
102D
102D
108D
108D
108D
E2
v
0
≠
0; u0=w0=ψα=ψβ=0
108D
108D
102D
108D
108D
E3
u
0
≠
0, v0≠0; w0=ψα=ψβ=0
108D
108D
108D
102D
102DAs mentioned before, we can know that the main focus and merits of this paper are to present a unified computational model of the functionally graded cylindrical, conical, spherical panels and shells of revolution. However, the previous studies are most confined to the signer geometric configuration, that is, panels and shells, the difficulty of which is that the admissible functions of the panels do not fit to the shells. As we all know, in addition to the external boundary conditions, the kinematic and physical compatibility should be satisfied at the common meridian ofβ=0 and 2π, if a complete shell of revolution needs considering [3, 53–55]. The kinematic compatibility conditions include the continuity of displacements. The physical compatibility conditions can only be the five continuous conditions for the generalized stress resultants. Thus, to consider the complete shell of revolution characterized by ϕ=2π, it is necessary to implement the kinematic and physical compatibility conditions between the two computational meridians of β=0 and 2π.The kinematic compatibility conditions are as follows:(15)u0α,0,t=u0α,2π,t,v0α,0,t=v0α,2π,t,w0α,0,t=w0α,2π,tψαα,0,t=ψαα,2π,t,ψβα,0,t=ψβα,2π,t.The physical compatibility conditions are as follows:(16)Nβα,0,t=Nβα,2π,t,Nαβα,0,t=Nαβα,2π,t,Qβα,0,t=Qβα,2π,tMβα,0,t=Mβα,2π,t,Mαβα,0,t=Mαβα,2π,t.In order to achieve the above requirements, the authors draw on the experience of two elastically coupled rectangular plates [56] to present a coupling spring technique. As one innovation point of present study, three groups of linear and two groups of coupling springs are applied to imitate the kinematic compatibility and physical compatibility conditions of functionally graded panels at the common meridian of β=0 and 2π when a complete shell of revolution is considered. Therefore, the potential energies (Ucp) stored in the five types of coupling springs can be defined as follows:(17)Ucp=12∫-h/2h/2∫0Lαkucuu0β=0-u0β=2π2+kvcvv0β=0-v0β=2π2+kwcww0β=0-w0β=2π2+Kαcαψαβ=0-ψαβ=2π2+Kβcβψββ=0-ψββ=2π2Bdαdz.It should be noted that the stiffness of corresponding springs used at the common meridian ofβ=0 and 2π of functionally graded panels is revalued to be zero automatically.


## 2.3. The Modified Fourier-Ritz Approach

The modified Fourier-Ritz approach was proposed by Jin et al. [49] to solve the free vibrations of laminated functionally graded shallow shells with general boundary conditions. The presented method consists of two main steps: firstly, the admissible functions of the structure elements are expanded into the modified Fourier series which consist of the two-dimensional (2D) Fourier cosine series and auxiliary functions to eliminate all the relevant discontinuities of the displacements and their derivatives at the edges regardless of boundary conditions; secondly, the natural frequencies of the structure elements are obtained by using the variational operation (Ritz method) in terms of the unknown the expanded coefficient. In contrast to most existing techniques, the modified Fourier-Ritz method can be universally applicable to a variety of boundary conditions including all the classical cases, elastic restraints, and their combinations without the need of making any changes to the solution procedure. Therefore, each of the referenced surface displacements or rotations of the normal of a shell can be written as an improved Fourier series as follows:(18a)u0α,β,t=∑m=0∞∑n=0∞Amnucos⁡λmαcos⁡λnβ+∑l=12χlβ∑m=0∞amlcos⁡λmα+∑l=12ζlα∑n=0∞bnlcos⁡λnβejωt(18b)v0α,β,t=∑m=0∞∑n=0∞Bmnvcos⁡λmαcos⁡λnβ+∑l=12χlβ∑m=0∞cmlcos⁡λmα+∑l=12ζlα∑n=0∞dnlcos⁡λnβejωt(18c)w0α,β,t=∑m=0∞∑n=0∞Cmnwcos⁡λmαcos⁡λnβ+∑l=12χlβ∑m=0∞emlcos⁡λmα+∑l=12ζlα∑n=0∞fnlcos⁡λnβejωt(18d)ψαα,β,t=∑m=0∞∑n=0∞Dmnαcos⁡λmαcos⁡λnβ+∑l=12χlβ∑m=0∞gmlcos⁡λmα+∑l=12ζlα∑n=0∞hnlcos⁡λnβejωt(18e)ψβα,β,t=∑m=0∞∑n=0∞Emnβcos⁡λmαcos⁡λnβ+∑l=12χlβ∑m=0∞kmlcos⁡λmα+∑l=12ζlα∑n=0∞qnlcos⁡λnβejωt,where λm=mπ/Lα, λn=nπ/Lβ, and Amnu, Bmnv, Cmnw, Dmnφ, and Emnθ are the Fourier coefficients of two-dimensional Fourier series expansions for the displacements functions, respectively. aml, bnl, cml, dnl, eml, fnl, gml, hnl, kml, and qnl are the supplemented coefficients of the auxiliary functions χlβ and ζl(α), where l=1,2. The two types of auxiliary functions χlβ and ζl(α) are selected to remove all the discontinuities potentially associated with the first-order derivatives at the boundaries. Thus, the function sets are capable of representing any free vibration motion of the shell. The two types of auxiliary functions are given as(19a)ζα=ζ1φ=Lα2πsin⁡πα2Lα+Lα2πsin⁡3πα2Lαζ2φ=-Lα2πcos⁡πα2Lα+Lα2πcos⁡3πα2Lα(19b)χβ=χ1β=Lβ2πsin⁡πβ2Lβ+Lβ2πsin⁡3πβ2Lβχ2β=-Lβ2πcos⁡πβ2Lβ+Lβ2πcos⁡3πβ2Lβ.It is easy to verify that(20a)ζ10=ζ1Lα=ζ1′Lα=0,ζ1′0=1(20b)ζ20=ζ2Lα=ζ2′0=0,ζ2′Lα=1(20c)χ10=χ1Lβ=χ1′Lβ=0,χ1′0=1(20d)χ20=χ2Lβ=χ2′0=0,χ2′Lβ=1The Lagrangian energy function (L) of the moderately thick functionally graded panels and shells of revolution can be written as(21)L=T-Us-Usp-Ucp.Substituting (9), (11), (14), and (17) into (21), then the Lagrangian expression is minimized by taking its derivatives with respect to these coefficients:(22)∂L∂ι=0,ι=Amnu,aml,bnl,Bmnv,cml,dnl,…,Emnβ,gml,hnl.Since the displacements and rotation components of the moderately thick functionally graded panels and shells of revolution are chosen asM and N to obtain the results with acceptable accuracy, a total of 5×(M+1)×(N+1)+10×(M+N+2) equations are obtained. They can be summed up in a matrix form:(23)K-ω2MH=0,where K, M, and H, respectively, represent the stiffness matrix, mass matrix, and vector of the unknown coefficients for the panel and shell. The detailed expressions for above matrices are given in Appendix. By solving (23), the frequencies (or eigenvalues) of moderately thick functionally graded panels and shells of revolution can be readily obtained and the mode shapes can be yielded by substituting the corresponding eigenvectors into series representations of displacement components.


## 3. Numerical Results and Discussion

In this section, some results and discussions about the free vibration of four-parameter functionally graded moderately thick cylindrical, conical, spherical panels and shells of revolution are presented to verify the accuracy and flexibility of the proposed unified approach. Unless mentioned otherwise, the panels and shells under consideration are assumed to be made of a mixture of ZnO2 (Ec=168 GPa, µc=0.3, ρc=5700 kg/m3) and Al (Em=70 GPa, µm=0.3, ρm=2707 kg/m3). In the following frequencies results, the nondimensional frequency parameters of the cylindrical, conical, and spherical panels and shells are expressed as follows: cylindrical meridian: Ω=ωL2/hρc/Ec; conical meridian: Ω=ωL02/hρc/Ec; spherical meridian: Ω=ωR2/hρc/Ec.


### 3.1. Convergence Study and Formulation Validation

From the theoretical formulations, we can know that the computational accuracy relies on a limited number of terms in the displacement expressions in actual calculation. In addition, when considering the shells of revolution, the results strongly rely on the coupling parameters. Thus, one important issue is the choice of the appropriate truncation terms and coupling parameters. The nondimensional frequency parameter convergence of the clamped shells with respect to different coupling parameters is examined in Table2. The geometrical parameters and power-law exponents for the functionally graded shells used in the study are as follows: cylindrical shell: R=1 m,L=4 m, h=0.1 m, and p=0; conical shell: R1=0.5 m, L0=4/cosα0R1, h=0.1 m, α0=40°, and p=0; spherical shell: R=1 m, φ0=π/6, φ0=π/2, h=0.1 m, and p=0. All coupling parameters are defined with the same value and defined as follows: kucu=kvcv=kwcw=Kαcα=Kβcβ=ϗ. It is obvious that the frequency parameters rapidly converge as the coupling parameter increases. In the absence of the coupling parameters or for small coupling parameters, pseudovibration characteristic may appear in the present solution. This implies that small coupling parameters cannot impose the interface continuity conditions in a proper manner. Thus, based on the existed analysis, ϗ=1014 has been adopted to present all the results in the following analysis.Table 2
Convergence of frequency parameters of the single curved shells with different coupling parametersϗ.
Shape
Mode
101
103
105
107
108
1010
1012
1014
1016
1018
Cylindrical shells
1
10.043
10.043
10.058
10.680
11.659
16.469
16.779
16.780
16.780
16.780
2
10.051
10.051
10.062
11.315
14.581
16.681
16.779
16.780
16.780
16.780
3
16.515
16.515
16.518
16.685
16.857
17.399
17.711
17.714
17.714
17.714
4
17.203
17.203
17.204
17.215
17.236
17.399
17.711
17.714
17.714
17.714
5
17.564
17.564
17.566
17.744
19.153
21.296
21.381
21.381
21.381
21.381
6
19.644
19.644
19.645
19.685
19.762
21.374
21.381
21.381
21.381
21.381
Conical shells
1
10.376
10.376
10.391
11.164
12.259
17.946
18.161
18.162
18.162
18.162
2
10.392
10.392
10.418
12.431
16.393
17.985
18.161
18.162
18.162
18.162
3
17.786
17.786
17.790
17.990
18.164
18.915
19.280
19.286
19.286
19.286
4
18.553
18.553
18.557
18.676
18.693
18.994
19.280
19.286
19.286
19.286
5
18.725
18.725
18.726
19.009
20.736
21.794
21.855
21.855
21.855
21.855
6
20.595
20.595
20.602
21.018
21.415
21.802
21.855
21.855
21.855
21.855
Spherical shells
1
9.766
9.761
9.757
9.867
10.088
11.761
12.034
12.038
12.038
12.038
2
9.766
9.764
9.769
10.017
10.809
11.827
12.036
12.038
12.038
12.038
3
12.046
12.046
12.046
12.055
12.086
12.163
12.221
12.224
12.224
12.224
4
12.117
12.117
12.118
12.133
12.153
12.185
12.223
12.224
12.224
12.224
5
12.308
12.308
12.319
12.321
12.340
12.672
13.084
13.089
13.089
13.089
6
12.493
12.493
12.493
12.523
12.665
12.947
13.085
13.089
13.089
13.089Next, the convergence studies of the cylindrical, conical, and spherical panels and shells will be discussed. Figure5 shows the convergence studies of the first three frequency parameters for the structure elements for complete clamped FG panels and shells with different truncated numbers M and N. The geometrical dimensions of the shells are the same as Table 2 and the panels are also in accordance with Table 2 except the circumferential angle ϕ=120°. From the figure, we can know that the proposed unified approach has fast convergence and good stability. In view of the excellent numerical behavior of the current solution, the truncation numbers will be simply set as M=N=15 in the following calculations.Figure 5
Variation of the frequency parameters Ω versus the truncated numbersM and N for complete clamped FG panels and shells: (a) cylindrical panels; (b) cylindrical shells; (c) conical panels; (d) conical shells; (e) spherical panels; (f) spherical shells.
(a)
(b)
(c)
(d)
(e)
(f)The validation of the unified computational model versus about the different geometric parameters is shown in Tables3–8. The details regarding the geometry of the structures considered are indicated as follows:(1)
Cylindrical panels (FFCF):R=1 m, L=2 m, h=0.1 m, and ϕ=120° (Table 3).(2)
Cylindrical shells (FC):R=1 m, L=2 m, and h=0.1 m (Table 4).(3)
Conical panels (SCSC):R1=1 m, L0=2 m, h=0.1 m, α0=40°, and ϕ=120° (Table 5).(4)
Conical shells (FC):R1=0.5 m, L0cos⁡α0=2 m, h=0.1 m, and α0=40° (Table 6).(5)
Spherical panels (FFCF):R=2 m, h=0.1 m, φ0=30°, φ1=90°, and ϕ=120° (Table 7).(6)
Spherical shells:R=1 m, h=0.05 m, φ0=22.5°, and φ1=90° (Table 8).Table 3
Comparison of the first nine frequencies (Hz) of functionally graded cylindrical panels with different power-law indexp.
p
Method
Mode numbers
1
2
3
4
5
6
7
8
9
0
Ref [52]
61.02
94.82
153.13
241.39
275.83
291.45
355.14
450.76
511.92
Ref [8]
61.06
94.92
153.11
241.75
276.37
291.94
355.07
450.82
512.48
Present
61.02
94.82
153.12
241.60
276.22
291.64
355.09
450.72
512.31
0.6
Ref [52]
59.24
93.14
148.51
234.79
267.06
287.85
343.22
443.06
495.89
Ref [8]
59.13
93.03
148.14
234.60
266.97
287.63
342.38
442.09
495.31
Present
59.09
92.93
148.15
234.45
266.83
287.34
342.39
441.99
495.15
1
Ref [52]
58.96
92.65
147.94
233.64
265.86
286.16
342.08
440.72
493.67
Ref [8]
58.82
92.48
147.49
233.31
265.61
285.77
341.04
439.49
492.80
Present
58.78
92.38
147.50
233.16
265.47
285.48
341.05
439.39
492.63
5
Ref [52]
60.09
91.69
152.02
236.75
272.42
279.34
354.89
436.34
505.35
Ref [8]
59.96
91.54
151.57
236.42
272.19
279.06
353.83
435.18
504.56
Present
59.92
91.44
151.59
236.29
272.04
278.78
353.84
435.09
504.37
20
Ref [52]
59.19
90.29
149.49
233.27
268.37
275.23
348.80
429.63
497.87
Ref [8]
59.16
90.30
149.29
233.32
268.57
275.41
348.32
429.19
497.87
Present
59.12
90.20
149.30
233.20
268.43
275.13
348.33
429.10
497.69
50
Ref [52]
58.20
89.55
146.55
229.79
263.58
273.98
341.04
425.83
489.06
Ref [8]
58.21
89.60
146.45
230.00
263.95
274.32
340.79
425.66
489.36
Present
58.17
89.51
146.46
229.87
263.81
274.04
340.80
425.57
489.19Table 4
Comparison of the first nine frequencies (Hz) of functionally graded cylindrical shells with different power-law indexp.
p
Method
Mode number
1
2
3
4
5
6
7
8
0
Ref [1]
152.93
152.93
220.06
220.06
253.78
253.78
383.55
383.56
Ref [9]
152.89
152.89
219.97
219.97
253.79
253.79
383.44
383.44
Present
152.89
152.89
220.12
220.12
253.77
253.77
383.95
383.95
0.6
Ref [1]
152.25
152.25
219.86
219.86
252.17
252.17
383.39
383.40
Ref [27]
152.02
152.02
219.54
219.54
251.91
251.91
382.87
382.87
Ref [9]
152.07
152.07
219.59
219.59
251.96
251.96
382.95
382.95
Present
152.08
152.08
219.73
219.83
251.95
251.95
383.46
383.46
1
Ref [1]
151.77
151.77
219.56
219.56
251.14
251.14
382.97
382.97
Ref [9]
151.52
151.52
219.19
219.19
250.81
250.81
382.35
382.35
Present
151.53
151.53
219.33
219.33
250.80
250.80
382.85
382.85
5
Ref [1]
148.97
148.97
218.87
218.88
244.40
244.40
382.46
382.47
Ref [27]
148.50
148.50
218.16
218.16
243.73
243.73
381.26
381.26
Ref [9]
148.53
148.53
218.21
218.21
243.76
243.76
381.33
381.33
Present
148.54
148.54
218.35
218.35
243.75
243.75
381.82
381.82
20
Ref [1]
146.46
146.46
215.90
215.90
239.84
239.84
377.34
377.34
Ref [27]
146.21
146.21
215.50
215.50
239.54
239.54
376.69
376.69
Ref [9]
146.24
146.24
215.55
215.55
239.57
239.57
376.76
376.76
Present
146.25
146.25
215.69
215.69
239.56
239.56
377.23
377.23Table 5
Comparison of the first nine frequencies (Hz) of the functionally graded conical panels with different power-law indexp.
p
Method
Mode number
1
2
3
4
5
6
7
8
9
F
G
M
I
(
a
=
1
/
b
=
0.5
/
c
=
2
/
p
)
0.6
Ref [3]
80.10
110.93
158.60
195.20
259.10
273.52
321.39
360.77
396.83
Ref [10]
80.15
111.03
158.61
195.49
259.31
273.52
321.38
361.16
397.06
Present
80.14
110.95
158.62
195.57
259.18
273.62
321.42
361.38
397.12
1
Ref [3]
79.89
110.56
158.15
194.72
258.36
272.88
320.38
359.84
395.97
Ref [10]
79.91
110.62
158.10
194.94
258.48
272.82
320.28
360.10
396.11
Present
79.91
110.54
158.11
195.02
258.36
272.91
320.31
360.33
396.18
5
Ref [3]
78.92
108.45
156.07
192.86
254.58
270.76
315.52
356.30
393.68
Ref [10]
78.86
108.39
155.87
192.87
254.45
270.51
315.12
356.20
393.55
Present
78.86
108.31
155.88
192.95
254.33
270.60
315.16
356.42
393.62
20
Ref [3]
77.81
106.72
153.93
190.43
250.80
267.51
311.11
351.98
389.27
Ref [10]
77.83
106.78
153.90
190.65
250.92
267.46
311.03
352.24
389.42
Present
77.83
106.70
153.91
190.73
250.80
267.56
311.07
352.46
389.48
50
Ref [3]
76.57
105.58
151.56
187.03
247.32
262.39
306.75
345.73
381.29
Ref [10]
76.63
105.69
151.60
187.34
247.57
262.43
306.81
346.17
381.56
Present
76.63
105.62
151.61
187.43
247.45
262.53
306.85
346.39
381.63
F
G
M
I
I
(
a
=
1
/
b
=
0.5
/
c
=
2
/
p
)
0.6
Ref [3]
79.87
110.71
158.27
194.75
258.47
272.88
320.85
360.10
395.93
Ref [10]
80.03
110.96
158.48
195.29
258.99
273.14
321.24
360.94
396.49
Present
80.03
110.88
158.49
195.37
258.86
273.23
321.27
361.16
396.56
1
Ref [3]
79.54
110.22
157.64
194.02
257.39
271.89
319.56
358.81
394.58
Ref [10]
79.73
110.51
157.90
194.63
257.99
272.21
320.05
359.76
395.23
Present
79.73
110.43
157.91
194.71
257.86
272.31
320.08
359.98
395.30
5
Ref [3]
78.23
107.82
155.00
191.47
252.65
268.67
313.79
354.27
390.76
Ref [10]
78.49
108.21
155.40
192.25
253.46
269.17
314.54
355.53
391.65
Present
78.49
108.13
155.41
192.33
253.34
269.26
314.58
355.75
391.71
20
Ref [3]
77.49
106.44
153.43
189.81
249.93
266.54
310.31
351.07
387.92
Ref [10]
77.67
106.70
153.67
190.37
250.47
266.83
310.75
351.94
388.52
Present
77.66
106.63
153.68
190.45
250.35
266.92
310.78
352.16
388.59
50
Ref [3]
76.43
105.46
151.35
186.76
246.94
261.96
306.41
345.34
380.70
Ref [10]
76.56
105.66
151.50
187.22
247.36
262.15
306.69
346.04
381.16
Present
76.56
105.59
151.52
187.30
247.25
262.25
306.73
346.26
381.24Table 6
Comparison of the first nine frequencies (Hz) of the functionally graded conical shells with different power-law indexp.
p
Method
Mode number
1
2
3
4
5
6
7
8
F
G
M
I
(
a
=
0
/
b
=
-
0.5
/
c
=
2
/
p
)
0.6
Ref [1]
208.92
208.92
230.11
230.11
284.73
284.74
321.51
321.51
Ref [29]
208.75
208.75
229.96
229.96
284.60
284.60
321.33
321.33
Ref [11]
208.58
208.58
230.06
230.06
284.67
284.67
321.28
321.28
Present
208.78
208.78
230.06
230.06
284.92
284.92
321.32
321.32
5
Ref [1]
204.81
204.81
223.84
223.84
275.52
275.53
316.64
316.64
Ref [29]
204.25
204.25
223.36
223.36
275.02
275.02
315.94
315.94
Ref [11]
203.36
203.36
223.72
223.72
275.26
275.26
315.48
315.48
Present
204.28
204.28
223.46
223.46
275.35
275.35
315.93
315.93
20
Ref [1]
204.89
204.89
227.33
227.33
282.68
282.69
312.50
312.50
Ref [29]
204.27
204.27
226.78
226.78
282.10
282.10
311.71
311.71
Ref [11]
203.29
203.29
227.17
227.17
282.35
282.35
311.15
311.15
Present
204.30
204.30
226.88
226.88
282.42
282.42
311.71
311.71
F
G
M
I
I
(
a
=
0
/
b
=
-
0.5
/
c
=
2
/
p
)
0.6
Ref [1]
208.49
208.49
229.65
229.65
284.17
284.17
321.18
321.18
Ref [29]
208.54
208.54
229.70
229.70
284.25
284.25
321.30
321.30
Ref [11]
208.74
208.74
229.61
229.61
284.21
284.21
321.33
321.33
Present
208.57
208.57
229.80
229.80
284.58
284.58
321.30
321.30
5
Ref [1]
202.87
202.87
221.78
221.78
273.02
273.02
315.18
315.18
Ref [29]
203.30
203.30
222.16
222.16
273.47
273.47
315.81
315.81
Ref [11]
204.15
204.15
221.67
221.67
273.14
273.14
315.72
315.72
Present
203.34
203.34
222.26
222.26
273.80
273.80
315.81
315.81
20
Ref [1]
202.60
202.60
224.87
224.87
279.65
279.66
310.83
310.83
Ref [29]
203.09
203.09
225.30
225.30
280.17
280.17
311.53
311.53
Ref [11]
203.98
203.98
224.73
224.73
279.80
279.80
311.39
311.39
Present
203.12
203.12
225.40
225.40
280.49
280.49
311.53
311.53Table 7
Comparisons of frequencies (Hz) of the functionally graded spherical panels with different power-law indexp.
p
Method
Mode number
1
2
3
4
5
6
7
8
9
F
G
M
I
(
a
=
1
/
b
=
0
/
c
/
p
)
0.6
Ref [2]
71.12
94.44
147.32
173.88
237.31
315.92
407.09
412.60
442.60
Ref [10]
71.01
94.37
147.25
173.87
237.25
315.55
406.65
412.62
441.76
Present
71.09
94.36
147.37
174.05
237.21
315.66
406.80
412.48
441.77
1
Ref [2]
70.78
94.02
146.63
173.14
236.17
314.53
405.19
410.79
440.35
Ref [10]
70.63
93.89
146.47
173.02
235.96
313.96
404.50
410.55
439.17
Present
70.71
93.88
146.59
173.21
235.91
314.09
404.67
410.40
439.17
5
Ref [2]
71.68
96.03
148.75
177.58
237.24
321.15
409.51
420.41
435.62
Ref [10]
71.54
95.89
148.59
177.46
237.03
320.58
408.82
420.09
434.56
Present
71.62
95.88
148.69
177.67
237.00
320.69
408.99
420.00
434.54
20
Ref [2]
70.59
94.55
146.50
174.82
233.56
316.21
403.23
413.98
428.88
Ref [10]
70.57
94.57
146.59
175.00
233.75
316.16
403.21
414.32
428.53
Present
70.64
94.56
146.69
175.20
233.71
316.28
403.37
414.24
428.51
50
Ref [2]
69.54
92.95
144.28
171.59
230.57
310.83
397.57
406.64
425.16
Ref [10]
69.57
93.01
144.45
171.87
230.90
310.98
397.80
407.27
425.06
Present
69.64
92.98
144.56
172.07
230.86
311.10
397.96
407.60
425.06
F
G
M
I
I
(
a
=
1
/
b
=
0
/
c
/
p
)
0.6
Ref [2]
70.59
93.62
146.22
172.31
235.55
313.20
403.88
409.05
439.88
Ref [10]
70.81
93.98
146.83
173.10
236.58
314.28
405.31
410.95
441.00
Present
70.89
93.97
146.96
173.29
236.55
314.41
405.48
410.80
441.02
1
Ref [2]
70.12
92.99
145.26
171.15
233.94
311.11
401.15
406.32
436.81
Ref [10]
70.39
93.41
145.95
172.05
235.11
312.37
402.82
408.46
438.20
Present
70.48
93.40
146.08
172.21
235.08
312.48
402.97
408.30
438.21
5
Ref [2]
71.04
95.02
147.45
175.58
235.02
317.81
405.72
416.02
432.13
Ref [10]
71.30
95.41
148.12
176.44
236.12
318.99
407.26
417.98
433.45
Present
71.38
95.45
148.23
176.64
236.10
319.11
407.43
418.05
433.45
20
Ref [2]
70.33
94.13
145.98
174.00
232.64
314.84
401.67
412.16
427.43
Ref [10]
70.47
94.36
146.40
174.57
233.35
315.49
402.55
413.44
428.05
Present
70.55
94.37
146.50
174.77
233.33
315.61
402.71
413.34
428.03
50
Ref [2]
69.43
92.74
144.04
171.22
230.16
310.22
396.86
405.84
424.51
Ref [10]
69.54
92.94
144.41
171.73
230.79
310.77
397.61
407.00
424.96
Present
69.60
92.91
144.48
171.87
230.69
310.81
397.66
406.77
424.84Table 8
Comparisons of frequencies (Hz) of the functionally graded spherical shells with different power-law indexp.
p
n
FF
SS
CC
Ref [9]
Ref [29]
Present
Ref [9]
Ref [29]
Present
Ref [9]
Ref [29]
Present
F
G
M
I
(
a
=
1
/
b
=
0
/
c
/
p
)
0.6
0
777.60
777.61
777.60
786.87
786.84
786.84
875.72
875.58
875.57
1
771.92
771.92
771.91
827.85
827.82
827.82
910.70
910.57
910.56
2
49.89
49.85
49.89
886.90
886.85
886.88
894.67
894.60
894.65
3
133.68
133.53
133.68
881.65
881.61
881.64
889.45
889.41
889.43
4
243.64
243.34
243.66
914.98
914.97
914.98
927.36
927.34
927.35
5
0
754.66
754.66
754.66
773.02
772.99
772.99
865.53
865.40
865.40
1
748.83
748.82
748.83
812.64
812.61
812.62
899.65
899.53
899.54
2
51.61
51.57
51.61
862.47
862.42
862.46
871.46
871.41
871.44
3
137.94
137.80
137.95
858.21
858.18
858.20
868.66
868.63
868.65
4
250.90
250.61
250.92
895.25
895.23
895.25
910.61
910.59
910.60
F
G
M
I
I
(
a
=
1
/
b
=
0
/
c
/
p
)
0.6
0
777.04
777.04
777.03
767.95
767.93
767.92
873.85
873.71
873.70
1
771.53
771.53
771.52
815.18
815.16
815.15
908.81
908.68
908.67
2
49.64
49.60
49.64
887.94
887.89
887.92
893.80
893.73
893.78
3
133.03
132.88
133.03
885.56
885.52
885.55
888.32
888.28
888.30
4
242.47
242.16
242.48
919.44
919.43
919.44
925.77
925.75
925.76
5
0
753.73
753.73
753.73
749.88
749.86
749.86
862.76
862.63
862.63
1
748.18
748.17
748.17
796.86
796.84
796.85
896.89
896.77
896.77
2
51.29
51.25
51.29
864.43
864.39
864.42
870.08
870.03
870.07
3
137.10
136.95
137.10
863.25
863.22
863.24
866.87
866.84
866.84
4
249.38
249.09
249.42
900.65
900.63
900.64
908.16
908.14
908.16In order to check the present method, the numerical results reported by Su et al. [7–9], Qu et al. [27], and Tornabene et al. [1, 3, 52] are also given in the above tables for comparison. From the comparisons, we can see a consistent agreement of present results taken from the current proposed unified approach and referential data. Besides, Tables 3–8 also show that it is appropriate to define the classical boundary conditions in terms of the boundary spring rigidities. Based on the formulation validation, some new results of the FG cylindrical, conical, and spherical panels and shells with classical-elastic and elastic restrain boundary condition are shown in Tables 9–14. Also, for any given modal frequency, the corresponding mode shape can be easily determined by submitting the solved eigenvector into (22). Thus, some select mode shapes of that are performed in Figures 6–8.Table 9
First six frequency parameters Ω of the functionally graded cylindrical panels with different boundary conditions.
Mode number
Boundary condition
CCCC
CSCS
CFCF
SFSF
CE1CE1
SE3SE3
E1E1E1E1
E2E2E2E2
E3E3E3E3
F
G
M
I
(
a
=
1
/
b
=
0
/
c
/
p
=
1
)
1
22.789
20.619
9.583
7.697
19.726
22.630
16.566
20.715
22.590
2
27.804
24.696
10.383
8.580
20.255
27.568
18.123
21.746
27.534
3
36.438
35.286
18.131
17.363
31.356
36.345
18.522
30.208
36.213
4
37.573
36.550
20.548
17.596
32.096
37.507
19.672
32.569
37.387
5
41.149
39.062
21.822
17.925
34.202
40.949
26.910
34.497
40.931
6
46.167
40.997
22.101
19.693
37.734
45.832
29.461
40.940
45.731
F
G
M
I
I
(
a
=
1
/
b
=
0
/
c
/
p
=
1
)
1
22.647
21.220
9.573
7.769
19.572
22.537
16.544
20.610
22.479
2
27.583
24.165
10.343
8.702
20.231
27.275
17.995
21.659
27.227
3
36.294
35.914
18.098
16.773
31.000
36.261
18.522
29.997
36.119
4
37.443
36.288
20.346
17.801
31.957
37.342
19.667
32.458
37.208
5
40.975
40.410
21.806
18.006
34.164
40.917
26.922
34.355
40.901
6
45.926
40.416
22.067
19.001
37.796
45.749
29.431
40.726
45.637Table 10
First six frequency parameters Ω of the functionally graded cylindrical shells with different boundary conditions.
Mode number
Boundary condition
CC
CS
CF
SF
CE1
SE3
E1E1
E2E2
E3E3
F
G
M
I
(
a
=
1
/
b
=
0
/
c
/
p
=
1
)
1
16.879
16.337
6.8893
6.8050
16.040
16.314
13.812
16.200
16.827
2
16.879
16.337
6.8893
6.8050
16.040
16.314
14.403
16.200
16.827
3
17.745
17.237
9.7991
9.7427
16.052
17.215
14.403
17.185
17.698
4
17.745
17.237
9.7991
9.7427
16.052
17.215
15.248
17.185
17.698
5
21.597
21.168
11.507
11.428
19.597
21.149
15.248
20.972
21.554
6
21.597
21.168
11.507
11.428
19.597
21.149
15.464
20.972
21.554
F
G
M
I
I
(
a
=
1
/
b
=
0
/
c
/
p
=
1
)
1
16.780
15.876
6.8620
6.5223
15.953
15.840
13.812
16.102
16.699
2
16.780
15.876
6.8620
6.5223
15.953
15.840
14.374
16.102
16.699
3
17.714
16.904
9.6811
9.5146
16.011
16.871
14.374
17.155
17.642
4
17.714
16.904
9.6811
9.5146
16.011
16.871
15.147
17.155
17.642
5
21.381
20.683
11.508
11.177
19.594
20.656
15.147
20.754
21.315
6
21.381
20.683
11.508
11.177
19.594
20.656
15.463
20.754
21.315Table 11
First six frequency parameters Ω of the functionally graded conical panels with different boundary conditions.
Mode number
Boundary condition
CCCC
CSCS
CFCF
SFSF
CE1CE1
SE3SE3
E1E1E1E1
E2E2E2E2
E3E3E3E3
F
G
M
I
(
a
=
1
/
b
=
0.5
/
c
=
2
/
p
)
1
24.598
22.827
10.062
8.130
21.649
22.861
18.335
23.804
24.476
2
27.546
25.319
11.116
9.246
22.310
25.382
19.905
24.825
27.392
3
35.766
34.653
19.734
17.732
30.047
32.676
25.501
34.596
35.612
4
39.938
36.473
21.090
18.147
35.305
36.417
26.760
35.304
39.668
5
41.194
38.498
23.559
20.537
36.797
38.274
29.303
37.316
41.058
6
42.731
40.899
24.102
20.650
41.696
39.241
32.699
38.101
42.355
F
G
M
I
I
(
a
=
1
/
b
=
0.5
/
c
=
2
/
p
)
1
24.562
23.102
10.060
8.2037
21.610
22.706
18.325
23.773
24.442
2
27.506
25.226
11.104
9.3506
22.300
25.232
19.866
24.793
27.338
3
35.737
35.024
19.725
17.659
29.983
32.942
25.502
34.572
35.608
4
39.882
36.565
21.053
17.835
35.291
36.496
26.767
35.263
39.626
5
41.166
38.973
23.545
20.502
36.737
38.180
29.240
37.282
41.021
6
42.646
40.733
24.085
20.688
41.606
39.057
32.684
38.059
42.281Table 12
First six frequency parameters Ω of the functionally graded conical shells with different boundary conditions.
Mode number
Boundary condition
CC
CS
CF
SF
CE1
SE3
E1E1
E2E2
E3E3
F
G
M
I
(
a
=
1
/
b
=
0.5
/
c
=
2
/
p
)
1
18.648
17.326
3.1641
3.0306
15.986
18.457
15.709
18.184
18.591
2
18.648
17.328
3.1641
3.0306
15.986
18.460
15.709
18.186
18.594
3
19.583
18.595
4.6352
4.4011
16.670
19.183
16.583
19.147
19.526
4
19.583
18.595
4.6352
4.4011
16.670
19.183
16.583
19.148
19.526
5
22.612
20.687
5.3016
5.2968
19.945
22.529
18.536
21.878
22.540
6
22.612
20.707
5.3016
5.2968
19.945
22.549
18.536
21.898
22.560
F
G
M
I
I
(
a
=
1
/
b
=
0.5
/
c
=
2
/
p
)
1
18.618
17.080
3.1594
2.9584
15.977
18.386
15.699
18.154
18.552
2
18.618
17.080
3.1594
2.9584
15.977
18.386
15.699
18.154
18.554
3
19.572
18.443
4.6350
4.2895
16.640
19.099
16.553
19.137
19.507
4
19.572
18.443
4.6350
4.2895
16.640
19.099
16.553
19.137
19.507
5
22.564
20.403
5.2888
5.2801
19.943
22.468
18.535
21.834
22.483
6
22.564
20.403
5.2888
5.2801
19.943
22.468
18.535
21.834
22.503Table 13
First six frequency parameters Ω of the functionally graded spherical panels with different boundary conditions.
Mode number
Boundary condition
CCCC
CSCS
CFCF
SFSF
CE1CE1
SE3SE3
E1E1E1E1
E2E2E2E2
E3E3E3E3
F
G
M
I
(
a
=
1
/
b
=
1
/
c
=
4
/
p
)
1
13.652
13.168
10.001
9.2907
12.374
12.924
5.215
11.613
13.567
2
14.297
14.090
10.057
9.3816
12.916
13.716
5.400
13.010
14.243
3
17.060
16.157
12.518
10.917
15.840
14.095
7.290
13.117
16.827
4
17.423
17.179
13.685
11.347
16.595
16.611
7.398
13.345
17.256
5
20.023
18.796
14.830
11.965
16.892
17.493
10.310
14.562
19.753
6
20.561
19.503
14.882
12.895
19.200
19.648
13.078
15.491
20.344
F
G
M
I
I
(
a
=
1
/
b
=
1
/
c
=
4
/
p
)
1
13.639
13.182
9.9925
9.4012
12.367
12.968
5.216
11.607
13.560
2
14.286
14.054
10.051
9.4953
12.902
13.764
5.401
13.001
14.234
3
17.037
16.214
12.508
10.878
15.817
14.029
7.282
13.110
16.801
4
17.401
17.167
13.668
11.330
16.599
16.660
7.399
14.553
17.243
5
19.993
18.755
14.805
11.985
16.867
17.486
10.295
15.478
19.726
6
20.531
19.495
14.860
12.923
19.170
19.644
13.073
15.824
20.314Table 14
First six frequency parameters Ω of the functionally graded spherical shells with different boundary conditions.
Mode number
Boundary condition
CC
CS
CF
SF
CE1
SE3
E1E1
E2E2
E3E3
F
G
M
I
(
a
=
1
/
b
=
1
/
c
=
4
/
p
)
1
12.293
11.727
1.7433
1.4432
6.5531
11.970
5.6226
10.036
12.242
2
12.293
11.727
1.7433
1.4432
6.5531
11.970
6.3197
10.838
12.242
3
12.552
11.918
2.7986
2.2795
7.2370
12.233
6.3199
10.838
12.488
4
12.552
11.918
2.7986
2.2795
8.2529
12.233
7.6911
12.060
12.488
5
13.316
12.855
3.0960
3.0509
8.2529
12.992
7.6912
12.060
13.286
6
13.316
12.855
3.0960
3.0510
10.393
12.992
8.6638
12.141
13.286
F
G
M
I
I
(
a
=
1
/
b
=
1
/
c
=
4
/
p
)
1
12.284
11.760
1.7406
1.4246
6.5481
12.002
5.6204
10.032
12.238
2
12.284
11.760
1.7406
1.4246
6.5481
12.002
6.3145
10.833
12.238
3
12.539
11.946
2.7974
2.2396
7.2331
12.251
6.3145
10.833
12.481
4
12.539
11.946
2.7974
2.2396
8.2452
12.251
7.6835
12.055
12.481
5
13.310
12.899
3.0896
3.0439
8.2452
13.032
7.6835
12.055
13.285
6
13.310
12.899
3.0896
3.0439
10.381
13.032
8.6633
12.132
13.285Figure 6
Mode shapes for cylindrical panels and shells with different boundary conditions.Figure 7
Mode shapes for conical panels and shells with different boundary conditions.Figure 8
Mode shapes for spherical panels and shells with different boundary conditions.


### 3.2. Parameter Studies

Although the above results of the FG cylindrical, conical, and spherical panels and shells with classical-elastic and elastic restrain boundary conditions can be served as benchmark data for the future numerical methods, it does not meet authors’ ambition. The goal of this paper is to provide some useful research works for the designer or engineer to avoid the unpleasant, inefficient, and structurally damaging resonant by given a unified computational model of the FG cylindrical, conical, and spherical panels and shells of revolution. So, in the this subsection, the parameter studies including the boundary restraint rigidityΓ, thickness h, circumferential angle ϕ, power-law exponent p, and material parameters a, b, and c will be conducted.Figure9 shows the effects of the all kinds of boundary restraint parameters on the frequency parameters of FG panels and shells with elastic boundary conditions. Five nondimensional boundary restraint parameters Γϐ (ϐ=u,v,w,α, and β) are defined as ratios of corresponding spring stiffness to the reference bending stiffness D; that is, Γu=ku/D, Γv=kv/D, Γw=kw/D, Γα=Kα/D, and Γβ=Kβ/D. The geometry properties of the panels and shells studied are the same as Tables 3–8. The panels and shells are free at edges β=constant, clamped at α=0 while the α=Lα is elastically restrained by all the five groups of boundary springs, of which only one group is with the variable stiffness and the others are assumed to be zeros. According to Figure 9, we can see that the change of the boundary elastic restraint parameter Γ has little effect on the frequency parameter Ω when it is smaller than 10-2D. However, when it is increased in a certain range, the frequency parameters increase rapidly as the elastic parameters increase, approach their utmost, and remain unchanged when Γ approaches infinity. It is noted that the certain range with respect to different kinds of elastic restraint parameters is different. Figures 10 and 11 show the influence of the fundamental frequency parameters versus thickness h for the FG panels and shells with different shear correction factors, respectively. It is obvious that the frequency parameters of the FG panels and shells with respect to the classical and elastic boundary condition increase when increasing the thicknesses. In addition, the figures also shows that the shear correction factor κ have a significant impact on the vibration behavior of FG panels and shells and a larger shear correction factor will result in higher frequency parameters. Next, Figure 12 presents the first four frequency parameters for the FG panels versus the circumference angles with the CCCC and E3E3E3E3 boundary conditions. From the figure, we can see that the frequency parameters rapidly decrease while the circumference angle changed from 5° to 75°, and out of this range the frequency parameters slowly minish and lastly almost remain unchanged. In addition, we also found that the variation of the frequency parameters has nothing to do with the modal order. Then, the effects of the power-law index p and the material parameters a, b, and c are studied as well. Figure 13 shows the fundamental frequency versus of the power-law index p for FG panels and shells with clamped boundary conditions. The material distribution using in the study is FGMI(a/b/c/p) and FGMII(a/b/c/p): cylindrical meridian: FGMI(a=1/b=0.5/c=2/p) and FGMII(a=1/b=0.5/c=2/p); conical meridian: FGMI(a=0/b=-0.5/c=2/p) and FGMII(a=0/b=-0.5/c=2/p); spherical meridian: FGMI(a=1/b=1/c=4/p) and FGMII(a=1/b=1/c=4/p). From the figure, we can see that the geometrical construction has effect on the vibration characteristic of the FG panels and shells. For case of the cylindrical meridian, the fundamental frequency firstly rapidly decreases and then increases and finally becomes decreased versus the power-law index p increased. However, for the conical and spherical meridian, the frequency parameters decrease monotonously while increasing the power-law index p. In addition to that, the figure also shows that the frequency parameters of the FGMI structure elements are always higher than the FGMI case. In Figure 14, the influence of the combination relations between the boundary condition and power-law index p on the vibration behavior for the FG panes and shells is reported. And to simplify the study, the research selects cylindrical panels as the example. From the figure, we can know that the variation trend of the fundamental frequency of the FG panes and shells with CFCF, E3E3E3E3, and SSSS is firstly rapidly decreased and then increased and finally becomes decreased versus the power-law index p increase. However, the frequency parameters increase rapidly as the power-law index p increases, approaches their utmost, and remains unchanged when p is more than a threshold. Thus, we can get that the boundary condition plays an indispensable role in the vibration behavior of the FG panes and shells. Finally, the influence of the material parameters a, b, and c will be exhibited in the next example. Figure 15 depicts the variations of the first frequencies of the clamped supported FGMI panels and shells with different material types: Type 1: a=1 and b=c=0; Type 2: a=1, b=0.5, and c=2; Type 3: a=0, b=-0.5, and c=2; Type 4: a=1, b=1, and c=4. From the figure, it is obvious that the material parameters a, b, and c have a significant influence on the fundamental frequencies for the FG panels and shells.Figure 9
Variation of the fundamental frequency parametersΩ versus the elastic restraint parameters for FG panels and shells: (a) cylindrical panels; (b) cylindrical shells; (c) conical panels; (d) conical shells; (e) spherical panels; (f) spherical shells.
(a)
(b)
(c)
(d)
(e)
(f)Figure 10
Fundamental frequency parameters versus thicknessh for the CCCC FG panels and shells with different shear correction factors: (a) cylindrical panels; (b) cylindrical shells; (c) conical panels; (d) conical shells; (e) spherical panels; (f) spherical shells.
(a)
(b)
(c)
(d)
(e)
(f)Figure 11
Fundamental frequency parameters versus thicknessh for the E3E3E3E3 FG panels and shells with different shear correction factors: (a) cylindrical panels; (b) cylindrical shells; (c) conical panels; (d) conical shells; (e) spherical panels; (f) spherical shells.
(a)
(b)
(c)
(d)
(e)
(f)Figure 12
Variation of the first four frequency parametersΩ versus the circumferential angle for FG panels: (a) cylindrical panels; (b) conical panels; (c) spherical panels.
(a)
(b)
(c)Figure 13
Variations of the first frequencies of the clamped supportedFGMI and FGMII: (a) cylindrical panels; (b) cylindrical shells; (c) conical panels; (d) conical shells; (e) spherical panels; (f) spherical shells.
(a)
(b)
(c)
(d)
(e)
(f)Figure 14
Variations of the first frequencies of theFGMI and FGMII cylindrical panels with different boundary conditions.
(a)
CFCF
(b)
SSSS
(c)
E1E1E1E1
(d)
E3E3E3E3Figure 15
Variations of the first frequencies of the clamped supportedFGMI panels and shells with different material types: Type 1: a=1 and b=c=0; Type 2: a=1, b=0.5, and c=2; Type 3: a=0, b=-0.5, and c=2; Type 4: a=1, b=1, and c=4.
(a)
Cylindrical panels
(b)
Cylindrical shells
(c)
Conical panels
(d)
Conical shells
(e)
Spherical panels
(f)
Spherical shells


## 3.1. Convergence Study and Formulation Validation

From the theoretical formulations, we can know that the computational accuracy relies on a limited number of terms in the displacement expressions in actual calculation. In addition, when considering the shells of revolution, the results strongly rely on the coupling parameters. Thus, one important issue is the choice of the appropriate truncation terms and coupling parameters. The nondimensional frequency parameter convergence of the clamped shells with respect to different coupling parameters is examined in Table2. The geometrical parameters and power-law exponents for the functionally graded shells used in the study are as follows: cylindrical shell: R=1 m,L=4 m, h=0.1 m, and p=0; conical shell: R1=0.5 m, L0=4/cosα0R1, h=0.1 m, α0=40°, and p=0; spherical shell: R=1 m, φ0=π/6, φ0=π/2, h=0.1 m, and p=0. All coupling parameters are defined with the same value and defined as follows: kucu=kvcv=kwcw=Kαcα=Kβcβ=ϗ. It is obvious that the frequency parameters rapidly converge as the coupling parameter increases. In the absence of the coupling parameters or for small coupling parameters, pseudovibration characteristic may appear in the present solution. This implies that small coupling parameters cannot impose the interface continuity conditions in a proper manner. Thus, based on the existed analysis, ϗ=1014 has been adopted to present all the results in the following analysis.Table 2
Convergence of frequency parameters of the single curved shells with different coupling parametersϗ.
Shape
Mode
101
103
105
107
108
1010
1012
1014
1016
1018
Cylindrical shells
1
10.043
10.043
10.058
10.680
11.659
16.469
16.779
16.780
16.780
16.780
2
10.051
10.051
10.062
11.315
14.581
16.681
16.779
16.780
16.780
16.780
3
16.515
16.515
16.518
16.685
16.857
17.399
17.711
17.714
17.714
17.714
4
17.203
17.203
17.204
17.215
17.236
17.399
17.711
17.714
17.714
17.714
5
17.564
17.564
17.566
17.744
19.153
21.296
21.381
21.381
21.381
21.381
6
19.644
19.644
19.645
19.685
19.762
21.374
21.381
21.381
21.381
21.381
Conical shells
1
10.376
10.376
10.391
11.164
12.259
17.946
18.161
18.162
18.162
18.162
2
10.392
10.392
10.418
12.431
16.393
17.985
18.161
18.162
18.162
18.162
3
17.786
17.786
17.790
17.990
18.164
18.915
19.280
19.286
19.286
19.286
4
18.553
18.553
18.557
18.676
18.693
18.994
19.280
19.286
19.286
19.286
5
18.725
18.725
18.726
19.009
20.736
21.794
21.855
21.855
21.855
21.855
6
20.595
20.595
20.602
21.018
21.415
21.802
21.855
21.855
21.855
21.855
Spherical shells
1
9.766
9.761
9.757
9.867
10.088
11.761
12.034
12.038
12.038
12.038
2
9.766
9.764
9.769
10.017
10.809
11.827
12.036
12.038
12.038
12.038
3
12.046
12.046
12.046
12.055
12.086
12.163
12.221
12.224
12.224
12.224
4
12.117
12.117
12.118
12.133
12.153
12.185
12.223
12.224
12.224
12.224
5
12.308
12.308
12.319
12.321
12.340
12.672
13.084
13.089
13.089
13.089
6
12.493
12.493
12.493
12.523
12.665
12.947
13.085
13.089
13.089
13.089Next, the convergence studies of the cylindrical, conical, and spherical panels and shells will be discussed. Figure5 shows the convergence studies of the first three frequency parameters for the structure elements for complete clamped FG panels and shells with different truncated numbers M and N. The geometrical dimensions of the shells are the same as Table 2 and the panels are also in accordance with Table 2 except the circumferential angle ϕ=120°. From the figure, we can know that the proposed unified approach has fast convergence and good stability. In view of the excellent numerical behavior of the current solution, the truncation numbers will be simply set as M=N=15 in the following calculations.Figure 5
Variation of the frequency parameters Ω versus the truncated numbersM and N for complete clamped FG panels and shells: (a) cylindrical panels; (b) cylindrical shells; (c) conical panels; (d) conical shells; (e) spherical panels; (f) spherical shells.
(a)
(b)
(c)
(d)
(e)
(f)The validation of the unified computational model versus about the different geometric parameters is shown in Tables3–8. The details regarding the geometry of the structures considered are indicated as follows:(1)
Cylindrical panels (FFCF):R=1 m, L=2 m, h=0.1 m, and ϕ=120° (Table 3).(2)
Cylindrical shells (FC):R=1 m, L=2 m, and h=0.1 m (Table 4).(3)
Conical panels (SCSC):R1=1 m, L0=2 m, h=0.1 m, α0=40°, and ϕ=120° (Table 5).(4)
Conical shells (FC):R1=0.5 m, L0cos⁡α0=2 m, h=0.1 m, and α0=40° (Table 6).(5)
Spherical panels (FFCF):R=2 m, h=0.1 m, φ0=30°, φ1=90°, and ϕ=120° (Table 7).(6)
Spherical shells:R=1 m, h=0.05 m, φ0=22.5°, and φ1=90° (Table 8).Table 3
Comparison of the first nine frequencies (Hz) of functionally graded cylindrical panels with different power-law indexp.
p
Method
Mode numbers
1
2
3
4
5
6
7
8
9
0
Ref [52]
61.02
94.82
153.13
241.39
275.83
291.45
355.14
450.76
511.92
Ref [8]
61.06
94.92
153.11
241.75
276.37
291.94
355.07
450.82
512.48
Present
61.02
94.82
153.12
241.60
276.22
291.64
355.09
450.72
512.31
0.6
Ref [52]
59.24
93.14
148.51
234.79
267.06
287.85
343.22
443.06
495.89
Ref [8]
59.13
93.03
148.14
234.60
266.97
287.63
342.38
442.09
495.31
Present
59.09
92.93
148.15
234.45
266.83
287.34
342.39
441.99
495.15
1
Ref [52]
58.96
92.65
147.94
233.64
265.86
286.16
342.08
440.72
493.67
Ref [8]
58.82
92.48
147.49
233.31
265.61
285.77
341.04
439.49
492.80
Present
58.78
92.38
147.50
233.16
265.47
285.48
341.05
439.39
492.63
5
Ref [52]
60.09
91.69
152.02
236.75
272.42
279.34
354.89
436.34
505.35
Ref [8]
59.96
91.54
151.57
236.42
272.19
279.06
353.83
435.18
504.56
Present
59.92
91.44
151.59
236.29
272.04
278.78
353.84
435.09
504.37
20
Ref [52]
59.19
90.29
149.49
233.27
268.37
275.23
348.80
429.63
497.87
Ref [8]
59.16
90.30
149.29
233.32
268.57
275.41
348.32
429.19
497.87
Present
59.12
90.20
149.30
233.20
268.43
275.13
348.33
429.10
497.69
50
Ref [52]
58.20
89.55
146.55
229.79
263.58
273.98
341.04
425.83
489.06
Ref [8]
58.21
89.60
146.45
230.00
263.95
274.32
340.79
425.66
489.36
Present
58.17
89.51
146.46
229.87
263.81
274.04
340.80
425.57
489.19Table 4
Comparison of the first nine frequencies (Hz) of functionally graded cylindrical shells with different power-law indexp.
p
Method
Mode number
1
2
3
4
5
6
7
8
0
Ref [1]
152.93
152.93
220.06
220.06
253.78
253.78
383.55
383.56
Ref [9]
152.89
152.89
219.97
219.97
253.79
253.79
383.44
383.44
Present
152.89
152.89
220.12
220.12
253.77
253.77
383.95
383.95
0.6
Ref [1]
152.25
152.25
219.86
219.86
252.17
252.17
383.39
383.40
Ref [27]
152.02
152.02
219.54
219.54
251.91
251.91
382.87
382.87
Ref [9]
152.07
152.07
219.59
219.59
251.96
251.96
382.95
382.95
Present
152.08
152.08
219.73
219.83
251.95
251.95
383.46
383.46
1
Ref [1]
151.77
151.77
219.56
219.56
251.14
251.14
382.97
382.97
Ref [9]
151.52
151.52
219.19
219.19
250.81
250.81
382.35
382.35
Present
151.53
151.53
219.33
219.33
250.80
250.80
382.85
382.85
5
Ref [1]
148.97
148.97
218.87
218.88
244.40
244.40
382.46
382.47
Ref [27]
148.50
148.50
218.16
218.16
243.73
243.73
381.26
381.26
Ref [9]
148.53
148.53
218.21
218.21
243.76
243.76
381.33
381.33
Present
148.54
148.54
218.35
218.35
243.75
243.75
381.82
381.82
20
Ref [1]
146.46
146.46
215.90
215.90
239.84
239.84
377.34
377.34
Ref [27]
146.21
146.21
215.50
215.50
239.54
239.54
376.69
376.69
Ref [9]
146.24
146.24
215.55
215.55
239.57
239.57
376.76
376.76
Present
146.25
146.25
215.69
215.69
239.56
239.56
377.23
377.23Table 5
Comparison of the first nine frequencies (Hz) of the functionally graded conical panels with different power-law indexp.
p
Method
Mode number
1
2
3
4
5
6
7
8
9
F
G
M
I
(
a
=
1
/
b
=
0.5
/
c
=
2
/
p
)
0.6
Ref [3]
80.10
110.93
158.60
195.20
259.10
273.52
321.39
360.77
396.83
Ref [10]
80.15
111.03
158.61
195.49
259.31
273.52
321.38
361.16
397.06
Present
80.14
110.95
158.62
195.57
259.18
273.62
321.42
361.38
397.12
1
Ref [3]
79.89
110.56
158.15
194.72
258.36
272.88
320.38
359.84
395.97
Ref [10]
79.91
110.62
158.10
194.94
258.48
272.82
320.28
360.10
396.11
Present
79.91
110.54
158.11
195.02
258.36
272.91
320.31
360.33
396.18
5
Ref [3]
78.92
108.45
156.07
192.86
254.58
270.76
315.52
356.30
393.68
Ref [10]
78.86
108.39
155.87
192.87
254.45
270.51
315.12
356.20
393.55
Present
78.86
108.31
155.88
192.95
254.33
270.60
315.16
356.42
393.62
20
Ref [3]
77.81
106.72
153.93
190.43
250.80
267.51
311.11
351.98
389.27
Ref [10]
77.83
106.78
153.90
190.65
250.92
267.46
311.03
352.24
389.42
Present
77.83
106.70
153.91
190.73
250.80
267.56
311.07
352.46
389.48
50
Ref [3]
76.57
105.58
151.56
187.03
247.32
262.39
306.75
345.73
381.29
Ref [10]
76.63
105.69
151.60
187.34
247.57
262.43
306.81
346.17
381.56
Present
76.63
105.62
151.61
187.43
247.45
262.53
306.85
346.39
381.63
F
G
M
I
I
(
a
=
1
/
b
=
0.5
/
c
=
2
/
p
)
0.6
Ref [3]
79.87
110.71
158.27
194.75
258.47
272.88
320.85
360.10
395.93
Ref [10]
80.03
110.96
158.48
195.29
258.99
273.14
321.24
360.94
396.49
Present
80.03
110.88
158.49
195.37
258.86
273.23
321.27
361.16
396.56
1
Ref [3]
79.54
110.22
157.64
194.02
257.39
271.89
319.56
358.81
394.58
Ref [10]
79.73
110.51
157.90
194.63
257.99
272.21
320.05
359.76
395.23
Present
79.73
110.43
157.91
194.71
257.86
272.31
320.08
359.98
395.30
5
Ref [3]
78.23
107.82
155.00
191.47
252.65
268.67
313.79
354.27
390.76
Ref [10]
78.49
108.21
155.40
192.25
253.46
269.17
314.54
355.53
391.65
Present
78.49
108.13
155.41
192.33
253.34
269.26
314.58
355.75
391.71
20
Ref [3]
77.49
106.44
153.43
189.81
249.93
266.54
310.31
351.07
387.92
Ref [10]
77.67
106.70
153.67
190.37
250.47
266.83
310.75
351.94
388.52
Present
77.66
106.63
153.68
190.45
250.35
266.92
310.78
352.16
388.59
50
Ref [3]
76.43
105.46
151.35
186.76
246.94
261.96
306.41
345.34
380.70
Ref [10]
76.56
105.66
151.50
187.22
247.36
262.15
306.69
346.04
381.16
Present
76.56
105.59
151.52
187.30
247.25
262.25
306.73
346.26
381.24Table 6
Comparison of the first nine frequencies (Hz) of the functionally graded conical shells with different power-law indexp.
p
Method
Mode number
1
2
3
4
5
6
7
8
F
G
M
I
(
a
=
0
/
b
=
-
0.5
/
c
=
2
/
p
)
0.6
Ref [1]
208.92
208.92
230.11
230.11
284.73
284.74
321.51
321.51
Ref [29]
208.75
208.75
229.96
229.96
284.60
284.60
321.33
321.33
Ref [11]
208.58
208.58
230.06
230.06
284.67
284.67
321.28
321.28
Present
208.78
208.78
230.06
230.06
284.92
284.92
321.32
321.32
5
Ref [1]
204.81
204.81
223.84
223.84
275.52
275.53
316.64
316.64
Ref [29]
204.25
204.25
223.36
223.36
275.02
275.02
315.94
315.94
Ref [11]
203.36
203.36
223.72
223.72
275.26
275.26
315.48
315.48
Present
204.28
204.28
223.46
223.46
275.35
275.35
315.93
315.93
20
Ref [1]
204.89
204.89
227.33
227.33
282.68
282.69
312.50
312.50
Ref [29]
204.27
204.27
226.78
226.78
282.10
282.10
311.71
311.71
Ref [11]
203.29
203.29
227.17
227.17
282.35
282.35
311.15
311.15
Present
204.30
204.30
226.88
226.88
282.42
282.42
311.71
311.71
F
G
M
I
I
(
a
=
0
/
b
=
-
0.5
/
c
=
2
/
p
)
0.6
Ref [1]
208.49
208.49
229.65
229.65
284.17
284.17
321.18
321.18
Ref [29]
208.54
208.54
229.70
229.70
284.25
284.25
321.30
321.30
Ref [11]
208.74
208.74
229.61
229.61
284.21
284.21
321.33
321.33
Present
208.57
208.57
229.80
229.80
284.58
284.58
321.30
321.30
5
Ref [1]
202.87
202.87
221.78
221.78
273.02
273.02
315.18
315.18
Ref [29]
203.30
203.30
222.16
222.16
273.47
273.47
315.81
315.81
Ref [11]
204.15
204.15
221.67
221.67
273.14
273.14
315.72
315.72
Present
203.34
203.34
222.26
222.26
273.80
273.80
315.81
315.81
20
Ref [1]
202.60
202.60
224.87
224.87
279.65
279.66
310.83
310.83
Ref [29]
203.09
203.09
225.30
225.30
280.17
280.17
311.53
311.53
Ref [11]
203.98
203.98
224.73
224.73
279.80
279.80
311.39
311.39
Present
203.12
203.12
225.40
225.40
280.49
280.49
311.53
311.53Table 7
Comparisons of frequencies (Hz) of the functionally graded spherical panels with different power-law indexp.
p
Method
Mode number
1
2
3
4
5
6
7
8
9
F
G
M
I
(
a
=
1
/
b
=
0
/
c
/
p
)
0.6
Ref [2]
71.12
94.44
147.32
173.88
237.31
315.92
407.09
412.60
442.60
Ref [10]
71.01
94.37
147.25
173.87
237.25
315.55
406.65
412.62
441.76
Present
71.09
94.36
147.37
174.05
237.21
315.66
406.80
412.48
441.77
1
Ref [2]
70.78
94.02
146.63
173.14
236.17
314.53
405.19
410.79
440.35
Ref [10]
70.63
93.89
146.47
173.02
235.96
313.96
404.50
410.55
439.17
Present
70.71
93.88
146.59
173.21
235.91
314.09
404.67
410.40
439.17
5
Ref [2]
71.68
96.03
148.75
177.58
237.24
321.15
409.51
420.41
435.62
Ref [10]
71.54
95.89
148.59
177.46
237.03
320.58
408.82
420.09
434.56
Present
71.62
95.88
148.69
177.67
237.00
320.69
408.99
420.00
434.54
20
Ref [2]
70.59
94.55
146.50
174.82
233.56
316.21
403.23
413.98
428.88
Ref [10]
70.57
94.57
146.59
175.00
233.75
316.16
403.21
414.32
428.53
Present
70.64
94.56
146.69
175.20
233.71
316.28
403.37
414.24
428.51
50
Ref [2]
69.54
92.95
144.28
171.59
230.57
310.83
397.57
406.64
425.16
Ref [10]
69.57
93.01
144.45
171.87
230.90
310.98
397.80
407.27
425.06
Present
69.64
92.98
144.56
172.07
230.86
311.10
397.96
407.60
425.06
F
G
M
I
I
(
a
=
1
/
b
=
0
/
c
/
p
)
0.6
Ref [2]
70.59
93.62
146.22
172.31
235.55
313.20
403.88
409.05
439.88
Ref [10]
70.81
93.98
146.83
173.10
236.58
314.28
405.31
410.95
441.00
Present
70.89
93.97
146.96
173.29
236.55
314.41
405.48
410.80
441.02
1
Ref [2]
70.12
92.99
145.26
171.15
233.94
311.11
401.15
406.32
436.81
Ref [10]
70.39
93.41
145.95
172.05
235.11
312.37
402.82
408.46
438.20
Present
70.48
93.40
146.08
172.21
235.08
312.48
402.97
408.30
438.21
5
Ref [2]
71.04
95.02
147.45
175.58
235.02
317.81
405.72
416.02
432.13
Ref [10]
71.30
95.41
148.12
176.44
236.12
318.99
407.26
417.98
433.45
Present
71.38
95.45
148.23
176.64
236.10
319.11
407.43
418.05
433.45
20
Ref [2]
70.33
94.13
145.98
174.00
232.64
314.84
401.67
412.16
427.43
Ref [10]
70.47
94.36
146.40
174.57
233.35
315.49
402.55
413.44
428.05
Present
70.55
94.37
146.50
174.77
233.33
315.61
402.71
413.34
428.03
50
Ref [2]
69.43
92.74
144.04
171.22
230.16
310.22
396.86
405.84
424.51
Ref [10]
69.54
92.94
144.41
171.73
230.79
310.77
397.61
407.00
424.96
Present
69.60
92.91
144.48
171.87
230.69
310.81
397.66
406.77
424.84Table 8
Comparisons of frequencies (Hz) of the functionally graded spherical shells with different power-law indexp.
p
n
FF
SS
CC
Ref [9]
Ref [29]
Present
Ref [9]
Ref [29]
Present
Ref [9]
Ref [29]
Present
F
G
M
I
(
a
=
1
/
b
=
0
/
c
/
p
)
0.6
0
777.60
777.61
777.60
786.87
786.84
786.84
875.72
875.58
875.57
1
771.92
771.92
771.91
827.85
827.82
827.82
910.70
910.57
910.56
2
49.89
49.85
49.89
886.90
886.85
886.88
894.67
894.60
894.65
3
133.68
133.53
133.68
881.65
881.61
881.64
889.45
889.41
889.43
4
243.64
243.34
243.66
914.98
914.97
914.98
927.36
927.34
927.35
5
0
754.66
754.66
754.66
773.02
772.99
772.99
865.53
865.40
865.40
1
748.83
748.82
748.83
812.64
812.61
812.62
899.65
899.53
899.54
2
51.61
51.57
51.61
862.47
862.42
862.46
871.46
871.41
871.44
3
137.94
137.80
137.95
858.21
858.18
858.20
868.66
868.63
868.65
4
250.90
250.61
250.92
895.25
895.23
895.25
910.61
910.59
910.60
F
G
M
I
I
(
a
=
1
/
b
=
0
/
c
/
p
)
0.6
0
777.04
777.04
777.03
767.95
767.93
767.92
873.85
873.71
873.70
1
771.53
771.53
771.52
815.18
815.16
815.15
908.81
908.68
908.67
2
49.64
49.60
49.64
887.94
887.89
887.92
893.80
893.73
893.78
3
133.03
132.88
133.03
885.56
885.52
885.55
888.32
888.28
888.30
4
242.47
242.16
242.48
919.44
919.43
919.44
925.77
925.75
925.76
5
0
753.73
753.73
753.73
749.88
749.86
749.86
862.76
862.63
862.63
1
748.18
748.17
748.17
796.86
796.84
796.85
896.89
896.77
896.77
2
51.29
51.25
51.29
864.43
864.39
864.42
870.08
870.03
870.07
3
137.10
136.95
137.10
863.25
863.22
863.24
866.87
866.84
866.84
4
249.38
249.09
249.42
900.65
900.63
900.64
908.16
908.14
908.16In order to check the present method, the numerical results reported by Su et al. [7–9], Qu et al. [27], and Tornabene et al. [1, 3, 52] are also given in the above tables for comparison. From the comparisons, we can see a consistent agreement of present results taken from the current proposed unified approach and referential data. Besides, Tables 3–8 also show that it is appropriate to define the classical boundary conditions in terms of the boundary spring rigidities. Based on the formulation validation, some new results of the FG cylindrical, conical, and spherical panels and shells with classical-elastic and elastic restrain boundary condition are shown in Tables 9–14. Also, for any given modal frequency, the corresponding mode shape can be easily determined by submitting the solved eigenvector into (22). Thus, some select mode shapes of that are performed in Figures 6–8.Table 9
First six frequency parameters Ω of the functionally graded cylindrical panels with different boundary conditions.
Mode number
Boundary condition
CCCC
CSCS
CFCF
SFSF
CE1CE1
SE3SE3
E1E1E1E1
E2E2E2E2
E3E3E3E3
F
G
M
I
(
a
=
1
/
b
=
0
/
c
/
p
=
1
)
1
22.789
20.619
9.583
7.697
19.726
22.630
16.566
20.715
22.590
2
27.804
24.696
10.383
8.580
20.255
27.568
18.123
21.746
27.534
3
36.438
35.286
18.131
17.363
31.356
36.345
18.522
30.208
36.213
4
37.573
36.550
20.548
17.596
32.096
37.507
19.672
32.569
37.387
5
41.149
39.062
21.822
17.925
34.202
40.949
26.910
34.497
40.931
6
46.167
40.997
22.101
19.693
37.734
45.832
29.461
40.940
45.731
F
G
M
I
I
(
a
=
1
/
b
=
0
/
c
/
p
=
1
)
1
22.647
21.220
9.573
7.769
19.572
22.537
16.544
20.610
22.479
2
27.583
24.165
10.343
8.702
20.231
27.275
17.995
21.659
27.227
3
36.294
35.914
18.098
16.773
31.000
36.261
18.522
29.997
36.119
4
37.443
36.288
20.346
17.801
31.957
37.342
19.667
32.458
37.208
5
40.975
40.410
21.806
18.006
34.164
40.917
26.922
34.355
40.901
6
45.926
40.416
22.067
19.001
37.796
45.749
29.431
40.726
45.637Table 10
First six frequency parameters Ω of the functionally graded cylindrical shells with different boundary conditions.
Mode number
Boundary condition
CC
CS
CF
SF
CE1
SE3
E1E1
E2E2
E3E3
F
G
M
I
(
a
=
1
/
b
=
0
/
c
/
p
=
1
)
1
16.879
16.337
6.8893
6.8050
16.040
16.314
13.812
16.200
16.827
2
16.879
16.337
6.8893
6.8050
16.040
16.314
14.403
16.200
16.827
3
17.745
17.237
9.7991
9.7427
16.052
17.215
14.403
17.185
17.698
4
17.745
17.237
9.7991
9.7427
16.052
17.215
15.248
17.185
17.698
5
21.597
21.168
11.507
11.428
19.597
21.149
15.248
20.972
21.554
6
21.597
21.168
11.507
11.428
19.597
21.149
15.464
20.972
21.554
F
G
M
I
I
(
a
=
1
/
b
=
0
/
c
/
p
=
1
)
1
16.780
15.876
6.8620
6.5223
15.953
15.840
13.812
16.102
16.699
2
16.780
15.876
6.8620
6.5223
15.953
15.840
14.374
16.102
16.699
3
17.714
16.904
9.6811
9.5146
16.011
16.871
14.374
17.155
17.642
4
17.714
16.904
9.6811
9.5146
16.011
16.871
15.147
17.155
17.642
5
21.381
20.683
11.508
11.177
19.594
20.656
15.147
20.754
21.315
6
21.381
20.683
11.508
11.177
19.594
20.656
15.463
20.754
21.315Table 11
First six frequency parameters Ω of the functionally graded conical panels with different boundary conditions.
Mode number
Boundary condition
CCCC
CSCS
CFCF
SFSF
CE1CE1
SE3SE3
E1E1E1E1
E2E2E2E2
E3E3E3E3
F
G
M
I
(
a
=
1
/
b
=
0.5
/
c
=
2
/
p
)
1
24.598
22.827
10.062
8.130
21.649
22.861
18.335
23.804
24.476
2
27.546
25.319
11.116
9.246
22.310
25.382
19.905
24.825
27.392
3
35.766
34.653
19.734
17.732
30.047
32.676
25.501
34.596
35.612
4
39.938
36.473
21.090
18.147
35.305
36.417
26.760
35.304
39.668
5
41.194
38.498
23.559
20.537
36.797
38.274
29.303
37.316
41.058
6
42.731
40.899
24.102
20.650
41.696
39.241
32.699
38.101
42.355
F
G
M
I
I
(
a
=
1
/
b
=
0.5
/
c
=
2
/
p
)
1
24.562
23.102
10.060
8.2037
21.610
22.706
18.325
23.773
24.442
2
27.506
25.226
11.104
9.3506
22.300
25.232
19.866
24.793
27.338
3
35.737
35.024
19.725
17.659
29.983
32.942
25.502
34.572
35.608
4
39.882
36.565
21.053
17.835
35.291
36.496
26.767
35.263
39.626
5
41.166
38.973
23.545
20.502
36.737
38.180
29.240
37.282
41.021
6
42.646
40.733
24.085
20.688
41.606
39.057
32.684
38.059
42.281Table 12
First six frequency parameters Ω of the functionally graded conical shells with different boundary conditions.
Mode number
Boundary condition
CC
CS
CF
SF
CE1
SE3
E1E1
E2E2
E3E3
F
G
M
I
(
a
=
1
/
b
=
0.5
/
c
=
2
/
p
)
1
18.648
17.326
3.1641
3.0306
15.986
18.457
15.709
18.184
18.591
2
18.648
17.328
3.1641
3.0306
15.986
18.460
15.709
18.186
18.594
3
19.583
18.595
4.6352
4.4011
16.670
19.183
16.583
19.147
19.526
4
19.583
18.595
4.6352
4.4011
16.670
19.183
16.583
19.148
19.526
5
22.612
20.687
5.3016
5.2968
19.945
22.529
18.536
21.878
22.540
6
22.612
20.707
5.3016
5.2968
19.945
22.549
18.536
21.898
22.560
F
G
M
I
I
(
a
=
1
/
b
=
0.5
/
c
=
2
/
p
)
1
18.618
17.080
3.1594
2.9584
15.977
18.386
15.699
18.154
18.552
2
18.618
17.080
3.1594
2.9584
15.977
18.386
15.699
18.154
18.554
3
19.572
18.443
4.6350
4.2895
16.640
19.099
16.553
19.137
19.507
4
19.572
18.443
4.6350
4.2895
16.640
19.099
16.553
19.137
19.507
5
22.564
20.403
5.2888
5.2801
19.943
22.468
18.535
21.834
22.483
6
22.564
20.403
5.2888
5.2801
19.943
22.468
18.535
21.834
22.503Table 13
First six frequency parameters Ω of the functionally graded spherical panels with different boundary conditions.
Mode number
Boundary condition
CCCC
CSCS
CFCF
SFSF
CE1CE1
SE3SE3
E1E1E1E1
E2E2E2E2
E3E3E3E3
F
G
M
I
(
a
=
1
/
b
=
1
/
c
=
4
/
p
)
1
13.652
13.168
10.001
9.2907
12.374
12.924
5.215
11.613
13.567
2
14.297
14.090
10.057
9.3816
12.916
13.716
5.400
13.010
14.243
3
17.060
16.157
12.518
10.917
15.840
14.095
7.290
13.117
16.827
4
17.423
17.179
13.685
11.347
16.595
16.611
7.398
13.345
17.256
5
20.023
18.796
14.830
11.965
16.892
17.493
10.310
14.562
19.753
6
20.561
19.503
14.882
12.895
19.200
19.648
13.078
15.491
20.344
F
G
M
I
I
(
a
=
1
/
b
=
1
/
c
=
4
/
p
)
1
13.639
13.182
9.9925
9.4012
12.367
12.968
5.216
11.607
13.560
2
14.286
14.054
10.051
9.4953
12.902
13.764
5.401
13.001
14.234
3
17.037
16.214
12.508
10.878
15.817
14.029
7.282
13.110
16.801
4
17.401
17.167
13.668
11.330
16.599
16.660
7.399
14.553
17.243
5
19.993
18.755
14.805
11.985
16.867
17.486
10.295
15.478
19.726
6
20.531
19.495
14.860
12.923
19.170
19.644
13.073
15.824
20.314Table 14
First six frequency parameters Ω of the functionally graded spherical shells with different boundary conditions.
Mode number
Boundary condition
CC
CS
CF
SF
CE1
SE3
E1E1
E2E2
E3E3
F
G
M
I
(
a
=
1
/
b
=
1
/
c
=
4
/
p
)
1
12.293
11.727
1.7433
1.4432
6.5531
11.970
5.6226
10.036
12.242
2
12.293
11.727
1.7433
1.4432
6.5531
11.970
6.3197
10.838
12.242
3
12.552
11.918
2.7986
2.2795
7.2370
12.233
6.3199
10.838
12.488
4
12.552
11.918
2.7986
2.2795
8.2529
12.233
7.6911
12.060
12.488
5
13.316
12.855
3.0960
3.0509
8.2529
12.992
7.6912
12.060
13.286
6
13.316
12.855
3.0960
3.0510
10.393
12.992
8.6638
12.141
13.286
F
G
M
I
I
(
a
=
1
/
b
=
1
/
c
=
4
/
p
)
1
12.284
11.760
1.7406
1.4246
6.5481
12.002
5.6204
10.032
12.238
2
12.284
11.760
1.7406
1.4246
6.5481
12.002
6.3145
10.833
12.238
3
12.539
11.946
2.7974
2.2396
7.2331
12.251
6.3145
10.833
12.481
4
12.539
11.946
2.7974
2.2396
8.2452
12.251
7.6835
12.055
12.481
5
13.310
12.899
3.0896
3.0439
8.2452
13.032
7.6835
12.055
13.285
6
13.310
12.899
3.0896
3.0439
10.381
13.032
8.6633
12.132
13.285Figure 6
Mode shapes for cylindrical panels and shells with different boundary conditions.Figure 7
Mode shapes for conical panels and shells with different boundary conditions.Figure 8
Mode shapes for spherical panels and shells with different boundary conditions.


## 3.2. Parameter Studies

Although the above results of the FG cylindrical, conical, and spherical panels and shells with classical-elastic and elastic restrain boundary conditions can be served as benchmark data for the future numerical methods, it does not meet authors’ ambition. The goal of this paper is to provide some useful research works for the designer or engineer to avoid the unpleasant, inefficient, and structurally damaging resonant by given a unified computational model of the FG cylindrical, conical, and spherical panels and shells of revolution. So, in the this subsection, the parameter studies including the boundary restraint rigidityΓ, thickness h, circumferential angle ϕ, power-law exponent p, and material parameters a, b, and c will be conducted.Figure9 shows the effects of the all kinds of boundary restraint parameters on the frequency parameters of FG panels and shells with elastic boundary conditions. Five nondimensional boundary restraint parameters Γϐ (ϐ=u,v,w,α, and β) are defined as ratios of corresponding spring stiffness to the reference bending stiffness D; that is, Γu=ku/D, Γv=kv/D, Γw=kw/D, Γα=Kα/D, and Γβ=Kβ/D. The geometry properties of the panels and shells studied are the same as Tables 3–8. The panels and shells are free at edges β=constant, clamped at α=0 while the α=Lα is elastically restrained by all the five groups of boundary springs, of which only one group is with the variable stiffness and the others are assumed to be zeros. According to Figure 9, we can see that the change of the boundary elastic restraint parameter Γ has little effect on the frequency parameter Ω when it is smaller than 10-2D. However, when it is increased in a certain range, the frequency parameters increase rapidly as the elastic parameters increase, approach their utmost, and remain unchanged when Γ approaches infinity. It is noted that the certain range with respect to different kinds of elastic restraint parameters is different. Figures 10 and 11 show the influence of the fundamental frequency parameters versus thickness h for the FG panels and shells with different shear correction factors, respectively. It is obvious that the frequency parameters of the FG panels and shells with respect to the classical and elastic boundary condition increase when increasing the thicknesses. In addition, the figures also shows that the shear correction factor κ have a significant impact on the vibration behavior of FG panels and shells and a larger shear correction factor will result in higher frequency parameters. Next, Figure 12 presents the first four frequency parameters for the FG panels versus the circumference angles with the CCCC and E3E3E3E3 boundary conditions. From the figure, we can see that the frequency parameters rapidly decrease while the circumference angle changed from 5° to 75°, and out of this range the frequency parameters slowly minish and lastly almost remain unchanged. In addition, we also found that the variation of the frequency parameters has nothing to do with the modal order. Then, the effects of the power-law index p and the material parameters a, b, and c are studied as well. Figure 13 shows the fundamental frequency versus of the power-law index p for FG panels and shells with clamped boundary conditions. The material distribution using in the study is FGMI(a/b/c/p) and FGMII(a/b/c/p): cylindrical meridian: FGMI(a=1/b=0.5/c=2/p) and FGMII(a=1/b=0.5/c=2/p); conical meridian: FGMI(a=0/b=-0.5/c=2/p) and FGMII(a=0/b=-0.5/c=2/p); spherical meridian: FGMI(a=1/b=1/c=4/p) and FGMII(a=1/b=1/c=4/p). From the figure, we can see that the geometrical construction has effect on the vibration characteristic of the FG panels and shells. For case of the cylindrical meridian, the fundamental frequency firstly rapidly decreases and then increases and finally becomes decreased versus the power-law index p increased. However, for the conical and spherical meridian, the frequency parameters decrease monotonously while increasing the power-law index p. In addition to that, the figure also shows that the frequency parameters of the FGMI structure elements are always higher than the FGMI case. In Figure 14, the influence of the combination relations between the boundary condition and power-law index p on the vibration behavior for the FG panes and shells is reported. And to simplify the study, the research selects cylindrical panels as the example. From the figure, we can know that the variation trend of the fundamental frequency of the FG panes and shells with CFCF, E3E3E3E3, and SSSS is firstly rapidly decreased and then increased and finally becomes decreased versus the power-law index p increase. However, the frequency parameters increase rapidly as the power-law index p increases, approaches their utmost, and remains unchanged when p is more than a threshold. Thus, we can get that the boundary condition plays an indispensable role in the vibration behavior of the FG panes and shells. Finally, the influence of the material parameters a, b, and c will be exhibited in the next example. Figure 15 depicts the variations of the first frequencies of the clamped supported FGMI panels and shells with different material types: Type 1: a=1 and b=c=0; Type 2: a=1, b=0.5, and c=2; Type 3: a=0, b=-0.5, and c=2; Type 4: a=1, b=1, and c=4. From the figure, it is obvious that the material parameters a, b, and c have a significant influence on the fundamental frequencies for the FG panels and shells.Figure 9
Variation of the fundamental frequency parametersΩ versus the elastic restraint parameters for FG panels and shells: (a) cylindrical panels; (b) cylindrical shells; (c) conical panels; (d) conical shells; (e) spherical panels; (f) spherical shells.
(a)
(b)
(c)
(d)
(e)
(f)Figure 10
Fundamental frequency parameters versus thicknessh for the CCCC FG panels and shells with different shear correction factors: (a) cylindrical panels; (b) cylindrical shells; (c) conical panels; (d) conical shells; (e) spherical panels; (f) spherical shells.
(a)
(b)
(c)
(d)
(e)
(f)Figure 11
Fundamental frequency parameters versus thicknessh for the E3E3E3E3 FG panels and shells with different shear correction factors: (a) cylindrical panels; (b) cylindrical shells; (c) conical panels; (d) conical shells; (e) spherical panels; (f) spherical shells.
(a)
(b)
(c)
(d)
(e)
(f)Figure 12
Variation of the first four frequency parametersΩ versus the circumferential angle for FG panels: (a) cylindrical panels; (b) conical panels; (c) spherical panels.
(a)
(b)
(c)Figure 13
Variations of the first frequencies of the clamped supportedFGMI and FGMII: (a) cylindrical panels; (b) cylindrical shells; (c) conical panels; (d) conical shells; (e) spherical panels; (f) spherical shells.
(a)
(b)
(c)
(d)
(e)
(f)Figure 14
Variations of the first frequencies of theFGMI and FGMII cylindrical panels with different boundary conditions.
(a)
CFCF
(b)
SSSS
(c)
E1E1E1E1
(d)
E3E3E3E3Figure 15
Variations of the first frequencies of the clamped supportedFGMI panels and shells with different material types: Type 1: a=1 and b=c=0; Type 2: a=1, b=0.5, and c=2; Type 3: a=0, b=-0.5, and c=2; Type 4: a=1, b=1, and c=4.
(a)
Cylindrical panels
(b)
Cylindrical shells
(c)
Conical panels
(d)
Conical shells
(e)
Spherical panels
(f)
Spherical shells


## 4. Conclusions

In this paper, a unified computational model is presented for the vibration analysis of the four-parameter functionally graded moderately thick cylindrical, conical, spherical panels and shells of revolution with general boundary condition based on first-order shear deformation theory. Applying the modified Fourier-Ritz approach, the admissible functions of the structure elements are expanded into the improved Fourier series which consist of the two-dimensional (2D) Fourier cosine series and auxiliary functions to eliminate all the relevant discontinuities of the displacements and their derivatives at the edges regardless of boundary conditions and then solve the natural frequencies by means of the Ritz method. In order to obtain the unified computational model of the functionally graded cylindrical, conical, spherical panels and shells of revolution, as one merit of this paper, the coupling spring technology is introduced to ensure the kinematic and physical compatibility at the common meridian, if a complete shell of revolution needs considering. The good convergence, excellent accuracy, and reliability of the present unified computational model are checked and validated by the comparison with the results presented by other contributors. Through the parameter studies, the vibration behavior of the FG cylindrical, conical, spherical panels and shells is not only related to the boundary conditions, but also related to their self-attribute. Comparing with the existing works, the present unified computational model contains the following highlights:(
1
) The proposed unified computational model is appropriate for problems which is the four-parameter functionally graded moderately thick cylindrical, conical, spherical panels and shells of revolution with general boundary condition.(
2
) The change of the boundary conditions can be easily achieved by just varying the stiffness of the boundary restraining springs along all the edges of plates without making any changes in the solution procedure.



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*Source: 9183924-2017-10-31.xml*