Free vibration and vibrational displacements analysis of thick elastically supported laminated curved panels with power-law distribution functionally graded layers and finite length via 2D GDQ method

Abstract

This paper deals with free vibration and vibrational displacements analysis of thick laminated curved panels with finite length resting on two-parameter elastic foundations based on the three-dimensional elasticity theory. Because of using two-dimensional generalized differential quadrature method, the present approach makes possible vibration analysis of cylindrical panels with two opposite axial edges simply supported and arbitrary boundary conditions including free, simply supported and clamped at the curved edges. The material properties vary continuously through the layers’ thickness according to a three-parameter power-low distribution. It is assumed that the inner surfaces of the functionally graded sheets are metal rich, while the outer surfaces of the layers can be metal rich, ceramic rich or made of a mixture of two constituents. The benefit of using the considered power-law distribution is to illustrate and present useful results arising from symmetric and asymmetric profiles. The effects of two-parameter elastic foundation modulus, geometrical and material parameters together with the boundary conditions on the frequency parameters of the laminated functionally graded panels are investigated. The obtained results show that the outer functionally graded material layers have significant effect on the vibration behavior of cylindrical panels. This study serves as a benchmark for assessing the validity of numerical methods or two-dimensional theories used to analysis of laminated curved panels.

Keywords

Natural frequency analysis FG laminated structures finite length curved panels vibration Pasternak foundation vibrational displacements

Introduction

Functionally graded (FG) cylindrical panels, as important structural components, have widely been used in different branches of engineering such as mechanical, energy and aerospace engineering. However, in comparison with the isotropic and conventional laminated cylindrical panels, the literature on the free vibration analysis of FG cylindrical panels is relatively scarce. In addition, in the most of the existing researches in this regard, single-layer FG cylindrical panels have been analyzed. In the following, some of these research works are briefly reviewed. Due to the mismatch of stiffness properties between the face sheets and the core, sandwich plates and panels are susceptible to face sheet/core debonding, which is a major problem in sandwich construction, especially under impact loading [1]. Various material profiles through the FG plate and panel thickness can be illustrated by using three-parameter power-low distribution. In fact, by using this power-low distribution, it is possible to study the influence of the different kinds of material profiles. Recently, Viola and Tornabene [2] used three-parameter power-low distribution to study the dynamic behavior of FG parabolic panels of revolution. Though there are research works that reported on general sandwich structures, very little work has been done to consider the vibration behavior of FGM sandwich structures [3,4]. Li et al. [5] studied free vibrations of FGSW rectangular plates with simply supported and clamped edges. Zenkour [6,7] presented a two-dimensional solution to study the bending, buckling and free vibration of simply supported FG ceramic-metal sandwich plates. Kamarian et al. [8] studied free vibration of FGSW rectangular plates with simply supported edges and rested on elastic foundations using differential quadratic method. The natural frequencies of FGM circular cylindrical shells are investigated [9], which was later extended to cylindrical shells under various end supporting conditions [10]. Patel et al. [11] carried out the vibration analysis of FG shell using a higher order theory. Pradyumna and Bandyopadhyay [12] studied the free vibrations analysis of FG curved panels by using a higher order finite element formulation. Free vibration and dynamic instability of FGM cylindrical panels under combined static and periodic axial forces were studied by using a proposed semi-analytical approach [13]. Elastic response analysis of simply supported FGM cylindrical shell under low-velocity impact was presented by Gang et al. [14]. Vibrations and wave propagation velocity in a FG hollow cylinder were studied by Shakeri et al. [15]. They assumed the shell to be in plane strain condition and subjected to an axisymmetric dynamic loading. The free vibration of simply supported, fluid-filled cylindrically orthotropic FG cylindrical shells with arbitrary thickness was investigated by Chen et al. [16]. Recently, Tornabene [17] used four-parameter power-law distribution to study the dynamic behavior of moderately thick FG conical and cylindrical shells and annular plates. In his study, the two-constituent FG isotropic shell consisted of ceramic and metal, and the generalized differential quadrature method (GDQ) was used to discretize the governing equation. Static and free vibration analyses of continuously graded fiber-reinforced cylindrical shells using generalized power-law distribution were presented by Sobhani Aragh and Yas [18]. Also, these authors [19] investigated three-dimensional free vibration of FG fiber orientation and volume fraction of cylindrical panels. Paliwal et al. [20,21] have investigated the free vibration of whole buried cylindrical shells with simply supported ends in contact with Winkler and Pasternak foundations using direct solution to the governing classical shell theory equations of motion. Yang et al. [22] have investigated the behavior of whole buried pipelines subjected to sinusoidal seismic waves by the finite element method. Cai et al. [23] have investigated free vibration of a cylindrical panel supported on Kerr foundation. Kerr model can be reduced to either a Pasternak model or a Winkler one by selecting certain values of foundation parameters. Gunawan et al. [24] examined the free vibrations of cylindrical shells partially buried in elastic foundations based on the finite element method. The shells are discretized into cylindrical finite elements, and the distribution of the foundation in the circumferential direction is defined by the expansion of Fourier series. Farid et al. [25] have studied three-dimensional temperature-dependent free vibration analysis of functionally graded material (FGM) curved panels resting on two-parameter elastic foundation subjected in thermal environment. The curved panels were made of isotropic material, and in order to discretize the governing equations, the differential quadrature method (DQM) in the thickness direction and the trigonometric functions in longitudinal and tangential directions in conjunction of the three-dimensional form of the Hamilton’s principle have been used. Free vibration and stability of FG shallow shells according to a two-dimensional higher order deformation theory were investigated by Matsunaga [26]. Civalek [27] has investigated the nonlinear dynamic response of doubly curved shallow shells resting on Winkler–Pasternak elastic foundation using the harmonic differential quadrature (HDQ) and finite differences (FD) methods. Hong and Lee [28] have presented a spectral element model for a modified FGM axial bar model, wherein non-uniform lateral contraction in the thickness direction is taken into account. We assume that material properties of the modified FGM axial bar model vary in the radial direction according to the power-law distribution. Marin et al. [29 –31] have investigated different kinds of elasticity problem considering dipolar and micropolar bodies. Tornabene et al. [32] studied free vibrations of free-form doubly curved shells made of FGMs using higher order equivalent single-layer theories. The partial differential system of equations was solved by using the GDQ method. Tornabene and Ceruti [33] studied a mixed static and dynamic optimization of four-parameter FGM doubly curved shells and panels. The two-constituent FG shell consists of ceramic and metal, and the volume fraction profile of each lamina varies through the thickness of the shell according to a generalized power-law distribution. Tornabene et al. [34] used GDQ method to study the dynamic behavior of FGMs and laminated doubly curved shells and panels of revolution with a free-form meridian. The First-order Shear Deformation Theory (FSDT) was used to analyze the abovementioned moderately thick structural elements. Tornabene and Viola [35,36] investigated free vibration of three- and four-parameter FG parabolic panels and shells of revolution. For the discretization of the system equations the GDQ method had been used. Numerical results concerning FG parabolic panels and shells showed the influence of the three parameters of the power-law distribution on their mechanical behavior.

To the authors’ best knowledge, there exists no study in the open literature for free vibration of thick Functionally Graded Sandwich (FGSW) curved panels resting on Pasternak foundations. Furthermore, this paper is motivated by the lack of studies in the technical literature concerning the effect of the parameters of power-law distributions on the vibration behavior of FG laminated curved panels. Frequency parameters are obtained by using numerical technique termed the GDQ method, which leads to a generalized eigenvalue problem. The DQM is found to be a simple and efficient numerical technique for vibration analysis of structures [37,38].

FG sandwich properties

Consider an FGSW curved panel rested on two-parameter elastic foundations as shown in Figure 1. A cylindrical coordinate system (r, θ, z) is used to label the material point of the panel. The inner surface is continuously in contact with an elastic medium that acts as an elastic foundation represented by the Winkler/Pasternak model with K_w and K_g being Winkler and shear coefficients of Pasternak foundation, respectively.

The panel has continuous grading of fiber reinforcement through radial direction. h, h_c and h_s are the thickness of panel, core and face sheets, respectively. In the present work, the fiber volume fraction of laminated curved panel is assumed as follows

V c = {1 - [(\frac{(r - R) + 0.5 h}{h s}) + b (1 - \frac{(r - R) + 0.5 h}{h s}) c] p - 0.5 h \leq r - R \leq - 0.5 h + h s 0 - 0.5 h + h s \leq r - R \leq 0.5 h - h s 1 - [(\frac{- (r - R) + 0.5 h}{h s}) + b (1 - \frac{- (r - R) + 0.5 h}{h s}) c] p 0.5 h - h s \leq r - R \leq 0.5 h

(1)

where the power-law index p

(0 \leq p \leq \infty)

and the parameters b and c dictate the fiber variation profile through the radial direction of the panel. According to the abovementioned relation, the core of sandwich panel and the inner surfaces of the FG sheets are metal rich. The outer surfaces of the sheets can be metal rich, ceramic rich or made of a mixture of two constituents. The through-thickness variations of the fiber volume fraction of sandwich panel are shown in Figures 2 to 5. The relevant material properties for the constituent materials are as follows [15]

Metal (Aluminum, Al)

E m = 70 \times 109 Pa, ρ m = 2702 K g / m 3, υ m = 0.3

-Ceramic (Alumina, Al₂O₃)

E c = 380 \times 109 Pa, ρ c = 3800 K g / m 3, υ c = 0.3

Governing equations

The mechanical constitutive relation that relates the stresses to the strains is as follows

[σ r σ θ σ z τ z θ τ rz τ r θ] = [C 11 C 12 C 13 000 C 12 C 22 C 23 000 C 13 C 23 C 33 000000 C 44 000000 C 55 000000 C 66] [ɛ r ɛ θ ɛ z γ z θ γ rz γ r θ]

(2)

In the absence of body forces, the governing equations are as follows

\frac{\partial σ r}{\partial r} + \frac{1}{r} \frac{\partial τ r θ}{\partial θ} + \frac{\partial τ rz}{\partial z} + \frac{σ r - σ θ}{r} = ρ \frac{\partial 2 u r}{\partial t 2}, \frac{\partial τ r θ}{\partial r} + \frac{1}{r} \frac{\partial σ θ}{\partial θ} + \frac{\partial τ θ z}{\partial z} + \frac{2 τ r θ}{r} = ρ \frac{\partial 2 u θ}{\partial t 2}, \frac{\partial τ rz}{\partial r} + \frac{1}{r} \frac{\partial τ θ z}{\partial θ} + \frac{\partial σ z}{\partial z} + \frac{τ rz}{r} = ρ \frac{\partial 2 u z}{\partial t 2}

(3)

Strain–displacement relations are expressed as

ɛ r = \frac{\partial u r}{\partial r}, ɛ θ = \frac{u r}{r} + \frac{1}{r} \frac{\partial u θ}{\partial θ}, ɛ z = \frac{\partial u z}{\partial z}, γ θ z = \frac{\partial u θ}{\partial z} + \frac{1}{r} \frac{\partial u z}{\partial θ}, γ rz = \frac{\partial u r}{\partial z} + \frac{\partial u z}{\partial r}, γ r θ = \frac{1}{r} \frac{\partial u r}{\partial θ} + \frac{\partial u θ}{\partial r} - \frac{u θ}{r}

(4)

where u_r, u_θ and u_z are radial, circumferential and axial displacement components, respectively. Upon substitution of equation (4) into (2) and then into (3), the equations of motion in terms of displacement components with infinitesimal deformations can be written as

[F 1 r F 1 θ F 1 z F 2 r F 2 θ F 2 z F 3 r F 3 θ F 3 z] {u r u θ u z} = {ρ \frac{\partial 2 u r}{\partial t 2} ρ \frac{\partial 2 u θ}{\partial t 2} ρ \frac{\partial 2 u z}{\partial t 2}}

(5)

where coefficients

F ij

are given in Appendix 1.

The boundary conditions at the concave and convex surfaces, r = r_i and r_o, respectively, can be described as follows

τ rz = τ r θ = 0, σ r = {- k w u r + k g {\frac{\partial 2 u r}{\partial z 2} + \frac{1}{r 2} \frac{\partial 2 u r}{\partial θ 2}} at r = r i 0 at r = r o

(6)

In this investigation, three different types of classical boundary conditions at edges z = 0 and L_z of the finite panel can be stated as follows:

Simply supported (S)

U r = U θ = σ z = 0

(7)

Clamped (C):

U r = U θ = U z = 0

(8)

Free (F)

σ z = σ z θ = σ zr = 0

(9)

Solution procedure

For the curved panels with simply supported at one pair of opposite edges, the displacement components can be expanded in terms of trigonometric functions in the direction normal to these edges. In this work, it is assumed that the edges θ = 0 and θ = Φ are simply supported. Hence

u r (r, θ, z, t) = \sum_{m = 1}^{\infty} U r (r, z) \sin (\frac{m π}{Φ} θ) e i ω t, u θ (r, θ, z, t) = \sum_{m = 1}^{\infty} U θ (r, z) \cos (\frac{m π}{Φ} θ) e i ω t, u z (r, θ, z, t) = \sum_{m = 1}^{\infty} U z (r, z) \sin (\frac{m π}{Φ} θ) e i ω t

(10)

where m is the circumferential wave number, ω is the natural frequency and i (=

\sqrt{- 1}

) is the imaginary number. Substituting displacement components from equation (10) into equation (5), and then using GDQ method to discretize the equations of motion (equation (5)), one can get the following equations (a brief review of GDQ method is given in Appendix 2)

In the r direction

(c 11) ij \sum_{n = 1}^{N r} B_{in}^{r} U rnj + (c 12) ij (\frac{m π}{Φ r_{i}^{2}} U θ ij - \frac{m π}{Φ r i} \sum_{n = 1}^{N r} A_{in}^{r} U θ nj + \frac{1}{r i} \sum_{n = 1}^{N r} A_{in}^{r} U rnj - \frac{U rij}{r_{i}^{2}}) + (c 13) ij \sum_{n = 1}^{N r} \sum_{v = 1}^{N z} A_{jv}^{z} A_{in}^{r} U znv + (\frac{\partial c 11}{\partial r}) ij \sum_{n = 1}^{N r} A_{in}^{r} U rnj + (\frac{\partial c 12}{\partial r}) ij (\frac{1}{r i} U rij - \frac{m π}{Φ r i} U θ ij) + (\frac{\partial c 13}{\partial r}) ij \sum_{n = 1}^{N z} A_{jn}^{z} U zin + \frac{(c 66) ij}{r i} (- \frac{m π}{Φ} \sum_{n = 1}^{N r} A_{in}^{r} U θ nj - \frac{1}{r i} (\frac{m π}{Φ}) 2 U rij + \frac{m π}{Φ r i} U θ ij) + (c 55) ij (\sum_{n = 1}^{N z} B_{jn}^{z} U rin + \sum_{n = 1}^{N r} \sum_{v = 1}^{N z} A_{jv}^{z} A_{in}^{r} U znv) + \frac{1}{r i} ((c 11) ij \sum_{n = 1}^{N r} A_{in}^{r} U rnj + (c 12) ij (\frac{U rij}{r i} - \frac{m π}{Φ r i} U θ ij) + (c 13) ij \sum_{n = 1}^{N z} A_{jn}^{z} U zin - (c 12) ij \sum_{n = 1}^{N r} A_{in}^{r} U rnj - (c 22) ij (\frac{U rij}{r i} - \frac{m π}{Φ r i} U θ ij) - (c 23) ij \sum_{n = 1}^{N z} A_{jn}^{z} U zin) = - ρ ij ω 2 U rij

(11)

In the θ direction

(c 66) ij (- \frac{1}{r_{i}^{2}} \frac{m π}{Φ} U rij + \frac{m π}{Φ r i} \sum_{n = 1}^{N r} A_{in}^{r} U rnj + \sum_{n = 1}^{N r} B_{in}^{r} U θ nj + \frac{U θ ij}{r_{i}^{2}} - \frac{1}{r i} \sum_{n = 1}^{N r} A_{in}^{r} U θ nj) + (\frac{\partial c 66}{\partial r}) ij (\frac{m π}{Φ r i} U rij + \sum_{n = 1}^{N r} A_{in}^{r} U θ nj - \frac{U θ ij}{r i}) + \frac{1}{r i} ((c 12) ij \frac{m π}{Φ} \sum_{n = 1}^{N r} A_{in}^{r} U rnj + (c 22) ij (\frac{m π}{Φ r i} U rij - \frac{1}{r i} (\frac{m π}{Φ}) 2 U θ ij) + (c 23) ij \frac{m π}{Φ} \sum_{n = 1}^{N z} A_{jn}^{z} U zin) + (c 44) ij (\sum_{n = 1}^{N z} B_{jn}^{z} U θ in + \frac{m π}{Φ r i} \sum_{n = 1}^{N z} A_{jn}^{z} U zin) + \frac{2 (c 66) ij}{r i} (\frac{m π}{Φ r i} U rij + \sum_{n = 1}^{N r} A_{in}^{r} U θ nj - \frac{U θ ij}{r i}) = - ρ ij ω 2 U θ ij

(12)

In the z direction

(c 55) ij (\sum_{n = 1}^{N r} \sum_{v = 1}^{N z} A_{jv}^{z} A_{in}^{r} U rnv + \sum_{n = 1}^{N r} B_{in}^{r} U znj) + (\frac{\partial c 55}{\partial r}) ij (\sum_{n = 1}^{N z} A_{jn}^{z} U rin + \sum_{n = 1}^{N r} A_{in}^{r} U znj) + \frac{(c 44) ij}{r i} (- \frac{m π}{Φ} \sum_{n = 1}^{N z} A_{jn}^{z} U θ in - \frac{1}{r i} (\frac{m π}{Φ}) 2 U zij) + (c 13) ij \sum_{n = 1}^{N r} \sum_{v = 1}^{N z} A_{jv}^{z} A_{in}^{r} U rnv + (c 23) ij (\frac{1}{r i} \sum_{n = 1}^{N z} A_{jn}^{z} U rin - \frac{m π}{Φ r i} \sum_{n = 1}^{N z} A_{jn}^{z} U θ in) + (c 33) ij \sum_{n = 1}^{N z} B_{jn}^{z} U zin + \frac{(c 55) ij}{r i} (\sum_{n = 1}^{N z} A_{jn}^{z} U rin + \sum_{n = 1}^{N r} A_{in}^{r} U znj) = - ρ ij ω 2 U zij

(13)

In the abovementioned equations i = 2,…, N_r−1 and j = 2,…, N_z−1. $A_{ij}^{r}$ , $A_{ij}^{z}$ and $B_{ij}^{r}$ , $B_{ij}^{z}$ are the first- and second-order GDQ weighting coefficients in the r- and z-directions, respectively.

Substituting displacement components from equation (10) into equation 6, and then using GDQ method to discretize the boundary conditions, one can get the following equations

\sum_{n = 1}^{N z} A_{jn}^{z} U rin + \sum_{n = 1}^{N r} A_{in}^{r} U znj = 0, \frac{m π}{Φ r i} U rij + \sum_{n = 1}^{N r} A_{in}^{r} U θ nj - \frac{U θ ij}{r i} = 0, (c 11) ij \sum_{n = 1}^{N r} A_{in}^{r} U rnj + (c 12) ij (\frac{U rij}{r i} - \frac{m π}{Φ r i} U θ ij) + (c 13) ij \sum_{n = 1}^{N z} A_{jn}^{z} U zin {k w U rij - k g (\sum_{n = 1}^{N z} B_{jn}^{z} U rin - (\frac{m π}{Φ r i}) 2 U rij)} δ 1 i = 0

(14)

where i = 1 at r = r_i and i = N_r at r = r_o, and j = 1,2,…,N_z; also δ_ij is the Kronecker delta. By following the same procedure the boundary conditions at z = 0 and L_z stated in equations (7) to (9) become

U rij = U θ ij = 0, (c 13) ij \sum_{n = 1}^{N r} A_{in}^{r} U rnj + (c 23) ij (\frac{U rij}{r i} - \frac{m π}{Φ r i} U θ ij) + (c 33) ij \sum_{n = 1}^{N z} A_{jn}^{z} U zin = 0

(15)

U rij = U θ ij = U zij = 0

(16)

(c 13) ij \sum_{n = 1}^{N r} A_{in}^{r} U rnj + (c 23) ij (\frac{U rij}{r i} - \frac{m π}{Φ r i} U θ ij) + (c 33) ij \sum_{n = 1}^{N z} A_{jn}^{z} U zin = 0, \sum_{n = 1}^{N z} A_{jn}^{z} U θ in + \frac{m π}{Φ r i} U zij = 0, \sum_{n = 1}^{N z} A_{jn}^{z} U rin + \sum_{n = 1}^{N r} A_{in}^{r} U znj = 0

(17)

In the above equations i = 2,…, N_r−1; also j = 1 at z = 0 and j = N_z at z = L_z.

In order to carry out the eigenvalue analysis, the domain and boundary nodal displacements should be separated. In vector forms, they are denoted as {d} and {b}, respectively. Based on this definition, the discretized form of the equations of motion and the related boundary conditions can be represented in the matrix form as

Equations of motion, equations (11) to (13)

[[K db] [K dd]] {{b} {d}} - ω 2 [M] {d} = {0}

(18)

Boundary conditions, equation (14) and equations (15) to (17)

[K bd] {d} + [K bb] {b} = {0}

(19)

Eliminating the boundary degrees of freedom in equation (18) using equation (19), this equation becomes

[K] - ω 2 [M] {d} = {0}

(20)

where

[K] = [K dd] - [K db] [K bb] - 1 [K bd]

. The above eigenvalue system of equations can be solved to find the natural frequencies and mode shapes of the curved panel.

Numerical results and discussion

To verify the proficiency of presented method and three-parameter model for volume fraction of FG materials, several numerical examples are carried out for comparisons. The results of the presented formulations are given in the form of convergence studies with respect to N_r and N_z, the number of discrete points distributed along the radial and axial directions, respectively. To validate the proposed approach, its convergence and accuracy are demonstrated via different examples. The obtained natural frequencies based on the three-dimensional elasticity formulation are compared with those of the power series expansion method for both FGM curved panels with and without elastic foundations [12,25,26]. In these studies, the material properties of FGMs are assumed as follows:

Metal (Aluminum, Al)

E m = 70 \times 109 Pa, ρ m = 2702 K g / m 3, υ m = 0.3

Ceramic (Alumina, Al₂O₃)

E c = 380 \times 109 Pa, ρ c = 3800 K g / m 3, υ c = 0.3

Subscripts m and c refer to the metal and ceramic constituents, which denote the material properties of the outer and inner surfaces of the panel, respectively. To validate the analysis, results for FGM cylindrical shells are compared with similar ones in the literature, as shown in Table 1. The comparison shows that the present results agreed well with those in the literatures. Besides fast rate of convergence of the method is quite evident, and it is found that only thirteen grid points (N_r = N_z = 13) along the radial and axial directions can yield accurate results. Further validation of the present results for isotropic FGM cylindrical panel is shown in Table 2. In this table, a comparison is made for different L_z/R and L_z/h ratios, and as it is observed, there is good agreement between the results.

Table 1.

Comparison of the normalized natural frequency of an FGM composite curved panel with four edges simply supported $(Ω 11 = ω 11 R Φ \sqrt{ρ m h / D}, D = E m h 3 / 12 (1 - υ_{m}^{2}))$ .

p (volume fraction index)		R/L_z
p (volume fraction index)		0.5	1	5	10	50
0	N_r = N_z = 5	69.9774	52.1052	42.7202	42.3717	42.2595
	N_r = N_z = 7	69.9722	52.1052	42.7158	42.3718	42.2550
	N_r = N_z = 9	69.9698	52.1003	42.7159	42.3700	42.2553
	N_r = N_z = 11	69.9700	52.1003	42.7160	42.3677	42.2552
	N_r = N_z = 13	69.9700	52.1003	42.7160	42.3677	42.2553
	Pradyumna and Bandyopadhyay [12]	68.8645	51.5216	42.2543	41.908	41.7963
0.2	N_r = N_z = 5	65.1470	47.9393	39.1282	38.8010	38.7020
	N_r = N_z = 7	65.4449	48.0456	39.1008	38.7366	38.6834
	N_r = N_z = 9	65.4526	48.1340	39.0836	38.7568	38.6581
	N_r = N_z = 11	65.4304	48.1340	39.0835	38.7568	38.6580
	N_r = N_z = 13	65.4304	48.1340	39.0835	38.7568	38.6581
	Pradyumna and Bandyopadhyay [12]	64.4001	47.5968	40.1621	39.8472	39.7465
0.5	N_r = N_z = 5	60.1196	43.5539	36.1264	35.8202	34.7341
	N_r = N_z = 7	60.2769	43.7128	36.1401	35.7964	35.0677
	N_r = N_z = 9	60.3574	43.7689	36.0944	35.7890	35.7032
	N_r = N_z = 11	60.3574	43.7688	36.0943	35.7891	35.7032
	N_r = N_z = 13	60.3574	43.7689	36.0944	35.7891	35.7032
	Pradyumna and Bandyopadhyay [12]	59.4396	43.3019	37.287	36.9995	36.9088
1	N_r = N_z = 5	54.1034	38.5180	31.9860	30.7065	30.6336
	N_r = N_z = 7	54.6039	39.1477	32.1140	31.6982	31.5397
	N_r = N_z = 9	54.7141	39.1620	32.0401	31.7608	31.6877
	N_r = N_z = 11	54.7141	39.1621	32.0401	31.7608	31.6878
	N_r = N_z = 13	54.7141	39.1621	32.0401	31.7608	31.6877
	Pradyumna and Bandyopadhyay [12]	53.9296	38.7715	33.2268	32.9585	32.875
2	N_r = N_z = 5	46.9016	34.7702	27.6657	27.4295	27.3725
	N_r = N_z = 7	47.9865	34.6980	27.5733	27.3389	27.2669
	N_r = N_z = 9	48.5250	34.6852	27.5614	27.3238	27.2663
	N_r = N_z = 11	48.5250	34.6851	27.5614	27.3239	27.2663
	N_r = N_z = 13	48.5250	34.6851	27.5614	27.3239	27.2662
	Pradyumna and Bandyopadhyay [12]	47.8259	34.3338	27.4449	27.1789	27.0961

Table 2.

Comparison of the normalized natural frequency of an FGM composite curved panel for various L_Z/R and L_Z/h ratios.

			p (volume fraction index)
			0	0.5	1	4	10
L_Z/h = 2		L_Z/R = 0.5
	Matsunaga [26]		0.9334	0.8213	0.7483	0.6011	0.5461
	Farid et al. [25]		0.9187	0.8013	0.7263	0.5267	0.5245
	N_r = N_z = 5		0.9342	0.8001	0.7149	0.5878	0.5133
	N_r = N_z = 7		0.9249	0.8011	0.7250	0.5783	0.5298
	N_r = N_z = 9		0.9250	0.8018	0.7253	0.5790	0.5301
	N_r = N_z = 11		0.9249	0.8017	0.7253	0.5789	0.5300
	N_r = N_z = 13		0.9250	0.8018	0.7252	0.5790	0.5301
	Matsunaga [26]	L_Z/R = 1	0.9163	0.8105	0.7411	0.5967	0.5392
	Farid et al. [25]		0.8675	0.7578	0.6875	0.5475	0.4941
	N_r = N_z = 5		0.8942	0.7531	0.6746	0.5741	0.4913
	N_r = N_z = 7		0.8851	0.7671	0.6912	0.5599	0.5074
	N_r = N_z = 9		0.8857	0.7666	0.6935	0.5531	0.5065
	N_r = N_z = 11		0.8857	0.7667	0.6934	0.5531	0.5063
	N_r = N_z = 13		0.8856	0.7667	0.6935	0.5532	0.5064
L_Z/h = 5		L_Z/R = 0.5
	Matsunaga [26]		0.2153	0.1855	0.1678	0.1413	0.1328
	Farid et al. [25]		0.2113	0.1814	0.1639	0.1367	0.1271
	N_r = N_z = 5		0.2230	0.1997	0.1542	0.1374	0.1373
	N_r = N_z = 7		0.2176	0.1823	0.1624	0.1362	0.1233
	N_r = N_z = 9		0.2130	0.1817	0.1639	0.1374	0.1296
	N_r = N_z = 11		0.2128	0.1816	0.1640	0.1377	0.1296
	N_r = N_z = 13		0.2129	0.1817	0.1640	0.1374	0.1295
	Matsunaga [26]	L_Z/R = 1	0.2239	0.1945	0.1769	0.1483	0.1385
	Farid et al. [25]		0.2164	0.1879	0.1676	0.1394	0.1286
	N_r = N_z = 5		0.2066	0.1765	0.1567	0.1476	0.1409
	N_r = N_z = 7		0.2133	0.1843	0.1688	0.1377	0.1288
	N_r = N_z = 9		0.2154	0.1848	0.1671	0.1392	0.1301
	N_r = N_z = 11		0.2155	0.1847	0.1675	0.1392	0.1299
	N_r = N_z = 13		0.2155	0.1847	0.1671	0.1392	0.1302

As another example, the convergence and accuracy of the method are investigated by evaluating the first three natural frequency parameters of the FG curved panel resting on Pasternak foundations. The non-dimensional forms of the elastic foundation coefficients are defined as K_w = k_wR/G_c and K_g = k_g/(G_cR) in which G_c is the shear modulus of elasticity of the ceramic layer. The results are prepared for different thickness-to-mean radius ratios and different values of the differential quadrature (DQ) grid points along the radial and axial directions, respectively, are shown in Table 3. Also, one can see that excellent agreement exists between the results.

Table 3.

Comparison of the first three non-dimensional natural frequency parameters of panel on an elastic foundation $(ϖ mn = ω mn h \sqrt{ρ C / E C}, P = 1, Φ = 60 \circ, N r = N z = 13)$ .

L_Z/R	h/R	K_w, K_g		$ϖ 11$	$ϖ 22$	$ϖ 33$
1	0.1	1, 0.1	Present	0.2201	0.4411	0.6462
			Farid et al.[25]	0.2200	0.4403	0.6427
		100, 10	Present	0.2241	0.4475	0.6679
			Farid et al.[25]	0.2243	0.4475	0.6681
	0.5	1, 0.1	Present	0.8043	1.8601	2.9796
			Farid et al.[25]	0.8041	1.8599	2.9796
		100, 10	Present	0.9500	1.8964	2.9956
			Farid et al.[25]	0.9503	1.8963	2.9956
2	0.1	1, 0.1	Present	0.1715	0.3430	0.5121
			Farid et al.[25]	0.1712	0.3434	0.5122
		100, 10	Present	0.174	0.3477	0.5202
			Farid et al.[25]	0.174	0.3475	0.5200
	0.5	1, 0.1	Present	0.5769	1.3408	2.1825
			Farid et al.[25]	0.5772	1.3409	2.1827
		100, 10	Present	0.7664	1.4034	2.2027
			Farid et al.[25]	0.7664	1.4037	2.2023

After demonstrating the convergence and accuracy of the present method, parametric studies for three-dimensional vibration analysis of thick FG sandwich curved panels with considering a three-parameter power-low distribution, length-to-mean radius ratio, elastic coefficients of foundation and different combinations of free, simply supported and clamped boundary conditions along the axial direction of the curved panel, are computed. The boundary conditions of the panel are specified by the letter symbols; for example, S-C-S-F denotes a curved panel with edges θ = 0 and Φ simply supported (S), edge z = 0 clamped (C), and edge z = L_z free (F).

The non-dimensional natural frequency, Winkler and shearing layer elastic coefficients are as follows

Ω mn = ω mn 10 h \sqrt{ρ c / E c}, K w = \frac{k w R}{G c}, K g = \frac{k g}{G c R}

(21)

where ρ_c, E_c and G_c represent the mass density, Young’s modulus and shear modulus of the ceramic, respectively.

The influence of the index p on the natural frequency is shown in Figures 6 and 7. According to these figures, the frequency parameter of laminated panels with rich ceramic layers is more than the natural frequency parameter of the limit cases of homogeneous layers of metal. It should be noticed that with the increase of ceramic volume fraction, the frequency parameter of the panels does not increase necessarily, so by considering suitable amounts of power-law index p $(0 \leq p \leq \infty)$ and the parameters b and c, one can get dynamic characteristics similar or better than the isotropic ceramic limit case for laminated FG curved panels.

Figure 6 shows that for p > 3 and $(0 \leq b < 1)$ , the discrepancy between the natural frequencies of the panels increases with the increase of p for different types of boundary conditions. For p < 1 and $(0 \leq b < 1)$ , increasing of parameter b does not have a significant effect on the amount of natural frequency parameter of panel.

As can be seen from Figure 7, with the increase of p, the discrepancy between the natural frequencies of the panels for different amounts of parameter c sharply decreases.

In Figure 8, the effects of variation of circumferential wave numbers (m) on the frequency parameters of S-C-S-C sandwich curved panel for different values of L_z/R ratio are demonstrated. According to Figure 8, the general behavior of the frequency parameters of sandwich panels for all L_z/R ratios is that the frequency parameters converge only in the range beyond that of the fundamental frequency parameters. This means that the effects of the L_z/R ratios are more prominent at low circumferential wave numbers, particularly those in the range before that of the fundamental frequency parameters, than at high circumferential wave numbers. It is also seen from Figure 8 that the frequency parameter decreases rapidly with the increase of the length-to-mean radius ratio (L_z/R) and then remains almost unaltered for the long cylindrical panel.

Figure 9 shows the effect of Winkler elastic coefficient on the non-dimensional natural frequencies of S-C-S-C FG layers panel for different values of shearing layer elastic coefficients. The non-dimensional natural frequency sharply increases in the most effective range of Winkler foundation stiffness (from 10³ to 10⁴) because in this range the stiffness of the panel is added to the foundation stiffness and it results in increasing the total stiffness of the structure (panel and foundation), so the frequency parameter suddenly increases. It is observed, for the large values of Winkler elastic coefficient, that the shearing layer elastic coefficient has less effect and the results become independent of it; in other words, the non-dimensional natural frequencies converge with increasing Winkler foundation stiffness. It should be noted that this behavior is also observed at other boundary conditions; but, for the sale of brevity, we consider only this type of boundary condition here.

The influence of shearing layer elastic coefficient on the fundamental frequency parameters is shown in Figure 10. One can see that the Winkler elastic coefficient has little effect on the fundamental frequency parameters at different values of shearing layer elastic coefficient. This behavior is also observed at other boundary conditions; but, for the sale of brevity, we consider only this type of the boundary condition.

In the following discussion, we turn our attention to investigation of the distributions of normalized modal radial (η_r = (r−R)/h), circumferential and axial (T_z = z/L_z) displacements associated with the natural frequencies for FG curved panels. The modal displacement (u_r, u_θ, u_z) of the FG curved panels in vibration is normalized by dividing its maximum absolute value, denoted by (U_rr, U _θθ , U_zz). The through-the-length variation of the mode shape of the non-dimensional radial displacement for S-C-S-S and S-C-S-C curved panels with respect to different values of mid-radius to thickness ratio (R/h) is plotted in Figures 11 and 12 corresponding to the first vibrating mode (m,s) = (1,1).

Figures 13 and 14 present the effect of the R/h ratio on the mode shape of the axial displacement of S-F-S-C and S-C-S-C curved panel, respectively. The second and third mode shape of axial displacement of S-C-S-C curved panel are also depicted in Figures 15 and 16.

Figures 17 to 19 present the effect of R/h ratio on the first three mode shapes of circumferential displacement of S-C-S-C curved panel resting on a two-parameter elastic foundation.

Conclusion remarks

In this research work, free vibration and vibrational displacements analysis of a thick finite FG layers panel resting on a two-parameter elastic foundation are investigated based on three-dimensional theory of elasticity. The elastic foundation is considered as a Pasternak model with adding a shear layer to the Winkler model. Three complicated equations of motion for the curved panel under consideration are semi-analytically solved by two-dimensional generalized differential quadrature (2D GDQ) method. Using the 2D GDQ method along the radial and axial directions allows one to deal with curved panel with arbitrary thickness distribution of material properties and also to implement the effects of the elastic foundations as a boundary condition on the lower surface of the curved panel efficiently and in an exact manner. The material properties vary continuously through the layers’ thickness according to a three-parameter power-low distribution. It is assumed that the inner surfaces of the FG sheets are metal rich, while the outer surfaces of the layers can be metal rich, ceramic rich or made of a mixture of two constituents. The effects of different geometrical parameters, the elastic foundation parameters, different profiles of fiber volume fraction and three parameters of power-law distribution on the vibration characteristics of the laminated curved panels are investigated. From this study, some conclusions can be made:

The non-dimensional natural frequency parameter sharply increases in the most effective range of Winkler foundation stiffness (from 10³ to 10⁴) because in this range the stiffness of the panel is added to the foundation stiffness and it results in increasing the total stiffness of the structure (panel and foundation), so the frequency parameter suddenly increases.

It is observed, for the large values of Winkler elastic coefficient, that the shearing layer elastic coefficient has less effect and the results become independent of it; in other words, the non-dimensional natural frequencies converge with increasing Winkler foundation stiffness.

Results show that the frequency parameter of laminated panels with ceramic layers rich is more than the natural frequency parameter of the limit cases of homogeneous layers of metal. It should be noticed that with the increase of ceramic volume fraction, the frequency parameter of the panels does not increase necessarily, so by considering suitable amounts of power-law index p $(0 \leq p \leq \infty)$ and the parameters b and c, one can get dynamic characteristics similar or better than the isotropic ceramic limit case for laminated FG curved panels.

It is observed that for p > 3 and $(0 \leq b < 1)$ , the discrepancy between the natural frequencies of the panels, increase with the increase of p for different types of boundary conditions. For p < 1 and $(0 \leq b < 1)$ , increasing of parameter b does not have significant effect on the amount of natural frequency parameter of panel.

It can be found that the general behavior of the frequency parameters of sandwich panels for all L_z/R ratios is that the frequency parameters converge only in the range beyond that of the fundamental frequency parameters. This means that the effects of the L_z/R ratios are more prominent at low circumferential wave numbers, particularly those in the range before that of the fundamental frequency parameters, than at high circumferential wave numbers.

It is shown that the variation of Winkler elastic coefficient has little effect on the fundamental frequency parameters at different values of shearing layer elastic coefficient. It is clear that in all cases, with increasing the shearing layer elastic coefficient of the foundation, the frequency parameters increase to some limit values. It is observed for the large values of shearing layer elastic coefficient; the results become independent of it.

Footnotes

Declaration of Conflicting Interests

The author(s) declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.

Funding

The author(s) received no financial support for the research, authorship, and/or publication of this article.

Appendix 1

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