Closed-form Solution for Freely Vibrating Functionally Graded Thick Doubly Curved Panel-A New Generic Approach

Abstract

Today, double curvature shell panels are the main parts of each design because their geometrical characteristics provide high strength to weight ratio, aerodynamic form and beauty for the structures such as boats, submarines, automobiles and buildings. Also, functionally graded materials which present multiple properties such as high mechanical and heat resistant, simultaneously, have attracted designers. So, as the first step of any dynamic analysis, this paper concentrates on presenting a high precision and reliable method for free vibration analysis of functionally graded doubly curved shell panels. To this end, panel is modeled based on third order shear deformation theory and both of the Donnell and Sanders strain-displacement relations. A new set of potential functions and auxiliary variables are proposed to present an exact Levy-type close-form solution for vibrating FG panel. The validity and accuracy of present method are confirmed by comparing results with literature and finite element method. Also, effect of various parameters on natural frequencies are studied which are helpful for designers.

Keywords:
Free vibration; exact solution; doubly curved panel; functionally graded material; third order displacement field

1 INTRODUCTION

Growing need of humans to optimal use of material, energy, and time forces engineers to combine multiple useful engineering ideas in their designs. These considerations lead to design of complex structures. One of these ideas is to construct a material with variable properties between its top and bottom layers. The main goal of this invention was to combine heat resistant of ceramic with mechanical strength of steel in order to use in aircraft and spacecraft structures. Also, continuous variation of mechanical properties overcome delamination problems in composites. This material was produced in the national aerospace agency of Japan which is known today as functionally graded material (Hirai et al., 1988). Functionally graded materials (FGMs) are non-homogeneous isotropic, orthotropic and even anisotropic materials whose mechanical properties vary through one, two or three directions. Applications of these materials in different geometries such as beam, plate, cylinder, cylindrical panels, and doubly curved panels have been investigated in the past years. To this end, various geometrical theories such as (Love, 1927), (Donnell, 1934), (Sanders, 1959), (Novozhilov, 1959), (Flugge, 1962), and etc have been used. Also, in order to improve results accuracy, different displacement fields such as classical, first order shear deformation (Mindlin, 1951;Reissner, 1945), and third order shear deformation (Reddy and Liu, 1985) theories have been applied in models. Among the numerous research paper published in this area, papers presented by (Wu et al., 1998), (Singh, 1999), (Messina, 2003), (Chaudhuri et al., 2005), (Redekop, 2006), (Biglari and Jafari, 2010), and (Fazzolari and Carrera, 2013) are examples of papers which studied vibrational behavior of composite structures, especially doubly curved panels. Also, Papers provided by (Zahedinejad et al., 2010), (Alibeigloo and Chen, 2010), (Vel, 2010), Hashemi et al. (2012), Kiani et al. (Kiani et al., 2012;Kiani et al., 2013), Civalek et al. (Akgöz and Civalek, 2013;Civalek, 2005), (Su et al., 2014), and (Sayyaadi et al., 2014) are good examples of researches which used various geometrical theories to analyze free vibration of FG structures such as cylindrical and doubly curved panels. These analysis shows that higher order displacement fields gives more accurate results respect to the lower ones but higher ones are computationally expensive. Additionally, Tornabene et al. (Tornabene, 2009,2011a,b;Tornabene et al., 2014;Tornabene et al., 2012;Tornabene et al., 2013;Viola et al., 2013) have developed generalized differential quadrature (GDQ) to analyzed various FG structures. Furthermore, Zhang and Liew with their coworkers have applied mesh-less methods to analyze different FG shells (Lei et al., 2014;Liew et al., 2014;Liew et al., 2011;Zhang et al., 2014a,b;Zhang et al., 2014c;Zhu et al., 2014).

Our literature survey shows that all related papers used first or third order shear deformation theories. But they either preferred numerical methods such as Rayleigh-Ritz, GDQ, and FE methods or presented Navier's solution for FG doubly curved panels. Indeed, there is no paper which presents exact solution for third order FG doubly curved shell panels. Therefore, in this paper our success in eliminating this defect is presented. In the present paper, a new exact closed-from solution is proposed for freely vibrating functionally graded doubly curved panel. To this end, equations of motion are derived by combining third order shear deformation theory with Donnell and Sanders strain fields. A new set of potential functions which satisfies Levy-type boundary conditions are applied to the equations of motion to decouple partial differential equations. By applying Levy-type boundary conditions to the equations, natural frequencies of FG doubly curved panel are found. Validity and accuracy of the present method are verified by comparing them with literature and finite element results. Also, the effect of different geometrical and material parameters changes on natural frequencies are investigated in various figures.

2 MATHEMATICAL FORMULATIONS

2.1 Material and geometrical assumptions

Consider a doubly curved shallow shell with rectangular base (as shown inFigure 1). A curvilinear coordinate system (x ₁, x ₂, x ₃) is used, with the x ₁ and x ₂ plane coinciding with the middle surface of the shell in its initial, undeformed configuration and the x ₃ axis normal to it. The principal radii of curvature R₁ and R₂ are assumed to be constant along the shell. a and b are the curvilinear lengths of the edges and h is the shell thickness. The displacements of an arbitrary point of coordinates (x ₁, x ₂) on the middle surface of the shell are denoted by u, v and w, in the x ₁, x ₂ and x ₃ directions, respectively. The shell has two opposite edges simply supported along x ₂ axis (i.e. along the edges x ₁ = 0 and x ₁ = a) while the other two edges may be free, simply supported or clamped.

Figure 1
Geometry of a FG doubly curved shell panel

The material properties of the shell are assumed to vary through the thickness according to a power-law distribution of the volume fractions of the two materials as ceramic and metal. Poisson's ratio n is assumed to be constant and is taken as 0.3. Young's modulus and mass density are assumed to vary continuously through the shell thickness as

In which the subscripts m and c represent the metallic and ceramic constituents, respectively, and the volume fraction V_f may be given by

where p is the power law index and takes only positive values. Typical values for metal and ceramics used in the FG shell are listed inTable 1.

Table 1
Material properties of the used FG shell.

Material	Properties
	E (GPa)	ν	ρ (Kg/m3)
Aluminum (Al)	70	0.3	2700
Alumina (Al2O3)	380	0.3	3800
Zirconia (ZrO2)	200	0.3	5700

2.2 Constitutive Relations Based on the TSDT

According to the third order shear deformation shell theories of Donnell and Sanders (TSDT) in which the in-plane displacements are expanded as cubic functions of the thickness coordinate and the transverse deflection is constant through the shell thickness, the displacement field is used as follows (Reddy, 2004)

where ψ₁, ψ₂ denote the rotations of normal to mid-plane about the x ₂ and x ₁ axes, respectively and t is the time. The strain-displacement relations are given as (Reddy, 2004)

where

where c ₁ = 0 and c ₁ = 1 correspond to Donnell and Sanders theories, respectively.

Based on Hooke's law and using Eqs. (3-5), the stress-displacement relations are defined as

where a comma followed by 1 and 2 denotes the differentiation with respect to x ₁ and x ₂ coordinates.

2.3 Non-dimensional Equations of Motion

Based on Hamilton's principle, the partial differential equations of motion of vibrating FG doubly curved shell panels based on TSDT are calculated by applying part by part integration technique:

where the overdot denotes time derivative. The stress resultants N_i,, Pi (i = 1,2,6) ,(i = 1,2) and the inertia asIi I_i ( i = 1, 7), J ₃, J ₅,,, Îi, Îii (i = 4 ) are defined as

For brevity, the non-dimensional parameters are introduced as

where ω is the natural frequency of the shell (rad/s).

Assuming that the five coordinates ψ1, ψ2, u, v and w vary harmonically with respect to the time variable t (for the free vibration analysis) and using the non-dimensional parameters defined in Eqs. (10), Eqs. (7a-c) can be expressed as

Where

And

2.4 Exact Solution Procedure

The eight auxiliary functions f1, f2, f3, f4 and φ1, φ2, φ3, φ4 are introduced as

Using these auxiliary functions, the equations of motion of Eqs. (11a-e) may now be expressed as

Where yi (i = 1 - 28) are constants governed easily by material properties, non-dimensional frequency β and structural geometry.

2.4.1 Obtaining of w

Elimination of f1, f2, f3, f4 and then φ1, φ2, φ3, φ4 between Eqs. (16a-e) and simplifying terms, the following equation is obtained

Where

and ai (i = 1, 2, 3,...27, 28) are constants and will not be expressed due to the space limits.

Since the Levy-type FG doubly curved shell panel will be used to analyze the free vibration of the shell, the displacement components (,,,) will be represented as

Where

Substituting Eq. (19e) into Eq. (17) gives

where the W1, W2, W3, W4, W5 and W6 are defined as

and,,,,andare the roots of the following sixth-order equation

The coefficients b_i (i = 1,2,3,4,5,6,7) - are given as

Using of Eqs. (16a-e), the introduced functions φ₁, φ₂, φ₃, φ₄ can be given by

where the constants of d_i, e_i, g_i h_i, i = (1,2,3,...20,21) can be obtained using Eqs. (16a-e).

2.4.2 Solution of u, υ, ψ1 and ψ2

Substituting Eqs. (19a-e), (21), (22a-e) and (25a-d) into Eqs. (16a-e), the auxiliary functions f ₁, f ₂, f ₃ and , f ₄ may be then expressed as

where the coefficients of C_i (i = 1,2,3,...23,24) may be derived by submitting Eqs. (26a-d) into Eqs. (16a-d) as well as using Eqs. (22a-e).

Finally, the exact closed-form displacement field of the shell according to the TSDT is obtained by substituting Eqs. (26a-d) into Eqs. (14a-d).

2.4.3 Solution of W 1, W 2, W 3, W 4, W 5 and W 6

According to Eqs. (22a-e), the solutions for the W_i (i = 1,2,3,.4, 5, 6) can be written as

Where

2.5 The Classical Boundary Conditions

The classical boundary conditions of the TSDT may be obtained for an edge parallel to, for example, X ₁-axis as

By changing subscripts 1 and 2 in Eqs.(29a-c), the different boundary conditions are obtained for the edges X ₁ = 0 and X ₁ = 1. The present solution procedure has been applied in our previous paper for other geometry which can help researchers in understanding and developing this method (Hosseini-Hashemi and Fadaee, 2011).

3 RESULTS AND DISCUSSION

3.1 Comparison study

In this section, in order to confirm validity and accuracy of present method two tables are presented, firstly. Next, effect of geometrical parameters changes on natural frequencies are studied in various plots. Material properties used in the tables and figures are indicated in table 1. It should be mentioned that in tables natural frequencies are compared with traditional finite element software results. In this case, panels are modeled by dividing its cross section to subsections with constant properties, firstly. Number of layers depends on the thickness and material power index. Next, models are meshed by using a 3D brick element containing 20 nodes with three degree of freedom in each node. Convergency analysis is done by increasing number of layers and number of elements. Finally, optimum number of elements is selected for the analysis.

The accuracy of Donnell and Sanders models are indicated as a percentage as follow:

Also, for simplicity the letters S, C and F are used as symbols of simply, clamped and free boundary conditions, respectively.

First six natural frequencies of FG cylindrical shell panel are tabulated intable 2for different thickness ratios as τ = 0.075,0.1,0.125. Donnell, Sanders and FEM methods are used to compute frequencies under SCSC boundary conditions.

Table 2
Comparison of first six natural frequencies (Hz) for SCSC functionally graded (Al/Al₂O₃) cylindrical shell panels (δ₁ = 6/π, δ₂ = ∞, η = 1, R₁ = 1(m), p = 1).

τ = h/a	Method	Mode sequence
		1	2	3	4	5	6
0.075	Present (TSDTD)*	1588.7	2679.6	3508.5	4540.6	4903.4	6052.7
	Present (TSDTS)**	1583.2	2675.3	3503.1	4524.6	4874.1	6048.9
	FEM (3D)	1580.9	2673.6	3501.2	4520.1	4869.7	6043.1
	Diff (%) (TSDTD)	-0.493	-0.224	-0.208	-0.453	-0.692	-0.159
	Diff (%) (TSDTS)	-0.145	-0.064	-0.054	-0.099	-0.090	-0.096
0.1	Present (TSDTD)	1970.2	3422.5	4353.4	5664.7	6170.0	7402.8
	Present (TSDTS)	1965.1	3414.2	4345.5	5641.8	6128.0	7397.7
	FEM (3D)	1962.2	3412.0	4340.1	5640.1	6122.1	7390.1
	Diff (%) (TSDTD)	-0.408	-0.308	-0.306	-0.436	-0.782	-0.172
	Diff (%) (TSDTS)	-0.148	-0.064	-0.124	-0.030	-0.096	-0.103
0.125	Present (TSDTD)	2328.8	4160.6	5079.2	6607.2	7241.5	8476.6
	Present (TSDTS)	2318.7	4123.9	5068.9	6577.3	7187.4	8470.6
	FEM (3D)	2313.3	4061.3	5061.2	6570.0	7180.1	8463.2
	Diff (%) (TSDTD)	-0.670	-2.445	-0.356	-0.566	-0.855	-0.158
Diff (%) (TSDTS)	-0.233	-1.541	-0.152	-0.111	-0.102	-0.087

• *Donnell third order shear deformation theory; **Sanders third order shear deformation theory

According totable 2, the present method captures natural frequencies, precisely. Also, both of the Donnell and Sanders theories have good agreement with the finite element results but Sanders one is more accurate than Donnell one. It is because of curvature which is considered in the strain relations in the Sanders theory. Furthermore,table 2shows that increasing the thickness ratio τ increases the natural frequencies. Increasing the panel rigidity is the main reason of such behavior.

As another comparison, intable 3, fundamental natural frequency parametersof a simply-supported doubly curved FG panel are compared with literature (Farid et al., 2010) for different deep ratio δ₁ = 5,10 and power index p. panel geometrical properties are considered as τ = 0.1, η = 1, δ₂ = ∞, D_m = E_mh ³/(12(1 - ν²⁾). Reference (Farid et al., 2010) solved three dimensional model of FG doubly curved panel by using differential quadrature method.Table 3confirms validity and accuracy of present method for computing FG doubly curved panel frequencies. Additionally,table 3shows that increasing power law index reduces natural frequencies. This behavior depends on the arrangement of inner and outer materials of the panel. Fortable 3aluminum/alumina composition is considered as the outer and inner layers. So, increasing index p grows aluminum effect on the natural frequencies which has lower rigidity respect to the alumina.

Table 3

3.2 Effect of Side Ratio δ on the Fundamental Natural Frequency

Variations of fundamental natural frequency parameter β against side ratio changes are plotted inFigure 2. A FG square doubly curved panel with material properties as Al / Al ₂ O ₃ and geometrical properties as t = 0.15,0.25,p = 1 are considered. Figures are plotted for three boundary conditions as SCSC, SSSS, and SFSF.Figures 2aandbshow that increasing side ratio δ₁ reduces fundamental natural frequency. ButFigure 2cshows reverse behavior which increasing side ratio δ₁ raises fundamental natural frequency. Also, increasing side ratio δ₁ and δ₂ has similar effect on the fundamental natural frequency. This is attributed to the fact that the geometrical (essential) boundary conditions such as the displacements u, v, w and the slopes ψ₁ and ψ₂, which are applied in the simply-supported and clamped edges, have direct relation with the curvature of the panel. So, increasing the side ratios δ₁ which decreases curvature of the panel reduces effect of geometrical boundary conditions. Indeed, geometrical boundary conditions have significant influences on the vibration of the doubly curved shell panel respect to the natural boundary conditions such as the stress resultants N_i , M_i (i=1,2,6and Q_i (i=1,2

In addition,Figures 2aandbshow that increasing thickness ratio τ reduces effect of side ratio increment on the fundamental natural frequency. In this case, increasing the thickness ratio τ reduce effect of curvature on the natural frequency. So, for thicker panels, changing side ratios δ₂ has not significant effect on the fundamental natural frequency.

Figure 2
Variation of the fundamental frequency parameters β of Al/Al₂O₃ square doubly curved shell panel against radius to side ratios δ₁ and δ₂ for (a) SCSC (b) SSSS (c) SFSF boundary conditions when τ = 0.15, 0.25 and p = 1.

3.3 Effect of Aspect Ratio η on the Fundamental Natural Frequency

Figure 3shows effect of aspect ratio η on the fundamental natural frequency parameter of FG doubly curved panel. This effect is studied for three boundary conditions as SCSC, SSSS, SFSF and two material compositions as Al / Al ₂ O ₃ and Al / ZrO ₂. By increasing aspect ratio η, panel transforms into strip which edges with simply-supported conditions place on longer side of strip. So, as shown inFigure 3a, increasing aspect ratio decreases effect of clamped edges which it reduce natural frequency. Also, inFigure 3b, increasing aspect ratio increases flexibility which it reduce natural frequency. But inFigure 3c, natural frequency increases because increasing aspect ratio increases ratio between lengths of simply-supported and free edges. Therefore, effect of aspect ratio on the panel rigidity and natural frequency depends on type of boundary conditions applied in the edges.

Figure 3
Variation of the fundamental frequency parameters β of Al/Al₂O₃ and Al/ZrO₂ doubly curved shell panels against aspect ratios η for (a) SCSC (b) SSSS (c) SFSF boundary conditions when τ = 0.2, δ₁ = 5, δ₂ = 10 and p = 1.

3.4 Effect of power law index p on natural frequency

Figure 4shows effect of power law index p on the first and second fundamental natural frequencies parameters β for a square FG doubly curved panel. Two material properties and three boundary conditions are considered as indicated in figures. It can be concluded forfigure 4that increasing power law index decreases natural frequencies for such material compositions, generally. Variations start from frequencies of an isotropic material such as aluminum and reach to frequencies of an isotropic material with properties of Al / Al ₂ O ₃ and Al / ZrO ₂. Furthermore, frequency variations against power law index are sharper for lower values than higher values.

Figure 4
Effect of the power law index p on the first and second modes of FG square doubly curved shell panel for (a) SCSC (b) SSSS (c) SFSF boundary conditions when τ = 0.2, δ₁ = 4, δ₂ = 8.

4 CONCLUSIONS

According to the Donnell and Sanders third order shear deformation type theories, a new exact close-form solution is presented to analyze free vibration of functionally graded doubly curved panel with Levy-type boundary condition. Validity and accuracy of the present method are confirmed by comparing its results with literature and finite element results. It is observed that Sanders theory has more accuracy respect to the Donnell theory. Also, effects of various geometrical and material parameters such as thickness, aspect ratio, and side ratio on natural frequency of FG panel are investigated. It is observed that effects of side ratio and aspect ratio on natural frequency depend on type of boundary conditions of the edges. In addition, it is shown that effect of power law index on natural frequencies is dependent to the material composition in the inner and outer layer of FG panel.

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Publication Dates

• Publication in this collection
  Sept 2015

History

• Received
  15 Sept 2014
• Reviewed
  20 Nov 2014
• Accepted
  21 Nov 2014