I investigate the vibration and buckling analysis of functionally graded material (FGM) structures, using a modified 8-node shell element. The properties of FGM vary continuously through the thickness direction according to the volume fraction of constituents defined by sigmoid function. The modified 8-ANS shell element has been employed to study the effect of power law index on dynamic analysis of FGM plates with various boundary conditions and buckling analysis under combined loads, and interaction curves of FGM plates are carried out. To overcome shear and membrane locking problems, the assumed natural strain method is employed. In order to validate and compare the finite element numerical solutions, the reference results of plates based on Navier’s method, the series solutions of sigmoid FGM (S-FGM) plates are compared. Results of the present study show good agreement with the reference results. The solutions of vibration and buckling analysis are numerically illustrated in a number of tables and figures to show the influence of power law index, side-to-thickness ratio, aspect ratio, types of loads, and boundary conditions in FGM structures. This work is relevant to the simulation of wing surfaces, aircrafts, and box structures under various boundary conditions and loadings.
Kongju National University1. Introduction
Functionally graded material (FGM) is a special kind of composites in which the material properties vary continuously and smoothly from one surface to the other. One of the main advantages of FGM is that it mitigates acute stress concentrations and singularities at intersections between interfaces usually presented in laminated composites. Chung and Chi [1] proposed a sigmoid FGM, which is composed of two power law functions to define a new volume fraction and indicated that the use of a sigmoid FGM can significantly reduce the stress intensity factors of a cracked body. Recent work on the bending, vibration, buckling, and transient analysis of FGM plates can be founded in Han et al. [2, 3] and Jung and Han [4]. Recently, the works on FGM and shear deformation theories with the thickness stretching effect are employed and developed by researchers (Belabed et al. [5], Hamidi et al. [6], Lee et al. [7], and Han et al. [8]).
It should be noted that they only investigated structural behaviors of simply supported FGM plates. Thus, needs exist for the development of shell finite element which is simple to use for vibration and buckling analysis FGM plates with arbitrary boundary conditions.
When compressive loads are applied onto most structures including FGM plates, they tend to buckle or are subjected to dynamic loads during their operation. Understanding the natural frequency and buckling behavior is an important issue from design perspective. Consequently, numerous studies on vibration and buckling of various plates can be found in literatures. For proper use of FGM plates as various structural components, their dynamic and stability response should be studied. To the best of the author’s knowledge, there are no solutions for structural stability response of FGM plates under combined compressive, tensile, and shear loads based on shear deformation theory of plate.
Bucalem and Bathe [9] improved the MITC8 shell elements [10] and concluded that while it performed quite effectively in some cases, in a few analyses the element presented a very stiff behavior rendering. In 8-node shell element [11, 12], the keeping of locking phenomena was found to continue through numerical solutions on the standard test problem of Macneal and Harder [13]. In order to improve the 8-node ANS shell element, a new combination of sampling points is adopted. Recently, Han et al. [14] presented modified 8-ANS shell element using the new interpolation functions and combination of sampling points for the assumed natural strain.
However, a few literatures have been found on the dynamic analysis of FGM plates with various boundary conditions and structural stability analysis under combined compressive, tensile, and shear loads. In the present work modified 8-ANS shell element has been employed to study the effect of power law index on dynamic analysis of FGM plates with various boundary conditions and buckling analysis under combined compressive, tensile, and shear loads. To validate the present 8-ANS shell element models, the numerical examples are studied and compared with those results from the references. The solutions of vibration and buckling analysis are numerically illustrated in a number of tables and figures to show the influence of power law index, side-to-thickness ratio, aspect ratio, types of loads, and boundary conditions in FGM structures.
2. Modified 8-ANS Finite Element2.1. Kinematics of Shell
The displacement u- of an arbitrary point of the shell (see Figure 1) for the first-order shear deformation theory can be expressed(1)u-ξα=uξα+ξ3φξα,where φ is vector of rotation at the midsurface of shell.
Kinematics of the first-order shear deformation theory.
A three-dimensional Green’s strain tensor in the linear case (infinitesimal strain theory) is given by (2)2Eij=u-,i·gj+u-,j·gi,where a comma ,i=∂/∂ξi is partial differentiation and gi is a triad of base vectors for the spatial coordinates ξi at the surfaces (ξ3 = const.) parallel to the midsurface of shell. If the displacement equation (1) is substituted into (2), the strain-displacement relations are obtained. From these strain-displacement relations, kinematics in different curvilinear coordinates can be acquired and expressed through the physical components in the matrix form(3)E=BU,where U are physical components of displacement u and rotation φ as follows:(4)UT=u1,u2,u3,ϕ1,ϕ2,ϕ3.
The shell theory presented above is the so-called first-order shear deformation theory with six degrees of freedom.
2.2. Various Enhanced Strain Interpolation Patterns
In this study, the ordinary 8 nodes of Lagrangian displacement interpolations are used and the various combinations of assumed natural strain interpolation functions are employed for the very efficient 8-node shell element. Figure 2 shows various patterns of sampling points that can be used for membrane, in-plane shear, and out-of-plane shear strain interpolations for the new 8-ANS finite element. Based on Figure 2, the β pattern is used for membrane and the δ pattern and γ6∗ pattern are used for in-plane and out-of-plane shear, respectively. The interpolation functions by Polit et al. [16] are used in the γ6∗ patterns. In the γ6∗ patterns, the strain component of center point is replaced by the mean of the components at points S1 and S2 (Bathe and Dvorkin, [10]).
Four possible patterns of sampling points for 8-node ANS shell element.
Pattern β
Pattern δ
Pattern γ6∗
3. Material Properties of the FGM
An FGM can be defined by the variation in the volume fractions. In this paper, the sigmoid function is used for FGM structures. The volume fraction using two power law functions which confirm smooth distribution of stresses is defined by(5a)Vf1t=1-12h/2-th/2pfor0≤t≤h2,(5b)Vf2t=12h/2+th/2pfor-h2≤t≤0,where subscripts 1 and 2 represent the two materials used and p is the power law index, which indicates the material variation profile through the thickness. The material properties of the S-FGM using the rule of mixture can be expressed as follows:(6a)Ht=Vf1tH1+1-Vf1tH2for0≤t≤h2,(6b)Ht=Vf2tH1+1-Vf2tH2for-h2≤t≤0.
4. Equilibrium Equation
By using virtual work principle, the equilibrium equation is obtained based on the membrane (N), bending (M), and transverse shear resultant forces (Q) as follows:(7)∫δEαβmTN+δEαβbTM+δEα3sTQdA≡δuTKLu=∫f·δudV,where Eαβm, Eαβb, and Eα3s are membrane, bending, and transverse shear strain components, KL is the linear stiffness matrix, and f is the body force.
5. Buckling and Vibration Analysis
When the equation is employed to estimate buckling loads, the stability condition may be simplified by(8)KLua+λcrGua=0,where ua is the vector of the nodal value of the displacements, λcr is the buckling load parameter and denotes the proportional increase in load needed to reach neutral equilibrium, and G is the geometric stiffness matrix. Applying to the structure a reference loading Nref and carrying out a generalized linear static analysis, (8) represents the standard eigenvalue problem. The lowest eigenvalue λcr in (8) is associated with buckling load. Therefore, the buckling load can be obtained by(9)Ncr=λcrNref.
The consistent mass is used to formulate the mass matrices for the FGM shell element. The mass matrix is determined using interpolation functions as follows: (10)M=∫VρNTNdV,where N is a matrix of shape functions.
Unlike (8), the governing equations of motion for free vibration analysis are of the form(11)KLua+Mu¨a=0,where the superposed dot denotes differentiation with respect to time.
6. Numerical Results6.1. Patch Test
Firstly, the patch tests proposed by Simo et al. [15] are investigated. In Figure 3, the boundary conditions and loading types are presented, simultaneously. The normalized solutions of nodal displacements on the right edges are shown in Table 1. The nondimensional form is expressed as follows: (12)Normalizedsolution=PresentsolutionReferencesolution.
Results of patch test under bending, shear, and tension.
Patch tests
Bending
Shear
Tension
Reference solutions
θy=MLEI=0.12×10-4
w=6SL5GA=0.312×10-5
u=TLEA=1.0×10-6
Normalized solutions
1.000
1.000
0.992
Mesh for patch test (Simo et al. [15]). Length of the square L=10; Young’s modulus E=1.0×107; Poisson’s ratio ν=0.3; and thickness h=1.0 and boundary displacement conditions for patch tests.
To validate the present 8-ANS finite element with FGM, a sigmoid FGM plate with geometrical properties is shown in Figure 4. The material properties are given by (13)E1=151×109Pa,ρ1=3000kg/m3,E2=70×109Pa,ρ2=2707kg/m3,ν1=ν2=0.3,where E1, ρ1, ν1 and E2, ρ2, ν2 express the property of the top and bottom faces of the plate, respectively. Equation (13) is used in computing the numerical values of all cases.
Geometry of FGM plates.
The nondimensional form of the results is defined by(14)ϖ-=ϖa2hρ1E2.
Table 2 shows the nondimensional natural frequency of S-FGM simply supported plates for convergence test. It is noticed that present 8-ANS finite element shows an excellent agreement to the result by analytical solution.
Normalized nondimensional natural frequency of S-FGM plate (power law index: p=10).
Nodes per side
4-node shell element (see [2])b
Ratio ([2]/exact)
Present
Ratio (present/exact)
5
8.076
1.105
7.717
1.056
9
7.517
1.029
7.351
1.006
17
7.366
1.008
7.323
1.002
33
7.329
1.003
—
—
Analytical solutiona
7.307
—
—
—
aResult is computed using Navier’s method with first-order shear deformation theory, independently.
bResults are computed using the quasi-conforming 4-ANS finite element, independently.
It is shown that the natural frequency of pure metal plate is smaller than that of pure ceramic plate in Table 3. The natural frequencies of the functionally graded material plates are intermediate to that of the metal and ceramic plates. Table 3 shows that numerical results of vibration analysis are reduced by increasing the power law p.
Nondimensional natural frequency of simply supported FGM plates (a/h=100).
Material parameter (p)
Navier solution
Mode number
1
2
3
Ref. [2]
Present
Ref. [2]
Present
Ref. [2]
Present
Pure ceramic
8.992a
9.041a
8.772
22.705a
21.940
22.705a
21.940
p=1
7.518
7.555
7.526
18.992
18.819
18.993
18.820
p=2
7.419
7.457
7.430
18.745
18.575
18.747
18.576
p=5
7.333
7.373
7.348
18.533
18.365
18.535
18.367
p=10
7.307
7.348
7.323
18.470
18.302
18.472
18.304
Pure metal
6.123a
6.148a
6.287
15.459a
15.726
15.459a
15.726
aResults are calculated by ρ=(ρ1+ρ2)/2.
Table 4 shows the numerical results of FGM plate for which p=10. In this example, the natural frequency is normalized with respect to the plate width a, thickness h, density ρ1, and elastic modulus E2 for various rectangular plate aspect ratios. As the plate aspect ratio increases, the natural frequency reduces and approaches 3.69.
Nondimensional fundamental frequency of simply supported FGM plates with various aspect ratio (p=10).
Solutions
Aspect ratio (b/a)
0.5
2.0
5.0
10.0
Navier solution
18.258
4.568
3.800
3.691
Ref. [2]
18.346
4.593
3.817
3.706
Present
18.276
4.577
3.803
3.692
6.2.2. FGM Plate with Arbitrary Edges
For convenience, a four-letter notation is used to describe the boundary conditions of the edges (see Figure 5). For example, CFSF indicates that first edge is clamped (C), second edge is free (F), third edge is simply supported (S), and the last is free (F). The natural frequencies of FGM CFFF plates are investigated and presented in Table 5. The results are expressed in the nondimensional form using (15). Numerical results show that the natural frequencies are reduced by increasing the power law index p. The results also confirm that power law index has significant effect on the dynamic response of FGM plates:(15)ϖ-=ϖa2hρ1E2×10.
Nondimensional natural frequency of FGM CFFF plates (a/h=100).
Material parameter (p)
Mode number
1
2
3
4
Ref. [2]
Present
Ref. [2]
Present
Ref. [2]
Present
Ref. [2]
Present
Pure ceramic
15.814a
15.430
38.767a
37.780
97.148a
94.570
124.28a
120.80
p=1
13.222
13.228
32.415
32.389
81.236
81.074
103.92
103.56
p=2
13.047
13.052
31.986
31.960
80.166
80.000
102.54
102.19
p=5
12.896
12.901
31.617
31.590
79.244
79.073
101.36
101.00
p=10
12.851
12.856
31.507
31.480
78.969
78.797
101.01
100.65
Pure metal
10.767a
11.060
26.395a
27.079
66.144a
67.785
84.615a
86.584
aResults of [2] are calculated by ρ=(ρ1+ρ2)/2.
Geometry of FGM cantilever plates.
In Table 6, the natural frequencies of FGM plates with arbitrary boundary conditions are presented. Four arbitrary values of the power law index p are examined. As expected, results show that the natural frequencies are reduced by increasing the power law index p.
Nondimensional natural frequency of FGM plates (a/h=100).
Material parameter (p)
Boundary conditions
CFFF
SSFF
SSSS
CCFF
CCSS
Pure ceramic
15.430
54.302
87.718
98.502
128.59
p=1
13.228
46.643
75.264
84.446
110.25
p=2
13.052
46.078
74.304
83.327
108.79
p=5
12.901
45.591
73.478
82.364
107.53
p=10
12.856
45.446
73.231
82.077
107.15
Pure metal
11.060
38.922
62.873
70.603
92.176
Based on present study, comprehensive results of natural frequency of FGM plates are also illustrated in Figure 7 for different boundary conditions. In each boundary condition, five different power law indices are considered. In Figure 8, two different values of side-to-thickness ratio are examined. In addition, five arbitrary values of the power law index are examined. These new results can be used for comparison with further FG plate models.
For validation, the stability analysis results of S-FGM simply supported plates (see Figure 4) using Navier’s method are compared with present 8-ANS finite element. The material properties and nondimensional form are used as shown in Section 6.2.1 and (14), respectively. It is shown that the pure ceramic plate has the largest buckling load and the pure metal plate has the smallest one in Table 7. The buckling loads of the FGM plates are intermediate to that of the metal and ceramic plates.
Nondimensional buckling loads of FGM simply supported plates N-cr=Ncrb2/E2h3.
Material parameter (p)
Navier solution
Ref. [2]
Present
Pure ceramic
7.794
7.828
7.797
p=1
5.448
5.484
5.466
p=2
5.305
5.346
5.331
p=5
5.183
5.229
5.216
p=10
5.147
5.195
5.182
Pure metal
3.613
3.629
3.615
The buckling loads versus the plate aspect ratio are presented in Table 8. There, for large plate aspect ratios (i.e., b/a≥2.0), the plate buckles into a single half wave in the x-direction. As the plate aspect ratio decreases, the plate buckles with increasing half waves in the x-direction.
Nondimensional buckling loads of FGM simply supported plates N-cr=Ncrb2/E2h3,p=10.
Solutions
Aspect ratio (b/a)
0.5
2.0
5.0
10.0
Navier solution
5.138
8.043
34.800
131.28
Ref. [2]
5.211
8.127
34.969
131.87
Present
5.165
8.110
34.859
131.37
6.3.2. FGM Cantilever Plate
In Table 9, the stability analysis results of S-FGM cantilever plates (see Figure 6) with various aspect ratio are presented. The results are presented in the nondimensional form. Numerical results show that the buckling loads are reduced by increasing the power law index p. The results also confirm that power law index has significant effect on the buckling loads of FGM cantilever plates. The stability analysis results of S-FGM cantilever plates under various loading types are investigated in Table 10. As expected, numerical results show that the buckling loads are reduced by increasing the power law index p and also confirm that loading types have very significant effect on the buckling loads of FGM cantilever plates.
Nondimensional buckling loads of FGM plates N-cr=Ncrb2/E2h3×10.
Material parameter (p)
Aspect ratio (b/a)
1
2
5
10
Ref. [2]
Present
Ref. [2]∗
Present
Ref. [2]∗
Present
Ref. [2]∗
Present
Pure ceramic
4.693
4.679
9.551
19.040
24.160
120.75
48.449
485.42
p=1
3.281
3.271
6.677
13.310
16.890
84.411
33.872
339.32
p=2
3.194
3.185
6.501
12.959
16.445
82.185
32.981
330.38
p=5
3.121
3.111
6.351
12.660
16.067
80.293
32.224
322.77
p=10
3.099
3.090
6.307
12.572
15.955
79.733
32.000
320.52
Pure metal
2.176
2.169
4.428
8.8265
11.200
55.979
22.460
225.03
∗Results of [2] are calculated by N-cr=Ncra2/E2h3b/a×10.
Nondimensional buckling loads of FGM plates N-cr=Ncrb2/E2h3×10.
Material parameter (p)
Types of combined loading
Compression
Shear + comp.
Pure shear
Shear + tension
Present
Present
Present
Present
Pure ceramic
4.679
4.279
15.135
95.116
p=1
3.271
2.991
10.580
66.495
p=2
3.185
2.912
10.302
64.745
p=5
3.111
2.845
10.065
63.257
p=10
3.090
2.826
9.9946
62.816
Pure metal
2.169
1.984
7.0163
44.094
FGM cantilever plates under combined loads.
Effect of power law index on the nondimensional natural frequency of FGM plate with arbitrary edges.
Effect of power law index on the nondimensional natural frequency of CCFF FGM plate with variation of side-to-thickness ratio.
Based on present study, comprehensive results of buckling loads of FGM plates under combined loads are also illustrated in Figures 9 and 10 for CFFF boundary conditions. The influence of in-plane load direction on the relationship between critical shear and in-plane loading is studied, when acting in combination. It is noticed that the tension may produce positive stiffness and the FGM plate becomes stronger than when it is subjected to compression.
Buckling load of FGM plates: combined compressive, tensile, and shear loading.
Buckling load of FGM plates with various power law index: combined compressive and shear loading.
In Figure 10, the natural frequencies of FGM plates under combined loading are investigated. Four arbitrary values of the power law index p are examined. As expected, results show that the buckling loads are increased by decreasing the power law index p.
7. Concluding Remarks
The natural frequency and buckling response have been studied for FGM plates. Extensive results obtained from computations refer to different loading, different geometry, different boundaries, and different power law indices. The advanced finite element analysis based on the modified 8-node ANS formulation shows the significance of various boundary conditions and loading conditions for FGM plates. From this study, a number of conclusions have been founded.
It is shown that the natural frequencies are reduced by increasing the power law index p. The results also confirm that power law index has significant effect on the dynamic response of FGM plates.
Dynamic response of FGM plates is affected by its boundary conditions. Clamped edges always produce a higher performance of the FGM plates than simply supported edges.
It is noticed that the tension may produce positive stiffness and the FGM plate becomes stronger than when it is subjected to compression. For combined shear and compressive loading the stability envelopes are symmetric about the line Nxy.
The suitable selection of sampling point used in ANS method is very important for vibration and buckling behavior of FGM plates. It is noticed that locking phenomenon occurs in the results of reference when the plates become very thin. This phenomenon may lead us to a conclusion that the suitable selection of sampling points prevents the locking problem from occurring in vibration and buckling analysis of either thick FGM plates or very thin ones.
In order to design the FGM plates under the in-plane shear loading, the present formulation and results may serve as benchmark for future guidelines and may be extended to dynamic instability analysis of various FGM structures. The numerical results of present study may serve as benchmark for future guidelines in designing FGM plates under compressive, tension, shear, and combined loading with arbitrary boundary conditions. Also, the present theory should provide engineers with the capability for the design of various FGM plates and shells.Competing Interests
The author declares that there are no competing interests regarding the publication of this paper.
Acknowledgments
This work was supported by the research grant of the Kongju National University in 2015.
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