Journal of Science and Technology 52 (2C) (2014) 240-249
FREE VIBRATION ANALYSIS OF FUNCTIONALLY GRADED
SANDWICH BEAMS BASED ON A HIGHER-ORDER SHEAR
DEFORMATION THEORY
Nguyen Ba Duy1, Nguyen Trung Kien2,*
1
Faculty of Civil Engineering, Thu Dau Mot University, 06 Tran Van On Street, Phu Hoa District,
Thu Dau Mot City, Binh Duong Province, Viet Nam
2
Faculty of Civil Engineering and Applied Mechanics, University of Technical Education Ho Chi
Minh City, 1 Vo Van Ngan Street, Thu Duc District, Ho Chi Minh City, Vietnam
*
Email:
[email protected]
Received: 13 November, 2013; Accepted for publication: 07 September, 2014
ABSTRACT
This paper presents free vibration analysis of functionally graded sandwich beams
using a higher-order shear deformation theory. The face layers of sandwich beam are assumed to
have isotropic metal-ceramic material distribution, and Young’s modulus and mass density are
assumed to vary according to power-law form in terms of the volume fractions of the
constituents. The core layer is still homogeneous and made of an isotropic metallic material.
Governing equations of motion are derived from the Hamilton’s principle. Navier-type solution
for simply-supported beams is developed to solve the problem. Numerical results are obtained
for sandwich beams with homogeneous softcore to investigate effects of the power-law index,
span-to-height ratio and ratio of layer thickness on the natural frequencies.
Keywords: functionally graded sandwich beams, free vibration, beam theory.
1. INTRODUCTION
Increase of the application of sandwich structures in aerospace, marine, civil construction
led to the development of functionally graded (FG) sandwich structures to overcome the material
discontinuity found in classical sandwich materials. With the wide application of FG sandwich
structures, understanding vibration of FG sandwich structures becomes an important task.
Thanks to the advantage that no shear correction factor is needed, the higher-order shear
deformation theory is widely used for vibration of sandwich plates. Based on this theory, though
many works on these problems are available in the open literature [1, 2, 3], only representative
samples are cited here, while detailed discussions can be found in [4]. Although there are several
studies of the behavior of FG sandwich plates, research on vibration of FG sandwich beams is a
few in number. Bhangale and Ganesan [5] studied vibration and buckling analysis of a FG
sandwich beam having constrained viscoelastic layer in thermal environment by using finite
element formulation. Amirani et al. [6] used the element free Galerkin method for free vibration
analysis of sandwich beam with FG core. Bui et al. [7] investigated transient responses and
240
Free vibration analysis of functionally graded sandwich beams based on a higher-order shear …
natural frequencies of sandwich beams with inhomogeneous FG core using a truly meshfree
radial point interpolation method. Vo et al. [8] proposed a finite element model for vibration and
buckling analysis of functionally graded sandwich beams based on a refined shear deformation
theory.
This paper aims to present free vibration analysis of functionally graded sandwich beams
using a higher-order shear deformation theory. The face layers of sandwich beam are assumed to
have isotropic metal-ceramic material distribution, and Young’s modulus and mass density are
assumed to vary according to power-law form in terms of the volume fractions of the
constituents. The core layer is still homogeneous and made of an isotropic metallic material.
Governing equations of motion are derived from the Hamilton’s principle. Navier-type solution
for simply-supported beams is developed to solve the problem. Numerical results are obtained
for sandwich beams with homogeneous softcore to investigate effects of the power-law index,
span-to-height ratio and ratio of layer thickness on the natural frequencies.
2. KINEMATICS
Consider a FG sandwich beam composed of three layers as shown in Fig. 1. The x-, y-, and
z-axes are taken along the length (L), width (b), and height (h) of the beam, respectively. The
face layers of the sandwich beam are made of an isotropic material with material properties
varying smoothly in the z-direction only. The core layer is made of an isotropic homogeneous
metallic material. The vertical positions of the bottom and top, and of the two interfaces between
the layers are denoted by h0 = − h/2 , h1 , h2, h3 = h/2, respectively. For the brevity, the ratio of
the thickness of each layer from bottom to top is denoted by the combination of three numbers,
i.e. “1-0-1”, “1-2-1”, “2-1-2”, and so on.
Figure 1. Geometry of the sandwich beam with functionally graded faces.
The displacement field of the present theory can be obtained as ([9]):
∂wb ( x, t )
∂w ( x, t )
− f ( z) s
∂x
∂x
W ( x, z , t ) = wb ( x, t ) + ws ( x, t )
U ( x, z , t ) = u ( x, t ) − z
(1a)
(1b)
where u is the axial displacement, wb and wb are the bending and shear components of transverse
displacement along the mid-plane of the beam. The non-zero strains are given by:
∈x =
∂U
=∈0x + zκ xb + f κ xs
∂x
(2a)
241
Nguyen Ba Duy, Nguyen Trung Kien
γ xz =
∂W ∂U
+
= gγ xz0
∂x
∂z
(2b)
where
4 z
f = z
3 h
2
df
z
g =1−
= 1− 4
dx
h
(3a)
2
(3b)
and ∈0x , γ xz0 , κ xb and κ xs are the axial strain, shear strain and curvatures in the beam,
respectively, defined as:
∈0x = u, x
(4a)
γ xz0 = ws , x
(4b)
κ xb = −wb, xx
(4c)
κ = −ws , xx
(4d)
s
x
where the comma indicates the differentiation with respect to the subscript that follows.
3. VARIATIONAL FORMULATION
In order to derive the equations of motion, Hamilton’s principle is used:
t2
δ ∫ (ℜ − ℧)dt = 0
t1
(5)
where ℧ and ℜ denote the strain energy and kinetic energy, respectively. The variation of the
strain energy can be stated as:
l
3 hn ( n )
(σ x δ ∈x +σ xz( n )δγ xz )dz dxdy
∑
∫
−b / 2
hn−1
n =1
δ ℧ = ∫∫
0
b/ 2
l
= ∫ ( N xδ ∈0x + M xbδκ xb + M xsδκ xs + Qxδγ xz0 )dx
0
(6)
where N x , M xb , M xs and Qx are the axial force, bending moments and shear force,
respectively, defined as:
3
hn
n =1
hn−1
N x = ∑ ∫ σ x( n )bdz
3
hn
n =1
hn −1
3
hn
n =1
hn −1
M xb = ∑ ∫ σ x( n ) zbdz
M xs = ∑ ∫ σ x( n ) fbdz
242
(7a)
(7b)
(7c)
Free vibration analysis of functionally graded sandwich beams based on a higher-order shear …
3
hn
n =1
hn−1
Qx = ∑ ∫ σ xz( n ) gbdz
(7d)
The variation of the kinetic energy is obtained as:
3 hn ( n ) ɺ ɺ ɺ ɺ
∑ ∫ ρ (UδU + W δW )dz dxdy
0 ∫− b / 2
n =1 hn−1
δℜ = ∫
l
b/ 2
l
= ∫ [δ uɺ ( I 0uɺ − I1wɺ b, x − I f wɺ s , x ) + δ wɺ b I 0 (wɺ b + wɺ s ) + δ wɺ b, x (− I1uɺ + I 2 wɺ b, x + I fz wɺ s , x )
0
(8)
+δ wɺ s I 0 (wɺ b + wɺ s ) + δ wɺ s , x (− I f uɺ + I fz wɺ b, x + I f 2 wɺ s , x )]dx
where the differentiation with respect to the time t is denoted by dot-superscript convention;
ρ ( n ) is the mass density of the each layer and I 0 , I1 , I 2 , I f , I fz and I f 2 are the inertia
coefficients, defined by:
3
hn
n =1
hn−1
( I 0 , I1 , I 2 , I f , I fz , I f 2 ) = ∑ ∫
ρ ( n) (1, z, z 2 , f , fz, f 2 )bdz
(9)
By substituting Eqs. (6) and (8) into Eq. (5), and integrating by parts versus both space and
time variables, and collecting the coefficients of δ u , δ wb , and δ ws , the following equations
of motion of the functionally graded sandwich beam are obtained:
ɺɺb, x − I f w
ɺɺs , x
δ u : N x , x = I 0uɺɺ − I1w
(10a)
ɺɺb + w
ɺɺs ) + I1uɺɺ, x − I 2 w
ɺɺb , xx − I fz w
ɺɺs , xx
δ wb : M xb, xx = I 0 ( w
(10b)
ɺɺb + w
ɺɺs ) + I f uɺɺ, x − I fz w
ɺɺb , xx − I f w
ɺɺs , xx
δ ws : M xs, xx + Qx , x = I 0 ( w
2
(10c)
4. CONSTITUTIVE EQUATIONS
The effective material properties for each layer, like Young’s modulus E and mass density ρ,
can be expressed as:
P ( n ) ( z ) = Pc + ( Pm − Pc )Vm( n )
(11)
where Pc and Pm denote the material property of ceramic and metal located at the top and
bottom surfaces, and at the core, respectively. The volume fraction of metal Vm( n ) through the
thickness of the sandwich beam faces follows a simple expression:
(1)
m
V
z − h0
=
h1 − h0
p
for z ∈ [ h0 , h1 ]
Vm( 2 ) = 1 for z ∈ [h1 , h2 ]
(3)
m
V
z − h3
=
h2 − h3
p
for z ∈ [ h2 , h3 ]
(12a)
(12b)
(12c)
where p is a power-law index, which is positive. The metal material distribution through the
beam thickness for various values of p is plotted in Fig. 2.
243
Nguyen Ba Duy, Nguyen Trung Kien
Figure 2. Distribution of metallic material through the thickness of (1-2-1) FG sandwich
beams with respect to the power-law index p.
The stress-strain relations for FG sandwich beams are given by:
σ x( n ) ( x, z ) = E ( n ) ( z ) ∈x ( x, z )
σ xz( n ) ( x, z ) =
E ( n) ( z)
γ xz ( x, z ) = G ( n ) ( z )γ xz ( x, z )
( n)
2[1 + υ ( z )]
(13a)
(13b)
The constitutive equations for beam forces and beam strains are obtained by using Eqs. (2),
(7) and (13):
Nx A
b
M x B
s= s
M x B
Q 0
x
B
Bs
D
Ds
0
Ds
Hs
0
0
0 ∈x
b
0 κ x
0 κ xs
As γ xz0
(14)
where the components of the stiffnesses of FG sandwich beams are given by:
3
hn
n =1
hn−1
( A, B, D, B s , D s , H s ) = ∑ ∫ (1, z, z 2 , f , zf , f 2 ) E ( n )bdz
3
hn
n =1
hn−1
As = ∑ ∫
g 2G ( n )bdz
(15a)
(15b)
By substituting Eqs. (4) and (14) into Eq. (10), the explicit form of the governing equations
of motion can be expressed as:
244
ɺɺb, x − I f w
ɺɺs , x
Au, xx − Bwb, xxx − B s ws , xxx = I 0uɺɺ − I1w
(16a)
ɺɺb + w
ɺɺs ) + I1uɺɺ, x − I 2 w
ɺɺb, xx − I fz w
ɺɺs , xx
Bu, xxx − Dwb, xxxx − D s ws , xxxx = I 0 ( w
(16b)
ɺɺb + w
ɺɺs ) + I f uɺɺ, x − I fz w
ɺɺb, xx − I f 2 w
ɺɺs , xx
B s u, xxx − D s wb , xxxx − H s ws , xxxx + As ws , xx = I 0 ( w
(16c)
Free vibration analysis of functionally graded sandwich beams based on a higher-order shear …
Eq. (16) is the most general form for the vibration analysis of FG sandwich beams, and the
dependent variables, u, wb and ws are fully coupled.
5. ANALYTICAL SOLUTIONS
The above equations of motion are analytically solved for free vibration problems. The
Navier solution procedure is used to determine the analytical solutions for a simply-supported
sandwich beam. The solution is assumed to be of the form:
∞
u ( x, t ) = ∑ ur cos α xeiωt
(17a)
r =1
∞
wb ( x, t ) = ∑ wbr sin α xeiωt
(17b)
r =1
∞
ws ( x, t ) = ∑ wsr sin α xeiωt
(17c)
r =1
where ω is the natural frequency, i = −1 the imaginary unit, α = rπ / L , ur , wbr and wsr are
wave amplitudes to be determined. Substituting Eqs. (17a - 17c) into Eq. (16), the following
eigenvalue problem is obtained:
k11
k12
k13
k12
k22
k23
k13
m11
2
k23 − ω m12
m13
k33
m12
m22
m23
m13 ur 0
m23 wbr = 0
m33 wsr 0
Equation (18) can be rewritten under a compact form
components of the stiffness matrix
K
(K − ω M ) U
and the mass matrix
U r = [ur wbr wsr ] are explicitly given as follows:
(18)
2
M
r
= 0 where the
associated with
T
k11 = Aα 2 , k12 = − Bα 3 , k13 = − B sα 3 , k22 = Dα 4 , k23 = D sα 4 , k33 = H sα 4 + Asα 2
m11 = I 0 , m12 = − I1α , m13 = − I f α , m22 = I 0 + I 2α 2 , m23 = I 0 + I fzα 2 , m33 = I 0 + I f 2 α 2
(19)
The solution of this eigenvalue problem will allow to calculate the natural frequencies and
vibration modes of FG sandwich beams.
6. NUMERICAL RESULTS
This section gives numerical examples to investigate the natural frequencies of simplysupported FG sandwich beams. Unless mentioned otherwise, Al/Al2O3 FG sandwich beams with
homogeneous softcore for two values of span-to-height ratio, L/h = 5 and 20, are considered.
Young’s modulus and mass density of aluminum are Em = 70GPa,ν m = 0.3 and
ρ m = 2702kg / m3 and Ec = 380GPa,ν c = 0.3 and ρc = 3960kg / m3 for alumina at the top and
bottom surfaces. For convenience, the following non-dimensional natural frequencies is used:
245
Nguyen Ba Duy, Nguyen Trung Kien
ω=
ω L2
ρm
h
Em
(20)
Figure 3: Effect of the power-law index and span-to-height ratio on the fundamental frequency of
Al/Al2O3 FG beams with homogeneous softcore.
Table 1. The first three non-dimensional natural frequencies of simply-supported Al/Al2O3 beams.
L/h Mode
5
1
2
3
20
1
2
3
246
Reference
Present
Nguyen et al. [10]
Simsek [11]
Thai and Vo (TBT) [12]
Present
Nguyen et al. [10]
Thai and Vo (TBT) [12]
Present
Nguyen et al. [10]
Thai and Vo (TBT) [12]
Present
Nguyen et al. [10]
Simsek [11]
Thai and Vo (TBT) [12]
Present
Nguyen et al. [10]
Thai and Vo (TBT) [12]
Present
Nguyen et al. [10]
Thai and Vo (TBT) [12]
p
0
5.1527
5.1525
5.1525
5.1527
17.8813
17.8711
17.8812
34.2100
34.1449
34.2097
5.4603
5.4603
5.4603
5.4603
21.5732
21.5732
21.5732
47.5930
47.5921
47.5930
0.5
4.4102
4.4075
4.4083
4.4107
15.4567
15.4250
15.4588
29.8331
29.7146
29.8382
4.6506
4.6504
4.6514
4.6511
18.3942
18.3912
18.3962
40.6480
40.6335
40.6526
1
3.9904
3.9902
3.9902
3.9904
14.0100
14.0030
14.0100
27.0981
27.0525
27.0979
4.2051
4.2051
4.2051
4.2051
16.6344
16.6344
16.6344
36.7679
36.7673
36.7679
2
3.6264
3.6344
3.6344
3.6264
12.6406
12.7120
12.6405
24.3154
24.4970
24.3152
3.8361
3.8368
3.8368
3.8361
15.1619
15.1715
15.1619
33.4689
33.5135
33.4689
5
3.4012
3.4312
3.4312
3.4012
11.5432
11.8157
11.5431
21.7162
22.4642
21.7158
3.6485
3.6509
3.6509
3.6485
14.3746
14.4110
14.3746
31.5781
31.7473
31.5780
10
3.2816
3.3135
3.3134
3.2816
11.0241
11.3073
11.0240
20.5565
21.3219
20.5561
3.5390
3.5416
3.5416
3.5390
13.9263
13.9653
13.9263
30.5370
30.7176
30.5369
Free vibration analysis of functionally graded sandwich beams based on a higher-order shear …
For verification purpose, the first three natural frequencies of FG beams with different
values of span-to-height ratio and power-law index are given in Table 1. The results obtained
from the present theory are compared with those of Nguyen et al. [10] based on first-order shear
deformation beam theory (FSBT), Simsek [11] and Thai and Vo [12] based on higher-order shear
deformation beam theory (TBT). It can be seen that the present model is in excellent agreement
with earlier works. As expected, an increase of the power-law index makes FG beams more
flexible, which leads to a reduction in natural frequencies. This holds irrespective of the
consideration of shear effects. Moreover, in order to verify the efficiency of the present study in
predicting the vibration responses of FG sandwich beams, Table 2 presents the comparison of
the fundamental frequency of Al/Al2O3 sandwich beams for various values of the power-law
index and four cases of thickness ratio of layers. It can be seen from this table that the present
solutions are similar to those of [8] based on a finite element model, except some minor
differences for
p = 0.5.
Table 2. The non-dimensional fundamental frequencies of simply-supported Al/Al2O3
sandwich beams with homogeneous softcore.
L/h
5
p
Reference
0
Present
Vo et al. (TBT) [8]
Present
Vo et al. (TBT) [8]
Present
Vo et al. (TBT) [8]
Present
Vo et al. (TBT) [8]
Present
Vo et al. (TBT) [8]
Present
Vo et al. (TBT) [8]
Present
Vo et al. (TBT) [8]
Present
Vo et al. (TBT) [8]
Present
Vo et al. (TBT) [8]
Present
Vo et al. (TBT) [8]
Present
Vo et al. (TBT) [8]
Present
Vo et al. (TBT) [8]
0.5
1
2
5
10
20
0
0.5
1
2
5
10
p
1-0-1
2.6773
2.6773
4.4443
4.4427
4.8525
4.8525
5.0945
5.0945
5.1880
5.1880
5.1848
5.1848
2.8371
2.8371
4.8595
4.8579
5.2990
5.2990
5.5239
5.5239
5.5645
5.5645
5.5302
5.5302
1-1-1
2.6773
2.6773
4.1850
4.1839
4.5858
4.5858
4.8740
4.8740
5.0703
5.0703
5.1301
5.1301
2.8371
2.8371
4.6306
4.6294
5.1160
5.1160
5.4410
5.4410
5.6242
5.6242
5.6621
5.6621
1-2-1
2.6773
2.6773
3.9930
3.9921
4.3663
4.3663
4.6459
4.6459
4.8564
4.8564
4.9326
4.9326
2.8371
2.8371
4.4169
4.4160
4.8938
4.8938
5.2445
5.2445
5.4843
5.4843
5.5575
5.5575
2-1-2
2.6773
2.6773
4.3059
4.3046
4.7178
4.7178
4.9970
4.9970
5.1603
5.1603
5.1966
5.1966
2.8371
2.8371
4.7473
4.7460
5.2216
5.2217
5.5113
5.5113
5.6382
5.6382
5.6452
5.6452
Furthermore, in order to investigate the effects of the power-law index and span-to-height
ratio on the natural frequencies, different types of symmetric FG sandwich beams are considered.
Numerical results are tabulated in table 3 and plotted in Fig. 3. In general, as the power-law
index increases, the natural frequencies increase for sandwich beams with homogeneous softcore.
This is due to the fact that higher values of power-law index correspond to high portion of
ceramic in comparison with the metal part for homogeneous softcore.
247
Nguyen Ba Duy, Nguyen Trung Kien
Table 3. The nondimensional natural frequencies of Al/Al2O3 sandwich beams with
homogeneous softcore..
L/h
Mode
5
1
2
3
20
1
2
3
Ratio of layer
thickness
1-0-1
1-1-1
1-2-1
2-1-2
1-0-1
1-1-1
1-2-1
2-1-2
1-0-1
1-1-1
1-2-1
2-1-2
1-0-1
1-1-1
1-2-1
2-1-2
1-0-1
1-1-1
1-2-1
2-1-2
1-0-1
1-1-1
1-2-1
2-1-2
p
0
2.6773
2.6773
2.6773
2.6773
9.2909
9.2909
9.2909
9.2909
17.7751
17.7751
17.7751
17.7751
2.8371
2.8371
2.8371
2.8371
11.2093
11.2093
11.2093
11.2093
24.7289
24.7289
24.7289
24.7289
0.5
4.4443
4.1850
3.9930
4.3059
14.5621
13.4460
12.8355
13.9147
26.6188
24.2347
23.1416
25.1824
4.8595
4.6306
4.4169
4.7473
19.0614
18.1125
17.2778
18.5851
41.5970
39.3631
37.5526
40.4406
1
4.8525
4.5858
4.3663
4.7178
15.9330
14.5406
13.7464
15.1484
29.1639
25.9687
24.4327
27.2875
5.2990
5.1160
4.8938
5.2216
20.7919
19.9718
19.0826
20.4231
45.3940
43.2784
41.2842
44.3799
2
5.0945
4.8740
4.6459
4.9970
16.9348
15.4382
14.4701
16.1356
31.2837
27.5509
25.5391
29.1783
5.5239
5.4410
5.2445
5.5113
21.7104
21.2371
20.4145
21.5740
47.5166
46.0103
44.0553
46.9385
5
5.1880
5.0703
4.8564
5.1603
17.5884
16.2202
15.1156
16.9250
32.9978
29.1401
26.6663
30.9506
5.5645
5.6242
5.4843
5.6382
21.9248
21.9860
21.3459
22.1202
48.1669
47.7391
46.0589
48.2846
10
5.1848
5.1301
4.9326
5.1966
17.7648
16.5445
15.4047
17.2150
33.6204
29.8897
27.2351
31.7167
5.5302
5.6621
5.5575
5.6452
21.8172
22.1607
21.6432
22.1774
48.0232
48.2030
46.7381
48.5068
8. CONCLUSIONS
This paper presented free vibration analysis of functionally graded sandwich beams using a
higher-order shear deformation theory. The face layers of sandwich beam are assumed to have
isotropic metal-ceramic material distribution, and Young’s modulus and mass density are
assumed to vary according to power-law form in terms of the volume fractions of the
constituents. The core layer is still homogeneous and made of an isotropic metallic material.
Governing equations of motion are derived from the Hamilton’s principle. Navier-type solution
for simply-supported beams is developed to solve the problem. Numerical results are obtained
for sandwich beams with homogeneous softcore to investigate effects of the power-law index,
span-to-height ratio and thickness ratio of layers on the natural frequencies. The present model is
found to be appropriate and efficient in analyzing vibration of FG sandwich beams.
Acknowledgements. This research is funded by Vietnam National Foundation for Science and Technology
Development (NAFOSTED) under Grant No. 107.02-2012.07.
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