Volume- 6, Issue-10, Oct.-2018, http://iraj.in
A MODIFIED SIMPLE SHEAR DEFORMATION THEORYFOR VIBRATION ANALYSISOF COMPOSITE STIFFENED PLATE
1A. NAYAN, 2T. Y. KAM
1, 2Mechanical Engineering Department, National Chiao Tung University, Taiwan Email: 1ahmad.nayan@gmail.com, 2tykam@mail.nctu.edu.tw
Abstract - A modified simple first-order shear deformation theory (MSFSDT) is proposed to analyze the free vibration of stiffened laminated composite plates. In the proposed theory, the shear rotation of the conventional first-order shear deformation theory is expressed in terms of the derivatives of the total and shear displacements of the plate. In such a way, the displacement degrees of freedom are reduced from 5 to 4.The Ritz method is used to perform the free vibration analysis of the stiffened plates. A number of examples have been given to illustrate the feasibility and accuracy of the proposed method for predicting the modal characteristics of composite plates with various boundary conditions.
Keywords - Composite plate, shear deformation, Ritz method, vibration
I. INTRODUCTION
Because of their good stiffness and lightweight properties,composite materials have been used to fabricate plate type structures in many industries such as aerospace, aircraft, marine, automotive, and building construction. In engineering applications, weight saving is always an important issue. Therefore, in many cases, for reducing weight, composite plates are stiffened with beams. A stiffened composite plate can havesatisfactory mechanical properties if the layers of the plate and the stiffeners are properlydesigned. In designing composite plates, it is desired to have an appropriate and efficient method that can predict the accurate mechanical behavior of the plates. Therefore, the development of simple and effective methods for analyzing the mechanical behavior of composite plates has long attracted the attention of many researchers.In recent years, many methods and theories have been proposed to analyze composite plates. For instance, comprehensive reviews of methods for vibration analysis of laminated composite and sandwich plates have been conducted by Khandam at el[1] and Sayyad and Ghugal[2]. It is noted that theClassical Lamination Theory (CLT) is only suitable for small thickness-to-width ratio. Reissner[3] and Mindlin[4]
proposed the first-order shear deformation (FSDT) to take into account the shear deformation effects. Kim and Cho[5]have proposed an enhance FSDT to attain better mechanical behavior prediction of laminate and sandwich plates. Kam and his associates have adopted various plate theories to study the vibro-acoustics of composite plates [6]–[8]. Recently, a simple FSDT, has been developed by Thai and Choi[9] to analyze various plates and beams whit different boundary and loading conditions. In this paper, a modified simple first-order shear deformation theory (MSFSDT) is proposed to analyze the free vibration of laminated composite plate with general boundary conditions. The Ritz method is used to analyze the vibration behavior of
the composite laminated plates. A number of composite plates with different boundary conditions and fiber orientations have been studied using the proposed method.
II. PLATE VIBRATION FORMULATED USING MSFSDT
2.1 Laminated Plate Formulation
The square laminated composite plate of size a (length) × b (width) × h (thickness) is elastically restrained along the plate periphery by distributed springs with translational and rotational spring constant intensities K and K , respectively, and stiffened by the four beams as shown in Figure 1.
The x−y plane of the reference coordinate is located at the mid-plane of the plate. Herein, the displacement field based on the MSFSDT is given as
, , , ( )
, , , ( )
, , ,
o p s
p o p
o p s
p o p
p o p
w w
u x y z u x y z
x x
w w
v x y z v x y z
y y
w x y z w x y
(1)
where u , v , and w are the displacements in x, y, and z directions, respectively; u and v , are mid-plane displacements; w is transverse displacement component due to shear deformation. It is assumed that both the plate and beams have same shear rotation.
(a) Stiffener location and edge boundary condition
Volume- 6, Issue-10, Oct.-2018, http://iraj.in
(b) Coordinates and shear rotation Fig 1. Stiffened laminated composite plate
The strain-displacement relations of the structure are expressed as
2 2
2 2
2 2
2 2
2 2
2
op op s
x p
op op s
y p
op op op s
xy p
op s op s
xz p
yz
u w w
u z
x x x x
v w w
v z
y y y y
u v w w
u v
y x y x z x y x y
w w w w
u w
z x z x x x x
v
op s op s
p
w w w w
w z
z y y y y y
(2
The stress-strain relations in the reference coordinate, x-y, are given as
11 12 16
12 22 26
44 45
45 55
16 26 66
0 0
0 0
0 0 0
0 0 0
0 0
x x
y y
yz yz
xz xz
xy xy
Q Q Q
Q Q Q
Q Q
Q Q
Q Q Q
(3)
The expressions for determining the above stiffness coefficients can be found in [10].
The strain energy, U , and kinetic energy, Tp, of the plate are written, respectively,as
U = ∫ σ ε +σ ε +τ γ +τ γ + τ γ dV (4)
and
2 2 2
1
p
2
Vp p p p p pT u v w dV
(5) Regarding the deformation of the beam type stiffener, the displacement field in y-direction is:
four edges of the plate are represented usingK and K as spring constant intensities in the longitudinal and rotational directions, respectively, where the subscript istands for the edge under consideration.
The proper assignments of the spring constant intensities can lead to free, clamped, or simple support boundary condition.
The strain energy,U , stored in the elastic restraints is written as
3
1 2 2 2 2 4 2
0 0
0 0 0 0
2 2
2 2
3
1 2 4
0 0 0 0
0 0 0 0
2 2 2 2
2 2 2 2
b b L a a
L L L
s x x a y y b
b b a a
R
R R R
x x y y
K
K K K
U w dy w dy w dx w dx
K
K w dy K w dy w dx K w dx
y y
x x
10) The total strain energy U and total kinetic energy T of the stiffened plate are written, respectively, as
1 N
p bi s
i
U U U U
(11)
Volume- 6, Issue-10, Oct.-2018, http://iraj.in
and
1 N
p bi
i
T T T
(12) whereN is number of stiffeners.
2.2 Ritz Method for Vibration Analysis
The Ritz method is used to study the free vibration of the plate. The displacements of the plate are
expressed as
sin
, , ,
sin
, , ,
sin
, , ,
sin
, , ,
op op o
s s
u x y t U x y t v x y t V x y t w x y t W x y t w x y t W x y t
(13) with
ˆ ˆ
1 1
ˆ ˆ
ˆ 1 ˆ
1
ˆ ˆ
ˆ 1 ˆ
1
ˆ ˆ
ˆ ˆ
1 1
, , ,
,
A B
ij i j
i j
C D
ij i j
j B
i A
I J
ij i j
j D
i C
M N
s ij i j
i I j J
U C
V C
W C
W C
(14)
whereC are unknown constants; A, B, C, D, I, J, M, N denote the numbers of terms in the series. Legendre’s polynomials,ϕand ψ, are used to represent the characteristic functions. Let ξ= 2x/a−1and η = 2y/b−1. The normalized characteristic functions, for instance, ϕ(ξ), are given as
ϕ (ξ) = 1,
−1≤ ξ ≤1 ϕ (ξ) =ξ, for n ≥3,
ϕ (ξ) = (2n−3)ξ×ϕ (ξ)−(n−2)
×ϕ (ξ) (n−1)
15)
Extremization of the functional Π = T – U gives the following eigenvalue problem.
[K− ω M]C = 0 (1)
(16) whereK and M are structural stiffness and mass matrices; ω is circular frequency. The solution of the
above eigenvalue problem can lead to the determination of the natural frequencies and mode shapes of the plate.
III. RESULTS AND DISCUSSION
The proposed method is first applied to analyze the free vibration of a simply supported laminated composite plate with thickness to width ratio h /a = 0.006. The material properties of a lamina areE = 2.45E , G = G = 0.48E ,G = 0.48E , ν = 0.23 , ρ= 8000 . Table 1 shows the dimensionless natural frequencies,
ω= ρh ω a D⁄ where D =
E h ⁄[12(1− ν ν )],of theplate.When comparing with the results obtained based on CLT, it shows that thepresent method can produce very good results. The dimensionless natural frequencies of the plate with thickness-to-width ratio (h /a = 0.3) are listed in Table 2.It is noted that thepresent method is also able to produce acceptable results as compared with those obtained based on the higher order shear deformable theory. The dimensionless natural frequencies of the laminated composite plate with fixed boundary condition are shown in Table 3. Again good results can be attained. Finally,consider the vibration of anelastically restrained composite plate with or without stiffeners. The plate of size a = b = 100 mm,thickness h = 0.6 mmand density of 1754 kg/m3 comprises 4 layers and four stiffeners. The
stiffener thickness may beof
h = 2 mm, 4 mm, or 6 mm. The length (L ) and width of the stiffeners are 20 mmand 0.15 mm, respectively. The elastic constants of the plate are:
E 115 (GPa)
E 7.84 (GPa)
ν 0.3
ν 0.02
G 4.1 (GPa)
G 0.68 (GPa)
G 4.1 (GPa) The elastic constants of the stiffeners are E = 15.47 GPa and ν= 0.33. The edge spring constant intensities are KL = 1929N/m2 and KR = 0.The dimensionless natural frequencies of the plates are listed in Table 4 in comparison with those using other methods. It is noted thatthe results obtained based on CPT and MSFSDT are in good agreement with or without stiffeners
.
Table 1Dimensionless natural frequencies of laminated square plates with simply supported boundary condition ( = . and ⁄ = . )
Ply angle Source
Modes
1 2 3 4 5 6
(0o, 0o, 0o) Present TSDT [11]
CPT [11]
Chow [12]
Leissa[13]
15.168 15.22 15.17 15.19 15.19
33.237 33.76 33.32 33.31 33.30
44.363 44.79 44.51 44.52 44.42
60.643 61.11 60.78 60.79 60.78
64.416 66.76 64.79 64.55 64.53
90.063 91.69 90.42 90.31 90.29 (15o, -15o, 15o) Present 15.393 34.018 43.797 60.694 66.516 91.240
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TSDT CPT Chow Leissa
15.45 15.40 15.37 15.43
34.54 34.12 34.03 34.09
44.25 43.96 43.93 43.80
61.36 60.91 60.80 60.85
68.68 66.92 66.56 66.67
92.99 91.76 91.40 91.40 (30o, -30o, 30o) Present
TSDT CPT Chow Leissa
15.851 15.92 15.87 15.86 15.90
35.755 36.28 35.92 35.77 35.86
42.505 43.00 42.70 42.48 42.62
61.236 62.05 61.53 61.27 61.45
71.4958 73.55 71.10 71.41 71.71
85.507 87.37 86.31 85.67 85.72 (45o, -45o, 45o) Present
TSDT CPT Chow Leissa
16.082 16.15 16.10 16.08 16.14
36.818 37.33 37.00 36.83 36.93
41.672 42.20 41.89 41.67 41.81
61.604 62.45 61.93 61.65 61.85
76.799 78.96 77.99 76.76 77.04
79.749 81.55 80.11 79.74 80.00
Table 2Dimensionless natural frequencies of laminated square thick plates with simply support boundary condition ( = and ⁄ = . )
Ply angle Source
Modes
1 2 3 4 5 6
(0o, 0o, 0o) Present TSDT [11]
CPT [12]
11.47 11.71 14.16
21.28 22.13 38.03
24.16 25.38 48.18
27.79 32.58 49.11
31.08 35.79 64.55
33.84 40.89 73.01 (15o, -15o, 15o) Present
TSDT CPT
11.57 11.84 14.37
21.53 22.38 37.54
24.05 25.29 47.93
28.46 32.74 51.03
31.13 36.13 65.39
34.18 40.56 74.67 (30o, -30o, 30o) Present
TSDT CPT
11.79 12.10 14.80
22.07 22.97 30.65
23.79 25.03 36.44
30.01 33.07 48.34
31.25 37.06 54.91
35.03 39.66 65.47 (45o, -45o, 45o) Present
TSDT CPT
11.90 12.24 15.02
22.38 23.36 35.75
23.61 24.80 48.65
31.12 33.24 59.08
31.31 38.20 61.04
34.85 38.55 77.19 (0o, 90o, 0o) Present
TSDT CPT
11.46 11.73 14.16
21.43 22.27 28.88
24.09 25.37 37.72
31.08 32.64 48.18
32.05 35.95 50.12
34.10 40.89 65.10
Table 3 Dimensionless natural frequencies of laminated square plates with clamped boundary condition ( = . and ⁄ = . )
Ply angle Source Modes
1 2 3 4 5 6
(0o, 0o, 0o) Present TSDT [11]
CPT [11]
Chow [12]
29.069 30.02 29.27 29.13
50.743 54.68 51.21 50.82
67.188 70.41 67.94 67.29
85.493 89.36 86.25 85.67
87.098 92.58 87.97 87.14
118.340 123.6 119.3 118.6 (15o, -15o, 15o) Present
TSDT CPT Chow
28.878 29.85 29.07 28.92
51.35 55.25 51.83 51.43
65.822 69.14 66.55 65.92
87.09 88.53 85.17 84.55
89.66 94.92 90.56 89.76
119.04 124.3 120.0 119.3 (30o, -30o, 30o) Present
TSDT
28.50 29.51
53.06 56.84
62.59 66.17
83.68 87.83
95.10 100.5
114.06 118.9
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CPT Chow
28.69 28.55
53.57 53.15
63.26 62.71
84.43 83.83
96.15 95.21
115.5 114.1 (45o, -45o, 45o) Present
TSDT CPT Chow
28.31 29.34 28.50 28.38
54.55 58.19 55.11 54.65
60.34 64.14 60.94 60.45
83.51 87.67 84.25 83.65
101.89 107.38 103.2 102.0
105.52 110.6 106.7 105.6
Table 4 Dimensionless natural frequencies of laminated square plate with and without stiffeners Modes
Stiffeners Method 1 2 3 4 5 6 7 8
No stiffener
CPT 60.825 93.212 94.420 148.339 170.941 267.171 375.949 482.685 MSFSDT 60.822 93.205 94.418 148.327 170.899 267.076 375.543 482.202
h = 2 mm
CPT 57.073 92.722 93.879 148.300 181.529 268.436 365.353 481.956 MSFSDT 57.040 92.715 93.865 148.217 181.122 268.044 364.963 481.432
h = 4 mm
CPT 57.729 92.706 93.908 148.292 194.456 274.762 364.836 481.796 MSFSDT 57.075 92.662 93.755 148.203 193.004 274.291 364.541 481.256
h = 6 mm
CPT 57.866 92.676 93.901 148.277 197.531 277.813 363.984 481.510 MSFSDT
57.818 92.538 93.879 148.165 197.229 277.384 363.917 481.594 CONCLUSIONS
A new simple first-order shear deformation theory has been presented and applied to the free vibration analysis of laminated composite plates with different boundary conditions. The proposed first-order shear deformation theory only requires to use four displacement components to formulate the displacement field of the plate with the consideration of shear deformation in the plate thickness direction. A number of numerical examples have been given to demonstrate the feasibility and accuracy of the proposed method in predicting the natural frequencies of laminated composite plates with different boundary conditions and aspect ratios. It has been shown that the results obtained using the present method can match those available in the literature well.
ACKNOWLEDGEMENT
This research was financially supported by the Ministry of Science and Technology (106-2221-E-009-136), Taiwan.
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