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Vibration of cross-ply laminated composite plates subjected to initial in-plane stresses
Thin-Walled Structures 40 (2002) 557–571 www.elsevier.com/locate/tws
Vibration of cross-ply laminated composite plates subjected to initial in-plane stresses Hiroyuki Matsunaga ∗ Department of Architecture, Setsunan University 17-8, Ikeda-nakamachi, Neyagawa, Osaka 572-8508, Japan Received 13 July 2001; received in revised form 3 December 2001; accepted 12 December 2001
Abstract Natural frequencies, modal displacements and stresses of cross-ply laminated composite plates subjected to initial in-plane stresses are analyzed by taking into account the effects of higher-order deformations and rotatory inertia. By using the method of power series expansion of displacement components, a set of fundamental dynamic equations of a two-dimensional higher-order theory for rectangular laminates is derived through Hamilton’s principle. Several sets of truncated approximate theories can be derived to solve the eigenvalue problems of a simply supported laminated plate. After examining the convergence properties of the lowest natural frequency, only the numerical results for M=5, which are considered to be sufficient with respect to the accuracy of solutions, are presented. Numerical results are compared with those of the published existing three-dimensional theory and FEM solutions. The modal displacement and stress distributions in the thickness direction are plotted in figures. The buckling stresses can be obtained in terms of the natural frequencies of the laminates without initial in-plane stresses. 2002 Elsevier Science Ltd. All rights reserved. Keywords: Higher-order theory; Cross-ply; Composite; Laminated plate; Vibration; Modal displacement and stress
1. Introduction Stiff, strong and lightweight composite materials are being widely used in many structural members, such as multilayered composite beams, plates and shells. ∗
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Vibration problems of laminated composite plates have attracted the attention of many researchers up to the present. The mechanical behaviors of laminated composite plates are strongly dependent on the degree of orthotropy of individual layers, the low ratio of transverse shear modulus to the in-plane modulus and the stacking sequence of laminates. It has been known that classical laminated plate theories, based on the Kirchhoff hypothesis, are inadequate for predicting the gross response characteristics of moderately thick laminated composite plates and/or highly anisotropic laminated composite plates. The inaccuracy is due to neglecting the transverse shear and normal strains in the laminate. In order to take into account the effects of low ratio of transverse shear modulus to the in-plane modulus, a number of first-order shear deformation theories have been developed. However, since in the Mindlin-type first-order shear deformation plate theory the transverse shear strains are assumed to be constant in the thickness direction, shear correction factors have to be incorporated to adjust the transverse shear stiffness for studying the static or dynamic problems of plates. Although the Mindlin-type first order shear deformation theory is quite accurate for the gross responses, such as natural frequencies of moderately thick laminates, the accuracy of solutions will be strongly dependent on predicting better estimates for the shear correction factors. It has been shown that the Mindlin-type first-order shear deformation theories are inadequate for the accurate prediction of the modal displacements and interlaminar stresses of laminated composite plates (see, for example, [1]). Without requiring the specification of a shear correction coefficient in the Mindlintype first-order shear deformation plate theory, various higher-order plate theories have been developed. In order to obtain the accurate predictions of the gross response characteristics such as natural frequencies, a number of contributions based on the three-dimensional elasticity theory have been made to analyze laminated composite plates [2-4]. However, accurate solutions based on the three-dimensional elasticity theory are often computationally expensive. Several approximately refined two-dimensional higher-order theories have been proposed to analyze the response characteristics of plates. Three excellent reviews of refined theories of laminated composite plates have been presented by Noor and Burton [5], Reddy [6] and Leissa [7]. In recent years, the layer-wise theories and individual layer theories have been presented to obtain more accurate information at the ply level. Although such theories have been proved to be one of the best alternatives to three-dimensional elasticity theories, these theories require numerous unknowns for multilayered plates and are often computationally expensive in obtaining accurate solutions. The number of unknowns in a laminate (which is dependent on the number of layers) will increase dramatically as the number of layers increases. A number of single-layer (global) higher-order plate theories that includes the effects of transverse shear and normal deformations has been published in the literature. Although various models of higher-order displacement fields have been considered, most of these theories are the third-order theories in which the in-plane displacements are assumed to be a cubic expression of the thickness coordinate and the out-of-plane displacement to be a quadratic expression at most. For isotropic plates, a two-dimensional higher-order theory has been developed, and has been
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applied to the statics and dynamics of thin and thick plates by Matsunaga [8-10]. Natural frequencies and buckling stresses for isotropic plates subjected to initial inplane stresses have been analyzed by using the approximate two-dimensional higherorder theories. Remarkable effects of transverse shear and normal deformations have been predicted in the results. General higher-order theories of laminates, which take into account the complete effects of transverse shear, normal deformations and rotatory inertia, have been investigated recently for the vibration and stability problems of multilayered composite plates [11]. This paper presents a global higher-order theory for analyzing natural frequencies, modal displacements and stresses for cross-ply laminated composite plates subjected to initial in-plane stresses. The complete effects of shear, normal deformations and rotatory inertia can be taken into account within the approximate two-dimensional theory. Several sets of the governing equations of truncated approximate theories are applied to the analysis of vibration problems of a simply supported multilayered elastic plate. Based on the power series expansions of displacement components, a fundamental set of equations of a two-dimensional higher-order plate theory is derived through Hamilton’s principle. Natural frequencies of a cross-ply laminated composite plate subjected to initial in-plane stresses are obtained by solving the eigenvalue problem numerically. Convergence properties of the present numerical solutions have been shown to be accurate for the natural frequencies, with respect to the order of approximate theories in the previous work [11]. A comparison of the present results is made with previously published results of the three-dimensional layerwise theory [1] and FEM [12] solutions. The buckling stresses of cross-ply laminated composite plates subjected to initial in-plane stresses are computed by increasing the absolute values of compressive stresses until the natural frequency vanishes. For multilayered plates, the distribution of modal displacements and stresses in the thickness direction has been obtained accurately at the ply level. The modal transverse stresses have been obtained by integrating the three-dimensional equations of motion in the thickness direction, and satisfying the continuity conditions at layer interfaces and stress boundary conditions on the top and bottom surfaces. There exist no results in the open literature for the natural frequencies of multilayered composite plates with a large number of layers (e.g. 50 layers). The present theory has the advantage of predicting natural frequencies of multilayered composite plates without increasing the unknowns involved as the number of layers increases.
2. Higher-order theory for laminated composite plates A cross-ply laminated composite plate of uniform thickness h, having a rectangular plan a×b, is shown in Fig. 1. Introducing a Cartesian reference coordinate system xi (I=1,2,3) on the middle plane of the plate, the dynamic displacement components are expressed as ⌼a ⬅ ⌼a(xi;t), ⌼3 ⬅ ⌼3(xi;t)
(1)
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Fig. 1.
K-layers cross-ply laminated composite plates and coordinates.
where t denotes time. Greek lower case subscripts are assumed to range over the integers 1,2. The displacement components may be expanded into power series of the thickness coordinate x3 as follows:
冘 ⬁
na ⫽
冘 ⬁
(n)
na(x3)n, n3 ⫽
n⫽0
(n)
n3(x3)n
(2)
n⫽0
where n=0,1,2,…,⬁. Based on this expression for the displacement components, a set of linear fundamental equations of a two-dimensional higher-order plate theory can be obtained as follows. 2.1. Strain-displacement relations Strain components may also be expanded as follows:
冘 ⬁
gab ⫽
冘 ⬁
(n)
g ab(x3)n, ga3 ⫽
n⫽0
(n)
g a3(x3)n, g33 ⫽
n⫽0
冘 ⬁
(n)
g 33(x3)n
(3)
n⫽0
and strain-displacement relations can be written as (n) (n) (n ⫹ 1) (n) (n) 1 (n) 1 g ab ⫽ ( n a,b ⫹ n b,a), g a3 ⫽ {(n ⫹ 1)其 n a ⫹ n 3,a, g 33 ⫽ 2 2
(n)
(n ⫹ 1)
(n ⫹ 1) n3
(4)
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where a comma denotes partial differentiation with respect to the coordinate subscript that follows. 2.2. Equations of motion Introducing stress components sab, sa3 and s33, Hamilton’s principle is applied to derive the equations of dynamic equilibrium and natural boundary conditions of the plate. In order to treat vibration problems of a plate subjected to initial in-plane distributed uniformly in x3-direction, additional terms due to these stresses sab 0 stresses, which are assumed to remain unchanged during vibration are taken into consideration. The boundary conditions are assumed to be traction-free on the top and bottom surfaces of the plate. Hamilton’s principle for the present problem may be expressed for an arbitrary time interval t1 to t2 as follows:
冕冕 t2
{sabdgab ⫹ 2sa3dga3 ⫹ s33dg33 ⫹ s0ab(n˙ l,adn˙ l,b ⫹ n˙ 3,adn˙ 3,b)
(5)
V
t1
⫺r(˙ ad˙ a ⫹ ˙ 3d˙ 3)其dV dt ⫽ 0 where the over dot indicates partial differentiation with respect to time, and r=r(z) denotes the mass density, while dV denotes the volume element. By performing the variation as indicated in Eq. (5), the equations of motion on the middle plane of a plate are obtained as follows: (n) ab
N ,a⫺n Q ⫹ (n)
冘 ⬁
(n⫺1) b
(n ⫹ m ⫹ 1)(m) b
m⫽0
冘
冘
⬁
(n⫺1)
Qa,a⫺n T ⫹
冘 ⬁
(n ⫹ m ⫹ 1) (m) a 0 b,al m⫽0
s
⫺
(n ⫹ m ⫹ 1) (m) ab 3,ab 0
s
⬁
⫺
m⫽0
r
¨ ⫽ 0
(n ⫹ m ⫹ 1)(m) 3
r
¨ ⫽ 0
(6)
(7)
m⫽0
where n, m=0,1,2, . . . ,⬁ and
冘
hk+1n+m+1⫺hkn+m+1 (n+m+1) ⫽ ,r n⫹m⫹1 k⫽1 K
sab 0
(n+m+1)
⫽ sab 0
冘 K
k⫽1
r(k)
hk+1n+m+1⫺hkn+m+1 n⫹m⫹1
(8)
(k) and hk denote the initial in-plane stress, the mass density The parameters sab 0 , r and thickness coordinate of the lower side of kth layer, respectively, and K denotes the total number of layers in the laminates. The stress resultants are defined as follows:
冘 K
(N
ab
(n)
,Q
(n)
a
,T ) ⫽ (n)
k⫽1
a3 33 (sab (k) ,s(k),s(k))
hk+1n+1⫺hkn+1 n⫹1
(9)
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2.3. Boundary conditions The boundary conditions are assumed to be traction-free on the upper and lower surfaces of the plate. For equations of boundary conditions along the boundaries on the middle plane, the following quantities
冘 ⬁
na(n) or nb[Nab
(n)
⫹
(n ⫹ m ⫹ 1)(m) bl a 0 l
(n)
(n)
n , ], n 3 or nb[Q b
s
(10)
m⫽0
冘 ⬁
⫹
(n ⫹ m ⫹ 1)(m) 3 ab 0 a
n, ]
s
m⫽0
are to be prescribed. 2.4. Constitutive relations For elastic and orthotropic materials, the constitutive relations for the kth layer of a laminated plate can be written as abln ab33 a3 a3l3 33 33ln 3333 sab (k) ⫽ C(k) gln ⫹ C(k) g33, s(k) ⫽ C(k) gl3, s(k) ⫽ C(k) gln ⫹ C(k) g33 ijkl
The material constants C as follows:
(11)
(i, j, k, l=1,2,3) for an individual layer can be expressed
(k) (k) 1122 (k) (k) (k) 1133 (k) (k) (k) C1111 ⫽ E(k) ⫽ E(k) ⫽ E(k) (k) 1 (1⫺n23 n32 ) / m, C(k) 1 (n21 ⫹ n31 n23 ) / m, C(k) 1 (n31 ⫹ n21 n32 ) / m, (k) (k) (k) 2222 (k) (k) 2233 (k) (k) (k) ⫽ E(k) ⫽ E(k) ⫽ E(k) C2211 (k) 2 (n12 ⫹ n32 n13 ) /m, C(k) 2 (1⫺n31 n13 ) / m, C(k) 2 (n32 ⫹ n12 n31 ) / m, (k) (k) (k) 3322 (k) (k) (k) 3333 (k) (k) C3311 ⫽ E(k) ⫽ E(k) ⫽ E(k) (k) 3 (n13 n12 n23 ) / m, C(k) 3 (n23 ⫹ n21 n13 ) / m, C(k) 3 (1⫺n12 ⫹ n21 ) / m,
(12)
1313 1212 C2323 ⫽ 2G(k) ⫽ 2G(k) ⫽ 2G(k) (k) 23 , C(k) 13 , C(k) 12
(k) (k) (k) (k) (k) (k) (k) (k) (k) where m ⫽ 1⫺n12 n21 ⫺n23 n32 ⫺n31 n13 ⫺2n21 n32 n13 . The parameters Ea(k) and E3(k) are (k) (k) Young’s moduli, while nab and na3 are Poisson’s ratios. The first suffix of n denotes the direction of stress, and the second denotes that of expansion and contraction. The following relations can be established by the reciprocal theorem:
E1n21 ⫽ E2n12, E1n31 ⫽ E3n13, E2n32 ⫽ E3n23
(13)
Stress resultants can be derived from Eqs. (9a) and (11) and Eqs. (3a) and (4) in terms of the expanded displacement components. The equations of motion, (6) and (7), can also be expressed in terms of the expanded displacement components. 2.5. Mth order approximate theory Since the fundamental equations mentioned above are complicated, approximate theories of various orders may be considered for the present problem. A set of the following combination of Mth order (Mⱖ1) approximate equations is proposed as follows:
冘
2M⫺1
na ⫽
m⫽0
冘
2M⫺2
n (x ) , n3 ⫽ (m) a
3 m
m⫽0
3 m n(m) 3 (x )
(14)
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where m=0,1,2,3, … The total number of the unknown displacement components is (6M⫺1), which is not dependent on the number of layers in a laminate. In the above cases of M=1, an assumption of plane strains is inherently imposed. Another set of the governing equations of lowest order approximate theory (M=1) is derived on the basis of an assumption that the normal stress s33 is zero (which is known as Mindlin-type laminated plate theory), with the shear correction coefficient 2 ⫽ 1.
3. Navier solution for simply supported plate In the following analysis, the Cartesian reference coordinate x=x1, y=x2 and z=x3 and displacement components u=n1, n=n2 and w=n3 are followed. For a simply supported rectangular laminated plate, the boundary conditions (10) can be expressed for the x-constant edges, u,x(n) ⫽ 0, n(n) ⫽ 0, w(n) ⫽ 0
(15)
and for the y-constant edges, u(n) ⫽ 0, n,y(n) ⫽ 0, w(n) ⫽ 0
(16)
Following the Navier solution procedure, displacement components that satisfy the boundary conditions (15) and (16) may be expressed as
冘冘 冘冘 ⬁
u
(n)
⬁
rpx spy ⫽ u cos sin ·eiwt, n(n) ⫽ a b r ⫽ 1s ⫽ 1 ⬁
w(n) ⫽
(n) rs
⬁
(n) wrs sin
r ⫽ 1s ⫽ 1
rpx spy iwt sin ·e a b
冘冘 ⬁
⬁
r ⫽ 1s ⫽ 1
(n) nrs sin
rpx spy iwt cos ·e a b
(17)
(18)
where r, s=1,2,3,…,⬁ and w denotes the circular frequency; i is the imaginary unit. The equations of motion (6) and (7) are rewritten in terms of the generalized displace(n)
(n)
(n)
ment components urs, nrs and wrs. The dimensionless natural frequency and initial in-plane stress are defined as follows: ⍀ ⫽ wh冑r(1) / E2(1), ⌳ ⫽ s0ab / E(1) 2
(19)
4. Eigenvalue vibration problem Using Eqs. (17a) and (18), the equations of motion, (6) and (7) (expressed in terms of displacement component) can be rewritten by collecting the coefficients for the generalized displacements of any fixed values r and s. The generalized displacement vector {U其 for the Mth order approximate theory is expressed as
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(20)
For free vibration problems, the equations of motion can be expressed as the following eigenvalue problem: ([K]⫺⍀2[M]){U其 ⫽ {0其
(21)
where [K] denotes the stiffness matrix and [M] denotes the mass matrix. 5. Determination of modal stress distributions The modal in-plane stresses for the kth layer can be derived in terms of the expanded displacement components by introducing the strain-displacement relations (4) into the constitutive relations (11). The modal transverse stresses for the kth layer are obtained by integrating the three-dimensional equations of motion in the thickness direction, starting from the lower (or upper) surface of the laminates as follows:
冕
x3
sa3 (k) ⫽ ⫺
冕
x3
(sab ¨ a)dx3 ⫹ Ca(k), s33 (k),b⫺r (k) ⫽ ⫺
hk
(sa3 ¨ 3)dx3 ⫹ C3(k) (k),a⫺r
(22)
hk
a and C3(k), are constants obtained from the stress-continuity conditions at where C(k) layer interfaces of the layer. If the boundary conditions for transverse stresses are prescribed on one of the outer surfaces of the plate, the stress boundary conditions on the other surface can be satisfied through the equations of motion, (6) and (7). Because of the discontinuity of the in-plane stresses at layer interfaces, the integration is performed in a piecewise manner. The modal stress components in the kth layer of cross-ply laminated composite plates can be expressed as follows:
冘 ⬁
sab (k) ⫽
(n) (n) [Dablngln ⫹ Dab33g33 ](k)(x3)n
(23)
n⫽0
冘 ⬁
sa3 (k) ⫽
[(x3)n+1⫺hkn+1] (n) (n) (n) [(Dablngln ⫹ Dab33g33 ),b ⫹ rw2na ](k) n⫹1 n⫽0
(24)
3 ⫹ sa3 (k⫺1)兩x =hk
冘 ⬁
s33 (k) ⫽
冋
(n) (n) [(Dablngln ⫹ Dab33g33 ),ab ⫹ rw2n,a(n) a ](k)
n⫽0
册 冘 ⬁
(x3)n+2⫺hkn+2 (n ⫹ 1)(n ⫹ 2)
3 n+1 ⫺hkn+1] a3 hkn+1(x3⫺hk) (n)[(x ) ⫺rw2 n3 ⫺s,a(k⫺1)兩[x3⫺h ] ⫺ k n⫹1 n⫹1 n⫽0 3 ⫹ s33 (k⫺1)|x =hk
(25)
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6. Numerical studies and results 6.1. Numerical examples Natural frequencies of cross-ply laminated composite plates with simply supported edges are determined. The orthotropic material for each layer are (k) (k) (k) (k) (k) (k) , G23 and G12 . Poisson’s ratios are given by n13 , n23 and n12 , and E1(k), E2(k), E3(k), G13 the other Poisson’s ratios can be obtained by the reciprocal theorem. The material properties are assumed to be the same in all the layers, but the fibre orientations may be different among the layers. The fibre orientations of the different laminae alternate between 0° and 90° with respect to the x-axis. The thickness of each layer is identical for the 0° and 90° layers in the laminates. Both symmetric and skewsymmetric laminations with respect to the middle plane are considered. In the symmetrical laminates having an odd number of layers, the 0° layers are at the outer surfaces of the laminate. The mass density is assumed to be uniform in the thickness direction, i.e. r(1), r(2), …, r(k) are identical. All the numerical results are shown in dimensionless quantities. It has been noticed in previous work [11] that results for M=2–5 converge accurately enough in the practical range of thickness parameter a/h. Only the results for M=5 are given in the present numerical examples. 6.2. Comparison of the lowest natural frequency with those of existing theories Table 1 shows a comparison of the lowest natural frequencies of cross-ply square laminates [0°/90°/0°/…] with other solutions for various values of the degree of orthotropy of the individual layers E1/E2 and various numbers of layers. For this numerical example, the material properties of the individual layers are taken from Noor and Burton [1], and Kant and Kommineni [12]: E1/E2=open, E3=E2, G12=G13=0.5E2, G23=0.35E2, n12=n13=n23=0.3. In Table 2, the lowest natural frequencies of square [0°/90°] and [0°/90°/90°/0°] laminates for various values of the thickness parameter a/h are compared with the previously published results. The material properties of the individual layers are taken from Wu and Chen [13], Nosier et al. [14] and Cho et al. [15]: E1/E2=40, E3=E2, G12=G13=0.6E2, G23=0.5E2, n12=n13=n23=0.25. Although a slight difference may be seen in the natural frequencies between the present results and other layerwise theories, this difference can be attributed to the degree of continuity conditions of displacements at the interface between layers. In equivalent single-layer theories, as in the present higher-order theories, the continuity conditions of displacement components are usually higher than C1. On the contrary, the C0 continuous displacement components are used in the layer-wise theories. The effect of the different continuity conditions of displacement components at the interface between the layers is very important in predicting the true mechanical characteristics of laminates. The present results for natural frequencies, obtained by a global higher-order approximate theory (M=5), are considered to be accurate enough for general crossply laminated composite plates with a small number of unknowns.
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Table 1 Comparison of natural frequency with other results (⍀, a/b=1, a/h=5, r=s=1, M=5; Stacking sequence: [0°/90°/0°/…]) No. of layers Solutiona
2
4
6
10
3
5
7
9
A B C A B C A B C A B C A B C A B C A B C A B C
E1/E2 3
10
15
30
40
0.2392 0.2388 0.2389 0.2493 0.2495 0.2491 0.2517 0.2517 0.2519 0.2530 0.2531 0.2536 0.2516 – 0.2513 0.2529 0.2528 0.2527 0.2533 0.2534 0.2535 0.2535 0.2536 0.2540
0.2671 0.2675 0.2669 0.3063 0.3002 0.3063 0.3164 0.3171 0.3169 0.3220 0.3224 0.3237 0.3109 – 0.3070 0.3195 0.3201 0.3173 0.3222 0.3224 0.3218 0.3234 0.3248 0.3244
0.2815 0.2809 0.2812 0.3307 0.3306 0.3309 0.3441 0.3442 0.3449 0.3518 0.3519 0.3542 0.3344 – 0.3278 0.3470 0.3470 0.3437 0.3514 0.3520 0.3505 0.3533 0.3535 0.3544
0.3117 0.3117 0.3116 0.3726 0.3725 0.3731 0.3914 0.3918 0.3926 0.4027 0.4028 0.4066 0.3739 – 0.3616 0.3931 0.3935 0.3876 0.4005 0.4004 0.3990 0.4040 0.4047 0.4058
0.3256 0.3236 0.3255 0.3887 0.3899 0.3893 0.4092 0.4100 0.4106 0.4220 0.4220 0.4265 0.3892 – 0.3745 0.4102 0.4121 0.4040 0.4190 0.4204 0.4173 0.4231 0.4237 0.4252
a A, 3-D solution (n23=0.49), Noor and Burton [1]; B, FEM solution, Kant and Kommineni [12]; C, Present solution (M=5)
6.3. Modal stress distributions in cross-ply laminated composite plates The interlaminar stresses are the dominant effects on the vibration response and delamination phenomena of multilayered composite plates. Therefore, the determination of modal stresses under vibration is one of the important research topics in the analysis of the mechanical behaviour of composite structures. Figs. 2 and 3 show the distributions of modal displacements and stresses associated with the fundamental frequency for two- and three-layer composite plates of the degree of orthotropy of the individual layers: E1/E2=30, E3=E2, G12=G13=0.6E2, G23=0.5E2, n12=n13=n23=0.25. Each of the response quantities in the figures is divided by its maximum absolute value. Note that the in-plane shear stress Sxy is continuous between the adjacent layers due to the orthogonal symmetry of both the displacements and material properties. The corresponding natural frequencies are shown in the captions of the figures.
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Table 2 Comparison of the lowest natural frequency with previously published results (⍀×(a/h)2, a/b=1, r=s=1) Layers
[0°/90°]
Solutiona
A B C D E [0°/90°/90°/0°] A B E
a/h 2
5
10
20
25
50
100
4.956 4.959 4.935 4.939 4.810 5.321 5.317 5.923
8.530 8.527 8.518 8.521 8.388 10.688 10.682 10.673
10.338 10.337 10.333 10.335 10.270 15.072 15.069 15.066
11.037 11.037 11.036 11.036 11.016 17.637 17.636 17.535
11.132 11.132 11.131 11.132 11.118 18.056 18.055 18.054
11.264 11.264 11.263 11.263 11.260 18.670 18.670 18.670
11.297 11.297 11.297 11.297 11.296 18.835 18.835 18.835
a A, Present solution (M=5); B, Wu et al. [13] (Local higher-order theory); C, Nosier et al. [14] (3-D elasticity solution); D, Nosier et al. [14] (Layerwise plate theory); E, Cho et al. [15] (Individual-layer plate theory)
Fig. 2. Displacement and stress distributions of cross-ply laminated composite plates (K=2: [0°/90°], r=s=1, a/b=1, ⍀=1.2250 (a/h=2), 0.3250 (a/h=5), 0.09623 (a/h=10)).
6.4. Fundamental natural frequencies with respect to number of layers Fig. 4 shows the variation of fundamental natural frequencies of cross-ply laminated composite plates with respect to the number of layers. The material properties of the individual layers are given by E3=E2, G12=G13=0.5E2, G23=0.6E2, n12=n13=n23=0.25. Since the total number of unknowns of the present global higherorder theory does not increase as the number of layers increases, multilayered com-
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H. Matsunaga / Thin-Walled Structures 40 (2002) 557–571
Fig. 3. Displacement and stress distributions of cross-ply laminated composite plates (K=3: [0°/90°/0°], r=s=1, a/b=1, ⍀=1.2764 (a/h=2), 0.3936 (a/h=5), 0.1358 (a/h=10)).
Fig. 4.
Natural frequency vs number of layers (a/b=1, r=s=1, [0°/90°/0°/90°/…]).
posite plates with a large number of layers can be analyzed without difficulty. For small numbers of layers, the natural frequency is influenced largely by the number of layers and the stacking sequences. However, for large number of layers, the natural frequency does not change and approaches a constant value. The feature of the ani-
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sotropy effects becoming stabilized as the number of layers increases can also be noticed for higher frequencies.
6.5. Natural frequencies of a square plate subjected to uniaxial initial in-plane stress The minimum natural frequencies of a square plate subjected to uniaxial (=0) initial in-plane stress in the x-direction are plotted versus the initial in-plane stresses in Fig. 5. The material properties of the individual layers are given by E1/E2=30, E3=E2, G12=G13=0.5E2, G23=0.35E2, n12=n13=n23=0.25. For the thickness parameter a/h=5, frequency curves are plotted for several displacement modes of r with s=1. When the natural frequencies go to zero, the initial in-plane stresses reduce to the buckling stresses of the plate. In Fig. 5, since the two sets of results look almost identical, the numerical values of natural frequency ⍀0 of laminates without initial in-plane stress, and of the buckling stresses ⌳b, are tabulated in the figures to improve the distinction between two- and three-layer laminates. In the case of simply supported cross-ply laminated composite plates subjected to initial in-plane stresses, the natural frequency ⍀ can be expressed in terms of the natural frequency ⍀0 of laminates without initial in-plane stresses, from a comparison of the equations of motion, as follows:
再 冉 冊 冎冉 冊
⍀2 ⫽ ⍀20 ⫹ p2 r2 ⫹
Fig. 5.
a 22 h 2 s ⌳ b a
(26)
Natural frequency vs in-plane stress curves (a/b=1, a/h=5, s=1, r=1⫺5, =0, E1/E2=30).
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The buckling stress ⌳b is related to the natural frequency ⍀0 as follows: ⌳b ⫽
⫺⍀02 p2{r2 ⫹ (a / b)2s2}(h / a)2
(27)
7. Conclusions It has been shown that the present global higher-order theory can provide accurate results for natural frequencies, modal displacements and stresses for general crossply laminated composite plates subjected to initial in-plane stresses. It should be pointed out that the total number of unknowns of the present approximate higherorder theory is (6M⫺1), which is not dependent on the number of layers in any multilayered plates. The present theory has the advantage of predicting natural frequencies and buckling stresses of cross-ply multilayered composite plates without increasing the unknowns involved as the number of layers increases. The distribution of modal stresses in the thickness direction has also been obtained accurately at the ply level.
Acknowledgements The author wishes to recognize the support of ‘Japan Society for the Promotion of Science’ grant-in-aid for scientific research (C)(2) No. 12650591.
References [1] Noor AK, Burton WS. Stress and free vibration analyses of multilayered composite plates. Compos Struct 1989;11:183–204. [2] Srinivas S, Rao AK. Bending, vibration and buckling of simply supported thick orthotropic rectangular plates and laminates. Int J Solids Struct 1970;6:1463–81. [3] Srinivas S, Joga Rao CV, Rao AK. An exact analysis for vibration of simply supported homogeneous laminated thick rectangular plates. J Sound Vib 1970;12:187–99. [4] Noor AK. Free vibrations of multilayered composite plates. AIAA J 1973;11(7):1038–9. [5] Noor AK, Burton WS. Assessment of shear deformation theories for multilayered composite plates. Appl Mech Rev 1989;42(1):1–13. [6] Reddy JN. A review of refined theories of laminated composite plates. Shock Vib Digest 1990;22(7):3–17. [7] Leissa AW. A review of laminated composite plate buckling. Appl Mech Rev 1987;40(5):575–91. [8] Matsunaga H. The application of a two-dimensional higher-order theory for the analysis of a thick elastic plate. Comput Struct 1992;45(4):633–48. [9] Matsunaga H. Free vibration and stability of thick elastic plates subjected to in-plane forces. Int J Solids Struct 1994;31(22):3113–24. [10] Matsunaga H. Buckling instabilities of thick elastic plates subjected to in-plane stresses. Comput Strut 1997;62(1):205–14. [11] Matsunaga H. Vibration and stability of cross-ply laminated composite plates according to a global higher-order plate theory. Compos Struct 2000;48(4):231–44.
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[12] Kant T, Kommineni JR. Large amplitude free vibration analysis of cross-ply composite and sandwich laminates with a refined theory and C0 finite elements. Comput Struct 1994;50:123–34. [13] Wu C-P, Chen W-Y. Vibration and stability of laminated plates based on a local high order plate theory. J Sound Vib 1994;177(4):503–20. [14] Nosier A, Kapania RK, Reddy JN. Free vibration analysis of laminated plates using a layerwise theory. AIAA J 1993;31(12):2335–46. [15] Cho KN, Bert CW, Striz AG. Free vibrations of laminated rectangular plates analyzed by higher order individual-layer theory. J Sound Vib 1991;145(3):429–42.
Vibration of cross-ply laminated composite plates subjected to initial in-plane stresses Hiroyuki Matsunaga ∗ Department of Architecture, Setsunan University 17-8, Ikeda-nakamachi, Neyagawa, Osaka 572-8508, Japan Received 13 July 2001; received in revised form 3 December 2001; accepted 12 December 2001
Abstract Natural frequencies, modal displacements and stresses of cross-ply laminated composite plates subjected to initial in-plane stresses are analyzed by taking into account the effects of higher-order deformations and rotatory inertia. By using the method of power series expansion of displacement components, a set of fundamental dynamic equations of a two-dimensional higher-order theory for rectangular laminates is derived through Hamilton’s principle. Several sets of truncated approximate theories can be derived to solve the eigenvalue problems of a simply supported laminated plate. After examining the convergence properties of the lowest natural frequency, only the numerical results for M=5, which are considered to be sufficient with respect to the accuracy of solutions, are presented. Numerical results are compared with those of the published existing three-dimensional theory and FEM solutions. The modal displacement and stress distributions in the thickness direction are plotted in figures. The buckling stresses can be obtained in terms of the natural frequencies of the laminates without initial in-plane stresses. 2002 Elsevier Science Ltd. All rights reserved. Keywords: Higher-order theory; Cross-ply; Composite; Laminated plate; Vibration; Modal displacement and stress
1. Introduction Stiff, strong and lightweight composite materials are being widely used in many structural members, such as multilayered composite beams, plates and shells. ∗
Tel.: +81-72-839-9129; fax: +81-72-838-6599. E-mail address: [email protected] (H. Matsunaga).
0263-8231/02/$ - see front matter 2002 Elsevier Science Ltd. All rights reserved. PII: S 0 2 6 3 - 8 2 3 1 ( 0 2 ) 0 0 0 1 2 - 5
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Vibration problems of laminated composite plates have attracted the attention of many researchers up to the present. The mechanical behaviors of laminated composite plates are strongly dependent on the degree of orthotropy of individual layers, the low ratio of transverse shear modulus to the in-plane modulus and the stacking sequence of laminates. It has been known that classical laminated plate theories, based on the Kirchhoff hypothesis, are inadequate for predicting the gross response characteristics of moderately thick laminated composite plates and/or highly anisotropic laminated composite plates. The inaccuracy is due to neglecting the transverse shear and normal strains in the laminate. In order to take into account the effects of low ratio of transverse shear modulus to the in-plane modulus, a number of first-order shear deformation theories have been developed. However, since in the Mindlin-type first-order shear deformation plate theory the transverse shear strains are assumed to be constant in the thickness direction, shear correction factors have to be incorporated to adjust the transverse shear stiffness for studying the static or dynamic problems of plates. Although the Mindlin-type first order shear deformation theory is quite accurate for the gross responses, such as natural frequencies of moderately thick laminates, the accuracy of solutions will be strongly dependent on predicting better estimates for the shear correction factors. It has been shown that the Mindlin-type first-order shear deformation theories are inadequate for the accurate prediction of the modal displacements and interlaminar stresses of laminated composite plates (see, for example, [1]). Without requiring the specification of a shear correction coefficient in the Mindlintype first-order shear deformation plate theory, various higher-order plate theories have been developed. In order to obtain the accurate predictions of the gross response characteristics such as natural frequencies, a number of contributions based on the three-dimensional elasticity theory have been made to analyze laminated composite plates [2-4]. However, accurate solutions based on the three-dimensional elasticity theory are often computationally expensive. Several approximately refined two-dimensional higher-order theories have been proposed to analyze the response characteristics of plates. Three excellent reviews of refined theories of laminated composite plates have been presented by Noor and Burton [5], Reddy [6] and Leissa [7]. In recent years, the layer-wise theories and individual layer theories have been presented to obtain more accurate information at the ply level. Although such theories have been proved to be one of the best alternatives to three-dimensional elasticity theories, these theories require numerous unknowns for multilayered plates and are often computationally expensive in obtaining accurate solutions. The number of unknowns in a laminate (which is dependent on the number of layers) will increase dramatically as the number of layers increases. A number of single-layer (global) higher-order plate theories that includes the effects of transverse shear and normal deformations has been published in the literature. Although various models of higher-order displacement fields have been considered, most of these theories are the third-order theories in which the in-plane displacements are assumed to be a cubic expression of the thickness coordinate and the out-of-plane displacement to be a quadratic expression at most. For isotropic plates, a two-dimensional higher-order theory has been developed, and has been
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applied to the statics and dynamics of thin and thick plates by Matsunaga [8-10]. Natural frequencies and buckling stresses for isotropic plates subjected to initial inplane stresses have been analyzed by using the approximate two-dimensional higherorder theories. Remarkable effects of transverse shear and normal deformations have been predicted in the results. General higher-order theories of laminates, which take into account the complete effects of transverse shear, normal deformations and rotatory inertia, have been investigated recently for the vibration and stability problems of multilayered composite plates [11]. This paper presents a global higher-order theory for analyzing natural frequencies, modal displacements and stresses for cross-ply laminated composite plates subjected to initial in-plane stresses. The complete effects of shear, normal deformations and rotatory inertia can be taken into account within the approximate two-dimensional theory. Several sets of the governing equations of truncated approximate theories are applied to the analysis of vibration problems of a simply supported multilayered elastic plate. Based on the power series expansions of displacement components, a fundamental set of equations of a two-dimensional higher-order plate theory is derived through Hamilton’s principle. Natural frequencies of a cross-ply laminated composite plate subjected to initial in-plane stresses are obtained by solving the eigenvalue problem numerically. Convergence properties of the present numerical solutions have been shown to be accurate for the natural frequencies, with respect to the order of approximate theories in the previous work [11]. A comparison of the present results is made with previously published results of the three-dimensional layerwise theory [1] and FEM [12] solutions. The buckling stresses of cross-ply laminated composite plates subjected to initial in-plane stresses are computed by increasing the absolute values of compressive stresses until the natural frequency vanishes. For multilayered plates, the distribution of modal displacements and stresses in the thickness direction has been obtained accurately at the ply level. The modal transverse stresses have been obtained by integrating the three-dimensional equations of motion in the thickness direction, and satisfying the continuity conditions at layer interfaces and stress boundary conditions on the top and bottom surfaces. There exist no results in the open literature for the natural frequencies of multilayered composite plates with a large number of layers (e.g. 50 layers). The present theory has the advantage of predicting natural frequencies of multilayered composite plates without increasing the unknowns involved as the number of layers increases.
2. Higher-order theory for laminated composite plates A cross-ply laminated composite plate of uniform thickness h, having a rectangular plan a×b, is shown in Fig. 1. Introducing a Cartesian reference coordinate system xi (I=1,2,3) on the middle plane of the plate, the dynamic displacement components are expressed as ⌼a ⬅ ⌼a(xi;t), ⌼3 ⬅ ⌼3(xi;t)
(1)
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Fig. 1.
K-layers cross-ply laminated composite plates and coordinates.
where t denotes time. Greek lower case subscripts are assumed to range over the integers 1,2. The displacement components may be expanded into power series of the thickness coordinate x3 as follows:
冘 ⬁
na ⫽
冘 ⬁
(n)
na(x3)n, n3 ⫽
n⫽0
(n)
n3(x3)n
(2)
n⫽0
where n=0,1,2,…,⬁. Based on this expression for the displacement components, a set of linear fundamental equations of a two-dimensional higher-order plate theory can be obtained as follows. 2.1. Strain-displacement relations Strain components may also be expanded as follows:
冘 ⬁
gab ⫽
冘 ⬁
(n)
g ab(x3)n, ga3 ⫽
n⫽0
(n)
g a3(x3)n, g33 ⫽
n⫽0
冘 ⬁
(n)
g 33(x3)n
(3)
n⫽0
and strain-displacement relations can be written as (n) (n) (n ⫹ 1) (n) (n) 1 (n) 1 g ab ⫽ ( n a,b ⫹ n b,a), g a3 ⫽ {(n ⫹ 1)其 n a ⫹ n 3,a, g 33 ⫽ 2 2
(n)
(n ⫹ 1)
(n ⫹ 1) n3
(4)
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561
where a comma denotes partial differentiation with respect to the coordinate subscript that follows. 2.2. Equations of motion Introducing stress components sab, sa3 and s33, Hamilton’s principle is applied to derive the equations of dynamic equilibrium and natural boundary conditions of the plate. In order to treat vibration problems of a plate subjected to initial in-plane distributed uniformly in x3-direction, additional terms due to these stresses sab 0 stresses, which are assumed to remain unchanged during vibration are taken into consideration. The boundary conditions are assumed to be traction-free on the top and bottom surfaces of the plate. Hamilton’s principle for the present problem may be expressed for an arbitrary time interval t1 to t2 as follows:
冕冕 t2
{sabdgab ⫹ 2sa3dga3 ⫹ s33dg33 ⫹ s0ab(n˙ l,adn˙ l,b ⫹ n˙ 3,adn˙ 3,b)
(5)
V
t1
⫺r(˙ ad˙ a ⫹ ˙ 3d˙ 3)其dV dt ⫽ 0 where the over dot indicates partial differentiation with respect to time, and r=r(z) denotes the mass density, while dV denotes the volume element. By performing the variation as indicated in Eq. (5), the equations of motion on the middle plane of a plate are obtained as follows: (n) ab
N ,a⫺n Q ⫹ (n)
冘 ⬁
(n⫺1) b
(n ⫹ m ⫹ 1)(m) b
m⫽0
冘
冘
⬁
(n⫺1)
Qa,a⫺n T ⫹
冘 ⬁
(n ⫹ m ⫹ 1) (m) a 0 b,al m⫽0
s
⫺
(n ⫹ m ⫹ 1) (m) ab 3,ab 0
s
⬁
⫺
m⫽0
r
¨ ⫽ 0
(n ⫹ m ⫹ 1)(m) 3
r
¨ ⫽ 0
(6)
(7)
m⫽0
where n, m=0,1,2, . . . ,⬁ and
冘
hk+1n+m+1⫺hkn+m+1 (n+m+1) ⫽ ,r n⫹m⫹1 k⫽1 K
sab 0
(n+m+1)
⫽ sab 0
冘 K
k⫽1
r(k)
hk+1n+m+1⫺hkn+m+1 n⫹m⫹1
(8)
(k) and hk denote the initial in-plane stress, the mass density The parameters sab 0 , r and thickness coordinate of the lower side of kth layer, respectively, and K denotes the total number of layers in the laminates. The stress resultants are defined as follows:
冘 K
(N
ab
(n)
,Q
(n)
a
,T ) ⫽ (n)
k⫽1
a3 33 (sab (k) ,s(k),s(k))
hk+1n+1⫺hkn+1 n⫹1
(9)
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2.3. Boundary conditions The boundary conditions are assumed to be traction-free on the upper and lower surfaces of the plate. For equations of boundary conditions along the boundaries on the middle plane, the following quantities
冘 ⬁
na(n) or nb[Nab
(n)
⫹
(n ⫹ m ⫹ 1)(m) bl a 0 l
(n)
(n)
n , ], n 3 or nb[Q b
s
(10)
m⫽0
冘 ⬁
⫹
(n ⫹ m ⫹ 1)(m) 3 ab 0 a
n, ]
s
m⫽0
are to be prescribed. 2.4. Constitutive relations For elastic and orthotropic materials, the constitutive relations for the kth layer of a laminated plate can be written as abln ab33 a3 a3l3 33 33ln 3333 sab (k) ⫽ C(k) gln ⫹ C(k) g33, s(k) ⫽ C(k) gl3, s(k) ⫽ C(k) gln ⫹ C(k) g33 ijkl
The material constants C as follows:
(11)
(i, j, k, l=1,2,3) for an individual layer can be expressed
(k) (k) 1122 (k) (k) (k) 1133 (k) (k) (k) C1111 ⫽ E(k) ⫽ E(k) ⫽ E(k) (k) 1 (1⫺n23 n32 ) / m, C(k) 1 (n21 ⫹ n31 n23 ) / m, C(k) 1 (n31 ⫹ n21 n32 ) / m, (k) (k) (k) 2222 (k) (k) 2233 (k) (k) (k) ⫽ E(k) ⫽ E(k) ⫽ E(k) C2211 (k) 2 (n12 ⫹ n32 n13 ) /m, C(k) 2 (1⫺n31 n13 ) / m, C(k) 2 (n32 ⫹ n12 n31 ) / m, (k) (k) (k) 3322 (k) (k) (k) 3333 (k) (k) C3311 ⫽ E(k) ⫽ E(k) ⫽ E(k) (k) 3 (n13 n12 n23 ) / m, C(k) 3 (n23 ⫹ n21 n13 ) / m, C(k) 3 (1⫺n12 ⫹ n21 ) / m,
(12)
1313 1212 C2323 ⫽ 2G(k) ⫽ 2G(k) ⫽ 2G(k) (k) 23 , C(k) 13 , C(k) 12
(k) (k) (k) (k) (k) (k) (k) (k) (k) where m ⫽ 1⫺n12 n21 ⫺n23 n32 ⫺n31 n13 ⫺2n21 n32 n13 . The parameters Ea(k) and E3(k) are (k) (k) Young’s moduli, while nab and na3 are Poisson’s ratios. The first suffix of n denotes the direction of stress, and the second denotes that of expansion and contraction. The following relations can be established by the reciprocal theorem:
E1n21 ⫽ E2n12, E1n31 ⫽ E3n13, E2n32 ⫽ E3n23
(13)
Stress resultants can be derived from Eqs. (9a) and (11) and Eqs. (3a) and (4) in terms of the expanded displacement components. The equations of motion, (6) and (7), can also be expressed in terms of the expanded displacement components. 2.5. Mth order approximate theory Since the fundamental equations mentioned above are complicated, approximate theories of various orders may be considered for the present problem. A set of the following combination of Mth order (Mⱖ1) approximate equations is proposed as follows:
冘
2M⫺1
na ⫽
m⫽0
冘
2M⫺2
n (x ) , n3 ⫽ (m) a
3 m
m⫽0
3 m n(m) 3 (x )
(14)
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where m=0,1,2,3, … The total number of the unknown displacement components is (6M⫺1), which is not dependent on the number of layers in a laminate. In the above cases of M=1, an assumption of plane strains is inherently imposed. Another set of the governing equations of lowest order approximate theory (M=1) is derived on the basis of an assumption that the normal stress s33 is zero (which is known as Mindlin-type laminated plate theory), with the shear correction coefficient 2 ⫽ 1.
3. Navier solution for simply supported plate In the following analysis, the Cartesian reference coordinate x=x1, y=x2 and z=x3 and displacement components u=n1, n=n2 and w=n3 are followed. For a simply supported rectangular laminated plate, the boundary conditions (10) can be expressed for the x-constant edges, u,x(n) ⫽ 0, n(n) ⫽ 0, w(n) ⫽ 0
(15)
and for the y-constant edges, u(n) ⫽ 0, n,y(n) ⫽ 0, w(n) ⫽ 0
(16)
Following the Navier solution procedure, displacement components that satisfy the boundary conditions (15) and (16) may be expressed as
冘冘 冘冘 ⬁
u
(n)
⬁
rpx spy ⫽ u cos sin ·eiwt, n(n) ⫽ a b r ⫽ 1s ⫽ 1 ⬁
w(n) ⫽
(n) rs
⬁
(n) wrs sin
r ⫽ 1s ⫽ 1
rpx spy iwt sin ·e a b
冘冘 ⬁
⬁
r ⫽ 1s ⫽ 1
(n) nrs sin
rpx spy iwt cos ·e a b
(17)
(18)
where r, s=1,2,3,…,⬁ and w denotes the circular frequency; i is the imaginary unit. The equations of motion (6) and (7) are rewritten in terms of the generalized displace(n)
(n)
(n)
ment components urs, nrs and wrs. The dimensionless natural frequency and initial in-plane stress are defined as follows: ⍀ ⫽ wh冑r(1) / E2(1), ⌳ ⫽ s0ab / E(1) 2
(19)
4. Eigenvalue vibration problem Using Eqs. (17a) and (18), the equations of motion, (6) and (7) (expressed in terms of displacement component) can be rewritten by collecting the coefficients for the generalized displacements of any fixed values r and s. The generalized displacement vector {U其 for the Mth order approximate theory is expressed as
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H. Matsunaga / Thin-Walled Structures 40 (2002) 557–571 (2M⫺1) (0) (2M⫺2) {U其T ⫽ {u(0) ;nrs ,...,n(2M⫺1) ;w(0) 其 rs ,...,urs rs rs ,...,wrs
(20)
For free vibration problems, the equations of motion can be expressed as the following eigenvalue problem: ([K]⫺⍀2[M]){U其 ⫽ {0其
(21)
where [K] denotes the stiffness matrix and [M] denotes the mass matrix. 5. Determination of modal stress distributions The modal in-plane stresses for the kth layer can be derived in terms of the expanded displacement components by introducing the strain-displacement relations (4) into the constitutive relations (11). The modal transverse stresses for the kth layer are obtained by integrating the three-dimensional equations of motion in the thickness direction, starting from the lower (or upper) surface of the laminates as follows:
冕
x3
sa3 (k) ⫽ ⫺
冕
x3
(sab ¨ a)dx3 ⫹ Ca(k), s33 (k),b⫺r (k) ⫽ ⫺
hk
(sa3 ¨ 3)dx3 ⫹ C3(k) (k),a⫺r
(22)
hk
a and C3(k), are constants obtained from the stress-continuity conditions at where C(k) layer interfaces of the layer. If the boundary conditions for transverse stresses are prescribed on one of the outer surfaces of the plate, the stress boundary conditions on the other surface can be satisfied through the equations of motion, (6) and (7). Because of the discontinuity of the in-plane stresses at layer interfaces, the integration is performed in a piecewise manner. The modal stress components in the kth layer of cross-ply laminated composite plates can be expressed as follows:
冘 ⬁
sab (k) ⫽
(n) (n) [Dablngln ⫹ Dab33g33 ](k)(x3)n
(23)
n⫽0
冘 ⬁
sa3 (k) ⫽
[(x3)n+1⫺hkn+1] (n) (n) (n) [(Dablngln ⫹ Dab33g33 ),b ⫹ rw2na ](k) n⫹1 n⫽0
(24)
3 ⫹ sa3 (k⫺1)兩x =hk
冘 ⬁
s33 (k) ⫽
冋
(n) (n) [(Dablngln ⫹ Dab33g33 ),ab ⫹ rw2n,a(n) a ](k)
n⫽0
册 冘 ⬁
(x3)n+2⫺hkn+2 (n ⫹ 1)(n ⫹ 2)
3 n+1 ⫺hkn+1] a3 hkn+1(x3⫺hk) (n)[(x ) ⫺rw2 n3 ⫺s,a(k⫺1)兩[x3⫺h ] ⫺ k n⫹1 n⫹1 n⫽0 3 ⫹ s33 (k⫺1)|x =hk
(25)
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6. Numerical studies and results 6.1. Numerical examples Natural frequencies of cross-ply laminated composite plates with simply supported edges are determined. The orthotropic material for each layer are (k) (k) (k) (k) (k) (k) , G23 and G12 . Poisson’s ratios are given by n13 , n23 and n12 , and E1(k), E2(k), E3(k), G13 the other Poisson’s ratios can be obtained by the reciprocal theorem. The material properties are assumed to be the same in all the layers, but the fibre orientations may be different among the layers. The fibre orientations of the different laminae alternate between 0° and 90° with respect to the x-axis. The thickness of each layer is identical for the 0° and 90° layers in the laminates. Both symmetric and skewsymmetric laminations with respect to the middle plane are considered. In the symmetrical laminates having an odd number of layers, the 0° layers are at the outer surfaces of the laminate. The mass density is assumed to be uniform in the thickness direction, i.e. r(1), r(2), …, r(k) are identical. All the numerical results are shown in dimensionless quantities. It has been noticed in previous work [11] that results for M=2–5 converge accurately enough in the practical range of thickness parameter a/h. Only the results for M=5 are given in the present numerical examples. 6.2. Comparison of the lowest natural frequency with those of existing theories Table 1 shows a comparison of the lowest natural frequencies of cross-ply square laminates [0°/90°/0°/…] with other solutions for various values of the degree of orthotropy of the individual layers E1/E2 and various numbers of layers. For this numerical example, the material properties of the individual layers are taken from Noor and Burton [1], and Kant and Kommineni [12]: E1/E2=open, E3=E2, G12=G13=0.5E2, G23=0.35E2, n12=n13=n23=0.3. In Table 2, the lowest natural frequencies of square [0°/90°] and [0°/90°/90°/0°] laminates for various values of the thickness parameter a/h are compared with the previously published results. The material properties of the individual layers are taken from Wu and Chen [13], Nosier et al. [14] and Cho et al. [15]: E1/E2=40, E3=E2, G12=G13=0.6E2, G23=0.5E2, n12=n13=n23=0.25. Although a slight difference may be seen in the natural frequencies between the present results and other layerwise theories, this difference can be attributed to the degree of continuity conditions of displacements at the interface between layers. In equivalent single-layer theories, as in the present higher-order theories, the continuity conditions of displacement components are usually higher than C1. On the contrary, the C0 continuous displacement components are used in the layer-wise theories. The effect of the different continuity conditions of displacement components at the interface between the layers is very important in predicting the true mechanical characteristics of laminates. The present results for natural frequencies, obtained by a global higher-order approximate theory (M=5), are considered to be accurate enough for general crossply laminated composite plates with a small number of unknowns.
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Table 1 Comparison of natural frequency with other results (⍀, a/b=1, a/h=5, r=s=1, M=5; Stacking sequence: [0°/90°/0°/…]) No. of layers Solutiona
2
4
6
10
3
5
7
9
A B C A B C A B C A B C A B C A B C A B C A B C
E1/E2 3
10
15
30
40
0.2392 0.2388 0.2389 0.2493 0.2495 0.2491 0.2517 0.2517 0.2519 0.2530 0.2531 0.2536 0.2516 – 0.2513 0.2529 0.2528 0.2527 0.2533 0.2534 0.2535 0.2535 0.2536 0.2540
0.2671 0.2675 0.2669 0.3063 0.3002 0.3063 0.3164 0.3171 0.3169 0.3220 0.3224 0.3237 0.3109 – 0.3070 0.3195 0.3201 0.3173 0.3222 0.3224 0.3218 0.3234 0.3248 0.3244
0.2815 0.2809 0.2812 0.3307 0.3306 0.3309 0.3441 0.3442 0.3449 0.3518 0.3519 0.3542 0.3344 – 0.3278 0.3470 0.3470 0.3437 0.3514 0.3520 0.3505 0.3533 0.3535 0.3544
0.3117 0.3117 0.3116 0.3726 0.3725 0.3731 0.3914 0.3918 0.3926 0.4027 0.4028 0.4066 0.3739 – 0.3616 0.3931 0.3935 0.3876 0.4005 0.4004 0.3990 0.4040 0.4047 0.4058
0.3256 0.3236 0.3255 0.3887 0.3899 0.3893 0.4092 0.4100 0.4106 0.4220 0.4220 0.4265 0.3892 – 0.3745 0.4102 0.4121 0.4040 0.4190 0.4204 0.4173 0.4231 0.4237 0.4252
a A, 3-D solution (n23=0.49), Noor and Burton [1]; B, FEM solution, Kant and Kommineni [12]; C, Present solution (M=5)
6.3. Modal stress distributions in cross-ply laminated composite plates The interlaminar stresses are the dominant effects on the vibration response and delamination phenomena of multilayered composite plates. Therefore, the determination of modal stresses under vibration is one of the important research topics in the analysis of the mechanical behaviour of composite structures. Figs. 2 and 3 show the distributions of modal displacements and stresses associated with the fundamental frequency for two- and three-layer composite plates of the degree of orthotropy of the individual layers: E1/E2=30, E3=E2, G12=G13=0.6E2, G23=0.5E2, n12=n13=n23=0.25. Each of the response quantities in the figures is divided by its maximum absolute value. Note that the in-plane shear stress Sxy is continuous between the adjacent layers due to the orthogonal symmetry of both the displacements and material properties. The corresponding natural frequencies are shown in the captions of the figures.
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Table 2 Comparison of the lowest natural frequency with previously published results (⍀×(a/h)2, a/b=1, r=s=1) Layers
[0°/90°]
Solutiona
A B C D E [0°/90°/90°/0°] A B E
a/h 2
5
10
20
25
50
100
4.956 4.959 4.935 4.939 4.810 5.321 5.317 5.923
8.530 8.527 8.518 8.521 8.388 10.688 10.682 10.673
10.338 10.337 10.333 10.335 10.270 15.072 15.069 15.066
11.037 11.037 11.036 11.036 11.016 17.637 17.636 17.535
11.132 11.132 11.131 11.132 11.118 18.056 18.055 18.054
11.264 11.264 11.263 11.263 11.260 18.670 18.670 18.670
11.297 11.297 11.297 11.297 11.296 18.835 18.835 18.835
a A, Present solution (M=5); B, Wu et al. [13] (Local higher-order theory); C, Nosier et al. [14] (3-D elasticity solution); D, Nosier et al. [14] (Layerwise plate theory); E, Cho et al. [15] (Individual-layer plate theory)
Fig. 2. Displacement and stress distributions of cross-ply laminated composite plates (K=2: [0°/90°], r=s=1, a/b=1, ⍀=1.2250 (a/h=2), 0.3250 (a/h=5), 0.09623 (a/h=10)).
6.4. Fundamental natural frequencies with respect to number of layers Fig. 4 shows the variation of fundamental natural frequencies of cross-ply laminated composite plates with respect to the number of layers. The material properties of the individual layers are given by E3=E2, G12=G13=0.5E2, G23=0.6E2, n12=n13=n23=0.25. Since the total number of unknowns of the present global higherorder theory does not increase as the number of layers increases, multilayered com-
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Fig. 3. Displacement and stress distributions of cross-ply laminated composite plates (K=3: [0°/90°/0°], r=s=1, a/b=1, ⍀=1.2764 (a/h=2), 0.3936 (a/h=5), 0.1358 (a/h=10)).
Fig. 4.
Natural frequency vs number of layers (a/b=1, r=s=1, [0°/90°/0°/90°/…]).
posite plates with a large number of layers can be analyzed without difficulty. For small numbers of layers, the natural frequency is influenced largely by the number of layers and the stacking sequences. However, for large number of layers, the natural frequency does not change and approaches a constant value. The feature of the ani-
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sotropy effects becoming stabilized as the number of layers increases can also be noticed for higher frequencies.
6.5. Natural frequencies of a square plate subjected to uniaxial initial in-plane stress The minimum natural frequencies of a square plate subjected to uniaxial (=0) initial in-plane stress in the x-direction are plotted versus the initial in-plane stresses in Fig. 5. The material properties of the individual layers are given by E1/E2=30, E3=E2, G12=G13=0.5E2, G23=0.35E2, n12=n13=n23=0.25. For the thickness parameter a/h=5, frequency curves are plotted for several displacement modes of r with s=1. When the natural frequencies go to zero, the initial in-plane stresses reduce to the buckling stresses of the plate. In Fig. 5, since the two sets of results look almost identical, the numerical values of natural frequency ⍀0 of laminates without initial in-plane stress, and of the buckling stresses ⌳b, are tabulated in the figures to improve the distinction between two- and three-layer laminates. In the case of simply supported cross-ply laminated composite plates subjected to initial in-plane stresses, the natural frequency ⍀ can be expressed in terms of the natural frequency ⍀0 of laminates without initial in-plane stresses, from a comparison of the equations of motion, as follows:
再 冉 冊 冎冉 冊
⍀2 ⫽ ⍀20 ⫹ p2 r2 ⫹
Fig. 5.
a 22 h 2 s ⌳ b a
(26)
Natural frequency vs in-plane stress curves (a/b=1, a/h=5, s=1, r=1⫺5, =0, E1/E2=30).
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The buckling stress ⌳b is related to the natural frequency ⍀0 as follows: ⌳b ⫽
⫺⍀02 p2{r2 ⫹ (a / b)2s2}(h / a)2
(27)
7. Conclusions It has been shown that the present global higher-order theory can provide accurate results for natural frequencies, modal displacements and stresses for general crossply laminated composite plates subjected to initial in-plane stresses. It should be pointed out that the total number of unknowns of the present approximate higherorder theory is (6M⫺1), which is not dependent on the number of layers in any multilayered plates. The present theory has the advantage of predicting natural frequencies and buckling stresses of cross-ply multilayered composite plates without increasing the unknowns involved as the number of layers increases. The distribution of modal stresses in the thickness direction has also been obtained accurately at the ply level.
Acknowledgements The author wishes to recognize the support of ‘Japan Society for the Promotion of Science’ grant-in-aid for scientific research (C)(2) No. 12650591.
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