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On the damping analysis of FRP laminated composite plates
Composite Structures 57 (2002) 169–175 www.elsevier.com/locate/compstruct
On the damping analysis of FRP laminated composite plates Yoshiki Ohta a
a,*
, Yoshihiro Narita a, Kenta Nagasaki
b
Department of Mechanical Systems Engineering, Hokkaido Institute of Technology, 7-15 Maeda, Tein-ku, Sapporo 006-8585, Japan b Graduate School, Hokkaido Institute of Technology, 7-15 Maeda, Tein-ku, Sapporo 006-8585, Japan
Abstract This paper presents the damping analysis of fiber reinforced plastics laminated composite plates. For this purpose, the maximum strain and kinetic energies of a cross-ply laminated plate are evaluated analytically based on the three-dimensional theory of elasticity. The displacements of the simply supported rectangular plates are expanded into the polynomial forms with respect to a thickness coordinate, and then governing equations are formulated by using the Ritz’s method. In the numerical calculations, natural frequencies and modal damping ratios are calculated for the plates with different stacking sequence and thickness ratios. The validity of the assumption of deformations and the applicability of the other plate theories (e.g. classical lamination theory (CLT), first-order shear deformation theory (FSDT) and higher-order shear deformation theory) to the laminated thick plates are discussed by comparing the numerical results obtained by the present method with the CLT and the FSDT solutions. Ó 2002 Elsevier Science Ltd. All rights reserved.
1. Introduction Recently fiber reinforced plastics (FRP) are being increasingly used in the structural applications, because the technical merits of high strength to weight ratio and high stiffness to weight ratio of FRP composite materials are outstanding. It is also known that the anisotropic property in FRP materials considerably affects the dynamic characteristics (e.g. vibration and buckling) of structural elements such as plates and shells. Therefore a large number of theoretical and experimental studies have been published on the dynamic problems of FRP laminated composite plates and shells. In most of the analyses found in the literatures, the laminated plate is macroscopically modeled as a thin plate of general anisotropy, and is analyzed by using the classical lamination theory (CLT) (e.g. Vinson et al. [1]) based on the Kirchhoff’s hypotheses, where transverse shear deformations and extension in the thickness direction of the plate are ignored. However the CLT cannot give accurate results for laminated thick plates, because the transverse shear deformations cannot be ignored due to the relatively smaller shear elastic moduli of FRP composite materials than the other elastic moduli. Thus the analyses including the effect of shear de*
Corresponding author. Tel.: +81-11-688-2284; fax: +81-11-6813622. E-mail address: [email protected] (Y. Ohta).
formations are required for the laminated thick plates. Yang et al. [2], Whitney et al. [3] and Dong et al. [4] analyzed laminated thick plates and shells based on the first-order shear deformation theory (FSDT), in which in-plane displacements are assumed to vary lineally through the thickness. However, the FSDT cannot satisfy the free surface conditions on transverse shear stresses on the top and bottom surfaces of the plate and thus requires a shear correction factor multiplied to the transverse shear stiffness of the plate. Reddy [5] proposed the higher-order shear deformation theory (HSDT), wherein in-plane displacements are expressed in the polynomial forms of third order. This theory satisfies the free surface conditions exactly and does not require the shear correction to the transverse shear stiffness. There are few papers discussing the displacement functions in the evaluation of frequency and damping ratio of FRP laminated composite plates although these theories have been developed. The purpose of this paper is to present the damping analysis of FRP laminated composite thick plates. For this purpose, the maximum strain and kinetic energies of an FRP cross-ply laminated plate are evaluated analytically based on the three-dimensional theory of elasticity, and the displacements of the rectangular plate, which is simply supported at all edges, are expanded into the polynomial forms of arbitrary order with respect to a thickness coordinate. A governing equation is derived by using the energy approach of minimizing the
0263-8223/02/$ - see front matter Ó 2002 Elsevier Science Ltd. All rights reserved. PII: S 0 2 6 3 - 8 2 2 3 ( 0 2 ) 0 0 0 8 0 - 6
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Lagrangian defined by the two energies (Ritz’s method), and the present problem is reduced to a complex eigenvalue problem by introducing the complex elastic moduli. On the other hand, two governing equations for the laminated plate are also formulated by using the CLT and the FSDT, respectively. In numerical calculations, natural frequencies and modal damping ratios are obtained for the laminated plates with different stacking sequence and thickness ratios by using different sets of displacements. Finally not only the validity of the assumption of deformations but also the applicability of the plate theories (CLT, FSDT and HSDT) applied to the FRP laminated thick plate are discussed by comparing with the numerical results obtained from the present method, the CLT and the FSDT analyses.
2. Theoretical analysis We consider a N-layer laminated composite rectangular plate (a b H ), which is simply supported at all edges. In the present analysis, the Cartesian coordinate system (x; y; z) is taken at the middle surface of the laminated plate as shown in Fig. 1. The maximum displacements (amplitude) of the plate in the x, y and z directions are denoted by u, v and w, respectively. Based on the three-dimensional theory of elasticity, the maximum strain and kinetic energies of the plate are expressed as follows: Z 1 Umax ¼ ðrx ex þ ry ey þ rz ez þ syz cyz þ szx czx þ sxy cxy ÞdV 2 V Z 1 q x2 ðu2 þ v2 þ w2 ÞdV Tmax ¼ 2 V m ð1Þ where qm is a mean mass density of FRP composite material and x means a circular frequency. For the cross-ply laminated plate, the stress–strain relation in the kth layer is given by
8 9ðkÞ 2 rx > Q11 > > > > > > 6 ry > > > > > 6 < = 6 rz ¼6 6 s > > yz > > 6 > > > > 4 s > > zx > > ; : sxy
Q12 Q22
Q13 Q23 Q33
0 0 0 Q44
0 0 0 0 Q55
Sym:
3ðkÞ 8 9 ex > 0 > > > > > > ey > 0 7 > > > 7 > = < e 0 7 z 7 c > 0 7 > 7 > > > yz > > c 0 5 > > > > ; : zx > cxy Q66 ð2Þ
ðkÞ
where elastic coefficients Qij are calculated [1] by using Young’s moduli, shearing elastic moduli, Poisson’s ratios of FRP material and the direction of fiber in the k-th layer. Assuming small amplitude vibration, the strains are related to the displacements by ou ov ow ; ey ¼ ; ez ¼ ox oy oz ow ov ou ow þ ; czx ¼ þ ; cyz ¼ oy oz oz ox
ex ¼
ov ou cxy ¼ þ ox oy
ð3Þ
For the generality of the present analysis, the following nondimensional quantities are introduced: x y z n¼ ; f¼ ; d¼ a b H =2 H b h¼ ; l¼ a a ð4Þ u v w ¼ u ¼ ; v ¼ ; w H H H 4 2 q Ha x k2 ¼ m ðfrequency parameterÞ D0 where D0 means a representative bending stiffness D0 ¼ E2 H 3 =12ð1 m12 m21 Þ in the analysis. (Suffix 1 and 2 denote the major material-symmetry direction and the inplane transverse directions, respectively.) When the laminated plate is simply supported at all edges: v ¼ w ¼ 0 at n ¼ 0; 1 ðx ¼ 0; aÞ u ¼ w ¼ 0 at f ¼ 0; 1 ðy ¼ 0; bÞ
ð5Þ
displacements can be expressed as follows: uðn; f; dÞ ¼ cos mpn sin npf
I X
Ui di1
i¼1
vðn; f; dÞ ¼ sin mpn cos npf
J X
Vj dj1
ð6Þ
j¼1
ðn; f; dÞ ¼ sin mpn sin npf w
K X
Wk dk1
k¼1
Fig. 1. Geometry and coordinates of an laminated composite plate.
where m and n are the half wave numbers in the x, y direction, respectively, and Ui , Vj and Wk are unknown coefficients. By substituting Eqs. (2)–(4) and (6) into Eq. (1), and minimizing the Lagrangian Tmax Umax with respect to the unknown coefficients Ui , Vj and Wk for a stationary value:
Y. Ohta et al. / Composite Structures 57 (2002) 169–175
oðTmax Umax Þ oðTmax Umax Þ oðTmax Umax Þ ¼ ¼ ¼0 oUa oVb oWc ða ¼ 1; 2; . . . ; I; b ¼ 1; 2; . . . ; J ; c ¼ 1; 2; . . . ; KÞ ð7Þ following governing equation is derived. 22 3 Aia Bjb Ckc 44 Djb Ekb 5 Sym: Fkc 2 338 9 Lia 0 0 < Ui = Ljb 0 55 Vj ¼ 0 k2 4 : ; Sym: Lkc Wk
ð8Þ
The elements in the coefficient matrix of Eq. (8) are given by m2 p2 00 2 n2 p2 00 q11ia þ 2 q11 q 55ia þ h 2 2l2 66ia mnp2 00 ðq12jb þ q00 Bjb ¼ 66jb Þ 2l mp 01 ðq Ckc ¼ þ q10 13kc Þ h 55kc m2 p2 00 2 n2 p2 00 q66jb þ 2 q11 q Djb ¼ 44jb þ h 2 2l2 22jb np 01 ðq Ekc ¼ q10 23kc Þ lh 44kc 2 2 m p 00 2 n2 p2 00 q55kc þ 2 q11 þ q Fkc ¼ h 33kc 2 2l2 44kc Z h2 1 m1 n1 Lmn ¼ g g dg 2 1 Z oðiÞ gm1 oðjÞ gn1 qijklmn ¼ Qkl dg odðiÞ odðjÞ
G12 ¼ G12 ð1 þ jg12 Þ;
g1 ¼ 0:0015;
ð9Þ
E2 ¼ E2 ð1 þ jg2 Þ; G23 ¼ G23 ð1 þ jg23 Þ
By using the analytical schemes developed in the previous section, numerical studies are carried out for cross-ply laminated square plates with different stacking sequence and thickness ratios. In numerical calculations, natural frequencies and modal damping ratios of the plates are obtained by using many sets of displacement functions (Eq. (6)) with different terms of I, J and K and by using the CLT and FSDT solutions. This means that frequencies and damping ratios are obtained by employing the displacement functions which have polynomial forms of (I 1)th and (K 1)th order for in-plane and transverse displacements, respectively. A graphite/ epoxy, which is highly orthotropic FRP material, is chosen and the material properties used in numerical calculations are m12 ¼ 0:25;
In the present analysis, complex elastic moduli with loss factors E1 ¼ E1 ð1 þ jg1 Þ;
3. Numerical results and discussions
E1 =E2 ¼ 20;
Aia ¼
ð10Þ
are introduced to evaluate not only natural frequencies but also modal damping ratios, and thus the present problem is reduced to a complex eigenvalue problem. In usual manner the real part of each eigenvalue yields a frequency parameter (natural frequency), and the ratio of imaginary part to real part of each eigenvalue gives a modal damping ratio of the laminated plates. For discussion on the assumption of deformations in vibration analysis, the laminated plate is modeled based on the CLT and the FSDT, respectively, and frequency parameters and modal damping ratios of the plate are also calculated numerically from the two kinds of solutions. However, the formulation of each analysis is not written here due to the limitation of the space of pages.
171
G12 =E2 ¼ 0:65;
G23 =E2 ¼ 0:5;
m23 ¼ 0:25 g2 ¼ 0:01;
g12 ¼ g23 ¼ 0:016
The stacking sequence is expressed by using the notation [a1 =a2 =a3 =a4 = ]. For instance, [0/90/0] denotes a 3layered plate which has fiber angles 0°, 90° and 0° from lower to upper layer in the plate. Table 1 shows the comparisons of numerical results obtained by the present Ritz’s method and by Ohta et al. [6], and the Ritz’s solutions are also compared in Table 2 with those obtained from the CLT and the FSDT analyses. The FSDT solutions in the table are calculated by using the shear correction factor j2 ¼ 5=6. As shown in these tables the comparisons show an excellent agreement although the CLT gives slightly higher frequencies than the others, and thus it demonstrates the validity of the present method and the analyses based on the CLT and FSDT in this paper. Table 3 presents the first three frequency parameters of 3-layered thick plates [0/90/0] obtained by employing different sets of terms I, J (¼I) and K. The CLT and the FSDT solutions are also shown in this table, and the frequencies underlined in this table correspond to the HSDT solutions, because in the HSDT in-plane and transverse displacements are assumed in the polynomial forms of third and zero order, respectively. As seen in Table 1 Comparisons of frequency parameters of laminated composites plates (l ¼ 1:0, [0/90/0], I ¼ J ¼ 4, K ¼ 3) h
Mode
Present
0.2
(1; 1) (1; 2) (2; 2)
32.42 52.10 83.87
Ohta et al. [6] 32.43 52.32 83.13
0.05
(1; 1) (1; 2) (2; 2)
46.54 72.93 168.2
46.56 72.99 168.4
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Y. Ohta et al. / Composite Structures 57 (2002) 169–175
Table 2 Comparisons of frequency parameters of laminated composites plates (h ¼ 0:005, I ¼ J ¼ 8, j2 ¼ 5=6) Stacking sequence
Theory
l ¼ 0:5
[0]
Present FSDTa CLTb
68.61 68.61 68.64
45.03 45.03 45.05
[0/90]
Present FSDTa CLTb
88.23 88.24 88.27
22.07 22.07 22.07
[0/90/0]
Present FSDTa CLTb
75.73 75.73 75.76
44.31 44.31 44.33
[0/90/0/0/90/0]
Present FSDTa CLTb
a b
109.0 109.1 109.1
l ¼ 2:0
39.73 39.73 39.75
Frequency parameters obtained by FSDT. Frequency parameters obtained by CLT.
Table 3 Frequency parameters of a laminated composite plate (l ¼ 1:0, h ¼ 0:2, [0/90/0], I ¼ J , j2 ¼ 5=6) I
K 1
2
3 a
4
5
the terms I and K giving the converged frequencies depend on vibration mode. The differences in frequencies among the present solutions (I ¼ 8, K ¼ 5), FSDT and HSDT solutions are found to be within 2%, and it seems that the FSDT and the HSDT give reasonable frequencies even when the plate is rather thick (h ¼ 0:2). Table 4 gives the fundamental frequency parameters of the 6-layered plates [0/90/0/0/90/0] with different thickness ratios (h ¼ 0:2, 0.1, 0.05, 0.01). Many terms of I and K in displacement functions are required to obtain accurate frequencies for the thick plate, but for the thin plate (h ¼ 0:01) the converged frequencies can be obtained by taking only I ¼ 2 and K ¼ 3 terms. For the plates with h 5 0:05, the FSDT gives accurate frequencies and the difference in frequencies are within 0.02%. The CLT also gives reasonable results, in which the differences are within 3%, when the thickness ratio of the plate is less than h ¼ 0:05. Table 5 presents the funda-
Table 4 Frequency parameters of laminated composite plates (l ¼ 1:0, [0/90/0/ 0/90/0], Mode (1; 1), I ¼ J , j2 ¼ 5=6) K
I
1
4
5
34.87 (FSDT ) 48.44 (CLT ) 36.40 36.40 36.23 36.40 36.40 36.23 34.97 34.97 34.89 34.97 34.97 34.89 34.76 34.76 34.67 34.76 34.76 34.67 34.74 34.74 34.65
36.23 36.23 34.89 34.89 34.67 34.67 34.65
36.23 36.23 34.88 34.88 34.66 34.66 34.64
43.47 (FSDT) 48.44 (CLT) 44.41 44.41 44.14 44.41 44.41 44.14 43.76 43.76 43.53 43.76 43.76 43.53 43.65 43.65 43.42 43.65 43.65 43.42 43.64 43.64 43.41
44.14 44.14 43.53 43.53 43.42 43.42 43.41
44.14 44.14 43.52 43.52 43.42 43.42 43.41
h ¼ 0.05: 47.02 (FSDT) 48.44 (CLT) 2 47.56 47.56 47.23 3 47.56 47.56 47.23 47.35 47.35 47.04 4 5 47.35 47.35 47.04 6 47.32 47.32 47.01 7 47.32 47.32 47.01 8 47.32 47.32 47.01
47.23 47.23 47.04 47.04 47.01 47.01 47.01
47.23 47.23 47.04 47.04 47.01 47.01 47.01
h ¼ 0.01: 48.38 (FSDT) 48.44 (CLT) 2 48.74 48.74 48.39 3 48.74 48.74 48.39 4 48.73 48.73 48.38 5 48.73 48.73 48.38 6 48.73 48.73 48.38 7 48.73 48.73 48.38 8 48.73 48.73 48.38
48.39 48.39 48.38 48.38 48.38 48.38 48.38
48.39 48.39 48.38 48.38 48.38 48.38 48.38
b
Mode (1, 1): 32.65 2 34.28 3 34.28 4 32.54 5 32.54 6 32.53 7 32.53 8 32.51
(FSDT ) 48.44 (CLT ) 34.28 34.07 34.28 34.07 32.54 32.42 32.54 32.42 32.53 32.41 32.53 32.41 32.51 32.39
34.07 34.07 32.42 32.42 32.41 32.41 32.39
34.07 34.07 32.42 32.42 32.40 32.40 32.38
Mode (1, 2): 52.62 2 55.18 3 55.18 4 52.41 5 52.41 6 52.24 7 52.24 8 52.12
(FSDT) 75.76 (CLT) 55.18 54.58 55.18 54.58 52.41 52.10 52.41 52.10 52.24 51.91 52.24 51.91 52.12 51.78
h ¼ 0.2: 2 3 4 5 6 7 8
54.58 54.58 52.10 52.10 51.91 51.91 51.78
54.58 54.58 52.06 52.06 51.86 51.86 51.73
Mode (2, 2): 83.85 2 89.59 3 89.59 4 83.98 5 83.98 6 83.90 7 83.90 8 83.79
(FSDT) 193.8 (CLT) 89.59 89.11 89.59 89.11 83.98 83.87 83.98 83.87 83.90 83.78 83.90 83.78 83.79 83.66
h ¼ 0.1: 2 3 4 5 6 7 8
89.11 89.11 83.87 83.87 83.78 83.78 83.66
89.11 89.11 83.79 83.79 83.69 83.69 83.58
a b
Frequency parameters obtained by FSDT. Frequency parameters obtained by CLT.
the table the accuracy of the solutions depends on the number of terms I and K, and frequency parameters become to have lower values as the number of terms increases. For the fundamental mode (1; 1) the frequencies converge within three significant figures by using the terms I = 4 and K = 3, and it is also found that
a b
2 a
3 b
Frequency parameters obtained by FSDT. Frequency parameters obtained by CLT.
Y. Ohta et al. / Composite Structures 57 (2002) 169–175 Table 5 Frequency parameters of laminated composite plates (l ¼ 1:0, h ¼ 0:2, Mode (1; 1), I ¼ J , j2 ¼ 5=6) I
K
Table 6 Modal damping ratios of a laminated composite plate (l ¼ 1:0, h ¼ 0:2, [0/90/0], I ¼ J , j2 ¼ 5=6) I
1
2 a
3
4
5
b
173
K 1
2
3 a
4
5
b
[0]: 32.65 (FSDT ) 48.44 (CLT ) 2 34.30 34.30 34.10 3 34.30 34.30 34.10 4 32.95 32.95 32.82 5 32.95 32.95 32.82 6 32.95 32.95 32.82 7 32.95 32.95 32.82 8 32.95 32.95 32.82
34.10 34.10 32.82 32.82 32.82 32.82 32.82
34.10 34.10 32.81 32.81 32.81 32.81 32.81
Mode (1, 1) 0.0106 (FSDT ) 0.0036 (CLT ) 2 0.0099 0.0099 0.0099 3 0.0099 0.0099 0.0099 4 0.0105 0.0105 0.0105 5 0.0105 0.0105 0.0105 6 0.0105 0.0105 0.0105 7 0.0105 0.0105 0.0105 8 0.0105 0.0105 0.0105
0.0099 0.0099 0.0105 0.0105 0.0105 0.0105 0.0105
0.0099 0.0099 0.0105 0.0105 0.0105 0.0105 0.0105
[0/90]: 26.99 (FSDT) 31.87 (CLT) 2 27.84 27.84 3 27.61 27.61 4 27.32 27.32 5 27.18 27.18 6 27.13 27.13 7 27.13 27.13 8 27.12 27.12
27.55 27.31 27.07 26.93 26.88 26.88 26.87
27.55 27.31 27.07 26.93 26.88 26.88 26.87
27.55 27.31 27.07 26.93 26.88 26.88 26.87
Mode (1, 2): 0.0109 (FSDT) 0.0066 (CLT) 2 0.0104 0.0104 0.0104 3 0.0104 0.0104 0.0104 4 0.0112 0.0112 0.0111 5 0.0112 0.0112 0.0111 6 0.0113 0.0113 0.0113 7 0.0113 0.0113 0.0113 8 0.0114 0.0114 0.0114
0.0104 0.0104 0.0111 0.0111 0.0113 0.0113 0.0114
0.0104 0.0104 0.0112 0.0112 0.0113 0.0113 0.0114
48.44 (CLT) 34.28 34.07 34.28 34.07 32.54 32.42 32.54 32.42 32.53 32.41 32.53 32.41 32.51 32.39
34.07 34.07 32.42 32.42 32.41 32.41 32.39
34.07 34.07 32.42 32.42 32.40 32.40 32.38
Mode (2, 2): 0.0134 (FSDT) 0.0036 (CLT) 2 0.0130 0.0130 0.0130 3 0.0130 0.0130 0.0130 4 0.0132 0.0132 0.0132 5 0.0132 0.0132 0.0132 6 0.0132 0.0132 0.0132 7 0.0132 0.0132 0.0132 8 0.0133 0.0133 0.0133
0.0130 0.0130 0.0132 0.0132 0.0132 0.0132 0.0133
0.0130 0.0130 0.0132 0.0132 0.0132 0.0132 0.0133
[0/90/0]: 32.65 (FSDT) 2 34.28 3 34.28 4 32.54 5 32.54 6 32.53 7 32.53 8 32.51
[0/90/0/90/0]: 34.87 (FSDT) 48.44 2 36.40 36.40 3 36.40 36.40 34.97 34.97 4 5 34.97 34.97 6 34.76 34.76 7 34.76 34.76 8 34.74 34.74 a b
(CLT) 36.23 36.23 34.89 34.89 34.67 34.67 34.65
a
36.23 36.23 34.89 34.89 34.67 34.67 34.65
36.23 36.23 34.88 34.88 34.66 34.66 34.64
Frequency parameters obtained by FSDT. Frequency parameters obtained by CLT.
mental frequency parameters of the laminated plates with different stacking sequence. Although the similar tendencies can be observed, many terms of I and K are required to obtain the accurate frequencies only for unsymmetrical laminates [0/90] and 6-layered plate [0/90/0/0/90/0]. Tables 6 and 7 present the modal damping ratios for the case of Tables 3 and 4, respectively. Similarly the accuracy of the damping solutions depends on the number of terms I and K, and modal damping ratios thus become to be higher values as the number of terms increases. However, the FSDT and HSDT solutions give more accurate damping ratios independent of the thickness of the plates than the case of frequency parameters. Fig. 2 indicates the nondimensional frequency parameters obtained from present converged solution (I ¼ J ¼ 4, K ¼ 3), HSDT (I ¼ J ¼ 4, K ¼ 1), FSDT and
b
Damping ratios obtained by FSDT. Damping ratios obtained by CLT.
CLT for four symmetrically 8-layered cross-ply thick plates (h ¼ 0:1) with different stacking sequence, and each frequency in the figure is normalized by each CLT solution. As seen in the results, not only the converged solution but also HSDT, FSDT and CLT give similar, reasonable solutions although there is the difference within 13% in frequencies depending on the stacking sequence. Fig. 3 indicates the nondimensional modal damping ratios for the case of Fig. 2, and each damping ratio in the figure is normalized by each converged solution (I ¼ J ¼ 4, K ¼ 3). It is found from these results that there is a greater difference between CLT solutions and the other solutions compared with the case of frequency shown in Fig. 2, and that the difference in damping ratio depends on the stacking sequence of the plate.
4. Conclusions The main purpose of this paper is to study the assumptions in deformation for the vibration analysis of laminated composite plates. For that purpose, the threedimensional analysis was developed for simply supported, cross-ply laminated rectangular plates, and three displacements were assumed in the polynomials of
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Y. Ohta et al. / Composite Structures 57 (2002) 169–175
Table 7 Modal damping ratios of laminated composite plates (l ¼ 1:0, [0/90/0/ 0/90/0], Mode (1; 1), I ¼ J , j2 ¼ 5=6) I
K 1
2
4
5
) 0.0036 (CLT ) 0.0086 0.0086 0.0086 0.0086 0.0093 0.0093 0.0093 0.0093 0.0094 0.0094 0.0094 0.0094 0.0094 0.0094
0.0086 0.0086 0.0093 0.0093 0.0094 0.0094 0.0094
0.0086 0.0086 0.0093 0.0093 0.0094 0.0094 0.0094
h ¼ 0.1: 0.0059 (FSDTa ) 0.0036 (CLTb ) 2 0.0055 0.0055 0.0055 3 0.0055 0.0055 0.0055 4 0.0058 0.0058 0.0058 5 0.0058 0.0058 0.0058 6 0.0058 0.0058 0.0059 7 0.0058 0.0058 0.0059 8 0.0059 0.0059 0.0059
0.0055 0.0055 0.0058 0.0058 0.0059 0.0059 0.0059
0.0055 0.0055 0.0058 0.0058 0.0059 0.0059 0.0059
h ¼ 0.05: 0.0043 (FSDTa ) 0.0036 (CLTb ) 2 0.0040 0.0040 0.0040 3 0.0040 0.0040 0.0040 4 0.0041 0.0041 0.0041 5 0.0041 0.0041 0.0041 6 0.0041 0.0041 0.0042 7 0.0041 0.0041 0.0042 8 0.0041 0.0041 0.0042
0.0040 0.0040 0.0041 0.0041 0.0042 0.0042 0.0042
0.0040 0.0040 0.0041 0.0041 0.0042 0.0042 0.0042
h ¼ 0.01: 0.0036 (FSDTa ) 0.0036 (CLTb ) 2 0.0034 0.0034 0.0035 3 0.0034 0.0034 0.0035 4 0.0034 0.0034 0.0035 5 0.0034 0.0034 0.0035 6 0.0034 0.0034 0.0035 7 0.0034 0.0034 0.0035 8 0.0034 0.0034 0.0035
0.0035 0.0035 0.0035 0.0035 0.0035 0.0035 0.0035
0.0035 0.0035 0.0035 0.0035 0.0035 0.0035 0.0035
a
h ¼ 0.2: 0.0092 (FSDT 2 0.0086 3 0.0086 4 0.0093 5 0.0093 6 0.0094 7 0.0094 8 0.0094
a b
3 b
Damping ratios obtained by FSDT. Damping ratios obtained by CLT.
Fig. 3. Nondimensional damping ratios of laminated composite plates (l ¼ 2:0, h ¼ 0:1, mode (1; 1), j2 ¼ 5=6).
arbitrary order with respect to a thickness coordinate. A frequency equation was derived by substituting the displacements into the functional and minimizing it. For comparison, the plate theories (FSDT and CLT) were also applied to this problem. Frequency parameters and modal damping ratios were calculated numerically by the present method and by two plate theories, respectively, and the effect of difference in employed displacements on the accuracy of the solutions are discussed from the comparison of the results obtained. The following conclusions were reached from the numerical experiments. (1) In the present three-dimensional analysis, more accurate frequencies and damping ratios are calculated by taking Ið¼ J Þ = 4 and K = 3 terms for relatively thicker plate (h ¼ 0:2). (2) For the plates with the thickness h 5 0:05, the FSDT gives reasonable results, showing differences in frequencies only within 2% from the converged values. (3) The accuracy of modal damping ratios are not more sensitive to the displacement functions employed in the analysis than the case of natural frequency. (4) The difference in damping ratios between CLT solutions and the other solutions is larger than the one in natural frequency, and it depends on the stacking sequence of the plate greatly.
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Fig. 2. Nondimensional frequencies of laminated composite plates (l ¼ 2:0, h ¼ 0:1, mode (1; 1), j2 ¼ 5=6).
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[5] Reddy JN. A simple higher-order theory for laminated composite plates. ASME J Appl Mech 1984;51(4):745. [6] Ohta Y, Narita Y, A comparison of the multi-layer and lamination theories in vibration analysis of laminated composite plates, ASME PVP-vol. 279: Developments in a progressing technology 1994;21.
On the damping analysis of FRP laminated composite plates Yoshiki Ohta a
a,*
, Yoshihiro Narita a, Kenta Nagasaki
b
Department of Mechanical Systems Engineering, Hokkaido Institute of Technology, 7-15 Maeda, Tein-ku, Sapporo 006-8585, Japan b Graduate School, Hokkaido Institute of Technology, 7-15 Maeda, Tein-ku, Sapporo 006-8585, Japan
Abstract This paper presents the damping analysis of fiber reinforced plastics laminated composite plates. For this purpose, the maximum strain and kinetic energies of a cross-ply laminated plate are evaluated analytically based on the three-dimensional theory of elasticity. The displacements of the simply supported rectangular plates are expanded into the polynomial forms with respect to a thickness coordinate, and then governing equations are formulated by using the Ritz’s method. In the numerical calculations, natural frequencies and modal damping ratios are calculated for the plates with different stacking sequence and thickness ratios. The validity of the assumption of deformations and the applicability of the other plate theories (e.g. classical lamination theory (CLT), first-order shear deformation theory (FSDT) and higher-order shear deformation theory) to the laminated thick plates are discussed by comparing the numerical results obtained by the present method with the CLT and the FSDT solutions. Ó 2002 Elsevier Science Ltd. All rights reserved.
1. Introduction Recently fiber reinforced plastics (FRP) are being increasingly used in the structural applications, because the technical merits of high strength to weight ratio and high stiffness to weight ratio of FRP composite materials are outstanding. It is also known that the anisotropic property in FRP materials considerably affects the dynamic characteristics (e.g. vibration and buckling) of structural elements such as plates and shells. Therefore a large number of theoretical and experimental studies have been published on the dynamic problems of FRP laminated composite plates and shells. In most of the analyses found in the literatures, the laminated plate is macroscopically modeled as a thin plate of general anisotropy, and is analyzed by using the classical lamination theory (CLT) (e.g. Vinson et al. [1]) based on the Kirchhoff’s hypotheses, where transverse shear deformations and extension in the thickness direction of the plate are ignored. However the CLT cannot give accurate results for laminated thick plates, because the transverse shear deformations cannot be ignored due to the relatively smaller shear elastic moduli of FRP composite materials than the other elastic moduli. Thus the analyses including the effect of shear de*
Corresponding author. Tel.: +81-11-688-2284; fax: +81-11-6813622. E-mail address: [email protected] (Y. Ohta).
formations are required for the laminated thick plates. Yang et al. [2], Whitney et al. [3] and Dong et al. [4] analyzed laminated thick plates and shells based on the first-order shear deformation theory (FSDT), in which in-plane displacements are assumed to vary lineally through the thickness. However, the FSDT cannot satisfy the free surface conditions on transverse shear stresses on the top and bottom surfaces of the plate and thus requires a shear correction factor multiplied to the transverse shear stiffness of the plate. Reddy [5] proposed the higher-order shear deformation theory (HSDT), wherein in-plane displacements are expressed in the polynomial forms of third order. This theory satisfies the free surface conditions exactly and does not require the shear correction to the transverse shear stiffness. There are few papers discussing the displacement functions in the evaluation of frequency and damping ratio of FRP laminated composite plates although these theories have been developed. The purpose of this paper is to present the damping analysis of FRP laminated composite thick plates. For this purpose, the maximum strain and kinetic energies of an FRP cross-ply laminated plate are evaluated analytically based on the three-dimensional theory of elasticity, and the displacements of the rectangular plate, which is simply supported at all edges, are expanded into the polynomial forms of arbitrary order with respect to a thickness coordinate. A governing equation is derived by using the energy approach of minimizing the
0263-8223/02/$ - see front matter Ó 2002 Elsevier Science Ltd. All rights reserved. PII: S 0 2 6 3 - 8 2 2 3 ( 0 2 ) 0 0 0 8 0 - 6
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Lagrangian defined by the two energies (Ritz’s method), and the present problem is reduced to a complex eigenvalue problem by introducing the complex elastic moduli. On the other hand, two governing equations for the laminated plate are also formulated by using the CLT and the FSDT, respectively. In numerical calculations, natural frequencies and modal damping ratios are obtained for the laminated plates with different stacking sequence and thickness ratios by using different sets of displacements. Finally not only the validity of the assumption of deformations but also the applicability of the plate theories (CLT, FSDT and HSDT) applied to the FRP laminated thick plate are discussed by comparing with the numerical results obtained from the present method, the CLT and the FSDT analyses.
2. Theoretical analysis We consider a N-layer laminated composite rectangular plate (a b H ), which is simply supported at all edges. In the present analysis, the Cartesian coordinate system (x; y; z) is taken at the middle surface of the laminated plate as shown in Fig. 1. The maximum displacements (amplitude) of the plate in the x, y and z directions are denoted by u, v and w, respectively. Based on the three-dimensional theory of elasticity, the maximum strain and kinetic energies of the plate are expressed as follows: Z 1 Umax ¼ ðrx ex þ ry ey þ rz ez þ syz cyz þ szx czx þ sxy cxy ÞdV 2 V Z 1 q x2 ðu2 þ v2 þ w2 ÞdV Tmax ¼ 2 V m ð1Þ where qm is a mean mass density of FRP composite material and x means a circular frequency. For the cross-ply laminated plate, the stress–strain relation in the kth layer is given by
8 9ðkÞ 2 rx > Q11 > > > > > > 6 ry > > > > > 6 < = 6 rz ¼6 6 s > > yz > > 6 > > > > 4 s > > zx > > ; : sxy
Q12 Q22
Q13 Q23 Q33
0 0 0 Q44
0 0 0 0 Q55
Sym:
3ðkÞ 8 9 ex > 0 > > > > > > ey > 0 7 > > > 7 > = < e 0 7 z 7 c > 0 7 > 7 > > > yz > > c 0 5 > > > > ; : zx > cxy Q66 ð2Þ
ðkÞ
where elastic coefficients Qij are calculated [1] by using Young’s moduli, shearing elastic moduli, Poisson’s ratios of FRP material and the direction of fiber in the k-th layer. Assuming small amplitude vibration, the strains are related to the displacements by ou ov ow ; ey ¼ ; ez ¼ ox oy oz ow ov ou ow þ ; czx ¼ þ ; cyz ¼ oy oz oz ox
ex ¼
ov ou cxy ¼ þ ox oy
ð3Þ
For the generality of the present analysis, the following nondimensional quantities are introduced: x y z n¼ ; f¼ ; d¼ a b H =2 H b h¼ ; l¼ a a ð4Þ u v w ¼ u ¼ ; v ¼ ; w H H H 4 2 q Ha x k2 ¼ m ðfrequency parameterÞ D0 where D0 means a representative bending stiffness D0 ¼ E2 H 3 =12ð1 m12 m21 Þ in the analysis. (Suffix 1 and 2 denote the major material-symmetry direction and the inplane transverse directions, respectively.) When the laminated plate is simply supported at all edges: v ¼ w ¼ 0 at n ¼ 0; 1 ðx ¼ 0; aÞ u ¼ w ¼ 0 at f ¼ 0; 1 ðy ¼ 0; bÞ
ð5Þ
displacements can be expressed as follows: uðn; f; dÞ ¼ cos mpn sin npf
I X
Ui di1
i¼1
vðn; f; dÞ ¼ sin mpn cos npf
J X
Vj dj1
ð6Þ
j¼1
ðn; f; dÞ ¼ sin mpn sin npf w
K X
Wk dk1
k¼1
Fig. 1. Geometry and coordinates of an laminated composite plate.
where m and n are the half wave numbers in the x, y direction, respectively, and Ui , Vj and Wk are unknown coefficients. By substituting Eqs. (2)–(4) and (6) into Eq. (1), and minimizing the Lagrangian Tmax Umax with respect to the unknown coefficients Ui , Vj and Wk for a stationary value:
Y. Ohta et al. / Composite Structures 57 (2002) 169–175
oðTmax Umax Þ oðTmax Umax Þ oðTmax Umax Þ ¼ ¼ ¼0 oUa oVb oWc ða ¼ 1; 2; . . . ; I; b ¼ 1; 2; . . . ; J ; c ¼ 1; 2; . . . ; KÞ ð7Þ following governing equation is derived. 22 3 Aia Bjb Ckc 44 Djb Ekb 5 Sym: Fkc 2 338 9 Lia 0 0 < Ui = Ljb 0 55 Vj ¼ 0 k2 4 : ; Sym: Lkc Wk
ð8Þ
The elements in the coefficient matrix of Eq. (8) are given by m2 p2 00 2 n2 p2 00 q11ia þ 2 q11 q 55ia þ h 2 2l2 66ia mnp2 00 ðq12jb þ q00 Bjb ¼ 66jb Þ 2l mp 01 ðq Ckc ¼ þ q10 13kc Þ h 55kc m2 p2 00 2 n2 p2 00 q66jb þ 2 q11 q Djb ¼ 44jb þ h 2 2l2 22jb np 01 ðq Ekc ¼ q10 23kc Þ lh 44kc 2 2 m p 00 2 n2 p2 00 q55kc þ 2 q11 þ q Fkc ¼ h 33kc 2 2l2 44kc Z h2 1 m1 n1 Lmn ¼ g g dg 2 1 Z oðiÞ gm1 oðjÞ gn1 qijklmn ¼ Qkl dg odðiÞ odðjÞ
G12 ¼ G12 ð1 þ jg12 Þ;
g1 ¼ 0:0015;
ð9Þ
E2 ¼ E2 ð1 þ jg2 Þ; G23 ¼ G23 ð1 þ jg23 Þ
By using the analytical schemes developed in the previous section, numerical studies are carried out for cross-ply laminated square plates with different stacking sequence and thickness ratios. In numerical calculations, natural frequencies and modal damping ratios of the plates are obtained by using many sets of displacement functions (Eq. (6)) with different terms of I, J and K and by using the CLT and FSDT solutions. This means that frequencies and damping ratios are obtained by employing the displacement functions which have polynomial forms of (I 1)th and (K 1)th order for in-plane and transverse displacements, respectively. A graphite/ epoxy, which is highly orthotropic FRP material, is chosen and the material properties used in numerical calculations are m12 ¼ 0:25;
In the present analysis, complex elastic moduli with loss factors E1 ¼ E1 ð1 þ jg1 Þ;
3. Numerical results and discussions
E1 =E2 ¼ 20;
Aia ¼
ð10Þ
are introduced to evaluate not only natural frequencies but also modal damping ratios, and thus the present problem is reduced to a complex eigenvalue problem. In usual manner the real part of each eigenvalue yields a frequency parameter (natural frequency), and the ratio of imaginary part to real part of each eigenvalue gives a modal damping ratio of the laminated plates. For discussion on the assumption of deformations in vibration analysis, the laminated plate is modeled based on the CLT and the FSDT, respectively, and frequency parameters and modal damping ratios of the plate are also calculated numerically from the two kinds of solutions. However, the formulation of each analysis is not written here due to the limitation of the space of pages.
171
G12 =E2 ¼ 0:65;
G23 =E2 ¼ 0:5;
m23 ¼ 0:25 g2 ¼ 0:01;
g12 ¼ g23 ¼ 0:016
The stacking sequence is expressed by using the notation [a1 =a2 =a3 =a4 = ]. For instance, [0/90/0] denotes a 3layered plate which has fiber angles 0°, 90° and 0° from lower to upper layer in the plate. Table 1 shows the comparisons of numerical results obtained by the present Ritz’s method and by Ohta et al. [6], and the Ritz’s solutions are also compared in Table 2 with those obtained from the CLT and the FSDT analyses. The FSDT solutions in the table are calculated by using the shear correction factor j2 ¼ 5=6. As shown in these tables the comparisons show an excellent agreement although the CLT gives slightly higher frequencies than the others, and thus it demonstrates the validity of the present method and the analyses based on the CLT and FSDT in this paper. Table 3 presents the first three frequency parameters of 3-layered thick plates [0/90/0] obtained by employing different sets of terms I, J (¼I) and K. The CLT and the FSDT solutions are also shown in this table, and the frequencies underlined in this table correspond to the HSDT solutions, because in the HSDT in-plane and transverse displacements are assumed in the polynomial forms of third and zero order, respectively. As seen in Table 1 Comparisons of frequency parameters of laminated composites plates (l ¼ 1:0, [0/90/0], I ¼ J ¼ 4, K ¼ 3) h
Mode
Present
0.2
(1; 1) (1; 2) (2; 2)
32.42 52.10 83.87
Ohta et al. [6] 32.43 52.32 83.13
0.05
(1; 1) (1; 2) (2; 2)
46.54 72.93 168.2
46.56 72.99 168.4
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Y. Ohta et al. / Composite Structures 57 (2002) 169–175
Table 2 Comparisons of frequency parameters of laminated composites plates (h ¼ 0:005, I ¼ J ¼ 8, j2 ¼ 5=6) Stacking sequence
Theory
l ¼ 0:5
[0]
Present FSDTa CLTb
68.61 68.61 68.64
45.03 45.03 45.05
[0/90]
Present FSDTa CLTb
88.23 88.24 88.27
22.07 22.07 22.07
[0/90/0]
Present FSDTa CLTb
75.73 75.73 75.76
44.31 44.31 44.33
[0/90/0/0/90/0]
Present FSDTa CLTb
a b
109.0 109.1 109.1
l ¼ 2:0
39.73 39.73 39.75
Frequency parameters obtained by FSDT. Frequency parameters obtained by CLT.
Table 3 Frequency parameters of a laminated composite plate (l ¼ 1:0, h ¼ 0:2, [0/90/0], I ¼ J , j2 ¼ 5=6) I
K 1
2
3 a
4
5
the terms I and K giving the converged frequencies depend on vibration mode. The differences in frequencies among the present solutions (I ¼ 8, K ¼ 5), FSDT and HSDT solutions are found to be within 2%, and it seems that the FSDT and the HSDT give reasonable frequencies even when the plate is rather thick (h ¼ 0:2). Table 4 gives the fundamental frequency parameters of the 6-layered plates [0/90/0/0/90/0] with different thickness ratios (h ¼ 0:2, 0.1, 0.05, 0.01). Many terms of I and K in displacement functions are required to obtain accurate frequencies for the thick plate, but for the thin plate (h ¼ 0:01) the converged frequencies can be obtained by taking only I ¼ 2 and K ¼ 3 terms. For the plates with h 5 0:05, the FSDT gives accurate frequencies and the difference in frequencies are within 0.02%. The CLT also gives reasonable results, in which the differences are within 3%, when the thickness ratio of the plate is less than h ¼ 0:05. Table 5 presents the funda-
Table 4 Frequency parameters of laminated composite plates (l ¼ 1:0, [0/90/0/ 0/90/0], Mode (1; 1), I ¼ J , j2 ¼ 5=6) K
I
1
4
5
34.87 (FSDT ) 48.44 (CLT ) 36.40 36.40 36.23 36.40 36.40 36.23 34.97 34.97 34.89 34.97 34.97 34.89 34.76 34.76 34.67 34.76 34.76 34.67 34.74 34.74 34.65
36.23 36.23 34.89 34.89 34.67 34.67 34.65
36.23 36.23 34.88 34.88 34.66 34.66 34.64
43.47 (FSDT) 48.44 (CLT) 44.41 44.41 44.14 44.41 44.41 44.14 43.76 43.76 43.53 43.76 43.76 43.53 43.65 43.65 43.42 43.65 43.65 43.42 43.64 43.64 43.41
44.14 44.14 43.53 43.53 43.42 43.42 43.41
44.14 44.14 43.52 43.52 43.42 43.42 43.41
h ¼ 0.05: 47.02 (FSDT) 48.44 (CLT) 2 47.56 47.56 47.23 3 47.56 47.56 47.23 47.35 47.35 47.04 4 5 47.35 47.35 47.04 6 47.32 47.32 47.01 7 47.32 47.32 47.01 8 47.32 47.32 47.01
47.23 47.23 47.04 47.04 47.01 47.01 47.01
47.23 47.23 47.04 47.04 47.01 47.01 47.01
h ¼ 0.01: 48.38 (FSDT) 48.44 (CLT) 2 48.74 48.74 48.39 3 48.74 48.74 48.39 4 48.73 48.73 48.38 5 48.73 48.73 48.38 6 48.73 48.73 48.38 7 48.73 48.73 48.38 8 48.73 48.73 48.38
48.39 48.39 48.38 48.38 48.38 48.38 48.38
48.39 48.39 48.38 48.38 48.38 48.38 48.38
b
Mode (1, 1): 32.65 2 34.28 3 34.28 4 32.54 5 32.54 6 32.53 7 32.53 8 32.51
(FSDT ) 48.44 (CLT ) 34.28 34.07 34.28 34.07 32.54 32.42 32.54 32.42 32.53 32.41 32.53 32.41 32.51 32.39
34.07 34.07 32.42 32.42 32.41 32.41 32.39
34.07 34.07 32.42 32.42 32.40 32.40 32.38
Mode (1, 2): 52.62 2 55.18 3 55.18 4 52.41 5 52.41 6 52.24 7 52.24 8 52.12
(FSDT) 75.76 (CLT) 55.18 54.58 55.18 54.58 52.41 52.10 52.41 52.10 52.24 51.91 52.24 51.91 52.12 51.78
h ¼ 0.2: 2 3 4 5 6 7 8
54.58 54.58 52.10 52.10 51.91 51.91 51.78
54.58 54.58 52.06 52.06 51.86 51.86 51.73
Mode (2, 2): 83.85 2 89.59 3 89.59 4 83.98 5 83.98 6 83.90 7 83.90 8 83.79
(FSDT) 193.8 (CLT) 89.59 89.11 89.59 89.11 83.98 83.87 83.98 83.87 83.90 83.78 83.90 83.78 83.79 83.66
h ¼ 0.1: 2 3 4 5 6 7 8
89.11 89.11 83.87 83.87 83.78 83.78 83.66
89.11 89.11 83.79 83.79 83.69 83.69 83.58
a b
Frequency parameters obtained by FSDT. Frequency parameters obtained by CLT.
the table the accuracy of the solutions depends on the number of terms I and K, and frequency parameters become to have lower values as the number of terms increases. For the fundamental mode (1; 1) the frequencies converge within three significant figures by using the terms I = 4 and K = 3, and it is also found that
a b
2 a
3 b
Frequency parameters obtained by FSDT. Frequency parameters obtained by CLT.
Y. Ohta et al. / Composite Structures 57 (2002) 169–175 Table 5 Frequency parameters of laminated composite plates (l ¼ 1:0, h ¼ 0:2, Mode (1; 1), I ¼ J , j2 ¼ 5=6) I
K
Table 6 Modal damping ratios of a laminated composite plate (l ¼ 1:0, h ¼ 0:2, [0/90/0], I ¼ J , j2 ¼ 5=6) I
1
2 a
3
4
5
b
173
K 1
2
3 a
4
5
b
[0]: 32.65 (FSDT ) 48.44 (CLT ) 2 34.30 34.30 34.10 3 34.30 34.30 34.10 4 32.95 32.95 32.82 5 32.95 32.95 32.82 6 32.95 32.95 32.82 7 32.95 32.95 32.82 8 32.95 32.95 32.82
34.10 34.10 32.82 32.82 32.82 32.82 32.82
34.10 34.10 32.81 32.81 32.81 32.81 32.81
Mode (1, 1) 0.0106 (FSDT ) 0.0036 (CLT ) 2 0.0099 0.0099 0.0099 3 0.0099 0.0099 0.0099 4 0.0105 0.0105 0.0105 5 0.0105 0.0105 0.0105 6 0.0105 0.0105 0.0105 7 0.0105 0.0105 0.0105 8 0.0105 0.0105 0.0105
0.0099 0.0099 0.0105 0.0105 0.0105 0.0105 0.0105
0.0099 0.0099 0.0105 0.0105 0.0105 0.0105 0.0105
[0/90]: 26.99 (FSDT) 31.87 (CLT) 2 27.84 27.84 3 27.61 27.61 4 27.32 27.32 5 27.18 27.18 6 27.13 27.13 7 27.13 27.13 8 27.12 27.12
27.55 27.31 27.07 26.93 26.88 26.88 26.87
27.55 27.31 27.07 26.93 26.88 26.88 26.87
27.55 27.31 27.07 26.93 26.88 26.88 26.87
Mode (1, 2): 0.0109 (FSDT) 0.0066 (CLT) 2 0.0104 0.0104 0.0104 3 0.0104 0.0104 0.0104 4 0.0112 0.0112 0.0111 5 0.0112 0.0112 0.0111 6 0.0113 0.0113 0.0113 7 0.0113 0.0113 0.0113 8 0.0114 0.0114 0.0114
0.0104 0.0104 0.0111 0.0111 0.0113 0.0113 0.0114
0.0104 0.0104 0.0112 0.0112 0.0113 0.0113 0.0114
48.44 (CLT) 34.28 34.07 34.28 34.07 32.54 32.42 32.54 32.42 32.53 32.41 32.53 32.41 32.51 32.39
34.07 34.07 32.42 32.42 32.41 32.41 32.39
34.07 34.07 32.42 32.42 32.40 32.40 32.38
Mode (2, 2): 0.0134 (FSDT) 0.0036 (CLT) 2 0.0130 0.0130 0.0130 3 0.0130 0.0130 0.0130 4 0.0132 0.0132 0.0132 5 0.0132 0.0132 0.0132 6 0.0132 0.0132 0.0132 7 0.0132 0.0132 0.0132 8 0.0133 0.0133 0.0133
0.0130 0.0130 0.0132 0.0132 0.0132 0.0132 0.0133
0.0130 0.0130 0.0132 0.0132 0.0132 0.0132 0.0133
[0/90/0]: 32.65 (FSDT) 2 34.28 3 34.28 4 32.54 5 32.54 6 32.53 7 32.53 8 32.51
[0/90/0/90/0]: 34.87 (FSDT) 48.44 2 36.40 36.40 3 36.40 36.40 34.97 34.97 4 5 34.97 34.97 6 34.76 34.76 7 34.76 34.76 8 34.74 34.74 a b
(CLT) 36.23 36.23 34.89 34.89 34.67 34.67 34.65
a
36.23 36.23 34.89 34.89 34.67 34.67 34.65
36.23 36.23 34.88 34.88 34.66 34.66 34.64
Frequency parameters obtained by FSDT. Frequency parameters obtained by CLT.
mental frequency parameters of the laminated plates with different stacking sequence. Although the similar tendencies can be observed, many terms of I and K are required to obtain the accurate frequencies only for unsymmetrical laminates [0/90] and 6-layered plate [0/90/0/0/90/0]. Tables 6 and 7 present the modal damping ratios for the case of Tables 3 and 4, respectively. Similarly the accuracy of the damping solutions depends on the number of terms I and K, and modal damping ratios thus become to be higher values as the number of terms increases. However, the FSDT and HSDT solutions give more accurate damping ratios independent of the thickness of the plates than the case of frequency parameters. Fig. 2 indicates the nondimensional frequency parameters obtained from present converged solution (I ¼ J ¼ 4, K ¼ 3), HSDT (I ¼ J ¼ 4, K ¼ 1), FSDT and
b
Damping ratios obtained by FSDT. Damping ratios obtained by CLT.
CLT for four symmetrically 8-layered cross-ply thick plates (h ¼ 0:1) with different stacking sequence, and each frequency in the figure is normalized by each CLT solution. As seen in the results, not only the converged solution but also HSDT, FSDT and CLT give similar, reasonable solutions although there is the difference within 13% in frequencies depending on the stacking sequence. Fig. 3 indicates the nondimensional modal damping ratios for the case of Fig. 2, and each damping ratio in the figure is normalized by each converged solution (I ¼ J ¼ 4, K ¼ 3). It is found from these results that there is a greater difference between CLT solutions and the other solutions compared with the case of frequency shown in Fig. 2, and that the difference in damping ratio depends on the stacking sequence of the plate.
4. Conclusions The main purpose of this paper is to study the assumptions in deformation for the vibration analysis of laminated composite plates. For that purpose, the threedimensional analysis was developed for simply supported, cross-ply laminated rectangular plates, and three displacements were assumed in the polynomials of
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Y. Ohta et al. / Composite Structures 57 (2002) 169–175
Table 7 Modal damping ratios of laminated composite plates (l ¼ 1:0, [0/90/0/ 0/90/0], Mode (1; 1), I ¼ J , j2 ¼ 5=6) I
K 1
2
4
5
) 0.0036 (CLT ) 0.0086 0.0086 0.0086 0.0086 0.0093 0.0093 0.0093 0.0093 0.0094 0.0094 0.0094 0.0094 0.0094 0.0094
0.0086 0.0086 0.0093 0.0093 0.0094 0.0094 0.0094
0.0086 0.0086 0.0093 0.0093 0.0094 0.0094 0.0094
h ¼ 0.1: 0.0059 (FSDTa ) 0.0036 (CLTb ) 2 0.0055 0.0055 0.0055 3 0.0055 0.0055 0.0055 4 0.0058 0.0058 0.0058 5 0.0058 0.0058 0.0058 6 0.0058 0.0058 0.0059 7 0.0058 0.0058 0.0059 8 0.0059 0.0059 0.0059
0.0055 0.0055 0.0058 0.0058 0.0059 0.0059 0.0059
0.0055 0.0055 0.0058 0.0058 0.0059 0.0059 0.0059
h ¼ 0.05: 0.0043 (FSDTa ) 0.0036 (CLTb ) 2 0.0040 0.0040 0.0040 3 0.0040 0.0040 0.0040 4 0.0041 0.0041 0.0041 5 0.0041 0.0041 0.0041 6 0.0041 0.0041 0.0042 7 0.0041 0.0041 0.0042 8 0.0041 0.0041 0.0042
0.0040 0.0040 0.0041 0.0041 0.0042 0.0042 0.0042
0.0040 0.0040 0.0041 0.0041 0.0042 0.0042 0.0042
h ¼ 0.01: 0.0036 (FSDTa ) 0.0036 (CLTb ) 2 0.0034 0.0034 0.0035 3 0.0034 0.0034 0.0035 4 0.0034 0.0034 0.0035 5 0.0034 0.0034 0.0035 6 0.0034 0.0034 0.0035 7 0.0034 0.0034 0.0035 8 0.0034 0.0034 0.0035
0.0035 0.0035 0.0035 0.0035 0.0035 0.0035 0.0035
0.0035 0.0035 0.0035 0.0035 0.0035 0.0035 0.0035
a
h ¼ 0.2: 0.0092 (FSDT 2 0.0086 3 0.0086 4 0.0093 5 0.0093 6 0.0094 7 0.0094 8 0.0094
a b
3 b
Damping ratios obtained by FSDT. Damping ratios obtained by CLT.
Fig. 3. Nondimensional damping ratios of laminated composite plates (l ¼ 2:0, h ¼ 0:1, mode (1; 1), j2 ¼ 5=6).
arbitrary order with respect to a thickness coordinate. A frequency equation was derived by substituting the displacements into the functional and minimizing it. For comparison, the plate theories (FSDT and CLT) were also applied to this problem. Frequency parameters and modal damping ratios were calculated numerically by the present method and by two plate theories, respectively, and the effect of difference in employed displacements on the accuracy of the solutions are discussed from the comparison of the results obtained. The following conclusions were reached from the numerical experiments. (1) In the present three-dimensional analysis, more accurate frequencies and damping ratios are calculated by taking Ið¼ J Þ = 4 and K = 3 terms for relatively thicker plate (h ¼ 0:2). (2) For the plates with the thickness h 5 0:05, the FSDT gives reasonable results, showing differences in frequencies only within 2% from the converged values. (3) The accuracy of modal damping ratios are not more sensitive to the displacement functions employed in the analysis than the case of natural frequency. (4) The difference in damping ratios between CLT solutions and the other solutions is larger than the one in natural frequency, and it depends on the stacking sequence of the plate greatly.
References
Fig. 2. Nondimensional frequencies of laminated composite plates (l ¼ 2:0, h ¼ 0:1, mode (1; 1), j2 ¼ 5=6).
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