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Analysis of natural frequencies of composite plates by an RBF-pseudospectral method
Composite Structures 79 (2007) 202–210 www.elsevier.com/locate/compstruct
Analysis of natural frequencies of composite plates by an RBF-pseudospectral method A.J.M. Ferreira a
a,*
, G.E. Fasshauer
b
Departamento de Engenharia Mecnica e Gesto Industrial, Faculdade de Engenharia da Universidade do Porto, Rua Dr. Roberto Frias, 4200-465 Porto, Portugal b Department of Applied Mathematics, Illinois Institute of Technology, Chicago 60616, USA Available online 21 February 2006
Abstract A study of free vibration of symmetric composite plates is presented. The analysis is based on an innovative numerical scheme, where collocation by radial basis functions and pseudospectral methods are combined to produce highly accurate results. Numerical results are presented and discussed for various thickness-to-length ratios. 2006 Elsevier Ltd. All rights reserved. Keywords: Radial basis functions; Pseudospectral methods; Free vibration; Composite plates
1. Introduction Vibration of thick and thin composite plates is an important subject in the design of mechanical, civil and aerospace applications. The thickness of most parts of composite plates makes the transverse shear and the rotatory inertia not negligible as in classical theories. Therefore the first-order shear deformation theory for plates should be considered in general analysis. The analysis of free vibration of isotropic and composite plates is best performed by numerical techniques. The differential quadrature method [1–3], the boundary characteristic orthogonal polynomials [4] and the pseudospectral method [5] were used in recent years. The finite element method also proved to be very adequate for this type of problems. The number of references is too large to list here. More recently the free vibration analysis of Timoshenko beams and Mindlin plates by Kansa’s non-symmetric radial basis function (RBF) collocation method was performed by Ferreira [6–13]. *
Corresponding author. Tel.: +351 22 957 8713; fax: +351 22 953 7352. E-mail addresses: [email protected] (A.J.M. Ferreira), fasshauer@ iit.edu (G.E. Fasshauer). 0263-8223/$ - see front matter 2006 Elsevier Ltd. All rights reserved. doi:10.1016/j.compstruct.2005.12.004
In the present work we illustrate for the first time the application of the combination of radial basis functions and pseudospectral methods to the eigenvalue analysis of symmetric composite plates. The method proves to be elegant and of very high quality when compared to analytical solutions and Kansa’s collocation method. The paper is organized as follows. In Section 2 our approach to pseudospectral methods is discussed. In Section 3 the free vibration of symmetric composite plates are presented and results discussed in Section 4. Finally conclusions are presented in Section 5. 2. The pseudospectral method Based on recent work by Fornberg, Schaback and others establishing a close connection between radial basis function and polynomial interpolants (see, e.g., [14–16]) various authors have suggested using RBFs in ‘‘pseudospectral mode’’ to solve partial differential equations (PDEs) (see, e.g., [17–21]). There are several potential advantages when using RBFs instead of traditional polynomial methods. First, RBFs allow us to work with irregular grids, and on domains exhibiting irregular geometries. And secondly, it has been
A.J.M. Ferreira, G.E. Fasshauer / Composite Structures 79 (2007) 202–210
shown that, while the limiting case of flat RBFs leads to polynomial interpolants, one can actually achieve more accurate results with RBFs that have not quite converged to their polynomial limit (see, e.g., [14,15]). Thus, RBFPS methods are potentially more flexible and more accurate than polynomial methods. Since we presented a fairly detailed exposition of the RBF-PS method in our recent paper [19] we will only give a brief introduction here. More details about the RBF-PS method can be obtained from either [18,19] or any of the other papers mentioned above. The starting point for the RBF-PS method is the representation of (the spatial part of) the approximate solution uh of a given PDE by a linear combination of smooth (radial) basis functions /j, j = 1, . . . , N, i.e., uh ðxÞ ¼
N X
cj /j ðxÞ;
x 2 Rd ;
ð1Þ
j¼1
where /j(x) = U(kx xjk) with a single univariate basic function U. For the numerical experiments in this paper we used the inverse multiquadric 1 UðrÞ ¼ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi . 2 1 þ ðrÞ
ð2Þ
This function is infinitely smooth and positive definite on Rd . As pointed out above, the shape parameter can be used to influence the accuracy of the numerical method. The (flat) limiting case ! 0 usually corresponds to a polynomial, and a small (positive) shape parameter usually yields the best accuracy (however, at the cost of low numerical stability). How to find this ‘‘optimal’’ shape parameter remains an open research question. An important feature of pseudospectral methods is the fact that one usually is content with obtaining an approximation to the solution on a discrete set of grid points X ¼ fxi ; i ¼ 1; . . . ; N g. One of several ways to implement the pseudospectral method is via so-called differentiation matrices, i.e., one finds a matrix L such that at the grid points xi we have uL ¼ Lu.
ð3Þ
Here u = [uh(x1), . . . , uh(xN)]T is the vector of values of uh at the grid points and uL ¼ ½Luh ðx1 Þ; . . . ; Luh ðxN ÞT , where L is the (spatial) linear differential operator of the PDE under consideration. The differentiation matrix L can be interpreted as a discretized differential operator that maps function values to ‘‘derivative values’’ on the computational grid X. An expression for the differentiation matrix L of (3) can be obtained in two steps. First, we evaluate (1) at the grid points xi, i = 1, . . . , N. This results in uh ðxi Þ ¼
N X
cj /j ðxi Þ;
i ¼ 1; . . . ; N ;
j¼1
or in matrix-vector notation
ð4Þ
203
u ¼ Ac;
ð5Þ T
where c = [c1, . . . , cN] is the coefficient vector, the evaluation matrix A has entries Aij = /j(xi), and u is as before. Next, we apply the (linear) differential operator L to (1). This leads to Luh ðxÞ ¼
N X
cj L/j ðxÞ.
ð6Þ
j¼1
If we again evaluate at the grid points xi then we get in matrix-vector notation uL ¼ AL c;
ð7Þ
where uL and c are as above, and the matrix AL has entries AL;ij ¼ L/j ðxÞx¼xi . It is now straightforward to obtain the desired formula for L. We simply solve Eq. (5) for c and substitute the result into (7). This gives us uL ¼ AL A1 u;
ð8Þ
so that the differentiation matrix L corresponding to (3) is of the form L ¼ AL A1 .
ð9Þ
We enforce boundary conditions directly by manipulating the corresponding rows of L as described, e.g., in [22]. This results in a modified differentiation matrix LC. As pointed out in [18,19] invertibility of the differentiation matrix LC is equivalent to the invertibility of the system matrix for Kansa’s non-symmetric RBF collocation method, and thus to be treated with care (since certain rare point configurations exist for which the matrix may be singular). 3. Free vibration analysis of symmetric laminated plates In this paper we consider a laminated plate of length a, width b and thickness h, with the coordinate system at the midplane of the laminate. We also consider that each layer of the laminate is orthotropic with respect to the x- and yaxes and all layers are of equal thickness. Each layer direction is defined by the angle with the x-axis. The modulus of elasticity for a layer parallel to fibres is E11 and the modulus of elasticity for a layer perpendicular to fibres is E22. The displacement field in the first-order shear deformation theory (FSDT) considers rotations hx(x, y) and hy(x, y) about the y- and x-axes that are independently interpolated due to uncoupling between inplane displacements and bending displacements for symmetrically laminated plates, and can be expressed as uðx; y; zÞ ¼ u0 ðx; yÞ þ zhx ðx; yÞ;
ð10Þ
vðx; y; zÞ ¼ v0 ðx; yÞ þ zhy ðx; yÞ; wðx; y; zÞ ¼ w0 ðx; yÞ;
ð11Þ ð12Þ
where u, v and w are the displacements in the x-, y- and zdirections, respectively, u0, v0 and w0 are the corresponding midplane displacements, and hx and hy are the rotations in the xz- and yz-planes.
204
A.J.M. Ferreira, G.E. Fasshauer / Composite Structures 79 (2007) 202–210
The Green–Lagrange strain tensor is defined as ou ; ox ov y ¼ ; oy
ð13Þ
x ¼
xy ¼
ð14Þ
ou ov þ ; oy ox
ð15Þ
ou ow þ ; oz ox ov ow . yz ¼ þ oz oy
ð16Þ
xz ¼
ð17Þ
ohx ; ox ohy ; y ¼ z oy ohx ohy þ xy ¼ z ; oy ox
ð18Þ
x ¼ z
ð19Þ ð20Þ
ow0 ; ox ow0 . yz ¼ hy þ oy
ð21Þ
xz ¼ hx þ
ð22Þ
Neglecting rz for each layer, the stress–strain relations in the orthotropic local coordinate system can be expressed as 8 9 2 38 9 Q11 Q12 r1 > 0 0 0 > > > > > > e1 > > > 6 > > > > > 7> > > > > Q r Q 0 0 0 e > > > > 7 6 2 2 22 < = 6 12 7< = 7 6 ð23Þ 0 Q33 s12 ¼ 6 0 0 0 7 c12 ; > > > > > > > 7> 6 > > > > 0 0 0 Q s 0 c > > > > 5 4 23 > 23 > 44 > > > > ; ; : > : > 0 0 0 0 Q55 s31 c31 where subscripts 1 and 2 are respectively the fibre and the normal to fibre inplane directions, 3 is the direction normal to the plate, and the reduced stiffness components, Qij are given by E1 ; 1 m12 m21
Q33 ¼ G12 ;
Q12
Q16
0
Q22
Q26
0
Q26 0
Q66 0
0 Q44
0
0
Q45
38 9 > > > exx > > > > 7> eyy > > > 0 7> = < 7 c 0 7 7> xy >. 7> > cyz > > > Q45 5> > > > ; : > czx Q55 0
ð25Þ
Now we evaluate the bending and shear stiffness components, respectively, Dij and Aij, as Z h 2 ðAij ; Dij Þ ¼ ð1; z2 ÞQij dz. ð26Þ h2
Although it would be easy to consider general laminates, we only consider in this paper symmetric laminates. Therefore we can disregard contributions of u0, v0 and strains can be expressed as
Q11 ¼
9 2 8 rxx > Q11 > > > > > > > 6 > > r > Q = 6 < yy > 6 12 sxy ¼ 6 6 Q16 > > > > 6 > > > > 4 0 > syz > > > ; : szx 0
Q22 ¼
Q44 ¼ G23 ;
E2 ; 1 m12 m21
Q12 ¼ m21 Q11
Q55 ¼ G31 ;
m21 ¼ m12
E2 ; E1
ð24Þ
in which E1, E2, m12, G12, G23 and G31 are material properties of the lamina. By performing an appropriate coordinate transformation, the stress–strain relations in the global x–y–z coordinate system can be obtained as [23]
Note that because of the laminated scheme the integral should be replaced by a summation across the thickness direction. The equations of motion for the free vibration of laminated plates [23,24] can be expressed by the dynamic version of the principle of virtual displacements as o2 hx o2 hy o2 hy o2 hx o2 hx þ 2D16 þ D66 2 þ D16 2 þ ðD12 þ D66 Þ 2 ox ox oxoy oxoy oy o2 hy ow ow o2 hx kA55 hx þ ¼ I2 2 ; þ D26 2 kA45 hy þ oy ox oy ot
D11
o2 hx o2 hy o2 hx o2 hy o2 hx þ 2D26 þ D26 2 þ D66 2 þ ðD12 þ D66 Þ 2 ox ox oxoy oxoy oy o2 hy ow ow o2 hy kA45 hx þ ¼ I2 2 ; þ D22 2 kA44 hy þ oy ox oy ot o ow ow kA45 hy þ þ kA55 hx þ ox oy ox o ow ow o2 w kA44 hy þ þ kA45 hx þ ¼ I0 2 ; oy oy ox ot
ð27Þ
D16
ð28Þ
ð29Þ
where k is the shear correction factor, and Ii are the mass inertias defined as [23] Z h Z h 2 2 I0 ¼ q dz; I 2 ¼ qz2 dz. ð30Þ h2
h2
Here q and h denote the density and the total thickness of the composite plate, respectively. The bending moments Mx, My and Mxy and the shear forces Qx and Qy are expressed as functions of the displacement gradients and the material constitutive equations by ohx ohy ohx ohy M x ¼ D11 þ D12 þ D16 þ ; ð31Þ ox oy oy ox ohx ohy ohx ohy þ D22 þ D26 þ ; ð32Þ M y ¼ D12 ox oy oy ox ohx ohy ohx ohy þ D26 þ D66 þ ; ð33Þ M xy ¼ D16 ox oy oy ox ow ow þ kA45 hy þ ; ð34Þ Qx ¼ kA55 hx þ ox oy ow ow þ kA55 hy þ . ð35Þ Qy ¼ kA45 hx þ ox oy For free vibration problems we assume harmonic solution in terms of displacements w, hx, hy in the form
A.J.M. Ferreira, G.E. Fasshauer / Composite Structures 79 (2007) 202–210
wðx; y; tÞ ¼ W ðx; yÞeixt ;
ð36Þ
ixt
ð37Þ
ixt
ð38Þ
hx ðx; y; tÞ ¼ Wx ðx; yÞe ; hy ðx; y; tÞ ¼ Wy ðx; yÞe ;
where x is the frequency of natural vibration. Substituting the harmonic expansion into the equations of motion (27)– (29) we obtain the following equations in terms of the amplitudes W, Wx, Wy o2 Wx o2 Wy o2 Wy o2 Wx o2 Wx þ 2D16 þ D66 þ D16 þ ðD12 þ D66 Þ 2 2 ox ox ox oy ox oy oy 2 2 o Wy oW oW kA55 Wx þ þ D26 kA45 Wy þ oy ox oy 2
205
method. We consider various stacking sequences, boundary conditions and thickness-to-side ratios. We consider both square and rectangular plates. In order to compare with other sources, eigenvalues are expressed in terms defined of the non-dimensional frequency parameter x, as sffiffiffiffiffiffi qh ¼ ðxb =p Þ ; x D0 2
2
D11
2
ð39Þ
¼ I 2 x Wx ;
ð40Þ
ð41Þ
E11 =E22 ¼ 40; G23 ¼ 0:5E22 ; m12 ¼ 0:25; m21 ¼ 0:00625.
The boundary conditions for an arbitrary edge with simply supported, clamped or free edge conditions are as follows: (a) Simply supported • SS1, w = 0; Mn = 0; Mns = 0. • SS2, w = 0; Mn = 0; hs = 0. (b) Clamped, w = 0; hn = 0; hs = 0. (c) Free, Qn = 0; Mn = 0; Mns = 0. In previous equations, the subscripts n and s refer to the normal and tangential directions of the edge, respectively; Mn, Mns and Qn represent the normal bending moment, twisting moment and shear force on the plate edge; hn and hs represent the rotations about the tangential and normal coordinates at the plate edge. The stress resultants on an edge whose normal is represented by n = (nx, ny) can be expressed as M n ¼ n2x M x þ 2nx ny M xy þ n2y M y ; M ns ¼
ðn2x
n2y ÞM xy
nx ny ðMy MxÞ;
D0 ¼ E22 h3 =12ð1 m12 m21 Þ. Also, a constant shear correction factor k = p2/12 is used for all computations. The examples considered here are limited to thick symmetric cross-ply laminates with layers of equal thickness. The material properties for all layers of the laminates are identical:
o2 Wx o2 Wy o2 Wx o2 Wy o2 Wx þ 2D þ D D16 þ D þ ðD þ D Þ 66 12 66 26 26 ox2 ox2 ox oy ox oy oy 2 2 o Wy oW oW kA45 Wx þ þ D22 kA44 Wy þ oy ox oy 2 ¼ I 2 x2 Wy ; o oW oW kA45 Wy þ þ kA55 Wx þ ox oy ox o oW oW kA44 Wy þ þ kA45 Wx þ ¼ I 0 x2 W . þ oy oy ox
where
ð42Þ ð43Þ
Qn ¼ nx Qx þ ny Qy ;
ð44Þ
hn ¼ nx hx þ ny hy ; hs ¼ nx hy ny hx ;
ð45Þ ð46Þ
where nx and ny are the direction cosines of a unit normal vector at a point at the laminated plate boundary [23,24]. 4. Results for composite plates Here we analyze the natural frequencies and vibration modes of laminated composite plates by the RBF-PS
G13 ¼ G12 ¼ 0:6E22 ;
We consider SSSS (simply supported on all boards) and CCCC (clamped on all boards) boundary conditions, for their practical interest. It is important to mention that with this method it is very easy to include any type of essential or natural boundary conditions. However both essential and natural boundary conditions must be included. The convergence pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffistudy of frequency parameters ¼ ðxb2 =p2 Þ ðqh=D0 Þ for three-ply (0/90/0) simply x supported SSSS rectangular laminates is performed in Table 1, while the corresponding convergence study for CCCC rectangular laminate is performed in Table 2. It can be seen that a faster convergence is obtained for higher t/b ratios irrespective of a/b ratios. In both SSSS and CCCC cases the results converge well to Liew [25] results. The effect of thickness-to-length ratio on the fundamenpffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi ¼ ðxb2 =p2 Þ ðqh=D0 Þ for a tal frequency parameters x simply supported square four-ply laminate (0/90/90/0) is examined in Table 3. Our present solution considers inverse multiquadrics with a shape parameter ¼ pffiffi N pðffiffiN Þ=80. For thinner plates we consider ¼ N ðN Þ=120. The reason for this is that the problem becomes more ill-conditioned and we need higher accuracy (given by a lower shape parameter). Present solutions compare very well with those of Reddy and Phan [26] who used a higher-order shear deformation theory and Liew [25] who used a p-Ritz solution. Present results show very small deviation only for very thick laminates. The effect of thickness in the fundamental frequencies is clearly shown here. In Tables p 4–6 we ffi investigate frequency parameters ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ¼ ðxb2 =p2 Þ ðqh=D0 Þ for three-ply, five-ply and eightx
206
A.J.M. Ferreira, G.E. Fasshauer / Composite Structures 79 (2007) 202–210
Table 1 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ¼ ðxb2 =p2 Þ ðqh=D0 Þ for three-ply (0/90/0) simply supported SSSS rectangular laminates Convergence study of frequency parameters x Modes a/b
t/b
Grid
1
2
3
4
5
6
7
8
1
0.001
13 · 13 15 · 15 17 · 17 19 · 19
6.6228 6.6257 6.6260 6.6180
9.4368 9.4519 9.4435 9.4368
16.3957 16.2876 16.2246 16.2192
25.1363 25.1185 25.1124 25.1131
25.5315 26.4952 26.4925 26.4938
26.4889 26.6441 26.6477 26.6667
30.4345 30.2880 30.3251 30.2983
38.2722 37.7580 37.7661 37.7850
6.6252
9.4470
16.2051
25.1146
26.4982
26.6572
30.3139
37.7854
3.5934 3.5934 3.5934 3.5934 3.5934
5.7683 5.7683 5.7683 5.7683 5.7683
7.3968 7.3969 7.3968 7.3968 7.3968
8.6870 8.6870 8.6870 8.6870 8.6870
9.1442 9.1444 9.1444 9.1444 9.1444
11.2077 11.2078 11.2078 11.2078 11.2078
11.2217 11.2218 11.2218 11.2218 11.2218
12.1160 12.1161 12.1162 12.1161 12.1162
3.5939
5.7691
7.3972
8.6876
9.1451
11.2080
11.2225
12.1166
2.3790 2.3511 2.4262 2.3670
6.6162 6.6368 6.5945 6.6331
6.7161 6.7586 6.6310 6.6691
9.4318 9.4973 9.3494 9.4676
14.4301 14.1298 14.3323 14.2921
14.4846 14.4600 14.5476 14.3915
15.9727 16.0620 16.1477 16.1009
16.3840 16.5252 16.1968 16.1009
2.3618
6.6252
6.6845
9.4470
14.2869
16.3846
16.1347
16.2051
1.9387 1.9387 1.9387 1.9387
3.5934 3.5934 3.5934 3.5934
4.8750 4.8750 4.8750 4.8750
5.4849 5.4851 5.4851 5.4851
5.7683 5.7683 5.7683 5.7683
7.1168 7.1170 7.1170 7.1170
7.3964 7.3970 7.3969 7.3968
8.5969 8.5969 8.5969 8.5969
1.9393
3.5939
4.8755
5.4855
5.7691
7.1177
7.3972
8.5973
Liew (p-Ritz) [25] 0.20
11 · 11 13 · 13 15 · 15 17 · 17 19 · 19
Liew (p-Ritz) [25] 2
0.001
13 · 13 15 · 15 17 · 17 19 · 19
Liew (p-Ritz) [25] 0.20
13 · 13 15 · 15 17 · 17 19 · 19
Liew (p-Ritz) [25]
Table 2 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ¼ ðxb2 =p2 Þ ðqh=D0 Þ for three-ply (0/90/0) clamped CCCC rectangular laminates Convergence study of frequency parameters x Modes a/b
t/b
Grid
1
2
3
4
5
6
7
8
1
0.001
13 · 13 15 · 15 17 · 17 19 · 19
14.2138 14.6918 14.5866 14.8138
17.669 18.4741 17.4065 17.6181
25.5193 26.9611 24.5479 24.1145
38.0121 37.6121 35.3335 36.0900
39.3376 39.3560 39.1869 39.0170
40.7548 40.9241 41.4113 40.8323
48.8639 48.9940 44.3108 44.9457
70.8403 52.2651 53.0840 49.0715
14.6655
17.6138
24.5114
35.5318
39.1572
40.7685
44.7865
50.3226
4.4465 4.4465 4.4467 4.4463
6.6418 6.6420 6.6418 6.6419
7.6995 7.6995 7.6995 7.6995
9.1848 9.1848 9.1848 9.1839
9.7377 9.7377 9.7376 9.7376
11.3990 11.3990 11.3990 11.3994
11.6434 11.6435 11.6434 11.6420
12.4655 12.4655 12.4654 12.4651
4.4468
6.6419
7.6996
9.1852
9.7378
11.3991
11.6439
12.4658
4.9869 5.0970 2.4262 2.3670
10.0330 10.4052 6.5945 6.6331
13.1204 10.6097 6.6310 6.6691
15.1032 14.3575 9.3494 9.4676
19.9742 18.4830 14.3323 14.2921
24.6265 18.9482 14.5476 14.3915
28.9221 19.7608 16.1477 16.1009
33.0874 21.1550 16.1968 16.1009
2.3618
6.6252
6.6845
9.4470
14.2869
16.3846
16.1347
16.2051
1.9387 1.9387 1.9387 1.9387
3.5934 3.5934 3.5934 3.5934
4.8750 4.8750 4.8750 4.8750
5.4849 5.4851 5.4851 5.4851
5.7683 5.7683 5.7683 5.7683
7.1168 7.1170 7.1170 7.1170
7.3964 7.3970 7.3969 7.3968
8.5969 8.5969 8.5969 8.5969
1.9393
3.5939
4.8755
5.4855
5.7691
7.1177
7.3972
8.5973
Liew (p-Ritz) [25] 0.20
13 · 13 15 · 15 17 · 17 19 · 19
Liew (p-Ritz) [25] 2
0.001
13 · 13 15 · 15 17 · 17 19 · 19
Liew (p-Ritz) [25] 0.20
Liew (p-Ritz) [25]
13 · 13 15 · 15 17 · 17 19 · 19
ply rectangular laminates with SSSS and CCCC boundary conditions. We consider the first eight frequency parameters with various aspect and thickness ratios. We
do not compare with other sources, but readers can verify that present results are in very close agreement with results of Liew [25].
A.J.M. Ferreira, G.E. Fasshauer / Composite Structures 79 (2007) 202–210
207
Table 3 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ¼ ðxb2 =p2 Þ ðqh=D0 Þ for a simply supported square four-ply laminate (0/ Effect of thickness-to-length ratio on the fundamental frequency parameters x 90/90/0) t/b
0.01
0.02
0.04
0.05
0.08
0.10
0.20
0.25
0.50
Reddy and Phan [26] Liew (p-Ritz) [25] Present results
6.578 6.606 6.6012
6.475 6.549 6.5438
6.330 6.338 6.3300
6.196 6.193 6.1844
5.708 5.677 5.6641
5.355 5.311 5.2960
3.854 3.807 3.7903
3.331 3.295 3.2796
1.956 1.929 1.9180
Table 4 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ¼ ðxb2 =p2 Þ ðqh=D0 Þ for three-ply (0/90/0) rectangular laminates with SSSS and CCCC boundary conditions Frequency parameters x Modes B.C.
a/b
t/b
3
4
5
6
7
8
SSSS
1
0.001 0.050 0.100 0.150 0.200 0.001 0.050 0.100 0.150 0.200
6.6226 6.1365 5.1652 4.2741 3.5934 2.3511 2.3221 2.2210 2.0848 1.9387
9.5306 8.8846 7.7549 6.6657 5.7683 6.6368 6.1361 5.1652 4.2741 3.5934
16.4255 15.1061 12.9129 9.4875 7.3968 6.7586 6.4785 6.0163 5.4428 4.8750
21.6731 19.3527 13.0478 10.8227 8.6870 9.4973 8.8835 7.7547 6.6656 5.4852
25.0307 20.6609 14.3735 10.8246 9.1444 14.1298 12.1487 9.0208 6.8683 5.7683
26.5070 24.0653 17.7849 13.8027 11.2078 14.4600 13.5782 10.7943 8.5950 7.1170
30.4155 24.3377 19.4998 14.6650 11.2218 16.0620 13.9489 11.8480 9.4871 7.3965
37.7401 31.0194 21.0501 15.5895 12.1162 16.5252 15.1050 12.9126 10.0820 8.5969
0.001 0.050 0.100 0.150 0.200 0.001 0.050 0.100 0.150 0.200
14.6918 10.9530 7.4107 5.5481 4.4465 5.0970 4.7775 4.1407 3.5395 3.0452
18.4741 14.0235 10.3930 8.1467 6.6420 10.4052 8.8358 6.6158 5.1811 4.2481
26.9611 20.3851 13.9124 9.9039 7.6995 10.6097 9.8448 8.3534 6.9263 5.7916
37.6121 23.1953 15.4281 11.6209 9.1848 14.3575 12.5070 9.8939 7.4236 5.9042
37.6121 24.9771 15.8046 12.0249 9.7377 18.4830 14.7012 9.9660 7.9354 6.5347
39.3560 29.2354 19.5706 14.6439 11.3990 18.9482 17.3208 12.4394 9.5758 7.6885
40.9241 29.3618 21.4885 14.9106 11.6435 19.7608 17.6708 13.6617 9.8560 7.7280
48.9940 36.2616 21.6199 16.1227 12.4655 21.1550 19.4302 14.1197 11.2299 9.1754
2
CCCC
1
2
1
2
Table 5 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ¼ ðxb2 =p2 Þ ðqh=D0 Þ for five-ply (0/90/0/90/0) rectangular laminates with SSSS and CCCC boundary conditions Frequency parameters x Modes B.C.
a/b
t/b
2
3
4
5
6
7
8
SSSS
1
0.001 0.050 0.100 0.150 0.200 0.001 0.050 0.100 0.150 0.200
6.6184 6.2237 5.3966 4.5811 3.9130 3.4097 3.3605 3.1641 2.9041 2.6310
13.7442 12.6569 10.5241 8.5751 7.0803 6.5120 6.2233 5.3966 4.5811 3.9130
23.0420 18.2915 12.7094 9.4002 7.4027 12.2291 11.3460 8.8821 6.9230 5.6097
26.4671 21.5864 15.6520 11.8834 9.4679 13.3916 11.5708 9.4992 7.7835 6.4529
28.1753 24.0215 17.9089 13.6241 10.8185 13.7545 12.6562 10.5239 8.5750 7.0802
36.6880 29.9092 20.4687 14.3910 11.0526 18.1865 16.1365 12.7089 9.3999 7.4024
49.1036 33.5073 21.3551 15.9159 12.5098 23.0930 18.2891 12.7125 10.0334 8.1431
51.1558 35.5265 22.4384 16.1283 12.5295 26.8986 21.5903 15.6531 11.8838 9.2173
0.001 0.050 0.100 0.150 0.200 0.001 0.050 0.100 0.150 0.200
14.4337 11.5785 8.1664 6.1120 4.8279 7.5611 6.9345 5.6265 4.4909 3.6566
18.7244 18.5068 12.9490 9.5821 7.5339 11.1165 9.8001 7.4626 5.7694 4.6327
35.8179 22.6809 13.9913 10.0141 7.7699 17.4333 14.7794 10.3319 7.7193 6.1015
40.9718 27.0134 17.2787 12.4715 9.7082 19.2374 15.9510 11.5057 8.6345 6.8399
40.9718 29.4758 19.5291 14.1885 11.0603 21.3109 17.4812 12.5413 9.3846 7.4233
52.5993 35.5447 21.1658 14.7616 11.2888 24.2316 20.7812 13.7094 9.9550 7.7551
66.2199 36.4352 22.6652 16.2923 12.6459 31.9367 20.9977 14.4607 10.7071 8.4255
66.2199 39.3569 23.4906 16.5381 12.7084 36.4012 25.6764 17.0595 12.3089 9.4919
2
CCCC
1
2
1
208
A.J.M. Ferreira, G.E. Fasshauer / Composite Structures 79 (2007) 202–210
Table 6 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ¼ ðxb2 =p2 Þ ðqh=D0 Þ for eight-ply (0/90/0/90)s rectangular laminates with SSSS and CCCC boundary conditions Frequency parameters x Modes B.C.
a/b
t/b
2
3
4
5
6
7
8
SSSS
1
0.001 0.050 0.100 0.150 0.200 0.001 0.050 0.100 0.150 0.200
6.6148 6.2674 5.5034 4.7051 4.0244 3.9453 3.8359 3.5462 3.1859 2.8308
15.8073 14.1850 11.3234 8.9701 7.2832 6.5314 6.2669 5.5033 4.7051 4.0244
21.6703 17.6952 12.6366 9.4599 7.4848 12.6020 11.2255 8.8411 6.9870 5.6928
26.4531 22.0135 16.0976 12.1654 9.6364 14.6609 13.1511 10.4566 8.2568 6.6929
33.4402 26.9372 18.9226 13.9935 10.9642 15.8207 14.1843 11.3232 8.9700 7.2832
40.1156 31.9240 20.4521 14.4859 11.1554 19.3743 17.1091 12.6361 9.4595 7.4845
47.8620 32.7265 22.1297 16.2275 12.6481 21.7475 17.6923 13.3045 10.3572 8.3223
50.6101 35.3623 22.7706 16.3827 12.6977 26.6222 22.0172 16.0987 11.9633 9.3078
0.001 0.050 0.100 0.150 0.200 0.001 0.050 0.100 0.150 0.200
14.5934 11.8554 8.4047 6.2450 4.9015 8.6914 7.7628 6.0089 4.6604 3.7366
25.3037 19.9841 13.4640 9.8065 7.6548 11.5251 10.2038 7.6943 5.8836 4.6899
33.9140 22.5157 14.1032 10.0991 7.8286 17.4447 14.7821 10.4473 7.7948 6.1484
40.4159 27.7837 17.6385 12.6436 9.8091 22.9715 17.8081 12.1039 8.8785 6.9648
47.1988 31.9119 20.2179 14.4547 11.1745 24.8652 19.0746 13.0527 9.5957 7.5326
56.2557 36.0981 21.2564 14.8657 11.3751 26.8336 20.7354 13.7684 10.0226 7.8082
66.2642 37.3787 23.2322 16.5204 12.7501 34.4222 21.9484 14.8664 10.8841 8.5232
67.6171 39.6694 23.7684 16.7072 12.8238 34.4222 26.4059 17.3496 12.3855 9.5582
2
CCCC
1
2
1
The results show that frequency parameters decrease with increasing a/b ratio. These data also show that an increase in thickness ratio results in decreasing of frequency parameters. This is due to the effects of shear deformation and rotary inertia. These effects are more pronounced in higher modes. Also higher frequencies are obtained with
CCCC boundaries, as expected because higher constraints produces higher frequencies. The present method produces very stable and well defined frequency modes, as shown in Figs. 1–3. The eigenmodes for the other examples have similar characteristics.
Fig. 1. Mode shapes (1–9) for three-ply (0/90/0) simply supported SSSS square laminates, a/b = 1, t/b = 0.2, 15 · 15 nodal grid.
A.J.M. Ferreira, G.E. Fasshauer / Composite Structures 79 (2007) 202–210
209
Fig. 2. Mode shapes (1–9) for three-ply (0/90/0) clamped CCCC square laminates a/b = 1, t/b = 0.2, 15 · 15 nodal grid.
Fig. 3. Mode shapes (1–9) for three-ply (0/90/0) simply supported SSSS rectangular laminates a/b = 2, t/b = 0.2, 15 · 15 nodal grid.
5. Conclusions A study of free vibration of isotropic and composite plates was presented. Equations of motion and boundary
conditions for symmetric laminated composite plates were presented. Numerical results were presented and discussed for various thickness-to-length ratios. The analysis is based on an innovative numerical scheme, where collocation by
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radial basis functions and pseudospectral methods are combined to produce highly accurate results. This method allows the use of non-rectangular geometries and promotes a more flexible framework for pseudospectral methods. Various aspect ratios, thickness ratios and boundary conditions were considered for isotropic and various types of laminates. The present method showed very high accuracy for both isotropic and composite plates. References [1] Laura PAA, Gutierrez RH. Analysis of vibrating Timoshenko beams using the method of differential quadrature. Shock Vib 1993;1(1):89–93. [2] Bert CW, Malik M. Differential quadrature method in computational mechanics: a review. Appl Mech Rev 1996;49(1):1–28. [3] Liew KM, Han JB, Xiao ZM. Vibration analysis of circular Mindlin plates using differential quadrature method. J Sound Vib 1997;205(5):617–30. [4] Liew KM, Hung KC, Lim MK. Vibration of Mindlin plates using boundary characteristic orthogonal polynomials. J Sound Vib 1995;182(1):77–90. [5] Lee J, Schultz WW. Eigenvalue analysis of Timoshenko beams and axisymmetric Mindlin plates by the pseudospectral method. J Sound Vib 2004;269:609–21. [6] Ferreira AJM. Free vibration analysis of Timoshenko beams and Mindlin plates by radial basis functions. Int J Comput Methods 2005;2(1):15–31. [7] Ferreira AJM, Roque CMC, Martins PALS. Radial basis functions and higher-order theories in the analysis of laminated composite beams and plates. Compos Struct 2004;66:287–93. [8] Ferreira AJM. A formulation of the multiquadric radial basis function method for the analysis of laminated composite plates. Compos Struct 2003;59:385–92. [9] Ferreira AJM, Roque CMC, Martins PALS. Analysis of composite plates using higher-order shear deformation theory and a finite point formulation based on the multiquadric radial basis function method. Composites: Part B 2003;34:627–36. [10] Ferreira AJM. Thick composite beam analysis using a global meshless approximation based on radial basis functions. Mech Adv Mater Struct 2003;10:271–84.
[11] Ferreira AJM. Analysis of composite plates using a layerwise deformation theory and multiquadrics discretization. Mech Adv Mater Struct 2005;12(2):99–112. [12] Ferreira AJM. Polyharmonic (thin-plate) splines in the analysis of composite plates. Int J Mech Sci 2005;46:1549–69. [13] Ferreira AJM, Batra RC, Roque CMC, Qian LF, Martins PALS. Static analysis of functionally graded plates using third-order shear deformation theory and a meshless method. Compos Struct 2005;69:449–57. [14] Driscoll TA, Fornberg B. Interpolation in the limit of increasingly flat radial basis functions. Comput Math Appl 2002;43:413–22. [15] Larsson E, Fornberg B. Theoretical and computational aspects of multivariate interpolation with increasingly flat radial basis functions. Comput Math Appl 2005;49:103–30. [16] Schaback R. Multivariate interpolation by polynomials and radial basis functions. Constr Approx 2005;21:293–317. [17] Fasshauer GE. RBF collocation methods as pseudospectral methods. In: Kassab A, Brebbia CA, Divo E, Poljak D, editors. Boundary elements XXVII. Southampton: WIT Press; 2005. p. 47–56. [18] Fasshauer GE. RBF collocation methods and pseudospectral methods, preprint 2004. [19] Ferreira AJM, Fasshauer GF. Natural frequencies of shear deformable beams and plates by an RBF-pseudospectral method, submitted for publication. [20] Platte RB, Driscoll TA. Eigenvalue stability of radial basis function discretizations for time-dependent problems. Comput Math Appl, to appear. [21] Sarra SA. Integrated multiquadric radial basis function approximation methods. Comput Math Appl, to appear. [22] Trefethen LN. Spectral Methods in Matlab. Philadelphia, PA: SIAM; 2000. [23] Reddy JN. Mechanics of laminated composite plates. New York: CRC Press; 1997. [24] Liew KM, Huang YQ, Reddy JN. Vibration analysis of symmetrically laminated plates based on FSDT using the moving least squares differential quadrature method. Comput Meth Appl Mech Eng 2003;192:2203–22. [25] Liew KM. Solving the vibration of thick symmetric laminates by Reissner/Mindlin plate theory and the p-Ritz method. J Sound Vib 1996;198(3):343–60. [26] Reddy JN, Phan PH. Stability and vibration of isotropic, orthotropic and laminated plates according to a higher order shear deformation theory. J Sound Vib 1985;98:157–70.
Analysis of natural frequencies of composite plates by an RBF-pseudospectral method A.J.M. Ferreira a
a,*
, G.E. Fasshauer
b
Departamento de Engenharia Mecnica e Gesto Industrial, Faculdade de Engenharia da Universidade do Porto, Rua Dr. Roberto Frias, 4200-465 Porto, Portugal b Department of Applied Mathematics, Illinois Institute of Technology, Chicago 60616, USA Available online 21 February 2006
Abstract A study of free vibration of symmetric composite plates is presented. The analysis is based on an innovative numerical scheme, where collocation by radial basis functions and pseudospectral methods are combined to produce highly accurate results. Numerical results are presented and discussed for various thickness-to-length ratios. 2006 Elsevier Ltd. All rights reserved. Keywords: Radial basis functions; Pseudospectral methods; Free vibration; Composite plates
1. Introduction Vibration of thick and thin composite plates is an important subject in the design of mechanical, civil and aerospace applications. The thickness of most parts of composite plates makes the transverse shear and the rotatory inertia not negligible as in classical theories. Therefore the first-order shear deformation theory for plates should be considered in general analysis. The analysis of free vibration of isotropic and composite plates is best performed by numerical techniques. The differential quadrature method [1–3], the boundary characteristic orthogonal polynomials [4] and the pseudospectral method [5] were used in recent years. The finite element method also proved to be very adequate for this type of problems. The number of references is too large to list here. More recently the free vibration analysis of Timoshenko beams and Mindlin plates by Kansa’s non-symmetric radial basis function (RBF) collocation method was performed by Ferreira [6–13]. *
Corresponding author. Tel.: +351 22 957 8713; fax: +351 22 953 7352. E-mail addresses: [email protected] (A.J.M. Ferreira), fasshauer@ iit.edu (G.E. Fasshauer). 0263-8223/$ - see front matter 2006 Elsevier Ltd. All rights reserved. doi:10.1016/j.compstruct.2005.12.004
In the present work we illustrate for the first time the application of the combination of radial basis functions and pseudospectral methods to the eigenvalue analysis of symmetric composite plates. The method proves to be elegant and of very high quality when compared to analytical solutions and Kansa’s collocation method. The paper is organized as follows. In Section 2 our approach to pseudospectral methods is discussed. In Section 3 the free vibration of symmetric composite plates are presented and results discussed in Section 4. Finally conclusions are presented in Section 5. 2. The pseudospectral method Based on recent work by Fornberg, Schaback and others establishing a close connection between radial basis function and polynomial interpolants (see, e.g., [14–16]) various authors have suggested using RBFs in ‘‘pseudospectral mode’’ to solve partial differential equations (PDEs) (see, e.g., [17–21]). There are several potential advantages when using RBFs instead of traditional polynomial methods. First, RBFs allow us to work with irregular grids, and on domains exhibiting irregular geometries. And secondly, it has been
A.J.M. Ferreira, G.E. Fasshauer / Composite Structures 79 (2007) 202–210
shown that, while the limiting case of flat RBFs leads to polynomial interpolants, one can actually achieve more accurate results with RBFs that have not quite converged to their polynomial limit (see, e.g., [14,15]). Thus, RBFPS methods are potentially more flexible and more accurate than polynomial methods. Since we presented a fairly detailed exposition of the RBF-PS method in our recent paper [19] we will only give a brief introduction here. More details about the RBF-PS method can be obtained from either [18,19] or any of the other papers mentioned above. The starting point for the RBF-PS method is the representation of (the spatial part of) the approximate solution uh of a given PDE by a linear combination of smooth (radial) basis functions /j, j = 1, . . . , N, i.e., uh ðxÞ ¼
N X
cj /j ðxÞ;
x 2 Rd ;
ð1Þ
j¼1
where /j(x) = U(kx xjk) with a single univariate basic function U. For the numerical experiments in this paper we used the inverse multiquadric 1 UðrÞ ¼ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi . 2 1 þ ðrÞ
ð2Þ
This function is infinitely smooth and positive definite on Rd . As pointed out above, the shape parameter can be used to influence the accuracy of the numerical method. The (flat) limiting case ! 0 usually corresponds to a polynomial, and a small (positive) shape parameter usually yields the best accuracy (however, at the cost of low numerical stability). How to find this ‘‘optimal’’ shape parameter remains an open research question. An important feature of pseudospectral methods is the fact that one usually is content with obtaining an approximation to the solution on a discrete set of grid points X ¼ fxi ; i ¼ 1; . . . ; N g. One of several ways to implement the pseudospectral method is via so-called differentiation matrices, i.e., one finds a matrix L such that at the grid points xi we have uL ¼ Lu.
ð3Þ
Here u = [uh(x1), . . . , uh(xN)]T is the vector of values of uh at the grid points and uL ¼ ½Luh ðx1 Þ; . . . ; Luh ðxN ÞT , where L is the (spatial) linear differential operator of the PDE under consideration. The differentiation matrix L can be interpreted as a discretized differential operator that maps function values to ‘‘derivative values’’ on the computational grid X. An expression for the differentiation matrix L of (3) can be obtained in two steps. First, we evaluate (1) at the grid points xi, i = 1, . . . , N. This results in uh ðxi Þ ¼
N X
cj /j ðxi Þ;
i ¼ 1; . . . ; N ;
j¼1
or in matrix-vector notation
ð4Þ
203
u ¼ Ac;
ð5Þ T
where c = [c1, . . . , cN] is the coefficient vector, the evaluation matrix A has entries Aij = /j(xi), and u is as before. Next, we apply the (linear) differential operator L to (1). This leads to Luh ðxÞ ¼
N X
cj L/j ðxÞ.
ð6Þ
j¼1
If we again evaluate at the grid points xi then we get in matrix-vector notation uL ¼ AL c;
ð7Þ
where uL and c are as above, and the matrix AL has entries AL;ij ¼ L/j ðxÞx¼xi . It is now straightforward to obtain the desired formula for L. We simply solve Eq. (5) for c and substitute the result into (7). This gives us uL ¼ AL A1 u;
ð8Þ
so that the differentiation matrix L corresponding to (3) is of the form L ¼ AL A1 .
ð9Þ
We enforce boundary conditions directly by manipulating the corresponding rows of L as described, e.g., in [22]. This results in a modified differentiation matrix LC. As pointed out in [18,19] invertibility of the differentiation matrix LC is equivalent to the invertibility of the system matrix for Kansa’s non-symmetric RBF collocation method, and thus to be treated with care (since certain rare point configurations exist for which the matrix may be singular). 3. Free vibration analysis of symmetric laminated plates In this paper we consider a laminated plate of length a, width b and thickness h, with the coordinate system at the midplane of the laminate. We also consider that each layer of the laminate is orthotropic with respect to the x- and yaxes and all layers are of equal thickness. Each layer direction is defined by the angle with the x-axis. The modulus of elasticity for a layer parallel to fibres is E11 and the modulus of elasticity for a layer perpendicular to fibres is E22. The displacement field in the first-order shear deformation theory (FSDT) considers rotations hx(x, y) and hy(x, y) about the y- and x-axes that are independently interpolated due to uncoupling between inplane displacements and bending displacements for symmetrically laminated plates, and can be expressed as uðx; y; zÞ ¼ u0 ðx; yÞ þ zhx ðx; yÞ;
ð10Þ
vðx; y; zÞ ¼ v0 ðx; yÞ þ zhy ðx; yÞ; wðx; y; zÞ ¼ w0 ðx; yÞ;
ð11Þ ð12Þ
where u, v and w are the displacements in the x-, y- and zdirections, respectively, u0, v0 and w0 are the corresponding midplane displacements, and hx and hy are the rotations in the xz- and yz-planes.
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The Green–Lagrange strain tensor is defined as ou ; ox ov y ¼ ; oy
ð13Þ
x ¼
xy ¼
ð14Þ
ou ov þ ; oy ox
ð15Þ
ou ow þ ; oz ox ov ow . yz ¼ þ oz oy
ð16Þ
xz ¼
ð17Þ
ohx ; ox ohy ; y ¼ z oy ohx ohy þ xy ¼ z ; oy ox
ð18Þ
x ¼ z
ð19Þ ð20Þ
ow0 ; ox ow0 . yz ¼ hy þ oy
ð21Þ
xz ¼ hx þ
ð22Þ
Neglecting rz for each layer, the stress–strain relations in the orthotropic local coordinate system can be expressed as 8 9 2 38 9 Q11 Q12 r1 > 0 0 0 > > > > > > e1 > > > 6 > > > > > 7> > > > > Q r Q 0 0 0 e > > > > 7 6 2 2 22 < = 6 12 7< = 7 6 ð23Þ 0 Q33 s12 ¼ 6 0 0 0 7 c12 ; > > > > > > > 7> 6 > > > > 0 0 0 Q s 0 c > > > > 5 4 23 > 23 > 44 > > > > ; ; : > : > 0 0 0 0 Q55 s31 c31 where subscripts 1 and 2 are respectively the fibre and the normal to fibre inplane directions, 3 is the direction normal to the plate, and the reduced stiffness components, Qij are given by E1 ; 1 m12 m21
Q33 ¼ G12 ;
Q12
Q16
0
Q22
Q26
0
Q26 0
Q66 0
0 Q44
0
0
Q45
38 9 > > > exx > > > > 7> eyy > > > 0 7> = < 7 c 0 7 7> xy >. 7> > cyz > > > Q45 5> > > > ; : > czx Q55 0
ð25Þ
Now we evaluate the bending and shear stiffness components, respectively, Dij and Aij, as Z h 2 ðAij ; Dij Þ ¼ ð1; z2 ÞQij dz. ð26Þ h2
Although it would be easy to consider general laminates, we only consider in this paper symmetric laminates. Therefore we can disregard contributions of u0, v0 and strains can be expressed as
Q11 ¼
9 2 8 rxx > Q11 > > > > > > > 6 > > r > Q = 6 < yy > 6 12 sxy ¼ 6 6 Q16 > > > > 6 > > > > 4 0 > syz > > > ; : szx 0
Q22 ¼
Q44 ¼ G23 ;
E2 ; 1 m12 m21
Q12 ¼ m21 Q11
Q55 ¼ G31 ;
m21 ¼ m12
E2 ; E1
ð24Þ
in which E1, E2, m12, G12, G23 and G31 are material properties of the lamina. By performing an appropriate coordinate transformation, the stress–strain relations in the global x–y–z coordinate system can be obtained as [23]
Note that because of the laminated scheme the integral should be replaced by a summation across the thickness direction. The equations of motion for the free vibration of laminated plates [23,24] can be expressed by the dynamic version of the principle of virtual displacements as o2 hx o2 hy o2 hy o2 hx o2 hx þ 2D16 þ D66 2 þ D16 2 þ ðD12 þ D66 Þ 2 ox ox oxoy oxoy oy o2 hy ow ow o2 hx kA55 hx þ ¼ I2 2 ; þ D26 2 kA45 hy þ oy ox oy ot
D11
o2 hx o2 hy o2 hx o2 hy o2 hx þ 2D26 þ D26 2 þ D66 2 þ ðD12 þ D66 Þ 2 ox ox oxoy oxoy oy o2 hy ow ow o2 hy kA45 hx þ ¼ I2 2 ; þ D22 2 kA44 hy þ oy ox oy ot o ow ow kA45 hy þ þ kA55 hx þ ox oy ox o ow ow o2 w kA44 hy þ þ kA45 hx þ ¼ I0 2 ; oy oy ox ot
ð27Þ
D16
ð28Þ
ð29Þ
where k is the shear correction factor, and Ii are the mass inertias defined as [23] Z h Z h 2 2 I0 ¼ q dz; I 2 ¼ qz2 dz. ð30Þ h2
h2
Here q and h denote the density and the total thickness of the composite plate, respectively. The bending moments Mx, My and Mxy and the shear forces Qx and Qy are expressed as functions of the displacement gradients and the material constitutive equations by ohx ohy ohx ohy M x ¼ D11 þ D12 þ D16 þ ; ð31Þ ox oy oy ox ohx ohy ohx ohy þ D22 þ D26 þ ; ð32Þ M y ¼ D12 ox oy oy ox ohx ohy ohx ohy þ D26 þ D66 þ ; ð33Þ M xy ¼ D16 ox oy oy ox ow ow þ kA45 hy þ ; ð34Þ Qx ¼ kA55 hx þ ox oy ow ow þ kA55 hy þ . ð35Þ Qy ¼ kA45 hx þ ox oy For free vibration problems we assume harmonic solution in terms of displacements w, hx, hy in the form
A.J.M. Ferreira, G.E. Fasshauer / Composite Structures 79 (2007) 202–210
wðx; y; tÞ ¼ W ðx; yÞeixt ;
ð36Þ
ixt
ð37Þ
ixt
ð38Þ
hx ðx; y; tÞ ¼ Wx ðx; yÞe ; hy ðx; y; tÞ ¼ Wy ðx; yÞe ;
where x is the frequency of natural vibration. Substituting the harmonic expansion into the equations of motion (27)– (29) we obtain the following equations in terms of the amplitudes W, Wx, Wy o2 Wx o2 Wy o2 Wy o2 Wx o2 Wx þ 2D16 þ D66 þ D16 þ ðD12 þ D66 Þ 2 2 ox ox ox oy ox oy oy 2 2 o Wy oW oW kA55 Wx þ þ D26 kA45 Wy þ oy ox oy 2
205
method. We consider various stacking sequences, boundary conditions and thickness-to-side ratios. We consider both square and rectangular plates. In order to compare with other sources, eigenvalues are expressed in terms defined of the non-dimensional frequency parameter x, as sffiffiffiffiffiffi qh ¼ ðxb =p Þ ; x D0 2
2
D11
2
ð39Þ
¼ I 2 x Wx ;
ð40Þ
ð41Þ
E11 =E22 ¼ 40; G23 ¼ 0:5E22 ; m12 ¼ 0:25; m21 ¼ 0:00625.
The boundary conditions for an arbitrary edge with simply supported, clamped or free edge conditions are as follows: (a) Simply supported • SS1, w = 0; Mn = 0; Mns = 0. • SS2, w = 0; Mn = 0; hs = 0. (b) Clamped, w = 0; hn = 0; hs = 0. (c) Free, Qn = 0; Mn = 0; Mns = 0. In previous equations, the subscripts n and s refer to the normal and tangential directions of the edge, respectively; Mn, Mns and Qn represent the normal bending moment, twisting moment and shear force on the plate edge; hn and hs represent the rotations about the tangential and normal coordinates at the plate edge. The stress resultants on an edge whose normal is represented by n = (nx, ny) can be expressed as M n ¼ n2x M x þ 2nx ny M xy þ n2y M y ; M ns ¼
ðn2x
n2y ÞM xy
nx ny ðMy MxÞ;
D0 ¼ E22 h3 =12ð1 m12 m21 Þ. Also, a constant shear correction factor k = p2/12 is used for all computations. The examples considered here are limited to thick symmetric cross-ply laminates with layers of equal thickness. The material properties for all layers of the laminates are identical:
o2 Wx o2 Wy o2 Wx o2 Wy o2 Wx þ 2D þ D D16 þ D þ ðD þ D Þ 66 12 66 26 26 ox2 ox2 ox oy ox oy oy 2 2 o Wy oW oW kA45 Wx þ þ D22 kA44 Wy þ oy ox oy 2 ¼ I 2 x2 Wy ; o oW oW kA45 Wy þ þ kA55 Wx þ ox oy ox o oW oW kA44 Wy þ þ kA45 Wx þ ¼ I 0 x2 W . þ oy oy ox
where
ð42Þ ð43Þ
Qn ¼ nx Qx þ ny Qy ;
ð44Þ
hn ¼ nx hx þ ny hy ; hs ¼ nx hy ny hx ;
ð45Þ ð46Þ
where nx and ny are the direction cosines of a unit normal vector at a point at the laminated plate boundary [23,24]. 4. Results for composite plates Here we analyze the natural frequencies and vibration modes of laminated composite plates by the RBF-PS
G13 ¼ G12 ¼ 0:6E22 ;
We consider SSSS (simply supported on all boards) and CCCC (clamped on all boards) boundary conditions, for their practical interest. It is important to mention that with this method it is very easy to include any type of essential or natural boundary conditions. However both essential and natural boundary conditions must be included. The convergence pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffistudy of frequency parameters ¼ ðxb2 =p2 Þ ðqh=D0 Þ for three-ply (0/90/0) simply x supported SSSS rectangular laminates is performed in Table 1, while the corresponding convergence study for CCCC rectangular laminate is performed in Table 2. It can be seen that a faster convergence is obtained for higher t/b ratios irrespective of a/b ratios. In both SSSS and CCCC cases the results converge well to Liew [25] results. The effect of thickness-to-length ratio on the fundamenpffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi ¼ ðxb2 =p2 Þ ðqh=D0 Þ for a tal frequency parameters x simply supported square four-ply laminate (0/90/90/0) is examined in Table 3. Our present solution considers inverse multiquadrics with a shape parameter ¼ pffiffi N pðffiffiN Þ=80. For thinner plates we consider ¼ N ðN Þ=120. The reason for this is that the problem becomes more ill-conditioned and we need higher accuracy (given by a lower shape parameter). Present solutions compare very well with those of Reddy and Phan [26] who used a higher-order shear deformation theory and Liew [25] who used a p-Ritz solution. Present results show very small deviation only for very thick laminates. The effect of thickness in the fundamental frequencies is clearly shown here. In Tables p 4–6 we ffi investigate frequency parameters ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ¼ ðxb2 =p2 Þ ðqh=D0 Þ for three-ply, five-ply and eightx
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Table 1 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ¼ ðxb2 =p2 Þ ðqh=D0 Þ for three-ply (0/90/0) simply supported SSSS rectangular laminates Convergence study of frequency parameters x Modes a/b
t/b
Grid
1
2
3
4
5
6
7
8
1
0.001
13 · 13 15 · 15 17 · 17 19 · 19
6.6228 6.6257 6.6260 6.6180
9.4368 9.4519 9.4435 9.4368
16.3957 16.2876 16.2246 16.2192
25.1363 25.1185 25.1124 25.1131
25.5315 26.4952 26.4925 26.4938
26.4889 26.6441 26.6477 26.6667
30.4345 30.2880 30.3251 30.2983
38.2722 37.7580 37.7661 37.7850
6.6252
9.4470
16.2051
25.1146
26.4982
26.6572
30.3139
37.7854
3.5934 3.5934 3.5934 3.5934 3.5934
5.7683 5.7683 5.7683 5.7683 5.7683
7.3968 7.3969 7.3968 7.3968 7.3968
8.6870 8.6870 8.6870 8.6870 8.6870
9.1442 9.1444 9.1444 9.1444 9.1444
11.2077 11.2078 11.2078 11.2078 11.2078
11.2217 11.2218 11.2218 11.2218 11.2218
12.1160 12.1161 12.1162 12.1161 12.1162
3.5939
5.7691
7.3972
8.6876
9.1451
11.2080
11.2225
12.1166
2.3790 2.3511 2.4262 2.3670
6.6162 6.6368 6.5945 6.6331
6.7161 6.7586 6.6310 6.6691
9.4318 9.4973 9.3494 9.4676
14.4301 14.1298 14.3323 14.2921
14.4846 14.4600 14.5476 14.3915
15.9727 16.0620 16.1477 16.1009
16.3840 16.5252 16.1968 16.1009
2.3618
6.6252
6.6845
9.4470
14.2869
16.3846
16.1347
16.2051
1.9387 1.9387 1.9387 1.9387
3.5934 3.5934 3.5934 3.5934
4.8750 4.8750 4.8750 4.8750
5.4849 5.4851 5.4851 5.4851
5.7683 5.7683 5.7683 5.7683
7.1168 7.1170 7.1170 7.1170
7.3964 7.3970 7.3969 7.3968
8.5969 8.5969 8.5969 8.5969
1.9393
3.5939
4.8755
5.4855
5.7691
7.1177
7.3972
8.5973
Liew (p-Ritz) [25] 0.20
11 · 11 13 · 13 15 · 15 17 · 17 19 · 19
Liew (p-Ritz) [25] 2
0.001
13 · 13 15 · 15 17 · 17 19 · 19
Liew (p-Ritz) [25] 0.20
13 · 13 15 · 15 17 · 17 19 · 19
Liew (p-Ritz) [25]
Table 2 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ¼ ðxb2 =p2 Þ ðqh=D0 Þ for three-ply (0/90/0) clamped CCCC rectangular laminates Convergence study of frequency parameters x Modes a/b
t/b
Grid
1
2
3
4
5
6
7
8
1
0.001
13 · 13 15 · 15 17 · 17 19 · 19
14.2138 14.6918 14.5866 14.8138
17.669 18.4741 17.4065 17.6181
25.5193 26.9611 24.5479 24.1145
38.0121 37.6121 35.3335 36.0900
39.3376 39.3560 39.1869 39.0170
40.7548 40.9241 41.4113 40.8323
48.8639 48.9940 44.3108 44.9457
70.8403 52.2651 53.0840 49.0715
14.6655
17.6138
24.5114
35.5318
39.1572
40.7685
44.7865
50.3226
4.4465 4.4465 4.4467 4.4463
6.6418 6.6420 6.6418 6.6419
7.6995 7.6995 7.6995 7.6995
9.1848 9.1848 9.1848 9.1839
9.7377 9.7377 9.7376 9.7376
11.3990 11.3990 11.3990 11.3994
11.6434 11.6435 11.6434 11.6420
12.4655 12.4655 12.4654 12.4651
4.4468
6.6419
7.6996
9.1852
9.7378
11.3991
11.6439
12.4658
4.9869 5.0970 2.4262 2.3670
10.0330 10.4052 6.5945 6.6331
13.1204 10.6097 6.6310 6.6691
15.1032 14.3575 9.3494 9.4676
19.9742 18.4830 14.3323 14.2921
24.6265 18.9482 14.5476 14.3915
28.9221 19.7608 16.1477 16.1009
33.0874 21.1550 16.1968 16.1009
2.3618
6.6252
6.6845
9.4470
14.2869
16.3846
16.1347
16.2051
1.9387 1.9387 1.9387 1.9387
3.5934 3.5934 3.5934 3.5934
4.8750 4.8750 4.8750 4.8750
5.4849 5.4851 5.4851 5.4851
5.7683 5.7683 5.7683 5.7683
7.1168 7.1170 7.1170 7.1170
7.3964 7.3970 7.3969 7.3968
8.5969 8.5969 8.5969 8.5969
1.9393
3.5939
4.8755
5.4855
5.7691
7.1177
7.3972
8.5973
Liew (p-Ritz) [25] 0.20
13 · 13 15 · 15 17 · 17 19 · 19
Liew (p-Ritz) [25] 2
0.001
13 · 13 15 · 15 17 · 17 19 · 19
Liew (p-Ritz) [25] 0.20
Liew (p-Ritz) [25]
13 · 13 15 · 15 17 · 17 19 · 19
ply rectangular laminates with SSSS and CCCC boundary conditions. We consider the first eight frequency parameters with various aspect and thickness ratios. We
do not compare with other sources, but readers can verify that present results are in very close agreement with results of Liew [25].
A.J.M. Ferreira, G.E. Fasshauer / Composite Structures 79 (2007) 202–210
207
Table 3 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ¼ ðxb2 =p2 Þ ðqh=D0 Þ for a simply supported square four-ply laminate (0/ Effect of thickness-to-length ratio on the fundamental frequency parameters x 90/90/0) t/b
0.01
0.02
0.04
0.05
0.08
0.10
0.20
0.25
0.50
Reddy and Phan [26] Liew (p-Ritz) [25] Present results
6.578 6.606 6.6012
6.475 6.549 6.5438
6.330 6.338 6.3300
6.196 6.193 6.1844
5.708 5.677 5.6641
5.355 5.311 5.2960
3.854 3.807 3.7903
3.331 3.295 3.2796
1.956 1.929 1.9180
Table 4 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ¼ ðxb2 =p2 Þ ðqh=D0 Þ for three-ply (0/90/0) rectangular laminates with SSSS and CCCC boundary conditions Frequency parameters x Modes B.C.
a/b
t/b
3
4
5
6
7
8
SSSS
1
0.001 0.050 0.100 0.150 0.200 0.001 0.050 0.100 0.150 0.200
6.6226 6.1365 5.1652 4.2741 3.5934 2.3511 2.3221 2.2210 2.0848 1.9387
9.5306 8.8846 7.7549 6.6657 5.7683 6.6368 6.1361 5.1652 4.2741 3.5934
16.4255 15.1061 12.9129 9.4875 7.3968 6.7586 6.4785 6.0163 5.4428 4.8750
21.6731 19.3527 13.0478 10.8227 8.6870 9.4973 8.8835 7.7547 6.6656 5.4852
25.0307 20.6609 14.3735 10.8246 9.1444 14.1298 12.1487 9.0208 6.8683 5.7683
26.5070 24.0653 17.7849 13.8027 11.2078 14.4600 13.5782 10.7943 8.5950 7.1170
30.4155 24.3377 19.4998 14.6650 11.2218 16.0620 13.9489 11.8480 9.4871 7.3965
37.7401 31.0194 21.0501 15.5895 12.1162 16.5252 15.1050 12.9126 10.0820 8.5969
0.001 0.050 0.100 0.150 0.200 0.001 0.050 0.100 0.150 0.200
14.6918 10.9530 7.4107 5.5481 4.4465 5.0970 4.7775 4.1407 3.5395 3.0452
18.4741 14.0235 10.3930 8.1467 6.6420 10.4052 8.8358 6.6158 5.1811 4.2481
26.9611 20.3851 13.9124 9.9039 7.6995 10.6097 9.8448 8.3534 6.9263 5.7916
37.6121 23.1953 15.4281 11.6209 9.1848 14.3575 12.5070 9.8939 7.4236 5.9042
37.6121 24.9771 15.8046 12.0249 9.7377 18.4830 14.7012 9.9660 7.9354 6.5347
39.3560 29.2354 19.5706 14.6439 11.3990 18.9482 17.3208 12.4394 9.5758 7.6885
40.9241 29.3618 21.4885 14.9106 11.6435 19.7608 17.6708 13.6617 9.8560 7.7280
48.9940 36.2616 21.6199 16.1227 12.4655 21.1550 19.4302 14.1197 11.2299 9.1754
2
CCCC
1
2
1
2
Table 5 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ¼ ðxb2 =p2 Þ ðqh=D0 Þ for five-ply (0/90/0/90/0) rectangular laminates with SSSS and CCCC boundary conditions Frequency parameters x Modes B.C.
a/b
t/b
2
3
4
5
6
7
8
SSSS
1
0.001 0.050 0.100 0.150 0.200 0.001 0.050 0.100 0.150 0.200
6.6184 6.2237 5.3966 4.5811 3.9130 3.4097 3.3605 3.1641 2.9041 2.6310
13.7442 12.6569 10.5241 8.5751 7.0803 6.5120 6.2233 5.3966 4.5811 3.9130
23.0420 18.2915 12.7094 9.4002 7.4027 12.2291 11.3460 8.8821 6.9230 5.6097
26.4671 21.5864 15.6520 11.8834 9.4679 13.3916 11.5708 9.4992 7.7835 6.4529
28.1753 24.0215 17.9089 13.6241 10.8185 13.7545 12.6562 10.5239 8.5750 7.0802
36.6880 29.9092 20.4687 14.3910 11.0526 18.1865 16.1365 12.7089 9.3999 7.4024
49.1036 33.5073 21.3551 15.9159 12.5098 23.0930 18.2891 12.7125 10.0334 8.1431
51.1558 35.5265 22.4384 16.1283 12.5295 26.8986 21.5903 15.6531 11.8838 9.2173
0.001 0.050 0.100 0.150 0.200 0.001 0.050 0.100 0.150 0.200
14.4337 11.5785 8.1664 6.1120 4.8279 7.5611 6.9345 5.6265 4.4909 3.6566
18.7244 18.5068 12.9490 9.5821 7.5339 11.1165 9.8001 7.4626 5.7694 4.6327
35.8179 22.6809 13.9913 10.0141 7.7699 17.4333 14.7794 10.3319 7.7193 6.1015
40.9718 27.0134 17.2787 12.4715 9.7082 19.2374 15.9510 11.5057 8.6345 6.8399
40.9718 29.4758 19.5291 14.1885 11.0603 21.3109 17.4812 12.5413 9.3846 7.4233
52.5993 35.5447 21.1658 14.7616 11.2888 24.2316 20.7812 13.7094 9.9550 7.7551
66.2199 36.4352 22.6652 16.2923 12.6459 31.9367 20.9977 14.4607 10.7071 8.4255
66.2199 39.3569 23.4906 16.5381 12.7084 36.4012 25.6764 17.0595 12.3089 9.4919
2
CCCC
1
2
1
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Table 6 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ¼ ðxb2 =p2 Þ ðqh=D0 Þ for eight-ply (0/90/0/90)s rectangular laminates with SSSS and CCCC boundary conditions Frequency parameters x Modes B.C.
a/b
t/b
2
3
4
5
6
7
8
SSSS
1
0.001 0.050 0.100 0.150 0.200 0.001 0.050 0.100 0.150 0.200
6.6148 6.2674 5.5034 4.7051 4.0244 3.9453 3.8359 3.5462 3.1859 2.8308
15.8073 14.1850 11.3234 8.9701 7.2832 6.5314 6.2669 5.5033 4.7051 4.0244
21.6703 17.6952 12.6366 9.4599 7.4848 12.6020 11.2255 8.8411 6.9870 5.6928
26.4531 22.0135 16.0976 12.1654 9.6364 14.6609 13.1511 10.4566 8.2568 6.6929
33.4402 26.9372 18.9226 13.9935 10.9642 15.8207 14.1843 11.3232 8.9700 7.2832
40.1156 31.9240 20.4521 14.4859 11.1554 19.3743 17.1091 12.6361 9.4595 7.4845
47.8620 32.7265 22.1297 16.2275 12.6481 21.7475 17.6923 13.3045 10.3572 8.3223
50.6101 35.3623 22.7706 16.3827 12.6977 26.6222 22.0172 16.0987 11.9633 9.3078
0.001 0.050 0.100 0.150 0.200 0.001 0.050 0.100 0.150 0.200
14.5934 11.8554 8.4047 6.2450 4.9015 8.6914 7.7628 6.0089 4.6604 3.7366
25.3037 19.9841 13.4640 9.8065 7.6548 11.5251 10.2038 7.6943 5.8836 4.6899
33.9140 22.5157 14.1032 10.0991 7.8286 17.4447 14.7821 10.4473 7.7948 6.1484
40.4159 27.7837 17.6385 12.6436 9.8091 22.9715 17.8081 12.1039 8.8785 6.9648
47.1988 31.9119 20.2179 14.4547 11.1745 24.8652 19.0746 13.0527 9.5957 7.5326
56.2557 36.0981 21.2564 14.8657 11.3751 26.8336 20.7354 13.7684 10.0226 7.8082
66.2642 37.3787 23.2322 16.5204 12.7501 34.4222 21.9484 14.8664 10.8841 8.5232
67.6171 39.6694 23.7684 16.7072 12.8238 34.4222 26.4059 17.3496 12.3855 9.5582
2
CCCC
1
2
1
The results show that frequency parameters decrease with increasing a/b ratio. These data also show that an increase in thickness ratio results in decreasing of frequency parameters. This is due to the effects of shear deformation and rotary inertia. These effects are more pronounced in higher modes. Also higher frequencies are obtained with
CCCC boundaries, as expected because higher constraints produces higher frequencies. The present method produces very stable and well defined frequency modes, as shown in Figs. 1–3. The eigenmodes for the other examples have similar characteristics.
Fig. 1. Mode shapes (1–9) for three-ply (0/90/0) simply supported SSSS square laminates, a/b = 1, t/b = 0.2, 15 · 15 nodal grid.
A.J.M. Ferreira, G.E. Fasshauer / Composite Structures 79 (2007) 202–210
209
Fig. 2. Mode shapes (1–9) for three-ply (0/90/0) clamped CCCC square laminates a/b = 1, t/b = 0.2, 15 · 15 nodal grid.
Fig. 3. Mode shapes (1–9) for three-ply (0/90/0) simply supported SSSS rectangular laminates a/b = 2, t/b = 0.2, 15 · 15 nodal grid.
5. Conclusions A study of free vibration of isotropic and composite plates was presented. Equations of motion and boundary
conditions for symmetric laminated composite plates were presented. Numerical results were presented and discussed for various thickness-to-length ratios. The analysis is based on an innovative numerical scheme, where collocation by
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radial basis functions and pseudospectral methods are combined to produce highly accurate results. This method allows the use of non-rectangular geometries and promotes a more flexible framework for pseudospectral methods. Various aspect ratios, thickness ratios and boundary conditions were considered for isotropic and various types of laminates. The present method showed very high accuracy for both isotropic and composite plates. References [1] Laura PAA, Gutierrez RH. Analysis of vibrating Timoshenko beams using the method of differential quadrature. Shock Vib 1993;1(1):89–93. [2] Bert CW, Malik M. Differential quadrature method in computational mechanics: a review. Appl Mech Rev 1996;49(1):1–28. [3] Liew KM, Han JB, Xiao ZM. Vibration analysis of circular Mindlin plates using differential quadrature method. J Sound Vib 1997;205(5):617–30. [4] Liew KM, Hung KC, Lim MK. Vibration of Mindlin plates using boundary characteristic orthogonal polynomials. J Sound Vib 1995;182(1):77–90. [5] Lee J, Schultz WW. Eigenvalue analysis of Timoshenko beams and axisymmetric Mindlin plates by the pseudospectral method. J Sound Vib 2004;269:609–21. [6] Ferreira AJM. Free vibration analysis of Timoshenko beams and Mindlin plates by radial basis functions. Int J Comput Methods 2005;2(1):15–31. [7] Ferreira AJM, Roque CMC, Martins PALS. Radial basis functions and higher-order theories in the analysis of laminated composite beams and plates. Compos Struct 2004;66:287–93. [8] Ferreira AJM. A formulation of the multiquadric radial basis function method for the analysis of laminated composite plates. Compos Struct 2003;59:385–92. [9] Ferreira AJM, Roque CMC, Martins PALS. Analysis of composite plates using higher-order shear deformation theory and a finite point formulation based on the multiquadric radial basis function method. Composites: Part B 2003;34:627–36. [10] Ferreira AJM. Thick composite beam analysis using a global meshless approximation based on radial basis functions. Mech Adv Mater Struct 2003;10:271–84.
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