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Free vibration of generally laminated composite plates with various edge support conditions
CompositeStructures29 (1994)249-258
© 1994Elsevier Science Limited Printedin GreatBritain.All fightsreserved 0263-8223/94/$7.00 ELSEVIER
Free vibration of generally laminated composite plates with various edge support conditions Gin Boay Chai _ SchoolofMechanicalandProduction Engineering, Nanyang Technological University, NanyangAvenue, Singapore 2263, Republic of Singapore The well known and versatile Rayleigh-Ritz method is employed to study the free vibration behaviour of generally laminated composite plates with various edge support conditions. Classical edge support conditions such as simplysupported, clamped or a combination of both are prescribed along the edges of the plate and the effect of these on the natural frequencies of the plate is presented and discussed. Thorough comparisons with exact solutions for some classes of laminated composite plates are presented to verify the theoretical predictions of the present analysis. Experimental results applied to rectangular C-S-C-S laminated composite plates with symmetric stacking sequence are also presented for verification.
NOTATION
INTRODUCTION
Length, width and total thickness of plate, respectively Extensional stiffness matrix [A] Stretching/sheafing coupling stiffAI6,A26 nesses [B] Coupling stiffness matrix BII, B22, BI2 Stretching/bending coupling stiffnesses Stretching/twisting coupling stiffBI6, B26 nesses Shearing/twisting coupling stiffness 866 Bending stiffness matrix [D] Bending/twisting coupling stiffD16, D26 nesses Bending stiffness parameter Do defined in text Transverse elastic modulus E22 Mx, My, Mxy Moment resultants Nx, Ny, Nxy Stress resultants Coefficients in the shape function Win. Cartesian co-ordinates x, y
In structural applications where materials with high strength-to-weight and stiffness-to-weight ratios are required, composite materials would no doubt be the preferred choice over conventional metallic materials. In particular laminated composite materials allow structures to be tailored to suit loading environment. One of the main disadvantages of laminated composite materials though is the existence of coupling responses. The degree of these coupling effects depends very much on the stacking sequence of the lamination. It is known that the existence of coupling responses in a laminated composite structure increases deflections in thin structures under load, and hence reduces the natural frequencies of the structure. 1,2 The coupling responses of Bij in the constitutive equations for a laminated composite plate can be eliminated by stacking the layers symmetrically. The stretching/shearing (AI6 and A26 ) and bending/twisting (916 and D26 ) coupling responses can be avoided by stacking plies antisymmetrically. The specially orthotropic condition occurs when no coupling responses are present and this is achieved by having all plies at 0 ° or 90 ° or symmetric cross-ply. Closed form exact solutions have been published for the free vibration frequencies of specially orthotropic, antisymmetric angle-ply and antisymmetric cross-ply laminated plates with
a, b, t
0 EOx,EO, Exy Kx, l~y, l~xy /9 (1) Q
Middle surface strains Middle surface curvatures Mass of the plate per unit area Natural frequency of the plate Non-dimensionalised natural frequency of the plate as defined in text 249
250
G. B. Chai
simply-supported edges. L2 However in many practical structural applications, the laminated composite plates may or may not have symmetric stacking sequence and have a combination of 0 °, 90 ° and 45 ° ply angle. In these instances some forms of coupling responses are always present. A closed form solution for this type of laminate is virtually impossible, but an approximate method using the energy approach has been proposed by Chamis 3 and Ashton 4 for the case of buckling and bending problems of symmetric and unsymmetric laminated composite plates. There are many other researchers in the field of free vibration of composite structures, overviews of the vast amount of relevant literature available were presented by Leissa 5 and Kapania and Raciti.6 A recent publication by Leissa and Narita 7 applies the Ritz method to simplysupported symmetrically laminated composite plates using a 144-term double-sine series to postulate the deflected shape. They found that the 144-term solution is necessary generally to give converged accurate values of the natural frequencies for the specific cases studied. Lin and King s presented exact and accurate converged solutions for antisymmetric laminated composite plates with various support conditions using an asymptotic method. In a recent contribution, Baharlou and Leissa 9 used polynomials as displacement functions in the application of the Ritz method and converged solutions were obtained for various combinations of clamped, simply-supported and free edges of generally laminated composite plates. The procedures presented in these two contributions 8,9 permitted all three displacement components (u, v, w) to have independent freedom, whereas the present method in this contribution eliminates components u and v. Though many solutions are available on the free vibration behaviour of laminated composite plates, they are applicable only to specific classes of laminated composite plate or specific boundary conditions. The present work describes an approximate method based on the versatile Rayleigh-Ritz energy approach to determine the free vibration frequencies of generally laminated composite plates with various edge support conditions. In addition experiments were performed to verify the predicted results for a rectangular C-S-C-S laminated carbon-fibre reinforced plastic plate of symmetric stacking sequence.
RAYLEIGH-RITZ APPROACH FOR LAMINATED COMPOSITE PLATES From the classical lamination theory, ~ the constitutive relationships for a laminated panel in full matrix form are:
"] Ny
Nxy
~,411 AI2 AI6 BII BI2 BI6
=
AI2 Ai6 Btt BI2 Bj6" A22 A26 BI2 B22 B26 A26 A66 BI6 B26 B66 Bi2 BI6 Dll DI2 DI6 B22 B26 Dl2 D22 D26 B26 B66 DI6 D26 D66
I !
(1) and this can be abbreviated in a simple compact format:
where [e °] and [r] are middle surface strains and curvatures, respectively. A more useful and practical form is the partially inverted form of eqn (2):
[;: ;:][:]
(3)
where [A*] = [A] -', [B*] = - [A]- l[B], [H*I=[BI[A] -' and[D*]=[D]-[B][A]-'[B]. The matrix [D*] is called the reduced bending stiffness matrix introduced by Chamis 3 and Ashton. 4 The strain energy of a laminated rectangular composite plate can be written in the form: 2 0 V~ = ~ , , { N x e O + Nvcy0 + gxyexy
+ M~rx + Myry + M~yr~y} dx dy Substitution of eqn (3)into eqn (4) yields:
(4)
Free vibration of laminatedplates
251
1 ~b~a
Us=2 J0 J0 {A~IN2 + A2*21~y+A6*6N2y °"
+ 2A'~2NxNy + 2A*6NxNxy + 2A'~6NyNxy +O*l r x2 + Dzzry * 2 * 2 + D66Kxy +2D~z~xry +2D~6KxKxy+2D'~6KyKxy}dxdy (5) The above eqn (5) suggests that the in-plane and bending energy of laminates with coupling responses Bi/can be solved in an uncoupled form, i.e.
[e°]=[A*][N]
V = Omax - Tmax
(7)
where Um~x is the strain energy of bending and Tm~xis the kinetic energy of the plate's mass. They are defined as:
.(ow)lowl
Uma~=-~JoJolDa,[-~xZ) +2D12 ~ x 2 iOy2] + 406"6/0--x-0y) + O2*2~y2]
,
0w
C
C-S-C-S
C-C-S-S
Coordinates
system and definition of support conditions of plate.
M N
w ( x , y ) = Z Z Wm.Xm(X) Y.(y)
(10)
m n
where Xm(X ) and Y.( y) are appropriate functions that satisfy the geometrical boundary conditions of the plate, and Wm. are the unknown coefficients of the functions. The appropriate functions for various cases of rectangular plates with different combinations of support conditions are given in Table 1. For example if a plate with simply supported-clamped-simply supported-clamped edges (S-C-S-C) is required to be analysed, then the chosen deflection function obtained from Table 1 will be:
wlx, yl=
.(o2w)Co2w
+4D16 ~ x 2
S
S
Fig. 1.
Thus the usual goveming equation for strain energy of bending of a symmetrically laminated composite plate can be used by replacing the flexural stiffness D 0 with the reduced bending stiffness D*. The total vibrating energy of a laminated composite plate of Fig. 1 can be written as:
C
C
(6)
[M]=[D*] [r]
+ 4D26
-----'~ X a
M
N
z
z
m = 1,2,3...
n = 1,2,3...
,
(marx I \Ul
kOxay] 0w
dx dy
(8)
Substituting the assumed function of eqn (10) into eqn (7)gives: V - - Umnpq - ~o) 2 Tmnp q
2 j0 j0
w 2}dx dy
(9)
It is a fact that fundamental frequencies obtained using the Rayleigh-Ritz method are always higher than the exact values since the plate's mode shape is postulated by a finite number of terms in the shape functions which inherently increase the rigidity of the plate. The accuracy of the Rayleigh-Ritz method therefore depends on the selection of compatible shape functions, such as:
(12)
where Umnpq = E ~ Z Z Wmn Wpq m n p q
* xx DII X m
* + 2D,2(XmX ~xx Y. Y~q) + D :(XmX. Y y) * x x y y + 4D66(XmX Y.Y ) + 4D16(XmX e Y, YY) xx
x
* x YYYqYY)}dx dy + 4026(SmX~
xx
G.B. Chai
252
Table 1. Trigonometric function used in postulating the deformed shape
Two opposite edges
X,,,(x)
S-S
s,n~--}
C-S
sm ~aa sln~-a ]
sin
sln/~- ]
,l=2,4,6.. .
C-C
sin
sin ~-
s'nv/-]
m=1,2,3... ,1= 1,2,3...
Y,,( y)
• {mzrxl
• [narYl
sm/~- ]
(1
Zmnpq=EZ~2 WmnWpq m n p q
sm/~- ]
(XmXpY,,yq)dxdy )
where superscripts x and xx refer to the first and second derivatives of the function with respect to x, respectively. And superscripts y and yy refer to the first and second derivatives of the function with respect to y, respectively. Based on the principle of minimum potential energy, eqn (12) is minimized with respect to the unknown coefficients to give a series of homogeneous simultaneous equations: ~V --=0;
~ Wll
OV OV --=0;...--=0
~ W12
O Wmn
(13)
Differentiating eqn (13) with respect to the unknown coefficients a second time and rearranging the format: [K]{ W}= 702[S] {W}
(14)
where [K] is the stiffness matrix, [S] is the mass matrix and {W} is the vector of the unknown coefficients. Specifying the required number of M terms and N terms in the postulated deflection function of eqn (10), the minimization process yields a set of M x N linear homogeneous simultaneous equations (13). Then eqn (14) will give M x N stiffness and mass matrices. A standard numerical eigensolution routine by Lindfield and Penney ~° was used to extract the eigenvalues and their corresponding eigenvectors to obtain the natural frequencies of the plate and their corresponding mode shapes, respectively. NUMERICAL R E S U L T S AND D I S C U S S I O N
Most of the results to be presented are based on the material properties for graphite/epoxy:
m = 1,2,3... n= 1,2,3...
El~=130 GPa, E22=9 GPa, G12=4"8 GPa and v12=0"28. It will be mentioned when different material properties are used. The non-dimensional natural frequencies mentioned in the figures and some tables to be presented are defined by the parameter
if2 wa 2~fV/D,, =
(15)
where E~t 3 O0
12(1 - v12v21)
For the classical cases of antisymmetric and symmetric cross-ply S-S-S-S laminated plates, exact solutions are available and can be found in Jones l and Whitney.2 The non-dimensional natural frequencies for these cases which are obtained using a four-term solution of the present approach are tabulated in Table 2 for the first four modes. These results are identical to those obtained using the exact solution of Jones 1 and Whitney2 (not shown in the table). Another classical class of the S-S-S-S laminated composite plate is the antisymmetric angleply case, for which a closed form exact solution is again available. L2 The non-dimensional frequencies for the first four modes of natural vibration of an antisymmetric angle-ply laminated square plate obtained using a four-term solution are shown in Fig. 2, together with the results obtained using the exact solution. L2 The figure shows variation in ply-angle and the number of layers stacked antisymmetrically. Apart from the results for the two layers, the present solution agrees very well with the exact solution. For the three classical cases presented above, the Rayleigh-Ritz results for a four-term solution and a 49-term solution are identical thus confirming that the solution has already converged using
Free vibration of laminatedplates
253
Table 2. Non-dimensional frequencies obtained using a four-term solution for square S - S - S - S cross-ply laminated composite plates NL a
Antisymmetric cross-ply
2 4 6 8 10 12 14 16 18 20
Symmetric cross-ply
Mode 1
Mode 2
Mode 3
Mode 4
Mode 1
7.7985 10.3384 10.743 1 10.881 1 10.944 5 10"9787 10.999 3 11.0126 11.021 8 11"0283
21.1006 28.801 7 30.011 8 30"423 9 30.612 8 30.7149 30"776 3 30"8161 30-8434 30.8628
21.1006 28.801 7 30.011 8 30.423 9 30.612 8 30.7149 30"776 3 30.8161 30-8434 30.8628
31.1942 41.3536 42.972 2 43.524 5 43.777 8 43.9148 43"997 1 44.0505 44.087 1 44.1132
. 11"056 1 10.743 1 11"056 1 10.944 5 11.056 1 10"999 3 11-0561 11-021 8 11-056 1
.
Mode 2
Mode 3
Mode 4
. 21"1576 30.011 8 26-507 4 30.612 8 28.0649 30.776 3 28"8122 30"8434 29.251 3
. 38"3097 30.011 8 34.822 9 30.612 8 33.5803 30.776 3 32.9414 30"8434 32-552 1
41"3892 42.972 2 44.224 5 43.777 8 44.2245 43.997 I 44.2245 44.087 1 44.2245
aThe number of layers in the laminated composite plate.
15. --
3,I i"
Present
~ 3s
• 14-
~ 13-
32
ff. 12-
~
•°
~H .~_
~
•
~'e
"~ 2 8 '
_e
~ 10
°o
oo
°°o o
o* °°
- -,~,~--
~, 26'
.7
o=
z
10
20
30
40 50 Ry Angle
60
70
80
•
31-
Exact [ I ]
~o 5 2 :E
Jk
27
~ 44
25 23
i
21 L5
Present
--
8
:"
17 15
o
1~
~o
~o
~
~o
~o
~o
~o
~o
Ry Angle
Fig. 2.
40
~ a6
J~
19
~o
3'o
~
56-
~II I L
29-
|
~
90
3533-
g
2422
9
o
30-
4'o
~o
6'o
7'o
do
9o
Ply Angle
%
--
Present
32 28
o
ib
,;o
~o
~o
;o
;o
~o
80
90
Ply Angle
First four natural modes of an antisymmetric angle-ply square laminated composite plate.
four terms in the postulated deflection function. It should be noted that the cusps or sharp points shown in Fig. 2 for the higher frequencies are actually crossing points for the frequencies, which would be obvious if all of these frequencies were plotted together on a single graph. Results obtained using the present 144-term solution for several cases of a two-ply square anti° symmetric laminated plate are compared with published accurate results. 8,9 The material properties used to generate these results are obtained
from Baharlou and Leissa: 9 Eli~E22=40,
G12/
E22 = 0"5, v12 = 0"25. T h e results are s u m m a r i s e d
in Tables 3 and 4 for a two-ply square antisymmetric angle-ply laminated composite plate, and in Tables 5 and 6 for a two-ply square antisymmetric cross-ply laminated composite plate. The results published by Whitney ~1 using the double Fourier series method are also included for comparison in Tables 4 and 5. It can be seen that the results using the present analysis agree reasonably well with the accurate solution. 8,9 The
G. B. Chai
254
Table 3. Fundamental frequencies wa2~oftwo-plysquareantisymmetricangle-plylaminatedS-C-S-Cplates 0°
$3C3
5 15 25 35 45 55 65 75 85
$3C1
$3C2
Present 144-term
Exact
Polynom
Exact
Polynom
Exact
Polynom
Ref. 8
Ref. 9
Ref. 8
Ref. 9
Ref. 8
Ref. 9
36"248 -23"137 -19'395 17"661 16'300 16'176 18"776
36"248 26"614 23"139 21"131 19"407 17"677 16"301 16-191 18"779
36-682 . 23"229 . 19"394 17"617 15"825 15"326 17"978
36"683 . 23-230 . 19"395 17"692 16"312 16"177 18"777
36"683
Table 4. Fundamental frequencies t a a 2 ~ 0°
36'683 .
. 23"230
.
. 19'396 17'649 16"078 15'746 18"424
23'230 19"396 17'694 16"313 16"178 18-777
of two-ply square antisymmetric angle-ply laminated C-C-C-C plates
C1
C3
Present 144-term
Asymptotic
Polynom
Fourier
Asymptotic
Polynom
Fourier
Ref. 8
Ref. 9
Ref. 11
Ref. 8
Ref. 9
Ref. 11
36"880 27"378 24"317 23"189 22"838
37"283 27"834 24"750 23"631 23-309
37"582 28"052 24"940 23"799 23-488
36"401 27"156 24"215 23"111 22"835
36"836 27"562 24"643 23"602 23"347
39"458 31"337 27-517 25'316 25'316
5 15 25 35 45
Table 5. Fundamental frequencies wa2fp-/E22t 3 of two-ply square antisymmetric cross-ply laminated C-C-C-C plates Aspect ratio a/b
Asymptotic
Fourier
Polynom
Present
Ref. 8
Ref. 11
Ref. 9
144-term
1 2 3 4 5
23"638 17.111 16"640 16.543 16.509
24.527 21.171 21.171 17'200 16.974
24.033 17"297 16.782 16"673 16-636
24.037 17.299 16.784 16.675 16.638
Table 6. Fundamental frequencies w a Z ~ of two-ply square antisymmetric cross-ply laminated plates with various edge supports Boundary conditions $2-$2-$2-$2 C2-$2-C2-$2 C2-C2-C2-C2 C1-C1-C1-C 1
35'274 25'606 22'716 21'039 19'404 17"569 15"818 14"937 17'526
Polynomial
Present
Ref. 9
144-term
11"17 18"72 24'01 24'03
11-164 18"727 24"037 24"037
boundary conditions S1, $2, $3, C1, C2 and C3 depend on the restriction of the displacement components and they are discussed in detail by Baharlou and Leissa. 9 For a single layer laminated composite plate or symmetric laminated composite plate (other than the cross-ply laminate), no known exact solutions
35"848 26"563 24"218 23"483 23'316
are available for comparison. However Leissa and Narita 7 have presented comprehensive results for the case of S - S - S - S symmetrically laminated composite plates using a 144-term Rayleigh-Ritz solution. The method of solution presented in this contribution is identical to the approach used by Leissa and Narita 7 for the S - S - S - S condition, the frequencies obtained are thus identical for this case and these results are therefore not presented here. For support conditions other than the S - S - S - S case and for a single layer laminated square composite plate, the results for the first four modes of the natural frequency are plotted as a function of increasing ply angle and these are shown in Figs 3-8 for various support conditions. It can be seen that for plates with opposite sides having the same support conditions (S-S-S-S, C - C - C - C , C - S - C - S ) the curves are symmetrical about the 45 ° ply angle. As for the other plates, the curves do not indicate any symmetry. Another point to note in these figures is the orthotropic curves which are obtained by assuming that the bending-twisting coupling stiffnesses (O16 and D26 ) are zero. The solution based on the orthotropic assumption is shown to converge at a faster rate than the actual anisotropic solution as shown in Fig. 9 for a single 45°-ply laminated square
Free vibration of laminated plates
255
70
60
f-~x
....
/
\
- -
//
50
/ /
/
/ / / /I
OrthoVopic Anisotropic
\ \
! / / ! / / / /
\ \
x \ x x \ \ \
p'~\\ 60
//
E
\
\
/
/
\
\ \
/
X X X
/\ '\\
\\
\\
/Z,
"-4,, //
E
\\
,,
\\\
///
g z
\\
i1,'11
3O
//
/ ....//
/¢/
\\
\
\
\\
\\\\\
i111
20
\X
Orthotropic Anisotropic
"\\
30
Z
\\
ii I
~
/
//
/
~, 50
\ \
...... k. //
/~
/
// \ \\ \
.... - -
\\\\ 20
//~ ~
"~.
_~/~/ 0
Fig. 3.
~
,,,,
10
10
_
;,,,,,,,, 20
0
,,
40 0 60 Ply angle (degree)
70
80
10
l ~ l ' I ' I * I I ' I 10 20 30 40 50 60 70 Ply angle (degree)
90
Natural frequencies of a single layer square S-S-S-S laminated composite plate.
Fig. 5.
I 80
90
Natural frequencies of a single layer square C-S-C-S laminated composite plate.
70
.... 60
/ / / / j / / / /
/
/¢ /
- -
\
\\
\ \
\ \
/ / /
\
\
60
\ \
//
50
.... - -
Orthotropic Anisotropic
/
/ / /
\\ \
j~--",\
//
/
f/----\,
\ \ \ \
\ \ \ \
// /
|
~_/_
-
~ 50
///
,,
~
\
\\\
/
//
/ ii
/
II
//
/
\
~Z 30
I ~S
\\ "~
"i
\ \k
/
/,-/\\
//
Orthotropic Anisotropic
\\
/// \
\\ \
z
//
30
//
\\\
\\
\
\\ \\ \\\
\\\X 2O
,o Fig. 4.
20
0
, ,10, ,
20
' ' ' ' ' 40 ' ' " o "50o '
30
60 Ply angle (degree)
7
8
,o
90
Natural frequencies of a single layer square S-C-S-S laminated composite plate.
Fig. 6.
0
'
' ' 'o 10 2
' ' ' ' ' ' ' 'o' 30 40 50 60 7 Ply angle (degree)
' 80
90
Natural frequencies of a single layer square C-C-S-S laminated composite plate.
G. B. Chai
256
~
70
60
//I
//
/
/\
\\\
/
/
/
/
/
/
/
\
k \ \
....
Orthotropic
- -
Anisotropic
\ \
//
\
~\ \ \
]/
\N
///'
so
\\>jl
E 40
///-
Z
\X\
//
X XX \\
/.///
z
///
30
/
x
xX
20
Fig. 7.
I
I
10
20
i
I
30
I
I
40 50 60 Ply angle (degree)
I
i
I
70
'
I
80
90
Natural frequencies of a single layer square C - C - C - S laminated composite plate.
80
60
Orthotropic
- -
Anisotropic
\\ \ ~\\ \
!/ III /
70
....
~'\\\vl/I ]
\\\ / / / / //
\\
/// so
NN\
/
z
// X
\
C - C - C - C composite plate. However the orthotropic assumption used for an anisotropic composite plate always gives non-conservative results, as shown in Figs 3-9. Theoretical results for a single 45°-ply laminated square C - C - C - C composite plate published by Whitney 2 are compared with results obtained using the present 144-term solution. Whitney used beam eigenfunctions to postulate the plate's out-of-plane deformation and the Rayleigh-Ritz method to solve for the natural modes. The results obtained by Whitney are given in Table 7 using 1-, 9-, 25- and 49-term solutions for the first mode, and frequencies for higher modes obtained using a 49-term solution are indicated in a footnote to Table 7. These results are based on material properties given in Whitney, 2 i.e. E~l =31 x 106 psi, E 2 2 = 2 " 7 x 106 psi, G~2=0.75 x 106 psi, vt2=0"28, t=0"0424 in., a = b = 2 in. and density = 1.92 × 10 -4 lb-sec2/in 4. It can be seen that at least a 144-term solution is required for convergence up to two decimal places for the first two modes and 0nly one decimal for the next two modes. Theoretical results are applied to rectangular 14-layer graphite/epoxy laminated composite plates with symmetric lay-up of [+ 45/0/0/90/0/ 0 / - 45/0]s, the nominal thickness of the plate is taken as 1.75 mm. The laminated composite plates are anisotropic with the flexural stiffness ratios of Dl6/Di1 = D26/DII = 0"1289. The natural frequencies in Hz are summarized in Table 8 for two aspect ratios. In addition, experimental results and finite element method results are also shown for comparison. Details of the experimental setup, test procedures and the finite element analysis
\\
//
\\ 50
25
G
z
~
\N\\\\\\
ii/I/11
//
\\\
"O. 'Q"- 0
-o- • 0
G- 43. ~
24
0
a3 ~ g 22 g E
17.
i
30
~4o E
[] ---~---
~\
......~...
\
Mode 1, Ortho Mode 1, Aniso Mode 2, Ortho
z
20
21
z
20
~
35
10
Fig. 8.
i 10
'
I 20
I 30
'
I I ' 1 40 50 60 Ply angle (degree)
i 70
'
~ 80
' 90
Natural frequencies of a single layer square C - C - C - C laminated composite plate,
30 0
I 2
= 4
I i 6 8 Number of M=N terms
r 10
I 12
19 14
Fig. 9. Convergence rate for the first two modes of a single layer square C - C - C - C laminated composite plate.
Free vibration of laminated plates
257
Table 7. Comparison between f~ obtained using present theory with published results 2 for a single 45°-ply C-C-C-C laminated square plate Ma
Present Rayleigh-Ritz solution Mode 1 Aniso Ortho
1 2 3 4 5 6 7 8 9 10 11 12
24.2869 21.3973 20.9239 20.5530 20"3762 20.2823 20.2164 20"1808 20"1522 20-1357 20.1216 20.1129
Mode 2 Aniso Ortho
24.2869 24.2869 23.3640 23.3640 23.1626 23.1626 23-0939 23.0939 23.0642 23.0642 23.0493 23.0493
. 39.0791 35.2879 35-0393 34.4008 34"3302 34"1409 34.1115 34.0360 34.0220 33"9859 33-9785
. . 49.0098 48.0566 47"1296 46"8796 46.3501 46.5423 46.4501 46.4081 46.3658 46.3442 46.3221
Whitneyb
Mode 3 Aniso Ortho . 57-2428 50-6339 48-4278 47-6429 47.1390 46.8614 46-6724 46-5505 46"4629 46-4016 46-3555
Mode 4 Aniso Ortho
. 49-0098 48-0566 47"1296 46"8796 46.3501 46-5423 46.4501 46"4081 46.3658 46-3442 46.3221
. 79-9169 54.6863 52"3517 51"5701 50"9995 50"8003 50"5800 50"5040 50.4023 50"366 1 50.3135
Mode 1 Aniso Ortho 23.51 -20"82 -20"57 -20.51 ------
79-0867 79"0867 75.6995 75"6995 74.6796 74.6796 74.2677 74.2677 74.0706 74.0706 73.9647
23.51 -23"27 -23.24 -23'23 ------
aThe number of M -- N terms used in the postulated function. hWhitney 2 higher mode results using a 49-term anisotropic solution: fl for Mode 2 is 35.01, Mode 3 is 47.07, Mode 4 is 52.21.
Table 8. Natural frequencies in Hz for C - S - C - S panels of Ref. 12 with stacking sequence of I+ 4 5 , 0 , 0 , 9 0 , 0 , - 45,0]s and DI6/ DI I = D 2 6 / D l l = 0"128 92 M =N
458 x 90 x 1-75 mm Mode 1
Mode2
481.77 477.31 475.77 474.54 474.17 473.71 473.57 473.36 473.29 473.18 473.07
. 572.81 562.23 557.32 555.02 553.43 552.58 551.85 551.46 551.06 550.62
Exp
473.00
Ort
480.75
FEM
471.60
1 2 3 4 5 6 7 8 9 10 12
458 x 120 x 1"75 mm
Mode 3
Mode4
Mode5
Mode 1
Mode 2
. 1825.45 720.31 702.13 696.25 691.98 689.94 688.16 687.23 686.31 685.29
. 1912.83 1817-83 922-79 900-94 892-11 887-51 883.99 882.15 880.38 878.44
-1895-57 1816.95 1180.73 1155.20 1144.67 1139.22 1134.89 1132.56 1129.12
314.54 310.30 308.83 307.74 307.39 307.00 306.88 306.70 306.64 306.54 306.45
. 410-00 400.82 396.22 394.33 392-89 392.22 391.58 391.64 390-94 390.57
606.00
734.00
903.00
1104.00
328-00
569"73
717"00
922.56
1184.91
313"45
547.50
687.23
868.20
1 134"89
305"50
388"20
.
Mode 3
Mode4
Mode5
. 1150.64 562.90 548.69 542.76 539-60 537.61 536.33 535.44 534.79 533.96
. 1239.70 1143.16 769.94 754.17 745.51 742.28 739.00 737.72 736.13 734-62
-1223.90 1142.51 1028.30 1012-52 1001.37 998.19 993.75 992.44 989.55
430.00
543.00
703.00
888.00
406.99
560.00
770.39
1036.25
529.00
725"50
973.50
.
Exp ~ Experimental data from Ref. 12. Ort ~ Whitney 2 solution assuming specially orthotropic conditions. FEM ~ Finite Element results from Ref. 12.
a r e g i v e n in a r e c e n t p a p e r . ~2 A l s o s h o w n in T a b l e 8 a r e t h e results o b t a i n e d u s i n g a c l o s e d f o r m s o l u t i o n 2 w h i c h is b a s e d o n t h e specially o r t h o t r o p i c a s s u m p t i o n . A s c a n b e s e e n f r o m t h e table, o v e r a l l t h e o r t h o t r o p i c results a r e c o n s i s t e n t l y a b o u t 5 % h i g h e r t h a n t h e results f o r the a n i s o t r o p i c case. O n t h e w h o l e t h e p e r c e n t a g e difference between the experimental data and a 144-term solution ranged from -4"50% to
11.44%.
CONCLUSION T h e results o f the p r e s e n t m e t h o d a g r e e d v e r y well with t h e e x a c t s o l u t i o n f o r the specific cases of symmetric cross-ply, antisymmetric cross-ply and antisymmetric angle-ply laminated square c o m p o s i t e plates s u b j e c t e d to f r e e v i b r a t i o n . T h e r e s o n a n t f r e q u e n c i e s f o r single l a y e r l a m i n a t e d s q u a r e c o m p o s i t e p l a t e s i n d i c a t e d that a sufficient n u m b e r o f t e r m s m u s t b e u s e d in t h e p o s t u l a t e d
258
G. B. Chai
trigonometric series function to give a c o n v e r g e d accurate solution; insufficient terms and orthotropic a s s u m p t i o n w o u l d lead to an unconservative solution. H o w e v e r the results obtained with the present a p p r o a c h seem to give a lower and thus m o r e conservative solution than the exact solution b e t w e e n ply angle 5°-35 ° and 5 5 ° - 8 5 ° for the case of a two-ply antisymmetric angle-ply laminated square plate as s h o w n in Fig. 2, Tables 3 and 4. Experimental and theoretical studies on a 14layered laminated rectangular c o m p o s i t e plate with symmetric stacking sequence show that for the laminates c o n s i d e r e d which have a low value of D16/Dll and O26/Dll the coupling responses can be neglected. T h e percentage difference is only about + 5% if these bending/twisting coupling responses are ignored. A m e t h o d of solution, using the classical Rayleigh-Ritz m e t h o d , was p r e s e n t e d to analyse the free vibration of generally laminated c o m posite plates. T h e effect of various s u p p o r t conditions are included in the theoretical formulation. For all the cases c o n s i d e r e d the results seem to indicate that the m e t h o d can be applied to laminated c o m p o s i t e plates in general.
ACKNOWLEDGEMENT This w o r k has b e e n carried out with the financial s u p p o r t of the Ministry of F i n a n c e (Singapore) u n d e r the research fund M O F R & D 5/89.
REFERENCES I. Jones, R. M., Mechanics of Composite Materials. McGraw-Hill, New York, 1975. 2. Whitney, J. M., Structural Analysis of Laminated Anisotropic Plates. Technomic, Lancaster, PA, USA, 1987. 3. Chamis, C. C., Buckling of anisotropic composite plates. Journal of Structural Division (ASCE), 95 (1969) 2119-39. 4. Ashton, J. E., Approximate solutions for unsymmetrically laminated plates. Journal of Composite Materials, 3 (1969) 189-91. 5. Leissa, A. W., An overview of composite plate buckling. Composite Structures, 4( 1) (1987) 1-29. 6. Kapania, R. K. & Raciti, S., Recent advances in analysis of laminated beams and plates Part lI: Vibrations and wave propagation. A1AA Journal, 27(7) (1989) 935-46. 7. Leissa, A. W. & Narita, Y., Vibration studies for simply supported symmetrically laminated rectangular plates. Composite Structures, 12 ( 1989 ) 113- 32.
8. Lin, C. C. & King, W. W., Free transverse vibration of rectangular unsymmetrically laminated plates. Journal of Sound and Vibration, 36 ( 1974) 91-103.
9. Baharlou, B. & Leissa, A. W., Vibration and buckling of generally laminated composite plates with arbitrary edge conditions. International Journal of Mechanical Sciences, 29 (8)(1993) 545-55. 10. Linfield, G. R. & Penny, J. E. T., Microcomputers in Numerical Analysis. Ellis Horwood, Chichester, UK, 1989. 11. Whitney, J. M., The effect of boundary conditions on the response of laminated composites. Journal of Composite Materials, 4 (1970) 192-203. 12. Chai, G. B., Chin, S. S., Lim, T. M. & Hoon, K. H., Vibration analysis of laminated composite plates: TVholography and finite element method. Composite Structures, 23 (1993) 273-83.
© 1994Elsevier Science Limited Printedin GreatBritain.All fightsreserved 0263-8223/94/$7.00 ELSEVIER
Free vibration of generally laminated composite plates with various edge support conditions Gin Boay Chai _ SchoolofMechanicalandProduction Engineering, Nanyang Technological University, NanyangAvenue, Singapore 2263, Republic of Singapore The well known and versatile Rayleigh-Ritz method is employed to study the free vibration behaviour of generally laminated composite plates with various edge support conditions. Classical edge support conditions such as simplysupported, clamped or a combination of both are prescribed along the edges of the plate and the effect of these on the natural frequencies of the plate is presented and discussed. Thorough comparisons with exact solutions for some classes of laminated composite plates are presented to verify the theoretical predictions of the present analysis. Experimental results applied to rectangular C-S-C-S laminated composite plates with symmetric stacking sequence are also presented for verification.
NOTATION
INTRODUCTION
Length, width and total thickness of plate, respectively Extensional stiffness matrix [A] Stretching/sheafing coupling stiffAI6,A26 nesses [B] Coupling stiffness matrix BII, B22, BI2 Stretching/bending coupling stiffnesses Stretching/twisting coupling stiffBI6, B26 nesses Shearing/twisting coupling stiffness 866 Bending stiffness matrix [D] Bending/twisting coupling stiffD16, D26 nesses Bending stiffness parameter Do defined in text Transverse elastic modulus E22 Mx, My, Mxy Moment resultants Nx, Ny, Nxy Stress resultants Coefficients in the shape function Win. Cartesian co-ordinates x, y
In structural applications where materials with high strength-to-weight and stiffness-to-weight ratios are required, composite materials would no doubt be the preferred choice over conventional metallic materials. In particular laminated composite materials allow structures to be tailored to suit loading environment. One of the main disadvantages of laminated composite materials though is the existence of coupling responses. The degree of these coupling effects depends very much on the stacking sequence of the lamination. It is known that the existence of coupling responses in a laminated composite structure increases deflections in thin structures under load, and hence reduces the natural frequencies of the structure. 1,2 The coupling responses of Bij in the constitutive equations for a laminated composite plate can be eliminated by stacking the layers symmetrically. The stretching/shearing (AI6 and A26 ) and bending/twisting (916 and D26 ) coupling responses can be avoided by stacking plies antisymmetrically. The specially orthotropic condition occurs when no coupling responses are present and this is achieved by having all plies at 0 ° or 90 ° or symmetric cross-ply. Closed form exact solutions have been published for the free vibration frequencies of specially orthotropic, antisymmetric angle-ply and antisymmetric cross-ply laminated plates with
a, b, t
0 EOx,EO, Exy Kx, l~y, l~xy /9 (1) Q
Middle surface strains Middle surface curvatures Mass of the plate per unit area Natural frequency of the plate Non-dimensionalised natural frequency of the plate as defined in text 249
250
G. B. Chai
simply-supported edges. L2 However in many practical structural applications, the laminated composite plates may or may not have symmetric stacking sequence and have a combination of 0 °, 90 ° and 45 ° ply angle. In these instances some forms of coupling responses are always present. A closed form solution for this type of laminate is virtually impossible, but an approximate method using the energy approach has been proposed by Chamis 3 and Ashton 4 for the case of buckling and bending problems of symmetric and unsymmetric laminated composite plates. There are many other researchers in the field of free vibration of composite structures, overviews of the vast amount of relevant literature available were presented by Leissa 5 and Kapania and Raciti.6 A recent publication by Leissa and Narita 7 applies the Ritz method to simplysupported symmetrically laminated composite plates using a 144-term double-sine series to postulate the deflected shape. They found that the 144-term solution is necessary generally to give converged accurate values of the natural frequencies for the specific cases studied. Lin and King s presented exact and accurate converged solutions for antisymmetric laminated composite plates with various support conditions using an asymptotic method. In a recent contribution, Baharlou and Leissa 9 used polynomials as displacement functions in the application of the Ritz method and converged solutions were obtained for various combinations of clamped, simply-supported and free edges of generally laminated composite plates. The procedures presented in these two contributions 8,9 permitted all three displacement components (u, v, w) to have independent freedom, whereas the present method in this contribution eliminates components u and v. Though many solutions are available on the free vibration behaviour of laminated composite plates, they are applicable only to specific classes of laminated composite plate or specific boundary conditions. The present work describes an approximate method based on the versatile Rayleigh-Ritz energy approach to determine the free vibration frequencies of generally laminated composite plates with various edge support conditions. In addition experiments were performed to verify the predicted results for a rectangular C-S-C-S laminated carbon-fibre reinforced plastic plate of symmetric stacking sequence.
RAYLEIGH-RITZ APPROACH FOR LAMINATED COMPOSITE PLATES From the classical lamination theory, ~ the constitutive relationships for a laminated panel in full matrix form are:
"] Ny
Nxy
~,411 AI2 AI6 BII BI2 BI6
=
AI2 Ai6 Btt BI2 Bj6" A22 A26 BI2 B22 B26 A26 A66 BI6 B26 B66 Bi2 BI6 Dll DI2 DI6 B22 B26 Dl2 D22 D26 B26 B66 DI6 D26 D66
I !
(1) and this can be abbreviated in a simple compact format:
where [e °] and [r] are middle surface strains and curvatures, respectively. A more useful and practical form is the partially inverted form of eqn (2):
[;: ;:][:]
(3)
where [A*] = [A] -', [B*] = - [A]- l[B], [H*I=[BI[A] -' and[D*]=[D]-[B][A]-'[B]. The matrix [D*] is called the reduced bending stiffness matrix introduced by Chamis 3 and Ashton. 4 The strain energy of a laminated rectangular composite plate can be written in the form: 2 0 V~ = ~ , , { N x e O + Nvcy0 + gxyexy
+ M~rx + Myry + M~yr~y} dx dy Substitution of eqn (3)into eqn (4) yields:
(4)
Free vibration of laminatedplates
251
1 ~b~a
Us=2 J0 J0 {A~IN2 + A2*21~y+A6*6N2y °"
+ 2A'~2NxNy + 2A*6NxNxy + 2A'~6NyNxy +O*l r x2 + Dzzry * 2 * 2 + D66Kxy +2D~z~xry +2D~6KxKxy+2D'~6KyKxy}dxdy (5) The above eqn (5) suggests that the in-plane and bending energy of laminates with coupling responses Bi/can be solved in an uncoupled form, i.e.
[e°]=[A*][N]
V = Omax - Tmax
(7)
where Um~x is the strain energy of bending and Tm~xis the kinetic energy of the plate's mass. They are defined as:
.(ow)lowl
Uma~=-~JoJolDa,[-~xZ) +2D12 ~ x 2 iOy2] + 406"6/0--x-0y) + O2*2~y2]
,
0w
C
C-S-C-S
C-C-S-S
Coordinates
system and definition of support conditions of plate.
M N
w ( x , y ) = Z Z Wm.Xm(X) Y.(y)
(10)
m n
where Xm(X ) and Y.( y) are appropriate functions that satisfy the geometrical boundary conditions of the plate, and Wm. are the unknown coefficients of the functions. The appropriate functions for various cases of rectangular plates with different combinations of support conditions are given in Table 1. For example if a plate with simply supported-clamped-simply supported-clamped edges (S-C-S-C) is required to be analysed, then the chosen deflection function obtained from Table 1 will be:
wlx, yl=
.(o2w)Co2w
+4D16 ~ x 2
S
S
Fig. 1.
Thus the usual goveming equation for strain energy of bending of a symmetrically laminated composite plate can be used by replacing the flexural stiffness D 0 with the reduced bending stiffness D*. The total vibrating energy of a laminated composite plate of Fig. 1 can be written as:
C
C
(6)
[M]=[D*] [r]
+ 4D26
-----'~ X a
M
N
z
z
m = 1,2,3...
n = 1,2,3...
,
(marx I \Ul
kOxay] 0w
dx dy
(8)
Substituting the assumed function of eqn (10) into eqn (7)gives: V - - Umnpq - ~o) 2 Tmnp q
2 j0 j0
w 2}dx dy
(9)
It is a fact that fundamental frequencies obtained using the Rayleigh-Ritz method are always higher than the exact values since the plate's mode shape is postulated by a finite number of terms in the shape functions which inherently increase the rigidity of the plate. The accuracy of the Rayleigh-Ritz method therefore depends on the selection of compatible shape functions, such as:
(12)
where Umnpq = E ~ Z Z Wmn Wpq m n p q
* xx DII X m
* + 2D,2(XmX ~xx Y. Y~q) + D :(XmX. Y y) * x x y y + 4D66(XmX Y.Y ) + 4D16(XmX e Y, YY) xx
x
* x YYYqYY)}dx dy + 4026(SmX~
xx
G.B. Chai
252
Table 1. Trigonometric function used in postulating the deformed shape
Two opposite edges
X,,,(x)
S-S
s,n~--}
C-S
sm ~aa sln~-a ]
sin
sln/~- ]
,l=2,4,6.. .
C-C
sin
sin ~-
s'nv/-]
m=1,2,3... ,1= 1,2,3...
Y,,( y)
• {mzrxl
• [narYl
sm/~- ]
(1
Zmnpq=EZ~2 WmnWpq m n p q
sm/~- ]
(XmXpY,,yq)dxdy )
where superscripts x and xx refer to the first and second derivatives of the function with respect to x, respectively. And superscripts y and yy refer to the first and second derivatives of the function with respect to y, respectively. Based on the principle of minimum potential energy, eqn (12) is minimized with respect to the unknown coefficients to give a series of homogeneous simultaneous equations: ~V --=0;
~ Wll
OV OV --=0;...--=0
~ W12
O Wmn
(13)
Differentiating eqn (13) with respect to the unknown coefficients a second time and rearranging the format: [K]{ W}= 702[S] {W}
(14)
where [K] is the stiffness matrix, [S] is the mass matrix and {W} is the vector of the unknown coefficients. Specifying the required number of M terms and N terms in the postulated deflection function of eqn (10), the minimization process yields a set of M x N linear homogeneous simultaneous equations (13). Then eqn (14) will give M x N stiffness and mass matrices. A standard numerical eigensolution routine by Lindfield and Penney ~° was used to extract the eigenvalues and their corresponding eigenvectors to obtain the natural frequencies of the plate and their corresponding mode shapes, respectively. NUMERICAL R E S U L T S AND D I S C U S S I O N
Most of the results to be presented are based on the material properties for graphite/epoxy:
m = 1,2,3... n= 1,2,3...
El~=130 GPa, E22=9 GPa, G12=4"8 GPa and v12=0"28. It will be mentioned when different material properties are used. The non-dimensional natural frequencies mentioned in the figures and some tables to be presented are defined by the parameter
if2 wa 2~fV/D,, =
(15)
where E~t 3 O0
12(1 - v12v21)
For the classical cases of antisymmetric and symmetric cross-ply S-S-S-S laminated plates, exact solutions are available and can be found in Jones l and Whitney.2 The non-dimensional natural frequencies for these cases which are obtained using a four-term solution of the present approach are tabulated in Table 2 for the first four modes. These results are identical to those obtained using the exact solution of Jones 1 and Whitney2 (not shown in the table). Another classical class of the S-S-S-S laminated composite plate is the antisymmetric angleply case, for which a closed form exact solution is again available. L2 The non-dimensional frequencies for the first four modes of natural vibration of an antisymmetric angle-ply laminated square plate obtained using a four-term solution are shown in Fig. 2, together with the results obtained using the exact solution. L2 The figure shows variation in ply-angle and the number of layers stacked antisymmetrically. Apart from the results for the two layers, the present solution agrees very well with the exact solution. For the three classical cases presented above, the Rayleigh-Ritz results for a four-term solution and a 49-term solution are identical thus confirming that the solution has already converged using
Free vibration of laminatedplates
253
Table 2. Non-dimensional frequencies obtained using a four-term solution for square S - S - S - S cross-ply laminated composite plates NL a
Antisymmetric cross-ply
2 4 6 8 10 12 14 16 18 20
Symmetric cross-ply
Mode 1
Mode 2
Mode 3
Mode 4
Mode 1
7.7985 10.3384 10.743 1 10.881 1 10.944 5 10"9787 10.999 3 11.0126 11.021 8 11"0283
21.1006 28.801 7 30.011 8 30"423 9 30.612 8 30.7149 30"776 3 30"8161 30-8434 30.8628
21.1006 28.801 7 30.011 8 30.423 9 30.612 8 30.7149 30"776 3 30.8161 30-8434 30.8628
31.1942 41.3536 42.972 2 43.524 5 43.777 8 43.9148 43"997 1 44.0505 44.087 1 44.1132
. 11"056 1 10.743 1 11"056 1 10.944 5 11.056 1 10"999 3 11-0561 11-021 8 11-056 1
.
Mode 2
Mode 3
Mode 4
. 21"1576 30.011 8 26-507 4 30.612 8 28.0649 30.776 3 28"8122 30"8434 29.251 3
. 38"3097 30.011 8 34.822 9 30.612 8 33.5803 30.776 3 32.9414 30"8434 32-552 1
41"3892 42.972 2 44.224 5 43.777 8 44.2245 43.997 I 44.2245 44.087 1 44.2245
aThe number of layers in the laminated composite plate.
15. --
3,I i"
Present
~ 3s
• 14-
~ 13-
32
ff. 12-
~
•°
~H .~_
~
•
~'e
"~ 2 8 '
_e
~ 10
°o
oo
°°o o
o* °°
- -,~,~--
~, 26'
.7
o=
z
10
20
30
40 50 Ry Angle
60
70
80
•
31-
Exact [ I ]
~o 5 2 :E
Jk
27
~ 44
25 23
i
21 L5
Present
--
8
:"
17 15
o
1~
~o
~o
~
~o
~o
~o
~o
~o
Ry Angle
Fig. 2.
40
~ a6
J~
19
~o
3'o
~
56-
~II I L
29-
|
~
90
3533-
g
2422
9
o
30-
4'o
~o
6'o
7'o
do
9o
Ply Angle
%
--
Present
32 28
o
ib
,;o
~o
~o
;o
;o
~o
80
90
Ply Angle
First four natural modes of an antisymmetric angle-ply square laminated composite plate.
four terms in the postulated deflection function. It should be noted that the cusps or sharp points shown in Fig. 2 for the higher frequencies are actually crossing points for the frequencies, which would be obvious if all of these frequencies were plotted together on a single graph. Results obtained using the present 144-term solution for several cases of a two-ply square anti° symmetric laminated plate are compared with published accurate results. 8,9 The material properties used to generate these results are obtained
from Baharlou and Leissa: 9 Eli~E22=40,
G12/
E22 = 0"5, v12 = 0"25. T h e results are s u m m a r i s e d
in Tables 3 and 4 for a two-ply square antisymmetric angle-ply laminated composite plate, and in Tables 5 and 6 for a two-ply square antisymmetric cross-ply laminated composite plate. The results published by Whitney ~1 using the double Fourier series method are also included for comparison in Tables 4 and 5. It can be seen that the results using the present analysis agree reasonably well with the accurate solution. 8,9 The
G. B. Chai
254
Table 3. Fundamental frequencies wa2~oftwo-plysquareantisymmetricangle-plylaminatedS-C-S-Cplates 0°
$3C3
5 15 25 35 45 55 65 75 85
$3C1
$3C2
Present 144-term
Exact
Polynom
Exact
Polynom
Exact
Polynom
Ref. 8
Ref. 9
Ref. 8
Ref. 9
Ref. 8
Ref. 9
36"248 -23"137 -19'395 17"661 16'300 16'176 18"776
36"248 26"614 23"139 21"131 19"407 17"677 16"301 16-191 18"779
36-682 . 23"229 . 19"394 17"617 15"825 15"326 17"978
36"683 . 23-230 . 19"395 17"692 16"312 16"177 18"777
36"683
Table 4. Fundamental frequencies t a a 2 ~ 0°
36'683 .
. 23"230
.
. 19'396 17'649 16"078 15'746 18"424
23'230 19"396 17'694 16"313 16"178 18-777
of two-ply square antisymmetric angle-ply laminated C-C-C-C plates
C1
C3
Present 144-term
Asymptotic
Polynom
Fourier
Asymptotic
Polynom
Fourier
Ref. 8
Ref. 9
Ref. 11
Ref. 8
Ref. 9
Ref. 11
36"880 27"378 24"317 23"189 22"838
37"283 27"834 24"750 23"631 23-309
37"582 28"052 24"940 23"799 23-488
36"401 27"156 24"215 23"111 22"835
36"836 27"562 24"643 23"602 23"347
39"458 31"337 27-517 25'316 25'316
5 15 25 35 45
Table 5. Fundamental frequencies wa2fp-/E22t 3 of two-ply square antisymmetric cross-ply laminated C-C-C-C plates Aspect ratio a/b
Asymptotic
Fourier
Polynom
Present
Ref. 8
Ref. 11
Ref. 9
144-term
1 2 3 4 5
23"638 17.111 16"640 16.543 16.509
24.527 21.171 21.171 17'200 16.974
24.033 17"297 16.782 16"673 16-636
24.037 17.299 16.784 16.675 16.638
Table 6. Fundamental frequencies w a Z ~ of two-ply square antisymmetric cross-ply laminated plates with various edge supports Boundary conditions $2-$2-$2-$2 C2-$2-C2-$2 C2-C2-C2-C2 C1-C1-C1-C 1
35'274 25'606 22'716 21'039 19'404 17"569 15"818 14"937 17'526
Polynomial
Present
Ref. 9
144-term
11"17 18"72 24'01 24'03
11-164 18"727 24"037 24"037
boundary conditions S1, $2, $3, C1, C2 and C3 depend on the restriction of the displacement components and they are discussed in detail by Baharlou and Leissa. 9 For a single layer laminated composite plate or symmetric laminated composite plate (other than the cross-ply laminate), no known exact solutions
35"848 26"563 24"218 23"483 23'316
are available for comparison. However Leissa and Narita 7 have presented comprehensive results for the case of S - S - S - S symmetrically laminated composite plates using a 144-term Rayleigh-Ritz solution. The method of solution presented in this contribution is identical to the approach used by Leissa and Narita 7 for the S - S - S - S condition, the frequencies obtained are thus identical for this case and these results are therefore not presented here. For support conditions other than the S - S - S - S case and for a single layer laminated square composite plate, the results for the first four modes of the natural frequency are plotted as a function of increasing ply angle and these are shown in Figs 3-8 for various support conditions. It can be seen that for plates with opposite sides having the same support conditions (S-S-S-S, C - C - C - C , C - S - C - S ) the curves are symmetrical about the 45 ° ply angle. As for the other plates, the curves do not indicate any symmetry. Another point to note in these figures is the orthotropic curves which are obtained by assuming that the bending-twisting coupling stiffnesses (O16 and D26 ) are zero. The solution based on the orthotropic assumption is shown to converge at a faster rate than the actual anisotropic solution as shown in Fig. 9 for a single 45°-ply laminated square
Free vibration of laminated plates
255
70
60
f-~x
....
/
\
- -
//
50
/ /
/
/ / / /I
OrthoVopic Anisotropic
\ \
! / / ! / / / /
\ \
x \ x x \ \ \
p'~\\ 60
//
E
\
\
/
/
\
\ \
/
X X X
/\ '\\
\\
\\
/Z,
"-4,, //
E
\\
,,
\\\
///
g z
\\
i1,'11
3O
//
/ ....//
/¢/
\\
\
\
\\
\\\\\
i111
20
\X
Orthotropic Anisotropic
"\\
30
Z
\\
ii I
~
/
//
/
~, 50
\ \
...... k. //
/~
/
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Fig. 3.
~
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40 0 60 Ply angle (degree)
70
80
10
l ~ l ' I ' I * I I ' I 10 20 30 40 50 60 70 Ply angle (degree)
90
Natural frequencies of a single layer square S-S-S-S laminated composite plate.
Fig. 5.
I 80
90
Natural frequencies of a single layer square C-S-C-S laminated composite plate.
70
.... 60
/ / / / j / / / /
/
/¢ /
- -
\
\\
\ \
\ \
/ / /
\
\
60
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.... - -
Orthotropic Anisotropic
/
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f/----\,
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20
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7
8
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90
Natural frequencies of a single layer square S-C-S-S laminated composite plate.
Fig. 6.
0
'
' ' 'o 10 2
' ' ' ' ' ' ' 'o' 30 40 50 60 7 Ply angle (degree)
' 80
90
Natural frequencies of a single layer square C-C-S-S laminated composite plate.
G. B. Chai
256
~
70
60
//I
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40 50 60 Ply angle (degree)
I
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70
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80
90
Natural frequencies of a single layer square C - C - C - S laminated composite plate.
80
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C - C - C - C composite plate. However the orthotropic assumption used for an anisotropic composite plate always gives non-conservative results, as shown in Figs 3-9. Theoretical results for a single 45°-ply laminated square C - C - C - C composite plate published by Whitney 2 are compared with results obtained using the present 144-term solution. Whitney used beam eigenfunctions to postulate the plate's out-of-plane deformation and the Rayleigh-Ritz method to solve for the natural modes. The results obtained by Whitney are given in Table 7 using 1-, 9-, 25- and 49-term solutions for the first mode, and frequencies for higher modes obtained using a 49-term solution are indicated in a footnote to Table 7. These results are based on material properties given in Whitney, 2 i.e. E~l =31 x 106 psi, E 2 2 = 2 " 7 x 106 psi, G~2=0.75 x 106 psi, vt2=0"28, t=0"0424 in., a = b = 2 in. and density = 1.92 × 10 -4 lb-sec2/in 4. It can be seen that at least a 144-term solution is required for convergence up to two decimal places for the first two modes and 0nly one decimal for the next two modes. Theoretical results are applied to rectangular 14-layer graphite/epoxy laminated composite plates with symmetric lay-up of [+ 45/0/0/90/0/ 0 / - 45/0]s, the nominal thickness of the plate is taken as 1.75 mm. The laminated composite plates are anisotropic with the flexural stiffness ratios of Dl6/Di1 = D26/DII = 0"1289. The natural frequencies in Hz are summarized in Table 8 for two aspect ratios. In addition, experimental results and finite element method results are also shown for comparison. Details of the experimental setup, test procedures and the finite element analysis
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z
20
21
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20
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10
Fig. 8.
i 10
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I 30
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I I ' 1 40 50 60 Ply angle (degree)
i 70
'
~ 80
' 90
Natural frequencies of a single layer square C - C - C - C laminated composite plate,
30 0
I 2
= 4
I i 6 8 Number of M=N terms
r 10
I 12
19 14
Fig. 9. Convergence rate for the first two modes of a single layer square C - C - C - C laminated composite plate.
Free vibration of laminated plates
257
Table 7. Comparison between f~ obtained using present theory with published results 2 for a single 45°-ply C-C-C-C laminated square plate Ma
Present Rayleigh-Ritz solution Mode 1 Aniso Ortho
1 2 3 4 5 6 7 8 9 10 11 12
24.2869 21.3973 20.9239 20.5530 20"3762 20.2823 20.2164 20"1808 20"1522 20-1357 20.1216 20.1129
Mode 2 Aniso Ortho
24.2869 24.2869 23.3640 23.3640 23.1626 23.1626 23-0939 23.0939 23.0642 23.0642 23.0493 23.0493
. 39.0791 35.2879 35-0393 34.4008 34"3302 34"1409 34.1115 34.0360 34.0220 33"9859 33-9785
. . 49.0098 48.0566 47"1296 46"8796 46.3501 46.5423 46.4501 46.4081 46.3658 46.3442 46.3221
Whitneyb
Mode 3 Aniso Ortho . 57-2428 50-6339 48-4278 47-6429 47.1390 46.8614 46-6724 46-5505 46"4629 46-4016 46-3555
Mode 4 Aniso Ortho
. 49-0098 48-0566 47"1296 46"8796 46.3501 46-5423 46.4501 46"4081 46.3658 46-3442 46.3221
. 79-9169 54.6863 52"3517 51"5701 50"9995 50"8003 50"5800 50"5040 50.4023 50"366 1 50.3135
Mode 1 Aniso Ortho 23.51 -20"82 -20"57 -20.51 ------
79-0867 79"0867 75.6995 75"6995 74.6796 74.6796 74.2677 74.2677 74.0706 74.0706 73.9647
23.51 -23"27 -23.24 -23'23 ------
aThe number of M -- N terms used in the postulated function. hWhitney 2 higher mode results using a 49-term anisotropic solution: fl for Mode 2 is 35.01, Mode 3 is 47.07, Mode 4 is 52.21.
Table 8. Natural frequencies in Hz for C - S - C - S panels of Ref. 12 with stacking sequence of I+ 4 5 , 0 , 0 , 9 0 , 0 , - 45,0]s and DI6/ DI I = D 2 6 / D l l = 0"128 92 M =N
458 x 90 x 1-75 mm Mode 1
Mode2
481.77 477.31 475.77 474.54 474.17 473.71 473.57 473.36 473.29 473.18 473.07
. 572.81 562.23 557.32 555.02 553.43 552.58 551.85 551.46 551.06 550.62
Exp
473.00
Ort
480.75
FEM
471.60
1 2 3 4 5 6 7 8 9 10 12
458 x 120 x 1"75 mm
Mode 3
Mode4
Mode5
Mode 1
Mode 2
. 1825.45 720.31 702.13 696.25 691.98 689.94 688.16 687.23 686.31 685.29
. 1912.83 1817-83 922-79 900-94 892-11 887-51 883.99 882.15 880.38 878.44
-1895-57 1816.95 1180.73 1155.20 1144.67 1139.22 1134.89 1132.56 1129.12
314.54 310.30 308.83 307.74 307.39 307.00 306.88 306.70 306.64 306.54 306.45
. 410-00 400.82 396.22 394.33 392-89 392.22 391.58 391.64 390-94 390.57
606.00
734.00
903.00
1104.00
328-00
569"73
717"00
922.56
1184.91
313"45
547.50
687.23
868.20
1 134"89
305"50
388"20
.
Mode 3
Mode4
Mode5
. 1150.64 562.90 548.69 542.76 539-60 537.61 536.33 535.44 534.79 533.96
. 1239.70 1143.16 769.94 754.17 745.51 742.28 739.00 737.72 736.13 734-62
-1223.90 1142.51 1028.30 1012-52 1001.37 998.19 993.75 992.44 989.55
430.00
543.00
703.00
888.00
406.99
560.00
770.39
1036.25
529.00
725"50
973.50
.
Exp ~ Experimental data from Ref. 12. Ort ~ Whitney 2 solution assuming specially orthotropic conditions. FEM ~ Finite Element results from Ref. 12.
a r e g i v e n in a r e c e n t p a p e r . ~2 A l s o s h o w n in T a b l e 8 a r e t h e results o b t a i n e d u s i n g a c l o s e d f o r m s o l u t i o n 2 w h i c h is b a s e d o n t h e specially o r t h o t r o p i c a s s u m p t i o n . A s c a n b e s e e n f r o m t h e table, o v e r a l l t h e o r t h o t r o p i c results a r e c o n s i s t e n t l y a b o u t 5 % h i g h e r t h a n t h e results f o r the a n i s o t r o p i c case. O n t h e w h o l e t h e p e r c e n t a g e difference between the experimental data and a 144-term solution ranged from -4"50% to
11.44%.
CONCLUSION T h e results o f the p r e s e n t m e t h o d a g r e e d v e r y well with t h e e x a c t s o l u t i o n f o r the specific cases of symmetric cross-ply, antisymmetric cross-ply and antisymmetric angle-ply laminated square c o m p o s i t e plates s u b j e c t e d to f r e e v i b r a t i o n . T h e r e s o n a n t f r e q u e n c i e s f o r single l a y e r l a m i n a t e d s q u a r e c o m p o s i t e p l a t e s i n d i c a t e d that a sufficient n u m b e r o f t e r m s m u s t b e u s e d in t h e p o s t u l a t e d
258
G. B. Chai
trigonometric series function to give a c o n v e r g e d accurate solution; insufficient terms and orthotropic a s s u m p t i o n w o u l d lead to an unconservative solution. H o w e v e r the results obtained with the present a p p r o a c h seem to give a lower and thus m o r e conservative solution than the exact solution b e t w e e n ply angle 5°-35 ° and 5 5 ° - 8 5 ° for the case of a two-ply antisymmetric angle-ply laminated square plate as s h o w n in Fig. 2, Tables 3 and 4. Experimental and theoretical studies on a 14layered laminated rectangular c o m p o s i t e plate with symmetric stacking sequence show that for the laminates c o n s i d e r e d which have a low value of D16/Dll and O26/Dll the coupling responses can be neglected. T h e percentage difference is only about + 5% if these bending/twisting coupling responses are ignored. A m e t h o d of solution, using the classical Rayleigh-Ritz m e t h o d , was p r e s e n t e d to analyse the free vibration of generally laminated c o m posite plates. T h e effect of various s u p p o r t conditions are included in the theoretical formulation. For all the cases c o n s i d e r e d the results seem to indicate that the m e t h o d can be applied to laminated c o m p o s i t e plates in general.
ACKNOWLEDGEMENT This w o r k has b e e n carried out with the financial s u p p o r t of the Ministry of F i n a n c e (Singapore) u n d e r the research fund M O F R & D 5/89.
REFERENCES I. Jones, R. M., Mechanics of Composite Materials. McGraw-Hill, New York, 1975. 2. Whitney, J. M., Structural Analysis of Laminated Anisotropic Plates. Technomic, Lancaster, PA, USA, 1987. 3. Chamis, C. C., Buckling of anisotropic composite plates. Journal of Structural Division (ASCE), 95 (1969) 2119-39. 4. Ashton, J. E., Approximate solutions for unsymmetrically laminated plates. Journal of Composite Materials, 3 (1969) 189-91. 5. Leissa, A. W., An overview of composite plate buckling. Composite Structures, 4( 1) (1987) 1-29. 6. Kapania, R. K. & Raciti, S., Recent advances in analysis of laminated beams and plates Part lI: Vibrations and wave propagation. A1AA Journal, 27(7) (1989) 935-46. 7. Leissa, A. W. & Narita, Y., Vibration studies for simply supported symmetrically laminated rectangular plates. Composite Structures, 12 ( 1989 ) 113- 32.
8. Lin, C. C. & King, W. W., Free transverse vibration of rectangular unsymmetrically laminated plates. Journal of Sound and Vibration, 36 ( 1974) 91-103.
9. Baharlou, B. & Leissa, A. W., Vibration and buckling of generally laminated composite plates with arbitrary edge conditions. International Journal of Mechanical Sciences, 29 (8)(1993) 545-55. 10. Linfield, G. R. & Penny, J. E. T., Microcomputers in Numerical Analysis. Ellis Horwood, Chichester, UK, 1989. 11. Whitney, J. M., The effect of boundary conditions on the response of laminated composites. Journal of Composite Materials, 4 (1970) 192-203. 12. Chai, G. B., Chin, S. S., Lim, T. M. & Hoon, K. H., Vibration analysis of laminated composite plates: TVholography and finite element method. Composite Structures, 23 (1993) 273-83.