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Vibration of point-supported rectangular composite plates
Composites Science and Technology 53 (105) 325-332 @) 1995 Elsevicr Science Limited ELSEVIER
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0266-3538(95)00004-6
VIBRATION
OF POINT-SUPPORTED RECTANGULAR COMPOSITE PLATES
Serge Abrate Department o,f Mechanical and Aerospace
Engineering and Engineering Rolla, Missouri 65401. USA
(Received 7 April 1994: revised version received 22 November
Mechanics,
University of Missouri-Rolla,
1994: accepted 9 December
1994)
cantilever plates with point supports along the free edges5 or inside.6 German’ revised some of the results presented earlier3.4 using the superposition method. The finite element method was used by Rao et al.’ and Raju and Amba-Rae’ to analyze the free vibration of rectangular plates with four-point supports located symmetrically on the diagonals. The Rayleigh-Ritz method was used by a number of investigators. Laura and Cortinez”’ used the Rayleigh-Schmidt technique and provided a one-term approximation for plates with two adjacent edges supported and a point support at the opposite corner. Kim and co-workers”.” used the Rayleigh-Ritz method with orthogonal polynomial approximation functions and the Lagrange multiplier techniques to study the free flexural vibrations of thin isotropic plates. Bhat” used the same approach and provided results for a simply-supported square plate with central point support and for a free square plate with four-point supports symmetrically located on the diagonals. Aksu and Felemban” studied corner-pointsupported square Mindlin plates and presented the first 18 natural frequencies for ratios of plate thickness to length of the side ranging from 0400.5 to 0.2. It can be assumed that for the lowest value of that ratio, the shear deformation and rotary inertia effects are negligible. Narita” analyzed rectangular plates resting on internal line supports that do not reach the edges of the plate. This type of problem is difficult to handle by variational approximation methods where the approximation functions are selected to satisfy both the essential boundary conditions along the edges of the plate and the zero displacement constraints along the internal line supports. The zero-displacement constraint along those supports was enforced by using a Lagrange-multiplier/Fourier-series approach. Young and Dickinson” studied the free vibrations of plates with straight and curved line supports. The objective of the present study is to develop a method for analyzing the free vibration of rectangular composite plates with arbitrary support conditions
Abstract A general approach is presented to study the free vibrations of rectangular symmetrically laminated composite plates with point supports by using the Rayleigh- Ritz method and the Lagrange multiplier technique for enforcing the zero displacement constraints at the support locations. Polynomial approximation functions are used and the constitutive relationships are written in terms of four lamination parameters. With these lamination parameters, the number of design variables is reduced to a minimum, which is useful for design optimization purposes, while all symmetric lay-ups are. considered. This paper illustrates how the lamination parameters can be used for optimal design of vibrating plates. Results are presented for several cases and the effect of aspect ratio. material properties and lay-up are fully investigated. Plates on point supports and with straight and curved internal line supports are considered. For each case, the lay-up that maximizes the first natural frequency of the plate is determined. Keywords: vibration, plates, optimum design, lamination parameters, point supports
INTRODUCTION The free vibration of isotropic rectangular plates with point supports has been studied extensively and most of the early work in the area has been reviewed by Leissa. ’ Kerstens* used the intermediate problem technique and presented results for corner-supported plates with various aspect ratios and also for plates with various other support positions. German’ used the superposition method to study the vibration of rectangular plates with four-point supports symmetrically distributed along the edges. Results are given for corner-supported plates with aspect ratios ranging from 1 to 3. GoTman used the same approach for rectangular plates with four-point supports symmetrically distributed along the diagonals and presented results for several support locations. Saliba also used the superposition method to study the vibration of 325
326
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along the edges and any number of internal point supports. The Rayleigh-Ritz method with polynomial approximation functions will be used and the zero displacement constraints at the support locations will be enforced using the Lagrange multiplier technique. It will be shown that internal line supports can be accurately modelled by replacing them by a number of equally spaced point supports. Symmetrically laminated composite plates will be considered. and the constitutive equations are written in terms of four non-dimensional lamination parameters. The effect of plate aspect ratio and thickness, material properties and lay-up on the natural frequencies of the plate are examined for several support conditions. For each case, the lay-up that maximizes the first natural frequency is determined. FORMULATION The transverse motion of symmetrically laminated plates is completely uncoupled from the in-plane motion and the free vibration of such plates is governed by the variational eigenvalue problem
where w is the transverse displacement, w the natural frequency, {K} = (-w,,,, -w,,,., -2~,,,)~ are the plate curvatures, u a test function that satisfies the essential boundary conditions of the problem and {K}= (-u, ~.~,-u,,.?., -2u.,,,)‘. The plate constitutive equations are given by @f) = [D](K)
(2)
where the c, are constants to be determined. polynomial approximation functions +j(x, Y) = x”‘Y”(x - a)a( y - b)”
In the (6)
the exponents m and n start from 0, 1, or 2 depending on whether the edges x = 0 and y = 0 are free, simply-supported or clamped. Similarly, CYand p are 0. 1, or 2 depending on whether the edges x = 0 and y = 0 are free, simply-supported or clamped. Taking p terms in the x direction and 9 terms in the y direction, we will then have a p X q approximation to the solution (p X q = IV).The Lagrange multiplier method is used to enforce r constraint equations of the form w(x,, Yk) = 0
(7)
which correspond to zero displacement at r point supports located at arbitrary positions (x,, yk). The modified functional for the problem is given by K(W, A) = I(w) +
c hkW(Xk,yh)
(8)
h=I
where I(w) is the functional given in eqn (8), and the A/, are the Lagrange multipliers. With the Rayleigh-Ritz method, we obtain the matrix eigenvalue problem
The matrix K, has dimension (N X r) and each column corresponds to one of the constraints being enforced. For example, if constraint k is w(xk, yk) = 0, then KM(i, k) = (b,(~~, yk). Equation (9) is an (N + r) x (N + r) eigenvalue problem in which the stiffness and mass matrices are given by
or, more explicitly, the moment resultants are related to the plate curvatures by
the Di, values being the bending rigidities of the plate. This weak formulation (eqn (1)) corresponds to the minimum of the quadratic functional
where the first term represents the maximum strain energy and the second term the maximum kinetic energy of the plate during harmonic motion. In this investigation, rectangular plates with one edge along the x axis will be considered. An N-term approximation for the transverse displacement is taken as
The eigenvalues and eigenvectors are obtained from eqn (9) using the inverse iteration method. The bending rigidities in eqn (3) can be written as
&I 022 012 Qih D16 D2h
j=l
I
=h’ 12
Vibration of point-supported
in terms of the laminate thickness invariants being defined as
h, the stiffness
Table 1. Natural frequencies of isotropic square plates with four-point supports on the diagonal
u, = [Q,, + 2(Q,2 + 2Qd + Qd4 u2
=
a
[Q, I - Q22]/2
0
UJ = [Q I I - 2(Q,2 + 2Qd + Q,,]/4 u, =
[Q,, + 2(Q,2 - 2Qd
& = [Q,, -2&u
Source
6x6 8X8
(13)
9x9
Ref. 13 Ref. 8 Ref. 14
+ Q,,]/4
+ Q,,]/4
and the lamination parameters
327
rectangular composite plates
Ref. 2
are defined as 0.2
6x6 8X8
9x9 Ref. 13 Ref. 8
l,,, = $ /_+r’2 z2 cos’ 28 dz
7.1118 7.1109 7.1109 7.1131 7.11089 7.13502 7.15
15.7716 15.7703 15.7703 15.7716 15.77 15.7659 15.64
15.7716 15.7703 15.7703 15.7716 15.77 15.7659 15.64
19.7257 19.5963 19.5961 19.7257 19.596 194481 19.49
19.726 19.596 19.4766 20.4901 19.5962
23.716 23.060 23.0354 27.53784 -
33.701 32.521 32.4766 34.4325
33.701 32.521 32.4766 37.4325 -
13.6740 13.3302 13.3182 13.28356
16.7411 14.6713 14.6548 -
16.7411 14.6713 14.6556 -
19.7257 17.7990 17.7966 -
12.2808 11.7274 11.6136 11.53535
14.6757 12.0974 11.8762 -
14.6757 12.0977 11.9035 --
19.4321 16.3607 16.3584 -
hi2
(14)
0.4
9x9 Ref. 8
h/2
z2
sin 28 cos 28 dz
0.4667
The effect of ply orientation and ply thickness distribution through the thickness are included in those lamination parameters. For an orthotropic lamina with elastic moduli E, in the fiber direction and E2 in the transverse direction, an in-plane shear modulus Cl2 and a Poisson ratio v12, the ply stiffness coefficients are defined as
Q,, = E,ltl
-
v,2~2,),
Q22
=
E2lU
-
~12~2,)
(15) QM = G,2
Q,2 = v,2Q22t
6X6 8X8
<, , = $ I_+*” z2 sin 28 dz
Note that v12E2 = v2, E,. For general laminated plates, a maximum of 12 parameters are needed to describe the effect of lamination, but since we are considering only symmetric laminates, only the four parameters in eqn (11) are needed. It can be shown that - 1 YG& I 1 and [$s [,,, % 1 so that, for all laminates, the bending rigidities D,, , D22. D,2 and DM depend only on the two parameters l9 and c,,, which vary in the domain
6x6 8X8
9x9 Ref. 8
between the parabola f,,, = l: and the line [,,, = 1. Each symmetric angle-ply laminate is represented by a point on the parabola, cross-ply laminates are represented by a point on the line <,,, = 1. The parameters c,, and 1,2 affect only the Dlh and D,, terms and we shall see that these terms can be neglected for laminates with many layers. The advantages of using this stiffness invariant formulation are that the effect of laminate thickness, lay-up and material properties are separated (eqn 12), and only four parameters are required to describe all symmetric laminates regardless of the number of plies. NUMERICAL
a
Plates with point supports
I_ -
$_
+ b
Pb
-
It-d cya
EXAMPLES
l-J Lya
Fig. 1. Rectangular plate with four-point supports located
symmetrically on the diagonals.
on the diagonals
Square plates with four-point supports located symmetrically on the diagonals (Fig. 1) are considered first. For isotropic plates with a Poisson’s ratio v = 0.3, the present results (Table 1) are in good agreement with those reported in previous studies.‘,‘” The natural frequencies non-dimensionalized as are R= (phw2a4/D)‘12 where D = Eh’/[12(1 - v’)]. When 9 X 9 terms are used in the displacement approximation (eqn (5)), the first four natural frequencies are consistently lower than those of Ref. 13. Only the first natural frequency is given in Ref. 8. The variation of the first natural frequency of corner-supported (a = 0) graphite/epoxy square plates with many plies is shown in Fig. 2 as a function of the lamination parameters & and l,,,. The elastic properties of graphite/epoxy are
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328
0.8
0.6
510 O-4 0.2
0
I
0
I
I
0.2
0.4
0.6
I
0.8
1
b
fig. 2. Variation of first natural frequency of cornersupported, square graphite/epoxy plate with many plies.
taken as E, = 181 GPa, E2 = 10.3 GPa, vIZ = 0.028, G,, = 7.17 GPa. For composite plates, frequencies are non-dimensionalized as R = (phw2a4/D’)“’ where E,h3
D’ =
12(1 -If&$) Since the plate is symmetric with respect to both diagonals, only the c9 > 0 half of the lamination space needs to be studied, and Fig. 2 shows that the first natural frequency is a maximum for some point along & = 0. A line search in that direction indicates that
0.8
the first natural frequency of square graphite/epoxy plate is a maximum when &, = 0 and l,,j = O-44. For an aspect ratio a/b = 2, the entire lamination space must be considered (Fig. 3) and the first natural frequency is a maximum when &, = 0.67 and l,,, = 0.68. Table 2 gives the first four natural frequencies for cornersupported, angle-ply laminated plates with aspect ratios of 1 and 2 and fiber orientations varying in 15” increments. Figure 4 shows the complex interaction between the first six modes as fiber orientation changes for symmetric angle-ply laminates (a/b = 2). Square graphite/epoxy laminates with many plies and supports located so that (Y= O-2 are considered next. The first natural frequency is a maximum when l9 = 0 and l,,, = 1 which corresponds to a family of cross-ply laminated lay-ups. When (Y= O-04, the first natural frequency is maximized when CC,= 0 and cl0 = 0.91 for a square plate, when i9 = 0.785 and 5,() = 1 for a plate with a/b = 2, and when iV = ilo = 1 with a/b = 3. In this study, laminates with many layers are considered and the D,, and DZh are neglected. It is known that, as the number of plies in the laminate becomes large, the effect of these bending-twisting coupling terms become negligible. Results in Table 3, for graphite/epoxy plates with four-point supports symmetrically located on the diagonals (a = 0.4), indicate that the effect of these terms is small for laminates with more than eight plies. Glass/epoxy plates with four supports symmetrically located on the diagonals with (Y= 0.4 are studied next in order to show the influence of material properties on the natural frequencies and on the optimal lay-up. For this material system, the four elastic constants are E, = 38.6 GPa, Ez = 8.27 GPa, V ,2 = 0.26, G,, = 4.14 GPa. For plates with an aspect ratio a/b = 1, the optimum lay-up is a cross-ply laminate with c4 = 0 and ilo = 1. When a/b = 1.5, the Table 2. Natural frequencies for graphite/epoxy angle-ply laminated plates with point supports at the four corners
0.6
Angle
Source
R,
Cl2
Q,
%
0.4
a/h = 1 0
6X6 8X8 8X8 8X8
2.2109 2.2088 2.3353 3.1423 4.4913
4.9077 4.907 1 7.0994 10.2952 7.1040
7.2109 7.1971 8.2990 10.3258 llS838
12.3494 11.8192 12.6703 10.8134 11M38
8X8 8X8 8X8 8X8 8X8 8X8 8X8
6.2544 7.0267 6.7941 4.2347 2.8935 2.4270 2.3311
12.3305
12.5708 16.1695 20.7300 22.5749 28.7178 13.5494 9.1019
21.3335 19.3864 21S225 28.7922 19.8406 21.5650 19.7521
"10
XXX
15 30 45
0.2
0 -1
Fig. 3. Variation
supported,
-0.5
0
0.5
of first natural frequency rectangular (a/h = 2) graphite/epoxy many plies.
1
of cornerplate with
a/b =2 0 1.5 30 4s 60 75 90
15.8296 19.1968 19.2803 12.5744 9.6777 8.9554
Vibration of point-supported
329
rectangular composite plates
Table 4. Natural frequencies of a square isotropic plate with poiut supports (a) at the mid-point of each side and (b) at all four comers and the mid-point of all four sides otherwise free
Source (a)
6X6 8X8 9x9
Ref. 12 Ref. 11 @)
0
15
30
45
80
75
8X8 9x9
Ref. 11
90
Fiber orientation Fig. 4. Interaction between the first six modes of corner-supported angle-ply laminated graphite/epoxy rectangular plates (a/b = 2).
first natural frequency is maximized when & = 0.77 and i,. = 1, while for a/b = 2, the optimum lay-up is a 0” unidirectional laminate. With glass/epoxy, the optimum lay-up is always a cross-ply laminate whereas it is not always so with graphite/epoxy. The material properties also affect the optimal values of the lamination parameters.
Q,
fiz
Q,
Q
13.4687 13.4682 13.4682 13.468 13.4682
19.0362 18.4080 18.1544 18.403 18.0307
20.1644 19.2274 19.029 19.226 18.9339
20.1644 19.2274 27.3141 27.361 27.0488
18.412 18.150
35.7681 35.3580
38.4323 38.4323
62.3082 60.943 1
18.037
35.1727
38.4316
60.5821
when a/b = 3 and 19= 0.40 and [,o=O*18 when a/b = 2. For the a/b = 2 or 3 cases, the lower left region and the upper right region correspond to different mode shapes and, at the optimum point, the first two natural frequencies are equal. The first four natural frequencies for a square isotropic plate with point supports at the corners and at the mid-point of each side (Table 4) are in good agreement with results reported earlier.” For a graphite/epoxy plate with the same combination of point supports the first natural frequency is a maximum for a *45” angle-ply lay-up.
Plates with point slipports along the edges
Table 4 gives the first six natural frequencies for square isotropic plates with point supports located at the mid-point of each side, The results are in good agreement with those reported previously.“*‘* For square graphite/epoxy plates with point supports at the mid-point of each side, the first natural frequency is a maximum with a *45” angle-ply laminate (39 = 51,)= 0). F or a/b = 2 and 3, the optimum lay-up is near the parabola ljo = 4; but is not an angle-ply laminate (Figs 5 and 6). The optimal values of the lamination parameters are & = 0.58 and {,. = 044 Table 3. Natural frequencies for a k27” angle-ply Iaminated plate with four-point supports on the diagonal (a = O-4)
Plates with intermediate
straight line supports
The present approach can also be used to model plates with internal line supports by enforcing the zero displacement constraint at a number of points along
t
,
10
0.8
0.6
5 10 0.4
Number of terms 0.2
Eight plies 7x7
8X8 9x9
4.6536 4.6509 46488
69093 6.0153 6.0087
10.064 9.3121 9-1847
11.348 10.968
4.6609 4%608 4.6592
6.9835 6.0768 6.0734
10.818 10.063 10.052
12.604 12.080 12.078
12.511 0
Many plies 7x7 8X8 9x9
Fig. 5. Variation
of the first natural frequency of a rectangular graphite/epoxy plate with point supports at the mid-point of each side (a/b = 2).
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330
Table 5. Natural frequencies of square isotropic plate with internal line supports Number of points 0.6
Number of terms
R,
Q,
Q,
Q,
4
5X5
14.749
23.891
27.841
36.511
5
5X5 6X6
15.034 14.874
24.323 23.931
47.823 30.749
90.587 40.086
6
6X6 7x7
15.040 14.771
24.133 23.591
47.823 33.112
60.389 44.361
7
7x7 8x8
14.880 14.807
23,748 23.598
46.497 35.133
60.335 47.416
8
8x7 9x7 10x7
14.880 14.871 14.871
23.748 23.709 23.709
46.497 46.352 46.352
57.766 57.766 57.755
9
9x7 10x7
14.871 14.871
23.709 23.709
46.352 46.352
57.766 57.755
0.4
0.2
0 -1
-0):5
6
0.5
1
Fig. 6. Variation of the first natural frequency rectangular graphite/epoxy plate with point supports mid-point of each side (a/b = 3).
of a at the
points is quite similar while being more straightforward. For this example, when a graphite/ epoxy laminate with many plies is used, the first natural frequency of the plate is a maximum when a 0” unidirectional laminate is used independently of the aspect ratio. Young and Dickinson” studied the free vibrations of rectangular SSSS isotropic plates (a/b = 2) with a straight line support making an angle 8 with the x direction and going through the center of the plate. That is, the transverse displacements are constrained to be zero on the line given by discrete
those lines. In the example first treated by Narita,‘” a rectangular isotropic plate is supported on two line segments (Fig. 7). The first four natural frequencies are calculated as the number of point supports on each line segment varies from four to nine when y = 6 = O-8 and a/b = 1. Table 5 shows that, as the number of points increases up to eight, the natural until the line support is frequencies increase adequately simulated for these first five frequencies. The effect of the number of terms in the displacement approximation is also investigated and when eight points are used for each line segment, a 9 X 7 approximation is sufficient. Results compare well with othersI but it must be noted that, in Ref. 15, a fixed number of terms was used to enforce the support conditions with a 7 X 7 polynomial approximation and it appeared that the solution had not yet converged. Considering that in Ref. 15 each term in the Fourier series represents some weighted average of the constraint equation along the line support, the present approach of imposing zero displacement constraints at
Fig. 7. Rectangular
plate with two supports.
partial
internal
line
tan 0 With the present method, enforcing the constraints at Table 6. Natural frequencies of SSSS rectangular plate (a/b =2) with oblique straight line support passing through the center of the plate at an angle 0 from the x axis 0
Number of points
Source
Q
90
5
9x9 Ref. 16
78.957 78.957
95.469 94.584
197.39 197.39
197.58 197.39
75
5
11 x5 Ref. 16
81.086 81.20
95.990 96.28
193.85 193.84
200.35 199.76
60
5
11 x6 Ref. 16
87.47 87.60
99.55 99.40
189.3 189.08
204.4 203.4
45
5
11 x5 Ref. 16
99.79 100.2
109.29 110.04
190.83 191.0
217.6 214.92
Vibration of point-supported
five points equally spaced on the line, the first four natural frequencies are in good agreement with those of Kim and Dickinsonr6 as shown in Table 6. For graphite/epoxy plates with a line support at 60” and an aspect ratio of 2, the first natural frequency is a maximum when a +51.65” angle-ply laminate is used. When the line support is at 45” (a/b = 2), the first natural frequency is again a maximum when an angle-ply laminate is used. In this case, l9 = -0.36 and lrO = &,2 which correspond to a ~t55.55” lay-up. SCSS, CSSS and CCSS rectangular graphite/epoxy plates with an aspect ratio of 2 and an oblique straight line support at 45” are also considered. In each case, the first natural frequency is a maximum for an angle-ply laminate. Optimum values of the fiber orientations are & = -0.29, -0.45 and 0.34 for CSSS, SCSS and CCSS plates respectively. Plates with curved internal line supports Young and Dickinson’6 also studied the free vibrations of square SSSS isotropic plates with an internal line support given by
F- 0x 2
(17)
a
Table 7 shows that using eight points, equally spaced in the x direction, to enforce the zero displacement constraint the present results agree well with those of Ref. 16. For such a plate, using graphite/epoxy, the first natural frequency is maximized when an angle-ply laminate with a fiber orientation of k39.8” (& = O-18) is used. CONCLUSIONS A general approach for studying the free vibration of rectangular, symmetrically laminated, composite plates with point supports has been presented. Plates with arbitrary combinations of support conditions along the edges can be analyzed. An arbitrary number of point supports can be introduced and they can be arranged to model internal line supports. Close agreement with results presented by previous investigators for isotropic plates is demonstrated for several examples involving point supports, and
Table 7. Natural frequencies of square isotropic SSSS plate with internal support at y/b = 1 - (X/U)*
Number of
Source
R,
9x9 9x9 9x9 Ref. 16
38909 38.922 38.924 38.920
points 6 7 8
77.249 77.399 77.457 77.404
88.399 88.549 88.582 88.556
106.40 106.55 106.61 106.41
rectangular composite plates
331
internal line supports that are either curved or straight. The free vibrations of symmetrically laminated composite plates are studied with the constitutive equations expressed in terms of four lamination parameters. These four non-dimensional parameters describe all symmetrical laminates accounting for all possible distributions of fiber orientation and layer thickness through the thickness of the laminate. As the number of plys becomes large, only two parameters are needed to describe all symmetric laminates. With this stiffness invariant formulation, the search for the lay-up that maximizes the first natural frequency is made easier since the number of design variables is reduced to a minimum. In each case, the optimum combination of design parameters can be obtained without prior restriction on fiber orientations or ply thickness distributions as opposed to previous studies where a particular lay-up was selected in advance and the fiber orientation of one group of plies is optimized, or cases where the fiber orientations are selected and the ply thicknesses are to be optimized.” The present approach also shows that the optimum design is not unique since many laminates have the same combination of lamination parameters even though fiber orientations and ply location through the thickness are different. In this study, symmetrically laminated composite plates are considered and it is shown that for plates with more than eight plies, the effect of the bending-twisting coupling coefficients is negligible. The material properties can have a significant effect on the dynamic behavior of the plate and result in drastically different optimum designs as shown by comparing the results obtained for graphite/epoxy and glass/epoxy. Results are presented for several point-supported plates. plates with partial internal line supports, oblique line supports and curved line supports. These results are new and should be of interest to those involved in designing composite structures and they show the versatility of the present approach in modeling composite plates with various support conditions. REFERENCES 1. Leissa, A. W., Vibration of Plates, NASA SP 160, 1969. 2. Kerstens, J. G. M., Vibration of rectangular plates supported at an arbitrary number of points. J. Sound Vibration, 65 (1979) 493-504. 3. Gorman, D. J., Free vibration analysis of rectangular plates with symmetrically distributed point supports along the edges. J. Sound Vibration, 73 (1980) 563-74. 4. Gorman, D. J., An analytical solution for the free vibration analysis of rectangular plates resting on symmetrically distributed point supports. J. Sound Vibration, 79 (1981) 561-74. 5. Saliba, H. T., Free vibration analysis of rectangular cantilever plates with symmetrically distributed point
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332
supports along the edges. J. Sound Vibration, 94 (1984) 381-95. 6. Saliba,
H. T., Free vibration analysis of rectangular cantilever plates with symmetrically distributed lateral point supports. J. Sound Vibration, 127 (1988) 77-89. 7. Gorman, D. J., A note on the free vibration analysis of rectangular plates resting on symmetrically distributed point supports. J. Sound Vibration, l31(1989) X5-19. 8. Rao, G. V., Raju, I. S. & Amba-Rao, C. L., Vibration of point supported plates. J. Sound Vibration, 29 (1973) 387-91. 9. Raju, I. S. & Amba-Rao,
C. L., Free vibration of a square plate symmetrically supported at four points along the diagonals. J. Sound Vibration, 90 (1983)
291-7.
10. Laura, P. A. A. & Cortinez, V. H., Fundamental frequency of point supported square plates carrying concentrated masses. J. Sound Vibration, 100 (1985) 456-S. 11. Kim, C. S. & Dickinson, S. M., The flexural vibration of rectangular plates with point supports. J. Sound Vibration, 117 (1987) 249-61.
12. Kim, C. S., Young, P. G. & Dickinson, S. M., On the flexural vibration of rectangular plates approached by using simple polynomials in the Rayleigh-Ritz method. J. Sound Vibration, 143 (1990) 379-94. 13. Bhat, R. B., Vibration of rectangular
plates on point and line supports using characteristic orthogonal polynomials in the Rayleigh-Ritz method. J. Sound
Vibration, 149 (1991) 170-2. 14. Aksu, G. & Felemban, M. B., Frequency
analysis of corner point supported Mindlin plates by a finite difference energy method. J. Sound Vibration, 158
(1992) 531-44. 15. Narita, Y., Vibration
analysis of a rectangular plate resting on line supports by the Lagrange multiplierFourier expansion approach. J. Appl. Mech., 53 (1986)
469-70. 16. Young, P. G. 8 Dickinson,
S. M., On the free flexural vibration of rectangular plates with straight or curved internal line supports. J. Sound Vibration, 162 (1993)
123-35. 17. Abrate,
S., Optimum design of laminated shells. Cornp. Struct., 29 (1994) 269-86.
plates and
Printed in Northern Ireland. All rights reserved 02~-3S3X/Y5/$0Y.50
0266-3538(95)00004-6
VIBRATION
OF POINT-SUPPORTED RECTANGULAR COMPOSITE PLATES
Serge Abrate Department o,f Mechanical and Aerospace
Engineering and Engineering Rolla, Missouri 65401. USA
(Received 7 April 1994: revised version received 22 November
Mechanics,
University of Missouri-Rolla,
1994: accepted 9 December
1994)
cantilever plates with point supports along the free edges5 or inside.6 German’ revised some of the results presented earlier3.4 using the superposition method. The finite element method was used by Rao et al.’ and Raju and Amba-Rae’ to analyze the free vibration of rectangular plates with four-point supports located symmetrically on the diagonals. The Rayleigh-Ritz method was used by a number of investigators. Laura and Cortinez”’ used the Rayleigh-Schmidt technique and provided a one-term approximation for plates with two adjacent edges supported and a point support at the opposite corner. Kim and co-workers”.” used the Rayleigh-Ritz method with orthogonal polynomial approximation functions and the Lagrange multiplier techniques to study the free flexural vibrations of thin isotropic plates. Bhat” used the same approach and provided results for a simply-supported square plate with central point support and for a free square plate with four-point supports symmetrically located on the diagonals. Aksu and Felemban” studied corner-pointsupported square Mindlin plates and presented the first 18 natural frequencies for ratios of plate thickness to length of the side ranging from 0400.5 to 0.2. It can be assumed that for the lowest value of that ratio, the shear deformation and rotary inertia effects are negligible. Narita” analyzed rectangular plates resting on internal line supports that do not reach the edges of the plate. This type of problem is difficult to handle by variational approximation methods where the approximation functions are selected to satisfy both the essential boundary conditions along the edges of the plate and the zero displacement constraints along the internal line supports. The zero-displacement constraint along those supports was enforced by using a Lagrange-multiplier/Fourier-series approach. Young and Dickinson” studied the free vibrations of plates with straight and curved line supports. The objective of the present study is to develop a method for analyzing the free vibration of rectangular composite plates with arbitrary support conditions
Abstract A general approach is presented to study the free vibrations of rectangular symmetrically laminated composite plates with point supports by using the Rayleigh- Ritz method and the Lagrange multiplier technique for enforcing the zero displacement constraints at the support locations. Polynomial approximation functions are used and the constitutive relationships are written in terms of four lamination parameters. With these lamination parameters, the number of design variables is reduced to a minimum, which is useful for design optimization purposes, while all symmetric lay-ups are. considered. This paper illustrates how the lamination parameters can be used for optimal design of vibrating plates. Results are presented for several cases and the effect of aspect ratio. material properties and lay-up are fully investigated. Plates on point supports and with straight and curved internal line supports are considered. For each case, the lay-up that maximizes the first natural frequency of the plate is determined. Keywords: vibration, plates, optimum design, lamination parameters, point supports
INTRODUCTION The free vibration of isotropic rectangular plates with point supports has been studied extensively and most of the early work in the area has been reviewed by Leissa. ’ Kerstens* used the intermediate problem technique and presented results for corner-supported plates with various aspect ratios and also for plates with various other support positions. German’ used the superposition method to study the vibration of rectangular plates with four-point supports symmetrically distributed along the edges. Results are given for corner-supported plates with aspect ratios ranging from 1 to 3. GoTman used the same approach for rectangular plates with four-point supports symmetrically distributed along the diagonals and presented results for several support locations. Saliba also used the superposition method to study the vibration of 325
326
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along the edges and any number of internal point supports. The Rayleigh-Ritz method with polynomial approximation functions will be used and the zero displacement constraints at the support locations will be enforced using the Lagrange multiplier technique. It will be shown that internal line supports can be accurately modelled by replacing them by a number of equally spaced point supports. Symmetrically laminated composite plates will be considered. and the constitutive equations are written in terms of four non-dimensional lamination parameters. The effect of plate aspect ratio and thickness, material properties and lay-up on the natural frequencies of the plate are examined for several support conditions. For each case, the lay-up that maximizes the first natural frequency is determined. FORMULATION The transverse motion of symmetrically laminated plates is completely uncoupled from the in-plane motion and the free vibration of such plates is governed by the variational eigenvalue problem
where w is the transverse displacement, w the natural frequency, {K} = (-w,,,, -w,,,., -2~,,,)~ are the plate curvatures, u a test function that satisfies the essential boundary conditions of the problem and {K}= (-u, ~.~,-u,,.?., -2u.,,,)‘. The plate constitutive equations are given by @f) = [D](K)
(2)
where the c, are constants to be determined. polynomial approximation functions +j(x, Y) = x”‘Y”(x - a)a( y - b)”
In the (6)
the exponents m and n start from 0, 1, or 2 depending on whether the edges x = 0 and y = 0 are free, simply-supported or clamped. Similarly, CYand p are 0. 1, or 2 depending on whether the edges x = 0 and y = 0 are free, simply-supported or clamped. Taking p terms in the x direction and 9 terms in the y direction, we will then have a p X q approximation to the solution (p X q = IV).The Lagrange multiplier method is used to enforce r constraint equations of the form w(x,, Yk) = 0
(7)
which correspond to zero displacement at r point supports located at arbitrary positions (x,, yk). The modified functional for the problem is given by K(W, A) = I(w) +
c hkW(Xk,yh)
(8)
h=I
where I(w) is the functional given in eqn (8), and the A/, are the Lagrange multipliers. With the Rayleigh-Ritz method, we obtain the matrix eigenvalue problem
The matrix K, has dimension (N X r) and each column corresponds to one of the constraints being enforced. For example, if constraint k is w(xk, yk) = 0, then KM(i, k) = (b,(~~, yk). Equation (9) is an (N + r) x (N + r) eigenvalue problem in which the stiffness and mass matrices are given by
or, more explicitly, the moment resultants are related to the plate curvatures by
the Di, values being the bending rigidities of the plate. This weak formulation (eqn (1)) corresponds to the minimum of the quadratic functional
where the first term represents the maximum strain energy and the second term the maximum kinetic energy of the plate during harmonic motion. In this investigation, rectangular plates with one edge along the x axis will be considered. An N-term approximation for the transverse displacement is taken as
The eigenvalues and eigenvectors are obtained from eqn (9) using the inverse iteration method. The bending rigidities in eqn (3) can be written as
&I 022 012 Qih D16 D2h
j=l
I
=h’ 12
Vibration of point-supported
in terms of the laminate thickness invariants being defined as
h, the stiffness
Table 1. Natural frequencies of isotropic square plates with four-point supports on the diagonal
u, = [Q,, + 2(Q,2 + 2Qd + Qd4 u2
=
a
[Q, I - Q22]/2
0
UJ = [Q I I - 2(Q,2 + 2Qd + Q,,]/4 u, =
[Q,, + 2(Q,2 - 2Qd
& = [Q,, -2&u
Source
6x6 8X8
(13)
9x9
Ref. 13 Ref. 8 Ref. 14
+ Q,,]/4
+ Q,,]/4
and the lamination parameters
327
rectangular composite plates
Ref. 2
are defined as 0.2
6x6 8X8
9x9 Ref. 13 Ref. 8
l,,, = $ /_+r’2 z2 cos’ 28 dz
7.1118 7.1109 7.1109 7.1131 7.11089 7.13502 7.15
15.7716 15.7703 15.7703 15.7716 15.77 15.7659 15.64
15.7716 15.7703 15.7703 15.7716 15.77 15.7659 15.64
19.7257 19.5963 19.5961 19.7257 19.596 194481 19.49
19.726 19.596 19.4766 20.4901 19.5962
23.716 23.060 23.0354 27.53784 -
33.701 32.521 32.4766 34.4325
33.701 32.521 32.4766 37.4325 -
13.6740 13.3302 13.3182 13.28356
16.7411 14.6713 14.6548 -
16.7411 14.6713 14.6556 -
19.7257 17.7990 17.7966 -
12.2808 11.7274 11.6136 11.53535
14.6757 12.0974 11.8762 -
14.6757 12.0977 11.9035 --
19.4321 16.3607 16.3584 -
hi2
(14)
0.4
9x9 Ref. 8
h/2
z2
sin 28 cos 28 dz
0.4667
The effect of ply orientation and ply thickness distribution through the thickness are included in those lamination parameters. For an orthotropic lamina with elastic moduli E, in the fiber direction and E2 in the transverse direction, an in-plane shear modulus Cl2 and a Poisson ratio v12, the ply stiffness coefficients are defined as
Q,, = E,ltl
-
v,2~2,),
Q22
=
E2lU
-
~12~2,)
(15) QM = G,2
Q,2 = v,2Q22t
6X6 8X8
<, , = $ I_+*” z2 sin 28 dz
Note that v12E2 = v2, E,. For general laminated plates, a maximum of 12 parameters are needed to describe the effect of lamination, but since we are considering only symmetric laminates, only the four parameters in eqn (11) are needed. It can be shown that - 1 YG& I 1 and [$s [,,, % 1 so that, for all laminates, the bending rigidities D,, , D22. D,2 and DM depend only on the two parameters l9 and c,,, which vary in the domain
6x6 8X8
9x9 Ref. 8
between the parabola f,,, = l: and the line [,,, = 1. Each symmetric angle-ply laminate is represented by a point on the parabola, cross-ply laminates are represented by a point on the line <,,, = 1. The parameters c,, and 1,2 affect only the Dlh and D,, terms and we shall see that these terms can be neglected for laminates with many layers. The advantages of using this stiffness invariant formulation are that the effect of laminate thickness, lay-up and material properties are separated (eqn 12), and only four parameters are required to describe all symmetric laminates regardless of the number of plies. NUMERICAL
a
Plates with point supports
I_ -
$_
+ b
Pb
-
It-d cya
EXAMPLES
l-J Lya
Fig. 1. Rectangular plate with four-point supports located
symmetrically on the diagonals.
on the diagonals
Square plates with four-point supports located symmetrically on the diagonals (Fig. 1) are considered first. For isotropic plates with a Poisson’s ratio v = 0.3, the present results (Table 1) are in good agreement with those reported in previous studies.‘,‘” The natural frequencies non-dimensionalized as are R= (phw2a4/D)‘12 where D = Eh’/[12(1 - v’)]. When 9 X 9 terms are used in the displacement approximation (eqn (5)), the first four natural frequencies are consistently lower than those of Ref. 13. Only the first natural frequency is given in Ref. 8. The variation of the first natural frequency of corner-supported (a = 0) graphite/epoxy square plates with many plies is shown in Fig. 2 as a function of the lamination parameters & and l,,,. The elastic properties of graphite/epoxy are
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328
0.8
0.6
510 O-4 0.2
0
I
0
I
I
0.2
0.4
0.6
I
0.8
1
b
fig. 2. Variation of first natural frequency of cornersupported, square graphite/epoxy plate with many plies.
taken as E, = 181 GPa, E2 = 10.3 GPa, vIZ = 0.028, G,, = 7.17 GPa. For composite plates, frequencies are non-dimensionalized as R = (phw2a4/D’)“’ where E,h3
D’ =
12(1 -If&$) Since the plate is symmetric with respect to both diagonals, only the c9 > 0 half of the lamination space needs to be studied, and Fig. 2 shows that the first natural frequency is a maximum for some point along & = 0. A line search in that direction indicates that
0.8
the first natural frequency of square graphite/epoxy plate is a maximum when &, = 0 and l,,j = O-44. For an aspect ratio a/b = 2, the entire lamination space must be considered (Fig. 3) and the first natural frequency is a maximum when &, = 0.67 and l,,, = 0.68. Table 2 gives the first four natural frequencies for cornersupported, angle-ply laminated plates with aspect ratios of 1 and 2 and fiber orientations varying in 15” increments. Figure 4 shows the complex interaction between the first six modes as fiber orientation changes for symmetric angle-ply laminates (a/b = 2). Square graphite/epoxy laminates with many plies and supports located so that (Y= O-2 are considered next. The first natural frequency is a maximum when l9 = 0 and l,,, = 1 which corresponds to a family of cross-ply laminated lay-ups. When (Y= O-04, the first natural frequency is maximized when CC,= 0 and cl0 = 0.91 for a square plate, when i9 = 0.785 and 5,() = 1 for a plate with a/b = 2, and when iV = ilo = 1 with a/b = 3. In this study, laminates with many layers are considered and the D,, and DZh are neglected. It is known that, as the number of plies in the laminate becomes large, the effect of these bending-twisting coupling terms become negligible. Results in Table 3, for graphite/epoxy plates with four-point supports symmetrically located on the diagonals (a = 0.4), indicate that the effect of these terms is small for laminates with more than eight plies. Glass/epoxy plates with four supports symmetrically located on the diagonals with (Y= 0.4 are studied next in order to show the influence of material properties on the natural frequencies and on the optimal lay-up. For this material system, the four elastic constants are E, = 38.6 GPa, Ez = 8.27 GPa, V ,2 = 0.26, G,, = 4.14 GPa. For plates with an aspect ratio a/b = 1, the optimum lay-up is a cross-ply laminate with c4 = 0 and ilo = 1. When a/b = 1.5, the Table 2. Natural frequencies for graphite/epoxy angle-ply laminated plates with point supports at the four corners
0.6
Angle
Source
R,
Cl2
Q,
%
0.4
a/h = 1 0
6X6 8X8 8X8 8X8
2.2109 2.2088 2.3353 3.1423 4.4913
4.9077 4.907 1 7.0994 10.2952 7.1040
7.2109 7.1971 8.2990 10.3258 llS838
12.3494 11.8192 12.6703 10.8134 11M38
8X8 8X8 8X8 8X8 8X8 8X8 8X8
6.2544 7.0267 6.7941 4.2347 2.8935 2.4270 2.3311
12.3305
12.5708 16.1695 20.7300 22.5749 28.7178 13.5494 9.1019
21.3335 19.3864 21S225 28.7922 19.8406 21.5650 19.7521
"10
XXX
15 30 45
0.2
0 -1
Fig. 3. Variation
supported,
-0.5
0
0.5
of first natural frequency rectangular (a/h = 2) graphite/epoxy many plies.
1
of cornerplate with
a/b =2 0 1.5 30 4s 60 75 90
15.8296 19.1968 19.2803 12.5744 9.6777 8.9554
Vibration of point-supported
329
rectangular composite plates
Table 4. Natural frequencies of a square isotropic plate with poiut supports (a) at the mid-point of each side and (b) at all four comers and the mid-point of all four sides otherwise free
Source (a)
6X6 8X8 9x9
Ref. 12 Ref. 11 @)
0
15
30
45
80
75
8X8 9x9
Ref. 11
90
Fiber orientation Fig. 4. Interaction between the first six modes of corner-supported angle-ply laminated graphite/epoxy rectangular plates (a/b = 2).
first natural frequency is maximized when & = 0.77 and i,. = 1, while for a/b = 2, the optimum lay-up is a 0” unidirectional laminate. With glass/epoxy, the optimum lay-up is always a cross-ply laminate whereas it is not always so with graphite/epoxy. The material properties also affect the optimal values of the lamination parameters.
Q,
fiz
Q,
Q
13.4687 13.4682 13.4682 13.468 13.4682
19.0362 18.4080 18.1544 18.403 18.0307
20.1644 19.2274 19.029 19.226 18.9339
20.1644 19.2274 27.3141 27.361 27.0488
18.412 18.150
35.7681 35.3580
38.4323 38.4323
62.3082 60.943 1
18.037
35.1727
38.4316
60.5821
when a/b = 3 and 19= 0.40 and [,o=O*18 when a/b = 2. For the a/b = 2 or 3 cases, the lower left region and the upper right region correspond to different mode shapes and, at the optimum point, the first two natural frequencies are equal. The first four natural frequencies for a square isotropic plate with point supports at the corners and at the mid-point of each side (Table 4) are in good agreement with results reported earlier.” For a graphite/epoxy plate with the same combination of point supports the first natural frequency is a maximum for a *45” angle-ply lay-up.
Plates with point slipports along the edges
Table 4 gives the first six natural frequencies for square isotropic plates with point supports located at the mid-point of each side, The results are in good agreement with those reported previously.“*‘* For square graphite/epoxy plates with point supports at the mid-point of each side, the first natural frequency is a maximum with a *45” angle-ply laminate (39 = 51,)= 0). F or a/b = 2 and 3, the optimum lay-up is near the parabola ljo = 4; but is not an angle-ply laminate (Figs 5 and 6). The optimal values of the lamination parameters are & = 0.58 and {,. = 044 Table 3. Natural frequencies for a k27” angle-ply Iaminated plate with four-point supports on the diagonal (a = O-4)
Plates with intermediate
straight line supports
The present approach can also be used to model plates with internal line supports by enforcing the zero displacement constraint at a number of points along
t
,
10
0.8
0.6
5 10 0.4
Number of terms 0.2
Eight plies 7x7
8X8 9x9
4.6536 4.6509 46488
69093 6.0153 6.0087
10.064 9.3121 9-1847
11.348 10.968
4.6609 4%608 4.6592
6.9835 6.0768 6.0734
10.818 10.063 10.052
12.604 12.080 12.078
12.511 0
Many plies 7x7 8X8 9x9
Fig. 5. Variation
of the first natural frequency of a rectangular graphite/epoxy plate with point supports at the mid-point of each side (a/b = 2).
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330
Table 5. Natural frequencies of square isotropic plate with internal line supports Number of points 0.6
Number of terms
R,
Q,
Q,
Q,
4
5X5
14.749
23.891
27.841
36.511
5
5X5 6X6
15.034 14.874
24.323 23.931
47.823 30.749
90.587 40.086
6
6X6 7x7
15.040 14.771
24.133 23.591
47.823 33.112
60.389 44.361
7
7x7 8x8
14.880 14.807
23,748 23.598
46.497 35.133
60.335 47.416
8
8x7 9x7 10x7
14.880 14.871 14.871
23.748 23.709 23.709
46.497 46.352 46.352
57.766 57.766 57.755
9
9x7 10x7
14.871 14.871
23.709 23.709
46.352 46.352
57.766 57.755
0.4
0.2
0 -1
-0):5
6
0.5
1
Fig. 6. Variation of the first natural frequency rectangular graphite/epoxy plate with point supports mid-point of each side (a/b = 3).
of a at the
points is quite similar while being more straightforward. For this example, when a graphite/ epoxy laminate with many plies is used, the first natural frequency of the plate is a maximum when a 0” unidirectional laminate is used independently of the aspect ratio. Young and Dickinson” studied the free vibrations of rectangular SSSS isotropic plates (a/b = 2) with a straight line support making an angle 8 with the x direction and going through the center of the plate. That is, the transverse displacements are constrained to be zero on the line given by discrete
those lines. In the example first treated by Narita,‘” a rectangular isotropic plate is supported on two line segments (Fig. 7). The first four natural frequencies are calculated as the number of point supports on each line segment varies from four to nine when y = 6 = O-8 and a/b = 1. Table 5 shows that, as the number of points increases up to eight, the natural until the line support is frequencies increase adequately simulated for these first five frequencies. The effect of the number of terms in the displacement approximation is also investigated and when eight points are used for each line segment, a 9 X 7 approximation is sufficient. Results compare well with othersI but it must be noted that, in Ref. 15, a fixed number of terms was used to enforce the support conditions with a 7 X 7 polynomial approximation and it appeared that the solution had not yet converged. Considering that in Ref. 15 each term in the Fourier series represents some weighted average of the constraint equation along the line support, the present approach of imposing zero displacement constraints at
Fig. 7. Rectangular
plate with two supports.
partial
internal
line
tan 0 With the present method, enforcing the constraints at Table 6. Natural frequencies of SSSS rectangular plate (a/b =2) with oblique straight line support passing through the center of the plate at an angle 0 from the x axis 0
Number of points
Source
Q
90
5
9x9 Ref. 16
78.957 78.957
95.469 94.584
197.39 197.39
197.58 197.39
75
5
11 x5 Ref. 16
81.086 81.20
95.990 96.28
193.85 193.84
200.35 199.76
60
5
11 x6 Ref. 16
87.47 87.60
99.55 99.40
189.3 189.08
204.4 203.4
45
5
11 x5 Ref. 16
99.79 100.2
109.29 110.04
190.83 191.0
217.6 214.92
Vibration of point-supported
five points equally spaced on the line, the first four natural frequencies are in good agreement with those of Kim and Dickinsonr6 as shown in Table 6. For graphite/epoxy plates with a line support at 60” and an aspect ratio of 2, the first natural frequency is a maximum when a +51.65” angle-ply laminate is used. When the line support is at 45” (a/b = 2), the first natural frequency is again a maximum when an angle-ply laminate is used. In this case, l9 = -0.36 and lrO = &,2 which correspond to a ~t55.55” lay-up. SCSS, CSSS and CCSS rectangular graphite/epoxy plates with an aspect ratio of 2 and an oblique straight line support at 45” are also considered. In each case, the first natural frequency is a maximum for an angle-ply laminate. Optimum values of the fiber orientations are & = -0.29, -0.45 and 0.34 for CSSS, SCSS and CCSS plates respectively. Plates with curved internal line supports Young and Dickinson’6 also studied the free vibrations of square SSSS isotropic plates with an internal line support given by
F- 0x 2
(17)
a
Table 7 shows that using eight points, equally spaced in the x direction, to enforce the zero displacement constraint the present results agree well with those of Ref. 16. For such a plate, using graphite/epoxy, the first natural frequency is maximized when an angle-ply laminate with a fiber orientation of k39.8” (& = O-18) is used. CONCLUSIONS A general approach for studying the free vibration of rectangular, symmetrically laminated, composite plates with point supports has been presented. Plates with arbitrary combinations of support conditions along the edges can be analyzed. An arbitrary number of point supports can be introduced and they can be arranged to model internal line supports. Close agreement with results presented by previous investigators for isotropic plates is demonstrated for several examples involving point supports, and
Table 7. Natural frequencies of square isotropic SSSS plate with internal support at y/b = 1 - (X/U)*
Number of
Source
R,
9x9 9x9 9x9 Ref. 16
38909 38.922 38.924 38.920
points 6 7 8
77.249 77.399 77.457 77.404
88.399 88.549 88.582 88.556
106.40 106.55 106.61 106.41
rectangular composite plates
331
internal line supports that are either curved or straight. The free vibrations of symmetrically laminated composite plates are studied with the constitutive equations expressed in terms of four lamination parameters. These four non-dimensional parameters describe all symmetrical laminates accounting for all possible distributions of fiber orientation and layer thickness through the thickness of the laminate. As the number of plys becomes large, only two parameters are needed to describe all symmetric laminates. With this stiffness invariant formulation, the search for the lay-up that maximizes the first natural frequency is made easier since the number of design variables is reduced to a minimum. In each case, the optimum combination of design parameters can be obtained without prior restriction on fiber orientations or ply thickness distributions as opposed to previous studies where a particular lay-up was selected in advance and the fiber orientation of one group of plies is optimized, or cases where the fiber orientations are selected and the ply thicknesses are to be optimized.” The present approach also shows that the optimum design is not unique since many laminates have the same combination of lamination parameters even though fiber orientations and ply location through the thickness are different. In this study, symmetrically laminated composite plates are considered and it is shown that for plates with more than eight plies, the effect of the bending-twisting coupling coefficients is negligible. The material properties can have a significant effect on the dynamic behavior of the plate and result in drastically different optimum designs as shown by comparing the results obtained for graphite/epoxy and glass/epoxy. Results are presented for several point-supported plates. plates with partial internal line supports, oblique line supports and curved line supports. These results are new and should be of interest to those involved in designing composite structures and they show the versatility of the present approach in modeling composite plates with various support conditions. REFERENCES 1. Leissa, A. W., Vibration of Plates, NASA SP 160, 1969. 2. Kerstens, J. G. M., Vibration of rectangular plates supported at an arbitrary number of points. J. Sound Vibration, 65 (1979) 493-504. 3. Gorman, D. J., Free vibration analysis of rectangular plates with symmetrically distributed point supports along the edges. J. Sound Vibration, 73 (1980) 563-74. 4. Gorman, D. J., An analytical solution for the free vibration analysis of rectangular plates resting on symmetrically distributed point supports. J. Sound Vibration, 79 (1981) 561-74. 5. Saliba, H. T., Free vibration analysis of rectangular cantilever plates with symmetrically distributed point
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332
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H. T., Free vibration analysis of rectangular cantilever plates with symmetrically distributed lateral point supports. J. Sound Vibration, 127 (1988) 77-89. 7. Gorman, D. J., A note on the free vibration analysis of rectangular plates resting on symmetrically distributed point supports. J. Sound Vibration, l31(1989) X5-19. 8. Rao, G. V., Raju, I. S. & Amba-Rao, C. L., Vibration of point supported plates. J. Sound Vibration, 29 (1973) 387-91. 9. Raju, I. S. & Amba-Rao,
C. L., Free vibration of a square plate symmetrically supported at four points along the diagonals. J. Sound Vibration, 90 (1983)
291-7.
10. Laura, P. A. A. & Cortinez, V. H., Fundamental frequency of point supported square plates carrying concentrated masses. J. Sound Vibration, 100 (1985) 456-S. 11. Kim, C. S. & Dickinson, S. M., The flexural vibration of rectangular plates with point supports. J. Sound Vibration, 117 (1987) 249-61.
12. Kim, C. S., Young, P. G. & Dickinson, S. M., On the flexural vibration of rectangular plates approached by using simple polynomials in the Rayleigh-Ritz method. J. Sound Vibration, 143 (1990) 379-94. 13. Bhat, R. B., Vibration of rectangular
plates on point and line supports using characteristic orthogonal polynomials in the Rayleigh-Ritz method. J. Sound
Vibration, 149 (1991) 170-2. 14. Aksu, G. & Felemban, M. B., Frequency
analysis of corner point supported Mindlin plates by a finite difference energy method. J. Sound Vibration, 158
(1992) 531-44. 15. Narita, Y., Vibration
analysis of a rectangular plate resting on line supports by the Lagrange multiplierFourier expansion approach. J. Appl. Mech., 53 (1986)
469-70. 16. Young, P. G. 8 Dickinson,
S. M., On the free flexural vibration of rectangular plates with straight or curved internal line supports. J. Sound Vibration, 162 (1993)
123-35. 17. Abrate,
S., Optimum design of laminated shells. Cornp. Struct., 29 (1994) 269-86.
plates and