Vibration
./ournalofSoundand
(1987) 119(l).
184188
VIBRATIONS OF ORTHOTROPIC SQUARE VARIABLE THICKNESS (PARABOLIC‘
PLATES HAVING VARIATION)
1. INTRODU<‘TION
Fibre reinforced plastics (FRP) are finding increasing applications in aerospace structures due to their high strength to weight and high stiffness to weight ratios. Orthotropic plates of varying thickness are used for some aerospace structural components. Determination of vibration behaviour of such components is necessary for their design. Chelladurai et al. [l] have done limited studies using a finite element method for square and rectangular orthotropic plates of constant thickness for two boundary conditions, obtaining the fundamental frequency. In the work being reported here orthotropic plates of variable thickness (parabolic variation along the xdirection) were studied for four boundary conditions: CCCC, CCFF, SSSS and FCFF (see Table 1 for the notation). The effect of fibre orientation, boundary conditions and degree of thickness variation on the fundamental frequency were studied by using a RayleighRitz method.
TABLE
Boundary conditions Sample no.
1 considered
Plan form and edge condition
: Notation
I
III3 1
1
cccc
I
CCFF
f1
II
3
ssss
L_J
4
FCFF
2.
A rectangular The bending
KAYL.tlCiHRITZ
Mt:THOI>
plate element with coordinate directions strain energy of the plate is given by LJ=;
[Cl’[Dl[CJ
is shown
in Figure
dx, d.r,
1.
(1)
184 0022,460X/87/220184+05
$03.00/O
0
1987 Academic
Press Limited
LETTERS
TO THF.
,
185
EDITOR
‘7i
Eizzl! ,I’
,/.’
/’
,/
/’
9
,/e ,/‘,T’
z
J
.Y
,,/”
Figure
1. Sketch of plate geometry.
where [C] = [a2w/ax2 a’w/rly’ a2w/i)xdy], kinetic energy of the plate is given by h
0
510
0
(t is time). 3.
PARABOLIC
VARIATION
The volume of the plate is kept constant. variation in the xdirection is given by
(21
OF
THICKNESS
The thickness
of the plate having
A series solution
satisfying
Figure
METHOD
boundary
2. Parabolic
OF
SOLUTION
conditions
variation
is taken
of thickness
a parabolic
(3)
n(x/a)‘]/(l0.3377),
where n is a constant whose value varies between 0 and 1 ( 7 = 0.0 represents thickness plate; n = 1.0 represents a maximum variation in thickness). 4.
[2]1. The
(1) and (2), LJ and T can be evaluated.
By using equations
t’ = ?,Jl
in reference
$pt’ti’ dx dy,
T= where ti = awlat
is as defined
and [II]
9s
in xdirection
a constant
186
LETTERS
TO THE
EDITOR
where C&(X) represents the beam function satisfying the plate boundary condition in the xdirection and qn(y) represents the beam function satisfying the plate boundary condition in the ydirection. In this study the conventional beam functions were used. The assumed displacement function (4) is then substituted in the energy equation. Integrations have been carried out by a Gaussian quadrature scheme. The variation of the energy equations is carried out to obtain the standard eigenvalue problem. The resulting eigenvalue problem has been solved by a simultaneous iteration technique. 5.
NUMERICAL
EXAMPLES
AND
DISCUSSION
This RayleighRitz formulation, as described, was used to study the effect of fibre orientation and boundary conditions on the frequencies of square orthotropic plates of variable thickness made of graphite/epoxy, the material properties of which are given in Table 2 [3]. In order to validate the present formulation, the fundamental frequency parameter (A, =pt’p:L4/m) of a clamped square plate for which solutions are available in references [l] and [4] were obtained and are shown in Table 3. Our values are close to those of references [l] and [4]. The effect of fibre orientation on the fundamental frequency parameter 4 for various boundary conditions and degree of thickness variation (77) was analyzed. The values of 6 obtained are tabulated in Table 4. 6.
BOUNDARY
CONDITION
1 (CCCC)
For 77= 0.0, 6 decreases for 19= 0” to 45”, becoming minimum at 6 = 45”; then it increases for 0 = 45” to 90”. Values of fi are symmetric with respect to 8 = 45”. For n = 0.2, the trend is the reverse of that for n = 0.0, 6 being maximum at 0 = 45”. For n = 0.4 and 0.6, &i, fi rs t increases . for 0 = 0” to 15”, then it decreases for 0 = 15” to 45”
TABLE
Material properties
El
El
Composites
(GPa)
(GPa)
Glass/epoxy Kevlar/epoxy Boron/epoxy Graphite/epoxy
38.6 76.0 204.0 181.0
8.27 5.50 18.50 10.30
0.26 0.34 0.23 0.28
Material Glass/epoxy Kevlar/epoxy Boron/epoxy Graphite/epoxy
0” 0” 0” 0”
____ Present
[3] Specific gravity
4.14 2.30 5.59 7.17
1.80 1.46 2.00 1.60
3
square plate (constant
Fibre orientation
composites
VIZ
TABLE
Clamped
2
of typical unidirectional
thickness)
a
~____ solution
31.75 44.87 42.94 47.24
Reference 38.54 45.72 43.46 48.34
____ [l]
Reference 38.58 45.83 43.54 48.46
[4]
LEIIERS
TO THE EDITOR TABLE
187
4
Parabolic variation of thickness in xdirection
0
0.0
0.2
0.4
15 30 45 60 75 90
47.24 47.14 46.93 46.83 46.93 47.14 47.24
46.96 46.98 47.10 48.60 47.32 46.98 46.96
2 (CCFF)
0 15 30 45 60 75 90
8.35 11.64 12.94 13.80 12.97 11.60 8.33
3 (SSSS)
0 15 30 45 60 75 90
4 (FCFF)
0 15 30 45 60 75 90
BC 1 (CCCC)
(degrees) 0
0.6
0.8
I! 0
47.45 50.82 48.08 47.77 48.06 50.82 47.45
48.95 53.21 49.21 49.01 49.24 53.21 48.95
51.69 51.58 51.31 51.12 51.33 51.58 51.69
55.87 55.60 54.93 54.19 54.86 55.60 55.87
9.12 11.47 13.41 13.94 12.77 10.37 7.94
10.3 12.55 14.22 14.36 12.80 IO.12 7.61
11.75 14.16 16.03 15.25 13.18 10.09 7.44
14.05 16.60 17.75 16.88 14.15 10.48 7.58
17.68 20.49 21.33 19.69 16.00 11.54 P.28
22.58 24.47 28.15 29.42 28.15 24.46 22.58
22.79 24,62 27.92 29.43 27.91 24.60 22.78
23.20 25.03 28.32 29.92 28.29 24.97 23.14
23.88 25.77 29.19 30.72 29.10 25.62 23.74
24.91 26.97 30.64 30.72 30.43 27.90 24.44
X.39 28 72 32.81 34.49 32.32 2X.11 35.46
6.48 6.32 5.45 3.99 2.59 1.85 1.72
7.06 6.95 6.10 4.49 2.92 2.08 1.94
8.52 8.64 7.95 6.00 3.93 2.81 2.61
9.66 9.90 9.36 7.19 4.55 341 3.17
11.52 11.86 11.43 8.96 6.00 4.32 4.02
7.71 7.70 6.90 5.14 3.3’5 2.39 2.23

and follows a symmetric pattern for 0 = 45 to 90”. For n = 0.8 and 1.0, JAY follows same pattern as for n = 0.0, but the values of 4 are higher. 7.
BOUNDARY
CONDITION
the
2 (CCFF)
For n = 0.0,0.2 and 0.4, a increases for 0 = 0” to 45”, becoming maximum at 0 = 45”; then it decreases for 0 = 45” to 90”. For n = 0.0, values of a are symmetric about 8 = 45”, but for n = 0.2 and 0.4, symmetry does not exist. For n = 0.6,O.S and 1.0 4 is maximum at 0 = 30”. 8.
For n = 0.0 to 1.0, 4 H = 0” to 45”, becoming
BOUNDARY
CONDITION
3 (SSSSI
follows the same pattern, which is as follows. &, increases maximum at 8 = 45”; then it decreases for 0 = 45” to 90”. 9.
BOUNDARY
CONDITION
for
4 (FCFF)
For n = 0.0, 0.2 and 0.4, fi decreases as 0 increases from 0” to 90”. For n = 0.6, 0.X and 1.0, 6 increases for 0 = 0” to 15”; then it decreases for 0 = 15” to 90”.
LETTEKS
158
TO
THI. EDITOR
10. C‘ONC‘LUDING
REMARKS
1. For parabolic
variation in thickness along the xdirection for a thin orthotropic square plate, the fundamental frequency parameter ~‘5; increases as n (the degree of thickness variation) is increased from 0.0 to 1 .O for almost all boundary conditions and fibre orientations. 2. The fibre angle at which \ii;h; is maximum for a given boundary condition is also dependent on the degree of thickness variation. 3. The pattern of variation of & with fibre angle is dependent on the degree ot thickness variation (7) for most of the boundary conditions. 4. From Table 4, it is possible to choose the fibre angle (0) that gives maximum vhy for a given boundary condition and degree of thickness variation. S. K. MALHorR.4
Fibre Reinforced Plastics Research Centre. Department of Applied Mechanics, Department qf Mechanical Engineering, Indian Institute of Technology, Madras600 036, India (Received
8 September
N. G.~NESAN
M. A. VtLUSWAMI
1987) REFERENCES
B. P. SHASTKV and G. V. RAO 1984 Fibre Science and Technolog?; 21, 7381. Effect of fibre orientation on the vibration behaviour of orthotropic rectangular plates. R. M. JONES 1975 Mechanics of Composite Materials. New York: McGrawHill. S. W. TSAI and H. T. HOHN 1979 AFML Technical Report TR78201. Introduction to composite materials, Volumes I and II. S. G. LEKHNITKII 1956 Anisotropic Plates. London: Gordon and Breach. T. CHELLADURAI,
APPENI)IX:
NOTATION
lengths of sides of rectangular plate (a = b = L in present radian frequency bending strain energy transverse displacement in the zdirection rectangular coordinates mass per unit area plate thickness average thickness of the plate E,(t’J3/12(1  v,2v.,l E,( t’j?/12(1  Z’,>V~,I degree of thickness variation
study)