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In:. J. Solids Srmcrures Vol. 30, No. 24, PP. 3357-3379, 1993 Printed in Great Britain
0020-7683/93 S6.00+ NJ 0 1993 Pergamoo Press Ltd
A CONTINUUM THREE-DIMENSIONAL VIBRATION ANALYSIS OF THICK RECTANGULAR PLATES
K. M. LIEW, K. C. HUNG and M. K. LIM Dynamics and Vibration Centre, School of Mechanical and Production Engineering,
Nanyang Technological University, Nanyang Avenue, Singapore 2263, Republic of Singapore
(Received 9 February 1993; in revisedform 28 June 1993)
Ahatraet-This paper pmsents a ennui ~dim~sion~ Ritz fo~u~tion for the vibration analysis of homogeneous, thick, rectangular plates with arbitrary combinations of boundary con- straints. This model is formulated on the basis of the linear, three-dimensional, small deformation elasticity theory to predict the vibratory responses of these thick rectangular plates. The dis- placement fields in the transverse and in-plane directions are expressed by sets of orthogonally generated polynomial functions. These shape functions are intrinsically a product of a class of orthogonal polynomial functions and a basic function which are chosen to satisfy the essential geometric boundary conditions at the outset. Sets of frequency data for plates with various aspect ratios and thickness ratios have been presented. These data are used to examine the merits and limitations of the classical plate theory and Mindlin plate theory by direct comparisons. Finally, using the three-dimensional continuum approach, sets of first known deformed mode shapes have been generated thus helping to understand the vibratory motion. Furthermore, these results may also serve as the benchmark to further research into the refined plate theories.
1. INTRODU~ION
To date, closed-form analytical solutions to the eigenvalue problems associated with the elastic continuum are limited to only simple cases. To seek practical solutions, many researchers have resorted to numerical approximations. As known, for elastic continuums of various geometries and boundary conditions, the Ritz energy method can provide accurate solutions, however, it depends largely on the choice of global admissible functions.
Well known existing functions have been expressed in terms of finite series such as trigonometric functions (Young, 1950; Warburton, 1954), power series expansion (Narita, 1985) and hill functions (Kao, 1973). They have been used with great success in the past, mainly for thin plates. Lately, sets of self-generating polynomial functions have been proposed and implemented in the Ritz procedure for various thin plate problems (Bhat, 1985 ; Liew et al., 1990). Liew and his co-workers (1993) have recently extended the polynomial-based Ritz formulation to Mindlin’s plates. The advantage of these polynomials, as pointed out (Liew et al., 1990), is mainly due to the ease of generation and numerical implementation that greatly enhance the computational efficiency of the Ritz method. With this merit, it is envisaged that the formidable efforts involved in three- dimensional computations can be drastically reduced.
The complexity of thr~-dimensional analysis has led to the development of refined plate theories (Levinson, 1980; Reddy, 1984). These theories aim to reduce the dimensions of problems (and thus the determinant size of the eigenvalue equation) from three to two by addressing the quantities of interest, such as membrane forces, bending moments and shear forces, in terms of certain averages over the displacement across the smaller dimension, i.e. thickness dimension. These simplifi~tions are inherently erroneous and result in erroneous responses of structures. The discrepancies become significant when the thickness dimension increases to that comparable to the other leading plate dimensions,
Very often, plates used in industrial situations are moderately thick or very thick. They can be found in many engineering applications such as civil, structural, military and marine structures. Due to the practical importance, solution methods that can provide reliable and accurate results will be of tremendous help to our understanding of their dynamic characteristics. Moreover, in order to assess the accuracy of simple plate theories, accurate three-dimensional solutions to these problems will be of advantage if available for direct comparison. Such closed-form exact solutions have been derived notably by Srinivas et al.
3357
3358 K. M. LIEW et al.
(1970) and Wittrick (1987) for simply supported thick rectangular plates. For other boundary conditions, solutions based on three-dimensional elasticity theory are available only for cantilevered parallelepipeds due to the excellent work of Leissa and Zhang (1983).
Three-dimensional analysis of thick plates subject to other support conditions is not found in current literature. This study attempts to fill this apparent void by providing sets of first known frequency data and three-dimensional deformed mode shapes for thick plates subject to different boundary constraints. The natural vibratory motions have been manifested into three-dimensional deformed mesh plots to depict vividly the contribution of each displacement component to the deformed geometry. This study covers five practical sets of boundary conditions, various aspect ratios and side-to-thickness ratios. The reported deflection contour plots and three-dimensional deformed mode shapes serve to enhance our understanding and visualization of the vibratory characteristics of these plates. The present solutions supplement the existing database, and further serve as the benchmark for researchers to examine the merits of new development in refined plate theories.
2. THEORETICAL FORMULATION
2.1. Three-dimensional elasticity theory Consider a homogeneous, thick, rectangular plate, as shown in Fig. 1, bounded by
the edges --a/2 < x < a/2, -b/2 < y ,< b/2 and -t/2 G z < t/2. Stress free surfaces are assumed at z = -t/2 and t/2. The origin of the coordinate system is taken to be at the geometric center of the plate. To determine the vibration frequencies and mode shapes of the plate, the Ritz energy procedure is followed. The deflections are decomposed into three orthogonal displacement components, U, P and w parallel to the x-, y- and z-directions, respectively.
For small amplitude vibratory motion, the strain energy of elastic plates in terms of the displacement fields, {u, v, w)’ can be expressed as :
ILI = ; [s]TD[s]dV, s U
where [E] is the strain tensor :
a/ax 0 0
0 wy 0 0 0 a/&
ajay alax 0
a/az 0 alax
0 ajaz fv%
(1)
(2)
7 X35 Reference coordinates and dimensions of a homogeneous, thick rectangular Fig. 1.
Three-dimensional vibration analysis
For an isotropic material, the elasticity matrix D is given by :
E(l -v)
D = (1+v)(l-29
symmetric
1 v/(1-v) v/(1-v) 0 0 0
1 v/(1-v) 0 0 0
1 0 0 0
(1-2v) o o 2(1-v)
(1-2v) o 2(1-v)
where E is the modulus of elasticity and v is Poisson’s ratio. The kinetic energy for free vibration is given by,
T = f (ti*+ti*+r+*)da, s ”
(1-2v)
2(1-v)
3359
> (3)
(4)
where ti, ti and 3 are the corresponding velocity components in the x-, y- and z-directions, respectively, and p is the mass density per unit volume.
The boundary conditions for the stress free surfaces at z = -t/2 and t/2 can be expressed in terms of the spatial displacement components as follows,
(W
at z( - t/2 ; t/2), (5b)
at z( - t/2 ; t/2),
where e is the dilatation given by :
e = (&; $; 7${u,,,,)T, and A and G are the Lame constants given by :
vE E
“=(l+v)(l-2v); G=2(,+v).
(5c)
(6)
(7)
The displacement vectors {u, v, w} T are each expressed in terms of functions in x, y, z and C:
u = u(x, y, z, f) = U(x, y, z)eiw”,
u = U(x, y, z, 4) = V(x, y, z)eio’,
w = w(x, y, z, 7) = W(x, y, z)eio’,
(84 @b) (84
in which F is time and w denotes the frequency of vibration.
3360
2.2. The Ritz energy method
K. M. LIEW et al.
For simplicity and generality, a nondimensional coordinate system is introduced :
(=i; q=% and cc:, (9)
where a and b are the length and width of the rectangular planform (see Fig. I), and t is the thickness dimension of the plate. The in-plane and transverse deflection amplitude
functions, u(5, V, 0, k’(5, r, 0 and W<, rl, i) in eqns (8) can be approximated by a set of separable orthogonal polynomial functions in 5, q and [, respectively,
I J K
P Q R
W5~'I~O = c c Cc,,,~,(r)rc/,(rl)xw,(r), uw
where cuijk, Gmn and G,, are the undetermined coefficients and 4, $ and x are the corresponding polynomial functions. The functions are basically sets of orthogonally gen- erated polynomials to be discussed in due course.
Let the maximum energy functional of the thick rectangular plate be,
E = Unax-TInax, (11)
in which Q,,,, the maximum strain energy, and T,,,, the maximum kinetic energy, occur respectively at maximum deflection and maximum velocity in a vibratory cycle.
Substituting the spatial displacement functions given in eqn (10) into the energy functional and minimizing with respect to the unknown coefficients according to the Ritz procedure :
a[F/a&jk = 0,
awk, = 0,
aqac,,,, = 0,
leads to the governing eigenvalue equation :
Ll [Ku,1 Ll
1 [ FIU”1 0 0
iKP”l Knvl --x2 PLI 0 symmetric KWVI symmetric [MA
in which [K,] is the stiffness matrix ; wij] is the diagonal mass matrix ; and {C,>, (C,> and {C,} are the column vectors of the unknown coefficients which can be expressed in the following forms :
{G> =
C UI I I
C ull2
C UIIK
C ul2l
C ulJK
C IA211
C ulJK
; {C”} =
C L'lll
C VII2
C 111 IN
C “121
C VIMN
C 021 I
C uLMN
and (C,} =
C WI I I
C WI12
Cw,IR
C WI21
C wlQR
C Iv211
C wPQR
(14)
WI
Three-dimensional vibration anaiysis 3361
The expressions of the various elements in the stiffness matrix [K] and mass matrix are given by :
(13
wid
(W
(15i)
where r, s = 0,l and subscripts a, /3 represent the corresponding displacement fields in U, tt and W. Solving the characteristic eigenvalue problem defined by eqn (13) yields the fre- quency parameter, X = ou(p/E) “’ This parameter can be expressed in the form of . A = (ob2/z2)( pt/D) ‘I2 by multiplying 1 with the following factor,
1 =-${12(1-v~)j”zz. (17)
2.3. Or~~~~~n~~~un~ii~ns and boundary conditions The set of polynomial functions representing the spatial displacements of the con-
tinuum is orthogonalized via the Gram-Schmidt process (Chihara, 1978). This is illustrated for (p(t) as follows,
cba+~(r) = (st~)-e,)~,(5)-BR#k_1(r); k= 1,2,3,..., (18)
3362
where
K. M. LFEW et al.
(194
and
fO.5
J _o,5 A&3 d< =- -k - s 0.5 (19b)
_o,54:~d5)d5
The polynomial r#~,,(Lj) is defined as zero. Note that the coefficients, Ok and & for the subsequent term can be computed from the two previous polynomials.
The set of polynomials generated satisfies the following orthogonality condition :
(20)
where 6, is the Kronecker delta and the value of ni, depends on the normalization used. The generating function, g(t), is chosen such that the higher terms continue to satisfy the geometric boundary conditions. Polynomials in the y- and z-directions can also be con- structed by following a similar procedure.
In this analysis, the boundary conditions are chosen such that xz- and yz-planes are the symmetry planes. This symmetry consideration permits the classification of the vibration modes into four distinct symmetry classes. Namely, doubly symmetric (SS) modes, symmetric-antisymmetric modes (SA), antisymmetric-symmetric modes (AS) and double antisymmetric modes (AA) about these xz- and yz-planes. The apparent computational advantage of this classification is that it reduces the determinant size of the resulting eigenvalue equation to that manageable by smaller computers.
The basic function in the z-direction, x(c), is chosen to be unity [x(c) = 11. It satisfies the essential geometric requirement of the stress free surfaces at z = -r/2 and t/2. In x- or y-direction, it depends on the boundary constraints and symmetry classes of vibration, the basic functions in these directions take on different forms. For a stress free surface on a straight edge, x = constant, the boundary requirements are ;
C,Y = 0, r,,. = 0 and z.,.; = 0. (21)
On the other hand, for a simple support on the same straight edge, the corresponding boundary conditions are :
w = 0, v=O and gI=O. (22)
Finally, for a fully clamped edge, the boundary conditions are :
w = 0, v=O and u=O. (23)
In the Ritz formulation, it is sufficient to choose admissible functions that satisfy only the essential geometric boundary conditions. Tables 1 (a) and 1 (b) summarize the respective basic functions used in each of the symmetry classes at different boundary conditions.
Three-dimensional vibration analysis 3363
Tabie l(a). Notations for various boundary conditions and the corresponding basic functions in the x-direction
Boundary conditions
Free-frm (F-F)
Simply-simply supports (S-S)
Clamped-clamped (C-C) 4,,(T) = 63-0.25r 9.,(E) = 524.25 (b,,(5) = 53-0.25< (p,,(e) = 52-0.25 c&,(t) = t3-0.25< &,(<) = ('-0.25
Table 1 (b). Basic functions for U, V and W components at different symmetry classes
Generating function g(t ; tj ; <) = ((I; rj’; i)
3. VERIFICATION AND NUMERICAL APPLICATIONS
The three-dimensional continuum method described in the previous section has been applied to compute the nondimensional frequency parameters, 2 = (~~2/~2)(~~/D)1~2 and vibration mode shapes for the homogeneous, thick, rectangular plates subject to arbitrary combinations of boundary constraints. The notation for boundary conditions, for instance, SFCF denotes a rectangular plate with edges x = -a/2, y = -b/2, x = a/2 and y = b/2 having the simply supported, free, clamped, and free support conditions, respectively. This boundary convention is used throughout the present study.
3.1. Convergence and comparison of eigenvalues Table 2(a) presents the convergence studies on the eigenvalues ,I for simply supported
square plates. The rate of convergence is examined for the plates with thickness ratios, t/b = 0.01, 0.1 and 0.5. The numbers of terms used in the triple infinite series of U, V and W are stepped steadily from 4 x 4 x 3 to higher terms to demonstrate the monotonic downward convergence behavior of 1.
A careful scrutiny of the convergence table reveals that as the thickness ratio increases, the number of terms needed for the polynomial function in the thickness direction (z- direction) increases proportionally. For a relatively thin plate (t/b = 0.01), 4 terms in the z-direction are sufficient to give converged results up to five significant figures. However, it needs as high as seven terms for a plate with thickness ratio t/b = 0.5 to achieve the same degree of accuracy. On the other hand, the number of terms needed in the shape functions along the plate surface (x- and y-directions) reduces drastically from 7 x 7 to 4 x 4 terms as the plate thickness dimension becomes significant. Consequently, the overall determinant size involved in the solution of thicker rectangular plates becomes smaller (det = 336 for a plate with t/b = 0.5 as compared to det = 558 for a plate with t/b = 0.01). To investigate the convergence behaviors with respect to the boundary conditions, frequency parameters for a moderately thick square plate are computed (t/b = 0.1). The rate of convergence for each boundary condition is presented in Table 2(b). The convergence patterns are found to be invariant with respect to the boundary constraints.
From the preceding convergence studies, it was established that for very thin plates (t/b = O.Ol), 9 x 9 x 4 terms were needed to achieve a reasonable convergence. For plates
Thi
ckne
ss
ratio
, t/b
Tab
le
2(a)
. C
onve
rgen
ce
of f
requ
ency
pa
ram
eter
s,
1 =
(ob*
/r?)
(pt/~
) “’
for
a th
ick
squa
re
plat
e w
ith
SSSS
bo
unda
ry
cond
ition
,
Sym
met
ry
clas
ses
and
mod
e nu
mbe
r T
erm
s in
D
et
X>Y
,Z
size
ss
-1
ss-2
ss
-3
SA-l
t SA
-2P
SA-3
t A
A-l
A
A-2
A
A-3
0.01
4X
4X3
144
1.99
94
9.98
64
9.98
64
4.99
61
12.9
75
17.6
31
7.99
00
20.5
63
4X4X
4 19
2 1.
9993
9.
9847
9.
9847
4.
9957
12
.972
17
.625
7.
9889
20
.555
4X
4X5
240
1.99
93
9.98
47
9.98
47
4.99
57
12.9
72
17.6
25
7.98
89
20.5
55
5X5X
4 30
0 1.
9993
9.
9826
9.
9826
4.
9956
12
.971
16
.971
7.
9888
19
.950
6x
6x4
432
1.99
93
9.98
26
9.98
26
4.99
56
12.9
71
16.9
50
7.98
88
19.9
31
7X7X
4 58
8 1.
9993
9.
9826
9.
9826
4.
9956
12
.971
16
.950
7.
9888
19
.930
8X
8X4
768
1.99
93
9.98
26
9.98
26
4.99
56
12.9
71
16.9
50
7.98
88
19.9
30
9X9X
4 97
2 1.
9993
9.
9826
9.
9826
4.
9956
12
.971
16
.950
7.
9888
19
.930
0.1
4X4X
3 14
4 1.
9403
8.
7672
8.
7672
4.
6546
6.
5234
11
.037
7.
1754
13
.047
5X
5X4
300
1.93
42
8.66
31
8.66
31
4.62
24
6.52
34
10.8
82
7.10
38
13.0
47
6x6x
4 43
2 1.
9342
8.
663
1 8.
6631
4.
6224
6.
5234
10
.882
7.
1038
13
.047
5X
5X5
375
1.93
42
8.66
17
8.66
17
4.62
22
6.52
34
10.8
79
7.10
30
13.0
47
5x5~
6 45
0 1.
9342
8.
6617
8.
6617
4.
6222
6.
5234
10
.879
7.
1030
13
.047
5X
5X7
525
1.93
42
8.66
17
8.66
17
4.62
22
6.52
34
10.8
79
7.10
30
13.0
47
5x5x
8 60
0 1.
9342
8.
6617
8.
6617
4.
6222
6.
5234
10
.879
7.
1030
13
.047
5X
5X9
675
1.93
42
8.66
17
8.66
17
4.62
22
6.52
34
10.8
79
7.10
30
13.0
47
0.5
4X4X
3 14
4 1.
2964
1.
8451
2.
9678
1.
3047
2.
4255
2.
9174
2.
6094
2.
6094
4X
4X4
192
1.26
13
1.84
51
2.93
37
1.30
47
2.34
40
2.91
74
2.60
94
2.60
94
5X5X
4 30
0 1.
2613
1.
8451
2.
9337
1.
3047
2.
3449
2.
9174
2.
6094
2.
6094
4X
4X5
240
1.25
90
1.84
51
2.93
26
1.30
47
2.33
16
2.91
74
2.60
94
2.60
94
4x4x
6 28
8 1.
2590
1.
8451
2.
9325
1.
3047
2.
3312
2.
9174
2.
6094
2.
6094
4X
4X7
336
1.25
90
1.84
51
2.93
25
1.30
47
2.33
12
2.91
74
2.60
94
2.60
94
4X4X
8 38
4 1.
2590
1.
8451
2.
9325
1.
3047
2.
3312
2.
9174
2.
6094
2.
6094
4X
4X9
432
1.25
90
1.84
51
2.93
25
1.30
47
2.33
12
2.91
74
2.60
94
2.60
94
20.5
63
20.5
55
20.5
55
19.9
50
19.9
31
19.9
30
19.9
30
19.9
30
?;:
13.0
47
5:
13.0
47
13.0
47
F E
13.0
47
13.0
47
2
13.0
47
e
13.0
47
13.0
47
3.24
89
3.13
61
3.13
61
3.10
93
3.10
81
3.10
80
3.10
80
3.10
80
t Fo
r a
thic
k sq
uare
pl
ate
with
SS
SS
boun
dary
co
nditi
on,
the
freq
uenc
y pa
ram
eter
s,
rZ, f
or
SA a
nd
AS
mod
es
are
iden
tical
.
Tab
le 2
(b).
Con
verg
ence
of
fre
quen
cy
para
met
ers,
1
= (
~&~/
r?)(
pr/D
) ‘I
*, fo
r a
thic
k sq
uare
pla
te w
ith d
iffe
rent
bo
unda
ry
cond
ition
s (r
/b =
0.1
)
Bou
ndar
y T
erm
s in
co
nditi
ons
-G_Y
,r
Sym
met
ry c
lass
es a
nd m
ode
num
ber
Det
si
ze
ss-1
ss
-2
ss-3
SA
-1
SA-2
SA
-3
AS-
1 A
s-2
AS-
3 A
A-l
A
A-2
A
A-3
SFSF
4X
4X3
144
5X5X
4 30
0 6X
6X4
432
SXSX
S 37
s 5X
5X6
450
5X5X
1 52
s 5x
5~8
600
5X5X
9 67
5 C
FCF
4X4X
3 14
4 Sx
Sx4
300
6x6x
4 43
2 5X
5X5
37s
5x5~
6 45
0 5X
5X7
52s
5x5~
8 60
0 5X
5X9
675
scsc
4X
4X3
144
5X5X
4 30
0 6x
6x4
432
SxSx
5 37
s 5x
5~6
450
sxsx
7 52
s 5x
5~8
600
5X5X
9 67
5 cc
cc
4X4X
3 14
4 Sx
Sx4
300
6x6x
4 43
2 5X
5X5
375
5x5~
6 45
0 5X
5X7
52s
5x5~
8 60
0 5x
sx9
675
0.95
91
3.46
3s
7.89
39
3.71
58
6.39
60
13.0
47
1.56
78
4.94
00
6.79
32
4.38
14
6.52
34
9.56
34
0.95
71
3.43
70
7.80
07
3.69
22
6.31
85
12.8
69
1.56
OS
4.
9400
6.
7297
4.
3459
6.
5234
9.
4286
0.
9571
3.
4362
7.
8001
3.
6920
6.
3168
12
.854
1.
5604
4.
9400
6.
7293
4.
3456
6.
5234
9.
4275
0.
957
1 3.
4361
7.
8000
3.
6920
6.
3 16
5 12
.854
1.
5653
4.
9400
6.
7293
4.
3455
6.
5234
9.
4274
0.
9571
3.
4361
7.
7991
3.
6919
6.
3162
12
.850
1.
5603
4.
9400
6.
7287
4.
3454
6.
5234
9.
4259
0.
9571
3.
4361
7.
7990
3.
6919
6.
3161
12
.849
1.
5603
4.
9400
6.
7287
4.
3454
6.
5234
9.
4257
0.
9571
3.
4361
7.
1990
3.
6919
6.
3161
12
.849
1.
5603
4.
9400
6.
7287
4.
4354
6.
5234
9.
4257
0.
9571
3.
4361
7.
7990
3.
6919
6.
3 16
1 12
.849
1.
5603
4.
9400
6.
7287
4.
3454
6.
5234
9.
4257
2.13
05
3.97
77
9.96
13
5.48
16
7.50
64
10.6
48
2.48
21
5.9.
547
7.05
30
5.93
49
10.2
66
10.9
65
2.10
76
3.92
68
9.74
18
5.39
30
7.36
74
10.6
40
2.45
16
5.95
15
6.97
08
5.83
46
10.0
80
10.9
62
2.10
58
3.92
46
9.73
62
5.38
94
7.36
25
10.6
37
2.44
98
5.95
02
6.96
95
5.83
09
10.0
76
10.9
62
2.10
51
3.92
40
9.73
43
5.38
80
7.36
11
10.6
35
2.44
91
5.94
96
6.96
92
5.82
94
10.0
7s
10.9
62
2.10
54
3.92
40
9.73
09
5.38
68
7.35
93
10.6
36
2449
3 5.
9sO
O
6.96
84
5.82
82
10.0
72
10.9
62
2.10
51
3.92
34
9.72
78
5.38
60
7.35
82
10.6
36
2.44
89
5.95
00
6.96
78
5.82
73
10.0
71
10.9
62
2.10
50
3.92
34
9.72
78
5.38
59
7.35
82
10.6
36
2.44
89
5.9s
OO
6.
9678
5.
8273
10
.071
10
.961
2.
1050
3.
9234
9.
7276
5.
3859
7.
3581
10
.636
2.
4489
5.
9500
6.
9678
5.
8272
10
.070
10
.961
2.
7490
8.
9733
10
.624
5.
0539
6.
5234
12
.455
6.
1549
ll
S21
11.6
58
8.18
62
13.0
47
15.4
92
2.72
12
8.85
56
10.3
99
5.00
31
6.52
34
12.1
95
6.05
90
11.4
59
11.5
18
8.05
82
13.0
47
15.4
90
2.71
98
8.85
51
10.3
95
5.00
22
6.52
34
12.1
91
6.05
61
11.4
58
11.5
17
8.05
58
13.0
47
15.4
90
2.71
92
8.85
49
10.3
93
5.00
19
6.52
34
12.1
90
6.05
49
11.4
57
11.5
17
8.05
49
13.6
47
15.4
90
2.71
93
8.85
32
10.3
89
5.00
15
6.52
34
12.1
84
6.05
32
11.4
52
11.5
17
8.05
24
13.0
47
15.4
90
2.71
89
8.85
28
10.3
86
5.00
11
6.52
34
12.1
81
6.05
23
11.4
51
11.5
17
8.05
14
13.0
47
15.4
90
2.71
89
8.85
28
10.3
86
5.00
11
6.52
34
12.1
81
6.05
23
11.4
51
11.5
17
8.05
13
13.0
47
15.4
90
2.71
88
8.85
28
10.3
86
5.00
11
6.52
34
12.1
81
6.05
22
11.4
51
il.5
17
8.05
13
13.0
47
15.4
90
3.36
75
10.7
46
10.8
47
6.46
08
12.5
31
13.0
18
6.46
08
12.5
31
13.0
18
9.08
70
14.8
71
17.7
62
3.32
54
10.5
13
10.6
13
6.35
36
12.5
25
12.7
24
6.35
36
12.5
25
12.7
24
8.91
47
14.8
71
17.3
0s
3.32
3 1
10.5
08
10.6
08
6.35
01
12.5
23
12.7
19
6.35
01
12.5
23
12.7
19
8.91
04
14.8
70
17.2
99
3.32
22
10.5
07
10.6
06
6.34
87
12.5
22
12.7
18
6.34
87
12.5
22
12.7
18
8.90
89
14.8
70
17.2
97
3.32
23
10.5
02
10.6
03
6.34
70
12.5
22
12.7
10
6.34
70
12.5
22
12.7
10
8.90
49
14.8
70
17.2
80
3.32
16
10.4
99
10.5
98
6.34
58
12.5
22
12.7
06
6.34
58
12.5
22
12.7
06
8.90
31
14.8
70
17.2
75
3.32
16
10.4
99
10.5
98
6.34
57
12.5
22
12.7
06
6.34
57
12.5
22
12.7
06
8.90
31
14.8
70
17.2
75
3.32
15
10.4
99
LO
.598
6.
3457
12
.522
12
.706
6.
3457
12
.522
12
.706
8.
9030
14
.870
17
.275
3366 K. M. LIEW et al.
with other thickness ratios, it has been found that similar convergence could be attained with polynomials of 6 x 6 x 8 terms. These established polynomial sets are used throughout the present computations.
TO validate the accuracy of the present method, comparison studies have been carried out for cases where the solutions for exact three-dimensional analysis and simple plate theories, notably the classical plate theory (CPT) and the shear deformable plate theory (SDPT) are available. For this purpose, thick rectangular plates with simply supported (SSSS) and cantilevered (CFFF) boundaries have been considered. Table 3(a) lists the computed frequency parameters for the simply supported plate together with the known solutions of approximate theories and existing exact three-dimensional analysis. The fre- quency data for the classical plate theory are obtained from the work of Leissa (1973) and the Mindlin theory’s results are quoted from the work of Dawe et al. (1985) that used a shear correction factor K = 7c2/12. The three-dimensional results are extracted from the excellent work of Srinivas et al. (1970) who pioneered in the exact three-dimensional analysis of simply supported thick plates. From Table 3(a), it is observed that at thickness ratio t/b = 0.01, the frequency parameters obtained from the approximate theories and the present three-dimensional analysis are in close agreement. At a higher thickness ratio, t/b = 0.1, the classical plate theory, however, tends to give much higher eigenvalues than those of the Mindlin theory and the present three-dimensional analysis. This is attributed to the assumption made in the classical plate theory that a line normal to the plate middle surface remains normal during the deformation. This assumption neglects the effects of through thickness shear deformation and rotary inertia which results in over-estimating the plate stiffness. On the other hand, in the Mindlin plate theory, the shear deformation effects are ingeniously accounted for in the formulation. A shear correction factor, however, is introduced to achieve a more accurate shear strain distribution through the plate thick- ness. While in the present three-dimensional analysis and the exact three-dimensional analysis of Srinivas et al. (1970), this effect has been implicitly incorporated in the elasticity formulation. This has resulted in excellent agreement between the results by the present method with those of Srinivas et al. (1970).
For the cantilevered thick rectangular plate, the frequency parameters are compared with those presented by Leissa and Zhang (1983). In this study, the symmetry classification adopted by Leissa and Zhang (1983) has been followed. The first five frequency parameters given in Table 3(b) for each symmetry class are compared with those values of Leissa and Zhang (1983). It can be seen that excellent agreement has also been obtained between both methods.
3.2. Results and discussion Having established the rate of convergence and degrees of accuracy of the present
formulation, the three-dimensional continuum method is applied to compute the non- dimensional frequency parameters, 2 = (~b*/?)(pt/D) “* for rectangular plates with vari- , ous geometric configurations and boundary conditions: SSSS, SFSF, CFCF, SCSC and CCCC. Numerical results are presented in Tables 4(a)-(e) for plates with aspect ratios ranging from 0.5-2.0; and side-to-thickness ratios covering the range of 0.014.5. The influences of plate aspect ratio and thickness ratio on the vibration frequencies are examined. The first three nondimensional frequency parameters, 1, corresponding to each distinct symmetry class were calculated. For all these cases where Poisson’s ratio is needed, it is taken to be v = 0.3.
From these tabulated results, it is observed that for plates with prescribed boundary condition and aspect ratio, the nondimensional frequency parameter, 2, decreases as the side-to-thickness ratio, t/b, increases. Conversely, for plates with constant thickness, the frequency parameter decreases for plates with higher aspect ratio, a/b. Comparing the frequency parameters, 1, for plates with different boundary conditions, it is evident that plates with more constraints imposed on the boundaries have a higher value of A. For example, the frequency parameters of the SFSF plate are generally many times higher than the corresponding modes of the CCCC plate with the same geometrical specifications.
Tab
le 3
(a).
Com
pari
son
of f
requ
ency
pa
ram
eter
s,
il =
(wb’
/n’)
(pt/~
) ‘P
for
a t
hick
squ
are
plat
e w
ith S
SSS
boun
dari
es
,
Mod
e se
quen
ce
num
ber
l/b
Met
hod
of s
olut
ion
1 (S
S-1)
2
(SA
-1)
3 (A
S-l)
4
(AA
- 1)
5
(SS-
2)
6 (S
S-3)
0.01
0.1
Cla
ssic
al t
heor
y?
2.00
0 5.000
5.000
8.00
0 10
.006
10
.000
S
DP
T R
itz m
etho
df
I.99
9 4.
995
4.99
5 7.
988
9.98
1 9.
98 l
Pres
ent
thre
e-di
men
sion
al
anal
ysis
1.
9993
4.
9956
4.
9956
7.
9888
9.
9826
9.
9826
C
lass
ical
the
ory?
2.
000
5.00
0 5.
000
8.00
0 lO
.O@
O
10.0
00
SD
PT
Ritz
met
hodS
1.
931
4.60
5 4.
605
7.06
4 8.
605
8.60
5 T
hree
-dim
ensi
onal
ex
act
anal
ytic
al
solu
tion§
1.
9342
4.
6222
4.
6222
7.
1030
8.
6618
8.
6618
Pr
esen
t th
ree-
dim
ensi
onal
an
alys
is
1.93
42
4.62
22
4.62
22
7.10
30
8.66
17
8.66
17
fLei
ssa
(197
3).
.$D
awe
et a
l. (1
985)
with
a s
hear
cor
rect
ion
fact
or
K =
n2/
12.
§Sri
niva
s ef
al.
(197
0).
Tab
le 3
(b).
Com
pari
son
of f
requ
ency
pa
ram
eter
s,
I=
(~~‘
/~‘~
(~f’
/D)
‘j2,
for
a ca
ntile
vere
d th
ick
squa
re
plat
e w
ith f
/b’
= 0
.5
Sym
met
ry
clas
ses
Sour
ce o
f da
ta
I
Mod
e se
quen
ce
num
ber
ss
SA
AS
AA
Lei
ssa
and
Zha
ngt
1.06
89
1.87
33
Pres
ent
thr~
-dim
ensi
onal
an
alys
isj
1.06
87
1.87
22
Lei
ssa
and
Zha
ng
0.29
96
1.11
44
Pres
ent
thre
e-di
men
sion
al
anal
ysis
0.
2976
1.
1087
L
eiss
a an
d Z
hang
0.
4469
1.
1882
Pr
esen
t th
ree-
dim
ensi
onal
an
alys
is
0.44
49
1.18
63
Lei
ssa
and
Zha
ng
0.52
79
1.48
63
Pres
ent
thre
e-di
men
sion
al
anal
ysis
0.
5263
I .
4795
2 3
2.14
24
2.13
94
1.52
53
1.52
22
2.05
45
2.03
69
2.30
28
2.27
72
4
2.92
00
2.81
93
2.27
26
2.25
36
2.85
13
2.70
05
2.54
29
2.49
19
5
3.17
84
3.12
60
2.53
27
2.36
08
3.19
62
3.16
16
2.99
16
2.84
82
tThe
va
lues
pr
esen
ted
in t
his
case
are
con
vert
ed
from
th
e fr
eque
ncy
resu
lts
for
a pa
ralle
lepi
ped
of c
on-
figu
ratio
n C
in
the
pape
r by
Lei
ssa
and
Zha
ng
(198
3).
$For
th
is c
ase,
the
fol
low
ing
tra~
fo~a
tion
has
been
don
e:
a’ =
t,
b’ =
b a
nd r
’ =
a.
Thi
s is
to
enab
le
the
com
pari
son
betw
een
the
pres
ent
t&e-
dim
ensi
onal
an
alys
is
resu
lts
and
that
of
the
ref
eren
ce
for
each
sym
met
ry
clas
s.
Tab
le
4(a)
. Fr
eque
ncy
para
met
ers,
a
= (~
~‘/~
‘)(~
~~~)
“’
for
a th
ick
plat
e w
ith
SSSS
bo
unda
ry
cond
ition
,
Asp
ect
Thi
ckne
ss
ratio
, a/
b ra
tio,
r/b
ss-I
ss
-2
ss-3
SA
-I
Sym
met
ry
clas
ses
and
mod
e nu
mbe
r
SA-2
SA
-3
AS-
l A
S-2
AS-
3 A
A-l
A
A-2
A
A-3
0.5
0.01
0.
1 0.
2 0.
3 0.
4 0.
5
1.0
0.01
0.
1 0.
2 0.
3 0.
4 0.
5
1.5
0.01
0.
1 0.
2 0.
3 0.
4 0.
5
2.0
0.01
0.
1 0.
2 0.
3 0.
4 0.
5
4.99
56
12.9
71
28.8
54
16.9
50
24.8
92
40.7
10
7.98
88
4.62
22
10.8
79
14.5
87
6.52
34
13.6
44
18.6
58
7.10
30
3.89
91
1.29
34
8.09
26
3.26
17
9.76
36
9.78
51
5.65
24
3.24
06
4.86
23
6.20
99
2.17
45
6.52
34
7.34
3 1
4.34
89
2.72
54
3.64
67
4.97
05
I .63
09
4.39
12
4.89
26
3.26
17
2.33
12
2.91
74
3.88
74
1.30
47
2.91
74
3.91
40
2.60
94
1.99
93
9.98
26
9.98
26
4.99
56
12.9
71
16.9
50
4.99
56
1.93
42
8.66
17
8.66
17
4.62
22
6.52
34
10.8
79
4.62
22
1 .I1
58
4.61
27
6.68
68
3.26
17
3.89
91
7.29
34
3.26
17
I .589
5 3.
0752
5.
1152
2.
1745
3.
2406
4.
8623
2.
1745
I.
4131
2.
3064
3.
7727
1.
6309
2.
7254
3.
6467
1.
6309
1.
2590
1.
8451
2.
9325
1.
3047
2.
3312
2.
9174
1.
3047
1.
4441
4.
9956
9.
4289
2.
7764
8.
0996
10
.758
4.
4410
1.
4096
4.
6222
7.
8402
2.
6538
6.
5234
7.
1916
4.
1415
1.
3209
3.
8991
3.
9201
2.
3747
3.
2617
5.
4362
2.
1745
I .2
088
2.61
34
3.24
06
2.07
31
2.17
45
3.62
41
1.44
96
1.09
54
1.96
00
2.72
54
1.63
09
1.80
64
2.71
81
1.08
72
0.99
12
1.56
80
2.33
12
1.30
47
1.58
48
2.17
45
0.86
98
I .249
1 3.
2482
7.
2408
1.
9993
4.
9956
9.
9826
4.
2468
I .2
237
3.08
25
6.50
03
1.93
42
4.62
22
6.52
34
3.26
17
1.15
55
2.71
97
3.64
67
1.77
58
3.26
17
3.89
91
1.63
09
1.06
69
2.34
43
2.43
11
I .58
9S
2.17
45
3.07
52
1.08
72
0.97
48
1.82
33
2.02
3 1
1.41
31
1.63
09
2.30
64
0.81
54
0.88
81
1.45
87
1.76
24
1.25
90
I .304
7 1.
8451
0.
6523
19.9
30
39.7
24
19.9
30
31.8
23
51.5
35
13.0
47
IS.5
97
13.0
47
15.5
96
22.6
10
6.52
34
9.22
55
6.52
34
10.9
01
13.0
47
4.51
23
6.1
SO3
4.34
89
8.10
58
8.45
28
3.69
72
4.61
27
3.26
17
5.22
13
6.37
71
3.10
80
3.69
02
2.60
94
3.69
02
5.21
87
12.9
71
16.9
50
7.98
88
19.9
30
19.9
30
6.53
24
10.8
79
7.10
30
13.0
47
13.0
47
3.89
91
7.29
34
5.65
24
6.52
34
6.52
34
3.24
06
4.86
23
4.34
89
4.34
89
4.51
23
2.72
54
3.64
67
3.26
17
3.26
17
3.69
72
2.33
12
2.91
74
2.60
94
2.60
94
3.10
80
7.98
88
IS.0
71
5.71
19
11.0
90
17.7
23
4.34
89
7.10
30
5.28
34
8.69
79
9.49
88
3.53
98
5.65
24
4.34
89
4.38
18
6.52
34
2.97
19
4.34
89
2.89
93
3.59
67
4.34
89
2.51
68
3.26
17
2.17
45
2.99
98
3.26
17
2.16
31
2.60
94
1.73
96
2.55
14
2.60
94
6.24
32
10.2
32
4.99
56
7.98
88
12.9
71
3.97
1s
5.67
85
4.62
22
6.52
34
7.10
31
3.41
IO
4.
6644
3.
2617
3.
8991
5.
6524
2.
8747
3.
2617
2.
1745
3.
2406
4.
3489
2.
4409
2.
4463
1.
6309
2.
7254
3.
2617
1.
9570
2.
1018
1.
3047
2.
3312
2.
6094
Asp
ect
Thi
ckne
ss
ratio
, a/
b ra
tio,
t/b
ss-1
Tab
le
4(b)
. Fr
eque
ncy
para
met
ers,
i
= (~
b*/z
’)(~
t/D
) “*
for
a
thic
k pl
ate
with
SF
SF
boun
dary
co
nditi
on
,
Sym
met
ry
clas
ses
and
mod
e nu
mbe
r
ss-2
ss
-3
SA-I
SA
-2
SA-3
A
S-I
AS-
2 A
S-3
AA
-1
AA
-2
AA
-3
0.5
0.01
0.
1 0.
2 0.
3 0.
4 0.
5
1.0
0.01
0.
1 0.
2 0.
3 0.
4 0.
5
1.5
0.01
0.
1 0.
2 0.
3 0.
4 0.
5
2.0
0.01
0.
1 0.
2 0.
3 0.
4 0.
5
3.94
22
7.14
59
17.0
96
15.8
34
19.3
55
29.9
91
4.72
68
11.2
03
24.9
24
16.6
36
23.8
01
3.69
19
6.31
61
13.0
99
12.8
49
13.0
47
15.0
98
4.34
54
9.42
57
11.4
01
6.52
34
13.3
46
3.18
88
5.05
95
6.54
57
6.52
34
9.27
28
10.5
64
3.66
51
5.69
81
7.10
55
3.26
17
9.54
97
2.70
10
4.05
72
4.35
98
4.34
89
7.00
17
7.24
82
3.05
00
3.79
67
5.44
30
2.17
45
6.52
34
2.30
19
3.26
54
3.32
27
3.26
17
4.07
71
5.22
13
2.56
65
2.84
57
4.22
16
1.63
09
4.39
12
1.98
72
2.60
71
2.77
60
2.60
94
2.60
94
3.69
02
2.19
38
2.27
50
3.25
37
1.30
47
2.91
74
0.97
55
3.71
18
8.89
85
3.94
22
7.14
59
15.8
34
1.63
16
7.60
74
9.70
65
4.72
68
11.2
03
0.95
71
3.43
61
7.79
90
3.69
19
6.31
61
12.8
49
1.56
03
4.94
00
6.72
87
4.34
54
6.52
34
0.91
20
2.96
50
4.61
27
3.18
88
5.05
95
6.52
34
1.43
09
2.46
97
5.36
1 I
3.26
17
3.66
51
0.85
23
2.51
68
3.07
52
2.70
10
4.05
72
4.34
89
1.28
55
1.64
62
4.08
72
2.17
45
3.05
00
0.78
83
2.14
69
2.30
64
2.30
19
3.26
17
3.26
54
1.14
66
1.23
44
3.06
07
1.63
09
2.56
65
0.72
61
1.84
51
1.85
02
1.98
72
2.60
71
2.60
94
0.98
74
1.02
23
2.44
21
1.30
47
2.19
38
0.43
03
2.95
72
3.94
22
1.74
27
4.66
87
7.02
59
0.97
14
4.72
68
6.86
64
2.46
53
7.83
10
0.42
63
2.78
35
3.69
19
1.68
82
4.25
69
6.30
55
0.93
47
2.80
75
4.34
54
2.33
45
6.52
34
0.41
65
2.46
39
3.18
88
1.56
05
3.58
23
5.09
32
0.87
44
1.40
37
3.66
51
2.08
77
3.26
17
0.40
24
2.13
86
2.27
17
1.40
82
2.97
84
3.47
27
0.80
41
0.93
57
3.05
00
1.82
72
2.17
45
0.38
54
1.70
26
1.85
75
1.26
11
2.50
07
2.60
29
0.70
18
0.73
22
2.43
46
1.59
55
1.63
09
0.36
71
1.36
08
1.62
53
1.13
03
2.08
04
2.12
80
0.56
14
0.66
34
1.94
53
1.30
47
1.40
01
0.24
09
2.20
96
2.66
83
0.97
55
3.71
18
3.94
22
0.69
55
2.95
42
6.59
73
1.63
16
4.72
68
0.23
96
2.12
45
2.53
38
0.95
71
3.43
61
3.69
19
0.67
04
1.80
41
2.78
00
1.56
03
4.34
54
0.23
63
1.93
18
2.27
05
0.91
20
2.96
50
3.18
89
0.63
19
0.90
21
2.45
16
1.43
09
2.46
97
0.23
15
1.71
38
1.73
09
0.85
23
2.51
68
2.70
11
0.58
66
0.60
14
2.11
78
1.28
55
1.64
62
0.22
55
1.29
75
1.51
30
0.78
83
2.14
69
2.30
19
0.45
10
0.53
89
1 x30
7 1.
1466
1.
2344
0.
2185
1.
0372
1.
3409
0.
7261
1.
8451
1.
8502
0.
3608
0.
4918
1.
5947
0.
9874
1.
0223
37.9
58
17.8
04
9.78
51
7.17
80
+ 4.
8926
$
3.91
40
I cz
16.6
36
EJ
9.42
57
“1
5.69
81
8.
3.79
67
i 2.
8457
s.
2.
2750
$
8.59
19
k 6.
9192
8
3.55
39
5 2.
3686
$.
1.
7759
g.
1.42
03
7.60
74
4.94
00
3.26
17
2.17
45
1.63
09
1.30
47
Asp
ect
Thi
ckne
ss
ratio
, a/
h ra
tio,
t/b
S
-I
Tab
le
4(c)
. Fr
eque
ncy
para
met
ers,
i
= (*
~~‘~
~*)(
~~/~
) “’
for
a th
ick
plat
e w
ith
CFC
F bo
unda
ry
cond
ition
.
Sym
met
ry
clas
ses
and
mod
e nu
mbe
r
ss-2
ss
-3
SA-I
SA
-2
SA-3
A
S-l
AS-
2 A
S-3
AA
-l
AA
-2
AA
-3
0.5
0.01
9.
0106
11
.107
19
.556
0.
1 1.
2293
8.
5641
14
.376
0.
2 5.
0464
5.
8943
7.
9491
0.
3 3.
1236
4.
3710
5.
2917
0.
4 2.
9186
3.
4558
3.
9580
0.
5 2.
3917
2.
8549
3.
1536
I.0
0.01
2.
2482
4.
4083
12
.153
0.
1 2.
1050
3.
9234
9.
7276
0.
2 1.
7996
3.
1909
5.
8790
0.
3 1.
4999
2.
6089
3.
9111
0.
4 1.
2582
2.
1845
2.
9226
0.
5 1.
0723
1.
8710
2.
3245
1.5
0.01
0.
9962
3.
1862
5.
3948
0.
1 0.
9662
2.
9499
4.
8158
0.
2 0.
8900
2.
5456
3.
8222
0.
3 0.
7971
2.
1730
3.
0331
0.
4 0.
7064
1.
8718
2.
4780
0.
5 0.
6262
1.
6336
2.
0845
2.0
0.01
0.
5589
2.
7687
3.
028
I 0.
1 0.
5492
2.
6076
2.
8300
0.
2 0.
5224
2.
3071
2.
4174
0.
3 0.
4863
2.
0014
2.
0265
0.
4 0.
4470
1.
6983
I .
7567
0.
5 0.
4086
1.
4556
1.
5471
24.7
583
16.4
894
10.1
690
7.16
33
5.41
07
4.33
51
6.19
72
5.38
59
4.10
62
3.16
47
2.53
21
2.09
51
2.74
73
2.56
66
2.18
71
1.81
89
1.52
34
1.29
67
1.54
11
I.48
11
1.33
67
1.17
18
1.02
05
0.89
25
27.4
62
36.4
67
9.45
30
14.3
48
26.8
71
25.3
3%
31.1
16
43.6
40
17.9
75
21.5
21
7.48
81
10.8
13
12.4
65
16.7
95
19.9
84
21.7
21
10.7
92
11.2
04
5.18
93
6.23
56
7.46
74
10.3
94
10.8
86
I 1.9
34
7.20
55
7.58
83
3.81
80
4.15
86
5.54
62
7.26
53
7.35
40
7.95
45
5.49
09
5.68
78
2.98
62
3.11
98
4.30
02
5.45
37
5.65
36
5.96
60
4.43
36
4.54
83
2.44
17
2.49
62
3.38
77
4.36
86
4.56
90
4.77
31
8.85
31
18.1
71
2.67
43
8.06
S3
12.8
13
6.79
85
12.5
60
20.7
54
7.35
81
10.6
36
2.44
89
5.95
00
6.96
78
5.82
72
10.0
70
10.9
61
5.33
13
5.46
25
2.03
63
2.97
70
5.43
25
4.40
84
5.48
24
6.59
29
3.56
11
4.20
65
1.66
39
1.98
58
4.26
08
3.40
31
3.65
51
4.39
99
2.67
41
3.38
43
I .37
64
1.49
0 1
3.35
81
2.73
73
2.74
1 i
3.30
28
2.14
19
2.71
96
1.16
01
I.19
26
2.60
78
2.19
28
2.28
01
2.64
39
5.30
58
8.92
31
1.38
54
6.05
11
7.00
44
3.32
50
9.05
35
9.61
78
4.69
77
7.07
65
1.30
78
3.76
20
5.31
92
3.04
64
7.16
85
7.71
99
3.54
51
3.78
23
I.15
76
1.88
25
4.16
92
2.54
19
3.58
42
4.99
30
2.36
73
3.05
90
0.99
89
I .25
S9
3.29
38
2.09
14
2.38
93
3.33
1 I
1 .I7
78
2.53
33
0.85
85
0.94
25
2.68
62
1.74
55
1.79
18
2.49
98
I.42
38
2.14
48
0.74
19
0.75
44
2.25
31
I .43
33
I .48
54
2.00
07
4.01
82
5.01
21
0.90
99
3.66
43
6.65
54
2.08
45
5.69
85
7.81
48
3.65
75
4.53
24
0.86
77
2.65
60
3.35
96
I.95
73
5‘07
07
5.21
62
2.65
56
3.07
20
0.78
84
I .32
9l
2.80
90
1.71
35
2.60
83
4.04
62
1.77
28
2.56
14
0.69
94
0.88
68
2.31
83
I .46
72
1.73
89
2.86
76
1.33
11
2.16
48
0.61
47
0.66
56
I .94
09
I.25
81
I .30
42
2.15
14
1.06
60
1.85
93
0.52
38
0.54
00
1.65
59
I .04
34
1.08
90
1.72
15
Asp
ect
Thi
ckne
ss
ratio
, a/
b ra
tio,
r/b
ss-
I
Tab
le
4(d)
. Fr
eque
ncy
para
met
ers,
1
= (c
ob2/
n2)(
pr/D
)“*,
f o
r a
thic
k pl
ate
with
SC
SC
boun
dary
co
nditi
on
Sym
met
ry
clas
ses
and
mod
e nu
mbe
r
ss-2
ss
-3
SA-I
SA
-2
SA-3
A
S-1
AS-
2 A
S-3
AA
-l
AA
-2
AA
-3
0.5
0.01
0.
1 0.
2 0.
3 0.
4 0.
5
1.0
0.01
0.
1 0.
2 0.
3 0.
4 0.
5
1.5
0.01
0.
1 0.
2 0.
3 0.
4 0.
5
2.0
0.01
0.
1 0.
2 0.
3 0.
4 0.
5
5.54
34
5.00
11
4.08
24
3.32
94
2.11
22
2.35
86
2.93
51
2.71
88
2.29
20
1.90
61
1.60
85
1.38
36
2.53
99
2.36
61
2.00
40
1.66
35
1.39
66
1.19
45
2.41
57
2.25
62
1.91
63
1.59
07
1.33
29
1.13
69
15.6
50
12.1
81
8.48
98
6.34
51
4.88
24
3.78
34
10.3
39
8.85
28
6.13
14
4.07
98
3.04
94
2.42
59
5.54
34
5.00
11
4.08
24
3.18
49
2.38
60
1.90
54
3.96
05
3.63
11
3.03
02
2.51
20
2.09
30
1.67
31
33.6
53
19.7
49
9.86
90
6.56
25
5.02
62
4.14
44
13.0
64
10.3
86
6.75
64
5.26
88
4.26
00
3.51
14
12.4
03
9.56
15
4.78
02
3.32
94
2.77
22
2.35
86
7.67
57
6.76
64
4.18
89
2.79
20
2.11
75
1.81
90
17.2
08
6.52
34
3.26
17
2.17
45
1.63
09
1.30
47
5.54
34
5.00
11
3.26
17
2.17
45
1.63
09
1.30
47
3.55
74
3.27
42
2.74
47
2.17
45
1.63
09
1.30
47
2.93
51
2.71
88
2.29
20
1.90
61
1.60
85
1.30
47
26.7
48
13.7
49
9.78
51
6.52
34
4.39
12
2.91
74
15.6
50
6.52
34
4.08
24
3.32
94
2.77
22
2.35
86
8.50
53
6.52
34
3.26
11
2.28
13
1.92
69
1.65
83
5.54
34
5.00
11
3.26
17
2.17
45
1.63
09
1.38
36
44.6
41
19.2
42
9.79
42
7.35
52
4.89
26
3.91
40
17.2
08
12.1
81
8.48
98
6.34
51
4.39
12
2.91
74
13.7
20
7.42
93
5.80
69
4.59
59
3.73
11
2.91
74
10.3
39
6.52
34
4.08
24
3.32
94
2.77
23
2.35
86
9.57
27
8.05
13
6.02
68
4.66
70
3.77
00
3.10
06
7.02
28
6.05
22
4.59
49
3.56
33
2.87
83
2.31
06
6.58
66
5.70
58
4.34
00
3.35
57
2.70
03
2.20
23
6.43
78
5.58
77
4.25
3 1
3.28
41
2.63
75
2.18
71
23.6
81
15.4
90
7.74
75
5.16
63
3.87
53
3.14
78
14.1
74
11.4
51
5.76
49
3.84
66
2.88
68
2.40
48
9.57
27
8.05
13
5.49
09
3.66
51
2.75
12
2.24
88
8.05
26
6.86
72
5.18
89
3.63
95
2.73
22
2.19
26
40.3
50
17.0
20
11.2
30
7.92
46
5.62
07
4.05
26
21.0
52
11.5
17
8.27
44
5.78
45
4.31
10
3.40
78
16.1
74
10.9
64
6.02
68
4.66
70
3.77
00
3.05
09
11.6
10
9.59
20
5.45
15
4.03
21
3.26
54
2.63
94
20.8
68
13.0
47
6.52
34
4.34
89
3.26
17
2.60
94
9.57
27
8.05
13
6.02
68
4.34
89
3.26
17
2.60
94
7.65
68
6.55
48
4.96
26
3.85
57
3.12
16
2.51
14
7.02
28
6.05
22
4.59
49
3.56
33
2.87
83
2.31
06
34.6
98
15.9
40
10.9
93
8.14
06
5.22
13
3.69
02
20.8
68
13.0
47
6.52
34
4.66
70
3.77
00
3.10
06
12.4
06
10.1
78
6.27
05
4.18
27
3.13
83
2.60
94
9.57
21
9.05
13
5.76
49
3.84
66
2.88
68
2.40
48
56.5
11
23.3
57
13.0
47
8.45
28
6.39
63
5.21
87
23.6
81
15.4
90
7.74
75
5.16
63
3.87
53
3.14
78
20.8
68
12.5
31
6.52
34
4.34
89
3.26
17
2.61
23
14.1
74
11.5
04
6.02
68
4.34
89
3.26
17
2.60
94
Asp
ect
Thi
ckne
ss
ratio
, a/
b ra
tio,
t/b
ss-1
Tab
le
4(e)
. Fr
eque
ncy
para
met
ers,
3,
=
(wb*
/n’)
(pt/D
) ‘/’
fo
r a
thic
k pl
ate
with
C
CC
C
boun
dary
co
nditi
on
,
Sym
met
ry
clas
ses
and
mod
e nu
mbe
r
ss-2
ss
-3
SA-I
SA
-2
SA-3
A
S-l
AS-
2 A
S-3
AA
-l
AA
-2
AA
-3
0.5
0.01
9.
9497
18
.092
35
.148
25
.803
0.
1 7.
9005
13
.338
22
.805
17
.106
0.
2 5.
5132
8.
9311
12
.380
10
.591
0.
3 4.
0896
6.
5367
8.
1891
7.
4983
0.
4 3.
2202
5.
1244
6.
0222
5.
6509
0.
5 2.
6471
4.
2077
4.
6085
4.
5247
1.0
0.01
3.
6492
13
.315
13
.379
1.
4352
0.
1 3.
3215
10
.498
10
.598
6.
3451
0.
2 2.
1261
7.
3241
7.
4249
4.
7725
0.
3 2.
2158
5.
4605
5.
5398
3.
6101
0.
4 1.
8330
4.
3261
4.
3868
2.
9412
0.
5 1.
5496
3.
5174
3.
6235
2.
4396
1.5
0.01
2.
1393
6.
7420
12
.682
4.
2284
0.
1 2.
5431
5.
8780
10
.109
3.
8264
0.
2 2.
1476
4.
5495
7.
0982
3.
1151
0.3
1.78
14
3.56
16
5.12
84
2.51
98
0.4
1.49
39
2.89
65
3.83
22
2.01
94
0.5
1.27
48
2.42
62
3.04
35
1.75
58
2.0
0.01
2.
4933
4.
5396
8.
8403
3.
2278
0.
1 2.
3261
4.
1014
7.
5448
2.
9728
0.
2 I .
975
1 3.
3346
5.
7012
2.
4842
0.
3 1.
6420
2.
6998
4.
1330
2.
0487
0.
4 1.
3783
2.
2328
3.
0951
1.
7133
0.
5 1.
1770
1.
8906
2.
4688
1.
4600
33.5
23
21.3
43
11.2
92
7.53
21
5.76
86
4.66
76
16.6
82
12.5
22
6.21
35
4.18
79
3.14
31
2.51
51
10.2
09
8.50
57
4.82
72
3.22
15
2.41
71
1.93
48
6.41
86
5.65
14
4.21
11
2.80
91
2.10
77
1.68
66
49.6
39
12.8
74
25.5
54
46.1
65
28.6
43
40.5
18
60.8
31
22.5
27
9.93
67
16.8
44
17.7
40
18.7
05
23.6
36
24.8
87
13.2
35
6.85
31
8.43
09
11.4
64
11.6
34
11.8
36
14.3
02
8.92
45
5.06
51
5.62
29
8.25
69
7.89
79
8.27
74
9.51
49
6.68
69
3.97
96
4.21
69
5.86
84
5.92
94
6.37
82
7.10
29
5.33
30
3.26
64
3.37
23
4.28
87
4.75
04
5.13
61
5.55
89
21.2
66
7.43
52
16.6
82
21.2
66
10.9
53
24.4
41
24.5
43
12.7
06
6.34
57
12.5
22
12.7
06
8.90
30
14.8
70
17.2
75
8.69
80
4.71
25
6.21
35
8.69
80
6.41
85
7.43
70
9.13
42
6.43
22
3.61
01
4.18
79
6.43
22
4.84
93
4.95
91
6.08
60
5.03
22
2.94
12
3.14
31
5.03
22
3.72
00
3.85
80
4.55
44
3.65
45
2.43
96
2.51
51
3.65
45
2.97
65
3.19
26
3.62
85
14.0
27
6.70
02
10.4
41
18.0
42
8.08
35
13.7
68
21.9
63
9.64
35
5.78
99
8.60
10
11.5
68
6.85
31
10.9
36
12.6
55
6.25
71
4.39
54
5.19
75
6.28
70
5.13
34
6.33
21
7.74
59
4.11
86
3.39
24
3.87
10
4.79
01
3.95
25
4.22
36
5.16
37
3.82
23
2.12
32
2.90
59
3.82
98
3.16
89
3.17
55
3.86
92
3.12
83
2.26
17
2.32
57
3.09
77
2.53
59
2.64
17
3.08
94
11.7
81
6.48
36
8.43
48
12.5
19
7.20
10
10.2
05
15.3
64
8.41
58
5.62
21
7.13
17
10.1
24
6.18
23
8.45
77
11.8
03
4.43
50
4.21
64
5.33
56
5.63
17
4.61
63
5.90
89
6.22
18
3.51
06
3.30
01
3.76
05
4.11
25
3.61
40
3.94
29
4.75
99
2.86
59
2.64
78
2.82
31
3.30
88
2.90
85
2.95
91
3.57
54
2.40
68
2.19
85
2.25
94
2.67
79
2.36
84
2.42
17
2.85
81
Three-dimensional vibration analysis 3313
A set of vibration mode shapes for thick square plates with thickness ratio, t/b = 0.2, are presented in Figs 2-6. These correspond respectively to SSSS, SFSF, CFCF, SCSC and CCCC boundary conditions. The normalized transverse deflection (W-mode shape) and in-plane deflections (U- and V-mode shapes) are presented in shaded contour plots. The shaded portions in these figures represent regions with negative deflection amplitude while the unshaded portions represent regions with positive deflection amplitude. The lines of demarcation form the nodal lines for that particular mode.
By dividing the plates into regular meshes and assigning the vectorial displacement components U, V and W to each mesh coordinate, sets of three-dimensional deformed mode shapes are obtained. These diagrams express vividly the vibratory motion of the elastic plate at each mode of vibration. For all the boundary constraints examined in this study, it is observed that in general, the first doubly symmetric mode is predominantly an out-of-plane flexural motion in the z-direction. In addition, the through thickness deformations can also be seen for plates with edges that are free or simply supported. These through thickness deformations take on the form of either thickness-twisting or thickness- shearing motions, depending on the shapes of the deformed geometries. Figures 2(a) and 2(b) depict the deformed modes of a simply supported plate. It is observed that the symmetric flexural motions occur at the first and third SS modes. The second SS mode is predominantly an in-plane stretching and contracting motion in the x- and y-directions. Other flexural modes can be identified in the second SA mode (also in the second AS mode because of symmetry) and the first AA mode, respectively.
It is interesting to examine the vibratory motion of plates with free edges. Figures 3 and 4 depict the vibratory motion of plates corresponding to SFSF and CFCF boundary conditions. Comparing the first three SS modes of vibration for both cases, it is observed that the deformations exhibit similar characteristics. For both boundary conditions, the fundamental SS mode is an out-of-plane deflection in the z-direction forming an arc shaped deformed geometry. The prominent in-plane motion is found in the third SS mode. A more complex coupling of both in-plane and out-of-plane deflections are manifested in the higher SA, AS and AA modes of vibration.
Vibration mode shapes for plates with lesser degrees of freedom are shown in Figs 5 and 6, respectively, with SCSC and CCCC boundary conditions. Modes with distinct thickness deformation in the x-direction (u-mode) can be observed for the plate with SCSC boundaries. For the fully clamped plate, however, the through thickness motions are completely restrained at the boundaries. The in-plane motions in the x-direction (u-mode of vibration) dominate at the second SS and AA modes as well as in the first SA mode of vibration for the plate with SCSC boundary condition. The vibratory motions for the fully clamped plate, however, are mainly out-of-plane motion in the z-direction with certain degrees of in-plane sketching within the plate domain.
4. CONCLUSIONS
A comprehensive study on the vibration analysis of thick rectangular plates based on the linear, three-dimensional elasticity theory has been presented. The analysis is performed using the polynomial-based Ritz energy approach. Sets of orthogonally generated poly- nomial functions were used to describe the spatial displacement components in U, Y and W. For this study, no simplifying assumption has been made on the displacement fields and on the strain distribution across the thickness, thus the method is capable of providing accurate frequency solutions to the natural vibration of moderately thick to very thick plates.
Convergence tests and comparison studies have been carried out to establish the rate of convergence and validate the degrees of accuracy of the present three-dimensional formulation. These studies demonstrate the computational efficiency and accuracy of the method in analysing the free vibration of thick rectangular plates. Extensive numerical results for homogeneous, thick plates with different combinations of boundary conditions, aspect ratios and thickness ratios have been presented. It is deduced that the frequency
K. M. LIEW et al.
(a)
(b)
Fig. 2. Displacement contour plots and three-dimensional deformed geometry for the SSSS thick square plate with t/b = 0.2 : (a) SS and SA modes ; (b) AS and AA modes.
Three-dimensional vibration analysis
(4
(b) Fig. 3. Displacement contour plots and three-dimensional deformed geometry for the SFSF thick
square plate with t/b = 0.2 : (a) SS and SA modes ; (b) AS and AA modes.
3376 K. M. LIEW et al.
(b)
Fig. 4. Displacement contour plots and three-dimensional deformed geometry for the CFCF thick square plate with t/b = 0.2 : (a) SS and SA modes ; (b) AS and AA modes.
Three-dimensional vibration analysis
(b)
Fig. 5. Displacement contour plots and three-dimensional deformed geometry for the SCSC thick square plate with t/b = 0.2 : (a) SS and SA modes ; (b) AS and AA modes.
3378 K. M. LIEW et al.
(4
(b)
Fig. 6. Displacement contour plots and three-dimensional deformed geometry for the CCCC thick square plate with t/b = 0.2 : (a) SS and SA modes ; (b) AS and AA modes.
Three-dimensional vibration analysis 3379
parameters increase for plates with lower aspect ratio and smaller thickness ratio. The presence of more boundary constraints has increased the flexural stiffness, thus increasing the vibration frequencies.
Vivid representation of the vibration mode shapes are manifested in shaded contour plots and three-dimensional deformed mesh geometry. Through these deformed mode shapes, we have achieved a better understanding of the dynamic characteristics of these thick rectangular plates. These modes encompass the flexural, thickness twist and thickness shear motions which could also be identified in the approximate theories. In addition, the present three-dimensional elasticity formulation is capable of predicting those modes of vibration in which the approximation plate theories are found inadequate to predict. This concluding remark, in effect, agrees with the observation reported in the pioneering work of Srinivas et al. (1970).
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Bhat, R. B. (1985). Natural frequencies of rectangular plates using characteristic orthogonal polynomials in the Rayleigl-Ritz method. J. Sound Vibr. 114,493499.
Chihara, T. S. (1978). An Introduction to Orthogonal Polynomials. Gordon and Breach, New York. Dawe, D. J., Horsington, R. W., Kamtekar, A. G. and Little, G. H. (1985). Aspects of the Analysis of Plate
Structures. Clarendon Press, Oxford. Dawe, D. J. and Roufaeil, 0. L. (1980). Rayleigh-Ritz vibration analysis of Mindlin plates. J. Sound Vibr. 69,
3455359. Hutchinson, J. R. and Zillmer, S. D. (1983). Vibration of a Free Rectangular Parallelepiped. ASME J. Appl.
Mech. 50, 1233130. Kao, R. (1973). Approximate solutions by utilizing hill functions. Comput. Structs 3,397-412. Leissa, A. W. (1973). The free vibration of rectangular plates. J. Sound Vibr. 31,257-293. Leissa, A. W. and Zhang, Z. D. (1983). On the three-dimensional vibrations of the cantilevered rectangular
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7,343-350. Liew, K. M., Lam, K. Y. and Chow, S. T. (1990). Free vibration analysis of rectangular plates using orthogonal
plate function. Comput. Structs 34,79-85. Liew, K. M., Xiang, Y. and Kitiporchai, S. (1993). Transverse vibration of thick rectangular plates--I: com-
prehensive sets of boundary conditions. Comput. Structs (in press). Narita, Y. (1985). The effect of point constraints on transverse vibration of cantilevered plates. J. Sound Vibr.
102,305-3 13. Reddy, J. N. (1984). A simple higher-order theory for laminated composite plates. ASME J. Appl. Mech. 51,
745-752. Srinivas, S., Rae: C. V. Joga and Rao, A. K. (1970). An exact analysis for vibration of simple-supported
homogeneous and laminated thick rectangular plates. J. Sound Vibr. 12, 187-199. Warburton, G. B. (1954). The vibration of rectangular plates. Proc. Institution of Mechanical Engineers, 168,
371-384. Wittrick, W. H. (1987). Analytical three-dimensional elasticity solutions to some plate problems, and some
observations on Mindlin’s plate theory. Int. J. Solids Structs 23,441-. Young, D. (1950). Vibration of rectangular plates by the Ritz method. ASME J. Appl. Mech. 17,448453.
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