Canadian Journal on Science and Engineering Mathematics Vol. 3 No. 4, May 2012
Rayleigh-Ritz Method For Free Vibration of
Mindlin Trapezoidal Plates
Mohamed A. El-Sayad* and Saad S.A. Ghazy*
Department of Engineering Mathematics and Physics, Faculty of Engineering,
Alexandria University, Alexandria, Egypt
E-mail: mel_sayad@hotmail.com , s_ghazy2000@yahoo.com
Xing*, and Bo Liu [3]. Free vibration analysis
of orthotropic rectangular plates with variable
thickness and general boundary conditions was
studied by M. Huang and et al [4]. Chopra and
Durvasula have investigated the free
oscillatory motion of simply supported
symmetric and asymmetric trapezoidal plates
by applying Galerkin's method in [5, 6]. Orris
and Petyt [7] used the finite element method to
study the free vibration of simply supported,
clamped triangular and trapezoidal plates.
Frequency analysis of trapezoidal plates and
membrance using discrete singular convolution
was established by Ö. Civalek and M. Gürses
[8]. Nagaya applied the integral equation
technique to investigate the free vibration of
plates of arbitrary shapes that have free and
simply supported mixed boundary conditions
[9]. Srinivasan and Babu [10] used the integral
equation method to study the free vibration of
cantilevered quadrilateral and trapezoidal
plates. K. Maruyama et al. presented the results
of the experimental study of the free traverses
vibration of clamped trapezoidal plates [11].
Bert and Malik [12] applied the differential
quadrature method to study the free lateral
oscillations of plates of irregular domains. For
trapezoidal plates that have variable thickness,
there is a little amount of work related to the
free vibration analysis of such plates. Laura et
al. [13] applied the Rayleigh-Ritz method to
investigate the free vibration of tapered
cantilevered trapezoidal plates. In [14, 15], the
authors has examined the problem of
transverse vibration of plates which have spanwise quadratic thickness variation. The finite
element method was applied and the results for
cantilevered trapezoidal plates were presented.
Three different cases, which are the linear, the
quadratic and the exponential thickness
variations were considered. The static and free
vibration analysis of a non-homogeneous
moderately thick plate using the meshless local
radial point interpolation method was
examined by P. Xia and et al. [16]. The Free
vibration and transient response of thick and
thin plates using the finite element method
were presented by T. Rock and E. Hinton [17].
Free vibration analysis of plates using leastsquare-based on finite difference method was
studied by M. Huang and et al [18]. An edge-
Abstract
In the present paper, the free vibration of
moderately thick trapezoidal plates has been
studied. The analysis is based on the Mindlin
shear deformation theory. The solutions are
determined using the pb-2 Rayleigh-Ritz method.
The transverse displacement and the rotations of
the plate are approximated by Ritz functions
defined as two dimensional polynomials of the
trapezoidal domain variables and a basic
function that satisfied as essential boundary
conditions. Three different arrangements of
boundary conditions are considered which are
the cantilevered, the simply supported and the
clamped edge conditions. The effects of both,
transverse shear and rotary inertia are
accounted. Convergence of the solutions is
verified by considering polynomials of several
subsequent degrees till the results converge. The
present results are compared with those available
in the open literature which indicates good
agreement between the present results and those
previously published. A set of tabulated results
for a wide range of variation of both thickness to
root width (H/a) and the trapezoid angle  for
each of the three different cases of boundary
conditions are presented.
Keywords: Free vibrations, Transverse shear,
Rayleigh-Ritz
method
Rotary
inertia,
Trapezoidal plates.
I.
Introduction
Many aircraft wings can be modeled as either
trapezoidal or quadrilateral plates. The free
vibration analysis of such models is a
necessary prerequisite to design them to
operate under different loading conditions.
Based on classical thin plate theory and several
different approximate mathematical methods,
there exists a reasonable amount of work
related to the vibration of thin trapezoidal
plates of constant thickness. The Free vibration
analysis of initially deflected stiffened plates
for various boundary conditions was studied by
S. Kitipornchai and et al [1]. Vibration analysis
of conical panels using the method of discrete
singular convolution was presented by Ö.
Civalek [2]. High-accuracy differential
quadrature finite element method and its
application to free vibrations of thin plate with
curvilinear domain was established by Y.
159
Canadian Journal on Science and Engineering Mathematics Vol. 3 No. 4, May 2012
based smoothed finite element method (ESFEM) with stabilized discrete shear gap
technique for analysis of Reissner–Mindlin
plates was applied by H. N. Xuan and et al
[19]. Transverse Vibration of Mindlin Plates
on Two-Parameter Foundations by Analytical
Trapezoidal p-Elements was presented by A.
Leung and B. Zhu [20]. An efficient
differential quadrature methodology for free
vibration analysis of arbitrary straight-sided
quadrilateral thin plates was presented by G.
Karami and P. Malekzadeh [21]. Polynomial
and harmonic differential quadrature methods
for free vibration of variable thickness thick
skew plates were demonstrated by P.
Malekzadeh and G. Karami [22]. Free
vibration analysis of plates using least-squarebased on finite difference method was studied
by C. Shu and et al [23].
U max
2

  θX  θY 




1

Y 
 X
  (D 
2A 
 θX  θY 1
 2(1  ν){


 X Y
4




2 
  θX  θY  


 }
X  
 Y

2
2

W
W 

(2)
 kGH  θ X 
   θY 
  ) dA

X

Y








1
1


Tmax  ω 2  ρHW 2 
ρH 3 θ2X  θ 2Y  dA (3)
2
12

A

In the present study, the effects of both the
transverse shear deformation and the rotary
inertia on the free vibration characteristics of
plates are accounted for The Mindlin plate
theory [24] is employed. The pb2 (Two
dimensional polynomial and a basic function)
Rayleigh-Ritz method is applied. Three
different cases of edge conditions that are the
cantilevered, the simply supported and the
fully clamped are considered. Convergence of
the present solutions is demonstrated through
using polynomials of several subsequent
degrees. The results for moderately thick
trapezoidal plates are not available in the open
literature. So, the results for isosceles
triangular Mindlin plates are obtained as
special cases from trapezoidal plates and then
compared with those presented by other
researchers. Also, the results for thin
trapezoidal plates that are obtained as special
solutions from those concerning thick plates
are found to be in good agreement with the
previously published results. The effects of
variation of the thickness to root width ratio
(h/a), the trapezoid angle () and the aspect
ratio of the plate  on the frequency
coefficients are studied.

in which W is the transverse displacement, x
is the rotation about the Y-axis, Y is the
rotation about the X-axis,  is the natural
frequency of the plate,  is the Poisson's ratio,
 is the density of the plate, k is the shear
correction factor, D is the flexural rigidity of
the plate [D = E H3 / 12(1-2)] .Where, E is the
modulus of elasticity, G is the modulus of
rigidity [G = E / 2 ( 1 + )] and A is the plate
surface area.
In the following formulation, the X,Y
Coordinates, the thickness H and the transverse
displacement of the plate middle surface W are
normalized by a characteristic length which is
the plate root width a, (x = X/a, y = Y/a, h =
H/a and w = W/a). The lateral displacement
and the rotations will be approximated by a set
of pb-2 Ritz functions in the x-y plane as
follows:
n
w(x, y) =  c i Φi (x, y)  {c}T {Φ}
(4-a)
i 1
n
II. Mathematical formulation
x (x, y) =  d i Ψ xi (x, y)  {d}T { Ψ x }
(4-b)
y (x, y) =  ei Ψ yi (x, y)  {e}T { Ψy}
(4-c)
i 1
n
A thick isotropic symmetric trapezoidal plate
of uniform thickness H, length b and root
width a are considered. The geometry of the
plate is shown in fig. 1. Following Karunasena
et al. [25], the energy functional  for a
Mindlin plate can be written in terms of the
maximum strain energy Umax and the
maximum kinetic energy Tmax as:
(1)
Π  U max  Tmax ,
Where:
i 1
Where {c}, {d}, {e} are the unknown
coefficients vectors containing ci, di, ei, which
are the unknown coefficients of the Ritz
functions as respective elements, {}, {x},
{ y} are the Ritz functions vectors associated
to w, x y respectively. The respective
elements are i, xi, yi and T denotes the
transpose of a matrix or a vector. The Ritz
function are defined over the domain of the
160
Canadian Journal on Science and Engineering Mathematics Vol. 3 No. 4, May 2012
plate by the products of basic functions that
must satisfy the
geometric boundary
conditions, and complete two dimensional
polynomials, which will be assumed here to
have the same degrees, as follows:
{} = P(x,y) {f}
(5-a)
{x} = Q(x,y) {f}
(5-b)
{y} = R(x,y) {f},
(5-c)
Where P(x,y), Q(x,y) and R(x,y) are the basic
functions that satisfy the essential boundary
conditions associated to w, x , y,
respectively, they are chosen according to the
plate edge conditions as follows:
For a plate which is cantilevered along the yaxis:
P (x,y) = Q (x, y) = R (x,y) = x.
For a simply supported plate:
P (x,y) = x (x- ) (y-cx+ 0.5) (y + cx- 0.5),
Q (x, y) = 1, R (x, y) = x (x- ).
For a fully clamped plate:
P (x,y) = Q (x, y) = R (x, y)
= x(x-) (y- cx + 0.5) (y + cx – 0.5),
In which  is the plate aspect ratio ( = b/a) and
c = tan, where  is the trapezoid angle. The
elements of the vector {f} are those of
complete two dimensional polynomials of x, y
that may have variable degrees. As an
example, for a polynomial of degree p = 4, the
components of {f} are given by:
Where, the sub matrices inside [K] and [M] are
defined by:
K cc 
Kc d
Kc e


In which  = ωa 2 ρH/D
T

K de     x

A

M cd
 M cc
[ M ]  
M dd
symmetric
M ce 
M de 
M ee 
(9)



 dA



y
T


 dA


(1  ) x Y
2
y
x
T
Mcc = h  φφ dA ,
T
(10-f)
(11-a)
A
Mcd = 0,
(11-b)
Mce = 0,
(11-c)
1 3
T
M dd  ρh  Ψ x Ψ x dA
(11-d)
12
A
Mde = 0,
(11-e)
1
T T
M ee 
(11-f)
 Ψ y Ψ y dA,
12 A
Where, Ᾱ is the non-dimensional area (dᾹ =
dx dy). Substituting from eqn.(5) into eqns.
(10, 11), for each of the three different cases of
boundary conditions ad carrying out the
associated integration over the domain of the
trapezoidal plate, the elements of both the
overall stiffness and the mass matrices are
evaluated. Setting the first variation of the
energy functional in eq. (6) to zero results in
the following eigen value problem.
(12)
K  λ 2 Mq 0
It must be mentioned that each of the overall
stiffness and mass matrices in eq. (12) are
symmetric and order 3n x 3n. Using EISPACK
routines [26], eq. (12) is solved for the natural
(7)
(8)
T
T
 y y
(1  ) y Y



2
x
x
 y y
K ee    6k (1  )
A
y y T
h2


is the normalized
k ce 
k de 
k ee 
(10-c)
(10 - e)
natural frequency coefficient and
k cc
k cd


[K]  
k dd
symmetric
(10-b)
(10-d)
T
x Y
(6)
{q}T = <{c}T{d}T {e}T >
(10-a)
6 k(1  ν)  Φ

x T dA
A
x
h2
6 k(1  ν)  Φ T

Y dA
A
y
h2
 x x
(1  ) x x




x

x
2
y Y

K dd    6k(1  )
A
x x T
2
h


For a two dimensional polynomial of degree =
p, the total number for elements n of the
vector {f} are given by: n = (p+1) (p+2)/2.
Substituting from eq. (4) into eqs. (1-3), the
energy functional  can be written as:
1
(q) T K  λ 2 M {q}
2
h2
  Φ  ΦT  Φ  ΦT 
 dA

A 
 y  y 
 x x
,
{f}T = {1 x y x2 xy y2 x3
x4 x3y x2y2 xy3 y4}

6 k(1  ν)
161



 dA



Canadian Journal on Science and Engineering Mathematics Vol. 3 No. 4, May 2012
frequency coefficient  for the considered
plates.
0.001,  = 0, =1) and moderately thick
trapezoidal plates (H/a = 0.1,  = 10,  =1) are
presented. For each case of boundary
conditions, the results that correspond to four
successive values of polynomial degrees are
computed. From the expressions of the basic
functions which satisfy the essential boundary
conditions, one can expect that, for simply
supported and clamped plates, two dimensional
polynomials of degrees (p) whose values are
less than those considered for the cantilevered
plates will result in efficiently accurate
solutions. So, for cantilevered plates, the
results that correspond to values of p, starting
from p = 7 up to p = 10 are given, while, for
both simply supported and clamped plates, the
values of p are taken to be in the range from p
= 6 to p = 9. As shown in table 1, the results
for both thin square plates and thick
trapezoidal plates are monotonically converged
to their accurate values as the degrees of the
polynomials increase.
III. Numerical solutions and discussion
Three different cases of boundary conditions,
which are, the cantilevered, the simply
supported and the fully clamped plates will be
considered. In each case, the convergence of
the solutions is demonstrated through using
polynomials of successive different degrees till
the convergence is achieved. The accuracy of
results is checked by comparisons with the
previously published solutions which are
available for both the isosceles triangular
Mandolin plates and the thin trapezoidal plates.
Finally, a series of tabulated results, for thick
trapezoidal plates, are given. To execute
correct comparisons, the Poisson‟s ratio is
taken to be 0.3 and the shear corrections factor
is considered to be k = 5/6 = 0.833 in all the
following calculations, which are the same
values which were considered in [25, 27, 28].
In table 1, the first four natural frequency
coefficients for both thin square plates (H/a =
Table 1
Convergence of results for square and trapezoidal plates
Boundary
condition
H/a

CFFF*
0.001
0
0.1
10
0.001
0
0.1
10
0.001
0
0.1
10
SSSS+
CCCC++
*
+
++
P
1
2
3
4
7
8
9
10
Ref. [27]
7
8
9
10
6
7
8
9
Exact
6
7
8
9
6
7
8
9
Exact
6
7
8
9
3.481
3.477
3.475
3.474
3.475
3.862
3.861
3.861
3.861
19.745
19.739
19.739
19.739
19.739
23.125
23.109
23.065
23.063
35.999
35.999
35.988
35.988
35.990
40.202
40.199
40.198
40.197
8.522
8.519
8.517
8.514
8.513
11.454
11.449
11.443
11.443
49.510
49.509
49.348
49.348
49.374
49.501
49.217
49.081
49.074
74.286
73.431
73.431
73.412
73.800
68.093
68.073
68.066
68.064
21.317
21.311
21.305
21.301
21.301
20.990
20.985
20.981
20.979
49.512
49.511
49.348
49.348
49.374
59.728
59.638
59.614
59.602
74.286
73.431
73.431
73.412
73.800
80.853
80.845
80.841
80.840
27.213
27.202
27.201
27.200
27.205
33.968
33.886
33.877
33.861
92.554
79.398
79.398
79.317
78.957
88.023
85.672
85.170
85.097
108.587
108.586
108.262
108.262
108.270
106.879
106.720
106.680
106.674
One edge is e\clamped and the other three ones are free.
The four edges are simply supported.
The four edges are clamped
accuracy. For certain values of the trapezoid
angle  and its aspect ratio, the symmetric
trapezoidal shape becomes an isosceles
triangular one. To demonstrate the accuracy of
the present solutions for thick plates, the
The present results for thin square plates are
found to be in good agreement with those
obtained by thin plate theory solutions. For
moderately thick trapezoidal plates, the present
results are new in literature, but the good
convergence of such results demonstrates their
162
Canadian Journal on Science and Engineering Mathematics Vol. 3 No. 4, May 2012
available in [27, 28]. As can be shown, they
agree well with both of the two sets of the
available results and the maximum percentage
difference between the three sets of different
solutions does not exceed 0.5%.
problem of the isosceles triangular Mindlin
plates is solved.
In table 2, the first four natural frequency
coefficients for isosceles triangular plates
( = 30,  = 3/2) are determine. The
present results are compared with those
Table 2:
Comparison of results for isosceles triangular Mindilin plates
Boundary
condition
CFF
H/a
Ref.
1
2
3
1
0.001
Present
[28]
[25]
Present
[28]
[25]
Present
[28]
8.920
8.922
8.922
8.647
8.646
8.646
8.414
8.413
35.089
35.092
35.086
31.435
31.410
31.435
28.445
28.457
38.482
38.485
38.482
34.836
34.809
34.839
31.877
31.831
89.603
89.598
89.606
75.522
75.398
75.522
65.116
64.987
Present
[28]
Present
[28]
52.637
52.634
42.332
42.311
123.081
122.836
85.873
85.326
123.081
122.836
85.873
85.326
210.941
210.550
128.943
128.365
Present
[28]
Present
[28]
99.031
99.023
64.696
64.591
189.025
188.998
104.841
104.741
189.025
188.998
104.841
104.741
295.403
295.247
145.984
145.330
0.1
0.15
SSS
0.001
0.15
CCC
0.001
0.15
In table 3, the results for both thin and
moderately thick cantilevered trapezoidal
plates, which correspond to three different
values of aspect ratios are given. For each
aspect ratio, three different values of H/a are
considered and for each value of H/a, the
solutions are determined for three different
values of . The present results, for thin plate
(H/a = 0.001,  = 1) are found to be in good
agreement with those obtained in [28] by thin
plate theory solution. The variation of the
natural frequency coefficients  against the
three varying parameters , H/a,  is found to
be as follows: For certain values of  and H/a,
the increase of  leads to a corresponding
increases of . The increase of the value of H/a
tends to decrease the value of , while the
values of both  and  remain stationary. The
increase of , for stationary of both  and H/a
tends to the decreases of . Such behavior of 

H/a
0.001
1
Ref. [14]
0.2
P
5
10
15
5
10
15
5
10
against the three varying parameters may be
explained as follows: For a single degree of
freedom vibrating system, the natural
frequency is given by 2 = K/M, where K is
the stiffness and M is the mass. For the same
value of K, 2 is inversely proportional to M.
For a freely vibrating plate, the stiffness is
mainly dependant on its boundary conditions,
while the mass of the plate is depending on the
plate thickness and the plate surface area. As 
increase, the area and hence, the mass of the
plate decreases. The increase of both  and H
leads to an increase of the mass of the plate. In
tables 4, 5, the results for simply supported and
clamped Mindlin trapezoidal plates are
presented. The variation of the natural
frequency coefficient  against each of the
three varying parameters is found to be nearly
the same as that of the case of the cantilevered
plates.
Table 3
Results for cantilevered trapezoidal Mindlin plates
1
2
3
3.663
10.072
21.770
3.909
12.207
22.216
4.261
15.297
22.790
3.663
10.070
21.768
3.910
12.207
22.217
4.262
15.300
22.793
3.520
8.575
17.978
3.750
10.211
18.352
163
4
33.823
37.641
43.255
33.813
37.640
43.266
25.759
28.173
Canadian Journal on Science and Engineering Mathematics Vol. 3 No. 4, May 2012
0.001
0.5
0.1
0.2
0.001
2.0
0.1
0.2

H/a
0.001
1.0
0.1
0.2
0.001
0.5
0.1
0.2
0.001
2.0
0.1
0.2

H/a
0.001
1.0
0.1
0.2
0.001
0.5
0.1
0.2
15
4.071
12.486
18.794
31.547
5
10
15
5
10
15
5
10
15
0
5
10
0
5
10
0
5
10
14.314
14.672
15.075
13.799
14.136
14.512
12.638
12.929
13.249
0.8607
0.9740
1.197
0.8558
0.9688
1.190
0.8480
0.9596
1.177
23.160
25.161
27.551
21.177
22.899
24.943
18.145
19.475
21.032
3.703
5.2952
5.970
3.548
5.027
5.861
3.318
4.620
5.614
45.367
50.852
57.475
39.919
44.530
50.062
32.507
35.913
39.421
5.363
5.569
8.377
5.274
5.476
7.841
5.058
5.252
6.980
85.278
89.143
90.277
71.757
73.455
74.304
51.976
52.542
53.173
12.053
14.958
15.426
11.444
14.045
15.042
10.455
12.605
13.761
ϴ
5
10
15
5
10
15
5
10
15
5
10
15
5
10
15
5
10
15
0
5
10
0
5
10
0
5
10
Table 4
Results for simply supported trapezoidal Mindlin plates
1
2
3
21.777
51.445
57.208
24.684
54.761
67.093
28.966
60.836
78.677
20.480
46.561
51.503
23.060
49.073
59.595
26.860
53.662
68.801
18.293
38.587
42.168
20.389
40.322
47.851
23.441
43.440
54.048
50.304
82.755
136.77
51.503
87.438
146.78
53.064
93.377
158.07
45.651
71.409
111.04
46.567
74.907
117.88
47.720
79.296
125.58
37.975
55.698
80.862
38.612
58.010
84.978
39.421
60.882
89.562
12.337
19.739
32.082
16.643
25.100
37.509
22.098
36.865
54.950
11.902
18.690
29.886
15.949
23.480
34.453
20.939
33.977
48.463
11.083
16.812
25.956
14.612
20.744
29.341
18.783
29.050
39.615
4
87.456
100.96
110.82
74.994
85.083
92.159
58.030
64.558
69.270
168.79
170.31
173.33
133.82
134.67
136.33
94.798
95.291
96.240
41.946
55.037
66.811
38.926
49.676
59.433
33.167
40.583
47.767
ϴ
5
10
15
5
10
15
5
10
15
5
10
15
5
10
15
5
10
15
Table 5
Results for clamped trapezoidal Mindlin plates
1
2
3
39.813
76.539
85.303
45.555
81.829
100.52
54.241
92.378
118.45
35.643
64.316
70.517
40.198
68.066
80.841
46.820
75.010
92.369
28.699
47.745
51.347
31.674
50.072
57.313
35.853
54.169
63.851
99.461
131.96
189.72
100.85
137.86
202.73
102.80
145.96
219.02
77.734
99.837
135.95
78.737
103.61
143.18
80.086
108.45
151.57
53.744
67.981
89.514
54.421
70.190
93.424
55.312
72.940
97.841
4
120.05
139.08
148.87
94.686
106.86
113.04
66.614
73.219
77.150
257.07
258.50
261.05
166.57
167.38
168.70
102.02
102.55
103.49
164
Canadian Journal on Science and Engineering Mathematics Vol. 3 No. 4, May 2012
0.001
2.0
0.1
0.2
0
5
10
0
5
10
0
5
10
24.579
33.380
43.086
22.971
30.228
38.083
19.240
24.548
29.956
31.829
42.763
62.587
29.132
38.236
53.245
24.205
30.281
40.023
44.820
46.846
83.135
40.182
48.315
69.030
32.473
37.618
50.126
63.968
75.667
100.41
54.982
62.233
80.644
41.499
47.311
57.106
Trapezoids”, Journal of Sound and Vibrations, Vol. 20,
(1972), pp. 125-134.
IV. Conclusions
The free lateral vibrations of thin and
moderately thick trapezoidal plates that have
three different arrangements of boundary
conditions are analyzed by applying the pb-2
Rayleight-Ritz method. From the preceding
analysis, it is possible to conclude the
following remarks:
1.
The convergence of the present
solutions is achieved through, using some
successive values of the degrees of the two
dimensional polynomials contained in the Ritz
functions. The accuracy of the present results
is demonstrated by comparisons with most of
those available in the open literature.
2.
The effects of variations of the plate
aspect ratio, the trapezoid angle and the
thickness to root width ratio on the frequency
coefficients has been investigated. The results
indicate that, the frequency coefficients
increase as the trapezoid angle increase, while
they decrease against the increase of both the
thickness and the aspect ratio.
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