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Exact Solutions for Vibration of Multi-Span Rectangular Mindlin Plates Y. Xiang
Center for Construction Technology and
Research,
University of Western Sydney,
Penrith South DC NSW 1797, Australia
email: y.xiang@uws.edu.au
G. W. Wei
Department of Computational Science,
National University of Singapore,
Singapore 117543
email: cscweigw@nus.edu.sg
Exact Solutions for Vibration
of Multi-Span Rectangular
Mindlin Plates
This paper presents the rst-known exact solutions for the vibration of multi-span rect-
angular Mindlin plates with two opposite edges simply supported. The Levy type solution
method and the state-space technique are employed to develop an analytical approach to
deal with the vibration of rectangular Mindlin plates of multiple spans. Exact vibration
frequencies are obtained for two-span square Mindlin plates with varying span ratios and
two-, three- and four-equal-span rectangular Mindlin plates. The inuence of the span
ratios, the number of spans and plate boundary conditions on the vibration behavior of
square and rectangular Mindlin plates is examined. The presented exact vibration results
may serve as benchmark solutions for such plates. DOI: 10.1115/1.1501083
1
Introduction
Multi-span rectangular plates are important structural compo-
nents that are widely used in various engineering applications, i.e.,
aeroplane surfaces, slabs in house construction, bridge decks and
glass window panels. The problem of free vibration of multi-span
rectangular plates or plates with internal line supports has at-
tracted the attention of many researchers. Veletsos and Newmark
1 pioneered the research in this area by studying the vibration
behavior of a two-span rectangular plate. Since then, many re-
searchers have carried out investigations on the topic using vari-
ous analytical and numerical 214 methods. Among these re-
searchers, Azimi et al. 7 obtained exact solutions for simply
supported multi-span plates by using the receptance method. Most
recently, Xiang et al. 15 proposed an analytical approach for the
vibration analysis of multi-span rectangular plates with two oppo-
site edges simply supported. By utilizing the Levy solution
method and the state-space technique, their approach is able to
nd exact solutions for the vibration of multi-span rectangular
plates involving free and clamped edges 15 .
All the aforementioned studies are for plates based on the clas-
sical thin plate theory the Kirchhoff plate theory . The investiga-
tions on the vibration of multi-span thick rectangular plates are
relatively scarce 16 . It is well-known that the classical thin plate
theory overestimates the vibration frequencies when a plate be-
comes thick or a plate vibrates at higher frequencies. This over-
estimation is due to the fact that the thin plate theory neglects the
effects of transverse shear deformation and rotary inertia in a
plate. To overcome this problem, one may employ the rst order
shear-deformable plate theory the Mindlin plate theory 17
or
the higher order plate theory the Reddy plate theory 18
for
analysis of thick plates. Using the pb-2 Ritz method, Liew et al.
19 studied the vibration of rectangular Mindlin plates with in-
ternal line supports being either parallel to the edges or in diago-
nal directions. They extended their studies later to the vibration of
rectangular Mindlin plates with intermediate stiffeners 20 and
skew Mindlin plates with internal line supports 21 . Kong and
Cheung 22 investigated the vibration of shear-deformable plates
with internal line supports by the nite layer approach.
It is apparent that there are no exact solutions for the vibration
of multi-span thick rectangular plates in the open literature. The
purpose of this work is to ll the current gap in this area by
introducing an analytical method for dealing with multi-span rect-
angular Mindlin plates and providing exact vibration solutions for
these plates. We shall consider a multi-span rectangular Mindlin
plate with two opposite edges simply supported while the other
two edges may take an arbitrary combination of boundary condi-
tions. The Levy solution method and the state-space technique
15,2327 are employed to establish the proposed analytical ap-
proach for the vibration analysis of the Mindlin plate. This ap-
proach is used to generate exact vibration solutions. These solu-
tions are indeed valuable and are tabulated as they serve as
important benchmark values for checking the convergence, valid-
ity and accuracy of potential numerical methods for the analysis
of multi-span thick plates.
2
Mathematical Modelling
Consider an isotropic rectangular Mindlin plate of length aL,
width L and thickness h as shown in Fig. 1. The plate is simply
supported on the two edges parallel to the x axis. There are (n
1) internal line supports in the x direction that divide the plate
into n spans see Fig. 1 . The internal line supports enforce zero
transverse displacement along the line supports in the plate. The
origin of the coordinates x, y is set at the middle point of the
plate bottom edge. The problem at hand is to determine the vibra-
tion frequencies of the multi-span rectangular Mindlin plate.
We take a typical span in the plate to derive the Levy type
solution. The governing differential equations based on the Mind-
lin plate theory 14 for the i</i>-th span in harmonic vibration can be
derived as:
2
Gh
x
w
i
x
x
i
y
w
i
y
y
i
h
2
w
i
0
(1)
D
x
x
i
x
i
y
y
1
D
2
y
y
i
x
x
i
y
2
Gh
w
i
x
x
i
h
3
12
2
x
i
0
(2)
D
y
y
i
y
x
i
x
1
D
2
x
y
i
x
x
i
y
2
Gh
w
i
y
y
i
h
3
12
2
y
i
0
(3)
where the superscript i(
1,2, . . . ,n) denotes the i</i>-th span in the
plate, E is Youngs modulus, G
E/ 2(1
) is the shear modu-
lus,
is Poissons ratio,
2
is the shear correction factor, D
Eh
3
/ 12(1
2
)
is the exural rigidity of the plate,
is the
Contributed by the Technical Committee on Vibration and Sound for publication
in the J
OURNAL OF
V
IBRATION AND
A
COUSTICS
. Manuscript received April 1 2001;
Revised May 2002. Associate Editor: T. Wickert.
Copyright
©
2002 by ASME
Journal of Vibration and Acoustics
OCTOBER 2002, Vol. 124
Õ
545 mass density of the plate,
is the vibration frequency of the plate,
w is the transverse displacement, and
x
and
y
are rotations in the
y and x directions.
The essential and natural boundary conditions for the two par-
allel edges at y
0 and y
L in the i</i>-th span are
w
i
0,
(4<i>a)
M
y
i
0,
(4<i>b)
x
i
0
(4<i>c)
where M
y
i
is the bending moment and is dened by
M
y
i
D
y
i
y
x
i
x
(5)
The general Levy-type solution approach is employed to solve
the governing differential equations for the i</i>-th span 15,26,27 .
The displacement elds can be expressed as
w
i
x,y
x
i
x,y
y
i
x,y
w
i
x sin m y
L
x
i
x sin m y
L
y
i
x cos m y
L
(6)
where
w
i
(x),
x
i
(x) and
y
i
(x) are unknown functions to be de-
termined. Equation 6 satises the simply supported boundary
conditions on edges y
0 and y
L as dened in Eqs. 4<i>a4<i>c .
Substituting Eq. 6 into Eqs. 13 , the following differential
equation system can be derived:
i
H
i
i
(7)
where
i
w
i
(
w
i
)
x
i
(
x
i
)
y
i
(
y
i
)
T
, the prime
repre-
sents the derivative with respect to x and H
i
is a 6
6 matrix with
the following non-zero elements:
H
12
i
H
34
i
H
56
i
1
(8)
H
21
i
m /L
2
2
Gh
h
2
2
Gh
(9)
H
24
i
1,
(10)
H
25
i
m
L
(11)
H
42
i
2
Gh
D
,
(12)
H
46
i
m /L
1
2
(13)
H
43
i
D 1
m /L
2
/2
2
Gh
h
3
2
/12
D
(14)
H
61
i
m /L
2
Gh
D 1
/2 ,
(15)
H
64
i
m /L
1
1
(16)
H
65
i
D m /L
2
2
Gh
h
3
2
/12
D 1
/2
(17)
A general solution of Eq. 7 can be obtained as
i
e
H</b><i>x
c
i
(18)
where c
i
is a constant column vector that can be determined by the
plate boundary conditions of the two edges parallel to the y axis
and/or interface conditions between adjacent spans and e
H</b><i>x
is the
general matrix solution of Eq. 7 . The detailed procedure in de-
termining Eq. 18 has been given in 26 and 27 .
Each of the two edges parallel to the y axis may have the
following edge conditions
M
x
i
0,
(19<i>a)
M
x y
i
0,
(19<i>b)
Q
x
i
0,
if the edge is free
(19<i>c)
w
i
0,
(20<i>a)
M
x
i
0,
(20<i>b)
y
i
0,
if the edge is simply supported
(20<i>c)
w
i
0,
(21<i>a)
x
i
0,
(21<i>b)
y
i
0,
if the edge is clamped
(21<i>c)
where i takes the value 1 or n, M
x
i
, M
x y
i
and Q
x
i
are bending
moment, twist moment and transverse shear force in the plate,
respectively, and dened by
M
x
i
D
x
i
x
y
i
y
(22)
M
x y
i
D 1 2
x
i
y
y
i
x
(23)
Q
x
i
2
Gh
w
i
x
x
i
(24)
To ensure the continuity and the internal line support condi-
tions, the essential and natural boundary conditions for the inter-
face between the i</i>-th and the (i
1)-th spans are dened as:
w
i
0,
(25<i>a)
w
i
1
0,
(25<i>b)
x
i
x
i
1
,
(25<i>c)
y
i
y
i
1
,
(25<i>d)
M
x
i
M
x
i
1
,
(25<i>e)
Fig. 1
Geometry and coordinate system for a Levy plate with n
À
1 internal line supports
546
Õ
Vol. 124, OCTOBER 2002
Transactions of the ASME M
x y
i
M
x y
i
1
(25<i>f)
In view of Eq. 18 , a homogeneous system of equations can be
derived by implementing the bounda