Table Of ContentDROWEROF
Having been involved myself with the study of plate vibrations for some four decades, I am
especially pleased to see this monograph which is based entirely upon the Mindlin Theory. A
tremendous amount of research has taken place on plate vibrations. My own monograph,
completed in 1967 and published in 1969, presented frequencies and mode shapes taken from
approximately 500 references. Since then at least 2000 additional relevant publications have
appeared; however, the vast majority of these, on the order of 90 percent I would estimate,
consider only thin plates. The present book deals with thick plates, although thin plates are
also accounted for.
A plate is typically considered to be thin when the ratio of its thickness to representative
lateral dimension (e.g., circular plate diameter, square plate side length) is 1/20 or less. In
fact, most plates used in practical applications satisfy this criterion. This usually permits one
to use classical, thin plate theory to obtain a fundamental (i.e., lowest) frequency with good
accuracy. However, the second frequency of a plate with thickness ratio of 1/20, determined
by thin plate theory, will not be accurate. It will be somewhat too high. And the higher
frequencies will typically be much too high- too high to be of practical value.
The inaccuracies described above are largely eliminated by use of the Mindlin theory, for it
does include the effects of additional plate flexibility due to shear deformation, and additional
plate inertia due to rotations (supplementing the translational inertia). Both effects decrease
the frequencies. There are still other effects not accounted for by the Mindlin theory (e.g.,
stretching in the thickness direction, warping of the normals to the midplane), but these are
typically unimportant for the lower frequencies until _yrev thick plates are encountered. For
such situations a three-dimensional analysis should be used. Some of my students and I have
made such analyses during the past three decades.
As for all boundary value problems, a few exact solutions for plate vibration problems exist in
rectangular and polar coordinates, applicable to some cases of rectangular, circular, annular
and sectorial plates. But for the vast majority of problems, including these shapes,
approximate solutions must be found. The present monograph uses the well-known Ritz
method exclusively. Displacements are assumed in the form of algebraic polynomials which
satisfy the geometric boundary conditions, a procedure that has been effectively used for at
least a half century. If this is done properly, then one can approach the exact frequencies and
mode shapes as closely as desired as sufficient polynomial terms are utilized.
The present monograph lays out the Mindlin theory briefly, including the appropriate
equations in polar, rectangular, and skew coordinates. In each chapter it is shown how the
procedure (Ritz method, with algebraic polynomial displacement functions) may be applied
straightforwardly to solve a host of free vibration problems. In a series of researches during
the past decade, resulting in numerous published papers, the authors have developed further
the algebraic equation manipulation procedures previously initiated by others (most notably
vi
Professor Yoshihiro Narita of Sapporo, Japan), which results in the rather general computer
programs displayed throughout this book.
The authors go one step further by presenting extensive numerical results - frequencies for
circular, annular, sectorial, elliptical, triangular, parallelogram and trapezoidal plates, all
according to the Mindlin theory. This serves several purposes:
It permits one to verify the correctness of the computer programs listed, and of one's
utilization of them.
.2 Considerable new, previously unpublished, frequency data are presented.
The effects of shear deformation and rotary inertia may be readily seen. The tables
typically give frequencies for thin, as well as thicker, plates.
Using the accurate, benchmark data presented, one may determine the accuracy of
.
other approximate methods, especially finite element codes which take various forms.
I congratulate the authors for having taken the time and effort to produce this work. It should be useful
ot many persons.
Arthur Leissa
Columbus, Ohio
March ,1 1998
vii
ECAFERP
Ever since Chladni in 1787 observed nodal pattems on square plates at their resonant
frequencies, there has been a tremendous research interest in the subject of plate vibrations.
To date, abundant thin plate vibration solutions based on the Kirchhoff plate assumptions are
available in the literature. A good reference source on this subject may be found in the
monograph entitled "Vibration of Plates" by Professor Leissa of The Ohio State University.
This invaluable document, initially published by NASA in 1969 and recently reprinted by the
Acoustical Society of America in 1993 due to a great demand, presents mainly vibration
results based on the Kirchhoffplate theory. This classical plate theory, however, overpredicts
all the vibration frequencies for thick plates, and the higher frequencies for thin plates, as it
neglects the effects of transverse shear deformation and rotary inertia. This shortcoming of
the Kirchhoff theory forced researchers to develop more refined plate theories. As a result,
we now have many such theories ranging from the first-order shear deformation plate theory
of Mindlin to higher-order plate theories such as the one proposed by Reddy.
Recent advances into this subject of plate vibration have focused more on these shear
deformable plates that are somewhat complicated to analyse. This trend in plate research is
further fueled by the availability of powerful digital computers that permit large numbers of
variables to be processed within a relatively short time.
Over the last several years, the four authors have jointly conducted research into the
analysis of vibrating Mindlin plates as a collaborative project between Nanyang
Technological University, The National University of Singapore, and The University of
Queensland. The research was prompted by the fact that there is a dearth of vibration results
for Mindlin plates when compared to classical thin plate solutions. To generate the vibration
results, the authors have successfully employed the Ritz method for general plate shapes and
boundary conditions. The Ritz method, once thought to be awkward for general plate
analysis, can be automated through suitable trial functions (for displacements) that satisfy the
geometric plate boundary conditions a priori. This work has been well-received by
academicians and researchers, as indicated by the continual requests of the authors' papers
and the Ritz software codes. The present monograph is written with the view to share this so-
called p-Ritz method for the vibration analysis of Mindlin plates and its software codes with
the research community. To the authors' knowledge, the monograph contains the first
published Ritz plate software codes of its kind.
Since it is a voluminous task to provide engineers and researchers with vibration
solutions of Mindlin plates of various shapes and boundary conditions, the software codes
listed in this monograph enable easy generation of the required results. The classical plate
solutions can be readily computed from the software by setting the plate thickness to a small
value. As it is more convenient to handle certain plate shapes in their natural coordinate
systems, four versions of the p-Ritz software are given. The software code VPRITZP1 is
based on a one-dimensional polar coordinate system for solving axisymmetric plate problems.
VPRITZP2 allows the analysis of nonaxisymmetric plates in polar coordinates. VPRITZRE
is based on the rectangular Cartesian coordinate system, and the software can be used to
analyse any plate shape whose edges are defined by polynomial functions. Finally
VPRITZSK is based on the skew coordinate system that is expedient for handling
viii
parallelogram plates and plates having parallel oblique edges. Although the software codes
are written for isotropic plates, they may be readily modified for laminated plates and for
complicating effects such as initial stress effects, foundation effects, etc. The programs can
also be modified to accommodate the bending and buckling analyses of Mindlin plates.
The authors would like to thank Professor A.W. Leissa for providing useful comments
and for writing the Foreword to this book, Dr C.W. Lim for generating the vibration mode
shapes, Dr K.K. Ang for useful comments on the software code, Mr W.H. Traves for proof-
reading the manuscript, and finally to our wives for their patience and support.
K.M. Liew
Nanyang Technological University, Singapore
C.M. Wang
The National University of Singapore, Singapore
Y. Xiang
The University of Westem Sydney Nepean, Australia
S. Kitipornchai
The University of Queensland, Australia
RETPAHC ENO
NOITCUDORTNI
1.1 BACKGROUND OF VIBRATION
Vibration, in mechanics, is the to and fro motion of an object. Examples of vibration abound
in nature as nearly everything vibrates; though some vibrations may be too low or too weak
for detection. Vibration can be felt by the sense of touch when a vehicle passes by or felt by
our eardrums when a guitar string is being plucked. Large vibrations occur during
earthquakes and when the ocean level rises and falls causing tides.
Vibration can be exploited for useful tasks, such as the use of a vibrator to massage the
body, to compact loose soil, to increase the workability of wet concrete and to shake sugar,
pepper and salt from their containers. On the other hand, vibration can cause discomfort for
people and problems for machines. Too much vibration can cause people to loose
concentration and to fall sick. In machines, vibration causes wear and tear and can even cause
the malfunctioning of the machine.
In view of the intensive use of structural components in various engineering
disciplines, especially in the aerospace, marine and construction sectors, a thorough
understanding of their vibratory characteristics is of paramount importance to design
engineers in order to ensure a reliable and lasting design. The negligence of considering
vibration as a design factor can lead to excessive deflections and failures. An unforgettable
incident showing the destructive nature of vibration is the dramatic collapse of Tacoma
Narrows Bridge in 1940 that was captured on film. Other failures due to wind induced
vibrations include collapse of chimneys, water tanks, transmission towers, etc. There are also
failures triggered by seismic shocks leading to large-scale destruction of cities such as the
1995 Kobe earthquake. The vibration design aspect is even more important in micro-
machines such as electronic packaging, micro-robots, etc. because of their enhanced
sensitivities to vibration.
1.2 PLATE VIBRATION
The study of plate vibration dates back to the early eighteenth century, with the German
physicist, Chladni (1787), who observed nodal patterns for a flat square plate. In his
experiments on the vibrating plate, he spread an even distribution of sand which formed
regular patterns as the sand accumulated along the nodal lines of zero vertical displacements
upon induction of vibration.
In the early 1800s, Sophie Germain, a French mathematician, obtained a differential
equation for transverse deformation of plates by means of calculus of variations. However,
she made the error of neglecting the strain energy due to the warping of the plate midplane.
The correct version of the governing differential equation, without its derivation, was found
posthumously among Lagrange's notes in 1813. Thus, Lagrange has been credited as being
2 Sec. 1.2 Plate Vibration
the first to use the correct equation for thin plates. Navier (1785-1836) derived the correct
differential equation of rectangular plates with flexural resistance. Using trigonometric series
introduced by Fourier around that time, Navier was able to readily determine the exact
bending solutions for simply supported rectangular plates. Poisson (1829) extended Navier's
work to circular plates. The extended plate theory that considered the combined bending and
stretching actions of a plate has been attributed to Kirchhoff (1850). His other significant
contribution is the application of the virtual displacement method for solving plate problems.
Lord Rayleigh (1877) presented a theory to explain the phenomenon of vibration
which to this day has been used to determine the natural frequencies of vibrating structures.
Based on the plate assumptions made by Kirchhoff (1850) and Rayleigh's theory, early
researchers used analytical techniques to solve the vibration problem. For example, Voigt
(1893) and Carrington (1925) successfully derived the exact vibration frequency solutions for
a simply supported rectangular plate and a fully clamped circular plate, respectively. Ritz
(1909) most probably was one of the early researchers to solve the problem of the freely
vibrating plate which does not have an exact solution. He showed how to reduce the upper
bound frequencies by including more than a single trial (admissible) function and performing
a minimization with respect to the unknown coefficients of these trial functions. The method
became known as the Ritz method.
Improving on the Kirchhoff plate theory, Hencky (1947) and Reissner (1945)
proposed a first order shear deformation plate theory to cater to thick plates where the effect
of transverse shear deformation is significant and thus cannot be neglected. Mindlin (1951)
presented a variational approach for deriving the governing plate equation for free vibration of
first-order shear deformable plates and incorporated the effect of rotary inertia. The first
order shear deformation plate theory of Mindlin, however, requires a shear correction factor to
compensate for the error due to the assumption of a constant shear strain (and thus constant
shear stress) through the plate thickness that violates the zero shear stress condition at the free
surfaces. The correction factors not only depend on material and geometric parameters but
also on the loading and boundary conditions. The study of Wittrick (1973) indicates that it
may be impossible, in general, to obtain the shear correction factors for a general orthotropic
plate.
A more refined plate theory that does away with the shear correction factor is that
proposed by Reddy (1984, 1997). His third-order shear deformation theory ensures that the
zero shear stress condition at the free surfaces of the plate is satisfied at the outset. Even
higher-order theories have also been proposed, but the tractability of solving the equations
become too difficult to warrant the relatively small improvement in the accuracy of the plate
solutions.
Owing to the technological advances in recent years, plate elements are commonly
selected as design components in many engineering structures because of their ability to resist
loads by two-dimensional structural action. With the evolution of light plate-structures,
tremendous research interests in vibration of plates are generated. From what has been done,
plates of almost any conceivable shape, support and loading conditions have been
investigated. Along with this, various analytical and numerical methods have been proposed.
Of these, the finite element method is the most commonly used because of its versatility to
handle any plate shape and boundary conditions. The use of the finite element method started
around the mid of 1950s. In 1956, Turner, Clough, Martin and Topp introduced the method,
which allows the numerical solution of complex plate and shell problems in an efficient way.
Numerous contributions in this field are also due to Argyris (1960) and Zienkiewicz (1977).
In recent years, especially, the Ritz method has been shown to be an efficient alternative to
other numerical techniques for the free vibration analysis of plates of arbitrary shape and
Chap. 1 Introduction 3
boundary conditions. This is made possible by using a set of geometrically generated Ritz
functions that automatically satisfy the geometric boundary conditions. The development of
the Ritz method for a single general plate brings us a step closer towards the ultimate goal of
generalizing the Ritz method for the analysis of complicated plate-structures.
1.3 ABOUT THIS MONOGRAPH
This monograph provides software codes for the free vibration analysis of thick plates. The
computer programs were written based on the p-Ritz method. The Mindlin plate theory was
employed to incorporate the effects of transverse shear deformation and rotary inertia.
Chapter 2 presents the plate theory of Mindlin and its underlying assumptions. The
goveming plate equations and the boundary conditions were derived using the Hamilton's
principle. Also highlighted herein is the existence of an exact relationship between the
frequencies of the classical thin (Kirchhoff) plates and Mindlin plates of polygonal shape and
simply supported for all the straight edges. The relationship allows one to obtain the vibration
frequencies of Mindlin plates from widely available Kirchhoff plate solutions. The shear
correction factor required in the Mindlin plate theory is discussed. The implementation of
Mindlin plate theory into the Ritz method is detailed.
Chapter 3 gives the Ritz formulation for plate analysis in a polar coordinate system.
Such a polar coordinate formulation is expedient for circular, annular and sectorial plates.
Some exact solutions for circular and annular Mindlin plates are given for validation
purposes. The software codes VPRITZP 1 and VPRITZP2 are provided with illustrative input
and output files. The former code is designed for the analysis of axisymmetric plates and the
latter for non-axisymmetric plates. Tables of vibration frequencies are also given for sectorial
and annular sectorial plates with the view to providing benchmark results.
Chapter 4 details the Ritz formulation in a rectangular coordinate system. This
coordinate system is the most commonly used as it can readily handle almost any plate shape.
The software code VPRITZRE is given with illustrative examples. Benchmark vibration
frequencies are also presented for isosceles triangular plates, trapezoidal plates and elliptical
plates.
Chapter 5 fumishes the Ritz formulation in a skew coordinate system. This coordinate
system is useful for plates with oblique parallel edges. The software code VPRITZSK is
given with an example. Vibration frequencies for skew plates with various boundary
conditions are also presented.
Finally, Chapter 6 discusses the treatment of various complicating effects such as
inplane stresses, an elastic foundation, presence of stiffeners, non-uniform thickness,
line/curved/loop intemal supports, point supports, mixed boundary conditions, re-entrant
comers, perforations and sandwich construction.
RETPAHC OWT
MINDLIN PLATE THEORY
DNA RITZ METHOD
2.1 MINDLIN PLATE TIIEORY
In the well-known classical thin plate or Kirchhoffplate theory for vibration, the following
assumptions have been made (Kirchhoff 1850):
(cid:12)9 No deformation occurs in the midplane of the plate;
(cid:12)9 Transverse normal stress is not allowed;
(cid:12)9 Normals to the undeformed midplane remain straight and normal to the deformed
midplane and unstretched in length; and
(cid:12)9 The effect of rotary inertia is negligible.
The assumption regarding normals to the midplane remaining normal to the deformed
plane amounts to neglecting the effect of transverse shear deformation. This effect, together
with the rotary inertia effect, become important when the plate is relatively thick or when
accurate solutions for higher modes of vibration are desired. Wittrick (1987) pointed out that
excluding edge effects, the error in the Kirchhoff plate theory is O(hZ/fl 2 ) where h is the
thickness and t,/ is a typical half-wavelength of the vibrating plate. If the Kirchhoff plate
theory is used, the frequency responses are overpredicted. A more refined plate theory is thus
necessary for thick plate analysis. There have been many thick (shear deformable) plate
theories proposed with the implicit objective of reducing the error to less than O(h 2/t~2 ).
Reissner (1944, 1945) proposed the simplest thick plate theory by introducing the
effect of transverse shear deformation through a complementary energy principle. Unlike
Reissner's work, Mindlin (1951) presented a first-order theory of plates where he accounted
for shear deformation in conjunction with a shear correction factor. In this theory, the first
two Kirchhoff assumptions are maintained. To allow for the effect of transverse shear
deformation, the theory relaxes the normality assumption so that
Normals to the undeformed midplane remain straight and unstretched in length
but not necessarily normal to the deformed midplane.
This assumption implies a non-zero transverse shear strain, but it also leads to the
statical violation of zero shear stress at the free surfaces since the shear stress becomes
constant through the plate thickness. To compensate for this error, Mindlin (1951) proposed a
shear correction factor cI 2 to be applied to the shear force. Besides, Mindlin (1951) modified
the fourth assumption so that the
6 Sec. 2.1 Mindlin Plate Theory
(cid:12)9 Effect of rotary inertia is included
In the literature, vibrating plates based on the first-order shear deformation plate
theory assumptions are widely referred to as Mindlin plates.
2.1.1 DISPLACEMENT COMPONENTS
In the Mindlin plate theory, the displacement components are assumed to be given by:
u(x, y,z,t) = zNx(x, y,t ) (2.1a)
v(x, y,z,t) = Zp'y (x, y,t) (2.1b)
~(x, y,z,t) = w(x, y,t) (2.1c)
where t is the time, u, v are the inplane displacements, w the transverse displacement and
~x, ~y the bending rotations of a transverse normal about the y and x axes, respectively, as
shown in Fig. 2.1. The notation that xN represents the rotation about the y-axis and vice
versa may be confusing to some and in addition they do not follow the right-hand rule.
However, these notations will be used herein because of their extensive use in the open
literature. Note that by setting r =-dw/dx and p,y =- dw/dy, the Kirchhoff plate theory
may be recovered. For higher-order plate theories, higher-order polynomials are used in the
expansion of the displacement components through the thickness of plate (see, for example,
papers by Nelson and Lorch 1974, Lo et al. 1977, Levinson 1980, Reddy 1984, Lim et al.
1989). Notable among them is the one proposed by Reddy (1984, 1997) who derived the
displacement field by imposing zero transverse shear strain condition at the free surfaces a
priori to the expanded inplane displacements up to the third power for the thickness
coordinate. Based on this displacement field, he derived variationally consistent equations
and boundary conditions for the so-called Reddy (third-order shear deformation) plate theory.
v%c ~x
ax
LZ x
Fig. 2.1 Rotations of the normals
Chap. 2 Mindlin Plate Theory and Ritz Method 7
2.1.2 STRAIN-DISPLACEMENT RELATIONS
In view of Eq. (2.1), the linear components of the engineering (non-tensorial) strains can be
expressed as
,~3c x
~x~ = z ~ (2.2a)
&
s :Z Y'/~( Oy (2.2b)
ezz =0 (2.2c)
yx'} : Z + (2.2d)
OW
(2.2e)
7" = x'~ +
aX
OW
Y yz --" yCb~ -" (2.20
cry
where c a , Cyy,Czz, are the normal strains and Yxy, Yxz, Yyz, the shearing strains.
2.1.3 STRESS RESULTANT-DISPLACEMENT RELATIONS
Based on the above strain-displacement relations and assuming a plane stress distribution in
accordance with Hooke's law, the stress-resultants are obtained by integrating the stresses:
M xx = Crxxz dz =
J-h~2 d-h~2 I_V 2
(2.3a)
Myy = J-h~2 o'yyz dz = d-h~2 1-v
=D +v (2.3b)
f h/2 f h/2
M y~ = %, z dz = G T y~ z dz
(cid:12)9 l-h~2 ,l-h~2
D(I- v) &,x +
(2.3c)
(2.3d)
J-h/2 rxz dz = K Gh ~'x +
