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EFFECT OF TRANSVERSE SHEAR DEFORMATION ON THE BENDING OF
ELASTIC PLATES
BY
EKANEM, UDOM KING
PG/M.ENG/09/50565
DEPARTMENT OF CIVIL ENGINEERING, FACULTY OF ENGINEERING,
UNIVERSITY OF NIGERIA, NSUKKA.
JULY, 2012
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EFFECT OF TRANSVERSE SHEAR DEFORMATION ON THE BENDING OF
ELASTIC PLATES
BY
EKANEM, UDOM KING
PG/M.ENG/09/50565
A MASTER DEGREE PROJECT SUBMITTED TO THE DEPARTMENT OF CIVIL
ENGINEERING, FACULTY OF ENGINEERING,
UNIVERSITY OF NIGERIA, NSUKKA.
IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE AWARD OF
MASTER OF ENGINEERING DEGREE (M. ENG) IN CIVIL ENGINEERING
(STRUCTURAL ENGINEERING)
JULY, 2012
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CERTFICATION
I, EKANEM, UDOM KING (PG/M.ENG/09/50565) hereby certify that this research work
“Effect of transverse shear deformation on the bending of Elastic plates” is original to me and
has not been submitted elsewhere for the award of a diploma or degree.
__________________________ _______________________
EKANEM, UDOM KING DATE
(PG/M.ENG/09/50565)
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APPROVAL
This work “Effect of Transverse Shear Deformation on the Bending of Elastic Plates” is hereby
approved as a satisfactory research work for the award of a master degree (M. Eng) in the
Department of Civil Engineering, Faculty of Engineering, University of Nigeria, Nsukka.
_______________________________ __________________
ENGR. PROF. N. N. OSADEBE DATE
(PROJECT SUPERVISOR)
______________________________ __________________
DATE
(EXTERNAL EXAMINER)
________________________________ __________________
ENGR. PROF. O. O. UGWU DATE
HEAD, CIVIL ENGINEERING
DEPARTMENT, UNIVERSITY OF
NIGERIA, NSUKKA.
________________________________ __________________
ENGR. PROF. T. C. MADUEME DATE
DEAN, FACULTY OF ENGINEERING,
UNVIERSITY OF NIGERIA, NSUKKA
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DEDICATION
This work is dedicated to my uncle, Chief Ime Effiong Ekanem.
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ACKNOWLEDGEMENT
I wish to express my sincere gratitude and appreciation to my supervisor, Engr. Professor
N. N. Osadebe, for guiding and encouraging me through out the period this research work lasted,
who also made everything clear and easy for me. His pain taking supervision and intellectual
assistance will ever remain fresh in my mind. Thank you Prof. for ever being there for me.
I am especially thankful to the external supervisor for proof-reading my work. I sincerely
appreciate his scientific and technical input to this work. I thankfully acknowledge my
indebtedness to all the members of staff of the Department of Civil Engineering, University of
Nigeria, Nsukka, who had in one way or the other impacted knowledge in me and especially my
course lecturers in the program; The Dean, Faculty of Engineering, University of Nigeria,
Nsukka, in the person of Engr. Professor T. C. Madueme, the Head, Department of Civil
Engineering, University of Nigeria, Nsukka, in the person of Engr. Professor O. O. Ugwu, Engr.
Professor J. C. Agunwamba, Engr. Dr. C. U. Nwoji, Engr. Dr. F. O. Okafor, Engr. Dr. B. O.
Mama, Engr. Dr. H. N. Onah, Engr. Adamou A. and Dr. C. C. Nnaji to mention but a few.
I also express my sincere thanks to my colleagues in the program who were always there
to criticize and this criticism had contributed to the success of this work. My profound gratitude
goes to my family members for their inspiration and encouragement. Above all, I thank the
Almighty God for my Life, His kindness and infinite mercy and for making me who I am today.
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ABSTRACT
Thin plates are initially flat structural members bound by two parallel planes called faces.
A plate resists transverse loads by means of bending extensively. The first satisfactory theory of
plate bending is associated with Navier and later Kirchhoff. These theories, however, just as for
any other approximation theory, are deficient as they do not take into account the effect of
transverse shear deformation on the bending of elastic plates.
This work, which demonstrates the transverse shear effect on the bending of elastic
plates, presents a refined theory which takes into account the transverse shear deformation on the
plate deformation. Resissner‟s and Mindlin‟s theories were evolved together with Helmholtz
equation. The differential equation of plate with shear effect was obtained using the Fourier
Double Trigonometry. The obtained equation was used for a simply supported rectangular plate
subjected to uniformly distributed load. The results showed that for thin and fairly thin plates,
transverse shear force has an effect on the plate deformation.
This work is therefore aimed at putting clarity on some burning issues associated with
bending of elastic plate according to Kirchhoff‟s hypotheses. Kirchhoff‟s work neglects
transverse shear effect. This research work has established that Transverse shear deformation has
an effect on the bending of elastic plate.
viii
TABLE OF CONTENTS
Certification - - - - - - - - - i
Approval - - - - - - - - - ii
Dedication - - - - - - - - - iii
Acknowledgement - - - - - - - - iv
Abstract - - - - - - - - - v
Table of Contents - - - - - - - - vi
List of Figures - - - - - - - - viii
List of Symbols - - - - - - - - - ix
CHAPTER ONE: INTRODUCTION
1.1 General Introduction - - - - - - - 1
1.2 Statement of Problem - - - - - - - 3
1.3 Objectives of Study - - - - - - - 5
1.4 General methodology - - - - - - - 6
1.5 Significance of Study - - - - - - - 6
1.6 Scope of Work - - - - - - - 7
1.7 Limitations of Study - - - - - - - 7
CHAPTER TWO: LITERATURE REVIEW
2.1 Introduction - - - - - - - - 8
2.2 Historical Background of Plates - - - - - 9
2.3 Types of Plates - - - - - - - 18
2.4 Refined Theory of Thin and Moderately Thick Plate - - 22
2.5 General Behaviour of Plates - - - - - - 23
2.6 Stress at a point - - - - - - - 24
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2.7 Strains and Displacements - - - - - - 25
2.8 Constitute Equations - - - - - - - 30
2.9 The Fundamentals of the Small Deflection plate Bending Theory - 31
2.9.1 Strain- Curvature Relations (Kinematic Equations) - - 31
2.9.2 Stresses, Stress Resultants and Stress Couples - - 36
CHAPTER THREE: METHODOLOGY
3.1 Governing Equation for Deflection of Plates in Cartesian Coordinates 43
3.2 Boundary Conditions - - - - - - - 46
3.3 The Effect of Transverse shear Deformation - - - - 50
3.4 Refined Theory of Bending Plates - - - - - 52
3.5 The Governing Equations of the Refined Plate Bending Theory - 53
3.6 Boundary Conditions - - - - - - - 61
3.7 Application of the Refined Theory by Double Trigonometric Series
(Navier Solution) - - - - - - - 64
3.8 Stresses Generated due to the Deflection in Equation 3.84 - - 74
CHAPTER FOUR: RESULTS AND DISCUSSION
4.1 Results - - - - - - - - 77
4.2 Discussion of Results - - - - - - - 84
CHAPTER FIVE: CONCLUSION AND RECOMMENDATIONS
5.1 Conclusion - - - - - - - - 86
5.2 Recommendation - - - - - - - 87
References - - - - - - - - - 88
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LIST OF FIGURES
Figure 2.1 : Flat Slab
Figure 2.2 : Thick Plates Internal forces
Figure 2.3 : Load-Free Plate
Figure 2.4 : Displacements on Elastic Body
Figure 2.5 : Strains in a parallelepiped
Figure 2.6 : Deformation of a parallelepiped
Figure 2.7 : Section of a Plate
Figure 2.8 : Plate Curvature
Figure 2.9 : Components of Stress
Figure 2.10 : Moments and the Shear Forces Acting on the Plate Element
Figure 3.1 : Boundary Conditions of a Plate
Figure 3.2 : Statically Equivalent Replacement of Couples of Horizontal Forces by
couples of vertical forces
Figure 3.3 : Simply supported rectangular plate under uniform distributed load
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LIST OF SYMBOLS
a and b: Rectangular plate dimensions
h : Plate thickness
E : Young‟s modulus
D : Flexural rigidity
P : Distributed load intensity
w : Deflection
4
: Biharmonic operator
G : Shear modulus
: Poisson‟s ratio
Q : Shear force
C : Shear stiffness
: Potential function
: Stream function
τ : Shear stress
σ : Normal stress
: Shear strain
ε : Normal strain
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CHAPTER ONE
INTRODUCTION
1.1 General Introduction
Thin plates are initially flat structural members bound by two parallel planes, called
faces. The distance between the plane faces is called the thickness (h) of the plate. The plate
thickness is assumed to be small compared with other characteristic dimensions of the faces
(length, width, diameter, etc). Geometrically, plates are bound either by straight or curve
boundaries. The static or dynamic loads carried by plates are predominantly perpendicular to the
plate faces. These loads are referred to as transverse loads on the plates.
The load-carrying action of a plate is similar, to a certain extent, to that of beams or
cables; thus, plates can be approximated by a grid work of an infinite number of beams or by a
network of an infinite number of cables, depending on the flexural rigidity of the structures.
This two-dimensional structural action of plate results in lighter structures, and therefore
offers numerous economic advantages. The plate, being originally flat, develops shear forces,
bending and twisting moments to resist transverse loads. Because the loads are generally carried
in both directions and because the twisting rigidity in isotropic plates is quite significant, a plate
is considerably stiffer than a beam of comparable span and thickness. Hence, thin plates combine
light weight and form efficiency with high load-carrying capacity, economy and technological
effectiveness.
In the light of the above advantage, thin plates are extensively used in all fields of
engineering. Thus, plates are used in architectural structures, bridges, hydraulic structures,
pavements, containers, airplanes, missiles, ships, instruments, machine parts, etc.
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A plate resists transverse loads by means of bending, exclusively. The first satisfactory
theory of bending plates is associated with Navier, who considered plate thickness in the general
plate equation as a function of rigidity, D. He also introduced an exact method which
transformed the differential equation into algebraic expressions by use of Fourier trigonometric
series. In 1850 Kirchhoff (Ventsel, E. and T. Krauthammer, 2001) published an important thesis
of thin plates. In this thesis, Kirchhoff stated two independent basic assumptions that are now
widely accepted in the plate-bending theory and are known as “Kirchhoff‟s hypothesis”. He
pointed out that there exist only two boundary conditions on a plate edge.
Kirchhoff‟s hypothesis permitted the creation of the classical bending theory of thin
plates which for more than a century has been the basis for the calculation and design of
structures in various areas of engineering and has yielded important theoretical and numerical
results. However, just as for any other approximation theory, Kirchhoff‟s theory has some
drawbacks and deficiencies. The most important assumption of Kirchhoff‟s plate theory is that
normal to the middle surface remains normal to the deflected mid-plane and straight. Since this
theory neglects the deformation caused by transverse shear, it would lead to considerable errors
if applied to moderately thick plates. For such plates, Kirchhoff‟s classical theory under-
estimates deflections and over-estimates frequencies and buckling loads.
Numerous researchers have attempted to refine Kirchhoff‟s theory and such attempts
continue to this day. E. Reissner (Reissner, E, 1944 and 1945) made the most important advance
in this direction. Reissner‟s theory takes into account the influence of the transverse shear
deformation on the deflection of the plate and leads to a sixth-order system of governing
differential equations, and accordingly, to three boundary conditions on the plate edge. Here, it is
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unnecessary to introduce the effective transverse shear force. Reissner‟s theory is free from the
drawbacks of Kirchhoff‟s theory.
The correct interpretation of Reissner‟s theory is complicated substantially by the fact
that this theory involves the variational procedure to derive the governing equations, which was
essentially based on the use of Kirchhoff‟s theory to approximate the stress distribution over the
thickness of the plate. Therefore, another approach is presented below for obtaining the
governing differential equation of the refined plate bending theory, which takes into account the
transverse shear deformation. According to this approach, the above equations are derived from
the equations of the theory of elasticity and contain physical hypothesis. This approach was
developed by Vasil‟ev (Vasil‟ev, V. V., 1998) and we follow the outline given in this reference
1.2 Statement of Problem
Kirchhoff‟s hypotheses which permitted the creation of the classical Kirchhoff‟s bending
theory of thin plates, for more than a century, has been the basis for calculation and design of
structures in various areas of engineering and has yielded important theoretical and numerical
results. However, just as for any other approximation theory, Kirchhoff‟s theory has some
drawbacks and deficiencies, two of which are:
(a) A well-known disagreement exists between the order of governing differential equation
of plate obtained by using the Kirchhoff‟s hypotheses and the number of the boundary
conditions on the plate free edge. As a result, the boundary conditions of the classical
theory take into account only two characteristics on the free edge of the plate rather than
three characteristic corresponding to the reality. Of the two conditions, only the first
condition (imposed on the bending moment) has a clear physical interpretation. The
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reduction of the twisting moment to transverse shear force is not justified in the general
case (Donnel, L. H., 1976). As a result of the replacement of the transverse shear force,
Qx (Qy) and twisting moment Mxy by their combination, effective shear force Vx (Vy), the
self-balanced tangential stresses remain at the free edge of a plate and the concentrated
forces arise at the corner points of a rectangular or polygonal plate when this reduction is
used. The role of the latter forces is still not clear (Donnel, L. H., 1976 and Alfutov, N.
A., 1992).
(b) Certain formal contradictions take place between the Kirchhoff‟s plate theory and the
three-dimensional equations of elasticity. The most important of the above contradictions
is associated with Hooke‟s law for the transverse shear stresses: τxz and τyz
That is,
2.1....................................................................
1.1...................................................................
G
G
yz
yz
xz
xz
Where,
G = Modulus of Elasticity in Shear
xz and yz are transverse shear deformations
xz and yz are transverse shear stresses.
In fact, the deformations yzxz and are absent according to the hypotheses of the classical
theory. However, the stresses yzxz and cannot be equal to zero, as it would be expected from
equations 1.1 and 1.2, because the shear forces Qx and Qy, which are resultants of the above-
mentioned stresses, are necessary for an equilibrium of the plates differential element.
xvi
Notice that as a result of an inaccuracy of Kirchhoff‟s theory, we cannot guarantee that
the stress distribution predicted by this theory will agree well with the actual stresses in the
immediate vicinity of the plate edge. Hence, the latter statement acquires a practical importance
of a refinement of the classical plate theory for plate fields neighboring to a boundary or to
opening whose diameter (or another typical dimension), is not too large compared with the plate
thickness.
1.3. Objectives of Study
The refined plate theory presented in this study aims at putting clarity on some burning
issues associated with bending of elastic plate according to Kirchhoff‟s hypotheses. These
burning issues in this study are:
(i) Influence of shear deformations on the state of stress in plate bending problems
(ii) Resolutions of the contradictions and drawbacks of the classical Kirchhoff‟s theory.
(iii) Establishment of elastic plate bending refined equation for plates under the influence of
shear deformations.
(iv) Application of the refined theory of plate bending problems with transverse shear
deformations.
(v) Numerical comparative analysis of Kirchhoff‟s equation of plate and the equation of plate
where as the influence of shear deformations exist.
xvii
1.4 General Methodology
The approach to the realization of the objectives of the study will be initiated by
reviewing the varying hypothesis developed by Kirchhoff (Ventsel, E. and T. Krauthammer,
2001) and Vasil‟ev (Vasil,ev, V. V., 1998). Following the outlines of these hypotheses, the
varying governing equations of plate bending will be obtained. Therefore, prompting the
discussion of their differences.
Efforts would be further made to obtain the solution of the governing differential
equations of the two plate hypotheses following their respective hypotheses. For the Vasil‟ev
refined theory, the so-called potential function of displacement field shall be introduced in
obtaining its governing differential equation.
The boundary conditions of these hypotheses shall equally be looked at and the
drawbacks of Kirchhoff‟s boundary conditions taken into account in the Vasil‟ev theory, leading
to clearer expressions.
These equations obtained from the two hypotheses shall be subjected to numerical
applications for comparative analysis, while discussions and conclusions will be outlined.
1.5 Significance of Study
This study will hopefully be relevant in the following respects:
(a) For the calculation and design of structures in various areas of engineering.
(b) It will have valuable implication for the analysis of the state of stress in plate fields with
cracks, holes whose diameter is so small as to be of the order of magnitude of the plate
thickness, concentrated load, etc.
xviii
(c) Also, it has been observed that the effect of the shear strains is more pronounced in
orthotropic plates than isotropic plates. Hence, this study has the potential of generating
further research in the analysis of orthotropic plates, because it has been observed that the
effect of the shear strains would be more on orthotropic plates.
1.6 Scope of Work
This work is delimited to the shear deformation effects of elastic isotropic rectangular
plates. Nevertheless, the equation obtained from such a plate throws up solution for orthotropic
plate situation. Example of such orthotropic plates is the grid-work system, which is similar to an
isotropic plate with holes.
The equations established in the study are associated with the assumptions, equations,
experiments and results of previous work done on elastic materials. No further attempts would be
made in experimenting the properties of elastic materials now known as Hooke‟s materials. The
plate is seen to be laterally loaded, thus neglecting the buckling effects.
1.7 Limitations of Study
The results of this work will be affected by the choice of only the first series of the shape
function of the deflected surface w(x,y); thus, resulting to small deviation from the exact results,
with very negligible percentage error. However, the improvements of such results are very
possible by selection of terms as many as possible in the shape function which entails very
complex mathematics.
xix
CHAPTER TWO
LITERATURE REVIEW
2.1 Introduction
Plates are straight, plane, two or three-dimensional structural components of which one
dimension, referred to as thickness, h, is much smaller than other dimensions. Geometrically
they are bound either by straight or curve lines. Like their counterparts, the beams, they are not
only serving as structural components but can also form complete structures such as slab bridges,
for example. Statically, plates have free, simply supported and fixed boundary conditions,
including elastic supports and elastic restraints or, in some cases, even point supports, figure 2.1.
The static and dynamic loads carried by plates are predominantly perpendicular to the plate
surface. These external loads are carried by internal bending and torsional moments and by
transverse shear force.
Since the load-carrying action of plates resembles to a certain extent, that of beams,
plates can be approximated by grid-works of beams. Such an approximation, however, arbitrarily
breaks the continuity of the structure and usually leads to incorrect results unless the actual two
dimensional behavior of plates is correctly accounted for.
The two-dimensional structural action of plates results in lighter structures and therefore,
offers economical advantages. Furthermore, numerous structural configurations require partial or
even complete enclosure that can easily be accomplished by plates, without the use of additional
covering resulting in further savings in material and labor costs. Consequently, plates and plate-
type structures have gained special importance and notably increased applications in recent
years. A large number of structural elements in engineering structures can be classified as plates.
Typical examples in civil engineering structures are floor and foundation slabs, lock-gates, thin
xx
retaining walls, bridge decks and slab bridges. Plates are also indispensable in ship building and
aerospace industries, the wings and a large part of the fuselage of an aircraft, for example,
consist of a slightly curved plate skin with an array of stiffened ribs.
Figure 2.1 Flat slab
The hull of a ship, its decks and its super structure are further examples of stiffened plate
structures. The majority of plate structures is analyzed by applying the governing equations of
the theory of elasticity.
2.2 Historical Background of Plates
Although the ancient Egyptians, Greeks and Romans already employed finely cut stone
slabs in their monumental buildings in addition to the most widely used tomb stones, there is a
fundamental difference between these ancient applications of slab and those of plates in modern
engineering structure. That is, the ancient builders established the slab dimensions and the load-
carrying capacity by “rule of thumb” handed down from generation to generation whereas
column (point)
support
xxi
nowadays engineers determine plate dimensions by applying various proven scientific methods
(Szilard, R., 2004).
The history of the evolution of scientific plate theories and pertinent solution techniques
is quite fascinating. While the development of structural mechanics as a whole commenced with
the investigation of static problem (T. Dhunter, I. and Pearson, K. A., 1960), the first analytical
and experimental studies on plates were devoted almost exclusively to free vibrations.
The first mathematical approach to the membrane theory of very thin plates was
formulated by L. Euler (1707-1783) in 1766 (Szilard, R., 2004). Euler solved the problems of
free vibrations of rectangular, triangular and circular elastic membranes by using the analogy of
two systems of stretched strings perpendicular to each other. His student, Jacques Bernoulli;
(1759-1789), extended Euler‟s analogy to plates by replacing the net of strings with a grid-work
of beams having only bending rigidity. Since the torsional resistance of the beams was not
included in the so-obtained differential equation of plates, he found only general resemblance
between his theory and experiments but no close agreement.
A real impetus to the research of plate vibrations, however, was given by the German
Physicist E. F. F. Chladni (1756-1827) (Szilard, R., 2004). In his book on acaustics, according to
Szilard, Chaldni described diverse experiments with vibrating plates. Chladni discovered various
modes of free vibrations. In his experiments he used evenly distributed powder that formed
regular patterns after introducing vibrations. The powder accumulated along the nodal lines,
where no displacement occurred. In addition, he was able to determine the frequencies
corresponding to these vibration patterns. Invited by the French Academy of Science in 1909, he
demonstrated his experiments in Paris. Chladni‟s presentation was also attended by Emperor
Napoleon, who was duly impressed by his demonstration. Following Napoleon‟s suggestion, the
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French Academy invited applications for a price essay dealing with the mathematical theory of
plate vibrations substantiated by experimental verification of the theoretical results. Since, at
first, no papers were submitted, the delivery date had to be extended twice. Finally in October
1811, on the closing day of the application, the Academy received only one paper, entitle
“Reserches sur la theorie des surfaces elastiques,” written by the mathematician Mlle Germain
(S. Germain, “L‟etat des sciences et des Lettres,” Paris 1833).
Sophie Germain (1776-1831), according to Szilard (Szilard, R., 2004), was indeed a
colorful personality of her time. Already as a young girl, Mlle Germain began to study
mathematics in all earnest to escape the psychological horrors created by the excesses of French
Revolution. She even corresponded with the greatest mathematicians of her time, including
Lagrange, Gauss, and Legendre, using the Pseudonym La Blanc continuing, Szilard [8] hinted
that she might have used the Pseudonym since female mathematicians were not taken seriously
in her time. In 1806, when the French army occupied Braunsch weigh in Germany, where Gauss
lived at the time, she personally intervened by General pernetty on behalf of the city and
Professor Gauss to eliminate the imposed fines.
As reported by Szilard (Szilard, R., 2004) in his monograph, Germain in her first work on
the theory of plate vibration, used (following Euler‟s previous work on elastic curves) a strain
energy approach. But in evaluating the strain energy using the virtual work technique, she made
a mistake and obtained an erroneous differential equation for the free vibration of the plate in the
following form (Szilard, R., 2004):
1.2042
6
24
62
2
2
yx
z
yx
z
t
z
xxiii
Where Z(x,y,t) represents the middle surface of the plate in motion expressed in x, y, z
Cartesian coordinate system, t is the time and 2 denotes a constant containing physical
properties of the vibrating plate. This constant was however, not clearly defined in her paper.
Lagrange who was one of the judges, noticed this mathematical error and corrected it. The so-
obtained differential equation now correctly describing the free vibrations of plates reads:
2.2..................................................02
t
Z4
4
22
4
4
42
2
2
y
Z
yx
Z
x
ZK
Since the judges were not entirely satisfied with Germain‟s work, they proposed the
subject again. In October 1813, Mlle. Germain entered the now correct equation [8] but left out
the precise definition of the constant k2. She won the prize in 1816. But the judges criticized her
new definition of the constant k2 since she had thought that it contains the fourth power of the
plate thickness instead of the correct value of h3. Although her original works are very hard to
read and contain some dubious mathematical and physical reasoning, but she must, nevertheless
be admired for her courage, devotion and persistence. The claim of priority for writing the first
valid differential equations belongs – without any doubt to her.
Next, Szilard (Szilard, R., 2004) wrote that the mathematician L. D. Poisson (1781-1840)
made an attempt to determine the correct value of the constant k2
in the differential equation 2.2
of plate vibration. By assuming, however, that the plate particles are located in the middle plane,
he erroneously concluded that this constant is proportional to the square of the plate thickness
and not to the cube. Later, in 1828, Poisson extended the use of Navier‟s equation to lateral
vibration of circular plates. The boundary conditions of the problem formulated by Poisson,
however, are applicable only to thick plates. Three boundary conditions derived by Poisson had
been the subject of much controversy and were the subject of further investigation (.Ventsel, E.
and T. Krauthammer, 2001)
xxiv
Finally, the famous engineer and scientist L. Navier (1785-1836) can be credited with
developing the first correct differential equation of plates subjected to distributed, static lateral
loads Pz (x, y), (Szilard, R., 2004 and Ventsel, E. and T. Krauthammer, 2001). The task, which
Navier set himself, was nothing less than the introduction of rigorous mathematical methods into
structural analysis. In his brilliant lectures, which he held in Paris at the prestigious Ecole
Polytechnique on Structural Mechanics, Navier integrated for the first time the isolated
discoveries of his predecessors and the results of his own investigations into a unified system.
Consequently, the publications of his textbook, Lecons (Navier, L. M. H., 1819) on this subject
was an important milestone in the development of modern structural analysis.
Navier applied Bernoulli‟s hypotheses which were already successfully used for treating
bending of beams, adding to them the two-dimensional actions of strains and stresses,
respectively. In his paper on this subject (published in 1823), he correctly defined the governing
differential equation of plates subjected to static, lateral loads, Pz as:
3.2.....................................................,2
4
4
22
4
4
4
yxPy
w
yx
w
x
wD z
Where D denotes the flexural rigidity of the plate, which is now proportional to the cube of the
plate thickness, where as w (x,y) represents the deflected middle surface.
For the solution of certain boundary value problems of rectangular plates, Navier
introduced a method that transforms the plate differential equation into algebraic equations. His
approach is based on the use of double trigonometric series introduced by Fourier during the
same decade. This so-called forced solution of the plate differential equation (2.3) yields
mathematically correct values to various problems with relative ease provided that the boundary
conditions of plates are simply supported. He also developed a valid differential equation for
xxv
lateral buckling of plates subjected to uniformly distributed compressive forces along the
boundary. He failed, however to obtain a solution to this more difficult problem, Szilard (Szilard,
R., 2004) reported. Navier‟s further theoretical works established connections between elasticity
and hydrodynamics based on a “molecular hypothesis,” to which he was as firmly attached as
Poisson (Dugas, R., 1988).
The high-quality engineering education given at the Ecole Polytechnique set the
standards for other European countries during the nineteenth century. The German
Polytechniques, established soon after the Napoleonic wars, followed the very same plan as the
French. The engineering training began with two years of courses in Mathematics, Mechanics
and Physics and concluded with pertinent design courses in the third and fourth years,
respectively such a thorough training produced a succession of brilliant scientists in both
countries engaged in developing the science of engineering in general and that of strength of
materials in particular.
In Germany, publication of Kirchhoff‟s book entitled lectures on Mathematical Physics,
Mechanics (in German) [8] created a similar impact on engineering science as that of Navier‟s
Lecons in France. Gustav R. Kirchhoff (1824 – 1887), developed the first complete theory of
plate bending. In his earlier paper on this subject, published in 1850, he summarized first, the
previous work done by the French scientists in this field, but he failed to mention Navier‟s
above-discussed achievement. Based on Bernoulli‟s hypotheses for beams, Kirchhoff derived the
same differential equation for plate bending as Navier, however, using a different energy
approach. His very important contribution to plate theory was the introduction of supplementary
boundary forces. These “equivalent shear forces” replace, in fact, the torsional moments at the
plate boundaries. Consequently, all boundary conditions could now be stated in function of
xxvi
displacements and their derivation with respect to x and y. Furthermore, Kirchhoff is considered
to be the founder of the extended plate theory, which takes into account the combined bending
and stretching. In analyzing large deflection of plates, he found that non-linear terms could no
longer be neglected. His other significant contributions are the development of a frequency
equation of plates and the introduction of virtual displacement methods for solution of various
plate problems. Kirchhoff‟s book (Kirchhoff, G., 1876) was translated into French by Clebsch.
His translation contains numerous valuable comments by Saint-Venant, the most important being
the extension of the differential equation of plate bending, which considers, in a mathematically
correct manner, the combined action of bending and stretching.
Another famous textbook that deals with the abstract mathematical theory of plate bending is
Love‟s principal work, a treatise on the mathematical theory of Elasticity (Love, A. E. H., 1926).
In addition to an extensive summary of the achievements made by his already mentioned
predecessors, Love considerably extends the rigorous plate theory by applying solutions of two-
dimentional problems of elasticity to plates.
Around the turn of the century, shipbuilders changed their construction methods by
replacing wood with structural steel. This change in the structural material was extremely fruitful
for the development of various plate theories (8 and 9). Russian scientists made a significant
contribution to naval architecture by being the first to replace ancient shipbuilding traditions by
mathematical theories of elasticity. Especially Krylov (1863 -1945) (Krylov, A., 1898) and his
student Boobnov (Boobnov, I. G., 1902, 1912, 1914 and 1953) contributed extensive to the
theory of plates with flexural and extensional rigidities. Because of the existing language barrier
(Szilard, R., 2004) the western world was slow to recognize these achievements and make use of
them. It is to Timoshenko‟s credit that the attention of the western scientists was gradually
xxvii
directed towards Russian research in the field of theory of elasticity. Among Timoshenko‟s
numerous important contributions are the solution of circular plats considering large deflections
(Timoshenko, S. P., 1915) and the formulation of Elastic stability problems (Ventsel, E. and T.
Krauthammer, 2001).
Foppl, in his book on engineering mechanics, first published in 1907, had already treated
the nonlinear theory of plates, Szilard (Szilard, R., 2004) narrated. The final form of the
differential equation of the large-deflection theory, however, was developed by the Hungarian
scientist Von Karman (Von Karman, T. H., ettal, 1932) who in his latter works also investigated
the problem of effective width and the post buckling behaviours of plates, Szilard (Zsilard, R.,
2004) remarked.
The book of another Hungarian engineer-scientist, Nadai, according to Szilard was
among the first devoted exclusively to the theory of plates. In addition to analytical solutions of
various important plate problems of the engineering practice, Szilard deposited that Nadai also
used the finite difference technique to obtain numerical results where the analytical methods
failed.
Westergaard (Westergaard, H. M., 1923) and Scheicher (Schleicher, E., 1926) investigated
problems related to plates on elastic foundation. Prescott, in his book Applied Elasticity
(Prescott, J., 1924) introduced a more accurate theory for plate bending by considering the
strains in the middle surface. The polish scientist Huber investigated orthotropic plates subjected
to non-symmetrical distributed loads and edge moments.
The development of modern aircraft industry provided another strong impetus towards more
rigorous analytical investigations of various plate problems. Plates subjected to, for example, in-
plane force, post buckling behaviour and stiffened plates were analyzed by many scientists
xxviii
(Szilard, R., 2004). Of the numerous researchers whose activities fall between the two world
wars, only Wagner, Levy, Bleich and Federhofer are mentioned here.
The most important of Kirchhoff‟s assumption of plate theory is that normal to the
middle surface remains normal to the deflected mid-plan and straight. Since this theory neglects
the deformation caused by the transverse shear, it would lead to considerable errors if applied to
moderately thick plates. For such plates, Kirchhhoff‟s classical theory underestimates deflections
and overestimates frequencies and buckling loads.
Reissner and Mindlin arrive at some what different theories for moderately thick plates to
eliminate the above-mentioned deficiency of the classical plate theory. The theory developed by
Reissner (Reissner, E., 1954) includes the effects of shear deformation and normal pressure by
assuming uniform shear stress distribution through the thickness of the plate. Applying his
theory, three instead of two boundary conditions must be satisfied on the edge. Of these three
displacement boundary conditions, one involves deflection and the other two represent normal
and tangential rotations respectively. Mindlin (Mindlin, R. D., 1951) also improved the classical
plate theory for plate vibrations by considering, in addition to the effect of shear deformation,
that of the rotary inertia. In Mindlin‟s derivation, displacements are treated as primary variables.
It was necessary, however, to introduce a correction factor to account for the prediction of
uniform shear stress distribution.
In addition to the analysis of moderately thick plates, Reissner‟s and Mindlin‟s plate
theories received a great deal of attention in recent years for the formulation of reliable and
efficient finite elements for thin plates. Since in both theories displacements and rotations are
independent and slope continuity is not required, developments of finite elements are greatly
facilitated.
xxix
Direct application of these higher-order theories to thin-plate finite elements, however, often
induced so-called shear locking behaviour, which first had to be overcome before such elements
could be used. To alleviate this undesirable effect, a refined theory is applied to the solution of
thin plate with transverse shear deformations in this work.
2.3 Types of Plates
In all structural analysis, the engineer is forced, due to the complexity of any real structure, to
replace the structure by a simplified analysis model equipped only with those important
parameters that mostly influence its static or dynamic response to loads. In plate analysis such
idealizations concern.
1. The geometry of the plate and its supports.
2. The behaviour of the material used, and
3. The type of loads and their method of application.
A rigorous elastic analysis would require, for instance that the plate should be considered as a
three-dimensional continuum. Needless to say, such an approach is highly impractical since it
would create almost insurmountable mathematical difficulties. Even if a solution could be found,
the resulting costs would be, in most cases, prohibitively high. Consequently, in order to
rationalize the plate analyses, we distinguish among four different differential equations.
The four plate-types might be categorized, to some extent, using the ratio of thickness to
governing length (h/l). Although, the boundaries between these individual plate types are some
what fuzzy, we can attempt to subdivide plates into the following major categories:
xxx
1. Stiff Plates (h/l = 1/50 - 1/10) are thin plates with flexural rigidity, carrying loads two
dimensionally, mostly by internal (bending and torsional) moments and by transverse
shear, generally in a manner similar to beams. In engineering practice, plate is
understood to be stiff plate unless otherwise specified.
2. Membranes (h/l < 1/50) are very thin plates without flexural rigidity, carrying loads by
axial and central shear forces. This load-carrying action can be approximated by a
network of stressed cable. Since, because of their extreme thinness, their moment
resistance is of negligible order.
3. Moderately Thick Plates (h/l = 1/10 – 1/5) are in many respects similar to stiff plates,
with the notable exception that the effects of transverse shear forces on the normal stress
components are also taken into account.
4. Thick Plates (h/l > 1/5) have an internal stress condition that resembles that of three
dimensional continua.
There is however, a considerable “gray” area between stiff plates and membranes, namely, if we
do not limit the deflection of stiff plates, we obtain so-called flexible plates, which carry the
external loads by the combined action of internal moments, transverse and central shear forces
and axial forces (figure 2.2 a to d). Consequently elastic plate theories distinguish sharply
between plates having small and large deflections. Plates having large deflections are avoided,
for the most part, in general engineering practice since they might create certain problems in
their analysis as well as in their use. The safety-driven and weight-conscious aerospace and
submarine-building industries are forced, however, to disregard these disadvantages since such
plates possess considerably increased load-carrying capacities.
xxxi
yz
(a) Plate
xy
(b) Membrane
Figure 2.2 (a and b): Internal forces in plates.
yzM
y
yx
Torsional
moment
Transverse shear
Bending moment
Torsional moment
h
Pz = lateral load
x
Pz
h
Central
force Axial force
Myz
Mzx
xxxii
(c) Plate with flexural and extension rigidities
(d) Thick Plate
Figure 2.2 (c and d): Internal forces in various types of plate elements.
dy =1
Pz h
Pz
h
y yz
yx
Myz
Myz
xy
xz
x
y
yz
yx
xz
x
xy
xxxiii
2.4 Refined Theories for Thin and moderately Thick Plates
Although classical plate theory yields sufficiently accurate results for thin plates, its
accuracy decreases with increasing thickness of the plate. Exact three-dimensional elasticity
analysis of some plate problems indicates that its error is on the order of plate thickness square.
Such an inherent limitation of classical plate theory for moderately thick plates necessitates the
development of more refined theories in order to obtain reliable results for the behaviour of these
structures.
Experiments have shown that Kirchhoff‟s classical plate theory underestimates
deflections and over-estimates natural frequencies and buckling loads for moderately thick
plates. These discrepancies are due to the neglect of the effect of transverse shear strains, since,
(following elementary beam theory) it is assumed that normal to the middle plane remain straight
and normal to the deflected mid-plane.
Nowadays, many plate theories exist that account for the effects of transverse shear
strains. The earliest attempt is due to M. Levy who in his pioneering work began to search for
solutions using the elasticity theory. More recently, primarily two approaches are used to take
into account the transverse shear deformations. In the first one, which is attributed to Reissner,
stresses are treated as primary variables. In the second approach, attributed to Mindlin,
displacements are treated as unknowns. Both approaches provide for further development.
Consequently, more sophisticated so-called higher-order theories have also been introduced in
the last decades. These higher-order theories have some complexities and therefore making room
for further work. It is in this regard that the background of this work was formed.
xxxiv
2.5 General Behaviour of Plates
Consider a load-free plate, shown in figure 2.3 in which the x-y plane coincides with the plate‟s
midplane and the z-coordinate is perpendicular to it and is directed upwards. The fundamental
assumption of the linear elastic, small-deflection theory of bending for thin plates may be stated
as follows.
Figure 2.3: Load-free plates
1. The material of the plate is elastic, homogenous and isotropic
2. The plate is initially flat
3. The deflection of the mid-plane is small compared with the thickness of the plate. The
slope of the deflected surface is therefore very small and the square of the slope is a
negligible quantity in comparison with unity.
4. The straight line, initially normal to the middle plane before bending remains straight and
normal to the middle surface during the deformation and the length of such element is not
altered. This means that the vertical shear strain yzzx and are negligible and the normal
strain z may also be omitted. This assumption is referred to as the hypothesis of straight
normal.
z
x
y
xxxv
5. The stress normal to the middle plane z is small compared with the other stress
components and may be neglected in the stress-strain relations.
6. Since the displacements of a plate are small, it is assumed that the middle surface remains
unstrained after bending.
Many of these assumptions known as „Kirchhoff‟s hypotheses are analologous to those
associated with the simple bending theory of beams. These assumptions result in the reduction of
a three dimensional plate problem to a two dimensional one.
Consequently, the governing plate equation can be derived in a concise and straight forward
manner. The plate bending theory based on the above assumption is referred to as classical or
Kirchhoff‟s plate theory.
2.6 Stress at a Point. (Stress tensor)
From theory of elasticity background, for an elastic body of any general shape subjected
to external loads which are in equilibrium, then for a material point anywhere in the interior of
the body, if we assign a cartesian coordinate frame with x,y,z axes, the stress tensor, Ts, is given
as:
Ts = 4.2....................................................
zzyzx
yzyyx
xzxyx
For equation 2.4, the reciprocity law of shear stresses holds; i.e.
5.2.......................................,, zyyzzxxzyxxy
xxxvi
Thus, only the six stress components out of nine in the stress tensor (2.4) are independent. The
stress tensor, Ts, completely characterizes the three-dimensional state of stress at a point of
interest.
For elastic stress analysis of plates, the dimensional state of stress is of special importance. In
this case, ;0 xzyzz
The 2-dimensional stress tensor has a form:
yxxy
yyx
xyx
s
where
T
,
6.2.................................................................................................
2.7 Strains and Displacements
Assume that the elastic body shown in figure 2.4 is supported in such a way that rigid body
displacements (translations and rotations) are prevented. Thus, this body deforms under the
action of external forces and each of its points has small elastic displacement. For example, a
point M had the coordinates, x, y and z in initial undeformed state. After deformation, this point
moved into position M1 and its coordinates become the following x
1 = x +u,
y1 = y + v, z
1 = z + w, where u, v, and w are projections of the displacement vector of point
M, vector MM1, on the coordinates axes x, y, z. In general case, u, v and w are function of x, y
and z.
xxxvii
Figure 2.4: Displacement on Elastic body
Again, consider an infinitesimal element in the form of parallelepiped enclosing point of
interest M. Assuming that a deformation of thin parallelepiped is small; we can represent it in the
form of the six simplest deformations shown in figure 2.5. The first three deformations shown in
figure 2.5 a, b and c define the elongation (or contraction) of edges of the parallelepiped in the
direction of the coordinate axes and can be defined as:
7.2......................................................;;
dz
dz
dy
dy
dx
dxzyx
The three other deformations shown in figure 2.5 d, e and f are referred to as shear strains
because they define a distortion of an initially right angle between the edges of the
parallelepiped. They are denoted by ., yzxzxy and The subscripts indicates the coordinate
planes in which the shear strains occur. Let us determine, for example, the shear strain in the x-y
coordinate plane. Consider the projection of the parallelepiped shown in figure 2.5 d, on this
coordinate plane.
z
M1(x+u,y+v,z+w)
.
. u
w v M(x,y,z)
y
x
xxxviii
(a)
(b)
z
x
dx dy
dz
y
dxdx x )(
z
dx
x
dy
dz
y
)(dydyy
z
dx
y
)(dzdzz
dy
dz
x
(c)
xxxix
Figure 2.5: Strains in parallelepiped
z
y
xy
x
(d)
z
y
yz
(e)
x
z
zx
y (f)
x
xl
Strains in equation 2.7 are called the normal or linear strains.
The increments dx can be expressed by the second term in the Taylor series, i.e.
,/ dxyudx etc. Thus, we can write Equation 2.7 as
az
w
y
v
x
uzyx 8.2....................................................,,
Figure 2.6 shows this projection in the form of the rectangle before deformation (ABCD) and
after deformation (A1B
1C
1D
1). The angle BAD in figure 2.6 deforms to the angle B
1A
1D
1, the
deformation being the angle ,111 thus, the shear is
).........(............................................................111 axy
Or it can be determined in terms of the in-plane displacement, u and v as follows:
y
v
y
u
x
ux
v
dydy
vy
dyy
u
dxx
ux
dxx
v
xy
11
Figure 2.6: Deformation of a Parallelepiped
A
u
dx B dx
x
uu
x(u)
dxx
vv
B1
1 A
1
11
v
D C dy
y
vv
dyy
uu
D1
C1
y(v)
xli
Since we have confined ourselves to the case of very small deflections, we may omit the
quantities x
u
and y
v
in the denominator of the last expression, as being negligibly small
compared with unity.
Finally, we obtain: )(......................................................... by
u
x
vxy
Similarly, we can obtain xz and zy . Thus, the shear strains are given by:
b
y
w
z
v
x
w
z
u
x
v
y
u
yz
xz
xy
8.2..................................................................................
Similarly to the stress tensor (2.4) at a given point, we can define a strain tensor as
9.2..................................................................
2
1
2
1
2
1
2
2
1
2
1
zzyzx
yzyyx
xzxyx
D
IT
The strain tensor is also symmetric because of the reciprocity of shear strains.
10.2.............................................................,, zyyzzxxzyxxy
2.8 Constitutive Equations
The constitutive equations relate the stress components to strain components. For the linear
elastic range, the equations represent the generalized Hooke‟s law. In this case of a
3-dimensional body, the constitutive equations are given by:
xlii
a
E
E
E
yxz
z
zxy
y
zyx
x
11.2......................................................................
b
G
G
G
yz
yz
xz
xz
xy
xy
11.2....................................................................................
;
Where, E, and G are the modulus of elasticity, Poisson‟s ratio, and the shear modulus
respectively. The following relationship exists between E and G:
12.212
EG
2.9 The fundamentals of the Small Deflection Plate Bending Theory
The foregoing assumptions introduced in section 2.5 make it possible to derive the basic
equation of classical or Kirchhoff‟s bending problems in terms of displacement.
2.9.1 Strain-Curvature Relations (Kinematic Equations)
Let u, v and w be components of the displacement vector of points in the middle surface
of the plate occurring in the x, y and z directions respectively. The normal component of the
displacement vector, w (called the deflection), and the lateral distributed load, P are positive in
the downward direction. As it follows from assumption (4) of section 2.5;
13.2...........................................................................................0,0,0 xzyzz
xliii
Integrating the expressions (2.8) for xzyzz and , , and taking into account equation 2.13, we
obtain;
14.2............................................................................................
),(
),(
),(
yxww
yxvy
wzv
yxux
wzu
z
z
z
Where, uz, v
z and w
z are displacements of points at a distance z from the middle surface. Based
on assumption (6) of section 2.5, we conclude that u = v = 0. Thus, equation 2.14 has the
following form in the context of Kirchhoff‟‟s theory:
15.2...........................................................................................
),(
yxww
y
wzv
x
wzu
z
z
z
Figure 2.7 shows a section of the plate by a plane parallel to 0xz, y = const. before and after
deformation. Consider a segment AB in the positive z direction. We focus on an arbitrary point B
which initially lies at a distance Z from the undeformed middle plane (from the point A). During
the deformation, point A displaces a distance w parallel to the original Z direction to point A1.
Since the transverse shear deformations are neglected according to Kirchooff‟s hypothesis, the
deformed position of point B must lie on the normal to the deformed middle plane erected at
point A, assumption(4). Its final position is denoted by B1. Due to assumption (4) and (5), the
distance Z between the above mentioned points during deformation remains unchanged and is
also equal to Z. We can represent the displacement uz and v
z ,equation 2.14 in the form:
xliv
16.2............................................................................................................
yzv
zu
z
x
z
Where,
17.2.................................................................................................................
y
wy
x
wx
Substituting equation 2.15 into the first two equation of 2.8a and into the first equation 2.8b, we
have;
18.2.................................................................................................
22
2
2
2
2
yx
wz
y
wz
x
wz
z
xy
z
y
z
x
Where, the superscripts z refers to the in-plane strain component at a point of the plate located at
a distance z from the middle surface since the middle surface deformations are neglected due to
the assumption (6). From here on, their superscripts will be omitted for all the strain and stress
components at points across the plate thickness. The second derivatives of the deflection on the
right-hand side of equations 2.18 have a certain geometrical meaning. Let a section MNP
represent some plane curve in which the middle surface of the deflected plate is intersected by a
plane y = const.
Due to the assumption (3) (section 2.7), this curve is shallow and the square of the slope
angle may be regarded as negligible compared with unity, i.e., .1
x
w
xlv
Then, the second derivative of the deflection 2
2
x
w
will define approximately the
curvature of the section along the x-axis, Xx.
Similarly 2
2
y
w
defines the curvature of the middle surface Xy along the y-axis. The
curvatures Xx and Xy characterize the phenomenon of bending of the middle surface in planes
parallel to the 0xz and 0yz coordinate planes respectively. They are referred to as bending
curvatures and are defined by
2
2
2
2
1
19.2................................................................................................
1
y
w
pX
x
w
pX
y
y
x
x
Figure 2.7: Section of a Plate
0 A
B
z
x,u
w w
A1 B1
θy u
z
θx
z,w
xlvi
We consider a bending curvature positive, if it is convex downward; i.e. in the positive
direction of the z-axis. The negative sign is taken in equation 2.19 since, for example, for the
deflection convex downward curve MNP (see figure 2.8,) the second derivative
negativeisx
w2
2
The curvature 2
2
x
w
can be also defined as the rate of change of the angle
x
wx
with
respect to distance x along this curve.
Figure: 2.8 Plate curvature
By analogy with the torsion theory of rods, the derivative yx
w
2
which defines the warping of the
middle surface at a point with coordinates x and y is called the twisting curvature with respect to
the x and y axes and is denoted by Xxy. Thus,
0
x
y
y
dy
z
M
M1 x
w
yx
w
yx
w
w
N
N1
P
P1
xlvii
20.2......................................................................................1 2
yx
w
PXX
xy
yxxy
Taking into account equations 2.19 and 2.20, we can write equation 3.15 as follows:
21.2..........................................................................2;; xyxyyyyxxx zXzXzX
2.9.2 Stresses, Stress Resultants and Stress Couples
In the case of a three – dimensional state of stress, stress and strain are related by the
equations 2.11 of the generalized Hooke‟s law. As was mentioned earlier, Kirchhoff‟s
assumptions of section 2.5 brought us to equation 2.13. From a mathematical stand-point this
means that the new three equations 2.13 are added to the system of governing equations of the
theory of elasticity. So, the later becomes over-determined and, therefore, it is necessary to also
drop three equations.
As a result, the three relations out of six of Hooke‟s law are discarded.
Moreover, the normal stress component .0z Solving equations 2.11 for stress components
,, xyyx and yields
22.2...................................................................................................1
1
2
2
xyxy
xyy
yxx
G
E
E
The stress components are shown in figure 2.9
xlviii
Figure: 2.9 Components of Stress
Introducing the plate curvature equations 2.19 and 2.20 and using equations 2.21, the above
equations appear as follows:
23.2..............................11 2
2
2
2
22
y
w
x
wEzXX
Ezyxx
dx
σx+ dxx
x
dxx
yx
yx
dy
dz
z
x
xz
xy
yx
yz
y
2h
2h
dx
x z
dz 2h
zx
σx
xlix
25.2......................................................11
24.2..................................11
2
22
2
2
2
2
22
yx
wEzX
Ez
x
w
y
wEzXX
Ez
xyxy
xyy
It is seen from equation 2.23 to 2.25 that Kirchhoff‟s assumptions have led to a completely
defined law of variation of the stresses through the thickness of the plate. Therefore, as in the
theory of beams, it is convenient to introduce instead of the stress components at a point
problems, the total statically equivalent forces and moments applied to the middle surface, which
are known as the stress resultants and stress couples are referred to as the shear forces, Qx and
Qy, as well as the bending and twisting moments Mx, My and Mxy respectively. Thus, Kirchhoff‟s
assumptions have reduced the three-dimensional plate straining problem to the two-dimensional
problem of straining the middle surface of the plate. Referring to figure 2.9, we can express the
bending and twisting moments, as well as the shear forces, in terms of stress components, i.e.
27.2........................................................................................................
26.2..............................................................................................
2
2
2
2
h
hyz
xz
y
x
h
h
xy
y
x
xy
y
x
Q
Q
and
zdz
M
M
M
Because of the reciprocity law of shear stresses (τxy = τyx), the twisting moments on
perpendicular faces of an infinitesimal plate element are identical, i.e.: Myx = Mxy
The sign convention for the shear forces and the twisting moments is the same as that for the
shear stresses. A positive bending moment is one which results in positive (tensile) stresses in the
l
bottom half of the plate. Accordingly, all the moments and the shear forces acting on the element
in figure 2.10 are positive. Note that the relations 2.26 and 2.27 determine the intensities of
moments and shear forces, i.e. moments and forces per unit length of the plate mid plane.
Figure 2.10: Moments and shear forces acting on the plate element.
Substituting equations 2.23 to 2.25 into equation 2.26 and integrating over the plate thickness,
we derive the following formulas for the stress resultants and couples in terms of the curvatures
and deflection:
0
x z y
dy
dx Mx Qx
Mxy
Qy
Myx My
P 2h
2h
dyy
MM
y
y
dyy
y
y
dyy
MM
yx
yx
dxx
QQ x
x
dxx
MM
xy
xy
dxx
MM x
x
li
88)1(3
88)1(3
3)1(
1
33
22
2
2
2
33
22
2
2
2
2/
2/
3
22
2
2
2
2
2
2
22
22
2
2
2
hhE
y
w
x
w
hhE
y
w
x
w
zE
y
w
x
w
dzy
w
x
wEz
zdzM
h
h
h
h
h
h xx
28.2.................................................................................)1(12 2
2
2
2
2
3
y
w
x
wEh
2
2
h
h yy zdzM
dzx
w
y
wEzh
h
2
2
2
22
22
2
1
2/
2/
3
22
2
2
2
3)1(
h
h
zE
x
w
y
w
88)1(3
33
22
2
2
2 hhE
x
w
y
w
88)1(3
33
22
2
2
2 hhE
x
w
y
w
lii
29.2...........................................................................................)1(12 2
2
2
2
2
3
x
w
y
wEh
2
2
h
h xyxy zdzM
platetheofrigidityflexuraltheisDEh
where
yx
wEh
hh
yx
wE
hh
yx
wE
z
yx
wE
dzyx
wEz
h
h
h
h
)1(12
,
30.2.................................................................................................)1(12
88)1(3
88)1(3
3)1(
)1(
2
3
23
332
332
2
2
32
22
2
2
i.e
DEhEh
)1()1(12)1(12
3
2
3
From 3.25 to 3.27,
liii
we have:
33.2................................................................................................................)1(
)1(
)1(
)1(
)1(12
32.2......................................................................................................
31.2.......................................................................................................
2
2
23
2
2
2
2
2
2
2
2
yx
wD
yx
wD
yx
wEhM
x
w
y
wDM
y
w
x
wDM
xy
y
x
liv
CHAPTER THREE
METHODOLOGY
3.1 Governing Equation for Deflection of Elastic Plates in Cartesian Coordinates
Consider equilibrium of an element dx x dy of the plate subject to a vertical distributed
load of intensity P(x,y) applied to an upper surface of the plate as shown in figure 2.10. Since the
stress resultants and stress couples are assumed to be applied to the middle plane of this element,
a distributed load P(x,y) is transferred to the mid plane. Note that as the element is very small,
the force and moment components may be considered to be distributed uniformly over the mid
plane of the plate element. For the system of force and moments shown in figure 2.10 the
following conditions of equilibrium may be set up:
(a) Shear forces equilibrium
The components of shear forces along the z- direction is given as
0
0
Pdxdydxdyy
Qdxdy
x
Q
PdxdydxQdxdyy
QQdyQdydx
yx
y
y
yx
x
x
x
Divide through by dxdy
1.3................................................................................................0
P
y
Q
x
Q yx
(b) Moment Equilibrium
Projecting forces (moments) along x-direction:
lv
02
.2
)(
2
)(
)0(
22
dxqdxdy
dxQ
dxdy
y
dxdydxx
QQdyQdxMdxdy
y
MMdyMdydx
x
MM
y
y
y
x
xxyx
yx
yxx
x
x
Note: square of small values 0
2.3.................................................................
0
x
yxx
x
yxx
Qy
M
x
M
dydxQdydxx
Mdxdy
x
M
Projecting forces (moments) along y-direction
022
)(
2
)(
)0(
22
dyqdxdy
dyQ
dydx
x
dxdydyy
QQdxQdyMdydx
x
MMdxMdxdy
y
MM
x
x
x
y
yyxy
xy
xyy
y
y
lvi
3.3.........................................................................
0
y
M
x
MQ
dxdyQdxdyx
Mdxdy
y
M
yxy
y
y
xyy
Putting equations 3.32 and 3.33 into equation 3.31
4.3............................................................2
2
22
2
2
2
222
2
2
Py
M
yx
M
x
M
Py
M
yx
M
yx
M
x
M
Py
M
x
M
yy
M
x
M
x
yxyx
yxyxyx
yxyyxx
Note; Mxy = Myx
Substituting the expressions Mx, My and Mxy given in 3.28, 3.29 and 3.30 into 3.34
5.3..................................................................2
2
2
2
2)1(
2
4
4
22
4
4
4
22
4
4
4
22
4
22
4
22
4
4
4
22
2
2
222
2
2
2
2
2
2
D
P
D
P
y
w
yx
w
x
w
Pyx
wD
y
wD
yx
wD
yx
wD
yx
wD
x
wD
Px
w
y
wD
yyx
wD
yxy
w
x
wD
x
This is the governing differential equation for the deflections of thin plate bending analysis based
on Kirchhoff‟s assumptions. This equation was obtained by Lagrange in 1811.
Equation 3.5 may be rewritten as follows:
6.3.........................................................................................422
D
Pww
Where,
lvii
7.3.........................................................................2
)(4
4
22
4
4
44
ydyxx
is commonly called the biharmonic operator.
3.2 Boundary Conditions
The boundary conditions are the known conditions on the surface of the plate which must
be prescribed in advance in order to obtain the solution of equation 3.5 corresponding to a
particular problem.
Such condition include the load P (x,y) on the upper and lower faces of the plate;
however, the load has been taken into account in the formulation of the general problem of
bending of plates and it enters in the right-hand side of equation 3.5. It remains to clarify the
conditions on the cylindrical surface, i.e at the edges of the plate, depending on the fastening or
supporting conditions. For a plate, the boundary conditions must be satisfied at each edge.
We consider below the following boundary conditions:
(1) Clamped, or built-in, or fixed edge y = 0 see figure 3.1
At the clamped edge y = 0, the deflection and slope are zero.
i.e
8.3........................................................................0;0 00
yyyy
y
ww
lviii
Figure 3.1: Boundary conditions of a plate
B Fixed edge
x
Simply supported edge
A b
a
Free edge
B
y
A
Beam edge
Plan view
Beam edge
h
x
a
z
Section A-A
y
z
h
b
Section B-B
lix
(2) Simply supported edge x = a
At these edges, the deflection, w and bending moment Mx are both zero, i.e.,
9.3................................................................0,02
2
2
2
axxaxy
w
x
wDMw
The first of these equations implies that along the edge x = a all the derivatives of w with respect
to y are zero, i.e., if x = a and w = 0, then 02
2
y
w
y
w
It follows that conditions expressed by equation 3.9 may appear in the following equivalent
forms:
10.3.................................................................................0,02
2
axax
x
ww
(3) Free Edge, y = b
Suppose that the edge y = b is perfectly free. Since no stress acts over their edge, then it is
reasonable to equate all the stress resultants and stress couples occurring at points of this edge to
zero, i.e.,
11.3..............................................................................0,0,0 byyxbyybyy MQM
These conditions were formulated by Poisson. Figure 3.10a shows two adjacent elements, each
of length dx belonging to the edge y = b. It is seen that, a twisting moment Myxdx acts on the
left-hand element while the right-hand element is subjected to .dxdxx
MM
yx
yx
These
elements are resultant couples produced by a system of horizontal shear stresses .yx Replace
them by couples of vertical forces Myx and
lx
Myx + dxx
M yx
with the moment arm dx having the same moment (figure 3.2b), i.e. as if
rotating the above-mentioned couples of horizontal forces through 900.
x
MM
yx
yx
yx yx
m
n
z
(a)
. o
. o
m
n
dxx
MM
yx
yx
yxM
dxM yx
dx dx
dx dx
dx dx
m
n
dxx
M yx
(b)
(c)
Figure 3.2 Statically equivalent replacement of couples of horizontal forces by couples of vertical forces.
x
lxi
It is known that the above-mentioned replacement is quite legal for such a body because it does
not disturb the equilibrium conditions and any moment may be considered as a free vector.
Forces Myx and Myx + dxx
M yx
act along the line mn (figure 3.2b) in opposite directions. Having
done this for all elements of the edge y = b we see that at the boundaries of two neighboring
elements, a single unbalanced force dxxM yx / is applied at points of the middle plane (figure
3.2c).
Thus, we have established that the twisting moment Myx is statically equivalent to a distributed
shear force of an intensity Myx/ x along the edge y = b, for a smooth boundary. Proceeding
from this, Kirchhoff proposed that the three boundary conditions at the free edge be combined
into two by equating to zero the bending moment My and the so-called effective shear force per
unit length, Vy. The latter is equal to the sum of the shear force Qy plus an unbalanced force
Myx/ x, which reflects the influence of the twisting moment Myx (for the edge y = b). Now,
we arrive at the following two conditions at the free edge:
aVM byybyy 12.3...................................................................................0,0
Where, bx
MQV
xy
yy 12.3..............................................................................
3.3 Effect of Transverse shear Deformation
Kirchhoff‟s hypotheses, discussed in section 2.5 permitted the creation of the classical
(Kirchhoff‟s) bending theory of thin plates which for more than a century has been the basis for
the calculation and design of structures in various areas of engineering and has yielded important
theoretical and numerical results. However, just as for any other approximation theory,
Kirchhoff‟s theory has some obvious drawbacks and deficiencies, two of which are:
lxii
(1) A well-known disagreement exists between the order of equation 3.5 and the number of the
boundary conditions on the plate free edge. As a result the boundary conditions of the
classical theory take into account only two characteristics on the free edge of the plate rather
than three characteristics corresponding to the reality. Of the two conditions of Equation
3.12a, only the first condition (imposed on the bending moment) has a clear physical
interpretation. The reduction of the twisting moment to a transverse shear force is not
justified in the general case. As a result of the replacement of the transverse shear force
Qx(Qy) and twisting moment Mxy by their combination, effective shear force Vx(Vy), the self-
balanced tangential stresses remain at the free edge of a plate and the concentrated forces
arise at the corner points of a rectangular or polygonal plate when this reduction is used. The
role of the latter forces is still not clear.
(2) Certain formal contradictions take place between the Kirchhoff‟s plate theory and the three-
dimensional equations of elasticity. The most important of the above contradictions is
associated with Hooke‟s law (Equation 2.11b) for the transverse shear stresses yxxz and .
In fact, the deformations xz and yz are absent according to the hypotheses of the classical
theory. However, the stresses yxxz and cannot be equal to zero, as it would be expected
from equation (2.11b), because the shear forces Qx and Qy which are resultants of the above-
mentioned stresses are necessary for an equilibrium of the plate differential element.
Notice that as a result of an inaccuracy of Kirchhoff‟s theory, we cannot guarantee that the
stress distribution predicted by this theory will agree well with the actual stresses in the
immediate vicinity of the plate edge. The latter statement acquires a practical importance of a
refinement of the classical plate theory for plate fields neighboring to a boundary or to
lxiii
openings whose diameter (or another typical dimension) is not too large compared with the
plate thickness.
3.4 Refined Theory of Bending of Plates
Numerous researchers have attempted to refine Kirchhoff‟s theory and such attempts
continue to this day. E. Reissner made the most important advance in this direction. Reissner‟s
theory takes into account the influence of the transverse shear deformation on the deflection of
the plate and leads to a sixth-order system of governing differential equations, and accordingly,
to three boundary conditions on the plate edge. Here it is unnecessary to introduce the effective
transverse shear force. Reissner‟s theory is free from the drawbacks of Kirchhoff‟s theory
discussed in section 3.3.
The correct interpretation of Reissner‟s theory is complicated substantially by the fact
that this theory involves the variational procedure to derive the governing equations, which was
essentially based on the use of Kirchhoff‟s theory to approximate the stress distribution over the
thickness of the plate.
Therefore, another approach is presented below for obtaining the governing differential
equations of the refined plate bending theory, which takes into account the transverse shear
deformations. According to this approach, the above equations are derived from the equations of
the theory of elasticity and certain physical hypotheses. This approach was developed by
Vasil‟ev.
lxiv
3.5 The Governing Equations of the Refined Plate Bending Theory
Let us introduce again familiar hypotheses:
1. The straight line normal to the middle plane of the plate does not change its length and
remains rectilinear when the plate is subjected to bending.
2. The transverse normal stress z is small compared with the stresses x and y and
3. The middle plane remains unstrained subsequent to bending.
The only difference between these hypotheses and Kirchhoff‟s discussed in section 3.1 is
that the former hypotheses do not require the straight line of the plate to be orthogonal to
the bent mid-surface of the plate. In this case, the straight line normal to the mid-surface,
called also the normal element acquires three independent degrees of freedom
corresponding to the deflection w(x,y) and the angles of rotation x and y which are not
related to the deflection, w as it has taken place in Kirchhoff‟s theory.
It has been shown in section 2.9.1 that the displacement components over the plate
thickness, according to the above-mentioned hypotheses, obey the following law (see
equations 2.14 to 2.17).
13.3..........................................).........,(),,(),,( yxwwyxzvyxzu yx
By repeating the derivation of relations 3.28-3.30 discussed in section 3.5.2, we obtain
14.3..............................................................................12
1
;;
xyDM
xyDM
yxDM
yx
xy
yy
y
yx
x
lxv
Note that if the angles of rotation are related to the deflections by relations of the type of
2.17, the bending and twisting moment equation 3.14 will coincide with the relations 2.31 to
2.33 derived for the classical Kirchhoff‟s theory.
The stresses xyyx and ,, are related to the stress resultants Mx, My and Mxy by relations
2.26 and the transverse shear forces, Qx and Qy are resultant of the transverse shear stresses
yzxz and (see equation 2.27). As mentioned previously, the transverse shear force cannot
be obtained directly by integrating equation 2.27 in Kirchhoff‟s theory because of the
hypotheses 2.1 adapted in this theory. However, the refined theory makes it possible to
obtain Qx and Qy in terms of the deflection, w and the angles of rotation xx and by
integrating equation 2.27. Substituting for xz and yz from equation 2.11b into equation
2.27 and using the second and third relations 2.8b and 3.13, we obtain the following
dz
y
w
z
v
x
w
z
u
GQ
Q
dzG
G
dzQ
Q
h
hy
x
h
hyz
xz
h
hyz
xz
y
x
2
2
2
2
2
2
lxvi
15.3..........................................................................................................
22
2
2
2
2
2
2
y
wyGh
x
wxGh
h
y
wyG
x
wxG
hh
y
wyG
x
wxG
z
y
wy
x
wx
G
y
wzyz
x
wzxz
G
zy
wv
zx
wu
GQ
Q
h
h
h
h
h
h
y
x
Hence,
17.3................................................................................................
16.3................................................................................................
y
wyGhQ
x
wxGhQ
y
x
Thus, within the framework of the theory, one can obtain the transverse shear forces as the
stress resultants of the shear stress yzxz and But, from equations 3.16 and 3.17,
lxvii
Let Gh = C; 18.3............................;y
w
x
wyyxx
Thus, 19.3..................................................; yyxx CQCQ
And C describes the shear stiffness of the plate in the planes xz and yz. Considering the
equilibrium of the plate element as shown in figure 2.10, we obtained equations 3.1, 3.2 and 3.3.
Substituting for the stress resultants from the relations 3.14 and 3.19 into Equations 3.1 to 3.3,
we arrive at the system of equations for ., wandyx It is possible to simplify this system. To
this end, we use equation 3.4. By substituting the moments according to equation 3.14 into the
above equation, we obtain; from 3.4,
Py
M
yx
M
x
M yxyx
2
22
2
2 2
Using equation 4.2
20.3...............................................................
12
2
2
2
2
2
2
2
2
3
3
2
3
2
3
3
3
2
3
3
3
2
3
2
3
2
3
2
3
2
3
3
3
2
22
2
2
D
P
yxyx
D
P
yxyyxx
D
P
yyxyxx
D
P
yxyyxxyyxyxxyx
Pxyy
D
xyyx
D
yxD
x
yx
yxyx
yyxx
xyyxyxyx
xyyxyx
But,
lxviii
22.3...................................................................
21.3............................................................
2
2
2
2
2
22
OPyx
D
D
P
yx
operatorLaplaceyx
yx
yx
Let us introduce the so-called potential function ),( yx of the displacement field in a plane z =
const. which satisfies the following relations:
23.3.........................................................................................................;yx
yx
Using this function, we can introduce equation 4.10 to the form:
24.3.....................................................................................................................
0
0
0
0
22
22
2
2
2
22
2
2
2
22
2
PD
PD
Pyx
D
Pyx
D
yyxxD
Using equations 3.2 and 3.3, and substituting there the expressions 3.14 and 3.19 for moment and
forces together with relations 3.23, we obtain:
25.3.................................................................................;0;0 2 wC
Df
y
f
x
f
lxix
From the two equations 3.25, it follows that f = const. The value of this constant is unessential
for the potential function and hence, can be assumed to be equal to zero without loss of
generality. Then, f = 0 and we obtain the solution from equation 3.25:
26.3........................................................................................................2 C
Dw
where the function is given by equation 3.23. This potential is sometimes referred to as the
penetrating potential, because it describes solutions that penetrate into the plate domain.
However, this potential cannot describe completely the bending behaviour of the plate. It does
not take into account the rotation of the plate element in its own plane, which can be described
by the so-called stream function, . This function may be introduced as follows:
27.3..........................................................................................0;;
w
yyyx
It can be easily verified that the relations 3.27 represent the solution of the homogenous equation
3.22 for P = 0. Substituting the expression 3.14 for the moments and expressions 3.19 and 3.18
for shear forces into equation 3.2 -3.3 and using 3.27, we obtain.
28.3........................................................,0,0 2
kfy
f
x
f
where,
29.3.............................................................................................)1(
22
D
Ck
By repeating the reasoning that led to equation 3.26, we arrive at the Helmholtz equation:
30.3.....................................................................................................022 k
Since the plate problem is assumed to be linear, we can use the method of superposition and
represent the angles of rotation of the normal element of the plate as follows:
lxx
31.3................................................................;xyyx
yx
Where, the functions and are defined by equations 3.23 and 3.27 respectively. Equation
3.24 and 3.30 represent the sixth-order system of governing differential equations of the plate
bending theory by taking into account the transverse shear deformations. This theory is
sometimes referred to as the shear or refined plate theory. The deflection w is given by equation
3.26. Taking into account the relation 3.14 and 3.31, we can express the bending and twisting
moments, as well as shear forces, in terms of the following functions and :
yxyxDM x
2
2
2
2
2
)1(
33.3...........................;.........;
32.3................................................2
11
)1(
2
2
2
22
2
2
2
2
2
xC
yDQ
yC
xDQ
yxyxDM
yxxyDM
yx
xy
y
As C (i.e, the transverse shear deformation is ignored), it follows from equation 3.26 and
3.30 that = 0 and w = and the resulting system of equations 3.24 and 3.30 degenerates into
the governing differential equation 3.5 of the Kirchhoff‟s theory. Now we can proceed to
determining the shear stresses yzxzand . If we substitute for the transverse shear strains
yzxzand from equation 2.8b, together with the relations 3.13, into the constitutive equation
2.11b for the transverse shear stresses, we obtain: from the second and third relations of 2.11b
lxxi
GG
yz
yz
xz
xz
,
Thus,
yz
xz
yz
xz
G
Using equation 2.8b,
y
w
z
v
x
w
z
u
G
yz
xz
Invoking 3.13
y
wz
z
x
wz
z
G
y
x
yz
xz
Differentiating by parts:
y
w
zz
z
z
x
w
zz
z
z
G
y
y
x
x
yz
xz
0
0
lxxii
34.3................................................................
y
y
x
yz
xz
w
x
w
G
Thus,
36.3.............................................................................
35.3..............................................................................
y
wG
x
wG
yyz
xxz
These stresses do not satisfy the following static boundary conditions
0
37.3..........................................................................
0
2
2
hzyz
hzxz
,
because the theory under consideration is based on the displacement approximation in the form
of equation 3.13
3.6 Boundary Conditions
As mentioned earlier, equations 3.24 and 3.30 form a sixth – order system of differential
equations; hence, the corresponding boundary value problem requires three boundary conditions.
To be specific, we consider the typical boundary conditions for the edge x = const.
(a) The edge x = const is perfectly fixed. For this edge the boundary conditions are
u = 0, v = 0, w = 0 …………………………………………………3.38
lxxiii
Using equation 3.13 and 3.31, we can rewrite these boundary conditions in terms of the functions
, and as follows:
39.3.........................................0;;
C
D
xyyx
(b) The edge x = const. is free
This type of boundary conditions implies that the stresses xzx , and xz must vanish. For the
bending problems of thin plates described by the refined theory presented above, these
requirements can be reduced to the following conditions:
40.3.............................................................0;0;0 xxyx QMM
Applying the relations 3.32 and 3.33 we can rewrite these boundary conditions in terms
of the functions , and as follows:
41.3................................................................0
02
1
0)1(
2
2
2
22
2
2
2
2
2
yxC
D
yxyx
yxyx
Let us transform the second boundary condition 3.41 Eliminating 2
2
x
with the use of 3.30,
differentiating with respect to y (along the plate edge), and eliminating with the use of the
third condition 3.41 we obtain
42.3....................................................0)1()2(2
2
2
2
2
2
yC
D
yxx
lxxiv
If ,0 Cand then the third condition 3.42 disappears and the first boundary condition
3.41 and equation 3.41 become equivalent to the conditions Mx = 0 and Vx = 0 in Kirchhoff‟s
theory.
Note that the condition Vx = 0 is obtained without reduction of the twisting moment to a force
performed in Kirchhoff‟s theory.
(c) The edge x = const. is simply supported.
The shear plate bending theory allows two types of boundary conditions for a simply supported
edge in contrast to Kirchhoff‟s theory.
The first case corresponds to a plate whose contour is supported by the diaphragms that are
absolutely rigid in their own planes. We then have
w = 0, Mx = 0, y = 0 ………………………………………………………3.43
Using the relations 4.20, we can represent these conditions in terms of functions and after
some transformations, as follows:
44.3..............................................................................................0,0,02
2
xx
The second possible case of a simply supported edge corresponds to a plate free-resting on a
supporting contour. The corresponding boundary conditions for this edge are:
w = 0, Mx = 0, Mxy = 0…………………………….. 3.45
Again, using equation 3.32, we can rewrite these conditions in terms of the following
functions and :
0 C
D
lxxv
46.3...........................................................02
1
0)1(
2
2
2
22
2
2
2
2
2
yxyx
yxyx
3.7 Application of the Refined Theory by Double Trigonometric Series (Navier
Solution)
In order to illustrate an application of this theory, let us consider the classical commonly
encountered problem: a simply supported rectangular plate with sides a and b (0 < x < a; 0 < y <
b) subjected to a uniformly distributed load of intensity P. see figure 4.1.
Fig. 3.3: Simply supported rectangular plate under uniform distributed load
This problem has been analyzed comprehensively by Kirchhoff‟s theory. The deflected surface
of the plate was given by equation 3.47.
...3,1 ...3,1
222
647.3...............................................
sinsin16
),(m n
b
n
a
mmn
b
yn
a
xm
D
Pyxw
a
b
P(xy)
y
x
lxxvi
Let us now consider the solution of this problem using the plate bending theory discussed above.
To satisfy the boundary condition 3.45, we seek the solution in the form of the series:
1 .1
1 1
49.3....................................coscos
48.3.......................................sinsin
m n
nmmn
m n
nmmn
yx
yx
Where,
50.3..........................................................,b
n
a
mnm
Using the Helmholtz equation:
51.3..................................................................022 k
Where,
)1(
22
D
Ck
By substituting 3.48 into 3.51, we obtain:
1 .11 .1
2
2
2
2
2
1 .11 .1
2
2
2
2
2
2
2
2
2
2
22
0coscoscoscos
0coscoscoscos
0
0
m n
mn
m n
mn
m n
nmmn
m n
nmmn
b
yn
a
xmk
b
yn
a
xm
yx
yxkyxyx
kyx
k
lxxvii
52.3............................................................0coscos
coscoscoscos
0coscos
coscoscoscos
1 .1
2
1 .1
2
1 .1
2
1 .1
2
1 .12
2
1 .12
2
m n
mn
mn
m nm n
mn
m n
mn
m n
mn
m n
mn
b
yn
a
xmk
b
yn
a
xm
b
m
b
yn
a
xm
a
m
b
yn
a
xmk
b
yn
a
xm
yb
yn
a
xm
x
Dividing equation 3.52 by
b
yn
a
xm
nm
coscos
.11
55.3..........................................................................................................0
...3,2,1
...3,2,1
54.3.............................................................................................0
53.3.....................................................................................0
2
22
2
22
n
mu
kb
n
a
mBut
kb
n
a
m
mn
mn
Thus, we obtain that;
56.3...................................................................................................................................0
lxxviii
Inserting 3.48 into 3.24, we have;
61.3..................................................................................................;
60.3......................sinsin2
sinsin
sinsin2sinsin
59.3...................................................................sinsin
sinsin2sinsin
sinsin2
58.3..........................................sinsin
57.3...................................................................sinsin
22
.11
4224
.11
4
.11
22
.11
4
1 .14
4
1 .122
4
1 .14
4
1 .14
4
22
4
4
4
1 .12
2
2
2
2
2
2
2
1 .1
22
22
hb
ng
a
mLet
D
P
b
yn
a
xm
b
n
b
n
a
m
a
m
D
P
b
yn
a
xm
b
n
b
yn
a
xm
b
n
a
m
b
yn
a
xm
a
m
D
P
b
yn
a
xm
y
b
yn
a
xm
yxb
yn
a
xm
x
D
P
b
yn
a
xm
yyxx
Pyxyxyx
D
PyxD
PD
mn
nm
mn
nm
mn
nm
mn
nm
m n
mn
m n
mn
m n
mn
m n
mn
m n
nmmn
m n
nmmn
lxxix
Then, equation 3.60 becomes;
63.3........................................................................................................2)(
62.3..................................................................sinsin2
222
1 .1
22
hghghgBut
D
P
b
yn
a
xmhghg
m
mn
n
Hence, we express equation 3.62 as:
1 .1
264.3.............................................................................sinsin
m n
mnD
P
b
yn
a
xmhg
Using the expressions in 3.61, we obtain 3.64 as:
1 .1
222
65.3.......................................................sinsinm n
mnD
P
b
yn
a
xm
b
n
a
m
Expressing the lateral load, P(x,y) in the form of Fourier series:
1 .1
66.3..............................................................................sinsin),(m n
mnb
yn
a
xmPyxP
Where, Pmn is the coefficient of lateral load in the series. Then equation 3.65 becomes
68.3.....sinsinsinsin
67.3........sinsin1
sinsin
1 .11 .1
222
1 .11 .1
222
b
yn
a
xmP
D
P
b
yn
a
xm
b
n
a
m
b
yn
a
xmP
Db
yn
a
xm
b
n
a
m
m n
mn
m n
mn
mn
m n
mn
m n
mn
Dividing equation 3.68 through by
1 .1
sinsinm n b
yn
a
xm
lxxx
We have;
71.3.........................................................................................................
70.3............................................................................................:
69.3............................................................................................
222
222
222
nm
mnmn
mnmn
mnmn
D
P
b
n
a
mD
P
D
P
b
n
a
m
Then using relation (3.48), into 3.26, we obtain:
1 .1
2
1 .1
2
1 .1
1 .12
2
1 .12
2
1 .1
1 1 .12
2
2
2
.1
1 .11 .1
2
2
sinsinsinsin
sinsin
sinsinsinsin
sinsin
sinsinsinsin
sinsinsinsin
m n
nmmnn
m n
nmmnm
nm
m n
mn
m n
nmmn
m n
nmmn
nm
m n
mn
m m n
nmmn
n
nmmn
m n
nmmn
m n
nmmn
yxyxC
D
yxw
yxy
yxxC
D
yxw
yxyxC
Dyxw
yxC
Dyxw
C
Dw
lxxxi
1 .1
22
1 .11 .1
22
1 .11 .1
22
1 .1
72.3.............................................................sinsin1
sinsinsinsin
sinsinsinsin
sinsin
m n
nmmnnm
m n
nm
m n
nmmnnmmn
m n
nmmn
m n
nnmmnm
m n
nmmn
yxC
Dw
yxC
Dyxw
yxyxC
D
yxw
Inserting 3.71 into 3.72
b
n
a
m
where
D
yxPC
D
w
n
m
nm
m
nm
n
mnnm
73.3........................................................
sinsin1
222
1 .1
22
The coefficient Pmn can be determined as follows:
Multiplying equation 3.66 by
,sinsinb
yn
a
xm we have;
74.3...................................................sinsinsinsin),(1 .1
22
m n b
yn
a
xm
b
yn
a
xmyxP
lxxxii
Integrating 3.74 twice within the boundary as x = 0 → a and y = 0 → b we obtain:
75.3..............sinsin
sinsinsinsin),(
1 10 0
22
,
0,0
22
1 10 00 0
dyb
yndx
a
xmPP
b
nb
ynCos
a
ma
xmCos
dxdyb
yn
a
xmPdxdy
b
yn
a
xmyxP
m n
a b
mn
ba
m n
a b
mn
a b
From the trigonometry,
76.3............................................................................................................2coscossin
1cossin
22
22
AAA
AA
From 3.76,
77.3...........................................................................................................2
12cossin
2cos1sin2
2cos)sin1(sin
sin1cos
2
2
22
22
AA
AA
AAA
andAA
lxxxiii
Using the similarity in 3.77, equation 3.75 becomes as follows:
79.3.........................................................2
sin2
2sin
24
coscos
2
2sin
2
2sin
4coscos
78.3.....................................................2
12
cos
2
12
cos
coscos
00
00
2
00
,
0,0
2
0 01 1
,
0,0
ba
mn
ba
ba
mn
ba
a b
m n
mn
ba
yb
yn
n
bx
a
xm
m
aP
b
yn
a
xm
mn
Pab
y
b
nb
yn
x
a
ma
xm
P
b
yn
a
xm
mn
Pab
dyb
yn
dxa
xm
P
b
yn
n
b
a
xm
m
aP
For a non-trivial solution, the values of m and n are series of odd numbers. i.e.:
80.3............................................................................. …………………… 111,3,5,7,9, =n
……………………111,3,5,7,9, = m
Then, 3.79 becomes:
bn
n
bam
m
aPnm
mn
Pab mn
2sin2
2sin24
1cos1cos2
For values of m and n we have:
lxxxiv
82.3...........................................................................................1616
81.3......................................................................................................4
4
422
004
1111
22
2
2
2
mn
P
abmn
PabP
abP
mn
Pab
baP
mn
Pab
baP
mn
Pab
mn
mn
mn
mn
Inserting 3.82 into 3.73, we have
b
n
a
m
where
yxC
D
mnD
Pw
D
yxmn
P
C
D
w
n
m
m n
nmnm
nm
nm
m n
nmnm
,
83.3.......................................sinsin116
sinsin16
1
1 1
22
2222
222
1 12
22
84.3...........sinsin116
.11
22
222
6b
yn
a
xm
b
n
a
m
C
D
b
n
a
mmnD
Pw
nm
lxxxv
3.8 Stresses Generated due to the Deflection in Equation 3.84
From equation 3.32,
85.3................................................................................
2
1)1(
)1(
)1(
2
2
2
22
2
2
2
2
2
2
22
2
yxyxDM
yxyxDM
yxyxDM
xy
y
x
Taking inference from 3.56
0
Equation 3.85 reduces as:
88.3.......................................................................................................)1(
87.3.........................................................................................................
86.3...........................................................................................................
2
22
2
22
2
yxDM
xyDM
yxDM
xy
y
x
Using equation 3.48;
i.e.:
89.3....................................................................................................sinsin1 .1
yx nm
m n
mn
Where
lxxxvi
92.3..........................coscossinsin
91.3..........................sinsinsinsin
90.3..........................sinsinsinsin
1 .11 .1
22
1 .1
2
1 .12
2
2
2
1 .1
2
1 .12
2
2
2
yxyxyxxy
yxyxyy
yxyxxy
nm
m n
mnnmnm
m n
mn
nm
m n
mnnnm
m n
mn
nm
m n
mnmnm
m n
mn
Using 3.90 through 3.92 into 3.86 through 3.88, we have:
93.3....................................................................sinsin
sinsinsinsin
1 .1
22
1 .1
2
1 .1
2
m n
nmmnnmx
nm
m n
mnnnm
m n
mnmx
yxDM
yxyxDM
Inserting 3.71 into 3.93, we have,
94.3.....................................................................sinsin
)(
sinsin)(
1 .1222
22
1 .1222
22
m n
nmmn
nm
nm
x
m n
nmmn
nm
nm
x
yxPM
yxPD
DM
Using the expression of Pmn of 3.82 in the equation 3.94 we have:
95.3....................................................................sinsin
)(16
1 .12222
22
m n
nm
nm
nm
x yxmn
PM
Similarly, 3.87 becomes:
lxxxvii
96.3.................................................................sinsin
)(16
1 .12222
22
m n
nm
nm
mn
y yxmn
PM
From 3.88 and 3.92
97.3............................................................coscos)1(1 .1
m n
nmmnnmxy yxDM
Using 3.71 and 3.82 simultaneously in 3.97 we obtain
1 .12222
1 .1222
2
98.3......................................................coscos)1(16
coscos)1(
16
m n
nm
nm
nmxy
m n
nmnm
nm
xy
yxmn
PM
yxD
mn
PD
M
lxxxviii
CHAPTER FOUR
RESULTS AND DISCUSSION OF RESULT
4.1 Results
In other to illustrate an application of this theory, let us consider the classically commonly
encountered problem: a simply supported rectangular plate with sides a and b subjected to a
uniformly distributed load of intensity P. Then using equation 3.84, taking only the first term of
the series, that is, m = n = 1, we have 3.84 as
max
22
2
2
22
6
),(2
,2
1.4......................................sinsin11
111
16
w
yxw
by
axat
b
yn
a
xm
baC
D
baD
Pw
Thus, we obtain
2.4..........................................................11
11
16
2sin
111
1
16
22
2
2
22
6
max
2
22
2
2
22
6
max
baC
D
ba
aD
Pw
baC
D
ba
aD
Pw
lxxxix
Numerical values
(i) Flexural Rigidity, D:
)1(12 2
3
EhD
(ii) Shear stiffness, C:
GhC
Then, equation 4.2 reduces as
222
22
2
22
36
2
max
222
23
2
222
36max
11
)1(121
11
)1(192
11
)1(121
11
112
16
baG
Eh
baEh
Pw
Gh
ba
Eh
ba
Eh
Pw
xc
3.4.....................................................)1(6
)1(11
111
)1(192
11
)1()1(2
121
11
)1(192
2
22
22
2
22
36
2
max
222
22
2
22
36
2
bah
baEh
Pw
baE
Eh
baEh
P
Assumptions
Poisson ratio, = 0.2
Elastic modulus, E, for steel = 205000N/mm2
= 205kN/mm
2
= 205kN/10-6
m2
= 205 x106kNm
-2
However, according to Navier, the solution of maximum deformation for such a plate is given as:
2
222
362
22
6
max
11
)1(12
16
11
16
ba
Eh
P
baD
Pw
4.4.........................................................................................11
)1(1922
22
36
2
baEh
P
xci
Figure 4.1: Plot of Deformation against plate thickness under constant lateral load for
Thin plate
PRESENT STUDY BASED ON REFINED THEORY
KIRCHHOFFS THEORY BASED ON NAVIER SOLUTION
xcii
Figure 4.2: Plot of Deformation against plate thickness under constant lateral load for
Membranes
PRESENT STUDY BASED ON REFINED THEORY
KIRCHHOFFS THEORY BASED ON NAVIER SOLUTION
xciii
Table 4.1: Variable deformation of elastic plate with thickness for thin plate (stiff plate)
S/N
Load, P
(KN)
Poisson
ratio
µ
Modulus
of
elasticity
MPa
Thickness
of plate
(m)
Kirchhoff’s
based on Navier
solution
(Deformation, m)
Present study
based Refined
theory
(Deformation, m)
Percentage deformation
difference (%)
1. 150 0.2 205 0.025 1.352E-01 1.352E-01 0.00E+00
2. 150 0.2 205 0.050 1.691E-02 1.690E-02 1.00E-03
3. 150 0.2 205 0.075 5.015E-03 5.007E-03 8.00E-04
4. 150 0.2 205 0.100 2.118E-03 2.113E-03 5.00E-04
5. 150 0.2 205 0.125 1.086E-03 1.082E-03 4.00E-04
6. 150 0.2 205 0.150 6.296E-04 6.259E-04 3.70E-04
7. 150 0.2 205 0.175 3.973E-04 3.942E-04 3.10E-04
8. 150 0.2 205 0.200 2.668E-04 2.641E-04 2.70E-04
9. 150 0.2 205 0.225 1.879E-04 1.855E-04 2.40E-04
10. 150 0.2 205 0.250 1.374E-04 1.352E-04 2.20E-04
11. 150 0.2 205 0.275 1.036E-04 1.016E-04 2.00E-04
12. 150 0.2 205 0.300 8.005E-05 7.824E-05 1.81E-04
13. 150 0.2 205 0.325 6.321E-05 6.154E-05 1.67E-04
14. 150 0.2 205 0.350 5.082E-05 4.927E-05 1.55E-04
15. 150 0.2 205 0.375 4.151E-05 4.006E-05 1.45E-04
16. 150 0.2 205 0.400 3.437E-05 3.301E-05 1.36E-04
17. 150 0.2 205 0.425 2.880E-05 2.752E-05 1.28E-04
18. 150 0.2 205 0.450 2.439E-05 2.318E-05 1.21E-04
19. 150 0.2 205 0.475 2.085E-05 1.971E-05 1.14E-04
20. 150 0.2 205 0.500 1.799E-05 1.690E-05 1.09E-04
Other Parameters used Short span, a = 4m
Long span, b = 4m
xciv
Table 4.2: Variable deformation of elastic plate with thickness for Membranes S/N
Load, P
(KN)
Poisson
ratio
µ
Modulus
of
elasticity
MPa
Thickness
of plate
(m)
Kirchhoff’s
based on Navier
solution
(Deformation, m)
Present study
based Refined
theory
(Deformation, m)
Percentage deformation
difference (%)
1. 150 0.2 205 0.005 1.690E+01 1.690E+01 0.00E+00
2. 150 0.2 205 0.010 2.113E+00 2.113E+00 0.00E+00
3. 150 0.2 205 0.015 6.260E-01 6.259E-01 0.00E+00
4. 150 0.2 205 0.020 2.641E-01 2.641E-01 0.00E+00
5. 150 0.2 205 0.025 1.352E-01 1.352E-01 0.00E+00
6. 150 0.2 205 0.030 7.826E-02 7.824E-02 0.00E+00
7. 150 0.2 205 0.035 4.929E-02 4.927E-02 0.00E+00
8. 150 0.2 205 0.040 3.302E-02 3.301E-02 0.00E+00
9. 150 0.2 205 0.045 2.319E-02 2.318E-02 0.00E+00
10. 150 0.2 205 0.050 1.691E-02 1.690E-02 0.00E+00
11. 150 0.2 205 0.055 1.271E-02 1.270E-02 0.00E+00
12. 150 0.2 205 0.060 9.789E-03 9.780E-03 0.00E+00
13. 150 0.2 205 0.065 7.701E-03 7.692E-03 0.00E+00
14. 150 0.2 205 0.070 6.167E-03 6.159E-03 0.00E+00
15. 150 0.2 205 0.075 5.015E-03 5.007E-03 0.00E+00
16. 150 0.2 205 0.080 4.133E-03 4.126E-03 0.00E+00
17. 150 0.2 205 0.085 3.446E-03 3.440E-03 0.00E+00
18. 150 0.2 205 0.090 2.904E-03 2.898E-03 0.00E+00
19. 150 0.2 205 0.095 2.470E-03 2.464E-03 0.00E+00
20. 150 0.2 205 0.100 2.118E-03 2.113E-03 0.00E+00
Other Parameters used Short span, a = 4m
Long span, b = 4m
xcv
4.2 Discussion of Results
The refined theory presented in this work can be used in predicting the deformation of
elastic plates. The derived governing differential equation contains the shear stiffness term as
shown in equation 4.3. The shear stiffness term makes it possible to predict the effect of shear
deformation on elastic plates especially where the thickness of the plate increases; fairly thin or
moderately thick plates. The shear stiffness term is missing in Kirchhoff-Navier equation as
indicated in equation 4.4. This omission points out the difference between Kirchhoff‟s theory
and the present refined theory.
The curves obtained from the two equations; that with shear deformation effect and that
without shear deformation, have the same shape. These are shown in figures 4.1 and 4.2. The
shapes of the curves depict the decrease in deformation of elastic structural material when the
flexural rigidity is increased, that is at increasing thickness. This is a factor that is usually
considered in the design of steel and reinforced concrete elements. The behaviour of the plate
under constant lateral load is equally shown in tables 4.1 and 4.2.
Figures 4.1 and 4.2 show the plots of deformations of Navier-Kirchhoff‟s theory and that
of present study based on the refined theory on the same scale against the plate thickness. The
plate thickness is very small compared with its other characteristic dimensions. The deformation
with transverse and that without transverse shear interfaced at the beginning and get slightly
separated as the plate thickness increases. Kirchhoff‟s theory under estimates deflection and
over estimates natural frequencies and buckling loads. The fact that the curves obtained from the
two theories have the same shape shows that Navier-Kirchhoff theory holds as well as the
present theory (Refined theory) which takes into account the effect of transverse shear
deformation.
xcvi
These similarities and differences are also shown in tables 4.1 and 4.2 as they also
display varying percentage difference in the deformations of the elastic plates.
xcvii
CHAPTER FIVE
CONCLUSION AND RECOMMENDATION
5.1 Conclusion
This research work, which demonstrates the effect of transverse shear deformation on the
bending of elastic plates, has successfully clarified the burning issues associated with Navier and
Kirchhoff‟s hypotheses. Their hypotheses neglect the deformation caused by transverse shear. In
this project, the derived governing equation of the refined plate bending theory shows that
transverse shear deformation has an effect on the bending of elastic plates.
In the light of this achievement, we therefore conclude from the results of the analysis as
follows:
a) Transverse shear force has an effect on the deformation of elastic plates.
b) Kirchhoff‟s and Navier theories neglect the effect of transverse shear force on the
deformation of an elastic plate.
c) Kirchhoff‟s and Navier equation for deformation of plate lacked the shear stiffness term
that predicts the effect of shear deformation on elastic plates, especially where the
thickness of the plate is fairly thin.
d) The effect of transverse shear forces on the deformation of simply supported plate under
uniformly distributed load is presented as the refined theory.
e) The expression of this deformation is also obtained as equation 4.3.
f) This equation derived is adequate for the solution of elastic plates.
xcviii
5.2 Recommendation
From the conclusion above, the following recommendations are made:
a) The Kirchhoff-Navier equation as well as the obtained refined theory equation could be
used for elastic plates.
b) The effect of transverse shear deformation should be taken into account when
determining the deflection of elastic plates.
c) The refined theory presented in this work should be used in predicting the deformation of
elastic plates.
d) The refined theory should be used in the solution of deformations in plates as the theory
gives equal and accurate results even for membranes and thin plates.
e) Further work is recommended on the effect of transverse shear deformation on plates
with different support conditions and circular plates.
xcix
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