i

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

ii

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

iii

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)

iv

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

v

DEDICATION

This work is dedicated to my uncle, Chief Ime Effiong Ekanem.

vi

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.

vii

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

ix

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

x

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

xii

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.

xiii

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

xiv

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

xxii

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

QQ

y

y

dyy

MM

yx

yx

dxx

QQ xx

dxx

MM

xy

xy

dxx

MM xx

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 yyzdzM

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 xyxyzdzM

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

QQ

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

QQ

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

QQ

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.........................................................................................422D

Pww

Where,

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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

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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

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(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 accordi