Theories and applications of plate analysis classical numerical and engineering methods

Theories and Applications
of Plate Analysis

Theories and Applications of Plate Analysis: Classical, Numerical and Engineering Methods. R. Szilard
Copyright © 2004 John Wiley & Sons, Inc.


Theories and Applications
of Plate Analysis
Classical, Numerical and Engineering Methods

Rudolph Szilard, Dr.-Ing., P.E.
Professor Emeritus of Structural Mechanics
University of Hawaii, United States
Retired Chairman, Department of Structural Mechanics
University of Dortmund, Germany

JOHN WILEY & SONS, INC.


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Library of Congress Cataloging-in-Publication Data
Szilard, Rudolph, 1921Theories and applications of plate analysis : classical, numerical and engineering
methods / by Rudolph Szilard.
p. cm.
Includes bibliographical references and index.
ISBN 0-471-42989-9 (cloth)
1. Plates (Engineering) I. Title.
TA660.P6S94 2003
624.1 7765—dc21
2003043256
Printed in the United States of America.
10 9 8 7 6 5 4 3 2 1

To the memory of my father,
Dipl.-Ing. Rudolph Seybold-Szilard, senior,
who encouraged and inspired my career in structural mechanics


Contents

Preface

xvii

Symbols

xxi

I Introduction
II Historical Background
PART I

1

Elastic Plate Theories and Their Governing
Differential Equations

10

21

23

1.1

Classical Small-Deflection Theory of Thin Plates*1

23

1.2

Plate Equation in Cartesian Coordinate System*

26

1.3

Boundary Conditions of Kirchhoff’s Plate Theory*

35

1.4

Differential Equation of Circular Plates*

42

1.5

Refined Theories for Moderately Thick Plates

45

1.6

Three-Dimensional Elasticity Equations for Thick
Plates

53

Membranes

57

1.7
1

Plate Theories and Analytical Solutions of
Static, Linear-Elastic Plate Problems

1

Asterisks (∗ ) indicate sections recommended for classroom use.

vii


viii

Contents

1.8

Summary*

60

Problems*

61

2 Exact and Series Solutions of Governing
Differential Equations

62

2.1

Rigorous Solution of Plate Equation

62

2.2

Solutions by Double Trigonometric Series
(Navier’s Approach)*

69

Solutions by Single Trigonometric Series
´

(Levy’s
Method)*

75

2.4

Further Examples of Series Solutions

83

2.5

´
Extensions of Navier’s and Levy’s
Methods

92

2.6

Method of Images

97

2.7

Plate Strips

99

2.8

Rigorous Solution of Circular Plates Subjected to
Rotationally Symmetric Loading*

110

Solutions of Membrane Problems

116

2.10

Series Solutions of Moderately Thick Plates

120

2.11

Summary*

126

Problems*

127

2.3

2.9

3 Further Plate Problems and Their Classical
Solutions

129

3.1

Plates on Elastic Foundation*

129

3.2

Plates with Variable Flexural Rigidity

139

3.3

Simultaneous Bending and Stretching

147

3.4

Plates of Various Geometrical Forms

150

3.5

Various Types of Circular Plates

156

3.6

Circular Plate Loaded by an Eccentric
Concentrated Force

161

Plates with Edge Moments

165

3.7


Contents

3.8

Solutions Obtained by Means of Superposition

168

3.9

Continuous Plates

173

Summary

179

Problems

180

3.10

4

Energy and Variational Methods for Solution
of Lateral Deflections
Introduction and Basic Concepts*

181

4.2

Ritz’s Method*

187

4.3

Galerkin’s Method and Its Variant by Vlasov*

196

4.4

Further Variational and Energy Procedures

212

4.5

Techniques to Improve Energy Solutions

226

4.6

Application of Energy Methods to Moderately
Thick Plates

231

Summary*

234

Problems*

235

PART II Numerical Methods for Solution of Static,
Linear-Elastic Plate Problems

6

181

4.1

4.7

5

ix

Finite Difference Methods

237
247

5.1

Ordinary Finite Difference Methods*

247

5.2

Improved Finite Difference Methods

276

5.3

Finite Difference Analysis of Moderately Thick
Plates

303

5.4

Advances in Finite Difference Methods

312

5.5

Summary and Conclusions*

314

Problems*

315

Gridwork and Framework Methods
6.1

Basic Concepts*

317
317


x

Contents

6.2

Equivalent Cross-Sectional Properties

320

6.3

Gridwork Cells and Their Stiffness Matrices*

328

6.4

Computational Procedures for Gridworks
6.4.1 Procedures Using Commercially Available
Programs*
6.4.2 Guidance for Gridwork Programming

336

337
343

Summary and Conclusions*

361

Problems*

362

6.5

7 Finite Element Method

364

7.1

Introduction and Brief History of the Method*

364

7.2

Engineering Approach to the Method*

370

7.3

Mathematical Formulation of Finite Element
Method*
7.3.1 Consideration of Total System*
7.3.2 Formulation of Element Stiffness Matrices*

380
380
383

7.4

Requirements for Shape Functions*

389

7.5

Various Shape Functions and Corresponding
Element Families
7.5.1 Polynomials and Their Element Families
7.5.2 Hermitian Elements
7.5.3 Other Element Families

392
393
399
403

7.6

Simple Plate Elements*
7.6.1 Rectangular Element with Four Corner
Nodes*
7.6.2 Triangular Element with Three Corner
Nodes*

406

7.7

Higher-Order Plate Elements
7.7.1 Rectangular Element with 16 DOF
7.7.2 Discrete Kirchhoff Triangular Element

418
418
423

7.8

Computation of Loads and Stress Resultants*

434

7.9

Moderately Thick Plate Elements

446

Thick-Plate Elements

453

7.10

406
411


Contents

8

9

xi

7.11

Numerical Integration

458

7.12

Modeling Finite Element Analysis*

463

7.13

Programming Finite Element Analysis*

465

7.14

Commercial Finite Element Codes*

469

7.15

Summary and Conclusions*

472

Problems*

474

Classical Finite Strip Method

475

8.1

Introduction and Basic Concepts

475

8.2

Displacement Functions for Classical FSM

477

8.3

Formulation of the Method

481

8.4

Outline of Computational Procedures

489

8.5

Summary and Conclusions

494

Problems

495

Boundary Element Method

496

9.1

Introduction

496

9.2

Basic Concepts of Boundary Element Method

497

PART III

Advanced Topics

10 Linear Considerations

505
507

10.1

Orthotropic Plates

507

10.2

Laminated and Sandwich Plates
10.2.1 Classical Laminated Plate Theory
10.2.2 Sandwich Plates
10.2.3 Moderately Thick Laminated Plates

530
531
534
539

10.3

Analysis of Skew Plates

546

10.4

Thermal Bending of Plates

561


xii

Contents

10.5

Influence Surfaces

571

10.6

Continuous Plates Supported by Rows of Columns

578

10.7

Additional Topics Related to FEM
10.7.1 Various Convergence Tests
10.7.2 Elements with Curved Sides

590
590
594

10.8

Extensions of Classical Finite Strip Method
10.8.1 Spline Finite Strip Method
10.8.2 Computed Shape Functions
10.8.3 Finite Strip Formulation of Moderately Thick
Plates

597
598
605

Summary and Conclusions

611

Problems*

612

10.9

11 Nonlinear Aspects

606

614

11.1

Large-Deflection Analysis

614

11.2

Numerical Methods for Geometrically Nonlinear

Analysis
11.2.1 Various Finite Element Procedures

624
637

11.3

Material Nonlinearity
11.3.1 Nonlinear Stress-Strain Relationships
11.3.2 Computational Procedures

645
645
648

11.4

Combined Geometrical and Material Nonlinearities

656

11.5

Reinforced-Concrete Slabs

662

11.6

Summary and Conclusions

672

Problems

672

PART IV Engineering Solution Procedures

673

12

675

Practical Design Methods
12.1

Need for Engineering Solution Procedures*

675

12.2

Elastic Web Analogy for Continuous Plate
Systems*

676

Simplified Slope-Deflection Method*

689

12.3


Contents

12.4

13

xiii

Moment Distribution Applied to Continuous
Plates*

700

12.5

Practical Analysis of RC Floor Slabs

710

12.6

Equivalent Frame Method Applied to Flat Slabs*

718

12.7

Other Practical Design Methods
12.7.1 Approximate Analysis of Bridge Decks
12.7.2 Simplified Treatments of Skew Plates
12.7.3 Degree-of-Fixity Procedure

727
727
730
733

12.8

Summary and Conclusions*

739

Problems*

740

Yield-Line Method

742

13.1

Introduction to Yield-Line Method*

742

13.2

Work Method*

751

13.3

Equilibrium Method*

758

13.4

Further Applications of Yield-Line Analysis*

763

13.5

Yield Lines due to Concentrated Loads*

770

13.6

Summary and Conclusions*

781

Problems*

782

PART V

Dynamic Analysis of Elastic Plates

14 Classical and Energy Methods in Dynamic
Analysis

785

787

14.1

Introduction to Structural Dynamics*

787

14.2

Differential Equations of Lateral Motion*

802

14.3

Free Flexural Vibration of Plates*

804

14.4

Free Transverse Vibration of Membranes

810

14.5

Energy Methods for Determination of Natural
Frequencies*

815

Natural Frequencies Obtained from Static
Deflections*

824

14.6


xiv

Contents

14.7

Forced Transverse Vibration of Rectangular
Plates*

830

14.8

Free Vibration of Moderately Thick Plates

839

14.9

Summary and Conclusions*

842

Problems*

843

15 Numerical Methods in Plate Dynamics
15.1

845

Solution of Differential Equation of Motion by
Finite Differences*

845

Application of Finite Element Method to Plate
Dynamics*
15.2.1 Matrix Equations of Free Vibrations*
15.2.2 Mass Matrix*
15.2.3 Forced Vibrations*

856
856
860
870

15.3

Damping of Discrete Systems

883

15.4

Slab Bridges under Moving Loads

889

15.5

Large-Amplitude Free-Vibration Analysis

895

15.6

Summary and Conclusions*

899

Problems*

902

15.2

PART VI Buckling of Plates

903

16 Fundamentals of Stability Analysis

905

16.1

Basic Concepts*

905

16.2

Equilibrium Method*

911

16.3

Energy Methods in Stability Analysis*

919

16.4

Finite Differences Solution of Plate Buckling*

928

16.5

Finite Element and Gridwork Approach to Stability
Analysis*

938

16.6

Dynamic Buckling

946

16.7

Buckling of Stiffened Plates

953

16.8

Thermal Buckling

961


Contents

16.9

xv

Buckling of Moderately Thick Plates

963

16.10

Postbuckling Behavior

966

16.11

Inelastic Buckling and Failure of Plates

978

16.12

Summary and Conclusions*

982

Problems*

983

Appendix A.1

Fourier Series

985

Appendix A.2

Conversion from One Poisson Ratio
to Another

999

Appendix A.3

Units

1001

Appendix A.4

About the CD

1003

Index

A.4.1

Plate Formulas

1003

A.4.2

WinPlatePrimer Program System

1004
1015


Preface

This monograph represents a completely reworked and considerably extended version

of my previous book on plates.1 It is based on the courses taught and the pertinent
research conducted at various universities in the United States and Germany, combined with my many years of experience as a practicing structural engineer. Like
its predecessor, this new version intends, at the same time, to be a text and reference book. Such dual aims, however, put any author in a difficult position since the
requirements of text and reference books are different. The global success of the first
version indicates, however, that such an approach is justified. In spite of the number
of books on plates, there is no single book at the present time that is devoted to the
various plate theories and methods of analysis covering static, dynamic, instability
and large-deflection problems for very thin, thin, moderately thick and thick plates.
The author hopes that this comprehensive monograph will serve as a text and reference book on these highly diverse subjects. Thus, the main objectives of this book
are as follows:
1. To serve as an introductory text to the classical methods in various
plate theories.
2. To acquaint readers with the contemporary analytical and numerical methods
of plate analysis and to inspire further research in these fields.
3. To serve as a reference book for practicing engineers not only by giving them
diverse engineering methods for quick estimates of various plate problems but
also by providing them with a user-friendly computer program system stored
on a CD-ROM for computation of a relatively large spectrum of practical
plate problems. In addition, the accompanying CD-ROM contains a collection
of readily usable plate formulas for solutions of numerous plate problems that
often occur in the engineering practice.
Requirements of a Textbook. A textbook must clearly formulate the fundamentals and present a sufficient number of illustrative examples. Thus throughout the
text the mathematical modeling of physical phenomena is emphasized. Although
1
Szilard, R., Theory and Analysis of Plates: Classical and Numerical Methods, Prentice-Hall, Upper
Saddle River, New Jersey, 1974.

xvii

xviii

Preface

the occasionally complicated mathematical theories of plates cannot be simplified,
they can certainly be presented in a clear and understandable manner. In addition, a
large number of carefully composed figures should make the text graphically more
descriptive. Rather than attempt the solutions of specific problems, the author has
introduced generally applicable analytical, numerical and engineering methods for
solution of static, dynamic and stability problems of plates. Since experience is the
best teacher, numerous worked examples illustrate the applications of these methods.
All the numerical examples are computed by using the modernized metric system as
defined by the International System of Units (Syst`eme international d’unit´es).
Although higher mathematics is essential to the analytical solution of most plate
problems, the mathematical prerequisites of the book are relatively modest. Merely
the familiarity with differential-integral calculus and matrix algebra is assumed, and
all further required mathematical tools are systematically developed within the text.
The sections dealing with the methods of higher analysis are treated as integral parts
of the text. The same procedure has been followed with such other prerequisites as
the theory of elasticity, structural dynamics, limit design and so forth. This approach
has resulted in a self-contained text on plates that can be used without consulting
related works.
Working knowledge of the fundamentals of the classical methods is considered
mandatory in spite of its serious limitations. As in most fields of mathematical physics,
exact analytical solutions can be obtained only for the simplest cases. For numerous
plate problems of great practical importance, the classical methods either fail to
yield a solution or yield a solution too complicated for practical application. Here,
the approximate and numerical methods offer the only reasonable approach. The
“exact” solutions, however, perform an important function because they provide the
benchmark against which all other solution techniques are tested.

With the present widespread availability of powerful desktop computers, there
has been a real revolution in the numerical analysis of plate problems. From the
various computer-oriented solution techniques, the finite difference, the gridwork,
finite element and finite strip methods have been treated extensively. The reader will
also find a short introduction to the recently emerging boundary element method.
The actual coding of the computerized solutions of plate problems is considered
to be outside the scope of this book. The numerical solutions of plate problems,
however, are formulated so that either standard computer programs can be used or
they can be easily programmed by utilizing readily available subroutines of numerical
analysis procedures. In order to facilitate the numerical solution of certain problems,
numerous finite difference stencils and finite element stiffness and mass matrices are
given in explicit forms. Furthermore, plate programs of practical interest are also
stored on the CD-ROM that accompanies this book. These include, in addition to
the FORTRAN source codes, the executable forms of these computer codes for static
and dynamic analysis of plates.
Of the analytical approaches, the energy methods are treated more extensively than
others because the author believes that their relative simplicity, efficiency and almost
universal use warrant this emphasis.
Sections marked with asterisks (*) in the table of contents are recommended for
classroom use in a one-semester course on plates for graduate students of civil,
mechanical, aeronautical, architectural, mining and ocean engineering and for students
of engineering mechanics and naval architecture. The material presented, however, is
sufficient for a two-semester course; preferably one semester of directed reading


Preface

xix

would be offered following the first semester of formal classroom presentation.

Exercises to be worked out by the students are included at the end of most chapters.
They are listed in order of ascending difficulty.
Use of the Book by Practicing Engineers. Although the requirements of practicing engineers are different from those discussed above, there are also numerous
overlapping areas. Practicing engineers must deal with “real-life” plate problems.
Consequently, they require a much broader coverage than that usually given in
“Analysis of Plates and Shells” textbooks. The present book, however, intends to
satisfy this important need by covering a large spectrum of plate problems and their
solution procedures. Plate analysis has undergone considerable changes during the
past decades. These changes were introduced by (a) proliferation of powerful—yet
relatively inexpensive—personal computers and (b) the development of computeroriented numerical analysis techniques such as gridwork and finite element methods,
to name the most important ones. Consequently, nowadays the practicing engineer
will apply a suitable computer code to analyze plate structures for their static or
dynamic behaviors and determine their stability performance under the given loads.
However, to be able to use such contemporary analysis methods properly, he or she
must have basic knowledge of pertinent plate theories along with the underlying principles of these numerical solution techniques. All these fundamental requirements for
a successful computer-based plate analysis are amply covered in this book. Furthermore, it is of basic importance that the engineer properly idealizes the plate structures
which are in essence two-dimensional continua replaced by equivalent discrete systems in the numerical approach. This idealization process includes definition of plate
geometry along with the existing support conditions and the applied loads. It also
incorporates the discretization process, which greatly influences the obtainable accuracy. Although proper idealization of a real structure is best learned under the personal
guidance of an experienced structural engineer, numerous related guidelines are also
given throughout in this book. To start a numerical analysis, ab ovo the plate thickness
is required as input. For this purpose, this work contains various engineering methods. Using these, the required plate thickness can be determined merely by simple
“longhand” computations. After obtaining a usable estimate for plate behavior under
the applied load, the engineer can use a computer to compute more exact numerical
results. For this purpose, interactive, easy to use computer programs covering the most
important aspects of plate analysis are stored on the companion CD-ROM attached to
the back cover of the book. This CD-ROM contains a finite element program system,
WinPlatePrimer, which not only solves important static and dynamic plate problems
but also teaches its users how to write such programs by using readily available
subroutines. Consequently, next to the executable files the corresponding FORTRAN

source codes are also listed. The finite elements used in these programs have excellent
convergence characteristics. Thus, good results can be obtained even with relatively
crude subdivisions of the continuum. To validate the computer results, the practicing
engineer needs, again, readily usable simple engineering approaches that can provide
valuable independent checks. It is also important that he or she knows the effectiveness and economy not only of these approximate solution techniques but also of all
methods presented here. These important aspects are also constantly emphasized. As
mentioned earlier, explicitly given structural matrices and finite difference stencils
allow the practicing engineer to develop his or her own computer programs to solve
some special problems not covered in commercially available program systems. In
general, strong emphasis is placed on practical applications, as demonstrated by an


xx

Preface

unusually large number of worked problems many of them taken directly from the
engineering practice. In addition, considering the needs of practicing engineers, the
book is organized so that particular topics may be studied by reading some chapters
before previous ones are completely mastered. Finally, it should be mentioned that
the practicing engineer has often to deal with such plate problems for which solutions
are already available in the pertinent technical literature. For this reason, a collection of the 170 most important plate formulas is given on the companion CD-ROM
attached to the back cover of the book. These formulas, along with the closed-form
solutions of certain plate problems presented in this book, can also be used to test
commercially available computer codes for their effectiveness and accuracy.
Guide to the Reference System of the Book. The mathematical expressions are
numbered consecutively, in parentheses, throughout each section carrying the pertinent section number before the second period. Equation (2.7.4), for example, refers
to Equation 4 in Section 2.7.
The author realizes that a complete volume could be written on each of the chapters
treated. Liberal inclusion of bibliographical references extends the comprehensiveness

of this book. The numbering system used for references is similar to that of equations.
The numbers in brackets refer to bibliographical references, again with the pertinent
section number as prefix. Numbers in brackets without a prefix indicate general
reference books on plates and are listed after the Introduction. References pertinent
to the history of development of various plate theories carry the prefix II. References
with prefixes “A” refer to the Appendixes.
Finally, the author is particularly indebted to Dr. L. Dunai, Professor, Technical University of Budapest, Hungary, and his co-workers (N. Kov´acs, Z. K´osa and
´ anyi) for developing the WinPlatePrimer program system. My thanks are also
S. Ad´
due to my wife, Ute, for her continuous encouragement and support in writing this
book and for editing the manuscript and checking the page proofs.


Symbols

The following symbols represent the most commonly used notations in this book.
Occasionally, the same symbols have been used to denote more than one quantity;
they are, however, properly identified when first introduced.
a, b
ai , bi
aij , bij
a, b, c
{a}, {a}T or [a]T
A
A, B, C, . . .
[A], [B], . . .
B
c1 , c2 , . . .
d i , de,i
d, de

D
Dx , Dy
Dt
D, [D]
E
E, [E]
f
fi (·)
g
G, Gxy
h
i, j, k, l
I
Itx , Ity

Plate dimensions in X and Y directions, respectively
Coefficients
Elements of matrices A and B, respectively
Column matrices or vectors, respectively
Column and row matrices, respectively
Area, constant
Matrices
Matrices
Effective torsional rigidity of orthotropic plate, constant
Constants or numerical factors
Elements of the displacement vectors d, de , respectively
Displacement vector (global/element)
Flexural rigidity of plate [D = Eh3 /12(1 − ν 2 )]
Plate flexural rigidities associated with X and Y directions,
respectively

Torsional rigidity of plate
Dynamical matrix of vibrating structural system, and representing
pertinent differentiations
Young’s modulus of elasticity
Elasticity matrix
Frequency of a vibrating structural system (Hz)
Function
Acceleration of gravity (≈9.81 m/s2 )
Shear moduli
Thickness of plate
Indices and/or positive integers (1, 2, 3, . . .)
Moment of inertia
Geometrical torsional rigidities of beams
xxi


xxii

Symbols

k
k ij , ke,ij
K, Ke
lx , ly , l
L(·), L(·)
m, n
mij
mT
mu , mu
mx , my

mxy
mr , mϕ
mrϕ
M
M, Me
Mx , My , Mt
n
ncr
nT
nx , ny
nxy
pu
N, [N ]
px , py , pz

PX , PY , PZ
qr , qϕ
q x , qy
r, ϕ, z
r0 , r1
R, ϕ, Z
R, [R]
t
T
T, [T ]
u, v, w
U
vx , v y
V
w H , wP

We , Wi
x, y, z

Modulus of elastic foundation, numerical factor
Elements of stiffness matrix (global/element)
Stiffness matrix (global/element)
Span lengths
Differential operators
Positive integers (1, 2, 3, . . .)
Elements of consistent mass matrix M in global reference system
Thermal equivalent bending moment
Ultimate bending moments per unit length
Bending moments per unit length in X, Y, Z Cartesian coordinate
system
Twisting moment per unit length in X, Y, Z Cartesian coordinate
system
Radial and tangential bending moments per unit length in r, ϕ, Z
cylindrical coordinate system
Twisting moment per unit length in r, ϕ, Z cylindrical coordinate
system
Moment-sum, =(mx + my )/(1 + ν)
Mass matrix (global/element)
Concentrated and/or external moments
Normal to boundary, index
Critical (buckling) load
Thermal force per unit length acting in X, Y plane
Normal forces per unit length acting in X, Y plane
Shear forces per unit length acting in X, Y plane
Ultimate load
Matrix of shape functions

Load components per unit area in X, Y, Z Cartesian coordinate
system
Concentrated forces in X, Y, Z Cartesian coordinate system
Transverse shear forces per unit length in r, ϕ, Z cylindrical
coordinate system
Transverse shearing forces in X, Y, Z Cartesian coordinate system
Cylindrical coordinates
Radii
Cylindrical coordinate system
Rotational matrix
Time or tangent to boundary
Temperature or kinetic energy
Transformation matrix
Displacement components in X, Y, Z directions
Strain energy
Lateral edge forces per unit length associated with X and Y
directions, respectively
Volume or potential of external forces
Homogeneous and particular solutions of plate equation,
respectively
Work of external and internal forces, respectively
Cartesian coordinates


Symbols

X, Y, Z
α, β, ϑ
αm , βm
αT

γ , γxy
δ, δi,j
δ(·)
εT , εx , εy
εi (x, y)
η, ξ
λ
λi
ϑ, θ, ϕ, φ
λ1 , λ2 , . . .
λcr
κx , κy , κxy , χ
v, vx , vy , vxy
ρ
σx , σy
σu
τ, τxy
ϕ, ϕi (·), ϕi (·)
(x, y)
χ
ω

xxiii

Coordinate axes of Cartesian coordinate system
Angles
Constants
Coefficient of thermal expansion
Shear strain, shear strain in X, Y plane
Displacement, flexibility coefficients

Variational symbol
Thermal strain, normal strains in X and Y directions, respectively
Error function
Oblique coordinates
Finite difference mesh width ( x = y = λ)
Lagrangian multiplier
Angles
Eigenvalues
Critical load factor
Curvatures of deflected middle surface
Poisson ratios
Total potential energy
Dimensionless quantity (r/r0 )
Normal stresses in X and Y direction, respectively
Ultimate stress
Shear stresses
Angle, functions
Stress function
Warping of deflected middle surface
Circular (angular frequency of free vibration (rad/s))

∇ 2 (·) =

∂ 2 (·) ∂ 2 (·)
+
;
∂x 2
∂y 2

∇ 4 (·) =

∂ 4 (·)
∂ 4 (·)
∂ 4 (·)
+
2
+
∂x 4
∂x 2 ∂y 2
∂y 4

∇r =

∂ 2 (·)
1 ∂ 2 (·) 1 ∂(·)
+ 2·
+ ·
2
∂r
r
∂φ 2
r ∂r

The various boundary conditions are shown in the following manner:
Section
Free edge
Simple support

Clamped edge

Point support

Elastic support

Plan view


I
Introduction

Plates are straight, plane, two-dimensional structural components of which one dimension, referred to as thickness h, is much smaller than the other dimensions. Geometrically they are bound either by straight or curved lines. Like their counterparts, the
beams, they not only serve 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 (Fig. I.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 forces.
Since the load-carrying action of plates resembles to a certain extent that of beams,
plates can be approximated by gridworks 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 components in engineering structures can be classified as plates. Typical
examples in civil engineering structures are floor and foundation slabs, lock-gates,
thin retaining walls, bridge decks and slab bridges. Plates are also indispensable in
shipbuilding 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. The hull of a ship, its deck and its superstructure are further examples of stiffened
plate structures. Plates are also frequently parts of machineries and other mechanical
devices. Figure I.2 schematically illustrates some of these industrial applications.
This book deals with the various plate analysis techniques which, of course, cannot be learned without a well-founded knowledge in the underlying plate theories.
The main objective to any structural analysis is to ensure that the structure under
investigation shall have an adequate safety factor against failure within reasonable

Theories and Applications of Plate Analysis: Classical, Numerical and Engineering Methods. R. Szilard
Copyright © 2004 John Wiley & Sons, Inc.

1


2

Introduction

p

g

Figure I.1

Various boundary conditions for plates.

economical bounds. Furthermore, the structure shall be serviceable when subjected
to design loads. A part of serviceability can be achieved, for example, by imposing
suitable limitations on deflections.
The majority of plate structures is analyzed by applying the governing equations
of the theory of elasticity. Consequently, a large part of this book presents various

elastic plate theories and subsequently treats suitable analytical and numerical solution
techniques to determine deflections and stresses.
As already mentioned in the Preface, “exact” solutions of the various governing
differential equations of plate theories can only be obtained for special boundary and
load conditions, respectively. In most cases, however, the various energy methods
can yield quite usable analytical solutions for most practical plate problems. Nowadays, with widespread use of computers, a number of numerical solution techniques
have gained not only considerable importance but, as in the case of the finite element
method, also an almost exclusive dominance. All numerical methods treated in this
book are based on some discretization of the plate continuum. The finite difference
and the boundary element methods apply mathematical discretization techniques for
solution of complex plate problems, whereas the gridwork, finite element and finite
strip methods use physical discretizations based on engineering considerations. Since


Introduction

3

the results obtained by the different computer-oriented numerical approaches always
require independent checks, engineering methods, capable of giving rough approximations by means of relatively simple “longhand” computations, are regaining their
well-deserved importance. In addition, engineering methods can also be used for preliminary design purposes to determine the first approximate dimensions of plates. In
addition to static plate problems, all the above-mentioned solution techniques also
treat pertinent dynamic and elastic stability problems.
However, these methods, based on elastic theories, have certain limitations. The
most important of these is that they do not give accurate indication of the factor of
safety against failure. Partly due to this limitation, there is a tendency to replace the
elastic analysis by ultimate load techniques. On the other hand, since this method
cannot always deal with all the problems of serviceability, the author recommends
that, if required, an elastic analysis should be augmented by a failure assessment
using the ultimate load approach.

Slab

Slab

(a1) Reinforced concrete slabs in buildings

Plate

(a2) Steel bridge deck

Slab bridge
(a) Use of plates in construction industry

Figure I.2

Use of plates in various fields of engineering.


4

Introduction

A
A

(b1) Merchant ship
Deck plate

C
Plane of symmetry

Plate
Steel plates

(b2) Section A–A
(b) Use of plates in shipbuilding

Figure I.2

(continued)

In all structural analyses 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.
2.
3.

the geometry of the plate and its supports,
the behavior of the material used, and
the type of loads and their way 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 plate categories with inherently different structural behavior and,
hence, different governing differential equations. The four plate-types might be categorized, to some extent, using their ratio of thickness to governing length (h/L).