The boundary element method for plate analysis

The Boundary

Element Method for

Plate Analysis

The Boundary

Element Method for

Plate Analysis

by

John T Katsikadelis

National Technical University of Athens, Athens, Greece

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14 15 16 17 18 10 9 8 7 6 5 4 3 2 1

To my wife Efi for her loving patience and support

Contents

Foreword........................................................................................................viii

Preface..............................................................................................................xi

1 Preliminary Mathematical Knowledge ............................................... 1

1.1 Introduction............................................................................................. 1

1.2 Gauss-Green Theorem............................................................................ 2

1.3 Divergence Theorem of Gauss............................................................... 3

1.4 Green’s Second Identity ......................................................................... 4

1.5 Adjoint Operator..................................................................................... 5

1.6 Dirac Delta Function .............................................................................. 6

1.7 Calculus of Variations; Euler-Lagrange Equation............................... 11

1.8 References............................................................................................. 18

Problems ...................................................................................................... 19

2 BEM for Plate Bending Analysis....................................................... 21

2.1 Introduction........................................................................................... 22

2.2 Thin Plate Theory................................................................................. 23

2.3 Direct BEM for the Plate Equation...................................................... 40

2.4 Numerical Solution of the Boundary Integral Equations .................... 61

2.5 PLBECON Program for Solving the Plate Equation with

Constant Boundary Elements ............................................................... 72

2.6 Examples............................................................................................... 79

2.7 References ........................................................................................... 109

Problems .................................................................................................... 110

3 BEM for Other Plate Problems....................................................... 113

3.1 Introduction......................................................................................... 114

3.2 Principle of the Analog Equation....................................................... 115

3.3 Plate Bending Under Combined Transverse and Membrane

Loads; Buckling.................................................................................. 118

3.4 Plates on Elastic Foundation .............................................................. 141

3.5 Large Deflections of Thin Plates ....................................................... 152

3.6 Plates with Variable Thickness .......................................................... 166

3.7 Thick Plates ........................................................................................ 177

3.8 Anisotropic Plates............................................................................... 187

3.9 Thick Anisotropic Plates .................................................................... 196

3.10 References......................................................................................... 203

Problems .................................................................................................... 207

vi

4 BEM for Dynamic Analysis of Plates.............................................. 211

4.1 Direct BEM for the Dynamic Plate Problem..................................... 212

4.2 AEM for the Dynamic Plate Problem................................................ 217

4.3 Vibrations of Thin Anisotropic Plates ............................................... 237

4.4 Viscoelastic Plates .............................................................................. 241

4.5 References ........................................................................................... 251

Problems .................................................................................................... 253

5 BEM for Large Deflection Analysis of Membranes...................... 257

5.1 Introduction......................................................................................... 257

5.2 Static Analysis of Elastic Membranes ............................................... 259

5.3 Dynamic Analysis of Elastic Membranes.......................................... 270

5.4 Viscoelastic Membranes..................................................................... 275

5.7 References ........................................................................................... 283

Problems .................................................................................................... 285

Appendix A: Derivatives of r and Kernels, Particular Solutions

and Tangential Derivatives................................................. 287

Appendix B: Gauss Integration................................................................. 297

Appendix C: Numerical Integration of the Equations of Motion ......... 313

Index.............................................................................................................. 325

Companion Website for this book: http://booksite.elsevier.com/9780124167391

Contents vii

Foreword

At first glance this book may appear to describe yet another highly specialised

method applied to the solution of plate problems, namely the one the author calls

Analog Equation Method (AEM). Nothing could be further from the truth.

Professor Katsikadelis has instead presented for the first time a generalised

and consistent BEM for all types of plate analysis. This has been possible only

because of his brilliant interpretation of the principle of virtual work.

The first two introductory chapters set the basis for the subsequent

treatment.

After having set up the basic principles of boundary elements (BEM) in an

elegant and consistent manner in the first chapter, the reader acquires the nec￾essary knowledge to understand how these principles can be employed in sub￾sequent chapters to solve many different problems.

This basic theory is then used to formulate the direct BEM for the analysis of

thin plates. The benefits of having previously described the fundamentals of the

method in a clear manner then become evident. Once the basic integral equa￾tions are derived, the author then demonstrates how they can be applied to write

a computer programme, which results are validated through a series of

comparisons.

The beauty of the approach followed by the author is that it describes how

the mathematical process gives rise to equations which can be reduced to com￾putational form for solving realistic engineering problems.

The above introductory chapters, important as they are, pale into insignifi￾cance in comparison with the rest of the book, where a series of most novel con￾cepts are described. The author starts by describing the analysis of plates under

membrane and bending forces, which leads to the equations governing buckling,

large deflections and post buckling of plates. Important as these cases are, the

most significant aspect is that they are solved using an original methodology

based on the author’s Analog Equation Method, which leads to the full analysis

of a wide range of plate problems [1,2].

Few developments in Boundary Elements have been as significant as this

idea of Professor Katsikadelis’ and, as with all truly original ideas, it is striking

in its simplicity and elegance.

To understand the AEM properly we have to refer to the basic idea behind

the principle of virtual work as defined by Aristotle who stated that the behav￾iour of physical systems could be expressed in terms of “potentiality” and “actu￾ality”. In other words, Aristotle set up the principle of virtual “potentialities” or

what we now call the principle of virtual work. While an “actual” field function

is to satisfy the equations giving the problem, a “virtual” function can be more

general. Usually we assume that the virtual function satisfies the same equation

viii

as those governing the actual field, or sometimes a reduced version of those

equations as in the case of the Dual Reciprocity Method [3]. Professor Katsida￾delis instead gave a much wider interpretation to the virtual functions – one that

would have pleased Aristotle – by stating that they do not necessarily need to

satisfy the same type of governing equations of the actual problem, provided

that they have the necessary degree of continuity (in the case of plate bending

fourth order for instance).

The resulting Method (AEM) when combined with the use of the localised

particular solutions proposed by the Dual Reciprocity Method, opens up a huge

range of possibilities to Boundary Elements, some of which are presented in

this book.

The part dealing with the time and non-linear analysis of plates for instance

leads to a series of original formulations based on the AEM. The possibility of

solving problems with membrane as well as bending forces can now be fully

exploited for cases like dynamic buckling, including flutter instability and a

series of applications of fundamental importance in aerospace engineering

for instance. Extensions to the case of membranes, non linear materials and

large deformations follow effortlessly.

Throughout the book the reader will find a clarity of exposition and consis￾tency which allows the progression from simple to more complex problems in a

stepwise fashion. This results in obtaining a full comprehension of the basic

principles and how they are applied to obtain practical solutions in a way that

is frequently missing in the current engineering sciences literature.

The fact that this book centres on the concept of the AEM developed by the

author does not imply any restrictions as the AEM can be interpreted to be the

most general version of the principle of virtual work developments ever pre￾sented in science and engineering.

Professor Katsikadelis’ Method effortlessly transforms a series of complex

problems into alternative problems which can be solved in BEM form using

simple fundamental solutions.

An added advantage of the AEM is that it allows for the solution of a given

set of problems, in the case of plates for instance, using the same type of com￾puter programme. This generality will allow boundary elements to become

more widely used for plates and shells, types of problems for which the method

still lags behind the less accurate but more versatile finite element method.

Those interested in knowing more about the many contributions of Professor

Katsikadelis to the solution of a wide variety of engineering problems and the

development of many different ideas, ought to refer to my own appraisal of his

work in reference [4]. It was while compiling that paper that I came to fully

appreciate his work, including his many contributions to the analysis of plates.

The contents of the present book represent without doubt, a major develop￾ment in engineering sciences.

Carlos A. Brebbia

Foreword ix

REFERENCES

[1] J.T. Katsikadelis, The analog equation method – a powerful BEM-based solution tech￾nique for solving linear and non-linear engineering problems, in: C.A. Brebbia (Ed.),

Boundary Element Methods XVI, Computational Mechanics Publications,

Southampton and Boston, 1994, pp. 167–182.

[2] Katsikadelis, J.T., Nerantzaki, M.S., The boundary element method for nonlinear prob￾lems. Eng. Anal. Bound. Elem. 23 (5), 365–373.

[3] D. Nardini, C.A. Brebbia, New approach to free vibration analysis using boundary ele￾ments, in: C.A. Brebbia (Ed.), Boundary Element Methods in Engineering, Springer

Verlag, Berlin and Computational Mechnics Publications, Southampton and Boston,

1982, pp. 312–326.

[4] C.A. Brebbia, In praise of John Katsikadelis, in: E.J. Sapountzakis (Ed.), Recent Devel￾opments in Boundary Element Methods, WIT Press, Southampton and Boston, 2010,

pp. 1–16.

x Foreword

Preface

This book presents the Boundary Element Method, BEM, for the static and

dynamic analysis of plates and membranes. It is actually a continuation of

the book Boundary Elements: Theory and Applications by the same author

and published by Elsevier in 2002. The latter was well received as a textbook

by the relevant international scientific community, which is ascertained by the

fact that it was translated into three languages, Japanese by the late Prof. Masa

Tanaka of the Shinshu University, Nagano (Asakura, Tokyo 2004), in Russian

by the late Prof. Sergey Aleynikov of the Voronezh State Architecture and Civil

Engineering University (Publishing House of Russian Civil Engineering

Universities, Moscow 2007), and in Serbian by Prof. Dragan Spasic of the

University of Novi Sad (Gradjevinska Κnjiga, Belgrade 2011).

The success of the first book on the BEM encouraged me to prepare this

second book on the BEM for plate analysis. Though there is extensive literature

on BEM for plates published in journals, there hasn’t been any book published

on this subject to date, either as a monograph or as a textbook. To my knowl￾edge, there are only two edited books with contributions of various authors on

different plate problems. These books are addressed to researchers and are not

suitable for introducing students or even scientists to the subject. Some books on

BEM contain the application of the method to plates as a concise chapter aim￾ing, rather, on the completeness of their book, than the presentation of material

necessary to understand the subject.

The main reasons for not writing a book on plates at an earlier time include

the following:

1. The basic plate problem, i.e., the problem for thin Kirchhoff plates, is

described by the biharmonic differential operator whose treatment with

the BEM requires special care, both in deriving the boundary integral equa￾tions and in obtaining their numerical solution. Thus, a comprehensive pre￾sentation of the material to the student is a tedious task and demands a great

effort from the author.

2. Different plate problems (e.g., plates on elastic foundation, plates under

simultaneous membrane loads, anisotropic plates, etc.) are described by dif￾ferent fourth-order partial differential equations (PDEs) that require the

establishment of the fundamental solution, in general not possible, and, thus,

different formulations for the derivation of the boundary integral equations

and special numerical treatment is needed to obtain results.

3. The difficulties in applying the conventional BEM become insurmountable

when plates with variable thickness and dynamic or nonlinear plate prob￾lems must be treated.

xi

The above reasons have discouraged potential authors from writing a book on

plates. Many have envisioned it as a digest of BEM formulations for plate prob￾lems rather than as an efficient computational method for practical plate anal￾ysis and design.

During the last 20 years, intensive research has been carried out in an effort to

overcome the above shortcomings, especially to alleviate the BEM from establish￾ing a fundamental solution for each plate problem. Several techniques have been

developed to cope with the problem. The DRM (Dual Reciprocity Method) has

enabled the BEM to efficiently solve static and dynamic engineering problems.

Although this method is quite general, it produces boundary-only solutions for

those cases where a linear operator with a well-known fundamental solution could

be extracted from the full governing equation. However, this is not always pos￾sible. The AEM introduced in 1994 overcomes all restrictions of the DRM and

enables the BEM to efficiently solve any problem. It is based on the concept (prin￾ciple) of the analog equation according to which a problem governed by a linear or

nonlinear differential equation of any type (elliptic, parabolic, or hyperbolic) can

be converted into a substitute problem described by an equivalent linear equation

of the same order as the original equation having a simple known fundamental

solution and subjected to a fictitious source, unknown in the first instance. The

value of this source can be established using the BEM. By applying this idea,

coupled linear or nonlinear equations can be converted into uncoupled linear ones.

This method is employed to solve all plate problems discussed in the present book.

As any plate problem is described by a single fourth-order PDE or coupled

with two second-order PDEs in the presence of membrane forces, the classical

plate equation and two Poisson’s equations serve as substitute equations. Both

types of equations have simple known fundamental solutions and can be readily

solved by the conventional direct BEM. A major advantage of the AEM is that

the computer program for the classical plate problem can be used to solve any

particular plate problem. The research of the author has highly contributed to

this end. Most of the material presented in this book can be found in the journal

articles written by the author and his colleagues. The AEM renders the BEM an

efficient computational method for practical plate analysis.

The material in this book is presented systematically and in detail so the

reader can follow without difficulty. A chapter on preliminary mathematical

knowledge makes the book self-contained. A special feature of the book is that

it connects theoretical treatment and numerical analysis. The comprehensibility

of the material has been tested with the author’s students for several years.

Therefore, it can be used as a textbook. The book contains five chapters:

Chapter 1 gives a brief, elementary description of the basic mathematical

tools that will be employed throughout the book in developing the BEM, such as

Green’s reciprocal identity and Dirac’s delta function. This chapter concludes

with a section on calculus of variations, which provides the reader with an effi￾cient mathematical tool to derive the governing differential equation together

with the associated boundary conditions in complicated structural systems from

xii Preface

stationary principles of mechanics. Comprehension of these mathematical

concepts helps readers feel confident in their subsequent application.

Chapter 2 presents the direct BEM for the static analysis of thin plates under

bending. First, the essential elements of the Kirchhoff plate are discussed. Then,

the BEM is formulated in terms of the transverse displacement of the middle

surface. The integral representation of the solution and the boundary integral

equations are derived in clear, comprehensible steps. Emphasis is on the numer￾ical implementation of the method. A computer program is developed for the

complete analysis of plates of arbitrary shape and arbitrary boundary conditions.

The program is explained thoroughly and its structure is developed systemati￾cally, so the reader can be acquainted with the logic of writing the BEM code in

the computer language of preference. The method is illustrated by analyzing

several plates.

Chapter 3 presents the BEM for the analysis of more complex plate prob￾lems appearing in engineering practice. First, there is a discussion of plate bend￾ing under the combined action of membrane forces, which applies to buckling of

plates. Then, it follows the analysis of plates resting on any type of elastic foun￾dation, and the large deflections of plates and their postbuckling response. Plates

with variable thickness are discussed with application to plate-thickness optimi￾zation for maximization of plate stiffness or buckling load. Thick plates are also

studied in this chapter, which concludes with the treatment of thin and thick

anisotropic plates. As all problems in this chapter are solved by the AEM, its

application and numerical implementation are described in detail. Several

example problems are solved to demonstrate the efficiency of the solution

procedure.

Chapter 4 develops the BEM for linear and nonlinear dynamic analysis of

plates, such as free and forced vibrations with or without membrane forces,

buckling of plates using the dynamic criterion, and flutter instability of plates

under nonconservative loads. Both isotropic and anisotropic plates are analyzed.

Plates under aerodynamic loads such as the wings of aircrafts are also discussed.

The chapter ends with the application of the BEM to static and dynamic analysis

of viscoelastic plates described with differential models of integer and

fractional order.

Chapter 5 presents the BEM for the static and dynamic analysis of flat elas￾tic and viscoelastic membranes undergoing large deflections. First, the non￾linear PDEs governing the response of the membrane are derived in terms of

the three displacements together with the associated boundary conditions.

The resulting boundary and initial boundary value problems are solved by

the BEM in conjunction with the principle of the analog equation. Several mem￾branes, elastic and viscoelastic, of various shapes under static and dynamic

loads are analyzed.

The book also includes three appendices. Appendix A gives useful formulas

for the differentiation of the kernel functions and the expressions of tangential

derivatives necessary for the treatment of boundary quantities on curvilinear

Preface xiii

boundaries. Appendix B presents the Gauss integration for the numerical eval￾uation of line and domain integrals. Finally, Appendix C describes the time

integration method employed for the solution of linear and nonlinear equations

of motion.

In closing, the author wishes to express his sincere thanks to his former stu￾dent and colleague Dr. A.J. Yiotis for carefully reading the manuscript, his sug￾gestions for constructive amendments and for his overall contribution to

minimizing the oversights of the text. Warm thanks, also, to Dr. Nikos G.

Babouskos, former student and colleague of the author, not only for the careful

reading of the manuscript and his suggestions for the improvement of the book,

but also for his assistance in developing the computer programs and in produc￾ing the numerical results for the examples, most of which are contained in joint

publications with the author of the book.

J.T. Katsikadelis

Athens

December 2013

xiv Preface

Chapter | one

Preliminary

Mathematical

Knowledge

CHAPTER OUTLINE

1.1 Introduction .................................................................................................................... 1

1.2 Gauss-Green Theorem .................................................................................................2

1.3 Divergence Theorem of Gauss ..................................................................................3

1.4 Green’s Second Identity .............................................................................................4

1.5 Adjoint Operator ...........................................................................................................5

1.6 Dirac Delta Function ....................................................................................................6

1.7 Calculus of Variations; Euler-Lagrange Equation ...............................................11

1.7.1 Euler-Lagrange Equation ..............................................................................................12

1.7.2 Natural Boundary Conditions ....................................................................................... 15

1.7.3 Functional Depending on a Function of Two Variables ......................................15

1.7.4 Examples ........................................................................................................................... 17

Example 1.1 ........................................................................................................................17

Example 1.2 .......................................................................................................................17

1.8 References ....................................................................................................................18

Problems ...............................................................................................................................19

1.1 INTRODUCTION

In this chapter, some mathematical relations required for the development and

understanding of the boundary element method (BEM) are presented. Although

these relations could have been included in an appendix, they are placed here to

show the reader their important role in the theoretical foundation and develop￾ment of the BEM. They will be used often throughout the book and particularly

for the transformation of the differential equations, which govern the response

of physical systems within a domain, into integral equations on the boundary.

The chapter ends with a section on calculus on variations, which provides the

The Boundary Element Method for Plate Analysis

© 2014 John T. Katsikadelis. Published by Elsevier Inc. All rights reserved.

1