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Introductory Finite Element Method 1st Edition
Chandrakant S. Desai Digital Instant Download
Author(s): Chandrakant S. Desai, TribikramKundu
ISBN(s): 9780849302435, 0849302439
Edition: 1
File Details: PDF, 4.70 MB
Year: 2001
Language: english

FINITEELEMENT
METHOD
Introductory

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Chandrakant S. Desai
Tribikram Kundu
FINITEELEMENT
METHOD
Introductory
Boca Raton London New York Washington, D.C.
CRC Press

This book contains information obtained from authentic and highly regarded sources. Reprinted material
is quoted with permission, and sources are indicated. A wide variety of references are listed. Reasonable
efforts have been made to publish reliable data and information, but the author and the publisher cannot
assume responsibility for the validity of all materials or for the consequences of their use.
Neither this book nor any part may be reproduced or transmitted in any form or by any means, electronic
or mechanical, including photocopying, microfilming, and recording, or by any information storage or
retrieval system, without prior permission in writing from the publisher.
The consent of CRC Press LLC does not extend to copying for general distribution, for promotion, for
creating new works, or for resale. Specific permission must be obtained in writing from CRC Press LLC
for such copying.
Direct all inquiries to CRC Press LLC, 2000 N.W. Corporate Blvd., Boca Raton, Florida 33431.
Trademark Notice: Product or corporate names may be trademarks or registered trademarks, and are
used only for identification and explanation, without intent to infringe.
Visit the CRC Press Web site at www.crcpress.com
© 2001 by CRC Press LLC
No claim to original U.S. Government works
International Standard Book Number 0-8493-0243-9
Library of Congress Card Number 2001017466
Printed in the United States of America 1 2 3 4 5 6 7 8 9 0
Printed on acid-free paper
Library of Congress Cataloging-in-Publication Data
Desai, C. S. (Chandrakant S.), 1936-
Introductory finite element method / Chandrakant S. Desai, Tribikram Kundu.
p. cm.
Includes bibliographical references and index.
ISBN 0-8493-0243-9 (alk. paper)
1. Finite element method. I. Kundu, T. (Tribikram). II. Title.
TA347.F5 .D48 2001
620′.001′1535--dc21 2001017466
0618fm/frame Page 4 Wednesday, April 11, 2001 9:27 AM

Dedication
To
Our wives: Patricia Desai
and Nupur Kundu
Children: Maya and Sanjay; Ina and Auni
and Parents: Sankalchand Desai and Kamala Desai;
Makhan Lal Kundu and Sandhya Rani Kundu

Preface
The finite element method has gained tremendous attention and popularity.
The method is now taught at most universities and colleges, is researched
extensively, and is used by the practicing engineers, industry, and government
agencies. The teaching of the method has been concentrated at the postgrad-
uate level. In view of the growth and wide use of the method, however, it
becomes highly desirable and necessary to teach it at the undergraduate level.
There are a number of books and publications available on the finite ele-
ment method. It appears that almost all of them are suitable for the advanced
students and require a number of prerequisites such as theories of constitu-
tive or stress-strain laws, mechanics, and variational calculus. Some of the
introductory treatments have presented the method as an extension of matrix
methods of structural analysis. This viewpoint may no longer be necessary,
since the finite element method has reached a significant level of maturity
and generality. It has acquired a sound theoretical basis, and in itself has been
established as a general procedure relevant to engineering and mathematical
physics. These developments permit its teaching and use as a general tech-
nique from which applications to topics such as mechanics, structures, geo-
mechanics, hydraulics, and environmental engineering arise as special cases.
It is therefore essential that the method be treated as a general procedure and
taught as such.
This book is intended mainly for the undergraduate and beginning gradu-
ate students. Its approach is sufficiently elementary so that it can be intro-
duced with the background of essentially undergraduate subjects. At the
same time, the treatment is broad enough so that the reader or the teacher
interested in various topics such as stress-deformation analysis, fluid and
heat flow, potential flow, time-dependent problems, diffusion, torsion, and
wave propagation can use and teach from it. The book brings out the intrinsic
nature of the method that permits confluence of various disciplines and pro-
vides a distinct and rather novel approach for teaching the finite element
method at an elementary level. The book can be used for any student with no
prior exposure to the finite element method. The prerequisites for under-
standing the material will be undergraduate mathematics, strength of mate-
rials, and undergraduate courses in structures, hydraulics, geotechnical
engineering, and matrix algebra. Introductory knowledge of computer pro-
gramming is desirable but not necessary. The text is written in such a way
that no prior knowledge of variational principles is necessary. Over a period
of the last 30 years or so, the authors have taught, based on these prerequi-
sites, an undergraduate course and a course for user groups composed of
beginners. This experience has shown that undergraduates or beginners

equipped with these prerequisites, available to them in the undergraduate
curricula at most academic institutions, can understand and use the material
presented in this book.
The first chapter presents a rather philosophical discussion of the finite ele-
ment method and often defines various terms on the basis of eastern and
western concepts from antiquity. The second chapter gives a description of
the eight basic steps and fundamental principles of variational calculus.
Chapters 3 to 5 cover one-dimensional problems in stress-deformation anal-
ysis and steady and time-dependent flow of heat and fluids. The fundamen-
tal generality of the method is illustrated by showing the common
characteristics of the formulation for these topics and by indicating the fact
that their governing equations are essentially similar. The generality is fur-
ther established by including computer codes in Chapter 6 that can solve dif-
ferent types of problems.
Understanding and using the finite element method are closely linked with
the use of the computer. It is the belief of the authors that strictly theoretical
teaching of the method may not give the student an idea of the details and
the ranges of applicability of the technique. Consequently this text endeavors
to introduce the student, gradually and simultaneously with the theoretical
teaching, to the use and understanding of computer codes. The codes pre-
sented in Chapter 6 are thoroughly documented and detailed so that they can
be used and understood without difficulty. Details of these codes, designed
for the beginner, are given in Appendix 3. It is recommended that these or
other available codes be used by the student while learning various topics in
this book.
Chapter 7 introduces the idea of higher-order approximation for the problem
of beam bending and beam-column. One-dimensional problems in mass trans-
port (diffusion-convection) and wave propagation are covered in Chapters 8
and 9, respectively. These problems illustrate, by following the general proce-
dure, formulations for different categories of time-dependent problems.
Chapter 10 presents the basic finite element formulation for two- and three-
dimensional problems. Then in Chapters 11 to 14 different types of two-
dimensional problems are presented. The chapters on Torsion (Chapter 11)
and Other Field Problems (Chapter 12) have been chosen because they
involve only one degree-of-freedom at a point. Chapters 13 and 14 cover two-
dimensional stress-deformation problems involving two and higher degrees-
of-freedom at a point.
The text presents the finite element method by using simple problems. It
must be understood, however, that it is for the sake of easy introduction that
we have used relatively simple problems. The main thrust of the method, on
the other hand, is for solving complex problems that cannot be easily solved
by the conventional procedures.
For a thorough understanding of the finite element method, it is essential
that the students perform hand calculations. With this in mind, most chapters
include a number of problems to be solved by hand calculations. They also
include problems for home assignments and self-study.

The formulations have been presented by using both the variational and
residual procedures. In the former, the potential, complementary, hybrid, and
mixed procedures have been discussed. In the residual procedures, main
attention has been given to Galerkin’s method. A number of other residual
methods are also becoming popular. They are described, therefore, in
Appendix 1, which gives descriptions, solutions, and comparisons for a
problem by using a number of methods: closed-form, Galerkin, collocation,
subdomain, least squares, Ritz, finite difference, and finite element.
Formulations by the finite element method usually result in algebraic
simultaneous equations. Detailed description of these methods is beyond the
scope of this book. Included in Appendix 2, however, are brief introductions
to the commonly used direct and iterative procedures for the solution of alge-
braic simultaneous equations.
Appendix 3 presents details of a number of computer codes relevant to var-
ious topics in the text.
The book can be used for one or two undergraduate courses. The second
course may overlap with or be an introductory graduate course. Although a
number of topics have been covered in the book, a semester or quarter
course could include a selected number of topics. For instance, a quarter
course can cover Chapters 1 to 6, and then one or two topics from the
remaining chapters. For a class interested in mechanics and stress-deforma-
tion analyses, the topics can be Beam Bending and Beam-Column
(Chapter 7), and Two-Dimensional Stress Deformation (Chapters 10 and 13).
If time is available (in the case of a semester course), Chapter 9 on One-
Dimensional Stress Wave Propagation, Chapter 11 on Torsion, and/or
Chapter 14 on Multicomponent Systems can be added. A class oriented
toward field problems and hydraulics can choose one or more of Chapters 8,
9, 11, and 12 in addition to Chapters 1 to 6 and 10.
We would like to express special appreciation to Jose Franscisco Perez Avila
for his assistance in the manuscript preparation and Shashank Pradhan for
implementing some of the computer codes and preparation of user’s manuals.
We realize that it is not easy to write at an elementary level for the finite ele-
ment method with so many auxiliary disciplines. The judgment of this book
is better left to the readers.
Many natural systems can be considered continuous or interconnected,
and their behavior is influenced by a large number of parameters. In order to
understand such a system, we must understand all the parameters. Since this
is not possible we make approximations, by selecting only the significant of
them and neglecting the others. Such a procedure allows understanding of
the entire system by comprehending its components taken one at a time.
These approximations or models obviously involve errors, and we strive con-
tinuously to improve the models and reduce the errors.
Chandrakant S. Desai
Tribikram Kundu
Tucson, Arizona

Authors
Chandrakant S. Desai is a Regents’ Profes-
sor and Director of the Material Modeling
and Computational Mechanics Center,
Department of Civil Engineering and Engi-
neering Mechanics, University of Arizona,
Tucson. He was a Professor in the Depart-
ment of Civil Engineering, Virginia Poly-
technic Institute and State University,
Blacksburg, from 1974 to 1981, and a
Research Civil Engineer at the U.S. Army
Engineer Waterways Experiment Station,
Vicksburg, MS from 1968 to 1974.
Dr. Desai has made original and signifi-
cant contributions in basic and applied
research in material modeling and testing,
and computational methods for a wide
range of problems in civil engineering,
mechanics, mechanical engineering, and
electronic packaging. He has authored/edited 20 books and 18 book chap-
ters, and has been the author/coauthor of over 270 technical papers. He was
the founder and General Editor of the International Journal for Numerical and
Analytical Methods in Geomechanics from 1977 to 2000, and he has served as a
member of the editorial boards of 12 journals.
Dr. Desai has also been a chair/member of a number of committees of var-
ious national and international societies. He is the President of the Interna-
tional Association for Computer Methods and Advances in Geomechanics.
Dr. Desai has also received a number of recognitions: Meritorious Civilian
Service Award by the U.S. Corps of Engineers, Alexander von Humboldt Stif-
tung Prize by the German Government, Outstanding Contributions Medal in
Mechanics by the International Association for Computer Methods and
Advances in Geomechanics, Distinguished Contributions Medal by the
Czech Academy of Sciences, Clock Award by ASME (Electrical and Electronic
Packaging Division), Five Star Faculty Teaching Finalist Award, and the
El Paso Natural Gas Foundation Faculty Achievement Award at the Univer-
sity of Arizona, Tucson.

Professor T. Kundu received his bachelor
degree in mechanical engineering from the
Indian Institute of Technology, Kharagpur
in 1979. His M.S. and Ph.D. were in the field
of mechanics from the Mechanical and
Aerospace Engineering Department of the
University of California, Los Angeles in
1980 and 1983, respectively. He joined the
University of Arizona as an assistant pro-
fessor in 1983 and was promoted to full
professor in 1994.
Dr. Kundu has made significant and orig-
inal contributions in both basic and applied
research in computational mechanics and
nondestructive evaluation (NDE) of mate-
rials by ultrasonic and acoustic microscopy
techniques. He is editor or coeditor of 8 books, coauthor of a textbook, author
of a book chapter, and author/coauthor of over 130 technical papers; half of
those have been published in refereed journals. He is a Fellow of ASME
(American Society of Mechanical Engineers) and ASCE (American Society of
Civil Engineers). He has received a number of awards, including the Presi-
dent’s Gold Medal from IIT, the UCLA Alumni Award, the Humboldt Fel-
lowship from Germany, and the Best Paper Award from the International
Society for Optical Engineering (SPIE). He has extensive research collabora-
tions with international and U.S. scientists. He has spent 21 months as an
Alexander von Humboldt Scholar in the Department of Biology, J. W. Goethe
University, Frankfurt, Germany. He has also spent several months as a visit-
ing professor at a number of other institutes — Department of Mechanics,
Chalmers University of Technology, Gothenberg, Sweden; Acoustic Micros-
copy Center, Semienov Institute of Chemical Physics, Russian Academy of
Science, Moscow; Department of Civil Engineering, EPFL (Swiss Federal
Institute of Technology in Lausanne), Switzerland; Department of Mechani-
cal Engineering, University of Technology of Compiegne, France; Materials
Laboratory, University of Bordeaux, France; LESiR Laboratory, Ecole Nor-
male Superior (ENS), Cachan, France; Aarhus University Medical School,
Aarhus, Denmark; Wright-Patterson Material Laboratory, Dayton, OH.

Contents
1 Introduction ..................................................................................... 1
Basic Concept...........................................................................................................1
Process of Discretization ........................................................................................3
Subdivision .....................................................................................................3
Continuity .......................................................................................................4
Convergence ...................................................................................................4
Bounds.............................................................................................................5
Error .................................................................................................................6
Principles and Laws................................................................................................7
Cause and Effect....................................................................................................10
Important Comment....................................................................................10
Review Assignments ............................................................................................10
Home Assignment 1 .................................................................................... 11
Home Assignment 2 ....................................................................................12
References ............................................................................................... 12
2 Steps in the Finite Element Method ............................................ 13
Introduction ...........................................................................................................13
General Idea...........................................................................................................13
Step 1. Discretize and Select Element Configuration.................14
Step 2. Select Approximation Models or Functions...................16
Step 3. Define Strain (Gradient)-Displacement (Unknown)
and Stress–Strain (Constitutive) Relationships .............18
Step 4. Derive Element Equations ................................................19
Energy Methods ...........................................................................................19
Stationary Value ...........................................................................................20
Potential Energy...........................................................................................20
Method of Weighted Residuals..................................................................22
Element Equations .......................................................................................25
Step 5. Assemble Element Equations to Obtain Global or
Assemblage Equations and Introduce Boundary
Conditions ..........................................................................26
Boundary Conditions ..................................................................................27
Step 6. Solve for the Primary Unknowns ....................................29
Step 7. Solve for Derived or Secondary Quantities....................29
Step 8. Interpretation of Results....................................................29
Introduction to Variational Calculus..................................................................30
Definitions of Functions and Functionals.................................................30
Variations of Functions................................................................................31
Stationary Values of Functions and Functionals .........................36

∆fx to ∆f Conversion .....................................................................................37
More on the Stationary Value of a Functional — Physical
Interpretation....................................................................................38
Natural and Forced Boundary Conditions...............................................39
Two-Dimensional Problems .......................................................................41
Summary ................................................................................................................47
Problems.................................................................................................................48
References ............................................................................................... 50
3 One-Dimensional Stress Deformation ........................................ 53
Introduction ...........................................................................................................53
Step 1. Discretization and Choice of Element
Configuration .....................................................................53
Explanation of Global and Local Coordinates..................................................54
Local and Global Coordinate System for the One-Dimensional Problem .....55
Step 2. Select Approximation Model or Function
for the Unknown (Displacement)....................................56
Generalized Coordinates ............................................................................57
Interpolation Functions........................................................................................60
Relation between Local and Global Coordinates .............................................61
Variation of Element Properties.................................................................61
Requirements for Approximation Functions ....................................................62
Step 3. Define Strain-Displacement and Stress–Strain
Relations..............................................................................64
Stress–Strain Relation...........................................................................................65
Step 4. Derive Element Equations ................................................66
Principle of Minimum Potential Energy............................................................66
Functional for One-Dimensional Stress Deformation Problem ............69
Total Potential Energy Approach...................................................69
Variational Principle Approach......................................................71
Expansion of Terms...............................................................................................73
Integration..............................................................................................................75
Comment.......................................................................................................77
Step 5. Assemble Element Equations to Obtain Global
Equations ............................................................................77
Direct Stiffness Method........................................................................................80
Boundary Conditions ...........................................................................................82
Types of Boundary Conditions ..................................................................83
Homogeneous or Zero-Valued Boundary Condition .............................83
Nonzero Boundary Conditions..................................................................84
Step 6. Solve for Primary Unknowns: Nodal
Displacements ....................................................................85
Step 7. Solve for Secondary Unknowns; Strains and
Stresses ................................................................................88
Strains and Stresses...............................................................................................88
Step 8. Interpretation and Display of Results .............................90

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[24] Vrt. Gesch. der deut. Litter. seit Lessings Tod von J. Schmidt,
s. 301, 446, 518. III 14. Neure Geschichte der poetischen
Nationalliteratur von Gervinus, II s. 595, 668. Die deutsche
Nationallitteratur in der ersten Hälfte des 19:n Jahrhunderts von E.
Gottschall, I s. 164 ja seur.

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repertoarissa: Björnstjerne-Björnson (2), Musset (2), Legouvé (2),
Banville (1), Sardou (2), Augier (1), Mosenthal (1), Brachvogel (1),
Börjesson (2); kuinka kuivalta näyttää sen rinnalla tämän vuoden
repertoari, jossa tavataan: Sardou (1), Björnstjerne-Björnson (1),
Grillparzer (1), Börjesson (1). Ei kenenkään myös sopine sanoa, että
hra Åhmanin näyttelijäseura oli köyhempi hyvistä taiteilijoista, ja
yhteisnäytteleminen nyt ja silloin tuskin sietänee vertailemista.

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