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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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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
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© 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.
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

Formulation by Galerkin’s Method....................................................................92
Explanation and Relevance of Interpolation Functions .........................92
Comment.....................................................................................................101
Computer Implementation................................................................................101
Other Procedures for Formulation ...................................................................101
Comment.....................................................................................................102
Complementary Energy Approach ..................................................................102
Comment.....................................................................................................105
Mixed Approach..................................................................................................105
Variational Method ....................................................................................105
Residual Methods ......................................................................................106
Variational Method ....................................................................................106
Comment.....................................................................................................109
Galerkin’s Method .....................................................................................109
Comment..................................................................................................... 110
Bounds.................................................................................................................. 112
Comment..................................................................................................... 113
Advantages of the Finite Element Method...................................................... 113
Problems............................................................................................................... 114
References ............................................................................................. 120
4 One-Dimensional Flow............................................................... 123
Theory and Formulation....................................................................................123
Governing Equation ..................................................................................123
Finite Element Formulation......................................................................124
Step 1. Choose Element Configuration......................................124
Step 2. Choose Approximation Function...................................124
Step 3. Define Gradient-Potential Relation and
Constitutive Law .............................................................125
Step 4. Derive Element Equations ..............................................127
Variational Approach.................................................................................127
Evaluation of [k] and {Q} ..........................................................................128
Step 5. Assemble............................................................................129
Step 6. Solve for Potentials...........................................................129
Step 7. Secondary Quantities.......................................................131
Step 8. Interpret and Plot Results ...............................................132
Formulation by Galerkin’s Method.........................................................132
Forced and Natural Boundary Conditions for Flow Problems...........133
Problems...............................................................................................................135
Bibliography ......................................................................................... 138
5 One-Dimensional Time Dependent Flow: Introduction to
Uncoupled and Coupled Problems ............................................ 139
Uncoupled Case ..................................................................................................139
Initial Stress.................................................................................................142
Residual Stresses ........................................................................................142
Comment.....................................................................................................144

Time-Dependent Problems................................................................................144
Governing Equation ..................................................................................144
Step 1. Discretize and Choose Element Configuration............146
Step 2. Choose Approximation Model.......................................146
Step 3. Define Gradient-Temperature and Constitutive
Relation .............................................................................147
Layered Media............................................................................................151
Solution in Time .........................................................................................151
Step 4. Derivation by Galerkin’s Method ..................................153
Step 5. Assembly for Global Equations......................................155
Boundary Conditions ................................................................................157
Step 6. Solve for Primary Unknowns.........................................159
Second Time Increment.............................................................................160
Step 7. Compute the Derived or Secondary Quantities...........161
One-Dimensional Consolidation ......................................................................162
Computer Code...................................................................................................165
Problems...............................................................................................................169
References ............................................................................................. 175
6 Finite Element Codes: One- and Two-Dimensional
Problems....................................................................................... 177
One-Dimensional Code......................................................................................177
Deformation, Flow, and Temperature/Consolidation Problems........177
Philosophy of Codes...........................................................................................178
Stages ....................................................................................................................178
Stage 1. Input Quantities..........................................................................178
Stage 2. Initialize .......................................................................................180
Stage 3. Compute Element Matrices.......................................................180
Stage 4. Assemble......................................................................................181
Stage 5. Concentrated Forces...................................................................181
Stage 6. Boundary Conditions.................................................................181
Stage 7. Time Integration .........................................................................181
Stage 8. Solve Equations...........................................................................181
Stage 9. Set {R}t = {H} = {R}t + ∆t.................................................................181
Stage 10. Output Quantities.....................................................................182
Explanation of Major Symbols and Arrays .....................................................182
User’s Guide for Code DFT/C-1DFE...............................................................184
Two-Dimensional Code......................................................................................187
Stress Deformation and Field Problems .................................................187
User’s Guide for Plane-2D.................................................................................188
Program Input to be Prepared by User...................................................188
General Comments ........................................................................188
Conventions....................................................................................188
Units.................................................................................................188

0. Title Record ............................................................................................188
1. Problem Parameters Record (Input Set 1) .........................................189
2. Material Property Specifications (Input Set 2)..................................189
3. Nodal Point Specifications (Input Set 3)............................................189
Automatic Node Generation........................................................190
4. Element Specification (Input Set 4).....................................................190
Automatic Element Generation ...................................................190
5. Surface Loading (Pressure) Specifications (Input Set 5)..................191
6. Blank Line (required if more than one data set)...............................192
Sample Problems for Plane-2D .........................................................................192
Plane Stress .................................................................................................192
User’s Guide for Field-2D..................................................................................195
Sample Problems for FIELD-2D........................................................................197
References ............................................................................................. 198
7 Beam Bending and Beam-Column ............................................. 199
Introduction .........................................................................................................199
Step 1. Discretize and Choose Element Configuration............199
Step 2. Choose Approximation Model.......................................200
Comment on Requirements for Approximation Function...................204
Step 3. Define Strain-Displacement and Stress-Strain
Relationships....................................................................204
Energy Approach .......................................................................................205
Derivation Using Galerkin’s Method......................................................207
Steps 5 to 8.......................................................................................210
Closed Form Solutions ..............................................................................212
Secondary Quantities.................................................................................212
Comment.....................................................................................................215
Mesh Refinement .......................................................................................215
Higher-Order Approximation..................................................................216
Beam-Column......................................................................................................219
Step 1................................................................................................219
Step 2................................................................................................219
Steps 3 to 5.......................................................................................221
Comment.....................................................................................................222
Other Procedures of Formulation.....................................................................222
Complementary Energy Approach .........................................................222
Mixed Approach.........................................................................................224
Problems...............................................................................................................228
References ............................................................................................. 237
8 One-Dimensional Mass Transport ............................................. 239
Introduction .........................................................................................................239
Finite Element Formulation...............................................................................239

Step 4. Derivation of Element Equations...................................240
Step 5. Assembly ...........................................................................243
Solution in Time .........................................................................................244
Convection Parameter vx....................................................................................244
Comment.....................................................................................................245
Comment.....................................................................................................247
References.............................................................................................................248
Bibliography ......................................................................................... 248
9 One-Dimensional Stress Wave Propagation ............................. 251
Introduction .........................................................................................................251
Step 5. Assemble Element Equations .........................................255
Time Integration.........................................................................................255
Boundary and Initial Conditions......................................................................258
Boundary Conditions .........................................................................................261
Damping......................................................................................................261
Problems...............................................................................................................263
References.............................................................................................................264
Bibliography ......................................................................................... 264
10 Two- and Three-Dimensional Formulations............................. 267
Introduction .........................................................................................................267
Two-Dimensional Formulation.........................................................................267
Triangular Element .............................................................................................267
Requirements for the Approximation Function ....................................270
Integration of [B]T[B]..................................................................................272
Quadrilateral Element........................................................................................273
Approximation Model for Unknown......................................................273
Requirements for the Approximation Function ....................................276
Secondary Unknowns ...............................................................................276
Integration of [B]T[B]..................................................................................280
Numerical Integration...............................................................................280
Three-Dimensional Formulation ......................................................................284
Tetrahedron Element ..........................................................................................285
Brick Element.......................................................................................................286
Problems...............................................................................................................288
References ............................................................................................. 294
11 Torsion .......................................................................................... 295
Introduction .........................................................................................................295
Finite Element Formulation (Displacement Approach) ................................296
Step 3. Gradient-Unknown Relation and Constitutive Law....297

Step 5. Assembly ...........................................................................301
Twisting Moment .......................................................................................307
Comparisons of Numerical Predictions and Closed Form Solutions..........309
Twisting Moment .......................................................................................309
Shear Stresses..............................................................................................309
Comment..................................................................................................... 311
Stress Approach................................................................................................... 311
Shear Stresses..............................................................................................316
Twisting Moment .......................................................................................317
Step 8. Interpretation and Plots...................................................318
Comparisons...............................................................................................319
Bounds.........................................................................................................320
Comparisons...............................................................................................323
Comparisons...............................................................................................326
Review and Comments ......................................................................................327
Hybrid Approach................................................................................................328
Step 4. Element Equations ...........................................................330
Element Stiffness Matrix ...........................................................................335
Inner Elements............................................................................................336
Computation of Boundary Shear Stresses..............................................337
Assembly.....................................................................................................337
Shear Stresses..............................................................................................341
Twisting Moment .......................................................................................342
Mixed Approach..................................................................................................345
Evaluation of Element Matrices and Load Vector.................................347
Static Condensation ............................................................................................354
Problems...............................................................................................................356
References ............................................................................................. 360
12 Other Field Problems: Potential, Thermal, Fluid, and
Electrical Flow.............................................................................. 361
Introduction .........................................................................................................361
Potential Flow......................................................................................................362
Derivation of the Governing Equation ...................................................366
Step 5. Assembly ...........................................................................369
Evaluation of {Q}........................................................................................369
Stream Function Formulation ...........................................................................371
Secondary Quantities.................................................................................372

Thermal or Heat Flow Problem ........................................................................377
Seepage.................................................................................................................380
Electromagnetic Problems .................................................................................384
Computer Code...................................................................................................384
Problems...............................................................................................................385
References.............................................................................................................392
Bibliography ......................................................................................... 393
13 Two-Dimensional Stress-Deformation Analysis...................... 395
Introduction .........................................................................................................395
Plane Deformations ............................................................................................395
Plane Stress Idealization ...........................................................................395
Plane Strain Idealization ...........................................................................397
Axisymmetric Idealization .......................................................................398
Strain-Displacement Relations.................................................................399
Requirements for Approximation Function...........................................402
Plane Stress Idealization ...........................................................................402
Evaluation of [k] and {Q} ..........................................................................405
Triangular Element ....................................................................................408
Comment on Convergence .......................................................................417
Computer Code...................................................................................................418
Comment.....................................................................................................419
Partial Results.............................................................................................422
Comment.....................................................................................................424
Comment.....................................................................................................427
Problems...............................................................................................................428
References ............................................................................................. 436
14 Multicomponent Systems: Building Frame and
Foundation ................................................................................... 437
Introduction .........................................................................................................437
Various Components ..........................................................................................437
Beam-Column.............................................................................................437
Plate or Slab ................................................................................................441
Membrane Effects ......................................................................................441
Bending........................................................................................................441
Assembly.....................................................................................................446
Representation of Foundation..................................................................446
Computer Code...................................................................................................447
Transformation of Coordinates .........................................................................455
Problems...............................................................................................................457
References ............................................................................................. 458

Appendix 1 Various Numerical Procedures: Solution of Beam
Bending Problem ......................................................... 459
Introduction .........................................................................................................459
Various Residual Procedures.............................................................................460
Beam Bending by Various Procedures.............................................................461
Collocation ..................................................................................................463
Subdomain Method ...................................................................................464
Least-Squares Method...............................................................................465
Galerkin’s Method .....................................................................................466
Ritz Method ................................................................................................466
Comment.....................................................................................................467
Finite Element Method..............................................................................467
Finite Difference Method ..........................................................................468
Comparisons of the Methods ............................................................................469
References ............................................................................................. 470
Appendix 2 Solution of Simultaneous Equations ......................... 471
Introduction .........................................................................................................471
Methods of Solution ...........................................................................................472
Gaussian Elimination ................................................................................472
Back Substitution .......................................................................................474
Banded and Symmetric Systems..............................................................474
Solution Procedure..............................................................................................475
Iterative Procedures...................................................................................477
Comment.....................................................................................................481
References.............................................................................................................482
Appendix 3 Computer Codes........................................................... 483
Introduction .........................................................................................................483
Advanced Codes .................................................................................................483
Other Codes on Diffusion-Convection ...................................................484
Other Codes on Flow Problems...............................................................485
Reference............................................................................................... 486
Index ..................................................................................................... 487

1
1
Introduction
Basic Concept
In its current form the finite element (FE) method was formalized by civil
engineers. The method was proposed and formulated previously in different
manifestations by mathematicians and physicists.
The basic concept underlying the finite element method is not new. The
principle of discretization is used in most forms of human endeavor. Perhaps
the necessity of discretizing, or dividing a thing into smaller manageable
things, arises from a fundamental limitation of human beings in that they
cannot see or perceive things surrounding them in the universe in their
entirety or totality. Even to see things immediately surrounding us, we must
make several turns to obtain a jointed mental picture of our surroundings.
In other words, we discretize the space around us into small segments, and
the final assemblage that we visualize is one that simulates the real contin-
uous surroundings. Usually such jointed views contain an element of error.
In perhaps the first act toward a rational process of discretization, man
divided the matter of the universe into five interconnected basic essences
(Panchmahabhuta), namely, sky or vacuum, air, water, earth, and fire, and
added to them perhaps the most important of all, time, by singing
Time created beings, sky, earth,
Time burns the sun and time will bring
What is to come. Time is the master
of everything.1*
We conceived the universe to be composed of an innumerable (perhaps
finite) number of solar systems, each system composed of its own stars,
planets, and galaxies. In our solar system we divided the planet earth into
interconnected continents and oceans. The plate of earth we live on is com-
posed of interconnected finite plates.
* The superscript number indicates references at the end of the chapter.
0618c01/frame Page 1 Wednesday, April 4, 2001 8:22 PM

2 Introductory Finite Element Method
When man started counting, the numeral system evolved. To compute the
circumference or area of a circle, early thinkers drew polygons of progres-
sively increasing and decreasing size inside and outside the circle, respec-
tively, and found the value of π to a high degree of accuracy. In (civil)
engineering we started buildings made of blocks or elements (Figure 1.1).
When engineers surveyed tracts of land, the tract was divided into smaller
tracts, and each small tract was surveyed individually (Figure 1.2). The con-
necting of the individual surveys provided an approximate survey of the entire
tract. Depending on the accuracy of the survey performed, a closing error
would be involved. In aerial photography a survey of the total area is obtained
by matching or patching together a number of photographs.
FIGURE 1.1
Building column composed of blocks or elements.
FIGURE 1.2
Discretization in surveying and closure error in survey of subdivided plot.
Block
or element

Introduction 3
For stress analysis of modern framed structures in civil engineering clas-
sically, methods such as slope deflection and moment distribution were used.
The structure was divided into component elements, each component was
examined separately, and (stiffness) properties were established (Figure 1.3).
The parts were assembled so that the laws of equilibrium and physical
condition of continuity at the junctions were enforced.
Although a system or a thing could be discretized in smaller systems,
components, or finite elements, we must realize that the original system itself
is indeed a whole. Our final aim is to combine the understandings of indi-
vidual components and obtain an understanding of the wholeness or con-
tinuous nature of the system. In a general sense, as the modern scientific
thinking recognizes, which the Eastern philosophical and metaphysical con-
cepts had recognized in the past, all systems or things are but parts of the
ultimate continuity in the universe!
The foregoing abstract and engineering examples make us aware of the
many activities of man that are based on discretization.
Process of Discretization
Discretization implies approximation of the real and the continuous. We use
a number of terms to process the scheme of discretization such as subdivision,
continuity, compatibility, convergence, upper and lower bounds, stationary
potential, minimum residual, and error. As we shall see later, although these
terms have specific meanings in engineering applications, their conception
has deep roots in man’s thinking. In the following we discuss some of these
terms; a number of figures and aspects have been adopted from Russell.2
Subdivision
Zeno argued that space is finite and infinitely divisible and that for things
to exist they must have magnitudes. Figure 1.4(a) shows the concept of finite
FIGURE 1.3
Discretization of engineering structure: (a) Actual structure, (b) Discretized structure, and (c)
Idealized one-dimensional model.
(a) (b) (c)

space. If the earth were contained in space, what contained the space in
turn?2 Figure 1.4(b) illustrates this idea for the divisibility of a triangle into
a number of component triangles.
Continuity
Aristotle said that a continuous quantity is made up of divisible elements.
For instance, other points exist between any two points in a line, and other
moments exist between two moments in a period of time. Therefore, space
and time are continuous and infinitely divisible,2 and things are consecutive,
contiguous, and continuous (Figure 1.5).
These ideas of finiteness, divisibility, and continuity allow us to divide
continuous things into smaller components, units, or elements.
Convergence
For evaluating the approximate value of π, or the area of a circle, we can
draw polygons within (Figure 1.6(a)) and around (Figure 1.6(b)) the circle.
As we make a polygon, say, the outside one, smaller and smaller, with a
FIGURE 1.4
Finiteness and divisibility: (a) Infinite space and (b) Infinitely divisible triangle. (From Ref. 2,
by permission of Aldus Books Limited, London.)
(a) (b)

Introduction 5
greater number of sides, we approach the circumference or the area of the
circle. This process of successively moving toward the exact or correct solu-
tion can be termed convergence.
The idea is analogous to what Eudoxus and Archimedes called the method
of exhaustion. This concept was used to find areas bounded by curves; the
available space was filled with simpler figures whose areas could be easily
calculated. Archimedes employed the method of exhaustion for the parabola
(Figure 1.7); here by inscribing an infinite sequence of smaller and smaller tri-
angles, one can find the exact numerical formula for the parabola. Indeed, an
active practitioner of the finite element method soon discovers that the pursuit
of convergence of a numerical procedure is indeed fraught with exhaustion!
In the case of the circle (Figure 1.6), convergence implies that as the inside
or outside polygon is assigned an increasingly greater number of sides, we
approach or converge to the area of the circle. Figure 1.6(c) shows the plots
of successive improvement in the values of the area of the circle from the
two procedures: polygons of greater sides drawn inside and outside. We can
see that as the number of sides of the polygons is increased, the approximate
areas converge or approach or tend toward the exact area.
Physical models: The student can prepare pictorial or physical (cardboard
or plastic) models to illustrate convergence from the example of the area of
the circle.
Bounds
Depending on the course of action that we take from within or from outside,
we approach the exact solution of the area of the circle. However, the value
FIGURE 1.5
Concepts of continuity: (a) Consecutive, (b) Contiguous, and (c) Continuous. From Ref. 2, by
permission of Aldus Books Limited, London.
(a)
(b)
(c)

from each method will be different. Figure 1.6(c) shows that convergence
from within the circle gives a value lower than the exact, while that from
outside gives a value higher. The former yields the lower bound and the
latter, the upper bound. Figure 1.8 depicts the process of convergence to
upper and lower bound solutions.
Error
It should be apparent that discretization involves approximation. Conse-
quently, what we obtain is not the exact solution but an approximation to
that solution. The amount by which we differ can be termed the error. For
example, the areas (or perimeters) of the polygons inscribed in the circle
(Figure 1.6) are always less than the area (or perimeter) of the circle, and the
areas (or perimeters) of the circumscribed polygons are always greater than
the area (or perimeter) of the circle. The difference between the approxima-
tion and the exact perimeter is the error, which becomes smaller and smaller
FIGURE 1.6(a)
Convergence and bounds for approximate area of circle: (a) Polygons inside circle.
A(4)= 1.125 units
A(6)= 1.462
A(8)= 1.591 A(9)= 1.628
A(7)= 1.540
A(5)= 1.322
(a)

Introduction 7
as the number of sides of the polygons increases. We can express error in
the area as
(1.1)
where A* = the exact area, A = approximate area, and ε = error.
Principles and Laws
To describe the behavior of things or systems around us, we need to establish
laws based on principles. A law can be a statement or can be expressed by
FIGURE 1.6(b)
Convergence and bounds for approximate area of circle: (b) Polygons outside circle.
A(4) = 2.250
A(6) = 1.945
A(8) = 1.864 A(9) = 1.843
A(7) = 1.896
A(5) = 2.043
(b)
A A
* ,
− = ε

a mathematical formula. Principles have often been proposed by intuition,
hypothesized, and then proved. Newton’s second law states
(1.2a)
or
(1.2b)
where F = force, m = mass, and a = acceleration or second derivative of
displacement u with respect to time t. The principle is that at a given time
the body is in dynamic equilibrium and a measure of energy contained in
the body assumes a stationary value.
A simple statically loaded linear elastic column (Figure 1.9(a)) follows the
principle that at equilibrium, under given load and boundary constraints,
the potential energy of the system assumes a stationary (minimum) value
and that the equation governing displacement is
FIGURE 1.6(c)
Convergence for approximate area of circle (also see Figures 1.6a and 1.6b).
2.0
1.0
0.0
4 5 6 7 8 9 10 11 12 13
Number of sides
(c)
Area
Outside polygons
Inside polygons
Exact area = 1.767
F ma
=
F m
d u
dt
=
2
2
,

Introduction 9
(1.3)
where v = displacement in the vertical y direction, P = applied load, L =
length of the column, A = cross-sectional area, and E = modulus of elasticity.
FIGURE 1.7
Concept of convergence or exhaustion. (From Ref. 2, by permission of Aldus Books Limited,
London.)
FIGURE 1.8
Concept of bounds.
v
PL
AE
= ,
Parabola
Upper
bound
Exact solution
Lower
bound

Cause and Effect
The essence of all investigations is the examination and understanding of causes
and their effects. The effect of work is tiredness and that of too much work is
fatigue or stress. The effect of load on a structure (Figure 1.9(b)) is to cause
deformations, strains, and stresses; too much load causes fatigue and failure.
When studying finite element methods, our main concern is the cause and
effect of forcing functions (loads) on engineering systems.
The foregoing offers a rather abstract description of ideas underlying the
process of discretization, inherent in almost all human endeavors. Compre-
hending these ideas significantly helps us to understand and extend the
finite element concept to engineering; that is the goal of this text.
Important Comment
Although we have presented the descriptions in this book by using simple
problems, we should keep in mind that the finite element method is powerful
and popular because it allows solution of complex problems in engineering
and mathematical physics. The complexities arise due to factors such as
irregular geometries, nonhomogeneities, nonlinear behavior, and arbitrary
loading conditions. Hence, after learning the method through simple exam-
ples, computations, and derivations, our ultimate goal will be to apply it to
complex and challenging problems for which conventional procedures are
not available or are very difficult.
Review Assignments
In the beginning stages of the study, it may prove very useful to assign the
student homework that requires review of some of the basic laws, principles,
FIGURE 1.9
Structures subjected to loads (causes): (a) Column and (b) Body or structure.
P
P
L
(a) (b)
A = area of
cross section

Introduction 11
and equations. This will facilitate understanding the method and also reduce
the necessity of reviews by the teacher. The following are two suggested
home assignments that cover topics from many undergraduate curricula.
Home Assignment 1
1. Define:
(a) Stress at a point
(b) Strain
(c) Hooke’s law
2. Define:
(a) Principal stresses and strains
(b) Invariants of stresses and strains
3. (a) Define potential energy as a sum of strain energy and potential
of applied loads.
(b) Give examples for analysis in (civil)engineering in which you
have used the concept of potential energy.
4. Define:
(a) Darcy’s law and coefficient of permeability
(b) Coefficient of thermal conductivity and coefficient of thermal
expansion
5. Derive the Laplace equation for steady-state flow.
6. Derive the fourth-order differential equation governing beam
bending,
where w = displacement, p = applied load, and ksw = support reaction, k, =
spring constant, E = modulus of elasticity, I = moment of inertia, and x =
coordinate along beam axis. See Figure 1.10.
FIGURE 1.10
d
dx
EI
d w
dx
p k w
s
2
2
2
2





 = − ,
z, w
p
ks

Home Assignment 2
1. Define:
(a) Determinant
(b) Row, column, and rectangular matrices
(c) Matrix addition and subtraction
(d) Matrix multiplication
(e) Inverse of a matrix
(f) Transpose of a matrix
(g) Symmetric matrix
(h) Sparsely populated and banded matrices
2. Define:
(a) A set of algebraic simultaneous equations
(b) Describe Gaussian elimination with respect to the following
equations,
and find the value of x1 and x2.
3. Define:
(a) Total derivative
(b) Partial derivative for one variable and two variables and the
chain rule of differentiation
References
1. The Upnishads — Praise of Time, 1964. See available translations of The Upnishads
from Sanskrit to English, e.g., The Upnishads by Swami Nikhilananda, Harper
Torchbooks, New York.
2. Russell, B., 1959. Wisdom of the West, Crescent Books, Inc., Rathbone Books
Ltd., London.
2 3 14
4 5 10
1 2
1 2
x x
x x
+ =
+ =
,
,

13
2
Steps in the Finite Element Method
Introduction
Formulation and application of the finite element method are considered to
consist of eight basic steps. These steps are stated in this chapter in a general
sense. The main aim of this general description is to prepare for complete
and detailed consideration of each of these steps in this and subsequent
chapters. At this stage, the reader may find the general description of the
basic steps in this chapter somewhat overwhelming. However, when these
steps are followed in detail with simple illustrations in the subsequent chap-
ters, the ideas and concepts will become clear.
Mathematical foundations of the variational formulation and the residual
formulation (Galerkin’s method) are given in more detail after a brief
description of the eight steps in the finite element method. A good compre-
hension of these procedures is necessary for a thorough understanding of
the derivation of the element equations.
General Idea
Engineers are interested in evaluating effects such as deformations, stresses,
temperature, fluid pressure, and fluid velocities caused by forces such as
applied loads or pressures and thermal and fluid fluxes. The nature of
distribution of the effects (deformations) in a body depends on the charac-
teristics of the force system and of the body itself. Our aim is to find this
distribution of the effects. For convenience, we shall often use displacements
or deformations u (Figure 2.1) in place of effects. Subsequently, when other
problems such as heat and fluid flow are discussed they will involve distri-
bution of temperature and fluid heads and their gradients.
We assume that it is difficult to find the distribution of u by using conven-
tional methods and decide to use the finite element method, which is based
on the concept of discretization, as explained in Chapter 1. We divide the

body into a number of smaller regions (Figure 2.1(a)) called finite elements.1,2
A consequence of such subdivision is that the distribution of displacement
is also discretized into corresponding subzones (Figure 2.1(b)). The subdi-
vided elements are now easier to examine as compared to the entire body
and distribution of u over it.
For stress-deformation analysis of a body in equilibrium under external
loading, the examination of the elements involves derivation of the stiff-
ness–load relationship. To derive such a relationship, we make use of the
laws and principles governing the behavior of the body. Since our primary
concern is to find the distribution of u, we contrive to express the laws and
principles in terms of u. We do this by making an advance choice of the
pattern, shape, or outline of the distribution of u over an element. In choosing
a shape, we follow certain rules dictated by the laws and principles. For
example, one law says that the loaded body, to be reliable and functional,
cannot experience breaks anywhere in its regime. In other words, the body
must remain continuous. Let us now describe in detail various steps involved
in the foregoing qualitative statements.
Step 1. Discretize and Select Element Configuration
This step involves subdividing the body into a suitable number of small
bodies, called finite elements. The intersections of the sides of the elements
are called nodes or nodal points, and the interfaces between the elements
are called nodal lines and nodal planes. Often we may need to introduce
additional node points along the nodal lines and planes (Figure 2.1(b)).
An immediate question that arises is how small should the elements chosen
be? In other words, how many elements would approximate the continuous
FIGURE 2.1
Distribution of displacement u, temperature T, or fluid head ϕ. (a) Discretization of two-
dimensional body and (b) Distribution of ue over a generic element e.
Nodal line
Finite element
Node
(a) (b)
y
Distribution of u(x, y)
for entire body
x
1
2
3
4
Additional
node
Corner or
primary node
u
(T,ϕ)
e
ue (x, y)

Steps in the Finite Element Method 15
medium as closely as possible? This depends on a number of factors, which
we shall discuss.
What type of element should be used? This will depend on the charac-
teristics of the continuum and the idealization that we may choose to use.
For instance, if a structure or a body is idealized as a one-dimensional line,
the element we use is a line element (Figure 2.2(a)). For two-dimensional
bodies, we use triangles and quadrilaterals (Figure 2.2(b)); for three-dimen-
sional idealization, a hexahedron with different specializations (Figure 2.2(c))
can be used.
Although we could subdivide the body into regular-shaped elements in
the interior (Figure 2.3), we may have to make special provisions if the
FIGURE 2.2
Different types of elements: (a) One-dimensional element, (b) Two-dimensional elements, and
(c) Three-dimensional elements.
Two-dimensional body
Three-dimensional body Hexahedron element
Quadrilateral and
triangular elements
One-dimensional body Line elements
2 unknowns at
each node Quadrilateral
Triangle
(a)
(b)
(c)
1 2
u1 u2
x

boundary is irregular. For many cases, the irregular boundary can be approx-
imated by a number of straight lines (Figure 2.3). On the other hand, for
many other problems, it may be necessary to use mathematical functions of
sufficient order to approximate the boundary. For example, if the boundary
shape is similar to a parabolic curve, we can use a second-order quadratic
function to approximate that boundary. The concept of isoparametric ele-
ments that we shall discuss later makes use of this idea. It may be noted that
inclusion of irregular boundaries in a finite element formulation poses no
great difficulty.
Step 2. Select Approximation Models or Functions
In this step, we choose a pattern or shape for the distribution (Figure 2.1) of
the unknown quantity that can be a displacement and/or stress for stress-
deformation problems, temperature in heat flow problems, fluid pressures
and/or velocity for fluid flow problems, and both temperature (fluid pres-
sure) and displacement for coupled problems involving effects of both flow
and deformation.
The nodal points of the element provide strategic points for writing math-
ematical functions to describe the shape of the distribution of the unknown
quantity over the domain of the element. A number of mathematical func-
tions such as polynomials and trigonometric series can be used for this
purpose, especially polynomials because of the ease and simplification they
provide in the finite element formulation. If we denote u as the unknown,
the polynomial interpolation function can be expressed as
(2.1)
Here u1, u2, u3,…, um are the values of the unknowns at the nodal points and
N1, N2,…, Nm are the interpolation functions; in subsequent chapters we shall
give details of these functions. For example, in the case of the line element
with two end nodes (Figure 2.2(a)) we can have u1 and u2 as unknowns or
degrees of freedom and for the triangle (Figure 2.2(b)) we can have u1, u2,…, u6
FIGURE 2.3
Discretization for irregular boundary.
Original body
Discretized body
u N u N u N u N u
m m
= + + + …+
1 1 2 2 3 3 .

as unknowns or degrees of freedom if we are dealing with a plane deforma-
tion problem where there are two displacements at each node.
A degree of freedom can be defined as an independent (unknown) dis-
placement that can occur at a point. For instance, for the problem of one-
dimensional deformation in a column (Figure 2.2(a)), there is only one way
in which a point is free to move, that is, in the uniaxial direction. Then a
point has one degree of freedom. For a two-dimensional problem
(Figure 2.2(b)), if deformations can occur only in the plane of the body (and
bending effects are ignored), a point is free to move only in two independent
coordinate directions; thus a point has two degrees of freedom. In Chapter 7,
when bending is considered, it will be necessary to consider rotations or
slopes as independent degrees of freedom.
We note here that after all the steps of the finite element method are
accomplished, we shall find the solution as the values of the unknowns u at
all the nodes, that is u1, u2,…, um. To initiate action toward obtaining the
solution, however, we have assumed a priori or in advance a shape or pattern
that we hope will satisfy the conditions, laws, and principles of the problem
at hand.
The reader should realize that the solution obtained will be in terms of the
unknowns only at the nodal points. This is one of the outcomes of the
discretization process. Figure 2.4 shows that the final solution is a combina-
tion of solutions in each element patched together at the common bound-
aries. This is further illustrated by sketching a cross section along A–A. It
can be seen that the computed solution is not necessarily the same as the
FIGURE 2.4
Approximate solution as patchwork of solutions over elements: (a) Assemblage, (b) Neighboring
elements, and (c) Section along A–A.
v
y
x
(a) (b)
u(x,y)
u4(x,y)
u3(x,y)
u2(x,y)
A
u1(x,y) 1
A
2
3
4
(c)
Common boundary
Common boundary
Finite element
approximation
Exact
Equal

exact continuous solution shown by the solid curve. The statement in
Chapter 1 that discretization yields approximate solutions can be visualized
from this schematic representation (Figure 2.4). Obviously, we would like
the discretization to be such that the computed solution is as close as possible
to the exact solution; that is, the error is a minimum.
Step 3. Define Strain (Gradient)-Displacement (Unknown)
and Stress–Strain (Constitutive) Relationships
To proceed to the next step, which uses a principle (say, the principle of
minimum potential energy) for deriving equations for the element, we must
define appropriate quantities that appear in the principle. For stress-defor-
mation problems one such quantity is the strain (or gradient) of displace-
ment. For instance, in the case of deformation occurring only in one direction
y (Figure 2.5(a)), the strain εy assumed to be small is given by
(2.2)
where v is the deformation in the y direction. For the case of fluid flow in
one direction, such a relation is the gradient gx of fluid head (Figure 2.5(b)):
(2.3)
Here ϕ is the fluid head or potential and gx is the gradient of ϕ, that is, rate
of change of ϕ with respect to the x coordinate.
In addition to the strain or gradient, we must also define an additional
quantity, the stress or velocity; usually, this is done by expressing its rela-
tionship with the strain. Such a relation is called a stress–strain law. In a
generalized sense, it is a constitutive law and describes the response or effect
(displacement, strain) in a system due to applied cause (force). The
stress–strain law is one of the most vital parts of finite element analysis.
FIGURE 2.5
Problems idealized as one-dimensional: (a) One-dimensional stress-deformation and (b) One-
dimensional flow.
εy
dv
dy
= ,
g
d
dx
x =
ϕ
.
(a)
(b)
Variation
of v
εy gradient
or slope of v
P
v
Variation of ϕ gx gradient or slope
of ϕ
x,ϕ
y

Unless it is defined to reflect precisely the behavior of the material or the
system, the results from the analysis can be of very little significance. As an
elementary illustration, consider Hooke’s law, which defines the relationship
of stress to strain in a solid body:
(2.4a)
where σy = stress in the vertical direction and Ey = Young’s modulus of
elasticity. If we substitute εy from Equation 2.2 into Equation 2.4a, we have
the expression for stress in terms of displacements as
(2.4b)
One of the other simple linear constitutive laws is Darcy’s law for fluid
flow through porous media:
(2.4c)
where kx = coefficient of permeability, vx = velocity, and gx = gradient. In
electrical engineering the corresponding law is Ohm’s law.
Step 4. Derive Element Equations
By invoking available laws and principles, we obtain equations governing
the behavior of the element. The equations here are obtained in general terms
and hence can be used for all elements in the discretized body.
A number of alternatives are possible for the derivation of element equations.
The two most commonly used are the energy methods and the residual methods.
Use of the energy procedures requires knowledge of variational calculus.
At this stage of our study of the finite element method, we shall postpone
detailed consideration of variational calculus and in a somewhat less rigorous
manner introduce the ideas simply through the use of differential calculus.
Energy Methods
These procedures are based on the idea of finding consistent states of bodies
or structures associated with stationary values of a scalar quantity assumed
by the loaded bodies. In engineering, usually this quantity is a measure of
energy or work. The process of finding stationary values of energy requires
use of the mathematical disciplines called variational calculus involving use
of variational principles. In this introductory book, detailed treatment of
variational calculus is considered to be not warranted. However, a description
of the energy approach used in the book is given below, and an introductory
description of variational calculus is given at the end of this chapter.
σ ε
y y y
E
= ,
σy y
E
dv
dy
= .
v k g
x x x
= − ,

Within the realm of energy methods, there are a number of methods and
variational principles, e.g., the principle of stationary potential and comple-
mentary energies, Reissner’s mixed principle, and hybrid formulations,
which are commonly used in finite element applications.3-6
Stationary Value
In simple words, the term stationary can imply a maximum, minimum, or
saddle point of a function F(x) (Figure 2.6). Under certain conditions, the
function may simply assume a minimum or a maximum value. To find the
point of a stationary value, we equate the derivative of F to zero:
(2.5)
Potential Energy
In the case of stress-deformation analysis, the function F is often represented
by one of the energy functions stated previously. For instance, we can define
F to be the potential energy in a body under load. If the body, say, a simple
column under the given support conditions (Figure 2.5(a)), is linear and
elastic and if it is in equilibrium, it can be shown that the column will assume
minimum potential energy. To comply with the commonly used notation we
denote potential energy by the symbol Πp, where the subscript denotes
potential energy.
The potential energy is defined as the sum of the internal strain energy U
and the potential of the external loads Wp; the latter term denotes the capacity
of load P to perform work through a deformation v of the column. Therefore,
(2.6a)
FIGURE 2.6
Stationary values of a function.
Maximum
Minimum
Neutral
X
F
F(x)
dF
dx
= 0.
Πp p
U W
= + .

When we apply the principle of minimum potential energy, we essentially
take the derivative (or variation) of Πp and equate it to zero. We assume that
the load remains constant while taking the derivative; then
(2.6b)
The symbol δ denotes variation of the potential energy Πp. As indicated sub-
sequently in Equation 2.9, we can interpret the variation or change as composed
of a series of partial differentiation of Πp. Here we use the relation between the
variation in potential of external loads and in work done by the loads as
(2.6c)
Note that the negative sign in Equations 2.6b and 2.6c arises because the
potential of the external loads in Equation 2.6a is lost through the work done
by the external load.
The fact that for linear, elastic bodies in equilibrium the value of Πp is a
minimum can be verified by showing that the second derivative or variation
of Πp is greater than zero; that is,
(2.7)
Proof of Equation 2.7 can be found in advance treatments on energy methods
and is not included in this text. The symbol δ is a compact symbol used to
denote variation or a series of partial differentiations. For our purpose, we
shall interpret it simply as a symbol that denotes derivatives of Πp with
respect to the independent coordinates or unknowns in terms of which it is
expressed. For example, if
(2.8)
where u1, u2,…, un are the total number of unknowns (at the nodes), then
δ Πp = 0 implies
(2.9)
Here n = total number of unknowns.
δ δ δ
Πp p
U W
= − = 0.
δ δ
W Wp
= − .
δ δ δ
2 2 2
0
Πp p
U W
= − > .
Π Π
p p n
u u u
= …
( )
1 2
, , , ,
∂
∂
=
∂
∂
=
Π
Π
p
p
u
u
1
2
0
0
,
,
.
.
.
∂
∂
=
Πp
n
u
0.

In subsequent chapters we shall illustrate the use of the principle of sta-
tionary energy and other energy principles for finite element formulations
of various problems.
Method of Weighted Residuals
One of the two major alternatives for formulating the finite element method
is the method of weighted residuals (MWR). A number of schemes are
employed under the MWR, among which are collocation, subdomain, least
squares, and Galerkin’s methods.3,7 For many problems with certain math-
ematical characteristics (discussed in later chapters), Galerkin’s method
yields results identical to those from variational procedures and is closely
related to them. Galerkin’s method has been the most commonly used resid-
ual method for finite element applications.
The MWR is based on minimization of the residual left after an approxi-
mate or trial solution is substituted into the differential equations governing
a problem. As a simple illustration, let us consider the following differential
equation:
(2.10a)
where u* is the unknown, x is the coordinate, t is the time, and f(x) is the
forcing function. In mathematical notation, Equation 2.10a is written as
(2.10b)
where
is the differential operator.
We are seeking an approximate solution to Equation 2.10 and denote an
approximate or trial function u for u* as
(2.11)
Here ϕ1, ϕ2,…, ϕn are known functions chosen in such a way as to satisfy the
homogeneous boundary conditions; ϕ0 is chosen to satisfy the essential,
∂
∂
−
∂
∂
= ( )
2
2
u
x
u
t
f x
* *
,
Lu f
* ,
=
L
x t
≡
∂
∂
−
∂
∂
2
2
u i i
i
n
n n
= +
= + + + … +
=
∑
ϕ α ϕ
ϕ α ϕ α ϕ α ϕ
0
1
0 1 1 2 2 .

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"Well! You see? Are you convinced? Didn't I say she was putting
on?"
Throwing back his shoulders, he burst into a laugh that rang with
contempt. The tango was over, so was its spell.
The bully became the bully again. He approached his sweetheart,
pushed her toward his comrades, who were sitting at their table
waiting to see how it would all end.
"Now get out of here!" he said, turning to Monsalvat; "but before
you go, I'll tell you who I am. It's to your interest, friend—just a
moment—we might meet again—take a look at me!"
He was serious now. His right hand slipped through the opening of
his tuxedo and rested on his belt. Then he announced solemnly:
"I am Dalmacio Arnedo, 'Pampa Arnedo,' as they call me."
Monsalvat started. Instinctively he raised a hand, but immediately let
it fall. The five of the patota made a rush for him. At the same
moment someone shouted: "The police!"
The cabaret seethed in confusion. Then suddenly an anxious calm
fell on the room, a forced appearance of peaceableness, prearranged
for the dull eyes of authority.
From the first there had been among the onlookers a certain number
who took sides with Monsalvat. His manner toward the patota won
him sympathizers. Some of them felt that the man had the strength
to support his assurance. The girl herself aroused pity even though
no one had had the courage to speak up in her defence. Two or
three of these most sympathetic, or most prudent, individuals had
called for the police, to have help on hand in case of an outbreak
from the rowdies.
As the alarm was given the members of the patota hurried back to
their places. Monsalvat, facing Arnedo, exclaimed:
"You rotter!"

Pampa Arnedo, safely seated at his table, answered with a sinister
smile, while his friends beside him made noises with their lips,
grimaced, and began offering toasts, simulating exaggerated
merriment. Nacha looked pityingly at her protector. Who was this
man? What did he want of her?
The police after a rapid glance around the room decided that "law
and order" were still quite intact, and with solemn prudence went
out again. Monsalvat returned to his table and paid his reckoning.
The Duck began to sing the well-known tune from a popular variety
show: "He's going now, he's going now!..."
The other members of the patota, and even some neutrals, joined in
the chorus, "Now, now, he's going now!" Monsalvat, as he got up,
saw that the girl, too, was singing and laughing. He paused a
moment, reproachfully it seemed, his eyes dimmed with tears. Then
quietly, without haste, he left the cabaret, while the fellow who had
burlesqued Nacha's weeping broke out again with his "Oh, oh, oh!"

CHAPTER II
Monsalvat had come to a crossroad in his life. For nearly forty years
he had gone straight ahead, never hesitating as to which turning to
take. But now, as though a complete transformation had occurred
within him, he seemed a stranger to himself, and he did not know
where this stranger was going.
Heretofore he had lived without criticizing the world of which he was
a part—which means that he had been fairly happy. But during the
past few months he had come to view life and himself from a critical
point of view, and he had reached the conclusion that as human
beings go, he was one of the unfortunates.
He and his sister Eugenia were illegitimate children. His father, of the
aristocracy, and rich with many millions, had, some five years before,
died suddenly without leaving a will. Fernando was intelligent and
had something of his father's manner and bearing; and as the
legitimate heirs of the Monsalvat fortune were all girls, Fernando was
given a good education while his father was still alive. In order to
keep him away from his mother, an ignorant, irresponsible woman of
the immigrant class, the boy was sent to a boarding school. It was
only during vacations that he saw her. Fernando remembered his
father's visits, the discussions with his mother, the admonitions he
himself always received. Once his father had taken the boy to one of
his ranches near Buenos Aires, a piece of property as big as an
entire state, on which were marvelous forests, a house as
magnificent as a palace, and paddocks full of splendid bulls and
woolly sheep. More clearly than anything else, he remembered how

his father took him along almost stealthily, and replied evasively
when a friend, on the train, asked who the child with him was. Later,
at boarding school, some boys who knew his father's legitimate
family, enlightened him as to his own birth.
When he left college he took up law. He was an excellent student;
and even before any regular admission to the bar, he was filling a
place in the office of a well-known lawyer. Later he became this
man's partner, made money, and won recognition. For a scruple he
left the law office and went to Europe, remaining there two years.
When he returned he was thirty-two. No longer wishing to continue
in his profession, he finally obtained a consulship to an Italian city. It
was now six months since he had returned, after seven years'
absence, to settle permanently in his own country.
Fernando's mother was still living. She was ill, and aged; indeed,
although not yet seventy, she seemed quite decrepit. Her son saw
little of her. She lived with a mulatto servant in a rather poor
neighborhood, in an apartment house facing Lezama Park. Of his
own sister he had seen little.
Monsalvat had lived as do most decent men of his social position. He
had worked hard in his law office, and as consul had rendered
services of distinction. From boyhood, books had been his chief
companions. He had taken up sociology, and from time to time he
got an article published. His opinions were respected and discussed
in certain intellectual circles. Though not socially inclined, and in
spite of his timidity and lack of confidence, he frequented the clubs
and theatres and race courses of Buenos Aires. He was not often
present at more private social affairs, for the circumstances of his
birth prevented his receiving invitations from certain quarters. While
a student he lived on an allowance from his father. Now, on his
return from Europe, he found himself possessed of no other income
than three hundred pesos monthly from a piece of property which
his father had given him upon his passing his law examinations.
The knowledge of his illegitimacy had exercised an incalculable
influence on his character and general outlook on life. When he was

a student certain youths of good family had made it plain that they
did not desire his friendship; and later he had been socially snubbed
on several occasions. He was, however, inclined to exaggerate the
number of these slights. If an acquaintance failed to notice him, as
he passed along the street, he believed the omission an intentional
offense. If, at a dance, a girl chanced to refuse his proffered arm, he
was beset always with the same thought.... "She does not dare to be
seen with me.... She knows!..." If he received in his examinations a
lower mark than the one he thought he must have earned, he did
not for a moment doubt that it was the stigma of his birth which was
to blame. Not a day passed that he did not at some moment revert
to this preoccupation. He bore society no grudge; on the contrary, it
seemed to him quite natural that, dominant ideas being what they
are, he should be thought less of. Nevertheless he felt humiliated,
with a vague consciousness that his value as a social being was
diminished by a misfortune beyond his control.
All this, of course, tended to isolate him, and confirmed his
tendencies toward bookishness. He had no real friends. He felt
himself to be quite alone in life—alone spiritually, that is; for social
relations in abundance could not fail a man of his intellect and
professional position, whose character, moreover, was above
reproach, and who, in spite of an outward coldness and an almost
savage shyness that frequently took possession of him, was a kind
and likeable sort of fellow.
This sense of solitude was tempered, if at all, by one or two
experiences in love. His dealings with women were not those current
among the young men of his generation. Gossip attributed,
nevertheless, sentimental affairs to him, some of them with women
of prominence in the life of the Capital. For Monsalvat, as his
acquaintances noted, knew how to please. There was something
that appealed to women in the soft inflections of his voice, and in
the deep seriousness of his eyes. But the secret of his successes
probably lay in the fact that he awakened in women that compassion
which is so ruinous to them—so much so that Monsalvat was quite
as often the pursued as the pursuer. Two or three times he had

thought himself in love—mistakenly, as he soon discovered; and
women for their part had loved him, and with passion. But these
affairs were, after all, nothing but passing gratifications of the
instinct of playfulness—little love episodes at best.
In other respects his life might have been considered a model and
an exception. He was courteous and simple in manner, with no
violent dislikes for anyone. Kind, always ready to do a good turn, he
pushed considerateness even to extremes. He lived scrupulously
within his means. He never paid court to those in whose power it
was to further his advancement. He never indulged in petty
disloyalties toward his friends nor paid off injury with injury. His
relations with people were always sincere and free from intrigue. A
useful and an honest fellow Fernando Monsalvat might have been
considered by anyone. Yet, these several months past, he had been
coming to the conclusion that he had lived in a useless sort of way,
that his life had been selfish, mediocre, barren of any good. He was
most of all ashamed of his articles on moral and social subjects, all
of them colored with "class" prejudice, mere reflections of the
conventional, insincere, and rankly individualistic standards which
pervaded the University, and which never failed of approval from
climbing politicians as well as from the cultured élite. Monsalvat
despised himself for having lived and thought like any other man of
his social group. What real good had he ever accomplished? He had
lived for himself alone; worked for the money that work might bring
him; written to gratify an instinct of vanity, a desire for prominence,
for applause. Now he endured a hidden torment: he was disgusted
with himself, with society, and even with life, repenting, in his soul's
secret, of so many wasted years.
To generous spirits, such moral crises are natural; moments are sure
to come when they must view their own conduct critically; and at
such junctures they loathe their sterile past. But how many ever
succeed in changing the direction of their lives? Most of us stifle this
moral unrest in the depths of our consciousness; discontented and
pessimistic, we go on living a life we hate, tempering the noble
impulses that beset our guilty consciences with considerations of

personal, even petty, interests that bid us take things as we find
them. This latter was the case with Monsalvat.
Two trifling events of his days in Paris had cast a gloom over his
outlook on life.
Convinced that he ought to put an end to his solitude, he decided to
marry; and he paid court to a girl of good family with whom he had
been on pleasantly cordial terms in Rome. But no sooner did the
family and the girl herself become aware of Monsalvat's intentions,
than all friendliness on their part vanished. An officious friend
intimated to Monsalvat—he never knew whether at the girl's own
request, or that of her parents—that his attentions were not desired.
Later, at the hotel where he was stopping, he made the
acquaintance of another fellow countrywoman. Friendship and
flirtation followed. Monsalvat became interested to the point of
believing himself in love. He made an offer of marriage and was
contemptuously rejected, as though such an idea on his part were in
itself an insult. In situations of this kind Monsalvat did not suffer so
much on his own account; it was not shame of being what he was
that hurt him, but a deepening sense of the injustice inherent in
people and in things.
He had given barely a thought to the imperfection, the inequalities,
of the world he was living in. Full of his own thoughts, his own
books, his own pleasures, he had paid no attention to the cry of
anguish rising from the depths of the social order—as an
established, an immutable order he had accepted it all along.
The fact that not till he had felt them himself had he opened his
eyes to the flagrant injustices of society aroused a deep self-
reproach in Monsalvat. It seemed to him that at the bottom of his
new opinions purely selfish motives lay. On the other hand, it was to
the universal, the human aspects of his own case that he gave his
attention. Besides, does not selfishness play a little part in our
striving toward the greatest ends?

It was some six months before the scenes in the cabaret, that
Fernando Monsalvat, disheartened and disillusioned, had arrived in
Buenos Aires. At first it startled him to find himself judging people
and institutions so mercilessly. Why did he see everything in its
darkest colors? Had he become an incorrigible cynic? Eventually he
came to understand that the severe judgments he was formulating
were the natural consequence of the critical spirit now aroused
within him. In the complex motivation of the finest, noblest, most
heroic gestures of men, how many small, unconfessable impulses
always have their play?
One afternoon chance revealed to him in vivid colors the degree to
which his life had been self-centered. The taxi in which he happened
to be riding came to a standstill at a turning in Lavalle Square. A
crowd was coming toward him, singing. It was a Sunday afternoon.
He noticed that all the doors of the neighborhood were closed. The
singing came nearer, swelling up from the street, rising above the
tree tops. It was an irritated, exasperated, tumultuous mob which
was approaching; and a song which both alarmed and attracted him
was resounding from hundreds of mouths, its spirit typified in the
red flag waving above the multitude. He got out of the taxi, and at
that moment a bugle sounded. The mob fell in on itself like a
punctured balloon. There was a volley of rifle shots, and in the
confusion he could see the police charging blindly with their swords.
The song continued, however, for a time; then the regimented
violence of the Law was stronger than the impulsive violence of the
Internationale. The rabble broke into the side streets and dispersed.
The swords of the police eagerly sought out the wretches crouching
for shelter in the doorways. Other wretches were in headlong flight,
their eyes wide with terror. No one was paying any attention to the
dead or wounded. Doors and windows remained closed and silent.
To Monsalvat, sick with indignation, his soul flaming in outrage, this
very silence seemed a horrible complicity in a crime.
His transformation, however, was purely an inner one. To be sure, he
had somewhat changed his manner of living: he no longer went to
his club nor to parties; he avoided most of his former friends. But,

after all, what had he actually done these six months past? Had he
perchance even discovered the road he really wanted to take? He
was ceaselessly tormented by these questions, which plunged him
for hours at a time into inconclusive meditations.
On one point he was resolved: he would not resume his practice of
law. What need had he to earn money? To save it up? To spend it on
amusements? At any rate, he might give it away. But to whom, and
how? A friend, a successful lawyer, who had a high opinion of
Monsalvat's judicial learning, proposed making him a partner in his
firm; but Monsalvat did not accept the offer. He thought, finally, he
would prefer a clerkship in the Department of Foreign Relations,
where his seven years as consul would count, and where, too, he
was already looked upon with great favor. The Minister had promised
him a post and the appointment would be coming along almost any
day.
Meanwhile he roamed the streets, gloomy and preoccupied, fleeing
from his acquaintances and the Centennial festivities of the
fashionable quarters to wander through the tenement districts and
the slums. Sometimes he would join the spectators of some street
entertainment; and as he listened to the talk of those about him, or
spoke to them, men and women, it surprised him to feel suddenly so
much at home with these poor people, so at one with them; till he
remembered that through his mother—born of laborers who had
worked their way up to the shopkeeping class—he, too, was pueblo,
very much pueblo, a true child of the proletariat.
One day he went to see the building—a small tenement—on the
income from which he was living. The house was a loathsome
plague-spot in which some fifteen wretched families lodged. How
was it that it had never before occurred to him to look this house up,
he wondered, disgusted with himself. And why had his agent never
reported such conditions? Then he remembered that he had visited
the property in person several times before his second trip to
Europe; save that then all this poverty and squalor seemed to him a
natural, even an excellent, thing! Was it not just this sort of

surroundings which pricked the ambition of these laboring people,
spurred them to work their way up to the comfort they had learned
through hard experience to appreciate? Was not this very misery the
first rung on the ladder of progress in this blessed country of
opportunity, where "no one need be poor unless he chooses to be"?
Monsalvat thought with shame of his earlier adherence to "economic
liberalism," a toothless theory, surely invented by the rich that they
might continue to exploit the poor! How much he would have given
now never to have written those fine articles of his! He went away
resolved to mortgage the tenement, and put the money into
improvements which would make the building sanitary at least.
The people of his old world, his men friends especially, made fun of
his new views. He had not been talking much of his recent mental
struggles; but his aloofness, coupled with a few articles of his giving
voice to the protest within him, annoyed not a few of the
distinguished persons who had been wont to applaud him.
Something had gone wrong inside this man; and society commented
on the change without forbearance. Some said he was crazy, others
thought there was something off with his liver or his spleen. More
than one of his old admirers looked at him with a kind of fear. What
was he going to do next? Perhaps break with all established
institutions.
Monsalvat, however, was nobody's enemy. Feelings of revolt could
not live long in his heart, but became transformed, soon after birth,
into a nameless anguish, a physical and moral uneasiness. He hated
only himself. His rebellion was a rebellion only against his own
selfish years.
What was it he wanted now? What was he looking for? What road
was he going to choose? He did not know. Around him he felt a
great emptiness that was ever growing greater. Wherever he went a
sense of infinite loneliness accompanied him. He spent hours
pondering the future. Meanwhile he had grown strangely sensitive
emotionally; and it seemed as though the moment had come when
his outward life, as well, must undergo its transformation.

One night idle curiosity led him to a cabaret. He knew little of this
form of diversion. The "show" entertained him; the tangos and the
orchestra stirred his emotions. This place of amusement seemed to
be a note of color in the bleak immensity of Buenos Aires. On the
other hand, he felt more alone than ever before. In all that dancing,
in all that music, he found, he scarcely knew why, the same sadness
which was in his soul. At times when the mandola wailed in a
crescendo from the depth of some vulgar popular tune—fraught with
all the coarseness and abjections of the tenements of the city—he
seemed to hear in it a cry of loneliness, despair, and bitterness rising
from the dregs of life itself.
It was on that night that his eyes first met Nacha's. They looked at
one another with surprise, and with a shade of embarrassment, as
though they knew one another. The girl lost her composure, lowered
her eyes, twisted her fingers nervously. For two hours Monsalvat
lingered in the cabaret, persisting in this flirtation. He did not
understand why he had never liked loose women; indeed, it all
seemed to him rather absurd—though the girl did have pretty eyes!
Perhaps she was not what she seemed! Perhaps she might some day
love him, chance permitting. Perhaps his loneliness would be more
bearable if a woman like her were there to sympathize with him.
When she left the cabaret, he followed in a taxi. With her
companion, she went into a house. Monsalvat concluded that she
lived there. He got out of the taxi, and loitered about in the middle
of the dark street. She came out on the balcony for a moment,
casting two or three rapid glances in his direction.
A few nights later Monsalvat returned to the cabaret. He did not find
her there. His loneliness again became unbearably acute, and his
restlessness intolerable. It seemed to him more than ever imperative
that he find some purpose in life again, some clear comprehension
of his mission and destiny.
A few days later the scene in the cabaret occurred.

CHAPTER III
It was one o'clock when Monsalvat came out of the cabaret. As he
stepped out on the sidewalk the cold, waiting thief like at the door,
leapt at his throat and face. He turned up the collar of his overcoat
and walked slowly away, careless of direction, his eyes following the
sidewalk in front of him as a wheel follows a groove.
At the first street corner he paused. People were leaving theatres
and cafés, whirling away into the dark in taxis and automobiles. The
trams were crowded. The cross-streets, of unpretentious apartment
houses and second-rate shops, all darkened and asleep, were poorly
lighted; but at its southern end, the center of the capital's night life
dusted the sky with a golden sheen. Monsalvat turned in that
direction, walking on mechanically till he came out on the brilliantly
illuminated avenue. Through the immense plate glass windows of
the cafés he could see the multitudes of little tables, and topping
them, hundreds of human torsos gesticulating under thick waves of
cigarette smoke, pierced with colored lights; while through the
opening and closing doors, tango music broke in irregular surges,
now strong, now weak. The street corners were sprinkled with men
stragglers or survivors from larger groups of joy-seekers. Automobile
horns, conversations in every tongue, the bells of blocked street-
cars, rent the lurid glow with resounding, impatient clangor. But in
spite of all the animation and illumination of the theatre district, the
merry-making had not the enthusiasm of the earlier hours. Only that
irreducible minimum of vitality remained, that residue of joy-thirst,
which survives evenings of revelry, clinging tenaciously to the later

hours, and scattering over the after-midnight streets a pervading
sense of weariness.
Indifferent to the animation of these glittering thoroughfares,
concentrated on his own inner misery, bewildered in the maze of
conflicting emotions within him, Monsalvat went on his way, but
walking more and more slowly now. He tried to analyze the thoughts
and sensations that were tormenting him; but the effort served only
to exasperate his distress. He had never suffered like this. All he
knew for the moment was that his heart, with an impulsiveness
which he felt certain was quite disinterested, had gone out to a girl
he saw doomed, the victim of her own will to live and of the evil
nature of others. How cowardly, futile, he had felt himself in the
presence of her helplessness and humiliation! And then something
overwhelming, imperious, had seemed to stir in his being, filling him
with a courage strangely unfamiliar to him, lifting him from his chair,
and throwing him forward against the girl's tormentors. But had he
not played the simple fool—in public? Had not even Nacha joined in
the mockery as he left the room, proving incapable of loyalty even
toward the man who had defended her? Then that final thrust of the
bully: "Take a good look at me! I am Dalmacio Arnedo! Pampa
Arnedo!" In the days of his thoughtless prosperity as a student and
man of promise, Fernando had thought little of the sister, Eugenia
Monsalvat, who shared his own position in his father's family. A
touch of shame and sorrow had come to him when he learned that
she had left her—and his—mother's home—disappearing from even
that penumbra of respectability, to live as the mistress of a man
named Arnedo. So this was the man, thus crossing his path a second
time, rising before him leering and insulting, and pronouncing his
own name as a symbol of redoubled scorn for the name of
Monsalvat! And that sister, again! Had he done anything to prevent
her fall, in the first place, or to redeem her, now that she had fallen?
He was still walking slowly down the avenue of white lights when he
felt a touch on his arm. It was Hamilcar Torres, one of the most
intimate of his few intimate acquaintances.

"Give me a few moments, Monsalvat. Let's go in here, shall we?"
They entered one of the large cafés. The orchestra here, composed
of girls, was playing a languid gypsy waltz, the music and the
musicians, in combination, evoking expressions of melting languor
on the faces of the males who were assembled there, most of them,
at this advanced hour, gazing about in stupid rapture over wine
glasses that were being filled and filled again.
"It was I who sent for the police," said Torres, when they had taken
a table. He brought out the words very deliberately, marking the
syllables, and in a tone calculated to emphasize the allusion, though
his manner at once changed from a mood of reproving seriousness
to one of amusement, and bantering knowingness.
Torres was a physician; his strikingly white teeth, crisp curly hair,
eyebrows prominent over deep-set black eyes, suggested a trace of
African blood in his veins. Under a thick black mustache, rather
handsomely set against rosy, smooth-shaven cheeks, he smiled
continuously, sometimes sadly, sometimes ironically, sometimes with
affected malevolence and shrewdness.
Monsalvat did not reply. The doctor, turning sideways to the table,
crossed his long legs, and, thrusting them far beyond the limits of
the space which might reasonably be allowed to each patron of the
café, obstructed all passage near him.
"I followed along after you," he said, shifting uneasily on his chair
and turning his head so as to face Monsalvat, "because I wanted to
put you on your guard. You've got to be careful with these people,
old man! I know them—they won't stop at anything—and I saw that
you ... and the girl ... well ... er ... eh?"
His right finger pointed, on the query, to his own right eye, then he
waggled it at Monsalvat. Again his face varied from a rather
exaggerated severity to a knowing smile; and turning his head so
that it was once more in line with his body, and he had to look
sideways at Monsalvat, he added:

"No need to deny it, my boy! After all, the girl is pretty enough! But
—be careful.... When women like that get a hold of a fellow...!"
"Aren't you putting it rather strongly, Torres? I have a feeling that
this particular girl is not of just the kind that...."
"Just the kind that what?" snapped the doctor, still eyeing Fernando
sidewise, and with a mocking smile. "You don't know her!"
Then facing Monsalvat, and mustering a choleric frown for the
occasion, he added impressively in a mysterious and earnest tone of
voice, as if revealing something from a transcendental source:
"More than one man has gone to the dogs on that girl's account!"
Whereupon, with an air of philosophical indifference, he settled back
to his former comfortable position.
Monsalvat was not convinced. Nacha's gentle eyes seemed to refute
the miserable innuendos Torres was making. And yet, supposing it
were all true? What then? A wave of passionate curiosity swept over
Monsalvat. He wanted to know more. He must know more! Yet he
said nothing. He could not bring out the question that was hanging
on his lips. Torres divined what his friend was thinking, and pleased
to be able to show how intimately he knew the ins and outs of life in
Buenos Aires, he began:
"This Arnedo fellow—Pampa, as they call him—is real low-life, the
kind who wouldn't hesitate to put a bullet through your body, or
forge your name. Two or three times he has come near going to jail.
And you saw how he treats the girl! An out and out bully!"
"What's her name? Who is she?" interrupted Monsalvat, with ill-
concealed eagerness.
"She's known as Lila about town; but her real name is Ignacia
Regules—Nacha, as most people call her for short. Her mother kept
a student's boarding house—still does, for that matter. I knew her
mother ... because once...."
"Keep to Nacha, won't you?"

"I see; you want to hear all about the girl! That's the important
subject!" The doctor looked slyly at Monsalvat, enjoying the latter's
confusion at this sudden self-betrayal. "I'll tell you something of
what I know—not all, of course. I'm obliged to keep the most
interesting parts to myself. Well, this Nacha, while still living in her
mother's boarding house, fell in love with a student and ran away
with him. He kept her a couple of years or so; then he left her, and
at a very critical juncture—she was in the hospital, with a child that,
fortunately, did not live. When she came out she took a job in a
store. Probably she was willing enough to live a decent life, but the
bad example of some of her girl friends was too much for her. She
began to earn ten times more than what she got in the store—in a
different way."
Torres winked as he now looked at Monsalvat.
"And how do you know all this?" the latter inquired.
"My dear fellow, that is something I don't tell."
The doctor did not wish to modify the effect of his story by simply
stating that Nacha had known a friend of his, and once, when she
was ill and Torres had been attending her, she had given him her
whole story. Torres enjoyed mystification for its own sake, and
preferred, just for the fun of it, to keep Monsalvat on edge a little
longer.
And this game, for that matter, was working well. In utter distress,
Monsalvat stared fixedly, now at his friend, now at the orchestra,
now at the unknown faces about the great hall. But he did not see
what was before his eyes. His mind was filled with the image of his
own sister, abandoned to misfortune, perhaps now a common
woman of the streets; of his mother weeping her life out over her
own and her daughter's shame; of Nacha Regules, caught in the
brutal clutches of Pampa Arnedo; and finally of his own past self,
happy, free to travel, flirting with handsome women, courting literary
fame, lounging at his club, or attending fashionable parties! While he
had been idling thoughtlessly along in this relative but still gilded

luxury, Eugenia Monsalvat was falling lower and lower in the social
scale! His sister! But not his sister, only! Millions of women were
enduring a misery like hers! And a world of well-nourished,
"successful" men and women went gaily on its way, indifferent to the
ceaseless suffering of these other women, proud of its money, and
its easy virtue, robbing the poor of sisters and daughters, buying
them, corrupting them, enjoying life.
"And then?" asked Monsalvat, noticing that Torres was studying him,
and eager to learn everything he could about the life of this girl, who
seemed to him at that moment to represent all the unfortunate
women of the earth in her person.
"Well, she left the store—you would never guess why! She wanted to
be 'respectable'! She took up some kind of work, I forget what; but
eventually she drifted into a café, as a waitress. Can you imagine
'respectable'—and a café waitress!"
Monsalvat, more and more irritated at his companion's flippancy,
suggested that these attempts of Nacha to work and to be
"respectable" were certainly nothing against her. She might be a
good girl, after all!
"Good? Of course! These girls are all good—almost all, at least. We
do judge them harshly, I realize. If they do wrong, it is without
knowing exactly that it is wrong. And some of them really have a
high moral code—for instance...."
Torres was not smiling now. Memories of the numberless poor
creatures he had known, memories of extraordinary cases of
generosity, and loyalty, and even heroism, for the moment drove his
superficial cynicism from him.
Monsalvat was not interested however, obsessed as he was by the
image of Nacha, who seemed to be appealing to him to rescue her.
And rescue her he would! He would save her from her present tragic
situation, from fearful hours awaiting her in the future, and from the
memory of frightful hours of the past. An idea that he must see her,
speak to her again, somehow, somewhere, took possession of him.

But how? And where? And supposing he should meet her again?
What would he say to her? He did not know; but his determination
was not shaken on that account. He would see her—and save her;
not for her own sake, nor because he was himself an "unfortunate"
in society; nor because she was beautiful, and his eyes had dwelt
upon her; but for love of his sister rather, for the sake of his own
real self!
"These poor girls are simply victims of conditions, I suppose,"
continued Torres. "Nacha told me once that wherever she went, in
shops, or workrooms, or business offices, the men were after her.
And it's true, isn't it? We men, even the best of us, are a bad lot. I'd
like to know how a girl who hasn't enough to eat, and who lives in
the worst sort of surroundings, can resist temptation, especially
when it comes in the form of a good-looking fellow who offers to
take her out of the hell she is living in.... No, they are not to
blame...."
Meanwhile the "Merry Widow" waltz floated languidly through the
thick air of the café like a maze of shimmering diaphanous silk or
impalpable tulle. But to Monsalvat it seemed that this music was
winding itself about him, body and soul, a merciless bandage which
bound him tighter and tighter, treacherously increasing the pain it
promised to soothe. The sadness dwelling at the core of all worldly
pleasures fell from each musical phrase, each bar, each note, on the
heavy air of the café. Music in such places as this always distressed
Monsalvat. Tonight his whole being was an open wound, over which
the ceaselessly moving grind of the music grated until he wondered
that he did not scream with pain. Was his own record absolutely
clean? Had he, too, not bought favors from women—be it, indeed,
with flattery and favors returned? And where were those women
now?
Had they, too, by selling themselves, lost all right to the world's
respect, the right to be treated as human beings, to be pitied? His
fault? He despised himself utterly. Only the violence of his self-
reproach gave him the strength to bear his pain.

"And then what?" he queried, rousing himself from his abstraction.
Torres, who had been silent for a time, now answered the question
that came almost mechanically from Monsalvat's lips, and told all he
knew of Nacha's history. Outstanding in her checkered career had
been her love affair with the young poet, Carlos Riga. Together they
had endured the most frightful poverty in the Argentine bohemia.
Nacha had left him finally, driven away by sheer hunger—and the
thought that perhaps her being always with him was an unjust
burden on her penniless lover. In these circumstances she'd
concluded that it was no use trying to be a "decent" girl; and she
had gone off "on her own," taking up with a man—who was soon
followed by another—better able to support her. One day the idea
came into her head that exclusive devotion to any one protector
meant a sort of unfaithfulness to Riga, whom she really adored.
From that moment she gave herself up to the roving life of the
cabarets and places of amusement. It was during this time that she
met Arnedo. He found her pretty, intelligent, admired the ease of
manner she had acquired in her mother's boarding house, was
impressed by the smatterings of culture she had absorbed from Riga
and other young writers she had known in Riga's company—in short,
decided that Nacha was the jewel he was looking for—a girl he could
"flash" on Capitol sportdom, and "show off" as his "woman" among
people appreciative of such display.
"A horrible story!" exclaimed Monsalvat, gloomily. "Can there be
many girls like that?"
"Thousands of them. And I really know something about it.... I have
long been a police physician. My dissertation was on that very
subject!" And he lectured at length on the theme, sparing no details
of the traffic which has made Buenos Aires famous as a market of
human flesh.
Monsalvat could not speak meanwhile. He was thinking of his sister,
trying to picture to himself what her lot must be. He saw her in the
abandonment that followed her disgrace, struggling not to lose her
grip on life, failing, struggling again to evade the deeper

degradations of the outcast she saw below her; and finally sinking in
the loathsome mire, dragged into its depths, by a trader's claws,
perhaps, tortured, enslaved, and—who could say!—dead! He
listened with speechless intentness. "What a ghastly nightmare this
world is!" He stammered at last:
"And what is being done to remedy all this?"
"What is there to do, my dear fellow? We would have to destroy
everything and construct society anew!"
At these words Monsalvat seized his friend's arm with violence; his
eyes were moist with emotion and his voice rang with a strange
solemnity, as he said slowly:
"Exactly! Exactly! Well, everything is being destroyed, and a new
society is coming into being!"
Torres assented, as far as his facial muscles were concerned,
responding to the suggestiveness of Monsalvat's moral earnestness,
to the emotion which his friend's vision of a great and approaching
Good stirred in his own sluggish depths. He even went so far as to
nod.... Then came reaction. His inner, his real self recovered from
the momentary spell of Monsalvat's ingenuous and lyric optimism.
One look about at the café's noisy and drunken hilarity, and the man
of generous instincts disappeared, giving place again to the man of
the world, the man like any other man, stamped with all the ideas
and sentiments of his kind. To Torres the words Monsalvat had
spoken, his Quixotic theories, his grief over things that were not only
irremediable and accepted, but even sanctioned, and necessary,
began to appear ridiculous, and speaking as a doctor, trained to seek
the origin of all human abnormalities in overstrung nerves and
disturbed physical or mental equilibrium, he replied lightly and
skeptically as before:
"The problem, you see, is too complex ... there is no solution
really...."
Monsalvat did not hear him. Another voice was filling his ears, a
voice from a thousand throats, convicting him of his own

responsibility, too, for the world's crimes. His heart seemed to him a
mournful, hollow, and despairing bell; his eyes saw the world as a
scene ready set for tragedy—the tragedy, first, of his mother,
deceived, suffering all her life, and handing on suffering to her
children; then his sister's; then Nacha's. In an eternal chorus of
tears rose the lamentations of the lost women of the earth, the
weeping of their parents, their brothers; the cries of the children
they were driven to destroy; their own screams of shame, and
clamorings of hunger.
"Why, man, what's the matter with you?" asked Torres finally.
"Hadn't we better be going? It's three o'clock."
Monsalvat nodded and got up. He took leave of Torres at once, on
the pretext he did not feel well, and started off for the South End,
toward the Avenida de Mayo, where he lived.
He went to bed at once upon reaching his rooms. But he could not
sleep. He did not know why it was; but the sound of the shots that
had brought down some of the human creatures in the mob at
Lavalle Square, and the song they had sung, became interwoven
with one of the cabaret tangos he had just been hearing. This
strange music haunted his ears and drove sleep far from him. Later,
when he had fallen into a kind of half slumber, there came towards
him a procession of frightful figures, howling and groaning louder
and louder as they approached; and he knew that this procession
was Humanity. It was already dawn when he began to sleep—
uneasily and for only a little while. But even this semblance of
slumber brought with it a nightmare. A monstrous phantom, covered
with gold, silks, and precious stones, its jaws those of an apocalyptic
beast, its claws, too, dripping blood, was there before him, in his
room, although scarcely contained by it. The monster approached
his bed, showing its fangs, about to devour him; and this monster,
with its charnel house of a belly, where lay countless generations of
the world's unfortunates, was Injustice.
Monsalvat got up late. He was quiet now. At last there was new life
within him. Everything had new life, new meaning. What this new

life was he could not have said. But he knew that within him there
was now a sense of clearness where before there had been nothing
but confusion and obscurity.
He breakfasted and went out, thinking, rather vaguely, that he would
go to his mother's. But, as he walked on, he turned in another
direction. Moving absent-mindedly, yielding to a new sweet sense of
inner calm, he seemed not to notice the streets along which he
passed. When he came to himself, he noted that he was within a
few yards of Nacha's house. Without hesitating, certain now that he
was doing the right thing, he went up the steps and rang the bell.

CHAPTER IV
Nacha had not been able to sleep. Rarely, even in her unhappy life,
had she spent so bad a night. On arriving home from the cabaret,
Arnedo had gone to bed in silence. This Indian-like taciturnity of his
always terrified her. Dread of the man's violence, fear of being once
more abandoned, and forced to return to her former precarious
circumstances, mingled with the anxieties the day had brought her.
Carlos Riga, she had only that morning learned, was dying in a
hospital ward. Yet curiously, what tortured her more than grief for
her former lover or fear for her own life, was the uneasiness aroused
within her by the memory of how she had treated that unknown
man who had so chivalrously come to her defense in the cabaret. He
had been ready to risk his life for her, and she had rewarded him
with a laugh, a laugh half of fear, half of distraction; but to him it
must have seemed one of treacherous mockery.
Into her heart that night a new, a strangely engrossing uneasiness
had come, a presentiment which she could not have explained, but
which she knew she must conceal from Arnedo as though it were a
crime. It was a sense of impending evil, an accumulation of
forebodings—reminiscences that the news of Riga's condition had
brought up, memories of the evening itself; bits of her own past;
pictures, which her frightened imagination painted, of a terrible
future—a future with at best such poor, such ill-nourished, such
unsubstantial hopes—all blending into a vague conviction that Fate
had decreed some great misfortune for her.

How she longed for the relief of slumber! She would need to look
fresh and happy when she faced Arnedo the following day. This
preoccupation filled her insomnia with a sort of hectic frenzy.
To destroy all traces of the hours of torment she was enduring, she
imagined herself digging little graves for them, and burying them
one by one under a dust of forgetfulness. Meanwhile, in her desire
for the dawn, she turned on the light every few minutes to see what
time it was. Four o'clock, half-past four, five! Never had a night
lasted so long! She thought the clock must be slow, and got up to
see if there were any signs of coming day. Darkness was still
unbroken. Only a faint glow in the depths of the sky seemed to
presage the possibility of morning. How she hungered for light in
that overwhelming darkness!
And meanwhile the image of the man in the cabaret haunted her. He
looked at her so strangely! No one else had ever looked at her in
just this fashion. There was not in his eyes that desire which she
saw in the eyes of other men. It was something else, something
else! Especially from the moment when all the café had turned on
her! Why had he gazed at her so persistently? A few nights before,
in this same cabaret, her eyes had met those of this man. She had
not been able to keep from looking at him; she had not been able to
avoid his gaze when he looked her way. And then he had followed
her home—doubtless to find out where she lived. She had seen him
lingering there in the street and had stepped out on the balcony for
a moment.... Who was he? Did he want to take her from Arnedo, to
have her for himself? Why should he wish to defend her when his
doing so could only injure her? He was to blame, in large measure,
for Pampa's bad humor. As for Pampa, she hated him; but she could
not leave him. He had broken her spirit. He could insult her and
knock her about; but instead of turning against him, she would
become more submissive and obedient than before. Why? How
strange life was! She would never understand herself. At times it
seemed as though another being dwelt within, forcing her to do
things she could not otherwise account for. Why else, for example,
should she have behaved so meanly, so contemptibly, towards this

man who had defended her; who, clearly, was interested in her; who
was, perhaps, in love with her? Why? Why? That whole long night
she had tried not to think of this stranger; but to no avail. There was
something about the way he held himself, something in his eyes,
and in the words he spoke, which set him apart from everyone else
she knew. And this distinction fascinated her. With what spirit he had
faced that hostile gang! Something was drawing her towards him. It
would frighten her to meet him again—yet she longed for just such
an encounter. Why should she want to see him? She did not know!
She refused to know!
Only the memory of the poet who had been her lover softened the
pain of that unending night. He at least was good! He was loyal! He
was compassionate! His heart knew the most beautiful words in the
world with which to console; he had developed her intelligence,
taught her to bow her head to irremediable injustice. Only this,
perhaps, had saved her from the hard, cynical desperation of other
women who had, like her, been overcome by wrong. And now he
was dying. He was perhaps already dead. She had seen a report of
his illness in a newspaper the night before; and the shock of it had
left her helpless to disguise the sadness which possessed her as she
sat with the others in the cabaret.
She felt responsible in a certain way for Riga's death. Had she not
abandoned him at the very moment when he most needed her
support? And why had she behaved so? Why was there this
incessant contradiction in her life? She had run away from home at
the very time when she had become most attached to her mother
and her sister. She had loved Riga passionately, and she had fled
from him. She felt sympathy and admiration for the man in the
cabaret, and she had mocked him. Why did she always act in this
unaccountable way? Then Riga took entire possession of her
thoughts, and she lived over again the time that had elapsed
between their first meeting and her tragic abandonment of him.
It was in her mother's boarding house that they had begun their
friendship. Later, after her misfortune, she learned of the poet's

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