The Finite Element Method Fifth edition Volume 1: The Basis.Professor O.C. Zienkiewicz docx

The Finite Element Method
Fifth edition
Volume 1: The Basis
Professor O.C. Zienkiewicz, CBE, FRS, FREng is Professor Emeritus and Director
of the Institute for Numerical Methods in Engineering at the University of Wales,
Swansea, UK. He holds the UNESCO Chair of Numerical Methods in Engineering
at the Technical University of Catalunya, Barcelona, Spain. He was the head of the
Civil Engineering Department at the University of Wales Swansea between 1961
and 1989. He established that department as one of the primary centres of ®nite
element research. In 1968 he became the Founder Editor of the International Journal
for Numerical Methods in Engineering which still remains today the major journal
in this ®eld. The recipient of 24 honorary degrees and many medals, Professor
Zienkiewicz is also a member of ®ve academies ± an honour he has received for his
many contributions to the fundamental developments of the ®nite element method.
In 1978, he became a Fellow of the Royal Society and the Royal Academy of
Engineering. This was followed by his election as a foreign member to the U.S.
Academy of Engineering (1981), the Polish Academy of Science (1985), the Chinese
Academy of Sciences (1998), and the National Academy of Science, Italy (Academia
dei Lincei) (1999). He published the ®rst edition of this book in 1967 and it remained
the only book on the subject until 1971.
Professor R.L. Taylor has more than 35 years' experience in the modelling and simu-
lation of structures and solid continua including two years in industry. In 1991 he was
elected to membership in the U.S. National Academy of Engineering in recognition of
his educational and research contributions to the ®eld of computational mechanics.
He was appointed as the T.Y. and Margaret Lin Professor of Engineering in 1992
and, in 1994, received the Berkeley Citation, the highest honour awarded by the
University of California, Berkeley. In 1997, Professor Taylor was made a Fellow in
the U.S. Association for Computational Mechanics and recently he was elected
Fellow in the International Association of Computational Mechanics, and was
awarded the USACM John von Neumann Medal. Professor Taylor has written sev-
eral computer programs for ®nite element analysis of structural and non-structural

systems, one of which, FEAP, is used world-wide in education and research environ-
ments. FEAP is now incorporated more fully into the book to address non-linear and
®nite deformation problems.
Front cover image: A Finite Element Model of the world land speed record (765.035mph) car THRUST
SSC. The analysis was done using the ®nite element method by K. Morgan, O. Hassan and N.P. Weatherill
at the Institute for Numerical Methods in Engineering, University of Wales Swansea, UK. (see K. Morgan,
O. Hassan and N.P. Weatherill, `Why didn't the supersonic car ¯y?', Mathematics Today, Bulletin of the
Institute of Mathematics and Its Applications, Vol. 35, No. 4, 110±114, Aug. 1999).
The Finite Element
Method
Fifth edition
Volume 1: The Basis
O.C. Zienkiewicz, CBE, FRS, FREng
UNESCO Professor of Numerical Methods in Engineering
International Centre for Numerical Methods in Engineering, Barcelona
Emeritus Professor of Civil Engineering and Director of the Institute for
Numerical Methods in Engineering, University of Wales, Swansea
R.L. Taylor
Professor in the Graduate School
Department of Civil and Environmental Engineering
University of California at Berkeley
Berkeley, California
OXFORD AUCKLAND BOSTON JOHANNESBURG MELBOURNE NEW DELHI
Butterworth-Heinemann
Linacre House, Jordan Hill, Oxford OX2 8DP
225 Wildwood Avenue, Woburn, MA 01801-2041
A division of Reed Educational and Professional Publishing Ltd
First published in 1967 by McGraw-Hill
Fifth edition published by Butterworth-Heinemann 2000
# O.C. Zienkiewicz and R.L. Taylor 2000

All rights reserved. No part of this publication
may be reproduced in any material form (including
photocopying or storing in any medium by electronic
means and whether or not transiently or incidentally
to some other use of this publication) without the
written permission of the copyright holder except
in accordance with the provisions of the Copyright,
Designs and Patents Act 1988 or under the terms of a
licence issued by the Copyright Licensing Agency Ltd,
90 Tottenham Court Road, London, England W1P 9HE.
Applications for the copyright holder's written permission
to reproduce any part of this publication should
be addressed to the publishers
British Library Cataloguing in Publication Data
A catalogue record for this book is available from the British Library
Library of Congress Cataloguing in Publication Data
A catalogue record for this book is available from the Library of Congress
ISBN 0 7506 5049 4
Published with the cooperation of CIMNE,
the International Centre for Numerical Methods in Engineering,
Barcelona, Spain (www.cimne.upc.es)
Typeset by Academic & Technical Typesetting, Bristol
Printed and bound by MPG Books Ltd
Dedication
This book is dedicated to our wives Helen and Mary
Lou and our families for their support and patience
during the preparation of this book, and also to all of
our students and colleagues who over the years have
contributed to our knowledge of the ®nite element
method. In particular we would like to mention

Professor Eugenio On
Ä
ate and his group at CIMNE for
their help, encouragement and support during the
preparation process.

Contents
Preface xv
1. Some preliminaries: the standard discrete system 1
1.1 Introduction 1
1.2 The structural element and the structural system 4
1.3 Assembly and analysis of a structure 8
1.4 The boundary conditions 9
1.5 Electrical and ¯uid networks 10
1.6 The general pattern 12
1.7 The standard discrete system 14
1.8 Transformation of coordinates 15
References 16
2. A direct approach to problems in elasticity 18
2.1 Introduction 18
2.2 Direct formulation of ®nite element characteristics 19
2.3 Generalization to the whole region 26
2.4 Displacement approach as a minimization of total potential energy 29
2.5 Convergence criteria 31
2.6 Discretization error and convergence rate 32
2.7 Displacement functions with discontinuity between elements 33
2.8 Bound on strain energy in a displacement formulation 34
2.9 Direct minimization 35
2.10 An example 35
2.11 Concluding remarks 37

References 37
3. Generalization of the ®nite element concepts. Galerkin-weighted residual
and variational approaches 39
3.1 Introduction 39
3.2 Integral or `weak' statements equivalent to the dierential equations 42
3.3 Approximation to integral formulations 46
3.4 Virtual work as the `weak form' of equilibrium equations for
analysis of solids or ¯uids 53
3.5 Partial discretization 55
3.6 Convergence 58
3.7 What are `variational principles'? 60
3.8 `Natural' variational principles and their relation to governing
dierential equations 62
3.9 Establishment of natural variational principles for linear,
self-adjoint dierential equations 66
3.10 Maximum, minimum, or a saddle point? 69
3.11 Constrained variational principles. Lagrange multipliers and
adjoint functions 70
3.12 Constrained variational principles. Penalty functions and the least
square method 76
3.13 Concluding remarks 82
References 84
4. Plane stress and plane strain 87
4.1 Introduction 87
4.2 Element characteristics 87
4.3 Examples ± an assessment of performance 97
4.4 Some practical applications 100
4.5 Special treatment of plane strain with an incompressible material 110
4.6 Concluding remark 111
References 111

5. Axisymmetric stress analysis 112
5.1 Introduction 112
5.2 Element characteristics 112
5.3 Some illustrative examples 121
5.4 Early practical applications 123
5.5 Non-symmetrical loading 124
5.6 Axisymmetry ± plane strain and plane stress 124
References 126
6. Three-dimensional stress analysis 127
6.1 Introduction 127
6.2 Tetrahedral element characteristics 128
6.3 Composite elements with eight nodes 134
6.4 Examples and concluding remarks 135
References 139
7. Steady-state ®eld problems ± heat conduction, electric and magnetic
potential, ¯uid ¯ow, etc. 140
7.1 Introduction 140
7.2 The general quasi-harmonic equation 141
7.3 Finite element discretization 143
7.4 Some economic specializations 144
7.5 Examples ± an assessment of accuracy 146
7.6 Some practical applications 149
viii Contents
7.7 Concluding remarks 161
References 161
8. `Standard' and `hierarchical' element shape functions: some general
families of
C
0
continuity 164

8.1 Introduction 164
8.2 Standard and hierarchical concepts 165
8.3 Rectangular elements ± some preliminary considerations 168
8.4 Completeness of polynomials 171
8.5 Rectangular elements ± Lagrange family 172
8.6 Rectangular elements ± `serendipity' family 174
8.7 Elimination of internal variables before assembly ± substructures 177
8.8 Triangular element family 179
8.9 Line elements 183
8.10 Rectangular prisms ± Lagrange family 184
8.11 Rectangular prisms ± `serendipity' family 185
8.12 Tetrahedral elements 186
8.13 Other simple three-dimensional elements 190
8.14 Hierarchic polynomials in one dimension 190
8.15 Two- and three-dimensional, hierarchic, elements of the `rectangle'
or `brick' type 193
8.16 Triangle and tetrahedron family 193
8.17 Global and local ®nite element approximation 196
8.18 Improvement of conditioning with hierarchic forms 197
8.19 Concluding remarks 198
References 198
9. Mapped elements and numerical integration ± `in®nite' and `singularity'
elements 200
9.1 Introduction 200
9.2 Use of `shape functions' in the establishment of coordinate
transformations 203
9.3 Geometrical conformability of elements 206
9.4 Variation of the unknown function within distorted, curvilinear
elements. Continuity requirements 206
9.5 Evaluation of element matrices (transformation in , , 

coordinates) 208
9.6 Element matrices. Area and volume coordinates 211
9.7 Convergence of elements in curvilinear coordinates 213
9.8 Numerical integration ± one-dimensional 217
9.9 Numerical integration ± rectangular (2D) or right prism (3D)
regions 219
9.10 Numerical integration ± triangular or tetrahedral regions 221
9.11 Required order of numerical integration 223
9.12 Generation of ®nite element meshes by mapping. Blending functions 226
9.13 In®nite domains and in®nite elements 229
9.14 Singular elements by mapping for fracture mechanics, etc. 234
Contents ix
9.15 A computational advantage of numerically integrated ®nite
elements 236
9.16 Some practical examples of two-dimensional stress analysis 237
9.17 Three-dimensional stress analysis 238
9.18 Symmetry and repeatability 244
References 246
10. The patch test, reduced integration, and non-conforming elements 250
10.1 Introduction 250
10.2 Convergence requirements 251
10.3 The simple patch test (tests A and B) ± a necessary condition for
convergence 253
10.4 Generalized patch test (test C) and the single-element test 255
10.5 The generality of a numerical patch test 257
10.6 Higher order patch tests 257
10.7 Application of the patch test to plane elasticity elements with
`standard' and `reduced' quadrature 258
10.8 Application of the patch test to an incompatible element 264
10.9 Generation of incompatible shape functions which satisfy the

patch test 268
10.10 The weak patch test ± example 270
10.11 Higher order patch test ± assessment of robustness 271
10.12 Conclusion 273
References 274
11. Mixed formulation and constraints± complete ®eld methods 276
11.1 Introduction 276
11.2 Discretization of mixed forms ± some general remarks 278
11.3 Stability of mixed approximation. The patch test 280
11.4 Two-®eld mixed formulation in elasticity 284
11.5 Three-®eld mixed formulations in elasticity 291
11.6 An iterative method solution of mixed approximations 298
11.7 Complementary forms with direct constraint 301
11.8 Concluding remarks ± mixed formulation or a test of element
`robustness' 304
References 304
12. Incompressible materials, mixed methods and other procedures of
solution 307
12.1 Introduction 307
12.2 Deviatoric stress and strain, pressure and volume change 307
12.3 Two-®eld incompressible elasticity (u±p form) 308
12.4 Three-®eld nearly incompressible elasticity (u±p±"
v
form) 314
12.5 Reduced and selective integration and its equivalence to penalized
mixed problems 318
12.6 A simple iterative solution process for mixed problems: Uzawa
method 323
x Contents
12.7 Stabilized methods for some mixed elements failing the

incompressibility patch test 326
12.8 Concluding remarks 342
References 343
13. Mixed formulation and constraints ± incomplete (hybrid) ®eld methods,
boundary/Tretz methods 346
13.1 General 346
13.2 Interface traction link of two (or more) irreducible form
subdomains 346
13.3 Interface traction link of two or more mixed form subdomains 349
13.4 Interface displacement `frame' 350
13.5 Linking of boundary (or Tretz)-type solution by the `frame' of
speci®ed displacements 355
13.6 Subdomains with `standard' elements and global functions 360
13.7 Lagrange variables or discontinuous Galerkin methods? 361
13.8 Concluding remarks 361
References 362
14. Errors, recovery processes and error estimates 365
14.1 De®nition of errors 365
14.2 Superconvergence and optimal sampling points 370
14.3 Recovery of gradients and stresses 375
14.4 Superconvergent patch recovery ± SPR 377
14.5 Recovery by equilibration of patches ± REP 383
14.6 Error estimates by recovery 385
14.7 Other error estimators ± residual based methods 387
14.8 Asymptotic behaviour and robustness of error estimators ± the
Babus
Ï
ka patch test 392
14.9 Which errors should concern us? 398
References 398

15. Adaptive ®nite element re®nement 401
15.1 Introduction 401
15.2 Some examples of adaptive h-re®nement 404
15.3 p-re®nement and hp-re®nement 415
15.4 Concluding remarks 426
References 426
16. Point-based approximations; element-free Galerkin ± and other
meshless methods 429
16.1 Introduction 429
16.2 Function approximation 431
16.3 Moving least square approximations ± restoration of continuity
of approximation 438
16.4 Hierarchical enhancement of moving least square expansions 443
16.5 Point collocation ± ®nite point methods 446
Contents xi
16.6 Galerkin weighting and ®nite volume methods 451
16.7 Use of hierarchic and special functions based on standard ®nite
elements satisfying the partition of unity requirement 457
16.8 Closure 464
References 464
17. The time dimension ± semi-discretization of ®eld and dynamic problems
and analytical solution procedures 468
17.1 Introduction 468
17.2 Direct formulation of time-dependent problems with spatial ®nite
element subdivision 468
17.3 General classi®cation 476
17.4 Free response ± eigenvalues for second-order problems and
dynamic vibration 477
17.5 Free response ± eigenvalues for ®rst-order problems and heat
conduction, etc. 484

17.6 Free response ± damped dynamic eigenvalues 484
17.7 Forced periodic response 485
17.8 Transient response by analytical procedures 486
17.9 Symmetry and repeatability 490
References 491
18. The time dimension ± discrete approximation in time 493
18.1 Introduction 493
18.2 Simple time-step algorithms for the ®rst-order equation 495
18.3 General single-step algorithms for ®rst- and second-order equations 508
18.4 Multistep recurrence algorithms 522
18.5 Some remarks on general performance of numerical algorithms 530
18.6 Time discontinuous Galerkin approximation 536
18.7 Concluding remarks 538
References 538
19. Coupled systems 542
19.1 Coupled problems ± de®nition and classi®cation 542
19.2 Fluid±structure interaction (Class I problem) 545
19.3 Soil±pore ¯uid interaction (Class II problems) 558
19.4 Partitioned single-phase systems ± implicit±explicit partitions
(Class I problems) 565
19.5 Staggered solution processes 567
References 572
20. Computer procedures for ®nite element analysis 576
20.1 Introduction 576
20.2 Data input module 578
20.3 Memory management for array storage 588
20.4 Solution module ± the command programming language 590
20.5 Computation of ®nite element solution modules 597
xii Contents
20.6 Solution of simultaneous linear algebraic equations 609

20.7 Extension and modi®cation of computer program FEAPpv 618
References 618
Appendix A: Matrix algebra 620
Appendix B: Tensor-indicial notation in the approximation of elasticity
problems 626
Appendix C: Basic equations of displacement analysis 635
Appendix D: Some integration formulae for a triangle 636
Appendix E: Some integration formulae for a tetrahedron 637
Appendix F: Some vector algebra 638
Appendix G: Integration by parts in two and three dimensions
(Green's theorem) 643
Appendix H: Solutions exact at nodes 645
Appendix I: Matrix diagonalization or lumping 648
Author index 655
Subject index 663
Contents xiii
Volume 2: Solid and structural mechanics
1. General problems in solid mechanics and non-linearity
2. Solution of non-linear algebraic equations
3. Inelastic materials
4. Plate bending approximation: thin (Kirchho) plates and
C
1
continuity require-
ments
5. `Thick' Reissner±Mindlin plates ± irreducible and mixed formulations
6. Shells as an assembly of ¯at elements
7. Axisymmetric shells
8. Shells as a special case of three-dimensional analysis ± Reissner±Mindlin
assumptions

9. Semi-analytical ®nite element processes ± use of orthogonal functions and `®nite
strip' methods
10. Geometrically non-linear problems ± ®nite deformation
11. Non-linear structural problems ± large displacement and instability
12. Pseudo-rigid and rigid±¯exible bodies
13. Computer procedures for ®nite element analysis
Appendix A: Invariants of second-order tensors
Volume 3: Fluid dynamics
1. Introduction and the equations of ¯uid dynamics
2. Convection dominated problems ± ®nite element approximations
3. A general algorithm for compressible and incompressible ¯ows ± the characteristic
based split (CBS) algorithm
4. Incompressible laminar ¯ow ± newtonian and non-newtonian ¯uids
5. Free surfaces, buoyancy and turbulent incompressible ¯ows
6. Compressible high speed gas ¯ow
7. Shallow-water problems
8. Waves
9. Computer implementation of the CBS algorithm
Appendix A. Non-conservative form of Navier±Stokes equations
Appendix B. Discontinuous Galerkin methods in the solution of the convection±
diusion equation
Appendix C. Edge-based ®nite element formulation
Appendix D. Multi grid methods
Appendix E. Boundary layer ± inviscid ¯ow coupling
Preface
It is just over thirty years since The Finite Element Method in Structural and
Continuum Mechanics was ®rst published. This book, which was the ®rst dealing
with the ®nite element method, provided the base from which many further develop-
ments occurred. The expanding research and ®eld of application of ®nite elements led
to the second edition in 1971, the third in 1977 and the fourth in 1989 and 1991. The

size of each of these volumes expanded geometrically (from 272 pages in 1967 to the
fourth edition of 1455 pages in two volumes). This was necessary to do justice to a
rapidly expanding ®eld of professional application and research. Even so, much ®lter-
ing of the contents was necessary to keep these editions within reasonable bounds.
It seems that a new edition is necessary every decade as the subject is expanding and
many important developments are continuously occurring. The present ®fth edition is
indeed motivated by several important developments which have occurred in the 90s.
These include such subjects as adaptive error control, meshless and point based
methods, new approaches to ¯uid dynamics, etc. However, we feel it is important
not to increase further the overall size of the book and we therefore have eliminated
some redundant material.
Further, the reader will notice the present subdivision into three volumes, in which the
®rst volume provides the general basis applicable to linear problems in many ®elds whilst
the second and third volumes are devoted to more advanced topics in solid and ¯uid
mechanics, respectively. This arrangement will allow a general student to study
Volume 1 whilst a specialist can approach their topics with the help of Volumes 2 and
3. Volumes 2 and 3 are much smaller in size and addressed to more specialized readers.
It is hoped that Volume 1 will help to introduce postgraduate students, researchers
and practitioners to the modern concepts of ®nite element methods. In Volume 1 we
stress the relationship between the ®nite element method and the more classic ®nite
dierence and boundary solution methods. We show that all methods of numerical
approximation can be cast in the same format and that their individual advantages
can thus be retained.
Although Volume 1 is not written as a course text book, it is nevertheless directed at
students of postgraduate level and we hope these will ®nd it to be of wide use. Math-
ematical concepts are stressed throughout and precision is maintained, although little
use is made of modern mathematical symbols to ensure wider understanding amongst
engineers and physical scientists.
In Volumes 1, 2 and 3 the chapters on computational methods are much reduced by
transferring the computer source programs to a web site.

1
This has the very substan-
tial advantage of not only eliminating errors in copying the programs but also in
ensuring that the reader has the bene®t of the most recent set of programs available
to him or her at all times as it is our intention from time to time to update and expand
the available programs.
The authors are particularly indebted to the International Center of Numerical
Methods in Engineering (CIMNE) in Barcelona who have allowed their pre- and
post-processing code (GiD) to be accessed from the publisher's web site. This
allows such dicult tasks as mesh generation and graphic output to be dealt with
eciently. The authors are also grateful to Dr J.Z. Zhu for his careful scrutiny and
help in drafting Chapters 14 and 15. These deal with error estimation and adaptivity,
a subject to which Dr Zhu has extensively contributed. Finally, we thank Peter and
Jackie Bettess for writing the general subject index.
OCZ and RLT
1
Complete source code for all programs in the three volumes may be obtained at no cost from the
publisher's web page: />xvi Preface
1
Some preliminaries: the standard
discrete system
1.1 Introduction
The limitations of the human mind are such that it cannot grasp the behaviour of its
complex surroundings and creations in one operation. Thus the process of sub-
dividing all systems into their individual components or `elements', whose behaviour
is readily understood, and then rebuilding the original system from such components
to study its behaviour is a natural way in which the engineer, the scientist, or even the
economist proceeds.
In many situations an adequate model is obtained using a ®nite number of well-
de®ned components. We shall term such problems discrete. In others the subdivision

is continued inde®nitely and the problem can only be de®ned using the mathematical
®ction of an in®nitesimal. This leads to dierential equations or equivalent statements
which imply an in®nite number of elements. We shall term such systems continuous.
With the advent of digital computers, discrete problems can generally be solved
readily even if the number of elements is very large. As the capacity of all computers
is ®nite, continuous problems can only be solved exactly by mathematical manipula-
tion. Here, the available mathematical techniques usually limit the possibilities to
oversimpli®ed situations.
To overcome the intractability of realistic types of continuum problems, various
methods of discretization have from time to time been proposed both by engineers
and mathematicians. All involve an approximation which, hopefully, approaches
in the limit the true continuum solution as the number of discrete variables
increases.
The discretization of continuous problems has been approached dierently by
mathematicians and engineers. Mathematicians have developed general techniques
applicable directly to dierential equations governing the problem, such as ®nite dif-
ference approximations,
1;2
various weighted residual procedures,
3;4
or approximate
techniques for determining the stationarity of properly de®ned `functionals'. The
engineer, on the other hand, often approaches the problem more intuitively by creat-
ing an analogy between real discrete elements and ®nite portions of a continuum
domain. For instance, in the ®eld of solid mechanics McHenry,
5
Hreniko,
6
Newmark
7

, and indeed Southwell
9
in the 1940s, showed that reasonably good solu-
tions to an elastic continuum problem can be obtained by replacing small portions
of the continuum by an arrangement of simple elastic bars. Later, in the same context,
Argyris
8
and Turner et al.
9
showed that a more direct, but no less intuitive, substitu-
tion of properties can be made much more eectively by considering that small
portions or `elements' in a continuum behave in a simpli®ed manner.
It is from the engineering `direct analogy' view that the term `®nite element' was
born. Clough
10
appears to be the ®rst to use this term, which implies in it a direct
use of a standard methodology applicable to discrete systems. Both conceptually and
from the computational viewpoint, this is of the utmost importance. The ®rst
allows an improved understanding to be obtained; the second oers a uni®ed
approach to the variety of problems and the development of standard computational
procedures.
Since the early 1960s much progress has been made, and today the purely mathe-
matical and `analogy' approaches are fully reconciled. It is the object of this text to
present a view of the ®nite element method as a general discretization procedure of con-
tinuum problems posed by mathematically de®ned statements.
In the analysis of problems of a discrete nature, a standard methodology has been
developed over the years. The civil engineer, dealing with structures, ®rst calculates
force±displacement relationships for each element of the structure and then proceeds
to assemble the whole by following a well-de®ned procedure of establishing local
equilibrium at each `node' or connecting point of the structure. The resulting equa-

tions can be solved for the unknown displacements. Similarly, the electrical or
hydraulic engineer, dealing with a network of electrical components (resistors, capa-
citances, etc.) or hydraulic conduits, ®rst establishes a relationship between currents
(¯ows) and potentials for individual elements and then proceeds to assemble the
system by ensuring continuity of ¯ows.
All such analyses follow a standard pattern which is universally adaptable to dis-
crete systems. It is thus possible to de®ne a standard discrete system, and this chapter
will be primarily concerned with establishing the processes applicable to such systems.
Much of what is presented here will be known to engineers, but some reiteration at
this stage is advisable. As the treatment of elastic solid structures has been the
most developed area of activity this will be introduced ®rst, followed by examples
from other ®elds, before attempting a complete generalization.
The existence of a uni®ed treatment of `standard discrete problems' leads us to the
®rst de®nition of the ®nite element process as a method of approximation to con-
tinuum problems such that
(a) the continuum is divided into a ®nite number of parts (elements), the behaviour of
which is speci®ed by a ®nite number of parameters, and
(b) the solution of the complete system as an assembly of its elements follows pre-
cisely the same rules as those applicable to standard discrete problems.
It will be found that most classical mathematical approximation procedures as well
as the various direct approximations used in engineering fall into this category. It is
thus dicult to determine the origins of the ®nite element method and the precise
moment of its invention.
Table 1.1 shows the process of evolution which led to the present-day concepts of
®nite element analysis. Chapter 3 will give, in more detail, the mathematical basis
which emerged from these classical ideas.
11ÿ20
2 Some preliminaries: the standard discrete system
Table 1.1
ENGINEERING MATHEMATICS

Trial
functions
Finite
differences
Variational
methods
Rayleigh 1870
11
Ritz 1909
12
ÿ
ÿ
"
Weighted
residuals
Gauss 1795
18
Galerkin 1915
19
Biezeno±Koch 1923
20
ÿ
ÿ
"
Richardson 1910
15
Liebman 1918
16
Southwell 1946
1

ÿÿ
"
Structural
analogue
substitution
Hreniko 1941
6
McHenry 1943
5
Newmark 1949
7
ÿ
ÿ
"
Piecewise
continuous
trial functions
Courant 1943
13
Prager±Synge 1947
14
Zienkiewicz 1964
21
Direct
continuum
elements
Argyris 1955
8
Turner et al. 1956
9

ÿ
ÿ
"
ÿÿ
"
ÿ
ÿ
"
Variational
finite
differences
Varga 1962
17
PRESENT-DAY
FINITE ELEMENT METHOD
1.2 The structural element and the structural system
To introduce the reader to the general concept of discrete systems we shall ®rst
consider a structural engineering example of linear elasticity.
Figure 1.1 represents a two-dimensional structure assembled from individual
components and interconnected at the nodes numbered 1 to 6. The joints at the
nodes, in this case, are pinned so that moments cannot be transmitted.
As a starting point it will be assumed that by separate calculation, or for that matter
from the results of an experiment, the characteristics of each element are precisely
known. Thus, if a typical element labelled (1) and associated with nodes 1, 2, 3 is
examined, the forces acting at the nodes are uniquely de®ned by the displacements
of these nodes, the distributed loading acting on the element (p), and its initial
strain. The last may be due to temperature, shrinkage, or simply an initial `lack of
®t'. The forces and the corresponding displacements are de®ned by appropriate com-
ponents (U, V and u, v) in a common coordinate system.
Listing the forces acting on all the nodes (three in the case illustrated) of the element

(1) as a matrixy we have
q
1

q
1
1
q
1
2
q
1
3
8
>
<
>
:
9
>
=
>
;
q
1
1

U
1
V

1

; etc: 1:1
y
y
x
x
p
p
Y
4
X
4
V
3
U
3
1
2
3
4
5
6
3
1
2
Nodes
(1)
(1)
(2)

(3)
(4)
A typical element (1)
Fig. 1.1
A typical structure built up from interconnected elements.
yA limited knowledge of matrix algebra will be assumed throughout this book. This is necessary for
reasonable conciseness and forms a convenient book-keeping form. For readers not familiar with the subject
a brief appendix (Appendix A) is included in which sucient principles of matrix algebra are given to follow
the development intelligently. Matrices (and vectors) will be distinguished by bold print throughout.
4 Some preliminaries: the standard discrete system
and for the corresponding nodal displacements
a
1

a
1
a
2
a
3
8
>
<
>
:
9
>
=
>
;

a
1

u
1
v
1

; etc: 1:2
Assuming linear elastic behaviour of the element, the characteristic relationship will
always be of the form
q
1
 K
1
a
1
 f
1
p
 f
1
"
0
1:3
in which f
1
p
represents the nodal forces required to balance any distributed loads acting
on the element and f

1
"
0
the nodal forces required to balance any initial strains such as
may be caused by temperature change if the nodes are not subject to any displacement.
The ®rst of the terms represents the forces induced by displacement of the nodes.
Similarly, a preliminary analysis or experiment will permit a unique de®nition of
stresses or internal reactions at any speci®ed point or points of the element in
terms of the nodal displacements. De®ning such stresses by a matrix r
1
a relationship
of the form
r
1
 Q
1
a
1
 r
1
"
0
1:4
is obtained in which the two term gives the stresses due to the initial strains when no
nodal displacement occurs.
The matrix K
e
is known as the element stiness matrix and the matrix Q
e
as the

element stress matrix for an element (e).
Relationships in Eqs (1.3) and (1.4) have been illustrated by an example of an ele-
ment with three nodes and with the interconnection points capable of transmitting
only two components of force. Clearly, the same arguments and de®nitions will
apply generally. An element (2) of the hypothetical structure will possess only two
points of interconnection; others may have quite a large number of such points. Simi-
larly, if the joints were considered as rigid, three components of generalized force and
of generalized displacement would have to be considered, the last of these correspond-
ing to a moment and a rotation respectively. For a rigidly jointed, three-dimensional
structure the number of individual nodal components would be six. Quite generally,
therefore,
q
e

q
e
1
q
e
2
F
F
F
q
e
m
8
>
>
>

<
>
>
>
:
9
>
>
>
=
>
>
>
;
and a
e

a
1
a
2
F
F
F
a
m
8
>
>
>

<
>
>
>
:
9
>
>
>
=
>
>
>
;
1:5
with each q
e
i
and a
i
possessing the same number of components or degrees of freedom.
These quantities are conjugate to each other.
The stiness matrices of the element will clearly always be square and of the form
K
e

K
e
ii
K

e
ij
ÁÁÁ K
e
im
F
F
F
F
F
F
F
F
F
K
e
mi
ÁÁÁ ÁÁÁ K
e
mm
2
4
3
5
1:6
The structural element and the structural system 5
in which K
e
ii
, etc., are submatrices which are again square and of the size l Âl, where l

is the number of force components to be considered at each node.
As an example, the reader can consider a pin-ended bar of uniform section A and
modulus E in a two-dimensional problem shown in Fig. 1.2. The bar is subject to a
uniform lateral load p and a uniform thermal expansion strain
"
0
 T
where  is the coecient of linear expansion and T is the temperature change.
If the ends of the bar are de®ned by the coordinates x
i
, y
i
and x
n
, y
n
its length can be
calculated as
L 

x
n
ÿ x
i

2
y
n
ÿ y
i


2

q
and its inclination from the horizontal as
  tan
ÿ1
y
n
ÿ y
i
x
n
ÿ x
i
Only two components of force and displacement have to be considered at the
nodes.
The nodal forces due to the lateral load are clearly
f
e
p

U
i
V
i
U
n
V
n

8
>
>
>
<
>
>
>
:
9
>
>
>
=
>
>
>
;
p
ÿ
ÿsin 
cos 
ÿsin 
cos 
8
>
>
>
<
>

>
>
:
9
>
>
>
=
>
>
>
;
pL
2
and represent the appropriate components of simple reactions, pL=2. Similarly, to
restrain the thermal expansion "
0
an axial force ETA is needed, which gives the
L
n
i
y
x
C
p
V
i
(v
i
)

U
i
(u
i
)
x
i
y
i
β
E
1
A
1
Fig. 1.2
A pin-ended bar.
6 Some preliminaries: the standard discrete system
components
f
e
"
0

U
i
V
i
U
n
V

n
8
>
>
>
<
>
>
>
:
9
>
>
>
=
>
>
>
;
"
0
ÿ
ÿcos 
ÿsin 
cos 
sin 
8
>
>
>

<
>
>
>
:
9
>
>
>
=
>
>
>
;
ETA
Finally, the element displacements
a
e

u
i
v
i
u
n
v
n
8
>
>

>
<
>
>
>
:
9
>
>
>
=
>
>
>
;
will cause an elongation u
n
ÿ u
i
cos  v
n
ÿ v
i
sin . This, when multiplied by
EA=L, gives the axial force whose components can again be found. Rearranging
these in the standard form gives
K
e
a
e


U
i
V
i
U
n
V
n
8
>
>
>
>
>
<
>
>
>
>
>
:
9
>
>
>
>
>
=
>

>
>
>
>
;

The components of the general equation (1.3) have thus been established for the
elementary case discussed. It is again quite simple to ®nd the stresses at any section
of the element in the form of relation (1.4). For instance, if attention is focused on
the mid-section C of the bar the average stress determined from the axial tension
to the element can be shown to be
r
e
%  
E
L
ÿcos ; ÿsin ; cos ;sin a
e
ÿ ET
where all the bending eects of the lateral load p have been ignored.
For more complex elements more sophisticated procedures of analysis are required
but the results are of the same form. The engineer will readily recognize that the so-
called `slope±de¯ection' relations used in analysis of rigid frames are only a special
case of the general relations.
It may perhaps be remarked, in passing, that the complete stiness matrix obtained
for the simple element in tension turns out to be symmetric (as indeed was the case
with some submatrices). This is by no means fortuitous but follows from the principle
of energy conservation and from its corollary, the well-known Maxwell±Betti
reciprocal theorem.


EA
L
cos
2
 sin  cos  ÿcos
2
 ÿsin  cos 
sin  cos  sin
2
 ÿsin  cos  ÿsin
2

ÿcos
2
 ÿsin  cos  cos
2
 sin  cos 
ÿsin  cos  ÿsin
2
 sin  cos  sin
2

2
6
6
6
6
4
3
7

7
7
7
5
u
i
v
i
u
n
v
n
8
>
>
>
>
<
>
>
>
>
:
9
>
>
>
>
=
>

>
>
>
;
The structural element and the structural system 7
The element properties were assumed to follow a simple linear relationship. In
principle, similar relationships could be established for non-linear materials, but
discussion of such problems will be held over at this stage.
The calculation of the stiness coecients of the bar which we have given here will
be found in many textbooks. Perhaps it is worthwhile mentioning here that the ®rst
use of bar assemblies for large structures was made as early as 1935 when Southwell
proposed his classical relaxation method.
22
1.3 Assembly and analysis of a structure
Consider again the hypothetical structure of Fig. 1.1. To obtain a complete solution
the two conditions of
(a) displacement compatibility and
(b) equilibrium
have to be satis®ed throughout.
Any system of nodal displacements a:
a 
a
1
F
F
F
a
n
8
>

<
>
:
9
>
=
>
;
1:7
listed now for the whole structure in which all the elements participate, automatically
satis®es the ®rst condition.
As the conditions of overall equilibrium have already been satis®ed within an ele-
ment, all that is necessary is to establish equilibrium conditions at the nodes of the
structure. The resulting equations will contain the displacements as unknowns, and
once these have been solved the structural problem is determined. The internal
forces in elements, or the stresses, can easily be found by using the characteristics
established a priori for each element by Eq. (1.4).
Consider the structure to be loaded by external forces r:
r 
r
1
F
F
F
r
n
8
>
<
>

:
9
>
=
>
;
1:8
applied at the nodes in addition to the distributed loads applied to the individual
elements. Again, any one of the forces r
i
must have the same number of components
as that of the element reactions considered. In the example in question
r
i


X
i
Y
i

1:9
as the joints were assumed pinned, but at this stage the general case of an arbitrary
number of components will be assumed.
If now the equilibrium conditions of a typical node, i, are to be established, each
component of r
i
has, in turn, to be equated to the sum of the component forces
contributed by the elements meeting at the node. Thus, considering all the force
8 Some preliminaries: the standard discrete system

components we have
r
i

X
m
e 1
q
e
i
 q
1
i
 q
2
i
ÁÁÁ 1:10
in which q
1
i
is the force contributed to node i by element 1, q
2
i
by element 2, etc.
Clearly, only the elements which include point i will contribute non-zero forces,
but for tidiness all the elements are included in the summation.
Substituting the forces contributing to node i from the de®nition (1.3) and noting
that nodal variables a
i
are common (thus omitting the superscript e), we have

r
i


X
m
e 1
K
e
i 1

a
1


X
m
e 1
K
e
i 2

a
2
ÁÁÁ
X
m
e 1
f
e

i
1:11
where
f
e
 f
e
p
 f
e
"
0
The summation again only concerns the elements which contribute to node i. If all
such equations are assembled we have simply
Ka  r ÿf 1:12
in which the submatrices are
K
ij

X
m
e 1
K
e
ij
f
i

X
m

e 1
f
e
i
1:13
with summations including all elements. This simple rule for assembly is very
convenient because as soon as a coecient for a particular element is found it can
be put immediately into the appropriate `location' speci®ed in the computer. This
general assembly process can be found to be the common and fundamental feature of
all ®nite element calculations and should be well understood by the reader.
If dierent types of structural elements are used and are to be coupled it must be
remembered that the rules of matrix summation permit this to be done only if
these are of identical size. The individual submatrices to be added have therefore to
be built up of the same number of individual components of force or displacement.
Thus, for example, if a member capable of transmitting moments to a node is to be
coupled at that node to one which in fact is hinged, it is necessary to complete the
stiness matrix of the latter by insertion of appropriate (zero) coecients in the
rotation or moment positions.
1.4 The boundary conditions
The system of equations resulting from Eq. (1.12) can be solved once the
prescribed support displacements have been substituted. In the example of Fig. 1.1,
where both components of displacement of nodes 1 and 6 are zero, this will mean
The boundary conditions 9