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

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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

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

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 , , 

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

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

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

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.

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

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

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

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

(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
>
=
>

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

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

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

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

>
>
>
:
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

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

>
>
>
>
>
;

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.

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

>
<
>
:
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
>
<

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

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