The Finite Element MethodFifth editionVolume 1: The BasisProfessor O.C. Zienkiewicz, CBE, FRS, FREng is Professor Emeritus and Directorof the Institute for Numerical Methods in Engineering at the University of Wales,Swansea, UK. He holds the UNESCO Chair of Numerical Methods in Engineeringat the Technical University of Catalunya, Barcelona, Spain. He was the head of theCivil Engineering Department at the University of Wales Swansea between 1961and 1989. He established that department as one of the primary centres of niteelement research. In 1968 he became the Founder Editor of the International Journalfor Numerical Methods in Engineering which still remains today the major journalin this eld. The recipient of 24 honorary degrees and many medals, ProfessorZienkiewicz is also a member of ve academies an honour he has received for hismany contributions to the fundamental developments of the nite element method.In 1978, he became a Fellow of the Royal Society and the Royal Academy ofEngineering. 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 ChineseAcademy of Sciences (1998), and the National Academy of Science, Italy (Academiadei Lincei) (1999). He published the rst edition of this book in 1967 and it remainedthe 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 waselected to membership in the U.S. National Academy of Engineering in recognition ofhis educational and research contributions to the eld of computational mechanics.He was appointed as the T.Y. and Margaret Lin Professor of Engineering in 1992and, in 1994, received the Berkeley Citation, the highest honour awarded by theUniversity of California, Berkeley. In 1997, Professor Taylor was made a Fellow inthe U.S. Association for Computational Mechanics and recently he was electedFellow in the International Association of Computational Mechanics, and wasawarded the USACM John von Neumann Medal. Professor Taylor has written sev-eral computer programs for nite element analysis of structural and non-structuralsystems, 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 andnite deformation problems.Front cover image: A Finite Element Model of the world land speed record (765.035 mph) car THRUSTSSC. The analysis was done using the nite element method by K. Morgan, O. Hassan and N.P. Weatherillat 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 theInstitute of Mathematics and Its Applications, Vol. 35, No. 4, 110114, Aug. 1999).The Finite ElementMethodFifth editionVolume 1: The BasisO.C. Zienkiewicz, CBE, FRS, FREngUNESCO Professor of Numerical Methods in EngineeringInternational Centre for Numerical Methods in Engineering, BarcelonaEmeritus Professor of Civil Engineering and Director of the Institute forNumerical Methods in Engineering, University of Wales, SwanseaR.L. TaylorProfessor in the Graduate SchoolDepartment of Civil and Environmental EngineeringUniversity of California at BerkeleyBerkeley, CaliforniaOXFORD AUCKLAND BOSTON JOHANNESBURG MELBOURNE NEW DELHIButterworth-HeinemannLinacre House, Jordan Hill, Oxford OX2 8DP225 Wildwood Avenue, Woburn, MA 01801-2041A division of Reed Educational and Professional Publishing LtdFirst published in 1967 by McGraw-HillFifth edition published by Butterworth-Heinemann 2000# O.C. Zienkiewicz and R.L. Taylor 2000All rights reserved. No part of this publicationmay be reproduced in any material form (includingphotocopying or storing in any medium by electronicmeans and whether or not transiently or incidentallyto some other use of this publication) without thewritten permission of the copyright holder exceptin accordance with the provisions of the Copyright,Designs and Patents Act 1988 or under the terms of alicence issued by the Copyright Licensing Agency Ltd,90 Tottenham Court Road, London, England W1P 9HE.Applications for the copyright holder's written permissionto reproduce any part of this publication shouldbe addressed to the publishersBritish Library Cataloguing in Publication DataA catalogue record for this book is available from the British LibraryLibrary of Congress Cataloguing in Publication DataA catalogue record for this book is available from the Library of CongressISBN 0 7506 5049 4Published with the cooperation of CIMNE,the International Centre for Numerical Methods in Engineering,Barcelona, Spain (www.cimne.upc.es)Typeset by Academic & Technical Typesetting, BristolPrinted and bound by MPG Books LtdDedicationThis book is dedicated to our wives Helen and MaryLou and our families for their support and patienceduring the preparation of this book, and also to all ofour students and colleagues who over the years havecontributed to our knowledge of the nite elementmethod. In particular we would like to mentionProfessor Eugenio On ate and his group at CIMNE fortheir help, encouragement and support during thepreparation process.ContentsPreface xv1. Some preliminaries: the standard discrete system 11.1 Introduction 11.2 The structural element and the structural system 41.3 Assembly and analysis of a structure 81.4 The boundary conditions 91.5 Electrical and uid networks 101.6 The general pattern 121.7 The standard discrete system 141.8 Transformation of coordinates 15References 162. A direct approach to problems in elasticity 182.1 Introduction 182.2 Direct formulation of nite element characteristics 192.3 Generalization to the whole region 262.4 Displacement approach as a minimization of total potential energy 292.5 Convergence criteria 312.6 Discretization error and convergence rate 322.7 Displacement functions with discontinuity between elements 332.8 Bound on strain energy in a displacement formulation 342.9 Direct minimization 352.10 An example 352.11 Concluding remarks 37References 373. Generalization of the nite element concepts. Galerkin-weighted residualand variational approaches 393.1 Introduction 393.2 Integral or `weak' statements equivalent to the dierential equations 423.3 Approximation to integral formulations 463.4 Virtual work as the `weak form' of equilibrium equations foranalysis of solids or uids 533.5 Partial discretization 553.6 Convergence 583.7 What are `variational principles'? 603.8 `Natural' variational principles and their relation to governingdierential equations 623.9 Establishment of natural variational principles for linear,self-adjoint dierential equations 663.10 Maximum, minimum, or a saddle point? 693.11 Constrained variational principles. Lagrange multipliers andadjoint functions 703.12 Constrained variational principles. Penalty functions and the leastsquare method 763.13 Concluding remarks 82References 844. Plane stress and plane strain 874.1 Introduction 874.2 Element characteristics 874.3 Examples an assessment of performance 974.4 Some practical applications 1004.5 Special treatment of plane strain with an incompressible material 1104.6 Concluding remark 111References 1115. Axisymmetric stress analysis 1125.1 Introduction 1125.2 Element characteristics 1125.3 Some illustrative examples 1215.4 Early practical applications 1235.5 Non-symmetrical loading 1245.6 Axisymmetry plane strain and plane stress 124References 1266. Three-dimensional stress analysis 1276.1 Introduction 1276.2 Tetrahedral element characteristics 1286.3 Composite elements with eight nodes 1346.4 Examples and concluding remarks 135References 1397. Steady-state eld problems heat conduction, electric and magneticpotential, uid ow, etc. 1407.1 Introduction 1407.2 The general quasi-harmonic equation 1417.3 Finite element discretization 1437.4 Some economic specializations 1447.5 Examples an assessment of accuracy 1467.6 Some practical applications 149viii Contents7.7 Concluding remarks 161References 1618. `Standard' and `hierarchical' element shape functions: some generalfamilies of C0 continuity 1648.1 Introduction 1648.2 Standard and hierarchical concepts 1658.3 Rectangular elements some preliminary considerations 1688.4 Completeness of polynomials 1718.5 Rectangular elements Lagrange family 1728.6 Rectangular elements `serendipity' family 1748.7 Elimination of internal variables before assembly substructures 1778.8 Triangular element family 1798.9 Line elements 1838.10 Rectangular prisms Lagrange family 1848.11 Rectangular prisms `serendipity' family 1858.12 Tetrahedral elements 1868.13 Other simple three-dimensional elements 1908.14 Hierarchic polynomials in one dimension 1908.15 Two- and three-dimensional, hierarchic, elements of the `rectangle'or `brick' type 1938.16 Triangle and tetrahedron family 1938.17 Global and local nite element approximation 1968.18 Improvement of conditioning with hierarchic forms 1978.19 Concluding remarks 198References 1989. Mapped elements and numerical integration `innite' and `singularity'elements 2009.1 Introduction 2009.2 Use of `shape functions' in the establishment of coordinatetransformations 2039.3 Geometrical conformability of elements 2069.4 Variation of the unknown function within distorted, curvilinearelements. Continuity requirements 2069.5 Evaluation of element matrices (transformation in , , coordinates) 2089.6 Element matrices. Area and volume coordinates 2119.7 Convergence of elements in curvilinear coordinates 2139.8 Numerical integration one-dimensional 2179.9 Numerical integration rectangular (2D) or right prism (3D)regions 2199.10 Numerical integration triangular or tetrahedral regions 2219.11 Required order of numerical integration 2239.12 Generation of nite element meshes by mapping. Blending functions 2269.13 Innite domains and innite elements 2299.14 Singular elements by mapping for fracture mechanics, etc. 234Contents ix9.15 A computational advantage of numerically integrated niteelements 2369.16 Some practical examples of two-dimensional stress analysis 2379.17 Three-dimensional stress analysis 2389.18 Symmetry and repeatability 244References 24610. The patch test, reduced integration, and non-conforming elements 25010.1 Introduction 25010.2 Convergence requirements 25110.3 The simple patch test (tests A and B) a necessary condition forconvergence 25310.4 Generalized patch test (test C) and the single-element test 25510.5 The generality of a numerical patch test 25710.6 Higher order patch tests 25710.7 Application of the patch test to plane elasticity elements with`standard' and `reduced' quadrature 25810.8 Application of the patch test to an incompatible element 26410.9 Generation of incompatible shape functions which satisfy thepatch test 26810.10 The weak patch test example 27010.11 Higher order patch test assessment of robustness 27110.12 Conclusion 273References 27411. Mixed formulation and constraints complete eld methods 27611.1 Introduction 27611.2 Discretization of mixed forms some general remarks 27811.3 Stability of mixed approximation. The patch test 28011.4 Two-eld mixed formulation in elasticity 28411.5 Three-eld mixed formulations in elasticity 29111.6 An iterative method solution of mixed approximations 29811.7 Complementary forms with direct constraint 30111.8 Concluding remarks mixed formulation or a test of element`robustness' 304References 30412. Incompressible materials, mixed methods and other procedures ofsolution 30712.1 Introduction 30712.2 Deviatoric stress and strain, pressure and volume change 30712.3 Two-eld incompressible elasticity (up form) 30812.4 Three-eld nearly incompressible elasticity (up"v form) 31412.5 Reduced and selective integration and its equivalence to penalizedmixed problems 31812.6 A simple iterative solution process for mixed problems: Uzawamethod 323x Contents12.7 Stabilized methods for some mixed elements failing theincompressibility patch test 32612.8 Concluding remarks 342References 34313. Mixed formulation and constraints incomplete (hybrid) eld methods,boundary/Tretz methods 34613.1 General 34613.2 Interface traction link of two (or more) irreducible formsubdomains 34613.3 Interface traction link of two or more mixed form subdomains 34913.4 Interface displacement `frame' 35013.5 Linking of boundary (or Tretz)-type solution by the `frame' ofspecied displacements 35513.6 Subdomains with `standard' elements and global functions 36013.7 Lagrange variables or discontinuous Galerkin methods? 36113.8 Concluding remarks 361References 36214. Errors, recovery processes and error estimates 36514.1 Denition of errors 36514.2 Superconvergence and optimal sampling points 37014.3 Recovery of gradients and stresses 37514.4 Superconvergent patch recovery SPR 37714.5 Recovery by equilibration of patches REP 38314.6 Error estimates by recovery 38514.7 Other error estimators residual based methods 38714.8 Asymptotic behaviour and robustness of error estimators theBabus ka patch test 39214.9 Which errors should concern us? 398References 39815. Adaptive nite element renement 40115.1 Introduction 40115.2 Some examples of adaptive h-renement 40415.3 p-renement and hp-renement 41515.4 Concluding remarks 426References 42616. Point-based approximations; element-free Galerkin and othermeshless methods 42916.1 Introduction 42916.2 Function approximation 43116.3 Moving least square approximations restoration of continuityof approximation 43816.4 Hierarchical enhancement of moving least square expansions 44316.5 Point collocation nite point methods 446Contents xi16.6 Galerkin weighting and nite volume methods 45116.7 Use of hierarchic and special functions based on standard niteelements satisfying the partition of unity requirement 45716.8 Closure 464References 46417. The time dimension semi-discretization of eld and dynamic problemsand analytical solution procedures 46817.1 Introduction 46817.2 Direct formulation of time-dependent problems with spatial niteelement subdivision 46817.3 General classication 47617.4 Free response eigenvalues for second-order problems anddynamic vibration 47717.5 Free response eigenvalues for rst-order problems and heatconduction, etc. 48417.6 Free response damped dynamic eigenvalues 48417.7 Forced periodic response 48517.8 Transient response by analytical procedures 48617.9 Symmetry and repeatability 490References 49118. The time dimension discrete approximation in time 49318.1 Introduction 49318.2 Simple time-step algorithms for the rst-order equation 49518.3 General single-step algorithms for rst- and second-order equations 50818.4 Multistep recurrence algorithms 52218.5 Some remarks on general performance of numerical algorithms 53018.6 Time discontinuous Galerkin approximation 53618.7 Concluding remarks 538References 53819. Coupled systems 54219.1 Coupled problems denition and classication 54219.2 Fluidstructure interaction (Class I problem) 54519.3 Soilpore uid interaction (Class II problems) 55819.4 Partitioned single-phase systems implicitexplicit partitions(Class I problems) 56519.5 Staggered solution processes 567References 57220. Computer procedures for nite element analysis 57620.1 Introduction 57620.2 Data input module 57820.3 Memory management for array storage 58820.4 Solution module the command programming language 59020.5 Computation of nite element solution modules 597xii Contents20.6 Solution of simultaneous linear algebraic equations 60920.7 Extension and modication of computer program FEAPpv 618References 618Appendix A: Matrix algebra 620Appendix B: Tensor-indicial notation in the approximation of elasticityproblems 626Appendix C: Basic equations of displacement analysis 635Appendix D: Some integration formulae for a triangle 636Appendix E: Some integration formulae for a tetrahedron 637Appendix F: Some vector algebra 638Appendix G: Integration by parts in two and three dimensions(Green's theorem) 643Appendix H: Solutions exact at nodes 645Appendix I: Matrix diagonalization or lumping 648Author index 655Subject index 663Contents xiiiVolume 2: Solid and structural mechanics1. General problems in solid mechanics and non-linearity2. Solution of non-linear algebraic equations3. Inelastic materials4. Plate bending approximation: thin (Kirchho) plates and C1 continuity require-ments5. `Thick' ReissnerMindlin plates irreducible and mixed formulations6. Shells as an assembly of at elements7. Axisymmetric shells8. Shells as a special case of three-dimensional analysis ReissnerMindlinassumptions9. Semi-analytical nite element processes use of orthogonal functions and `nitestrip' methods10. Geometrically non-linear problems nite deformation11. Non-linear structural problems large displacement and instability12. Pseudo-rigid and rigidexible bodies13. Computer procedures for nite element analysisAppendix A: Invariants of second-order tensorsVolume 3: Fluid dynamics1. Introduction and the equations of uid dynamics2. Convection dominated problems nite element approximations3. A general algorithm for compressible and incompressible ows the characteristicbased split (CBS) algorithm4. Incompressible laminar ow newtonian and non-newtonian uids5. Free surfaces, buoyancy and turbulent incompressible ows6. Compressible high speed gas ow7. Shallow-water problems8. Waves9. Computer implementation of the CBS algorithmAppendix A. Non-conservative form of NavierStokes equationsAppendix B. Discontinuous Galerkin methods in the solution of the convectiondiusion equationAppendix C. Edge-based nite element formulationAppendix D. Multi grid methodsAppendix E. Boundary layer inviscid ow couplingPrefaceIt is just over thirty years since The Finite Element Method in Structural andContinuum Mechanics was rst published. This book, which was the rst dealingwith the nite element method, provided the base from which many further develop-ments occurred. The expanding research and eld of application of nite elements ledto the second edition in 1971, the third in 1977 and the fourth in 1989 and 1991. Thesize of each of these volumes expanded geometrically (from 272 pages in 1967 to thefourth edition of 1455 pages in two volumes). This was necessary to do justice to arapidly 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 andmany important developments are continuously occurring. The present fth edition isindeed motivated by several important developments which have occurred in the 90s.These include such subjects as adaptive error control, meshless and point basedmethods, new approaches to uid dynamics, etc. However, we feel it is importantnot to increase further the overall size of the book and we therefore have eliminatedsome redundant material.Further, the reader will notice the present subdivision into three volumes, in which therst volume provides the general basis applicable to linear problems in many elds whilstthe second and third volumes are devoted to more advanced topics in solid and uidmechanics, respectively. This arrangement will allow a general student to studyVolume 1 whilst a specialist can approach their topics with the help of Volumes 2 and3. 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, researchersand practitioners to the modern concepts of nite element methods. In Volume 1 westress the relationship between the nite element method and the more classic nitedierence and boundary solution methods. We show that all methods of numericalapproximation can be cast in the same format and that their individual advantagescan thus be retained.Although Volume 1 is not written as a course text book, it is nevertheless directed atstudents 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 littleuse is made of modern mathematical symbols to ensure wider understanding amongstengineers and physical scientists.In Volumes 1, 2 and 3 the chapters on computational methods are much reduced bytransferring the computer source programs to a web site.1This has the very substan-tial advantage of not only eliminating errors in copying the programs but also inensuring that the reader has the benet of the most recent set of programs availableto him or her at all times as it is our intention from time to time to update and expandthe available programs.The authors are particularly indebted to the International Center of NumericalMethods in Engineering (CIMNE) in Barcelona who have allowed their pre- andpost-processing code (GiD) to be accessed from the publisher's web site. Thisallows such dicult tasks as mesh generation and graphic output to be dealt witheciently. The authors are also grateful to Dr J.Z. Zhu for his careful scrutiny andhelp 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 andJackie Bettess for writing the general subject index.OCZ and RLT1Complete source code for all programs in the three volumes may be obtained at no cost from thepublisher's web page: http://www.bh.com/companions/femxvi Preface1Some preliminaries: the standarddiscrete system1.1 IntroductionThe limitations of the human mind are such that it cannot grasp the behaviour of itscomplex surroundings and creations in one operation. Thus the process of sub-dividing all systems into their individual components or `elements', whose behaviouris readily understood, and then rebuilding the original system from such componentsto study its behaviour is a natural way in which the engineer, the scientist, or even theeconomist proceeds.In many situations an adequate model is obtained using a nite number of well-dened components. We shall term such problems discrete. In others the subdivisionis continued indenitely and the problem can only be dened using the mathematicalction of an innitesimal. This leads to dierential equations or equivalent statementswhich imply an innite number of elements. We shall term such systems continuous.With the advent of digital computers, discrete problems can generally be solvedreadily even if the number of elements is very large. As the capacity of all computersis nite, continuous problems can only be solved exactly by mathematical manipula-tion. Here, the available mathematical techniques usually limit the possibilities tooversimplied situations.To overcome the intractability of realistic types of continuum problems, variousmethods of discretization have from time to time been proposed both by engineersand mathematicians. All involve an approximation which, hopefully, approachesin the limit the true continuum solution as the number of discrete variablesincreases.The discretization of continuous problems has been approached dierently bymathematicians and engineers. Mathematicians have developed general techniquesapplicable directly to dierential equations governing the problem, such as nite dif-ference approximations,1.2various weighted residual procedures,3.4or approximatetechniques for determining the stationarity of properly dened `functionals'. Theengineer, on the other hand, often approaches the problem more intuitively by creat-ing an analogy between real discrete elements and nite portions of a continuumdomain. For instance, in the eld of solid mechanics McHenry,5Hreniko,6Newmark7, and indeed Southwell9in the 1940s, showed that reasonably good solu-tions to an elastic continuum problem can be obtained by replacing small portionsof the continuum by an arrangement of simple elastic bars. Later, in the same context,Argyris8and Turner et al.9showed that a more direct, but no less intuitive, substitu-tion of properties can be made much more eectively by considering that smallportions or `elements' in a continuum behave in a simplied manner.It is from the engineering `direct analogy' view that the term `nite element' wasborn. Clough10appears to be the rst to use this term, which implies in it a directuse of a standard methodology applicable to discrete systems. Both conceptually andfrom the computational viewpoint, this is of the utmost importance. The rstallows an improved understanding to be obtained; the second oers a uniedapproach to the variety of problems and the development of standard computationalprocedures.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 topresent a view of the nite element method as a general discretization procedure of con-tinuum problems posed by mathematically dened statements.In the analysis of problems of a discrete nature, a standard methodology has beendeveloped over the years. The civil engineer, dealing with structures, rst calculatesforcedisplacement relationships for each element of the structure and then proceedsto assemble the whole by following a well-dened procedure of establishing localequilibrium at each `node' or connecting point of the structure. The resulting equa-tions can be solved for the unknown displacements. Similarly, the electrical orhydraulic 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 thesystem 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 dene a standard discrete system, and this chapterwill 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 atthis stage is advisable. As the treatment of elastic solid structures has been themost developed area of activity this will be introduced rst, followed by examplesfrom other elds, before attempting a complete generalization.The existence of a unied treatment of `standard discrete problems' leads us to therst denition 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 ofwhich is specied 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 wellas the various direct approximations used in engineering fall into this category. It isthus dicult to determine the origins of the nite element method and the precisemoment of its invention.Table 1.1 shows the process of evolution which led to the present-day concepts ofnite element analysis. Chapter 3 will give, in more detail, the mathematical basiswhich emerged from these classical ideas.11202 Some preliminaries: the standard discrete systemTable 1.1ENGINEERING MATHEMATICSTrialfunctionsFinitedifferencesVariationalmethodsRayleigh 187011Ritz 190912"WeightedresidualsGauss 179518Galerkin 191519BiezenoKoch 192320"Richardson 191015Liebman 191816Southwell 19461"StructuralanaloguesubstitutionHreniko 19416McHenry 19435Newmark 19497"Piecewisecontinuoustrial functionsCourant 194313PragerSynge 194714Zienkiewicz 196421DirectcontinuumelementsArgyris 19558Turner et al. 19569"""VariationalfinitedifferencesVarga 196217PRESENT-DAYFINITE ELEMENT METHOD1.2 The structural element and the structural systemTo introduce the reader to the general concept of discrete systems we shall rstconsider a structural engineering example of linear elasticity.Figure 1.1 represents a two-dimensional structure assembled from individualcomponents and interconnected at the nodes numbered 1 to 6. The joints at thenodes, 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 matterfrom the results of an experiment, the characteristics of each element are preciselyknown. Thus, if a typical element labelled (1) and associated with nodes 1, 2, 3 isexamined, the forces acting at the nodes are uniquely dened by the displacementsof these nodes, the distributed loading acting on the element (p), and its initialstrain. The last may be due to temperature, shrinkage, or simply an initial `lack oft'. The forces and the corresponding displacements are dened 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 matrix| we haveq1=q11q12q13

q11 = U1V1 . etc. (1.1)yyxxppY4X4V3U31 23 45 6312Nodes(1)(1)(2)(3)(4)A typical element (1)Fig. 1.1 A typical structure built up from interconnected elements.| A limited knowledge of matrix algebra will be assumed throughout this book. This is necessary forreasonable conciseness and forms a convenient book-keeping form. For readers not familiar with the subjecta brief appendix (Appendix A) is included in which sucient principles of matrix algebra are given to followthe development intelligently. Matrices (and vectors) will be distinguished by bold print throughout.4 Some preliminaries: the standard discrete systemand for the corresponding nodal displacementsa1=a1a2a3

a1 = u1v1 . etc. (1.2)Assuming linear elastic behaviour of the element, the characteristic relationship willalways be of the formq1= K1a1 f1p f10 (1.3)in which f1p represents the nodal forces required to balance any distributed loads actingon the element and f10 the nodal forces required to balance any initial strains such asmay 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 denition ofstresses or internal reactions at any specied point or points of the element interms of the nodal displacements. Dening such stresses by a matrix r1a relationshipof the formr1= Q1a1r10 (1.4)is obtained in which the two term gives the stresses due to the initial strains when nonodal displacement occurs.The matrix Keis known as the element stiness matrix and the matrix Qeas theelement 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 transmittingonly two components of force. Clearly, the same arguments and denitions willapply generally. An element (2) of the hypothetical structure will possess only twopoints 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 andof 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-dimensionalstructure the number of individual nodal components would be six. Quite generally,therefore,qe=qe1qe2FFFqem

and ae=a1a2FFFam

(1.5)with each qei and ai possessing the same number of components or degrees of freedom.These quantities are conjugate to each other.The stiness matrices of the element will clearly always be square and of the formKe=Keii Keij KeimFFFFFFFFFKemi Kemm

(1.6)The structural element and the structural system 5in which Keii, etc., are submatrices which are again square and of the size l l, where lis 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 andmodulus E in a two-dimensional problem shown in Fig. 1.2. The bar is subject to auniform lateral load p and a uniform thermal expansion strain0 = cTwhere c is the coecient of linear expansion and T is the temperature change.If the ends of the bar are dened by the coordinates xi, yi and xn, yn its length can becalculated asL =

[(xn xi)2 (yn yi)2[

and its inclination from the horizontal asu = tan1 yn yixn xiOnly two components of force and displacement have to be considered at thenodes.The nodal forces due to the lateral load are clearlyfep =UiViUnVn

p= sin ucos usin ucos u

pL2and represent the appropriate components of simple reactions, pL2. Similarly, torestrain the thermal expansion 0 an axial force (EcTA) is needed, which gives theLniyxCpVi (vi)Ui (ui)xiyiE1 A1Fig. 1.2 A pin-ended bar.6 Some preliminaries: the standard discrete systemcomponentsfe0 =UiViUnVn

0= cos usin ucos usin u

(EcTA)Finally, the element displacementsae=uiviunvn

will cause an elongation (un ui) cos u (vn vi) sin u. This, when multiplied byEAL, gives the axial force whose components can again be found. Rearrangingthese in the standard form givesKeae=UiViUnVn

cThe components of the general equation (1.3) have thus been established for theelementary case discussed. It is again quite simple to nd the stresses at any sectionof the element in the form of relation (1.4). For instance, if attention is focused onthe mid-section C of the bar the average stress determined from the axial tensionto the element can be shown to bere- o = EL[cos u. sin u. cos u. sin u[ae EcTwhere all the bending eects of the lateral load p have been ignored.For more complex elements more sophisticated procedures of analysis are requiredbut the results are of the same form. The engineer will readily recognize that the so-called `slopedeection' relations used in analysis of rigid frames are only a specialcase of the general relations.It may perhaps be remarked, in passing, that the complete stiness matrix obtainedfor the simple element in tension turns out to be symmetric (as indeed was the casewith some submatrices). This is by no means fortuitous but follows from the principleof energy conservation and from its corollary, the well-known MaxwellBettireciprocal theorem.= EALcos2u sin u cos u cos2u sin u cos usin u cos u sin2u sin u cos u sin2ucos2u sin u cos u cos2u sin u cos usin u cos u sin2u sin u cos u sin2u

uiviunvn

The structural element and the structural system 7The element properties were assumed to follow a simple linear relationship. Inprinciple, similar relationships could be established for non-linear materials, butdiscussion of such problems will be held over at this stage.The calculation of the stiness coecients of the bar which we have given here willbe found in many textbooks. Perhaps it is worthwhile mentioning here that the rstuse of bar assemblies for large structures was made as early as 1935 when Southwellproposed his classical relaxation method.221.3 Assembly and analysis of a structureConsider again the hypothetical structure of Fig. 1.1. To obtain a complete solutionthe two conditions of(a) displacement compatibility and(b) equilibriumhave to be satised throughout.Any system of nodal displacements a:a =a1FFFan

(1.7)listed now for the whole structure in which all the elements participate, automaticallysatises the rst condition.As the conditions of overall equilibrium have already been satised within an ele-ment, all that is necessary is to establish equilibrium conditions at the nodes of thestructure. The resulting equations will contain the displacements as unknowns, andonce these have been solved the structural problem is determined. The internalforces in elements, or the stresses, can easily be found by using the characteristicsestablished a priori for each element by Eq. (1.4).Consider the structure to be loaded by external forces r:r =r1FFFrn

(1.8)applied at the nodes in addition to the distributed loads applied to the individualelements. Again, any one of the forces ri must have the same number of componentsas that of the element reactions considered. In the example in questionri =

XiYi

(1.9)as the joints were assumed pinned, but at this stage the general case of an arbitrarynumber of components will be assumed.If now the equilibrium conditions of a typical node, i, are to be established, eachcomponent of ri has, in turn, to be equated to the sum of the component forcescontributed by the elements meeting at the node. Thus, considering all the force8 Some preliminaries: the standard discrete systemcomponents we haveri =me =1qei = q1i q2i (1.10)in which q1i is the force contributed to node i by element 1, q2i 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 denition (1.3) and notingthat nodal variables ai are common (thus omitting the superscript e), we haveri =

me =1Kei 1

a1

me =1Kei 2

a2 me =1fei (1.11)wherefe= fep fe0The summation again only concerns the elements which contribute to node i. If allsuch equations are assembled we have simplyKa = r f (1.12)in which the submatrices areKi j =me =1Kei jfi =me =1fei(1.13)with summations including all elements. This simple rule for assembly is veryconvenient because as soon as a coecient for a particular element is found it canbe put immediately into the appropriate `location' specied in the computer. Thisgeneral assembly process can be found to be the common and fundamental feature ofall nite element calculations and should be well understood by the reader.If dierent types of structural elements are used and are to be coupled it must beremembered that the rules of matrix summation permit this to be done only ifthese are of identical size. The individual submatrices to be added have therefore tobe 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 becoupled at that node to one which in fact is hinged, it is necessary to complete thestiness matrix of the latter by insertion of appropriate (zero) coecients in therotation or moment positions.1.4 The boundary conditionsThe system of equations resulting from Eq. (1.12) can be solved once theprescribed 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 meanThe boundary conditions 9the substitution ofa1 = a6 =

00

which is equivalent to reducing the number of equilibrium equations (in this instance12) by deleting the rst and last pairs and thus reducing the total number of unknowndisplacement components to eight. It is, nevertheless, always convenient to assemblethe equation according to relation (1.12) so as to include all the nodes.Clearly, without substitution of a minimum number of prescribed displacements toprevent rigid body movements of the structure, it is impossible to solve this system,because the displacements cannot be uniquely determined by the forces in such asituation. This physically obvious fact will be interpreted mathematically as thematrix K being singular, i.e., not possessing an inverse. The prescription of appropri-ate displacements after the assembly stage will permit a unique solution to beobtained by deleting appropriate rows and columns of the various matrices.If all the equations of a system are assembled, their form isK11a1 K12a2 = r1 f1K21a1 K22a2 = r2 f2etc.(1.14)and it will be noted that if any displacement, such as a1 = "a1, is prescribed then theexternal `force' r1 cannot be simultaneously specied and remains unknown. Therst equation could then be deleted and substitution of known values of a1 made inthe remaining equations. This process is computationally cumbersome and thesame objective is served by adding a large number, cI, to the coecient K11 andreplacing the right-hand side, r1 f1, by "a1c. If c is very much larger than otherstiness coecients this alteration eectively replaces the rst equation by the equa-tionca1 = c"a1 (1.15)that is, the required prescribed condition, but the whole system remains symmetricand minimal changes are necessary in the computation sequence. A similar procedurewill apply to any other prescribed displacement. The above artice was introduced byPayne and Irons.23An alternative procedure avoiding the assembly of equationscorresponding to nodes with prescribed boundary values will be presented inChapter 20.When all the boundary conditions are inserted the equations of the system can besolved for the unknown displacements and stresses, and the internal forces in each ele-ment obtained.1.5 Electrical and uid networksIdentical principles of deriving element characteristics and of assembly will be foundin many non-structural elds. Consider, for instance, the assembly of electricalresistances shown in Fig. 1.3.10 Some preliminaries: the standard discrete systemIf atypical resistance element, ij, is isolatedfromthe systemwe canwrite, byOhm's law,the relation between the currents entering the element at the ends and the end voltages asJei = 1re (Vi Vj)Jej = 1re (Vj Vi)or in matrix form

JeiJej

= 1re 1 11 1

ViVj

which in our standard form is simplyJe= KeVe(1.16)This form clearly corresponds to the stiness relationship (1.3); indeed if an exter-nal current were supplied along the length of the element the element `force' termscould also be found.To assemble the whole network the continuity of the potential (V) at the nodes isassumed and a current balance imposed there. If Pi now stands for the external inputof current at node i we must have, with complete analogy to Eq. (1.11),Pi =nj =1me =1Kei jVj (1.17)where the second summation is over all `elements', and once again for all the nodesP = KV (1.18)in whichKi j =me =1Kei jijPiijJj ,VjJi ,VireFig. 1.3 A network of electrical resistances.Electrical and uid networks 11Matrix notation in the above has been dropped since the quantities such as voltageand current, and hence also the coecients of the `stiness' matrix, are scalars.If the resistances were replaced by uid-carrying pipes in which a laminar regimepertained, an identical formulation would once again result, with V standing forthe hydraulic head and J for the ow.For pipe networks that are usually encountered, however, the linear laws are ingeneral not valid. Typically the owhead relationship is of a formJi = c(Vi Vj)(1.19)where the index lies between 0.5 and 0.7. Even now it would still be possible to writerelationships in the form (1.16) noting, however, that the matrices Keare no longerarrays of constants but are known functions of V. The nal equations can onceagain be assembled but their form will be non-linear and in general iterative techniquesof solution will be needed.Finally it is perhaps of interest to mention the more general form of an electricalnetwork subject to an alternating current. It is customary to write the relationshipsbetween the current and voltage in complex form with the resistance being replacedby complex impedance. Once again the standard forms of (1.16)(1.18) will beobtained but with each quantity divided into real and imaginary parts.Identical solution procedures can be used if the equality of the real and imaginaryquantities is considered at each stage. Indeed with modern digital computers it ispossible to use standard programming practice, making use of facilities availablefor dealing with complex numbers. Reference to some problems of this class will bemade in the chapter dealing with vibration problems in Chapter 17.1.6 The general patternAn example will be considered to consolidate the concepts discussed in this chapter.This is shown in Fig. 1.4(a) where ve discrete elements are interconnected. Thesemay be of structural, electrical, or any other linear type. In the solution:The rst step is the determination of element properties from the geometric materialand loading data. For each element the `stiness matrix' as well as the correspond-ing `nodal loads' are found in the form of Eq. (1.3). Each element has its own iden-tifying number and specied nodal connection. For example:element 1 connection 1 3 42 1 4 23 2 54 3 6 7 45 4 7 8 5Assuming that properties are found in global coordinates we can enter each `sti-ness' or `force' component in its position of the global matrix as shown in Fig.1.4(b), Each shaded square represents a single coecient or a submatrix of typeKij if more than one quantity is being considered at the nodes. Here the separatecontribution of each element is shown and the reader can verify the position of12 Some preliminaries: the standard discrete systemthe coecients. Note that the various types of `elements' considered here present nodiculty in specication. (All `forces', including nodal ones, are here associatedwith elements for simplicity.)The second step is the assembly of the nal equations of the type given by Eq. (1.12).This is accomplished according to the rule of Eq. (1.13) by simple addition of allnumbers in the appropriate space of the global matrix. The result is shown inFig. 1.4(c) where the non-zero coecients are indicated by shading.As the matrices are symmetric only the half above the diagonal shown needs, infact, to be found.All the non-zero coecients are conned within a band or prole which can becalculated a priori for the nodal connections. Thus in computer programs onlythe storage of the elements within the upper half of the prole is necessary, asshown in Fig. 1.4(c).The third step is the insertion of prescribed boundary conditions into the nalassembled matrix, as discussed in Sec. 1.3. This is followed by the nal step.The nal step solves the resulting equation system. Here many dierent methodscan be employed, some of which will be discussed in Chapter 20. The general1 23 4 567 812 34 51 2 3 4 5a+ + + ++ + + = +a{f } [ K]=BAND(c)(b)(a)Fig. 1.4 The general pattern.The general pattern 13subject of equation solving, though extremely important, is in general beyond thescope of this book.The nal step discussed above can be followed by substitution to obtain stresses,currents, or other desired output quantities.All operations involved in structural or other network analysis are thus of anextremely simple and repetitive kind.We can now dene the standard discrete system as one in which such conditionsprevail.1.7 The standard discrete systemIn the standard discrete system, whether it is structural or of any other kind, we ndthat:1. A set of discrete parameters, say ai, can be identied which describes simulta-neously the behaviour of each element, e, and of the whole system. We shall callthese the system parameters.2. For each element a set of quantities qei can be computed in terms of the systemparameters ai. The general function relationship can be non-linearqei = qei (a) (1.20)but in many cases a linear form exists givingqei = Kei 1a1 Kei 2a2 fei (1.21)3. The system equations are obtained by a simple additionri =me =1qei (1.22)where ri are system quantities (often prescribed as zero).In the linear case this results in a system of equationsKa f = r (1.23)such thatKi j =me =1Kei j fi =me =1fei (1.24)from which the solution for the system variables a can be found after imposingnecessary boundary conditions.The reader will observe that this denition includes the structural, hydraulic, andelectrical examples already discussed. However, it is broader. In general neitherlinearity nor symmetry of matrices need exist although in many problems thiswill arise naturally. Further, the narrowness of interconnections existing in usualelements is not essential.While much further detail could be discussed (we refer the reader to specic booksfor more exhaustive studies in the structural context2426), we feel that the generalexpose given here should suce for further study of this book.14 Some preliminaries: the standard discrete systemOnly one further matter relating to the change of discrete parameters need bementioned here. The process of so-called transformation of coordinates is vital inmany contexts and must be fully understood.1.8 Transformation of coordinatesIt is often convenient to establish the characteristics of an individual element in acoordinate system which is dierent from that in which the external forces anddisplacements of the assembled structure or system will be measured. A dierentcoordinate system may, in fact, be used for every element, to ease the computation.It is a simple matter to transform the coordinates of the displacement and forcecomponents of Eq. (1.3) to any other coordinate system. Clearly, it is necessary todo so before an assembly of the structure can be attempted.Let the local coordinate system in which the element properties have been evalu-ated be denoted by a prime sux and the common coordinate system necessary forassembly have no embellishment. The displacement components can be transformedby a suitable matrix of direction cosines L asa/ = La (1.25)As the corresponding force components must perform the same amount of work ineither system|qTa = q/ Ta/ (1.26)On inserting (1.25) we haveqTa = q/ TLaorq = LTq/ (1.27)The set of transformations given by (1.25) and (1.27) is called contravariant.To transform `stinesses' which may be available in local coordinates to globalones note that if we writeq/ = K/a/ (1.28)then by (1.27), (1.28), and (1.25)q = LTK/Laor in global coordinatesK = LTK/L (1.29)The reader can verify the usefulness of the above transformations by reworkingthe sample example of the pin-ended bar, rst establishing its stiness in its lengthcoordinates.| With ( )Tstanding for the transpose of the matrix.Transformation of coordinates 15In many complex problems an external constraint of some kind may be imagined,enforcing the requirement (1.25) with the number of degrees of freedom of a and a/being quite dierent. Even in such instances the relations (1.26) and (1.27) continueto be valid.An alternative and more general argument can be applied to many other situationsof discrete analysis. We wish to replace a set of parameters a in which the systemequations have been written by another one related to it by a transformationmatrix T asa = Tb (1.30)In the linear case the system equations are of the formKa = r f (1.31)and on the substitution we haveKTb = r f (1.32)The new system can be premultiplied simply by TT, yielding(TTKT)b = TTr TTf (1.33)which will preserve the symmetry of equations if the matrix K is symmetric. However,occasionally the matrix T is not square and expression (1.30) represents in fact anapproximation in which a larger number of parameters a is constrained. Clearly thesystem of equations (1.32) gives more equations than are necessary for a solutionof the reduced set of parameters b, and the nal expression (1.33) presents a reducedsystem which in some sense approximates the original one.We have thus introduced the basic idea of approximation, which will be the subjectof subsequent chapters where innite sets of quantities are reduced to nite sets.A historical development of the subject of nite element methods has been pre-sented by the author.27.28References1. R.V. Southwell. Relaxation Methods in Theoretical Physics. Clarendon Press, 1946.2. D.N. de G. Allen. Relaxation Methods. McGraw-Hill, 1955.3. S.H. Crandall. Engineering Analysis. McGraw-Hill, 1956.4. B.A. Finlayson. The Method of Weighted Residuals and Variational Principles. AcademicPress, 1972.5. D. McHenry. A lattice analogy for the solution of plane stress problems. J. Inst. Civ. Eng.,21, 5982, 1943.6. A. Hreniko. Solution of problems in elasticity by the framework method. J. Appl. Mech.,A8, 16975, 1941.7. N.M. Newmark. Numerical methods of analysis in bars, plates and elastic bodies, inNumerical Methods in Analysis in Engineering (ed. L.E. Grinter), Macmillan, 1949.8. J.H. Argyris. Energy Theorems and Structural Analysis. Butterworth, 1960 (reprinted fromAircraft Eng., 19545).9. M.J. Turner, R.W. Clough, H.C. Martin, and L.J. Topp. Stiness and deection analysisof complex structures. J. Aero. Sci., 23, 80523, 1956.16 Some preliminaries: the standard discrete system10. R.W. Clough. The nite element in plane stress analysis. Proc. 2nd ASCE Conf. on Electro-nic Computation. Pittsburgh, Pa., Sept. 1960.11. Lord Rayleigh (J.W. Strutt). On the theory of resonance. Trans. Roy. Soc. (London), A161,77118, 1870.12. W. Ritz. U ber eine neue Methode zur Lo sung gewissen Variations Probleme der math-ematischen Physik. J. Reine Angew. Math., 135, 161, 1909.13. R. Courant. Variational methods for the solution of problems of equilibrium and vibra-tion. Bull. Am. Math. Soc., 49, 123, 1943.14. W. Prager and J.L. Synge. Approximation in elasticity based on the concept of functionspace. Q. J. Appl. Math., 5, 24169, 1947.15. L.F. Richardson. The approximate arithmetical solution by nite dierences of physicalproblems. Trans. Roy. Soc. (London), A210, 30757, 1910.16. H. Liebman. Die angena herte Ermittlung: harmonischen, functionen und konformerAbbildung. Sitzber. Math. Physik Kl. Bayer Akad. Wiss. Munchen. 3, 6575, 1918.17. R.S. Varga. Matrix Iterative Analysis. Prentice-Hall, 1962.18. C.F. Gauss, See Carl Friedrich Gauss Werks. Vol. VII, Go ttingen, 1871.19. B.G. Galerkin. Series solution of some problems of elastic equilibrium of rods and plates'(Russian). Vestn. Inzh. Tech., 19, 897908, 1915.20. C.B. Biezeno and J.J. Koch. Over een Nieuwe Methode ter Berekening van Vlokke Platen.Ing. Grav., 38, 2536, 1923.21. O.C. Zienkiewicz and Y.K. Cheung. The nite element method for analysis of elasticisotropic and orthotropic slabs. Proc. Inst. Civ. Eng., 28, 471488, 1964.22. R.V. Southwell. Stress calculation in frame works by the method of systematic relaxationof constraints, Part I & II. Proc. Roy. Soc. London (A), 151, 5695, 1935.23. N.A. Payne and B.M. Irons, Private communication, 1963.24. R.K. Livesley. Matrix Methods in Structural Analysis. 2nd ed., Pergamon Press, 1975.25. J.S. Przemieniecki. Theory of Matrix Structural Analysis. McGraw-Hill, 1968.26. H.C. Martin. Introduction to Matrix Methods of Structural Analysis. McGraw-Hill, 1966.27. O.C. Zienkiewicz. Origins, milestones and directions of the nite element method. Arch.Comp. Methods Eng., 2, 148, 1995.28. O.C. Zienkiewicz. Origins, milestones and directions of the nite element method Apersonal view. Handbook of Numerical Analysis, IV, 365. Editors P.C. Ciarlet and J.L.Lions, North-Holland, 1996.172A direct approach to problemsin elasticity2.1 IntroductionThe process of approximating the behaviour of a continuum by `nite elements'which behave in a manner similar to the real, `discrete', elements described in theprevious chapter can be introduced through the medium of particular physical appli-cations or as a general mathematical concept. We have chosen here to follow the rstpath, narrowing our view to a set of problems associated with structural mechanicswhich historically were the rst to which the nite element method was applied. InChapter 3 we shall generalize the concepts and show that the basic ideas are widelyapplicable.In many phases of engineering the solution of stress and strain distributions inelastic continua is required. Special cases of such problems may range from two-dimensional plane stress or strain distributions, axisymmetric solids, plate bending,and shells, to fully three-dimensional solids. In all cases the number of inter-connections between any `nite element' isolated by some imaginary boundariesand the neighbouring elements is innite. It is therefore dicult to see at rstglance how such problems may be discretized in the same manner as was describedin the preceding chapter for simpler structures. The diculty can be overcome (andthe approximation made) in the following manner:1. The continuum is separated by imaginary lines or surfaces into a number of `niteelements'.2. The elements are assumed to be interconnected at a discrete number of nodalpoints situated on their boundaries and occasionally in their interior. InChapter 6 we shall show that this limitation is not necessary. The displacementsof these nodal points will be the basic unknown parameters of the problem, justas in simple, discrete, structural analysis.3. A set of functions is chosen to dene uniquely the state of displacement within each`nite element' and on its boundaries in terms of its nodal displacements.4. The displacement functions now dene uniquely the state of strain within anelement in terms of the nodal displacements. These strains, together with anyinitial strains and the constitutive properties of the material, will dene the stateof stress throughout the element and, hence, also on its boundaries.5. A system of `forces' concentrated at the nodes and equilibrating the boundarystresses and any distributed loads is determined, resulting in a stiness relationshipof the form of Eq. (1.3).Once this stage has been reached the solution procedure can follow the standard dis-crete system pattern described earlier.Clearly a series of approximations has been introduced. Firstly, it is not always easyto ensure that the chosen displacement functions will satisfy the requirement of dis-placement continuity between adjacent elements. Thus, the compatibility conditionon such lines may be violated (though within each element it is obviously satiseddue to the uniqueness of displacements implied in their continuous representation).Secondly, by concentrating the equivalent forces at the nodes, equilibrium conditionsare satised in the overall sense only. Local violation of equilibrium conditions withineach element and on its boundaries will usually arise.The choice of element shape and of the form of the displacement function forspecic cases leaves many opportunities for the ingenuity and skill of the engineerto be employed, and obviously the degree of approximation which can be achievedwill strongly depend on these factors.The approach outlined here is known as the displacement formulation.1.2So far, the process described is justied only intuitively, but what in fact has beensuggested is equivalent to the minimization of the total potential energy of the systemin terms of a prescribed displacement eld. If this displacement eld is dened in asuitable way, then convergence to the correct result must occur. The process is thenequivalent to the well-known RayleighRitz procedure. This equivalence will beproved in a later section of this chapter where also a discussion of the necessary con-vergence criteria will be presented.The recognition of the equivalence of the nite element method to a minimizationprocess was late.2.3However, Courant in 19434| and Prager and Synge5in 1947 pro-posed methods that are in essence identical.This broader basis of the nite element method allows it to be extended to other con-tinuumproblems where a variational formulation is possible. Indeed, general proceduresare nowavailable for a nite element discretization of any problemdened by a properlyconstitutedset of dierential equations. Such generalizations will be discussed in Chapter3, and throughout the book application to non-structural problems will be made. It willbe foundthat the processes describedinthis chapter are essentially anapplicationof trial-function and Galerkin-type approximations to a particular case of solid mechanics.2.2 Direct formulation of nite element characteristicsThe `prescriptions' for deriving the characteristics of a `nite element' of a continuum,which were outlined in general terms, will now be presented in more detailedmathematical form.| It appears that Courant had anticipated the essence of the nite element methodin general, andof a triangularelement in particular, as early as 1923 in a paper entitled `On a convergence principle in the calculus of varia-tions.' Ko n. Gesellschaft der Wissenschaften zuGo ttingen, Nachrichten, Berlin, 1923. He states: `We imagine amesh of triangles covering the domain . . . the convergence principles remain valid for each triangular domain.'Direct formulation of nite element characteristics 19It is desirable to obtain results in a general form applicable to any situation, butto avoid introducing conceptual diculties the general relations will be illustratedwith a very simple example of plane stress analysis of a thin slice. In this a divisionof the region into triangular-shaped elements is used as shown in Fig. 2.1. Relation-ships of general validity will be placed in a box. Again, matrix notation will beimplied.2.2.1 Displacement functionA typical nite element, e, is dened by nodes, i, j, m, etc., and straight line boundaries.Let the displacements u at any point within the element be approximated as a columnvector, u :u ~ u =kNkaek = [Ni. Nj. F F F[aiajFFF

e= Nae(2.1)in which the components of N are prescribed functions of position and aerepresents alisting of nodal displacements for a particular element.yxmijeui (Vi)ui (Ui) t = txtyFig. 2.1 A plane stress region divided into nite elements.20 A direct approach to problems in elasticityIn the case of plane stress, for instance,u =

u(x. y)v(x. y)

represents horizontal and vertical movements of a typical point within the elementandai =

uivi

the corresponding displacements of a node i.The functions Ni, Nj, Nm have to be chosen so as to give appropriate nodaldisplacements when the coordinates of the corresponding nodes are inserted inEq. (2.1). Clearly, in general,Ni(xi. yi) = I (identity matrix)whileNi(xj. yj) = Ni(xm. ym) = 0. etc.which is simply satised by suitable linear functions of x and y.If both the components of displacement are specied in an identical manner thenwe can writeNi = NiIand obtain Ni from Eq. (2.1) by noting that Ni = 1 at xi, yi but zero at othervertices.The most obvious linear function in the case of a triangle will yield the shape of Niof the form shown in Fig. 2.2. Detailed expressions for such a linear interpolation aregiven in Chapter 4, but at this stage can be readily derived by the reader.The functions N will be called shape functions and will be seen later to play a para-mount role in nite element analysis.2.2.2 StrainsWith displacements known at all points within the element the `strains' at any pointcan be determined. These will always result in a relationship that can be written inmjiyxNi1Fig. 2.2 Shape function Ni for one element.Direct formulation of nite element characteristics 21matrix notation as|e ~ e = Su(2.2)where S is a suitable linear operator. Using Eq. (2.1), the above equation can beapproximated ase ~ e = Ba(2.3)withB = SN(2.4)For the plane stress case the relevant strains of interest are those occurring in theplane and are dened in terms of the displacements by well-known relations6whichdene the operator S:e =xyxy

=0u0x0v0y0u0y 0v0x

=00x. 00. 00y00y. 00x

uv

With the shape functions Ni, Nj, and Nm already determined, the matrix B caneasily be obtained. If the linear form of these functions is adopted then, in fact, thestrains will be constant throughout the element.2.2.3 StressesIn general, the material within the element boundaries may be subjected to initialstrains such as may be due to temperature changes, shrinkage, crystal growth,and so on. If such strains are denoted by e0 then the stresses will be caused by thedierence between the actual and initial strains.In addition it is convenient to assume that at the outset of the analysis the body isstressed by some known system of initial residual stresses r0 which, for instance, couldbe measured, but the prediction of which is impossible without the full knowledge ofthe material's history. These stresses can simply be added on to the general denition.Thus, assuming general linear elastic behaviour, the relationship between stresses andstrains will be linear and of the formr = D(e e0) r0(2.5)where D is an elasticity matrix containing the appropriate material properties.| It is known that strain is a second-rank tensor by its transformation properties; however, in this bookwe will normally represent quantities using matrix (Voigt) notation. The interested reader is encouragedto consult Appendix B for the relations between tensor forms and matrix quantities.22 A direct approach to problems in elasticityAgain, for the particular case of plane stress three components of stress correspond-ing to the strains already dened have to be considered. These are, in familiar notationr =oxoytxy

and the D matrix may be simply obtained from the usual isotropic stressstrainrelationship6x (x)0 = 1E ox iE oyy (y)0 = iE ox 1E oyxy (xy)0 = 2(1 i)E txyi.e., on solving,D = E1 i21 i 0i 1 00 0 (1 i)2

2.2.4 Equivalent nodal forcesLetqe=qeiqejFFF

dene the nodal forces which are statically equivalent to the boundary stresses anddistributed body forces on the element. Each of the forces qei must contain thesame number of components as the corresponding nodal displacement ai and beordered in the appropriate, corresponding directions.The distributed body forces b are dened as those acting on a unit volume ofmaterial within the element with directions corresponding to those of the displace-ments u at that point.In the particular case of plane stress the nodal forces are, for instance,qei =

UiVi

ewith components U and V corresponding to the directions of u and v displacements,and the distributed body forces areb =

bxby

in which bx and by are the `body force' components.Direct formulation of nite element characteristics 23To make the nodal forces statically equivalent to the actual boundary stresses anddistributed body forces, the simplest procedure is to impose an arbitrary (virtual)nodal displacement and to equate the external and internal work done by the variousforces and stresses during that displacement.Let such a virtual displacement be caeat the nodes. This results, by Eqs (2.1) and(2.2), in displacements and strains within the element equal tocu = Ncaeand ce = Bcae(2.6)respectively.The work done by the nodal forces is equal to the sum of the products of the indi-vidual force components and corresponding displacements, i.e., in matrix languagecaeTqe(2.7)Similarly, the internal work per unit volume done by the stresses and distributedbody forces isceTr cuTb (2.8)or|caT(BTr NTb) (2.9)Equating the external work with the total internal work obtained by integratingover the volume of the element, Ve, we havecaeTqe= caeT

VeBTrd(vol)

VeNTb d(vol)

(2.10)As this relation is valid for any value of the virtual displacement, the multipliersmust be equal. Thusqe=

VeBTr d(vol)

VeNTb d(vol)(2.11)This statement is valid quite generally for any stressstrain relation. With the linearlaw of Eq. (2.5) we can write Eq. (2.11) asqe= Keae fe(2.12)whereKe=

V eBTDBd(vol)(2.13a)andfe=

V eNTb d(vol)

V eBTDe0 d(vol)

V eBTr0 d(vol)(2.13b)| Note that by the rules of matrix algebra for the transpose of products(AB)T= BTAT24 A direct approach to problems in elasticityIn the last equation the three terms represent forces due to body forces, initialstrain, and initial stress respectively. The relations have the characteristics of thediscrete structural elements described in Chapter 1.If the initial stress system is self-equilibrating, as must be the case with normalresidual stresses, then the forces given by the initial stress term of Eq. (2.13b) areidentically zero after assembly. Thus frequent evaluation of this force component isomitted. However, if for instance a machine part is manufactured out of a block inwhich residual stresses are present or if an excavation is made in rock whereknown tectonic stresses exist a removal of material will cause a force imbalancewhich results from the above term.For the particular example of the plane stress triangular element these characteris-tics will be obtained by appropriate substitution. It has already been noted that the Bmatrix in that example was not dependent on the coordinates; hence the integrationwill become particularly simple.The interconnection and solution of the whole assembly of elements follows thesimple structural procedures outlined in Chapter 1. In general, external concentratedforces may exist at the nodes and the matrixr =r1r2FFFrn

(2.14)will be added to the consideration of equilibrium at the nodes.A note should be added here concerning elements near the boundary. If, at theboundary, displacements are specied, no special problem arises as these can be satis-ed by specifying some of the nodal parameters a. Consider, however, the boundaryas subject to a distributed external loading, say"t per unit area. A loading term on thenodes of the element which has a boundary face Aewill now have to be added. By thevirtual work consideration, this will simply result infe=

AeNT"t d(area)(2.15)with the integration taken over the boundary area of the element. It will be notedthat "t must have the same number of components as u for the above expression tobe valid.Such a boundary element is shown again for the special case of plane stressin Fig. 2.1. An integration of this type is sometimes not carried out explicitly.Often by `physical intuition' the analyst will consider the boundary loading to berepresented simply by concentrated loads acting on the boundary nodes and calculatethese by direct static procedures. In the particular case discussed the results will beidentical.Once the nodal displacements have been determined by solution of the overall`structural' type equations, the stresses at any point of the element can be foundfrom the relations in Eqs (2.3) and (2.5), givingr = DBae De0 r0(2.16)Direct formulation of nite element characteristics 25in which the typical terms of the relationship of Eq. (1.4) will be immediatelyrecognized, the element stress matrix beingQe= DB (2.17)To this the stressesr0 = De0 and r0 (2.18)have to be added.2.2.5 Generalized nature of displacements, strains, and stressesThe meaning of displacements, strains, and stresses in the illustrative case of planestress was obvious. In many other applications, shown later in this book, this termi-nology may be applied to other, less obvious, quantities. For example, in consideringplate elements the `displacement' may be characterized by the lateral deection andthe slopes of the plate at a particular point. The `strains' will then be dened as thecurvatures of the middle surface and the `stresses' as the corresponding internalbending moments (see Volume 2).All the expressions derived here are generally valid provided the sum product ofdisplacement and corresponding load components truly represents the externalwork done, while that of the `strain' and corresponding `stress' components resultsin the total internal work.2.3 Generalization to the whole region internal nodalforce concept abandonedIn the preceding section the virtual work principle was applied to a single element andthe concept of equivalent nodal force was retained. The assembly principle thusfollowed the conventional, direct equilibrium, approach.The idea of nodal forces contributed by elements replacing the continuousinteraction of stresses between elements presents a conceptual diculty. However,it has a considerable appeal to `practical' engineers and does at times allow an inter-pretation which otherwise would not be obvious to the more rigorous mathematician.There is, however, no need to consider each element individually and the reasoning ofthe previous section may be applied directly to the whole continuum.Equation (2.1) can be interpreted as applying to the whole structure, that is,u = ~Na (2.19)in which a lists all the nodal points and~Ni = Nei (2.20)when the point concerned is within a particular element e and i is a point associatedwith that element. If point i does not occur within the element (see Fig. 2.3)~Ni = 0 (2.21)26 A direct approach to problems in elasticityMatrix "B can be similarly dened and we shall drop the bar superscript, consideringsimply that the shape functions, etc., are always dened over the whole region V.For any virtual displacement ca we can now write the sum of internal and externalwork for the whole region ascaTr =

VcuTb dV

AcuT"t dA

VceTr dV (2.22)In the above equation ca, cu, and ce can be completely arbitrary, providing theystem from a continuous displacement assumption. If for convenience we assumethey are simply variations linked by the relations (2.19) and (2.3) we obtain, on sub-stitution of the constitutive relation (2.5), a system of algebraic equationsKa f = r(2.23)whereK =

VBTDBdV(2.24a)andf =

VNTb dV

ANT"t dA

VBTDe0 dV

VBTr0 dV(2.24b)The integrals are taken over the whole volume V and over the whole surface area Aon which the tractions are given.It is immediately obvious from the above thatKi j =Kei j fi =fei(2.25)by virtue of the property of denite integrals requiring that the total be the sum of theparts:

V( ) dV =

V e( ) dV (2.26)yxiNi~Fig. 2.3. A `global' shape function "NiGeneralization to the whole region internal nodal force concept abandoned 27The same is obviously true for the surface integrals in Eq. (2.25). We thus see that the`secret' of the approximation possessing the required behaviour of a `standard dis-crete system of Chapter 1' lies simply in the requirement of writing the relationshipsin integral form.The assembly rule as well as the whole derivation has been achieved withoutinvolving the concept of `interelement forces' (i.e., qe). In the remainder of thisbook the element superscript will be dropped unless specically needed. Also nodierentiation between element and system shape functions will be made.However, an important point arises immediately. In considering the virtual workfor the whole system [Eq. (2.22)] and equating this to the sum of the elementcontributions it is implicitly assumed that no discontinuity in displacement betweenadjacent elements develops. If such a discontinuity developed, a contribution equalto the work done by the stresses in the separations would have to be added.Smoothing zoneududxd2udx2Fig. 2.4 Differentiation of a function with slope discontinuity (C0 continuous).28 A direct approach to problems in elasticityPut in other words, we require that the terms integrated in Eq. (2.26) be nite.These terms arise from the shape functions Ni used in dening the displacement u[by Eq. (2.19)] and its derivatives associated with the denition of strain [viz. Eq.(2.3)]. If, for instance, the `strains' are dened by rst derivatives of the functionsN, the displacements must be continuous. In Fig. 2.4 we see how rst derivatives ofcontinuous functions may involve a `jump' but are still nite, while second derivativesmay become innite. Such functions we call C0 continuous.In some problems the `strain' in a generalized sense may be dened by secondderivatives. In such cases we shall obviously require that both the function N andits slope (rst derivative) be continuous. Such functions are more dicult to derivebut we shall make use of them in plate and shell problems (see Volume 2). Thecontinuity involved now is called C1 continuity.2.4 Displacement approach as a minimization of totalpotential energyThe principle of virtual displacements used in the previous sections ensured satis-faction of equilibrium conditions within the limits prescribed by the assumeddisplacement pattern. Only if the virtual work equality for all, arbitrary, variationsof displacement was ensured would the equilibrium be complete.As the number of parameters of a which prescribes the displacement increases with-out limit then ever closer approximation of all equilibrium conditions can be ensured.The virtual work principle as written in Eq. (2.22) can be restated in a dierent formif the virtual quantities ca, cu, and ce are considered as variations of the real quantities.Thus, for instance, we can writec

aTr

VuTb dV

AuT"t dA

= cW (2.27)for the rst three terms of Eq. (2.22), where W is the potential energy of the externalloads. The above is certainly true if r, b, and "t are conservative (or independent ofdisplacement).The last term of Eq. (2.22) can, for elastic materials, be written ascU =

VceTr dV (2.28)where U is the `strain energy' of the system. For the elastic, linear material describedby Eq. (2.5) the reader can verify thatU = 12

VeTDe dV

VeTDe0 dV

VeTr0 dV (2.29)will, after dierentiation, yield the correct expression providing D is a symmetricmatrix. (This is indeed a necessary condition for a single-valued U to exist.)Thus instead of Eq. (2.22) we can write simplyc(U W) = c() = 0 (2.30)in which the quantity is called the total potential energy.Displacement approach as a minimization of total potential energy 29The above statement means that for equilibrium to be ensured the total potentialenergy must be stationary for variations of admissible displacements. The nite ele-ment equations derived in the previous section [Eqs (2.23)(2.25)] are simply thestatements of this variation with respect to displacements constrained to a nitenumber of parameters a and could be written as00a =00a100a2FFF

= 0 (2.31)It can be shown that in stable elastic situations the total potential energy is not onlystationary but is a minimum.7Thus the nite element process seeks such a minimumwithin the constraint of an assumed displacement pattern.The greater the degrees of freedom, the more closely will the solution approximatethe true one, ensuring complete equilibrium, providing the true displacement can, inthe limit, be represented. The necessary convergence conditions for the nite elementprocess could thus be derived. Discussion of these will, however, be deferred tosubsequent sections.It is of interest to note that if true equilibrium requires an absolute minimum of thetotal potential energy, , a nite element solution by the displacement approach willalways provide an approximate greater than the correct one. Thus a bound on thevalue of the total potential energy is always achieved.If the functional could be specied, a priori, then the nite element equationscould be derived directly by the dierentiation specied by Eq. (2.31).The well-known Rayleigh8Ritz9process of approximation frequently used inelastic analysis uses precisely this approach. The total potential energy expressionis formulated and the displacement pattern is assumed to vary with a nite set ofundetermined parameters. A set of simultaneous equations minimizing the totalpotential energy with respect to these parameters is set up. Thus the nite elementprocess as described so far can be considered to be the RayleighRitz procedure.The dierence is only in the manner in which the assumed displacements areprescribed. In the Ritz process traditionally used these are usually given byexpressions valid throughout the whole region, thus leading to simultaneousequations in which no banding occurs and the coecient matrix is full. In the niteelement process this specication is usually piecewise, each nodal parameterinuencing only adjacent elements, and thus a sparse and usually banded matrix ofcoecients is found.By its nature the conventional Ritz process is limited to relatively simple geo-metrical shapes of the total region while this limitation only occurs in nite elementanalysis in the element itself. Thus complex, realistic, congurations can be assembledfrom relatively simple element shapes.A further dierence in kind is in the usual association of the undetermined param-eter with a particular nodal displacement. This allows a simple physical interpretationinvaluable to an engineer. Doubtless much of the popularity of the nite elementprocess is due to this fact.30 A direct approach to problems in elasticity2.5 Convergence criteriaThe assumed shape functions limit the innite degrees of freedom of the system, andthe true minimum of the energy may never be reached, irrespective of the neness ofsubdivision. To ensure convergence to the correct result certain simple requirementsmust be satised. Obviously, for instance, the displacement function should be able torepresent the true displacement distribution as closely as desired. It will be found thatthis is not so if the chosen functions are such that straining is possible when theelement is subjected to rigid body displacements. Thus, the rst criterion that thedisplacement function must obey is as follows:Criterion 1. The displacement function chosen should be such that it does notpermit straining of an element to occur when the nodal displacements are causedby a rigid body motion.This self-evident condition can be violated easily if certain types of function are used;care must therefore be taken in the choice of displacement functions.A second criterion stems from similar requirements. Clearly, as elements getsmaller nearly constant strain conditions will prevail in them. If, in fact, constantstrain conditions exist, it is most desirable for good accuracy that a nite size elementis able to reproduce these exactly. It is possible to formulate functions that satisfy therst criterion but at the same time require a strain variation throughout the elementwhen the nodal displacements are compatible with a constant strain solution. Suchfunctions will, in general, not show good convergence to an accurate solution andcannot, even in the limit, represent the true strain distribution. The second criterioncan therefore be formulated as follows:Criterion 2. The displacement function has to be of such a form that if nodaldisplacements are compatible with a constant strain condition such constantstrain will in fact be obtained. (In this context again a generalized `strain' denitionis implied.)It will be observed that Criterion 2 in fact incorporates the requirement of Criterion 1,as rigid body displacements are a particular case of constant strain with a value ofzero. This criterion was rst stated by Bazeley et al.10in 1965. Strictly, both criterianeed only be satised in the limit as the size of the element t