Introduction to Finite Element Analysis Using MATLAB® and Abaqus

487 Pages·2013·13.97 MB·English

by  Khennane, Amar

Mathematics Khennane
Introduction to
Introduction to
Finite Element Analysis Using MATLAB® and Abaqus
Finite Element Analysis
“A very good introduction to the finite element method with a balanced treatment of
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theory and implementation.” n
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— F. Albermani, Reader in Structural Engineering, Using MATLAB® and
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The University of Queensland, Australia o
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There are some books that target the theory of the finite element, while others focus s
i c Abaqus
on the programming side of things. Introduction to Finite Element Analysis Using n t
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MATLAB® and Abaqus accomplishes both. This book teaches the first principles of o

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the finite element method. It presents the theory of the finite element method while M

maintaining a balance between its mathematical formulation, programming implemen- t
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tation, and application using commercial software. The computer implementation is T
carried out using MATLAB, while the practical applications are carried out in both L F
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MATLAB and Abaqus. MATLAB is a high-level language specially designed for dealing
with matrices, making it particularly suited for programming the finite element meth- B it
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®
od, while Abaqus is a suite of commercial finite element software.
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e
Introduction to Finite Element Analysis Using MATLAB® and Abaqus introduces d
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and explains theory in each chapter, and provides corresponding examples. It offers
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introductory notes and provides matrix structural analysis for trusses, beams, and
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frames. The book examines the theories of stress and strain and the relationships be- a t

tween them. The author then covers weighted residual methods and finite element ap- q A
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proximation and numerical integration. He presents the finite element formulation for n
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plane stress/strain problems, introduces axisymmetric problems, and highlights the a
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theory of plates. The text supplies step-by-step procedures for solving problems with y
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Abaqus interactive and keyword editions. The described procedures are implemented
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as MATLAB codes, and Abaqus files can be found on the CRC Press website.
Amar Khennane
K16894
6000 Broken Sound Parkway, NW
Suite 300, Boca Raton, FL 33487
711 Third Avenue
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an informa business
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Introduction to
Finite Element Analysis
Using MATLAB® and
Abaqus
©2013byTaylor&FrancisGroup,LLC
© 2013 by Taylor & Francis Group, LLC
Introduction to
Finite Element Analysis
Using MATLAB® and
Abaqus
Amar Khennane
Boca Raton London New York
CRC Press is an imprint of the
Taylor & Francis Group, an informa business
©2013byTaylor&FrancisGroup,LLC
MATLAB® is a trademark of The MathWorks, Inc. and is used with permission. The MathWorks does not warrant the
accuracy of the text or exercises in this book. This book’s use or discussion of MATLAB® software or related products
does not constitute endorsement or sponsorship by The MathWorks of a particular pedagogical approach or particular
use of the MATLAB® software.
CRC Press
Taylor & Francis Group
6000 Broken Sound Parkway NW, Suite 300
Boca Raton, FL 33487-2742
© 2013 by Taylor & Francis Group, LLC
CRC Press is an imprint of Taylor & Francis Group, an Informa business
No claim to original U.S. Government works
Version Date: 20130220
International Standard Book Number-13: 978-1-4665-8021-3 (eBook - PDF)
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Contents
ListofFigures ........................................................................................ xiii
ListofTables .........................................................................................xxv
Preface .............................................................................................. xxvii
Author................................................................................................ xxix
Chapter1 Introduction...............................................................................1
1.1 Prologue ............................................................................1
1.2 FiniteElementAnalysisandtheUser.............................................1
1.3 AimoftheBook....................................................................2
1.4 BookOrganization .................................................................2
Chapter2 BarElement...............................................................................5
2.1 Introduction.........................................................................5
2.2 One-DimensionalTrussElement..................................................5
2.2.1 FormulationoftheStiffnessMatrix:TheDirectApproach .............5
2.2.2 Two-DimensionalTrussElement.........................................7
2.3 GlobalStiffnessMatrixAssembly.................................................9
2.3.1 Discretization..............................................................9
2.3.2 Elements’StiffnessMatricesinLocalCoordinates......................9
2.3.3 Elements’StiffnessMatricesinGlobalCoordinates................... 10
2.3.3.1 Element1 ..................................................... 11
2.3.3.2 Element2 ..................................................... 11
2.3.3.3 Element3 ..................................................... 12
2.3.4 GlobalMatrixAssembly................................................ 12
2.3.4.1 OnlyElement1IsPresent.................................... 13
2.3.4.2 OnlyElement2IsPresent.................................... 13
2.3.4.3 OnlyElement3IsPresent.................................... 13
2.3.5 GlobalForceVectorAssembly......................................... 14
2.4 BoundaryConditions............................................................. 15
2.4.1 GeneralCase............................................................. 15
2.5 SolutionoftheSystemofEquations ............................................ 16
2.6 SupportReactions................................................................ 17
2.7 Members’Forces................................................................. 18
2.8 ComputerCode:truss.m ......................................................... 19
2.8.1 DataPreparation......................................................... 20
2.8.1.1 NodesCoordinates............................................ 20
2.8.1.2 ElementConnectivity......................................... 20
2.8.1.3 MaterialandGeometricalProperties ........................ 20
2.8.1.4 BoundaryConditions......................................... 20
2.8.1.5 Loading ....................................................... 21
2.8.2 ElementMatrices........................................................ 21
2.8.2.1 StiffnessMatrixinLocalCoordinates....................... 21
2.8.2.2 TransformationMatrix ....................................... 22
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vi Contents
2.8.2.3 StiffnessMatrixinGlobalCoordinates...................... 22
2.8.2.4 “Steering”Vector ............................................ 22
2.8.3 AssemblyoftheGlobalStiffnessMatrix............................... 23
2.8.4 AssemblyoftheGlobalForceVector.................................. 23
2.8.5 SolutionoftheGlobalSystemofEquations........................... 23
2.8.6 NodalDisplacements.................................................... 23
2.8.7 ElementForces .......................................................... 23
2.8.8 ProgramScripts.......................................................... 24
2.9 Problems.......................................................................... 27
2.9.1 Problem2.1.............................................................. 27
2.9.2 Problem2.2.............................................................. 32
2.10 AnalysisofaSimpleTrusswithAbaqus........................................ 35
2.10.1 OverviewofAbaqus .................................................... 35
2.10.2 AnalysisofaTrusswithAbaqusInteractiveEdition.................. 36
2.10.2.1 Modeling...................................................... 36
2.10.2.2 Analysis....................................................... 51
2.10.3 AnalysisofaTrusswithAbaqusKeywordEdition.................... 57
Chapter3 BeamElement .......................................................................... 63
3.1 Introduction....................................................................... 63
3.2 StiffnessMatrix................................................................... 63
3.3 UniformlyDistributedLoading.................................................. 67
3.4 InternalHinge .................................................................... 71
3.5 ComputerCode:beam.m......................................................... 73
3.5.1 DataPreparation......................................................... 73
3.5.1.1 NodesCoordinates............................................ 73
3.5.1.2 ElementConnectivity......................................... 74
3.5.1.3 MaterialandGeometricalProperties ........................ 74
3.5.1.4 BoundaryConditions......................................... 74
3.5.1.5 InternalHinges................................................ 74
3.5.1.6 Loading ....................................................... 75
3.5.1.7 StiffnessMatrix............................................... 76
3.5.2 AssemblyandSolutionoftheGlobalSystem
ofEquations ............................................................. 76
3.5.3 NodalDisplacements.................................................... 76
3.5.4 ElementForces .......................................................... 77
3.6 Problems.......................................................................... 81
3.6.1 Problem3.1.............................................................. 81
3.6.2 Problem3.2.............................................................. 84
3.6.3 Problem3.3.............................................................. 87
3.7 AnalysisofaSimpleBeamwithAbaqus ....................................... 90
3.7.1 InteractiveEdition....................................................... 90
3.7.2 AnalysisofaBeamwithAbaqusKeywordEdition ..................103
Chapter4 RigidJointedFrames..................................................................107
4.1 Introduction......................................................................107
4.2 StiffnessMatrixofaBeam–ColumnElement .................................107
4.3 StiffnessMatrixofaBeam–ColumnElementinthePresence
ofHingedEnd...................................................................107
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Contents vii
4.4 GlobalandLocalCoordinateSystems .........................................108
4.5 GlobalStiffnessMatrixAssemblyandSolutionforUnknown
Displacements...................................................................109
4.6 ComputerCode:frame.m.......................................................109
4.6.1 DataPreparation........................................................109
4.6.1.1 NodesCoordinates...........................................110
4.6.1.2 ElementConnectivity........................................110
4.6.1.3 MaterialandGeometricalProperties .......................110
4.6.1.4 BoundaryConditions........................................110
4.6.1.5 InternalHinges...............................................111
4.6.1.6 Loading ......................................................111
4.6.2 ElementMatrices.......................................................112
4.6.2.1 StiffnessMatrixinLocalCoordinates......................112
4.6.2.2 TransformationMatrix ......................................113
4.6.2.3 StiffnessMatrixinGlobalCoordinates.....................113
4.6.2.4 “Steering”Vector............................................113
4.6.2.5 ElementLoads...............................................113
4.6.3 AssemblyoftheGlobalStiffnessMatrix..............................113
4.6.4 SolutionoftheGlobalSystemofEquations..........................114
4.6.5 NodalDisplacements...................................................114
4.6.6 ElementForces .........................................................114
4.7 AnalysisofaSimpleFramewithAbaqus......................................124
4.7.1 InteractiveEdition......................................................124
4.7.2 KeywordEdition........................................................132
Chapter5 StressandStrainAnalysis.............................................................135
5.1 Introduction......................................................................135
5.2 StressTensor.....................................................................135
5.2.1 Definition ...............................................................135
5.2.2 StressTensor–StressVectorRelationships............................137
5.2.3 TransformationoftheStressTensor...................................139
5.2.4 EquilibriumEquations .................................................139
5.2.5 PrincipalStresses.......................................................140
5.2.6 vonMisesStress........................................................141
5.2.7 NormalandTangentialComponentsoftheStress
Vector...................................................................141
5.2.8 Mohr’sCirclesforStress...............................................143
5.2.9 EngineeringRepresentationofStress .................................144
5.3 DeformationandStrain .........................................................144
5.3.1 Definition ...............................................................144
5.3.2 LagrangianandEulerianDescriptions.................................145
5.3.3 Displacement ...........................................................146
5.3.4 DisplacementandDeformationGradients ............................147
5.3.5 GreenLagrangeStrainMatrix.........................................148
5.3.6 SmallDeformationTheory.............................................149
5.3.6.1 InfinitesimalStrain ..........................................149
5.3.6.2 GeometricalInterpretationoftheTermsoftheStrain
Tensor ........................................................150
5.3.6.3 CompatibilityConditions ...................................152
5.3.7 PrincipalStrains ........................................................152
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viii Contents
5.3.8 TransformationoftheStrainTensor...................................153
5.3.9 EngineeringRepresentationofStrain .................................153
5.4 Stress–StrainConstitutiveRelations............................................154
5.4.1 GeneralizedHooke’sLaw..............................................154
5.4.2 MaterialSymmetries ...................................................155
5.4.2.1 SymmetrywithrespecttoaPlane ..........................155
5.4.2.2 SymmetrywithrespecttoThreeOrthogonalPlanes.......157
5.4.2.3 SymmetryofRotationwithrespecttoOneAxis...........157
5.4.3 IsotropicMaterial.......................................................158
5.4.3.1 ModulusofElasticity........................................160
5.4.3.2 Poisson’sRatio...............................................160
5.4.3.3 ShearModulus...............................................160
5.4.3.4 BulkModulus................................................160
5.4.4 PlaneStressandPlaneStrain ..........................................162
5.5 SolvedProblems.................................................................163
5.5.1 Problem5.1.............................................................163
5.5.2 Problem5.2.............................................................164
5.5.3 Problem5.3.............................................................167
5.5.4 Problem5.4.............................................................168
5.5.5 Problem5.5.............................................................170
5.5.6 Problem5.6.............................................................171
5.5.7 Problem5.7.............................................................172
5.5.8 Problem5.8.............................................................174
Chapter6 WeightedResidualMethods ..........................................................175
6.1 Introduction......................................................................175
6.2 GeneralFormulation ............................................................175
6.3 GalerkinMethod ................................................................176
6.4 WeakForm.......................................................................178
6.5 IntegratingbyPartoverTwoandThreeDimensions(GreenTheorem)......179
6.6 RayleighRitzMethod...........................................................183
6.6.1 Definition ...............................................................183
6.6.2 FunctionalAssociatedwithanIntegralForm.........................183
6.6.3 RayleighRitzMethod..................................................183
6.6.4 ExampleofaNaturalFunctional ......................................185
Chapter7 FiniteElementApproximation........................................................191
7.1 Introduction......................................................................191
7.2 GeneralandNodalApproximations............................................191
7.3 FiniteElementApproximation .................................................193
7.4 BasicPrinciplesfortheConstructionofTrialFunctions......................195
7.4.1 CompatibilityPrinciple.................................................195
7.4.2 CompletenessPrinciple ................................................196
7.5 Two-DimensionalFiniteElementApproximation ............................197
7.5.1 PlaneLinearTriangularElementforC0Problems....................197
7.5.1.1 ShapeFunctions .............................................197
7.5.1.2 ReferenceElement...........................................199
7.5.1.3 AreaCoordinates ............................................202
7.5.2 LinearQuadrilateralElementforC0Problems .......................203
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