A First Course in the Finite Element Method, 5th ed.

1000 Pages·2008·29.61 MB·English

by  Daryl L. Logan

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AFirstCourseintheFiniteElementMethod, (cid:2)c 2012CengageLearning
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A F C
IRST OURSE
F
IN THE INITE
E M d
LEMENT ETHOD
Fifth Edition
Daryl L. Logan
University of Wisconsin–Platteville
Australia • Brazil • Japan • Korea • Mexico • Singapore • Spain • United Kingdom • United States
Copyright 2010 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s).
Editorial review has deemed that any suppressed content does not materially affect the overall learning experience. Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it.
d
C
ONTENTS
Preface xi
1 Introduction 1
ChapterObjectives 1
Prologue 1
1.1 BriefHistory 2
1.2 IntroductiontoMatrixNotation 4
1.3 RoleoftheComputer 6
1.4 GeneralStepsoftheFiniteElementMethod 7
1.5 ApplicationsoftheFiniteElementMethod 15
1.6 AdvantagesoftheFiniteElementMethod 23
1.7 ComputerProgramsfortheFiniteElementMethod 25
References 27
Problems 29
2 Introduction to the Stiffness (Displacement) Method 31
ChapterObjectives 31
Introduction 31
2.1 DefinitionoftheSti¤nessMatrix 32
2.2 DerivationoftheSti¤nessMatrixforaSpringElement 32
2.3 ExampleofaSpringAssemblage 38
2.4 AssemblingtheTotalSti¤nessMatrixbySuperposition
(DirectSti¤nessMethod) 40
2.5 BoundaryConditions 42
2.6 PotentialEnergyApproachtoDeriveSpringElementEquations 56
SummaryEquations 65
References 66
Problems 66
iii
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iv d Contents
3 Development of Truss Equations 72
ChapterObjectives 72
Introduction 72
3.1 DerivationoftheSti¤nessMatrixforaBarElement
inLocalCoordinates 73
3.2 SelectingApproximationFunctionsforDisplacements 79
3.3 TransformationofVectorsinTwoDimensions 82
3.4 GlobalSti¤nessMatrixforBarArbitrarilyOrientedinthePlane 85
3.5 ComputationofStressforaBarinthex-yPlane 90
3.6 SolutionofaPlaneTruss 92
3.7 TransformationMatrixandSti¤nessMatrixforaBar
inThree-DimensionalSpace 100
3.8 UseofSymmetryinStructure 109
3.9 Inclined,orSkewed,Supports 112
3.10 PotentialEnergyApproachtoDeriveBarElementEquations 118
3.11 ComparisonofFiniteElementSolutiontoExactSolutionforBar 129
3.12 Galerkin’sResidualMethodandItsUsetoDerivetheOne-Dimensional
BarElementEquations 133
3.13 OtherResidualMethodsandTheirApplicationtoaOne-Dimensional
BarProblem 136
3.14 FlowchartforSolutionofThree-DimensionalTrussProblems 141
3.15 ComputerProgramAssistedStep-by-StepSolutionforTrussProblem 141
SummaryEquations 144
References 145
Problems 146
4 Development of Beam Equations 166
ChapterObjectives 166
Introduction 166
4.1 BeamSti¤ness 167
4.2 ExampleofAssemblageofBeamSti¤nessMatrices 177
4.3 ExamplesofBeamAnalysisUsingtheDirectSti¤nessMethod 179
4.4 DistributedLoading 192
4.5 ComparisonoftheFiniteElementSolutiontotheExactSolution
foraBeam 205
4.6 BeamElementwithNodalHinge 211
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Contents d v
4.7 PotentialEnergyApproachtoDeriveBeamElement
Equations 218
4.8 Galerkin’sMethodforDerivingBeam
ElementEquations 221
SummaryEquations 223
References 224
Problems 225
5 Frame and Grid Equations 235
ChapterObjectives 235
Introduction 235
5.1 Two-DimensionalArbitrarilyOrientedBeamElement 235
5.2 RigidPlaneFrameExamples 239
5.3 InclinedorSkewedSupports—FrameElement 258
5.4 GridEquations 259
5.5 BeamElementArbitrarilyOrientedinSpace 277
5.6 ConceptofSubstructureAnalysis 290
SummaryEquations 296
References 298
Problems 299
6 Development of the Plane Stress
and Plane Strain Stiffness Equations 328
ChapterObjectives 328
Introduction 328
6.1 BasicConceptsofPlaneStressandPlaneStrain 329
6.2 DerivationoftheConstant-StrainTriangularElement
Sti¤nessMatrixandEquations 334
6.3 TreatmentofBodyandSurfaceForces 349
6.4 ExplicitExpressionfortheConstant-Strain
TriangleSti¤nessMatrix 354
6.5 FiniteElementSolutionofaPlaneStressProblem 356
6.6 RectangularPlaneElement(BilinearRectangle,Q4) 367
SummaryEquations 373
References 376
Problems 377
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vi d Contents
7 Practical Considerations in Modeling;
Interpreting Results; and Examples
of Plane Stress–Strain Analysis 384
ChapterObjectives 384
Introduction 384
7.1 FiniteElementModeling 385
7.2 EquilibriumandCompatibilityofFiniteElementResults 398
7.3 ConvergenceofSolution 402
7.4 InterpretationofStresses 405
7.5 StaticCondensation 407
7.6 FlowchartfortheSolutionofPlaneStress–StrainProblems 411
7.7 ComputerProgram-AssistedStep-by-StepSolution,OtherModels,
andResultsforPlaneStress–StrainProblems 411
References 417
Problems 420
8 Development of the Linear-Strain Triangle Equations 437
ChapterObjectives 437
Introduction 437
8.1 DerivationoftheLinear-StrainTriangularElement
Sti¤nessMatrixandEquations 437
8.2 ExampleLSTSti¤nessDetermination 442
8.3 ComparisonofElements 445
SummaryEquations 448
References 448
Problems 449
9 Axisymmetric Elements 452
ChapterObjectives 452
Introduction 452
9.1 DerivationoftheSti¤nessMatrix 452
9.2 SolutionofanAxisymmetricPressureVessel 463
9.3 ApplicationsofAxisymmetricElements 469
SummaryEquations 474
References 476
Problems 476
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