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有限单元法：基本原理及其在土木水利机械和航天工程中的应用（精）
作者：朱伯芳 著
出版社：清华大学出版社
出版时间：2020-06-01
ISBN：9787302542650
定价：¥198.00
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内容简介
本书系统地阐述了有限单元法的基本原理及其在工程问题中的应用，包括弹性力学平面问题和空间问题、薄板、薄壳、厚板、厚壳、弹性稳定、塑性力学、大位移、断裂、动力反应、徐变、岩土力学、混凝土与钢筋混凝土、流体力学、热传导、工程反分析、仿真计算、网络自动生成、误差估计及自适应技术。该书内容丰富、取材新颖，概念清晰，可供土木、水利、机械、航空、力学专业的设计、科研人员使用，并可用做高等院校有关专业的教材。
作者简介
朱伯芳，中国工程院院士，1951年毕业于上海交通大学土木系后，参加我国首批三座大坝设计，1957年调至中国水科院，1995年当选院士，发表论文210篇出版著作10本，曾获国家自然科学奖1项，国家科技进步奖2项。国际大坝会议奖1项，部奖8项，建立了混凝土温度应力，拱坝优化等新学科。首次把有限元引入混凝土温度应力和水坝仿真计算。
目录
Preface i
About the Author iii
1 Introduction to Finite Element Method and Matrix Analysis of Truss 1
1.1 IntroductiontoFiniteElementMethod 1
1.2 TrussAnalysisOverview 5
1.3 Sti.nessMatrixofHorizontalBarElement 8
1.4 Sti.nessMatrixofInclinedBarElement 10
1.5 CoordinateTransformation 11
1.6 NodalEquilibriumEquationandGlobalSti.nessMatrix 14
1.7 TreatmentofBoundaryConditions 15 Bibliography 23
2 Plane Problems in Theory of Elasticity 25
2.1 DiscretizationofContinuousMedium 25
2.2 DisplacementFunction 28
2.3 ElementStrain 30
2.4 InitialStrain 31
2.5 ElementStress 32
2.5.1 IsotropicBody:PlaneStress 32
2.5.2 IsotropicBody:PlaneStrain 33
2.5.3 AnisotropicBody 34
2.6 EquivalentNodalForceandElementSti.nessMatrix 35
2.7 NodalLoads 40
2.7.1 EquivalentNodalLoadsofDistributedBoundaryForces 41
2.7.2 NodalLoadsofUniformVolumeForce 41
2.7.3 NodalLoadsDuetoPotentialofVolumeForce 42
2.7.4 NodalLoadsCausedbyInitialStrain 43
2.8 NodalEquilibriumEquationandGlobalSti.nessMatrix 43
2.9 EstablishtheGlobalSti.nessMatrixbytheCodingMethod 48
2.10 CalculationExample 51
2.10.1 StressConcentrationneartheCircularHole 51
2.10.2 StressAnalysisofIBeamwithaHoleinWeb 51
2.10.3 StressAnalysisoftheConcreteGravityDam 51 Bibliography 51
3 Element Analysis 53
3.1 PrincipleofVirtualDisplacement 53
3.2 ElementDisplacement 56
3.3 ElementStrainandStress 57
3.4 NodalForceandElementSti.nessMatrix 57
3.5 NodalLoad 59
3.5.1 DistributedVolumeForce 60
3.5.2 DistributedSurfaceForce 60
3.5.3 InitialStrainandInitialStress 61
3.6 ApplicationExamplesofthePrincipleofVirtualDisplacements:BeamElement 61
3.7 StrainEnergyandComplementaryStrainEnergy 64
3.8 PrincipleofMinimumPotentialEnergy 65
3.9 MinimumComplementaryEnergyPrinciple 69
3.10 HybridElement 70
3.11 HybridElementExample:PlaneRectangularElement 73
3.12 MixedEnergyPrinciple 75
3.13 CompositeElement 77 Bibliography 79
4 Global Analysis 81
4.1 NodalEquilibriumEquation 81
4.2 ApplicationofthePrincipleofMinimumPotentialEnergy 82
4.3 TheLowLimitPropertyoftheSolutionofMinimumPotentialEnergy 84
4.4 TheConvergenceofSolutions 85
4.5 AnalysisoftheSubstructure 88
4.5.1 MultipleSubstructures 89
4.5.2 CondensationoftheInternalDegreesofFreedomofSubstructures 90
4.5.3 CoordinateTransformation 90 Bibliography 91
5 High-Order Element of Plane Problem 93
5.1 RectangularElements 93
5.2 AreaCoordinates 97
5.3 High-OrderTriangularElement 100
5.3.1 6-NodeQuadraticTriangularElement 100
5.3.2 10-Node3-OrderTriangularElement 101
5.3.3 3-Node18DOFTriangularElement 102 Bibliography 104
6 Axisymmetrical Problems in Theory of Elasticity 105
6.1 StressesDuetoAxisymmetricalLoads 105
6.1.1 DisplacementFunction 105
6.1.2 ElementStrains 106
6.1.3 ElementStress 108
6.1.4 ElementSti.nessMatrix 109
6.1.5 NodalLoads 110
6.2 AntisymmetricalLoad 110 Bibliography 114
7 Spatial Problems in Theory of Elasticity 115
7.1 ConstantStrainTetrahedralElements 115
7.1.1 DisplacementFunction 115
7.1.2 ElementStrain 117
7.1.3 ElementStress 118
7.1.4 Sti.nessMatrixoftheElement 119
7.1.5 NodalLoad 120
7.2 VolumeCoordinates 121
7.3 High-OrderTetrahedralElements 122
7.3.1 10-NodeLinearStrainTetrahedralElements 122
7.3.2 20-NodeTetrahedralElement 123 Bibliography 124
8 Shape Function, Coordinate Transformation, Isoparametric Element, and In.nite Element 125
8.1 De.nitionofShapeFunctions 125
8.2 One-DimensionalShapeFunctions 126
8.3 Two-DimensionalShapeFunction 127
8.4 Three-DimensionalShapeFunction 130
8.5 CoordinateTransformation 136
8.5.1 PlaneCoordinateTransformation 142
8.5.2 SpatialCoordinateTransformation 144
8.6 DisplacementFunction 145
8.7 ElementStrain 147
8.8 Sti.nessMatrix 151
8.9 NodalLoads 153
8.10 DegradationofIsoparametricElements 155
8.10.1 Degradationof4-NodePlaneIsoparametricElements 155
8.10.2 Degradationofan8-NodeSpaceIsoparametricElement 158
8.10.3 DegradationofHigh-OrderElements 160
8.11 NumericalIntegration 161
8.11.1 One-DimensionalGaussQuadratureFormula 162
8.11.2 Two-DimensionalandThree-DimensionalGaussQuadrature Formulas 163
8.12 SelectionoftheNumericalIntegrationOrder 164
8.12.1 ConditionsforNonsingularityoftheGlobalSti.nessMatrix[K] 164
8.12.2 IntegralOrderEnsuringtheCalculationPrecision 165
8.12.3 ReducedIntegrationandSelectedIntegration 167
8.13 StressRe.nementandStressSmoothing 168
8.13.1 StressRe.nement 168
8.13.2 StressSmoothing 169
8.14 ElementalFormandLayout 173
8.14.1 E.ectoftheElementalShapeonStrain 173
8.14.2 E.ectofEdgeNodeSpacingonStrain 175
8.14.3 Intensi.cationofComputingMeshofIsoparametricElements 175
8.15 InconsistentElements 176
8.16 PatchTest 179
8.17 Triangular,Tetrahedral,andPrismaticCurved-SideElements 183
8.18 VectorComputationinIsoparametricElements 187
8.18.1 DirectionCosine 188
8.18.2 ScalarProduct 188
8.18.3 VectorProduct 188
8.18.4 In.nitesimalAreainCurvilinearCoordinateSystem 189
8.18.5 In.nitesimalAreaofSpatialCurvedSurface 190
8.18.6 SpatialIn.nitesimalVolumes 191
8.19 NumericalExamplesofIsoparametricElements 191
8.20 In.niteElements 192
8.20.1 Two-DimensionalIn.niteElements 192
8.20.2 Three-DimensionalIn.niteElements 196 Bibliography 199
9 Comparison and Application Instances of Various Planar and Spatial Elements 201
9.1 ComparisonandSelectionofVariousPlanarElements 201
9.2 ComparisonandSelectionofVariousSpatialElements 205
9.3 AnalysisofStressesinArchDam 209
9.3.1 ComparisonofDi.erentComputationMethods 210
9.3.2 TheE.ectofFoundationDeformationontheDisplacementandStressofArchDam 212
9.4 AnalysisofStressinButtressDam 215
9.5 AnalysisofSpatialE.ectofGravityDam 217
9.6 AnalysisofSpatialE.ectofEarthDam 217
9.7 AnalysisofStressonTunnelLining 220 Bibliography 221
10 Elastic Thin Plate 223
10.1 BendingofElasticThinPlate 223
10.2 RectangularThinPlateElement 228
10.2.1 DisplacementFunction 229
10.2.2 Sti.nessMatrix 231
10.2.3 NodalLoad 232
10.2.4 Example 233
10.2.4.1 SquareThinPlateSupportedbyFourEdges 233
10.2.4.2 SquareThinPlateSupportedbyCornerPoints 233
10.3 TriangularThinPlateElement 235
10.3.1 DisplacementFunction 235
10.3.2 Sti.nessMatrixandNodalLoad 238
10.3.3 SmoothingCurvature 238
10.3.4 Example 239
10.3.4.1 TheSquarePlateBearingConcentratedandDistributedLoads 239
10.3.4.2 TheDistortionoftheSquarePlate 239
10.4 PlateElementwithCurvedBoundaryandDe.ectionandRotationDe.nedRespectively 241
10.4.1 BeamElementConsideringtheShearingDeformation 241
10.4.2 CurvedPlateElementwiththeDe.ectionandRotationInterpolatedRespectively 245
10.5 ThePlateonElasticFoundation 248
10.5.1 PlateonWinklerFoundation 248
10.5.2 PlateonElasticHalfSpace 249 Bibliography 252
11 Elastic Thin Shell 255
11.1 ElementSti.nessMatrixinLocalCoordinateSystem 255
11.2 CoordinateTransformation:GlobalSti.nessMatrix 259
11.3 DirectionCosineofLocalCoordinate 261
11.4 Curved-SurfaceShellElement 264
11.5 ShellSupportedorReinforcedbyCurvedBeam 268
11.6 Example 271 Bibliography 271
12 Axisymmetric Shell 273
12.1 LinearElement 273
12.2 CurvedElement 277 Bibliography 280
13 Problems in Fluid Mechanics 281
13.1 RelationbetweenStressandStrainforNewtonianFluids 281
13.1.1 Stress–StrainRelationsforSolids 281
13.1.2 Stress–RateandStrainRelationsforFluid 282
13.2 EquationofMotion 283
13.3 ContinuityEquation 284
13.4 EnergyEquation 284
13.5 StateandViscosityEquations 284
13.6 FundamentalEquationsforSteadySeepageFlowandTheirDiscretization 285
13.6.1 GeneralizedDarcyLaw 285
13.6.2 FundamentalEquations 287
13.6.3 DiscretizationoftheProblems 287
13.7 FreeSurfaceCalculationforSeepageAnalysis 290
13.7.1 MethodofMeshRevision 290
13.7.2 MethodofRevisionoftheConductivityMatrix 290
13.7.3 ResidualVelocityMethod 291
13.7.4 InitialVelocityMethod 294
13.8 SubstitutionoftheCurtainofDrainageHolesbytheSeepingLayerforSeepageAnalysis 296
13.9 UnsteadySeepageFlow 300
13.10 DynamicWaterPressureduringEarthquake 301
13.11 InviscidFluidFlowFormulatedbyPotentialFunction Φ 303
13.11.1 BasicEquations 303
13.11.2 TheFlowaroundObjectswithoutLift 306
13.11.3 TheFlowaroundObjectswithLift 307
13.12 PotentialFlowFormulatedbyStreamFunction .. 307
13.12.1 BasicEquations 307
13.12.2 TheFlowaroundObjectswithoutLift 308
13.12.3 TheFlowaroundObjectswithLift 310
13.13 FlowontheFreeSurface 312
13.14 ViscousandNon-NewtonianFlow 316
13.14.1 SolutionoftheStokesEquation 316
13.14.2 SolutionoftheNavier–StokesEquations 317
Bibliography 318
14 Problems in Conduction of Heat in Solids 321
14.1 Di.erentialEquation:InitialandBoundaryConditionsforConductionofHeatinSolids 321
14.2 VariationalPrincipleforConductionofHeatinSolids 322
14.2.1 Euler’sEquation 322
14.2.2 VariationalPrincipleofProblemofHeatConduction 322
14.3 DiscretizationofContinuousBody 323
14.4 FundamentalEquationsforSolvingUnsteadyTemperatureFieldby
FEM 324
14.5 Two-DimensionalUnsteadyTemperatureField,TriangularElements 327
14.6 IsoparametricElements 329
14.6.1 Two-DimensionalIsoparametricElements 329
14.6.2 Three-DimensionalIsoparametricElements 331
14.7 ComputingExamplesofUnsteadyTemperatureField 331
14.8 TemperatureFieldofMassConcretewithPipeCooling 332
14.8.1 ConcreteCylinderCooledbyWaterPipe 332
14.8.2 MassConcreteCooledbyWaterPipe 334
14.8.3 MassConcreteCooledbyWaterPipewithPrecise ..(..) 334
Bibliography 335
15 Methods for Nonlinear Finite Element Analysis 337
15.1 IncrementalMethod 338
15.1.1 MethodofStartingPointSti.ness 338
15.1.2 MethodofMidpointSti.ness 339
15.2 IterativeMethod 342
15.2.1 DirectIterativeMethod 342
15.2.2 NewtonMethod 343
15.2.3 Modi.edNewtonMethod 344
15.2.4 Quasi-NewtonMethod 345
15.2.5 TheCalculationof {Ψn} andInitialStressMethodandInitialStrain Method 347
15.3 MixedMethod 349
15.4 ApplicationofSubstructureMethodinNonlinearAnalysis 349 Bibliography 351
16 Problems in Theory of Plasticity 353
16.1 One-DimensionalStress–StrainRelation 353
16.2 DecomposeofStressTensorandStressInvariant 355
16.3 Haigh–WestergaardStressSpace 357
16.3.1 GeometricCharacteristicsofStressSpace 357
16.3.1.1 TheHydrostaticStressAxis 357
16.3.1.2 .. Plane 358
16.3.1.3 Line L′ ParalleltotheLine L 358
16.3.1.4 ThePlaneParallelto .. Plane 358
16.3.2 TheGeometricExpressionofAnyPoint 358
16.3.3 PrincipalStresses 361
16.4 DecomposeofStrainTensor 362
16.5 CriterionofYield 363
16.5.1 TrescaYieldCriterion 364
16.5.2 MisesYieldCriterion 365
16.5.3 Mohr–CoulombYieldCriterion 367
16.5.4 Drucker–PragerYieldCriterion 368
16.5.5 LadeYieldCriterion 370
16.5.6 Bresler–PisterYieldCriterion 370
16.5.7 OttosenYieldCriterion 371
16.5.8 Hsieh–Ting–ChenFour-ParameterCriterion 371
16.5.9 Mohr–CoulombCriterionwiththeMaximumTensileStress 372
16.5.10 Willam–WarnkeCriterionwithThreeandFiveParameters 373
16.5.10.1 Willam–WarnkeCriterionwithThreeParameters 374
16.5.10.2 Willam–WarnkeCriterionwithFiveParameters 376
16.5.11 Zhang–LuYieldCriterion 378
16.6 StrainHardening 379
16.6.1 IsotropicStrainHardeningModel 380
16.6.2 FlowingStrainHardeningModel 381
16.6.3 MixedStrainHardeningModel 381
16.7 CriterionofLoadingandUnloading 382
16.7.1 LoadingandUnloadingofIdealPlasticMaterial 382
16.7.2 LoadingandUnloadingofStrainHardenedMaterials 382
16.7.3 StrainSoftening,BrittleFailure,andResidualStrength 383
16.8 TheFiniteElementMethodinElastic–PlasticIncrementalTheory 384
16.8.1 TheElastoplasticMatrixofIncrementalTheory 384
16.8.2 SymmetricExpressionofNonassociatedElastic–PlasticSti.nessMatrix 386 {}
..F
16.8.3 TheCalculationof 387
....
16.8.4 E.ectiveStress,E.ectivePlasticStrain,andCalculationof ..F/.... 389
16.8.4.1 TheE.ectiveStress ..i 389
16.8.4.2 TheE.ectivePlasticStrain ..i 389
16.8.4.3 TheCalculationof ..F/.... 390
16.8.5 SingularPointsontheYieldSurface 391
16.8.6 NumericalCalculationMethod 392
16.8.6.1 TheDisplacementIncrement 392
16.8.6.2 TentativeStress 392
16.8.6.3 TheScaleFactor 393
16.8.6.4 ThePlasticStressIncrement 394
16.8.6.5 StressBacktotheYieldSurface 395
16.8.6.6 CalculationSteps 396
16.8.7 Example 396
16.9 FiniteElementMethodintheFullVariableTheoryofPlasticity 397
16.9.1 BasicAssumptionofFullVariableTheory 397
16.9.2 TheStress–StrainRelationshipofYiliuxin 398
16.9.3 TheElastic–PlasticMatrixofFullVariableTheory 399
16.10 PracticalSimpli.edModelsforNonlinearProblemofMaterial 399
16.10.1 IsotropicModelContainingOne-VariableModulus E(t) 400
16.10.2 IsotropicModelContainingTwo-VariableModulus K(t)and G(t) 400
16.10.3 OrthotropicModelandtheEquivalentUniaxialStrain 401
16.10.3.1 OrthotropicConstitutiveRelations 401
16.10.3.2 EquivalentUniaxialStrain 403
16.10.4 TheApproximateCalculationofStrainSoftening 404 Bibliography 404
17 Creep of Concrete and its In.uence on Stresses and Deformations of Structures 407
17.1 Stress–StrainRelationofConcrete 407
17.1.1 Stress–StrainRelationofConcreteunderActionofStressinOne Direction 408
17.1.2 Stress–StrainRelationUnderComplexStressConditions 411
17.1.3 ModulusofElasticityofConcrete E(.. ) 413
17.1.4 UnitCreepofConcrete 414
17.1.5 FormulaforPreliminaryDesign 416
17.2 In.uenceofCreeponStressesandDeformationsofLinearElastocreepingBody 416
17.3 AnalysisofElastocreepingStressesofConcreteStructure 419
17.3.1 TheCalculationofStrainIncrementunderUniaxialStress 420
17.3.1.1 TheElasticStrainIncrement 420
17.3.1.2 TheIncrementofCreepStrainWhen C(t,..)= ..(..)[1. e.r(t . .. )] 420
∑
.rj(t . .. )]
17.3.1.3 TheIncrementofCreepStrainWhen C(t,..)= ..j(.. )[1. e422
17.3.2 TheCalculationofStrainIncrementsunderComplexStressConditions 423
17.3.3 EquilibriumEquations 423
17.4 CompoundLayerElementfortheSimulationAnalysisofConcreteDams 424 Bibliography 429
18 Stress Analysis for Viscoelastic and Visco-Plastic Bodies 431
18.1 TheStress–StrainRelationofViscoelasticBodyundertheActionofUnidirectionalStress 431
18.1.1 TheStress–StrainRelationofIdealElasticBody(HookeBody) 431
18.1.2 TheStress–StrainRelationofIdealPlasticBody:TheDashpot 431
18.1.3 MaxwellBody 431
18.1.4 KelvinBody 432
18.1.5 Standardthree-ComponentViscoelasticBody 433
18.1.6 KelvinChain 433
18.1.7 TheStress–StrainRelationWhenStressChangeswithTime 434
18.2 TheStress–StrainRelationundertheActionofComplexStresses 434
18.2.1 TheStress–StrainRelationWhenPoisson’sRatioIsConstant 434
18.2.2 Di.erentLawforVolumeDeformationandShearDeformation 435
18.3 StressAnalysisofViscoelasticBody 436
18.3.1 StressAnalysisofViscoelasticBodywithConstantPoisson’sRatio 437
18.3.2 StressAnalysisofViscoelasticBodywithDi.erentLawsforVolumeDeformationandShearDeformation 437
18.4 E.ectiveModulusMethodandEquivalentTemperatureMethodforSimpleHarmonicTemperatureCreepStressAnalysisofConcreteatLateAgesandViscoelasticBody 439
18.5 StressAnalysisforVisco-PlasticBodies 441
18.5.1 Viscoelastic–PlasticProblemsunderActionofOne-Dimensional Stress 441
18.5.2 Viscoelastic–PlasticProblemswithComplexStressStates 444
18.5.3 Visco-PlasticStrainIncrement 446
18.5.4 StressAnalysisofViscoelastic–PlasticBodies 446
18.5.5 TheChoiceofTimeInterval Δtn 448
18.6 CombinedViscoelastic–PlasticModels 449 Bibliography 451
19 Elastic Stability Problem 453
19.1 GeometricalSti.nessMatrixoftheBeamElement 453
19.2 GeometricalSti.nessMatrixofPlateElements 457
19.3 GlobalAnalysis 459
19.4 CasesofBeamSystem 461
19.5 ComputingExamplesofElasticStabilityofThinPlateSystem 462
19.5.1 RectangularThin-PlateElement 462
19.5.2 TriangularThin-PlateElements 464 Bibliography 465
20 Problems in Analysis of Structures with Large Displacement 467
20.1 TheBasicMethodforGeometricalNonlinearProblems 467
20.1.1 BasicFormulas 467
20.1.2 TheSolution 469
20.1.3 TheElasticStabilityProblem 470
20.2 ThePlateElementofLargeDe.ection 471
20.3 Three-DimensionalSolidElementofLargeDisplacement 476
20.4 DoubleNonlinearity:ElastoplasticLargeDisplacementProblem 478 Bibliography 478
21 Problems in Fracture Mechanics 481
21.1 Introduction 481
21.2 DirectMethod 484
21.2.1 DisplacementMethod 484
21.2.2 StressMethod 486
21.3 J-Integral Method 486
21.4 EnergyMethod,FlexibilityMethod,andBuecknerFormula 490
21.4.1 EnergyReleaseRate G andtheRelatedFormulas 490
21.4.2 FlexibilityMethod 491
21.4.3 EnergyMethod 492
21.4.4 BuecknerFormula 492
21.5 Sti.nessDerivativeMethod 494
21.5.1 PlaneProblem 494
21.5.2 AxialSymmetricalProblem 495
21.5.3 SpaceProblem 497
21.6 SingularElementoftheCrackTip 499
21.6.1 TriangularSingularElement 499
21.6.2 CircleSingularElement 500
21.6.3 HybridSingularElement 500
21.7 SingularIsoparametricElement(1/4LengthMidpointMethod) 502
21.7.1 RectangularSingularIsoparametricElement 502
21.7.2 TriangularDegeneratedSingularIsoparametricElement 503
21.8 BluntCrackZoneModel 506
21.9 Elastic–PlasticFracture 509
21.10 ExtendedFiniteElementMethodforFractureAnalysis 512 Bibliography 514
22 Problems in Structural Dynamics 515
22.1 EquationsofMotion 515
22.2 MassMatrix 516
22.2.1 ConsistentMassMatrix 517
22.2.2 LumpedMassMatrix 517
22.2.3 SeveralTypicalElementMassMatrices 518
22.2.3.1 BeamElement 518
22.2.3.2 PlaneConstantStrainTriangularElements 518
22.2.3.3 RectangularPlateElement 520
22.2.4 ComparisonofTwoMassMatrices 520
22.3 DampingMatrix 522
22.3.1 DampingofSingleFreedomSystem 522
22.3.2 DampingofSystemofMultidegreeofFreedom 523
22.4 NaturalFrequencyandVibrationModeofStructure 526
22.4.1 NaturalFrequencyandVibrationMode 526
22.4.2 OrthogonalityofModes 529
22.4.3 FreeVibrationEquationofStructureRepresentedbyFlexibilityMatrix 531
22.4.4 E.ectsofZeroMass 532
22.4.5 StaticCondensation 532
22.5 ModeSuperpositionMethodforAnalyzingtheStructureofForcedVibration 535
22.6 DynamicResponseofStructureundertheActionofEarthquakeSolvingbyVibrationModeSuperpositionMethod 536
22.7 VectorIterationMethodforComputingtheNaturalFrequencyandVibrationMode 538
22.7.1 InverseIterationMethod:TheCalculationofLowestFrequencyandVibrationMode 539
22.7.2 ModeClearance:CalculationofOtherFrequenciesandModes 541
22.7.3 Shifting:ToImprovetheConvergenceSpeed 544
22.7.4 PositiveIterativeMethod:CalculationoftheMaximumFrequencyandVibrationMode 545
22.8 EnergyMethodforComputingtheNaturalFrequenciesofStructure 545
22.8.1 RayleighEnergyMethod 546
22.8.2 RitzEnergyMethod 547
22.9 SubspaceIterationMethodforComputingtheNaturalFrequenciesandVibrationModesofStructure 548
22.9.1 SubspaceIterationMethod 549
22.9.2 Modi.edSubspaceIterationMethod 553
22.10 RitzVectorSuperpositionMethodforSolvingForcedVibrationofStructure 554
22.11 Modi.edRitzVectorSuperpositionMethod 556
22.12 DynamicSubstructureMethod 557
22.13 DirectIntegrationMethodforSolvingtheEquationofMotion 560
22.13.1 LinearAccelerationMethod 561
22.13.2 WilsonMethod(.. Method) 563
22.13.3 NewmarkMethod 564
22.13.4 CalculationStability,Precision,andtheSelectionofTimeStep 566
22.13.4.1 ComputationalStability 567
22.13.4.2 CalculationAccuracy 567
22.13.4.3 TheSelectionoftheTimeStep Δt 569
22.14 CoupledVibrationofSolidandFluid 570
22.15 SeismicStressofGravityDam 571
22.16 SeismicStressofButtressDam 574
22.17 VibrationofArchDam 575
22.18 SeismicStressofEarthDam 575
22.19 SeismicStressesofCylindricalShell 577
22.20 NonlinearDynamicResponsesofUndergroundStructures 578 Bibliography 580
......
About the Author iii
1 Introduction to Finite Element Method and Matrix Analysis of Truss 1
1.1 IntroductiontoFiniteElementMethod 1
1.2 TrussAnalysisOverview 5
1.3 Sti.nessMatrixofHorizontalBarElement 8
1.4 Sti.nessMatrixofInclinedBarElement 10
1.5 CoordinateTransformation 11
1.6 NodalEquilibriumEquationandGlobalSti.nessMatrix 14
1.7 TreatmentofBoundaryConditions 15 Bibliography 23
2 Plane Problems in Theory of Elasticity 25
2.1 DiscretizationofContinuousMedium 25
2.2 DisplacementFunction 28
2.3 ElementStrain 30
2.4 InitialStrain 31
2.5 ElementStress 32
2.5.1 IsotropicBody:PlaneStress 32
2.5.2 IsotropicBody:PlaneStrain 33
2.5.3 AnisotropicBody 34
2.6 EquivalentNodalForceandElementSti.nessMatrix 35
2.7 NodalLoads 40
2.7.1 EquivalentNodalLoadsofDistributedBoundaryForces 41
2.7.2 NodalLoadsofUniformVolumeForce 41
2.7.3 NodalLoadsDuetoPotentialofVolumeForce 42
2.7.4 NodalLoadsCausedbyInitialStrain 43
2.8 NodalEquilibriumEquationandGlobalSti.nessMatrix 43
2.9 EstablishtheGlobalSti.nessMatrixbytheCodingMethod 48
2.10 CalculationExample 51
2.10.1 StressConcentrationneartheCircularHole 51
2.10.2 StressAnalysisofIBeamwithaHoleinWeb 51
2.10.3 StressAnalysisoftheConcreteGravityDam 51 Bibliography 51
3 Element Analysis 53
3.1 PrincipleofVirtualDisplacement 53
3.2 ElementDisplacement 56
3.3 ElementStrainandStress 57
3.4 NodalForceandElementSti.nessMatrix 57
3.5 NodalLoad 59
3.5.1 DistributedVolumeForce 60
3.5.2 DistributedSurfaceForce 60
3.5.3 InitialStrainandInitialStress 61
3.6 ApplicationExamplesofthePrincipleofVirtualDisplacements:BeamElement 61
3.7 StrainEnergyandComplementaryStrainEnergy 64
3.8 PrincipleofMinimumPotentialEnergy 65
3.9 MinimumComplementaryEnergyPrinciple 69
3.10 HybridElement 70
3.11 HybridElementExample:PlaneRectangularElement 73
3.12 MixedEnergyPrinciple 75
3.13 CompositeElement 77 Bibliography 79
4 Global Analysis 81
4.1 NodalEquilibriumEquation 81
4.2 ApplicationofthePrincipleofMinimumPotentialEnergy 82
4.3 TheLowLimitPropertyoftheSolutionofMinimumPotentialEnergy 84
4.4 TheConvergenceofSolutions 85
4.5 AnalysisoftheSubstructure 88
4.5.1 MultipleSubstructures 89
4.5.2 CondensationoftheInternalDegreesofFreedomofSubstructures 90
4.5.3 CoordinateTransformation 90 Bibliography 91
5 High-Order Element of Plane Problem 93
5.1 RectangularElements 93
5.2 AreaCoordinates 97
5.3 High-OrderTriangularElement 100
5.3.1 6-NodeQuadraticTriangularElement 100
5.3.2 10-Node3-OrderTriangularElement 101
5.3.3 3-Node18DOFTriangularElement 102 Bibliography 104
6 Axisymmetrical Problems in Theory of Elasticity 105
6.1 StressesDuetoAxisymmetricalLoads 105
6.1.1 DisplacementFunction 105
6.1.2 ElementStrains 106
6.1.3 ElementStress 108
6.1.4 ElementSti.nessMatrix 109
6.1.5 NodalLoads 110
6.2 AntisymmetricalLoad 110 Bibliography 114
7 Spatial Problems in Theory of Elasticity 115
7.1 ConstantStrainTetrahedralElements 115
7.1.1 DisplacementFunction 115
7.1.2 ElementStrain 117
7.1.3 ElementStress 118
7.1.4 Sti.nessMatrixoftheElement 119
7.1.5 NodalLoad 120
7.2 VolumeCoordinates 121
7.3 High-OrderTetrahedralElements 122
7.3.1 10-NodeLinearStrainTetrahedralElements 122
7.3.2 20-NodeTetrahedralElement 123 Bibliography 124
8 Shape Function, Coordinate Transformation, Isoparametric Element, and In.nite Element 125
8.1 De.nitionofShapeFunctions 125
8.2 One-DimensionalShapeFunctions 126
8.3 Two-DimensionalShapeFunction 127
8.4 Three-DimensionalShapeFunction 130
8.5 CoordinateTransformation 136
8.5.1 PlaneCoordinateTransformation 142
8.5.2 SpatialCoordinateTransformation 144
8.6 DisplacementFunction 145
8.7 ElementStrain 147
8.8 Sti.nessMatrix 151
8.9 NodalLoads 153
8.10 DegradationofIsoparametricElements 155
8.10.1 Degradationof4-NodePlaneIsoparametricElements 155
8.10.2 Degradationofan8-NodeSpaceIsoparametricElement 158
8.10.3 DegradationofHigh-OrderElements 160
8.11 NumericalIntegration 161
8.11.1 One-DimensionalGaussQuadratureFormula 162
8.11.2 Two-DimensionalandThree-DimensionalGaussQuadrature Formulas 163
8.12 SelectionoftheNumericalIntegrationOrder 164
8.12.1 ConditionsforNonsingularityoftheGlobalSti.nessMatrix[K] 164
8.12.2 IntegralOrderEnsuringtheCalculationPrecision 165
8.12.3 ReducedIntegrationandSelectedIntegration 167
8.13 StressRe.nementandStressSmoothing 168
8.13.1 StressRe.nement 168
8.13.2 StressSmoothing 169
8.14 ElementalFormandLayout 173
8.14.1 E.ectoftheElementalShapeonStrain 173
8.14.2 E.ectofEdgeNodeSpacingonStrain 175
8.14.3 Intensi.cationofComputingMeshofIsoparametricElements 175
8.15 InconsistentElements 176
8.16 PatchTest 179
8.17 Triangular,Tetrahedral,andPrismaticCurved-SideElements 183
8.18 VectorComputationinIsoparametricElements 187
8.18.1 DirectionCosine 188
8.18.2 ScalarProduct 188
8.18.3 VectorProduct 188
8.18.4 In.nitesimalAreainCurvilinearCoordinateSystem 189
8.18.5 In.nitesimalAreaofSpatialCurvedSurface 190
8.18.6 SpatialIn.nitesimalVolumes 191
8.19 NumericalExamplesofIsoparametricElements 191
8.20 In.niteElements 192
8.20.1 Two-DimensionalIn.niteElements 192
8.20.2 Three-DimensionalIn.niteElements 196 Bibliography 199
9 Comparison and Application Instances of Various Planar and Spatial Elements 201
9.1 ComparisonandSelectionofVariousPlanarElements 201
9.2 ComparisonandSelectionofVariousSpatialElements 205
9.3 AnalysisofStressesinArchDam 209
9.3.1 ComparisonofDi.erentComputationMethods 210
9.3.2 TheE.ectofFoundationDeformationontheDisplacementandStressofArchDam 212
9.4 AnalysisofStressinButtressDam 215
9.5 AnalysisofSpatialE.ectofGravityDam 217
9.6 AnalysisofSpatialE.ectofEarthDam 217
9.7 AnalysisofStressonTunnelLining 220 Bibliography 221
10 Elastic Thin Plate 223
10.1 BendingofElasticThinPlate 223
10.2 RectangularThinPlateElement 228
10.2.1 DisplacementFunction 229
10.2.2 Sti.nessMatrix 231
10.2.3 NodalLoad 232
10.2.4 Example 233
10.2.4.1 SquareThinPlateSupportedbyFourEdges 233
10.2.4.2 SquareThinPlateSupportedbyCornerPoints 233
10.3 TriangularThinPlateElement 235
10.3.1 DisplacementFunction 235
10.3.2 Sti.nessMatrixandNodalLoad 238
10.3.3 SmoothingCurvature 238
10.3.4 Example 239
10.3.4.1 TheSquarePlateBearingConcentratedandDistributedLoads 239
10.3.4.2 TheDistortionoftheSquarePlate 239
10.4 PlateElementwithCurvedBoundaryandDe.ectionandRotationDe.nedRespectively 241
10.4.1 BeamElementConsideringtheShearingDeformation 241
10.4.2 CurvedPlateElementwiththeDe.ectionandRotationInterpolatedRespectively 245
10.5 ThePlateonElasticFoundation 248
10.5.1 PlateonWinklerFoundation 248
10.5.2 PlateonElasticHalfSpace 249 Bibliography 252
11 Elastic Thin Shell 255
11.1 ElementSti.nessMatrixinLocalCoordinateSystem 255
11.2 CoordinateTransformation:GlobalSti.nessMatrix 259
11.3 DirectionCosineofLocalCoordinate 261
11.4 Curved-SurfaceShellElement 264
11.5 ShellSupportedorReinforcedbyCurvedBeam 268
11.6 Example 271 Bibliography 271
12 Axisymmetric Shell 273
12.1 LinearElement 273
12.2 CurvedElement 277 Bibliography 280
13 Problems in Fluid Mechanics 281
13.1 RelationbetweenStressandStrainforNewtonianFluids 281
13.1.1 Stress–StrainRelationsforSolids 281
13.1.2 Stress–RateandStrainRelationsforFluid 282
13.2 EquationofMotion 283
13.3 ContinuityEquation 284
13.4 EnergyEquation 284
13.5 StateandViscosityEquations 284
13.6 FundamentalEquationsforSteadySeepageFlowandTheirDiscretization 285
13.6.1 GeneralizedDarcyLaw 285
13.6.2 FundamentalEquations 287
13.6.3 DiscretizationoftheProblems 287
13.7 FreeSurfaceCalculationforSeepageAnalysis 290
13.7.1 MethodofMeshRevision 290
13.7.2 MethodofRevisionoftheConductivityMatrix 290
13.7.3 ResidualVelocityMethod 291
13.7.4 InitialVelocityMethod 294
13.8 SubstitutionoftheCurtainofDrainageHolesbytheSeepingLayerforSeepageAnalysis 296
13.9 UnsteadySeepageFlow 300
13.10 DynamicWaterPressureduringEarthquake 301
13.11 InviscidFluidFlowFormulatedbyPotentialFunction Φ 303
13.11.1 BasicEquations 303
13.11.2 TheFlowaroundObjectswithoutLift 306
13.11.3 TheFlowaroundObjectswithLift 307
13.12 PotentialFlowFormulatedbyStreamFunction .. 307
13.12.1 BasicEquations 307
13.12.2 TheFlowaroundObjectswithoutLift 308
13.12.3 TheFlowaroundObjectswithLift 310
13.13 FlowontheFreeSurface 312
13.14 ViscousandNon-NewtonianFlow 316
13.14.1 SolutionoftheStokesEquation 316
13.14.2 SolutionoftheNavier–StokesEquations 317
Bibliography 318
14 Problems in Conduction of Heat in Solids 321
14.1 Di.erentialEquation:InitialandBoundaryConditionsforConductionofHeatinSolids 321
14.2 VariationalPrincipleforConductionofHeatinSolids 322
14.2.1 Euler’sEquation 322
14.2.2 VariationalPrincipleofProblemofHeatConduction 322
14.3 DiscretizationofContinuousBody 323
14.4 FundamentalEquationsforSolvingUnsteadyTemperatureFieldby
FEM 324
14.5 Two-DimensionalUnsteadyTemperatureField,TriangularElements 327
14.6 IsoparametricElements 329
14.6.1 Two-DimensionalIsoparametricElements 329
14.6.2 Three-DimensionalIsoparametricElements 331
14.7 ComputingExamplesofUnsteadyTemperatureField 331
14.8 TemperatureFieldofMassConcretewithPipeCooling 332
14.8.1 ConcreteCylinderCooledbyWaterPipe 332
14.8.2 MassConcreteCooledbyWaterPipe 334
14.8.3 MassConcreteCooledbyWaterPipewithPrecise ..(..) 334
Bibliography 335
15 Methods for Nonlinear Finite Element Analysis 337
15.1 IncrementalMethod 338
15.1.1 MethodofStartingPointSti.ness 338
15.1.2 MethodofMidpointSti.ness 339
15.2 IterativeMethod 342
15.2.1 DirectIterativeMethod 342
15.2.2 NewtonMethod 343
15.2.3 Modi.edNewtonMethod 344
15.2.4 Quasi-NewtonMethod 345
15.2.5 TheCalculationof {Ψn} andInitialStressMethodandInitialStrain Method 347
15.3 MixedMethod 349
15.4 ApplicationofSubstructureMethodinNonlinearAnalysis 349 Bibliography 351
16 Problems in Theory of Plasticity 353
16.1 One-DimensionalStress–StrainRelation 353
16.2 DecomposeofStressTensorandStressInvariant 355
16.3 Haigh–WestergaardStressSpace 357
16.3.1 GeometricCharacteristicsofStressSpace 357
16.3.1.1 TheHydrostaticStressAxis 357
16.3.1.2 .. Plane 358
16.3.1.3 Line L′ ParalleltotheLine L 358
16.3.1.4 ThePlaneParallelto .. Plane 358
16.3.2 TheGeometricExpressionofAnyPoint 358
16.3.3 PrincipalStresses 361
16.4 DecomposeofStrainTensor 362
16.5 CriterionofYield 363
16.5.1 TrescaYieldCriterion 364
16.5.2 MisesYieldCriterion 365
16.5.3 Mohr–CoulombYieldCriterion 367
16.5.4 Drucker–PragerYieldCriterion 368
16.5.5 LadeYieldCriterion 370
16.5.6 Bresler–PisterYieldCriterion 370
16.5.7 OttosenYieldCriterion 371
16.5.8 Hsieh–Ting–ChenFour-ParameterCriterion 371
16.5.9 Mohr–CoulombCriterionwiththeMaximumTensileStress 372
16.5.10 Willam–WarnkeCriterionwithThreeandFiveParameters 373
16.5.10.1 Willam–WarnkeCriterionwithThreeParameters 374
16.5.10.2 Willam–WarnkeCriterionwithFiveParameters 376
16.5.11 Zhang–LuYieldCriterion 378
16.6 StrainHardening 379
16.6.1 IsotropicStrainHardeningModel 380
16.6.2 FlowingStrainHardeningModel 381
16.6.3 MixedStrainHardeningModel 381
16.7 CriterionofLoadingandUnloading 382
16.7.1 LoadingandUnloadingofIdealPlasticMaterial 382
16.7.2 LoadingandUnloadingofStrainHardenedMaterials 382
16.7.3 StrainSoftening,BrittleFailure,andResidualStrength 383
16.8 TheFiniteElementMethodinElastic–PlasticIncrementalTheory 384
16.8.1 TheElastoplasticMatrixofIncrementalTheory 384
16.8.2 SymmetricExpressionofNonassociatedElastic–PlasticSti.nessMatrix 386 {}
..F
16.8.3 TheCalculationof 387
....
16.8.4 E.ectiveStress,E.ectivePlasticStrain,andCalculationof ..F/.... 389
16.8.4.1 TheE.ectiveStress ..i 389
16.8.4.2 TheE.ectivePlasticStrain ..i 389
16.8.4.3 TheCalculationof ..F/.... 390
16.8.5 SingularPointsontheYieldSurface 391
16.8.6 NumericalCalculationMethod 392
16.8.6.1 TheDisplacementIncrement 392
16.8.6.2 TentativeStress 392
16.8.6.3 TheScaleFactor 393
16.8.6.4 ThePlasticStressIncrement 394
16.8.6.5 StressBacktotheYieldSurface 395
16.8.6.6 CalculationSteps 396
16.8.7 Example 396
16.9 FiniteElementMethodintheFullVariableTheoryofPlasticity 397
16.9.1 BasicAssumptionofFullVariableTheory 397
16.9.2 TheStress–StrainRelationshipofYiliuxin 398
16.9.3 TheElastic–PlasticMatrixofFullVariableTheory 399
16.10 PracticalSimpli.edModelsforNonlinearProblemofMaterial 399
16.10.1 IsotropicModelContainingOne-VariableModulus E(t) 400
16.10.2 IsotropicModelContainingTwo-VariableModulus K(t)and G(t) 400
16.10.3 OrthotropicModelandtheEquivalentUniaxialStrain 401
16.10.3.1 OrthotropicConstitutiveRelations 401
16.10.3.2 EquivalentUniaxialStrain 403
16.10.4 TheApproximateCalculationofStrainSoftening 404 Bibliography 404
17 Creep of Concrete and its In.uence on Stresses and Deformations of Structures 407
17.1 Stress–StrainRelationofConcrete 407
17.1.1 Stress–StrainRelationofConcreteunderActionofStressinOne Direction 408
17.1.2 Stress–StrainRelationUnderComplexStressConditions 411
17.1.3 ModulusofElasticityofConcrete E(.. ) 413
17.1.4 UnitCreepofConcrete 414
17.1.5 FormulaforPreliminaryDesign 416
17.2 In.uenceofCreeponStressesandDeformationsofLinearElastocreepingBody 416
17.3 AnalysisofElastocreepingStressesofConcreteStructure 419
17.3.1 TheCalculationofStrainIncrementunderUniaxialStress 420
17.3.1.1 TheElasticStrainIncrement 420
17.3.1.2 TheIncrementofCreepStrainWhen C(t,..)= ..(..)[1. e.r(t . .. )] 420
∑
.rj(t . .. )]
17.3.1.3 TheIncrementofCreepStrainWhen C(t,..)= ..j(.. )[1. e422
17.3.2 TheCalculationofStrainIncrementsunderComplexStressConditions 423
17.3.3 EquilibriumEquations 423
17.4 CompoundLayerElementfortheSimulationAnalysisofConcreteDams 424 Bibliography 429
18 Stress Analysis for Viscoelastic and Visco-Plastic Bodies 431
18.1 TheStress–StrainRelationofViscoelasticBodyundertheActionofUnidirectionalStress 431
18.1.1 TheStress–StrainRelationofIdealElasticBody(HookeBody) 431
18.1.2 TheStress–StrainRelationofIdealPlasticBody:TheDashpot 431
18.1.3 MaxwellBody 431
18.1.4 KelvinBody 432
18.1.5 Standardthree-ComponentViscoelasticBody 433
18.1.6 KelvinChain 433
18.1.7 TheStress–StrainRelationWhenStressChangeswithTime 434
18.2 TheStress–StrainRelationundertheActionofComplexStresses 434
18.2.1 TheStress–StrainRelationWhenPoisson’sRatioIsConstant 434
18.2.2 Di.erentLawforVolumeDeformationandShearDeformation 435
18.3 StressAnalysisofViscoelasticBody 436
18.3.1 StressAnalysisofViscoelasticBodywithConstantPoisson’sRatio 437
18.3.2 StressAnalysisofViscoelasticBodywithDi.erentLawsforVolumeDeformationandShearDeformation 437
18.4 E.ectiveModulusMethodandEquivalentTemperatureMethodforSimpleHarmonicTemperatureCreepStressAnalysisofConcreteatLateAgesandViscoelasticBody 439
18.5 StressAnalysisforVisco-PlasticBodies 441
18.5.1 Viscoelastic–PlasticProblemsunderActionofOne-Dimensional Stress 441
18.5.2 Viscoelastic–PlasticProblemswithComplexStressStates 444
18.5.3 Visco-PlasticStrainIncrement 446
18.5.4 StressAnalysisofViscoelastic–PlasticBodies 446
18.5.5 TheChoiceofTimeInterval Δtn 448
18.6 CombinedViscoelastic–PlasticModels 449 Bibliography 451
19 Elastic Stability Problem 453
19.1 GeometricalSti.nessMatrixoftheBeamElement 453
19.2 GeometricalSti.nessMatrixofPlateElements 457
19.3 GlobalAnalysis 459
19.4 CasesofBeamSystem 461
19.5 ComputingExamplesofElasticStabilityofThinPlateSystem 462
19.5.1 RectangularThin-PlateElement 462
19.5.2 TriangularThin-PlateElements 464 Bibliography 465
20 Problems in Analysis of Structures with Large Displacement 467
20.1 TheBasicMethodforGeometricalNonlinearProblems 467
20.1.1 BasicFormulas 467
20.1.2 TheSolution 469
20.1.3 TheElasticStabilityProblem 470
20.2 ThePlateElementofLargeDe.ection 471
20.3 Three-DimensionalSolidElementofLargeDisplacement 476
20.4 DoubleNonlinearity:ElastoplasticLargeDisplacementProblem 478 Bibliography 478
21 Problems in Fracture Mechanics 481
21.1 Introduction 481
21.2 DirectMethod 484
21.2.1 DisplacementMethod 484
21.2.2 StressMethod 486
21.3 J-Integral Method 486
21.4 EnergyMethod,FlexibilityMethod,andBuecknerFormula 490
21.4.1 EnergyReleaseRate G andtheRelatedFormulas 490
21.4.2 FlexibilityMethod 491
21.4.3 EnergyMethod 492
21.4.4 BuecknerFormula 492
21.5 Sti.nessDerivativeMethod 494
21.5.1 PlaneProblem 494
21.5.2 AxialSymmetricalProblem 495
21.5.3 SpaceProblem 497
21.6 SingularElementoftheCrackTip 499
21.6.1 TriangularSingularElement 499
21.6.2 CircleSingularElement 500
21.6.3 HybridSingularElement 500
21.7 SingularIsoparametricElement(1/4LengthMidpointMethod) 502
21.7.1 RectangularSingularIsoparametricElement 502
21.7.2 TriangularDegeneratedSingularIsoparametricElement 503
21.8 BluntCrackZoneModel 506
21.9 Elastic–PlasticFracture 509
21.10 ExtendedFiniteElementMethodforFractureAnalysis 512 Bibliography 514
22 Problems in Structural Dynamics 515
22.1 EquationsofMotion 515
22.2 MassMatrix 516
22.2.1 ConsistentMassMatrix 517
22.2.2 LumpedMassMatrix 517
22.2.3 SeveralTypicalElementMassMatrices 518
22.2.3.1 BeamElement 518
22.2.3.2 PlaneConstantStrainTriangularElements 518
22.2.3.3 RectangularPlateElement 520
22.2.4 ComparisonofTwoMassMatrices 520
22.3 DampingMatrix 522
22.3.1 DampingofSingleFreedomSystem 522
22.3.2 DampingofSystemofMultidegreeofFreedom 523
22.4 NaturalFrequencyandVibrationModeofStructure 526
22.4.1 NaturalFrequencyandVibrationMode 526
22.4.2 OrthogonalityofModes 529
22.4.3 FreeVibrationEquationofStructureRepresentedbyFlexibilityMatrix 531
22.4.4 E.ectsofZeroMass 532
22.4.5 StaticCondensation 532
22.5 ModeSuperpositionMethodforAnalyzingtheStructureofForcedVibration 535
22.6 DynamicResponseofStructureundertheActionofEarthquakeSolvingbyVibrationModeSuperpositionMethod 536
22.7 VectorIterationMethodforComputingtheNaturalFrequencyandVibrationMode 538
22.7.1 InverseIterationMethod:TheCalculationofLowestFrequencyandVibrationMode 539
22.7.2 ModeClearance:CalculationofOtherFrequenciesandModes 541
22.7.3 Shifting:ToImprovetheConvergenceSpeed 544
22.7.4 PositiveIterativeMethod:CalculationoftheMaximumFrequencyandVibrationMode 545
22.8 EnergyMethodforComputingtheNaturalFrequenciesofStructure 545
22.8.1 RayleighEnergyMethod 546
22.8.2 RitzEnergyMethod 547
22.9 SubspaceIterationMethodforComputingtheNaturalFrequenciesandVibrationModesofStructure 548
22.9.1 SubspaceIterationMethod 549
22.9.2 Modi.edSubspaceIterationMethod 553
22.10 RitzVectorSuperpositionMethodforSolvingForcedVibrationofStructure 554
22.11 Modi.edRitzVectorSuperpositionMethod 556
22.12 DynamicSubstructureMethod 557
22.13 DirectIntegrationMethodforSolvingtheEquationofMotion 560
22.13.1 LinearAccelerationMethod 561
22.13.2 WilsonMethod(.. Method) 563
22.13.3 NewmarkMethod 564
22.13.4 CalculationStability,Precision,andtheSelectionofTimeStep 566
22.13.4.1 ComputationalStability 567
22.13.4.2 CalculationAccuracy 567
22.13.4.3 TheSelectionoftheTimeStep Δt 569
22.14 CoupledVibrationofSolidandFluid 570
22.15 SeismicStressofGravityDam 571
22.16 SeismicStressofButtressDam 574
22.17 VibrationofArchDam 575
22.18 SeismicStressofEarthDam 575
22.19 SeismicStressesofCylindricalShell 577
22.20 NonlinearDynamicResponsesofUndergroundStructures 578 Bibliography 580
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