Analysis of Structures: An Introduction Including Numerical Methods

626 Pages·2011·12.89 MB·English

by  Antony Waas

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ANALYSIS OF
STRUCTURES
P1:TIX/XYZ P2:ABC
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ANALYSIS OF
STRUCTURES
AN INTRODUCTION INCLUDING
NUMERICAL METHODS
JoeG.Eisley
AnthonyM.Waas
CollegeofEngineering
UniversityofMichigan,USA
A John Wiley & Sons, Ltd., Publication
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Thiseditionfirstpublished2011
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LibraryofCongressCataloging-in-PublicationData
Eisley,JoeG.
Analysisofstructures:anintroductionincludingnumericalmethods/JoeG.Eisley,AnthonyM.Waas.
p. cm.
Includesbibliographicalreferencesandindex.
ISBN978-0-470-97762-0(cloth)
1.Structuralanalysis(Engineering)–Mathematics. 2.Numericalanalysis. I.Waas,AnthonyM. II.Title.
TA646.W332011
624.1(cid:3)71–dc22
2011009723
AcataloguerecordforthisbookisavailablefromtheBritishLibrary.
PrintISBN:9780470977620
E-PDFISBN:9781119993285
O-bookISBN:9781119993278
E-PubISBN:9781119993544
MobiISBN:9781119993551
Typesetin9/11ptTimesbyAptaraInc.,NewDelhi,India
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Wewouldliketodedicatethisbooktoourfamilies.
ToMarilyn,PaulandSusan
—Joe
ToDayamal,Dayani,SheharaandMichael
—Tony
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Contents
AbouttheAuthors xiii
Preface xv
1 ForcesandMoments 1
1.1 Introduction 1
1.2 Units 1
1.3 ForcesinMechanicsofMaterials 3
1.4 ConcentratedForces 4
1.5 MomentofaConcentratedForce 9
1.6 DistributedForces—ForceandMomentResultants 19
1.7 InternalForcesandStresses—StressResultants 27
1.8 RestraintForcesandRestraintForceResultants 32
1.9 SummaryandConclusions 33
2 StaticEquilibrium 35
2.1 Introduction 35
2.2 FreeBodyDiagrams 35
2.3 Equilibrium—ConcentratedForces 38
2.3.1 TwoForceMembersandPinJointedTrusses 38
2.3.2 SlenderRigidBars 44
2.3.3 PulleysandCables 49
2.3.4 Springs 52
2.4 Equilibrium—DistributedForces 55
2.5 EquilibriuminThreeDimensions 59
2.6 Equilibrium—InternalForcesandStresses 62
2.6.1 EquilibriumofInternalForcesinThreeDimensions 65
2.6.2 EquilibriuminTwoDimensions—PlaneStress 69
2.6.3 EquilibriuminOneDimension—UniaxialStress 70
2.7 SummaryandConclusions 70
3 Displacement,Strain,andMaterialProperties 71
3.1 Introduction 71
3.2 DisplacementandStrain 71
3.2.1 Displacement 72
3.2.2 Strain 72
3.3 Compatibility 76
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3.4 LinearMaterialProperties 77
3.4.1 Hooke’sLawinOneDimension—Tension 77
3.4.2 Poisson’sRatio 81
3.4.3 Hooke’sLawinOneDimension—ShearinIsotropicMaterials 82
3.4.4 Hooke’sLawinTwoDimensionsforIsotropicMaterials 83
3.4.5 GeneralizedHooke’sLawforIsotropicMaterials 84
3.5 SomeSimpleSolutionsforStress,Strain,andDisplacement 85
3.6 ThermalStrain 89
3.7 EngineeringMaterials 90
3.8 FiberReinforcedCompositeLaminates 90
3.8.1 Hooke’sLawinTwoDimensionsforaFRPLamina 91
3.8.2 PropertiesofUnidirectionalLamina 94
3.9 PlanfortheFollowingChapters 96
3.10 SummaryandConclusions 98
4 ClassicalAnalysisoftheAxiallyLoadedSlenderBar 99
4.1 Introduction 99
4.2 SolutionsfromtheTheoryofElasticity 99
4.3 DerivationandSolutionoftheGoverningEquations 109
4.4 TheStaticallyDeterminateCase 116
4.5 TheStaticallyIndeterminateCase 129
4.6 VariableCrossSections 136
4.7 ThermalStressandStraininanAxiallyLoadedBar 142
4.8 ShearingStressinanAxiallyLoadedBar 143
4.9 DesignofAxiallyLoadedBars 145
4.10 AnalysisandDesignofPinJointedTrusses 149
4.11 WorkandEnergy—Castigliano’sSecondTheorem 153
4.12 SummaryandConclusions 162
5 AGeneralMethodfortheAxiallyLoadedSlenderBar 165
5.1 Introduction 165
5.2 Nodes,Elements,ShapeFunctions,andtheElementStiffnessMatrix 165
5.3 TheAssembledGlobalEquationsandTheirSolution 169
5.4 AGeneralMethod—DistributedAppliedLoads 182
5.5 VariableCrossSections 196
5.6 AnalysisandDesignofPin-jointedTrusses 202
5.7 SummaryandConclusions 211
6 Torsion 213
6.1 Introduction 213
6.2 TorsionalDisplacement,Strain,andStress 213
6.3 DerivationandSolutionoftheGoverningEquations 216
6.4 SolutionsfromtheTheoryofElasticity 225
6.5 TorsionalStressinThinWalledCrossSections 229
6.6 WorkandEnergy—TorsionalStiffnessinaThinWalledTube 231
6.7 TorsionalStressandStiffnessinMulticellSections 239
6.8 TorsionalStressandDisplacementinThinWalledOpenSections 242
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Contents ix
6.9 AGeneral(FiniteElement)Method 245
6.10 ContinuouslyVariableCrossSections 254
6.11 SummaryandConclusions 255
7 ClassicalAnalysisoftheBendingofBeams 257
7.1 Introduction 257
7.2 AreaProperties—SignConventions 257
7.2.1 AreaProperties 257
7.2.2 SignConventions 259
7.3 DerivationandSolutionoftheGoverningEquations 260
7.4 TheStaticallyDeterminateCase 271
7.5 WorkandEnergy—Castigliano’sSecondTheorem 278
7.6 TheStaticallyIndeterminateCase 281
7.7 SolutionsfromtheTheoryofElasticity 290
7.8 VariableCrossSections 300
7.9 ShearStressinNonRectangularCrossSections—ThinWalledCrossSections 302
7.10 DesignofBeams 309
7.11 LargeDisplacements 313
7.12 SummaryandConclusions 314
8 AGeneralMethod(FEM)fortheBendingofBeams 315
8.1 Introduction 315
8.2 Nodes,Elements,ShapeFunctions,andtheElementStiffnessMatrix 315
8.3 TheGlobalEquationsandtheirSolution 320
8.4 DistributedLoadsinFEM 327
8.5 VariableCrossSections 341
8.6 SummaryandConclusions 345
9 MoreaboutStressandStrain,andMaterialProperties 347
9.1 Introduction 347
9.2 TransformationofStressinTwoDimensions 347
9.3 PrincipalAxesandPrincipalStressesinTwoDimensions 350
9.4 TransformationofStraininTwoDimensions 354
9.5 StrainRosettes 356
9.6 StressTransformationandPrincipalStressesinThreeDimensions 358
9.7 AllowableandUltimateStress,andFactorsofSafety 361
9.8 Fatigue 363
9.9 Creep 364
9.10 OrthotropicMaterials—Composites 365
9.11 SummaryandConclusions 366
10 CombinedLoadingsonSlenderBars—ThinWalledCrossSections 367
10.1 Introduction 367
10.2 ReviewandSummaryofSlenderBarEquations 367
10.2.1 AxialLoading 367
10.2.2 TorsionalLoading 369
10.2.3 BendinginOnePlane 370
10.3 AxialandTorsionalLoads 372
10.4 AxialandBendingLoads—2DFrames 375
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10.5 BendinginTwoPlanes 384
10.5.1 WhenI isEqualtoZero 384
yz
10.5.2 WhenI isNotEqualtoZero 386
yz
10.6 BendingandTorsioninThinWalledOpenSections—ShearCenter 393
10.7 BendingandTorsioninThinWalledClosedSections—ShearCenter 399
10.8 StiffenedThinWalledBeams 405
10.9 SummaryandConclusions 416
11 WorkandEnergyMethods—VirtualWork 417
11.1 Introduction 417
11.2 IntroductiontothePrincipleofVirtualWork 417
11.3 StaticAnalysisofSlenderBarsbyVirtualWork 421
11.3.1 AxiallyLoading 421
11.3.2 TorsionalLoading 426
11.3.3 BeamsinBending 427
11.3.4 CombinedAxial,Torsional,andBendingBehavior 430
11.4 StaticAnalysisof3Dand2DSolidsbyVirtualWork 430
11.5 TheElementStiffnessMatrixforPlaneStress 433
11.6 TheElementStiffnessMatrixfor3DSolids 436
11.7 SummaryandConclusions 437
12 StructuralAnalysisinTwoandThreeDimensions 439
12.1 Introduction 439
12.2 TheGoverningEquationsinTwoDimensions—PlaneStress 440
12.3 FiniteElementsandtheStiffnessMatrixforPlaneStress 445
12.4 ThinFlatPlates—ClassicalAnalysis 452
12.5 ThinFlatPlates—FEMAnalysis 455
12.6 ShellStructures 459
12.7 StiffenedShellStructures 466
12.8 ThreeDimensionalStructures—ClassicalandFEMAnalysis 470
12.9 SummaryandConclusions 477
13 AnalysisofThinLaminatedCompositeMaterialStructures 479
13.1 IntroductiontoClassicalLaminationTheory 479
13.2 StrainDisplacementEquationsforLaminates 480
13.3 Stress-StrainRelationsforaSingleLamina 482
13.4 StressResultantsforLaminates 486
13.5 CLTConstitutiveDescription 489
13.6 DeterminingLaminaeStress/Strains 492
13.7 LaminatedPlatesSubjecttoTransverseLoads 493
13.8 SummaryandConclusion 498
14 Buckling 499
14.1 Introduction 499
14.2 TheEquationsforaBeamwithCombinedLateralandAxialLoading 499
14.3 BucklingofaColumn 504
14.4 TheBeamColumn 512
14.5 TheFiniteElementMethodforBendingandBuckling 515
14.6 BucklingofFrames 524
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Contents xi
14.7 BucklingofThinPlatesandOtherStructures 524
14.8 SummaryandConclusions 527
15 StructuralDynamics 529
15.1 Introduction 529
15.2 DynamicsofMass/SpringSystems 529
15.2.1 FreeMotion 529
15.2.2 ForcedMotion—Resonance 540
15.2.3 ForcedMotion—Response 547
15.3 AxialVibrationofaSlenderBar 548
15.3.1 SolutionsBasedontheDifferentialEquation 548
15.3.2 SolutionsBasedonFEM 560
15.4 TorsionalVibration 567
15.4.1 TorsionalMass/SpringSystems 567
15.4.2 DistributedTorsionalSystems 568
15.5 VibrationofBeamsinBending 569
15.5.1 SolutionsoftheDifferentialEquation 569
15.5.2 SolutionsBasedonFEM 574
15.6 TheFiniteElementMethodforallElasticStructures 577
15.7 AdditionofDamping 577
15.8 SummaryandConclusions 582
16 Evolutioninthe(Intelligent)DesignandAnalysisofStructuralMembers 583
16.1 Introduction 583
16.2 EvolutionofaTrussMember 584
16.2.1 Step1.SlenderBarAnalysis 584
16.2.2 Step2.RectangularBar—PlaneStressFEM 585
16.2.3 Step3.RectangularBarwithPinHoles—PlaneStressAnalysis 586
16.2.4 Step4.RectangularBarwithPinHoles—SolidBodyAnalysis 587
16.2.5 Step5.AddMaterialAroundtheHole—SolidElementAnalysis 588
16.2.6 Step6.BossesAdded—SolidElementAnalysis 590
16.2.7 Step7.ReducingtheWeight—SolidElementAnalysis 591
16.2.8 Step8.BucklingAnalysis 592
16.3 EvolutionofaPlatewithaHole—PlaneStress 592
16.4 MaterialsinDesign 594
16.5 SummaryandConclusions 594
A MatrixDefinitionsandOperations 595
A.1 Introduction 595
A.2 MatrixDefinitions 595
A.3 MatrixAlgebra 597
A.4 PartitionedMatrices 598
A.5 DifferentiatingandIntegratingaMatrix 598
A.6 SummaryofUsefulMatrixRelations 599
B AreaPropertiesofCrossSections 601
B.1 Introduction 601
B.2 CentroidsofCrossSections 601
B.3 AreaMomentsandProductofInertia 603
B.4 PropertiesofCommonCrossSections 609