Thermal Stresses

508 Pages·2018·7.011 MB·English

by  Bill B. Lund, Naotake Noda, Yoshinobu Tanigawa

THERMAL STRESSES

THERMAL STRESSES
2nd Edition
Naotake Noda
DepartmentofMechanicalEngineering
ShizuokaUniversity
Hamamatsu,Japan
Richard B. Hetnarski
DepartmentofMechanicalEngineering
RochesterInstituteofTechnology
Rochester,NewYork,USA
Yoshinobu Tanigawa
DepartmentofMechanicalSystemsEngineering
OsakaPrefectureUniversity
Sakai,Japan
New York London
•
DeniseT.Schanck,VicePresident
RobertH.Bedford,Editor
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Publishedin2003by
Taylor&Francis
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PublishedinGreatBritainby
Taylor&Francis
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Copyright©2003byTaylor&Francis.
PrintedintheUnitedStatesofAmericaonacid-freepaper.
Allrightsreserved.Nopartofthisbookmaybereprintedorreproducedorutilizedinanyformorby
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10 9 8 7 6 5 4 3 2 1
LibraryofCongressCataloging-in-PublicationData
Noda,Naotake.
Thermalstresses/editedbyNaotakeNoda,RichardB.Hetnarski,&YoshinobuTanigawa.–2nded.
p. cm.
Includesbibliographicalreferencesandindex.
ISBN1-56032-971-8
1.Thermalstresses. I.Hetnarski,RichardB. II.Tanigawa,Yoshinobu. III.Title.
TA418.58.N632002
620.1(cid:1)121—dc21
2002066209
CONTENTS
PrefacetotheSecondEdition xi
PrefacetotheFirstEdition xiii
1 ThermalStressesinBars 1
1.1 StressandStrain 1
1.1.1 StressandStrain 1
1.1.2 Hooke’sLaw 3
1.1.3 FreeThermalExpansionandFreeThermalStrain 4
1.1.4 ActionofExternalForceandTemperatureChange 5
1.2 ThermalStressesinClampedBars 5
1.2.1 ConstantTemperatureChange 5
1.2.2 Non-UniformTemperatureChange 13
1.3 ThermalStressesinPartiallyRestrainedBars 15
1.4 ThermalStressesinBarsUnderBending 22
Problems 24
2 ThermalStressesinBeams 29
2.1 ThermalStressesinBeams 29
2.1.1 StressesinBeamsSubjectedtoMechanicalLoads 29
2.1.2 ThermalStressesinClampedBeams 31
2.1.3 ThermalStressesinRectangularBeams 32
2.2 GeneralTechniqueforThermalStressesinBeams 34
2.3 ThermalStressesinCompositeBeams 39
2.3.1 ThermalStressesinaTwo-LayeredBeam 39
2.3.2 ThermalStressesinTwoBeamsClampedatEachEnd 46
v
vi | Contents
2.3.3 ThermalStressesinMulti-LayeredCompositeBeams 52
2.4 ThermalStressesinNonhomogeneousBeams 54
2.5 ThermalStressesatArbitraryTemperature 56
2.5.1 HomogeneousBeams 56
2.5.2 NonhomogeneousBeams 59
2.6 ThermalDeflectioninBeams 62
2.7 CurvedBeams 65
2.8 ThermalShearingStressesinThin-WalledBeams 68
2.9 ThermalStressesinBeamsonanElasticFoundation 71
Problems 76
3 HeatConduction 81
3.1 HeatConductionEquation 81
3.2 BoundaryandInitialConditions 89
3.2.1 BoundaryConditions 90
3.2.2 InitialCondition 93
3.3 One-DimensionalHeatConductionProblemsinCartesianCoordinates 93
3.3.1 One-DimensionalHeatConductionEquation 93
3.3.2 One-DimensionalTemperatureintheSteadyState 94
3.3.3 One-DimensionalTransientTemperature;SeparationofVariables 96
3.3.4 One-DimensionalTransientTemperature;LaplaceTransform 102
3.4 One-DimensionalHeatConductionProblemsinCylindricalCoordinates 111
3.4.1 HeatConductionEquationinCylindricalCoordinates 111
3.4.2 One-DimensionalTemperatureintheSteadyState 112
3.4.3 One-DimensionalTransientTemperatureinaSolidCylinder 115
3.4.4 One-DimensionalTransientTemperatureinaHollowCylinder 118
3.5 One-DimensionalHeatConductionProblemsinSphericalCoordinates 123
3.5.1 HeatConductionEquationinSphericalCoordinates 123
3.5.2 One-DimensionalTemperatureintheSteadyState 124
3.5.3 One-DimensionalTransientTemperatureinaSolidSphere 127
3.5.4 One-DimensionalTransientTemperatureinaHollowSphere 129
Problems 134
4 BasicEquationsofThermoelasticity 137
4.1 StressinaCartesianCoordinateSystem 137
4.1.1 EquationsofEquilibrium 137
4.1.2 CoordinateTransformationofStress 142
4.2 StraininaCartesianCoordinateSystem 144
4.2.1 ComponentsofStrain 144
4.2.2 CompatibilityConditions 145
4.2.3 CoordinateTransformationofStrain 146
4.3 GoverningEquationsofThermoelasticity 147
4.3.1 ConstitutiveEquations 147
4.3.2 DisplacementEquations 149
4.3.3 CompatibilityEquationsinTermsofStress 150
4.3.4 BoundaryConditions 151
4.3.5 Body-ForceAnalogy 152
4.4 GeneralSolutionofNavier’sEquations 153
4.4.1 ThermoelasticDisplacementPotential 153
Contents | vii
4.4.2 DisplacementPotentials 156
4.4.3 GeneralSolutionsforthePotentialFunction 160
4.5 CylindricalCoordinateSystem 162
4.5.1 StressinaCylindricalCoordinateSystem 162
4.5.2 StraininaCylindricalCoordinateSystem 163
4.5.3 GeneralizedHooke’sLawinaCylindrical
CoordinateSystem 165
4.5.4 Navier’sEquationsofThermoelasticityina
CylindricalCoordinateSystem 165
4.6 SphericalCoordinateSystem 168
4.6.1 StressinaSphericalCoordinateSystem 168
4.6.2 StraininaSphericalCoordinateSystem 169
4.6.3 GeneralizedHooke’sLawinaSphericalCoordinateSystem 171
4.6.4 Navier’sEquationsofThermoelasticityina
SphericalCoordinateSystem 172
4.7 MultiplyConnectedBodies 173
4.7.1 BasicEquationsforMultiplyConnectedBodies 173
4.7.2 TemperatureDistributionswithZeroThermalStress 176
Problems 178
5 PlaneThermoelasticProblems 179
5.1 PlaneStrainandPlaneStress 179
5.1.1 Introduction 179
5.1.2 PlaneStrain 180
5.1.3 PlaneStress 183
5.1.4 GoverningEquationsofPlaneProblems 187
5.2 ThermalStressFunction 189
5.2.1 ThermalStressFunction 189
5.2.2 DisplacementsbyThermalStressFunction 190
5.2.3 BoundaryConditionsinTermsofThermalStressFunction 192
5.2.4 MultiplyConnectedBodies 193
5.2.5 TemperatureDistributionwithZeroThermalStress 196
5.2.6 GeneralSolutionofThermalStressFunction 198
5.3 ComplexVariableMethod 203
5.3.1 ComplexPresentationofThermalStressFunction 203
5.3.2 ComplexRepresentationofStressandDisplacement 205
5.3.3 GeneralFormulaeforMultiplyConnectedBodies 208
5.3.4 ConformalMappingoftheFormulaeofPlaneTheory 211
5.4 PotentialMethod 214
Problems 218
6 ThermalStressesinCircularCylinders 221
6.1 One-DimensionalProblems 221
6.1.1 DisplacementTechnique 222
6.1.2 ThermalStressFunctionTechnique 233
6.1.3 ThermoelasticPotentialTechnique 234
6.1.4 ComplexVariableTechnique 236
6.1.5 ANoteonReciprocalTheoryTechnique 240
6.1.6 DislocationTechnique 240
viii | Contents
6.2 PlaneProblems 244
6.2.1 SteadyStateThermalStressesinaHollowCylinder 244
6.2.2 TransientThermalStressesinaSolidCylinder 250
6.3 Two-DimensionalProblems(AxisymmetricProblems) 259
6.3.1 FundamentalEquationsforTwo-DimensionalProblems 259
6.3.2 SteadyStateThermalStressesinaSemi-InfiniteBody 261
6.3.3 SteadyStateThermalStressesinaLongCircularCylinder 266
6.3.4 TransientThermalStressesinaThickPlate 269
6.4 Three-DimensionalProblems 273
6.4.1 SolutionMethodsinThree-DimensionalProblems 273
6.4.2 SteadyStateThermalStressesinaSemi-InfiniteBody 276
6.4.3 SteadyStateThermalStressesinaLongCircularCylinder 284
Problems 293
7 ThermalStressesinSphericalBodies 295
7.1 ThermalStressesinSphericalBodies 295
7.2 One-DimensionalProblems 295
7.2.1 AHollowSphere 297
7.2.2 ASolidSphere 301
7.2.3 InfiniteBodies 304
7.3 Two-DimensionalProblems(AxisymmetricProblems) 309
7.4 IllustrativeExamplesofAxisymmetricProblemsinSphericalBodies 314
7.4.1 FundamentalSolutionofTemperatureField 314
7.4.2 ComponentsofDisplacementandStressinTermsofDisplacement
Functionsϕandψ 316
Problems 341
8 ThermalStressesinPlates 345
8.1 ThermalStressinaPlateDuetoUniform
ThermalLoad 345
8.2 BasicEquationsforaRectangularPlate 355
8.2.1 FundamentalRelationsofThermalBending 355
8.2.2 BoundaryConditionsofthePlate 360
8.3 FundamentalSolutionsforRectangularPlates 365
8.3.1 SimplySupportedRectangularPlate 365
8.3.2 ARectangularPlatewithTwoOppositeEdgesSimplySupported
andtheOtherTwoEdgesClamped 367
8.3.3 ARectangularPlatewithBuilt-inAllEdges 371
8.4 BasicEquationsforaCircularPlate 376
8.4.1 FundamentalRelationsofThermalBending 376
8.4.2 BoundaryConditionsforaCircularPlate 379
8.4.3 AxisymmetricThermalBendingProblems 380
8.5 FundamentalSolutionsforCircularPlates 381
8.5.1 AxisymmetricProblems 381
8.5.2 Non-axisymmetricProblems 387
Problems 392
9 ThermallyInducedInstability 397
9.1 InstabilityofBeam-Column 397
Contents | ix
9.1.1 Introduction 397
9.1.2 Beam-ColumnProblemsbyAxiallyImposedExternalLoad 399
9.2 InstabilityofPlate 409
9.2.1 BasicEquationsforaRectangularPlate 409
9.2.2 IllustrativeExamplesforaRectangularPlate 413
9.2.3 BasicEquationsforaCircularPlate 419
9.2.4 IllustrativeExamplesforaCircularPlate 424
Problems 429
10ThermodynamicsofThermoelasticity 431
10.1 ThePrincipleofEnergyConservation 431
10.2 TheSecondLawofThermodynamics 434
10.2.1 ReversibleandIrreversibleProcesses 434
10.2.2 CycleofCarnot 435
10.2.3 Entropy 437
10.3 ThermodynamicFunctions 439
10.4 FundamentalDifferentialEquationsofThermoelasticity 446
10.4.1 HeatConductionEquation 446
10.4.2 EquationofMotion 447
10.4.3 SolutionofEquationofMotion 449
10.5 TheVariationalTheoremsofThermoelasticity 451
10.6 UniquenessTheorem 454
10.7 ReciprocalTheorem 457
10.7.1 ReciprocalTheorem 457
10.7.2 PracticalProblem 463
Problems 466
A InverseLaplaceTransform 469
B BesselFunctions 473
C LegendreFunctions 479
BooksonThermalStresses 485
Index 489