Applied Elasticity and Plasticity

565 Pages·2018·18.471 MB·English

by  Mumtaz Kassir

Applied
MECHANICAL ENGINEERING
Kassir
Applied Elasticity
Elasticit y
and Plasticity
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Applied Elasticity and Plasticity is a comprehensive work that p
introduces graduate students and professionals in civil, mechanical,
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aeronautical, and metallurgical engineering to the basic theories of elas-
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ticity, plasticity, and their practical applications. Based on experimental Plasticit y
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data of static tension tests of material, several elastic and plastic stress-
strain relations are derived, and commonly used yield criteria and strain
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hardening rules are discussed as well.
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Analysis of conventional, deviatoric, and mathematical stress and strain a
in two and three dimensions is presented. Analytical applications include s
torsion and bending of structural components subjected to various load-
t Mumtaz Kassir
ings, thick-walled cylindrical and spherical vessels subjected to internal i
c
and external pressures, stress-concentrations around holes, stress-
intensity factors in structural components containing circular, elliptical, i
t
and many more concepts important for professionals and students alike. 
y
Features
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• Motivates the study of elasticity and plasticity concepts with a wide
n
range of engineering examples 
d
• Combines a thorough introduction to both symbolic and numerical

problem solving in mechanics 
P
• Includes finite element examples in ANSYS for elastic problems and l
a
ABAQUES for plasticity problems 
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• Progresses from two-dimensional to three-dimensional mechanics
t
in a clear, systematic manner 
i
c
• Offers over 80 detailed examples and over 200 assignment problems
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y
K33338
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an informa business
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w w w . c r c p r e s s . c o m
Applied Elasticity and
Plasticity

Applied Elasticity and
Plasticity
Mumtaz Kassir
CRCPress
Taylor&FrancisGroup
6000BrokenSoundParkwayNW,Suite300
BocaRaton,FL33487-2742
©2018byTaylor&FrancisGroup,LLC
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Contents
Preface..............................................................................................................................................xi
Author...........................................................................................................................................xiii
1. Introduction..............................................................................................................................1
1.1 AxialExtension...............................................................................................................1
1.2 BendingofSymmetricBeams.......................................................................................8
1.3 TorsionofCircularMembers......................................................................................14
Problems..................................................................................................................................16
ListofElementaryFormulas................................................................................................21
References................................................................................................................................22
2. AnalysisofStress..................................................................................................................23
2.1 TheStressataPoint.....................................................................................................24
2.2 EquationsofEquilibrium.............................................................................................26
2.3 StressVector(Traction)................................................................................................28
2.4 StressesonanInclinedPlane......................................................................................28
2.5 TransformationofStress..............................................................................................34
2.6 PrincipalStresses..........................................................................................................38
2.6.1 PrincipalNormalStresses...............................................................................38
2.6.2 PrincipalShearStresses...................................................................................42
2.7 StressDeviatorTensor.................................................................................................47
2.8 OctahedralStresses.......................................................................................................50
2.9 Mohr’sStressCircles....................................................................................................52
Problems..................................................................................................................................58
References................................................................................................................................65
3. AnalysisofStrain..................................................................................................................67
3.1 DisplacementataPoint...............................................................................................67
3.2 NormalandShearStrains...........................................................................................69
3.2.1 2DStrain............................................................................................................70
3.2.2 3DStrain............................................................................................................75
3.3 CompatibilityConditions............................................................................................77
3.4 Dilatation.......................................................................................................................79
3.5 TransformationofStrain..............................................................................................79
3.5.1 2DStrain............................................................................................................80
3.5.2 3DStrain............................................................................................................81
3.6 StrainTensor.................................................................................................................85
3.6.1 MathematicalStrain.........................................................................................86
3.7 PrincipalStrains............................................................................................................88
3.8 OctahedralStrains........................................................................................................92
3.9 Mohr’sCircleofStrain.................................................................................................92
3.10 StrainGages...................................................................................................................95
Problems..................................................................................................................................98
References..............................................................................................................................103
v
vi Contents
4. Stress–StrainRelations.......................................................................................................105
4.1 ElasticBehavior...........................................................................................................105
4.1.1 Hooke’sLaw...................................................................................................106
4.1.2 IncompressibleMaterial................................................................................110
4.1.3 EquationsofEquilibriuminTermsofDisplacements..............................110
4.2 StrainEnergy...............................................................................................................113
4.2.1 DilatationalandDistortionalStrainEnergyDensities..............................117
4.2.2 ImpactLoading...............................................................................................119
4.3 PlasticBehavior...........................................................................................................120
4.3.1 StaticTensionTest..........................................................................................120
4.3.2 BasicStress–StrainRelations.........................................................................121
4.3.3 ModelsofUniaxialStress–StrainRelations................................................123
4.3.4 TrueStressandStrain....................................................................................135
4.4 YieldCriterion.............................................................................................................138
4.4.1 MaximumShearingStressCriterion............................................................139
4.4.2 MaximumDistortionEnergyCriterion.......................................................140
4.4.3 YieldSurfacesinHaigh–WesregaardStressSpace...................................145
4.4.4 StrainHardeningRules.................................................................................146
4.4.4.1 IsotropicHardeningRule...............................................................147
4.4.4.2 KinematicHardeningRule............................................................147
4.4.4.3 IndependentHardeningRule........................................................148
4.5 PlasticStress–StrainRelations...................................................................................153
4.5.1 Stress–StrainRelationsfor
PerfectlyPlasticMaterial...............................................................................154
4.5.2 DeformationTheoryofPlasticity(J —Material)........................................155
2
Problems................................................................................................................................159
References..............................................................................................................................167
5. TorsionofPrismaticBars..................................................................................................169
5.1 Introduction.................................................................................................................169
5.2 TorsionofaCylindricalBar......................................................................................170
5.2.1 Elastic-PerfectlyPlasticMaterial..................................................................171
5.2.2 Work–HardeningMaterial............................................................................176
5.3 TorsionofNoncircularBars......................................................................................179
5.4 ElasticSolutions..........................................................................................................180
5.4.1 PolynomialSolution.......................................................................................183
5.4.2 FourierSeriesSolutions.................................................................................191
5.4.3 ApproximateSolution...................................................................................196
5.5 PlasticSolutions..........................................................................................................198
5.5.1 AxialDisplacement........................................................................................200
5.5.2 OvalCrossSection.........................................................................................202
5.5.3 FullyPlasticTorque.......................................................................................206
5.6 MembraneAnalogy....................................................................................................208
5.6.1 TorsionofThin-WalledMembers................................................................211
5.6.2 TorsionofThin-WalledTubes......................................................................214
5.7 TorsionofRestrainedStructuralSections...............................................................217
Problems................................................................................................................................220
References..............................................................................................................................225
Contents vii
6. Two-DimensionalProblems.............................................................................................227
6.1 PlaneStress..................................................................................................................227
6.2 PlaneStrain..................................................................................................................229
6.3 StressFunction............................................................................................................231
6.4 PolynomialSolutions.................................................................................................234
6.4.1 CantileverBeamLoadedbyaConcentratedForce...................................235
6.4.2 BendingofaSimplySupportedBeambyUniformLoad........................239
6.4.3 CantileverBeamSubjectedtoHydrostaticPressure.................................243
6.5 Saint-Venant’sPrinciple.............................................................................................245
6.6 SolutionsinPolarCoordinates.................................................................................248
6.7 SolutionsoftheCompatibilityEquation.................................................................252
6.7.1 AxiallySymmetricProblems........................................................................254
6.7.2 WedgeProblems.............................................................................................265
6.7.3 Semi-InfiniteSolid..........................................................................................270
6.8 RotatingCircularDisks..............................................................................................275
6.8.1 SolidDisk........................................................................................................277
6.8.2 AnnularDisk...................................................................................................280
6.8.3 AngularRotationofaThinRingorCylinder............................................284
6.8.4 DisksofVariableThickness..........................................................................285
6.8.5 RotatingDiskofUniformStrength..............................................................290
Problems................................................................................................................................291
References..............................................................................................................................299
7. OtherElastoplasticProblems............................................................................................301
7.1 Thick-WalledCylindricalTubes...............................................................................301
7.1.1 CylindricalTubeSubjectedtoInternalPressure........................................303
7.1.2 ExternalPressureOnly..................................................................................306
7.1.3 Elastic–PlasticExpansionofaCylinderSubjectedto
InternalPressure.............................................................................................307
7.1.4 ResidualStresses............................................................................................314
7.1.5 CompoundCylinders....................................................................................315
7.1.6 InfluenceofStrainHardeningoftheMaterial...........................................319
7.2 Thick-WalledSphericalShell....................................................................................322
7.2.1 SphericalShellSubjectedtoInternalPressure...........................................324
7.2.2 InfluenceofChangesinGeometryoftheSphere......................................328
7.2.3 Work-HardeningMaterial............................................................................329
7.3 StressConcentrationFactors.....................................................................................332
7.3.1 InfinitePlateContainingaCircularHole...................................................332
7.3.2 InfinitePlateContaininganEllipticalHole................................................339
7.4 StructuralMembersContainingCracks..................................................................340
7.4.1 PlasticZonenearCrackTip..........................................................................343
Problems................................................................................................................................345
References..............................................................................................................................349
8. ThermalStresses.................................................................................................................351
8.1 ThermalStressesinUniformCircularDisks..........................................................356
8.1.1 SolidDisk........................................................................................................357
8.1.2 AnnularDisk...................................................................................................357
8.1.3 CombinedThermalandRotationalStresses..............................................358
viii Contents
8.2 ThermalStressesinCylindricalVessels..................................................................361
8.2.1 Steady-StateHeatFlow.................................................................................363
8.2.2 ThermalStressesinPressurizedVesselsSubjecttoSteady-State
HeatFlow........................................................................................................367
8.2.3 InitiationofDuctileYield..............................................................................368
8.2.4 Elastic–PlasticExpansion..............................................................................368
8.3 ThermalStressesinThick-WalledSphericalShell.................................................370
8.3.1 Steady-StateTemperatureDistributioninSphericalVessels...................372
8.3.2 InitiationofYield............................................................................................373
8.3.3 Elastic–PlasticExpansion..............................................................................374
8.3.4 CombinedSteady-StateHeatFlowandInternalPressure.......................376
Problems................................................................................................................................378
References..............................................................................................................................381
9. Three-DimensionalProblems...........................................................................................383
9.1 ElongationofaPrismaticBarduetoItsOwnWeight..........................................383
9.2 3DAxisymmetricDeformationsofSolids...............................................................388
9.2.1 PolarCoordinatesandLegendreFunctions...............................................390
9.3 RotationofaCircularDisk........................................................................................393
9.4 ForceataPointinanInfiniteSolid..........................................................................396
9.5 StressConcentrationaroundaSphericalCavity....................................................401
9.6 ConcentratedForceonBoundaryofaSemi-InfiniteSolid...................................404
9.7 UniformLoadDistributedoveraPartoftheBoundaryofa
Semi-InfiniteSolid......................................................................................................408
9.7.1 UniformLoadDistributedoveraCircularAreaofRadiusa..................409
9.7.2 UniformPressureDistributedoverSquares
andRectangles................................................................................................410
9.8 ContactProblemsinThreeDimensions..................................................................411
9.8.1 SolutionDependingonOneHarmonicFunction......................................412
9.8.2 GeneralSolutionof3DContactProblems..................................................415
9.8.3 ContactProblemsforSolidsContainingCavities.....................................420
9.8.4 ContactbetweenTwoSphericalBodies:HertzProblem..........................428
9.9 ASemi-InfinitePlaneCutorCrackinanInfiniteBody........................................433
9.9.1 PlaneCrackProblem.....................................................................................433
9.9.2 SymmetricNormalLoads.............................................................................435
9.9.3 AntisymmetricShearLoadingNormaltoCrackEdge.............................439
9.9.4 AntisymmetricShearLoadingParalleltoCrack-Edge.............................443
9.10 DeformationofaCircularCylinder.........................................................................448
9.10.1 PressureBandofFiniteLengthActingona
LongCylinder...............................................................................................451
Problems................................................................................................................................452
References..............................................................................................................................454
10. Three-DimensionalThermalStresses.............................................................................457
10.1 GeneralEquations......................................................................................................457
10.2 ThinCircularDisk(PlaneStress)..............................................................................459
10.3 LongCircularCylinder(PlaneStrain).....................................................................460
10.3.1 SolidCylinder...............................................................................................462
10.3.2 CylinderwithaConcentricCentralHole.................................................464
Contents ix
10.4 ThermalStressesinaSphere.....................................................................................466
10.4.1 SolidSphere..................................................................................................467
10.4.2 SpherewithaSphericalHoleattheCenter..............................................468
10.5 ThermoelasticDisplacementPotential.....................................................................469
10.5.1 PlaneStrainCase..........................................................................................471
10.5.2 PlaneStress(ThinPlate)..............................................................................471
10.6 StructuralMembersContainingFlawsorCracks..................................................474
10.6.1 CircularCrack...............................................................................................474
10.6.2 ExternalCircularCrack...............................................................................483
10.6.3 EllipticalCrack..............................................................................................487
10.6.3.1 SymmetricalCase:ConstantTemperatureon
CrackSurface...............................................................................489
10.6.3.2 Skew-SymmetricCase:InsulatedEllipticalCrack..................492
10.6.3.3 SymmetricalCase:PolynomialTemperature
SpecifiedonCrackSurface.........................................................495
10.6.3.4 Skew-SymmetricCase:PolynomialTemperature
GradientSpecifiedonCrackSurface........................................498
10.6.4 Semi-InfinitePlaneCrack............................................................................503
Problems................................................................................................................................515
References..............................................................................................................................517
11.SolutionstoSelectedProblems.........................................................................................519
Index.............................................................................................................................................539