Theory of Elasticity Aldo Maceri Theory of Elasticity 123 Prof.Dr.-Ing.AldoMaceri UniversitáRomaTre DepartimentodiIngegneriaMeccanicae Industriale ViadellaVascaNavale,79 00146Roma Italy [email protected] ISBN978-3-642-11391-8 e-ISBN978-3-642-11392-5 DOI10.1007/978-3-642-11392-5 SpringerHeidelbergDordrechtLondonNewYork LibraryofCongressControlNumber:2010923688 ©Springer-VerlagBerlinHeidelberg2010 Thisworkissubjecttocopyright.Allrightsarereserved,whetherthewholeorpartofthematerialis concerned,specificallytherightsoftranslation,reprinting,reuseofillustrations,recitation,broadcasting, reproductiononmicrofilmorinanyotherway,andstorageindatabanks.Duplicationofthispublication orpartsthereofispermittedonlyundertheprovisionsoftheGermanCopyrightLawofSeptember9, 1965,initscurrentversion,andpermissionforusemustalwaysbeobtainedfromSpringer.Violations areliabletoprosecutionundertheGermanCopyrightLaw. Theuseofgeneraldescriptivenames,registerednames,trademarks,etc.inthispublicationdoesnot imply,evenintheabsenceofaspecificstatement,thatsuchnamesareexemptfromtherelevantprotective lawsandregulationsandthereforefreeforgeneraluse. Coverdesign:eStudioCalamarS.L. Printedonacid-freepaper SpringerispartofSpringerScience+BusinessMedia(www.springer.com) Preface TheTheoryofelasticitystudiesthebehaviorofthosebodiesthatrecovertheirini- tial state when the causes which produce deformations are removed. Its results constitute the foundations of the Theory of structures and then are of maximum importanceforengineers. The Theory of elasticity moves freely within an unified mathematical frame- workthatprovidestheanalyticaltoolsforcalculatingstressesanddeformationsin astrainedelasticbody.Alltheelasticproblemscanbeexactlyanalyzedemploying the classical Mathematical analysis, with the exception of the unilateral problems forwhichtheemploymentoftheFunctionalanalysisismandatory. TheTheoryofelasticitywasfoundedbythefamousmathematicianCauchyinthe eighteenth-century.Duringitshistoricaldevelopmentthisscientificsectorproposed tothemathematiciansvariousproblemsthathavecontributedorentirelygenerated thedevelopmentofcomplexmathematicaltheories,astheVariationalcalculusand theFiniteelementmethod. Thematteranalyzedinthisbookis –three-dimensional problems (Chap. 1), and particularly the problem of Saint Venant(Chap.1), –two-dimensionalproblems,aspanels,plates,shells(Chap.3), –one-dimensionalproblems,asropes,beams,arches(Chap.4), –thermalstressproblems(Chap.5), –stabilityproblems(Chap.6), –anisotropicproblems,thatconstitutethebasictoolfortheanalysisofstructuresin compositematerial(Chap.7), –nonlinearelasticproblems,asfiniteelasticityandunilateralproblems(Chap.8). InthisbookIhaveconstantlykeptinmindthepracticalapplicationofthetheo- reticalresults.SoIhavealwaystriedtogivetoengineers,inasimpleform,aclear indication of the necessary fundamental knowledge of the Theory of elasticity. In thepastsometechniquesofcalculationweredevelopedforparticularelasticprob- lemsthatcannotbeorganizedinmathematicaltheoriesbutareextremelysimpleto apply.Suchtechnicaltheorieshavealwaysfurnishedresultsexperimentallyverified v vi Preface withgoodapproximationandthenamongthemIhavepresentedthosethatarestill usefultoolsofverificationintheStructuraldesign. Throughout the analysis of the elastic problems my constant focus has been to achieve the maximum clarity and because of this I have sacrificed various bright discussions.Ihave developed thetreatment of thesubjects inclassicalway, butto thelightofthemodernMathematicaltheoryoftheelasticityandwithmoreaccented relief to the connections with the Thermodynamics. Just for this, to give a clear justification of the fundamental equation of the Thermoelasticity I have applied a technique of analysis proper of the Fluid dynamics. However in the discussion of theunilateralproblems,wheretheFunctionalanalysisiscompulsory,Ihaverelated indetailsthemathematicalaspectsofthetheoreticalanalysis. Roma,Italy AldoMaceri October2009 Contents 1 TheThree-DimensionalProblem . . . . . . . . . . . . . . . . . . . . 1 1.1 AnalysisofStrain . . . . . . . . . . . . . . . . . . . . . . . . . . 1 1.1.1 ComponentsofDisplacement . . . . . . . . . . . . . . . . 1 1.1.2 InfinitesimalDeformation . . . . . . . . . . . . . . . . . . 2 1.1.3 ElongationandShearingStrain . . . . . . . . . . . . . . . 4 1.1.4 SmallDeformations . . . . . . . . . . . . . . . . . . . . . 5 1.1.5 ComponentsofStrain . . . . . . . . . . . . . . . . . . . . 9 1.1.6 PrincipalDirectionofStrain . . . . . . . . . . . . . . . . 14 1.1.7 InvariantsofStrain . . . . . . . . . . . . . . . . . . . . . 21 1.1.8 PlaneStateofStrain . . . . . . . . . . . . . . . . . . . . . 23 1.1.9 EquationsofCompatibility . . . . . . . . . . . . . . . . . 24 1.1.10 MeasurementofStrain . . . . . . . . . . . . . . . . . . . 25 1.2 AnalysisofStress . . . . . . . . . . . . . . . . . . . . . . . . . . 27 1.2.1 StressVector. . . . . . . . . . . . . . . . . . . . . . . . . 27 1.2.2 NormalStress–ShearingStress . . . . . . . . . . . . . . 29 1.2.3 ComponentsofStress . . . . . . . . . . . . . . . . . . . . 30 1.2.4 Symmetry of τ – Differential Equations ofEquilibrium–Cauchy’sBoundaryConditions. . . . . . 31 1.2.5 SymmetryofStressVector . . . . . . . . . . . . . . . . . 38 1.2.6 Relations Between Normalor Shearing Stress andComponentsofStress. . . . . . . . . . . . . . . . . . 39 1.2.7 PrincipalDirectionofStress . . . . . . . . . . . . . . . . 40 1.2.8 InvariantsofStress . . . . . . . . . . . . . . . . . . . . . 42 1.2.9 Mohr’sCircle . . . . . . . . . . . . . . . . . . . . . . . . 43 1.2.10 Mohr’sPrincipalCircles . . . . . . . . . . . . . . . . . . 57 1.2.11 DeterminationoftheMaximumNormalStress orShearingStressbytheMohr’sPrincipalCircles . . . . . 61 1.2.12 PlaneStateofStress . . . . . . . . . . . . . . . . . . . . . 63 1.2.13 UniaxialStateofStress . . . . . . . . . . . . . . . . . . . 65 1.2.14 MeasurementofStress . . . . . . . . . . . . . . . . . . . 66 1.3 PrincipleofVirtualWorks . . . . . . . . . . . . . . . . . . . . . 66 1.3.1 PrincipleofVirtualWorks . . . . . . . . . . . . . . . . . 66 vii viii Contents 1.4 RelationsBetweenStressandStrain . . . . . . . . . . . . . . . . 70 1.4.1 TensileBreakingTest . . . . . . . . . . . . . . . . . . . . 70 1.4.2 HomogeneousandIsotropicMaterials–Navier’sRelations 75 1.4.3 BoundsfortheElasticModulus. . . . . . . . . . . . . . . 78 1.5 TheElasticEquilibriumProblem . . . . . . . . . . . . . . . . . . 80 1.5.1 ClassicalFormulations . . . . . . . . . . . . . . . . . . . 80 1.5.2 VariationalFormulations . . . . . . . . . . . . . . . . . . 90 1.6 StrainEnergy . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95 1.6.1 ElementsofThermodynamics . . . . . . . . . . . . . . . 95 1.6.2 ThermodynamicsoftheProblemoftheElasticEquilibrium 101 1.6.3 StrainWork . . . . . . . . . . . . . . . . . . . . . . . . . 103 1.6.4 TheElasticPotential . . . . . . . . . . . . . . . . . . . . 107 1.6.5 WorkTheorems . . . . . . . . . . . . . . . . . . . . . . . 110 1.7 StrengthCriterions . . . . . . . . . . . . . . . . . . . . . . . . . 115 1.7.1 StructuralSafety. . . . . . . . . . . . . . . . . . . . . . . 115 1.7.2 TheMaximumShearingStressCriterion . . . . . . . . . . 116 1.7.3 TheOctahedralShearingStressCriterion. . . . . . . . . . 117 1.7.4 TheEnergeticCriterion . . . . . . . . . . . . . . . . . . . 118 1.7.5 TheIntrinsicCurveCriterion . . . . . . . . . . . . . . . . 124 2 TheProblemofSaintVenant . . . . . . . . . . . . . . . . . . . . . . 127 2.1 GeometryofAreas . . . . . . . . . . . . . . . . . . . . . . . . . 127 2.1.1 Centroid . . . . . . . . . . . . . . . . . . . . . . . . . . . 127 2.1.2 InertiaCentroidalEllipse . . . . . . . . . . . . . . . . . . 131 2.1.3 Antipolarity . . . . . . . . . . . . . . . . . . . . . . . . . 141 2.1.4 InertiaCentroidalKernel . . . . . . . . . . . . . . . . . . 159 2.2 TheProblemofSaintVenant . . . . . . . . . . . . . . . . . . . . 164 2.2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . 164 2.2.2 StateofStress . . . . . . . . . . . . . . . . . . . . . . . . 169 2.3 Bending . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 173 2.3.1 RightBendingofAxisx . . . . . . . . . . . . . . . . . . 173 2.3.2 RightBendingofAxisy . . . . . . . . . . . . . . . . . . 187 2.3.3 DeviatedBending . . . . . . . . . . . . . . . . . . . . . . 190 2.4 AxialLoad . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 195 2.4.1 CentroidalAxialLoad. . . . . . . . . . . . . . . . . . . . 195 2.4.2 NonCentroidalAxialLoad . . . . . . . . . . . . . . . . . 201 2.4.3 MaterialNonResistanttoTraction . . . . . . . . . . . . . 215 2.5 Torsion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 219 2.5.1 TheExactSolution . . . . . . . . . . . . . . . . . . . . . 219 2.5.2 TheCircularCrossSection . . . . . . . . . . . . . . . . . 226 2.5.3 TheStressConcentration . . . . . . . . . . . . . . . . . . 232 2.5.4 ClosedThinWalledCrossSection . . . . . . . . . . . . . 240 2.5.5 OpenThinWalledCrossSection . . . . . . . . . . . . . . 250 2.5.6 NonUniformTorsion . . . . . . . . . . . . . . . . . . . . 264 2.6 Shear . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 278 2.6.1 TheExactSolution . . . . . . . . . . . . . . . . . . . . . 278 Contents ix 2.6.2 TheApproximateSolution . . . . . . . . . . . . . . . . . 294 2.6.3 TheCircularCrossSection . . . . . . . . . . . . . . . . . 304 2.6.4 OpenThinWalledCrossSection . . . . . . . . . . . . . . 308 2.6.5 ClosedThinWalledCrossSection . . . . . . . . . . . . . 351 3 TheTwo-DimensionalProblems . . . . . . . . . . . . . . . . . . . . 359 3.1 Panels . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 359 3.1.1 TheProblemofthePanel . . . . . . . . . . . . . . . . . . 359 3.1.2 RectangularPanels . . . . . . . . . . . . . . . . . . . . . 364 3.1.3 CircularPanels . . . . . . . . . . . . . . . . . . . . . . . 370 3.1.4 EffectofaHole . . . . . . . . . . . . . . . . . . . . . . . 387 3.2 Plates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 392 3.2.1 SmallDeflectionsofThinPlates . . . . . . . . . . . . . . 392 3.2.2 ThinPlatesonElasticFoundation . . . . . . . . . . . . . 402 3.3 Shells . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 403 3.3.1 Membranes . . . . . . . . . . . . . . . . . . . . . . . . . 403 3.3.2 ThinShells . . . . . . . . . . . . . . . . . . . . . . . . . 411 3.4 PlaneStrainProblems . . . . . . . . . . . . . . . . . . . . . . . . 422 3.4.1 PlaneStrainProblems . . . . . . . . . . . . . . . . . . . . 422 4 TheOne-DimensionalProblems . . . . . . . . . . . . . . . . . . . . 427 4.1 Ropes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 427 4.1.1 TheFunicular . . . . . . . . . . . . . . . . . . . . . . . . 427 4.1.2 TheRopes . . . . . . . . . . . . . . . . . . . . . . . . . . 430 4.2 Beams . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 433 4.2.1 TheDeflectedBeam. . . . . . . . . . . . . . . . . . . . . 433 4.2.2 TheAnalogyofMohr . . . . . . . . . . . . . . . . . . . . 449 4.2.3 PrincipleofVirtualWorks . . . . . . . . . . . . . . . . . 462 4.2.4 StrainEnergy . . . . . . . . . . . . . . . . . . . . . . . . 463 4.2.5 DeflectedBeamsonElasticFoundation . . . . . . . . . . 481 4.3 Arches . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 482 4.3.1 ArcheswithSmallCurvature . . . . . . . . . . . . . . . . 482 4.3.2 ArcheswithGreatCurvature . . . . . . . . . . . . . . . . 488 5 Thermoelasticity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 493 5.1 MechanicsofContinuousMedia . . . . . . . . . . . . . . . . . . 493 5.1.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . 493 5.1.2 ClassicalThermodynamics . . . . . . . . . . . . . . . . . 494 5.1.3 TheEquationsofBalance . . . . . . . . . . . . . . . . . . 501 5.1.4 ThermodynamicsoftheIrreversibleProcesses . . . . . . . 503 5.2 FluidDynamics . . . . . . . . . . . . . . . . . . . . . . . . . . . 506 5.2.1 TheMathematicalModel . . . . . . . . . . . . . . . . . . 506 5.2.2 TheCharacteristicNumbers . . . . . . . . . . . . . . . . 510 5.2.3 NonDissipativeFlows . . . . . . . . . . . . . . . . . . . 524 5.2.4 DissipativeFlows . . . . . . . . . . . . . . . . . . . . . . 537 5.3 MechanicsofSolids . . . . . . . . . . . . . . . . . . . . . . . . . 552 5.3.1 TheDynamicThermoelasticProblem . . . . . . . . . . . 552 x Contents 5.3.2 TheThermoelasticDissipation . . . . . . . . . . . . . . . 562 5.3.3 TheUncoupledThermoelasticProblem . . . . . . . . . . 563 5.3.4 Thermoelasticity . . . . . . . . . . . . . . . . . . . . . . 566 5.3.5 TheOne-DimensionalProblem . . . . . . . . . . . . . . . 573 6 Stability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 593 6.1 StabilityoftheElasticEquilibrium . . . . . . . . . . . . . . . . . 593 6.1.1 TheBuckling . . . . . . . . . . . . . . . . . . . . . . . . 593 6.1.2 TheUltimateStrength . . . . . . . . . . . . . . . . . . . . 595 6.2 EnergyMethod . . . . . . . . . . . . . . . . . . . . . . . . . . . 596 6.2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . 596 6.2.2 SomeElementaryApplication . . . . . . . . . . . . . . . 598 6.3 StaticMethod . . . . . . . . . . . . . . . . . . . . . . . . . . . . 607 6.3.1 TheBeamAxiallyLoaded . . . . . . . . . . . . . . . . . 607 6.3.2 ApproximateAnalysis . . . . . . . . . . . . . . . . . . . 607 6.3.3 ExactAnalysis . . . . . . . . . . . . . . . . . . . . . . . 618 6.3.4 EffectoftheImperfections . . . . . . . . . . . . . . . . . 622 6.3.5 LimitSlenderness . . . . . . . . . . . . . . . . . . . . . . 624 6.3.6 OtherWaysofBuckling. . . . . . . . . . . . . . . . . . . 627 6.4 SecondTypeInstability . . . . . . . . . . . . . . . . . . . . . . . 629 6.4.1 TheSnapping . . . . . . . . . . . . . . . . . . . . . . . . 629 7 Anisotropy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 635 7.1 TheThree-DimensionalAnisotropicProblem . . . . . . . . . . . 635 7.1.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . 635 7.1.2 ConstituentLinks . . . . . . . . . . . . . . . . . . . . . . 636 7.1.3 TheAnisotropicElasticBody . . . . . . . . . . . . . . . . 639 7.1.4 EnergeticAspects . . . . . . . . . . . . . . . . . . . . . . 641 7.1.5 TheAnisotropicSaintVenant’sProblem . . . . . . . . . . 642 7.2 TheMacroscopicAnisotropy . . . . . . . . . . . . . . . . . . . . 649 7.2.1 CompositeMaterials . . . . . . . . . . . . . . . . . . . . 649 7.2.2 StructuralAnisotropy . . . . . . . . . . . . . . . . . . . . 656 8 NonlinearElasticity . . . . . . . . . . . . . . . . . . . . . . . . . . . 663 8.1 NonlinearProblems . . . . . . . . . . . . . . . . . . . . . . . . . 663 8.1.1 TheNonlinearityCauses . . . . . . . . . . . . . . . . . . 663 8.2 FiniteDeformations . . . . . . . . . . . . . . . . . . . . . . . . . 664 8.2.1 Three-DimensionalProblem . . . . . . . . . . . . . . . . 664 8.2.2 LargeDeflectionsofThinPlates . . . . . . . . . . . . . . 669 8.3 UnilateralProblems . . . . . . . . . . . . . . . . . . . . . . . . . 673 8.3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . 673 8.3.2 ContactProblems . . . . . . . . . . . . . . . . . . . . . . 674 8.3.3 UnilateralConstraints . . . . . . . . . . . . . . . . . . . . 684 Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 705 AuthorIndex . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 707 SubjectIndex . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 709