Theory of Elasticity

TheTheoryofelasticitystudiesthebehaviorofthosebodiesthatrecovertheiri- tial state when the causes which produce deformations are removed. Its results constitutethefoundationsofthe Theory of structuresandthenareofmaximum importanceforengineers. The Theory of elasticity moves freely within an uni ed mathematical fra- workthatprovidestheanalyticaltoolsforcalculatingstressesanddeformationsin astrainedelasticbody. Alltheelasticproblemscanbeexactlyanalyzedemploying theclassicalMathematicalanalysis,withtheexceptionoftheunilateralproblems forwhichtheemploymentoftheFunctionalanalysisismandatory. TheTheoryofelasticitywasfoundedbythefamousmathematicianCauchyinthe eighteenth-century. Duringitshistoricaldevelopmentthisscienti csectorproposed 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,as niteelasticityandunilateralproblems(Chap. 8). InthisbookIhaveconstantlykeptinmindthepracticalapplicationoftheth- reticalresults. SoIhavealwaystriedtogivetoengineers,inasimpleform,aclear indicationofthenecessaryfundamentalknowledgeoftheTheoryofelasticity. In thepastsometechniquesofcalculationweredevelopedforparticularelasticpr- lemsthatcannotbeorganizedinmathematicaltheoriesbutareextremelysimpleto apply. Suchtechnicaltheorieshavealwaysfurnishedresultsexperimentallyveri ed v vi Preface withgoodapproximationandthenamongthemIhavepresentedthosethatarestill usefultoolsofveri cationintheStructuraldesign. Throughouttheanalysisoftheelasticproblemsmyconstantfocushasbeento achievethemaximumclarityandbecauseofthisIhavesacri cedvariousbright discussions. Ihavedevelopedthetreatmentofthesubjectsinclassicalway,butto thelightofthemodernMathematicaltheoryoftheelasticityandwithmoreaccented relief to the connections with the Thermodynamics. Just for this, to give a clear justi cationofthefundamentalequationoftheThermoelasticityIhaveapplieda techniqueofanalysisproperoftheFluiddynamics. Howeverinthediscussionof theunilateralproblems,wheretheFunctionalanalysisiscompulsory,Ihaverelated indetailsthemathematicalaspectsofthetheoreticalanalysis. Roma,Italy AldoMaceri October2009 Contents 1 The Three-Dimensional Problem. . . . . . . . . . . . . . . . . . . . 1 1. 1 AnalysisofStrain. . . . . . . . . . . . . . . . . . . . . . . . . . 1 1. 1. 1 ComponentsofDisplacement. . . . . . . . . . . . . . . . 1 1. 1. 2 In nitesimalDeformation. . . . . . . . . . . . . . . . . . 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. 10MeasurementofStrain . . . . . . . . . . . . . . . . . . . 25 1. 2 AnalysisofStress. . . . . . . . . . . . . . . . . . . . . . . . . . 27 1. 2. 1 StressVector. . . . . . . . . . . . . . . . . . . . . . . . . 27 1. 2. 2 NormalStress–ShearingStress . . . . . . . . . . . . . . 29 1. 2. 3 ComponentsofStress . . . . . . . . . . . . . . . . . . . . 30 1. 2. 4 Symmetryof? –DifferentialEquations ofEquilibrium–Cauchy’sBoundaryConditions. . . . . . 31 1. 2. 5 SymmetryofStressVector. . . . . . . . . . . . . . . . . 38 1. 2. 6 RelationsBetweenNormalorShearingStress andComponentsofStress. . . . . . . . . . . . . . . . . . 39 1. 2. 7 PrincipalDirectionofStress . . . . . . . . . . . . . . . . 40 1. 2. 8 InvariantsofStress . . . . . . . . . . . . . . . . . . . . . 42 1. 2. 9 Mohr’sCircle. . . . . . . . . . . . . . . . . . . . . . . . 43 1. 2. 10Mohr’sPrincipalCircles . . . . . . . . . . . . . . . . . . 57 1. 2. 11 DeterminationoftheMaximumNormalStress orShearingStressbytheMohr’sPrincipalCircles. . . . . 61 1. 2. 12PlaneStateofStress. . . . . . . . . . . . . . . . . . . . . 63 1. 2. 13UniaxialStateofStress. . . . . . . . . . . . . . . . . . . 65 1. 2. 14MeasurementofStress . . . . . . . . . . . . . . . . . . . 66 1. 3 PrincipleofVirtualWorks . . . . . . . . . . . . . . . . . . . . . 66 1. 3. 1 PrincipleofVirtualWorks . . . . . . . . . . . . . . . . .