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'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'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 ...