双语版材料力学第六章

1MechanicsofMaterials23§6–4Determinedeflectionsandanglesofrotationofthebeambytheprincipleofsuperposition§6–5ChecktherigidityofthebeamCHAPTER6DEFORMATIONINBENDING§6–6Strainenergyofthebeaminbending§6–7Methodtosolvesimplestaticallyindeterminateproblemsofthebeam§6–8Strainenergyinbendingbeam§6–1Summary§6–2Approximatedifferentialequationofthedeflecturecurveofthebeamanditsintegral§6–3Methodofconjugatebeamtodeterminethedeflectandtheangleofrotationofthebeam4§6–1概述§6–2梁的挠曲线近似微分方程及其积分§6–3求梁的挠度与转角的共轭梁法§6–4按叠加原理求梁的挠度与转角§6–5梁的刚度校核第六章弯曲变形§6–6梁内的弯曲应变能§6–7简单超静定梁的求解方法§6–8梁内的弯曲应变能5§6－１SUMMARYStudyrange：Calculationofthedisplacementofthestraightbeamwithequalsectioninsymmetricbending.Studyobject：①Dorigiditycheckforthebeam；②Solveproblemsaboutstaticallyindeterminatebeams（complementaryequationsaresuppliedbytheconditionsofdeformationofthebeam)6§6－１概述研究范围：等直梁在对称弯曲时位移的计算。研究目的：①对梁作刚度校核；②解超静定梁（变形几何条件提供补充方程）。71).Deflection：Thedisplacementofthecentroidofasectioninadirectionperpendiculartotheaxisofthebeam.Itisdesignatedbytheletterv.Itispositiveifitsdirectionisthesameasf，ornegative.3、Therelationbetweentheangleofrotationanddeflectioncurve：1、Twobasicquantitiesoftomeasuredeformationofthebeam(1)ddtgfxf小变形PxvCC1f2).Angleofrotation：Theanglebywhichcrosssectionturnswithrespecttoitsoriginalpositionabouttheneutralaxis.itisdesignatedbytheletter.Itispositiveiftheangleofrotationrotatesintheclockwisedirection,ornegative.2、deflectioncurve：Thecurvewhichtheaxisofthebeamwastransformedintoafterdeformationiscalleddeflectioncurve.Itsequationisv=f(x)8健身增肌二次发育WeiXinTaoBao91.挠度：横截面形心沿垂直于轴线方向的线位移。用v表示。与f同向为正，反之为负。2.转角：横截面绕其中性轴转动的角度。用表示，顺时针转动为正，反之为负。二、挠曲线：变形后，轴线变为光滑曲线，该曲线称为挠曲线。其方程为：v=f(x)三、转角与挠曲线的关系：一、度量梁变形的两个基本位移量(1)ddtgfxf小变形PxvCC1f10§6-2APPROXIMATEDIFFERENTIALEQUATIONOFTHEDEFLECTURECURVEOFTHEBEAMANDITSINTEGRALzzEIxM)(11、ApproximatedifferentialequationofthedeflectioncurvezzEIxMxf)()(Formula(2)isapproximatedifferentialequationofthedeflectioncurve.EIxMxf)()(……（2）)()1()(1232xffxfSmalldeformationfxM00)(xffxM00)(xf11§6-2梁的挠曲线近似微分方程及其积分zzEIxM)(1一、挠曲线近似微分方程zzEIxMxf)()(式（2）就是挠曲线近似微分方程。EIxMxf)()(……（2）)()1()(1232xffxf小变形fxM00)(xffxM00)(xf12)()(xMxfEIForthestraightbeamwiththesameshapeandequalsectionarea,approximatedifferentialequationofthedeflectioncurvemaybewrittenasthefollowingform：2、Determinetheequationofthedeflectioncurve(elasticcurve))()(xMxfEI1d))(()(CxxMxfEI21d)d))((()(CxCxxxMxEIf1).integralofthedifferentialequation2).BoundaryconditionsofthedisplacementPABCPD13)()(xMxfEI对于等截面直梁，挠曲线近似微分方程可写成如下形式：二、求挠曲线方程（弹性曲线）)()(xMxfEI1d))(()(CxxMxfEI21d)d))((()(CxCxxxMxEIf1.微分方程的积分2.位移边界条件PABCPD14Discussion：①Fittothethinnerandlongerbeamthatmadeupfromlinearelasticmaterialwhenitsdeformationisofplanarbendingandsmaller.②Maybeappliedtodeterminethedisplacementsofthebeamwiththesamesectionshapeandequalsectionareaactingvariousloads.③Integrateconstantsmaybedeterminedbythegeometricconditions相容（boundaryconditions、continuityconditions).④Advantages：Rangethatitbeappliediswide，maydeterminedirectlyaccuracysolution；Defects：Complicatedcalculation.Displacementconditionsatthesupports：Continuityconditions：Slidingconditions：0Af0Bf0Df0DCCffCCCrightCleftorCrightCleftffor15讨论：①适用于小变形情况下、线弹性材料、细长构件的平面弯曲。②可应用于求解承受各种载荷的等截面或变截面梁的位移。③积分常数由挠曲线变形的几何相容条件（边界条件、连续条件）确定。④优点：使用范围广，可以编程求出较精确的数值解；缺点：计算比较繁琐。支点位移条件：连续条件：光滑条件：0Af0Bf0Df0DCCffCC右左或写成CC右左或写成CCff16Example1Determinetheelasticcurves、maximumdeflectionsandmaximumanglesofrotationofthefollowingbeams.Setupthecoordinatesandwriteoutthebendingmomentequation：)()(LxPxMWriteoutthedifferentialequationandintegrateitDeterminatetheintegralconstantsbytheboundaryconditions)()(xLPxMfEI12)(21CxLPfEI213)(61CxCxLPEIf061)0(23CPLEIf021)0()0(12CPLfEIEI322161;21PLCPLCSolution：PLxf17例1求下列各等截面直梁的弹性曲线、最大挠度及最大转角。建立坐标系并写出弯矩方程)()(LxPxM写出微分方程的积分并积分应用位移边界条件求积分常数)()(xLPxMfEI12)(21CxLPfEI213)(61CxCxLPEIf061)0(23CPLEIf021)0()0(12CPLfEIEI322161;21PLCPLC解：PLxf18Writeouttheequationoftheelasticcurveandplotitscurve3233)(6)(LxLxLEIPxfEIPLLff3)(3maxEIPLL2)(2maxThemaximumdeflectionandthemaximumangleofrotationxfPL19写出弹性曲线方程并画出曲线3233)(6)(LxLxLEIPxfEIPLLff3)(3maxEIPLL2)(2max最大挠度及最大转角xfPL20Solution：Setupthecoordinatesandwriteoutthebendingmomentequation)(0)0()()(LxaaxaxPxM112)(21DCxaPfEI21213)(61DxDCxCxaPEIf)(0)0()(LxaaxxaPfEIxfPLaWriteoutthedifferentialequationandintegrateit21解：建立坐标系并写出弯矩方程)(0)0()()(LxaaxaxPxM写出微分方程的积分并积分112)(21DCxaPfEI21213)(61DxDCxCxaPEIf)(0)0()(LxaaxxaPfEIxfPLa22Determinetheintegralconstantsbyboundaryconditions061)0(23CPaEIf021)0(12CPaEI32221161;21PaDCPaDC)()(afaf)()(aa11DC2121DaDCaCPLaxf23应用位移边界条件求积分常数061)0(23CPaEIf021)0(12CPaEI32221161;21PaDCPaDC)()(afaf)()(aa11DC2121DaDCaCPLaxf24Writeouttheequationofth