英汉双语材料力学11PPT推荐.ppt
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5111变形能的普遍表达式变形能的普遍表达式一、能量原理:
一、能量原理:
二、杆件变形能的计算:
1.1.轴向拉压杆的变形能计算:
轴向拉压杆的变形能计算:
弹性体内部所贮存的变形能,在数值上等于外力所作的功,即利用这种功能关系分析计算可变形固体的位移、变形和内力的方法称为能量方法。
62.2.Calculationofthestrainenergyofrodsintorsion:
3.3.Calculationofstrainenergyofrodsinbending:
orDensityofthestrainenergy:
orDensityofthestrainenergy:
72.2.扭转杆的变形能计算:
扭转杆的变形能计算:
3.3.弯曲杆的变形能计算:
弯曲杆的变形能计算:
83、Generalexpressionsofthestrainenergy:
Strainenergyisindependentoftheorderofloading.Deformationsduetomutuallyindependentloadmaybesummedupeachother.Forslendercolumns,thestrainenergyduetoshearingforcesmaybeneglected.Deflectionfactorofshear9三、变形能的普遍表达式:
三、变形能的普遍表达式:
变形能与加载次序无关;
相互独立的力(矢)引起的变形能可以相互叠加。
细长杆,剪力引起的变形能可忽略不计。
10Solution:
Inenergymethod(workdonebyexternalforcesisequaltothestrainenergy)DetermineinternalforcesDetermineinternalforcesABendingmoment:
Torque:
Example1Asemicirclerodasshowninthefigureislieinhorizontalplane.AverticalforcePactatitspointA.DeterminethedisplacementofpointAinverticaldirection.PROQMNMTAAPNBjjTO11MN例例11图示半圆形等截面曲杆位于水平面内,在A点受铅垂力P的作用,求A点的垂直位移。
解:
用能量法(外力功等于应变能)求内力APROQMTAAPNBjjTO12WorkdonebyexternalforcesisequaltothestrainenergyWorkdonebyexternalforcesisequaltothestrainenergyStrainenergyStrainenergy:
Letthen13外力功等于应变能变形能:
14ExampleExample2DeterminethedeflectionofpointCbytheenergymethod,wherethebeamisofequalsectionandstraight.Solution:
WorkdonebyexternalWorkdonebyexternalforcesisequaltothestrainenergyforcesisequaltothestrainenergyByusingsymmetryweget:
Thinking:
Forthedistributedload,canwedeterminethedisplacementofpointCbythismethod?
qCaaAPBfLet15例例2用能量法求C点的挠度。
梁为等截面直梁。
外力功等于应变能应用对称性,得:
思考:
分布荷载时,可否用此法求C点位移?
qCaaAPBf16112MOHRSTHEOREM(METHODOFUNITFORCE)DeterminethedisplacementfAofanarbitrarypointA.1、Provementofthetheorem:
aAFigfAq(x)FigcA0P=1q(x)fAFigbA=1P017112莫尔定理莫尔定理(单位力法单位力法)求任意点A的位移fA。
一、定理的证明:
aA图fAq(x)图cA0P=1q(x)fA图bA=1P018Mohrstheorem(methodofunitforce)2、GeneralformofMohrstheorem19莫尔定理莫尔定理(单位力法单位力法)二、普遍形式的莫尔定理二、普遍形式的莫尔定理203、WhatwemustpayattentiontoasweapplyMohrstheorem:
CoordinateofCoordinateofM0(x)mustbecoincidewiththatofM(x).Foreachsegmentthecoordinatemaybesetupfreely.MohrsMohrsintegrationmustintegrationmustbethroughthewholestructure.bethroughthewholestructure.M0:
Theinternalforceofthestructureasweactageneralizedunitforcealongthedirection,ofthegeneralizeddisplacementthatistobedetermined,wheretheappliedforceistakenout.M(x):
Theinternalforceofthestructureactedbyoriginalloads.TheproductoftheappliedgeneralizedunitforceandthegeneralizedTheproductoftheappliedgeneralizedunitforceandthegeneralizeddisplacementtobedetermineddeterminedmustbeofthedimensionofworkdisplacementtobedetermineddeterminedmustbeofthedimensionofwork.21三、使用莫尔定理的注意事项:
三、使用莫尔定理的注意事项:
M0(x)与M(x)的坐标系必须一致,每段杆的坐标系可自由建立。
莫尔积分必须遍及整个结构。
M0去掉主动力,在所求广义位移广义位移广义位移广义位移点,沿所求广义位移广义位移广义位移广义位移的方向加广义单位力广义单位力广义单位力广义单位力时,结构产生的内力。
M(x):
结构在原载荷下的内力。
所加广义单位力与所求广义位移之积,必须为功的量纲。
22Example33DeterminethedisplacementandtheangleofrotationofpointCbytheenergymethod.Solution:
PlotthediagramofthestructureactedbytheunitloadPlotthediagramofthestructureactedbytheunitloadDeterminetheinternalforceBAaaCqBAaaC0P=1x23例例33用能量法求C点的挠度和转角。
画单位载荷图求内力BAaaCqBAaaC0P=1x24SymmetrySymmetryDeformationBAaaC0P=1BAaaCqx()25变形BAaaC0P=1BAaaCqx()26Determinetheangleofrotation.Setupthecoordinateagain(asshowninthefigureDeterminetheangleofrotation.Setupthecoordinateagain(asshowninthefigure)qBAaaCx2x1BAaaCMC0=1d)()()()()(00)(00+=aBCaABxEIxMxMdxEIxMxM=027求转角,重建坐标系(如图)qBAaaCx2x1BAaaCMC0=1d)()()()()(00)(00+=aBCaABxEIxMxMdxEIxMxM=028Solution:
PlotthediagramofPlotthediagramofthestructureactedbyaunitloadthestructureactedbyaunitloadDeterminetheinternalforce51020A300P=60NBx500Cx151020A300Bx500C=1P0Example4Example4Afoldingrodisshowninthefigure.AbearingisatpositionAandtherodmayrotatefreelyinthebearingbutcannotmoveupanddown.Knowing:
E=210Gpa,G=0.4E,DeterminetheverticaldisplacementofpointB.29例例44拐