资源描述：
《计算力学教学内容.ppt》由会员上传分享，免费在线阅读，更多相关内容在教育资源-天天文库。
1、计算力学Forlinearlyelasticdeformationsofalinearspring,Figure1.5Force-deflectionrelationforalinearelasticspring.Theslopeoftheforce-deflectionlineisthespringconstantsuchthatTherefore,theworkrequiredtodeformsuchaspringis(1.40)springconstant,i.e.,stiffnessStrainenergyMechanicalwork2weobse
2、rvethattheworkandresultingelasticpotentialenergyarequadraticfunctionsofdisplacementUtilizingEquation1.28,(1.41)thisresultisconvertedtoadifferentformasfollows:(1.42)whereisthetotalvolumeofdeformedmaterial[1.28]thestrainenergycanbewrittenas3thequantityisstrainenergyperunitvolume,als
3、oknownasstrainenergydensity.Ingeneral,foruniaxialloading,thestrainenergyperunitvolumeisdefinedby(1.43)Equation1.43representstheareaundertheelasticstress-straindiagram.Strainenergy4Castigliano’sFirstTheoremForanelasticsysteminequilibrium,thepartialderivativeoftotalstrainenergywithr
4、especttodeflectionatapointisequaltotheappliedforceinthedirectionofthedeflectionatthatpoint.Consideranelasticbodysubjectedtoforcesforwhichthetotalstrainenergyisexpressedas(1.44)whereisthedeflectionatthepointofapplicationofforce,inthedirectionofthelineofactionoftheforce.5Ifallpoints
5、ofloadapplicationarefixedexceptone,say,i,andthatpointismadetodeflectaninfinitesimalamountbyanincrementalinfinitesimalforce,thechangeinstrainenergyis(1.45)whereitisassumedthattheoriginalforceisconstantduringtheinfinitesimalchange.TheintegralterminEquation1.45involvestheproductofinf
6、initesimalquantitiesandcanbeneglectedtoobtain(1.46)ignored6whichinthelimitasapproacheszerobecomes(1.47)Review:Castigliano’sFirstTheoremForanelasticsysteminequilibrium,thepartialderivativeoftotalstrainenergywithrespecttodeflectionatapointisequaltotheappliedforceinthedirectionofthed
7、eflectionatthatpoint.ThefirsttheoremofCastiglianoisapowerfultoolforfiniteelementformulation,asisnowillustratedforthebarelement.7CombiningEquations1.30,1.31,and1.43,thetotalstrainenergyforthebarelementisgivenby(1.48)ApplyingCastigliano’stheoremwithrespecttoeachdisplacementyields(1.
8、49)(1.50)whichareobservedtobeiden
