土木工程专业翻译平衡方程.docx
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土木工程专业翻译平衡方程.docx
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土木工程专业翻译平衡方程
外文原文
7.2EquilibriumEquations
7.2.1EquilibriumEquationandVirtualWorkEquation
ForanyvolumeVofamaterialbodyhavingAassurfacearea,asshowninFigure7.2,ithasthefollowingconditionsofequilibrium:
FIGURE7.2Derivationofequationsofequilibrium.
Atsurfacepoints
Atinternalpoints
Wherenirepresentsthecomponentsofunitnormalvectornofthesurface;Tiisthestressvectoratthepointassociatedwithn;σji,jrepresentsthefirstderivativeofσijwithrespecttoxj;andFiisthebodyforceintensity.Anysetofstressesσij,bodyforcesFi,andexternalsurfaceforcesTithatsatisfiesEqs.(7.1a-c)isastaticallyadmissibleset.
Equations(7.1bandc)maybewrittenin(x,y,z)notationas
and
Whereσx,σy,andσzarethenormalstressin(x,y,z)directionrespectively;τxy,τyz,andsoon,arethecorrespondingshearstressesin(x,y,z)notation;andFx,Fy,andFzardthebodyforcesin(x,y,z,)direction,respe-
ctively.
Theprincipleofvirtualworkhasprovedaverypowerfultechniqueofsolvingproblemsandprovidingproofsforgeneraltheoremsinsolidmechanics.Theequationofvirtualworkusestwoindependentsetsofequilibriumandcompatible(seeFigure7.3,whereAuandATrepresentdisplacementandstressboundary),asfollows:
compatibleset
equilibriumset
or
whichstatesthattheexternalvirtualwork(δWext)equalstheinternalvirtualwork(δWint).
HeretheintegrationisoverthewholeareaA,orvoluneV,ofthebody.
Thestressfieldδij,bodyforcesFi,andexternalsurfaceforcesTiareastaticallyadmissiblesetthatsatisfiesEqs.(7.1a–c).Similarly,thestrainfieldεij﹡andthedisplacementui﹡areacompatiblekinematics
setthatsatisfiesdisplacementboundaryconditionsandEq.(7.16)(seeSection7.3.1).Thismeanstheprincipleofvirtualworkappliesonlytosmallstrainorsmalldeformation.
Theimportantpointtokeepinmindisthat,neithertheadmissibleequilibriumsetδij,Fi,andTi(Figure7.3a)northecompatiblesetεij﹡andui﹡(Figure7.3b)needbetheactualstate,norneedtheequilibriumandcompatiblesetsberelatedtoeachotherinanyway.Intheotherwords,thesetwosetsarecompletelyindependentofeachother.
7.2.2EquilibriumEquationforElements
Foraninfinitesimalmaterialelement,equilibriumequationshavebeensummarizedinSection7.2.1,whichwilltransferintospecificexpressionsindifferentmethods.AsinordinaryFEMorthedisplacementmethod,itwillresultinthefollowingelementequilibriumequations:
FIGURE7.4Planetrussmember–endforcesanddisplacements.(Source:
Meyers,
V.J.,MatrixAnalysisofStructures,NewYork:
Harper&Row,1983.Withpermission.)
Where{
}eand{
}earetheelementnodalforcevectoranddisplacementvector,respectively,while{
}eiselementstiffnessmatrix;theoverbarheremeansinlocalcoordinatesystem.
Intheforcemethodofstructuralanalysis,whichalsoadoptstheideaofdiscretization,itisprovedpossibletoidentifyabasicsetofindependentforcesassociatedwitheachmember,inthatnotonlyaretheseforcesindependentofoneanother,butalsoallotherforcesinthatmemberaredirectlydependentonthisset.Thus,thissetofforcesconstitutestheminimumsetthatiscapableofcompletelydefiningthestressedstateofthemember.Therelationshipbetweenbasicandlocalforcesmaybeobtainedbyenforcingoverallequilibriumononemember,whichgives
Where[L]=theelementforcetransformationmatrixand{P}e=theelementprimaryforcesvector.ItisimportanttoemphasizethatthephysicalbasisofEq.(7.5)ismemberoverallequilibrium.
Takeaconventionalplanetrussmemberforexemplification(seeFigure7.4),onehas
FIGURE7.5Coordinatetransformation.
and
whereEA/l=axialstiffnessofthetrussmemberandP=axialforceofthetrussmember.
7.2.3CoordinateTransformation
ThevaluesofthecomponentsofvectorV,designatedbyv1,v2,andv3orsimply,areassociatedwiththechosensetcoordinateaxes.OftenitisnecessarytoreorientthereferenceaxesandevaluatenewvaluesforthecomponentsofVinthenewcoordinatesystem.AssumingthatVhascomponentsviandvi′intwosetsofright-handedCartesiancoordinatesystemsxi(old)andxi′(new)havingthesameorigin(seeFigure7.5),and
aretheunitvectorsofxiandxi′,respectively.Then
Where
thatis,thecosinesoftheanglesbetweenxi′andxjaxesforiandjrangingfrom1to3;and[α]=(lij)3×3iscalledcoordinatetransformationmatrixfromtheoldsystemtothenewsystem.
Itshouldbenotedthattheelementsoflijormatrix[α]arenotsymmetrical,lij≠lji.Forexample,l12isthecosineofanglefromx1′tox2andl21isthatfromx2′tox1(seeFigure7.5).Theangleisassumedtobemeasuredfromtheprimedsystemtotheunprimedsystem.
Foraplanetrussmember(seeFigure7.4),thetransformationmatrixfromlocalcoordinatesystemtoglobalcoordinatesystemmaybeexpressedas
whereαistheinclinedangleofthetrussmemberwhichisassumedtobemeasuredfromtheglobaltothelocalcoordinatesystem.
7.2.4EquilibriumEquationforStructures
Fordiscretizedstructure,theequilibriumofthewholestructureisessentiallytheequilibriumofeachjoint.Afterassemblage,
ForordinaryFEMordisplacementmethod
Forforcemethod
where{F}=nodalloadingvector;[K]=totalstiffnessmatrix;{D}=nodaldisplacementvector;[A]=totalforcestransformationmatrix;{P}=totalprimaryinternalforcesvector.
Itshouldbenotedthatthecoordinatetransformationforeachelementfromlocalcoordinatestotheglobalcoordinatesystemmustbedonebeforeassembly.
Intheforcemethod,Eq.(7.11)willbeadoptedtosolveforinternalforcesofastaticallydeterminatestructure.Thenumberofbasicunknownforcesisequaltothenumberofequilibriumequationsavailabletosolveforthemandtheequationsarelinearlyindependent.Forstatically
unstablestructures,analysismustconsidertheirdynamicbehavior.Whenthenumberofbasicunknownforcesexceedsthenumberofequilibriumequations,thestructureissaidtobestaticallyindeterminate.Inthiscase,someofthebasicunknownforcesarenotrequiredtomaintainstructuralequilibrium.Theseare“extra”or“redundant”forces.Toobtainasolutionforthefullsetofbasicunknownforces,itisnecessarytoaugmentthesetofindependentequilibriumequationswithelastic
behaviorofthestructure,namely,theforce–displacementrelationsofthestructure.Havingsolvedforthefullsetofbasicforces,wecandeterminethedisplacementsbybacksubstitution.
7.2.5InfluenceLinesandSurfaces
Inthedesignandanalysisofbridgestructures,itisnecessarytostudytheeffectsintriguedbyloadsplacedinvariouspositions.Thiscanbedoneconvenientlybymeansofdiagramsshowingtheeffectofmovingaunitloadacrossthestructures.Suchdiagramsarecommonlycalledinfluencelines(forframedstructures)orinfluencesurfaces(forplates).Observethatwhereasamomentorsheardiagramshowsthevariationinmomentorshearalongthestructureduetosomeparticularpositionofload,aninfluencelineorsurfaceformomentorshearshowsthevariationofmomentorshearataparticularsectionduetoaunitloadplacedanywherealongthestructure.
Exactinfluencelinesforstaticallydeterminatestructurescanbeobtainedanalyticallybystaticsalone.FromEq.(7.11),thetotalprimaryinternalforcesvector{P}canbeexpressedas
bywhichgivenaunitloadatonenode,theexcitedinternalforcesofallmemberswillbeobtained,andthusEq.(7.12)givestheanalyticalexpressionofinfluencelinesofallmemberinternalforcesfordiscretizedstructuressubjectedtomovingnodalloads.
Forstaticallyindeterminatestructures,influencevaluescanbedetermineddirectlyfromaconsiderationofthegeometryofthedeflectedloadlineresultingfromimposingaunitdeformationcorrespondingtothefunctionunderstudy,basedontheprincipleofvirtualwork.Thismaybetterbedemonstratedbyatwo-spancontinuousbeamshowninFigure7.6,wheretheinfluencelineofinternalbendingmomentatsectionMBisrequired.
FIGURE7.6Influencelineofatwo-spancontinuousbeam.
FIGURE7.7DeformationofalineelementforLagrangianandEluerianvariables.
CuttingsectionBtoMBexposeandgiveitaunitrelativerotationδ=1(seeFigure7.6)andemployingtheprincipleofvirtualworkgives
Therefore,
whichmeanstheinfluencevalueofMBequalstothedeflectionv(x)ofthebeamsubjectedtoaunitrotationatjointB(representedbydashedlineinFigure7.6b).Solvingforv(x)canbecarriedouteasilyreferringtomaterialmechanics.
中文译文
7.2平衡方程
7.2.1平衡方程和虚功方程
对于任何有一定体积的材料都有一个表面积,如图7.2所示,它具有以下平衡条件:
在表面的点:
图7.2平衡方程的推导
在内部的点
其中,ni表示n表面的单位法向量;Ti表示与n相关的向量点应力;σji,j表示σij关于xj的一阶导数;而Fi表示体积力。
任何一系列满足方程(7.1a)-(7.1c)的应力σij、体积力Fi、表面力Ti都是一个静态的容许集。
方程(7.1b和7.1c)可以写成如下所示(x,y,z)的形式,
和
其中,σx,σy,和σz分别是(x,y,z)方向的正应力,τxy和τy等表示(x,y,z)中的剪应力;Fx,Fy和Fz分别表示(x,y,z)方向的体积力
虚功原理被证明是一个解决问题的非常有效的方法,它在固体力学领域为一般性定理提供了证明。
虚功方程采用两套独立的平衡集和兼容集(见图7.3,其中Au和AT分别表示位移边界和应力边界),如下所示:
图7.3虚功方程的两独立集
相容集
平衡集
或是
它表明外力虚功(δWext)等于内力虚功(δWint)。
这个集成包括了物体的整个面积或体积。
应力场δij,体积力Fi和外部表面力Ti是一个满足方程(7.1a-7.1c)的静态容许集。
相似的,应变场εij﹡和位移ui﹡是一个满足位移边界条件和方程(7.16)(见7.3.1节)的兼容的运动学集。
这意味着虚功原理仅适用于小应变或变形小的情况。
需要注意的重要一点是,无论容许均衡集δij,Fi,和Ti(图7.3a),还是兼容集εij﹡和ui﹡都不需要明确的状态,也不需要平衡集和兼容集以任何方式彼此相关。
换句话说,这两个集合是完全相互独立的。
7.2.2单元的平衡方程
对于一个无穷小单元,平衡方程已经在7.2.1节中总结,这可以转化成不同方法中的具体表达式。
正如在普通有限元法、位移法中,它可以导出以下单元平衡方程:
图7.4平面桁架端承力和位移(来源:
Meyers,V.J.,《结构矩阵分析》,1983年纽约Harper&Row出版授权社出版)
其中,{
}e和{
}e分别表示单元节点力向量和位移向量,而{
}e表示单元刚度矩阵;这里的上划线表示局部坐标系。
在力法的结构分析中采用了离散化的方法,这被证明可以用来确定一套与各构件相关联的基本独立的力,在其中不仅这些力彼此之间相互独立,而且构件中的所有其他的力直接依赖于本集。
因此,这些力构成的最小集能够完全定义构件的受力状态。
基本力与局部力的关系可以通过乘以整体平衡的一个构件来获得,
如下所示:
其中,[L]表示单元力的变换矩阵,{P}e表示单元基本的向量力。
需要强调的是,物理基本方程(7.5)是所有平衡的组成
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