桥梁设计外文翻译.docx
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桥梁设计外文翻译.docx
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桥梁设计外文翻译
外文文献翻译
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),andaretheunitvectorsofxiandxi′,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
wher
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