毕设翻译3196106.docx
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毕设翻译3196106.docx
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毕设翻译3196106
CHAPTER6
Arches
TheRomanswerethefirsttousearchesasmajorstructuralelements,employingthem,mainlyinsemicircularform,inbridgeandaqueductconstructionandforroofsupports,particularlythebarrelvault.Theirchoiceofthesemicircularshapewasduetotheeasewithwhichsuchanarchcouldbesetout.Generallythesearches,asweshallsee,carriedmainlycompressiveloadsandwerethereforeconstructedfromstoneblocks,orvoussoirs,wherethejointswereeitherdryorusedweakmortar.DuringtheMiddleAges,Gothicarches,distinguishedbytheirpointedapex,wereusedtoalargeextentintheconstructionofthegreatEuropeancathedrals.Thehorizontalthrustdevelopedatthesupports,orspringings,andcausedbythetendencyofanarchtoflatten’underloadwasfrequentlyresistedbyj?
yiug
buttresses.Thistypeofarchwasalsousedextensivelyinthel9thcentury.
Inthe18thcenturymasonryarcheswereusedtosupportbridgesoverthelargenumberofcanalsthatwerebuiltinthatperiod.Manyofthesebridgessurvivetothepresentdayandcarryloadsunimaginedbytheirdesigners.
Todayarchesareusuallymadeofsteelorofreinforcedorprestressedconcreteandcansupportbothtensileaswellascompressiveloads.Theyareusedtosupportbridgedecksandroofsandvaryinspanfromafewmetresinaroofsupportsystemtoseveralhundredmetresinbridges.AlineexampleofasteelarchbridgeistheSydneyharbourbridgeinwhichthedeckissupportedbyhangerssuspendedfromthearch(seeFigs1.6(a)and(b)forexamplesofbridgedeckssupportedbyarches).Archesareconstructedinavarietyofforms.Theircomponentsmaybestraightorcurved,butgenerallyfallintotw’ocategories.Thefirst,whichweshallconsiderinthischapter,isthethree-pinnedarchwhichisstaticallydeterminate,whereasthesecond,thetwo-pinnedarch,isstaticallyindeterminateandwillbeconsideredin
Chapterl6.
Initiallyweshallexaminethemannerinwhicharchescarryloads.
6.1Thelineararch
Thei’eisadirectrelationshipbetweentheactionofaflexiblecableincarryingloadsandtheactionofan:
irch.InSection5.lwedeterminedthetensileforcesinthesegmentsoflightweiJ;htcablescarryingconcentratedloadsandsawthatthegeometryofacablechangeclunderdifferentloadingsystems;hence,forexample,thetwogeometriesofthesamecableinFigs5.2(a)and(b).
LetussupposethatthecableinFig.5.2(a)ismadeupofthreebarsorlinksAC,CDandDBhingedtogetheratCandDandpinnedtothesupportsatAandB.Iftheloadingremainsunchangedthedeflectedshapeofthethree-linkstructurewillbeidenticaltothatofthecableinFig.5.2(a)andisshowninFig.6.1(a).Furthermorethetensioninalinkwillbeexactlythesameasthetensioninthecorrespondingsegmentofthecable.Nowsupposethatthethree-linkstructureofFig.6.1(a)isinvertedasshowninFig.6.1(b)andthattheloadsiV,andW,›zeappliedasbefore.InthissituationtheforcesinthelinkswillbeidenticalinmagnitudetothoseinFig.6.1(a)butwillnowbecompressiveasopposedtotensile;thestructureshowninFig.6.1(b)ispatentlyanarch.
Thesameargumentcanbeappliedtoanycableandloadingsystemsothattheinternalforcesinanarchmaybededucedbyanalysingacablehavingexactlythesameshapeandcarryingidenticalloads,afactfirstrealizedbyRobertHookeinthe17thcentury.AsintheexampleinFig.6.1theinternalforcesinthearchwillhavethesamemagnitudeasthecorrespondingcableforcesbutwillbecompressive,nottensile.
Itisobviousfromtheabovethattheinternalforcesinthearchactalongtheaxesofthedifferentcomponentsandthatthearchisthereforenotsubjectedtointernalshearforcesandbendingmoments;anarchinwhichtheinternalforcesarepurelyaxialiscalledalineararch.Wealsodeduce,fromSection5.2,thattheinternalforcesinanarchwhoseshapeisthatofaparabolaandwhichcarriesauniformhorizontallydistributedloadarepurelyaxial.Further,itwillnowhavebecomeclearwhytheinternalmembersofabowstringtruss(Section4.1)carryingloadsofequalmagnitudealongitsupperchoraljointscarryzeroforce.
Thereis,however,amajordifferencebetweenthebehaviourofthetwostructuresinFigs6.l(a)and(b).AchangeinthevaluesoftheloadsIV,and+willmerelyresultinachangeinthegeometryofthestructureinFig.6.I(a),whereastheslightestchangesinthevaluesofIV,andIV.inFig.6.1(b)willresultinthecollapseofthearchasamechanism.InthisparticularcasecollapsecouldbepreventedbyreplacingthepinnedjointatC(orD)byarigidjointasshowninFig.6.2.The
forcesinthemembersremainunchangedsincethegeometryofthestructureisunchanged,butthearchisnowstableandhasbecomeathree-pinnedarchwhich,asweshallsee,isstaticallydeterminate.
(b)
Fig.6.1Equivalenceofcableandarchstructures
Fig.6.2Linearthree-pinnedarch
Thelineararch121
IfnowthepinnedjointatDwasreplacedbyarigidjoint,theforcesinthememberswouldremainthesame,butthearchhasbecomeatwo-pinnedarch.Inthiscase,becauseofthetensioncableequivalence,thearchisstaticallydeterminate.Itisimportanttorealize,however,thattheaboveargumentsonlyapplyforthesetofloadsIY,and+2whichproducethepanicularshapeofcableshowninFig.6.1(a).Iftheloadswererepositionedorchangedinmagnitude,thetwo-pinnedarchwouldbecomestaticallyindeterminateandwouldprobablyceasetobealineararchsothatbendingmomentsandshearforceswouldbeinduced.Thethree-pinnedarchofFig.6.2wouldalsobecomenon-lineariftheloadswererepositionedorchangedin
magnitude.
Intheabovewehaveignoredtheeffectonthegeometryofthearchcausedbytheshorteningofthemembers.Theeffectofthisonthethree-pinnedarchisnegligiblesincethepinscanaccommodatethesmallchangesinanglebetweenthememberswhichthiscauses.Thisisnotthecaseinatwo-pinnedarchorinanarchwithnopinsatall(ineffectaportalframe)sothatbendingmomentsandshearforcesareinduced.However,solongastheloads(WeandW;inthiscase)remainunchangedinmagnitudeandposition,thecorrespondingstressesare‘secondary’andwillhavelittleeffectontheaxialforces.
Thelineararch,inwhichtheinternalforcesarepurelyaxial,isimportantforthestructuraldesignersincethelineararchshapegivesthesmalleststresses.If,however,thethrustlineisnotaxial,bendingstressesareinducedandthesecancausetensionontheinnerorouterfaces(theintradosandextrados)orthearch.Inamasonryarchinwhichthejointsareeitherdryormadeusingaweakmortar,thiscanleadtocrackingandpossiblefailure.Furthermore,ifthethrustlineliesoutsidethefacesofthearch,instabilityleadingtocollapsecanalsooccur.WeshalldeduceinSection9.2thatfornotensiontobedevelopedinarectangularcross-section,thecompressiveforceonthesectionmustliewithinthemiddlethirdofthesection.
Insmall-spanarchbridges,thesefactorsarenotofgreatimportancesincethegreatestloadsonthearchcomefromvehiculartraffic.Theseloadsvarywiththesizeofthevehicleanditspositiononthebridge,sothatitisgenerallyimpossibleforthedesignertoachievealineararch.Ontheotherhand,inlarge-spanarchbridges,theself-weightofthearchformsthemajorportionoftheloadthearchhastocarry.InSection5.2wesawthatacableunderitsownweighttakesuptheshapeofacatenary.Itfollowsthattheidealshapeforanarchofconstantthicknessisaninvertedcatenary.However,intheanalysisofthethree-pinnedarchweshallassumeageneralcaseinwhichshearforcesandbendingmoments,aswellasaxialforces,arepresent.
6.2Thethree-pinnedarch
Athree-pinnedarchwouldbeusedinsituationswherethereisapossibilityofsupportdisplacement;this,inatwo-pinnedarch,wouldinduceadditionalstresses.Intheanalysisofathree-pinnedarchthefirststep,generally,istodeterminethesupportreactions.
Supportreactions-supportsonsamehorizontallevel
ConsiderthearchshowninFig.6.3.Itcarriesaninclinedconcentratedload,ID,atagivenpointD,ahorizontaldistance‹zfromthesupportpointA.Theequationoftheshapeofthearchwillgenerallybeknownsothatthepositionofspecifiedpointsonthearch,sayD,canbeobtained.Weshallsupposethatthethirdpinispositionedatthecrown,C,ofthearch,althoughthisneednotnecessarilybethecase;theheightorriseofthearchish.
ThesupportsatAandBarepinnedbutneithercanbearollersupportorthearchwouldcollapse.Therefore,inadditiontothetwoverticalcomponentsofthereactionsatAandB,therewillbehorizontalcotSA,HaB.H•Usthere
arefourunknowncomponentsofreactionbutonlythreeequationsofoverall
equilibrium(Eqs(2.10))sothatanadditionalequationisrequired.ThisisobtainedfromthefactthatthethirdpinatCisunabletotransmitbendingmomentsalthough,obviously,itisabletotransmitshearforces.
Thus,fromtheoverallverticalequilibriumofthearchinFig.6.3,wehave
andfromthehorizontalequilibrium
Nowtakingmomentsabout,say,B,
fig,L—IVcoso(L—a)—Wsinixlie-0
Fig.6.3Three-pinnedarch
(6.2)
(6.3)
TheinternalmomentatCiszerosothatwecantakemomentsaboutCofforcestotheleftorrightofC.AslightlysimplerexpressionresultsbyconsideringforcestotheleftofC;thus
2
’A.V=0
(6.4)
Equations(6.1)—(6.4)enablethefourcomponentsofreactiontobe
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