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7Rigid-FrameStructuresArigid-framehigh-risestructuretypicallycomprisesparallelororthogonallyarrangedbentsconsistingofcolumnsandgirderswithmomentresistantjoints.Resistancetohorizontalloadingisprovidedbythebendingresistanceofthecolumns,girders,andjoints.Thecontinuityoftheframealsocontributestoresistinggravityloading,byreducingthemomentsinthegirders.Theadvantagesofarigidframearethesimplicityandconvenienceofitsrectangularform.Itsunobstructedarrangement,clearofbracingmembersandstructuralwalls,allowsfreedominternallyforthelayoutandexternallyforthefenestration.Rigidframesareconsideredeconomicalforbuildingsofupto'about25stories,abovewhichtheirdriftresistanceiscostlytocontrol.If,however,arigidframeiscombinedwithshearwallsorcores,theresultingstructureisverymuchstiffersothatitsheightpotentialmayextendupto50storiesormore.Aflatplatestructureisverysimilartoarigidframe,butwithslabsreplacingthegirdersAswitharigidframe,horizontalandverticalloadingsareresistedinaflatplatestructurebytheflexuralcontinuitybetweentheverticalandhorizontalcomponents.Ashighlyredundantstructures,rigidframesaredesignedinitiallyonthebasisofapproximateanalyses,afterwhichmorerigorousanalysesandcheckscanbemade.Theproceduremaytypicallyincludethefollowingstages:1.Estimationofgravityloadforcesingirdersandcolumnsbyapproximatemethod. 2.Preliminaryestimateofmembersizesbasedongravityloadforceswitharbitraryincreaseinsizestoallowforhorizontalloading.3.Approximateallocationofhorizontalloadingtobentsandpreliminaryanalysisofmemberforcesinbents.4.Checkondriftandadjustmentofmembersizesifnecessary.5.Checkonstrengthofmembersforworstcombinationofgravityandhorizontalloading,andadjustmentofmembersizesifnecessary.6.Computeranalysisoftotalstructureformoreaccuratecheckonmemberstrengthsanddrift,withfurtheradjustmentofsizeswhererequired.Thisstagemayincludethesecond-orderP-Deltaeffectsofgravityloadingonthememberforcesanddrift..7.Detaileddesignofmembersandconnections.Thischapterconsidersmethodsofanalysisforthedeflectionsandforcesforbothgravityandhorizontalloading.Themethodsareincludedinroughlytheorderofthedesignprocedure,withapproximatemethodsinitiallyandcomputertechniqueslater.StabilityanalysesofrigidframesarediscussedinChapter16.7.1RIGIDFRAMEBEHAVIORThehorizontalstiffnessofarigidframeisgovernedmainlybythebendingresistanceofthegirders,thecolumns,andtheirconnections,and,inatallframe,bytheaxialrigidityofthecolumns.Theaccumulatedhorizontalshearaboveanystoryofarigidframeisresistedbyshearinthecolumnsofthatstory(Fig.7.1).Theshearcausesthestory-heightcolumnstobendindoublecurvaturewithpointsofcontraflexureatapproximatelymid-story-heightlevels.Themomentsappliedtoajointfromthecolumnsaboveandbelowareresistedbytheattachedgirders,whichalsobendindoublecurvature,withpointsofcontraflexureatapproximatelymid-span.Thesedeformationsofthecolumnsandgirdersallowrackingoftheframeandhorizontaldeflectionineachstory.Theoveralldeflectedshapeofarigidframestructureduetorackinghasashearconfigurationwithconcavityupwind,amaximuminclinationnearthebase,andaminimuminclinationatthetop,asshowninFig.7.1.Theoverallmomentoftheexternalhorizontalloadisresistedineachstorylevelbythecoupleresultingfromtheaxialtensileandcompressiveforcesinthecolumnsonoppositesidesofthestructure(Fig.7.2).Theextensionandshorteningofthecolumnscauseoverallbendingandassociatedhorizontaldisplacementsofthestructure.Becauseofthecumulativerotationuptheheight,thestorydriftduetooverallbendingincreaseswithheight,whilethatduetorackingtendstodecrease.Consequentlythecontributiontostorydriftfromoverallbendingmay,in.theuppermoststories,exceedthatfromracking.Thecontributionofoverallbendingtothetotaldrift,however,willusuallynotexceed10%ofthatofracking,exceptinverytall,slender,,rigidframes.Thereforetheoveralldeflectedshapeofahigh-riserigidframeusuallyhasashearconfiguration.Theresponseofarigidframetogravityloadingdiffersfromasimplyconnectedframeinthecontinuousbehaviorofthegirders.Negativemomentsareinducedadjacenttothecolumns,andpositivemomentsofusuallylessermagnitudeoccurinthemid-spanregions.Thecontinuityalsocausesthemaximumgirdermomentstobesensitivetothepatternofliveloading.Thismustbeconsideredwhenestimatingtheworstmomentconditions.Forexample,thegravityloadmaximumhoggingmomentadjacenttoanedgecolumnoccurswhenliveloadactsonlyontheedgespanandalternateotherspans,asforAinFig.7.3a.Themaximumhoggingmomentsadjacenttoaninteriorcolumnarecaused,however,whenliveloadactsonlyonthespansadjacenttothecolumn,asforBinFig.7.3b.Themaximummid-spansaggingmomentoccurswhenliveloadactsonthespanunderconsideration,andalternateotherspans,asforspansABandCDinFig.7.3a.Thedependenceofarigidframeonthemomentcapacityofthecolumnsforresistinghorizontalloadingusuallycausesthecolumnsofarigidframetobelargerthanthoseofthecorrespondingfullybracedsimplyconnectedframe.Ontheotherhand,whilegirdersinbracedframesaredesignedfortheirmid-spansaggingmoment,girdersinrigidframesaredesignedfortheend-of-spanresultanthoggingmoments,whichmaybeoflesservalue.Consequently,girdersinarigidframemaybesmallerthaninthecorrespondingbracedframe.Suchreductionsinsizealloweconomythroughthelowercostofthegirdersandpossiblereductionsinstoryheights.Thesebenefitsmaybeoffset,however,bythehighercostofthemorecomplexrigidconnections.7.2APPROXIMATEDETERMINATIONOFMEMBERFORCESCAUSEDBYGRAVITYLOADSIMGArigidframeisahighlyredundantstructure;consequently,anaccurateanalysiscanbemadeonlyafterthemembersizesareassigned.Initially,therefore,membersizesaredecidedonthebasisofapproximateforcesestimatedeitherbyconservativeformulasorbysimplifiedmethodsofanalysisthatareindependentofmemberproperties.Twoapproachesforestimatinggirderforcesduetogravityloadingaregivenhere.7.2.1GirderForces—CodeRecommendedValuesInrigidframeswithtwoormorespansinwhichthelongerofanytwoadjacentspansdoesnotexceedtheshorterbymorethan20%,andwheretheuniformlydistributeddesignliveloaddoesnotexceedthreetimesthedeadload,thegirdermomentandshearsmaybeestimatedfromTable7.1.ThissummarizestherecommendationsgivenintheUniformBuildingCode[7.1].Inothercasesaconventionalmomentdistributionortwo-cyclemomentdistributionanalysisshouldbemadeforalineofgirdersatafloorlevel.7.2.2Two-CycleMomentDistribution[7.2].Thisisaconciseformofmomentdistributionforestimatinggirdermomentsinacontinuousmultibayspan.ItismoreaccuratethantheformulasinTable7.1,especiallyforcasesofunequalspansandunequalloadingindifferentspans.Thefollowingisassumedfortheanalysis:1.Acounterclockwiserestrainingmomentontheendofagirderispositiveandaclockwisemomentisnegative.2.Theendsofthecolumnsatthefloorsaboveandbelowtheconsideredgirderarefixed.3.Intheabsenceofknownmembersizes,distributionfactorsateachjointaretakenequalto1/n,wherenisthenumberofmembersframingintothejointintheplaneoftheframe.Two-CycleMomentDistribution—WorkedExample.Themethodisdemonstratedbyaworkedexample.InFig,7.4,afour-spangirderAEfromarigid-framebentisshownwithitsloading.Thefixed-endmomentsineachspanarecalculatedfordeadloadingandtotalloadingusingtheformulasgiveninFig,7.5.ThemomentsaresummarizedinTable7.2.Thepurposeofthemomentdistributionistoestimateforeachsupportthemaximumgirdermomentsthatcanoccurasaresultofdeadloadingandpatternliveloading.Adifferentloadcombinationmustbeconsideredforthemaximummomentateachsupport,andadistributionmadeforeachcombination.ThefivedistributionsarepresentedseparatelyinTable7.3,andinacombinedforminTable7.4.DistributionsainTable7.3arefortheexteriorsupportsAandE.ForthemaximumhoggingmomentatA,totalloadingisappliedtospanABwithdeadloadingonlyonBC.Thefixed-endmomentsarewritteninrows1and2.Inthisdistributiononly.theresultingmomentatAisofinterest.Forthefirstcycle,jointBisbalancedwithacorrectingmomentof-(-867+315)/4=-U/4assignedtoMBAwhereUistheunbalancedmoment.Thisisnotrecorded,buthalfofit,(-U/4)/2,iscarriedovertoMAB.Thisisrecordedinrow3andthenaddedtothefixed-endmomentandtheresultrecordedinrow4.ThesecondcycleinvolvesthereleaseandbalanceofjointA.Theunbalancedmomentof936isbalancedbyadding-U/3=-936/3=-312toMBA(row5),implicitlyaddingthesamemomenttothetwocolumnendsatA.Thiscompletesthesecondcycleofthedistribution.TheresultingmaximummomentatAisthengivenbytheadditionofrows4and5,936-312=624.ThedistributionforthemaximummomentatEfollowsasimilarprocedure.DistributionbinTable7.3isforthemaximummomentatB.ThemostsevereloadingpatternforthisiswithtotalloadingonspansABandBCanddeadloadonlyonCD.TheoperationsaresimilartothoseinDistributiona,exceptthattheTfirstcycleinvolvesbalancingthetwoadjacentjointsAandCwhilerecordingonlytheircarryovermomentstoB.Inthesecondcycle,Bisbalancedbyadding-(-1012+782)/4=58toeachsideofB.Theadditionofrows4and5thengivesthemaximumhoggingmomentsatB.Distributionscandd,forthemomentsatjointsCandD,followpatternssimilartoDistributionb.ThecompletesetofoperationscanbecombinedasinTable7.4byinitiallyrecordingateachjointthefixed-endmomentsforbothdeadandtotalloading.Thenthejoint,orjoints,adjacenttotheoneunderconsiderationarebalancedfortheappropriatecombinationofloading,andcarryovermomentsassigned.totheconsideredjointandrecorded.Thejointisthenbalancedtocompletethedistributionforthatsupport.MaximumMid-SpanMoments.Themostsevereloadingconditionforamaximummid-spansaggingmomentiswhentheconsideredspanandalternateotherspansandtotalloading.Aconcisemethodofobtainingthesevaluesmaybeincludedinthecombinedtwo-cycledistribution,asshowninTable7.5.Adoptingtheconventionthatsaggingmomentsatmid-spanarepositive,amid-spantotal;loadingmomentiscalculatedforthefixed-endconditionofeachspanandenteredinthemid-spancolumnofrow2.Thesemid-spanmomentsmustnowbecorrectedtoallowforrotationofthejoints.Thisisachievedbymultiplyingthecarryovermoment,row3,attheleft-handendofthespanby(1+0.5D.F.)/2,andthecarryovermomentattheright-handendby-(1+0.5D.F.)/2,whereD.F.istheappropriatedistributionfactor,andrecordingtheresultsinthemiddlecolumn.Forexample,thecarryovertothemid-spanofABfromA=[(1+0.5/3)/2]x69=40andfromB=-[(1+0.5/4)/2]x(-145)=82.Thesecorrectionmomentsarethenaddedtothefixed-endmid-spanmomenttogivethemaximummid-spansaggingmoment,thatis,733+40+82=8ColumnForcesThegravityloadaxialforceinacolumnisestimatedfromtheaccumulatedtributarydeadandlivefloorloadingabovethatlevel,withreductionsinliveloadingaspermittedbythelocalCodeofPractice.Thegravityloadmaximumcolumnmomentisestimatedbytakingthemaximumdifferenceoftheendmomentsintheconnectedgirdersandallocatingitequallybetweenthecolumnendsjustaboveandbelowthejoint.Tothisshouldbeaddedanyunbalancedmomentduetoeccentricityofthegirderconnectionsfromthecentroidofthecolumn,alsoallocatedequallybetweenthecolumnendsaboveandbelowthejoint.第七章框架结构高层框架结构一般由平行或正交布置的梁柱结构组成,梁柱结构是由带有能承担弯矩作用节点的梁、柱组成。具有抗弯能力的梁、柱和节点共同作用抵抗水平荷载。连续框架可降低梁的跨中弯矩而有利于抵抗重力荷载。框架结构有简捷和便于采用矩形体系的优点。由于这种布置形式没有斜支撑和结构墙体,因此,没有不便利之处,内部可以自由布置,外部可以自由设计门、窗。框架结构对于25层以内的建筑是经济的,超过25层由于要限制其位移而花费的代价高,显得很不经济。如果框架及剪力墙及芯筒相结合,刚度能够大幅度提高,可以建造50层以上的建筑。板柱结构及框架结构非常相似,不同之处仅是用板代替了梁。和框架结构一样,板柱结构是通过其水平和竖向构件之间的连续抗弯作用来抵抗水平和竖向荷载。对于高次超静定框架结构,应根据近似分析进行初步设计,随后进行精确分析和校核。分析过程一般包括以下几步:1.按近似方法确定梁和柱所受重力荷载;2.初步确定在重力荷载作用下构件的截面尺寸,考虑水平荷载的作用进行构件截面尺寸的任意调整;3.将水平荷载分配到各梁柱结构上,对这些结构构件的内力进行初步分析;4.检验位移并对构件截面尺寸做必要的调整;5.按最不利的重力荷载和水平荷载组合检验构件强度,做必要的构件截面尺寸调整;6.为了更精确地验算构件强度和位移,利用计算机对结构进行整体分析,需要时则近一步调整构件截面尺寸。这一阶段中应包括考虑重力荷载对构件内力和位移产生的Ρ一△二阶效应;7.构件和节点的详细设计。本章讨论在重力和水平荷载作用下结构的变形和内力分析方法。这些方法基本上按照设计过程中的次序介绍,首先是近似法,然后介绍计算机分析技术。框架结构的稳定性分析将在第十六章中讨论。7.1框架结构的性能框架结构的侧向刚度主要取决于梁、柱及节点的抗弯能力,在较高的框架中主要取决于柱子的轴向刚度。作用于框架任一层间的水平集中剪力由该层柱子的抗剪能力抵抗(图7.1)。剪力使框架结构每层的柱产生双曲率弯曲,其反弯点大约在层高的中间部位。上、下柱引起的作用于节点处的弯矩由相邻梁承担,该梁、柱的变形引起框架的整体变形,使各层间产生水平位移。在水平推力作用下结构的整体变形和剪力图如图7.1所示,其凹面朝向风荷载作用方向,最大倾角在基底附近,最小倾角在顶端。外部水平荷载产生的总弯矩由各层间两个边柱中的轴向拉、压力组成的力矩抵抗(图7.2)。柱子的伸、缩引起结构的整体弯曲变形,并产生相应的水平位移。因为转角沿建筑高度累加,所以整体弯曲变形引起的层间位移随高度增加而增加,而剪切变形引起的层间位移随高度的增加而减小。其结果在建筑的最顶部整体弯曲对层间位移的贡献会大大超过剪切变形对层间位移的贡献。但是,整体弯曲变形对总位移的贡献及剪切变形对总位移的贡献之比不会超过10%,除非在极高或细长的框架中。因此,高层框架结构变形型式为剪切型。从梁的连接受力性能来看,高层建筑采用的刚性节点连续的框架不同于一般简单连接的普通框架。梁在柱边附近产生负弯矩,跨中正弯矩值常常很小。这种连续性能使梁中最大弯矩对活荷载的作用方式非常敏感。如果能够估计出产生最不利弯矩的因素,则必须加以认真的考虑。例如,重力荷载作用下梁在边柱附近产生的最大负弯矩只会在活荷载作用于边跨和相间跨时才能发生,如图7.3a中的A点。而梁在内柱附近产生的最大负弯矩只会在活荷载作用于相邻跨时才能发生,如图7.3a中的B点。当活荷载作用于本跨和相间跨时,梁的跨中正弯矩最大,如图7.3a中的AB和CD跨。框架的尺寸取决于柱子在水平荷载作用·下的抗弯强度,这往往会使框架柱的截面尺寸大于相应全对角支撑简单连接框架的柱截面尺寸。另外,框架支撑结构中的梁被设计为只具有跨中正弯矩,而框架结构的梁则被设计为端部为负弯矩和跨中为正弯矩,跨中弯矩值较小。因此,框架结构中梁的截面尺寸会小于相应的框架支撑结构中梁的截面尺寸。梁截面的减小将会降低其造价,有时可以降低层高,经济效益明显。但是,由于刚性节点的处理相当复杂,代价较高,使上述经济优势被削弱。7.2重力荷载作用下构件内力的近似计算框架结构是多次超静定结构,因此,只有在确定了构件截面尺寸后才能进行精确分析。所以,在初步设计阶段,可根据传统的公式和不考虑构件特征值的简化分析法近似确定构件中的内力,以此为基础确定构件的截面尺寸。下面将讨论在重力荷载作用下构件内力计算的两种方法。7.2.1梁的内力—规范推荐值对于两跨以上的框架结构,当任何相邻两跨中的长跨不超过短跨的20%跨度,同时设计均布活荷载不超过3倍的恒载时,梁的弯矩和剪力可以按表7.1确定。表中各数值是根据统一建筑规范【7.1】中的推荐值给出。对于其它情况,可按照楼面连续梁采月传统弯矩分配法或两次循环弯矩分配法进行分析确定。7.2.2弯矩分配法【7.2】弯矩分配

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