Unit 34TheConceptofStaticalDeterminacyText1.Introduction Astructurewhichcanbefullysolvedfromtheequationsofequilibriuminthiswayissaidtobestaticallydeterminate.ThestructureinFig.1,whichhasfourexternalreactions,cannotbesolvedbythismethodbecausethenumberofunknownreactionsisgreaterthanthenumberofequationswhichcanbederivedbyconsideringtheequilibriumoftheexternalforcesystem.ThestructureinFig.2isalsoinsolublebyequilibriumduetothefactthatthenumberofinternalforceswhichitcontainsisgreaterthanthenumberofindependentequationswhichcanbederivedbyconsideringonlytheequilibriumofallpossible'free-body-diagrams'.Thesestructuresaresaidtobestaticallyindeterminate.Fig.34.1 Theframework(a)isstaticallydeterminate.Framework(b)isstaticallyindeterminate.Fig34.2 FrameworkisstaticallyindeterminateText Structurescanthereforebesubdividedintotwocategories,thosewhicharestaticallydeterminateandthosewhicharestaticallyindeterminate.Thetwotypesbehaveinsignificantlydifferentwaysinresponsetoloadandthedecisionastowhichshouldbeadoptedinaparticularsituationisanimportantaspectofstructuraldesign.Moststructuralgeometriescanbeproducedineitherformandthedesignerofastructuremusttakeaconsciousdecisionastowhichtypeisappropriate.Thechoiceaffectsthedetailedgeometryofthestructureandcaninfluencetheselectionofthestructuralmaterial.2.Thecharacteristicsofstaticallydeterminateandstaticallyindeterminatestructures 2.1Internalforces InFig.3twoindependentstaticallydeterminatestructures,ABCandADC,areshown.Theyhappentosharethesamesupports.AandC,butineveryotherrespecttheyareindependent.IfhorizontalloadsofPand2PareappliedtojointsBandD,respectively,thestructureswillresisttheseinternalforcesandreactionswillbedeveloped,allofwhichcanbecalculatedfromtheequationsofequilibrium,andtheelementswillundergoaxialstrain,themagnitudesofwhichwilldependontheelasticityofthematerialandthesizesoftheelementcross-sections.BothjointsBandDwillsufferla-Text-teraldeflectionsbutthesewillnotaffecttheinternalforcesintheelements,whichwillbesolelydependentontheexternalloadsandonthegeometriesofthearrangement(toafirstapproximation).Fig34.3 Thepatternofinternalforcesinastaticallyindeterminatestructuredependsonthepropertiesoftheelementsaswellasontheoverallgeometryofthearrangement. Ifafifthelementisadded,whichconnectsjointsBandD,thesystembecomesstaticallyindeterminate.ThetwojointsarenowconstrainedtodeflectbythesameamountunderallloadconditionsandifthetwoloadsareappliedasbeforetheextentoftheresultingelongationorcontractionoftheelementswillnotbethesameasoccurredwhenthejointsBandDwerefreetodeflectindependently.Thismeansthatthejointwhichpreviouslydeflectedlesswillbepulledorpushedfurtherthanbeforeandthereversewilloccurtotheotherjoint.AtransferofloadwillthereforeTextoccuralongtheelementBDandthiswillalterthepatternofinternalforcesinthewholeframe.Theamountofloadtransfer,andthereforeofchangetotheinternalforcesystem,willdependonthedifferencebetweenthedeflectionswhichoccurredtothetwojointsinthestaticallydeterminateforms.Thisisdeterminedbytherigidityoftheelements,sothedistributionofinternalforcesinthestaticallyindeterminatestructureisthereforedependentonthepropertiesoftheelementsaswellasontheoverallgeometryoftheframeandthemagnitudesoftheexternalloads.Theelementpropertiesmustthereforebetakenintoaccountintheanalysisofthisstructure.Thisisgenerallytrueofstaticallyindeterminatestructuresandisoneoftheimportantdifferencesbetweenstaticallydeterminateandstaticallyindeterminatestructures.

Thefactthatelementpropertieshavetobeconsideredintheanalysisofstaticallyindeterminatestructuresmakestheiranalysismuchmorecomplicatedthanthatofequivalentstaticallydeterminatestructures;inparticular,itrequiresthattherigidityoftheelementsbetakenintoaccount.Asthiscanonlybedoneoncetheelementdimensionshavebeendecidedandamaterialselected,itmeansthatthedesigncalculationsforstaticallyindeterminatestructuresmustbecarriedoutonatrialanderrorbasis.Asetofelementsizesmustbeselectedinitiallytoallowtheanalysistobecarriedout.Oncetheinternalforceshavebeencalculatedthesuitabilityofthetrialsizescanbeassessedbycalculatingthestresswhichwilloccurinthem.Theelementsizesmustnormallybealte-Text-redtosuittheparticularinternalforceswhichoccurandthiscausesachangeinthepatternoftheinternalforces.Afurtheranalysisisthenrequiredtocalculatethenewinternalforces,followedbyafurtherrevisionoftheelementdimensions.Thesequencemustbecontinueduntilsatisfactoryelementsizesareobtained.Cyclesofcalculationsofthistypeareroutineincomputer-aideddesign. Bycomparison,thecalculationsforstaticallydeterminatestructuresaremuchmorestraightforward.Theinternalforcesintheelementsdependsolelyontheexternalloadsandontheoverallgeometryofthestructure.Theycanthereforebecalculatedbeforeanydecisiononelementdimensionsorastructuralmaterialhasbeentaken.Oncetheinternalforcesareknown,amaterialcanbechosenandappropriateelementdimensionsselected.Thesewillnotaffectthepatternoftheinternalforcesandsoasinglesequenceofcalculationsissufficienttocompletethedesign.

2.2Efficiencyintheuseofmaterial Theefficiencywithwhichstructuralmaterialisusedisnormallygreaterwithstaticallyindeterminatestructuresbecausethepresenceofalargernumberofconstraintsallowsamoredirecttransmissionofloadstothefoundationsandamoreevensharingofloadbyalloftheelements.Thebenefitsofstaticalindeterminacyinthisrespectaremosteasilyseeninrelationtostructureswithrig-Text-idjoints,inwhichtheresultingstructuralcontinuitycausessmallerbendingmomentstooccurthanarepresentinequivalentstaticallydeterminatestructuresunderthesameloadconditions.Asbeforethedifferencesbetweenthetwotypesofstructurecanbeappreciatedbystudyingverysimpleexamples.

2.3The'1ack-of-fit'problem Withthepossibleexceptionofinsitureinforcedconcretestructures,moststructuresareprefabricatedtosomeextentsothattheirconstructiononsiteisaprocessofassembly.Asprefabricatedcomponentscanneverbeproducedwithpreciselythecorrectdimensions,thequestionof'lack-of-fit'andofthetolerancewhichmustbeallowedforthisisanecessaryconsiderationinstructuraldesign.Itcanaffectthedecisiononwhethertouseastaticallydeterminateorindeterminateform,becausethetoleranceofstaticallydeterminatestructuresto'lack-of-fit'ismuchgreaterthanthatofstaticallyindeterminatestructures.ThearrangementinFig.7(a)isstaticallydeterminatewhilethatinFig.7(b)isanequivalentstaticallyindeterminateform.Itwillbeassumedthattheframesareassembledfromstraightelements,thatthestructuralmaterialissteelandthatthehinge-typejointsaremadebybolting.Theelementswouldbefabricatedinasteelfabricationworkshopandallboltho-Text-leswouldbepre-drilled.However,itwouldbeimpossibletocuttheelementstoexactlythecorrectlength,ortodrilltheboltholesinexactlythecorrectpositions；therewouldalwaysbesomesmallerrornomatterhowmuchcarewastakeninthefabricationprocess.Fig7 The'lack-of-fit'problem. Theinitialstagesoftheassemblywouldbethesameforbothformsandmightconsistofboltingthebeamstothetopsofthetwocolumns.Theresultingarrangementswouldstillbemechanismsatthisstageandanydiscrepancieswhichexistedbetweenthelengthofthenextelementtobeinserted,thatisthefirstdiagonalelement,andthelengthofthespaceintowhichitmustfitcouldbeeliminatedbyswayingtheassemblyuntilthedistancebetweenthejointswasexactlythesameasthelengthoftheelement.TextTheinsertionofthefirstdiagonalelementwouldcompletetheassemblyofthestaticallydeterminateform.Tocompletethestaticallyindeterminateformtheseconddiagonalmustbeadded.Ifanydiscrepancyexistsbetweenthelengthofthisandthedistancebetweenthejointstowhichitmustbeattached,thedistancecannotnowbeadjustedeasilybymovingthepartlyassembledframebecauseitisnowastructureandwillresistanyforcewhichisappliedtoitinanattempttoalteritsshape.Asignificantforcewouldthereforehavetobeappliedtodistorttheframebeforethefinalelementcouldbeinserted.Thiswouldproducestressintheelements,whichwouldtendtorestoretheframetoitsoriginalshapewhentheforcewasreleasedaftertheinsertionofthefinalelement.Thepresenceoftheseconddiagonalelementintheframewouldpreventitfromreturningtoitsoriginalshape,however,andtheresultwouldbethatalloftheelementsintheframewouldfinallycarryapermanentstressasaresultofthe'lack-of-fit'.Thiswouldbeadditionaltoanystresswhichtheyhadtocarryasaresultoftheapplicationoftheframe’slegitimateload. Theperformanceinrespectof'lack-of-fit'isanimportantdifferencebetweenstaticallydeterminateandstaticallyindeterminatestructures.Staticallydeterminatestructurescanbeassembledfairlyeasilydespitethefactthatitisimpossibletofabricatestructuralcomponentswitha-Text-bsoluteaccuracyasanydiscrepancywhichexistsbetweentheactualdimensionsofcomponentsandtheirintendeddimensionscannormallybeaccommodatedduringtheconstructionprocess.Thisdoes,ofcourse,resultinafinalstructuralgeometrywhichisslightlydifferentfromtheshapewhichwasplanned,butthelevelofaccuracyreachedinthefabricationisnormallysuchthatanydiscrepancyisundetectabletothenakedeyedespitebeingsignificantfromthepointofviewoftheintroductionof'lack-of-fit'stresses. Inthecaseofstaticallyindeterminatestructuresevensmalldiscrepanciesinthedimensionscanleadtodifficultiesinassemblyandtheproblembecomesmoreacuteasthedegreeofindeterminacyisincreased.Ithastwoaspectsfirstly,thereisthedifficultyofactuallyconstructingthestructureiftheelementsdonotfitperfectly,andsecondly,thereisthepossibilitythat'lack-of-fit'stressesmaybedeveloped,whichwillreduceitscarryingcapacity.Theproblemisdealtwithbyminimisingtheamountof'lack-of-fit’whichoccursandalsobydevisingmeansof'adjusting'thelengthsoftheelementsduringconstruction(forexamplebyuseofpackingplates).Bothoftheserequirethathighstandardsareachievedinthedetaileddesignofthestructures,inthemanufactureofitscomponentsandalsointhesettingoutofthestructureonsite.Aconsequenceofthe'lack-of-fit'problem,therefore,isthatboththedesignandtheconstructionofstaticallyindeterminatestructuresaremoredifficultandthereforemoreexpensivethanthoseofequivalentstaticallydeterminatestructures.Text

2.4Thermalexpansionand'temperature'stresses ItwasseeninSection2.3thatinthecaseofstaticallyindeterminatestructuresstressescanbeintroducedintotheelementsiftheydonotfitperfectlywhenthestructureisassembled.Evenifperfectfitweretobeachievedinitially,however,anysubsequentalterationtothedimensionsofelementsduetothermalexpansionorcontractionwouldleadtothecreationofstress.Suchstressisknownas'temperature'stress.Itdoesnotoccurinstaticallydeterminatestructures,inwhichsmallchangesindimensionsduetothermalexpansionareaccommodatedbyminoradjustmentstothestructuresshapewithouttheintroductionofstress. Thermalexpansionmustbeconsideredinthedesignofmoststaticallyindeterminatestructuresandtheelementsmadestrongenoughtoresisttheresultingadditionalstresswhichwilloccur.Thisdependsontherangeoftemperaturetowhichthestructurewillbeexposedandonthecoefficientofthermalexpansionofthematerial.Itisafactorwhichobviouslyreducestheloadcarryingcapacityandthereforeefficiencyofstaticallyindeterminatestructures.Text3.Designconsiderationsinrelationtostaticaldeterminacy Moststructuralgeometriescanbeproducedineitherastaticallydeterminateorastaticallyindeterminateformdependingonhowtheconstituentelementsareconnectedtogether.Thequestionofwhichshouldbeadoptedinaparticularcaseisoneofthefundamentalissuesofthedesignprocessandthedecisionisinfluencedbythefactorswhichhavebeenconsideredabove.Themainadvantageofstaticallyindeterminatestructuresisthattheyallowamoreefficientuseofmaterialthanequivalentstaticallydeterminateforms.Itisthereforepossibletoachievelongerspansandcarryheavierloadsthanwithstaticallydeterminateequivalents.Theprincipaldisadvantageofstaticallyindeterminatestructuresarethattheyaremorecomplextodesignandmoredifficulttoconstructthanstaticallydeterminateequivalents;thesefactorsusuallymakethemmoreexpensivedespitetheirgreaterefficiency.Otherdisadvantagesarethepossibilitiesof'lack-of-fit'and'temperature'stressesandthegreatersusceptibilityofstaticallyindeterminatestructurestodamageasaresultofdifferentialsettlementoffoundations.Thesevariousfactorsareweighedagainsteachotherbythedesignerofastructurewhomustdecidewhichtypeismoresuitableinanindividualcase.Text Thedecisionastowhichmaterialshouldbeusedforastructureisoftenrelatedtothedecisionondeterminacy.Reinforcedconcreteisidealforstaticallyindeterminatestructuresduetotheeasewithwhichcontinuitycanbeachievedwithoutthedisadvantageofthe'lack-of-fit'problemandalsotoitslowcoefficientofthermalexpansion,whichresultsintemperaturestressesbeinglow.Mostreinforcedconcretestructuresarethereforedesignedtobestaticallyindeterminate. Theuseofsteelforstaticallyindeterminatestructures,ontheotherhand,canbeproblematicalduetothe'lack-of-fit'problemandtotherelativelyhighcoefficientofthermalexpansionofthematerial.Steelthereforetendstobeusedforstaticallydeterminatestructuresratherthanforstaticallyindeterminatestructuresunlesstheparticularadvantagesofindeterminacyarespecificallyrequiredinconjunctionwiththeuseofsteel.Steelandtimberareinfactparticularlysuitableforstaticallydeterminatestructuresduetotheeasewithwhichhinge-typejointscanbeproducedinthesematerials.Text Usuallythecircumstancesofaparticularbuildingwilldictatethechoiceofstructuretypeandmaterial.Ifabuildingisofsmallormoderatelylargesizewithnoverylargespansthenthesimplicityofthestaticallydeterminateformwillnormallyfavouritsuse. Ifveryhighstructuralefficiencyisrequiredtoachievelongspansorsimplytoprovideanelegantstructuralformthenthismightfavourtheuseofstaticalindeterminacyinconjunctionwithastrongmaterialsuchassteel.Theresultingstructurewouldbeexpensive,however.Whererelativelyhighefficiencyisrequiredtocarryveryheavyloadsthenastaticallyindeterminatestructureinreinforcedconcretemightbethebestchoice.Ifastructureistobeplacedonasiteonwhichdifferentialsettlementislikelytooccur,theuseofastaticallydeterminateforminconjunctionwithasuitablematerialsuchastimberorsteelwouldprobablybeappropriateThedecisiononthetypeofstructureisthereforetakeninconjunctionwiththedecisiononstructuralmaterial,andbotharedependentontheindividualcircumstancesofthebuildingconcerned.equilibrium n.均衡；平靜；保持平衡的能力insoluble adj.不能解決的indeterminate adj.不確定的；模糊的；含混的axial adj.軸的；軸向的elasticity n.彈性；彈力；靈活性arrangement n.布置；整理；準(zhǔn)備deflect vi.轉(zhuǎn)向；偏斜vt.使轉(zhuǎn)向；使偏 斜；使彎曲elongation n.伸長；伸長率；延伸率；延長contraction n.收縮，緊縮；縮寫式；害病rigidity

n.[物]硬度，[力]剛性；嚴格，刻

板；僵化；堅硬equivalent

adj.等價的，相等的；同意義的dimension

n.方面;[數(shù)]維；尺寸；次元；容

積suitability

n.適合；適當(dāng)；相配constraint

n.[數(shù)]約束；局促，態(tài)度不自然；

強制transmission

n.傳動裝置，[機]變速器；傳遞；

傳送；播送continuity

n.連續(xù)性；一連串insitu

在原地，就地；在原來位置NewWordsandPhrasesprefabricatevt.預(yù)先制造；預(yù)先構(gòu)思；預(yù)制構(gòu)件fabricatevt.制造；偽造；裝配discrepancyn.不符；矛盾；相差insertionn.插入；嵌入；插入物legitimateadj.合法的；正當(dāng)?shù)?；合理的；正統(tǒng)的coefficientn.[數(shù)]系數(shù)；率；協(xié)同因素susceptibilityn.敏感性dictatevt.命令；口述；使聽寫NewWordsandPhrasesNotes（1）Moststructuralgeometriescanbeproducedineitherformandthedesignerofastructuremusttakeaconsciousdecisionastowhichtypeisappropriate. 大多數(shù)的幾何結(jié)構(gòu)可以由兩種形式中任一種生成，結(jié)構(gòu)設(shè)計者必須采取主觀決定哪些類型是合適的。（2）Thefactthatelementpropertieshavetobeconsideredintheanalysisofstaticallyindeterminatestructuresmakestheiranalysismuchmorecomplicatedthanthatofequivalentstaticallydeterminatestructures;inparticular,itrequiresthattherigidityoftheelementsbetakenintoaccount. 事實上，在超靜定結(jié)構(gòu)的分析中考慮單元的屬性使其分析遠比等效的靜定結(jié)構(gòu)復(fù)雜；特別是，其要求考慮單元的剛度。（3）Asprefabricatedcomponentscanneverbeproducedwithpreciselythecorrectdimensions,thequestionof'lack-of-fit'andofthetolerancewhichmustbeallowedforthisisanecessaryconsiderationinstructuraldesign. 由于預(yù)制組件無法用精確的尺寸制作出來，因此結(jié)構(gòu)設(shè)計中的要考慮配合與間隙的問題。Notes（4）Suchstressisknownas'temperature'stress.Itdoesnotoccurinstaticallydeterminatestructures,inwhichsmallchangesindimensionsduetothermalexpansionareaccommodatedbyminoradjustmentstothestructuresshapewithouttheintroductionofstress. 這樣的應(yīng)力被稱為'溫度'應(yīng)力。它不會發(fā)生在靜定結(jié)構(gòu)中，而在靜定結(jié)構(gòu)中，由于熱膨脹引起的尺寸很小的改變會因結(jié)構(gòu)S形微小的調(diào)整而不引入應(yīng)力得到解決。（5）Steelandtimberareinfactparticularlysuitableforstaticallydeterminatestructuresduetotheeasewithwhichhinge-typejointscanbeproducedinthesematerials. 鋼鐵和木材實際上是特別適合靜定結(jié)構(gòu)，因為鉸鏈?zhǔn)疥P(guān)節(jié)很容易用這些材料制作。1.Structurescanthereforebesubdividedintotwocategories,thosewhicharestaticallydeterminateandthosewhicharestaticallyindeterminate.Thetwotypesbehavein