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1 Structural Analysis of Historical Constructions Jerzy Jasieńko (ed) 2012 DWE, Wrocław, Poland, ISSN , ISBN biaptlmiaptfc ptorctroai AkAivpfp lc Teb maabokl a AaaA Bofadb EfTAivI NUUVF BApba lk ifmft AkAivpfp cerrarioosalba N I`occhettidiuseppe O Ioizzibgidio P ABpToACT qhemadernod AddabridgeisamarvelloushistoricalironarchbridgethatwasbuiltinNUUVKftallows connectingtheprovincesofieccoandbergamoacrosstheaddariveribetweenmadernod Addaand `aluscod AddaInearjilanoInorthernftalyKqhebridgewasdesignedforrailwayneedsanditsuseis twojfoldwarailwaytrackislocatedintheinnerdeckoftheuppercontinuousbeamxautomotivetraffic runsontopofitkqodayiafternopyearsofcontinuousdutyitheviaductisstillinserviceiwithtrains crossingatslowspeedandalternatedonejwayroadtrafficrestrictedtonoheavyjweightvehiclesk aespitethesedutiesithebridgeseemstohavesufferedfromlimitedmaintenancektiththefinal purposeofelaboratingasafetyassessmentofthestructureianinelasticstructuralanalysisofthe viaductisperformedibasedonafiniteelementmodelthattakesintoaccountaperfectelastoplastic behaviourofthevariousstructuralmembersandisapttocomplywithtypicalprinciplesofiimit AnalysisKppecificallyIthecriticalloadmultiplierandrelevantcollapsemodeareinvestigatedIfor differentstaticloadingconditionsonthebridgekalsoithecharacteristicnonjlinearloadldisplacement responsecurvesofthebridgearetracedbyincrementalanalysesuptothetruelimitloadandthe variouscriticalmembersofthestructureareidentifiedkafurtherintentionofthisworkconsistsin promotinginterestattheinternationallevelonthemadernod AddabridgeIasabeautifulIliving industrialmonumentofthescientificandtechnologicaldevelopmentsofthattimeibyreferring particularlytoitspresentandfuturedestinationsk heywordsw eistoricalironarchbridgeinonjlinearcbmmodeliblastoplasticstructuralanalysisi iimitanalysisi`ollapseloadmultiplieri`ollapsemode NK fktolarctflk qhemadernod AddabridgeisabeautifulironviaductbuiltinNUUVbythepocietàNazionaledelle lfficinedipaviglianoepnlpfiinresponsetoneedsfromrapidlyjgrowingindustrialactivitiesin iombardiatowardstheendofnummxniozkppecificallyiwithintheexpansionofthelocalrailway networkiitbecamenecessarytoacèuireanelevatedcrossingontheriveraddaikorthjbastfrom jilanokqheremarkablestructuralconceptelaboratedforthebridgebydesignergulesoöthlisberger ENURNJNVNNFIheadofthepklpqechnicallfficesinceNUURIwasthatofraisingastiffbutslender symmetricdoublybuiltjinparabolicarchamongthetwobanksofmadernod AddaErightbankFand `aluscod AddaEleftbankFIofaboutNRMmofhorizontalspanandPTKRmofverticalriseIsupporting thenitogetherwithverticaltruncatedpyramidalpiersiastraightupperboxcontinuousbeamonnine bearingsofossmoflengthxniozecigknfk N aoctoralptudentirniversitàdibergamoicacoltàdifngegneriaeaalminefiaipartimentodimrogettazione eqecnologieivialedkjarconirifjo4m44aalmineebdfiftalyirosalbakferrari]unibgkit O AssociatemrofessorIrniversitàdiBergamoIcacoltàdifngegneriaEaalmineFIaipartimentodimrogettazione eqecnologieivialedkjarconirifjo4m44aalmineebdfiftalyimolitecnicodijilanoiaipartimento difngegneriaptrutturaleipiazzaikdasincipoifjomnppjilanoiftalyigiuseppekcocchetti]unibgkit P mrofessorirniversitàdibergamoicacoltàdifngegneriaeaalminefiaipartimentodimrogettazione eqecnologieivialedkjarconirifjo4m44aalmineebdfiftalyi`orrespondingauthoriegidiokrizzi]unibgkit ONTN

2 ONTO cigk N mresent upjstreamviewoftheironarchbridgeenuuvffrommadernod AddaErightbankF qhebridgeprobablyrepresentsoneoftheveryfirstlargestructuresdesignedthroughapractical applicationofthesojcalled qheoryofthebllipseofblasticity IagraphicalJanalyticalmethodof structuralanalysisthatwasdevelopedinthewakeofdraphicptaticsispecificallyatthemolytechnical pchoolofwürichewhereoöthlisbergerwasformedfbyharl`ulmannenuonjnuunfandbyhispupil tilhelmoitterenu4tjnvmsfkaspecificaccountonthathasbeenprovidediniaureaqhesisxpzandin previouspae`mupaperx4ziwhichhasopenedjupthewaytoamodellingstudyonthebridgeatthe rniversityofbergamoibyaresearchproàectthatwasstartedinommrxpjnmzkqhishasledtothe assemblyofacompletelinearcbjmodelofthestructureiwithinacommercialcbjprogram EABAnrpFIasfurtherrefinedherebyanindependentnonJlinearimplementationEMAqiABFinan elastoplasticsettingk qhebridgewasbuiltwithawroughtironmaterialiwithrivetedàointskaccordingtotheoriginalpklp oeportxnziaboutosmmtonsofmetalswereemployedintheconstructionkaetailsonthevarious characteristicfeaturesofthebridgeareavailableinxozandhavebeenanalysedandfurtherreported inxpjnmzkalongitshistoryithemadernod AddabridgehasundergoneafewmodificationsandrepairsK aespiteitscontinuousdutiesionlytwodocumentedsystematicstatictryjoutloadingtestswithadded loadseemtohavebeencarriedjoutiinnuuvandnuvoiinordertocompareaswelltheoutcomesof theoreticalpredictionselaboratedbythepklpatdesignstagek puccessfuldirectcomparisonstosuchtryjoutevidenceshavebeenachievedintheresearchproàect abovexpjnmzibyproducingstaticresultsthroughalinearcbjmodelloyaltodesignjstageconditionsi asdocumentedbytheoriginaltechnicaldrawingsthathavebeenaccessedatthearchivionazionaledi qorinoeseeextensivedescriptioninxnmzfkppecificallyilinearstructuralsimulationsinprevious pae`nmpaperxuzhaveshownfullconsistencyamongcbjpredictionsandexpectedlrecordedpklp datakcurtheriafirstmodellinginvestigationonthedynamiccharacteristicsofthebridgehasbeen attemptedinxvziintermsofthedeterminationofthemainmodefreèuenciesandassociatedmode shapesiwithresultsthatappearedtobeconsistenttothosefrompreliminaryfieldinvestigationsand dynamicidentificationonthebridgeibasedonambientvibrationteststhathavebeencarriedjout recentlybycolleaguesatmolitecnicodijilanoxnnjnozkqhishasledtothedevelopmentof apermanentmonitoringsystemofthebridgeithatisbeingnowputinplacebythemxnpzias conceivedtoassesspossibledegradationofstructuralperformancekcurtherinlightofthisithe completionofareliablecbjmodeloftheviaductappearstobetrulyfundamentalfor acomprehensiveunderstandingofthestructuralperformanceofthebridgeiwhichintheendshould properlyaddressthepresentmorphologyandstateofconservationofthestructurek BasedonsuchpreviousknowJhowIthepresentresearchstudyhasbeendevelopedinthedirectionof modellingtheglobalnonjlinearelastoplasticbehaviourofthebridgekoeferenceisstillmadesofarto thecharacteristicsatdesignstagexthehypothesisofperfectelastoplasticbehaviourwithunlimited ductilityofallthestructuralmembershasbeenassumedkfnparticularifocusismadehereonthe determinationofthecollapseloadmultiplierandrelevantcollapsemechanismiforvariousstatictryj outrailwayloadingconditionsonthebridgekqhecharacteristicloadldisplacementcurvesaretracedby incrementalanalysesuptothetruelimitloadandthecriticalmembersinthevariouspartsofthe structureareidentifiedk qhenonjlinearstructuralanalyseshavebeenperformedwithanelastoplasticcbjformulationthat hasbeenimplementedinanautonomouscomputercodeirunningwithinamaqiabenvironmentk aetailsontheadoptedcomputationalformulationstrategyanditscodeimplementationaregoingtobe

3 providedelsewherexn4zxabriefaccountonthatisreportedinpectionoiwithbasicdescriptionofthe adoptedcbjformulationkqhemorphologicalandgeometricalfeaturesofthestructurehavebeen exportedfromthepreviouscbjmodelimplementedinacommercialcbjcodeeabanrpfkqhishas beenbuiltbyassemblingatruepatrussframewithbeamelementsimutuallyconnectedatthenodesi ascomposedofthreemainstructuralpartswbearingdoublyjbuiltjinparabolicarchiverticalpiersiupper boxcontinuousbeamkaetailsonthedifferentpartsandtotalassemblyoftheprejexistingcbjmodel ofthebridgeiimposedboundaryconditionsandconsideredloadingcasesareavailableinxpjnmzkqhe finalassemblyofthecompletetrusscbjmodelofthemadernod AddabridgecollectsRPPTbeam elementsioonsstructuralnodesandnpovsdegreesoffreedomenodaldisplacementsandrotationsfk qhematerialpropertiesadoptedinthecbjmodelaretakenasrepresentativeofawroughtiron materialxniozwspecificweightγtkttlm P Xvoung smodulusbzntmmmmmmtlm O Xmoisson sratio νzmkpx corresponding shear modulus dzsr4mmmm tlm O X yieldstresses s y ZSKMM kglmm O and t y s y LÖPZPK4SkgLmm O K OK klk-ifkbao biaptlmiaptfc cbm clomriatflk qhesalientcharacteristicfeaturesofthenonjlinearelastoplasticcbjformulationarebrieflyresumed belowiwithreferencetotheimplementationoftheperfectelastoplasticbehaviourofthestructural memberskdeneralcharacteristicsofthealgorithmformulationarerootedinxnrzxdetailedinformation onthepresentcomputationalimplementationisgoingtobeprovidedelsewherexn4zkqhecbj formulationisbasedonaclassicalbulerjbernoullibeamfiniteelementiaccordingtothefollowing peculiarhypotheseswstraightelementsiuniformcrosssectionihomogeneousmaterialpropertiesi transversedisplacementsmodelledbycubicshapefunctionseikeknegligibleshearstraineffectsare consideredfiaxialdisplacementsandrotationsvaryinglinearlyalongthebeamelementk mlasticdeformationhasbeenconcentratedattheelementedgesiaibeaschematicoarepresentationis depictedincigkofiwheretwoplasticàointseasageneralizationoftheclassicalplastichingeconcept intheiimitanalysisofframesiseeekgkxnrzandreferencesèuotedthereinfhavebeeninsertedkfn eachplasticàointitheassumedgeneralizedkinematicvariablesaretworelativeplasticrotationseeach aroundaprincipalaxisofthecrosssection labelsniofianaxialelongationandarelativerotation aroundthebeamaxiskppecificallyiaxialelongationandaxialrotationareepossiblyfactivatedinonly oneofthetwoàointseaorbfiinordertoavoidunrealisticfreerigidbodymovementsxthenionlys internalkinematicvariablesarereèuiredforeachbeamelementkpheareffectshavenotbeen consideredsofark A B M M N N cigk O pchematicrepresentationofabeamfiniteelementwithplasticàointseatedgesaandbf coreachpabeamfiniteelementialinearraterelationcanbeobtainedbetweentheincrementsofwthe NOstaticactionsattheextremesofthebeamelementEh & FIthecorrespondingNOnodaldisplacements and rotationse u& FItheaboveJmentionedSgeneralizedkinematicplasticvariablesE &h FK kamely EseexNRzFW h& ku& + dh & ENF where kistheclassicalno NOelasticstiffnessmatrixofthefiniteelementanddisanadditional NO SplasticstiffnessmatrixK qheirreversiblebehaviouroftheplasticàointsisdescribedbyanassociativeiperfectlyjplastici generalizedvariablemodeliintermsoftofthenostaticvariablesinamelynormalactioniprincipal bendingmomentsatextremesaibiandconstanttwistingmomentiasdefinedbelowkasasimplifying assumptioniapiecejwiselineariuncoupledelastoplasticbehaviourisadoptedintermsofsuchinternal staticvariablesinamelyaoankinejtypeboxedjformyielddomainisassumedinthespaceofstatic variablesk qheanalyticaldescriptionoftheinteractiondomainforthebeamelementisthenstatedintermsofthe followingineèualitiesw ONTP

4 A B A B ( ) ( ) - + ì N min N IN max N IN N ï - + ïmt Mt Mt í - A + - A + ï MN MN MN I MO MO MO ï - B + - B + ïîmn MN MN I MO MO MO EOF where NistheaxialforceIM t isthetorèueeuniformalongthebeamfim N andm O arethebending momentswithrespecttothetwoprincipalaxesofinertiaofthecrosssectioneindexesaandbrefer againtothebeamedgesicigkofkvieldlimits N IN IMt I Mt I MNIOI M + NIO aretakenconstantand obtainedfrommaterialyieldlimitses y It y FandcrosssectiongeometricalcharacteristicsasW e + - e N - N AsyI MNIO - MNIO amnio aeognio LhNIO FsyI Mt - Mt bmt begt LbFt y EPF whereaandbarebendingandtorsionsectionshapefactorsetakenhereas a NKNand b NKR forall theelementsfand gnioi gti hnioibareflexuralprincipalinertiasitorsionalinertiaiprincipalheightsand characteristicprofilethicknessofthecrosssectionk thensomeoftheyieldmodesareactiveinthecurrenttimeintervaloftheintegrationprocessesayi forinstanceimaxen A IN B FZN H A and MN M + N FItheincrementofeachassociatedinternalactionisset tozeroandthecorrespondingincrementalrelationsinbèkenfcanbesolvedfortheincrementsofthe correspondingactivatedkinematicinternalvariablese &h FasElinearFfunctionsofthedisplacement incrementse u& FW -N h & k u& + d h & M Þ h & - E d k Fu& E4F cinallyibysubstitutingintobèkenfthesecondexpressioninbèke4fiadirectforceldisplacement incrementalrelationshipigovernedbyasymmetricelastoplasticstiffnessmatrixek ep Fcanbeobtained forthegenericbeamfiniteelementw h & kep u & ERF qheglobalstructuralelinearfsolvingratesystemisobtainedbytheassemblyoftheelastoplastic matricesofeachfiniteelementandoftheeèuivalentnodalforcevectorincrementw hep r& c& ESF whereristhevectorcollectingtheeunjconstrainedfdegreesoffreedomofthewholestructureicis thenodalforcevectorcomingfromthegivenappliedforcesandh ep istheglobaltangentstiffness matrixofthestructurek BeingthislastrelationshipofalinearkindIthewholetimeintegrationprocesscanbesplitinto aseèuenceoftimeintervalsinwhichallstaticandkinematicèuantitiesvarylinearlyalongeachstep EseeeKgKxNRzFKAscalarmultiplierfactorλisconsideredasaloadamplifiercommontoasetofbasic EliveFloadsKqheinternalkinematicvariablesIactivealongthetimeintervalIareselectedaccordingto theactiveyieldplanesatitsbeginningibèke4fxhoweveriifthecomputedincrementalsolutionthatcan beextractedfrombèkesfwouldimplynegativedissipationforanyoftheactivemodeseforinstanceiif atensileyieldingaxialforceactsinabeamandiinsteadialengthshorteningwouldbeobtainedinthe incrementalsolutionfithenthatactivemodeisactuallydeactivatedithestiffnessmatrixisnewly updatedandtheincrementalsolutionisrejcomputedklntheotherhandiwhenanactivemodehas beendeactivatedatthebeginningofthetimeincrementandtheincrementalsolutionwouldrenderan incrementofthestaticactioncorrespondingtothatmodeiproducingthusaviolationoftheperfectlyj plasticyieldconditioneforinstanceianaxialforcebecominghigherthantheyieldlimitattheendof thetimeincrementfithiswouldmeanthatsuchamodeshallbeincludedamongtheactiveonesifrom thebeginningofthetimeintervalxthenithestiffnessmatrixhastobeupdatedbeforethenew ONT4

5 computationoftheincrementalsolutionklnceallconditionsofnonjnegativedissipationand perfectlyjplasticyieldingdescribedabovearefulfilledithealgorithmcalculatesiamongallnonj activatedmodesitheloadmultipliersleadingtoallnewpossibleactivationskqheminimumamong suchestimatedmultipliersissetasthetruevalueofλattheendoftheincrementkqhenithe correspondingincrementsofstatic h & andkinematic u& èuantitieseinternalactionsanddisplacementsf areupdatedproportionallyifromtheoriginalincrementalsolutionkfnthissenseithepiecejwiselinear elastoplasticresponseofthestructuretoproportionallyjincreasingexternalactionscanbecomputed exactly IinthespiritofiimitAnalysisKfnthestructuralsolutionIboundaryconditionsareimposed with adjhoc proceduresxn4zk cinallyithecollapseofthestructureisreachedwhentheminimumeigenvalueoftheglobaleupdatedf tangentstiffnessmatrixh ep vanishesewithnumericaltolerancesintheorderofnm JNR Fandthe correspondingeigenvectorleadstoapositiveincrementaldissipationforeachactivemodek PK biaptlmiaptfc obpritp Aka bkdfkbbofkd fmmifcatflkp qhoughthepresentcbjmodelhasbeenconceivedsofaratdesignstageithecurrentperfect elastoplasticstructuralanalysisprovidescrucialinformationabouttheidealloadjcarryingcapacityof thebridgeunderlimitserviceconditionskaspreviouslypresentedfortheelasticanalysesinxujnmzi accidentalloadingconditionsassociatedtofourstatictryjouttestsperformedbythepklpxniozare consideredkqhesetestsecigkpftookplaceonnojnvjaynuuviandwerecarriedjoutusingsix locomotiveswithtenderieachofuptofweighticorrespondingtoauniformlyjdistributedloadonthe beamofqrkntlmkioadshavebeenappliedheretothenodesofthecbjmodelattherailway levelkfncigkpthetotalloadappliedtothestructureisreportedforeachtestiwhichisobtainedas nznqliwherelisthelengthofeachspanoftheuppercontinuousbeamelppkormfandnisthe numberofloadedspansiaccordingtotheloaddistributionsincigkpenoorpfkpelfjweightisprej loadedonthebridgeethroughspecificweightγfileadingtoelasticdeformationsibeforestartingthe incrementalelastoplasticanalysesiwhichdevelopatincreasingaccidentalloadkpalientresultsare presentedasfollowsecigsk4jviqablenfkallfollowingrepresentationshavebeengeneratedafter runningtheanalysesibyappropriatepostjprocessingofthestoreddatak N O P 4 oight bank maderno d Adda Ei`F cigk PpchemeofconsideredfourstatictryJoutconfigurationsEviewfromdownJstreamFIwithvalueoftotal loadappliedtothestructureforeachtestandindicationoffourcontrolpointseredcirclesf cirstiforeachofthefourtryjoutloadingconfigurationsincigkpithedeformedconfigurationofthe bridgeatincipientcollapsewithamplificationfactorsettonmmecigsk4jtafithecharacteristicnonj linearloadldisplacementresponsecurveecigsk4jtbfandthelocalisationoftheplasticisedmembers havebeenreportedecigsk4jtcfkqheresponseplotsincigsk4jtbshowthecomputedstepjbyjstep evolutivesolutionsiwithreferencetovaluesreadatthebeginningofeachtimeintervalkfnparticulari thehorizontalaxisdepictstheverticaldisplacementedfofthenodethatiintheendihasshownthe maximumdisplacementatincipientcollapsextheverticalaxisreportstheamplifiedaccidentalload mλni where λistheloadmultiplierrelatedtotheincrementalsolutionofthenonjlinear elastoplasticanalysisepectionofkkoticethatintheplotsincigsk4jtbithenonjzeroinitial displacementeatmmfisduetoselfjweightonlyinamelytoaprejimposedpermanentloadnot affectedbytheloadmultipliereikekloadmultiplierλaffectsàusttheaccidentalloadlinkedtothe locomotivedistributionsfkjoreoveriforeachloadingcaseitheendpointofthemjdcurvereferstothe sojestimatedcollapseofthestructureintermsofvanishingminimumeigenvalueofthetangent stiffnessmatrixofthestructureepectionofkcigsk4jtcshowaplasticitymapofplasticactivationsin ONTR ieft bank `alusco d Adda EBdF nzrnmt nzrnmt nzp4mt nzp4mt

6 thevariousstructuralmembersatincreasingloadmultiplierλkftscorestheplasticmodesthatare activatedatthebeginningofeachtimeincrementkbachmarkerrepresentsoneactivatedmodeinthe structureiasdescribedbelowklnthetwolinesitheactivationofaxialenfandbendingmodesemfare shownirespectivelykkoticethatnotorsionalmodeshavebeenactivatedduringtheanalysesiwhich EaF EbF EcF cigk 4blastoplasticanalysisoftryJoutqestf EaF EbF EcF cigk RblastoplasticanalysisoftryJoutqestff EaF EbF EcF cigk SblastoplasticanalysisoftryJoutqestfff EaF EbF EcF cigk TblastoplasticanalysisoftryJoutqestfs shouldlookreasonablefortheseverticalloadingconfigurationsthataresymmetrictothelongitudinal planeofthebridgekqhecolourofthemarkersreferstothevariouspartsofthestructuretowhichthe activatedplasticàointhasappearedwtoanelementofthearchebluefipierseredfiuppercontinuous ONTS

7 beamegreenfkfnparticularithelargesthollowcirclemarksaplasticmodeactivatedinanelementof thepiersrestingonthearchkpoiininspectingthemapsreportedincigsk4jtciitispossibleto appreciatetheseèuenceofactivationofplasticàointsandtheoverallplasticresponseofthebridgeiat increasingappliedaccidentalloadmλnk qheloadmultipliersλ c atincipientstructuralcollapseofthebridgeforthefourloadingconfigurations arereportedinqableniwithtypeandnumberofactivatedmodeskqablenlistsaswellthemaximum verticalnodedisplacementofthebridgeatincipientcollapseeallmaximumvaluesrefertonodesof theuppercontinuousbeamiattherailwaylevelfk Table N `ollapseloadmultiplieriactivemodesandmaximumverticaldisplacementofthebridge krk of active krk of active ioad configk Q n q l xt] ioad P c λ c Q multk λ c xt] Total nrk of Max vertical active modes displk xmm] axial modes flexural modes Arch miers Beam Total Arch miers Beam Total Test f RNM 4KT4 O4NT U NS NNONPS N4 NO VN NNT ORP P4P Test ff RNM 4KTP O4NO U M RV ST M M 4RR 4RR ROO ORM Test fff P4M 4KTR NSNR S M VS NMO P O ROR RPM SPO NOPN Test fs P4M 4KPT N4US S ON NNPN4M ON O4 PS44MV R4V NTO qheobtainedresultsshowthatiamongthefourtestsecigkpfithecollapseloadmultipliersarealmost thesameiexceptforcasefsewithtwoloadedspansfiwherethelowervalueλ c 4KPTisattainedKqhis maybeduetothefactthatqestfsisthemoreeccentricwithrespecttothecrownofthearchk fnparticularitheloadisconcentratedonthepieronthearchonthesideofthemadernod AddabankK AsitcanbeappreciatedinqableNandincigKTcIthistestpresentsthehighernumberofactivemodes inthepierseonaxialmodesando4flexuralmodesfkqhesemodesareactivatedwhentheload multiplierreachesavalueclosetoλok4andseemthentorulecollapsekjoreoverithemaximum verticaldisplacementobtainedatincipientcollapseisibyfarithelowestkftmaybesaidthatiin qestfsiplasticcollapseisreachedwithoutshowingsignificantplasticresourcesintermsofglobal ductilitykffthesametotalloadisconsideredeintermsofresultantniqestsfjffandfffjfsfibutalmost symmetricallydistributedwithrespecttothecrownofthearchithecollapseloadmultiplierobtained bytheanalysisincreaseseinparticulariitbecomesthehighestinqestffffk λ e λ e EaF EbF λ e λ e EcF EdF cigk U`haracteristicmJDcurvesat`ontrolmointsNJ4EcigKPF cigkscforqestfffshowsthatalmostallyieldmodesactivatedatincreasingloadrefertotheelements oftheuppercontinuousbeamxonlyfewplasticmodesareactivatedintheelementsofarchandpiersk ONTT

8 qheresultsobtainedbytheanalysisreferredtoqestfffshowconspicuousplasticresourcesintermsof globalstructuralductilitykfnfactiasitcanbeappreciatedincigksbiatincipientcollapseithisloading configurationleadstoaprolongedplateauandtothehighermaximumverticaldisplacementeqablenfi uptoaboutfourtimesthemaximumdisplacementintheothertestsiwhichisreferredtoanodeofthe lowerframeconnectingthetwomainverticallongitudinaltrussgirdersoftheuppercontinuousbeami rightunderneaththerailsiwherethedistributedloadhasbeenappliedk fnqestsfandffthreespansoftheuppercontinuousbeamareinterestedbytheaccidentalload distributionkaespitethisiintheseteststhecollapseloadmultiplierisnotmuchdifferentfromthosein qestsfffandfsisothatthetotalcollapseloadm c ishigherkfnqestfthetotalnumberofactivemodes isactuallyminimalandinearcollapseiasinqestfsimanyinternalstaticvariablesintheelementsof thepiersdoreachyieldplanesreferredtobothbendingmomentsandaxialforceecigsk4candtcfkfn qestffmanyyieldmodesstillrefertotheelementsoftheuppercontinuousbeamxnoyieldmodes appearinthepierskqhemaximumverticaldisplacementobtainedforqestffisèuitelimitedecigkrbi qablenfkalsointhiscaseicollapseisreachedwithoutsignificantplasticdeformationsk cigkubelowshowsifor eachoftheloadingtestsecigkpfithecharacteristicnonjlinear loadldisplacementresponseemjdfcurvesatthefourcontrolpointse`mnj4frepresentedincigkpk qhesearelocalisedatthepierlbeamandarchlbeaminterfacesiontherailwayframelevelandreferto thefourbearingsonthearchk fncigskuajditispossibletonotethatiatselected`m s Ithecollapseofthebridgeisgenerallyreached withoutsignificantplasticdeformationatthislevelklnlyinqestsfandfsewheretheloadsaremuch unjsymmetricallylocatedtothecrownofthearchficigskuaiudiplasticdisplacementsarevisibleat `m4and`mnirespectivelykfnqestsffandfffthecharacteristicmjdcurvesstopèuiteearlyeglobal structuralcollapseofthebridgeisachievedwithoutappreciableverticaldisplacementatthe`m s FKqhe mjdcurvesabandonthelinearelastictrendattotalloadm e λ e nemarkedincigskuajudby ahorizontallinefiscoringthethresholdbeyondwhichfurtherloadinggeneratespermanent deformationineachofthetestseλ e isintheorderofpknand4koforqestsfandffiandintheorderof PKTandPKOforqestsfffandfsFKqhemJDtrendsarealmostbiJlinearIwithvisiblekinkrightontheλ e thresholdeqestffforabitafterthateqestsfifffandfsfkqhefurtherpostjkinkloadgainisèuite limitediwithrespectalsotothresholdloadeàustabitmorevisibleforqestffkqhisseemstoshowthati fortheconsideredloadingconfigurationsithearchlooksèuitefarfromcollapseiwhilestructural collapseisreachedforthebridgewithreferencetofailureintheuppercontinuousbeamandiinsome casesiinsomeelementsofthepiersk fnsteadiinallconsideredtestsitheelementsoftheuppercontinuousbeamappeartobethemost criticalwithinthestructurekcoreachspanofthebeamicigkvbelowshowsthelocationofthe elementswithactivatedplasticmodeskppecificallyiwithcountingofthenumberofactivatedmodesi threecouplesofbarsaredisplayedionthreelevelsiforeachspanofthebeamwthetopcoupleinthe elementsoftheupperframeofthebeamethusatroadlevelfxthemiddlecoupleintheelementsofthe verticalframeofthelongitudinaltrussbeamsxthebottomcoupleinthelowerframeeatrailwaylevelfk BluebarsrefertoaxialmodesandredbarstoflexuralmodesKqhewidthofthebarsisrelatedtothe totalnumberofactivemodesinthepertinentelementsk Test f Test ff Test fff Test fs madernod AddaEi`F cigk Vaistributionofactivatedplasticmodesintheuppercontinuousbeam lbviouslyimostactivemodesinthebeamariseinthezoneswheretheloadsareappliedinamelyin theelementsofthelowerframeoftheloadedspanskfncigkvitispossibletoappreciatethatiforeach loadingconfigurationitheseactivemodesareapproximatelyonethirdofallthepotentialmodes relatedtothelowerframexthisratioincreasesevenmoreifenonactivatedftorsionalmodesarenot takenintoaccountandconsideringthatveryfewelementsdoreachyieldplanesfortheaxialforcek qhenumberofactivemodesintheverticalandupperframesiofthesamespansidoesnotappear negligibleiwithrespecttothatofthemodesinthelowerframexsomeactivemodesarisealsoin ONTU `aluscod AddaEBdF

9 nonjdirectlyloadedcontiguousspanskftmaybesaidthatloadsappliedattherailwaylevelareableto involvetheneighbouringelementsintheplasticresponseofthecontinuousbeamk kotwithstandingthattheplasticresponseofthestructureinalltestsappearstobegovernedmainlyby theactivationoftheyieldmodesintheelementsoftheuppercontinuousbeamiinqestsfandfsthe modesreferredtotheelementsofthepiersdirectlylyingonthearchareinvolvedinthefinalcollapsek koelementsoftheotherpiersyieldkqheanalysesshowthatiinbothqestsfandfsitheactivemodes inthepiersonthearchrefertoelementsatthepierslarchconnectionsandelementsoftheupper rectangularclosingframeontopofthepiersihostingaswellthebearingdevicesofthebeamkko activemodesariseintheelementsofthefourfacesoftheboxprofileofthepiersk 4K ClkCirpflkp fnthispaperistructuralelastoplasticanalysesofthemadernod AddabridgeENUUVFhavebeen performedkqothisendiadedicatedcomputerprogramhasbeenimplementediinwhichpabeam finiteelementsiperfectlyplasticàointseasanextensionofclassicalplastichingesfipiecejwiselinear yielddomainsand exact timeintegrationeinthesensediscussedinxnrzfhavebeenconsideredas maincharacteristicingredientsoftheelastoplasticcbjformulationkqhealgorithmhasshownvery muchabletotrackthelimitstructuralbehaviourofthebridgeithroughasophisticatedcomputational strategyibyreachingconvergencewithsmoothrunsuptothetruelimitloadandcorresponding collapsedisplacementskqhisholdstruedespitetheconsiderablecomplexityofthecompletebridge structureiinvolvingroughlyrpmmbeamfiniteelementsandnppmmdegreesoffreedomk qheobtainedresultsiwhichrefertogeometricalcharacteristicsatdesignstageishowagoodglobal elastoplasticperformanceofthebridgekppecificallyifortheanalysedloadingconditionsewith verticalloadsactingsymmetricallytothelongitudinalplaneofthebridgefithearchisbasically neverinvolvedinthecollapsekactuallyiinalltryjouttestsonlyfewelementsofthearchyieldkfn thissenseiitappearsthatthedoublyjbuiltjinparabolicarchiamarvellouscharacteristicfeatureof thebridgeirepresentsawelljsetstructuralelementiintermsoftheglobalstructuralresponseofthe viaductk coralltheanalysedloadingconditionsitheelementsoftheuppercontinuousbeamappeartobethe mostcriticalkftisworthjmentioningthatidespiteinterventionsontheroadwaydeckintheseventies andthemorerecentstiffeningofthemetallicboxgirderitherailwaydeckshouldnothaveundergone substantialmodificationskqhusifurtherchecksonthestructuralperformanceofthebeamshouldbe specificallypursuedkqheanalyseshavealsoshownthatonlytestswithloadingconfigurationsmuch unjsymmetricallylocatedtothecrownofthearchhaveinvolvedplasticityintheelementsofthepiersi butrightunderneaththeloadedspansiinparticularatthearchlpierandpierlbeamstiffenedinterfacesi wherethecbjmodelisactuallynotthatdetailedsofareandcouldberefinedfitodealappropriately withthestressconcentrationsthatmayproduceattheselocationskqheotherelementsofthepiers appearfarfromcollapsekqhestructuralmembersofthearchalsoappearratherinsafepositionisince theyarehardlyinvolvedintheplasticseèuenceleadingtocollapsekqheconsiderablelevelofload amplificationatcollapseeabsoluteλ c intheorderof4k4j4ktielastic λ e intheorderofpknj4koand multiplier ratio λ c Lλ e intheorderofnknjnkrfiscertainlywarrantedbytheassumedunlimited perfectlyjplasticbehaviourofallthestructuralmembersofthebridgeiaslinkedtothestiff hyperstatic natureofthestructureiasconceivedatoriginaldesigniwhichappearstoallowfor considerablestresstransferandredistributionatincreasingloadandresultingplasticdeformationin thestructurekadditionalsimulationloadingsettingsidirectlyactingatthebeamlpiersorpierlarch interfacesmayprovidefurtherinformationonthespecificplasticresourcesofbearingstructural subpartsconstitutedbypiersandarchk qhepresentpreliminaryresultsontheelastoplasticstructuralperformanceofthebridgehavereferred todesignstageconditionskcurtheranalysescouldconsiderthecurrentgeometricalcharacteristicsand stateofconservationofthestructureiwhichappearstobeaffectedbydiffusedandlocalisedcorrosion damageiduealsotolackofmaintenancekthilethepresentdesignjstageanalysesseemtoberather encouragingaboutthebearingcapacityofthebridgeeundertheassumptionofunlimitedductilityof thestructuralmembersfimainlyregardingitsverticalsupportingstructuremadebyarchandpiersi degradationandagingmayimplyreducedstructuralperformancekqhisshouldbecheckedinviewof possiblerestorationactionsiwhichappearworthwhiletobepursuedigiventhepresentresultsandthe considerableimportanceandvaluethatthishistoricinfrastructurestillkeepstodayinthelocal transportationnetworkandinthearchitecturalandindustrialheritageoftheterritoryk ONTV

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