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DynamicCollapseAnalysisofReticulatedShellStructureswithSubstructuresLiHong-mei,WangJun-lin,RenXiao-qiang,SunJian-hengCollegeofUrbanandRuralConstruction,AgriculturalUniversityofHebei,Baoding071001,ChinaLuWeiEngineeringandTechnicalcollegeofHebei,Cangzhou061001,Chinaxqren@126.
comAbstract—Dynamiccollapseanalysisisanimportantresearchsubjectforlargespansinglelayerreticulatedshellstructures.
Inthispaper,thedynamiccollapsebehaviorofthesinglelayerreticulatedshellwithsubstructurewhichsupportsthereticulatedshellisinvestigatedundertheearthquakeactions.
Intheanalysis,thegeometricimperfections,thematerialandthegeometricnonlinearofthestructuresareconsidered.
Theeffectsofthedifferentstiffnessofsubstructuretothecollapseearthquakeaccelerationsandtheplasticmemberdistributionofthereticulatedshellareinvestigated.
Keywords—Singlelayersphericalreticulatedshell;dynamiccollapse;Substructures;plasticityratioI.
INTRODUCTIONReticulatedshellstructureisabasictypeofthelargespatialstructures.
Anditiswidelyusedinengineeringduetoitsattractivearchitecturalperformanceandthegoodloadbearingcapacity.
Becausethemembraneforceisthemainresistanceforceofthereticulatedshellstructuresunderloads,thestabilitybehaviorofthistypestructureisacontrollingfactorintheanalysisanddesign.
Thestabilitybehaviorincludesstaticstabilityanddynamicstability.
Inthepastdecades,thestaticstabilityofthereticulatedshellstructureshasbeenextensivelystudied,andalotofresearchresultshavebeengot[1-4].
Inrecentyears,thedynamiccollapseofthereticulatedshellcausedbytheearthquakeactionalsoattractsalotofresearchers,andaseriesoftheinvestigationresultshavebeenpresented[5-8].
Butuptonow,mostofthedynamiccollapseanalysispapersconsideredonlythereticulatedshellitselfandneglectedthesupportingframestructures,namelysubstructures.
Inpractical,mostspatialstructureshaveasupportingframeorcalled,"substructure".
Duringanearthquake,theeffectsofseismicgroundmotionsactonthebaseofthesesubstructuresandthentheseeffectsaretransmittedupintothemainreticulatedshellstructure.
Inthisrespect,anaccurateandrealisticinvestigationofthebehaviorofearthquakeresistantspatialstructureswouldbeachievedifthereticulatedshellstructureandthesupportingframe(substructure)areconsideredasanintegralwhole.
Todate,thereareonlyafewpaperspublishedconcerningthisissue[9-11].
Thispaperconsidersthereticulatedsphericalshellstructureandthesubstructuresasanintegralwholeandinvestigatesthedynamiccollapsebehaviorofthereticulatedshellunderearthquakeactions.
Intheanalysis,theinitialgeometricimperfectionstogetherwithgeometricandmaterialnonlinearitiesareallincluded,andthereticulatedsphericalshellswithsubstructuresofthedifferentstiffnessareanalyzedtodemonstratetheeffectsofstiffnessonthedynamiccollapseofthestructures.
Fig.
1.
K8reticulatedshellFig.
2.
K8reticulatedshellwithsubstructureII.
RETICULATEDSHELLMODELSANDCOLLAPSEANALYSISMETHODThewidelyusedK8reticulatedsphericalshell,asshowninFig.
1andFig.
2,isusedasthemodelstructureinthenumericalanalysis.
Themodelreticulatedshellhasaspanof50mandriseof10mwhichgivethestructurearisetospanratioof0.
2.
ThesteelframeshowninFig.
2isusedasthesubstructuretosupportthemainreticulatedsphericalshellstructure.
Themainreticulatedsphericalshellisrigid-jointedwiththesubstructure.
Thesubstructurehasaheightof8mandisalsorigid-jointedwiththebase.
Auniformlydistributedloadof1.
3kN/m2wasassumedtobeappliedoverthedome.
ThesteelmaterialusedforthemembersofboththedomeandsubstructurewasQ235withamodulusofelasticityE=206MPa,Poissonratioν=0.
26,yieldstrengthfy=235MPaandthematerialdensityis7850kg/m3.
Allofthematerialwasassumedtobeperfectlyelastic-plasticinbehavior.
TheRayleighdampingisusedinthenumericalanalysisandadampingratioof0.
02wasassumed.
Threetypeoftubularcross-sectionsareappliedforthemembersofthereticulatedsphericalshell,andtheyareΦ108*4,Φ83*4andΦ70*4respectivelyaccordingtotheinternalforceofmembersarisingfromstaticanalysis.
Theringbeamofthesubstructureismadeofsteelwitha'I'section250*250*10(flange)*8(web)cross-section.
Thecrosssectionsofthemembersofthestructurearealsotubularcrosssectionsandtheirdimensionisgiveninthefollowingsection.
ThenumericalanalysisofthestructuresiscarriedoutbyusingthefiniteelementanalysissoftwareANSYS[12].
IntheanalysisbyANSYS,thePIPE20elementisusedforallthetubularmembers.
Thiselementtypecandealwithboththegeometricandmaterialnonlinearbehaviorofthestructure.
Themembersofthemaindomeandthesubstructureareallrigidlyconnected.
Tomodeltheweightofthestructurefortheseismicanalysis,three-dimensionalMASS21elementsareusedtoconcentratetheweightofthestructureontothecorrespondingnodes.
ThethreedimensionalEl-Centroearthquakeaccelerationtimeseriesisselectedastheinputacceleration,inwhichthethreepeakaccelerationsofthetimeseriesinbothhorizontalandverticaldirectionsareax=2.
1014m/s2,ay=3.
4170m/s2,az=-2.
0635m/s2,respectively[13].
Tensecondtimehistorydurationisusedsothatallthepeakaccelerationsareincludedintheanalysis.
Forthemaindomestructure,avalueofD/300fortheinitialgeometricimperfectionwasconsidered,andthefirstbucklingmodeisemployedforthedistributionoftheimperfection.
Inthenumericalanalysis,theBudinsky-Roth[14]criterionisusedtodeterminethedynamiccollapseaccelerationofthemainreticulatedshellstructure.
Byusingthiscriterion,theseismicaccelerationincreasesgraduallybythesamefactorinthreedirectionswhilethecycleofthetimeseriesiskeptunchanged.
Thedynamicresponseofthestructureismonitoredunderincreasingacceleration,andasuddenincreaseofdisplacementduetoaverysmallincreaseinthemagnitudeoftheaccelerationisconsideredasanindicationofthedynamiccollapseofthestructure.
III.
DYNAMICCOLLAPSEANALYSISOFTHERETICULATEDSHELLWITHSUBSTRUCTURETodemonstratetheeffectofthesubstructuretothedynamiccollapseofthemainstructure,thereticulatedsphericalshellwithoutsubstructureisanalyzedfirstly.
Intheanalysis,thereticulatedsphericalshellispinconnectedwiththebase,andallthethreetranslationaldisplacementsoftheboundarynodesofthereticulatedstructuresarerestrained.
Fig.
3.
MaximumdisplacementofthereticulatedshellwithoutsubstructureFig.
4.
Dynamicresponseofthemaximumdisplacementofnode91Fig.
5.
Dynamicresponseofthemaximumdisplacementofnode91ThenumericaldynamicanalysisresultsofthereticulatedsphericalshellwithoutconsideringthesubstructureareshowninFig.
3,Fig.
4andFig.
5.
Theresultalsoshowsthatthemaximumdisplacementoccursintheverticaldisplacementofnode91.
Fig.
3showsthevariationofthemaximumnodedisplacementofthereticulatedshellwiththeearthquakepeakacceleration.
Thefigureindicatesthatwhentheearthquake0510152025050100150200250300350400Displacement/mmSeismicaccelerate/m/s2Time/sDisplacement/mTime/sDisplacement/mpeakaccelerationincreasesfrom3.
4m/s2to11.
9m/s2,themaximumdisplacementincreasesfrom50mmto157mm.
Thedisplacementincreasesnearlylinearlywithearthquakepeakacceleration.
Whentheearthquakepeakaccelerationincreasesfrom11.
9m/s2to13.
2m/s2,themaximumdisplacementincreasesto206mmfrom157mm,whichismuchlargerthantheincreasingratiooftheearthquakeacceleration.
Fig.
4showsthatwhentheearthquakeaccelerationis11.
9m/s2,thedynamicresponseofthemaximumdisplacementmaintainsthecharacterofvibratingatitsinitialvibrationequilibriumposition.
Fig.
5showsthatwhentheearthquakeaccelerationreaches13.
2m/s2,thedynamicresponseofthemaximumdisplacementdeviatesfromitsinitialvibrationequilibriumposition.
BaseontheBudinsky-Rothcriterion,thecollapseaccelerationofthestructureisbetween11.
9m/s2and13.
2m/s2,andtheaveragenumber12.
6m/s2istakenasthedynamiccollapseaccelerationofthereticulatedsphericalshellwithoutasubstructure.
Whenthesubstructureisconsidered,thesteelframeshowninFig.
2isusedasthesubstructure.
ThetubularcrosssectionofФ194*8isadoptedforallthecolumnsofthesubstructure.
ThenumericalanalysisresultsareshowninFig.
6andFig.
7.
Themaximumdisplacementundertheactionofearthquakeoccursintheverticaldisplacementofnode53insteadofnode91whenthesubstructureisnotconsidered.
Fig.
6showsthemaximumdisplacementofnode53underdifferentpeakacceleration.
Whenthepeakaccelerationincreasesfrom3.
4m/s2to9.
2m/s2,themaximumdisplacementincreasefrom67mmto129mm,andwhenthepeakaccelerationincreasesfrom9.
2m/s2to9.
5m/s2only,themaximumdisplacementincreasesto144mmrapidly.
Fig.
7showsthatthedynamicresponseofnode53hasseriouslydeviatesfromitsinitialvibrationequilibriumpositionwhenthepeakaccelerationreaches9.
5m/s2.
BasedontheBudinsky-Rothcriterion,thedynamiccollapseaccelerationofthereticulatedsphericalshellwithsubstructureofthecrosssectionФ194*8is9.
2m/s2,whichisless24.
6%thanthecollapseaccelerationwithoutsubstructure.
Fig.
6.
MaximumdisplacementofthereticulatedshellwithsubstructureFig.
7.
Dynamicresponseofthemaximumdisplacementofnode53IV.
EFFECTOFTHESTIFFNESSOFTHESUBSTRUCTURETheaboveanalysisclearlyshowsthatthecollapseaccelerationdecreaseslargelywhenthesubstructureisconsidered.
Toillustratetheeffectofadifferentstiffnessofthesubstructuretothecollapseaccelerationofthemainreticulatedshellstructure,afurtheranalysisofadifferentcrosssectionofthesubstructureiscarriedout.
Inthenumericalanalysis,thetubularcrosssectionofΦ245*10,Φ152*6isusedrespectivelyforallthecolumnofthesubstructure.
Fig.
8showsthemaximumdynamicdisplacementofthereticulatedshellwithsubstructure'scrosssectionofΦ245*10,Φ152*6andΦ194*8respectively.
Thefigureshowsthatwhenthedynamicaccelerationisless4m/s2,thedifferentstiffnessofthesubstructurehaslittleeffecttothemaximumdisplacementofthemainreticulatedshell.
Themaximumdisplacementofthemainreticulatedshellincreaseswiththedecreaseofthestiffnessofthesubstructurewhenthedynamicaccelerationislargerthan4m/s2.
TableIalsoclearlyshowsthatthedynamiccollapseaccelerationofthemainreticulatedshelldecreaseswiththeweakenedofthesubstructure.
WhenthetubularcrosssectionofΦ245*10,Φ194*8andΦ152*6isusedasthecolumnofthesubstructurerespectively,thedynamiccollapseaccelerationreduced19.
0%,24.
6%and35.
7%correspondinglycomparingwiththedynamiccollapseaccelerationofthemainstructurewithoutconsideringthesubstructure.
Themaximumdisplacementisaffectedlittlebythestiffnessofthesubstructurewhenthemainreticulatedshellcollapses.
Fig.
8.
Effectofthestiffnessofsubstructure051015050100150200250300350Displacement/mmSeismicaccelerate/m/s2Time/sDisplacement/m0501001502002503003500246810121416Φ152*6Φ194*8Φ245*10Displacement/mmSeismicaccelerate/m/s2TABLEI.
EFFECTOFSTIFFNESSOFSUBSTRUCTURE.
SectionofcolumnΦ245*10Φ194*8Φ152*6Dynamiccollapseacceleration(m/s2)10.
29.
28.
1Reducedratio19.
0%24.
6%35.
7%Maximumdisplacement(mm)158144157V.
THEPLASTICITYMEMBERSDISTRIBUTIONOFTHEMAINRETICULATEDSHELLSTRUCTUREWiththeincreaseofthedynamicacceleration,somemembersofthereticulatedshellwillreachintoplasticityfromelasticity,andthiswillaffectthedynamiccollapseaccelerationofthestructure.
Todemonstratehowthestiffnessofthesubstructureaffectstheplasticitydevelopmentofthememberofthemainstructure,theinvestigationofthewholeprocessoftheplasticitydevelopmentofmembersunderincreasingdynamicaccelerationispresentedbyFig.
9andFig.
10.
Fig.
9showstherationofplasticitymemberofwithoutconsideringthesubstructureandconsideringthesubstructureofdifferentstiffness.
Thefigureshowsthatforthesamedynamicacceleration,theratioofplasticitymemberofthereticulatedshellwithsubstructureismuchhigherthanthatofthereticulatedshellwithoutsubstructureandthattherationofplasticitymemberincreasesrapidlywiththedecreaseofthestiffnessofthesubstructure.
Whenthedynamicaccelerationis3.
4m/s2,1.
5%ofthemembersofthereticulatedshellwithasubstructureofΦ152*6hasreachedintoplasticity,butnoplasticitymembersappearfortheotherconditions.
Whenthedynamicaccelerationreaches5.
1m/s2,theplasticityratioofthememberofthereticulatedshellwithasubstructureofΦ152*6increasesto4.
6%,andthereticulatedshellwithoutsubstructurehasnoplasticitymemberstill.
Thenwiththeincreaseofthedynamicacceleration,theplasticitymembersappearforreticulatedshellofallconditions,andtheplasticityratioofmembersalsoincreases.
Theplasticityratioofmemberschangesfrom14%to16.
5%accordingtodifferentsupportconditionwhenthedynamiccollapseofthemainreticulatedshelloccurs.
Theinvestigationindicatesthatwhenmoreandmoremembersreachintoplasticitybehavior,thestiffnessofthemainreticulatedshellisreduced,andwhichfinallycausesthecollapseofthestructure.
Theplasticitymembersofthemainreticulatedshellwiththeweakersubstructureappearmuchmoreearlyandtheratioofplasticitymemberincreasemuchfasterthanthatofthereticulatedshellwithstrongersubstructureandwithoutsubstructure.
Therefore,thedynamiccollapseaccelerationofthereticulatedshellwithweakersubstructureismuchlessthanthatofthereticulatedshellwithstrongersubstructureandwithoutsubstructure.
Fig.
9.
Theplasticratioofthememberofreticulatedshellwithandwithoutsubstructure.
(a)a=5.
1m/s2(b)a=6.
8m/s2(c)a=8.
5m/s2(d)a=10.
2m/s2(e)a=11.
9m/s2Fig.
10.
DevelopmentProcessoftheplasticitymembersofthereticulatedshell051015202502468101214Proportionofplasticmembers/%Seismicaccelerate/m/s2withoutsubstructureΦ245*10Φ194*8Φ152*6Fig.
10showsthedevelopmentprocessofplasticitymembersofthemainreticulatedshellwithasubstructureoftubularcrosssectionΦ194*8,anditclearlydemonstratesthatwiththeincreaseofthedynamicacceleration,themoreandmoremembersofthereticulatedshellreachintoplasticitybehaviorfromelasticitybehavior.
VI.
CONCLUSIONThispaperinvestigatestheeffectofsubstructuretothedynamiccollapseofthereticulatedshell.
Intheanalysis,thegeometricimperfections,thematerialandthegeometricnonlinearofthestructuresareconsidered.
Theeffectsofthedifferentstiffnessofsubstructuretothecollapseearthquakeaccelerationsandtheplasticmemberdistributionofthereticulatedshellarealsoinvestigated.
(1)Thesubstructurewillreducethedynamiccollapseaccelerationsofthemainreticulatedshellstructure,andwhenthedynamiccollapseofthereticulatedshellstructureisanalyzed,themainstructureandthesubstructureshouldbeconsideredasanintegralwhole.
(2)Thedynamiccollapseaccelerationreducedwiththedecreaseofthestiffnessofthesubstructure.
Thisindicatesthatthestiffnessofthesubstructureshouldhaveacertainstiffnesstoensurethatthemainreticulatedshellhasenoughearthquakeresistancecapability(3)Theplasticitymembersofthemainreticulatedshellwiththeweakersubstructureappearmuchmoreearlyandtheplasticityratioofmembersalsoincreasemuchfasterthanthatofthereticulatedshellwithstrongersubstructureandwithoutsubstructure.
Therefore,thedynamiccollapseaccelerationofthereticulatedshellwithweakersubstructureismuchlessthanthatofthereticulatedshellwithstrongersubstructureandwithoutsubstructure.
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