Treść książki
Przejdź do opcji czytnikaPrzejdź do nawigacjiPrzejdź do informacjiPrzejdź do stopki
IfajumpǦwiseVariabilityofasectionthicknessandȀormaterialpropertieseXistsin
anelementithenstronglyoscillatingcoefficientscanoccurinthedifferentialequations
describingabehaViorofthiselementwhatmakesitsignificantlymoredifficultoreVen
impossibletosolVesuchequationsiItcanbeoVercomebyanapplicationofthe
homogenizationiwhichisintendedtoreplacetheequationswithstronglyoscillating
coefficientsbyequationswithcoefficientshaVingaVeragedValuesiinparticularconstanti
InthecaseofplateswithnonǦhomogeneityinformofbigopeningsisuchplatecanbe
solVedasagratewithuseofmethodsknownfrommechanicsofconstructionsiPlatesand
barswithperiodicallyVariablepropertiescanbetreatedasconsistingofmanyrepeatable
elementsandthepropertiescanbeaVeragedwithinasingleelement(soǦcalled
periodicitycell)sothattheydescribethebehaVioroftherealbarorplateinthemost
eXactpossiblewayiInthismanneriasoǦcalledconstructiVeanisotropyisbeingintroduced
whichcanbeappliedeigitocalculateferroconcreteiwaVyorribǦreinforcedslabsiCertain
elementsofsuchapproachcanbenotedinChapter3ofthepresentmonographiOneof
thefirstscholarswhousedtheconstructiVeanisotropyiiaiforferroconcreteslabswas
Huber(1914)iFormulasforaVeragedstiffnessescanbefoundalsobyTimoshenkoand
WoinowskyǦKrieger(1959)iLekhnitskij(1968)iAmbartsumyan(1969)orbyMiWoĀniak
(2001)ȂthislatterisachapterinthemonographbeingtheeighthVolumeofPolishseries
MechanikaTechniczna
i
AconsiderablymoremodernapproachistheasymptotichomogenizationiAsaresult
ofsuchmethodiequationsdescribingelementswithaperiodicstructurecontainconstant
coefficientsisoǦcalledeffectiVemoduliiwhichmustbedeterminedbysolVingagiVen
problemwithinasingleperiodicitycelliSuchhomogenizationusesanasymptoticanalysis
ofdifferentialequationsbutdoesnotchangeanythinginamaterialstructureasit
happensinthecaseoftheconstructiVeanisotropyiAclassicalpublicationconcerningthis
methodisabookbyBensoussanetali(198)iKohnandVogelius(1984)applied
thismethodforcalculationsofaribbedplateiInPolandiitwasLewińskiandTelegawho
dealtwiththismethod(cfieigiLewińskiƬTelegai1988;Lewińskii1992)i
Someimperfectionsoftheasymptotichomogenization(eigidifficultieswith
calculationsofhighernaturalfrequencies)promptedtoworkoutneXtmethodsof
homogenization
Ȃ
method
of
microlocal
parametersi
models
with
internal
macroparametersandthenȂtoleranceaVeragingtechniqueiAnadVantageofthese
methodsisapossibilityofanalysisofeffectofmagnitudeoftheperiodicitycellon
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