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attemptbyconcerningcalculationofstaticandstabilityproblemsofframesmadeofbars
withVariablestiffnessiTheauthorspresentedtheretransformationformulasofthe
displacementmethodideriVedforthebarshaVingabendingandtorsionalstiffness
eXpressedasapowerfunctionofthesecondithethirdandtheforthdegreeiAsitresults
fromthepublication(andwhatiscommonlyknown)itheVariabilityofproperties
eXpressedeVenȂseeminglyȂsimplefunctionscreatesmultipleproblemsinsolVing
equationsgoVerningagiVenphenomenoniThereasonisthatVariablecoefficientsoccur
intheequationsiresultingeXactlyfromtheVariabilityofthesecondareamomentofthe
crosssectionareaortheVariabilityofYoungmodulusiAneXamplecanbeanequationof
Vibrationsofabeamwiththecrosssectionwherethewidth(iieiadimensiontransVerse
tothedeflection)changeslinearlyȂitdenotesalinearVariabilityofthesecondarea
momentbuteVensuchasimplefunctioncausesthattheBesselfunctionsoccurinthe
solutionimakingitslightlypracticalinapplicationiAsimilarcaseisfortheequationof
deflectedaXisofabuckledbarhaVingasecondareamomentVaryingaccordingtothe
thirdpowerofalengthcoordinate(whentheheightiiieithedimensionparallelto
thedirectionofdeflectionichangeslinearly)Ȃthatiswhyinthetransformationformulas
inKrynickiandMazurkiewicz(1966)theBesselfunctionsoccuriDifficultiesresultingin
solVingequationswithVariablecoefficientsweretriedtooVercomebyKacner(1996)
whopresentedawayofcalculationofbarsandplateswithVariablefeaturesbyuseof
trigonometricseriesiThepresentedmethodiseffectiVeihoweVersuchintricatethatits
practicalapplicationisVeryproblematiciAllthegreaterdifficultiesarebeingintroduced
byelementswithjumpǦwiseandȀorperiodicalVariabilityofpropertiesi
ThereismanywaysofcopingwiththesedifficultiesiThemostobViousoneseems
tobeanapproXimationconsistinginreplacinganelementwithVariableproperties
withasetofappropriatelyselectedandarrangedelementswithconstantpropertiesi
Inaimtoobtainasatisfyingaccuracyiitisnecessarytoassumeahighnumberof
componentsinscribedintotheapproXimatedelementiwhatinturneVokesalarge
computationalefforti
AnotherwayisanapplicationofcertainmethodsofsolVing(eigienergeticmethodsi
Fouriertransform)whichgiVeformallyeXactsolutionsbutrequireusingFourierseries
todescribeabardeflectionlineorplatedeflectionsurfaceiSuchapproachislesssuitable
incaseofstructuresconsistingofseVeralelementswithVariablepropertiesi
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