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multi-segmentbeamsiobtaining
generalsolutionsof
Frobeniusmethodandsubmittedthetabu
TheanalysisisVeryeXtensiVe-theresultsencompasssiXteencombinationsofsupports(eVen
unstable
stiff
Duethis
the
alsodeterminedapproXimately
pot
theshap
deflectedatauniformcontinuousstatic
withV
practicalapplicationiSuchapproachwasappliedtoanalyzeofVibrationsofasolidandhollow
truncate
parabola(JaworskiƬSzlachetkai2017)aswellasconVeXparabola(Szlachetkaetalii2017)i
ential(elastic)andkineticenergyofaVibratingbariUsingthis
itemsserVingasareferencefortheresultsobtained
nessVariability(dependingonthefourthandthirdpowerofthelongitudinalcoordinate)i
Naguleswaran(1994)obtainedaneXactsolutionfordouble-taperedbeamsusingthe
Thefirstfrequencyofnatural(transVersal)VibrationsofVariablecrosssection
ariablecrosssect
eofthebaraXisdeflectedduringVibrationisthesameasashapeoftheaXisofabeam
dcone(conicalpipe)withgeneratriceshaVingtheshapeofstraightlineandconcaVe
iassuchwith
eXtensiVityandaccuracyithis
aone-stepbeamanditeratiVemethodi
bothendsfreeoroneendfreeandtheothersliding)andtwotypesof
ionsandobtainedresultswhoseaccuracyseemstobesufficientfor
Ȃusingthe
eXactresults
itemcanbeacknowledgedasbenchmarkiItisoneof
loadiJaworskietali(2015)analyzedcantileVerbars
Rayleighmethodconsistingincomparisonofthe
latedresults
byusingtransfer
inth
ischapter(cfiSection2i6)i
forVarioustypesofbeamsi
methodandassumingthat
matriXmethodithe
barscanbe
eXact
Accordingtothesepapersidifferencesbetweentheresultsobtainedbymeansofthisapproach
andthos
fromthe
proVidedthe
toV
offleXuralstiffnessofthebarandthatofaXialloadsactingonthebarwereeXpressedaspower
functionsoreXponentialfunctionsiTheeXtractedeXactsolutio
Besselfunctionsandsupergeometricseriesieapalasetali(2005)presentedatheoretical
numericalanalysisofthin
ariousaXialloadsiincludingconcentratedandVariablydistributedones
Thedeflectionlineafterbucklingandthe
eobtained
Euler-BernoullidifferentialequationofbeamdeflectioniQiushengetali(1995)
eXactsolut
withthe
ionsforstability
-walledtap
finiteelementmethod(FEM)donoteXceed3Ψi
eredbeamcolumnssubjecte
analysisofbarswithVaryingcrosssectionssubjected
bucklingcriticalforcealsocanbedetermined
nswereeXpressedintermsof
dto
abendingmomentand
iThedistribution
and
aXialforceiAstandardFEMcodewasusedforanumericalestimationofadimensionless
criticalbucklingload(inthepap
the
with
elementsanddoesnotproVideappropriateFEMmatricesiCoçkunandAtay(2009)usedthe
criticalbucklingforceofanaXiallytaperedbeamcanbecalculatedasforauniformbeam
anadditionalcorrectionfactori
erireferredto
ThepaperihoweVeriislimitedonlytothin-walled
ascriticalloadmultiplier)iItwasassumedthat
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