Introduction to Aircraft Structural Analysis (Elsevier Aerospace Engineering)

8.3 Effect of Initial Imperfections 263

Extensiveexperimentscarriedoutonaluminiumalloycolumnsbytheaircraftindustryinthe1940s
showed that the actual buckling load was approximately equal to the tangent modulus load. Shanley
(1947)explainedthatforcolumnswithsmallimperfections,increasesofbothaxialloadandbending
occursimultaneously.Hethenshowedanalyticallythatafterthetangentmodulusloadisreached,the
strainontheconcavesideofthecolumnincreasesrapidly,whilethatontheconvexsidedecreasesslowly.
Thelargedeflectioncorrespondingtotherapidstrainincreaseontheconcaveside,whichoccurssoon
afterthetangentmodulusloadispassed,meansthatitisonlypossibletoexceedthetangentmodulus
loadbyasmallamount.Itfollowsthatthebucklingloadofcolumnsisgivenmostaccuratelyforpractical
purposesbythetangentmodulustheory.
Empiricalformulaehavebeenusedextensivelytopredictbucklingloads,althoughinviewofthe
closeagreementbetweenexperimentandthetangentmodulustheory,theywouldappearunnecessary.
Severalformulaeareinuse;forexample,theRankine,Straight-line,andJohnson’sparabolicformulae
aregiveninmanybooksonelasticstability[Ref.1].

8.3 EffectofInitialImperfections......................................................................

Obviously,itisimpossibleinpracticetoobtainaperfectlystraighthomogeneouscolumnandtoensure
thatitisexactlyaxiallyloaded.Anactualcolumnmaybebentwithsomeeccentricityofload.Such
imperfectionsinfluencetoalargedegreethebehaviorofthecolumnwhich,unliketheperfectcolumn,
beginstobendimmediatelytheaxialloadisapplied.
Letussupposethatacolumn,initiallybent,issubjectedtoanincreasingaxialloadPasshownin
Fig.8.9.Inthiscase,thebendingmomentatanypointisproportionaltothechangeincurvatureofthe
columnfromitsinitialbentposition.Thus,

EI

d^2 v

dz^2

−EI

d^2 v 0

dz^2

−Pv (8.22)

which,onrearranging,becomes

d^2 v

dz^2

+λ^2 v=

d^2 v 0

dz^2

(8.23)

Fig.8.9

Initially bent column.