Introduction to Aircraft Structural Analysis (Elsevier Aerospace Engineering)

254 CHAPTER 8 Columns

IftheperfectcolumnofFig.8.1issubjectedtoacompressiveloadP,onlyshorteningofthecolumn
occursnomatterwhatthevalueofP.However,ifthecolumnisdisplacedasmallamountbyalateral
loadF,then,atvaluesofPbelowthecriticalorbucklingload,PCR,removalofFresultsinareturn
ofthecolumntoitsundisturbedposition,indicatingastateofstableequilibrium.Atthecriticalload,
thedisplacementdoesnotdisappear,and,infact,thecolumnwillremaininanydisplacedpositionas
long as the displacement is small. Thus, the buckling loadPCRis associated with a state ofneutral
equilibrium.ForP>PCR,enforcedlateraldisplacementsincreaseandthecolumnisunstable.
Considerthepin-endedcolumnABofFig.8.2.Weassumethatitisinthedisplacedstateofneutral
equilibriumassociatedwithbucklingsothatthecompressiveloadPhasattainedthecriticalvaluePCR.
Simplebendingtheory(seeChapter15)gives

EI

d^2 v

dz^2

=−M

or

EI

d^2 v

dz^2

=−PCRv (8.1)

Fig.8.1

Definition of buckling load for a perfect column.

Fig.8.2

Determination of buckling load for a pin-ended column.