Introduction to Aircraft Structural Analysis (Elsevier Aerospace Engineering)

8.1 Euler Buckling of Columns 257

Fig.8.4

Behavior of a perfect pin-ended column.

However,aswehaveseen,forvaluesofPbelowPCRthecolumnisinstableequilibrium,whereasfor
P>PCRthecolumnisunstable.Aplotofloadagainstlateraldeflectionatmidheightwouldtherefore
have the form shown in Fig. 8.4, where, at the pointP=PCR, it is theoretically possible for the col-
umntotakeoneofthreedeflectionpaths.Thus,ifthecolumnremainsundisturbed,thedeflectionat
midheight would continue to be zero but unstable (i.e., the trivial solution of Eq. (8.3),v=0), or, if
disturbed,thecolumnwouldbuckleineitheroftwolateraldirections;thepointatwhichthispossible
branchingoccursiscalledabifurcationpoint;furtherbifurcationpointsoccuratthehighervaluesof
PCR( 4 π^2 EI/l^2 ,9π^2 EI/l^2 ,...).

Example 8.1
A uniform column of lengthLand flexural stiffnessEIis simply supported at its ends and has an
additionalelasticsupportatmidspan.Thissupportissuchthatifalateraldisplacementvcoccursatthis
point,arestoringforcekvcisgeneratedatthepoint.Deriveanequationgivingthebucklingloadofthe
column.Ifthebucklingloadis4π^2 EI/L^2 ,findthevalueofk.Also,iftheelasticsupportisinfinitely
stiff,showthatthebucklingloadisgivenbytheequationtanλL/ 2 =λL/2,whereλ=

√

P/EI.

ThecolumnisshowninitsdisplacedpositioninFig.8.5.Thebendingmomentatanysectionofthe
columnisgivenby

M=Pv−

kvc

2

z

sothat,bycomparisonwithEq.(8.1),

EI

d^2 v

dz^2

=−Pv+

kvc

2

z