Introduction to Aircraft Structural Analysis (Elsevier Aerospace Engineering)

8.3 Effect of Initial Imperfections 265

whichgivesonrearranging

δ=PCR

δ

P

−A 1 (8.28)

andweseethatagraphofδplottedagainstδ/Phasaslope,intheregionofthecriticalload,equalto
PCRandaninterceptequaltotheinitialcentraldeflection.ThisisthewellknownSouthwellplotfor
theexperimentaldeterminationoftheelasticbucklingloadofanimperfectcolumn.
Timoshenko[Ref.1]alsoshowedthatEq.(8.27)maybeusedforaperfectlystraightcolumnwith
smalleccentricitiesofcolumnload.

Example 8.2
The pin-jointed column shown in Fig. 8.10 carries a compressive loadPapplied eccentrically at a
distanceefromtheaxisofthecolumn.Determinethemaximumbendingmomentinthecolumn.

Thebendingmomentatanysectionofthecolumnisgivenby

M=P(e+v)

Then,bycomparisonwithEq.(8.1),

EI

d^2 v

dz^2

=−P(e+v)

giving

d^2 v

dz^2

+μ^2 v=−

Pe

EI

(μ^2 =P/EI) (i)

Fig.8.10

Eccentrically loaded column of Example 8.2.