Introduction to Aircraft Structural Analysis (Elsevier Aerospace Engineering)

270 CHAPTER 8 Columns

The deflection curve is then obtained from Eq. (8.38) by substitutingλafor sinλa(sinceλais now
verysmall)andMBforWa.Thus,

v=

MB

P

(

sinλz

sinλl

−

z

l

)

(8.40)

Inasimilarway,wefindthedeflectioncurvecorrespondingtoMAactingalone.SupposethatWmoves
toward A such that the productW(l−a)=constant=MA. Then, as (l−a)tends to zero, we have
sinλ(l−a)=λ(l−a),andEq.(8.39)becomes

v=

MA

P

[

sinλ(l−z)

sinλl

−

(l−z)

l

]

(8.41)

The effect of the two moments acting simultaneously is obtained by superposition of the results of
Eqs.(8.40)and(8.41).Hence,forthebeam-columnofFig.8.13,

v=

MB

P

(

sinλz

sinλl

−

z

l

)

+

MA

P

[

sinλ(l−z)

sinλl

−

(l−z)

l

]

(8.42)

Equation(8.42)isalsothedeflectedformofabeam-columnsupportingeccentricallyappliedendloads
atAandB.Forexample,ifeAandeBaretheeccentricitiesofPattheendsAandB,respectively,then
MA=PeA,MB=PeB,givingadeflectedformof

v=eB

(

sinλz

sinλl

−

z

l

)

+eA

[

sinλ(l−z)

sinλl

−

(l−z)

l

]

(8.43)

Otherbeam-columnconfigurationsfeaturingavarietyofendconditionsandloadingregimesmay
beanalyzedbyasimilarprocedure.

8.5 EnergyMethodfortheCalculationofBucklingLoadsinColumns...........................

Thefactthatthetotalpotentialenergyofanelasticbodypossessesastationaryvalueinanequilibrium
statemaybeusedtoinvestigatetheneutralequilibriumofabuckledcolumn.Inparticular,theenergy
methodisextremelyusefulwhenthedeflectedformofthebuckledcolumnisunknownandhastobe
“guessed”.
First,weshallconsiderthepin-endedcolumnshowninitsbuckledpositioninFig.8.14.Theinternal
orstrainenergyUofthecolumnisassumedtobeproducedbybendingactionaloneandisgivenby
thewell-knownexpression

U=

∫l

0

M^2

2 EI

dz (8.44)