Introduction to Aircraft Structural Analysis (Elsevier Aerospace Engineering)

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272 CHAPTER 8 Columns


WehaveseeninChapter7thatexactsolutionsofplatebendingproblemsareobtainablebyenergy
methodswhenthedeflectedshapeoftheplateisknown.Anidenticalsituationexistsinthedetermination
ofcriticalloadsforcolumnandthinplatebucklingmodes.Forthepin-endedcolumnunderdiscussion,
adeflectedformof


v=

∑∞

n= 1

Ansin

nπz
l

(8.49)

satisfiestheboundaryconditionsof


(v)z= 0 =(v)z=l= 0

(

d^2 v
dz^2

)

z= 0

=

(

d^2 v
dz^2

)

z=l

= 0

andiscapable,withinthelimitsforwhichitisvalidandifsuitablevaluesfortheconstantcoefficients
Anarechosen,ofrepresentinganycontinuouscurve.Weare,therefore,inapositiontofindPCRexactly.
SubstitutingEq.(8.49)intoEq.(8.48)gives


U+V=

EI

2

∫l

0


l

) 4

(∞


n= 1

n^2 Ansin

nπz
l

) 2

dz


PCR

2

∫l

0


l

) 2

(∞


n= 1

nAncos

nπz
l

) 2

dz

(8.50)

TheproducttermsinbothintegralsofEq.(8.50)disappearonintegration,leavingonlyintegratedvalues
ofthesquaredterms.Thus,


U+V=

π^4 EI
4 l^3

∑∞

n= 1

n^4 A^2 n−

π^2 PCR
4 l

∑∞

n= 1

n^2 A^2 n (8.51)

AssigningastationaryvaluetothetotalpotentialenergyofEq.(8.51)withrespecttoeachcoefficient
Aninturn,thentakingAnasbeingtypical,wehave


∂(U+V)
∂An

=

π^4 EIn^4 An
2 l^3


π^2 PCRn^2 An
2 l

= 0

fromwhich


PCR=

π^2 EIn^2
l^2

asbefore.

We see that each term in Eq. (8.49) represents a particular deflected shape with a corresponding
critical load. Hence, the first term represents the deflection of the column shown in Fig. 8.14, with
PCR=π^2 EI/l^2 .ThesecondandthirdtermscorrespondtotheshapesshowninFig.8.3,havingcritical
loadsof4π^2 EI/l^2 and9π^2 EI/l^2 andsoon.Clearly,thecolumnmustbeconstrainedtobuckleintothese
morecomplexforms.Inotherwords,thecolumnisbeingforcedintoanunnaturalshape,isconsequently

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