Introduction to Aircraft Structural Analysis (Elsevier Aerospace Engineering)

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

(π

l

) 4

(∞

∑

n= 1

n^2 Ansin

nπz

l

) 2

dz

−

PCR

2

∫l

0

(π

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