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