256 CHAPTER 8 Columns
Fig.8.3
Buckling loads for different buckling modes of a pin-ended column.
gyrationofthecrosssectionsincethecolumnwillbendaboutanaxisaboutwhichtheflexuralrigidity
EIisleast.Alternatively,ifbucklingispreventedinallbutoneplane,thenEIistheflexuralrigidityin
thatplane.
Equations(8.5)and(8.6)maybewrittenintheform
PCR=
π^2 EI
le^2(8.7)
and
σCR=π^2 E
(le/r)^2, (8.8)
whereleistheeffectivelengthofthecolumn.Thisisthelengthofapin-endedcolumnthatwouldhave
thesamecriticalloadasthatofacolumnoflengthl,butwithdifferentendconditions.Thedetermination
ofcriticalloadandstressiscarriedoutinanidenticalmannertothatforthepin-endedcolumnexcept
thattheboundaryconditionsaredifferentineachcase.Table8.1givesthesolutionintermsofeffective
length for columns having a variety of end conditions. In addition, the boundary conditions referred
to the coordinate axes of Fig. 8.2 are quoted. The last case in Table 8.1 involves the solution of a
transcendentalequation;thisismostreadilyaccomplishedbyagraphicalmethod.
Letusnowexaminethebucklingoftheperfectpin-endedcolumnofFig.8.2ingreaterdetail.We
haveshown,inEq.(8.4),thatthecolumnwillbuckleatdiscretevaluesofaxialloadandthatassociated
witheachvalueofbucklingloadthereisaparticularbucklingmode(Fig.8.3).Thesediscretevaluesof
bucklingloadarecalledeigenvalues,theirassociatedfunctions(inthiscasev=Bsinnπz/l)arecalled
eigenfunctions,andtheproblemitselfiscalledaneigenvalueproblem.
Further,supposethatthelateralloadFinFig.8.1isremoved.Sincethecolumnisperfectlystraight,
homogeneousandloadedexactlyalongitsaxis,itwillsufferonlyaxialcompressionasPisincreased.
This situation, theoretically, would continue until yielding of the material of the column occurred.
Table 8.1
Ends le/l Boundary Conditions
Both pinned 1.0 v= 0 atz= 0 andl
Both fixed 0.5 v= 0 atz= 0 andz=l,dv/dz= 0 atz=l
One fixed, the other free 2.0 v= 0 and dv/dz= 0 atz= 0
One fixed, the other pinned 0.6998 dv/dz= 0 atz= 0 ,v= 0 atz=landz= 0