274 CHAPTER 8 Columns
Integratingandsubstitutingthelimits,wehave
U+V=
17
35
PCR^2 v^20 l
2 EI
−
3
5
PCR
v^20
l
Hence,
∂(U+V)
∂v 0
=
17
35
P^2 CRv 0 l
EI
−
6 PCRv 0
5 l
= 0
fromwhich
PCR=
42 EI
17 l^2
=2.471
EI
l^2
This value of critical load compares with the exact value (see Table 8.1) ofπ^2 EI/ 4 l^2 =2.467EI/l^2 ;
theerror,inthiscase,isseentobeextremelysmall.Approximatevaluesofcriticalloadobtainedby
theenergymethodarealwaysgreaterthanthecorrectvalues.Theexplanationliesinthefactthatan
assumeddeflectedshapeimpliestheapplicationofconstraintsinordertoforcethecolumntotakeup
an artificial shape. This, as we have seen, has the effect of stiffening the column, with a consequent
increaseincriticalload.
Itwillbeobservedthatthesolutionfortheprecedingexamplemaybeobtainedbysimplyequating
theincreaseininternalenergy(U)totheworkdonebytheexternalcriticalload(−V).Thisisalways
the case when the assumed deflected shape contains a single unknown coefficient, such asv 0 in the
precedingexample.
8.6 Flexural–TorsionalBucklingofThin-WalledColumns.........................................
Insomeinstances,thin-walledcolumnsofopencrosssectiondonotbuckleinbendingaspredictedbythe
Eulertheorybuttwistwithoutbending,orbendandtwistsimultaneously,producingflexural–torsional
buckling.Thesolutiontothistypeofproblemreliesonthetheoryforthetorsionofopensectionbeams
subjectedtowarping(axial)restraint.Initially,however,weshallestablishausefulanalogybetween
thebendingofabeamandthebehaviorofapin-endedcolumn.
Thebendingequationforasimplysupportedbeam,carryingauniformlydistributedloadofintensity
wyandhavingCxandCyasprincipalcentroidalaxesis
EIxx
d^4 v
dz^4
=wy (seeChapter15) (8.52)
Also,theequationforthebucklingofapin-endedcolumnabouttheCxaxisis(seeEq.(8.1))
EIxx
d^2 v
dz^2
=−PCRv (8.53)