Introduction to Aircraft Structural Analysis (Elsevier Aerospace Engineering)

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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)
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