Introduction to Aircraft Structural Analysis (Elsevier Aerospace Engineering)

262 CHAPTER 8 Columns

whereErisknownasthereducedmodulus,gives

ErI

d^2 v

dz^2

+Pv= 0

ComparingthiswithEq.(8.2),weseethatifPisthecriticalloadPCR,then

PCR=

π^2 ErI

le^2

(8.17)

and

σCR=

π^2 Er

(le/r)^2

(8.18)

Theprecedingmethodforpredictingcriticalloadsandstressesoutsidetheelasticrangeisknownasthe
reducedmodulustheory.FromEq.(8.13),wehave

E

∫d^1

0

y 1 dA−Et

∫d^2

0

y 2 dA= 0 (8.19)

which,togetherwiththerelationshipd=d 1 +d 2 ,enablesthepositionofnntobefound.
ItispossiblethattheaxialloadPisincreasedatthetimeofthelateraldisturbanceofthecolumn
suchthatthereisnostrainreversalonitsconvexside.Therefore,thecompressivestressincreasesover
thecompletesectionsothatthetangentmodulusappliesoverthewholecrosssection.Theanalysisis
thenthesameasthatforcolumnbucklingwithintheelasticlimitexceptthatEtissubstitutedforE.
Hence,thetangentmodulustheorygives

PCR=

π^2 EtI

le^2

(8.20)

and

σCR=

π^2 Et

(le/r^2 )

(8.21)

By a similar argument, a reduction inPcould result in a decrease in stress over the whole cross
section.Theelasticmodulusappliesinthiscase,andthecriticalloadandstressaregivenbythestandard
Eulertheory,namely,Eqs.(8.7)and(8.8).
In Eq. (8.16),I 1 andI 2 are together greater thanI, whileEis greater thanEt. It follows that the
reducedmodulusErisgreaterthanthetangentmodulusEt.Consequently,bucklingloadspredictedby
thereducedmodulustheoryaregreaterthanbucklingloadsderivedfromthetangentmodulustheory,so
thatalthoughwehavespecifiedtheoreticalloadingsituationswherethedifferenttheorieswouldapply,
therestillremainsthedifficultyofdecidingwhichshouldbeusedfordesignpurposes.