Introduction to Aircraft Structural Analysis (Elsevier Aerospace Engineering)

8.1 Euler Buckling of Columns 255

sothatthedifferentialequationofbendingofthecolumnis

d^2 v

dz^2

+

PCR

EI

v= 0 (8.2)

Thewell-knownsolutionofEq.(8.2)is

v=Acosμz+Bsinμz (8.3)

whereμ^2 =PCR/EIandAandBareunknownconstants.Theboundaryconditionsforthisparticular
casearev=0atz=0andl.Thus,A=0and

Bsinμl= 0

Foranontrivialsolution(i.e.,v
=0),then

sinμl=0orμl=nπ wheren=1,2,3,...

giving

PCRl^2

EI

=n^2 π^2

or

PCR=

n^2 π^2 EI

l^2

(8.4)

NotethatEq.(8.3)cannotbesolvedforvnomatterhowmanyoftheavailableboundaryconditionsare
inserted.Thisistobeexpected,sincetheneutralstateofequilibriummeansthatvisindeterminate.
Thesmallestvalueofbucklingload—inotherwords,thesmallestvalueofPwhichcanmaintain
thecolumninaneutralequilibriumstate—isobtainedbysubstitutingn=1inEq.(8.4).Hence,

PCR=

π^2 EI

l^2

(8.5)

OthervaluesofPCRcorrespondington=2,3,...,are

PCR=

4 π^2 EI

l^2

,

9 π^2 EI

l^2

,...

Thesehighervaluesofbucklingloadcausemorecomplexmodesofbucklingsuchasthoseshownin
Fig.8.3.Thedifferentshapesmaybeproducedbyapplyingexternalrestraintstoaveryslendercolumn
atthepointsofcontraflexuretopreventlateralmovement.Ifnorestraintsareprovided,thentheseforms
ofbucklingareunstableandhavelittlepracticalmeaning.
Thecriticalstress,σCR,correspondingtoPCR,is,fromEq.(8.5)

σCR=

π^2 E

(l/r)^2

, (8.6)

whereristheradiusofgyrationofthecross-sectionalareaofthecolumn.Theterml/risknownasthe
slendernessratioofthecolumn.Foracolumnthatisnotdoublysymmetrical,ristheleastradiusof