Introduction to Aircraft Structural Analysis (Elsevier Aerospace Engineering)

268 CHAPTER 8 Columns

WemayrewriteEq.(8.32)intermsoftheEulerbucklingloadPCR=π^2 EI/l^2 forapin-endedcolumn.
Hence,

Mmax=

wl^2

π^2

PCR

P

(

1 −sec

π

2

√

P

PCR

)

(8.33)

AsPapproachesPCR,thebendingmoment(anddeflection)becomesinfinite.However,thepreceding
theoryisbasedontheassumptionofsmalldeflections(otherwise,d^2 v/dz^2 wouldnotbeacloseapproxi-
mationforcurvature)sothatsuchadeductionisinvalid.Theindicationis,though,thatlargedeflections
willbeproducedbythepresenceofacompressiveaxialloadnomatterhowsmallthetransverseload
mightbe.
Now,letusconsiderthebeam-columnofFig.8.12withhingedendscarryingaconcentratedload
Watadistanceafromtheright-handsupport.For

z≤l−aEI

d^2 v

dz^2

=−M=−Pv−

Waz

l

(8.34)

andfor

z≥l−aEI

d^2 v

dz^2

=−M=−Pv−

W

l

(l−a)(l−z) (8.35)

Writing

λ^2 =

P

EI

Eq.(8.34)becomes

d^2 v

dz^2

+λ^2 v=−

Wa

EIl

z

thegeneralsolutionofwhichis

v=Acosλz+Bsinλz−

Wa

Pl

z (8.36)

Fig.8.12

Beam-column supporting a point load.