Introduction to Aircraft Structural Analysis (Elsevier Aerospace Engineering)

264 CHAPTER 8 Columns

whereλ^2 =P/EI.Thefinaldeflectedshape,v,ofthecolumndependsontheformofitsunloadedshape,
v 0 .Assumingthat

v 0 =

∑∞

n= 1

Ansin

nπz

l

(8.24)

andsubstitutinginEq.(8.23),wehave

d^2 v

dz^2

+λ^2 v=−

π^2

l^2

∑∞

n= 1

n^2 Ansin

nπz

l

Thegeneralsolutionofthisequationis

v=Bcosλz+Dsinλz+

∑∞

n= 1

n^2 An

n^2 −α

sin

nπz

l

whereBandDareconstantsofintegrationandα=λ^2 l^2 /π^2 .Theboundaryconditionsarev=0atz= 0
andl,givingB=D=0,fromwhich

v=

∑∞

n= 1

n^2 An

n^2 −α

sin

nπz

l

(8.25)

Notethatincontrasttotheperfectcolumn,weareabletoobtainanontrivialsolutionfordeflection.
Thisistobeexpected,sincethecolumnisinstableequilibriuminitsbentpositionatallvaluesofP.
Analternativeformforαis

α=

Pl^2

π^2 EI

=

P

PCR

(seeEq.(8.5))

Thus,αisalwayslessthanoneandapproachesunitywhenPapproachesPCRsothatthefirsttermin
Eq.(8.25)usuallydominatestheseries.Agoodapproximation,therefore,fordeflectionwhentheaxial
loadisintheregionofthecriticalloadis

v=

A 1

1 −α

sin

πz

l

(8.26)

oratthecenterofthecolumn,wherez=l/ 2

v=

A 1

1 −P/PCR

(8.27)

inwhichA 1 isseentobetheinitialcentraldeflection.Ifcentraldeflectionsδ(=v−A 1 )aremeasured
fromtheinitiallybowedpositionofthecolumn,thenfromEq.(8.27)weobtain

A 1

1 −P/PCR

−A 1 =δ