Introduction to Aircraft Structural Analysis (Elsevier Aerospace Engineering)

8.6 Flexural–Torsional Buckling of Thin-Walled Columns 279

Anassumedbuckledshapegivenby

u=A 1 sin

πz

L

v=A 2 sin

πz

L

θ=A 3 sin

πz

L

(8.71)

inwhichA 1 ,A 2 ,andA 3 areunknownconstants,satisfiestheprecedingboundaryconditions.Substituting
foru,v,andθfromEqs.(8.71)intoEqs.(8.61),(8.62),and(8.70),wehave

(

P−

π^2 EIxx

L^2

)

A 2 −PxSA 3 = 0

(

P−

π^2 EIyy

L^2

)

A 1 +PySA 3 = 0

PySA 1 −PxSA 2 −

(

π^2 E

L^2

+GJ−

I 0

A

P

)

A 3 = 0

⎫

⎪⎪

⎪⎪

⎪⎪

⎪⎪

⎪⎬

⎪⎪

⎪⎪

⎪⎪

⎪⎪

⎪⎭

(8.72)

FornonzerovaluesofA 1 ,A 2 ,andA 3 ,thedeterminantofEqs.(8.72)mustequalzero;thatis,
∣ ∣ ∣ ∣ ∣ ∣
0 P−π^2 EIxx/L^2 −PxS
P−π^2 EIyy/L^20 PyS
PyS −PxS I 0 P/A−π^2 E/L^2 −GJ

∣ ∣ ∣ ∣ ∣ ∣

= 0 (8.73)

Therootsofthecubicequationformedbytheexpansionofthedeterminantgivethecriticalloadsfor
theflexural–torsionalbucklingofthecolumn;clearlythelowestvalueissignificant.
In the case where the shear center of the column and the centroid of area coincide—that is, the
columnhasadoublysymmetricalcrosssection—xS=yS=0,andEqs.(8.61),(8.62),and(8.70)reduce,
respectively,to

EIxx

d^2 v

dz^2

=−Pv (8.74)

EIyy

d^2 u

dz^2

=−Pu (8.75)

E

d^4 θ

dz^4

(

GJ−I 0

P

A

)

d^2 θ

dz^2

= 0 (8.76)

Equations(8.74),(8.75),and(8.76),unlikeEqs.(8.61),(8.62),and(8.70),areuncoupledandprovide
threeseparatevaluesofbucklingload.Thus,Eqs.(8.74)and(8.75)givevaluesfortheEulerbuckling
loadsaboutthexandyaxes,respectively,whereasEq.(8.76)givestheaxialloadwhichwouldproduce
puretorsionalbuckling;clearlythebucklingloadofthecolumnisthelowestofthesevalues.Forthe
columnwhosebuckledshapeisdefinedbyEqs.(8.71),substitutionforv,u,andθinEqs.(8.74),(8.75),
and(8.76),respectively,gives

PCR(xx)=

π^2 EIxx

L 2

PCR(yy)=

π^2 EIyy

L^2

PCR(θ)=

A

I 0

(

GJ+

π^2 E

L^2

)

(8.77)