Introduction to Aircraft Structural Analysis (Elsevier Aerospace Engineering)

8.6 Flexural–Torsional Buckling of Thin-Walled Columns 277

pointBwillbedisplacedtoB′′.ThehorizontalmovementofBinthexdirectionisthen

uB=u+B′F=u+B′B′′cosβ

But

B′B′′=S′B′θ=SBθ

Hence

uB=u+θSBcosβ

or

uB=u+(yS−yB)θ (8.63)

Similarly,themovementofBintheydirectionis

vB=v−(xS−xB)θ (8.64)

Therefore,fromEqs.(8.63)and(8.64)andreferringtoEqs.(8.55)and(8.56),weseethatthecompressive
loadontheelementδsatB,σtδs,isequivalenttolateralloads

−σtδs

d^2

dz^2

[u+(yS−yB)θ]inthexdirection

and

−σtδs

d^2

dz^2

[v−(xS−xB)θ]intheydirection

ThelinesofactionoftheseequivalentlateralloadsdonotpassthroughthedisplacedpositionS′ofthe
shearcenterand,therefore,produceatorqueaboutS′leadingtotherotationθ.Supposethattheelement
δsatBisofunitlengthinthelongitudinalzdirection.ThetorqueperunitlengthofthecolumnδT(z)
actingontheelementatBisthengivenby

δT(z)=−σtδs

d^2

dz^2

[u+(yS−yB)θ](yS−yB)

+σtδs

d^2

dz^2

[v−(xS−xB)θ](xS−xB) (8.65)

IntegratingEq.(8.65)overthecompletecrosssectionofthecolumngivesthetorqueperunitlength
actingonthecolumn;thatis,

T(z)=−

∫

Sect

σt

d^2 u

dz^2

(yS−yB)ds−

∫

Sect

σt(yS−yB)^2

d^2 θ

dz^2

ds

+

∫

Sect

σt

d^2 v

dz^2

(xS−xB)ds−

∫

Sect

σt(xS−xB)^2

d^2 θ

dz^2

ds (8.66)