Introduction to Aircraft Structural Analysis (Elsevier Aerospace Engineering)

8.6 Flexural–Torsional Buckling of Thin-Walled Columns 275

DifferentiatingEq.(8.53)twicewithrespecttozgives

EIxx

d^4 v

dz^4

=−PCR

d^2 v

dz^2

(8.54)

ComparingEqs.(8.52)and(8.54),weseethatthebehaviorofthecolumnmaybeobtainedbyconsider-
ingitasasimplysupportedbeamcarryingauniformlydistributedloadofintensitywygivenby

wy=−PCR

d^2 v

dz^2

(8.55)

Similarly,forbucklingabouttheCyaxis

wx=−PCR

d^2 u

dz^2

(8.56)

Considernowathin-walledcolumnhavingthecrosssectionshowninFig.8.16andsupposethat
thecentroidalaxesCxyareprincipalaxes(seeChapter15);S(xS,yS)istheshearcenterofthecolumn
(seeChapter16),anditscross-sectionalareaisA.Duetotheflexural–torsionalbucklingproduced,say,
byacompressiveaxialloadP,thecrosssectionwillsuffertranslationsuandvparalleltoCxandCy,
respectively,andarotationθ,positiveanticlockwise,abouttheshearcenterS.Thus,duetotranslation,
CandSmovetoC′andS′,andthen,duetorotationaboutS′,C′movestoC′′.Thetotalmovementof
C,uC,inthexdirectionisgivenby

uc=u+C′D=u+C′C′′sinα(S′Cˆ′C′′ 90 ◦)

But

C′C′′=C′S′θ=CSθ

Hence

uC=u+θCSsinα=u+ySθ (8.57)

Also,thetotalmovementofCintheydirectionis

vC=v−DC′′=v−C′C′′cosα=v−θCScosα

sothat

vC=v−xsθ (8.58)

Sinceatthisparticularcrosssectionofthecolumnthecentroidalaxishasbeendisplaced,theaxialload
Pproducesbendingmomentsaboutthedisplacedxandyaxesgiven,respectively,by

Mx=PvC=P(v−xSθ) (8.59)

and

My=PuC=P(u+ySθ) (8.60)