Introduction to Aircraft Structural Analysis (Elsevier Aerospace Engineering)

276 CHAPTER 8 Columns

Fig.8.16

Flexural–torsional buckling of a thin-walled column.

Fromsimplebeamtheory(Chapter15)

EIxx

d^2 v

dz^2

=−Mx=−P(v−xSθ) (8.61)

and

EIyy

d^2 u

dz^2

=−My=−P(u+ySθ), (8.62)

whereIxxandIyyarethesecondmomentsofareaofthecrosssectionofthecolumnabouttheprincipal
centroidal axes,Eis Young’s modulus for the material of the column, andzis measured along the
centroidallongitudinalaxis.
TheaxialloadPonthecolumnwill,atanycrosssection,bedistributedasauniformdirectstressσ.
Thus, the direct load on any element of lengthδsat a point B(xB,yB)isσtdsacting in a direction
parallel to the longitudinal axis of the column. In a similar manner to the movement of C to C′′,the