Page 197 - Aircraft Stuctures for Engineering Student

6.1 2 Flexural-torsional buckling of thin-walled columns 181

Differentiating Eq. (6.66) twice with respect to z gives

d4v d2v
EIxx- = -P CR Q (6.67)
dz4
Comparing Eqs (6.65) and (6.67) we see that the behaviour of the column may be
obtained by considering it as a simply supported beam carrying a uniformly
distributed load of intensity wJ given by

(6.68)

Similarly, for buckling about the Cy axis

d2u
w, = -PCR 7 (6.69)
dz
Consider now a thin-walled column having the cross-section shown in Fig. 6.21 and
suppose that the centroidal axes Cxy are principal axes (see Section 9.1); S(xs,yS) is
the shear centre of the column (see Section 9.3) and its cross-sectional area is A. Due
to the flexural-torsional buckling produced, say, by a compressive axial load P the
cross-section will suffer translations u and v parallel to Cx and Cy respectively and
a rotation 8, positive anticlockwise, about the shear centre S. Thus, due to translation,
C and S move to C’ and S’ and then, due to rotation about S’, C’ moves to C”. The

Fig. 6.21 Flexural-torsional buckling of a thin-walled column.