Page 199 - Aircraft Stuctures for Engineering Student

6.1 2 Flexural-torsional buckling of thin-walled columns 183
or

(6.76)
UB=U+(YS-YB)@
Similarly the movement of B in the y direction is
vg = v - (xs - xB)6 (6.77)

Therefore, from Eqs (6.76) and (6.77) and referring to Eqs (6.68) and (6.69), we
see that the compressive load on the element 6s at B, at&, is equivalent to lateral
loads
d’
-at&- [u + (ys - YB)e] in the x direction
dz2
and

d2
-at&- [v - (xs - xB)O] in the y direction
dz2
The lines of action of these equivalent lateral loads do not pass through the displaced
position S’ of the shear centre and therefore produce a torque about S’ leading to the
rotation 8. Suppose that the element 6s at B is of unit length in the longitudinal z
direction. The torque per unit length of the column ST(z) acting on the element at
B is then given by
d2
6T(z) = - at6sdZ, [U + (YS -vB)e](.h -YB)

d2
+ d6s-[V - (xs - xB)e](xs - XB) (6.78)
dz2
Integrating Eq. (6.78) over the complete cross-section of the column gives the torque
per unit length acting on the column, i.e.

Expanding Eq. (6.79) and noting that a is constant over the cross-section, we obtain

(6.80)