Page 200 - Aircraft Stuctures for Engineering Student

184 Structural instability
Equation (6.80) may be rewritten

In Eq. (6.81) the term Ixx + Iyy + A(4 + y;) is the polar second moment of area Io of
the column about the shear centre S. Thus Eq. (6.81) becomes

P d28
(6.82)

Substituting for T(z) from Eq. (6.82) in Eq. (11.64), the general equation for the
torsion of a thin-walled beam, we have
d2v d2u
PXST + Pys- - 0 (6.83)
dz dz2 -
Equations (6.74), (6.75) and (6.83) form three simultaneous equations which may be
solved to determine the flexural-torsional buckling loads.
As an example, consider the case of a column of length L in which the ends are
restrained against rotation about the z axis and against deflection in the x and y
directions; the ends are also free to rotate about the x and y axes and are free
to warp. Thus u = v = 8 = 0 at z = 0 and z = L. Also, since the column is free to
rotate about the x and y axes at its ends, M, = My = 0 at z = 0 and z = L, and
from Eqs (6.74) and (6.75)

d2v d2u
- - 0 at z = 0 and z = L
=
=
dz2 dz2
Further, the ends of the column are free to warp so that
d28
_-
0 at z = 0 and z = L (see Eq. (11.54))
dz2 -
An assumed buckled shape given by
7rZ 7rZ 7rz
,
,
u = AI sin - 21 = A2 sin - 8 = A3 sin - (6.84)
L L L
in which Al, A2 and A3 are unknown constants, satisfies the above boundary
conditions. Substituting for u, v and 8 from Eqs (6.84) into Eqs (6.74), (6.75) and
(6.83), we have
1
2EIXX
(P-~)A~-PX~A~=O

(P-9)A1+PysA3=O (6.85)