## 1

2 j d2v tyBds + a-^xs ids d20 2 txBds - ""t^s

Sect az~

(Sect

Sect txBds tx I ds

Equation (6.80) may be rewritten 42,. a2.

In Eq. (6.81) the term Ixx + Iyy + + j|) is the polar second moment of area /0 of the column about the shear centre S. Thus Eq. (6.81) becomes

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

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, Mx from Eqs (6.74) and (6.75)

Further, the ends of the column are free to warp so that d20

— = 0 at z = 0 and z = L (see Eq. (11.54)) An assumed buckled shape given by

J-t Lt JLj

in which Ay, A2 and A3 are unknown constants, satisfies the above boundary conditions. Substituting for u, v and 0 from Eqs (6.84) into Eqs (6.74), (6.75) and (6.83), we have

fi?ET

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