3.2.1 Equal End Moments

A simply supported beam which is bent in its stiffer principal plane by equal and opposite end moments is shown in Fig. 3.4(a). The beam is elastic and of uniform doubly symmetric I-section. The beam supports prevent both lateral deflection and twist, but the flange ends are free to warp.

Figure 3.4: Buckling of a simply supported I-beam

The beam will buckle at an elastic critical moment M _e when a deflected and twisted equilibrium position, such as that shown in Fig. 3.4, is possible. The differential equilibrium equations of minor axis bending and torsion of the beam in this position are

(3.1)

and

(3.2)

in which each prime (') indicates one differentiation with respect to z. In these equations, M _e ? is the lateral bending moment induced by the twisting ? of the beam and M _eu' is the torque induced by the lateral deflection u, while the right-hand rule is used to determine the signs of the twist and of the external moment and torque. These equations can also be obtained from the more general Eq. (2.178) by substituting M _xo = M _e and (equal and opposite major axis end moments), P = 0 (no axial load), M _yo = = 0 (no minor axis end moments), and K = 0 (doubly symmetric cross-section with P = 0). When these equations are...