The lateral instability of yielded mild steel beams of rectangular cross-section

The critical load causing secondary failure of a deep beam by lateral buckling may be calculated by standard methods for those cases in which the beam behaves elastically under the applied load. When, however, the load is sufficiently great to cause partial yield of the beam, these methods give an estimate for the critical load which is too high. In the present paper the phenomenon of lateral buckling in deep mild steel beams of rectangular cross-section is studied from both a theoretical and an experimental standpoint. The paper is divided into three parts. In part I the critical lateral buckling load is shown to depend on the flexural rigidity of the beam about its weaker principal axis while the applied load, causing flexure about its stronger principal axis, is held constant. The dependence of this rigidity on the extent to which the beam has yielded is calculated, and the results are confirmed by tests on beams of rectangular and circular cross-section. It is also shown that the critical load depends on the initial torsional rigidity of the beam, defined as the initial slope of the torque against angle of twist per unit length relation for torsion about the longitudinal axis of the beam while the applied bending load is held constant. In part II it is first shown that in a beam which has partially yielded the shear force due to the variation of the applied bending moment along the length of the beam is carried entirely in the central elastic core of the beam. Using the theory of combined elastic and plastic deformation, it is then shown that the initial torsional rigidity remains constant at its value for elastic torsion, and experimental evidence in favour of this conclusion is presented. Using the results of parts I and II, the conditions causing lateral instability in deep mild steel beams of rectangular cross-section are determined in part III. For a beam bent by pure terminal couples these conditions may be deduced directly, but for the cases of beams subjected to central concentrated loads and of cantilevers a step by step solution of the governing differential equation is necessary. Experimental confirmation is given for the case of pure bending.