Transverse-longitudinal bending of rheonomic rods

In 1, 2], an energy method for the determination of critical buckling times is developed for rods subjected to compression in the conditions of longitudinal bending. In this case, for given compressive loads, the bending moments in the rod cross-sections depend only on the current deflection of the rod axis. In contrast to longitudinal bending, in the case of transverse-longitudinal bending the bending moment in general depends not only on the deflection but also on the axial coordinate and the reaction forces in the supports. Depending on the rod fixing conditions, the problems of transverse-longitudinal bending can be categorized as statically determinate or statically indeterminate. In the latter case, the derivation of equilibrium conditions for a rod segment is complicated by the indefiniteness of the reactions in the rod buckling process. In the current paper, the energy method developed in 1, 2] is extended to a class of statically indeterminate transverse-longitudinal bending problems. To determine the redundant variables, it is proposed to use the principle of minimum of additional dissipation.