Closed Form Solution in the Buckling Optimization Problem of Twisted Shafts

The counterpart for Euler’s buckling problem is Greenhill’s problem, which studies the forming of a loop in an elastic beam under torsion. In the context of twisted shafts, the optimal shape of the beam along its axis is searched. A priori form of the cross-section remains unknown. For the solution of the actual problem, the stability equations take into account all possible convex and simply connected shapes of the cross-section. The cross-sections are similar geometric figures related by a homothetic transformation with respect to a homothetic center on the axis of the beam and vary along its axis. The distribution of material along the length of a twisted shaft is optimized so that the beam is of the constant volume and will support the maximal moment without spatial buckling. The applications of the variational method for stability problems are illustrated in this manuscript.