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 context twisted shafts, optimal shape along its axis searched. A priori form cross-section remains unknown. For solution actual stability equations take into account all possible convex and simply connected shapes cross-section. cross-sections are similar geometric figures related by homothetic transformation with respect to center on vary axis. distribution material length shaft optimized so that constant volume will support maximal moment without spatial buckling. applications variational method problems illustrated this manuscript.