Stability optimization for a simultaneously twisted and compressed rod

Purpose The authors search the optimal distribution of bending flexure along axis rod. For solution actual problem, stability equations take into account all possible convex, simply connected shapes cross-section. study cross-sections with equal principal moments inertia. are similar geometric figures related by a homothetic transformation respect to center on rod and vary its axis. cross-section that delivers maximum or minimum for critical eigenvalue must be determined among domains. form is known an equilateral triangle. material length twisted compressed optimized so support maximal moment without spatial buckling, presuming volume remains constant admissible rods. static Euler's approach applicable supported (hinged), conservative axial compressing force. Design/methodology/approach optimization problems rods studied in this manuscript. complement buckling problem Greenhill's which studies forming loop elastic bar under simultaneous torsion compression (Greenhill, 1883). Findings determining solution, directly compare different lengths using invariant factors. simultaneously stated closed form. Research limitations/implications (1) linear applied. (2) No nonlinear postbuckling effects were accounted. (3) moment-free, ideal boundary conditions both ends assumed. Practical implications One most common design cases mechanical engineering concurrent twisting straight members. closed-form allows immediate estimation effect axes rotors industrial automotive engineering. Social application lighter material-saving structural elements allow saving fabrication resources, reducing mass vehicles industry machines. systematic usage assists stabilization global energy balance Earth. Originality/value Albeit governing ordinary differential linear, optimality leads nonlinearity final equations. one mathematically hardest tasks mathematics. presents terms higher transcendental functions.