Stability optimization for a simultaneously twisted and compressed rod

Abstract

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