The applications of this method for stability problems in the context of twisted and compressed rods are demonstrated 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). We 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. We study the cross-sections with equal principle 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. For determining the optimal solution, we 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 in terms of the higher transcendental functions.

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Posted 24 Nov, 2021

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Posted 24 Nov, 2021

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The applications of this method for stability problems in the context of twisted and compressed rods are demonstrated 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). We 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. We study the cross-sections with equal principle 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. For determining the optimal solution, we 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 in terms of the higher transcendental functions.

Figure 1

Figure 2

Figure 3

Figure 4

Figure 5

Figure 6

Figure 7

Figure 8

Figure 9

Figure 10

The full text of this article is available to read as a PDF.

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