2018, Vol. 5, No. 2. - go to content...

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DOI: 10.15862/09SATS218 (https://doi.org/10.15862/09SATS218)

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Zemtsova O.G., Shein A.I. Research of the boundary hypersurface of the stability region in the axes of rigidity. Russian Journal of Transport Engineering. 2018; 5(2). Available at: https://t-s.today/PDF/09SATS218.pdf (in Russian). DOI: 10.15862/09SATS218

Research of the boundary hypersurface of the stability region in the axes of rigidity

Zemtsova Olga Grigorevna
Penza state university of architecture and construction, Penza, Russia
E-mail: zemtsova-og@yandex.ru

Shein Alexander Ivanovich
Penza state university of architecture and construction, Penza, Russia
E-mail: shein-ai@yandex.ru

Abstract. The stability region of the optimization problem is the set of those points that correspond to the rigidity of the system, under which the equilibrium is stable with respect to possible deviations of the given system. The stability region is separated from the instability region by a boundary hypersurface. The paper gives an earlier analytical solution of the problem of optimizing the rigidity of frame systems from the stability condition. In this case, the necessary first-order conditions do not determine the nature of the vector of the found values of the linear stiffnesses. As is well known, in the case of convexity of the hypersurface of the stability region in a given interval, the resolving equations give the global minimum.

In connection with this, this paper is devoted to the study of the convexity of the hypersurface of the domain of stability of a system in the axes of rigidity. This study essentially reduces to finding the sign-definite angular minors from the second differentials of the Lagrange function of the problem of optimizing the rigidity of the elements of frame systems. In this paper, we prove that the boundary hypersurface of the domain of stability of a system is convex and the conditionally stationary point x* realizes a global minimum on a given interval. The proof was carried out by the authors by justifying the fact that the quadratic form of the second differentials of the Lagrange function is positive definite.

The solution given in the article refers to multi-storey overpasses, which are free and/or non-free frame systems. For such systems, the most typical is the exhaustion of the load-bearing capacity in the form of loss of stability. Thus, the analytical solution obtained by the authors of the problem of optimal design of the transport rack provides ready-made formulas for the design of structures of this type.

Keywords: stability; stability region; instability region; boundary hypersurface; convexity of a hypersurface; global minimum; optimization; analytical solution

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