Nonlinear oscillations of a prestressed concrete bridge beam subjected to harmonic perturbation in the conditions of indeterminacy of parameters




forced oscillations of prestressed concrete beam, membership function, perturbation frequency, the theory of fuzzy numbers


This paper deals with the nonlinear oscillations of a prestressed reinforced concrete beam firmly attached to two supports. The beam is subjected to a harmonic force. The calculations of such beams are associated with a number of uncertainties in the initial data. This publication is devoted to questions of their correct accounting.

For a long period of time in mechanics, to tack into account some uncertainties, they have been using the probability theory for modeling and such theory dominates. It have been proven that the probability theory can solve a lot of problems but nevertheless it has some weaknesses. In particular, the lack of statistical information or incomplete information does not adequately reflect the real object of study in a mathematical model. Recently, many researchers have noted that the uncertainty in construction is not only stochastic in nature, and this provides an impetus for the introduction of new developing methods and theories of soft computing. Among them, theories of fuzzy and rough sets, the reliability of which has already been proven in solving control problems, etc. They are the most popular and effective theories now.

For the beam under consideration, the amplitude of beam oscillations is determined, provided that its parameters are indeterminate (fuzzy) and vary within certain limits. An example of determining the amplitude of the oscillation of the 33-meter-long prestressed beam designed by Soyuzdorproekt is studied. The membership function for the amplitude of the beam transverse oscillations using the theory of fuzzy numbers is constructed. The influence analysis of the fuzziness of the disturbance frequency value on the amplitude of oscillations is performed.

It has been revealed that even a small indeterminacy in the frequency setting can cause the beam damage, although there will not yet be any damage when setting the accurate frequency. Thus for the value, the corresponding value  of the right endpoint of the amplitude interval exceeds the maximum acceptable value of 0.076 m, although the modal value of the amplitude does not exceed the acceptable value. Therefore, when calculating the amplitude of structural oscillations, the interval endpoints of the frequency variation should be taken into account, and not its modal value. Analysis of the table shows that further increase in the oscillations frequency leads to resonance, because it moves beyond the acceptable limits both the endpoints of the interval of undetermined amplitude, and the modal value.

Author Biographies

Stanislav Baiev, Prydniprovska State Academy of Civil Engineering and Architecture

Doctor of technical science, Professor of the chair of higher mathematics

Denys Volchok, Prydniprovska State Academy of Civil Engineering and Architecture

Phd, Associate professor of thechair of structural mechanics and strength of materials


M.G. Bondar. Nelinejnye statsionarnye kolebaniya (Nonlinear stationary oscillations). Science idea, Kiev, 1974. (in Russian)

L. Hong, J. Q. Sun. Double crises in fuzzy chaotic systems, Int. J.Dynam. Control, Springer-Verlag, Berlin, 32-40, 2013, DOI: 10.1007/s40435-013-0004-2

L. Hong, J. Jiang, J. Q. Sun. Crises in Chaotic Pendulum with Fuzzy Uncertainty, Journal of Applied Nonlinear Dynamics, 4(3), 215-221, 2015, DOI: 10.5890/JAND.2015.09.001

B. Liu. Theory and Practice of Uncertain Programming, Springer-Verlag, Berlin, 2009.