“In the real world, there are some phenomena which seem irregular. These phenomena can be either destructive or useful and practical”, said Dr. Asil Gharebaghi, Associate Professor of Civil Engineering department at K. N. Toosi University of Technology. If these dynamic systems follow some special rules, they are known as chaotic systems. Systems with chaotic behaviors are very sensitive to initial conditions. This means that a little difference between two very close initial conditions results in a considerable difference by the passage of time. Thus, a chaotic phenomenon is unpredictable. Having nonlinear equations is the main reason for such behavior. In this research, some three-dimensional buildings are excited by the Kobe earthquake and the result will be analyzed.
In nonlinear systems, sometimes we observe behavior that we do not understand all its reasons. Usually, it is supposed that simple systems have simple answers too. But not all of them work in this way.
Chaotic systems are like these. In most cases, such systems have definite and certain behaviors, but irregular responses. In this case, by discovering the governing equations, it may be possible to control their behaviors in each moment. As mentioned before, two responses, with two initial conditions, of a chaotic system could not be compared with each other. Thus, controlling the first system may not result in controlling the second system.
Many researchers including Henry Poincare, Mary Cartwright, and John Littlewood have done researches in this field and have proposed simple equations. The most important and effective device in the development of Chaos Theory is a computer because it is easy and appropriate to solve these equations with them.
This research is an effort to answer that if there is any possibility of chaotic behavior in common structural buildings under earthquake excitation.
For detecting a Chaotic system four features will be analyzed:
• Non-periodic behavior
• Extremely sensitive to the initial conditions
• Complicated orbit, which distributes in the phase plane by the passage of time
• Governing equations which are not too much complicated
One important property of time series is called the maximum Lyapunov exponent. A Lyapunov exponent (λ) is a parameter to understand if an equation is sensitive to the initial conditions or not. If the system’s λ is positive, it may be chaotic, and if not it is probably not chaotic. The number of Degrees of Freedom (DoF) sets the number of λ. For a simple time series, one can compare the results of their algorithm with a famous chaotic system of equation(s) like Logistic, Henon, and Lorentz.
After designing three buildings in compliance with Iranian design specifications, the Kobe earthquake is applied to the buildings in two directions perpendicular to each other and the results will be recorded on all floors. The structures will be analyzed by a nonlinear OPENSEES solver. Besides, the number of floors is two, four, and six.
After receiving data from OPENSEES software, the maximum Lyapunov exponents for all buildings are computed in search of positive values. This shows that contrary to what was thought, even though the duration of the Kobe earthquake was less than one minute, these steel structures, which are designed based on Iranian design specifications may experience chaotic behavior. It is worth mentioning that λ is bigger on the lower floors. The more positive λ is, the more chaotic behavior can be observed in the system. This is especially important when these buildings face natural and more powerful loadings and are tested in irregular architectural plans. Despite the above mentioned three buildings, if the system has linear behavior, then it has negative λ, so there is not any concern about observing chaotic behavior in linear systems. All in all, investigating the effect of chaotic behavior on the loading and design criteria of structures affected by lateral loadings is one of the new areas for further research.
More details on this topic and how to do calculations are available in the following article. All rights go to this reference:
S. Asil Gharebaghi, M. Shirzad, “Chaotic behavior of 3D shear buildings under 2D earthquake excitation”, 3rd International Conference & 4th national Conference on Civil Engineering, Architecture and Urban Design, Tabriz, Iran, Sep. 5-7, 2018.