Dealing with Uncertainty in Earthquake Engineering: a Discussion on the Application of the Theory of Open Dynamical Systems

Abstract

Earthquakes in relation to structural engineering have often been addressed from a primary statistical-probabilistic perspective. The use of ‘primary’ indicates that randomness has been artificially introduced within the variables of investigation. Alternative views have been advanced by a number of researchers that have classified earthquakes as chaotic from an ontological point of view (Strogatz 1994). Their arguments are founded in the high degree of non-linearity and complexity of the equations ruling the corresponding wave dynamics. Nevertheless, the sensitivity of large time behavior of dynamical systems to variations in initial conditions, the Chaos Paradigm (Lorenz 1963), appears as a by-product of a deeper insight on natural phenomena, named the Theory of Open Dynamic Systems – ODS (von Bertalanffy, 1950a,b). An open system is currently defined as the relation between a part of Nature, the main system, inside which our observations are made, and its surrounding (unobserved) environment. Since the observed system and the environment are both in motion, as well as their exchanges of matter, energy, and information in general, there is a dynamic interaction. As such, an important consideration for the mathematical modeling of open systems is to determine the different time and space scales associated with the above collection of evolutions, which compose the whole. In a classical context, the laws governing the dynamics of mechanical systems are described by Newton’s equations. Hence, once the initial conditions are fixed, the evolution of states (positions and moments) is fully determined. This is an example of a closed or isolated system. Unfortunately, it is impossible to completely isolate a part of Nature; therefore a closed system cannot exist in reality. This is particularly true when studying seismic phenomena. To the knowledge of the authors, no literature exists applying ODS to structural dynamics. In this paper, a preliminary philosophical discussion related to the use of these concepts in earthquake engineering in general and structural dynamics in particular, is presented. Focus is placed on discussing the uncertainty involved in predicting the response of structures subject to seismic events. Using an elastic, damped single degree of freedom (SDOF) system model, which represents in the most simplistic way a structure, it is shown that when applying stochastic perturbations which represent the interaction of the open system (structure) with the environment (earth) in the form of a Lévy process, the dynamic response of that system under the same recorded ground motion exhibits significant variations when compared to the classical (unique) response, but maintain the same order of magnitude. Important differences can also be appreciated when compared the proposed approach to one which includes white noise only.