Keywords: robust optimization, particle swarm, support vector machines, Monte Carlo simulation, stochastic equivalent linearization, earthquake engineering.

The design of earthquake-resistant structures is generally carried out by means of code procedures that incorporate the large uncertainties associated to the seismic phenomenon in a partial manner, namely that associated to the peak ground acceleration. The randomness of the time history and the uncertainties in material properties are normally not considered. This is due to the difficulties linked to a full probabilistic analysis in earthquake engineering. The estimation of failure probabilities can be performed using either random vibration theories or Monte Carlo simulation [1]. In the former case, rough, usually very inaccurate estimates of failure probabilities can be obtained with methods that do not require large computational costs but a sound knowledge of the involved techniques by the analyst; for these reasons nonlinear random vibration methods are rarely applied in the design practice. In the latter case it is necessary to perform several nonlinear analysis using a different set of acceleration history and structural parameters in each of them. In this case the conceptual framework is not so involved but the computational costs are very large.

Consequently, despite the clear code specification of the probability of the peak ground acceleration exceeding a threshold, codes usually do not require the design to meet similar thresholds for the probability of failure of the structural system in different failure modes. This implies that the introduction of the probabilistic considerations into earthquake-resistant design is normally limited only to the decision on the peak value of the ground motion. This restriction is in marked contrast to the large uncertainties associated with earthquake input and response.

In this paper a method for advancing a step further into the consideration of probabilistic concepts in practical earthquake resistant design is proposed. To this end the desirable but difficult goal of estimating the failure probabilities, requiring a full probabilistic design, is abandoned and instead it is proposed to adopt the more practical goal of controlling the second-order dynamic response. This means renouncing to the reliability-based design and assuming a robust design [2] for the earthquake load case. This is due to the fact that, whereas the estimation of failure probabilities in earthquake resistant design faces the difficulties mentioned above, the estimation of nonlinear second order responses can be easily obtained using well established nonlinear random vibration methods. Therefore, the robust design can be formulated by linking this technique to an optimization method yielding the best solution without significant computational effort.

To this end the nonlinear analysis of the structure is carried out by means of the random vibration procedure known as equivalent linearization method [1], which has found some practical applications in earthquake engineering [3]. The optimal solution is sought by minimizing a cost function formulated in terms of probabilistic second-order responses that incorporate the many uncertainty sources present in the model. The search space is explored with a combination of two relevant computational learning tools, which are the particle swarm optimization method [4] and the support vector machines [5]. The first is employed for searching the optimal solution while the latter is a classification (or pattern recognition) technique that is very useful for discriminating feasible from infeasible solutions while reducing the search space. In fact, at a difference with respect to other pattern classification methods the support vector machines provide the auxiliary tool of a margin band around the discrimination function, which allows a drastic simplification of the number of solver calls when using random methods of structural optimization and reliability [6].

The procedure is illustrated with an example concerning a four-story building for which the maximal flexibility is sought under the constraint that the standard deviations of the drift displacements do not exceed specified threshold values. It is shown that the procedure converges very rapidly.

1

J.B. Roberts and P. D. Spanos, "Random Vibration and Statistical Linearization", John Wiley and Sons, Chichester, 1990.

2

I. Doltsinis and A. Kang, "Robust design of structures using optimization methods", Computer Methods in Applied Mechanics and Engineering, 193, 2221 - 2237, 2004. doi:10.1016/j.cma.2003.12.055

3

Y.G. Zhao, T. Ono and H. Idota, "Response uncertainty and time-variant reliability analysis for hysteretic MDF structures", Earthquake Engineering and Structural Dynamics, 28, 1187 - 1213, 1999. doi:10.1002/(SICI)1096-9845(199910)28:10<1187::AID-EQE863>3.0.CO;2-E

4

J. Kennedy and R. C. Eberhart, "Swarm Intelligence", Morgan Kaufmann, San Francisco, 2001.

5

V.N. Vapnik, "Statistical Learning Theory", John Wiley and Sons, New York, 1998.

6

J.E. Hurtado, "Structural Reliability. Statistical Learning Perspectives", Heidelberg, Springer, 2004.

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