Sensitivity of First-Excursion Probabilities for Nonlinear Stochastic Dynamical Systems

Quantification of the performance of structural systems subject to dynamic loading is of paramount interest in several fields of engineering and particularly in the case of earthquake engineering. Knowledge on the performance of a structure during seismic events allows taking design decisions that ensure its serviceability and safety throughout its life. Nonetheless, quantification of performance is a challenging task as there is always uncertainty on future loadings that affect a structure during its lifetime. Structural reliability has emerged as a discipline that allows accounting for the unavoidable effects of uncertainty over performance. Thus, probability theory is used to describe the uncertainty associated with different relevant parameters that affect performance by means of random variables, random fields, and/or stochastic processes. In this manner, uncertainty is propagated from these input parameters to the responses of interest such as displacements, accelerations, forces, etc. A particularly useful way to measure the effects of uncertainty in the dynamic response of structural systems is the so-called first-excursion probability. This probability is widely used in stochastic structural dynamics and measures the chances that one or more structural responses exceed a prescribed threshold level within the duration of a dynamical excitation (Soong and Grigoriu 1993). First-excursion probability estimation is particularly challenging as characterization of uncertain loading usually comprises stochastic processes whose discrete representation can involve hundreds or even thousands of random variables. Similarly, the number of possible failure criteria involved can be extremely large as well, i.e., there can be several responses of interest that must be controlled at a large number of discrete time instants. Hence, several different techniques have been proposed in order to estimate first-excursion probabilities. Among these, methods based on simulation (such as the Monte Carlo method and its more advanced variants) have been shown to be the most appropriate approach to compute these probabilities (Schueller et al. 2004). Although first-excursion probability provides a most useful way to rationally account for the effects of uncertainty on structures subject to stochastic loading, it is certainly not the only metric that should be taken into account when designing a system. In fact, it is also of interest analyzing the sensitivity of the probability with respect to variations in the properties of the structural system. For example, determining the variation in probability due to a change in the size of a structural member can provide useful information to increase the safety level or to identify the most influential design parameters. Nonetheless, estimation of the sensitivity of first-excursion probabilities for dynamical