System reliability analysis of fatigue-induced, cascading failures using critical failure sequences identified by selective searching technique

Many structural systems are subjected to the risk of cascading system-level failures initiated by local failures. For efficient reliability analysis of such complex system problems, many research efforts have been made to identify critical failure sequences with significant likelihoods by an event-tree search coupled with system reliability analyses; however, this approach is time-consuming or intractable due to repeated calculations of the probabilities of innumerable failure modes, which often necessitates using heuristic assumptions or simplifications. Recently, a decoupled approach was proposed (Kim 2009; Kurtz et al. 2010): critical failure modes are first identified in the space of random variables without system reliability analyses or an event-tree search, then an efficient system reliability analysis is performed to compute the system failure probability based on the identified modes. In order to identify critical failure modes in the decreasing order of their relative contributions to the system failure probability, a simulation-based selective searching technique was developed by use of a genetic algorithm. The system failure probability was then computed by a multiscale system reliability method that can account for statistical dependence among the component events as well as among the identified failure modes (Song & Kang 2009; Song & Ok 2010). This paper presents this decoupled approach in detail and demonstrates its applicability to complex bridge structural systems that are subjected to the risk of cascading failures induced by fatigue. Using a recursive formulation for describing limit-states of local fatigue cracking, the system failure event is described as a disjoint cut-set event (Lee & Song 2010). Critical cut-sets, i.e. failure sequences with significant likelihood are identified by the selective searching technique using a genetic algorithm. Then, the probabilities of the cut-sets are estimated by use of a sampling method. Owing to the mutual exclusiveness of the cut-sets, the lower-bound on the system cascading failure probability is obtained by a simple addition of the estimated probabilities of the identified cut-sets. A numerical example of a bridge structure demonstrates that the proposed search method skillfully identifies dominant failure modes contributing most to the system failure probability, and the system failure probability is accurately estimated with statistical dependence fully considered. An example bridge with 97 truss elements is considered to investigate the applicability of the method to realistic large-size structures. The efficiency and accuracy of the method are demonstrated through comparison with brute-force Monte Carlo simulations. (Galambos 1990; Henwadi & Frangopol 1994). Cutset events Ck are also statistically dependent since they share common or statistically dependent component events; hence, a system reliability analysis method must account for statistical dependence at both levels, i.e. among failure modes and among component events, to accurately evaluate the systemlevel risk. For efficient system reliability evaluation, most of the existing failure-mode-based approaches employ approximation methods such as bounding formulas (Ditlevsen 1979; Feng 1989; Park 2001) or response surfaces (Zhao & Ono 1998). While these may enable rapid estimation, they are not flexible in including types and amount of available information on components or in accounting for statistical dependence. To overcome these issues, a new bounding approach was developed by use of linear programming (Song & Der Kiureghian 2003) and was further developed for multi-scale analysis (Der Kiureghian & Song 2008); however, solving such linear programming problems may cause computational or numerical issues when the feasible domain of linear programming is small or the system event consists of a large number of components. Another issue present in system reliability analysis is that innumerable failure modes often exist, because real structures are highly redundant and the failures of members re-define the limit-states of the remaining members due to stress re-distribution. These issues make it intractable to enumerate all possible limit states for system reliability analysis especially for complex structures with a large number of structural elements. To overcome these difficulties, some methods using an event tree (Murotsu et al. 1984; Karamchandani 1987; Srividya & Ranganathan 1992) have been developed to identify only the failure modes with significant likelihood (Moses & Stahl 1978; Murotsu et al. 1984; ThoftChristensen & Murotsu 1986; Ranganathan & Deshpande 1987). The system failure probability is then calculated using the probabilities and statistical dependence of the identified failure modes; however, while evaluating the contributions of individual failure modes to the search process, component and system reliability analyses need to be performed repeatedly, requiring high computational cost for structures with large amounts of redundancy. In order to deal with these issues, Kim (2009) proposed a new framework for risk assessment that decouples the identification process of the dominant failure modes from the process for evaluating the probabilities of failure modes and the system event. This dichotomy reduces the need for repeated component and system reliability analyses in the failure mode searching process. First, dominant failure modes are obtained by a simulation-based selective searching technique using a genetic algorithm, which identifies cascading fatigue failure modes rapidly. Then, the probabilities of the failure modes identified by the selective search and the corresponding system failure probability are computed by system reliability analyses. While brute-force Monte-Carlo simulation of failure sequences could provide the system reliability accurately given sufficient time to converge, the selective searching method provides not only the system failure probability but also critical failure modes without prior knowledge of the system response. In this paper, the selective searching method is applied to a bridge structural system subjected to the risk of fatigue-induced cascading failures. Using an efficient characterization of fatigue-induced failure modes developed by Lee & Song (2010), cascading failure events are described as mutually exclusive (or disjoint) cut-set events, making the system failure probability simply the sum of the probabilities of all identified critical failure modes. This paper first introduces the simulation based selective searching technique, followed by a summary of the efficient formulation of fatigue-induced failure modes and methods used for calculating the probabilities of the identified cut-sets. The proposed risk assessment framework is then demonstrated by a large-size planar-truss bridge structure. 2 SELECTIVE SEARCHING TECHNIQUE FOR DOMINANT FAILURE MODES Most of the methods developed to identify failure modes of structural systems can be placed into the following two types of approaches (Shao & Murotsu 1999): the so-called probabilistic approach, which includes the branch and bound method (Murotsu et al. 1984; Thoft-Christensen & Murotsu 1986; Karamchandani 1987) and simulation based techniques (Grimmelt & Schueller 1982; Rashedi 1983; Moses & Fu 1988; Ditlevsen & Bjerager 1989; Melchers 1994); and the so-called deterministic approach, which includes the incremental loading method (Moses & Stahl 1978; Moses 1982; Lee 1989), the βunzipping approach (Thoft-Christensen & Murotsu 1986), the methods based on mathematical programming (Corotis & Nafday 1989), or methods employing heuristic techniques (Xiao & Mahadevan 1994; Shetty 1994). In general, the probabilistic approach is considered theoretically rigorous but computationally costly, whereas the deterministic approach is computationally efficient but has the risk of overlooking important failure modes (Shao & Murotsu 1999). To remedy these issues, Shao & Murotsu (1999) proposed an improved simulation-based selective searching technique in which a genetic algorithm (GA) (Holland 1975; Goldberg 1989) is used to find the few most dominant failure modes that contribute the most to the system failure probability. Noting that GA works with a population of multiple searching points, Kim (2009) extended the approach to capture multiple failure modes at once. The proposed searching method differs from the one proposed by Shao & Murotsu (1999) by the two distinct GA strategies: searching direction and elitism, as explained below. Consider an n-dimensional random variable space x which represents possible realizations of uncertain quantities in a system reliability problem. Through a nonlinear transformation determined by the joint probability distribution model of the corresponding random vector X, one can obtain the space of uncorrelated standard normal variables u, i.e. u = T(x) (for details, see Der Kiureghian 2005). For a graphical reference, see an example in Figure 1 below: