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Further development of matrix-based system reliability method and applications to structural systems

Won-Hee Kang, Young-Joo Lee, Junho Song, Bora Gencturk

2012
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Structure and Infrastructure Engineering
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In efforts to estimate the risk and reliability of a complex structure or infrastructure network, it is often required to evaluate the probability of a "system" event, i.e. a logical function of multiple component events. Its sensitivities with respect to design parameters are also useful in decision-making processes for more reliable systems and in reliability-based design optimization. The recently developed, matrixbased system reliability (MSR) method can compute the probabilities of general
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... bilities of general system events including series, parallel, cut-set and link-set systems and their parameter sensitivities, by use of efficient matrix-based procedures. When the component events are statistically dependent, the method transforms the problem into an integral in the space of random variables which cause the statistical dependence, termed as the common source random variables (CSRVs). One can identify CSRVs by fitting a generalized Dunnett-Sobel (DS) model to a given correlation coefficient matrix. This paper introduces two further developments of the MSR method: First, for efficient evaluation, it is proposed that the integral in the CSRV space can be performed using the first-or second-order reliability methods. Second, a new matrix-based procedure is developed to compute the sensitivity of the system failure probability with respect to the parameters that affect the correlation coefficients between the components. In addition, an extensive parametric study is performed to investigate the effect of the error in fitted generalized DS model on the accuracy of the estimates by the MSR method. The further developed MSR method is demonstrated by two examples: system reliability analysis of a three-story Daniels system structure, and finite element reliability analysis of a bridge pylon system. Recently, a matrix-based system reliability (MSR) method was developed to compute the probabilities of general system events, including series, parallel, cut-set and link-set systems, and their parameter sensitivities by use of efficient matrix-based procedures (Kang et al. 2008, Song and Kang 2009). When one needs to identify the sources of the statistical dependence between component events, the correlation coefficient matrix of random variables representing components, e.g. safety margins, can be fitted by a Dunnett-Sobel (DS) class correlation coefficient matrix (Dunnett and Sobel 1955). This approach is used to achieve conditional independence between components, which enables us to utilize efficient matrix-based procedures originally developed for independent components (Song and Kang 2009). The MSR method also allows us to compute the sensitivities of system probability with respect to its parameters through matrix-based procedures (Song and Kang 2009). The method can be used to obtain the probability distribution functions of the uncertain number of failed components and the network flow quantities. Given the observed events, the conditional probabilities of component/system failures can also be evaluated. These conditional probabilities are useful for quantifying the relative importance of the components with respect to a system event of interest. These merits of the MSR method have been demonstrated through its applications to risk assessment of structural systems (Song and Kang 2008, 2009), post-hazard connectivity analysis of lifeline networks (Kang et al. 2008, Song and Ok, 2010), post-hazard network flow capacity analysis (Lee et al. 2010), and system reliability based design/topology optimization (Nguyen et al. 2010). These application examples helped identify the following limitations of the method: (1) If a large number of CSRVs are needed for accurate fitting by use of a generalized DS correlation model (Song and Kang 2009), the numerical integration in the CSRV space can be inefficient. Therefore, for rapid risk assessment, one may need to tolerate a certain level of fitting error. However, the impacts of the fitting error on the accuracy of the system failure probabilities are not known in general; (2) If direct numerical integration is used in the CSRV space, the method becomes inefficient as the number of CSRVs are increased for more accurate evaluation; and (3) Matrix-based procedures for obtaining parameter sensitivities of system failure probability are needed also for parameters that affect the correlation coefficients between the components. In this paper, after a brief review of the MSR method, the effect of the error in the fitted DS correlation matrix on the accuracy of the MSR method is first investigated through an extensive parametric study using Monte Carlo simulations (MCS). The MSR formulations for evaluating the multivariate normal integral of a general system event and its parameter sensitivities are also discussed. Then, two further developments of the MSR method are introduced: First, a method is proposed to evaluate the integral in a large-dimensional CSRV space in an efficient manner by use of the first-or secondorder reliability methods (FORM/SORM). Second, a new matrix-based procedure is developed to compute the sensitivity of the system failure probability with respect to parameters that affect the correlation coefficients between components. The further

doi:10.1080/15732479.2010.539060
fatcat:taawfdq4uzchbn5xzeynlq2cta