System reliability and sensitivity under statistical dependence by matrix-based system reliability method

A matrix-based system reliability (MSR) method has been recently proposed to compute the probabilities of general system events efficiently by simple matrix operations. The proposed matrix-based framework describes both a system event and the likelihood of its component events by vectors that are obtained by efficient matrix-based procedures. The probability of the system event is computed by the inner product of the two vectors. Therefore, the method is uniformly applicable to any type of system events including series, parallel, cut-set and link-set systems. In the case when one has incomplete information on component probabilities and/or on the statistical dependence between components, the matrix-based framework enables us to obtain the narrowest bounds on the system probability by linear programming. Various importance measures and conditional probabilities are also efficiently estimated by the proposed method. This paper presents the MSR method and further develops it in terms of statistical dependence and parameter sensitivity of system reliability. First, a method is developed to use the MSR method for systems with statistically dependent components. The correlation coefficients between the basic random variables or the component safety margins are represented by a Dunnett-Sobel class correlation matrix to identify the source of the statistical dependence and to make use of the matrix-based procedure developed for independent components. Second, a new matrix-based procedure is proposed to calculate the sensitivities of system reliability with respect to parameters. This paper demonstrates the MSR method and these further developments by two numerical examples of structural systems. First, the system fragility of a bridge structure is computed based on the analytical fragility models of the bridge components and the correlation coefficients between the seismic demands at different components. In the second example, the MSR method is used to estimate the probability of the collapse of a statically indeterminate structure subjected to an abnormal load. The sensitivities of the probability with respect to the means and standard deviations of uncertain member capacities are estimated for an optimal upgrade of the structural system.