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

doi:10.1016/j.strusafe.2008.06.012

Structural Safety

Volume 31, Issue 2, March 2009, Pages 148-156

https://doi.org/10.1016/j.strusafe.2008.06.012 Get rights and content

Abstract

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.

Introduction

The failure of a structure is often a complex “system” event that is a Boolean (or logical) function of other “component” events such as the occurrences of structural failure modes or the failures of constituent members or substructures. For reasonable decision-making on structural designs, retrofits, repairs and damage mitigations, it is essential to accurately estimate the likelihood of the system failure event by system reliability analysis. However, most of risk quantification efforts for structural systems have been made by component reliability analyses. For example, structural fragility models are often developed based on a single component failure event defined in terms of a parameter representing the system status such as “engineering demand parameter” [1], [2]. This is mostly because computing the probability of such a system event is often costly or infeasible due to the complexity of the system and/or the lack of information.

Theoretical bounding formulas [3], [4] have been widely used for computing bounds on the probabilities of series and parallel system events in terms of the component probabilities and the low-order joint probabilities. The first-order system reliability method [4] transforms system reliability problems to multi-normal calculations by use of the results of component reliability analyses. It is applicable to series and parallel systems directly, and to cut-set and link-set systems in conjunction with bounding formulas. These existing system reliability analysis methods are not flexible in incorporating various types and amount of available information on components and their statistical dependence. Moreover, the complexity of a system event makes the reliability computations more complicated and/or time-consuming. Song and Der Kiureghian [5] proposed a method for computing bounds on system failure probability by use of linear programming (LP). This “LP bounds” method subdivides the sample space of component events into mutually exclusive and collectively exhaustive events and describes the system probability and the available information by use of vectors representing subdivided areas. Then, the lower and upper bounds of the system failure probability are obtained by solving LP problems. This matrix-based framework of system reliability analysis provides the narrowest possible bounds on the probability of any general systems with significantly enhanced flexibility in incorporating available information. However, when the complete information is available, i.e., all the joint component failure probabilities can be provided as constraints for the LP, solving these over-constrained LP often causes numerical issues.

In order to make use of the matrix-based framework of the LP bounds method even in the case when complete information is available, Song and Kang [6] recently proposed a matrix-based system reliability (MSR) method. Just as the LP bounds method, the MSR method is uniformly applicable to general systems, but the reliability is computed by a simple matrix calculation instead of solving an LP. Matrix-based procedures were newly developed for obtaining vector representations of system events and marginal/joint component probabilities of the MSR framework in an efficient and convenient manner. The procedure developed for identifying system events can be used for the LP bounds method as well. The proposed method was demonstrated through numerical examples of structural systems [6]. The MSR method was later used to estimate the reliability of a bridge transportation network [7] and natural gas network [8]. The probabilities of complex system events such as disconnection between cities or nodes were computed efficiently by the MSR method based on the seismic vulnerability of its constituent bridges or gas pipelines. Also computed are the probability mass function of the number of failed components and the conditional probabilities of failures given observed failure events. These conditional probabilities are useful for quantifying the relative importance of the network components with respect to system events of interest. The numerical examples demonstrated the versatility of the MSR method in dealing with various complex system events and its flexibility in incorporating various types and amount of available information.

This paper further develops the MSR methodology in two ways. First, a method is developed to use the MSR framework even in the case the sources of the statistical dependence between component events are not explicitly identified. The correlation matrix of basic random variables or component safety margins (or factors) is approximately represented by a Dunnett–Sobel class correlation coefficient matrix to describe the statistical dependence by use of one or few random variables. This enables us to use the matrix-based procedure originally developed for computing the probability vector of independent components for components under statistical dependence as well. Second, a new method is developed to calculate the sensitivities of system probability with respect to parameters in a convenient manner. The paper demonstrates the MSR method and its further developments by two numerical examples of structural systems. In the first example, the system fragility of a bridge structure is efficiently computed based on the analytical fragilities of the bridge components and the correlation coefficients between the seismic demands. The probabilities of various system events and the conditional probabilities are also computed conveniently. In the second numerical example, the MSR method is used to estimate the probability of the collapse of a statically indeterminate truss structure subjected to an abnormal load. The sensitivities of the collapse probability with respect to the means and standard deviations of uncertain member capacities are computed to help decision-making regarding design changes or uncertainty management.

Section snippets

Matrix-based system reliability method

Consider a system event whose ith component, i = 1, … , n has two distinct states, e.g., the failure and survival. The sample space can be subdivided into m = 2ⁿ mutually exclusive and collectively exhaustive (MECE) events. These are named the “basic” MECE events and denoted by e_j, j = 1, … , m. Then, any system event can be represented by an “event” vector c whose jth element is 1 if e_j belongs to the system event and 0 otherwise. Let p_j = P(e_j), j = 1, … , m, denote the probability of e_j. Due to the mutual

MSR analysis under statistical dependence between component events

When component events are statistically dependent, it may be a daunting task to construct the probability vector p because the basic MECE events can not be computed simply by products of probabilities of components and their complementary events. However, in many structural system reliability problems, we can achieve conditional independence between component events given outcomes of a few random variables representing the sources of “environmental dependence” or “common source effects”.

Let X

Sensitivity of system reliability by MSR method

For a decision-making based on risk quantification, it is essential to estimate the sensitivities of failure probability P_f with respect to design parameters. For example, the sensitivities with respect to the means of parameters can help achieve optimal designs while those with respect to the standard deviations are critical in efforts for uncertainty management or quality control. The following sensitivity-based importance measures are often used to compare the relative importance of the

System fragility of a bridge structure

“Fragility” is defined as the conditional probability that a structure will exceed a specified limit state for a given level of loading intensity. Nielson and DesRoches [21], [22] developed analytical fragility curves of major components of highway bridges (see Fig. 1 for illustrations of major bridge components). Let C_i and D_i, respectively denote the seismic capacity and demand of the ith component of a bridge. Assuming both follow the lognormal distribution, the safety factor F_i = ln C_i − ln D_i is

Summary and conclusions

The matrix-based system reliability (MSR) method can compute the probabilities of general system events efficiently by simple matrix operations. Both a system event and the likelihood of its component events are described by vectors that are obtained by efficient matrix-based procedures. The method is uniformly applicable to any type of system events including series, parallel, cut-set and link-set systems. The MSR method can estimate various importance measures and conditional probabilities

Acknowledgments

This work was supported by the Mid-America Earthquake Center (National Science Foundation, Award No.: EEC-97010785) and Caterpillar Inc. (RPS No.: 06-077). This support is gratefully acknowledged.

References (24)

M. Hohenbichler et al.
First-order concepts in system reliability
Struct Saf
(1983)
W.-H. Kang et al.
Matrix-based system reliability method and applications to bridge networks
Reliab Eng Syst Saf
(2008)
A. Der Kiureghian et al.
Multi-scale reliability analysis and updating of complex systems by use of linear programming
Reliab Eng Syst Saf
(2008)
A. Der Kiureghian et al.
Availability, reliability and downtime of systems with repairable components
Reliab Eng Syst Saf
(2007)
R.H. Sues et al.
System reliability and sensitivity factors via the MPPSS method
Probab Eng Mech
(2005)
Cornell CA, Krawinkler H. Progress and challenges in seismic performance assessment. PEER Center News, Spring 2000....
A. Der Kiureghian
Non-ergodicity and PEER’s framework formula
Earthquake Eng Struct Dyn
(2005)
O. Ditlevsen
Narrow reliability bounds for structural systems
J Struct Mech
(1979)
J. Song et al.
Bounds on system reliability by linear programming
J Eng Mech ASCE
(2003)
Song J, Kang W-H. Matrix-based system reliability method and applications to structural systems. In: Proceedings of the...

Song J, Ok S-Y, Chang L. Rapid risk assessment and decision support for urban infrastructure networks by matrix-based...

Song J, Der Kiureghian A. Component importance measures by linear programming bounds on system reliability. In:...

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

Abstract

Introduction

Section snippets

Matrix-based system reliability method

MSR analysis under statistical dependence between component events

Sensitivity of system reliability by MSR method

System fragility of a bridge structure

Summary and conclusions

Acknowledgments

Struct Saf

Reliab Eng Syst Saf

Reliab Eng Syst Saf

Reliab Eng Syst Saf

Probab Eng Mech

Non-ergodicity and PEER’s framework formula

Earthquake Eng Struct Dyn

Narrow reliability bounds for structural systems

J Struct Mech

Bounds on system reliability by linear programming

J Eng Mech ASCE