Example 368

A parallel system consists of three components where the probability of failure of each individual component is pf = 0.01. What is the probability of failure of the system if the failure modes are independent? If they are perfectly correlated?

Answer

For a parallel system consisting of three components whose individual probabilities of failure are pf = 0.01, the system failure probability is upper bounded (first-order) by pfjparauei = min(pf1, pf2, pf3) = 0.01 if the three failure modes are perfectly correlated, and it is lower bounded (first-order) by parallel = (0.01)(0.01)(0.01) = 0.000001 if the three failure modes are independent. As indicated in this simplified example, there can be huge errors if correlation is neglected.

36.4.3 General Systems

Many general engineering systems can be modeled as a combination of series and parallel systems. For example, a series of y parallel systems where each parallel system a has za components would have a probability of failure expressed as

where gab(X) identifies the limit state equation for a specific component in the system model. Most complex systems can be sequentially broken down into simpler equivalent subsystems. The reliabilities of a series subsystem and parallel subsystem are solved individually as described above using the reliabilities and direction cosines at the points of failure of individual components. An example of how a seriesparallel system is reduced is shown in Figure 36.5. The system shown contains six components (1-6). Initially, the two parallel subsystems are reduced to equivalent components (7 and 8) that formed part of a series system. The series system is then reduced to a single equivalent component (9). The correlation of the equivalent components is computed using equivalent alpha vectors that are a function of the equivalent direction cosines as described in Estes and Frangopol (1998). Any structural system that can be modeled as a combination of series and parallel components can be analyzed.

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