## 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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