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3. Develop quantitative models of the performance of individual components. 4. Identify mechanical interactions among component failures and failure modes. 5. Investigate statistical or probabilistic correlations, if any, among component failures and failure modes. 6. Integrate component performance models, interactions, and correlations within an overall system performance model. 7. Calculate numerical results for system reliability.
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22.2 Dependencies Among Component Failures
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The interdependencies of component failures or failure modes, whether caused by mechanical interaction as in step four or by correlation as in step ve above, are extremely important. Consider the design of Figure 22.1, in which one tank in a patio is surrounded by a rewall to contain leaks. Presume that the annual probability of the tank failing and spilling its contents into the patio is pT = 0.01 and that the over ow capacity of the patio is suf cient to retain the full volume of one tank. For oil to leak out of the patio, the tank must fail, and then the rewall must fail, too. Let the probability of the rewall failing given an oil load behind it be pF = 0.01. The joint probability of both the tank and rewall failing, presuming the probabilities independent, is the product, Pr{oil loss} = PT PF = 0.0001, a fairly small number. However, what if liquefaction of the site caused by seismic ground shaking had an annual probability of occurring of 0.001, and should liquefaction occur, both the tank and rewall would fail The probability of this failure is then 0.001. While the probability of liquefaction is a small contributor to the annual risk of tank failure alone, the probabilistic dependence it causes between tank and rewall failure increases the annual probability of loss of oil off the site (system failure) by a factor of ten. Dependencies in component failure probabilities can arise in at least three ways: 1. Mechanical interaction among failure modes (e.g. the tank fails and in so doing uproots the soil under the rewall, and the wall then fails, too). 2. Probabilistic correlation (e.g. a common initiating event affects both the tank and rewall).
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Figure 22.1 Typical tank farm geometry.
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EVENT TREE REPRESENTATIONS
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3. Statistical correlation (e.g. uncertainty about the consolidation coef cient of the foundation soils affects the performance of the tank and rewall in the same way; excessive settlement of each occurs together). The last two categories are often called common mode failures.
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22.3 Event Tree Representations
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The most common way of decomposing a geotechnical risk assessment is by building an event tree, which we described in 20. An event tree starts with some initiating event, and then considers all possible chains of events that could lead from the rst event. Each chain of events leads to some performance of the system. Some of these chains of events lead to adverse outcomes; some do not. For each event in the tree, a probability is assessed presuming the occurrence of all the events preceding it in the tree, that is, a conditional probability. The total probability for a particular chain of events or path through the tree is found by multiplying the sequences of conditional probabilities. In this way, we can build an event tree for the oil storage patio of Figure 22.1. This event tree is shown in the upper part of Figure 22.2. The rst event is loss of oil from the tank. This occurs with probability p. If the tank leaks, then either the re wall retains the spilled oil or it does not. Let the probability that the re wall fails to retain the oil be, q. Note, this probability q depends on whether the tank leaks or not. The pressure of
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