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ponded oil against the rewall presumably makes the wall more likely to fail, compared to the case without oil pressure. If system failure is de ned as loss of oil off the site, then the only end node in the event tree that includes a failure is that for which both the tank has spilled its oil and the re wall does not retain the oil. Thus, the probability of system failure is p q. Now consider the initiating event that seismic ground shaking leads to liquefaction of the soils underlying the patio. This event tree is shown in the lower part of Figure 22.2. Once liquefaction occurs, the assumption is made that both the tank and the re wall fail. So, in this case, rather than the two failures being probabilistically independent, they are correlated through the occurrence of a common event which causes each of them to fail at the same time. An event tree serves as a simple way of showing the interrelationship of events in a system failure. An event tree can be used to decompose a problem at different levels of detail. Bury and Kreuzer (1986) and Vick (1997) describe in simple terms how event trees can be structured for gravity dams. Usually, analytical calculations or judgment are more easily applied to smaller components, and research suggests that more detailed decomposition, within reason, enhances the accuracy of calculated failure probabilities. One reason, presumably, is that the more detailed the event tree is, the less extreme the conditional probabilities that need to be calculated or estimated (Vick 2002). Whitman (1984) describes a simple event tree that is part of the risk assessment for erosion in an earthen dam. The issue being addressed is scour in the channel downstream of the dam, caused by large releases over the concrete spillway. The spillway is capable of passing large ows, and the natural channel below may be eroded by these large discharges. Headward erosion of the channel may undermine the spillway basin and then possibly undermine the spillway itself. If the spillway fails, this may lead to breaching of the dam directly, or to erosion of an adjacent earth embankment that in turn could lead to breaching of the dam. The initiating event in this case is a ood discharge of some speci ed range of magnitudes centered on the probable maximum ood, PMF. The subsequent events leading from this initiating event are shown on the event tree of Figure 22.3. In sequence, these potential events are: 1. The natural downstream channel erodes back to the stilling basin, causing scour holes of various depths. 2. The foundation of the stilling basin collapses as a result of a scour hole. 3. Collapse of the stilling basin leads to undermining of the spillway and consequent breaching of the earthen dam. 4. Collapse of the stilling basin leads to erosion of an adjacent earthen embankment and thus to breaching of the dam. The probability of the initiating event, occurrence of the PMF ood, is established from hydrologic studies. In this case, the probability of the PMF within the design life of the dam was estimated to be 10 4 . Based on model hydraulic tests, it was concluded that the natural channel was certain to erode under the discharge of the PMF. Thus, the branch probability at the rst node after the initiating event was taken to be 1.0. Using stability calculations and other hydraulic model tests, the various other branch probabilities were estimated and lled into the event tree. Each branch probability is conditional on
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