CONCLUSIONS FROM THE JAMES BAY STUDY

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the associated uncertainty can be a signi cant contributor to the overall uncertainty. Since bias is often ignored in theoretical treatments of analytical procedures and it is dif cult to quantify, the engineer must often rely on judgment to establish its contribution. The strength of the reliability analysis is not that one can get a better estimate of each of these uncertainties but that one can deal with them explicitly and coherently. Uncertainties in soil properties yield a lower bound estimate of the probability of failure, not the absolute probability of failure. That requires a more elaborate probabilistic risk analysis involving fault trees or other methods of evaluating risk due to all contingencies (e.g. Whitman 1984). However, for most practical applications the calculation of relative probability of failure is suf cient for parametric analysis. A 1% chance that the whole slope will fail is different from a near certainty that one per cent of a long embankment will fail, and reliability analysis allows such a distinction.

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Analysis of the James Bay Embankments

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The analysis of the James Bay embankments illustrates a multi-step procedure for evaluating the reliability index. Once the uncertainties in the soil properties, geometry, loads, and other contributing factors have been established, it is necessary to propagate their effects through the stability analysis. Equation (14.2) provides one way to do so. The FOSM approach is a powerful approximate method to deal with the calculations. The Taylor series combined with a divided difference approximation to the derivatives has proven effective. In the analysis, the effects of both spatial and systematic uncertainty are considered. The averaging effect tends to reduce greatly the contribution of the spatial uncertainty. In the case of the multi-stage construction spatial uncertainty contributes so little that it could almost be ignored without affecting the results. The small embankment with a short failure surface has a large uncertainty in F due to spatial variation in undrained strength. This means that several small failures along the length of the embankment are likely. The multi-stage embankment with a long failure surface has almost all the uncertainty in systematic error, which leads to a small probability of failure along a long section of the embankment. When the autocorrelation distance is not known, the analysis can at least give point estimates of the reliability index without providing information on the length of embankment likely to fail. The example also illustrates the contribution of model error, and the dif culty in determining the magnitude of such bias effects. However the solution is carried out, it is important to deal rationally with the spatial variability of the parameters. In particular, the averaging effect on variations that are due to inherent spatial scatter and are not systematic must be accounted for, or the resulting probabilities of failure will be too high. The technique used here is one appropriate method.

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Improvements in the James Bay Analysis

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The preceding sections describe the reliability analysis of the James Bay dikes as it was actually carried out. Several factors were not included in the analysis, and some of the procedures could have been improved upon. This is typical of most engineering calculations; they are done under the pressures of the job and with the tools available

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FIRST ORDER SECOND MOMENT (FOSM) METHODS

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at the time. Nevertheless, we should mention some areas where the procedures could be modi ed. First, the analysis ignored the possible correlation between the parameters and variables that entered into the calculation. Little additional effort is needed to incorporate correlation in an FOSM analysis. Equation (14.2) includes all the necessary terms, and more rows can be added to Tables 14.5 and 14.6 to incorporate the correlated terms. In the case of the James Bay dikes there was not enough information available to include correlation effects. Furthermore, the effect of most reasonable correlations would be to increase the reliability and decrease the probability of failure. Therefore, ignoring the correlation effects was conservative in this case, but, if data were available to establish the correlation coef cients, their inclusion would certainly be a meaningful extension of the analysis. The analysis of the multi-stage dike relied on a Morgenstern-Price analysis of a failure that incorporated a long, at middle section. More recent work indicates that a failure mechanism involving a curved or wavy middle section can give lower factors of safety in some cases (Ladd 1996). The analysis found the minimum factor of safety for the various modes of failure. Further iterations could have been introduced to nd the failure mode that gave the lowest value of instead of the lowest value of F . It does not seem that the overall results would have changed very much, but it would be a desirable improvement in methodology, especially if the results were very near the margin for a decision. The procedure for reducing the contribution of the spatial variations by averaging their effects over the length of the failure surface assumed that the spatial correlation for the strength parameters applied to all the variables. This is probably not the case. A better procedure would be to apply the averaging correction to each variable separately.

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