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14.3.4.2 Multi-stage embankment
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Table 14.6 shows the results for the multi-stage dike. The failure surface in this case is large (see Figure 14.2) and mostly horizontal, so the appropriate autocorrelation distance for the shear strength is also large. The reduction ratio for spatial variation of shear strength
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Table 14.7 Summary of reliability results and effects of model error. (Christian, J. T., Ladd, C. C. and Baecher, G. B., 1994, Reliability Applied to Slope Stability Analysis, Journal of Geotechnical Engineering, ASCE, Vol. 120, No. 12, pp. 2180 2207, reproduced by permission of the American Society of Civil Engineers) Var[F ] Case (No. Stages) Sgl., H = 6 m w/o mod. err. with mod. err. Sgl., H = 12 m w/o mod. err. with mod. err. Multi, H = 23 m w/o mod. err. with mod. err.
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= E[F ] 1 Var[F ] 1.84 1.96 2.66 2.60 3.69 3.19
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0.074 0.086 0.029 0.041 0.0134 0.0244
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E[F ] with model error = E[F ] w/o model error 1.05 V [e] = (0.07 increased E[F ])2
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0 friction density D crust D till Parameter cu (M) cu (L)
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Figure 14.3 Variance components for single-stage dikes. (Christian, J. T., Ladd, C. C. and Baecher, G. B., 1994, Reliability Applied to Slope Stability Analysis, Journal of Geotechnical Engineering, ASCE, Vol. 120, No. 12, pp. 2180 2207, reproduced by permission of the American Society of Civil Engineers.)
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becomes 0.07. Because the overall uncertainty is much lower in this case, the reliability index, at 3.69, is the largest of all the cases. As in the single-stage cases, the strength of the lacustrine clay and the density are the most important contributors to the overall
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uncertainty. Since very little of the failure surface passes through the marine clay, its uncertainty contributes little. The failure surface passes about equally through the intact and consolidated lacustrine clays, and each contributes about equally to the uncertainty. Although the coef cients of variation are larger for the intact clay, the mean strength is smaller. The contributions of the different parameters are illustrated in Figure 14.4, which again shows the importance of the lacustrine clay. This gure shows the breakdown of the uncertainty between spatial and systematic contributions and also shows the important effect of averaging the spatial uncertainty.
14.3.4.3 Model error
Even if the soil parameters have been estimated correctly, the predictions of an engineering model can be expected to deviate from reality. Estimating the error introduced by this factor can be dif cult; the most effective approach is to rely on empirical observations when they exist. An alternative is to use formal analysis. In the present case three sources of model error were considered: three-dimensional failure compared to two-dimensional analysis (i.e. end effects), failure to nd the most critical failure surface in the analysis, and numerical and rounding error. Azzouz et al. (1983) examined the contributions of end effects in eighteen case histories. They found that the ratio between the three-dimensional and the plane strain F was 1.11 with a standard deviation of 0.06. These numbers were rounded to 1.1 and 0.05. Failure to nd the lowest factor of safety is considered to overestimate F by about 5% with a standard deviation of about 5%. The third factor was incorporated by assuming that it contributed 2% to the variability of the results. The combined effect of these three sources is to increase E[F ] by about 5% and to create an
0.014 0.012 Variance Contribution 0.01 0.008 0.006 0.004 0.002 0 t (M) Intact t (L) t (M) t (L) f g fill Consolidated