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Table 14.5 Variance composition, single-stage dike, H = 12 m. (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) Variance Parameter ll Dcrust Dtill cu (M) cu (L) F / xi 0.01 0.06 0.008 0.056 0.0137 0.0215 Systematic 3.0 1.0 0.04 1.0 7.6 24.9 Total 4.0 2.0 1.0 1.0 47.6 99.7 V [F ] = Spatial V [F ] = Vave [F ] at R = 0.2 = = = ( F / xi )2 Var(xi ) Systematic 0.0003 0.0036 nil 0.0031 0.0014 0.0115 Total 0.0004 0.0072 0.0001 0.0031 0.0089 0.0461
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0.0199 0.0658 0.0459 0.0199 + 0.0092 = 0.0291 (1.453 1.0)/ 0.0291 2.66
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The numbers in Tables 14.1 14.4 are the result of the rst three steps. Tables 14.5 and 14.6 show the results of the last three steps for two speci c cases. In these tables column 3 contains the systematic variances, and column 4 the total variances. The differences are the spatial variances. Similarly, column 5 contains the contribution of the systematic variances to the variance in F , and column 6 the total variances. Again the differences are the spatial contributions. Columns 5 and 6 are summed to obtain the systematic variance in F and the total variance in F . Their difference is the spatial variance, which is reduced as explained below. First, however, it should be noted that the procedure minimizes F over the range of failure surfaces and then calculates . It does not minimize directly. The minimum F surface and the minimum surface may not be identical (Lacasse and Nadim 1996). Oka and Wu (1990) observed that, in a situation with two different but likely modes of failure, reliability calculations based on one mode only can be seriously in error. This turned out not to be the situation here.
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The results for the single stage dike built with one berm to a height of 12 m, without the model error, are displayed in Table 14.5. The circular arc analyses gave, for the critical failure circle, E[F ] = 1.453. The spatial variations tend to average out over the failure surface. By integrating the statistically varying shear strength over the failure surface it can be shown (see 9) that the effective spatial variance of the average shear strength is 2ro V [cu,ave ] (14.8) V [cu ] L
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Table 14.6 Variance composition, multi-stage dike H = 23 m. (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) Variance Parameter ll Intact clay (M) (L) Consolidated clay (M) (L) 0.0021 0.009 52.7 62.0 111.5 124.4 V [F ] = Spatial V [F ] = Vave [F ] at R = 0.07 = = = 0.0002 0.0050 0.0122 0.0005 0.0101 0.0018 0.012 13.3 26.3 41.5 88.5 nil 0.0038 0.0001 0.0127 F / xi 0.0088 0.055 Systematic 3.0 1.0 Total 4.0 2.0 ( F / xi )2 Var(xi ) Systematic 0.0002 0.0030 Total 0.0003 0.0061
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0.0298 0.0176 0.0122 + 0.0012 = 0.0134 (1.427 1.0)/ 0.0134 3.69
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where ro is the autocorrelation distance and L is the length of the curve of the intersection of the failure surface with the plane of analysis. For the geometry of this case the ratio becomes 0.2. This is applied to the spatial V [F ] of 0.046. For the case with a single stage to 6 m, E[F ] = 1.50, the critical failure circle is much smaller, and the reduction ratio for the spatial variation in Equation (14.8) becomes 0.7. This leads to signi cantly larger variation in the computed factor of safety and a much lower value of (Table 14.7). Figure 14.3 shows graphically how the different parameters contribute to the variance in the factor of safety for the two single stage cases. The bars represent the sum of the systematic uncertainty and the spatial uncertainty reduced due to averaging over the failure surface. The importance of the uncertainty in the strength of the lacustrine clay is obvious. The density of the ll and the strength of the marine clay are less important. The 6 m dike has a much smaller than does the 12 m dike (1.84 versus 2.66), even though its factor of safety is slightly higher (1.50 versus 1.45). The reduced reliability index re ects a somewhat larger systematic uncertainty and a much larger spatial variability.
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