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Table 14.1 Soil pro le and ll property uncertainties for reliability analysis. (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 Variable Depth of crust Depth to till Fill density Fill friction Expected value 4m 18.5 m 20 kN/m3 30 Spatial 0.96 0.0 1.0 1.0 Systematic 0.04 1.0 1.0 3.0 Total 1.0 1.0 2.0 4.0
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and the soil properties were better known, and the major uncertainty was in the properties of the ll and the partially consolidated clay. The depth of the crust and the depth to the till were both treated as uncertain variables. From the results of eld exploration, the uncertainties listed in Table 14.1 were established. The separation of the uncertainty into spatial and systematic components is important because averaging over the failure surface affects the contribution of spatial variability. The depth of the crust affects the undrained strength pro le, and its spatial variance was estimated from the eld vane data across the entire axis of the dike. The systematic statistical uncertainty for the vane soundings was estimated from Equation (14.5) with n = 27. Uncertainty in the depth to the till affects the depth of the failure surfaces in the analysis. Since the critical surfaces are nearly tangent to the top of the till for H = 12 m, this uncertainty does not affect the average across the pro le but is a systematic uncertainty even though the estimate of variance re ects spatial variance along the cross-section. That is, there is only one critical failure surface at any location. The location of this surface and its associated F are dictated by the uncertain depth of the till at that location, but the uncertainty in the depth of the till comes about by the variation in depth over the whole project site. The estimate of systematic error re ects the effects of a nite number of borings. Uncertainty in the density of the ll was taken to be composed equally of spatial and systematic components. This assumption and estimated magnitudes of the variances were based on judgment. The assumption that E[ ] = 30 re ected Canadian practice.
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14.3.3 Uncertainty in Shear Strength of Foundation Clay
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Table 14.2 presents the uncertainty in cu used for the single stage analyses, based on the eld vane data for the marine and lacustrine clays. The relatively small coef cient of variation selected for the bias in for the marine clay ( = 0.075) re ected detailed comparison with results from laboratory CKo U shear tests with different modes of failure, described by Lefebvre et al. (1988). This bias was doubled for the lacustrine clay, for which a similarly detailed comparison had not been made. The other components of spatial and systematic uncertainty were developed from the techniques described above. For example, the statistical coef cient of variation for the marine clay ([(0.24)2 /62]1/2 = 0.03) follows from Equation (14.5). As the total variance due to several independent
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Table 14.2 Undrained shear strength uncertainty based on vane shear data. (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) Clay Field vane statistics Number of tests Mean, kPa Data scatter, COV Spatial variability, COV Systematic error, COV: Statistical Correction factor: Sub-total Total COV Marine 62 34.5 0.236 0.183 0.030 0.075 0.08 0.20 Lacustrine 37 31.2 0.272 0.272 0.045 0.15 0.16 0.32
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variates is the sum of the individual variances, coef cients of variation and standard deviations are combined by taking the square root of the sum of the squares of the individual components. For the multi-stage case, the shear strengths were established from a combination of undrained strength ratios and the in situ stress history. For the overconsolidated intact clay beyond the limits of the vertical drains, no consolidation was assumed, and cu / p ratios were applied to the p pro le shown in Figure 14.2. For the clay within the limits of the vertical drains, 80% consolidation was assumed under berm number 2, and the cu / vc ratios were applied to vc = v 0 + 192 kPa (= 0.80 12 m 20 kN/m3 ). Therefore, it is necessary to deal with uncertainty in both cu / p or cu / vc and p or vc . Morgenstern Price wedge stability analyses used anisotropic undrained strength pro les for compression, direct simple shear, and extension modes of failure. The undrained shear strength ratios were obtained from CKo U triaxial compression and extension and Geonor direct simple shear tests run using the recompression technique and treated for strain compatibility to account for the effects of progressive failure (Ladd et al. 1983; Ladd 1991). Table 14.3 shows best estimates of these ratios. These were not intended to be conservative. In this table the subscript C indicates compression failure, D direct simple shear failure on a horizontal plane, and E extension failure. Table 14.4 presents the uncertainties in the undrained strength ratios, the vertical preconsolidation and consolidation stresses, and the resulting values of undrained shear strength, . Some uncertainties were developed by combining engineering judgment, experience, and experimental data. As in Table 14.3, these are not conservative values of parameters, but best estimates.
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