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(14.1)
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where the x s are the values of the variables X that enter into the calculation of F . As is shown in the following example, a modeling error can be added to these expressions. The function F could be any relevant function, but for present purposes it is the factor of safety calculated by some appropriate numerical method. Once the mean and variance have been calculated it is a simple matter to compute the reliability index : = E[F ] 1 F (14.3)
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The usual practice is then to assume that the factor of safety is Normally distributed and to compute the probability of failure from . The First Order Second Moment (FOSM) procedures that follow are easier to demonstrate than to describe, so the bulk of this chapter is a detailed presentation of the application of FOSM methods to a speci c problem of the reliability of the dikes for the James Bay Project (Christian et al. 1994).
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Reliability and Statistics in Geotechnical Engineering Gregory B. Baecher and John T. Christian 2003 John Wiley & Sons, Ltd ISBN: 0-471-49833-5
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14.1 The James Bay Dikes
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The James Bay hydroelectric project required the construction of approximately fty kilometers of dikes on soft, sensitive clays (Ladd et al. 1983; Ladd 1991). When the stability of an embankment in such circumstances is evaluated, professional opinions often diverge regarding the strengthening of foundation soil during construction, selecting appropriate factors of safety, and evaluating strength properties. The owners formed an international committee to address and resolve the issues of the stability of the slopes. In particular, the engineers had to determine appropriate factors of safety to be used in the design of dikes for different parts of the project ranging in height from 6 23 m. Ladd et al. (1983) provide details of the use of the mean rst-order reliability method to evaluate the single- or multi-stage construction of a typical dike, whose cross-section is shown in Figure 14.1. The goals of the analysis included understanding the relative safety of different designs, obtaining insight about the in uence of different parameters, and establishing consistent criteria for preliminary designs. The rst design cases to be considered were the construction of embankments in a single stage, either to a height of 6 m without berms or 12 m with one berm. Undrained shear strength values were obtained from eld vane tests, and the stability analyses were done using the simpli ed Bishop circular arc method of slices. The second design case called for multi-stage construction. Following the installation of vertical drains, the rst stage would be built to a height of 12 m. Then the zone with the drains would be allowed to consolidate to 80% of primary consolidation, and a second stage would be built to a height of 23 m. In this case the undrained shear strength would be obtained from consolidated-undrained shear tests combined with knowledge of the stress histories (Ladd
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160 140 120 Y axis (m) 100 30 80 Till Limit of vertical drains Foundation clay Critical wedge - stage 2 60 Fill properties: g = 20 kN/m3 f = 30 Note: All slopes are 3 horizontal to 1 vertical. Y 123 m Berm 2 Berm 1 Stage 1 Stage 2
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Figure 14.1 Cross-section of typical dike. (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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INDEX PROPERTIES SOIL 0 10 20 30 Ip (%) PROFILE 0 1 2 3 IL 0 Crust kN m3 Mean from 8 FV FIELD VANE STRESS HISTORY s AND s (kPa) vo p 100 200
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0 2 4 6 Depth, Z (m)
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cu (FV), (kPa) 20 40 60 0
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