Justi cation for Improvement Vertical Cut in Cohesive Soil in Visual Studio .NET

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Justi cation for Improvement Vertical Cut in Cohesive Soil
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Example 13.1 described the reliability analysis of a vertical cut in cohesive soil expressed in terms of the margin of safety M M = c H /4
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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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(16.1)
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THE HASOFER LIND APPROACH (FORM)
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where c is the cohesion, is the unit weight, and H is the height of the cut. For a 10 m deep cut and uncorrelated cohesion and unit weight, the following values were used: H = 10 m c = 100 kPa = 20 kN/m3 Then c = 30 kPa (16.2)
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= 2 kN/m3
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M = 100 (20)(10)/4 = 50 kPa
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2 M = (30)2 + (2)2 (10/4)2 = 925(kPa)2 = 50/ 925 = 1.64
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(16.3)
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If the cohesion and unit weight are both Normally distributed, any linear combination of them, such as M, must also be normally distributed, and pf = P [M 0] = ( ) = 5.01 10 2 (16.4)
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Note that, because the variables are normally distributed and M is a linear combination of them, these results are exact. Now, let us solve the problem using FOSM. The derivatives are M = 1 and c
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2 2 2 = (1)2 (30)2 +
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M H = 4
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(16.5)
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The value of M is unchanged at 50 kPa. The FOSM estimate for M is
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2 M
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M c
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c2 +
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M
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10 4
(2)2 = 925 kPa
(16.6)
which is identical to the exact result. The rest of the calculation gives once again = 1.64 and pf = 5.01 10 2 . This is what one would expect from a linear combination of Normal variables. However, the problem could also have been stated in terms of the factor of safety F = 4c H and F 4c 1 = H 2 (16.7) (16.8)
F 4 = c H F = 0.020 c F
2 F
If the evaluations are done at the mean values of the variables, then F = 0.100 (16.9) (16.10) F
2 2 = (0.020)2 (30)2 + (0.100)2 (2)2 = 0.4
4 c = 2.00 H F c
c2 +
(16.11) (16.12) (16.13)
= (2.0 1.0)/ 0.4 = 1.58 pf = 3.81 10
JUSTIFICATION FOR IMPROVEMENT VERTICAL CUT IN COHESIVE SOIL
The calculation of the probability of failure assumes F is Normally distributed, which is certainly not the case. However, it is also not clear what the correct distribution of F is. This example shows that, even for a very simple problem, the FOSM procedure gives different results for calculations based on M and on F even though the failure conditions (M = 0 and F = 1) are mathematically identical. Discrepancies and inaccuracies can be expected to become larger as problems become more complicated. Figure 16.1 illustrates the differences between the two computational approaches. Values of c are plotted along the horizontal axis, and values of H /4 are plotted along the vertical axis. The large black dot represents the mean values of both variables. The heavy black line represents the failure condition (M = 0 or F = 1). Points above it and to the left are unsafe, and those below it and to the right are safe. The dashed lines represent constant values of M. The solid lines represent constant values of F . The lines for M = 0 and F = 1 coincide. Reliability analysis is an attempt to quantify how close the system is to failure, that is how close the black dot is to the heavy failure line. The FOSM analysis based on M does this by treating M as a surface whose contours are the dashed lines. Then the most rapid line of descent to M = 0 is the negative gradient of the surface (the positive gradient is uphill). The negative gradient is identi ed on the gure, but, because the contours are parallel, the same gradient would be found anywhere on the M surface. When the same procedure is applied to the F surface, whose contours are the solid lines, the gradient does not represent the shortest distance to the failure condition. Furthermore, different gradients are found at different points. In particular, if we had been able to evaluate the gradient at a point on the line representing the failure condition (F = 1), it would have had the same orientation as the gradient from the M surface and the results from the two approaches would have been the same. The failure to achieve this result is a de ciency of the FOSM approach. Furthermore, even if we did nd the shortest distance from the black dot to the failure surface and if we knew the probability densities of the variables c and , we would not
Figure 16.1 Factor of safety reliability compared to margin of safety reliability. Both methods start from the same point (shown as a large solid dot), but they go in different directions and meet the failure line at different points.
THE HASOFER LIND APPROACH (FORM)
know the distribution of F . The computation of probability of failure requires that we assume that F has one of the common distributions, such as the Normal or LogNormal distributions. The Hasofer Lind approach deals with both these problems.