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Table 15.1 Results from point-estimate method for vertical cut Point 1 2 3 4 c (kPa) 70 130 70 130 (kN/m3 ) 18 18 22 22 M (kPa) 25 85 15 75 p pM pM 2 234.375 903.125 28.125 2109.375 3275.000 = 27.84
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0.375 9.375 0.125 10.625 0.125 1.875 0.375 28.125 Sum = 50.000 Var = 3275 (50)2 = 775 3 H = 10 m, c = 100 kPa, c = 30 kPa, c = 20 kN/m , c = 2 kN/m3
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is 50. Similarly, the products of the entries in column 5 and the square of the entries in column 4 give the entries in column 7, and the sum of these gives the estimate of E[M 2 ]. The difference between this and the square of the estimate of the mean gives the estimate of the variance. The square root of the variance is the standard deviation. The calculations in the next-to-last row show that this gives 27.84. The values of the mean and the standard deviation of M agree exactly with the theoretical values. These give a reliability index = 1.796, and, if M is Normally distributed, a probability of failure pf = 3.6 10 2 . When the correlation coef cient is set to zero, M is unchanged, and M in both the exact and Rosenblueth calculations becomes 30.41 kPa. Then = 1.644, and pf = 5.0 10 2 .
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15.3.2 Example Culmann Failure Mechanism along a Plane
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A more complicated example is the margin of safety of a slope against sliding on a single plane, otherwise known as the Culmann analysis (Taylor 1948). The margin of safety is M =c+ 1 H sin( ) cos [tan tan ] 2 sin (15.29)
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where is the inclination of the failure plane, is the inclination of the front of the slope, H is the vertical distance from the crest to the toe of the slope, and the other variables have the conventional soil mechanics meanings. If the only uncertain variables are the unit weight, the cohesion, and the tangent of the friction angle ( , c, and tan ), and if these variables are mutually independent, the mean and variance of M can be expressed exactly. For the following values of the parameters: H = 10 m, = 20 , = 26 c = 5 kPa, c = 0.5 kPa tan = 0.267949( = 15 ), tan = 0.018756( the exact analytical results are M = 2.633 kPa M = 0.722 kPa (15.31)
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= 22 kN/m3 , = 2.2 kN/m3 = 0.07) (15.30)
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RELATION TO ORTHOGONAL POLYNOMIAL QUADRATURE
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Table 15.2 Point 1 2 3 4 5 6 7 8
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Results from point-estimate method for Culmann slope c (kPa) 5.5 5.5 4.5 4.5 5.5 5.5 4.5 4.5 tan( ) 0.286706 0.249193 0.286706 0.249193 0.286706 0.249193 0.286706 0.249193 M (kPa) 3.405 2.388 2.405 1.388 3.786 2.954 2.786 1.954 p pM pM 2 1.4493 0.7128 0.7230 0.2408 1.7917 1.0908 0.9702 0.4773 7.4559 = 0.7222
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(kN/m3 ) 24.2 24.2 24.2 24.2 19.8 19.8 19.8 19.8
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0.125 0.4256 0.125 0.2985 0.125 0.3006 0.125 0.1735 0.125 0.4733 0.125 0.3693 0.125 0.3483 0.125 0.2443 Sum = 2.6333 Var = 7.4559 (2.6333)2 = 0.52160 H = 10 m, = 20 deg, = 26 deg
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Table 15.2 shows the calculations for Rosenblueth s method with three independent variables. The arrangement is similar to that of Table 15.1. In this case all the weights are the same (0.125) because the variables are uncorrelated. The results are the same as the exact values to the limit of computational precision. In both cases = 3.647, and pf = 1.3 10 4 . It is much easier to carry out the Rosenblueth point estimates than to derive the exact expressions for the mean and variance of M. Further, in the cases not considered here that the variables were correlated or that some of the other xed variables were uncertain, an analytical expression would be very dif cult or impossible to derive, and an approximate approach would be necessary. These results inspire some con dence in the point-estimate method.
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