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app17_slope_rel - Application Example 17(Reliability...

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Application Example 17 (Reliability Analysis) STABILITY OF NATURAL SLOPES The possible instability of natural slopes is a major geologic hazard in many regions of the world. The degree of stability of a slope may be expressed through the factor of safety F, which is the ratio between a soil resistance measure and the associated applied load. Hence values of F larger than 1 are associated with stability and values smaller than 1 indicate unstable conditions. For natural slopes with a shallow soil layer (so-called infinite slopes; see Figure 1), the factor of safety is given by (Skempton and DeLory, 1957): ' c + [ γ z (z z w ) γ ] cos 2 α tan φ ' (1) w F = γ sin z α cos α Eq. 1 can be simplified to: F = c' tan φ ' m γ w tan φ ' (2) γ zsin α cos α + tan α γ tan α where ' c is the cohesion of the soil φ ' is the angle of internal friction of the soil γ is the unit weight of the soil γ w is the unit weight of water α is the inclination of the slope to the horizontal z is the depth below the ground surface z w is the depth of the water table below the ground surface m is the degree of saturation of the soil, m = z w z z
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-5 0 5 10 15 20 0 5 10 15 20 25 ) ) z w z SURFACE ROCK -20 -15 -10 -5 x (m y (m SOIL BEDROCK INTERFACE/FAILURE SOIL Figure 1. Infinite Slope Geometry and Definition of Parameters As indicated above, the failure criterion is F < 1. For example, a slope with the parameters in Table 1 has F= 1.15 and is stable. z (m) 5 z w (m) 2.5 m 0.5 c' (kN/m 2 ) 25 φ ' 30 α 35 γ (kN/m 2 ) 20 γ w (kN/m 2 ) 9.81 Table 1. Hypothetical slope parameters However, if the parameters are uncertain, there may be a nonzero probability of slope failure. Of the variables in Table 1, the most uncertain ones are the strength parameters ' c and φ ' , with
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