The evaluation of the probability of rock wedge failure using the point estimate method

Abstract

In a probabilistic analysis of rock slope stability, the Monte Carlo simulation technique has been widely used to evaluate the probability of slope failure. While the Monte Carlo simulation technique has many advantages, the technique requires complete information of the random variables in stability analysis; however, in practice, it is difficult to obtain complete information from a field investigation. The information on random variables is usually limited due to the restraints of sampling numbers. This is why approximation methods have been proposed for reliability analyses. Approximation methods, such as the first-order second-moment method and the point estimate method, require only the mean and standard deviation of the random variable; therefore, it is easy to utilize when the information is limited. Usually, a single closed form of the formula for the evaluation of the factor of safety is needed for an approximation method. However, the commonly used stability analysis method of wedge failure is complicated and cumbersome and does not provide a simple equation for the evaluation of the factor of safety. Consequently, the approximation method is not appropriate for wedge failure. In order to overcome this limitation, a simple equation, which is obtained from the maximum likelihood estimation method for wedge failure, is utilized to calculate the probability of failure. A simple equation for the direct estimation of the safety factors for wedge failure has been empirically derived from failed and stable cases of slope, using the maximum likelihood estimation method. The developed technique has been applied to a practical example, and the results from the developed technique were compared to the results from the Monte Carlo simulation technique.

Probability Methods for Stability Design of Open Pit Rock Slopes: An Overview

“…Evaluation of the points is chosen at the mean plus and mean minus one standard deviation for each variable, resulting in 2 n evaluations for n random variables. A weighting value ρ is used at each evaluation point to ensure the expected (mean) value and standard deviation of the input parameters are recovered [19,39,59,60,81,82]. The method accounts for a maximum of three statistical moments, namely mean, variance or standard deviation, and skewness.…”

“…The reliability-based design approach has been developed and utilized to address the shortcomings of the deterministic approach and considers uncertainties explicitly [14,34]. Several applications of solution methods using Monte Carlo simulation (e.g., [35][36][37]), the point estimate method (e.g., [19,38,39]), and the response surface method (RSM) have been reported in the literature (e.g., [32,40,41]). [7] explored the mean firstorder reliability method, which is a simplification of the more general first-order reliability method.…”

Probability Methods for Stability Design of Open Pit Rock Slopes: An Overview

“…Evaluation of the points is chosen at the mean plus and mean minus one standard deviation for each variable, resulting in 2 n evaluations for n random variables. A weighting value ρ is used at each evaluation point to ensure the expected (mean) value and standard deviation of the input parameters are recovered [19,39,59,60,81,82]. The method accounts for a maximum of three statistical moments, namely mean, variance or standard deviation, and skewness.…”

“…The reliability-based design approach has been developed and utilized to address the shortcomings of the deterministic approach and considers uncertainties explicitly [14,34]. Several applications of solution methods using Monte Carlo simulation (e.g., [35][36][37]), the point estimate method (e.g., [19,38,39]), and the response surface method (RSM) have been reported in the literature (e.g., [32,40,41]). [7] explored the mean firstorder reliability method, which is a simplification of the more general first-order reliability method.…”

Probabilistic modeling of rainfall-induced shallow landslide using a point-estimate method

“…Chen et al (2007) applied the first-order second-moment method to assess the probability of an infinite slope failure. Park et al (2012) used the point-estimate method to evaluate the failure probability of a rock wedge. Generally, the point-estimate method is simpler and more flexible than the first-order second-moment method, particularly when the model is complicated or lacks analytical solutions (Tung and Yen 2005).…”

Seismic analytical reliability analysis of triangular rock wedge slope stability

“…Utilizing the widely used spreadsheet platform, Low [30] conducted a reliability analysis and developed a reliability-based design for a two-dimensional joint rock slope employing the first-order reliability method. Additionally, Park et al [31] derived an equation for wedge failure through the point estimation method.…”

Probability Methods for Stability Design of Open Pit Rock Slopes: An Overview

“…Evaluation of the points is chosen at the mean plus and mean minus one standard deviation for each variable, resulting in 2 n evaluations for n random variables. A weighting value ρ is used at each evaluation point to ensure the expected (mean) value and standard deviation of the input parameters are recovered [19,39,59,60,81,82]. The method accounts for a maximum of three statistical moments, namely mean, variance or standard deviation, and skewness.…”

“…The reliability-based design approach has been developed and utilized to address the shortcomings of the deterministic approach and considers uncertainties explicitly [14,34]. Several applications of solution methods using Monte Carlo simulation (e.g., [35][36][37]), the point estimate method (e.g., [19,38,39]), and the response surface method (RSM) have been reported in the literature (e.g., [32,40,41]). [7] explored the mean firstorder reliability method, which is a simplification of the more general first-order reliability method.…”

Probabilistic modeling of rainfall-induced shallow landslide using a point-estimate method

“…Chen et al (2007) applied the first-order second-moment method to assess the probability of an infinite slope failure. Park et al (2012) used the point-estimate method to evaluate the failure probability of a rock wedge. Generally, the point-estimate method is simpler and more flexible than the first-order second-moment method, particularly when the model is complicated or lacks analytical solutions (Tung and Yen 2005).…”

Seismic analytical reliability analysis of triangular rock wedge slope stability

“…Utilizing the widely used spreadsheet platform, Low [30] conducted a reliability analysis and developed a reliability-based design for a two-dimensional joint rock slope employing the first-order reliability method. Additionally, Park et al [31] derived an equation for wedge failure through the point estimation method.…”

Probability Methods for Stability Design of Open Pit Rock Slopes: An Overview

“…Evaluation of the points is chosen at the mean plus and mean minus one standard deviation for each variable, resulting in 2 n evaluations for n random variables. A weighting value ρ is used at each evaluation point to ensure the expected (mean) value and standard deviation of the input parameters are recovered [19,39,59,60,81,82]. The method accounts for a maximum of three statistical moments, namely mean, variance or standard deviation, and skewness.…”

“…The reliability-based design approach has been developed and utilized to address the shortcomings of the deterministic approach and considers uncertainties explicitly [14,34]. Several applications of solution methods using Monte Carlo simulation (e.g., [35][36][37]), the point estimate method (e.g., [19,38,39]), and the response surface method (RSM) have been reported in the literature (e.g., [32,40,41]). [7] explored the mean firstorder reliability method, which is a simplification of the more general first-order reliability method.…”

Probabilistic modeling of rainfall-induced shallow landslide using a point-estimate method

“…Chen et al (2007) applied the first-order second-moment method to assess the probability of an infinite slope failure. Park et al (2012) used the point-estimate method to evaluate the failure probability of a rock wedge. Generally, the point-estimate method is simpler and more flexible than the first-order second-moment method, particularly when the model is complicated or lacks analytical solutions (Tung and Yen 2005).…”

Seismic analytical reliability analysis of triangular rock wedge slope stability

“…Utilizing the widely used spreadsheet platform, Low [30] conducted a reliability analysis and developed a reliability-based design for a two-dimensional joint rock slope employing the first-order reliability method. Additionally, Park et al [31] derived an equation for wedge failure through the point estimation method.…”