Applications Of The Point Estimation Method For Stochastic Rock Slope Engineering

The point estimation method can be applied to the safety factor (SF) equation for any specified rock slope failure mode (such as plane shear, step path, or wedge) to obtain reliable estimates of the mean and standard deviation of the SF probability distribution. A gamma probability density function is recommended for modeling this probability distribution, because it allows only for positive values and is flexible enough to provide symmetrical shapes and right-skewed, exponential-type shapes for the SF distribution. The mean and standard deviation define this distribution, which then can be integrated numerically from 0 to 1 to obtain the probability of sliding, PS (portion of the SF distribution where SF < 1.0). The overall probability of failure, PF, for the potential slope failure mass is the joint probability that the rock discontinuities are long enough to allow kinematic failure (PL) and that sliding occurs along the rock discontinuities (PS); that is, PF = PSPL. This method for estimating the probability of sliding is extremely efficient computationally, and thus, expedites slope stability simulation routines used by NIOSH software to stochastically describe rock slope behavior and assist the engineer in designing catch benches for large rock slopes. Enhanced bench design translates into increased operational efficiency and safer working conditions in open pit mines and quarries. Taylor series expansion [2], Fourier analysis [3], and statistical point estimation [4]. Initial versions of the NIOSH bench stability software relied on the Fourier method [1] due to its numerical efficiency and its capability to provide a discretized, general output pdf (probability density function) for the factor of safety, rather than relying on a specific model for the pdf (e.g., a normal or lognormal pdf). However, our recent experience with probabilistic studies of rock slope stability has indicated that the safety factor pdf tends to behave like a slightly right-skewed gamma pdf or a left-truncated normal pdf (truncated at zero, because the safety factor realistically cannot take on negative values). The right skew apparently is caused by a combination of the positive-only gamma shape of the input pdf for the shear strength (along the sliding plane) and the exponential-type shape of the input pdf for the fracture waviness. In this context, the waviness is measured on a scale of about 1-2 meters and is defined as the average dip of the fracture minus its minimum dip, as presented by Call and others [5]. Thus, assuming that the output pdf for the safety factor takes on the form of a gamma pdf, then the point estimation method clearly has considerable computational advantage even over the Fourier method (which relies heavily on mathematical manipulations of discretized pdf’s [3]). 2. POINT ESTIMATION METHOD When a random variable of interest can be expressed in an equation as the result of a mathematical operation of other random variables, then the point estimation method developed by Rosenblueth [6] provides a direct computational procedure to obtain moment estimates for that random variable. In particular, these statistical moments are the mean (i.e., the first moment about the origin) and the variance (i.e., the second moment about the mean). Geotechnical engineering applications of this method have been around for several decades, and recent publications [2, 4] have clearly presented such work. The particular shape of any pdf used for any input random variable is not critical to the analysis, because the pdf is represented by the mean and two hypothetical point masses located at plus and minus one standard deviation (s) from the mean (μ). Consequently, required inputs for a probabilistic rock slope stability analysis are: 1) a defined performance function (i.e., safety factor equation), 2) estimated value for each input attribute if it is assumed to have negligible variability, and 3) estimated mean μ and standard deviation s of each input attribute treated as a random variable. Typical attributes for a rock slope failure mass with a defined geometry would include shear strength, rock mass unit weight, and fracture waviness. Calculation steps are presented below for the point estimation method using two random variables X1 and X2 in a performance function to obtain the mean and variance of F, the factor of safety. 1. Calculate the output value of F using the performance function evaluated with the values of mean-plus-one-s.d. for each of the two variables. F++ = fn[(μ1 + s1), (μ2 + s2)] (1a) Repeat for other combinations, as follows: F= fn[(μ1 s1), (μ2 s2)] (1b) F+ = fn[(μ1 + s1), (μ2 s2)] (1c) F+ = fn[(μ1 s1), (μ2 + s2)] (1d) 2. Calculate the point-mass “weights” [6]. P++ = P= (1/4)(1 + ρ12) (2a) P+ = P+ = (1/4)(1 ρ12) (2b) where ρ12 = correlation coefficient between input variables X1 and X2. 3. Calculate the expectation (mean, μF) of F [6]. E(F) = P++ F++ + PF+ P+ F+ + P+ F+ (3) 4. Calculate the variance (sF) of F. Var(F) = E(F 2 ) – [E(F)] (4) where E(F 2 ) is calculated using Eq. (3) with F 2 terms substituted for the F terms. 5. The standard deviation (sF) of F then is calculated by taking the square root of sF. For three input variables X1, X2, X3, there are eight calculations in Step 1, and the point-mass weights in Step 2 are given by [6]: P+++ = P= (1/8)(1 + ρ12 + ρ23 + ρ31) (5a) P+ = P++ = (1/8)(1 ρ12 + ρ23 ρ31) (5b) P++ = P+ = (1/8)(1 + ρ12 ρ23 ρ31) (5c) P+ + = P + = (1/8)(1 ρ12 ρ23 + ρ31) (5d) Eq. (3) for E(F) is extended from a summation of four terms to a summation of eight terms for this case. The sF value subsequently can be calculated using Eq. (4), after first using eight F 2 terms in the extended Eq. (3) to calculate E(F 2 ). 3. PLANE SHEAR ANALYSIS A probabilistic procedure based on the point estimation method can be applied to the twodimensional slope stability analysis used for the plane shear failure mode (Fig.1). For a plane shear mass with a defined geometry (i.e., slope angle δ, dip of failure plane α, height of the failure mass H), the SF equation is assumed to contain two random variables, the shear strength and the waviness of the geologic discontinuity. Other input terms, such as the rock-mass density, which is used to calculate the weight of the potential failure mass, are treated as constants. The plane shear SF equation is: ) sin( ) tan( α τ σ W r L L n F + = ( ) ) tan( ) sin( ) sin( r W L W L n ⋅ ⋅ + = α τ α σ (6) where: τ = shear strength, r = waviness σn = effective normal stress, α = average dip of sliding surface, L = length of sliding surface, and W = weight of slide mass. The point estimation method can be applied to this expression if we have reliable estimates of the mean and standard deviation of each random variable.