Rock slope kinematic analysis for planar failure: A probabilistic approach

The probabilistic kinematic analysis for rock slope has been conducted on one slope along USAID road in Aceh Province, Indonesia. This research aims to develop the modelled probability density function (PDF) and determine the probabilistic of planar failure occurrence (Pop). The geometry of discontinuity planes (dip and dip directions) and slope geometry (slope angle and slope face) were collected from our previous study. One slope with planar failure criteria was selected. The Monte Carlo simulations were performed in generating 100 new random values in 100 time iterations to produce modelled PDFs for the geometry of discontinuity plans based on statistical parameters of field observed data. The probabilistic of each experimental PDFs were computed to produce the probabilistic of planar failure occurrence. The result reveals that the distribution of dip and dip directions for experimental PDFs are considered Beta and Normal distributions. The statistical parameters produced in the model are almost likely similar to observed data. It means the model that was developed are reliable and conscientious. The rule of Φ < βj < βs and dip directions (αj) within ±20o to slope face (αs) are utilised as the boundaries to calculate the probabilistic of planar failure occurrence (Pop) which revealing 0.26.


Background
Rock slope failure analysis for mining and civil projects has attracted rock engineers to develop numerous methods to analyse the rock slope stability, including the rock slope kinematic analysis, limit equilibrium, numerical modelling, empirical approach, and rock mass classification [1][2][3][4][5]. In a conventional approach, those methods merely concern a single input parameter value known as deterministic, and it previously applies in deterministic slope kinematic analysis. Nevertheless, the rocks' physical, mechanical, and structural properties vary from point to point and naturally in rock [6]. Thus, the conventional approach is required to be improved obviously to the probability approach considering the uncertainty of geological structural data collected from the surface of rock slope to increase the reliable result in slope design and prevention measure.
The rock slope kinematic analysis approach studies the typology of slope failure without any consideration of the force working on the slope [2,5,7]. The type of potential failures determined in the rock slope kinematic approach was based on the stereography interpretation, which has been successfully conducted by numerous researchers [2,5,7]. However, their approaches are * Corresponding author: ibnu@unsyiah.ac.id assumed as a deterministic approach picking the mean values from stereography as a single value. The probabilistic rock slope kinematic analysis has been introduced by numerous researchers [8][9][10], considering the effect of variability of joint orientations. Furthermore, the probability of planar failure due to variability of structural features have been conducted by Rahim et al. [11] and Farhan and Rai [12] and give reliable results. The comprehensive study on how the variability of discontinuity plane drives the planar failure of rock slope is elaborated and scrutinised in this paper. Hence, this study aims to develop the modelled probability density function (PDF) and determine the probabilistic of planar failure occurrence (Pop) at USAID highways rock slope in Aceh province, Indonesia.
Overall, this paper discusses the deterministic kinematic analysis, which previously has been conducted by Rusydy et al. [5] for Slope 1, revealing the planar failure potential at the joint set (J2). The data were then re-analysed by performing Monte Carlo simulation to generate a large number of random values based on statistical parameters from the joint set (J2). The probabilistic kinematic analysis was conducted to determine the probabilistic of planar failure occurrence

Methodology
This research was conducted at the rock slopes along USAID highways side, build after Indian Ocean Tsunami hit Aceh province. During construction, a multitude of rock slopes have been exposed; hence, some slopes are vulnerable and need to be investigated and analysed. The dip and dip direction of discontinuity planes (e.g. joints, bedding) on the rock slopes were recording directly employing geological compass by Rusydy et al. [2,5,7,13]. The dip and dip direction discontinuity planes data were plotted in stereographic projection to determine the number of planes most likely having a similar direction with the slope face, and the dips of planes are higher than friction angle (Ф) but lower than slope angle. The Alejano et al. [14] suggested method was utilised in this study to determine the basic friction angle.

Rock slope kinematic analysis
Rock slope kinematic analysis is associated with the rock slope movement process without considering the forces working the rock slope. This approach is introduced by Hoek & Bray [15] and Goodman [16]. The analysis and calculation are stands on the stereography projection, which are projected in the dip (βj ) and dip direction of discontinuity planes (αj ), which is 3D into the 2D model [2,5,7]. Other input data required in the kinematic analysis is friction angle (Φ) determined by employing the tilt testing suggested by Alejano et al. [14].
The Orient software developed by Vollmer [17] was utilised in this research to produce a stereography projection model (see Fig. 2). The mean of plane orientations (βj and αj ) are determined from Orient software as the joint sets. Those joint sets combine with the orientation of slope (βs and αs) and friction angle (Φ) are deployed to determine the typology of failures (e.q. planar, wedge, toppling). Using the mean of plane orientation without considering the variability of those orientations is known as the deterministic approach.
Ten rock slopes have been investigated by Rusydy et al. [2] and Rusydy et al. [5], employing the scan-line method. Four from ten slopes are potentially had planar failure refer to deterministic kinematic analysis. Nonetheless, this paper merely discusses one slope as an example to introduce the probabilistic approach in analysing the planar failure probabilities.
Planar failure is a common failure typology in rock slope stability study; nevertheless, it requires numerous geometric circumstances to occur. Wyllie and Mah [18] noted that the planar failure occurs when the dip direction of joint planes (αj ) are within ±20° to the slope face (αs). The following condition to planar failure to occur is when the dip of joint (βj) is lower than slope angle (βs), yet it must be higher than friction angle (Φ), or in another way it can be written as Φ < βj < βs. The last circumstances is a present of tension crack in the upper part of the slope, as denoted in Fig. 1. In this study, those circumstances play a crucial role as the boundaries in determining probabilistic of planar failure occurrence except the presence of tension crack due to it is undefined in the field.

Probabilistic kinematic analysis
Probabilistic analysis is performed when the field data are insufficient or burdensome to assign a single value to calculate the model. Probabilistic describe the degree of belief in the truth of circumstances [19]. Hence, this study revealed probabilistic values between 0 and 1 to express the degree of belief for planar failure to occur. Obregon and Mitri [20] argued that the probability of failure for slope stability could be computed by encountering two models (series and parallel) influenced by numerous parameters. The series model is described as a single line from starting point to the end. If one parameter fails, it could trigger the rock slope failure. While in the parallel system, it requires all failure components to trigger the rock slope failure. The  Joint Slope Face Ф planar failure in rock slope is more likely similar to the parallel system requiring at least two geometric circumstances to slope to have planar failure as mentioned previously (Φ < βj < βs) as the first component and the second component is dip direction (αj ) within ±20° to the slope face (αs), please see Fig. 3. According to Obregon and Mitri [20], the probability of all components are expressed as P[p1 Ո p2 … Ո pN]. It means the failure could happen in a parallel system when all components fail, and in probability analysis, they have an intersection relationship.
The probabilistic of failure (Pf) for parallel system failure in rock slope can be simplified as denoted in Equation 1. This rule assumes all the parameters (N) governing the rock failure are independent and has their probabilities (Pi).
Naturally, the values of dip direction (αj ), dip (βj), and friction angle (Φ) vary from site to site, from point to point while the slope angle (βs) and slope face (αs) have a single value. The variability αj , βj, and Φ lead to uncertainty; accordingly, the probabilistic kinematic analysis approach is necessary to cope with this variability. The probabilistic density function (PDF) of αj and βj were generated at the first stages utilising the R programming software, while the Φ is assuming as a single value. The example of PDFs and histograms in this study is denoted in Fig. 4.

Monte carlo simulation
The Monte Carlo simulation was run in this study to generate large numbers of dips and dip directions. This simulation is the most common approach in stochastic analysis in that the random values to the model is generated from sampled statistical distribution or frequency distributions like histograms and probability density function (PDF) [21,22].
The number of observations for the joint set (J2) is merely 22 data, and it is insufficient to perform the probabilistic analysis directly from the PDF of field observed data. Thus, the Monte Carlo simulation was performed to generate large new random values based on statistic characterisation from field observed data (βj, αj ) distributions (see Fig. 3). 100 new data have been generated in 100x iteration for dips and dip directions using R data analysis software. As denoted in Fig. 4, the dip of joints has a Beta distribution typology while the dip direction is recognised as Normal distribution typology. In performing the simulation for Beta distribution, it requires the Shape1, and Shape2 from observed data. The EnvStats package in R programming software developed by Millard [23] was employed to determine the Shape1 and Shape2 values from field observed data. Due to the dip direction data distribution is assumed as a normal distribution, it merely requires the mean (μ) and standard deviation (σ) values.

Results
The observed data distributions are the key in developing the experimental model utilising the Monte Carlo simulation; accordingly, the result of experimental models varies from 100x of iteration, and probabilistic kinematic analysis for J2 discusses.

The experimental models
The EnvStats R package was employed for Beta distribution and revealed the Shape1 value for a dip of joints is 7.753, and Shape2 is 3.324 from observed data (J2). Furthermore, the dip direction was recognised as Normal distribution, having 287.136 of μ and 20.185 of σ. The experimental distribution was developed by employing those statistical parameters. In developing experimental distribution, 100 times iteration was run for 100 data, and all those simulations are plotted in one histogram and PDF as denoted in Fig.   5. Even though the range of the dip from the observed data is 34 to 87, the Monte Carlo simulation reveals bit higher data than expected. Yet, this simulation result is reliable, and the first quarter (Q1) and the third quarter (Q3) are similar to observed data. The distribution of dip direction experimental histogram and PDF is considered as normal distribution where the median and mean value are almost likely similar.

Probabilistic kinematic
After all simulated models reveal reliable results, the next stage is computing the probabilistic planar failure occurrence (Pop) employing Equation 2. The geometric circumstances for planar failure are played an essential role as the boundaries in calculating the probabilistic to dip and dip direction in the PDF model.
According to Rusydy et al. [5], the investigated slope has 27 o of friction angle (Φ ) and the slope angle (βs) is 70 o ; accordingly, those data are utilised as boundaries in a dip of joint PDF curve to estimate the area between them. The area of 0.47 represents the probability of the first component as denoted in Fig. 7a; obviously, this area is calculated in R programming software. The second component as shown in Fig. 7b     The connection between the first component and second component is recognised as an intersection probability relationship. The total probability of two components (0.47 and 0.56) are computed by multiplying those components; hence, the probabilistic of planar failure occurrence (Pop) in this study is 0.26. All parameters are employing in simulation as denoted in Table 1.

Conclusion
The slope in this study area has structural geology features from bedding, joints, and minor fault crossing the rock faces. The planar failure mainly occurs due to the dip direction of joints higher or lower than 20 o from the slope face, and the dips are higher than the internal friction angle yet lower than the slope angle. Those geometric rules for planar failure obviously can be utilised as the boundaries in calculating the probability in PDF curves either for observed data or experimental/simulated data.
The experimental PDFs were developed based on the statistical parameters from observed data, and they must be scrutinised carefully to produce reliable results. This study identifies two typologies of data distributions, beta and normal which are required shape1, shape2 for beta and mean, the standard deviation for normal distribution.
This merely conducted probabilistic kinematic analysis for planar failure in one rock slope. A multitude of rock slopes needs to be investigated by applying probability analysis to deal with the uncertainty and variability of rock properties. It is suggested to conduct more comprehensive research and cover more rock slopes in the future to mitigate rock slope failure alongside roads or probably in mining areas.