📄 earth_science

Seismic analytical reliability analysis of triangular rock wedge slope stability

This study proposes the Jointly Distributed Random Variables (JDRV) method as a computationally efficient and accurate analytical solution for the seismic reliability analysis of triangular rock wedge slopes, demonstrating that treating all geotechnical and geometric parameters as stochastic variables yields more conservative failure probabilities and identifying the sliding surface friction angle as the most critical sensitivity factor.

Original authors: Ali johari, A. Duzandeh, SS. Kalantari Kouhbanani, M.A. Najafgholipour

Published 2026-06-25

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CC BY 4.0

Original authors: Ali johari, A. Duzandeh, SS. Kalantari Kouhbanani, M.A. Najafgholipour

Original paper licensed under CC BY 4.0 (https://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

Imagine a mountain made of giant, triangular blocks of rock. Sometimes, one of these blocks is on the verge of sliding down the mountain like a heavy box on a ramp. Engineers need to know: Will it slide? How likely is it to slide?

Usually, they try to calculate a "Safety Factor"—a number that tells them how safe the rock is. But nature is messy. The rock isn't perfectly uniform, the ground might shake during an earthquake, and the angles of the cracks aren't always exact. Because of these uncertainties, a simple "yes or no" answer isn't enough. We need to know the probability of failure.

This paper introduces a new, smarter way to calculate that probability, comparing it to the old, heavy-handed methods.

The Problem: The "Guess-and-Check" Game

To figure out the odds of a rock slide, researchers usually use a method called Monte Carlo Simulation (MCS). Think of this like a massive game of "Guess the Weight."

Imagine you have a bag of marbles representing all the possible rock conditions (some heavy, some light, some slippery, some rough). To get an answer, you have to pull out a marble, calculate the safety, put it back, pull out another, calculate again, and do this millions of times to get a reliable average.

The Downside: It's like trying to find a specific grain of sand on a beach by picking up every single grain one by one. It takes a huge amount of time and computer power.

The Solution: The "Mathematical Shortcut" (JDRV)

The authors propose a new method called Jointly Distributed Random Variables (JDRV).

Instead of playing the "Guess-and-Check" game millions of times, JDRV is like having a super-accurate map of the beach. Instead of picking up every grain of sand, the map tells you exactly where the heavy grains and light grains are clustered. It uses advanced math (calculus and probability theory) to calculate the final answer directly, without needing to run millions of simulations.

The Paper's Claims about JDRV:

Speed: It's much faster. It doesn't need to run millions of trials; it solves the equation directly.
Accuracy: It gives results that are statistically very similar to the "Gold Standard" (Monte Carlo) but with much less effort.
The "All-Hands" Approach: The study emphasizes that you must treat everything as uncertain. If you only guess the rock's weight but assume the earthquake strength is fixed, your answer will be too optimistic. When they treated the rock weight, the friction, the geometry, and the earthquake shaking all as random variables, the rock looked less safe (higher chance of failure) than when they ignored some uncertainties.

The "Seismic" Twist

The paper specifically looks at what happens when an earthquake hits (seismic stability). They modeled the earthquake shaking not as a fixed number, but as a random event that could be mild or strong.

The Finding: When you add the randomness of earthquakes to the randomness of the rock itself, the "Safety Factor" distribution changes. The JDRV method handles this complex mix of variables smoothly, whereas older methods might struggle or take forever to compute.

The "Friction" Factor

The researchers also played a game of "What if?" to see which variable matters the most. They tweaked the numbers slightly to see what caused the biggest change in safety.

The Result: The friction angle (how "grippy" the sliding surface is) was the most important factor. If the rock is slippery, the whole system becomes unstable. If it's rough and grippy, it holds firm. This is the "tipping point" variable.

The Comparison: JDRV vs. The Old Guard

The paper compares their new "Map" (JDRV) against two other methods:

Monte Carlo (The "Grain-by-Grain" method): JDRV matches its accuracy but is much faster.
FOSM (First Order Second Moment): This is another shortcut method. The paper claims JDRV is statistically similar to this but offers a more complete picture of the probability distribution (the shape of the risk curve).

The Bottom Line

The paper argues that we don't need to wait days for a computer to crunch millions of numbers to know if a rock slope is safe. By using the JDRV method, engineers can get a highly accurate, detailed picture of the risk (including the chance of an earthquake causing a slide) in a fraction of the time.

In short: They found a way to predict the odds of a rock slide that is as accurate as the slow, brute-force methods but runs as fast as a shortcut, provided you remember to account for the fact that everything in nature is a little bit random.

Technical Summary: Seismic Analytical Reliability Analysis of Triangular Rock Wedge Slope Stability

Problem Statement
Rock slope stability in mountainous regions is inherently stochastic due to uncertainties in geotechnical parameters, geometrical configurations, and seismic loading. While deterministic methods exist, they often fail to quantify the probability of failure associated with these variables. Previous research has addressed uncertainties in fracture geometry or rock material characteristics separately, or utilized numerical simulation methods like Monte Carlo Simulation (MCS) which, while robust, can be computationally expensive. A significant gap remains in the simultaneous investigation of uncertainties in geometrical, seismic, and rock parameters using an analytical approach that avoids the high computational costs of numerical simulations. This study addresses the need for an efficient, accurate analytical method to assess the reliability of triangular rock wedges under planar sliding conditions subjected to seismic forces.

Methodology
The study proposes the application of the Jointly Distributed Random Variables (JDRV) method to derive the probability density function (PDF) of the safety factor for rock slopes. The methodology involves the following components:

Mathematical Framework: The JDRV method utilizes theorems regarding the transformation of random variables (Theorems 1–4) to analytically integrate the PDFs of input variables within the slope stability model. This allows for the direct derivation of the safety factor's PDF without the need for repeated random sampling.
Stochastic Modeling: Six input parameters are treated as random variables:
- Geometric Parameters: Triangular wedge coordinates ( $y, z$ ) and the resulting failure surface angle ( $\beta$ ) and length ( $L_{ab}$ ).
- Rock Parameters: Apparent cohesion ( $c$ ), friction angle ( $\phi$ ), and unit weight ( $\gamma$ ).
- Seismic Parameter: Horizontal pseudo-static coefficient ( $k_h$ ).
- Distributions: The rock and geometric parameters are modeled using truncated normal distributions, while the seismic coefficient is modeled using an exponentially truncated distribution.
Comparative Analysis: The JDRV results are benchmarked against two other methods:
1. Monte Carlo Simulation (MCS): Used as a reference standard with 108 simulations to generate empirical PDFs and Cumulative Distribution Functions (CDFs).
2. First Order Second Moment (FOSM): An approximate method using Taylor series expansion to estimate the mean and variance of the safety factor.
Sensitivity Analysis: Two approaches are employed to evaluate parameter influence:
1. Increasing the Coefficient of Variation (COV) of all six stochastic parameters.
2. Varying the mean values of specific parameters (e.g., friction angle, geometric coordinates) while holding others constant.

Key Results

Accuracy and Agreement: The study demonstrates that the JDRV method yields results statistically similar to the FOSM method and shows excellent agreement with MCS. Specifically, the CDFs generated by JDRV and MCS exhibit a "perfect match" across all three tested scenarios (rock parameters only; rock + seismic; and rock + seismic + geometric parameters).
Computational Efficiency: A primary finding is the significant reduction in computational time and cost offered by JDRV compared to MCS. While MCS requires thousands of iterations to converge, JDRV achieves convergence with fewer iterations due to its analytical nature.
Statistical Performance: The JDRV method provides stable mean and standard deviation results regardless of the number of simulations, offering higher precision in estimating the safety factor's PDF compared to the fitting required for MCS outputs (e.g., epsilon-skew-normal distribution).
Impact of Stochastic Variables: When all variables (geometric, seismic, and rock parameters) are treated as stochastic simultaneously, the analysis reveals a lower reliability index and a higher probability of failure compared to analyses that fix certain variables. This underscores the risk of underestimating failure probability by ignoring geometric or seismic uncertainties.
Sensitivity Findings: The sensitivity analysis identifies the friction angle of the sliding surfaces as the most influential parameter affecting rock slope stability.

Significance and Claims
The paper claims that the JDRV method serves as a superior analytical tool for the reliability analysis of rock wedge slopes under seismic conditions. Its significance lies in its ability to:

Simultaneously account for uncertainties in geometry, seismicity, and rock mechanics within a single analytical framework, a capability often lacking in previous studies.
Reduce computational burdens significantly compared to MCS while maintaining high accuracy in determining safety factor probability density functions.
Provide a robust alternative to approximate methods like FOSM by capturing the full shape of the PDF rather than just the first two moments.

The authors emphasize that treating all relevant variables as stochastic parameters is critical for realistic risk assessment, as neglecting these uncertainties can lead to an overestimation of slope stability. The study concludes that the JDRV method, by combining fundamental relations with Jacobian equations, offers a reliable, efficient, and accurate approach for seismic stability analysis of triangular rock wedges.