The core of the work is a reliability analysis method for the permanent displacement of soil slopes subjected to sequential ground motions. Permanent displacement, the irreversible sliding deformation that accumulates when seismic forces momentarily exceed the resisting strength of the slope material, is a standard yardstick for judging whether a slope remains serviceable after an earthquake. The authors argue that judging this quantity from mainshock shaking alone systematically misrepresents the hazard, because aftershocks add cumulative damage to soil that has already been degraded by the first event. Their solution couples a carefully constructed stochastic description of sequence-type ground motions with the probability density evolution method, a computational framework for nonlinear stochastic dynamics originally developed by Li and co-workers and previously extended to complex slope reliability problems.

Building the seismic input was the first technical hurdle. The team assembled 96 recorded mainshock–aftershock pairs drawn from 15 seismic events in the NGA-West2 strong-motion database maintained by the Pacific Earthquake Engineering Research Center. Selection criteria were strict: the aftershock had to be the largest-magnitude event within twelve months of the mainshock, both events needed magnitudes of at least 5.0 and peak ground accelerations of at least 0.05 g, both records had to come from the same component of the same station, the site shear-wave velocity in the upper 30 meters had to fall between 100 and 700 meters per second, and rupture distances could not exceed 80 kilometers. Each sequence was configured as a mainshock followed by a twenty-second quiet interval and then the aftershock, preserving the fully non-stationary character of real sequences.

Rather than treating these recordings as isolated deterministic inputs, the researchers modeled them as zero-mean, fully non-stationary stochastic processes using an evolutionary power spectral density framework. Frequency-domain energy distribution curves were fitted for each ground motion using an optimal square approximation criterion and least-squares regression, yielding a parameter vector that defines each evolving spectrum. This random-function dimension-reduction approach generates representative ground-motion time histories together with their associated probabilities, forming a complete probabilistic set that can feed directly into the probability density evolution method. That is a crucial advantage over Monte Carlo simulation, which demands enormous sample counts and often leaves the probability structure of the sample set incomplete.

A second, equally consequential decision concerned how to measure the intensity of a ground-motion sequence. Conventional practice leans on peak ground acceleration, but a single peak value says little about the sustained, damage-accumulating character of repeated shaking. The team therefore screened 21 candidate intensity measures spanning three families: peak-based parameters such as peak ground velocity and displacement, spectral measures such as response spectral acceleration and acceleration spectral intensity, and cumulative measures including Arias intensity, significant duration, and several root-mean-square and cumulative-absolute quantities. Correlation analysis against the simulated slope responses produced a clear winner: cumulative absolute velocity, or CAV, showed the strongest correlation with permanent displacement, with a correlation coefficient of 0.872, and captured the cumulative aftershock contribution better than any peak-based alternative.

With the seismic input and intensity measure defined, the mechanical side of the analysis proceeded on a two-layer soil slope model 150 meters long and 30 meters high, built in the finite-difference platform FLAC3D with a mesh refined to roughly half a meter in the potential sliding zone. The soil, a clay from southwestern China, was represented by a Mohr–Coulomb constitutive model with a tensile cutoff, and local damping with a 5 percent critical damping fraction, corresponding to a damping coefficient of 0.157, reproduced energy dissipation during wave transmission. Because FLAC3D avoids assembling a global stiffness matrix, it is computationally efficient for the many nonlinear dynamic simulations that a probabilistic treatment requires. Ninety-six scaled sequences were run at three CAV levels of 12, 28, and 40 meters per second, generating the response ensemble needed for density evolution.

The results reveal how strongly and how variably sequences load a slope. Average permanent displacements rose from 0.067 meters at a CAV of 12 meters per second to 0.328 meters at 28 and 0.633 meters at 40 meters per second, values comparable to those produced by mainshock-only shaking at peak ground accelerations of 0.4, 0.5, and 0.6 g in earlier work. Even with CAV as the intensity measure, the displacement results remained scattered, a reminder that frequency content and duration also matter and that no single parameter fully predicts slope response. That residual variability is precisely why the authors insist on a probabilistic treatment: the probability density evolution method propagates the uncertainty through the dynamics rather than averaging it away.

The probability distributions that emerged are strikingly non-standard. The probability density curves of permanent displacement showed bimodal or multimodal shapes that cannot be described by normal or lognormal distributions, underscoring how sequence effects reshape the statistics of slope response. Reliability estimates depended heavily on the chosen failure threshold. At a CAV of 12 meters per second, displacements ranged from 0 to 0.25 meters and reliability was 48.5 percent against a low-level threshold of 0.05 meters. At 28 meters per second the range widened to 0 to 1.00 meters and reliability fell to 45.5 percent at the medium threshold of 0.25 meters, while at 40 meters per second the range stretched to 0 to 2.00 meters and reliability dropped to 31.7 percent at the high threshold of 0.50 meters.

Comparisons against a conventional PGA-based analysis highlight the practical payoff of the new approach. At the 50 percent cumulative probability level, CAV-based displacements were approximately 0.07, 0.32, and 0.55 meters for the three intensity levels, against roughly 0.20, 0.50, and 0.70 meters for the corresponding PGA cases, and the PGA-based curves showed heavier large-displacement tails. Overall, switching from PGA to CAV reduced the dispersion of predicted displacements by about 40 percent on average, producing tighter, more informative reliability estimates. The authors caution that their results apply to clayey soil slopes of the kind found in southwestern China and to shallow crustal earthquake sequences, and that soil-parameter and geometric variability were not yet included, an extension they expect will further improve engineering applicability.

For disaster prevention, the message is clear: the aftershock is not a footnote but a first-order player in landslide hazard. By embedding sequence effects into a rigorous probabilistic framework and identifying cumulative absolute velocity as the intensity measure that best tracks cumulative damage, the study gives engineers a defensible way to quantify how likely a slope is to exceed critical displacement limits when the ground refuses to stay still. As seismic safety requirements tighten worldwide, methods that treat mainshock–aftershock sequences as the coupled, cumulative phenomena they truly are may well become the standard against which slope stability is judged.

Subject of Research: Probabilistic reliability analysis of soil slope permanent displacement under mainshock–aftershock earthquake sequences

Article Title: Reliability analysis method for soil slopes permanent displacement under mainshock–aftershock sequences

Article References: Wang, T., Zhang, C., Zhang, J., Chen, S., & Dai, Z. (2026). Reliability analysis method for soil slopes permanent displacement under mainshock–aftershock sequences. Natural Hazards and Earth System Sciences, 26(9), 4611-4619. https://doi.org/10.5194/nhess-26-4611-2026

Image Credits: AI Generated

DOI: 10.5194/nhess-26-4611-2026

Keywords: aftershocks, mainshock–aftershock sequences, slope stability, permanent displacement, cumulative absolute velocity, probability density evolution method, reliability analysis, landslides, ground motion intensity measures, FLAC3D, seismic hazard, soil slopes

The research focuses on what mining engineers call the goaf-side narrow coal pillar, or GSNCP, a thin ribbon of coal left standing between an active tunnel and a mined-out void. In extra-thick coal seams averaging eleven meters, these pillars are routinely riddled with legacy roadways from earlier, less regulated mining eras. When a new roadway, known as a goaf-side entry, is driven alongside such a compromised pillar, the results on site have been alarming: severe rib heaving, roof convergence, rib collapse, and even complete roof caving in zones where abandoned roadways cluster together. Traditional support methods such as bolting and grouting, the authors note, fail entirely under these conditions.

To understand why, the team built a large three-dimensional numerical model using FLAC3D 7.0, spanning 500 by 500 meters laterally and 300 meters in depth, discretized into half-meter cubic elements with more than 7.6 million cells in the critical study area alone. The model reproduced the actual excavation sequence of the mine, including dynamic compaction of the adjacent goaf using double-yield elements and stepwise extraction of the working face at one-meter cycles. The coal was governed by a strain-softening criterion while the overlying strata followed the Mohr-Coulomb failure criterion, allowing the researchers to track how stress redistributes as abandoned roadways slice the pillar into segments.

The simulations revealed a striking sensitivity to the spacing between abandoned roadways. When intervening coal ribs between two roadways were narrower than five meters, the segments entered a low-strength bearing state and simply shed their load onto neighboring pillars, triggering severe rib heaving. At a width of seven meters, the internal abutment pressure peaked at a dangerous 27.9 megapascals in a single sharp spike, a condition the authors describe as an ultra-high strength bearing state in which the pillar risks internal plasticization and fracturing. Only when spacing exceeded nine meters did the stress curve split into a healthier bimodal shape, with the peak dropping to 23.1 megapascals and the load distributed across a broad high-strength bearing zone.

The second half of the puzzle lies overhead. Using the classical OX break theory of main roof behavior, the team developed a theoretical model linking the arc-shaped triangular roof block that forms above the goaf edge to the abandoned roadways hollowing out the pillar beneath it. This triangular block, whose dimensions follow the periodic weighting span of the main roof, rotates and subsides around a fracture line after the roof breaks. The researchers identified nine possible relative positions among the solid coal rib, the triangular block, and the abandoned roadways, and derived moment-balance equations for each configuration to determine when the block remains in equilibrium and when it fails.

The analysis produced two instability coefficients with clear physical meaning. When the coefficient K2 exceeds one, the triangular block undergoes rotational deformation instability, crushing the rock at its corners. When the coefficient K1 falls between zero and one, the block slides instead. Crucially, the calculations showed that as more abandoned roadways accumulate beneath a single triangular block, both coefficients decrease simultaneously, meaning the weakened pillar can no longer supply the support force needed to prevent sliding. Because at most three abandoned roadways can fit beneath one block, the team showed that the failure mode shifts from rotation to sliding as roadway density increases, with the block’s gravitational load transferring through the roadway roof to the solid coal rib and producing uncontrollable deformation, roof fracturing, or even roof cutting.

These theoretical predictions were tested in a 1:100 scale physical analogue model built from layered sand, lime, and gypsum mixtures matched to the site’s borehole geology, with hydraulic jacks applying 34.6 kilopascals to reproduce the overburden load at the mine’s 250-meter depth. The experiments confirmed the mechanism in all three fracture-line scenarios: with backfill in place, the pillar and roof stayed intact, but removing the backfill reproduced the full disaster pattern of pillar crushing and severe roof collapse observed in the field. The observed diagonal shear fractures, angled at roughly 55 to 60 degrees, matched the theoretical shear angle predicted by the Mohr-Coulomb criterion for coal with internal friction angles between 20 and 30 degrees.

Having diagnosed the disease, the researchers prescribed a cure that is as economical as it is effective: backfill the abandoned roadways with the very coal excavated during roadway driving. Laboratory tests determined the optimal recipe, mixing crushed coal, P425 Portland cement, and water in a ratio of 6 to 1 to 0.7. With a particle gradation of roughly 60 percent fines under five millimeters, the solidified mixture achieved a uniaxial compressive strength of 9.8 megapascals, about 80 percent of the raw coal strength of 12.3 megapascals, while still meeting the pumping requirements of mining concrete pumps. The mixture is pumped into mining-purpose backfill bags that prevent leakage into the goaf, with roof bolts fitted with end caps to avoid puncturing the bags.

Numerical simulations of the backfilled system showed a clear strength threshold. At just 20 percent of coal strength, the backfill carried little load and stress remained dangerously concentrated in the pillar. At 60 percent, the backfill began sharing roof load, though stress zones remained isolated. Only at 80 percent of coal strength did the stresses within the pillar and backfill merge into a single continuous high-strength bearing zone without peak eccentricity. The backfill also exerts lateral confinement on the flanking coal segments, creating what the authors call a strong-weak-strong configuration that boosts the pillar’s effective bearing capacity by approximately 27.9 percent when a single roadway among three is filled.

Field deployment at the 110203 longwall face validated the approach under production conditions. Borehole peeping through the pillar rib in the abandoned roadway concentration zone showed high internal integrity after treatment, and convergence monitoring recorded roughly 16 centimeters of rib-to-rib closure and 14 centimeters of roof-to-floor deformation at backfilled locations, figures notably smaller than in untreated areas. The study’s broader message for the industry is twofold: roadway density in protective pillars should be carefully controlled during mine planning, and where legacy roadways already compromise pillar integrity, solidified excavated coal offers a low-cost, locally sourced route to restoring load-bearing capacity and keeping goaf-side entries safe for mining.

Subject of Research: Instability of goaf-side narrow coal pillars beneath concentrated abandoned roadways in extra-thick coal seams and solidification-backfilling control

Article Title: Instability mechanism of narrow coal pillars in extra-thick coal seam within concentrated abandoned roadway areas and solidification-backfilling control of roadway excavation coal mass

Article References: Instability mechanism of narrow coal pillars in extra-thick coal seam within concentrated abandoned roadway areas and solidification-backfilling control of roadway excavation coal mass. (n.d.). https://doi.org/10.1016/j.rineng.2026.112978

Image Credits: AI Generated

DOI: 10.1016/j.rineng.2026.112978

Keywords: coal mining, narrow coal pillar, abandoned roadways, goaf-side entry, roof stability, triangular block, numerical simulation, backfilling, Portland cement, strata control, FLAC3D, mine safety