The research team, led by Ala Ameryan, Mansour Ghalehnovi and Mohsen Rashki, tackled a problem that has haunted reliability engineers for decades: the sheer computational expense of evaluating a performance function. In realistic engineering settings, each evaluation may require running a full finite element model of a structure, a calculation that can take minutes or hours on high-performance computers. The gold standard, Monte Carlo Simulation, demands millions of such evaluations to reliably estimate a small failure probability, making it utterly impractical for real-world design work. Alternative approaches such as the First Order Reliability Method, known as FORM, are cheap but stumble badly when the boundary between safety and failure is strongly curved or irregular, sometimes failing outright to locate the critical design point.

The new method builds on two earlier lines of research. The first is the Kriging surrogate model, a statistical technique borrowed from geostatistics that constructs a probabilistic approximation of an expensive function. Kriging has a remarkable property: it not only predicts the value of the function at any point but also quantifies its own uncertainty, reporting a variance that signals how much it trusts each prediction. The second ingredient is active learning, embodied in the well-known AK-MCS method. Instead of blindly sampling the expensive function everywhere, AK-MCS uses a so-called U-learning function to hunt for the sample points where the surrogate is most uncertain about whether the structure is safe or failed. Each such point is evaluated with the true model and added to the training set, so the surrogate grows smarter precisely where it matters most: along the limit state surface that separates survival from collapse.

AK-MCS alone, however, struggles with rare events. When the failure probability is extremely small, the Monte Carlo candidate pool must be enormous, and the surrogate must classify millions of points. The authors’ solution was to fuse this active learning machinery with the First Order Control Variates Method, or FOCM, a technique they had previously developed for high-dimensional, highly nonlinear problems. FOCM works by locating the most probable failure point, linearizing the performance function around it, and then using the control variates principle to correct the bias introduced by that linearization. The correction factor, denoted lambda, is estimated by a modest simulation and multiplies the first-order failure probability to recover the true value. Crucially, FOCM is largely insensitive to the dimensionality of the problem, a property that most surrogate methods lack.

The mathematical elegance of AK-FOCM lies in how the two components divide the labor. The active learning Kriging model concentrates computational effort on the narrow, critical band around the limit state surface, aggressively filtering out the vast safe and failed regions where classification is trivial. Once the surrogate has converged, FOCM takes over: the design point is refined, the performance function is linearized without ever computing explicit partial derivatives, and a correction factor absorbs the truncation error. The authors justify a key simplification, the decoupling of a statistical weighting term from the failure indicators, using the concentration of measure phenomenon in high-dimensional spaces, which guarantees that the residual error shrinks as the number of random variables grows. For rare events, an n-sigma scaling trick artificially inflates the failure probability to a measurable level and then maps it back analytically, sidestepping the statistical wall that defeats ordinary Monte Carlo.

The validation campaign was extensive, spanning twenty mathematical and engineering benchmark problems. In the first test, a two-dimensional exponential function so nonlinear that both FORM and AK-MCS failed completely, AK-FOCM delivered a reliability index of 3.99 against the Monte Carlo reference of 4.01, an error of just half a percent, using only 89 function evaluations. The original FOCM needed 1,818 evaluations for comparable accuracy, meaning the new integration cut the computational cost by 95.1 percent. In a second, severely nonlinear example with exponentially distributed variables, AK-FOCM reached the correct answer with 63 evaluations, roughly 22 percent of what FOCM required, while AK-MCS produced an error three times larger.

The method then faced genuine engineering challenges. A conical shell structure loaded by compression and bending, whose dominant failure mode is buckling, was analyzed with six random variables following lognormal, normal and Gumbel distributions. Monte Carlo demanded one hundred million samples; AK-FOCM matched the reference reliability index of 4.79 with only 115 function calls, whereas AK-MCS failed to converge even with twenty times more initial samples. A fourth case study examined the static strength of an automobile front axle beam, an I-section component carrying the entire weight of the vehicle’s front end. Here AK-FOCM again reproduced the Monte Carlo reliability index of 2.06 with 92 evaluations, outperforming FOCM’s 123 calls while matching its accuracy.

Because surrogate methods involve random initial sampling, the team also ran a rigorous statistical robustness analysis. Thirty independent runs of the hardest nonlinear example produced a mean reliability index of 2.514 with a coefficient of variation below one percent, demonstrating that the method is remarkably insensitive to the luck of the initial sample draw. Sensitivity tests showed that varying the initial training set size from 10 to 30 points changed the final answer by less than half a percent, and enlarging the Monte Carlo candidate pool beyond a critical threshold drove the variance of the estimate essentially to zero. Boxplots of the required function calls across repeated runs showed not only a lower median cost for AK-FOCM but also tighter variability, indicating predictable, stable performance rather than occasional lucky runs.

The authors are candid about the boundaries of their framework. If the initial design of experiments completely misses a small, isolated failure region, the active learning loop could converge prematurely and underestimate risk. Severely discontinuous or highly oscillatory performance functions may violate the assumption that the linearization error can be fully compensated by the correction factor. And although FOCM itself tolerates high dimensions, the Kriging surrogate remains vulnerable to the curse of dimensionality, so problems with thousands of random variables would still degrade its fidelity and inflate its overhead. The method has so far been tested on static structural problems, and the authors call for future comparisons against emerging deep learning-based reliability frameworks and validation on full-scale, time-variant dynamic finite element models.

Even with those caveats, the implications for engineering practice are substantial. Reliability analysis has long been trapped in a trade-off between methods that are accurate but unaffordable and methods that are affordable but fragile. By coupling a self-aware surrogate that learns where to look with a dimensionally robust variance-reduction core, AK-FOCM offers a way to estimate the probability of catastrophic rare events at a fraction of the traditional cost. For designers of critical infrastructure, vehicles and aerospace structures, that could mean running reliability checks routinely during design rather than as a final, expensive afterthought, ultimately translating abstract probability mathematics into safer buildings, safer cars and safer machines in the physical world.

Subject of Research: A hybrid active learning Kriging and first order control variates method for efficient structural reliability analysis of rare failure events

Article Title: AK-FOCM: Integration of active learning Kriging with first order control variates reliability method

Article References: Ameryan, A., Ghalehnovi, M., & Rashki, M. (2026). AK-FOCM: Integration of active learning Kriging with first order control variates reliability method. Results in Engineering, 32, Article 113197. https://doi.org/10.1016/j.rineng.2026.113197

Image Credits: AI Generated

DOI: Not provided

Keywords: structural reliability, AK-FOCM, active learning, Kriging, Monte Carlo simulation, control variates, failure probability, surrogate modeling, finite element analysis, FORM, rare events, computational efficiency

The research team focused on tension-type prestressed anchors, the workhorse of early slope reinforcement in China. Unlike compression anchors, whose grout stays squeezed and therefore rarely cracks, tension anchors place their grout in a state that invites cracking. Those cracks become highways for groundwater, oxygen, chloride, and sulfate ions to reach the steel tendon, which is itself held under high stress. The result is a particularly vicious form of deterioration: stress corrosion that can transform a ductile steel strand into a brittle one, quietly stripping away the load-carrying capacity that engineers counted on at design time.

To capture this decay mathematically, the researchers built on an electrochemical corrosion rate model originally developed by Hong, adapting it to anchor cables. Their equation links the corrosion rate to ambient temperature, relative humidity, the thickness of the grout cover wrapping the tendon, and the compressive strength of the grout itself. The physics is intuitive once unpacked: warmer conditions accelerate the electrochemical reactions through Arrhenius-type kinetics, moisture drives the cathodic half of the corrosion cell, and a thicker, stronger grout cover slows the ingress of aggressive species. Because fully enclosed anchors corrode far more slowly than those exposed to wet-dry cycling, the team applied a correction factor, effectively assuming the grout remains intact until corrosion products expand and crack it from within.

From this corrosion rate, the researchers derived time-varying expressions for three competing failure modes. The first is debonding at the mortar-rock interface, where the entire grout column pulls out of the borehole. The second is slippage at the tendon-grout interface, where corrosion products and chemical degradation of the cement matrix erode the bond that transfers load from steel to rock. The third is tensile yielding of the free segment, as the effective cross-section of the steel shrinks and its yield strength drops. Crucially, the anchorage system behaves as a series, or weakest-link, system: whichever of the three resistances is smallest at any given moment governs the anchor’s capacity, and failure of that single link means failure of the whole.

The study’s most striking mechanistic finding is that this weakest link does not stay put. In the early decades of service, the mortar-rock interface typically controls the anchoring force. But as corrosion progresses, expanding rust products crack the grout, allowing water, oxygen, chloride, and sulfate ions to flood the tendon surface. The tendon-grout bond then collapses far faster than the other resistances, and control of the anchoring force shifts to that degrading interface. In the case study, a twelve-meter anchor’s tendon-grout bond resistance plummeted from 2,714 kilonewtons to just 291 kilonewtons over a century, while the free segment’s yield resistance fell more gently from 1,316 to 1,033 kilonewtons. This mechanism transition is what produces the alarming acceleration of anchoring-force decay in the later service life.

To translate anchor degradation into slope-scale risk, the team embedded their time-dependent anchor capacities into a pseudo-static limit equilibrium framework based on the horizontal slice method and the Mohr-Coulomb strength criterion. Seismic demand was not applied uniformly: following the Chinese seismic design code for hydraulic structures, the horizontal acceleration coefficient was distributed trapezoidally along the slope height, rising from the toe to the crest to reflect the well-documented topographic amplification of ground motion. A genetic algorithm searched for the critical slip surface, and the First-Order Reliability Method, implemented through the JC algorithm with equivalent normalization of non-normal variables, computed failure probabilities at each service age. Peak ground acceleration was treated as a Gumbel-distributed extreme-value random variable, consistent with standard probabilistic seismic hazard practice.

The case study examined a 28-meter rock slope cut into homogeneous slate at 47 degrees, reinforced with seven rows of high-strength strands and analyzed under four seismic intensity levels ranging from 0.1 g to 0.3 g. The results trace a clear three-act drama. For the first forty years, failure probability stays essentially flat, governed mainly by seismic parameters rather than corrosion. Then, between roughly fifty and seventy years, the curves bend upward: under 0.1 g shaking, failure probability jumps from 8.3 percent to 17.5 percent, a 110 percent increase, while under 0.3 g it climbs from about 50.7 to 59.1 percent. In the final decades, the system enters what the authors call a high-risk state, with failure probability exceeding 38.5 percent at 0.1 g and reaching 67.8 percent at 0.3 g by year one hundred.

Perhaps the most counterintuitive insight concerns how seismic intensity and corrosion interact. The absolute gap in failure probability between the 0.3 g and 0.1 g scenarios over a century is about 29.3 percentage points, but the relative difference shrinks dramatically, from 7.7-fold to 1.8-fold. In other words, long-term corrosion accumulation erodes system reliability most severely in low-seismicity environments, where a corroded anchor system has little residual margin to absorb even modest shaking. The authors also benchmarked their results against Chinese code target reliability indices, finding that under a 0.2 g scenario the failure probability surpasses 40 percent after about 85 years, a reliability index below 0.25 that would be unacceptable for most structures, while early-life indices above 1.0 are only marginally adequate even for low-consequence applications.

The sensitivity analysis delivers a rare piece of genuinely actionable engineering guidance. Increasing the grout cover thickness over the tendon from 20 millimeters to 50 millimeters cuts the hundred-year failure probability by 42 percent, raising the reliability index from roughly 0.31 to 0.82. Thin covers show accelerated failure probability growth beginning around thirty years, whereas thicker covers delay that inflection to around fifty years. Conversely, every 10-degree-Celsius rise in average temperature or 20 percent rise in relative humidity boosts the failure probability growth rate by 15 to 25 percent, a sobering prospect for coastal and monsoon-climate regions and for a warming world. Among all parameters examined, the environmental corrosion coefficient exerts the strongest influence on reliability, followed by humidity and temperature, confirming that durability, not seismic demand alone, dominates long-term performance.

The authors are candid about the limits of their framework. They assume uniform rather than pitting corrosion, treat corrosion across the anchor array as statistically independent rather than spatially correlated, and model fully saturated conditions without the strengthening effect of matric suction in unsaturated soil, all choices that make their baseline estimates conservative. Even so, the practical implications are hard to ignore. The fifty-year inflection point offers a quantitative target for maintenance scheduling, pointing to interventions such as secondary grouting or anchor supplementation well before reliability collapses. For major projects with design lives beyond half a century, the study argues, durability design must incorporate environmental acceleration factors and hygrothermal coupling, and slopes in hot, humid regions with inadequate grout cover may fall below acceptable reliability thresholds well within their intended service life. What emerges is a clear message for infrastructure owners: the anchors holding back the world’s slopes are aging on a schedule that static, time-invariant safety checks were never designed to see.

Subject of Research: Time-variant reliability analysis of corroded prestressed anchor-reinforced slopes under seismic loading

Article Title: Time-variant reliability of anchored slopes considering coupled corrosion and seismic effects

Article References: Wang, H., Meng, X., Yang, X., Jia, J., Tu, B., Zhang, L., Pan, X., & Han, Q. (2026). Time-variant reliability of anchored slopes considering coupled corrosion and seismic effects. Discover Geoscience, 4(1), Article 339. https://doi.org/10.1007/s44288-026-00700-5

Image Credits: AI Generated

DOI: 10.1007/s44288-026-00700-5

Keywords: anchored slopes, prestressed anchors, corrosion, seismic loading, time-variant reliability, slope stability, geotechnical engineering, failure probability, grout cover thickness, limit equilibrium method, FORM, infrastructure durability