The method, called IOPER, short for Inverse-Optimized Pushover with Energy Redistribution, was developed by Jose Mendoza, Alfredo Canelas, and Berardi Sensale. Its central idea is deceptively simple: instead of guessing how seismic demand should be distributed over the height of a building, let the structure itself reveal the answer. In the first stage, the analysts run a conventional displacement-controlled pushover using a modal force pattern, usually based on the dominant mode shape, which in their benchmark buildings accounted for roughly 78 percent of the effective modal mass. As the structure is pushed, the software records the hysteretic energy dissipated at every node, obtained by numerically integrating the enclosed area of each force-deformation loop. That energy is then aggregated story by story, producing a vertical profile of where the building actually absorbed its inelastic demand.

What happens next is the methodological heart of IOPER. The dissipated energy at each level is multiplied by a residual rigidity coefficient, computed through a logarithmic scheme that penalizes stories with high cumulative hysteretic demand. The logarithmic form is deliberately sublinear: it compresses peak energy values while preserving the contrast between stories, mirroring the saturation behavior seen in dynamic analyses, where stiffness loss slows even as energy demand keeps climbing. Damaged stories therefore retain residual capacity and continue participating in the global response rather than being artificially erased. The resulting weighted vector is normalized and exported as the lateral force pattern for a second pushover run on the original, undegraded model. A third iteration repeats the process using the energy traces of the second run, tracing progressively advanced redistribution scenarios. The output is not a single curve but a triplet of capacity curves, each representing a distinct interpretive layer: modal response, energy-induced redistribution at an initial degraded state, and redistribution at a final degraded state.

This architecture is deliberately external to the simulation engine. Where fully adaptive pushover methods, such as the Force-Based Adaptive Pushover procedure of Antoniou and Pinho, recalibrate the load pattern at every analysis step inside the solver using updated mode shapes and spectral accelerations, IOPER performs its adaptivity outside, through MATLAB scripting that reads SeismoStruct output files, integrates hysteretic loops, computes degradation coefficients, and writes new force vectors back into the model. The authors argue that this decoupling is a feature rather than a compromise. Many commercial platforms restrict access to internal state data, lack restart capabilities, or bury their adaptive logic in undocumented code. By making every redistribution decision explicit, auditable, and reproducible through plain text files, IOPER offers a quasi-adaptive alternative that practitioners can inspect, modify, and trust, without iterative eigenvalue calculations at every step.

The validation campaign is substantial. The team tested the framework on two benchmark steel moment-resisting frames originally developed for the ASCE structural control problem: a nine-story building with a square footprint 45.73 meters per side and a first-mode period of 2.28 seconds, and a twenty-story tower rising 80.77 meters with a fundamental period of 3.89 seconds. Both frames qualify as vertically irregular. The nine-story structure shows a stiffness drop at the second level to less than 25 percent of the base value, a classic soft-story condition, while the twenty-story building combines a stiffness reduction below 21 percent at the third floor with a mass jump exceeding 150 percent of the floor below. These irregularities matter because they are precisely the conditions under which fixed load patterns fail, and where elevation-sensitive redistribution strategies earn their keep.

The dynamic benchmark comprised seven real ground motion records, including El Centro, Kobe, Northridge, Landers, Hollister, Loma Prieta, and Imperial Valley, spectrally matched to the Eurocode 8 design spectrum for Ground Type B and scaled to four intensity levels ranging from 0.45 g to 0.90 g. Nonlinear time history analyses used a fiber-based distributed plasticity model with 150 fibers per section, force-based elements, a co-rotational formulation for geometric nonlinearity, and the implicit Hilber-Hughes-Taylor integration scheme. Against this demanding reference, IOPER held its own. In the elastic range, all static methods converged on similar stiffness, but beyond it the differences emerged. The ASCE 41 uniform-load procedure rose monotonically to 7182 kilonewtons on the nine-story frame, blind to degradation, while FAP and IOPER saturated near 6479 and 6518 kilonewtons respectively, closer to the physically plausible post-elastic behavior of the dynamic envelope, which peaked at 7509 kilonewtons. Notably, both adaptive-family methods exhibited a capacity recovery phase beyond 0.6 meters of roof displacement, with IOPER closely following the shape of the incremental dynamic analysis mean curve.

Quantitative metrics reinforced the visual comparisons. At mid- and upper-story nodes, the researchers measured the mean absolute deviation of the IOPER curve from the pointwise median of the NLTHA ensemble and the proportion of drift points at which IOPER fell inside the absolute dynamic envelope. Deviations stayed below 0.05 in normalized base shear across all examined nodes, and the ensemble inclusion ratio ranged from about 71 to 91 percent, peaking at 90.8 percent for the mid-story node of the twenty-story frame. For total drift, root-mean-square errors against NLTHA dropped below 11 percent at the highest intensities, with FAP reaching 1.62 percent on the nine-story frame at 0.90 g. The study also found a meaningful correlation between the energy content of the scaled accelerograms and IOPER’s accuracy: records with high displacement and velocity demand, such as Loma Prieta and Imperial Valley, gave the redistribution logic richer energy traces to work with, improving predictions at severe intensities.

The framework is not without honest limitations, which the authors state plainly. Because the structural model is reset at each iteration, plasticity is not accumulated inside the solver; degradation lives entirely in the external energy metrics and rigidity coefficients. IOPER therefore cannot capture transient amplification, record-to-record variability, or instantaneous modal reshaping, and its accuracy degrades for interstory drift, particularly in upper stories of tall frames, where global errors exceeded 40 percent at 0.90 g on the nine-story building and 60 percent on the twenty-story tower. A refined metric restricted to stories meeting a 15 percent agreement threshold revealed that static methods perform well at lower elevations, with errors as low as 2.8 to 6.4 percent, but diverge near the roof, a persistent dynamic amplification phenomenon that the authors flag as a priority for future work through height-dependent corrections and transient-sensitive formulations.

Those caveats aside, the significance of the work lies in what it demonstrates about the philosophy of seismic assessment. IOPER treats a building’s response not as a geometric abstraction but as an accumulated energy narrative, in which the history of hysteretic dissipation shapes the forces applied next. It recontextualizes modal information as a baseline rather than a straitjacket, and shows that externally scripted control can emulate adaptive behavior without internal feedback loops, challenging the assumption that realism requires embedding adaptivity deep inside the solver. The authors position the framework for preliminary fragility assessment, redistribution-sensitive detailing, and comparative evaluation of modal activation, and note its compatibility with the spirit of ASCE 41 and Eurocode 8, both of which acknowledge stiffness degradation and force redistribution even though neither prescribes energy-based load patterns. Future extensions outlined in the paper include element-wise degradation models, calibration against experimental data for reinforced concrete and composite systems, integration with site-specific hazard spectra, extension to three-dimensional frames with torsional coupling, and evaluation under near-fault pulse-type motions. For a discipline caught between the fidelity of dynamic simulation and the tractability of static methods, IOPER offers a transparent middle path, one in which every force applied to the structure carries the memory of the energy it has already dissipated.

Subject of Research: A quasi-adaptive pushover framework using hysteretic energy redistribution for seismic assessment of steel moment-resisting frames

Article Title: IOPER: Inverse-optimized pushover with energy redistribution: A framework for enhanced control and degraded stiffness-aware seismic analysis

Article References: Mendoza, J., Canelas, A., & Sensale, B. (2026). IOPER: Inverse-optimized pushover with energy redistribution: A framework for enhanced control and degraded stiffness-aware seismic analysis. Results in Engineering, 32, Article 113209. https://doi.org/10.1016/j.rineng.2026.113209

Image Credits: AI Generated

DOI: 10.1016/j.rineng.2026.113209

Keywords: pushover analysis, seismic assessment, hysteretic energy, steel moment frames, nonlinear static analysis, stiffness degradation, NLTHA, ASCE 41, Eurocode 8, adaptive load patterns, structural engineering, earthquake engineering

The core idea behind the framework is staged, or incremental, seismic retrofit. Rather than committing to a single, massive construction campaign, an owner can spread the work over multiple phases, strengthening the most critical parts of the building first and deferring less urgent interventions until funds become available. The concept is not new; the Federal Emergency Management Agency has published incremental rehabilitation guidelines for schools, hospitals, offices, and apartment buildings since the early 2000s. What has been missing, the authors argue, is a systematic method for deciding the sequence of those interventions at the level of individual stories, optimized against explicit economic and safety metrics. Their framework fills that gap by combining probabilistic loss estimation with optimization and multi-criteria decision analysis.

The researchers propose two complementary approaches. The first, which they call the rigorous approach, employs a Greedy search algorithm as a zero-order optimization tool. Greedy algorithms work by making the locally best choice at each step, and here the method groups and prioritizes economically viable stories for retrofit one stage at a time. At every stage, the algorithm evaluates candidate retrofit packages using three financial metrics: expected annual loss, which captures the average yearly economic damage a building is likely to suffer; net present value of benefit, which discounts future risk reduction into today’s money; and marginal benefit, which measures how much additional value each successive retrofit stage delivers. Crucially, all three metrics can be computed both with and without accounting for fatality losses, allowing decision-makers to see how the ranking of interventions changes when human lives are assigned an economic value.

Underpinning these metrics is a probabilistic seismic loss engine built on iterated cloud analyses. Cloud analysis is a technique in performance-based earthquake engineering in which a structural model is subjected to a suite of ground motion records of varying intensities, and the resulting engineering demand parameters, such as interstory drift ratios and floor accelerations, are regressed against ground motion intensity measures. By fitting these demand-intensity relationships probabilistically, the method captures both the aleatory variability of ground motions and the uncertainty in structural response. The resulting fragility and vulnerability functions feed directly into loss calculations consistent with the FEMA P-58 performance assessment methodology, giving the economic metrics a defensible, hazard-consistent foundation rather than relying on simplified empirical rules.

The second, simplified approach is designed for practitioners who lack the computational resources or time for repeated loss analyses. It prioritizes stories based on cost efficiency using the Technique for Order of Preference by Similarity to Ideal Solution, or TOPSIS, a well-established multi-criteria decision-making method. TOPSIS ranks alternatives by measuring their geometric distance from an ideal positive solution, which maximizes all beneficial criteria, and from a negative ideal solution, which minimizes them. In this application, the alternatives are stories of the building, and the criteria are engineering demand parameters obtained from a single cloud analysis, a pushover analysis, and a response spectrum analysis. Pushover analysis pushes a nonlinear structural model laterally to reveal its strength, stiffness, and yielding sequence, while response spectrum analysis captures modal demands efficiently. Together these three analyses supply a rich but tractable set of indicators for each story.

Because the criteria in the TOPSIS framework carry different degrees of importance, the researchers needed a principled way to weight them. They applied two independent weighting schemes: the Analytical Hierarchy Process, a subjective method introduced by Thomas Saaty in which experts make pairwise comparisons to express relative priorities, and the Entropy-Based Method, an objective technique rooted in Claude Shannon’s information theory that assigns weights according to the dispersion of criterion values across alternatives. Using both methods side by side allows the analysts to test whether the resulting story rankings are sensitive to how the weights are chosen, an important robustness check in any multi-criteria study.

To demonstrate the framework, the authors applied both approaches to two case study buildings: a nine-story and a three-story steel moment-resisting frame. In these structures, the staged rehabilitation targeted the beam-column connections, the welded joints that are notorious failure points in older steel frames, as dramatically illustrated by the 1994 Northridge earthquake. The results were striking: the rigorous Greedy search approach and the simplified TOPSIS approach yielded similar story groups and similar priorities for both buildings. This convergence matters because it suggests that engineers who cannot afford the full probabilistic machinery can still arrive at nearly the same retrofit sequence using a handful of standard analyses and a spreadsheet-implementable decision method.

The practical implications extend well beyond steel moment frames. Although the demonstration focused on connection rehabilitation, the underlying decision-making architecture is agnostic to structural system and material. It could, in principle, guide the staged strengthening of reinforced concrete frames, masonry buildings, or any structure subjected to comparable rehabilitation strategies. For public agencies managing portfolios of schools and hospitals, for insurance companies pricing retrofit incentives, and for owners of aging commercial buildings, the framework provides a defensible way to allocate scarce funds across years or decades. It also aligns with a broader shift in earthquake engineering from prescriptive code compliance toward risk-informed decision-making, in which the value of an intervention is measured in avoided losses rather than in abstract performance targets.

The study also contributes a subtle but important insight: the ranking of retrofit stages can shift depending on whether fatality losses are included in the economic calculus. When the value of a statistical life is folded into expected annual loss and net present value calculations, stories whose failure would endanger many occupants may rise in priority even if their direct repair costs would be modest. By reporting metrics both ways, the framework makes this trade-off explicit rather than hiding it inside a single number, giving stakeholders the information they need to weigh economic efficiency against life safety in a transparent manner. The authors report no external funding for the work, and both the rigorous and simplified procedures are presented as flexible yet robust tools that support risk-informed, economically optimized, and practically implementable retrofit strategies.

As cities from Tehran to Los Angeles grapple with vast stocks of seismically deficient buildings, tools like this one may prove decisive in converting awareness into action. The promise of the framework lies in its pragmatism: it does not demand unlimited budgets or heroic assumptions, but instead meets owners where they are, offering a mathematically sound path to safer buildings one stage at a time. If widely adopted, staged retrofit guided by such risk-based prioritization could steadily chip away at urban seismic vulnerability, ensuring that every dollar spent on strengthening delivers the maximum possible reduction in losses and, ultimately, in lives saved when the next major earthquake strikes.

Subject of Research: Risk-based multi-criteria decision-making for sequencing multi-stage seismic retrofit of buildings

Article Title: Multi-stage seismic retrofit of buildings: a risk-based multi criteria decision-making procedure

Article References: Orumiyehei, A., & Dolatshahi, K. M. (2026). Multi-stage seismic retrofit of buildings: a risk-based multi criteria decision-making procedure. Bulletin of Earthquake Engineering. https://doi.org/10.1007/s10518-026-02673-9

Image Credits: AI Generated

DOI: 10.1007/s10518-026-02673-9

Keywords: seismic retrofit, earthquake engineering, decision-making, TOPSIS, Greedy search, expected annual loss, cost-benefit analysis, steel moment frames, cloud analysis, incremental rehabilitation, structural safety, risk assessment

The appeal of friction dampers lies in their physics. Conventional steel frames absorb earthquake energy by yielding, which means the structure itself accumulates damage with every loading cycle, a phenomenon engineers call cyclic deterioration or pinching. An RFD behaves differently. Governed by Coulomb’s law of dry friction, it produces stable, rectangular hysteretic loops whose enclosed area, and therefore the energy dissipated per cycle, remains constant no matter how many times the ground shakes. The device consists of a central rotating plate sandwiched between side plates with friction discs, all clamped by a pre-tensioned bolt. When inter-story drift forces the plates to rotate against one another, friction resists the motion with a constant moment, protecting the primary frame. Experimental work dating back to Mualla and Belev in 2002 showed reductions of up to 89 percent in lateral displacement and 41 percent in base shear under optimized configurations.

Simulating such devices accurately, however, is notoriously difficult. The abrupt transition between sticking and slipping is mathematically discontinuous, and the two common workarounds both introduce error. Regularization smooths the discontinuity with an artificial curve, adding fake compliance exactly where the response should be sharp, while equivalent-damping approaches replace the device with a viscous dashpot that cannot reproduce the rate-independent, rectangular hysteretic loop. Both require case-specific calibration, which would be hopeless at the scale of hundreds of thousands of simulations. The authors instead built on the Non-Smooth Dynamics method, reformulating Coulomb friction as a Linear Complementarity Problem solved at each timestep with a Projected Successive Over-Relaxation algorithm. This formulation reproduces the exact friction law with no tuning parameters, and because each step typically converges in fewer than five iterations, sweeping thousands of structural configurations becomes computationally feasible. Crucially, it also means the training labels fed to the machine learning models contain pure physics rather than artifacts of a smoothing choice.

The dataset behind the study is staggering in scope. The team generated 1,000 shear-type frames ranging from 3 to 20 stories, with randomized Latin Hypercube sampling covering fundamental periods from 0.3 to 3.0 seconds, floor masses from 150 to 350 tonnes, and four different mass and stiffness distributions. Each frame was fitted with roughly 50 different damper configurations, varying the number of devices, their vertical placement, and friction moments between 25 and 500 kilonewton-meters. Eight real seismic records, including El Centro 1940, Kobe 1995, and Northridge 1994, were applied at their original historical intensities. In total, 433,984 nonlinear simulations were run, each quantified with a composite index called CR2 that combines peak roof displacement, peak roof acceleration, and peak base shear, each normalized by the uncontrolled response, so that lower values mean better performance.

Six regression models were then trained to predict seismic performance from the design variables, and the comparison itself delivered a scientific finding. Linear models achieved a cross-validated coefficient of determination of only 0.64 for the typical scenario, while XGBoost reached 0.907 and Random Forest 0.904, cutting prediction error nearly in half. The gap proves that the relationship between damper design and seismic response is fundamentally nonlinear, with threshold effects and interactions that simple equations cannot capture. The nonlinearity deepens under severe ground motion: for the conservative 90th-percentile scenario, linear models managed only 0.31, while Random Forest reached 0.786. To prevent data leakage, the team used structure-grouped cross-validation, ensuring models were always evaluated on buildings they had never seen, and verified accuracy on 200 entirely held-out structures with calibrated 90 percent prediction intervals.

The most striking practical insight concerns coverage density, the fraction of floors equipped with dampers. Feature importance analysis identified this single variable as the dominant predictor of performance, and the response surface revealed clear diminishing returns. By formulating the threshold as an explicit optimization over the trained surrogate, the researchers found that the median coverage at which 80 percent of the maximum achievable gain is reached sits at about 0.83, or 0.86 when read from the empirical envelope across the whole population. Resolved by building height, the picture inverts common intuition: low-rise frames of 3 to 5 stories require full coverage, mid-rise frames of 6 to 12 stories need at least 86 percent, and high-rise frames of 13 to 20 stories need only about 73 percent, with none requiring full instrumentation. The mechanical explanation is elegant, since tall buildings concentrate drift demand in a subset of stories that can be covered selectively, whereas the flat drift profile of a low-rise frame leaves no inexpensive stories to omit.

The analysis also uncovered a counterintuitive failure mode. A small number of oversized dampers, high friction moments paired with low coverage, can leave a frame worse off than no dampers at all. In this lock-up regime, the devices never reach their slip threshold under the imposed demand, act as rigid links, and locally stiffen the structure, amplifying floor accelerations and base shear. These configurations account for just 1.4 percent of the dataset but nearly 10 percent of the total prediction error, and more than 80 percent of them occur in frames of five stories or fewer. The practical lesson is a lower bound on coverage for any given device capacity: coverage cannot be traded freely against friction moment. The study identifies an optimal friction moment window of roughly 300 to 460 kilonewton-meters for mid- and high-rise frames, with a lower optimum near 250 kilonewton-meters for short buildings.

Robustness across seismic scenarios proved to be another strength of the framework. When the researchers compared the optimal damper count recommended under typical conditions with that recommended under the conservative 90th-percentile scenario, 67.9 percent of the 1,000 structures received identical recommendations, and in the remaining cases the median discrepancy was just one damper. The conservative scenario never suggested fewer devices, only occasionally one or two more, meaning designers can optimize for typical demand and then verify against severe conditions with minimal adjustment. This multi-record approach, summarizing performance with 10th-percentile, median, and 90th-percentile statistics consistent with the FEMA P-58 framework, addresses a long-standing gap in earlier studies that optimized dampers against a single ground motion.

The authors are candid about the limits of their results. The structural models are elastic shear frames, and a substantial fraction of the simulated demand levels actually lie at or beyond the yield limit of real steel frames, so the coverage recommendation should be read as an upper bound for quasi-elastic serviceability design rather than a validated rule for ductile response. The design space is a factorial sweep rather than a representative building stock, and the eight-record suite carries sampling uncertainty of about 0.10 in CR2 units, which exceeds the surrogate’s own prediction error of 0.070 by roughly half. That last point may be the study’s most consequential methodological message: enlarging the ground motion database, not refining the machine learning model, is what would most sharpen future design guidance. Future extensions include soil-structure interaction, three-dimensional frames with torsional coupling, nonlinear material behavior, floor-specific friction moments, and cost-based optimization.

Even with those caveats, the work marks a genuine shift in how earthquake protection devices might be designed. Instead of running expensive simulations case by case, engineers can now query a trained surrogate that exposes the entire response surface across damper placement, capacity, and building geometry in milliseconds, complete with calibrated uncertainty bounds. The finding that taller buildings need proportionally fewer instrumented floors, that a coverage threshold near 86 percent captures most of the achievable benefit for mid-rise construction, and that a handful of oversized devices can actively harm a structure, together give practitioners an evidence-based starting point that previously required months of computation. As machine learning surrogates spread through earthquake engineering, this study offers both a template, exact physics in the labels, grouped validation, honest uncertainty, and a reminder that the biggest gains sometimes come from questioning design intuition.

Subject of Research: Machine learning surrogate modeling of nonlinear seismic response of steel moment frames with rotational friction dampers

Article Title: A machine learning-based surrogate model for analyzing the nonlinear seismic response of steel moment frames with rotational friction dampers

Article References: Ponce-López, E., & Yáñez-Godoy, H. (2026). A machine learning-based surrogate model for analyzing the nonlinear seismic response of steel moment frames with rotational friction dampers. Machine Learning with Applications, 26, Article 101030. https://doi.org/10.1016/j.mlwa.2026.101030

Image Credits: AI Generated

DOI: 10.1016/j.mlwa.2026.101030

Keywords: rotational friction dampers, machine learning, seismic response, steel moment frames, XGBoost, non-smooth dynamics, earthquake engineering, surrogate model, coverage density, structural optimization, energy dissipation, time-history analysis