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 research is rooted in performance-based earthquake engineering, a framework that treats structures not as objects that simply stand or fall, but as systems whose performance can be predicted probabilistically. At the heart of this framework sits the intensity measure, a scalar descriptor of ground shaking, such as peak ground acceleration or spectral acceleration at a given period, that serves as the bridge between hazard analysis and structural response prediction. The quality of an intensity measure is judged by its efficiency, meaning how tightly it correlates with the engineering demand parameters that describe structural and operational damage, and by its sufficiency, meaning how well the predicted response remains independent of other ground-motion characteristics. An intensity measure that is both efficient and sufficient allows engineers to build accurate probabilistic seismic demand models with fewer costly simulations, which is precisely where the new study makes its contribution.

The particular system the researchers examined, known as the simply-supported-bridge-vehicle coupled system, is among the most common bridge forms on high-speed railway networks, especially on challenging routes such as the Sichuan-Tibet Railway, where lines must cross regions laced with active strike-slip faults. Simply supported spans rest on bearings that allow rotation and, to a degree, translation, which makes them economical and constructible but also vulnerable when the ground beneath them lurches in two directions at once. When a train is present, the problem becomes even more intricate, because the vehicle, the track, and the bridge form a dynamically coupled system in which the running safety of the train depends on the deformation of the deck, and the vibration of the deck is in turn influenced by the moving masses of the vehicles above it. Past investigations by these and other research groups have shown that near-fault pulse-type ground motions can compromise derailment resistance and that fault rupture itself imposes permanent, quasi-static displacements that no amount of dynamic damping can absorb.

Strike-slip faulting introduces a uniquely punishing combination of effects. As the fault ruptures, the ground on either side shears horizontally past the other, and a structure crossing the fault trace is forced to accommodate the offset. In the near-fault zone, two signature phenomena dominate: the fling step, a permanent, often unidirectional displacement pulse produced by tectonic deformation, and forward directivity, a strong long-period velocity pulse that arrives when the rupture front propagates toward the site at nearly the speed of the shaking itself. These effects are inherently directional, aligned with the fault-parallel and fault-normal orientations, so the structural response depends critically on the angle at which the bridge crosses the fault. Compounding the challenge, recorded ground motions close to strike-slip surface ruptures are scarce, forcing analysts to work with limited datasets in which the choice of intensity measure carries outsized consequences for the reliability of the resulting risk estimates.

To tackle this problem, the team developed a modified intensity measure selection framework for cloud analysis, a widely used statistical technique in which a family of ground motion records, each scaled or unscaled, is run through the structural model and the resulting demands are regressed against candidate intensity measures in logarithmic space. The innovation lies in the normalization of the intensity measures, which sharpens the comparison of efficiency across candidates whose raw numerical ranges differ by orders of magnitude. By normalizing before evaluating statistical performance, the framework reduces distortions that can arise in regression diagnostics and produces a fairer ranking of alternatives. The researchers then applied the framework across a battery of candidate measures drawn from the standard toolbox of earthquake engineering, including peak ground velocity, peak spectral displacement, peak spectral velocity, and measures defined from individual ground motion components as well as geometric-mean combinations, testing each against six representative engineering demand parameters spanning the bridge and the running vehicles.

The verdict from thousands of coupled dynamic analyses is strikingly clear: under the coupled fling-step and forward-directivity demands of crossing strike-slip faulting, velocity- and displacement-based spectral measures outperform the acceleration-based measures that have traditionally dominated fragility studies. Specifically, the peak spectral displacement, SDmax, and peak spectral velocity, SVmax, emerged as the top performers for constructing probabilistic seismic demand models of the coupled system. This makes physical sense. Long-period velocity pulses and permanent displacement offsets, the hallmarks of near-fault strike-slip motion, resonate most directly with displacement-type demands such as bearing displacement, pier drift, and the deck deformations that govern train running safety. Peak ground acceleration, by contrast, emphasizes high-frequency content that is relatively less consequential for these long-period, quasi-static-dominated failure modes, and its correlation with demand weakens accordingly.

Equally important is the finding about directionality. The study shows that intensity measures computed from the fault-parallel component of ground motion perform consistently well in integrated assessments across all six engineering demand parameters, reflecting the dominant role of the shearing displacement imposed along the fault trace. When the researchers turned to specific engineering scenarios defined by the fault-bridge crossing angle, they identified scenario-specific optima: for a 90-degree crossing, the geometric mean of peak spectral displacement across the two horizontal components, SDmax,GM, proved best, while for a shallower 45-degree crossing, the fault-parallel peak spectral displacement, SDmax,FP, took the top spot. In both cases the chosen measures delivered a balanced combination of efficiency and sufficiency, giving engineers a defensible, defensible-to-auditor basis for record selection and fragility construction tailored to the actual geometry of a proposed crossing.

The practical implications reach well beyond academic statistics. High-speed rail corridors in tectonically active regions, from southwest China to Turkey, California, and Taiwan, increasingly must traverse fault zones because alternative routings are economically or geographically impossible. The 1999 Kocaeli and Duzce earthquakes in Turkey and the Chi-Chi earthquake in Taiwan famously collapsed or displaced simply supported spans whose unseated girders traced the fault rupture across their alignments. By identifying which ground-motion descriptors most faithfully capture the demand imposed on a coupled bridge-train system, the new framework enables more economical and more trustworthy fragility assessment, supporting decisions about bearing seat widths, restrainers, isolation systems, and operational speed limits during seismic events. Because cloud analysis with unscaled records is computationally expensive, the improved efficiency of the recommended measures also translates directly into fewer simulations required for a given confidence level, a meaningful saving when each coupled vehicle-track-bridge analysis involves extensive nonlinear computation.

Methodologically, the study also contributes a reusable template. The normalization-based cloud analysis framework is not tied to any particular bridge form, and the authors’ evaluation metrics, which weigh efficiency, sufficiency, and practicality across multiple demand parameters simultaneously, can be redeployed for continuous girders, cable-stayed spans, and suspension bridges crossing faults, where prior work by the same community has documented severe track-bridge interaction and long-span dynamic amplification. The research was supported by the National Natural Science Foundation of China, the Department of Education of Guangdong Province, the Foshan Science and Technology Bureau, and Hunan University of Science and Technology, and drew on the strong-motion database of the Pacific Earthquake Engineering Research Center’s Next Generation Attenuation-West2 project. As high-speed rail networks push deeper into seismically hostile mountains and basins, the humble task of choosing the right number to describe a ground motion, once treated as a technical footnote, now stands revealed as one of the decisive levers for keeping trains, bridges, and passengers safe when the ground itself refuses to hold still.

Subject of Research: Selection of optimal seismic intensity measures for railway simply-supported-bridge-vehicle coupled systems subjected to crossing strike-slip faulting.

Article Title: Analysis and selection of seismic intensity measures for railway simply-supported-bridge–vehicle coupled systems subjected to crossing-strike-slip faulting

Article References: Zhou, T., Li, Z., Jiang, L., & Wen, T. (2026). Analysis and selection of seismic intensity measures for railway simply-supported-bridge–vehicle coupled systems subjected to crossing-strike-slip faulting. Bulletin of Earthquake Engineering. https://doi.org/10.1007/s10518-026-02676-6

Image Credits: AI Generated

DOI: 10.1007/s10518-026-02676-6

Keywords: seismic intensity measures, railway bridge, strike-slip fault, vehicle-bridge interaction, probabilistic seismic demand, fling step, forward directivity, cloud analysis, running safety, high-speed railway, fault crossing, earthquake engineering