The research, published in the Bulletin of Earthquake Engineering, was carried out by Mustafa Al-Mashaykhi and Pathmanathan Rajeev of Swinburne University of Technology in Melbourne, together with Kushan Wijesundara and Sahan Bandara of the University of Peradeniya in Sri Lanka. Their focus was a workhorse of steel construction: the concentrically braced frame, or CBF, a structural system in which diagonal steel braces carry lateral forces through axial tension and compression. CBFs are popular because they are stiff, economical and efficient at controlling drift, yet their braces are notoriously vulnerable to the kind of repeated inelastic cycling that long-duration earthquakes deliver.

The problem with drift-based assessment, the authors explain, is that it captures the peak deformation a frame experiences but says nothing about how that deformation was reached or how many times the structure yielded along the way. When a brace buckles in compression and stretches back in tension, cycle after cycle, the steel accumulates low-cycle fatigue damage even if the maximum drift never changes. Two ground motions can produce identical peak drifts while leaving the structure in radically different states — one largely intact, the other with fractured braces and a frame on the verge of collapse. Classic post-earthquake surveys, including studies of the 1994 Northridge and 1995 Hyogo-ken Nanbu earthquakes, documented exactly this kind of cumulative damage in steel buildings, motivating engineers to look beyond single-deformation measures.

To capture that cumulative effect, the team turned to an energy-based damage index, building on earlier work by Rajeev and Wijesundara that applied the continuous wavelet transform to steel braced frames. The wavelet approach, rooted in mathematical techniques first formalised by Dennis Gabor in 1946 and developed extensively since, allows the seismic energy absorbed by the structure to be computed as a function of time. Instead of treating the ground motion as a single peak event, the method decomposes the response record into localised wave components, making it possible to track how hysteretic energy — the energy dissipated through plastic deformation — accumulates over the full duration of shaking. The resulting damage index blends the maximum deformation demand with the total dissipated energy, echoing the influential Park and Ang damage model of 1985 and later energy-based formulations by researchers such as Bojorquez, Teran-Gilmore and Fajfar, who explicitly linked ductility demands to low-cycle fatigue.

The mechanics of the study were ambitious in scale. The team designed eight different CBF steel structures, varying both building height and bracing configuration — the geometric arrangements of diagonals, such as diagonal and chevron patterns, that determine how forces flow through the frame. Each frame was then subjected to thirty different ground motion records spanning a wide range of seismic intensities and hazard levels, drawn from European strong-motion databases. Using the open-source simulation platform OpenSees, with brace models capable of capturing cyclic inelastic buckling as characterised by Uriz, Filippou and Mahin, the researchers ran Incremental Dynamic Analysis, the method introduced by Vamvatsikos and Cornell in 2002, in which each record is scaled to progressively higher intensities until the structure reaches defined limit states.

From this mountain of nonlinear simulations, the team built probabilistic seismic demand models — statistical relationships between ground-motion intensity and structural demand — for both the conventional drift measure and the new energy-based damage index. These models then fed into fragility curves, the standard tool of seismic risk assessment, which express the probability that a structure will exceed a given damage state at a given level of shaking. Fragility curves are what engineers and insurers ultimately rely on to estimate expected losses and to decide where retrofit money should be spent, so their statistical quality matters enormously.

The headline finding is that the energy-based approach produces fragility estimates with lower statistical dispersion. In practical terms, the record-to-record variability — the scatter that arises when different ground motions, even at the same intensity, push a structure to very different levels of damage — is reduced when damage is measured by accumulated hysteretic energy rather than peak drift. Lower dispersion means tighter, more confident fragility curves, and tighter curves mean more reliable estimates of collapse risk. For a profession that must make safety decisions under deep uncertainty, reducing this scatter is not a cosmetic improvement; it directly changes the width of the confidence intervals within which real buildings are judged safe or unsafe.

Equally significant is where the two approaches agree and where they diverge. At lower damage levels, corresponding to the Immediate Occupancy performance state — the condition in which a building remains safe to enter and use immediately after a quake — the drift-based and energy-based methods yield broadly comparable fragility predictions. This makes sense: at light damage, the structure has barely yielded, little hysteretic energy has been dissipated, and peak deformation is a reasonable proxy for the state of the frame. But at the high damage end, at the Collapse Prevention limit state, where the structure is barely standing and braces have undergone many large inelastic cycles, the two methods diverge substantially. The energy-based index, sensitive to those cumulative inelastic hysteretic cycles, paints a different — and, the authors argue, more physically faithful — picture of how close a CBF really is to collapse under severe, long-duration shaking.

The implications reach beyond braced steel frames. Energy-based damage indices have been gaining traction across earthquake engineering, with recent work applying energy concepts to mainshock-damaged buildings, to integrated displacement-and-energy design procedures, and to the plastic design of steel moment frames that explicitly account for cumulative damage. The new study adds an important probabilistic leg to that edifice: it demonstrates, quantitatively and across a family of real frame designs, that an energy-based damage measure delivers a more stable estimate of seismic performance precisely where it matters most — near collapse. For code writers, that suggests energy demands deserve a place alongside drift limits in performance-based design frameworks, particularly for structural systems whose components, like steel braces, degrade through repeated buckling and yielding.

There are, of course, practical hurdles before energy-based fragility becomes routine practice. Computing hysteretic energy requires nonlinear time-history analysis and the wavelet-based signal processing that underpins the damage index, tools that are more demanding than the elastic drift checks familiar from everyday design offices. The authors note that all data, models and code supporting the study are available from the corresponding author on reasonable request, which should help other researchers replicate and extend the method to other structural systems, longer-duration motions and site classes not covered here. But the direction of travel is clear. As earthquake engineers push toward more reliable, performance-based assessment of the buildings people depend on, measuring the total energy a quake pumps into a structure — not just how far it sways — may become the difference between a fragility curve that looks reassuring and one that tells the truth about collapse risk.

Subject of Research: Energy-based seismic fragility assessment of concentrically braced steel frames

Article Title: Energy-based seismic fragility for concentrically braced steel frames

Article References: Al-Mashaykhi, M., Rajeev, P., Wijesundara, K., & Bandara, S. (2026). Energy-based seismic fragility for concentrically braced steel frames. Bulletin of Earthquake Engineering. https://doi.org/10.1007/s10518-026-02717-0

Image Credits: AI Generated

DOI: 10.1007/s10518-026-02717-0

Keywords: seismic fragility, concentrically braced frames, energy-based damage index, inter-storey drift, incremental dynamic analysis, wavelet transform, hysteretic energy, collapse prevention, probabilistic seismic demand, steel structures, low-cycle fatigue, earthquake engineering

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