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