⚡ electrical engineering

Influence of the response modification factor on the bidirectional seismic performance of reinforced-concrete buildings

This study utilizes FEMA P695 methodology and Incremental Dynamic Analysis on 3D reinforced-concrete building models to demonstrate that bidirectional seismic excitation significantly reduces collapse capacity, necessitating a reduction in the response modification factor (R) from 8 to 4–5, while validating an efficient 2DOF framework for accurate bidirectional collapse assessment.

Original authors: Marco F. Garcia, Renzo J. Mejia, Victor I. Fernandez-Davila

Published 2026-08-14

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CC BY 4.0

Original authors: Marco F. Garcia, Renzo J. Mejia, Victor I. Fernandez-Davila

Original paper licensed under CC BY 4.0 (https://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

Imagine you are building a sandcastle on the beach. You know the tide is coming, and you want your castle to survive the waves without needing to be made of solid concrete, which would be heavy, expensive, and boring. So, you build it with a little bit of "wiggle room." You design it so that when a big wave hits, the castle can bend, squish, and maybe even lose a few turrets, but it won't collapse completely. This is the secret sauce of modern earthquake engineering: we don't just build things to be unbreakable; we build them to be tough.

To figure out how much "wiggle room" we can safely give a building, engineers use a special number called the Response Modification Factor (or just R). Think of R as a "discount coupon" for the strength you need to build. If a building is very flexible and can dance through an earthquake without falling, you get a big discount (a high R number), meaning you can use less steel and concrete. If it's stiff and brittle, the discount is small. But here's the tricky part: most of the time, we test these buildings in our computer simulations by shaking them only from the left or only from the right, like a single giant hand pushing a toy car. In the real world, however, the ground doesn't just push left or right; it pushes in a chaotic, swirling dance, shaking the building from both sides at the same time. This paper asks a simple but critical question: Does our "discount coupon" still work when the earthquake is shaking the building from two directions at once?

This research, led by Marco F. Garcia and his team, dives deep into this two-way shaking problem using a method called Incremental Dynamic Analysis. Imagine you have a video game where you can slowly turn up the volume of an earthquake until the building finally breaks. The researchers did this for real-life-style buildings, but they also created a super-fast, simplified version of the building to see if it could predict the breakage just as well as the heavy, slow computer model. They wanted to know if the "discount" (the R factor) needs to be smaller when the ground is shaking in two directions, and if their fast, simplified model could save engineers a massive amount of time.

The team studied four different sizes of reinforced-concrete buildings: a tiny 1-story house, a 2-story home, a 4-story apartment, and a tall 8-story tower. They designed these buildings according to strict Peruvian safety codes, which already assume the buildings can bend and sway. Then, they subjected these digital twins to 30 different pairs of real earthquake records from Peru and Chile. They ran two types of tests: one where the ground shook only in one direction (like a single push) and another where it shook in two directions simultaneously (like a chaotic swirl).

The results were a bit of a wake-up call. When the buildings were shaken from both sides at once, their ability to survive dropped significantly. The researchers found that the "safety margin" (called the Collapse Margin Ratio) shrank by anywhere from 6.6% to 30.1% compared to the single-direction tests. In plain English, the buildings were much more likely to collapse when the earthquake hit them from two angles. Because of this, the "discount coupon" they were using (an R value of 8, which is quite generous) turned out to be too optimistic. To keep the buildings safe under these two-way shaking conditions, the researchers found they had to lower the discount, reducing the R factor to between 4 and 5. This means the buildings would need to be built stronger, with more steel and concrete, to survive the same level of earthquake.

But there was a silver lining, and it came in the form of a clever shortcut. The researchers also built "simplified" versions of these buildings, using a model with just two moving parts (a 2DOF system) instead of the thousands of parts in the full 3D model. They wanted to see if this "lite" version could predict the collapse point as accurately as the heavy-duty simulation. The answer was a resounding yes. For most of the buildings, the simplified model predicted the collapse strength with an error of less than 6%. It was like using a quick sketch to predict exactly when a complex machine would break, and the sketch was almost perfect. This is huge news for engineers because running the full, detailed simulation for every possible earthquake scenario takes hours of computer time, while the simplified model does it in seconds.

The study also looked at how "wobbly" the results were. They found that shorter buildings (1, 2, and 4 stories) were very sensitive to the specific details of each earthquake; some quakes made them fail easily, while others they survived, leading to a wide range of outcomes. However, the tallest building (8 stories) was surprisingly consistent; it didn't matter which specific earthquake hit it, it tended to fail at roughly the same intensity. This suggests that the taller the building, the less it cares about the "personality" of the earthquake and the more it just cares about the raw strength of the shaking.

In the end, this paper tells us that we can't ignore the "two-way dance" of earthquakes. If we keep using the old, high-discount rules, our buildings might be a bit too fragile when the ground starts shaking from all sides. But, thanks to this research, we now have a reliable, super-fast tool (the 2DOF model) that can help us figure out exactly how strong our buildings need to be, ensuring they can survive the real-world chaos of a two-directional earthquake without needing to be built out of unbreakable concrete.

Technical Summary: Influence of the Response Modification Factor on the Bidirectional Seismic Performance of Reinforced-Concrete Buildings

Problem Statement
While the response modification factor ( $R$ ) is a cornerstone of modern seismic design, allowing for the reduction of elastic forces based on inelastic capacity, its calibration has historically relied on simplified unidirectional (UNI) single-degree-of-freedom (SDOF) systems. This approach limits applicability to real-world reinforced-concrete (RC) buildings, which are subjected to simultaneous bidirectional (BID) seismic excitation. Existing literature suggests that BID interaction can significantly alter inelastic demands, reduce plastic rotation capacities, and modify collapse mechanisms compared to UNI analyses. However, a systematic gap remains in quantifying how BID excitation affects the collapse margin of RC buildings designed with current code-prescribed $R$ values, and in validating whether simplified bidirectional models can serve as reliable, computationally efficient substitutes for complex three-dimensional (3D) nonlinear models within a probabilistic collapse assessment framework.

Methodology
This study employs the FEMA P695 methodology and Incremental Dynamic Analysis (IDA) to evaluate the seismic performance of four 3D RC building archetypes (1-, 2-, 4-, and 8-story) designed according to Peruvian seismic regulations (MVCS, 2026). The buildings were subjected to 30 pairs of subduction earthquake ground motions from Peru and Chile, scaled to represent the Maximum Considered Earthquake (MCE).

The analysis proceeded in two parallel tracks:

Detailed 3D Modeling: Nonlinear time-history response analyses (NLTHA) were conducted using OpenSees. The models utilized concentrated plasticity for beams (IMK hysteretic model) and distributed plasticity for columns (fiber-section approach), incorporating bond-slip effects at the base. Collapse was defined by a 10% reduction in lateral stiffness or specific strain limits.
Equivalent 2DOF Modeling: To address computational limitations, equivalent two-degree-of-freedom (2DOF) systems were developed. These systems utilized an elliptical bidirectional yield surface and an IMK-based hysteretic model to capture coupled bidirectional behavior. The 2DOF models were calibrated against the fundamental mode of the 3D structures.

The study evaluated the Collapse Margin Ratio (CMR) and Adjusted Collapse Margin Ratio (ACMR) for both UNI and BID excitations. An iterative process was performed to calibrate the response modification factor ( $R$ ), testing values from 8 down to 4 to determine the minimum $R$ required to meet FEMA P695 acceptance criteria for Performance Groups (PGs).

Key Results

Impact of Bidirectional Excitation: BID seismic excitation consistently reduced the collapse capacity of the RC buildings compared to UNI analyses. The Collapse Margin Ratio (CMR) decreased by 6.6% to 30.1% across the different building heights. Specifically, the 4-story building experienced the most significant reduction (30.1%), while the 1-story building saw a 6.6% reduction.
Performance Group Compliance: Using the code-prescribed design value of $R=8$ , neither Performance Group (PG1: short-period; PG2: intermediate-period) met the FEMA P695 acceptance criteria. PG1 achieved only 93.3% of the required ACMR, and PG2 achieved 90.4%.
Calibration of $R$ : To satisfy the acceptance criteria under BID excitation, the response modification factor required reduction. The study found that compliance was achieved by lowering $R$ to a range of 4 to 5, depending on the structural system. For instance, PG1 required $R=5$ , while PG2 required $R=4$ .
Effectiveness of 2DOF Models: The proposed 2DOF framework successfully reproduced the median collapse capacity ( $S_{a,median}$ ) of the detailed 3D models with high accuracy. For 2-, 4-, and 8-story buildings, the relative errors in $S_{a,median}$ were generally below 6%. The largest discrepancies occurred in the 1-story building at lower $R$ values (underestimation of ~20–25%). However, the 2DOF models showed greater variability in predicting the logarithmic standard deviation ( $\beta$ ), with errors ranging up to 68% for the 8-story building, indicating that while the median capacity is well-captured, the dispersion of collapse fragility is less accurately represented by the simplified model.
Sensitivity to Record-to-Record Variability: The study observed that shorter-period buildings (1S, 2S, 4S) exhibited higher logarithmic standard deviations ( $\beta \approx 0.50–0.74$ ), indicating greater sensitivity to ground motion characteristics. In contrast, the 8-story building showed lower $\beta$ values ( $0.25–0.35$ ), suggesting its collapse capacity is less sensitive to record-to-record variability.

Significance and Claims
The paper claims that neglecting bidirectional seismic interaction leads to an overestimation of the collapse capacity of RC buildings, potentially resulting in unsafe designs if standard unidirectional $R$ factors are applied without adjustment. The primary contribution is the demonstration that for RC buildings in high-seismic zones, the standard $R=8$ is insufficient to meet probabilistic collapse safety targets when BID effects are considered; a reduction to $R=4–5$ is necessary.

Furthermore, the study validates the proposed 2DOF methodology as a computationally efficient tool for preliminary collapse assessment and $R$ calibration. By reducing the computational time per ground-motion record from hours (3D) to seconds (2DOF), the framework enables rapid parametric studies. The authors assert that this approach offers a practical alternative for engineers to perform bidirectional collapse assessments and calibrate response modification factors, reserving detailed 3D IDA analyses for final validation. The findings align with and support a previously proposed formulation for $R$ that explicitly accounts for bidirectional effects, reinforcing the need to integrate bidirectional interaction into seismic design methodologies.

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