Development of fragility curves for tuned mass damper inerter (TMDI) systems under pulse-like ground motions from the 2024 Noto earthquake (Mw 7.6)

Beerabbi Eshapurada, Madhu; Reddy, K. S. K. Karthik

doi:10.3389/fbuil.2025.1746271

ORIGINAL RESEARCH article

Front. Built Environ., 29 January 2026

Sec. Earthquake Engineering

Volume 11 - 2025 | https://doi.org/10.3389/fbuil.2025.1746271

This article is part of the Research TopicResilient and Intelligent Technologies: Advances in Earthquake Engineering and Structural Health MonitoringView all articles

Development of fragility curves for tuned mass damper inerter (TMDI) systems under pulse-like ground motions from the 2024 Noto earthquake (M_w 7.6)

Madhu Beerabbi Eshapurada

K. S. K. Karthik Reddy*

School of Civil Engineering, Vellore Institute of Technology, Vellore, Tamil Nadu, India

Enhancing seismic resilience in the built environment is essential for safeguarding structures in earthquake-prone regions. Inerter-based vibration control systems represent an important advancement in earthquake engineering, yet their effectiveness under pulse-like, near-fault ground motions has not been thoroughly evaluated. This study investigates the seismic response of building structures equipped with tuned mass damper inerter (TMDI) systems when subjected to intense, pulse-like excitations, with reference to the 2024 Noto earthquake in Japan (M_w 7.6). Through comprehensive simulation of low, medium, and high-rise structural models, spanning fundamental periods of 0.5, 1.0, and 2.0 s, the research assesses TMDI performance across varying mass ratios. Results indicate notable reductions in seismic vulnerability and damage potential, with performance gains validated against HAZUS fragility benchmarks for comparable structures demonstrating up to 145% improvement in median PGA for slight damage and 167% for moderate damage states. The findings underscore the potential of TMDIs as scalable, innovative vibration control solutions, contributing to safer and more sustainable built environments in near-fault seismic regions worldwide.

1 Introduction

Frequent earthquakes worldwide have made seismic risk evaluation and mitigation a critical and essential aspect of structural engineering (Kazemi et al., 2024; Nandu et al., 2025). The two important considerations in vibration control are, finding the earthquake response of the structure and improving its resilience to earthquake (Jamshidiha and Yakhchalian, 2019). Various methods have been proposed to evaluate the structures for its seismic effects, including dynamic analysis techniques such as response spectrum analysis and time history analysis (Zentner et al., 2017). These approaches help engineers understand how structures respond to seismic forces under different conditions, thereby allowing them to assess the potential vulnerabilities of buildings and infrastructure. To enhance the resilience against earthquakes, engineers often incorporate vibration control devices into their designs. These devices, such as base isolators and dampers, play a crucial role in mitigating the effects of seismic activity by absorbing and dissipating energy from ground motion (Ma et al., 2021). Effective implementation of vibration control devices not only aids in the design phase but also enhances the retrofitting of existing structures, thereby improving their performance during seismic events (Wu et al., 2022). This integration minimizes risks of life and property, ensuring that buildings can withstand the forces generated by earthquakes while maintaining structural integrity (Chen et al., 2022).

The fundamental types of vibration control systems in structures include passive, active, semi-active, and hybrid systems (Elias and Matsagar, 2017; Çalım et al., 2023). These systems are designed to reduce structural vibrations by absorbing and dissipating vibrational energy from the primary structure through various mechanisms (Li et al., 2024). Each type employs different control strategies and technologies to enhance the structural stability and performance under dynamic loading conditions. Among these systems, passive vibration control systems have received considerable attention for controlling structural vibration (Elias and Matsagar, 2017; Long et al., 2023). This is because passive systems are simpler, cost-effective, require less maintenance compared to the other systems and they do not require an external energy source for activation (Elias and Matsagar, 2017; Jamshidiha and Yakhchalian, 2019; Song et al., 2023). In contrast, the other three systems often include controllable force devices, sensors, and controllers for activation during earthquakes (Elias and Matsagar, 2017; Islam and Jangid, 2021; Ma et al., 2021). Research has been conducted on different types of passive dampers over the years by incorporating various combinations of other passive dampers, including viscous dampers, tuned mass dampers, friction dampers, and many more, for structural response mitigation. This research has led to a better understanding of the effectiveness of passive dampers in reducing structural vibrations and improving the overall safety of buildings and other structures (Málaga-Chuquitaype et al., 2019; De Domenico et al., 2020; Rajana et al., 2023). In addition to component-based damping devices, base isolation has become a significant passive seismic protection strategy (Mazza et al., 2022; Mazza et al., 2024). Base-isolated systems effectively reduce the forces transmitted to the structure by decoupling the superstructure from ground motion. However, when exposed to near-fault pulse ground motions, base-isolated buildings can experience significant displacement demands despite their proven efficiency (Mazza and Labernarda, 2017; Mazza and Labernarda, 2021). While several previous studies have evaluated passive control systems using sets of natural ground motions with diverse characteristics, the specific effects of pulse-like motions remain insufficiently explored (Brando et al., 2015; De Matteis et al., 2018). Such vulnerabilities highlight the importance of investigating the behavior of passive vibration systems under pulse-type of ground motions.

The successful implementation of passive vibration control strategies has been consistently achieved using devices such as the tuned mass damper (TMD), which has been extensively proven to be an effective solution in various applications. The TMD system operates based on the principle of resonance, where the secondary mass is tuned to vibrate at frequencies close to the natural frequency of the primary structure (Li et al., 2023b; Rajana et al., 2023). When the primary structure vibrates at its natural frequency, the secondary mass also vibrates at its natural frequency, creating a force that counteracts the primary vibration. Tuned mass dampers are widely employed as passive dampers in tall structures, high-rise buildings, and bridges across the globe to manage sway and improve stability, especially in regions susceptible to dynamic vibrations. These systems are essential for addressing the resonance and stability challenges of such structures (Xu et al., 2022; Zhang et al., 2023b). Prominent examples of structures equipped with these dampers include the Statue of Unity in India, Taipei 101 in Taiwan, Akashi Kaikyo Bridge in Japan, and Sydney TV Tower in Australia (Larsen and Toubro Limited, 2025; Wang et al., 2019). Tuned mass dampers in the aforementioned structures have proven to be better structural vibration control systems because of their efficiency in vibration control, relatively simple design, and implementation.

However, the TMD system also has some limitations. It is only effective in reducing the amplitude of primary vibrations at specific frequencies, and it may not be effective in reducing the amplitude of vibrations at other frequencies. Another important design consideration in TMD is the mass ratio of the additional tuned mass to the primary structure mass (Kaveh et al., 2020). A higher mass ratio can provide greater energy transfer and more effective vibration reduction, but it also increases the cost, complexity of the system and additional weight in the design of structure (Lazar et al., 2016; Barkale and Jangid, 2022). To overcome these shortcomings, many concepts have been evolved for real-time control of structural vibration based on the actual vibration frequency (Weber et al., 2022). In recent years, there has been a growing interest in using the TMD system in conjunction with other passive damper systems to address the limitations of such systems. For example, the TMD system was used in conjunction with a seismic isolation system to provide additional protection against seismic activity. The TMD system was also used in conjunction with a tuned liquid damper system to provide additional protection against wind-induced vibrations. Research has shown that incorporating an inerter element into tuned mass dampers has become a notable area of focus, and it achieves a reduction in the additional weight that TMDs contribute to structures (Fabrizio et al., 2019; Xu et al., 2023; Akbari et al., 2024; Yue and Han, 2024).

The inerter is a two-node mechanical element, which was first introduced by Smith (2002). The proposed model improves the vibration control performance by using the relative acceleration between its two nodes. The primary function of the inerter element is to create a force directly proportional to the relative acceleration between its two terminals, with the proportionality factor referred to as ‘inertance (b)’ and expressed in kilograms (Smith, 2002; Faraj et al., 2019; Ma et al., 2021; Tiwari et al., 2021). It is capable of generating a substantial inertial mass effect with its small physical mass. It produces inertia because of its rotating mechanism, which makes it harder to resist motion while being lightweight. Research studies indicate that a system incorporating an inerter can replicate a dynamic effect equivalent to 60 to 200 times its physical mass, depending on the dimensions of its flywheel (Papageorgiou and Smith, 2005; Marian and Giaralis, 2014; Giaralis and Taflanidis, 2016). Over the years, many researchers have proposed various inerter configurations and studied their effects on structural vibration control. Inerter dampers are beneficial in buildings, automobiles, and mechanical systems where incorporating significant physical mass and large-size dampers would be unfeasible (Zhang and Cao, 2022). Inerter is effective because it mimics the impact of a heavy mass without actually increasing the physical weight. The inerter damper achieves the same effect as TMDs via rotational mechanics (Marian and Giaralis, 2014; Xu et al., 2019; Li et al., 2023a).

In comparison with conventional TMDs, studies have proven that inerter-based damper systems have better resonance suppression and reduced peak displacements (Giaralis and Taflanidis, 2018; Jangid, 2024). Inerter-based devices have consistently outperformed classic TMDs in comparative studies. For instance (Pietrosanti et al., 2020), showed that, inerter dampers produce greater vibration suppression with lower mass ratios, making them ideal for structures in which adding significant physical mass is impractical or undesirable. In addition, in contrast to TMDs, which are primarily tuned to a single frequency, the mass magnification effect of the inerter allows the system to target multiple vibration modes (Li et al., 2023b). Research findings demonstrate that inerter dampers systems maintain their effectiveness across a variety of structural and excitation uncertainties, including variations in ground motion characteristics, structural properties, and modeling inaccuracies. The robustness of these systems makes them reliable and practical solutions for practical applications (Shi and Zhu, 2018; Zhang et al., 2023b). To alleviate seismic and wind vibrations, these inerter dampers are also being thoroughly studied in MDOF systems. The deployment of multiple tuned inerter dampers (TIDs) across various floors has shown greater improvement in structural performance by reducing peak displacements, inter-story drifts, and base shear forces (Wen et al., 2017; Giaralis and Taflanidis, 2018; Zhang et al., 2023a). Furthermore, extensive research has focused on optimizing the design and performance of inerter systems, particularly in terms of their placement and tuning parameters. Advanced optimization techniques, such as genetic algorithms and particle swarm optimization, and other techniques are employed to identify optimal configurations that maximize the vibration control efficiency (Li et al., 2023c; Elias and Djerouni, 2024). Inerter dampers are remarkably effective at minimizing structural responses under seismic excitation. For example, a study on a 12-story structure equipped with multiple TIDs reported up to 54% reduction in peak displacement and inter-story drift (Lara-Valencia et al., 2024).

Previous research on inerter dampers has primarily focused on optimizing their parameters to enhance their efficiency in structural vibration control (Marian and Giaralis, 2014; Giaralis and Taflanidis, 2018; Wu et al., 2022; Zhang et al., 2023b; Ma et al., 2024; Pandit et al., 2024). However, a comprehensive study of their performance has not been explored extensively. It is essential to understand the efficiency of inerter dampers under various seismic intensities for their effective application. Research into hybrid systems that combine inerters with other damping mechanisms continues to show potential for addressing a wider range of dynamic challenges, especially in areas at risk of significant earthquakes. Additionally, the risk assessment of inerter damper systems in seismic performance plays a critical role in the performance-based seismic design of resilient structures by evaluating how effectively inerter-based dampers reduce vulnerability to varying intensities. Thus, the present study provides a comprehensive vulnerability assessment of an SDOF system that incorporates an inerter element by employing incremental dynamic analysis and fragility curve development to evaluate its performance when subjected to ground motion records.

In this study, an incremental dynamic analysis procedure was used to assess the seismic performance of inerter dampers in an SDOF system to evaluate their behavior under increasing earthquake intensities. The IDA results are then used to generate fragility curves that quantify the likelihood of exceeding various damage states. This approach offers deeper insight into the effectiveness of inerter dampers in limiting damage, thereby aiding in the design of resilient structures and in making informed decision-making for seismic risk assessment. The seismic performance of the inerter-based system was assessed across low, medium, and high-rise structural representations addressing various fundamental periods and dynamic characteristics, in order to enhance the study’s practical relevance and scope. This expanded analysis enables a more comprehensive understanding of the fragility trends and control efficiency of the damper system across various building types.

This study’s main objective is to investigate the seismic fragility of a tuned mass inerter damper system subjected to pulse-like ground motions. To better understand the seismic vulnerability of the proposed system, the damage probability under various inertance ratios was compared. The fragility plots were developed by considering peak relative displacement as an engineering demand parameter, with peak ground acceleration (PGA) and spectral acceleration (Sa (T1, 5%)) as intensity measures. Fragility analysis is a powerful tool for understanding the behavior of structures subjected to ground motions with divergent characteristics (Sharma et al., 2021; Veggalam et al., 2021). The present research mainly examines how the effects of varying mass ratios affect the structural response, considering pulse-like ground motions. Record-to-record variability in pulse-like ground motions is considered to assess the resulting variability in nonlinear structural response across different excitation intensities. By capturing the nonlinear behavior and damage probabilities across different excitation intensities, this study quantifies the potential risk reduction of inerter damper systems and provides insights into their efficacy in structural resilience improvement.

The remaining framework of the paper is organized as follows. Section 2 describes the ground motion selection; classification of the Noto earthquake used in the current study and its characteristics. Section 3 explains the TMD modeling and its validation against the reference study. Section 4 deals with the incremental dynamic analysis for the inerter-based system, followed by the development of fragility curves for different damage states along with the HAZUS median fragility comparison. Discussion of results are presented in Section 5 and followed by conclusion in Section 6.

2 Ground motion selection and classification

Ground motions of magnitude 7.6 (M_w) were selected from Japan’s Noto earthquake that occurred in January 2024 (National Research Institute for Earth Science and Disaster Resilience, 2019). The records were obtained from Japan’s K-NET strong-motion seismograph network (National Research Institute for Earth Science and Disaster Resilience, 2019). To prepare the data for analysis, a high-pass Butterworth filter of 4th order was applied to the velocity time histories to remove the low-frequency noise content and baseline drift. A cutoff frequency of 0.1 Hz was adopted, which was selected to suppress long-period components while preserving the relevant dynamic response of the signal. The Butterworth filter was selected due to its smooth passband response, which helps preserve the original signal shape (Xue et al., 2020). Consequently, this filtering process was necessary to ensure the accuracy of the results. These velocity pulses were extracted using an open-source platform that employed Baker’s pulse classification method (Shahi and Baker, 2014; Papazafeiropoulos and Plevris, 2018; Karthik Reddy et al., 2021; Wani et al., 2024; Kumar et al., 2025). A total of 22 pulse ground motion records were extracted as pulse-like ground motions. Among these, 13 records were from the north-south components and 9 records were from the East-West components. In this classification, the largest velocity pulse was extracted from each record using wavelet transform method. The 4^th-order Daubechies wavelet was used as the mother wavelet in this process. Due to the limited availability of pulse-like ground motions in the east-west direction, this study considers only north south components for subsequent analysis.

Extracted pulse with corresponding station codes are shown in Figure 1, while characteristics of pulse motions are listed in Table 1.

Figure 1

Seismograph charts labeled ISK001 to ISK007 display velocity over time in centimeters per second. Each chart features two lines: black for original ground motion and red for the largest wavelet. Inset graphs emphasize a specific section of the data. Seven graphs display seismic data labeled ISK008, ISK015, NIG014, NIG025, TYM002, TYM005, and TYM006. Each graph shows original ground motion in black and largest wavelet in red, with velocity on the vertical axis and time on the horizontal axis. Inserts highlight specific time segments for detailed analysis.

Figure 1. Largest velocity pulses extracted from Japan K-NET ground motion records corresponding to a magnitude of 7.6 Noto earthquake.

Table 1

Table 1. Parameters of thirteen pulse-like ground motions from the Japan Noto earthquake of magnitude 7.6 (M_w) in the NS direction providing the details of pulse records.

3 Structural modelling and validation

3.1 Tuned mass damper inerter (TMDI)

This research utilized a tuned mass damper with an inerter (TMDI) system to enhance the seismic response of a single-degree-of-freedom (SDOF) structure as discussed in the section 1. The TMDI represents an advancement of the traditional tuned mass damper shown in Figure 2a, incorporating an inerter component that generates a force proportional to the relative acceleration between its two ends. The addition of an inerter improves energy dissipation and increases the effectiveness of dynamic control without significantly increasing the additional system’s physical mass. The TMDI consists of a tuned secondary mass, spring, damper dashpot, and inerter element characterized by its inertance ‘b’ which is shown in Figure 2b. Consequently, the kinetic equations for both the systems are given in Equations 1, 2, respectively (Wu et al., 2022). System tuning was performed at the fundamental frequency of the structure to ensure optimal performance during seismic activity. The mass ratio, defined as the ratio of the auxiliary mass to the primary structure mass, is crucial for evaluating the dynamic performance of the TMDI. To systematically assess its influence, a range of mass ratios was considered, enabling a comprehensive evaluation of the TMDI performance across different configurations.

M_{1} {\ddot{x}}_{1} + C_{1} {\dot{x}}_{1} + K_{1} x_{1} = - M_{1} r_{1} {\ddot{x}}_{g} (1)

M_{2} {\ddot{x}}_{2} + C_{2} {\dot{x}}_{2} + K_{2} x_{2} = - M_{2} r_{2} {\ddot{x}}_{g} (2)

Figure 2

Diagram showing two mechanical systems. Part (a) features two masses, m1 and m2, connected by springs k1 and k2, with damping coefficients C1 and C2. Part (b) adds to the system with additional spring k21 and k22 and a damper b, creating a more complex structure. Both diagrams indicate ground excitation ẋg.

Figure 2. Schematic diagrams illustrating the configurations of passive control systems used in this study: (a) Tuned Mass Damper (TMD) and (b) Tuned Mass Damper Inerter (TMDI) (Wu et al., 2022).

Where,

M 1 = [\begin{array}{c} m_{1} & 0 \\ 0 & m_{2} \end{array}]; C 1 = [\begin{array}{c} c_{1} + c_{2} & {- c}_{2} \\ {- c}_{2} & c_{2} \end{array}]; K 1 = [\begin{array}{c} k_{1} + k_{2} & {- k}_{2} \\ {- k}_{2} & k_{2} \end{array}]

M 2 = [\begin{array}{c} m_{1} & 0 & 0 \\ 0 & m_{2} + b & - b \\ 0 & - b & b \end{array}]; C 2 = [\begin{array}{c} c_{1} & 0 & 0 \\ 0 & c_{2} & - c_{2} \\ 0 & - c_{2} & c_{2} \end{array}]; K 2 = [\begin{array}{c} k_{1} + k_{21} + k_{22} & {- k}_{21} & {- k}_{22} \\ - k_{21} & k_{21} & 0 \\ k_{22} & 0 & k_{22} \end{array}]

x 1 = [\begin{array}{c} x_{1} \\ x_{2} \end{array}]; r 1 = [\begin{array}{c} 1 \\ 1 \end{array}]; x 2 = [\begin{array}{c} x_{1} \\ x_{2} \\ x_{b} \end{array}]; r 2 = [\begin{array}{c} 1 \\ 1 \\ 0 \end{array}]

In Equation 1, the matrices M₁, C_1, and K₁ are the mass, damping, and stiffness matrices of the TMD system. Here, $m_{1}$ , $c_{1}$ , and $k_{1}$ correspond to structural mass, damping, and stiffness, while $m_{2}$ , $c_{2},$ and $k_{2}$ are mass, damping, and stiffness of the attached TMD.

In Equation 2, the matrices M₂, C_2, and K₂ are the mass, damping, and stiffness matrices of the TMDI system. Here, $k_{21} a n d k_{22}$ are the stiffnesses of the springs utilized to connect the inerter element. x₁ and x₂ correspond to the ground displacement, while, ${\ddot{x}}_{g}$ corresponds to the input ground acceleration, and r is the influence factor.

3.2 Numerical model verification

The numerical models developed for the tuned mass damper and tuned mass damper inerter systems were validated against the results presented in (Wu et al., 2022). The structural properties, damper parameters, and ground motion input, which was the El Centro earthquake, were used as specified in the study. Using these models, the time history responses, specifically, the relative displacement of the primary structure and its absolute acceleration, were simulated. The computed responses were compared to those reported in the reference article. The comparisons showed a strong agreement in both displacement and acceleration time histories. Validation plots showing the response comparisons are provided in Figures 3a, 2b to illustrate the close match between the two results. The generated model yielded a maximum relative displacement of 0.1699 m, while the reference study recorded 0.1645 m, resulting in a 3.31% difference. Similarly, the maximum absolute acceleration from the current model is 1.604 m/s², whereas 1.565 m/s² is reported in the reference study, which shows a 2.51% difference. These minor variations demonstrate that the suggested model accurately depicts the dynamic behavior mentioned in the reference study.

Figure 3

Illustration showing two graphs comparing system responses. Graph (a) displays relative displacement in meters over 50 seconds. Graph (b) shows absolute acceleration in meters per second squared over the same period. Both graphs have two lines: a solid blue line labeled

Figure 3. Comparison of time history responses of the developed TMDI model with results from the reference article subjected to El Centro earthquake. (a) Relative displacement of the primary structure (left). (b) Absolute acceleration of the primary structure (right) (Wu et al., 2022).

4 Incremental dynamic analysis and fragility curve development

Incremental Dynamic Analysis is a key approach in performance-based seismic analysis technique used to evaluate seismic performance across a broad range of earthquake intensities (Vamvatsikos and Allin Cornell, 2002; Tidke and Adhikary, 2021). In this method, a set of ground motion records is progressively scaled to multiple intensity levels, and the non-linear structural response is computed for each scaled ground motion. This generates a series of peak response points corresponding to different intensity measures, allowing the construction of IDA curves. While IDA is a robust method, its results can be influenced by the selection and scaling of ground motions, which may introduce variability in demand and collapse predictions. Additionally, the approach requires significant computational effort due to the large number of nonlinear time history analysis (Vamvatsikos and Allin Cornell, 2002). In this study, each pulse-like ground motion record was scaled from 0 to 2 at an increment of 0.1 m/s², ensuring a comprehensive assessment of structural response across a range of seismic intensities. The selection of intensity measures and engineering demand parameters plays an important role in analyzing the seismic performance of structures. In particular, intensity measure selection is very critical because IMs are strongly correlated with earthquake risks and the resulting structural response. There are about 25–30 IMs are proposed in many studies based on their correlation, efficiency, practicality, and proficiency (Jamshidiha and Yakhchalian, 2019; Irslan Khalid et al., 2023; Sun et al., 2023). Peak ground acceleration (PGA) and spectral acceleration at the fundamental time period of the structure (Sa T1, 5%) were considered in this study because these IMs are versatile, effective, and strongly correlated with the structural response (Aghani and Farahmand-Tabar, 2024). The response of the proposed system was then evaluated using the peak relative displacement as the engineering demand parameter.

Fragility curves were subsequently developed from the IDA results to quantify the probability of exceeding specific damage thresholds at a given level of seismic intensity (Bakalis and Vamvatsikos, 2018). As mentioned in previous studies, 10–20 ground motion records are sufficient to estimate seismic demand in structures through incremental dynamic analysis (Bakalis and Vamvatsikos, 2018; Luo et al., 2018; Sharma et al., 2021). In this study, pulse-like ground motions of thirteen records in the NS direction were considered (as mentioned in the Section 2) to develop the fragility curves. The fragility function P () given in Equation 3 is the probability of exceeding a damage state in any structure for a given ground motion intensity measure (IM). This is typically modeled as a lognormal cumulative distribution function $φ$ () (Zentner et al., 2017; Tidke and Adhikary, 2021).

P (E D P \geq {E D P}_{T h} | I M) = φ (\frac{\ln (I M) - \ln (θ)}{σ}) (3)

Here, EDP is engineering demand parameter, EDP_Th is the threshold value of EDP associated with the specific damage limit state, and IM is the intensity measure parameter, which is PGA and Sa (T1, 5%) considered in this study. θ is the median of the fragility function (the IM level with a 50% probability of collapse) and σ is the standard deviation of lnIM (also referred to as the dispersion of IM) (Baker and Eeri, 2015). The combination of IDA and fragility analysis provides a comprehensive understanding of structural performance, enabling probabilistic seismic risk assessment of TMDI systems.

5 Results and discussion

Evaluating the seismic performance of structural systems can be efficiently conducted through fragility analysis, providing a probabilistic approach to determine the likelihood of surpassing predetermined performance thresholds under seismic forces. The performance of an inerter-based tuned mass damper system was assessed using Japan ground motion records from Noto earthquake of magnitude 7.6 (M_w). Investigations have been conducted into the incorporation of an inerter device that generates an inertia force proportional to relative acceleration to assess its impact on enhancing seismic performance. Incremental Dynamic Analysis (IDA), a non-linear analysis method, was used. In the IDA analysis, the structural system was subjected to scaled versions of ground motion records to capture its nonlinear dynamic properties.

IDA curves provide a crucial tool for engineers for assessing the performance of a system under seismic loading. By analyzing the relationship between engineering demand parameters and intensity measures, these curves enable the identification of specific damage states and track the corresponding system responses. This information is then used to construct fragility curves, which offer a probabilistic representation of the system’s likelihood of reaching or exceeding various damage states as a function of seismic intensity.

Fragility curves were derived using lognormal distribution fitting based on the IDA results to represent the variability in structural response associated with record-to-record difference in pulse-like ground motions. It is recognized that seismic performance may also be influenced by various source of uncertainty, including source-to-source variability, uncertainties in structural properties, and uncertainties related to damper properties. These sources of uncertainty are not explicitly considered in the present study and are considered as a part of future scope of work. Previous investigations have shown that uncertainties in damper parameters generally have negligible influence on overall seismic response compared to ground motion variability; therefore, they are often excluded from fragility-based assessments (Vamvatsikos and Cornell, 2004; Masi et al., 2021; Cocco et al., 2024). The consideration of PGA and Sa (T1, 5%) as intensity measures and peak relative displacement as an engineering demand parameter, as discussed in section 4, provides a comprehensive understanding of the structural response under seismic loading. The response spectra for ground motions considered is provided in Figure 4. Employing Sa (T1, 5%) as the spectral acceleration at the fundamental period, this parameter serves as a reliable intensity measure (IM) for fragility plotting. By employing Sa (T1, 5%) in this manner, a direct correlation can be established between the structural response and the dominant characteristics of the ground motion. This approach significantly enhances the reliability and physical significance of the fragility curves, thereby ensuring that they accurately reflect the structural vulnerabilities in the context of pulse-like ground motion events.

Figure 4

Graph depicting spectral acceleration (m/s²) against time period (seconds) for various data sets labeled ISK001 to TYM006 and the mean. Lines show fluctuation peaks around 0.5 to 1 second, then gradually decline. A legend identifies each line.

Figure 4. 5% damped linear response spectra for a set of thirteen selected pulse-like ground motions recorded in the North-South direction along with mean response spectra across all motions.

5.1 IDA responses of TMDI systems under pulse-like ground motions

The IDA plot shown in Figures 5a–c illustrates the seismic performance of low, medium and high-rise structures, each equipped with a TMDI tuned for mass ratios ranging from 0.6 to 1.0 (from top left to bottom center). The x-axis represents the peak relative displacement (EDP), while the y-axis shows the corresponding peak ground acceleration (PGA) levels. Each curve in the plot corresponds to a different pulse-like ground motion record that is subjected to incremental scaling, as discussed in section 4. The plots demonstrate a general trend where the peak displacement increases with higher PGA levels. At lower PGA levels, the response is relatively stable and linear, indicating effective vibration control by the TMDI. However, as the PGA increases, some curves exhibit sudden jumps or steeper slopes, suggesting the nonlinear response or reduced control effectiveness at higher intensity levels. The spread between the individual ground motion curves becomes more noticeable at higher displacements, highlighting record-to-record variability in structural response.

Figure 5

Five line graphs display the relationship between Peak Relative Displacement and Peak Ground Acceleration (PGA) for low-rise structures with Tn equals zero point five seconds, at varying mass ratios: μ equals 0.6, 0.7, 0.8, 0.9, and 1. Each graph includes multiple lines representing different conditions, identified in the legend, indicating varying trends across the graphs. Graphs show the relationship between peak ground acceleration (PGA) and peak relative displacement for medium-rise buildings with a natural period of one second. Five subplots represent different mass ratios (μ) from 0.6 to 1.0. Each graph displays multiple colored lines, corresponding to different ground motion records. Five line graphs show peak ground acceleration (PGA) versus peak relative displacement for different values of μ: 0.6, 0.7, 0.8, 0.9, and 1. Each graph features multiple data series, represented by differently colored lines labeled ISK001 to TYM006, indicating various scenarios. The x-axis shows displacement in meters, and the y-axis shows PGA in meters per second squared. These plots are for high-rise buildings with natural period of two seconds.

Figure 5. Incremental dynamic analysis results showing peak relative displacement responses plotted against peak ground acceleration for TMDI system. Results are presented for mass ratios of 0.6, 0.7, 0.8, 0.9, and 1.0 (from top left to bottom center) for (a) Low-Rise structure, (b) medium-rise structure, and (c) high-rise structure.

The IDA plots of low, medium and high-rise structures presented in Figures 6a–c demonstrate a clear trend as the mass ratio increases from 0.6 to 1.0 (from top left to bottom center), when spectral acceleration (Sa) is used as the intensity measure. The IDA plots become more representative of the structure’s dynamic characteristics, since Sa (T1, 5%) is evaluated at the fundamental period of the structure.

Figure 6

Five graphs display the relationship between spectral acceleration ($Sa$) and peak relative displacement for various models (ISK, NIG, TYM) at different mass ratios (μ = 0.6, 0.7, 0.8, 0.9, 1.0). Each graph shares a legend identifying different line styles for each model. All graphs show a trend of increasing spectral acceleration with higher displacement for low-rise buildings with a natural period ($Tn$) of 0.5 seconds. Five line graphs display the relationship between spectral acceleration and peak relative displacement for different mass ratios (μ = 0.6 to μ = 1.0). Each graph plots multiple datasets labeled from ISK001 to TYM006. The bottom label indicates “Medium-Rise, Tn = 1.0s.” Five line graphs show spectral acceleration versus peak relative displacement for various data series labeled ISK001 to TYM006, at different mass ratios (μ = 0.6, 0.7, 0.8, 0.9, 1.0). Each graph displays a unique range for displacement and acceleration. The analysis focuses on high-rise buildings with a period of Tn = 2.0 seconds.

Figure 6. Incremental dynamic analysis results showing peak relative displacement responses plotted against spectral acceleration for TMDI system. Results are presented for mass ratios of 0.6, 0.7, 0.8, 0.9, and 1.0 (from top left to bottom center) for (a) Low-Rise. correspond to system with increasing mass ratios of 0.6, 0.7, 0.8, 0.9, 1.0 respectively. Results are shown for: (a) low-rise structure, (b) medium-rise structure, and (c) high-rise structure.

5.2 Comparison of fragility curves for TMD and TMDI systems

This section presents a comparative fragility analysis of structures with TMD and TMDI as shown in Figures 7a–c, Figures 8a–c. Using both PGA and Sa (T1, 5%) as intensity measures, fragility curves are developed to estimate the likelihood of exceeding different damage states under increasing seismic intensity. Structural performance levels and drift-based damage thresholds considered in this study for all three structural typologies (low-, mid-, and high-rise) are given in Table 2 as per FEMA 356 guidelines (American Society of Civil Engineers, 2000). By examining performance trends, the analysis seeks to assess the relative efficacy of TMD and TMDI systems across a range of building heights and mass ratios. TMDI systems outperforms conventional TMD when the mass ratio is reduced from 1.0 to 0.6, particularly in low and mid-rise structures. This performance advantage is attributed to the inertial amplification provided by the inerter and enables TMDI systems to maintain control effectiveness under limited mass conditions.

Figure 7

Graphs showing the probability of exceedance versus peak ground acceleration (PGA) for different scenarios of low-rise structures with Tn = 0.5s. Each graph corresponds to different damping mass ratios (μ = 0.6, 0.7, 0.8, 0.9, 1.0) and compares traditional and modified damping devices (TMD and TMDI) across damage states (DS1 to DS4). Curves indicate increasing exceedance probabilities with higher PGA. Five graphs showing the probability of exceedance against peak ground acceleration (PGA) in meters per second squared for medium-rise structures with various damping factors mass ratios (μ = 0.6, 0.7, 0.8, 0.9, 1.0). Each graph compares models TMD and TMDI across four damage states (DS1 to DS4) using different colored lines. The probability increases with higher PGA, with TMDI models generally showing lower probabilities compared to TMD. Five graphs show the probability of exceedance versus peak ground acceleration (PGA) for varying damping ratios mass ratios (μ = 0.6, 0.7, 0.8, 0.9, 1.0) in a high-rise building with Tn = 2.0s. Each graph contains multiple curves for TMD (Tuned Mass Damper) and TMDI (Tuned Mass Damper with Inerter) across four damage states (DS1 to DS4).

Figure 7. Comparison of probability of exceedance curves for TMD and TMDI systems across damage states DS1 to DS4 as a function of peak ground acceleration (PGA). Results are shown for mass ratios ranging from 0.6 to 1.0 for: (a) low-rise structure, (b) medium-rise structure, and (c) high-rise structure.

Figure 8

Graphs showing the probability of exceedance versus spectral acceleration (Sa) in meters per second squared for a low-rise structure with a natural period of 0.5 seconds. Four subplots represent different μ values: 0.6, 0.7, 0.8, and 0.9. Each plot compares TMD and TMDI configurations across four damage states (DS1, DS2, DS3, DS4), indicated by different colored lines. The probability curves for each configuration vary, showing different exceedance likelihoods relative to Sa. Probability exceedance graphs for a medium-rise structure with varying mass ratios (μ = 0.6, 0.7, 0.8, 0.9, 1) show TMD and TMDI models compared across four damage states (DS1 to DS4). The horizontal axis indicates spectral acceleration (Sa) in meters per second squared, while the vertical axis shows the probability of exceedance. Five probability of exceedance graphs for high-rise structures with different mass ratios (μ = 0.6, 0.7, 0.8, 0.9, 1.0) and spectral acceleration (Sa) on the x-axis. Each graph shows curves for various TMD-DS combinations.

Figure 8. Comparison of probability of exceedance curves for TMD and TMDI systems across damage states DS1 to DS4 as a function of Spectral acceleration (Sa). Results are shown for mass ratios ranging from 0.6 to 1.0 for: (a) low-rise structure, (b) medium-rise structure, and (c) high-rise structure.

Table 2

Table 2. Structural performance levels and drift-based damage thresholds.

5.3 Fragility assessment of TMDI for different mass ratios and damage states

Fragility curves were developed for TMDI systems across a range of mass ratios from 0.1 to 1.0. Initial analysis revealed that decreasing the mass ratio from 1.0 to approximately 0.6 consistently led to improved fragility performance, indicating enhanced structural resilience. However, for mass ratios below 0.6, a reverse trend was observed where fragility performance deteriorated with further reductions in mass ratio.

Therefore, in the interest of clarity and to highlight the most effective configurations, fragility plots are presented only for mass ratios ranging from 1.0 to 0.6. This selected range captures the behavior where the TMDI system provides maximum benefit. The observed trend can be attributed to the fact that while higher mass ratios generally improve the energy dissipation and control effectiveness, excessively reducing the mass ratio below a critical threshold reduces the inertial capacity of the TMDI, thereby diminishing its effectiveness against seismic demands. Further, the statistical metrics pertaining to fragility curves, including median values and standard deviations, are presented in Tables 3, 4 for intensity measures PGA and Sa respectively. Each table includes values corresponding to low, medium, and high-rise structures.

Table 3

Table 3. Median and standard deviation values of damage states (slight, moderate, severe and collapse) across mass ratios 0.1 to 1.0 for low, medium and high-rise structures using PGA as intensity measure.

Table 4

Table 4. Median and standard deviation values of damage states (slight, moderate, severe and collapse) across mass ratios 0.1 to 1.0 for low, medium and high-rise structures using Sa (T1, 5%) as intensity measure.

The fragility curves shown in Figures 9a–c have been constructed for low, medium, and high-rise structures equipped with a Tuned Mass Damper Inerter (TMDI) system. These curves utilize Peak Ground Acceleration (PGA) as the intensity measure to represent slight, moderate, severe, and collapse damage states. The illustrations indicate a consistent and progressive decline in the probability of exceedance as the mass ratio diminishes from 1.0 to 0.6 across all three structural types. This trend suggests that a reduction in the mass ratio within this specified range enhances the seismic resilience of the system by mitigating damage probability under increasing ground motion intensities. In the case of low-rise structures, the drop in exceedance probability is particularly steep, especially for slight and moderate damage states, indicating a high sensitivity to inertial damping. Mid-rise structures also demonstrate a clear performance improvement with a decreasing mass ratio, though the gradient of improvement is more gradual. For high-rise structures, a comparable declining trend is observed; however, in the context of the collapse state, an exception is noted at mass ratio 0.7, where the fragility curve flattens slightly, which implies a marginal reduction in the probability of collapse. Overall, the results highlight the robust influence of optimized mass ratio selection in improving seismic performance across different structure types.

Figure 9

Graphs show probability of exceedance for different damage levels (slight, moderate, severe, collapse) of a low-rise structure with a natural period Tn of 0.5 seconds. Each subplot displays curves for various friction coefficients mass ratios (μ = 0.6, 0.7, 0.8, 0.9, 1.0) plotting peak ground acceleration (PGA) against probability. Higher friction coefficients generally indicate higher probabilities at lower PGAs. Four graphs depict the probability of exceedance versus PGA (m/s²) for various damage levels: slight, moderate, severe, and collapse, for a period Tn = 1.0 seconds. Each graph shows curves for different mass ratios (μ = 0.6, 0.7, 0.8, 0.9, 1.0). As the ductility decreases, the probability curves shift downward, indicating lower probabilities of exceedance at higher ductility levels for each damage state. Four graphs show the probability of exceedance against peak ground acceleration (PGA) for different damage levels in high-rise buildings. Each graph represents slight, moderate, severe, and collapse damage. Curves vary based on the mass ratios (μ) from 1.0 to 0.6. As PGA increases, the probability of exceedance rises, with different growth rates for each damage level and μ value.

Figure 9. Fragility curves for TMDI system using peak ground acceleration (PGA) as intensity measure (IM) and peak relative displacement as engineering demand parameter (EDP). The plots show the probability of exceeding slight, moderate, severe, and collapse damage states. Results are shown for mass ratios ranging from 0.6 to 1.0 for: (a) low-rise structure, (b) medium-rise structure, and (c) high-rise structure.

Figures 10a–c represents the fragility curves for low, medium, and high-rise structures respectively, considering spectral acceleration as intensity measure. Each subfigure displays curves corresponding to different mass ratios across four damage states. It shows the general trend of these fragility curves with respect to mass ratio remains consistent with that observed for PGA across all three structural types. However, when spectral acceleration is considered, the curves for all building types exhibit a noticeable leftward shift, indicating that the probability of exceedance reaches higher values at lower intensity levels compared to PGA. This leftward shift is particularly evident in the slight and moderate damage states, signifying that structures display increased vulnerability under spectral acceleration, with damage likely to initiate at lower Sa values. The effect is more pronounced in low-rise and medium-rise buildings due to their sensitivity to higher-frequency ground motions, whereas high-rise structures, while still showing a similar shift, may respond differently due to their dominant period and dynamic characteristics. Overall, the use of Sa as an IM highlights a higher fragility of the system across all structural configurations compared to when PGA is used.

Figure 10

Four line graphs illustrate the probability of exceedance versus spectral acceleration for different damage levels in lowrise buildings: slight, moderate, severe, and collapse. Each graph shows lines for values of μ from 0.6 to 1.0, with probability increasing with higher spectral acceleration. The layout emphasizes how damage severity correlates with acceleration changes. Four graphs show probabilities of building damage types—slight, moderate, severe, collapse—versus spectral acceleration for medium-rise structures. Each graph displays curves for different ductility values (μ = 1, 0.9, 0.8, 0.7, 0.6), illustrating varying probabilities of exceedance as acceleration increases up to 8 meters per second squared. Graphs displaying probability of exceedance versus spectral acceleration for high-rise structures with Tn = 2.0 seconds. Slight, moderate, severe, and collapse damage are represented. Five curves per graph denote different ductility levels (μ): 1, 0.9, 0.8, 0.7, and 0.6, with μ = 1 showing the highest probability of exceedance across all damage levels.

Figure 10. Fragility curves for TMDI system using spectral acceleration (Sa) as intensity measure (IM) and peak relative displacement as engineering demand parameter (EDP). The plots show the probability of exceeding slight, moderate, severe, and collapse damage states. Results are shown for mass ratios ranging from 0.6 to 1.0 for: (a) low-rise structure, (b) medium-rise structure, and (c) high-rise structure.

Similar trends have been reported in previous studies, where the use of Sa, being a structure-specific and period-sensitive intensity measure, tends to more accurately capture the resonant amplification effects in structures, leading to earlier exceedance of damage thresholds. This outcome emphasizes the sensitivity of fragility estimations to the choice of intensity measure and highlights the importance of careful IM selection based on the structural period characteristics and dynamic response features (Mazza and Labernarda, 2017; Irslan Khalid et al., 2023).

5.4 Comparison with HAZUS median fragilities

The median fragility curves from this study are compared with HAZUS median fragilities for corresponding structural types, as shown in Figure 11. The HAZUS median values for PGA are taken from Table-28, and those for spectral acceleration are taken from Table-22, corresponding to the building types considered in this study (Hazus Earthquake Model Technical Manual Hazus, 2020). Figure 1a presents the comparison for low, medium, and high-rise structures across all four damage states using PGA as the intensity measure, while Figure 1b shows the same comparison using Sa (T1, 5%) as the intensity measure. The result demonstrates that incorporating an inerter damper significantly enhances the performance of the TMDI system across all four damage states when PGA is considered as IM. Across all structural heights and damage states, the TMDI system yields higher median PGA values compared to the HAZUS benchmarks. The improvement is especially notable for low-rise and mid-rise structures, where slight and moderate damage states exhibit median PGA values exceeding HAZUS thresholds by substantial margins, indicating enhanced protection against frequent and moderate seismic events. Although improvements are still evident in high-rise structures, they are slightly less pronounced. The peak improvements in median PGA reach up to 145% for slight damage and 167% for moderate damage, particularly around a mass ratio of 0.6, emphasizing the effectiveness of optimized TMDI configurations.

Figure 11

Six line graphs display the relationship between mass ratio and median values for PGA and spectral acceleration at different periods (Tn = 0.5, 1.0, and 2.0 seconds). Each graph compares four data sets (TMDI - DS 1 to 4) with distinct colored lines and markers. Horizontal dashed lines indicate reference levels for different HAZUS DS. The graphs in columns represent PGA and spectral acceleration for each period. The x-axis shows mass ratio (μ), and the y-axis shows the respective median values. Legend differentiates the data sets by color and style.

Figure 11. Comparison of TMDI-based fragility curves with median fragility curves defined in HAZUS for low-rise, mid-rise and high-rise concrete structures. (a) Peak ground acceleration (PGA) as intensity measure (first row). (b) Spectral acceleration (Sa) as intensity measures (second row).

However, when Sa is used as the IM (Figure 11b), the fragility results show a different trend. Except for the slight damage state in low- and medium-rise structures, the TMDI curve lies below the HAZUS median values across all structural types and damage states. This pattern indicates that, under spectral acceleration demands, TMDI configurations show comparatively higher sensitivity to ground motion intensity than HAZUS baseline predictions. Furthermore, the consistently higher median values of HAZUS demonstrate the more conservative nature of the HAZUS fragility curves relative to those derived for TMDI-equipped systems. Despite this shift in trend, the Sa-based assessment still shows significant performance improvements. The TMDI system produces a prominent increase in median capacity, with the largest improvements reaching 136% in the slight damage state and 60% in the moderate damage state when spectral acceleration is considered as the IM. Overall, these findings confirm that TMDI systems enhance seismic resilience, particularly when evaluated using PGA. Nonetheless, the contrast in performance when Sa-based assessment emphasizes the need for further study into IM selection and its influence on fragility behavior in performance-based seismic design.

6 Conclusion

In the present investigation, a tuned mass damper inerter (TMDI) system was modeled to evaluate its efficacy in improving its structural performance subjected to pulse-like ground motions excitations. The study encompassed low, medium, and high-rise structures to assess how the performance of the TMDI system varies with building height. Initially, a non-linear time history analysis IDA was conducted to calculate the collapse capacities of the system across a range of mass ratios. Utilizing the resultant peak relative displacement responses, a performance-based seismic assessment was executed through fragility analysis. The probability of exceedance for four different damage states was computed and fitted using cumulative distribution function. Ultimately, for evaluation and comparison the median fragility values derived from the analysis were juxtaposed with fragility metrics defined by HAZUS. Based on the findings, the following conclusions are drawn:

1. Comparisons of fragility curves show that TMDI systems outperform conventional TMDs when the mass ratio is reduced from 1.0 to 0.6, particularly in low and mid-rise structures, owing to the enhanced inertial effect provided by the inerter damper.

2. The TMDI system was found to enhance seismic resilience across all structural types, with fragility curves showing consistent reduction in the probability of damage exceedance as the mass ratio decreased from 1.0 to 0.6. The most significant improvements were observed at a mass ratio of 0.6, though a local irregularity was noted at 0.7 in the collapse state, suggesting sensitivity to dynamic response characteristics.

3. Fragility curves using PGA as the IM exhibit substantial increase in median values across all damage states when compared to HAZUS median fragilities. At mass ratio 0.6, improvement reached up to 145% for slight damage and 167% for moderate damage, indicating that TMDI systems significantly outperform conventional expectations for seismic performance.

4. Compared to the HAZUS median fragilities for similar structure, the TMDI system delivers improved performance suggesting that TMDI equipped structures can achieve higher seismic resilience than predicted by conventional empirical models.

5. When Sa (T1, 5%) was used as the IM, fragility curves consistently shifted toward lower Sa values, indicating earlier exceedance probabilities. This shift observed across all structural types and suggests increased vulnerability under spectral acceleration, highlighting the sensitivity of fragility outcomes to the choice of IM.

6. While TMDI configurations tend to show higher fragility under Sa-based assessments relative to HAZUS thresholds, a mass ratio of 0.7 yielded notably improved performance for DS1, DS2, and DS4. This highlights the opportunity for parameter tuning to improve TMDI effectiveness under Sa demands.

7. The findings show that the TMDI system significantly enhances capacity, with maximum improvements of 136% at slight damage and 60% at moderate damage when spectral acceleration is considered as an intensity measure.

8. Overall, the study confirms that TMDI equipped structures offer superior seismic resilience under PGA-based assessments across various structural types. However, the discrepancy observed for Sa as IM underscores the importance of carefully selecting IMs when evaluating seismic fragility and designing TMDI systems for performance-based seismic design.

Data availability statement

The original contributions presented in the study are included in the article/Supplementary Material, further inquiries can be directed to the corresponding author.

Author contributions

MB: Conceptualization, Validation, Methodology, Data curation, Writing – original draft. KR: Supervision, Investigation, Conceptualization, Writing – review and editing, Validation.

Funding

The author(s) declared that financial support was not received for this work and/or its publication.

Conflict of interest

The author(s) declared that this work was conducted in the absence of any commercial or financial relationships that could be construed as a potential conflict of interest.

Generative AI statement

The author(s) declared that generative AI was not used in the creation of this manuscript.

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Supplementary material

The Supplementary Material for this article can be found online at: https://www.frontiersin.org/articles/10.3389/fbuil.2025.1746271/full#supplementary-material

Footnotes

^{Abbreviations:}TMD, Tuned mass damper; TMDI, Tuned mass damper inerter; IDA, Incremental dynamic analysis; PGA, Peak ground acceleration; Sa (T1, 5%), Spectral acceleration at fundamental period of structure T1, with 5% critical damping; M_w, Moment magnitude; N-S, North-South; E-W, East-West; SDOF, Single-degree-of-freedom; IM, Intensity measure; PGV, Peak ground velocity; R_jb, Joyner-Boore distance; T_p, Pulse period; C1H, High-rise concrete moment frame; DS, Damage State; μ, Mass ratio; θ, Median; σ, Standard Deviation; T_n, Natural time period of primary structure; PI, Pulse Indicator; R_epi, Epicentral Distance.

References

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