# Topological Optimization and Robust Performance Zones of Rotational Friction in Seismic Retrofit: A Probabilistic Non-Smooth Dynamics Approach

Emiliano Ponce L, Miguel A. Perez Lara Y Hernandez, Humberto Yáñez-Godoy, L. Francisco Pérez-Moreno, Ivan Fermin Arjona Catzim

<figure><img src="image_1.png" /></figure>

## ► To cite this version:

Emiliano Ponce L, Miguel A. Perez Lara Y Hernandez, Humberto Yáñez-Godoy, L. Francisco Pérez-Moreno, Ivan Fermin Arjona Catzim. Topological Optimization and Robust Performance Zones of Rotational Friction in Seismic Retrofit: A Probabilistic Non-Smooth Dynamics Approach. Current Journal of Applied Science and Technology, 2026, 45 (1), pp.150-170. hal-05472825

HAL Id: hal-05472825

https://hal.science/hal-05472825v1

Submitted on 23 Jan 2026

HAL is a multi-disciplinary open access
archive for the deposit and dissemination of sci-
entific research documents, whether they are pub-
lished or not. The documents may come from
teaching and research institutions in France or
abroad, or from public or private research centers.

L'archive ouverte pluridisciplinaire HAL, est
destinée au dépôt et à la diffusion de documents
scientifiques de niveau recherche, publiés ou non,
émanant des établissements d'enseignement et de
recherche français ou étrangers, des laboratoires
publics ou privés.

<figure><img src="image_2.png" /></figure>

Distributed under a Creative Commons CC BY 4.0 - Attribution - International License
<figure><img src="image_3.png" /></figure>

# Topological Optimization and Robust Performance Zones of Rotational Friction in Seismic Retrofit: A Probabilistic Non-Smooth Dynamics Approach

Emiliano Ponce L<sup>a*</sup>, Miguel A. Perez Lara y Hernandez<sup>a</sup>, Humberto Yáñez-Godoy<sup>b</sup>, L. Francisco Pérez-Moreno<sup>a</sup> and Ivan Fermin Arjona Catzim<sup>a</sup>

## Authors' contributions

This work was carried out in collaboration among all authors. Authors EPL, HYG, MAPLyH, LFPM and IFAC contributed equally to this work in conceptualization, methodology, validation, formal analysis, investigation, data curation, writing, review and editing. All authors read and approved the final manuscript.

## Article Information

DOI: https://doi.org/10.9734/cjast/2026/v45i14658

### Open Peer Review History:

This journal follows the Advanced Open Peer Review policy. Identity of the Reviewers, Editor(s) and additional Reviewers, peer review comments, different versions of the manuscript, comments of the editors, etc are available here: https://pr.sdiarticle5.com/review-history/149711

Original Research Article

Received: 18/11/2025

Published: 22/01/2026

<footnote><sup>a</sup> Universidad Autónoma de Querétaro, Facultad de Ingeniería, Cerro de Las Campanas s/n, Ciudad Universitaria, Querétaro, Qro., México.</footnote>

<footnote><sup>b</sup> University of Bordeaux, Arts et Metiers Institute of Technology, CNRS, Bordeaux INP, I2M, UMR 5295, F-33400, Talence, France</footnote>

<footnote>*Corresponding author: E-mail: Eponce08@alumnos.uaq.mx;</footnote>
Abstract

This study aims to identify optimal topological configurations of Rotational Friction Dampers (RFDs) for the seismic retrofit of a 10-story shear building, using a probabilistic performance-based approach. The non-linear stick-slip behavior of RFDs was modeled using the Non-Smooth Dynamics method, integrated within a Monte Carlo framework to account for uncertainties in both design parameters and seismic excitations. The sensitivity of the seismic response was analyzed across a broad design space, evaluating randomized configurations subjected to five diverse seismic excitations. Statistical analysis revealed that the quantity and slip load capacity of RFDs are the governing parameters, with structural performance improving as both increased, converging towards high frictional moments (350–450 kN·m). However, a distinct saturation point was identified at 7 dampers, beyond which the system exhibits diminishing returns. Topological analysis further demonstrated that damper installation in the lower stories is critical to control the soft-story mechanisms, while upper levels allow for greater topological flexibility. The study concludes that engineers should target "robust performance zones" rather than a singular optimal configuration, providing a flexible design framework that accommodates seismic uncertainty and enhances structural resilience.

Keywords: Seismic risk mitigation; energy dissipation devices; steel moment-resisting frames; computational structural dynamics; structural resilience.

1. Introduction

The accelerated growth of the population in urban centers has led to the need for vertical development as a solution to social demand (Montejano-Castillo et al., 2020). Inevitably, this development entails a series of structural challenges, among which the effects on structures subjected to seismic loads must be addressed; therefore, strategies are required that reduce structural demand and increase resilience under severe and variable excitations, while always seeking to save material resources without compromising occupant safety (Perez-Martinez et al., 2025).

Among current solutions, the use of passive control systems has gained particular interest thanks to their low manufacturing cost and the simplicity of installation and maintenance (Titirla, 2024). Among the multiple passive control systems, friction dampers stand out; these have been widely studied, and multiple solutions can be found in the literature (Artar & Carbas, 2022; Mousavi et al., 2023; Mualla & Belev, 2002; Z. Zhang et al., 2023).

One of the devices belonging to this category is the rotational friction damper (RFD), due to its stable hysteretic behavior and post-seismic resistance, stands out as a viable solution (Mualla & Belev, 2002), as experimental tests have shown a significant reduction in hysteretic energy demand on the primary structural system.

The performance of these devices depends largely on the number of devices installed, their location, and their resisting force; in turn, the evaluation criteria depend on reductions in maximum relative displacements, maximum acceleration, and base shear of the analyzed structure (Wong & Johnson 2009; Mualla & Belev, 2002). However, other studies suggest complementary metrics depending on the type of damper as well as the structural system analyzed (Artar & Carbas, 2022; Gao et al., 2023; Zhao et al., 2022).

On the other hand, different studies have highlighted the need for a model capable of faithfully representing the nonlinear behavior inherent to RFDs; for this reason, the concept of Non-Smooth Dynamics (NSD) has been developed and integrated into time-history analysis, which is necessary to evaluate the behavior of these devices.

Currently, strategies for the design and implementation of dampers in buildings in general result in models with low efficiency or high computational demand (Aydin, 2013; Jarrahi et al., 2020). In addition, they adopt a deterministic approach, resulting in unique solutions for a given set of input parameters (Ampiah & Zhao, 2020; Jarrahi et al. 2020; R. Zhang & Soong, 1992). However, recent studies have demonstrated the existence of groups of equivalent solutions (Aydin, 2013; Namorado et al., 2022).
Therefore, this research sets as its object of study the relationship between the location and number of RFDs with respect to the performance criteria proposed by (Mualla & Belev, 2002) and (Namorado et al., 2022), with the aim of identifying systematic placement trends within the structural system.

To address this question, a probabilistic framework is proposed that integrates the use of a time-history analysis algorithm incorporating NSD logic, a broad topological exploration of configurations, and an evaluation under different seismic records, to identify robust zones and transferable design rules.

In particular, this work proposes a procedure based on NSD and Monte Carlo-type simulation applied to a benchmark 10-story shear-building structure, with the purpose of: (a) mapping performance in the joint space of quantity-location-capacity, (b) identifying robust performance zones through balanced criteria, and (c) identifying trends regarding preferred installation locations and saturation thresholds. The remainder of the article is organized as follows: Section 2 describes the device model and the numerical-statistical framework employed; Section 3 presents the results and their discussion; finally, Section 4 summarizes the conclusions and design implications.

## 2. Materials and Methods

This section details the principles and configuration of the RFD, as well as the NSD method for its numerical evaluation. It also presents the numerical framework developed to evaluate the robust placement of friction

dampers. The section describes the benchmark structure, the Monte Carlo statistical model used in this study, and finally, the procedure adopted to obtain the results.

### 2.1 Numerical Modeling of Rotational Friction Dampers

The RFD illustrated in Fig. 1 is modeled using an ideal Coulomb friction law, as shown in Fig. 7. Unlike viscous dampers, an RFD exhibits frequency-independent but highly nonlinear behavior due to the stick-slip phenomenon, as depicted in Fig. 2. To accurately capture this nonlinear behavior, the Non-Smooth Dynamics (NSD) method is employed, following the approach developed and validated in (Namorado et al., 2022).

According to (Namorado et al., 2022), the effects of the RFD on structural response modify the formulation of the general equation of motion given in Eq. (1), leading to the inclusion of the terms W and $Q_f$, resulting in Eq. (2).

$$M \Delta a + C \Delta v + K \Delta u = -M \ddot{s}(t) \quad (1)$$

$$M \Delta a + C \Delta v + K \Delta u + W Q_f = -M \ddot{s}(t) \quad (2)$$

Where M, C, K are the mass, damping, and stiffness matrices, respectively, and $\Delta a, \Delta v, \Delta u, \ddot{s}(t)$ represent acceleration, velocity, displacement, and seismic ground acceleration, respectively. The inclusion of the terms W and $Q_f$ corresponds to the coupling matrix and the generalized force vector, which is expressed in Eq. (3):

<figure><img src="image_4.png" /><figcaption>Fig. 1. Components of RFD (Mualla & Belev, 2002)</figcaption></figure>
<figure><img src="image_5.png" /><figcaption>Fig. 2. Principle of action of RFD (Mualla & Belev, 2002)</figcaption></figure>

$$Q_f = \begin{cases} -M_{f1}\text{Sign}(\theta_1') \\ -M_{f2}\text{Sign}(\theta_2') \\ \vdots \\ -M_{fn}\text{Sign}(\theta_n') \end{cases} \qquad (3)$$

Thus, the effects introduced by the damper are incorporated through the term $-M_{fi} \text{Sign}(\theta_i')$, which is later expressed more conveniently for practical purposes in terms of the resisting force $F_{RFD,j}(t)$.

The resistive force, $F_{RFD,j}(T)$, generated by the damper installed at the j-th story is defined by the relative velocity between adjacent floors, $(\dot{u}_j - \dot{u}_{j-1})$. This relationship is mathematically described in Eq. (4) using the signum function as follows:

$$F_{RFD,j}(T) = -F_{SLIP,j} \cdot \text{Sign}(\dot{u}_j - \dot{u}_{j-1}) \qquad (4)$$

where $F_{SLIP,j}$ is the slip force capacity, related to the frictional moment $M_f$ and the lever arm $H_A$ by the relationship: $F_{SLIP,j} = \frac{M_f}{h_a}$. The state of the damper is determined at each time step: if the external force is lower than $F_{SLIP,j}$, the damper remains in a "stick" state; otherwise, it enters a "slip" mode, dissipating energy through a rectangular hysteretic loop as shown in Fig. 3.

## 2.2 Performance criteria

In order to establish a reference of the performance of the RFD, Mualla defines a criterion (Mualla & Belev, 2002), described as follows in Eq. (5) – (9):

$$R_1 = \frac{X_{NfMax}}{X_{N0Max}} \qquad (5)$$

$$R_2 = \frac{\dot{X}_{NfMax}}{\dot{X}_{N0Max}} \qquad (6)$$

$$R_3 = \frac{F_{bf}}{F_{b0}} \qquad (7)$$

$$CR_1 = \sqrt{R_1^2 + R_2^2} \qquad (8)$$

$$CR_2 = \sqrt{R_1^2 + R_2^2 + R_3^2} \qquad (9)$$

Where $X_{NfMax}$ denotes the peak displacement of the upper (N-th) floor relative to the ground, $\ddot{X}_{NfMax}$ is the peak absolute acceleration of the upper floor, and $F_b$ is the peak seismic base shear. Subscripts $X_{N0Max}$, $\ddot{X}_{N0Max}$, $f_0$ refer to the frictionally undamped structure cases. Therefore, $R_1$, $R_2$, and $R_3$ represent normalized peak-response ratios.

It must be considered that minimizing drift often penalizes accelerations or base shear, two combined efficiency criteria $CR_1$ and $CR_2$ are used to identify equilibrated designs that simultaneously control multiple response measures rather than optimizing a single metric.

## 2.3 Benchmark Structural Model

To evaluate the efficiency of the proposed optimization framework, a ten-story benchmark shear-building model was adopted (Fig. 4), originally reported by Wong and Johnson (Wong & Johnson, 2009). This structure represents a medium-rise building and is modeled with rigid floor diaphragms and lumped floor masses, resulting in a reduced number of degrees of freedom compatible with the NSD-based time-integration method.
<figure><img src="image_6.png" /><figcaption>Fig. 3. Hysteretic loop of RFD with Mf = 200 N-m</figcaption></figure>

The structural properties are characterized by a non-uniform stiffness distribution along the height but a constant floor mass (Jarrahi, Asadi, Khatibinia, Etedali, et al., 2020; Wong & Johnson, 2009). Each floor mass is taken as 218.9 tons. Inherent structural damping is represented through Rayleigh damping, calibrated to achieve a 3% damping ratio in the first two vibration modes. The story stiffnesses $k_i$ for each floor are defined to reproduce the dynamic characteristics of the reference benchmark, using an equivalent inverted pendulum model. To evaluate the robustness of the ensuing conclusions under stiffness uncertainty, two additional configurations are considered, identical to the benchmark except for a uniform scaling of all story stiffnesses by -30% and +30%, respectively.

In order to conduct a validation of the use of NSD, the benchmark model seismic response was compared with the research (Jarrahi, Asadi, Khatibinia, Etedali, et al., 2020), resulting in the nearly identical lateral displacement profile. As can been compared in Fig. 5 a) and b). Also, a time history analysis was conducted on SAP2000 with the 1940 earthquake El Centro. The results, presented in Fig. 6, demonstrate that the proposed NSD algorithm produces an almost identical response to the commercial software, confirming the accuracy of the integration scheme.

## 2.4 Probabilistic Optimization Framework

To study the influence of damper quantity and story-wise placement while accounting for record-to-record variability, a Monte Carlo simulation (MCS) framework was adopted to explore the global response space of the

benchmark structure. Each damper layout (topology) is represented by a binary vector $z \in \{0,1\}^{10}$, where $z_i = 1$ indicates the presence of an RFD at story i and $z_i = 0$ its absence. For a ten-story structure, this yields $2^{10} = 1024$ admissible topological configurations.

To keep the study computationally tractable yet statistically representative, 280 distinct configurations were selected via uniform random sampling from the 1024 possible combinations. The SBM was analyzed under three stiffness scenarios, a base case, and two additional cases obtained by scaling globally a ±30%. For each topology, the friction moment $M_f$ was swept from 25 to 500 kN-m in increments of 25 kN-m, covering regimes from low-capacity to near-locked behavior. Each triad ($z, M_f, K$) was evaluated using five unscaled earthquake records: El Centro (1940), Kobe (1995), Loma Prieta (1989), Northridge (1994), and Trinidad. The integration time step was $\Delta t = 0.01$ s for Kobe, Loma Prieta, Northridge, and Trinidad, and $\Delta t = 0.02$ s for El Centro.

For each analysis, the nonlinear equation of motion (Eq. 2) was integrated using a Non-Smooth Dynamics (NSD) solver implemented in Python and accelerated with Numba, following the NSD formulation presented in (Namorado et al., 2022). Overall, the experimental campaign comprises 84,000 nonlinear time-history analyses, producing a comprehensive database spanning damper quantity, placements, friction capacities $M_f$, and seismic excitations. The statistical post-processing and comparative assessments derived from this dataset are presented in Section 3.
<figure><img src="image_7.png" /><figcaption>Fig. 4. Ten-story benchmark shear-building model (SBM)</figcaption></figure>
<figure><img src="image_8.png" /><figcaption>Fig. 5. Relative displacement profile of 10 story SBM subjected to Northridge a) analyzed with NSD b) analyzed with open sees</figcaption></figure>

<figure><img src="image_9.png" /><figcaption>Fig. 6. Comparison of Time history analysis of El Centro Earthquake with NSD method and SAP2000</figcaption></figure>
# 3. Results and Discussion

## 3.1 Influence of Damper Quantity on Seismic Performance

Figs 7 to 11 present box-and-whisker plots illustrating the relationship between the number of friction dampers and the performance coefficients ($R_1, R_2, R_3, CR_1$, and $CR_2$). Each plot includes red dashed lines that serve as critical reference benchmarks:

* For $R_1, R_2, R_3$: The line at ratio = 1 ("No reduction") represents the performance of the structure without any improvement from dampers, indicating that the damped response is equivalent to the undamped case.

* For $CR_1$: The line at 1.414 ("No reduction" baseline) represents the baseline performance where the combined effect of displacement and acceleration reductions equals that of the undamped structure.

* For $CR_2$: The line at approximately 1.73 ("Uncontrolled") represents the performance of an uncontrolled structure under seismic excitation, serving as an upper bound for ineffective configurations.

Across all plots, a clear trend emerges: seismic performance improves with an increasing number of dampers. The median values of the

performance coefficients consistently show enhancement as the number of dampers increases, indicating that, on average, adding more dampers improves the seismic performance of the structure.

However, significant variability and outliers are notably present, particularly in the range of 3 to 7 dampers. This variability suggests that factors beyond mere quantity—specifically the topological arrangement and the slip load capacity ($M_f$)—play a critical role in the structural response. Configurations with 3 to 7 dampers show a wide performance range and extreme outliers, indicating a high sensitivity to both the placement of dampers and the frictional moment capacity. This highlights that certain configurations can underperform despite having an adequate number of devices.

In contrast, configurations with 8 or more dampers exhibit more robust and stable performance, with less variability and fewer outliers. This indicates that beyond 7 dampers, the system reaches a saturation point where the addition of more dampers has a marginal effect on performance. At this stage, the exact arrangement becomes less critical, suggesting that the system achieves a state of robustness where performance is less sensitive to specific variations in damper placement.

<figure><img src="image_10.png" /><figcaption>Fig. 7. Distribution of roof displacement ratio ($R_1$) as a function of the number of dampers</figcaption></figure>
<figure><img src="image_11.png" /><figcaption>Fig. 8. Distribution of roof acceleration ratio ($R_2$) as a function of the number of dampers</figcaption></figure>

<figure><img src="image_12.png" /><figcaption>Fig. 9. Distribution of base shear ratio ($R_3$) as a function of the number of dampers</figcaption></figure>

<figure><img src="image_13.png" /><figcaption>Fig. 10. Distribution of combined displacement/acceleration index ($CR_1$) as a function of the number of dampers</figcaption></figure>
<figure><img src="image_14.png" /><figcaption>Fig. 11. Distribution of global performance index (CR₂) as a function of the number of dampers in top 5% topological configurations</figcaption></figure>

## 3.2 Performance of Top 5% Configurations

To verify if the trend of improved performance with an increasing number of dampers persists under optimal conditions, the top 5% of the best-performing configurations were isolated, as shown in Figs 12 and 13. These figures present box-and-whisker plots illustrating the distribution of the performance indices $CR_1$ and $CR_2$ as a function of the number of dampers. Two red dashed lines are included to provide reference benchmarks: the line at 1.414 ("No reduction") represents the baseline performance of the structure without any dampers, indicating no improvement in seismic response, while the line at approximately 1.73 ("Uncontrolled") represents the performance of the uncontrolled structure under seismic excitation, serving as an upper bound for ineffective configurations.

The elite subset of configurations demonstrates a clear trajectory of performance enhancement with increasing damper count. However, a notable stabilization occurs once the quantity reaches 7 dampers, beyond which further increases yield only marginal improvements. This plateau in performance suggests that the system

reaches a state of robustness at around 7 dampers, where it becomes less sensitive to specific variations in damper placement. The exact arrangement of dampers has a reduced impact on the overall seismic performance, indicating that configurations with 7 or more dampers are inherently more stable and reliable.

The median values of $CR_1$ and $CR_2$ for configurations with 7 or more dampers fall well below the "No reduction" baseline, indicating a significant improvement in seismic performance. This demonstrates that such configurations consistently achieve performance levels that effectively mitigate seismic response. Conversely, configurations with fewer than 7 dampers exhibit a wide performance range with extreme outliers, often approaching or exceeding the "Uncontrolled" benchmark. This indicates a high dependency on precise placement and frictional moment capacity to avoid ineffective designs, underscoring the critical importance of careful optimization when working with a limited number of dampers. Small changes in their arrangement can lead to significant differences in seismic performance, making the optimization process crucial for configurations with fewer than 7 dampers.
<figure><img src="image_15.png" /><figcaption>Fig. 12. Distribution of combined displacement/acceleration index (CR₁) as a function of the number of dampers in top 5% topological configurations</figcaption></figure>

<figure><img src="image_16.png" /><figcaption>Fig. 13. Distribution of global performance index (CR₂) as a function of the number of dampers</figcaption></figure>

### 3.3 Influence of Damper Location

The specific influence of damper location on seismic performance is examined through Figs. 14 and 15, which present Spearman correlation coefficients between damper presence at each story and the performance metrics CR₁ and CR₂. These correlations are calculated separately for the global dataset and the top 5% elite configurations, providing insights into the relationship between damper placement and structural performance.

The Spearman correlation coefficient (ρ) measures the monotonic relationship between damper presence at a specific story and the

performance metrics. Negative values indicate that damper presence at a particular story is associated with improved performance, while positive values suggest that damper presence may be associated with less effective performance. The magnitude of the correlation indicates the strength of this relationship.

For the global dataset (all configurations):

* Figure 14 (CR₁): Strong negative correlations are observed at stories 1, 4, 5, and 6, with story 1 showing the most pronounced negative correlation. This suggests that damper placement at these stories generally
improves performance in terms of combined displacement and acceleration reduction.

* Fig. 15 (CR<sub>2</sub>): Strong negative correlations extend to stories 1, 4, 5, 6, and 9, with story 1 again showing the strongest effect. The inclusion of story 9 suggests that damper placement at this upper story may also contribute to improved performance when considering the additional factor of base shear.

For the top 5% elite configurations:

* Fig. 14 (CR<sub>1</sub>): Negative correlations are more pronounced and extend to stories 1, 2, 3, 5, and 6. This indicates that in the most effective configurations, damper placement at these lower and middle stories is consistently associated with improved performance. Conversely, stories 7 to 10 present positive Spearman correlation coefficients, suggesting that damper placement at these upper levels may be less beneficial or even detrimental to performance when considering only displacement and acceleration metrics.

* Fig.15 (CR<sub>2</sub>): Negative correlations are observed at stories 1, 2, 3, 4, 5, and 6, reinforcing the importance of lower and middle stories. Interestingly, stories 9 and 10 show positive correlations, suggesting that in these elite configurations, damper placement at the uppermost stories may be less beneficial or even detrimental to performance when considering all three

metrics (displacement, acceleration, and base shear).

The transition from negative to positive correlations in the upper stories (particularly stories 9 and 10 in the CR<sub>2</sub> elite configurations) highlights an important design consideration. While lower stories consistently benefit from damper placement, the upper stories may require more careful consideration, with their effectiveness depending on the specific performance metrics being optimized and the overall damper configuration. These correlation patterns suggest that:

1. Stories 1, 2, 3, 5, and 6 are consistently the most effective locations for damper placement, particularly for the elite configurations.

2. Story 1 plays a particularly critical role in improving seismic performance, likely due to its position at the base of the structure and its influence on the soft-story mechanism.

3. The effectiveness of damper placement in upper stories (7-10) is more variable and depends on the specific performance metrics being considered.

4. The transition from negative to positive correlations in upper stories for elite configurations suggests that optimal damper placement strategies may need to avoid or carefully consider placement in these upper levels when aiming for comprehensive performance improvement across all metrics.

<figure><img src="image_17.png" /><figcaption>Fig. 14. Story-wise influence of damper placement on CR<sub>1</sub>: Spearman correlation for the global set and the Top 5% configurations</figcaption></figure>
<figure><img src="image_18.png" /><figcaption>Fig. 15. Story-wise influence of damper placement on $CR_2$: Spearman correlation for the global set and the Top 5% configurations</figcaption></figure>

## 3.4 Optimal Performance Zones

Figs 16 and 17 summarize the performance relative to both damper quantity and slip load capacity ($M_f$). These contour plots reveal a clear "valley" of optimal performance, concentrated around configurations with 8 dampers and slip loads ranging from 350 to 500 kN·m, where the performance metrics $CR_1$ and $CR_2$ reach their minimum values, indicating the most effective reduction of seismic response. Detailed analysis of the performance contours reveals several key observations:

* For $CR_1$ performance (Fig. 16):

* The optimal performance zone (dark green area) is clearly centered around 8 dampers with $M_f$ values between 350-500 kN·m.

* Configurations with fewer than 5 dampers consistently show poor performance (yellow/orange/red zones) regardless of the slip load capacity.

* Performance significantly degrades when slip loads exceed optimal ranges, particularly for configurations with fewer than 3 dampers.

* For $CR_2$ performance (Fig. 17):

* A similar optimal zone emerges, but with minimum $CR_2$ values achieved with 8 dampers and slip loads in the 350-450 kN·m range.

* The performance degradation becomes more pronounced for configurations with fewer than 6 dampers.

These results demonstrate that global optimization is controlled by performance regions

rather than a single deterministic configuration. The identified performance valley indicates that multiple combinations of damper number and slip load capacity lead to near-optimal seismic performance, enabling more flexible and robust design strategies. The heatmaps presented in Figs 18 and 19 further substantiate this observation, indicating that optimal performance is predominantly concentrated around configurations employing 8 dampers among the top 5% of elite designs.

## 3.5 Preferred Damper Locations in Elite Configurations

Figs 20 and 21 examine the tendency for preferred damper locations within the top 5% elite configurations, providing detailed insights into spatial arrangement patterns. Both $CR_1$ and $CR_2$ metrics show a clear increase in damper placement probabilities as the number of dampers ($N_d$) increases, particularly for $N_d \ge 6$.

The results highlight stories 1, 2, and 6 as the most consistently preferred locations, with story 1 systematically emerging as the most critical level across all configurations. Stories 3 and 5 also exhibits high placement probabilities, especially for highly redundant configurations $N_d \ge 8$.

Contrary to a purely unfavorable behavior, the upper stories, 7 to 10 become increasingly selected in configurations with a large number of dampers, $N_d \ge 8$, indicating that these levels actively contribute to performance in highly redundant structural layouts rather than being detrimental.
<figure><img src="image_19.png" /><figcaption>Fig. 16. Heatmap of CR₁ versus Mf and number of dampers</figcaption></figure>
<figure><img src="image_20.png" /><figcaption>Fig. 17. Heatmap of $CR_2$ versus $M_f$ and number of dampers</figcaption></figure>
<figure><img src="image_21.png" /><figcaption>Fig. 18. Heatmap of CR₁ versus Mf and number of dampers for top performance 5%</figcaption></figure>
<figure><img src="image_22.png" /><figcaption>Fig. 19. Heatmap of CR₂ versus Mf and number of dampers for top performance 5%</figcaption></figure>
<figure><img src="image_23.png" /><figcaption>Fig. 20. Preferred damper locations in the Top 5% configurations: presence probability by story and number of dampers for CR₁</figcaption></figure>
<figure><img src="image_24.png" /><figcaption>Fig. 21. Preferred damper locations in the Top 5% configurations: presence probability by story and number of dampers for CR<sub>2</sub></figcaption></figure>

The middle stories, 3 and 5, display a relatively balanced probability distribution as redundancy increases, reflecting significant topological flexibility. This flexibility allows multiple damper layouts to achieve robust seismic performance.

The heatmaps corroborate that the global performance optimum is achieved around 8 dampers, with a relatively homogeneous spatial distribution for highly redundant configurations. Nevertheless, this apparent uniformity is accompanied by strong preferential tendencies at stories 1, 2, 3 and 5, confirming their critical role in controlling the dynamic structural response.

## 4. Conclusion

This study presented a comprehensive probabilistic framework for optimizing the placement and capacity of Rotational Friction Dampers (RFD) in shear buildings. By integrating

the Non-Smooth Dynamics (NSD) method with a Monte Carlo simulation covering the entire design space, the following conclusions are drawn:

1. Dominance of quantity and saturation effects: Damper quantity governs structural performance, with significant benefits observed up to approximately seven devices. Beyond this threshold, additional dampers yield limited improvements in average response and mainly contribute to reduced variability. From a design perspective, configurations with seven to eight dampers represent an efficient balance between performance gains and resource utilization, as they approach structural saturation while avoiding unnecessary system complexity.

2. Preference for high slip loads: Optimal seismic mitigation is achieved when dampers are designed with slip load capacities in the
range of 350–450 kN·m. Damper placement should be prioritized at stories 1, 2, 3, 5 and 6, with story 1 being critical for global response control.

3. Robustness in upper-story configurations (7 to 10): Upper stories become effective only in highly redundant configurations. Beyond seven to eight devices, the system becomes more robust and less sensitive to precise placement, offering valuable constructability and design flexibility.

4. Robustness of the probabilistic approach: The probabilistic framework adopted in this study demonstrates a high level of robustness. Despite the inherent variability associated with seismic records and structural configurations, the results consistently reveal stable performance trends and well-defined optimal regions in the design space. Rather than converging towards a single deterministic solution, the approach identifies families of near-optimal solutions, enhancing the reliability and practical applicability of the method. Moreover, the probabilistic treatment reduces sensitivity to local fluctuations and modeling uncertainties, especially in highly redundant configurations. This confirms the ability of the proposed methodology to provide stable, reliable, and transferable design insights under uncertainty.

5. Recommendations for future research: To assess the generality of these findings, future studies should apply the proposed probabilistic framework to a wider range of structural configurations, including buildings with different heights, stiffness distributions, and geometric or structural irregularities. This will allow verification of whether the identified topological design rules can be extrapolated to broader classes of structural systems.

**Disclaimer (Artificial Intelligence)**

Author(s) hereby declare that NO generative AI technologies such as Large Language Models (ChatGPT, COPILOT, etc) and text-to-image generators have been used during writing or editing of this manuscript.

**Acknowledgements**

The first author would like to acknowledge the support provided by the SECIHTI scholarship, which enabled the completion of this work.

**Competing Interests**

Authors have declared that no competing interests exist.

**References**

Ampiah, A., & Zhao, X. (2020). Optimal Placement of Friction Dampers in High Rise Buildings Under Seismic Excitation. En H. Okada & S. N. Atluri (Eds.), *Computational and Experimental Simulations in Engineering* (Vol. 75, pp. 547–556). Springer International Publishing. https://doi.org/10.1007/978-3-030-27053-7_47

Artar, M., & Carbas, S. (2022). Optimum sizing design of steel frame structures through maximum energy dissipation of friction dampers under seismic excitations. *Structures*, *44*, 1928–1944. https://doi.org/10.1016/j.istruc.2022.08.119

Aydin, E. (2013). A simple damper optimization algorithm for both target added damping ratio and interstorey drift ratio. *Earthquakes and Structures*, *5*(1), 83–109. https://doi.org/10.12989/EAS.2013.5.1.083

Gao, J., Wang, C.-L., Meng, S., & Zeng, B. (2023). Performance evaluation of friction dampers considering different metal pairs and loading rates. *Journal of Constructional Steel Research*, *204*, 107859. https://doi.org/10.1016/j.jCSR.2023.107859

Jarrahi, H., Asadi, A., Khatibinia, M., & Etedali, S. (2020). Optimal design of rotational friction dampers for improving seismic performance of inelastic structures. *Journal of Building Engineering*, *27*, 100960. https://doi.org/10.1016/j.jobe.2019.100960

Jarrahi, H., Asadi, A., Khatibinia, M., Etedali, S., & Samadi, A. (2020). Simultaneous optimization of placement and parameters of rotational friction dampers for seismic-excited steel moment-resisting frames. *Soil Dynamics and Earthquake Engineering*, *136*, 106193. https://doi.org/10.1016/j.soildyn.2020.106193

Montejano-Castillo, M., Moreno-Villanueva, M., & Espinosa-Jiménez, E. (2020). Mapping Vertical Urban Growth in Mexico City in a Seismic Risk Context. *International Journal of Safety and Security Engineering*, *10*(1), 97–103. https://doi.org/10.18280/ijsse.100113

Mousavi, H., Sabbagh Yazdi, S. R., & Almohammad-Albakkar, M. (2023). A novel
method for efficient design of frame
structures equipped with nonlinear viscous
dampers by using computational results of
cylindrical friction damper. Australian
Journal of Structural Engineering, 24(1),
50-66.
https://doi.org/10.1080/13287982.2022.208
8055

Mualla, I. H., & Belev, B. (2002). Performance of steel frames with a new friction damper device under earthquake excitation. *Engineering Structures, 24*(3), 365-371.
https://doi.org/10.1016/S0141-0296(01)00102-X

Namorado, B., Pinto Da Costa, A., & Simões, F. M. F. (2022). Modelling rotational friction damping devices using the Non-Smooth Dynamics method. *Engineering Structures, 259*, 114118.
https://doi.org/10.1016/j.engstruct.2022.114118

Perez-Martinez, J. L., Horta-Rangel, J. M., Perez-Moreno, L. F., & Arjona-Catzim, I. F. (2025). Influence of Geometrical Configuration on the Dynamic Response of 5-Story Diagrid Structures. *Current Journal of Applied Science and Technology, 44*(11), 102-118.
https://doi.org/10.9734/cjast/2025/v44i1146

Titirla, M. D. (2024). Dynamic response of structures with a friction-yield damper.

Journal of Physics: Conference Series,
2647(25), 252016.
https://doi.org/10.1088/1742-
6596/2647/25/252016

Wong, K. K., & Johnson, J. (2009). Seismic Energy Dissipation of Inelastic Structures with Multiple Tuned Mass Dampers. *Journal of Engineering Mechanics, 135*(4), 265-275.
https://doi.org/10.1061/(ASCE)0733-9399(2009)135:4(265)

Zhang, R., & Soong, T. T. (1992). Seismic Design of Viscoelastic Dampers for Structural Applications. *Journal of Structural Engineering, 118*(5), 1375-1392.
https://doi.org/10.1061/(ASCE)0733-9445(1992)118:5(1375)

Zhang, Z., Zhan, Y., Shen, H., Qian, H., & Sheng, P. (2023). Seismic performance of self-centering braced rocking frame with novel self-centering friction dampers characterized by low-secondary stiffness. *Structures*, 57, 105225.
https://doi.org/10.1016/j.istruc.2023.105225

Zhao, G., Ma, Y., Yang, Z., Wang, Q., & Li, Y. (2022). Friction mechanism and experimental investigation of the response-amplified friction damper. *Structural Control and Health Monitoring, 29*(7).
https://doi.org/10.1002/stc.2953

**Disclaimer/Publisher's Note:** The statements, opinions and data contained in all publications are solely those of the individual author(s) and contributor(s) and not of the publisher and/or the editor(s). This publisher and/or the editor(s) disclaim responsibility for any injury to people or property resulting from any ideas, methods, instructions or products referred to in the content.

© Copyright (2026): Author(s). The licensee is the journal publisher. This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Peer-review history:
The peer review history for this paper can be accessed here:
https://pr.sdiarticle5.com/review-history/149711