Single-step reliability-based optimization of Visco-frictional multiple tuned mass dampers (VFMTMDs) for minimizing the seismic failure probability of buildings using CIOA

Abstract

This paper proposes a single-step reliability-based optimization framework for the design of Visco-Frictional Multiple Tuned Mass Dampers (VFMTMDs), which integrate both viscous and frictional damping mechanisms within a single device, unlike traditional approaches that consider only one type of damping. The proposed methodology simultaneously optimizes the number, placement, and mechanical parameters of VFMTMDs to minimize the probability of structural failure under seismic loading, explicitly accounting for the inherent uncertainties of ground motion. The optimization process employs the Circle-Inspired Optimization Algorithm (CIOA), a state-of-the-art metaheuristic developed by the authors. A ten-story benchmark building under seven actual seismic accelerograms served as a case study. The results of three independent optimization runs produced identical solutions, confirming the robustness and repeatability of the proposed approach. The optimal configuration consisted of two VFMTMDs installed on the top two floors, each with a mass ratio of only 1.5%. The optimized design achieved a 96.85% reduction in the probability of structural failure and a more than 35% increase in the seismic acceleration associated with the average fragility, demonstrating a substantial improvement in seismic reliability. Overall, the findings validate the proposed methodology as a robust and efficient framework for the optimal design of VFMTMDs, significantly enhancing the seismic performance of buildings in earthquake-prone regions.

1. Introduction

Seismic events continue to pose a critical threat to the safety, serviceability, and performance of civil infrastructure, often causing significant structural damage, economic losses, and endangering human life. The fundamental goal of earthquake engineering is to design buildings that can withstand these dynamic forces, ensuring resilience and reliability. Traditional seismic design methodologies, typically based on deterministic analyses and idealized energy dissipation models, often fail to account for the intrinsic uncertainties associated with seismic excitation and the nonlinear behavior of damping devices. These limitations highlight the need for advanced control strategies and probabilistic design frameworks that explicitly incorporate uncertainty and realistic device behavior into the optimization process.

Among various vibration mitigation techniques, passive control devices such as Viscous Dampers (e.g., Dai et al., ¹ Cao et al., ² Ijmulwar and Patro, ³ Pollini ⁴ ), Friction Dampers (e.g., Nabid et al., ⁵ Miguel et al.,^6–10 Sanghai and Pawade, ¹¹ Ontiveros-Pérez et al.^12–15), Metallic Dampers (e.g., Sanati and Karamodin ¹⁶ , Guo et al., ¹⁷ Henao-Leon et al. ¹⁸ ), and Tuned Mass Dampers (e.g., Brandão et al.,^19–21 Vellar et al., ²² Ontiveros-Pérez and Miguel, ²³ Fadel Miguel et al.,^24–26 Rossato and Miguel, ²⁷ Brito and Miguel, ²⁸ Miguel and Santos, ²⁹ Miguel and Souza ³⁰ ), as well as their variations such as Tuned Mass Dampers Inerter and Tuned Inerter Dampers (e.g., Miguel et al. ³¹ and Costa and Miguel ³² ) can be cited.

According to Cao, ³³ Tuned Mass Dampers (TMDs) are particularly popular due to their simplicity, low maintenance, and effectiveness in reducing dynamic responses. However, conventional single TMDs are typically tuned to a single mode and rely solely on linear viscous damping, limiting their performance under broadband excitation and strong non-stationary seismic loads. In response, Multiple Tuned Mass Dampers (MTMDs) have been developed, extending the frequency bandwidth and improving robustness by distributing absorbers across multiple modes or locations in the structure (e.g., Yamaguchi and Harnpornchai, ³⁴ Igusa and Xu ³⁵ ). Even so, most MTMD configurations still assume idealized viscous damping, overlooking realistic dissipative mechanisms such as friction-induced hysteresis and amplitude-dependent behavior.

Thus, over the years, several enhancements have been proposed to improve the performance of TMDs, including the development of variants such as the Friction Tuned Mass Damper (FTMD) and Friction Multiple Tuned Mass Dampers (FMTMDs), in which the traditional viscous damper is replaced by a friction-based mechanism (Restelatto ³⁶ ). In this line of research, Salimi et al. ³⁷ emphasized the advantages of FTMDs over conventional TMDs, particularly for low mass ratios, highlighting the superior efficiency of friction-based damping mechanisms. Pisal and Jangid ³⁸ confirmed the effectiveness of FTMDs in suppressing vibrations in SDOF systems under both harmonic and seismic excitations. Pisal ³⁹ demonstrated that FMTMDs significantly outperform a single FTMD in reducing structural displacement in buildings subjected to various earthquakes. Kim and Lee ⁴⁰ proposed an optimal design for FMTMDs by applying a statistical linearization method to address the inherent nonlinearity of friction. Love and Haskett ⁴¹ investigated the practical effects of friction in TMDs used in tall buildings, underscoring the importance of the lock-out phenomenon and noting that the linearized model predictions do not agree with the nonlinear simulations. In the context of integral bridges, Labbafi et al. ⁴² showed that FTMDs outperform conventional TMDs, especially at lower levels of seismic intensity. Besharatian et al.^43,44 developed an optimization approach for FTMDs based on Particle Swarm Optimization (PSO), demonstrating improved performance under earthquake loads through optimally tuned parameters. Lin et al. ⁴⁵ experimentally verified the effectiveness of FMTMDs in high-rise buildings, showing that well-designed friction mechanisms can significantly reduce structural vibrations. Khatibinia et al. ⁴⁶ revealed that optimized FTMDs effectively reduce seismic vulnerability in steel moment-resisting frames by promoting a more uniform distribution of damage along the height of the structure. Furthermore, Djerouni and Garcia ⁴⁷ compared the performance of FTMDs and TMDs in buildings using a hybrid PSO-GWO algorithm and found that FTMDs are more effective in high-rise buildings.

All these recent studies highlight the growing interest in and proven effectiveness of FTMDs and FMTMDs as advanced passive solutions for seismic and dynamic vibration control, underscoring their practical efficiency, optimization potential, and robustness. These systems offer economical and high-performance alternatives to conventional vibration control strategies in modern structural engineering. However, most of these studies rely on simplified models, for instance, using statistical linearization to represent the nonlinear behavior of friction, and/or neglecting the viscous damping of the TMD, replacing it entirely with frictional dissipation. This highlights the need for models that realistically account for both friction-induced nonlinearity and the simultaneous action of viscous and frictional dissipation within the same device.

To bridge this gap, it is important to explore non-classical damping devices that combine viscous and frictional elements within a single unit, leading to the development of Visco-Frictional Tuned Mass Dampers (VFTMDs). These hybrid devices exploit both velocity-dependent damping and Coulomb-like stick-slip hysteresis, resulting in superior energy dissipation and adaptability across varying seismic intensities. When deployed in multi-unit configurations, named Visco-Frictional Multiple Tuned Mass Dampers (VFMTMDs), they offer the potential for improved control over a wider range of structural modes and loading scenarios. Despite their promising capabilities, the optimal design of VFMTMDs remains complex due to the high-dimensional design space and nonlinear frictional behavior.

Additionally, most prior studies on TMDs, MTMDs, and friction-based variants, such as FTMDs, have primarily relied on deterministic optimization or simplified two-stage reliability frameworks, which can lead to suboptimal solutions, especially when nonlinearities affect structural behavior. Moreover, while reliability-based design optimization (RBDO) has gained power in structural engineering, its application to nonlinear VFMTMD systems remains limited. The literature reveals three persistent gaps: (i) a lack of integrated treatment of VFMTMD behavior under uncertainty, (ii) limited adoption of single-step RBDO approaches that directly embed failure probability into the optimization process, and (iii) underutilization of efficient metaheuristic algorithms for handling the complexity of nonlinear RBDO problems.

Addressing these gaps, this paper proposes a comprehensive single-step reliability-based optimization framework for designing Visco-Frictional Multiple Tuned Mass Dampers. The objective is to directly minimize the seismic failure probability of buildings by simultaneously optimizing the number of devices, their placement within the structure, and their mechanical parameters (stiffness, viscous damping, and friction forces). This is achieved within a single optimization loop that integrates reliability assessment, eliminating the bias introduced by sequential or surrogate-based approaches.

To solve this complex problem efficiently, the proposed methodology employs the Circle-Inspired Optimization Algorithm (CIOA), a robust and powerful metaheuristic algorithm recently developed by two authors of this paper (Souza and Miguel, ⁴⁸ Souza ⁴⁹ ). CIOA is particularly well-suited for constrained, multimodal problems and has demonstrated superior performance compared to traditional algorithms such as Genetic Algorithms and Particle Swarm Optimization (Souza and Miguel, ⁴⁸ Souza, ⁴⁹ Miguel and Souza, ³⁰ Miguel et al. ³¹ ).

In summary, this study makes several key contributions: (i) Integration of hybrid damping mechanisms in VFMTMDs, (ii) Probabilistic modeling of seismic excitation and failure probability, (iii) Single-step RBDO formulation for direct reliability optimization, and (iv) Application of the CIOA to optimize the VFMTMD design. This framework offers improved robustness, reduced approximation bias, and enhanced risk mitigation capabilities. A numerical case study demonstrates the effectiveness of this approach in reducing seismic failure probability under more realistic assumptions.

The remainder of this paper is structured as follows: Section 2 describes the problem and the mathematical foundations. Section 3 details the single-step reliability-based optimization proposed. Section 4 presents the CIOA and its implementation, along with a flowchart of the entire proposed methodology. Section 5 details the case study and discusses numerical results. Finally, Section 6 concludes the paper with key findings.

2. Problem statement and mathematical foundations

The dynamic response of buildings equipped with VFMTMDs under seismic excitation is modeled using a coupled multi-degree-of-freedom (MDOF) system, where both the primary structure and the auxiliary dampers are represented explicitly.

The governing equation of motion for the coupled system is given by:

$\mathbf{M}\overrightarrow{\ddot{\mathit{z}}}\left(t\right)+\mathbf{C}\overrightarrow{\dot{z}}\left(t\right)+\mathbf{K}\overrightarrow{z}\left(t\right)=\overrightarrow{F_{exc}}\left(t\right)+\overrightarrow{F_{fric}}\left(t\right),$

(1)

where $\mathbf{M}$ , $\mathbf{C}$ , and $\mathbf{K}$ are the (n + N _VFTMD ) × (n + N _VFTMD ) global mass, damping, and stiffness matrices, respectively, of the system (including both the structure and the VFMTMDs); $\overrightarrow{z}\left(t\right)$ , $\overrightarrow{\dot{z}}\left(t\right)$ , and $\overrightarrow{\ddot{z}}\left(t\right)$ are the (n + N _VFTMD ) vectors of displacements, velocities, and accelerations, respectively; $\overrightarrow{F_{exc}}\left(t\right)$ is the (n + N _VFTMD ) external excitation force vector (derived from seismic input); and $\overrightarrow{F_{fric}}\left(t\right)$ represents the (n + N _VFTMD ) nonlinear friction force vector produced by the VFMTMDs. The total number of degrees of freedom (DOFs) in the system is N _DOF = n + N _VFTMD , in which n is the number of structural DOFs, and N _VFTMD is the number of VFTMDs.

The external excitation force vector is derived from the ground acceleration as:

$\overrightarrow{F_{exc}}\left(t\right)=-\mathbf{MB}\overrightarrow{\ddot{\mathit{y}}}\left(t\right),$

(2)

where $\overrightarrow{\ddot{y}}\left(t\right)$ represents the w-dimensional vector of the ground acceleration, and the $\mathbf{B}$ matrix, of size (n + N _VFTMD ) × w, covers the cosine directors, being w the number of ground motion directions.

Each VFTMD introduces a nonlinear friction force that depends on the relative velocity between the floor of the structure where the damper is installed and the damper mass itself. Thus, the friction force associated with the $j^{th}$ damper is defined as:

$F_{fric}^{\left(j\right)}\left(t\right)={Fa}_{VFTMD}^{\left(j\right)}\text{sign}\left({\dot{z}}_{floo{r}_{\left(j\right)}}\left(t\right)-{\dot{z}}_{VFTM{D}_{\left(j\right)}}\left(t\right)\right),$

(3)

where ${Fa}_{VFTMD}^{\left(j\right)}$ is the absolute value of the friction force for the $j^{th}$ VFTMD, and ${\dot{z}}_{floo{r}_{\left(j\right)}}\left(t\right)$ and ${\dot{z}}_{VFTM{D}_{\left(j\right)}}\left(t\right)$ are the velocities of the floor and the damper mass, respectively.

These friction forces are then assembled into a global friction force vector $\overrightarrow{F_{fric}}\left(t\right)$ , where each damper contributes forces in opposite directions to the associated structural DOF and its corresponding damper DOF:

$\overrightarrow{F_{fric}}\left(t\right)={\displaystyle \sum }_{j=1}^{N_{VFTMD}}{F}_{fric}^{\left(j\right)}\left(t\right)\left(e_{{VFTMD}_j}-e_{{floor}_{\left(j\right)}}\right),$

(4)

where $e_{{VFTMD}_j}$ and $e_{{floor}_{\left(j\right)}}$ are unit vectors corresponding to the DOFs of the $j^{th}$ VFTMD and the structural floor to which it is attached, respectively.

This formulation ensures that the friction force acts as an internal force pair: it resists the relative sliding between the damper and the structure. The sign function introduces a non-smooth, piecewise nonlinearity into the system, making the overall problem nonlinear and path-dependent. Due to the nature of dry friction, the force does not depend on the magnitude of the velocity, but only on its direction. This behavior is updated incrementally at each time step, based on the relative velocity computed from the previous time step, which ensures proper modeling of stick-slip transitions that are typical in friction-damped systems.

A computational routine using Matlab ⁵⁰ was elaborated by the authors to solve equation (1). However, unlike previous works that used the explicit method of Central Finite Differences,^51–59 in this paper the dynamic system is integrated in time using the Newmark method (Rao ⁶⁰ ), which allows for an implicit time integration scheme suitable for nonlinear problems. The friction forces are computed and updated iteratively at each time step based on the relative velocities from the previous iteration, and are added to the right-hand side of the system of equations. This results in an efficient and stable solution process that captures both viscous and frictional damping effects.

3. Single-step reliability-based optimization

This paper proposes a single-step reliability-based optimization framework that accounts for uncertainty in engineering design by treating performance criteria probabilistically rather than deterministically. Achieving an appropriate balance between structural safety and cost efficiency remains a key challenge, particularly in the reliability-based optimal design of vibration control systems under seismic loading. While previous studies by Ontiveros-Pérez et al. ¹² and Castaldo et al.^61,62 have addressed friction dampers, and Ontiveros-Pérez and Miguel ²³ and Miguel et al. ³¹ addressed conventional and inerter-based tuned mass dampers, respectively, research on reliability-oriented optimization of advanced systems such as VFMTMDs is still limited. Hence, this section introduces a framework aimed at minimizing the probability of seismic-induced failure and enhancing overall structural reliability, with subsequent subsections detailing the failure probability estimation, the design variables and constraints, and the mathematical formulation of the proposed optimization problem.

It is important to mention that the probabilistic framework considers uncertainty only in seismic demand (PGA variability). Uncertainties in structural properties, device parameters, and modeling assumptions are not included in the present formulation.

3.1. Estimation of failure probability

The main objective of this study is to minimize the probability of structural failure ( $P_f$ ) in buildings subjected to real seismic events. To evaluate seismic risk, it is essential to estimate the potential damage caused by earthquakes, for which fragility curves play a key role. These curves express the probability of failure as a function of the seismic event’s intensity, represented by the Peak Ground Acceleration (PGA). In this work, fragility curves are developed through Incremental Dynamic Analysis (IDA), where real seismic records are progressively scaled to increasing PGA levels to analyze the structural response under growing seismic demands.

A widely accepted failure criterion is adopted, defining failure as the condition in which either the maximum interstory drift ( $d_{\mathit{max}}$ ) exceeds a prescribed limit or the maximum displacement ( $z_{\mathit{max}}$ ) surpasses an allowable threshold. Both parameters are obtained from the displacement vector $\overrightarrow{z}\left(t\right)$ , computed by solving equation (1), as described in Section 2. Accordingly, following the recommendations of technical standards such as ASCE/SEI 7-22, ⁶³ the random variable $D$ is defined as:

$D=\{\begin{array}{l}\hspace{0.17em}1,\text{if}\hspace{0.17em}\left[\left|d_{\mathit{max}}\right|>0.02h\hspace{1em}\text{or}\hspace{1em}\left|z_{\mathit{max}}\right|>0.02H\right]\\ \hspace{0.17em}0,\text{if}\hspace{0.17em}\left[\left|d_{\mathit{max}}\right|\le 0.02h\hspace{0.5em}\text{and}\hspace{1em}\left|z_{\mathit{max}}\right|\le 0.02H\right]\end{array},$

(5)

where $D=1$ indicates structural failure, and $D=0$ means no failure. $h$ is the interstory height, and $H$ is the total building height.

By subjecting the structure to a set of normalized seismic records and applying the failure criterion of equation (5), it is possible to determine the number of instances in which the system fails. This allows the construction of the fragility curve, which expresses the conditional probability of failure for a given PGA, as defined by equation (6):

$P_D\left(\text{failure}|PGA\right)=P_D\left(PGA\right)=P\left[D=1\right]=P\left[\left|d_{\mathit{max}}\right|>0.02h\hspace{1em}\text{or}\hspace{1em}\left|z_{\mathit{max}}\right|>0.02H\right].$

(6)

From the relative frequency of failures, the statistical moments are computed, and a Lognormal distribution is fitted to describe the data. The cumulative distribution function of the Lognormal distribution, given in equation (7), defines the fragility curve:

$P_D\left(PGA\right)=\frac{1}{\sqrt{2\pi }{\zeta }_{PGA}}\underset{0}{\overset{\max \left(PGA\right)}{{\displaystyle \int }}}\frac{1}{pga}\mathit{exp}\left[\frac{-{\left(\ln \left(pga\right)-{\lambda }_{PGA}\right)}^2}{2{\zeta }_{PGA}^2}\right]\mathrm{d}pga,\text{for}\hspace{0.17em}PGA\hspace{0.17em}>\hspace{0.17em}0,$

(7)

where ${\lambda }_{PGA}$ and ${\zeta }_{PGA}$ are the Lognormal parameters, computed as:

${\lambda }_{PGA}=\mathrm{E}\left(lnPGA\right)=\mathit{ln}\left(\mathrm{E}\left(PGA\right)\right)-0.5{\zeta }_{PGA}^2,$

(8)

${\zeta }_{PGA}^2=\text{Var}\left(lnPGA\right)=\mathit{ln}\left[1+{\left(\frac{\mathrm{\sigma }\left(PGA\right)}{\mathrm{E}\left(PGA\right)}\right)}^2\right].$

(9)

Finally, the failure probability ( $P_f$ ) is calculated as the integral of the product between the fragility function $P_D\left(PGA\right)$ and the seismic risk function $f_{Amax50}\left(PGA\right)$ , over a 50-year period:

$P_f=\underset{0}{\overset{\max \left(PGA\right)}{{\displaystyle \int }}}{P}_D\left(PGA\right)f_{Amax50}\left(PGA\right)\mathrm{d}pga.$

(10)

In summary, the proposed reliability-based framework consists of three main stages: (i) normalization of selected seismic accelerograms by PGA, followed by the evaluation of structural failure at each level; (ii) construction of the fragility curve by determining the relative frequency of failure for each PGA level; and (iii) computation of the overall failure probability through the integration of the fragility function with the seismic risk model over a 50-year period.

3.2. Design variables and constraints

To achieve optimal system performance, it is crucial to carefully define both the design variables and the associated constraints. Unlike previous studies that focus solely on device parameters, this work considers not only the parameters (stiffness, damping constants, and friction forces) but also the quantity and placement of the VFMTMDs as design variables, aiming to minimize the probability of structural failure. In this formulation, the number and location of VFMTMDs within the building are treated as discrete variables, while the stiffness (k _VFTMD ), damping constants (c _VFTMD ), and friction forces (Fa _VFTMD ) of each device are treated as continuous variables.

The problem constraints include the number of available installation positions (NP _VFTMD ), the maximum number of VFTMDs allowed per floor (MaxN _VFTMDFloor ), the lower ( $k_{VFTMD}^{\mathit{min}}$ ) and upper ( $k_{VFTMD}^{\mathit{max}}$ ) bounds of the stiffness, the lower ( $c_{VFTMD}^{\mathit{min}}$ ) and upper ( $c_{VFTMD}^{\mathit{max}}$ ) bounds of the damping constants, and the lower ( ${Fa}_{VFTMD}^{\mathit{min}}$ ) and upper ( ${Fa}_{VFTMD}^{\mathit{max}}$ ) bounds of the friction forces. The total mass of the VFMTMDs is obtained by multiplying the mass ratio ( $\alpha$ , which is assumed to be a constant value) by the total building mass ( $M_{tb}$ ). The mass of each individual VFTMD (m _VFTMD ) is then determined by equally dividing the total mass of the VFMTMDs by the number of installed devices. The quantity-position vector of VFMTMDs, denoted as $\overrightarrow{P}$ , contains entries of 0 (no VFTMD on that floor) or 1 (one VFTMD on that floor). For simplicity, all design variables are grouped into a single design vector, expressed as $\overrightarrow{v}=\left[\overrightarrow{P},{\overrightarrow{k}}_{VFTMD},{\overrightarrow{c}}_{VFTMD},{\overrightarrow{Fa}}_{VFTMD}\right]$ .

3.3. Formulating the proposed optimization problem

As discussed earlier, the methodology seeks to simultaneously optimize the number, placement, and parameters of the VFMTMDs to minimize the probability of failure ( $P_f$ , equation (10)). Following the procedure described in the last subsections, the optimization problem can be expressed as:

$Find:\overrightarrow{v}=\left[\overrightarrow{P},{\overrightarrow{k}}_{VFTMD},{\overrightarrow{c}}_{VFTMD},{\overrightarrow{Fa}}_{VFTMD}\right]$

$Minimize:P_f\left(\overrightarrow{v}\right)$

$Subjected\hspace{0.17em}to:\text{Number}\hspace{0.17em}\text{of}\hspace{0.17em}\text{available}\hspace{0.17em}\text{installation}\hspace{0.17em}\text{positions}={NP}_{VFTMD,}$

$\text{Maximum}\hspace{0.17em}\text{number}\hspace{0.17em}\text{of}\hspace{0.17em}\text{VFTMDs}\hspace{0.17em}\text{per}\hspace{0.17em}\text{floor}={MaxN}_{VFTMDFloor,}$

(11)

${k_{VFTMD}^{\mathit{min}}\le k}_{VFTM{D}_{\left(j\right)}}\le k_{VFTMD}^{\mathit{max}},\forall \hspace{0.5em}VFTMD\hspace{0.17em}j,$

${c_{VFTMD}^{\mathit{min}}\le c}_{VFTM{D}_{\left(j\right)}}\le c_{VFTMD}^{\mathit{max}},\forall \hspace{0.5em}VFTMD\hspace{0.17em}j,$

${Fa}_{VFTMD}^{\mathit{min}}\le {Fa}_{VFTM{D}_{\left(j\right)}}\le {Fa}_{VFTMD}^{\mathit{max}},\forall \hspace{0.5em}VFTMD\hspace{0.17em}j,$

$m_{VFTMD}=\alpha \frac{M_{tb}}{N_{VFTMD}}.$

The proposed reliability-based optimization problem is notably complex for several reasons: (i) it involves a dynamic system; (ii) it accounts for uncertainties in seismic excitation; (iii) its objective function may be non-convex and multimodal; and (iv) it includes both discrete and continuous design variables.

Due to these challenges, specialized optimization methods are required, particularly metaheuristic algorithms (Yang^64,65), which offer several advantages: (i) they do not rely on gradient information, making them suitable for problems with difficult or undefined gradients; (ii) they can efficiently handle non-convex or discontinuous objective functions; (iii) when properly tuned, they reduce the risk of convergence to local minima; (iv) they can address mixed-variable optimization problems; and (v) they provide a set of alternative optimal solutions, giving designers flexibility in decision-making (Miguel and Fadel Miguel, ⁶⁶ Fadel Miguel et al. ⁶⁷ ).

Among the various metaheuristic methods, the Circle-Inspired Optimization Algorithm (CIOA), recently proposed by Souza and Miguel, ⁴⁸ has shown strong performance in prior studies (e.g., Souza and Miguel, ⁴⁸ Souza ⁴⁹ ). Therefore, the CIOA, described in detail in the next section, is employed to solve the optimization problem presented in this research.

4. Circle-inspired optimization algorithm (CIOA)

The CIOA constitutes a recent, robust, and computationally efficient metaheuristic optimization algorithm introduced by the authors (Souza and Miguel, ⁴⁸ Souza ⁴⁹ ). Despite its straightforward and easily implementable structure, the algorithm represents a modern and versatile optimization approach, capable of effectively handling complex problems such as the one addressed in this study. Accordingly, the CIOA is employed herein to solve the proposed optimization task.

The CIOA is grounded in fundamental equations derived from the trigonometric circle, which enables a conceptually simple and easily interpretable framework. This design allows the algorithm to be implemented with minimal coding effort. As a result, it demonstrates rapid convergence behavior, substantially decreasing the computational time required to identify optimal solutions. The algorithm includes only two user-defined parameters: the angle θ, which determines the arc along which a search agent moves, and the Glob _It parameter, which specifies the proportion of iterations preceding a transition to a strictly local search phase. Based on extensive empirical evaluations conducted by the authors (Souza and Miguel, ⁴⁸ Souza ⁴⁹ ), recommended values are θ = 17° and Glob _It = 0.85.

The formulation of the CIOA can be outlined as follows: once the parameters θ and Glob _It are defined by the user, a radius vector $\overrightarrow{r}$ is constructed, where each element $r_j$ is computed according to equation (12):

$c_r=\frac{\sqrt{U_b-L_b}}{N_{ag}}\hspace{1em}and\hspace{1em}{r}_j=\frac{c_r.j^2}{N_{ag}},1\le j\le N_{ag},$

(12)

being $L_b$ and $U_b$ the lower and upper bounds of the design variables, respectively; $c_r$ is an auxiliary constant used in the formulation, and $N_{ag}$ denotes the total number of search agents.

Within the main loop of the algorithm, global and local search processes are executed concurrently. At each iteration, the solutions obtained by all search agents are ranked. Consequently, the agent that achieves the $j^{th}$ best solution in iteration $k$ will have its position updated in iteration $k+1$ according to equations (13) and (14):

$x_{2i}\left(k+1\right)=x_{2i}\left(k\right)-{rand}_1{r}_j\hspace{0.17em}\sin \left(k\theta \right)+{rand}_2{r}_j\hspace{0.17em}\sin \left(\left(k+1\right)\theta \right),$

(13)

$x_{2i-1}\left(k+1\right)=x_{2i-1}\left(k\right)-{rand}_3{r}_j\hspace{0.17em}\cos \left(k\theta \right)+{rand}_4{r}_j\hspace{0.17em}\cos \left(\left(k+1\right)\theta \right),$

(14)

in these expressions, $2i$ and $2i-1$ denote even and odd indices, respectively, corresponding to coordinate positions in the solution space. The variables ${rand}_1$ , ${rand}_2$ , ${rand}_3$ , and ${rand}_4$ represent independent random numbers drawn from a uniform distribution in the interval [0, 1].

If a variable $x_i$ exceeds its predefined bounds, its value is reassigned to the corresponding component of the best-performing search agent from the most recent iteration. Additionally, whenever the product $k\theta$ surpasses an integer multiple of 360°, the radius vector $\overrightarrow{r}$ is updated according to:

${\overrightarrow{r}}_{new}=0.99\overrightarrow{r},$

(15)

where ${\overrightarrow{r}}_{new}$ denotes the recalculated radius vector. This update mechanism progressively reduces the search radius, promoting convergence by intensifying the local search as the algorithm evolves.

When the ratio between the current iteration and the total number of iterations reaches the value defined by the Glob _It parameter, the algorithm transitions into an exclusively local search phase. At this stage, all search agents are reinitialized with the coordinates corresponding to the best solution identified up to that point. Furthermore, the lower and upper bounds of each design variable are refined to $L_{{b1}_i}$ and $U_{{b1}_i}$ , respectively, computed according to equation (16):

$U_{{b1}_i}=x_{i_{best}}+\frac{U_b-L_b}{N_{ag}}\hspace{1em}and\hspace{1em}{L}_{{b1}_i}=x_{i_{best}}-\frac{U_b-L_b}{N_{ag}},$

(16)

here, $x_{i_{best}}$ denotes the value of the $i^{th}$ variable in the best solution found so far, and $U_b$ and $L_b$ are the original upper and lower bounds of the design variables. This adjustment narrows the search space around the current optimum, thereby enhancing local exploitation during the final phase of the algorithm.

Once the local search phase is initiated, the same update equations used in the main optimization loop (equations (13) and (14)) are applied, now considering the adjusted variable bounds defined in equation (16). During this phase, if a design variable $x_i$ assumes a value such that $L_{{b1}_i}<x_i<L_b$ or $U_{{b1}_i}>x_i>U_b$ , its value is automatically corrected to the original boundary, i.e., $x_i=L_b$ or $x_i=U_b$ , respectively. This constraint-handling mechanism ensures that all variables remain within a feasible and controlled search region during the intensified local exploration.

The pseudo-code of the CIOA is presented in Figure 1. For a more comprehensive understanding of the algorithm, readers are referred to Souza and Miguel, ⁴⁸ and Souza. ⁴⁹ An application of the CIOA to the optimization of vibration control systems in footbridges is discussed in Ref. 30, while ³¹ presented an application in buildings. Open-source implementations of the algorithm are available in both Matlab and Python and can be freely accessed through the following repositories:

Figure 1.

Pseudocode representation of the CIOA.

• Matlab version: https://github.com/oapsouza/CIOA-Circle-Inspired-Optimization-Algorithm-Matlab-version

• Python version: https://github.com/oapsouza/CIOA-Circle-Inspired-Optimization-Algorithm-Python-version

Accordingly, the comprehensive flowchart outlining the entire proposed reliability-based optimal design methodology for VFMTMDs is presented in Figure 2.

Figure 2.

Flowchart of the proposed methodology for the reliability-based optimal design of VFMTMDs.

5. Case study

To demonstrate the effectiveness of the proposed single-step reliability-based optimal design methodology for VFMTMDs, aimed at minimizing structural failure probability through fragility analysis under real earthquake records over a 50-year service life, a ten-story benchmark structure, previously analyzed by various researchers for different objectives (e.g., Hadi and Arfiadi, ⁶⁸ Lee et al., ⁶⁹ Bekdaş and Nigdeli, ⁷⁰ Mohebbi et al. ⁷¹ ), is adopted as a case study. For the implementation of the proposed approach, custom computational algorithms were developed by the authors using Matlab.

As illustrated in Figure 3, the structure is modeled as a shear-type building, containing ten potential locations for the installation of dampers (NP _VFTMD = 10), with a maximum allowance of one VFTMD per floor (MaxN _VFTMDFloor = 1). The total height of the building is 37 meters ( $H$ = 37m), with each story having a uniform height of 3.7 meters ( $h$ = 3.7m).

Figure 3.

Ten-story shear-type building model with ten potential locations for the placement of the VFMTMDs, one per floor.

Each of the ten stories is modeled with a concentrated mass of 360×10³kg, an equivalent lateral stiffness of 650×10⁶N/m, and an equivalent damping coefficient of 6.2×10⁶Ns/m. Based on these parameters, the solution of the eigenvalue problem yields the natural frequencies of the structure as: 1.01, 3.01, 4.94, 6.76, 8.43, 9.91, 11.18, 12.19, 12.92, and 13.37Hz.

A set of 7 real earthquake ground motion records was selected. These seismic data were sourced from the Center for Engineering Strong Motion Data (CESMD), and their key features are summarized in Table 1.

Table 1.

Key characteristics of the seven seismic records selected for this study.

Name	Origin time (UTC)	Magnitude	Epicentral distance (km)	Ground PGA (g)
Coalinga	1983/05/02 - 23:43:00	6.5Ml	57.3	0.114
Isleton	2023/10/18 - 16:29:14	4.2Mw	24.5	0.016
Loma Prieta	1989/10/18 - 00:04:00	6.9Mh	97.7	0.160
Parkfield	2004/09/28 - 17:15:24	6.0Ml	11.9	0.629
Pleasant Hill	2019/10/15 - 05:33:42	4.5Mw	28.3	0.009
San Simeon	2003/12/22 - 19:15:56	6.5Ml	261.0	0.004
South Napa	2014/08/24 - 10:20:44	6.0Mw	41.5	0.036

Figure 4 presents the principal segments of the accelerograms normalized to 0.30g for each of the seven selected seismic records, accompanied by their respective power spectral densities and acceleration response spectra computed at 5% damping.

Figure 4.

(a) Normalized accelerograms of the seven selected real earthquake records at 0.30g, (b) Corresponding power spectral densities, and (c) Acceleration response spectra with 5% damping.

Following the approach of Ontiveros-Pérez, ⁷² who assessed seismic risk using PGA records, the Lognormal distribution is employed to characterize the seismic risk over a 50-year period ( $f_{Amax50}\left(PGA\right)$ ), as explained in Section 3. This probabilistic model estimates the exceedance probabilities within the 50-year period, yielding an expected PGA value of E(PGA) = 0.2523g and a standard deviation of σ(PGA) = 0.0904g.

Accordingly, following the methodology outlined in Section 3, the selected seismic accelerograms were normalized by PGA in increments of 0.01g, ranging from 0.01g to 3.0g. As explained in Section 2, the Newmark integration method, with a time step of 0.02s, was then applied to solve equation (1) and obtain the displacement vector $\overrightarrow{z}\left(t\right)$ . This approach enables the computation of the maximum interstory drift ( $d_{\mathit{max}}$ ) and the maximum displacement ( $z_{\mathit{max}}$ ). Subsequently, the failure probability $P_f$ over a fifty-year period is determined using equation (10).

Initially, the failure probability of the uncontrolled building (i.e., without VFMTMDs) is evaluated. Next, an optimization procedure is implemented to minimize this probability. The optimization problem considers as design variables the number, placement, and mechanical properties of the VFMTMDs, i.e., the stiffness coefficients (k _VFTMD ), damping constants (c _VFTMD ), and friction forces (Fa _VFTMD ) associated with each device.

Figure 3 shows ten potential locations for the installation of VFMTMDs (NP _VFTMD = 10), with a maximum allowance of one device per floor (MaxN _VFTMDFloor = 1). The total mass allocated to the VFMTMDs is defined as the product of the mass ratio ( $\alpha$ = 0.03) and the total building mass ( $M_{tb}$ = 3,600,000kg). Consequently, the mass of each individual VFTMD is obtained by equally dividing the total VFMTMD mass by the number of installed devices. The lower and upper bounds for the stiffness coefficients (k _VFTMD ), damping constants (c _VFTMD ), and friction forces (Fa _VFTMD ), are defined as [0 - 2.0×10⁷]N/m, [0 - 1.3×10⁶]Ns/m, and [0 - 1.0×10⁶]N, respectively.

To perform the single-step reliability-based optimization of the quantities, positions, and parameters of the VFMTMDs (equation (11)), the CIOA was employed, utilizing 80 search agents and 400 iterations, resulting in a total of 32,000 function evaluations. The CIOA control parameters were set to θ = 17° and Glob _It = 0.85, in accordance with the recommendations of Souza and Miguel.4 ⁴⁸ Three independent runs were carried out, and the corresponding results are presented in Table 2.

Table 2.

Summary of the optimization results.

Run	Quantities and positions $\overrightarrow{P}$	VFTMD mass m (kg)	VFTMD parameters k (N/m); c (Ns/m); Fa (N);	Probability of failure P f	Reduction (%)
-	Uncontrolled [0 0 0 0 0 0 0 0 0 0]	-	-	1.2137×10-2	-
1	[0 0 0 0 0 0 0 0 1 1]	m VFTMD = 54×103	k VFTMD9 = 1.4029×106	3.8247×10-4	96.85
	​	​	k VFTMD10 = 2.4194×106	​	​
	​	​	c VFTMD9 = 3.0093×104	​	​
	​	​	c VFTMD10 = 4.7684×104	​	​
	​	​	Fa VFTMD9 = 1.6704×104	​	​
	​	​	Fa VFTMD10 = 3.4949×104	​	​
2	[0 0 0 0 0 0 0 0 1 1]	m VFTMD = 54×103	k VFTMD9 = 1.4037×106	3.8247×10-4	96.85
	​	​	k VFTMD10 = 2.4201×106	​	​
	​	​	c VFTMD9 = 3.0101×104	​	​
	​	​	c VFTMD10 = 4.6593×104	​	​
	​	​	Fa VFTMD9 = 1.8025×104	​	​
	​	​	Fa VFTMD10 = 3.5303×104	​	​
3	[0 0 0 0 0 0 0 0 1 1]	m VFTMD = 54×103	k VFTMD9 = 1.4040×106	3.8247×10-4	96.85
	​	​	k VFTMD10 = 2.4158×106	​	​
	​	​	c VFTMD9 = 3.0078×104	​	​
	​	​	c VFTMD10 = 4.6182×104	​	​
	​	​	Fa VFTMD9 = 1.9101×104	​	​
	​	​	Fa VFTMD10 = 3.5612×104	​	​

Table 2 summarizes the failure probability of the structure, as well as the optimized VFMTMD quantities, locations, and mechanical parameters, for both the uncontrolled and optimized configurations.

Table 2 illustrates the success of the proposed approach in optimizing VFMTMDs. As can be seen in this table, the optimal number of VFTMDs is 2, and the best position of these VFTMDs is on the last two stories (9^th and 10^th stories). The optimal VFTMD parameters (stiffness coefficients, damping constants, and friction forces) are also shown in Table 2 and in which it is also possible to see that the mass of each one of the two VFTMDs is only 1.5% of the total mass of the building. It is interesting to note that the results of the three independent runs are very similar, confirming the robustness of the method. It is also important to highlight that after installing the optimized VFMTMDs, the probability of failure is reduced by 96.85%.

Figure 5 presents the fragility curves for the structure in its original configuration (without VFTMD) and after the implementation of the optimal VFMTMD design, corresponding to Run 1 in Table 2. This configuration includes two VFTMDs installed at the 9^th and 10^th stories, respectively.

Figure 5.

Fragility curves for the structure without VFTMD (red curve, P _f = 1.2137×10^-2) and with the implementation of the two optimized VFTMDs (blue curve, P _f = 3.8247×10^-4).

As illustrated in Figure 5, the implementation of the optimized VFMTMD configuration leads to a reduction in the structural vulnerability under seismic loading, as indicated by the displacement of the fragility curve toward higher acceleration levels. Specifically, the seismic acceleration associated with the average fragility (0.5) increases from 0.7172g to 0.9718g, which means an enhancement of 35.5%, thereby confirming the effectiveness of the proposed design strategy.

Figure 6 presents the failure probability functions of the building for the uncontrolled case (red curve) and the optimized configuration corresponding to Run 1 in Table 2 (blue curve).

Figure 6.

Failure probability functions for the structure without VFTMD (red curve, P _f = 1.2137×10^-2) and with the implementation of the two optimized VFTMDs (blue curve, P _f = 3.8247×10^-4).

An analysis of the ordinate axis in Figure 6 reveals that the highest value of the failure probability function for the uncontrolled case (red curve) is 38.5 times greater than that of the optimized case (blue curve). This substantial reduction undoubtedly shows the effectiveness of the optimal VFMTMD installation in diminishing the likelihood of the structure reaching the limit state, as reflected by the marked decrease in the area under the blue curve.

In terms of computational effort, each of the three complete independent runs took approximately 16 hours on a desktop equipped with an AMD Ryzen 5 PRO 5650G processor and 16 GB of RAM.

To further illustrate the effectiveness of the proposed methodology, Figure 7(a) and (b) present the interstory drift at the first story and the displacement at the top, respectively, for a single seismic record, the South Napa earthquake at a PGA of 0.30g, comparing the uncontrolled case (red curve) with the optimized VFMTMD configuration (blue curve). Additionally, Figure 8(a) and (b) represent the maximum interstory drift and maximum displacement per story, respectively, before and after the installation of the optimized VFMTMD system (Run 1 in Table 2).

Figure 7.

Structural response in terms of displacement for the South Napa earthquake at a PGA of 0.30g, comparing the uncontrolled case without VFTMD (red curve) and the optimized VFMTMD configuration (blue curve). (a) Interstory drift at the first story, (b) Displacement at the top level.

Figure 8.

Structural response per story for the South Napa earthquake at a PGA of 0.30g, comparing the building without VFTMD (red curve) and with the optimized VFMTMD configuration (blue curve). (a) Maximum interstory drift, (b) Maximum displacement.

As illustrated in Figures 7 and 8, the implementation of the optimized VFMTMDs resulted in a 44.6% reduction in the maximum interstory drift at the first story and a 39.3% reduction in the maximum displacement at the top level, demonstrating the effectiveness of the proposed control strategy.

It is important to emphasize that, for this specific earthquake, the South Napa, reductions of approximately 45% and 40% were observed in the maximum interstory drift and the maximum displacement, respectively. Nevertheless, since the optimization process is grounded in a reliability-based approach, aimed at minimizing the probability of failure rather than directly targeting the reduction of the maximum interstory drift or the maximum displacement, and the inherent uncertainties of dynamic loading are addressed through the evaluation of multiple earthquakes rather than a single deterministic seismic record, it is reasonable to infer that had the optimization been tailored exclusively to this specific earthquake, the observed reductions would likely have been even more significant.

5.1. Comparison with MTMDs and FMTMDs

To demonstrate the superiority of the proposed VFMTMDs over traditional MTMDs and FMTMDs, this section compares the probability of failure associated with their respective optimized designs.

The comparison was conducted using Run 1 in Table 2, and the results are presented in Table 3. For consistency, the CIOA was applied to optimize the MTMDs and FMTMDs using the same number of search agents (80) and iterations (400) as in the VFMTMD optimization.

Table 3.

Comparison of results obtained with optimized VFMTMDs, MTMDs, and FMTMDs.

​	Optimized quantities and positions	Mass (kg)	Optimized parameters k (N/m); c (Ns/m); Fa (N);	Probability of failure
VFMTMDs	[0 0 0 0 0 0 0 0 1 1]	m VFTMD = 54×103	k VFTMD9 = 1.4029×106	3.8247×10-4
	​	​	k VFTMD10 = 2.4194×106	​
	​	​	c VFTMD9 = 3.0093×104	​
	​	​	c VFTMD10 = 4.7684×104	​
	​	​	Fa VFTMD9 = 1.6704×104	​
	​	​	Fa VFTMD10 = 3.4949×104	​
MTMDs	[0 0 0 0 0 0 0 0 1 1]	m TMD = 54×103	k TMD9 = 1.4101×106	4.1115×10-4
	​	​	k TMD10 = 2.3998×106	​
	​	​	c TMD9 = 3.1112×104	​
	​	​	c TMD10 = 4.7001×104	​
FMTMDs	[0 0 0 0 0 0 0 0 1 1]	m FTMD = 54×103	k FTMD9 = 1.4148×106	4.1074×10-4
	​	​	k FTMD10 = 2.3895×106	​
	​	​	Fa FTMD9 = 1.9101×105	​
	​	​	Fa FTMD10 = 3.3571×105	​

As shown in Table 3, the results obtained with optimized MTMDs and FMTMDs are identical to those achieved with optimized VFMTMDs regarding both the optimal number of devices and their placement. All three designs indicate that the optimal configuration consists of two devices installed on the ninth and tenth floors of the building.

Since the number of devices is the same in all three cases, each device has the same mass, corresponding to a mass ratio of 1.5%. In terms of stiffness, the three device types also exhibited similar results. A comparison between the damping coefficients of the MTMD and VFMTMD configurations indicates close agreement. In contrast, a comparison of the friction forces in the FMTMD and VFMTMD configurations shows that the friction forces in the FMTMD case are approximately one order of magnitude greater than those in the VFMTMD case, likely compensating for the absence of viscous damping in the FMTMD configuration.

The probability of failure obtained with MTMDs and FMTMDs is 7.50% and 7.39%, respectively, higher than that achieved with VFMTMDs, highlighting the superior performance of VFMTMDs compared to both MTMDs and FMTMDs.

6. Conclusions

This study proposed the integration of both viscous and frictional damping mechanisms within a single device, the Viscous-Frictional Multiple Tuned Mass Dampers (VFMTMDs), differing from previous works that typically consider only one type of damping. A comprehensive, single-step, reliability-based methodology was developed for the modeling and optimal design of VFMTMDs, involving the simultaneous optimization of their number, placement, and mechanical parameters to minimize the probability of structural failure under seismic loading, while accounting for ground motion uncertainties. Computational routines were created to assess the dynamic structural response and estimate failure probabilities through fragility curves. These routines were integrated with the Circle-Inspired Optimization Algorithm (CIOA), a state-of-the-art metaheuristic particularly suitable for solving complex optimization problems of this nature.

As a case study, the proposed methodology was employed to design VFMTMDs for a ten-story benchmark building, with the objective of minimizing the probability of structural failure over a 50-year service life, using a suite of seven recorded ground motions. Based on the results obtained, the following conclusions can be drawn:

• The three independent optimization runs resulted in identical failure probabilities (3.8247×10^-4), demonstrating the robustness and repeatability of the proposed methodology;

• In all runs, the vector $\overrightarrow{P}$ resulted in identical values, indicating that the optimal number of VFMTMDs was two, with their optimal placement on the top two floors of the building;

• In this most effective configuration, comprising two VFMTMDs installed on the top two floors, each device had a mass ratio of only 1.5%;

• The optimized parameters of the VFMTMDs were also consistent across the three runs, further confirming the robustness and reliability of the approach;

• The three optimized runs exhibited a substantial decrease of 96.85% in the probability of structural failure, significantly enhancing the building’s seismic reliability and validating the effectiveness of the proposed procedure;

• The seismic acceleration associated with the average fragility increased by more than 35% in the optimized case, showing again the success of the proposed design strategy;

• The comparison between the VFMTMDs proposed in this study and MTMDs and FMTMDs demonstrated that VFMTMDs were more effective in reducing the probability of failure.

Overall, the results confirm that the proposed methodology constitutes a robust and efficient framework for the optimal design of VFMTMDs, demonstrating its potential to significantly enhance structural performance.