Seismic Response Evaluation of Isolated Bridges Equipped with Fluid Inerter Damper

Abstract

This research investigates the seismic behavior of continuous-span base-isolated bridges integrated with fluid inerter damper (FID) through a linear analytical framework under recorded earthquake excitations. The resisting mechanism of the FID is modelled as a combination of inertial and viscous forces, which are functions of the relative acceleration and velocity between connected nodes. Linear time-history simulations and a series of parametric analyses are conducted to examine how variations in inertance, damping ratio, and installation location affect key seismic response parameters, including deck acceleration, bearing displacement, and substructure base shear. Comparative analyses with conventional viscous dampers and isolation alone establish the relative effectiveness of FID. Analysis indicates that FID effectively reduces deck accelerations through apparent mass amplification, suppresses bearing displacements via viscous damping, and redistributes seismic forces depending on placement strategies. An optimum inertance range is identified that minimizes accelerations without amplifying base shear, with abutment-level placement proving most effective for pier shear control, while intermediate placement provides balanced reductions. Overall, FID consistently outperforms viscous dampers and conventional isolation, underscoring their potential as an advanced inerter-based solution for both new bridge design and retrofit applications.

1. Introduction

Bridges are essential elements of transportation networks, ensuring uninterrupted connectivity and economic stability. Their seismic performance is vital to post-earthquake functionality, as bridge failures can paralyze entire regions. Traditional seismic design philosophies have emphasized increasing the strength and ductility of bridge components to prevent collapse during major earthquakes. While this approach ensures life safety, it often results in excessive inelastic deformation, residual displacement, and costly post-event repairs [1]. Consequently, the emphasis in seismic engineering has shifted from purely strength-based design to performance-based seismic design, which focuses on controlling dynamic responses through energy dissipation and system modification rather than stiffness enhancement.

Among the various seismic control technologies, passive systems are widely used due to their simplicity, reliability, cost effectiveness, and independence from external power sources [2,3]. Various supplemental damping devices—such as viscous, friction, and viscoelastic dampers—have been widely implemented in both building and bridge structures to dissipate seismic energy and reduce excessive vibrations [4]. Among passive control strategies, base isolation is recognized as one of the most efficient techniques for seismic protection of bridges [5,6]. It functions by decoupling the superstructure from ground motion, thereby elongating the structural period and reducing the transmitted forces [7]. Early pioneering studies [8] established the concept, while subsequent studies demonstrated that isolation systems are highly effective in minimizing pier base shear, leading to considerable reductions in transmitted seismic forces [9].

Despite these advantages, base isolation suffers from excessive isolator displacements during strong, near-fault, or long-duration earthquakes, potentially causing bearing overstress or unseating [10,11]. It is also important to note that the effectiveness of isolation systems can be compromised under certain conditions; recent studies have shown that pre-existing ground displacements or foundation movements may reduce isolator deformation capacity and alter their intended behavior, thereby increasing the seismic fragility of isolated bridges [12]. To mitigate this issue, supplemental damping devices such as viscous and friction dampers have been incorporated into isolation systems [13,14]. Although effective at reducing isolator displacement, these dampers introduce a trade-off: increasing damping reduces displacement but increases deck acceleration and transmitted forces [15]. This limitation has driven the development of inertial-based damping systems, capable of simultaneously controlling both acceleration and displacement [16].

A significant advancement in vibration control was achieved with the development of the inerter, introduced by Smith [17]. As a two-node mechanical component, the inerter provides resistance by responding to the relative acceleration between the points at which it is attached. Thus, it provides an apparent mass amplification effect without adding physical mass, allowing for significant improvement in vibration control efficiency. Subsequent studies [18,19] verified the concept experimentally and paved the way for practical mechanical configurations. When integrated with conventional springs and damping elements, the inerter can be configured into various vibration-control systems; examples include the clutching inerter damper, tuned mass damper–inerter, and tuned inerter damper [20,21,22]. These hybrid devices have been shown to achieve enhanced energy dissipation and broader frequency suppression compared to conventional dampers [23,24].

Rack-and-pinion, ball-screw, and hydraulic inerter mechanisms have been introduced, and these inerter mechanisms can produce inertial forces multiple times the magnitude of their physical mass [25,26,27,28]. These devices are particularly suitable for bridges, where increasing physical mass is undesirable but additional apparent inertia is beneficial [29,30,31]. Studies have confirmed that inerter-based systems can effectively reduce deck accelerations, isolator displacements, and base shear simultaneously [32].

The FID represents a significant advancement in this field, combining the principles of hydraulic inertance and viscous damping in a single device [33]. Structurally, an FID consists of a piston–cylinder mechanism with coiled helical fluid channels. When subjected to motion, the piston forces the fluid through the helical path, generating both inertial resistance (acceleration-dependent) and viscous damping (velocity-dependent) forces [34,35]. This dual mechanism enables simultaneous control of accelerations and displacements. The performance of the FID can be tuned through two key parameters: the inertance ratio and the damping ratio, which were validated through experiments that the helical-channel design yields stable inertial effects suitable for large-scale structures [36]. Further research on semi-active FIDs employing magnetorheological fluids [37] has shown that real-time damping adjustment can enhance adaptability to varying seismic intensities.

Despite encouraging findings from previous studies, the integration of FID within bridge isolation systems remains relatively unexplored. Most existing research has focused on simplified single-degree-of-freedom representations, offering limited insights into the complex dynamic behavior of multi-span bridge systems. Consequently, the influence of key design parameters—such as damping ratio, inertance ratio, and device placement—on overall seismic performance has not been systematically quantified. Moreover, the redistribution of seismic forces between abutments and piers resulting from FID installation is still insufficiently understood. Given these limitations, there is a need for a comprehensive evaluation of FID-equipped isolated bridges that accounts for realistic multi-span configurations and recorded ground motions. Such an investigation is essential to identify the governing parameters that influence structural responses and to develop rational guidelines for the design and use of inerter-based control devices in bridges.

To address the lack of detailed insight in previous literature, this study focuses on advancing the understanding of how FIDs influence the seismic response of isolated bridge systems. It examines how inertance, damping, and placement of FID affect deck acceleration, isolator displacement, and base shear. The outcomes are expected to offer design-oriented insights for performance-based seismic design, encourage the practical adoption of FIDs in bridge isolation systems, and support the development of resilient, next-generation bridge infrastructure.

2. Analytical Modelling of Isolated Bridge System Integrated with FID

The schematic layout of the seismically isolated bridge analyzed in this work is depicted in Figure 1a. A continuous multi-span deck is supported on piers and abutments through seismic isolation bearings, and the bridge is arranged straight in plan without skew. FIDs are positioned at both the abutments and the pier locations to provide deck isolation and supplemental energy dissipation. A simplified variant of the helical fluid inerter is the internal-helix fluid inerter, as illustrated in Figure 1b, considered as FID. This configuration comprises a piston with an integrated helical groove on its outer surface moving inside a fluid-filled cylinder. During structural vibration, the motion of the piston drives the working fluid through the helical flow channel, generating the inertial effect as the fluid circulates during piston motion. The inertial force is proportional to the acceleration difference between the damper terminals. Simultaneously, viscous resistance generated by the fluid provides additional energy dissipation. Compared with conventional dampers, the FID thus simultaneously enhances both energy dissipation and apparent inertia, leading to reduced acceleration and displacement responses under dynamic excitation.

In conducting the seismic evaluation, a set of modelling assumptions is employed. The deck is assumed to be longitudinally restrained through transverse diaphragm action and represented as a rigid body with an equivalent mass $m_d$. The abutment is idealized as a rigid support, whereas the pier is modelled as a flexible structural component. The piers are assumed to remain elastic, as the isolation system limits their deformation within elastic ranges. This standard approach for isolated bridges, where seismic demands are primarily governed by isolator deformation [38,39], enables focused evaluation of the FID while avoiding uncertainties from pier yielding. Each pier is assumed to possess a uniform cross-section along its height and is idealized as a lumped-mass model, providing a single horizontal degree of freedom at each of the $n$ discretized nodes [32]. The pier foundations are considered fixed on hard rock or stiff soil, neglecting soil–structure interaction. For bridges on stiff soils or with light substructures, fixed-base models produce responses similar to those with soil–structure interaction. Significant interaction effects occur mainly on soft soils, where foundation flexibility influences time periods and force distribution [39]. The isolator is modelled using equivalent linear stiffness and damping values [38,39]. The FID produces a force proportional to the acceleration difference between its terminals. The bridge model is analyzed under horizontal ground excitations acting in both the longitudinal and transverse directions, whereas vertical motion is disregarded. Impact interaction between the deck and abutments is not considered, assuming sufficient expansion gaps are provided to prevent pounding. The adopted modelling procedure follows the general framework established in previous benchmark analyses of base-isolated bridge systems [32]. The analytical model does not account for vehicle–bridge interaction, and no vehicular traffic loads are included in the seismic simulations. Although this simplification isolates the role of the FID, it is noted that moving traffic can influence the modal properties of bridges. Recent studies have shown that vehicle passage may modify frequency content, affect modal scaling, and alter operational mode shapes [40,41].

The base-isolated bridge model with the FID, as presented in Figure 1c, follows the fundamental configuration adopted in previous studies [32,42]. When stiffness and damping characteristics are defined in the longitudinal and transverse directions, the resultant analytical model represents a simplified version of the benchmark bridge configuration established in previous studies [43]. Since the abutment is modelled as a rigid support, an FID installed at this location is effectively grounded to the ground and responds only to the input ground acceleration. In contrast, an FID is grounded to the flexible pier top, which undergoes both ground excitation and pier dynamic response. As a result, the inertial force developed by the pier-level FID depends on the combined pier acceleration, whereas the abutment-level FID responds solely to ground motion. For an isolated bridge subjected to seismic excitation, the governing motion equations are given by:

$$m_d{\ddot{u}}_d+F_b^a+F_f^a+F_b^p+F_f^p=-m_d{\ddot{u}}_g$$

(1)

$$[m_p]\{{\ddot{u}}_p\}+[c_p]\{{\dot{u}}_p\}+[k_p]\{u_p\}-\{\psi \}(F_b^p+F_f^p)=-[m_p]\{r\}{\ddot{u}}_g$$

(2)

where $m_d$ denotes the mass of the bridge deck, ${\ddot{u}}_d$ represents the deck acceleration relative to the ground, $F_b^a$ and $F_b^p$ correspond to the isolation (or bearing) forces acting at the abutment and pier levels, respectively, while $F_f^a$ and $F_f^p$ denote the forces generated by the FID at these locations, and ${\ddot{u}}_g$ is the ground acceleration induced by earthquake excitation. The matrices $[m_p]$, $[c_p]$, and $[k_p]$ denote the mass, damping, and stiffness characteristics of the free-standing pier, respectively, each having dimensions of $n\times n$. Here, $n$ represents the total number of discretized nodes distributed along the height of the pier. The displacement vector of the pier is given by $\{u_p\}=\{u_1,u_2,\dots ,u_n{\}}^T$, where $u_i$ indicates the horizontal displacement of the $i^{th}$ pier node relative to the ground, and the superscript $T$ denotes the transpose. The position vector $\{\psi \}=\{\mathrm{0,0},\dots ,1{\}}^T$ identifies the top node of the pier, while $\{r\}=\{\mathrm{1,1},\dots ,1{\}}^T$ represents the influence coefficient vector, each having dimensions $n \times 1$.

For the pier considered as a free-standing cantilever, the fundamental vibration period, denoted as $T_p$, is given by

$$T_p={\displaystyle \frac{2\pi }{(1.875{)}^2}}\sqrt{{\displaystyle \frac{{\overline{m}}_p h^4}{EI}}}$$

(3)

Here, ${\bar{m}}_p$ denotes the mass per unit length of the pier, $h$ is the total pier height, and $EI$ represents its flexural rigidity. This expression represents the fundamental period of a uniform cantilever beam undergoing lateral vibration, as originally described by Clough and Penzien [44].

The mass ratio $\gamma$, is defined as the ratio between the total mass of the pier and that of the bridge deck, and can be expressed as follows:

$$\gamma ={\displaystyle \frac{{\overline{m}}_p h}{m_d}}$$

(4)

The overall behavior of the isolation system is characterized by three principal parameters: the isolation period $\left(T_b\right)$, the damping ratio of the isolator $\left({\xi }_b\right)$, and the inertance ratio $\left(\beta \right)$ and the FID incorporated into the bridge model. These parameters are expressed as follows:

$$T_b=2\pi \sqrt{{\displaystyle \frac{m_d}{k_b^a+k_b^p}}}$$

(5)

$$2{\xi }_b{\omega }_b={\displaystyle \frac{c_b^a+c_b^p}{m_d}}$$

(6)

$$\mathrm{\beta }={\displaystyle \frac{b}{m_d}}$$

(7)

Here, $T_b$ denotes the isolation period; $k_b^a$ and $c_b^a$ represent the isolation system’s effective stiffness and viscous damping characteristics at the abutment supports, respectively; while $k_b^p$ and $c_b^p$ correspond to the same properties assigned to the pier locations. The isolation circular frequency is given by ${\omega }_b=2\pi /T_b$, while ${\xi }_b$ denotes the isolation damping ratio. The parameter $\beta$ is defined as the inertance ratio, representing the ratio of the total inertance of the FID, $b$, to the deck mass, $m_d$.

Of the total inertance $b$, a portion $\alpha b$ is assigned to the abutment, and the remaining $(1-\alpha )b$ is assigned to the pier. The coefficient $\alpha$ is referred to as the placement factor, where $\alpha =0$ corresponds to the case where the FID is installed exclusively at the pier, and $\alpha =1$ represents installation only at the abutment. Similarly, $\alpha =0.5$ corresponds to the case where 50% of the total inertance is assigned to the abutment and the remaining 50% is assigned to the pier.

$$F_f^a=\alpha b{\ddot{u}}_d+2{\alpha \xi }_f{\omega }_b{m}_d{\dot{u}}_d$$

(8)

$$F_f^p=\left(1-\alpha \right)b{\ddot{u}}_{dp}+2{\left(1-\alpha \right)\xi }_f{\omega }_b{m}_d{\dot{u}}_{dp}$$

(9)

where $u_{dp} {(u}_d-u_{1)}$ represents the displacement of the bridge deck referenced to the pier top and ${\xi }_f$ represents the damping ratio of the viscous force of the FID.

The pier damping matrix is constructed using the pier’s mode shapes together with its associated natural frequencies, assuming a uniform modal damping ratio of 2% for all vibration modes. The pier is discretized into five nodes $(n=5)$, with an overall height of $h=10 \mathrm{m}$. The mass ratio between the pier and the deck is maintained at a constant value of $(\gamma =0.15)$. Both the pier and abutment bearings are modelled as having the same dynamic characteristics $(k_b^a=k_b^p,\hspace{0.25em}{c}_b^a=c_b^p)$ and are designed in accordance with the corresponding bridge deck weight.

Considering the modelling assumptions and parameter definitions introduced earlier, the dynamic response of the isolated bridge fitted with the FID can be interpreted through six essential governing parameters. These include the inertance ratio of the FID $\left(\beta \right)$, the placement factor $\left(\alpha \right)$, the isolation period $\left(T_b\right)$, the isolation damping ratio $\left({\xi }_b\right)$, the FID damping ratio $\left({\xi }_f\right)$, and the fundamental vibration period of the pier $\left(T_p\right)$. The effectiveness of the FID in controlling seismic response is examined through four principal performance indicators. One of these is the absolute deck acceleration, which accounts for both the structural response and the imposed ground motion, and is expressed as ${\ddot{u}}_d^a={\ddot{u}}_d+{\ddot{u}}_g$. The relative displacement between the bridge deck and the upper end of the pier is expressed as $u_{dp}$, and this quantity is referred to as the pier bearing displacement. The base shear at the abutment is given by $V_a=F_b^a+F_f^a$, while the base shear at the pier is calculated as

$$V_p=F_b^p+F_f^p-\sum _{i=1}^n{m}_i({\ddot{u}}_i+{\ddot{u}}_g)$$

The base shear responses at the abutment and pier locations are presented in normalized form with respect to the deck weight ${(W}_d=m_{d}g)$. The seismic demand transmitted through the structure is primarily influenced by the absolute acceleration of the deck and the resulting base shears developed at the abutments and piers. In contrast, the relative displacement accommodated by the isolation bearings is a crucial measure of the effectiveness and serviceability of the isolation system.

3. Seismic Response Results of the Isolated Bridge Equipped with FID

This section presents the seismic response evaluation of the base-isolated bridge system, integrating the FID, is analyzed using recorded earthquake ground motions. Four representative acceleration records are selected to encompass a broad range of frequency content and intensity levels: (i) the 1940 El Centro earthquake, north–south component; (ii) the 1989 Loma Prieta earthquake, N00E component recorded at the Los Gatos Presentation Center; (iii) the 1994 Northridge earthquake, N90E component recorded at the Sylmar Station; and (iv) the 1995 Kobe earthquake, N00E component obtained from the Japan Meteorological Agency.

The selected ground motion set encompasses a broad intensity range, with PGAs of 0.34 g (El Centro), 0.57 g (Loma Prieta), 0.60 g (Northridge), and 0.83 g (Kobe). A uniform time window of 30 s is applied to each record to facilitate consistent comparison, and their time-history traces are provided in Figure 2.

The dynamic response of the isolated bridge system equipped with the FID, shown in Figure 1c, is obtained by numerically solving the governing equations of motion presented in Equations (1) and (2). The analysis primarily aims to evaluate how variations in the FID parameters; specifically, inertance, damping ratio, and placement factor—are significant in determining the seismic response of the isolated bridge. All structural and isolation characteristics, aside from those associated with the FID, are assumed constant to distinguish the contribution of the FID parameters. The fixed parameters are chosen as follows: fundamental period of the pier ($T_p=0.1\hspace{0.17em}\mathrm{s}$), isolation damping ratio (${\xi }_b=0.1$), and isolation period ($T_b=2.0\hspace{0.17em}\mathrm{s}$).

Figure 3 illustrates how the peak deck acceleration varies with changes in the inertance ratio and FID damping ratio across different earthquake records and device placement configurations. The results indicate that the deck acceleration initially decreases with increasing inertance and reaches a minimum before rising again at higher $\beta$ values. This observation indicates the presence of an optimal inertance ratio that leads to a minimum deck acceleration response in the bridge system. The optimal $\beta$ value corresponding to the minimum acceleration typically lies in the range of 0.2 to 0.5, depending on the FID damping ratio, placement factor, and earthquake characteristics.

The influence of FID damping and placement factor on peak deck acceleration is generally modest across the considered earthquake records. However, for the Loma Prieta (1989) earthquake, the peak deck acceleration shows a comparatively stronger sensitivity to variations in damping ratio than observed in the other ground motions.

In Figure 4, the variation in the peak pier bearing displacement (i.e., the peak deck displacement referenced to the pier top) with the inertance ratio and the FID damping ratio is presented for a range of earthquake inputs and placement configurations. The results show that the peak bearing displacement decreases as the inertance ratio increases and is further reduced at higher values of FID damping. At a damping ratio of ${\xi }_f=0.3$, the influence of $\beta$ on peak bearing displacement becomes relatively minor across most earthquake records. However, for the El Centro (1940) earthquake, the bearing displacement exhibits a comparatively stronger dependence on $\beta$, indicating a more pronounced inertial interaction.

At larger damping ratios, the control of peak bearing displacement is primarily governed by FID damping rather than inertance. The placement factor shows negligible influence on the bearing displacement across all considered earthquake records and damping ratios.

In Figure 5, the variation in the peak abutment base shear with inertance ratio and FID damping ratio is presented for several earthquake inputs and placement cases. A continuous decrease in abutment base shear is noted when the FID is positioned at the pier as the inertance increases. Conversely, when the FID is positioned either at the abutment or distributed between the abutment and pier levels ($\alpha =0.5$ and $\alpha =1$), a reduction in base shear is observed with increasing inertance, followed by a minimum and a gradual increase thereafter. This behavior signifies an optimal inertance ratio at which the abutment base shear is minimized.

For most earthquake records, the optimum $\beta$ corresponding to the minimum abutment base shear is attained for inertance ratios in the range of 0.2 to 0.4, depending on the FID damping ratio and excitation characteristics. However, under the El Centro (1940) earthquake, the abutment base shear initially remains nearly constant and later increases with increasing inertance, showing a distinct deviation from other ground motions. The influence of FID damping on the abutment base shear becomes less significant as the placement factor increases from 0 to 1. Hence, the damping effect of the FID is more pronounced when the device is installed at the pier location.

Figure 6 illustrates how the peak pier base shear varies with the inertance ratio and FID damping ratio under different earthquake records and device placement configurations. In the configuration where the FID is located solely at the abutment, variations in the inertance ratio have minimal influence on the pier base shear. In contrast, when the device is positioned at the pier or distributed between both locations, the pier base shear initially decreases, exhibiting minor fluctuations near the minimum value, and subsequently increases with further increments in inertance. These fluctuations are attributed to the dynamic interaction and superposition between the vibrational responses of the pier segments and the bridge deck.

The influence of FID damping on the pier base shear becomes increasingly significant as the placement factor varies from 0 to 1 across the considered earthquake records. Consequently, the effect of FID damping is most pronounced when the device is positioned at or near the abutment location, reflecting the higher rigidity and reduced deformability of the abutment supports.

Overall, the results indicate that in FID-integrated isolated bridges, reducing one base shear component (either pier or abutment) may lead to a corresponding increase in the other. Given that bridge abutments are generally more rigid than piers, placement of the FID at the abutment level is recommended to effectively control pier base shear without compromising overall system stability.

A comparative evaluation of the seismic performance of the FID and a conventional viscous damper (VD) under varying parameter combinations ($\alpha$, $\beta$, and ${\xi }_f$) and earthquake excitations are summarized in

Table 1. Peak Seismic Response of the Isolated Bridge Equipped with FID and VD under Different Earthquake Records.

El-Centro, 1940
	${\ddot{u}}_d^a$ (g)		$u_{dp}$ (mm)		Va/Wd		Vp/Wd
BIS	0.155		146.3		0.078		0.093
	FID	VD	FID	VD	FID	VD	FID	VD
BIS + FID (α = 0, ${\xi }_f$ = 0.15, β = 0.45)	0.121	0.132	102.4	108.3	0.055	0.058	0.083	0.091
BIS + FID (α = 0.5, ${\xi }_f$ = 0.2, β = 0.25)	0.114	0.127	94.2	97.0	0.058	0.064	0.068	0.078
BIS + FID (α = 1, ${\xi }_f$ = 0.1, β = 0.2)	0.118	0.135	120.2	118.1	0.075	0.076	0.073	0.078
Loma Prieta, 1989
	${\ddot{u}}_d^a$ (g)		$u_{dp}$ (mm)		Va/Wd		Vp/Wd
BIS	0.558		530.4		0.283		0.258
	FID	VD	FID	VD	FID	VD	FID	VD
BIS + FID (α = 0, ${\xi }_f$ = 0.15, β = 0.45)	0.311	0.425	299.6	344.2	0.159	0.184	0.215	0.266
BIS + FID (α = 0.5, ${\xi }_f$ = 0.2, β = 0.25)	0.288	0.393	275.1	296.8	0.147	0.199	0.184	0.208
BIS + FID (α = 1, ${\xi }_f$ = 0.1, β = 0.2)	0.316	0.443	351.4	386.1	0.176	0.253	0.198	0.193
Northridge, 1994
	${\ddot{u}}_d^a$ (g)		$u_{dp}$ (mm)		Va/Wd		Vp/Wd
BIS	0.353		339.5		0.180		0.197
	FID	VD	FID	VD	FID	VD	FID	VD
BIS + FID (α = 0, ${\xi }_f$ = 0.15, β = 0.45)	0.224	0.272	209.5	228.9	0.111	0.121	0.146	0.180
BIS + FID (α = 0.5, ${\xi }_f$ = 0.2, β = 0.25)	0.195	0.253	192.6	200.5	0.100	0.129	0.133	0.141
BIS + FID (α = 1, ${\xi }_f$ = 0.1, β = 0.2)	0.242	0.284	253.4	254.5	0.123	0.160	0.139	0.144
Kobe, 1995
	${\ddot{u}}_d^a$ (g)		$u_{dp}$ (mm)		Va/Wd		Vp/Wd
BIS	0.340		317.0		0.172		0.181
	FID	VD	FID	VD	FID	VD	FID	VD
BIS + FID (α = 0, ${\xi }_f$ = 0.15, β = 0.45)	0.232	0.275	165.0	215.3	0.089	0.117	0.171	0.192
BIS + FID (α = 0.5, ${\xi }_f$ = 0.2, β = 0.25)	0.206	0.251	155.3	186.7	0.107	0.126	0.150	0.158
BIS + FID (α = 1, ${\xi }_f$ = 0.1, β = 0.2)	0.208	0.285	189.5	237.9	0.139	0.162	0.136	0.153

. For consistency of energy-dissipation capacity, the VD is modelled using the same damping ratio (${\xi }_f$) but no inertance contribution ($\beta =0$). This comparison is therefore intended as a baseline representation of a traditional velocity-dependent damping device rather than an optimized VD configuration. The selected FID parameters are those that achieve notable reductions in all key response quantities—deck acceleration, pier bearing displacement, and abutment and pier base shears—across the considered earthquake records. The results consistently show that the inclusion of the FID provides markedly greater reductions in seismic responses compared with the VD, illustrating the added effectiveness of the inertance component.

The bridge responses are evaluated for four configurations: (i) a base-isolated system (BIS) without any supplemental damping; (ii) BIS + FID ($\alpha =0$, ${\xi }_f=0.15$, $\beta =0.45$), in which the FID is installed exclusively at the pier; (iii) BIS + FID ($\alpha =0.5$, ${\xi }_f=0.20$, $\beta =0.25$), where the FID is distributed between the abutment and the pier; and (iv) BIS + FID ($\alpha =1$, ${\xi }_f=0.10$, $\beta =0.20$), representing the case where the FID is installed solely at the abutment.

A comparative assessment of the results presented in Table 1 reveals that incorporating the FID consistently reduces all major seismic response parameters relative to both BIS and the non- optimized VD configurations. Specifically, deck accelerations are reduced by 22–49% relative to the BIS, and by an additional 7–31% compared to the VD. Pier bearing displacements exhibit reductions of 18–50% with respect to the BIS, along with further 3–21% reductions over the VD. Similarly, the abutment base shear decreases by 21–47% relative to the BIS and by 5–33% compared to the VD, while the pier base shear shows comparatively smaller reductions of 11–32% and 3–22% with respect to the BIS and VD, respectively.

The dynamic response histories, which include the deck acceleration, the relative displacement at the pier bearing, and the base shear demands at both the abutment and the pier—under the four recorded earthquake excitations (El Centro, 1940; Loma Prieta, 1989; Northridge, 1994; and Kobe, 1995) are presented in Figure 7, Figure 8, Figure 9 and Figure 10, respectively, for the considered bridge configurations.

4. Conclusions

A comprehensive assessment of the seismic response of the base-isolated bridge incorporating the FID has been carried out using four recorded earthquake excitation histories. Based on the numerical simulations and parametric analyses, the following conclusions are drawn:

• Deck acceleration decreases with increasing inertance ratio up to an optimum range between 0.2 and 0.5, beyond which further increase results in amplification. The influence of inerter damping and placement factor remains comparatively limited across the examined earthquake records.
• Pier bearing displacement decreases with increasing inertance and is further reduced at higher inerter damping levels. At larger damping values, the response control becomes predominantly governed by damping effects, while the placement factor exerts negligible influence across the examined earthquake records.
• For pier-level placement of FID, the abutment base shear decreases steadily with increasing inertance and is further reduced at higher inerter damping. Conversely, for abutment-level placement, the abutment base shear decreases up to an optimum inertance and then increases, remaining largely insensitive to variations in inerter damping.
• For abutment-level placement of FID, the pier base shear remains largely insensitive to variations in inertance, although the effect of inerter damping becomes more pronounced at this location. For pier-level or combined placements, the pier base shear initially decreases with increasing inertance and subsequently increases at higher inertance values. These fluctuations are attributed to the dynamic interaction between the pier and the deck.
• The distribution of seismic forces is highly sensitive to the placement of FID, and achieving balanced performance requires the joint optimization of inertance, inerter damping, and placement configuration. Abutment placement is generally more effective for reducing pier base shear due to the greater stiffness of the abutments.
• For the considered cases, the FID consistently outperforms both conventional viscous dampers and the base-isolated bridge alone, consistently minimizing deck acceleration, limiting bearing displacement, and lowering base shear demands at both the abutments and piers, regardless of earthquake input or device placement. This performance advantage is further corroborated by the time-history response analyses for all cases.

This study is restricted to numerical analysis, as no laboratory or field validation of the proposed FID has yet been performed. Experimental verification, through prototype development and physical testing, may be considered in future research as the device concept matures.