Seismic Damage Pattern Analysis of Long-Span CFST Arch Bridges Based on Damper Configuration Strategies

Abstract

Variations in damper configuration strategies have a direct impact on the seismic damage patterns of long-span deck-type concrete-filled steel tube (CFST) arch bridges. This study developed an analysis and evaluation framework to identify the damage category, state, and progression sequence of structural components. The framework aims to investigate the influence of viscous dampers on the seismic response and damage patterns of long-span deck-type CFST arch bridges under near-fault pulse-like ground motions. The effects of different viscous damper configuration strategies and design parameters on seismic responses of long-span deck-type CFST arch bridges were systematically investigated, and the preferred configuration and parameter set were identified. The influence of preferred viscous damper configurations on seismic damage patterns of long-span deck-type CFST arch bridges was systematically analyzed through the established analysis and evaluation frameworks. The results indicate that a relatively optimal reduction in bridge response can be achieved when viscous dampers are simultaneously installed at both the abutments and the approach piers. Minimum seismic responses were attained at a damping exponent α = 0.2 and damping coefficient C = 6000 kN/(m/s), demonstrating stability in mitigating vibration effects on arch rings and bearings. In the absence of damper implementation, the lower chord arch foot section is most likely to experience in-plane bending failure. The piers, influenced by the coupling effect between the spandrel construction and the main arch ring, are more susceptible to damage as their height decreases. Additionally, the end bearings are more prone to failure compared to the central-span bearings. Implementation of the preferred damper configuration strategy maintains essentially consistent sequences in seismic-induced damage patterns of the bridge, but the peak ground motion intensity causing damage to the main arch and spandrel structure is significantly increased. This strategy enhances the damage-initiation peak ground acceleration (PGA) for critical sections of the main arch, while concurrently reducing transverse and longitudinal bending moments in pier column sections. The proposed integrated analysis and evaluation framework has been validated for its applicability in capturing the seismic damage patterns of long-span deck-type CFST arch bridges.

1. Introduction

Since the 20th century, frequent seismic events have been observed worldwide, including the Northridge earthquake in California, the Rudbar earthquake in Iran, and the Wenchuan earthquake in western China. CFST arch bridges have been widely applied in mountainous and canyon regions, owing to the efficient synergy of their material properties, as well as structural forms and construction methods that overcome topographical constraints. Some CFST bridges in mountainous areas are located in high seismic intensity zones and are subjected to the challenge of strong seismic actions. While the CFST arch bridge was initially developed in Europe and North America, its application has been more extensively promoted and rapidly advanced in China. China’s first CFST arch bridge, the Wangcang Donghe Bridge in Sichuan Province, was completed in 1990 in a high-seismic region of western China, subsequently triggering a rapid evolution of CFST arch bridge technology nationwide [1]. However, the construction history of CFST arch bridges in China is relatively short, and they have not yet been subjected to the test of a major earthquake, which poses significant challenges for the seismic performance evaluation and vibration mitigation design of the numerous existing and newly built long-span deck-type CFST arch bridges in service [2]. During the 2008 Wenchuan earthquake, multiple CFST arch bridges were subjected to severe damage, including piers cracking, buckling arch ribs, and damage to connection components, revealing near-fault earthquakes’ destructive effects [3]. Structures in near-fault regions are often subjected to severe damage due to the potentially devastating ground motions. To address the above issues, it is indispensable to have effective seismic mitigation methods and seismic damage analysis and evaluation frameworks.

In the context of seismic mitigation for long-span arch bridges, an extended KDamper (EKD) concept is proposed by Konstantinos A. Kapasakalis et al. [4] for implementation as a seismic base absorber. A dedicated constrained optimization problem was formulated to determine the optimal system parameters, and the effectiveness of EKD as a potential seismic retrofit option was ultimately verified. Hoehler et al. [5] investigated the seismic retrofit of the Kiskiminetas River bridge. In their study, both linear and nonlinear simulation results were compared, and the seismic performance of the strengthened arch structure was evaluated. Deng et al. [6] analyzed variations in the capacity demand ratio of key components under different viscous damper parameters. The study investigated the energy dissipation efficiency and rational parameter design of dampers under seismic action, proposing optimal parameter settings to enhance structural resilience. As for the seismic damage evaluation of long-span deck-type arch bridges, Xia et al. [7] analyzed the failure mechanisms and ductility of high axial–load ratio components by establishing quantitative seismic performance indices. This work provides critical data to address the lack of codified provisions for CFST members in current bridge design specifications. Zhang et al. [8] used N–M correlation curves and capacity demand ratios to investigate the seismic performance of the main arch ring and the piers on the arch, respectively. The results show that arch piers usually suffer damage before the main arch ring. Xu et al. [9] proposed an approach to analyzing the static-dynamic sequences considering various factors. It was found that the construction process of long-span deck-type CFST arch bridges has a significant impact on their seismic damage outcomes. Current research predominantly focuses on seismic reduction methods and structural seismic damage assessment as isolated domains. Some of them compare the seismic performance of a specific bridge before and after the implementation of a damping scheme. Few scholars have analyzed the seismic damage patterns of long-span deck-type CFST arch bridges after implementing rational shock absorption measures. In fact, such damage patterns can provide valuable references for bridges of the same type under similar site conditions. Furthermore, a more reasonable and effective seismic damage assessment method for long-span deck-type CFST arch bridges, which is a kind of complex spatial structure, is also worth exploring further.

From the perspective of seismic mitigation schemes and the seismic damage patterns, this paper focuses on a long-span deck-type CFST arch bridge under the influence of near-fault ground-shaking. Then, the analysis and evaluation framework and refined finite element analysis modeling are established by taking into account material and geometric nonlinearities. Based on this model, the effects of damper configuration and parameter selection on the seismic response of the structure were investigated. Finally, the influence of the optimized scheme on the seismic damage patterns of long-span deck-type CFST arch bridges was analyzed. In this study, a damage pattern identification framework is proposed, and the typical seismic damage characteristics of long-span CFST arch bridges subjected to near-fault earthquakes are systematically revealed. The findings, validated through case studies, are recommended as references for similar CFST arch bridge seismic reduction designs and research for seismic damage patterns.

2. Seismic Damage Analysis and Evaluation Framework

Current seismic damage pattern analysis methodologies for long-span deck-type CFST arch bridges are characterized by low approach consistency and a lack of well-defined key component ineffectiveness criteria. The seismic damage analysis and evaluation framework proposed in this paper, with a view to identifying the damage categories and states of the structure under different ground motion intensities, embeds the component condition assessment based on the capacity demand ratios method (C/D) into the incremental dynamic analysis (IDA) curve. Then, the IDT-based damage sequence identification method is introduced to establish a framework for seismic damage patterns that integrates damage category, state, and order analysis. Compared with the existing seismic damage analysis methods, this framework can accurately identify the damage modes of the components, study the seismic damage pattern for the destruction styles, and obtain accurate analysis results.

2.1. Damage Analysis Methods

In this paper, nonlinear time history analysis using incremental dynamic analysis is used to investigate the seismic damage patterns of long-span deck-type CFST arch bridges in near-fault earthquake zones under the deployment of a viscous damper scheme. It combines the capacity demand ratio method, initial damage time, and damage indices of key bridge components.

(1)

Capacity Demand Ratios Method

The capacity to demand ratio (CDR) method, popularized by the U.S. Federal Highway Administration (FHWA), is primarily applied to the seismic strengthening of bridges. This method assesses the seismic safety of bridge structures by analyzing the capacity-to-demand ratios of key components, revealing the correlation between damage states and seismic intensity. If the C/D value is less than 1.0, it indicates that the member may be in a hazardous condition under seismic action. The capacity-to-demand ratio is calculated as:

$$r_i={\displaystyle \frac{C_i}{{\displaystyle \sum N{S}_i+D_{\mathrm{EQ}}}}}$$

(1)

where C_i is the seismic capacity of the structural element itself; ${\displaystyle \sum N{S}_i}$ is the response of the structure to non-seismic loads; D_EQ is the response of the structure due to seismic action alone.

This assessment does not involve interactions between members, but rather focuses on the performance of each member, making it a more conservative evaluation method, which is more appropriate for long-span deck-type arch bridges.

(2)

Incremental Dynamic Analysis

Incremental dynamic analysis (IDA) is widely recognized in seismic engineering as a parametric analysis method for evaluating structural seismic capacity and investigating collapse. The method uses the amplitude modulation coefficient SF to calibrate the intensity of selected seismic waves. The results obtained from the capacity-to-demand ratio method are utilized as the dependent variable of IDA curves, which not only maps the correlation between the ground vibration intensity and the structural response, but also reveals the development path of the structure from elasticity to plasticity, and even the complete collapse process.

(3)

Initial Damage Time

The selection of an appropriate damage index is fundamental to the quantitative assessment of the structure’s performance state, and there is a time-based sequence of damage or reaching a certain limit state under seismic action. Building on this, initial damage time (IDT) is adopted to represent the time when the structure first reaches the damage state in the seismic time domain. The concept of IDT can be effectively applied to classify damage categories within individual structures and chronologically sequence damage initiation across different structural systems under seismic action. A lower IDT implies early damage and weaker seismicity, while a higher IDT reflects stronger seismic resistance. Figure 1 illustrates the whole process of damage analysis by the seismic damage analysis and evaluation framework.

2.2. Indicator Selection

The selection and quantification of damage indicators for long-span deck-type CFST arch bridges help to identify the structural damage state, enabling the focus on the most critical factors in structural damage under complex seismic actions [10]. Building on this foundation, the methodology for determining each of the key structural damage indicators is described in detail below.

The main arch ribs in the case-study bridge are constructed with concrete-filled steel tubular sections of doubly symmetric dumbbell-shaped cross-sections. Given their direction-dependent load-bearing capacities, the P-M₂-M₃ yield surface method [2] (PMM method) is employed for section capacity analysis. The initial damage state of a component is identified when its time-response curve first exceeds the boundary of the P-M₂-M₃ interaction surface, with the corresponding IDT and $f_{\mathrm{IDT}}\left({\mathrm{P},\mathrm{M}}_2{,\mathrm{M}}_3\right)$. The procedure for calculating the P-M₂-M₃ yield surface is as follows:

(1)

Fiber division of the component cross-section according to the section form and material distribution

(2)

The fiber finite element method is used to calculate the cross-section P-M₂-M₃ correspondence, the direction of bending calculations to 20° as a gradient from 0° to 180°, and each control cross-section to form 10 analysis angles.

(3)

The analyzed data are compiled and plotted within a 3D Cartesian coordinate system to form the yield surface.

Arch piers differ significantly from conventional piers in their connection configurations: the base of arch piers is rigidly connected to the main arch rib, whereas conventional piers are embedded in the foundation substructure. This distinct connection detail results in substantially weaker boundary constraints for spandrel piers compared to traditional piers [10]. Based on the literature [8], the equivalent yield curvature of the section is adopted as the damage index for the arch piers in this study. Combined with the incremental dynamic analysis (IDA) method, the capacity–demand ratio C/D is used to judge the yielding state of the piers. When C/D > 1, it indicates that the pier is undamaged; conversely, when C/D < 1, the pier is considered to be damaged.

Given that the bearing is prone to serious damage under strong earthquakes, such as dislocation and falling beams, studying their damage patterns is of significant theoretical and practical importance for optimizing seismic mitigation measures and ensuring structural safety. Therefore, bearing displacement and bearing shear force are adopted as damage indices following Reference [11], with the ultimate displacement $D_{\mathrm{u}}$ and shear strength $V_u$ selected as damage threshold values. Analytical model: The general layout of the reference bridge case is arranged as: Left approach spans: (51 + 2 × 66) m T-beam continuous girders; Main span: 500 m concrete-filled steel tubular (CFST) arch bridge; Right approach spans: (2 × 66 + 51) m T-beam continuous girders. The centerlines of the arch rib, upper chord, and lower chord steel tubes are defined by a catenary curve, with a calculated arch rise of 105 m and a rise-to-span ratio of 1/4.76. The main beam of the building on the arch is divided into three couples, using long couples of continuous steel box girders with a span of 10 × 40.8 m, and the bearings are made of spherical steel bearings.

2.3. Finite Element Modeling

In recent years, numerous studies have demonstrated that considering both geometric and material nonlinearities is essential for capturing the true seismic response of structural systems. For instance, Reference [12] employed a higher-order nonlinear modeling approach to analyze seismic responses, highlighting the role of coupling effects and restoring force models. Reference [13] proposed a novel safety evaluation method that accelerates material nonlinear analysis by incorporating sectional constitutive behavior into the geometric nonlinear framework. Therefore, in this study, both geometric and material nonlinearities are considered in the modeling process to accurately represent the behavior of the bridge under investigation.

A three-dimensional finite element model of the entire bridge was established using the finite element analysis software SAP2000 (v25). The model takes into account both material and geometric nonlinearities [14]. In the finite element model, the connections between the main beam and bearings, as well as between the bearings and piers, are simulated using rigid links (Body constraints) in SAP2000, and the arch piers are rigidly connected to the arch ribs. The vertical behavior of bearings is modeled using linear elastic link elements (accounting for internal force effects), while the horizontal response is simulated with elastoplastic (Wen model) elements, and all remaining structural components and connections are idealized as linear elastic members [15,16]. The CFST arch rib cross-sections are simulated using a cross-section discretization tool, with material constitutive defined by a parabolic strain-hardening model. The bridge deck pavement and other secondary permanent loads are simulated using nodal loads and uniformly distributed loads [17]. The expansion joint gap at the junction pier was modeled using a gap unit to simulate the boundary nonlinearity [18], with the gap taken as 0.06 m and the collision stiffness taken as 10⁶ kN/m. When considering the material properties corresponding to different fibers, the kinematic constitutive model is employed [19]. Fiber plastic hinges are set in the arch ribs and piers on the arch to consider the material nonlinearity, and the model considers the P-Δ effect. Rigid arch bridges typically have a fundamental frequency between 2.5 Hz and 5.3 Hz. In contrast, this bridge has a fundamental frequency of 0.24 Hz and a natural period of 4.20 s, indicating low stiffness and a flexible structural behavior. The finite element model of the whole bridge is shown in Figure 2.

2.4. Selection of Ground Motions

Currently, the region within 20–30 km of the fault rupture plane is generally defined as the near-fault region in seismic engineering practice [20,21]. Hayden et al. [22] proposed using peak-to-peak velocity (PPV) and normalized cumulative squared velocity (NCSV) to characterize the pulse characteristics of pulse-like ground motion. Figure 3 presents a comparative analysis of NCSV distributions between near-fault pulse-like ground motion and ordinary ground motion. The NCSV at a specific time $t_m$ in the seismic record is calculated as:

$$NCSV(t_m)={\displaystyle \sum _{i=1}^m{V}_i^2}/{\displaystyle \sum _{i=1}^n{V}_i^2}$$

(2)

where $t_m$ represents the m-th time step; $V_i$ represents the velocity at the i-th time step, with a total of n time steps in the entire ground motion record.

As shown in Figure 3, during the pulse period of an earthquake, pulse-type ground motions exhibit a significant increase in NCSV within the PPV period compared to non-pulse-type ground motions, demonstrating distinctly different growth patterns.

Based on previous research [21,22,23], seven near-fault pulse-like ground motions (No. 1 to No. 7) were selected from the PEER ground motion database. Some of the parameters of these seismic waves are shown in

Table 1. Basic information of selected ground motions.

Serial Number	Earthquake Record	Events	Year	Magnitude	Distance (km)	Vs30 (m/s)	PGA (g)
No. 1	RSN180	Imperial Valley-06	1979	6.53	3.95	205.63	0.5941
No. 2	RSN1085	Northridge-01	1994	6.69	5.74	370.52	0.4486
No. 3	RSN1161	Kocaeli Turkey	1999	7.5	10.9	196.23	0.3429
No. 4	RSN1489	Chi-Chi_Taiwan	1999	7.62	3.76	487.27	0.5370
No. 5	RSN1493	Chi-Chi_Taiwan	1999	7.62	5.95	454.55	0.3644
No. 6	RSN6962	Darfield_New Zealand	2010	7.78	1.54	295.74	0.5066
No. 7	RSN8606	El Mayor-Cucapah	2010	6.80	6.30	5.38	0.6830

.

According to the seismic design code in China [24], the structural damage severity of long-span deck-type arch bridges may be underestimated if only unidirectional seismic excitation is considered. When performing nonlinear time–history dynamic analysis, adopting a three-dimensional input method of longitudinal (X) + transverse (Y) + vertical (Z) directions is more suitable for the three-dimensional dynamic coupling characteristics of long-span deck-type CFST arch bridges. Meanwhile, the long-span deck-type CFST arch bridge has a higher seismic risk in the transverse direction; thus, in the subsequent amplitude modulation analysis of seismic waves, the Y-direction is selected as the main direction of input, and the seismic input ratio is defined as longitudinal: transverse: vertical = 0.65:1:0.3.

3. Deployment and Parameters Comparison of FVDs

Viscous dampers, classified as velocity-dependent energy dissipation devices with inherently stiffness-free characteristics, offer superior energy dissipation capacity and stable hysteretic performance. They can function effectively even when structural displacement responses are small. By the reciprocating motion of the piston, the liquid flows through the small holes at the top of the piston to generate a damping force. The greater the relative velocity, the better the energy dissipation performance of the damper. It has been widely used in major engineering structures all over the world [25]. This paper takes viscous dampers as external damping elements to explore their reasonable deployment and selected parameter criteria in a long-span deck-type CFST arch bridge. The damping force–velocity relationship of viscous dampers is theoretically expressed as:

$$F=\mathrm{C}\cdot sgn(v)\cdot |v{|}^{\alpha }$$

(3)

where C denotes the damping coefficient, experimentally determined through cyclic loading tests; sgn(·) represents the signum function defining the directionality of damping forces; α indicates the velocity exponent governing nonlinearity.

The Maxwell model has been commonly used in SAP2000 to simulate viscous dampers [26]. The model is composed of a spring unit and a damper unit in series, see Figure 4. The mechanical expression for the Maxwell model is:

$$F=K_{\mathrm{d}}\cdot d_k=C\cdot \mathrm{sgn}\left(d_d\right)\cdot \left|{\displaystyle \frac{d_d}{{\nu }_0}}\right|$$

(4)

where $K_d$ represents the spring stiffness; $d_k$ denotes the spring deformation; $d_d$ indicates the damper deformation; ${\nu }_0$ stands for the reference velocity, typically set to 1 m/s. When analyzing the seismic reduction in viscous dampers, a higher value of spring stiffness (generally recommended to be 10⁸ kN/m) will be set to ignore the effect of the elastic deformation of the damper itself.

The response analysis results presented in this section are averaged across seven near-fault pulse-like ground motions, aiming to holistically assess the dynamic responses of the long-span deck-type CFST arch bridge under seismic excitation and quantify the seismic reduction effectiveness of the damper deployment method and selected parameters.

3.1. Reasonable Configuration Strategies of FVDs

Long-span deck-type CFST arch bridges exhibit a longer fundamental vibration period and greater in-plane flexibility compared to out-of-plane stiffness, while the vibration coupling between piers and arch ribs of different heights varies considerably due to the influence of higher-order vibration patterns of piers and arch ribs. To fully leverage the energy dissipation capacity of viscous dampers, this study conducts a comprehensive evaluation of structural compatibility within the long-span deck-type CFST arch bridge system, and a relatively better deployment method is determined, i.e., the viscous dampers are installed in the abutment, Pier ① and the combination, respectively (see Figure 5 for details).

In order to clarify the influence of viscous dampers on the seismic response of a long-span deck-type CFST arch bridge, the influence rate of viscous dampers on the seismic response of the structure is defined as: $\mathrm{R}=\left({\mathrm{R}}_{\mathrm{z}}{-\mathrm{R}}_{\mathrm{w}}\right)/{\mathrm{R}}_{\mathrm{w}}\times 100\%$. Here, the average of the maximum value of the response envelope of the structure without dampers is the average of the maximum value of the response envelope of the structure with damping.

(1)

Effect of damper configuration strategies on seismic response of arch ribs

The arch ribs of long-span deck-type CFST arch bridges, as a typical compression structure, exhibit more intense spatial dynamic responses under near-fault pulse-like ground motions. The arch foot, as a seismically vulnerable region, is highly susceptible to buckling, while the internal force amplification and vibration response of the lower chord are more pronounced compared to the upper chord [27]. Due to space limitations, the lower chord of the main arch is selected as a representative component to analyze the seismic response of a typical main arch section of long-span deck-type CFST arch bridges under near-fault ground motions.

Figure 6 presents the mean standard deviation of the envelope extrema for the axial force and bending moment of the lower chord in the arch ring. The results indicate that the overall internal force distribution of the arch ring follows the same pattern across different damper layout configurations. Specifically, the axial force response exhibits a “U”-shaped distribution, while the in-plane bending moment response presents a multi-peak distribution. Figure 6a shows that the peak axial force of the arch ring occurs at the arch foot under all three damper layout configurations, with their respective influence rates on the seismic response of the arch foot axial force being −1.66%, −0.57%, and −1.72%. Figure 6b similarly indicates that the peak in-plane bending moment of the arch ring occurs at the arch foot under different damper layout configurations, with the seismic response influence rates at the arch foot section being −0.97%, −1.07%, and −1.95%, respectively. The above analysis indicates that the addition of viscous dampers has a limited impact on the seismic response of the main arch ring in long-span deck-type CFST arch bridges. This is primarily because the release of seismic energy first impacts the arch ring, which bears the primary structural forces. When viscous dampers are installed on the abutment and approach pier, their activation lags behind the initial accumulation of the arch ring’s seismic response. Energy dissipation by the dampers begins only after the seismic energy has propagated to the spandrel structure. Overall, Scheme 3 demonstrates the greatest advantage compared to the other two schemes, achieving the most effective reduction in both the axial force response and the peak in-plane bending moment of the arch ring. Scheme 1 ranks second, while Scheme 2 has the least impact.

(2)

Effect of damper configuration strategies on seismic response of piers

Figure 7 illustrates the influence rate R of the seismic response of pier internal forces under the three damper configurations (Discrete points and average value of the influence rate R). As observed in Figure 7a, the longitudinal shear force influence rate R in the piers is unevenly distributed. Notably, the influence rate in the taller piers on both sides is even found to exceed zero after the dampers are installed. For the shortest piers, No. 4 and No. 7, the seismic response reduction is relatively significant when dampers are arranged according to Scheme 1, with shear force influence rates of −0.59% and −0.97%, respectively. In contrast, under Scheme 2, the influence rate R of the shorter piers exceeds zero, indicating an actual increase in seismic response. Scheme 3 also reduces the shear force response of No. 4 and No. 7 piers, though its effectiveness is inferior to that of Scheme 1. Figure 7b shows that the moment influence rate R of the same pier varies across different damper configurations, with values that can be either positive or negative. Scheme 1 effectively reduces the seismic response of the piers in a stable manner, whereas the other two schemes tend to amplify the longitudinal bending moment response in most cases. When dampers are arranged according to Scheme 1, the reduction rates of the bending moment response for Piers 1# to 4# are 2.05%, 1.08%, 1.05%, and 0.88%, respectively. This indicates that the taller the pier, the more effective the damping system is in reducing its seismic response. For tall piers in flexible arch bridges, their small cross-sectional dimensions, great height, and high flexibility make them particularly sensitive to higher-order vibration configurations. Additionally, plastic hinge zones are more likely to form in the upper and middle sections of the piers, and the installation of dampers may further exacerbate this effect. Therefore, for the tallest piers, the installation of dampers may instead increase the peak shear force and bending moment. Overall, arranging dampers according to Scheme 1 provides the most stable reduction in pier seismic response.

(3)

Effect of damper configuration strategies on seismic response of bearings

Figure 8 shows the analysis results of the bearing shear force and displacement response influence rates under the three damper configurations (due to the large number of bearings and the concentration of the discrete points around the average value, only the average value is presented). As shown in Figure 8a, except for the bearing at the center of the central-span, the shear force response influence rates of other bearings are close to zero, indicating that the dampers have minimal impact on the shear force at these bearings. The shear force response influence rates of the mid-span bearing 6 under the three damper layout configurations are −2.74%, −0.44%, and −2.91%, respectively, while those of the side-span bearing 6 are −2.42%, −1.04%, and −2.76%, respectively. This confirms that the dampers effectively reduce seismic forces on the mid-span bearing. Among the three configurations, Scheme 3 demonstrates the best seismic performance, followed by Scheme 1, with Scheme 2 being the least effective. Figure 8b further confirms the advantage of Scheme 3 in reducing the displacement response of the side-span bearings. For the mid-span bearing 6, the influence rates under the three damper layout configurations are −3.00%, −1.79%, and −5.37%, respectively. Similarly, for the beam-end bearing 11, the displacement response influence rates are −3.69%, −2.31%, and −4.22%, respectively. Once again, Scheme 3 proves to be the most effective.

3.2. Optimization of Damper Parameters

After determining the viscous damper deployment, more specific parameters for the viscous dampers need to be selected. Existing studies indicate that the seismic mitigation effectiveness of viscous dampers is influenced by the damping coefficient C and the damping exponent α [6]. The parameter combinations of dampers significantly influence the seismic response of the bridge, and precise tuning of these parameters can effectively enhance seismic mitigation performance. Based on the characteristics of this bridge and the empirical selection of viscous damper parameters for similar CFST arch bridges [28,29,30], the damping coefficient C is determined within the range of 1000–6000 kN/(m/s), while the damping exponent α is set between 0.2 and 0.8. The specific parameter values of the viscous dampers are listed in

Table 2. Parameters of viscous dampers.

Damping Exponent α	0.2	0.3	0.5	0.7	0.8
Damping coefficient C (kN/(m/s))	1000	1000	1000	1000	1000
	2000	2000	2000	2000	2000
	3000	3000	3000	3000	3000
	4000	4000	4000	4000	4000
	5000	5000	5000	5000	5000
	6000	6000	6000	6000	6000

.

Regarding the impact of parameter variations on the arch ring section, the envelope values of the seismic response at the lower chord arch foot section under different damper parameters are shown in Figure 9. As shown in Figure 9a,b, when the damping exponent α remains constant, the axial force and bending moment responses at the lower chord arch foot section decrease linearly with increasing damping coefficient C. The minimum response values occur when α = 0.2 (the lowest value) and C = 6000 (the highest value). Additionally, for a fixed damping coefficient C, the seismic response is consistently minimized when α = 0.2. When the damping exponent α is set to 0.2 and the damping coefficient C increases from 1000 to 6000, the axial force response decreases by 2.8%, 2.7%, 2.5%, 2.3%, and 2.2%, respectively, showing a declining trend in the reduction rate. Conversely, when the damping coefficient is fixed at 6000 and the damping exponent α increases from 0.2 to 0.8, the bending moment response at the lower chord arch foot section increases by 0.62%, 0.93%, 0.99%, 0.93%, 0.89%, and 0.89%, When the damping coefficient C is 3000, the increase in bending moment response is the most significant, indicating that at this point, the bending moment response is relatively more sensitive to variations in the damping exponent α. From the above, it can be seen that for the seismic response of the lower chord arch foot section, the better values of the selected parameters of the damper are represented by the damping index α being taken as 0.2 and the damping coefficient C being taken as 6000.

Regarding the impact of parameter variations on the piers, the sections with the maximum longitudinal and transverse responses are selected as the study objects. The envelope values of the seismic response at the key pier sections under different damper parameters are shown in Figure 10. As observed from the figure, when the damping exponent α remains constant, the longitudinal bending moment response gradually decreases as the damping coefficient C increases. Specifically, when C increases from 1000 to 2000, the bending moment response drops rapidly, with a 0.5% reduction. However, when C exceeds 2000, the decrease in bending moment response becomes more gradual. The transverse bending moment response exhibits different trends depending on the damping exponent α. When α is set to its maximum value of 0.8, and the damping coefficient C increases from 1000 to 2000, the peak bending moment response decreases significantly by 60.42%. Beyond this point, the response values tend to stabilize. When the damping exponent α is at its minimum value of 0.2, the pier bending moment decreases as the damping coefficient C increases, but only when C < 4000. When C ≥ 4000, the bending moment begins to increase with further increases in C. In contrast, when α = 0.5, the bending moment response exhibits an irregular distribution rather than a consistent trend. This phenomenon may be attributed to the nonlinear sensitivity of the viscous damper model. At higher damping levels, a sudden redistribution of structural response or local resonance effects may occur, leading to changes in the bending moment of the column section. Considering the seismic response of the critical pier, the selected parameters are determined as follows: damping exponent α = 0.2 and damping coefficient C = 6000 kN/(m/s).

Regarding the impact of parameter variations on the bearing, Figure 11 shows the seismic response envelope values for bearing 1 under different damper parameters. The figure indicates that, when the damping exponent is low, increasing the damping coefficient results in a more stable reduction in the shear force response of the bearing. However, when α is set to its maximum value of 0.8, further increases almost no longer reduce the bearing’s shear force and displacement response values. When the damping exponent α is set to 0.3 and the damping coefficient increases from 1000 to 6000, the shear force response reduction rates at side-span bearing 1 are 0.0027%, 0.0029%, 0.0014%, 0.0034%, and 0.0027%, respectively, as the damping coefficient increases. The displacement response reduction rates at the mid-span bearing are 0.33%, 0.34%, 0.17%, 0.41%, and 0.32%, respectively. When the damping exponent is set to 0.3 and the damping coefficient C is set to 6000, the minimum shear force and displacement responses are obtained for both the side-span bearing and mid-span bearing 1. At this point, if the damping coefficient is set to 0.2, the response values are almost identical to those at α = 0.3, C = 6000. The observed pattern is generally as follows. When the damping exponent is fixed, increasing the damping coefficient results in a nonlinear decrease in both the shear force and displacement response values of the bearings. The damper has a negligible impact on reducing the shear force response but has a moderate impact on reducing the displacement response. Considering the seismic response of the critical bearings, the optimal damper parameters are determined as follows: damping exponent α = 0.2 or 0.3, and damping coefficient C = 6000.

The analysis results from the above charts effectively illustrate the different response patterns of key structures as the damper layout configuration and damper parameters vary. The damper layout has a limited effect on reducing the seismic response of the arch ring and piers, while it is more effective in reducing the seismic response of certain bearings. Considering all three aspects, Scheme 3 is selected as the final layout scheme. The seismic response of the bridge structure is generally positively correlated with the damping exponent and negatively correlated with the damping coefficient. Considering the specific impact of both parameters on the responses of multiple bridge structures, the optimal damper parameters are determined as follows: damping exponent α = 0.2 and damping coefficient C = 6000.

4. Effect of FVD on Seismic Damage Patterns

4.1. Seismic Damage Categories and States of Arch Rings

In this study, the PGA of the selected ground motions is adjusted in 0.1 g increments from 0.1 g to 1.2 g, using scaling factors (SF) with identical increments, to investigate the influence of dampers on the seismic damage patterns of the main arch rib in the long-span deck-type CFST arch bridge and explore the effectiveness of dampers in mitigating arch rib responses. Limited to space, this paper selects a representative section at the lower chord arch foot section for analysis.

Figure 12 presents the IDA curves of the axial force-bending moment (P-M₂ and P-M₃) at the arch foot section under seismic wave No. 1 without dampers, demonstrating the cumulative structural damage effects and dynamic evolution process under seismic excitation. As shown in Figure 12a,b, the seismic response curves of the arch foot section simultaneously exceed the P-M yield lines at PGA = 0.6 g. The initial damage time (IDT) can differentiate between in-plane (P-M₃) and out-of-plane (P-M₂) bending damage: Out-of-plane bending damage (P-M₂): IDT = 26.98 s; In-plane bending damage (P-M₃): IDT = 19.12 s. The shorter IDT for in-plane bending indicates that in-plane bending damage initiates before out-of-plane damage at the arch foot section. Due to the weaker transverse seismic resistance compared to longitudinal seismic resistance at the lower chord arch foot section, Figure 13 presents the IDA curves of the P-M₃ curves for the lower chord arch foot section under seismic wave No. 1. When PGA = 0.6 g and no dampers are installed, the seismic response curve of the chord section at the arch foot exceeds the P-M₃ yield surface boundary, indicating a high-risk state of this section under the seismic wave. After damper installation, the response curve no longer exceeds the P-M₃ yield surface boundary, which demonstrates that the dampers reduce the damage probability of the lower chord arch foot section under seismic wave No. 1.

Using the PGA of ground motions at the initial structural damage as a reference, the damage-initiation PGA values of the lower chord arch foot section of the main arch rib under seven ground motions are summarized and compared (Figure 14) based on the presence or absence of dampers. The influence of dampers on the damage-initiation PGA of the lower chord arch foot section varies under different ground motions. Under seismic waves No. 2, No. 4, No. 5, and No. 7, the installation of dampers exerts almost no influence on the damage-initiation PGA of the arch foot section. Under seismic waves No. 1, No. 3, and No. 6, the damage-initiation PGA values of the lower chord arch foot section are 0.6 g, 0.6 g, and 0.4 g when dampers are not installed. In contrast, with dampers installed, the corresponding values increase to 0.9 g, 0.8 g, and 0.7 g, demonstrating that dampers enhance the damage-initiation PGA of the lower chord arch foot section.

4.2. Seismic Damage Categories and States of Piers

Under seismic actions, the response patterns of the left and right half-span piers of the arch bridge are essentially symmetrical. Therefore, Figure 15 presents the IDA curves of the curvature capacity-to-demand (C/D) ratios for key sections of pier (1#–4#). As the seismic acceleration (PGA) increases, the curvature capacity-to-demand ratio (C/D value) of the key pier sections exhibits a monotonic decreasing trend. When the PGA reaches a certain threshold, the rate of decrease in the C/D value begins to slow down. Among them, Piers 1# and 2# exhibit slightly higher longitudinal bending resistance compared to Piers 3# and 4#. Pier 4# is more susceptible to transverse bending failure, whereas Pier 2# demonstrates stronger transverse bending resistance, only yielding under high-PGA seismic excitations. Comprehensive observations of the transverse and longitudinal curvatures reveal that: Pier 1# yields at PGA = 0.8 g; Pier 2# yields at PGA = 0.7 g; Pier 3# yields at PGA = 0.5 g; Pier 4#, with a C/D value slightly greater than 1 at PGA = 0.1 g, is on the verge of yielding. As the PGA increases, the sequence of bending damage occurrence in the arch piers is as follows: Pier 4# → Pier 3# → Pier 2# → Pier 1#. The closer a pier is to the Central-Span, the higher the risk of bending damage. This is attributed to the direct connection between the spandrel structure and the main arch rib, which induces dynamic coupling through their mutual vibrations. The shorter the pier height of the piers, the stronger the coupling effects, making pier 4# (with the minimum pier height) most susceptible to damage. Consequently, pier 4# should be prioritized in seismic design and retrofitting efforts.

Based on the analytical results above, the key sections of piers 3# and 4# are selected as the research objects before and after damper installation. Figure 16 shows the impact of dampers on the capacity-to-demand ratio (C/D) IDA curves for piers 3# and 4#. From Figure 16a, it can be observed that without dampers, Pier 3# enters a damage state at PGA = 0.6 g, with a longitudinal curvature C/D = 0.89 (less than 1). After damper installation, the longitudinal curvature C/D ratio of Pier 3# remains below 1 even at PGA = 0.9 g. These results demonstrate that the installation of dampers significantly enhances the seismic performance of the pier in the longitudinal direction. As shown in Figure 16b, Pier 4# is highly susceptible to transverse bending failure under tri-directional seismic excitation. After damper installation, the PGA at which Pier 4# enters the damage state is increased from 0.1 g to 0.4 g, indicating that dampers also have a certain seismic mitigation effect in the transverse direction. Consequently, pier 4# should be prioritized in seismic design and retrofitting efforts. Analytical results demonstrate that damper installation not only effectively delays longitudinal bending failure in pier 3# but also enhances the transverse stability of pier 4#. The research findings highlight the potential of viscous dampers in improving pier resistance to both transverse and longitudinal seismic-induced damage.

4.3. Seismic Damage Categories and States of Bearings

To investigate the damage patterns of spherical steel bearings across spans in the long-span deck-type arch bridge, Figure 17 presents the IDA curves of the average capacity-to-demand (C/D) ratios for shear forces and displacements of mid-span bearings 1–11 and side-span bearings 1–11 on the left connecting girders under seven pulse-type ground motions. Based on the damage indices (C/D ratios) defined above for bearing assessment, fixed bearings are verified against shear forces, while sliding bearings are verified against displacements in this section. Therefore, Figure 17 presents the capacity-to-demand (C/D) ratios for different bearing responses. Regarding the damage categories of bearings, as shown in Figure 17a, only the end bearings 1 and 11 enter the damage state, initiating damage at PGA = 0.4 g, with all other bearings remaining in a safe state. The mid-span end bearings are the most susceptible to lateral sliding failure, whereas the mid-span center bearings are the least likely to experience such failure. As shown in Figure 17b, among the 11 bearings in the side-span, a total of 5 experienced damage. Failure progression is initially observed in end bearing 1 and 11 at PGA = 0.3 g, followed by central-span proximal bearing 5 failing at PGA = 1.1 g. Ultimately, right-span proximal bearings 9 and 10 exhibited transverse shear failure at PGA = 1.2 g. From Figure 17c, it can be seen that as PGA increased from 0.1 g to 0.2 g, the longitudinal displacement C/D values of all bearings reached their maximum. With a further increase in PGA, these values monotonically decreased. When PGA reached 0.4 g, the longitudinal C/D value of bearing 9 became slightly less than 1, making it the first to enter the damaged state. At a PGA of 0.7 g, two-thirds of the spherical steel bearings across the bridge successively experienced longitudinal sliding failure. By the time the PGA reached 1.2 g, all bearings had entered the damaged state one after another. From Figure 17d, it can be seen that only the central-span bearing had a longitudinal shear C/D value slightly less than 1 at a PGA of 1.2 g, indicating that the bearing is damaged.

Research findings indicate that bearings closer to the end regions exhibit poorer transverse seismic performance, while those near the central-span demonstrate weaker longitudinal seismic resistance. Compared to the central-span and end bearings, the bearings located between them perform relatively better in both longitudinal and transverse seismic resistance. End bearings are the most likely to reach the damaged state, with the end bearings of the edge span first experiencing transverse shear failure, followed by the mid-span end bearings experiencing transverse sliding failure. This phenomenon may be attributed to the exclusive installation of end bearings on abutments and approach piers; the relatively high rigidity of abutments significantly restricted transverse space for the bearings. The central-span, being identified as the primary load-bearing zone in arch bridge configurations, is subjected to substantial longitudinal inertial forces under seismic excitation. This mechanical characteristic leads to comparatively reduced longitudinal seismic resistance capacity.

To more clearly illustrate the PGA corresponding to bearing failure, Figure 18a shows the PGA at which transverse bearings fail under shear effects, while Figure 18b presents the PGA at which longitudinal bearing displacements exceed the design limit.

Figure 19 presents the impact of dampers on the IDA curve of the capacity-to-demand ratio (C/D) for the critical bearing (end bearing 1). In terms of the seismic damage state of the bearings, the influence of longitudinally installed dampers on transverse shear force IDA curves was found to be insignificant, both before and after the installation of dampers. The side-span bearing 1 enters the damage state at a PGA of 0.3 g. The transverse shear force C/D values were measured as 0.999 and 0.998, indicating that the shear capacity of the bearings approached the ultimate limit under these conditions. The mid-span bearing 1 undergoes transverse sliding failure at a PGA of 0.5 g, with transverse displacement C/D values of 0.797 and 0.621, respectively. It is indicated that the longitudinally installed dampers have certain limitations in their impact on the transverse displacement of bridge bearings, but are slightly more effective than controlling shear forces.

4.4. Damage Sequence of Whole Bridge

Potential seismic vulnerabilities in the main arch ring are identified as including arch foot sections, L/4 sections, L/8 sections, crown sections, along with spherical steel bearings at each pier column and deck slab span (bearings not depicted in Figure 20). These critical components have been systematically organized and summarized into the damage sequence illustrated in Figure 20. Due to duration discrepancies among the seven selected ground motions, seismic responses at key sections of arch ribs cannot be evaluated using the identical methodology applied to pier columns and bearings (i.e., averaging procedures applied to seven ground motions for seismic assessment). Therefore, for determining the damage of key sections of the arch rib, the PGA value at which damage occurs most frequently is selected as the damage PGA for that section. When occurrence frequencies are identical, the representative value is determined by selecting the smaller PGA. As shown in Figure 20a, End Bearing 1 and the shortest pier 4# first entered damage states, corresponding to a PGA of 0.3 g. Subsequently, damage initiation occurred simultaneously in pier 3# and the lower chord crown section when PGA reached 0.5 g. With a further increase in PGA, damage initiation was observed sequentially at Bearing 3 and the lower chord arch foot section when PGA attained 0.6 g, followed by Bearings 4/5, Pier 2#, upper chord arch foot, and lower chord L/8 section as PGA reached 0.7 g. With progressive intensification of ground motion, damage progression was sequentially triggered in Pier 1#, lower chord L/4 section, bearing 2, upper chord L/8 section, and mid-span bearing 6 at PGAs of 0.8 g, 0.9 g, 1.0 g, and 1.2 g, respectively.

The PGA values corresponding to the initiation of structural damage and the critical sections of the entire bridge after damper installation are indicated in the bridge schematic, as shown in Figure 20b. As evidenced in the diagram, the implementation of viscous dampers in long-span deck-type CFST arch bridges under near-fault pulse-like ground motions achieves a significant reduction in damage probability for the main arch ring and piers. However, the seismic mitigation efficiency demonstrates limited effectiveness for bearings, and localized response amplification phenomena may even be observed in certain bearing components. The post-damper damage progression path of the retrofitted structure is sequentially characterized by initial failure at end bearing 1, followed by short pier 4#, mid-span bearing 1, upper chord L/4 section, other main arch ring sections, and ultimately shorter pier 3#. The damage progression path of the damper-retrofitted structure remains essentially consistent with that of the original configuration. However, the spandrel structure demonstrates significantly higher vulnerability to damage compared to the main arch ring.

5. Conclusions

(1)

Under the three viscous damper configurations, the average values of the response envelope for the lower chord arch foot section, piers, and bearings of the long-span deck-type CFST bridge all decreased to varying extents. From the perspective of mitigating seismic responses at arch foot and bearings, the preferred placement strategy for viscous dampers should be implemented simultaneously at abutments and approach piers. At this point, the viscous dampers can significantly reduce the shear force and displacement responses of the central-span bearings, effectively lowering the peak values of the main arch axial force response and the shear force response of the short piers. The seismic mitigation effectiveness of bearings in long-span deck-type CFST arch bridges is significantly influenced by viscous damper configuration parameters. When α = 0.2 and C = 6000, the viscous damper achieves a better damping effect.

(2)

The seismic damage analysis and evaluation framework proposed in this study employs IDA and IDT methodologies to evaluate the seismic capacity of long-span deck-type CFST arch bridges through structural collapse mechanisms and damage progression sequences. This framework enables the evaluation of the effectiveness of dampers in reducing damage to critical structural components of the arch bridge.

(3)

The primary seismic-induced damage of long-span deck-type CFST arch bridges is manifested in the in-plane and out-of-plane bending failure of the arch rib, shear and sliding damage of end bearings, and bending failure of the relatively short piers in the central-span region.

(4)

Implementation of viscous dampers in the optimized configuration exerts negligible influence on seismic damage categories of critical components in long-span deck-type CFST arch bridges. A significant enhancement of structural damage-initiation PGA is achieved through this layout configuration, with damage to the main arch being predominantly concentrated at arch foot sections of both upper and lower chords, lower chord crown sections, and upper chord L/4 sections. A considerable reduction in damage probability for the main arch ring and pier is effectively achieved, with concurrent enhancement of C/D ratios under identical PGA levels, but the damping effect on bearings remains considerably limited. The seismic damage path of the optimized structure is generally consistent with that of the original structure; however, the superstructure on the arch is more prone to damage compared to the main arch rib.

(5)

The practical application of viscous dampers in CFST arch bridges can significantly reduce the response of critical components, such as the arch springing section and main girder displacement. However, it may also have adverse effects on other parts of the structure, such as the spandrel structure. Therefore, when applying the optimized damper parameters and configurations proposed in this study, both the damping effectiveness on the target control components and the potential response amplification in other regions should be comprehensively considered.