Investigation of the working mechanism of a quasi-floating hierarchically sacrificial seismic system for small and mid-span girder bridges

Abstract

Seismic resistance systems for small and mid-span girder bridges often lacks hierarchically repeatable earthquake resistance, leading to challenging and time-consuming post-earthquake repairs. This research introduces a novel quasi-floating seismic resistance system (QFSRS) with hierarchically sacrificial components to enable multiple instances of earthquake resistance and swift post-earthquake restoration. Finite element modeling, a numerical probabilistic approach, and earthquake-simulating shake-table tests identified highly sensitive parameters from the QFSRS to establish theoretical equations describing the mechanical model and working mechanism of the system. The results indicate that the working mechanism of the QFSRS under seismic conditions aligns with the theoretical design, featuring four hierarchically sacrificial seismic stages. Specifically, under moderate (0.3g) or higher seismic conditions, QFSRS reduced relative displacement between piers and beams by 55.15% on average. The strain at pier bases increased 6.17% across all seismic scenarios, significantly enhancing bridge seismic performance. The QFSRS provides resilient and restorable earthquake resistance for girder bridges.

Graphical abstract

Keywords

hierarchically sacrificial seismic resistance / alternative retainers / quasi-floating seismic system / post-earthquake bridge restoration

1 Introduction

During seismic activity, shear keys in small and medium-span girder bridges exhibit a notably higher failure rate compared to pier columns, with numerous plate-type rubber bearings sustaining complete damage [1]. Investigations have revealed that the primary cause of the girder unseating and plate-type rubber bearing sliding is shear key failure [2]. Consequently, when evaluating the seismic performance of bridges, the lateral restraint provided by shear keys is crucial [3]. And optimizing the design methods and structures of shear keys can effectively reduce the risk of main beam collapse [4–6].

Several studies [7–13] have introduced a variety of energy-absorbing or replaceable new types of shear keys through improved structural design, utilizing self-weight and precast concrete, among other methods. These innovations aim to enhance the shear key’s load-bearing capacity, eliminate the post-earthquake residual displacement of the bridge’s superstructure, and improve the shear key’s post-earthquake recovery capabilities. However, these shear keys often face challenges in simultaneously achieving high load-bearing capacity and rapid post-earthquake restoration. Furthermore, solely enhancing the shear key’s load-bearing capacity offers minimal assistance in the post-earthquake repair of bridges. Inspired by structural energy absorption and sacrificial energy dissipation [14,15], a novel replaceable energy-dissipating retainer (REDR) [16,17] has been developed that combines high load-bearing capacity with rapid post-earthquake repair capabilities. Li and Dong et al. [18–20] proposed that the design concept of shear keys should progressively shift from structural seismic resistance and vibration reduction to post-earthquake recoverable and functionally recoverable design principles while also considering the role of plate-type rubber bearings.

Interactions among various bridge components, such as piers, bearings, and shear keys, significantly influence the seismic response of a bridge [21]. The failure of shear keys and the sliding of bearings can affect the seismic response of the bridge’s substructure [22]. Collisions between shear keys and the main beam can amplify the seismic response at the base of the piers [23], while the sliding of plate-type rubber bearings can reduce the seismic response of the piers [24] but weakening the confinement effect of concrete shear keys [25]. Some researchers have suggested that under seismic conditions, it is preferable for the plate-type rubber bearings of small and mid-span girder bridges to fail first, followed by the shear keys [26]. In comparison to solely enhancing the seismic performance of a single bridge component, researchers have attempted to create various new seismic systems by combining different bridge components [27–30]. These systems aim to replicate real-world engineering conditions better and often exhibit superior seismic performance.

Research on seismic resistance for small and mid-span girder bridges is increasing from solely enhancing the seismic performance of a single shear key component to attempting to combine various bridge components into new seismic systems. Despite these advancements, a persistent challenge remains: many of these seismic systems are designed for one-time use. This limitation hinders their ability to offer graded, repeatable earthquake resistance, subsequently leading to prolonged post-earthquake repair processes. We aim to 1) propose a novel seismic resistance system, the quasi-floating seismic resistance system (QFSRS), for small and mid-span girder bridges; 2) investigate the working mechanism of the new QFSRS; 3) evaluate its seismic performance. To achieve these objectives, we establish a finite element model of bridges to extract highly sensitive parameters of the QFSRS, derive theoretical equations for its working mechanism, conduct earthquake simulation shake table tests to verify its working mechanism, and perform data analysis and discussion.

2 Quasi-floating seismic resistance system

2.1 Design of quasi-floating seismic resistance system

The QFSRS consists of plate-type rubber bearings or sliding energy-dissipating bearings in conjunction with the REDR [16,17], as illustrated in Fig.1. The basic performance characteristics of the system are presented in Tab.1. This novel seismic system integrates frictional energy dissipation, sliding isolation, and sacrificial energy dissipation. This system aims to prevent catastrophic damage to bridge structures during earthquakes by employing the alternative link or overall hierarchically sacrificial energy dissipation of the bearings or the REDR. After an earthquake, the seismic resistance capacity of the bridge can be quickly restored by replacing the sacrificial components locally.

The QFSRS operates in four phases under seismic forces, defined as follows. Assuming an initial relative displacement X_{\text{dl}} of zero between the pier and the beam of the model bridge, the gap between the REDR and the beam is denoted as a variable X_{\text{d}}. The ultimate sliding distance of the bearing is represented by X_{\mathrm{z}}, the structural energy dissipation capacity by E_1, the seismic energy imparted to the structure by E_2, the sliding energy dissipation of the bearing by E_3, the alternative link sacrificial energy dissipation of the REDR segment by E_4, and the overall sacrificial energy dissipation of the REDR by E_5.

In the first phase, E_1\ge E_2.

In the second phase, , X_{\text{d}}⩾X_{\text{dl}}, E_2-E_1=E_3.

In the third phase, , , E_2-E_1=E_3+E_4.

In the fourth phase, , , E_2-E_1=E_3+E_4+E_5{X}_{\text{d}}⩾X_{\text{dl}}=0.

Four-Phase Working Diagram of the QFSRS is shown in Fig.2.

2.2 Identification of highly sensitive parameters of quasi-floating seismic resistance system

A prestressed concrete simply supported T-beam bridge with dimensions of 2 × 20 m is chosen as the prototype bridge, as shown in Fig.3, consisting of 7 T-beams. The beam height is 1.5 m, the bridge width is 16.25 m, and the substructure consists of double-column solid piers with a diameter of 1.2 m and a height of 10 m. The cap beam has a height of 1.6 m.

Referring to principles such as multi-scale analysis [31,32], the main beam, cap beam, and piers are modeled using beam elements. The plate-type rubber bearings are divided into three elements for simulation, with vertical connections represented by spring elements and horizontal and longitudinal connections, as well as the REDR, modeled using spring elements, as shown in Fig.4. The elastic modulus of the concrete material is denoted as E=3.5\times {10}^4 N/mm², the mass density as p=2500 kg/m³, and the Poisson’s ratio as v=0.2. The finite element model is depicted in Fig.5.

We conducted sensitivity analysis through a numerical probabilistic approach [33], based on the finite element model, employing a finite element random generator to obtain random samples of input parameters and analyzing the probability outputs. Sensitivity is obtained through post-processing techniques.

First, let’s define vectors, a=\left\{A_1,C_1,F_1,K_1,\dots ,b_{\text{n}}\right\}, and A_1,C_1,F_1,K_1,\dots ,b_{\text{n}} as the parameters of the model, where each parameter is independent and follows an arbitrary distribution. We define the random parameter a_{\text{k}}=\{a_{\text{k1}},a_{\text{k2}},a_{\text{k3}},\dots ,a_{\text{kn}}\}, n_{\text{k}}\in ⟨0,1⟩ as a set of uniformly distributed random numbers. By substituting n_{\text{k}}, the cumulative distribution function F_{\text{a}}\left(a_{\text{k}}\right) can be obtained. Assuming n_{\text{k}j}=s_j{p}^{-1}, based on the recursive relationship:

(1)s_j=b{s}_{j-1}+c-k_{j-1}p,

(2)a_{\text{k}j}=F_{\text{a}}^{-1}\left(n_{\text{k}j}\right),

where b and c are positive integers, k_{j-1} represents the integer part of \left(b{s}_{j-1}+c\right)m^{-1}, and s_j and s_{j-1} are the current and previous seed values of j=1,2,\dots ,p in the recursion, respectively.

The summary statistics of generated input variables a_{\text{k}} can be computed in terms of the mean value through the expected value estimator.

(3)⟨a_{\text{k}}⟩=E\left[a_{\text{k}}\right]=\sum _{j=1}^p{a}_{\text{k}j}{p}^{-1}.

The value of standard deviation through the standard deviation estimator is calculated as

(4){\sigma }_{\text{x},i\text{v}}=\sqrt{V_{\text{br}}\left(a_{\text{k}}\right)}=\sqrt{\sum _{j=1}^p{\left(a_{\text{k}j}-⟨a_{\text{k}}⟩\right)}^2{(p-1)}^{-1}},

where V_{\text{br}}\left(a_{\text{k}}\right) represents the variance estimate of the input variable a_k.

Next, if we wish to output a sample of N, it can be evaluated by solution of the boundary value problem for each value of a_{\text{k}}=\left\{a_{\text{k1}},a_{\text{k2}},a_{\text{k3}},\dots ,a_{\text{kn}}\right\} such that the output sample is N\left[a_{\text{k}}\right]=\left[N_{\text{k1}}\left[a_{\text{k}}\right],N_{\text{k2}}\left[a_{\text{k}}\right],\dots ,N_{\text{k}j}\left[a_{\text{k}}\right]\right]. Output summary statistics can be then obtained from Eqs. (1) and (2) by replacing input a_{\text{k}} with output N\left[a_{\text{k}}\right] parameter.

At this point, the correlation coefficient r between the input variable and the output variable can be measured to assess their correlation:

(5)r=\frac{\mathrm{C}\mathrm{o}\mathrm{v}\left(a_{\text{k}},N\left[a_{\text{k}}\right]\right)}{\sqrt{\mathrm{V}\mathrm{b}\mathrm{r}\left(a_{\text{k}}\right)\mathrm{V}\mathrm{b}\mathrm{r}\left(N\left[a_{\text{k}}\right]\right)}},

where \mathrm{C}\mathrm{o}\mathrm{v}\left(a_{\text{k}},N\left[a_{\text{k}}\right]\right) represents the covariance estimate between the input variable a_{\mathrm{k}} and the output variable N\left[a_{\text{k}}\right].

Finally, r\in (-1,1), if r=1, it indicates a correlation between the input variable a_{\text{k}} and the output variable N\left[a_{\text{k}}\right]. If r=-1, it signifies a perfect negative correlation between the input variable a_{\text{k}} and the output variable N\left[a_{\text{k}}\right]. And if r=0, it implies that the input variable a_{\text{k}} and the output variable N\left[a_{\text{k}}\right] are completely uncorrelated.

Finally, based on the aforementioned method of selection, when considering the relative displacement X_{\text{dl}} between the bridge piers and beams as the target, the highly sensitive parameters are identified as follows: stiffness of the bearing K_{\text{z}}, coefficient of friction of the bearing u_{\text{z}}, yield bearing capacity of the REDR F_1, ultimate bearing capacity F_2, and peak ground acceleration of the seismic wave A_{max}.

2.3 Establishing working mechanism equations of quasi-floating seismic resistance system

According to the basic principles of structural dynamics, selecting the highly sensitive parameters, the QFSRS can be simplified into a mass model, considering the bridge pier, plate-type rubber bearing, beam, and the REDR.

It is evident that the dynamic equations of the QFSRS correspond to four distinct states across its working phases: pre-sliding sacrificial of the plate-type rubber bearing, post-sliding sacrificial of the plate-type rubber bearing, the alternative link sacrificial and overall sacrificial of the REDR. These states are illustrated in Fig.6.

According to the D’Alembert’s principle, the dynamic equilibrium equations for the structural system in the four phases are given by Eqs. (6)–(10).

Phase I:

(6)\{\begin{array}{l}{m}_{\text{d}}{\ddot{u}}_{\text{d}}(t)+c_{\text{d}}{\dot{u}}_{\text{d}}(t)+k_{\text{d}}{u}_{\text{d}}(t)-c_{\text{z}}{\dot{u}}_{\text{z}}(t)-k_{\text{z}}{u}_{\text{z}}(t)=-m_{\text{d}}{\ddot{u}}_{\text{g}}(t),\\ m_{\text{l}}\left[{\ddot{u}}_{\text{d}}(t)+{\ddot{u}}_{\text{z}}(t)\right]+c_{\text{z}}{\dot{u}}_{\text{z}}(t)+k_{\text{z}}{u}_{\text{z}}(t)=-m_{\text{l}}{\ddot{u}}_{\text{g}}(t),\end{array}

Phase II:

(7)\{\begin{array}{l}{m}_{\text{d}}{\ddot{u}}_{\text{d}}(t)+k_{\text{d}}{u}_{\text{d}}(t)+c_{\text{d}}{\dot{u}}_{\text{d}}(t)-F_{\text{c}}=-m_{\text{d}}{\ddot{u}}_{\text{g}}(t),\\ m_{\text{l}}\left[{\ddot{u}}_{\text{d}}(t)+{\ddot{u}}_{\text{h}}(t)\right]+F_c=-m_{\text{l}}{\ddot{u}}_{\text{g}}(t),\end{array}

(8)

Phase III:

(9)\{\begin{array}{rl}& m_d{\ddot{u}}_{\text{d}}(t)+k_{\text{d}}{u}_{\text{d}}(t)+c_{\text{d}}{\dot{u}}_{\text{d}}(t)-F_{\text{c}}-k_1\left[u_{\text{l}}(t)-u_{\text{d}}(t)-X_{\text{d}}\right]\\ & -c_k\left[{\dot{u}}_l(t)-{\dot{u}}_{\text{d}}(t)\right]=-m_{\text{d}}{\ddot{u}}_{\text{g}}(t),\\ & m_{\text{l}}{\ddot{u}}_{\text{d}}(t)+m_{\text{l}}{\ddot{u}}_{\text{h}}(t)+F_{\text{c}}+k_1\left[u_{\text{l}}(t)-u_{\text{d}}(t)-X_{\text{d}}\right]\\ & +c_{\text{k}}\left[{\dot{u}}_l(t)-{\dot{u}}_{\text{d}}(t)\right]=-m_{\text{l}}{\ddot{u}}_{\text{g}}(t),\end{array}

Phase IV:

(10)\{\begin{array}{rl}& m_{\text{d}}{\ddot{u}}_{\text{d}}(t)+k_{\text{d}}{\mu }_{\text{d}}(t)+c_{\text{d}}{\dot{u}}_{\text{d}}(t)-F_{\text{c}}-k_2\left[u_{\text{l}}(t)-u_{\text{d}}(t)-X_{\text{d}}\right]\\ & -c_{\text{k}}\left[{\dot{u}}_{\text{l}}(t)-{\dot{u}}_{\text{d}}(t)\right]=-m_{\text{d}}{\ddot{u}}_{\text{g}}(t),\\ & m_{\text{l}}{\ddot{u}}_{\text{d}}(t)+m_{\text{l}}{\ddot{u}}_{\text{h}}(t)+F_{\text{c}}+k_2\left[u_{\text{l}}(t)-u_{\text{d}}(t)-X_{\text{d}}\right]\\ & +c_{\text{k}}\left[{\dot{u}}_{\text{l}}(t)-{\dot{u}}_{\text{d}}(t)\right]=-m_{\text{l}}{\ddot{u}}_{\text{g}}(t),\end{array}

where m_{\text{d}} and m_{\text{l}} are the equivalent masses of the bridge piers and upper beam structure, respectively, K_{\text{d}} and c_{\text{d}} are the equivalent stiffness and damping of the bridge piers, c_{\text{z}} is damping of the bearings. u_{\text{d}}(t) and u_{\text{l}}(t) are the displacements of the piers and the beam relative to the foundation, u_{\text{z}}(t) is shear deformation of the bearings. u_{\text{g}}(t) is horizontal seismic ground motion experienced by the foundation. u_{\text{h}}(t) is sliding displacement of the beam on top of the bearings. F_{\text{c}} is dynamic friction force between the beam and the bearings. k_1 and k_2 are the collision stiffnesses between the beam and the alternative link of REDR, and between the beams and the overall structure of REDR, respectively, c_{\text{k}} is damping of the collision unit.

The formulations provided represent the dynamic equations for the four phases of the QFSRS. The displacement, acceleration, and other response values of feature points in the bridge structure, such as the main beams and pier tops, can be determined by solving these equations.

3 Experimental verification of working mechanism of quasi-floating seismic resistance system

3.1 Similitude requirements

Based on dimensional analysis and practical considerations, organic glass will be selected as the material for the scaled bridge model. The reasons for choosing acrylic are threefold: 1) the original materials cannot be used due to experimental size limitations; 2) acrylic meets the experimental requirements; 3) acrylic is easy to process. The similarity coefficient of geometric length, denoted as S_{\text{l}}, is determined to be 1:20. The density similarity coefficient, denoted as S_{\text{P}}, is determined to be 1:2. The elastic modulus similarity coefficient, denoted as S_{\text{a}}, is determined to be 1:13.27. The acceleration similarity coefficient, denoted as, is determined to be 3:1 based on Eq. (11). Consequently, the similarity ratios between the model and prototype for various parameters can be obtained, as shown in Tab.2.

(11)\frac{S_{\text{E}}}{S_{\text{P}}{S}_{\text{a}}{S}_{\text{l}}}=1.

To ensure the similarity between the gravitational and inertial effects of the model structure and the prototype structure, this study compensates for the effects arising from the insufficient mass of the model structure by adding extra mass blocks.

By assuming S_{\text{g}}=1, it follows that S_{\text{E}}=S_{\text{p}}{S}_{\text{l}}, and through corresponding derivations, S_{\text{m}}=S_{\text{p}}{S}_{\text{l}}^3=S_{\text{E}}{S}_{\text{l}}^2 can be obtained. Let the mass of the model structure be m_{\text{m}}, and the mass of the applied artificial mass block be m_{\text{r}}. The total mass of the model is the sum of the two. To satisfy the condition of mass similarity, let the mass of the prototype model structure be m_{\text{p}}, which leads to the relationship expressed in Eq. (12).

(12)m_{\text{r}}=S_{\text{E}}{S}_{\text{l}}^2{m}_{\text{p}}-m_{\text{m}}.

Accordingly, the required artificial mass is determined to be 8.65 kg.

3.2 Design of the scaled model

A plate-type rubber bearing is installed beneath each T-beam to simulate the sliding isolation effect under load, with its bottom and top surfaces in contact with the bearing pad and the surface of the main beam, respectively. The bridge piers are constructed as solid cylindrical piers with a diameter of 60 mm. The bottom of the piers is securely connected to the organic glass plate, which is pre-drilled with bolt holes. Reliable connections are established by bolting the organic glass plate to the vibration table surface.

The REDR uses stainless steel material utilizing 3D printing technology [34,35]. It is placed at a 3 mm interval from the beam. The schematic diagram illustrating the relevant dimensions is presented in Fig.7.

3.3 Experimental program

1) Input ground motion

Three types of seismic waves have been selected, taking into account the duration, acceleration amplitude, and spectral components of seismic records: Imperial Valley (Class I site), El-Centro (Class II site), and Taft (Class III site). Among them, the seismic acceleration time history and corresponding Fourier spectrum values for the 150 gal condition are shown in Fig.8. In contrast, the other conditions can be referred to in Tab.3, with differences in peak accelerations.

2) Experimental cases

The parameters of the vibration table used in this experiment are presented in Tab.3.

To satisfy the similarity relationship, the seismic waves used in the experiment must undergo time compression based on the original seismic time history. The duration is compressed to 0.129 times the original seismic record, and the seismic acceleration amplitudes are adjusted to 150, 300, 600, 900, and 1200 gal, respectively. Due to the unidirectional loading capability of the vibration table and the consideration of seismic effects in the longitudinal (X-direction) and transverse (Y-direction) directions separately for a straight bridge, this experiment focuses only on transverse loading. It includes two conditions: A (without REDRs) and B (with the REDRs). The other conditions are consistent and can be seen in Tab.4.

The experimental setup includes acceleration measurement points a, displacement measurement points L, and strain measurement points ε. In addition to these traditional measurement methods, a non-contact displacement and strain measurement system is also employed to collect data from the same points. This is done to prevent data loss and facilitate data comparison and validation. The on-site setup and layout of measurement point are depicted in Fig.9.

4 Results and analyses

4.1 Analysis of displacement response

The data from Load Case A (without the use of the QFSRS) and Load Case B (with the use of the QFSRS) indicate that the maximum difference between the displacement measurement points at the left and right ends of the same condition is less than 1.28 mm, with a maximum difference relative to either end of the beam not exceeding 5.00%. Similar patterns are observed for the displacement measurement data between the top of the left and right piers and between the middle of the left and right piers. The maximum difference in displacement data between measurement points from the same condition is less than 1.40 mm (maximum proportion of 3.69%). The average value of the two measurement points from the exact location is taken as the final displacement value to facilitate analysis.

During Load Cases 1–3, Load Cases A and B exhibit similar displacement values and trends at each measurement point, as shown in Fig.10. There is a significant overlap in the symbols of the two conditions, indicating that the QFSRS has no effect during this phase. During Load Cases 4–6, symbols are separated, and the overlapping area decreases in Phase 5, where the data from the two conditions differ by 0.20 mm (12.5% out of the relative displacement between Pier B and the beam). This difference indicates that the plate-type rubber bearings in the QFSRS have slipped from their original positions. As shown in Fig.11, during Load Cases 7–9, there is no overlap in the symbol of the relative displacement values between the piers and the beam for the two conditions. The maximum difference has increased by 9.5 times compared to Phase 5, indicating the activation of the REDRs. The data indicate that when the peak acceleration of the seismic wave reaches 600 gal, the bridge test model with the QFSRS exhibits a smaller relative displacement between the piers and the beam compared to the model without this system.

As shown in Fig.12, during Load Cases 10–15, the difference between the peak relative displacement of the piers and the beam in Load Cases A and B becomes more pronounced as the peak acceleration of the seismic motion increases. The QFSRS is fully functional at this point. The average relative displacement between piers and beams of bridges using the QFSRS decreases by 55.15%. As shown in Fig.13, the curves of relative displacement changes between piers and beams for Load Cases A and B gradually separate.

4.2 Analysis of strain response

The peak strain values and growth trends at each measurement point in Load Cases A and B do not exhibit significant differences in magnitude. They follow the pattern where the maximum strain response occurs at the bottom of the piers, followed by the middle of the piers, and with the minimum strain response observed in the main beam.

As shown in Fig.14, during Phases 1–9, the peak strain values for both conditions increase with the increasing peak acceleration of the seismic motion. During Load Cases 10–15, the difference in peak strain values at the bottom of the piers between the two conditions begins to widen. Except for the left bottom of Pier B in Load Case B10, the peak strain values at the bottom of the piers in Load Case B are generally greater than those in Load Case A. During this Load Case, the average increase in the absolute peak strain values at the bottom of the piers in Load Case B, compared to Load Case A, is 248.29 × 10⁻⁶. In all seismic cases, there is an average increase of 6.17% in strain response at the pier bases.

4.3 Comparison of experimental and theoretical results

By comparing the displacement response of the main beam and the strain response at the bottom of the piers in Load Cases A and B, it is observed that Load Case B exhibits a different structural response in four distinct phases as the peak acceleration of the seismic motion increases. These phases correspond to the four operational phases of the QFSRS described in Subsection 2.1. The experimental results are consistent with the theoretical analysis.

As shown in Fig.15 and Fig.16, when the peak acceleration of the seismic motion is 150 gal, the absolute difference in relative displacement between the piers and the beam for both conditions approaches zero, and the lines representing the absolute peak strain values at the bottom of the piers intersect and overlap many times. At this point, the inertial forces of the main beam are almost entirely dissipated by the structure itself, and the QFSRS has not yet functioned. This state corresponds to the first phase.

When the peak acceleration of the seismic motion reaches 300 gal, the absolute difference in relative displacement between the piers and the beam for both conditions shows a linear trend, and the lines representing the absolute peak strain values at the bottom of the piers intersect and overlap frequently. The overall trend of the lines becomes similar, with a rising pattern characterized by higher values in the middle and lower values on the sides. During this phase, the plate-type rubber bearings have slipped from their original positions, but the sliding displacement is limited. The peak relative displacement between the piers and the beam is 1.7 mm, and the REDRs have not yet functioned. The sliding of the plate-type rubber bearings effectively dissipates the inertial forces of the main beam. This state corresponds to the second phase.

When the peak acceleration of the seismic motion is 600 gal, the absolute difference in relative displacement between the piers and the beam for both conditions starts to deviate from zero, and the trend shows fluctuations. The lines representing the absolute peak strain values at the bottom of the piers still intersect and overlap, but the spacing between the lines of the two conditions begins to increase. Compared to the earthquake peak accelerations of 150–300 gal, it is now possible to distinguish the four peak variation curves of the left and right bottom of the piers for both conditions in the graph. However, the overall trend remains similar, and the difference between the two conditions remains relatively small.

During this phase, the plate-type rubber bearings start to sacrifice energy but have not reached their maximum displacement. The REDRs have just functioned. The data shows that although the relative displacement between the piers and the beam in Load Cases B7–B9 decreases compared to Load Cases A7–A9, only the displacement values of the main beam in Phases B8 and B9 show a significant decrease compared to Phases A8 and A9. This decline indicates that the REDRs have just functioned. The main beam collides with the REDRs, resulting in a rebound displacement in the opposite direction. At the same time, there is a plate-type rubber bearing to dissipate the inertial forces of the rebound displacement of the main beam. As a result, the main beam displacement decreases slightly, but there is no significant increase in strain at the bottom of the piers. During this phase, the inertial forces of the main beam are mainly dissipated by the plate-type rubber bearings, and the function of the REDRs dissipates the remaining inertial forces. This state corresponds to the third phase.

When the earthquake peak acceleration is 900 to 1200 gal, the absolute difference in relative displacement between the piers and the beam in both conditions shows severe fluctuations. The lines representing the absolute peak strain values at the bottom of the piers no longer exhibit significant intersections or overlaps at the same measurement points. The spacing between the lines of the two conditions further widens, and the peak strain values at various measurement points in Load Case B are all higher than those in Load Case A. This phenomenon is particularly evident when the earthquake peak acceleration is 1200 gal, where the peak strain values at the bottom of the piers in Load Case B show a significant increase compared to Load Case A.

During this phase, the QFSRS is fully functioning. The relative displacement between the piers and the beam is significantly reduced, and the displacement of the main beam is restricted. However, a portion of the inertial forces from the main beam still inevitably transmit to the lower structure of the bridge test model. This transmission increases strain response at the bottom of the piers in Load Cases B10–B15 compared to Load Cases A10–A15. This state corresponds to the fourth phase.

5 Discussions

The adoption of the QFSRS significantly influences the seismic response of small and mid-span girder bridges. This difference primarily arises from the hierarchically sacrificial mechanism of the plate-type rubber bearings and REDR within the seismic resistance system.

In the seismic performance design, it is only after the function of the REDR that the relative displacement between piers and beams under condition B begins to decrease significantly (Fig.13). The difference in relative displacement between piers and beams between the two cases widens (Fig.15). This trend aligns with similar studies [36], where the installation of conventional shear key on bridges led to a 20.4% reduction in main beam displacement and an 18.7% increase in base shear. Moreover, the installation of a sliding shear key resulted in a 21.4% reduction in main beam displacement and a 28.8% increase in base shear. However, in contrast to similar research [36], in the fourth phase of this study’s QFSRS, the average relative displacement between piers and beams decreases significantly by 55.15% (Fig.12). In all seismic cases, there is an average increase of 6.17% in strain response at the pier bases. (Fig.14 and Fig.16). The system exhibits excellent restraint capability, with minimal increases in pier base response, aligning with the industry-admired design concept that entails not only effective control of main beam lateral displacement but also the avoidance of significantly increasing lateral seismic forces on bridge piers [26]. Similar results were confirmed in studies [19,37]. The significant reduction of seismic demand on the bridge substructure is mainly due to the dissipation of seismic energy through the hysteretic behavior of the steel shear keys. Replacing traditional shear keys with high-energy dissipation steel shear keys can enhance the seismic performance of bridges.

In post-earthquake repair design, the test results (Fig.15 and Fig.16) verify the working mechanism of the QFSRS as designed (Fig.2). This study introduces a novel idea for enhancing post-earthquake restoration performance by increasing the number of cycles of operation of the seismic resistance system. Moreover, the QFSRS, comprising REDR and plate-type rubber bearings, is easier to repair after operation. Similar research [38] has indicated that seismic damage to interior shear keys is easier to detect and repair than other load-carrying members, such as abutments, cap beams, and piles.

Currently, this study has applied the QFSRS to existing bridges (Fig.17). However, due to experimental limitations, the range of test conditions was limited, resulting in an insufficient sample size of data. Factors such as varying numbers of spans, different span lengths, clearances between REDR and the main beam, the frequency of collisions between REDR and the main beam, and collision forces were not considered. Furthermore, there is still a lack of data on the performance of the QFSRS in earthquake scenarios because the system has not yet been subjected to seismic testing in engineering applications. All of these aspects require further research and investigation.

Due to their small span and simple structure, small and mid-span girder bridges are often overlooked and can suffer severe damage under earthquake conditions. However, small and mid-span girder bridges represent the most widely distributed and numerous types of bridges in the world. This study aims to optimize the seismic design of small and mid-span girder bridges, enhance their seismic performance, and reduce post-earthquake repair costs. It offers a framework and reference for research related to ensuring smooth and safe traffic during the critical 72 h following rare earthquakes, as well as for conserving repair materials and manpower and reducing carbon emissions in the aftermath of small to moderate earthquakes.

6 Conclusions

This research proposes a QFSRS for small and mid-span girder bridges consisting of plate-type rubber bearings and REDR. A three-dimensional finite element model was established using a prototype bridge from an engineering project. Highly sensitive parameters of the QFSRS were extracted using a numerical probabilistic approach. Based on these parameters, theoretical equations for the working mechanism of the QFSRS were established. Shake-table tests, simulating earthquake conditions, were conducted to verify the working mechanism of the QFSRS. The primary conclusions are as follows:

1) The QFSRS for small and mid-span girder bridges is proposed, characterized by hierarchically sacrificial components, enabling graded, repeatable earthquake resistance and reduced post-earthquake recovery time.

2) The highly sensitive parameters of the QFSRS are stiffness of the bearings K_{\text{z}}, coefficient of friction of the bearings u_{\text{z}}, yield bearing capacity of the REDRs F_1, ultimate bearing capacity F_2, and peak ground acceleration of the seismic wave A_{max}.

3) The established theoretical equations for the working mechanism reveal four distinct phases of the QFSRS, corresponding to the pre-sliding sacrificial stage of the plate-type rubber bearing, the post-sliding sacrificial stage of the plate-type rubber bearing, the alternative link sacrificial stage, and the overall sacrificial stage of the REDR.

4) The QFSRS operates in four phases under earthquake, allowing for hierarchically sacrificial seismic resistance. The theoretical prediction of the working mechanism aligns with the experimental results. In the first phase, the plate-type rubber bearings experience no sliding, and the bridge structure itself dissipates seismic energy. In the second phase, sliding occurs in the plate-type rubber bearings, and seismic energy dissipation mainly relies on bearing sliding. In the third phase, the sliding of the plate-type rubber bearings continues to increase, and collisions occur between the main beam and the REDR. Seismic energy dissipation primarily occurs through the sacrificial behavior of the bearings and the alternative link of the REDR. In the fourth phase, the sliding of the plate-type rubber bearings reaches the limit, and continuous collisions occur between the main beam and the REDR with seismic energy dissipation relying on the overall sacrificial behavior of the bearings and the REDR.

5) The seismic performance of bridges equipped with the QFSRS has been enhanced under moderate (0.3g) or higher seismic conditions, QFSRS reduced relative displacement between piers and beams by 55.15% on average. Strain at pier bases increased 6.17% across all seismic scenarios, significantly enhancing bridge seismic performance.

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