Seismic performance of a continuous bridge with winding rope device activated by a fluid viscous damper

Abstract

The seismic requirements of piers with fixed bearings (the fixed pier) for continuous girder bridges are relatively high, while the potential seismic capabilities of piers with sliding bearings (the sliding piers) are not fully utilized. To solve this contradiction, a new type of winding rope shock absorption device activated by a fluid viscous damper (WRD-D) was proposed. The WRD-D was installed on the top of the sliding piers, and the both ends of a fluid viscous damper were connected to the superstructure by winding ropes. During an earthquake, the damping force rises with the increase of relative speed between the sliding piers and the superstructure, activating the WRD-D and producing large frictional resistance, subsequently causing the sliding piers and the fixed pier to bear the seismic load cooperatively. In this study, the working mechanism of the WRD-D was researched. The shaking table test of a scaled continuous girder bridge model employing the WRD-D was conducted. The test results reveal that the WRD-D can effectively reduce the seismic requirements of the fixed pier and the superstructure displacements.

Keywords

continuous girder bridge shaking table test winding rope device fluid viscous damper seismic performance

Introduction

A single fixed pier is usually set in a continuous girder bridge to accommodate the influence of expansion due to thermal strain, creep, and shrinkage of concrete on the substructure. This means the longitudinal seismic forces of the whole continuous girder bridge are mostly transmitted to foundation through the only fixed pier. This inevitably increases the seismic demand of fixed pier on the longitudinal direction, and the large longitudinal displacement of superstructure could cause severe seismic damage, such as the expansion joints failure, the bearing unseating, even the beam falling of approach bridges (Han et al., 2009; Jay Lin et al., 2008; Li et al., 2008a, 2008b, 2017; Liu, 2018). To address this issue, extensive studies on the seismic design of continuous girder bridges have been performed.

Considerable attention has been paid to the research and development of seismic isolation devices (e.g., laminated rubber bearings and friction pendulum bearings), which replace the fixed bearings with isolation bearings, thereby prolonging the structural period, and dissipating seismic energy under an earthquake action (Kunde, 2003; Peng et al., 2011). Economical laminated rubber bearing composed of alternate layers of rubber and steel sheets has been extensively adopted for large scale bridge construction because of their significant energy dissipation and seismic response reduction abilities (Alam et al., 2012; Steelman et al., 2013, 2016). However, an investigation report (Jay Lin et al., 2008) from the Wen-chuan earthquake demonstrated that the sliding of laminated rubber bearings led to the failure of shear keys, bearings, and span collapse resulting from excessive superstructure displacement. Relevant experiments have also shown that bearing sliding could result in the significant displacement of bearings and the unseating damage (Xiang et al., 2018). Friction pendulum bearing has better vertical bearing capacity, excellent stability, and self-returning function. However, it has a low tensile capacity and the center of gravity of the superstructure is raised under normal service conditions. The energy dissipation capacity of friction pendulum bearing mainly depends on the magnitude of vertical dead loads (Iemura et al., 2005; Roussis et al., 2003; Tsopelas et al., 1996; Zou et al., 2019) and are therefore unsuitable in cases with relatively small vertical dead loads. On the other hand, the friction pendulum bearing installed on sliding piers of continuous girder bridges will generate relative slip in a normal service condition, and the accumulated friction loss problem has attracted the attention of the maintenance department.

Passive energy dissipation dampers can effectively dissipate the seismic energy transferring from the superstructure to the substructure, so protecting the bridge piers under an earthquake action. The application of shape memory alloy in the seismic design of bridges has shown that it could significantly improve the seismic performance of bridge piers through re-centering, thereby significantly reducing permanent damage (Hedayati Dezfuli and Shahria Alam, 2013; Ozbulut and Hurlebaus, 2010; Shrestha et al., 2017). However, the high cost and demanding machining of shape memory alloys remain limiting factors to large scale implementation (Xiang et al., 2019). Fluid viscous dampers are commonly used as energy dissipation devices in the longitudinal direction of long span bridges and can also be used in small to medium span bridges as unseating prevention restrainers (Hwang and Tseng, 2005; Yi et al., 2018, 2019). Kelly (Kelly et al., 1972) earlier had proposed metallic yielding damper as passive control device in structural engineering, used mostly in buildings. Subsequently, an innovative isolation system for highway bridges composed of typical laminated-rubber bearings and X steel dampers was developed. Both the test and numerical results showed that the system can effectively control the relative displacement between the superstructure and the piers, simultaneously protecting the piers from severe damage (Li et al., 2016).

These seismic isolation devices can prolong the period and reduce the seismic response of bridges, but at the cost of the larger relative displacement between the piers and superstructure. Moreover, the shock absorption devices can consume the seismic energy, reducing the displacement of the superstructure and protecting bearings and substructures from severe damage. However, although these devices are used in continuous girder bridges, the fixed pier still bear the main longitudinal horizontal seismic force independently and the potential seismic capability of the existing sliding piers is not fully utilized under an earthquake action. To achieve a better balance of internal forces and displacements between the superstructure and piers in continuous girder bridges, a more reasonable seismic design of continuous girder bridge is required. Therefore, the winding rope device activated by a fluid viscous damper (WRD-D) was proposed, based on research into the collaborative loading of sliding piers and fixed piers in continuous girder bridges (Zhang et al., 2017; Fang et al., 2019). In practical seismic design, the WRD-D should be installed on the top of sliding piers and connected to the superstructure by winding ropes. The ordinary sliding bearings would bear vertical loads, while the WRD-D would bear horizontal service-level loads and seismic forces. The WRD-D can not only meet the displacement requirements of structure under the normal service condition, but also make the sliding piers and the fixed pier bear the horizontal seismic loads cooperatively under an earthquake action. The device has the advantage of low cost, good durability, and the convenience of inspection and maintenance. In this study, the mechanism principle of the WRD-D was studied, and the dynamic performance experiment of the fluid viscous damper was performed. Moreover, a shaking table test of 1/30 scaled models was conducted, and the experimental results were studied to verify the seismic performance of the WRD-D.

Design of the WRD-D

WRD-D configuration

Figure 1 (a) presents the configuration of the WRD-D, which includes the following components: the fluid viscous damper, the back plate, the friction shaft, the side plate, the winding rope, the bottom plate, the tension force adjusting bolt, the anchor bolt, and the bracket. The entire device was fixed to the top of the sling pier by the bottom plate and the anchor bolts, and connected to the main girder by the winding ropes. The left section view “I-I” of the WRD-D is shown in Figure 1 (b). The two side plates were inserted into the reserved card slots of the bottom plate and fixed to the bottom plate by bolts. The two back plates were precisely designed to be caught in the card slots reserved by the side plates. The viscous damper was fixed in the slot between the two back plates. One end of the winding rope was connected to the viscous damper. After winding a few turns around the friction shaft, the other end was connected to the main girder by the tension force adjusting bolt, and a certain pre-tension force was applied on the winding ropes. In practical application design, the winding rope can be made of high-strength steel cable. The friction coefficient between the winding rope and the steel friction shaft is measured by corresponding tests.

(1) fluid viscous damper; ②back plate; ③friction shaft; ④side plate ⑤winding rope; ⑥bottom plate; ⑦winding rope force adjusting bolt; ⑧anchor bolt; ⑨bracket; ⑩sling pier; ⑪girder

Figure 1.

Configuration of the WRD-D. (a) Front section view. (b) Left section view of Ⅰ-Ⅰ.

Mechanics principle of the WRD-D

When the WRD-D is applied to an actual bridge, the WRD-D can not only meet the thermal strain requirements of bridges under normal operating conditions but also make the sliding piers and the fixed pier bear earthquake loads together under an earthquake action. The mechanical principle of WRD-D can be divided into three states: (1) an initial static state; (2) a slow gross slip state under temperature loads; and (3) a fast-gross slip state under an earthquake action. When the winding rope slips against the fixed circular cylinder and the friction fully develops, the tension forces at both ends of the winding rope under slipping limit conditions should satisfy the Euler formula, as equation (1).

T_{1} = T_{2} \exp (μ_{k} θ)

(1)

where θ is the contact angle, μ_k is the coefficient of the kinetic friction between the contacting surfaces, T₁ is the pull force at the high-tension side of the winding rope, and T₂ is the hold force at the low-tension side of the winding rope.

(1) In the initial static state, it is assumed that piers and superstructure are in a state of static equilibrium when the WRD-D is installed as shown in Figure 2 (a). The pretension forces of the winding ropes at both ends of the girder brackets, as well as the static friction forces generated by the two friction shafts, are considered equal (F₁ = F₂ and f_μ1 = f_μ2, respectively). The static friction forces can be calculated from the tension forces on both ends of the winding ropes, as in equation (2).

Figure 2.

Mechanical analysis diagram of WRD-D at the initial static state. (a) The initial state of WRD-D before gross slip. (b) The detail drawing of loaded winding rope on the left friction shaft of WRD-D.

f_{μ 1} = f_{μ 2} = F_{1} - F_{0} = F_{2} - F_{0}

(2)

where F₀ is the initial tension force of winding ropes at both ends of viscous damper (low-tension side), F₁ and F₂ are the pretension forces of winding ropes generated by the rope force adjusting bolts (high-tension side), and f_μ1 and f_μ2 are the static friction forces of the winding ropes.

Before the state of gross slip throughout contact region, the tension force of winding ropes in the contact region can be determined by considering the stretch ability of winding ropes only (Lubarda, 2015). The force equilibrium condition of the winding ropes (left friction shaft of WRD-D in Figure 2 (a)) was shown in Figure 2 (b). The angle θ_s is the slip region, where the friction force is transmitted. The angle (2πm-θ_s) is the adhesion region, where no friction force and slip are generated (Gerbert, 1991; Wu et al., 2019). When the winding ropes slip over the friction shafts, the tension force at the ends of winding rope satisfies Euler formula, as the equation (1). The Euler’s value of tension force at the high-tension side is F^E = F₀ exp(μ_k·2πm). Before the winding ropes slip over the friction shafts, the tension force F₁ is less than the Euler’s value F^E. Under the tension force F₀ < F₁ = F₂ < F^E, and the tension force of the winding rope T(φ) is constant within the inactive contact range (0 < φ < 2πm-θ_s), because the friction force vanishes. Elsewhere the state of impending slip extends within the slip range (2πm-θ_s < φ < 2πm), as equation (3)

T (φ) = {\begin{cases} F_{0} 0 \leq φ \leq 2 π m - θ_{s} \\ F_{0} \exp {μ [φ - (2 π m - θ_{s})]} 2 π m - θ_{s} \leq φ \leq 2 π m \end{cases}

(3)

where m is the number of rope winding turns, and μ is the coefficient of static friction between the contacting surfaces.

In particular, the tension force of winding ropes is as equation (4), so that the slip angle θ_s can be determinate as equation (5).

F_{1} = F_{2} = F_{0} \exp (μ θ_{s})

(4)

θ_{s} = \frac{1}{μ} \ln \frac{F_{1}}{F_{0}}

(5)

Consequently, equation (3) can be rewritten as equation (6)

T (φ) = {\begin{cases} F_{0} 0 \leq φ \leq 2 π m - θ_{s} \\ F_{1} \exp [μ (φ - 2 π m)] 2 π m - θ_{s} \leq φ \leq 2 π m \end{cases}

(6)

(2) Under the action of temperature loads, the slow gross slip occurs between the sliding piers and the superstructure and thus the friction fully developed, the tension force at the both ends of winding ropes in slipping limit conditions should satisfy the Euler formula. However, the relative movement speed is small, and the viscous damping force F_d(∆v(t)) is zero, as equation (7) (Hwang and Tseng, 2005). The equilibrium diagram of WRD-D at a certain moment is shown in Figure 3. The frictional resistance F_RT of WRD-D under the temperature loads action is shown in equation (8), and the friction forces of the winding ropes at both sides of WRD-D are constant as shown in equation (9) and equation (10).

Figure 3.

Mechanical analysis diagram of WRD-D under temperature loads.

F_{d} (Δ v (t)) = C {(Δ v (t))}^{α}

(7)

where C is the damping coefficient of viscous damper, α is the damping index, which is between 0 and 1, and ∆v(t) is the relative speed between the sliding piers and the superstructure.

F_{R T} = f_{μ t 1} + f_{μ t 2}

(8)

f_{μ t 1} = F_{1} (\exp (2 π m μ_{k}) - 1)

(9)

f_{μ t 2} = (F_{1} + f_{μ t 1}) (\exp (2 π m μ_{k}) - 1)

(10)

Consequently, equation (8) can be rewritten as

F_{R T} = F_{1} (\exp (4 π m μ_{k}) - 1)

(11)

where F_RT is the frictional resistance of the WRD-D under temperature loads, and F_T2 (t) is the tension force of winding ropes at the ends (corresponding to the inertial force of the girder applied under temperature loads); moreover, f_μt1 and f_μt2 represent the friction forces generated by two friction shafts.

(3) Under an earthquake action, when the relative displacement between the superstructure and the sliding piers occurs, the friction force of winding ropes increases with the increase of the damping force, the viscous damping force increase with the increase of the speed between the superstructure and sliding piers. The mechanical equilibrium diagram of WRD-D at a certain moment is consistent with Figure 3. When the winding ropes slip over the friction shafts, the frictional resistance F_RE(t) of WRD-D is shown in equation (12), and the friction forces of the winding ropes at both sides of WRD-D are shown in equation (13) and equation (14).

F_{R E} (t) = f_{μ E 1} + f_{μ E 2} + F_{d} (Δ v (t))

(12)

f_{μ E 1} = F_{1} (\exp (2 π m μ_{k}) - 1)

(13)

f_{μ E 2} = (F_{1} + f_{μ E 1} + F_{d} (Δ v (t))) (\exp (2 π m μ_{k}) - 1)

(14)

Consequently, equation (12) can be rewritten as

F_{R E} (t) = F_{1} (\exp (4 π m μ_{k}) - 1) + C {(Δ v (t))}^{α} \exp (2 π m μ_{k})

(15)

where f_μE1 and f_μE2 are the friction forces generated by two friction shafts under an earthquake action; moreover, F_RE(t) is the frictional resistance of WRD-D under an earthquake action, and F_E2(t) is the tension force of the winding ropes at the ends (corresponding to the inertial force of the girder applied under an earthquake action).

When the winding ropes tension force F_E2(t) is greater than frictional resistance F_RT under the slow gross slip state of WRD-D, the displacement between the superstructure and sliding piers, and the gross slip of winding ropes occur, as equation (16). The viscous damping force F_d(∆v(t)) increases with the increase of relative speed between the superstructure and sliding piers, thereby amplifying the frictional resistance F_RE(t) of WRD-D, as equation (15). A frictional sliding connection between the superstructure and sliding piers is established to bear earthquake loads.

F_{E 2} (t) > F_{R T} = F_{1} (\exp (4 π m μ_{k}) - 1)

(16)

Shaking table test

Design of test model

Similarity ratio design

A medium span continuous girder bridge was selected as the prototype structure for the test model, with a span arrangement of 45 m+72 m+45 m and hollow column piers made of C40 concrete for the bridge. The height of the middle pier and side pier is 45 m and 27 m, respectively. The design basis of the prototype in terms of loading conditions and seismicity is not related to the shaking table test. The test model is designed in scale according to the prototype bridge, but not in strict accordance with the similarity ratio because of size limitations of the shaking table and existing test conditions.

The shaking table test was conducted in Beijing University of Technology, China. The specimen was scaled at 1/30 from the prototype bridge. Moreover, the elastic modulus similarity ratio was 6.34, because the piers of the test model were simulated with Q235 steel (the elastic modulus is 2.06×10⁵ N/mm², and the yield strength is 235 N/mm²), but the prototype bridge is a RC structure. Based on these requirements, the original and derived similarity ratios are shown in Table 1.

Table 1.

Similarity ratios of the scale model to the prototype bridge.

Physical property	Physical quality	Parameter equation	Similarity ratio
Geometric characteristics	Length, l	$S_{l}$	1/30
	Area, A	$S_{A} = S_{l}^{2}$	1/30²
	Moment of inertia, I	$S_{I} = l^{4}$	1/30⁴
	Flexural rigidity, EI	$S_{E I} = E l^{4}$	7.83×10^–6
	Tensile rigidity, EA	$S_{E A} = E l^{2}$	7.04×10^–3
Material properties	Modulus of elasticity, E	$S_{E} = S_{σ}$	6.34
	Strain, ε	$S_{ε} = 1$	1
	Stress, σ	$S_{σ}$	6.34
	Equivalent quality density, ρ	$S_{ρ} = S_{E} / (S_{a} S_{l})$	190.15
	Poisson ratio, μ	$S_{μ} = 1$	1
Dynamic characteristics	Mass, ρl³	$S_{m} = S_{ρ} S_{l}^{3}$	7.04×10^–3
	Time, t	$S_{t} = {(S_{m} / S_{k})}^{0.5}$	0.18
	Frequency, f	$S_{f} = 1 / S_{t}$	8.66
	Rigidity, k	$S_{k} = S_{E} S_{l}$	0.21
	Velocity, v	$S_{v} = S_{l} / S_{t}$	0.18
	Acceleration, a	$S_{a} = 1$	1

According to the geometric similarity ratio, the span of the test model was 1.5 m+2.4 m+1.5 m (5.4 m). The span of continuous girder bridge was only to simulate the equivalent inertia behavior of the bridge superstructure. Therefore, the main girder was not strictly scaled on the 1/30 ratio, but reduced in length with the similarity ratio of 1/45. The span of the test model was adjusted to 0.995 m+1.64 m+0.995 m (3.63 m).

Superstructure and pier design

The shaking table with a dimension of 3 m × 3 m and a payload of 200 kN was used to support the test model, as shown in Figure 4 (a). Pier 2 is the only fixed pier, while the others are sliding piers. The sliding piers were fixed to the extended steel beam by a connecting device of a steel box component with high rigidity. The height of middle piers and side piers were 1.5 m and 0.9 m, respectively. All the piers were fixed to the steel beam, which is bolted on the shaking table. Due to the limitations of test conditions, the interaction between the WRD-D connection system and abutment was not considered. Figure 4 (b) and (c) show details of the designed piers and the main girder. The supplemented mass was attached to corresponding components to increase the structure’s density and produce a realistic static load and inertial force. The middle pier of the model had a weight of 120 kg, including 20 kg self-weight and 100 kg supplemented mass. The side pier of the model had a weight of 54 kg, including 14 kg self-weight and 40 kg supplemented mass. The supplemented mass of the girder was 1.0 × 10³ kg.

Figure 4.

Shaking-table test model (unit: mm). (a) Elevation of test model. (b) Side view of sliding pier 1 and of fixed pier 2. (c) Cross section of girder and piers. (d) The setup of the test model.

The aim of this test was to analyze the WRD-D seismic performance in continuous girder bridges. So, the piers were simulated with Q235 steel. The section area of piers satisfied the similarity ratios in terms of tensile and flexural stiffness. The setup of the test model is shown in Figure 4 (d). As presented in Figure 5, the fixed support conditions of longitudinal and transverse movements between the fixed pier and the main girder were simulated by combining a hinged bearing. The sliding bearings were simulated by installing a lubricated PTFE plate on square steel, which were fixed at both sides of the WRD-D bottom plate.

Figure 5.

Fixed bearing and sliding bearing in the test model. (a) Fixed bearing. (b) Sliding bearing.

The WRD-D connection system

Figure 6 presents a setup of the WRD-D connection system. Due to the limitations of the test model size and conditions, the winding rope was simulated by a solid high-strength polyethylene fiber rope in the test model. The friction coefficient between the fiber rope and the steel friction shaft was set as 0.1. The viscous damper in WRD-D is to activate the winding rope device under an earthquake action, hence its size can be small. In the process of designing the test model, the reasonable parameters of the viscous damper were determined by numerical simulation. The damping coefficient and damping index of the viscous damper in the test model were 2000 (N·(s/m)^α) and 0.5, respectively. The number of winding turns of ropes was four, namely, the contact angle of ropes winding around the shaft was θ = 8π. Moreover, the pretension force of winding ropes was 1.0 kN.

Figure 6.

Setup of the WRD-D in the test model.

Hysteretic behavior of fluid viscous damper

A dynamic performance experiment was performed to investigate the ultimate displacement, the maximum damping force, the damping coefficient, and the damping index of the fluid viscous damper. The experiment loading adopted the displacement control method, inputting the simple harmonic load, and the fundamental frequency (f = 3.65 Hz) of the test model was selected as the loading frequency. The design displacement of the viscous damper was U₀ = 30 mm. The loading displacement amplitudes of the hysteretic test were 0.1U₀, 0.2U₀, 0.5U₀, 0.7U₀, and 1.0U₀, respectively, with ten cycles per step. This study enumerates the hysteretic curves with a loading displacement magnitude of 0.2U₀, 0.5U₀, and 0.7U₀ as shown in Figure 7. Figure 7 (a) presents the hysteretic curves for the viscous dampers, showing the considerable energy dissipation capability of the viscous damper. It should be noted that the hysteretic curves are not full due to the insufficient oil in the cylinder after many tests of the viscous damper. Figure 7 (b) presents the force-velocity curves of the viscous damper, indicating that the viscous damper is nonlinear. The damping coefficient and damping index are C = 2000 ((N·s/m)^α) and α = 0.5, respectively, calculated from the experiments.

Figure 7.

The test curves of the fluid viscous damper. (a) Hysteretic curves of the viscous damper. (b) Force-velocity curves of the viscous damper.

Influence of the speed on the resistance of the WRD-D

Based on the mechanical principle of the WRD-D, we analyzed the influence of different speeds on its resistance by the theoretical analysis. The loading of the WRD-D was done through the displacement control method, inputting a simple harmonic load with the displacement amplitude u₀ = 100 mm, and the fundamental frequency of the shaking table test model was selected as the loading frequency (f = 3.65 Hz). The parameters of WRD-D are the same as the section “The WRD-D connection system.”

Figure 8 presents the force-velocity and the hysteretic curves of the different parts of the WRD-D. The stiffness of the viscous damper was not considered. Both the viscous damping force and the friction force of the winding ropes increase with the increase of the loading speed as shown in Figure 8 (a) and (b), thereby increasing the total frictional resistance of WRD-D as shown in Figure 8 (c). The friction force of the winding ropes is composed of two parts, one part is a constant value related to the number of winding turns and the pre-tension force, and the other part is a variable value related to the speed. Figure 8 (d) presents the hysteretic curves of the WRD-D. It can be seen that the viscous damping force accounts for a small proportion of the total resistance force of WRD-D, but the friction force of the winding ropes accounts for a large proportion of the total resistance force of WRD-D. Because the viscous damper of WRD-D aims to activate the winding ropes, the size of the viscous damper can be small to reduce the cost.

Figure 8.

The force-velocity and hysteretic curves of the WRD-D. (a) Viscous damper. (b) Frictional force of the winding ropes. (c) Total resistance of the WRD-D. (d) Hysteretic curves of the WRD-D.

Ground motion

The bridge models were subjected to four ground motions in the longitudinal direction: the Qian an wave, EL-Centro wave, Taft wave, and Tianjin wave. The peak ground acceleration (PGA) of each ground motion varied from 0.1 g to 0.8 g. The time axis of the ground motion is compressed to square root of 1/30 according to the similarity ratio, but this value is too small to generate effective excitation for the test model. Therefore, by increasing the counterweight of superstructure, the period ratio of the test model and prototype bridge is close to 1/3. This approach is acceptable because the shaking table test aims to study the seismic performance of WRD-D, rather than the seismic response of a project. The amplitude of the ground motion could be linearly scaled up and down to create a wide range of PGAs. Figures 9(a) to (d) presents the time history of each earthquake at a PGA of 0.6 g. Figure 9 (e) shows the acceleration spectrum of each ground motion with a PGA of 0.6 g and a spectrum derived from the Codes for Chinese Guidelines for Specification for Seismic Design of Highway Bridges.

Figure 9.

Acceleration histories and spectra of the input ground motion with a compressed time axis and a scaled PGA of 0.6 g. (a) Qian an wave. (b) EL-Centro wave. (c) Taft wave. (d) Tianjin wave. (e) Spectral acceleration.

Experimental scenarios

Two test models were separately subjected to a series of seismic waves in a longitudinal direction. Model 1 had no WRD-Ds installed on the sliding piers, while Model 2 had WRD-Ds installed on the sliding piers 1, 3 and 4. The other design parameters of the two models were the same. Considering the site type of seismic waves, I, II, III, IV (from stiff to soft) as prescribed in the Chinese Guidelines for Specification for Seismic Design of Highway Bridges, the four seismic waves (Qian an wave, EL-Centro wave, Taft wave, and Tianjin wave) were input in sequence according to the PGA. It should be noted that there was a deviation between the output PGA and input PGA due to the extended beam and the shaking table.

To reduce the influence of the loading sequence in the shaking table test, the PGAs were applied from a small to large value (i.e., from 0.1 g to 0.8 g). Between the runs, white noise motion with a low amplitude 0.05 g PGA was input to identify the dynamic characteristics of the test model. Moreover, to study the influence of the different number of winding turns on the bridge response, comparison tests of the number of winding turns were performed. PGAs of 0.4 g, 0.6 g and 0.8 g were applied.

Test Results

Model 1 is presented as a benchmark, and the test results between the Models 1 and 2 were compared to evaluate the effectiveness of the WRD-D. For a better understanding of the WRD-D seismic reduction effect, the seismic reduction rate λ of the fixed pier internal force (the longitudinal bending moment and shear force at the bottom of piers) and the girder displacement, the magnification η of the sliding piers internal force (the longitudinal bending moment and shear force at the bottom of piers), and the acceleration amplification factor R_Ax are defined as follows

λ = \frac{R_{1, \max} - R_{2, \max}}{R_{1, \max}}

(17)

η = \frac{R_{2 i, \max} - R_{1 i, \max}}{R_{1 i, \max}}

(18)

R_{A x} = \frac{A_{\max}}{A_{g}}

(19)

where R_1,max is the maximum seismic response (internal force and girder displacement) of Model 1, and R_2,max is the maximum seismic response (internal force and girder displacement) of Model 2, R_1i,max is the maximum internal force of the sliding pier i in Model 1, and R_2i,max is the maximum internal force of the sliding pier i in Model 2, A_max is the maximum acceleration measured on the main girder, and A_g is the output maximum acceleration on the shaking table.

Horizontal acceleration response

Figure 10 shows the peak acceleration of the main girder in both models. Figure 11 presents the acceleration amplification factor of the main girder for both models.

Figure 10.

Superstructure acceleration response under. (a) Qian an wave. (b) EL-Centro wave. (c) Taft wave. (d) Tianjin wave.

Figure 11.

Superstructure acceleration amplification factor under. (a) Qian an wave. (b) EL-Centro wave. (c) Taft wave. (d) Tianjin wave.

It can be seen from the Figure 10 that the acceleration responses of Model 2 are significantly increased compared to Model 1 after the WRD-Ds are activated and functioned. With the increase of input PGA, the peak acceleration of the main girder increases accordingly, indicating that WRD-Ds make the sliding piers and the fixed pier cooperatively loaded under an earthquake action.

As presented in the Figure 11, the amplification factors decrease with the increase of the PGA; Model 2 has a larger amplification factor than Model 1 under the same PGA, indicating that the acceleration amplification factor can be amplified after the WRD-Ds are activated.

It can also be seen from Figure 11 that the main girder acceleration amplification factor firstly decreases significantly and then becomes less sensitive with the increase of the PGA. This can be reasonably explained as follows: In Model 1, when the PGA is low, the structural stiffness is larger and small relative displacements occur between some sliding piers and the main girder, so the acceleration amplification factors are large. As the PGA increases, the relative displacements between the sliding piers and the main girder occur and the structural stiffness is reduced, so the acceleration amplification factor is relatively reduced. In Model 2, when the PGA is low, the WRD-Ds are not activated because the relative displacements between the sliding piers and the main girder do not occur. When the WRD-Ds are activated at a certain PGA, the gross slip of winding ropes occurs, and the winding ropes dissipate seismic energy through the frictional resistance. Therefore, the acceleration amplification factor first decreases and then remains relatively stable.

Displacement response

Figure 12 shows the comparison of the maximum displacements of the main girder for both models under the three earthquake waves. Because the main girder displacement response is very small under the action of the Qian an wave, its girder displacements are not presented in Figure 12. The main girder displacements in Model 2 are significantly reduced compared to Model 1, indicating that the WRD-D can effectively reduce the displacement response and avoid earthquake damage such as the bearing failure and span collapse. It should be noted that when the PGA is less than 0.2 g, the main girder displacements for both models are essentially the same, as the WRD-Ds of Model 2 have not been activated and the overall relative displacements between the superstructure and sliding piers do not occurred for the both models.

Figure 12.

Main girder displacement under. (a) EL-Centro wave. (b) Taft wave. (c) Tianjin wave.

Figure 13 present the main girder displacement time-history curves of two models under the Tianjin wave. The displacement response of Model 2 is significantly smaller than that of Model 1 when the PGA ≥ 0.4 g. The period of Model 2 is significantly shorter than Model 1 after the WRD-Ds are activated, because the large frictional resistance generated by the WRD-D increases the overall stiffness of Model 2.

Figure 13.

Time history of main girder displacement under Tianjin wave. (a) PGA=0.4 g. (b) PGA=0.6 g.

The displacement time-history curves between the main girder and the sliding pier 3 in Model 2 are shown in Figure 14. The relative displacement remains essentially at zero under the 0.2 g PGA, illustrating that the WRD-D on the sliding pier 3 has established lock connection with the main girder and cooperatively loaded with the fixed pier.

Figure 14.

Displacement time history of sliding pier 3 and girder for Model 2. (a) EL-Centro wave. (b) Taft wave.

Table 2 gives the seismic reduction rate of the main girder displacements under three earthquake waves. Under the action of the Taft and EL-Centro waves, the seismic reduction rate of the main girder displacement is small or even negative when the PGA is less than 0.2 g. This is because the WRD-Ds are not activated and the relative displacements between the main girder and sliding piers do not occur for both models when the PGA is small. Moreover, the proportion of test interference factors affecting the structure response is relatively large when the PGA was small. However, the absolute error of the girder displacement between the Model 1 and Model 2 was small and acceptable. When the PGA is greater than 0.2 g, the WRD-Ds are activated, and the gross slip occurs. The frictional resistance of WRD-D increases with the relative velocity between the sliding piers and the main girder. Therefore, the main girder displacements seismic reduction rate increases with the increase of the PGA.

Table 2.

Seismic reduction rate of the main girder displacement.

PGA (g)	Seismic reduction rate λ (%)
PGA (g)	Taft	EL-Centro	Tianjin
0.1	8.27	−78.73	18.59
0.2	−30.57	−6.54	34.66
0.4	46.18	35.46	40.54
0.6	52.66	54.58	47.74
0.8	51.66	61.46	49.12

Force response

Comparison of the peak micro-strains measured at the piers bottom of the Models 1 and 2 is presented in Figure 15. As shown in Figure 15 (a) and (b), under the EL-Centro earthquake wave, the fixed pier strains of Model 2 are significantly reduced in comparison with Model 1, and the strains of sliding piers are significantly increased, indicating that WRD-Ds can make the sliding piers and fixed pier bear earthquake loads cooperatively under the earthquake action. When the PGA ≤ 0.2 g, the fixed pier strains of Model 2 are all slightly larger than that of Model 1 under the EL-Centro wave. Because the gross slip of WRD-D does not occur, and the frictional resistance of WRD-D increase the lateral stiffness of Model 2. When the PGA > 0.2 g, the WRD-Ds are all activated, the gross slip occurs and the frictional resistance dissipate the seismic energy, so the fixed pier internal force response is reduced.

Figure 15.

Micro-strain under EL-Centro wave. (a) fixed pier. (b) sliding piers.

Table 3 gives the maximum absolute micro-strain at the piers bottom, and the seismic reduction rate of micro-strain when the PGA = 0.4 g. The seismic reduction rates of the fixed pier strains are all greater than 20% under three earthquake waves. The strain amplification rates of the sliding pier 1 are obviously greater than that of the sliding pier 3 because the stiffness of the sliding pier 1 is larger than that of the sliding pier 3.

Table 3.

Seismic reduction effect of micro-strain under different earthquake waves.

PGA (g)	Earthquake wave	Model	Micro-strain of pier 1	Amplification rate η (%)	Micro-strain of pier 2	Seismic reduction rate λ (%)	Micro-strain of pier 3	Amplification rate η (%)
	Taft	Model 1	85.5	200.0	748.3	28.7	173.4	41.2
	Taft	Model 2	256.4	200.0	533.5	28.7	244.8	41.2
0.4	EL-Centro	Model 1	74.6	262.9	596.8	24.0	166.2	68.0
	EL-Centro	Model 2	270.7	262.9	453.7	24.0	279.2	68.0
	Tianjin	Model 1	67.6	770.4	1379.0	27.5	216.1	152.8
	Tianjin	Model 2	588.4	770.4	999.2	27.5	546.3	152.8

Figure 16 presents the micro-strain comparison of Model 1 fixed pier and all piers of Model 2 under three earthquake waves. When the PGA > 0.2 g, the internal force responses of all piers in Model 2 are smaller than that of the fixed pier in Model 1. That means the overall internal force response of Model 2 is decreased when the WRD-Ds are activated. As for Model 2, even though the internal force of the sliding piers increases to some extent, the internal force of the fixed pier is still the largest.

Figure 16.

Micro-strain comparison of Model 1 fixed pier and Model 2 piers. (a) EL-Centro wave. (b) Taft wave. (c) Tianjin wave.

It should be noted that when the PGA is larger, the fixed pier internal forces of the Models 1 and 2 both enter the yielding state. Under the action of the Tianjin wave, when the PGA is 0.4 g, the maximum micro-strain of the fixed pier in the Model 1 was 1379 με, which slightly exceeds the yield micro-strain value of 1140 με. When the PGA is 0.8 g, the maximum micro-strain of the fixed pier in the Models 1 and 2 is 3452 με and 1930 με, respectively. However, Q235 steel has a large strain flow range in the yielding stage, which can reach up to 1500–25000 με. This minor yielding has little impact on the capacity of piers.

The influence of winding turns on the WRD-D seismic performance

The number of winding turns has an obvious effect on the frictional resistance of WRD-D, so it is necessary to study the influence of the different number of winding turns on the WRD-D seismic performance. The structural seismic responses with different number of winding turns in Model 2 were compared and analyzed, and the number of winding turns varies from 1 to 4.

Figure 17 presents the maximum displacement of the main girder in the Models 1 and 2 with different numbers of winding turns. The “n” in the legend “Model 2-n” represents the number of winding turns. The main girder displacements of Model 2 with the different number of winding turns are significantly reduced compared to Model 1. Under the action of the EL-Centro wave, the seismic reduction rate of the girder displacement in Model 2 is between 52% and 73%, but the number of winding turns has no obvious effect on the WRD-D seismic reduction rate. Under the action of the Tianjin wave, the seismic reduction rate of girder displacement for Model 2 is between 35% and 56% and the seismic reduction rate of girder displacement increases with the increase of winding turns, but the overall increasing trend is not obvious.

Figure 17.

Displacement of main girder with different winding turns under. (a) EL-Centro wave. (b) Tianjin wave.

Figures 18 and 19 present the maximum micro-strain of the piers in the Models 1 and 2 with different numbers of winding turns. As shown in Figure 18, the fixed pier strains of Model 2 with the different number of winding turns are significantly reduced compared to Model 1. However, the number of winding turns has no obvious effect on the internal force seismic reduction rate of the fixed pier. The seismic reduction rates of internal forces in Model 2 are 44%–79% under the action of the EL-Centro wave and 37%–57% under the action of the Tianjin wave. The reason for this can be explained as follows: The viscous dampers used for different windings turns are the same. Under the same earthquake wave, the WRD-D has a higher demand for damping force when the number of windings turns is small. As the number of winding turns increases, the demand for the damping force decreases. In addition, the instantaneous relaxation and hardening of the winding ropes during the test also reduces the seismic reduction effect of the WRD-D. Therefore, the number of windings turns has no obvious influence on the shock absorption effect of the WRD-D.

Figure 18.

Micro-strain of fixed pier 2 with different winding turns under. (a) EL-Centro wave. (b) Tianjin wave.

Figure 19.

Micro-strain of sliding pier 1 with different winding turns under. (a) EL-Centro wave. (b) Tianjin wave.

As shown in Figure 19, the strains on the sliding pier 1 in Model 2 are significantly larger than that in Model 1. Under the action of the EL-Centro and Tianjin waves, the strains on the sliding pier 1 are obviously increased with the increase of winding turns. It should be noted that there must be a reasonable number of winding turns. When the number of winding turns is too large, it may be detrimental to the sliding piers, and the overall seismic response of the structure will also increase accordingly. As shown in Figure 19, under the action of the Tianjin wave (PGA = 1.0 g), the sliding pier 1 with four winding turns of winding ropes is in the yielding state. In summary, the recommended number of winding turns is three.

Conclusion

A shock absorption device named WRD-D was proposed, which can utilize the potential capability of the sliding piers and reduce the seismic demand of the fixed pier through making the sliding piers equipped with WRD-D and the fixed piers bear the seismic loads together. The shaking table test based on the three-span continuous girder bridge with unequal pier heights was conducted to study the seismic performance of the WRD-D. The main conclusions of this study are as follows:

(1) The WRD-D performs well in reducing the seismic response of the fixed pier internal force and the superstructure displacement once activated. However, when the PGA is low, the seismic response of the fixed pier internal force and the superstructure displacement in Model 2 is greater than Model 1 because the WRD-Ds are not activated, and the overall stiffness of Model 2 is larger than Model 1.

(2) The WRD-D can significantly increase the internal force response of the sliding piers, but the peak internal force of the sliding piers is still smaller than that of the fixed pier in Model 2. Therefore, the WRD-D can enable the sliding piers and the fixed pier to bear the seismic loads cooperatively and improve the seismic performance of continuous girder bridge effectively.

(3) The number of winding turns has little effect on the seismic reduction of the fixed pier internal force and the superstructure displacement, while has significant effect on the seismic response of the sliding piers. The larger of the number of winding turns, the bigger of the seismic response of the sliding piers.

(4) In order to apply the WRD-D to seismic design of continuous girder bridges, the following works need to be further studied and improved such as the selection and optimization of viscous damper parameters, the number of winding turns, the optimal ratio of the WRD-D resistance force between the friction element and the viscous element under different structures, and how to make the WRD-D meet both the longitudinal and transverse seismic reduction demands simultaneously.

Footnotes

Acknowledgements

The authors acknowledge the research supports from the financial support of National Natural Science Foundation of China (Grant number 51778022), which made the work presented in this paper possible. Thanks to Luoyang Shuang Rui Special Equipment Co., Ltd. for supporting this test.

Declaration of conflicting interests

The author(s) declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.

Funding

The author(s) disclosed receipt of the following financial support for the research, authorship, and/or publication of this article: This research was sponsored by the National Natural Science Foundation of China (Grant No. 51778022).

ORCID iD

Lijun Su

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