Mechanical Behaviors of a Buckling-Plate Self-Centering Friction Damper

Abstract

In order to improve the resilience of structures subjected to strong earthquakes, a buckling-plate self-centering friction damper (BPSCFD) with low post-yielding stiffness is proposed, which consists of a group of post-buckling plates and a self-centering variable friction mechanism. The damper is intended to not only reduce the peak and residual deformation of structures, but also to limit the additional internal force of the structural elements. Through theoretical derivation and finite element simulation, the hysteretic damping and self-centering characteristics of BPSCFDs are studied. In order to examine the seismic performance of the BPSCFDs, the dampers are employed to retrofit a double-columns bridge bent, and the corresponding elastic-plastic time history analysis is conducted. The results show that the force-displacement relationships of BPSCFDs with different parameter combinations are characterized by typical flag-shaped self-centering hysteretic loops and low post-yielding stiffness, and the dampers can effectively reduce the peak and residual deformation of the bridge bent without increasing the peak acceleration and base shear. The research results could supply a guideline for the design and application of the damper.

1. Introduction

Conventional energy dissipation dampers (e.g., metallic dampers [1,2,3]; viscous dampers [4,5]) can effectively mitigate the seismic responses of structures through dissipating the earthquake energy. However, such dampers are not ideal for residual displacement control. Recent research shows that the self-centering energy dissipation devices can effectively reduce the seismic responses and residual deformation of structures subjected to strong earthquakes. The self-centering devices usually consist of self-centering components and energy dissipation components in function. In the existing research, the self-centering components majorly include prestressed tendons, preloaded disc springs, and shape memory alloys [6,7,8,9,10,11,12,13,14]. The energy dissipation components mainly include steel core yield braces (e g. BRB), friction-based dampers, and viscous fluid dampers [15,16,17].

In recent years, high-efficiency energy-consuming variable friction self-centering devices were proposed and attracted wide attention. In 2018, Hashemi et al. [18] proposed a novel resilient sliding friction (RSF) damper composed of a couple of corrugated outer plates, a slotted inner plate, high-strength bolt, and disc springs. In these dampers, the lateral pressure generated by the disc springs can be converted into the positive normal force of the friction surface, and the restoring force to the equilibrium position, through a special mechanism. Therefore, the dampers can achieve the function of self-centering and friction energy dissipation at the same time. Based on a similar mechanism, different self-centering friction dampers have been proposed [19,20,21,22,23]. Fang et al. [24,25] proposed a self-centering friction damper composed of SMA friction ring springs or high elastic steel friction ring springs. The research results suggest that the proposed dampers have stable self-centering capacity and hysteretic energy dissipation performance. Zhang et al. [26] and Qiu et al. [27] employed SMA as a restoring element in RSF, and the experimental and numerical results show that the novel device displayed significant self-centering and hysteretic energy dissipation performance. Xue et al. [28] studied, experimentally and numerically, the mechanical behaviors of RSF and its retrofitted RC double-column bridge bents. The results showed that RSF can greatly improve the resilient performance of RC double-column bridge bents. The above research indicates that these self-centering friction dampers possess significant resilient performance and simple configuration due to the special design of the inclined friction surface, which has a wide application prospect.

The current research shows that the installation of displacement-dependent dampers in structures could increase the additional internal force of the structural members adjacent to the dampers. Dong et al. [29,30] proposed a novel self-centering BRB (SC-BRB) composed of a group of disc springs and a traditional BRB in parallel and conducted experimental and numerical investigations of the mechanical behavior of the damper and its application for retrofitting a double-column bridge bent. Compared with ordinary BRBs, SC-BRBs can improve the self-centering ability of the bridge bent, but could increase the peak acceleration and base shear of structures, in which the acceleration can be measured by the novel wireless accelerometer [31]. Xiang et al. [32] compared the seismic mitigation performance of piston-based self-centering Braces (PBSCBs), BRBs and viscous dampers for retrofitting a double-column bridge. The results show that the PBSCBs have a superior self-centering performance, but their restoring force demand is significantly higher than that of BRBs for the same displacement target, which suggests that PBSCBs could lead to a greater base shear of the bridge bent. In order to reduce the additional internal force caused by the self-centering dampers, Yousef-Beik et al. [33] proposed a novel self-centering brace with zero post-yield stiffness (SC-ZSB). The research results show that the SC-ZSB system can effectively control the peak and residual displacement of the structure without significantly increasing the base shear of the structures. Therefore, the self-centering damper with zero post-yield stiffness is beneficial for limiting the additional internal force of the self-centering retrofitted structures. Asfaw and Ozbulut [34] placed SMA rods in grouted steel tubes to form a novel buckling restrained brace, and eight sets of specimens were fabricated to verify the hysteretic performance of the energy dissipation braces. The experimental results show that the self-centering energy dissipation braces have low post-buckling stiffness and good flag-shaped hysteretic behavior. Previous studies show that the self-centering damper with low post-buckling stiffness has a relatively large equivalent damping ratio, which can effectively reduce the additional internal force of the structural members. However, the related research is still relatively limited, and the existing self-centering dampers with low post-buckling stiffness commonly have a complex mechanism or high material costs.

In this study, a novel self-centering friction damper with low post-yield stiffness composed of a group of buckling plates and a self-centering friction device is proposed, which is abbreviated to BPSCFD for convenience. The BPSCFD is desired to reduce the peak and residual deformation of structures without significantly increasing the additional internal force of the structural members. The research on the mechanical properties and seismic performance of BPSCFDs is organized as follows. In Section 2, the configuration and working principle of the damper are introduced. In Section 3, the theoretical axial force-deformation relationship of the buckling-plates is derived and verified using the finite element method. In Section 4, the force-displacement relationship of the whole damper is studied using theoretical and numerical methods. In Section 5, the effect of the key parameters on the mechanical behaviors of the damper is investigated. In Section 6, an application case of BPSCFDs for retrofitting a double-column bridge bent is investigated to examine the seismic mitigation performance of the damper through elastic-plastic time history analysis. Finally, the research conclusions are summarized in Section 7. The research shows that the proposed BPSCFD damper is characterized by a simple configuration, low cost, stable damping, and significant self-centering performance, and the results could supply a guideline for the design and application of the damper.

2. Configuration and Working Principle of the BPSCFD

Configuration: The configuration of the proposed BPSCFD is shown in Figure 1. It is predominantly composed of two symmetrically arranged corrugated friction plates, several couples of wedge-shaped sliders, several groups of buckling-plates, a couple of left and right frames, a driving frame, an actuator rod, and several connecting screws. The couple of corrugated friction plates and the couple of the left and right frames are connected by high strength screws. The submit of the wedge-shaped sliders initially matches the vertex of the corrugated friction plates. As shown in Figure 1, there are two groups of buckling plates installed in this damper. Each buckling plate is inserted into the reserved gaps of the upper and lower wedge-shaped sliders, respectively, to form the fixed constraint at two ends. According to this connecting method, two groups of buckling-plates are installed between the upper and lower wedge-shaped sliders and fixed firmly by fixing screws through the bolt holes on the wedge-shaped sliders and buckling plates. Meanwhile, the four wedge-shaped sliders are inserted into the rectangular slotted holes on the driving frame, which can slide freely in the slotted holes in the vertical direction. The actuator rod is connected with a driving frame on the right and goes through the hole on the right frame. According to the damping force demand, multiple buckling plates can be installed in parallel between the corresponding wedge-shaped sliders. In the practice of design and application, the parameters of the damper can be measured conveniently by the wireless sensors [35].

Working principle: As shown in Figure 2a, when the damper is loaded in tension, the actuator rod actuates the driving frame move in the right direction, and the driving frame further forces the wedge-shaped sliders to slide along the right slopes. At this moment, each slider will suffer the driving force from the driving frame, the resisting force from the buckling-plate, the normal force of the slope, and the friction from the slope. With the increase in the deformation, the distance between the two corresponding wedge-shaped sliders in the transverse direction gradually decreases; thus, the deformation of the buckling-plate between the two corresponding wedge-shaped sliders will increase, which results in a gradual increase in the resisting force of the buckling-plate, the supporting force and the friction force. When the damper is unloaded, the restoring force from the buckling-plates drives the wedge-shaped sliders to slide along the slope, then further actuates the driving frame and actuator rod to return to the original balance position to achieve the self-centering function. When the damper is in compression, the movement process of each component is opposite.

3. Elastic Buckling Analysis of the Buckling Plates

The buckling plates are the key components for supplying the restoring force of the damper and the normal force of the friction surface. In this section, the theoretical and finite element analysis of the elastic buckling behavior of the plates is conducted, which supplies a foundation for the force-displacement relationship of the whole damper.

3.1. Theoretical Derivation

According to the connection of the buckling-plates in BPSCFD, the buckling-plates are fixed at two ends; thus, the mechanical model of the buckling-plate under compression can be modelled as shown in Figure 3. Before buckling of the plate, the buckling-plates are under axial compression, when the axial force-deformation relationship can be expressed by Hooke’s law as Equation (1).

$$F_{\mathrm{b}}=\frac{EA}{l_{\mathrm{r}}}\delta$$

(1)

where $F_{\mathrm{b}}$, E, A, $l_{\mathrm{r}}$ and $\delta$ are the axial pressure, elastic modulus, cross-sectional area, length and axial deformation of the buckling-plate, respectively.

According to the Euler formula, the buckling critical pressure of the buckling-plate is

$$F_{\mathrm{cr}}=\frac{{\pi }^{2}E{I}_Z}{{(0.5l_{\mathrm{r}})}^2}$$

(2)

where $I_Z$ is the minimum section inertia moment of the buckling-plate. The critical deformation ${\delta }_{\mathrm{cr}}$ corresponding to $F_{\mathrm{cr}}$ is determined by the Formula (1). According to reference [36], the equation of the deflection curve of the buckling-plate can be obtained as follows:

$$w=B(\cos \frac{2\pi }{l_{\mathrm{r}}}\sqrt{\frac{F_{\mathrm{b}}}{F_{\mathrm{cr}}}}x-1)$$

(3)

According to Figure 3, the projection length of the buckling-plate in the axial direction after buckling is $l_{\mathrm{r}}-\delta$. According to Equation (3), the maximum deflection $w_{\max }=\left|-2B\right|$ is achieved when $x=\frac{l_{\mathrm{r}}-\delta }{2}$ and $\cos \frac{2\pi }{l_{\mathrm{r}}}\sqrt{\frac{F_{\mathrm{b}}}{F_{\mathrm{cr}}}}\frac{l_{\mathrm{r}}-\delta }{2}=-1$, thus the axial force-displacement relationship can be obtained as follows:

$$F_{\mathrm{b}}={F_{\mathrm{cr}}/\left(1-\frac{\delta }{l_{\mathrm{r}}}\right)}^2$$

(4)

Combined with Equation (1), the axial force-displacement relationship of the plate before and after buckling can be rewritten into the piecewise function as Equation (5).

$$F_{\mathrm{b}}=\{\begin{array}{c}\frac{EA}{l_{\mathrm{r}}}\delta \\ {F_{\mathrm{cr}}/\left(1-\frac{\delta }{l_{\mathrm{r}}}\right)}^2\end{array}\begin{array}{c},\\ \\ ,\end{array}\begin{array}{c}\delta \le {\delta }_{\mathrm{cr}}\\ \\ \delta >{\delta }_{\mathrm{cr}}\end{array}$$

(5)

As shown in Figure 3, the axial deformation can be expressed as Equation (6),

$$\delta ={\displaystyle {\int }_0^l\left(\mathrm{d}s-\mathrm{d}x\right)}$$

(6)

where

$$\mathrm{d}s=\sqrt{{(\mathrm{d}x)}^2+{(\mathrm{d}\omega )}^2}\approx \left(1+\frac{1}{2}{\left(\frac{\mathrm{d}\omega }{\mathrm{d}x}\right)}^2\right)\mathrm{d}x$$

(7)

According to Equations (5)–(7), the B in Equation (3) can be obtained as Equation (8).

$$B=\frac{l_{\mathrm{r}}\sqrt{10F_{\mathrm{b}}\pi \sqrt{F_{\mathrm{b}}{F}_{\mathrm{cr}}}(40+l_{\mathrm{r}}{}^4-40\sqrt{F_{\mathrm{cr}}/F_{\mathrm{b}}})[4\pi \sqrt{F_{\mathrm{b}}/F_{\mathrm{cr}}}-\sin (4\pi \sqrt{F_{\mathrm{b}}/F_{\mathrm{cr}}})]}}{10F_{\mathrm{b}}\pi [4\pi \sqrt{F_{\mathrm{b}}/F_{\mathrm{cr}}}-\sin (4\pi \sqrt{F_{\mathrm{b}}/F_{\mathrm{cr}}})]}$$

(8)

According to the differential equation of the bending deformation, the bending moment in the mid-span cross section of the buckling-plate is maximum, namely

$$M_{\max }=E{I}_{\mathrm{Z}}{\ddot{w}}_{\max }=4B{\pi }^2{F}_{\mathrm{b}}E{I}_{\mathrm{Z}}/(l_{\mathrm{r}}{}^2{F}_{\mathrm{cr}})$$

(9)

To ensure that the material is elastic, the maximum bending stress should be less than the yield strength of the material, i.e.,

$${\sigma }_{\max }=\frac{M_{\max }}{W_{\mathrm{Z}}}<{\sigma }_s$$

(10)

where ${\sigma }_{\max }$ is the maximum bending stress, $W_{\mathrm{Z}}$ is the section modulus in the bending of the cross section, and ${\sigma }_{\mathrm{s}}$ is the yield strength of the material. Combined with Equations (5), (8), (9), and (10), the maximum axial deformation ${\delta }_{\max }$ of the buckling-plate can be determined under the premise that the material is elastic.

The theoretical axial force-deformation of the buckling plates with fixed constraint at two ends is derived, and the maximum axial deformation when the material is elastic can be determined according to the above method.

3.2. Finite Element Simulation

In order to verify the axial force-displacement relationship of the buckling-plate, as in Equations (1), (4) and (5), the mechanical behavior of buckling-plate with fixed both ends is investigated numerically. The 40SiMnVBE spring steel is selected as the buckling-plate material [37]. The strength limit of the material is 1930 MPa, the yield strength is 1760 MPa, the elongation is 11.5%, the elastic modulus is 200 GPa, the Poisson’s ratio is 0.3, and the density is 7850 kg/m³. The geometric parameters of the buckling-plate are tabulated in

Table 1. Geometric parameters of buckling-plates.

Group	Length (mm)	Width (mm)	Thickness (mm)
01	350, 400, 450	100	2
02	400	100	1.5, 2, 2.5

. According to the material properties, the finite element simulation of the buckling-plate is carried out. The elastic-plastic model is employed to simulate the buckling-plate material, and the SOLID186 element is selected for meshing. One end of the buckling-plate is fixed, while another end is only allowed to move in the axial direction, to simulate the fixed support at both ends.

Figure 4a shows the stress cloud diagram of the buckling-plate when its dimension is 400 mm length, 100 mm width, 2.5 mm thickness, as the maximum stroke is 15 mm. The results show that the maximum stress occurs at the middle of the plate. In order to show the stress at the middle of the plate, Figure 4b presents the variation of the maximum stress with $\delta$. The results show that the stress increases with the increase in the $\delta$, and the stress a has positive correlation with the thickness of the plate and a negative correlation with the length of the plate. Meanwhile, the ${\delta }_{\max }$ of the plate is 14.5 mm under the premise that the material is elastic, which is near to the theoretical maximum deformation 14.8 mm determined by Equations (5), (8), (9), and (10) in Section 3.1. In order to verify the theoretical model of the force-displacement relationship, Figure 4c compares the numerical results of the axial force-displacement relationship with the theoretical results. The results show that theoretical analysis agrees well with the finite element results, which demonstrates that the two methods can simulate the axial force-deformation relationship of the buckling plate reasonably.

In this section, the finite element method was employed to demonstrate the theoretical axial force-deformation relationship proposed in Section 3.1. The results supply a foundation for the following analysis.

4. Force-Displacement Relationship of BPSCFD

In this section, the mechanical behavior of the damper is studied. Firstly, the force-displacement relationship of the damper is derived, and the key parameters evaluating the hysteretic performance and self-centering ability of the damper are defined in Section 4.1. Then, the finite element simulation of the damper is carried out to obtain the numerical force-displacement relationship of the damper to verify the theoretical model in Section 4.2.

4.1. Theoretical Model

According to the working principle of the damper, when the damper is loaded, the actuating rod drives the wedge-shaped sliders through the driving frame. As shown in Figure 5a, each wedge-shaped slider is subjected to the resisting force $F_{\mathrm{b}}$ of the buckling-plate, the normal force $F_{\mathrm{N}}$ and the friction force $F_{\mathrm{f}}$ from the slope, and the driving force $F_{\mathrm{vf}}$ from the driving frame. When the damper is unloaded, the recovery of the buckling plate could drive the wedge-shaped slider to return to the original balance position, when the wedge-shaped slider is subjected to the resisting force $F_{\mathrm{b}}$ of the buckling-plate, the normal force $F_{\mathrm{N}}$ and the friction force $F_{\mathrm{f}}$ of the slope, and reaction force $F_{\mathrm{vf}}$ of the driving frame, as shown in Figure 5b. According to the Coulomb friction model, the friction can be expressed as

$$\begin{array}{c}{F}_{\mathrm{f}}=\mu F_{\mathrm{N}}\end{array}$$

(11)

The resisting force $F_{\mathrm{b}}$ of the buckling-plate is dependent on its axial deformation. When the displacement of the wedge-shaped slider relative to the initial position is $u$ (positive to the right), the compression deformation of the buckling-plate $\delta$ is equal to $2\left|u\right|\tan \beta$. According to the axial force-displacement relationship of the buckling plates, as in Equation (5) in Section 3.1, $F_{\mathrm{b}}$ can be rewritten as

$$F_{\mathrm{b}}=\{\begin{array}{c}2\left|u\right|\tan \beta \frac{EA}{l_{\mathrm{r}}}\\ {F_{\mathrm{cr}}/\left(1-\frac{2\left|u\right|\tan \beta }{l_{\mathrm{r}}}\right)}^2\end{array}\begin{array}{c},\\ \\ ,\end{array}\begin{array}{c}\left|u\right|\tan \beta \le {\delta }_{\mathrm{cr}}\\ \\ \left|u\right|\tan \beta >{\delta }_{\mathrm{cr}}\end{array}$$

(12)

According to Figure 5, the equilibrium equations of the wedge-shaped slides under loading and unloading cases can be written as Equation (13).

$$\{\begin{array}{l}\sum F_x=0F_{\mathrm{vf}}-\mathrm{sgn}(u\dot{u})F_{\mathrm{f}}\cos \beta -F_{\mathrm{N}}\sin \beta =0\\ \sum F_y=0F_{\mathrm{b}}+\mathrm{sgn}(u\dot{u})F_{\mathrm{f}}\sin \beta -F_{\mathrm{N}}\cos \beta =0\end{array}$$

(13)

where $\dot{u}$ represents the relative speed of the slider, $\mathrm{sgn}()$ is a symbolic function. When $\mathrm{sgn}(u\dot{u})=1$, the damper is in loading. When $\mathrm{sgn}(u\dot{u})=-1$, the damper is in unloading. According to Equations (6)–(8), the driving force for each wedge-shaped slider can be expressed as

$$\begin{array}{c}{F}_{\mathrm{vf}}=\frac{F_{\mathrm{b}}\left(\mathrm{sgn}(u\dot{u})\mu +\tan \beta \right)}{1-\mathrm{sgn}\left(u\dot{u}\right)\mu \tan \beta }\end{array}$$

(14)

It should be noted that the self-locking phenomenon should be avoided to assure that the damper can work normally. According to Equation (14), the parameter conditions of avoiding the self-locking phenomenon are obtained as follows:

$$\begin{array}{c}\mu \end{array}<\tan \beta <1/\mu$$

(15)

Generally, $\mu <1$, thus the upper limit of the inequality is not easy to meet.

Figure 1 shows that there are four same wedge-shaped sliders in a damper, thus the damping force of the damper $F_{\mathrm{D}}=4F_{\mathrm{vf}}$, namely

$$F_{\mathrm{D}}=\left[\frac{4F_{\mathrm{b}}\left(\mathrm{sgn}\left(u\dot{u}\right)\mu +\tan \beta \right)}{1-\mathrm{sgn}\left(u\dot{u}\right)\mu \tan \beta }\right]\mathrm{sgn}\left(u\right)$$

(16)

Figure 6 is a schematic diagram of the force-displacement relationship of the damper, where $F_{\mathrm{y}}$ and $u_y$ are the yield force and yield displacement of the BPSCFD, respectively; $F_{\mathrm{r}}$ is the restoring force as the deformation returns to zero, $F_{\mathrm{m}}$ and $u_{\mathrm{m}}$ represent the maximum force and displacement of the damper, respectively; $K_{\mathrm{l}}$ and $K_2$ represent the pre-yield and post-yield stiffness of BPSCFD, respectively. According to the above theoretical analysis, the parameters $F_{\mathrm{y}}$, $u_y$, $F_{\mathrm{r}}$, $K_1$, and $K_2$ can be calculated as follows.

$$\begin{array}{c}{F}_{\mathrm{y}}=\frac{4F_{\mathrm{cr}}\left(\tan \beta +\mu \right)}{1-\mu \tan \beta }\end{array}$$

(17)

$$u_y={\delta }_{\mathrm{cr}}/2\tan \beta$$

(18)

$$F_{\mathrm{r}}=\frac{4F_{\mathrm{cr}}\left(\tan \beta -\mu \right)}{1+\mu \tan \beta }$$

(19)

$$\begin{array}{c}{K}_1=\frac{F_{\mathrm{y}}}{u_y}\end{array}$$

(20)

$$\begin{array}{c}{K}_2=\frac{F_{\mathrm{m}}-F_{\mathrm{y}}}{u_{\mathrm{m}}-u_y}\end{array}$$

(21)

The ratio of $K_2$ to $K_1$ is defined as post-yield stiffness ratio, namely

$$\begin{array}{c}\alpha =\frac{K_2}{K_1}\end{array}$$

(22)

According to Figure 6, the ratio of $F_{\mathrm{r}}$ to $F_{\mathrm{y}}$ is defined as the self-centering factor $\rho$.

$$\begin{array}{c}\rho =\frac{F_{\mathrm{r}}}{F_{\mathrm{y}}}\end{array}$$

(23)

The case $\rho >0$ indicates that the damper has a complete self-centering ability, and $\rho <0$ indicates that the damper does not have a complete self-centering ability.

The equivalent damping ratio ${\xi }_{\mathrm{eq}}$ of the damper can determined by Equation (24) to quantify the damping characteristics of the damper [8,27].

$$\begin{array}{c}{\xi }_{\mathrm{eq}}=\frac{W_{\mathrm{H}}}{4\mathsf{\pi }{W}_{\mathrm{E}}}\end{array}$$

(24)

where $W_{\mathrm{H}}$ is defined as the energy dissipation of the damper within a loading and unloading period, and $W_{\mathrm{E}}$ is the maximum elastic potential energy, as shown in Figure 6.

4.2. Finite Element Simulation of BPSCFD

In order to verify the theoretical force-displacement relationship of the damper in Section 3.1, the finite element method is used to simulate the BPSCFD in this section. The size of the specimen of the damper is shown in Figure 7a. In the finite element model, the buckling-plates are made of 40SiMnVBE spring steel, and the other components are made of Q345 steel. Frictional contact is employed to simulate the contact interface between the corrugated friction plate and the wedge-shaped slider, with a friction coefficient of 0.2. Frictionless contact is employed to simulate the contact interface between the wedge-shaped slider and the driving frame, and the contact interface between the actuator and the right frame. Bonded contact is used to simulate the connecting surfaces of the other components. A fixed constraint is applied on the left end of the frame, and a displacement load is applied on the right end of the actuator rod. The geometric and material parameters used in the finite element simulation of the BPSCFD are tabulated in

Table 2. Material parameters of BPSCFD.

Material	$\mathbf{Dimension}({\mathit{l}}_{\mathbf{r}}\times \mathit{h}\times \mathit{b})$	Elastic Modulus	Poisson’s Ratio	Yield Stress
buckling-plate	$400\times 100\times 2$	200 GPa	0.3	1760 GPa
frames	---	200 GPa	0.3	345 GPa

.

Figure 7b shows the stress cloud diagram of the model when the compression deformation of the BPSCFD is maximum. The results show that the maximum stress occurs in the vertex of the corrugated friction plates and the edge of the friction surface of the wedge-shaped sliders. When the maximum displacement load is 20 mm, the maximum equivalent stress at the friction plate and the edge of the wedge-shaped slider is 279.47 MPa, which is much smaller than the yield strength of Q345 steel. At this moment, the axial displacement of the buckling-plates is 10.72 mm. According to the ${\delta }_{\max }$ of the buckling-plates, the buckling-plates is still in the elastic stage. The results show that each component is elastic, which ensures that the whole damper is in an undamaged state within the given amplitude range. In Figure 7c, the theoretical and finite element results of the force-displacement relationship of the damper are compared. The results show that the theoretical results agree well with the finite element calculation results, the force-displacement relationship of the damper presents a typical flag-shaped hysteresis model with low post-yield stiffness [27], and the hysteresis curves could reflect the hysteretic energy dissipation performance [38]. In addition, the numerical force-displacement curves show obvious fluctuation in the initial stage of loading and unloading, which is attributed to the stick-slip effects of the friction surface [39].

The theoretical force-displacement relationship of the BPSCFDs is proposed and verified by finite element simulation in this section. The results show that the theoretical model agrees well with the finite element results, which suggests that the two methods can accurately simulate the force-displacement relationship of the proposed damper.

5. Parametric Analysis

In this section, the effects of the key parameters on the hysteretic performance, energy dissipation capacity and self-centering ability of the damper are investigated. Firstly, the comparison between the different self-centering elements (linear spring and buckling-plate) are discussed in Section 5.1, and then the effects of the geometric parameters of the buckling plates, the inclination angle, and the friction coefficient of the friction surface on the mechanical property of the damper are studied in Section 5.2.

5.1. Effect of the Type of Self-Centering Element

In this section, the mechanical behavior of BPSCFDs is compared with the variable friction self-centering damper, in which a linear compression spring is employed as its self-centering element (VFLSCDs). The two dampers are compared in two cases: (1) the slope friction coefficient $\mu$, slope angle $\beta$, and yield force $F_{\mathrm{y}}$ are the same; (2) The slope friction coefficient $\mu$, slope angle $\beta$ and maximum force $F_{\mathrm{m}}$ are the same. Figure 8 shows the comparison of the force-displacement curves and the equivalent damping ratio between two dampers. The results show that the post-yield stiffness of the BPSCFD is 66% smaller than the VFLSCDs, and the equivalent damping ratio is 6% larger than the VFLSCDs in the first case. In the second case, the post-buckling stiffness of the BPSCFD is 71% smaller than the VFLSCDs, and the equivalent damping ratio of the BPSCFD is 19% larger than the VFLSCDs. Meanwhile, the difference in the equivalent damping ratio increases with the increase in the displacement amplitude. According to the displacement-based seismic design method, the low post-buckling stiffness and high equivalent damping ratio of the damper could reduce the damping force demand for the same design displacement target of structures, thereby reducing the additional internal forces of the structural components adjacent to the self-centering dampers [32].

5.2. Effects of Key Parameters

According to Equations (11)–(24), the self-centering performance and damping characteristics of BPSCFD mainly depend on the friction coefficient $\mu$, the inclination angle $\beta$, the minimum cross-sectional inertia moment $I_Z$ and the length $l_{\mathrm{r}}$ of the buckling-plate. In this section, the finite element method is employed to investigate the effects of the parameters on the damping and self-centering performance of the damper.

According to Equations (17), (19) and (23), the parameters $l_{\mathrm{r}}$ and $I_Z$ have no effect on the self-centering coefficient of the damper. Therefore, the influences of $l_{\mathrm{r}}$ and $I_Z$ only on the equivalent damper ratio and post-buckling stiffness of the damper are discussed here. Given $\mu =0.2$, $\beta ={15}^{°}$ and $I_Z=66{\mathrm{mm}}^4$, Figure 9a presents the hysteresis curves of the damper with different $l_{\mathrm{r}}$. The results show that $F_{\mathrm{y}}$ and the area of hysteretic loops increase significantly with the decrease in $l_{\mathrm{r}}$. Figure 9b presents the influences of $l_{\mathrm{r}}$ on the equivalent damping ratio ${\xi }_{\mathrm{eq}}$ and the post-buckling stiffness ratio $\alpha$. The results show that: ${\xi }_{\mathrm{eq}}$ increases, but $\alpha$ decreases with the increase in $l_{\mathrm{r}}$. When $l_{\mathrm{r}}$ increases from 350 mm to 450 mm, ${\xi }_{\mathrm{eq}}$ increases by 4.2% and $\alpha$ decreases by about 88.8%. Note that $\alpha$ is basically ${10}^{-4}$ order of magnitude in the considered range, which suggests that the post-yield stiffness ratio of the damper remains insignificant. This property of the damper should be attributed to the constant force characteristics of the post-buckling plates, which could result an insignificant variation in the friction with the increase in the displacement.

According to Equation (2), $I_Z$ plays an important role in the critical pressure of the buckling-plate. The influences of $I_Z$ on the force-displacement hysteresis curve, damping and stiffness characteristics of the damper are shown in Figure 10. Figure 10a shows that the $F_{\mathrm{y}}$ and $F_{\mathrm{r}}$ of the damper increases with the increase in $I_Z$. Figure 10b shows the influences of $I_Z$ on ${\xi }_{\mathrm{eq}}$ and $\alpha$. The results show that $\alpha$ increases obviously with the increase in $I_Z$, while ${\xi }_{\mathrm{eq}}$ is nearly independent on $I_Z$.

The above results show that $I_Z$ and $l_{\mathrm{r}}$ have no significant effect on the damping and stiffness characteristics of the BPSCFD. Therefore, it is not necessary to consider the influences of the geometric parameters of the buckling-plate on its damping and stiffness characteristics in the procedure of the buckling plate design. In other words, the geometrical parameters and the number of parallel plates can be determined based on the theoretical analysis in Section 3.1, only considering the self-centering function, the stroke and the restoring force demand of the damper.

With the exception of the geometrical parameters, the friction coefficient $\mu$ and slope angle $\beta$ could have a significant influence on the mechanical behavior of the damper, according to Equations (11)–(24). Figure 11 presents the influences of the friction coefficient $\mu$ and slope angle $\beta$ on the mechanical characteristics of the damper. Figure 11a shows that $F_{\mathrm{y}}$ and the hysteresis loop area increases significantly, but $F_{\mathrm{r}}$ decreases with the increase in $\mu$. When $\mu =0.3$, $F_{\mathrm{r}}$ is negative, meaning that the damper has lost the self-centering capacity. Figure 11b shows that $F_{\mathrm{y}}$ and $F_{\mathrm{r}}$ of the damper increase obviously with the increase in $\beta$. In order to quantitatively investigate the influences of $\mu$ and $\beta$ on the self-centering and damping performance of the damper, Figure 12a,b present the variation in the self-centering coefficient $\rho$ and equivalent damping ratio ${\xi }_{\mathrm{eq}}$ with $\mu$ and $\beta$. The results show that the self-centering coefficient $\rho$ decreases with the increase in $\mu$, and it increases with the increase in $\beta$. The equivalent damping ratio ${\xi }_{\mathrm{eq}}$ increases with the increase in $\mu$, but decreases with the increase in $\beta$. Specifically, when $\mu$ increases from 0.05 to 0.25, $\rho$ decreases by 127% and ${\xi }_{\mathrm{eq}}$ increases by 159%; when $\beta$ increases from 10° to 30°, $\rho$ decreases by 150% and ${\xi }_{\mathrm{eq}}$ decreases by 59%. The results further show that increasing the friction coefficient can significantly increase the damping performance of the damper, whereas it significantly weakens its self-centering performance; increasing the slope angle can effectively increase the self-centering performance of the damper, but reduces its damping performance. In engineering applications, the balance between the equivalent damping ratio and the self-centering performance should be sought according to the engineering demand through adjusting the parameters.

The effects of the key parameters, including the friction coefficient $\mu$, the inclination angle $\beta$, the cross-section inertia moment $I_Z$ and the length $l_{\mathrm{r}}$ of the buckling-plate, on the hysteretic curves, damping and self-centering characteristics of the BPSCFDs were investigated in this section. The results show that the parameters $I_Z$ and $l_{\mathrm{r}}$ have significant effects on the damping force of the BPSCFD, but insignificant effects on its self-centering and damping characteristics, while the parameters $\mu$ and $\beta$ have significant effects on the damping force, self-centering and damping characteristics of the damper.

6. An Application Case

To examine the seismic mitigation performance of the proposed BPSCFD, the BPSCFDS are employed to retrofit a RC double-column bridge bent of a typical three-span continuous highway bridge in Vancouver [32]. The configuration of the double-column bridge bent retrofitted with BPSCFDs is presented in Figure 13a, which represents the transversal scheme of the bridge. The bridge superstructure is composed of three I-shaped RC girders with an effective height of 200 cm. The width of the superstructure is 1220 cm, accommodating two vehicle lanes plus sidewalks. According to the general continuous bridge [40], the bridge is supported onto the abutments by the expansion bearings, and the superstructure is fixed to the cap beam, as shown in Figure 13a. The substructure is composed of double columns and a bent cap. The columns’ height is 7 m, the diameter is 1.5 m, the distance between the two columns is 6.6 m. The cross-section of the columns is shown in Figure 13b, which shows that the columns have a longitudinal reinforcement of $28\varnothing 30\mathrm{mm},$ with a reinforcement ratio of 1.12%. The peak strength of the concrete is 30 MPa, and the yield strength of the steel bars is 400 MPa. The bent cap has a 11.4 m length, 1.8 m width and 2.6 m height. The weight of a span of the superstructure is 920 t and the weight of the double-column bent is 90 t. In order to simplify the research, the double-column bridge bent consists of a double-column bent, and a span superstructure, as shown in Figure 13a, is employed to substitute the whole bridge in the seismic analysis. Previous research shows that the substitution is reasonable to obtain reliable results [32].

The BPSCFDs are installed in the double-column bridge bent through a rigid support, as shown in Figure 13a. The finite model of the prototype of the bridge bent is built in OpenSees, as shown in Figure 14a, in which the elastic Beam Column element is employed to simulate the bent cap, and the nonlinear fiber column element is used to simulate the RC columns. The pushover analysis is conducted to obtain the pushover curves, as shown in Figure 15, which shows that the peak force $F_{\mathrm{p}}$ and deformation $u_{\mathrm{p}}$ of the double-column bridge bent is 3660 kN and 38.5 mm, respectively. In order to evaluate the performance of the BPSCFDs with limited size, the yield force of the damper $F_{\mathrm{y}}$ is assumed to be approximately 25% of $F_{\mathrm{p}}$, namely $F_{\mathrm{y}}\approx 915 \mathrm{kN}$. According to Equations (1), (2) and (5), the critical force $F_{\mathrm{cr}}$ and deformation ${\mathsf{\delta }}_{\mathrm{cr}}$ of the buckling plates with $400\mathrm{mm}\times 400\mathrm{mm}\times 2\mathrm{mm}$ fixed at two ends are 13.2 kN and 0.03 mm, and the post-buckling stiffness is 4.2 kN/m. The friction coefficient and the inclination of the friction surface of the dampers is set to be 0.15 and $15°$, respectively. According to Equations (17)–(21), the yielding force, yield deformation, initial stiffness and post-yield stiffness of the BPSCFDs with one buckling-plate are 22.92 kN, 0.06 mm, 3.7 × 10⁵ kN/m and 4.2 kN/m, respectively. As $F_{\mathrm{y}}\approx 915 \mathrm{kN}$, the dampers should contain 40 buckling-plates. As shown in Figure 13a, there are two BPSCFDs installed in the bridge bent; thus, one BPSCFD should contain 20 buckling-plates. Finally, the parameters of the damper $F_{\mathrm{y}}=457.5 \mathrm{kN}$, $K_1=7.5e6 \mathrm{kN}/\mathrm{mm}$, $K_2=78.5 \mathrm{kN}/\mathrm{mm}$, $F_{\mathrm{r}}=120 \mathrm{kN}$ were employed in the finite model.

The finite model of the bridge bent with the BPSCFDs in OpenSees is shown in Figure 14b, in which the BPSCFDs is modeled by the Two Node Link element assigned with Self Centering material. The hysteretic coefficient $\mathsf{\beta }$ of the Self Centering model in Figure 14b can be calculated to be 0.73. Three ground motions were selected from the earthquake database of Pacific Earthquake Engineering Research, and their information is tabulated in

Table 3. Information of ground motions.

No.	Earthquake	Year	Station	Magnitude	Epicentral Distance (km)	Vs30 (m/s)
1	Imperial Valley-06	1983	Delta	6.53	7.69	242.05
2	Coalinga-01	1999	Pleasant Valley P.P.-bldg	6.36	16.04	257.38
3	Chi-Chi_Taiwan	2010	CHY036	7.62	11.86	233.14

. The peak acceleration of the ground motion is adjusted to e 2 m/s², meaning that the seismic fortification is equivalent to 8 degrees, according to the China seismic code [41]. The acceleration spectra of the adjusted ground motions are presented in Figure 16, which suggest that the ground motions have complex frequency components.

Figure 17 presents the comparison of the seismic responses between the prototype and bridge retrofitted with BPSCFDs, subjected to three ground motions, respectively. Figure 17a shows that the peak displacement of the retrofitted bridge has a significant reduction compared to the prototype, and the mean reduction ratio is 54%. Meanwhile, the residual displacement is nearly eliminated in the retrofitted bridge, as shown in Figure 17b, which suggests that the BPSCFDs have a significant self-centering performance. Figure 17c,d show that the mean peak acceleration and base shear of the retrofitted bridge are a little smaller than the prototype. The results indicate that the BPSCFDs has a certain advantage in terms of limiting the peak acceleration and base shear when they effectively reduce the peak and residual displacement compared to the existing self-centering damper with a bigger post-yield stiffness ratio [9].

In order to show the difference of the seismic performance of the different structures directly, the displacement time-history curves and hysteretic curves of the prototype and retrofitted bridge bent under Coalinga-01 ground motion are compared in Figure 18. The results also show that the BPSCFDs has the desired flag-shaped hysteretic behavior, and the retrofitted bridge has significant reductions in the peak and residual displacement. The displacement time-history curves also show that the BPSCFDs can quickly eliminate the vibration of the structure, which could be attributed to the high equivalent damping ratio of the damper with low post-yield stiffness and its early activation. In addition, the displacement response of the retrofitted bridge has better symmetry than the prototype, which could be beneficial to control the residual displacement.

7. Conclusions

In order to improve the resilience of structures subjected to strong earthquakes, a buckling-plate self-centering friction damper (BPSCFD) with low post-yielding stiffness is proposed, which is desired not only to reduce the peak and residual deformation of structures, but also to limit the additional internal force of the structural elements adjacent to the damper. Through theoretical and finite element analysis, the major conclusions were obtained as follows:

(1)

The BPSCFD has concise configuration and low cost. Its theoretical force-displacement relationship is derived and demonstrated by the finite element simulation results, which present typical flag-shaped self-centering characteristics and stable hysteretic loops.

(2)

Compared with the variable friction self-centering damper with a linear spring employed as the self-centering element, the BPSCFD has lower post-yielding stiffness and a larger equivalent damping ratio when they have same restoring force demand.

(3)

The length and inertia moment of the cross-section of the buckling-plate have significant effects on the yielding force of the BPSCFD, but insignificant effects on the self-centering and damping characteristics of the damper. Meanwhile, the damping and self-centering characteristics of the damper mainly depend on the friction coefficient and the slope angle of the friction interface.

(4)

The elastic-plastic history analysis of the retrofitted bridge and prototype shows that the BPSCFD can effectively reduce the peak and residual displacement of a bridge, without increasing the acceleration and base shear, which could be attributed to the low post-yield stiffness of the damper. Meanwhile, the vibration of a structure can be quickly eliminated due to the smaller yield deformation and relative equivalent damping ratio.