# Development and Analysis of the Control Effect of a Reid Damper with Self-Centering Characteristics

Authors: Ying-jie Kang; Ling-yun Peng
Journal: Shock and Vibration
 (2018)

---

## Abstract

To improve the recoverability of structures following an earthquake, a Reid friction damper with self-centering characteristics is proposed and its hysteretic behavior is studied by theoretical analysis and experimental research. The main parameters of the damper are the equivalent stiffness and energy dissipation coefficient. Based on a 10-story steel frame structure, 10 energy dissipation design schemes using the proposed Reid damper are proposed. The additional equivalent damping ratios of the 10 schemes are equal, whereas the energy dissipation coefficients of the dampers are different. The vibration control effects of the energy dissipation structures are analytically investigated under four earthquake loads. The experimental results of the friction damper are in good agreement with the theoretical results, and the hysteretic behavior of the damper follows that of a typical Reid model. The seismic response and structural damage can be reduced using any of the 10 design schemes; however, the effects are different. When the energy dissipation coefficient is in the range of 0.1–0.3, the control effect on the interstory drift is better; however, the structural acceleration response and damping force of the dampers increase. When the energy dissipation coefficient is in the range of 0.6–1.0, the energy dissipation effect of the dampers is good; however, the self-centering ability is poor. Therefore, the optimum range of the energy dissipation coefficient of a Reid damper intended for energy dissipation structures should be 0.3–0.6.


---

## Body

## 1. Introduction

Currently, the performance-based seismic design has been widely recognized, and scholars from various countries have conducted numerous studies on the same [1]. Scholars in the United States and Japan have proposed the resilient city as a general direction for cooperation in earthquake engineering [2]. Under the premise that a structure meets the performance target, a postearthquake structure can be quickly repaired and restored to its normal function. This has become one of the important research directions for the sustainable development of earthquake resistant projects. The difficulty in restoring earthquake-hit structures and the associated economic cost is directly related to the residual deformation of the structures. Therefore, restricting their residual deformation has become a key point for structural recovery [3].Metal-yielding energy dissipaters, such as buckling-restrained braces (BRBs), can be used to significantly improve the seismic performance of an entire structure; however, the downside is the higher residual deformation following a strong earthquake [4]. In recent years, the development of energy dissipating devices with good energy dissipation and self-centering characteristics has become a hot topic [5–10]. A self-centering damper has two parts: an energy dissipation part and a restoring force part. The energy dissipation part is similar to that of a conventional damper, performing functions such as metal-yielding energy dissipation [5, 6], friction energy dissipation [7, 8, 11–13], and viscous energy dissipation [9]. The restoring force part mainly comprises shape memory alloys (SMAs) [5, 11], dish springs [7, 9], synthetic fiber material [12, 13], and prestressed steel strand [14]. The SMA is used to provide resilience for the dampers, owing to its shape memory property. Another important property of SMAs is pseudoelasticity (superelasticity), which implies that the large strain produced during loading will be recovered and the energy will be dissipated simultaneously. Based on the excellent properties of SMA, many SMA dampers with self-centering characteristics have been designed [10, 15–17].For self-centering dampers, the idealized bilinear elastic model is generally used to simulate the restoring force, whereas the bilinear elastoplastic model is used to simulate the dissipated energy. Hence, the hysteretic models of self-centering dampers are flag-shaped [18] or improved flag-shaped [6, 19–21], which have attracted significant attention. Another model that can describe the self-centering dampers is the Reid model [22–24]. This model is used to describe the damping force, which is proportional to the displacement amplitude and in phase with the velocity [25]. One of its important characteristics is that it can consider the varying amplitude damping force with the increase in the structural deformation and can better meet the demand of shock absorption under earthquakes of varying intensity. Currently, the damper that can realize the Reid model mainly comprises a variable friction device and a restoring force device. The restoring force device uses the same parts as those in other self-centering dampers. The main methods of achieving variable friction include piezoelectricity [26], variable contact surface friction coefficient [27], and variable friction contact surface [28]. Nims et al. developed an energy dissipating restraint (EDR), whose damping force is a combination of spring elasticity and friction, and the friction is proportional to the displacement [29, 30]. The force-displacement hysteretic curve of the EDR is consistent with the Reid model.Nims et al. studied the vibration control effect of Reid dampers in a single degree of freedom system by numerical simulation [29] and carried out shaking table tests of a simple steel structure with Reid dampers [30]. The results show that Reid dampers are effective to control the vibration response of the structure, but the control effect for the multidegree of the freedom structure and the structure in the nonlinear stage is not involved. In this paper, the hysteresis curves and energy dissipation characteristics of a Reid model are first analyzed. A Reid friction damper is then proposed and studied by theoretical analysis and experimental research. Based on a 10-story steel frame structure, 10 energy dissipation design schemes using the Reid dampers are determined. The additional equivalent damping ratios of the 10 schemes are equal, whereas the energy dissipation coefficients of the dampers are different. Compared to an uncontrolled structure, the vibration control effects on the displacement, acceleration, energy dissipation, structural damage, and residual deformation of the energy dissipation structures are analytically investigated under four earthquake loads. Finally, based on the research results, the design of the Reid damper for energy dissipation structures is proposed.


## 2. Reid Hysteresis Model

Figure1(a) shows the force-displacement relationship of the Reid hysteretic model. The two diagonal triangles, which are symmetric about the origin, in the first and third quadrants constitute the hysteresis loop. It is assumed that the transition between the loading and unloading states is instantaneous. For the Reid hysteresis model, the relationship between the damping force fd and the displacement ud can be expressed as follows:(1)fd=kr1ud,udu˙d>0,fd=kr3ud,udu˙d<0,where kr1 and kr3 denote the loading stiffness and unloading stiffness, respectively, and u˙d denotes the velocity.Figure 1
Reid hysteretic loop.As shown in Figures1(b) and 1(c), the Reid model can be decomposed into a linear hysteretic damping model, whose force is proportional to the displacement amplitude and in phase with the velocity, and a linear spring model.Accordingly, the relationship between the damping force and displacement can be rewritten as follows:(2)fd=1+ηkreud,udu˙d>0,fd=1−ηkreud,udu˙d<0,where kre and η denote the equivalent elastic stiffness and energy dissipation coefficient, respectively, and η is less than 1.Furthermore, from Equations (1) and (2), it follows that(3)kre=kr1+kr32,η=kr1−kr3kr1+kr3.The Reid hysteretic loop shows that the damper will automatically return to its initial position after unloading and will not generate residual deformation. With regard to the energy dissipation capacity, the energy dissipated when repeatedly loading a cycle for a displacementud can be obtained as follows:(4)Wrd=2ηkreud2.


## 3. Theoretical Analysis, Construction Method, and Experimental Study of Reid Damper

### 3.1. Theoretical Analysis of Reid Damper

According to the classical friction theory, the sliding friction magnitude is related to only the magnitude of the pressureFN and friction coefficient μ of the contact surface. The greater the pressure and friction coefficient, the greater the sliding friction. The formula for calculating the frictional force of the constant friction damper is as follows:(5)Ff=μFN,u˙d>0,Ff=−μFN,u˙d<0.The frictional force of the constant friction damper is independent of the displacement but is in phase with the velocity. To realize a linear hysteretic damping model, the friction magnitude should vary linearly with respect to the displacement. In current piezoelectric friction dampers, the positive force between the frictional contact surfaces can be changed by applying a voltage. However, they are semiactive devices, and the construction is more complex.Figure2 shows the schematic of the passive variable friction damper, which can realize the Reid model. In Figure 2, μ0 denotes the friction coefficient between the sliding block and the friction plate, μ1 denotes the friction coefficient between the sliding block and the extrusion block, and θ denotes the angle between the two frictional contact surfaces. During the loading, the interaction force between the extrusion block and the sliding block will increase, consequently, increasing the friction between the sliding block and the friction plate.Figure 2
Schematic of the passive variable friction damper.At the same time, the compression force acting on the spring increases. During the unloading, the interaction force between the extrusion block and the sliding block and the friction between the sliding block and the friction plate decrease. The transition between the loading and unloading stages of the damper is assumed to be instantaneous. As shown in Figure3(a), the sliding and extrusion blocks are isolated. During the loading, the damping force is equal to the sum of the spring compression reaction force and the frictional force. During the unloading, the damping force is equal to the difference between the spring compression reaction and the frictional force.Figure 3
Force balance of the passive variable friction damper: (a) overall force balance of the slider and extrusion block; (b) force balance of the isolated sliding and extrusion blocks.
(a)
(b)The damping force is obtained as follows:(6)Fd=Fs+Ff,loading,Fd=Fs−Ff,unloading,where Fd, Fs, and Ff denote the damping force, spring elastic force, and frictional force, respectively.For the loading stage, as shown in Figure3(b), the force balance relationship can be obtained by considering the extruded block as an isolated body:(7)Fs=2N1sinθ+f1cosθ,Fd=2N2sinθ+f2cosθ,where N1 and f1 denote the positive pressure and frictional force acting on the contact surface between the extrusion block and the sliding block on the left side, respectively, and similarly, N2 and f2 denote the same for the right side.With the sliding block considered as an isolated body, the frictional force and positive pressure can be obtained by balancing the forces as follows:(8)Ff=2N2sinθ+f2cosθ−N1sinθ−f1cosθ,N=N1cosθ−f1sinθ+N2cosθ−f2sinθ,where N denotes the positive pressure acting on the contact surface between the friction plate and the sliding block.The relationships between the positive pressure and the frictional force are as follows:Ff=2μ0N, f1=μ1N1, and f2=μ1N2. Combined with Equations (6)–(8), the damping force can be obtained as follows:(9)Fd=sinθ+μ1cosθ+μ0cosθ−μ1sinθsinθ+μ1cosθ−μ0cosθ−μ1sinθFs.Equation (9) can be rewritten as follows:(10)Fd=1−μ0μ1tanθ+μ1+μ01+μ0μ1tanθ+μ1−μ0Fs.The damping force for the unloading stage can be obtained using the same method:(11)Fd=1−μ0μ1tanθ−μ1+μ01+μ0μ1tanθ−μ1−μ0Fs.As the slip deformation between the extrusion block and the sliding block is relatively small, its contribution to the axial deformation of the damper can be ignored. The following equations are obtained by substitutingFs=ksud into Equations (10) and (11):(12)kr1=1−μ0μ1tanθ+μ1+μ01+μ0μ1tanθ+μ1−μ0ks,kr3=1−μ0μ1tanθ−μ1+μ01+μ0μ1tanθ−μ1−μ0ks.To simplify the design, it is assumed that the friction coefficients between the sliding block and the extrusion block and that between the sliding block and the friction plate are equal, i.e.,μ0=μ1=μ. Therefore, we have the following equation:(13)kre=1−μ21+μ2ks,η=2μ1+μ2μ1−μ2tanθ.Figure4(a) shows the hysteresis curves of the dampers for a spring stiffness (ks) of 100 kN/m, a friction coefficient (μ) of 0.15, and angles (θ) of π/9, π/6, and π/3. Figure 4(b) shows the variation curve of the energy dissipation coefficient η with respect to θ for different values of the friction coefficient. The following are the analysis results:(1)
From the theoretical analysis of the simplified mechanical model of the variable friction damper, we prove that the damper can realize the Reid damping hysteretic model(2)
Under the same friction coefficientμ, the energy dissipation coefficient η decreases with the increase in θ; and under the same θ, the energy dissipation coefficient η increases with the increase in the friction coefficient μ(3)
The equivalent stiffnesskre of the damper is slightly lower than the spring stiffness ksFigure 4
Hysteretic loop and parameter relationship of passive variable friction dampers: (a) damper theory hysteretic loop; (b) variation in the energy dissipation coefficientη with θ.
(a)
(b)


### 3.2. Construction Scheme of Reid Damper

Figure5 shows the detailed construction scheme of the Reid variable friction damper, established on the basis of the above theoretical analysis on friction dampers. The main components include a friction plate, sliding blocks, extrusion blocks, a spring, a transmission rod, and end plates (which are used to fix the entire damper). It needs to be added that the spring does not require prestress, which is one of the characteristics of the damper. In addition, if the spring is prepressed, the damper hysteresis curve can realize the double flag type. The damper is a symmetric structure. When the intermediate transmission rod moves from the initial position to the right, only the right-side extrusion block, sliding block, and spring start to bear the force. When the middle transmission rod moves from the initial position to the left, only the left-side extrusion block, sliding block, and spring start to bear the force.Figure 5
Detailed construction of the Reid variable friction damper: (a) three-dimensional model of the damper; (b) profile of the damper.
(a)
(b)


### 3.3. Performance Experiment on Damper

Based on the above damper design, a small-scale Reid variable friction damper is established using Q235 steel. The height, width, and total length of the damper test specimen are 70, 79, and 277 mm, respectively. The angleθ between the two frictional contact surfaces is π/6. The friction coefficients μ1 between the sliding block and the friction plate and that between the sliding block and the extrusion block (μ2) are equal, i.e., 0.12. The compression springs used in the damper are red mold springs made according to Japanese Industrial Standards. They are made of high-performance chrome alloy steel. The outer diameter of the spring is 30 mm, the inner diameter is 15 mm, the length is 50 mm, the compression capacity is 16 mm in 300,000 cycles, and the bearing capacity is 1800 N.As shown in Figure6(a), the spring performance test results show that the spring force is linear with the displacement, the elastic stiffness is 100 kN/m, and the ultimate displacement can reach 20 mm. After the unloading is completed, the spring returns to the initial position and the residual deformation is substantially zero. In summary, the mold spring performance is stable and can provide better self-recovery ability. The mold spring used in this test is small in size, and the damper applied in the engineering structure often requires a spring with a large bearing capacity. Therefore, it is necessary to process and manufacture larger-sized mold springs and perform corresponding performance tests to meet engineering needs. Figure 6(b) shows the test specimen, test loading device, and sensor layout.Figure 6
Performance test of the spring and the damper: (a) performance test of the spring; (b) loading scheme of the damper performance test.
(a)
(b)The low-frequency reciprocating test is loaded for a displacement amplitude of 15 mm. Figure7 shows 30 hysteresis loop curves of the dampers under cyclic loading. The theoretical result obtained using Equation (12) is represented using dotted lines. The experimental results are in good agreement with the theoretical results, proving that the design scheme of the Reid damper is feasible.Figure 7
Force-displacement hysteretic curves of the damper.


## 3.1. Theoretical Analysis of Reid Damper

According to the classical friction theory, the sliding friction magnitude is related to only the magnitude of the pressureFN and friction coefficient μ of the contact surface. The greater the pressure and friction coefficient, the greater the sliding friction. The formula for calculating the frictional force of the constant friction damper is as follows:(5)Ff=μFN,u˙d>0,Ff=−μFN,u˙d<0.The frictional force of the constant friction damper is independent of the displacement but is in phase with the velocity. To realize a linear hysteretic damping model, the friction magnitude should vary linearly with respect to the displacement. In current piezoelectric friction dampers, the positive force between the frictional contact surfaces can be changed by applying a voltage. However, they are semiactive devices, and the construction is more complex.Figure2 shows the schematic of the passive variable friction damper, which can realize the Reid model. In Figure 2, μ0 denotes the friction coefficient between the sliding block and the friction plate, μ1 denotes the friction coefficient between the sliding block and the extrusion block, and θ denotes the angle between the two frictional contact surfaces. During the loading, the interaction force between the extrusion block and the sliding block will increase, consequently, increasing the friction between the sliding block and the friction plate.Figure 2
Schematic of the passive variable friction damper.At the same time, the compression force acting on the spring increases. During the unloading, the interaction force between the extrusion block and the sliding block and the friction between the sliding block and the friction plate decrease. The transition between the loading and unloading stages of the damper is assumed to be instantaneous. As shown in Figure3(a), the sliding and extrusion blocks are isolated. During the loading, the damping force is equal to the sum of the spring compression reaction force and the frictional force. During the unloading, the damping force is equal to the difference between the spring compression reaction and the frictional force.Figure 3
Force balance of the passive variable friction damper: (a) overall force balance of the slider and extrusion block; (b) force balance of the isolated sliding and extrusion blocks.
(a)
(b)The damping force is obtained as follows:(6)Fd=Fs+Ff,loading,Fd=Fs−Ff,unloading,where Fd, Fs, and Ff denote the damping force, spring elastic force, and frictional force, respectively.For the loading stage, as shown in Figure3(b), the force balance relationship can be obtained by considering the extruded block as an isolated body:(7)Fs=2N1sinθ+f1cosθ,Fd=2N2sinθ+f2cosθ,where N1 and f1 denote the positive pressure and frictional force acting on the contact surface between the extrusion block and the sliding block on the left side, respectively, and similarly, N2 and f2 denote the same for the right side.With the sliding block considered as an isolated body, the frictional force and positive pressure can be obtained by balancing the forces as follows:(8)Ff=2N2sinθ+f2cosθ−N1sinθ−f1cosθ,N=N1cosθ−f1sinθ+N2cosθ−f2sinθ,where N denotes the positive pressure acting on the contact surface between the friction plate and the sliding block.The relationships between the positive pressure and the frictional force are as follows:Ff=2μ0N, f1=μ1N1, and f2=μ1N2. Combined with Equations (6)–(8), the damping force can be obtained as follows:(9)Fd=sinθ+μ1cosθ+μ0cosθ−μ1sinθsinθ+μ1cosθ−μ0cosθ−μ1sinθFs.Equation (9) can be rewritten as follows:(10)Fd=1−μ0μ1tanθ+μ1+μ01+μ0μ1tanθ+μ1−μ0Fs.The damping force for the unloading stage can be obtained using the same method:(11)Fd=1−μ0μ1tanθ−μ1+μ01+μ0μ1tanθ−μ1−μ0Fs.As the slip deformation between the extrusion block and the sliding block is relatively small, its contribution to the axial deformation of the damper can be ignored. The following equations are obtained by substitutingFs=ksud into Equations (10) and (11):(12)kr1=1−μ0μ1tanθ+μ1+μ01+μ0μ1tanθ+μ1−μ0ks,kr3=1−μ0μ1tanθ−μ1+μ01+μ0μ1tanθ−μ1−μ0ks.To simplify the design, it is assumed that the friction coefficients between the sliding block and the extrusion block and that between the sliding block and the friction plate are equal, i.e.,μ0=μ1=μ. Therefore, we have the following equation:(13)kre=1−μ21+μ2ks,η=2μ1+μ2μ1−μ2tanθ.Figure4(a) shows the hysteresis curves of the dampers for a spring stiffness (ks) of 100 kN/m, a friction coefficient (μ) of 0.15, and angles (θ) of π/9, π/6, and π/3. Figure 4(b) shows the variation curve of the energy dissipation coefficient η with respect to θ for different values of the friction coefficient. The following are the analysis results:(1)
From the theoretical analysis of the simplified mechanical model of the variable friction damper, we prove that the damper can realize the Reid damping hysteretic model(2)
Under the same friction coefficientμ, the energy dissipation coefficient η decreases with the increase in θ; and under the same θ, the energy dissipation coefficient η increases with the increase in the friction coefficient μ(3)
The equivalent stiffnesskre of the damper is slightly lower than the spring stiffness ksFigure 4
Hysteretic loop and parameter relationship of passive variable friction dampers: (a) damper theory hysteretic loop; (b) variation in the energy dissipation coefficientη with θ.
(a)
(b)


## 3.2. Construction Scheme of Reid Damper

Figure5 shows the detailed construction scheme of the Reid variable friction damper, established on the basis of the above theoretical analysis on friction dampers. The main components include a friction plate, sliding blocks, extrusion blocks, a spring, a transmission rod, and end plates (which are used to fix the entire damper). It needs to be added that the spring does not require prestress, which is one of the characteristics of the damper. In addition, if the spring is prepressed, the damper hysteresis curve can realize the double flag type. The damper is a symmetric structure. When the intermediate transmission rod moves from the initial position to the right, only the right-side extrusion block, sliding block, and spring start to bear the force. When the middle transmission rod moves from the initial position to the left, only the left-side extrusion block, sliding block, and spring start to bear the force.Figure 5
Detailed construction of the Reid variable friction damper: (a) three-dimensional model of the damper; (b) profile of the damper.
(a)
(b)


## 3.3. Performance Experiment on Damper

Based on the above damper design, a small-scale Reid variable friction damper is established using Q235 steel. The height, width, and total length of the damper test specimen are 70, 79, and 277 mm, respectively. The angleθ between the two frictional contact surfaces is π/6. The friction coefficients μ1 between the sliding block and the friction plate and that between the sliding block and the extrusion block (μ2) are equal, i.e., 0.12. The compression springs used in the damper are red mold springs made according to Japanese Industrial Standards. They are made of high-performance chrome alloy steel. The outer diameter of the spring is 30 mm, the inner diameter is 15 mm, the length is 50 mm, the compression capacity is 16 mm in 300,000 cycles, and the bearing capacity is 1800 N.As shown in Figure6(a), the spring performance test results show that the spring force is linear with the displacement, the elastic stiffness is 100 kN/m, and the ultimate displacement can reach 20 mm. After the unloading is completed, the spring returns to the initial position and the residual deformation is substantially zero. In summary, the mold spring performance is stable and can provide better self-recovery ability. The mold spring used in this test is small in size, and the damper applied in the engineering structure often requires a spring with a large bearing capacity. Therefore, it is necessary to process and manufacture larger-sized mold springs and perform corresponding performance tests to meet engineering needs. Figure 6(b) shows the test specimen, test loading device, and sensor layout.Figure 6
Performance test of the spring and the damper: (a) performance test of the spring; (b) loading scheme of the damper performance test.
(a)
(b)The low-frequency reciprocating test is loaded for a displacement amplitude of 15 mm. Figure7 shows 30 hysteresis loop curves of the dampers under cyclic loading. The theoretical result obtained using Equation (12) is represented using dotted lines. The experimental results are in good agreement with the theoretical results, proving that the design scheme of the Reid damper is feasible.Figure 7
Force-displacement hysteretic curves of the damper.


## 4. Structural Analysis Model and Energy Dissipation Design Using Reid Dampers

### 4.1. Design Parameters of Structural Analysis Model

A 10-story steel frame structure is selected as the analysis model. The structure is designed according to the Chinese code [29]. The precautionary intensity of the structure is 8 degrees, and the design peak of the ground motion is 0.2 g. The site condition is category II, and the classification of the design earthquake is the first group. The characteristic period of the site is 0.35 s. The frame is simplified to a two-dimensional plane model, as shown in Figure 8.Figure 8
Analysis model of the steel frame structure.The steel frame structure has five 8 m spans. The height of each layer is 4 m, and the total height is 40 m. The same square steel tubes are used for the frame columns of each layer of the structure, and the same type-I steel is used for the frame beams. Figure8 shows the specific parameters of the section of the frame. The entire structure is made of Q345 steel, and the additional mass of each floor is 320 t. The structure satisfies the assumption of the rigid diaphragm. In other words, the horizontal degrees of freedom of the nodes on the same floor are coupled.


### 4.2. Energy Dissipation Design Using Reid Dampers

As shown in Figure9, the dampers are diagonally arranged on the outer frames. The relationship between the interstory displacement Δui of story i and the axial displacement Δudi of the dampers on story i is given as follows:(14)Δudi=Δui∗cosθd,where θd denotes the angle between the dampers and the frame beam.Figure 9
Placement of dampers.The equivalent damping ratio of the dampers attached to the structure can be obtained as follows:(15)ξeq=Wd4πEs,where ξeq denotes the equivalent damping ratio, Wd denotes the energy dissipated by the dampers under the specified displacement, and Es denotes the structural elastic potential energy.The additional equivalent damping ratio of the entire structure is determined based on the seismic performance requirements. The energy dissipation requirements of the dampers are obtained to optimize the design parameters of the damper. Suppose the additional equivalent damping ratio of each story isξadd, according to Equation (15), it can be obtained as follows:(16)ξadd=Wdi4πEsi,where Wdi and Esi denote the energy dissipated by the dampers and the elastic potential energy of story i, respectively. Esi is calculated as follows:(17)Esi=12kiΔ2ui,where ki denotes the shear stiffness of story i.When the Reid dampers are used, the energy dissipated by the dampers on story i is obtained as follows:(18)Wdi=n⋅2ηikreiΔ2udi,where n denotes the number of dampers on story i and krei and ηi denote the equivalent elastic stiffness and energy dissipation coefficient on story i, respectively.Combined with Equations (14)–(18), the Reid damper parameters for story i can be calculated as follows:(19)ηikrei=πξaddkincos2θd.Taking the stiffness ratio asλi=krei/ki, we have the following equation:(20)ηiλi=πξaddncos2θd.According to Equation (20), with the increase in the damping ratio, the energy dissipation coefficient is inversely proportional to the stiffness ratio. For the selected steel frame structure, 10 different schemes are established with an additional damping ratio of 0.05. Table 1 lists the mechanical parameters of each scheme of the Reid dampers. Table 2 lists the detailed damper parameters kre for each story of all schemes.Table 1
Mechanical parameters of the Reid dampers used in the 10 schemes.
1
2
3
4
5
6
7
8
9
10
η
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1.0
λ
0.98
0.49
0.33
0.25
0.20
0.16
0.14
0.12
0.11
0.10Table 2
Detailed damper parameterskre for each story of all schemes (unit: kN/m).
Story
Structural shear stiffness
Scheme
1
2
3
4
5
6
7
8
9
10
1
1.37E5
1.34E5
6.73E4
4.48E4
3.36E4
2.69E4
2.24E4
1.92E4
1.68E4
1.49E4
1.35E4
2
6.81E4
6.69E4
3.34E4
2.23E4
1.67E4
1.34E4
1.11E4
9.55E3
8.36E3
7.43E3
6.69E3
3
5.75E4
5.64E4
2.82E4
1.88E4
1.41E4
1.13E4
9.41E3
8.06E3
7.06E3
6.27E3
5.65E3
4
5.37E4
5.27E4
2.64E4
1.76E4
1.32E4
1.05E4
8.79E3
7.53E3
6.59E3
5.86E3
5.27E3
5
5.22E4
5.12E4
2.56E4
1.71E4
1.28E4
1.02E4
8.54E3
7.32E3
6.40E3
5.70E3
5.13E3
6
5.10E4
5.01E4
2.50E4
1.67E4
1.25E4
1.00E4
8.34E3
7.15E3
6.26E3
5.56E3
5.01E3
7
4.85E4
4.76E4
2.38E4
1.59E4
1.19E4
9.52E3
7.93E3
6.80E3
5.95E3
5.29E3
4.76E3
8
4.69E4
4.60E4
2.30E4
1.53E4
1.15E4
9.21E3
7.67E3
6.58E3
5.75E3
5.12E3
4.61E3
9
4.84E4
4.75E4
2.38E4
1.58E4
1.19E4
9.50E3
7.92E3
6.79E3
5.94E3
5.28E3
4.75E3
10
5.52E4
5.42E4
2.71E4
1.81E4
1.35E4
1.08E4
9.03E3
7.74E3
6.77E3
6.02E3
5.42E3


### 4.3. Finite Element Model of the Structure

The finite element software ABAQUS is used to establish the plane analysis model of the steel structure, and the nonlinear time history analysis of the structure model under earthquake is carried out. The frame beams and columns are simulated using two-dimensional fiber beam elements (B21). The horizontal degrees of freedom of the nodes on the same story are coupled using the “rigid” command to simulate the rigid baffle assumption. The load acting on the structure is converted to the equivalent density of the frame beam. The constitutive relationship of the steel is simulated using the bilinear hardening model. The initial elastic modulus of steel is 2.06E11 Pa, the elastic modulus after yielding is 1.03E9 Pa, and the yield strength is 345 Mpa. The first three order frequencies of the structure are 0.33, 1.02, and 1.90 Hz, respectively. Figure10 shows the corresponding vibration modes. The Rayleigh damping model is employed for the structural damping. The mass damping coefficient α and stiffness damping coefficient β are calculated using Equation (21). The damping ratio ξ of the structural model is 0.04. ω1 and ω2 denote the first and second order circular frequencies of the structure, respectively:(21)α=2ω1ω2ξ/ω1+ω2,β=2ξ/ω1+ω2.Figure 10
Vibration modes of the steel frame structure.The Reid damper model is implemented using the user material subroutine interface (UMAT) provided by ABAQUS. Figure11 shows the test results of the Reid damper model η=0.6,kre=1E4kN/m.Figure 11
Test result of Reid element subroutine.


### 4.4. Earthquake Records and Analysis Conditions

To evaluate the proposed control strategies, two far-field and two near-field earthquake historical records are selected:El Centro, Hachinohe, Northridge, and Kobe [32]. Table 3 lists the detailed information of the earthquake records. Figure 12 shows the acceleration-time histories of the earthquakes. Additionally, this vibration control study considers various levels of earthquake records, including 0.3 g and 0.51 g, which correspond to fortification earthquake and rare earthquake, respectively.Table 3
Information about earthquake records.
Type
Code
Earthquake
Station
Component
Year
Far field
El centro
Imperial valley
El centro
N-S
1940
Hachinohe
Tokachi-oki
Hachinohe city
N-S
1968
Near field
Northridge
Northridge
SCH
N-S
1994
Kobe
Hyogo-ken
KJMA
N-S
1995Figure 12
Acceleration-time histories of the earthquakes: (a) far field; (b) near field.
(a)
(b)


## 4.1. Design Parameters of Structural Analysis Model

A 10-story steel frame structure is selected as the analysis model. The structure is designed according to the Chinese code [29]. The precautionary intensity of the structure is 8 degrees, and the design peak of the ground motion is 0.2 g. The site condition is category II, and the classification of the design earthquake is the first group. The characteristic period of the site is 0.35 s. The frame is simplified to a two-dimensional plane model, as shown in Figure 8.Figure 8
Analysis model of the steel frame structure.The steel frame structure has five 8 m spans. The height of each layer is 4 m, and the total height is 40 m. The same square steel tubes are used for the frame columns of each layer of the structure, and the same type-I steel is used for the frame beams. Figure8 shows the specific parameters of the section of the frame. The entire structure is made of Q345 steel, and the additional mass of each floor is 320 t. The structure satisfies the assumption of the rigid diaphragm. In other words, the horizontal degrees of freedom of the nodes on the same floor are coupled.


## 4.2. Energy Dissipation Design Using Reid Dampers

As shown in Figure9, the dampers are diagonally arranged on the outer frames. The relationship between the interstory displacement Δui of story i and the axial displacement Δudi of the dampers on story i is given as follows:(14)Δudi=Δui∗cosθd,where θd denotes the angle between the dampers and the frame beam.Figure 9
Placement of dampers.The equivalent damping ratio of the dampers attached to the structure can be obtained as follows:(15)ξeq=Wd4πEs,where ξeq denotes the equivalent damping ratio, Wd denotes the energy dissipated by the dampers under the specified displacement, and Es denotes the structural elastic potential energy.The additional equivalent damping ratio of the entire structure is determined based on the seismic performance requirements. The energy dissipation requirements of the dampers are obtained to optimize the design parameters of the damper. Suppose the additional equivalent damping ratio of each story isξadd, according to Equation (15), it can be obtained as follows:(16)ξadd=Wdi4πEsi,where Wdi and Esi denote the energy dissipated by the dampers and the elastic potential energy of story i, respectively. Esi is calculated as follows:(17)Esi=12kiΔ2ui,where ki denotes the shear stiffness of story i.When the Reid dampers are used, the energy dissipated by the dampers on story i is obtained as follows:(18)Wdi=n⋅2ηikreiΔ2udi,where n denotes the number of dampers on story i and krei and ηi denote the equivalent elastic stiffness and energy dissipation coefficient on story i, respectively.Combined with Equations (14)–(18), the Reid damper parameters for story i can be calculated as follows:(19)ηikrei=πξaddkincos2θd.Taking the stiffness ratio asλi=krei/ki, we have the following equation:(20)ηiλi=πξaddncos2θd.According to Equation (20), with the increase in the damping ratio, the energy dissipation coefficient is inversely proportional to the stiffness ratio. For the selected steel frame structure, 10 different schemes are established with an additional damping ratio of 0.05. Table 1 lists the mechanical parameters of each scheme of the Reid dampers. Table 2 lists the detailed damper parameters kre for each story of all schemes.Table 1
Mechanical parameters of the Reid dampers used in the 10 schemes.
1
2
3
4
5
6
7
8
9
10
η
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1.0
λ
0.98
0.49
0.33
0.25
0.20
0.16
0.14
0.12
0.11
0.10Table 2
Detailed damper parameterskre for each story of all schemes (unit: kN/m).
Story
Structural shear stiffness
Scheme
1
2
3
4
5
6
7
8
9
10
1
1.37E5
1.34E5
6.73E4
4.48E4
3.36E4
2.69E4
2.24E4
1.92E4
1.68E4
1.49E4
1.35E4
2
6.81E4
6.69E4
3.34E4
2.23E4
1.67E4
1.34E4
1.11E4
9.55E3
8.36E3
7.43E3
6.69E3
3
5.75E4
5.64E4
2.82E4
1.88E4
1.41E4
1.13E4
9.41E3
8.06E3
7.06E3
6.27E3
5.65E3
4
5.37E4
5.27E4
2.64E4
1.76E4
1.32E4
1.05E4
8.79E3
7.53E3
6.59E3
5.86E3
5.27E3
5
5.22E4
5.12E4
2.56E4
1.71E4
1.28E4
1.02E4
8.54E3
7.32E3
6.40E3
5.70E3
5.13E3
6
5.10E4
5.01E4
2.50E4
1.67E4
1.25E4
1.00E4
8.34E3
7.15E3
6.26E3
5.56E3
5.01E3
7
4.85E4
4.76E4
2.38E4
1.59E4
1.19E4
9.52E3
7.93E3
6.80E3
5.95E3
5.29E3
4.76E3
8
4.69E4
4.60E4
2.30E4
1.53E4
1.15E4
9.21E3
7.67E3
6.58E3
5.75E3
5.12E3
4.61E3
9
4.84E4
4.75E4
2.38E4
1.58E4
1.19E4
9.50E3
7.92E3
6.79E3
5.94E3
5.28E3
4.75E3
10
5.52E4
5.42E4
2.71E4
1.81E4
1.35E4
1.08E4
9.03E3
7.74E3
6.77E3
6.02E3
5.42E3


## 4.3. Finite Element Model of the Structure

The finite element software ABAQUS is used to establish the plane analysis model of the steel structure, and the nonlinear time history analysis of the structure model under earthquake is carried out. The frame beams and columns are simulated using two-dimensional fiber beam elements (B21). The horizontal degrees of freedom of the nodes on the same story are coupled using the “rigid” command to simulate the rigid baffle assumption. The load acting on the structure is converted to the equivalent density of the frame beam. The constitutive relationship of the steel is simulated using the bilinear hardening model. The initial elastic modulus of steel is 2.06E11 Pa, the elastic modulus after yielding is 1.03E9 Pa, and the yield strength is 345 Mpa. The first three order frequencies of the structure are 0.33, 1.02, and 1.90 Hz, respectively. Figure10 shows the corresponding vibration modes. The Rayleigh damping model is employed for the structural damping. The mass damping coefficient α and stiffness damping coefficient β are calculated using Equation (21). The damping ratio ξ of the structural model is 0.04. ω1 and ω2 denote the first and second order circular frequencies of the structure, respectively:(21)α=2ω1ω2ξ/ω1+ω2,β=2ξ/ω1+ω2.Figure 10
Vibration modes of the steel frame structure.The Reid damper model is implemented using the user material subroutine interface (UMAT) provided by ABAQUS. Figure11 shows the test results of the Reid damper model η=0.6,kre=1E4kN/m.Figure 11
Test result of Reid element subroutine.


## 4.4. Earthquake Records and Analysis Conditions

To evaluate the proposed control strategies, two far-field and two near-field earthquake historical records are selected:El Centro, Hachinohe, Northridge, and Kobe [32]. Table 3 lists the detailed information of the earthquake records. Figure 12 shows the acceleration-time histories of the earthquakes. Additionally, this vibration control study considers various levels of earthquake records, including 0.3 g and 0.51 g, which correspond to fortification earthquake and rare earthquake, respectively.Table 3
Information about earthquake records.
Type
Code
Earthquake
Station
Component
Year
Far field
El centro
Imperial valley
El centro
N-S
1940
Hachinohe
Tokachi-oki
Hachinohe city
N-S
1968
Near field
Northridge
Northridge
SCH
N-S
1994
Kobe
Hyogo-ken
KJMA
N-S
1995Figure 12
Acceleration-time histories of the earthquakes: (a) far field; (b) near field.
(a)
(b)


## 5. Analysis of Vibration Control Effect

The main evaluation criteria include structural deformation, level acceleration, energy consumption ratio, structural damage, and state of the Reid dampers. Additionally, the control effect of the Reid dampers on the structural residual deformation is analyzed.


### 5.1. Control Effect on Structural Deformation

Figure13 shows the interstory drift ratio results of the different controlled schemes and that of the uncontrolled structure under the condition that the peak value of the earthquake is 0.3 g. The interstory drifts of the uncontrolled structure under the excitation of different earthquakes are clearly different. There is an obvious deformation mutation between the upper and lower stories of the structure, particularly under the action of the Kobe earthquake. The maximum peak interstory drift ratios of the uncontrolled structure, mainly in the third and eighth story, are 1/75, 1/50, 1/111, and 1/73 under the four groups of earthquakes, respectively. The different schemes have an effective control effect on the structural deformation response and show some discreteness. Overall, the deformation distributions of the different controlled structures are similar, thus effectively solving the deformation mutation problem of the structure. This is most obvious in the Kobe earthquake condition.Figure 13
Control effect of structural interstory drift under 0.3 g peak value conditions (dashed lines indicate different schemes, blue solid lines indicate the mean value of the different schemes, and green solid lines indicate the uncontrolled structures): (a) El Centro; (b) Hachinohe; (c) Kobe; (d) Northridge.
(a)
(b)
(c)
(d)Figure14 shows the interstory drift ratio results of the different controlled schemes and that of the uncontrolled structure under the condition that the peak value of the earthquake is 0.51 g. Compared to the 0.3 g condition, the interstory drift response of the uncontrolled structure is aggravated, and the overall deformation shape of the structure is consistent. The maximum peak interstory drift ratios of the uncontrolled structure increase to 1/49, 1/38, 1/68, and 1/50.Figure 14
Control effect of structural interstory drift under 0.51 g peak value conditions: (a) El Centro; (b) Hachinohe; (c) Kobe; (d) Northridge.
(a)
(b)
(c)
(d)Figure15 shows the reduction rates of the maximum interstory drift ratios of the different schemes, wherein Figure 15(a) represents the 0.3 g condition, and Figure 15(b) represents the 0.51 g condition. Overall, the control effect of the maximum interstory drift ratio is good, and the average reduction rates of the four earthquakes for each scheme are in the range of 21–33% under the 0.3 g conditions. Compared to the 0.3 g conditions, the control effect of the maximum interstory drift ratio is similar under the 0.5 g conditions. However, the control effect is poor, and the average reduction rates of the four earthquakes for each scheme are in the range of 15–24%. Nevertheless, the control effect of the first scheme is the best. However, the equivalent stiffness of the first scheme added to the structure is clearly higher, thus significantly increasing the acceleration response. The reasons for the same are explained in the next section.Figure 15
Reduction rates of the structural interstory drift ratios for each scheme (η=0.1∼1.0 are the code names of the 10 schemes, respectively): (a) 0.3 g conditions; (b) 0.51 g conditions.
(a)
(b)Figure16 shows the time histories of the top-story displacements of the sixth scheme controlled structure and uncontrolled structure under the action of the El Centro earthquake. The displacement amplitudes of the uncontrolled structures are 0.354 and 0.546 m, respectively, under the 0.3 g and 0.51 g conditions, whereas those of the controlled structure are 0.318 and 0.478 m, respectively, and the corresponding reduction rates are 10.2% and 12.5%. Further, by comparing the displacement time responses of the controlled and uncontrolled structures, we found that the displacement amplitude of the controlled structure can be effectively suppressed in the early stages of the earthquake, and the displacement response of the structure can be quickly attenuated to reach the static state in the later stages of the earthquake.Figure 16
Top-story displacement-time histories of the structures under the action of the El Centro wave: (a) 0.3 g conditions; (b) 0.51 g conditions.
(a)
(b)


### 5.2. Control Effect on Acceleration Response

The control effects on the acceleration responses of each story in the controlled and uncontrolled structures are similar under the 0.3 g and 0.51 g conditions. Therefore, only the contrast results under the 0.51 g conditions are given, as shown in Figures17, and 18 shows the reduction rates of the maximum story accelerations for the different schemes. For the first scheme, the equivalent stiffness of the Reid damper attached to the structure is clearly higher, resulting in a significant increase in the acceleration response of the structure. With respect to the average value of the acceleration reduction rates of the four earthquakes, the acceleration responses of the nine other schemes do not significantly increase.Figure 17
Control effect of structural maximum acceleration under 0.51 g conditions (dashed lines indicate different schemes, blue solid lines indicate the mean value of different schemes, and green solid lines indicate uncontrolled structures): (a) El Centro; (b) Hachinohe; (c) Kobe; (d) Northridge.
(a)
(b)
(c)
(d)Figure 18
Reduction rates of structural maximum accelerations.


### 5.3. Analysis of Energy Consumption Ratio

For the uncontrolled structure, the input energy of the earthquake is largely reflected in the structural damping and plastic deformation of the structure. With the increase in the earthquake intensity, the input energy increases, and the energy dissipation associated with the structural plastic deformation increases. The structural damage is directly related to the energy dissipation associated with the structural plastic deformation. The greater the energy dissipation associated with the plastic deformation, the greater the structural damage. Figure19 shows the energy-time history curves of the controlled structure and sixth scheme controlled structure under the action of the 0.51 g El Centro wave. With the installation of the Reid dampers, the plastic energy consumption of the structure decreases from 796.4 kJ to 339.5 kJ, reducing by 57.4%. This shows that the plastic damage of the structures can be effectively reduced using the Reid dampers.Figure 19
Energy-time history curves of the structure under the action of the 0.51 g El Centro wave (EP denotes the structural potential energy, EK denotes the structural kinetic energy, ED denotes the structural plastic damage-associated energy dissipation, EV denotes the damping energy dissipation, EW denotes the earthquake input energy, and EDamper denotes the Reid damper energy dissipation): (a) uncontrolled structure; (b) controlled structure (η=0.6).
(a)
(b)The dynamic characteristics of the structure can change depending on the arrangement of the Reid dampers in the structure. Even under the action of the same earthquake, the input energy of the earthquake is different. Therefore, to compare the effects of the different schemes on the energy consumption of the structure, the ratios of the damping energy, plastic deformation energy, and energy consumption of the damper to the total input energy are qualitatively evaluated. Figure20 shows the energy consumption ratios of the controlled and uncontrolled structures under the 0.51 g El Centro wave and 0.51 g Northridge wave. On the one hand, the plastic damage energy consumption ratio of the uncontrolled structure is greater than that of the controlled structure; on the other hand, the greater the energy dissipation factor of the Reid dampers, the greater the energy consumption of the dampers. The total energy dissipated by the Reid dampers, designed for an additional damping ratio of 0.05, is slightly lower than that of Rayleigh damping (0.04). This is largely because the damping ratio of the higher order modes in the Rayleigh damping model is greater than 0.04.Figure 20
Proportional relationship between structural damping energy dissipation, plastic deformation energy dissipation, and damper energy dissipation (E-damper, E-plastic, and E-Rayleigh denote the energy dissipation ratio of the Reid dampers, structural plastic deformation, and structural Rayleigh damping, respectively,η=0.1∼1.0 indicate the code names of the 10 schemes, respectively, and UC indicates the uncontrolled structure): (a) Hachinohe; (b) Northridge.
(a)
(b)


### 5.4. Structural Damage and State of Reid Dampers

The damage degree of the entire structural component is evaluated in terms of the strain state of the fiber element, using which the beam and column are simulated. When the maximum strain of the component exceeds the yield strain of the material, the yield of the component is determined, and a plastic hinge is formed. The damage degree is determined from the ratio of the maximum strain to the yield strain. Whenεy<ε<2εy, the damage degree is level 1, indicated using blue-colored plastic hinges; when 2εy<ε, the damage degree is level 2, indicated using green-colored plastic hinges.Figure21(a) shows the damage state of the uncontrolled structure under the 0.51 g El Centro wave. Plastic hinges are observed in 84% of the frame beams. The main damage degree is level 1, and the damage degree of the beams on story 7 is level 2. Figure 21(b) shows the damage state of the sixth scheme controlled structure under the 0.51 g El Centro wave. The proportion of frame beams that exhibit a plastic hinge is 66%, and the main damage degree is level 1. The Reid dampers in their particular layout could effectively reduce the degree of structural damage.Figure 21
Damage degree of integral structure component: (a) uncontrolled structure; (b) controlled structure (η=0.6).
(a)
(b)Under the premise of ensuring the damping effect, the lower the maximum damping force, the fewer the design requirements for the joints. For this purpose, the maximum damping forces of the Reid dampers under different schemes are investigated. Figure22 shows the maximum damping forces of the Reid dampers on the first, fifth, eighth, and tenth stories under the action of the 0.51 g El Centro wave. The maximum output force of the dampers decreases with the increase in the energy dissipation coefficient. When the energy dissipation coefficient is too low, such as 0.1, the excessive damping force is not conducive to the structural design.Figure 22
Statistics of maximum damping force of dampers under the action of the 0.51 g El Centro wave.Figure23 shows the hysteretic curve of the Reid dampers on the first, fifth, and tenth stories under the action of the 0.51 g El Centro wave. From the comparison of the results shown in Figure 23(a) and 23(b), for the damper on the same story, if the energy dissipation coefficient of the Reid damper is larger, the hysteresis curve of the damper will be fuller. In addition, it can be seen that the dampers of each stories will return to the original position after being completely unloaded, thereby controlling the residual deformation of the structure.Figure 23
Hysteretic curve of Reid dampers in the structure under the action of the 0.51 g El Centro wave: (a)η=0.3; (b) η=0.6.
(a)
(b)


### 5.5. Residual Interstory Drift

This section mainly analyzes the control effect of the Reid dampers on the residual interstory drift of the structure. The limit of residual interstory drift is set to 0.5%, and the structure cannot be repaired if the value exceeds this limit [33]. The analysis results show that the maximum residual interstory drift of the uncontrolled structure is 1.47% under the action of the 0.51 g Hachinohe wave, and the maximum residual interstory drifts under the other three seismic wave conditions are lower than 0.5%. Figure 24 shows the maximum residual interstory drifts of the first, third, fifth, seventh, and tenth stories of the controlled structure and those of the uncontrolled structure under the 0.51 g Hachinohe wave condition. The residual deformation of each story of the uncontrolled structure is greater than the limit value. All the shock absorption schemes using the Reid dampers have better control effect on the residual deformation of the structure; however, the control effect of each scheme is different. The trend is that the control effect decreases with the increase in the energy dissipation coefficient. This is largely because of the decrease in the equivalent stiffness coefficient with the increase in the energy dissipation coefficient of the damper. Moreover, the elastic resilience provided by the damper decreases, thus reducing the self-centering capacity of the structure. In summary, when a Reid damper is used to control the residual interstory drift of a structure, the equivalent stiffness of the damper must be guaranteed.Figure 24
Analysis results of structural residual interstory drift.


### 5.6. Shock Absorption Design when Using Reid Dampers

The control effects of the 10 types of Reid damper shock absorption schemes with the same additional damping ratio are compared. The lower the energy dissipation factor of the Reid damper, the greater the equivalent stiffness coefficient.First, the deformation control effect of the structure decreases with the increase in the energy dissipation coefficient, and the control effect is similar when the energy dissipation coefficient is in the range of 0.3–1.0. In this range, no obvious amplification in the acceleration response of the structure is observed, and the energy dissipation capacity of the dampers is good, thus effectively controlling the plastic damage energy of the structures.Second, when the energy dissipation coefficient is in the range of 0.1–0.6, the damper has a relatively high equivalent stiffness and good self-centering ability, thus effectively controlling the residual deformation of the structure. However, a greater damping force will act on the damper because of the excessive elastic stiffness, making it difficult to design connections and joints.In summary, when a Reid damper is designed for energy dissipation, the energy dissipation coefficient of the damper should be in the range of 0.3–0.6.


## 5.1. Control Effect on Structural Deformation

Figure13 shows the interstory drift ratio results of the different controlled schemes and that of the uncontrolled structure under the condition that the peak value of the earthquake is 0.3 g. The interstory drifts of the uncontrolled structure under the excitation of different earthquakes are clearly different. There is an obvious deformation mutation between the upper and lower stories of the structure, particularly under the action of the Kobe earthquake. The maximum peak interstory drift ratios of the uncontrolled structure, mainly in the third and eighth story, are 1/75, 1/50, 1/111, and 1/73 under the four groups of earthquakes, respectively. The different schemes have an effective control effect on the structural deformation response and show some discreteness. Overall, the deformation distributions of the different controlled structures are similar, thus effectively solving the deformation mutation problem of the structure. This is most obvious in the Kobe earthquake condition.Figure 13
Control effect of structural interstory drift under 0.3 g peak value conditions (dashed lines indicate different schemes, blue solid lines indicate the mean value of the different schemes, and green solid lines indicate the uncontrolled structures): (a) El Centro; (b) Hachinohe; (c) Kobe; (d) Northridge.
(a)
(b)
(c)
(d)Figure14 shows the interstory drift ratio results of the different controlled schemes and that of the uncontrolled structure under the condition that the peak value of the earthquake is 0.51 g. Compared to the 0.3 g condition, the interstory drift response of the uncontrolled structure is aggravated, and the overall deformation shape of the structure is consistent. The maximum peak interstory drift ratios of the uncontrolled structure increase to 1/49, 1/38, 1/68, and 1/50.Figure 14
Control effect of structural interstory drift under 0.51 g peak value conditions: (a) El Centro; (b) Hachinohe; (c) Kobe; (d) Northridge.
(a)
(b)
(c)
(d)Figure15 shows the reduction rates of the maximum interstory drift ratios of the different schemes, wherein Figure 15(a) represents the 0.3 g condition, and Figure 15(b) represents the 0.51 g condition. Overall, the control effect of the maximum interstory drift ratio is good, and the average reduction rates of the four earthquakes for each scheme are in the range of 21–33% under the 0.3 g conditions. Compared to the 0.3 g conditions, the control effect of the maximum interstory drift ratio is similar under the 0.5 g conditions. However, the control effect is poor, and the average reduction rates of the four earthquakes for each scheme are in the range of 15–24%. Nevertheless, the control effect of the first scheme is the best. However, the equivalent stiffness of the first scheme added to the structure is clearly higher, thus significantly increasing the acceleration response. The reasons for the same are explained in the next section.Figure 15
Reduction rates of the structural interstory drift ratios for each scheme (η=0.1∼1.0 are the code names of the 10 schemes, respectively): (a) 0.3 g conditions; (b) 0.51 g conditions.
(a)
(b)Figure16 shows the time histories of the top-story displacements of the sixth scheme controlled structure and uncontrolled structure under the action of the El Centro earthquake. The displacement amplitudes of the uncontrolled structures are 0.354 and 0.546 m, respectively, under the 0.3 g and 0.51 g conditions, whereas those of the controlled structure are 0.318 and 0.478 m, respectively, and the corresponding reduction rates are 10.2% and 12.5%. Further, by comparing the displacement time responses of the controlled and uncontrolled structures, we found that the displacement amplitude of the controlled structure can be effectively suppressed in the early stages of the earthquake, and the displacement response of the structure can be quickly attenuated to reach the static state in the later stages of the earthquake.Figure 16
Top-story displacement-time histories of the structures under the action of the El Centro wave: (a) 0.3 g conditions; (b) 0.51 g conditions.
(a)
(b)


## 5.2. Control Effect on Acceleration Response

The control effects on the acceleration responses of each story in the controlled and uncontrolled structures are similar under the 0.3 g and 0.51 g conditions. Therefore, only the contrast results under the 0.51 g conditions are given, as shown in Figures17, and 18 shows the reduction rates of the maximum story accelerations for the different schemes. For the first scheme, the equivalent stiffness of the Reid damper attached to the structure is clearly higher, resulting in a significant increase in the acceleration response of the structure. With respect to the average value of the acceleration reduction rates of the four earthquakes, the acceleration responses of the nine other schemes do not significantly increase.Figure 17
Control effect of structural maximum acceleration under 0.51 g conditions (dashed lines indicate different schemes, blue solid lines indicate the mean value of different schemes, and green solid lines indicate uncontrolled structures): (a) El Centro; (b) Hachinohe; (c) Kobe; (d) Northridge.
(a)
(b)
(c)
(d)Figure 18
Reduction rates of structural maximum accelerations.


## 5.3. Analysis of Energy Consumption Ratio

For the uncontrolled structure, the input energy of the earthquake is largely reflected in the structural damping and plastic deformation of the structure. With the increase in the earthquake intensity, the input energy increases, and the energy dissipation associated with the structural plastic deformation increases. The structural damage is directly related to the energy dissipation associated with the structural plastic deformation. The greater the energy dissipation associated with the plastic deformation, the greater the structural damage. Figure19 shows the energy-time history curves of the controlled structure and sixth scheme controlled structure under the action of the 0.51 g El Centro wave. With the installation of the Reid dampers, the plastic energy consumption of the structure decreases from 796.4 kJ to 339.5 kJ, reducing by 57.4%. This shows that the plastic damage of the structures can be effectively reduced using the Reid dampers.Figure 19
Energy-time history curves of the structure under the action of the 0.51 g El Centro wave (EP denotes the structural potential energy, EK denotes the structural kinetic energy, ED denotes the structural plastic damage-associated energy dissipation, EV denotes the damping energy dissipation, EW denotes the earthquake input energy, and EDamper denotes the Reid damper energy dissipation): (a) uncontrolled structure; (b) controlled structure (η=0.6).
(a)
(b)The dynamic characteristics of the structure can change depending on the arrangement of the Reid dampers in the structure. Even under the action of the same earthquake, the input energy of the earthquake is different. Therefore, to compare the effects of the different schemes on the energy consumption of the structure, the ratios of the damping energy, plastic deformation energy, and energy consumption of the damper to the total input energy are qualitatively evaluated. Figure20 shows the energy consumption ratios of the controlled and uncontrolled structures under the 0.51 g El Centro wave and 0.51 g Northridge wave. On the one hand, the plastic damage energy consumption ratio of the uncontrolled structure is greater than that of the controlled structure; on the other hand, the greater the energy dissipation factor of the Reid dampers, the greater the energy consumption of the dampers. The total energy dissipated by the Reid dampers, designed for an additional damping ratio of 0.05, is slightly lower than that of Rayleigh damping (0.04). This is largely because the damping ratio of the higher order modes in the Rayleigh damping model is greater than 0.04.Figure 20
Proportional relationship between structural damping energy dissipation, plastic deformation energy dissipation, and damper energy dissipation (E-damper, E-plastic, and E-Rayleigh denote the energy dissipation ratio of the Reid dampers, structural plastic deformation, and structural Rayleigh damping, respectively,η=0.1∼1.0 indicate the code names of the 10 schemes, respectively, and UC indicates the uncontrolled structure): (a) Hachinohe; (b) Northridge.
(a)
(b)


## 5.4. Structural Damage and State of Reid Dampers

The damage degree of the entire structural component is evaluated in terms of the strain state of the fiber element, using which the beam and column are simulated. When the maximum strain of the component exceeds the yield strain of the material, the yield of the component is determined, and a plastic hinge is formed. The damage degree is determined from the ratio of the maximum strain to the yield strain. Whenεy<ε<2εy, the damage degree is level 1, indicated using blue-colored plastic hinges; when 2εy<ε, the damage degree is level 2, indicated using green-colored plastic hinges.Figure21(a) shows the damage state of the uncontrolled structure under the 0.51 g El Centro wave. Plastic hinges are observed in 84% of the frame beams. The main damage degree is level 1, and the damage degree of the beams on story 7 is level 2. Figure 21(b) shows the damage state of the sixth scheme controlled structure under the 0.51 g El Centro wave. The proportion of frame beams that exhibit a plastic hinge is 66%, and the main damage degree is level 1. The Reid dampers in their particular layout could effectively reduce the degree of structural damage.Figure 21
Damage degree of integral structure component: (a) uncontrolled structure; (b) controlled structure (η=0.6).
(a)
(b)Under the premise of ensuring the damping effect, the lower the maximum damping force, the fewer the design requirements for the joints. For this purpose, the maximum damping forces of the Reid dampers under different schemes are investigated. Figure22 shows the maximum damping forces of the Reid dampers on the first, fifth, eighth, and tenth stories under the action of the 0.51 g El Centro wave. The maximum output force of the dampers decreases with the increase in the energy dissipation coefficient. When the energy dissipation coefficient is too low, such as 0.1, the excessive damping force is not conducive to the structural design.Figure 22
Statistics of maximum damping force of dampers under the action of the 0.51 g El Centro wave.Figure23 shows the hysteretic curve of the Reid dampers on the first, fifth, and tenth stories under the action of the 0.51 g El Centro wave. From the comparison of the results shown in Figure 23(a) and 23(b), for the damper on the same story, if the energy dissipation coefficient of the Reid damper is larger, the hysteresis curve of the damper will be fuller. In addition, it can be seen that the dampers of each stories will return to the original position after being completely unloaded, thereby controlling the residual deformation of the structure.Figure 23
Hysteretic curve of Reid dampers in the structure under the action of the 0.51 g El Centro wave: (a)η=0.3; (b) η=0.6.
(a)
(b)


## 5.5. Residual Interstory Drift

This section mainly analyzes the control effect of the Reid dampers on the residual interstory drift of the structure. The limit of residual interstory drift is set to 0.5%, and the structure cannot be repaired if the value exceeds this limit [33]. The analysis results show that the maximum residual interstory drift of the uncontrolled structure is 1.47% under the action of the 0.51 g Hachinohe wave, and the maximum residual interstory drifts under the other three seismic wave conditions are lower than 0.5%. Figure 24 shows the maximum residual interstory drifts of the first, third, fifth, seventh, and tenth stories of the controlled structure and those of the uncontrolled structure under the 0.51 g Hachinohe wave condition. The residual deformation of each story of the uncontrolled structure is greater than the limit value. All the shock absorption schemes using the Reid dampers have better control effect on the residual deformation of the structure; however, the control effect of each scheme is different. The trend is that the control effect decreases with the increase in the energy dissipation coefficient. This is largely because of the decrease in the equivalent stiffness coefficient with the increase in the energy dissipation coefficient of the damper. Moreover, the elastic resilience provided by the damper decreases, thus reducing the self-centering capacity of the structure. In summary, when a Reid damper is used to control the residual interstory drift of a structure, the equivalent stiffness of the damper must be guaranteed.Figure 24
Analysis results of structural residual interstory drift.


## 5.6. Shock Absorption Design when Using Reid Dampers

The control effects of the 10 types of Reid damper shock absorption schemes with the same additional damping ratio are compared. The lower the energy dissipation factor of the Reid damper, the greater the equivalent stiffness coefficient.First, the deformation control effect of the structure decreases with the increase in the energy dissipation coefficient, and the control effect is similar when the energy dissipation coefficient is in the range of 0.3–1.0. In this range, no obvious amplification in the acceleration response of the structure is observed, and the energy dissipation capacity of the dampers is good, thus effectively controlling the plastic damage energy of the structures.Second, when the energy dissipation coefficient is in the range of 0.1–0.6, the damper has a relatively high equivalent stiffness and good self-centering ability, thus effectively controlling the residual deformation of the structure. However, a greater damping force will act on the damper because of the excessive elastic stiffness, making it difficult to design connections and joints.In summary, when a Reid damper is designed for energy dissipation, the energy dissipation coefficient of the damper should be in the range of 0.3–0.6.


## 6. Conclusion

By conducting a theoretical analysis and a performance experiment, we developed a passive self-centering friction damper, which can realize the Reid model, and analyzed the control effect of a steel frame structure with Reid dampers. The following are the conclusions drawn from this study:(1)
The performance experiment results of the passive self-centering friction damper proposed in this paper are in good agreement with the theoretical analysis results. This shows that the damper can realize the Reid model. A theoretical basis for the design of the damper is provided.(2)
Considering the control effect on the structural displacement response, acceleration response, damper energy dissipation effect, failure mode, and residual deformation, it is suggested that the energy dissipation coefficient should be in the range of 0.3–0.6 when Reid dampers are used for energy dissipation.The Reid damper has the potential to replace viscous dampers that are employed in tuned mass damper structures and isolation structures. This research will be carried out in the future.



---

*Source: 1946818-2018-10-08.xml*