Hysteresis analysis of pre-pressed spring self-centering energy dissipation braces using different models

Abstract

The ability of an idealized piecewise-linear restoring force model and a nonlinear mechanical model to describe the hysteretic performances of the pre-pressed spring self-centering energy dissipation braces was evaluated based on experimental data. The hysteretic behaviors predicted by these two proposed models were compared with the experimental results of a typical prototype brace, and the results demonstrated that the two models can explain the brace force-time responses, and that the nonlinear mechanical model is more effective in describing the stiffness transition and energy dissipation of the brace. The two proposed models can be used for the design of the pre-pressed spring self-centering energy dissipation brace specimens, and the nonlinear mechanical model may be more useful for designing the structures with the pre-pressed spring self-centering energy dissipation braces. An orthogonal experiment was applied to analyze the influences of the key parameters on the performances of pre-pressed spring self-centering energy dissipation braces based on the nonlinear mechanical model. The results indicate that the friction slip force of energy dissipation mechanism, the pre-pressed force of self-centering mechanism, and the post-activation stiffness significantly affect the hysteretic performances and equivalent viscous damping ratios of the bracing system, while the changes in other parameters only produce slight effects. The determination of the pre-pressed force of the self-centering mechanism should be coordinated with the friction slip force of the energy dissipation mechanism to achieve a better hysteretic performance of the pre-pressed spring self-centering energy dissipation brace.

Keywords

combination disc springs hysteretic behavior nonlinear mechanical model piecewise-linear restoring force model self-centering energy dissipation brace

Introduction

Self-centering energy dissipation (SCED) bracing systems are constantly developed to reduce structural damage and minimize residual deformation that can occur in structures during earthquakes. Tensioning elements and shape memory alloys (SMAs) have been widely used in the design of the SCED bracing systems due to their good self-centering performances (Dong et al., 2017; El-Attar et al., 2008; Li et al., 2018; Liu et al., 2011; Olsen et al., 2008). Christopoulos et al. (2008) proposed an SCED brace that used friction devices to dissipate energy, and a set of tensioning elements to provide self-centering capability. Miller et al. (2011, 2012) proposed a new type of self-centering buckling-restrained brace that employed several SMA bars to provide self-centering forces. Qiu and Zhu (2017) and Qiu et al. (2018) proposed an SCED bracing system that combined the SMA wires with the friction devices.Xu et al. (2016a, 2016b, 2019) proposed a new type of SCED brace that used friction devices to dissipate energy and two groups of combination disc springs for self-centering. In addition, in order to further analyze the seismic performances of structures with SCED braces, various restoring force models have been proposed to portray the flag-shaped hysteretic behaviors of the SCED bracing system. Ma and Yam (2011) proposed a mathematical restoring force model for SCED dampers that used a Bouc–Wen model to predict the behaviors of the energy dissipation mechanisms and a rigid-elastic model to describe the performances of the self-centering mechanisms (SCMs). Erochko et al. (2013, 2015) proposed an approach that used the nonlinear incremental stiffness method to characterize the hysteretic behaviors of the SCED brace, and this model took fabrication tolerances into account. Wiebe and Christopoulos (2011) used Bézier curves to model the gradual stiffness transitions of the flag-shaped hysteretic behaviors to improve the accuracy of the SCED element models. Zhou et al. (2016) proposed an elastoplastic rheological model to predict the hysteretic performances of self-centering buckling-restrained braces.

The pre-pressed spring self-centering energy dissipation (PS-SCED) bracing system, which makes full use of the special characteristics of the disc springs for self-centering and the friction devices for energy dissipation, has been developed and experimentally verified (Xu et al., 2016a, 2016b; Xiao et al., 2019). Xu et al. (2016b) have proposed an idealized piecewise-linear restoring force model to describe the flag-shaped hysteretic behaviors of the PS-SCED bracing system. A new nonlinear mechanical model that takes full advantage of the superior numerical adjustability of the Bouc–Wen model was also developed to describe the restoring force, stiffness transition, and energy dissipation of the brace in more detail (Xu et al., 2018). A coefficient that considers the contact friction between the combination disc springs was introduced in the proposed model. The abilities of the two models to describe the hysteretic performances of the PS-SCED brace were assessed based on the experimental results of one typical prototype brace. The parametric analysis of the PS-SCED brace was also conducted using the nonlinear mechanical model according to the orthogonal experiments.

Behaviors and restoring force models of the PS-SCED brace

Figure 1 shows the configuration and the hysteretic response of a PS-SCED brace, which is comprised of an inner tube, an outer tube, an SCM, and a friction energy dissipation mechanism (FEDM). The SCM and FEDM consist of two groups of pre-pressed combination disc springs and eight friction devices, respectively. Flag-shaped hysteretic behaviors with negligible residual deformation and good energy dissipation capability are performed by the PS-SCED brace due to the mutual cooperation between the FEDM and SCM.

Figure 1.

Configuration and corresponding hysteretic response of the PS-SCED brace.

The brace response in tension is divided into four regions, as shown in Figure 2. At the beginning of Region I, the brace force F is below the activation force P_a. The inner tube and the outer tube remain relatively static, and there is only very slight elastic deformation in this region. Hence, the brace deforms under a large pre-activation stiffness K₁, and its force varies approximately linearly with the excitation displacement U. At this stage, the static friction force is provided by the FEDM, and the SCM is still in the initial compression state.

Figure 2.

PS-SCED brace force response analysis.

After the brace force exceeds the activation level, the constant friction slip force and increased self-centering force are presented by the FEDM and SCM, respectively. These forces also cause brace force to increase with a smaller post-activation stiffness K₂ in Region II. In this region, the relative displacement between the inner and outer tubes continues to increase with the increase of excitation displacements, causing the tube members to no longer provide axial forces for the brace. Only the activated friction devices in FEDM and further compressed combination disc springs in SCM provide axial forces for the brace. The velocity of the brace continues to decrease until the maximum displacement is arrived, and the velocity will be zero at the end of Region II.

In Region III, the FEDM starts to change the direction, and the velocity begins to increase in the negative direction. In this region, the inner and outer tubes remain relatively static as in Region I. Therefore, the brace again deforms under a large stiffness. The friction force of FEDM continues to decrease or even to be negative. However, the force of SCM is always maintained within a large positive range in this region. Hence, even though the total brace force is decreasing, the direction of the force does not change and the value still remains positive. In Region IV, the brace starts to return to the zero displacement position, and its velocity continues to increase. It is noted that, due to the self-centering force provided by the SCM, the zero brace force and maximum velocity will occur when the displacement is equal to zero.

The mechanical behavior of the brace in compression is similar to that in tension, and almost no residual deformation can be observed when the pre-pressed force P₀ of the SCM is larger than the friction slip force F₀ of the FEDM. Although small residual deformation occurs when the force P₀ is less than F₀, this residual deformation can still be reduced or even eliminated when the friction slip force of the FEDM is removed.

An idealized piecewise-linear restoring force model has been developed by Xu et al. (2016b) to describe the PS-SCED brace force in each region. This model assumes that the force and displacement of the brace in the corresponding region are linearly correlated, and the equations governing the brace force in each region under tension are

F = {\begin{matrix} K_{1} \cdot U (t) Region I \\ K_{1} \cdot u_{y} + K_{2}^{+} (U (t) - u_{y}) Region II \\ K_{1} (u_{y} + U (t) - U_{max} (t)) + K_{2}^{+} (U_{max} (t) - u_{y}) Region III \\ K_{1} (u_{y} - {u'}_{y}) + K_{2}^{+} (U_{max} (t) - u_{y}) - K_{2}^{-} (U_{max} (t) - {u'}_{y} - U (t)) Region IV \end{matrix}

(1)

where U_max is the maximum target deformation, and u_y and $u'_{y}$ are the activation deformation of the brace in the loading and unloading stages, respectively. Similarly, $K_{2}^{+}$ and $K_{2}^{-}$ are the post-activation stiffness of the brace in loading and unloading stages, respectively, by considering the influences of the contact friction between the combination disc springs. In order to simplify the model, the smoothness of transition between the adjacent regions is ignored. However, seismic analysis has demonstrated that an abrupt stiffness decrease or increase at a high velocity always induces a substantial increase in peak floor acceleration of the structures (Wiebe and Christopoulos, 2011). Considering that the smoothness of stiffness transition is closely related to the peak floor acceleration responses of the structures equipped with PS-SCED elements, Xu et al. (2018) proposed a new nonlinear mechanical model, in which two Bouc–Wen models are connected in parallel to describe the SCM and FEDM, respectively, as shown in Figure 3. The equation governing the brace force F is

\begin{matrix} F (t) = α_{1} \cdot k_{f 1} \cdot U_{h} (t) + (1 - α_{1}) \cdot k_{f 1} \cdot z_{f} (t) \cdot u_{y 1} \\ + α_{2} \cdot k_{s 1} \cdot U_{h} (t) + (1 - α_{2}) \cdot k_{s 1} \cdot z_{s} (t) \cdot u_{y 2} \end{matrix}

(2)

where $α_{1}$ is the ratio of post- to pre-activation stiffness of FEDM. The value of $α_{2}$ is the ratio of post- to pre-activation stiffness of SCM, and is defined as

α_{2} = {\begin{matrix} \frac{k_{l 1} \cdot K_{2}}{(k_{l 1} - K_{2}) \cdot k_{s 1}} ({\overset{\cdot}{U}}_{h} (t) \cdot U_{h} (t) > 0) \\ \frac{μ \cdot k_{l 1} \cdot K_{2}}{(k_{l 1} - μ \cdot K_{2}) \cdot k_{s 1}} ({\overset{\cdot}{U}}_{h} (t) \cdot U_{h} (t) < 0) \end{matrix}

(3)

where µ is a coefficient that is used to describe the contact friction between the combination disc springs, and can be predicted according to the configuration and the smoothness of the disc springs’ contact surface (China Standards Press, 2005). The elastic stiffness of the connecting ends of the brace is k_l₁. The initial stiffness and activation deformation, respectively, of FEDM are k_f₁ and u_y₁. Similarly, k_s₁ and u_y₂ are the initial stiffness and activation deformation of SCM, respectively. The value of U_h is the displacement of FEDM or SCM. The evolutionary variables z_f(t) and z_s(t) are expressed as follows

\begin{matrix} {\overset{\cdot}{z}}_{f} (t) = \frac{1}{u_{y 1}} [1 - (γ_{1} + β_{1} \cdot sign (z_{f} (t) \cdot {\overset{\cdot}{U}}_{h} (t))) \cdot {| z_{f} (t) |}^{n_{1}}] \\ \cdot {\overset{\cdot}{U}}_{h} (t) \end{matrix}

(4)

\begin{matrix} {\overset{\cdot}{z}}_{s} (t) = \frac{1}{u_{y 2}} [1 - (γ_{2} + β_{2} \cdot sign (z_{s} (t) \cdot {\overset{\cdot}{U}}_{h} (t))) \cdot {| z_{s} (t) |}^{n_{2}}] \\ \cdot {\overset{\cdot}{U}}_{h} (t) \end{matrix}

(5)

where β₁, β₂, γ₁, γ₂, n₁, and n₂ are the parameters that define the shape of hysteretic curves. The values of parameters β₁ and β₂ reflect the energy dissipation level of the PS-SCED braces (Domaneschi, 2015; Ikhouane et al., 2007), γ₁ and γ₂ adjust the unloading path in the hysteretic behaviors, and n₁ and n₂ relate to the smoothness of transitions during the cyclic loading process. The variation of parameters n₁, n₂, γ₁, and γ₂ cannot significantly change the brace responses, and thus, the theoretical values of these parameters can be predicted according to their physical meanings. On the contrary, the variations in the energy dissipation parameters β₁ and β₂ have obvious influences on the hysteretic responses of the brace. Therefore, the genetic algorithm can be selected to identify these parameters based on the experimental data.

Figure 3.

Nonlinear mechanical model of the PS-SCED bracing system.

Comparisons of different restoringforce models

Comparative analysis

To assess the abilities of these two restoring force models to portray the PS-SCED brace performance, the proposed idealized piecewise-linear restoring force model and the nonlinear mechanical model are employed to describe the brace responses under sinusoidal displacement excitations with increasing amplitudes. The experimental data of the PS-SCED brace-I from Xu et al. (2016a) were used. The brace test system and the displacement excitation measured by a displacement meter are shown in Figure 4. The brace specimen consisted of a 1.2-m-long box-shaped outer tube with dimensions of 215 mm × 215 mm × 6 mm, and a 1.68-m-long cylindrical inner tube with a thickness of 10 mm. Steel materials with nominal yield strengths of 345 and 235 MPa were selected for the inner and outer tubes. Thirty-six sections of disc springs, which were made of 50CrVA, were employed to provide self-centering force for the brace. The outer diameter, inner diameter, and height of each disc spring were 200, 112, and 16.2 mm, respectively. Thus, the pre- and post-activation stiffness of the brace specimen was 245.79 and 38.25 kN/mm. In addition, the force P₀ and F₀ were set to 270 and 200 kN, respectively.

Figure 4.

Experiment of the PS-SCED brace: (a) test system and (b) the measured displacement excitations.

Figures 5 and 6 show the comparisons of the detailed responses predicted by these two modes for the selected brace specimen, and the parameters for the nonlinear mechanical model were chosen as α₁ = 0, β₁ = 0.96, β₂ = 0, γ₁ = 0.12, γ₂ = 1, µ = 0.9, n₁ = 1.16, and n₂ = 0.62. The parameter α₂ in the loading and unloading stages was calculated as 0.051 and 0.046, respectively. It is indicated that the two proposed restoring force models can provide accurate representations of the force-time history responses and hysteretic behaviors of the brace, suggesting that the idealized piecewise-linear restoring force model and the nonlinear mechanical model can be used to predict the restoring force and hysteretic performances for the design of the PS-SCED brace. In addition, by comparing the close-up of the time history of brace force near the time of 1000 to 1100 s, as shown in Figure 5, and the stiffness transitions in hysteretic behaviors, as shown in Figure 6, it is evident that the idealized piecewise-linear restoring force model failed to capture the gradual force transition in the force-time history responses and gradual stiffness transition during loading and unloading stages in the hysteretic responses. The nonlinear mechanical model displays obvious improvement over the idealized model. It better matches the brace responses, especially the gradual transition that occurs in the real brace responses due to the reasonable values of the parameters n₁ and n₂.

Figure 5.

Comparisons of the force-time history responses of PS-SCED brace.

Figure 6.

Comparisons of predicted and experimental hysteretic responses.

Figure 7 shows the energy dissipation comparison of the predicted and experimental results under different excitation amplitudes. Due to the unsteady sliding between the inner and outer tubes, the energy dissipation values predicted by these two models are slightly greater than the experimental results when the FEDM is initially activated. A better agreement is achieved by these two restoring force models after a few repeated cycles due to the smoother sliding of the friction devices. Under the largest displacement amplitudes, the experimental energy dissipation in tension and compression is 5.00 and 7.02 kJ, respectively. The difference between the energy dissipation calculated from the idealized piecewise-linear restoring force model and the experimental results is 17.80% in tension and 16.68% in compression, respectively, and the respective values from the nonlinear mechanical model are 0.40% and 7.35%. This indicates that the energy dissipation is better described by the nonlinear mechanical model. It is also suggested that the nonlinear mechanical model may be more useful for the design of the structures with the PS-SCED braces.

Figure 7.

Comparisons of energy dissipation between predictions and experimental results.

Parametric analysis

PS-SCED braces are expected to provide sufficient lateral stiffness, energy dissipation, and self-centering capabilities for buildings to improve the seismic performances of the structures. Therefore, it is necessary to design proper parameters for the PS-SCED braces. The friction slip force F₀, pre-pressed force P₀, pre-activation stiffness K₁, and post-activation stiffness K₂ are the four key parameters that have a significant influence on the shape of the hysteretic cycles, energy dissipation, and self-centering performances of the PS-SCED brace. In addition, although the contact friction between the combination disc springs can provide additional energy dissipation for the brace, it also reduces the unloading stiffness of the brace in Region IV. Therefore, it is necessary to analyze the influence of the friction coefficient µ of the combination disc springs on the hysteretic performances of the PS-SCED brace.

Based on the nonlinear mechanical model, the parametric analysis of the PS-SCED brace was performed using an orthogonal experiment method, and the initial design values of these five key parameters of the PS-SCED brace-I were used as the first-level values. F₀, P₀, K₁, and K₂ changed up to 20% compared to their first-level values, and µ changed up to 10% compared to its first-level value to obtain the second- to the fifth-level values, as listed in Table 1. In addition, a five-level orthogonal table was selected as the basis for the experimental design, and a total of 25 groups of experiments were conducted. Sinusoidal displacement excitation was selected, and the displacement amplitudes were 2, 3, 4, …, and 12mm, respectively. The equivalent viscous damping ratio ξ was taken into account and calculated as

ξ = \frac{E_{D}}{4 π E_{SO}}

(6)

where E_SO and E_D are the maximum elastic strain energy and energy dissipated for the cycle, respectively.

Table 1.

Five-level values of design parameters.

Level	Factor
Level	F ₀ (kN)	P ₀ (kN)	K ₁ (kN/mm)	K ₂ (kN/mm)	µ
1	200	270	245.8	38.0	0.90
2	180	243	221.2	34.2	0.85
3	160	216	196.6	30.4	0.80
4	220	297	270.4	41.8	0.95
5	240	324	295.0	45.6	1.00

Table 2 lists the maximum equivalent viscous damping ratio ξ and corresponding design parameters for each test, where $k_{i} (i = 1, 2, \dots, 5)$ is the mean value of the maximum equivalent viscous damping ratio corresponding to the parameters F₀, P₀, K₁, K₂, and µ at the ith level. The range of k_i is defined as the difference in value between the maximum and the minimum of k_i, and the respective values corresponding to the parameters F₀, P₀, K₁, K₂, and µ were 2.56, 2.70, 1.42, 1.71, and 0.93, respectively. This indicates that P₀ and F₀ are the main factors affecting the equivalent viscous damping ratio of the PS-SCED brace, followed by K₂ and K₁. The change of µ has the least influence. The average response of the equivalent viscous damping ratio corresponding to each factor at different levels is shown in Figure 8, and Figures 9 to 13 show the hysteretic responses of the PS-SCED brace and corresponding equivalent viscous damping ratios calculated from the changed parameters. As the parameter P₀ or K₂ increased, the axial forces of the brace increased, but because they had no obvious effect on the energy dissipation capability of the PS-SCED brace, the equivalent viscous damping ratio of the PS-SCED brace decreased. When parameters P₀ and F₀ were 216 and 240 kN, the average residual deformation of the brace was 1.8 mm, and this value decreased with the increase of P₀. No obvious residual deformation was observed when P₀ was approximately equal to F₀, such as in Test-18 and Test-22, as shown in Figures 12 and 13. Therefore, the self-centering capability of the PS-SCED brace can be improved by increasing the pre-pressed force of the combination disc springs. In addition, the value of P₀ is also closely related to the stroke of the combination disc springs. Thus, the deformability of the PS-SCED brace is also limited when the self-centering capability increases. However, this phenomenon can be avoided by adjusting the thickness, quantity, and combination of the disc springs to meet the design requirements for the pre-activation stiffness and deformability of the brace.

Table 2.

Equivalent viscous damping ratio corresponding to the design parameters for each test.

Test No.	Factor					Result
Test No.	F ₀ (kN)	P ₀ (kN)	K ₁ (kN/mm)	K ₂ (kN/mm)	µ	Maximum (ξ)
1	200	270	245.8	38.0	0.90	16.97
2	200	243	221.2	34.2	0.85	17.92
3	200	216	196.6	30.4	0.80	19.01
4	200	297	270.4	41.8	0.95	16.05
5	200	324	295.0	45.6	1.00	15.17
6	180	270	221.2	30.4	0.95	16.58
7	180	243	196.6	41.8	1.00	15.24
8	180	216	270.4	45.6	0.90	17.27
9	180	297	295.0	38.0	0.85	16.30
10	180	324	245.8	34.2	0.80	15.75
11	160	270	196.6	45.6	0.85	14.31
12	160	243	270.4	38.0	0.80	16.88
13	160	216	295.0	34.2	0.95	17.83
14	160	297	245.8	30.4	1.00	15.17
15	160	324	221.2	41.8	0.90	13.63
16	220	270	270.4	34.2	1.00	18.03
17	220	243	295.0	30.4	0.90	20.04
18	220	216	245.8	41.8	0.85	19.18
19	220	297	221.2	45.6	0.80	16.27
20	220	324	196.6	38.0	0.95	15.39
21	240	270	295.0	41.8	0.80	18.87
22	240	243	245.8	45.6	0.95	17.62
23	240	216	221.2	38.0	1.00	18.55
24	240	297	196.6	34.2	0.90	17.16
25	240	324	270.4	30.4	0.85	18.40
k ₁	17.02	16.95	16.94	16.82	17.01	\
k ₂	16.23	17.54	16.59	17.34	17.22
k ₃	15.56	18.37	16.22	17.84	17.36
k ₄	17.78	16.19	17.33	16.59	16.69
k ₅	18.12	15.67	17.64	16.13	16.43
Range	2.56	2.70	1.42	1.71	0.93

Figure 8.

Average response of equivalent viscous damping ratio corresponding to each factor at different levels.

Figure 9.

Comparisons of (a) hysteretic responses and (b) equivalent viscous damping ratios of the PS-SCED braces from Test-1 to Test-5.

Figure 10.

Comparisons of (a) hysteretic responses and (b) equivalent viscous damping ratios of the PS-SCED braces from Test-6 to Test-10.

Figure 11.

Comparisons of (a) hysteretic responses and (b) equivalent viscous damping ratios of the PS-SCED braces from Test-11 to Test-15.

Figure 12.

Comparisons of (a) hysteretic responses and (b) equivalent viscous damping ratios of the PS-SCED braces from Test-16 to Test-20.

Figure 13.

Comparisons of (a) hysteretic responses and (b) equivalent viscous damping ratios of the PS-SCED braces from Test-21 to Test-25.

Fuller hysteretic curves and better energy dissipation were achieved by the PS-SCED brace with a larger F₀, and increased equivalent viscous damping ratios were also observed by increasing this parameter. However, it is noted that the self-centering capability was gradually weakened with the increase of F₀, and obvious residual deformation was observed when F₀ was greater than P₀. Therefore, the relative size between P₀ and F₀ is an important factor to determine the occurrence of the residual deformation, and the determination of these two values should be coordinated to achieve a better hysteretic performance.

Figures 9 to 13 show that the changes of K₁ do not have significant influences on the energy dissipation and restoring force of the PS-SCED brace, but the activation deformation of the brace decreases with the increase of K₁. A smaller activation deformation means that the FEDM and SCM will dissipate energy earlier and provide self-centering capability for the brace, and a larger equivalent viscous damping ratio may be achieved at the low-amplitude displacement responses.

The parameter µ has a slight but not significant effect on the hysteretic performance and equivalent viscous damping ratio of the PS-SCED brace, but the additional energy dissipation provided by the contact friction between the combination disc springs results in the increase of the equivalent viscous damping ratio of the brace as µ decreases. Although the changes of parameters K₁ and µ have little impact on the brace hysteresis, these changes may have obvious influences on the overall responses of the structures with the PS-SCED braces.

Conclusion

The PS-SCED brace presents distinctive flag-shaped hysteretic behaviors during the cyclic loading process. This article assessed the abilities of the proposed idealized piecewise-linear restoring force model and nonlinear mechanical model to describe the hysteretic performance of the brace. The configuration of this bracing system was described and the concepts of these two models were presented according to the mechanism of the PS-SCED brace. Because of the simplified calculation, the idealized piecewise-linear restoring force model has high computational efficiency but ignores the smoothness of stiffness transition. The nonlinear mechanical model is different from the idealized piecewise-linear restoring force model and employs two parallel Bouc–Wen models to describe the hysteretic behaviors of the main bracing mechanism. Comparisons of the brace responses and energy dissipation between the experiments and the predictions by these two restoring force models were conducted. The results indicated that the two proposed models can effectively predict the force-time responses of the brace and can be used for the design of the PS-SCED brace. The nonlinear mechanical model can describe the brace responses in more detail and can take into account the gradual stiffness transition that occurs in the real responses. The energy dissipation is also better described by the nonlinear mechanical model, and this model may be more useful for the design of the structures with the PS-SCED braces. The parametric analysis for the PS-SCED brace was performed using the nonlinear mechanical model based on orthogonal experiments. The analysis results indicated that the friction slip force of FEDM, pre-pressed force, and post-activation stiffness have significant effects on the hysteretic performances and the equivalent viscous damping ratios of the bracing system, while the changes in pre-activation stiffness and the friction coefficient of combination disc springs only cause slight effects. The relative size between pre-pressed force of the combination disc springs and the friction slip force of the FEDM is the key factor in determining the occurrence of the residual deformation, and the determination of these two values should be coordinated to achieve a better hysteretic performance of the PS-SCED brace.

Footnotes

Declaration of Conflicting Interests

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

Funding

The author(s) disclosed receipt of the following financial support for the research, authorship, and/or publication of this article: The writers gratefully acknowledge the partial support of this research by the Fundamental Research Funds for the Central Universities of China under Grant No. 2017YJS156, the National Natural Science Foundation of China under Grant No. 51578058, and Beijing Natural Science Foundation of China under Grant No. 8172038.

ORCID iD

Long-He Xu

References

China Standards Press (2005) Disc Spring GB/T 1972-2005. Beijing, China: China Standards Press.

Christopoulos

Tremblay

Kim

, et al. (2008) Self-centering energy dissipative bracing system for the seismic resistance of structures: development and validation. Journal of Structural Engineering: ASCE 134(1): 96–107.

Domaneschi

(2015) Structural control of cable-stayed suspended bridges. Doctoral Dissertation, University of Pavia, Pavia, Italy.

Dong

Han

, et al. (2017) Performance of an innovative self-centering buckling restrained brace for mitigating seismic responses of bridge structures with double-column piers. Engineering Structures 148: 47–62.

El-Attar

Saleh

El-Habbal

, et al. (2008) The use of SMA wire dampers to enhance the seismic performance of two historical Islamic minarets. Smart Structures and Systems 4(2): 221–232.

Erochko

Christopoulos

Tremblay

(2015) Design, testing and detailed component modeling of a high-capacity self-centering energy-dissipative brace. Journal of Structural Engineering: ASCE 141(8): 04014193.

Erochko

Christopoulos

Tremblay

, et al. (2013) Shake table testing and numerical simulation of a self-centering energy dissipative braced frame. Earthquake Engineering & Structural Dynamics 42(11): 1617–1635.

Ikhouane

Mañosa

Rodellar

(2007) Dynamic properties of the hysteretic Bouc-Wen model. Systems & Control Letters 56(3): 197–205.

Shu

Liu

, et al. (2018) Research and development of an innovative self-centering energy dissipation brace. Structural Design of Tall and Special Buildings 27(15): e1514.

10.

Liu

Zhu

, et al. (2011) Displacement-based design approach for highway bridges with SMA isolators. Smart Structures and Systems 8(2): 173–190.

11.

Yam

MCH

(2011) Modelling of a self-centring damper and its application in structural control. Journal of Constructional Steel Research 67(4): 656–666.

12.

Miller

Fahnestock

Eatherton

(2011) Self-centering buckling-restrained braces for advanced seismic performance. In: Proceedings of the structures congress 2011, Las Vegas, NV, 14–16 April.

13.

Miller

Fahnestock

Eatherton

(2012) Development and experimental validation of a nickel–titanium shape memory alloy self-centering buckling-restrained brace. Engineering Structures 40: 288–298.

14.

Olsen

Van der Eijk

Zhang

(2008) Numerical analysis of a new SMA-based seismic damper system and material characterization of two commercial NiTi-alloys. Smart Structures and Systems 4(2): 137–152.

15.

Qiu

Zhu

(2017) Performance-based seismic design of self-centering steel frames with SMA-based braces. Engineering Structures 130: 67–82.

16.

Qiu

Zhang

, et al. (2018) Seismic behavior of properly designed CBFs equipped with NiTi SMA braces. Smart Structures and Systems 41(4): 479–491.

17.

Wiebe

Christopoulos

(2011) Using Bézier curves to model gradual stiffness transitions in nonlinear elements: application to self-centering systems. Earthquake Engineering & Structural Dynamics 40(14): 1535–1552.

18.

Xiao

(2019) Seismic performance and damage analysis of RC frame–core tube building with self-centering braces. Soil Dynamics and Earthquake Engineering 120: 146–157.

19.

Fan

(2016a) Development and experimental verification of a pre-pressed spring self-centering energy dissipation brace. Engineering Structures 127: 49–61.

20.

Fan

(2018) Hysteretic analysis model for pre-pressed spring self-centering energy dissipation braces. Journal of Structural Engineering: ASCE 144(7): 04018073.

21.

Fan

, et al. (2016b) Hysteretic behavior studies of self-centering energy dissipation bracing system. Steel and Composite Structures 20(6): 1205–1219.

22.

Liu

(2019) Cyclic behaviors of steel plate shear wall with self-centering energy dissipation braces. Journal of Constructional Steel Research 153: 19–30.

23.

Zhou

Xie

Meng

, et al. (2016) Hysteretic performance analysis of self-centering buckling restrained braces using a rheological model. Journal of Engineering Mechanics: ASCE 142(6): 04016032.