#### Abstract

Self-centering bracing systems, by which residual deformations of structures after earthquakes can be minimized, are considered effective solutions to achieve seismic resilience. In this paper, a parametric study on the seismic response of intermediate and high-rise steel-framed buildings with novel self-centering tension-only braces (SC-TOBs) is numerically conducted. Three key parameters, the stiffness degradation factor, the activation strain, and the initial axial stiffness of the SC-TOBs, are investigated to explore the design space for the SC-TOB frames (SC-TOBFs) because of their unique tunability compared with traditional bracing systems. Identical steel frames equipped with buckling restrained braces (BRBs) are also designed and examined for comparison purposes. The results indicate that increasing the stiffness degradation factor can improve the second stiffness of SC-TOBFs and successfully make the distribution of interstory drifts more uniform; an increase in the activation strain leads to a larger activation deformation of SC-TOBFs, but it has a very limited effect on the interstory drifts; increasing the initial axial stiffness appropriately is beneficial to reduce the interstory drifts of the low stories. The lateral behavior of SC-TOBFs is comparable to that of BRB frames when a lower activation strain and a higher initial axial stiffness are selected. Furthermore, when a higher stiffness degradation factor and a lower initial axial stiffness are selected simultaneously, the seismic action on SC-TOBFs can be effectively reduced, and a relatively uniform distribution over the building height can be obtained. The SC-TOBFs are considered to be a type of performance-tunable structure, and tuning can be achieved by varying a frame’s adjustable parameters.

#### 1. Introduction

Controlling inelastic ductility to soften the seismic response and to dissipate the hysteretic energy is a basic concept for the seismic design of structures. However, inelastic ductility can also result in the concentration of damage in local parts of a structure and produce residual deformations (e.g., [1–3]), leading to undesirable influences on the resulting structure in terms of prohibitive costs of rehabilitation for structural and nonstructural components, interruption of building function, and a high collapse risk due to the *P*-delta effects [4–8].

Braced frames are lateral resisting systems that have been commonly used in steel buildings for earthquake resistance. To reduce or eliminate residual deformations, Christopoulos et al. [9] proposed a self-centering energy dissipative (SCED) brace, which provided a restoring force by pretensioned (PT) aramid fiber-reinforced polymer (AFRP) elements and energy dissipation through a friction mechanism. Quasistatic and dynamic tests demonstrated that the SCED brace had satisfactory recentering and energy dissipation capacities, and its bracing system could self-center within the target design drift. To confirm the seismic performance of the SCED braces within structures, Erochko et al. [7] carried out a shake table test on a 3-story steel frame braced with SCED braces and performed numerical simulations for comparison. To improve the recentering capability of self-centering braces (SCBs), the enhanced elongation telescoping SCED (T-SCED) brace and the dual-core SCB (DC-SCB), both of which incorporate multiple self-centering systems, were developed independently [10, 11]. A full-scale one-story one-bay steel frame with DC-SCBs was tested to examine its seismic performance, and the results showed that the initial axial stiffness of the DC-SCB would decrease significantly from the influence of fabrication errors [12, 13].

Buckling restrained braces (BRBs), which have symmetric hysteretic behaviors and excellent energy dissipation capacity, have been widely used in recent years. However, this bracing system tends to induce large residual deformations after an earthquake [14, 15]. To address this drawback, Liu and Wu [16] proposed the self-centering BRB (SC-BRB) using PT steel strands to provide a restoring force. Chou et al. [17] proposed a dual-core self-centering sandwiched BRB (SC-SBRB) combing the self-centering property of a DC-SCB and the energy dissipation of a sandwiched BRB. Multiple cyclic tests demonstrated that the SC-SBRB exhibited appreciable self-centering, deformation, and energy dissipation capacities. Zhou et al. [18, 19] used basalt fiber-reinforced polymer for the PT tendons and developed a dual-tube SC-BRB, which exhibited a good flag-shaped hysteretic performance and self-centering capacity. Xie et al. [20] improved the dual-tube SC-BRB configuration by adding a rubber cushion so as to reduce the negative influence of the fabrication error. Other feasible solutions for braces to achieve self-centering, such as prepressed springs, and energy dissipation, such as magnetorheological fluid devices, can be found in the literature [21–23]. In addition, shape memory alloys (SMAs), characterized by superelasticity to recenter and dissipate energy on their own, have been employed to develop various types of SMA-based braces [24–26].

Unlike the abovementioned relatively rigid braces, tension-only braces (TOBs) are flexible bracing members, which can enable the full use of high-strength materials without buckling under compression, leading to a mitigated seismic response of the braced structures due to a prolonged fundamental period. Thus, TOBs have many applications in buildings in areas of low seismicity [27–29]. However, because of their severe pinched hysteresis and inferior energy dissipation capacity, TOBs are prohibited as the sole lateral resistant system in areas of high seismicity [30, 31]. Regardless of these drawbacks, a lot of efforts have been made to expand the application of TOBs. For example, Mousavi and Zahrai [32] proposed a preslacked cable brace (PSCB), and their numerical study indicated that PSCBs could eliminate the strength degradation of the braced nonductile frame. Thereafter, Mousavi and Zahrai [33] proposed a slack-free connection (SFC), by which the pinching of the TOBs could be completely avoided and energy dissipation capacity was thereby significantly improved. Zahrai et al. [34] proposed a hybrid TOB (HTOB), which has a stable hysteresis with tunable postyield stiffness. Mehrabi et al. [35] proposed a TOB system with a precompressed spring, which enables both diagonal bracings to be constantly in tension. Experimental and analytical investigations validated the enhanced lateral performance of the cable braced frames in terms of strength and ductility.

In light of these studies, to take advantage of the seismic resilience of SCBs and the seismic mitigation of TOBs, a novel self-centering TOB (SC-TOB) has been developed and numerically verified by Chi et al. [36, 37]. To further investigate how the SC-TOBs can be implemented in structures to improve seismic performance, a parametric study on the seismic response of 9- and 16-story steel-framed buildings, which can be considered typical of intermediate and high-rise buildings, with SC-TOBs is numerically conducted through pushover analysis. Three key parameters including the stiffness degradation factor, the activation strain, and the initial axial stiffness of the SC-TOBs, are investigated thoroughly because of their unique characteristics and tunability compared with traditional bracing systems. Identical steel frames equipped with BRBs are also designed and examined for comparison purposes.

#### 2. Configuration and Mechanics of the SC-TOB

A schematic of the SC-TOB [36, 37] showing its basic function is shown in Figure 1. The brace mainly consists of three parts: a high-strength steel (HSS) cable as a bracing element, a frictional device (FD) to dissipate seismic energy, and PT tendons to produce a full self-centering hysteresis. One end of the PT tendons is anchored to the blocking plate, and the other end passes around the pulley and connects with the FD after a certain pretension is imposed.

The mechanics of the SC-TOB can be explained using the analytical model presented in Figure 2: the PT tendons, FD, and HSS cable are idealized as springs with axial stiffnesses , , and , respectively, and the frictional resistance of the FD is *F*; the pretension of the PT tendons is . The blocking plate *R* is used to balance the pretension and restrict the left movement of the FD.

The hysteretic behavior of the SC-TOB is illustrated in Figure 3. When the value of the lateral load *P* is less than the sum of the pretension force and the frictional resistance (i.e., ), only the HSS cable works (Stage o-a), and the initial axial stiffness of the SC-TOB is :where , , and are Young’s modulus, cross-sectional area, and original length of the HSS cable, respectively.

As *P* increases to , the energy dissipative mechanism provided by FD is activated. Defining the load *P* at Event *a* as the activation load ,

At Stage a-b, the stiffness of the SC-TOB decreases significantly from to the postactivation stiffness, , given by

As the cross-sectional area of the friction device is much greater than that of the cable and tendon, while the length of the friction device is much smaller than that of the cable and tendon, i.e., , equation (3) can be revised as

When unloading begins at Event *b*, the friction will first reduce gradually from *F* to zero and then increase in the opposite direction to −*F* at Event *c*. During this stage, only the HSS cable works, so the stiffness of the SC-TOB is recovered to . As *P* further unloads (Stage c-d), the SC-TOB is capable of returning to its initial position by the sufficient restoring force produced by the PT tendons as long as is no less than . During this stage, the stiffness of the SC-TOB is reduced again to the postactivation stiffness . With continued unloading, the tension force of the HSS cable decreases to zero after load removal and the stiffness of the SC-TOB during Stage d-o is once again recovered to .

A structure incorporating the SC-TOBs in a frame bay is described in Figure 4, in which the solid line indicates an active HSS cable and the dotted line indicates a loose one. The pulleys mounted on the beam near the beam-column connections, which are not part of an SC-TOB, are used to guide the HSS cables.

#### 3. Building Design and Modeling

##### 3.1. Building Design

The 9- and 16-story prototype buildings, which have an identical plan configuration and a constant story height of 3.9 m, are braced with BRBs or SC-TOBs, as shown in Figure 5, in which both types of bracing elements are denoted by the dotted lines. Figure 6 illustrates the elevation view of the 9-story SC-TOB frames (SC-TOBFs) and BRB frames (BRBFs), in which all the beams are pinned to the columns. The load information considered in this design is listed in Table 1.

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In view of the symmetrical plan dimensions of the prototype buildings, the torsion effect is neglected and thereby the lateral force will be equally distributed to the corresponding braced frames, allowing for a 2D analysis to be performed in lieu of 3D analysis for efficiency and simplicity purposes. The modal analysis comparison results, as shown in Table 2, confirms that the 2D models will suffice for the following investigation.

##### 3.2. Simulation of SC-TOB

The structural analysis program SAP2000 [38] is utilized for simulating the SC-TOB. A multilinear elastic element (MEE) is used to model the PT tendons behavior, as shown in Figure 7(a). The force-deformation relationship is nonlinear, but it is elastic. This means that the element loads and unloads along the same curve, and no energy is dissipated. The friction mechanism is modeled by using a multilinear plastic element (MPE), as shown in Figure 7(b). Note that for both elements, only the second-stage curves with a softened stiffness are valid to model the required behaviors of the PT tendons and FD, so a negligible first-stage deformation has to be specified. The HSS cable could have been simulated using the linear elastic frame element (LEFE), but a severe numerical oscillation would occur when the MEE or MPE is connected directly with LEFE due to incompatibility between linear and nonlinear elements. Therefore, the MEEs with a sufficiently long first-stage curve are also used herein for the cables to ensure that the cable would always work within this stage. The target hysteresis and integrated model of the SC-TOB are shown in Figures 7(c) and 7(d), respectively.

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The pulley is simulated using five hinged frame elements within the dotted circle, as shown in Figure 8, in which nodes 1, 2, and 3 are located around the center point O. Because the pulley is almost a rigid body in real structures, the axial stiffness of each frame element has to be specified sufficiently large. A “body constraint” is specified to node O and its vertical projection O’ on the beam to ensure that both the nodes would move together as a 3D rigid body.

The combination of all the elements associated with the SC-TOB simulation is illustrated in Figure 9, in which C’ and D’ on the beam are the vertical projection of nodes C and D, respectively. Body constraints are also specified for C and C’, and D and D’, except that the translational degree of freedom of node C is released.

##### 3.3. Simulation of BRB

The MPE with kinematic hysteresis is introduced to simulate the BRB behavior [38], as shown in Figure 10, in which , , , and are the yield capacity, yield deformation, axial stiffness, and postyield stiffness ratio of the steel core, respectively. In this paper, is taken as 3%. The design axial strength of the brace, , is determined under the frequently occurred earthquake (FOE) condition, and according tothe cross-sectional area of the steel core, , can be calculated, where is the yield stress of the core. The yield capacity of the brace, , can be calculated bywhere is the overstrength factor of the core. The maximum axial strength of the brace, , can be calculated bywhere is the strain-hardening adjustment factor.

##### 3.4. Design Information of the Prototype Buildings

The criterion used for the SC-TOBFs design is that all member force demands and the story drift must satisfy the design objectives under the design basis earthquake. With the increase of earthquake intensity, the seismic forces induced in the load-resisting elements such as beams, columns, and braces in the braced bays increase nonlinearly, whereas that in the unbraced bays remain nearly unchanged. Hence, different limits of “demand-capacity ratio” are specified: 0.5 and 0.8 for the elements in braced and unbraced frames, respectively, so as to ensure that all of them remain essentially elastic or achieve full self-centering even under the most severe load condition in this analysis. The BRB components are designed according to the design codes [39–42]. The step-by-step design procedure of the SC-TOB could be referred to the literature [37], and the ratio of and is set at 1.05 for all SC-TOBs throughout this paper. The geometric and material properties of all the elements are listed in Tables 3–7. A strength check is performed through the structure after every pushover procedure, confirming that all the structural elements remain elastic without any damage in the analyses of the following section.

#### 4. Parametric Study

##### 4.1. Lateral Load Distribution

As illustrated in Figure 11, two lateral load distributions, the parabolic distribution (denoted as “LD-1”) and the uniform distribution (denoted as “LD-2”), are adopted as suggested in ASCE 7-10 [31]. These distributions can be expressed aswhere is the lateral load increment assigned to floor level , is the base shear increment of the structure, and are the building weights located on floor level and , respectively, and are the heights from the base to floor level and , respectively, is the total number of stories, for , for , and linear interpolation is used to select the values of between .

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An incremental static procedure for both load distributions is performed until a target displacement, which is represented by the total drift angle of the prototype buildings, (as expressed in equation (10)), is exceeded:where is the horizontal displacement of the control node located at the center of mass of the roof and is the total height of the structure. In this paper, is taken as 2% corresponding to the limit prescribed by the building codes [41, 42].

##### 4.2. Effects of Stiffness Degradation Factor *r*

The stiffness degradation factor, , as defined by the ratio of the postactivation stiffness to the initial axial stiffness of the SC-TOB, is expressed as

If is too large, the seismic forces induced in the brace and adjacent structural members will increase rapidly, which is neither economical nor safe. But if is too small (e.g., 3% for BRB [43]), damage concentration would be induced at certain stories, thus limiting the capacity of the structure to redistribute the demand along the height. Furthermore, the *P*-delta effects will also elevate the collapse risk. Hence, varying from 4% to 10% is investigated in this section.

###### 4.2.1. Base Shear Response

The effects of on the base shear response of SC-TOBFs with 9 and 16 stories are shown in Figures 12 and 13, respectively. With increasing from 4% to 10%, the second stiffness of the structure increases significantly, and the negative corresponding to is improved. The changes in have no effect on the first stiffness , the activation load , and the activation total drift angle of the structure. The SC-TOBFs exhibit a smaller than BRBFs do, because SC-TOBs can make full use of high-strength materials such that the seismic response of structures is mitigated because of a prolonged fundamental period. Besides, unlike BRBFs for which the stiffness degrades due to the yielding of the steel core, of SC-TOBFs is determined by , which makes significantly lower than and plays a similar role in yielding or ductility. This provides a flexible design space for the SC-TOBFs to achieve a required postactivation performance by varying .

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###### 4.2.2. Interstory Drift Response

The effects of on the interstory drift of SC-TOBFs with 9 and 16 stories are shown in Figures 14 and 15, respectively. Basically, both SC-TOBFs and BRBFs show the same tendency in terms of interstory drifts, decreasing from bottom to top. With the increase of , the interstory drifts of the lower part of the SC-TOBFs decrease gradually while the drifts of the upper part increase, indicating that the distribution of drifts over the building could be improved by selecting a relatively large .

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##### 4.3. Effects of Activation Strain

The activation strain, , defined as the strain of the HSS cable when an SC-TOB reaches its activation load , can be expressed aswhere is the activation deformation of the SC-TOB corresponding to . Since determines the deformation state of the structure when the SC-TOB starts to work, it permits designers to advance or delay the activation by specifying a suitable value accordingly. For BRBs, the activation strain depends on the yield strain of their steel core, which is made of Q160LY with yield strain of 0.07% in this paper.

###### 4.3.1. Base Shear Response

The effects of on the base shear response of SC-TOBFs with 9 and 16 stories are shown in Figures 16 and 17, respectively. With the increase of , the activation deformation and activation load of the structure increase gradually. However, the changes in have no effect on and of the structure. Due to the limitations of material property for the core, BRBFs have a constant . Compared with BRBFs, SC-TOBFs can flexibly control the deformation state when the structures enter the postactivation stage by adjusting the value of .

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###### 4.3.2. Interstory Drift Response

The effects of on the interstory drifts of SC-TOBFs with 9 and 16 stories are shown in Figures 18 and 19, respectively. For the SC-TOBFs, the changes in have a very limited effect on the distribution of interstory drifts of each story. This is primarily because , governing the activation deformation state of the structure, is very small compared with the strain of the SC-TOB at the end of analysis, such that it tends to display a negligible impact on the final interstory drift response.

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##### 4.4. Effects of Initial Axial Stiffness

To investigate the effects of on SC-TOBFs, a stiffness amplification factor is introduced aswhere is the initial axial stiffness of a specific SC-TOB selected from Tables 5 and 6 and is the amplified stiffness of the corresponding brace. Note that should remain constant in this section.

###### 4.4.1. Base Shear Response

The effects of on the base shear response of SC-TOBFs with 9 and 16 stories are shown in Figures 20 and 21, respectively. With increasing from 1.0 to 1.3, and increase continuously, while remain identical as designed. BRBFs have a larger than SC-TOBFs, because BRBs tend to adopt low-yield steel for the brace core, resulting in larger cross sections.

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###### 4.4.2. Interstory Drift Response

The effects of on the interstory drifts of SC-TOBFs with 9 and 16 stories are shown in Figures 22 and 23, respectively. Under the parabolic distribution of loads (LD-1), the interstory drifts of low stories decrease as increases, but the effect on the middle and upper stories is not remarkable. Under the uniform distribution (LD-2), the effect is negligible.

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#### 5. Discussion

From the results presented in Section 4, it can be found that the first-stage lateral behavior of SC-TOBFs, in terms of , , and , is jointly determined by and . This section deals with the coupling effects of these two parameters on SC-TOBFs based on the premise that is set as constant as possible for each lateral load distribution with different building heights.

##### 5.1. Base Shear Response

The coupling effects of and on the base shear response of SC-TOBFs with 9 and 16 stories are shown in Figures 24 and 25, respectively. As the values of and increase and decrease, respectively, the first-stage curves of the SC-TOBFs are more and more close to that of the BRBFs, indicating that by selecting the appropriate and , SC-TOBFs can exhibit a similar first-stage performance as BRBFs.

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##### 5.2. Interstory Drift Response

The coupling effects of and on the interstory drifts of SC-TOBFs with 9 and 16 stories are shown in Figures 26 and 27, respectively. As the values of and increase and decrease, respectively, the drifts of the low stories decrease gradually, while the drifts of high stories increase, similarly to the effects. Compared with BRBFs, SC-TOBFs can achieve a comparable deformation performance by selecting the appropriate and .

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#### 6. Conclusions

A parametric study on the seismic response of 9- and 16-story steel-framed buildings, which can be considered typical of intermediate and high-rise buildings, with SC-TOBs is numerically conducted, and the results are compared with those of BRBFs. The effects of the stiffness degradation factor , the activation strain , and the initial axial stiffness of the SC-TOBs on the lateral behavior of the SC-TOBFs are investigated thoroughly to explore the design space. The following conclusions can be drawn from this study:(1)An increase in leads to a larger second stiffness of the structure, , but changes in have no significant effect on the activation load or the activation deformation , of the structure. A relatively large is suggested to improve the distribution of drifts over the building height.(2)With the increase in , and increase gradually, but has no effect on the first stiffness of the structure , and , and only a slight effect on the interstory drift distribution.(3)With the increase in , and increase continuously, and the interstory drifts of the lower part of the building under the parabolic distribution of loads are reduced.(4)The coupling effects of an increasing and a decreasing are similar to the effects. It is advantageous to select a large and a small simultaneously to make the drift distribution more uniform.(5)The SC-TOBFs are considered to be a type of performance-tunable structure, and tuning can be achieved by varying a frame’s adjustable parameters. The first-stage lateral behavior of SC-TOBFs is comparable to that of BRBFs when a lower and a higher are selected, and a required second-stage behavior can be obtained by specifying a suitable .

#### Data Availability

The data used to support the findings of this study are available from the corresponding author upon request.

#### Conflicts of Interest

The authors declare that there are no conflicts of interest regarding the publication of this paper.

#### Acknowledgments

This research was financially supported by the National Natural Science Foundation of China (Grants numbers 51708482 and 51578478), the China Postdoctoral Science Foundation (Grant number 2017M621593), and the Six Talent Peaks Project in Jiangsu Province (Grant number JZ-035).