SCB can effectively reduce the residual displacement and maximum instantaneous displacement of SMF. However, it will also simultaneously amplify the acceleration response, leading to an increase in the seismic loss of the acceleration-sensitive non-structural components. Therefore, how to design SCB according to the seismic demand of the structure and make full use of SCB is of great significance. Generally, there are four key parameters describing the mechanical properties of SCB: initial elastic stiffness *K*SCB, post-activation stiffness coefficient *α*, activation strength *F*a,SCB, and energy dissipation coefficient *β*. The larger the value of *β*, the more the hysteretic energy dissipated by SCB, the smaller the acceleration and displacement responses of the structure are. Under the premise of ensuring the structure returns to the initial position, the larger the *β* is, the more favorable the structure. Therefore, this section will only focus on the effects of the initial elastic stiffness *K*SCB, the post-activation stiffness coefficient *α*, and the activation strength *F*a,SCB on the seismic responses of the structure.

## 5.1. Initial elastic stiffness

Based on the 8 equivalent SDOF models of SMF with different fundamental period established in Section 4.1, 9 different initial stiffnesses of the SCB are set and the other parameters of the models remain unchanged. That is, *K*SCB/*K*0 = [0.2, 0.4, 0.6, 0.8, 1.0, 1.2, 1.4, 1.6, 2.0]. Where, *K*0 is the initial stiffness of SMF and *K*SCB represents the initial stiffness of SCB system. A total of 72 equivalent SDOF models of the SCBF are thus obtained. Based on the 22 far-field ground motion records selected in this paper as the seismic input, the seismic responses of the SCBFs with different initial stiffness are obtained by 1584 NTHA. To make the results more intuitive, the seismic responses of the SCBFs are normalized by the seismic responses of the corresponding SMF. The specific normalized results are discussed below.

Normalized acceleration responses of the SCBFs are shown in Fig. 8a. All the values of normalized acceleration are greater than 1, indicating that the SCB system increases the acceleration response of the SMF. The shorter the fundamental period of the SMF, the more significantly the initial stiffness of the SCB system will increase the acceleration. For the SMF with the fundamental period of 0.2s, the SCB amplifies the acceleration by 50 ~ 69%; while for SMF with the fundamental period of 4.0s, the SCB only amplifies the acceleration by 1 ~ 2%. For SMF with the fundamental periods of 0.2s and 0.4s, the acceleration response of SCBF increases linearly with the increase of the initial stiffness of the SCB system. For the SMF with the fundamental period equal to or greater than 0.6s, the acceleration response increases rapidly first, and then the increase rate gradually becomes slower. Moreover, for SMF with the fundamental period greater than or equal to 2s, when *K*SCB/*K*0 ≥ 1.0, even if the initial stiffness of the SCB system continues to increase, the acceleration response of SCBF will not continue to increase.

The normalized displacement responses of SCBFs are shown in Fig. 8b. All values of normalized displacement are less than 1, indicating that the SCB system can effectively reduce the displacement response of the SMF. And the shorter the fundamental period of the SMF, the more significant the displacement reduction is achieved by SCB. For SMF with the fundamental period of 4s, the SCB reduces the structural displacement by 5%~21%, and for SMF with the fundamental period of 0.2s, the SCB reduces structural displacement by 33%~66%. As the initial stiffness of the SCB system increases, the displacement reduction effect increases gradually, but when *K*SCB>1.2*K*0, the rate of displacement reduction becomes increasingly stable.

The normalized residual displacement responses of SCBFs are shown in Fig. 8c. All values of the normalized residual displacement are much less than 1, indicating that the SCB system can effectively reduce the residual displacement of the SMF. And with the increase of the initial stiffness of the SCB, the residual displacement decreases gradually, but if *K*SCB/*K*0 > 0.6, the rate of the reduction of the residual displacement tends to be gentle.

The normalized hysteretic energy dissipated by frames of SCBF is shown in Fig. 8d. Numerous indicators are used in the damage evaluation of steel structures and energy-related damage indicators are common damage indicators (De Domenico and Hajirasouliha 2021; Rajeev and Wijesundara 2014), illustrating that the hysteretic energy of the steel structure can reflect the damage degree of the structure to a certain extent. Higher amounts of hysteresis energy indicate more serious damage of the steel structure. Figure 8d shows that with the increase of the initial stiffness of the SCB system, the hysteretic energy of the frame in SCBF decreases gradually, indicating that the damage to frames is gradually reduced. The shorter the fundamental period of the structure, the less the hysteretic energy of beams and columns of SCBF. Note that partial values of the normalized hysteretic energy in Fig. 8d are greater than 1. For example, if the fundamental period of SMF equals 4.0s and the *K*SCB/*K*0 ≤ 0.8, the normalized hysteretic energy values are greater than 1. This means that the damage to beams and columns of SCBF is greater than the damage to beams and columns of SMF. The main function of SCB in the SCBF is to control the seismic response and reduce the damage. Therefore, in the actual design, the SCB system should be designed reasonably so that the damage of the steel frame in SCBF will not exceed the original SMF.

Overall, a rise increase in the initial stiffness of SCB will increase the acceleration response and reduce the displacement and residual displacement response of the SCBF and the damage of the steel frame. Considering acceleration, displacement, residual displacement and damage together, SCB can boost the seismic performance of SMF when *K*SCB/*K*0 = 0.6 ~ 1.2, and the longer the fundamental period of the steel frame, the greater the initial stiffness of the SCB should be.

## 5.2. Post-activation stiffness coefficient

Based on the 8 equivalent SDOF models of SMF with different fundamental period established in Section 4.1, 6 different post-activation stiffness coefficients of SCB are set and the other parameters of the models remain unchanged. That is, *α*=[0.01, 0.02, 0.05, 0.1, 0.2, 0.3]. A total of 48 equivalent SDOF models of SCBF are thus obtained. Based on the 22 far-field ground motion records selected in this paper as the seismic input, the seismic responses of SCBFs with different post-activation stiffness coefficients are obtained by 1056 NTHA. The normalized seismic responses of the SCBFs are shown in Fig. 9.

Figure 9a illustrates that with the increase of the post-activation stiffness, the acceleration responses of SCBFs increase linearly, and the shorter the fundamental period, the more obvious the increase of acceleration. When the fundamental period of SMF is 0.2s, the amplification of the acceleration is between approximately 1.58 and 2.48, while if the fundamental period of SMF is 4.0s, the amplification of the acceleration is only between 1.01 and 1.05. As illustrated in Fig. 9b, the post-activation stiffness has little effect on the displacement response of the SCBF. With the increase of the post-activation stiffness, the displacement response of SCBF decreases slightly, and the longer the period, the smaller the influence. Figure 9c displays that all normalized residual displacement values are less than 0.1, indicating that the SCB can significantly reduce the residual deformation of the structure, and the residual displacement decreases slightly with the increase of the post-activation stiffness coefficient. As shown in Fig. 9d, with the increase of the post-activation stiffness coefficient, the hysteretic energy of the steel frame in SCBF increases linearly, except for the structure with the fundamental period of 0.2s. The longer the fundamental period, the more obviously the hysteresis energy increases. Therefore, in order to effectively control the damage of beams and columns, the post-activation stiffness coefficient should not be too large. For example, when the fundamental period of steel frame is 4.0s, the post-activation stiffness coefficient should not be greater than 0.05, and when the period of steel frame is 3.0s, the post-activation stiffness coefficient should not exceed 0.1.

Overall, the post-activation stiffness coefficient mainly affects acceleration response and the damage of beams and columns of the SCBF. The larger the post-activation stiffness coefficient, the larger the acceleration response, and the more serious the damage of the steel frame. And the post-activation stiffness has little effect on the displacement and residual displacement of the SCBF. Conclusively, the SCB can make the steel frame achieve a better performance when the value of the post-activation stiffness coefficient does not exceed 0.1.

## 5.3. Activation strength

Based on the 8 equivalent SDOF models of SMF with different fundamental period established in Section 4.1, 9 different activation strength of SCBs are set and the other parameters of the models remain unchanged. That is, *F*a,SCB/*F*y=[0.2, 0.4, 0.6, 0.8, 1.0, 1.2, 1.4, 1.6, 1.8]. Where, *F*y is the yield strength of SMF and *F*a,SCB represents the activation strength of SCB. A total of 72 equivalent SDOF models of the SCBFs are thus obtained. Based on the 22 far-field ground motions selected in this paper as the seismic input, the seismic responses of the SCBFs with different *F*a,SCB are obtained by 1584 NTHA. And the normalized seismic responses of the SCBFs are displayed in Fig. 10.

As illustrated in Fig. 10a, *F*a,SCB has a significant effect on the acceleration response of the SCBF. As *F*a,SCB increases, the acceleration response of the SCBF increases linearly. The shorter the fundamental period of the steel frame, the more significant the influence of the activation strength on the acceleration. When the fundamental period of the steel frame is 4.0s, the maximum magnification of acceleration is only about 1.04, and when the fundamental period is 0.2s, the magnification of acceleration reaches a range of 1.26 ~ 1.97. The greater the acceleration response of the structure, the larger the seismic loss of the acceleration-sensitive non-structural components. Therefore, from the perspective of controlling the acceleration, the activation strength of the SCB system should not be too high. Figure 10b displays that as the activation strength of the SCB increases, the displacement response of SCBF decreases rapidly, and the shorter the fundamental period of the steel frame, the more significant the influence of activation strength on displacement. When the fundamental period of the steel frame is 4.0s, the displacement response of SCBF is reduced by 12%~21%, and when the fundamental period of the steel frame is 0.2s, the displacement response of the SCBF is reduced by 31%~70%. Therefore, from the perspective of controlling the displacement response, the greater the activation strength of the SCB, the smaller the structural displacement response. Figure 10c shows that *F*a,SCB has a significant effect on the residual displacement of the SCBF. With the increase of the activation strength, the residual displacement of the SCBF decreases rapidly. And when *F*a,SCB/*F*y >0.8, the rate of decrease of the residual displacement gradually becomes gentle. Figure 10d indicates that when the fundamental period of the steel frame does not exceed 0.4 s, the damage of the steel frame in the SCBF gradually decreases with the increase of the activation strength, and all the damage of the steel frames is reduced compared to the damage of the original SMF. When the fundamental period of the steel frame is more than 0.4s, the damage of the steel frame decreases first and then rises with the increase of the activation strength. Particularly for the long-period structure, the damage of the steel frame even exceeds the damage of the original SMF. For example, when *F*a,SCB/*F*y >1.2 at the fundamental period of 4.0 s, the values of the normalized hysteresis energy of the steel frame in the SCBF exceed 1, indicating that the damage of the steel frame exceeds the damage of the corresponding SMF.

Overall, the activation strength has a significant effect on the acceleration, displacement, residual displacement of the SCBF and the damage of steel frame. As *F*a,SCB increases, the acceleration response increases gradually, and the displacement and residual displacement response gradually decrease. For short-period structures, the damage of the steel frame is gradually reduced, while for the medium- or long-period structures, the damage of the steel frame is first reduced and then increased. It can be concluded that, the SCB can enhance the seismic performance of the SMF when *F*a,SCB/*F*y = 0.6 ~ 1.2.