## 5 Conclusions  
This manuscript assesses the impact of ground motion scaling on the additional damage caused by aftershocks. The evaluation is conducted by quantifying the bias introduced through the scaling of aftershock ground motions. The research employs a variety of single-degree-of-freedom (SDOF) systems that display different types of hysteretic behaviors, including stiffness degradation, pinching, and strength deterioration. Additionally, a dataset of 274 recorded mainshock-aftershock (MSAS) ground motions obtained from global crustal earthquakes is employed to simulate the structural damage under these MSAS ground motions. Eight different intensity measures, including acceleration-related (i.e., \(PGA\), \(I_{\rm A}\), and \(I_{\rm C}\)), velocity-related (i.e., \(PGV\)), displacement-related (i.e., \(d_{\rm rms}\) and \(PGD\)),  
Figure 10: Potential bias arising from the scaling of ground motion for systems operating within the entire period range.  
and spectra-related (i.e., \(S_{\rm a}\) and \(SI\)), are selected as alternative scale indicators for aftershock records. The primary discoveries can be summarized as follows:  
1. For systems in the short-period region, using \(\
abla S_{\rm a}\) as the scale indicator would introduce the biases within 30% for a scale factor as large as 10.0, while the corresponding bias increases to 50% when \(\
abla PGA\) is used as a scale indicator.
2. For systems in the medium-period region, the biases are generally within 20% even for a scale factor of 10.0, when \(\
abla SI\) and \(\
abla I_{\rm C}\) are used as the scaled indicators, and corresponding biases for \(\
abla PGV\) and \(\
abla S_{\rm a}\) are generally within 30%. The superiority of \(\
abla I_{\rm C}\) as the scale indicator can be explained by two points: the first is that the dimension of \(I_{\rm C}\) is between acceleration and velocity, and the second is that the computation of \(I_{\rm C}\) incorporates the effects of duration.
3. In the long-period region, the biases resulting from the scaling of aftershock ground motion with parameters such as \(\
abla SI\), \(\
abla PGV\), and \(\
abla I_{\rm C}\) are typically below 30% when the scale factor is less than 10.0. However, the bias associated with the parameter \(\
abla S_{\rm a}\) increases to 40%, which is still lower than the biases observed for displacement-related intensity measures (e.g., \(\
abla PGD\)).
4. After studying the mean bias in the whole-period region, it can be found that the performance of \(\
abla SI\), \(\
abla PGV\), \(\
abla I_{\rm C}\), and \(\
abla S_{\rm a}\) is superior to the other four scale indicators, and they can control the mean bias within 30% even for scale factors as large as 10.0. Aftershock ground motion scaling tends to introduce higher bias in the additional damage of EPP systems than degrading systems.
5. Scaling aftershock ground motions up (i.e., \(SF>1.0\)) using \(\
abla S_{\rm a}\) tends to overestimate the additional damage to structures.  
This study has some limitations that need to be further addressed in future research. Several SDOF systems were employed in this study to assess the impact of scaling ground motion on additional damage. The investigation needs to be expanded to multi-degree-of-freedom systems for more practical applications.  
Conceptualization, B.Z.; Software, B.Z.; Formal analysis, J.H.; Investigation, C.Y.; Resources, B.Z. and Q.K.; Data curation, J.H. and Q.K.; Writing--original draft, B.Z.; Writing--review & editing, J.H. and W.W.; Visualization, C.Y.; Supervision, W.W. All authors have read and agreed to the published version of the manuscript.  
This investigation was supported by Shanghai Municipal Science and Technology Commission Project (Grant No. 22 dz1200200 and 22 dz1201400) and Shanghai Municipal Finance Project (Grant No. 090301). This support is greatly appreciated.  
Not applicable.  
Data Availability Statement: Data are contained within the article.  
The authors declare no conflict of interest.