## 2 Ground Motions and Structures  
This research study examines a total of 274 recorded MSAS ground motions obtained from various global crustal seismic sequences [16]. The ground motions are sourcedfrom four prominent strong motion databases, namely the Pacific Earthquake Engineering Research Center (PEER) Next Generation Attenuation (NGA) relationships database, the European Strong-Motion dataset, the Center for Engineering Strong Motion Data (CESMD), and the China Earthquake Networks Center. To ensure consistency and comparability, the dataset used in this study excludes records from soft soil sites (i.e., average shear--wave velocity over the top 30 m of the site Vs30 is smaller than 180 m/s). Furthermore, only the strongest aftershock record is chosen from each MSAS sequence, based on the criterion of having the highest _PGA_. By using a diverse range of recorded MSAS ground motions from reputable databases, this study aims to provide a comprehensive analysis of the impact of scaling aftershock ground motions on the additional damage experienced by structures. The exclusion of soft soil sites and the selection of the strongest aftershock records contribute to a focused investigation and enhance the reliability and relevance of the findings. In this manuscript, we employ inelastic SDOF systems as a means to simulate the response of structures under the influence of ground motions. The period of the inelastic system spans from 0.1 to 3.0 s, with an increment of 0.1 s. A viscous damping ratio of 5% is presumed for the system. The lateral strength of the system is ascertained by means of the strength reduction factor \(R\), which is computed using the subsequent equation:  
\[R=\frac{m\cdot S_{\text{a, ms}}}{F_{\text{y}}} \tag{1}\]  
where \(m\) is the mass of the system, which is assumed to be 1.0 herein; \(S_{\text{a,ms}}\) is the spectral acceleration of the mainshock ground motion; and \(F_{\text{y}}\) is the yield strength of the inelastic SDOF system. To examine the impacts of various yield strengths, three different values of the strength reduction factor \(R\) (i.e., \(R\) = 2, 4, and 6) are taken into account in this study. Four distinct hysteretic systems are chosen for analysis: (1) the elastic-perfectly-plastic (EPP) system, (2) the modified Clough (MC) system, (3) the pinching (PH) system [50], and (4) the stiffness strength degradation (SSD) system [51]. The objective of employing these different hysteretic systems is to investigate the impact of stiffness degradation, pinching, and strength deterioration on the bias introduced by aftershock ground motion scaling. For a comprehensive understanding of the hysteretic systems and their parameter assumptions, please refer to the detailed description provided in Reference [16]. In this study, the modified Park-Ang damage index [51] is used to quantitatively assess the cumulative damage caused by the MSAS ground motions. This damage index is defined as follows:  
\[DI=\frac{x_{\text{m}}-x_{\text{y}}}{x_{\text{u}}-x_{\text{y}}}+\beta\frac{E_{ \text{H}}}{x_{\text{u}}F_{\text{y}}}=\frac{\mu-1}{\mu_{\text{u}}-1}+\beta \frac{E_{\text{H}}}{F_{\text{y}}\mu_{\text{u}}x_{\text{y}}} \tag{2}\]  
where \(\mu\) is the ductility demand; \(x_{\text{u}}\) is the ultimate deformation capacity of the structure under monotonic loadings; \(\mu_{\text{u}}\) is the corresponding ultimate ductility capacity of the structure under monotonic loadings, defined as the ratio of the structure's maximum deformation under monotonic loadings to its yield displacement, and given a value of 10 in this study; \(\beta\) is a positive dimensionless parameter to scale the effect of hysteretic energy dissipation on the final damage of the structure; and \(E_{\text{H}}\) is the hysteretic energy dissipation of structure under ground motion. The median value of \(\beta\) (i.e., \(\beta\) = 0.15) [52] is used in this study.