## 3 Methodology  
This study aims to quantify the impact of ground motion scaling on additional damage by assessing the bias introduced through the scaling of aftershock records. The methodology for computing this bias is outlined in this section. To quantify the bias, the additional damage induced by aftershocks is compared between the original unscaled aftershock records and the scaled aftershock records. The variation in additional damage, quantified by \(\
abla DI\) in this study, versus relative intensity \(\
abla IM\) of the aftershock record is firstly studied. Figure 1 shows the variation of \(\
abla DI\) versus \(\
abla IM\) for a system with \(T\) = 1.0 s and \(R\) = 4. The results for \(\
abla S_{\rm a}\) and \(\
abla SI\) are both shown in Figure 1. It can be seen that there is no additional damage (i.e., \(\
abla DI\) = 1.0) when the intensity of the aftershock record is small (e.g., \(\
abla IM\leq 0.3\)). Then the additional damage gradually increases with the increase in \(\
abla IM\). Based on the aforementioned observations, a theoretical model is proposed to elucidate the correlation between the extent of additional damage and the relative intensity of aftershocks:  
\[\lambda(\
abla DI|\
abla IM)= \begin{cases}1.0&\
abla IM\leq\
abla IM_{1}\\ 1.0\cdot\left(\frac{\
abla IM}{\
abla IM_{1}}\right)^{b}&\
abla IM>\
abla IM_{1 }\end{cases}  \tag{5}\]  
where \(\lambda(\
abla DI|\
abla IM)\) is the median additional damage on the condition of \(\
abla IM\); and \(\
abla IM_{1}\) is the intensity threshold, beyond which aftershocks begin to induce the additional damage. Equation (5) is first fitted for the unscaled database (i.e., benchmark database in this manuscript), and the regressed model is used as the benchmark model. Figure 1 also shows the fitted model (i.e., the benchmark model represented by the red straight line) for the system with \(T\) = 1.0 s and \(R\) = 4. Then each aftershock record in the benchmark database is scaled up (i.e., \(SF\) > 1.0) gradually with a series of factors from 1.5 to 10.0, and the corresponding values of \(\
abla DI\) are computed with the scaled ground motions. The scaled aftershock ground motions with \(SF\), whose \(\
abla IM\) is below the scale threshold (determined from benchmark database), are used to regress the model. In this study, the primary emphasis is placed on the case of scaling up the aftershock records. This is because in previous studies, the scaling of aftershock records has typically been performed to investigate the additional damage experienced by mainshock-damaged structures. Aftershock records with small intensities that do not induce additional damage are considered less critical for engineering structures and are therefore not the main focus of this research. Therefore, we scale up each aftershock record, i.e., \(SF\) > 1.0. The disparities between the anticipated \(\
abla DI\) value of the benchmark model and the model created from scaled ground motions (referred to as the \"scaled model\" hereafter) indicate the bias introduced by scaling aftershock ground motions in terms of the median. The bias is computed as:  
\[B=\frac{1}{n}\sum_{i=1}^{n}\left|1-\frac{\hat{\lambda}_{i,\ \rm scaled}|SF_{i}}{\hat{ \lambda}_{i,\ \rm unscaled}}\right| \tag{6}\]where \(B\) is the bias introduced by aftershock ground motion scaling with an given scale factor \(SFj\); \(n\) is the number of scaled aftershock ground motions, whose \(\
abla IM\) is below the scale threshold; \(\hat{\lambda}_{i,\text{ unscaled}}\) is the median \(\
abla DI\) predicted by the benchmark model, while \(\hat{\lambda}_{i,\text{ scaled}|SFj}\) is the median \(\
abla DI\) predicted by the scaled model with the given scale factor \(SFj\). The flowchart in Figure 2 illustrates the process of calculating the bias as defined in Equation (6). Figure 1: Variation in additional damage versus relative intensity of the aftershock record for a system with \(T=1.0\) s and \(R=4\): (**a**) \(\
abla S_{\text{a}}\); (**b**) \(\
abla SI\).