## 1 Introduction  
Mainshock-aftershock (MSAS) sequences, which involve subsequent destructive aftershocks, have been observed in numerous historical earthquakes [1, 2, 3, 4]. For example, notable MSAS sequences occurred during the 2008 M7.9 Wenchuan earthquake in Sichuan, China, and the 2010 M7.0 Darfield earthquake in New Zealand. In these cases, the time intervals between the mainshock and the subsequent destructive aftershocks were relatively short, leaving insufficient time for the damaged structures to undergo repairs before being subjected to further aftershocks. As a result, the damaged structures tend to experience more severe damage during the aftershocks. The current seismic codes employed worldwide do not adequately account for the effects of aftershocks. Consequently, structures designed according to these codes may experience unacceptable levels of damage after an earthquake sequence. Since the current seismic codes do not consider the influence of aftershocks, they fail to address the increased vulnerability of structures to subsequent events in MSAS sequences. This highlights the need for improved understanding and incorporation of aftershock effects in seismic design practices to ensure more resilient and safe structures. The impact of MSAS on the seismic behavior of structures has garnered significant attention in recent years [2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47]. The objective of these studies is to quantify the additional damage caused by aftershocks and propose methods to incorporate their effects into performance evaluation and seismic design. Numerous investigations [2; 3; 4; 5; 6; 7; 8; 9; 10; 11; 12; 13; 14; 15; 16; 17; 18] have focused on the inelastic response spectra of MSAS sequences, aiming to develop a straightforward tool for integrating aftershocks into seismic design or conducting rapid performance assessments. Additionally, many studies [19; 20; 21; 22; 23; 24; 25; 26; 27; 28; 29; 30; 31; 32; 33; 34; 35; 36; 37; 38; 39; 40; 41; 42; 43; 44; 45; 46; 47] have used multiple-degree-of-freedom (MDOF) structures to simulate the damage progression of structures under MSAS sequences and evaluate the vulnerability and economic losses induced by such sequences. One of the key considerations regarding the impact of aftershocks is the production of MSAS ground motions. Given the limited availability of recorded MSAS ground motions, numerous studies have resorted to generating artificial MSAS ground motions [5; 6; 7; 8; 9; 10; 11; 12; 13; 14; 15; 16; 17; 18; 19; 20; 21; 22; 23; 24; 25; 26; 27; 28; 29; 30; 31; 33; 34; 42; 43]. This involves the random selection of aftershock seed and appropriate amplitude scaling. Additionally, several studies have used recorded MSAS ground motions [9; 10; 11; 12; 13; 14; 15; 16; 27; 28; 36; 44; 45; 46; 47]. However, regardless of whether artificial or recorded MSAS ground motions were employed in these studies, the aftershock ground motions were generally scaled to a specific intensity level. For example, [PERSON] and [PERSON] [5] and [PERSON] [6] scaled the peak ground acceleration (_PGA_) of the aftershock record to 0.8526 or 0.7767 times the _PGA_ of its mainshock record. [PERSON] et al. [7] scaled the _PGA_ of aftershock records to different levels (from 0.047 \(g\) to 0.2 _g_) based on the prediction results of the ground motion prediction equation. The relative _PGA_ of the aftershock record (defined as the ratio of _PGA_ of the aftershock record to that of the mainshock record) was scaled from 0.5 to 1.5 to study the effects of aftershocks [12; 13], while the range was further broadened (i.e., from about 0.17 to 6.0) in Ref. [14]. Similarly, the relative spectral acceleration (\(S_{\mathrm{a}}\)) of the aftershock record (defined as the ratio of \(S_{\mathrm{a}}\) of the aftershock record to that of the mainshock record) was scaled from 0.5 to 1.0 in Refs. [11; 16; 17; 41].