## 1 Introduction  
Past earthquake events have evidenced that strong aftershocks may follow the mainshocks. On 12 May 2008, an M-7.9 earthquake struck Wenchuan, China, then strong aftershocks followed the mainshock, eight of which range from M 6.0 to 6.5 [1, 2]. An M-7.8 earthquake occurred in Nepal on 25 April 2015, and hundreds of aftershocks were recorded, including an M 7.3 aftershock [3] within a month after the mainshock. Strong aftershocks may also follow a mainshock within a very shorter period. On 26 October 2015, an M-7.5 earthquake occurred in Afghanistan, and an M-4.8 aftershock was recorded only 40 min after the mainshock [4]. These aftershocks have a high potential to lead to more severe damage to civil structures, especially if the structures are damaged or sustain residual deformation during the mainshock. It is indeed meaningful to investigate the effects of mainshock-aftershock (M-A) events on the probability of failure of engineering structures, including buildings, bridges, offshore platforms, or other types of structures. Abundant research records are found that focus on the behavior of building, bridge, or offshore platforms structures under M-A sequences [5; 6; 7; 8; 9; 10]. [PERSON] et al. evaluated the seismic risk of mainshock-damaged bridges further subjected to aftershocks, in which the seismic risk of aftershock to bridges was estimated based on fragility modeling using the Omori-law-based hazard model for the aftershocks [5]. In reference [6], seismic fragility analysis was conducted by scaling the mainshocks to reach a designated damage state; with introducing five seconds of rest time, the aftershock was applied on the damaged columns. [PERSON] et al. studied the behavior of offshore platforms under seismic sequences and pointed out that M-A sequences cause noteworthy cumulative damage to platforms [10]. In reference [11], the inelastic seismic demands of wood-frame houses were estimated under seismic sequences. Their findings indicated that aftershocks resulted in the median of the inelastic seismic demands increasing about 5-20% if a moderate degree of structural response is induced. [PERSON] et al. studied the probability of collapse of steel structures under seismic sequences. They pointed out that the collapse capacity of a structure may decrease significantly when the structure is subject to a very strong mainshock, and thus that a small aftershock may induce the collapse of the damaged structure [12]. In their work, the collapse probability of differently damaged structures under aftershocks was investigated; however, their findings cannot reflect the effects of mainshock intensity on the probability of collapse. In reference [13], the seismic performance of a reinforced concrete containment (RCC) building under M-A sequences was investigated, and the results implied that aftershocks have great effects on the maximum peak accelerations, maximum peak displacements, and accumulated damage of RCC. [PERSON] et al. studied the seismic risk of old non-ductile reinforced-concrete (RC) framed buildings before and after retrofitting with isolation devices under M-A sequences. Their findings revealed that the seismic risk at higher damage levels can be reduced significantly by using isolation devices [14]. By neglecting aftershocks, it can cause considerable underestimation of the combined seismic risk. [PERSON] and [PERSON] pointed out that aftershock is one of the important contributors to the total seismic losses of light-frame wood buildings under earthquake sequences [15]. [PERSON] and [PERSON] found that the amplitude and frequency contents of M-A sequences have a significant influence on the structural damage pattern of steel moment-resisting building frames with welded moment-resistant connections [16]. [PERSON] and [PERSON] demonstrated that the frequency contents between mainshock and aftershock are weakly correlated [17].