## 4 SDOF Systems  
A total of eight different hysteretic models are considered herein to simulate the nonlinear behavior of SDOF systems under MS-AS sequences. Similar SDOF model assumptions and considerations have been used in [41]. Figure 4 shows the hysteretic rules of the considered models, where the force and displacement are both normalized by the yield force and yield displacement, respectively. A polygonal type of hysteretic model is adopted in this study for illustration. Compared to the smooth type of hysteretic models (e.g., Bouc-Wen models [55; 56; 57; 58], which have the advantage of using a set of differential or algebraic equations), the polygonal type of hysteretic model requires less parameters, which can also include both strength and stiffness deterioration. The SDOF systems partly or totally w/wo considering \(P\)-\(\Delta\) effect, pinching, and damage-induced deterioration are used to account for the varying hysteretic behaviors of the SDOF system. Table 4 provides the considered eight SDOF models, where Delta is the parameter controlling the \(P\)-\(\Delta\) effect; PinchX and PinchY are the pinching factors for strain (or deformation) and stress (or force) during reloading, respectively; and Damage1 and Damage2 denote the damage due to ductility and energy, respectively, which are used to account for the deterioration due to damage.  
Among the considered models, the basic model, which is numbered as Model-1, is an elastic-plastic-with-hardening model (see Figure 4a). Based on the basic model, three important factors, including the \(P\)-\(\Delta\) effect, pinching effect, and unloading degradation, are partly or totally incorporated in Models 2-8 to reflect more complex nonlinear behaviors of structures under seismic loading. In particular, Models 2-4 represent the bilinear hysteretic models by individually considering the \(P\)-\(\Delta\) effect, pinching effect, and unloading degradation, respectively (see Figure 4b-d). For Models 5-7, two of the above-stated influence factors are accounted for in the hysteretic behavior, as shown in Figure 4e-g, respectively. Besides, Model-8 has the most complex hysteretic behavior since it collectively incorporates the \(P\)-\(\Delta\) effect, pinching effect, and unloading degradation (see Figure 4h). It is noteworthy that the considered Models 2-8 are used for parametric study to examine the effect of hysteretic models of SDOF systems on the obtained results.  
Figure 4: \(Cont.\)A total of 44 fundamental vibration periods ranging from 0.2 s to 6.0 s are considered for each SDOF system, consisting of 29 periods between 0.2 s and 3.0 s (with an increment of 0.1 s), and 15 periods between 3.2 s and 6.0 s (with an interval of 0.2 s). Four yield strength reduction factors, i.e., \(R\) = 2, 3, 4, and 5, are considered to account for the varying ductility levels of structures. Moreover, a constant damping coefficient \(\xi\) = 0.05 is utilized for all the considered SDOF systems according the past available studies, such as [15; 21]. It is noteworthy that the analysis process of this paper is also suitable for the cases with other \(\xi\) values. However, the discussion on the effect of damping ratios is beyond this study and will be discussed in the future work. Consequently, a total of 1408 (8 hysteretic models \(\times\) 44 vibration periods \(\times\) 4 ductility levels) SDOF systems are used in subsequent calculations.