## 6 Effect of Hysteretic Models  
Two aspects of investigations are conducted on the effect of hysteretic models, including hysteretic model types and \(R\) levels. Figure 12 shows the calculated \(\rho\) values between the concerned four IM-DI pairs, including \(\text{ln}IM^{\text{MS}}\) vs. \(\text{ln}DI_{1}\text{{}^{\text{MS}-\text{AS}}}\), \(\text{ln}IM^{\text{AS}}\) vs. \(\text{ln}DI_{1}\text{{}^{\text{MS}-\text{AS}}}\), \(\text{ln}IM_{1}\text{{}^{\text{MS}-\text{AS}}}\) vs. \(\text{ln}DI_{1}\text{{}^{\text{MS}-\text{AS}}}\), and \(\text{ln}IM_{2}\text{{}^{\text{MS}-\text{AS}}}\) vs. \(\text{ln}DI_{2}\text{{}^{\text{MS}-\text{AS}}}\), using all the considered eight types of models with a constant \(R\) value of 2. As seen from this figure, the correlation between \(\text{ln}IM_{2}\text{{}^{\text{MS}-\text{AS}}}\) and \(\text{ln}DI_{2}\text{{}^{\text{MS}-\text{AS}}}\) is significantly higher than those of the other three IM-DI pairs regardless of the hysteretic model types. Among the acceleration-, velocity-, and displacement-related IMs, \(I_{A}\), \(v_{ts}\), and \(PGD\) are the most optimal IMs to formulate \(IM_{2}\text{{}^{\text{MS}-\text{AS}}}\), respectively, due to the associated high correlation with \(DI_{2}\text{{}^{\text{MS}-\text{AS}}}\). The observations stated above are consistent with those obtained in Figure 9. Moreover, varying hysteretic models does not cause significant differences in the obtained \(\rho\) values. Therefore, the effect of hysteretic model types on the obtained correlation results is limited and has no significant effect over the response for the current purposes.  
Figure 11: The optimal IMs for damage potential prediction of MS–AS sequences at the entire period range.  
Figure 10: Comparison of correlation between related to different IM categories.  
To examine the effect of strength reduction ratio, the systems simulated by the basic model with varying \(R\) values, i.e., \(R=2\)-\(5\), are adopted for illustration. Figure 13 compares the calculated \(\rho\) values between In\(IM^{\text{MS}}\) vs. In\(DI_{1}\)MS-AS, In\(IM^{\text{AS}}\) vs. In\(DI_{1}\)MS-AS, In\(IM_{1}\)MS-AS vs. In\(DI_{1}\)MS-AS, and In\(IM_{2}\)MS-AS vs. In\(DI_{2}\)MS-AS. It is clear that varying \(R\) values does not change the observations obtained in Figures 10 and 11, including: (1) In\(IM_{2}\)MS-AS shows better performance than \(IM^{\text{MS}}\), \(IM^{\text{AS}}\), and \(IM_{1}\)MS-AS due to its higher correlation with \(DI_{2}\)MS-AS; and (2) \(I_{A}\), \(v_{ts}\), and \(PGD\) are the most optimal IMs to formulate \(IM_{2}\)MS-AS among the acceleration-related, velocity-related and displacement-related IMs, respectively.  
Figure 14 illustrates the effect of changing \(R\) on the \(\rho\) values between \(I_{A,2}\), \({}^{\text{MS-AS}}\) and \(DI_{2}\), \(v_{rs,2}\), \({}^{\text{MS-AS}}\) and \(DI_{2}\), \({}^{\text{MS-AS}}\), and \(PGD_{2}\), \({}^{\text{MS-AS}}\) and \(DI_{2}\), \({}^{\text{MS-AS}}\) at the entire period range. As observed, the change of \(R\) values yields a slightly larger difference on the \(\rho\) values than that of varying hysteretic models, especially for the two IM-DI pairs in terms of \(v_{rs,2}\) and \(DI_{2}\), \({}^{\text{MS-AS}}\), and \(PGD_{2}\), \({}^{\text{MS-AS}}\) and \(DI_{2}\), \({}^{\text{MS-AS}}\). This is because a system with larger \(R\) value is easier to suffer nonlinear behavior under a MS excitation, leading to a more complex effect of the AS on the structural damage.