## 2 Selection of IMs and DMs  
The reasonable selection of IMs and DMs can significantly enhance the accuracy of IDA-based evaluations of seismic performance in structural engineering. Generally, IMs are categorized into scalar and vector types [15]. Scalar-type IMs are expressed using a single parameter, such as PGA, peak ground velocity (PGV), and spectral acceleration (\(S_{\text{a}}\)). Vector-type IMs are defined using two or more parameters, mitigating the risk of inaccuracies resulting from the limitations of a single parameter. In the context of this study, we considered 32 IMs, comprising 25 scalar and 7 vector types, which span various subcategories like acceleration, velocity, displacement, and spectral types [16; 17]. Detailed specifications for these parameters are provided in Appendix A.  
Currently, the division of seismic performance levels for underground structures is predominantly determined by the maximum inter-story drift ratio (\(\theta_{\text{max}}\)) [18], adhering to criteria established for above-ground structures. However, this approach may not be fully appropriate due to the different load-transfer mechanisms that are a function of the surrounding soil in underground constructions. Therefore, this paper broadens the scope of DMs to encompass three dimensions: force, deformation, and energy. Specifically, we considered 7 DMs at both the structural layer and key component layer (such as the central column) to provide a more comprehensive understanding of the dynamic response characteristics of underground structures. In this respect, \(\theta_{\text{max}}\), energy ratio \(R_{\text{E}}\), and two-parameter \(D_{\text{e}}\) are utilized as DMs at the structural layer. Meanwhile, column end displacement ratio \(\theta_{\text{c}}\), bending moment \(M_{\text{C}}\), shear force \(F_{\text{SC}}\), and axial force \(F_{\text{NC}}\) at the bottom of the central column are employed as DMs at the component layer.  
The energy ratio \(R_{\text{E}}\) serves as a useful metric, representing the ratio of energy dissipated through structural plastic damage to the total energy input to the structure during seismic activity. The formal definition for this measure is as follows:  
\[R_{E}=\frac{E_{p}^{e}}{\sum E_{n}} \tag{1}\]  
where \(E_{p}^{e}\) represents the structural plastic damage energy dissipation; and \(\sum E_{n}\) denotes the total input energy of the structure, including structural compression damage energy dissipation, structural plastic damage energy dissipation, and structural strain damage energy dissipation.  
Furthermore, a two-parameter DM offers the advantage of accounting for the synergistic effects of both structural deformation and energy. In 1985, [PERSON] et al. [19] conducted an analysis on post-earthquake concrete column and beam elements, introducing a composite DM that combined deformation and hysteretic energy dissipation. This concept has since gained widespread acceptance. Subsequently, [PERSON] et al. [20] and [PERSON] et al. [21] extended the two-parameter DM to masonry and steel structures, respectively. In the current study, we adapt this two-parameter concept to the seismic performance evaluations for underground structures. Specifically, we propose a linear combination measure based on inter-story displacement and \(R_{\text{E}}\), aiming to assess its suitability for such structures. The definition for this measure is provided as follows:  
\[D_{\text{e}}=\frac{X_{\text{m}}}{X_{\text{u}}}+\frac{E_{\text{p}}^{\text{e}}} {\sum E_{n}} \tag{2}\]  
where \(X_{\text{m}}\) stands for the maximum inter-story displacement of the structure; \(X_{\text{u}}\) signifies the maximum inter-story displacement corresponding to the collapse limit state of the structure, which can be obtained from reference [1], that is, the corresponding \(X_{\text{u}}\) is 0.0825 m.