## 2 Incremental Dynamic Analysis Method Based on Double-Index  
The IDA method amplifies the same seismic wave in a certain proportion and inputs the ground motion into the building structure step by step. By obtaining the maximum response curve of the structure under different amplifying proportion coefficients of seismic wave, the IDA curve is drawn. Considering the diversity of ground motions, multiple ground motions are generally used for IDA of structures to reduce the discreteness and draw IDA curve clusters. The selection of ground motion Intensity Measure (IM) and structural Damage Measure (DM) is an important part of the IDA. The IM mostly selects peak ground acceleration (PGA), peak ground velocity (PGV), and elastic acceleration response spectrum \(S_{a}(T_{1},5\%)\). Recently, extensive work has been carried out to propose more suitable IMs for different structures, such as vector-valued IMs and advanced scalar IMs [40; 41], which represent broader period ranges of structures [42; 43; 44]. The DM, also called the structural performance parameter, usually selects maximum inter-story drift ratio, node rotation, and maximum base shear. According to different structures and research fields, different and multiple indices can be selected. For specific selection methods, please refer to [PERSON] and [PERSON] [14]. The double-index IDA method requires two DMs to be selected based on one IM. The selected ground motion records are with the amplitude modulation, which are input into the established structural model in the form of time-history curves, and the nonlinear dynamic analysis is performed. Under the same IM, two DM values are extracted (in this paper, the maximum inter-story drift ratio and residual inter-story drift ratio were selected), and a number of data points were marked on the coordinate axis with DM as the abscissa and IM as the ordinate. The logarithmic data points were simulated by the cubic Bezier interpolation function, and IDA curve clusters of two types of DMs were drawn. Under the same IM, the two DMs are assumed to be lognormal in two dimensions. The specific steps are shown in Figure 1. After selecting two DMs, IDA was performed for the two DMs and two IDA curve clusters were obtained. Then, the DMs of the two structures under the same ground motion intensity were extracted, and the correlation between the two sets of data was considered for analysis. Two DM values can be output simultaneously under the same amplitude modulation coefficient, so the computational efficiency of the double-index IDA method is not affected. In order to obtain a smooth IDA curve, the cubic Bezier interpolation function is used to simulate the data points composed of IM and DM, and the approximate formula is used to obtain the function values at different points on the curve. The approximate formula of the cubic Bessel interpolation is as follows [45]:  
\[F(t)=(1-t)^{3}P_{1}+3t(1-t)^{2}P_{2}+3t^{2}(1-t)P_{3}+t^{3}P_{4}, \tag{1}\]  
where \(F(t)\) is the expression of the curve between every two points, and the value range of \(t\) is \([0,1]\). \(P_{1}\), \(P_{2}\), \(P_{3}\), and \(P_{4}\) are the four control points of the curve. When drawing the quantile value curve, the DM statistics method is adopted. According to the principle of statistics, it is assumed that IM presents a lognormal distribution of the conditional probability when DM = \(x\), which can be expressed as follows:  
\[\ln(\,\mathrm{IM}|\mathrm{DM}\,=x)\,\sim\,\mathrm{N}\Big{(}\mu,\sigma^{2}\Big{)}, \tag{2}\]  
Figure 1: Steps of IDA method based on double index. where \(x\) represents a number that a DM can get, \(\mu=\ln_{\eta\text{IM}|\text{DM}=x}\) is the mean value of the normal distribution, and \(\sigma=\beta_{\text{IM}|\text{DM}=x}\) is the standard deviation of the normal distribution.