## 4 Selection of Seismic _IMs_ for the PCA Algorithm
### Removal of Weakly Correlated IMs  
The PCA algorithm is ideal for reducing the dimensionality of variables with a strong correlation; however, the PCA algorithm is unlikely to achieve satisfactory dimensionality reduction results when the absolute value of the correlation coefficient between variables is less than 0.3. Consequently, this study will remove the weakly correlated _IMs_, although the correlation coefficient still represents the correlation between two _IMs_. However, at present, there are no clear evaluation criteria by which to judge which of the two _IMs_ should be removed. Therefore, the correlation between the _IMs_ and the maximum cross-section curvature values of high piers under seismic loading is used as the evaluation standard in this study. Based on the evaluation standard, the _IM_ that has less correlation coefficient value with the maximum cross-sectional curvature values among the two weakly correlated _IMs_ would be removed. In this section, the key cross-sections selected for analysis include the bottom cross-sections of Pier 1# and Pier 2# (cross-sections 1-1 and 6-6), the one-quarter-height cross-sections of Pier 1# and Pier 2# (cross-sections 2-2 and 7-7), the mid-height cross-sections of Pier 1# and Pier 2# (cross-sections 3-3 and 8-8), the three-quarter-height cross-sections of Pier 1# and Pier 2# (cross-sections 4-4 and 9-9), and the top cross-sections of Pier 1# and Pier 2# (cross-sections 5-5 and 10-10), as shown in Figure 3. This study will then conduct nonlinear dynamic time-history analyses (NDTHA) for a five-span continuous rigid-frame bridge (as shown in Section 2.2) under the influence of the following four pulse-like ground motion excitation calculation cases:  
1. A longitudinal earthquake;
2. A transverse earthquake;
3. A biaxial earthquake (longitudinal direction + transverse direction);
4. A triaxial earthquake (longitudinal direction + transverse direction + vertical direction). According to the GB 50011-2010 code [31], the PGA of triaxial ground motions is regulated in the following ratio: longitudinal direction:transverse direction:vertical direction = 1:0.85:0.65. By recording the maximum cross-sectional curvature values of each key cross-section for each calculated case, this study calculates the Spearman correlation coefficient values between each _IM_ of each pulse-like ground motion (selected in Section 2.1) and the maximum cross-sectional curvature of each key cross-section, using Equation (9). Furthermore, upon analyzing the correlation coefficient values between the _IM_s and the key cross-section curvature values of high piers, it was found that the correlation coefficient values corresponding to the same _IM_ were inconsistent, and some of the correlation coefficient values were even dispersed to a large extent, as shown in Figure 7. Due to space limitations, Figure 7 only represents the correlation coefficient values between 5 _IMs_, comprising PGA, PGV, PGD, PGV/PGA, and PGD/PGV, and the cross-sectional curvature values of Pier 1# and Pier 2#:  
\[\rho=1-\frac{6\sum\limits_{k=1}^{n}\left(IM_{i,k}-\varphi_{l,m,k}\right)^{2}}{ n(n^{2}-1)} \tag{9}\]  
where \(\varphi_{l,m,k}\) is the cross-sectional curvature value of the \(l\)-th \((l=1,2,\ldots,10)\) key cross-section under the influence of \(k\)-th \((k=1,2,\ldots,100)\) of ground motion of the \(m\)-th \((m=1,2,3,4)\) calculated case. It can be seen from Figure 7 that the mean value of the correlation coefficients corresponding to PGV is the largest and the discreteness is the smallest, yet the mean values of the correlation coefficients of the other 4 _IMs_ are all smaller than PGV, and the dispersion is relatively large. This is especially applicable to the mean value of the correlation coefficients corresponding to PGA, which is commonly used in seismic fragility analysis. It is the smallest value among the 5 _IMs_ and the discreteness of the correlation coefficient values is the largest.