## 3 Case Study
### IGM Selection for the NLRHA of RC Shear Wall Structures  
To investigate the capability of the MGMS on selecting more proper IGMs for the NLRHA of RC shear wall structures, two spectra-matching-based IGM selection methods were employed to select IGMs from the seed motion sets. These two IGM selection procedures are uniformly weighted closest spectra matching procedure (UW-CM), which is widely used in the research and engineering practice, and variably weighted closest spectra matching procedure (VW-CM), which was recently developed in [28]. The UW-CM selects seven IGMs that have the lowest MSE, for which the weight value is 1.0 for all the period points within the period range of 0.2 TI and 2 TI, between the response spectra of the IGMs and target spectrum. The VW-CM method also selects seven IGMs with the lowest MSE. Still, the weight value for different periods varies to take into account the contribution of \begin{table}
\begin{tabular}{c c c c c} \hline \hline \multirow{2}{*}{**Motion Set**} & **Target** & \multirow{2}{*}{**Scaling Factor**} & **The Period Range for** & **Structures** \\  & **Spectrum** & & & **Minimizing the MSE** & **Adopted for** \\ \hline
1 & DS 1 & 1.0–5.0 & 1.0 s–5.0 s & W1, W2, W3 \\
2 & DS 2 & 1.0–5.0 & 1.0 s–5.0 s & W1, W2, W3 \\
3 & W1 CMS 1 & 3.0–5.0 & 0.125 s–1.250 s & W1 \\
4 & W1 CMS 2 & 3.0–5.0 & 0.125 s–1.250 s & W1 \\
5 & W2 CMS 1 & 3.0–5.0 & 0.366 s–3.660 s & W2 \\
6 & W2 CMS 2 & 3.0–5.0 & 0.366 s–3.660 s & W2 \\
7 & W3 CMS 1 & 3.0–5.0 & 0.478 s–4.781 s & W3 \\
8 & W3 CMS 2 & 3.0–5.0 & 0.478 s–4.781 s & W3 \\ \hline \hline \end{tabular}
\end{table} 
Table 4: IGMs scaling criteria. Figure 2: Target spectra for the IGM selection and scaling. the different modes to the structural seismic responses. The VW-CM method calculates the MSE of motion \(j\) with the following equation:  
\[MSE_{j}^{VW}=\sum_{i=1}^{n}\left[\frac{a_{i}\sum_{T=0.5(T_{i+1}+T_{i})or1.5T_{i} }^{0.5(T_{i+1}+T_{i})or1.5T_{i}}\left(S_{a_{j}}(T)-S_{a}^{t}(T)\right)^{2}}{ \sum_{i=1}^{n}\alpha_{i}}\right] \tag{11}\]  
where \(S_{aj}(T)\) and \(S_{a}^{t}(T)\) are the spectrum acceleration value of motion \(j\) and target spectrum, respectively; \(T_{i}\) is period of ith mode; \(\alpha_{i}\) is the modal mass coefficient of \(i\)th mode; \(n\) is the number of modes that ensure the sum of modal mass coefficients is not less than 90%. Since the whole set of seed IGMs have similar response spectra and the broadest range of FCCTD, the mean seismic demand of NLRHA with the entire set of seed IGMs is the most reliable seismic demand. Thus, in this study, the mean seismic demand computed with all the seed IGMs of the set was employed as the reference seismic demand, and the average spectrum of IGMs of the whole motion set was adopted as the target spectrum, when adopting CM procedures to select IGMs from each seed motion set. The mean seismic demands from NLRHA with IGMs selected with different IGM selection methods were compared to investigate the rationality of the selected IGMs. NLRHA with motion sets 1 and 2 were conducted for all the shear wall structures. IGMs of sets 3-8 were adopted for the NLRHA of wall structures whose fundamental mode period is the same as the conditioning period of the target CMS of the set. Table 4 summarizes the motion sets for the NLRHA of different wall structures. It is known from the selection process that the MGMS procedure ensures that the response spectra of the selected IGMs match well with the mean spectrum of the set, and the chosen IGMs can cause the most significant interaction of the first two dominated modes. Although both UW-CM and VW-CM procedures ensure that the response spectra of the selected IGMs match best with the mean spectrum of the set, the VW-CM considers the different contributions of modes to the seismic response of structures.