## 2 Modal-Based Ground Motion Selection Procedure for RC Shear Wall Structures
### Proposed Modal-Based Ground Motion Selection Procedure for RC Shear Wall Structures  
The modal mass coefficient \(\alpha_{n}\), which can be calculated using Equation (4), is the primary factor that affects the number of modes considered in the MGMS procedure. \[\alpha_{n}=\frac{m_{eff}}{\sum_{l=1}^{N}m_{l}}=\frac{1}{\sum_{l=1}^{N}m_{l}} \frac{L_{n}^{2}}{M_{n}} \tag{4}\]  
For most structures with symmetric plane arrangements, the sum of modal mass coefficients of the first three modes is higher than 90% [29]. Thus, the MGMS included the first three modes to account for the effect of IGMs on the multi-mode interaction. At the same time, it is found that for RC shear wall structures with a symmetric plan, the modal mass coefficients of the first two modes are larger than the sum of the modal mass coefficients of the higher-order modes [31,34]. At the same time, the modal mass coefficients of other higher modes are smaller than the sum of the modal mass coefficientsof the higher-order modes. The relations of the modal mass coefficients of RC shear wall structures with a symmetric plan can be expressed as:  
\[\alpha_{i}\geq\sum_{i+1}^{N}\alpha_{n}\text{ for }i=1,\;2 \tag{5}\]  
\[\alpha_{i}<\sum_{i+1}^{N}\alpha_{n}\text{ for }i=3,\;4,\ldots,\;N-2 \tag{6}\]  
The large modal mass coefficient values of the first two modes show the overwhelming domination of the first two modes in the vibration of the RC shear wall structures. Thus, considering the effect of IGMs on the interaction of the first two dominated modes is more reasonable in selecting IGMs for the NLRHA of RC shear walls with a symmetric plan. Otherwise, the higher-order mode effect will be exaggerated in the IGM selection, and the reliability of the NLRHA of shear wall structures will be affected. Accounting for the domination of the first two modes in the vibration of the RC shear wall structures, the MGMS procedure was modified by considering the first two modes' interaction in the IGMs selection. After selecting twenty seed IGMs through a spectrum-matching procedure, the NL-ESDOFs of the first two modes were developed through the modal pushover analysis procedure:  
1. Apply lateral force whose distribution is \(m\phi_{n}\), where \(n=1,\,2\), to the structure separately, and generate two sets of curves of base shear-roof displacement (\(V_{bn}-u_{rn}\)). Convert the (\(V_{bn}-u_{rn}\)) curves to bilinear curves. 2. Transfer the (\(V_{bn}-u_{rn}\)) bilinear curves to the (\(F_{sn}/L_{n}-D_{n}\)) pushover curves of the NL-ESDOFs using Equation (7): \[\frac{F_{sn}}{L_{n}}=\frac{V_{bn}}{M_{n}^{*}}\text{ and }D_{n}=\frac{u_{rn}}{ \Gamma_{n}\phi_{rn}}\] (7)
3. The vibration period of NL-ESDOFs can be calculated as: \[T_{n}^{*}=2\pi\sqrt{\frac{L_{n}D_{ny}}{F_{sny}}}\text{ and }D_{ny}=\frac{u_{ny}}{ \Gamma_{n}\phi_{rn}}\] (8)
4. Develop the NL-ESDOFs model using the base shear-top displacement relations of (\(F_{sn}/L_{n}-D_{n}\)) curves and \(T_{n}^{*}\) as the vibration period of NL-ESDOFs. After obtaining the model of the NL-ESDOFs, NLRHA of NL-ESDOFs are conducted with twenty seed IGMs to compute the roof displacement time histories and the peak absolute roof displacement of the NL-ESDOFs \(u_{jn}(t_{k})\) and \(\hat{u}_{jn}=\underset{t_{k}\in T_{0}}{\text{max}}\big{|}u_{jn}(t_{k})\big{|}\). Where \(n=1,\,2\), is the mode ID; \(j=1,\,2,\,\ldots,\,20\), is the ID of the seed IGMs; \(t_{k}\) is the time point within the duration of the IGM \(j\); \(T_{j0}\). The level of multi-mode interaction caused by IGM j was then quantified by the combination factor \(\beta_{j}\), which is the largest value of the factor \(\beta_{jk}\) that establishes the following inequalities simultaneously for all \(t_{k}\in T_{j0}\):  
\[\Big{(}\big{|}u_{j1}(t_{k})\big{|}\geq|\Gamma_{1}|\beta_{jk}\hat{u}_{j1}\Big{)} \tag{9}\]  
\[\Big{(}\big{|}u_{j2}(t_{k})\big{|}\geq\beta_{jk}\hat{u}_{j2}\Big{)} \tag{10}\]  
The \(\beta_{j}\) shall range from 0 to \(1/|\Gamma_{1}|\).