## 4 Results and Discussion
### Effects of the Higher Sloshing Modes on the Sloshing Response
#### 4.4.5 Sloshing Response at the Corner of the Tank
This procedure was repeated for the spectral accelerations at the other four sloshing periods. The R\({}^{2}\) scores representing the relationship between the spectral accelerations at the first five sloshing periods and the sloshing responses at \"c\" are shown in Table 13.  
As shown in Table 13, it is evident that not only was the maximum corner sloshing greatly affected by the spectral acceleration of the external excitations at the first sloshing period but the value of this acceleration at the two other odd sloshing periods (the 3 rd and the 5 th sloshing periods) also affected the maximum corner sloshing. \begin{table}
\begin{tabular}{c c c c c c} \hline \hline
**S1** & **1 st** & **2 nd** & **3 rd** & **4 th** & **5 th** \\ \hline Major component & 0.93 & 0.04 & 0.74 & 0.19 & 0.83 \\ Combined component & 0.92 & 0.25 & 0.81 & 0.31 & 0.8 \\ \hline \hline \end{tabular}
\end{table} 
Table 13: Relationship between corner sloshing responses and the spectral accelerations for the major and combined components at the first five sloshing periods—S1. Figure 14: Corner sloshing response as a function of spectral acceleration at the first sloshing period for major, minor, and combined components—S1. Corner Sloshing and Spectral Acceleration--S2  
As with S1, the relationship between the maximum sloshing response and the spectral acceleration at the sloshing period was examined for S2, when the horizontal earthquake components rotated 90 degrees around the vertical axis. As a result, R\({}_{\text{dir}}\) decreased to 0.78. Initially, this relationship was evaluated between the maximum corner sloshing response and the spectral acceleration at the first sloshing period of the major, minor, and combined components. Figure 15 shows these relationships as well as their R\({}^{2}\) scores. According to Figure 15, the effect of the minor earthquake component on the corner sloshing increases when the two horizontal components rotate around the vertical axis (from S1 to S2). As a result, it can be seen that decreasing R\({}_{\text{dir}}\) from 1.29 to 0.78 increases the effect of the minor component. As the role of the minor component becomes more significant, the impact of the major component on corner sloshing decreases, and the combined component becomes predominant. Table 14 shows the sensitivity of the corner sloshing to the spectral accelerations at the first five sloshing periods for S2. Figure 15: Corner sloshing response as a function of spectral acceleration at the first sloshing period of the major, minor, and combined components—S2. Based on Table 14, spectral accelerations at higher sloshing modes, even the odd ones, had a negligible effect on corner sloshing, contrary to the behavior observed under the first loading condition (S1). This shows that changing R\({}_{\text{dir}}\) can not only change the effect of the major and minor components on the corner sloshing as the critical sloshing location but also the effect of higher modes on the corner sloshing response in such a way that the role of higher modes diminished while the role of minor components becomes more important. When switching from S1 to S2, it is essential to consider that the distance between the earthquake horizontal components and point \"c\" changes.