## 5 Summary and Conclusions  
In this study, a rectangular concrete liquid storage tank was numerically modeled using the Lagrangian-Lagrangian method and subjected to seven three-dimensional near-fault earthquakes under two different loading conditions. Under the first condition (S1), the major and the minor horizontal earthquake components acted along the longer and shorter tank walls, respectively. Under the second loading condition (S2), these earthquake components rotated 90 degrees around the vertical direction. Under each loading condition, the relationship between the maximum sloshing response at three different locations, at the corner and in the middle of the tank's shorter and longer walls, and the values of spectral acceleration at the first five sloshing periods were studied. Further, the relationship between the PGAs of earthquakes and the maximum sloshing responses for the minor earthquake horizontal components was studied. Due to the fact that switching the loading condition from S1 to S2 changes the ratio between the length along which the major and minor horizontal earthquake components act, R\({}_{\text{dir}}\), the effect of changing this ratio on sloshing response, particularly its sensitivity to higher sloshing modes, was examined. The results are listed as follows:  
1. Under both loading conditions and all seven earthquakes, the maximum sloshing response occurred at the corner of the tank, which is considered to be the critical sloshing location in rectangular liquid storage tanks. Under the considered earthquakes and the two loading conditions, the sloshing response was up to 50% higher at the corner of the tank than in the middle of the walls. 2. Regardless of the value of R\({}_{\text{dir}}\), the maximum sloshing response that occurred in the middle of the longer and shorter tank walls, perpendicular to the seismic excitation direction, was highly affected by the value of the input seismic excitation's spectral acceleration at the first convective period (the 1 st convective mode). 3. In this case study, the sensitivity of sloshing response to higher sloshing modes depended on R\({}_{\text{dir}}\). As shown in this paper, when the major horizontal earthquake component acted along the longer wall (S1), the role of higher sloshing modes on the sloshing response in the middle of the shorter wall, perpendicular to the major horizontal earthquake component, and at the corner of the tank, the critical sloshing location, became significant, according to the 0.95 and 0.70 R\({}^{2}\) scores for the 3 rd and 5 th sloshing modes. On the other hand, when the major horizontal earthquake component acted along the shorter tank wall (S2), only the value of spectral acceleration at the first sloshing period affected the maximum sloshing response in the middle of the longer wall, perpendicular to the major component, and at the corner. Therefore, the value of R\({}_{\text{dir}}\) seems to be an important parameter regarding the sensitivity of the sloshing response to the sloshing higher modes. \begin{table}
\begin{tabular}{c c c c c c} \hline \hline
**S2** & **1 st** & **2 nd** & **3 rd** & **4 th** & **5 th** \\ \hline Major component & 0.89 & 0.11 & 0.19 & 0.11 & 0.71 \\ Combined component & 0.96 & 0.38 & 0.39 & 0.08 & 0.76 \\ \hline \hline \end{tabular}
\end{table} 
Table 14: Relationship between corner sloshing and spectral accelerations of the major and combined components at the first five sloshing periods—S2. As R\({}_{\rm dir}\) increases, the major horizontal earthquake component dominates the maximum sloshing response at the corner of the tank. In other words, by decreasing R\({}_{\rm dir}\), the role of the minor horizontal earthquake component becomes more significant. As mentioned above, at the lower R\({}_{\rm dir}\) ratio, only the spectral acceleration at the first sloshing period became important.