## 4 Results and Discussion
### Effects of the Higher Sloshing Modes on the Sloshing Response  
As explained above, the tank was subjected to horizontal and vertical earthquake components under two different loading conditions. First, the major and minor earthquakes' horizontal components acted along the longer and shorter tank walls, respectively (S1). Under the second condition (S2), these two earthquake components rotated 90 degrees around the vertical earthquake component so that the major and minor earthquake components acted along the shorter and longer tank walls, respectively. Major earthquake components were scaled to 0.4 g, and minor earthquake components, as well as vertical earthquake components, were scaled according to the scale factor calculated for the major components. Thus, while the PGAs of all major components were equal to 0.4 g, those of the minor components were not equal. Under each loading condition, the relationship between spectral accelerations at the first five sloshing periods of vibration of the major and minor earthquake horizontal components and the maximum sloshing responses measured perpendicular to the direction of the considered earthquake component (points a and b--Figure 7) was individually studied. Finally, the relationship between these accelerations and the maximum sloshing response at the corner of the tank (point c), the critical sloshing location for rectangular liquid storage tanks [(38; 78)], was studied.  
A linear regression line was drawn for each graph representing the relationships mentioned above. Additionally, the \(R^{2}\) score, i.e., the coefficient of determination, was used to evaluate the impact of the spectral acceleration of external seismic excitations at sloshing modes on this response. This coefficient is used to evaluate the performance of a linear regression line.  
Figure 8: Sloshing response at “a”,” b”, and “c” under the first and the second loading conditions.  
\begin{table}
\begin{tabular}{c c c c} \hline \hline \multicolumn{3}{c}{**In the Middle of the Shorter Wall—a (S1)**} & \multicolumn{2}{c}{**In the Middle of the Longer Wall—b (S2)**} \\ \hline Event & Normalized sloshing response & Event & Normalized sloshing response \\ ChiChi-122 & 1 & Kocaeli & 1 \\ IMV-\#6 & 0.62 & IMV-\#6 & 0.82 \\ Kocaeli & 0.46 & IMV-\#7 & 0.64 \\ IMV-\#7 & 0.44 & ChiChi-122 & 0.41 \\ Northridge & 0.22 & ChiChi-129 & 0.25 \\ IMV-EDA & 0.21 & IMV-EDA & 0.23 \\ ChiChi-129 & 0.17 & Northridge & 0.19 \\ \hline \hline \end{tabular}
\end{table}
Table 12: Sorted normalized sloshing responses of the tank under the considered earthquakes for both loading conditions.  
regression line. In other words, it shows how the sloshing response is affected by the value of spectral accelerations at the first five sloshing periods (Equation (14)):  
\[R^{2}=\left[\frac{n\sum xy-\sum x\sum y}{\sqrt{\left[n\sum x^{2}-\left(\sum x \right)^{2}\right]\left[n\sum y^{2}-\left(\sum y\right)^{2}\right]}}\right]^{2} \tag{14}\]  
where \(R^{2}\) is the coefficient of determination, and \(x\) and \(y\) are the real values and the predicted sloshing, respectively. A high R\({}^{2}\) score indicates that there is a strong correlation between spectral accelerations at different modes and the sloshing response. On the other side, lower R\({}^{2}\) scores indicate that the relationship is weaker, and other factors may be contributing to the sloshing response.