## 2 Governing Equation for a Liquid Storage Tank Considering Material Nonlinearity
### Nonlinear Dynamic Responses to a Three-Directional Earthquake Ground Motion  
It was observed that the elastic earthquake responses of rectangular liquid storage tanks are influenced significantly by the incident angles of bi-directional input ground motion [27]. Therefore, the effects of incident angles on the inelastic earthquake responses of the fluid-structure interaction system considered here will be examined with three-directional input ground motion. To evaluate the effects of the incident angle of earthquake ground motion, the three-directional motion in Figure 3 is rotated. In order to define a new \(\hat{x}\hat{y}\hat{z}\) coordinate system, the original \(xyz\) system is rotated by an angle of \(\theta\) as shown in Figure 13. The components \(\tilde{u}_{gx}^{0}(t)\), \(\tilde{u}_{gy}^{0}(t)\), and \(\tilde{u}_{gz}^{0}(t)\) in Figure 3 are applied in the \(\hat{x},\hat{y}\), and \(\hat{z}\) directions. The components \(\tilde{u}_{gx}^{\theta}(t)\), \(\tilde{u}_{gy}^{0}(t)\), and \(\tilde{u}_{gz}^{\theta}(t)\) of input ground motion for the incident angle \(\theta\) can then be determined as follows:  
\[\left\{ \begin{array}{l}\tilde{u}_{gx}^{\theta}(t)\\ \tilde{u}_{gy}^{0}(t)\\ \tilde{u}_{gz}^{\theta}(t)\end{array} \right\}=\left[ \begin{array}{ccc}\cos \theta&\sin\theta&0\\ -\sin\theta&\cos\theta&0\\ 0&0&1\end{array} \right]\left\{ \begin{array}{l}\tilde{u}_{gx}^{0}(t)\\ \tilde{u}_{gy}^{0}(t)\\ \tilde{u}_{gz}^{0}(t)\end{array} \right\} \tag{15}\]  
Figure 11: Relative displacement when the horizontal components of the input ground motion are considered only. Figure 10: Sloshing heights when the incident angle of the input ground motion is zero. The values considered here for the incident angle \(\theta\) range from \(0^{\circ}\) to \(170^{\circ}\) in increments of \(10^{\circ}\). The variations of the maximum displacement of the structure, the maximum hydrodynamic pressure, and the maximum sloshing height are shown in Figure 14a,c, respectively. It can be observed that the earthquake responses are significantly affected by the incident angle. However, no consistent trend can be observed in the responses. Therefore, it is necessary to consider the directionality of the input ground motion for reliable predictions of the earthquake responses of this system. Figure 12: Tensile damage variable \(d_{t}\) when the horizontal components of the input ground motion are considered only: (a) 1.51 s; (b) 1.91 s; (c) 2.60 s; (d) 9.46 s.  
Figure 13: \(\hat{x}\hat{y}\hat{z}\) coordinate system for the incident angle \(\theta\). It can be observed in Figure 14a-c that the responses are significantly affected at the incident angles of 50\({}^{\circ}\), 100\({}^{\circ}\), and 140\({}^{\circ}\). Therefore, detailed examinations of the responses at these angles are provided. The time histories for displacement are shown in Figure 15. Similar to when the incident angle is zero, several times in Table 3 are chosen to examine the evolution of the tensile damage variable \(d_{t}\). It can be observed that Times A and B are similar for the considered incident angles. However, Times C and D are very different for these cases. These different times mean that the earthquake responses of a rectangular liquid storage tank are influenced significantly by the incident angle of earthquake ground motions. The evolution of the tensile damage variable \(d_{t}\) can be observed in Figures 16-18 for the corresponding incident angles. The evolution outcomes of the damage-related variables show very similar patterns in the analyses. However, it should be noted that at Times C and D, at which one long-sided wall or both walls are damaged, are different in these analyses. It can be concluded from these observations that a rigorous inelastic analysis must be considered to predict the earthquake responses of a rectangular liquid storage tank accurately during bi-/three-directional earthquake ground motions.