## 2 Governing Equation for a Liquid Storage Tank Considering Material Nonlinearity  
A governing equation for a rectangular liquid storage tank is derived in this section. The fluid contained in a tank is assumed to be an ideal, incompressible, and inviscid fluid. An equation of motion for such a fluid can be expressed as follows:  
\[\
abla^{2}p=\frac{\partial^{2}p}{\partial x^{2}}+\frac{\partial^{2}p}{ \partial y^{2}}+\frac{\partial^{2}p}{\partial z^{2}} \tag{1}\]where \(p(x,y,z,t)\) is the hydrodynamic pressure of the fluid. The boundary condition on the interfaces of a structure is expressed as:  
\[\frac{\partial p}{\partial n}=-\rho\ddot{u}_{n}^{t} \tag{2}\]  
where \(n\) denotes the outward direction normal to the fluid, \(\partial p/\partial n\) is a directional derivative in the direction of \(n\), \(\rho\) is the density of fluid, and \(\ddot{u}_{n}^{t}\) is a total acceleration of the fluid in the direction of \(n\). The boundary condition on the free surface is given by:  
\[\frac{\partial p}{\partial z}=-\frac{1}{g}\ddot{p} \tag{3}\]  
where \(g\) is the gravitational acceleration. A solution for the governing Equation (1) with boundary conditions (2) and (3) can be obtained by adopting the finite-element approach [28]. The discretized governing equation for the fluid in this case can be written as:  
\[\mathbf{G}_{fs}\ddot{\mathbf{P}}+\mathbf{H}\mathbf{P}=\mathbf{Q} \tag{4}\]  
\[\mathbf{G}_{fs}=\frac{1}{g}\int\mathbf{N}^{T}\mathbf{N}dS \tag{5}\]  
\[\mathbf{H}=\int\mathbf{B}^{T}\mathbf{B}dV  
The governing equation for the fluid-structure system can be obtained from Equations (4) and (10). \[\left[ \begin{array}{cc}\mathbf{M}&0\\ \rho\mathbf{S}&\mathbf{G}_{f_{S}}\end{array} \right]\left\{\begin{array}{c} \begin{array}{c}\ddot{\mathbf{U}}\\ \ddot{\mathbf{p}}\end{array} \right\}+\left[ \begin{array}{cc}\mathbf{C}&0\\ 0&0\end{array} \right]\left\{ \begin{array}{c}\dot{\mathbf{U}}\\ \dot{\mathbf{p}}\end{array} \right\}+\left\{ \begin{array}{cc}\mathbf{f}_{int} \big{(}\varepsilon,\dot{\varepsilon}\big{)}\\ 0\end{array} \right\}+\left[ \begin{array}{cc}0&-\mathbf{S}^{T}\\ 0&\mathbf{H}\end{array} \right]\left\{ \begin{array}{c}\mathbf{U}\\ \mathbf{p}\end{array} \right\}=\left\{ \begin{array}{c}-\mathbf{M}\ddot{\mathbf{ U}}_{g}\\ -\rho\mathbf{S}\dot{\mathbf{U}}_{g}\end{array} \right\} \tag{14}\]  
The dynamic responses of a rectangular liquid storage tank can be obtained by solving Equation (14). The internal force \(\mathbf{f}_{int}\big{(}\varepsilon,\dot{\varepsilon}\big{)}\) in Equation (14) is considered to be elastic when the level of input ground motion is low. An inelastic internal force must be considered, however, when inelastic behavior of the tank structure is dominant due to strong earthquake ground motion. In this study, the nonlinear stress-strain relationship will be considered with regard to inelastic behavior. The dynamic earthquake responses of a concrete tank, which is reinforced with steel, are determined by considering the inelastic behavior in reinforced concrete. The concrete damaged plasticity model will be used to reflect the material nonlinearity. This model reflects all inelastic behaviors of concrete both in tension and compression, including damage characteristics. It was proposed by [PERSON] et al. [29] and modified by [PERSON] and [PERSON] [30; 31]. The accuracy of this model has been verified by [PERSON] et al. [26], [PERSON] and [PERSON] [32], and [PERSON] et al. [33]. 3 Nonlinear Earthquake Responses of a Liquid Storage Tank to a Three-Directional Input Ground Motion  
Liquid storage tanks are commonly used for the storage of water, oil, and other liquids, including chemicals. Damage to these tanks is a major concern because it can lead to serious social and economic losses. Therefore, many researchers have investigated the dynamic responses of liquid storage tanks. In this study, the nonlinear earthquake responses of the rectangular liquid storage tank shown in Figure 1 will be examined for various incident angles of three-directional input ground motion.