## 2 Governing Equation for a Liquid Storage Tank Considering Material Nonlinearity
The tank here is roofless and assumed to be fixed onto rigid ground, i.e., soil-structure interaction is not considered in this study. The detailed material properties of the structure and the fluid contained within it are given in Table 1. The parameters of the concrete damaged plasticity model are given in Table 2. Figure 2 shows the plasticity behavior of concrete. Figure 1: Rectangular liquid storage tank: (**a**) schematic view; (**b**) finite element model. The earthquake responses of the system are obtained by the finite element analysis code ABAQUS [34]. The rectangular liquid storage tank is modeled as shown in Figure 1b. The structure is represented by the S4 elements. The *REBAR LAYER command is used to consider reinforcement. The fluid and interface elements, which are governed by Equation (4), are implemented using user-defined elements in ABAQUS. Rayleigh damping with \(\alpha=1.311\text{s}^{-1}\) and \(\beta=1.685\times 10^{-3}\text{s}\) is employed for damping matrix \(\mathbf{C}\) in Equation (5). The three-directional El Centro earthquake ground motion in Figure 3a is considered as the input ground motion. The ground motion is the far-field motion recorded at the Imperial Valley Irrigation District station, El Centro, California during the Imperial Valley Earthquake of May 18, 1940. The response spectra for the components and the corresponding V/H ratios are also shown in Figure 3b. The V/H ratios of the peak ground accelerations are 0.6184 and 0.9069 for the \(x\) and \(y\) components, respectively. With the aforementioned numerical model and earthquake ground motion, dynamic analyses are conducted for various incident angles of motion. The effects of static loads due to the self-weight of the structure and the hydrostatic pressure are considered before the dynamic analyses. Earthquake responses are calculated using the constant acceleration method with a time increment of 0.002 s. \begin{table}
\begin{tabular}{c c c} \hline
**Property** & **Value** \\ \hline \multirow{8}{*}{Structure} & Density & 2300 kg/m\({}^{3}\) \\  & Poisson’s Ratio & 0.17 \\  & Young’s Modulus & 38.527 GPa \\  & Compressive yield stress & 13.0 MPa \\  & Compressive ultimate strength & 24.1 MPa \\  & Inelastic strain at compressive ultimate strength & 0.001 \\  & Concrete & Tension failure strength & 2.9 MPa \\  & Inelastic tensile behavior & Table 2 \\  & Dilation angle & 36.31\({}^{\circ}\) \\  & Flow potential eccentricity & 0.1 \\  & Ratio of initial equibiaxial and uniaxial compressive yield stresses & 1.16 \\  & Parameter \(\rho\)[29] or \(K_{c}\)[34] & 0.667 \\  & Viscosity parameter & 0 \\ \hline \multirow{4}{*}{Reinforcement} & Cross-sectional area & 0.00306 m\({}^{2}\) \\  & Young’s modulus & 2.0 GPa \\ \cline{1-1}  & Plastic modulus & 0.345 GPa \\ \cline{1-1}  & Ratio of volume of steel to that of concrete & 1.6576 \(\times\) 10\({}^{-3}\) \\ \hline Liquid & Density & 1000 kg/m\({}^{3}\) \\ \hline \end{tabular}
\end{table} 
Table 1: Material properties of a rectangular liquid tank. \begin{table}
\begin{tabular}{c c c} \hline
**Cracking Strain** & **Tensile Stress (MPa)** & **Damage Parameter** \\ \hline \multirow{2}{*}{\(0\)} & 2.9 & 0 \\  & \(6.6185\times 10^{-5}\) & 1.9439 & 0.3812 \\  & \(1.2286\times 10^{-5}\) & 1.3031 & 0.6171 \\  & \(1.7343\times 10^{-5}\) & 0.8735 & 0.7631 \\  & \(2.2019\times 10^{-4}\) & 0.5855 & 0.8534 \\  & \(2.2647\times 10^{-4}\) & 0.3925 & 0.9093 \\  & \(3.0808\times 10^{-4}\) & 0.2631 & 0.9439 \\  & \(3.5105\times 10^{-4}\) & 0.1763 & 0.9653 \\  & \(3.9414\times 10^{-4}\) & 0.1182 & 0.9785 \\  & \(4.3774\times 10^{-4}\) & 0.0792 & 0.9867 \\  & \(4.8217\times 10^{-4}\) & 0.0531 & 0.9918 \\ \hline \end{tabular}
\end{table} 
Table 2: Inelastic tensile behavior of concrete.