## 2 Seismic Analysis of Liquid Storage Tanks  
### Mechanical Model  
The seismic behavior of liquid storage tanks is complex due to the fluid-structure interaction inherent in their design. When subjected to ground motion, the dynamic response involves both the tank walls and the liquid within. This interaction creates significant computational challenges, particularly when the confined fluid and its intricate contact dynamics with the tank shell are explicitly modeled using _FEM_. These challenges are exacerbated in performance-based earthquake engineering (PBEE) frameworks, where numerous nonlinear analyses are required to capture the full range of seismic responses.  
To address these computational demands, simplified surrogate models, such as the 2 DoF spring-mass system, are commonly employed. These models provide an efficient alternative for representing the dynamic behavior of liquid storage tanks, capturing both the impulsive and convective components of motion. The impulsive component accounts for the liquid that moves rigidly with the tank wall, while the convective component represents the sloshing motion of the liquid free surface. Such models are widely used for both anchored and unanchored tanks and have been validated in previous studies [16; 35].  
In this study, a 2 DoF spring-mass model is utilized to simplify the representation of the tank's dynamic behavior. As illustrated in Figure 2, the model consists of two masses (impulsive and convective) connected by springs and dampers, representing the stiffness and damping properties of the tank-liquid system. The equations of motion for this system are given as follows:  
\[m_{i}\ddot{x_{i}}+(c_{i}+c_{c})\dot{x_{i}}-c_{c}\dot{x_{c}}+(k_{i}+k_{c})x_{i}- k_{c}x_{c}=0 \tag{1}\]  
\[m_{c}\ddot{x_{c}}-c_{c}\dot{x_{i}}+c_{c}\dot{x_{c}}-k_{c}x_{i}-k_{c}x_{c}=0 \tag{2}\]  
where \(m_{i}\) and \(m_{c}\) are the impulsive and convective masses, \(k_{i}\) and \(k_{c}\) are the impulsive and convective stiffness values, \(c_{i}\) and \(c_{c}\) are the impulsive and convective damping coefficients, \(x_{i}\) and \(x_{c}\) are the displacements of the impulsive and convective masses, and \(\dot{x}\) and \(\ddot{x}\) denote the first and second time derivatives of displacement.  
The coupling between these differential equations highlights the interdependence of impulsive and convective motions, which significantly influences the tank's overall seismic response. In a matrix representation, Equations (1) and (2) are expressed as follows:  
\[\text{M}\;\ddot{x}(t)+\text{C}\;\dot{x}(t)+\text{K}\;x(t)=0 \tag{3}\]  
where M is the mass matrix, C the damping matrix, K the stiffness matrix, and \(x\) is the displacement vector, given as follows:  
\[\text{M}=\begin{bmatrix}m_{i}&0\\ 0&m_{c}\end{bmatrix}\]  
Figure 1: Proposed framework for developing fragility curves of fluid storage tanks.  
\[\mathrm{C}=\begin{bmatrix}c_{i}+c_{c}&-c_{c}\\ -c_{c}&c_{c}\end{bmatrix}\]  
\[\mathrm{K}=\begin{bmatrix}k_{i}+k_{c}&-k_{c}\\ -k_{c}&k_{c}\end{bmatrix}\]  
\[x=\begin{cases}x_{i}\\ x_{c}\end{cases}\]  
The 2 DoF model balances computational efficiency with accuracy, making it a practical choice for developing fragility curves. By capturing the primary dynamic characteristics of the tank-liquid system, the model serves as a reliable surrogate for more computationally demanding finite element analyses, particularly when requiring large-scale parametric studies or probabilistic seismic assessments.