## 3 Structural Model of the Liquid Storage Tank  
The three-dimensional finite element modelling and analysis of a cylindrical liquid tank is a complicated process and requires computational effort mainly due to the interaction of tank and fluid. Besides, the MF, which is made of different materials, increases its modelling and analysing complexity. Therefore, the use of a simple and accurate model with the capability to determine the response of a liquid storage tank under seismic ground motions is of interest. As an alternative approach of finite element modelling, the simplified mass-spring model, which is accepted by standard codes such as API 650 [36], ASCE 7-16 [37], and Eurocode 8 [38] can be employed to model and analyse of tank containers. This simplified mass-spring model is generally based on the work conducted by [PERSON] [39]. He evaluated the effect of hydrodynamic actions in liquid storage tanks, assuming that the tank wall has a rigid behaviour. According to [PERSON]'s model, the hydrodynamic response of a fluid tank system is determined by the superposition of two components, namely convective and impulsive components. The convective component is generated by convective mass (\(m_{c}\)), and the impulsive component is produced by impulsive mass (\(m_{i}\)). The convective mass refers to the portion of the liquid in the upper part of the tank near the free surface, which experiences a long period of sloshing motion during dynamic loading. Conversely, the portion of the filling liquid near the base of the liquid storage tank accelerating in unison with the tank wall is represented by impulsive mass. [PERSON] and [PERSON] [40] modified the simplified mass-spring model of [PERSON] to consider the tank wall flexibility. [PERSON] et al. [41] offered a simple and accurate procedure to assess the seismic responses of liquid storage tanks. They combined the first impulsive modal mass with the higher impulsive modal mass and the first convective modal mass with the higher convective modal mass. As a result of such a combination, the tank liquid system was represented by two modes only. Figure 3 demonstrates a cylindrical liquid tank mounted on an \(N\)-layer MF. The geometrical properties of the tank are the height of the liquid (\(H\)), the thickness of the wall of the tank (\(t_{w}\)) and the radius of the tank (\(R\)). In this paper, the mass-spring model suggested by [PERSON] et al. is used for the dynamic time history analysis of the liquid storage tank (Figure 4). The impulsive and convective masses (\(m_{i}\) and \(m_{c}\)) are connected to the wall of the tank by linear springs whose stiffnesses are \(k_{i}\) and \(k_{c}\), respectively. The stiffness of springs is influenced by the properties of the filling fluid, the tank material, and the geometric properties of the tank. The damping coefficients corresponding to the impulsive and convective masses are shown by \(c_{i}\) and \(c_{c}\), respectively. The damping coefficient and stiffness corresponding to impulsive and convective masses are expressed as follows:  
\[c_{i}=2\zeta_{i}m_{i}\times\frac{2\pi}{T_{i}} \tag{15}\]  
\[c_{c}=2\zeta_{c}m_{c}\times\frac{2\pi}{T_{c}} \tag{16}\]  
Figure 4: Simplified model of the liquid storage tank. Figure 3: A liquid storage tank mounted on an MF. \[k_{i}=m_{i}\times\frac{4\pi^{2}}{T_{i}{}^{2}} \tag{17}\]  
\[k_{c}=m_{c}\times\frac{4\pi^{2}}{T_{c}{}^{2}} \tag{18}\]  
where \(\zeta_{i}\) and \(\zeta_{c}\) are the damping ratios of the impulsive and convective masses. \(T_{i}\) and \(T_{c}\) are natural periods corresponding to impulsive and convective responses, given by [41]:  
\[T_{i}=C_{i}\frac{H\sqrt{R}}{\sqrt{t_{w}/R}\times\sqrt{E_{s}}} \tag{19}\]  
\[T_{c}=C_{c}\sqrt{R} \tag{20}\]  
where \(E_{s}\) is the modulus of elasticity of the tank material and \(\rho_{w}\) is the mass density of filling fluid, respectively.