## 4 Results and Discussion
### Modal Analysis/Verification of Simulation  
The vibration parameters of a tank include the natural frequencies and modal shapes, which are essential parameters in its analysis. Determining these parameters in the first step can be very helpful in interpreting the behavior of the tank. In addition, modal analysis has been used to determine these characteristics. This type of analysis can also be a starting point for other studies, such as time history analysis. The first reason is to verify the model by extracting the first frequency from the modal analysis and comparing it with the values of the code. The second reason is to calculate the \(\alpha\) and \(\beta\) coefficients.  
For Equations (8) and (9), the convective mode period determined from the water-filled model was compared with the values from the ACI regulations to validate the model. According to the ACI [36] regulation, the following results were obtained: HL = 5 m, L = 30.63 m, and \(\lambda\) = 3.8352 for a convective period of 9.07 s. The software outputs gave a convective time of 9.04 s, which is almost the same, and the agreement between the theoretical and software values shows the accuracy of the modeling. In dynamic analysis, the main objective is to solve the equation of motion for the successive accelerations of the structure. The damping matrix for the overall system's equation of motion combines the damping matrices for the viscous fluid components and the structures. For the damping of structures, [PERSON]'s damping assumption was used. The damping is linearly dependent on mass and stiffness. In this case, Equation (10) is obtained:  
\[[\text{C}]=\alpha[\text{M}]+\beta[\text{K}]+\sum\text{mi}=1\ [\text{Cfi}] \tag{10}\]where [CFi] is the damping matrix of element i, the viscous fluid, and m is the number of fluid elements [40]. Based on the damping coefficients proposed in the ACI 350 regulations, 5% of the coefficients \(\alpha\) = 0.0228 and \(\beta\) = 0.0087 were determined for shock damping and 0.5% of the coefficients \(\alpha\) = 0.0228 for vibration damping, respectively [41].  
To facilitate comparison and establish validation, results from a laboratory-scaled model incorporating a rigid floor within a three-dimensional space have been utilized [51]. In the specified laboratory example, a scaled-down model undergoes testing on a shaking table subject to an El Centro earthquake component, with a time record scaled up by a factor of four and a peak acceleration value of 0.22 g. Such acceleration is visually represented in Figure 6.  
The analysis covers the time history of acceleration at the middle section of the elongated wall, alongside the time history of hydrodynamic pressure at the intersection between the floor slab and the central section of the wall's expansive face. The introduction should articulate the study's originality, and any constraints or limitations found in previous research should be noted.  
For verification, the results of the model and the reference laboratory model [51] are compared in Figures 7 and 8. The maximum acceleration and hydrodynamic pressure values are 2.41 m/s\({}^{2}\) and 1.4 kPa for the experimental model and 2.25 m/s\({}^{2}\) and 1.31 kPa for the current model under the mentioned accelerogram. The relatively decent adaptation of the results is proof of the accuracy of the modeling.  
Figure 8: Time history of hydrodynamic pressure in the interface between the floor slab and the middle section of the wall. (**a**) Laboratory model; (**b**) model.  
Figure 6: Time history of the acceleration component of the El Centro earthquake [51].  
Figure 7: The time history of the measured acceleration at the top of the middle section (**a**) Laboratory model; (**b**) model.