# Understanding the Seismic Resilience of Metallic Cylindrical Tanks Through Parametric Analysis
## 3 Case Studies, Material, Geometry and FEM  
The case study structures analyzed in this research are vertical cylindrical tanks with both flexible and rigid walls, which were all perfectly anchored at their base. To ensure a comprehensive study, ten different geometric configurations have been considered. The slenderness ratios (\(H/R\)) for the case study steel tanks were selected to represent both squat tanks (with 0.5 \(<H/R<\) 1.5) and slender tanks (with 1.5 \(<H/R<\) 6), allowing for an analysis of a wide range of tank geometries. The radius-to-thickness (\(R/t\)) ratio was also varied to investigate the effects of both rigid walls (with \(R/t\) close to 100) and flexible walls (with \(R/t\) close to 2000). Table 2 provides a summary of all the geometric configurations in terms of the \(H/R\) and \(R/t\) ratios considered for the FE analysis. The ten case study tanks were designed to contain water with a fixed radius (\(R\)) of 1.83 m. The height (\(H\)) of the tanks ranged from 0.91 m to 10.97 m, covering the full range of slenderness ratios (0.5 \(<H/R<\) 6.0). The tanks were made of aluminum with a density of 2700 kg/m\({}^{3}\), an elastic modulus of 71,000 MPa, and a yield stress of 100 MPa. The shell thickness (\(t\)) of the tanks varied between 0.91 mm and 18.29 mm and was kept constant along the entire height of the tank. To avoid unrealistic geometrical configurations in terms of the \(R/t\) ratio, preliminary designs were performed, and only significant \(R/t\) ratios were selected for each \(H/R\) case study tank (see Table 3). Figure 4 illustrates the FE models used for the geometrically and materially nonlinear analyses, which include fluid-structure interaction capabilities in LS-DYNA. These models were designed to simulate the full range of seismic behaviors and provide insight into the dynamic response of the tanks under earthquake loading. In the proposed FE approach, the motion of the fluid was described using compressible Navier-Stokes equations, which were implemented \begin{table}
\begin{tabular}{c c c c} \hline \hline
**Parameter** & **Experimental** & **FEM** & **[\%]** \\ \hline Max radial disp at shell top [mm] & 3.32 & 3.40 & 2.41 \\ Max meridional compression [N/m] & 63,396 & 62,205 & 1.88 \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Comparison of experimental and numerical results. Figure 3: Haroun experimental tank FEM (**a**); and deformed shapes at different time steps (**b**). through a mesh of eight-noded, one-point integration ALE solid elements. For modeling the tank wall, four-noded, two-point integration shell elements were selected, as described in [44]. This type of shell element provides a computationally efficient solution, as it is based on a combined co-rotational and velocity-strain formulation. To model the wall's material behavior, a bilinear elastic-plastic constitutive law (MAT_003) was employed, incorporating both isotropic and kinematic hardening. This approach was used to capture the cyclic permanent deformations exhibited by the tank wall under seismic excitation. In this study, null material was used to simulate the fluid, which is conservatively assumed to be water. Consequently, the bulk modulus of the fluid was assumed to be \(2.25\times 10^{9}\) N/m\({}^{2}\). The resulting FE models, based on different tank geometries, vary in complexity, with the number of nodes ranging from 6162 to 25,034, depending on the specific tank configuration being analyzed. This range reflects the model's ability to handle different tank sizes and geometries while maintaining computational efficiency. Before conducting a series of case study analyses, the acceleration history of the input motion used in this research was the same as that employed by [PERSON] and [PERSON] [42] for their experimental test. The acceleration history and response spectrum of the input motion, used in both the experimental tests and the numerical simulations of the case studies, are provided in Figures 5a and 5b, respectively.