## 3 Simulation Model for Numerical Analyses
### Material Constitutive Models  
The construction material of the representative tank is defined as steel with the designation P355 NL2, a fine-grained, weldable steel, normalized material with high toughness at cold temperatures, used, for example, for highly stressed welded joints and pressure vessels. The mechanical properties, which are considered for the material of all components of the representative tank structure, are summarized in Table 1. Nominal material values according to DIN EN 10028-3 [51] at room temperature are considered. The force-displacement behavior of the steel can be described using a simple bilinear material formulation. A bilinear material constitutive model represents the behavior of steel by dividing the stress-strain curve into two linear segments. The first segment represents the elastic behavior of the steel until it reaches the yield point. The second segment represents the plastic behavior of the steel after the yield point is exceeded (Figure 7). The simplified model uses these two segments to accurately simulate the nonlinear response of steel structures under varying loads. Hardening effects are considered with the tangent modulus that can be determined using the following equation:  
\[\text{Et}=\frac{f_{ik}-f_{yk}}{(\epsilon_{B}-0.002)-\frac{f_{yk}}{E}}, \tag{5}\]  
with the parameters being explained in Table 1. Such a bilinear material formulation is one of the simplest ways of mapping nonlinear material behavior. A further formulation captures the transition from the elastic to the plastic region with multiple points of the load-deformation curve. It also considers strain rate effects, where materials exhibit higher resistances at faster loading rates. Especially for highly dynamic loads that generate structural responses in the yield limit range, material formulations that take strain rate effects into account are often used. This approach is more complex but provides a more accurate representation of the material's behavior than the bilinear formulation and can lead to higher utilization of the material properties in the simulation for loads that stress the steel around the yield point and into plastic deformation ranges. \begin{table}
\begin{tabular}{c c c} \hline \hline
**Name** & & **Value** \\ \hline Tensile strength & \(\text{f}_{\text{uk}}\) & 490 MPa \\ Yield strength & \(\text{f}_{\text{yk}}\) & 355 MPa 1  \\ Ultimate strain & \(\text{e}_{\text{B}}\) & 22\% \\ Modulus of elasticity & E & 212 GPa \\ Density & \(\rho\) & 7.82 t/m3  \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Mechanical properties of steel P355 NL2. \begin{table}
\begin{tabular}{c c} \hline \hline
**Name** & & **Value** \\ \hline Hardening parameter & \(\beta\) & 0.5 \\ Material parameter & A & 345 MPa \\ Material parameter & B & 51 MPa \\ Material parameter & N & 0.26 \\ Material parameter & C & 0.014 \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Values adopted for constitutive models for numerical simulation. One widely used example is the simplified Johnson-Cook model that determines the yield stress \(\sigma_{y}\) according to Equation (6):  
\[\sigma_{y}=\Big{(}\text{A}+\text{B}\cdot\overline{\varepsilon}^{\text{p}^{n}} \Big{)}\Big{(}1+\text{C}\cdot\ln\dot{\varepsilon}^{*}\Big{)}. \tag{6}\]  
A, B, N, and C are material parameters, \(\overline{\varepsilon}^{\text{p}}\) is the effective plastic strain, and \(\dot{\varepsilon}^{*}\) is the normalized plastic strain rate. Material failure is defined by a critical plastic strain. If the plastic strain exceeds the critical strain, the correspondent element has no further strength. In the current problem, the load level is below the threshold for failure.