# Numerical Analysis for Critical Structures Protection against Blast Loading Using Metallic Panels
## 3 Mathematical Model and Model Hypothesis  
Protective panel models consist of two steel plates joined with connectors. The models introduced in this paper are called the Combined Protection System (CPS), which is an extension of a previous model for a protective panel investigated in [44]. The panel design used in this study consisted of two steel plates, each 20 mm in thickness with a steel yield stress of \(4.0\times 10^{5}\) kPa. The panel dimensions were 3 m tall, 3 m wide, and 350 mm thick. Moreover, the panel was supported at the bottom due to the limitations of deformation in the middle; hence, the bottom fixed support helped in reducing the deformation in the middle of the panel. The parameters were adopted from [44]. Furthermore, the effective charge weight was 50 kg of TNT. The standoff distance between the explosive charge and panels was set to 1 m. The ANSYS AUTODYN finite element analysis package was deployed to simulate the blast wave propagation, due to its ability to study air-structure interactions. The material  
Figure 1: Pressure vs. time plot for blast wave. properties for the TNT and air were retrieved from the standard AUTODYN library. The modeling of the air was carried out using 3D Euler equations, while the explosive charge was typically modeled using the Jones Wilkens Lee (JWL) equation of state, which can be written as Equation (3) [28; 45; 46]  
\[P=C_{1}\bigg{(}1-\frac{w}{r_{1}v}\bigg{)}e^{-r_{1}v}+C_{2}\bigg{(}1-\frac{w}{r_ {2}v}\bigg{)}e^{-r_{2}v}+\frac{we}{v} \tag{3}\]  
where \(P\) denotes the hydrostatic pressure, \(v\) denotes the specific volume, and \(e\) denotes the specific internal energy. \(C_{1}\), \(r_{1}\), \(C_{2}\), \(r_{2}\), and \(w\) are the material constants. The values of constants \(C_{1}\), \(r_{1}\), \(C_{2}\), \(r_{2}\), and \(w\) for many common explosives have been previously determined from dynamic experiments. The values of the constants for TNT explosives are available in AUTODYN as shown in Table 1[28; 46; 47]. In addition, the air is modeled by an ideal gas equation of state, which can be expressed as in Equation (4)  
\[P=C_{1}(\gamma-1)\rho e \tag{4}\]  
where, \(\gamma\) denotes the heat specific ratio and \(\rho\) denotes the density. The standard constants of air were taken from AUTODYN material library, that is, air density, where \(\rho=1.225\) kg/m\({}^{3}\) and \(\gamma=1.4\). The air initial internal energy was assumed to be \(2.068\times 10^{5}\) kJ/kg [46]. The material of the steel panel was selected from the ANSYS library. Additionally, the linear equation of state and strength model was applied. The yield stress of the steel panel was assumed to be \(4\times 10^{8}\) (Pa). The solver used for steel plates was the shell element. The air element was modeled with dimensions of 4 m, 3 m, and 3 m, representing length, width, and height, respectively. Moreover, movable gauges to measure deformation were added, where the critical gauge position was placed at the mid-point of the panel. During the analysis there was a raised governing variable paradox either pressure or deformation to be considered as governing variable as clarified in [PERSON], [PERSON] et al. [44]. Furthermore, deformation was selected to be the governing value. Eventually, the boundary condition was set as fixed free from the lower cross-section of the panel. Figure 2 illustrates the model setup. For accuracy, the model validation was based on the previous work of the authors of [44]. By comparing the results of the verification model and the reference model it was found that the correlation of the results was very strong. The displacement peak value in the reference model was 82 mm and was 79 mm in the verification model [44]. \begin{table}
\begin{tabular}{c c c c c c} \hline \hline
**Parameters** & **C\({}_{1}\)/MPa** & **C\({}_{2}\)/MPa** & \(r_{1}\) & \(r_{2}\) & \(w\) \\ \hline Value & \(3.7377\times 10^{5}\) & \(3.7471\times 10^{3}\) & 4.15 & 0.9 & 0.35 \\ \hline \hline \end{tabular}
\end{table} 
Table 1: TNT explosive parameters. Figure 2: Model setup.