## 3 Development of Methodology Using Finite Element Analysis
### Finite Element Modelling  
The AUTODYN software package was used to evaluate the degree of protection of the K9 self-propelled artillery position. There are several material options provided for the modeling of the structure elements in AUTODYN. In this study, the RHT (Riedel-Hiermaier-Thoma) dynamic damage concrete model in which the strain hardening and the third invariant dependence were considered is applied to present the behavior of the concrete during blasting loading. The RHT model is expressed by three strength surfaces defined as failure surface, elastic limit surface, and residual surface [34; 35]. The elastic strength surface can reflect material strain hardening behavior. The residual strength surface can express the strength of completely crushed concrete under ultimate blasting loading. In addition, the concrete in the constitutive relation of FE model is considered as a quasi-brittle material with different behavior in compression and tension under blasting loading. A typical stress-strain curve for concrete is shown in Figure 4.  
Figure 4: Typical uniaxial compression and tensile stress–strain curve for concrete. For steel reinforcement in a 3D stress space, the Johnson and Cook constitutive model was used to describe the steel behavior inside the concrete subjected to large strains and high strains with high temperature [36]. A bilinear elastoplastic stress-strain curve for steel is shown in Figure 5.  
Air was modeled as an ideal gas with a density of 1.225 kg/m\({}^{3}\), an initial internal energy of kJ/kg and adiabatic exponent (gamma) of 1.4. Table 2 presents the structural specifications and material properties used in FE model. To model in AUTODYN, ANSYS Space-Claim was used to design the geometry of the model to analyze the air layers, walls, and slabs, as shown in Figure 6. The basic condition of the structure was set to fixed support, the joint condition of the heterogeneous materials of the concrete and reinforcing \begin{table}
\begin{tabular}{c c c} \hline \hline
**Category** & \multicolumn{2}{c}{**Parameter**} \\ \hline \multirow{6}{*}{Concrete} & Compressive Strength & 30 MPa \\  & Elastic Modulus & 28,000 MPa \\  & Poisson’s ratio & 0.19 \\  & Bulk Modulus & 35.27 GPa \\  & Density & 2.75 g/cm\({}^{3}\) \\ \hline \multirow{6}{*}{Reinforcement} & Yield Strength & 400 MPa \\  & Elastic Modulus & 232,800 MPa \\  & Poisson’s ratio & 0.28 \\ \cline{1-1}  & Density & 7.83 g/cm\({}^{3}\) \\ \cline{1-1}  & Bulk modulus & \(1.59\times 10^{8}\) kPa \\ \cline{1-1}  & Shear modulus & \(8.18\times 10^{7}\) kPa \\ \cline{1-1}  & Diameter & HD13, HD16, HD19 \\ \hline \multirow{6}{*}{Structure size} & Width & 37.30 m \\  & Length & 20.35 m \\ \cline{1-1}  & Height & 6.94 m \\ \cline{1-1}  & Wall Thickness & 600 mm, 500 mm, 430 mm \\ \hline \multirow{6}{*}{Air} & Density & 1.225 kg/m\({}^{3}\) \\  & Gamma & 1.40 \\ \cline{1-1}  & Initial Internal Energy & 206,800 kJ/kg \\ \hline \multirow{6}{*}{TNT} & Density & 1.630 \\ \cline{1-1}  & Detonation velocity & \(6.93\times 10^{3}\) m/s \\ \cline{1-1}  & Energy/unit volume & \(6.00\times 10^{6}\) kJ/m\({}^{3}\) \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Material properties used in FE model. Figure 5: Stress–strain curve for steel reinforcement. steel was assumed to be sufficiently integrated, and the corresponding effect on flexure was examined rather than the effect on shearing. Considering the air boundary conditions for the analysis, the five surfaces excluding the bottom surface were set to flow out, and the bottom surface touching the ground was set to reflection with a coefficient of 1.8 [37]. This was to reflect the consequent energy losses to displace the ground, i.e., for cratering [1,37]. The blast pressure was measured at a total of 47 sensor locations to evaluate the structural response of the protective structure. Figure 7 presents the sensor locations.