## 2 Analytical Model
### Numerical Simulation  
Numerical simulations of the explosion response of steel tunnels were conducted using hydrodynamic finite element code LS-DYNA. Arbitrary Lagrangian-Eulerian (ALE) algorithm for the fluid-structure interaction (FSI) model of the LS-DYNA was adopted. The structure model was discretized into a finite number of sections, and the conservation and constitutive equations were solved. There are four materials in the numerical models: air, explosives, steel tunnels and soil ground. The constitutive models and material parameters used in this paper are shown in Table 1 and described below [11]. For air material, the MAT_NULL equation was used and combined with a linear polynomial equation of state (EOS) as shown in Equations (2) and (3), respectively. \[\mathrm{P}=\mathrm{C}_{0}+\mathrm{C}_{1}\mathrm{\mu}+\mathrm{C}_{2}\mathrm{ \mu}^{2}+\mathrm{C}_{3}\mathrm{\mu}^{3}+(\mathrm{C}_{4}+\mathrm{C}_{5}\mathrm{ \mu}+\mathrm{C}_{6}\mathrm{\mu}^{2})\mathrm{E}_{0} \tag{2}\]  
\[\mathrm{E}_{0}=\mathrm{\rho}_{0}\mathrm{C}_{V}\mathrm{T} \tag{3}\]  
where P is air pressure, \(\mathrm{\rho}_{0}\) is the initial density of air, \(\mathrm{C}_{V}\) is the specific heat of air at constant volume and T is the initial temperature of air. \(\mathrm{E}_{0}\) is the initial internalenergy per unit volume, and \(\mathrm{\mu}\) is the volumetric strain. The linear polynomial equation of state may be used to model air by setting \(\mathrm{C}_{1}\), \(\mathrm{C}_{2}\), \(\mathrm{C}_{3}\), and \(\mathrm{C}_{6}\) to zero and setting \(\mathrm{C}_{4}=\mathrm{C}_{5}=\gamma-1\),where \(\gamma=1.4\) is the ratio of the specific heat of air. The equation of state can be further simplified for perfect gas as follows:  
\[\mathrm{P}=(\gamma-1)\frac{\mathrm{\rho}}{\mathrm{\rho}_{0}}\mathrm{E}_{0} \tag{4}\]  
Figure 2: External tunnel explosion experiments. where \(\rho\) is the current density of air. For highly explosive materials, including C4, MAT_HIGH_EXPLOSIVE_BURN was used. The Jones-Wilkins-Lee (JWL) equation of state was necessary to simulate the behavior of explosive wave propagation and is expressed assfollows:  
\[\mathrm{P}=\mathrm{A}\bigg{(}1-\frac{\omega}{\mathrm{R}_{1}\mathrm{V}}\bigg{)} \mathrm{e}^{-\mathrm{R}_{1}\mathrm{V}}+\mathrm{B}\bigg{(}1-\frac{\omega}{ \mathrm{R}_{2}\mathrm{V}}\bigg{)}\mathrm{e}^{-\mathrm{R}_{2}\mathrm{V}}+\frac{ \omega\mathrm{E}_{0}}{\mathrm{V}} \tag{5}\]  
where A, B, R\({}_{1}\), R\({}_{2}\) and \(\omega\) are coefficients for a specific explosive; P is pressure; V is the relative volume; and E\({}_{0}\) is the initial internalenergy per unit volume. If A and B are set to zero and \(\omega\) is \(\gamma-1\), Equation (5) represents the equation of state for an ideal gas [12]. In the FSI model as shown in Figure 3, both the explosive and the air were modeled with Eulerian meshes, while the tunnel and soil ground were modeled with Lagrangian meshes. To obtain accurate solutions, one Lagrangian element should cover two Eulerian elements when coupling the two meshes [16]. The mesh lengths of air and C4 charges were 1 cm, while those of the tunnel and the ground model were 2 cm. The tunnel structure and ground were made out of used shell elements. The air and explosive charges were represented by solid elements. The Constrained Lagrangian in Solid command was adopted for the coupling between Lagrangian elements and Eulerian elements.