# Development of Energy-Based Impact Formula--Part I: Penetration Depth
## 3 Development of New Energy-Based Penetration Formula
### Penetration Depth Formula  
The penetration depth can be determined using the energy equilibrium. The kinetic energy (\(E_{K}\)) basically equals the sum of the deformed energy of a projectile (\(E_{DP}\)), the elastic penetrated energy of a concrete panel (\(E_{EP}\)), the overall deformed energy of a slab (\(E_{SD}\)), and the spalling-resistant energy of a concrete panel (\(E_{SP}\)), as shown in Equation (117), and in special cases, the tunneling-resistant energy (\(E_{T}\)) will be added to Equation (117). As the value to be obtained from the impact formula is the penetration depth, the equation of energies can be summarized as shown in Equation (118). If the energy concentration effect (\(\alpha_{1}\alpha_{2}\)) is less than 1, the spalling-resistant energy may exceed the kinetic energy, which does not satisfy the law of conservation of energy. Therefore, when the value of the energy concentration effect is less than 1, it is set to be equal to 1, to satisfy the law of energy conservation (Equation (119)). By contrast, when the energy concentration effect exceeds 1, \(E_{SP}\) is smaller than \(E_{K}\). As a result, to satisfy the law of energy conservation, the overflow energy produces the tunneling depth. Assuming that the cone shape is hemispherical, to simplify the equation (Figure 9), the final penetration depth can be summarized as shown in Equations (120) to (123). In Equation (116), the tunnel energy is obtained, and the tunneling depth can be obtained by summarizing the tunneling depth (\(x_{t}\)) following Equations (124) to (126). Thus, the penetration depth when \(\alpha_{1}\alpha_{2}\) is over 1 is expressed as shown in Equation (127).  
\[E_{K}=E_{DP}+E_{EP}+E_{SD}+E_{SP}+(E_{T}) \tag{117}\]  
\[E_{K}-E_{DP}-E_{EP}-E_{SD}=E_{SP}\leq E_{K} \tag{118}\]\[(E_{K}-E_{DP}-E_{EP}-E_{SD})=\] \[0.7f^{\prime}_{c,dyn}\bigg{[}\pi\Big{(}d+x_{pe}\tan\theta_{s} \Big{)}\frac{x_{p}}{\cos\theta_{s}}+\frac{\pi\alpha_{p}^{2}}{4}\bigg{]}\alpha_{ 1}\alpha_{2}\frac{V_{\text{sc}}}{\lambda_{pe}}\leq E_{K} \tag{119}\] \[x_{pe}=\sqrt[3]{E_{SP}\frac{6}{8\times 0.7\times f^{\prime}_{c,dyn} \times\alpha_{1}\alpha_{2}\alpha_{3}\times\pi}}\ (1\leq\alpha_{1}\alpha_{2})\] (120) \[\alpha_{1}=\left(\frac{V_{imp}}{340}\right)^{0.15}\] (121) \[\alpha_{2}=\ln\left(0.8+\frac{l_{p}}{d_{p}}\right)\] (122) \[\alpha_{3}=\left(\frac{h}{d}\right)^{0.1}+0.2\] (123) \[(E_{K}-E_{DP}-E_{EP}-E_{SD})-(E_{K}-E_{DP}-E_{EP}-E_{SD})\frac{1}{ \alpha_{1}\alpha_{2}}=E_{T}\] (124) \[E_{T}=\pi d_{p}(2x_{t}^{2}+0.25d_{p}x_{t}+x_{t}^{2}N_{p}\tau_{dyn})\] (125) \[x_{t}=\frac{\sqrt{\frac{E_{T}}{\pi d_{p}}(2+N_{p}\tau_{dyn})+0.01 6d_{p}^{2}}-0.125d_{p}}{2+N_{p}\tau_{dyn}}\] (126) \[x_{pe}+x_{t}=\sqrt[3]{E_{SP}\frac{6}{8\times 0.7\times f^{ \prime}_{c,dyn}\times\alpha_{1}\alpha_{2}\alpha_{3}\times\pi}}+\frac{\sqrt{ \frac{E_{T}}{\pi d_{p}}(2+N_{p}\tau_{dyn})+0.016d_{p}^{2}-0.125d_{p}}}{2+N_{ p}\tau_{dyn}}(\text{for}\ \alpha_{1}\alpha_{2}>1) \tag{127}\]  
where \(f^{\prime}_{c,dyn}\) is the compressive strength of the concrete affected by the dynamic load; \(x_{pe}\) is the penetration depth (mm); \(\alpha_{1}\) is the velocity effect factor; \(\alpha_{2}\) is the projectile length factor; \(\alpha_{3}\) is the factor related to the panel thickness and the diameter of a projectile; \(d_{p}\) is the diameter of a projectile (mm); \(x_{t}\) is the length of the tunnel (mm); \(N_{p}\) is the nose shape factor; \(\tau\) is the bond stress (MPa); and \(M_{p}\) is the mass of a projectile (kg).