# Development of Energy-Based Impact Formula--Part I: Penetration Depth
## 3 Development of New Energy-Based Penetration Formula
### Involved Energies in Impact Mechanism
#### 3.2.5 Spalling-Resistant Energy (E\({}_{\text{SP}}\))  
Some impact energy will transfer to a concrete panel, which becomes a part of the concrete panel spall. The spalling shape was normally considered a truncated cone in previous experimental studies. As shown in Figure 8, the idealized concrete failure and spalling-resistant energy are determined from the kinetic energy of the spalled concrete that also has mass (kg) and velocity (m/s), as shown in Equation (99), and the spalled failure is related to drag force (\(F_{dmg}\)) (Equations (100) to (102)) [7]. This kinetic energy is the same as that in the case where the forces acting on the area and volume of the spalled concrete are multiplied (Equation (103)), and is known as energy density (energy/volume). \[E_{SP}=\frac{1}{2}M_{sp}V_{sp}^{2} \tag{99}\]  
\[F_{dmg}=\frac{\rho V_{sp}^{2}}{2}A_{sp} \tag{100}\]  
\[F_{sp}=\frac{\rho_{sp}V_{sp}^{2}}{2}A_{sp}=\frac{V_{sp}^{2}}{2}A_{sp}\frac{M_{ sp}}{Vol_{sp}} \tag{101}\]  
\[F_{sp}\frac{Vol_{sp}}{A_{sp}}=\frac{M_{sp}}{2}V_{sp}^{2}=E_{SP} \tag{102}\]  
\[E_{SP}=F_{sp}\frac{Vol_{sp}}{A_{sp}} \tag{103}\]  
where \(E_{SP}\) is the spalling-resistant energy (J); \(M_{sp}\) is the mass of the spalled concrete cone (kg); \(F_{sp}\) is the force between the spalled cone and a concrete slab (N); \(V_{sp}\) is the velocity of the spalled concrete cone (m/s); \(Vol_{sp}\) is the volume of the idealized concrete cone (mm\({}^{3}\)); \(A_{sp}\) is the projected area of the idealized concrete cone (mm\({}^{2}\)); and \(\rho_{sp}\) is the density of the spalled cone (kg/mm\({}^{3}\)). The volume and area can be obtained from Equations (104) to (107). \[Vol_{sp}=\frac{\pi x_{pe}}{12}\big{(}4x_{pe}^{2}\tan^{2}\theta_{sp}+6d_{p}x_{pe }\tan\theta_{sp}+3d_{p}^{2}\big{)} \tag{104}\]  
\[A_{sp,big}=\frac{\pi}{4}\big{(}d+2x_{p}\tan\theta_{sp}\big{)}^{2} \tag{105}\]  
\[A_{sp,Little}=\frac{\pi d^{2}}{4} \tag{106}\]\[A_{sp,lateral}=\pi\big{(}d_{p}+x_{p}\tan\theta_{sp}\big{)}\frac{x_{p}}{\cos\theta_{ sp}} \tag{107}\]  
where \(Vol_{sp}\) is the volume of the spalled cone (mm\({}^{3}\)); \(x_{pe}\) is the penetration depth (mm); \(d_{p}\) is the diameter of a projectile (mm); \(A_{sp,big}\) is the top area of the spalled cone in Figure 8 (mm\({}^{2}\)); \(A_{sp,left}\) is the base area of the spalled cone in Figure 8 (mm\({}^{2}\)); \(A_{sp,left}\) is the lateral area of the spalled cone in Figure 8 (mm\({}^{2}\)); and \(\theta_{sp}\) is the slant angle of the lateral surface (degree). Finally, the spalling-resistant energy formula can be derived as shown in Equations (108) to (111), and the concrete spalling resistance force (\(F_{sp}\)) is defined using the shear stress controlled by compression, which is a spalled cone area. The reinforcing bar's action is neglected because the spalled cone is pressured under compression in the spalling condition. When the projectile has a long length or hyper velocity, the energy per unit area increases; hence, these effects should be applied to the penetration prediction formula. \(\alpha_{1}\) is the velocity effect factor applied at a velocity greater than 340 m/s. The value of 340 m/s is Mach 1, and if the projectile velocity exceeds 340 m/s, it will be hypersonic. This can lead to a totally different shock phenomenon [47]. Depending on the speed, it can be assumed that the speed of transferring the projectile energy to the concrete will also vary. In addition, if the projectile is slender, the energy per unit area will be increased. The effect factor takes into account the fact that the projectile length is \(\alpha_{2}\). In other words, energy transference velocity and energy per unit area should be considered even with the same kinetic energy.