# Development of Energy-Based Impact Formula--Part I: Penetration Depth
## 3 Development of New Energy-Based Penetration Formula
### Concept of New Formula
#### 3.1.2 Forces Acting in Impact Mechanism  
Considering the experimental results and the existing theoretical models, it can be assumed that the penetration depth is affected by compressive strength, while the scabbing depth is affected by tensile strength. In addition, the slope of the spalled and scabbed cones is in the form of shear failure. Therefore, it is shown that spalling failure is shear failure affected by compressive strength (Figure 3), while scabbing failure is shear failure affected by tensile strength (Figure 4). In addition, it is assumed that shear and compressive stresses act on the interface between a spalled concrete cone and a reinforced concrete panel (Figure 3). Shear and tension stresses act on the interface between a scabbed concrete cone and a panel (Figure 4). Based on the punching slab research [27, 28, 29, 30], the shear stress capacities controlled by compression or tension are defined as Equations (26) to (29), respectively. Here, \(v_{nc,st}(z)\) indicates the shear stress capacity controlled by compression under static loading, while \(v_{nt,st}(z)\) indicates that controlled by tension under static loading. As shear strength is defined under static loading, dynamic material properties should be applied considering dynamic conditions (Equations (30) to (33)). \[v_{nc,st}(z) = \sqrt{f^{\prime}_{c,st}(f^{\prime}_{c,st}-\sigma_{u,st}(z))}\ ( \text{MPa, mm}) \tag{26}\] \[v_{nc,st}(z) = \sqrt{f^{\prime}_{t,st}(f^{\prime}_{t,st}+\sigma_{u,st}(z))}\ ( \text{MPa, mm})\] (27) \[\sigma_{u,st}(z) = f^{\prime}_{c,st}\bigg{[}2\bigg{(}\frac{\alpha z}{c_{u}}\bigg{)} \!-\!\bigg{(}\frac{\alpha z}{c_{u}}\bigg{)}^{2}\bigg{]}\ ( \text{MPa, mm})\] (28) \[f^{\prime}_{t,st} = f_{t,st}(1-\sigma_{u,st}(z)/f^{\prime}_{c,st})\ (\text{MPa})\] (29) \[v_{nc,dyn}(z) = \sqrt{f^{\prime}_{c,dyn}(f^{\prime}_{c,dyn}-\sigma_{u,dyn}(z))} \ (\text{MPa, mm})\] (30) \[v_{nc,dyn}(z) = \sqrt{f^{\prime}_{t,dyn}(f^{\prime}_{t,dyn}+\sigma_{u,dyn}(z))} \ (\text{MPa, mm})\] (31) \[\sigma_{u,dyn}(z) = f^{\prime}_{c,dyn}\bigg{[}2\bigg{(}\frac{\alpha z}{c_{u}}\bigg{)} \!-\!\bigg{(}\frac{\alpha z}{c_{u}}\bigg{)}^{2}\bigg{]}\ ( \text{MPa, mm}) \tag{32}\]  
Figure 2: Energies involved in impact mechanism. \[f^{\prime}_{t,dyn}=f_{t,dyn}(1-\sigma_{u,dyn}(z)/f^{\prime}_{c,dyn})\ (\text{MPa}) \tag{33}\]  
where \(v_{nc,st}\)(z) and \(v_{nc,dyn}\)(z) are the static shear stress capacity (MPa) controlled by static compression and the dynamic shear stress capacity (MPa) controlled by dynamic compression, respectively; \(v_{nt,st}\)(z) and \(v_{nt,dyn}\)(z) are the static shear stress capacity (MPa) controlled by the static tension and the dynamic shear stress capacity (MPa) controlled by the dynamic tension, respectively; \(z\) is the distance from the neutral axis (mm); \(f^{\prime}_{c,st}\) and \(f^{\prime}_{c,dyn}\) are the static compressive stress of concrete and the dynamic compressive stress of concrete (MPa), respectively; \(f_{t,st}\) and \(f_{t,dyn}\) are the static tensile stress of concrete and the dynamic tensile stress of concrete (MPa), respectively; \(\sigma_{u,st}\) and \(\sigma_{u,dyn}\) are the static compressive stress (MPa) and the dynamic compressive stress affected by strain rate, respectively; \(\sigma_{u,st}\) and \(\sigma_{u,dyn}\) are related to the distance \(z\) using the parabolic stress-strain relationship of concrete; \(\alpha\) is the ratio of the current compressive strain \(\alpha\varepsilon_{o}\) to the strain \(\varepsilon_{o}\) related to the uniaxial concrete compressive strength; \(\alpha\varepsilon_{o}\) is the current compressive strain at the extreme compression fiber of the cross-section; and \(c_{u}\) is the depth of the concrete compression zone (mm). The \(f^{\prime}_{t,st}\) and \(f^{\prime}_{t,dyn}\) are the effective static tensile strength reduced by the transverse compressive stress (MPa) [31] and the effective dynamic tensile strength reduced by the transverse compressive stress (MPa), respectively.