# Development of Energy-Based Impact Formula--Part I: Penetration Depth
## 2 Literature Review  
Prior to 1943, the Ordnance Department of the US Army and the Ballistic Research Laboratory (BRL) conducted multiple impact tests on concrete structures [5]. Based on their experimental data, the Army Corps of Engineers [8] developed the penetration formula (Equation (1)), where the term \(M_{p}\)/\(d_{p}\)\({}^{3}\) indicates the missile caliber density. It was based on the regression analyses made on the experimental data for 37, 75, 76.2, and 155 mm steel cylindrical missiles. \[\frac{x_{pe}}{d_{p}}=\frac{3.5\times 10^{-4}}{\sqrt{f_{c}^{7}}}\Bigg{(}\frac{M _{p}}{d_{p}^{3}}\Bigg{)}\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\! \!\!\!\!\!\!\!\!\!\!\  
was proposed with the new \(K\) value. For now, the NDRC formula can be recognized as the modified NDRC formula.  
\[G=3.8\times 10^{-5}\frac{N_{p}M_{p}}{d_{p}\sqrt{f_{c}^{\prime}}}\bigg{(}\frac{V_{ imp}}{d_{p}}\bigg{)}^{1.8} \tag{2}\]  
\[G=\bigg{(}\frac{x_{pe}}{2d_{p}}\bigg{)}^{2}\text{ for }\frac{x_{pe}}{d_{p}} \leq 2\text{ or }G=\frac{x_{pe}}{d_{p}}-1\text{ for }\frac{x_{pe}}{d_{p}}>2 \tag{3}\]  
where \(G\) is the impact function; \(x_{pe}\) is the penetration depth (m); \(N_{p}\) is the nose shape factor (0.72, 0.84, 1.0, and 1.14 for flat, hemispherical, blunt, and very sharp noses, respectively); \(M_{p}\) is the mass of a projectile (kg); \(d_{p}\) is the diameter of a projectile (m); \(V_{imp}\) is the impact velocity of a projectile (m/s); and \(f_{c}^{\prime}\) is the compressive strength of concrete (Pa). [PERSON]'s group studied the safety of nuclear power plants with respect to missiles [11,12], focusing on the local impact damages to the concrete with respect to turbine missiles. [PERSON] and [PERSON] [12] proposed penetration equations using the impact factor (I), which comprises \(M_{p}\), \(N_{p}\), \(V_{imp}\), \(d_{p}\), and \(f_{c}^{\prime}\) (Equation (4)). [PERSON] and [PERSON] [12] modified the suggested penetration depth and formulae (Equations (5) to (7)) after further analyzing the additional experimental data obtained using small projectiles, such as bullets. The penetration depth (\(x_{pe}\)) could be predicted for all missile types, from bullets to turbines. In the formula, the value of \(N_{p}\) is taken from the modified NDRC formula (Equation (2)) [9]. \[I=\frac{M_{p}N_{p}V_{imp}^{2}}{d_{p}^{3}f_{c}^{\prime}} \tag{4}\]  
\[\frac{x_{pe}}{d_{p}}=-0.0308+0.2251I\text{ for }0.3\leq I\leq 4.0 \tag{5}\]  
\[\frac{x_{pe}}{d_{p}}=+0.6740+0.0567I\text{ for }4.0<I\leq 21.0 \tag{6}\]  
\[\frac{x_{pe}}{d_{p}}=1.1875+0.0299I\text{ for }21.0<I\leq 455 \tag{7}\]  
where \(x_{pe}\) is the penetration depth (m); \(N_{p}\) is the nose shape factor (0.72, 0.84, 1.0, and 1.14 for flat, hemispherical, blunt, and very sharp noses, respectively); \(M_{p}\) is the mass of a projectile (kg); \(d_{p}\) is the diameter of a projectile (m); \(V_{imp}\) is the impact velocity of a projectile (m/s); and \(f_{c}^{\prime}\) is the compressive strength of concrete (Pa). [PERSON] [13] proposed an impact formula using the force-penetration model. This formula considers the strain rate effect on concrete tensile strength (\(f_{t}\)) using the dynamic increase factor (\(S_{Hughes}\), Equation (8)), which is defined by introducing an impact factor (I, Equations (9) and (10)) that is similar to that used by [PERSON] and [PERSON] [12]. The dimensionless impact factor (I) represents the concrete penetrability and comprises \(M_{p}\), \(V_{imp}\), \(d_{p}\), and \(f_{t}\). While other formulae use concrete compressive strength (\(f_{c}^{\prime}\)), this formula is characterized by the use of the tensile strength of concrete (\(f_{t}\)). The values of \(N_{p}\) were set as 1, 1.12, 1.26, and 1.39 for flat, blunt, spherical, and very sharp noses, respectively. The penetration depth formula was derived from \(I\), \(S_{Hughes}\), and \(N_{p}\), as shown in Equation (11), and is applicable only when \(I\) is smaller than 3500.