# Development of Energy-Based Impact Formula--Part I: Penetration Depth
## 1 Introduction  
Concrete is a primary material used in building structures and has contributed significantly to the development of the construction industry [1; 2; 3]. It became the main axis of the construction industry over a hundred years ago (before the beginning of the First World War), which led to an increased interest in protecting human life and the safety of building structures. There has also been a considerable interest in the impact resistance of concrete against conventional weapons. In the 1980s, the research on the blast and impact resistance of nuclear power plants was intensively carried out, and the research on the protection performance of structures against terrorism has been continued. To secure safety, blast and explosion design is important for preventing a building collapse and for minimizing local impact damage. Particularly, small fragments from explosions or conventional weapons (e.g., bullets and missiles) or car/airplane crashes can cause local damage, which also poses a safety threat. The local damage may cause the entire building to collapse, due to the lack of structural integrity, ductility, and irregularities that prevent the propagation of local damage [4]. There are several manuals or guidelines that recommend impact formulae for assessing local damage to reinforced concrete panels. The US Army recommends the Army Corps of Engineers (ACE) formula for evaluating the local damage to concrete panels in accordance with TM-5-855-1 [5], while the US Air Force recommends the modified National Defense Research Committee (NDRC) formula for evaluating the penetration depth in accordance with ESL-TR-87-57 (1987) [5]. The British Army recommends the UK Atomic Energy Authority (UKAEA) formula in accordance with the British Army Manual [5]. The ACE formula, the modified NDRC formula, and the UKAEA formula weredeveloped in 1946, 1976, and 1990, respectively [5]. However, because these recommended formulae are old, they do not reflect the characteristics of the present concrete matrices. Various materials have been developed to overcome the weakness of concrete. Concrete exceeding 100 MPa is widely used in the construction industry. Various fibers have been added to increase the ductility and energy dissipation capacity of the concrete, and blast furnace slag was added to improve the fluidity and strength. Concrete with various properties has been produced by varying the concrete mixture and methods [6]. In particular, new concrete matrices exhibit improved tensile strength due to the presence/addition of steel fibers. For most high-strength concrete, steel fibers are used, which greatly improve the concrete's tensile characteristics. There are few impact formulae that can be applied to high-strength concrete, and there is no impact formula that reflects this increased tensile strength due to steel fibers, except for the [PERSON] formula. Therefore, it is necessary to develop a new impact formula that can be applied to high-strength concrete and reflects its increased tensile strength due to steel fibers. In addition, the existing impact formulae are inaccurate when the ratio of a projectile's diameter to a panel's thickness is low. For these reasons, it is necessary to develop a new impact formula that is highly accurate even in cases involving a low projectile diameter-to-panel thickness ratio. However, most of the models have been derived from experiments that used a limited number of variables. The force-penetration depth relationship was commonly used. In addition, there were some errors in the predicted values assigned to different experimental results. Therefore, a new theoretical model, rather than that based on the experimental results, and a theoretical framework are required to develop a new impact formula. This paper proposes a new impact formula based on the theoretical framework proposed by [PERSON] et al. [7], and the study on the impact mechanism, impact force, strain rate, and coefficient related to the penetration is extended.