## 1 Introduction  
Concrete and steel materials are two of the most common construction materials [1; 2; 3; 4]. Concrete or concrete-steel composite plate members are often exposed to high strain-rate loading, such as blasts and impacts [5], and there are impact formulae for estimating the local damage caused by high-velocity impact loading. These impact formulae are classified into penetration depth, scabbing limit, and perforation limit formulae, where the scabbing and perforation limits indicate the minimum thicknesses required to prevent scabbing and perforation, respectively [6; 7; 8]. Penetration depth, in particular, has been extensively studied, and over 27 formulae have been proposed, so far, for predicting local damage [9; 10; 11; 12]. However, despite the importance of scabbing and perforation limit formulae in assessing secondary risk, they are fewer than the penetration depth formulae proposed so far. Secondary critical threats, such as chemical and biological weapons, conventional weapons, and soldiers, can enter concrete structures through perforated holes, even if a building does not collapse. When a spalling/scabbing failure occurs due to impact loadings, the fragments/debris can fly into people/things in concrete structures, resulting in secondary physical and personal damage. As the thickness of the concrete panels increases, the safety of the concrete structures against external threats is enhanced. However, the concrete thickness cannot be increased to an unlimited extent and increasing it can be very expensive. Therefore, the scabbing and perforation limits are as important as the penetration depth.  
To estimate the damage level of the concrete member caused by impact loadings, various studies have been conducted. Most studies first focused on deriving the penetration depth using the force-penetration relation, and then derived the scabbing and perforation limit formulae from the regression analysis conducted using the relations between penetration depth and low boundary of scabbing and perforation [17]. In other words, there are hardly any scabbing and perforation limit formulae obtained using the theoretical background.  
[PERSON] et al. [12] proposed energy-based penetration depth based on the theoretical background. [PERSON] and [PERSON] [22] defined the impact mechanism and energies involved and proposed the penetration depth formula using the energy conservation law. The impact mechanism involves seven types of energy: kinetic energy (\(K_{E}\)), deformed energy of the projectile (\(E_{DP}\)), elastically deformed energy of the concrete panel (\(E_{EP}\)), overall deformed energy of the concrete panel (\(E_{SD}\)), spalling-resistant energy (\(E_{SP}\)), tunneling-resistant energy (\(E_{T}\)), and scabbing-resistant energy (\(E_{SC}\)). Of these, scabbing-resistant energy occurs in the case where the target concrete panel is insufficient to absorb the spalling-resistant energy and/or tunneling-resistant energy.  
In this paper, the condition of scabbing failure, including perforation failure, is theoretically defined using the energy-depth relation and scabbing-resistant energy. Also, the scabbing depth, scabbing limit, and perforation limit formulae underpinned by a theoretical framework are proposed rather than that based on the experimental results. In particular, this study is important in that the scabbing depth formula is proposed, since there have been no prior existing formulae to estimate scabbing depth. The proposed impact formulae are then compared with the existing data.