## 1 Introduction
For assessment of local denting of cylindrical shells, only a few analytical studies were provided by [PERSON] et al. [10; 11; 12], [PERSON] et al. [13], and [PERSON] et al. [14]. They focused on deriving the simplified formulae to predict the quasi-static denting behaviors of stiffened cylinders. Simplified formulae are the most rapid tools for predicting the impact strength of cylinder structures, especially during the initial design stage and the risk assessment. In these studies, the cylindrical shell is assumed as perfectly plastic material and neglected the elastic strains in the damage areas. It is also assumed that the outside of the damage regions is rigid, and that the plastic damage zone is localized and finite. However, in real structural behaviors, the dynamic strain rate effects and inertial forces should be considered for assessing the extent of local denting damage accurately [15]. Furthermore, in these formulations, the force-displacement and deformation profile is not smooth because of sudden jumps in the vicinity of the stiffeners and it does not match well the experimental observations. Because these researches are purely theoretical, they are required to be verified with reliable test data. Detailed explanations of these formulations are given in Section 5. More recently, [PERSON] et al. [2] improved the existing formulations proposed by [PERSON] and [PERSON] [13] for the local denting behavior of ring-stiffened cylinders. He revised the circumferential bending resistance by including the effect of the ring stiffeners and changing the fully plastic bending moment of the cylindrical shell by an equivalent bending moment per length. This simple formulation has good accuracy when compared to the limited experimental data. However, more studies are needed to validate this analytical approach with a wide range of geometry and reliable test data. Another approximate formulation to forecast the permanent local denting damage of the ring-stiffener cylinder was reported by [PERSON] et al. [9]. These formulations are derived using the FEA results of rigorous case studies on the real design full-scale ring-stiffened cylinders. It has high accuracy when compared with available test models. However, there are several shortcomings existing in these formulations. First, the impact location was considered in these formulations only at mid-bay of the ring-stiffeners and perpendicular to the axial direction. In actual cases, the impact position is varying on the cylinder structures. It is clear that the impact behavior of the stiffened cylinder was sharply improved when impact locations were close to ring stiffener [7]. Thus, the effects of various impact locations need to consider predicting the extent of damage to cylinder structures. Secondly, the damage deformations were only generated by a rigid knife-edge indenter with a rounded tip, but the striking mass in a real case is not only a knife-edge indenter but also a hemispherical or rectangular indenter type. Therefore, to get accuracy of the extent of damage prediction, all the information on striking mass types should be investigated. To overcome these shortcomings, the proposed formulations developed in this study for assessing the permanent local dent depth, include all parameters such as impact location factor, impact angle factor, and indenter shape factor. To the best of the authors' knowledge, until recently, there was no reliable equation or code recommendation for assessing the permanent damage of ring- or/and stringer-stiffened cylinders under dynamic mass impact loading. Therefore, it is sorely needed to propose the new formulations to predict the extent of local dent depth of these structures, and this study's objective is to fill that gap. Nowadays, finite element modeling (FEM) is an excellent tool for predicting ship collisions and groundings. Numerical assessment of cylinder structures under collision loading conditions has been performed by many researchers [16; 17; 18; 19; 20; 21; 22; 23].