## 1 Introduction  
When constructing a structure in the civil engineering field, a joining method that has structural continuity between concrete members and other members (reliable stress transmission) is very important, and joining with anchor bolts is one of the typical methods not only in new structures but also in existing structures. In addition, anchor bolts are used to connect the seismic retrofitting members, such as bridge restrainers [1] and shear walls, to existing concrete structures. Anchor bolts are also applied in railway sleepers [2], modular wall constructions [3], tunnel roofs [4], and nuclear-related facilities [5], and they are expected to continue to be used in the future. However, only a simple evaluation has been made for the ultimate strength when a pull-out load is applied to the anchor bolt. Therefore, there is a concern that the pull-out strength of bolts will be significantly reduced due to improper installation conditions and the aging deterioration of concrete materials. In general, fracture modes of anchor bolts subjected to a pull-out load are mainly classified into bolt fracture, concrete body fracture, and bond fracture. Table 1 shows the typical fracture modes assumed when designing anchor bolts and shows the equations for calculating the pull-out strength of each fracture type. Basically, it is designed using the pull-out strength of the fracture mode, which has the lowest strength among the assumed these fracture modes shown in Table 1.  
However, when installing new anchor bolts in an existing concrete structure, it may be difficult to secure sufficient anchor embedding depth and spacing between adjacent anchor bolts, depending on the arrangement of existing reinforcing bars inside. Furthermore, it is well known that the actual fracture mode of anchor bolt joint cannot be clearly classified into three types, and that a combined fracture mode in which cone fracture and adhesive fracture might occur depending on the installation conditions. For example, Figure 1a [6] shows the fracture of an anchor bolt in the pull-out test conducted by the authors, and a clear combined fracture mode in which cone fracture and adhesion fracture are mixed is obtained. Thus, a simple evaluation based on the strength comparison of each mode shown in Table 1 may not be applicable. Therefore, it is very important to establish a method that can predict the accurate fracture mode and pull-out strength of anchor bolts with various material properties, structural specifications, and loading conditions. In addition, when installing anchor bolts in an existing concrete structure where complex rebars arrangements are already embedded inside, it is often difficult to secure sufficient embedding depth, anchor bolt spacing, and distance from the edge of the structural member depending on the position of the existing internal reinforcing bars. Figure 1b shows an example of damage to an anchor bolt joint with a bridge collapse prevention device attached by the earthquake force. This accident is a typical fracture due to the insufficient distance from the edge, and this installation situation is not included in Table 1 either. Thus, in order to design and apply anchor bolts to structure members with sufficient structural safety, the installation situations and the damage patterns which are not included in the existing specifications are also needed to be taken into consideration. In this study, minimum allowable embedding depth, minimum bolt spacing, and minimum distance from the edge of structural member were evaluated, respectively, and proper reinforcement for the lack of distance from the edge of structural member was considered. Under these backgrounds, many researchers have already conducted the research on the pull-out fracture of anchor bolts.