## 4 Anchor Bolt Pull-Out Analysis
### Effect of Installation Conditions on the Fracture Mode  
Next, in order to make a basic consideration of the effects of anchor bolt installation conditions (bolt diameter, embedding depth, joint strength between bolt and concrete) on pull-out strength and fracture pattern, an analysis model with the same dimensions as the specimens used in the previous study [6] was examined. An analysis model was created in which anchor bolts were embedded in the center of a concrete block, and analysis was performed by constraining the vertical displacement of the particles at the four corners, as shown in Figure 10. Here, assuming anchor bolts used to attach seismic retrofitting members to concrete blocks, we examined load-bearing performance when the embedding depth is not sufficient or when chemical adhesives with different adhesive strengths are used. Table 3 shows the material parameters used in the analysis, and Table 4 shows the condition parameters for anchor installation. In this study, 0.5 m/s was selected as the pull-out speed given to the anchor bolts according to the maximum speed level assumed for ordinary seismic wave. First, the crack growth process of concrete with the increase of pull-out displacement of anchor bolts was investigated, and the adhesive stress distribution that changed, at the \begin{table}
\begin{tabular}{c c c} \hline \hline \multirow{2}{*}{**Case**} & **Maximum Bond Stress** & **Anchor Bolt Diameter \(d\)** & **Anchor Bolt Embedment** \\  & **\(\tau\) (MPa)** & **(mm)** & **Depth \(l_{e}\) (mm)** \\ \hline
1 & 8 & 16 & 48 \\
2 & 12 & 16 & 48 \\
3 & 16 & 16 & 48 \\
4 & 12 & 8 & 48 \\
5 & 12 & 24 & 48 \\
6 & 12 & 16 & 32 \\
7 & 12 & 16 & 64 \\ \hline \hline \end{tabular}
\end{table} 
Table 4: Analysis cases. \begin{table}
\begin{tabular}{c c c} \hline \hline  & **Steel** & **Concrete** \\ \hline Compressive strength (MPa) & 408.2 & 48.0 \\ Tensile strength (MPa) & 408.2 & 3.1 \\ Poisson’s ratio & 0.30 & 0.20 \\ Density (kg/m\({}^{3}\)) & 79,000 & 2350.0 \\ Young’s modulus (MPa) & 214,000 & 21,430 \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Material constants used in the analysis. \begin{table}
\begin{tabular}{c c c} \hline \hline  & **Steel** & **Concrete** \\ \hline Compressive strength (MPa) & 408.2 & 48.0 \\ Tensile strength (MPa) & 408.2 & 3.1 \\ Poisson’s ratio & 0.30 & 0.20 \\ Density (kg/m\({}^{3}\)) & 79,000 & 2350.0 \\ Young’s modulus (MPa) & 214,000 & 21,430 \\ \hline \hline \end{tabular}
\end{table} 
Table 3: Material constants used in the analysis with different installation conditions. same time, was analytically considered. Figure 11 show the final crack pattern of Case 1, 2, and 3, respectively. With the increase of adhesive stress limit, the fracture pattern changed from bond fracture, bond-cone combined fracture, to cone fracture. As shown in Figure 11, there are many small cracks around the anchor bolt, and a major crack can be found at the bottom of the anchor bolt. In the case of adhesive stress limit is 16 MPa, the major crack developed to the surface of the concrete, forming a cone shaped fracture pattern. The bond stress development are shown in Figure 12. Due to the mechanical symmetry of the loading condition, the resulting figure shows only zoomed half of the analysis area. Focusing on the result of Figure 12a,b with the adhesive stress limit set to 8 MPa, when the pull-out displacement reaches \(x=75\) um, fine cracks are seen near the anchor bolt, and the bond stress rises to about 5 MPa in the entire circumference of the anchor bolt. When pull-out displacement \(x=225\) um, a major crack that could lead to cone fracture can be found at the bottom of the anchor bolt. In addition, adhesive fracture was confirmed in the concrete area with a depth of 10 to 15 mm around the anchor bolt. Regarding the adhesive stress distribution, it was found that the stress decreased to zero at the bottom and at about 10-15 mm depth of the concrete. This phenomenon indicate that the bond stress can be influenced by cracks.