## 3 Local Deformation Characteristics of the Bolt  
From the analysis of the numerical simulation in the previous chapter, it was determined that the local deformation area of the bolt was approximately the same as the damaged area of the anchoring agent. With the dislocation of the structural surface, the anchoring agent near the structural surface was damaged, which provided space for the deformation of the bolt, so in this range, it was no longer constrained by the anchoring agent and was in a free state. A schematic diagram of the bolt deformation at the bolt-structural face intersection is shown in Figure 13.  
Figure 13: Schematic diagram of rock bolt deformation at structural surface. \(a\) is the length of the anchoring agent damaged area after the dislocation of the structural surface, \(a^{\prime}\) is the original length of the anchoring agent damage area, and \(l\) is the shear displacement of the structural surface dislocation, which is known by the geometric relationship:  
\[a^{\prime}=\sqrt{a^{2}-l^{2}} \tag{3}\]  
It was determined that the strain \(\varepsilon\) of the local deformation area of the bolt is:  
\[\varepsilon=\frac{\Delta a}{a^{\prime}}=\frac{a-a^{\prime}}{a^{\prime}} \tag{4}\]  
The strain \(\varepsilon\) in the area where the local deformation of the anchoring agent occurred was not uniformly distributed, as shown in Figure 13. Taking the \(OA\) section for analysis, the strain at the location of point \(O\) at the structural surface was the largest, and the strain at the location of point \(A\) at the distal end was the smallest. Assuming that the strain in the range was linearly distributed, the strain distribution in the local deformation area of the bolt is:  
\[\varepsilon(x)=2\varepsilon\bigg{(}1-\frac{2}{a}x\bigg{)}\bigg{(}0\leq x\leq \frac{a}{2}\bigg{)} \tag{5}\]  
The axial stress distribution in this local deformation region is:  
\[\sigma(x)=2\varepsilon E\bigg{(}1-\frac{2}{a}x\bigg{)}\bigg{(}0\leq x\leq \frac{a}{2}\bigg{)} \tag{6}\]  
At the same time, with the dislocation of the structural surface, the bolt originally anchored at an angle of \(90^{\circ}\) was deformed near the structural surface, making the anchoring angle less than \(90^{\circ}\). So, the axial force can be decomposed into normal force \(F_{\mathrm{n}}(x)\) and tangential force \(F_{\mathrm{s}}(x)\), and, from the geometric relationship, it can be obtained as follows:  
\[F_{n}(x)=A\cdot\sigma(x)\cdot\frac{a^{\prime}}{a} \tag{7}\]  
\[F_{\mathrm{s}}(x)=A\cdot\sigma(x)\cdot\frac{l}{a} \tag{8}\]  
Therefore, the normal force and tangential force of the bolt at the structural surface could be determined by knowing the anchor bolt cross-sectional area (\(A\)), the damage length of the anchoring agent (\(a\)), and the shear displacement of the structural surface (\(l\)). According to the fitting formula of the shear displacement of the structural surface-damage length of the anchoring agent in the previous chapter, the relevant mechanical parameters of the bolts in the different rock structural surfaces were obtained as shown in Table 4.  
The calculated axial stresses of the bolt at the structural faces of the coal, sandstone, and granite rock when the bolt yielded were 212.76 MPa, 196.54 MPa, and 182.09 MPa, respectively, and the axial stresses obtained from numerical simulations were 221.36 MPa, 201.78 MPa, and 188.69 MPa, respectively. The axial stress distribution along the length of the bolt is shown in Figure 14, which shows that the calculated results of the axial stress of the bolt are in general agreement with the numerical simulation results. The bolt deformation areas in the simulation results are slightly larger than the bolt deformation