## 2 Deformation Characteristics of Rock Joint
### Analysis of Calculation Examples  
The compressive strength of a certain granite is 250 MPa; \(E\), \(\mu\), \(G\) are \(2\times 10^{4}\) MPa, 0.2, 8.3 \(\times\) 10\({}^{3}\) MPa, respectively [33]. For the granite bolted joint, the radius of the bolt \(R\) is 5 mm, the uniform load \(F\) is 5.2 \(\times\) 10\({}^{5}\) N/mm, and the overall displacement \(x_{0}\) is 5.4 mm at the interface between the bolt and the granite. By putting the preceding test data into Equation (24), we can obtain the analytical solution of the compression-shear side displacement along the radial distribution of the bolted rock joint as follows:  
\[u_{r}=[5.4+\frac{15.663}{\pi}(\frac{25}{r^{2}}-1)]\cos\theta \tag{25}\]  
Figure 5a shows the variation of \(u_{r}\) with distance \(r\) at different angles \(\theta\). As shown in Figure 5a, the value of \(u_{r}\) at different \(\theta\) decreases with the increase of \(r\). When \(r\) varies in the range of 5-30 mm (\(R\)-6\(R\)), the decrease trend of \(u_{r}\) with \(r\) is very obvious, from 5.40 mm (0\({}^{\circ}\)), 4.68 mm (30\({}^{\circ}\)), 3.82 mm (45\({}^{\circ}\)), 2.70 mm (60\({}^{\circ}\)) plunged to 0.55 mm (0\({}^{\circ}\)), 0.48 mm (30\({}^{\circ}\)), 0.39 mm (45\({}^{\circ}\)), 0.28 mm (60\({}^{\circ}\)), respectively, and an 89.8% drop of the initial radial displacement can be observed. When \(r\) is greater than 30 mm, the change of \(u_{r}\) tends to be gentle with the increase of \(r\). Figure 5b shows the change rate of \(u_{r}\) with the increase of \(r\), in which the rate of change goes to 0 after \(r\) is greater than 6\(R\). Therefore, when the value of \(r\) is greater than six times the radius of the bolt (i.e., 6\(R\), 30 mm), the bolt basically does not have any influence on the rock joint at the compression shear side, i.e., the influence distance of the bolt on the rock joint at the compression shear side is approximately 6\(R\).  
When \(r\) equals \(6R\), by comparing the value of \(u_{r}\), corresponding to \(\theta\) in Figure 5a (\(0^{\circ}\), \(30^{\circ}\), \(45^{\circ}\) and \(60^{\circ}\)), we can see that the maximum radial displacement \(u_{max}\) is 0.55 mm. The determination of this value is significant for the analysis of the mechanical properties of bolted rock joints. This radial displacement only accounts for 10.2% of the initial radial displacement \(u_{0}\) (5.40 mm). Assuming that the radial displacement at any point of the bolted rock joint is smaller than 0.55 mm, that is, the ratio to the initial radial displacement \(u_{0}\) is smaller than 10.2%, we can consider that the bolt has few influences on the rock joint at this point. According to the preceding assumptions, when \(u_{r}\) is 0.55 mm, \(\theta\) is \(0^{\circ}\), \(15^{\circ}\), \(30^{\circ}\), \(45^{\circ}\), \(60^{\circ}\), \(75^{\circ}\), the corresponding \(r\) is 30.00, 28.13, 23.69, 18.45, 13.46, and 8.53, respectively (Figure 6 gives these points), and the adjacent points are connected. Based on this, the action range diagram of the bolt at the compression-shear side of the rock joint can be drawn, as shown in Figure 6. The figure shown in the red curve in Figure 6 approximates an ellipse, and the radial displacements of all points inside the ellipse are greater than 0.55 mm, which means that the rock joint within the ellipse region can be significantly affected by the bolt.  
Figure 5: Variation of \(u_{r}\) in radial displacement. (**a**) Radial displacements at different distances. (**b**) Radial displacement decreases with radius.