## 2 Theoretical Basis
### Shift Support Characteristic Curve  
Shaft excavation construction is typically carried out using parallel excavation-support operations, as shown in Figure 4. During the shaft excavation process, the excavation face exerts a constraint effect equivalent to that of the setting support structure on the surrounding rock over a certain range behind the excavation. This constrained effect size is therefore reflected by the LDP, and the support structure is generally set at a distance \(L\) from the excavation surface. At time \(t_{1}\) for section Y in Figure 4, when the excavation is carried forward to excavation 1 at a distance \(L_{1}\), the Y section is partially reduced by the constraint effect of the excavation surface and the rock stress is partially released behind the excavation surface. This produces an initial radial displacement \(u_{in}\). The support structure is set after the rock stress is released, and the rock in section Y acts on the pressure \(P_{I1}=0\) on the support structure. An additional point of support \(P_{i}\) is required to suppress the rock and produce the same radial displacement \(u_{in}\) without the effect of excavation surface confinement. The distance between excavation face 2 and section Y is \(L_{2}\). As the excavation face continues to advance to excavation face 2 (\(t_{2}\) in Figure 4), the constraint effect of the excavation face on section Y is greatly weakened owing to expansion of the spacing. The radial displacement of the surrounding rock at section Y and the pressure acting on the supporting structure continue to increase to \(u_{2}\) and \(P_{I2}\), respectively. When the excavation face advances to excavation face 3 at a distance of \(L_{3}\) from section Y at time \(t_{3}\) (Figure 4), the excavation face no longer has a limiting effect on section Y and the stresses of the surrounding rock and supporting structure at section Y are adjusted to a stable state. The stable radial displacement of the surrounding rock is \(u_{eq}\) and the stable pressure of the supporting structure is \(P_{eq}\). As shown in Figure 5, the supporting pressure of the supporting characteristic curve can be expressed as:  
\[P=K\cdot u \tag{18}\]  
where \(K\) is the support stiffness coefficient (i.e., slope of the SCC). When \(u_{in}<u<u_{el}\), the supporting structure is in the elastic stage; when \(u_{el}<u<u_{\text{max}}\), the supporting structure is in the yielding stage and \(K=0\); when \(u>u_{\text{max}}\), the support system is destroyed. For end-anchored anchor rods and anchor cables:  
\[ \begin{array}{c}K_{bol}=\frac{1}{S_{c}S_{l}[\frac{t_{\text{Lab}}}{\mu\pi^{2}E_{ l}}+Q]}\\ P_{\text{max},bol}=\frac{T_{\text{max}}}{S_{c}S_{l}}\end{array} \Bigg{\}} \tag{19}\]  
where \(K_{bol}\) is the supporting stiffness of the anchor rod, \(P_{\text{max},bol}\) is the maximum bearing capacity of the anchor rod system; \(E_{bol}\) is the elastic modulus of the anchor rod material,  
Figure 4: Stress release process in shaft excavation. Figure 5: Characteristic curves of the wall rock and supporting structure. \(R\) is the excavation radius of the circular roadway; \(S_{c}\) and \(S_{I}\) are the anchor rod spacing and row spacing, respectively; \(L_{bol}\) is the length of the free end of the anchor rod; \(\phi\) is the diameter of the anchor rod; \(T_{\text{max}}\) is pull-out resistance of the anchor rod; \(Q\) is a constant related to the force and deformation characteristics of the anchor rod body, backing plate, and anchor head. For concrete support:  
\[K_{shot}=\frac{E_{con}}{(1+z_{con})}\frac{\left[R^{2}-(R-t_{shot})^{2}\right] }{(1-2z_{con})R^{2}+(R-t_{shot})^{2}}\frac{1}{R} \tag{20}\] \[P_{\text{max,}shot}=\frac{1}{2}\sigma_{c}\left[1-\frac{(R-t_{shot })^{2}}{R^{2}}\right]\]  
where \(K_{shot}\) is the stiffness of the concrete support, \(P_{\text{max,}shot}\) is the maximum bearing capacity of the concrete structure, \(t_{shot}\) is the shotcrete thickness, and \(E_{con}\) and \(v_{con}\) are the elastic modulus and Poisson's ratio of the shotcrete, respectively.