Analytical solutions for bolted tunnel structures with voids behind the lining

Abstract

Understanding the mechanical behaviour of tunnels reinforced by end-anchored bolts and lining with voids behind the lining is crucial in determining the appropriate supporting measures. In this study, we constructed a geometric model for a tunnel with a single or double void. We used the substructure method and point matching method to obtain the elastic analytical solutions for the model. We examined the stress, internal force, and deformation characteristics of the tunnel and support structure under the influence of single and double voids. To validate the analytical model, we established a numerical model using the finite difference method. Additionally, we compared and analyzed the mechanical behaviour of tunnels reinforced by bolts and lining versus lining-only, and determined the contribution factor of the bolts.

KEYWORDS:

• Voids; substructure method; point matching method; internal force
• bolt contribution factor

1. Introduction

Tunnel and underground excavation projects are typically extensive and intricate, involving numerous uncertainties during construction [1]. It is common for various problems to arise during the excavation and support phases. Void defects common occur during the construction and operation of underground tunnels. The presence of voids or cavities often causes the detachment between rock and lining, which leads to localized stress concentration [2]. Due to voids or cavities, the supporting capability of the lining cannot be fully utilized, which seriously affects the safety and stability of the tunnel and can even result in the cracking and failure of the tunnel lining. Therefore, it is essential to explore the interaction between the surrounding ground and the lining and the load transfer mechanism under such conditions.

The typical methods of studying the effects of voids and cavities include numerical simulation, physical model testing, and theoretical analysis [2–4]. Meguid [5,6] used finite element analysis (FEA) to reveal the distribution of internal forces in the supporting structure caused by two-dimensional (2D) and three-dimensional (3D) voids under elastoplastic conditions. Augarde [7] simulated and analyzed the ground subsidence caused by spherical voids based on the elastic-plastic theory. Wang [8] used the Midas software to simulate the shaft and bending moment distribution in the lining, considering the effects of chamber shape, the location, size, depth of the void, and the lateral stress coefficient. Quan [9] found that with the concentration of the bending moment and shear stress, the reduction in axial pressure appeared on the lining structure at the cavities due to local collapses. Yu [10] and Li [11] revealed the deformation and stress distribution of the tunnel lining under the impact of voids using physical model tests. Zhang [2] used numerical simulations and laboratory experiments to analyze the bending moment and axial force of the tunnel lining with double voids and revealed the damage evolution of the lining influenced by voids. Li [3] derived the analytical solution of the stress distribution for an elliptical void by using the complex variable function and further used the classical elastic-plastic theory to determine the plastic boundary. Liao [12] established a stress solution at any point for an elliptical tunnel with voids under far-field stress. Based on Bobet and Yasuda [13–15] used the substructure method and the point matching method to obtain the elastic solutions for tunnels with voids behind lining under 2D and 3D seismic loading.

Although existing studies have explored the impact of voids on the tunnel lining, the mechanical characteristics and load transfer mechanism from the rock to the combined bolt and lining system under such conditions remain unclear. As rock bolt is one of the primary initial support methods, it is of significance to study the impact of voids on the effect of anchoring. In this paper, based on the solutions of Yasuda [14] and Bobet [16], the bolts, lining, and surrounding ground an integrated support system was considered. Subsequently, the interaction and load transfer mechanism between the surrounding ground and the support structure under the influence of voids was explored. The results can serve as a theoretical foundation for studying the quality of the support structure under such conditions.

2. Model development

2.1. Basic assumptions

For the convenience of theoretical analysis, the following assumptions are made: (1)Due to the fact that the longitudinal axis of underground structures is much larger than the horizontal dimension, it can be analyzed as a plane strain problem; (2)The forces from the end-anchored bolts are two concentrated forces of equal magnitude and opposite direction, regarded as distributed forces [16,17]; (3) Under the combined effect of primal earth stress and anchoring, the surrounding rock anchors of the tunnel are always in an elastic state; (4)The tunnel is deeply buried, and the cross-section is circular; (5)The surrounding ground is homogeneous and isotropic; (6)Infinitesimal deformations; (7)The deformations of the surrounding ground and rock bolts are consistent.

2.2. Analytical solutions

Due to gravity or construction quality, gaps, voids, or cavities often exist between the lining and the surrounding ground in tunnels, which can seriously affect the safety and stability of the tunnels [8]. In this work, we develop a mechanical model for the end-anchored bolts and lining in the tunnel with voids behind the lining. Based on the Yasuda’s [14] solution, the elastic solution for the tunnel is derived.

Figure 1 shows a schematic of a tunnel with a void, in which σ₀ is the initial in situ stress; r is the tunnel radius; R is the radius from the centre of the tunnel to the reinforced area and the unreinforced area; h is the lining thickness; E_r, μ_r, and G_r are the elastic modulus, Poisson's ratio, and shear modulus of the rock; ρ is the radial distance in the polar coordinates. In this study, tensile stress is positive and compressive stress is negative.

Figure 1. Schematic of a tunnel supported by end-anchored bolts and lining with a void behind the lining.

According to the substructure method [18] and the point matching method [14], for a tunnel supported by end-anchored bolts and lining with a void behind the lining, the support model can be decomposed into two different problems: rock stress under the actions of end-anchored bolts and lining in hydrostatic stress condition (Figure 2) and lining stress under the action of end-anchored bolts (Figure 3). Furthermore, rock stress supported by end-anchor bolts and lining in hydrostatic stress conditions (Figure 2(a)) can be equivalent to the superposition of rock stress under only the initial in situ stress (Figure 2(b)) [16] and rock stress under only the bolts and lining support (Figure 2(c)).

Figure 2. Rock under the actions of end-anchored bolts and lining in an infinite medium. Fig. (a) is equivalent to the superposition of Figs (b) and (c).

Figure 3. Lining stress when the rock is reinforced by end-anchored bolts.

The displacement of the surrounding ground consists of two parts: the displacement induced by the initial in situ stress and the displacement from the support of end-anchored bolts and lining [14], which can be written as: (1) $u_{(1)}=u_{(1)}^1+u_{(1)}^2$(1) where u₍₁₎ is the rock displacement under combined actions of in situ stress and support from the lining and end-anchored bolts; u¹₍₁₎ is the displacement caused by the in-situ stress; u2 (1) is the displacement caused by the lining and end-anchored bolts. Superscripts 1 and 2 indicate the in-situ stress and lining support, and subscripts (1) and (2) represent the bolted rock and the lining, respectively. f 1 (1) is the stress of the bolted rock under in situ stress; f 2 (1) is the rock stress on the contact surface between the rock and the lining; f₍₂₎ is the lining stress on the contact surface; u₍₂₎ is the displacement of the contact surface between the lining and the rock.

3. Approximate solutions for rock supported by end-anchored bolts and lining with the void behind the lining

3.1. Elastic solution for rock reinforced by end-anchored bolts under in situ stress

A simplification for the rock reinforced by end-anchored bolts under in situ stress is shown in Figure 2(b) [16]. Since the simplified geometry is axisymmetric, its stress function is independent of θ. Therefore, considering the deformation compatibility between the bolts and the rock, we can obtain the elastic solution for the anchored rock based on the stress boundary conditions and displacement boundary conditions. The stress and displacement can be expressed as follows.

The radial stress of the rock in the bolt reinforced area (r<ρ<R): (2) ${\sigma }_{\rho \text{re}}^1=\frac{-\frac{1}{2}\left({\sigma }_0\left(\begin{array}{l}-k(1+{\mu }_{\text{r}})\left({\mu }_{\text{r}}-\frac{1}{2}\right)r^4+k(1+{\mu }_{\text{r}})\left({\mu }_{\text{r}}-\frac{1}{2}\right)R{r}^3+(2k(1+{\mu }_{\text{r}})\left({\mu }_{\text{r}}-\frac{3}{4}\right){\rho }^2+E_{\text{r}}({\mu }_{\text{r}}-1)R)r^2-2k(1+{\mu }_{\text{r}})\left({\mu }_{\text{r}}-\frac{3}{4}\right){\rho }^2\mathrm{Rr}-R{\rho }^2{E}_{\text{r}}({\mu }_{\text{r}}-1)\end{array}\right)R\right)}{\left(\begin{array}{l}-\frac{1}{4}k(1+{\mu }_{\text{r}})r^3+(-{\mu }_{\text{r}}^2+1)\mathrm{kR}{r}^2+k(1+{\mu }_{\text{r}})\left({\mu }_{\text{r}}-\frac{3}{4}\right)R^{2}r+\frac{1}{2}{R}^2{E}_{\text{r}}({\mu }_{\text{r}}-1)\end{array}\right){\rho }^2}$(2) The tangential stress of the rock in the bolt reinforced area (r<ρ<R): (3) ${\sigma }_{\theta \text{re}}^1=\frac{\frac{1}{2}\left({\sigma }_0\left(\begin{array}{l}-k(1+{\mu }_{\text{r}})\left({\mu }_{\text{r}}-\frac{1}{2}\right)r^4+k(1+{\mu }_{\text{r}})\left({\mu }_{\text{r}}-\frac{1}{2}\right)R{r}^3+(-2k(1+{\mu }_{\text{r}})\left({\mu }_{\text{r}}-\frac{3}{4}\right){\rho }^2+E_{\text{r}}({\mu }_{\text{r}}-1)R)r^2+2k(1+{\mu }_{\text{r}})\left({\mu }_{\text{r}}-\frac{3}{4}\right){\rho }^2\mathrm{Rr}+R{\rho }^2{E}_{\text{r}}({\mu }_{\text{r}}-1)\end{array}\right)R\right)}{\left(\begin{array}{l}-\frac{1}{4}k(1+{\mu }_{\text{r}})r^3+(-{\mu }_{\text{r}}^2+1)\mathrm{kR}{r}^2+k(1+{\mu }_{\text{r}})\left({\mu }_{\text{r}}-\frac{3}{4}\right)R^{2}r+\frac{1}{2}{R}^2{E}_{\text{r}}({\mu }_{\text{r}}-1)\end{array}\right){\rho }^2}$(3) The radial displacement of the rock in the bolt reinforced area (r<ρ<R): (4) $u_{\rho \text{re}}^1=\frac{1}{2}\frac{\begin{array}{c}(k(R-r)(1+{\mu }_{\text{r}})\left({\mu }_{\text{r}}-\frac{1}{2}\right)(\mathrm{Rr}+{\rho }^2)+R^2{E}_{\text{r}}({\mu }_{\text{r}}-1))(1+{\mu }_{\text{r}})r{\sigma }_0\end{array}}{\begin{array}{c}(\mathrm{kr}(R-r)(1+{\mu }_{\text{r}})\left(\left({\mu }_{\text{r}}-\frac{3}{4}\right)R+\frac{1}{4}r\right)+\frac{1}{2}{R}^2{E}_{\text{r}}({\mu }_{\text{r}}-1))E_{\text{r}}\rho \end{array}}$(4) The radial stress of the rock in the unreinforced area (ρ>R): (5) ${\sigma }_{\rho \text{unre}}^1=-\frac{\begin{array}{c}2(-k(1+{\mu }_{\text{r}})\left({\mu }_{\text{r}}-\frac{1}{2}\right)R^2-R{E}_{\text{r}}({\mu }_{\text{r}}-1)+k(1+{\mu }_{\text{r}})\left({\mu }_{\text{r}}-\frac{1}{2}\right)r^2)R{r}^2{\sigma }_0\end{array}}{\begin{array}{c}((-4k(1+{\mu }_{\text{r}})+\left({\mu }_{\text{r}}-\frac{3}{4}\right)r-2E_{\text{r}}({\mu }_{\text{r}}-1))R^2+k(4{\mu }_{\text{r}}^2-4)r^{2}R+k(1+{\mu }_{\text{r}})r^3){\rho }^2\end{array}}+{\sigma }_0$(5) The tangential stress of the rock in the unreinforced area (ρ>R): (6) ${\sigma }_{\theta \text{unre}}^1=\frac{\begin{array}{c}2(-k(1+{\mu }_{\text{r}})\left({\mu }_{\text{r}}-\frac{1}{2}\right)R^2-R{E}_{\text{r}}({\mu }_{\text{r}}-1)+k(1+{\mu }_{\text{r}})\left({\mu }_{\text{r}}-\frac{1}{2}\right)r^2)R{r}^2{\sigma }_0\end{array}}{\begin{array}{c}((-4k(1+{\mu }_{\text{r}})\left({\mu }_{\text{r}}-\frac{3}{4}\right)r-2E_{\text{r}}({\mu }_{\text{r}}-1))R^2+k(4{\mu }_{\text{r}}^2-4)r^{2}R+k(1+{\mu }_{\text{r}})r^3){\rho }^2\end{array}}+{\sigma }_0$(6) The radial displacement of the rock in the unreinforced area (ρ>R): (7) $u_{\rho \text{unre}}^1=\frac{1}{2}\frac{\begin{array}{c}(k(1+{\mu }_{\text{r}})\left({\mu }_{\text{r}}-\frac{1}{2}\right)R^2+R{E}_{\text{r}}({\mu }_{\text{r}}-1)-k(1+{\mu }_{\text{r}})\left({\mu }_{\text{r}}-\frac{1}{2}\right)r^2)(1+{\mu }_{\text{r}})R{r}^2{\sigma }_0\end{array}}{\begin{array}{c}(\left(k(1+{\mu }_{\text{r}})\left({\mu }_{\text{r}}-\frac{3}{4}\right)r+\frac{1}{2}{E}_{\text{r}}({\mu }_{\text{r}}-1)\right)R^2+k(-{\mu }_{\text{r}}^2+1)r^{2}R-\frac{1}{4}k(1+{\mu }_{\text{r}})r^3)E_{\text{r}}\rho \end{array}}$(7) where, ${\sigma }_{\rho \text{re}}^1$ and ${\sigma }_{\theta \text{re}}^1$ are the radial and tangential stresses of the rock in the reinforced area; ${\sigma }_{\mathrm{\rho un}\text{re}}^1$, ${\sigma }_{\mathrm{\theta un}\text{re}}^1$ are the radial and tangential stresses of the rock in the unreinforced area; $u_{\mathrm{\rho un}\text{re}}^1$ and $u_{\mathrm{\theta un}\text{re}}^1$ are the radial displacement of the rock in the reinforced area and unreinforced area, respectively. “re” and “unre” represent abbreviations for reinforced area and unreinforced area, respectively.

Without lining, the radial displacement of the rock reinforced by end-anchored bolts can be expressed as: (8) $u_{\rho \text{re}}^1|{}_{\rho =r}=U_{\rho 0(1)}^1$(8) where the equation exists only in the 0th order of the Fourier series expansion.

Figure 4 shows the interaction between an end-anchored bolt and the rock. The axial force of bolt T on the rock can be converted into a uniformly distributed stress, P₀, at the tunnel perimeter. The distributed stress between the reinforced and unreinforced areas is rP₀/R. The total stress at ρ = R^- is the sum of stress rP₀/R and the radial stress σ1 ρ1|_ρ = R^-. (9) $P_0=\frac{T}{S_{\theta }{S}_{\text{Z}}}=\frac{K_{\text{b}}\mathrm{\Delta L}}{S_{\theta }{S}_{\text{Z}}}=k(u_{\rho \text{re}}^1|{}_{\rho =r}-u_{\rho \text{re}}^1|{}_{\rho =R})$(9) where P₀ is the uniformly distributed stress of the bolt acting on the inner wall of the tunnel; K_b is the deformation stiffness of the free end of the bolt; ΔL is the elongation of the free end of the bolt; S_θ and S_Z are the circumferential spacing of the bolts and the spacing between bolts along the tunnel axis, respectively; and k = K_b/(S_θS_z) is constant.

Figure 4. Schematic of the interaction between the bolt and the rock in the reinforced area.

3.2. Elastic solution for rock reinforced by end-anchored bolts and lining

In an infinite medium, the secondary stress field under the action of any symmetrically distributed reaction force f 2 (1) in a circular tunnel can be derived by the corresponding stress function as the following [19], the mechanical model of which is shown in Figure 2(c). (10) $\begin{array}{rl}\varphi & =a_0\ln \rho +a_1{\rho }^{-1}\cos \theta \\ & +\sum _{n=2}^{\mathrm{\infty }}(a_n{\rho }^{-n}+b_n{\rho }^{-n+2})\cos \mathrm{n\theta }\end{array}$(10) where a₀ and a₁ are constants.

According to the stress function, the corresponding stresses and displacement can be expressed as (11) $\begin{array}{c}{\sigma }_{\rho (1)}^2=a_0{\rho }^{-2}-2a_1{\rho }^{-3}\cos \theta \\ +\sum _{n=2}^{\mathrm{\infty }}\{-a_n(n^2+n){\rho }^{-n-2}-b_n(n^2+n-2){\rho }^{-n}\}\\ \cos \mathrm{n\theta }\\ {\sigma }_{\theta (1)}^2=-a_0{\rho }^{-2}+2a_1{\rho }^{-3}\cos \theta \\ +\sum _{n=2}^{\mathrm{\infty }}\{a_n(n^2+n){\rho }^{-n-2}+b_n(n^2-3n+2){\rho }^{-n}\}\\ \cos \mathrm{n\theta }\\ {\sigma }_{\mathrm{\rho \theta }(1)}^2=-2a_1{\rho }^{-3}\sin \theta \\ +\sum _{n=2}^{\mathrm{\infty }}\{-a_n(n^2+n){\rho }^{-n-2}-b_n(n^2-n){\rho }^{-n}\}\cos \mathrm{n\theta }\end{array}\}$(11) $u_{(1)}^2$: (12) $\begin{array}{c}2G_{\text{r}}{u}_{\rho (1)}^2=-a_0{\rho }^{-1}+a_1{\rho }^{-2}\cos \theta \\ +\sum _{n=2}^{\mathrm{\infty }}\{a_{n}n{\rho }^{-n-1}+b_n(n-4{\mu }_{\text{r}}+2){\rho }^{-n+1}\}\cos \mathrm{n\theta }\\ 2G_{\text{r}}{u}_{\theta (1)}^2=a_1{\rho }^{-2}\sin \theta \\ +\sum _{n=2}^{\mathrm{\infty }}\{a_{n}n{\rho }^{-n-1}+b_n(n+4{\mu }_{\text{r}}-4)r^{-n+1}\}\sin \mathrm{n\theta }\end{array}\}$(12) According to Equs. (11) and (12), the displacement of the rock at the contact surface u2 (1) and the reaction force of the lining f 2 (1) can be expressed in the Fourier series as [14]: (13) $\begin{array}{rl}& \begin{array}{c}{u}_{\rho (1)}^2|{}_{\rho =r}=\sum _{n=0}^{\mathrm{\infty }}{U}_{\mathrm{\rho n}(1)}^2\cos \mathrm{n\theta }\\ u_{\theta (1)}^2|{}_{\rho =r}=\sum _{n=0}^{\mathrm{\infty }}{U}_{\mathrm{\theta n}(1)}^2\sin \mathrm{n\theta }\end{array}\}\end{array}$(13) (14) $\begin{array}{rl}& \begin{array}{c}{f}_{\rho (1)}^2=-{\sigma }_{\rho (1)}^2|{}_{\rho =r}=\sum _{n=0}^{\mathrm{\infty }}{F}_{\mathrm{\rho n}(1)}^2\cos \mathrm{n\theta }\\ f_{\theta (1)}^2=-{\sigma }_{\mathrm{\rho \theta }(1)}^2|{}_{\rho =r}=\sum _{n=0}^{\mathrm{\infty }}{F}_{\mathrm{\theta n}(1)}^2\sin \mathrm{n\theta }\end{array}\}\end{array}$(14) where U_{2 ρn(1)}, U_{2 θn(1)}, F_{2 ρn(1)}, and F_{2 θn(1)} are constants, and the following expression is derived by Shiba [20]: (15) $\left(\begin{array}{c}{F}_{\mathrm{\rho n}(1)}^2{F}_{\mathrm{\theta n}(1)}^2\end{array}\right)=\left(\begin{array}{ccc}{K}_{\mathrm{\rho n}(1)}& K_{\mathrm{\rho \theta n}(1)}{K}_{\mathrm{\rho \theta n}(1)}& K_{\mathrm{\theta n}(1)}\end{array}\right)\left(\begin{array}{c}{U}_{\mathrm{\rho n}(1)}^2{U}_{\mathrm{\theta n}(1)}^2\end{array}\right)$(15) where, (16) $\begin{array}{rl}& n=0,K_{\rho 0(1)}=\frac{2G_{\text{r}}}{r}(F_{\rho 0(1)}^2=F_{\theta 0(1)}^2=0)\\ & n=1,K_{\rho 1(1)}=K_{\mathrm{\rho \theta }1(1)}=K_{\theta 1(1)}=\frac{2G_{\text{r}}}{r}(F_{\rho 1(1)}^2=F_{\theta 1(1)}^2,\\ & U_{\rho 1(1)}^2=U_{\theta 1(1)}^2)\\ & n\ge 2,K_{\mathrm{\rho n}(1)}=K_{\mathrm{\theta n}(1)}=\frac{2G_{\text{r}}\{2n+1-2{\mu }_{\text{r}}(n+1)\}}{r(3-4{\mu }_{\text{r}})},\\ & K_{\mathrm{\rho \theta n}(1)}=\frac{2G_{\text{r}}\{n+2-2{\mu }_{\text{r}}(n+1)\}}{r(3-4{\mu }_{\text{r}})}\end{array}$(16) The coefficient matrix on the right side of Equ. (15) is the rock stiffness matrix.

3.3. Elastic solutions for the lining under the action of rock reinforced by end-anchored bolts

Figure 3 shows the lining force under the action of rock reinforced by end-anchored bolts. In calculating the elastic solution for the lining with a void behind, Yasuda [14] simplified the lining to an elastic cylindrical shell, assumed the lining thickness h to be much smaller than the tunnel radius r, and considered the shell radius to be equal to the tunnel radius r. Flügge [21] derived the relationships between displacements u_ρ(2), u_θ(2) and f_ρ(2), f_θ(2) on the outer surface of the shell as: (17) $\begin{array}{rl}\left(\begin{array}{c}{f}_{\rho (2)}\\ f_{\theta (2)}\end{array}\right)& =\left(\begin{array}{cc}{\displaystyle \frac{D}{r^2}}+{\displaystyle \frac{K}{r^4}}\left(1+2{\displaystyle \frac{{\mathrm{\partial }}^2}{\mathrm{\partial }{\theta }^2}}+{\displaystyle \frac{{\mathrm{\partial }}^4}{\mathrm{\partial }{\theta }^4}}\right)& {\displaystyle \frac{D}{r^2}}{\displaystyle \frac{\mathrm{\partial }}{\mathrm{\partial }\theta }}\\ -{\displaystyle \frac{D}{r^2}}{\displaystyle \frac{\mathrm{\partial }}{\mathrm{\partial }\theta }}& -{\displaystyle \frac{D}{r^2}}{\displaystyle \frac{{\mathrm{\partial }}^2}{\mathrm{\partial }{\theta }^2}}\end{array}\right)\\ & \times \left(\begin{array}{c}{u}_{\rho (2)}\\ u_{\theta (2)}\end{array}\right)\end{array}$(17) where D and K represent the tensile stiffness and bending stiffness of the lining, respectively; E₁ is the modulus of elasticity; and μ₁ is the Poisson's ratio. (18) $\begin{array}{c}D={\displaystyle \frac{E_{\text{l}}h}{1-{\mu }_{\text{l}}^{\text{2}}}}\\ K={\displaystyle \frac{E_{\text{l}}{h}^3}{12(1-{\mu }_{\text{l}}^{\text{2}})}}\end{array}\}$(18) Using Fourier series expansion, the displacements u₍₂₎ and the surface force f₍₂₎ can be expressed as: (19) $\begin{array}{rl}& \begin{array}{c}{u}_{\rho (2)}=\sum _{n=0}^{\mathrm{\infty }}{U}_{\mathrm{\rho n}(2)}\cos \mathrm{n\theta }\\ u_{\theta (2)}=\sum _{n=0}^{\mathrm{\infty }}{U}_{\mathrm{\theta n}(2)}\sin \mathrm{n\theta }\end{array}\}\end{array}$(19) (20) $\begin{array}{rl}& \begin{array}{c}{f}_{\rho (2)}=\sum _{n=0}^{\mathrm{\infty }}{F}_{\mathrm{\rho n}(2)}\cos \mathrm{n\theta }\\ f_{\theta (2)}=\sum _{n=0}^{\mathrm{\infty }}{F}_{\mathrm{\theta n}(2)}\cos \mathrm{n\theta }\end{array}\}\end{array}$(20) where U_ρn(2), U_θn(2), F_ρn(2), and F_θn(2) are constants. Their relationship can be expressed as: (21) $\left(\begin{array}{c}{F}_{\mathrm{\rho n}(2)}\\ F_{\mathrm{\theta n}(2)}\end{array}\right)=\left(\begin{array}{cc}{K}_{\mathrm{\rho n}(2)}& K_{\mathrm{\rho \theta n}(2)}\\ K_{\mathrm{\rho \theta n}(2)}& K_{\mathrm{\theta n}(2)}\end{array}\right)\left(\begin{array}{c}{U}_{\mathrm{\rho n}(2)}\\ U_{\mathrm{\theta n}(2)}\end{array}\right)$(21) and (22) $\begin{array}{l}{K}_{\mathrm{\rho n}(2)}={\displaystyle \frac{D}{r^2}}+{\displaystyle \frac{K}{r^2}}(n^2-1{)}^2{K}_{\mathrm{\rho \theta n}(2)}=n{\displaystyle \frac{D}{r^2}},n\ge 0K_{\mathrm{\theta n}(2)}=n^2{\displaystyle \frac{D}{r^2}}\end{array}\}$(22) where the coefficient matrix on the right side of Equ. (21) is the nth-order stiffness matrix of the lining.

3.4. Elastic solution for tunnel supported by end-anchored bolts and lining with single and double voids behind the lining

The boundary condition for the void between the rock and the lining is: (23) $f_{\rho (1)}^2=f_{\theta (1)}^2=f_{\rho (2)}=f_{\theta (2)}$(23) The boundary condition for the contact between the rock and the lining is: (24) $\begin{array}{lll}\begin{array}{l}{u}_{\rho 1(1)}^1|{}_{\rho =r}+u_{\rho (1)}^2|{}_{\rho =r}=u_{\rho (2)},\end{array}& \begin{array}{l}{u}_{\theta 1(1)}^1|{}_{\rho =r}+u_{\theta (1)}^2|{}_{\rho =r}={u_{\theta }}_{(2)}\end{array}{f}_{\rho (1)}^2+f_{\rho (2)}=0,& f_{\theta (1)}^2+f_{\theta (2)}=0\end{array}\}$(24) Due to the existence of void and non-void sections, the analytical solution cannot fully meet the boundary conditions. We use the point matching method proposed by Yasuda [14], that is, several points are randomly selected along the tunnel perimeter (except for the points at the intersections between the void and the contact area). These points are then substituted to Equs. (23) and (24) to obtain the infinite series expression of stress and displacement along the tunnel perimeter. Considering the symmetry of the tunnel, only half of the tunnel is analyzed. According to Yasuda's solution process [14], the left half of the tunnel is divided into N (N = 16,64,256) aliquots (0°≤θ≤180°), so the points are isometric (θn(°) = 180n/N, 0 ≤ n ≤ N). The first N + 1 terms of the infinite series (from N = 0 to N) represent the approximate solution.

In the left half of the tunnel, the angle is defined as from −180° to 0° clockwise from the bottom of the tunnel to the vault. The void angle refers to the angle of the non-contact section between the lining and the rock at the tunnel vault. When the void angle is 60° (Since only the left half is considered in the simulation, the void angle ranges only from −30° to 0°.) and N = 16, the number of points at the void location is i (i = 30/(180/N)):

In the void section of the tunnel (0 ≤ i ≤ 2): (25) $\begin{array}{l}{f}_{\rho (1)}^2=0:\\ \sum _{n=0}^{16}[(K_{\mathrm{\rho n}(1)}{U}_{\mathrm{\rho n}(1)}^1+K_{\mathrm{\rho \theta n}(1)}{U}_{\mathrm{\theta n}(1)}^1)\cos n{\theta }_i]=0\\ f_{\theta (1)}^2=0\\ \sum _{n=0}^{16}[(K_{\mathrm{\rho \theta n}(1)}{U}_{\mathrm{\rho n}(1)}^1+K_{\mathrm{\theta n}(1)}{U}_{\mathrm{\theta n}(1)}^1)\sin n{\theta }_i]=0\\ f_{r(2)}=0\\ \sum _{n=0}^{16}[(K_{\mathrm{\rho n}(2)}{U}_{\mathrm{\rho n}(2)}+K_{\mathrm{\rho \theta n}(2)}{U}_{\mathrm{\theta n}(2)})\cos n{\theta }_i]=0\\ f_{\theta (2)}=0\\ \sum _{n=0}^{16}[(K_{\mathrm{\rho \theta n}(2)}{U}_{\mathrm{\rho n}(2)}+K_{\mathrm{\theta n}(2)}{U}_{\mathrm{\theta n}(2)})\sin n{\theta }_i]=0\end{array}\}$(25) In the contact sections (non-void sections) of the tunnel (3 ≤ i ≤ 16): (26) $\begin{array}{l}{u}_{\mathrm{\rho n}(1)}^1|{}_{\rho =r}+u_{\mathrm{\rho n}(1)}^2|{}_{\rho =r}=u_{\mathrm{\rho n}(2)}:\\ \sum _{n=0}^N(U_{\mathrm{\rho n}(1)}^1+U_{\mathrm{\rho n}(1)}^2)\cos n{\theta }_i=\sum _{n=0}^N{U}_{\mathrm{\rho n}(2)}\cos n{\theta }_i\\ u_{\mathrm{\theta n}(1)}^1|{}_{\rho =r}+u_{\theta (1)}^2|{}_{\rho =r}={u_{\theta }}_{(2)}:\\ \sum _{n=0}^N(U_{\mathrm{\theta n}(1)}^1+U_{\mathrm{\theta n}(1)}^2)\sin n{\theta }_i=\sum _{n=0}^N{U}_{\mathrm{\theta n}(2)}\sin n{\theta }_i\\ f_{\mathrm{\rho n}(1)}^2+f_{\mathrm{\rho n}(2)}=0:\\ \sum _{n=0}^N(K_{\mathrm{\rho n}(1)}{U}_{\mathrm{\rho n}(1)}^2+K_{\mathrm{\rho \theta n}(1)}{U}_{\mathrm{\theta n}(1)}^2)\cos n{\theta }_i+\\ \sum _{n=0}^N(K_{\mathrm{\rho n}(2)}{U}_{\mathrm{\rho n}(2)}+K_{\mathrm{\rho \theta n}(2)}{U}_{\mathrm{\theta n}(2)})\cos n{\theta }_i=0\\ f_{\mathrm{\theta n}(1)}^2+f_{\mathrm{\theta n}(2)}=0:\\ \sum _{n=0}^N(K_{\mathrm{\rho \theta n}(1)}{U}_{\mathrm{\rho n}(1)}^2+K_{\mathrm{\theta n}(1)}{U}_{\mathrm{\theta n}(1)}^2)\sin n{\theta }_i\\ +\sum _{n=0}^N(K_{\mathrm{\rho \theta n}(2)}+K_{\mathrm{\theta n}(2)}{U}_{\mathrm{\theta n}(2)})\sin n{\theta }_i=0\end{array}\}$(26) When the left half of the tunnel is divided into 64 or 256, the calculation process is consistent with Equs. (25) and (26) and therefore will not be repeated here. Matlab is used to solve the equations, and it is assumed that U_θ0(1) = U_θ0(2) = U_θN(1) = U_θN(2)= 0. Letting K_ρ1(1)=  K_θ1(1)= 4G_r/r and K_ρθ1(1)= 0 makes it easier to find the solution without rigid body displacement in the vertical direction.

3.5. Tangential stress of the lining

The axial force and bending moment of the lining can be expressed as: (27) $\begin{array}{l}{n}_{\theta }=\sum _{n=0}^{\mathrm{\infty }}{N}_{\mathrm{\theta n}}\cos n{\theta }_i\\ m_{\theta }=\sum _{n=0}^{\mathrm{\infty }}{M}_{\mathrm{\theta n}}\cos n{\theta }_i\end{array}\}$(27) where, (28) $\begin{array}{l}{N}_{\mathrm{\theta n}}={\displaystyle \frac{D}{r}}(U_{\mathrm{\rho n}(2)}+n{U}_{\mathrm{\theta n}(2)})+{\displaystyle \frac{K}{r^3}}(1-n^2)U_{\mathrm{\rho n}(2)}{M}_{\mathrm{\theta n}}={\displaystyle \frac{K}{r^2}}(1-n^2)U_{\mathrm{\rho n}(2)}\end{array}\}$(28) The circumferential stress σ_θ(2) on the lining surface can be approximated by n_θ and m_θ: (29) $\begin{array}{l}{\sigma }_{\theta (2)}\approx {\displaystyle \frac{1}{h}}{n}_{\theta }\pm {\displaystyle \frac{h}{2I}}{m}_{\theta }\\ I={\displaystyle \frac{h^3}{12}}\end{array}\}$(29) where I is the offset distance of the lining, and ± represents the inner and outer surfaces of the lining.

4. Model convergence and numerical verification

To better understand stress and displacement distributions in the bolted rock and the lining influenced by voids, we analyze the effects of rock properties and void size. The properties of the rock, end-anchored bolts and lining are listed in Table 1, and the in-situ stress σ₀ of the rock is −3.0MPa.

Table 1. Material properties.

Parameter	Elastic Modulus (Pa)	Poisson's ratio	Radius (m)	Thickness (m)
Hard rock	3.0 × 10^9	0.30	4.00	–
Soft rock	5.0 × 10^8	0.30	4.00	–
Rock bolt	2.0 × 10^11	0.20	–	–
Lining	30.0 × 10^9	0.20	4.00	0.50

Without voids behind the lining, the closed-form solutions for the tunnel and lining at ρ = r are provided in Table 2. Under hydrostatic stress, the stress and displacement of the rock reinforced by end-anchored blots and lining, and the axial force and bending moment of the lining are all first-order series, that is, $u_{\rho (1)}^1{|}_{\rho =r}$ = $U_{{\rho }_0(1)}^1$ and $f_{\rho (1)}^2{|}_{\rho =r}$ = $K_{{\rho }_0(1)}^{}{U}_{{\rho }_0(1)}^2$+ $K_{{\theta }_0(1)}^{}{U}_{{\theta }_0(1)}^2$. It can be seen from the analytical solution that the radial stress ${\sigma }_{\rho (1)}^1$ of the rock is −1.92MPa, and the total stress of the end-anchored rock ${\sigma }_{\rho (1)}^2$ ( = $-f_{\rho (1)}^2$) is −1.90M Pa at the location of ρ = r. Because the bolt reinforces the rock, the absolute value of the radial stress is greater than the absolute value of the total stress of the end-anchored anchored rock. The difference in stress is exactly equal to the bolt stress P₀ after equilibrium. The changes in stress and displacement indicate that the role of rock bolts is relatively small in the reinforcement of hard rock, where the lining plays the dominant role in ensuring the stability of the rock.

Table 2. Numerical solutions for lining and tunnel reinforced by end-anchored bolts.

	$u_{\rho (1)}^1{|}_{\rho =r}$(m)	$u_{\rho (1)}^{}{|}_{\rho =r}$(m)	${\sigma }_{\rho (1)}^{}{|}_{\rho =r}$ (MPa)	$f_{\rho (1)}^2{|}_{\rho =r}$ (MPa)
Hard rock	−5.16 × 10^−3	−1.91 × 10^−3	−1.92	−1.90
Soft rock	−2.96 × 10^−2	−2.65 × 10^−3	−2.90	−2.88
Soft rock without bolt reinforcement (Yasuda's solution)	−3.12 × 10^−2	−2.79 × 10^−2	−2.83	−2.83

For both soft and hard rocks, the radial stress σ_ρ and tangential stress σ_θ approach the in-situ stress of −3.0MPa along the radial direction of the tunnel. The graph in Figure 5 shows that the radial and tangential stresses of the rock system reinforced with bolts and lining are lower than those supported only by lining. Additionally, the displacement of the former is lower than the latter at the tunnel perimeter. The reason is the interaction between the bolt and the rock. Specifically, the bolt is subjected to tensile stress, and the stress applied to the rock in the reinforced area is compressive. Thus, the radial stress of the rock in the reinforced area increases and is greater than the Yasuda solution. The radial stress in the unreinforced area decreases; the displacement of the rock in the reinforced area decreases, and that in the unreinforced area increases. It is worth noting that with the reduction of the elastic modulus of the rock, the effect of the bolts changes the overall stress states in both reinforced and unreinforced areas. At ρ = R, the slopes of the displacement change are not equal.

Figure 5. Evolution of tunnel stress and displacement in the radial direction.

4.1. Mechanical model for tunnel supported by end-anchored bolts and lining with a single void behind the lining

The reaction force f_ρ(2) from the rock to the lining on the contact surface when a void exists is shown in Figure 6, where the void angle is 60°. With the presence of the void, the approximate solution becomes more and more accurate as the number of selected points and the number of series terms increase. Due to the stress concentration at the intersection between the void and non-void areas, the reaction force f_ρ(2) in the lining converges slowly. The presence of voids will cause stress redistribution on the lining, and the stress concentration then quickly decreases with distance away from the void into the surrounding ground. It is worth noting that a small number of areas show reduced stress near the stress concentration area. The farther away from the void, the less the stress is affected. For tunnels with voids behind the lining, careful attention must be paid to interface stress concentration areas that could affect tunnel stability.

Figure 6. Evolution of the reaction force on the lining from the rock reinforced by end-anchored bolts and lining.

When N = 256, in the case of hard rock, the reaction forces on the lining near the interface between void and non-void areas are −16.8 MPa and −17.0 MPa for the bolts and lining supported rock and lining-only supported rock, respectively. Therefore, the stress concentration reduced by the bolts is 1.18%. In the case of soft rock, the reaction forces on the lining under the two support conditions are −18.6 MPa and −19.5 MPa, respectively, with a stress concentration reduction rate of 4.6%. These reductions show that the anchoring effect can decrease the stress concentration under the influence of the void.

At the void location, the top of the lining moves up (towards the rock), and the lining near the non-void area (±30^o) moves down (away from the rock) and accelerates, the displacement at ±30^o is 2.88 × 10⁻³ m. As the distance from the void increases, the displacement gradually converges and stabilizes to a specific value, and it is −1.85 × 10⁻³ m at the bottom of the tunnel. The approximate evolution of the lining radial displacement u_ρ(2) is shown in Figure 7. Due to the existence of the void, the additional load is induced on the nearby surrounding ground, resulting in the elliptical tunnel shape at the void position. The displacement u_ρ(2) converges very quickly because the lining stiffness is positively correlated with the order of series n. As the order increases, the high order Fourier series becomes less sensitive, and the radial displacement of the lining decreases.

Figure 7. Evolution of the lining displacement in tunnel reinforced by end-anchored bolts and lining.

In the case of hard rock, the displacements of the lining at the bottom of the arch are −1.84 × 10⁻³ m and −1.86 × 10⁻³ m for the bolts and lining supported rock and lining-only supported rock, respectively. The lining displacements at the vault in the void area are 3.88 × 10⁻³ m and 3.92 × 10⁻³ m for the two support cases, respectively. The displacement reduction rates by the bolts are 1.07% and 1.02%, respectively. In the case of soft rock, the displacements of the lining at the bottom of the arch are −2.45 × 10⁻³ m and −2.59 × 10⁻³ m for the bolts and lining supported rock and lining-only supported rock, respectively. The lining displacements at the vault in the void area are 1.58 × 10⁻³ m and 1.67 × 10⁻³ m for the two support cases, respectively. Due to the reinforcing effect of the bolts, the displacements are reduced by 5.41% and 5.39%, respectively. The evolution of the lining displacement of the two cases is shown in Figure 16(a,b). These data indicate that rock bolts not only inhibit the displacement of the lining without voids but also reduce the deformation of the lining towards the tunnel when voids are present. Moreover, the rock bolts play a larger reinforcing role in soft rocks.

Figure 8 exhibits the evolution of the approximate solution for the axial force n_θ and the bending moment m_θ of the lining under elastic conditions. Due to the presence of the void, the axial force decreases, and the axial force at the bottom of the tunnel is about 2.3 times that near the void. At a distance far from the void, that is, [−180°, −60°] or [60°, 180°], the bending moment of the lining is less affected by the void. At this point, the surface of the lining away from the rock is in tension, while the surface close to the rock is in compression. Moreover, the absolute value of the bending moment is much larger than that under the condition without voids. Affected by the bending moment of the void in the range of [−15°, 15°], the surface of the lining away from the rock is in tension and the surface of the lining close to the rock is in compression. Furthermore, the absolute value of the bending moment is significantly increased, and the maximum bending moment at the void is 19.0 times that at the bottom of the tunnel. The existence of the void significantly changes the internal force distribution of the tunnel support structure, which reduces the axial force and changes the direction of the bending moment, thereby reducing the support ability of the structure. The axial force and bending moment of the lining converge very quickly because the axial force n­ and the bending moment m_θ are functions of displacements u_ρ(2) and u_θ(2), and their sensitivities decrease as the order of the Fourier series increases.

Figure 8. Evolution of the axial force and bending moment for tunnel reinforced by end-anchored bolts and lining: (a) lining axis force n_θ, and (b) lining bending moment m_θ.

In the case of hard rock, for the bolts and lining supported tunnel and lining-only supported tunnel, the axial forces at the bottom of the arch are −7.85 × 10⁶ N and −7.93 × 10⁶ N, and the bending moments are −52.7 kN·m and −53.2 kN·m, respectively. Near the void boundary, the lining axial forces are −3.36 × 10³ kN and −3.40 × 10³ kN, and the bending moments are 8.12 × 10² kN·m and 8.20 × 10³ kN·m, respectively. At the vault near the void, the axial forces are −3.18 × 10³ kN and −3.21 × 10³ kN, and the bending moments are 1.00 × 10³ kN·m and 1.01 × 10³ kN·m, respectively. The reduction rate in axial force and bending moment from the bolts is about 1%. In the case of soft rock, for the bolts and lining supported tunnel and lining-only supported tunnel, the axial forces at the arch bottom are −6.68 × 10³ kN and −7.14 × 10³ kN, and the bending moments are −1.48 × 10² kN·m and −1.56 × 10² kN·m, respectively. Near the void boundary, the axial forces are −3.72 × 10³ kN and −3.92 × 10³ kN, and the bending moments are 1.03 × 10³ kN·m and 1.09 × 10³ kN·m, respectively. At the vault near the void, the axial forces are −4.10 × 10³ kN and −4.32 × 10³ kN, and the bending moments are −2.30 × 10³ kN·m and −2.43 × 10³ kN·m, respectively. With the reinforcement of bolts, the axial force is reduced by 6.44%, 5.10%, and 5.09%, and the bending moment is reduced by 5.12%, 5.50%, and 5.35%, respectively. Therefore, the anchor reinforcement plays a role in reducing the axial force and bending moment and strengthens the structural stability of the lining to a certain extent.

Figure 9 shows the evolution of the approximate solution for the distribution of circumferential stress σ_θ(2) on the outer surface of the lining when a void exists. The void significantly influences the distribution of circumferential stresses in the lining. At the edge of the void, the circumferential stress of the lining increases dramatically. At the void, the circumferential stress of the lining increases significantly, and the direction reverse phenomenon occurs, and the maximum absolute value of the circumferential stress takes place at the void centre. The elevated and direction reversed circumferential stress may lead to cracking or even failure of the lining. Consistent with the radial displacement of the lining, the circumferential stress on the outer lining surface converges very quickly due to the increase in stiffness and the decrease in sensitivity as the order of the Fourier series increases.

Figure 9. Evolution of the approximate solution for the circumferential stress of the outer lining surface.

In the case of hard rock, for the bolts and lining supported tunnel and lining-only supported tunnel, the evolutions of lining circumferential stress are shown in Figure 16(c,d). The reduction rates of the bolts to the circumferential stress near the void are 0.7% and 4.9% for hard and soft rock conditions, respectively. Apparently, the end-anchored bolts have a limited effect in inhibiting the circumferential stress for the lining, compared with the results presented above. Therefore, the anchoring effect of the bolts can reduce the circumferential stress concentration to a small extent because it acts more in the radial direction of the tunnel.

Figure 10 shows the evolution of the approximate solution for the radial stress of the rock and the overall stress of the end-anchored tunnel at ρ = r. They are of equal magnitude to f_ρ(2) in Figure 6 but in opposite directions. The radial stress of the rock inside the void is 0, and significant stress concentration occurs at the contact surface between the void and the surrounding ground. The radial stress of the substructure is smaller than the radial stress of the rock, which indicates that the rock bolts play a reinforcing role and to a certain extent enhance the stability of the tunnel. However, it should be noted that the rock bolt is treated as a uniformly distributed stress during the calculation process, which magnifies the overall reinforcing effect but underestimates its effect in the area near the bolt. Consistent with Figure 6, the substructure and the surrounding ground do not converge even at N = 256, so increasing the series order does not affect its sensitivity.

Figure 10. Evolution of the overall stress and radial stress of the rock reinforced by end-anchored bolts: (a) hard rock reinforced by end-anchored bolts ρ = r, (b) hard rock ρ = r.

In Figure 10(a), when N = 256, the evolution of the radial stress in the rock reinforced by end-anchored bolts at the radial distance ρ = r does not converge. When N = 64, the radial stress converges at the radial distance of ρ = 1.05r, as shown in Figure 11. Through comparison, it is found that increasing the radial distance does not change the overall radial stress distribution trend of the surrounding ground with a void. As the radial distance increases, and the stress concentration value at the interface between the void and the surrounding ground gradually reaches a fixed value as the order N increases. Compared with the location at ρ = r, the stress concentration is lower, indicating that the effect of the void gradually weakens as the radial distance increases.

Figure 11. Radial stress evolution of the rock reinforced by end-anchored bolts at ρ = 1.05r.

We select the order N = 64 to express the lining surface load. At the void above the tunnel vault, the surface load quickly returns to zero. The load increases steeply at the contact interface between the lining and the void, and then rapidly unloads to the surrounding lining as the distance from the void increases, as shown in Figure 12. From N = 16 to N = 64, the stiffness increases quickly as the series order increases, and eventually the function converges. It can be seen from the series order of the surface load that if the conditions of U_θ0(1)= U_θ0(2)= U_θN(1)= U_θN(2)= 0 and U_ρ1(1)= U_θ1(1) are satisfied, as the order increases, F_ρ(2) and F_θ(2) do not change significantly, but show regular fluctuations until they reach the N order. This explains the lack of rapid convergence of lining load shown in Figure 6.

Figure 12. Evolution of lining surface load and series order: (a) lining surface load at ρ = R; (b) surface load series order.

We select the order N = 256 to express the rock and lining displacements. Because of the void, the surface loads of the lining and the surrounding ground redistribute; the lining at the void near the tunnel vault moves upwards, and the rock moves downwards, showing an elliptical shape. The circumferential displacement at the bottom of the arch is close to 0, but due to the existence of the void, their distribution is centrally symmetric rather than axisymmetric. The tangential displacements of the rock and the lining at ρ = r are centrally symmetrically distributed, as shown in Figure 13(a). The order N = 64 is still selected in Figure 13(b) to express the displacement. When the order is 9, U_ρ(1), U_θ(1), U_ρ(2), and U_θ(2) all gradually approach 0. This indicates that displacement converges even when the order is small, explaining the rapid convergence of the rock and the lining displacement.

Figure 13. Evolution of lining displacement and displacement order: (a) lining displacement at ρ = r; (b) displacement order.

When N = 256, the evolution of the approximate solution of the lining circumferential stress σ_θ(2) is shown in Figure 14. The stress difference between the inner, middle and outer surfaces is determined by the bending moment of the lining. Combined with Figure 8, the stress redistribution when a void exists mainly depends on the axial force because the bending moment is small. Therefore, stress concentration occurs near the void, resulting in unfavourable stress distribution of the lining. The outer surface of the lining at the void is in tension, and the inner surface is in compression. When the tensile stress at the outer surface exceeds the tensile strength of the lining, it will cause the creation of cracks or even failure of the lining. The circumferential stress σ_θ(2) converges rapidly at N = 256 because it_­ is a function of the axial force n_θ and the bending moment m_θ, while both are determined by the displacements u_ρ(2) and u_θ(2). As the series order increases, the lining stiffness increases rapidly, so the convergence is rapid.

Figure 14. Evolution of the lining circumferential stress distribution.

Figure 15 shows the impact of void size on the circumferential stress distribution of the lining. Under the condition that the stiffness of the lining and the rock remains unchanged, increasing void size will significantly impact the lining stress distribution due to the interaction between the lining and the rock. The void affects the internal force of the lining, and there is a significant stress concentration on the outer surface. The expansion of the void causes a decrease in the circumferential stress of the lining. Stress concentration increases dramatically when the void changes from 30° to 60°and slightly weakens when it changes from 60° to 120°. Therefore, we infer that when the void reaches a certain size, the stress concentration will reach the maximum, and then enlarging the void will have a limited effect on the stress distribution of the rock and lining.

Figure 15. Evolution of tangential stress of the lining with different void sizes.

Figure 16 shows the comparison of displacement and stress between the Yasuda's solution and the approximate solution for the tunnel reinforced by end-anchored bolts and lining with a void behind the lining. Regardless of soft or hard rock, the lining displacement and tangential stress from Yasuda's solution are larger than that from the approximate solution.

Figure 16. Evolution of lining stress and displacement: (a) hard rock lining displacement; (b) soft rock lining displacement; (c) Tangential stress of hard rock lining; (d) Tangential stress of soft rock lining.

It is worth noting that under hard rock conditions, due to its high elastic modulus and good stability, the reinforcing effect of the rock bolt is minimal when lining support exists. Under soft rock conditions, the reinforcing effect of the bolt is greater, which plays an important role in enhancing the stability of the rock. The approximate solution developed in this study is in good agreement with Yasuda's solution, which suggests the fidelity of the theoretical model.

4.2. Mechanical model for tunnel supported by end-anchored bolts and lining with double voids behind the lining

During the construction of tunnels, voids also often appear near the spandrels on both sides. In this section, we study the effect of double voids on the tunnel structure through the model. Each void has an angle of 30°, and their positions are [−42°, −12°] and [12°, 42°], as shown in Figure 17. The lining forces are symmetrically distributed, and a strong stress concentration occurs at the void boundary, which is quickly unloaded at [−12°, 12°]. The stress at the lining-rock contact between the double voids increases sharply. At the tunnel vault (0°), the lining stress f_ρ(2) is −1.68 MPa. The lining stresses f_ρ(2) are −14.7 MPa at the boundaries of the voids (±12°). Therefore, the tunnel segment between the double voids is also an unstable segment, which requires reinforcement. In the case of hard rock, for the bolts and lining supported tunnel and lining-only supported tunnel, the lining stresses f_ρ(2) at the void boundaries (±12°) are −14.7 MPa and −14.9 MPa, respectively. In the case of soft rock, the lining stresses f_ρ(2) at the void boundaries (±12°) are −15.7 MPa and −16.6 MPa, respectively. For the two types of surrounding rocks, the reduction rates in lining stress from the bolts are 1.34% and 5.42%, respectively.

Figure 17. Evolution of the lining reaction force for the tunnel with double voids behind the lining.

Under hard rock conditions and with double voids, the displacement of the lining is 1.94 × 10⁻³ m at the tunnel vault (0°), 1.46 × 10⁻³ m at the void boundaries (±12°), and −1.91 × 10⁻³ m at the bottom (±180°). It can be seen from Figure 18 that in the range of [−12°, 12°], due to the influence of double voids, the lining displacement presents a V-shaped distribution, and in the range of [−60°, 60°], the lining displacement presents an M-shaped distribution. It is worth noting that the entire lining moves downward. In the case of hard rock, for the bolts and lining supported tunnel, the lining displacements in the voids (±27°), at the vault (0°), near the voids (±60°), and at the bottom are −5.89 × 10⁻⁴ m, −1.94 × 10⁻³ m, −2.15 × 10⁻³ m, and −1.91 × 10⁻³ m, respectively. For the lining-only supported tunnel, the lining displacements are −5.94 × 10⁻⁴ m, −1.96 × 10⁻³ m, −2.17 × 10⁻³ m, and −1.93 × 10⁻³ m, respectively. The reduction rate in lining displacement from the bolts is 1%. In the case of soft rock, for the bolts and lining supported tunnel, the lining displacements are 1.11 × 10⁻³ m, −8.92 × 10⁻⁴ m, −3.43 × 10⁻³ m, and 2.71 × 10⁻³ m, respectively. For the lining-only supported tunnel, the lining displacements are 1.17 × 10⁻³ m, −9.40 × 10⁻⁴ m, −3.61 × 10⁻³ m, and −2.85 × 10⁻³ m, respectively. The reduction rate in lining displacement from the bolts is 5%. It should be noted that under soft rock conditions, the lining at the void moves upward.

Figure 18. Evolution of the lining displacement for the tunnel with double voids behind the lining

As shown in Figure 19(a), the axial force is −6.48 × 10³ kN at the tunnel vault and −7.59 × 10³ kN at the tunnel bottom, which shows that the double voids seriously affect the stability of the tunnel vault [−12°,12°]. Double voids cause the axial force to increase sharply from the tunnel bottom to the void and to the vault, with local fluctuations up and down, and it is pronounced at the void boundaries. At ±12°, the axial force is reduced from −6.48 × 10³ to −6.08 × 10³ kN. The bending moment is 4.18 × 10² kN·m at the lining vault and −43.4 kN·m at the tunnel bottom, as shown in Figure 19(b). Double voids cause the bending moment of the lining vault to rise sharply and then reverse. At this point, the bending moment of the lining vault at [−12°, 12°] is positive, indicating that the surface of the lining close to the rock is in compression and the other side of the lining is in tension.

Figure 19. Evolution of the lining axial force and bending moment for the tunnel with double voids behind the lining. (a) lining axial force, (b) lining bending moment.

In the case of hard rock, for the bolts and lining supported tunnel, the lining axial forces in the voids (±27°), at the vault (0°), and at the tunnel bottom are −6.32 × 10³ kN, −6.48 × 10³ kN, and −7.59 × 10³ kN, respectively. The lining bending moments are −6.56 × 10² kN·m, 4.18 × 10² kN·m, and −43.4 kN·m, respectively. For the lining-only supported tunnel, the lining axial forces are −6.38 × 10³ kN, −6.54 × 10³ kN, and −7.66 × 10³ kN, respectively. The lining bending moments are −6.62 × 10² kN·m, 4.22 × 10² kN·m, and −43.8 kN·m, respectively. The reduction rate in the lining axial force and bending moment from the bolts is 1%. In the case of soft rock, for the bolts and lining supported tunnel, the lining axial forces are −8.92 × 10³ kN, −9.12 × 10³ kN, and −1.06 × 10⁴ kN, respectively. The lining bending moments are −1.15 × 10³ kN·m, 8.59 × 10², and 95.5 kN·m, respectively. The reduction rate in the lining axial force and bending moment from the bolts is 5%. Under soft rock conditions, both the axial force and bending moment increase. Therefore, the void segments of the lining and the segment between the two voids are all vulnerable parts of the tunnel.

The lining circumferential stress σ_θ(2) is a function of the lining axial force n_θ and the bending moment m_θ. The circumferential lining stress increases rapidly near the void, from negative to positive in the void, −16.3 MPa at the void boundaries (±42°), and 3.47 MPa at the voids (±27°), indicating that stress concentration occurs at the void boundaries, as shown in Figure 20. The circumferential stresses of the lining at the vault and at the bottom are −23.2 MPa and −14.1 MPa, respectively. The lining segment between the two voids is [−12°, 12°]. Due to the influence of double voids, the circumferential stress of this segment increases, which is 1.65 times that at the tunnel bottom. Therefore, double voids deteriorate the internal force distribution of the lining and should be the key monitoring locations for tunnel safety evaluation during the construction process.

Figure 20. Evolution of the lining circumference stress for tunnel with double voids behind the lining.

Figure 21 shows the evolution of the approximate solution for the radial stress of the rock and the overall stress of the end-anchored tunnel at ρ = 1.05r. The radial stresses of the rock at ±27° are not 0 and the radial stresses at ±12° are −1.69 MPa. The radial stresses of the rock at boundaries of ±12° and ±42° are −2.77 MPa, and −2.81 MPa, respectively. Stress concentration occurs at the void boundaries. However, with the increase of the distance from the void boundary, stress is quickly unloaded to the surrounding rock, and in some area the radial stress is even smaller than that at the tunnel bottom. The radial stress of the rock at the vault is −1.72 MPa, which is larger than that at the bottom of the tunnel, −1.69 MPa, because the lining segment between the double voids is small, resulting in the incomplete unloading of the rock stress.

Figure 21. Evolution of the rock radial stress for tunnel with double voids behind the lining (ρ = 1.05r).

Figure 22 shows the evolution of the lining surface stress and Series order for tunnel with double voids behind the lining. Due to the symmetrical distribution of the voids, the surface stress f_ρ(2) of the lining is symmetrically distributed, f_θ(2) is centrally symmetrically distributed, and the surface stress at the voids is 0. The lining stiffness increases rapidly with the increase of the order. F_ρ(2) is −1.78 × 10⁶ Pa, 1.57 × 10⁵ Pa and 5.19 × 10⁴ Pa at the 1st, 2nd, and 3rd order, respectively. F_θ(2) is 0 Pa, 1.57 × 10⁵ Pa, and 9.65 × 10⁴ Pa at the 1st, 2nd, and 3rd order, respectively. When the order reaches 5, F_ρ(2) and F_θ(2) gradually show regular fluctuations and do not converge.

Figure 22. Evolution of the lining surface stress and Series order for the tunnel with double voids behind the lining. (a) lining surface stress at ρ = R, (b) Series order.

In the model with double voids, the circumferential displacement of the surrounding rock is N-shaped, the tangential displacement at the vault is 0, and the overall tangential displacement is centrally symmetrical. The rock at the voids deforms towards the lining, and the lining deforms towards the rock to accommodate the stress distribution. In the [−60°, 60°] range, the displacement is symmetrically distributed in a M-shape, as shown in Figure 23(a). In Figure 23(b), U_ρ(1), U_θ(1), U_ρ(2), and U_θ(2) all approach 0 at order N = 15. As the order rises, the lining stiffness increases dramatically, and the sensitivity decreases.

Figure 23. Evolution of lining and rock displacement for the tunnel with double voids behind the lining. (a) lining displacement at ρ = r, (b) Series order.

4.3. Comparative analysis of Mechanical model for tunnel supported by end-anchored bolts and lining with single and double voids behind the lining

Figure 24 shows the evolution of the lining reaction force for the end-anchored tunnel with single void and double voids behind the lining. Since in the case of double voids, each void range is 30°, the stress concentration at the boundaries of the double voids is weaker than the case with a single void of 60°. Compared with lining stress at the bottom of the tunnel, the lining stresses at the void boundaries with single and double voids are 1.81 and 1.61 times, respectively. The lining stresses at the tunnel bottom are −1.77 MPa and −1.72 MPa, respectively, in the single and double voids cases. Single void has a higher impact on the lining than double voids. The lining between the two voids is greatly affected by the presence of the voids. Moreover, the main tunnel segment influenced by single and double voids on the stress redistribution is [−60°, 60°].

Figure 24. Evolution of lining reaction force for the end-anchored tunnel with single and double voids behind the lining.

It can be seen from Figure 25 that the influence of a single void on lining displacement is much greater than that of double voids. The maximum displacements at the single void and double voids are 3.89 × 10⁻³ m and −5.72 × 10⁻⁴ m, respectively. In the case of a single void, the lining at the void deforms upward (toward the rock). In the case of double voids, the linings at the voids deform downward (away from the rock), and the lining displacement at the voids is smaller than that at the tunnel bottom. The displacements at the tunnel bottom under single and double voids are −1.85 × 10⁻³ m and −1.91 × 10⁻³ m, respectively. The difference between the maximum displacement and the displacement at the tunnel bottom in the single void case is 4.28 times that of the double voids case.

Figure 25. Evolution of lining displacement for the end-anchored tunnel with double voids behind the lining.

In the cases of single and double voids, the axial force shows an overall decreasing trend, and the bending moment at the voids presents an increasing trend. However, the variation range of axial force and bending moment under double voids is smaller than that under a single void, as shown in Figure 26. Compared with the axial force at the tunnel vault, the axial forces at the tunnel bottom are 2.34 and 1.17 times in the single and double voids cases, respectively. The influence range of single and double voids on the lining bending moment is [−60°, 60°], the bending moments at the void are 18.87 and 15.26 times, respectively, compared with that at the tunnel bottom. Similarly, the lining bending moments at the void boundaries are15.09 and 5.74 times, respectively. In the double voids case, the lining between voids is the most vulnerable segment of the tunnel, and its bending moment is 9.63 times that at the tunnel bottom.

Figure 26. Evolution of lining axial force and bending moment for the tunnel with single void and double voids behind the lining. (a) lining axial force, (b) bending moment.

From the comparative analysis of the single and double voids models in Figures 24–26, it can be seen that when the total void angle is the same, single void has a greater impact on lining stress, displacement, bending moment, and axial force than double voids. Therefore, timely protection measures should be taken. The tunnel structure with double voids space should be continuously monitored as a crucial aspect of the safety evaluation of the tunnel.

5. Numerical analyses on the tunnel rock reinforced by end-anchored bolts and lining with a void behind the lining

To further verify the correctness of the analytical solution, we use the finite difference method to calculate the displacement and stress in a tunnel reinforced by rock bolts and lining. The numerical model of the tunnel with bolts, a lining, and a void is shown in Figure 27.

Figure 27. FLAC3D numerical simulation model setup.

Because of the symmetry of the circular tunnel, the model size should be at least six times the circular tunnel size to effectively eliminate the boundary effects in numerical simulation [22]. For that reason, the total boundary length of the model is 80m, the tunnel radius is 4m, and the number of grids along the tunnel perimeter is 120. The number of lining CID components is 240, and there is a total of 8860 solid elements. The single void and double voids distributions are shown in Figures 6 and 17, respectively. Displacement constraints are set on the front, back, left, right, top, and bottom of the model, and in-situ stress of −3.0 MPa is applied to the left, right, top, and bottom of the model. Elastic constitutive model is applied in this study.

The surrounding ground is divided into a reinforced area and an unreinforced area. Because the reinforced area is the focus of this study, mesh refinement is performed in this area. Rock bolt is assumed to be a spring element in the model and is a pair of forces of equal magnitude and opposite direction during the simulation. Therefore, the anchoring segment is not considered when the bolts are added to the model. Cable elements are used for the bolts. For the simulation in the free segment of the bolt, the approximate effect can be achieved by setting the grout parameters small enough. The material parameters are listed in Table 3. The lining structure uses shell elements with a thickness of 0.5 m and an elastic modulus of 30 GPa.

Table 3. Rock bolt material properties.

Material property	xcarea/ ×10^−4m^2	emod/ ×10^11Pa	gr_coh/N	gr_per/m	gr_k/Pa
Value	3.0	2.0	1.0	0.05	1.0

Under hard rock conditions, Figure 27 shows the radial stresses and radial displacements of end-anchored rock, circumferential axial forces, and bending moments of the lining with single void and double voids behind the lining. Figures on the left are results from the single void case, and figures on the right are results from the double voids case. The radial stress of the rock gradually decreases near the void, and the stress concentration occurs at the intersection between the void and the lining, larger than −3.0 MPa. The radial displacement of the rock is largest at the void, and the closer to the lining inside the void, the smaller the displacement. As the distance from the void increases, the displacement of the rock increases first and then decreases, which is consistent with the results in Figures 13(a) and 23(a). It can be seen from Figure 28(c,d) that the impact of voids on the tunnel lining is mostly on the axial force, and the impact on the bending moment is insignificant.

Figure 28. Simulation results with single void (left column) and with double voids (right column). (a) rock radial stress, (b) rock radial displacement, (c) lining circumferential axial force, and (d) ling bending moment.

Figures 6 and 17 show the locations of the single void and double voids of the two circular tunnels. Since the voids are located above the vault, it has little effect on the radial stress at the bottom of the tunnel. We compare the simulated radial stress at the tunnel bottom with the approximate solution, as shown in Figure 29(a). The overall radial stress from the approximate solution is smaller than that from the simulated result, mainly due to the transformation of bolt force to the distributed stress on the rock during the simulation process. While this magnifies the overall reinforcing effect of the bolts, it weakens the localized effect.

Figure 29. Evolution of approximate and simulated solutions: (a) radial stress; (b) radial displacement.

In the approximation solution, the radial stress of the rock is −1.92 × 10⁶ Pa (Table 2), and the approximation solution completely neglects the influence of the rock and the void behind the lining. To maintain rock stability, the lining affected by the void needs to provide more significant support to the surrounding ground. Therefore, the simulated radial stress will be slightly larger than that from the approximate solution. In Figure 29(b), the difference between the radial displacement of the rock from the approximate solution and the simulated result is negligible. In general, the two solutions of radial stress and displacement are in good agreement, which again suggests the validity of the analytical model.

Figure 30 shows the evolution of lining bending moment and axial force from the approximate solutions and the simulated results for rock with double voids. The approximate solutions of the bending moment and axial force agree well with the simulated results. It once again verifies that the lining is dominantly influenced by the axial force and less impacted by the bending moment. The approximate solution of the lining bending moment is slightly smaller than the simulated result because the approximate solution does not consider the thickness of the voids, while a thickness of 0.05 m is considered in the simulation model. Increasing the thickness of the voids can also impact the accuracy of the approximate solution.

Figure 30. Evolution of approximate solution and simulated solution: (a) lining bending moment; (b) lining axial force.

6. Conclusions

In this paper, we developed approximate analytical solutions for bolted tunnel structures with voids behind the lining. We then analyzed the distribution of stress and displacement of the rock and the support structure by varying rock properties and void size. The following conclusions can be drawn from this study.

1. End-anchored bolts have a more significant reinforcing effect on soft rock than hard rock. The contributing factors of these bolts are 1% and 5% for hard rock and soft rock, respectively. End-anchored bolts can reduce the displacement and tangential stress at the void and alleviate the stress concentration near the void.

2. When a void exists between the rock and the lining, the interaction between these two media at the void location disappears, which causes stress transfer from the void location to both media. Stress transfer in soft rock is more intense than that in hard rock. The lining moves upwards at the void location, moves downwards at the nearby non-void sections, and gradually intensifies. The farther away from the void, the smaller the displacement at the non-void areas. The lining displacement far away from the void tends to remain unchanged. The overall displacement of the lining is distributed in an elliptical shape.

3. Under hydrostatic stress conditions, the lining is in a state of dominant axial force and low bending moment. The tangential stress on the outer surface of the void zone lining is tensile stress, while the tangential stress on the inner surface is compressive stress. When the tensile stress on the outer surface of the lining exceeds the tensile strength, cracks can be easily created on the lining surface, and even lining failure can occur. Rock bolt reinforcement can restrain the sudden change of axial force and bending moment, and can strengthen the stability of the lining to a certain extent.

4. When the overall void range remains constant, a single void has a greater impact on the stress, displacement, bending moment, and axial force of the lining compared to double voids. Thus, timely protection and repair should be conducted. With double voids, the surrounding rock and lining cannot transfer force within the range of double voids, causing the tunnel structure with double voids space to become unstable. As such, it must be monitored as a key aspect of tunnel safety evaluation.

5. Our analytical model for tunnels with end-anchored bolts and lining voids has limitations. It assumes rock bolts produce uniformly distributed stress, which may weaken the localized reinforcement. Additionally, the model is two-dimensional and cannot accurately represent stress along the tunnel’s direction. Thus, we aim to improve it to incorporate localized reinforcement from rock bolts and depict stress correctly in future research.

Acknowledgments

The authors would like to thank the editor and anonymous referees very much for their work on this paper.

Disclosure statement

No potential conflict of interest was reported by the author(s).

Additional information

Funding

This work was supported by National Natural Science Foundation of China: [Grant Number 52079077].

Appendix A

Since there is no explicit expression of the approximate solution for tunnels with voids, the elastic solutions of the stress and displacement for rock reinforced by end-anchored bolts and lining are provided as follows.

Radial stress of the rock in the reinforced area (r<ρ<R): (A.1) $\begin{array}{r}{\sigma }_{\rho 0\text{un}(1)}^1=\left(\begin{array}{l}\frac{-\frac{1}{2}\left({\sigma }_0\left(\begin{array}{l}-k(1+{\mu }_{\text{r}})\left({\mu }_{\text{r}}-\frac{1}{2}\right)r^4+k(1+{\mu }_{\text{r}})\left({\mu }_{\text{r}}-\frac{1}{2}\right)R{r}^3+(2k(1+{\mu }_{\text{r}})\left({\mu }_{\text{r}}-\frac{3}{4}\right){\rho }^2+E_{\text{r}}({\mu }_{\text{r}}-1)R)r^2-2k(1+{\mu }_{\text{r}})\left({\mu }_{\text{r}}-\frac{3}{4}\right){\rho }^2\mathrm{Rr}-R{\rho }^2{E}_{\text{r}}({\mu }_{\text{r}}-1)\end{array}\right)R\right)}{\left(\begin{array}{l}-\frac{1}{4}k(1+{\mu }_{\text{r}})r^3+(-{\mu }_{\text{r}}^2+1)\mathrm{kR}{r}^2+k(1+{\mu }_{\text{r}})\left({\mu }_{\text{r}}-\frac{3}{4}\right)R^{2}r+\frac{1}{2}{R}^2{E}_{\text{r}}({\mu }_{\text{r}}-1)\end{array}\right){\rho }^2}+\frac{1}{2}\frac{r^2(1+{\mu }_{\text{r}})\left(\begin{array}{l}k(R-r)(1+{\mu }_{\text{r}})\left({\mu }_{\text{r}}-\frac{1}{2}\right)(\mathrm{Rr}+r^2)+E_{\text{r}}({\mu }_{\text{r}}-1)R^2\end{array}\right)(D{r}^2+K){\sigma }_0}{\begin{array}{l}(\mathrm{kr}(R-r)(1+{\mu }_{\text{r}})\left(\left({\mu }_{\text{r}}-\frac{3}{4}\right)R+\frac{1}{4}r\right)+\frac{1}{2}{E}_{\text{r}}({\mu }_{\text{r}}-1)R^2)(E_{\text{r}}{r}^3+D(1+{\mu }_{\text{r}})r^2+K(1+{\mu }_{\text{r}})){\rho }^2\end{array}}\end{array}\right)\end{array}$(A.1) Tangential stress of the rock in the reinforced area (r<ρ<R): (A.2) $\begin{array}{r}{\sigma }_{\theta 0\text{un}(1)}^1=\left(\begin{array}{l}\frac{\frac{1}{2}\left({\sigma }_0\left(\begin{array}{l}-k(1+{\mu }_{\text{r}})\left({\mu }_{\text{r}}-\frac{1}{2}\right)r^4+k(1+{\mu }_{\text{r}})\left({\mu }_{\text{r}}-\frac{1}{2}\right)R{r}^3+(-2k(1+{\mu }_{\text{r}})\left({\mu }_{\text{r}}-\frac{3}{4}\right){\rho }^2+E_{\text{r}}({\mu }_{\text{r}}-1)R)r^2+2k(1+{\mu }_{\text{r}})\left({\mu }_{\text{r}}-\frac{3}{4}\right){\rho }^2\mathrm{Rr}+R{\rho }^2{E}_{\text{r}}({\mu }_{\text{r}}-1)\end{array}\right)R\right)}{\left(\begin{array}{l}-\frac{1}{4}k(1+{\mu }_{\text{r}})r^3+(-{\mu }_{\text{r}}^2+1)\mathrm{kR}{r}^2+k(1+{\mu }_{\text{r}})\left({\mu }_{\text{r}}-\frac{3}{4}\right)R^{2}r+\frac{1}{2}{R}^2{E}_{\text{r}}({\mu }_{\text{r}}-1)\end{array}\right){\rho }^2}-\frac{1}{2}\frac{r^2(1+{\mu }_{\text{r}})\left(\begin{array}{l}k(R-r)(1+{\mu }_{\text{r}})\left({\mu }_{\text{r}}-\frac{1}{2}\right)(\mathrm{Rr}+r^2)+E_{\text{r}}({\mu }_{\text{r}}-1)R^2\end{array}\right)(D{r}^2+K){\sigma }_0}{\begin{array}{l}(\mathrm{kr}(R-r)(1+{\mu }_{\text{r}})\left(\left({\mu }_{\text{r}}-\frac{3}{4}\right)R+\frac{1}{4}r\right)+\frac{1}{2}{E}_{\text{r}}({\mu }_{\text{r}}-1)R^2)(E_{\text{r}}{r}^3+D(1+{\mu }_{\text{r}})r^2+K(1+{\mu }_{\text{r}})){\rho }^2\end{array}}\end{array}\right)\end{array}$(A.2) Radial displacement of the rock in the reinforced area (r<ρ<R): (A.3) $\begin{array}{r}{u}_{\rho 0\text{un}(1)}^1=\left(\begin{array}{l}{\displaystyle \frac{1}{2}}{\displaystyle \frac{\begin{array}{c}(k(R-r)(1+{\mu }_{\text{r}})\left({\mu }_{\text{r}}-\frac{1}{2}\right)(\mathrm{Rr}+{\rho }^2)+R^2{E}_{\text{r}}({\mu }_{\text{r}}-1))(1+{\mu }_{\text{r}})r^2{\sigma }_0\end{array}}{\begin{array}{c}(\mathrm{kr}(R-r)(1+{\mu }_{\text{r}})\left(\left({\mu }_{\text{r}}-\frac{3}{4}\right)R+\frac{1}{4}r\right)+\frac{1}{2}{R}^2{E}_{\text{r}}({\mu }_{\text{r}}-1))E_{\text{r}}\rho \end{array}}}-{\displaystyle \frac{1}{2}}{\displaystyle \frac{\begin{array}{c}{r}^2{(1+{\mu }_{\text{r}})}^2\left(\begin{array}{l}k(R-r)(1+{\mu }_{\text{r}})\left({\mu }_{\text{r}}-\frac{1}{2}\right)(\mathrm{Rr}+r^2)+R^2{E}_{\text{r}}({\mu }_{\text{r}}-1)\end{array}\right)(D{r}^2+K){\sigma }_0\end{array}}{\begin{array}{l}(\mathrm{kr}(R-r)(1+{\mu }_{\text{r}})\left(\left({\mu }_{\text{r}}-\frac{3}{4}\right)R+\frac{1}{4}r\right)+\frac{1}{2}{R}^2{E}_{\text{r}}({\mu }_{\text{r}}-1))(E_{\text{r}}{r}^3+D(1+{\mu }_{\text{r}})r^2+K(1+{\mu }_{\text{r}}))E_{\text{r}}\rho \end{array}}}\end{array}\right)\end{array}$(A.3) Radial stress of the rock in the unreinforced area (ρ>R): (A.4) $\begin{array}{r}{\sigma }_{\rho 0\text{unre}(1)}^1=\left(\begin{array}{l}-{\displaystyle \frac{\begin{array}{c}2(-k(1+{\mu }_{\text{r}})\left({\mu }_{\text{r}}-\frac{1}{2}\right)R^2-R{E}_{\text{r}}({\mu }_{\text{r}}-1)+k(1+{\mu }_{\text{r}})\left({\mu }_{\text{r}}-\frac{1}{2}\right)r^2)R{r}^2{\sigma }_0\end{array}}{\begin{array}{c}(\left(-4k(1+{\mu }_{\text{r}})\left({\mu }_{\text{r}}-\frac{3}{4}\right)r-2E_{\text{r}}({\mu }_{\text{r}}-1)\right)R^2+k(4{\mu }_{\text{r}}^2-4)r^{2}R+k(1+{\mu }_{\text{r}})r^3){\rho }^2\end{array}}}+{\sigma }_0+\frac{1}{2}\frac{\begin{array}{c}{r}^2(1+{\mu }_{\text{r}})\left(\begin{array}{l}k(R-r)(1+{\mu }_{\text{r}})\left({\mu }_{\text{r}}-\frac{1}{2}\right)(\mathrm{Rr}+r^2)+E_{\text{r}}({\mu }_{\text{r}}-1)R^2\end{array}\right)(D{r}^2+K){\sigma }_0\end{array}}{\begin{array}{l}(\mathrm{kr}(R-r)(1+{\mu }_{\text{r}})\left(\left({\mu }_{\text{r}}-\frac{3}{4}\right)R+\frac{1}{4}r\right)+\frac{1}{2}{E}_{\text{r}}({\mu }_{\text{r}}-1)R^2)(E_{\text{r}}{r}^3+D(1+{\mu }_{\text{r}})r^2+K(1+{\mu }_{\text{r}})){\rho }^2\end{array}}\end{array}\right)\end{array}$(A.4) Tangential stress of the rock in the unreinforced area (ρ>R): (A.5) $\begin{array}{r}{\sigma }_{\theta 0\text{unre}(1)}^1=\left(\begin{array}{l}{\displaystyle \frac{\begin{array}{c}2(-k(1+{\mu }_{\text{r}})\left({\mu }_{\text{r}}-\frac{1}{2}\right)R^2-R{E}_{\text{r}}({\mu }_{\text{r}}-1)+k(1+{\mu }_{\text{r}})\left({\mu }_{\text{r}}-\frac{1}{2}\right)r^2)R{r}^2{\sigma }_0\end{array}}{\begin{array}{c}(\left(-4k(1+{\mu }_{\text{r}})\left({\mu }_{\text{r}}-\frac{3}{4}\right)r-2E_{\text{r}}({\mu }_{\text{r}}-1)\right)R^2+k(4{\mu }_{\text{r}}^2-4)r^{2}R+k(1+{\mu }_{\text{r}})r^3){\rho }^2\end{array}}}+{\sigma }_0-\frac{1}{2}\frac{\begin{array}{c}{r}^2(1+{\mu }_{\text{r}})\left(\begin{array}{l}k(R-r)(1+{\mu }_{\text{r}})\left({\mu }_{\text{r}}-\frac{1}{2}\right)(\mathrm{Rr}+r^2)+E_{\text{r}}({\mu }_{\text{r}}-1)R^2\end{array}\right)(D{r}^2+K){\sigma }_0\end{array}}{\begin{array}{l}(\mathrm{kr}(R-r)(1+{\mu }_{\text{r}})\left(\left({\mu }_{\text{r}}-\frac{3}{4}\right)R+\frac{1}{4}r\right)+\frac{1}{2}{E}_{\text{r}}({\mu }_{\text{r}}-1)R^2)(E_{\text{r}}{r}^3+D(1+{\mu }_{\text{r}})r^2+K(1+{\mu }_{\text{r}})){\rho }^2\end{array}}\end{array}\right)\end{array}$(A.5) Radial displacement of the rock in the unreinforced area (ρ>R): (A.6) $\begin{array}{r}{u}_{\rho 0\text{unre}(1)}^1=\frac{1}{2}\left(\begin{array}{l}{\displaystyle \frac{\begin{array}{c}(k(1+{\mu }_{\text{r}})\left({\mu }_{\text{r}}-\frac{1}{2}\right)R^2+R{E}_{\text{r}}({\mu }_{\text{r}}-1)-k(1+{\mu }_{\text{r}})\left({\mu }_{\text{r}}-\frac{1}{2}\right)r^2)(1+{\mu }_{\text{r}})R{r}^2{\sigma }_0\end{array}}{\begin{array}{c}(\left(k(1+{\mu }_{\text{r}})\left({\mu }_{\text{r}}-\frac{3}{4}\right)r+\frac{1}{2}{E}_{\text{r}}({\mu }_{\text{r}}-1)\right)R^2+k(-{\mu }_{\text{r}}^2+1)r^{2}R-\frac{1}{4}k(1+{\mu }_{\text{r}})r^3)E_{\text{r}}\rho \end{array}}}-\frac{1}{2}\frac{\begin{array}{c}{r}^2{(1+{\mu }_{\text{r}})}^2\left(\begin{array}{l}k(R-r)(1+{\mu }_{\text{r}})\left({\mu }_{\text{r}}-\frac{1}{2}\right)(\mathrm{Rr}+r^2)+R^2{E}_{\text{r}}({\mu }_{\text{r}}-1)\end{array}\right)(D{r}^2+K){\sigma }_0\end{array}}{\begin{array}{l}(\mathrm{kr}(R-r)(1+{\mu }_{\text{r}})\left(\left({\mu }_{\text{r}}-\frac{3}{4}\right)R+\frac{1}{4}r\right)+\frac{1}{2}{R}^2{E}_{\text{r}}({\mu }_{\text{r}}-1))(E_{\text{r}}{r}^3+D(1+{\mu }_{\text{r}})r^2+K(1+{\mu }_{\text{r}}))E_{\text{r}}\rho \end{array}}\end{array}\right)\end{array}$(A.6)