Calculation Method for Determining the Wall Displacement and Primary Support Bearing Capacity of Tunnels

Abstract

The primary support structure of a tunnel often needs various support methods, such as bolts, steel arch frames, and shotcrete. It is of great significance to quickly calculate the displacement of tunnel walls and the bearing capacity of primary supports for guiding construction and ensuring construction safety. In order to solve the above problems, this paper constructs a mechanical model of primary support and analyzes the displacement of a tunnel wall at different construction stages of primary support; based on the Mohr–Coulomb strength criterion, the displacement of the tunnel wall at different stages of the primary support is derived, and the calculation method and process for calculating the bearing capacity of the primary support are given. The accuracy of the proposed method is then verified by an example. Compared with previous methods, the method proposed in this paper considers the support effects of bolts, steel arches, and shotcrete separately and calculates the corresponding displacement of the tunnel wall, which is closer to the actual construction situation. In addition, the bearing capacity of the primary support can be calculated and analyzed. By comparing with the numerical simulation results, it has been proven that the proposed method has a relatively small calculation error. Compared with the numerical simulation results and previous analytical methods, the method proposed in this paper is simpler and faster in calculation; thus, it can quickly assist in the design of tunnel support structures.

1. Introduction

With the increasing demand for infrastructure construction in China, the construction of more and more tunnels has been put on the agenda, such as the Qichongcun Tunnel in Guizhou, Hanjialing Tunnel, Yangzong Tunnel, and Gongbei Tunnel of the Hong Kong–Zhuhai–Macao Bridge Project [1,2,3,4]. The safety of such tunnels during and after construction is lower than that of conventional tunnels. In order to ensure construction safety, the primary support of tunnels not only consists of bolts and shotcrete but steel arch frames or steel grids as well [1] (see Figure 1). The displacement of tunnel walls and the bearing capacity of the primary support are important indicators to evaluate the safety of tunnel structures and guide the tunnel support scheme. Generally, engineers will use numerical simulation software to perform calculations [5,6,7,8]. However, due to the computational speed and complexity of numerical simulation software, there is an urgent need to propose a simple and efficient calculation method.

Analytical methods are simple and convenient, and they are widely used in tunnel mechanics. Many scholars have studied the analytical methods of tunnel stress and displacement.

Kirsch (1898) solved the stress, strain, and displacement distribution of deep-buried circular tunnels using a plane strain model based on the elastic mechanics theory [9]. Fenner (1938) proposed the Fenner formula considering the plastic deformation of surrounding rock based on the Mohr–Coulomb constitutive model [10]. The function of a complex variable proposed by Muskhelishvili (1963) has been widely used by many researchers in tunnel mechanics [11]. Hu et al. (2006) and Lei et al. (2001) analyzed the stress distribution of unlined tunnels using this method [12]. Compared with previous studies using the theory of complex variables, it is more convenient to calculate the stress distribution of deep-buried tunnels in a non-hydrostatic stress field [13]. The analytical method of tunnel mechanics proposed above gives the solution of the stress and displacement of unlined tunnels but does not consider the effect of the lining on the surrounding rock.

Ren (2001) improved the Fenner formula by providing an elastic–plastic solution considering the lining based on the operational characteristics of hydraulic tunnels and compared it with the original Fenner formula; the results show that the improved Fenner formula can better calculate the plastic zone and stress distribution of hydraulic tunnels during operation [14]. Du et al. (2022) considered the variability of surrounding rock and lining performance and provided an analytical solution for pressure tunnels under long-term operation, which is of great significance for the study of the long-term durability of tunnel structures [15]. Li et al. (2020) provided analytical solutions for the stress and deformation of noncircular tunnel lining and surrounding rock based on the theory of complex variables, which can better analyze the interaction between the surrounding rock and the support structure [16]. Li et al. (2013) deduced the stress and displacement of the tunnel surrounding the rock in elastic semi-infinite space based on the analytical continuation method in the function of a complex variable, which solved the mechanical problems of shallow buried tunnels [17]. Fang et al. (2021) deduced the stress field and displacement field of multiple tunnels in deep strata by using the function of a complex variable and compared the contact mode between the surrounding rock and the lining during the solution [18]. Li et al. (2013) considered the stress release effect when deducing the stress and deformation of deep-buried tunnels [19]. The convergence constraint method, first proposed in 1978, has been a guiding method for tunnel support design until now. This method simulates the stress release and wall displacement of rock excavation through the ground response curve (GRC) and determines the optimal support timing and strength based on the support characteristic curve (SCC) [20]. Brown derived analytical solutions for the stress, strain, and displacement of tunnel surrounding rock based on the Hoek–Brown strength criterion and strain-softening model and obtained the GRC considering the plastic volume strain of the rock mass [21]. Alonso derived a one-dimensional numerical solution for the GRC curve of tunnels in strain-softening rock masses, which can also be degenerated into the GRC of the Tresca, Mohr–Coulomb, and Hoek–Brown strength criteria [22]. The above analytical method considers the effect of the lining on surrounding rock, but in order to facilitate the solution, the lining is regarded as a single material that is only concrete, and the supporting effect of bolts, steel arch frames, and shotcrete cannot be accurately solved.

Meng et al. (2015) used the Mohr–Coulomb strength criterion to derive the expressions of the stress, displacement, and radius of the plastic zone of a circular tunnel before and after bolt support. This is of great significance for solving the interaction between the surrounding rock and the support structure [23]. He et al. deduced the bearing capacity of surrounding rock after applying bolts. This method only considers the function of bolts to improve the mechanical properties of rock masses [24]. Zhou et al. (2021) deduced and analyzed the support load of bolts and secondary lining in rheological rock mass [25]. Sun et al. (2021) studied the interaction process of grouting bolts and the tunnel surrounding rock and solved the radius of the plastic zone, the displacement, and the stress field of the tunnel [26]. It can be seen that the analytical method has a good effect on the analysis of the supporting effect and bearing capacity of bolts.

In order to make up for the shortcomings of previous studies in considering tunnel displacement and primary support bearing capacity, this paper derives tunnel wall displacement and primary support bearing capacity based on elastic–plastic theory. A semianalytical solution is proposed to calculate the displacement of the tunnel wall and the bearing capacity of the primary support of a tunnel. Compared with previous studies, this paper considers the displacement of the tunnel wall more carefully, deduces the displacement of the tunnel wall solution after the application of bolts and the tunnel wall displacement solution after the application of a steel arch frame and shotcrete, and finally gives the calculation method of the bearing capacity of the primary support. The Xiabeishan No. 2 Tunnel was selected as a verification calculation example, and the accuracy of the calculation method proposed in this paper is verified through comparison with the numerical simulation results. Compared with the previous studies, the method proposed in this paper can directly reflect the displacement of the tunnel wall at different stages of primary support and the impact of different support methods on the bearing capacity of the primary support.

2. Problem Description

2.1. Basic Assumptions

This paper theoretically analyzes the displacement of a tunnel wall and the bearing capacity of the primary support of the tunnel. For the convenience of discussion, the following basic assumptions are made:

• The rock mass and supporting structure are homogeneous, continuous, and isotropic materials, and the mechanical behavior of the rock mass is linear elastic;
• The tunnel is deeply buried and bears uniform stress at infinity;
• The deformation of the bolts, steel arch frame, and shotcrete is elastic, and the installation is instantaneous;
• The bolts are fully grouted bolts of equal length, and the steel arch frame and shotcrete are closely attached to the surrounding rock;
• The position of the end of the anchor rod is set in the plastic zone according to the most unfavorable situation;
• Rock mass failure follows the Mohr–Coulomb model;
• The tunnel is circular and in a static water stress field;
• The steel arch and shotcrete are closely connected to the surrounding rock and undergo deformation together with the surrounding rock [4,5,6,7,8].

2.2. Mechanical Model

Based on the assumptions in Section 2.1, a mechanical model of the displacement of a tunnel wall and the bearing capacity of the primary support is established (see Figure 2). In order to visually illustrate the mechanical model in this article, half of the tunnel was taken for analysis. Assuming that the tunnel is in the hydrostatic stress field, it can be regarded as an axisymmetric problem. Wherein, P is the bearing capacity of the composite arch and P includes two components, the bearing capacity of the bolts P_s and the bearing capacity of the shotcrete and the steel arch frame P_c.

3. Theoretical Derivations

3.1. Bolts

3.1.1. Basic Assumptions and Mechanical Model

In many tunnels built in China, fully bonded bolts are used. Freeman put forward the neutral point theory of fully bonded bolts. This theory believes that fully bonded bolts rely on the adhesive between the rock and bolts to provide anchoring force, that is, adhesion force and friction force, so as to control the deformation of the surrounding rock [27]. Figure 3 shows a schematic diagram of the bolt and the bonding layer.

The construction sequence and process of tunnel bolts involves drilling holes in the tunnel wall and injecting a resin anchoring agent or mortar followed by the implantation of anchor rods (threaded steel bars). The bolts are connected to the tunnel surrounding rock as a whole through the bonding effect of the anchoring interface layer. The bearing capacity of the full-length adhesive bolt is also transmitted through the anchoring interface layer between the surrounding rock and the bolts. Based on this, the following assumptions are made:

• The bolt only produces axial deformation and is in an elastic state;
• The anchor agent is also in the elastic state, and the bolt bonding layer only produces shear deformation and can maintain a complete bond with the bolt and surrounding rock;
• The surrounding rock of the tunnel is in an elastic–plastic state, and the surrounding rock in the area where the bolt is located is in a plastic state.

In Figure 3, at the distance r from the tunnel center, a differential segment with a length of dr was selected for analysis, as shown in Figure 4.

3.1.2. Bearing Capacity Derivation

The static balance equation of the bolt:

$${\sigma }_b\cdot \frac{\pi }{4}{d}_b{}^2+\tau \cdot \pi \cdot d_b\cdot dr=({\sigma }_b+d{\sigma }_b)\cdot \frac{\pi }{4}{d}_b{}^2$$

(1)

where d_b is the diameter of the bolt, σ_b is the axial stress of the bolt, and τ is a function of r, which can be written as τ(r). By simplifying the above formula, we can obtain:

$$\frac{d{\sigma }_b}{dr}=\frac{4\tau (r)}{d_b}$$

(2)

It is assumed that the axial displacement u_b of the bolt is positive towards the inside of the tunnel, which is opposite to the direction of the polar coordinate system with the tunnel center as the origin. At this time, the axial strain of the bolt microsegment is

$${\epsilon }_b=-\frac{d{u}_b}{dr}$$

(3)

The axial stress of the bolt is

$${\sigma }_b=-E_b\frac{d{u}_b}{dr}$$

(4)

where E_b is the elastic modulus of the bolt.

The simultaneous Equations (2) and (4) can be obtained as

$$\frac{d^2{u}_b}{d{r}^2}=-\frac{4}{E_b{d}_b}\tau (r)$$

(5)

Use G_b to represent the shear modulus of the bolt agent, t_b to represent the thickness of the bolt agent, and u_rb to represent the radial displacement of the surrounding rock after the bolt is applied. At this time, the shear strain of the adhesive layer is (u_rb − u_b)/t_b. It can be seen that the shear stress of the bolt agent is

$$\tau (r)=\frac{G_b}{t_b}(u_{rb}-u_b)$$

(6)

The simultaneous Equations (5) and (6) can be obtained as

$$\frac{d^2{u}_b}{d{r}^2}-m^2{u}_b+m^2{u}_{rb}=0$$

(7)

which are obtained after solving

$$\begin{array}{l}{u}_b=C_1{e}^{Br}+C_2{e}^{-Br}-\frac{B{e}^{Br}}{2}{\displaystyle \int e^{-Br}{u}_{rb}}dr+\frac{B{e}^{-Br}}{2}{\displaystyle \int e^{Br}{u}_{rb}}dr\\ B=\sqrt{4G_b/E_b{d}_b{t}_b}\end{array}$$

(8)

To solve the above formula, it is necessary to solve the radial displacement of the surrounding rock. Assuming that the surrounding rock around the bolt is in a plastic state, according to the literature [28], the displacement of the axisymmetric plastic zone of the circular tunnel is

$$u=\frac{A}{r}$$

(9)

A is the integral constant. Here, u is replaced with u_rb:

$$u_{rb}=\frac{A_{rb}}{r}$$

(10)

u_rb is the displacement of the tunnel wall after the bolt is applied, and A_rb is the integral constant after the bolt is applied.

Substitute Equation (10) into Equation (8).

$$u_b=C_1{e}^{Br}+C_2{e}^{-Br}-\frac{A_{rb}B{e}^{Br}}{2}{\displaystyle \int \frac{e^{-Br}}{r}}dr-\frac{A_{rb}{B}^2{e}^{-Br}}{2}{\displaystyle \int \frac{e^{Br}}{r}}dr$$

(11)

According to the literature in [26], there are

$$\begin{array}{l}{\displaystyle \int \frac{e^{-Br}}{r}}dr\approx \frac{e^{-Br}}{-Br}{\displaystyle \sum _{n=0}^{\infty }\frac{{(-1)}^{n}n!}{{(Br)}^n}}\\ {\displaystyle \int \frac{e^{Br}}{r}}dr\approx \frac{e^{Br}}{Br}{\displaystyle \sum _{n=0}^{\infty }\frac{n!}{{(Br)}^n}}\end{array}$$

(12)

Since the general fully bonded bolt does not need to apply a backing plate, and the axial force at both ends of the bolt is 0, there are the following boundary conditions:

$${\left.\frac{d{u}_b}{dr}\right|}_{r=r_0}=0,{\left.\frac{d{u}_b}{dr}\right|}_{r=r_0+L}=0$$

(13)

where r₀ is the radius of the tunnel and L is the length of the bolt.

The simultaneous Equations (11)–(13) can be obtained as

$$C_1=A_{rb}{D}_1,C_2=A_{rb}{D}_2$$

(14)

where

$$\left.\begin{array}{l}{D}_1=\frac{e^{-B{r}_0}}{2}\frac{\gamma (r_0)-\gamma (r_0+L)e^{BL}}{e^{2BL}-1}\\ D_2=\frac{e^{B(r_0+L)}}{2}\frac{\gamma (r_0)e^{BL}-\gamma (r_0+L)}{e^{2BL}-1}\\ \gamma (r)=\frac{1}{r}{\displaystyle \sum _{n=0}^{\infty }\frac{{(-1)}^{n}n!-n!}{{(Br)}^n}}\end{array}\right\}$$

(15)

$${\sigma }_b(r)=-A_{rb}B{E}_b[D_1{e}^{Br}-D_2{e}^{-Br}+\frac{\gamma (r)}{2}]$$

(16)

$$\tau (r)=\frac{A_{rb}{G}_b}{t_b}[\frac{1}{r}-D_1{e}^{Br}-D_2{e}^{-Br}-\frac{1}{2r}{\displaystyle \sum _{n=0}^{\infty }\frac{{(-1)}^{n}n!+n!}{{(Br)}^n}}]$$

(17)

For the fully bonded bolt, when r = ρ (the radius of the neutral point), the maximum axial force of the anchor rod is

$$Q_{\max }=\frac{\pi }{4}{d}_b{}^2{\sigma }_b(\rho )$$

(18)

The radius of the neutral point adopts the Equation (19) calculation [29]:

$$\rho =\frac{L}{\ln (1+L/r_0)}$$

(19)

The displacement of the tunnel wall after the bolt support can be expressed as

$$u_1=\frac{A_{rb}}{r_0}$$

(20)

where u₁ is the displacement of the tunnel wall after the bolt support. Section 3.1.2 gives its solution method.

The simultaneous Equations (16), (18)–(20) can be obtained as

$$Q_{\max }=\frac{\pi }{4}{d}_b{}^2{E}_{b}B{r}_0{u}_1(D_1{e}^{B\rho }-D_2{e}^{-B\rho }+\frac{\gamma (\rho )}{2})$$

(21)

According to the literature in [29], Q_s = Q_max.

3.1.3. Displacement of the Tunnel Wall Derivation

The wedge element of the surrounding rock containing only one bolt was analyzed, and the supporting force of the bolt on the surrounding rock was simplified to the radial volume force f(r).

The resultant force dQ on the microend dr interface of the bolt is

$$dQ=-\pi d_b\tau (r)dr$$

(22)

The volume dV of this microend is

$$dV=Dr\theta dr$$

(23)

$$f(r)=\frac{dQ}{dV}=-\frac{\pi d_b}{D_b\theta }\times \frac{\tau (r)}{r}$$

(24)

where D_b and θ are the longitudinal spacing and circumferential included angle of the bolts along the tunnel, respectively.

According to Mindlin’s solution, assuming that the cement paste and the rock are elastic materials with the same properties, the rock is regarded as half space, the bolt is infinite, and the displacement of the rock mass at the orifice is equal to the total elongation of the bolt, the bond stress distribution equation of the fully bonded bolt is derived τ(r):

$$\tau (r)=\frac{p_{b}tr}{2\pi a_b}{e}^{-\frac{1}{2}t{r}^2}$$

(25)

$$t=\frac{1}{(1+\mu )(3-2\mu )a_b^2}(\frac{E_0}{E_b})$$

(26)

where p_b is the ultimate pulling force of the bolt, a_b is the radius of the bolt, and E₀ and μ are the elastic modulus and Poisson’s ratio of the soil, respectively.

p_b can be calculated by the following formula:

$$p_b=2\pi a_{b}L{\tau }_{ult}$$

(27)

where τ_ult is the ultimate shear strength of the rock mass.

The surrounding rock in the anchorage zone will meet the equilibrium differential equation:

$$\frac{d{\sigma }_r}{dr}+\frac{{\sigma }_r-{\sigma }_{\theta }}{r}+f(r)=0$$

(28)

where σ_r is the radial stress in the tunnel anchorage zone and σ_θ is the circumferential stress in the anchorage zone.

Replace Equation (28). In conjunction with the Mohr–Coulomb criterion, the following is obtained:

$$\frac{{\sigma }_r+C_b\cot {\phi }_b}{{\sigma }_{\theta }+C_b\cot {\phi }_b}=\frac{1-\sin {\phi }_b}{1+\sin {\phi }_b}$$

(29)

where φ_b and C_b are the internal friction angle and cohesion of the surrounding rock, respectively, through the following constant variation method:

$$\begin{array}{l}{\sigma }_r=r^{n-1}[A_1-{\displaystyle \int f(r)r^{1-n}}dr]-C_b\cot {\phi }_b\\ n=\frac{1-\sin {\phi }_b}{1+\sin {\phi }_b}\end{array}$$

(30)

where A₁ is the integral constant, which is determined by the boundary conditions, when r = r₀, σ_r = p_i, where p_i is the support force of the shotcrete layer. If the shotcrete layer is not applied, p_i = 0. Based on this, the following can be obtained:

$$A_1=\frac{p_i+C_b\cot {\phi }_b}{r_0^{n-1}}+{\displaystyle {\int }_{}^{r_0}f(r)}{r}^{1-n}dr$$

(31)

Replace Equation (31) and substitute Equation (30) and obtain

$${\sigma }_r=r^{n-1}[\frac{p_i+C_b\cot {\phi }_b}{r_0^{n-1}}+{\displaystyle {\int }_{}^{r_0}f(r)}{r}^{1-n}dr-{\displaystyle \int f(r)r^{1-n}}dr]-C_b\cot {\phi }_b$$

(32)

The anchorage zone can be considered as the support resistance p_i′ to the deep surrounding rock, and the tunnel radius can be considered as r = r₀ + L.

$$p_i\prime ={\sigma }_r(r_0+L)={(r_0+L)}^{n-1}[\frac{p_i+C_b\cot {\phi }_b}{r_0^{n-1}}+{\displaystyle {\int }_{r_0}^{L+r_0}f(r)}{r}^{1-n}dr]-C_b\cot {\phi }_b$$

(33)

Combine Equation (33) with Equations (24), (25) and (27) to obtain

$$p_i\prime ={\sigma }_r(r_0+L)={(r_0+L)}^{n-1}[\frac{p_i+C_b\cot {\phi }_b}{r_0^{n-1}}-\frac{2\pi a_b}{D_b\theta }·L·{\tau }_{ult}·t{\displaystyle {\int }_{r_0}^{L+r_0}{e}^{-\frac{1}{2}t{r}^2}}{r}^{1-n}dr]-C_b\cot {\phi }_b$$

(34)

$${\displaystyle {\int }_{r_0}^{L+r_0}{e}^{-\frac{1}{2}t{r}^2}}{r}^{1-n}dr=-\frac{(\mathrm{\Gamma }(-\frac{n-2}{2},\frac{(r_0{}^2+2L{r}_0+L^2)t}{2})-\mathrm{\Gamma }(-\frac{n-2}{2},\frac{r_0{}^{2}t}{2}))t^{\frac{n}{2}-1}}{2^{\frac{n}{2}}}$$

(35)

From this, the radius of plastic zone R_s0 of the tunnel supported by the system bolt is

$$R_{s0}=(r_0+L){[\frac{(P+C_b\cot {\phi }_b)(1-\sin {\phi }_b)}{p_i\prime +C_b\cot {\phi }_b}]}^{\frac{1-\sin {\phi }_b}{2\sin {\phi }_b}}$$

(36)

where P is the original rock stress of the tunnel.

The total displacement of the tunnel wall under the bolt support is

$$u_2=\frac{(P+C_b\cot {\phi }_b)\sin {\phi }_b{R}_{s0}^2}{2G_s{r}_0}$$

(37)

The released displacement of the surrounding rock before the bolt support is u₀. Considering the construction sequence, it can be concluded that

$$u_1=u_0-u_2$$

(38)

where u₀ can be determined by Equation (40).

$$u_0=\frac{(P+C_b\cot {\phi }_b)\sin {\phi }_b{R}^2}{2G_s{r}_0}$$

(39)

where R is the radius of the plastic zone without support, calculated by the following equation:

$$R=r_0{[\frac{P+C_b\cot {\phi }_b}{C_b\cot {\phi }_b}(1-\sin {\phi }_b)]}^{\frac{1-\sin {\phi }_b}{2\sin {\phi }_b}}$$

(40)

3.2. Steel Arch Frame and Shotcrete

3.2.1. Bearing Capacity Derivation

The bearing capacity of the inner support first needs to determine the circumferential axial force of the inner support. The circumferential axial force can be written as follows:

$$T=E_{f+c}{A}_{f+c}{\epsilon }_i$$

(41)

where E_f+cA_f+c is the equivalent axial compressive stiffness of the inner support structure, which is calculated according to the following Equation:

$$A_{f+c}=A_f+A_c$$

(42)

$$E_{f+c}=\frac{{\sqrt{[E_{c}s+(E_f/E_c-1)E_f{A}_f/d]}}^3}{{\sqrt{[E_c{s}^3+(E_f/E_c-1)E_f{J}_f/d]}}^{}}$$

(43)

where A_f is the cross-sectional area of the steel arch frame (m²); A_c is the cross-sectional area of concrete (m²); E_f is the elastic modulus of the steel arch frame (Pa); E_c is the elastic modulus of the shotcrete (Pa); s is the thickness of the shotcrete (m); and J_f is the section moment of inertia of the steel arch frame.

The displacement of the tunnel wall under the joint action of the shotcrete, the steel arch frame, and the bolts is recorded as u₃; then, Equation (42) can be recorded as

$$T=E_{f+c}{A}_{f+c}\frac{u_3}{r_0}$$

(44)

3.2.2. Displacement of the Tunnel Wall Derivation

The shotcrete and steel arch frame deform together with the surrounding rock under the pressure of the surrounding rock. Because the bolt is adopted, the surrounding rock pressure at this time is p_i′. The calculation Equation of the lining stress can be obtained from the thick-wall cylinder theory as follows:

$$\begin{array}{l}{\sigma }_r=\frac{p_i\prime }{(1/r_0{}^2-1/r_1{}^2)r^2}-\frac{p_i\prime }{(1/r_0{}^2-1/r_1{}^2)r_1{}^2}\\ {\sigma }_{\theta }=\frac{-p_i\prime }{(1/r_0{}^2-1/r_1{}^2)r^2}-\frac{p_i\prime }{(1/r_0{}^2-1/r_1{}^2)r_1{}^2}\end{array}$$

(45)

where r₁ is the inner radius of the lining.

According to this, the total displacement of the tunnel wall under the joint action of the shotcrete, the steel arch frame, and the bolts is

$$u_4=\frac{1+{\mu }_1}{E_{f+c}}{p}_i^{}\prime [\frac{r_1{}^2+(1-2{\mu }_1)r_0{}^2}{r_0{}^2-r_1{}^2}]r_0{}^{}$$

(46)

where μ₁ is the Poisson’s ratio of the lining; the Poisson’s ratio of the concrete is still used because of the small impact.

According to this, the tunnel wall displacement under the joint action of the shotcrete, the steel arch frame, and system anchor rod is

$$u_3=u_4-u_2$$

(47)

3.3. Bearing Capacity of the Primary Support Derivation

The circular tunnel can be regarded as an axisymmetric plane strain problem without shear stress. At this time, the axial stress and tangential stress of the surrounding rock are σ₁ and σ₃, respectively. The surrounding rock under combined support follows the Mohr–Coulomb criterion, including

$${\sigma }_1={\sigma }_3\frac{1+\sin {\phi }_b}{1-\sin {\phi }_b}+2C_b\frac{2\cos {\phi }_b}{1-\sin {\phi }_b}$$

(48)

Let K_r = (1 + sinφ_b)/(1 − sinφ_b), there is P_c = K_rP, and the support resistance P can be decomposed into the support force P_s provided by the bolt and the support resistance P_c provided by the steel arch frame and shotcrete layer. P_s can be written as

$$P_{\mathrm{s}}=\frac{Q_s}{D_l{D}_a}$$

(49)

where Q_s is the anchoring force of the bolt and D_l and D_a are the spacing and row spacing of the system, respectively. It is generally assumed that these two values are equal.

The support structure composed of the shotcrete and the steel arch frame provides support force by changing the circumferential axial force, so P_l can be written as

$$P_l=\frac{T}{r_0{L}_s}$$

(50)

where L_s is the length of the support formed by the shotcrete and the steel arch frame. Since the length of the semicircle arch can be expressed as

$$L_s=\pi r_0$$

(51)

The above equation can be written as

$$P_l=\frac{T}{\pi r_0^2}$$

(52)

Then, the support generated by the primary support can be written as Equation (53):

$$P=\frac{Q_s}{D_l{D}_a}+\frac{T}{\pi r_0^2}$$

(53)

where σ_T is the circumferential stress of the inner support, b_T is the longitudinal unit length of the tunnel, taken as 1m, and h is the thickness of the inner support. The bearing capacity of the bolt is further analyzed below. The thickness b of the surrounding rock composite arch is taken as follows:

$$b=\frac{L\tan \alpha -D_b}{\tan \alpha }$$

(54)

where L is the effective length of the bolt; α is the control angle of the bolt in the broken rock mass, usually 43–45°; and D_b is the row spacing of the bolts. Along the longitudinal unit length of the tunnel, the combined bearing capacity N of the surrounding rock composite arch is

$$N=Pb\frac{1+\sin {\phi }_b}{1-\sin {\phi }_b}+\frac{1}{2}k{b}^2$$

(55)

where k is the increasing slope of radial stress. When the surrounding rock is in an unstable weak and broken state, k = 0; L is the effective length of the anchor rod, and the above formula is simplified as

$$N=P\frac{1+\sin {\phi }_b}{1-\sin {\phi }_b}\frac{L\tan \alpha -D_b}{\tan \alpha }$$

(56)

If the circumferential axial force of the surrounding rock composite arch is N₀ and the differential segment of the composite arch along the circumferential direction is ds, then ds = (r₀ + b/2) dθ and perform the static balance solution:

$$\begin{array}{l}\sum Y=0;{\displaystyle {\int }_0^{\pi }q\sin \theta ds=2N_0}\\ N_0=(r_0+\frac{b}{2})q\end{array}$$

(57)

If you want to ensure the stability of the composite arch, you need to meet N ≥ N₀ to ensure the safety of the surrounding rock. When N = N₀, the composite arch reaches the limit equilibrium state. Bring this condition into Equation (57), and you can obtain

$$q=(\frac{Q_s}{D_l{D}_a}+\frac{T}{\pi r_0{}^2})\frac{2(1+\sin {\phi }_b)(L\tan \alpha -D_b)}{(1-\sin {\phi }_b)[(2r_0+L)\tan \alpha -D_b]}$$

(58)

In the above equation, q is the support force, and L is the length of the bolt in the broken surrounding rock. Expand the above equation to obtain:

$$q=\frac{2Q_s(1+\sin {\phi }_b)(L\tan \alpha -D_b)}{D_l{D}_a(1-\sin {\phi }_b)[(2r_0+L)\tan \alpha -D_b]}+\frac{2T(1+\sin {\phi }_b)(L\tan \alpha -D_b)}{\pi r_0{}^2(1-\sin {\phi }_b)[(2r_0+L)\tan \alpha -D_b]}$$

(59)

Simplify Equation (59):

$$q={\lambda }_{\mathrm{s}}{Q}_{\mathrm{s}}+{\lambda }_{\mathrm{T}}T$$

(60)

where

$${\lambda }_{\mathrm{s}}=\frac{2(1+\sin {\phi }_b)(L\tan \alpha -D_b)}{D_l{D}_a(1-\sin {\phi }_b)[(r_0+L)\tan \alpha -D_b]}$$

(61)

$${\lambda }_{\mathrm{T}}=\frac{2(1+\sin {\phi }_b)(L\tan \alpha -D_b)}{\pi r_0{}^2(1-\sin {\phi }_b)[(r_0+L)\tan \alpha -D_b]}$$

(62)

Substitute the Q_s and T obtained in Section 3.1.2 and Section 3.2.1, respectively, into Equation (60) to obtain the bearing capacity of the primary support.

The specific process for calculating the displacement of the tunnel wall and the bearing capacity of the primary support is shown in Figure 5.

4. Case Study and Analysis

4.1. Subsection

In order to verify the proposed calculation method, this article selects the DK215+312 section of the No. 2 tunnel in Xiabeishan as the analysis object. The basic parameters of the tunnel and strata are shown in

Table 1. Tunnel parameters.

b (m)	h (m)	r0 (m)	P (GPa)
26.3	16.3	10.65	0.0006168

and

Table 2. Ground parameters.

Cb (GPa)	φb (°)	Es (GPa)	Gs (GPa)	μ
0.00065	19	0.6	0.231	0.3

, respectively. Using the equivalent circle method to achieve the equivalence of noncircular tunnels, using Equation (63), multicenter circular tunnels can be effectively equivalent to circular tunnels. The schematic diagram of the equivalent method is shown in Figure 6.

$$r_0=\frac{h+b}{4}$$

(63)

In order to analyze the impact of different bolt lengths on the primary support bearing capacity, three different bolt lengths, 6, 8, and 10 m, were selected for verification. The anchor rod parameters are shown in

Table 3. Bolt parameters.

Eb (GPa)	Gb (GPa)	tb (m)	db (m)	Lb (m)	Db (m)	θ (°)
210	1.15	0.009	0.032	6	1.6	45
210	1.15	0.009	0.032	8	1.6	45
210	1.15	0.009	0.032	10	1.6	45

, the steel arch parameters are shown in

Table 4. Steel frame parameters.

Ef (GPa)	Af (GPa)	Db (m)	I (m4)
206	0.003955	1.6	0.000025

, and the parameters of the sprayed concrete are shown in

Table 5. Shotcrete parameters.

Es (GPa)	As (GPa)	μ	l (m)
21	0.636045	0.21	0.3

.

4.2. Comparative Analysis of Theory and Numerical Simulation

As shown in Figure 7, MIDAS GTS was used for modeling and FLAC3D 5.0 software was imported for numerical simulation; this improves the modeling speed and computational efficiency. Considering the symmetry of the knife model, only 1/4 of the actual model was established, thus improving the computational efficiency. The buried depth of the tunnel is 60 m, and the soil weight is 25.7. The bolt lengths of 6 m, 8 m, and 10 m were taken as an example to verify the accuracy of the method proposed in this study. The lining thickness is 0.3 m, and the bolts were simulated using the beam element. In order to improve computational efficiency, the grid sizes of excavated soil, near end soil, and far end soil were distinguished and presented in different colors during model construction.

The comparison of the theoretical calculation and numerical simulation results is shown in

Table 6. Comparison between the analytical results of the tunnel wall displacement and the numerical simulation results.

Comparison	Length of The Bolts (m)	u0 (mm)	u1 (mm)	u2 (mm)	u3 (mm)	u4 (mm)
Analytical	6	45.74	11.7	34.05	17.2	16.85
Numerical simulation		48.75	13.59	35.16	20.56	14.6
Error		3.01	1.89	1.11	3.36	2.25
Analytical	8	45.74	13.63	32.11	17.83	14.28
Numerical simulation		48.75	17.17	31.58	19.26	12.32
Error		3.01	3.54	0.53	1.43	1.96
Analytical	10	45.74	14.21	31.53	19.59	11.94
Numerical simulation		48.75	19.2	29.55	19.37	10.18
Error		3.01	4.99	1.98	0.22	1.76

.

The calculation results indicate that the method proposed in this paper has smaller errors and higher accuracy compared with the numerical simulation results.

4.3. Analysis of Primary Support Bearing Capacity

4.3.1. The Influence of the Bolts’ Length

The spacing between the bolts is set at 1.6 m, and the lining thickness is set at 0.3 m. The results of analyzing the tunnel wall displacement and bearing capacity under different bolt lengths using the method proposed in this article are shown in

Table 7. Analytical results of the wall displacement and bearing capacity of different bolt lengths.

Bolts’ Length	u0 (mm)	u1 (mm)	u2 (mm)	u3 (mm)	u4 (mm)	Qs (MPa)	T (MPa)	q (MPa)
6	45.74	11.7	34.05	17.2	16.85	0.074	0.57	0.411
8	45.74	13.63	32.11	17.83	14.28	0.096	0.59	0.473
10	45.74	14.21	31.53	19.59	11.94	0.121	0.64	0.574

.

According to the analytical results, as the length of the bolts increases, the displacement u₂ of the tunnel wall after applying the bolts gradually decreases. It can be seen that the longer the bolts, the more they can constrain the loosening of the surrounding rock. The parameters of the shotcrete and steel frame have not changed, but the surrounding rock u₄ of the tunnel wall after support also shows a gradual decreasing trend. It can be seen that the change in the bolt lengths can improve the constraint displacement ability of the entire primary support system, reflecting the concept of the synergistic effect of the bolts and shotcrete support system.

4.3.2. The Influence of Space between the Bolts

The length of the bolts was determined to be 6 m, and the results of analyzing the tunnel wall displacement and bearing capacity under different bolt spacings using the method proposed in this article are shown in

Table 8. Analysis results of the wall displacement and bearing capacity of different bolt spacings.

Bolts’ Spacing (m)	u0 (mm)	u1 (mm)	u2 (mm)	u3 (mm)	u4 (mm)	Qs (MPa)	T (MPa)	q (MPa)
1.4	45.74	13.5	32.24	18.49	13.75	0.092	0.62	0.524
1.6	45.74	11.7	34.05	17.2	16.85	0.074	0.57	0.411
1.8	45.74	9.53	36.21	17.09	19.12	0.052	0.55	0.374

.

From the analytical results, it can be seen that as the spacing between the bolts increases, the displacement u₂ of the tunnel wall after applying the bolts gradually increases. It can be seen that the smaller the spacing between the bolts, the more they can constrain the loosening of the surrounding rock. The parameters of the shotcrete and steel arch frame have not changed, but the surrounding rock u₄ of the tunnel wall after support also shows a gradually increasing trend, indicating that the change in the bolt spacing will also affect the bearing capacity of the subsequent support measures.

4.3.3. The Influence of the Thickness of the Lining

The length of the bolts was determined as 6 m, and the displacement and bearing capacity of the tunnel wall under different lining thicknesses was analyzed using the method proposed in this article, as shown in

Table 9. Analysis results of the tunnel wall displacement and bearing capacity with different lining thicknesses.

Lining Thicknesses (m)	u0 (mm)	u1 (mm)	u2 (mm)	u3 (mm)	u4 (mm)	Qs (MPa)	T (MPa)	q (MPa)
0.25	45.74	11.7	34.05	16.14	17.91	0.074	0.51	0.332
0.3	45.74	11.7	34.05	17.2	16.85	0.074	0.57	0.411
0.35	45.74	11.7	34.05	18.41	15.64	0.074	0.61	0.511

.

From the analysis results, it can be seen that as the thickness of the lining increases, the surrounding rock u₃ of the tunnel wall after support also shows a gradually increasing trend. It can be seen that increasing the thickness of the lining can improve the constraint displacement ability of the entire initial support system.

5. Conclusions

This article studies the wall displacement and bearing capacity of deep-buried tunnels at different stages of initial support. Based on the Mohr–Coulomb strength criterion, the tunnel wall displacement in the unsupported state, the tunnel wall displacement after system bolt support, the tunnel wall displacement after steel frame and shotcrete support, and the bearing capacity of the anchor, steel frame, and shotcrete were derived. The effectiveness of the proposed method was verified through FLAC3D numerical simulation, and the results show that the calculation results of the proposed method are accurate. The specific conclusions are as follows:

(1)

Based on the Mindlin solution, the solutions for the radius of the plastic zone and the displacement of the tunnel wall after implementation of the system bolt support were derived. This method simulates the tunnel model after the application of bolts by applying additional support forces, which is relatively similar to what occurs in engineering practice.

(2)

Based on the equivalent stiffness method, the solution for the displacement of the tunnel wall after the construction of the steel frame and shotcrete was derived. This method equates the stiffness of the steel frame with the shotcrete, making the solution more practical in engineering.

(3)

Based on the neutral point theory of bolts, a calculation method for the bearing capacity of the system bolts was derived. The calculation method for the primary support bearing capacity was derived based on the Mohr–Coulomb strength criterion, and the partial coefficient calculation methods for two types of support forces were determined, taking into account the cooperative effect of the primary support system.

(4)

The increase in the length and spacing of the system bolts, as well as the increase in the lining thickness, can enhance the displacement constraint effect on the surrounding rock. At the same time, the increase in the bolts’ length and spacing will also enhance the displacement constraint effect of the steel arch and shotcrete.