Elastoplastic Solution for Tunnel Composite Support Structures Based on Mohr–Coulomb Criterion

Abstract

Linings and anchor bolts, as common support structures, play a crucial role in ensuring the stability of tunnel surrounding rock. This study aims to investigate the elastic–plastic behavior of lining-anchor bolt support structures in deep circular tunnels under static water pressure. Based on the Mohr–Coulomb (M-C) criterion and equivalent substitution method, this study derives the solution process for the radius of the plastic zone and the elastic–plastic stress expressions under a given lining internal pressure. Furthermore, through examples, this study analyzes the radius of the plastic zone and the stress evolution curve of the lining-anchor bolt-supported surrounding rock system under three conditions: without anchoring, various elastic moduli, and different anchor bolt support angles. Finally, this study verifies the rationality of the elastic–plastic analytical solution for the lining-anchor bolt-supported surrounding rock structure through numerical simulation methods. The integrated lining-anchor bolt support is notably effective in inhibiting the expansion of the plastic zone, particularly in soft rock. Additionally, the elastic modulus and circumferential spacing of the anchor bolts significantly influence the plastic zone inhibition rate in soft rock.

1. Introduction

The complexity and unpredictability of the geological environment result in tunnel construction projects that are risky, expensive, and challenging [1,2]. The excavation of tunnels disrupts the pre-existing stress equilibrium in the surrounding rock, altering its initial stress field [3,4,5]. While the surrounding rock possesses an inherent load-bearing capacity, in soft rock conditions or under high geostress, it may fail to maintain equilibrium relying solely on this capacity, potentially resulting in lining cracks or rock collapses. These issues can significantly affect project schedules and the safety of construction personnel [6,7,8]. Therefore, the stability and support measures of tunnel engineering have always been of great concern to scholars.

To enhance load-bearing capacity, support structures are frequently employed in the strengthening of surrounding rocks during tunnel construction. The commonly used structures for this purpose include shotcrete, steel shapes, arch frames, and rock bolts [9,10,11,12]. Numerous varieties of support systems and their distinct mechanisms exist, and the performance of combined support systems differs from that of single-method supports. Researchers have developed composite support solutions tailored to the properties of both the surrounding rock and the support structure [13,14,15,16]. For instance, Sun and Zhang [17] proposed a compound support ring model with three elements, an anchor reinforcement ring, a primary lining ring, and a secondary lining ring, elucidating the role of each through experimental data. Sun et al. [18] discovered that shotcrete–anchor support bolsters the surrounding rock strength, promotes the development of pressure arches, and enhances the inherent load-bearing capability of the rock.

With the advent of the New Austrian Tunneling Method, modern support theories like the convergence–constraint method (CCM) have seen rapid advancement [19,20]. Wang et al. [21] used the convergence–confinement method (CCM) to study the support performance of steel–concrete composite support (SCCS) systems, and compared it with conventional support systems. An et al. [22] identified the key failure zones of a roadway based on the convergence–confinement method and established a relationship between the failure tendency coefficient of these critical zones and their distance from the working face. The convergence–constraint method (CCM), based on the continuous medium assumption theory, evaluates the stability of surrounding rocks by analyzing the interactions between support structures and the rock mass, effectively reflecting the characteristics of the interplay between surrounding rocks and support structures. It is widely applied in the design and construction of tunnel engineering. Studies based on the CCM take into account the reciprocal action of anchor bolts with rock and soil masses, assessing support efficacy using characteristic support curves. Zhao et al. [23] conducted laboratory modeling tests and theoretical analyses to investigate the stress and deformation behavior of bolts, jointed planes, anchored bodies, and rock masses under shear conditions. They further elucidated the interactions between rock mass and bolts through displacement curve analysis. Carlos Carranza-Torres and Engen, M. [24] conducted a detailed analysis of the equations necessary for constructing support characteristic curves for a combined support system. Ding et al. [25] compared the characteristic curves of the surrounding rock and initial support at different stress release rates. They summarized how the stress release rate affects the support load and determined the optimal excavation stress release rate through calculations of stable rock pressure. Other studies treat the anchor bolts and surrounding rock as a unit and assess support through characteristic rock curves. Tan [26] presented a finite difference method (FDM)-based solution and validated its practical advantages through the radial displacement and stress distribution curves of the surrounding rock. Zou et al. [27] proposed a new Ground Response Curve (GRC) analysis method and verified the validity of the method by surrounding rock characterization curves. Wang et al. [28] proposed an analytical method for calculating the active and passive reinforcement indices of anchored surrounding rock and verified the reinforcement mechanisms of different anchoring methods through physical simulation tests. Given the intricate dynamics of anchor bolt–rock mass interactions, a growing consensus is forming around treating anchor bolts as a means of reinforcing surrounding rock, rather than as standalone support structures [29,30,31].

The early Fenner [32] and Kastner [33] formulas provided analytical expressions for the support reaction forces in deeply buried circular chambers. They detailed the plastic zone range and the mechanical parameters of the surrounding rock, thus offering a theoretical foundation for managing significant deformation in the surrounding rock. Zhou et al. [34] derived a point stress method and approximate plastic condition method for calculating the plastic zone and performed finite element numerical calculations; Li et al. [35] determined the plastic zone radius and radial displacement of loess tunnels based on a modified Fenner formula. Currently, international and domestic analyses of tunnel surrounding rock elasticity and plasticity primarily rely on the following strength criteria: the Mohr–Coulomb (M-C) strength criterion, which considers the linear relationship between shear stress and normal stress [36,37,38,39]; the Hoek–Brown (H-B) criterion, which accounts for the nonlinear behavior of rock masses [40,41]; the Drucker–Prager (D-P) criterion, which is an extension of the Mohr–Coulomb criterion for plasticity [42,43,44]; and the unified strength theory (UST), which integrates various failure modes [45,46,47]. Due to its high applicability and reliability, the Mohr–Coulomb criterion is commonly employed in engineering design.

Although many scholars have proposed various yield criteria and elastic–plastic solutions for different conditions, research on support mechanisms has mainly focused on primary lining and anchor spray supports in shallow tunnels. There has been limited attention paid to elastic–plastic analytical solutions for lining-anchor support structures in deep circular tunnels. This study introduces a mechanical model for the surrounding rock structure reinforced by lining-anchor forces based on the Mohr–Coulomb criterion and the equivalent material method. It establishes the elastic–plastic equation for the surrounding rock in deep circular tunnels supported by a lining and anchors under hydrostatic pressure. Through case studies, this research analyzes the effects of different anchoring parameters—such as the presence or absence of anchoring, anchor bolt elastic modulus, and anchor support angle—on the tunnel’s plastic zone. The findings provide a vital theoretical basis to elucidate the stability and working mechanisms of lining-anchor support structures.

2. Elastic–Plastic Analysis of Lining-Anchor Support in Surrounding Rock Structures

2.1. Basic Assumptions

We assume that a circular tunnel exists in an infinitely long elastic medium, with the load being axisymmetrically distributed around the tunnel’s cross-section. The lining material is concrete, which is treated as a homogeneous, isotropic elastic–plastic material. The Mohr–Coulomb criterion is used to describe the plastic behavior of the rock. As depicted in Figure 1, the inner radius of the tunnel is denoted by ρ₀, the outer radius of the tunnel lining by ρ₁, and the radius of the tunnel’s anchoring zone by ρ₂. The pressure on the inner wall of the tunnel lining is represented by P, while the ground stress is indicated by P₀. The lining has an elastic modulus of E₁, a Poisson’s ratio of μ₁, cohesion of c₁, and an internal friction angle of φ₁. The rock mass has an elastic modulus of E₂, a Poisson’s ratio of μ₂, cohesion of c₂, and an internal friction angle of φ₂. It is assumed that the yield radius is ρ_p, with compressive stress being positive and tensile stress being negative.

2.2. Analytical Solution for Stress Within the Plastic Zone Located Within the Lining

2.2.1. Stress in the Plastic Zone

When the interface of the plastic zone is situated inside the tunnel lining, as depicted in Figure 1, we have ρ₀ ≤ ρ_p ≤ ρ₁. The Mohr–Coulomb criterion is expressed as follows:

$$\left(1-\sin \phi \right){\sigma }_{\rho }-\left(1+\sin \phi \right){\sigma }_{\theta }=2c\cos \phi$$

(1)

This leads to the differential equation form for σ_ρ:

$${\displaystyle \frac{\mathrm{d}{\sigma }_{\rho }}{\mathrm{d}\rho }}+{\displaystyle \frac{{\sigma }_{\rho }}{\rho }}{\displaystyle \frac{2\sin \phi }{\sin \phi +1}}+{\displaystyle \frac{2c\cos \phi }{\rho \left(\sin \phi +1\right)}}=0$$

(2)

By solving the differential equation, the stress within the plastic zone of the lining can be determined as follows:

$${\sigma }_{\rho }=-{\displaystyle \frac{c\left(3\cos \phi +\cos 3\phi \right)}{\sin \phi +\sin 3\phi }}+N_1{\rho }^{-{\displaystyle \frac{2\sin \phi }{1+\sin \phi }}}$$

(3)

where N₁ represents the integration constant, and the yield stress can be determined by the boundary conditions.

$${\left({\sigma }_{\rho }\right)}_{\rho ={\rho }_0}=P$$

(4)

By inserting the values of the tunnel lining’s cohesion (c= c₁) and friction angle (φ = φ₁) into Equation (3), we derive the following:

$$N_1=-{\displaystyle \frac{c_1\left(3\cos {\phi }_1+\cos 3{\phi }_1\right)+P\left(\sin {\phi }_1+\sin 3{\phi }_1\right)}{{\rho }_0^{-\frac{2\sin {\phi }_1}{1+\sin {\phi }_1}}\left(\sin {\phi }_1+\sin 3{\phi }_1\right)}}$$

(5)

This leads to the following formula for the stresses within the plastic zone of the tunnel lining:

$$\begin{array}{ll}{\sigma }_{\rho 1\mathrm{p}}& =\left({\displaystyle \frac{c_1\left(3\cos {\phi }_1+\cos 3{\phi }_1\right)+P\left(\sin {\phi }_1+\sin 3{\phi }_1\right)}{\left(\sin {\phi }_1+\sin 3{\phi }_1\right)}}\right)\\ & {\left({\displaystyle \frac{\rho }{{\rho }_0}}\right)}^{-{\displaystyle \frac{2\sin {\phi }_1}{1+\sin {\phi }_1}}}-c_1\left({\displaystyle \frac{3\cos {\phi }_1+\cos 3{\phi }_1}{\sin {\phi }_1+\sin 3{\phi }_1}}\right)\end{array}$$

(6)

$$\begin{array}{cc}{\sigma }_{\theta 1\mathrm{p}}& =-\left({\displaystyle \frac{c_1\left(\sin {\phi }_1-1\right)\left(3\cos {\phi }_1+\cos 3{\phi }_1\right)+P\left(\sin {\phi }_1+\sin 3{\phi }_1\right)}{\left(\sin {\phi }_1+\sin 3{\phi }_1\right)\left(\sin {\phi }_1-1\right)}}\right){\left({\displaystyle \frac{\rho }{{\rho }_0}}\right)}^{-{\displaystyle \frac{2\sin {\phi }_1}{1+\sin {\phi }_1}}}\\ & +c_1\left({\displaystyle \frac{\left(\sin {\phi }_1-2\sin 3{\phi }_1+3\right)\cos {\phi }_1+\left(\sin {\phi }_1-1\right)\cos 3{\phi }_1}{\left(\sin {\phi }_1+\sin 3{\phi }_1\right)\left(\sin {\phi }_1+1\right)}}\right)\end{array}$$

(7)

where σ_ρ1p and σ_θ1p represent the radial and circumferential stresses in the plastic zone of the lining, respectively. Equations (6) and (7) are the expressions for the radial and circumferential stresses in the plastic zone of a circular tunnel lining. These expressions are derived under the condition that the first principal stress is the radial stress, compressive stress is considered positive, and the yield condition is governed by the Mohr–Coulomb (M-C) criterion.

2.2.2. Stress in the Elastic Zone

At this stage, the plastic zone of the tunnel remains within the lining. When ρ > ρ_p, the lining is in the elastic phase, and similarly, the anchored zone and the original rock mass are also in the elastic phase. The stress and displacement expressions for the lining, anchored zone, and original rock mass, all in the elastic phase, are given by the following forms [48]:

Lining zone:

$$\begin{array}{l}{\sigma }_{\rho 1}={\displaystyle \frac{A_1}{{\rho }^2}}+2B_1\hspace{1em}\hspace{1em}{\sigma }_{\theta 1}=-{\displaystyle \frac{A_1}{{\rho }^2}}+2B_1\\ u_{\rho 1}={\displaystyle \frac{1-{\mu }_1^2}{E_1}}\left(-\left(1+{\displaystyle \frac{{\mu }_1}{1-{\mu }_1}}\right){\displaystyle \frac{A_1}{\rho }}+2\left(1-{\displaystyle \frac{{\mu }_1}{1-{\mu }_1}}\right)B_1\rho \right)\end{array}$$

(8)

Anchoring zone:

$$\begin{array}{l}{\sigma }_{\rho 2}={\displaystyle \frac{A_2}{{\rho }^2}}+2B_2\hspace{1em}\hspace{1em}{\sigma }_{\theta 2}=-{\displaystyle \frac{A_2}{{\rho }^2}}+2B_2\\ u_{\rho 2}={\displaystyle \frac{1-{\mu }_2^2}{E_{\mathrm{f}}}}\left(-\left(1+{\displaystyle \frac{{\mu }_2}{1-{\mu }_2}}\right){\displaystyle \frac{A_2}{\rho }}+2\left(1-{\displaystyle \frac{{\mu }_2}{1-{\mu }_2}}\right)B_2\rho \right)\end{array}$$

(9)

Original rock zone:

$$\begin{array}{l}{\sigma }_{\rho 3}={\displaystyle \frac{A_3}{{\rho }^2}}+2B_3\hspace{1em}\hspace{1em}{\sigma }_{\theta 3}=-{\displaystyle \frac{A_1}{{\rho }^2}}+2B_1\\ u_{\rho 3}={\displaystyle \frac{1-{\mu }_3^2}{E_2}}\left(-\left(1+{\displaystyle \frac{{\mu }_2}{1-{\mu }_2}}\right){\displaystyle \frac{A_1}{\rho }}+2\left(1-{\displaystyle \frac{{\mu }_2}{1-{\mu }_2}}\right)B_1\rho \right)\end{array}$$

(10)

These represent the general formulas for stress and displacement in the elastic zones. E₁ represents the elastic modulus of the lining material, while μ₁, μ₂, and μ₃ denote the Poisson’s ratios of the lining, the rock bolt–surrounding rock composite, and the original rock, respectively. σ_ρ1, σ_θ1, and u_ρ1 denote the radial stress, circumferential stress, and radial displacement in the elastic zone of the lining, respectively; σ_ρ2, σ_θ2, and u_ρ2 represent the radial stress, circumferential stress, and radial displacement in the elastic zone of the anchored zone, respectively; σ_ρ3, σ_θ3, and u_ρ3 correspond to the radial stress, circumferential stress, and radial displacement in the intact rock zone, respectively. A₁, A₂, A₃, B₁, B₂, and B₃ are the unknowns that need to be determined.

In the anchoring zone, when applying the Mohr–Coulomb criterion to avoid overly complicating the issue, the anchor bolts and surrounding rock are treated as a homogenized complex. By utilizing parameter equivalence formulas, the mechanical parameters of the anchoring zone are characterized.

The elastic modulus of the composite anchoring zone can be derived using the tensile strength equivalence principle, as indicated below:

$$E_{\mathrm{f}}={\displaystyle \frac{E_{\mathrm{b}}\mathsf{\pi }{r}_{\mathrm{b}}^2+E_2\left(s_1{s}_r-\mathsf{\pi }{r}_{\mathrm{b}}^2\right)}{s_1{s}_r}}$$

(11)

where E_b and E₂ represent the elastic moduli of the anchor bolts and the surrounding rock, respectively; s₁ and s_r are the inter-row and circumferential spacings of the anchor bolts; and r_b denotes the anchor bolt’s cross-sectional radius.

The friction angle (φ_f) and cohesion (c_f) of the composite anchoring zone are expressed as

$${\displaystyle \frac{1+\sin {\phi }_{\mathrm{f}}}{1-\sin {\phi }_{\mathrm{f}}}}=\left(1+{\alpha }_{\mathrm{f}}\right){\displaystyle \frac{1+\sin {\phi }_2}{1-\sin {\phi }_2}}{\displaystyle \frac{2c_{\mathrm{f}}\cos {\phi }_{\mathrm{f}}}{1-\sin {\phi }_{\mathrm{f}}}}=\left(1+{\alpha }_{\mathrm{f}}\right){\displaystyle \frac{2c_2\cos {\phi }_2}{1-\sin {\phi }_2}}$$

(12)

where c_f and φ_f represent the cohesion and friction angle in the anchoring zone of the equivalent material, respectively; α_f indicates the anchor bolt density factor.

These are the expressions for the anchoring zone’s mechanical parameters with the equivalent material. The stress, displacement boundary conditions, and contact conditions are outlined as follows:

$$\begin{array}{l}{\left({\sigma }_{\rho 1}\right)}_{\rho ={\rho }_{\mathrm{p}}}=q_{\rho \mathrm{p}}\\ {\left({\sigma }_{\rho 1}\right)}_{\rho ={\rho }_1}={\left({\sigma }_{\rho 2}\right)}_{\rho ={\rho }_1}\\ {\left({\sigma }_{\rho 2}\right)}_{\rho ={\rho }_2}={\left({\sigma }_{\rho 3}\right)}_{\rho ={\rho }_2}\\ {\left(u_{\rho 1}\right)}_{\rho ={\rho }_1}-{\left(u_{\rho 2}\right)}_{\rho ={\rho }_1}=0\\ {\left(u_{\rho 2}\right)}_{\rho ={\rho }_2}-{\left(u_{\rho 3}\right)}_{\rho ={\rho }_2}=0\\ {\left({\sigma }_{\rho 3}\right)}_{\rho \to \infty }=P_0\end{array}$$

(13)

where q_ρp denotes the stress in the plastic zone, and A₁, A₂, A₃, B₁, B₂, and B₃. are determined by solving six equilibrium equations. Substituting A₁, A₂, A₃, B₁, B₂, and B₃ into the stress expression yields the stress expression for the elastic zone (refer to Appendix A).

With the stress expression in the elastic zone, we can obtain

$${\sigma }_{\rho 1}+{\sigma }_{\theta 1}=4B_1$$

(14)

It is also satisfied by the relation at the elastic–plastic interface as follows:

$${\left({\sigma }_{\rho }\right)}_{\rho ={\rho }_{\mathrm{p}}}+{\left({\sigma }_{\theta }\right)}_{\rho ={\rho }_{\mathrm{p}}}=4B_1$$

(15)

ρ_p can be deduced based on the contact conditions at the elastic–plastic boundary:

$${\left({\sigma }_{\rho 1}\right)}_{\rho ={\rho }_{\mathrm{p}}}={\left({\sigma }_{\rho }\right)}_{\rho ={\rho }_{\mathrm{p}}}=q_{\rho \mathrm{p}}$$

(16)

Equation (16) is a transcendental equation with respect to ρ_p, which is solved using the iterative method. The plastic stress is determined based on the expression for the normal stress q_ρp at the elastoplastic interface and ξ₁ (see Appendix A for details). In this context, we set ρ_p to ρ₀ and the yield stress to P, thereby allowing the calculation of the first critical pressure, P₁.

2.3. Derivation of the Elastic–Plastic Stress Expression in the Anchoring Zone Where the Plastic Zone Is Present

At this stage, ρp ≥ ρ₁, and the stress within the plastic zone satisfies both the equilibrium differential equation and the yielding condition.

$${\displaystyle \frac{\mathrm{d}{\sigma }_{\rho }}{\mathrm{d}\rho }}+{\displaystyle \frac{{\sigma }_{\rho }-{\sigma }_{\theta }}{\rho }}=0$$

(17)

$${\tau }_t=c+{\sigma }_n\tan \phi$$

(18)

The plastic stress within the lining can be expressed as

$${\sigma }_{\rho 1}=\begin{array}{l}\left({\displaystyle \frac{c_1\left(3\cos {\phi }_1+\cos 3{\phi }_1\right)+P\left(\sin {\phi }_1+\sin 3{\phi }_1\right)}{\left(\sin {\phi }_1+\sin 3{\phi }_1\right)}}\right){\left({\displaystyle \frac{\rho }{{\rho }_0}}\right)}^{-{\displaystyle \frac{2\sin {\phi }_1}{-1+\sin {\phi }_1}}}\\ -c_1\left({\displaystyle \frac{3\cos {\phi }_1+\cos 3{\phi }_1}{\sin {\phi }_1+\sin 3{\phi }_1}}\right)\end{array}$$

(19)

$${\sigma }_{\theta 1}=\begin{array}{l}\left({\displaystyle \frac{c_1\left(\sin {\phi }_1+1\right)\left(3\cos {\phi }_1+\cos 3{\phi }_1\right)+P\left(\sin {\phi }_1+\sin 3{\phi }_1\right)}{\left(\sin {\phi }_1+\sin 3{\phi }_1\right)\left(\sin {\phi }_1-1\right)}}\right){\left({\displaystyle \frac{\rho }{{\rho }_0}}\right)}^{-{\displaystyle \frac{2\sin {\phi }_1}{-1+\sin {\phi }_1}}}\\ +c_1\left({\displaystyle \frac{\left(\sin {\phi }_1-2\sin 3{\phi }_1+3\right)\cos {\phi }_1+\left(\sin {\phi }_1+1\right)\cos 3{\phi }_1}{\left(\sin {\phi }_1+\sin 3{\phi }_1\right)\left(\sin {\phi }_1-1\right)}}\right)\end{array}$$

(20)

The contact condition of the elastic–plastic interface is ${\left({\sigma }_{\rho 2}\right)}_{\rho ={\rho }_{\mathrm{p}}}=q_{\rho \mathrm{p}}$:

$${\sigma }_{\rho 2}=-{\displaystyle \frac{c_{\mathrm{f}}\left(3\cos {\phi }_{\mathrm{f}}+\cos 3{\phi }_{\mathrm{f}}\right)}{\sin {\phi }_{\mathrm{f}}+\sin 3{\phi }_{\mathrm{f}}}}+N_2{\rho }^{-{\displaystyle \frac{2\sin {\phi }_{\mathrm{f}}}{-1+\sin {\phi }_{\mathrm{f}}}}}$$

(21)

The integration constant N₂ is solved for and can be found in Appendix A.

Upon substitution of the integration constant N₂ into Equation (21), the expression for plastic stress within the anchoring zone is provided in Appendix A.

3. Analysis of the Elastic–Plastic Response in the MC Lining-Anchor-Support Rock System

This section aims to facilitate a more intuitive understanding of the induced stress field and elastic–plastic zones in the rock mass due to the lining-anchor support. Utilizing the calculation parameters from

Table 1. Lining and tunnel dimensions for calculation.

Parameter	P0/Pa	E1/Pa	μ1	c1/MPa	φ1	ρ0/m	ρ1/m	ρ2/m
Value	1 × 107	2 × 1010	0.167	7	45°	4	5	8

[49,50] and

Table 2. Parameters of surrounding rock and anchor bolts for calculation.

Parameter	E2/Pa	μ2	c2/Pa	φ2	Eb/Pa	Sθ/°
Case 1	2 × 109	0.25	1 × 106	45°	1 × 1011	14.3°
Case 2	2 × 109	0.25	1 × 106	45°	2 × 1011	14.3°
Case 3	2 × 109	0.25	1 × 106	45°	3 × 1011	14.3°
Case 4	2 × 109	0.25	1 × 106	45°	2 × 1011	11.5°
Case 5	2 × 109	0.25	1 × 106	45°	2 × 1011	8.6°
Case 6	5 × 108	0.25	5 × 105	30°	1 × 1011	14.3°
Case 7	5 × 108	0.25	5 × 105	30°	2 × 1011	14.3°
Case 8	5 × 108	0.25	5 × 105	30°	3 × 1011	14.3°
Case 9	5 × 108	0.25	5 × 105	30°	2 × 1011	11.5°
Case 10	5 × 108	0.25	5 × 105	30°	2 × 1011	8.6°

[49,51], it conducts case analyses on scenarios with and without anchoring, varying anchor elastic moduli, and different anchor support angles.

3.1. Analysis of Elastic–Plastic Case Studies with and Without Anchoring

The anchoring parameters in the anchoring zone, as illustrated in Figure 2, include Eb = 2 × 10⁵ MPa, Sθ = 11.5°, the initial critical pressure P₁ = 3.1 MPa, and the secondary critical pressure P₂ = 18.8 MPa. In Phase I, when P < P₁, the first principal stress is the circumferential stress. The plastic zone is located within the lining, and the effect of the anchor bolts in suppressing the expansion of the plastic zone inside the lining is negligible. Additionally, the difference in the radius of the plastic zone between hard and soft rocks, with and without the anchoring zone, is negligible. In Phase II, when P1 < P < P₂, both the lining and the surrounding rock are in the elastic state. The installation of anchor bolts in soft rock increases the interval length by 9.32%, whereas in hard rock, the increase is 6.72%. When P > P₂, the first principal stress becomes radial stress. In Phase III, when 18.8 MPa < P < 20.5 MPa, the plastic zone remains within the lining. In Phase IV, when 20.58 MPa < P < 22.8 MPa, the radius of the plastic zone is 5 m, and the plastic zone has not yet expanded to the anchoring zone. In Phase V, when P > 22.8 MPa, the plastic zone expands into the anchoring zone. At this stage, the radius of the plastic zone under the influence of the anchoring bolts is significantly smaller than that of the unanchored zone. As the plastic zone expands, the disparity between the anchored and unanchored plastic zones diminishes. Under consistent anchoring parameters, the soft rock with the anchoring zone exhibits a maximum inhibition rate of 14.97% and a minimum of 4.92% against the expansion of the plastic zone, whereas the inhibition rates for hard rock are 5.35% and 3.35%, respectively. Soft rock exhibits a more pronounced response in the lining-anchor support system in terms of limiting the extent of the plastic zone.

It is observed that the trends in the evolution curves of the plastic zone radius are largely consistent, regardless of the presence or absence of anchoring. This confirms the accuracy of the theoretical calculation model for the plastic zone radius of the lining-anchor system in supporting surrounding rock under a specified internal lining pressure. Additionally, plastic zone radius and stress analyses were conducted for lining internal pressures of P = 2 MPa and P = 24 MPa. For the former, circumferential stress is the primary principal stress, with the plastic zone located inside the lining; for the latter, radial stress is dominant, and the plastic zone is within the anchoring zone.

Figure 3 illustrates the evolution curves of radial and circumferential stresses at different radial distances for P = 2 MPa. At this stage, the pressure P is less than the critical pressure P₁, and the initial density ρ₀ is less than the peak density ρ_p, which is in turn less than the ultimate density ρ₁. The presence or absence of an anchor support exerts a negligible influence on the expansion of the plastic zone within the lining, resulting in a variance of less than 2%. The internal radial stress within the lining shows no significant difference in its influence, whether anchor support is present or not. In soft rock, the radial stress within the lining is greater than that in hard rock. Within the anchoring zone, the radial stress decreases from the boundary of the lining and anchoring zone to the boundary with the original rock, and soft rock exhibits a higher overall radial stress than hard rock. Furthermore, circumferential stress within the lining exhibits significant fluctuations at different radial distances and undergoes a sharp increase at the interface between the lining and the surrounding rock. The peak circumferential stress occurs at the elastoplastic interface; in soft rock, the peaks are 74.30 MPa and 68.33 MPa, whereas in hard rock they are 54.93 MPa and 54.22 MPa, respectively. The circumferential stress at the interface of the lining and the anchoring zone in soft rock is 8.1 MPa and 6.11 MPa, compared to 10.43 MPa and 8.42 MPa in hard rock, indicating that the circumferential stress in the anchoring zone exceeds that in the non-anchoring zone.

Figure 4 illustrates the evolution curves of radial and circumferential stresses across different radial distances at a pressure of P = 24 MPa. In this scenario, where P > P₂ and ρp > ρ₁, the primary stress shifts from the circumferential to the radial stress. The lining zone has already yielded completely. Within the anchoring zone, i.e., at ρ₁ < ρ < ρ₂, there is a negligible distinction in radial stress between hard and soft rocks, regardless of the presence of anchoring. Radial stress generally demonstrates a declining trend, with soft rock exhibiting marginally higher radial stress than hard rock. Figure 4b illustrates that the circumferential stress within the plastic zone of the anchoring section decreases, achieving its minimum at the elastic–plastic interface of the anchoring zone. The circumferential stress in the plastic zone of the rock mass with anchor bolt support is lower than that in the rock mass without anchor bolt support. Additionally, in the elastic zone, the circumferential stress in soft rock exceeds that in hard rock. Under the influence of anchor bolt support, the soft rock’s plastic zone radius decreases by Δ2 = 0.68 m, while the reduction for hard rock is Δ1 = 0.27 m. The plastic zone radius containment rates for the soft rock and hard rock are 11.47% and 5.35%, respectively, highlighting the anchor bolt’s enhanced effectiveness in curtailing the expansion of the soft rock’s plastic zone.

3.2. Analysis of Elastoplastic Case Studies with Varying Anchor Elastic Moduli

To investigate the impact of variations in the elastic modulus on the composite support effect of rock bolts for tunnel linings, this paper selects three different elastic moduli for analysis. In Figure 5, the anchoring parameters for the anchoring zone are Eb = 1 × 10⁵ MPa, Eb = 2 × 10⁵ MPa, and Eb = 3 × 10⁵ MPa, with Sθ = 14.3°. At varying anchor elastic moduli, the soft rock’s initial critical pressures are 4.57 MPa, 4.73 MPa, and 4.84 MPa, respectively, whereas the secondary critical pressures are 19.71 MPa, 20.28 MPa, and 20.70 MPa. When P₁ < P<P₂, the plastic zone radius is 4 m, and the lining is in the elastic phase. The greater the elastic modulus of the anchoring bolts, the wider the internal pressure range of the lining represented by the interval [P₁, P₂], with a variation rate of 4.54%. For hard rock, the initial critical pressures are 3.11 MPa, 3.13 MPa, and 3.15 MPa, respectively, while the secondary critical pressures are 19.25 MPa, 19.46 MPa, and 19.66 MPa, with a variation rate of 2.24%. When P < P₁, the plastic zone is located inside the lining. As the elastic modulus of the anchor bolt increases, its impact on the expansion of the plastic zone in the lining remains minor, with a suppression rate of less than 1%. When P > P₂, the first principal stress transitions from circumferential to radial stress. The enhanced elastic modulus of the anchoring bolts exhibits a more pronounced effect on inhibiting the expansion of the plastic zone within the anchoring zone of soft rock compared to hard rock, with the plastic zone radius variation rate decreasing from 5.43% to 1.38%. Consequently, the anchoring performance of the bolts exhibits a more significant improvement when supporting soft rock than in hard rock.

Figure 6 illustrates the evolution curves of radial and circumferential stresses at various radial distances, under different anchor elastic moduli, with P = 2 MPa. In this scenario, P < P₁, ρ₀ < ρp < ρ₁, and the response of the plastic zone radius within the lining is minimal, regardless of the presence of anchor support. Figure 6a shows that the radial stress variation is not significantly affected by the increase in the anchor’s elastic modulus, and the radial stress in the lining overlying soft rock is larger than that in the overlying hard rock; in the anchoring zone, the radial stress decreases from the boundary between the lining and the anchoring zone to the boundary between the anchoring zone and the original rock. Conversely, as depicted in Figure 6b, the circumferential stress shows significant fluctuations within the lining at different radial distances, with a substantial jump at the interface between the lining and the surrounding rock. The peak circumferential stresses occur at the elastoplastic interface, registering 70.78 MPa, 72.52 MPa, and 73.41 MPa for soft rock, and 54.22 MPa, 54.93 MPa, and 54.93 MPa for hard rock, respectively. At the boundary between the lining and the anchoring zone where ρ = R+, the circumferential stresses are 6.77 MPa, 7.35 MPa, and 7.88 MPa for soft rock, and 8.73 MPa, 9.14 MPa, and 9.47 MPa for hard rock, with the differences in the radius of the plastic zone diminishing as the radial distance increases. Under the same anchoring parameters, the circumferential stress in hard rock exceeds that in soft rock. Overall, as the elastic modulus of the anchor increases, so does the circumferential stress in the anchoring zone.

Figure 7 illustrates the evolution curves of radial and circumferential stress at various radial distances for different anchor elastic moduli, under a constant pressure of P = 24 MPa. In this case, with P > P₂ and ρp > ρ₁, the primary principal stress shifts from circumferential to radial stress. Simultaneously, yielding occurs in the lining zone, with the radial and circumferential stresses being equal in magnitude. Within the anchoring zone, where ρ₁ < ρ < ρ₂, an increase in the anchor’s elastic modulus results in only minor differences in radial stress, and the maximum rate of boundary change remains below 1.5%. On a broad scale, radial stress is negatively correlated with increasing radial distance, with soft rock showing greater radial stress than hard rock. Figure 7b reveals that, within the elastic zone, the circumferential stress of soft rock is slightly higher than that of hard rock and exhibits a relatively larger rate of change. The circumferential stress within the plastic zone decreases, hitting the lowest point at the elastic–plastic interface in the anchoring zone. For both soft and hard rocks, augmenting the anchor’s elastic modulus does not alter the circumferential stress within the plastic zone. However, increasing the anchor’s elastic modulus results in a reduction of the plastic zone radius. For soft rock, the radius shrinks by 0.30 m (a change rate of 4.76%), while for hard rock, it decreases by 0.07 m (a change rate of 1.34%). This illustrates a particularly notable effect of the heightened anchor elastic modulus in inhibiting the expansion of the plastic zone in soft rock.

3.3. Elastic–Plastic Analysis of Different Anchor Support Angles

The anchoring parameters for the anchoring zone depicted in Figure 8 are Sθ = 14.3°, Sθ = 11.5°, Sθ = 8.6°, and Eb = 2 × 10⁵ MPa. At varying anchoring angles, the initial critical pressures for soft rock are 4.73 MPa, 4.87 MPa, and 4.95 MPa, respectively, whereas the secondary critical pressures are 20.28 MPa, 20.83 MPa, and 21.42 MPa, respectively. When the pressure P falls between P₁ and P₂, the radius of the plastic zone is 4 m, and the entire lining is in the elastic stage. Smaller anchor support angles increase the range of the internal pressure interval represented by [P₁, P₂]. Correspondingly, for hard rock, the initial critical pressures are 3.13 MPa, 3.16 MPa, and 3.07 MPa, while the secondary critical pressures are 19.46 MPa, 20.16 MPa, and 20.92 MPa, respectively. When P is less than P₁, the plastic zone remains inside the lining, and decreasing the angle of the anchor support exerts a minimal impact on the expansion of the plastic zone; conversely, when P exceeds P₂, the first principal stress shifts from circumferential stress to radial stress. Under hard rock conditions, reducing the anchor support angle markedly inhibits the expansion of the plastic zone, with the maximum inhibition rate of the plastic zone radius reaching 5.63%. The response in soft rock anchoring zones is even more pronounced. However, as the plastic zone continues to expand, the difference in the radius of the plastic zone shows a reduction trend. The maximum plastic zone radius inhibition rate is 8.14%, while the minimum rate is 3.60%. The anchoring action of the bolts thus proves to be significantly more effective in enhancing the stability of soft rock compared to hard rock.

Figure 9 illustrates the evolution curves of radial and circumferential stresses at different radial distances under various anchor support angles. At this stage, when P < P₁ and ρ₀ < ρp < ρ₁, the presence or absence of anchor support has a minimal impact on the expansion of the plastic zone within the lining. In Figure 9a, reducing the support angle does not significantly impact the change in radial stress. The radial stress in soft rock linings is greater than that in hard rock. Within the anchoring zone, the radial stress decreases from the boundary between the lining and the anchoring zone to the boundary between the anchoring zone and the original rock, with the radial stress in soft rock exceeding that in hard rock. In contrast, Figure 9b depicts considerable fluctuations in circumferential stress within the lining at various radial distances, with significant jumps at the interface between the lining and the surrounding rock. The peak circumferential stress is observed at the elastic–plastic interface, with values for soft rock being 72.52 MPa, 74.3 MPa, and 74.3 MPa and for hard rock being 54.93 MPa, 54.93 MPa, and 53.5 MPa, respectively. The circumferential stresses to the right of the interface between the lining and anchoring zone for soft rock are 7.35 MPa, 8.11 MPa, and 9.2 MPa, respectively, while for hard rock, they are 9.14 MPa, 10.43 MPa, and 12.31 MPa. With the same anchoring parameters, the circumferential stress in hard rock is higher than that in soft rock, and the circumferential stress in the anchoring zone shows an increasing trend as the anchor support angle decreases.

Figure 10 illustrates the evolution of radial and circumferential stress at various radial distances for different anchor support angles at P = 24 MPa. At this stage, P > P₂, ρp > ρ₁, and the primary principal stress transitions from circumferential to radial stress. The entire lining zone has yielded, with its radial and circumferential stresses equalizing. Within the anchoring zone, where ρ₁ < ρ < ρ₂, the radial stress changes marginally as the anchor support angle diminishes. The maximum rate of change at the boundary remains under 2.1%. Overall, radial stress decreases with increasing radial distance, and the radial stress in soft rock exceeds that in hard rock. Concurrently, in the plastic zone of the anchoring zone, circumferential stress trends downwards, reaching its minimum at the elastic–plastic interface. In the plastic zones of both soft and hard rock, the decrease in anchor support angle does not induce significant changes; however, the plastic zone radius in hard rock is clearly smaller than in soft rock. With decreasing anchor support angles, the plastic zone radius in soft rock shrinks by 0.44 m, while in hard rock it shrinks by 0.30 m. The respective plastic zone radius change rates are 6.89% for soft rock and 5.63% for hard rock. Thus, decreasing the anchor support angle more prominently curtails the expansion of the plastic zone in soft rock.

4. Numerical Simulation Verification of the Elastic–Plastic Response of the M-C Lining-Anchor-Support Rock System

4.1. Modeling

The FLAC3D numerical simulation method is utilized to validate the accuracy of the elastic–plastic computational results for the lining-anchor-supported rock structure when the plastic zone is within the lining. The model parameters are presented in

Table 3. Elastic parameters of lining and surrounding rock for calculation.

Parameter	E/Pa	μ1	c/Pa	φ
Lining	2 × 1010	0.167	7 × 106	45°
Hard rock	2 × 109	0.25	1 × 106	45°
Soft rock	5 × 108	0.25	5 × 105	30°

[49,50] and

Table 4. Anchor bolt parameters for calculation.

Parameter	Length/m	Radius/m	Eb/Pa	Sθ/°
Value	3	0.21	2 × 1011	11.4°

[49,51].

To ensure an accurate model verification, the selected model has a radius of 30 m, whereas the tunnel has a radius of 4 m. The model’s radius is six times that of the tunnel to reduce the impact of boundary constraints. The boundary conditions consist of an annular normal stress constraint and a normal displacement constraint applied to the free surface of the tunnel’s y-axis. At the outermost boundary of the model, an initial circumferential ground stress of P₀ = 10 MPa is applied, and an annular internal pressure of P = 2 MPa is imposed at the inner boundary of the lining. The parameters for the lining and surrounding rock are established according to the Mohr–Coulomb criterion. The rock bolt is 3 m in length, with a radius of 0.21 m and an elastic modulus of 2 × 10¹¹ Pa, along with a support angle of 11.4°. The constructed model is depicted in Figure 11.

4.2. Comparative Analysis of Numerical Simulation

The plastic radius determined in the numerical simulation is depicted in Figure 12. According to the analysis of the elastic–plastic analytical solution of the lining-anchor support structure for surrounding rock, when the internal pressure of the lining is 2 MPa, the primary stress is circumferential, and the radius of the plastic zone should be within the lining. As shown in Figure 12, the plastic zone is indeed located inside the lining and assumes a circular ring shape due to the influence of hydrostatic pressure. The computed plastic zone radius for hard rock is 4.28 m, with a theoretical calculation error of 3.38%; for soft rock, the computed radius is 4.38 m, matching the theoretical value; and the plastic zone radius in soft rock is slightly larger than in hard rock.

Figure 13 illustrates that the simulated circumferential stress in both scenarios is marginally lower than the theoretical values. The stress displays an initial increase followed by a decrease, reaching its maximum at the elastic–plastic boundary. Within the lining, the circumferential stress for soft rock is significantly larger than that for hard rock; however, in the anchoring zone, the circumferential stress for soft rock is marginally lower. The theoretical calculations exhibit a pronounced discontinuity at the interface between the lining and anchoring zone. This discontinuity manifests in the simulation as an abrupt increase in the rate of reduction of circumferential stress on either side of this interface, aligning closely with the theoretical model’s trends. Within the anchoring zone, circumferential stress increases and eventually aligns with the original ground stress level as the radial distance grows. In both computational scenarios, the maximum error for soft rock is 11.73%, while the minimum error is 1.00%; for hard rock, the errors are 9.85% and 1.00%, respectively. The calculation error for the soft rock supported by the lining-anchor support structure is slightly higher than that for hard rock. Additionally, the average error for hard rock is relatively small, amounting to 6.38%, compared to 7.66% for soft rock. The overall trend is consistent with the theoretical model. The errors in this study are relatively minor, with the average errors for both soft and hard rock kept within 10%. This further validates the effectiveness and reliability of the research method employed in this study.

5. Conclusions

This study, utilizing the M-C criterion and the equivalent substitution method, introduces an elastic–plastic expression for the surrounding rock in deep circular tunnels supported by a lining-anchor bolts. It provides an expression for the radius of the plastic zone and the elastic–plastic radial stress under a specified lining internal pressure. By analyzing the surrounding rock supported by a lining-anchor bolts with or without anchoring and varying the anchor elastic modulus and different anchor bolt support angles, the study further elucidates the evolution of the plastic radius and stress state of the deep circular tunnel under the given lining pressure. The accuracy of the theoretical calculations is validated through the FLAC3D finite difference method, offering theoretical support for the lining–anchor bolt support mechanism in deep circular tunnels. The key findings are as follows:

(1)

The radius of the plastic zone in the surrounding rock with lining-anchor bolt support is influenced by various factors. These include tunnel radius, lining thickness, the mechanical parameters of the lining and surrounding rock (such as friction angle, cohesion, elastic modulus, and Poisson’s ratio), and anchoring parameters (such as length of anchoring zone, anchor bolt elastic modulus, anchor bolt support angle, and anchor cross-sectional area);

(2)

In scenarios with or without an anchoring zone, under the action of the anchor bolt, when the specified lining internal pressure is less than the initial critical pressure, the primary stress is the circumferential stress, with the plastic zone located within the lining. The anchoring action does not significantly influence the expansion of the plastic zone, although it extends the Phase II interval length. When the specified lining internal pressure exceeds the secondary critical pressure, the first primary stress transforms into radial pressure. When the plastic zone extends to the anchoring zone, the anchor effectively inhibits plastic expansion under soft rock conditions. The plastic zone radius inhibition rate peaks at 14.97% and 4.92% in soft rock, and at 5.34% and 3.35% in hard rock;

(3)

When the plastic zone is inside the lining, an increase in the elastic modulus of the anchor bolts results in only a negligible change to the plastic zone. The anchoring effect increases the Phase II interval length. Once the plastic zone extends to the anchoring zone, the plastic zone radius negatively correlates with the elastic modulus of the anchor bolts. Particularly under soft rock conditions, the anchor bolt notably inhibits plastic zone expansion, with a maximum inhibition rate of 5.43%, compared to 1.38% in hard rock;

(4)

When the plastic zone is within the lining, variations in the anchor bolt support angle exert little influence on the expansion of the plastic zone within the lining. The anchoring effect increases the Phase II interval by over 5%. When the plastic zone extends to the anchoring zone, the radius of the plastic zone is positively correlated with the angle of the anchor bolt support. The anchoring effect is more pronounced in inhibiting the expansion of the plastic zone, particularly in soft rock conditions. The maximum inhibition rate is 8.14% in soft rock and 5.63% in hard rock;

(5)

The circumferential stress calculated via numerical simulation is less than that from the theoretical model, with an average error below 10%. The overall trend aligns with the theoretical model, further confirming its accuracy and reliability.