Numerical Simulation of Mechanical Characteristics and Safety Performance for Pre-Cracked Tunnel Lining with the Extended Finite Element Method

Abstract

The service performance of tunnel lining is affected by crack properties and development states. In this paper, numerical simulation models were established to investigate the mechanics characteristics and safety performance for lining structures under different cracks based on the extended finite element method (XFEM). Analyze multiple quantitative factors in simulation, including changes in crack location, crack length, and crack distribution range in the lining structure. The axial force and bending moment of the preset cracks in the lining structures were first studied. The maximum safety factor attenuation rate ($D_{kmax}$) was proposed to analyze the impact of longitudinal and annular cracks on the safety performance. The axial force at the vault of the lining arch is the most significantly affected by the combined longitudinal cracks at multiple locations. When the length of a longitudinal crack increases from 1 m to 6 m, the axial force value at the crack point decreases by 33.77%, 36.15%, and 11.32%. However, the bending moment value increases by 4.47 times, 2.50 times, and 1.69 times. Under the influence of longitudinal cracks in an “arch crown + arch shoulder”, “arch crown + arch waist”, and “arch crown + arch shoulder + arch waist”, the axial force in the arch vault increased by 21.55%, decreased by 17.52%, and decreased by 13.45%. The distribution pattern of the bending moment under the influence of circumferential cracks shows convexity at the arch shoulder and arch foot, and concavity at the arch waist and side walls. The safety factor scatter curve with longitudinal cracks shows a gradual transition from a “W” shape to a “U” shape. The safety factor curve with circumferential cracks presents an approximately symmetrical wave-shaped distribution.

1. Introduction

The scale and quality of highway construction have been improved due to the rapid improvement in China’s comprehensive national strength and the gradual expansion of transportation infrastructure [1,2,3,4]. In recent years, China has ranked first in the world in terms of the number and mileage of tunnels built [5,6]. The construction technology in the field of tunnel engineering tends to be standardized and mature [7,8,9,10]. It indicates that the workload of disease treatment may lead to high-cost consumption in the development stage where both construction and operation cycles are valued. Cracks, as one of the most common diseases in the lining of mountain tunnels, have a serious impact on the overall service state of the structure. In order to address the evaluation of the usage status and maintenance cycle of lining structures with cracks, it is significant to study the mechanical characteristics and safety performance of lining structures under the influence of cracks with different properties.

The main raw material for tunnel lining is concrete. The rich functions of the extended finite element method (XFEM) result in a fracture displacement field description suitable for concrete structures [11,12,13,14]. Especially, numerous studies on structural fracture mechanisms and propagation laws are based on the XFEM [15,16]. For example, Liu et al. [17] combined XFEM with model experiments to study the cracking mechanism and crack evolution law of lining concrete structures under geological biases. Zhang et al. [18] improved the XFEM by constructing a new fracture energy conversion equation to study the failure mechanism of the bending performance for shield tunnels under load and corrosion coupling, and the results were verified by full-scale experiments. Fu et al. [19] studied the structural stress distribution, crack development direction, and distribution law under the influence of an insufficient thickness of cavities and lining based on XFEM. Han et al. [20] achieved secondary development through the combination of tensile shear failure criteria and XFEM embedding to study the fracture mechanism of shield tunnels, considering factors such as support pressure, thickness of overlying rock layers, shear strength of strata, and mud viscosity. Zhao et al. [21] evaluated the performance of lining structures under complex defects based on geological radar non-destructive testing results using XFEM.

The above research results indicate that XFEM has been successfully applied in the study of mechanical properties of tunnel lining structures. And the mechanical characteristics and safety performance of the lining structure should be given more attention. Therefore, Min et al. [22] conducted model and experimental research on the mechanical properties of lining arches based on a crack model. Xu et al. [23] focused on analyzing the crack patterns of lining under temperature stress, bearing capacity performance, and safety performance based on similar model experiments.

Factors such as high and low temperatures, geological biases, rock loosening, and special loads are used as prerequisites for studying the cracking mode and mechanical properties of lining concrete [24,25,26,27]. However, under conventional factors, the lining structure may exhibit a crack service mode. Only after structural maintenance and reinforcement can the negative work with cracks be eliminated [28,29,30,31]. In addition, longitudinal cracks are often considered as influencing factors. It indicates that there seems to be a lack of relevant research on the influence of circumferential cracks.

The aim of this study is to explore the mechanical state and safety performance of lining structures in service with cracks under conventional conditions. The quantitative analysis method combined with XFEM is used to establish a load structure model for lining, especially for working conditions with annular cracks. Next, simulation was conducted on the pre-cracked lining under the influence of factors such as crack type, crack location, crack length, and crack distribution range, taking a tunnel in Shaanxi Province as the background. A quantitative comparison analysis of axial force and bending moment under the influence of cracks with different properties was performed. Finally, based on the calculation method of the safety factor for the lining structure, the key cross-sectional safety evolution law is given. Finally, the maximum safety factor attenuation rate is proposed to evaluate the impact of cracks with different properties on the service performance of structures.

2. Project Background

2.1. Project Overview

This study takes a tunnel in Shaanxi Province as an example, which is a long tunnel crossing a mountain ridge with a maximum burial depth of over 400 m. It is a control project between two county towns. The new Austrian tunneling method is used for tunnel excavation. The lining is composed of initial support, waterproof layer, and secondary lining. Especially, the construction lining of the V-grade surrounding rock section should be given the most attention. The composite lining design of the V-grade surrounding rock section is shown in Figure 1.

2.2. Monitoring of Lining Contact Pressure

Mature monitoring methods in China are used to monitor the contact pressure between secondary lining and initial support during tunnel construction. The monitoring points were selected at five points on the upper arch, taking into account the geological conditions, construction technology, and lining structure characteristics of the site, as shown in Figure 2. The soil pressure box was immediately buried after the initial support construction was completed. Moreover, the data obtained from on-site monitoring could be considered as a reasonable load application in the simulation.

2.3. On-Site Investigation of Lining Cracks

This paper is based on the tunnel project, which has been in service for more than 10 years since its completion. It has suffered from extensive lining damage, in particular, serious cracking of the lining. In response to the lining damage that seriously threatens the service performance of the tunnel structure, the investigation of the tunnel site mainly included lining cracking, tunnel water seepage, and lining concrete strength. Based on the on-site survey, it was found that the lining cracking form was complicated, in which the crack type was dominated by longitudinal cracks and annular cracks. The proportion of longitudinal cracks is 45.8%, and the proportion of annular cracks is 44.6%. The maximum length of longitudinal crack extension is 5 m, and the maximum length of annular cracks is 7 m. There are a certain number of diagonal cracks and Y-type cracks in the tunnel lining concrete, but the distribution range and length can be neglected compared with longitudinal and annular cracks.

From the distribution location of lining cracks, the probability of lining cracks at the arch vault, arch shoulder, and arch waist is not much different, and the cracks at the arch vault account for 27.4%, the cracks at the arch shoulder account for 29.2%, and the cracks at the arch waist account for 26.5%. In addition, the cracks at the side walls accounted for 16.9%. The type of crack and the distribution ratio of different positions are shown in Figure 3. Based on the on-site lining crack investigation, photos of longitudinal cracks and annular cracks were drawn, as shown in Figure 4.

It is not difficult to find that the types of lining cracks are mainly longitudinal cracks and annular cracks. Two types of cracks have an impact on the mechanical characteristics and safety status of the lining structure. Therefore, this paper mainly focuses on the force characteristics and safety performance of lining structures with different properties of longitudinal cracks and annular cracks under the condition of Grade-V surrounding rock.

3. Simulation Model of Tunnel Lining with Pre-Cracks

3.1. Extended Finite Element Method

The extended finite element method (XFEM) originates from the improvement in the finite element method. It solves the discontinuity of the fracture displacement field based on a shape function basis. The enhanced description of extended finite element nodes is shown in Figure 5. The function that can perform displacement interpolation at any point X in the model is expressed as the sum of FEM displacement $u^{FEM}$ and enhanced displacement $u^{CNR}$, as shown in Equation (1) [32,33]:

$$u^h(x)=u^{FEM}+u^{CNR}={\displaystyle \sum _{i=1}^n{N}_i(x)u_i+}{\displaystyle \sum _{k=1}^m{\varphi }_k(x)a_k}$$

(1)

In Equation (1), the subscript $i$ denotes the domain of influence of the node; $N_i(x)$ is the finite element shape function defined on the domain of influence of the node; $u_i$ is the degree of freedom of the node associated with $N_i(x)$; ${\varphi }_k(x)$ is an enrichment at discontinuous nodes; $a_k$ is an additional degree of freedom to the regular one; n is the number of conventional finite element nodes; and m is the number of enhanced nodes.

For the study of structural cracking, Equation (1) can be further rewritten as

$$u_{XFEM}(x)=u^{FEM}(x)+u^H(x)+u^{TIP}(x)$$

(2)

where $u^H(x)$ represents the displacement of the reinforcement nodes on both sides of the crack surface, and $u^{TIP}(x)$ represents the displacement of the reinforcement nodes at the crack tip.

The displacement expression of any point X within the unit is as follows [33]:

$$u_{XFEM}(x)={\displaystyle \sum _{j=1}^n{N}_j(x)u_j}+{\displaystyle \sum _{h=1}^{mh}{N}_h(x)H(x)a_h}+{\displaystyle \sum _{k=1}^{mt}{N}_k(x)}{\displaystyle \sum _{l=1}^4{F}_l(x)b_k^l}$$

(3)

Among them, n is the number of conventional finite element nodes, mh is the number of reinforced nodes on both sides of the crack surface, and mt is the number of pointed reinforced nodes; $u_j$ is the conventional finite element node degree of freedom vector, $a_h$ is the enhanced node degree of freedom vector on both sides of the crack surface, and $b_k^l$ is the enhanced node degree of freedom vector at the crack tip; $N_j$, $N_h$, and $N_k$ are the shape functions H(x) corresponding to node j, h, and k; H(x) is the value of the Heaviside function at point x, and F_l (x) is the value of the sharp enhancement function at point X.

After extending the finite element reinforcement to the crack surface and the element where the crack tip is located, it could result in only some nodes of adjacent elements having enhanced degrees of freedom. The enhanced shape function may no longer satisfy unit decomposition within these elements. In order to eliminate the influence of mixed elements on computational accuracy and convergence, many researchers have conducted in-depth research on structural fracture simulation by improving XFEM, such as, automatic X-FEM [34], the linear smoothed extended finite element method [35], and non-smooth equations that improve XFEM [36].

By selecting appropriate optimized analytical solutions for enrichment functions based on different application backgrounds and solution requirements, it is possible to reconstruct the singular field at the crack tip and achieve continuity of adjacent displacement fields in a finite element analysis. As mentioned above, compared to traditional methods, the extended finite element method can solve the difficulties of a high stress field at the crack tip and deformation concentration in high-density grid areas without being affected by geometric interiors and physical boundaries in calculations. It does not need to adapt to new grid division tasks caused by crack initiation and continuous expansion during the calculation process. Therefore, the application of the extended finite element method can solve the stress characteristics and safety performance research of lining under preset crack conditions.

3.2. XFEM Simulation Verification of Lining Cracking

The comparison between Mashimo indoor experiments [37] in Japan and XFEM simulation results was achieved by setting almost identical lining loading conditions. The convergence displacement, stress at the arch crown, and crack distribution expansion state that vary with load are used to evaluate the accuracy of the XFEM method by comparing the two methods. The specific comparison is shown in Figure 6.

The cracking mode of the lining is shown in Figure 6a based on the comparison results of the indoor loading experiment and numerical simulation. It indicates that the inner side of the arch crown and outer side of the arch shoulder are under tension under tensile deformation. As the relaxation load continues to increase, cracks controlled by tensile stress appear in the tension area. The results of the extended finite element method and indoor experiments are completely consistent.

The comparison results of the vertical displacement curve and crack development status at the arch crown with changes in load are given, as shown in Figure 6b. There is a certain degree of agreement between the two in terms of numerical changes. In summary, XFEM is reliable for simulating crack propagation in lining structures. It can accurately simulate the initiation and development of lining structures.

3.3. Model Establishment and Parameter Settings

The load structure method is used to construct a solid model of a cracked lining. The solid length of the tunnel model is 6 m, and the unit type is C3D8R in ABAQUS 2019 software. Among them, C represents a solid element, “3D” represents “three-dimensional”, “8” represents the number of nodes this element has, and “R” indicates that this element is a “reduced integral element”. The lining model is divided into three parts: arch crown, left and right arch feet, and inverted arch bottom to achieve control over different blocks, especially the application of loads.

The tunnel simulation model is shown in Figure 7, and the foundation spring is used to achieve the contact system between the surrounding rock and lining. The implementation method is to apply springs with stiffness and damping to each node in the model, as shown in Figure 7a,b. The load is applied in the simulation through surface pressure, and the load on different parts of the surface presents a vertical direction (see Figure 7c).

Consider the effect of geological strata on the tunnel structure as a load applied to the tunnel lining, calculate the internal force of the lining according to the calculation method of elastic foundation structures, and then proceed with structural interface design. The calculation is carried out using the action mode of the formation reaction spring, and the elastic resistance of the formation is obtained from the following equation:

$$F_n=K_n\cdot U_n$$

(4)

$$F_s=K_s\cdot U_s$$

(5)

where:

$$K_n=\{{}_{K_n^{-}\stackrel{}{{\displaystyle }}{U}_n<0}^{K_n^{+}\stackrel{}{{\displaystyle }}{U}_n\ge 0}$$

(6)

$$K_s=\{{}_{K_s^{-}\stackrel{}{{\displaystyle }}{U}_s<0}^{K_s^{+}\stackrel{}{{\displaystyle }}{U}_s\ge 0}$$

(7)

In the equations, $F_n$ and $F_s$ represent the normal and tangential elastic resistance, respectively. $K_n$ and $K_s$ are the corresponding elastic resistance coefficients of the surrounding rock. The layer spring unit can be set along the entire section or only at some nodes. When setting up layer spring elements along the entire cross-section, an iterative method is required for a deformation control analysis during the calculation process.

This study mainly focuses on the research of lining cracks under the condition of V-grade surrounding rock. The resistance coefficient of the surrounding rock is taken as 200 MPa/m, and the lining is made of C30 concrete with a thickness of 45 cm.

The calculation steps of the preset crack model based on the XFEM are as follows: The Maxps criterion is used as the cracking criterion for lining concrete. According to the parameters of C30 concrete, the maximum principal stress is set to 1.6 MPa. When the maximum principal stress in the model exceeds it, structural cracks initiate and expand. The evolution stage of structural damage follows the fracture energy criterion. The fracture energy is taken as G_fI = Gf_II = Gf_III = 100 N/m. The preset contact attribute between the two-dimensional plane and the lining is set to normal hard contact and tangential frictionless. Assign the set contact attributes to the cracks in the XFEM module, and set the overall model to the areas where the cracks may expand. Taking certain preset crack conditions as an example for illustration is shown in Figure 8.

The arch crown is preset with a single 5-m-long longitudinal crack as shown in Figure 8a. The combination of the arch crown and arch waist is preset as 4-m-long longitudinal cracks, as shown in Figure 8b. A single preset 30° annular crack in the arch shoulder is shown in Figure 8c. The double section arch waist is jointly preset with 30° annular cracks as shown in Figure 8d. Meanwhile, the physical and mechanical calculation parameter values used in the numerical simulation of C30 concrete are shown in

Table 1. Parameter values for tunnel lining concrete.

Material Name	Material Weight/(kN/m3)	Elastic Modulus/MPa	Poisson’s Ratio	Maximum Principal Stress/MPa	Viscosity Coefficient
Concrete	23.8	30,000	0.22	1.60	0.001

.

The determination of the bearing capacity of the lining surrounding rock is a complex and systematic problem, and the reasonable and scientific calculation of the load is the foundation of the numerical simulation calculation results. This study relies on on-site monitoring data of physical engineering for numerical simulation of mechanical loading, with the arch crown of q₁ = 155 kPa and arch shoulder of q₂ = 124 kPa. The arch waist and side wall (e = 93 kPa) are for loading. The diagram of load distribution calculation based on on-site monitoring is shown in Figure 9.

3.4. Simulate Specific Details

Based on the investigation and statistical results of engineering lining crack diseases mentioned earlier, quantitative control of preset parameters for longitudinal and annular cracks is carried out to achieve a control variable analysis of the impact for different crack characteristics, seen in

Table 2. Design table for simulation working conditions of crack properties.

Crack Properties	Crack Location	Parameter Settings
Preset single longitudinal crack	Arch Crown	Length: 1 m, 2 m, 3 m, 4 m, 5 m, 6 m
	Arch Shoulder	Length: 1 m, 2 m, 3 m, 4 m, 5 m, 6 m
	Arch Waist	Length: 1 m, 2 m, 3 m, 4 m, 5 m, 6 m
Preset combination of longitudinal cracks	Arch Crown + Arch Shoulder	Length: 1 m, 2 m, 3 m, 4 m, 5 m, 6 m
	Arch Crown + Arch Waist	Length: 1 m, 2 m, 3 m, 4 m, 5 m, 6 m
	Arch Crown + Arch Shoulder + Arch Waist	Length: 1 m, 2 m, 3 m, 4 m, 5 m, 6 m
Single section annular crack	Arch Crown	Distribution range: 30°, 60°, 90°
	Arch Shoulder	Distribution range: 30°, 60°
	Arch Waist	Distribution range: 30°
Double section annular crack	Arch Crown	Distribution range: 30°, 60°, 90°
	Arch Shoulder	Distribution range: 30°, 60°
	Arch Waist	Distribution range: 30°

.

The specific plans are divided into a single longitudinal crack, combined longitudinal cracks, a single section annular crack, and double section annular cracks. A single longitudinal crack is preset at the arch crown, arch shoulder, and arch waist, with a quantitative length setting of 1 m, 2 m, 3 m, 4 m, 5 m, and 6 m. The combined longitudinal cracks are divided into three types: “arch crown + arch shoulder”, “arch crown + arch waist”, and “arch crown + arch shoulder + arch waist”. The quantitative length settings are 1 m, 2 m, 3 m, 4 m, 5 m, and 6 m. The single section annular crack rotates counterclockwise from the arch crown, with preset values of 30°, 60°, and 90°. Double section annular cracks are achieved by replicating a set of new annular preset cracks on a single section working condition. The cloud atlas of crack units in the lining models is shown in Figure 10.

4. Extraction of Results and Safety Evaluation

4.1. Extraction of Calculation Results

The force characteristics of the lining structure are mainly characterized by the axial force and bending moment values. It is first necessary to define four local coordinate systems with different origins in the simulation models. The center of the upper arch, the circular shape of the left arch foot, the center of the right arch foot, and the center of the inverted arch form a coordinate system.

Noting the practical significance of the bending moments, the model is modeled according to the longitudinal direction of 1 m. Subsequently, it is also necessary to slice the mesh with the local coordinate systems and monitoring points laid out. The axial force and bending moment are calculated for each slice. The axial force components and bending moment components in the R, T, and Z directions can be obtained. Among them, the extracted grids with a set local coordinate system and monitoring points are shown in Figure 11. For the components of the calculated results in three directions, all axial forces in this study are F_T. The bending moment is M_Z, which is the annular bending moment of the lining. The tensile state corresponds to the positive direction of the T-axis, and vice versa. Determine the positive and negative bending moments in the Z-direction based on the right-hand spiral rule.

4.2. Calculation of Safety Factor for Lining Structure

In the complex formation of V-grade perimeter rock, the secondary lining not only serves as a safety reserve, but also bears part of the perimeter rock pressure. Therefore, the secondary lining as a load-bearing structure needs to be evaluated for safety.

In the calculation, it is necessary to classify the discussion according to the axial eccentricity. When the eccentricity and its bearing capacity are controlled by the compressive strength, the structural safety coefficient of concrete axial eccentric compression is calculated according to Equation (8) [38]:

$$KN\le \phi \alpha R_{a}bh$$

(8)

In Equation (8), K is the safety factor; $\phi$ is the longitudinal bending coefficient of the structure, which can be taken as 1 for the secondary lining of the tunnel; N is the axial force; b is the width of the section; h is the thickness of the section; and $R_a$ is the ultimate compressive strength of the lining concrete; $\alpha$ is the axial eccentricity influence coefficient, which can be calculated according to Equation (9) [38]:

$$\alpha =1+0.648(\frac{e_0}{h})-12.569{(\frac{e_0}{h})}^2+15.444{(\frac{e_0}{h})}^3$$

(9)

In addition, when the eccentricity distance $e_0>0.20h$, the bearing capacity is controlled by tensile strength. The tensile strength of the lining structure constructed under rectangular eccentric compression can be solved using Equation (10) [38]:

$$KN\le \phi \frac{1.75R_{1}bh}{\frac{6e_0}{h}-1}$$

(10)

In Equation (10), $R_1$ is the tensile ultimate strength of lining concrete; $e_0$ is the axial eccentricity; other symbols have the same meaning as before.

5. Result Analysis

5.1. Mechanical Characteristics of Lining with Longitudinal Crack

Comprehensively consider the impact of a single longitudinal crack and combined longitudinal cracks on the mechanical properties of the lining. Draw a variation diagram of the axial force extraction results of the lining structure cross-section, as shown in Figure 12. The axial forces of the lining under the influence of longitudinal cracks are preset at the arch crown, arch shoulder, and arch waist, as shown in Figure 12a–c. Under the influence of a single longitudinal crack, the axial force values of each part are negative, and the structure presents a compressive state. There are slight differences in the overall axial force mode when the preset position of a single longitudinal crack is different.

As the length of the longitudinal crack gradually increases, the curves show a regular change in numerical value. When the longitudinal crack at the arch crown is 1 m, the difference in axial force values at the arch crown, two arch shoulders, and two arch waists is not significant. However, the maximum axial force values are −472.61 kN and −472.31 kN at the two arch feet. As the length of longitudinal cracks increases, the bearing capacity of the arch gradually decreases. When the length of longitudinal cracks is 1 m and 6 m, the corresponding axial force values of the arch are −394.12 kN and −261.31 kN, which decrease by 33.77%, as shown in Figure 12a. Under the influence of a single longitudinal crack preset on the arch shoulder, the axial force at the crack area gradually decreases. The left upper arch gradually loses its bearing capacity due to cracks, while the right arch needs to “bear” the excess load on the left arch. The axial force values at the left arch shoulder, left arch waist, and right arch shoulder have all decreased. The change data are −432.23 kN decreased to −276.31 kN, −444.51 kN decreased to −301.26 kN, and −457.27 kN decreased to −307.57 kN. The reduction ranges are 36.15%, 36.07%, and a decrease of 32.74%.

However, as the length of the crack increases, the axial force at the arch crown gradually increases. When the crack increases from 1 m to 6 m, the axial force at the arch crown increases by 6.33%. The amplitude of axial force change is very small at the right arch waist and right wall (see Figure 12b). When a single longitudinal crack is preset at the arch waist, the axial force at the crack area gradually decreases with the increase in the longitudinal crack length. The value decreased from −486.85 kN to −431.76 kN, a decrease of 11.32%. The axial force of the arch gradually increases, from −395.64 kN to −452.97 kN. The increase in axial force at the left arch shoulder and left wall is uniform. But the axial force at the right arch foot is significantly greater than the axial force at the left arch foot. The maximum operating conditions for both are −580.93 kN and −520.02 kN, respectively, as shown in Figure 12c. As mentioned above, under the influence of a single preset longitudinal crack, the axial force at the crack location in the lining structure is significantly affected. It shows a trend of decreasing axial force value as the longitudinal length of the crack increases. When the length of the longitudinal crack increased from 1 m to 6 m, the axial force values at the crack site decreased by 33.77%, 36.15%, and 11.32%, respectively.

Consider complex longitudinal crack situations. A combination of longitudinal crack conditions was set in the numerical analysis. And draw the axial force variation curve, as shown in Figure 12d–f. The combination of preset longitudinal cracks at the arch crown and arch shoulder increased the axial force at the arch crown, right arch shoulder, and right arch waist by 21.55%, 20.62%, and 17.41%. The axial forces on the left arch shoulder and left arch waist decreased by 18.25% and 16.88%. When the longitudinal cracks are preset in the two parts of the arch crown and arch waist, the axial force of the arch crown and left arch waist decreases by 17.52% and 16.90%. When the three parts of the arch crown, arch shoulder, and arch waist are preset with longitudinal cracks, the axial force of the arch crown, left arch shoulder, and left arch waist changes from −358.52 kN to −31.30 kN, a decrease of 13.45%, from −358.52 kN to −316.19 kN, a decrease of 11.81, and from −383.51 k to −337.35 kN, a decrease of 12.03%. As mentioned above, the axial force at the arch top of the tunnel lining is the most significantly affected by the combined longitudinal cracks.

The lining bending moment corresponding to the preset cracks located at the arch crown, arch shoulder, and arch waist with a length of 1–6 m is shown in Figure 13a–c. When working with cracks in the lining, the arch crown, arch waist, and side walls show internal tension, while the arch shoulder and arch foot show external tension. The bending moment value of the inverted arch is smaller compared to other parts. The bending moment of the lining structure is affected by longitudinal cracks, which is opposite to the axial force situation. The occurrence of cracks could increase the bending moment value. And the lining arch foot is affected by the stress concentration effect, and its bending moment value is significantly greater than the bending moment value of other parts.

From the analysis of Figure 13a–c, it can be seen that under the influence of a single longitudinal crack, the bending moment of each section of the tunnel lining is distributed asymmetrically with the inverted arch as the center. When the crack is located at the arch crown, the bending moments of the arch crown and left arch shoulder change from 7.12 kN m to 13.40 kN m, and from −9.32 kN m to −21.10 kN m, increasing by 1.88 times and 2.26 times. When the crack is located at the arch shoulder, the bending moments of the arch crown and left arch shoulder change from 4.98 kN m to 13.43 kN m, and from −9.99 kN m to −24.97 kN m, increasing by 2.70 times and 2.50 times. When the crack is located at the arch waist, the bending moments of the arch crown and left arch waist change from 27.20 kN m to 38.12 kN m, and 9.56 kN m to 16.23 kN m, increasing by 1.40 times and 1.69 times.

Figure 13e–f represent the bending moment mode of the lining structure under the influence of combined longitudinal cracks. It is no different from the mode under the influence of a single longitudinal crack. The bending moments of the arch crown, arch waist, and side walls are positive, while the bending moments of other parts are negative. Under the influence of combined cracks at the arch crown and arch shoulder, the bending moment corresponding to the arch crown increases by 1.88 times, the bending moment of the left arch shoulder increases by 2.26 times, and the bending moment of the left arch waist increases by 2.32 times. It increases the crack length from 1 m to 6 m.

When a combination crack is preset at the arch crown and arch waist, the bending moment of the arch crown increases by 1.45 times, the bending moment of the left arch shoulder increases by 2.08 times, and the bending moment of the left arch waist increases by 1.91 times. When cracks are jointly preset at the arch crown, arch shoulder, and arch waist, the bending moment at the arch crown increases by 2.80 times, the bending moment at the left arch shoulder increases by 1.69 times, and the bending moment at the left arch waist increases by 2.12 times.

The increase in the distribution of longitudinal combined cracks results in a significant increase in the bending moment at the arch crown. Compared to the preset cracks in the two parts, the proportion of times the bending moment at the arch crown increases is 48.94% and 93.10%. In addition, special attention needs to be paid to the changes in bending moment values at the arch foot of the lining. Under the three working conditions, the bending moment at the left arch foot increases by 1.22 times, 1.29 times, and 1.47 times, and the bending moment at the right arch foot increases by 1.36 times, 1.49 times, and 1.50 times, respectively.

5.2. Mechanical Characteristics of Lining with Annular Crack

The bending moment results of the preset circumferential cracks on the arch under single and double cross-sections are plotted in Figure 14a–d. It indicates that when circumferential cracks appear on the lining arch, the larger the preset distribution angle of circumferential cracks, the smaller the bending moment value of the arch. When the range of a single cross-section increases from 30° to 60°, the bending moment fluctuations of the arch crown, left and right arch waists, and left and right arch feet are significant. It shows a numerical decrease from 38.93 kN/m to 16.97 kN/m, from 31.03 kN/m to 23.06 kN/m, and from 16.14 kN/m to 17.63 kN/m.

When the double section circumferential crack increases from 30° to 60°, the corresponding position values change from −19.48 kN m to −33.49 kN m, from −29.14 kN m to −23.57 kN m, from −48.93 kN m to −44.76 kN m, and from −44.78 kN m to −48.88 kN m. The proportion of changes is decreased by 72.9%, increased by 19.1%, decreased by 42.1%, increased by 8.5%, and increased by 9.2%.

Figure 15a–d are the bending moment diagrams corresponding to the preset circumferential cracks at the arch shoulder. The comparison results show that the tensile effect on the outer side of the arch shoulder is significantly amplified, but an increase in the distribution range could to some extent reduce the bending moment of the arch shoulder. When the distribution range of annular cracks on a single arch shoulder increases from 30° to 60°, the bending moment of the arch crown, left and right arch waists, and left and right arch feet increases from 12.91 kN/m to 26.56 kN/m, decreases from −30.03 kN/m to −23.19 kN/m, and from −44.78 kN/m to −48.88 kN/m. The changes are an increase of 51.4%, a decrease of 22.8%, an increase of 8.6%, a decrease of 32.5%, and an increase of 14.6%.

Comparing Figure 16a–d, it is found that when there are annular cracks in the lining arch waist and arch crown, the degree of concave lining walls intensifies, and the degree of asymmetric bending moments on the left and right sides of the lining section is more pronounced. The reason for the analysis is that compared to the above two parts, the circumferential cracks at the arch waist will affect the areas below the arch line of the tunnel lining.

The bending moment of the secondary lining under the influence of annular cracks shows an asymmetric distribution, showing a pattern of a “convex arch shoulder, concave arch waist, side wall, and convex arch foot” as shown in Figure 14, Figure 15 and Figure 16. According to the statistics of 12 different annular crack calculation conditions, the maximum positive bending moment occurs once at the arch crown, seven times at the left arch waist, and four times at the left wall. The maximum negative bending moment occurs seven times at the left foot arch and five times at the right foot arch. Therefore, under the influence of the preset circumferential direction, there is a probability of extreme bending moments occurring in the secondary lining section.

5.3. Stress Characteristics of Lining with Crack

The maximum principal stress of the lining is one of the characterizations of internal stress in the structure, and it is also the failure criterion used in this simulation. Conduct a post-processing analysis on the maximum principal stress cloud map of lining under the influence of cracks with different properties. The presence of cracks at the arch shoulder can alter the maximum principal stress mode of the lining.

As shown in Figure 17b,d, the presence of arch shoulder cracks increases the tensile zone along both sides of the crack, and the maximum principal stress increases in a negative direction. Under the action of load, the maximum principal stress on the outer side of the lining with longitudinal cracks at the arch crown first reaches the limit, and the extension of the inner crack at both ends is not deep. The tensile effect of longitudinal cracks at the arch shoulder and arch waist is obvious, and the maximum negative principal stress exceeds the limit. The crack tip extends and develops a certain distance along the longitudinal direction, as shown in Figure 17e,d. Stress concentration areas appear along the longitudinal lining along both sides of the crack. There is little difference in the maximum principal stress mode of the lining under the influence of longitudinal cracks and circumferential cracks in the arch crown and arch waist. However, the maximum principal stress of the lining has stronger continuity under the influence of circumferential cracks.

5.4. Safety Performance of Lining with Crack

We fully studied the changes in the safety performance of the lining under the influence of cracks. It was found that the safety factor at the center of the inverted arch is much greater than other key nodes of the cross-section. Therefore, the results at the center of the inverted arch are omitted. The scatter plot of the safety factor is drawn in a counterclockwise direction starting from the arch crown.

According to Figure 18, it can be observed that the impact of preset longitudinal cracks on the safety performance of the lining structure is complex. When longitudinal cracks are preset at the arch crown, the safety at the two arch feet is at the lowest level. When cracks are preset at the arch shoulder, the safety coefficients of the right arch shoulder and left arch waist are at a high level, with good safety reserves, and the fitted curve shows a “W” shape of first rising, then falling, and then rising. When cracks are preset at the arch waist, the safety coefficients of the right arch shoulder and right arch waist are greater than those of the rest of the section, and the fitted curve shows a trend of first rising, then decreasing, and then slowly rising. The rate of rise is smaller than the degree of cracks in the arch shoulder belt. See Figure 18a–c.

Under the influence of preset longitudinal combination cracks, the safety factor on both sides of the arch foot and the right-side wall of the lining is the most significant area. When there are cracks in both the arch crown and arch shoulder, except for the left arch shoulder and left arch waist, the safety factor of the lining is higher, and the safety factor values of other parts are similar. When cracks appear on both the arch crown and arch waist, the safety performance of the lining arch crown is at its highest level; see Figure 18d–f.

According to the changes in the safety factor of the lining under the influence of circumferential cracks, as shown in Figure 18g,h, when the number of preset circumferential cracks increases from the single section to double section, there is a certain variation in the safety factor. The safety factors of the right wall and right arch foot are the most affected, with the minimum safety factors increasing from 3.48 to 4.02, increasing by 15.52%, and decreasing from 3.68 to 2.98, decreasing by 19.02%.

5.5. Impact of Crack Properties on Safety Performance

The evolution laws of the lining safety coefficient under the influence of longitudinal cracks and annular cracks are respectively presented in the previous text. This study calculated the safety factor attenuation rate of 10 key nodes to compare the influence of crack characteristics. It can be calculated according to Equation (11):

$$D_{ki}=\frac{k_{max}-k_{min}}{k_{max}}\times 100\% (i=1, 2\dots 10)$$

(11)

where $D_{ki}$ is the attenuation rate of the safety coefficient of the key nodes of the lining; $k_{max}$ is the maximum safety factor of a critical node in the lining; $k_{min}$ is the minimum safety factor of a critical node in the lining.

Take the extreme value of the calculation results for each node to obtain the maximum safety factor attenuation rate, which is calculated according to Equation (12):

$$D_{kmax}=max(D_{k1}, D_{k2},\dots , D_{k10})$$

(12)

According to Equations (11) and (12), calculate the maximum attenuation rate of the lining safety coefficient for each working condition and summarize the relevant data table, as shown in

Table 3. Maximum attenuation rate of lining safety coefficient.

Crack Properties	Crack Location	Maximum Attenuation Rate of Safety Factor ${\mathit{D}}_{\mathit{kmax}}$
Longitudinal crack	Arch Crown	22.32%
Longitudinal crack	Arch Shoulder	32.97%
Longitudinal crack	Arch Waist	18.81%
Longitudinal crack	Arch Crown + Arch Shoulder	20.54%
Longitudinal crack	Arch Crown + Arch Waist	16.33%
Longitudinal crack	Arch Crown + Arch Shoulder + Arch Waist	12.70%
Annular crack	/	12.10%
Annular crack	/	18.95%

. On the basis of ignoring the absolute differences in safety coefficient values under different working conditions, the relative rate of safety coefficient attenuation under the influence of crack characteristics is mainly considered.

According to Table 3, the maximum attenuation rates of the safety factor under the influence of single section and double section circumferential cracks are 12.10% and 18.95%. The increase in the number of sections with annular cracks has a significant promoting effect on the rate of reduction in the safety factor. When setting longitudinal cracks on the arch shoulder, the maximum safety factor attenuation rate of the lining is 32.97%. It has the fastest rate of decrease in the safety factor among all operating conditions. Observing all working conditions except for the preset longitudinal cracks at the arch shoulder, it was found that the maximum attenuation rate of single section circumferential cracks was lower than the values under the other preset longitudinal crack working conditions.

Compared to circumferential cracks, longitudinal cracks have a more significant impact on the decreasing trend of lining safety performance. Especially when longitudinal cracks develop at the arch shoulder of the tunnel lining, annular cracks develop at both cross-sections, and annular cracks develop simultaneously at multiple cross-sections. Cracks in tunnel lining should be filled in a timely manner, and the overall structure of the lining should be reinforced and treated in a timely manner to slow down the rate of safety performance degradation of the lining.

The cosine amplitude method is used to study the sensitivity of the safety impact of lining structures under the influence of crack location, crack length, and crack distribution range. For array $K=\left[k_1, k_2,\cdots k_n\right]$, each element $k_i$ in this array is a vector of length m. Each data node is considered as any point in the m-dimensional space. The degree of influence $h_{ij}$ of two data points $k_i$ and $k_j$ is calculated according to Equation (13):

$$h_{ij}=\frac{{\displaystyle \sum _{k=1}^m\left(k_{ik}\times k_{jk}\right)}}{\sqrt{{\displaystyle \sum _{k=1}^m{k}_{ik}^2\times {\displaystyle \sum _{k=1}^m{k}_{jk}^2}}}}$$

(13)

For this study, single and combined working conditions corresponding to longitudinal cracks of different lengths were calculated, and single and double cross-sections with different distribution ranges (seeing Figure 19). It indicates that a single longitudinal crack has the most significant impact on the service safety performance of the lining, with an impact level of 0.86. The impact level of a single section annular crack is the smallest, at 0.46. The combined crack influence factor is 0.79, followed by the horizontal double section annular crack.

6. Conclusions

The mechanical characteristics and safety performance of a tunnel lining structure under service with cracks have been studied through XFEM, and the conclusions are as follows:

(1)

Cracks have a serious impact on the stress of the lining structure, causing it to be in an asymmetric state. The axial force, bending moment, and safety factor are severely affected by the pre-crack wave. When a single preset crack position is placed on the arch crown, arch shoulder, and arch waist, the axial force and bending moment exhibit opposite changes. After the crack length transitioned from 1 m to 6 m, the axial force values at the crack site decreased by 33.77%, 36.15%, and 11.32%. Meanwhile, the bending moment values at the crack site increased by 4.47 times, 2.50 times, and 1.69 times, respectively.

(2)

The combination of multiple longitudinal cracks has the most significant impact on the axial force at the arch crown. Under the three working conditions of an “arch crown + arch shoulder”, “arch crown + arch waist”, and “arch crown + arch shoulder + arch waist”, the increase is 21.55%, the decrease is 17.52%, and the decrease is 13.45%. The bending moment at the arch foot is the most heavily affected. The left and right arch feet increase by 1.22 times, 1.29 times, 1.47 times, 1.36 times, 1.49 times, and 1.50 times.

(3)

The circumferential cracks have a small impact on the axial force of the lining. However, the bending moment exhibits a pattern of “protruding outside the arch shoulder, concave inside the arch waist and side walls, protruding outside the arch foot”. In addition, the service safety scatter of cracked lining shows a gradual transition from a “W” shape to a “U” shape. The maximum safety factor attenuation rate of the lining is more significantly affected by longitudinal cracks, and the preset longitudinal crack condition of the arch shoulder is a maximum decrease rate of 32.97%. The increase in the number of sections with circumferential cracks will accelerate the rate of attenuation of the lining safety factor.