Study on the Fracture Resistance of Mixed Fiber Concrete Lining in a Reverse Fault Tunnel

Abstract

With the increasing complexity of engineering environments in tunnel construction, some projects are located in areas with adverse geological conditions, including areas prone to fault movements and other geological hazards. Fiber-reinforced concrete can significantly enhance structural performance in such challenging conditions. Currently, the application of high-performance steel-polypropylene fiber in tunnel linings and its anti-fault performance are quite limited. Therefore, this paper focuses on studying the application of steel-polypropylene mixed fiber-reinforced concrete in tunnel lining under reverse fault movements. In this paper, the constitutive relationship of mixed fiber-reinforced concrete is modified based on literature data. Utilizing the Najar formula, fiber-related variables were designed, and a three-dimensional numerical simulation model of a reverse fault tunnel was established. By designing various parameter combination cases, the structural response of tunnel linings was calculated and studied. then the influence patterns of mixed fiber parameters on the anti-fault performance of tunnel linings are summarized, the relationships between main variables are explored, and a reasonable value range for the two fibers is determined, which can provide guidance for practical engineering.

1. Introduction

Fiber concrete first appeared in a research report by American scholar H.F. Porter at the beginning of the last century, and it is widely positioned as a composite material composed of cementitious materials (cement stone, mortar, or concrete) and various types of fibers. Cementitious materials have advantages such as good compressive properties and high durability, but their low tensile strength and small elongation make them a brittle material, so the structural applicability of considering only ordinary cementitious materials in engineering and construction is limited, and the safety risks are large. Fiber in the history of human construction has a fairly long history [1], with high tensile strength, elongation, toughness, etc., so adding fiber to the cementitious material helps to improve its flexural and tensile properties while also helping to toughen the deformation resistance, cracking, and impact resistance, etc., which greatly broadens the applicability of the cementitious material of breadth and depth and therefore has more and more engineering applications, and there is a study on the fiber concrete. At the same time, the research on fiber concrete is still broadening, and the fiber type is also gradually rich, from the initial commonly used metal steel fiber development to today’s synthetic fibers, carbon fibers, and so on, and even a variety of fibers mixed came into being.

Currently, several scholars have studied concrete materials with steel fibers as the admixture, and their results show that the admixture of steel fibers can effectively improve the stability and strength of concrete, and the nature of the admixture, such as the shape, size, etc., will also affect the performance of the final concrete material. At the same time, a variety of materials simultaneously as an admixture will affect the performance of the concrete material to varying degrees. In 1975, the first organic fiber and inorganic fiber blending test was conducted by Walton [2] and others to investigate the tensile strength and impact resistance of this cement-based composite material. Walton and other first organic fibers and inorganic fibers for mixing test, the study of the tensile strength and impact resistance of the cementitious composites, which is also the first systematic research on mixed-fiber concrete. In 1987, Rossi [3,4] published a series of research results on mixed fibers, which, for the research of this kind of material fever, swept the world. Since then, many scholars have conducted research on mixed fiber concrete [5,6,7]. The results show that mixed fibers have a more significant improvement in the mechanical properties of cement composites. The current research on mixed fiber mainly includes steel fiber mixed with polypropylene fiber, steel fiber mixed with polyvinyl alcohol fiber, steel fiber mixed with basalt fiber, basalt fiber mixed with polypropylene fiber, polyvinyl alcohol fiber mixed, etc., which can improve all kinds of properties of the concrete material in different degrees and therefore has become a widely used construction material.

With the increasing complexity of the engineering environment in tunnel construction, some projects are located in unfavorable geological conditions, including areas prone to fault movement and other geological hazards. At present, some scholars have studied the anti-fault measures, which mainly propose two ways of anti-seismic joints and flexible joints, to study the anti-fault measures in tunnels. Current scholars on the application of fiber concrete in tunnel lining have been part of the study, such as Wang [8], etc. Through the reinforced concrete and basalt fiber concrete lining indoor mechanical properties model test, the study of basalt fiber concrete lining bearing characteristics found that basalt fibers help to improve the lining of the initial bearing performance, compared with reinforced concrete, which is more conducive to controlling the deformation of the tunnel of the weak surrounding rock. Liu [9] et al. conducted full-size tests of synthetic fiber concrete and conventional reinforced concrete lining to investigate the damage mechanism of synthetic fiber concrete tunnel lining and compared it with conventional reinforced concrete tunnel lining. The results showed that the incorporation of synthetic fibers delayed the onset of the yield state and led to higher yield loads compared to the conventional RC specification. [10] conducted a comparative analysis of seismic displacements, stress responses, and sidewall convergence of plain concrete, steel-fiber-reinforced concrete, and steel-basalt hybrid-fiber-reinforced concrete secondary lining structures passing through fault zones using FLAC3D to obtain that steel-basalt mixed fiber concrete secondary lining has better seismic performance than steel fiber concrete secondary lining. Ren [11] investigated the suppression of tunnel lining problems by multi-scale polypropylene fiber-reinforced concrete and the mechanical properties it used for secondary tunnel lining using numerical simulation and field tests. However, there is limited research on the fault resistance of fiber concrete in tunnel lining. Therefore, this paper focuses on the application of steel-polypropylene mixed fiber-reinforced concrete under reverse fault movement in tunnel lining.

In this paper, the ontological relationship of mixed fiber-reinforced concrete is revised. The damage coefficients and related parameters are calculated using Najar’s formula, with experimental design variables containing all factors. Then a three-dimensional numerical simulation model of a reverse fault tunnel was established to comparatively study the structural stress–strain and damage of the tunnel lining under different conditions of mixed fiber incorporation.

2. Methodology and Theoretical Background

2.1. Damage Variable

In order to study the fault-breaking resistance of the lining, it is necessary to introduce the damage variable of the material as a key parameter of the study, which can reflect the stiffness degradation of the material, play a deterministic role in the stress-strain relationship of the material, and determine the basic load-bearing performance and damage mode of the material. Research on the damage mechanics of materials initially started in the field of metals [12,13], and since then it has gradually developed into different fields. The damage mechanics of continuous media [14,15] consider that the damage to materials is due to the gradual expansion and penetration of microscopic holes and microcracks under external loading, so that the bearing capacity and mechanical properties of materials are weakened or even fail, which can be considered as damage to materials, and therefore the damage can be considered as the degradation of the material stiffness. Damage variable $d$ and effective stresses were first proposed by Rabotnov Y. N. [16] to quantify the damage to the material, which can be expressed by the following equation:

$$d=\left(A-A^{\prime }\right)/A$$

(1)

$${\sigma }_d=\sigma \left(1-d\right)$$

(2)

where $A$ is the area of the material before damage, $A^{\prime }$ is the area after damage, $\sigma$ is the total stress without considering damage, and ${\sigma }_d$ is the actual effective stress considering material damage.

The value range of damage variable $d$ is [0, 1], $d=0$ corresponds to a completely damage-free state, $d=1$ indicates that the material is completely damaged. As the value gradually increases, the degree of damage is also gradually larger, indicating that the more damage to the material, the more serious the degradation of the stiffness. The general concrete can be regarded as completely damaged when $d=0.9$ [17], according to which the stress-strain relationship is shown in the following equation:

$$\sigma =(1-d)E_0(\epsilon -{\tilde{\epsilon }}_{pl})$$

(3)

$$\tilde{\sigma }=\frac{\sigma }{(1-d)}=E_0(\epsilon -{\tilde{\epsilon }}_{pl})$$

(4)

where $\tilde{\sigma }$ is the effective stress, $E_0$ is the initial elastic modulus of the material, $\epsilon$ is the strain, and ${\tilde{\epsilon }}_{pl}$ is the equivalent plastic strain.

After decades of development, a number of scholars have further studied the damage variables according to the definition of damage variables and effective stresses, and currently, there are theoretical models such as the Najar damage model, the strain equivalence model, the Sidiroff energy equivalence model, and the Birtle–Mark formula to modify the calculation of damage variables. Among them, the damage factor calculation of Najar damage theory requires the full curve of material stress–strain, so the calculation is relatively cumbersome, but Hao [18,19] and Nan [19] show that the value obtained by using the Najar damage calculation method has universality and higher accuracy, so in this paper, the Najar damage theory is used to calculate the damage factor of concrete.

Najar defines the damage variables of brittle materials from the energy point of view [20], and the uniaxial compression mechanism shown in Figure 1 is used as an example to explain Najar’s damage theory. It is believed that the decrease in energy is due to the damage to the material, so the damage variable $d$ can be characterized by the change in energy; the specific expression is as follows:

$$d=\frac{W_0-W^{\prime }}{W_0}$$

(5)

$$W_0=\frac{1}{2}{E}_c{\epsilon }^2$$

(6)

$$W^{\prime }=\frac{1}{2}{E}_c^{\prime }{\epsilon }^2$$

(7)

where $W_0$ corresponds to the strain energy of the initial undamaged state in Figure 1, $W^{\prime }$ is the strain energy of Najar’s simplified linear calculation of the damage state, and $E_c$ is the compressive modulus of elasticity of concrete.

Najar damage theory considers the elastic-plastic strain energy and damage energy caused by the work done by an external force, which has high applicability for concrete materials, but it can be found that its simplification in calculating the strain energy of the damage state has a certain gap with the actual post-damage strain energy shown in Figure 1, so many scholars have used mathematical integration methods such as segment integral, Simpson’s integral, and Gaussian integral to correct the calculation of the energy of the damage state [21,22], and therefore the expression of the energy-based damage variable is widely used as follows:

$$d=\frac{\frac{1}{2}{E}_c{\epsilon }^2-\int \epsilon d\epsilon }{\frac{1}{2}{E}_c{\epsilon }^2}$$

(8)

Although the modified method is generally applicable to concrete materials, it requires the provision of pre-existing stress–strain curves and therefore has a high demand for a true reflection of the stress–strain curves.

2.2. Uniaxial Tensile and Compressive Curve

For concrete material, its uniaxial tensile and compression curve, i.e., the stress-strain curve under unidirectional tensile and compressive action, can reflect the development of the force of the material under the action of the external load, and the analysis of the curve can be used to find out the material’s force characteristics at various stages from a macroscopic point of view, which is a reflection of the basic mechanical properties, so this paper studies the uniaxial tensile and compressive curves of the steel-polypropylene mixed fiber concrete.

Experimental studies have been carried out on the uniaxial tensile and compressive curves of steel-polypropylene mixed fiber concrete, and mathematical expressions were fitted to the curve equations according to the existing studies, but most of the current studies do not take into account the effect of the grade of matrix concrete. Mei [23] carried out a study on the uniaxial compressive performance of concrete with different contents of steel fibers and polypropylene fibers and length-to-diameter ratios, and 34 sets of stress-strain full curves were obtained. First, the equations for the relationship between the peak stress of steel-polypropylene blended-fiber concrete and its corresponding strain and the peak stress of plain concrete and its corresponding strain were fitted, and the equations were selected based on the characteristics of the curves qualitatively, as shown in Equations (9) and (10) shown in the expression form of a stress–strain curve with parameters:

$$f_{ft}=f_{mt}\left(1+0.366{\lambda }_{sf}+0.277{\lambda }_{pf}\right)$$

(9)

$${\epsilon }_{ft}={\epsilon }_{mt}\left(1+0.498{\lambda }_{sf}+0.697{\lambda }_{pf}\right)$$

(10)

where $f_{mt}$, $f_{ft}$ is the peak stress of plain concrete and mixed fiber concrete, i.e., axial tensile strength, ${\epsilon }_{mt}$, ${\epsilon }_{ft}$ is the strain corresponding to the peak tensile stress of plain concrete and mixed fiber concrete, respectively, ${\lambda }_{sf}$, ${\lambda }_{pf}$ is the product of the characteristic values of steel fibers and polypropylene fibers, the length-to-diameter ratio, and the dosage, respectively: ${\lambda }_{sf}={\rho }_{sf}{l}_{sf}/d_{sf}$, ${\lambda }_{pf}={\rho }_{pf}{l}_{pf}/d_{pf}$.

The parameter calculation method was also quantitatively fitted by testing specific values:

$$y=\left\{\begin{array}{cc}{a}_{1}x+(1.5-1.25a_1)x^2+(0.25a_1-0.5)x^6\hfill & \hfill 0\le x<1\\ \frac{x}{{\alpha }_t{(x-1)}^{1.7}+x}\hfill & \hfill x>1\end{array}\right.$$

(11)

$$a_1=1.2\left(1+0.265{\lambda }_{sf}+0.277{\lambda }_{pf}\right)$$

(12)

$${\alpha }_t=\frac{0.312f_{mt}^2}{1+3.366{\lambda }_{sf}+3.858{\lambda }_{pf}}$$

(13)

In the formula, $x$, $y$ are strain and stress normalization parameters, $x=\mathsf{\epsilon }/{\epsilon }_t$, $\mathrm{y}=\mathsf{\sigma }/f_t$; $\epsilon$, $\sigma$ represent strain and stress, ${\epsilon }_t$ is the strain value corresponding to the peak stress, $f_t$ is the peak stress that is the axial tensile strength; $0\le x<1$ that is, the interval before the peak strain that is the ascending section, $x>1$ is the interval after the peak strain that is the descending section; $a_1$ is the parameter of the ascending section, the practical significance for the ratio of tangent modulus of elasticity at the origin of the curve to the peak at the cut-line modulus, ${\alpha }_t$ is the parameter of the descending section, and the bigger the value of which is the curve is the steeper, which influences the curve’s degree of fullness.

In this paper, the uniaxial tensile and compression equations for steel-polypropylene fiber concrete with a matrix concrete grade of C30 are modified, and the resulting uniaxial tensile full curve equations are as follows:

$$\left\{\begin{array}{c}{f}_{ft-c30}=1.38f_{mt}\left(1+0.366{\lambda }_{sf}+0.277{\lambda }_{pf}\right)\\ {\epsilon }_{ft-c30}=2.39{\epsilon }_{mt}\left(1+0.498{\lambda }_{sf}+0.697{\lambda }_{pf}\right)\\ a_{1-c30}=1.42\times 1.2\left(1+0.265{\lambda }_{sf}+0.277{\lambda }_{pf}\right)\\ {\alpha }_{t-c30}=1.51\times \frac{0.312f_{mt}^2}{1+3.366{\lambda }_{sf}+3.858{\lambda }_{pf}}\end{array}\right.$$

(14)

The uniaxial pressurized full curve equation is as follows:

$$\left\{\begin{array}{c}{f}_{fc-c30}=1.33f_{mc}\left(1+0.206{\lambda }_{sf}+0.388{\lambda }_{pf}\right)\\ {\epsilon }_{fc-c30}=1.5{\epsilon }_{mc}\left(1+0.705{\lambda }_{sf}+0.364{\lambda }_{pf}\right)\\ a_{2-c30}=1.3\times \left(28.2283-23.2771f_{fc}^{0.0374}+0.4772{\lambda }_{sf}-0.4917{\lambda }_{pf}\right)\\ {\alpha }_{c-c30}=0.54\times \left(1+0.3688f_{fc}^{-0.2846}+{\lambda }_{sf}+{\lambda }_{pf}\right)\end{array}\right.$$

(15)

Observe the expression of the modified axial tensile and axial compressive stress-strain full curve, which involves parameters such as steel fiber eigenvalues and polypropylene fiber eigenvalues ${\lambda }_{sf}$ and ${\lambda }_{pf}$, matrix concrete axial tensile strength and corresponding strains $f_{mt}$ and ${\epsilon }_{mt}$, and concrete matrix axial compressive strength and corresponding strains $f_{mc}$ and ${\epsilon }_{mc}$. The calculations of the first two eigenvalues are shown in Equations (9) and (10), which can be done by directly inputting the fiber mixing amount and the length-to-diameter ratio. For the peak stresses and strains, due to the lack of experimental data, it is necessary to give reference equations according to the specification and the research of various scholars. The values of $f_{mt}$, ${\epsilon }_{mc}$, and $f_{mc}$ are taken from the specification [24], ${\epsilon }_{mt}$ is taken from Bai’s research results [25]:

$$\left\{\begin{array}{c}{f}_{mt}=0.395f_{cu,k}^{0.55}\\ {\epsilon }_{mt}=66\times f_{mc}^{0.52}\times {10}^{-6}\\ f_{mc}=0.72f_{cu,k}\\ {\epsilon }_{mc}=(700+172f_{mc}^{0.5})\times {10}^{-6}\end{array}\right.$$

(16)

where, $f_{cu,k}$ for the matrix concrete cubic compressive strength, other meanings are the same as above.

2.3. Concrete Damage Plasticity Model

In this paper, the Najar damage model is simulated using the concrete damage plasticity model (CDP model), which assumes that the concrete failure mechanism is mainly based on uniaxial tensile cracking and uniaxial compressive cracking of the concrete material and that the evolution of the yield surface is controlled by two hardening variables associated with the compressive and tensile failure mechanisms (tensile and compressive equivalent plastic strains ${\tilde{\epsilon }}_{pl}$), which are simulated by using the CDP model with the aid of the ABAQUS software (version 2020). In this paper, the material properties are simulated using a CDP model of the material in ABAQUS software. Although the CDP model in ABAQUS has some assumptions of simplification, it can capture the stress-strain evolution of concrete materials in uniaxial tension and uniaxial compression, which can reflect the main mechanical properties of the material.

For the damage variable $d_t$, the calculation method has been described in Section 2.1, and the corresponding cracking strain ${\tilde{\epsilon }}_t^{ck}$ and compressive strain ${\tilde{\epsilon }}_c^{in}$ values are calculated according to the following formula:

$${\tilde{\epsilon }}_t^{pl}=\epsilon -{\epsilon }_{0t}^{el}; {\epsilon }_{0t}^{el}={\sigma }_t/E_t$$

(17)

$${\tilde{\epsilon }}_t^{pl}={\tilde{\epsilon }}_t^{ck}-\frac{d_t}{(1-d_t)}{\sigma }_t/E_t$$

(18)

$${\tilde{\epsilon }}_c^{in}=\epsilon -{\epsilon }_{0c}^{el}; {\epsilon }_{0c}^{el}={\sigma }_c/E_c$$

(19)

$${\tilde{\epsilon }}_c^{pl}={\tilde{\epsilon }}_c^{in}-\frac{d_c}{(1-d_c)}{\sigma }_c/E_c$$

(20)

The concrete damage plasticity model assumes that the model follows non-associated flow rule and its plastic potential function $G$ is expressed by the Drucker–Prager yield condition as follows:

$$G=\sqrt{{(\epsilon {\sigma }_{t0}\tan \psi )}^2+{\overline{q}}^2}-\overline{p}\tan \psi$$

(21)

where $\psi$ is the dilation angle measured in the p-q plane at high confining pressure, ${\sigma }_{t0}$ is the uniaxial tensile stress at failure, taken from the user-specified tension stiffening data, $\epsilon$ is a parameter, referred to as the eccentricity, that defines the rate at which the function approaches the asymptote (the flow potential tends to a straight line as the eccentricity tends to zero).

The yield function proposed by Lubliner and improved by Lee and Fenves is used in the calculation to account for different evolution of strength under tension and compression. The evolution of the yield surface is controlled by the hardening variables ${\tilde{\epsilon }}_{pl}$. In terms of effective stresses, the yield function takes the form as follows:

$$F=\frac{1}{1-\alpha }(\overline{q}-3\alpha \overline{p}+\beta ({\tilde{\epsilon }}^{pl})〈{\widehat{\overline{\sigma }}}_{\max }〉-\gamma 〈-{\widehat{\overline{\sigma }}}_{\max }〉)-{\overline{\sigma }}_c({\tilde{\epsilon }}_c^{pl})=0$$

(22)

With

$$\alpha =\frac{\left({\sigma }_{b0}/{\sigma }_{c0}\right)-1}{2\left({\sigma }_{b0}/{\sigma }_{c0}\right)-1}$$

(23)

$$\beta =\frac{{\overline{\sigma }}_c({\tilde{\epsilon }}_c^{pl})}{{\overline{\sigma }}_t({\tilde{\epsilon }}_t^{pl})}(1-\alpha )-(1+\alpha )$$

(24)

$$\gamma =\frac{3(1-K_c)}{2K_c-1}$$

(25)

where: ${\widehat{\overline{\sigma }}}_{\max }$ is the maximum principal effective stress, ${\sigma }_{b0}/{\sigma }_{c0}$ is the ratio of initial equibiaxial compressive yield stress to initial uniaxial compressive yield stress, $K_c$ is the ratio of the second stress invariant on the tensile meridian to that on the compressive meridian.

In this paper, several plasticity parameters were selected as default values for calculation, and the damage variables were calculated from Section 2.1 and entered into ABAQUS in tabular form.

3. Model Implementation

3.1. Statement of Geometry

Take a single tunnel as an engineering example, based on the simulation software ABAQUS, to establish the reverse fault fracture misalignment conditions of steel-polypropylene fiber concrete materials used in the tunnel secondary lining numerical calculation model, the tunnel section for the three-centered circle, and the specific dimensions as shown in Figure 2. The simulated fault condition is set as a fixed value, the width of the fault zone is 20 m, the dip angle of the fault is 60°, and the vertical misalignment displacement is 30 cm (horizontal misalignment is 17 cm). The tunnel width is 12.4 m, and the height is 9.9 m. Based on St. Venant’s theorem, the size of the model should take the boundary effect into consideration, and it is generally considered that the boundary is 3–5 times the diameter of the hole from the center of the hole [17,26]. Based on this, the lateral boundary of this paper takes 50 m, i.e., the model width is 100 m; the distance of the bottom boundary from the bottom of the tunnel takes 60 m, and the depth of the tunnel is 20 m, so the whole model is about 90 m high; the longitudinal length of the model is set to be 300 m and the lining of the tunnel includes the initial support and the secondary lining, with thicknesses of 0.15 m and 0.35 m, and the thickness of the whole support is 0.5 m. In the mesh delineation, faults, fracture zones, and the surrounding rocks near the lining are taken into account, and the lining is not as thick as it is. In the grid division, the faults and the surrounding rocks near the lining are encrypted, and the final model and grid division are shown in Figure 3.

3.2. Materials and Contact Properties

The model contains surrounding rock, fault fracture zone, and lining, and the lining contains initial support and secondary lining. The surrounding rock, fault fracture zone, and initial support adopt the Mohr–Coulomb principal model and the secondary lining selects the CDP model to simulate the fiber concrete material. The specific material parameters are shown in

Table 1. Parameters of materials of tunnel construction.

Materials	$\mathit{\rho }$ (kg/m3)	$\mathit{E}$ (GPa)	$\mathit{\mu }$	$\mathit{c}$ (MPa)	$\mathit{\phi }$ (°)
Rock (in class IV)	2200	7	0.3	0.5	35
Fault fracture zone	2000	5	0.3	0.25	25
Initial lining (in C25)	2400	27.5	0.2	12.5	51
Secondary lining (in C30)	2500	29	0.25	/	/

. Because the interaction between secondary lining and initial support is not considered in this paper, these two parts are set as bound contact in the model; frictional contact is used between the lining and surrounding rock, and the friction coefficient is set to 0.6; and the contact friction coefficient between sliding surfaces of fault fracture zones is set to 0.5 [27].

For concrete fiber admixture, the steel fiber and polypropylene fiber admixture and L/D ratio (length to diameter ratio of fibers) settings are shown in

Table 2. Characteristic parameters of fibers.

Steel	Admixture (${\rho }_{\mathrm{sf}}$)	0.5%	1.2%	1.9%
	L/D ratio (${\mathrm{l}}_{\mathrm{sf}}/{\mathrm{d}}_{\mathrm{sf}}$)	30	60	80
Polypropylene	Admixture (${\rho }_{\mathrm{pf}}$)	0.05%	0.1%	0.15%
	L/D ratio (${\mathrm{l}}_{\mathrm{pf}}/{\mathrm{d}}_{\mathrm{pf}}$)	396

, and the strain capacity of polypropylene is 15–35%, as suggested by Mei [23]. Based on the values of the variables identified in Table 2, several groups of variables were designed for calculation, including the change of each parameter under mixed and single doped, and plain concrete was designed as a control group.

Existing studies have shown [23,28] that the modulus of elasticity of concrete with blended fibers is slightly smaller than that of ordinary concrete, but the difference is negligible, so the modulus of elasticity of C30 concrete for the secondary lining here is still calculated according to ordinary concrete. The calculation of plastic variables for its concrete plastic damage model is given in Section 2.3.

3.3. Boundary Conditions and Simulation Process

In this section, numerical software is used to simulate the mechanical properties and damage distribution of secondary lining doped by different mixed fibers under fault misalignment conditions, which need to focus on the simulation of reverse fault misalignment. References [29,30] show the simulation is divided into three steps: (1) geostress equilibrium; (2) tunnel excavation. Here, the simulation of tunnel excavation is simplified and the softening modulus method [31] is used for one-step excavation; and (3) fault misalignment. In the first step to balance the ground stress, except for the top free boundary, the other five surfaces are set to normal constraints; in the third step to simulate the fault reference [32,33], the top of the model is still a free boundary, the displacement constraints are applied horizontally in line with the fault strike, the normal displacement constraints are set vertically, the normal constraints are set at the bottom of the lower plate, and the bottom of the left and right sides are respectively equal in size, with the center of the fault fragmentation zone as the dividing interface. With the center of the fault zone as the interface, the vertical and horizontal displacements of the left and right sides of the plate are equal in size and opposite in sign in order to realize the fault misalignment along the center of the fault zone as the sliding surface. The detailed fault simulation bottom constraints are shown in Figure 4.

4. Validation

Yang [34] mixed different types of steel fibers in different matrix strength concrete, studied the effect of steel fiber type on the axial tensile properties of concrete, the literature as a concrete axial tensile properties of the calculations, selected test specimen F3-3010 (C30 grade concrete, 1% shear-type steel fibers, with a L/D ratio of 55), according to the pre-modified [23] in Section 2.2, respectively, and modified curve prediction equations in Section 2.2, and the calculated results were compared with the test results, and the comparison results are shown in Figure 5.

Ni [35] took concrete matrix strength, steel fiber admixture, and steel fiber type as variables and conducted axial compression tests on 63 specimens to investigate the effect of steel fiber type and admixture on the axial compression performance of concrete with different matrix strengths. The literature was used as an example of concrete axial compression performance calculations, in which A1B1C1-B1.0 (C30 grade concrete, 1% shear-type steel fibers, with an L/D ratio of 40) was used. Calculations were performed according to the pre-modified [28] and post-modified curve prediction equations in Section 2.2, respectively, and the calculated results were compared with the test results, and the comparison results are shown in Figure 5.

From Figure 5, it can be seen that the modified prediction curve is consistent with Ni’s experimental data, and the modified curve expression has a high degree of coincidence with the test results in terms of both the value and the change trend.

The vertical displacement of the top of the lining obtained from the simulation of this paper is compared with Qiu’s monitoring data [32]. Due to the difference in the simulation method, it is necessary to offset the simulation results of this paper upward by a certain amount, and the final comparison results are shown in Figure 6.

Observing Figure 6, the simulation results are basically consistent with Qiu’s monitoring data [32]. Due to the inverse fault misalignment, the whole lining is “Z”-shaped bending shape, basically symmetric about the center of the fracture zone. Due to the simulation of this paper, the material’s own gravity and the fault sliding displacement are considered as two kinds of external loads, so there is a difference in the displacement value, but the difference is small, and it does not affect the overall law, so it can be considered that the model of this paper has reasonableness, and it can be calculated in the next step.

5. Calculation of the Reverse Fault Tunnel

In this paper, we mainly study the impact of inverse fault misalignment on tunnel lining misalignment performance under different mixed fiber admixture conditions, analyze the stress-strain and damage of lining during fault misalignment according to numerical simulation monitoring results, set up key lining monitoring points shown in Figure 7, monitor the maximum principal stress, longitudinal strain, and shear strain of the lining, including the secondary lining three indicators, and carry out a detailed analysis of the corresponding tensile and compressive injuries.

In this section, the parameters of the addition and L/D ratio of steel fiber and polypropylene fiber are selected as 0.5% (${\rho }_{\mathrm{sf}}$), 30 (${\mathrm{l}}_{\mathrm{sf}}/{\mathrm{d}}_{\mathrm{sf}}$), 0.05% (${\rho }_{\mathrm{pf}}$), 30 (${\mathrm{l}}_{\mathrm{pf}}/{\mathrm{d}}_{\mathrm{pf}}$), and Figure 8 gives the specific numerical variations of the maximum principal stress values, longitudinal strains, shear strains, compressive, and tensile damage variables distributed along the longitudinal direction of the tunnel in the five monitoring points.

From Figure 8a, it can be obtained that the influence range of reverse fault misalignment on the maximum principal stress of the lining is about 87.5 m in the vicinity of the fault zone, and the maximum stress values of the five key parts within the range fluctuate significantly, but the difference between the extremes of the values is not large, in which the extreme values of the arch crown and the arch haunch are relatively large, respectively, 2.79 MPa and 2.74 MPa, while the extreme value of the invert arch is relatively small, 2.22 MPa. At the same time, it can be found that the maximum principal stress extreme values of the arch crown and arch haunch are biased towards the hanging wall, in which the arch crown is subjected to large tensile stress in addition to the hanging wall, and a small area of compressive stress appears in the footwall; the maximum stress values of the arch springing, arch springing, and invert arch are biased towards the footwall, in which the arch springing is opposite to the arch crown and is subjected to a large range of tensile in the footwall, and part of the compressive area appears in the hanging wall.

Figure 8b shows that the longitudinal strain changes are concentrated in the range of 100 m near the reverse fault, and it can more clearly reflect the longitudinal strain of the lining structure in the hanging wall and footwall, which are subjected to different tensile and compressive properties. Among them, the arch crown and arch haunch are tensile in the fault zone near the hanging wall and compressive near the footwall, while the arch sidewall, arch springing, and invert arch are compressive in the fault zone near the hanging wall and tensile near the footwall. Among the several key parts, there are longitudinal tensile and compressive extremes in the arch crown (560 με in tensile and 1220 με in compressive), followed by the invert arch (320 με in tensile and 980 με in compressive), and there is not much difference in the strain extremes of the arch haunch, arch sidewall, and arch springing.

Figure 8c gives the specific value distribution of shear strain at the arch haunch, arch springing, and arch springing. It can be obtained that the fluctuation range of lining shear strain in the three parts is mainly concentrated at 75 m near the misalignment zone, i.e., the lining is obviously influenced by shear stress in this range, and the maximum values of shear strain are close to the fault zone. The maximum shear strain is found at arch springing (9380 με), the minimum at arch haunch (1910 με), and at the invert arch (7790 με).

In Figure 8d, the analysis can be obtained that the secondary lining pressure damage under the reverse fault misalignment is concentrated at 75 m near the fault zone, and the closer to the misalignment zone, the larger the damage value is. The pressure damage extreme value at different locations is different, from big to small order for the arch springing, arch sidewall, arch haunch, arch crown, and invert arch. The maximum value of the arch springing is 0.834, and the minimum value of the invert arch is 0.053, that is, the arch springing under the reverse fault fracture misalignment under the pressure damage is the most serious, and the invert arch under the pressure damage is relatively minor.

Figure 8e can be obtained in the fault fracture zone near the 80 m range of lining tensile damage is serious, away from the broken zone damage gradually decreases, and the arch and arch haunch tensile damage is located in the hanging wall; the other three parts of the damage location tend to be close to the footwall. Comparing the extreme values of tensile damage in five parts, the arch springing is the largest (0.854), the inverted arch is the second largest (0.783), the arch crown and inverted arch are not much different from each other, respectively, 0.501 and 0.371, and the damage to the arch haunch is the smallest (0.188), which can be ignored in comparison.

According to the calculation results, maximum principal stress and longitudinal strain at arch crown, shear strain at arch springing, compressive damage variable, and tensile damage variable at arch springing were selected as the monitoring data in Section 6, and the calculation results of the model were analyzed under different material conditions.

6. Parametric Study

Through numerical simulation, the effect of fiber compound parameters on the tunnel misalignment is calculated. In this section, the results of five selected monitoring points have been calculated and obtained, and based on this analysis, it is carried out to study the connection between the parameters of the fibers and the effect on the performance of the tunnel lining in terms of misalignment resistance.

6.1. Influences of Polypropylene Fiber (${\rho }_{\mathrm{pf}}$)

Figure 9 shows the results of the monitoring points under the condition of different additions of steel fiber (${\rho }_{\mathrm{pf}}$) and L/D ratios of steel fiber (${\mathrm{l}}_{\mathrm{sf}}/{\mathrm{d}}_{\mathrm{sf}}$), and with the gradual increase in the addition of polypropylene fiber (${\rho }_{\mathrm{pf}}$) from 0.05%, 0.10%, to 0.15%.

The analysis shows that as ${\rho }_{\mathrm{pf}}$ increases, the maximum principal stress at the arch crown increases uniformly in the extreme value, and this law does not change with the change of ${\rho }_{\mathrm{sf}}$ and ${\mathrm{l}}_{\mathrm{sf}}/{\mathrm{d}}_{\mathrm{sf}}$, i.e., the increase in ${\rho }_{\mathrm{pf}}$ in this range helps to improve the bearing capacity of the concrete. At the same time, shear strain at arch springing, the extreme value of the compressive damage variable at arch springing, and the extreme value of the tensile damage variable at arch springing all decrease accordingly. Further, Figure 10 shows the magnitude of change in each dependent variable as influenced by the independent variables.

As shown in Figure 10a, as ${\rho }_{\mathrm{sf}}$ or ${\mathrm{l}}_{\mathrm{sf}}/{\mathrm{d}}_{\mathrm{sf}}$ grows, the increase in ${\rho }_{\mathrm{pf}}$ causes a gradual decrease in the increase in the extreme value of the maximum principal stress at the arch crown, i.e., the effect of ${\rho }_{\mathrm{pf}}$ on the stress is related to ${\lambda }_{\mathrm{sf}}$. When ${\lambda }_{\mathrm{sf}}$ is 0.15, the maximum principal stress increases by 30.8% when ${\rho }_{\mathrm{pf}}$ is increased from 0.05% to 0.15%, while when ${\lambda }_{\mathrm{sf}}$ is increased to 1.52, the increase in the extreme value of the principal stress caused by the same change in ${\rho }_{\mathrm{pf}}$ is 9%, and the former is about three times as much as the latter, i.e., when ${\lambda }_{\mathrm{sf}}$ is lower, ${\rho }_{\mathrm{pf}}$ plays a role in increasing the principal stresses, and when ${\lambda }_{\mathrm{sf}}$ is increased, the effect is gradually weakened. This is because when ${\lambda }_{\mathrm{sf}}$ is less than 0.15, the influence of steel fibers is weaker; at this time, polypropylene fibers play a major role in stress enhancement, and with the increase in ${\lambda }_{\mathrm{sf}}$, the enhancement effect of steel fibers is stronger than that of polypropylene fibers, and therefore the influence of ${\rho }_{\mathrm{pf}}$ diminishes, from which it can be deduced that when ${\lambda }_{\mathrm{sf}}$ reaches a certain value, the stress change caused by the continued increase in ${\rho }_{\mathrm{pf}}$ is weak and even produces a negative effect.

Similarly, when ${\lambda }_{\mathrm{sf}}$ is small, the effect of ${\rho }_{\mathrm{pf}}$ on longitudinal and shear strain does not vary significantly with ${\lambda }_{\mathrm{sf}}$, while ${\lambda }_{\mathrm{sf}}$ at larger scales, the magnitude of the reduction diminishes with further increase. For longitudinal strains, when ${\lambda }_{\mathrm{sf}}$ reaches 0.57, the strain reduction corresponding to the increase in ${\rho }_{\mathrm{pf}}$ from 0.05% to 0.15% is 14.9%, whereas when ${\lambda }_{\mathrm{sf}}$ is increased to 1.52, the strain reduction for the same change in ${\rho }_{\mathrm{pf}}$ is 11.6%. This is because the polypropylene fibers act as the main toughening role when the steel fiber characteristics are not obvious, so the structural deformation is significantly affected by ${\rho }_{\mathrm{pf}}$, while the growth role of the polypropylene fibers gradually assumes a secondary role as the characteristics of the steel fibers become apparent, so the magnitude of the change is reduced. However, at this time, the interaction of the two fibers is weakly reflected in the strain relative to the stress. While for shear strains, its weakening effect is relatively slight even when ${\lambda }_{\mathrm{sf}}$ is large, which is because the polypropylene fibers assume the main role of shear resistance when the characteristics of the steel fibers are not obvious, and with the enhancement of the characteristics of the steel fibers, the role of the polypropylene fibers is weakened, and in terms of the magnitude of the reduction, the polypropylene fibers play a relatively limited role in shear resistance in comparison with the maximum principal stresses and the longitudinal strains.

Figure 10d,e, on the other hand, shows that there is no significant pattern in the magnitude of the extreme reduction of the compressive damage variable of the arch springing caused by ${\rho }_{\mathrm{pf}}$ with respect to the steel fiber parameters. It can be obtained that ${\rho }_{\mathrm{pf}}$ has a weaker effect on the tensile damage variable compared to the compressive damage variable, i.e., polypropylene fibers play a more significant role in compressive toughening performance than tensile bridging performance in concrete.

6.2. Influences of L/D Ratio of Steel Fiber (${\mathrm{l}}_{\mathrm{sf}}/{\mathrm{d}}_{\mathrm{sf}}$)

Figure 11 shows the results of the monitoring points, computed with ${\mathrm{l}}_{\mathrm{sf}}/{\mathrm{d}}_{\mathrm{sf}}$ increased from 30 to 80 under different ${\rho }_{\mathrm{sf}}$ and ${\rho }_{\mathrm{pf}}$.

The analysis shows that as ${\mathrm{l}}_{\mathrm{sf}}/{\mathrm{d}}_{\mathrm{sf}}$ increases, the maximum principal stress at the arch crown increases, but the magnitude of the increase is different; the longitudinal strain decreases; the extreme shear strain at arch springing decreases with the increase in ${\mathrm{l}}_{\mathrm{sf}}/{\mathrm{d}}_{\mathrm{sf}}$; the extreme compressive damage variable at arch springing decreases gradually; and the extreme tensile damage variable decreases gradually. Further, Figure 12 shows the magnitude of change in each dependent variable as influenced by the independent variables.

Figure 11a shows that the stress increase caused by the increase in ${\mathrm{l}}_{\mathrm{sf}}/{\mathrm{d}}_{\mathrm{sf}}$ is greater when ${\rho }_{\mathrm{sf}}$ is larger, and ${\rho }_{\mathrm{sf}}$ increases from 0.05% to 0.1% when the growth rate of enhancement is larger, and with the further increase in ${\rho }_{\mathrm{sf}}$, the enhancement is gradually weakened. That is, although the increase in ${\mathrm{l}}_{\mathrm{sf}}/{\mathrm{d}}_{\mathrm{sf}}$ will improve the bearing capacity of concrete, but the corresponding ${\rho }_{\mathrm{sf}}$ is too small will affect the improvement effect, too large may produce side effects, and is not conducive to the actual construction operation, so it is necessary to take a reasonable value; and with the increase in ${\rho }_{\mathrm{pf}}$, the corresponding increase in the magnitude of the slight weakening, that is, there is also an interaction between ${\rho }_{\mathrm{pf}}$ and ${\mathrm{l}}_{\mathrm{sf}}/{\mathrm{d}}_{\mathrm{sf}}$, it is not appropriate to take a larger value at the same time.

Figure 11b shows that the reduction of longitudinal and shear strain increases with the increase in ${\rho }_{\mathrm{sf}}$. For longitudinal strain, the trend of its reduction is basically uniform when ${\rho }_{\mathrm{sf}}$ is 0.5% and ${\mathrm{l}}_{\mathrm{sf}}/{\mathrm{d}}_{\mathrm{sf}}$ is increased from 30 to 60. When ${\rho }_{\mathrm{sf}}$ is increased, its reduction increases significantly when ${\mathrm{l}}_{\mathrm{sf}}/{\mathrm{d}}_{\mathrm{sf}}$ is increased in the small value range (30–60), and then decreases when it is increased in the large value range (60–80), which is similar to the magnitude of maximum principal stress. For shear strain, the reduction is more pronounced at larger ${\rho }_{\mathrm{sf}}$ and this pattern is largely independent of ${\rho }_{\mathrm{pf}}$. In terms of the magnitude change from Figure 12c, when ${\rho }_{\mathrm{sf}}$ increases from 0.5% to 1.2%, the role of ${\mathrm{l}}_{\mathrm{sf}}/{\mathrm{d}}_{\mathrm{sf}}$ changes significantly, and the difference in the effect of ${\mathrm{l}}_{\mathrm{sf}}/{\mathrm{d}}_{\mathrm{sf}}$ becomes weaker with the further increase in ${\rho }_{\mathrm{sf}}$, i.e., when ${\rho }_{\mathrm{sf}}$ varies in a low range, the increase in ${\mathrm{l}}_{\mathrm{sf}}/{\mathrm{d}}_{\mathrm{sf}}$ has a more significant shear strain suppression effect, i.e., for the suppression of shear strain deformation, the two main parameters of the steel fibers still have an interaction.

Figure 12d,e show that the reduction of damage variables is relatively insignificant when ${\rho }_{\mathrm{sf}}$ is low, and when ${\rho }_{\mathrm{sf}}$ increases from 0.5% to 1.2%, the damage suppression effect is significant after increasing ${\mathrm{l}}_{\mathrm{sf}}/{\mathrm{d}}_{\mathrm{sf}}$, and at this time, there are significant changes in the reduction of damage variables at tunnel crown when ${\mathrm{l}}_{\mathrm{sf}}/{\mathrm{d}}_{\mathrm{sf}}$ is increased in a low range interval (30–60). When ${\mathrm{l}}_{\mathrm{sf}}/{\mathrm{d}}_{\mathrm{sf}}$ is increased again, the reduction gets weaker, even causing negative effects, which means steel fiber parameters ${\rho }_{\mathrm{sf}}$ and ${\mathrm{l}}_{\mathrm{sf}}/{\mathrm{d}}_{\mathrm{sf}}$ are also unfavorable for the suppression effect of compression damage when they are too large.

6.3. Influences of Steel Fiber (${\rho }_{\mathrm{sf}}$)

Figure 13 shows the results of the monitoring points, computed with ${\rho }_{\mathrm{sf}}$ increases from 0.5% to 1.9% for different ${\mathrm{l}}_{\mathrm{sf}}/{\mathrm{d}}_{\mathrm{sf}}$ and ${\rho }_{\mathrm{pf}}$.

The analysis shows that as ${\rho }_{\mathrm{sf}}$ increases, the maximum principal stress at the arch crown increases, i.e., an increase in ${\rho }_{\mathrm{sf}}$ helps to increase the bearing capacity of the lining; the longitudinal strain at the arch crown decreases; the extreme shear strain at arch springing decreases; the extreme compressive damage variable at arch springing decreases; and the extreme tensile damage variable at arch springing decreases with an increase in ${\rho }_{\mathrm{sf}}$. Further, Figure 14 shows the magnitude of change in each dependent variable as influenced by the independent variables.

Figure 13a and Figure 14a show that the growth amplitude is more obvious when the value of ${\mathrm{l}}_{\mathrm{sf}}/{\mathrm{d}}_{\mathrm{sf}}$ is larger, but when ${\mathrm{l}}_{\mathrm{sf}}/{\mathrm{d}}_{\mathrm{sf}}$ increases from 30 to 60, the effect of ${\rho }_{\mathrm{sf}}$ is more obvious, and the amplitude change is gradually weakened when ${\mathrm{l}}_{\mathrm{sf}}/{\mathrm{d}}_{\mathrm{sf}}$ is increased again, i.e., although a larger value of ${\rho }_{\mathrm{sf}}$ is favorable for lining load bearing in a certain range, a negative interaction effect may occur when the corresponding ${\mathrm{l}}_{\mathrm{sf}}/{\mathrm{d}}_{\mathrm{sf}}$ is larger as well; the amplitude of the stress increase is weakened as ${\rho }_{\mathrm{pf}}$ increases, i.e., the values of ${\rho }_{\mathrm{pf}}$ and ${\rho }_{\mathrm{sf}}$ should not both be too large.

Figure 13b,c shows that the magnitudes of both compressive and shear strain reduction are greater at larger ${\mathrm{l}}_{\mathrm{sf}}/{\mathrm{d}}_{\mathrm{sf}}$, and when ${\mathrm{l}}_{\mathrm{sf}}/{\mathrm{d}}_{\mathrm{sf}}$ is larger, the effect of ${\rho }_{\mathrm{sf}}$ on the strain at the tunnel crown will be more obvious when it increases in the interval of the small value (0.5~1.2%), that is, when ${\mathrm{l}}_{\mathrm{sf}}/{\mathrm{d}}_{\mathrm{sf}}$ is larger, the ${\rho }_{\mathrm{sf}}$ has a more appropriate value, which is the best effect of strain control, and too small a value won’t achieve more significant effect, too large is not favorable to the economy. The eigenvalues are too significant to produce a negative effect when both ${\rho }_{\mathrm{sf}}$ and ${\mathrm{l}}_{\mathrm{sf}}/{\mathrm{d}}_{\mathrm{sf}}$ are larger, and therefore, it is recommended that ${\mathrm{l}}_{\mathrm{sf}}/{\mathrm{d}}_{\mathrm{sf}}$ be taken to be 60 in the case of general fault resistance.

Figure 13d and Figure 14d show that when ${\mathrm{l}}_{\mathrm{sf}}/{\mathrm{d}}_{\mathrm{sf}}$ is larger, the increase in ${\rho }_{\mathrm{sf}}$ will help the fibers to act as a stronger connection, which can delay the damage to the concrete matrix. And ${\rho }_{\mathrm{sf}}$ increases in the small value interval (0.5–1.2%) causing compressive damage variable reduction is greater, when ${\mathrm{l}}_{\mathrm{sf}}/{\mathrm{d}}_{\mathrm{sf}}$ is taken to be 80, ${\rho }_{\mathrm{sf}}$ is taken to be 1.2% is more reasonable, at this time, the compression damage suppression effect is better and more economical. However, for the tension damage variable, when ${\rho }_{\mathrm{sf}}$ is 0.19% and at the same time ${\mathrm{l}}_{\mathrm{sf}}/{\mathrm{d}}_{\mathrm{sf}}$ is 80, the reduction of the tensile damage variable of the arch springing is 12.2%, instead of decreasing, so it is more reasonable to consider the optimum effect of tensile damage, ${\mathrm{l}}_{\mathrm{sf}}/{\mathrm{d}}_{\mathrm{sf}}$ is taken as 60 when ${\rho }_{\mathrm{sf}}$ is 1.9%.

6.4. Influences of Single Fiber

Figure 15 gives the results of the monitoring points caused by the change of each fiber parameter in a single fiber, respectively.

The results show that in different conditions of each single fiber, due to the high elastic modulus and excellent toughness resistance, it can also help the concrete to carry more external force, and the maximum principal stress of the arch crown will also increase with the increase in the fiber addition or the fiber L/D ratio. There is also an interaction between ${\mathrm{l}}_{\mathrm{sf}}/{\mathrm{d}}_{\mathrm{sf}}$ and ${\rho }_{\mathrm{sf}}$, i.e., the magnitude of the stress change caused by ${\rho }_{\mathrm{sf}}$ (${\mathrm{l}}_{\mathrm{sf}}/{\mathrm{d}}_{\mathrm{sf}}$) is more significant when ${\mathrm{l}}_{\mathrm{sf}}/{\mathrm{d}}_{\mathrm{sf}}$ (${\rho }_{\mathrm{sf}}$) is larger. The effect on the longitudinal strain of the lining is not much different from that of compound doping, and the maximum value of longitudinal strain at the arch crown decreases with the increase in the single-doped fiber doping or the aspect ratio, and there is an interaction between ${\mathrm{l}}_{\mathrm{sf}}/{\mathrm{d}}_{\mathrm{sf}}$ and ${\rho }_{\mathrm{sf}}$. In the range set in this study, the increase in two kinds of fiber doping and L/D ratio is beneficial to slow down the longitudinal deformation of the top of the lining, but it is necessary to consider the deformation suppression effect and economic benefits.

Shear strain at arch springing will be reduced, but at the same time, can be found at this time ${\rho }_{\mathrm{sf}}$ and ${\mathrm{l}}_{\mathrm{sf}}/{\mathrm{d}}_{\mathrm{sf}}$ between the interaction effect is still obvious, that is, the two major characteristics of the steel fiber in the shear resistance of the single mixing parameters still have interaction.

The extreme value of the foot of arch compressive damage variable also decreases gradually. And there is an interaction between ${\rho }_{\mathrm{sf}}$ and ${\mathrm{l}}_{\mathrm{sf}}/{\mathrm{d}}_{\mathrm{sf}}$, when ${\mathrm{l}}_{\mathrm{sf}}/{\mathrm{d}}_{\mathrm{sf}}$ or ${\rho }_{\mathrm{sf}}$ is larger, the reduction of the foot of arch compressive damage variable with the increase in ${\rho }_{\mathrm{sf}}$ or ${\mathrm{l}}_{\mathrm{sf}}/{\mathrm{d}}_{\mathrm{sf}}$ corresponds to more obvious.

The extreme values of the tensile damage variable at invert arch are all reduced, and the magnitude of the change caused by the increase in ${\rho }_{\mathrm{sf}}$ (${\mathrm{l}}_{\mathrm{sf}}/{\mathrm{d}}_{\mathrm{sf}}$) is larger when ${\mathrm{l}}_{\mathrm{sf}}/{\mathrm{d}}_{\mathrm{sf}}$ (${\rho }_{\mathrm{sf}}$) is larger, i.e., there is still an interplay between ${\mathrm{l}}_{\mathrm{sf}}/{\mathrm{d}}_{\mathrm{sf}}$ and ${\rho }_{\mathrm{sf}}$ for the tensile damage.

7. Discussion

7.1. Influences of Fibers Compound

The properties of the materials are compared by considering single fiber, mixed fiber, and no fiber. Figure 16 gives the variation of maximum principal stress and longitudinal strain extremes at the arch crown, shear strains at the arch sidewall and arch springing, and compressive and tensile damage variables in the invert arch for different conditions.

The maximum principal stress extreme value at the arch crown is obviously larger than that of plain concrete after mixed fiber concrete, and the size of the extreme value always meets mixed fiber > single fiber > plain concrete, i.e., there is a synergistic effect of the two kinds of fibers in the matrix concrete, and it is more beneficial to improve the load-bearing capacity of the matrix concrete. The effect of the two single-doped fibers is related to the ${\lambda }_{\mathrm{sf}}$, and when the ${\lambda }_{\mathrm{sf}}$ < 0.15, the maximum value of the single polypropylene fiber is larger than single steel fiber. But when ${\lambda }_{\mathrm{sf}}$ further increase, the maximum value of single steel fiber is greater than that of single polypropylene fiber, and with continuous growth of ${\lambda }_{\mathrm{sf}}$, the difference between the two single-doped effect is significant. This is because the steel fiber characteristics are not obvious, polypropylene fiber plays a major bridging role, and ${\lambda }_{\mathrm{sf}}$ reaches a certain value, the steel fiber plays a stronger role, from which we can put forward a limit of the characteristics of the value of ${\lambda }_{\mathrm{sf}\text{-}1}$. In this case, the boundary eigenvalue ${\lambda }_{\mathrm{sf}\text{-}1}$ for distinguishing between single steel fiber and single polypropylene fiber for the strength of the stress effect is 0.15.

Due to the enhanced performance of mixed fiber, the extreme value of the longitudinal strain at the arch crown of fiber-enhanced concrete is much smaller than that of plain concrete, and the strain of mixed fiber is also smaller than that of single fiber, which indicates that the mixed fiber will inhibit the longitudinal deformation at the arch crown under the fault fracture misalignment, and the effect of compound admixture is better than that of single admixture. For the comparison of the effects of the two single-doped fiber strain suppressions, when ${\lambda }_{\mathrm{sf}}$ < 0.36, the single polypropylene fiber effect is better, with the increase in ${\lambda }_{\mathrm{sf}}$, the single steel fiber effect is gradually better. That is, for the inhibition effect of longitudinal strain, the boundary eigenvalue ${\lambda }_{\mathrm{sf}\text{-}1}$ of the two single-doped fibers is 0.36.

The shear strain of arch springing and invert arch is obviously smaller than that of plain concrete after mixed fiber, and the inhibition effect of mixed doped will be better than single doped, for the comparison of two single doped fibers, arch springing in ${\lambda }_{\mathrm{sf}}$ ≤ 0.36, single steel fiber inhibition effect is not better than single polypropylene fiber, and invert arch in ${\lambda }_{\mathrm{sf}}$ > 0.15, the effect of single steel fiber inhibition of shear strain will be better than single polypropylene fiber gradually, i.e., for shear strain, arch springing and invert arch exist different boundary characteristic value ${\lambda }_{\mathrm{sf}\text{-}1}$, it can be obtained that invert arch has a higher requirement for ${\lambda }_{\mathrm{sf}}$, so it is more sensitive than other parts. At the invert arch, there are different boundary characteristic value of ${\lambda }_{\mathrm{sf}\text{-}1}$, it can be obtained that invert arch for ${\lambda }_{\mathrm{sf}}$ requirements are higher, so more sensitive than other parts. Considering the economic effect, different part can be designed with different ratios. After analyzing, it can be obtained that the relationship between fiber parameters and shear strain at the invert arch and arch springing is not much different in mixed fiber and single fiber.

The extreme value of the compressive damage variable at the invert arch of mixed fiber is smaller than that of plain concrete, and the compressive damage inhibition effect of mixed fibers is much better than that of single fibers. For two different single-doped cases, when ${\lambda }_{\mathrm{sf}}$ < 0.15, the single polypropylene fiber effect is better, with ${\lambda }_{\mathrm{sf}}$ gradually increased, the effect of single steel fiber is gradually better, and when ${\lambda }_{\mathrm{sf}}$ increased again, this effect is more significant; that is, the eigenvalue ${\lambda }_{\mathrm{sf}\text{-}1}$ is 0.15 for the pressure damage inhibition effect. In the comparison of single polypropylene fiber and mixed fiber, it can be concluded that the damage inhibition effect of single polypropylene fiber is weak, and single polypropylene fiber is not recommended in general.

The tensile damage at arch springing after adding fibers is significantly relieved, and the effect of mixed fiber is better than single doping, but when ${\lambda }_{\mathrm{sf}}$ < 0.15, the inhibition effect of single polypropylene fibers is better than single steel fiber, and with the gradual increase in ${\lambda }_{\mathrm{sf}}$, the effect of single steel fibers gradually becomes better, i.e., for the inhibition of tensile damage, the boundary characteristic value ${\lambda }_{\mathrm{sf}\text{-}1}$ is 0.15. The effect of single polypropylene fiber to suppress tensile damage is poorer, and therefore it is not recommended to dope polypropylene fiber individually.

7.2. Correlation Analysis

Through the above analysis, it can be obtained that ${\rho }_{\mathrm{sf}}$, ${\mathrm{l}}_{\mathrm{sf}}/{\mathrm{d}}_{\mathrm{sf}}$ and ${\rho }_{\mathrm{pf}}$ have mutual influence on the extreme value of the maximum principal stress of the lining, and in order to further analyze the degree of influence of these three parameters and four combined variables (a total of seven variables), the Spearman’s correlation coefficient analysis in spss was used to obtain Figure 17. The theoretical distribution interval of the correlation coefficients in the table is [−1, 1], and in general, it is considered to have weak correlation in the range of 0.3~0.5, medium correlation in the range of 0.5~0.7, and strong correlation in the range of 0.7~0.9.

From Figure 17, it can be significantly observed that seven variables have a significant correlation with maximum principal stress and maximum longitudinal strain at arch crown, maximum shear strain at arch springing, compressive damage variable at arch springing, and tensile damage variable at arch springing. Among them, the four combined parameters are more relevant to the results than the three initial parameters.

For the maximum principal stress and longitudinal strain at the arch crown, the most significant influence is ${\rho }_{\mathrm{pf}}$·${\rho }_{\mathrm{sf}}$·${\mathrm{l}}_{\mathrm{sf}}/{\mathrm{d}}_{\mathrm{sf}}$, which indicates that the combined performance of steel-polypropylene fiber can significantly improve the bearing performance of the lining top.

For shear strain at arch sidewall, it can be found that the correlation coefficient of ${\rho }_{\mathrm{pf}}$ is the lowest, and lower relative to several other variables, i.e., considering ${\rho }_{\mathrm{pf}}$ alone has less influence on the shear strain; at the same time, it can be found that several variables containing a factor of ${\rho }_{\mathrm{sf}}$ have a higher correlation with the shear strain, and the ones that consider ${\lambda }_{\mathrm{sf}}$ and those that consider the combination of the three variables have a high degree of correlation with it, so it can be assumed that ${\rho }_{\mathrm{sf}}$ has a determinative effect on the lining shear resistance.

For the compressive damage variable and tensile damage variable at arch springing, the most important influencing factors are ${\rho }_{\mathrm{pf}}$·${\rho }_{\mathrm{sf}}$·${\mathrm{l}}_{\mathrm{sf}}/{\mathrm{d}}_{\mathrm{sf}}$ and ${\rho }_{\mathrm{sf}}$·${\mathrm{l}}_{\mathrm{sf}}/{\mathrm{d}}_{\mathrm{sf}}$, respectively, which are strongly correlated. In the compressive damage variable, ${\mathrm{l}}_{\mathrm{sf}}/{\mathrm{d}}_{\mathrm{sf}}$ and ${\rho }_{\mathrm{pf}}$ showed weak correlation, ${\rho }_{\mathrm{sf}}$ and ${\rho }_{\mathrm{pf}}$·${\mathrm{l}}_{\mathrm{sf}}/{\mathrm{d}}_{\mathrm{sf}}$ showed moderate correlation, which also indicates that steel-polypropylene mixed fiber is more effective in resisting compressive damage. In the tensile damage variable, ${\mathrm{l}}_{\mathrm{sf}}/{\mathrm{d}}_{\mathrm{sf}}$, ${\rho }_{\mathrm{sf}}$, ${\rho }_{\mathrm{pf}}$·${\mathrm{l}}_{\mathrm{sf}}/{\mathrm{d}}_{\mathrm{sf}}$ and ${\rho }_{\mathrm{sf}}$·${\rho }_{\mathrm{pf}}$ are all medium correlation, while ${\rho }_{\mathrm{pf}}$·${\rho }_{\mathrm{sf}}$·${\mathrm{l}}_{\mathrm{sf}}/{\mathrm{d}}_{\mathrm{sf}}$ correlation is only the second one, which explains that in the tensile damage, the steel fibers are more important than polypropylene fibers, and in order to inhibit the tensile damage, it is necessary to reasonably select the steel fiber parameters.

8. Conclusions

In this paper, steel-polypropylene mixed fibers are applied to tunnel lining the mechanical properties of the secondary lining of the tunnel were analyzed in various fiber parameters when the inclination angle is 60°, the displacement of vertical misalignment is 30 cm, and the distribution of the damages by five indicators, and the main conclusions obtained are as follows:

(1)

The fluctuation range of the secondary lining caused by reverse fault misalignment is concentrated within 80 m near the fault zone, where the steel-polypropylene mixed fibers can be used to focus on the reinforcement. The misalignment causes tensile at the arch crown and invert arch, as well as significant shear at the arch sidewall. Fiber parameters do not affect the basic distribution law of the mechanical properties of the lining, but the corresponding index extremes are closely correlated with the three parameters of ${\rho }_{\mathrm{pf}}$, ${\rho }_{\mathrm{sf}}$ and ${\mathrm{l}}_{\mathrm{sf}}/{\mathrm{d}}_{\mathrm{sf}}$.

(2)

With the increase in ${\rho }_{\mathrm{pf}}$, the stress, strain, and damage indicators decrease, and the law is more significant when ${\lambda }_{\mathrm{sf}}$ is small, where the characteristics of the steel fibers are relatively inconspicuous, and the polypropylene fibers play a major role, and assume stress-enhancing, deformation-inhibiting, and damage-weakening effect, and as ${\lambda }_{\mathrm{sf}}$ increases, the effect produced by ${\rho }_{\mathrm{pf}}$ change is weakened, even may produce a negative effect. In addition, the effect of ${\rho }_{\mathrm{pf}}$ is relatively weak for shear and inhibit tensile damage, it is recommended that in a reasonable range, ${\rho }_{\mathrm{pf}}$ to take a small value, ${\lambda }_{\mathrm{sf}}$ to take a relatively large value.

(3)

With the increase in ${\rho }_{\mathrm{sf}}$ or ${\mathrm{l}}_{\mathrm{sf}}/{\mathrm{d}}_{\mathrm{sf}}$, stress indicator gradually increases, strain factors gradually decrease, and damage indicators also decrease, and there is an interaction between ${\rho }_{\mathrm{sf}}$ and ${\mathrm{l}}_{\mathrm{sf}}/{\mathrm{d}}_{\mathrm{sf}}$, it is recommended ${\rho }_{\mathrm{sf}}$ takes 1.9% when ${\mathrm{l}}_{\mathrm{sf}}/{\mathrm{d}}_{\mathrm{sf}}$ takes 60, and ${\rho }_{\mathrm{sf}}$ takes 1.2%, when ${\mathrm{l}}_{\mathrm{sf}}/{\mathrm{d}}_{\mathrm{sf}}$ takes 80. For the serious tensile damage near fault zone, both ${\mathrm{l}}_{\mathrm{sf}}/{\mathrm{d}}_{\mathrm{sf}}$ and ${\rho }_{\mathrm{sf}}$ can be taken as larger values.

(4)

Fibers play an important role in anti-fault performance, and mixed fibers have with better effect than single fiber, and there existed a characteristic value ${\lambda }_{\mathrm{sf}\text{-}1}$ for the two fibers in single doped, less than ${\lambda }_{\mathrm{sf}\text{-}1}$, the effect of single polypropylene fiber is better, more than ${\lambda }_{\mathrm{sf}\text{-}1}$, the other become better. It is not recommended to use single polypropylene fiber for its weak shear deformation and damage inhibition effects. The value of ${\lambda }_{\mathrm{sf}\text{-}1}$ varies at different part, therefore, the ratio of each part can be designed according to actual engineering requirements.