Analytical Solution for Longitudinal Response of Tunnel Structures Under Strike-Slip Fault Dislocation Considering Tangential Soil–Tunnel Contact Effect and Fault Width

Abstract

The existence of fault zones in high-intensity earthquake areas has a serious impact on engineering structures, and the longitudinal response of tunnels crossing faults needs further in-depth research. To analyze the tangential contact effect between the surrounding rock and the tunnel lining, and the axial deformation characteristics of the tunnel structure, tangential foundation springs were introduced and a theoretical model for the longitudinal response of the tunnel under fault dislocation was established. Firstly, the tunnel was simplified as a finite-length beam. The normal and tangential springs were taken to represent the interaction between the soil and the lining. The fault’s free-field displacement was applied at the end of the normal foundation spring to simulate fault dislocation, and the differential equation for the longitudinal response of the tunnel structure was obtained. The analytical solution of the structural response was obtained using the Green’s function method. Then, the three-dimensional finite difference method was used to verify the effectiveness of the analytical model in this paper. The results show that the tangential contact effect between the surrounding rock and the lining has a significant impact on the longitudinal response of the tunnel structure. Ignoring this effect leads to an error of up to 35.33% in the peak value of the structural bending moment. Finally, the influences of the width of the fault zone, the soil stiffness of the fault zone, and the stiffness of the tunnel lining on the longitudinal response of the tunnel were explored. As the fault width increases, the internal force of the tunnel structure decreases. Increasing the lining concrete grade leads to an increase in the internal force of the structure. The increase in the elastic modulus of the surrounding rock in the fault area reduces the bending moment and shear force of the structure and increases the axial force. The research results can provide a theoretical basis for the anti-dislocation design of tunnels crossing faults.

1. Introduction

The western region of China is characterized by complex geological structures, active tectonic movements, and densely distributed active faults. Tunnel sites in this area are frequently subjected to high-intensity earthquakes [1,2], marked by their large magnitude, shallow epicenters, wide distribution, sudden occurrence, and severe seismic hazards. With the construction of major transportation projects and the Central Asia Natural Gas Pipeline in western China, engineering routes inevitably traverse large active fault zones in high seismic intensity areas. Most projects prioritize tunnel engineering solutions, aligning tunnels with the passive structural plates of fault zones or intersecting faults at large angles [3,4].

While tunnels are generally considered resilient to seismic activity, those crossing adverse geological zones such as active faults often suffer severe structural damage under the coupled effects of fault-induced deformation and strong earthquakes [5,6,7,8]. Investigating the mechanisms of fault dislocation on tunnel structures and enhancing the anti-dislocation capacity of long tunnels crossing active fault zones are critical for the construction of tunnels in seismically active and geologically complex mountainous regions [9].

Numerous domestic and international scholars have explored tunnel damage mechanisms under fault dislocation and seismic coupling using numerical simulations, model tests, and theoretical analyses. Xiong [10] conducted parametric sensitivity analyses via numerical simulations, identifying fault displacement, dip angle, tunnel depth, and the tunnel–fault intersection angle as key factors influencing structural damage (in descending order). Lin [11] used ABAQUS to study the effect of fault width on the dynamic response of tunnels crossing strike-slip faults. Wang [12] employed FLAC3D to analyze the forced responses and key parameters of tunnels under strike-slip faulting, validated by model tests. The results indicated that lining structures near fault–upper/–lower plate interfaces are prone to shear cracks. Xiao [13] designed strike-slip fault simulation devices for model tests, revealing the structural responses of tunnels with varying stiffnesses. Zhou [14] conducted anti-dislocation model tests on the Xianglushan Tunnel (of the Yunnan Water Diversion Project in China), analyzing rock fracturing and lining damage during fault movement.

Though model tests validate numerical and theoretical analyses, they are resource-intensive. Numerical simulations face the challenges of computational efficiency and accuracy. Theoretical analytical methods, which avoid these issues, play an irreplaceable role in understanding anti-dislocation mechanisms and practical engineering applications.

Jalali [15] proposed a displacement pattern-based method for articulated tunnel design across faults, providing theoretical guidance for the Karaj Tunnel. Sun [16] derived formulas for minimum segment numbers and maximum allowable widths in anti-dislocation design, assuming linear relative rotation at deformation joints. Yu et al. [17] developed an analytical solution for the longitudinal dynamic response of tunnels crossing soft rock–hard rock interfaces using elastic foundation beam theory. Peng Jiaming transformed fault dislocation into equivalent loads on foundation beams to derive deformation and internal force responses. Liu et al. [18] introduced the Pasternak two-parameter foundation model to account for shear stiffness effects. Tao et al. [19] applied complementary error functions to derive analytical solutions for underground pipelines under normal faulting. Chang et al. [20] developed a simplified longitudinal beam and spring tunnel model to assess the longitudinal response of tunnel structures, incorporating splaying and staggering deformations at flexible joints. Zhao et al. [21] proposed a mechanism and analysis method of a double-layer pre-support system for tunnels passing under large underground pipe galleries in water-rich sandy strata based on beam theory. Existing elastic foundation beam models simplify surrounding rock as independent normal springs (Figure 1a), ignoring tangential rock-lining interactions (Figure 1b), which significantly affect longitudinal responses. Under the action of external loads, the tunnel structure bends and undergoes tensile (compressive) deformation simultaneously, and horizontal displacements will occur at each point of the tunnel. These factors can all lead to serious deviations in the calculation results of the theoretical analytical methods.

Axial tunnel deformation under fault dislocation leads to horizontal displacements during bending. The omission of which results in substantial errors in theoretical predictions.

To address these limitations, this paper introduces tangential foundation springs to model rock–lining interactions and axial deformation. A theoretical model for a tunnel’s longitudinal responses under fault dislocation was established, enabling accurate analytical solutions via Green’s function. Key contributions include: (1) The integration of tangential contact effects into elastic foundation beam theory. (2) Validation through 3D finite difference modeling. (3) The quantitative analysis of parameters (fault width, elastic modulus, lining concrete grade) on structural responses.

2. Theoretical Modeling

This study employs elastic foundation beam theory to simulate the longitudinal response of tunnels crossing fault zones under strike-slip fault dislocation. Tangential foundation springs are introduced to account for the tangential contact effects between the surrounding rock and the lining, as well as the axial deformation of the tunnel structure. By adjusting the foundation spring coefficients to simulate the distinct geomechanical properties of the fault zone, fixed plate, and active plate, a theoretical analytical model for the longitudinal response of a tunnel under fault dislocation is established. The Green’s function method is applied to derive the longitudinal response of the tunnel structure subjected to fault-induced forced displacement. The schematic configuration of the theoretical model for tunnels crossing fault zones is illustrated in Figure 2.

2.1. Analytical Model for Longitudinal Response of Tunnels Crossing Fault Zones

To derive the analytical solution for the longitudinal response of tunnel structures under fault dislocation, this study proposes a simplified model using an elastic foundation beam to simulate the interaction between the surrounding rock and the tunnel structure, as shown in Figure 3.

The tunnel structure and surrounding rock are divided into three zones, which are the fault zone, the fixed block, and the moving block. The tunnel has a circular cross-section with constant outer radius $R$ and inner radius $r$. The bending stiffness of the tunnel is $E_{l}I$, where $E_l$ is the elastic modulus of the lining and $I$ is the moment of inertia of the cross-section. The surrounding rock is simplified into normal foundation springs and tangential foundation springs to simulate rock–lining interactions. The fault’s free-field displacement is applied at the end of the normal springs to account for fault dislocation effects.

Normal full-space foundation springs are adopted to simulate the normal interaction between the lining and the surrounding rock. According to Yu et al. [22], the stiffness of the full-space surrounding rock normal foundation springs is expressed by Equation (1):

$$K_h=\sqrt{2}{[{\displaystyle \frac{8{(1-v)}^2}{3-4\upsilon }}]}^{1.125}{\displaystyle \frac{E}{1-{\upsilon }^2}}\sqrt[8]{{\displaystyle \frac{E{b}^4}{E_{l}I}}}$$

(1)

where $E$ is the elastic modulus of the soil, $\nu$ is the Poisson’s ratio of the soil, and $b$ is the height of the tunnel ($b=2R$ for strike-slip faults).

Shear stress exists at the soil–tunnel interface, and the tangential contact lies between no slip and full slip [23,24,25]. By introducing a tangential foundation spring to simulate the contact effect between the surrounding rock and the tunnel, and to account for the effect of the vertical deflection caused by external forces on the axial deformation of the lining, the stiffness of the tangential foundation spring is derived as follows, based on Yu et al. [26]:

$$K_p=\xi K_h$$

(2)

where $\xi$ is the axial stiffness coefficient of the stratum and represents the normal foundational spring stiffness of the soil.

Assuming the spring’s reaction force (i.e., the tangential foundational reaction force) is proportional to the tangential relative displacement between the beam base and foundation, the normal reaction force $q_z$ and tangential foundational reaction force $q_x$ are expressed as:

$$\left\{\begin{array}{c}{q}_z=K_{h}w\left(x\right)\\ q_x=-K_p(u\left(x\right)-R{\displaystyle \frac{dw\left(x\right)}{dx}})\end{array}\right.$$

(3)

where $w\left(x\right)$ and $u\left(x\right)$ represent the tunnel displacement components in the z-direction (vertical) and x-direction (axial) along the beam axis, respectively; $R$ is the radius of the tunnel structure’s cross-section.

2.2. Governing Equations of Beam Element

The mechanical analysis is conducted on a differential beam element of length $dx$, as depicted in Figure 4.

Based on the force equilibrium of the differential beam, the following equations can be derived:

$$\left\{\begin{array}{c}{\displaystyle \frac{dQ\left(x\right)}{dx}}=q_z-p_z(x)\\ {\displaystyle \frac{dN\left(x\right)}{dx}}=q_x\\ {\displaystyle \frac{dM\left(x\right)}{dx}}=Q\left(x\right)+R{q}_x\end{array}\right.$$

(4)

We can assume that the horizontal axial force acts through the centroid of the beam’s cross-section (i.e., no additional bending moment is induced by the axial force), the axial force N is expressed as:

$$N=(EA){\displaystyle \frac{du}{dx}}$$

(5)

where $A$ is the cross-sectional area of the beam and $EA$ denotes the equivalent axial stiffness of the tunnel.

According to the constitutive equation of the Winkler elastic beam, the relationship between the deformation of the beam section and the bending moment is governed by Equation (6):

$$M\left(x\right)=-EI{\displaystyle \frac{d^{2}w}{d{x}^2}}$$

(6)

where $M$ is the bending moment of the beam, $I$ is the inertia moment of the tunnel cross-section, and $EI$ represents the longitudinal equivalent flexural rigidity of the tunnel structure.

By coupling Equations (3)–(6), the following system can be derived:

$$u^{\left(2\right)}-{\displaystyle \frac{K_p}{EA}}u+{\displaystyle \frac{K_{p}R}{EA}}{w}^{\left(1\right)}=0$$

(7)

$$w^{\left(4\right)}-{\displaystyle \frac{K_p{R}^2}{EI}}{w}^{\left(2\right)}+{\displaystyle \frac{K_h}{EI}}w+{\displaystyle \frac{K_{h}R}{EI}}{u}^{\left(1\right)}={\displaystyle \frac{p_z\left(x\right)}{EI}}$$

(8)

By performing a secondary decoupling of Equations (7) and (8), the following equations are obtained:

$$w^{\left(6\right)}+X_1{w}^{\left(4\right)}+X_2{w}^{\left(2\right)}+X_{3}w={\displaystyle \frac{{p_z}^{\left(2\right)}}{EI}}+{\displaystyle \frac{p_z{K}_p}{EI·EA}}$$

(9)

where $X_1$, $X_1$, and $X_3$ are constant coefficients, with their specific values given as follows:

$$\left\{\begin{array}{c}{X}_1={\displaystyle \frac{K_p}{EA}}+{\displaystyle \frac{K_p{R}^2}{EI}}\hfill \\ X_2={\displaystyle \frac{K_h}{EI}}-{\displaystyle \frac{{K_p}^2{R}^2}{EA·EI}}+{\displaystyle \frac{K_p{K}_h{R}^2}{EA·EI}}\hfill \\ X_3={\displaystyle \frac{K_p{K}_h{R}^2}{EA·EI}}\hfill \end{array}\right.$$

(10)

2.3. Green’s Function Method

By introducing the Dirac delta function $\delta \left(x\right)$ [22,26] to calculate the response of the beam at any position under a unit concentrated force, Equation (9) can be transformed into:

$$w^{\left(6\right)}+X_1{w}^{\left(4\right)}+X_2{w}^{\left(2\right)}+X_{3}w=X_4{\delta (x-x_0)}^{\left(2\right)}+X_5\delta (x-x_0)$$

(11)

where $X_4$ and $X_5$ are constant coefficients, with their specific values given as follows:

$$\left\{\begin{array}{c}{X}_4={\displaystyle \frac{1}{EI}}\\ X_5={\displaystyle \frac{K_p}{EI·EA}}\end{array}\right.$$

(12)

By applying the Laplace transform to Equation (11) separately, the following equations are obtained:

$$\begin{gathered} left=(s^{6}L\left(w\right)-s^{5}w\left(0\right)-s^4{w}^1\left(0\right)-s^3{w}^2\left(0\right)-s^2{w}^3\left(0\right)-s{w}^4\left(0\right)-w^5\left(0\right) \\ +X_1\left(s^{4}L\left(w\right)-s^{3}w\left(0\right)-s^2{w}^1\left(0\right)-s{w}^2\left(0\right)-w^3\left(0\right)\right)+X_2(s^{2}L\left(w\right)-sw\left(0\right)-w^1\left(0\right))+X_3 \end{gathered}$$

(13)

$$right=X_4{s}^2{e}^{-s{x}_0}+X_5{e}^{-s{x}_0}$$

(14)

In the equation, $w\left(0\right)$, $w^1\left(0\right)$, $w^2\left(0\right)$, $w^3\left(0\right)$, $w^4\left(0\right)$, and $w^5\left(0\right)$ are constants determined by the boundary conditions of the tunnel structure.

By solving Equations (13) and (14) simultaneously and combining like terms, the Laplace-transformed structural displacement function is derived as follows:

$$\widehat{w}\left(s,x_0\right)=L\left[w\left(x\right)\right]={\displaystyle \frac{{\alpha }_1+{\alpha }_2+{\alpha }_3+{\alpha }_4+{\alpha }_5+{\alpha }_6+{\alpha }_7}{s^6+X_1{s}^4+X_2{s}^2+X_3}}$$

(15)

where the parameter α is defined as follows:

$$\left\{\begin{array}{c}{\alpha }_1=(X_4{s}^2+X_5)e^{-s{x}_0}\\ {\alpha }_2={(s}^5+X_1{s}^3+X_{2}s)W(0)\\ \begin{array}{c}{{\alpha }_3=(s}^4+X_1{s}^2+X_2)W^1(0)\\ \begin{array}{c}{{\alpha }_4=(s}^3+X_{1}s)W^2(0)\\ {\alpha }_5={(s}^2+X_1)W^3(0)\\ {\alpha }_6=s{W}^4(0)\end{array}\\ {\alpha }_7=W^5(0)\end{array}\end{array}\right.$$

(16)

Let $s_1$, $s_2$, $s_3$, $s_4$, $s_5$, and $s_6$ be the six roots of the equation $s^6+X_1{s}^4+X_2{s}^2+X_3=0$. Define new parameters $A_i$ as follows:

$$\begin{gathered} A_1\left(x\right)={\displaystyle \frac{e^{s_{1}x}}{\left(s_1-s_2\right)\left(s_1-s_3\right)\left(s_1-s_4\right)\left(s_1-s_5\right)\left(s_1-s_6\right)}} \\ {A}_2\left(x\right)={\displaystyle \frac{e^{s_{2}x}}{\left(s_2-s_1\right)\left(s_2-s_3\right)\left(s_2-s_4\right)\left(s_2-s_5\right)\left(s_2-s_6\right)}} \\ {A}_3\left(x\right)={\displaystyle \frac{e^{s_{3}x}}{\left(s_3-s_1\right)\left(s_3-s_2\right)\left(s_3-s_4\right)\left(s_3-s_5\right)\left(s_3-s_6\right)}} \\ {A}_4\left(x\right)={\displaystyle \frac{e^{s_{4}x}}{\left(s_4-s_1\right)\left(s_4-s_2\right)\left(s_4-s_3\right)\left(s_4-s_5\right)\left(s_4-s_6\right)}} \\ {A}_5\left(x\right)={\displaystyle \frac{e^{s_{5}x}}{\left(s_5-s_1\right)\left(s_5-s_2\right)\left(s_5-s_3\right)\left(s_5-s_4\right)\left(s_5-s_6\right)}} \\ {A}_6\left(x\right)={\displaystyle \frac{e^{s_{6}x}}{\left(s_6-s_1\right)\left(s_6-s_2\right)\left(s_6-s_3\right)\left(s_6-s_4\right)\left(s_6-s_5\right)}} \end{gathered}$$

(17)

By performing the inverse Laplace transform on Equation (15), the expression of Green’s function for the displacement response of the tunnel structure under a concentrated force at any position can be derived as follows:

$$\begin{gathered} G\left(x,x_0\right)=L^{-1}\left\{\widehat{w}\left(s,x_0\right)\right\}=H\left(x-x_0\right){\phi }_1\left(x-x_0\right)+{\phi }_2\left(x-x_0\right)W\left(0\right)+{\phi }_3\left(x-x_0\right)W^1\left(0\right) \\ +{\phi }_4\left(x-x_0\right)W^2\left(0\right)+{\phi }_5\left(x-x_0\right)W^3\left(0\right)+{\phi }_6\left(x-x_0\right)W^4\left(0\right)+{\phi }_7\left(x-x_0\right)W^5\left(0\right) \end{gathered}$$

(18)

In the equation, $H\left(x\right)$ is the Heaviside step function, defined as the integral of the Dirac delta function $\delta \left(x\right)$ over the real number domain. Its explicit expression is:

$$H\left(x\right)=\left\{\begin{array}{c}1,x\ge 0\\ 0,x<0\end{array}\right.$$

(19)

The parameter ${\phi }_i$ satisfies the following condition:

$${\phi }_i(x)={\sum }_{j=1}^6{\alpha }_i{A}_j\left(x\right)$$

(20)

2.4. Calculation of Response of Tunnel Under Fault Dislocation

By integrating Green’s functions of the displacement responses under concentrated forces [22,26], the vertical dynamic response expression for any point on a tunnel structure resting on an elastic foundation subjected to arbitrary external loads is derived as follows:

$$W(x)={\int }_0^{L}u\left(x_0\right)G(x,x_0)d{x}_0$$

(21)

where $u\left(x_0\right)$ denotes the external load acting at any position $x_0$ on the beam. In this study, it specifically represents the fault dislocation displacement imposed on the tunnel structure.

The analytical expression for the displacement response of the tunnel structure under fault dislocation is given by Equation (22):

$$W\left(x\right)=\left\{\begin{array}{c}{W}_1\left(x\right),x\in (-\infty ,-{\displaystyle \frac{L}{2}}]\\ W_2\left(x\right),x\in (-{\displaystyle \frac{L}{2}},+{\displaystyle \frac{L}{2}}]\\ W_3\left(x\right),x\in (+{\displaystyle \frac{L}{2}},+\infty )\end{array}\right.$$

(22)

where $-{\displaystyle \frac{L}{2}}$ represents the coordinate of the interface between the fixed block and the fault zone, while ${\displaystyle \frac{L}{2}}$ denotes the coordinate of the interface between the active block and the fault zone.

The longitudinal response expression for the single-segment beam structure contains 6 unknowns, resulting in 18 unknowns in Equation (22). These unknowns can be solved using the continuity conditions and boundary conditions of the tunnel structure. At the interfaces between the fault zone and the fixed/active blocks, the deformations and internal forces of the tunnel structure satisfy the continuity conditions, as expressed in Equation (23):

$$\left\{\begin{array}{c}{W}_1\left(-{\displaystyle \frac{L}{2}}\right)=W_2\left(-{\displaystyle \frac{L}{2}}\right)\\ U_1\left(-{\displaystyle \frac{L}{2}}\right)=U_2\left(-{\displaystyle \frac{L}{2}}\right)\\ \begin{array}{c}{\theta }_1\left(-{\displaystyle \frac{L}{2}}\right)={\theta }_2\left(-{\displaystyle \frac{L}{2}}\right)\\ \begin{array}{c}{M}_1\left(-{\displaystyle \frac{L}{2}}\right)=M_2\left(-{\displaystyle \frac{L}{2}}\right)\\ Q_1\left(-{\displaystyle \frac{L}{2}}\right)=Q_2\left(-{\displaystyle \frac{L}{2}}\right)\end{array}\\ N_1\left(-{\displaystyle \frac{L}{2}}\right)=N_2\left(-{\displaystyle \frac{L}{2}}\right)\end{array}\end{array}\right.,\left\{\begin{array}{c}{W}_2\left(+{\displaystyle \frac{L}{2}}\right)=W_3\left(+{\displaystyle \frac{L}{2}}\right)\\ U_2\left(+{\displaystyle \frac{L}{2}}\right)=U_3\left(+{\displaystyle \frac{L}{2}}\right)\\ \begin{array}{c}{\theta }_2\left(+{\displaystyle \frac{L}{2}}\right)={\theta }_3\left(+{\displaystyle \frac{L}{2}}\right)\\ \begin{array}{c}{M}_2\left(+{\displaystyle \frac{L}{2}}\right)=M_3\left(+{\displaystyle \frac{L}{2}}\right)\\ Q_2\left(+{\displaystyle \frac{L}{2}}\right)=Q_3\left(+{\displaystyle \frac{L}{2}}\right)\end{array}\\ N_2\left(+{\displaystyle \frac{L}{2}}\right)=N_3\left(+{\displaystyle \frac{L}{2}}\right)\end{array}\end{array}\right.$$

(23)

where $W$, $U$, $\theta$, $M$, $Q$, and $N$, represent the normal deformation, axial deformation, rotation angle, bending moment, shear force, and axial force of the tunnel structure, respectively. These parameters can be calculated using Equation (24):

$$\left\{\begin{array}{c}U={\displaystyle \frac{-1}{X_1{X}_5}}(W^{\left(5\right)}+X_3{W}^{\left(3\right)}+\left(X_2{X}_5+X_4\right)W^{\left(1\right)}-X_4{p}_z^{\left(1\right)})\\ \begin{array}{c}\theta =W^{\left(1\right)}\\ \begin{array}{c}M=-X_4{W}^{\left(2\right)}\\ Q=-X_4{W}^{\left(1\right)}-R{K}_h(U-R{W}^{\left(1\right)})\end{array}\\ N=EA{U}^{\left(1\right)}\end{array}\end{array}\right.$$

(24)

According to Saint-Venant’s principle, the influence range of fault dislocation is limited. Therefore, the two ends of the entire tunnel can be treated as free boundaries (i.e., the bending moment M, shear force Q, and axial force N are zero at both ends). This leads to the following expressions for internal forces at the tunnel extremities:

$$\left\{\begin{array}{c}\begin{array}{c}\begin{array}{c}M\left(x\right)=0,x=-\infty |+\infty \\ Q\left(x\right)=0,x=-\infty |+\infty \end{array}\\ N\left(x\right)=0,x=-\infty |+\infty \end{array}\end{array}\right.$$

(25)

3. Verification of Theoretical Analysis Methods

3.1. Numerical Model

Cui [27] statistically analyzed tunnel structures damaged by fault dislocation during the Wenchuan earthquake. The results indicate that fault fracture zones are generally composed of Grade V soil, while the hanging wall and footwall consist of Grade IV or V soil, with fault widths typically ranging from 5 m to 40 m. Based on these findings, a 3D numerical simulation model with dimensions of 200 m in length, 100 m in height, and 100 m in width was established (as shown in Figure 5) to eliminate the boundary effects of the model.

The model comprises three components—the footwall, fault zone, and hanging wall. The width of the fault zone is 20 m, and it is located at the center, while the fixed block and moving block of equal length flank the fault zone. To accurately capture the fault zone’s impact on the tunnel response and enhance computational precision, mesh refinement was implemented along 20 m zones extending bilaterally from the fault fracture zone along the tunnel’s longitudinal axis, with a longitudinal element length of 0.5 m, as detailed in Figure 5c. A contact surface between the lining and the soil (Figure 5d) was created to simulate normal and tangential interactions, with contact surface parameters consistent with those in the theoretical analytical model.

According to references [28,29], the moving block and fixed block correspond to Grade IV soil, while the fault zone exhibits poorer mechanical properties as Grade V soil. The material parameters for both the soil and lining are summarized as shown in

Table 1. Surrounding rock and lining material parameters.

Item	Material	Elastic Modulus ($\mathbf{G}\mathbf{P}\mathbf{a}$)	Poisson’s Ratio	Density (kg/${\mathbf{m}}^3$)
Fault zone	Rock and soil	1.0	0.35	2000
Fixed block (or moving block)	Rock and soil	3.0	0.30	2300
Tunnel lining	Concrete	30.0	0.20	2400

.

The displacement pattern of the fault dislocation has an impact on the internal force of tunnels. Scholars have proposed various displacement patterns, including a dislocation mode [30], an S-shaped displacement model [19], and a linear displacement model [31]. Russo et al. [31] suggest that a linear displacement pattern is reasonable for predicting the longitudinal response of tunnels under fault dislocation, and this model has been widely adopted in engineering applications [31,32]. Therefore, this study adopts a linear displacement pattern to validate the theoretical model. Following Zhang et al.’s research [33], the active plate maintains a displacement ($\mathsf{\Delta }f=0.10\mathrm{m}$) along the X-direction, while the internal displacement within the fault zone increases linearly along the Y-axis. Taking the intersection of the tunnel axis and the central plane of the fault zone as the origin, the free-field displacement distribution along the tunnel axis is expressed as:

$$u=\left\{\begin{array}{c}{u}_{y1}=0,x\in (-\infty ,-{\displaystyle \frac{L}{2}}]\\ u_{y2}=∆f({\displaystyle \frac{1}{L}}x+{\displaystyle \frac{1}{2}}),x\in (-{\displaystyle \frac{L}{2}},+{\displaystyle \frac{L}{2}}]\\ u_{y3}=∆f,x\in (+{\displaystyle \frac{L}{2}},+\infty )\end{array}\right.$$

(26)

where $u_{y1}$, $u_{y2}$, and $u_{y3}$ represent the free-field displacements of the fixed block, fault zone, and moving block, respectively.

Fixed constraints were applied to the bottom surface of the footwall and the x-direction boundaries, while vertical displacement constraints were imposed on the hanging wall and fault zone. Calculations were performed to analyze the longitudinal response of the tunnel structure under fault dislocation.

3.2. Validation of the Theoretical Analytical Model

Numerical simulations of the longitudinal response of tunnel structures under fault dislocation were conducted using MATLAB R2023A and FLAC 3D 5.0, respectively. The results from these numerical simulations were compared with those from the analytical model for validation. By setting the tangential foundation spring stiffness $K_P$ to zero and neglecting the tangential contact effects between the surrounding rock and tunnel structure, the traditional calculation method for tunnel longitudinal response under fault dislocation was obtained.

The traditional theoretical analysis model in this paper contrasts with the research conducted by Yu et al. [34], where the analytical solution does not consider the coupling of tangential and normal interactions between the surrounding rock and the lining. The displacement response results calculated by the three methods (the proposed analytical model, the traditional method, and numerical simulation) are illustrated in Figure 6.

The traditional method neglects the axial deformation of tunnel structures and the tangential contact effect between the surrounding rock and the lining, thus failing to obtain the axial displacement of and axial force acting on tunnel structures. Under fault dislocation loading, the longitudinal normal displacement of tunnel structures exhibits an S-shaped distribution. The numerical simulation results show good agreement with the theoretical analytical solutions, consistent with conclusions from previous studies [18,19]. The proposed method introduces tangential foundation springs to consider the tangential contact effect between the surrounding rock and the lining, successfully capturing the axial deformation characteristics of tunnel structures as shown in Figure 6b, and demonstrating excellent consistency with the numerical simulation results.

Subsequently, a comparative analysis is conducted on the longitudinal structural responses from different calculation methods through internal force perspectives. The conversion formulas from stress to internal forces [35] are expressed as Equations (27)–(29):

$$M={\sum }_{i=1}^n{S}_{yy}{A}_{i}l$$

(27)

$$Q={\sum }_{i=1}^n{S}_{yx}{A}_i$$

(28)

$$N={\sum }_{i=1}^n{S}_{yy}{A}_i$$

(29)

where $S_{yy}$ represents the magnitude of stress in the yy-direction of the FLAC 3D element; $S_{yx}$ denotes the magnitude of stress in the yx-direction of the FLAC 3D element; $A_i$ is the area of the subdivided element; and $l$ indicates the distance from the element’s center to the neutral axis.

The internal force responses along the tunnel axis obtained from numerical simulation and theoretical analysis are shown in Figure 7. The differences in peak internal forces between the numerical simulation and the proposed analytical method are relatively minor, with a 6.55% variation for the bending moment, 2.64% for the shear force, and 1.47% for the axial force.

In contrast, the analytical model that neglects the surrounding rock–lining contact effect shows significant discrepancies compared to the numerical results, exhibiting errors of 35.33% in terms of the bending moment and 14.04% for the shear force. Furthermore, the traditional method completely fails to account for axial forces acting on tunnel structures.

Although certain discrepancies exist between the peak values of the bending moments and shear forces obtained by the proposed method and numerical solutions, their peak locations and variation trends demonstrate complete consistency. Notably, the proposed method successfully captures the axial forces acting on tunnel structures under fault dislocation, showing excellent agreement with the numerical simulations. The axial force peaks are observed precisely at the interfaces between the fixed wall/fault zone and active wall/fault zone, which aligns perfectly with the longitudinal strain distribution pattern at the left and right haunches of tunnel structures under strike-slip faulting as reported in Wang’s study [12], thereby further validating the theoretical reliability of this research.

To rigorously assess the rationality of the theoretical framework, correlation coefficients were employed to quantify the consistency between numerical simulations and analytical solutions regarding tunnel deformation and internal forces under fault dislocation. The correlation coefficient analysis serves to determine the degree of association between the numerical results and the analytical predictions.

$$R_{X,Y}={\displaystyle \frac{Cov(X,Y)}{\sqrt{Var\left[X\right]Var\left[Y\right]}}}$$

(30)

where $Cov(X,Y)$ represents the covariance between variables $X$ and $Y$; and $Var\left[X\right]$ and $Var\left[Y\right]$ denote the variances of variables $X$ and $Y$, respectively.

However, the correlation coefficient can only be used to evaluate the degree of correlation between the numerical simulation results and the analytical solutions, rather than quantifying the error between them. Therefore, it is necessary to select appropriate metrics to compare the errors between the numerical and analytical results. The 2-norm can be employed to assess the errors in tunnel horizontal displacement, shear force, and bending moment between the numerical and analytical solutions [36]. The calculation method for the 2-norm is as follows:

$$\mu ={\displaystyle \frac{\left|\right|X-Y\left|\right|}{\left|\right|X\left|\right|}}={\displaystyle \frac{\sqrt{\sum {(X-Y)}^2}}{\sqrt{\sum X^2}}}$$

(31)

where X and Y represent the reference data and comparison data, respectively.

As shown in

Table 2. Comparison of structures with different calculation methods.

Solution	Item	Normal Displacement	Axial Displacement	Bending Moment	Shear Force	Axial Force
Traditional solution	correlation coefficient	0.998	\	0.940	0.786	\
	2-norm	0.040	\	0.454	693	\
Proposed solution	correlation coefficient	0.999	0.992	0.996	0.982	0.994
	2-norm	0.033	0.127	0.143	0.187	0.126

, the structural deformation and internal force responses obtained by the proposed method in this paper exhibit high consistency with the simulation results of the three-dimensional finite difference method. In terms of the degree of correlation and the error magnitude, the proposed method outperforms traditional approaches. Moreover, it can account for the tangential contact effect between the surrounding rock and the lining, accurately capturing the distribution patterns of deformation and internal forces in tunnel structures under fault dislocation.

It is worth noting that flexible joints and segmented linings are crucial measures for seismic and anti-dislocation design in tunnels crossing faults. Under fault dislocation and seismic actions, flexible joints undergo tri-directional deformations—tension–compression, torsion, and shear. However, current analytical methods based on beam theory neglect the axial force and axial deformation of tunnel structures, limiting them to consideration of shear and torsional deformations at the joints while failing to account for tension–compression deformations, which deviates from real-world conditions. In contrast, the analytical method proposed in this paper can rapidly determine the axial force response of tunnel structures, providing richer information for the design of flexible joints for tunnels crossing faults.

Based on the theoretical analytical calculations of tunnel structural deformation and internal force responses, the deformation of a tunnel structure under strike-slip fault dislocation is illustrated in Figure 8.

An analysis of the stress conditions on the left haunch of the tunnel lining reveals the following: Due to fault dislocation, the tunnel lining undergoes non-uniform deformation along the longitudinal direction. Near the interface between the “fixed plate and fault”, the tunnel structure experiences longitudinal compression, while near the interface between the “fault and active plate”, it undergoes longitudinal tension. Within the fault zone, the lining transitions from compression to tension. Conversely, the stress conditions on the right haunch of the tunnel lining are opposite to those on the left haunch, and are primarily determined by the direction of fault dislocation.

4. Parametric Sensitivity Analysis

To investigate the influence of the width of fault zone, the stiffness of the fault zone, and the stiffness of the tunnel lining on the tunnel’s longitudinal response under fault dislocation, a parametric sensitivity analysis was conducted using the analytical method proposed, which explicitly accounts for the contact interaction between the surrounding rock and the lining.

4.1. Width of Fault Zone

This section focuses on analyzing the effect of fault zone width on the longitudinal response of the lining. Based on Cui’s [27] statistical study of fault-crossing tunnels damaged during the Wenchuan earthquake, where fault widths typically range from 5 m to 40 m, four fault widths—L = 10 m, 20 m, 30 m, and 40 m—are selected for comparative analysis. All other parameters remain unchanged and align with those defined in Section 3.1 of this study.

The structural deformation under fault dislocation with varying fault widths is shown in Figure 9. The deformation range of the tunnel structure induced by fault dislocation synchronizes with fault zone width variation. The peak axial displacement decreases with increasing fault width—taking the L = 10 m fault width as a baseline, the peak axial displacements decrease by 16%, 30%, and 41%, respectively. Under constant dislocation displacement, the compressive stress on the tunnel structures within the fault zones reduces as fault width increases.

The internal forces are illustrated in Figure 10. All peak internal forces decrease with increasing fault width. The bending moment and axial force show similar reduction magnitudes (approximately 30%, 50%, and 60%), while the shear force exhibits greater reductions (59%, 79%, and 86%). The decreasing rate of internal force peaks gradually slows with fault width expansion. This occurs because the fault fracture zone, being a relatively flexible area compared to active/fixed plates, has a lower foundation stiffness coefficient. As the fault width increases, tunnel bending induced by equivalent dislocation decreases, combined with a reduced constraint from the fracture zones, leading to decreased bending moments and shear forces.

Under different fault widths, the bending moment and axial force maintain consistent distribution patterns, with peaks at fault zone boundaries (fixed–active plate interfaces). The shear force distribution characteristics change with fault width expansion—for the L = 10 m width, the shear force peaks at the fault center; as the width increases, the shear forces within fault zones tend to equalize.

Consequently, tunnel structures are prone to tensile failure at fault zone boundaries and shear failure at fault centers. Smaller fault widths increase structural damage potential. Therefore, special attention should be paid to anti-fracture design at both fault zone boundaries and central regions for tunnels crossing faults. Critical reinforcement measures should be implemented at these vulnerable positions.

The structural deformation under fault dislocation with varying fault widths is shown in Figure 9. The deformation range of the tunnel structure induced by fault dislocation synchronizes with fault zone width variation. The peak axial displacement decreases with increasing fault width—taking the L = 10 m fault width as a baseline, the peak axial displacements decrease by 16%, 30%, and 41%, respectively. Under constant dislocation displacement, the compressive stress on the tunnel structures within fault zones reduces as fault width increases.

4.2. Elastic Modulus of Fault Zone

The mechanical properties of the surrounding rock in the fault zone play a critical role in the longitudinal response of tunnel structures under dislocation effects.

Theoretical model analysis indicates that the Poisson’s ratio of the surrounding rock in the fault zone has a minor impact on the tunnel’s response, whereas the elastic modulus of the fault zone significantly influences the interaction between the surrounding rock and the tunnel. Therefore, this section investigates the effect of the elastic modulus of the fault zone on the dynamic structural response. Based on literature research [29,37], the ratio of the elastic modulus between the fault zone and the upper/lower rock strata ranges from 2 to 5. To maintain consistency, the elastic modulus of the upper/lower rock strata is kept constant, while four elastic modulus values are selected for the fault zone: $E=1.5 \mathrm{G}\mathrm{P}\mathrm{a},E=1.0 \mathrm{G}\mathrm{P}\mathrm{a}$, $E=0.75 \mathrm{G}\mathrm{P}\mathrm{a}, \mathrm{a}\mathrm{n}\mathrm{d} E=0.6 \mathrm{G}\mathrm{P}\mathrm{a}$.

As shown in Figure 11, with changes in the elastic modulus of the rock surrounding the fault zone, the normal displacement of the structure remains almost unchanged, while the axial displacement shows a slight reduction, with a maximum reduction of 24%.

The internal force responses (Figure 12) indicate that the peak values of the bending moments and shear forces increase as the ratio of the elastic modulus between the fault zone and the adjacent regions decreases. The greater the difference in the elastic modulus of the surrounding rock between the fault zone and the adjacent regions, the more susceptible the tunnel structure becomes to damage. The tunnel structure near the interface between the fixed plate and the fault zone (or the fault zone and the active plate) is equivalent to crossing a transition zone between soft and hard surrounding rock.

Tunnel damage observed in the Wenchuan earthquake [38] revealed that transition zones between soft and hard rock, or abrupt changes in surrounding rock quality, where the tunnel experiences significant differences in rock constraints, are highly prone to failure. Furthermore, as the disparity in rock properties across the interface increases, the peak bending moments and shear forces of the tunnel structure also increase [17]. To mitigate these risks, it is crucial to implement anti-fault dislocation fortification measures, such as the progressive grouting method proposed in prior research [39]. By progressively grouting near the interfaces between the fault zone and adjacent regions, the difference in elastic modulus between the fault zone and the surrounding rock can be reduced, thereby minimizing the structural response of the tunnel.

4.3. Stiffness of Tunnel Lining

The stiffness ratio between the soil and lining is a crucial factor influencing the structural longitudinal response. Variations in tunnel structural dimensions and the concrete grade of the lining both alter this stiffness ratio, exhibiting similar effects on the ratio. This section focuses on analyzing the impact of the concrete lining grade on the tunnel’s response. Based on actual engineering practices [29], four concrete grades (C20, C30, C40, and C50) were selected with corresponding elastic moduli of 25.5 GPa, 30.0 GPa, 32.5 GPa, and 34.5 GPa, respectively. Other parameters remain consistent with Section 3.1 of this paper.

As shown in Figure 13, tunnel displacement under different lining concrete grades shows no significant variation in structural deformation. However, the internal force responses depicted in Figure 14 reveal that the peak values of the bending moment, shear force, and axial force all increase with concrete grade enhancement. The growth rates of the internal forces gradually decrease at 13%, 7%, and 5%, respectively, as the concrete grade elevates.

5. Conclusions

This study proposes a calculation method for the response of tunnels under fault dislocation. The elastic foundation beam theory is employed to reflect the interaction between the surrounding rock and the tunnel, where foundation spring parameters are modified to simulate fault zones and adjacent areas. By applying the fault displacement at the ends of normal springs to consider fault dislocation effects, structural longitudinal response expressions are derived through Green’s function solutions. A numerical model is subsequently established to validate the analytical approach. Finally, parameter sensitivity analyses are conducted using the analytical model to investigate the influences of the width of the fault zone, the stiffness of the soil in the fault zone, and the stiffness of the lining on the tunnel’s structural responses. The main conclusions are as follows:

(1)

A longitudinal response model for tunnels crossing fault zones is developed by introducing tangential foundation springs to account for surrounding rock–lining contact effects. Numerical validation demonstrates the critical influence of tangential contact effects on tunnel responses, revealing a 29% difference in peak bending moment compared with traditional models that neglect this mechanism.

(2)

Significant impacts on structural internal forces are observed: Increasing the fault zone’s width from 10 m to 40 m reduces peak internal forces by 86%. Conversely, decreasing the elastic modulus of the rock surrounding the fault zone increases the structural forces. Using higher concrete grades in tunnel linings elevates internal force peaks, indicating that increasing the lining–surrounding rock stiffness ratio adversely affects anti-dislocation performance.

(3)

Variations in fault width maintain consistent distribution patterns for bending moments and axial forces, with peaks occurring at fault boundary interfaces. Shear force distribution characteristics change with fault width—at the L = 10 m width, shear peaks appear in the fault’s center; wider fault zones exhibit more uniform shear distribution across fault regions.

The proposed analytical solution, considering tangential rock–lining interactions, enables the rapid acquisition of longitudinal deformation and internal force patterns in tunnels under fault dislocation. Compared with traditional analytical methods, this approach better aligns with practical engineering conditions and provides enhanced theoretical support for the anti-dislocation design of fault-crossing tunnels.