Longitudinal Seismic Analysis of Tunnels with Nonuniform Strata Considering the Effect of Karst

Abstract

The longitudinal direction of shield tunnels is prone to seismic damage due to excessive deformation under seismic action, especially in nonuniform stratum. In this paper, the longitudinal dynamic response of the tunnel under different seismic effects is calculated based on the longitudinal equivalent stiffness model using the stratigraphic load model, and the seismic indexes such as longitudinal corner, tube sheet and joint bolt stresses are verified. The calculation results show that the longitudinal seismic weakness of the shield tunnel is in the interface between soft and hard strata and karst development. The longitudinal axial force of the structure is larger during the longitudinal excitation of seismic waves, and the maximum bending moment is mainly in the vertical plane, i.e., the vertical bending moment. The axial force of the tunnel is smaller during the transverse excitation of seismic waves. The maximum bending moment is mainly the bending moment in the horizontal plane, i.e., the transverse moment. The South Lake section of the Two Lakes Tunnel has good seismic performance in the event of a rare earthquake with a 50-year exceedance probability of 2%. The investigation can guide the seismic design of the South Lake section of the Two Lakes Tunnel.

1. Introduction

The development of underground space is an increasingly vital trend in structural engineering in the 21st century. Initially, the engineering and academic communities thought that the layers of earth and rock had a shielding effect on seismic waves and that underground structures were constrained by the surrounding soil and had less vibration amplitude than surface structures, so the seismic design of underground structures was ignored. However, research data from several major earthquakes around the world have shown that underground structures can also produce more serious damage. Examples of such earthquakes include the 1989 Loma Prieta [1,2,3], 1994 Northridge [4,5], 1995 Kobe, 2008 Wenchuan, and 2011 Tohoku earthquakes [6,7,8,9,10,11,12,13,14]. In particular, the 1995 Kobe earthquake caused severe damage, including concrete cracking, joint movement, and longitudinal and transversal deformation to several subway tunnels and station caverns, as well as the collapse of the Dakai station [15,16,17]. The 1999 Chi-Chi, Taiwan, earthquake (Mw 7.6) caused damage to 49 tunnels within 60 km of the epicenter [18,19,20]. These seismic hazards have drawn the attention of the academic and engineering communities to the analysis of the seismic response of underground structures. Large diameter shield tunnels are an important form of underground structure. Compared to old methods such as mining and open cutting, the shield method has the advantage of a small construction area and is not affected by the climate and hydrological conditions of the ground, as it only requires the excavation of a working shaft [21]. Due to the special “tube sheet + joint” structure of shield tunnels, the transverse equivalent bending stiffness is usually several to several dozen times more efficient than the longitudinal equivalent bending stiffness, which makes them more susceptible to large deformations and thus to severe damage during earthquakes.

In engineering practice, the design of shield tunnels is usually controlled by cross-sectional forces and deformations, without much consideration of longitudinal indicators. Initially, research on the seismic resistance of underground structures was mainly focused on structures such as metro stations [22,23,24,25,26]. Azadi and Hosseini [27] studied the effect of soil liquefaction on the dynamic response of metro stations during earthquakes, and Unutmaz [28] studied the effect of burial depth on the seismic response of underground structures on this basis. Zhuang et al. [29] established a finite element model of the interaction of a soil–subsurface diaphragm and a wall–subsurface structure and studied the seismic response characteristics of subsurface diaphragm walls. Based on fluctuation theory, the unique dynamic response of a large underground frame structure adjacent to a fault is investigated by transforming ground vibrations into an equivalent force attached to a three-dimensional viscoelastic artificial boundary node. Masoud [30] and others carried out shaking table tests on the Tehran metro tunnel to compare the geodynamic response of the tunnel structure with that of the free-field strata in the amplified effect of vibration in the surface strata.

Subsequently, Yohsuke [31], Anne Lemnitzer [32] and others carried out large-scale shaking table tests to observe the structural damage pattern and strata cracking pattern and to study the effect of strata–structure interaction. Presently, the longitudinal seismic response of shield tunnels has been less studied and is mainly based on numerical simulations. Chen [33] used the generalized reaction displacement method to analyze the non-linear seismic response of a 2.7 km long Suai tunnel with segmental lining. In the longitudinal seismic response analysis, the engineering geology characteristics and nonlinear dynamic behavior of the Suai seabed soil, the nonuniform mesh layout of the free-field site, the artificial boundary conditions and nonuniform seismic input, simulation model, and the parameters of soil–tunnel interaction systems are considered in detail. Liu [34] proposed a longitudinal integral response deformation method for the analysis of the longitudinal seismic response of tunnel structures under asynchronous seismic excitation, and derived the longitudinal maximum deformation and the critical moment of the structure at the onset of internal forces. By comparing it with the seismic response of a shield tunnel structure in Beijing, the method has a high applicability.

Using numerical simulation software [35,36,37,38], Geng et al. [39] established a three-dimensional finite element analysis model and validated the equivalent stiffness model to simulate the seismic response characteristics of shield tunnels by evaluating the deformation of the model after applying seismic acceleration. Huang [40,41] derived an equivalent nodal force formulation for a two-dimensional planar P-wave with an arbitrary incidence angle based on the viscous-spring artificial boundary finite element method and applied it using the commercial software ABAQUS. Wu [42] and Liang et al. [43] changed the traditional Euler beam to the Timoshenko beam applied in the longitudinal equivalent stiffness model to investigate the longitudinal bending deformation as well as shear deformation of shield tunnels. Shao [44] simplified the shield tunnel into a line beam, simplified the stratum, established an equivalent continuous model with the same longitudinal deformation as the shield tunnel, and carried out a 3D finite element analysis of the longitudinal seismic response of a shield tunnel in Nanjing. Li [45] developed three-dimensional numerical simulation techniques for a site response to an asynchronous earthquake. Then, a 3D soil–tunnel structure interaction model was established to simulate the longitudinal response of the tunnel when subjected to an asynchronous earthquake. Saif Alzabeebee [46], using a carefully developed finite element model, investigated the combined effects of seismic shaking and pipe diameter, burial depth, installation conditions and seismic shaking intensity. I. Shahrour [47] presents an elastoplastic finite element analysis of the seismic response of tunnels constructed in soft soils, using a cyclic elastoplastic constitutive relation involving both isotropic and kinematic hardening to describe the behavior of the soil material. Miao [48] developed an accurate beam–spring model for a shield tunnel and carried out a longitudinal seismic response analysis of the shield tunnel based on the generalized reaction displacement method, taking into account the nonlinearity of the soil.

The results of the above studies do not take into account the influence of the complexity of the geological conditions of the tunnel on the seismic response characteristics of the structure, such as the tunnel crossing the junction of longitudinal soft and hard strata, the junction of transverse soft and hard rock formations, cavities, etc. Shield tunnels have the most severe seismic hazards precisely where the geological conditions change drastically or where shaped structures, such as joints, are located. In addition, in terms of calculation models, existing studies equate the tunnel to a straight beam with a small diameter, whereas very large diameter shield tunnels of 14 m and above are currently the mainstream designs used in tunnel construction, and the above results are of little relevance to engineering practice.

In this paper, the research object is a very-large-diameter shield tunnel in the Nanhu section of the “Two Lakes Tunnel” in Wuhan City, where the tunnel lining entity is generated by fitting a B-sample curve to the shield bored direction, refining the modeling of its stratigraphic conditions based on survey data, and studying the seismic dynamic response of the shield tunnel when it crosses karst and soft and hard nonuniform strata several times in the longitudinal direction. The seismic dynamic response of shield tunnels with multiple longitudinal crossings through karst and soft and hard nonhomogeneous strata is investigated, and seismic performance indicators such as ring joint tension, concrete and bolt stresses are verified. The results of the study are of great significance for the seismic design of large-diameter shield tunnels in complex geological conditions.

2. Project Overview

The Two Lakes Tunnel Project is a fundamental, functional and forward-looking major urban construction project for the long-term development of Wuhan according to the “Thirteen Five-Year Plan” finalized by the municipal government and the municipal party committee. It is located in the central city of Wuchang on the north—south axis. Figure 1 shows the geographical location of the Two Lakes Tunnel Project. The main line of the tunnel is 14.45 km in length, with a two-way six-lane standard to the south of Luoyu Road and a two-way four-lane standard for each of the two sublanes to the north of Luoyu Road, giving a total scale of eight lanes in both directions. All entrances and exits are built to a two-lane standard. The general section of the tunnel is divided into 13.9 m single-tube two-way four-lanes and 15.4 m single-tube two-way six-lanes. The tunnel section diagram is shown in Figure 2.

The onshore section of the tunnel is constructed by the open cut method, and the underwater section is constructed by the shield method. The South Lake section of the tunnel features a long tunnel distance, large diameter, large changes in burial depth, crossing uneven strata, crossing fracture zones, complex section forms, karst development and irregular structures of many working shafts. At the same time, the shield tunnel passes through the South Lake, crossing a variety of weak soil layers, such as silty powder clay and medium weathered mudstone, which have low strength and are easily deformed under pressure; the mudstone has low strength and is easily softened by water, especially the strongly weathered rock, which is fragmented and has poor self-stability. The physical and mechanical parameters of the strata vary greatly and are determined by complex geological conditions. The stratigraphic section is shown in Figure 3. The physical and mechanical parameters of each stratum are shown in

Table 1. Physical and mechanical parameters of the stratum.

Stratum	Elastic Modulus E (kN/m2)	Poisson’s Ratio λ	Volumetric Weight γ (kN/m3)
Miscellaneous Fill	8 × 104	0.32	18.5
Silty Clay	3 × 104	0.32	18.9
Clay	6.04 × 104	0.32	19.6
Muddy Siltstone	7 × 104	0.32	28
Breccia Limestone	1.4 × 106	0.26	25.9
Limestone	3 × 106	0.32	23
Limestone	1.3 × 106	0.27	25.8

. The project area mainly involves the Tanlu Seismic Statistical Area, the Middle Yangtze River Seismic Statistical Area and the North China Plain Seismic Statistical Area. The region has a moderately strong seismic activity level, and the trend of its seismic activity is estimated at the level of the seismically active period in the next hundred years, and the possibility of an M6 magnitude earthquake in the next hundred years cannot be excluded. The near-field area and adjacent zones have a moderate level of destructive seismicity, with a moderate frequency and weak intensity of modern seismicity.

The South Lake Tunnel XK8+115 to XK9+730 section is 1600 m long, with the characteristics of long tunnel distance, large diameter, large changes in burial depth, crossing uneven strata, crossing fracture zones, crossing caves, etc. It is representative of the South Lake section of the tunnel and is more suitable for calculation.

3. Seismic Wave Synthesis

3.1. Bedrock Synthetic Ground Shaking Time Intervals

In seismic response time history analysis, the selection of a suitable ground acceleration time course is of critical importance. Although the number of actual earthquake records has increased considerably over the last few decades, the site conditions at their recording locations can differ significantly from those of actual building sites. Instead, we need a set of ground shaking samples that satisfy the same statistical properties as the seismic response input. In this section, based on the ground shaking characteristic parameters of the project site, the triangular level superposition simulation is used to synthesize the bedrock ground shaking timescale. The computational model is

$$x^{″}(t)=f(t){\displaystyle \sum _{k=0}^n{C}_k\cos ({\omega }_{k}t+{\phi }_k)=f(t)a(t)}$$

(1)

$$a(t)={\displaystyle \sum _{k=0}^n{C}_k\cos ({\omega }_{k}t+{\phi }_k)}$$

(2)

where φ_k is the random phase angle uniformly distributed in the interval (0, 2_π); $C_k$ and ${\phi }_k$ are the amplitude and frequency of the _kth frequency component, respectively; $f(t)$ is the intensity envelope function, which is a deterministic function of time; and $a(t)$ is a smooth Gaussian process.

Equation (1) [49] shows that a synthetic acceleration time course $x^{″}(t)$ characterized by a nonsmooth stochastic process is the product of a Gaussian process $a(t)$ characterized by a smooth stochastic process and a certain intensity envelope function $f(t)$. The synthetic acceleration time course $x^{″}(t)$ must satisfy the following two conditions.

(1) The response spectrum calculated from the synthetic acceleration time course $x^{″}(t)$ is as closely consistent as possible with the given target response spectrum.

(2) The synthetic acceleration timescale $x^{″}(t)$ has the peak acceleration and total duration required for the design site.

The coefficients in Equation (2) [49] can be derived from a given power spectral density function $S\left({\omega }_k\right)$:

$$\{\begin{array}{c}{C}_k=\sqrt{4S\left({\omega }_k\right)\mathsf{\Delta }\omega }\\ {\omega }_k=2\pi k/T\\ \mathsf{\Delta }\omega =2\pi /T\end{array}$$

(3)

where $T$ is the total holding time of the smooth stochastic process. To adopt the acceleration response spectrum as the target spectrum for the artificial acceleration time course, the following approximation between the response spectrum and the power spectrum can be used instead of $S\left({\omega }_k\right)$ in Equation (3) [49]:

$$S\left({\omega }_k\right)=-\frac{\zeta }{\pi \omega }{\left[S_a^T\left(\omega \right)\right]}^2\frac{1}{\ln \left[\frac{-\pi }{\omega T}\ln (1-P)\right]}$$

(4)

where $S_a^T\left(\omega \right)$ is the response spectrum for a given target acceleration, $\zeta$ is the damping ratio, $T$ is the duration, and $P$ is the probability of exceeding the response.

The intensity envelope function $f(t)$ is related to the magnitude of the earthquake, the epicenter distance, the geological and tectonic background and the site soil conditions, reflecting the unsteady characteristics of ground shaking over time. In general, the intensity envelope function $f(t)$ is represented by the following segmental function.

$$f(t)=\{\begin{array}{c}{\left(t/t_1\right)}^2 0\le t\le t_1\hfill \\ 1 t_1\le t\le t_2 \hfill \\ e^{-c(t-t_2)} t_1\le t\le t_d \hfill \end{array}$$

(5)

where (0, $t_1$) is the rise section, ($t_1$, $t_2$) is the smooth section, ($t_2$, $t_d$) is the decay section, and $c$ is the decay coefficient.

Since the relationship between the response spectrum and the power spectrum expressed in Equation (4) [49] is approximate, the response spectrum calculated according to the initial time course is generally only approximate to the target response spectrum. To improve the fitting accuracy, iterative adjustment is needed, and the iteration is stopped when the difference between the calculated response spectrum and the target response spectrum is less than the control accuracy (taken as 5% in this paper).

The parameters $t_1$, $t_2$, $c$ and $T$ in the intensity envelope function $f(t)$ are given according to the statistical laws of the actual seismic record, and a number of empirical formulae for determining these parameters are available. Using the expected magnitude and expected epicenter distance obtained from the seismic hazard analysis and considering the seismotectonic environment of the project site, the statistical formulae proposed by Mcguire, Housner, Shannon and Huo Junrong were used, and the holding time parameters determined are shown in

Table 2. Bedrock ground-vibration time-holding parameters.

Beyond Probability	Intensity Envelope Intensity Parameters
	${\mathit{T}}_1(\mathit{s})$	${\mathit{T}}_{\mathit{s}}(\mathit{s})$	${\mathit{T}}_2(\mathit{s})$	$\mathit{C}$
10% over 50 years	3.0	11.5	14.5	0.15
2% over 50 years	3.0	13.4	16.4	0.15

.

According to the needs of the numerical calculation, three time intervals (each corresponding to three different sets of random phases) were synthesized with a probability of exceeding 10% and 2% for 50 years for a total of six acceleration time intervals. The sampling step of the time intervals is 0.02 s, the target acceleration response spectrum is taken from 0.04 to 6.0 s with 50 control points according to the principle of logarithmic equal spacing distribution, and the relative error between the response spectrum of the synthetic time intervals and the target spectrum is less than 5%.

3.2. Analysis of the Seismic Response of Site Soils

The results of the site seismic engineering geological condition survey indicate that the variation in medium properties and topography along the horizontal plane is not very significant within the area of the site. Therefore, this paper uses a one-dimensional site model to consider the influence of local site conditions on ground vibration characteristics. The 1D site model soil seismic response analysis, using the equivalent linearized analysis method of 1D soil shear dynamic response analysis, is based on the following basic principles.

It is assumed that the soil layer is horizontally layered and approximately infinitely extended, that the substrate and the soil medium are isotropic linear elastomers and that shear waves are incident vertically from the bedrock into the N-layered horizontal layer of the viscoelastic medium. Therefore, in each layer of the medium, the fluctuation equation must be satisfied.

$${\rho }_i\frac{{\partial }^2{u}_i\left(x,t\right)}{\partial t^2}=G_i\frac{{\partial }^2{u}_i\left(x,t\right)}{\partial x^2}+{\eta }_i\frac{{\partial }^3{u}_i\left(x,t\right)}{\partial x^2\partial t}$$

(6)

where $u_i\left(x,t\right)$ is the displacement function of the soil in layer $i$, ${\rho }_i$ is the density of the soil in layer $i$, $G_i$ is the shear modulus of the soil in layer $i$, and ${\eta }_i$ is the coefficient of adhesion of the soil in layer $i$.

The displacement continuity condition and the stress continuity function shall be satisfied at the interface between the soil layers, while at the surface (top surface of the first layer of soil), the zero stress condition shall be satisfied:

$$u_i\left(h_i,t\right)=u_{i+1}\left(0,t\right)$$

(7)

$${\tau }_i\left(h_i,t\right)={\tau }_{i+1}\left(0,t\right)$$

(8)

$${\tau }_i\left(0,t\right)=0$$

(9)

In the calculation, the transfer function of the vertical upward incidence of each soil layer with respect to the half-space is first calculated and then multiplied by the payoff spectrum of the incident wave from the bedrock to obtain the acceleration payoff spectrum of the top surface of each soil layer and the shear strain payoff spectrum of the middle surface of each soil layer, which are converted by the payoff inversion to obtain the corresponding acceleration time and shear strain time.

Given the nonlinear nature of soils, the dynamic shear modulus and the hysteresis damping ratio of each soil layer are functions of shear strain. Therefore, an initial shear modulus $G_0$ and a damping ratio $Z_0$ are assumed, and the corresponding shear strain values for each layer are calculated. Then, the maximum shear strain value is multiplied by a discount factor (0.65 in this paper) as the equivalent shear strain, and the corresponding shear modulus $G^{\prime }$ and damping ratio $Z^{\prime }$ are determined from the shear modulus curve and damping ratio curve, and $G^{\prime }$, $G_0$ and $Z^{\prime }$, $Z_0$ are compared. If the relative errors of the shear modulus and damping ratio are less than the permissible error (5% in this paper), the stress—strain relationship is considered to be satisfied; otherwise, the above calculation process is repeated with the new shear modulus $G^{\prime }$ and damping ratio $Z^{\prime }$ as the initial values until the relative errors are satisfied.

In accordance with the requirements of the seismic design of the project structure, half of the amplitude of the bedrock ground acceleration time (three samples each) for the 50-year exceedance probabilities of 10% and 2% are taken as the input of the basal incident wave to the 1D soil seismic response calculation model, and the seismic wave is assumed to be incident vertically from the bedrock surface. For each case, the acceleration timescales and their response spectra are calculated for the ground shaking response at the project site.

3.3. Determination of the Site Design Ground Vibration Parameters

The design ground vibration parameters for each project site, including the design ground vibration peak acceleration and acceleration response spectrum, are derived from the site soil response calculation results obtained in Section 3.2. The design ground vibration peak acceleration is determined by considering the sample peak and short-period acceleration response spectra, and the design acceleration response spectrum is determined by averaging the calculated acceleration response spectrum.

The design ground vibration acceleration response spectrum for the project site is taken as

$$S_a(T)=A_{\max }\beta (T)$$

(10)

$${\alpha }_{\max }=A_{\max }{\beta }_{\max }/g$$

(11)

where $A_{\max }$ is the design peak ground acceleration, ${\alpha }_{\max }$ is the maximum seismic impact factor, $g$ is the acceleration due to gravity, and $\beta (T)$ is the design ground acceleration amplification factor response spectrum, given in the form of the Seismic Code for Highway Engineering (JTG/T 2232-01-2019) [49]:

$$\beta (T)=\{\begin{array}{c}1+(\eta {\beta }_{\max }-1)T/T_1 0\le T\le T_1(s)\hfill \\ \eta {\beta }_{\max } T_1\le T\le T_g(s)\hfill \\ \eta {\beta }_{\max } {(T_g/T)}^{\gamma } T>T_g(s)\hfill \end{array}$$

(12)

where $T$ is the self-oscillation period of the structure, ${\beta }_{\max }$ is the maximum of the reaction spectrum, $T_1$ is the starting period of the plateau section of the maximum of the reaction spectrum, $T_g$ is the characteristic period, and $\gamma$ is the decay index of the falling section of the reaction spectrum.

$$\nu =\frac{\alpha Tg}{2\pi }$$

(13)

where $\nu$ is the peak design ground vibration velocity, $\alpha$ is the peak design ground vibration acceleration ($m/s^2$), and $T_g$ is the characteristic period. The peak displacement S is statistically taken to be $S/a=1/15$. $\eta$ is the damping adjustment factor, determined in accordance with Equation (14), $\zeta$ is the damping ratio, and $\gamma$ is the attenuation factor of the falling section of the curve, determined in accordance with Equation (15) [49].

$$\eta =1+\frac{0.05-\zeta }{0.08+1.6\zeta }$$

(14)

$$\gamma =1+\frac{0.05-\zeta }{0.3+6\zeta }$$

(15)

Equation (9) [49] is used in conjunction with the results of the calculated horizontal ground vibration acceleration response spectra (5% damping ratio) for 50-year exceedance probabilities of 10% and 2% at the project site to obtain the corresponding fitted curves for the ground vibration acceleration response spectra at the line project site. For convenience, the design ground vibration parameters at 15 m below ground level are given in a comprehensive manner.

3.4. Site Design Ground Vibration Acceleration Timescale

In accordance with the needs of seismic calculations for engineering structures (calculated using the time history analysis method), this section artificially synthesizes the site design ground acceleration time course based on the values of the engineering site design ground vibration parameters to be used as seismic input to the engineering structural dynamic response analysis calculations. Using the site design peak seismic acceleration, target acceleration response spectrum and time-step intensity envelope function parameters determined in Section 3.3, the site ground acceleration timescales were artificially synthesized as described in the previous method. According to the requirements of the numerical calculation, three samples of seismic acceleration time courses were synthesized for the project site at the 50-year exceedance probabilities of 10% and 2%, corresponding to three different sets of random phases. The sampling step was 0.02 s, the target acceleration response spectrum was taken as 57 control points in the range of 0.04 to 6.0 s according to the principle of logarithmic equal spacing distribution, and the relative error between the response spectrum and the target response spectrum was less than 5%. The resulting acceleration time samples of the three different phases were averaged to obtain seismic wave time ranges with 50-year exceedance probabilities of 10% and 2% corresponding to basic and rare earthquakes, respectively, as shown in Figure 4.

4. Analysis of the Longitudinal Seismic Performance of Shield Tunnels

4.1. Shield Stiffness Simplification

The shield tunnel structure consists of a number of tube pieces and joints assembled in a complex configuration. The connection of the shield tunnel generally needs to be simplified in the numerical modeling process. The most commonly used simplification models are the longitudinal beam spring model proposed by Jun Koizumi [50] and the longitudinal equivalent stiffness model proposed by Yukio Shibo [22]. The former longitudinal beam spring model simplifies the tunnel tube piece in the longitudinal direction as a straight beam and uses the spring to simulate the longitudinal joint to find the deformation and internal forces at the joint and joint face, but the joint spring stiffness is not easily determined in this method. The latter longitudinal equivalent stiffness model reduces the shield tunnel to a homogeneous beam of equal cross section and uses the longitudinal equivalent stiffness to simulate it, which is simpler and easier to calculate than the former. Therefore, the longitudinal equivalent stiffness model was chosen for this paper to simplify the shield tunnel.

The longitudinal equivalent stiffness model by Yukio Shibo divides the longitudinal deformation of the tunnel during an earthquake into axial tensile deformation and serpentine bending deformation, as shown in Figure 4. If the transverse circular section is assumed to be under planar strain, the axial compression of the tunnel considers only the compression of the concrete pipe piece, and the axial tension considers the joint tension of the longitudinal bolts and the concrete pipe piece. The equivalent beam of the shield tunnel has compressive stiffness ${\left(EA\right)}_{eq}^C$ and tensile stiffness ${\left(EA\right)}_{eq}^T$, as shown in Equations (16) and (17).

$${\left(EA\right)}_{eq}^C=E_C{A}_C$$

(16)

$${\left(EA\right)}_{eq}^T=\frac{E_C{A}_C}{1+\frac{E_C{A}_C}{l_S{K}_j}}$$

(17)

where $E_C$ is the modulus of elasticity of the concrete, $A_C$ is the cross-sectional area of the tube sheet ring, $K_j$ is the sum of the tensile stiffness of the longitudinal bolts, and $l_S$ is the width of the tube sheet ring.

As shown in Figure 5, the equivalent bending stiffness of the shield tunnel is ${\left(EI\right)}_{eq}$, defined as

$${\left(EI\right)}_{eq}=\frac{{\cos }^3\phi }{\cos \phi +\left(\frac{\pi }{2}+\phi \right)\sin \phi }{E}_C{I}_C$$

(18)

$$\phi +\cot \phi =\pi \left(\frac{1}{2}+\frac{K_i}{\raisebox{1ex}{$E_C{A}_C$}\!\left/ \!\raisebox{-1ex}{$L_S$}\right.}\right)$$

(19)

where $L_S$ is the moment of inertia of the tube section, and $\phi$ is the angle between the centerline of the tunnel and the neutral axis.

The shield tunnel of this project adopts a single-layer lining, concrete grade C60 tube piece with an inner diameter of 14.2 m, an outer diameter of 15.5 m, and a thickness of 0.65 m. The tube piece ring adopts the “7 + 2 + 1 (1/3 capping block)” blocking method, longitudinal joints use 40 M36 bolts, and ring joints use 56 M30 bolts. The ring is shown in Figure 6.

Table 3. Shield tunnel annular seam calculation parameters.

Parameters	Value
Tunnel O.D. D (m)	15.5
Thickness of tunnel t (m)	0.65
Tunnel loop width Ls (m)	2
Elastic Modulus of concrete Ec (kPa)	3.6 × 107
Diameter of bolt D (mm)	36
Length of bolt Lj (mm)	710
Number of bolts N	40
Elastic Modulus of bolt Ej (kPa)	2.1 × 108
$\phi$	1.0 (57.5°)
Second moment of area Ic (m4)	837.1
Total area of bolts Aj (m2)	0.04

shows the calculated parameters of the shield tunnel for this project, which, when substituted into Equations (16)–(19), is the equivalent reduction in the longitudinal flexural stiffness of the tunnel ${\eta }_M=0.056$.

4.2. Numerical Modeling

To evaluate the assessment of the internal forces and joint tensions in the shield tunnel under seismic action, a 3D refined model was built using the finite element software Midas GTS NX. To avoid boundary effects, the longitudinal length of the model was 1600 m, the width was 90 m, the height was 300 m and the shield diameter was 15.5 m. To realistically reflect the geometry of the shield tunnel, a shield tunnel section was first created, followed by a B-sample curve to fit the tunnel orientation, the apex of which was shifted to the center of the shield tunnel section to form a guide curve. To take into account the karst area in the formation and the convergence of the computational model, the karst surface area was simplified to a hexahedron for the simulation. The model is shown in Figure 7.

A mixed mesh (hexahedral and tetrahedral meshes) was used in the model, with the shield tunnel cell mesh size set to 3 and the karst cell mesh size controlled in the range of 5–10 to ensure nodal coupling. The tunnel lining was simulated using 2D plate cells, the interaction with the soil was modeled as an elastic spring subjected to compression forces only, and the stiffness of the plate cells was equated using Yukio Shibo’s longitudinal equivalent stiffness model. The soil ontology is modeled using the Mohr–Coulomb model. Fixed hinge supports are applied at the base of the model, and free field units are symmetrically applied around the model to avoid the reflection of seismic waves at the model boundary.

This study is based on the shield tunnel in the South Lake section of the Two Lakes Tunnel Project. The shield tunnel passes through soft and hard uneven soil layers and karst-developed rock formations, with extremely different physical and mechanical parameters, which belong to complex geological conditions, as shown in Table 1. The input direction of seismic waves directly affects the bending direction, axial force and bending moment distribution of the tunnel. In this study, two types of seismic waves, basic earthquakes and rare earthquakes, were input along the axial and cross-sectional directions of the tunnel, and a total of four dynamic analysis conditions are shown in

Table 4. Single-layer lining longitudinal seismic internal force calculation working conditions.

Operating Conditions	Types of Seismic Waves	Acceleration Peak	Direction of Mass Vibration
1	Basic	0.05 g	Longitudinal along the tunnel (X)
2	Rare	0.1 g	Longitudinal along the tunnel (X)
3	Basic	0.05 g	Lateral along the tunnel (Y)
4	Rare	0.1 g	Lateral along the tunnel (Y)

.

In shield tunnel design, the controlling indicators are the tunnel axial force and bending moment. The axial force direction is along the tunnel axis, divided into longitudinal pressure and longitudinal tension; the bending moment can be divided into a transverse moment and longitudinal moment.

Along the tunnel axis, a total of 13 monitoring points were placed at key locations such as stratigraphic junctions, karst development and large tunnel depths to monitor the axial force and bending moment inside the slab unit. The relationship between the monitoring points and the stratigraphic model is shown in Figure 8.

4.3. Analysis of Longitudinal Internal Forces in the Longitudinal Equivalent Stiffness Model Structures

The results of the calculations for Case 1 are shown in Figure 9. Under the longitudinal excitation of the seismic load, the extreme value of the longitudinal interring pressure in the shield tunnel is 26,587 kN, the extreme value of the longitudinal interring tension is 21,875 kN, the extreme value of the vertical bending moment is 728.31 kN·m, and the extreme value of the transverse bending moment is 166.382 kN·m. Both the extreme value of the axial force and the extreme value of the bending moment occur at the junction of the strata. From Figure 9, we can see that in working condition 1, the axial pressure is greater than the axial tension, and the longitudinal bending moment is greater than the transverse bending moment. The results of the calculations for all working conditions are shown in

Table 5. Calculation of the longitudinal internal forces of the structure.

Case	Longitudinal Interring Pressure Extremes (kN)	Longitudinal Interring Tension Extremes (kN)	Vertical Bending Moment (kN·m)	Transverse Bending Moment (kN·m)
①	26,587	21,875	728.31	120
②	28,226	22,875.6	1200	166.382
③	20,993.2	25,504.3	237.185	1546.64
④	22,317.9	26,833.9	244.742	1580.05

. From Table 5, we can see that when the seismic wave is loaded longitudinally, the axial pressure is greater than the axial tension, and the longitudinal bending moment is greater than the transverse bending moment.

The average values of the axial force and bending moment at each monitoring point under different working conditions are shown in Figure 10. $J_3$ is the junction of medium-weathered mudstone and medium-weathered tuff, which is a soft and hard rock formation and is a seismic weak point in the shield tunnel. $J_2$ is slightly smaller than $J_1$ and $J_{13}$ but slightly larger than $J_{12}$. This is because the burial depth of $J_2$ is greater than that of $J_1$ and $J_{13}$ is greater than that of $J_{12}$. The greater the burial depth is, the less the seismic damage. Therefore, the minimum values of the axial force and bending moment occur at $J_7$, where the burial depth is the largest and the surrounding rock stiffness is high. As shown in Figure 10, $J_6$, $J_8$ and $J_7$ are similar at the depth of burial, but the internal force values are greater than $J_7$ because the presence of karst at $J_6$ and $J_8$ has a negative impact on the seismic resistance of the shield tunnel. Therefore, $J_{10}$ and $J_{11}$ have a greater depth of burial than $J_{12}$ but have more karst below them, resulting in greater axial forces and bending moments than $J_{12}$. The development of karst above the shield tunnel also leads to increased seismic damage, as monitoring point $J_6$ has a greater depth of burial than $J_8$, but its internal forces are still greater than $J_8$. Therefore, karst development is a weak point for seismic resistance in shield tunnels and should be reinforced during their design.

4.4. Parametric Analysis

Figure 11 shows the axial forces at each monitoring point for different seismic effects. From Figure 11a,b, it can be seen that the axial forces under the action of the rare earthquakes are greater than those under the action of the basic earthquakes. In Figure 11a,b, the lines from top to bottom are the pressure when the seismic wave is input in the X direction, the tension when the seismic wave is input in the Y direction, the tension when the seismic wave is input in the X direction, and the pressure when the seismic wave is input in the Y direction, respectively, and the lines hardly cross each other. This shows that (1) the axial force when the seismic wave is input along the X direction is always greater than the axial force when the seismic wave is input along the Y direction; (2) when the seismic wave is input along the X direction, the axial force is dominated by pressure; when the seismic wave is input along the Y direction, the axial force is dominated by tension.

Figure 12 shows the bending moments at each monitoring point for different seismic wave input directions. From Figure 12a,b, it can be seen that the structural bending moment under the action of the rare earthquake is greater than that under the action of the basic earthquake. From Figure 12a, it can be seen that when the seismic wave is input in the X direction, the bending moment within the structure is mainly vertical bending moment, which is much larger than the transverse bending moment. From Figure 12b, it can be seen that when the seismic wave is input along the Y direction, the bending moment within the structure is mainly dominated by the transverse bending moment, and the transverse bending moment is much larger than the vertical bending moment.

From Figure 11 and Figure 12, it can be obtained that:

(1) The internal force of the structure under rare earthquakes is greater than that under basic earthquakes.

(2) When the seismic wave is excited longitudinally, the structural axial force is larger, the axial force is dominated by pressure, and the bending moment in the structure is mainly vertical bending moment; when the seismic wave is excited laterally, the structural axial force is smaller, the axial force is dominated by tension, and the bending moment in the structure is mainly transverse bending moment.

4.5. Evaluation of Circumferential Cracks in Shield Tunnels

4.5.1. Tunnel Structure Force Test

The longitudinal corner of the shield tunnel, the concrete stress of the tube sheet and the bolt tensile stress under the action of bending moment and axial force can be found according to the following equations, where the load superposition principle is used to calculate the extreme value of the tunnel stress. Here, for example, parameters for Case ① are calculated:

Longitudinal corner of the tunnel:

$$\theta =\frac{l_s}{E_C{I}_C}\times \frac{\cos \phi +(\frac{\pi }{2}+\phi )}{{\cos }^3\phi }\times M$$

(20)

Concrete bending compressive stress of the pipe sheet:

$${\sigma }_c=E_c\cdot {\epsilon }_c+\frac{F}{A_c}$$

(21)

$${\sigma }_c=\frac{M}{I_c}\cdot \frac{\cos \phi +(\frac{\pi }{2}+\phi )\cdot \sin \phi }{{\cos }^3\phi }\cdot (\frac{D}{2}-\gamma \cdot \sin \phi )+\frac{F}{A_c}$$

(22)

Concrete bending tensile stress of the pipe sheet:

$${\sigma }_t=E_c\cdot {\epsilon }_t+\frac{F}{A_c}$$

(23)

$${\sigma }_t=\frac{M}{I_c}\cdot \frac{\cos \phi -(\frac{\pi }{2}-\phi )\cdot \sin \phi }{{\cos }^3\phi }\cdot (\frac{D}{2}+r\cdot \sin \phi )+\frac{F}{A_c}$$

(24)

Bolt bending tensile stress:

$${\sigma }_{st}=\frac{k_j\cdot {\delta }_j}{I_c}+\frac{F}{A_j}=\frac{k_j}{A_j}\cdot \frac{\pi \sin \phi }{\cos \phi +(\frac{\pi }{2}+\phi )\cdot \sin \phi }\cdot (r+r\sin \phi )\cdot \theta +\frac{F}{A_j\cdot N}$$

(25)

The longitudinal turning angles and stress values of the shield tunnel under the action of longitudinal bending moments and axial forces in each working condition in the previous section are shown in

Table 6. Shield tunnel longitudinal turning angle and stress values.

Case		Corner (Rad)	Concrete Bending Compressive Stress of Pipe Sheet (Pa)	Concrete Bending Tensile Stress of Pipe Sheet (Pa)	Bolt Bending Tensile Stress (Pa)
Seismic Category	Stimulus Direction
Basic	X	8.6 × 10−7	9.96 × 105	6.97 × 105	5.19 × 108
Rare	X	1.4 × 10−6	1.15 × 106	1.82 × 105	1.35 × 108
Basic	Y	1.8 × 10−6	1.07 × 106	7.31 × 105	5.44 × 108
Rare	Y	1.9 × 10−6	1.13 × 106	7.46 × 105	5.56 × 108

.

4.5.2. Calculation of the Ring Seam Tension

Calculation of the shield tunnel ring seam tension is based on the principle of superposition.

The elongation of the bolt under maximum tension is given by Equation (26):

$${\delta }_f=\frac{N{j}_{\max }}{n{E}_j{A}_j}$$

(26)

The tension of the joint under the maximum bending moment is given by Equation (27):

$${\delta }_t=\frac{M{s}_{\max }}{E_c{A}_c\frac{\pi \sin \phi }{{\cos }^3\phi }(\frac{D}{2}+r\sin \phi )}$$

(27)

Using the principle of superposition, the ring seam tension is given by Equation (28):

$$\delta ={\delta }_f+{\delta }_t$$

(28)

Substituting the tunnel structural parameters in Table 2 and the maximum tension and bending moments from the numerical simulations into Equations (26)–(28) [49], the total annular seam tension is $\delta =1.88 \mathrm{mm}$.

From the above calculation and analysis, it can be seen that the ring seam opening in the shield section of the Two Lakes Tunnel is 1.88 mm at the occurrence of a horizontal earthquake with a longitudinal 50-year exceedance probability of 2%, and according to the Code for Seismic Design of Highway Tunnels (JTG/T 2232-01-2019) [49], the interring seam opening is within 2~3 mm, which can be considered as within the elastic limit, so that safety is basically not a concern.

5. Conclusions

In this paper, the structural longitudinal dynamic response of the shield tunnel in the South Lake Tunnel are investigated. The main parameters such as excitation directions and peak acceleration earthquakes were considered. Subsequently, the obtained internal force values were used to calculate seismic indicators, including the longitudinal seismic angle of the shield tunnel, stress of the tube sheet and the bolt tensile stress. The following conclusions are obtained:

(1)

The maximum values of internal forces in different directions of excitation and peak accelerations occur in the interface of the soft and hard strata, where the shield tunnel structure has weak longitudinal seismic resistance.

(2)

Strata with a higher elastic modulus can effectively mitigate the effect caused by seismic waves, while the karst phenomenon reduces the shielding of seismic waves by the strata. Therefore, karst development was a weak location for seismic resistance in shield tunnels.

(3)

When seismic waves are excited longitudinally, the longitudinal axial force of the structure is larger. The maximum bending moment is mainly in the vertical plane, while the horizontal bending moment is smaller. The tunnel axial forces are smaller when the seismic waves are excited transversely. The maximum bending moment is dominated by bending in the horizontal plane, i.e., the transverse moment, while the bending in the vertical plane, i.e., the vertical moment, is smaller.

(4)

In the case of a rare earthquake with a 50-year exceedance probability of 2% in the South Lake section of the Two Lakes Tunnel shield tunnel, the annular seam tension is 1.88 mm, which meets the longitudinal tension requirement.