Study on Longitudinal Stress Relaxation Effect and Reinforcement Technology of Segment Lining during Shield Docking

Abstract

Shield docking technology is widely used in underwater shield tunnels. As the technology is not perfect, the instability of the tunnel face and the stress relaxation of segment lining caused by shield dismantling need to be solved. Using a cross-sea tunnel project, the segment lining reinforcement technology with steel channel in the process of shield docking is studied. The longitudinal stress relaxation effect of segment lining after thrust unloading is analyzed, and the theoretical solution of segment circumferential joint opening, with or without steel channel reinforcement, is derived. The accuracy of the theoretical model and the shell connector numerical model is verified by comparison. Then, the influence of key parameters, such as reinforcement range, steel channel quantity and steel channel model, on reinforcement effect is discussed. The results show that: (1) with the increase in the range of reinforcement, the maximum longitudinal displacement and the opening of the circumferential joints decrease first and then increase; (2) with the increase in the quantity and model of the steel channel, the maximum longitudinal displacement and the opening of the circumferential joints gradually decrease, and the reduction rate also decreases. The farther away from the shield tail, the weaker the reinforcement effect on segment lining.

1. Introduction

With the rapid development of China’s rail transit engineering systems, shield tunneling technology has become more and more perfect, and underwater tunnels have gradually become the norm [1,2]. As for the cross-sea shield tunnel, due to difficulties such as large span, long construction period and complex stratum conditions, the shield docking technology has emerged because of necessity [3]. This technology can greatly shorten the construction period, reduce costs and increase social benefits, which is of great significance for underwater shield tunnel construction [4,5].

According to different construction methods, shield docking technology can be divided into the civil docking method and the mechanical docking method. In the civil docking method, the stratum of the docking section is strengthened by grouting reinforcement or freezing reinforcement, and then the shield is disassembled and the tunnel lining is applied. The mechanical docking method involves directly docking the specially designed shield, including two methods of mechanical shield docking (MSD) and concentric interlace docking (CID). Shield docking technology has high requirements of docking accuracy, tunneling face stability and structural safety. In the shield machine disassembly process, thrust unloading can cause problems such as instability of the tunneling face, stress relaxation of segment lining and joint leakage. A serious safety accident will be caused by slight carelessness [6]. At present, some scholars have carried out relevant studies on segment lining structures [7,8,9,10,11,12,13,14,15,16,17,18,19,20] and shield reinforcement technology [21,22,23,24,25,26,27,28,29,30].

In the field of segment lining structures, the structural stiffness of the whole ring will change with k_θ and further affect the transmission and distribution of structural internal forces and deformation as well, which has been found through model shaking table tests and two-dimensional finite element dynamic analysis [7]. A simplified method for evaluating the moment carrying of a segmental tunnel liner that involves examining the influence of segmental joints, the number of segments and the soil subgrade modulus on the bending moment carrying characteristics of a segmental tunnel has been proposed [8]. The longitudinal stress relaxation characteristics of a shield tunnel in soft soil are studied via field tests [12]. The behavior of the load transfer mechanism of a shield-constructed tunnel in the longitudinal direction has been analyzed, and the distribution of stress and deformation in the tunnel lining was obtained. Additionally, the effects of circumferential and radial stiffness on the behavior of deformation and internal forces of tunnel lining have been discussed in detail [13]. An integrated framework based on laboratory experiments, real-time monitoring and statistical theory has been developed. As a case study, this presented method was employed in a typical underwater shield tunnel to prevent leakage disasters [16]. A risk assessment methodology that aims to evaluate the risks of the circumferential cracks of preexisting shield tunnels was proposed to ensure the security of existing tunnels under deep excavation during construction [17]. Considering the stratum with multiple discontinuities, a method was proposed to evaluate the longitudinal deformation caused by potential disturbance after shield tunnel construction, which can better predict the deformation characteristics of a shield tunnel, such as settlement, dislocation and the opening of joints [18]. When a shield tunnel is subjected to earthquakes and longitudinal deformation of the stratum, the segment ring may open along the longitudinal direction, and the tensile characteristics and failure modes of circumferential joints and bolted joints can be analyzed [19,20]. The above studies are aimed at the phenomenon of the lining structure settlement, dislocation and opening of circumferential joints under the action of soil and water load or earthquake load, but they are not applicable to the calculation of circumferential joint opening after thrust unloading.

In the field of mechanical shield docking, the Trans-Tokyo Bay Highway was excavated using the slurry shield method. After the completion of tunneling, the two shields facing each other were docked together under a high water pressure at the depth of about 60 m under the sea level [21,22]. The test boring, the filling of high-concentration slurry, the freezing and forced thawing, etc., were carried out during underground shield docking. A newly developed relative position measuring system was provided. Applying this system, the docking of two shield machines was performed accurately within a few millimeters of relative distance deviation [23]. In addition, an insertion system, a cylindrical magnifying mortar shield machine (capable of underground mechanical docking of tunnels of different diameters) was developed [24]. In China, the feasibility of applying the freezing method in shield joint of the Qiongzhou Strait tunnel was studied [26]. Using the model test of freezing reinforcement for a shield junction, the influence law of strata deformation caused by the frost heave effect was obtained and the distribution characteristics of the temperature field in the soil stratum during the freezing process were analyzed [27,28]. In the process of the shield docking of a cable tunnel in Taiwan, broken seal plate bolts caused the ground to collapse. Ground improvement was suggested, even while using the concentric interlace docking method [29]. The possibility of long-distance and high-speed construction using a shield tunneling method was discussed through a case study on the planning of an undersea tunnel project in Osaka Bay. If the mechanical shield docking method were used, it would only take 40 months to construct the tunnel [30]. As a new technology for shield tunneling, the theoretical method of shield docking technology is not mature; in particular, the model and theory of longitudinal stress relaxation between segment rings after shield dismantling and thrust unloading are lacking. Most of the existing studies focus on the docking method and formation reinforcement, but there are few studies on the longitudinal reinforcement of the segment ring in the docking process. Currently, longitudinal reinforcement technology for segments mainly include adding steel channels, welding steel plates and erecting steel arch frames. These reinforcement measures are all empirical methods, and the influence of key parameters on the reinforcement effect has not been studied and summarized in depth.

This paper takes the Qiongzhou Strait undersea tunnel to be built in China as the research object. The purpose is to solve the urgent problems, such as the instability of the lining structure and the seawater leakage caused by the longitudinal stress relaxation effect during the docking process, relying on channel steel reinforcement technology. The mechanical mechanism of the longitudinal stress relaxation effect of the segment lining after thrust unloading is analyzed. The theoretical solution for circumferential joint opening with or without steel channel reinforcement is derived using the established mechanical model. Through the numerical simulation of the typical section of the supporting project, the results of the numerical simulation and the theoretical calculation are compared to verify each other. The opening of the circumferential joints is taken as the judgment standard, and then the effect of parameters, such as reinforcement range and the quantity and model of the steel channel, on reinforcement effect is discussed. On the basis of a sensitivity analysis of key parameters, the feasibility of the segmenting reinforcement technology for a steel channel is determined, and a reasonable reinforcement scheme for a steel channel is proposed.

2. Methodology

In the process of shield docking, the shield machine needs to be disassembled, and the longitudinal stress relaxation effect will occur in the segment lining after thrust unloading. Considering the stress process of segment lining, a new mechanical model will be established to derive the analytical solution of the circumferential joint opening with or without steel channel reinforcement after thrust unloading.

2.1. Establishment of the Mechanical Model

Segments are assembled into rings within the shield shell by the segment erector. In the assembly process, the sealing gaskets and rubber gaskets in the circumferential joints are compressed under shield thrust. The segment rings are then pushed out one by one as the shield drives forward. The compression force between rings is always equal to the shield thrust F_t, and the opening of circumferential joints is a fixed value x₀, as shown in Figure 1.

In the process of shield underground docking, the disassembly operation will cause shield thrust unloading. The sealing gaskets, rubber gaskets and longitudinal bolts between segment rings will be stretched longitudinally under the action of segment weight and friction force f, and the opening of the circumferential joint will change to x_i. The load characteristics and longitudinal compression forces of the segment ring change, eventually reaching a new equilibrium, as shown in Figure 2.

According to the above analysis, the force between segment rings undergoes two processes of compression and unloading during shield docking and disassembly. In the compression process, due to the extremely strict requirements on the formation condition and accuracy control during underground docking, the shield thrust always remains stable, and the initial opening of circumferential joints under thrust is fixed x₀. In the thrust unloading process, the compression force is released, which is resisted by lining–stratum friction and the opening of circumferential joints. The derivation process is complicated and difficult to understand if the opening of circumferential joints is calculated according to the directly released compression force.

In this paper, the above processes are considered with a new idea and the mechanical model is simplified. As shown in Figure 3, the thrust unloading process is considered to be a tension process connected by multiple blocks. To simplify the mechanical model, thrust unloading is assumed to apply an equal and opposite tensile force, and the circumferential joint is considered to be an incompressible spring. The release of the compression force between the ring circumferential joint is equivalent to the same tensile force, and the compression modulus of the ring’s circumferential joint is equivalent to the tensile stiffness of the spring. Each segment ring is assumed to be a separate block. The initial opening of circumferential joint x₀ is the initial distance between the blocks which are connected by a variable stiffness spring k.

The calculation mechanism in Figure 3 is simple and clear, and can be used to accurately calculate the theoretical value of the opening after thrust unloading without any reinforcement measures. In practical engineering, thrust unloading leads to the excessive opening of circumferential joints, which leads to waterproofing failure and even the instability of the lining structure. Therefore, longitudinal reinforcement, such as adding a steel channel, is usually carried out on the segment structure before thrust unloading. Steel channel reinforcement, connecting the first segment ring and the back segment ring, can greatly reduce the openings of circumferential joints. According to the reinforcement principle of steel channels, the steel channel is equivalent to the reinforcement spring K, and the segment ring load model under the reinforcement condition of steel channels is established, as shown in Figure 4.

Since the mechanical model of the circumferential joint opening after thrust unloading is not involved in the existing studies, a mechanical model is established, as shown in Figure 4, to analyze the longitudinal stress relaxation effect of the segment ring after shield docking and dismantling. In this model, considering the longitudinal steel channel reinforcement of the segment ring, the two processes of compression and unloading of the segment ring during the shield docking and dismantling are optimized into a multi-block tension process. The tedious interaction process involving compressive stress between rings and the tensile deformation of steel channel is simplified to the process of the variable stiffness springs between rings and the equivalent reinforcement springs jointly resisting the equivalent tensile force F_t. It is convenient to understand the complex mechanical relationship of the model and derive the theoretical solution of the circumferential joint opening after thrust unloading.

2.2. Basic Equation of Circumferential Joints

The segment of the underwater shield tunnel typically adopts two sealing gaskets, which are mainly composed of EPDM rubber, and a rubber gasket is set in the middle as a pressure gasket. The above mechanical analysis model considers thrust unloading as a reverse tensile process, which relies on the tensile deformation of sealing gaskets, rubber gaskets and longitudinal bolts to resist tension. The circumferential joint equivalent spring stiffness k_c is a piecewise function, including equivalent stiffness k_c1 composed of bolts and sealing gaskets and equivalent stiffness k_c2 composed of bolts, sealing gaskets and rubber gaskets. In the process of reverse stretching, the equivalent stiffness k_c2 is used to resist the tensile deformation at first. After the opening of circumferential joints increases to the point that the rubber gaskets are out of contact, the equivalent stiffness k_c1 will resist deformation until the sealing gaskets are out of contact.

The basic equation of circumferential joint structure is thus derived, and displacement is based on force:

$$\mathrm{x}(\mathrm{F})={\mathrm{x}}_0+\mathrm{\Delta }\mathrm{x}(\mathrm{F})=\{\begin{array}{l}{\mathrm{x}}_0+\frac{\mathrm{F}}{{\mathrm{k}}_{\mathrm{c}2}} \mathrm{F}<{\mathrm{F}}_{\mathrm{c}}\\ {\mathrm{x}}_0+\frac{{\mathrm{F}}_{\mathrm{c}}}{{\mathrm{k}}_{\mathrm{c}2}}+\frac{\mathrm{F}-{\mathrm{F}}_{\mathrm{c}}}{{\mathrm{k}}_{\mathrm{c}1}} \mathrm{F}>{\mathrm{F}}_{\mathrm{c}}\end{array}$$

(1)

Force is also based on displacement:

$${\mathrm{F}}_{\mathrm{k}}(\mathrm{\Delta }\mathrm{x})=\{\begin{array}{l}\mathrm{\Delta }\mathrm{x}{\mathrm{k}}_{\mathrm{c}2} \mathrm{\Delta }\mathrm{x}<\mathrm{\Delta }{\mathrm{x}}_{\mathrm{c}}\\ \mathrm{\Delta }{\mathrm{x}}_{\mathrm{c}}{\mathrm{k}}_{\mathrm{c}2}+(\mathrm{\Delta }\mathrm{x}-\mathrm{\Delta }{\mathrm{x}}_{\mathrm{c}}){\mathrm{k}}_{\mathrm{c}1} \mathrm{\Delta }\mathrm{x}>\mathrm{\Delta }{\mathrm{x}}_{\mathrm{c}}\end{array}$$

(2)

where F_c is the tension force that moves the rubber gaskets out of contact.

2.3. Theoretical Calculation of the Opening of Circumferential Joints

2.3.1. Unreinforced Condition

The mechanism of the thrust unloading segment lining load model established in Figure 4 is simple and clear. According to the above model, the number of rings affected by thrust unloading without any reinforcement n₀ is:

$${\mathrm{n}}_0=\mathrm{int}(\frac{{\mathrm{F}}_{\mathrm{t}}}{\mathrm{f}})$$

(3)

The number of rings which the rubber cushion is out of contact with n_c0 is:

$${\mathrm{n}}_{\mathrm{c}0}=\mathrm{int}(\frac{{\mathrm{F}}_{\mathrm{c}}}{\mathrm{f}})$$

(4)

The variation of the circumferential joints Δx_i is:

$$\mathrm{\Delta }{\mathrm{x}}_{\mathrm{i}}=\mathrm{\Delta }\mathrm{x}(\mathrm{F}-\mathrm{i}\mathrm{f}) (1\le \mathrm{i}\le {\mathrm{n}}_0)$$

(5)

$$\mathrm{\Delta }{\mathrm{x}}_{\mathrm{i}}=0 (\mathrm{i}>{\mathrm{n}}_0)$$

(6)

As shown in Figure 5, it is obvious that the variation of circumferential joints decreases in proportion. When the variation of circumferential joints is less than Δx_c, the variation of circumferential joints decreases by f/k_c2 per ring. In this case, the tensile force needs to overcome the bolts, sealing gaskets and rubber gaskets. When the variation of circumferential joints is greater than Δx_c, the variation of circumferential joints decreases by f/k_c1 per ring. In this case, the rubber gaskets are out of contact, and there are only bolts and sealing gaskets in the circumferential joints.

2.3.2. Steel Channel Reinforcement Condition

As for the segment ring load model established in Figure 4 under the condition of steel channel reinforcement, due to the presence of a steel channel, the longitudinal displacement of any segment ring will change the overall force. The longitudinal displacement of a segment ring must overcome both k_c, the circumferential joint spring stiffness, and K, the equivalent spring stiffness of the steel channel under the action of tension F_t, so the structural stress analysis is complicated.

Assuming that the reinforcement range is m, the segment lining is divided into the reinforced area and unreinforced area. The longitudinal mechanical balance of each block in the reinforcement area is analyzed. Firstly, determine n and n_c after thrust unloading under the condition of steel channel reinforcement:

$$\mathrm{n}=\mathrm{int}(\frac{{\mathrm{F}}_{\mathrm{k}}(\mathsf{\Delta }{\mathrm{x}}_1)}{\mathrm{f}})$$

(7)

$${\mathrm{n}}_{\mathrm{c}}=\mathrm{int}(\frac{{\mathrm{F}}_{\mathrm{k}}(\mathsf{\Delta }{\mathrm{x}}_1)-{\mathrm{F}}_{\mathrm{c}}}{\mathrm{f}})$$

(8)

When the reinforcement range m ≥ n₀, establish the longitudinal mechanical balance equation matrix:

$$\mathrm{K}•\mathrm{X}=\mathrm{F}$$

(9)

$$\left(\left[\begin{array}{cccc}-{\mathrm{F}}_{\mathrm{k}}(\mathrm{\Delta }\mathrm{x})& & & 0\\ {\mathrm{F}}_{\mathrm{k}}(\mathrm{\Delta }\mathrm{x})& -{\mathrm{F}}_{\mathrm{k}}(\mathrm{\Delta }\mathrm{x})& & \\ & \ddots & \ddots & \\ 0& & {\mathrm{F}}_{\mathrm{k}}(\mathrm{\Delta }\mathrm{x})& -{\mathrm{F}}_{\mathrm{k}}(\mathrm{\Delta }\mathrm{x})\end{array}\right]+\left[\begin{array}{c}{\mathrm{K}}_{1\times \mathrm{n}}\\ 0_{(\mathrm{n}-2)\times \mathrm{n}}\\ ⋮\\ {\mathrm{K}}_{1\times \mathrm{n}}\end{array}\right]\right)•\left[\begin{array}{c}\mathrm{\Delta }{\mathrm{x}}_1\\ \mathrm{\Delta }{\mathrm{x}}_2\\ ⋮\\ \mathrm{\Delta }{\mathrm{x}}_{\mathrm{n}}\end{array}\right]=\left[\begin{array}{c}\mathrm{f}-{\mathrm{F}}_{\mathrm{t}}\\ \mathrm{f}\\ ⋮\\ \mathrm{f}\end{array}\right]$$

(10)

The variation of circumferential joints Δx is:

$$\mathrm{\Delta }{\mathrm{x}}_{\mathrm{i}-1}-\mathrm{\Delta }{\mathrm{x}}_{\mathrm{i}}=\frac{\mathrm{f}}{{\mathrm{k}}_{\mathrm{c}1}} (2\le \mathrm{i}\le {\mathrm{n}}_{\mathrm{c}})$$

(11)

$$\mathrm{\Delta }{\mathrm{x}}_{{\mathrm{n}}_{\mathrm{c}}+1}=\mathrm{\Delta }{\mathrm{x}}_1-\frac{{\mathrm{F}}_{\mathrm{c}}-\mathrm{f}}{{\mathrm{k}}_{\mathrm{c}2}}-\frac{\mathrm{f}({\mathrm{n}}_{\mathrm{c}}+1)-{\mathrm{F}}_{\mathrm{c}}}{{\mathrm{k}}_{\mathrm{c}1}}$$

(12)

$$\mathrm{\Delta }{\mathrm{x}}_{\mathrm{i}}-\mathrm{\Delta }{\mathrm{x}}_{\mathrm{i}+1}=\frac{\mathrm{f}}{{\mathrm{k}}_{\mathrm{c}1}} ({\mathrm{n}}_{\mathrm{c}}+1\le \mathrm{i}\le \mathrm{n}-1)$$

(13)

$$\mathrm{\Delta }{\mathrm{x}}_{\mathrm{i}}=0 (\mathrm{i}>\mathrm{n})$$

(14)

$${\displaystyle \sum _{\mathrm{i}=1}^{\mathrm{n}-1}\mathrm{\Delta }{\mathrm{x}}_{\mathrm{i}}}=\mathrm{\Delta }{\mathrm{x}}_1{\mathrm{n}}_{\mathrm{c}}+\mathrm{\Delta }{\mathrm{x}}_{{\mathrm{n}}_{\mathrm{c}}+1}(\mathrm{n}-{\mathrm{n}}_{\mathrm{c}})-\frac{{\mathrm{n}}_{\mathrm{c}}({\mathrm{n}}_{\mathrm{c}}-1)\mathrm{f}}{2{\mathrm{k}}_{\mathrm{c}2}}-\frac{(\mathrm{n}-{\mathrm{n}}_{\mathrm{c}})(\mathrm{n}-{\mathrm{n}}_{\mathrm{c}}-1)\mathrm{f}}{2{\mathrm{k}}_{\mathrm{c}1}}$$

(15)

When the reinforcement range 0 < m < n₀, establish the longitudinal mechanical balance equation matrix:

$$\mathrm{K}•\mathrm{X}=\mathrm{F}$$

(16)

$$\left(\left[\begin{array}{cccc}-{\mathrm{F}}_{\mathrm{k}}(\mathrm{\Delta }\mathrm{x})& & & 0\\ {\mathrm{F}}_{\mathrm{k}}(\mathrm{\Delta }\mathrm{x})& -{\mathrm{F}}_{\mathrm{k}}(\mathrm{\Delta }\mathrm{x})& & \\ & \ddots & \ddots & \\ 0& & {\mathrm{F}}_{\mathrm{k}}(\mathrm{\Delta }\mathrm{x})& -{\mathrm{F}}_{\mathrm{k}}(\mathrm{\Delta }\mathrm{x})\end{array}\right]+\left[\begin{array}{c}{\mathrm{K}}_{1\times \mathrm{n}}\\ 0_{(\mathrm{m}-2)\times \mathrm{n}}\\ {\mathrm{K}}_{1\times \mathrm{n}}\\ 0_{(\mathrm{n}-\mathrm{m})\times \mathrm{n}}\end{array}\right]\right)•\left[\begin{array}{c}\mathrm{\Delta }{\mathrm{x}}_1\\ \mathrm{\Delta }{\mathrm{x}}_2\\ ⋮\\ \mathrm{\Delta }{\mathrm{x}}_{\mathrm{n}}\end{array}\right]=\left[\begin{array}{c}\mathrm{f}-{\mathrm{F}}_{\mathrm{t}}\\ \mathrm{f}\\ ⋮\\ \mathrm{f}\end{array}\right]$$

(17)

When m < n_c, the variation of circumferential joints Δx is:

$$\mathrm{\Delta }{\mathrm{x}}_{\mathrm{i}-1}-\mathrm{\Delta }{\mathrm{x}}_{\mathrm{i}}=\frac{\mathrm{f}}{{\mathrm{k}}_{\mathrm{c}1}} (2\le \mathrm{i}\le \mathrm{m})$$

(18)

When m > n_c, the variation of circumferential joints Δx is:

$$\mathrm{\Delta }{\mathrm{x}}_{\mathrm{i}-1}-\mathrm{\Delta }{\mathrm{x}}_{\mathrm{i}}=\frac{\mathrm{f}}{{\mathrm{k}}_{\mathrm{c}1}} (2\le \mathrm{i}\le {\mathrm{n}}_{\mathrm{c}})$$

(19)

$$\mathrm{\Delta }{\mathrm{x}}_{{\mathrm{n}}_{\mathrm{c}}+1}=\mathrm{\Delta }{\mathrm{x}}_1-\frac{{\mathrm{F}}_{\mathrm{c}}-\mathrm{f}}{{\mathrm{k}}_{\mathrm{c}2}}-\frac{\mathrm{f}({\mathrm{n}}_{\mathrm{c}}+1)-{\mathrm{F}}_{\mathrm{c}}}{{\mathrm{k}}_{\mathrm{c}1}}$$

(20)

$$\mathrm{\Delta }{\mathrm{x}}_{\mathrm{i}}-\mathrm{\Delta }{\mathrm{x}}_{\mathrm{i}+1}=\frac{\mathrm{f}}{{\mathrm{k}}_{\mathrm{c}1}} ({\mathrm{n}}_{\mathrm{c}}+1\le \mathrm{i}\le \mathrm{m}-1)$$

(21)

It can be seen from the relationship curve in Figure 6 that steel channel reinforcement can effectively reduce the variation of circumferential joints within the reinforcement range, and the reduction is related to the equivalent stiffness of the steel channel. There is a distinct change at the m-ring, and the variation of circumferential joints after m is equal to that no reinforcement. It is worth noting that the curve slope of the opening is the same before and after reinforcement, but the variation of circumferential joints in the unreinforced area does not change. The reason is that the method of adding a steel channel is equivalent to transferring the internal force of the segment lining and connecting the segment lining within the strengthened scope into a whole with greater strength to reduce the variation of circumferential joints. The steel channel reinforcement has no external force on the segment lining, so it will not affect the variation of circumferential joints in the unreinforced area.

3. Comparative Validation

The typical section of the project is selected to establish the numerical model. The results of the numerical simulation and theoretical calculation are compared and verified, which lays a foundation for further parametric analysis.

3.1. Numerical Simulation

3.1.1. Selection of Computing Model

Considering the characteristics of high water pressure, large diameter and longitudinal mechanical characteristics, the shell connector model is selected as the calculation model, which has the following advantages:

(1)

The segment lining adopts a thick shell element, which can be provided with a reinforcement layer.

(2)

The structure load is applied in the form of field load, and the distribution mode and size are more suitable for the actual project.

(3)

The connection mode of the numerical model is improved. The connector element is used to simulate the interaction between segments. Compared with the traditional spring connection and friction contact, the complex node coupling and other operations are omitted, which greatly saves modeling time and improves the simulation accuracy.

(4)

The connector unit shown in Figure 7 includes a radial spring element, a tangential spring element and a failure control, where the spring element can be set with nonlinear stiffness to model different tension compression stiffnesses and the bending stiffness of bolts.

3.1.2. Simulation of Segment Joints

The segment ring adopts the form of “7 + 1”, including four standard blocks, two adjacent blocks and one key block. There are two bolt joints at the longitudinal joint of each segment. Three bolt joints are provided at the circumferential joints of the standard block and the adjacent block, and only one bolt joint is provided at the circumferential joints of the key block. All nodes between segments are connected with connector units.

Since the opening of circumferential joints after thrust unloading is the focus of the study, the simulation of circumferential joints must be very accurate. For the connector unit at the non-bolted joint, the nonlinear compressive stiffness and linear shear stiffness are set to simulate the sealing gasket and rubber gasket in the circumferential joints, and the failure criterion is given to ensure that the connector unit at the non-bolted joint does not produce tensile stress. The connector unit at the bolted joint not only sets nonlinear compressive stiffness and linear shear stiffness, but also adds flexural stiffness and tensile stiffness to simulate the connection effect of bolts.

Table 1. Material parameters in circumferential joints.

Material	Compression Modulus	Poisson’s Ratio
Segment (C50)	34.5 Gpa	0.2
Bolt (Q235)	210 Gpa	0.3
Sealing gasket (EPDM)	7 Mpa	0.3
Rubber gasket (NBR)	25 Mpa	0.35

lists the parameters of each material.

In this simulation, according to engineering practice, a 20 mm interval is set at the segment circumferential joint, which is the total width before thrust is applied. The thickness of the sealing gasket is 20 mm, and the thickness of the rubber gasket is 4 mm. The sealing gasket is compressed within the initial 16 mm range, and the rubber gasket begins to be compressed within the 16 mm–20 mm range. When the compression exceeds 20 mm, the segments contact each other and begin to be compressed. Since the segment will not be compressed in the actual project, the compression stiffness of the connector element k_s in the circumferential joints is assumed to be the following piecewise function:

$${\mathrm{k}}_{\mathrm{s}}=\{\begin{array}{l}{\mathrm{k}}_{\mathrm{s}1} 20\le \mathrm{t}\le 4 \\ {\mathrm{k}}_{\mathrm{s}2} 4 \le \mathrm{t}\le 0\end{array}$$

(22)

Then, according to the following formula, the compressive modulus of the sealing gasket and the rubber gasket is converted into the compressive stiffness of the connector unit:

$$\mathrm{k}=\frac{\mathrm{E}\mathrm{S}}{\mathrm{l}}$$

(23)

where t is the opening of the circumferential joints, l is the length of the connector unit and S is the cross-sectional area of the material.

3.1.3. Numerical Simulation of Typical Sections

Relying on a sea-crossing tunnel project, the tunnel adopts the construction method of MSD. Two slurry balance shields with a diameter of 14 m are excavated towards each other and docked on the seabed. The shield tunnel lining is assembled from prefabricated segments with a thickness of 0.6 m, and the segments are connected by diagonal bolts. The stratum sections from top to bottom are silt, silty clay and strongly weathered tuff, and the excavation section is a single strongly weathered tuff stratum. The top of the tunnel is about 60 m above sea level, and the thickness of the covering soil is about 40 m. The formation section of the simulation section is shown in Figure 8, and the formation parameters are shown in

Table 2. Formation parameters.

Material	Dry Density (kg/m3)	Elasticity Modulus (MPa)	Poisson’s Ratio	Cohesion (kPa)	Frictional Angle (°)	Lateral Pressure Coefficient
Silt	1800	40	0.3	2	31.6	0.7
Silty clay	1600	20	0.25	22	16	0.65
Strongly weathered tuff	2200	2580	0.21	180	13	0.6

.

The above typical sections are selected for simulation, and it is assumed that the thrust load is removed after the steel channel reinforcement is completed. Then, the maximum longitudinal displacement of the segment lining and the opening of the circumferential joints are calculated and analyzed.

3.2. Analysis of Calculation Results

The opening of circumferential joints is significantly affected by thrust and stratum friction, but little by water and soil pressure. Considering factors such as the assembling position of segments and unsolidified synchronized grout in actual engineering, it is assumed in the simulation that the first four-ring segment is not affected by the formation friction. Figure 9 is a cloud diagram of the longitudinal displacement of the segment lining without any reinforcement measures after thrust unloading. In Figure 9, the left side is the front section of the tunnel. It can be seen from the figure that longitudinal stress relaxation occurs in the segment lining after thrust unloading. The whole tunnel has a longitudinal displacement towards the front section of the tunnel. Meanwhile, the circumferential joints change significantly, which is manifested as the opening of circumferential joints in the front section of the tunnel increases significantly.

Figure 10 is the relationship curve of the opening of circumferential joints after thrust unloading. The simulation results show that thrust unloading causes longitudinal stress relaxation in approximately 18 ring segments. The maximum longitudinal displacement reaches 73.33 mm, and the initial opening of the circumferential joints x₀ is 3.124 mm. The maximum opening is 20 mm in the first four rings, which then decreases by 0.071 mm one by one. After 18 rings, the openings of the circumferential joints are reduced to 3.124 mm and does not change.

3.3. Comparative Validation

According to the application situation in the actual project, this simulation selects 24 steel channels, with the steel channel model of No. 14 and the reinforcement range of 20 rings as the basic case. Figure 11 is the comparison diagram of the theoretical calculations and numerical simulation under the basic case. It can be seen that the two calculation results have the same change rule, but the numerical value is slightly different. The maximum opening of the theoretical calculation and the numerical simulation are 5.89 mm and 5.97 mm, respectively, with an error of 1.36%. The error of the results obtained by the two methods is extremely small by comparison. The accuracy of the two methods for calculating the opening of circumferential joints under the steel channel reinforcement is verified, which can be used for further parameter analysis and research. In the follow-up analysis and research, the method of numerical simulation is used to explore the effect of parameters such as reinforcement range, steel channel quantity and steel channel model on the opening of circumferential joints.

4. Parametric Analysis

4.1. Critical Opening of Circumferential Joints

In practical engineering, the addition of a steel channel is performed to solve the problem of the longitudinal stress relaxation of the segment lining. Therefore, when studying the influence of key parameters on the reinforcement effect, the opening of circumferential joints should be taken as the judgment standard. There are three main methods used to determine the critical opening of circumferential joints, and the minimum of the three methods is calculated and selected as the critical value.

(1)

According to the “Code for Shield Tunnel Construction and Acceptance (GB50446-2008)”, the critical opening of circumferential joints should not exceed 6 mm.

(2)

According to the actual project, the theoretical calculation is carried out as follows:

$$\mathsf{\delta }\ge \mathrm{B}·\mathrm{D}/({\mathsf{\rho }}_{\min }-\mathrm{D}/2)+{\mathsf{\delta }}_0+{\mathsf{\delta }}_1\ge 0.008$$

(24)

where: δ is the critical opening of the circumferential joints; B is the segment width; D is the tunnel diameter; ρ_min is the minimum curve radius of the tunnel longitudinal deformation curve; δ₀ is the circumferential clearance caused by production and construction errors, generally taken as 0.002 m; δ₁ is the circumferential clearance caused by mechanical assembly, generally taken as 0.002 m.

(3)

Under the condition of high water pressure in cross-sea tunnels, the leakage water of shield tunnels is closely related to the opening of circumferential joints, which can be determined according to the waterproof mechanism of sealing gaskets. The critical opening of this method is:

$$\mathsf{\delta }\ge \frac{{\mathsf{\sigma }}_{\mathrm{w}}\mathrm{l}}{\mathrm{E}}\ge 0.0084$$

(25)

where σ_w is the water pressure.

Given the results of the above three methods, the minimum value of 6 mm is selected as the critical opening of circumferential joints.

4.2. Effect of Reinforcement Range

Based on the numerical simulation of the above typical cross-sections, under the premise of keeping other parameters of the model unchanged, only the reinforcement range of the steel channel is changed. The equivalent spring stiffness in different cases is shown in

Table 3. Equivalent spring stiffness of steel channel under different reinforcement ranges.

Reinforcement Range	Reinforcement Length (m)	Equivalent Spring Stiffness (N/m)
4	8	1.34 × 109
8	16	6.71 × 108
12	24	4.48 × 108
16	32	3.36 × 108
20	40	2.69 × 108

.

Figure 12 shows that increasing the reinforcement range can effectively reduce the maximum longitudinal displacement of the segment lining. With the increase in the reinforcement range, the maximum longitudinal displacement decreases sharply at first, and then increases linearly with a slow increase rate. When the reinforcement range is eight rings, the maximum longitudinal displacement is the smallest, which is 10.01 mm. The effect is most obvious when the front four-ring segment is reinforced, and the maximum longitudinal displacement is greatly reduced. When the reinforcement range is greater than eight rings, the maximum longitudinal displacement increases slowly as the reinforcement range further increases.

It can be seen from Figure 13 that under the same reinforcement range, the farther the segment ring is from the shield tail, the smaller the opening of circumferential joints x_i. When the reinforcement range is less than five rings, since the first four-ring segments are not affected by the formation friction, they will be completely separated. With the increase in the reinforcement range, the equivalent spring stiffness K gradually decreases, and the reinforcement effect is weakened. When the reinforcement range is 20 rings, the maximum opening of the circumferential joints x_i is 5.97 mm.

As shown in Figure 14, the variation trend of the opening of circumferential joints x_i and the longitudinal maximum displacement is similar. The difference is that when the reinforcement range just reaches the ring, the opening of circumferential joints x_i suddenly changes to the minimum. The first four-ring segment has the best reinforcement effect, which is the position with the largest opening of circumferential joints. When the reinforcement reaches 20 rings, the opening of the circumferential joints is 5.97 mm, which is less than the critical opening of circumferential joints. Due to the existence of formation friction, the maximum opening of subsequent circumferential joints is only 3.94 mm. Steel channel reinforcement will change the opening of subsequent circumferential joints slightly, but the reinforcement effect is not obvious.

From the above analysis, it can be seen that the closer to the shield tail, the more significant the reinforcement effect is, and the change of the reinforcement range has no effect on the opening of circumferential joints after 18 rings. The reason for this phenomenon is that friction force between the segment lining and the soil leads to the limited influence range and the effect of thrust unloading on the segment lining. The method of adding a steel channel is equivalent to transferring the internal force of the segment lining, and no external force is applied, so the change of the segment reinforcement range will not affect the opening of circumferential joints after 18 rings. Considering engineering safety factors, if the equipment facilities and construction conditions are allowed, it is recommended to reinforce all segments within the impact range of thrust unloading, which not only meets the requirements for the opening of circumferential joints, but also ensures construction safety.

4.3. Effect of Quantity of Steel Channels

Given the numerical simulation of the above typical cross-sections, under the premise of keeping other parameters of the model unchanged, only the quantity of the steel channels is changed. The equivalent spring stiffness in different cases is shown in

Table 4. Equivalent spring stiffness of steel channels under different quantities.

The Quantity of Steel Channels	The Total Area (m2)	Equivalent Spring Stiffness (N/m)
8	1.71 × 10−2	8.95 × 107
16	3.41 × 10−2	1.79 × 108
24	5.16 × 10−2	2.69 × 108
32	6.82 × 10−2	3.58 × 108

.

Figure 15 shows that increasing the quantity of channels can effectively reduce the longitudinal displacement of the segment lining. With the increase in the quantity of steel channels, the maximum longitudinal displacement gradually decreases, and the reduction rate also decreases.

As shown in Figure 16 and Figure 17, the variation trend of the opening of circumferential joints is roughly the same under the different quantities of steel channels. With the increase in the quantity of steel channels, the opening of circumferential joints becomes smaller and smaller, and with the increase in the quantity of steel channels, the opening of circumferential joints decreases gradually, and the reduction rate also decreases. The reinforcement effect of the front four rings is more significant, which reduced the openings from 9.99 mm to 5.13 mm. The openings of subsequent circumferential joints was also slightly reduced, but the reinforcement effect is relatively weak. The openings of the first four rings are the largest. When the quantity of steel channels is 24, the maximum opening size is 5.97 mm. If the quantity of steel channels is less than 24, the opening of circumferential joints will exceed the critical value of 6 mm, which does not meet the specification requirements.

Through the above analysis, adding a steel channel is equivalent to several springs in parallel that resist the longitudinal tension of the segment lining. With the increase in the quantity of steel channels, the tensile stiffness of the reinforcement area becomes larger and larger; meanwhile, the ability to resist longitudinal tension becomes stronger and stronger. However, due to the large size of compressive stiffness of rubber gasket, the reinforcement effect on the segment ring far from the shield tail is weak. It can be seen from the simulation results that the engineering requirements can be met when the quantity of steel channels is greater than 24. Given the above analysis, and considering various factors such as the layout space, the degree of difficulty, and the cost-effectiveness of construction, when using No. 14 steel channel to reinforce the 20-ring segment ring, it is more appropriate to make sure that the quantity of steel channels is not less than 24.

4.4. Effect of Steel Channel Model

On the basis of the numerical simulation of the above typical cross-sections, and under the premise of keeping other parameters of the model unchanged, only the model of the steel channel is changed. The equivalent spring stiffness in different cases is shown in

Table 5. Equivalent spring stiffness of steel channels under different models.

The Model of Steel Channel	The Total Area (m2)	Equivalent Spring Stiffness (N/m)
10	3.06 × 10−2	1.61 × 108
12	3.69 × 10−2	1.94 × 108
14	5.11 × 10−2	2.69 × 108
16	6.04 × 10−2	3.17 × 108
18	7.03 × 10−2	3.69 × 108

.

As shown in Figure 18, the effect of steel channel models and quantity on longitudinal displacement is similar. Due to the change of the section of the steel channel of No. 14 and above, there is a sudden change in the maximum longitudinal displacement at No. 14. Changing the model of the steel channel can not only reduce the maximum longitudinal displacement, but also save the tunnel construction space. A reasonable matching of the model and the quantity of steel channels during reinforcement can help to achieve a better reinforcement effect.

As shown in Figure 19 and Figure 20, the change of the steel channel model and the increase in the quantity of steel channels have roughly the same reinforcement effect on the opening of each ring circumferential joint, and the relationship curves are also very similar. The openings of the first four rings are the largest. When the model of steel channels is No. 14, the maximum opening is 5.97 mm. If the model of steel channel is less than No. 14, the opening of circumferential joint will exceed the critical value of 6 mm, which does not meet the specification requirements.

Through the above analysis, the increase in the model and quantity of steel channels is equivalent to increasing the total cross-sectional area of the reinforced steel channels, so the effect of the two reinforcement methods is almost the same. Considering the safety and practicality of the project, when 24 steel channels are used to reinforce the 20-ring segment, it is more appropriate to make sure that the model of the steel channel is no less than No. 14.

5. Conclusions

Relying on an underground docking project of a cross-sea tunnel, this paper uses the methods of theoretical calculation and numerical simulation to study the longitudinal stress relaxation effect of segment lining and steel channel reinforcement technology after the thrust is unloaded in the process of shield docking and dismantling. The accuracy of theoretical calculations and numerical simulation is verified by comparing the openings of circumferential joints. Numerical simulation was used to further analyze the effect of key parameters, such as reinforcement range, model and quantity of steel channels, on the reinforcement effect after thrust unloading. The influence law and reinforcement mechanism of key parameters on the openings of circumferential joints are revealed, and the following results are obtained:

(1)

The reinforcement method of adding a steel channel is equivalent to transferring the internal force of the segment lining with several springs in parallel to resist the axial tension of the segment lining. This method can effectively reduce the stress relaxation effect of segment lining in the process of docking.

(2)

With the increase in the reinforcement range, the maximum longitudinal displacement and the openings of circumferential joints decrease first and then increase. The openings of circumferential joints are minimized when the reinforcement range just reaches the ring. When the reinforcement range is 20 rings, the maximum opening of circumferential joints reaches 5.97 mm.

(3)

The model and quantity of steel channels have similar effects on the reinforcement mechanism of the segment lining. With the increase in the model and quantity, the maximum longitudinal displacement and the opening of circumferential joint gradually decrease, and the reduction ratio also decreases. The farther away from the shield tail, the worse the reinforcement effect is.

(4)

In order to ensure that there is no sea water leakage in the process of the shield docking of the supporting engineering, it is recommended that the reinforcement range be 18–20 rings, the number of channels should be no less than 24 and the channel steel type should be no less than 14 when using channel steel reinforcement technology for the longitudinal reinforcement of segments.