2.2. Reinforcing Mechanism of Shield End Well
(1) Elastic sheet theory
According to the definition of the thin plate, the thickness of the thin plate is much smaller than the plane dimension of the thin plate, and the bending theory of the thin plate is similar to the elementary theory of beam bending. The practice has proved that ignoring some unimportant influencing factors and adopting simplified theoretical assumptions and boundary conditions can simplify the theoretical calculation process and will not have too much influence on the calculation results [
18]. In the research on the end reinforcement of shield tunnels, the reinforced soil can be simplified as an elastic thin plate, which can be regarded as a small burning problem for research. For rectangular thin plates with different boundary conditions, the theoretical solution can be obtained by superimposing the bending moment of the boundary with the solution of simply supported rectangular thin plates on four sides. Therefore, according to the actual situation of engineering construction and the relevant knowledge of plate and shell theory, the simply supported rectangular small deflection thin plate is used to simulate the reinforcement of soil, and the load on the plate is mainly the lateral pressure of soil distributed in the trapezoid. The formula for calculating the maximum bending stress and shear stress of the reinforced soil mass is obtained by using the Levy solution. Applying the calculation formula of the maximum bending stress and shear stress of the reinforced soil to engineering practice can check whether the strength of the reinforced soil meets the safety requirements. Such research and calculation results are partial to safety for the end reinforcement of thick plate, and there will be no engineering problems or accidents.
(2) Theoretical model of end soil reinforcement
In the analysis and research of the force mechanism of the reinforced soil at the starting and arriving ends of the shield, the reinforced soil is often simplified as a circular thin plate. The stress on the reinforced soil at the start of the shield is similar to that at the end of the shield, and the reinforced soil at the end is affected by the lateral combined water and soil force [
19]. The simplified mechanical model is shown in
Figure 1.
According to the basic knowledge of the plate theory of elastic mechanics, it is assumed that the diameter of the end reinforcement soil is D. According to the basic knowledge of soil mechanics, the thin circular plate is subjected to hydrostatic pressure and lateral earth pressure, and the combined force of the two is a trapezoidal load. The trapezoidal load acting on a circular section is an asymmetric problem, and the mathematical model is quite complex. It is very difficult to obtain an accurate solution that satisfies all the basic equations and boundary conditions. In previous mechanical models, the trapezoidal load is simplified to a uniformly distributed load, and then the axial symmetry knowledge of elastic mechanics is used to obtain the longitudinal reinforcement range that meets the strength criteria, as shown in
Figure 1.
(3) Elasticity Theory Model
Shield end reinforcement is an important part of shield starting and arriving technology. After the tunnel door is broken, the soil body at the end is exposed, the force balance of the stratum at the end is destroyed, the structure, load, and stress of the soil body at the end will change, and the collapse of the tunnel door will occur from time to time [
20]. Therefore, it is necessary to pay attention to the rationality of the reinforcement scheme for the soft soil layer at the end and the control of the reinforcement quality. It is very necessary to strengthen the front end of the excavation, as shown in
Figure 2.
The result of the solution after the trapezoidal load is simplified to a uniformly distributed load is dangerous. Given this, the pros and cons of traditional research methods are summed up through analysis and research of traditional mechanical models, and the equivalent mechanical model is found within the range of elasticity that is allowed. According to
Figure 2, the trapezoidal load is equivalent to the superposition of a uniformly distributed load and a triangular symmetric load, that is, the superposition of solving a symmetric problem.
The purpose of the stratum reinforcement at the end of the shield tunnel is to prevent the impact of vibration when the temporary enclosure structure is dismantled. Before the shield cutter head reaches the end of the face and establishes the earth pressure, it can stabilize the surrounding rock and prevent the loss of groundwater. The most common problems of soil reinforcement at the end in China are as follows:
(1) Improper design of the reinforcement range at the end, resulting in soil erosion at departure and arrival and engineering accidents such as water seepage, well flooding, and collapse. In the past, there were few studies on the scope of end reinforcement in China. The selection of the reinforcement range at the end was generally based on engineering experience, and the theoretical basis was weak [
21]. The reinforcement range of the end reinforcement design is usually 3.0 m above and 3.0 m below the bottom plate, and the longitudinal reinforcement length is usually taken as 6~8 m.
(2) The selection of the reinforcement method of the end is unreasonable, the stratum adaptability is poor, the reinforcement effect of the end is not ideal, and the stratum of the end is collapsed when the tunnel door is broken. To ensure the safe departure and arrival of the shield, the above two problems must be solved, so that the stratum after reinforcement should have good uniformity and integrity, the stratum can stabilize itself after the hole is cut out, and has a good waterproof and anti-penetration function. In addition, after the end reinforcement is completed, a drilling core test should be carried out to check the reinforcement effect, and the unconfined compressive strength of the cored sample should reach 0.8 MPa. Drilling holes in the reinforcement area to check the seepage volume, the permeability coefficient should not be greater than 1.0 × 10−5 cm/s and the total seepage volume should not be greater than 10 L/min.
In this paper, when studying the stability of the soil at the beginning and end of the shield, to simplify the calculation, an ideal sliding failure model of the soil at the end is established. According to whether the soil has cohesion, it is divided into cohesive soil and sandy soil, and two different mechanical models are established, respectively. The stability of the soil at the beginning and end of the shield is analyzed; the range of soil instability and failure is deduced; and the reasonable reinforcement range of the soil at the end is determined.
2.3. Method of Reinforcement Design of Shield End
(1) Reinforcement design for soil stability
An ideal soil sliding failure model can be established for the reinforcement design of soil stability. At the same time, soil is divided into cohesive soil and sandy soil based on its different properties, and two different mechanical models are established accordingly. Through in-depth analysis of the stability of the model, the failure range of soil instability is deduced, and the scientific and reasonable reinforcement range is determined according to the results.
For cohesive soil, because of its viscous force, the slice method is usually used in the calculation of the mechanical model during design. The specific process is as follows: the sliding soil body is vertically divided into several strips, and these soil strips are regarded as rigid bodies, and the sliding moment and anti-sliding moment of different soil strips are obtained. According to the basic principle of static equilibrium, the sliding moment and anti-slip moment are obtained.
In sandy soil, there is no cohesion in the soil. According to statistical data, the failure process is mostly sudden, and the slip surface forms a linear slip surface from the top to the bottom. According to the principle of earth pressure and centrifugal experiment, it can be known that the failure surface of sandy soil is the vertical sliding surface. Based on this, a corresponding destructive force model can be established to carry out the necessary calculations.
(2) Determination of soil range
Scope of lateral reinforcement: To carry out tunnel construction, excavation must be carried out. After the excavation of the soil mass, the original stress balance is destroyed, which also has a certain impact on the surrounding soil mass. The stress is concentrated around the tunnel wall. If the maximum shear stress exceeds the shear strength at this time, the surrounding soil will be damaged to a certain extent and gradually spread to the surrounding, forming a plastic loose circle. The existence of the plastic loose circle causes the surrounding stress to be significantly reduced, and the maximum stress concentration is shifted to the boundary area between the plastic circle and the elastic circle. Therefore, to ensure the stability of the lateral soil, the necessary reinforcement work must be carried out in advance.
Scope of longitudinal reinforcement: If the construction soil is anhydrous, only the requirements of strength and stability need to be considered during reinforcement. For soil with water, in addition to meeting the above requirements, the body size and water-stop requirements of the shield machine must also be considered.
(1) After the tunnel is excavated, the shield cutter head has not yet carried out construction on the top excavation surface. There are two main purposes of reinforcement at this time: first, to meet the requirements of strength and stability, and to prevent the end soil from being pulled, sheared, and damaged under the action of water and soil pressure and leading to overall instability; second, to meet the requirements of water-stop, prevent the water in the stratum from infiltrating into the construction area, and the shield working well, thereby causing soil erosion and causing damage to the lower layer, resulting in subsidence or excessive subsidence of the surface.
(2) When the shield machine starts construction at the bottom of the end, the relevant longitudinal reinforcement range should be calculated first according to the strength and stability. At the same time, comparing it with the length of the main body of the shield machine, the following two situations will occur. First, the longitudinal reinforcement range is smaller than the length of the shield mainframe. When this happens, the longitudinal reinforcement range can be calculated according to the following formula, according to the requirements of geometric criteria and actual engineering experience: L2 = 2 × 3 × B, where L2 is the length of the shield machine and B is the width of the segment. Second, the longitudinal reinforcement range is greater than the length of the shield mainframe. When this happens, just take the calculated longitudinal reinforcement range.
(3) End reinforcement range technology
The two main theories based on the calculation of the longitudinal reinforcement length at the end are the elastic thin plate theory and the slip instability theory. The elastic sheet theory and slip instability theory satisfy the strength and stability requirements, respectively. There are two types of elastic thin plates: circular and rectangular. Among them, the circular thin plate model is widely used in the calculation of the end reinforcement range. Specifically, it can be divided into two processing methods: the uniform load calculation model and the uniform load plus inverse triangular symmetrical load. When the diameter of the tunnel is less than 10 m, the results of the two calculation methods have little difference.
(1) The strength check formula and the minimum longitudinal reinforcement length are calculated according to the uniform load model
Minimum longitudinal reinforcement length:
In the formula, D is the diameter of the tunnel door of the working well; t is the longitudinal reinforcement range; and p is the lateral water and soil pressure acting on the center of the tunnel door. For sandy soil, the water pressure and earth pressure are calculated separately. The soil pressure is considered the static earth pressure; μ is the Poisson’s ratio of the reinforced soil; σt is the ultimate tensile strength of the reinforced soil, which is generally 10% of the ultimate compressive strength, that is, σt = qu/10; τc. To strengthen the ultimate shear strength of soil, the value is selected according to experience, τc = qu/6; k1, k2 are safety factors, generally 1.5.
(2) Longitudinal reinforcement length as specified by the Japanese JET GROUT Association (JJGA)
In the formula, safety factor K0 is taken as 1.5 to 2.0, and the calculation coefficient β is taken as 1.2. Other symbols have the same meaning as before.
The ideal global slip theory in the clay stratum holds that the reinforced soil may slide as a whole in the tunnel along a certain sliding plane under the combined action of the ground overload p and the upper soil. The hole diameter is the arc surface of the radius.
(3) The longitudinal reinforcement length of the end obtained by the ideal global slip instability theory
Anti-slip moment:
where
M is the sliding moment;
Md is the anti-sliding moment; Δ
c is the increased cohesion of the soil after improvement;
K is the anti-sliding safety factor;
θ is the angle between the reinforced soil and the sliding surface; γ
t is the tunnel range Inner soil weight.
(4) Lateral reinforcement technology
The lateral reinforcement range of end reinforcement is mainly to consider the disturbance range of the soil around the cave after the removal of the door and the enclosure structure. The theoretical basis is the limit equilibrium theory of soil disturbance. The lateral reinforcement range must be larger than the disturbance range (plastic circle) to ensure lateral stability.
The scope of reinforcement on the upper and lower sides of the tunnel:
Reinforcement range on the left and right sides of the tunnel:
where
k is the reinforcement safety factor.
Usually, soil layers include the following types: first, fill layer; second, silty soil layer; third, cohesive soil layer; fourth, silt layer, clay soil, etc. For this type of stratum, the reinforcement methods include the following two: cement-soil mixing piles and rotary jetting piles or freezing methods. When the nature of the soil layer is a round gravel layer, the key technology to be solved during reinforcement is the water-stopping effect of the reinforcement. Under normal circumstances, a high-reliability, plain silicon diaphragm wall is used to stop water. When the soil layer is mudstone, the usual method is to retain the soil with silicon piles outside the hole.
2.4. Research on the Strength of Soil Reinforcement at the End
The soil at the end of the shield when entering and exiting the hole generally needs to be reinforced, and the reinforced soil has a certain strength. Therefore, from this aspect, the reinforced soil will generally deform greatly before the strength failure occurs. In the end soil strength analysis, the reinforced end soil is generally assumed to be an elastic thin plate, but in practice, the thickness of the end soil is much larger, so this simplification is relatively safe. To prove the correctness of the simplification of the reinforced end soil, finite element analysis technology can be used to calculate the reinforced end soil, and the results can be compared with the actual situation. In addition, experimental methods can also be used to verify the simplified results.
According to the plate model theory stipulated by the JET GROUT Society (JJGA), when analyzing and studying the reinforced soil at the end of the shield, the soil at the end is generally assumed to be a thin cylindrical plate. In this model, the soil at the end is assumed to be an elastic circular plate supported by the surrounding elastic and subjected to the combined force of the longitudinal water and soil pressure of the tunnel. The maximum bending stress appears at the center of the circular plate, and the shear force at the support is the largest. According to the principle, the strength check formula can be obtained:
where
D is the diameter of the tunnel door;
t is the reinforcement thickness of the longitudinal soil;
P is the resultant force of the longitudinal water and soil pressure on the sealing door of the tunnel;
μ is the Poisson’s ratio of the soil;
σt is the ultimate tensile strength of the soil; left is the safety factor, which is generally taken as 1.5.
According to the thin circular plate theory, the required soil reinforcement thickness is:
According to the principle of elastic mechanics, the check formula for sheet shear force can also be obtained:
In the formula, τc is the critical shear strength of the soil at the reinforced end, usually τc = q/6; k is the safety factor, generally 1.5.
From Formula (13), it can be calculated that the soil reinforcement thickness that meets the shear resistance requirements of the soil is:
To make the reinforced soil meet the requirements of both the tensile strength and the shear strength of the soil, the thickness of the reinforced soil should meet the following formula:
Because the soil at the end does not have a high modulus like steel or concrete, it tends to undergo large deformations before failure, and the bearing capacity is not controlled by the strength. Rear stability is particularly important for end reinforcement.
The ground load is the moment caused by the self-weight of the overlying soil and the moment generated by the soil within the slip plane inside the hole n, which can be calculated according to the following formula: