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Due to the poor stability of the loose sandy soil layer, if the support force is not properly controlled during the construction process of the shield tunnel using the earth pressure balance method, it is easy to cause the ground to collapse or uplift. Therefore, understanding the support force of the excavation surface of shield tunneling in sandy soil layer is very vital to ensure the stability of the excavation surface. Firstly, it is assumed that the damaged soil is a three-dimensional wedge and a modified three-dimensional wedge in the active and passive failure modes, respectively. The shallow soil pressure theory and the soil plastic limit equilibrium theory are derived by analyzing the stress distribution on the damaged soil. The equation for revealing the inner essence between the support force of the shield excavation surface and excavation surface displacement under the condition of sand-covered soil is used. Secondly, the numerical simulation method analyzes the displacement of the excavation surface when the support force changes under different working conditions, and the relationship curve between the excavation surface support force and the shield tunneling displacement is obtained. The comparison and analysis between the numerical simulation calculation and the theoretical analysis indicate that the deduced calculation equation for the excavation surface support force based on the displacement earth pressure is reasonable.

With the acceleration of China’s urbanization process and the increasing urban population, the subway has become the first choice to relieve traffic pressure and promote urban development. Shield tunneling is extensively utilized in subway construction owing to its advantages of small environmental impact [

Many theories have been made on the support force of the shield excavation surface, which have formed a variety of analysis methods such as the limit analysis method, and plastic balance theory, even considering the grouting reinforcement effect [

In the limit equilibrium method, the soil’s sliding surface located before the excavation surface is assumed first, and then the solution is obtained by the equilibrium of each isolation body in the sliding surface. There are many models using limit equilibrium theory, and the three-dimensional wedge model is the most widely used. Vermeer et al. [

Liang et al. [

In view of the concept of displacement earth pressure and the plastic limit equilibrium theory of soil, this paper deduces the theoretical equation between the support force and the displacement of shield excavation surface under the shallow sandy soil layer’s condition. A numerical simulation method is then utilized to reveal the displacement of the excavation surface when the support force changes under different working conditions. The relationship curve between the support force and the corresponding displacement (i.e., deformation) of the excavation surface is deduced, and the equations are analyzed. The results are helpful to understand the relationship between the induced displacement and the corresponding support force and to guide the selection of the suitable support force.

Actually, the theory of displacement earth pressure means that there is a close association between the earth pressure (i.e., the stress in soil) and the induced displacement during the movement of the retaining structure, which also involves the coupling process of external force, seepage action [

Terzaghi [

In the light of the theory of displacement earth pressure, it is clear that the earth pressure of the retaining structure changes with its displacement. When the retaining structure moves in the direction of filling, the earth pressure increases, and it decreases when it moves away from the soil until the deformation of the soil body achieves the limit state. The retaining structure is defined as the nonlimit state from an initial static state to a continuous sliding surface (reaching limit state).

The displacement ratio is defined to describe the nonlimit state of shield tunneling as follows:_{a} is the displacement when reaching the active limit state, and _{p} is the displacement when reaching the passive limit state.

When the internal friction angle (_{m}) and external friction angle (_{m}) reach the maximum values of _{m} and _{m} and displacement ratio can be established.

The evolution of internal friction angle with displacement.

It is evident from Figure _{m} = _{0}, _{m} = _{0}; when the displacement of the retaining structure reaches the limit displacement, that is, _{a} (_{p}), _{m} = _{m} =

When 0 ≤ _{a} (_{p}), the calculation equations of _{m} and _{m} are written as follows, respectively:

From the above equations that when in the static state, _{d} = 0; when in the limit state, _{a} (or _{p}), displacement ratio _{d} = 1. When the action of the initial external friction angle (_{0}) of the retaining structure is not considered, the initial internal friction angle (_{0}) is simply expressed as_{0} is the static earth pressure coefficient, namely, _{0} = 1 − sin

When considering the influence of the initial internal friction angle (_{0}) of retaining structure, Chang [_{0} is generally taken as _{0}/2.

Because the interaction between the shield and excavation surface is very complex in practical engineering [

Based on the three-dimensional wedge model composed of two parts, this section deduces the equation of excavation surface support force owing to the concept of displacement-based earth pressure. Thus, the stress analysis mode of the wedge is shown in Figure

Force analysis of actively damaged wedge.

According to the basic assumption, the area of the middle surface (

If the perimeter of the rectangle (

Substituting equation (

The static equilibrium in the _{3}, _{3}, and _{3} are the supporting force, sliding friction, and cohesive friction of wedge side (_{2}, _{2}, and _{2} are the supporting force, sliding friction, and cohesive friction of the wedge inclined plane (_{p} and _{p} are the wedge sliding and cohesive friction, respectively; and

From the static equilibrium in the _{a} is the shield support force to the excavation surface.

By substituting equations (

The value of the internal friction angles _{0} and _{m} and the external friction angles _{0} and _{m} can be obtained by equations (

Assuming that the support force and earth pressure are balanced during shield tunneling, the active support force at the center point of the excavation surface can be obtained as follows:

Generally, the inclination angle of the sliding surface of the wedge can be assumed as 45° +

The calculation methods of overburden earth pressure include the proctor’s earth pressure theory, the loose pressure estimation method proposed by Terzaghi (1923), and the standard calculation method. Among these, the loose earth pressure theory given by Terzaghi is extensively utilized because of its rationality. Therefore, the calculation of overburden earth pressure is acquired by

The stress analysis of the wedge is shown in Figure

Force analysis of wedge.

The force on both sides of the wedge (_{3}) is

In equation (

Substituting equation (

The integral of equation (

Based on the three-dimensional wedge model, the failure pattern of the excavation surface is still assumed as a wedge, and the top of the wedge is changed into an inverted prism. The stress analysis of the wedge in front of the excavation surface is indicated in Figure

Force analysis of passively damaged wedge.

The calculation process of the passive support force is similar to that of the active support force. Therefore, in view of the equilibrium state in

In equation (_{0} and _{m} and external friction angles _{0} and _{m} can be seen from equations (_{m}; other parameters are calculated as follows:

Similarly, the passive support force at the center point of the excavation surface can be obtained as follows:

On the basis of the theory of displacement-based earth pressure in this paper, the inclination angle of the wedge sliding surface (

The inclination angle of the inverted prism (

Thus, the displacement-based lateral pressure coefficient is given by

The stress analysis of the inverted prism above the wedge is shown in Figure

Stress analysis of chamfered cylinder.

The geometric parameters of the inverted prism are defined: the lower surface of the inverted prism or the upper wedge surface is _{1}, and the volume of the inverted prism is

Based on the mechanical equilibrium of the inverted prism, the force _{1} of the inverted wedge prism is obtained as follows:_{5} is the force of the soil above the inverted prism in the failure height; _{4} and _{4} are the sliding friction and cohesive friction of the inverted prism, respectively; and

The corresponding calculation equations are as follows:

By substituting equation (_{1} can be obtained.

A numerical calculation based on the program FLAC3D is used to verify the established theoretical model in this paper. The FLAC3D is a numerical simulation software based on the difference method (Itasca company, USA). The geometric model scale is 20 × 30 × 24 m. Here, the diameter size of the tunnel section is 6 m. Due to the symmetry of the geometric model, half of the models are selected for calculation (Figure

Numerical calculation model.

The shield segment is assumed to be made of C50 reinforced concrete. The thickness of the segment is 35 cm and is considered a linear elastic material. Mohr–Coulomb’s yield criterion is used to describe the sandy soil layer. Shell element is used for the contact surface between segment and concrete, and the specific material parameters are shown in Table

Computational parameter.

Materials | Elastic modulus (MPa) | Poisson’s ratio | Density (kg/m^{3}) | Cohesion (kPa) | Tensile strength (kPa) |
---|---|---|---|---|---|

Soil | 20 | 0.35 | 1800 | 1 | 1 |

Shield segment | 30000 | 0.25 |

The sand soil layers with two buried depth ratios (i.e.,

Cases of numerical simulation.

Case | ||
---|---|---|

1 | 0.5 | 25 |

2 | 0.5 | 30 |

3 | 0.5 | 35 |

4 | 0.5 | 40 |

5 | 1 | 25 |

6 | 1 | 30 |

7 | 1 | 35 |

8 | 1 | 40 |

Case 1–Case 4 are taken as an example to obtain the influence of shield tunneling displacement on the active support force. The material calculation parameters and displacement ratio are substituted into equations (

Comparison of the influence of shield driving displacement on active support force: (a) theoretical results and (b) numerical results.

According to Figure

The same method is adopted to obtain the influence of shield tunneling displacement on the passive support force. The theoretical and numerical results shown in Figure

Comparison of the influence of shield driving displacement on the passive supporting force: (a) theoretical results and (b) numerical results.

Using equations (

Theoretical results of the ultimate supporting force.

Active ultimate support force (kPa) | Passive ultimate support force (kPa) | ||
---|---|---|---|

0.5 | 25 | 14.51 | 480 |

0.5 | 30 | 11.65 | 630 |

0.5 | 35 | 8.4 | 820 |

0.5 | 40 | 6.12 | 1120 |

1 | 25 | 17.77 | 850 |

1 | 30 | 12.56 | 950 |

1 | 35 | 9.57 | 1420 |

1 | 40 | 6.6 | 1880 |

The excavation surface support force is gradually increased or decreased with the same conditions and material parameters mentioned above until the center displacement of the tunnel excavation surface develops rapidly and reaches the limit state. The corresponding support force is selected as the limit support force calculated by numerical method. The ultimate support force calculated by theoretical and numerical methods is compared, as manifested in Figure

Comparison of ultimate support force obtained from theoretical and numerical calculation: (a) active ultimate support force and (b) passive ultimate support force.

From Figure

Based on the three-dimensional wedge assumption, the relationship of the support force and the corresponding displacement on the excavation surface of the shield in the shallow sandy soil layer is derived using the displacement earth pressure and the plastic limit equilibrium theories.

When the ratio of internal friction angle to buried depth is certain, the active support force decreases with increasing the displacement ratio, while the passive support force decreases with increasing the displacement ratio. Moreover, the difference between the theoretical calculation and the numerical simulation is very small.

When the buried depth ratio is constant, the active ultimate support force decreases with increasing the internal friction angle, while the passive ultimate support force increases with increasing the internal friction angle. From another perspective, as the internal friction angle is fixed, the active and passive ultimate support forces increase with the rise in the buried depth ratio. Overall, the above analyses indicate the rationality of the proposed displacement-based calculation theory for predicting the support force.

The data used to support the findings of this study are available from the corresponding author upon request.

The authors declare that they have no conflicts of interest.

This work was supported by the National Natural Science Foundation of China (52078031).

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