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By FE simulation with Mohr-Coulomb perfect elastoplasticity model, the relationship between the support pressure and displacement of the shield tunnel face was obtained. According to the plastic strain distribution at collapse state, an appropriate failure mechanism was proposed for upper bound limit analysis, and the formula to calculate the limit support pressure was deduced. The limit support pressure was rearranged to be the summation of soil cohesion

The key issue during shield tunneling is to keep the stability of tunnel face, and this generally depends on the support pressure which was applied on the tunnel face after soil excavation. The pressure must be controlled at least no less than its limit value which corresponds to the active failure state of the tunnel face. So far, various kinds of methods have been proposed to study this problem. Experimental methods, including physical modeling and centrifuge modeling [

When shield tunnel locates under the water table line, the soil excavation often induces underground water seepage and apt to cause the collapse of shield tunnel face. The face stability analysis under seepage condition was studied by numerical simulation [

In this paper, the stability of shield tunnel face was studied by elastoplasticity FE simulation; the collapse mechanism and limit support pressure in active failure state were obtained. Based on the numerical results, a failure mechanism was proposed and a 2D upper-bound limit analysis model was established, and the formula for calculating the limit support pressure was also deduced. Following the Terzaghi superposition method which has been commonly used in bearing capacity analysis, the limit support pressure was rearranged as the summation of soil cohesion, surcharge load, and soil gravity multiplied by their corresponding coefficients, and the varying characteristics of these coefficients with the depth-to-diameter ratio of tunnel and the friction angle of soil were studied in detail. The influence of seepage on the stability of shield tunnel under water table was also studied. The pore water pressure distribution and seepage force on the shield tunnel face were obtained by FE numerical simulation. After the calculation of seepage force on the failure area of tunnel face, the proposed upper bound limit analysis model was extended into seepage condition.

The relationship of deformation and support pressure of shield tunnel face was obtained by FE analysis with PLAXIS software, the constitutive model adopted is the widely used Mohr-Coulomb perfect elastoplasticity model. The tunnel diameter is ^{3}. Considering that dilatancy angle has no influence on the limit support pressure [

The finite element mesh for the stability analysis of shield tunnel.

The initial stress field induced by the soil weight and surcharge load was calculated. After the excavation of the soil during shield tunneling, the initial condition was recovered by applying lateral earth pressure with its value determined by

The relationship between the support pressure and displacement at center-point of the tunnel face (

The increments of displacement and plastic strain distributions at collapse state are shown in Figures

The displacement increment around the tunnel face at collapse state.

The equivalent plastic strain distribution around the tunnel face at collapse state.

In order to analyze the stability of the tunnel face, an appropriate failure mechanism needs to be proposed. According to the plastic strain distribution at collapse state obtained from numerical modeling in Section

The proposed failure mechanism.

Based on the failure mechanisms proposed in previous section, the upper bound solution of the limit support pressure could be obtained by equaling the power of external force and plastic dissipation energy [

The differential of the power of the weight of shearing zone

After integration, the power of the weight of shearing zone

The power of the weight of block

The power of the surcharge load is

The power of the support pressure on the tunnel face is

The internal energy dissipation along line

The internal energy dissipation along the line

The internal energy dissipation along line

By equaling the power of external forces to the internal energy dissipation, we get

By combining from (

As shown in Figure

The influence of the soil strength parameters on the limit support pressure (

The influence of cohesion

The influence of friction angle (

The coefficients of cohesion, surcharge load, and soil gravity in (

The relationship between

The relationship between

The relationship between

When shield tunnel locates under water table line, the soil excavation often induces seepage and causes the failure of tunnel face. It is necessary to establish a calculation model to estimate the influence of seepage on the stability of tunnel face. The key point of the stability analysis under seepage condition is the calculation of the seepage force. By assuming the underground water seepage to follow the Darcy law, the seepage equation in steady state is

FE simulation was employed to analyze the seepage characteristics of the tunnel face. In the simulation, two cases with tunnel diameter of 5 m and 10 m were studied; the water table varies from the top of the tunnel to three times of tunnel diameters. When tunnel diameter ^{−5} m/s; the pore water pressure distribution near the tunnel face obtained from numerical simulation is shown in Figure

The distribution of the pore water pressure around the tunnel face.

From the pore water pressure distribution, the water head difference between the failure line and the tunnel face could be obtained, and then the seepage force

The relationship between the average seepage force and the water table.

In order to study the seepage force in detail, the horizontal and vertical components of the seepage forces in area

The ratios of each component of average seepage force over hydrostatic force are shown in Figure

The relationship between the seepage force and the water table.

By including the power rate of the seepage force in upper bound limit analysis, the influence of seepage on the limit support pressure of tunnel face could be studied. The power rate of the seepage force

After the integration of (

The formula of (^{3} and 19 kN/m^{3}. For simplicity, the surcharge load is assumed to be zero. The calculated limit support pressures in dry sand case are 11.9 kPa (

The relationship between the total limit support pressure and the water table.

The face stability of shield tunnel was studied by elastoplasticity FE simulation; the equivalent plastic strain distribution and limit support pressure at collapse state were obtained. Based on the numerical results, a failure mechanism was proposed to study the face stability of shield tunnel by upper bound limit analysis. The calculating formula of the limit support pressure was rearranged to be the summation of cohesion, surcharge load, and soil gravity multiplied by corresponding coefficients. Parametric analysis showed the coefficient of cohesion increases with friction angle of the soil and decreases with the depth-to-diameter ratio of tunnel. The coefficients of surcharge load and soil gravity decrease with the friction angle of soil. Both coefficients decrease with the tunnel depth-to-diameter ratio only when the friction angle is less than an appropriate value; otherwise they are independent of the depth-to-diameter ratio and the coefficient of surcharge load goes to zero. The seepage analysis was conducted by FE simulation; the pore water pressure distribution and seepage force on the tunnel face were obtained. By adding the power rate induced by seepage force, the proposed upper bound limit analysis was extended to seepage condition. The results showed that a large part of the limit support pressure was used to equilibrate the seepage force, and the total limit support pressure varied almost linearly with the water table.

The authors declare that there is no conflict of interests regarding the publication of this paper.

The financial support by the National Science Foundation of China (NSFC through Grant no. 50908171) and Shanghai Municipal Science and Technology Commission (through Grant no. 13ZR1443800) is gratefully acknowledged.