Upper Bound Limit Analysis of Deep Tunnel Face Support Pressure with Nonlinear Failure Criterion under Pore Water Conditions

Abstract

Based on the nonlinear failure criterion and modified tangential technique, the upper bound solutions of the critical supporting pressure on the deep tunnel face were obtained under pore water pressure conditions. The influence of parameters on the critical supporting pressure and collapse range was investigated according to the unlimited block failure mechanism. It was found that the upper bound solutions of the critical supporting pressure increase with the growth of the nonlinear coefficient and pore water pressure coefficient. The collapse range of the tunnel face scales out with the increase in the nonlinear coefficient and shrinks with an increasing pore water pressure coefficient. Moreover, with the increase in the nonlinear coefficient, the impact strength on critical supporting pressure and collapse range declines gradually. According to the calculated results, both the pore water pressure and nonlinear criterion factors have negative impacts on the stability of the tunnel face. Thus, more attention should be paid to these parameters to ensure face stability in deep tunnel construction.

1. Introduction

The underground environment is complex, indeterminate, and irregular because of the combined effects of many factors, such as complex in situ stress field, composition of rock and soil, underground water, etc. Therefore, it is of great significance for deep tunnel construction to investigate the stability and stress fields under complicated environments.

Generally, theoretical analysis, test methods, and numerical simulation are common methods in tunnel engineering analysis. The classic theoretical analysis approach includes the limit equilibrium method [1,2] and limit analysis theorem [3,4]. In the limit analysis approach, it is assumed that rock and soil are ideal elastic–plastic materials and obey the associated flow rule in stability analysis. Researchers from various countries have carried out work on tunnels using the limit analysis theorem [5,6,7]. These investigations mainly include two aspects: the vault and side wall stability and the tunnel face stability. In this paper, the upper bound solutions of the tunnel face critical supporting pressure, which belongs to the latter one, were obtained.

The upper bound limit analysis theorem establishes the energy–work balance equation to quantify the stability of geotechnical engineering. For tunnel engineering, the balance equation includes the contributions of gravity, soil strength, pore water, support pressure, etc. All of these different effects act on a computational model known as the failure mechanism. The failure mechanism is crucial for an accurate solution. Concerning the study on the tunnel face’s critical supporting pressure and failure mechanism, Leca and Dormieux [8] proposed three simple rigid cone failure mechanisms for shallow tunnel faces and calculated the upper bound solution of the critical supporting pressure in clay and friction materials. Soubra [9] improved the failure mechanism by introducing a logarithmic spiral shear block consisting of enormous, small cones that were formulated with vertex angles 2φ and a circular bottom surface. Compared with the previous results, this new failure mechanism shows obvious advantages. To consider spatial variability of the soil shear strength parameters for determining the critical collapse pressure of tunnel faces, Mollon et al. [10] established new 2D and 3D failure mechanisms by means of applying spatial discretization and ‘point by point’ techniques.

The failure mechanisms of tunnel faces have been gradually improved, and corresponding influencing factors have been introduced into the analysis of these complicated issues. Water, as a predominant element affecting tunnel stability, has received much attention from researchers. Michalowski [11] first considered the pore water pressure as an external force acting on the soil particles and recognized that the work done by the pore water pressure is the sum of the power on the soil deformation and the power on the boundaries. Thus, the water factor could be considered in the limit analysis theorem. According to Michalowski’s research, Huang and Yang [12,13] derived the collapsing detaching curve functions of tunnel roofs and the corresponding change laws in square and circular tunnels, respectively, via the upper bound theorem and variation principle under the pore water pressure condition. Zhong et al. [14] and Hou et al. [15] calculated the upper-bound solutions of the tunnel’s critical supporting pressure. The influence of excavation velocity and seepage force on the stability of the tunnel face was investigated.

In the field of plasticity theory, the nonlinear failure criterion is widely adopted. Most of the previous investigations on tunnel stability were based on the Mohr–Coulomb linear failure criterion. However, plenty of tests and theoretical studies have already demonstrated that almost all failure envelopes of geotechnical materials obey the nonlinear rule and that the linear condition is merely a special case [16,17]. The limit analysis method and the limit equilibrium method have advantages and disadvantages in dealing with this problem. The limit equilibrium method requires solving the stress distribution along the sliding surface, but the nonlinear strength criterion complicates the calculation. The limit analysis method, on the other hand, can provide the upper bound solution by using a rational failure mechanism and is convenient to apply by coupling the nonlinear strength criterion with the tangent method.

To introduce the nonlinear failure criterion into the upper bound theorem, Yang and Yin [18] proposed the ‘tangent method’ to investigate the nonlinear characteristics of geo-materials with a nonlinear SQP algorithm for the optimization of practical projects. For tunnel engineering, Fraldi and Guarracino [19] derived the analytical solution of collapse mechanisms with arbitrary excavation profiles in plastic Hoek–Brown nonlinear rock masses. According to the power-law nonlinear failure criterion, Yang et al. [20] obtained the upper bound solutions of the critical supporting pressure in shallow tunnels with optimization theory, and they proposed the modified tangential technique in the meantime.

The excavation of deep tunnels under high pore water pressure conditions is still a challenging program for engineers. It is of analytical and practical importance to analyze the stability of deep tunnels with the nonlinear failure criterion. Therefore, under the consideration of pore water pressure and nonlinear failure criterion, the upper bound solutions of deep tunnel face support pressure with a new failure mechanism based on the modified tangential technique were investigated in this paper. The multi-tangent method takes into account the different stress states for different sliding blocks as well as the differences of the equivalent M-C parameters. The optimization approach was applied to determine the values of these parameters, reducing the complex calculation problem to a simple optional process. The proposed method provides a practical way to account for the strength nonlinearity of soil and can serve as a reference for subsequent works.

2. Calculation of Critical Supporting Pressure

2.1. Upper Bound Theorem under the Pore Water Pressure Condition

To investigate the stability of various geotechnical structures under the pore water pressure condition with limit analysis theory, Viratjandr and Michalowski [21] regarded the power generated by pore water pressure as an external power so that the water factor could be considered in the calculating process of energy dissipation in the upper bound theorem. They also held the view that the pore water pressure mainly acts on soil skeletons and the kinematically admissible velocity field boundaries. The specific processes of derivation were shown in Viratjandr’s paper, and the final expression for pore water pressure power can be expressed as

$$P_{\mathrm{u}}=-{\displaystyle {\int }_{V}u{\dot{\epsilon }}_{ii}\mathrm{d}V-}{\displaystyle {\int }_{\Gamma }u{n}_i{v}_i\mathrm{d}\Gamma }$$

(1)

where u is the pore water pressure, which can be obtained by multiplying pore water pressure coefficient r_u, soil weight γ, and vertical distance h from a random underground point to the water table, i.e., u = r_uγh. Because the distribution of the pore water pressure is difficult to describe analytically, r_u is often used in the analytical approach as a simplified and idealized quantity to reflect the impact of the distribution of the pore water pressure. The determination of r_u needs the calibration procedure. In this paper, the values of r_u were directly specified. ${\dot{\epsilon }}_{ii}$ means the volumetric strain rate, and V is the failure mechanism element volume. v_i denotes the velocity vector of the kinematically admissible velocity field. n_i represents the outward normal vector on the detaching surface, and Γ is the failure mechanism boundary.

On the right side of Equation (1), the first part indicates the power generated by pore water pressure acting on the soil skeleton, and the second part represents the power produced on the boundaries of the failure mechanism. Because the failure mechanism adopted in this paper consists of rigid blocks, the geometric deformation is negligible. Therefore, the volumetric strain rate is equal to zero, i.e., ${\dot{\epsilon }}_{ii}$ = 0, which shows that the pore water pressure power is completely composed of the part produced on the failure mechanism boundaries.

According to the upper bound theorem, if the hypothetical kinematical admissible velocity field satisfies the displacement boundary conditions, the loads that are obtained by making the external power equal to the energy dissipation rate should be no less than the actual results in the limit state. Thus, the formula of the upper bound theorem under the condition of pore water pressure can be acquired.

$${\displaystyle {\int }_v{\sigma }_{ij}{\dot{\epsilon }}_{ij}\mathrm{d}V\ge }{\displaystyle {\int }_{\Gamma }{T}_i{v}_i\mathrm{d}\Gamma }+{\displaystyle {\int }_v{X}_i{v}_i\mathrm{d}V}-{\displaystyle {\int }_{\Gamma }u{n}_i{v}_i\mathrm{d}\Gamma }$$

(2)

where σ_ij and ${\dot{\epsilon }}_{ij}$ are the stress tensor and strain rate tensor in a kinematically admissible velocity field, respectively. T_i represents the load acting on the boundary Γ, and X_i means the volume force.

2.2. Nonlinear Failure Criterion

2.2.1. Power-Law Nonlinear Failure Criterion

The power-law failure criterion describes the nonlinear relationship between normal stress and shear strength when geotechnical materials reach yield. Because of its simple expression and clear physical meanings, the power-law failure criterion was widely used to solve geotechnical problems [22,23]. The formula can be expressed as follows,

$$\tau =c_0{\left(1+{\displaystyle \frac{{\sigma }_{\mathrm{n}}}{{\sigma }_{\mathrm{t}}}}\right)}^{{\displaystyle \frac{1}{m}}}$$

(3)

where τ and σ_n are the shear strength and normal stress on the detaching surfaces, respectively. Parameter c₀ means the initial cohesion, and σ_t represents the tensile strength. Parameter m is the nonlinear coefficient determining the bending degree of the strength envelope. It is obvious that when the nonlinear coefficient m = 1.0, Equation (3) will be degraded into the Mohr–Coulomb failure criterion.

2.2.2. Tangent Method in Nonlinear Strength Criterion

Since the strength envelope of the nonlinear failure criterion is a curve, the intercept, and gradient cannot be applied directly to determine the strength parameters as the linear failure criterion does. In this case, Yang and Yin [18] introduced a tangent on the strength envelope of the nonlinear failure criterion and regarded its intercept c_t and gradient φ_t as the equivalent parameters in linear failure criterion conditions. Thus, the nonlinear strength parameters of the geotechnical materials were obtained.

As shown in Figure 1, the tangent function is

$$\tau =c_{\mathrm{t}}+{\sigma }_{\mathrm{n}}\tan {\phi }_{\mathrm{t}}$$

(4)

where c_t and tanφ_t represent the intercept and gradient of the tangent, respectively, and their formulas can be expressed as follows:

$$c_{\mathrm{t}}={\displaystyle \frac{m-1}{m}}\cdot c_0{\left({\displaystyle \frac{m{\sigma }_{\mathrm{t}}\tan {\phi }_{\mathrm{t}}}{c_0}}\right)}^{{\displaystyle \frac{1}{1-m}}}+{\sigma }_{\mathrm{t}}\tan {\phi }_{\mathrm{t}}$$

(5)

$$\tan {\phi }_{\mathrm{t}}={\displaystyle \frac{\mathrm{d}\tau }{\mathrm{d}{\sigma }_{\mathrm{n}}}}={\displaystyle \frac{c_0}{m{\sigma }_{\mathrm{t}}}}{\left(1+{\displaystyle \frac{{\sigma }_{\mathrm{n}}}{{\sigma }_{\mathrm{t}}}}\right)}^{{\displaystyle \frac{1-m}{m}}}$$

(6)

2.2.3. Modified Tangential Method

The ordinary approach to obtain the upper bound solutions under the nonlinear conditions is the single tangential method. This method is applied to search for an optimal tangent on the nonlinear strength curve, which can introduce a group of variables (c_t, φ_t) to make the failure mechanism reach the limit state. The single tangential method simplifies the nonlinear failure criterion to a linear one. However, in reality, the stress field in a different normal stress σ_n will lead to various values of variables (c_t, φ_t).

A modified tangential technique is proposed based on the single tangential method, with the same formulas of c_t and tanφ_t, i.e., Equations (5) and (6). The differences in the variables (c_t, φ_t) subjected to the different stress states are considered in the modified tangential technique, which introduces several optimal tangents to make the failure mechanism approach the limit state. In other words, the more optimal tangents with different (c_t, φ_t) are found on the strength envelope, the more accurate the stress field is. In this way, the modified tangential technique is a strict nonlinear analysis method.

2.3. Determination of Failure Mechanism

The analysis object of this paper is limited to the deep tunnel because of the difference in the failure mechanisms between the shallow and the deep tunnels. The failure of the deep tunnel ends in an arch shape, while the failure of the shallow tunnel extends to the ground surface. Herein, the depth of the tunnel is hypothetically deep enough to be considered a deep tunnel. Referring to the failure mechanism proposed by Lu and Wang [24], the one adopted herein with a number of rigid sliding blocks is shown in Figure 2.

The upper part of this mechanism is an isosceles triangle with a 2φ₀ vertex angle, and it has the vertical down direction v₀. Parameter φ₀ is an unknown variable that needs to be optimized, and it represents the internal frictional angle under its current stress state conditions. The lower part consists of rigid triangle blocks with vertex angles of the same size, ${\displaystyle \frac{\mathsf{\pi }}{2n}}$. The parameters φ_i and ${{\varphi }^{\prime }}_i$ represent internal frictional angles in each rigid triangle block and are determined by its stress states. They need to be optimized. h represents the vertical distance from the tunnel vault to the water table, and d means the tunnel span.

2.4. Process of Calculating Critical Supporting Pressure

The numerical relationships between each velocity vector can be determined by the trigonometric equations expressed in Figure 3. The velocity v_i and v_i−1,i represent the absolute and relative velocity of each rigid triangle sliding block, respectively.

$$\begin{array}{c}{v}_1={\displaystyle \frac{\sin \left(\frac{\mathsf{\pi }}{2}+{{\phi }^{\prime }}_1\right)}{\sin \left({\alpha }_1-{\phi }_1-{{\phi }^{\prime }}_1\right)}}{v}_0,\\ v_{01}={\displaystyle \frac{\sin \left(\frac{\mathsf{\pi }}{2}-{\alpha }_1+{\phi }_1\right)}{\sin \left({\alpha }_1-{\phi }_1-{{\phi }^{\prime }}_1\right)}}{v}_0,\\ v_i={\displaystyle \frac{\sin \left(\mathsf{\pi }-\frac{\mathsf{\pi }}{2n}-{\alpha }_{i-1}+{\varphi }_{i-1}+{{\varphi }^{\prime }}_i\right)}{\sin \left({\alpha }_i-{\varphi }_i-{{\varphi }^{\prime }}_i\right)}}{v}_{i-1},\\ v_{i-1,i}={\displaystyle \frac{\sin \left(\frac{\mathsf{\pi }}{2n}+{\alpha }_{i-1}-{\varphi }_{i-1}-{\alpha }_i+{\varphi }_i\right)}{\sin \left({\alpha }_i-{\varphi }_i-{{\varphi }^{\prime }}_i\right)}}{v}_{i-1}.\end{array}$$

(7)

The geometrical relationships between velocity discontinuous lines are:

$$\begin{array}{c}\overline{O{B}_n}={\displaystyle \frac{\sin \left(\mathsf{\pi }-{\alpha }_n-\frac{\mathsf{\pi }}{2n}\right)}{\sin {\alpha }_n}}\overline{OA},\\ {\overline{AB}}_n={\displaystyle \frac{\sin \frac{\mathsf{\pi }}{2n}}{\sin {\alpha }_n}}\overline{OA},\\ \overline{O{B}_i}={\displaystyle \frac{\sin \left(\mathsf{\pi }-{\alpha }_i-\frac{\mathsf{\pi }}{2n}\right)}{\sin {\alpha }_i}}\overline{O{B}_{i+1}},\\ \overline{B_i{B}_{i+1}}={\displaystyle \frac{\sin \frac{\mathsf{\pi }}{2n}}{\sin {\alpha }_i}}\overline{O{B}_{i+1}},\\ \overline{O{B}_0}=\overline{B_0{B}_1}={\displaystyle \frac{\sin \left(\frac{\mathsf{\pi }}{2}-{\phi }_0\right)}{\sin \left(2{\phi }_0\right)}}\overline{O{B}_1}.\end{array}$$

(8)

When calculating the pore water pressure power, we should set auxiliary lines, as shown in Figure 2, with dotted lines. Their formulas can be expressed by the length of velocity discontinuous lines.

$$\begin{array}{c}\overline{B_0{C}_0}=h-\overline{O{B}_0}\cdot \cos {\phi }_0,\\ \overline{B_i{C}_i}=h+\overline{O{B}_i}\cdot \sin \left[{\displaystyle \frac{\mathsf{\pi }}{2n}}\cdot \left(i-1\right)\right],\\ \overline{C_0{C}_1}=\overline{O{B}_0}\cdot \sin {\phi }_0,\\ \overline{C_i{C}_{i+1}}=\overline{B_i{B}_{i+1}}\cdot \sin \left[{\displaystyle \frac{\mathsf{\pi }}{2n}}\cdot \left(n+i-1\right)-{\alpha }_i\right],\\ \overline{A^{\prime }{C}_i}=\overline{O{B}_i}\cdot \cos \left[{\displaystyle \frac{\mathsf{\pi }}{2n}}\cdot \left(i-1\right)\right].\end{array}$$

(9)

The areas of the rigid sliding blocks and the polygons composed by auxiliary lines, respectively, are

$$\begin{array}{c}{S}_{{\overline{O{B}_{0}B}}_1}={\displaystyle \frac{1}{2}}{\overline{O{B}_0}}^2\cdot \sin 2{\phi }_0,\\ S_{\overline{O{B}_i{B}_{i+1}}}={\displaystyle \frac{1}{2}}\overline{O{B}_i}\cdot \overline{O{B}_{i+1}}\cdot \sin {\displaystyle \frac{\mathsf{\pi }}{2n}},\\ S_{{\overline{B_{0}B}}_1}={\displaystyle \frac{1}{2}}\left(\overline{B_0{C}_0}+h\right)\cdot \overline{C_0{C}_1},\\ S_{\overline{O{B}_i}}={\displaystyle \frac{1}{2}}\left(\overline{B_i{C}_i}+h\right)\cdot \overline{A^{\prime }{C}_i},\\ S_{\overline{B_i{B}_{i+1}}}={\displaystyle \frac{1}{2}}\left(\overline{B_i{C}_i}+\overline{B_{i+1}{C}_{i+1}}\right)\cdot \overline{C_i{C}_{i+1}}.\end{array}$$

(10)

The self-weight power can be expressed as

$$P_{\mathsf{\gamma }}=S_{{\overline{O{B}_{0}B}}_1}\cdot \gamma \cdot v_0\cdot +{\displaystyle \sum _{i=1}^n{S}_{\overline{O{B}_i{B}_{i+1}}}\cdot \gamma \cdot }{v}_i\cdot \cos \left[{\displaystyle \frac{\mathsf{\pi }}{2n}}\left(n+i-1\right)-{\alpha }_i+{\phi }_i\right]$$

(11)

Considering the pore water pressure as an external force, its power acting on the failure mechanism boundaries is

$$P_{\mathrm{u}}=r_{\mathrm{u}}r\cdot \left[2S_{{\overline{B_{0}B}}_1}\cdot v_0\cdot \sin {\phi }_0+{\displaystyle \sum _{i=1}^n{S}_{\overline{O{B}_i}}\cdot v_{i-1,i}\cdot \sin {{\phi }^{\prime }}_i}+{\displaystyle \sum _{i=1}^n{S}_{\overline{B_i{B}_{i+1}}}\cdot v_i\cdot \sin {\phi }_i}\right]$$

(12)

The critical supporting pressure power results in

$$P_{\mathrm{qrc}}=\overline{OA}\cdot v_n\cdot \sin \left(\mathsf{\pi }-{\displaystyle \frac{\mathsf{\pi }}{2n}}-{\alpha }_n-{\phi }_n\right)$$

(13)

The energy dissipation rate on the velocity discontinuous lines can be expressed as

$$P_{\mathsf{\nu }}=2\overline{O{B}_0}\cdot {c^{\prime }}_0\cdot v_0\cdot \cos {\phi }_0+{\displaystyle \sum _{i=1}^n\overline{O{B}_i}\cdot }{c^{\prime }}_i\cdot v_{i-1,i}\cdot \cos {{\phi }^{\prime }}_i+{\displaystyle \sum _{i=1}^n\overline{B_i{B}_{i+1}}}\cdot c_i\cdot v_i\cdot \cos {\phi }_i$$

(14)

where ${c^{\prime }}_0$ is the cohesion parameter corresponding to the internal frictional angle φ₀ on the upper part of the failure mechanism, which is different from the initial cohesion c₀.

According to the virtual principle, the power generated by external forces is equal to the internal energy dissipation rate, i.e., P_γ + P_u + P_qcr = P_v. By substituting the above formulas, the upper bound solutions of the critical supporting pressure are obtained.

$$q_{\mathrm{cr}}={\displaystyle \frac{P_{\mathsf{\gamma }}+P_{\mathrm{u}}-P_{\mathsf{\nu }}}{\overline{OA}\cdot v_n\cdot \sin \left(\mathsf{\pi }-\frac{\mathsf{\pi }}{2n}-{\alpha }_n+{\phi }_n\right)}}$$

(15)

2.5. Constraint Conditions

The significance of the constraint conditions is to ensure that the failure mechanism satisfies the geometric relationships and boundary conditions; that is, the sliding interfaces must be generated within a proper range and that the twisted distortions cannot occur in the rigid triangle blocks. According to the velocity vector relationships shown in Figure 3, the constraint conditions of the failure mechanism are

$$\left\{\begin{array}{l}0<{\alpha }_i<\mathsf{\pi },\\ 0<{\phi }_i, {{\phi }^{\prime }}_i<{\displaystyle \frac{\mathsf{\pi }}{2}},\\ {\alpha }_i<{\alpha }_{i+1},\\ 0<{\displaystyle \frac{\mathsf{\pi }}{2n}}+{\alpha }_{i-1}-{\phi }_{i-1}-{\alpha }_i+{\phi }_i<{\displaystyle \frac{\mathsf{\pi }}{2}},\\ \mathsf{\pi }-{\displaystyle \frac{\mathsf{\pi }}{2n}}-{\alpha }_{i-1}+{\phi }_{i-1}+{\phi }^{\prime }>0,\\ {\alpha }_i+\left(\mathsf{\pi }-{\alpha }_{i-1}-{\displaystyle \frac{\mathsf{\pi }}{2n}}\right)<\mathsf{\pi },\\ {\alpha }_i-{\alpha }_{i-1}-{\displaystyle \frac{\mathsf{\pi }}{2n}}<0,\\ \overline{O{B}_i}<\overline{O{B}_{i+1}}.\end{array}\right.$$

(16)

Because of the adoption of the modified tangential technique and the multi-block failure mechanism, plenty of parameters remain unknown in the process of calculation. Specifically, φ_i, ${{\varphi }^{\prime }}_i$, and α_i are unknown in each triangle sliding block. Adding the vertex angle φ₀ in the isosceles triangle, the sum of the unknown parameters reaches the number of 3n + 1. The value of them should be determined by optimization.

Moreover, it is necessary to note that the internal frictional angle φ₀ lacks mutual constraint conditions with other unknown parameters except that it should vary from 0 to ${\displaystyle \frac{\mathsf{\pi }}{2}}$. One may not obtain the global optimal solutions or results that reach convergence from the Matlab optimizing process. Therefore, the varying scope of parameter φ₀ is controlled artificially within a relatively smaller range compared with (0, ${\displaystyle \frac{\mathsf{\pi }}{2}}$). The controlling principle is based on the general corresponding relationships between parameters of geotechnical materials. In this paper, the variation range of φ₀ is controlled within 10 degrees, so that it could accelerate the speed of computing and ensure the correctness of the results.

3. Validation of the Proposed Method

Lu and Wang [24] investigated the stability of the shield tunnel face and calculated the critical supporting pressure by means of limit analysis and limit equilibrium theory, respectively. They found that when the soil internal frictional angle φ > 20°, the tunnel depth ratio has little effect on the value of critical supporting pressure, which is consistent with the finite element simulation results by Vermeer et al. [25]. The reason is that in a deep tunnel, the failure surface cannot extend to the ground due to the soil arching effect, and consequently, the increase of overlying soil thickness does not continuously increase the magnitude of the critical supporting pressure.

Identical parameters are used in this paper, γ = 17 kN/m³, d = 10 m, c₀ = 2 kPa, and the number of rigid sliding blocks refers to n = 9. Furthermore, since Lu and Wang [24] adopted the Mohr–Coulomb criterion and did not consider the pore water pressure factor, the nonlinear coefficient m and the pore water pressure coefficient r_u here should be equal to 0 and 1.0, respectively. The comparison results are shown in

Table 1. Comparison of the calculating solutions with the results by Lu and Wang [24].

Internal Frictional Angle, φ/°	Critical Supporting Pressure by Lu and Wang [24], q/kPa	Critical Supporting Pressure in This Paper, q/kPa
20	47.59	49.8192
25	35.38	36.6889
30	25.71	26.0332
35	18.69	18.7667
40	14.74	14.7491

.

It is clear from Table 1 that under the same parameter conditions, the two values of critical supporting pressure are fairly close to each other. Furthermore, with the increase in the internal frictional angle, the differences between these two results tend to diminish. When φ approaches 40 degrees, the difference is about 9 Pa.

To further validate the proposed method, a comparison was made between the results of the simulation method using the FLAC3D v7.0 software and the proposed method while considering the strength nonlinearity and the pore water pressure. The basic parameters were given as follows: γ = 22 kN/m³, c₀ = 20 kPa, σ_t = 40 kPa, h = 20 m, d = 10 m, r_u = 0.15. The water level was 20 m above the tunnel top. The fluid configuration was inactive in the simulation. Thus, the distribution of the pore water pressure was horizontal. To accommodate the conditions of the limit analysis, the water density was reduced to the product of the soil weight and the r_u. The deformation properties were chosen as the same as the article of Mollon et al. [10], i.e., E = 240 MPa and ν = 0.22. The profile of the numerical model is shown in Figure 4.

Because the nonlinear criterion expressed in the form of τ-σ is inappropriate for a numerical simulation process [26], the equivalent Mohr–Coulomb parameters were utilized instead. They were obtained using the traditional tangent method, with each black triangle having the same Mohr–Coulomb values. The critical support pressure of the simulation method was determined by the dichotomy method, as described below. First, the range of support pressure values was given, and then the mean value was applied to the tunnel face. After sufficient calculation cycles, the convergence ratio (1 × 10⁻⁴) was used to indicate whether the calculation had reached convergence. If convergence was reached, the upper bound was decreased to the mean value; otherwise, the lower bound was increased to the mean value. Finally, the critical state was obtained when the gap between the upper and the lower was less than the tolerance (1 kPa).

The results of the proposed method and the simulation method are presented in

Table 2. Comparison of the results of the critical support pressure q/kPa.

m	The Proposed Method	The Simulation Method	The Equivalent Mohr–Coulomb Parameters
			c/kPa	φ/°
1.0	59.3	65	21.4	27.2
1.4	86.9	90	20.9	23.3
1.8	105.2	116	20.7	19.4
2.2	117.5	125	21.0	17.6

, as well as the equivalent Mohr–Coulomb parameters. From Table 2, the critical support pressure increases with increasing m. In general, the results of the limit analysis method are smaller than those of the simulation method, which is consistent with the principle that the limit analysis method provides a boundary of the real solution. Moreover, the gaps are smaller than 10%, showing a good functionality of the proposed method in addressing the tunnel face problem.

Figure 5 shows the contour of the maximum shear strain increment, which can characterize the failure morphology of the tunnel face. The failure morphology of the simulation in the top region shows differences from the failure mechanism of the limit analysis. The differences are partly because the failure behavior of the simulation evolves with the calculation steps [27], and the presented plot is about a final state. On the contrary, the failure mechanism is related to an initial state. Moreover, the failure blocks in the limit analysis are rigid, and the top block is simplified as a triangle. Although there are differences in the failure morphology, the comparison of the results shows the feasibility of the proposed method.

4. Results and Discussion

4.1. Influence of Triangle Blocks Quantity on Critical Supporting Pressure

In the new failure mechanism, the number of triangle rigid blocks is variable, and their quantity directly impacts the magnitude and accuracy of the critical supporting pressure. Therefore, to study the influence of the triangle blocks quantity on critical supporting pressure, the corresponding parameters are h = 20 m, γ = 22 kN/m³, d = 10 m, r_u = 0.3, c₀ = 20 kPa, and σ_t = 40 kPa. Meanwhile, the values of nonlinear coefficients m are 1.0, 1.4, 1.8, 2.2, and 2.6, respectively. The upper bound solutions of critical supporting pressure with different nonlinear coefficients are calculated and shown in

Table 3. Critical supporting pressure with different triangle block numbers and nonlinear coefficients (unit: kPa).

Nonlinear Coefficient, m	Rigid Triangle Block Numbers, n
	1	2	3	4	5	6	7	8
1.0	116.4443	113.9599	115.6726	115.4698	114.8485	115.1002	114.4105	114.3245
1.4	134.9470	128.4984	127.2792	128.4241	127.5683	126.9878	126.9342	126.4257
1.8	148.2151	142.8174	138.7556	138.1773	138.4983	137.8676	136.9791	136.5215
2.2	156.3417	151.9835	148.2910	146.0036	145.5345	145.3620	144.6967	143.8646
2.6	161.7502	158.1695	154.7465	152.4798	150.8689	150.4718	149.9788	149.2259

.

Table 3 shows that when the failure mechanism has a few triangle blocks, ranging from 1 to 4, the values of the critical supporting pressure fluctuate. With the continuous increase in the number of triangle blocks, the critical supporting pressure value gradually decreases and stabilizes at the end. When the triangle block number reaches 6 and 7, it is found that under the conditions of distinct nonlinear coefficients, the differences between the values of critical supporting pressure remain within 1.0 kPa. Thus, the calculating results can be considered as converged, and the feasibility of the new failure mechanism is also proved.

4.2. Influence of the Pore Water Pressure Coefficient on Critical Supporting Pressure

To investigate the influence of pore water pressure on critical supporting pressure, the parameters chosen in calculations are as follows: h = 20 m, d = 10 m, γ = 22 kN/m³, c₀ = 20 kPa, σ_t = 40 kPa, n = 9, m varies from 1.0 to 2.6, and the pore water pressure coefficient r_u changes from 0.1 to 0.5. The upper-bound solutions of the critical supporting pressure are presented in Figure 6.

From Figure 6a, it is found that the value of the critical supporting pressure increases with the increasing coefficient r_u, but the increment gradually decreases. When m = 1.0, the critical supporting pressure presents a linear growth. It is consistent with the feature of the Mohr–Coulomb linear failure criterion, which shows the correctness of the calculated results in this paper. Moreover, the influence of nonlinear coefficient m on the critical supporting pressure gradually reduces with the increase in coefficient r_u. For example, in Figure 6b, when r_u = 0.5, all values of the critical supporting pressure with different m remain within the range of 190 to 200 kN.

4.3. Influence of the Underground Water Table Height on Critical Supporting Pressure

To study the influence of the underground water table height on critical supporting pressure, the parameters chosen in calculations are as follows: m = 2.0, d = 10 m, γ = 22 kN/m³, c₀ = 20 kPa, σ_t = 40 kPa, n = 9, r_u varies from 0.1 to 0.5, and the underground water table height h changes from 20 to 40 m. The upper-bound solutions of the critical supporting pressure are shown in Figure 5.

It can be found from Figure 7a,b that with the increase in the water table and the pore water pressure coefficient, the results of the critical supporting pressure showed an upward trend. Moreover, when the parameter h is fixed, the value of the critical supporting pressure tends to increase in a curve, while the results present linear growth when the parameter r_u is fixed.

4.4. Influence of the Tunnel Span on Critical Supporting Pressure

To investigate the influence of the tunnel span on critical supporting pressure, the parameters in calculations are chosen as follows: γ = 22 kN/m³, c₀ = 20 kPa, σ_t = 40 kPa, r_u = 0.3, h = 20 m, n = 9, m changes from 1.0 to 2.6, and the tunnel span d varies from 6 to 14 m. The upper-bound solutions of the critical supporting pressure are shown in Figure 6.

It can be seen from Figure 8a,b that the solutions of the critical supporting pressure increase gradually with the growth of the tunnel spans and the nonlinear coefficients. Furthermore, one can also see that when the value of the tunnel span remains small, such as d = 6 m, the changes in the nonlinear coefficient have almost no impact on the critical supporting pressure. The reason is that tunnels with smaller excavation section heights have better stability.

4.5. Influence of the Nonlinear Coefficient on Collapse Range

To estimate the tunnel face collapse range with different nonlinear coefficients, the parameters selected were: γ = 22 kN/m³, c₀ = 20 kPa, σ_t = 40 kPa, r_u = 0.2, h = 20 m, d = 10 m, n = 9. According to the optimized angles α_i, the collapse ranges are plotted in Figure 7 with nonlinear coefficients m varying from 1.0 to 1.8.

As shown in Figure 9, the collapse range of the tunnel face scales out with the increase in the nonlinear coefficient, but the expansion decreases gradually. It is consistent with the variations between the nonlinear coefficient and the critical supporting pressure.

Judging from the overall shape of the collapse range, the failure mode can be divided into three parts, the triangular section at the bottom, the curved section in the middle, and the isosceles triangle section on the top, as shown in Figure 9 with bold lines. Furthermore, the shape of the collapse range is similar to the tunnel face failure mechanisms proposed by Mollon et al. [10], which demonstrates that the new failure mechanism adopted in this paper is feasible.

4.6. Influence of the Pore Water Pressure Coefficient on Collapse Range

To investigate the collapse range of tunnel face with different pore water pressure coefficients, the parameters in the calculations are set as follows: γ = 22 kN/m³, c₀ = 20 kPa, σ_t = 40 kPa, m = 2.0, h = 20 m, d = 10 m, n = 9. Based on optimized angles α_i, the collapse ranges are plotted in Figure 10 with r_u varying from 0.1 to 0.5.

It shows from Figure 8 that the change law is opposite to that of m, i.e., the tunnel face collapse range shrinks gradually with the increase in pore water pressure coefficient, which is in agreement with the results of Huang et al. [13].

5. Conclusions

A new deep tunnel face failure mechanism with triangle blocks is put forward. According to the nonlinear failure criterion, the upper bound solutions of critical supporting pressure on a tunnel face are obtained by a modified tangential technique. Compared with previous theoretical results, the correctness of the computing process and the failure mechanism are validated. The formulas of the critical supporting pressure were deduced.

The influence of different parameters on the critical supporting pressure is investigated, and the collapse range of the tunnel face is described according to the calculating results. The problem is convergent as the triangle block number in the calculation increases. It is found that the upper bound solutions of critical supporting pressure increase with the growth of the nonlinear coefficient, pore water pressure coefficient, height of the underground water table, and tunnel span. The collapse range of the tunnel face scales out with the increase in the nonlinear coefficient; however, it shrinks with the increase in the pore water pressure coefficient. Moreover, with the increase in the nonlinear coefficient, the impact strength on critical supporting pressure and collapse range declines gradually.

To avoid the case that the optimizing process cannot converge, the range of parameter φ₀ was restricted based on the general corresponding relationship between parameters. Therefore, the failure mechanism adopted in this paper could be further improved to solve the problem without restricting the parameter φ₀.