1 Introduction

The potential of rectangular tunnels, which have high space utilization ratios, in urban underground construction has been observed. Rectangular tunnels are widely used in combined utility, roadway and subway lines. To avoid significant disturbance to the surroundings, most rectangular tunnels are built with a trenchless technique called box jacking. Recently, many rectangular tunnels with large cross sections have been successfully installed by jacking over Chinese cities. The world’s largest box tunnel with a width of 14.82 m and height of 9.45 m was recently successfully built in JiaXing, China [20]. Currently, boring machines are commonly used in tunnel box jacking to enhance the efficiency of excavation. Typically, the machine head is equipped with multiple spoke cutter heads that are driven independently. Soil cutting with multiple cutter heads causes unexpected fluctuations in support pressure, which increases the risk of face instability.

The working face stability of circular tunnels has been intensively investigated. Limit equilibrium analysis is widely used for analyzing face stability because its concept is familiar to engineers and intuitively clear. Most limit equilibrium analyses are performed based on the silo-wedge model (Horn [18]). Anagnostou and Kovári [2] used the model to investigate the working face stability of earth-pressure-balanced (EPB) shield tunnels under groundwater seepage. In this analysis, the seepage forces applied to the wedge block were calculated using Gausses’ theorem. Broere [4] modified the mode by dividing the wedge block into multiple slices to investigate slurry shield tunnel face instability in layered soils, and the slurry penetration-induced excess pore pressure on the face stability was evaluated. Anagnostou [3] developed the model by accounting for horizontal soil arching on the wedge block. Based on the model, face stability under complex conditions was explored [15, 39, 47]. The silo-wedge model was also applied to investigate the face instability of jacking tunnels [5], [19], [45].

The overburden of the soil column is important to the silo-wedge model and is intensively investigated. Nv et al. [34] employed the traditional silo-wedge model to investigate the impact of cover soil on both passive and active face failures, and the critical cover depth of a shield tunnel was identified. Chen et al. (2014) proposed a two-dimensional maximum principal stress arching model to explore the stress distribution in the soil column, and the height of failure was identified as a result. This model was verified and was in agreement with numerical simulations [8, 10, 41]. Later, Chen (9 established the concept of multi-arching that identifies the upper end-bearing arch, the middle friction, and the stability zone in the arching zone. Accordingly, a three-dimensional multi-arching model was proposed. This multi-arching model has been improved by exploring the stress-transfer mechanisms within the different arching zones [24], [44].

An upper-bound limit analysis is also employed to investigate the face instability. Constructing a kinematically admissible mechanism is the key to the upper-bound analysis. Two-dimensional mechanisms have been developed by many researchers [12, 35, 27]. Constructing a three-dimensional mechanism is more difficult. Leca and Dormieux [22] proposed a family of 3D mechanisms with conical blocks for face instability, and the model was developed for more extensive applications [28, 29, 26, 23]. The 3D mechanism construction was prompted when Mollon [30, 31, 32] applied a “point-by-point” spatial discretization technique in modeling. As a result, many kinematic models have been developed to investigate more complicated problems [16, 36, 37, 38, 40, 42]. Recently, the evolution of face failure was intensively investigated through experimental model tests (Chambon and Corte [6, 1, 20], Idinger [17], Liu [25], Chen [11], [12, 21, 22, 33]). Most of the tests observed the internal movement of soil, and the failure mechanism was revealed. The experimental tests verify the rationality of the silo-wedge model.

This paper investigates the working face stability of box tunnels, and in particular, this paper focuses on the situation in which the working face chamber exposed to free air due to obstruction removal. Inspired by the traditional model, the modified silo-wedge model is established for this problem. In the proposed model, the wedge and prim blocks are divided into sub-blocks, and the interactions between the blocks are taken into account. Based on the proposed model, the solution of the residual support pressure is derived through limit equilibrium analysis. The verification of the current model is performed by comparison to the traditional silo-wedge model. The current model is validated by investigating a box tunnel case in Suzhou, China.

2 Problem definition

2.1 Box tunneling machines and obstructions removing

Most box tunneling machines are designed to work in earth-pressure-balanced (EPB) mode. For high excavation performance, the faces are usually designed to be equipped with multiple spoke cutter heads, and each cutter head is driven independently (Fig. 1a). Usually, double-screw conveyors are installed behind the bulkhead to expel the cuttings (Fig. 1b). When the tunnel is driven at shallow depth, the presence of unexpected obstructions, such as boulders, steel plates, and steel pipe debris, might cause excavation problems. The obstructions entering the chamber easily obstruct the cutter heads. The obstructions must be removed to ensure the performance of the cutter heads.

Fig. 1
figure 1

Typical box tunneling machine

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In project T221 of the box tunnel in Singapore, the tunnel drive was suspended several times to remove the boulders and steel debris left in the working chamber. With the man-lock retrofitted to the boring machine (Fig. 2a), the working chamber was cleared by allowing workers to enter the chamber through the man-lock under compressed air. The compressed air serves as auxiliary support for face stability. As a result, the ground surface settlement was controlled, and the working face stability was ensured.

Fig. 2
figure 2

Project T221 in Singapore [13]

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However, for most cases, box tunnel machines are not equipped with man locks due to their high cost. The chamber clearance in these machines has to be carried out under open air, and the working face changes into open mode. In this situation, obstruction removal is riskier, and the ground becomes more vulnerable to instability. Recently, in the case of box tunnels in Suzhou, China, typical chamber obstruction removal under open air was successfully carried out. Figure 3a shows the working face configuration of the tunnel machine. The tunnel cross section has a width \(L\) of 6.9 m and height \(D\) of 4.2 m. Six independent spoke cutter heads are installed at the face for excavation, and double-screw conveyors are installed at the bulkhead. The conveyor gates have an identical diameter of 0.56 m, and the distance between the centers of circular gates is 2.4 m. The working parameters of the cutter heads and conveyors are listed in Table 1. When the tunnel machine was driven mainly in the mixed layers of miscellaneous fills and silty clay, boulders were encountered. Some boulders that entered the chambers through the opening between the spokes obstructed the cutter heads, and the tunnel drive had to be suspended to remove the boulders. First, the double conveyors were dismantled, and the bottom gates were kept open. Then, the cuttings were removed as much as possible to identify the boulders. The boulders were manually extracted out of the chamber through the bottom gates in the end. In this case, a number of large pieces of boulders were extracted from the chamber, and the largest piece had a length over 50 cm (Fig. 3b). In the entire work, the working face stability was sustained. More details of the case are presented in Sect. 5.

Fig. 3
figure 3

The box tunnel case in Suzhou, China

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Table 1 Working parameters of cutter heads and conveyors
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2.2 Partially supported working face

Figure 4 shows the working face partially supported for the particular situation. The rectangular cross section features a normalized ratio \(\lambda = {D \mathord{\left/ {\vphantom {D L}} \right. \kern-\nulldelimiterspace} L}\). In Fig. 4a, the distribution of the remains is illustrated. Due to the configuration of the machine head, the obstructions in the chamber have to be removed from the conveyor gate at the bottom. In this work, the conveyor is dismantled while the gate is open. The cuttings are expelled through the open gate before the obstructions can be removed smoothly. The cuttings in the middle chamber are influenced most significantly during cutting expelling, and in most likely cases, the middle part of the chamber is almost empty. On the other hand, for the neighboring parts, the cuttings are influenced much less, and the remains serve as the support for the working face. During the entire work, the pressure can be measured by earth pressure sensors for the face stability assessment. In Fig. 4b, the face is divided into three rectangular parts accordingly. It assumes that the middle part II with width \(B\) is completely unsupported and that the remaining Parts I and III are supported by the remaining parts in the chamber. The concept of an unsupported ratio \(\mu\) defined by \(\mu = {B \mathord{\left/ {\vphantom {B L}} \right. \kern-\nulldelimiterspace} L}\) is introduced for featuring a partially supported face. The pressure applied on Parts I and III is assumed to be equal and uniform, and the pressure is denoted by \(s\). The pressure \(s\) required for the face stability is investigated in the following section.

Fig. 4
figure 4

Partially supported working face

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3 Improved model for ground stability

3.1 Profile of box tunneling

Figure 5 presents a box tunnel driven in homogenous soil. The box tunnel has a cross section with a height of \(D\) and width of \(L\). At the face, the middle part with a width of \(B\) is unsupported, while the neighboring parts are supported by uniform pressure \(s\). The cover depth is denoted as \(C\) and normalized by diameter \(D\) to define a dimensionless parameter \({C \mathord{\left/ {\vphantom {C D}} \right. \kern-\nulldelimiterspace} D}\). The surcharge on the ground surface is denoted as \(\sigma_{s}\). In practice, when the tunnel drive is suspended for obstruction, the road traffic restrictions are suggested to be imposed. Thus, \(\sigma_{s}\) is assumed to be 0 kPa in this analysis. The soil is idealized as a rigid-plastic material obeying the Mohr–Coulomb failure condition with effective cohesion \(c^{\prime}\) and effective angle of internal friction \(\varphi^{\prime}\). The total stress concept is adopted, and the unit weight of soil is \(\gamma\).

Fig. 5
figure 5

Profile of box tunneling

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3.2 Traditional model

Figure 6 shows the profile of the traditional silo-wedge model. The traditional model consists of a wedge block (i) and a prism block (ii). The wedge block has a height of \(D\) and width of \(L\). This block is partially supported by the cutting remains in the chamber. On plane abkj, uniform pressure \(s\) is applied to the area abod and ghkj, and the remaining part with a width of \(B\) is entirely unsupported. As shown in Fig. 6b, this wedge block is loaded by its total weight \(G_{1}\), and its movement is resisted by the pressure \(s\). The slanted plane befk inclines to the horizontal at an angle of \(\theta\). At this slanted plane, there is the resultant shear force \(T_{1}\) acting parallel to the slanted plane, which results from the normal force \(N_{1}\), working perpendicular to the slanted plane. The side surfaces abe and fjk of the wedge are loaded by the shear forces \(T_{2}\) and \(T_{3}\), which are both parallel to the slant plane. The top face of the wedge is loaded by the overburden force \(P_{1}\) from the soil column. In the prism block, the distribution of vertical stress along the depth in the prism can be calculated by using Broere’s model [4]. In this model, the lateral earth pressure coefficient \(K\) is chosen as the at-rest lateral earth pressure coefficient \(K_{0}\), and the effective length of the arching is calculated by \(R_{{0}} = LD\cot \theta /2\left( {L + D\cot \theta } \right)\). Referring to previous studies [2, 4], the pressure \(s\) can be obtained based on the force equilibrium of the wedge.

Fig. 6
figure 6

The traditional silo-wedge model

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By introducing the notations

$$\kappa = \frac{{1 - e^{{ - K_{0} \tan \varphi^{{^{\prime } }} \frac{C}{{R_{{0}} }}}} }}{{K_{0} \tan \varphi^{\prime } }}$$
(1)
$$\xi^{ - } = \cos \theta (\sin \theta - \cos \theta \tan \varphi^{\prime } ),$$
(2)

and

$$\xi^{ + } = \sin \theta (\cos \theta + \sin \theta \tan \varphi^{\prime } )$$
(3)

the solution of pressure \(s\) can be written in the form of:

$$s = \gamma DN_{\gamma }^{0} - c^{\prime } N_{c}^{0} ,$$
(4)

where

$$N_{\gamma }^{0} = \frac{1}{{\left( {1 - \mu } \right)\xi^{ + } }}\left[ {\frac{{{2}\kappa R_{{0}} /D{ + }1}}{2}\xi^{ - } - \frac{{K_{0} \left( {{2}\kappa {{R_{{0}} } \mathord{\left/ {\vphantom {{R_{{0}} } L}} \right. \kern-\nulldelimiterspace} L}{ + }\lambda } \right)}}{3}\cos \theta \tan \varphi^{\prime } } \right]$$
(5)
$$N_{c}^{0} = \frac{1}{{\left( {1 - \mu } \right)\xi^{ + } }}\left[ {\kappa \xi^{ - } + 1 + \lambda \cos \theta \left( {1 - \frac{2}{3}K_{0} \kappa \tan \varphi^{\prime } } \right)} \right]$$
(6)

3.3 Modified model

Figure 7 shows the profile of the proposed model. This model is established by modifying the traditional silo-wedge model. A trapezoidal prism is used for silo modeling. The bottom corner at the top trapezoidal surface l’m’n’p’ is denoted by \(\alpha\), which is calculated by

$$\alpha = \arctan \frac{2D\cot \theta }{{L - B}}.$$
(7)
Fig. 7
figure 7

The modified model

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The slanted surface b’e’f’k’ inclines to the horizontal at an angle of \(\theta\). The modification is performed by dividing the wedge and prism blocks into three sub-blocks. The wedge block is a composite of three rigid blocks (blocks I, II and III), while the above prism block is a composite of three rigid blocks (blocks IV, V and VI). Blocks I and III are supported by pressure \(s\) subjected to the remains in the chamber, and the two are assumed to be stationary. Block II amid blocks I and III is unsupported, and its movement is inhibited by the neighboring blocks. In the prism, blocks IV and VI immediately above blocks I and III, respectively, are assumed to be stationary. Block V lies on the top surface of block II, and its movement is resisted by neighboring blocks IV and VI.

As shown in Fig. 8a, block II acts upon by (a) the resultant vertical force \(P^{\prime}_{2}\) of the overburden from block V onto the top surface o’e’f’g’; (b) the resultant normal forces \(\left( {N^{\prime}_{12} {, }N^{\prime}_{32} } \right)\) and shear forces \(\left( {T^{\prime}_{12} {, }T^{\prime}_{32} } \right)\) along the parallel lateral surfaces o’d’e’ and g’f’h’; (c) the resultant normal force \(N^{\prime}_{2}\) and shear force \(T^{\prime}_{2}\) along the slip surface e’d’h’f’; and (d) its weight \(G_{2}\).

Fig. 8
figure 8

The forces applied on the sub-blocks

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The vertical force \(P^{\prime}_{2}\) applied on surface o’e’f’g’ is calculated by:

$$P_{2}^{\prime } = \sigma_{v} BDcot\theta ,$$
(8)

where \(\sigma_{v}\) denotes the vertical stress. Respecting the arching effect within the silo, \(\sigma_{v}\) can be calculated by:

$$\sigma_{v} = \max \left\{ {0, \, \frac{{R_{{1}} \gamma - c^{\prime } }}{{K_{0} \tan \varphi^{\prime } }}\left( {1 - e^{{ - K_{0} \tan \varphi^{\prime}\frac{C}{{R_{{1}} }}}} } \right) + \sigma_{s \, } e^{{ - K_{0} \tan \varphi^{\prime}\frac{C}{{R_{{1}} }}}} } \right\},$$
(9)

where \(K_{0}\) is the at-rest lateral earth pressure coefficient (\(K_{0} = 1 - \sin \varphi^{\prime}\)) and \(R_{{1}}\) is the semi-length of the effective arching. Referring to Broere’s research [4], the semi-length \(R_{{1}}\) of three-dimensional arching is calculated by:

$$R_{{1}} = \frac{BD\cot \theta }{{2(B + D\cot \theta )}}$$
(10)

On slanted surface e’f’h’d’, the resultant normal force \(N^{\prime}_{2}\) is perpendicular to the surface, and the shear force \(T^{\prime}_{2}\) parallel to the surface results from \(N^{\prime}_{2}\). The relationship between \(N^{\prime}_{2}\) and \(T^{\prime}_{2}\) is presented as

$$T_{2}^{\prime } = \frac{{c^{\prime } BD}}{\sin \theta } + N_{2}^{\prime } \tan \varphi^{\prime } .$$
(11)

The self-weight \(G_{2}\) of block II is

$$G_{2} = \gamma V_{2} = \frac{\gamma }{2}BD^{2} \cot \theta$$
(12)

Due to the symmetry, on lateral surfaces o’e’d’ and g’f’h’, the resultant shear forces \(T^{\prime}_{12}\) and \(T^{\prime}_{32}\), respectively, result from normal forces and are equal (\(T^{\prime}_{12} = T^{\prime}_{32}\)).

It is obvious that the condition of force equilibrium in the x-direction is stratified due to symmetry. The equilibrium conditions in the y and z directions lead to:

$$\left\{ \begin{gathered} N_{{2}}^{\prime } \sin \theta = (T_{{2}}^{\prime } + 2T_{12}^{\prime } )cos\theta \hfill \\ P_{2} + G_{2} = (T_{{2}}^{\prime } + 2T_{{{12}}}^{\prime } )\sin \theta + N_{{2}}^{\prime } \cos \theta \hfill \\ \end{gathered} \right.$$
(13)

As shown in Fig. 8b, c., block I is acted upon by (a) the resultant vertical force \(P^{\prime}_{1}\) of the overburden from block IV onto the top surface o’a’e’; (b) the resultant normal force \(N^{\prime}_{21}\) and shear force \(T^{\prime}_{21}\) along the interface o’e’d’; (c) the resultant normal forces \(N^{\prime}_{l}\) and shear forces \(T^{\prime}_{l}\) along the lateral surfaces a’b’e’; (d) the resultant normal force \(N^{\prime}_{1}\) and shear force \(T^{\prime}_{1}\) along the slanted slip surfaces b’d’e’; (c) the support force \(S\) on o’a’b’d’; and (d) its weight \(G^{\prime}_{1}\). It should be noted that the forces \(\left( {N^{\prime}_{12} , \, T^{\prime}_{12} } \right)\) and \(\left( {N^{\prime}_{21} , \, T^{\prime}_{21} } \right)\) are applied to block I and II interactions, respectively (\(N^{\prime}_{12} = N^{\prime}_{21}\), \(T^{\prime}_{12} = T^{\prime}_{21}\)).

The resultant vertical force \(P^{\prime}_{1}\) applied on the top surface o’a’e’ is calculated by:

$$P_{1} = \frac{\gamma (L - B)CDcot\theta }{4}.$$
(14)

On slanted surface b’d’e’, the resultant normal force \(N^{\prime}_{1}\) is perpendicular to the surface, and the shear force \(T^{\prime}_{1}\) resulting from \(N^{\prime}_{1}\) is assumed to be parallel to Line d’e’. The relationship between \(N^{\prime}_{1}\) and \(T^{\prime}_{1}\) is presented as:

$$T_{1}^{\prime } = \frac{{c^{\prime } (L - B)D}}{4\sin \theta } + N_{1}^{\prime } \tan \varphi^{\prime } .$$
(15)

On lateral surface a’b’e’, the resultant normal force \(N^{\prime}_{l}\) perpendicular to the surface is calculated by:

$$N_{l}^{\prime } = \frac{\cot \theta }{{\sin \alpha }}\int_{0}^{D} {K_{{0}} } \sigma_{z} \left( {D - z} \right){\text{d}}z,$$
(16)

where \(\sigma_{z}\) is the vertical stress at a certain depth and is calculated by \(\sigma_{z} = \gamma \left( {C + z} \right)\).

Resulting from \(N^{\prime}_{l}\), the shear force \(T^{\prime}_{l}\) parallel to Line b’e’ is calculated by

$$T_{l}^{\prime } = \frac{{c^{\prime } D^{2} \cot \theta }}{2\sin \alpha } + N_{l}^{\prime } \tan \varphi^{\prime } .$$
(17)

The support force \(S\) acting on plane oabd is \(S = {{s\left( {L - B} \right)D} \mathord{\left/ {\vphantom {{s\left( {L - B} \right)D} 2}} \right. \kern-\nulldelimiterspace} 2}\). The weight \(G^{\prime}_{1}\) of Block I is

$$G_{1}^{\prime } = \frac{\gamma }{6}\left( {L - B} \right)D^{2} \cot \theta .$$
(18)

The equilibrium conditions in the y’ and z’ directions lead to:

$$\left\{ \begin{gathered} S + T_{1}^{\prime } \cos \theta + T_{l}^{\prime } \sin \alpha \cos \theta = N_{l}^{\prime } \cos \alpha + N_{1}^{\prime } \sin \theta + T_{21}^{\prime } \cos \theta \hfill \\ P_{1}^{\prime } + G_{1}^{\prime } + T_{21}^{\prime } \sin \theta = N_{1}^{\prime } \cos \theta + T_{l}^{\prime } \sin \alpha \sin \theta + T_{1}^{\prime } \sin \theta \, \hfill \\ \end{gathered} \right.$$
(19)

Combining equation settings (11) and (19), the pressure \(s\) is solved. After rearrangement, the pressure can be written as

$$s = \gamma DN_{\gamma }^{{1}} - c^{\prime } N_{c}^{1}$$
(20)

where

$$N_{\gamma }^{1} = \left[ {\left( {\frac{C}{2D} + \frac{1}{3}} \right){ + }\frac{{\left( {2\kappa R_{{1}} /D{ + }1} \right)\mu }}{{2\left( {1 - \mu } \right)}}} \right]\frac{{\xi^{ - } }}{{\xi^{ + } }}{ + }K_{0} \frac{{\left( {{{3C} \mathord{\left/ {\vphantom {{3C} L}} \right. \kern-\nulldelimiterspace} L} + \lambda } \right)}}{{3\left( {1 - \mu } \right)}}\left( {\cot \theta \cot \alpha - \frac{{\cos \theta \tan \varphi^{\prime } }}{{\xi^{ + } }}} \right)$$
(21)

and

$$N_{c}^{1} = \frac{1}{{\left( {1 - \mu } \right)\xi^{ + } }}\left[ {\mu \left( {\kappa \xi^{ - } + 1} \right){ + }\lambda \cos \theta } \right] + \frac{1}{{2\xi^{ + } }}$$
(22)

4 Stability analysis

4.1 Parametric analysis

In this part, a typical situation of \(D{\text{ = 6 m}}\) and \(\lambda { = 0}{\text{.5}}\) is considered. The unsupported ratio \(\mu\) varies from 1/4 to 3/4. For shallowly buried tunnels, the depth ratio \({C \mathord{\left/ {\vphantom {C D}} \right. \kern-\nulldelimiterspace} D}\) varies from 1 to 2, and the total stress concept is accepted for calculating the soil stress. The soil weight is chosen as \(\gamma = 20{\text{ kN/m}}^{3}\). The frictional angle φ′ varies from 15° to 45°.

Figure 9 shows the variation of \(N_{\gamma }^{{0}}\) and \(N_{\gamma }^{{1}}\) depending on \(\varphi^{\prime}\). For \({C \mathord{\left/ {\vphantom {C {D{ = 1}}}} \right. \kern-\nulldelimiterspace} {D{ = 1}}}\) (Fig. 9a), both \(N_{\gamma }^{{0}}\) and \(N_{\gamma }^{{1}}\) decrease nonlinearly with increasing \(\varphi^{\prime}\). At constant \(\varphi^{\prime}\), \(N_{\gamma }^{{1}}\) is obviously higher than \(N_{\gamma }^{{0}}\) when \(\mu \le {{1} \mathord{\left/ {\vphantom {{1} {2}}} \right. \kern-\nulldelimiterspace} {2}}\), and they become close to each other when \(\mu { = }{{3} \mathord{\left/ {\vphantom {{3} {4}}} \right. \kern-\nulldelimiterspace} {4}}\). In Fig. (b), similar variations in \(N_{\gamma }^{{0}}\) and \(N_{\gamma }^{{1}}\) at \({C \mathord{\left/ {\vphantom {C {D{ = 2}}}} \right. \kern-\nulldelimiterspace} {D{ = 2}}}\) can be observed. The comparison between Figs. 9a, b shows the influence of \({C \mathord{\left/ {\vphantom {C D}} \right. \kern-\nulldelimiterspace} D}\) on \(N_{\gamma }^{{0}}\) and \(N_{\gamma }^{{1}}\). The higher \({C \mathord{\left/ {\vphantom {C D}} \right. \kern-\nulldelimiterspace} D}\) is, the higher are the coefficients \(N_{\gamma }^{{0}}\) and \(N_{\gamma }^{{1}}\). Figure 10 shows the variations in \(N_{c}^{{0}}\) and \(N_{c}^{{1}}\) depending on \(\varphi^{\prime}\). For \({C \mathord{\left/ {\vphantom {C {D{ = 1}}}} \right. \kern-\nulldelimiterspace} {D{ = 1}}}\) (Fig. 10a), both \(N_{c}^{{0}}\) and \(N_{c}^{{1}}\) decrease nonlinearly with increasing \(\varphi^{\prime}\). Both \(N_{c}^{{0}}\) and \(N_{c}^{{1}}\) are virtually independent of \(\varphi^{\prime}\) when \(\mu \le {1 \mathord{\left/ {\vphantom {1 2}} \right. \kern-\nulldelimiterspace} 2}\), and they become more sensitive at higher \({B \mathord{\left/ {\vphantom {B L}} \right. \kern-\nulldelimiterspace} L}\). At constant \(\varphi^{\prime}\), \(N_{c}^{{0}}\) is higher than \(N_{c}^{{1}}\), and the discrepancy becomes more significant at higher \(\mu\). Similar variations in \(N_{\gamma }^{{0}}\) and \(N_{\gamma }^{{1}}\) are illustrated in Fig. 10b.

Fig. 9
figure 9

Variations in \(N_{\gamma }^{{0}}\) and \(N_{\gamma }^{{1}}\) depending on \(\varphi^{\prime}\)

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Fig. 10
figure 10

Variations in \(N_{c}^{{0}}\) and \(N_{c}^{{1}}\) depending on \(\varphi^{\prime}\)

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In this part, the typical cohesion-less soil (\({{c^{\prime}} \mathord{\left/ {\vphantom {{c^{\prime}} {\left( {\gamma D} \right) = 0}}} \right. \kern-\nulldelimiterspace} {\left( {\gamma D} \right) = 0}}\)) is considered for the worse cases. As a result, Eq. (20) is simplified as \(s = \gamma DN_{\gamma }^{{1}}\). Table 2 shows the variation in \(\theta\) and \(\alpha\) depending on \(\varphi^{\prime}\). It can be seen that with increasing \(\varphi^{\prime}\), \(\theta\) increases, while \(\alpha\) decreases. The increase in \(\theta\) indicates that the wedge slope becomes steeper. The decrease in \(\alpha\) indicates the shrinkage of the base area of the prism. The variations in \(\theta\) and \(\alpha\) imply the volume contraction of failure blocks. Predictably, the failure zone associated with the current model decreases with increasing \(\varphi^{\prime}\). At constants \(\varphi^{\prime}\) and \({C \mathord{\left/ {\vphantom {C D}} \right. \kern-\nulldelimiterspace} D}\), a higher \(\mu\) contributes to a lower \(\theta\) but a higher \(\alpha\). For particular (\(\varphi^{\prime} = 20^\circ\)), as \(\mu\) increases from 1/4 to 3/4, \(\theta\) decreases from 62.5° to 59.5°, while \(\alpha\) reversely increases from 34.7° to 67°. The decrease in \(\theta\) implies that the wedge inclination approaches the slope angle of \(45^\circ + {{\varphi^{\prime}} \mathord{\left/ {\vphantom {{\varphi^{\prime}} 2}} \right. \kern-\nulldelimiterspace} 2}\) in the traditional model. The change in \(\theta\) and \(\alpha\) indicates that the modified model resembles the traditional model at high \(\mu\). Moreover, at constants \(\varphi^{\prime}\) and \(\mu\), \(\theta\) and \(\alpha\) remain virtually constant with increasing \({C \mathord{\left/ {\vphantom {C D}} \right. \kern-\nulldelimiterspace} D}\). The failure model is virtually independent of the ratio \({C \mathord{\left/ {\vphantom {C D}} \right. \kern-\nulldelimiterspace} D}\).

Table 2 Variation in the optimal \(\theta\) and \(\alpha\) depending on \(\varphi^{\prime}\)
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4.2 Comparisons on the support pressure

Figure 11 shows the variation in normalized support pressure \({s \mathord{\left/ {\vphantom {s {\gamma D}}} \right. \kern-\nulldelimiterspace} {\gamma D}}\) depending on frictional angle \(\varphi^{\prime}\). In Fig. 11a, both the current and traditional models predict that \({s \mathord{\left/ {\vphantom {s {\gamma D}}} \right. \kern-\nulldelimiterspace} {\gamma D}}\) decreases in a similar pattern with increasing \(\varphi^{\prime}\). At constant \(\varphi^{\prime}\) and \(\mu\), the current mode predicts higher \({s \mathord{\left/ {\vphantom {s {\gamma D}}} \right. \kern-\nulldelimiterspace} {\gamma D}}\) than the traditional model does. \({s \mathord{\left/ {\vphantom {s {\gamma D}}} \right. \kern-\nulldelimiterspace} {\gamma D}}\) associated with the current model is safer for the design. Moreover, it can be seen that the discrepancy in \({s \mathord{\left/ {\vphantom {s {\gamma D}}} \right. \kern-\nulldelimiterspace} {\gamma D}}\) is significant at relative \(\mu = {1 \mathord{\left/ {\vphantom {1 4}} \right. \kern-\nulldelimiterspace} 4}\) and is reduced with increasing \(\mu\). This indicates that for scenarios with small \(\mu\), the current model is more capable of stability assessment than the traditional model. On the other hand, at high \(\mu\), the current model works similarly to the traditional model. In Fig. 11b (\({C \mathord{\left/ {\vphantom {C D}} \right. \kern-\nulldelimiterspace} D} = 2\)), a similar variation of \({s \mathord{\left/ {\vphantom {s {\gamma D}}} \right. \kern-\nulldelimiterspace} {\gamma D}}\) depending on \(\varphi^{\prime}\) can be observed. Compared to the scenario of \({C \mathord{\left/ {\vphantom {C D}} \right. \kern-\nulldelimiterspace} D} = 1\), both models predict higher \({s \mathord{\left/ {\vphantom {s {\gamma D}}} \right. \kern-\nulldelimiterspace} {\gamma D}}\) at \({C \mathord{\left/ {\vphantom {C D}} \right. \kern-\nulldelimiterspace} D} = 2\). At constant \(\mu\), the discrepancy between the models is more significant at higher \({C \mathord{\left/ {\vphantom {C D}} \right. \kern-\nulldelimiterspace} D}\). The current model outperforms the traditional model for cases with either low \(\mu\) or high \({C \mathord{\left/ {\vphantom {C D}} \right. \kern-\nulldelimiterspace} D}\).

Fig. 11
figure 11

Variation in \({s \mathord{\left/ {\vphantom {s {\gamma D}}} \right. \kern-\nulldelimiterspace} {\gamma D}}\) depending on \(\varphi^{\prime}\)

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Figure 12 shows the variation in \({s \mathord{\left/ {\vphantom {s {\gamma D}}} \right. \kern-\nulldelimiterspace} {\gamma D}}\) depending on \(\mu\) for typical \(\varphi^{\prime} = 30^\circ\). In Fig. 12a (\({C \mathord{\left/ {\vphantom {C D}} \right. \kern-\nulldelimiterspace} D} = 1\)), both models predict that \({s \mathord{\left/ {\vphantom {s {\gamma D}}} \right. \kern-\nulldelimiterspace} {\gamma D}}\) increases with increasing \(\mu\). It can also be found that at identical \({L \mathord{\left/ {\vphantom {L D}} \right. \kern-\nulldelimiterspace} D}\), the current model is less sensitive to \(\mu\) than the traditional model, and the discrepancy between the models becomes less significant with increasing \(\mu\). This result probably occurs because in the current model at a relatively low \(\mu\), the resistances from the neighboring blocks on the sliding block are significant, and this effect is reduced with increasing \(\mu\). For particular, at \(\mu = 0.8\), the current and traditional models predict similar results. The predicted pressure is lower than the at-rest horizontal earth pressure when \(\mu \le 0.5\). In this scenario, the current model works better than the traditional model by predicting the higher pressure, and the face stability could be assessed by comparing the measured pressure to the predicted pressure. However, when \(\mu > 0.5\), the required pressures are higher than the at-rest horizontal earth pressure, which means that the cuttings should be thrusted. The above result affirms that the models are more rational when \(\mu \le 0.5\). A similar variation in \({s \mathord{\left/ {\vphantom {s {\gamma D}}} \right. \kern-\nulldelimiterspace} {\gamma D}}\) at \({C \mathord{\left/ {\vphantom {C D}} \right. \kern-\nulldelimiterspace} D} = 2\) is presented in Fig. 12b. The comparison shows that when \(\mu\) varies from 0.2 to 0.6, the discrepancy between the models is more significant at higher \({C \mathord{\left/ {\vphantom {C D}} \right. \kern-\nulldelimiterspace} D}\).

Fig. 12
figure 12

Variation in \({s \mathord{\left/ {\vphantom {s {\gamma D}}} \right. \kern-\nulldelimiterspace} {\gamma D}}\) depending on \(\mu\) at typical \(\varphi { = 30}^\circ\)

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Figure 13 shows the variation of \({s \mathord{\left/ {\vphantom {s {\gamma D}}} \right. \kern-\nulldelimiterspace} {\gamma D}}\) depending on \({C \mathord{\left/ {\vphantom {C D}} \right. \kern-\nulldelimiterspace} D}\) for typical \(\varphi^{\prime} = 30^\circ\). It can be found that both models predict that \({s \mathord{\left/ {\vphantom {s {\gamma D}}} \right. \kern-\nulldelimiterspace} {\gamma D}}\) increases with increasing \({C \mathord{\left/ {\vphantom {C D}} \right. \kern-\nulldelimiterspace} D}\). At \(\mu = {1 \mathord{\left/ {\vphantom {1 4}} \right. \kern-\nulldelimiterspace} 4}\) and \(\mu = {1 \mathord{\left/ {\vphantom {1 2}} \right. \kern-\nulldelimiterspace} 2}\), the increase in \({s \mathord{\left/ {\vphantom {s {\gamma D}}} \right. \kern-\nulldelimiterspace} {\gamma D}}\) predicted by the current model is virtually linear and rises faster than that predicted by the traditional model. The current model is more sensitive to \({C \mathord{\left/ {\vphantom {C D}} \right. \kern-\nulldelimiterspace} D}\), probably because the increase in overburden on tetrahedral blocks with increasing \({C \mathord{\left/ {\vphantom {C D}} \right. \kern-\nulldelimiterspace} D}\) results in a higher \({s \mathord{\left/ {\vphantom {s {\gamma D}}} \right. \kern-\nulldelimiterspace} {\gamma D}}\). At \(\mu = {3 \mathord{\left/ {\vphantom {3 4}} \right. \kern-\nulldelimiterspace} 4}\), the variations in \({s \mathord{\left/ {\vphantom {s {\gamma D}}} \right. \kern-\nulldelimiterspace} {\gamma D}}\) predicted by both models are similar. This is consistent with the previous finding that both models have similar performance at a high ratio \(\mu\).

Fig. 13
figure 13

Influence of \({C \mathord{\left/ {\vphantom {C D}} \right. \kern-\nulldelimiterspace} D}\) on \({s \mathord{\left/ {\vphantom {s {\gamma D}}} \right. \kern-\nulldelimiterspace} {\gamma D}}\) (\(\varphi^{\prime} = 30^\circ\))

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5 Case study

Figure 14 shows the profile of the mentioned box tunnel project built in Suzhou, China. This urban underground passageway tunnel is an accessory structure of a metro station and has a cross section with an outside width of 6.9 m and height of 4.2 m. The tunnel is approximately 62.8 m long and was built by using the jacking method. Forty-two pieces of prefabricated linings with lengths of 1.5 m and thickness of 0.45 m are used for installation. The alignments of the tunnels incline to horizontal at 2.7 ‰, and the cover depths are approximately 4.5 m. The tunnel is driven mainly in mix layers of ① miscellaneous fill and ② clay and underlain with ③ and ④. The soil properties are listed in Table 3. The groundwater table is 1 m below the ground surface. A rectangular earth-pressure-balanced boring machine is used for tunneling. The working information of the tunneling machine is presented in Sect. 2.1. The unsupported ratio approximately equals \(\mu = 0.35\).

Fig. 14
figure 14

Profile of the project

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Table 3 Soil properties
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Table 4 Comparisons of the pressure in the chamber (kPa)
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The tunnel drive was suspended for obstruction removal when lining rings No. 28 and 29 were under thrusted. The suspension of excavation lasted for approximately 8 days. During the entire work, obstruction removal was performed under open air, and the working face was not fully unsupported. In addition, no preliminary ground improvement was used because of environmental restrictions. The variation in ground surface settlement can be observed. It is clear that the ground surface settlement developed rapidly for approximately 18 mm during this work, and the rate of ground settlement was less than 3 mm/day. The accumulative settlement and the rate of settlement are both under control, and the ground stability is sustained. Then, the settlement developed more slowly when the work was finished. The subsequent settlement is attributed to soil strength softening due to stress release and tail voids between the boring machine and the linings. Table 3 shows the comparisons of earth pressure in the chamber. In normal drilling, the earth pressure in the working chamber varies from 65 to 78 kPa. During the suspension period, the chamber is open and the support pressure \(s\) provided by the cutting remains in the chamber varies from 58 to 67 kPa. Although unexpected ground surface settlement develops, the pressure \(s\) serves to sufficiently retain the ground stability. The comparison indicates that the proposed model outperforms the traditional model and predicts higher and safer pressures. The pressure predicted by the proposed model is similar to the measured value. This comparison verifies the capability of the current model.

6 Conclusions

This paper performs an analytical investigation on the working face stability of box tunnels and, in particular, focuses on face stability during chamber obstruction removal. In this situation, the face is under open air and no longer fully supported. The face support is explored with respect to the cutting distribution in the working chamber. By modifying the traditional model, an analytical model is proposed. In the current model, the wedge block is divided into multiple blocks respecting the partially supported working face, and the interactions between the blocks are accounted for. Accordingly, the above prism block is also modified in the same way. Based on the proposed model, the solution of the support pressure is derived through limit equilibrium. The influences of factors on the face stability are discussed in parametric analysis. The comparison between the current and traditional models is carried out for verification. The results indicate that the current model outperforms the traditional model, especially when \(\mu \le 1/2\) and \({s \mathord{\left/ {\vphantom {s {\gamma D}}} \right. \kern-\nulldelimiterspace} {\gamma D}}\) predicted by the current model are safer. The discrepancy between the models in the prediction of \({s \mathord{\left/ {\vphantom {s {\gamma D}}} \right. \kern-\nulldelimiterspace} {\gamma D}}\) decreases with increasing \(\mu\). Finally, the current model is validated to investigate the working face stability of a box tunnel in Suzhou, China. The current and traditional models are applied for the investigation. The results predicted by the current model agree with the in situ measurements, indicating that the current model outperforms the traditional model in this case. This research also recommends the proper distance of double conveyors with respect to \(\mu \le 1/2\) in machine design in cases of chamber obstruction removal under open air.