Introduction

The earth pressure at the end of the shield is an important parameter in the design of large diameter shallow buried shield, and it is also a hot topic in the research of shallow buried shield. Such research results often assume the potential sliding surface of soil at the end of the shield, and carry out force analysis on the soil at the sliding surface. Based on the limit equilibrium theory and the virtual work principle, the calculation formula of earth pressure at the end of the shield under the assumption of rigid body and deformed body can be obtained respectively. At present, this kind of earth pressure calculation formula is mainly for the linear shield, that is, the end surface of the shield is perpendicular to the horizontal plane. However, in practical shield engineering, especially large diameter and long distance shield engineering, the deflection Angle of the tunnel is often changed. The Angle between the end of the shield and the horizontal plane near the starting well is usually less than 90°. The Angle between the end of the shield and the horizontal plane near the receiving well is usually greater than 90°. The middle section shield is in the horizontal state, and the Angle between the end surface and the horizontal surface is equal to 90°. The Angle between the end surface and the horizontal surface affects the position of the potential sliding surface of the unstable soil near the end and the distribution characteristics of the earth pressure at the end.

At present, the models for the earth pressure shield formula at the end of the earth are mainly based on the wedge slider model of limit equilibrium analysis model. For example, Horn1 proposed the 3D wedge-silo failure model for the first time. Based on the three-dimensional wedge model, Monnet et al.2 analyzed the influencing factors and variation rules of shield excavation face stability. Chen3 proposed a three-dimensional wedge-prism model considering prism height and earth arch effect. Fu Helin et al.4 proposed the"inverted trapezoid-prism"model of the square soil mass in front of curved shield tunnel excavation. Wang Guofu et al.5 put forward the limit equilibrium model of curved gradient wedge prism excavation face and derived the theoretical calculation formula of limit support force on curved shield tunnel excavation face. Huang Qi et al.6 established an improved three-dimensional CVF model for longitudinal inclined tunnel driven by pressurized shield, and obtained the threshold support force under excavation. The research on calculating the earth pressure at the end of the shield tunnel with the limit equilibrium analysis model of the three-dimensional curved sliding block model is mainly focused on the horizontal tunnel. Liu Keqi7 proposed a three-dimensional sliding fracture model of the palm surface of shield construction. Based on the active failure model of soil mass at the end of a horizontal shield tunnel. Zhang Mengxi8 established a three-dimensional limit equilibrium analysis model of a horizontal spherical fracture slider. Liu Yingnan9 established a three-dimensional failure model of logarithmic spiral and elliptic cylinder. Li Chunlin10 proposed a soil sliding model of a combined curved body with a semi-circular platform and a local sphere, and obtained the analytical formula of earth pressure at the end of a linear shield.

Based on the above analysis, at present, there are mainly plane sliding block model and surface failure model for the earth pressure calculation model of shield end. The calculation formula based on wedge planar sliding block mode is relatively simple, but it is different from the practical sliding block mode. The surface failure mode is closer to the engineering practice, and the theoretical calculation results are more accurate, but the processing of sliding surface load is relatively complicated, and the calculation formula is also relatively complicated. And the current research is mainly aimed at horizontal shield tunnel. In order to study the earth pressure at the end of the shield under different driving angles, the ultimate equilibrium analysis model of the earth pressure at the end of the shield was established based on the surface failure mode. The unified solution of the earth pressure at the end of the shield under the conditions of upward, horizontal and downward driving was obtained, and the influence of the driving Angle on the earth pressure at the end of the shield was analyzed. It provides a design idea for soil structure analysis at the end of shallow buried shield and theoretical support for soil pressure calculation at the end.

Construction of 3D surface slider model

In order to obtain the unified solution of the earth pressure at the end under different driving angles, the calculation models of the potential three-dimensional curved sliding soil near the end were established respectively. In order to simplify the calculation, the potential sliding soil model with the lower part of the sphere and the part of the semi-circle is used in this study. The geometric analysis models of sliding soil under the conditions of upward (β > 0), horizontal (β = 0) and downward (β < 0) tunneling are shown in Figs. 1a, b, 2a, b and 3a, b, respectively. The distribution dimensions of the model section under the three driving conditions are shown in Fig. 4. The geometric model of potential sliding soil at the end of the tunnel is divided into three parts: the tunnel bottom to the waist depth is a local sphere; The depth range of the two waists and the vault of the tunnel is a small semi-circular platform. The depth of the tunnel from the vault to the ground is mostly circular. When analyzing the ultimate equilibrium state of the soil mass at the end, the soil mass begins to break from the square part of the sphere surface in front of the excavation, gradually develops to the middle small circle and the upper large circle, and finally extends to the surface. The position A and B of the two waists on the end surface of the shield extend to the surface at a θ angle, and \(\uptheta =45^\circ +\frac{\varphi }{2}.\)

Fig. 1
figure 1

Geometric model for stability analysis of sliding soil mass at the end of the up-driving shield (β > 0).

Full size image
Fig. 2
figure 2

Geometric model for stability analysis of sliding soil mass at the end of horizontal shield (β = 0).

Full size image
Fig. 3
figure 3

Geometric model for stability analysis of sliding soil mass at downward shield end (β < 0).

Full size image
Fig. 4
figure 4

Geometric calculation model profile.

Full size image

Rationality analysis of model construction

ABAQUS finite element analysis software was used to numerically simulate the excavation process at different excavation angles under shallow burial conditions11,12. To verify the rationality of the 3D surface slider model. The numerical model is shown in Figure 5.

Fig. 5
figure 5

Numerical model of horizontal driving 1/2 (Software: ABAQUS 2020,  https://www.3ds.com/products-services/simulia/products/abaqus/).

Full size image

Model parameters: both length and height are 100 m, width 80 m, shield radius R=8 m, shield depth 20 m, elastic modulus 300 MPa, Poisson’s ratio 0.24,γ=18 KN/m2, The driving angles of the shield are respectivelyβ=0°、6°、−6°.

When the depth to diameter ratio of the face of the palm is 1, the influence of the active damage area of the face of the three driving angles is shown in Figs. 6, 7, 8. Figure 6, 7, 8(a) shows: when c = 10 kPa, \(\varphi\)=15°、20°、25°、30°, β = 0°、6°、−6°, The damage range of the soil cross section of the palm surface is distributed, and the damage range decreases gradually with the increase of the internal friction Angle, and the overall change range decreases gradually. This is basically consistent with the rule found in literature13. Figure 6(b) shows that when the excavation Angle β = 0°, the influence range of soil above the normal section is about 18 grids; Fig. 7(b) shows that when the excavation Angle β = 6°, the influence range of soil above the normal section is about 21 grids. Figure 8(b) shows that when the excavation Angle β = −6°, the influence range of soil above the normal section is about 16 grids. Under the three excavation angles, the difference between the soil above the normal section is between 10 and 30%, and the difference is not large. According to the comprehensive analysis in Figs. 6, 7, 8, under different driving angles, the soil influence area above the palm surface presents the damage form of a round table plus some spheres. When the depth to diameter ratio is 1, the change of driving Angle has little influence on the soil above the palm surface.

Fig. 6
figure 6

Active failure mode diagram of horizontal driving (Software: ABAQUS 2020,  https://www.3ds.com/products-services/simulia/products/abaqus/).

Full size image
Fig. 7
figure 7

Active failure mode diagram of upward driving (Software: ABAQUS 2020,  https://www.3ds.com/products-services/simulia/products/abaqus/).

Full size image
Fig. 8
figure 8

Active failure mode diagram of downward driving (Software: ABAQUS 2020,  https://www.3ds.com/products-services/simulia/products/abaqus/).

Full size image

The above numerical model verifies the rationality of the geometric model and the unified solution of shallow buried shield with different driving angles based on the limit equilibrium theory from the perspective of continuum mechanics.

The calculation formula of earth pressure at the end of the limit state is derived

Basic assumption

When the limit equilibrium method is used to deduce the earth pressure load at the end of the shield, the following assumptions are made:

  • 1) Ignore the influence of the friction force on the palm surface caused by the rotation of the cutter head and the friction force between the shield shell and the soil.

  • 2) The sliding soil in the model is a rigid body.

  • 3) The soil is uniform, isotropic soil sliding in the body causes damage to the soil, and the soil on the sliding surface conforms to the Mohr-Coulomb criterion.

  • 4)The shear strength formula of each unit on each sliding surface in the soil body is \(\uptau =c+\sigma tan\varphi\), c is the cohesion force (kPa)、\(\upsigma\) is the normal stress (kPa), and φ is the internal friction Angle (°).

  • 5)The failure mode of the soil above the horizontal plane where the two sides of the shield tunnel are located adopts passive failure mode, and the Angle between the fracture plane and the horizontal plane always keeps θ. θ is not affected by the driving Angle.

  • 6)The failure mode of sliding soil below the horizontal plane where the two waists of shield tunnel are located is controlled by β, and α changes with the change of β.

Calculation of earth pressure load at the end of excavation face

First, a semicircle is obtained by cutting the surface of the sliding area below the arch. The relation between the friction force f and the normal force N on the boundary of the section of the semicircle is obtained. As shown in Fig. 9, assuming that the soil pressure on its side is \({\text{p}}_{\text{a}}\), in order to obtain its resultant force in the X-axis direction, the area of its march is divided into:

$${\oint }_{L}{p}_{a}\mathit{cos}tds={\int }_{-\frac{\pi }{2}}^{\frac{\pi }{2}}{p}_{a}\mathit{cos}t\cdot Rdt=2R{p}_{a}$$
(1)

where, t is the integral Angle (°), R is the diameter of the shield excavation face, and other parameters are the same as above.

Fig. 9
figure 9

Schematic diagram of friction force on the sliding surface.

Full size image

If the resultant force of \({\text{p}}_{\text{a}}\), in the X-axis direction is N’, then: \({p}_{a}=\frac{{N}{\prime}}{2R}\), Further, the friction force on the boundary of the intercepted semicircular plane is obtained \({f}{\prime}\).\({f}{\prime}=\frac{{N}{\prime}}{2R}\pi R\mathit{tan}\varphi =\frac{{\pi N}{\prime}}{2}\mathit{tan}\varphi\). It can be seen that the relationship between f and N in the sliding region of the whole surface body is as follows:

$$f=\frac{N}{2R}\pi R\mathit{tan}\varphi =\frac{\pi N}{2}\mathit{tan}\varphi$$
(2)

where N is the normal force perpendicular to the surface of the slip surface (kN) and f is the friction force parallel to the surface of the slip surface (kN).

Suppose the tangential force on the surface of the sliding soil cracking surface is τ, then:

$$\tau =C+f=C+N\mathit{tan}\varphi$$
(3)

where, the cohesion force C is the product of the cohesion force c and the area of the sliding soil surface (kN).

The shield tunnel was driven upward (β>0)

The mechanical analysis of sliding soil beneath the tunnel vault is shown in Fig 10. According to the equilibrium column of sliding soil under the vault, the equilibrium equation is written as follows:

Fig. 10
figure 10

Schematic diagram of soil force analysis under tunnel vault (β > 0).

Full size image

Horizontal force balance:

$$P\mathit{cos}\beta ={N}_{1}\mathit{sin}(\frac{\pi }{4}+\frac{\beta }{2})-{T}_{1}\mathit{cos}(\frac{\pi }{4}+\frac{\beta }{2})+{N}_{2}\mathit{sin}\theta -{T}_{2}\mathit{cos}\theta$$
(4)

Vertical force balance:

$${P}_{v}+G={N}_{1}\mathit{cos}(\frac{\pi }{4}+\frac{\beta }{2})+{T}_{1}\mathit{sin}(\frac{\pi }{4}+\frac{\beta }{2})+{N}_{2}\mathit{cos}\theta +{T}_{2}\mathit{sin}\theta +P\mathit{sin}\beta$$
(5)
$${N}_{1}={N}_{2}\mathit{cos}(\frac{\varphi -\beta }{2})$$
(6)
$${T}_{2}={C}_{2}+\frac{\pi {N}_{2}}{2}\mathit{tan}\varphi$$
(7)
$${T}_{1}={T}_{2}\mathit{cos}(\frac{\varphi -\beta }{2})=({C}_{2}+\frac{\pi {N}_{2}}{2}\mathit{tan}\varphi )\mathit{cos}(\frac{\varphi -\beta }{2})$$
(8)

Eqs. (4), (5), (6), (7) and (8) are combined to solve the minimum support force to maintain the stability of the excavation face:

$$ \begin{aligned} P = & \left\{ { - 2\left[ {3 + {\text{Cos}}\left( {\beta - \varphi } \right) + 2\sin \frac{{\pi + 4\theta - 2\varphi }}{4}} \right.} \right. \\ & \left. { + 2\sin \frac{{\pi - 4\beta + 4\theta + 2\varphi }}{4}C_{2} + 2\left( {G + P_{v} } \right)} \right]\left[ {2\sin \theta - \pi \cos \theta \tan \varphi } \right. \\ & {{\left. {\left. { + \cos \frac{{\varphi - \beta }}{2}\left( {2\sin \frac{{\pi + 2\beta }}{4} - \pi \cos \frac{{\pi + 2\beta }}{4}} \right)\tan \varphi } \right]} \right\}} \mathord{\left/ {\vphantom {{\left. {\left. { + \cos \frac{{\varphi - \beta }}{2}\left( {2\sin \frac{{\pi + 2\beta }}{4} - \pi \cos \frac{{\pi + 2\beta }}{4}} \right)\tan \varphi } \right]} \right\}} {\left\{ {4\cos \left( {\beta - \theta } \right)} \right.}}} \right. \kern-\nulldelimiterspace} {\left\{ {4\cos \left( {\beta - \theta } \right)} \right.}} \\ & \left. { + 4\cos \frac{{\beta - \varphi }}{2}\sin \frac{{\pi + 2\beta }}{4} + \pi \left[ {\cos \frac{{\pi + 2\varphi }}{4} - 2\sin \left( {\beta - \theta } \right) + \sin \frac{{\pi - 4\beta + 2\varphi }}{4}} \right]\tan \varphi } \right\} \\ \end{aligned} $$
(9)

where, G is the dead weight of square sliding soil before excavation (kN), \(\text{G}=({\text{V}}_{1}+{\text{V}}_{2}+{\text{V}}_{3})\upgamma\), \(\upgamma\) is the soil weight \((\text{kN}/{\text{m}}^{3})\). P is the top thrust of shield machine head (kN). Pv is the force exerted by the overlying soil on the arch roof (kN). N2 is the supporting force of soil on the side of the semi-circular platform (kN). N1 is the supporting force of the soil around the local sphere acting on the side of the local sphere (kN); T1 is the friction force of the surrounding soil on the surface of the local sphere (kN). T2 is the friction force of the surrounding soil (kN). The remaining parameters are the same as above.

The expression of each variable is as follows:

The area S1 and volume V1 of some spheres are as follows:

$${S}_{1}=\frac{90^\circ -\beta }{360^\circ }\times 4\pi {R}^{2}=\frac{90^\circ -\beta }{90^\circ }\pi {R}^{2}$$
(10)
$${V}_{1}=\frac{90^\circ -\beta }{360^\circ }\times \frac{4}{3}\pi {R}^{3}=\frac{90^\circ -\beta }{270^\circ }\pi {R}^{3}$$
(11)

The side area S2 and volume V2 of the small semi-circular table are as follows:

$${S}_{2}=\frac{1}{2}\pi {R}^{2}\left(\text{cot}\theta +2\right)\sqrt{{\text{cot}}^{2}\theta +{\text{cos}}^{2}\beta }$$
(12)
$${V}_{2}=\frac{1}{6}\pi {R}^{3}\text{cos}\beta \left({\text{cot}}^{2}\theta +3\text{cot}\theta +3\right)$$
(13)

The side areas of the wedge S3 and V3 volumes are as follows:

$${S}_{3}=\frac{1}{2}{R}^{2}{\mathit{sin}}^{2}\beta \text{cot}\frac{\beta }{2}$$
(14)
$${V}_{3}=\frac{1}{3}{R}^{3}\mathit{sin}\beta \mathit{cos}\beta \left(2\mathit{cot}\theta +3\right)$$
(15)

As shown in Fig 11, the overlying soil force analysis of shield tunnel arch is composed of a mostly circular platform and a quadrilateral platform.

Fig. 11
figure 11

Overlying soil model and force analysis (β > 0).

Full size image

The soil model on top of the arch is shown in the figure. In order to simplify the vertical stress expression of the soil, Fig 11 is divided into two soils (a) and (b) for differential calculation. As shown in Fig. 11, stress analysis was carried out on most of the round table and four edge table above the arch. h is the buried depth (m) of the differential element body. H is the height (m) from the arch top to the ground. dw1 is the body weight of the differential element body of the semi-circular platform. dw2 is the body weight of the differential element body of the four-edge platform.Other parameters are the same as above.

According to the vertical force of the microelement body of the most circular table, the balance equation is set up and simplified:

$$\frac{d{\sigma }_{v1}}{dh}+\frac{2K\mathit{tan}\varphi (2+\pi \mathit{sin}\theta )}{\pi {B}_{1}{\prime}}{\sigma }_{v1}=\gamma -\frac{2c(2+\pi )}{\pi {B}_{1}{\prime}}$$
(16)

where: \({\upsigma }_{\text{v}}\) is the overlying soil pressure (kPa). Kis the lateral earth pressure coefficient. \({B}_{1}{\prime}\) is the width of sliding soil column at the calculation unit. The remaining parameters are the same as above.

$${B}_{1}{\prime}={B}_{1}+(H-h)\mathit{cot}\theta$$
(17)

Substituting, there is:

$$\frac{d{\sigma }_{v1}}{dh}+\frac{2K\mathit{tan}\varphi (2+\pi \mathit{sin}\theta )}{\pi \left[{B}_{1}+(H-h)\mathit{cot}\theta \right]}{\sigma }_{v1}=\gamma -\frac{2c\left(2+\pi \right)}{\pi \left[{B}_{1}+(H-h)\mathit{cot}\theta \right]}$$
(18)

Combined with the boundary conditions, when \(\text{h}=0\), \({\upsigma }_{\text{v}}=0\), the particular solution of the differential equation is obtained, and h = H is substituted into the particular solution:

$$\begin{aligned} \sigma _{{v1}} = & \cot \varphi \left( {H\cos \theta + B_{1} \sin \theta } \right)^{{ - \frac{{2K\left( {2 + \pi \sin \theta } \right)\tan \theta \tan \varphi }}{\pi }}} \left\{ {\left( {H\cos \theta + B_{1} \sin \theta } \right)^{{\frac{{2K\left( {2 + \pi \sin \theta } \right)\tan \theta \tan \varphi }}{\pi }}} \left[ {c\pi \left( {2 + \pi } \right)\cot \varphi } \right.} \right. \\ & \left. { + K\left( { - 4c - 2c\pi + \pi B_{1} \gamma } \right)\left( {2 + \pi \sin \theta } \right)\tan \theta } \right] - \left( {B_{1} \sin \theta } \right)^{{\frac{{2K\left( {2 + \pi \sin \theta } \right)\tan \theta \tan \varphi }}{\pi }}} \left[ {c\pi \left( {2 + \pi } \right)\cot \varphi } \right. \\ & \left. {\left. { + K\left( {2 + \pi \sin \theta } \right)(H\pi \gamma - 4c\tan \theta - 2c\pi \tan \theta + + \pi \gamma B_{1} \tan \theta )} \right]} \right\}/\left\{ {K\left( {2 + \pi \sin \theta } \right)\left[ { - \pi \cot \varphi } \right.} \right. \\ & \left. { \left. { + 2K\left( {2 + \pi \sin \theta } \right)\tan \theta } \right]} \right\} \\ \end{aligned} $$
(19)

According to the vertical force of the four-prismatic micro-element body, the equilibrium equation is set up and simplified:

$$\frac{d{\sigma }_{v2}}{dh}+\frac{K\mathit{sin}\theta \mathit{tan}\varphi }{{B}_{1}+(H-h)\mathit{cot}\theta }{\sigma }_{v2}=\gamma -\frac{c}{{B}_{1}+(H-h)\mathit{cot}\theta }$$
(20)

Combined with the boundary conditions, when \(\text{h}=0\), \({\upsigma }_{\text{v}}=0\), the particular solution of the differential equation is obtained, and h = H is substituted into the particular solution:

$$\begin{aligned} \sigma _{{v2}} = & \;\frac{1}{{K\left( {K - {\text{cot}}\theta {\text{cot}}\varphi {\text{csc}}\theta } \right)}}{\text{cot}}\theta {\text{cot}}\varphi {\text{csc}}\theta \left( {H{\text{cos}}\theta + B_{1} {\text{sin}}\theta } \right)^{{ - K{\text{sin}}\theta {\text{tan}}\theta {\text{tan}}\varphi }} \\ & \left\{ { - \left( {B_{1} {\text{sin}}\theta } \right)^{{K{\text{sin}}\theta {\text{tan}}\theta {\text{tan}}\varphi }} \left[ {HK\gamma + c{\text{cot}}\varphi {\text{csc}}\theta + K\left( { - c + B_{1} \gamma } \right){\text{tan}}\theta } \right]} \right. \\ & \left. { + \left( {H{\text{cos}}\theta + B_{1} {\text{sin}}\theta } \right)^{{K{\text{sin}}\theta {\text{tan}}\theta {\text{tan}}\varphi }} \left[ {c{\text{cot}}\varphi {\text{csc}}\theta + K\left( { - c + B_{1} \gamma } \right){\text{tan}}\theta } \right]} \right\} \\ \end{aligned}$$
(21)

The \({\upsigma }_{\text{v}}\) obtained from Eq. (19) is the stress on the upper surface of the large semi-circular table and is denoted by the symbol \({\upsigma }_{\text{v}1}\), while the \({\upsigma }_{\text{v}}\) obtained from Eq. (21) is the stress on the upper surface of the four-prism and is denoted by the symbol \({\upsigma }_{\text{v}2}\).

The vertical stress of the soil overlying the arch roof is the weighted average of the stress of the semi-circular \({\upsigma }_{\text{v}1}\) and the quad-prismatic \({\upsigma }_{\text{v}2}\):

(22)

At this time, the overlying soil pressure can be obtained when driving upward \({\text{P}}_{\text{v}}\)

(23)

By substituting Eq. (23) to Eq. (9), the average earth pressure load at the tunnel end under the condition of β>0 can be obtained P.

Horizontal tunneling of shield tunnel (β=0)

The mechanical analysis diagram of sliding soil under the tunnel vault is shown in Fig 12.

Fig. 12
figure 12

Schematic diagram of soil force analysis under tunnel vault (β = 0).

Full size image

Combining the combined surface body in the horizontal and vertical direction of the force balance can be obtained:

Horizontal force balance:

$$P={N}_{1}\mathit{sin}\frac{\pi }{4}-{T}_{1}\mathit{cos}\frac{\pi }{4}+{N}_{2}\mathit{sin}\theta -{T}_{2}c\mathit{os}\theta$$
(24)

Vertical force balance:

$${P}_{v}+G={N}_{1}\mathit{cos}\frac{\pi }{4}+{T}_{1}\mathit{sin}\frac{\pi }{4}+{N}_{2}\mathit{cos}\theta +{T}_{2}\mathit{sin}\theta$$
(25)
$${N}_{1}={N}_{2}\mathit{cos}\frac{\varphi }{2}$$
(26)
$${T}_{1}={T}_{2}\mathit{cos}\frac{\varphi }{2}=({C}_{2}+\frac{\pi {N}_{2}}{2}\mathit{tan}\varphi )\mathit{cos}\frac{\varphi }{2}$$
(27)

The expression \({\text{T}}_{2}\) is the same as Eq. (8). Similarly, Eqs. (7), (8), (24), (25) and (26) are combined to solve the excavation face support force.

$$ \begin{aligned} P = & \left\{ { - \left( {G + P_{v} } \right)\left[ { - 4sin\theta + 2\pi cos\theta {\text{tan}}\varphi + \sqrt 2 cos\frac{\varphi }{2}\left( { - 2 + \pi {\text{tan}}\varphi } \right)} \right]} \right. \\ & {{\left. { + 2C_{2} \left[ { - 3 - {\text{cos}}\varphi - 2\sqrt 2 cos\frac{\varphi }{2}\left( {cos\theta + sin\theta } \right)} \right]} \right\}} \mathord{\left/ {\vphantom {{\left. { + 2C_{2} \left[ { - 3 - {\text{cos}}\varphi - 2\sqrt 2 cos\frac{\varphi }{2}\left( {cos\theta + sin\theta } \right)} \right]} \right\}} {\left[ {4cos\theta + 2\pi sin\theta {\text{tan}}\varphi } \right.}}} \right. \kern-\nulldelimiterspace} {\left[ {4cos\theta + 2\pi sin\theta {\text{tan}}\varphi } \right.}} \\ & \left. { + \sqrt 2 cos\frac{\varphi }{2}\left( {2 + \pi {\text{tan}}\varphi } \right)} \right] \\ \end{aligned} $$
(28)

where \(\text{G}=({\text{V}}_{1}+{\text{V}}_{2})\upgamma\), \({{\text{C}}_{2}={\text{S}}_{2}\text{c }}\), \({\text{V}}_{1}\), \({\text{V}}_{2}\), \({\text{S}}_{1}\), \({\text{S}}_{2}\) are the volume and area of partial sphere and minor semicircle, respectively. Its expression is related to β. The expressions of S2 and V2 are the same as Eq. (12) and Eq. (13) respectively. C2 is the cohesive force (kN) on the corresponding plane.

The area and volume of some spheres:

$${S}_{1}=\frac{1}{4}\times 4\pi {R}^{2}=\pi {R}^{2}$$
(29)
$${V}_{1}=\frac{1}{4}\times \frac{4}{3}\pi {R}^{3}=\frac{1}{3}\pi {R}^{3}$$
(30)

The force analysis of soil overlying the shield tunnel is shown in Fig 11(a).

According to the stress balance of most of the micro-units on the circular table covering the soil above the vault, the balance equation is written and simplified as follows:

$$\frac{d{\sigma }_{v3}}{dh}+\frac{2K \mathit{tan}\varphi (2+\pi \mathit{sin}\theta )}{\pi \left[{B}_{1}+(H-h)\mathit{cot}\theta \right]}{\sigma }_{v3}=\gamma -\frac{2c\left(2+\pi \right)}{\pi \left[{B}_{1}+(H-h)\mathit{cot}\theta \right]}$$
(31)

Combined with the boundary conditions, when \(\text{h}=0\), \({\upsigma }_{\text{v}}=0\), the particular solution of the differential equation is obtained, and h = H is substituted into the particular solution:

$$ \begin{aligned} \sigma _{{v3}} = & {\text{cot}}\varphi \left( {H{\text{cos}}\theta + B_{1} {\text{sin}}\theta } \right)^{{ - \frac{{2K\left( {2 + \pi {\text{sin}}\theta } \right){\text{tan}}\theta {\text{tan}}\varphi }}{\pi }}} \left( {H{\text{cos}}\theta + B_{1} {\text{sin}}\theta } \right)^{{\frac{{2K\left( {2 + \pi {\text{sin}}\theta } \right){\text{tan}}\theta {\text{tan}}\varphi }}{\pi }}} \\ & \left[ {c\pi \left( {2 + \pi } \right){\text{cot}}\varphi } \right.\left. { + K\left( {2 + \pi {\text{sin}}\theta } \right)} \right]\left[ { - 2c\left( {2 + \pi } \right) + \pi \gamma B_{1} } \right]{\text{tan}}\theta - \left( {B_{1} {\text{sin}}\theta } \right)^{{\frac{{2K\left( {2 + \pi {\text{sin}}\theta } \right){\text{tan}}\theta {\text{tan}}\varphi }}{\pi }}} \\ & \left[ {c\pi \left( {2 + \pi } \right){\text{cot}}\varphi K\left( {2 + \pi {\text{sin}}\theta } \right)} \right]\left\{ {H\pi \gamma + \left[ { - 2c\left( {2 + \pi } \right) + \pi \gamma B_{1} } \right]{\text{tan}}\theta } \right\}/K\left( {2 + \pi {\text{sin}}\theta } \right) \\ & \left[ { - \pi {\text{cot}}\varphi + 2K\left( {2 + \pi {\text{sin}}\theta } \right){\text{tan}}\theta } \right] \\ \end{aligned} $$
(32)

In this case, the pressure load \({\text{P}}_{\text{v}}\) (kN) of the overlying soil under horizontal driving of the shield is:

$${P}_{v}={\sigma }_{v3}\times \frac{\pi {{B}_{1}}^{2}}{2}={\sigma }_{v3}\times \frac{\pi {\left[R\left(1+\text{cot}\theta \right)\right]}^{2}}{2}$$
(33)

By substituting Eq. (33) to Eq. (28), the average earth pressure at the end of the excavation face under the horizontal state of the shield driving direction can be obtained P.

The shield tunnel went down (\({\varvec{\beta}}<0\))

The mechanical analysis of sliding soil under the tunnel vault is shown in Fig. 13.

Fig. 13
figure 13

Schematic diagram of soil force analysis under the tunnel vault (β < 0).

Full size image

According to the soil force balance under the vault, the balance equation is written as follows:

Horizontal force balance:

$$P\mathit{cos}\beta ={N}_{1}\mathit{sin}(\frac{\pi }{4}-\frac{\beta }{2})-{T}_{1}\mathit{cos}(\frac{\pi }{4}-\frac{\beta }{2})+{N}_{2}\mathit{sin}\theta -{T}_{2}\mathit{cos}\theta$$
(34)

Vertical force balance:

$${P}_{v}+G+P\mathit{sin}\beta ={N}_{1}\mathit{cos}(\frac{\pi }{4}-\frac{\beta }{2})+{T}_{1}\mathit{sin}(\frac{\pi }{4}-\frac{\beta }{2})+{N}_{2}\mathit{cos}\theta +{T}_{2}\mathit{sin}\theta$$
(35)
$${N}_{1}={N}_{2}\mathit{cos}(\frac{\varphi +\beta }{2})$$
(36)
$${T}_{1}={T}_{2}\mathit{cos}(\frac{\varphi +\beta }{2})=({C}_{2}+\frac{\pi {N}_{2}}{2}\mathit{tan}\varphi )\mathit{cos}(\frac{\varphi +\beta }{2})$$
(37)

The \({\text{T}}_{2}\) expression is the same as Eq. (8). Similarly, Eqs. (7), (8), (31), (32) and (33) are combined to solve the excavation face support force.

$$\begin{aligned} P = & - \left[ {C_{1} sin\left( {\frac{{\pi + 2\beta }}{4}} \right) + C_{2} cos\theta } \right]\left\{ {2cos\theta + \pi sin\theta tan\varphi + cos(\frac{{\varphi + \beta }}{2})\left[ {2sin\left( {\frac{{\pi + 2\beta }}{4}} \right) + \pi {\text{cos}}\left( {\frac{{\pi + 2\beta }}{4}} \right)tan\varphi } \right]} \right\} \\ & - \left[ {G - C_{1} {\text{cos}}\left( {\frac{{\pi + 2\beta }}{4}} \right) - C_{2} sin\theta + P_{v} } \right]\left\{ {2sin\theta - \pi cos\theta tan\varphi + cos\left( {\frac{{\varphi + \beta }}{2}} \right)} \right.\left[ {2{\text{cos}}\left( {\frac{{\pi + 2\beta }}{4}} \right)} \right. \\ & {{\left. {\left. { - \pi sin\left( {\frac{{\pi + 2\beta }}{4}} \right)tan\varphi } \right]} \right\}} \mathord{\left/ {\vphantom {{\left. {\left. { - \pi sin\left( {\frac{{\pi + 2\beta }}{4}} \right)tan\varphi } \right]} \right\}} {\left\{ {2{\text{cos}}\left( {\theta + \beta } \right) + {\text{cos}}\frac{{\pi + 4\beta + 2\varphi }}{4} + {\text{sin}}\frac{{\pi + 2\varphi }}{4}} \right.}}} \right. \kern-\nulldelimiterspace} {\left\{ {2{\text{cos}}\left( {\theta + \beta } \right) + {\text{cos}}\frac{{\pi + 4\beta + 2\varphi }}{4} + {\text{sin}}\frac{{\pi + 2\varphi }}{4}} \right.}} \\ & \left. { + \frac{1}{2}\pi \left[ {{\text{cos}}\frac{{\pi + 2\varphi }}{4} + 2{\text{sin}}\left( {\theta + \beta } \right) + {\text{sin}}\frac{{\pi + 4\beta + 2\varphi }}{4}} \right]tan\varphi } \right\} \\ \end{aligned}$$
(38)

where \(\text{G}=({\text{V}}_{1}+{\text{V}}_{2}-{\text{V}}_{3})\upgamma\), \({\text{C}}_{2}=\left({\text{S}}_{2}-{\text{S}}_{3}\right)\text{c}\), \({\text{V}}_{1}\), \({\text{V}}_{2}\), \({\text{V}}_{3}\), \({\text{S}}_{1}\), \({\text{S}}_{2}\), \({\text{S}}_{3}\) Are the volume and area of a partial sphere, a small semicircle and a wedge, and their expressions are related to β.\(\text{The expressions of }{\text{S}}_{2}\), \({\text{S}}_{3}\), \({\text{V}}_{2}\), \({\text{V}}_{3}\) are the same as Eqs. (12), (14), (13), (15), \({\text{C}}_{2}\) Is the cohesive force on the corresponding plane \((\text{kN})\).

The area and volume of some spheres:

$${S}_{1}=\frac{90^\circ +\beta }{360^\circ }\times 4\pi {R}^{2}=\frac{90^\circ +\beta }{90^\circ }\pi {R}^{2}$$
(39)
$${V}_{1}=\frac{90^\circ +\beta }{360^\circ }\times \frac{4}{3}\pi {R}^{3}=\frac{90^\circ +\beta }{90^\circ }\pi {R}^{3}$$
(40)

The mechanical analysis diagram of the whole soil covering the tunnel arch and the soil on the four-edge platform are shown in Fig. 11(a) and (b) respectively.

dw1 is the weight of the differential element body of the overlying soil of the arch roof. dw2 is the weight of the differential element body body of the quadrangular platform.

The equilibrium equation written for the whole soil mass above the vault and the vertical direction of the four-edge platform is the same as that for the upward driving (β > 0), and then \({\upsigma }_{\text{v}4}\) and \({\upsigma }_{\text{v}5}\) are obtained. Where \({\upsigma }_{\text{v}4}\) and \({\upsigma }_{\text{v}5}\) are the same as Eq. (19) and Eq. (21). The vertical stress of the soil overlying the arch is the weighted average of the vertical stress \({\upsigma }_{\text{v}4}\) of the semi-circular platform and \({\upsigma }_{\text{v}5}\) of the quadrilateral platform:

(41)

The overlying soil pressure \({P}_{v}\) under the condition β < 0 can be obtained

(42)

By substituting Eq. (42) into Eq. (38), the end earth pressure load P under β<0 can be obtained.

Comparative analysis of formulas under different driving conditions: Comparative analysis of the calculation formulas of earth pressure load at the end of the shield under three different driving directions. When β=0 is applied to the calculation formulas of driving down (β<0) and driving up (β<0), the end earth pressure load P obtained is consistent with the calculation formulas of driving down (β<0) and driving up (β>0), indicating that the calculation formula of horizontal driving (β=0) is a special case of driving down (β<0) and driving up (β>0). When β is replaced by"-β"in the calculation formula of downward driving (β<0), the calculation formula obtained is the same as that of upward driving (β>0). It shows that the solution of earth pressure load at the end of shield tunnel under different driving directions is uniform.

Model validation and parameter sensitivity analysis

Model verification

In order to verify the rationality of the model in this paper, the calculation results of the formula are compared with the experimental results and the calculation results of other existing theories. In order to obtain the general rule under different soil weight and tunnel diameter, the parameters were treated without dimension. The ordinate was Pa/γD, and the ordinate was H/D. For the convenience of comparison, the internal friction Angle of 37° and the cohesion of 0 kPa and 0.5 kPa were selected as the relevant calculation parameters in this paper, and the results are shown in Fig. 14.

Fig. 14
figure 14

Comparison of calculation results of earth pressure at the end.

Full size image

Figure 14 shows that when the comparative cohesion is 0, Angnostou model and Liu K-Q model are higher than the calculation result of this model. This model is not much different from Mollon model and Chen model, but obviously higher than Leca model. Anagnostou model uses limit balance method to divide the limit support force of palm surface into water pressure and effective pressure, and builds a prismatic limit balance model14. The calculation result is closely related to the water head difference between ground and chamber. However, this paper establishes a 3D surface slider limit balance model without considering the influence of water pressure, and the calculation result is less than that of Anagnostou model. When H/D<1.5, this model, Chen model and Liu K-Q model all increase gradually, but the increase amplitude is different. When H/D>1.5, all models show that with the increase of depth to diameter ratio, the limiting support force of palm surface changes little. Leca model adopts the upper limit theorem of limit analysis, so the calculated limit support force of the palm surface is small and the safety factor is not high15. When the cohesion is 0 and 0.5, the change curve of the model is affected by the increase of cohesion, but the effect is not large.

To sum up, the calculated results of the model in this paper are consistent with those obtained by experiments and existing theories, which indicates that the calculated results of the model in this paper are reliable.

Parameter sensitivity analysis

Influence of soil internal friction Angle

Take tunnel diameter D=16 m, γ=18kN/m3, shield buried depth H=20 m, cohesion c=10kPa. Internal friction Angle φ=15°、20°、25°、30°、35°. The change curves of earth pressure at the lower end of different internal friction angles with different shield tunneling angles were calculated respectively, as shown in Fig. 15.

Fig. 15
figure 15

Earth pressure at the end varies with the Angle of internal friction.

Full size image

Figure 15 shows that when the depth to diameter ratio and cohesion are constant, the earth pressure at the end decreases with the increase of the internal friction Angle at different driving angles, and the change curve presents nonlinear characteristics. The rate of change decreases with the increase of internal friction Angle. The curvature of the curve of soil pressure with internal friction Angle is similar for different driving angles.

This observation aligns with Anagnostou’s findings in the stability analysis of excavation faces using earth pressure balance shields, where increased internal friction angle (φ) reduces the required support pressure while exhibiting nonlinear characteristics in its variation trend. The results also correspond with the pattern demonstrated in Leca’s model, where the ultimate support pressure decreases progressively with the augmentation of φ.

When φ>35° is drilled horizontally and downward, the soil pressure at the end tends to be 0 or negative, and the soil at the end is in a natural stable state. Upward driving will not result in earth pressure at the end being equal to 0 or negative, indicating that there will be no natural stable state in upward driving. This change law reflects the objective law of earth pressure change at the end, conforms to the engineering practice, and is reasonable.

Influence of soil cohesion

Take tunnel diameter D=16 m, γ=18kN/m3, internal friction Angle φ=20°, tunnel burial depth H=20, cohesion c 0-30kPa. The change curves of the earth pressure at the end of the shield under different cohesion conditions with different shield driving angles were calculated respectively, as shown in Fig. 16.

Fig. 16
figure 16

Earth pressure at the end varies with cohesion.

Full size image

As can be seen from Fig. 16, internal friction Angle and depth-diameter ratio remain unchanged. Under different driving angles, cohesion increases and earth pressure at the end decreases almost linearly. Compared with Fig. 15, under the same depth-diameter ratio, the curve variation amplitude under the influence of cohesion is smaller than that under the influence of internal friction Angle, indicating that the change of internal friction Angle has a greater influence on the earth pressure at the end of different driving angles.

Tunnel depth influence

Take tunnel diameter D=16 m. γ=18kN/m3. internal friction Angle φ=20°. cohesion c=10kPa. tunnel burial depth H=8、16、24、32、40. The change curves of the earth pressure at the end of the tunnel under different buried depths and different shield tunneling angles were calculated respectively, as shown in Figs. 17 and 18.

Fig. 17
figure 17

Change of earth pressure at the end with depth-diameter ratio (small Angle driving).

Full size image
Fig. 18
figure 18

Change of earth pressure at the end with depth-diameter ratio (large Angle driving).

Full size image

Figure 17 shows that under the condition of small-angle driving, when the internal friction Angle, cohesion force and shield diameter are constant, the earth pressure at the end increases with the increase of the depth-diameter ratio, and the change curve shows strong nonlinearity. When the depth to diameter ratio is 0.5 to 2.5, the earth pressure growth rate from upward to downward driving Angle is 138%, 135.9%, 133.7%, 131.4%, 129%, 126.4%, 123.8%, 120.9%, 118%, 114.8% and 111.1%, respectively. It is obvious that the depth to diameter ratio has a larger influence on the earth pressure at the heading head and a smaller influence on the earth pressure at the heading head.

Figure 18 shows that under the condition of large-angle driving, when the internal friction Angle, cohesion force and shield diameter are constant, the earth pressure at the end of upward driving and horizontal driving increases with the increase of depth diameter ratio, and the curve form changes from nonlinear to linear characteristics. When the driving Angle is greater than 30°, the earth pressure at the end gradually approaches 0, and the earth pressure at the end tends to be horizontal with the change of deep-to-diameter ratio, indicating that the change range of earth pressure at the end decreases with the deep-to-diameter ratio, and the change range gradually tends to 0. When the downward driving Angle is greater than 40°, the earth pressure at the end is less than 0, which is impossible, indicating that the soil is in a natural stable state. At this time, the earth pressure at the end decreases slightly with the increase of the depth-diameter ratio, and this change law has no practical significance. Upward driving will not result in zero or negative earth pressure at the end. The above variation law is consistent with the engineering practice, indicating that the formula reflects the objective law of earth pressure change at the end of the shield tunneling process, which is reasonable.

As demonstrated in Figs. 17 and 18, the functional curves exhibit the following variation patterns: Earth pressure during small-angle excavation displays pronounced nonlinear characteristics, while this nonlinearity progressively diminishes with increasing excavation angles.This phenomenon arises because downward tunneling induces a gradual reduction in face earth pressure with increasing driving angles, accompanied by enhanced self-stabilization capacity of the face soil mass. Consequently, the pressure variation transitions towards linear variation patterns. Conversely, upward tunneling exhibits amplified earth pressure magnitudes with increasing angles, where the improved soil self-stabilization effectively mitigates nonlinear deformation characteristics, resulting in less prominent nonlinear pressure variations.

Model engineering application

In order to better apply the model in the actual project, the application steps of establishing the model are shown in the Fig. 19.

Fig. 19
figure 19

Model engineering application diagram.

Full size image

During the initial phase of model implementation, comprehensive parameters investigation should be conducted, encompassing geotechnical parameters (Soil parameters: φ,c,γ; Geometric parameters: H,D; Excavation angle: β). Subsequently, model applicability assessment must be performed through engineering condition evaluation. For projects meeting the model’s prerequisites, appropriate computational formulas should be selected based on practical tunneling angle scenarios to establish the analytical framework. Following this, systematic validation through result verification and reliability analysis should be executed to confirm model effectiveness.

Conclusion

  • 1) Based on the limit equilibrium theory, a 3D curved slide analysis model was established to obtain the unified solution of the earth pressure at the end of the shield under the conditions of upward, horizontal and downward driving, and the changing law of the influence of driving Angle on the earth pressure at the end of the shield. Under the same conditions, the earth pressure at the end of the upward driving is the largest, followed by the horizontal driving, and the downward driving is the smallest. When the internal friction Angle of the soil mass is large, the earth pressure at the end of the horizontal and downward driving will appear negative. When the downward driving Angle is large, the earth pressure at the end will also appear negative. However, upward driving will not result in negative soil pressure at the end. It shows that the formula derived in this paper reflects the objective law of earth pressure change at the end of shield tunneling to a certain extent, and accords with the engineering practice.

  • 2) The earth pressure at the end of the shield decreases with the increase of the internal friction Angle, and the variation range decreases with the increase of the internal friction Angle. When the internal friction Angle is greater than 35°, the earth pressure at the horizontal and downward driving end is negative, and the earth pressure at the end is 0, which is in a natural stable state. The earth pressure at the end of the shield decreases linearly with the increase of cohesion, and the change range is less than the influence of internal friction Angle.

  • 3) Under the condition of small Angle tunneling, the earth pressure at the end of shield increases nonlinearly with the increase of depth to diameter ratio. Under the condition of large Angle tunneling, the earth pressure at the end of the shield increases with the increase of the depth to diameter ratio, and the variation range decreases with the increase of the depth to diameter ratio, and the variation law gradually presents a linear change. When the downward driving Angle is greater than 30°, the curve tends to be horizontal, and the variation amplitude of earth pressure at the end decreases greatly with the increase of depth diameter ratio. When the downward driving Angle is greater than 40°, the earth pressure at the end head is all negative, which is impossible to exist, and the earth pressure at the end head is in a natural stable state. In this case, the change law of the earth pressure at the end head with the depth diameter ratio reflected in the formula is meaningless.