Numerical analysis of ground displacement and segmental stress and influence of yaw excavation loadings for a curved shield tunnel

Computers and Geotechnics 118 (2020) 103325

Contents lists available at ScienceDirect

Computers and Geotechnics journal homepage: www.elsevier.com/locate/compgeo

Research Paper

Numerical analysis of ground displacement and segmental stress and influence of yaw excavation loadings for a curved shield tunnel

T

Mingju Zhang, Shaohua Li, Pengfei Li

⁎

The Key Laboratory of Urban Security and Disaster Engineering, Ministry of Education, Beijing University of Technology, Beijing 100124, China

ARTICLE INFO

ABSTRACT

Keywords: Yaw excavation Numerical simulation Curved tunnel Friction force

This paper describes the key influences of yaw excavation loadings on ground displacement and segmental stress for a curved shield tunnel. The influences are investigated through finite element models, the reliabilities of which are validated through comparisons to field data and analytical solutions. Multiple case studies of different curvature tunnels and their comparison to straight-line tunnels are presented. Under the dual action of overcutting and construction loadings, the surface settlement of the curved tunnel is larger than that of the straightline tunnel. The horizontal displacements at the inner and outer sides of the curved tunnel are asymmetric with respect to the tunnel axis. This asymmetry can increase significantly during yaw excavation of over one ring width. Yaw excavation loadings have a significant influence on the horizontal and vertical displacements of the ground within a span of shield length starting from the position of the hydraulic jacks until the back. The circumferential compressive stress, axial tensile stress, and axial compressive stress of newly installed segment of the curved tunnel are greater than those of the straight-line tunnel. Interestingly, the stress increments increase linearly with yaw severity. The results are of benefit to suggest improvements for practical construction procedures.

1. Introduction With increasing urbanization and need for complex transport infrastructure, underground tunnels have become progressively commonplace. The tunnels need to work around with constraints of both existing structures and structures that are about to be constructed. Typical obstacles include underground pile foundation of high-rise buildings and underground traffic rails. The need to maneuver through such obstacles has necessitated that some sections of the tunnel be built with very tight curvatures, which can complicate the selection of the appropriate underground tunnel shielding. Unlike straight-line tunnels, the excavating trajectories of curved tunnels are formed by a series of discontinuous straight sections that together, approximate different types of curves. Each section can be excavated by an earth pressure balance tunnel boring machine (EPB-TBM). In order to approximate a curve during curved tunnel excavation, each subsequent section is positioned at an angle with respect to the previously section. The change in direction can introduce unintentional yawing. Therefore, if the loadings are not controlled properly in the tunneling process, the extent of yawing (and thus increasing the angles between sections) can become more severe. If the construction loadings are not controlled

⁎

properly in the tunneling process, which can negatively impact the integrity of the surrounding earth or structures. In order to keep shield tunneling on the desired track and minimize disturbances to the surrounding ground [1–3] and local structures [4–6], a wide range of theoretical analyses, numerical simulations, and field tests have been performed. The published theoretical research covers the analytical prediction of ground displacement due to ground loss [7–13] as well as due to loadings generated during excavation [14–17]. A potential drawback of pure theoretical analysis is the difficulty in knowing exactly the magnitude and direction of loadings exerted onto the soil mass during construction (e.g. jacking forces). Such knowledge can only be obtained through field tests [18–21]. On the other hand, finite element modelling (FEM) has emerged as among the most effective and accurate methods for predicting ground displacement caused by shield excavation [22]. The influence of different construction parameters [23,24] on ground displacement and structural behaviors [25,26] can be estimated through FEM. In addition, numerical simulations of ground disturbances caused by construction loadings (e.g. face pressure, friction force, grouting pressure, etc.) [27,28] show that the friction force between skin of the shield and the surrounding soil is exerted on the soil indirectly via a contact pair between the shield

Corresponding author. E-mail addresses: [email protected] (M. Zhang), [email protected] (S. Li), [email protected] (P. Li).

https://doi.org/10.1016/j.compgeo.2019.103325 Received 13 August 2019; Received in revised form 14 October 2019; Accepted 24 October 2019 0266-352X/ © 2019 Elsevier Ltd. All rights reserved.

Computers and Geotechnics 118 (2020) 103325

M. Zhang, et al.

skin and the excavation interface [21]. Therefore, the reliability of this numerical approach in simulating the friction force remains to be verified. To realize yaw excavation along a curved trajectory [29], two key factors, which include overcutting and the control of hydraulic jacks, must be considered. These two key factors distinguish curved-line tunnels from straight-line tunnels. Concretely, the EPB-TBM shield needs to open the copy cutter to over-excavate the earth at the inner side of the curved tunnel [30]. The purpose of the latter factor is to generate a differential jacking force between the inner and outer sides [31] and to produce a yaw angle [32] between its original orientation and the axis of newly installed segment ring. For curved tunnels, the asymmetric boring around the axis of the tunnel will inevitably impact the tunnel structure as well as the surrounding ground. Therefore, it is necessary to investigate in further detail regarding the effects of different levels of yaw excavation on both the tunnel and its surroundings. The following study uses FEM to investigate the influence of yaw excavation on ground displacement and segmental stress for a curved shield tunnel. Numerical results are compared with field data and analytical solutions. The study also explores different construction cases (straight tunnels and tunnels with different curvature radii.

The geology of the Zhuhai area typically consists of composite strata with soft upper layers and hard lower layers. The average soil properties of the different layers are listed in Table 1. The tunnel axis is to be situated at 21.39 m below the surface (Fig. 1b). In the numerical simulation, the soil strata are simplified into two layers for the purpose of a more straightforward comparison to the analytical solution and efficient meshing. Specifically, the equivalent Young's modulus of soil layer 1 in Table 2 is obtained via the weighted average of the soft upper soil layers in Table 1 [33]. The other properties of soil layer 1 are in accordance with the values of the layer of silty clay. In addition, the properties of soil layer 2 are in accordance with the values of the layer of weathered granite. The classical Mohr-coulomb constitutive model is used to model the soil. 2.2. Numerical models Refs. [34,35] verified that including a tunnel length of at least 120 m can adequately eliminate boundary effects in numerical simulations. Thus the 3D FEM model of a straight-line shield tunnel generated using the FEM software ABAQUS is 120 m, as shown in Fig. 2a. In order to minimize the difference of boundary conditions between the curved tunnel model (Fig. 2b) and the straight-line tunnel model, the distance between tunnel axis and the edge of the model (including soil) are the same for both models. The boundary conditions used in the numerical models are as follows: the upper surface is free to move, while the bottom boundary is fixed. Furthermore, the lateral boundaries are specified as roller supports to constraint the normal (i.e. perpendicular to the boundaries) displacements.

2. Numerical simulations 2.1. Overview of physical model The inter-city railway from the Zhuhai urban area to the airport (i.e. between the “Wanzai North” station and the “Wanzai” station) includes a curved shield tunnel (curvature radius is 450 m, as shown in Fig. 1a). Markers are set on the ground surface at 10 m intervals along the curved trajectory of the tunnel.

2.3. Model characteristics The numerical models (Fig. 2) are all composed of solid elements. The EPB-TBM shield has a cylindrical shape [19], and its conicity [28] is ignored. Specific parameters for the shield body are listed in Table 3. Note that due to the presence of the cutter-head and the chamber, the shield elements have a stiffness 10 times higher than the elastic modulus of steel and deformations are ignored [21]. The universal segments (double-sided, wedge-shaped) are modeled using continuous elements. Since longitudinal and circumferential joints are not simulated, the reduction coefficient is assumed to be 0.75 [21] in order to take stiffness reduction into consideration. Thus, C50 concrete (elastic modulus of 25.9 GPa, Poisson ratio of 0.2 and density of 25 kN/m3) is selected to model the tunnel segment. The dimensions of the prefabricated segment ring has an outside diameter of 8.5 m, a wall thickness of 40 cm, and an average width of 1.6 m. Since the elastic modulus of the grout material is time-dependent [36], the filling grout (Fig. 2) between the excavation interface and outer wall of the tunnel segment is defined by annular concrete elements with two-phase elastic behaviors. An initial elastic modulus of 40 MPa is assigned to the filling grout at the tail of the shield in order to simulate the mechanical properties of fresh grout. On the other hand, hardened grout, which has an elastic modulus of 500 MPa is assigned to the grouted sections after the shield tail has passed seven rings. In the present curved shield tunnel model (Fig. 2b), the overcutting at the inner side of the curved tunnel are modeled using equivalent, semicircular zone elements with low modulus and thin thickness, which is different from the characteristics of straight-line tunnel (Fig. 2a). Fig. 3 demonstrates the mechanism of the over-excavation process for the articulated EPB-TBM shield, where LM1 and LM2 are the lengths of front shield body and rear shield body, respectively [37]. Considering promptly support after excavation, it is assumed that the overcutting ω is equal to one-third [38] of the theoretical value (Ga) [30,38]:

Fig. 1. (a) Plan view of project site; (b) Geological profile along the curved trajectory.

= 2

2 (Q + R)2 + L M2

3

(Q + R)

(1)

Computers and Geotechnics 118 (2020) 103325

M. Zhang, et al.

Table 1 Soil properties of the local Zhuhai area. Type of soil (i)

Average thickness (m)

Young’s modulus (MPa)

Poisson ratio (–)

Cohesion (kPa)

Density (kN/m3)

Friction angle (°)

Lateral pressure coefficient (–)

Artificial fill (1) Silty clay (2) Mucky silty clay (3) Silty-fine sand (4) Silty clay (5) Weathered granite (6)

2.0 13.5 2.2 1.8 11.3 –

9 48 29 78 48 3000

0.42 0.3 0.3 0.3 0.3 0.25

16 25 32 – 25 100

17.1 19.5 19 20.2 19.5 20

14 22 20 37 22 35

0.62 0.4 0.4 0.3 0.4 0.35

Table 2 Soil properties in the numerical simulations. Type of soil

Thickness (m)

Young’s modulus (MPa)

Poisson ratio (–)

Cohesion (kPa)

Density (kN/m3)

Friction angle (°)

Lateral pressure coefficient (–)

Layer 1 Layer 2

30.8 29.2

46 3000

0.3 0.25

25 100

19.5 20

22 35

0.4 0.35

(a) Straight-line shield tunnel

(b) Curved shield tunnel Fig. 2. Perspective view, zoom view, and boundary conditions of the numerical models.

3

Computers and Geotechnics 118 (2020) 103325

M. Zhang, et al.

Table 3 Parameters for the EPB-TBM shield. Diameter of the cylinder shape (m)

Thickness (cm)

Length (m)

Weight (kN)

Elastic modulus (GPa)

Poisson ratio (–)

8.78

11

11.2

6860

2450

0.2

2.4. Modelling construction loadings 2.4.1. Modelling face pressure Fig. 4 illustrates the construction loadings generated as EPB-TBM shield advances. It is assumed that the pressure exerted by the EPB-TBM shield on the excavation face varies linearly with elevation and ground density. The reference pressure at the axis of the tunnel is usually equal to 50% of the total horizontal geostatic stress (i.e. the sum of stresses at tunnel crown and stresses at the tunnel invert) [28]. On this basis, an additional thrust (about 15 kPa [15]) is exerted on the excavation face during actual construction; thus the face pressure at the tunnel axis is set to 182 kPa in this study. 2.4.2. Modelling grouting pressure The grouting pressure acting on the interface between the excavated ground and the outer wall of the installed segment is assumed to have a uniform distribution (σg), and is related to the ground overburden pressure at the tunnel crown (σvc) [6,35]: g

= 1.2·

vc

(2)

where σvc = γ(h–R), γ is the soil density, and h is the tunnel depth.

Fig. 3. Over-excavation of a curved tunnel.

2.4.3. Modelling yaw excavation loadings Loadings due to the yaw excavation of the curved shield tunnel are mainly composed of the jacking force as well as the friction force between the shield skin and the ground.

where Q is the radius of curvature of the curved tunnel, R is the external diameter of the EPB-TBM shield, LM2 = 2b, and b is equal to the width of one ring. Based on the listed parameters, the overcutting thickness is determined to be about 4 mm when the radius of curvature is 450 m. Compared with the tunnel diameter, the small elements needed to model the overcutting thickness is exceedingly small, which can complicate the meshing process when fine and coarse meshes are juxtaposed. Therefore, in the finite element models, the thickness of the partially over-excavated soil is set to ten times the calculated overcutting thickness, and its modulus is set to one tenth of the original value.

(1) Jacking force For the straight-line tunnel, the total force of hydraulic jacks (about 32.6 MN) is enforced equally on the circumferential surface of the installed segment [35]. Thus, the jacking force per unit area is about 3.204 MPa. For the curved tunnel, Fig. 5 illustrates the yaw excavation loadings at the moment when a new, pending segment is about to be installed. At this moment, the advancing of the shield is driven by the

Fig. 4. FEM of the advancement of the EPB-TBM shield.

4

Computers and Geotechnics 118 (2020) 103325

M. Zhang, et al.

2.5. Curved shield tunneling process The excavation of soil for 80 m (50 rings) along the tunnel axis is simulated. The process consists of a sequence of stages, with each stage following a set of procedures that result in the advancement of one ring width. Taking stage “n” (Fig. 4) as an example, the simulation of the curved shield tunnel excavation is conducted as follows:

• In the current stage of excavation, the material properties of the • • •

overcutting elements are changed when ground slice “n” is removed. At the same time, the pressure from the EPB-TBM shield is applied on the face of slice “n + 1”. With the advancement of the EPB-TBM shield from slices “n − 7” to “n − 1”, the shield and friction contact pair at slice “n” are activated when the shield slice “n − 7” is removed. Simultaneously, the prescribed friction force is applied on the outer torus of shield slice “n”. The segment installation is carried out by activating segment slice “n − 7”, and the prescribed jacking force is applied on segment slice “n − 7”. “Synchronous grouting” at the shield tail (slice “n − 7”) is achieved by applying the prescribed grouting pressure on both the ground and the segment after the removal of that acting on slice “n − 8”. In addition, the elements of the filling grout (elastic modulus of 40 MPa) are activated during this step. As the EPB-TBM shield advances, the elastic modulus of the hardened grout located at slice “n − 14” is increased to 500 MPa.

3. Numerical results and comparisons 3.1. Numerical results 3.1.1. Ground vertical displacement Fig. 6 shows the contours of the vertical displacement fields for both the straight-line and curved tunnel models after 50 excavation stages. For the straight-line tunnel (Fig. 6a), the ground immediately above the tunnel crown settles about 30 mm at the shield tail, and the tunnel invert rises about 24 mm. Moreover, the settlement trough gradually widens as the elevation increases, and the ground settlement gradually stabilizes as the distance from the excavation face increases [39]. The ground surface settlement near the back of excavation face at a distance of 30 rings (about 50 m) is about 13.9 mm. For the curved tunnel (Fig. 6b), the ground settlement immediately above the tunnel crown is slightly larger than that of the straight tunnel. Furthermore, the uplift just below the tunnel invert is slightly smaller. Note that the surface settlement of the curved tunnel is more extensive than that of straightline tunnel. For example, the surface settlement near the back of excavation face at a distance of 30 rings is 14.6 mm, which is about 0.7 mm larger than that of the straight-line tunnel. The increased settlement of the curved tunnel is mainly caused by the overcutting of the curved tunnel.

Fig. 5. Illustration of yaw excavation loadings: (a) plan view of Fig. 2b and (b) wedge angle of a segment ring.

jacking force acting on the cross section of the installed segment. The surface that is being acted upon is equally divided into the inner side and the outer side of the curved tunnel. According to field data, the average ratio of jacking force between the outer side and the inner side is close to 2. Based on this ratio, the total force of hydraulic jacks is adjusted over the corresponding area of the segments in the model. Consequently, the jacking force per unit area acting on the inner side and outer side of the circumference of the segment, corresponding to “Ji” and “Jo” in Fig. 5a, is 2.136 MPa and 4.272 MPa, respectively. In addition, the direction of jacking force is inclined with respect to the outer side of the curved tunnel by a yaw angle of α, where α is equal to the wedge angle of a segment ring (η) when the shield is advanced a distance of one ring width, as shown in Fig. 5b. (2) Friction force

3.1.2. Segmental stress Taking the newly installed segment ring that has just emerged from the shield tail as a case study, the circumferential and axial stresses of the ring are respectively shown in Figs. 7 and 8 for both straight-line and curved tunnels. The maximum tensile circumferential stress is small for both the straight-line and curved tunnels. On the other hand, the circumferential compressive stress, axial tensile stress, and axial compressive stress are much higher. These stresses are higher for the curved shield tunnel than for the straight-line tunnel. The high stresses for the curved shield tunnel suggest the increased possibility of adverse conditions, such as severe concrete cracking, splaying of circumferential or longitudinal joints, and dislocation.

It is difficult to measure the friction force between the shield skin and the surrounding soil in practice. Assuming mechanical equilibrium, the difference between the jacking force acting on the segment and the thrust acting on the excavation face is used to approximate the friction force. Likewise, the outer torus of the shield is also divided into inner and outer regions, with friction force per unit area of 48 kPa and 96 kPa, respectively (corresponding to “Fi” and “Fo” in Fig. 5a). The friction-sliding model is established by means of a contact pair consisting of the shield skin and the excavation interface. In ABAQUS, the Coulomb friction law is adopted, and the friction coefficient is 0.25 [21]. 5

Computers and Geotechnics 118 (2020) 103325

M. Zhang, et al.

Fig. 6. Contours of the vertical displacement field from FEM simulations: (a) straight-line shield tunnel and (b) curved shield tunnel.

3.2. Comparison of monitored field data and numerical results

position. Here, the advancing direction is clockwise when viewed from the plan view; l0 tracks the position along the curved axis in the clockwise position (arc length |l0| from point (0, Q, h)). For instance, l0 = −L refers to the back of excavation face at a distance of L. The negative sign in front of “L” indicates that the moving direction from point (0, Q, h) to this point is counter-clockwise (Fig. 10), and L is the length of EPB-TBM shield. The advancing trajectory of the EPB-TBM shield is assumed as a space torus C(x0, y0, z0), which can be represented by a standard equation or set of parametric equations:

The FEM result is compared to the monitored ground surface settlement along the curved tunnel axis in Fig. 9. As seen in the figure, the results obtained by FEM agree well with the monitored data. The results show that the ground settlement along the tunnel axis follows an “S” shape, which were predicted by experimental, analytical and numerical methods [18,21,22,28]. 3.3. Comparison of analytical and numerical solutions for friction force

x 0 2 + y0 2 )2 + (z 0

h) 2 = R2

(3)

In order to verify the accuracy of the simulated friction force between the shield skin and ground, this section presents an analytical solution that can estimate the friction force. While experimental data will be more suitable for verification purposes, such experimental data can be challenging and impractical to obtain.

(Q

3.3.1. Analytical solution To facilitate the analysis, a computational model is set up in advance. Fig. 10 presents the model of a curved tunnel, in which a EPBTBM shield advances along a curved axis from the point (Qcosθ, Qsinθ, h) to the position (0, Q, h), where θ = π/2 + l0/Q in the starting

With reference to the complicated spatial model shown in Fig. 10, the use of the traditional method described in [15,16] can be inefficient as it distinctly transforms the coordinates of the force functional point into point (0, 0, c) as specified in Mindlin solutions [14]. Thus, a set of general formulae that better handles the problem can be arrived upon

x 0 = (Q + R cos ) cos , y0 = (Q + R cos ) sin , z 0 = h + R sin ,

6

(4)

Computers and Geotechnics 118 (2020) 103325

M. Zhang, et al.

Fig. 7. Stress of straight-line tunnel segment: (a) circumferential stress (S22) and (b) axial stress (S33).

via the substitution of the specified point for an arbitrary point (x0, y0, z0). Through the equations, the ground settlements caused by a unit force acting at this arbitrary point in the x and y directions (Fig. 10) are respectively estimated by

sx =

(x x 0 ) 16 G (1 µ ) +

z

(3 z0 + R13

4(1 µ)(1 2µ) R2 (R2 + z + z 0 )

4µ )(z R23

z 0)

sy =

(y y0 ) 16 G (1 µ ) +

6z 0 z (z + z 0 ) R25

z

(3 z0 + R13

4µ )(z R23

4(1 µ)(1 2µ) R2 (R2 + z + z 0 )

z 0)

6z 0 z (z + z 0 ) R25 (6)

where R1 = [(x x 0) 2 + (y y0 )2 + (z z 0)2]1/2 , R2 = [(x x 0 )2 + (y y0 ) 2 + (z + z 0 ) 2]1/2 , G is elastic shear modulus of the soil, and μ is Poisson’s ratio. The rewritten Mindlin solutions are applied to the estimation of friction force mentioned above. The friction force is assumed to be

(5)

Fig. 8. Stress of curved tunnel segment: (a) circumferential stress (S22) and (b) axial stress (S33).

7

Computers and Geotechnics 118 (2020) 103325

M. Zhang, et al.

-1.0

Ground settlement (mm)

Ground settlement (mm)

-6 0 FEM Monitored Data

6 12 18 -50

-40

-30

-20

-10

0

10

20

30

0.0 0.5 1.0 -50

40

Distance from the tunnel face (m)

FEM Mindlin Solution

-0.5

-40

-30 -20 -10 0 10 20 Distance from the tunnel face (m)

30

40

Fig. 11. Ground surface settlement of curved tunnel caused by friction force.

Fig. 9. Comparison of measured and numerical ground surface settlement along the tunnel axis.

3.3.2. Comparison The following parameters are input into the analytical solution described in the previous section: Q = 450 m, h = 21.39 m, R = 4.39 m, b = 1.6 m, L = 11.2 m, μ = 0.3, G = 17.7 MPa, βs = 0.91, γ = 19.5 kN/ m3, K0 = 0.4, and δ = 8°. The analytical results obtained from these parameters agree well with the FEM results. Fig. 11 shows the ground settlement estimated from FEM and from the analytical Mindlin solution. The results are characterized by the centrosymmetry about the shield center. In other words, the ground uplift appears in front whereas ground settlement appears at the back of the shield tail. Both methods indicate peak values close to 0.7 mm, which further verifies the reliability of the developed numerical simulation in this study. With the numerical model verified for accuracy, the effects of friction force on ground settlement can be isolated by comparing a numerical model that applies friction versus a model that does not. 4. Analysis of yaw excavation loadings influence on ground displacement and segmental stress Due to an angle between the direction of the jacking force vector and the central axis of the preceding segment, lateral forces that can greatly impact the surrounding ground are generated. However, since hydraulic jacks act directly on the structure of the segment, there are limited empirical data that can be used to verify the lateral forces (location, magnitude, and direction). In the following sections, a validated FEM is used to investigate such impact. In order to highlight the effects of tunnel curvature, tunnel models with a radius of curvature of 300 m were developed. Three hundred meters is the minimum radius of curvature for universal wedge-shaped segments. The following numerical analyses keeps the total yaw excavation loadings (jacking force as well as friction force) unchanged and investigates their influence on horizontal ground displacements and segmental stress. Seven case studies (Table 4) were developed by changing the ratio of the inner and outer yaw excavation (Ji/Jo and Fi/ Fo in Fig. 5a) and by adjusting the angle between the jacking force

Fig. 10. Curved shield tunneling model.

distributed along the outer torus of the shield from the cross-section l0 = –L to the cross-section l0 = 0, which can be obtained by the following expression [40–42]:

f=

(7)

tan

s

where βs denotes the softening coefficient, σφ is the radial normal stress acting on the shield, σφ = σvsin2φ + σhcos2φ, σv = γh + γRsinφ, σh = K0σv, δ is the angle of skin friction. Friction force f is divided into two components fx = fcos(θ + π/2) and fy = fsin(θ + π/2), which are respectively parallel to the x-axis and y-axis. Then, the concentrated x and y forces acting on the surface elements of the space torus are fxR(Q + Rcosφ)dφdθ and fyR (Q + Rcosφ)dφdθ, respectively. Through double integration, the ground settlements at an arbitrary point (x, y, z) due to fx and fy can be expressed by 2

Sx = 2

2

Sy = 2

2 L Q

0

2 L Q

0

fx sx R (Q + Rcos )d d

Table 4 Seven case studies based on different yaw load ratios and angles.

(8)

fy sy R (Q + Rcos )d d

(9)

To sum up, the ground settlement S caused by friction force f is

S = Sx + S y

(10)

8

Condition

Ratio of yaw excavation loadings

Yaw angle (°)

I II III IV V VI VII

4:1 3:1 2:1 1:1 1:2 1:3 1:4

−1.8 −0.9 −0.3 0 0.3 0.9 1.8

Computers and Geotechnics 118 (2020) 103325

M. Zhang, et al.

0

l0= 2L l 0= 0 l0= –L l0= –2L l0= –4L

5

Depth (m)

10

l 0= L l 0= 0 l0= –L l0= –2L l0= –4L

Outer side

15

Inner side

20

-1.1

-0.7

Depth: 21.39 m

0.6

0.9

25 30 35 -1.2

-0.8

-0.4

0

0.4

1.2

0.8

0.4

0

Increments of horizontal displacement (mm)

-0.4

Fig. 12. Horizontal displacements along the depth (IV).

Fig. 14. Increments of horizontal displacements along the depth (VII – IV).

vector (reverse direction of advancement) and the axis of the prior segments (α in Fig. 5a). The cases include yaw excavation along straight-line trajectory (IV), curved trajectories with large curvature radii (I; II; III) and with small curvature radii (V; VI; VII). Taking condition III as an example, “2:1” means the magnitude of yaw excavation loadings at the inner side is two times that of the outer side; in the third column of Table 4, “−0.3” means that the yaw angle between the reverse direction of jacking force and the segment axis is one times the wedge angle of a segment ring (η in Fig. 5b), where the negative sign in front of “0.3” indicates the direction of advancement is inclined to the outer side.

results in Fig. 13 and the results in Fig. 12 are shown in Fig. 14. The values in Fig. 14 are defined as the increments of horizontal displacements caused by the variation of yaw excavation loadings (VYEL for short, which correspond to the extent of yaw excavation). As seen in Fig. 14, at cross-sections l0 = L, l0 = 0 and l0 = −4L, the increments of horizontal displacements caused by the VYEL at the inner side are oriented towards away from the tunnel (negative). On the other hand, at the outer side, the increments of horizontal displacements are oriented towards the tunnel (positive). It should be noted that the values above are small; however, for the ground at the cross-sections l0 = −L and l0 = −2L, the directions of the VYEL are reversed, and the increments are relatively large. For case VII, the VYEL produces horizontal force components that are directed towards the outer side of the curved tunnel. The forces in turn cause large displacement in the ground between cross-section l0 = −L (position of hydraulic jacks) and cross-section l0 = −2L. For the ground far away from the shield tail and also in front of the excavation face (i.e. at cross-sections l0 = −4L, l0 = 0 and l0 = L), the effects on displacement increments inclined to the inner side of curved tunnel are minor. The total horizontal displacements due to the dual action of overcutting and construction loadings (VII) are shown in Fig. 13. The differences between the results when advancing along a curved trajectory with large curvature radius and the straight-line benchmark (Fig. 12) (I–IV) in terms of increments of horizontal displacements caused by the VYEL are shown in Fig. 15. As seen in Fig. 15, the horizontal forces directed at the inner side of curved tunnel is produced by the VYEL when the shield is advanced along a curved trajectory with a large curvature radius. Hence, for different cross-sections, the influence of VYEL on the displacement increments near the buried position (both the inner and outer side of curved tunnel) are contrary to those shown in Fig. 14. In other words, horizontal displacements (directed at the inner side) occur in large

4.1. Influence on horizontal ground displacement The horizontal (i.e. perpendicular to the curved tunnel axis) displacements of the ground located at a horizontal distance of 9.39 m from the tunnel axis when advancing along straight-line trajectory (IV) are shown in Fig. 12. The figure shows five cross-sections selected along the tunnel axis. Positive values on the abscissa indicate the displacements inclined to the tunnel. As seen in Fig. 12, the horizontal displacements at the inner and outer sides of the curved tunnel are asymmetric with respect to the tunnel axis due to the influence of overcutting. Particularly, for both the cross-sections at l0 = −L and l0 = −2L, the horizontal displacements at the inner side are about 0.7 mm larger than those at the outer side. The reason for this phenomenon, which is observed at these two crosssections, is that under the action of construction loadings, the EPB-TBM shield and the segments generate a more pronounced pressing effect on the inner face of the curved tunnel (near the buried position). Fig. 13 shows the horizontal displacements for the case of the small curvature radius (VII), which are slightly different than those in Fig. 12. To illustrate the changes more clearly, the differences between the 0

Depth (m)

10 15

10

Inner side

20

Depth: 21.39 m

Outer side

25

l 0= L l 0= 0 l0= –L l0= –2L l0= –4L

Outer side

15

Inner side

20

1.2

1.7

-1.1

Depth: 21.39 m

-1.7

25 30

30 35 -12

l 0= L l 0= 0 l0= –L l0= –2L l0= –4L

5

Depth (m)

5

0

l0= L l0= 0 l0= –L l0= –2L l0= –4L

l0= L l0= 0 l0= –L l0= –2L l0= –4L

-6

0

6

0

Horizontal displacement (mm)

-6

35

12

Fig. 13. Horizontal displacements along the depth (VII).

-1

0

1

2

1

0

-1

Increments of horizontal displacement (mm)

-2

Fig. 15. Increments of horizontal displacements along the depth (I – IV). 9

Computers and Geotechnics 118 (2020) 103325

M. Zhang, et al.

Table 5 Peak and absolute values of horizontal displacements along the depth (mm). l0 = −4L

I II III IV V VI VII

l0 = −2L

l0 = −L

l0 = 0

l0 = L

Outer

Inner

|O–I|

Outer

Inner

|O–I|

Outer

Inner

|O–I|

Outer

Inner

|O–I|

Outer

Inner

|O–I|

−11.20 −11.17 −11.12 −11.03 −10.94 −10.89 −10.86

−11.19 −11.22 −11.26 −11.35 −11.44 −11.49 −11.52

0.01 0.05 0.14 0.32 0.50 0.60 0.65

−10.18 −10.35 −10.54 −10.83 −11.13 −11.34 −11.54

−12.12 −11.93 −11.73 −11.47 −11.23 −11.06 −10.92

1.94 1.58 1.20 0.65 0.10 0.28 0.63

−9.57 −9.79 −10.05 −10.49 −10.95 −11.26 −11.55

−12.21 −11.92 −11.62 −11.18 −10.77 −10.52 −10.32

2.64 2.14 1.57 0.69 0.18 0.74 1.22

−4.93 −4.89 −4.79 −4.58 −4.44 −4.43 −4.47

−4.74 −4.69 −4.69 −4.80 −4.98 −5.07 −5.11

0.19 0.20 0.10 0.22 0.54 0.64 0.64

−0.78 −0.78 −0.76 −0.70 −0.66 −0.64 −0.64

−0.64 −0.64 −0.65 −0.70 −0.76 −0.78 −0.79

0.15 0.14 0.11 0.00 0.10 0.14 0.15

Outer: peak values at the outer side of curved tunnel; Inner: peak values at the inner side of curved tunnel; |O–I| = |Outer–Inner|.

increments within a span of L starting from the position of the hydraulic jacks until the back. On the other hand, the increments of horizontal displacement are small and directed towards the outer side in positions far away from the shield tail and in front of the excavation face. For each of the operating conditions listed in Table 4, the peak values of horizontal displacements along the depth at different crosssections of the curved tunnel are summarized in Table 5, which listed the absolute values of differences between the peak values at the outer side and the inner side. The peak values of horizontal displacements at the cross-section l0 = L (i.e. in front of the excavation face) are small for all listed operating conditions. When advancing along straight-line trajectory (IV), all of the peak values at different cross-sections appear in the inner side of the curved tunnel. For instance, the inner value is 0.69 mm greater than the outer value at the cross-section l0 = −L. Based on this difference and the intensifying yaw when advancing along the curved trajectories with large curvature radii (from III to I), the maximum displacements at cross-sections l0 = 0 and l0 = −4L appear in the positions of transition from the inner side to the outer side. Similarly, for the cases of small curvature radii (from V to VII), the maximum displacements at cross-sections l0 = −L and l0 = −2L appear in the positions of transition from the inner side to the outer side. When significant yaw is present (I and VII), the maximum displacements are greater than those of condition IV. The above results can be used to suggest improvements for practical construction procedures. Specifically, for unavoidable yaw excavation of curved shield tunnels, it is advisable to plan the layout of the excavating trajectory in advance, and the extent of yaw excavation should be minimized when advancing along a curved trajectory. Otherwise, yaw excavation of over one ring width is likely to increase the asymmetry of horizontal displacements on both sides of the curved tunnel significantly. For instance, the absolute differences between the outer values and the inner values in conditions I and VII are 2.64 mm and 1.22 mm, which are about 2.0 mm and 0.5 mm greater than the 0.69 mm in condition IV, respectively.

4.2. Influence on ground vertical displacement The computed differences between the ground vertical displacements at a depth of 12 m below ground surface (i.e. 5 m above the tunnel crown) resulting from the VYEL of curved trajectories versus the straight-line trajectory (IV) are shown in Fig. 16. Similar to the findings on horizontal displacements, ground surface settlements also occur in large increments within a span of L starting from the position of the hydraulic jacks until the back, herein, only the results at the crosssection l0 = −L are presented. When advancing along the curved trajectories with large curvature radii (I–IV; II–IV; III–IV), the VYEL mainly leads to ground uplifts appear at the outer side of the curved tunnel, whereas ground settlements appear at the inner side. The values and positions of settlements are respectively larger and closer from the tunnel axis than those of uplifts. For the ground above the tunnel axis, the vertical displacements appear as settlement. Interestingly, the influence of VYEL on the increments of ground vertical displacement for trajectories with large curvature radii are approximately contrary to those of trajectories with small curvature radii (V–IV; VI–IV; VII–IV). Furthermore, the increments of ground vertical displacement increase with yaw severity. 4.3. Influence on segmental stress The maximum and minimum segmental stresses in different operating conditions are listed in Table 6. For all considered operating conditions, the minimum circumferential stresses (compressive) are large, which increase with yaw severity. However, all the maximum circumferential stresses are small. Moreover, the maximum and minimum values of segmental axial stress when advancing along curved trajectories are larger than when advancing along straight-line trajectory (IV), and increase with yaw severity. Fig. 17 shows the computed differences between the stresses resulting from the VYEL of curved trajectories versus the straight-line trajectory (IV). The increase of yaw leads to a nearly linear increase of stress increments (including the circumferential compressive stresses and the axial tensile and compressive stresses). Interestingly, the profile of stress increments for trajectories with large curvature radii (I–IV;

Increment of ground settlement (mm)

-0.2 Outer side

Inner side

0.0

Table 6 Segmental stress (MPa).

0.2 I-IV II-IV III-IV

0.4 0.6 -50

-40

-30

VII-IV VI-IV V-IV

-20 -10 0 10 20 30 Distance from the tunnel axis (m)

40

I II III IV V VI VII

50

Fig. 16. Increments of ground vertical displacement (l0 = −L). 10

Circumferential stress

Axial stress

S22 (MAX)

S22 (MIN)

S33 (MAX)

S33 (MIN)

0.28 0.27 0.22 0.1 0.02 −0.04 −0.09

−10.17 −10.14 −9.96 −9.37 −9.76 −9.92 −9.96

2.73 2.56 2.29 1.91 2.11 2.42 2.6

−5.96 −5.73 −5.26 −4.37 −5.06 −5.53 −5.76

Computers and Geotechnics 118 (2020) 103325

M. Zhang, et al.

actual operation. Otherwise, the yaw excavation of over one ring width can significantly increase the asymmetry of horizontal displacements on both sides of the curved tunnel. Declaration of Competing Interest The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper. Acknowledgements The financial support for this work has been provided by National Natural Science of Foundation of China (Grants No. 51538001, 51778025 and 51978019), which are gratefully acknowledged.

Fig. 17. Influence of yaw severity on increments of segmental stress.

References

II–IV; III–IV) shows substantial similarity to those of trajectories with small curvature radii (V–IV; VI–IV; VII–IV).

[1] Chakeri H, Unver B, Ozcelik Y. A novel relationship for predicting the point of inflexion value in the surface settlement curve. Tunn Undergr Space Technol 2014;43:266–75. [2] Zhao C, Lavasan AA, Barciaga T, et al. Model validation and calibration via back analysis for mechanized tunnel simulations – The Western Scheldt tunnel case. Comp Geotech 2015;69:601–14. [3] Liu C, Zhang Z, Regueiro RA. Pile and pile group response to tunnelling using a large diameter slurry shield – Case study in Shanghai. Comp Geotech 2014;59:21–43. [4] Do NA, Dias D, Oreste P, et al. Three-dimensional numerical simulation of a mechanized twin tunnels in soft ground. Tunn Undergr Space Technol 2014;42:40–51. [5] Hasanpour R, Schmitt J, Ozcelik Y, et al. Examining the effect of adverse geological conditions on jamming of a single shielded TBM in Uluabat tunnel using numerical modeling. J Rock Mech Geotech Eng 2017;9(6):1112–22. [6] Ninić J, Meschke G. Simulation based evaluation of time-variant loadings acting on tunnel linings during mechanized tunnel construction. Eng Struct 2017;135:21–40. [7] Peck RB. Deep excavations and tunneling in soft ground. State-of-the-art report. In: Proc 7th Int conf soil mechanics and found engineering, Mexico, 1969; p. 225–90. [8] Sagaseta C. Analysis of undrained soil deformation due to ground loss. Géotechnique 1987;37(3):301–20. [9] Lee KM, Powe RK, Lo KY. Subsidence owing to tunneling. I: estimating the gap parameter. Can Geotech J 1992;19:929–40. [10] Verruijt A, Booker JR. Surface settlements due to deformation of a tunnel in an elastic half plane. Géotechnique 1996;46(4):753–6. [11] Loganathan N, Poulos HG. Analytical prediction for tunneling-induced ground movement in clays. J Geotech Geoenviron Eng 1998;124(9):846–56. [12] Zhang Z, Huang M. Geotechnical influence on existing subway tunnels induced by multiline tunneling in Shanghai soft soil. Comp Geotech 2014;56:121–32. [13] Wan T, Li PF, Zheng H, Zhang MJ. An analytical model of loosening earth pressure in front of tunnel face for deep-buried shield tunnels in sand. Comp Geotech 2019;115:103170. [14] Mindlin RD. Force at a point in the interior of a semi-infinite solid. J. Appl. Phys. 1936;7(5):195–202. [15] Zhu JF, Xu RQ, Liu GB. Analytical prediction for tunnelling-induced ground movements in sands considering disturbance. Tunn Undergr Space Technol 2014;41:165–75. [16] Wei G, Pang SY, Zhang SM. Prediction of ground deformation induced by double parallel shield tunneling. Disaster Adv 2013;6:91–8. [17] Zhang Z, Zhang M, Jiang Y, Bai Q, Zhao Q. Analytical prediction for ground movements and liner internal forces induced by shallow tunnels considering nonuniform convergence pattern and ground-liner interaction mechanism. Soils Found 2017;57:211–26. [18] Guo W, Wang G, Bao Y, et al. Detection and monitoring of tunneling-induced riverbed deformation using GPS and BeiDou: a case study. Appl Sci 2019;9(13):2759. [19] Lambrughi A, Rodríguez LM, Castellanza R. Development and validation of a 3D numerical model for TBM-EPB mechanised excavations. Comp Geotech 2012;40:97–113. [20] Dias D, Kastner R. Movements caused by the excavation of tunnels using face pressurized shields—analysis of monitoring and numerical modeling results. Eng Geol 2013;152(1):17–25. [21] Zheng G, Lu P, Diao Y. Advance speed-based parametric study of greenfield deformation induced by EPBM tunneling in soft ground. Comp Geotech 2015;65:220–32. [22] Migliazza M, Chiorboli M, Giani GP. Comparison of analytical method, 3D finite element model with experimental subsidence measurements resulting from the extension of the Milan underground. Comp Geotech 2009;36(1–2):113–24. [23] Hasanpour R, Rostami J, Thewes M, et al. Parametric study of the impacts of various geological and machine parameters on thrust force requirements for operating a single shield TBM in squeezing ground. Tunn Undergr Space Technol 2018;73:252–60. [24] Li PF, Chen KY, Wang F, Li Z. An upper-bound analytical model of blow-out for a shallow tunnel in sand considering the partial failure within the face. Tunn Undergr Space Technol 2019;91:102989.

5. Conclusions This study presents in-depth numerical simulations of the whole excavation process for curved shield tunnels. The results of the simulations are in good agreement with field measurements. Furthermore, the influence of friction forces estimated by analytical methods are in good agreement with the numerical solutions. A series of numerical simulations were conducted to investigate the influence of yaw excavation loadings on ground displacements and segmental stresses. The following general conclusions are reached: Compared with the straight-line tunnel, the ground settlement for the curved tunnel just above the tunnel crown is slightly larger, whereas the uplift just below the tunnel invert is slightly smaller. During the construction of a curved tunnel, overcutting and yaw excavation loadings can cause, the surface settlement of the curved tunnel to be larger than that of straight-line tunnel. Moreover, the horizontal displacements at the inner and outer sides of curved tunnel are asymmetric with respect to the tunnel axis, especially for the ground at the back of the excavation face. The values of circumferential compressive stress, axial tensile stress, and axial compressive stress for a newly installed segment of a curved shield tunnel are greater than those of the straight-line tunnel. As a result, multiple adverse effects, such as large cracks in concrete, splaying of circumferential or longitudinal joints, and dislocation can occur during excavation. The horizontal force components produced by yaw excavation loadings have a significant influence on the ground within a span of shield length starting from the position of the hydraulic jacks until the back. However, the influence on the ground far away from the shield tail as well as the front of the excavation face is minor. When the yawing is severe, the maximum horizontal ground displacements occur at the outer side in the highly effected regions for trajectories with small curvature radii and the less effected regions for trajectories with large curvature radii. Furthermore, the increments of the ground horizontal and vertical displacements increase with the yaw severity in the highly effected regions. The circumferential compressive stress, axial tensile stress and axial compressive stress when advancing along curved trajectories are larger than when advancing along a straight-line trajectory. The stress increments increase linearly with the yaw severity. In addition, the stress increments for trajectories with large curvature radii are in agreement with those for trajectories with small curvature radii. Considering that greater ground horizontal displacement and segmental stress can easily occur from yaw excavation loadings, it is necessary to keep strict control of the extent of yaw excavation during the 11

Computers and Geotechnics 118 (2020) 103325

M. Zhang, et al. [25] Yu H, Cai C, Bobet A, Zhao X, Yuan Y. Analytical solution for longitudinal bending stiffness of shield tunnels. Tunn Undergr Space Technol 2019;83:27–34. [26] Ninić J, Freitag S, Meschke G. A hybrid finite element and surrogate modelling approach for simulation and monitoring supported TBM steering. Tunn Undergr Space Technol 2017;63:12–28. [27] Li PF, Wang F, Zhang CP, Li Z. Face stability analysis of a shallow tunnel in the saturated and multilayered soils in short-term condition. Comp Geotech 2019;107:25–35. [28] Kavvadas M, Litsas D, Vazaios I, et al. Development of a 3D finite element model for shield EPB tunnelling. Tunn Undergr Space Technol 2017;65:22–34. [29] Kasper T, Meschke G. On the influence of face pressure, grouting pressure and TBM design in soft ground tunnelling. Tunn Undergr Space Technol 2006;21(2):160–71. [30] Sigl O, Atzl G. Design of bored tunnel linings for Singapore MRT North East Line C706. Tunn Undergr Space Technol 1999;14(4):481–90. [31] Alsahly A, Stascheit J, Meschke G. Advanced finite element modeling of excavation and advancement processes in mechanized tunneling. Adv Eng Softw 2016;100:198–214. [32] Shen X, Yuan DJ, Jin DL. Influence of shield attitude change on shield–soil interaction. Appl Sci 2019;9(9):1812. [33] Wang HN, Wu L, Jiang MJ, Song F. Analytical stress and displacement due to twin tunneling in an elastic semi-infinite ground subjected to surcharge loads. Int J Numer Anal Meth Geomech 2018;42(6):809–28.

[34] Mollon G, Dias D, Soubra AH. Probabilistic analyses of tunneling-induced ground movements. Acta Geotech 2013;8(2):181–99. [35] Do NA, Dias D, Oreste P, et al. Three-dimensional numerical simulation for mechanized tunnelling in soft ground: the influence of the joint pattern. Acta Geotech 2014;9(4):673–94. [36] Kasper T, Meschke G. A 3D finite element simulation model for TBM tunnelling in soft ground. Int J Numer Anal Meth Geomech 2004;28(14):1441–60. [37] Chen J, Sugimoto M. Analysis on shield operational parameters to steer articulated shield. Japanese Geotech Soc Spec Publ 2016;2(42):1497–500. [38] Lo KY, Rowe RK. Predicting settlement due to tunnelling in clays. Proc tunnelling in soil and rock, geotech III conference. Reston, Va: ASCE; 1984. p. 46–76. [39] Chakeri H, Ozcelik Y, Unver B. Effects of important factors on surface settlement prediction for metro tunnel excavated by EPB. Tunn Undergr Space Technol 2013;36:14–23. [40] Alonso EE, Josa A, Ledesma A. Negative skin friction on piles: a simplified analysis and prediction procedure. Géotechnique 1984;34(3):341–57. [41] Zhang QQ, Li SC, Li LP, Chen YJ. Simplified method for settlement prediction of pile groups considering skin friction softening and end resistance hardening. Chinese J Rock Mech Eng 2013;32(3):615–24. [42] Chi SY, Chern JC, Lin CC. Optimized back-analysis for tunneling-induced ground movement using equivalent ground loss model. Tunn Undergr Space Technol 2001;16(3):159–65.

12