Experimental study on face instability of shield tunnel in sand

Tunnelling and Underground Space Technology 33 (2013) 12–21

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Experimental study on face instability of shield tunnel in sand Ren-peng Chen a, Jun Li b, Ling-gang Kong a,⇑, Lv-jun Tang a a

Key Laboratory of Soft Soils and Geoenvironmental Engineering of Ministry of Education, Department of Civil Engineering, Zhejiang University, 388 Yuhangtang Road, Hangzhou 310058, China b The Third Railway Survey Design Institute Group Corporation, Tianjin 300251, China

a r t i c l e

i n f o

Article history: Received 28 July 2011 Received in revised form 30 July 2012 Accepted 10 August 2012 Available online 16 November 2012 Keywords: Tunnel Face stability Arching effect Model test Numerical simulation

a b s t r a c t Face stability is critical for ground settlement and construction safety control in shield tunneling. In this paper, a series of 3D large-scale model tests with a tunnel of 1 m diameter were conducted in dry sand for various cover-to-diameter ratios C/D = 0.5, 1, and 2 (i.e., relative depth; C is the cover depth and D is the diameter of tunnel). Each test provided a measurement of the support pressure and the ground settlement with the advance of face displacement. The evolution of soil arching during face failure was investigated by monitoring the redistribution of earth pressure in front of the face in the test case of C/D = 2. In the displacement-controlled face failure tests in the medium density sands, the support pressure dropped steeply to the minimum value, then increased to a steady state with the continuing increase in the face displacement. Relationships between the support pressure and face displacement for various cover depths were also verified by the numerical analysis using the finite difference program, FLAC3D (Itasca, 2005). The limit support pressure increases with the increase of the relative depth C/D and then tends to be constant. A significant rotation of principal stress axes in the upward arches in the soil during face failure was found in the tests. A two-stage failure pattern is proposed based on the observation of earth pressure. The theoretical and empirical formulas for estimating limit support pressure were verified by the tests results. Ó 2012 Elsevier Ltd. All rights reserved.

1. Introduction The need for mechanized excavation of tunnels in cities has continuously increased in recent years, especially as a result of the number of tunnels being constructed for subways (Guglielmetti et al., 2008). Tunnels with low covers are often headed, using the advanced shield technique (slurry or EPB shield). Nevertheless, face collapse during the construction of shallow tunnels still occurs. In extreme cases, the collapse propagates up to the ground surface creating a significant surface subsidence. Hence, failure mechanisms and the estimation of the limit support pressures for shields continue to be an important research topic. In a predominately sandy soil, drained stability conditions should be considered. A number of authors have described failure mechanisms at the tunnel face, and have derived formulae to calculate the appropriate face support pressure based on the limit equilibrium method and the limit analysis method (e.g. Davis et al., 1980; Leca and Dormieux, 1990; Lee and Nam, 2001; Li et al., 2009; Mollon et al., 2009, 2010). Horn (1961) was among the first to present a sliding wedge mechanism for the given problem. The limit equilibrium method proposed by Horn (1961) has ⇑ Corresponding author. Tel.: +86 571 88208772; fax: +86 571 88208793. E-mail addresses: [email protected] (R.-p. Chen), [email protected] (J. Li), [email protected] (L.-g. Kong), [email protected] (L.-j. Tang). 0886-7798/$ - see front matter Ó 2012 Elsevier Ltd. All rights reserved. http://dx.doi.org/10.1016/j.tust.2012.08.001

been used extensively and extended in many subsequent works (e.g. Jancsecz and Steiner, 1994; Anagnostou and Kovari, 1994, 1996; Broere, 1998; Kirsch and Kolymbas, 2005; Anagnostou, 2012). Numerical methods were frequently used to investigate stability of the tunnel face (Vermeer et al., 2002; Li et al., 2009; Kim and Tonon, 2010, 2011; Anagnostou et al., 2011; Melis-Maynar and Medina-Rodríguez, 2005; Zhang et al., 2011; Chen et al., 2011a). As much attention should be paid to the selection of the proper constitutive model and parameters of the materials, numerical analyses are generally difficult to be adopted for routine design. The validity of numerical analyses should also be checked, either by in situ measurements or by laboratory model tests. Physical modeling has played an important role in the study of tunnel excavation. Several centrifuge model tests have been performed to investigate tunnel face stability (Atkinson et al., 1977; Mair, 1979; Kimura and Mair, 1981; Chambon and Corté, 1994; Al Hallak et al., 2000; Kamata and Mashimo, 2003; Plekkenpol et al., 2006; Meguid et al., 2008; Idinger et al., 2011). Most of the previous centrifuge tests were mainly focused on face support pressure and ground settlement. A few small-scale models at single gravity (i.e., 1 g) have also been conducted to provide insight into the failure mechanism at the tunnel face during support pressure reduction (Sterpi and Cividini, 2004; Kirsch, 2010; Messerli et al., 2010). Kirsch (2010) investigated the failure mechanisms

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2. Materials and methods 2.1. Test setup and instrument Three tests with various relative depths C/D = 0.5, 1, and 2 were conducted. As shown in Fig. 1, a 3D model of a tunnel heading rested firmly against a concrete block, and both of them were preinstalled in a model chamber with inner dimensions of 4 m  5 m  6 m (length  width  depth). The tunnel heading consisted of a 1 m long cylindrical steel shell with an inner diameter of 1 m and a wall thickness of 8 mm. The tunnel face was supported by a hard glass plate fixed on the steel frame with eight wheels. The rear of the frame was attached to a load cell and moved backwards by a screw jack. The displacement of support plate was recorded by two LVDTs. The gap between the plate and the shell was sealed with rubber ring and PVC membrane. The friction was reduced by applying Vaseline to the sealing ring and the wheels. The friction was about 3–5% of the support force from the calibration before the tests. Five earth pressure cells (EPCs) were installed on the surface of the plate to monitor the change of the earth pressure during

LVDT Ground surface

Sand

C Shield LVDT

Plate

6m

and the evolution of support pressure in dense and loose sands by two series of small-scale model tests with a 10 cm diameter tunnel. It was found that the overburden has a negligible influence on the extent and evolution of the failure zone. Different regularities of relationships of support pressure to face displacement in dense and loose sands were recorded. In reality, the construction of a tunnel induces a redistribution of stress associated with soil arching (Jancsecz and Steiner, 1994; Anagnostou and Kovari, 1996; Broere, 2001; Lee et al., 2006; Anagnostou, 2012). For most mechanical models, the support pressure at collapse can be accurately predicted only if the effects of soil arching are included. Broere (2001) investigated the influence of the relaxation length of the arching column on the vertical load acting on the top of the failure wedge, as the vertical load was thought to be a key parameter for the calculation of minimum support pressure. Lee et al. (2006) conducted a series of numerical simulations to evaluate the tunnel stability and arching effects that develop during tunneling in soft clayey soil. The boundaries of the arching zones were proposed based on the variations of arching ratio, which are expressed as the change of the vertical stress divided by the original vertical stress. Anagnostou (2012) analyzed the contribution of horizontal arching to the tunnel face stability. It was found that the horizontal arching has significantly influence on the limit support pressure. As there have been limited studies (especially the experimental study) on the effects of soil arching, a satisfactory understanding of the failure mechanism, together with a reliable way for calculating the support pressure, is still limited. To understand the limit support pressure, failure mechanism and the evolution of soil arching with increasing face displacement, a series of 3D large-scale model tests with a tunnel of 1 m diameter were conducted in dry sand for various cover-to-diameter ratios C/ D = 0.5, 1, and 2 (i.e., relative depth; C is the cover depth and D is the diameter of tunnel). The relationships between the support pressure and face displacement for various cover depths were first discussed, which were also verified by the numerical analysis using the finite difference program, FLAC3D (Itasca, 2005). Then, the development of the surface settlement during tunnel face failure was investigated. Next, for the cover depth C/D = 2, variations of the earth pressure in front of the tunnel face during tunnel face failure were emphatically analyzed to evaluate the soil arching effect. Finally, a two-stage failure pattern was proposed in this study. The theoretical and empirical formulae for estimating limit support pressure were verified using the tests results.

Load Cell

1m Screw Jack

1m

Base

3m 1m

1m

5m

4m Fig. 1. Setup of tunnel model.

experiments. Forty-eight earth pressure cells were placed in front of the tunnel face to measure the evolution of the soil stresses in the test of C/D = 2.0 (see Fig. 2). During testing, both vertical and horizontal earth pressures were recorded. It should be noted that all the earth pressure cells were carefully checked via soil calibration for properly registering the stress during the experiment. Moreover, the rotation of the EPC was strictly controlled during placement. The thin colored sand grids were placed on the soil surface. The settlements of the grids were recorded by thirteen LVDTs which were fixed to a stationary cruciform beam. For each LVDT, a contact plate was equipped at the tip of the probe. The detailed arrangements of earth pressure cells and the LVDTs are depicted in Fig. 2. 2.2. Soil preparation All tests in this study were performed with medium grained sand collected from the Yangtze River. A total of 100 tons of sand was oven-dried in a large rotary dryer, with the water content controlled between 0.3% and 0.4% after drying. The grain size distribution of the sand is shown in Fig. 3. The average grain size is 0.33 mm, the maximum void ratio is 0.70, and the minimum is 0.40. The ground was formed using the sand raining method. Two sand containers were adopted during the sand raining. The size of the large one is 2 m  1.5 m  0.5 m (length  width  depth), and the size of the small one is 0.5 m  0.5 m  0.5 m (length  width  depth). In the process of the sand raining, the large sand container was mainly used, while the small sand container was merely adopted around the positions of the pre-existing hard objects (i.e., base and ‘‘TBM’’) to minimize the shadowing effect (i.e., lower sand density areas around the pre-existing hard objects). It should be noted that the size of the sieve pore, and the distance between adjacent sieve pores of two containers are identical. If the falling height for two containers during soil raining is identical, the same sand density can be obtained. Fig. 4 shows the relationship between the falling height and the relative density of the sand, which was obtained from a series of calibration tests. From the figure, the density of the sand is a unique function of falling height as long as other variables remain the same. The desired relative density of the sand used in the model tests is Dr = 0.55 with a dry density of 16.5 kN/m3, which means that the sand is medium dense. Hence, the height between the bottom of the sand container, and the surface of each soil layer should be controlled to 1.8 m based on Fig. 4. The soil in the model chamber was prepared in 25 cm thick layers by pouring the sand from the sand container with perforated bottom. The procedure was presented as follows.

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R.-p. Chen et al. / Tunnelling and Underground Space Technology 33 (2013) 12–21

B

Relative Density Dr

Earth pressure cells 2.8

-2 -1

0

1

2

3 y/D

2.4

0.6

0.8

1.0

2.2 2.0 1.8 1.6 1.4 1.2 1.0

B

0.4

Maximum Density

Height of fall of sand /m

0

A -1

A

0.2

Tests Average

2.6

1

2

x horizontal y horizontal z vertical

0.0

Minimum Density

x/D

0.8

(a) Horizontal section

1400

1500

1600

1700

Dry Density rd

0.15D

1800

1900

/kg/m3

Fig. 4. Relationship between dry density of sand and fall height.

2.3. Tests procedure

3

2

1 0

(1) The sand with the desired weight was placed into the sand container. (2) The sand container was located in its planned position and on the height of 1.8 m above the soil surface. (3) The door of container with a link switch was opened to allow the sand fall into the chamber. To evaluate the density uniformity of the ground, a series of cone penetration tests (CPTs) were conducted after the sand raining. Fig. 5 shows the results of CPT for the three different positions (i.e., No. 1, No. 2 and No. 3) near the corner of the ground. As shown in Fig. 5, the mean value of the total cone resistance generally increases linearly with the increase of the depth. That is to say the density of the ground is uniform in general, although slight density variation within some layer indeed exists which may be due to the relatively large increment in sand raining.

1

0

LVDT

4

C B A B C -1

0

1

3 y/D

2

z/D

(b) Longitudinal section A-A 0.25D

3

2

LVDT

A

D E

-1

0

1

4

E D

z/D

-2

2

x/D

(c) Transverse section B-B Fig. 2. Setup of LVDTs and locations of earth pressure cells for C/D = 2.

The tunnel was first installed in the model chamber. Then the soil was poured layer by layer in the chamber. During the soil preparation, the earth pressure cells were installed on the designed positions. After the soil preparation, the LVDTs were installed on the soil surface. The retraction of the support plate was displacement-controlled. For the first 3.0 mm, the backward movement of plate was controlled in 0.1 mm/h. Thus, the load reduction at small backward displacements could be captured sufficiently. After that, the backward velocity increased gradually until a total displacement of 60 mm. It should be mentioned that the main reason

0.0

Mean value NO.2

0.5

80

NO.1 NO.3

1.0

60

Depth /m

Percentage Finer /%

100

d50=0.33mm

40

1.5 2.0

20

2.5 tunnel face

0 0.01

0.1

1

Grain Diametter /mm Fig. 3. Particle size distribution curve of sand.

10

3.0

0

10

20

30

40

50

60

70

80

Total cone resistance /MPa Fig. 5. Total cone resistance with depth.

90

100

R.-p. Chen et al. / Tunnelling and Underground Space Technology 33 (2013) 12–21

15

2.4. Numerical simulations of model tests

C=0.5, 1, 2m

D=1m

1m

3m

2. 5 m 1m

Fig. 6. Computational domain.

Fig. 7. Sketch of the layered sandy ground.

(1) Firstly, laboratory triaxial tests for the dry sand with the same relative density Dr = 0.55 and the different confining pressure r3 = 100 kPa, 200 kPa were conducted. It can be seen from Fig. 8, the E50 is 44.9 MPa and 66.4 MPa when r3 = 100 kPa, 200 kPa, respectively. (2) Secondly, Suppose E50 obeys Janbu’s (1963) formula, which means E50 = KPa(r3/Pa)n, where K and n are two undetermined parameters, Pa is standard atmospheric pressure. The values of K and n can be obtained by solving the equations based on the results of the tests (K = 449, n=0.56). (3) Lastly, the horizontal earth pressure r3 in the middle of each layer was calculated by assuming the earth pressure ratio at rest K0 = 1 sin u = 0.4. According to Janbu’s (1963) formula, as K, n and r3 were obtained, the E50 for each layer was determined (see Table 1).

900 σ3=100kPa

800 700

σ1 −σ3 (kPa)

σ3=200kPa

E50

600

1

500 400 300 200 100 0 0

3

6

9

12

Numerical simulations of face failure with C/D = 0.5, 1, and 2 were conducted with the 3D finite-difference program FLAC3D. The main purpose of the simulations was to verify the accuracy of the limit support pressure obtained by the experiment. Due to the symmetry of the tunnel, half of the circular tunnel was used to simulate the behaviors of the tunnel face during failure. Fig. 6 shows the computational domain of the numerical simulations. As shown in Fig. 6, the ground was represented by the ‘‘brick’’ mesh (i.e., brick-shaped mesh) and the ‘‘radcylinder’’ mesh (i.e., radially graded mesh around cylindrical-shaped tunnel) in FLAC3D. The tunnel lining was modeled with ‘‘liner’’ elements. The boundary conditions of the computational domain were as follows: the ground surface was free, the side surfaces had roller boundaries and the base was fixed. The soils were assumed to be elastic-perfectly plastic materials, conforming to the Mohr–Coulomb failure criterion. Although the MC model is not suitable for capturing shear band formation, which means the failure plane in front of the tunnel face may not be well revealed via this model, it is particularly suited to predict the limit support pressure of the tunnel face (Vermeer et al., 2002; Anagnostou et al., 2011).In the simulations, the stiffness of the lining was assumed to be infinite. The unit weight of the sand c is 16.5 kN/m3, the Poisson ratio m of the sand is 0.3, the cohesion of the sand is 0 kPa. The friction angle / and dilantancy friction angle w of the sand at the desired relative density is 37° and 8.4°, respectively, which were obtained from the laboratory direct shear tests. The lateral stress ratio at rest K0 was equal to 0.4 (K0 = 1 sin / where / = 37°). It should be noted that secant modulus E50 was adopted in the simulations. As the secant modulus is stress dependent, the ground was divided into layers of equal thickness of 0.5 m (see Fig. 7). The procedure of determining the secant modulus E50 for each layer was presented as follows.

15

ε1 (%) Fig. 8. Relationships between deviatoric stress and axial strain of the triaxial test.

why the displacement-controlled boundary was adopted is that it can capture strain softening effects of the material (e.g., the medium dense sand for this experiment) in the process of the tunnel face failure, which is not possible by using the stress-controlled boundary (Kirsch, 2010; Messerli et al., 2010). Besides that, in the process of experiment for relative depth C/D = 0.5, slightly rotation of the support plate was found due to insufficient stiffness of the loading system (i.e., the glass plate and the steel frame). Thus, before the experiments for C/D = 1.0 and C/D = 2.0 were to be preformed, the stiffness of the loading system was improved to avoid the rotation of the support plate.

The procedure of the calculations was presented as follows. Firstly, remove the volume elements inside the tunnel and activate the ‘‘liner’’ elements of the lining. At the same time, fix the velocity of all the grid points on the tunnel face in the direction of the tunnel axes. Secondly, perform an iteration process to make sure that the ground reaches an equilibrium state. Finally, retreat all the grid points on the tunnel face at the speed of 10 7 m/step until the support pressure decrease asymptotically to a stable value. 3. Results and discussions 3.1. Support pressure and face displacement Fig. 9 shows the curves of normalized face support pressure P/Po versus normalized face displacement d/D from model tests for

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R.-p. Chen et al. / Tunnelling and Underground Space Technology 33 (2013) 12–21

Table 1 Secant modulus of the sand for each layer in the simulation. The number of Layers

1

2

3

4

5

6

7

8

Secant modulus E50 (MPa)

4.51

8.34

11.1

13.41

15.43

17.27

18.96

20.54

1.0

1.0

C/D=0.5 C/D=1 C/D=2

0.9

P/Po

0.8

0.4

0.6

0.2

Scr0.5 (Pr0.5 )

0.0 0.0

0.5

1.0

0.3

3.2. Ground settlement

TUNNEL

D

1.5

Scr1 (Pr1 )

0.4

2.0

2.5

3.0

Scr2 (Pr2 )

Pu0.5

0.2

Pu1

0.1 0.0

C

0.6

0.7

0.5

GROUND

0.8

Pu2 0

5

10

15

20

ent cohesion in the wet sand results in the less limit support pressure compared to that in the dry sand. In Kirsch’s tests, the values of Pu/cD were found to be 0.06–0.07 for dense sands and 0.09–0.13 for loose sands with the range of C/D = 0.5–1.5. The model test results obtained from different researchers (Chambon and Corté, 1994; Kirsch, 2010) imply that limit support pressure does not vary with the cover depth when C/D P 1.0. The residual support pressure ratios Pr/cD in the current model tests are 0.125 for C/ D = 0.5, 0.134 for C/D = 1.0, and 0.142 for C/D = 2.0. The ratios coincide with Kirsch’s tests, in which the values of Pr/cD are from 0.08 to 0.18 for C/D = 0.25–2 in dense sand (Kirsch, 2010).

25

30

35

d/D×1000 Fig. 9. Normalized load–displacement curves of model tests.

three relative depths with C/D = 0.5, 1, and 2, where P is the load acting on the tunnel face, Po is the initial load on the tunnel face, and d is the face displacement. It should be noted that the loads in the experiment of C/D = 0.5 have been overestimated due to the friction between the wheel and the tunnel caused by the rotation of the plate (Chen et al., 2011b). In this paper, the measured load–displacement curve for C/D = 0.5 has been corrected according to the earth pressure measured by the earth pressure cell installed on the centre of the support plate. It can be seen from Fig. 9 that the face support pressure decreases steeply to the minimum value, Pu, called limit support pressure, at the face displacement of (0.2–0.26)%D, then increases to a steady value, Pr, called residual support pressure, at the face displacement of (1.2–1.8)%D. The ratio of Pu/Po for C/D = 2 is lower than those for C/D = 1 and C/D = 0.5. Fig. 10 shows comparison of normalized load–displacement curves of model tests and numerical simulations. As shown in Fig. 10, the numerical calculations almost accurately predict the load–displacement curves when the support pressure does not exceed the limit support pressure. After the limit support pressure is obtained, the numerical calculations in this paper cannot give the proper predictions. The concave shape of the load–displacement curves in the model tests was also observed by Kirsch (2010) from small-scale model tests (see Fig. 11). As pointed out by Kirsch (2010), the peak strength of dense sand is mobilized at the limit support pressure, after this minimum, the support pressure rises again due to the dilatancy of the sand. Table 2 summarizes Pu/cD, Pr/cD and the corresponding face displacements. As seen in Table 2, no further significant variation in Pu/cD for C/D = 1 and 2 other than C/D = 0.5 is found. The limit support pressure ratios Pu/cD tend to reach a steady state as C/ D P 1. Chambon and Corté (1994) conducted centrifuge tests to investigate tunnel face stability in sand by stress-controlled method, and found that values of Pu/cD = 0.04–0.05 for C/D = 0.5–4. It is noted that in the centrifuge tests by Chambon and Corté (1994), the sand was not fully dry and contained a little water. The appar-

Ground settlements were measured by the thirteen LVDTs placed on the ground surface (see Fig. 2). The relationships between the face displacement and the maximum ground settlement are shown in Fig. 12. At the beginning of the face displacement, the ground settlements were not sensitive to the face displacements, and no significant ground settlement was measured before Pu. The ground settlements significantly increase as they exceed a settlement, named as critical settlement, Scr. The face displacements related to Scr vary from 9.5 mm to 36.8 mm as C/D changes from 0.5 to 2. It was observed that the failure zone on the ground surface was an area of 1D width and 0.75D length in front of the tunnel face and the middle point of the subsidence area laid in the range of (0.25–0.3)D in front of the tunnel face. This observation implies that the increase of C/D does not significantly influence in the size of the final subsidence area. 3.3. Earth pressure and soil arching To investigate soil arching effect, 48 earth pressure cells were embedded above the tunnel crown in the test of C/D = 2.0 (see Fig. 2). Figs. 13 and 14 show the vertical and horizontal soil stresses along depth, respectively. Fig. 15 shows the earth pressure coefficient, K, along the depth, which is equal to the horizontal stress divided by the corresponding vertical stress. It should be mentioned that the horizontal earth pressure cells were located at 0.25D from the line A (see Fig. 2). Thus, Fig. 14 gives approximate estimations of horizontal earth pressure rh along the depth on the center-line. The black lines without marks (d = 0 mm) in Figs. 13 and 14 are theoretical distributions of the vertical and horizontal stresses along depth due to gravity. The coefficient of earth pressure at rest is taken as Ko = 1 sin/. As shown in Fig. 13, with the increase of the face displacement, the vertical stresses rv decrease from the bottom up in order, finally tends to be steady. As P = Pu, the notable decrease of the vertical stresses is within the depth of 1 m above the tunnel crown. As d = 27.64 mm in which the face displacement equals to the critical face displacement, the failure zone extends to the ground surface and the vertical stresses along depth are nearly constant. In Fig. 14, the horizontal stresses rh in x- and y-directions above the crown increase first, and then decrease steeply with the face displacement. Similar to rv, the horizontal stresses in both directions almost keep constant along depth. Similarly, the horizontal coefficients of earth pressure K (=rh/rv), which is shown in Fig. 15, increase first, and then decrease with the face displacement increasing. Larger steady values of rh and K in x-direction are observed in Figs. 14 and 15. Figs. 13–15 show that the development of the soil arching is related to the depth offset from the tunnel crown and the face displacement. The soil arch crown and failure zone move upward with the increase of the face displacement. The reason for different final K values may be attributed to that the development of the soil arching depends on the geometry of subsidence area (inferred from surface subsidence area).

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R.-p. Chen et al. / Tunnelling and Underground Space Technology 33 (2013) 12–21

10

1.0

GROUND

Test (C/D=0.5) FDM (C/D=0.5)

C

8

Parameters of Kirsch's tests Dry Sand: Relative density Dr= 0.83

0.9 0.8

Mean grain size d 50 = 0.58mm

TUNNEL

D

Critical state friction angle ϕc = 32.5°

0.7

Tunnel: Diameter D = 0.1m

0.6

P/Po

P kPa

6

4

C/D=0.5 (Kirsch 2010) C/D=1.0 (Kirsch 2010) C/D=0.5 (Experiment in this paper) C/D=1.0 (Experiment in this paper)

0.5 0.4

2

0.3 0.2

0

0

2

4

6

8

0.1

10

d/D×1000

0.0

(a) C/D=0.5

P kPa

14 C

10

D

10

15

20

25

30

35

40

Fig. 11. Comparisons of the normalized load–displacement curves between Kirsch’s tests (2010) and our model tests in the relatively dense sandy ground.

Test (C/D=1) FDM (C/D=1) TUNNEL

Table 2 Pu and Pr for various C/D by model tests.

8 6 4

C/D

Pu/cD

Pr/cD

d at Pu (mm)

d at Scr (mm)

0.5 1 2

0.065 0.076 0.072

0.125 0.134 0.142

1.99 2.08 2.08

9.46 20.33 36.77

Note: C = cover depth, D = tunnel diameter, c = unit weight, d = face displacement, Pu = limit support pressure, Pr = residual support pressure, Scr = critical settlement of soil surface.

2 0

5

d/D×1000

GROUND

12

0

0

2

4

6

8

10

d/D×1000

(b) C/D=1 18

80

Ground settlement S/mm

GROUND

16 C

Test (C/D=2) FDM (C/D=2)

14 D

TUNNEL

P kPa

12 10 8 6

50

30

0 4

6

8

10

d/D×1000

(c) C/D=2 Fig. 10. Comparison of normalized load–displacement curves of model tests and numerical simulation.

To further investigate the stress redistribution due to soil arching, two stress concentration ratios are defined. A vertical stress concentration ratio, kv, is defined as kv = rv/rvo, in which rvo is initial vertical earth pressure; while a horizontal stress concentration ratio, kh, is kh = rh/rho, in which rho is initial horizontal earth pressure. The soil zone with kv < 1 and kh < 1 is regarded as failure zone. The region of arch zone can be divided into the arch crown (kv < 1 and kh > 1) and the arch foot (kv > 1 and kh < 1). Larger kh are obtained in the arch crown as a result of the mobilized shear stresses; while the arch foot receiving loads transferred from arch crown have larger vertical stress concentration ratio (kv > 1). On the other hand, as a result of the release of stress in the failure zone, kv < 1

S cr0.5

20

2

2

C/D=0.5 C/D=1 C/D=2

40

10

0

TUNNEL

60

4

0

GROUND

s

70

0

10

S cr1

20

S cr2

30

40

50

60

Face displacement d mm Fig. 12. Relationship between face displacement and maximum subsidence on ground surface.

and kh < 1 can be found in the arch crown and foot, respectively. Arch crown and failure zone propagate upward with the increase of the face displacement. For P = Pu (d = 2.08 mm), two approximate boundaries, the solid line and dotted line (see Figs. 16 and 17), can be determined by the stress concentration ratios (kv and kh). The outer boundary of the arch zone is 1.5 D above tunnel crown. 4. Face failure pattern and limit support pressure According to the measurements of earth pressures and ground subsidence, failure patterns of tunnel face are illustrated in Fig. 18. A local failure zone first forms in front of the tunnel face, in which the soil becomes yielding and the vertical and horizontal

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R.-p. Chen et al. / Tunnelling and Underground Space Technology 33 (2013) 12–21

0.0 0.0

Ka K0 d=42.58mm (Pr2)

d=27.64mm (Scr2 )

0.5

d=27.64mm (Scr2 )

d= 3.98mm

Depth /m

Depth /m

0.5

1.0 d= 2.08mm (Pu2 ) d=0.91mm 1.5

d=0mm

d=21.83mm 1.0

d=8.96mm 1.5

d=2.08mm (Pu2) 2.0

0

5

10

15

20

25

30

35

40

2.0 0.0

Vertical stress /kPa

0.5

1.5

2.0

2.5

Kx

Fig. 13. Vertical stress (rv) along depth (C/D = 2, Line A).

(a) K in x direction

0.0

0.0

Ka K0 d=35.77mm

d=59.18mm (Pr2)

d =27.64mm(Scr2 )

0.5

d=27.64mm (Scr2 )

0.5

Depth /m

Depth /m

1.0

1.0 d=2.08mm(Pu2)

1.5

1.0

d=8.96mm 1.5

d=2.08mm (Pu2)

d=0mm

2.0

0

2

4

6

8

10

12

14

16

18

Horizontal stress of x direction /kPa

2.0 0.0

0.5

1.0

1.5

2.0

2.5

Ky

(a) σ h in x direction

(b) K in y direction

0.0

Fig. 15. Earth pressure coefficient (K) along depth (C/D = 2, Line A).

d=42.58mm

Depth /m

0.5

d= 27.64mm (Scr2)

1.0 d= 2.08mm (Pu2)

1.5 d=0mm

2.0

0

2

4

6

8

10

12

14

16

18

Horizontal stress of y direction /kPa

(b) σ h in y direction Fig. 14. Horizontal stress (rh) along depth (C/D = 2, Line A).

soil stresses around the tunnel face decrease (kv < 1 and kh < 1). The soil failure induces stress redistribution in the soil around the tunnel crown, that is, soil arching occurs above the failure area. Larger horizontal stress (kh > 1) and vertical stress (kv > 1) occur in arch crown and foot, respectively, and the principal stress axes rotate in the arch. The arch continues to sustain more overburden pressure until the minimum face load (i.e. limit support pressure Pu) is reached. Then a collapse, referred to here as a local collapse, occurs with the formation of a through shear surface (see Fig. 18a). Fig. 18a presents the comparison of the local failure zone given by the centrifuge model tests (Chambon and Corté, 1994) with that

obtained by our model tests. As shown in Fig. 18a, the upper boundary of the local failure zone revealed by the centrifuge model tests (Chambon and Corté, 1994) is higher than that proposed by our model tests. The reason is that the local failure zone proposed by the centrifuge model tests (Chambon and Corté, 1994) was determined slightly after the limit support pressure was reached. Moreover, there is no obvious settlement on ground surface at this moment. After the local collapse, with the increase of the face displacement, the migration of underlying soil into the tunnel face leads to the development of failure surfaces. Meanwhile, the arch zone propagates towards ground surface and cause the surface settlement. As a result, the settlements increase steeply as the settlements exceed Scr, which denotes a global collapse occurs (see Fig. 18b). In conclusion, face failure occurs accompanied by the evolution of arch in two stages: local collapse and global collapse. The results of the limit support pressures obtained from the current model tests were compared with the predicted results of the wedge model by Anagnostou and Kovari (1994), the modified wedge model considering the horizontal arching by Anagnostou (2012), the upper bound solution by Leca and Dormieux (1990) and Mollon et al. (2010), the empirical approach by Vermeer et al. (2002), and the finite difference method (FDM). The dimensionless factor Pu/cD is calculated for the following geometry and material parameters of the model tests, i.e., tunnel diameter D = 1 m; self-weight c = 16.5 kN/m3; cohesion c = 0 or c = 0.5 kPa (which was only adopted in the upper bound solutions as the sand was actually not fully dry); and, friction angle / = 37°. The predicted results are shown in Fig. 19. In Fig. 19, when C/D is less than

19

R.-p. Chen et al. / Tunnelling and Underground Space Technology 33 (2013) 12–21

Distance from tunnel face y/D

Distance from center line x/D -0.5 0.0

0.0

0.5

1.5

2.0 0.0

D

A 0.5

1.0

-1.0

-0.5

E

0.0

B

C

0.99

0.5 1.0

0.89

1.01

1.01

1.02

1.01

1.03

1.01

0.69

Failure 2.0 0.10

2.5

1.04

Arch foot

3.0

3.5

1.5

2.0 0.0

A 0.5

1.11

1.0

1.21

1.5

1.06

D

0.69

0.90 1.02

1.02

Tunnel face

λv=σv/σv0 d=0mm

(a) λ v in section B-B

-1.0

-0.5

C

E

0.0

B

0.92

0.91

0.86

0.93

0.79

1.00

Depth /m

Depth /m

3.5

1.5

C

1.09 Arch crown

Arch foot

1.14

1.04 0.90 Arch foot

0.99

0.77

1.08

0.72 0.88

0.87

0.23 Failure

1.5 2.0

3.0

d=2.08mm(Pu2 )

3.5

(b) λ h in section A-A

1.0, the experimental results show that Pu/cD increases with the increase of C/D, similar results are also obtained by Anagnostou and Kovari (1994). When C/D is more than 1.0, the results of the model tests show that C/D has almost no effect on Pu/cD, which is also confirmed by the results of Anagnostou and Kovari (1994), Anagnostou (2012), Leca and Dormieux (1990), Vermeer et al. (2002) and Mollon et al. (2010). Moreover, it can be seen that Anagnostou and Kovari’s (1994) model and Vermeer et al. ’s (2002) model are relatively safe methods to estimate the limit support pressure. We can also see that the modified wedge model considering the horizontal arching (Anagnostou, 2012) predicts the limit support pressure much more closer to the experimental results than the traditional wedge model (Anagnostou and Kovari, 1994). The results obtained from FDM simulations show a good agreement with those of the experiments. Notice that comparing to the experiments results, the upper bound solutions proposed by Leca and Dormieux (1990) and Mollon et al. (2010) overestimate Pu/cD when the sand is thought to be fully dry (c = 0). While in theory, the upper bound solutions should give the minimum value of Pu/cD compared to the other methods. Actually, the conflict between the results obtained from the upper bound solutions and the experiments is due to the not fully dried sand (water content between 0.3% and 0.4%). That means the sand of the experiments may have a little cohesion. When c = 0.5 kPa is adopted in the upper bound solutions, the normalized limit support pressure Pu/cD is 0.05 (Mollon et al., 2010) and 0.042 (Leca and Dormieux, 1990), which is much smaller than the experimental result.

1.07

Tunnel face

λh=σh/σh0

Fig. 16. Distribution of stress concentration ratios in section A–A (C/D = 2).

1.00

2.5

0.80

d=0mm

1.0

B

1.0

2.0

3.0

0.5

A

0.5

Failure

2.5

d=2.08mm( Pu2)

Face displacement d

Arch crown

0.36

0.95 1.02

0.10 Failure

Displacement from tunnel face y/D

Distance from center line x/D 1.0

1.01

1.09

(a) λ v in section A-A

0.5

0.92 1.01

3.0

3.5

0.0

0.89

2.0

d=2.08mm(Pu2 )

d=0mm

1.00 1.01 Arch foot

1.5

2.5

λv=σv/σv0

-0.5 0.0

1.5

C

Arch crown

Depth /m

Depth /m

1.5

1.0

B

0.99

Arch crown 1.0

0.5

A

0.72 0.93

λh=σh/σh0

d =0mm Face displacement d

d =2.08mm(Pu2 )

(b) λ h in section B-B Fig. 17. Distribution of stress concentration ratios in section B–B (C/D = 2).

5. Conclusion A series of model tests with various C/D ratios have been conducted to investigate the failure behaviors of a tunnel face in sand. Three-dimensional finite difference method was also employed to simulate the model tests. Based on the results of face pressure, ground settlements, and earth pressures in soil, the following conclusions can be drawn: 1. The failure mechanism is described as the evolution of soil arching by monitoring the variation of the soil stresses in the zone above the tunnel crown. Tunnel face failure has two stages: local collapse and global collapse. The boundaries of the arch zone are proposed by examining the stress concentration ratios (kv and kh) of the soil element surrounding the tunnel. When the local collapse occurs, the arch height is 0.5–1.5 times of the tunnel diameter. 2. When the relative depth C/D < 1, the normalized limit support pressure Pu/cD increases with the increase of the relative depth C/D. When C/D P 1, the normalized limit support pressure Pu/ cD is almost not affected by C/D. 3. The approaches proposed by Anagnostou and Kovari (1994) and Vermeer et al. (2002) overestimate the limit support pressure compared to the results from the experiments. However, the modified wedge model considering the horizontal arching (Anagnostou, 2012) and the FDM simulations in this paper provide an accurate estimation of the limit support pressure. The

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R.-p. Chen et al. / Tunnelling and Underground Space Technology 33 (2013) 12–21

Undisturbed zone λ v =1 λ h=1

Arch crown λ v <1 λ h>1

C Arch foot

h=1.5D h1=1D λ v <1 λ h <1

P=Pu

Acknowledgments

λ v >1 λ h<1

D

Failure zone (Chambon & Corte,1994) Failure zone (This paper)

Face displacement

(a) Local collapse

0.75D

1D

λ v <1 λ h<1

λ v =1 λ h=1

λ v >1 λ h<1

C

Residue arch foot

P
D

Failure zone (This paper)

Face displacement

(b) Global collapse Fig. 18. Schematic diagram of failure mechanism.

0.20

a b1 b2 0.15 c1 c2 d f

0.10

Anagnostou & Korari 1994 (C=0) Model tests(This paper) Leca & Dormieux 1990 (C=0) FDM (This paper, C=0) Leca & Dormieux 1990 (C=0.5kPa) Mollon et al. 2010 (C=0) Mollon et al. 2010 (C=0.5kPa) Vermeer et al. 2002 (C=0) Anagnostou 2012 (C=0)

a

d

c1

f

b1

0.00 0.0

c2

b2

0.05

0.5

1.0

1.5

The authors wish to thank the grand science and technology special project of Zhejiang province, China (Research Grant: 2011C13043), and Program for New Century Excellent Talents in University (Research Grant: NCET-08-0491) for financial support. The authors greatly appreciate professor Charles Wang Wai Ng for his proper comments. The authors would also like to thank the reviewers for their helpful comments and suggestions, on which the manuscript can be improved. References

Undisturbed zone

Pu/ γD

4. Surface settlement is not obvious at the beginning of face collapse, but after global collapse occurs, the surface settlement increases dramatically. For deep tunnels, both pressure control and settlement measurement should be required to judge the stability of tunnel face.

2.0

2.5

3.0

3.5

4.0

C/D Fig. 19. Relation between the limit support pressure and the cover depth.

upper bound solutions proposed by Leca and Dormieux (1990) and Mollon et al. (2010) show reasonable limit support pressure when little cohesion (c = 0.5 kPa) is considered due to the not fully dried sand.

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