The effect of earth pressure on the failure mode of high-speed railway tunnel linings

Engineering Failure Analysis 110 (2020) 104398

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The effect of earth pressure on the failure mode of high-speed railway tunnel linings

T

Ben-Guo Hea,b,c, , Hong-Pu Lia, Zhi-Qiang Zhangb ⁎

a b c

Key Laboratory of Ministry of Education on Safe Mining of Deep Metal Mines, Northeastern University, Shenyang 110819, China Key Laboratory of Transportation Tunnel Engineering, Ministry of Education, Southwest Jiaotong University, Chengdu 610031, China Department of Geological and Environmental Sciences, Ben-Gurion University of the Negev, Beer Sheva 84105 Israel

ARTICLE INFO

ABSTRACT

Keywords: Tunnel lining Earth pressure Structural safety Cracking pattern Model test

The effect of earth pressure on the mechanical behaviour of horseshoe-shaped tunnel linings was explored using 1:30 scaled models. The linings considered are typically used to carry 300 km/h high-speed trains on double tracks. A self-weight stress field causes the tunnel lining’s roof settlement to dominate over the horizontal deformation of the walls. In contrast, a horizontal tectonic stress field causes the knees of the lining to bend outwards and the walls to bend inwards. These differences must be taken into account when formulating the design parameters for the lining. The cracking pattern observed in the tunnel lining coincides with the bending distribution, but does not reflect the deformation very well when subjected to its own self-weight stress field. The most favourable mechanical behaviour of the tunnel lining occurs when the coefficient of lateral pressure K has values ranging from 0.6 to 1.2. When K is less than 0.6, flexural failure occurs at the intrados of the lining’s roof and the extradoses of the shoulders. However, when K is greater than 1.2, the knees of the lining become the most vulnerable locations. A series of destructive tests were also performed which yielded a high correlation between the cracking pattern and the coefficient of lateral pressure. The experimental results were further supported by performing nonlinear finite-element numerical modeling.

1. Introduction Earth pressure is one of the most important issues affecting the performance of deep underground structures and significantly influences the likelihood of a tunnel structure experiencing failure. Typically, a tunnel lining must be designed to withstand earth pressure so that there is a sufficient safety margin [1]. As a rule of thumb, when the ground load level exceeds 1 MPa, the tunnel lining may not be able to survive the total external earth pressure it is subjected to [2]. Cracks can form at various positions around tunnel linings because the earth pressure acting on them can be complex. It is, therefore, imperative to study the effect of ground pressure on the mechanical behaviour of the tunnel lining. How to interpret and identify the possible causes of cracks formed in linings due to earth pressure is an essential concern. With this knowledge, engineers can implement countermeasures at vulnerable positions to effectively prevent the failure of the lining. Naturally, cracking events are most often observed in tunnels built in regions where the earth pressure is high and constitute a significant threat to the structural soundness of the tunnel during operation. In the past few decades, campaigns have been undertaken to investigate the concentration of stress on the rock masses around openings, rather than the evolution of the failure process.

⁎

Corresponding author. E-mail address: [email protected] (B.-G. He).

https://doi.org/10.1016/j.engfailanal.2020.104398 Received 28 June 2019; Received in revised form 6 January 2020; Accepted 13 January 2020 Available online 21 January 2020 1350-6307/ © 2020 Elsevier Ltd. All rights reserved.

Engineering Failure Analysis 110 (2020) 104398

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These studies have greatly improved the understanding of the stability of tunnel linings. However, the effect of lateral pressure coefficient (i.e. the ratio of the horizontal to vertical earth pressure components, K) on the failure mode occurring in horseshoeshaped tunnel linings has not been considered in any great detail. It is becoming standard engineering practice to use field-monitoring technology to monitor, on a daily basis, the internal forces acting on a tunnel lining starting from the initial construction stage and extending into the long-term [3]. Such steps have helped maintain the safety of tunnel linings [4]. Nevertheless, the prohibitively high cost of field monitoring means that a limited number of earth pressure measurements are made in any specific engineering project; therefore, detailed failure modes are generally not obtained for the tunnel structures, especially during operational service [5]. Feng et al. investigated the causes of sidewall cracking in tunnels by means of field observations [6]. However, the effect of earth pressure on tunnel failure is rarely investigated. Typically, the external load has a dominant role in determining the damage (cracking) incurred by the reinforced concrete lining of a tunnel [7]. Using numerical models to estimate the mechanical behaviour of tunnel linings is a technique that is attracting widespread interest [8] and is useful for optimizing structural design parameters. For example, finite element (FE) methods have been utilized to aid in comprehending the elastic behaviour of tunnel linings induced by excavation [9]. Unfortunately, elastic theory does not reflect the cracking process very well, as this type of deformation is necessarily discontinuous. Using nonlinear techniques to analyse circular tunnel linings has also been attempted by many design practitioners [10] to take into account variables such as excavation size, overburden, and lining thickness [11]. However, the cross-sections of railway tunnels are rarely truly circular in form, so this approach cannot clarify the failure patterns occurring in the more commonly encountered horseshoe-shaped tunnel linings. The aforementioned efforts have considerably improved many aspects of our understanding of tunnel linings. However, the effect of the lateral pressure coefficient (K) on the mechanical behaviour of linings has still not been satisfactorily considered, especially in the context of tectonic stress fields in deep underground tunnels. In practice, different K values can lead to a variety of failure modes occurring in tunnel linings and very different cracking patterns appearing around the periphery of the tunnel of interest. Designers need to take these differences into account when formulating their lining parameters. However, to date, this issue has not been studied thoroughly enough, and further attention is needed. In general, examining how strongly earth pressure affects the structural stability of a tunnel lining can provide valuable insight into this challenging issue. The purpose of this paper is to clarify the effect of earth pressure on the mechanical behaviour and cracking patterns of standard horseshoe-shaped tunnel linings. In particular, we concentrate on the coefficient of lateral pressure, an issue that, quite surprisingly, has not been studied in-depth. It is expected that our quantitative evaluations will facilitate the optimization of the structural parameters in specific in situ stress conditions when the tunnel’s structure is planned. 1:30 scaled models were handled to procure the deformation profiles, internal forces stored, and cracking patterns produced in the lining when the structure is subjected to a wide range of lateral ground pressure coefficients. The results were further validated via nonlinear numerical modelling. The experimental results allow us to predict what failure mode will occur in the lining when it encounters a specific K value. As a result, specific countermeasures can be recommended for local sites in the lining that are vulnerable to failure, which should assist in preventing the failure of the lining. 2. Model preparation The Muzhailing Tunnel, which carries high-speed trains as part of the Lanzhou-Chongqing rail project, was embedded in carbonaceous slate under a maximum overburden depth of 728 m. It is located at the intersection of the Tibet Plate, North China Craton, and Yangtze Plate, as shown in Fig. 1. The in-situ stress measured at the Muzhailing Tunnel reveals that the maximum horizontal stress σH is 32.0 MPa, the minimum horizontal stress σh is 18.7 MPa, and the coefficient of lateral pressure K is 1.7–3 due to tectonic compression. As the tectonic stress is high and the carbonaceous slate has rather poor geological properties, the horizontal tunnel wall deformation (937 mm) is much more than the roof settlement encountered. Moreover, many cracks appeared in the tunnel lining during 6 months following the lining construction (Fig. 2). Cracks with widths between 1 and 5 cm could be observed, most of which are located at the intradoses of the lining. The 1054-m lining of the Muzhailing Tunnel thus needs to be removed and reconstructed. Physical tests were performed on 1:30 scaled models to investigate the effect of earth pressure on the behaviour of horseshoeshaped tunnel linings. The experimental results were subsequently verified using nonlinear FE numerical modeling. 2.1. Scale model similitude Tests using small-scale models are typically inexpensive, have good repeatability, and yield accurate measurements. According to the theory of scale model similitude, the mechanical behaviour of small-scale model linings subjected to a variety of earth pressure conditions gives a good representation of the corresponding performances of prototype linings [12]. Furthermore, according to the concepts of dimensionless analysis [13], when scaled models are used, it is necessary to scale other independent parameters in addition to geometrical sizes. Based on the similarity ratios used in this work for geometrical size, CL = 1/ 30, and the unit weight, Cγ = 1, the similarity ratios of the other physical parameters required are as follows:

2

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Siberia Plate

Tarim Basin North China Craton

Lanzhou-Chongqing Railway Project

Tibet Plate Yangtze Plate

Fig. 1. Map showing the location of the Muzhailing Tunnel (part of the Lanzhou-Chongqing railway project) which lies at the intersection of the Tibet Plate, North China Craton, and Yangtze Plate.

y x

z

(a)

(b)

Fig. 2. Earth pressure-induced cracks in: (a) the roof, and (b) the knee of the Muzhailing Tunnel lining in operational service half a year after completion of construction.

C = 1; C = CE = Cc = 1 30; CN = CV = 1 27000; CM = 1 810000,

(1)

where the similarity ratios Cφ, Cσ, CE, Cc, CN, and CM correspond to the friction angle (φ), stress (σ), Young’s modulus (E), cohesion (c), axial force (N), and bending moment (M), respectively. All of the monitoring data from the 1:30 scaled model tests were divided by the similarity ratios given in Eq. (1) to yield the corresponding values for an actual prototype tunnel lining. 2.2. Material preparation In reality, the linings are made of cast-in-place C30 concrete with a Young’s modulus (Ecm) of 33 GPa and compressive strength (fck) of 30 MPa [14]. To procure model linings with appropriate mechanical parameters (i.e. that satisfies the scale similitude requirements for C30 concrete), several cylinders were made using different proportions of water: gypsum and were subjected to uniaxial compression testing on a trial-and-error basis. The results showed that a 1 : 1.18 mixture of water and gypsum was most suitable for simulating C30 concrete. Diatomite was also added to the gypsum mix, but further discussion regarding this aspect is 3

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Table 1 Mechanical parameters for the model lining and actual C30 concrete. Parameter

Young′s modulus (GPa) Uniaxial compressive strength (MPa) a

C30 concretea

Model lining material Test value

Corresponding prototype value

1.02 0.96

30.6 28.8

33.0 30.0

Taken from Eurocode 2 [14].

beyond the scope of the present study. Table 1 provides a comparison of the mechanical parameters obtained for the model lining material and those of a real prototype C30 concrete lining [14] based on the stress similarity ratio Cσ = 1/30. In practice, the tunnel linings used in deep underground engineering projects are reinforced. It is difficult to find a material that exactly meets the similarity requirements of steel in terms of the elastic modulus Es. The compressive/tensile capacity of the steel rebars in the tunnel lining can be regarded to be the essential factor affecting their performance. Therefore, we consider the product of the Young’s modulus Es and cross-sectional area As of the steel rebar (Es As) to be the value that is needed to meet the similitude relationship for the prototype steel used in a real lining. A mixture of barite powder, fly ash, fine quartz sand, and engine oil was chosen to simulate the poor rock enveloping the tunnel lining in the model. The use of quartz sand allows the friction angle φ of the model ground to be readily adjusted, and engine oil was employed to regulate the internal cohesion c to an appropriate value based on the similarity ratios CL = 1/30, Cγ = 1, and those given in Eq. (1). The addition of some barite powder, which is utilized because of its high density characteristic, increases the density of the model ground so that it matches the specifications for ‘poor rock’ according to the RMR system [15]. Several (4–5) shear tests were performed to achieve the desired friction angle and cohesion of the model ground. Some typical results for the following different normal stresses are plotted in Fig. 3: σn = 50, 150, 250, and 350 kPa. The physical mechanical properties of the model ground and its corresponding prototype are also compared in Table 2. 2.3. Monitoring programme used in the small-scale model tests A single horseshoe-shaped tunnel of the type typically used in China to carry high-speed (300 km/h) trains on double tracks was selected to use in our tests in this work (Fig. 4a). In conventional tunnels built using the new Austrian tunnelling method [16], a steel formwork trolley (Fig. 4b) is placed in position after the spray application of C25 shot concrete and the completion of a waterproof layer. Then, liquid concrete is poured into the gap between the waterproof layer and the steel formwork trolley to cast the tunnel lining in place.

Shear stress τ (kPa)

300

50 kPa 150 kPa

250

250 kPa

200

350 kPa

150 100 50 0

0

1

2 3 4 Shear displacement ∆ (mm)

5

6

(a)

Shear stress τ (kPa)

300 240 180 y = 0.6987x + 5.7435 R² = 0.9932

120 60 0

0

100

200 300 Normal stress σn (kPa)

400

(b)

Fig. 3. Typical results obtained using the direct shear apparatus and the material used to model the ground: (a) results obtained for shear stress τ as a function of shear displacement Δ for several values of normal stress σn; (b) the relationship between shear stress τ and normal stress σn. 4

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Table 2 Mechanical parameters for the model ground and prototype.

Model ground Prototype Similarity ratio

Cohesion, c (kPa)

Friction angle, φ (°)

Young′s modulus, E (GPa)

Unit weight, γ (kN·m−3)

5.74 172.2 1/30

35.0 35.0 1

0.057 1.71 1/30

22 22 1

Ø22 rockbolts Ø6.5 wire mesh Steel rib

Centre line of tunnel O3

Waterproof layer C30 reinforced lining

1224

50

28

Centre line of track

C25 shotcrete

O1

460 1438

28

60

O2

(a) Tunnel lining

Steel formwork

(b) Fig. 4. Details of a horseshoe-shaped tunnel designed to carry 300 km/h high-speed trains on double tracks: (a) the design of the prototype tunnel lining (where all values are in cm); (b) a photograph of the steel formwork trolley used for casting a tunnel lining in place.

As might be expected, it is very difficult to simulate the entire casting procedure in the laboratory. However, this aspect of the simulation procedure is not essential, as the aim of this study is to analyse the mechanical behaviour of a tunnel lining under earth pressure. We, therefore, begin our experiments by excavating an opening in the centre of the model ground. Then, we place a precast model lining into this cavity. Finally, hydraulic jacks were used to impose artificial loads on the boundaries of the model ground to simulate the earth pressure. Pressure is applied until cracks appear in the surface of the model lining. Regarding the Muzhailing Tunnel (Fig. 2), the high far-field stress causes the cracks in the lining, and the squeezing ground where the lining is embedded cannot create the arching effect. Our model tests therefore focus on the effect of far-field stress on the tunnel lining and do not consider any excavation-induced arching or unloading effects. The gypsum material was first utilized to form the 1:30 scale model linings (Fig. 5a). Prior to testing, strain gauges were attached to the intrados and extrados of the precast model lining (Fig. 5b) to measure the change in the strain at various positions when the ground is subjected to different far-field loads. The strain gauges allow the axial force (N) and bending moment (M) to be derived as follows:

5

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41 cm

B.-G. He, et al.

48 cm

(a)

(b) Roof Hance

Deformation transducer

Shoulder

Spring line Knee Kne y x

Wall

Intrados Knee Extrados

(c)

Invert

(d)

Fig. 5. Details of the horseshoe-shaped linings employed: (a) a typical 1:30 scaled model lining, (b) attaching strain gauges to the intrados and extrados of the lining, (c) deformation transducers used to monitor the radial deformation of model lining, and (d) summary of the names of various positions around the tunnel lining.

N= M=

1 2 1 12

×E×(

in

×E×(

in

+

ex )

× b × t;

ex )

× b × t 2,

(2)

where εin (εex) denotes the strain change (with sign) measured at the intrados (extrados) of the lining, b signifies the unit length (set to 1 m here), and t is the thickness of the model lining. It should be noted that a positive strain represents compression, while a negative strain represents tension. After the model lining was put in position in the model ground, deformation transducers were installed against the inner surface of the lining to monitor the radial deformation of the lining induced by the artificial earth pressure (Fig. 5c). 2.4. Testing procedure The following steps were used to investigate the effect of earth pressure on the horseshoe-shaped model linings:

• Step 1: The model linings (Fig. 5a) were precast according to the procedures outlined in Sections 2.2 and 2.3. Thereafter, strain • •

gauges were attached to the intrados and extrados of the tunnel lining to be tested at the following positions: roof, hance, shoulder, spring line, wall, knee, and invert (Fig. 5d). Step 2: The tunnel space in the model ground (Fig. 6a) was excavated, and the model lining with attached strain gauges was positioned inside the opening (Fig. 6b-c). Step 3: Hydraulic jacks (Fig. 6a-c) capable of exerting radial loads on the boundaries of the model ground in the x-, y-, and zdirections were employed to simulate the earth pressure which is increased in a step-by-step manner until cracks appear in the model lining. A servo-hydraulic test system with load-control was used during the experiments; consequently, the components of the pressure exerted in the x-, y-, and z-directions can be independently controlled. In these tests, they are coordinated to increase simultaneously so that the K = σx/σy remains constant. More specifically, the stress increment in the y-direction Δσy was set to 25 kPa, and thus the stress increment in the x-direction corresponded to Δσx = 25 K kPa.

6

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Jacks to impose ground loads , x y

Precast lining

y

Hydraulic jacks Jacks to impose

x

z

Boundary of model ground (a)

(b)

Ground model material

(c)

Excavating tunnel space Hydraulic jacks

Precast model lining positioned in the model ground

Loading earth pressure on the boundaries of model ground (d)

Fig. 6. Details of the experimental apparatus and procedures used: (a) a schematic diagram of the equipment used to impose earth pressure on the model; (b) front view of the equipment showing the jacks used to impose stress increments of Δσz = ν(Δσx + Δσy); (c) top view of the actual equipment; (d) outline of the steps involved in the testing process. (Note that the system is designed so that the test model is lying horizontally for the sake of convenience. The effect of gravity, however, can be ignored, as the artificial earth pressure applied creates much greater loads.) Table 3 Ground stresses applied in the different cases to explore the behaviour of the tunnel lining. Case

Vertical stress σy (MPa)

Horizontal stress σx (MPa)

K = σx /σy

Research method

1 2 3 4 5 6

1.0 1.9 0.4 0.76 1.5 0.6

0.6 1.14 1.2 2.28 0.9 1.8

0.6 0.6 3.0 3.0 0.6 3.0

Model test Model test Model test Model test Numerical simulation Numerical simulation

Assuming no excavation-induced deformation occurs along the tunnel axis (plane strain condition), the strain in the z-direction is as follows: z

=

1 [ E

z

(

x

+

y )]

=0

(3)

so that Δσz = ν(Δσx + Δσy) is the required stress increment in the z-direction needed during the tests, where ν is Poisson′s ratio (Fig. 6b). These steps are further illustrated in Fig. 6d. It should be noted that the test system was designed so that the model would lie 7

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Table 4 Representative cases used to explore the effect of different K values. K (=σx/σy)

0.2

0.3

0.4

0.5

0.6

0.4

0.8

0.9

1.0

1.1

σx (kPa) σy (kPa) K (=σx/σy) σx (kPa) σy (kPa) K (=σx/σy) σx (kPa) σy (kPa)

89.4 447.2 1.2 219.1 182.6 2.2 296.6 134.8

109.5 365.1 1.3 228.0 175.4 2.3 303.3 131.9

126.5 316.2 1.4 236.6 169.0 2.4 309.8 129.1

141.4 282.8 1.5 244.9 163.3 2.5 316.2 126.5

154.9 258.2 1.6 253.0 158.1 2.6 322.5 124.0

167.3 239.0 1.7 260.8 153.4 2.7 328.6 121.7

178.9 223.6 1.8 268.3 149.1 2.8 334.7 119.5

189.7 210.8 1.9 275.7 145.1 2.9 340.6 117.4

200.0 200.0 2.0 282.8 141.4 3.0 346.4 115.5

209.8 190.7 2.1 289.8 138.0

Note: the product of σx and σy is the same in each of these scenarios (40,000 kPa2).

4.6 4 6

19.7

26.4

0.6

53.5

76.7 7

8.5 2.4

y x

0.6

1.8

1.00 MPa 1.90 MPa 2.6

5.9 0.4 .4 0.5

(a)

0.8

0.7

1.4

1.3

3.3

2.8 13.1

35.8 0.40 MPa

14.3

0.76 MPa

1.6

49.1

3.9 0.4 0.6

(b) Fig. 7. Radial deformation of the model lining due to the coefficient of lateral pressure. The deformations shown correspond to K (=σx/σy) values of: (a) 0.6, and (b) 3.0. All deformations are in mm and have been scaled up from the model values by dividing by the appropriate similarity ratio (CL = 1/30). Note also that the loads stated in the diagrams refer to σy values (so σx = Kσy specifies the x-component, and Δσz = ν(Δσx + Δσy) stipulates the z-component).

horizontally rather than vertically (Fig. 6a) as the model lining could be much more readily positioned in a horizontal experimental system. Thus, the weight of the model ground around the opening is not properly reflected in this horizontal system. However, the high earth pressure field imposed on the model ground is much greater than the effect of gravity so the latter can be reasonably ignored in the experimental domain used here. According to the published in-situ stress data (see Appendix), the coefficient of lateral pressure K mostly lies in the range 0.3–3.0. To obtain a better understanding of the mechanical behaviour of the tunnel lining, we selected typical pre-excavation stress states, 8

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Radial deformation (mm)

20 Roof

0

Hance shoulder

-20

Spring line Wall

-40

Knee -60 -80

Invert

0.0

0.4

0.8

1.2

1.6

2.0

Earth pressure component σy (MPa)

(a)

Radial deformation (mm)

10 Roof

0

Hance shoulder

-10

Spring line

-20

Wall Knee

-30

Invert

-40 -50

0.0

0.2 0.4 0.6 Earth pressure component σy (MPa)

0.8

(b)

Fig. 8. The radial deformations of different positions of the tunnel lining as a function of σy for K values corresponding to: (a) 0.6, and (b) 3.0. A negative deformation denotes an inward movement of the lining towards the tunnel axis.

i.e., K = 0.6 (self-weight stress) and K = 3.0 (tectonic stress), which are analysed in detail; the six cases performed are tabulated in Table 3. Regarding the failure modes of the tunnel lining, the analysis is more general, as the analysis method way is the same as that used in the aforementioned six cases in Table 3. We adopted 29 cases in Table 4, which cover a wider range of K = 0.2–3.0, in an attempt to capture the influence of the coefficient of lateral pressure K on the failure modes of tunnel lining. 3. Mechanical behaviour of linings subject to ground load 3.1. Lining deformation Because of the symmetry of the model and the applied artificial earth pressure (Fig. 6a-b), only the results for one half of the model lining need to be considered in the analyses that follow. As the similarity ratio for size is CL = 1/30, the monitored deformations need to be multiplied by a magnitude of 30 to reflect the actual deformation of a real tunnel lining. Fig. 7 illustrates the results obtained for two values of K. Thus, the deformation of the model lining is strikingly different when it encounters a self-weight stress field (K = 0.6) compared to a tectonic stress field (K = 3.0). The coefficient K clearly plays an important role in determining the deformation profile of the tunnel lining. In a self-weight stress field (K = 0.6), the earth pressure in the y-direction σy is almost twice that in the x-direction σx; therefore, the lining becomes flattened in this situation (Fig. 7a). The largest deformation occurs at the roof of the tunnel lining and corresponds to 26.4 mm (inwards into the tunnel) when σy reaches 1.00 MPa. For σy = 1.90 MPa, the roof settlement increases to 76.7 mm. Conversely, applying a tectonic stress field (K = 3.0) causes the most prominent deformation to be in the horizontal direction and occurs at the wall of the lining (Fig. 7b). This behaviour is completely different from that in the self-weight stress field (Fig. 7a). 9

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297

—615

724 1273

1121

445

y 239

— 328

x 86

112

1.00 MPa 1.90 MPa 24

417

554

-25

(a)

—85

15 2

—630

372 700

0.40 MPa 0.76 MPa

26

—997

306 1179

5 —971

—29

—1046

(b) Fig. 9. The bending moments (M) experienced by the model lining due to the earth pressure. The M values shown correspond to the K values of: (a) 0.6, and (b) 3.0. All moments are in kN·m (scaled up from the model values by dividing them by the similarity ratio CM), and a positive bending moment denotes tension on the inner side of the lining. Once again, the loads specified in the diagram are σy values.

The radial deformation of the model lining will clearly depend on the amplitude of the earth pressure experienced, and, hence, the K value. Fig. 8 presents the radial deformations found in different positions of the lining, where a negative value means that the lining deflects inwards into the tunnel space. Clearly, when K = 0.6, the inward deformations of the roof and hance are much greater than in the other positions. When σy exceeds 1.6 MPa, the radial deformation here starts to grow nonlinearly with respect to the earth pressure, and microcracks are likely to occur. On the other hand, the deformations in the x-direction are much larger than those in the y-direction when a tectonic stress field is applied (K = 3.0; Fig. 8b). The maximum deformations in this case develop at the wall and spring line and involve the lining bending inwards. It should be noted that the roof and hance bend slightly outwards at the same time, but the level of this deformation is so small as to be negligible (especially compared to that of the wall and spring line). 3.2. Internal force acting on the tunnel lining Using Eq. (2) and the data from the strain gauges, the bending moments experienced by the tunnel lining can be derived, as plotted in Fig. 9. It can be seen that the largest bending moments are experienced in the upper part of the tunnel lining when K = 0.6 (Fig. 9a). The maximum M value appears at the roof which tends to bend inwards while the shoulders bend outwards. The knees of the tunnel lining bend outwards as well. The bending that occurs at the walls and invert, however, is much less than that in the upper part of the lining. Interestingly, the distribution of the bending moment is not consistent with the radial deformation which is much smaller at the shoulder (Fig. 7a). In a tectonic stress field (K = 3.0; Fig. 9b), the knees bend outward and the walls bend inward, which is distinct from the behaviour in the self-weight stress field. Essentially, the mechanical behaviour of the tunnel lining may well depend on the coefficient of lateral pressure K. As Fig. 10a (K = 0.6) demonstrates, the bending moments experienced by the lining are proportional to the earth pressure provided σy ≤ 1.60 MPa. When σy > 1.60 MPa, the bending moments are redistributed however. For example, the bending moment experienced by the spring line increases linearly with continuously increasing earth pressure but decreases rapidly once the σy 10

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1500 Roof

Bending M (kN·m)

1000

Hance shoulder

500

Spring line

0

Wall

-500

Knee

-1000

Invert

-1500 0.0

0.5

1.0

1.5

2.0

Ground load σy (MPa)

(a)

1500 Roof

Bending M (kN·m)

1000

Hance shoulder

500

Spring line

0

Wall

-500

Knee

-1000

Invert

-1500 0.0

0.2

0.4

0.6

0.8

Ground load σy (MPa)

(b)

Fig. 10. The bending moments acting on different parts of the horseshoe-shaped tunnel lining as a function of σy for the coefficients of lateral pressure corresponding to: (a) 0.6; (b) 3.0.

exceeds 1.60 MPa, implying that the cracking of the model lining initiates. Recall that σx = Kσy specifies the x-component in the model test. For K = 3.0 (σx = 3σy; Fig. 10b), the bending of the tunnel lining increases linearly as the ground load is increased, provided σy ≤ 0.6 MPa. When σy > 0.6 MPa, the moments experienced by the wall, spring line, and knee decline, as the lining is once again beginning to fracture at this level of stress. Fig. 11 shows plots of the axial forces (N) experienced by different parts of the lining. In general, the force is fairly uniform along the entire circumference of the lining, especially when compared to the bending distribution (Fig. 9). The axial force distribution also varies slightly with K, but the maximum force appears to consistently occur at the invert in our model tests. For the self-weight stress field (K = 0.6; Fig. 11a), the axial force increases from the roof to the invert. On the other hand, for the tectonic stress field (K = 3.0; Fig. 11b), the force decreases initially from roof to wall and then increases to the invert. Moreover, in this case, the magnitude of the force at the roof is almost the same as that at the invert. Fig. 12 shows the axial force results plotted out for different parts of the lining. As can be appreciated from this figure, the axial force acting on the horseshoe-shaped tunnel lining is found to be compressive and linearly proportional to the earth pressure. The largest axial force occurs at the invert for K = 0.6, while the smallest value occurs at the lining roof. When K = 3.0, the spread in the range of the force experienced around the lining (i.e., the difference between the maximum and minimum values) decreases. Furthermore, the maximum axial force appears at the roof in this case, but the minimum force occurs at the wall. 3.3. Cracking patterns on the tunnel linings When the artificial earth pressure imposed by hydraulic jacks on the model has increased sufficiently, the lining will eventually succumb to failure. Fig. 13 shows the failure mode typically encountered when K = 0.6, which is characterized by flexural cracks first occurring in the intrados of the roof lining, followed by cracks occurring in the extrados of the shoulders. This behaviour is consistent with the bending moment distribution shown in Fig. 9a. However, it does not appear to match the deformation shown in Fig. 7a 11

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6114

11698 116 12296

6363

13397

7067 y x

7530

14453

1.00 MPa 7787

15159

1.90 MPa

8647

17142 9322

(a) 9111

18533 18 17691

8705

15484

7670

13717

6618

0.40 MPa 0.76 MPa

6127

7614

12736

15368 8883 83

18322

(b) Fig. 11. The axial forces acting on the tunnel lining due to earth pressure for the K values of (a) 0.6, and (b) 3.0 (all values in kN). The ground loads given in the figure again correspond to the σy values.

wherein the maximum displacement appears at the roof but the deformation at the shoulder is negligible. Therefore, the deformation, at times, cannot reflect the actual mechanical behaviour of the horseshoe-shaped tunnel lining. For K = 3.0, the first crack appeared in the knee of the model lining during our test process (Fig. 14). Then, the second crack formed in the intrados of the spring line, and finally, the third crack formed in the intrados of the wall. Therefore, the failure mode in this scenario is comparable to the bending moment distribution (Fig. 9b) and is also in line with the deformation distribution (Fig. 7b). In conclusion, the deformation that is often monitored in the course of tunnel engineering is not a reliable tool to assess the mechanical state of the tunnel lining in a self-weight stress field. Considering the high compressive strength of concrete, however, the bending moment M does seem to reflect well the failure mode of a horseshoe-shaped tunnel lining. 3.4. Numerical and experimental comparison 3.4.1. Nonlinear numerical modelling A nonlinear finite element method was employed to test the validity of our experimental results. In this work, we use ANSYS software. Steel reinforcements in concrete can be simulated using a ‘smeared’ element in which the steel reinforcement is discretized into the nearby concrete and averaged within the related SOLID65 elements [17]. Once cracking initiates, the relevant elements are assumed to maintain continuum; nevertheless, the properties of stiffness together with strength are adjusted to consider the effect of cracking, according to the defined evolving relationship between strain and stress [18]. The interaction between the tunnel lining and soft ground was simulated using compression-only LINK10 elements. According to Kolymbas [19], the normal LINK10 stiffness between the tunnel lining and ground (Ks) is related to the Young’s modulus of the 12

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20000

Axial force N (kN)

Roof 15000

Hance shoulder

10000

Spring line Wall

5000

Knee Invert

0 0.0

0.5

1.0 1.5 Ground load σy (MPa)

2.0

(a)

20000

Axial force N (kN)

Roof 15000

Hance shoulder

10000

Spring line Wall

5000

Knee Invert

0 0.0

0.2

0.4 0.6 Ground load σy (MPa)

0.8

(b)

Fig. 12. The axial force borne by different parts of the lining as a function of σy for the coefficients of lateral pressure: (a) 0.6, and (b) 3.0.

(a)

(b)

Fig. 13. Photographs of cracks in a 1:30 scaled model lining due to ground load when K = 0.6: (a) a flexural crack in the intrados of the roof of the lining, and (b) an oblique crack in the extrados of the shoulder of the lining.

ground (E), the Poisson ratio (ν), and the local radius of the tunnel lining (r) via the following expression:

Ks =

E (1 ) r (1 + )(1 2 )

(4)

Clough and Duncan [20] proposed that the shear stress at the interface between the tunnel lining and stratum is very small and 13

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Spring line Wall

Intrados

Knee

Fig. 14. Cracking pattern in the test model lining when K = 3.0.

may be negligible compared to the normal stress. Thus, shear elements were not adopted in our numerical model. To take into account the non-uniform thickness of the concrete lining (50 cm at the roof; 60 cm at the invert), three-dimensional 8-node nonlinear elements SOLID65 with von Mises plasticity were used to model the following nonlinear behaviour of the concrete: cracking under tension, crushing under compression, and plastic deformation [21]. The resulting model is shown in Fig. 15. The Young’s modulus and Poisson′s ratio of the C30 concrete are assumed to be 33 GPa and 0.20, respectively [14], and the mechanical parameters of the prototype ground are as shown in Table 2. The loading steps used during numerical modelling follow the order used in the aforementioned model tests as far as possible. That is, the ground load imposed on the tunnel lining increases in a step-by-step manner, while the K remains constant. 3.4.2. Numerical modelling results Fig. 16 demonstrates the principal stresses on the tunnel lining when the ground loads in the x-, y-, and z-directions amounted to 0.9, 1.5, and 0.9 MPa, respectively (so that K = 0.6; Poisson’s ratio ν = 0.375). The tensile stress is clearly concentrated at the intrados of the roof, as well as the extrados of the shoulder (Fig. 16a), which makes these regions the most prone to tension-induced failure. This agrees well with the bending moment distribution (Fig. 9a). The most unfavourable position in the lining occurs at the roof where the maximum tensile stress σ3 amounts to 1.89 MPa which is close to the tensile strength of C30 concrete, fctm = 2.90 MPa [14]. Interestingly, cracks similar to the ones found in our tests have been observed in field observations made in the bifurcation tunnel of the Huizhou Pumped Storage Power Station [2]. In that particular case, a wide range of cracks appeared in the roof of the tunnel lining, and a comprehensive analysis revealed that the underlying damage was caused by the tensile strain in the inner side of the lining. Moreover, Fig. 16b indicates that the maximum 14

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y

z

y x

x

z

(a)

y x

(b) Fig. 15. The nonlinear FE model employed in this work: (a) the entire numerical model subjected to three-dimensional normal stresses, i.e. σx, σy, and σz; (b) magnified view of the tunnel lining.

compressive stress (38.1 MPa) at the extrados of the lining roof exceeds the compressive strength of C30 concrete, fck = 30 MPa [14]. The occurrence of cracks in the tunnel lining can be revealed using the nonlinear FE ANSYS software by using a plasticity algorithm – the results are shown in Fig. 17 for the case K = 0.6. The ground load is increased gradually in a step-by-step manner until the nonlinear FE calculations do not converge, indicating that the lining is failing. As intuitively expected, cracks will occur if the stress exceeds the strength of the material. In the early stages of the failure process, moment-induced cracks begin to form in the intrados of the roof of the tunnel lining. Soon, others follow in the extradoses of the shoulder. Thus, the cracking pattern and cracking sequence predicted by our nonlinear numerical modelling appears to be in excellent agreement with the experimental results obtained in our tests on 1:30 scaled models (Fig. 13). Fig. 18 portrays the mechanical behaviour of a tunnel lining subjected to a tectonic stress field, i.e., σx = 1.8 MPa, σy = 0.6 MPa, and σz = 0.9 MPa (so K = 3.0; ν = 0.375). The maximum tensile stress appears at the intrados of the lining wall, up to 3.27 MPa (see Fig. 18a), which exceeds the tensile strength of C30 concrete, fctm = 2.90 MPa [14]. For that reason, tension failure takes place at the intrados of the lining wall. Likewise, the maximum compressive stress occurs at 35.5 MPa (see Fig. 18b), which surpasses the compressive strength of C30 concrete, fck = 30 MPa [14], so that the crushing failure appears at the intrados of the tunnel lining knee (Fig. 18c). The nonlinear finite element modelling results are capable of interpreting both the bending moment distribution in Fig. 9b and the cracking pattern in Fig. 14.

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y x

(a)

(b) Fig. 16. The principal stresses acting on the lining when σx = 0.90 MPa and σy = 1.50 MPa, i.e., K = 0.6: (a) the minimum stress σ3; (b) the maximum stress σ1 (all values in Pa: a positive value signifies tension, and a negative value denotes compression).

y x

Fig. 17. The occurrence of flexural cracks in the tunnel lining and cracking sequence (circled numbers) predicted using the nonlinear numerical FE modelling and plasticity algorithm. The coefficient of lateral pressure, K, has a value of 0.6 in this simulation.

4. Effect of the lateral pressure coefficient on the failure mode 4.1. The effect of K on the internal forces acting on the lining Under plane strain conditions, the strain energy equivalence principle gives an expression for the product of the stress components in the x- and y-directions, as follows [22]:

EnergyConstant =

1 x y 2 E

(5)

Using this equation, we can derive sets of σx and σy values, which all correspond to the same strain energy (Table 4). The reasons for the appropriate limits for K are briefly described in the Appendix. In each of the 29 scenarios tabulated, the product of σx and σy is 16

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Maximum tensile stress

(a)

Maximum compressive stress

(b) Intrados of lining wall

Extrados of lining knee

(c) Fig. 18. Mechanical behaviour of the tunnel lining subjected to a tectonic stress field of σx = 1.80 MPa and σy = 0.60 MPa, i.e., K = 3.0: (a) the minimum stress σ3 in Pa; (b) the maximum stress σ1 in Pa; (c) cracking events at the intrados of lining wall and the extrados of knee.

the same (40,000 kPa2), so the strain energy will be the same. Using the values shown in the table, we can, therefore, examine the effect that different K values have on a 1:30 scaled model subject to a constant strain energy. Fig. 19 reveals that the value of K significantly affects the internal forces acting on the model lining even though the strain energy in each of the cases is the same. When K < 0.7, the largest positive bending moment appears at the roof of the lining, while the largest negative moment occurs at the shoulders. Recall that a positive bending moment indicates tension in the intrados of the model lining (conversely, a negative value suggests tension in the extrados). As K increases, the moments at the roof and shoulder decrease in magnitude and arrive at their smallest values when K = 0.9 (Fig. 19a). Thereafter, they start to increase again. When K > 0.9, the largest positive moment appears at the spring line of the lining, while the largest negative bending moment is found at the knee. As shown in Fig. 19b, the axial force looks relatively uniform along the model lining for a specific value of K (compared to the bending moment). The general trend is that the axial force decreases at first as K increases, reaches a minimum value, and then recovers somewhat. When K < 0.9, the largest force occurs at the spring line and the smallest appears at the roof. Interestingly, the axial force reaches its lowest value and becomes almost the same around the lining when K is equal to 1.0. Once K > 1.0, the differences among the forces around the lining become quite small to the extent that they could be deemed identical.

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400 Roof Hance

Bending M (kN·m)

200

Shoulder Spring line

0

Wall Knee Invert

-200

-400 0.2

0.6

1.0

1.4 1.8 2.2 The coefficient of lateral pressure K

2.6

3.0

2.6

3.0

(a)

The coefficient of lateral pressure K 0.2 0

0.6

1.0

1.4

1.8

2.2

Roof

Axial force N (kN)

-1000

Hance Shoulder

-2000

Spring line Wall Knee

-3000

Invert

-4000

(b) Fig. 19. The internal forces on the tunnel lining plotted as a function of the coefficient of lateral pressure (K): (a) the bending moment (where a positive M value denotes tension on the inside of the lining); (b) axial force (N). The results from the model test have again been divided by the appropriate similarity ratio to yield the corresponding value for a prototype lining.

4.2. Evaluating the safety of the tunnel lining Fig. 20a represents a reinforced lining member with strain and stress distributions under both a bending moment M and axial force N [23]. The position of the plastic centroid depends on the stresses developed by the internal forces, so (Fcc h 2 + Fsc d + Fs d ) thatx¯ p = , (6)where Fcc represents the compression acting through the centroid of the stress block, Fsc is the (F + F + F ) cc

sc

s

compression on the centroid of the steel zone As , Fs denotes the tension or compression on the centroid of the steel area As, and the other quantities are as shown in Fig. 20a [14,23]. In light of the ultimate limit states specified in Eurocode 2 [14], the axial force should be cancelled out by the forces formed in the cross-section, so:

N=

0.567fck b × 0.8x + fsc As + fs As if 0.8x < h; 0.567fck bh + fsc As + fs As

if 0.8x

h,

(7)

where fck is the characteristic strength of a cylinder of concrete, fsc is the compressive stress in the steel zone As , and fs is the tensile/ compressive stress in the steel zone As [14,23]. Additionally, the bending moment at the plastic centroid is given by the following:

M=

0.567fck b × 0.8x (x¯ p 0.567fck bh (x¯ p

0.8x 2) + fsc As (x¯ p h 2) + fsc As (x¯ p

d)

d)

fs As (d

fs As (d

x¯ p )

x¯ p ) if 0.8x < h ; if 0.8x

h,

(8)

Eqs. (7) and (8) allow an M−N interaction diagram to be constructed for a single horseshoe-shaped tunnel lining capable of carrying 300 km/h high-speed trains on double tracks. This is plotted in Fig. 20b (the red closed curve). Due to the symmetry of the section of the reinforced tunnel lining, the bending moment signs do not need to be considered. Consequently, the internal forces on the lining at the various monitoring points fall within the M−N interaction diagram. It is evident that the K significantly affects the 18

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(a)

(b) Fig. 20. Diagrams illustrating the bending moments and axial forces in the ultimate limit state: (a) schematic diagram of a symmetrical section of the tunnel lining; (b) comparison of the M−N interaction diagram for the lining and internal forces monitored at various positions. (The M−N interaction diagram is based on the information in Eurocode 2. The signs of the bending moments do not need to be considered for the symmetrical section of the reinforced tunnel lining.)

safety of the lining. The tunnel lining is vulnerable when K = 0.2, especially the roof of the lining which is closest to the M−N interaction diagram. When K = 3.0, the knee can be seen to have a small margin of safety. Regarding K = 0.9, all of the monitored positions lie far from the M−N interaction diagram and should therefore be safe. To obtain a more quantitative assessment of the safety of the lining, a safety factor can be employed. For x > 0.55d, the safety factor is given by the following:

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5.0

Safety factor of lining F

4.0

Roof Hance

3.0

Shoulder 2.0

Spring line Wall

1.0

Knee

0.0 0.2

0.6

1.0

1.4

1.8

2.2

2.6

3.0

The coefficient of lateral pressure K Fig. 21. Plots of the safety factor (F) as a function of the coefficient of lateral pressure (K) for various positions on the reinforced lining. The most favourable range of K values is from 0.6 to 1.2. When K < 0.6, the position most susceptible to failure is the roof of the lining; when K > 1.2, the position most susceptible to failure is the knee.

Fig. 22. Photograph of an array of model linings used to investigate the effect of the coefficient of lateral pressure on the failure mode experienced.

F=

0.5Ra bd 2 + Rg Ag' (d

d) (9)

Ne

where Ra is the ultimate compressive strength of C30 concrete, Rg is the tensile/compressive strength of the steel rebar, e is an eccentricity, and the other quantities are as shown in Fig. 20a in accordance with Eurocode 2 [14]. The eccentricity is included in this expression because it affects the position of the neutral axis and, hence, the strains along with the stresses involved in the reinforcement [23]. For × ≤ 0.55d, the safety factor is given by the following:

F=

Rw bx + Rg (Ag'

Ag )

(10)

N

where Rw is the ultimate flexural compressive strength of concrete. These expressions for the safety factor F can be used to highlight the effect of the coefficient of lateral pressure on the mechanical behaviour of the tunnel lining. To this end, Fig. 21 shows plots of F vs. K for various positions of the tunnel lining. The figure clearly shows an overall general trend as K increases from 0.2; i.e., the safety factor first increases, peaks, and then decreases again (regardless of the lining position considered). Furthermore, when K < 0.6, the roof has the lowest safety factor, and the hance has the highest value. Similarly, when K > 1.0, the knee is the most vulnerable position, and the roof is the safest position. Fig. 21 shows that the most favourable range of K values is 0.6–1.2. In addition, when K < 0.6, the position most susceptible to failure is the roof; when K > 1.2, the position most susceptible to failure is the knee. 20

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Fig. 23. Cracking patterns observed in the model linings subjected to increasing earth pressure. The sequences shown correspond to K coefficients within the following ranges: (a) 0.2–0.5; (b) 0.6–0.8; (c) 0.9–1.0; (d) 1.1; (e) 1.2; (f) 1.3–3.0. The cracks in the intrados and extrados of the lining are represented using solid red and dashed blue lines, respectively. (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)

4.3. Cracking pattern A large number of model linings (Fig. 22) were tested using K values in the range 0.2–3.0 (at intervals of 0.1). In each case, the earth pressure components in the x-, y-, and z-directions were increased (keeping K constant with plain strain conditions, of course) until the model lining failed. The cracks in the linings were then examined. Fig. 23 illustrates the cracking patterns observed for different ranges of K values. The figure confirms that K plays an important role in determining the cracking positions on the model lining and the sequences in which they occur. It should be noted that we did not attempt to observe micro-fractures in the tests carried out on our models. When 0.2 ≤ K ≤ 0.5 (Fig. 23a), cracks first develop in the intrados of the model lining in the roof area where the positive bending 21

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moment bends the roof inwards. Thereafter, cracks form in the extradoses of the lining at the shoulders which bend outwards due to the negative bending moment experienced. When 0.6 ≤ K ≤ 0.8, cracks appear symmetrically and sequentially when the ground load is gradually increased (Fig. 23b). That is, cracks appear in the outer surface of the model lining shoulder quite soon after the moment-induced crack appears in the intrados of the roof. The third set of cracks then appear in the abutment areas between the knees and invert. Finally, the fourth set appear at the knees, and thereafter the ground load can no longer be resisted. At this point, the model lining utterly fails. When 0.9 ≤ K ≤ 1.0 (Fig. 23c), cracks only form in the abutment areas between the knees and invert; the other positions appear to remain safe. Similarly, cracking only occurs at the extradoses of the knees when K = 1.1 (Fig. 23d). When K = 1.2, two sets of cracks appear in the lining as following: the first set in the extradoses at the knees, followed by cracks in the intradoses at the spring lines (Fig. 23e). Finally, when 1.3 ≤ K ≤ 3.0, three sets of cracks appear sequentially. The first set forms in the knees (extradoses), the second in the spring lines (intradoses), and the third in the walls (intradoses). Overall, it is evident that the pattern of cracks that develops in such tunnel linings depends on the coefficient of lateral pressure K, provided that the pressure is sufficiently high. If the K is measured in-situ prior to construction, then the parts of the tunnel lining susceptible to failure can be predicted (using, for example, Fig. 23). Then, corresponding countermeasures can be prepared for these vulnerable positions of the lining to prevent the potential failure expected under the specific in-situ stress conditions encountered. 5. Conclusions (1) The cracking occurrence of horseshoe-shaped tunnel linings depends, to large extent, on the coefficient of lateral pressure (K). Tunnel designers need to take this into account when formulating the design parameters for the tunnel lining. During the design stage, when the structure of the tunnel is being planned, attention should be given to ascertaining which parts of the lining are potentially most susceptible to failure. (2) As K increases, the safety factors at all of the positions around the tunnel lining increase initially, peak, and then decrease. The most favourable range of K values is from 0.6 to 1.2. When K < 0.6, failure is most likely to occur at the roof; however, when K > 1.2, the knees are the most susceptible locations. (3) The value of K significantly affects the internal forces acting on the model lining. When K < 0.7, the largest positive bending moment appears at the roof of the lining, and the maximum negative bending moment occurs at the shoulders, assuming that a positive bending moment indicates tension in the intrados of the model lining. As K increases, the moments at the roof and shoulder decrease quickly. When K > 0.9, the largest positive moment appears at the spring line of the lining, while the largest negative bending moment is found at the knee. (4) The cracking pattern of the tunnel lining shows good correspondence with the bending moment distribution. However, it does not compare well with the deformation if a self-weight stress field is imposed. The deformation, therefore, does not always correctly reflect the failure mode of the lining. As a result of this, it is worth noting that monitoring the deformation of a tunnel lining may not reliably reflect the mechanical behaviour of the lining when it is located in a self-weight stress field. 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. Acknowledgments This work was supported by the National Key Research and Development Program of China (2018YFC0407006) and the Young Scientists Fund of the National Natural Science Foundation of China (51809038). The model tests were carried out at Southwest Jiaotong University, and the authors gratefully acknowledge Mr. Yun-Jun Liu, Mr. Guang-Jing Zhang, and Tian-Yu Chen for their unwavering support. Appendix:. The limiting values of K Brown and Hoek [24] examined a number of published in-situ stress values and used them to derive hyperbolic expressions for the limits of the K, as follows:

0.3 +

100 1500 < K < 0.5 + Z Z

(A1)

where Z is the overburden (in metres) and K is the ratio of the horizontal to vertical in-situ stresses (Fig. A1). Thus, K mostly lies in the range of 0.3–3.0.

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B.-G. He, et al. The coefficient of lateral pressure K 0.0

0.5

1.0

1.5

2.0

2.5

3.0

Overburden depth Z (m)

500

1000

K = 0.5 +

1500

Australia Canada

2000

U.S.A Southern Africa

2500 K = 0.3 +

Scandinavia Other regions

Fig. A1. Field measurement results illustrating the relationship between the overburden depth and the ratio of horizontal to vertical in-situ stress, K, that is, the coefficient of lateral pressure (modified from Brown and Hoek [24]).

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