Numerical modelling of tunnel face stability in homogeneous and layered soft ground

Tunnelling and Underground Space Technology 94 (2019) 103096

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Numerical modelling of tunnel face stability in homogeneous and layered soft ground Ahmed S.N. Alaghaa, a b

T

⁎,1

, David N. Chapmanb

Department of Engineering, University of Cambridge, Cambridge CB2 1PZ, UK Department of Civil Engineering, College of Engineering, University of Birmingham, Birmingham B15 2TT, UK

ARTICLE INFO

ABSTRACT

Keywords: Soft ground tunnelling Face stability Finite element Arching effect Layered ground

Soft ground tunnelling in urban areas is more frequently being performed using the shield method. Due to its great influence on both ground settlement and construction safety, face stability is one of the most critical problems in shield tunnelling. In this research, a series of 3 D finite element simulations were conducted using the Midas GTS NX software in order to determine the required collapse pressure of a tunnel face during tunnelling in homogeneous or layered soils. In terms of homogeneous ground, the effects of different soil strength parameters, cover-to-diameter ratios and tunnel diameters were investigated. Based on the numerical results of 140 analyses, a new design equation has been derived to calculate the required face collapse pressure during tunnelling in a purely '

frictional soil or a c ' soil above the groundwater table. The results of this equation are in close agreement with the results from available experimental tests and theoretical approaches, and hence the equation provides a very useful method for estimating face collapse pressure. Furthermore, the arching effect has been explicitly investigated, and the failure mechanisms ahead of the tunnel face have been presented for various cases. For layered ground, two stratification scenarios were considered. Each scenario comprised two strata; an upper and a lower stratum. The first scenario (Case 1) involved the upper stratum intersecting with the lower stratum at the tunnel crown, while in the second scenario (Case 2), both strata intersect at the tunnel axis. In each case, the shear strength parameters of the upper and lower strata were changed to study the influence of these variations on face collapse pressure. In Case 1, it was found that the face collapse pressure is far more sensitive to parameter variations in the lower stratum than those of the upper stratum. In Case 2, however, almost the same values of face collapse pressure were obtained for both sets of parameter variation. Furthermore, if the lower stratum is stronger than the upper stratum, the required face collapse pressure in Case 2 is greater than that in Case 1.

1. Introduction The progressive development of cities and the rising demand for public and sustainable transport have increased the need to construct tunnels in these metropolitan areas. The use of 'shield' tunnelling machines has grown dramatically in the last few decades within the construction industry for small to large infrastructure projects in soft ground (Bäppler, 2016). In this context, tunnelling shields can be used with either partial-face or full-face excavation. The latter is known as a tunnel boring machine or TBM. A key role of TBMs (which can be broadly divided into compressed-air shields, slurry shields and earth pressure balanced ‘EPB’ shields) is to provide a suitable support pressure at the tunnel face to maintain ground stability and/or to prevent water ingress (Chapman et al., 2010). The face support pressure is therefore one of the most important parameters in tunnelling

projects, and also plays a vital role in minimising ground movements during tunnel construction. Therefore, this pressure should not be too small to avoid collapse (active failure) or too large to avoid blow-out (passive failure) of the soil mass around the tunnel face (Li et al., 2009). Due to its practical importance, many approaches have been developed to analyse the stability of the tunnel face as well as to determine the minimum required face support pressure. All the available methods can be divided into three primary groups: (a) analytical methods; (b) experimental methods; and (c) numerical methods. Analytical methods are used to study the tunnel face stability in cohesive, frictional and cohesive – frictional soils within the frameworks of limit equilibrium and limit analysis. Since the early sixties, many authors have employed the limit equilibrium concept to evaluate the stability of the tunnel face. Horn (1961) introduced the first limit equilibrium failure

Corresponding author. E-mail addresses: [email protected] (A.S.N. Alagha), [email protected] (D.N. Chapman). 1 Formerly, Department of Civil Engineering, College of Engineering, University of Birmingham, Birmingham B15 2TT, UK. ⁎

https://doi.org/10.1016/j.tust.2019.103096 Received 13 February 2019; Received in revised form 20 August 2019; Accepted 20 August 2019 0886-7798/ © 2019 Elsevier Ltd. All rights reserved.

Tunnelling and Underground Space Technology 94 (2019) 103096

A.S.N. Alagha and D.N. Chapman

mechanism by assuming a three-dimensional sliding wedge-silo model. The first practical application of this model was proposed by Jancsecz and Steiner (1994). Their model took into account the effect of soil arching above the TBM. Two years later, Anagnostou and Kovari (1996) used a modified wedge-silo model to assess the face stability of a tunnel driven by an earth pressure balance machine in drained conditions. Broere (2001) extended the wedge stability model to investigate the influence of soil heterogeneity on the minimal face support pressure. Anagnostou (2012) presented a method to estimate the tunnel face stability in frictional/cohesive – frictional soils with particular attention to the effect of horizontal stresses. The developed model is consistent with the silo theory and is based on the method of slices that does not require an assumption for the vertical stress. Based on this method, a design equation for computing face support pressure at failure has been proposed. Qarmout et al. (2019) introduced a new mechanism to study the tunnel face stability in frictional soil based on the Kinematical Element Method (KEM). Their model consisted of two rigid blocks, a tetrahedron wedge block, and a triangular prism block, and the tunnel face was considered a triangle with the same area of the circular tunnel. For ' < 30° , the results of their model indicated a notable dependence, in contrary with the majority of other studies, of the limiting support pressure on the C /D ratio. This is probably due to the limited number of blocks they have used in their assumed mechanism. Nevertheless, their results were in good agreement with the published experimental, analytical and numerical methods. The other analytical approach is the limit analysis method, which is based on the theorems of soil plasticity (upper and lower bound theorems). Researchers (Davis et al., 1980; Leca and Dormieux, 1990; Li et al., 2009, 2019; Mollon et al., 2009, 2010, 2011, 2013; Tang et al., 2014; Ibrahim et al., 2015; Senent and Jimenez, 2015; Han et al., 2016; Zou et al., 2019a,b; amongst others) have extensively used limit analysis method over the years to assess tunnel face stability. Davis et al. (1980) evaluated the stability of an unlined tunnel face in purely cohesive soil assuming undrained conditions. Their study investigated face stability using both lower and upper bound approaches. Leca and Dormieux (1990) proposed a three-dimensional mechanism based on the motion of two rigid conical blocks in a frictional material. They derived lower and upper bound values for the maximum and minimum face support pressure of a shield tunnel. Mollon et al. (2009) developed an upper bound solution for the tunnel face stability in frictional/cohesive – frictional soils by enhancing the study of Leca and Dormieux (1990). They assumed a three-dimensional mechanism based on the motion of five truncated rigid conical blocks (multi-block mechanism). The main disadvantage of this study is that the intersection between the conical blocks and the tunnel face is an elliptical surface that does not perfectly fit with the circular face of the tunnel. Mollon et al. (2010) tackled this problem and considered the entire circular tunnel face by developing a three-dimensional failure surface point by point using a spatial discretization technique. Compared with the results of Mollon et al. (2009), the upper bound values were improved. It should be noted that the foregoing limit analysis studies were developed to evaluate tunnel face stability in homogeneous ground. Tang et al. (2014) proposed an upper bound solution to investigate tunnel face stability in layered soils by modifying the model developed by Leca and Dormieux (1990) for homogeneous soils. In this modified model, the soil across the tunnel face is homogeneous, while the soil above the tunnel crown is layered. The proposed failure mechanism of the crossed ‘homogeneous’ layer consists of two conical blocks, with the intersection of these blocks being an elliptical cross section inscribed to the circular tunnel face. In addition, the mechanism of the cover ‘layered’ soil involves rigid blocks in each layer. Finally, this study gives relatively small face support pressure values due to the simplified failure mechanism that has been used. Ibrahim et al. (2015) studied the tunnel face stability in purely frictional layered ground. Their upper bound solution is an extension for the rotational failure mechanism developed by Mollon et al. (2011). Their study considered the cases of two and three soil layers with a cover-to-diameter ratio of 1 or more to avoid outcropping of the failure mechanism at the ground

surface. In addition, they generated 3 D FEM simulations to compare the numerical results (both the collapse pressure and the failure patterns) with the results of the proposed mechanism, and good agreement between the results was achieved. Senent and Jimenez (2015) modified the model of Mollon et al. (2010) to investigate the tunnel face stability in layered ground. Their mechanism, which takes into account the whole area of the circular tunnel, consists of one block that rotates around an axis perpendicular to the vertical plane of symmetry of the tunnel. The mechanism also considers the possibility of partial collapse in stratified ground. The majority of these studies estimated the limit face support pressure for the case of an upper stratum (single or multiple layers) overlaying a lower stratum at the tunnel crown, however in reality, the scenario of multiple strata intersecting at the tunnel face is quite common, and therefore research is required to provide practical considerations for this strata scenario. Sloan (2013) compared the analytical solutions, in terms of their accuracy and applicability, and also highlighted the differences between them. This study concluded that although analytical methods can define the range (upper and lower bounds) of the true solution in geotechnical problems and can deal with complex loading and boundary conditions, they cannot deal with complicated soil models and cannot predict the interaction between the soil and existing structures during tunnelling, therefore, the best method in such cases is to conduct numerical modelling. Experimental tests have been widely used by researchers to assess tunnel face stability. Two main groups of small-scale model tests can be found in literature; 1-g model tests (e.g. Fellin et al., 2010; Kirsch, 2010; Berthoz et al., 2012; Chen et al., 2013; Lü et al., 2018) and centrifuge tests under n-g conditions (e.g. Mair, 1979; Chambon and Corté, 1994; Oblozinsky and Kuwano, 2006; Idinger et al., 2011; Soranzo et al., 2015). Chen et al. (2013) performed a series of 3 D large-scale (1-g) model tests to investigate the stability of the tunnel face in sandy soil collected from the Yangtze River. The shear strength parameters of this sand were ' = 37° and c' = 0.5 kPa , however, the extracted sand was dried in a large rotary dryer resulting in a water content between 0.3% and 0.4% with negligible or very little cohesion (not exceeding 0.5 kPa ) and hence they did not include this in their main analyses. For the tested overburden ratios (C / D = 0.5, 1 and 1.5) , the authors concluded that the overburden had almost no influence on the face support pressure when C /D 1. Chambon and Corté (1994) performed a series of centrifuge tests in Nantes LCPC using sandy soil with little cohesion. Various dia(D = 5, 10 and 13 m) and cover-to-diameter ratios meters (C / D = 0.5, 1, 2 and 4) were considered. The outcomes of their experiments revealed that the C /D ratio has little influence on the face support pressure at failure, and also the relationship between the limiting support pressure and the tunnel diameter was found to be linear. Numerical simulations have become a more commonplace tool for assessing tunnel face stability over the last few years. From the literature, numerical simulations can be divided into two approaches; continuum and discrete. Continuum numerical analyses are performed using either the Finite Element Method (FEM) or the Finite Difference Method (FDM) ; whereas, discrete numerical simulations can be conducted using the Discrete Element Method (DEM) . Vermeer et al. (2002) generated a series of 3 D FEM simulations to assess tunnel face stability in drained conditions. For friction angles ( ' 20°) , the overburden was shown to have no influence on the limiting support pressure. The effect of cohesion has also been studied by Vermeer et al. (2002) and a design equation to calculate the limiting support pressure derived. This equation is expressed in terms of non-dimensional coefficients, namely soil stability number (N ) and cohesion stability number (Nc ) . The derived value of N is valid for ( ' 20° and C / D 1), while the value of Nc is applicable for the same range( ' 20° and C / D 1) with the condition of no surface load is applied (i. e. surcharge = 0) . The experimental study of Chen et al. (2013) concluded that the empirical equation of Vermeer et al. (2002) overestimates the limiting support pressure. Therefore, a 2

Tunnelling and Underground Space Technology 94 (2019) 103096

A.S.N. Alagha and D.N. Chapman

Fig. 1. Details of the analyses conducted during this study.

similar extensive study with more accurate results is highly recommended. Ukritchon et al. (2017) examined the undrained tunnel face stability in clay with a linearly increasing shear strength with depth by means of 3 D FEM. The effects of cover-to-diameter ratio, overburden stress factor and linear strength gradient ratio on the undrained tunnel face stability were studied. Furthermore, a new design equation has been proposed to calculate the factor of safety FS of the undrained tunnel face stability in clay with homogeneous and linearly increasing with depth strength profiles. Many other studies that included 3 D FEM can be found in the literature (e.g. Ruse, 2004; Ibrahim et al., 2015; Soranzo et al., 2015; Zhang et al., 2017). The numerical analyses in the majority of these studies were mainly generated to assess proposed analytical methods rather than being a detailed numerical parametric study in their own right. The Finite Difference Method (FDM) has been increasingly employed by researchers (e.g. Mollon et al., 2009, 2010; Li et al., 2009; Liu and Yuan, 2013; Senent and Jimenez, 2015; Zhang et al., 2015) for evaluating tunnel face stability. It is interesting to note that all the aforementioned studies were generated using the finite difference code FLAC3D software (Itasca Consulting Group, 2009), which implies a good outcome can be obtained from this software. Zhang et al. (2015) conducted a series of 3 D FDM modelling to analyse the face stability of shallow circular tunnels in frictional/cohesive – frictional soils. The study considered two cover-to-diameter ratios

(C/D = 0.5 and 1) and four sets of shear strength parameters ( = 20°, 40°, 17° and 25°; c = 0, 0, 7 and 10 kPa, respectively ) . The face collapse pressure and the failure zones ahead of the tunnel face were extracted from the numerical modelling results for all cases. Furthermore, a simple technique was suggested to determine the boundary strip of the failure zones at collapse. In contrast, little attention to date has been paid to the discrete element method (DEM) . Zhang et al. (2011) produced a series of 2 D DEM analyses using the PFC2D software (Itasca Consulting Group, 2004) to estimate the induced ground displacements associated with slurry shield tunnelling in clay. They defined a support pressure ratio as the ratio between the face support pressure and the initial ground horizontal stress at the tunnel axis. They concluded that the tunnel face is reasonably stable for values between 0.8 and 1.5. It should be noted that the results of this study have not been compared with the available literature. Later in the same year, Chen et al. (2011) generated a 3 D DEM simulations using PFC3D (Itasca Consulting Group, 2008) to study face stability of shallow tunnels. This study was limited to sand, and the authors concluded the modelling process was very time consuming. It is apparent from the literature that although there have been numerous studies investigating face stability in soft ground tunnelling, there are a number of aspects of this problem where further research would be beneficial. Two of these are considered in this paper: the first 3

Tunnelling and Underground Space Technology 94 (2019) 103096

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Table 1 Geotechnical parameters for the soil layers in Phase 3. Case No.

Set No.

Layer Name

1

a

Bottom Layer

Variable

c0' = 0.5, 1.5, 2.5, 3.5, 4.5

Top Layer

Constant

c1' = 2.5

Bottom Layer

Constant

Top Layer

Variable

b

c ' (kPa)

(°)

c ' (kPa)

c0' = c0'

c1'

' 0

( 2, ...,2) – –

c0' = 2.5 c1' = 0.5, 1.5, 2.5, 3.5, 4.5

c1' = c1'

c0'

' 1

= 15, 20, 25, 30, 35 ' 1

= 20

' 0

= 20

= 15, 20, 25, 30, 35

( 2, ...,2) 2

a

b

Bottom Layer

Variable

c0' = 2.5

Top Layer

Constant

c1' = 2.5

Bottom Layer

Constant

Top Layer

Variable

c0' = 2.5

c1'

= 2.5

c0' = 0

' 0

–

=0

'a 0

=

' 0

' 1

' 1

' 0

– – ' 1

=

( 5, ...,15)

–

c1'

(°)

' 1

= 15, 20, 25, 30, 35 ' 1

= 20

' 0

= 20

= 15, 20, 25, 30, 35

' 0

' 0

=

' 1

( 5, ...,15) – – ' 1

=

' 1

' 0

( 5, ...,15)

is to develop a robust and accurate equation to estimate the limiting face support pressure based on numerical modelling when tunnelling in homogeneous ground; the second is to provide practical considerations when tunnelling in layered ground, and particularly when the upper and lower strata intersect at the tunnel axis, where there is limited information in the literature. The aim of this research was to conduct a series of 3 D finite element modelling using the Midas GTS NX software (Midas IT., 2016) to address the two limitations determined in the literature presented previously. The first was to derive an empirical equation for estimating the minimum face support pressure that should be provided by a TBMtunnelling through a

study, a total of 170 analyses were conducted using the Finite Elemnt Method utilising the Midas GTS NX software. This software has been used and validated by many researchers (e.g. Kim and Tonon, 2010; Ibrahim et al., 2015; Alzabeebee et al., 2018a, b, 2019) to investigate different geotechnical problems related to tunnels and buried infrastructure. Three phases have been considered; the first phase was to determine the limiting support pressure in homogeneous frictional soil with different diameters, cover-to-diameter ratios and friction angles. The second phase was to identify the effect of cohesion on the limiting support pressure. Finally, phase three was to study the influence of different stratification scenarios (Case 1 and Case 2) on face collapse pressure. The geotechnical parameters for Phases 1 and 2 are presented in Fig. 1. However, the numerical values for these parameters for Phase 3 are shown in Table 1. It should be noted that these values are ‘typical’ soft ground parameters and have been chosen based on the parameters of the existing studies involving soft ground tunnelling (e.g. Leca and Dormieux, 1990; Vermeer et al., 2002; Zhang et al., 2017; Qarmout et al., 2019). Table 1 shows that for each case, the shear strength parameters of both layers were divided into two sets. In the first set (Set a), the ' properties of the top layer were kept constant (c1' 1) , while the bottom layer was assigned different strength parameters ' ' (c0' c0' 0 and increments 0 ) . Conversely, in the second set (Set b), the top layer was assigned different strength parameters ' ' (c1' c1' 1 and increments 1) , and the parameters of the bottom ' ' layer were kept constant (c0 0 ) . It should be noted that when changing in friction angle, the cohesion remained constant at 2.5 kPa for both layers, whereas the friction angle remained constant at 20° while changing the cohesion. It should also be noted that the cohesion was kept constant in both sets for Case 2.

'

) soil (i.e. purely frictional material ( ' soil) or a cohesive – frictional (c' a soil with some cohesion) above the groundwater table. Different strength parameters for each soil type, a range of cover-to-diameter ratios and tunnel diameters have been considered. To produce an appropriate equation, the relationship between these three factors, i.e. strength, cover-to-diameter ratio and tunnel diameter, and face collapse pressure needed to be determined. The resulting equation has then been compared to available analytical, experimental and numerical studies as described previously in the literature. Finally, the importance of arching effect has been explicitly highlighted, and the failure patterns in front of the tunnel face in frictional soils have been presented for different geometrical and geotechnical cases. The second limitation of the previous literature was to offer practical considerations when tunnelling in layered ground by conducting numerical simulations for two stratification cases (Fig. 1, Phase 3). The first case is where one (upper) stratum above the tunnel crown overlays a single (lower) stratum, while the second scenario is where one (upper) stratum above the tunnel axis overlays a single (lower) stratum, again in both cases the groundwater table is well below the tunnel invert. For each case, the effects of changing the shear strength parameters of the top and bottom strata on the face support pressure at failure were determined and the two cases compared with each other. Subsequently, the results of both cases have been compared and practical considerations discussed. The outcomes of the first case were also compared with the available upper bound solutions. Finally, the effect of a soil layer immediately below the tunnel invert on the face collapse pressure has been studied. Again, this has been achieved by changing the strength parameters of this layer and then comparing the results with those obtained from the homogeneous case.

2.2. Material constitutive model In the finite element simulations, the soil was considered to behave as an elastic perfectly plastic (elasto-plastic) material conforming to the Mohr-Coulomb failure criterion. This constitutive model, although having limitations, has been assessed previously as being precise enough to determine the tunnel face support pressure at failure (Vermeer et al., 2002). There are five main parameters required in this constitutive model; elastic modulusE , Poisson’s ratio , friction angle ' , cohesion c ' and dilation angle . The unit weight and lateral earth pressure coefficient K o are also required as general parameters. The shear strength parameters ( ' and c') were presented in the previous section. The unit weight was considered constant ( = 18 kN/m3) in all analyses including the layers in Phase 3. Ruse (2004) proved that

2. Numerical modelling of the tunnel face stability 2.1. Simulation programme Fig. 1 provides an overview of the analyses conducted during this 4

Tunnelling and Underground Space Technology 94 (2019) 103096

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E , , K o and have no influence on face collapse pressure. Therefore, the values of E , , K o and were considered fixed at 100 MPa, 0.3, 0.5 and 0. 01° , respectively for all analyses. The concrete tunnel lining was modelled as linearly elastic. The elastic modulus, Poisson’s ratio, thickness and unit weight of the lining were taken as 20 GPa, 0.2, 20 cm and 24 kN/m3 , respectively.

until failure occurs (Mollon et al., 2013). To explain this concept, Fig. 3 shows an example from numerical modelling results. The point of maximum face displacement is known as the control point (Fig. 3a). The pressure-displacement curve is produced by recording the displacement of the control point at each increment of pressure decrease (Fig. 3b). As the applied support pressure decreases, the resulting control point displacement increases. When the curve becomes horizontal, this implies failure has taken place. Therefore, the minimum required face support pressure Pf should be taken immediately prior to reaching this failure point (Vermeer et al., 2002).

2.3. Numerical model description The geometry, boundary conditions, meshes and construction stages of all numerical analyses are discussed in this section. As shown in Fig. 2, due to symmetry of the geometry, only half of the tunnel was used to simulate the face behaviour at failure. The boundary dimensions of the model were carefully chosen so as not to affect the face collapse pressure, i.e. by conducting analyses with different boundary dimensions and comparing the results, and also based on the recommendations of Ruse (2004). For the displacement boundary conditions, all translations and rotations were restricted at the base of the model (fixed), and the motions in the transverse direction were constrained (rollers) on the vertical faces of the model. The groundwater table was assumed to be well below the tunnel invert and therefore it was not included in the analyses. Saturated conditions and water ingress into the tunnel face were therefore not considered in the analyses. Four-node (2 D) first-order shell elements were employed to model the tunnel lining, and eight-node (3 D) first-order solid elements were used for the ground. The tunnel excavation was simulated using a simplified single-step process, assuming that the section of the tunnel (1D in length) was excavated simultaneously. To simulate this excavation sequence in the software, two stages were adopted. The first simulation step involved determining the initial anisotropic in-situ stress condition. The second step involved removing (deactivating) the solid elements of the ground within the tunnel to simulate the tunnel excavation. Instantaneously, the shell elements for the tunnel lining were activated and the uniform face support pressure was applied normal to the excavation face. This method of modelling the tunnel excavation and construction process is in line with previous research in this subject (e.g. Vermeer et al., 2002; Li et al., 2009; Ibrahim et al., 2015).

2.5. Validation of the numerical model This section presents a validation of the proposed modelling technique used in this study. Four physical model tests (Chambon and Corté, 1994; Oblozinsky and Kuwano, 2006; Idinger et al., 2011; Chen et al., 2013) have been reproduced numerically. The same dimensions, soil parameters and boundary conditions from these physical models have been used in the numerical modelling. Table 2 compares the face collapse pressures of these physical tests with the numerical values obtained based on the proposed numerical procedure. The face support pressure, Pf , is expressed in a normalised form, Pf / D , which has been commonly used in previous studies. As can be seen from Table 2, the numerical values of Pf / D that have been obtained based on the proposed numerical technique are in close agreement with the results of all the physical model tests. Consequently, it can be argued that this numerical technique can be confidently used to analyse tunnel face stability problems. 3. Numerical modelling results and discussions 3.1. Homogeneous ground (Phase 1 & Phase 2) 3.1.1. Influence of tunnel diameter D on collapse pressure Pf (Frictional, ' , soil) The relationship between tunnel diameter, D , and face support pres40°) , is illustrated in sure at failure, Pf , for various friction angles ' (20° Fig. 4. It is clear that tunnel diameter has a significant influence on the face support pressure at failure. For a specific friction angle, as the tunnel diameter increases, the limiting support pressure increases accordingly. However, increasing the friction angle reduces the effect of diameter. In other words, the maximum slope (maximum diameter effect) is achieved at the minimum friction angle ( ' = 20°), whereas the minimum slope (minimum diameter influence) is obtained at the maximum friction angle ( ' = 40°). Friction angle also has a large impact on the limiting support pressure. Under a certain diameter, with the reduction in friction angle, there is an increasing support pressure requirement. For example, at D = 10 m , the required value of Pf is approximately 11 kPa for ' = 40° , however, this value increases to approximately 43 kPa for ' = 20° . It is worth noting that the relationship between tunnel diameter and face collapse pressure is linear (doubling the diameter necessitates doubling the supporting pressure at failure). The quality of this relationship is about 99.5% or above for all the friction angles. The slight deviation in this linear relationship is due to the accuracy of predicting the actual collapse pressure using the technique shown in Fig. 3.

2.4. Definition of face support pressure at failure Pf As per the majority of the existing numerical studies (e.g. Vermeer et al., 2002; Li et al., 2009; Mollon et al., 2013; Ibrahim et al., 2015), the stress-controlled method has been used to determine the required face support pressure at failure in this numerical study. The concept of this technique is based on gradually decreasing the applied pressure

3.1.2. Influence of cover-to-diameter ratio C /D on collapse pressure Pf (Frictional, ' , soil) Owing to the linear relationship between tunnel diameter and face support pressure at failure, the normalised face support pressure at failure Pf / D is well suited to the problem and can be used instead of Pf . Fig. 5a shows the relationship between Pf / D and C /D for different 40°) . The same values of Pf / D have been friction angles ' (10° achieved for all diameters D (5, 7 and 10 m), and therefore only the results for D = 10 m are shown herein by way of example. It is noticeable that for friction angles ' 20° , the cover-to-diameter ratio C /D

Fig. 2. Typical 3-D finite element mesh used in the analyses. 5

Tunnelling and Underground Space Technology 94 (2019) 103096

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Fig. 3. Definition of face support pressure at failure, (a) Example finite element mesh showing the control point on the tunnel face; (b) Example showing an applied support pressure versus displacement plot.

has no influence on the normalised collapse pressure Pf / D , and therefore the tunnel face stability is entirely independent of C /D . This is due to the arching effect within the ground above the tunnel crown, i.e. a stress arch is produced above the tunnel crown, which carries the ground regardless of its thickness (see next section). This threshold ( ' 20°) can be seen more clearly in Fig. 5b, showing the relationship between Pf / D and friction angle, ' , for different C /D values. Furthermore, this behaviour corresponds to many other studies (e.g. Ruse, 2004; Anagnostou, 2012; Zhang et al., 2017). In contrast, for a friction angle ' = 10° , the relationship between Pf / D and C /D is completely different, i.e. the normalised collapse pressure increases steadily with C /D . Zhang et al. (2017) obtained similar patterns numerically. This relationship is probably caused by the low friction angle, which makes it difficult for arching to develop (see next section). The effect of C /D on Pf / D decreases gradually between ' = 10° and ' = 18°, before having no effect at ' = 20° . Finally, based on the numerical results in Fig. 5, it is clear that for a specific cover-to-diameter ratio, the normalised collapse pressure increases with decreasing friction angle, which highlights the effect of friction angle on face stability as mentioned in the previous section. Leca and Dormieux (1990) defined the normalised face collapse pressure Pf / D as the weighting coefficient N , which can also be referred as the soil stability number or N = Pf / D in a purely frictional soil. Accordingly, the face support pressure at failure for a purely frictional soil ( 'soil) can be calculated using Eq. (1).

Fig. 4. Effect of tunnel diameter on face support pressure at failure.

average of all the diameters and all the C /D values for each friction ' = 20°, the average N values (of angle. For example, for C /D = 0.5, 1, and 4 ) 0.245, 0.246 and 0.243 are for D = 10, 7 and 5 m , respectively. The average of these three values is 0.245, which is the value plotted in Fig. 6. The N values for the other friction angles (i. e. ' = 25° , 30°, 35° and 40°) were calculated using the same technique. Consequently, based on the values in Fig. 6, the relationship between soil stability number N and friction angle was found to be represented by equation (2).

(1)

Pf = DN

N =

As demonstrated in Fig. 5, the value of N is totally dependent on the friction angle if ' 20° [i. e. N = f ( ')]. Therefore, the relationship between soil stability number N and friction angle ' is shown in Fig. 6, which was drawn from the outcomes of 140 analyses (Phase 1). It should be noted that the N values in Fig. 6 were calculated as an

1 8sin

0.12

'

(2)

Therefore, the face support pressure at failure for a purely frictional soil and for any value of C /D 0.5 can be calculated using Eq. (3).

Table 2 Validation of the proposed numerical technique against some available physical model test results. Physical model

Chambon and Corté (1994) (Centrifuge tests) Oblozinsky and Kuwano (2006) (Centrifuge tests) Idinger et al. (2011) (Centrifuge tests) Chen et al. (2011) (1-g model tests)

Model properties

Pf / D

D (m )

C /D

(kN/m3)

c ' (kPa)

Φ′(°)

Physical model

Numerical model (this study)

5

1 2 2 4 1 1.5 1 2

15.3

0–5(1)

38–42(1)

15.3

0

42

14.7

0

34

16.5

0

37

0.072 0.055 0.058 0.042 0.060–0.080(2) 0.088–0.130(2) 0.076 0.072

0.033–0.076 0.026–0.059 0.053 0.041 0.088 0.095 0.080 0.069

4 5 1

6

Tunnelling and Underground Space Technology 94 (2019) 103096

A.S.N. Alagha and D.N. Chapman

Fig. 7. Relationship between depth and vertical stress showing the effect of arching.

et al., 2019a). An understanding of the arching behaviour is thus required for the design of tunnels and particularly for investigating tunnel face stability. Fig. 7 is used to highlight the redistribution of the vertical stresses, and thus the development of arching, as the cover depth of the tunnel increases. For the shown friction angles ( = 10° and = 40°) , there is a notable reduction in the vertical stresses as a result of arching. However, the stress reduction is less prominent in the weaker soil ( = 10°). This is because the shearing resistance in the soil against the downward relative movement is low (due to the lower friction angle), which results in a relatively small decrease (around one half of the original stress ‘ v,1 92 kPa ’) in the vertical stress and a very high settlement (S1 = 10 cm) at the tunnel crown level. On the other hand, the stronger soil ( = 40° ) shows the opposite behavior. The shearing resistance in the soil is very large and thus the arching (stress reduction) is more noticeable (resulting in approximately nine times less than the original stress ‘ v,2 17 kPa ’). This will also lead to a smaller settlement at the tunnel crown (S2 = 1 cm) . Therefore, arching is a crucial design criterion in determining the tunnel face stability in frictional soil, and ignoring or underestimating the arching effect results in an overestimation of the required tunnel face support pressure.

Fig. 5. Effect of cover-to-diameter ratio on normalised support pressure at failure, (a) shows the relationship between Pf / D and C / D for different friction 40°) ; (b) shows the relationship between Pf / D and friction angles ' (10° ' angle, , for different C / D values.

3.1.4. Influence of cohesion c ' on collapse pressure Pf (c′-φ′ soil) Cohesion plays an essential role in reducing the required face collapse pressure as demonstrated by many researchers (e.g. Leca and Dormieux, 1990; Vermeer et al., 2002; Anagnostou, 2012). This previous research also theoretically proved that the tunnel geometry has no influence on the effect of cohesion. However, Fig. 8 is presented to highlight the effect of tunnel geometry on the face support pressure in cohesive – frictional soil based on the numerical results of this research. Fig. 8a shows the relationship between the face collapse pressure and cohesion for different diameters (D = 5, 7 and 10 m). It is evident that the relationship between collapse pressure and cohesion is linear for all diameters. The values at zero cohesion were obtained from Phase 1 results (frictional soil). Therefore, the slope of each line represents the effect of cohesion on the face collapse pressure. It is clear that the effect of cohesion is the same for all diameters (same slope for all diameters) meaning the diameter has no influence on the effect of cohesion on face support pressure. Furthermore, to study the effect of tunnel depth on the effect of cohesion, Fig. 8b compares the effect of cohesion with the cover-to-diameter ratio for different friction angles( ' = 20°, 30° and 40°) . The effect of cohesion, Pf , is calculated herein as the difference between the numerical results of the frictional soil case (with c '=0 kPa) and cohesive – frictional soil case (with c '=5 kPa) . It is noticeable that for a certain friction angle, the effect of cohesion is constant with increasing C /D , which implies the tunnel depth has no influence on the effect of cohesion. To conclude, the effect of cohesion is independent of tunnel geometry (i. e. D and C / D ) .

Fig. 6. Relationship between soil stability number and friction angle.

Pf = D

1 8sin

'

0.12

(3)

3.1.3. Arching effect in frictional soil In the previous section, the soil overburden above the tunnel crown was shown to have no influence on the face collapse pressure for friction 20° due to the soil arching effect. This section illustrates how angles arching is developed in frictional soil. Soil arching usually takes place in geotechnical structures such as tunnels, buried pipes, retaining walls and piles. When a void forms due to the construction of a tunnel, for example, at a certain depth of soil, the soil moves into the tunnel void, and hence there is differential deformation in the overburden soil. This relative movement induces a stress redistribution in the soil, known as ‘arching’, which reduces the applied stress on any tunnel lining structure (Zou 7

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Fig. 9. Effect of cohesion on face support pressure at failure for different friction values.

Fig. 8. Effect of tunnel geometry on the effect of cohesion, (a) the effect of tunnel diameter; (b) the effect of cover-to-diameter ratio. ' To calculate the value of Pf in a c ' soil, many authors (e.g. Vermeer et al., 2002; Mollon et al., 2010; Anagnostou, 2012) have applied the commonly used Eq. (4).

c 'Nc

Pf = DN

Fig. 10. Relationship between cohesion stability number and friction angle.

(4)

where c ' is the effective cohesion and Nc is cohesion stability number. The value of N was determined in the previous section. To find the value of Nc and to identify the effect of cohesion on face collapse pressure, the relationship between face support pressure at failure and 40°) is shown in cohesion for various values of friction angle ' (20° Fig. 9 (for the example case of D = 10 m and C / D = 1). One can clearly see that the relationship between collapse pressure and cohesion is linear for all values of friction angle. The slope of this line can be referred as the cohesion stability number Nc . It is also clear that under a specific friction angle, the collapse pressure decreases with increasing cohesion. Furthermore, bearing in mind the tunnel geometry has no influence on the effect of cohesion, the value of Nc (gradient of the line) is entirely dependent on the friction angle [Nc = f ( ')]. Fig. 10 shows the relationship between cohesion stability numberNc and friction angle ' . The values in this Figure were obtained from the slopes of each line in Fig. 9. Furthermore, the relationship between cohesion stability number and friction angle is shown in Eq. (5).

Nc =

1.1 sin

0.5

'

Note: A negative value obtained from Eq. (6) means no support pressure is needed. To check the accuracy of Eq. (6) in predicting the face collapse pressure compared with the actual values obtained from the numerical analyses, Table 3 compares the face support pressure values computed using Eq. (6) with the actual modelling results for some cases. It is clear that the difference in the face collapse pressure values is very small (less than 1 kPa ) and it is always conservative (i.e. the equation gives higher values than the numerical model). It should be noted that the maximum difference in all the analyses conducted in this research was 2 kPa . Furthermore, the %error values range between 0.45% and 3.4% in Table 3, however, these values could increase up to 12% when the face support pressure values are very small, as a small difference could result in a large %error . 3.1.5. Failure mechanisms in frictional soil This section presents the failure patterns of the tunnel face in frictional soils with different friction angles ( = 20° and 40°) and for all cover-to-diameter ratios (i. e. C / D = 0.5, 1, 2 and 4) . In Section 2.2, the values of E , , K o and were considered fixed for all analyses as they do not affect the face collapse pressure (Ruse, 2004). However, the displacements of the soil in front of the tunnel face are affected by these parameters. Therefore, Fig. 11 is presented to illustrate the failure mechanism ahead of the tunnel face. The failure patterns in this Figure were extracted as vertical cross sections along the tunnel axis from the 3 D numerical models. In all cases, the failure zone is bulb-shaped, which complies with the existing experimental (Chambon and Corté, 1994; Lü et al., 2018) and numerical (Zhang et al., 2015) results. This bulb-shape is less pronounced in the case of C /D = 4 , which is probably due to the extreme arching effect at this depth. Furthermore, for all

(5)

By combining Eqs. (2), (4) and (5), the general empirical equation ' for calculating face support pressure at failure in a c ' soil is given by Eq. (6).

Pf = D

1 8sin

'

0.12

c'

1.1 sin

'

0.5

(6)

This equation is valid for the following conditions: ' 40° and c' 0 1. For 20° 2. For D 10 m and C / D 0.5

8

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et al., 2010; Chen et al., 2013) in terms of normalised face support pressure (Pf / D ). Due to the uncertainty in the measurements of the friction angle and cohesion, Chambon and Corté, (1994) have given a possible 42° and c ' = 0 5 kPa) . range for their tested material ( ' = 38° Therefore, the shaded area in Fig. 14 is employed herein to validate the results of this research against the experimental results of Chambon and Corté (1994). It is clear that all values of the normalised collapse pressure obtained by Chambon and Corté (1994) exist within the shaded area, meaning good agreement between the outcomes of this research and the experimental results. Fellin et al. (2010) conducted a series (31 in total) of 1-g model tests to investigate the tunnel face stability in sand. One can see that their experimental results are in good agreement with the values calculated from Eq. (6) of this research. In addition, Fellin et al. (2010) conducted many tests for each cover-to-diameter ratio and therefore they determined the values of Pf / D as mean values with ± 0.01 precision. It is noteworthy that both studies of Chambon and Corté, (1994) and Fellin et al. (2010) concluded that the cover-to-diameter ratio has a little impact on the normalised face support pressure at failure. The experimental results of Chen et al. (2013) are slightly less than the outcomes of Eq. (6). Nevertheless, the tested sand was not fully dried and the water content ranged between 0.3% and 0.4% , meaning their sand might have had a little cohesion of around 0.5 kPa or less. Consequently, the experimental results of Chen et al. (2013) lie within the ' =37° and c ' = 0 0.5 kPa) . possible range of this research (i. e. Finally, based on the comparisons with the existing numerical, analytical and experimental studies in Figs. 12–14, it can be argued that the current numerical study provides at least as good a prediction compared to many previous studies and better than most.

Table 3 Comparisons of the face collapse pressure (D = 10 m) between Eq. (6) and the actual FEM values. C /D

Φ′(°)

c ' (kPa)

Pf ,modelling (kPa)

Pf ,Eq .(6) (kPa)

%error

0.5 1 2 4

20

0

43.5 43.5 44 44

44.2

1.58 1.58 0.45 0.45

0.5 1 2 4

40

0

13 13 13 13

13.4

3.01 3.01 3.01 3.01

1

20

2.5 5 7.5 10

36.5 30 23 16.5

37.4 30.6 23.8 17

2.40 1.96 3.36 2.94

values of C /D , the failure zones start from the tunnel invert (for = 20°) and from approximately 0.12D above the tunnel invert (for = 40°) , and then propagates, however for C /D 1, the failure area does not reach the ground surface due the soil arching effect. Similar patterns have been obtained by many researchers (e.g. Zhang et al., 2011, 2015; Lü et al., 2018). Finally, it is clear that under a certain C /D ratio, as the friction angle increases(from 20° to 40°) , the size of the geometry of the failure mechanism decreases. 3.1.6. Validation of the derived empirical equation (Eq. (6)) Fig. 12 compares the outcomes of Eq. (6) for frictional soils (in terms of normalised face support pressure or N values) with previous theoretical, experimental and numerical studies. It is clear that the limit equilibrium study of Anagnostou and Kovari (1996) overestimates the face collapse pressure. This is because the use of a simplified linear distribution for the vertical stresses along the slip surfaces of the wedge, and therefore neglecting the arching effect in calculating the horizontal stresses. However, the modified model of Anagnostou (2012) is in close agreement with the outcomes of Eq. (6) (this research). This is probably due to the consideration of the horizontal stress arching across the tunnel face. Similarly, the results of the Kinemetical Element Method (KEM) of Qarmout et al. (2019), which takes into account the effect of horizontal arching along the tunnel face, are very close to the outcomes of this research. The upper bound study of Leca and Dormieux (1990) gives smaller values than those obtained from Eq. (6) and the majority of other studies. This is perhaps because of their simplified failure mechanism as they have used an elliptic shape for the tunnel face, which has a different area from the circular shape. Furthermore, the calculated values from Eq. (6) agree very well with those achieved by the 1-g model tests of Kirsch (2010). Finally, the values obtained from the numerical study of Vermeer et al. (2002) are slightly higher than those calculated from Eq. (6) (this research). As can be seen from Fig. 13, the values of the non-dimensional coefficient, Nc , obtained from this numerical study are very close to the majority of the existing numerical (Vermeer et al., 2002) and analytical (Mollon et al., 2010; Anagnostou, 2012; Qarmout et al., 2019) studies. However, the limit equilibrium study of Anagnostou and Kovari (1996) gives extremely conservative values. It should be noted that the Nc values of Vermeer et al. (2002) and Anagnostou (2012) are identical. This is because they have used the same theoretical approach in deriving the cohesion stability number resulting in the same formula (Nc = cot '). Furthermore, the upper bound study of Mollon et al. (2010) gives reasonable results as an upper limit solution, this is because they considered the entire circular tunnel face rather than an elliptical cross section. Therefore, the comparisons in Figs. 12 and 13 provide the evidence to suggest the results from Eq. (6) are very consistent with those from previous studies. Fig. 14 primarily compares the outcomes of this research with the results of various physical model tests (Chambon and Corté, 1994; Fellin

3.2. Layered ground (Phase 3) 3.2.1. Upper soil stratum overlays a lower soil stratum at the tunnel crown (Case 1) 3.2.1.1. Influence of the lower stratum (Set a). This part of the study aimed to identify the influence of layering within the soil intersecting the tunnel face, in this case, the strata boundary is at the tunnel crown. In this set of analyses, the effect of the shear strength parameters of the lower stratum on support pressure at failure was investigated, while keeping the parameters of the upper stratum constant. The shear strength parameters for this case, and for all other cases, were presented in Table 1 (Section 2.1). Fig. 15a compares the results of this study with previous published results (Tang et al., 2014; Senent and Jimenez, 2015) for the face support pressure at failure with different ' values of 0 . The face collapse pressure declines nonlinearly for ' 5° to 15°. The decrease in the collapse increasing 0 from ' 5° and 0° is steeper than that for the pressure with 0 between interval 0° and 15° , which indicates that 0' has a greater effect on face collapse pressure if the lower stratum is weaker than the upper stratum. The upper bound results of Tang et al. (2014) are lower than those obtained in this research. This is probably because of their simplified failure mechanism, which considers the circular tunnel face as an elliptical cross section. However, the study of Senent and Jimenez (2015) gives slightly higher values than the outcomes of this research. The reason for this improvement, i.e. the increase in face support pressure values compared to the Tang et al. (2014) study, is that Senent and Jimenez (2015) used one rotating block, which fully fits the circular tunnel face. In Fig. 15b, the influence of c0' on the face collapse pressure is presented. The face support pressure at failure decreases linearly with increasing c0'. Likewise, the numerical results of this study are located between the results of Tang et al. (2014) and Senent and Jimenez (2015). This indicates that the approach of Tang et al. (2014) underestimates the face collapse pressure, and thus this solution can be considered as potentially unsafe. However, the study of Senent and 9

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Fig. 11. Vertical cross sections along the tunnel axis showing the failure patterns in front of the tunnel face for different cases.

Jimenez (2015) showed increased face support pressures at failure and can therefore be considered as a potentially safe analytical solution. Finally, a comparison between Fig. 15a and b shows that the collapse pressure influenced more by variations in friction angle than by variations in cohesion.

3.2.1.2. Influence of the upper stratum (Set b). This section presents the results from analyses where the parameters of the lower stratum are kept constant and the effect of variable strength parameters for the upper stratum on the face support pressure at failure are studied. ' ' Fig. 16a and b show the influence of 1 & c1 on collapse pressure, 10

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Fig. 12. Comparison of N values with analytical, experimental and numerical studies from the literature.

Fig. 15. Influence of strength parameter variations for a layered soil scenario ' on the collapse pressure (Phase 3, Case 1, Set a), (a) Influence of 0 ; (b) ' Influence of c0 .

respectively. It can be seen that from Fig. 16a, the face collapse pressure ' 5° to 15°. declines almost linearly with the increase of 1 from Also, the numerical results of this study are smaller than those obtained by Senent and Jimenez (2015) and larger than the results of Tang et al. (2014). In Fig. 16b, the face support pressure at failure drops linearly with increasing c1'. Similarly, the results of this numerical study are marginally smaller than those of Senent and Jimenez (2015) and larger than the results of Tang et al. (2014). To compare the influence of the lower stratum (Set a) with that of the upper stratum (Set b) on the face collapse pressure, Figs. 15a and 16a are combined in Fig. 17a, and similarly, Figs. 15b and 16b are ' combined in Fig. 17b. For the variations in friction angle ( 0' or , 1) Fig. 17a indicates that the collapse pressure decreases from 28 kPa to 7.5 kPa (a reduction of 20.5 kPa ) over the range of 0' (Set a), whereas it declines from 21.5 kPa to 17.5 kPa (a reduction of 4 kPa ) ' throughout the range of 1 (Set b). For the variation in cohesion ' ' ( c0 or c1) , Fig. 17b illustrates that the face collapse pressure drops from 23 kPa to 15 kPa (a reduction of 8 kPa ) over the range of c0' (Set a), while it decreases from 20.5 kPa to 18.5 kPa (a reduction of 2 kPa ) throughout the range of c1' (Set b). Consequently, it can be concluded that the face support pressure at failure is influenced more by the shear strength parameter variations in the lower stratum (Set a) than in the upper stratum (Set b) for the parameter variations investigated in this study.

Fig. 13. Comparison of Nc values with analytical and numerical studies from the literature.

3.2.2. Upper soil stratum intersects the lower soil stratum at the tunnel axis (Case 2) This case can happen in practice depending on the relative strata thicknesses and constraints on the restrictions on line and level of a

Fig. 14. Comparison between Eq. (6) and various experimental studies.

11

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Fig. 18. Comparison between the two sets of Case 2 in terms of the effect on the collapse pressure.

especially with respect to the variation in friction angle due to its greater influence. Firstly, a comparison between Sets a and b for Case 2 is presented in Fig. 18. It is noticeable that the collapse pressure de' ' creases nonlinearly with an increase in 0 or 1 . Interestingly, the ' curves of both sets are almost identical over the range of 0' or 1. This indicates that, in contrast to Case 1, both sets of shear strength variations give almost the same collapse pressure. Therefore, in practice it does not matter if the variations in the friction angle are either in the upper or in the lower stratum (i.e. the arrangement of the strong and the weak strata is not important). It should also be noted that similar results for this case could not be found in the literature. To compare Case 1 with Case 2, and to give some practical direction if any of these stratification cases is encountered during tunnelling, Figs. 15a and 18 are combined in Fig. 19. Set a has been chosen for this comparison because of its greater influence in Case 1. It is evident that the collapse pressure is more sensitive to the first stratification scenario ' (Case 1) than to Case 2. For the range of 0 > 0 , which means the lower stratum is stronger than the top stratum, the second stratification scenario (Case 2) requires more support pressure to be provided by the TBM than that required in Case 1. However, if Case 2 could not be avoided, for example, if Case 1 was the existing stratification scenario during excavation and then Case 2 was encountered, one should be aware that the required support pressure would have to be increased. ' ° To give a clearer example, for 0 = 15 , the required face support pressure for Case 1 is 7.5 kPa , however, this should be increased to 12 kPa if the second stratification scenario (Case 2) is encountered.

Fig. 16. Influence of strength parameter variations in the upper stratum on the collapse pressure (Phase 3, Case 1, Set b), (a) Influence of 1'; (b) Influence of c1'.

3.2.3. Influence of a soil strata intersection at the tunnel invert on collapse pressure Pf To study the effect of the soil below the tunnel invert on face support pressure at failure, the results of two different stratification cases were compared. The first case consisted of a homogeneous frictional soil (i.e. the same soil stratum is from the ground surface to well below the tunnel invert) with a friction angle of ' = 40° . The second case consisted of two frictional soil strata (one strong stratum above the tunnel invert with ' = 40° overlying a weaker stratum below the invert of the tunnel with ' = 15°). The diameter of the tunnel was chosen to be 7 m for the sake of comparison. Both cases resulted in the same face collapse pressure of 9 kPa meaning the soil stratum below the tunnel invert had no influence on the face collapse pressure. Fig. 17. Comparison between the two sets of Phase 1, Case 1 in terms of the ' ' or effect on the collapse pressure, (a) Influence of 1 ; (b) Influence of 0 c0' or c1'.

4. Conclusion A series of 3 D finite element simulations have been conducted using the Midas GTS NX software to determine the collapse pressure of a tunnel face in both homogeneous and layered grounds. For the homogeneous ground, various shear strength parameters, cover-to-diameter ratios and tunnel diameters were considered. A new design

tunnel. Therefore, this section aimed to compare the face collapse pressure for the case of a two strata intersecting at the tunnel axis (Case 2) with the results obtained from the previous section (Case 1), and 12

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4.2. For layered ground (1) In Case 1, the face collapse pressure is much more affected by the changing strength parameters of the lower stratum than those of the upper stratum over the same value range. However, in Case 2, variations in the parameters in both the upper and lower strata give almost the same face collapse pressure. (2) When the strength parameter variations are in the lower stratum, the face collapse pressure is more sensitive to these variations in Case 1 than in Case 2. (3) If the lower stratum is stronger than the upper stratum, more support pressure is needed in Case 2 than in Case 1. (4) No matter what the variations are in the upper stratum or in the lower stratum, the face support pressure is affected more by the change in the friction angle than in the cohesion. (5) The soil layer below the tunnel invert has no impact on the face collapse pressure for the cases investigated in this study.

Fig. 19. Comparison between the two stratification scenarios with variations in ' 0.

equation for estimating the required face collapse pressure in a purely ' frictional soil or a c ' soil has been developed. Additionally, the failure patterns in front of the tunnel face in frictional soil were extracted from the modelling results. In terms of the layered ground, two stratification scenarios were adopted. The first (Case 1), was where the upper soil stratum intersects with the lower soil stratum at the tunnel crown, whereas in the second scenario (Case 2) both strata intersect at the tunnel axis. The impact of the ground below the tunnel invert on face collapse pressure was also studied. Based on the numerical results of face support pressure at failure, the following conclusions can be drawn:

Notwithstanding these conclusions, this research has some limitations that are listed as follows: 1. The face stability analysis in clay under undrained conditions has not been studied. 2. The support pressure at the tunnel face was assumed to be uniform, however in reality, the face support pressure applied by TBMs (slurry shields and EPB shields) varies from the crown to the invert. 3. Only dry and unsaturated soil conditions have been considered in these analyses as the groundwater table was assumed to be well below the tunnel invert level. 4. The non-homogeneity within any of the strata has not been considered, i.e. the layers were assumed homogeneous throughout their depth.

4.1. For homogeneous ground (1) In frictional soils, the relationship between tunnel diameter and face collapse pressure is linear (i.e. doubling the diameter leads to a doubling of the collapse pressure). ' 40°) and under a (2) Over a given range of friction angles (20° specific diameter D , the maximum diameter effect corresponds with the minimum friction angle ( ' = 20°), likewise, the minimum effect of diameter corresponds with the maximum friction angle ( ' = 40°). (3) For friction angles ' 20° and C / D 0.5, the overburden stress has no influence on the face collapse pressure due to the influence of arching. (4) The arching effect in frictional soil has been explicitly investigated and its importance for face stability has been highlighted. (5) Under a certain tunnel geometry (diameter and cover-to-diameter ratio, for example: D = 10 m and C / D = 1), the increase in friction angle reduces the required face collapse pressure nonlinearly. (6) The cohesion has a large influence in reducing the face collapse pressure. This effect is independent of the tunnel geometry, however, it is highly dependent on the friction angle (the effect of cohesion decreases with increasing friction angle). (7) The final form of the derived equation was as follows:

Pf = D

1 8sin

0.12

'

c'

1.1 sin

'

Acknowledgements The first author is grateful to the Hani Qaddumi Scholarship Foundation (HQSF) which covered the tuition fees, as well as living expenses, during his master’s degree at the University of Birmingham. Appendix A. Supplementary material Supplementary data to this article can be found online at https:// doi.org/10.1016/j.tust.2019.103096. References Alzabeebee, S., Chapman, D.N., Faramarzi, A., 2019. Economical design of buried concrete pipes subjected to UK standard traffic loading. Proc. Inst. Civil Eng.-Struct. Build. 172 (2), 141–156. Alzabeebee, S., Chapman, D.N., Faramarzi, A., 2018a. Development of a novel model to estimate bedding factors to ensure the economic and robust design of rigid pipes under soil loads. Tunnel. Undergr. Space Technol. 71, 567–578. Alzabeebee, S., Chapman, D.N., Faramarzi, A., 2018b. A comparative study of the response of buried pipes under static and moving loads. Transp. Geotech. 15, 39–46. Anagnostou, G., 2012. The contribution of horizontal arching to tunnel face stability. Geotechnik 35 (1), 34–44. Anagnostou, G., Kovari, K., 1996. Face stability conditions with earth-pressure-balanced shields. Tunnel. Undergr. Space Technol. 11 (2), 165–173. Bäppler, K., 2016. New developments in TBM tunnelling for changing grounds. Tunnel. Undergr. Space Technol. 57, 18–26. Berthoz, N., Branque, D., Subrin, D., Wong, H., Humbert, E., 2012. Face failure in homogeneous and stratified soft ground: theoretical and experimental approaches on 1g EPBS reduced scale model. Tunnel. Undergr. Space Technol. 30, 25–37. Broere, W., 2001. Tunnel Face Stability and New CPT Applications. PhD thesis. Delft University Press, Netherlands. Chambon, P., Corte, J.F., 1994. Shallow tunnels in cohesionless soil: stability of tunnel face. J. Geotech. Eng. 120 (7), 1148–1165. Chapman, D.N., Metje, N., Stärk, A., 2010. Introduction to Tunnel Construction, first ed. Taylor and Francis, London. Chen, R.P., Li, J., Kong, L.G., Tang, L.J., 2013. Experimental study on face instability of shield tunnel in sand. Tunnel. Undergr. Space Technol. 33, 12–21.

0.5

This equation is valid for the following conditions: '

40°andc' 0 . 1. For 20° 2. For D 10 m and C / D 0.5. (8) The results using this equation agree very well with the available experimental and theoretical approaches, therefore, this equation can be confidently used in practice to quickly calculate the required tunnel face support pressure that should be provided during shield tunnelling. (9) In frictional soils, the size of the failure mechanism in front of the tunnel face decreases with increasing friction angle. 13

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