Face stability is a key factor for shallow tunnels excavated in urban areas, under severe environmental constraints and in many cases through challenging and difficult ground conditions.

Tunnel face instability may lead to unacceptable relaxation of the advance core (Lunardi, 2000) and also, in extreme cases, to partial or total face failure (ITA-AITES, 2006).

The factors that influence the face stability conditions are associated with the geometrical characteristics of the tunnel and the hydro-geological/geotechnical conditions of the ground.

The assessment of face stability conditions and necessary support pressure requires in principle three-dimensional (3D) numerical analyses. However, these analyses have a number of certain drawbacks mainly associated with complicated input preparation and output presentation, time consuming analysis, multiple simulation stages and incompatibility between improved accuracy and level of knowledge of round conditions (Kavvadas, 2005).

In conceptual studies and preliminary design stages where available geotechnical data are limited and in many cases parametric analyses have to be carried out, the use of simple and practical assessment tools would lead to a significant reduction in numerical effort/analytical studies and associated time and cost.

The outcome of the present work is a series of dimensionless graphs that can be used to preliminarily assess the support pressure that needs to be applied on the tunnel face to ensure stability conditions.

**Face stability assessment methods**

In the literature, there are many analytical methods available for assessing face stability conditions (Guglielmetti et. al., 2007). Most of these methods make use of limit equilibrium models as well as upper and lower bound theorems of plasticity.

In the present work, three different analytical methods have been used: a) the 3D wedge limit equilibrium model with side friction, b) the convergence-confinement method and c) a combination of the two.

*3D wedge limit equilibrium model with side friction – Silo load*

This method was originally introduced by Horn (1961) and has its basis on the silo theory by Janssen (1895). Anagnostou and Kovari (1994) have extended it and in the international literature it is now well known as Anagnostou & Kovari method.

The failure mechanism comprises a wedge ahead of the tunnel face and a prism above it (see figure 1, below).The loads acting on the wedge are:

•Its self weight,

•The resulting normal and shear forces along the failure surfaces ADE, BCF and ABFE,

•The face support applied on ABCD

•The overburden load acting on the interface CDEF.

The methodology for defining the necessary support pressure P is based on the wedge limit equilibrium and is described in detail in Anagnostou and Kovari 1994.

This method assumes that the ground on the wedge and the prism is homogenous. Anagnostou and Serafeimidis (2007) have extended the method to account for heterogeneous, layered ground on the tunnel face.

*Convergence-confinement method*

The excavation of a tunnel induces ground deformations and causes changes in the stress regime of the surrounding ground as a result of the gradual reduction of the in situ stress (Po). This stress relaxation is depicted in figure 2 and can be expressed as:

Where Po is the isotropic geostatic stress and ? is the deconfinement ratio.

Pi represents the self supporting mechanisms of the ground with the form of a fictitious internal support pressure.

Figure 3 shows the longitudinal deformation profile of an advancing tunnel, where the ratio of the crest settlement ur to the maximum crest settlement ur,max (ur/ur,max) can be related with the distance from the advancing face (x/R).

The stress state at the unsupported face of an advancing tunnel represents, in a simplified approach, uniaxial compression conditions (see figure 4).

According to Kavvadas (2005), the initial safety factor FSo of an unsupported tunnel face can be expressed as:

Where Ns=2Po/scm is the overload factor, scm=2*c*cosf/(1-sinf) is the ground strength and s1=(1-?)Po is the overburden stress at distance x=H*tan(45-f/2)/2 ahead of the face.

If support pressure s3 is applied on the face, the ground strength increases to sc=s3*tan2(45+f/2)+scm and the stress state at the face represents triaxial conditions (Kavvadas, 2005). Assuming that ? is not affected by the application of the support pressure, the improved safety factor FS is:

The critical value of the deconfinement ratio ?cr is defined as (Kavvadas, 2002):

where k=tan2(45+f/2) = 1 If ?cr=1 (elastic ground) or if {?cr <1 (elastoplastic ground) and ?=?cr}, then no plastic zone is developed around the tunnel (elastic conditions)

If ?cr <1 and ?>?cr (elastoplastic ground), then plastic zone is developed around the tunnel (plastic conditions)

3D wedge limit equilibrium model with side friction – Deconfinement load This method is based on the Anagnostou and Kovari method. The main differences pertain to the definition of the overburden load and the shear stresses on the failure surfaces (Kavvadas, 2002).

The overburden vertical stress sv acting on the interface CDEF (see figure 1) is defined as:

Where Po is the geostatic stress at tunnel level and ? is the deconfinement ratio at distance x=H*tan(45-f/2)/2 ahead of the face (see figure 4).

The shear stress acting along the failure surfaces is defined as:

Where K is the earth pressure coefficient, svo is the vertical in situ stress and F is the safety factor.

The nature of K depends primarily on the stress relaxation in the advance core. In front of the advance core, the stress state represents conditions at rest Ko (see figure 2, above). Due to the radial pre-convergences that start to develop in the advance core, the stress state could be assumed to represent active earth pressure conditions Ka. In the present work, however, Ko was adopted in the parametric analyses.

**Tunnel Stability Factor**

The Tunnel Stability Factor (TSF) has been proposed by (Mihalis et. al., 2001) for the assessment of behaviour of underground openings in weak rock conditions and is defined as:

Where scm is the ground strength, ? is the ground unit weight, H is the height of overburden and D is the equivalent tunnel diameter.

**Probabilistic analyses**

The probabilistic analyses were carried out with the software @Risk (www.palisade.com) and Monte Carlo simulation was used as the sampling technique for the uniformly distributed random variables.

The following cases were examined:

•Tunnel diameter D=4m-10m

•Overburden heights (ground level to tunnel crown) H=5m-30m

•c=5KPa-25KPa

•f=20º-35º

•Ko=1-sinf

•Dilatancy angle ?=0º-fº/6

•E=15MPa-60MPa

•v= 0,15-0,25

•?=18-22 KN/m3

According to Mollon et. al. (2009), the assumption of negative correlation between c and f gives greater reliability of the tunnel face stability. In the present work, however, c and f were assumed to be uncorrelated.

The ground water table Hw (see figure 1) was taken into account only in the Anagnostou and Kovari method and was treated both as random and independent variable.

Moreover, no seepage forces were considered on the tunnel face (zero hydraulic gradient).

**Safety considerations**

The analyses were performed for unity safety factor (FS or F=1). This approach is in accordance with the general framework of EC7 that requires Design Resistance Rd= Design Effect of Actions Ed.

In essence, the tunnel face can be considered as a vertical Slope. In EC7 the design requirements for the overall slope stability are defined through the GEO and STR Limit States that concern failure (ULS) or excessive deformation (SLS) of the ground and/or of any supporting structure, respectively.

The methods adopted here for the assessment of face stability conditions do not consider the structural contribution of the support systems but only the resulting pressure, hence the STR limit state is not relevant.

Moreover, the methods are based on failure mechanisms rather than deformation driven modes.

Consequently, the overall stability of the tunnel face can be considered here as a GEO ULS situation.

The ground parameters are considered design values in accordance with EC7 and the partial material factors depend on the Design Approach (DA) adopted.

**Results**

In the following graphs, the exponent a of the TSF was derived through the correlation coefficients of the input random variables for the output support pressure P, i.e. through the relative impact of the tunnel diameter D and the overburden height H to the support pressure P that has to be applied on the tunnel face to ensure stable conditions.

*3D wedge limit equilibrium model with side friction – Silo load*

The results for dry conditions (Hw=0) (Figure 5) present a very well determined trend of behaviour. It can be observed that the scatter increases with the reduction of the TSF. The smaller the TSF, the wider the range of required support pressure ratio P/c, thus the bigger the uncertainty and the lower the reliability. According to the graph, no face support pressure is required to ensure stability conditions for TSF values greater than 0.35.

In order to determine the exponent a of the TSF for the case of ground water conditions, a series of analyses were carried out for Hw={H, 3H/4, H/2, H/4, H/8). In the case of Hw?0, the TSF exponent a can be expressed as a function of Hw/H (see also figure 6):

In figure 7 the ground water table Hw was treated as a uniformly distributed random variable. It can be observed that the range of P/c is relatively wide for TSF values smaller than 0.5.

In order to reduce this range, Hw was treated as an independent variable (Mihalis et. al., 2009). Figure 8 shows the results for five different ratios of Hw/H.

The P/c range can be further reduced by considering ranges of f for specific ratios of Hw/H. Figure 9 shows the results for Hw=3H/4.

*Convergence–confinement method*

The results for elastic conditions are presented in figure 10. For the plastic conditions it was not possible to correlate the ratio P/c with the TSF. However, ? can be correlated with the overload factor Ns=2Po/scm. Figure 11 presents the results for different ranges of f.

The necessary support pressure P can then be estimated as a percentage of the in situ stress Po (see figure 12).

As ? increases, P decreases. However, high values of ? correspond to high values of Ns which in turn are associated with tunnel cross sectional instability. Consequently, the convergence – confinement method under plastic conditions should be used with extreme care for the assessment of face stability conditions and must be accompanied with cross sectional analysis to evaluate the overall behaviour and stability conditions of the tunnel. The results of the probabilistic analyses for elastic conditions are given in figure 13.

**Application**

The application concerns a typical tunnel of D=7m diameter and an overburden height above the tunnel crown of H=20m, excavated through a ground mass with cd=15Kpa, fd=26o and ?=20KN/m3. The deformability parameters E and v are both within the value range of the examined cases. According to figure 5, TSF=0.32 and P is close to zero for the case of the 3D wedge model-Silo load. Following the convergence-confinement method under elastic conditions (figure 10, below), we get TSF=0.12 and P/c=5.5~7.5, hence the necessary support pressure P must be in the range of 82.5-112.5KPa.

Finally, for the 3D wedge model-deconfinement load under elastic conditions (see figure 13, below), we get TSF=0.165 and P/c=3~4.5, hence the necessary support pressure P must be in the range of 45-67.5KPa.

In the ground water conditions, the ground water table is assumed to be Hw=15m above the tunnel crown. From equation 8 and for Hw/H=0.75, a=0.78 and therefore TSF=0.15. From figure 7, P/c=5~15 which means that P must be in the range of 75-225KPa. From figure 8, where Hw is treated as an independent variable, we get P/c=8-14 for the same situation and therefore P must be in the reduced range of 120-210KPa. By considering ranges of f for ratios of Hw/H (figure 9, for Hw=3H/4), we get P/c=10-13 and the range of P for the same situation can be reduced to 150-195KPa.

**Conclusions**

In the present contribution, probabilistic dimensionless graphs have been produced, based on three different analytical methods, for the preliminary assessment of face stability conditions and necessary support pressure for shallow tunnels in various homogenous soft grounds.

The results from all three methods present a well-determined trend with clear upper and lower value bounds.

The differences in the range of the necessary support pressure P can be attributed to the nature and basis of each analytical method.

The next step will be to statistically process the results and validate them through case histories and 3D numerical analysis in order to produce practical design charts.

These charts could be used in conceptual studies and preliminary design stages saving a significant amount of effort and resources.

Figure 1, Tunnel face failure mechanism (Anagnostou & Kovari, 1994) Figure 2, Gradual reduction of in situ stress Po relative to the distance from the advancing face of an unsupported tunnel (Hoek, 2007) Figure 3 Longitudinal deformation profile of an advancing tunnel (Kavvadas, 2005). Figure 4, Stress state in the advance core of an unsupported tunnel face (Kavvadas, 2005) Fig 5. 3D wedge – silo load – dry conditions (Hw=0) Fig 6. Hw/H vs. exponent a of the TSF for Hw=0 Fig 7. 3D wedge – silo load – ground water conditions (Hw=0) Fig 8. 3D wedge – silo load – ground water conditions (Hw=0) Fig 9. 3D wedge – silo load – ground water conditions (Hw=3H/4) Fig 10. Convergence – confinement (elastic) Fig 11. Convergence – confinement (plastic) Fig 12. Convergence – confinement (plastic) Fig 13. 3D wedge – deconfinement load – elastic conditions Form_1 Form_2 Form_3 Form_4 Form_5 Form_6 Form_7 Form_8