Numerical Investigation of Tunnel Face Sability using Forepoling or Fiberglass Nails

Pre-support of tunnel excavation faces using fiberglass nails or forepoling umbrellas aims to improve face stability in cases where an unsupported excavation face will develop uncontrollably large face extrusion, leading to face instability. The paper presents the results of a large set of parametric 3D numerical analyses of tunnel face excavation by the Finite Element Code Simulia Abaqus, using various degrees of reinforcement by fiberglass nails or forepoling umbrellas. The analyses use the average face extrusion as a measure of face stability, considering that face instability is associated with large face extrusions while the safety factor against face instability can be correlated with lower face extrusions in case of pre-supported tunnel faces. The results of the analyses are normalized and a set of semi-empirical formulae and design graphs are produced to calculate the safety factor of supported tunnel faces against instability and other useful quantities in tunnel design (average face extrusion, volume loss and deconfinement coefficient) as a function of ground strength, overburden depth and amount of face reinforcement. The analyses show that tunnel face reinforcement with FG nails is much more effective and less costly in securing face stability than the use of forepoling umbrellas. It is shown that even a coarse grid of FG nails can achieve better results than very heavy forepoling, and the difference in effectiveness is more pronounced in weaker ground and or deeper tunnels.


INTRODUCTION
Controlling face stability is very important in tunnelling, as cases of face instability are frequent, severely affect the cost and construction schedule of tunnels and can damage surface structures and utilities in shallow urban tunnels. In mechanized tunnelling with active face pressure (e.g. EPB and Slurry TBMs), the risk of face instability is controlled by the applied face pressure, which is usually adjusted empirically using past performance in "similar" conditions. In tunnelling with conventional techniques (SCL / NATM), face stability is also assessed empirically in most cases, by subjective comparison of the excavation face with past behaviour of faces under "similar" conditions (i.e., avoiding past failures), occasionally by simplified limit equilibrium analyses (e.g. Leca & Dormieux, 1990;Anagnostou & Kovari, 1996;Kim & Tonon, 2010), and rarely by numerical analysis or systematic measurements of axial face movements (face extrusion). When the risk of face instability is considered unacceptable in SCL / NATM tunnels, the size of the excavation face is reduced or active face support measures are applied, such as fiber-glass (FG) nailing, forepoling, or even leaving a ground wedge on the face. In practically all cases, the capacity of these measures is selected empirically: standard size forepoles (114/100mm spaced at 0.50m over a 120-degree crown arch) or FG nails at a "reasonable" spacing (one anchor per 2.5-5 m 2 of face area).
Forepoling umbrellas (FPU) include a set of stiff steel pipes installed ahead of the tunnel face, at the perimeter of the crown of the tunnel. The forepoles are 12m long steel tubes of diameter 60-170mm, installed with a typical spacing S = 40 -60 cm, at a slight inclination (5 to 7 degrees) with respect to the tunnel axis, and cover the upper part of the tunnel wall, at an angle of about 120 degrees. The steel pipes need to overlap along a length of 3 -5m in order to provide continuous face support as the tunnel face advances. The vertical load from the ground above the forepoles is carried by bending of the forepoles and is transferred to their rear support (steel sets of the temporary lining) and front end support (fixation in the ground ahead of the extruding face core), thus reducing the vertical pressure on the extruding core ahead of the tunnel face. The reduction of the ground load on the extruding core reduces face extrusion and radial wall convergence, thus improving face stability. The reduction of tunnel wall convergence by means of forepoling umbrellas has been studied by several researchers, using 2D and 3D numerical analyses (e.g. Oke et al, 2014).
Fiberglass nails (FGN) have also been used for the reinforcement of tunnel faces. FGN are inserted in holes drilled on the tunnel face at a regular grid, which are then fully cement grouted to ensure full bonding of the nails with the surrounding ground. Their length must be sufficient to permit anchoring beyond the extruding core ahead of the tunnel face. As FG nails are installed in regular distances as the tunnel face advances, a length overlap is crucial in controlling their effectiveness. FGN provide an equivalent pressure on the tunnel face equal to their total tensile force over the face area. Tension of the FGN develops by the tendency of the tunnel face to extrude; thus the FGN reduce face extrusion and radial wall convergence, improving face stability. Peila (1994) reports that use of fiberglass nails can reduce tunnel face deformations by up to 40%. Yoo (2002) based on 3D numerical analyses reports that the use of fiberglass nails can reduce tunnel face deformations by up to 70%. Ng and Lee (2002), based on 3D numerical analyses, also highlight the importance of face reinforcement by fiberglass nails in reducing tunnel face deformation and settlements at ground surface.
The effectiveness of various face support measures in improving tunnel face stability has been studied by several researchers, such as: Lunardi & Bindi, 2000;Kamata & Mashimo, 2003;Anagnostou & Serafeimidis, 2007;Kavvadas & Prountzopoulos, 2007;Juneja et al., 2010;Dias, 2011;Anagnostou & Perazzelli, 2015;Perazzelli & Anagnostou, 2020. Despite these efforts, the design of face support measures still remains largely empirical. The main reason of the extensive empiricism in assessing face stability, is that quantitative assessment of face stability requires the definition of a suitable "safety factor", and its calculation using complex three-dimensional (3D) numerical analyses with realistic constitutive models and suitably measured/estimated ground parameters. Although seemingly trivial, even the definition of a "safety factor" for face stability analyses is not always straight forward, let alone its numerical calculation. Georgiou et al (2021) have investigated face stability in tunnels with unsupported faces by performing a large set of parametric 3D finite element analyses of the excavation face. In this investigation, the "safety factor" against face instability was defined and calculated in various ground and tunnel conditions. The present study extends the methods developed by Georgiou et al (2021), to include tunnel faces supported with forepoling umbrellas (FPU) and fiber-glass nails (FGN). The investigation method is similar to that used for unsupported tunnel faces: a large set of parametric 3D finite element analyses of the excavation face is performed using the Finite Element Code Simulia Abaqus. In each case of ground conditions and overburden depth, a number of analyses are performed using various densities of FGN reinforcement or forepoling umbrellas to calculate the average face extrusion, which is obviously lower than that of the corresponding unsupported face. The results of the analyses are normalized and a set of semi-empirical formulae are produced to calculate the safety factor of the supported excavation face against instability and other useful quantities in tunnel design (average face extrusion, volume loss and deconfinement coefficient). As these assessments are based and extend on quantities defined in the paper by Georgiou et al (2021) for unsupported tunnel faces, a summary of these findings is given in the next section. Georgiou et al (2021) investigated face stability in tunnels with unsupported faces by performing 120 3D finite element analyses of the excavation face with the Finite Element Code Simulia Abaqus, varying ground conditions (in soils and weak rockmasses) and overburden depths. The analyses calculate the axial displacement (face extrusion) at all integration points on the tunnel face. These values are averaged over the tunnel face to give an "average face extrusion" (Uh) which is then normalized by the tunnel width (D) and a modulus-to-depth factor (E / po ) to give the dimensionless "face extrusion parameter" (Ωf):

FACE STABILITY IN TUNNELS WITH UNSUPPORTED FACE
where E is the elastic Young modulus of the ground (soil or rockmass) and pο = 0.5(1 + Ko)γΗ is the average overburden pressure at the tunnel axis (average of vertical and horizontal geostatic stresses).
The 120 values of the calculated face extrusion parameter (Ωf) were then plotted against various forms of strength-to-stress ratio, in search of an optimal correlation. The optimal correlation was achieved for the following dimensionless "face stability parameter" (Λf): (2) which combines ground strength (σcm), average overburden stress (po) and tunnel width (D). Georgiou et al (2021) have shown that this parameter can best describe the relevant parameters of the tunnel excavation face (face extrusion, volume loss, radial wall displacement, deconfinement ratio and safety factor against stability), in the sense that the calculated values in the FE analyses correlate well with the corresponding parameter (Λf). Control analyses have shown that the above formula can also be used in various tunnel shapes, including separate excavation of the top heading of a tunnel, via an equivalent tunnel size parameter D = 1.15 sqrt(A), where A is the section area of the tunnel or phase.
In soils, the ground strength was calculated by the Uniaxial Compressive Strength using the Mohr-Coulomb failure criterion: In weak or very fractured rockmasses, yielding was modelled by the Hoek-Brown failure criterion. In this case, "rockmass strength" (σcm) and "rockmass modulus" (E) were calculated by the following empirical formulae, in terms of the Geological Strength Index -GSI (Hoek & Diederichs 2006, Litsas et  In an unsupported tunnel face, the optimal correlation between face extrusion (Ωf) and face stability parameter (Λf) is given by the formula: This formula can be used to estimate the face extrusion parameter (Ωf) and, via equation 1, calculate the average face extrusion (Uh) for an unsupported tunnel face with given (Λf), i.e., a tunnel of size (D), overburden depth (H), in ground with strength (σcm).
At limiting face stability, equations (4), (1) and (2) give the limiting face extrusion and limiting ground strength in unsupported tunnel faces: where: (σcm)lim is the lowest ground strength to ensure limiting stability of an unsupported face for a tunnel with width (D) and overburden depth (H).
Since stable tunnel faces correspond to Λf > 1 and unstable faces to Λf < 1, the safety factor (SF) of a tunnel face (either unsupported or supported) can be defined by the ratio of the face stability parameter (Λf), defined by equation (2) and depending only on tunnel geometry and ground strength, to the corresponding limiting value (Λf)lim (corresponding to face extrusion (Ωf )lim = 1.4 for any tunnel face): For an unsupported tunnel face, (Λf)lim = 1 and thus (using equations 6 and 2): For supported tunnel faces, (Λf)lim < 1, due to the beneficial effect of the face support measures. Thus, the reduction of the limiting face stability parameter by the face support measures, increases the safety factor of an initially unstable unsupported tunnel face (Λf < 1), possibly rendering the tunnel face stable (if SF>1). The present paper calculates the decrease of the limiting face stability parameter by the face support measures, and thus calculates the safety factor of supported tunnel faces (from equation 6).

 
and the corresponding volume loss (VL), caused by (UR) which "squeezes" the core ahead of the tunnel face, giving face extrusion (Uh). Volume loss (VL) is defined as the reduction (ΔV) of the volume (V) of the extrusion core ahead of the tunnel face per unit volume of the core (VL = ΔV / V): Finally, the face stability parameter (Λf) is used to calculate the deconfinement ratio (λ) at the tunnel face by the formula: At limiting face stability (Λf = 1), the deconfinement ratio at the tunnel face is: λlim = 0.70.
In supported tunnel faces, quantities (UR), (VL) and (λ) can be calculated by the same formulae, replacing (Λf) by the corresponding limiting value (Λf)lim which depends on the face support measures.

NUMERICAL INVESTIGATION OF SUPPORTED TUNNEL FACES
The present study extends the above methods to include tunnel faces supported with forepoling umbrellas (FPU) and fiber-glass nails (FGN). The investigation method includes a large set of parametric 3D finite element analyses of the excavation face, using the Finite Element Code Simulia Abaqus. Figure 1 shows the shape of the tunnel section used in the analyses. Figure 2 shows the finite element mesh used in the analyses, with eight-node hexahedral finite elements. Due to symmetry about the vertical axis, only the left half of the model is analysed. Following a sensitivity analysis, the extent of the finite element mesh is sufficiently large (at least 3.5D in each direction) to minimize boundary effects in all directions. The conditions on the tunnel face are assessed after an initial tunnel advance by 5 to 7 D from the left (start) edge of the FE mesh, leaving a clear distance 3.5 to 5.5 D from the right end boundary (ahead of the tunnel face). For overburden depths H > 3D, additional overburden is simulated as a surface pressure, because ground stiffness and strength in this zone does not influence tunnel excavation. The tunnel section is excavated in a single phase (full face excavation) with excavation steps of 1m (equal to the size of the elements in the axial direction). In each excavation step, a relatively stiff (30cm thick) shotcrete liner is installed on the tunnel wall (full ring) 2m behind the excavation face. The shotcrete liner is modelled as linearly elastic 4-noded shell elements, with a relatively low concrete modulus equal to E = 15 GPa to account for concrete setting time during tunnel advance. Parametric analyses have shown that the thickness and stiffness of the shotcrete shell do not influence the magnitude of face extrusion (which is of main interest in the present study).
Forepoling tubes are modelled as horizontal beams spanning the tunnel crown at an angle 120 degrees (60 degrees at each side of the crown) with spacing S = 0.50m. The beams are very long, to avoid the need to install new forepoles every several meters of tunnel advance. Parametric analyses have shown that the replacement of the slightly inclined forepoles with horizontal beams does not influence the accuracy of the results. The beams are modelled as elasticperfectly plastic with elastic modulus E = 200 GPa, Poisson ratio ν = 0.25 and yield stress 235 MPa. The following six types of forepoling beams are used in the analysis (the lighter tubes aimed to confirm that the numerical solution converges to the unsupported face condition when the forepoling stiffness tends to zero): Fiberglass nails are modelled as horizontal truss elements, evenly spaced on the tunnel face. The nails are very long, to avoid the need to install new nails every several meters of tunnel advance. The nails are modelled as elastic -perfectly plastic with elastic modulus E = 40 GPa, Poisson ratio ν = 0.30, yield load Pu,FG = 200 kN and mobilisation factor 50% of the yield load (F=2). Three nail densities (n/A) are examined: one anchor per 1.0, 2.0 and 4.0 m 2 of tunnel face area. For each anchor density, the following dimensionless FGN density parameter (β) is defined: The numerical analyses include shallow tunnels in soils and deep tunnels in weak or heavily fractured rockmass. In each ground/overburden combination, a number of analyses are performed using various degrees of reinforcement by fiberglass nails (three nail densities) or forepoling umbrellas (six forepoling types) to calculate the average face extrusion of the supported tunnel face and compare it to the corresponding values of the unsupported face.
Shallow tunnels: In shallow tunnels, two cases of overburden depth (measured from the tunnel axis up to the ground surface), are examined: H = 20m (H/D=2) and H=30m (H/D=3), with ground unit weight γ = 20 kN/m 3 and two cases of the horizontal geostatic stress coefficient: Ko = 0.5 and 1.0 (to include the effect of the horizontal stress ratio). The ground is relatively stiff elastic -perfectly plastic, with elastic modulus (E), Poisson ratio v = 0.33, yielding according to the Mohr-Coulomb criterion (c = cohesion, φ = friction angle). Table 2 shows the six sets of ground parameters used in the analyses.  (3a). The E-modulus is rather large, because it represents the elastic response before plastic yielding. After yielding, stiffness is drastically reduced by the accumulation of plastic strains. The total number of numerical analyses of shallow tunnels was 24 (two tunnel depths, two Ko values and six sets of material parameters). Strength Index in the range GSI = 25 to 45 (three cases). The rockmass was assumed linearly elasticperfectly plastic, yielding according to the Generalised Hoek-Brown failure criterion (Hoek et al, 2002) with various parameters (mb, s, a). Table 3 shows the three sets of rockmass parameters used in the parametric analyses. The "rockmass strength" (σcm) and "rockmass modulus" (E) for the various GSI values are calculated by formulae (3b). The total number of numerical analyses for the deep tunnels was 18 (three tunnel depths, two Ko values, and three sets of material parameters).
Each numerical analysis calculates the axial displacement (face extrusion) at all integration points on the tunnel face. These values are averaged over the tunnel face to give an "average face extrusion" (Uh) which is then normalized by the tunnel width (D) and a modulus-to-depth factor (E / po ) to give the dimensionless "face extrusion parameter" (Ωf) defined in Equation (1). The face extrusion parameter of the unsupported tunnel face is denoted as (Ωfo) and the reduction of the face extrusion parameter with respect to the unsupported face is:  Table 1) reduce face extrusion more, while forepoles of very small stiffness give very small extrusion reductions, comparable with the unsupported face (ΔΩf = 0), confirming the robustness of the numerical algorithm. Larger values of the face stability parameter (Λf), corresponding to better ground strength / overburden stress combinations, cause smaller face extrusions for all forepoling types. For Λf > 2, face extrusion is very small even in unsupported faces (Ωfo < 0.5) and thus the additional decrease of face extrusion due to face reinforcement is negligible for all types of forepoles, rendering the use of face support by forepoling as practically useless.
In Figure 3b, the vertical axis is normalized with a forepoling stiffness parameter (α), defined by equation (12a). Parameter (α) was optimally selected to reduce the scatter of the face extrusion reductions (ΔΩf) shown in Figure 3a for the commonly used forepoles (do > 70mm); most of the data points deviating a lot from the best fit curve correspond to small diameter forepoles, not used in practical applications. The best fit curve in Figure 3b can be expressed by the formula (combined with equation 4):     Using Equation (1) Using Equation (14b), Figure 4 plots the relationship of the face extrusion parameter (Ωf) versus (Λf), for selected values of forepoling stiffness (α). Appreciable reduction of face extrusion occurs for Λf < 1, where the unsupported face has limiting stability. Figure 5 plots the limiting value of the face stability parameter (Λf)lim versus the forepoling stiffness parameter (α). The parameter (Λf)lim is the value of (Λf) causing face extrusion Ωf = 1.4 which, by analogy to the unsupported tunnel faces (equation 5), corresponds to limiting unstable tunnel faces. Thus, Figure  5 gives the limiting values of (Λf) where face instability occurs, with increasing stiffness of the forepoles. Stiffness values larger than α = 0.20 correspond to forepoling diameters larger than those typically used (up to about 200mm). For typical diameters of the forepoling tubes (d = 60 -120mm), the parameter α = 0.10 -0.15 which gives (Λf)lim = 0.90 -0.80 (i.e., 10-20% reduction with respect to the unsupported face).
Using equation (2), and the limiting values of (Λf) shown in Figure 5, equation (15) gives the limiting value of ground strength (σcm)lim , which is the lowest ground strength to ensure limiting face stability for a given tunnel size (D), overburden depth (H) and face support with forepoling stiffness parameter (α).  Figure 7a plots the calculated decrease of the face extrusion parameter (ΔΩf), normalized with the corresponding face extrusion of the unsupported face (Ωfo), versus the face stability parameter (Λf) (defined by equation 2) for tunnel faces supported with three densities of Fiber-Glass nails (FGN). Nail grids of higher density reduce face extrusion more, while larger values of the face stability parameter, corresponding to better ground strength / overburden stress combinations, cause smaller face extrusions for all grid densities. For Λf > 2, face extrusion is very small even in unsupported faces (Ωfo < 0.5) and thus the additional decrease of face extrusion due to face reinforcement is negligible for all FG grid densities, rendering the use of FG nails as practically useless.

ANALYSIS OF FACE SUPPORT WITH FIBERGLASS NAILS (FGN)
In Figure 7b, the vertical axis is normalized with a FGN density parameter (β), defined by equation (12b). Parameter (β) was optimally selected to reduce the scatter of the face extrusion reductions (ΔΩf) in Figure  7a. The best fit curve can be expressed by the formula (combined with equation 4):  Using Equation (16b), Figure 8 plots the relationship of the face extrusion parameter (Ωf) versus (Λf), for selected values of FGN densities (β). Appreciable reduction of face extrusion occurs for Λf < 1, where the unsupported face has limiting stability. For FGN density β > 0.675, the tunnel face is stable for practically any ground strength / overburden stress combinations (i.e., any value of Λf). Figure 9 plots the limiting value of the face stability parameter (Λf)lim versus the FGN density parameter (β). The parameter (Λf)lim is the value of (Λf) causing face extrusion Ωf = 1.4 which, by analogy to the unsupported tunnel faces (equation 5), corresponds to unstable tunnel faces. Thus, Figure 9 gives the limiting values of (Λf) where face instability occurs, with increasing FGN density. For FGN densities β > 0.92, the tunnel face is stable for practically any ground strength / overburden stress combinations (i.e., any value of Λf ). For typical FG nail densities (one nail per 12.5 ÷ 4 m 2 of tunnel face), the FG density parameter β = 0.50 -0.74, which gives (Λf)lim = 0.80 -0.50 (i.e., 20-50% reduction with respect to the unsupported face). Comparing these reductions with those of face support by forepoling tubes (10-20% reduction with respect to the unsupported face), it is concluded that the lightest FG nail density (1 nail per 12.5 m 2 of face area) achieves the same improvement of face stability as the heaviest forepoling tubes (diameter about 200mm), giving a clear advantage to face support by FG nails.
Using equation (2), and the limiting values of (Λf) shown in Figure 9, equation (17) gives the limiting value of ground strength (σcm)lim , which is the lowest ground strength to ensure limiting face stability for a given tunnel size (D), overburden depth (H) and face support with FGN density parameter (α).

OTHER RESULTS FOR SUPPORTED TUNNEL FACES
By analogy to the unsupported tunnel faces (equations 8, 9, 10) the following equations give the the radial displacement (UR) of the tunnel wall, the volume loss (VL) and the deconfinement ratio (λ) at the excavation face, for tunnels with supported faces, in terms of the face stability parameter (Λf) and the corresponding limiting value (Λf)lim which depends on the type and capacity of the face support ( figure 5 for forepoles and figure 9 for FG nails):  

APPLICATION EXAMPLE
The above formulae, produced by normalizing the results of many parametric finite element analyses of tunnel faces supported with forepoling or FG nails, can assist in the selection of suitable face support measures in unstable or marginally stable tunnel faces. Some typical examples are given below, with the objective to compare the effectiveness of forepoling versus FG nails in improving face stability. If the tunnel face is supported with very heavy forepoling tubes of diameter 168/154mm, spaced every 50cm on a top arch of 120 degrees (again 22 tubes in total), the forepoling stiffness parameter is α = 0.176 which which corresponds to a limiting value of the face stability parameter (Λf)lim = 0.761 and the safety factor of the tunnel face is SF = 1.21 > 1. This is the largest possible safety factor, since the heaviest possible forepoling was used. For this configuration, the radial displacement (UR) of the tunnel wall at the excavation face is UR / D = 4.3%, the volume loss is VL = 4.5% and the deconfinement ratio at the tunnel face is λ = 0.66. These values are very little lower than those corresponding to the 114/100mm forepoling, indicating that even very heavy forepoling does not increase the safety factor of the tunnel face significantly (SFo = 0.92 of the unsupported face, increased to SF = 1.21 for the most heavy forepoling).
Alternatively, if the tunnel face is supported with only n = 8 FG nails ( The difference in effectiveness of the FG nails is even larger in weak ground, where even the heaviest forepoling cannot achieve face stability. For example, in a tunnel excavated in heavily fractured rockmass with intact strength σci = 8 MPa, intact modulus Ei = 2 GPa, GSI = 20 at depth H =180m, the safety factor of the unsupported face is: SFo = 0.75 (unstable face). With the most heavy forepoling (168/154mm spaced at 50cm), the safety factor increases to SF = 0.98 and the face is still marginally stable. If the tunnel face is supported with FG nails, the same improvement (SF=0.98) can be achieved with only 8 FG nails (one nail every 8.5 m 2 of tunnel face). With 13 nails (one nail every 5.2 m 2 of tunnel face) the safety factor is increased to SF = 1.21 and with 22 nails (one nail every 3 m 2 of tunnel face) the safety factor is increased to SF = 1.97. In the last case, the radial displacement (UR) of the tunnel wall at the excavation face is UR / D = 2.8% , the volume loss is VL = 2.9% and the deconfinement ratio at the tunnel face is λ = 0.53. The advantage of FG nails in improving face stability in very poor rockmasses, compared to the use of forepoling umbrellas, is evident. Georgiou et al (2021) have investigated stability of unsupported tunnel faces by parametric 3D finite element analyses using the Finite Element (FE) Code Simulia Abaqus. The present study extends these methods to include tunnel faces supported with forepoling umbrellas (FPU) and fiber-glass nails (FGN). In each case of ground conditions and overburden depth, a number of FE analyses are performed using various densities of FGN reinforcement or forepoling umbrellas to calculate the average face extrusion, which is obviously lower than that of the corresponding unsupported face. The analyses use the average face extrusion as a measure of face stability, considering that face instability is associated with large face extrusions while the safety factor against face instability can be correlated with lower face extrusions in case of pre-supported tunnel faces. The results of the analyses are normalized and a set of semi-empirical formulae and design graphs are produced to calculate the safety factor of supported tunnel faces against instability and other useful quantities in tunnel design (average face extrusion, volume loss and deconfinement coefficient) as a function of ground strength, overburden depth and amount of face reinforcement.

CONCLUSIONS
The analyses show that tunnel face reinforcement with FG nails is much more effective and less costly in securing face stability than the use of forepoling umbrellas. It is shown that even a coarse grid of FG nails can achieve better results than very heavy forepoling, and the difference in effectiveness is more pronounced in weaker ground and or deeper tunnels.

ACKNOWLEDGEMENTS
The present work is part of the PhD thesis research of Mr. D. Georgiou. It was supported by scholarship funding from the Onassis Foundation and the Evgenides Foundation.             Relationship of (Ωf) versus (Λf) for selected values of the forepoling stiffness parameter (α). The thick black line corresponds to the unsupported tunnel face (α=0). The dashed line (Ωf = 1.4) corresponds to limiting face stability.

Figure 5
Correlation of the limiting value of the face stability parameter (Λf) versus the forepoling stiffness parameter (α). Values of α > 0.20 are not examined as they correspond to unrealistically stiff forepoling tubes (d > 200mm).

Figure 6
Correlation of the Safety Factor (SF) against face stability with the stability parameter (Λf) for selected values of the forepoling stiffness parameter (α).

Figure 7
a: Effect of Fiber-Glass Nail (FGN) density on the reduction of the average face extrusion, plotted versus the face stability parameter (Λf). b: Effect of Fiber-Glass Nail (FGN) density on the reduction of the average face extrusion, plotted versus the face stability parameter (Λf). The vertical axis is normalized with an optimally selected FGN density parameter (β).

Figure 9
Correlation of the limiting value of the face stability parameter (Λf) versus the FGN density parameter (β).

Figure 10
Correlation of the Safety Factor (SF) against face stability with the stability parameter (Λf) for selected values of the FGN density parameter (β).