Tunnel Face Reinforcement by Bolting-Numerical Modelling of Centrifuge Tests

This paper presents the results of numerical investigations on the deformation behaviour of a tunnel face reinforced by longitudinal pipes. The three dimensional finite difference calculations are based on series of reduced-scale centrifuge model tests (Al Hallak, 1999). Numerical analyses are developed and carried out with particular emphasis on the validation of the excavation method. Geomechanical parameters of two constitutive models are calibrated on triaxial and oedometric tests. The numerical models are then compared with the centrifuge model results. This comparison concerns especially the tunnel face extrusion, the ground surface settlements and the minimum pressure that guarantees the face equilibrium before failure. Bolting influence on these parameters is studied.


Introduction
The excavation of a tunnel produces a modification of the initial ground mass stress field.Deformations of soil (extrusion and preconvergence) are triggered in the core ahead the tunnel face, afterwards spread in the cavity and towards the surface.This happens especially in shallow depth tunnels.In urban zones, it is essential to control these ground surface settlements in order to avoid damages on neighbouring buildings.Several studies and real cases have shown that it is possible to control and limit the ground mass deformation response by acting on the core-face system rigidity (Lunardi, 2008).
In this aim, the application of fibre glass or steel bolts in tunnel has known, during these last twenty years, several applications.The preconfinement of the core with horizontal bolts has been generally combined with other methods, such as the realization of "pre-arches" using mechanical pre-cutting, sub-horizontal jet-grouting, steel or fibre glass elements (injected with grout) according to the type of ground and stress-strain conditions.This mixed technique assures, on one hand, stability of the excavation and safety of workers and, on other hand, it permits to control movements and to limit settlements on ground surface.
In order to optimize reinforcement technique, it is important to predict deformation response of the bolted ground mass to excavate.In addition to theoretical methods, experimental tests can be realized to study the threedimensional problem of tunnel digging.It is possible to study these phenomena by full-scale models or scale-down models.Because of cost and difficulties to realize tests on full-scale models, scale-down models are often preferred.This last type of physical models permit to better control the initial state of stress, they are faster and they can be used until failure.Nevertheless, different researches have dem-onstrated that scale-down models have to respect some similitude laws to guarantee mechanical behaviour similitude with the full-scale model.These similitude laws are based on dynamic equations, the law of mass conservation and the stress-strain laws of materials (Mandel, 1962;Weber, 1971;Garnier, 2001).In this context, different researchers have shown that experimental test results on scale-down models, submitted to gravity acceleration, reflect measures in situ with a better approximation.Centrifuge facility allows models to have a stress-strain behaviour in good agreement with reality.
The aim of this study is to develop and validate a numerical excavation simulation method of a tunnel reinforced using longitudinal pipes ahead the face.The following research is based on experimental tests realized on a centrifuge scale-down model (Al Hallak, 1999).A threedimensional numerical model is carried out using FLAC 3D in order to simulate the physical centrifuge model.Numerical results are then compared with scale-down outcomes.This comparison concerns especially the tunnel face extrusion, the ground surface settlements and the minimum pressure that guarantees the face equilibrium before failure.Bolting influence on these parameters is then studied.

Centrifuge tests on a scale-down model
The bolting face influence during tunnel excavation has been studied by Al Hallak (1999) on the stability and on movements of the surrounding soil mass.Centrifuge tests were realized using a scale-down model.
The physical model (Fig. 1) simulated a full scale tunnel having a 10 m diameter and an overburden equal to two times the diameter.The chosen scale factor was 1/50.Con-sequently, the scale-down model geometrical characteristics were the followings: • Diameter (D) = 200 mm; • Overburden (C) = 400 mm.
The rule, that guarantees the behaviour similitude between the scale-down model and the full-scale one, is the following: where g is the gravity and l represents a model dimension.Therefore, in order to respect this law, the studied model was submitted to an acceleration of 50 times the gravity.
The container and the tunnel lining were steel made and supposed to be not deformable.The tunnel face was made in a deformable latex membrane where air pressure can be applied.The ground mass, that enclosed the tunnel lining, was "Fontainebleau" sand, a fine homogeneous material often used in France for rheological investigations (D 50 = 0.2 mm et g = 15.7 kN/m 3 ).Several sensors placed in the model permitted to measure the: • Tunnel face displacements; • Surface Settlements; • Pressure on the latex membrane.
Two types of reinforcement were chosen: threaded steel bars or PVC sand-coated pipes.The latter permitted to simulate fibre glass reinforcement in a scale-down model.The bolting geometry adopted (diameter, length, density) was based on medium values normally used in tunnels construction and especially on the first Toulon tube tunnel (Dias et al., 2002).
The test was characterized by two phases: 1.The model was submitted to an acceleration increasing from 1 to 50 g.During this phase an air pressure was applied to the latex membrane in order to assure the face stability.Air pressure value in this phase was kept equal to the geostatic stress at the tunnel centreline (in this case, p 50g = 200 kPa).2. Once acceleration of 50 g was reached, the tunnel excation was simulated by decreasing the air pressure on the latex membrane until failure occurred.During this process, sensors measured soil displacements corresponding to decreasing pressure values.Different tests were realized varying reinforcement parameters such as the material composition, the number of bolts and their length (Table 1).
The tests showed that: • Reinforcement of the core ahead the tunnel face using longitudinal bolts reduces by an half the limit confinement face pressure and reduces significantly the extrusion and the ground surface settlements.• Increasing the bolting density permits to further reduce the face limit confinement pressure.• 130 mm length bars (0.7D) give the same resistance as longer bolts.This experimental study is the reference for our research.

Three-dimensional numerical model
Numerical simulations are carried out and compared to the above experimental results.A 3D finite difference numerical model was performed using FLAC 3D (Itasca, 2005).Geometrical characteristics and dimensions of the model are the same as the physical model.Nevertheless, only one half of the model (Fig. 2) is modeled due to its symmetry.The model is made of about 56000 zones (soil elements).
For the displacement boundary conditions, the bottom boundary is assumed to be fixed and the vertical boundaries are constrained in motion in the normal direction.An interface element is introduced around the cavity to allow the soil to slide on the lining.A friction angle between the tunnel lining and the sand is chosen to be equal to 2/3 rd of the sand friction angle.This value reproduces better the reality than a perfect adherence around the cavity wall.In Table 2, the principal interface characteristics are presented.
In the numerical simulation, the process is reproduced as in the experimental tests.The model is submitted to an acceleration equal to 50 g.A 200 kPa pressure is applied to the tunnel face.Then, tunnel digging is simulated decreasing pressure until failure occurres.
Surface settlements and face extrusion are analysed and compared with the experimental results.

Constitutive models and geotechnical parameters
Two constitutive models with an increasing complexity were chosen to take into account the behaviour of the Fontainebleau sand: • A linear elastic perfectly plastic model with a failure criterion of Mohr-Coulomb type; • An elastoplastic model with two mechanisms named CJS2 (Jenck & Dias, 2004;Maleki, 1998;Cambou & Jafari, 1988, Hejazi et al., 2008).The CJS2 model is a simplified version of the original CJS model developed by Cambou & Jafari (1988) for cohesionless soils.It is based on an elastic nonlinear part with two plasticity mechanisms: a deviatoric and an isotro-pic mechanism.It allows taking into account the nonlinearity of the behaviour at low stress level and the existence of dilatancy before the failure for dense or overconsolidated materials.The use of this model requires the determination of two elastic parameters, five deviatoric mechanism parameters and one isotropic mechanism parameter.The description of the model is given in Jenck et al. (2009).
The sand used in the scale-down test is considered as a pure frictional material.
The geotechnical model parameters were backanalysed from triaxial and oedometric tests on the Fontainebleau sand.
Triaxial tests were performed with three different confinement pressures: 30 kPa, 60 kPa and 90 kPa. Figure 3 illustrates the relationship between the deviatoric stress and the axial strain.Two triaxial tests results for a confinement pressure of 60 kPa are shown; the back analysis has been done on all the confinement pressures.A limit of these triaxial tests is the fact that the experimental applied confinement pressures are lower than the pressures values applied at the tunnel face during the centrifuge tests.The curves obtained with Mohr-Coulomb (M-C) and CJS2 simulations are compared with the experimental results.
In the M-C model, the initial Young modulus and plastic threshold values reflect experimental outcomes.However, it appears that this model simulates approximately the real soil behaviour due to its linearity.On the other hand the CJS2 model better represents the non linear soil behaviour since the model is based on a non linear elastic part and two plastic mechanisms.
The volumetric strain and the axial deformation are shown on Fig. 4. As observed regarding the deviatoric stress, the CJS2 model allows better representation of the real soil behaviour.
Oedometric tests were simulated using FLAC 3D .Loading and unloading cycles were applied.
Figure 5 illustrates the relationship between the mean pressure and strain.Curves obtained with M-C and CJS2 constitutive models are again compared with the experimental one.The M-C constitutive model linearity appears clearly in this graph.The Young modulus value for the M-C constitutive model was chosen equal to 33 MPa which corresponds to approximately the vertical stress at the upper part of the tunnel face.It is not possible using this constitutive model to distinguish loading and unloading phases, whereas the CJS2 model permits to better simulate ground response with accuracy.

Comparison between Numerical and Experimental Results
Physical tests were simulated using the constitutive model parameters adjusted to laboratory tests results.For tests n°I and n°II, no reinforcement was applied ahead the tunnel face.Afterwards, the bolting influence on ground movements was analysed.
Numerical results, such as face extrusion, surface settlements and limit equilibrium pressure until failure occurs, were compared to experimental results.

No reinforcement simulation -Reference case (Test II)
Figure 6 presents the confinement pressure vs. the tunnel centerline face extrusion.In an initial phase, when the pressure starts decreasing, the displacements are lim-  Plastic bulk modulus for the reference pressure P a : K 0 p (MPa) ited.Then extrusion increases until the limit pressure is reached and the face failure occurs.Numerical results show good agreement with the experimental ones.However, there is a significant difference between the two used constitutive model results.CJS2 model fit better the experimental results.
The relationship between surface settlements and the confinement pressure is shown in Fig. 7.The settlement refers to the surface point above the tunnel face (point C).
A considerable difference between numerical and physical results is noted for the M-C calculation.This difference increases with the internal pressure decrease.The numerical simulation underestimates the surface settlements.The calculation considering the CJS2 constitutive model is in better agreement with the experimental results.
The experimental limit confinement face pressure before failure occurs, is compared to the numerical one, as shown on Table 5.
In the numerical calculation, the limit confinement pressure is the last pressure for which the model reaches an equilibrium status.The equilibrium state in FLAC 3D is identified when the largest ratio of maximum unbalanced force to the average applied force is below a specified limit called "equilibrium ratio" (the default value is equal to 0.001).The unbalanced force indicates when a mechanical equilibrium state (or the onset of plastic flow) is reached for a static analysis.A model is in exact equilibrium if the net nodal force vector at each grid point is zero.The maximum nodal force vector is monitored in FLAC (called the unbalanced force) and will never reach exactly zero on a numerical analysis.The model is considered to be in equilibrium whenever the value of the maximum unbalanced force is small compared to the total applied forces in the problem (Itasca, 2005).In our calculations, equilibrium is reached when the unbalanced force ratio reaches 10 -8 (the default is set to 10 -5 ).Effective convergence must then be checked by verifying the grid-point velocities, which must be very small when static equilibrium is reached in our problem, less than 10 -11 m/ step.
Under the limit confinement pressure a failure occurs and a horizontal threshold appears (Fig. 6).
The limit pressure values, obtained with numerical simulations, are in good agreement with the physical values.The difference with the experimental value is respectively equal to 2.5% and 1.3% for the M-C case and the CJS2 one.
Figure 8 shows the failure mechanisms obtained with the two constitutive models.The failure mechanism is represented with the contour of velocity magnitude.
With the M-C constitutive model the failure mechanism has the same width but is higher than the CJS2 one.As it can be seen in Fig. 8, the failure only occurs in the upper half part of the tunnel face.The same phenomenon has been obtained by Mollon et al. (2013).This observation is important, some theoretical models (Mollon et al., 2009;Mollon et al., 2010;Dias et al., 2008) consider that the failure always concerns all the face.
The plastic zones at the failure moment are visible in Fig. 9 and are almost the same for M-C and CJS2.The classical chimney rupture appears and the horizontal extension width is about 100 mm in the numerical case against 67 mm measured in experimental test (Fig. 10).Such a rupture extension is reported by Chambon & Corté (1994).
The previous comparison between numerical calculation and experimental test shows CJS2 model allows reproducing better the physical outcomes.Therefore, all the   following studies are realized using only the CJS2 constitutive model.

Bolting reinforcement simulation
In this part, the bolting influence on ground movements is analysed.Physical test n°III, IV, V and VI were simulated.The interface between bolts and soil is modelled using zero cohesion and a frictional angle equal to 2/3 of the sand frictional angle.These interface behaviour properties are the key parameters allowing representation of the real bolting influence on face stability and soil displacements.Numerical studies of Sudret & De Buhan (1999), Bourgeois et al. (2001), Wong et al. (2000), Wong et al. (2004) and Wong et al. (2006) highlight this problem.
Other cases were also considered in order to study the relationship between the soil mass movements and the bolting characteristics such as the density, the bolt length and material.
In the physical model, 28 PVC sand-coated pipes were placed ahead the tunnel face in order to simulate the fibre glass reinforcement in a scale-down model.
The bolting characteristics used in the simulation of test III are shown in Table 6.
The bolts Young modulus is determined thanks to traction tests on the PVC pipes.
At prototype scale, the corresponding bolting density is about 0.4 bolt/m 2 .In numerical simulation only an half of tunnel is simulated as shown in Fig. 11.
Figure 12 presents the confinement pressure vs. the tunnel centerline face extrusion.Test n°II (without reinforcements) is compared with the test n°III.Both experimental outcomes and numerical results are shown.
Numerical calculation confirms the physical major result: the application of reinforcements in the core ahead  the face tunnel permits to reduce considerably the ground extrusion.Besides, the limit confinement pressure is reduced from 8 kPa to 5.5 kPa in the experimental test.A similar reduction interval is obtained with calculation.Numerical simulation corresponds with a good accuracy to physical outcomes.At limit pressure vicinity (p = 6 kPa), the difference between numerical and experimental extrusion is only of 5%.Therefore, the interface parameters chosen, between pipes and sol, correctly reflect the real ones.
The relationship between surface settlements at point C and the confinement pressure is shown in Fig. 13.As is in the no reinforcement case, the numerical simulation correctly predicts the measured surface settlements.
Nevertheless, calculation confirms that reinforcing the core ahead the tunnel face permits to reduce also the surface settlements, in addition to the extrusion and the limit confinement pressure.In Table 7 the decrease of these measures is shown.The numerical values are in good concordance with the physical ones.
Test n°IV differed from previous test only by the inclusions number.In this case 48 bolts were placed ahead the face tunnel with a corresponding bolting density of 0.6 bolt/m 2 at the prototype scale.
The comparison between the two different tests is shown in Fig. 14.Experimental results proved that increasing the bolts number permits to reduce the extrusion face movement and to decrease the limit confinement pressure from 5.5 kPa to 4 kPa.Even in this case, numerical simulation confirms these outcomes with a accuracy.At limit pressure, the difference between numerical and experimental extrusion is equal to 4%.
Nevertheless, the outcomes proximity between the test III and IV seems to indicate that there is a limit beyond which it is not effective to increase the bolts density.
Therefore, in order to study the relationship existing between the ground movement (extrusion and settlements) and the reinforcement density, a parametric study is realized varying only the inclusions number.
Figure 15 shows the maximum tunnel face extrusion vs. the bolts density.Due to gravity, this value is always located below the tunnel axis.The extrusion outcomes are compared at the same confinement pressure value of 8 kPa.This enables to present on the graph the case without reinforcement as well.Bolting experimental tests outcomes are also visible on the graph and the agreement with numerical results is confirmed.
First of all, the results show that a small reinforcement density (d = 0.2 bolt/m 2 ) is already able to reduce the centreline face extrusion of 34% in comparison to the no reinforcement case.
Besides, a significant curve trend change appears on the graph, highlighted by the intersection of two straight lines.The trend modification corresponds to a bolting den-      and Plaxis3D the first and second Toulon tube excavation.This outcome is also supported by the results of field measurements reported by Poma et al. (1995).The field measurements were carried out for a 18-m-diameter tunnel with a cover depth of 100 m constructed in a clay formation.Finally, Lunardi (2008) obtained a threshold value equal to 0.4 bolt/m 2 using a procedure based on the interpretation of the extrusion curves carried out from triaxial extrusion tests.
The reinforcement density influence on surface settlement (point C) is analysed as well (Fig. 16).Even in this case, a curve trend change of the surface settlement appears beyond a bolting density value equal to about 0.4 bolt/m 2 .Nevertheless, the trend modification seems to be less obvious than in extrusion movement.
Experimental test V maintained the same bolting characteristics (material, density) of test III, except the pipes length reduced from 300 mm to 130 mm.Unfortunately, during this test, sensors malfunctioned and the results could not be analyzed.Only the failure extension ahead the face tunnel showed that it was similar to test III.
As for the density parameter, the pipe length influence on ground movements was studied using numerical calculation.In addition to the test V simulation, other calculations are done, varying only the PVC pipes length (density = 0.4 bolt/m 2 ).The relationship between the tunnel centerline face extrusion and the bolts length is shown in Fig. 17.The extrusion results are compared at the same confinement pressure value of 8 kPa in order to present on the graph also the no reinforcement case.The experimental extrusion measured in test III is also visible on the graph and is in good agreement with the numerical curve trend.
The two straight lines highlight a significant curve trend change corresponding to a bolts length equal to about 0.4 D. This result suggests that the length of reinforcements has to be longer than 0.4 D in order to have a favourable effect on face stability.
This conclusion is in accordance with the current design practice, which adopts a minimum overlap length of approximately 0.3-0.4D (Peila et al., 1996).Yoo & Shin (2003) obtained a comparable critical value of pipes length in their 3D numerical investigations on the deformation behaviour of tunnel face reinforced with longitudinal pipes.A threshold value lightly higher (L c = 0.5 D) was found in the numerical study of Wong et al. (1997), Dias (1999) and Dias et al. (2002).This value has confirmed thanks to centrifuge experimental tests and 2D calculations (distinct element method) carried out by Kamata & Mashimo (2003).
The bolting length influence on surface settlement (point C) is analysed as well (Fig. 18).Even in this case, a significant settlement reduction appears with bolts longer than 0.4 D.
Test VI was the experimental test using threaded steel bolts instead of PVC pipes to reinforce the core ahead the tunnel face.
The bolting characteristics used in this simulation are shown in Table 8.
In addition to bolt diameter, another major difference in respect to PVC pipes is the Young modulus.The steel tension characteristic was adopted in the numerical calculations.Figure 17 -Axial face extrusion vs. bolts length.
In order to study the stiffness reinforcement influence on ground movements and limit pressure, results of numerical test VI are compared to another case simulation.This calculation using PVC reinforcements has the same characteristics as steel pipes (number, diameter, length...) but a different Young modulus (3 GPa).
Figure 19 presents the confinement pressure vs. the tunnel centerline face extrusion.The comparison between physical model results and numerical simulations shows that, until the limit pressure vicinity, numerical calculation seems to slightly overestimate the experimental extrusion values.Nevertheless, the gradient curves are similar.Be-sides, the limit numerical confinement pressure is exactly the same as physical one (p = 4.8 kPa).
Comparing the steel bolting case with the PVC one, it appears evident that the differences are not important.Although the extrusion with the steel bolts is smaller, the maximum difference is equal to 20%.Besides, the numerical calculation reaches the equilibrium at the same limit pressure value.
For the excavation geometry and soil characteristics analysed in this study, the increase of the bolt stiffness is irrelevant in respect to the face stability and ground movements.The surface settlements analysis confirms this conclusion.
The PVC pipes, with a Young modulus of 3 GPa, are effective against the stress in the soil caused by the tunnel excavation.The main parameters controlling the soil deformations and the face stability are the length bolts and the density reinforcement.

Conclusions and Perspectives
Numerical calculations were developed and carried out with the particular emphasis on the validation of the excavation simulation method, simulating various reducedscale centrifugal model tests (Al Hallak, 1999).Comparing numerical results to physical model results, the study focuses on the bolting influence on the face extrusion, the ground surface settlements and the limit confinement pressure.The influence of reinforcement characteristics such as density, length and stiffness are studied as well.Based upon the comparison with the centrifuge tests, some conclusions emerge.
An elastoplastic constitutive model with two mechanisms permits to simulate with accuracy the soil response in the case of a tunnel excavation.It is able to correctly represent the non linearities of the soil.The common linear elastic perfectly plastic M-C constitutive model is also tested.
Numerical simulation of the no reinforcement case, with CJS2 model, shows that the face extrusion displacements, the surface settlements and the limit pressure are in agreement with the experimental values.The M-C constitutive model appears to be too simple and is insufficient when dealing sith soil deformations.
Numerical simulations of bolting face tests are in agreement with physical results.The tunnel behaviour is especially well simulated at the limit pressure vicinity and in the failure zone.The interface properties, between pipes and soil, are the key parameters allowing representation of the real bolting influence.Numerical outcomes confirm that the reinforcement of the core ahead the face tunnel permits to reduce the limit confinement pressure, the face extrusion and the surface settlements.
Numerical analysis of bolting parameters shows that: • A small reinforcement density (d = 0.2 bolt/m 2 ) is already able to reduce substantially the centreline face extrusion in comparison with the no reinforcement case.A   significant further reduction of ground movements is evident until a bolts density equal to 0.4 bolt/m 2 .Beyond this value, the reinforcement efficacy decreases sensibly.
• The bolt length has to be longer than 0.4 D in order to have a favourable effect on face stability and ground movements (extrusion and surface settlements).• In relation to the soil characteristics and the excavation geometry analysed in this study, increasing bolt stiffness is irrelevant in respect to the face stability and ground movements.The PVC pipes are sufficiently effective to limit and control the strains caused by the tunnel excavation.• Same results on the influence of bolting density, bolt length have been obtained by others authors using onsite field measurements, laboratory experiments and numerical studies.

Figure 6 -
Figure 6 -Confinement pressure vs. axial face displacement, comparison between experimental and numerical results.

Figure 7 -
Figure 7 -Surface settlement (point C) vs. confinement pressure, comparison between experimental and numerical simulation results.
Figure 12 -Confinement pressure vs. axial face displacement, comparison between experimental and numerical results (Test II and Test III).

Figure 11 -
Figure 11 -PVC pipes repartition in test simulation n° III.

Figure 13 -
Figure 13-Surface settlement (point C) vs. confinement pressure, comparison between experimental and numerical simulation results (Test II and Test III).

Figure 19 -
Figure 19 -Confinement pressure vs. axial face displacement, comparison between experimental and numerical results (Test VI) and stiffness reinforcement influence.

Table 3 and
Table 4 summarize, respectively for Mohr-Coulomb and CJS2 models, parameters that fit the experimental curves.

Table 7 -Bolting influence.
Figure 14 -Confinement pressure vs. axial face displacement, comparison between experimental and numerical results (Test III and Test IV).