1

The application of Tunnel Stability Factor (TSF) in estimating the convergence mechanism and face stability in weak rock conditions

Ilias K. Michalis Dipl. Eng. NTUA (Civil), MSc, DIC, Senior Manager Tunnels, Qatar Rail

Spyridon Konstantis Dipl. Eng NTUA (Civil), MSc, MSc, CEng-MICE, Technical Manager, Qatar Rail

ABSTRACT: The contribution examines the influence of overburden height H and size D of underground openings, as well as of the in-situ rock-mass strength σcm, in weak rock conditions, with the introduction of the Tunnel Stability Factor (TSF): TSF=σcm/ γΗaD1-a. Tunnel convergences and tunnel face stability conditions were studied extensively, through parametric numerical and probabilistic analyses, covering a wide range of ground conditions and underground openings of different sizes, both in shallow and deep overburden heights. Based on the results of the aforesaid analyses, dimensionless plots were derived: (i) tunnel convergence / tunnel equivalent radius vs. TSF, (ii) plastic zone radius / tunnel equivalent radius vs, TSF (iii) required face pressure / cohesion of the rock mass vs TSF. These plots, in spite of the very wide range of the examined conditions, present a very good indication of the average trend and can be used as practical engineering tools for preliminary tunnel design purposes.

1. GENERAL

According to the traditional methods of tunnels design, critical decisions about the effectiveness of the chosen primary support measures and the adopted excavation schemes are based mainly on the results of the application of rock-mass classification systems. These classifications can be made on the basis of geological observations and reliable laboratory test results, with the use of well-known systems, such as the Rock Mass Rating (Bieniawski 1989) or Geological Strength Index (Hoek and Brown 1997, Marinos et al 2000, 2005) systems. The advantages of such classifications are directly related to: (i) the derivation of the mechanical properties of the in-situ rock-masses, and (ii) the qualitative estimations of rock-masses general behavior during the underground construction works.

Until recently and before the extensive use of finite element methods in tunnels designs, factors such as the initial geostatic stresses (i.e. p0=γH) and the tunnel’s size (i.e. equivalent diameter D) were practically ignored, although their impact on to tunnels performance, in similar geotechnical conditions, had been observed and recorded / monitored in a large number of underground projects Additionally,

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it has to be pointed out that the aforementioned factors play a very decisive role, especially in the design of those tunnels, which are excavated in weak rock conditions.

Weak rock tunneling presents some special challenges to the geotechnical and tunnel engineer, since misjudgments in the criticality of the predicted deformation and failure mechanisms and the design of the support systems may lead sometimes to very costly failures. Moreover, the increasing demand to construct tunnels of large size in relatively poor ground conditions - under shallow and significant overburden heights - dictates the need to adopt new approaches during the early stages of the tunnels designs. These approaches have to incorporate the influence of depth and size of underground excavations, together with the application of rock-mass classification systems.

The purpose of this paper is to present the application principles of the Tunnel Stability Factor (TSF) in estimating (and quantifying preliminary) any deformation and possible failure mechanisms in weak rock tunnelling, by combining the results of the use of rock-mass classifications with the overburden heights and the different sizes of the underground openings.

2. THE CONCEPT OF TUNNEL STABILITY FACTOR (TSF)

Hoek (1999) argued that the severity of the tunnels stability problems in weak rock conditions, depends upon the ratio of the in-situ rock-mass strength, σcm, to the in-situ stress level, p0=γΗ. The merit of this argument was verified by the construction of two dimensionless plots (Figure 1 and Figure 2), based on the results of a Monte Carlo analysis, in which the input parameters for the rock-mass strength and the tunnel deformation were varied at random in 2000 iterations.

The upper and lower bounds of the used parameters in Hoek’s analysis are as follows: (a) Intact rock strength, σci=1-30MPa, (b) Hoek-Brown constant, mi=5-12, (c) Geological Strength Index, GSI=10-35, (d) In-situ hydrostatic stress=2-20MPa and (e) Tunnel radius, r0=2-8m.

Figure 1 gives a plot of the ratio of plastic zone radius to tunnel radius (rp/r0) versus the ratio of rock-mass strength to the in-situ stress (σcm/p0). Figure 2 is a plot of the ratio of the tunnel convergence to tunnel radius (ui/r0) against the ratio of rock-mass strength to the in-situ stress (σcm/p0). In spite of the wide range of the examined conditions, the reasonably well-fitted curves of both diagrams can show a good indication of the average trend. It must be pointed out that the complete analysis corresponds to cases of unsupported tunnels.

An important conclusion derived from Figures 1 and 2 is that the dimensionless quantities rp/r0 and ui/r0, increase very rapidly, once the rock-mass strength falls below 20% of the in-situ stress. According to Hoek(1999), the limit of σcm/p0≤0.20 can be considered as a practical criterion for the general stability of a tunnel excavated in weak rock conditions.

Following Hoek’s original idea, to adopt the ratio σcm/p0, as the controlling parameter of the tunnels general stability conditions, Michalis et al (2001), (2009) and Konstantis (2011), (2012) have already proposed further extensions, by including the size of the underground openings in the derivation of a new controlling engineering index, the Tunnel Stability Factor (TSF). TSF is expressed mathematically by Equation 1:

The inclusion of size (equivalent diameter D) of the underground openings in TSF, results from the

practical experience, which can be summarized as follows. In similar geological / geotechnical environment and at the same depth form the ground surface, tunnels of different size, exhibit modes of deformational behavior and failures of different scale and degree of criticality.

)1(1 aD

acmTSF

3

Figure 1. Relationship between size of plastic zone and ratio of rock-mass strength to in-situ stress (Hoek 1999)

Figure 2. Tunnel deformation versus ratio of rock-mass strength to in-situ stress (Hoek 1999)

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3. THE USE OF TUNNEL STABILITY FACTOR (TSF) IN ASSESSING THE TUNNEL CONVERGENCE MECHANISM

3.1 Data of parametric analysis

Figure 3 summarizes graphically the most important features of the general pattern of the convergence mechanism of a tunnel advancing through a weak rock-mass. According to Figure 3: (i) the tunnel convergence reach its total value at a distance of one and one half tunnel diameters (1.5D) behind the face and (ii) the tunnel convergence starts one half a tunnel diameter (0.5D) ahead of the advancing face.

.

Figure 3. Pattern of the convergence mechanism of an advancing tunnel in weak rock-mass conditions

The previously described tunnel convergence mechanism was extensively examined through a

parametric analysis with the use of finite element method. The performed numerical analyses were focused on to the examination of deformation patterns and shear failures of the rock-mass, surrounding unlined tunnels of different sizes in various overburden heights. The examined cases covered a large number of tunneling conditions, in which the depth of underground operations and the quality of the surrounding rock-mass are given in Table 1.

The factors: (i) tunnel convergence / tunnel equivalent radius and (ii) plastic zone radius / tunnel equivalent radius were derived in all the examined cases and a unifying framework is then developed to account for the severity of tunnel stability problems in terms of TSF values. The rock-mass shear behavior was modeled with the use of the Generalized Hoek-Brown failure criterion (Hoek & Brown 1997), according to Equations 2, 3, 4, 5 and 6:

)2(5.25

02.0GSI

ecicm

5

where: σcm = in-situ rock mass strength, σci = uniaxial laboratory compressive strength of the intact rock; and GSI = Geological Strength Index (Marinos and Hoek , 2000, Marinos et al, 2005), σ1, σ3 = maximum and minimum principal stresses at failure; mb = value of the Hoek-Brown constant of the rock-mass (see Eq. 4); s, α = constants which depend upon the GSI value of the rock-mass (see

Equations 5, 6).

Table 1. Examined tunneling cases

H (m)

GSI 50 75 100 150 200 300 400 500 15 5 5 5 5 20 5 5 5 5 5 5 20 10 10 10 10 10 10 30 5 5 5 5 5 5 30 10 10 10 10 10 10 30 20 20 20 20 20 30 30 30 30 30

40 10 10 10 10 10 10

40 20 20 20 20

40 30 30 30

40 40 40

50 10 10 10 10 10 10 10 10

50 20 20 20 20 20

50 30 30 30 30

50 40 40 40

50 50 50 The values in cells determine the adopted strength σci in MPa and the mi value has been taken 10. Examined tunnel diameters D= 4m to 10m.

)3(3

31

sm

ci

bci

)4(28

100exp

GSImm ib

)5(5.09

100exp25

aand

GSIsGSIFor

)6(200

65.0025GSI

aandsGSIFor

6

The deformation modulus of the rock-masses E, was calculated by applying the following

mathematical expression (Equation 7):

Equation 7 is based upon a large number of observations and back-analyses of excavation behavior in poor quality rock-masses and has been proposed for the in-situ modulus of deformation of weak rocks (Hoek & Brown, 1997).

In all the numerical analyses, the initial geostatic stress field was of gravitational type with characteristic values of the earth pressure coefficient: Κ0 = 0.5, 0.75 and 1.0. In this paper only the results of the analyses for K0 = 1.0 is presented. Nevertheless, similar conclusions apply for K0 = 0.5 or 0.75, since the aforesaid range of the examined K0 values seems not to influence seriously the behavior of the analyzed unlined tunnel cases.

3.2 Evaluation of the results of the parametric analysis A total of 74 finite element analyses have been performed, where the tunnels overburden heights and depths, the GSI and σci values of the surrounding rock-masses are shown in Table 1.

A useful means of studying the general tunnel behavior trend is to investigate the variation of representative dimensionless quantities for all the examined cases. These quantities were chosen as the ratios ui/r0 and rp/r0,, where ui=the tunnel crown convergence; rp = the plastic zone radius; r0=the equivalent tunnel radius.

Figures 4 and 5 give plots of rp/r0 and ui/r0 in terms of the assumed GSI values (as per Table 1) respectively. The erratic nature of the variation of rp/r0 and ui/r0 in both plots indicates that only the geological / geotechnical environment can by no means be considered as the unique controlling parameter of the convergence behavior of an unlined tunnel excavated in weak rock mass conditions. Obviously, the need to consider the effects of the overburden height and the tunnel’s size in obtaining a robust framework for the tunnels convergence patterns is clear.

Figure 4. Ratio of plastic zone radius to the equivalent tunnel radius versus GSI

)7(exp)(10100

40

10

MPainressedforGPainE ci

GSI

ci

0.00

0.50

1.00

1.50

2.00

2.50

3.00

3.50

4.00

4.50

5.00

10 15 20 25 30 35 40 45 50 55

r p/r

o

GSI

7

Figure 5. Ratio of tunnel convergence to the equivalent tunnel radius versus GSI

The governing role of the proposed TSF in evaluating the tunnels convergence mechanism has

been verified with the construction of Figures 6 and 7, where the ratios rp/r0 and ui/r0 are plotted against the calculated TSF values. It is remarkable, that in spite of the very wide range of conditions included in the finite element analyses, the results in both figures tend to follow a very similar trend. In addition, the fitted curves are considered that can give a satisfactory indication of the average tunnels behavior, without including the effect of their primary support.

Figure 6. Ratio of plastic zone radius to the equivalent tunnel radius versus TSF

0

0.02

0.04

0.06

0.08

0.1

0.12

10 15 20 25 30 35 40 45 50 55

ui/r o

GSI

rp/r0 = 1,79xTSF-0,43

0.00

0.50

1.00

1.50

2.00

2.50

3.00

3.50

4.00

4.50

5.00

0.00 0.25 0.50 0.75 1.00 1.25 1.50 1.75 2.00 2.25 2.50

r p/r

o

TSF = σcm/{γH0.75D0.25}

8

Figure 7 gives a set of approximate guidelines on the criticality of tunnels stability that can be encountered for different levels of the ratio ui/r0. More specifically these levels can be considered that give a first estimate of tunnel squeezing problems.

It is noted that from Figure 7 it can be derived the following:

For TSF≤0.20, 5%<ui/r0≤10%: Severe tunnel convergences. Squeezing rock mass conditions.

For 0.20<TSF≤0.30, 2.5%<ui/r0≤5%: Major tunnel convergences. Very weak rock mass conditions.

For 0.30<TSF≤0.60, 1%<ui/r0≤2.5%: Average tunnel convergences. Weak rock mass conditions

For TSF>0.60, ui/r0 ≤1%: Minor tunnel convergences. Fair rock mass conditions

Figure 7. Ratio of tunnel convergence to the equivalent tunnel radius versus TSF

The aforementioned strain levels are suggested on the basis of experience. According to the examined cases, strain levels of less than 1% occur in relatively better rock-masses with GSI ≥ 50 for all the considered overburden heights. Another useful conclusion derived from Figures 6 and 7 is that when TSF falls below 0.30, both plastic zones and tunnels convergences increase substantially, indicating that unless the appropriate support is installed, collapse mechanisms are likely to occur.

The curves shown in Figures 6 and 7 are defined by Equations 8 and 9.

ui/r0 = 0,0053xTSF-1,31

0%

1%

2%

3%

4%

5%

6%

7%

8%

9%

10%

11%

12%

0.00 0.25 0.50 0.75 1.00 1.25 1.50 1.75 2.00 2.25 2.50

ui/r o

TSF = σcm/{γΗ0.75D0.25}

ui/r0 between 5% and 10%

Severe tunnel convergences

ui/r0 between 2.5% and 5% Major

tunnel convergences

ui/r0 between 1% and 2.5%

Average tunnel convergence s ui/r0 less than 1%

Minor tunnel convergences

)8(79.1

43.0

25.075.0

0

Dr

rcmp

)9(0053.0

31.1

25.075.0

0

DHr

u cmi

9

4. THE USE OF TUNNEL STABILITY FACTOR (TSF) IN ASSESSING TUNNEL FACE STABILITY CONDITIONS

4.1 Tunnel Face stability assessment methods Tunnel face instability may lead to unacceptable relaxation of the advance core (Lunardi 2000) and in extreme cases to partial or total face failure (ITA-AITES 2006). The factors that determine the face stability conditions are associated with the geometrical dimensions of the tunnel and the hydro-geological/geotechnical characteristics of the ground.

The assessment of face stability conditions and necessary support pressure is a 3 dimensional problem and a detailed solution requires in principal 3D numerical analyses. However, these analyses have a number of certain drawbacks mainly associated with complicated input preparation and output presentation, increased computational effort, multiple simulation stages and incompatibility between improved accuracy and level of knowledge of ground conditions (Kavvadas. 2005).

In the literature, there are many available analytical methods that can be used to assess the face stability conditions. A summary can be found in (Guglielmetti et al., 2007). Most of these methods are based on limit equilibrium models or the upper and lower bound theorems of plasticity and give satisfactory results especially when based on 3D failure mechanisms (Russo, 2003).

In the present paper three different analytical methods have been used: (i) the Anagnostou & Kovari method (Anagnostou & Kovari, 1994 & 1996), (ii) an approach based on the convergence-confinement method (Kavvadas, 2005) and (iii) a combination of the two afore said methods (Kavvadas, 2002).

4.2 Anagnostou & Kovari (A&K) method

This limit equilibrium method was originally introduced by Horn (1961) and has its basis on the silo theory by Janssen (1895). The failure mechanism comprises a 3D wedge on the face and a prism above it (Figure 8). The loads acting on the wedge are its self weight, the resulting normal and shear forces along the failure (slip) surfaces, the applied face support and the overburden silo load imposed by the overlying prism. The methodology for defining the necessary face support is based on the 3D wedge limit equilibrium and the process is iterative. The method can be used to assess the face stability conditions and the required face support pressure for both mechanised (TBM) and conventional (SCL) tunnelling. The original method of A&K assumes that the ground on the face wedge is homogenous. Anagnostou & Serafeimidis (2007) and Serafeimidis et. al. (2007) have extended the method to account for heterogeneous tunnel faces.

Figure 8. Tunnel Face Stability mechanism (after Horn, 1961)

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2.2 Convergence – Confinement (C-C) method

The excavation of a tunnel causes changes in the stress regime of the surrounding rock mass and induces deformations, as a result of the gradual reduction of the in situ stress Po. This stress relaxation can be expressed as:

)10()1( PoPi

where: Pi represents the self supporting mechanisms of the ground with the form of a fictitious internal support pressure, Po is the in situ geostatic stress at the tunnel axis level and λ is the deconfinement ratio (AFTES, 2001).

For the case of an unlined tunnel, Hoek (2007) describes the variation of λ along the tunnel axis as follows:

λ = 0, at one half a tunnel diameter (0.5D) ahead of the tunnel face. Initial (at rest K0) geostatic stess conditions apply.

λ =1 at about one and one half tunnel diameters (1.5D) behind the tunnel face. In the advance core of a tunnel with unsupported face, the geostatic lateral confining stress σ3 in

the direction of the tunnel advance is gradually reduced and becomes zero at the face. In a simplified approach, the stress state at the unsupported face area represents uniaxial compression conditions (Kavvadas 2005).In this case, the initial safety factor FS0 of the tunnel face can be conservatively expressed as:

)11(

1

2

1 NsFSo

cm

where: Ns=2P0/σcm is the overload factor, σcm=2ccosφ/(1-sinφ) is the in-situ rock mass strength, c = rock mass cohesion, φ = rock mass friction angle and σ1=(1-λ)Po is the relaxed overburden stress at distance x=Dtan(450-φ/2)/2, ahead of the tunnel face, assuming a circular tunnel with diameter D and a failure mechanism according to the A&K method.

If support pressure σ3 is applied on the face the in-situ rock mass strength increases to the value of σc=σ3tan2(45+φ/2)+σcm. Τhe stress state at the tunnel face area represents triaxial conditions and the improved safety factor FS becomes (Kavvadas, 2005):

)12(2

45tan3

)1(

1 2

1

PoFSoFS

c

Equation 12 assumes that value of the deconfinment ratio λ is not affected by the application of the face support pressure σ3. In a conventionally excavated tunnel, where the face is, for instance, reinforced with fibreglass nails, this assumption is acceptable, as in this case the crown settlement of the tunnel is only slightly reduced (Kavvadas, 2005). In a mechanically (TBM) excavated tunnel, however, λ depends on the confinement level applied by the TBM (Aristaghes & Autuori, 2003). The critical value of the deconfinement ratio λcr is defined as (Kavvadas, 2002):

)13(1

1

21

Ns

Ns

kcr

11

where k=tan2(45+φ/2) ≥ 1 and the following cases are considered:

If λcr≥1 or if {λcr <1 and λ≤λcr}, then no plastic zone is developed around the tunnel (Elastic conditions apply)

If λcr <1 and λ>λcr, then plastic zone is developed around the tunnel (Plastic conditions apply) This simplified approach is in principal conservative as it does not account for the shear resistance

at the lateral failure surfaces. Moreover, the C-C method is based on a plain strain analysis of a deep circular tunnel in isotropic stress conditions. If ground water exists, the water flow through the tunnel

face introduces de-stabilizing seepage forces with intensity , where is the hydraulic gradient and γw is the ground water unit weight, resulting in increased required face support pressure σ3 (Kavvadas, 2002). The seepage forces have also a significant effect on the deconfinement ratio λ as they modify both the Ground Reaction Curve and the Longitudinal Deformation Profile (Lee & Nam, 2006).

4.3 Combination of A&K and Convergence-Confinement method

In this method (Kavvadas, 2002), which is based on the A&K method, the overburden load acting on the wedge/prism interface is the relaxed overburden stress σ1=(1-λ)Po at distance x=Dtan(45-φ/2)/2 ahead of the face and the shear stress acting along the lateral failure surfaces is:

)14(tan SFKFSc vf

where: K = the earth pressure coefficient, σv = the vertical stress at tunnel level and FS = the safety factor.

The nature of K depends primarily on the stress relaxation occurring during the tunnel advance. As previously explained, although initial (at rest K0) geostatic stess conditions can be assumed, at a distance greater than one half a tunnel diameter (0.5D), 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. However in the present parametric analyses, K0 was adopted. The original method of A&K considers ‘Ko’(λκ)=0,4 for the calculation of the shear forces along the surfaces of the adopted three dimensional wedge failure mechanism (Figure 8).

In case of ground water conditions, where hydraulic head differences exist, effective stress analysis (drained conditions) has to be carried out and the seepage forces have to be taken also into account (Anagnostou & Kovari, 1996).

5. PROBABILISTIC ANALYSES

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

The following ranges of the random variables were examined:

Tunnel diameter D=4m-10m

Overburden height (from ground level to tunnel crown) H=5m-30m

Cohesion c=5KPa-25KPa

Friction angle φ=20o-35o

Earth pressure coefficient at rest Ko=(1-sinφ)=0.43-0.66

Dilation angle ψ=0ο-φο/6

Elasticity modulus E=15MPa-60MPa

Poisson’s ratio ν= 0,15-0,25

Ground unit weight γ=18-22 ΚΝ/m3

12

The assumed shear strength and deformability parameters have been assumed for heavily weathered rock mass conditions.

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

The ground water conditions Hw, taken into account only in the A&K method, were treated both as random and independent variable (Michalis et. al., 2009) and no destabilising seepage forces were considered on the tunnel face.

The analyses were performed for unity safety factor (FS=1). This approach is in accordance with the general framework of Eurocode 7 (EC7) that requires Design Resistance Rd≥ Design Effect of Actions Ed. In principle, the tunnel face can be considered as a vertical slope. In EC7 the design requirements for the overall slope stability are defined through the Geotechnical (GEO) and Structural (STR) Limit States that concern failure (Ultimate Limit State) or excessive deformation (Serviceability Limit State) 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 considered. 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 shear strength parameters c and φ are considered design values in accordance with EC7.

6. RESULTS

In the graphs presented below, the dimensionless parameter α≤1 of the TSF was derived through an iterative process for the maximization of the coefficient of determination R2 of the scatter plots. In this process, the correlation coefficients (relative contribution) of the random variables H and D for the support pressure P that has to be applied on the tunnel face to ensure stability conditions were used. The support pressure P(P’) is refered to the tunnel axis.

6.1 Anagnostou & Kovari (A&K) method

The results of the probabilistic analyses in dry conditions are presented in Figure 99. Despite the wide range of the random variables, the trend is very well defined with R2 values in the order of 97%. It can be observed that no face support pressure P is required to ensure stability conditions for TSF (α=0.07) > 0.35. Moreover, for TSF (a=0.07) <0.15, the increase rate of P/c is higher and the scatter of the results increases.

For the case of ground water conditions the definition of TSF was based on the results of probabilistic analyses, where ground water heights Hw={H, 3H/4, H/2, H/4, H/8) were assumed. The ground water height Hw is measured from the ground water table elevation to the tunnel crown, i.e. Hw=H means that the ground water table is on the ground surface. In the case of Hw≠0, the dimensionless parameter α of the TSF can be expressed as a function of Hw/H (Konstantis, 2011):

)15(9974.0,0,8833.3

0356.0ln 2 RHw

HHwa

In the results presented in Figure 1010, Hw was treated as a uniformly distributed random variable.

Even though the average trend is well defined, it can be observed that the range of P/c is relatively wide for TSF values smaller than 0.45 and the scatter increases with the reduction of the TSF. The smaller the TSF, the wider the range of required P/c, thus the bigger the uncertainty and the lower the reliability of this probabilistic approach.

13

Figure 9. A&K method – Dry Conditions

Figure 10 A&K method – Ground Water Conditions

In order to reduce the range of P/c for low values of TSF, Hw was also treated as an independent variable (Michalis et. al., 2009) and the results are presented in the following Figure 1111 for different ratios of Hw/H. The range of P/c can be further reduced by considering ranges of φ for ratios of Hw/H (Konstantis, 2011).

97,0,13,12exp69,35 2 RTSFcP

97,0,22,12exp2,26 2 RTSFcP

96,0,42,12exp82,19 2 RTSFcP

9,0,*458,100612,0 21437,1

RTSFcP

99,0,8833,30356,0

ln 2 RH

Hwa

14

Figure 11. A&K method – Ground Water Conditions for ratios of Hw/H

6.2 Convergence – Confinement (C-C) method under Elastic conditions

The results for the elastic conditions are presented in the following Figure 1212, where it can be observed that no face support pressure P is essentially required for TSF (α=0,98) greater than 0.6.

Figure 12 Convergence – Confinement method - Elastic Conditions

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.

6.3 Combination of A&K and C-C method under Elastic conditions The results are presented in the following Figure 13.

HHwRTSFcP

,96,0,69,000435,0 21222,1

43,96,0,995,000732,0 21336,1 HHwRTSFcP

2,96,0,6356,10135,0 21515,1 HHwRTSFcP

4,96,0,23,30233,0 2178,1 HHwRTSFcP

8,97,0,45,503245,0 21072,2 HHwRTSFcP

2520deg,98,0,2865,100264,0 210965,1

RTSFcP

3025deg,98,0,667,100445,0 211368,1

RTSFcP

3530deg,98,0,245,200852,0 21196,1

RTSFcP

15

Figure 13 Combination of A&K and C-C – Elastic conditions

7. CONCLUSIONS

The major conclusion of this paper is that the Tunnel Stability Factor (TSF=σcm/ γΗaD1-a) can play a governing role in assessing adequately both the tunnel convergence mechanism and tunnel face stability problems, in a variety of weak rock mass conditions, overburden heights and tunnels’ dimensions.

Despite the different base and assumptions of the analytical methods, the results present a very well determined trend and validate the importance and usefulness of the TSF. Dimensionless graphs with analytical expressions have been produced that can be used in conceptual studies and preliminary design stages, saving a significant amount of effort and resources.

8. REFERENCES

AFTES (2001). Recommendations on the Convergence-Confinement method. Version 1. Anagnostou, G. & Kovari, K. (1996). Face stability conditions with Earth Pressure Balanced shields. Tunnelling and Underground Space Technology, Vol. 11, No. 2, pp.165-173. Anagnostou, G. & Kovari, K. (1994). The face stability of slurry-shield driven tunnels. Tunnelling and Underground Space Technology, Vol. 9, No. 2, pp.165-174. Anagnostou, G. & Serafeimidis, K. (2007). The dimensioning of tunnel face reinforcement. In: Bartak, Hrdina, Romancov & Zlamal (eds) ‘‘Underground Space – the 4th Dimension of Metropolises’’, 291-296. Taylor & Francis Group, London. Aristaghes, P. & Autuori, P. (2003). Confinement efficiency concept in soft ground bored tunnels. (Re)Claiming the underground Space, Saveur (ed.). Swets and Zeitlinger, Lisse, ISBN 90 5809 542 8. Bieniawski, Z.T. (1989). Engineering Rock Mass Classifications. A complete manual for Engineers and Geologists in Mining, Civil, and Petroleum Engineering United States of America: John Wiley & Sons, Inc. Guglielmetti, V., Mahtab, A. & Xu, S. (2007).Geodata S.p.A., Turin, Italy. Mechanised Tunnelling in Urban Areas – Design methodology and construction control, Taylor & Francis. Hoek, E. (2007). Practical Rock Engineering.

2520deg,94,0,457,201155,0 2135,1

RTSFcP

3025deg,93,0,467,402507,0 21517,1

RTSFcP

3530deg,92,0,02,804334,0 2169,1

RTSFcP

16

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