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Convergence confinement analysis of a bolt-supported tunnel using the homogenization method Article  in  Canadian Geotechnical Journal · January 2011 DOI: 10.1139/t06-016

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3 authors: Henry K. K. Wong

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Convergence–confinement analysis of a boltsupported tunnel using the homogenization method Henry Wong, Didier Subrin, and Daniel Dias

Abstract: The behaviour of tunnels reinforced with radially disposed fully grouted bolts is investigated in this paper. Perfect bonding and ideal diffusion of bolt tension are assumed, so that the bolt tension can be assimilated to an equivalent uniaxial stress tensor. An analytical model of the convergence–confinement type is proposed that accounts for the delayed action of bolts due to ground decompression prior to bolt installation. This factor leads to nonsimultaneous yielding, and more generally, a different stress history for each constituent, requiring special treatments in the incremental elastoplasticity calculations. Nonetheless, the resulting model remains sufficiently simple, and an analytical solution is still accessible. Charts are provided to allow for parametric studies and quick preliminary designs. Comparisons with 3D numerical calculations show that the model gives precise results if the correct convergence at the moment of bolt installation is used as an “external” input parameter, validating the homogenization approach. An approximate methodology based on previous works is proposed to determine this parameter to render the proposed model “selfsufficient.” Its predictions are again compared to 3D numerical computations, and the results are found to be sufficiently accurate for practical applications. Key words: reinforcement, anisotropy, analytical, lining, yield, elastoplasticity. Résumé : Dans cet article, on étudie le comportement de tunnels armés de boulons complètement injectés disposés radialement. On suppose qu’il y a une parfaite adhésion et une diffusion idéale de la tension des boulons, de telle sorte que la tension des boulons puisse être assimilée à un tenseur équivalent de contraintes uniaxiales. On propose un modèle analytique de type convergence–confinement qui prend en compte le retard dans l’action des boulons dû à la décompression du terrain avant l’installation des boulons. Ce facteur conduit à des déformations plastiques non simultanées, et plus généralement à une histoire de contraintes différente pour chaque constituant, requérant des traitements spéciaux dans les calculs incrémentiels élastoplastiques. Néanmoins, le modèle résultant demeure suffisamment simple et une solution analytique demeure accessible. On fournit des graphiques pour permettre des études paramétriques et des conceptions préliminaires rapides. Des comparaisons avec des calculs numériques 3D montrent que le modèle donne des résultats précis si, au moment de l’installation des boulons, la bonne convergence est utilisée comme paramètre d’entrée extérieure, validant l’approche d’homogénéisation. On propose une méthodologie approximative basée sur des travaux antérieurs pour déterminer ce paramètre pour rendre le modèle autonome. Ses prédictions sont encore comparées aux calculs numériques 3D, et on trouve que les résultats sont suffisamment précis pour les applications pratiques. Mots clés : armature, anisotropie, analytique, revêtement, déformation plastique, élastoplasticité. [Traduit par la Rédaction]

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Introduction The philosophy of tunnel lining design was subjected to a fundamental revision with the advent of the new Austrian tunnelling method (Rabcewicz and Golser 1973) and the convergence–confinement theory (Egger 1973, Panet and

Guenot 1982, Brown et al. 1983). Instead of a mere source of loading, it was realized that the surrounding ground plays an active part for its proper support. The final equilibrium is therefore a result of the ground–structure interaction, which depends on the relative strength and stiffness of the ground and the lining, the initial geostatic pressure, as well as the

Received 26 January 2004. Accepted 9 December 2005. Published on the NRC Research Press Web site at http://cgj.nrc.ca on 6 April 2006. H. Wong1 and D. Subrin.2 Ecole Nationale des Travaux Publics de l’Etat, Département Génie Civil et Bâtiment (CNRS URA 1652), Laboratoire Géomatériaux, rue Maurice Audin, 69518 Vaulx-en-Velin CEDEX, France. D. Dias. Institut National des Sciences Appliquées de Lyon, Unité de Recherche en Génie Civil (Géotechnique), avenue Albert Einstein, 69621 Villeurbanne, France. 1 2

Corresponding author (e-mail: [email protected]). Present address: Centre d’Etudes Techniques de l’Equipement de Lyon, Laboratoire Régional des Ponts et Chaussées de Lyon, 25 avenue François Mitterrand, Case No. 1, 69674 Bron CEDEX, France.

Can. Geotech. J. 43: 462–483 (2006)

doi:10.1139/T06-016

© 2006 NRC Canada

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Fig. 1. Determination of the unconfinement at bolt installation λp (or equivalently Up).

precise instant of lining placement. Intricately related to this philosophy is the increasing use of bolts placed close to the tunnel head, often combined with a layer of shotcrete used as temporary lining. This technique (Pelizza and Peila 1993) presents the advantage of being very flexible (as the density can be modified punctually to account for local geotechnical conditions) and efficient compared to a classical steel frame support, which requires heavy equipment for installation. On the other hand, despite the efficiency observed in situ, the use of bolts is partially hindered by theoretical difficulties in design calculations to realistically take into account the reinforcement action of bolting (Peila et al. 1996). Attempts have been made, either by applying an arbitrary equivalent support pressure at the tunnel wall, or by an equally arbitrary increase, in an isotropic manner, of the cohesion of the massif (Grasso et al. 1991). Otherwise, some models have been proposed to account for bolts and their interaction with the ground, under perfect axisymmetry and bolts arranged in the radial direction, within the framework of the convergence–confinement method. Stille et al. (1989) considered a complex material behaviour, elastoplastic with softening in the post-yield domain, and took into account plastic dilation. In addition, an eventual sliding at the interface and the influence of a bolt–end plate were analysed. However, the variations of bolt density with radius and the elastic strains inside the yielded zone are neglected, and lining is always supposed to be installed at the moment of first yielding of ground. The latter hypothesis appears to be excessively restrictive on account of different situations that can occur in practice. In parallel to laboratory experiments (Indraratna 1993) and field measurements, Indraratna and Kaiser (1990) introduced a reinforced rock mass zone, with increased strength properties (cohesion and friction), taking account of a frictional interaction mechanism at the interface, while giving an arbitrary value to the anchorage length (determined by equilibrium considerations) and neglecting the increase in stiffness due to the reinforcement. Oreste and Peila (1996) presented an interesting computation procedure based on finite differences that analyses the effect of the unsupported span, the influence of the bolt density, and the presence of a bolt–end plate on the response curves. Comparisons to 2D numerical calculations on a typical cross section and in-situ data showed a good agreement with the computed results. Nevertheless, none of these approaches have considered the “upstream” influence of the lining sup-

port already disposed on the determination of the tunnel convergence at bolt installation. As shown in Fig. 1, these simplified approaches suppose that the lining support only changes the convergence profile after its installation at point B, and follows the curve BB′′. In fact, the presence of lining changes the overall convergence profile, even on the upstream part A′A prior to lining installation. The error committed is equal to U p′ – Up in Fig. 1. The hypothesis Up ~ U p′ was introduced by Corbetta (Corbetta et al. 1991) and bears his name. It is recognized that this hypothesis may lead to important errors in some cases (Bernaud and Rousset 1992). Moreover, neither of these approaches have been validated by 3D numerical simulations, which is the only way to precisely consider the three-dimensional effect of the tunnel face, and also to study exactly the ground–bolt interaction by individually simulating each bolt. As shown by the foregoing models (and especially in the case of passive fully grouted bolts as an internal reinforcement), the simulation of the inclusion and their interaction with the ground is rather complex. The heterogeneous mixture (ground plus bolts) tends to act as a composite material upon loading (Etienne 1985): the supplementary stiffness and strength globally reduce the displacements and strains at all points inside the medium, contrary to the confinement action concentrated at the tunnel wall in the case of classic steel or shotcrete linings. It is also worthwhile to mention that finite element analyses, assimilating bolts to linear 1D “bar elements,” underestimates the contribution of the bolts. Their action appears to be better diffused in reality than is predicted by numerical computations. Many finite element codes would also require the bolts to coincide with the element boundaries making the mesh generation difficult, although such difficulties do not exist for a few codes like FLAC (Itasca Consulting Group Inc. 1997). These observations motivated the development of theoretical models using the homogenization approach for periodical media (De Buhan 1986, De Buhan and Salençon 1993), presenting an interesting alternative of analysis. Assimilating the heterogeneous mixture to an equivalent homogeneous medium renders computations more efficient. In this case, analytical solutions become accessible under simplified geometry – see for example Wong et al. (2000a) for an analysis of face reinforcement. The classical convergence– confinement method (see Appendix A for a brief introduction) © 2006 NRC Canada

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can then be applied to the homogenized medium to construct a model for lining support design (radial bolting plus shotcrete for example). Within this framework, an analytical model was proposed by Greuell (Greuell 1993, Greuell et al. 1994) and numerically implemented by Bernaud et al. (1995). They considered a simple elastic – perfectly plastic material behaviour. The directional contribution of the bolts leads to an anisotropic behaviour. However, Greuell supposed that both constituents (ground and bolts) always yield simultaneously in such a way that the equivalent homogeneous medium is also elastic – perfectly plastic. Conceptually, the erroneous hypothesis of simultaneous yielding supposes that as soon as the ground yields, the bolt tension jumps at once to its ultimate value independent of the stress history of the bolts. On the other hand, in Fig. 2, it can be seen that bolt emplacement at an unconfinement rate λp (see Appendix A for definition) normally takes place after yielding of ground, under realistic conditions. Therefore, at that point, the ground has already acquired a certain amount of strain ε(λp) when the bolts are freshly installed. The error committed is conceptually represented by the triangle BCD in the stress–strain diagram in Fig. 2. According to Greuell’s model (Greuell 1993), with the stress being all of a sudden inside the yield envelope, the medium is hence in elastic unloading immediately after bolt installation (i.e., incrementally elastic behaviour). In reality, bolt tension can only increase gradually after installation as the ground deforms, consequently, the macroscopic yield criterion can only expand progressively after bold installation. The medium thus continues to behave plastically, with a gradual expansion of the yield surface illustrated by the dashed line in Fig. 2. The purpose of this paper is to develop a model that accounts for this gradual tension build-up. The proposed model can be seen as an improvement of the one by Greuell (1993). The difference in stress history of the two components leads macroscopically to some sort of strain hardening behaviour for the equivalent homogeneous medium without elastic unloading. In a previous paper (Wong and Larue 1998), the aforementioned assumption of simultaneous yielding was shown to overestimate quantitatively the strength contribution of bolts and lead to a wrong prediction of the evolution of convergence and plastic zones. In the present paper, as a complement to Wong et al. (2000b), a simple approach proposed by Corbetta et al. (1991) is used in a first step to estimate the unconfinement rate λp, given the distance d relative to the tunnel face, which allows for the use of the aforementioned approximation Up ~ U ′p illustrated in Fig. 1. We know that Up ~ U ′p is not always an acceptable assumption. Another empirical procedure is now needed to determine Up, which is an input parameter required by the model, to make it self-sufficient. To test the precision of this empirical determination of Up and the pertinence and precision of the homogenized representation of the composite (ground plus bolts), 3D numerical computations were performed. The results show that despite the simplicity of the proposed method, accounting for bolting support, the errors involved are acceptable under normal cases.

General assumptions All of the classic assumptions within the framework of the

Can. Geotech. J. Vol. 43, 2006 Fig. 2. Conceptual behaviour of the composit: assumptions of simultaneous or delayed yielding.

convergence–confinement method are admitted: small strains and quasi-static behaviour, perfectly circular tunnel of radius R, material behaviour and initial stress homogeneous and isotropic, plane strain condition, so that the problem is essentially 1D. The ground is supposed to be elastic – perfectly plastic and obeys Tresca’s criterion and its associated flow rule. In practical situations, the technique described in this paper is most often applied to grounds ranging from hard soils to soft rocks. As a minimum, stand-up time is required so that bolts can be installed under sufficiently “safe” conditions, very soft grounds are excluded (soft soils for example). In what follows, the in situ ground is characterized by three parameters: C, Es, and νs, respectively, cohesion, Young’s modulus, and Poisson’s ratio (elastic incompressibility is assumed: νs = 0.5). The suffix “s” refers to the ground (“s” for hard “soil”). The bolts, equally elastic – perfectly plastic, are supposed to be in a pure tensile state, with zero stiffness and strength with respect to moment and shear. Their mechanical behaviour can thus be described by four parameters (the suffix “b” refers to “bolts”): Eb, σyb, Sb, and db, respectively, Young’s modulus, yield stress, cross-sectional area, and density (number of bolts per square metre at the wall of the gallery), Tyb = Sbσyb being the yield strength of one bolt. The bolts are of infinite length and perfectly bonded to the surrounding ground, so that they share identical strain and displacement increments. The infinite length assumption is not unreasonable given the rapidly decreasing density with radial distance r. It has also been justified in Wong et al. (2000a) for the case of frontal bolts. Note that to include the finite bolt length would have made the model much more complex and less accessible to professional engineers. As a price to be paid for the simplicity of the analytical solution so obtained, it has to be mentioned that the hypotheses involved puts the following limits on the domain of validity of the model: (1) The hypotheses of homogeneity and isotropy of the initial stress field only applies to deep tunnels in isotropic ground. Any significant deviation from this hypotheses will of course lead to computational inaccuracies. This © 2006 NRC Canada

Wong et al.

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Fig. 3. Average force distribution in the bolts versus normalized radius from the tunnel axis r/R, far behind the tunnel face.

does not mean that the model cannot be applied, but the results have to be interpreted with care. (2) The present model is restricted to the case of short-term behaviour (undrained loading for example) of clayey or marly formations where the frictional component is not significant; in other words, the internal friction angle is close to zero. In particular, deep clay formations (with much stronger mechanical properties than surface clays), which are often a favourable host formation considered in underground nuclear waste storage sites, belong to this category. In addition, perfect plasticity is assumed so that no strength loss is accounted for when the ground exhibits large deformations. These restrictions can, however, be removed by the development of other models based on the Mohr–Coulomb criterion or taking into account a strain-softening behaviour of the ground. These models are nonetheless more complex and are still being studied at the time of this research. Perfect bonding between ground and bolts implies that the bolt axial strain and stress decreases with r, hence bolt tensions are maximum at the tunnel wall (see Fig. 3). This is in fact only a gross approximation. As observed on site for finite bond strengths (Freeman 1978), the tension should be zero at the wall and climb up to a maximum at some distance behind, before decreasing progressively to zero. For sufficiently high bond strengths, this approximation is acceptable, while for low bond strengths, the results require careful interpretation. The analytical treatment of debonding at the interface has been developed by Wong et al. (1999) for the case of face reinforcement and can be applied to the present case of radial bolting. All in all, the model constructed as such is very simple. When applied with caution and properly interpreted, it remains a very useful tool for sensitivity analyses and quick preliminary designs. Other than its logical transparency, which gives some intuitive insight into the machanical behaviour, its instantaneous character also allows an extensive

parametric study with little cost at a preliminary design stage, so that more time-consuming numerical studies can be performed on a final set of parameters.

Constitutive equations Cylindrical coordinates (r, θ, z) are naturally adopted, due to the cylindrical symmetry assumed, where r is the distance from the tunnel axis (r = R at the tunnel wall). The previous hypotheses imply that the displacement u is purely radial and stress σ and strain ε tensors are diagonal since the principal directions coincide with the coordinate axes [1]

⎡u(r)⎤ u(r) = ⎢ 0 ⎥ ; ⎥ ⎢ ⎢⎣ 0 ⎥⎦

0 0⎤ ⎡ε r (r) ⎢ ε (r) = 0 ε θ(r) 0 ⎥ ; ⎢ ⎥ 0 0 ⎥⎦ ⎢⎣ 0 0 0 ⎤ ⎡σ r (r) ⎢ 0 ⎥ σ θ(r) σ (r) = 0 ⎥ ⎢ 0 σ z (r)⎥⎦ ⎢⎣ 0

where the zero axial strain εz(r) = 0 comes from the plane strain condition. Here, an underscore denotes a vector and a double underscore denotes a second order tensor, which can be taken to be a square matrix in the present context. The dependence on time, t, is implicit and is not indicated to simplify the notation. Note that in the absence of time effects in the material behaviour, physical time has only a kinematic character and may be replaced by any monotone increasing function with time. The natural candidate here is the unconfinement rate λ. In the following, all quantities will be parametrized with respect to the space coordinate r and the “time” coordinate λ. The dot above any variable will denote partial differentiation with respect to λ, i.e., [2]

y& = ∂ λ y © 2006 NRC Canada

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Can. Geotech. J. Vol. 43, 2006

In this paper, only the face flow condition is considered (i.e., the three principal stresses are strictly unequal: σr > σz > σθ). In addition, tensile stress is taken to be positive. With perfect bonding being assumed, the tensile stress in the bolts only depends on the longitudinal bolt strain εb, in the direction parallel to the bolt axis, equal to the strain increment in the ground between the time of bolt installation and the current time. Under such conditions and within the framework of the homogenization method (De Buhan 1986, De Buhan and Salencon 1993), the stress tensor of the equivalent homogeneous material can be considered to be the sum of two components, due to ground and bolts, respectively [3]

σ = σs + σb

where σ s is the homogenized ground stress tensor and σ b the homogenized bolt stress tensor. The latter can be expressed in terms of the bolt strain εb, the bolt yield stress σyb and yield strain εyb = σyb /Eb, and two dimensionless parameters β and Ω defined in eq. [5]. [4]

⎡1 0 0 ⎤ σ b = σ o (r) ⎢0 0 0 ⎥ ⎥ ⎢ ⎢⎣0 0 0 ⎥⎦

[7]

e ε& =

1 + vs V σ& s − s tr (σ& s ) 1 Es Es

where tr(A) =

∑ Aii i

is the sum of the diagonal elements of a square matrix and 1 is the identity matrix. The domain of elastic behaviour is supposed to be defined by Tresca’s criterion [8]

fs(σs) = σ sr – σ sθ – 2C < 0

where σ sr > σ sz > σ sθ are the principal stresses of the ground. When the stress reaches the yield envelope and stays on it, i.e., [9]

fs(σs) = 0

then plastic strains occur and are given by the associative flow rule

[10]

⎡ζ& 0 0 ⎤ ⎥ ⎢ f ∂ ε& = ζ& s = ⎢0 − ζ& 0 ⎥ ∂σ s ⎢0 0 0 ⎥ ⎦ ⎣ p

(ζ& ≥ 0 is the plastic multiplier)

with [4a]

R σ o (r) = β E s ε b r

[4b]

σ o (r) = ΩC

R r

ε b < ε yb =

if

if

σ yb Eb

ε b ≥ ε yb

The tensor σ b is purely uniaxial (in the radial direction) since the shear and flexural contributions are neglected, which leads to an anisotropic behaviour of the composite, and σo(r) is simply the bolt tension, which depends on the bolt strain (Tb = SbEbεb ≤ Tyb = Sbσyb) divided by the tributary area S(r) per bolt at a distance r from the tunnel axis with S(r) = (r/R)/db, and db being the number of bolts per square metre at the tunnel wall (r = R). The essential dimensionless constants β and Ω represent the relative stiffness and strength contributions of the bolts, respectively, and are very useful for an intuitive appraisal of the “degree” of reinforcement by the design engineer [5]

β = db Sb

Eb , Es

Ω = db Sb

[11]

C

e p ε& = ε& + ε&

When the massif behaves elastically, the strain rates are related to the ground stress rate through Hooke’s law

f (σ) = σr – σθ – 2C – σo(r) < 0

where σo(r) is given either by eqs. [4a] or [4b] depending on the strain level. In that sense and contrary to Greuell (1993), a progressive tension build-up inside the bolts is therefore taken into account in the construction of the yield criterion of the equivalent composite (Fig. 2). Equation [10] remains unaltered. Finally, eq. [6] (taking into account eqs. [7] and [10] of the elastic and plastic strains, the decomposition eq. [3] of σ, and after elimination of σz using the plane strain condition εz = 0) leads to the following relationship between macroscopic strains and stresses, recalling that νs = 0.5 is assumed [12]

σ yb

Since the strain increments in the ground completely determine that of the bolts (because of the perfect bonding assumption), attention will be concentrated on the former in all subsequent analyses. The subscript “s” is therefore omitted concerning ground strains. For small strains, the total strain rate is assumed to be the sum of an elastic strain rate and a plastic strain rate [6]

On account of eqs. [3] and [4], the yield criterion eq. [8] of the ground can be written in a more convenient form in terms of the macroscopic stress σ

⎡ε& r ⎤ 3 ⎢ε& ⎥ = 4 E s ⎣ θ⎦

⎡ 1 −1⎤ ⎡σ& r − σ& o ⎤ & ⎡ 1 ⎤ ⎥ + ζ ⎢−1⎥ ⎢−1 1 ⎥ ⎢ σ& θ ⎦ ⎣ ⎦ ⎣ ⎦⎣

Fundamental equations As long as the bolts are elastic, eq. [4a] applies. The perfect bonding assumption, which implies ε& b = ε& r , and the compatibility relationships (εr = ∂ ru, εθ = u/r), lead to 3 (σ& r − σ& θ) + ζ& E s (4 + 3βR / r)

[13]

∂ ru& =

[14]

u& 3 = (σ& r − σ& θ) − ζ& r E s (4 + 3βR / r)

with ζ& > 0 in a plastic zone, and ζ& = 0 either in an elastic zone, or when elastic unloading occurs. © 2006 NRC Canada

Wong et al.

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Fig. 4. Evolution of plastic zones.

When bolts have yielded, σo(r) is given by eq. [4b], therefore σ& o = 0 (i.e., bolt tension remains constant during plastic flow of ground). In that case, the term β in the denominator of eqs. [13] and [14] should be cancelled (i.e., the bolts are “incrementally absent”). As a result of axisymmetry and plane strain assumptions, there is only one equilibrium equation [15]

r∂ rσr + (σr – σθ) = 0

The initial conditions at t = 0 before excavation are [16]

σ

∞

= – P∞1; ε

∞

= 0; u

∞

=0

crement from λp to λ. In consequence, the bolt strain is given by [20]

εb =

A ; r2

λ

A(λ) =

& λ′ )dλ′ ∫ B(

λp

where evidently A(λ) ≠ B(λ) hence εb ≠ εr (i.e., bolt and & =B & (i.e., ground radial strains are different), while we have A bolt and ground radial strain rates are identical) due to the assumption of perfect bonding. Substituting eqs. [19] and [15] into eq. [14] leads to & ⎞ R⎞⎛ A ⎛4 r ∂ rσ& r = E s ⎜ + β ⎟ ⎜⎜ − 2 + ζ& ⎟⎟ r ⎠⎝ r ⎝3 ⎠

where P∞ is the initial hydrostatic pressure, while the boundary conditions for t ≥ 0 are

[21]

[17a] σr(∞,t) = –P∞

when bolt stress is below their yield limit, and

[17b] σr(R,t) = –(1 – λ)P∞

[22]

The problem to be solved, as λ increases monotonously from 0 to 1, is completely defined by the yield criterion eq. [11] and eqs. [13]–[17], taking into account the remarks below eq. [14]. Thanks to the incompressibility assumption (νs = 0.5 and e p Tresca’s criterion), which leads to tr (&) ε = tr (&ε ) = tr (&ε ) = 0, we can deduce from eqs. [13] and [14] that [18]

∂ r u& + u&/ r = 0

hence the displacement and radial strain fields in the ground always simplify to the following: [19]

u& = −

& B ; r

u=−

B ; r

ε& r =

& B ; r2

εr =

B r2 λ

with

B(λ) =

∫ B& (λ′ )dλ′ 0

where B is an integration “constant” coming from eq. [18]; it only depends on the kinematic time λ and is independent of r. The integral expression of B in eq. [19] emphasizes the fact that the ground displacement depends on the entire loading history from the initial reference state λ = 0 to the present time λ > 0. On the contrary, the bolts come into action only at “time” λp; therefore they only experience strain in-

r ∂ rσ& r =

& ⎞ 4Es ⎛ A ⎜⎜ − 2 + ζ& ⎟⎟ 3 ⎝ r ⎠

when the bolts have yielded.

Scenarios For comprehension purposes, as well as for future reference, it would be helpful to describe the sequence of events as unconfinement λ progresses from 0 to 1; notably the evolution of the plastic zones, without proof. In reference to Fig. 4, for λ < λp, the medium is unreinforced and the solution is classical (see e.g., Panet 1995). The bolts are supposed to be installed after the ground has yielded (for λ = λe = C/P∞ ), which is indeed a realistic assumption as numerical simulations will further demonstrate. After bolt emplacement, growth of the plastic zone in the massif (bounded by x the radius of soil plasticity) slows down but continues. The stress tensor moves outwards and enlarges the yield envelope of the composite (macroscopic hardening) as the bolt tension increases progressively (Fig. 2). Hence in zone PLs /ELb (EL stands for elastic, PL for plastic), the ground has yielded but not the bolts. At a critical unconfinement λbp, the bolts also reach their yield limit, and the plastic zone PLs /PLb (where both constituents have yielded) appears and develops from the tunnel wall (bounded externally by w the radius of bolt plasticity). For some particular combinations © 2006 NRC Canada

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Can. Geotech. J. Vol. 43, 2006 Table 1. Transition values of the unconfinement λ. Configuration 1/2 2/3 3/4

4/5

Transition values of λ P λe = ∞ C λp → bolt installation ⎡⎛ ⎞ 2 ⎤ λ bp x 2Ω ⎥ Ω = 1 + ln ⎢⎜ p ⎟ + + ⎢⎜⎝ R ⎟⎠ λe 3β ⎥ 3 ⎣ ⎦ ⎛x λc = 1 + 2 ln ⎜ p ⎜R λe ⎝

⎛ 3β ⎞⎟ 2R + Ω ⎜1 − ⎟ ⎜ 3β − 2Ω ⎠ 3 xp ⎝

3β − 2Ω ⎞⎟ 3β ⎠⎟

Table 2. Equations determining the unknown plastic boundaries x and w. Configuration

Equations

2

⎛λ ⎞ ⎛x⎞ − 1⎟ ⎜ ⎟ = exp ⎜⎜ ⎟ λ R ⎝ ⎠ ⎝ e ⎠

3

4 and 5

2

2

⎛ xp ⎞ ⎛λ ⎞ ⎜ ⎟ = exp ⎜ p − 1⎟ → ⎜R⎟ ⎜λ ⎟ ⎝ ⎠ ⎝ e ⎠ 2⎤ 2 ⎡ ⎛x ⎞ ⎛x⎞ β ⎛x⎞ λ = 1 + 2 ln ⎜ ⎟ + ⎢⎜ ⎟ − ⎜ p ⎟ ⎥ ⎜R⎟ ⎥ λe ⎝ R ⎠ 2 ⎢⎝ R ⎠ ⎝ ⎠ ⎦ ⎣ 2 2 ⎡ ⎛ xp ⎞ ⎤ ⎛ ⎛x⎞ 2R ⎞ w 3β ⎢⎛ x ⎞ λ = 1 + 2 ln ⎜ ⎟ + Ω ⎜1 − ⎜ ⎟ − ⎜⎜ ⎟⎟ ⎥ ⎟; = 3w ⎠ R 2Ω ⎢⎝ R ⎠ λe ⎝ ⎝R⎠ ⎝ R ⎠ ⎥⎦ ⎣

of parameters, this zone PLs/PLb may erase entirely zone PLs /ELb at λ = λc, and a new zone ELs /PLb appears in which bolts have yielded but not the ground. The successive appearance of these five different configurations, separated by four critical “times” λe, λp, λbp, and λc, is illustrated in Fig. 4.

Resolution for each configuration The problem has to be solved for each of the five different configurations encountered (Fig. 4). For a particular configuration, a set of differential equations applies to each zone. For certain configurations, the equations are obtained after integrating from λ = 0 to the actual “time” λ, using the appropriate equations applicable in each particular “time” interval. Elimination allows one to single out the two main unknowns: the displacement u and the radial stress σr, which satisfies two differential equations. The assumption νs = 0.5 uncouples the system in all cases and leads to a single differential equation on σr, which hopefully admits an explicit solution (otherwise, no global analytical solution can be found) comprising unknown constants of integration. These unknown constants can be solved in each zone using the appropriate boundary conditions – yielding of ground or bolts, or condition at infinity. The unknown plastic boundaries are finally determined by imposing the continuity of σr at these points. The other quantities are then determined by simple substitution. For clarity of presentation, the detailed solution process is reported in Appendix B, while the detailed expressions (displacement, stresses, strains) in each zone and in each config-

uration are grouped together in Appendix C. Tables 1–4 summarize the key results obtained.

Empirical methods to determine the unconfinement ␭p at bolt installation Through the functional relationships obtained in Tables 1– 4, it appears that the structural responses depend on five dimensionless parameters, in other words [23]

U*, X, W, T *b = F (λ, λe, λp, β, Ω)

where λ is the unconfinement rate representing the loading parameter (which increases from 0 to 1); λe is the inverse of the load factor N = P∞ /C; λp is the unconfinement at bolt installation; β and Ω have already been defined in eq. [5]; and U* = (Es /C)[u(R)/R], X = x/R, W = w/R, and T *b = (db /C)Tb(R) are the convergence, the extent of the plastic zones of the ground and of the bolts, and the maximum bolt tension, respectively, all normalized to a dimensionless form. Contrary to the three parameters λe, β, and Ω that are imposed by the material properties and the geostatic pressure, the unconfinement λp (or equivalently the convergence at bolt installation Up) depends on the unsupported span before lining installation, which can, within a certain extent, be chosen by the designer. Two different methods will be used to assess this input parameter of the model. Corbetta’s method The first method, from Corbetta et al. (1991), supposes that the displacement profile of the lined tunnel follows that of the unlined tunnel before lining installation at a distance d © 2006 NRC Canada

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Table 3. Convergence U(R) at the wall of the gallery. Displacement at the tunnel wall U(R) = u(R)/R 3C λ − 2 Es λ e

Configuration 1 All others

−

3C 2 Es

⎛x⎞ ⎜ ⎟ ⎝R⎠

Fig. 5. Equivalent stiffness of bolt support (bolts supposed to remain elastic).

2

Table 4. Maximum tension in the bolts. Maximum dimensionless bolt tension T *b = (db /C)Tb(R)

Configuration

Zone

3, 4, and 5

PLs or EL

4 and 5

PLb

2 2 ⎡ ⎛ xp ⎞ ⎤ 3β ⎢⎛ x ⎞ ⎜ ⎟ ⎜ ⎟ −⎜ ⎟ ⎥ 2 ⎢⎝ R ⎠ ⎝ R ⎠ ⎥⎦ ⎣ Ω

from the tunnel face, both corresponding to curve BB′ in Fig. 1. An approximate expression of U p′ – the convergence at lining installation (point B) – can be found for example in Panet (1995) ⎛ d ⎞ ⎟U ∞ [24a] U p′ = α ⎜ ⎜ χ pR ⎟ ⎝ ⎠ ⎡ ⎛ 0.75 ⎞ 2 ⎤ ⎛d⎞ [24b] α ⎜ ⎟ = 0.25 + 0.75 ⎢1 − ⎜ ⎟ ⎥ ⎝R⎠ ⎢⎣ ⎝ 0.75 + d/R ⎠ ⎥⎦ ⎛ 3P ⎞ [24c] χ p = U ∞ ⎜⎜ ∞ ⎟⎟ ⎝ 2Es ⎠

−1

where U∞ is the free convergence of an unlined tunnel far behind the excavation face and λp is such that the point P (Up, 1–λp) is on the convergence curve (Fig. 5). The principal advantage of this method is its simplicity. However, the negligence of the “upstream” influence of the lining (i.e., AA′ instead of BB′ in Fig. 1) inevitably leads to an overestimation of the final convergence (B′′ instead of A′′). This method is used to check the empirical method proposed hereafter, which does account for the “upstream” influence of the lining. Proposed method based on Minh–Guo’s work The proposed method is based on Minh–Guo’s proposal (Nguyen Minh and Guo 1993), which is itself based on numerical parametric studies on classic “external” lining (not equal to internal reinforcement). The extrapolations involved render the proposed method entirely empirical; the only justification lies in its capacity to produce quick and acceptably accurate results. Minh–Guo’s method takes into account the dimensionless lining stiffness ksn (ksn = 3Ksn /2Es) in the determination of Up, through the simultaneous determination of the three unknowns (U ∞s , λ ∞s , Up), defined in Fig. 1, thanks to the following:

[25]

⎧ fM (U ∞s ) = 1 − λ ∞s ⎪ 3 ⎨T = φ(S ) = 0.55 + 0.45S − 0.42(1 − S ) ⎪(1 − λ ∞s )P∞ = K sn (U s∞ − U p ) ⎩ with

S = U ∞s / U ∞ ;

T = U p /U p′

where the function fM represents the convergence behaviour of the unreinforced medium (Fig. 1) and U p′ is determined from Corbetta’s method (eq. [24]). Eliminating λ ∞s and Up leads to the following implicit equation on the unique unknown S: [26]

⎡ ⎛ d ⎞⎤ ⎟⎥ fM (SU ∞ ) = χ p ksn ⎢S − φ(S )α ⎜ ⎜ χ pR ⎟⎥ ⎢⎣ ⎠⎦ ⎝

Contrary to pretensioned mechanical bolts (Labiouse 1991), which may be simulated by an active support pressure at the tunnel wall (as classical linings), grouted passive bolts act as an “internal” reinforcement modifying the shape of the convergence curve (Fig. 5). To derive a simplified method for the case of grouted bolt support, an indefinitely elastic behaviour for the bolts (i.e., Ω → ∞) is first considered. By equating on one hand the convergence at equilibrium of the medium supported by an elastic “external” lining (Point E in Fig. 5) and on the other hand the corresponding value due to elastic bolts (Point F), we deduce an equivalent lining stiffness (of the bolting support) k eq sn to be inserted into Minh–Guo’s method [27]

k eq sn =

β 1 E = db Sb b 2 2 Es

It is noteworthy that bolting, as an “internal” reinforcement of the medium, modifies the shape of the convergence curve (Fig. 5). This formulation allows one to determine the convergence at bolt placement Ud, and as a consequence, the unconfinement λd, which is then inserted into the analytical model, accounting for the possible yielding of bolts. The fact that Ud is determined considering elastic bolts inevitably leads to errors when substantial yielding of bolts take place. © 2006 NRC Canada

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Fig. 6. Unconfinement λp versus unsupported distance d/R for different values of λe: (a) Corbetta and (b) Minh–Guo’s method.

However, such an event is not likely to occur given that a large percentage of decompression has already taken place at the tunnel face and to a still larger proportion at the point where bolts can be installed. This last observation makes practical applications of the proposed method feasible.

Parametric study The dimensionless form and the restricted number of parameters of the structural responses in eq. [23] allow the construction of design charts covering a wide range of mechanical parameters. Range of dimensionless parameters For the mechanical parameters of the ground, of the type hard soil or soft rock (cf. general assumptions), the elastic modulus and the cohesion, respectively, lie typically in the intervals: 50 MPa < Es < 300 MPa and 0.1 MPa < C < 1 MPa. For the bolts (fully grouted steel bars), the elastic modulus is given by Eb = 200 000 MPa, with a yield stress in the order of σyb = 500 MPa. A typical Y25 bar corresponds to a cross-sectional area of Sb = 5 cm2, while the bolt density varies typically between 0.25 and 1.00 bolt/m2. This choice of physical parameters leads to the following intervals for the key dimensionless parameters β and Ω: 0.0833 < β < 2 (hence k eq sn < 1) and 0.0625 < Ω < 2.5. Moreover, the dimensionless overburden is supposed to lie within “realistic” values, in other words: N = P∞ /C = γH/C ≤ 5 (i.e., λe ≥ 0.2). The unconfinement at bolt installation λp depends on the unsupported span d, and is necessarily bounded from below. In the particular case of elastic behaviour for an unreinforced medium: λ(z = 0) = 0.25 at the tunnel face and λ(z = R/4) ≅ 0.55 at a quarter of a radius behind. The value of λp is even larger under elastoplastic behaviour. In Fig. 6, the variation of λp vs. d/R for different values of λe (inverse of the “load factor”) is plotted, for both Corbetta’s and Minh– Guo’s methods (for the latter, the most unfavourable value of k eq sn = 1 is chosen). In practical situations, λe usually lies inside the interval 0.2 < λe < 0.5, so that for d/R > 0.2: λp > 0.5 (as observed in the following numerical simulations), validating the hypothesis that yielding of ground occurs prior to bolt installation.

Preliminary design charts One of the advantages of such analytical models is that pertinent dimensionless parameters can be logically deduced. Moreover, the computational efficiency makes it easy to establish design charts that are very useful for preliminary design. Examples of such design charts are shown in Figs. 7–9, constructed for the particular overburden N = 3 and the unconfinement rate at bolt placement λp = 0.7. Figure 7a gives the variation of the convergence U* = (Es/C)@[u(R)/R] (and Fig. 7b gives the extent of the plastic zone X = x/R) against the bolt stiffness β for different bolt strengths Ω. Considering the design of a tunnel with a radius R = 5 m, in a medium such that P∞ = 0.3 MPa, C = 0.1 MPa, and Es = 100 MPa; if surface settlement considerations require a limitation of the convergence to values U* < 7 (i.e., u(R)/R < 0.7%), Fig. 7a leads one to take β = 0.5 and Ω > 1. For instance Y25 bars, with Sb = 5 cm2, Eb = 200 000 MPa, and σyb = 500 MPa, placed at d = 1 m (so that λd = 0.7 – Fig. 6 with Corbetta’s assumption) at every 2 m2 gives β = 0.5 and Ω = 1.25, which satisfies the above requirement. The corresponding extent of the decompressed zone amounts to X = 2.2, in other words, to a thickness 1.2 times the tunnel radius (Fig. 7b). In a more general case, Fig. 8 shows the evolution of the dimensionless convergence U* versus the unconfinement at bolt installation λp for different values of β and Ω, while Fig. 9 shows the corresponding variations for the maximum bolt tension T *b = (db /C)Tb(R) at the tunnel wall. The maximal admissible convergence, determined by settlement considerations, can be obtained by choosing the adequate couple of bolt reinforcement parameters β and Ω in Fig. 8, along with the corresponding value of the bolt tension in Fig. 9. As long as the bolts are elastic, bolt tension is given by the continuous lines for a particular value of β, while T *b = Ω when bolts have yielded.

Comparison of the analytical model to 3D numerical calculations The results of the analytical model, combined with the empirical approaches to determine the tunnel convergence at the moment of bolt installation, are compared to those issued from 3D numerical computations. The latter are carried out using a 3D finite difference code FLAC 3D (Itasca Con© 2006 NRC Canada

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Fig. 7. Evolution of (a) the dimensionless convergence U* and (b) the extent of the decompressed zone X versus bolt stiffness β for different values of bolt strength Ω with N = 3 and λp = 0.7.

Fig. 8. Dimensionless convergence U* versus unconfinement at bolt installation λp for different values of β and Ω with N = 3.

sulting Group Inc. 1997), which allows one to take into account the “exact” geometry of the tunnel, among others the lining behind the face, each of the inclusions individually and the nonlinearities of material behaviour, in particular the imperfect bonding between bolt and ground. The purpose of this comparison is to check the pertinence and the precision of the homogenized representation of the composite ground plus bolts, as well as the empirical determination of Up, the convergence of the tunnel at lining installation (Fig. 1). Note that this aim cannot be achieved by 2D numerical analysis since additional hypotheses would need to be introduced: either the convergence at lining installation would have to be guessed or the bolts would have to be replaced by membranes.

Geometry and advancing procedure For this study, a deep circular tunnel of radius R = 5 m in a homogeneous and isotropic medium is considered. On account of axisymmetry, only a quarter of the geometry is modelled. The medium is elastic – perfectly plastic, characterized by Tresca’s yield criterion and its associated flow rule. The bolts, disposed in the radial direction (Fig. 10), are simulated individually by linear 1D “bar elements,” equally elastic – perfectly plastic. To simulate perfect bonding, rheological parameters of the bolt–ground interface are as follows: shear stiffness 2 × 109 kN and maximum shear force 1 × 106 kN/mL. The excavation and the corresponding face advance take place in steps of 2 m. Bolts are installed after each excava© 2006 NRC Canada

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Fig. 9. Dimensionless maximum bolt tension T *b vs. unconfinement at bolt installation λp, for different values of β and Ω, with N = 3.

Fig. 10. Three-dimensional mesh and radial distribution of bolting.

tion step in two “rings” of 1 m apart (Fig. 11) with 32 bolts/ring (8 bolts for the 3D mesh, which models one quarter of the geometry) when the bolt density is db = 1 bolt/m2, 16 bolts/ring when db = 0.5 bolts/m2, or 16 bolts in only one ring at midspan when db = 0.25 bolt/m2. In the first configuration (double-ring), two different convergences occur at the location of each bolt ring, while only one single value can be accounted for in the analytical model. In such circumstances and for practical implementations of the method, it is suggested that this convergence Up (or equivalently the unconfinement rate λp) be calculated at the midpoint between the two rings of bolts (i.e., d = (d1 + d2)/2). This procedure overestimates the “initial convergence” for bolts in ring a, closer to the face (underestimation for ring b), while the inverse is true concerning the final bolt

tensions. This procedure is expected to give satisfactory results when the bolts are installed not “too” close to the face. Geomechanical parameters Table 5 summarizes the geomechanical parameters for the ground, shotcrete lining, and bolt support. In particular, the following parameters will be varied throughout the comparative study. The geostatic pressure is limited to realistic values so that the “load factor” N = P∞ /C is less than 5: P∞ = 0.3, 0.4, and 0.5 MPa (i.e., N = 3, 4, or 5). The Y25 steel bars are fully grouted over their entire length, with a density at the tunnel wall (r = R) of either 0.25, 0.5, or 1 bolt/m2. The distance to face at installation is either 1, 2, 3, or 5 m (i.e., d/R = 0.2, 0.4, 0.6, or 1). The medium possesses a © 2006 NRC Canada

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473 Table 5. Ground, shotcrete lining, and steel bolts properties. Parameters

Ground

Shotcrete lining

Steel bolts

Elastic modulus, E (MPa) Poisson’s ratio, ν Cohesion, C (MPa) Geostatic pressure (MPa) Thickness (m) Unsupported distance, d (m) Yield strength, σyb (MPa) Section, Sb (mm2) Density, db (bolt/m2)

50–100 0.3 0.1 0.3–0.4–0.5 NA NA NA NA NA

10 000 0.2 NA NA 0.1 3 NA NA NA

200 000 NA NA NA NA 1–2–3–5 500 500 0.25–0.5–1

Fig. 11. Convergence Up at bolt installation in the case of two rings.

Young’s modulus Es = 50 or 100 MPa, leading to Es/C = 500 or 1000. The 22 numerical simulations carried out for this comparative study are summarized in Table 6, along with the value of the relative convergence at equilibrium obtained numerically. Most of these cases consider a Poisson’s ratio of = 0.3 for the numerical calculation, while the value of ν MN s = 0.5 is imposed by the analytical model. We have deν MA s noted by “MN” the values used in numerical computations and “MA” in analytical modelling. This choice for Poisson’s ratio was justified by the fact that the influence of νs is considered to be insensitive by existing literature, while values of νs approaching 0.5 drastically slow down the numerical calculations. To validate this choice, which is expected to give similar results, a few computations were carried out with both values of νs (either 0.3 or 0.498). In particular, Fig. 12 shows the ratio of convergence between νs = 0.498 and νs = 0.3 against the load factor N = P∞ /C (obtained by varying the cohesion C). When the medium is not reinforced (case 3 versus 2), the error amounts to 7%, matching the analytical result shown in Fig. 12 for the load factor N = 3 (Point M), whereas a difference of only 2% is observed when bolting is installed (case 16 versus 9), matching the analytical difference for N ≅ 2.5 (Point N in Fig. 12). Numerical convergence profile Examples of convergence profile along the tunnel axis are plotted in Fig. 13 with or without bolting and (or) shotcrete lining. All profiles lie between the unreinforced elastic case (1) and the unreinforced plastic case (2). Using bolting alone

(cases 7, 9, or 13) or combined with a layer of shotcrete (case 17) lead to a reduction of convergence at equilibrium. Note that the dents in the curves are a numercial artefact, observed in other similar types of caculations (Bernaud and Rousset 1992), and they do not affect the validity of the results. Comparison of the final convergence at equilibrium The final convergences at equilibrium (i.e., far behind the tunnel face), obtained from the different approaches, are shown in Fig. 14a: (i) U MN ∞ , the numerical convergence at equilibrium listed in Table 6, serves as the reference value (the thickest dark line); MN (ii) U MA ∞ (Up = U p ) is the value of the analytical convergence, taking the convergence at bolt installation from the numerical calculation, over U MN ∞ ; (iii) U MA ∞ (Corbetta) is the analytical convergence using Corbetta’s method, over U MN ∞ ; and (iv) U MA ∞ (Minh–Guo) is the analytical convergence using Minh–Guo’s method, over U MN ∞ . To facilitate their comparison, the three analytical convergences are normalized with respect to the numerical convergence, taken as the reference, and plotted in Fig. 14b. The numerical model has first been validated on a few “unreinforced” cases (1–3), and the calculations were found to be consistent with analytical solutions available for any value of Poisson’s ratio. For the elastic reinforced case (4), analytical models only consider νs = 0.5. In this case, the analytical convergences overestimate the numerical reference (νs = 0.3), due to the influence of Poisson’s ratio, U MN ∞ which is fairly sensitive under elastic behaviour. The following “reinforced” cases (5–16) consider a fixed load factor N = P∞ /C = 3 and varying parameters of bolting support. Comparing the analytical results to the reference line No. 1 leads to the following remarks. When the real convergence at bolt emplacement is input as an external parameter, the analytical model correctly reproduces the final convergence (i.e., curve No. 2 is always close to the reference line No. 1), with a maximum error of 9%. This observation validates the homogenization assumption, at least for the bolt densities considered. Moreover, it is noteworthy that, using the empirical methods presented herein to determine the convergence at bolt placement, the analytical results tend to bracket the latter (i.e., lines 3 and 4 compared to line 2). Owing to the inherent assumptions of Corbetta’s © 2006 NRC Canada

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Can. Geotech. J. Vol. 43, 2006 Table 6. Comparison cases and relative convergences obtained with the numerical model. Case

N = P∞ /C

ν

Es /C

db (bolts/m2)

β

Ω

d/R

U ∞MN (%)

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22

3 (EL) 3 3 3 (EL) 3 3 3 3 3 3 3 3 3 3 3 3 3 4 4 4 5 5

0.300 0.300 0.498 0.300 0.300 0.300 0.300 0.300 0.300 0.300 0.300 0.300 0.300 0.300 0.300 0.498 0.300 0.300 0.300 0.300 0.300 0.300

1000 1000 1000 1000 1000 1000 1000 1000 1000 1000 1000 1000 1000 500 500 1000 1000 1000 1000 500 1000 1000

0 0 0 1.00 0.25 0.25 0.25 0.50 0.50 0.50 1.00 1.00 1.00 1.00 0.50 0.50 0.50a 0.25 1.00 1.00 1.00 1.00

0 0 0 1.00 0.25 0.25 0.25 0.50 0.50 0.50 1.00 1.00 1.00 2.00 1.00 0.50 0.50 0.25 1.00 2.00 1.00 1.00

0 0 0 2.500 0.625 0.625 0.625 1.250 1.250 1.250 2.500 2.500 2.500 2.500 2.500 1.250 1.250 0.625 2.500 2.500 2.500 2.500

× × × 0.2 0.2 0.4 0.6 0.2 0.6 1.0 0.2 0.4 0.6 0.2 0.6 0.6 0.6 0.2 0.4 0.2 0.2 0.4

0.38 1.17 1.08 0.33 0.89 0.92 0.93 0.73 0.84 0.90 0.59 0.68 0.74 1.01 1.53 0.83 0.52 2.10 1.59 2.38 2.03 2.96

a

Plus shotcrete.

Fig. 12. Evolution of the ratio U∞ (νs = 0.5)/ U∞ (νs = 0.3) versus the load factor N for an unreinforced medium.

method, it tends to overestimate the final convergence, while Minh–Guo’s method generally underestimates it. When bolting support alone is installed in a normally loaded medium, Corbetta’s method is expected to give satisfactory results with enough precision in practical situations. Figure 15 shows the variation of the analytical convergence at equilibrium with respect to the unsupported distance d/R, for N = 3, Es /C = 1000 and various bolt densities. The continuous lines were obtained using Corbetta’s assumption, while the dashed lines are based on Minh–Guo’s method. The points correspond to the numerically computed cases (5–13), which sit very close to Corbetta’s target curves. Things begin to diverge for N ≥ 4–5 (cases 18–22), and it should be noted that such a load factor is normally too high for bolting support alone. Corbetta’s method appears to lead to significant errors whenever a relatively stiff bolting or lin-

Fig. 13. Examples of convergence profile along the tunnel axis (see Table 6 for case No.).

ing system is applied close to the tunnel face. This is due to the negligence of the upstream influence of the lining, as shown by case 17 when a shotcrete layer is installed (k shot sn = 3.125 compared to k bolt = = 0.25). To account for evenβ/2 sn tual stiff supports or high load factors, it is suggested that the final convergence be evaluated as the average value of Corbetta’s and Minh–Guo’s results. Overall, the analytical model has been made “selfsufficient” when combined with the empirical methods proposed. Its predictions on tunnel convergence are generally in relatively good agreement with 3D numerical computations, except for limiting cases of high load factor, where bolts are unlikely to be used alone as lining support. Comparison of stress state and bolt tension The evolution of radial and ortho-radial stresses in the ground, at equilibrium far behind the tunnel face, are plotted © 2006 NRC Canada

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Fig. 14. Comparisons of the convergence at equilibrium from the analytical approaches versus the numerical reference for all the cases considered (see Table 6): (a) absolute and (b) relative values.

against the normalized radius r/R from the tunnel axis in Fig. 16 for the unreinforced case (3), the bolted reference case (9) (Corbetta’s method is assumed), and the case (17) when bolting and shotcrete are used together (the results are evaluated as the average of Corbetta’s and Minh–Guo’s methods). The analytical results closely reproduce the shape of the curves obtained numerically, at least at a certain distance from the tunnel wall. In particular, it can be seen from the ortho-radial stresses that the boundary of the ground plastic zone (where a sharp change of slope of σθ occurs), which reduces with increasing support, is correctly evalu-

ated. It is noteworthy that in the bolted case, the analytical radial stress differs from zero at the tunnel wall, due to the influence of the bolt contribution σo(r). The average force distribution in the bolts, at equilibrium far behind the tunnel face, is plotted in Fig. 3 for three cases with or without shotcrete (8, 9, and 17). It appears that the shapes of the curves are strongly different, with a numerically observed “skin effect” at the vicinity of the tunnel wall. Due to the homogenization approach and the perfect bonding assumption, the bolt strain εb(r) in the analytical model follows closely although it is not identical to the ground ra© 2006 NRC Canada

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Fig. 15. Convergences at equilibrium versus unsupported distance for N = 3, Es /C = 1000, and db = 0.25–0.5–1 bolts/m2.

Fig. 16. Radial and ortho-radial stresses distribution versus normalized radius from the tunnel axis r/R far behind the tunnel face.

dial strain εr(r) and therefore leads to an incorrect prediction that εb(r), and hence also the bolt stress σb(r), are decreasing functions of r. In reality, the bolt stress and strain start at zero at the tunnel if no end-plate is installed (assumed in the numerical computations), climb to a maximum, then decrease afterwards. Hence the negligence of this boundary effect in the analytical model leads to an overestimation of the maximum bolt tension.

Conclusions Within the convergence–confinement method and the ho-

mogenization approach, a simple analytical model is proposed to analyse the behaviour of a tunnel where the lining support is provided by radially disposed bolts grouted over their entire length. This model allows one to estimate the convergence of the tunnel wall as well as the extent of the plastic zone, taking into consideration the progressive tension build-up in the bolts in the course of face advance, as well as the difference of yield strains and the nonsimultaneous yielding of the constituents due to plastification of the ground prior to bolt installation. Such information is of prime importance to estimate the damage consequences of tunnelling on surface structures. In addition, sensitive grounds may © 2006 NRC Canada

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exhibit strength loss at large strains, due to decompression during excavation. The evaluation of tunnel displacements and strains helps to assess the possible occurence of such unstable situations. With regard to the mechanical behaviour, the contribution of the bolts on the improvement of the mechanical characteristics of the ground is represented in this model by two dimensionless parameters: β (the stiffness contribution) and Ω (the strength contribution). Bearing in mind that such improvements have a directional character, the homogenization approach leads to an overall anisotropic behaviour. The effect of the progressive build-up of bolt tension, directly visible from the dependence of the frontal displacement on bolt stiffness β, has been shown to be a determining factor. Otherwise, to make the analytical model self-sufficient, two empirical methods (based on previous works) are provided to determine the convergence at bolt placement, taking into account or not the influence on the upstream convergences of lining support. From the utility standpoint, a variable count shows that the structural behaviour at equilibrium only depends on four dimensionless parameters (N, d/R, β, Ω), hence the model possesses a great simplicity. The semi-explicit character of the equations allows one to clearly visualize the different events, as well as the causal links between various parameters (loading, stiffness, strength, mechanical response, etc.). This makes the model ideally suited to quantity estimations and parametric studies at the preliminary design stage. As a matter of fact, the parametric studies presented already constitute a very useful design aide. To appreciate the precision of the analytical model, its results are compared to those obtained by 3D numerical simulations (FLAC 3D). It is shown that when the convergence at bolt installation, Up, obtained numerically, is introduced as an “external” input parameter in the analytical model, the result is in good agreement with the final convergence given by the numerical calculation, thus validating the homogenization assumption. Two empirical approaches, due to Corbetta and Minh–Guo, were then introduced to estimate the value of Up, to be inserted into the analytical model to make it self-sufficient. In practical situations, when the support is only provided by bolting and for limited load factors (N < 4), Corbetta’s method is expected to give satisfactory results. The justification lies in the restricted stiffness of the bolted scheme, which does not greatly influence the convergence profile before support installation. For higher load factors or when a stiffer lining system is installed (e.g., bolting plus shotcrete layer), the real behaviour may lie between the predictions of both empirical methods (Corbetta or Minh–Guo). Nonetheless, the simplifying assumptions do put limits on the range of validity of the model. Extreme caution must be taken here against any hasty conclusions, on account of the hypotheses involved – elastic incompressibility, perfect plasticity, zero internal friction and dilation, infinite bolt length, and perfect bonding amongst others. In particular, a finite bond strength coupled with a finite anchorage length (all the more important as the bond strength is low) may significantly reduce the reinforcement effect and modify the overall behaviour as well. To remove the foregoing assumptions, the development of more complicated models taking into

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consideration the more complex material behaviours are currently under progress (Subrin 2002). Along with numerical validations, the comparisons of the analytical results with insitu measurements are being undertaken.

References Bernaud, D., and Rousset, G. 1992. La “nouvelle méthode implicite” pour l’étude du dimensionnement des tunnels. Revue Française de Géotechnique, 60: 5–26. Bernaud, D., De Buhan, P., and Maghous, S. 1995. Numerical simulation of the convergence of a bolt-supported tunnel through a homogenization method. International Journal for Numerical and Analytical Methods in Geomechanics, 19: 267–288. Brown, E.T., Bray, J.W., Ladanyi, B., and Hoek, E. 1983. Ground response curves for rock tunnels. Journal of Geotechnical Engineering, ASCE, 109(1): 15–39. Corbetta, F., Bernaud, D., and Nguyen Minh, D. 1991. Contribution à la méthode convergence–confinement par le principe de similitude. Revue Française de Géotechnique, 54: 5–11. De Buhan, P. 1986. Approche fondamentale du calcul à la rupture des ouvrage en sols renforcés, Thèse d’Etat, Université Paris VI, France. De Buhan, P., and Salençon, J. 1993. A comprehensive stability analysis of nailed structures. European Journal of Mechanics A/Solids, 12: 325–345. Egger, P. 1973. Influence du comportement post-rupture des roches sur le soutènement des tunnels. Ph.D. thesis, Karlsruhe University, Germany. Etienne, M. 1985. Rôle et emploi du boulonnage. Tunnels et Ouvrages Souterrains, 72: 225–240. Freeman, T.J. 1978. The behaviour of fully bonded rock bolts in the Kielder Experimental Tunnel. Tunnels and Tunnelling, June: 37–40. Grasso, P., Mahtab, A., Ferrero, A.M., and Pelizza, S. 1991. The role of cable bolting in ground reinforcement. In Proceedings of the International Conference on Soil and Rock Improvement in Underground Works, Milano, Italy, pp. 127–138. Greuell, E. 1993. Etude du soutènement des tunnels par boulons passifs dans les sols et les roches tendres, par une méthode d’homogénéisation. Ph.D. thesis, Ecole Polytechnique, Paris, France. Greuell, E., De Buhan, P., Panet, M., and Salençon, J. 1994. Comportement des tunnels renforcés par boulons passifs. In Proceedings of the 13th International Conference on Soil Mechanics and Foundation Engineering, New Delhi, India. Indraratna, B. 1993. Effect of bolts on failure modes near tunnel openings in soft rock. Géotechnique, 43(3): 433–442. Indraratna, B., and Kaiser, P.K. 1990. Analytical model for the design of grouted rock bolts. International Journal for Numerical and Analytical Methods in Geomechanics, 14: 227–251. Itasca Consulting Group Inc. 1997. FLAC 3D User’s Manual, Software Development Group, Minneapolis, Minn. Labiouse, V. 1991. Rockbolting in the rock-support interaction analysis. In Proceedings of the 7th International Congress of the Society of Rock Mechanics, Aachen, pp. 1321–1324. Nguyen Minh, D., and Guo, C. 1993. Sur un principe d’interaction massif-soutènement des tunnels en avancement stationnaire. In Proceedings of the Eurock’93 Conference. Oreste, P.P., and Peila, D. 1996. Radial passive rockbolting in tunnelling design with a new convergence–confinement model. International Journal of Rock Mechanics, Mining Sciences & Geomechanics, 33(5): 443–454. © 2006 NRC Canada

478 Panet, M. 1995. Le calcul des tunnels par la méthode convergence– confinement, Presses de l’Ecole Nationale des Ponts et Chaussées, Paris, France. Panet, M., and Guenot, A. 1982. Analysis of convergence behind the face of a tunnel. In Proceedings of Tunnelling’82 Conference, Brighton, England, pp. 197–204. Peila, D., Oreste, P.P., Pelizza, S., and Poma, A. 1996. Study of the influence of sub-horizontal fiber-glass pipes on the stability of a tunnel face. In Proceedings of North American Tunneling’96 Conference, Washington, D.C., pp. 425–432. Pelizza, S., and Peila, D. 1993. Soil and rock reinforcements in tunnelling. Tunnelling and Underground Space Technology, 8(3): 357–372. Rabcewicz, L.V., and Golser, J. 1973. The principles of dimensionning the supporting system for the “New Austrian Tunnelling Method”. Water Power, 25(3): 88–93. Stille, H., Holmberg, M., and Nord, G. 1989. Support of weak rock with grouted bolts and shotcrete. International Journal of Rock Mechanics, Mining Sciences & Geomechanics, 26(1): 99–113. Subrin, D. 2002. Etude théorique sur la stabilité et le comportement des tunnels renforcé par boulonnage. Ph.D. thesis, ENTPE-INSA de Lyon, France. Wong, H., and Larue, E. 1998. Modeling of bolting support in tunnels taking account of nonsimultaneous yielding of bolts and ground. In Proceedings of the International Conference on the Geotechnics of Hard Soils – Soft Rocks, Napoli, Italy 12– 14 October 1998. Edited by A. Evangelista and L. Picarelli. A.A. Balkema, Rotterdam, The Netherlands. pp. 1027–1038. Wong, H., Trompille, V., and Dias, D. 1999. Déplacements du front d’un tunnel renforcé par boulonnage prenant en compte le glissement boulon-terrain: approches analytique, numérique et données in situ. Revue Française de Géotechnique, 89: 13–28. Wong, H., Subrin, D., and Dias, D. 2000a. Extrusion movements of a tunnel head reinforced by finite length bolts–a closed form solution using homogenization approach. International Journal for Numerical and Analytical Methods in Geomechanics, 24(6): 533–565. Wong, H., Subrin, D., and Dias, D. 2000b. Behavior of a boltsupported tunnel, using convergence–confinement and homogenization approach. In Proceedings of the International Conference on Geotechnical and Geoenvironmental Engineering, Melbourne, Australia, 19–24 November 2000.

Appendix A. Classical convergence– confinement method Unlined tunnel Consider an unlined tunnel driven in an infinite homogeneous ground (Fig. A1). Stationary conditions are assumed so that the radial displacement u at a cross section S – fixed with respect to the ground – only depends on the distance z from the excavation face. Hence, the time evolution of u at section S as the tunnel face advances can be assessed by studying the spatial variation of u at increasing distances from a fixed tunnel face z. This duality between space and time is implicitly assumed in the convergence–confinement method. This assumption excludes creep behaviours as well as heterogeneous ground conditions (depth, geology, etc.) An essential hypothesis of this method is that the effect of tunnel face advance, provoking tunnel convergence u (or the dimensionless convergence: U = u/R), can be simulated by a decreasing internal fictive pressure Pf. Relative to a fixed

Can. Geotech. J. Vol. 43, 2006

section S, the pressure profile advances at the same speed V as the tunnel excavation. The fictive pressure Pf starts upstream at the geostatic pressure P∞ (where u = 0) and decreases downstream. At any intermediate state, when S is at a distance z from the face, the radial displacement u(z) is calculated using a 1D analytical model, assuming axisymmetry and plane strain conditions subject to a constant far field pressure P∞ at r = ∞ and internally to a pressure that decreases from P∞ to Pf(z). In other words, an internal unconfinement provokes tunnel convergence. Moreover, under linear elastic behaviour, U = u/R is proportional to (P∞ – Pf). This motivates the definition of the dimensionless “unconfinement rate” [A1]

λ=

P∞ − Pf P∞

Either Pf or λ can be used as the loading variable because Pf and λ are related by Pf = (1 – λ)P∞ . In practice λ is increasing and can more conveniently be regarded as the kinematic time. Under nonlinear elastic–plastic behaviour, the dependence of U on Pf is more complex, but can still be expressed in an analytical form for simple material behaviours, and we can write [A2]

U = fM(Pf)

where the function fM depends on the ground behaviour. Lined tunnel When a lining support is installed at a distance d from the tunnel face, a real contact pressure Ps develops between the lining and the ground. This reduces the unlined tunnel convergence profile U(z) to Us(z) (the suffix “s” stands for support). Restricting ourselves to linear elastic linings, the pressure Ps is proportional to the convergence experienced by the lining itself. In other words, at a distance z > d, we have [A3]

Ps = Ksn[Us(z) – Us(d)]

where Ksn is the stiffness of the lining. For example, for a shotcrete lining of thickness e (Panet 1995) [A4]

K sn =

Eb e 1 − ν 2b R

Three other important assumptions are: (i) with lining support, the fictive pressure profile Pf remains unchanged; (ii) the two convergence profiles with and without lining coincide over the unlined portion z ≤ d; and (iii) the convergence U at any distance z can be calculated by the 1D model upon replacement of the fictive pressure Pf by the total pressure Pf + Ps in eq. [A2]. With these assumptions, the lined tunnel convergence and the contact pressure verify the two following equations: [A5]

Us = fM(Pf + Ps)

[A6]

Ps = Ksn[Us(z) – Up]

where Ud = U(d) is the unlined tunnel convergence at a distance d from the tunnel face, which is an important input parameter of the convergence–confinement method. It can © 2006 NRC Canada

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Fig. A1. Convergence–confinement method. The tunnel face advance (a) and its effect on the convergence u at section S is simulated in (b) by an advancing fictive pressure profile; u is calculated by a 1D model (c) with decreasing internal pressure. With lining support, the support pressure adds to the fictive pressure to reduce convergence (d).

either be evaluated by previous experience or estimated using an empirical formula (Panet 1995). In practice, one is not interested in the transient states when z varies from d to ∞. Only the final equilibrium state – when the tunnel face is far from the section question – is important. Letting x tend to infinity, and noting that the fictive pressure then goes to zero, it can easily be seen that the convergence and support pressure at final equilibrium, denoted by Us∞ = Us(∞) and Ps∞ = Ps(∞), are solutions of the following set of equations: [A7]

U = fM(Pi)

[A8]

Pi = Ksn(U – Ud)

This set of equation admits a very simple graphic interpretation, as shown in Fig. A2, where instead of Pi, we have used the equivalent parameter Pi/P∞ = 1 – λ. This is nothing more than a simple change of scale upon division by P∞ .

Reference Panet, M. 1995. Le calcul des tunnels par la méthode convergence– confinement, Presses de l’Ecole Nationale des Ponts et Chaussées, Paris, France.

Appendix B. Resolution process for each configuration Attention is concentrated on the determination of the plastic boundaries – the fundamental unknowns of the problem – as well as the critical values of the loading parameter λ, which define the transition from one configuration to another.

Fig. A2. Graphical interpretation of eqs. [A7] and [A8].

Configurations 1 and 2: unreinforced medium The resolution for the unreinforced case is classical and can be found elsewhere (e.g., Panet 1995). The yield criterion is verified when the unconfinement λ reaches the critical value [B1]

λe =

C P∞

Above the unconfinement given by eq. [B1], a plastic zone appears near the tunnel wall, and the situation corresponds to configuration 2 as shown in Fig. B1. The plastic boundary x writes 2

[B2]

⎞ ⎛λ ⎛x⎞ − 1⎟⎟ ⎜ ⎟ = exp ⎜⎜ ⎝R⎠ ⎠ ⎝ λe

Configuration 3: bolts remain elastic Configuration 3 is shown in Fig. B2. © 2006 NRC Canada

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Can. Geotech. J. Vol. 43, 2006

Fig. B1. Configuration 2.

Fig. B3. Configuration 4.

Fig. B2. Configuration 3.

and the solution, after taking into account eq. [17b] 2

[B6]

Inside the plastic zone PLs (R < r < x), eliminating ε b and σθ from eq. [15] (i.e., r∂ r σr + (σr – σθ ) = 0) and the yield criterion eq. [11] (i.e., f(σ) = σr – σθ – 2C – σo(r) < 0) using [4a]

σ o (r) = β

R Eε r s b

if

ε b < ε yb =

σ yb

A ; r2

& λ′ )dλ′ ∫ B(

[B8]

λp

1 ⎞ ⎛ r ⎞ βE RA ⎛ 1 σ r = −(1 − λ)P∞ − 2C ln ⎜ ⎟ + s ⎜ 3 − 3 ⎟ R 3 r R ⎝ ⎠ ⎝ ⎠

In the outer (reinforced) elastic zone EL (x < r), ζ = 0. Eliminating σθ and u from eqs. [13], [15], and [19], i.e., [13]

3 (σ& r − σ& θ) + ζ& ∂ ru& = E s (4 + 3βR / r)

[15]

r∂ rσr + (σr – σθ) = 0

[19]

u& = −

& B ; r

u=−

B ; r

ε& r =

& B ; r2

εr =

B r2

λ

with

B(λ) =

∫ B& (λ′ )dλ′ 0

then integrating between the time of bolt installation λp and the current time λ gives 2

[B5]

A=

3C 2 (x − x 2p) 2Es

2 2 ⎡ ⎛ xp ⎞ ⎤ λ ⎛x⎞ β ⎛x⎞ = 1 + 2 ln ⎜ ⎟ + ⎢⎜ ⎟ − ⎜ ⎟ ⎥ ⎜R⎟ ⎥ λe ⎝ R ⎠ 2 ⎢⎝ R ⎠ ⎝ ⎠ ⎦ ⎣

This configuration ends when the yield limit of the bolts is reached at the tunnel wall, in other words

A r∂ rσ r + 2C + β E sR 3 = 0 r

which gives, on account of eq. [17b] (i.e., σr(R,t) = –(1 – λ)P∞ ) [B4]

[B7]

λ

A(λ) =

we obtain [B3]

where xp = x(λp) is the plastic boundary at bolt placement. The constant A is determined by the condition that the yield criterion is just reached at x+. Hence, substituting eq. [B6] for σr and that of σθ using eq. [15] into eq. [11] using eq. [4a], we get

The continuity of σr at x leads to the following equation for the determination of x, and all other quantities by substitution:

Eb

and [20] ε b =

⎛ xp ⎞ βR ⎞ 2E A ⎛ σ r = −P∞ + C ⎜ ⎟ + s2 ⎜1 + ⎟ ⎜ r ⎟ 3r ⎝ 2r ⎠ ⎝ ⎠

⎛ xp ⎞ E A r∂ rσ r + 2C ⎜ ⎟ + s 2 (4 + 3βR / r) = 0 ⎜ r ⎟ 3r ⎝ ⎠

[B9]

ε b (R) =

σ yb A = ε yb = R2 Eb

which gives the corresponding critical unconfinement λbp, on account of eqs. [B7] and [B8] [B10]

λ bp λe

⎡⎛ x ⎞ 2 ⎤ 2Ω ⎥ Ω p + = 1 + ln ⎢⎜ ⎟ + 3β ⎥ 3 ⎢⎜⎝ R ⎟⎠ ⎣ ⎦

Configuration 4 Configuration 4 is shown in Fig. B3. When λ reaches λbp , a new plastic zone PLs /PLb (R < r < w) appears at the tunnel wall and develops, in which both the ground and the bolts have yielded. Combining eqs. [15] and [11] using [4b]

σ o (r) = ΩC

R if ε b ≥ ε yb r

we get the following differential equation on σr [B11] r∂ rσ r + 2C + ΩC

R =0 r

and the solution, on account of eq. [17b] is © 2006 NRC Canada

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481

R⎞ ⎛ ⎛r⎞ [B12] σ r = − (1 − λ)P∞ − 2C ln ⎜ ⎟ − ΩC ⎜1 − ⎟ R r⎠ ⎝ ⎝ ⎠

Fig. B4. Configuration 5.

In the outer elastic zone (x < r), eq. [B6] still holds, while in the intermediate plastic zone PLs (w < r < x), eq. [B3] applies, which together with the continuity of σr at r = x gives (x 2 − x p2 ) ⎛r⎞ β [B13] σ r = −P∞ + C − 2C ln ⎜ ⎟ + RC r3 ⎝x⎠ 2 Then σθ can be obtained by the equilibrium eq. [15]. The yield condition of bolts at r = w+ leads to the following relationship [B14]

3β w = 2Ω R

Fig. B5. Integration paths used in configuration 5.

2⎤

2 ⎡ ⎛x ⎞ ⎢⎛⎜ x ⎞⎟ − ⎜ p ⎟ ⎥ ⎜R⎟ ⎥ ⎢⎝ R ⎠ ⎝ ⎠ ⎦ ⎣

Finally, continuity of σr at r = w results in the following equation, which together with eq. [B14], determine the unknowns x and w, and subsequently all others quantities: [B15]

λ 2R ⎞ ⎛x⎞ ⎛ = 1 + 2 ln ⎜ ⎟ + Ω ⎜1 − ⎟ λe R 3 w⎠ ⎝ ⎠ ⎝

From eq. [B14], we can deduce the eventual existence of an additional phase, with a plastic zone PLb, in which only the bolts have yielded, bounded by the zones ELs/ELb and PLs /PLb . Putting x = w in eq. [B14], we see that this occurs when the two plastic radii cross each other at the particular radius xc = x(λc), given by [B16]

3β 3β − 2Ω

xc xp = R R

which, clearly, is only possible if

& ⎞ R⎞⎛ A ⎛4 [B20] r ∂ rσ& r = E s ⎜ + β ⎟ ⎜⎜ − 2 ⎟⎟ r ⎠⎝ r ⎠ ⎝3

3β [B17] >1 2Ω Then λc is determined by eq. [B15], using eq. [B16] [B18]

⎛ xp λc = 1 + 2 ln ⎜ ⎜R λe ⎝

3β ⎞⎟ 3β − 2Ω ⎟⎠ 3β − 2Ω ⎞⎟ 3β ⎟ ⎠

Configuration 5 Configuration 5 is shown in Fig. B4. In the inner zone PLs/PLb (R < r < x), eq. [B12] obtained in configuration 4 applies. In the outer zone EL (w < r), eq. [B6] holds but A is no longer given by eq. [B7]. Instead, the yield condition of bolts at r = w+ gives the “current” value of A [B19] A(λ) =

Eb

from λp to λw(r) (i.e., from point a to b), the yield condition of bolts, εb[λw(r),r] = A[λw(r)]/r2 = σyb/Eb, leads to [B21] r∂ rσ r[λ w (r)] = r∂ rσ r (λ p ) −

⎛ 2R + Ω ⎜1 − ⎜ 3x p ⎝

σ yb

system of equations first of all from “point” a to b (Fig. B5), then from b to c. To begin with, denote by λw(r) the inverse function of w(λ) for λ > λc. Considering a radius r such that x(λ) < r < w(λ), then integrating eq. [21] with ζ = 0, i.e.,

w2

The determination of σr in zone ELs/PLb (x < r < w) requires two incremental calculations. We have to integrate the

ΩC ⎛ 4 R⎞ ⎜ +β ⎟ β ⎝3 r⎠

where ⎛ xp ⎞ r∂ rσ r (λ p ) = −2C ⎜ ⎟ ⎜ r ⎟ ⎝ ⎠

2

For the integration from λw(r) to λ (i.e., point b to c), [B22] r ∂ rσ& r =

& ⎞ 4Es ⎛ A ⎜⎜ − 2 ⎟⎟ 3 ⎝ r ⎠

instead of eq. [21] is used to account for the constant bolt stress after yielding; then after some calculations 2 ⎡ ⎛ x ⎞2 ⎤ 4Ω ⎛ w ⎞ R⎥ p [B23] r∂ rσ r (λ) = − C ⎢2 ⎜ ⎟ + + Ω ⎜ ⎟ 3β ⎝ r ⎠ r⎥ ⎢ ⎜⎝ r ⎟⎠ ⎣ ⎦

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Can. Geotech. J. Vol. 43, 2006

Integrating between r and w, and taking into account the continuity of σr at r = w results in the following expression valid for x < r < w: 2

⎛ xp ⎞ 2Ω ⎛ w ⎞ ⎛R 2 R ⎞ [B24] σ r = −P∞ + ΩC ⎜ − C⎜ ⎟ ⎟ + C ⎜⎜ ⎟⎟ + 3 r 3β ⎝ r ⎠ r w ⎝ ⎠ ⎝ ⎠

2

References

The yield criterion at x+ gives the first equation for the determination of the unknown boundaries x and w, identical to eq. [B14] of configuration 4. The continuity of σr at r = x gives the second equation 2

2Ω ⎛ w ⎞ 2 R ⎛ xp ⎞ λ ⎛x⎞ [B25] = 2 ln ⎜ ⎟ + Ω − Ω + ⎜ ⎟ + ⎜ ⎟ ⎟ ⎜ 3 w ⎝ x ⎠ 3β ⎝ x ⎠ λe ⎝R⎠

On account of eq. [B14], eq. [B25] turns out to be identical to eq. [B15]. Evolution of the boundary radii x and w are therefore governed by the same set of equations as in configuration 4 after their interception. The validity of this phase continues up to total unconfinement λ = 1.

Panet, M. 1995. Le calcul des tunnels par la méthode convergence– confinement, Presses de l’Ecole Nationale des Ponts et Chaussées, Paris, France.

2

Appendix C. Detailed solutions for each configuration Tables C1 to C5 contain detailed solutions for configurations 1 through 5, respectively.

Table C2. Detailed solution for configuration 2.

Table C1. Detailed solution for configuration 1. EL (R < r < ∞) 3C λ R 2 Es λ e r

u

−

σr

⎛R⎞ −P∞ + λP∞ ⎜ ⎟ ⎝r⎠

σθ

⎛R⎞ −P∞ − λP∞ ⎜ ⎟ ⎝r⎠

EL (x < r < ∞)

PLS (R < r < x)

2

2

3C x 2 2 Es r

u

−

2

σr

⎛r⎞ − (1 − λ )P∞ − 2C ln ⎜ ⎟ ⎝R⎠

⎛x⎞ −P∞ + C ⎜ ⎟ ⎝r⎠

2

2

σθ

⎡ ⎛ r ⎞⎤ − (1 − λ )P∞ − 2C ⎢1 + ln ⎜ ⎟⎥ ⎝ R ⎠⎦ ⎣

⎛x⎞ −P∞ − C ⎜ ⎟ ⎝r⎠

2

3C x 2 Es r

−

Table C3. Detailed solution for configuration 3. EL (x < r < ∞)

PLS (R < r < x) 2

3C x 2 2 Es r

u

−

σr

⎛1 ⎛r⎞ β 1 ⎞ − (1 − λ )P∞ − 2C ln ⎜ ⎟ + RC ( x 2 − x p2 ) ⎜ 3 − 3 ⎟ R ⎠ ⎝r ⎝R⎠ 2

( x 2 − x p2 ) ⎛x⎞ β −P∞ + C ⎜ ⎟ + RC 2 r3 ⎝r⎠

σθ

⎡ ⎛ r ⎞⎤ β ⎛2 1 ⎞ − (1 − λ )P∞ − 2C ⎢1 + ln ⎜ ⎟⎥ − RC ( x 2 − x p2 ) ⎜ 3 + 3 ⎟ R 2 r R ⎝ ⎠⎦ ⎝ ⎠ ⎣

( x 2 − x p2 ) ⎛x⎞ −P∞ − C ⎜ ⎟ − βRC r3 ⎝r⎠

3C x 2 Es r

−

2

2

Table C4. Detailed solution for configuration 4. PLS /PLb (R < r < w) 3C x 2 2 Es r

u

−

σr

⎞ ⎛R ⎛r⎞ − (1 − λ )P∞ − 2C ln ⎜ ⎟ + ΩC ⎜ − 1⎟ r R ⎠ ⎝ ⎝ ⎠ ⎡ ⎛ r ⎞⎤ − (1 − λ )P∞ − 2C ⎢1 + ln ⎜ ⎟⎥ − ΩC ⎝ R ⎠⎦ ⎣

σθ a

PLS (w < r < x)

EL (x < r < ∞)a

3C x 2 2 Es r ⎡ ( x 2 − x p2 ) ⎛ r ⎞⎤ β −P∞ + C ⎢1 − 2 ln ⎜ ⎟⎥ + RC r3 ⎝ x ⎠⎦ 2 ⎣ ⎡ ( x 2 − x p2 ) ⎛ r ⎞⎤ −P∞ − C ⎢1 + 2 ln ⎜ ⎟⎥ − βRC r3 ⎝ x ⎠⎦ ⎣

−

−

3C x 2 2 Es r

Equations for σr and σθ are formally identical to the expressions in Table C3. © 2006 NRC Canada

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483 Table C5. Detailed solution for configuration 5. PLS/PLb (R < r < x)a u

−

3C x 2 2 Es r

PLb (x < r < w) −

3C x 2 2 Es r

EL (w < r < ∞)b −

σr

⎛ R 2R ⎞ ⎛x⎞ −P∞ + C ⎜ ⎟ + ΩC ⎜ − ⎟ ⎝ r 3w ⎠ ⎝r⎠

σθ

⎛x⎞ 2R −P∞ − C ⎜ ⎟ − ΩC r w 3 ⎝ ⎠

3C x 2 2 Es r

2

a b

2

Equations for σr and σθ are formally identical to expressions in Table C4. Equations for σr and σθ are formally identical to expressions in Table C3.

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