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International Journal of Rock Mechanics & Mining Sciences journal homepage: www.elsevier.com/locate/ijrmms

Passive bolts reinforcement around a circular opening in strain-softening elastoplastic rock mass Cheng Hua TAN n ARCADIS 9, avenue Réaumur, 92354 Le Plessis-Robinson cedex, France

art ic l e i nf o Article history: Received 14 January 2016 Received in revised form 2 June 2016 Accepted 28 July 2016 Keywords: Convergence-Conﬁnement method Strain-softening behavior Mohr-Coulomb criterion Hoek-Brown criterion Passive bolts reinforcement

1. Introduction Rock bolts reinforcement, such as passive grouted bolts, have been widely used as a primary and effective support system to improve and stabilize the rock mass around an opening cavity such as tunnel, underground mine galleries etc. For the passive bolts embedded in deformable rock mass and closed to the excavation opening boundary, the deformation of rock mass transmits the shear stress to bolts via their contact surface so that axial tension develops in a bolt rod which retrains in return the deformation within rock mass and reduces the yield region around the excavation boundary. For a preliminary design of circular axisymmetric plane strain openings encountered in tunneling problem, an exact and closeform solution with Mohr-Coulomb (M-C) criterion is normally feasible. However, for many types of rock, particularly for a joint rock mass with poor or very poor quality, the Generalized HoekBrown (H-B) criterion is more suitable. The behaviors of rock materials generally evolve in three possible ways. Hard, good quality rock mass tends to show an elastic brittle plastic behavior, for which the properties of rock collapses instantly to residual values as soon as the peak mass strength is exceed. The very poor quality rock mass shows an elastic perfectly plastic behavior. For rock mass with average quality, it was n

Corresponding author. E-mail address: [email protected]

http://dx.doi.org/10.1016/j.ijrmms.2016.07.024 1365-1609/& 2016 Elsevier Ltd. All rights reserved.

commonly observed in ﬁeld and laboratory that it tends to show strain-softening behavior during the post-peak stage. Knowledge of post-failure behavior is signiﬁcant to several aspects of rock engineering design, including prediction of underground opening behavior of rock mass under static and seismic loads, installation of opening support and sequencing of mine extraction. For rock mass with elastic perfectly plastic or elastic brittle plastic behavior, a closed-form solution of such H-B criterion was recently formulated by Carranza-Torres1 for stresses and displacements around circular openings. The formulation is based on a transformation technique that required the use of numerical method to solve a non-linear equation for stresses and a secondorder differential equation for displacement. Sharan2,3 proposed an exact solution without using any transformation. However, the integration on the displacement needed still to be carried out numerically. Therefore, a numerical calculation framework is always necessary if the solution can be applied to M-C as well as to Generalized H-B criterions. As for as rock mass with strain-softening which is referred to as the progressive loss of strength when material is loaded beyond peak resistance, the fewer attempts based on the rigorous method in terms of incremental plastic strain have been made thus far because the difﬁculties of the experimental identiﬁcation and numerical modeling implementation. The constitutive nonlinearity of stress-strain and strength parameters during strainsoftening stage necessitates a robust numerical computational procedure in order to obtain a coherent and reliable solution. There exist rarely analytical attempts such as Zhang et al.4 closed-

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form solution, but its formulation is limited strictly to the classical H-B criterion for intact rock mass. Alonso et al.5 presented a numerical solution based on the self-similar approximation for the circular tunnel in strain-softening rock mass. Lee and Pietruszczak6 described the approximate solution by ﬁnite difference method. However, all these numerical approaches are limited to the analysis of classical circular opening and are not applicable to the case with reinforcement of passive bolts. In the reinforcement operation, the primary role of bolts is to restrain the rock mass deformation though the loads transfer mechanism between bolts and rock mass. However, the presence of bolts in the rock mass complicates ground solution signiﬁcantly. Among the researches concerning the reinforcement of passive bolts, the majority of studies have been made using the ﬁnite element modeling through the attempts of implementation of frictional model into the calculation codes as elastic rock-bolt bonded element in elastic-perfectly-plastic rock mass16 or bondslip rock bolt element in strain-softening rock mass.17 Among the very few studies published so far using the unclosed-form numerical method in strain-softening rock mass, Guan et al.18 proposed a numerical solution of great interest which is able to evaluate the effect of passive bolts on the stress and displacement behavior of rock around the tunnel. Unfortunately, their solution used a relatively simple and not representative value: major principal plastic strain as softening parameter, instead of shear plastic strain which is widely accepted among researches because rock mass is sheared to failure material. Moreover, their solution is limited only to the linear failure criterion Mohr-Coulomb. The use of analytical axisymmetric approaches for such problem as done in classic convergence-conﬁnement method has encountered many theoretical difﬁculties. This is because, unlike the other support systems, the passive bolt does not act independently of the rock mass but their strain-stress response is inter-connected via the frictionally coupling mechanism. A closedform solution for such a problem is therefore difﬁcult to obtain even for elastic perfectly plastic rock mass, not to mention the case of strain-softening behavior. A computational framework has been successfully developed to fully evaluate ground response for the reinforcement mechanics of passive bolt in the conventional tunneling. Based on the incremental plasticity using Finite Difference Method (FDM), the previous research work7 covered in detail the solution for the rock mass exhibiting elastic brittle plastic behavior with M-C as well as H-B criterions and illustrated clearly that the framework is able to capture quantitatively the key elements of passive bolt reinforcement observed in in-situ monitoring. In this paper, an extension of this developed computational framework to strain-softening behavior of rock mass with and without passive bolt reinforcement is presented. The results of the proposed solution to the classical circular opening problem during the process of strain-softening are ﬁrstly examined in great detail and veriﬁed against the published results from the literatures for M-C as well as Generalized H-B criterions. The reinforcement mechanism of passive bolts and the inﬂuence of various concerned properties are then demonstrated via case studies. Our research work is intended to present a uniﬁed computational framework which is general enough to be able applied not only to the classical circular opening excavated in elastoplastic rock masses but also to the analysis of reinforcement of passive bolts and then, if necessary, to be extended in the future to a tunnel excavated below the groundwater table and submitted to seepage force. This makes the convergence-conﬁnement method (CCM) more readily applicable to tunnel design, regardless of the type of failure criterions and associated elastoplastic behaviors used to deliver a guideline for a given rock mass.

2. Frictional model between passive bolt and rock mass 2.1. Equation of equilibrium A passive bolt embedded in deformable rock mass and closed to the excavation opening boundary is solicited by displacement of rock mass after its installation. It provides resistance to the movement of rock mass though shear stress developed axially in the contact surface with rock mass. When the shear stress exceeds the strength of the contact interface, slippage may occur and produce so called decoupling as shown in Fig. 1, where the displacement of surrounding rock eventually leads to the development of two distinct zones represented by “pick-up length” and “anchor length” along the bolt. These two zones are separated by a neutral point (or neutral section) where the shear stress is zero. Therefore, the shear stresses describing the rock-bolt interaction, rather than the grouting material itself, are of great importance in the overall resistance of a bolt reinforcement system. The bolt is represented by a one-dimensional rod and hence the equilibrium between the shear force and the axial force in the bolt rod can be established for a small section of the bolt dx :

dT = − τ = − k τ (ur − ub)dx T = EbτAb

dub dx

(1)

(2)

The equation of equilibrium is:

du ⎞ d ⎛ ⎜ EbτAb b ⎟ − k τub = − k τur dx ⎝ dx ⎠

(3)

where the subscripts b , r represent respectively the quantity for bolt and rock mass. τ is the shear force due to frictional interaction between bolt and rock mass, Ab is the area of cross-section of the bolt and Ebτ is the secant modulus of bolt (Fig. 2). Ab is considered as constant along its length and Ebτ is constant for the elastic bolt and is variable for the case of elastic perfectly plastic bolt which tensile stress is limited by σlim . The elongation of the bolt itself is neglected since it is much smaller in comparing to the strain of rock mass. The displacement of rock mass ur , decreasing monotonically with radial distance from the circular tunnel wall, induces the movement of bolt via frictional interaction mechanism. kτ is the stiffness of frictional interaction and can be determined from loaddisplacement response during laboratory pullout tests. The experimental studies of pull-out tests show that load-displacement curve is non-linear because of the de-bonding mechanism between passive bolt and rock mass. This behavior can be approached by a nonlinear frictional strength versus relative shear displacement Δu = (ur − ub) illustrated in Fig. 2, where τpeak , τres is the peak and residual friction of the interface respectively. The coefﬁcient ks represents the shear stiffness in bonding phase

Fig. 1. Frictional interactions between bolt and rock mass.

C.H. TAN / International Journal of Rock Mechanics & Mining Sciences 88 (2016) 221–234

223

Fig. 2. (a) Frictional interactions of bolt and rock mass; (b) Elastic perfectly plastic curve of bolt.

where the shear strength is mobilized linearly by Δu. The decoupling begins after the peak friction τpeak and a nonlinear frictional stiffness kτ is then used to describe the loss of shear strength with Δu. The passage between the peak strength τpeak and the residual strength τres is controlled by a softening slope. From the above equation of equilibrium, the displacement of bolt ub can be obtained only if the displacement of rock mass ur , the source of solicitation, is known beforehand. In the proposed solution, for each ﬁctive pressure applied to tunnel wall, ur is the increment of displacement in the reinforced rock since the bolts installation as described in detail in computation section. Therefore, a fully coupled iterative procedure is required for ur calculation so that the congruence of displacement occurring in the rock and in the bolt can be guaranteed.

Free-force boundary condition Tbn = 0 is for the other far bolt end ( i = n). The discretized formulation Eq. (4) gives a system of equations with n unknown displacements of bolt ubi(i = 1, .. . n). Because of the non-linear frictional behavior between bolt and rock mass as well as elastoplastic bolt behavior, an iterative calculation procedure is required to obtain the solution on bolt displacement ub along its length. The axial tensile force in the bolt can be then calculated from its discretized form:

2.2. Solution using FDM

3. Reinforcement model of passive bolts in conventional tunneling

Using Finite Difference method, the bolt is discretized n steps along its length with h = xi − xi − 1 as step length (i = 1 denoting the tunnel wall end), the equation of equilibrium Eq. (3) can be written with central difference schema as:

⎛1 ⎞ 1 Ab ⎜ Ebi+τ 1 + Ebiτ − Ebi−τ 1⎟ubi+ 1 − 2EbiτAb + k τh2 ubi ⎝4 ⎠ 4 ⎛ 1 i+1 ⎞ 1 + Ab ⎜ − Ebτ + Ebiτ + Ebi−τ 1⎟ubi− 1 = − k τh2uri ⎝ 4 ⎠ 4

(

)

(

)

ubi+ 1 − ubi− 1 xi + 1 − xi − 1

(7)

For a circular shape of tunnel in isotropic rock mass subjected to an isotropic stress ﬁeld po and under axisymmetric plane strain condition, it is possible to determine the stress and strain state when a support pressure applied to it tunnel wall. 3.1. Equation of equilibrium

(4)

In case of elastic bolt, the secant modulus remains constant Ebτ = Eb , Eq. (4) reduces to

EbAb ubi+ 1 − 2EbAb + k τh2 ubi + EbAb ubi− 1 = − k τh2uri

Tbi = EbτAb

(5)

such as described in the previous study.7 Two types of boundary condition can be applied to its two ends:

For an inﬁnitesimal volume in the radial direction as shown in Fig. 1 with the presence of a passive bolt, the rock mass is subjected to not only a radial stress σr and a circumferential stress σϑ but also an axial force T provided by bolt. The equation of equilibrium can be reduced from the static equilibrium condition of this volume:

dσr σ − σr R 1 dT = ϑ + i dr r SlSc r dr

(8)

1. without face plate bolt Free-force boundary condition Tb1 = Tbn = 0 is for two bolt ends ( i = 1 and i = n) 2. with face plated bolt at wall end ( i = 1)

where Ri is the radius of tunnel and Sl, Sc are bolt spacing in the longitudinal and circumferential direction respectively. If the bolt reinforcement is absent ( dT = 0), the Eq. (8) reduces to its classical form.

In this case, the reaction force Tb1 > 0 is developed at wall face ( i = 1):

3.2. Equation of elastic responses

Tb1 = Kpb(ur1 − ub1)

(6)

where Kpb denotes the composed stiffness describing the stiffness of the face plate on the rock mass and the stiffness between plate and bolt tie on the tunnel surface.

The elastic radial and circumferential strain is determined by known stress state according to the theory of elasticity:

εre =

1 + ν⎡ ⎣ (1 − ν )Δσr − νΔσϑ⎤⎦ E

(9)

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Kp, res =

1 + sin φres 1 − sin φres

UCSres =

2cres cos φres 1 − sin φres

where φres and cres are the residual friction angle and cohesion of rock mass respectively. The radial stress σr = pcrit at the elastoplastic interface r = Rpl

(

)

can be calculated from

pcrit =

2po − UCSpeak Kp, peak + 1

(16)

with

Kp, peak =

1 + sin φ peak 1 − sin φ peak

UCSpeak =

2cpeak cos φ peak 1 − sin φ peak

where φpeak and cpeak are the peak friction angle and cohesion of rock mass respectively. The rock mass will experience plastic response only if the internal pressure of tunnel wall falls below pcrit . The rock mass exhibiting strain-softening behavior is characterized by a transitional failure from the peak values cpeak, φpeak

Fig. 3. Softening behavior of rock mass.

εϑe =

1 + ν⎡ ⎣ (1 − ν )Δσϑ − νΔσr ⎤⎦ E

(10)

The relation of total strain and displacement can be expressed as

εr =

du dr

εϑ =

u r

(11)

(12)

3.3. Equations of elastoplastic responses In the plastic zone, the stresses state is limited by a failure criterion in which a strain-softening behavior of shear strength is allowed (Fig. 3):

f (σ , η) = 0

(13)

where η represents the so-called softening parameter and will be detailed in the following section. Two failure envelopes can be deﬁned: one for the peak strength and the other for residual strength. The plastic straining is limited by these two envelopes. The shear strength drops gradually from the peak value to residual value during plastic straining. In absence of softening phase in Fig. 3, the peak value collapses into residual value as soon as plastic straining takes place and the rock mass manifests an elastic brittle plastic behavior. If residual strength is identical to peak strength, it exhibits simply an elastic perfectly plastic behavior. Therefore, elastic perfectly plastic and elastic brittle plastic behaviors constitute the limiting cases of the strain-softening behavior. 3.3.1. Mohr-Coulomb criterion The Mohr-Coulomb failure criterion is written in case of σϑ > σr :

σϑ =

1 + sin φ 2c cos φ σr + 1 − sin φ 1 − sin φ

(14)

where φ and c are the friction angle and cohesion of rock mass respectively. These strength parameters are the function of softening parameter η in strain-softening rock mass. In the plastic zone, the criterion is expressed as:

σϑ = Kp, resσr + UCSres with

(15)

to residual values cres, φres . 3.3.2. Generalized Hoek-Brown criterion The Generalized Hoek-Brown failure criterion expressed in major and minor principal stresses σ1 and σ3 is 1,9:

⎛ ⎞a σ σ1 = σ3 + σci⎜ mb 3 + s⎟ ⎝ σci ⎠

(17)

where σci is the unconﬁned compressive strength and the coefﬁcients mb , s and a are semi-empirical parameters that characterize the rock mass. These strength parameters are the function of softening parameter η in strain-softening rock mass. In practice, these parameters are computed from the value GSI (Geological Strength Index) and the value D (disturbance factor of rock mass):

⎛ GSI − 100 ⎞ ⎟ mb = mi exp⎜ ⎝ 28 − 14D ⎠

(18)

⎛ GSI − 100 ⎞ ⎟ s = exp⎜ ⎝ 9 − 3D ⎠

(19)

a=

1 1 −( GSI/15) + e − e−( 30/3) 2 6

(

)

(20)

where mi is a material constant for the intact rock and D is a disturbance factor which should not be altered to a residual value as it used to calculate the peak parameters mb and s. The range of GSI and D are normally as follows: 0 ≤ D ≤ 1 and 10 ≤ GSI ≤ 100. The value D = 1 corresponds to a highly disturbed rock mass. Very weak rocks ( GSI < 30) manifest elastic perfectly plastic and no dilation behavior. Jointed intermediate rocks ( 40 < GSI < 50) assume strain-softening and small dilation behavior. Massive brittle rocks ( 70 < GSI < 90) represent high stress resulting in intact rock failure and practically all strength lost at failure.11 In the plastic zone, the failure criterion is expressed by the values of residual strength.

⎛ ⎞ares σ σ1 = σ3 + σci, res⎜⎜ mb, res 3 + sres⎟⎟ σci, res ⎝ ⎠

(21)

(

The radial stress σr = pcrit at the elastoplastic interface r = Rpl can be obtained from the values of peak strength 1.

)

C.H. TAN / International Journal of Rock Mechanics & Mining Sciences 88 (2016) 221–234

pcrit

⎡ ⎤ speak ⎥ ( (1 − apeak) / apeak) ⎢ = ⎢ Picr − σci, peak ⎥mb 1/ a : ⎢⎣ mb(, peakpeak) ⎥⎦

εrp + Kϕεϑp = 0 (22)

where the transformed value of Picr must be computed numerically (Newton-Raphson method by example) from the following nonlinear equation:

μ(

a Picr

)

+ 2Picr − 2So = 0

(23)

(1 − a

) / apeak

)

po

So =

+

(1 − apeak ) / apeak ) m( σ

ci, peak

b, peak

speak b, peak

2

1 + 16So ⎤ ⎥ 4 ⎥⎦

⎡ σci, peakmb: , peak ⎢ = ⎢1 − 16 ⎣ speakσci, peak − mb: , peak

(24)

(25)

3.3.3. Compatibility equation of displacement The strain in plastic zone is the sum of an elastic component and a plastic component. The plastic strains are governed by an appropriate ﬂow rule. Since the extent of plastic responses depends on the dilation characteristics of the failed rock mass, a nonassociated linear potential using M-C function is adopted:

(26)

(30)

εϑ =

u 1 + ν⎡ = εϑe + εϑp = ⎣ (1 − ν )Δσϑ − νΔσr ⎤⎦ + εϑp r E

(31)

)

(

)

(32)

3.3.4. Strain-softening behavior The rock mass exhibiting strain-softening behavior is characterized by a transitional plastic state from the peak values to residual values following a certain evolution rule (Fig. 3). A bilinear function is chosen generally in the literature by simplicity for the softening parameter η . Such choice is justiﬁed by the fact that experiments on rock samples shows a trend of linear decreasing strength parameters while plastic straining.

(33)

where ω ∈ (ωpeak , ωres ) refers to any one of the strength parameters of a given failure criterion and ωpeak , ωres represent its peak and residual value respectively. For Mohr-Coulomb criterion, ω can refer to the cohesion (c ), the friction angle (φ) and dilatancy angle (ϕ). For Hoek-Brown criterion, ω can be any of (σci, mb: , s, a) and dilatancy angle (ϕ). The elastic parameter E (or shear module G ) can be also degraded during plastic straining if necessary. η* is the critical value of η from which the residual plastic behavior starts and should be identiﬁed through experimental tests in the laboratory. The rock mass exhibits elastic brittle plastic behavior if η* = 0 and elastic perfectly plastic behavior if η* → ∞ (a big value η* ≥ 10 is enough in reality). It is widely acceptable that softening parameter η can be deﬁned as the deviatoric plastic strain:

η = εϑp − εrp

(34)

Combing with the compatibility equation of displacement (Eq. (29)), the softening parameter η can be expressed as

with

Kϕ =

1 + ν⎡ du = εre + εrp = ⎣ (1 − ν )Δσr − νΔσϑ⎤⎦ − Kϕεϑp dr E

ωpeak − ωres ⎧ η (η < η*) ⎪ ωpeak − η* ω=⎨ ⎪ (η ≥ η*) ⎩ ωres

⎤2 ⎛ speak ⎞ ⎥ po ⎜ ⎟ 1 + 16⎜ + 2: ⎟ ⎥ : mb, peak ⎠ ⎦ ⎝ σci, peakmb, peak

The rock mass will experience plastic response only if the internal pressure of tunnel wall falls below pcrit . The rock mass exhibiting strain-softening behavior is characterized by a transitional failure from the peak values : : : : to residual values σci, res, mb: , res , sres . σci, peak, mb: , peak , speak , apeak , ares

g = σ1 − Kϕ σ3

εr =

(

1/ a : m ( peak)

In this case, pcrit can be expressed directly through the values of peak strength:

pcrit

The total strains are

du u + Kϕ dr r 1⎡ = ⎣ 1 − ν 2 − νKϕ − ν 2Kϕ Δσr + Kϕ − Kϕν 2 − ν − ν 2 Δσϑ⎤⎦ E

A closed-form (exact) solution is only possible for a = 0.5 which corresponds to the original form of H-B failure criterion9 suitable only for intact hard rock:

⎡1 − Picr = ⎢ ⎢⎣

(29)

Summing εr with εϑ multiplied by the coefﬁcient Kϕ , the compatibility equation of displacement can be written as a function of known stress increments:

with

μ = mb(, peak peak

225

1 + sin ϕ 1 − sin ϕ

⎛u ⎞ η = (1 + Kϕ)εϑp = (1 + Kϕ)⎜ − εϑe⎟ ⎝r ⎠

where ϕ ∈ (ϕpeak , ϕres ) is the dilatancy angle of rock mass, reﬂecting the sensibility of deformation on the dilation of rock mass and can be function of softening parameter η in strain-softening rock mass. The plastic strains is then formulated through the ﬂow rule via plastic multiplier λ

εrp = λ

∂g ∂σr

(27)

εϑp = λ

∂g ∂σϑ

(28)

(35)

The value η depends on the displacements and stresses which in turn depend on η via strength parameters and dilatancy angle. This inter-dependence necessitates, of course, the implementation of a reliable iterative computational procedure so that a coherent and converged solution can be obtained. This explains partially the numerical complexity of the solution for rock mass exhibiting strain-softening behavior. 3.4. Elastoplastic solution using FDM

For an isotropic material, the principal axes of stress and strain increment coincide so that the compatibility equation of displacement can be expressed as

The elastoplastic problem with bolted reinforcement described above by the equation of bolt-rock interaction Eq. (3), the equation of equilibrium Eq. (8), the compatibility equation of displacement (Eqs. (11), 12) and (32) and the failure criterion can be only solved

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C.H. TAN / International Journal of Rock Mechanics & Mining Sciences 88 (2016) 221–234

εr, (i) = 2

ri + 1 r εϑ, (i + 1) − 2 i + 1 εϑ, (i) − εr, (i + 1) ri + 1 − ri ri + 1 − ri

(40)

Combing the incremental form of elastic strain and stress relation on the internal surface (i) and on the external surface (i þ1), we obtain the stress relation between two surfaces of annulus:

σr, (i) = Cbσϑ, (i) + Ccσϑ, (i + 1) − Cdσr, (i + 1)

(41)

with following four constants:

Ca =

Cb =

ri + 1 + ri ri + 1 − ri (1 + ν ) − Ca(1 − v2) (1 − ν 2) + (1 − Ca)(ν + v2)

Fig. 4. Discretization into annular rings used in stepwise FDM solution.

numerically. Therefore, a computation procedure with two interconnected algorithms via displacement distribution is elaborated in detail in the following sections. The numerical solution using a simple stepwise procedure based on Finite Difference Method is adopted to solve the equations of stresses and displacement. The stepwise solution consists of the discretization of rock mass along the passive bolt into annular rings of constant thickness starting from tunnel wall towards far bolt end by passing through unknown elastoplastic interface in rock mass as show in Fig. 4. Each annular ring is composed of an internal surface r = ri and an external surface r = ri + 1 on which the stresses, strains and radial displacements are evaluated through the equation of equilibrium and the failure criterion. 3.4.1. Equation of equilibrium The equation of equilibrium Eq. (8) for a thin annulus can be written in incremental term:

σr, (i + 1) − σr, (i) ri + 1 − ri

=

σϑ, (i + 1) + σϑ, (i) ri + 1 + ri +

−

σr, (i + 1) + σr, (i)

(36)

After certain arrangement, we have the relation of the stresses between the internal surface σ(i) and the external internal surface σ(i + 1) :

(

)

σr, (i) = Ceσr, (i + 1) − Cf σϑ, (i) + σϑ, (i + 1) +

Ri Ti + 1 − Ti SlSc ri

(37)

with two constants Ce and Cf

Ce =

Cd =

(1 + ν ) + Ca(1 − v2) (1 − ν 2) + (1 − Ca)(ν + v2) (1 + ν ) + Caν(1 + v) (1 − ν 2) + (1 − Ca)(ν + v2)

Equalizing the expression of σr , (i) Eq. (41) with the expression σr , (i) Eq. (37) obtained from the equation of equilibrium, the stresses on the external surface σr , (i + 1), σϑ, (i + 1) can be now determined from its known values on the internal surface σr , (i), σϑ, (i) :

1

σr, (i + 1) =

Ce − −

σϑ, (i + 1) =

Cf Cd Cc

⎡⎛ ⎛ C ⎞ CC ⎞ ⎢ ⎜ 1 + f ⎟σr, (i) + ⎜ Cf − f b ⎟σϑ, (i) ⎢⎣ ⎝ Cc ⎠ Cc ⎠ ⎝

Ri Ti + 1 − Ti ⎤ ⎥ ⎥⎦ SlSc ri

(42)

1⎡ ⎣ σr, (i) + Cdσr, (i + 1) − Cbσϑ, (i)⎤⎦ Cd

(43)

Once the stress state for a given annulus is obtained, the elastic strain state can be fully deﬁned from the incremental relation of strain and stress so that the radial displacement can be expressed:

ri + 1 + ri

Ti + 1 − Ti 2R i 1 SlSc ri + 1 + ri ri + 1 − ri

Cc =

ri + 1 2ri

ui = ri

1 + ν⎡ ⎤ ⎣ (1 − ν ) σϑ, (i) − po − ν σr, (i) − po ⎦ E

(

)

(

)

(44)

3.4.3. Stress and strain in plastic zone In plastic zone, the total strains are made up of elastic component and plastic component. The stress state is governed by the equation of equilibrium as well as by a failure criterion which deﬁnes the relationship between stress components. For a given failure criterion σϑ = F ( σr , η), the incremental form can be written on two surfaces of annulus:

(

σϑ, (i) = F σr, (i), η(i)

)

(45)

r − ri Cf = i + 1 2ri

(

σϑ, (i + 1) = F σr, (i + 1), η(i + 1) 3.4.2. Stress and strain in elastic zone For a given annulus, the incremental form of strain is written

εr =

1 εr, (i + 1) + εr, (i) 2

(

)

(38)

as well as via the relation of strain and displacement,

εr =

εϑ, (i + 1)ri + 1 − εϑ, (i)ri u − ui du = i+1 = dr ri + 1 − ri ri + 1 − ri

We obtain therefore

(39)

)

(46)

where F is a linear function of σr with the coefﬁcients φ and c depending on the softening parameter η in the M-C case, or a parabolic function of σr with the coefﬁcients σci , mb , s and a depending on the η in the H-B case. Substituting σϑ, (i + 1) in the expression σr , (i) of the equilibrium equation Eq. (37), we obtain the following equation which deﬁnes the stresses state relation between two surfaces of each annulus:

σr, (i) + Cf σϑ, (i) − Ceσr, (i + 1) + Cf F ( σr, (i + 1)) −

Ri Ti + 1 − Ti =0 SlSc ri

(47)

C.H. TAN / International Journal of Rock Mechanics & Mining Sciences 88 (2016) 221–234

It is an implicit expression for σr , (i + 1) . If the stresses on the internal surface σr , (i), σϑ, (i) are known, the radial stress on the external surface σr , (i + 1) can be always obtained by a numerical solution. The circumferential stress σϑ, (i + 1) is then easily evaluated from the given failure criterion. It is worth noting that Eq. (47), satisfying both the equilibrium equation and the failure criterion, is general enough to be applicable for any forms of elastoplastic criterions covering from elastic-perfectly-plastic and elastic-brittle-plastic to strain-softening behaviors. If the failure criterion is linear such as M-C case,

σϑ, (i + 1) = Kpσr, (i + 1) + UCS

(48)

an explicit form of σr , (i + 1) can be deduced from Eq. (39)

σr, (i + 1) =

⎡ R T − Ti ⎤ 1 ⎢ σr, (i) + Cf σϑ, (i) + Cf UCS − i i + 1 ⎥ Ce − Cf Kp ⎣ SlSc ri ⎦

(49)

The displacement in plastic zone can be obtained from the increment form of the compatibility equation Eq. (32):

u − ui u + ui du u + Kϕ = i + 1 + Kϕ i + 1 dr r ri + 1 − ri ri + 1 + ri

(50)

After making some arrangement, we obtain the displacement on the internal surface ui in function of its value on the external surface ui + 1 such as8:

ui = −

Ct C ui + 1 + z Cq CqE

(51)

with

1 Cq = − ri + 1 + ri ri + 1 − ri

Kϕ ri + 1 + ri

+

1 ri + 1 − ri

(

σr , (i + 1) + σr , (i)

)( − ν − ν )(

Cz = 1 − ν 2 − νKϕ − ν 2Kϕ

(

+ Kϕ − Kϕν 2

2

2

). −p)

− po

σϑ, (i + 1) + σϑ, (i)

o

2

3.4.4. Computation procedure of strain-softening The rock mass with strain-softening tends to lose its strength resistance once plastic straining occurs. As mention early, the complexity of strain-softening problem resides on the path dependence of the stress-strain behavior. It means that the stressstrain response depends not only on the current loading amplitude but also on its loading path history via the softening parameter η . In such case of constitutive nonlinearity, an iterative computational procedure is required so that the strength parameters can be derived from the assumed solution before the convergence is obtained. In the analysis of circular opening excavation, the load path constitutes by a series of decreasing ﬁctive internal pressures pk of tunnel from the value of initial stress in situ po (po ≥ pk ≥ 0) . Therefore, pk is the loading amplitude of the treated problem. If pk > pcrit , the rock mass is in elastic state so that we have always ηk = 0. When the condition

(p

k

≤ pcrit

turn depend on η via strength parameters and dilatancy angle. This nonlinear inter-dependence necessitates a rigorous iterative treatment in order to ﬁnd the correct stress-strain state. For each pk < pk − 1 of step k, the value ηk − 1 of step k-1 is used as initial guess value to evaluate stresses and displacements and then its updated k value ηnew is used in the iterative procedure until there is a convergence to assure fully F (σ , η) = 0 so that a coherent and consistent stress-strain state can be obtained for the step k (Fig. 5). 3.4.5. Computation procedure of strain-softening with passive bolt reinforcement The computation procedure is divided into two steps interconnected in order to assure the congruence between the displacement ﬁeld obtained in the reinforced rock mass and the displacement ﬁeld used to solicit the frictional model of bolt-rock. Step 1: Noting that pinst is the ﬁctive internal pressure when the passive bolts are installed, its corresponding displacement ﬁeld in rock mass along the bolt length Lb is noted as uinst (r ) calculated by the computation procedure illustrated in Ref. 7. For a given stage k with a ﬁctive internal pressure pk < pinst which generates an unbolted displacement ﬁeld uk (r ) = uub(r ) along the bolt length using the computation procedure illustrated in Fig. 5, a relative displacement ﬁeld with respect to uinst (r ) can be thus obtained

Δui (r ) = ⎡⎣ (uik (r ) − uiinst (r )⎤⎦ − ⎡⎣ (uk (Ri + Lb) − uinst (Ri + Lb)⎤⎦

for Ri ≤ r ≤ Ri + Lb (52)

Using ur (r ) = Δu(r ) as the known solicitation displacement for the frictional model described in the Section 2, the axial force Tb(r ) in the bolt can be obtained from Eq. (7) after the solution of the nonlinear system Eq. (4) via an iterative calculation by NewtonRaphson method. Step 2: The equation of equilibrium of rock mass is then solicited by the axial force Tb(r ) provided by the bolt. The radial stress at the

Kϕ

Ct =

227

) is veriﬁed, some

rock around the circular opening begins plastic straining which induces softening effect on the shear resistance for a given failure criterion. Smaller value of pk becomes, more plastic zone penetrates into the rock mass and more softening effect is mobilized. During this plastic straining process, the value of softening parameter η is calculated from displacements and stresses which in

tunnel wall is known σr (Ri ) = σr , (1) = pk , which serves as the boundary condition of the equilibrium equation on the internal surface of the ﬁrst annulus. Stress evaluation: The radial stress and circumferential stress with bolt reinforcement at each sequential annulus ( ri, ri + 1) are evaluated according to its position in elastic zone by (Eqs. (42) 43) or in plastic zone by Eq. (47). If the radial stress veriﬁes σr (r ) ≤ pcrit the ﬁrst time, note its position as the radius of elastoplastic interface ( r = Rpl ). Displacement evaluation: Once the stresses state has been fully deﬁned in the discretization zone, displacement ﬁeld ub(r ) with bolt reinforcement can be evaluated from r = Ri + Lb to r = Ri according to its position in elastic zone by Eq. (44) or in plastic zone by Eq. (51). k Softening evaluation: The softening parameters ηnew (r ) is evaluated at each sequential annulus ( ri, ri + 1) according to Eq. (35). k Congruence evaluation of η : If ηnew and ηk is compatible, go to

k and the Congruence evolution of ub . Otherwise, update ηk = ηnew repeat iterative calculation from Stress evaluation. Congruence evaluation of ub : The displacement ﬁeld with bolt reinforcement ub(r )is compared to uk (r ) in Step 1 used to solicit the frictional model. If the displacement ub(r ) and uk (r ) are compatible, the calculation passes on the next stage kþ 1 from Step 1 with a new pk + 1 < pk and repeats until the ﬁnal stage where the internal pressure p = 0 is achieved. Otherwise, the calculation remains in the stage k by updating

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Fig. 5. Flow chart of strain-softening computation procedure.

uk (r ) = ub(r ) and repeats from Step 1 for a new iterative computation of ub(r ). Please to refer to7 for the ﬂow chart of computation procedure for passive bolt reinforcement.

4. Veriﬁcation of the proposed solution Before applying the proposed FDM solution to the case of passive bolts reinforcement, we must ﬁrstly verify the pertinence of the proposed computation framework for the strain softening case of classical circular opening for which bolt force is simply null ( T = 0) so that all equations described above reduce to their classical forms. The errors in a nonlinear numerical solution can be of various natures: incorrect mathematical model itself, inappropriate numerical method, insufﬁcient convergence of spatial discretization, insufﬁcient convergence of an iterative procedure etc. In absence of experimental results to compare with and analytical solutions used as benchmark, the predictive accuracy of the proposed solution is veriﬁed according to the following methods: Using the solutions from other numerical methods. If the physics of the problem is properly modeled in all techniques used, the approximate results should have a clear trend and similarity, being of the same order of magnitude. Using the upper and lower bound solutions. The solution is veriﬁed in the asymptotic region through the limiting cases of the

strain-softening behavior for which associated analytical solutions are available. Using convergence tests to guarantee the quality of implemented iterative algorithm which assures numerical stability and converges effectively to a coherent, consistent and reliable solution. 4.1. Mohr-Coulomb criterion

A. Veriﬁcation 1: The following input parameters contain the mechanical parameters for comparison test with the approximate solution obtained by Lee et al. 6, i.e., Ri = 3m, po = 5MPa , E = 10000MPa , ν = 0.25, cpeak = 1.0MPa , φpeak = 30°, ϕpeak = 15°, cres = 0.7MPa , φres = 22°, ϕres = 0°. During the strain-softening, the cohesion (c ), the friction angle (φ) and dilatancy angle (ϕ) are simultaneously decreased from peak value to residual value. The results of the proposed solution are compared with the approximate solution obtained by Lee et al.6 as shown in Table 1 for different critical softening values η* when internal pressure p = 0. As can be observed in Table 1, a good agreement is obtained between both solutions. The softening solutions are bounded effectively by their limiting cases for which the analytical solutions are also included. With η* = 0.8 and η* = 1E − 05, the solution converges respectively to analytical elastic perfectly plastic and elastic brittle

C.H. TAN / International Journal of Rock Mechanics & Mining Sciences 88 (2016) 221–234

of equilibrium on the solution via softening parameter (plastic strains). In such case, the strength parameters need to be derived from the assumed solution. In order to verify the quality of the iterative computational procedure, two schemas of decreasing internal pressure p of tunnel wall pcrit = 9.134MPa ≥ p ≥ 0 are also presented in Fig. 6: one with the increment Δp = − 0.1 MPa and the other Δp = − 1.0 MPa . Two GRC responses are almost identical and the implemented iterative computational procedure converges efﬁciently to a coherent and consistent solution.

Table 1 Comparison with the approximate solution.6 Approximate solution6

This study

Type

ƞ*

Rp/Ro (Ur*E)/ (Ro*Po)

Brittle (Analy.)

0

Perfect (Analy.)

1.0E-05 1.0E-03 2.0E-03 8.0E-03 8.0E-02 8.0E-01 1

Rp/Ro (Ur*E)/ (Ro*Po)

1.74 1.68 1.58 1.42 1.40 1.39

3.20 3.19 2.98 2.27 2.12 2.12

1.74

3.20

1.74 1.68 1.59 1.43 1.39 1.39 1.39

3.20 3.02 2.69 2.23 2.13 2.11 2.11

(

RpRes/Ro

1.74 1.48 1.08

The proposed solution is also compared with the self-similar solution by Alonso et al.5 with the parameters shown below, which has been used by the closed-form solution4 and the approximate solution6 to verify the quality of their approach respectively: The input parameters are Ri = 3m , po = 20MPa, E = 10000MPa , ν = 0.25, cpeak = 1.0MPa , φpeak = 30°, ϕpeak = 3.75°,

cres = 0.7MPa , φres = 22°, ϕres = 3.75°. The plastic straining takes place when p ≤ pcrit = 9.134 MPa

( pcrit /p0 = 0.457). The GRC curve and evolution of total (Rp) and residual (RpRes) plastic radius are illustrated in Fig. 6. It is observed that the surrounding rock cannot reach the residual plastic state when p/p0 > 0.176. A fully comparable ﬁgure is not available here, but when p = 0 , a very close agreement is obtained for wall displacement ( ur = 116.79 mm) as well as for plastic radius

)

(

)

4.2. Generalized Hoek-Brown criterion

plastic one so that the previous research work7 becomes then a special cases of the current solution. For 8E − 03 ≤ η* ≤ 8E − 01, the plastic straining takes place only in softening regime when p ≤ pcrit = 1.634 MPa and there is no any plastic residual zone developed in the rock mass. B. Veriﬁcation 2:

(R

229

)

Rpres = 7.82 m p = 12.24 m , only the residual plastic radius deviates little bit, because of discretization effect most probably. The strong nonlinearity of η makes it very sensitive, particularly for the passage from residual to softening zone which progresses numerically by one annular ring. The constitutive nonlinearity with strain-softening behavior implies the dependence the strength parameters in the equation

A. Veriﬁcation 1: The proposed solution is veriﬁed against the closed-form solution4 which is limited strictly to the classical H-B criterion for intact rock mass ( α = 0.5). The material properties are Ri = 3m , po = 15MPa , E = 5700MPa , ν = 0.25, σci, peak = 30MPa , mb, peak = 2, speak = 0.004 , apeak = 0.5, ϕpeak = 15°, σci, res = 25MPa, mb, res = 0.6, sres = 0.002, ares = 0.5, ϕres = 5°, η* = 0.01. Fig. 7 shows GRC curve and evolution of plastic radius. When the internal pressure p = 0 , the proposed solution ﬁts very well with the closed-form solution and predicts practically the same displacement ( ur = 67.6 mm) and plastic radius Rp = 7.71 m . Only the residual plastic radius deviates little bit Rpres = 5.46 m for the reason described above. B. Veriﬁcation 2:

(

(

)

)

In order to verify the proposed solution in the case of Generalized H-B criterion ( α > 0.5), the following published data used by Sharan's exact solution3 as well as Carranza-Torres's solution1 is employed, where all strength parameters ( σci, mb: , s, a) degrade from peak values to residual values, i.e., Ri = 2m, po = 15MPa, E = 5700MPa , ν = 0.3, σci, peak = 30MPa , mb, peak = 1.7, speak = 0.0039, apeak = 0.55, ϕpeak = 0°, σci, res = 25MPa,

mb, res = 0.85, sres = 0.0019, ares = 0.6, ϕres = 0°. The solution used the same computational framework7 were found to be almost identical to both analytical solutions in case of elastic brittle plastic behavior as shown in Table 2, satisﬁed even for the dilating cases producing excessively large displacements. Now, the solution of strain-softening behavior is shown in Table 3 for various critical softening values η* for the internal pressure p = 0 . With η* = 0.3 and η* = 0.001, the solution converges respectively to analytical elastic perfectly plastic and elastic brittle plastic behaviors so that the previous research work7 becomes

Fig. 6. (a) Ground reaction curve; (b) Evolution of plastic radius in M-C strain-softening rock mass.

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Fig. 7. (a) Ground reaction curve; (b) Evolution of plastic radius in H-B strain-softening rock mass. Table 2 Comparison with analytical solutions for elastic brittle behavior.3,7 Brittle

Sharan (Exact solution)3

Tan (FDM solution)7

Po (MPa) p (MPa) Kψ

Rpl (m)

Rpl (m) Ur (mm) Error Rpl (%)

15 15 15 15

0 3 0 1

1 1 1.98 1.98

15

2.5

1

7.80 81.20 3.02 10.40 7.80 272.00 4.56 53.00 CarranzaTorres (RungeKutta)1 3.28 12.5

Ur (mm)

Error Ur (%)

7.78 3.02 7.80 4.56

81.56 10.44 273.69 52.96

0.26 0.44 0.00 0.38 0.00 0.62 0.00 0.08

3.28

12.53

0.02

0.24

Table 3 Solutions of strain-softening behavior in H-B rock mass.

Perfect

Brittle

properties is used for validating its implementation of Lee and Pietruszczak solution6, except for ν = 0.25 instead of ν = 0.3 and a ﬁxed σci = 25 MPa instead of varying σci = 25 − 30 MPa . Fig. 8 shows the comparison GRC of two solutions for various values of the critical deviatoric plastic strain η*. It is clearly observed that the results of two solutions are in good accordance within gradual strain-softening behavior. As expected, the results all tend to the Carranza-Torres solution1 in case of η* = 0, which corresponds to a perfectly brittle material with no strain-softening regime. In summary, the cross veriﬁcation against multiple published results in the literature shows that the extension of the proposed computational framework to strain-softening rock mass is coherent, consistent and reliable so that the implementation is believed to be accurate and sound so that he previous research work7 becomes a special cases of the current solution. It provides therefore a fully uniﬁed computational framework which can be applied to rock mass exhibiting commonly mechanical behaviors of rock mass from elastic-perfectly-plastic to elastic-brittle-plastic by passing through strain-softening behavior. 4.3. Solution of frictional model during pullout tests

This study

(p ¼0)

ƞ*

Rp (m)

Ur (mm)

RpRes (m)

1 0.3 0.2 0.05 0.025 0.02 0.012 0.008 0.007 0.005 0.003 0.001 0

4.09 4.13 4.17 4.45 5.25 5.61 6.42 6.94 7.06 7.34 7.62 7.74 7.78

20.73 21.26 21.55 24.94 35.38 41.21 54.70 64.20 66.69 72.19 77.80 80.72 81.56

2 2 2 2.24 2.64 3.81 5.05 5.45 6.26 7.22 7.74

then a special cases of the current solution. In order to illustrate further the validity of our approach, the proposed solution is compared with software solution of RocScience Inc, a company works intensively with several universities and industry leaders in rock mechanics modeling. The veriﬁcation example #4 is taken from its software RocSupport (an analysis tool for estimating support requirements of tunnels in weak rock). In its veriﬁcation manual10 the same set of rock

Field monitoring on the bolts in situ outlines the behavior of passive bolts in a deformed rock formation: rock deformation applies a load on the pick-up section (“pick-up length”) of the bolt which drags the anchor section (“anchor length”) of the bolt towards the underground opening as shown in Fig. 1. These two sections of the bolt are separated by a “neutral point” where shear stress at the interface between rock and bolt is zero. Unlike bolts in situ, bolts in a pullout test only have an anchor length. Therefore, the pullout tests are usually used to investigate simply the anchoring capacity of bolt. The proposed solution of frictional model between rock and bolts has been veriﬁed using the results obtained from pullout tests. For the frictional model using FDM schema (Eq. (4)) in such pullout test, the displacement of rock or concrete is negligible in comparison to the bolt displacement and the pullout force constitutes the only known load which is applied to bolt head. Therefore, the axial force Tb1 (pullout force) of the ﬁrst discretized point (i = 1) on bolt can be rewritten using Eq. (7):

Tb1 = Ebτ,1Ab

ub2 − ub0 x2 − x 0

From Eq. (53), the ﬁctive node (i = 0) can be expressed as:

(53)

C.H. TAN / International Journal of Rock Mechanics & Mining Sciences 88 (2016) 221–234

231

Fig. 8. Comparison with RocScience Inc. implementation of Lee and Pietruszczak solution for various critical softening values.

ub0 =

−2hTb1 + ub2 Ebτ,1Ab

(54)

Incorporating this relation (54) into Eq. (4) for (i = 1) and keeping all other conditions same, the discretized formulation gives a system of equations with n unknown displacements of bolt ubi(i = 1, .. . n) describing completely the behavior of embedded bolt subjected to a given pullout load (Tb1) during pullout tests. A. Veriﬁcation 1: In his pioneering work in studying the behavior if bolts under tensile loading, Farmer13 conducted a series of pullout tests of resin grouted rock bolts anchored in elastic concrete, limestone and chalk. In his tests using bolt length 350 mm and 500 mm, the tension load was applied after 24 h from the installation of bolt. Strain gauges along the bolt at regular interval were used to measure the axial strain of bolt during the tests. The bolt is subjected to a series of pullout loads, respectively. As Farmer13 mentioned, his theoretical solution predicted well that the axial strain of the bolt and shear stress of the interface decrease exponentially from the point of loading to the far end for low pullout force. However, the substantial differences begin to be observed for high axial loads (greater than 60 kN) and this differences ampliﬁes more and more with increased pullout force. He concluded that de-bonding mechanism might be the source of such difference and a non-linear frictional relation needed to be requested. The bolt length is 500 mm. The initial frictional stiffness of bolt interface ks calculated from load-displacement curve recorded by Farmer13 is founded to be 6000 MPa. The diameter and the elastic modulus of bolt are 20 mm and 180 GPa, respectively, as given by Farmer13. The other input parameters are τres = 0.5 MN /m and the softening τpeak = 0.7 MN /m, slope¼ 1000 MPa. Strain distributions along bolt length from the proposed model are shown in Fig. 9 in comparison with experimental gauge strains during the tests. It can be seen that the results of this study match well with the experimental results. The results of this study describe well the de-bonding

Fig. 9. Distribution of axial strain along the bolt length.

mechanism through the mobilized frictional stress. It is noted that the bolt starts de-bonding when the pullout force increases near to 80 kN and the de-bonding process progresses with the increased pullout force. The decoupling length arrives to 80 mm for pullout force of 120 kN and from which, shear stress resumes the exponential decay along the remained part of the embedded length. B. Veriﬁcation 2: The second pullout test veriﬁed here was carried out by Rong et al.14 The bolt was a threaded steel bar embedded in concrete. Strain gauges were attached at 5–10 mm intervals along the bon length to measure the axial stress variation in the bolt. The radius and the elastic modulus of bolt are 16 mm and 210 GPa, respectively. The parameters used for nonlinear frictional model are ks = 3500 MPa , τpeak = 0.524 MN /m, τres = 0.1 MN /m and the softening slope ¼400 MPa. Axial stress distributions along the bolt obtained from the numerical model are shown in Fig. 10 in comparison with the pullout tests.14 It can be seen that the proposed numerical model reproduces closely the experimental results. It is noted that the de-bonding starts when the pullout force

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Fig. 10. Comparison with experimental results: Axial stress along the bolt length, (b) Axial force - displacement.

increases near to 150kN and propagates along the bolt with increasing pullout force. Fig. 10(b) shows the good comparison of load-displacement curve between the experimental data and this study and the analytical solution proposed by Ren et al.15 for pullout test.

5. Application to passive bolt reinforcement Two cases studies are given to illustrate the application of the proposed solution to the reinforcement mechanics of passive bolts in conventional tunneling, one with Mohr-Coulomb criterion and the other with Hoek-Brown criterion. In the following examples, stress and displacement distribution around the tunnel are discussed in detail. For comparison purpose, the results without bolts are also presented and plotted as “Unbolted” against “Bolted” for the reinforced case. 5.1. Mohr-Coulomb criterion The case studied above in Section 4.1 Veriﬁcation 2 is used. The bolt and bolt-rock interaction properties are listed in Table 4. The results of unbolted case reveal a big radial displacement of tunnel wall, which can be reduced by the installation of lengthy bolts with intensive bolt density. Therefore, the bolting spacing used is 0.8 m 0.8 m along the circumferential and longitudinal direction respectively with bolt length Lb = 4 m of which the elastic limit is 550 MPa. The passive bolts are supposed to be installed as soon as possible near the tunnel face at the ﬁctive internal pressure pinst = 1.88 MPa . When the tunnel face advances, the ﬁctive internal pressure p decreases, the strength of the bolt is mobilized and a retaining effect tends to develop with the displacement release of rock mass around the tunnel wall. At p = 0 , the rock mass and the passive bolts carry together the totality of in-situ stress release, resulting in a substantial reduction of displacement. The presence of passive bolts fortiﬁes greatly the strength of rock mass and helps constrain the range of plastic radius from 12.24 m to 10.47 m and the convergence of tunnel wall reduces consequently from 116.80 mm to 83.33 mm (approximate 28% reduction). A bolt under tension compresses the rock, which provides the essence of reinforcement effect. Since the bolt is considered to be an elastic-perfectly-plastic material as shown in Fig. 2(b), the tensile strength plays an important part in the bolted response. With the elastic limit 550 MPa, the bolts work fully within its

Table 4 Properties of M-C rock mass for the application case. Tunnel and Rock Parameters Radius of tunnel (m) Initial stress (MPa) Young’s modulus (MPa) Poisson’s ratio Peak cohesion (MPa) Peak friction (deg.) Residual cohesion (MPa) Residual friction (deg.) Dilation (deg.) Critical softening ƞ*

Bolt Data 3 20 10,000 0.25 1 30 0.7 22 3.75 0.008

Diameter of bolts (m) Young’s modulus (GPa) Elastic limit (MPa) Sc (m) Sl (m) Bolt - Rock mass interaction Ks (MPa) Peak friction (MN/m) Residual friction (MN/m) Softening slope (MPa)

0.04 210 550 0.8 0.8 400 0.5 0.3 45

elastic regime. However, if the bolts of lower resistances 270 MPa have been installed, the allowable axial force is limited to 339 kN so that one part of bolt is yielded and forced to work plastically as shown in Fig. 11. In this case, the installed bolts provide less axial forces which can compress surrounding rock and there is naturally a reduction of reinforcement effect on the tunnel opening displacement. In the case of partially yielded bolts where the rigidity matrix of frictional model becomes heavily nonlinear owing to nonlinearity of frictional stiffness as well as of variable bolts stiffness, the implemented iterative computational procedure shows a quite satisfactory performance, except for a locally numerical oscillation on the transition between elastic and plastic regime as observed in Fig. 11. It is also worth noting from Fig. 11 the concepts of “pick-up length” and “anchor length” separated by a “neutral point” (zero friction) corresponding to the point of maximal axial force. The direction of frictional stress in these two zones is opposite, which conﬁrms the ﬁeld observations. The frictional stresses on the pickup length of the bolt pick up the load from the rock deformation and drag the bolt towards the tunnel opening, while frictional stresses on the anchor length try to anchor the bolt to the rock so that the reinforcement can be accomplished to restrain the deformation of rock mass. In the case of partially yielded bolt (270 MPa), a large neutral section instead of a single point is developed, indicating that this section of the bolt displaces in strictly conjunction with the surrounding rock mass (no relative displacements). It is noted that the inﬂuence of bolting pattern on the dimension of bolting intervention have been investigated in detail in Ref. [7].

C.H. TAN / International Journal of Rock Mechanics & Mining Sciences 88 (2016) 221–234

233

Fig. 12. Ground reaction curves in H-B strain-softening rock mass with two bolt reinforcement patterns.

Fig. 11. Distribution of axial force, friction and displacements along the bolt for p = 0 and bolt length¼ 4 m.

Table 5 Properties of H-B rock mass for the application case. Tunnel and rock parameters Radius of tunnel (m) Initial stress (MPa) Young’s modulus (MPa) Poisson’s ratio Peak Sigci (MPa) Peak mb Peak s Peak a Residual Sigci (MPa) Residual mb Residual s Residual a Peak dilation (deg.) Residual dilation (deg.) Critical softening ƞ*

6. Conclusion

Bolt data 3 15 5700 0.25 30 2 0.004 0.5 25 0.6 0.002 0.5 15 5 0.01

Diameter of bolts (m) Young’s modulus (GPa) Elastic limit (MPa) Sc (m) Sl (m) Bolt - Rock mass interaction Ks (MPa) Peak friction (MN/m) Residual friction (MN/m) Softening slope (MPa)

unbolted tunnel is 67.58 mm with a plastic radius 7.71 m. With the installation of Lb = 5 m long bolts, the convergence decreases to 47.39 mm (approximate 29% reduction) with a smaller extent of plastic radius 6.63 m. Fig. 12 shows the GRC curve. As can be seen, the bolt pattern plays a important part in the restreinte of opening displacement. With a bolting spacing 0.8 m 0.8 m, the wall displacement reduces to 40 mm with the plastic radius 6.20 m. If a support system (SCC) is installed after tunnel face advanced further away and the corresponding released wall displacement arrived at 36 mm, the intersection point bolted GRC generates an equilibrium pressure carried by the support much smaller than with unbolted GRC, so that the safety factor is thus much more improved.

0.04 210 635 1 1 400 0.5 0.3 45

5.2. Generalized Hoek-Brown criterion The case shown in Section 4.2 Veriﬁcation 1 is intended to illustrate the bolt reinforcement with H-B failure criterion. Table 5 lists the set of parameters of rock and bolt. The bolting spacing is 1.0 m 1.0 m along the circumferential and longitudinal direction respectively. The length of bolts is Lb = 5 m so that it exceeds slightly the plastic radius of unbolted case. The critical pressure of tunnel wall is pcrit = 5.70 MPa . The passive bolts are mounted at the ﬁctive internal pressure of pinst = 1.25 MPa , which corresponds to a distance ¼1.0 m behind the tunnel face according to Panet's longitudinal proﬁle.12 At the ﬁctive internal pressure p = 0 , the convergence of

Base on the previous research work, this paper presents an extension of the developed computational framework to strainsoftening behavior of rock mass with and without passive bolt reinforcement. The solution provides a framework to the ground and passive bolts responses under classical convergence – conﬁnement context and can be applied to the elastoplastic models such as Mohr-Coulomb and Generalized Hoek-Brown failure criterions, two reputed failure criterions widely used for tunneling excavation problems. A pullout test based nonlinear frictional model is developed to describe the load transfer between rock mass and elastoplastic passive bolts and the results are founded to be in closed agreement with two experimental data. An iterative computation procedure is thus developed to guarantee the congruence of displacement between bolts and rock mass of strainsoftening behavior. The proposed solution is ﬁrstly veriﬁed against the multiple published results of literatures for the unbolted case. The results obtained are in good agreement and show it is a coherent, consistent and reliable implementation. The application to the cases of passive bolt reinforcement with Mohr-Coulomb as well as HoekBrown failure criterions illustrates clearly that the solution is able to capture quantitatively the key elements of bolt reinforcement observed in in-situ monitoring. The efﬁcacy of the proposed solution makes it easy to obtain Ground Reaction Curve (GRC) of passive bolt reinforcement, which constitutes the key element of the Convergence – Conﬁnement Method in the tunnel design. Since this paper extents the solution to strain-softening behavior (much more general behavior of rock mass), the previous research work becomes then a special cases of the current

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solution. Therefore, this research work constitutes a uniﬁed solution framework covering fully mechanical behaviors of rock mass from elastic-perfectly-plastic to elastic-brittle-plastic by passing through shear strain-softening to failure behavior with and without passive bolt reinforcement.

5.

6.

7.

Acknowledgments 8.

This research was supported by the Direction of Technical Development of ARCADIS-FRENCH, which is gratefully acknowledged. The author would also like to acknowledge the contribution of his colleagues Mr. Olivier GIVET and Mr. Guillaume CHAMPAGNE de LABRIOLLE for their constructive comments.

9. 10. 11. 12.

References

13. 14.

1. Carranza-Torres C. Elasto-plastic solution of tunnel problems using the generalized form of the Hoek-Brown failure criterion. Int J Rock Mech Min Sci. 2004;1:629–639. 2. Sharan SK. Exact and approximate solutions for displacements around circular openings in elastic–brittle-plastic Hoek-Brown rock. Int J Rock Mech Min Sci. 2005;42:542–549. 3. Sharan SK. Analytical solutions for stress and displacements around a circular opening in a generalized Hoek-Brown rock. Int J Rock Mech Min Sci. 2008;45:78–85. 4. Zhang Q, Jiang BS, Wang SL, Ge XR, Zhang HQ. Elasto-plastic analysis of a

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