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Reinforcement mechanics of passive bolts in conventional tunnelling
ARTICLE IN PRESS
International Journal of Rock Mechanics & Mining Sciences 44 (2007) 625–636 www.elsevier.com/locate/ijrmms
Reinforcement mechanics of passive bolts in conventional tunnelling Zhenchang Guana,, Yujing Jiangb, Yoshihiko Tanabasib, Hongwei Huangc a
Graduate School of Science and Technology, Nagasaki University, 1-14 Bunkyo Machi, Nagasaki 852-8521, Japan b Department of Civil Engineering, Faculty of Engineering, Nagasaki University, Nagasaki, Japan c Department of Geotechnical Engineering, Tongji University, Shanghai, China Received 5 June 2006; received in revised form 3 October 2006; accepted 6 October 2006 Available online 11 January 2007
Abstract According to the conventional pullout tests for passive bolts, a spring–slider model is adopted in this paper to account for the interaction relationship between the bolt and the rock mass. Based on the incremental theory of plasticity and the plane strain axial symmetry assumption, an elasto-plastic ground and bolt responses analyses method is proposed. In addition, the validity of the proposed method is verified by numerical simulations. The reinforcement mechanics of passive bolts in conventional tunnelling is clearly demonstrated via an illustrative case study. Through parameter studies, the influence of the bolt properties on the reinforcement effect is highlighted. The proposed method provides a framework to the elasto-plastic ground response analyses with passive bolts reinforcement. It also permits incorporating other features about the bolt–rock interaction model or other failure criterion of the rock mass in further analyses. r 2006 Elsevier Ltd. All rights reserved. Keywords: Passive bolt; Ground reinforcement; Conventional tunnelling; CCM; Reinforced GRC
1. Introduction The new Austrian tunneling method (NATM) has changed the concept of conventional tunnelling from resisting the passive earth pressure to helping the ground support itself. Consequently, shotcrete lining and rock bolting are widely used in practice to help the surrounding ground to stabilize itself. The convergence confinement method (CCM), under the plane strain axial symmetry assumption, provides a classic and efficient tool for the support system design in conventional tunnelling. It consists of the longitudinal deformation profile (LDP), the support characteristic curve (SCC) and the ground reaction curve (GRC). The arrangement and the intersection of these three curves are considered, to compromise the complex nature near the tunnel face with a general albeit simplified approach. It enables the taking of some different constitutive laws of rock mass and some different Corresponding author.
E-mail address: [email protected] (Z. Guan). 1365-1609/$ - see front matter r 2006 Elsevier Ltd. All rights reserved. doi:10.1016/j.ijrmms.2006.10.003
types of supports (including shotcrete lining, steel sets and active bolts) into account [1–3]. However, the presence of passive bolts complicates the problem significantly. Generally, passive bolts refer to the continuously mechanically coupled bolt and the continuously frictionally coupled bolt, and are particularly represented by fully cement (or resin) grouted bolt and Swellex bolt [4]. Several categories of models, including analytical and numerical models, have been proposed to account for the interaction between the bolt and the rock mass. As far as the analytical models are concerned, the bolt together with the rock mass within its tributary area is generally considered as one entity, in which the bolt–rock interaction is embedded. Then given a free displacement distribution (the displacement of rock mass without bolting), one can compute the response of rock bolt itself, as well as the reinforced displacement of rock mass after bolting. The analytical models are represented by Li and Stillborg [5], Cai et al. [6] and others. From a mathematical viewpoint, it can be regarded as a ‘‘one-step’’ mapping from an initial state to a final state, while a plastic problem
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requires an iterative computing algorithm along a certain loading path. As far as the numerical models are concerned, there are two ways to simulate the bolt–rock interaction: explicit and implicit models. The explicit model simulates the interaction distinctly via a series of spring–sliders, so that it is suitable for finite-difference method and typically utilized in the numerical codes FLAC3D [7]. In the implicit model, the bolt together with the rock mass within its tributary area is integrated as one sub-element, in which the bolt–rock interaction is embedded. Then the sub-element can be inserted into existing codes and be treated as a common element. The implicit model is suitable for finiteelement method as demonstrated by Chen et al. [8] and others. While resorting to numerical methods, it is straightforward to deal with any ordinary constitutive laws of rock mass and any complicated boundary conditions, but with a drawback of being time consuming. After introducing the spring–slider model, a seminumerical ground response analyses method is proposed in this paper to account for the reinforcement mechanics of passive bolts in conventional tunnelling. Based on the incremental theory of plasticity and the plane strain axial symmetry assumption, equilibrium equations and compatibility equations are first deduced theoretically, and then solved numerically. The reinforcement mechanics of passive bolts in conventional tunnelling is clearly demonstrated via an illustrative case study. The validity of the proposed method is verified by numerical simulations. In addition, the influence of the bolt properties on the reinforcement effect is highlighted through parameter studies. 2. Problem description 2.1. The constitutive laws of rock mass For universality, the rock mass is assumed to exhibit strain-softening behavior, which can be reduced into a perfect elasto-plastic behavior or an elasto-brittle behavior in some special cases [9,10]. Although many researchers 1
elastic
plastic
have reported that strain-softening materials may experience instability and bifurcation during an unloading process [11,12], this paper restricts the discussion to its stable (or basic) solutions. Generally, the rock mass exhibiting strain-softening behavior is characterized by a transitional failure criterion f(sij, Z) and a plastic potential g(sij, Z), where Z is a softening parameter controlling the gradual transition from a peak failure criterion (or potential) to a residual one. In this paper, the rock mass is assumed to satisfy the linear Mohr–Coulomb criterion and the linear plastic potential. The softening parameter can be defined in different ways, but there has not been a commonly accepted one among researchers. The major principal plastic strain, ep1 , is employed as the softening parameter in this paper, because it is relatively simple and can be obtained easily from the results of uniaxial compression tests (Schematically represented in Fig. 1). Therefore, the failure criterion f and the plastic potential g can be formulated as follows f ¼ s 1 K p s 3 sc ¼ 0 8 1 2 p < s1 ðsc sc Þ1 c a with sc ¼ : s2 c
ð0pp1 paÞ ðp1 XaÞ
g ¼ s1 K c s3 ¼ 0.
Here, s1 and s3 are the major and minor principal stresses of rock mass. Kp and Kc are, respectively, the passive coefficient and the dilation factor of the rock mass, which are regarded as constants within the complete plastic region. sc is the compression strength, and transits gradually from s1c to s2c , according to the evolution of the major principal plastic strain ep1 . a is a shift point of the softening parameter that distinguishes the strain-softening region from the residual region. 2.2. The plane strain axial symmetry assumption Estimation of the support system required to stabilize a tunnel opening during excavation, especially in the vicinity -3
1p
E c2
K -d1p
v 1e
1e+
1
ð1Þ
(2)
c1
0
,
0
1e
d1p 1p
Fig. 1. Typical strain and stress behaviors of soft rock under uniaxial tests.
1
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627
P0
passive bolts
Re Ra (a)
tunnel face
plastic region elastic region
Lb
P0
Ra
Lz central line
plastic region
Pi
(b)
elastic region
passive bolts
Fig. 2. Schematic representation of (a) the cross-section and (b) the longitudinal profile in the conventional tunnelling with passive bolts reinforcement.
Pcri i ¼ sre ¼
2P0 s1c . Kp þ 1
(3)
dr discretized bolt node
(a)
shear spring Ks
cohesive slider cs grout
(b)
us
Ds
Fs dr Ks
bolt
t tansDs cs Ds
of the tunnel face, is essentially a four-dimensional problem. It not only concerns with three spatial dimensions but also one temporal dimension. The time effect, which corresponds to the advancing process of the tunnel face (or synonymously the unloading process of ground), is significant when a plastic region occurrs in the surrounding rock mass. The CCM, considering a long round-section tunnel excavated in the rock mass under a hydrostatic insitu stress, simplifies the three dimensions in space into one dimension in the radial direction via the plane strain axial symmetry assumption. This assumption is adopted in this paper. Furthermore, the paper assumes that the passive bolts are also mounted axial-symmetrically (i.e., pattern bolting) and the effect of each bolt can be averaged within its tributary area, so that the axial-symmetry assumption always holds for this problem. The schematic representation of the problem is illustrated in Fig. 2. In Fig. 2, Ra and Re are the radii of the tunnel opening and the plastic region; Lb is the length of the passive bolts; c and Lz are the bolting interval around the tunnel opening perimeter and the bolting space along the tunnel axis, respectively. P0 is the hydrostatic in-situ stress, and Pi is a fictitious inner pressure provided by the tunnel face and/or the shotcrete lining. The loading path, corresponding to the advancing of the tunnel face, refers to a monotonic decrease of the fictitious inner pressure from P0 (before excavation) to Pfin i (the final loading carried by the lining after the tunnel face advancing away). The radial stress at the elasto-plastic interface (r ¼ Re), sre, can be computed theoretically by Eq. (3) [13,14]. An important feature of the solution is that sre is a constant that only depends on the properties of rock mass itself and is independent of the position of the elasto-plastic interface. In other words, the rock mass will not experience a plastic region before the inner pressure falls below sre. Thus, it is synonymously named as the critical inner pressure, Pcri i . This position independent feature will help reduce computation effort significantly as presented below
(c)
t
Fig. 3. Schematic representation of (a) the spring-slider model, (b) shear force per unit length versus relative shear displacement, and (c) the bolt effective diameter.
3. The reinforcement mechanics of passive bolts in conventional tunnelling 3.1. The basic spring–slider model First, some pullout tests for passive bolts are reviewed briefly here [15,16], which could improve the understanding of the interaction relationship between the bolt and the rock mass. The failure of fully grouted bolts in pullout tests generally occurs at the bolt–grout interface (cementgrouted bolts) or at the grout–rock interface (resin-grouted bolts), since these interfaces are relatively weak and have only cohesion and frictional resistance. For fully frictional bolts, the failure can only occur at the bolt–rock interface. In some tests, while the bolts are mounted in a hard rock mass and subjected to high confinement pressure, shear failure of the grout itself could occur. These four failure modes presented above can be generalized into one spring–slider model, as illustrated in Fig. 3(a), since they are all cohesive and frictional shear failure in nature. The bolt itself can be discretized into a series of rigid-connected nodes along its axis direction (i.e., the radial direction in Fig. 2), where the elongation of the bolt itself is neglected since it is negligibly small compared
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to the deformation of rock mass. The shear behavior per unit length is jointly controlled by a shear spring that is characterized by its shear stiffness, Ks and a slider that is characterized by its cohesive strength, cs. The relationship between the shear force per unit length, Fs/dr and the relative shear displacement, Dus is illustrated in Fig. 3(b). Before decoupling, the shear force is mobilized by and proportional to the relative shear displacement. The maximum shear force per unit length is limited by the confinement stress, st, the friction coefficient in failure interface, tan fs, the cohesive strength in failure interface, cs, and the bolt effective diameter, Ds. After decoupling, the cohesive strength is lost, and the shear force per unit length falls to its residual value that is only controlled by st, tan fs, and Ds. Therefore, the Fs–Dus relationship, can be formulated as ( K s Dus dr ðK s Dus drpF max Þ s Fs ¼ (4-1) max , st tan fs pDs dr ðK s Dus dr4F s Þ ¼ ðst tan fs þ cs ÞpDs dr. F max s
(4-2)
Here, the bolt effective diameter Ds may equal the diameter of bolt itself, the diameter of grout hole, or the average of the former two, corresponding to the bolt–grout, the grout–rock or the inside–grout shear failure modes, respectively, as shown in Fig. 3(c). The properties involved in this model, Ks, tan fs, cs, and Ds, can be evaluated from conventional pullout tests (see details in Appendix). Dividing the shear force per unit length by the bolt effective perimeter, the shear stress around the bolt, ts, can be formulated as Eq. (4-3). Integrating the shear force per unit length along the bolt axis, the axial force of bolt can be obtained via Eq. (4-4). ( K Du s s ðK s Dus drpF max Þ Fs s pDs ts ¼ ¼ , (4-3) st tan fs ðK s Dus dr4F max Þ pDs dr s Z Fn ¼
F s dr.
(4-4)
3.2. The passive bolts in conventional tunnelling While applying the passive bolts to ground reinforcement in conventional tunnelling, Freeman first proposed the concepts of the neutral point, the pickup length and the anchor length in passive bolts, according to the in-site monitoring data [17]. At the neutral point r, the relative shear displacement and the shear force are zero, and the tensile axial force in the bolt reaches its peak value. The pickup length refers to the section from the near end (on the tunnel wall) to the neutral point, where the shear force picks up the load from the rock and drags the bolt towards the tunnel. The anchor length refers to the section from the neutral point to the far end (seated deep in the rock), where the shear force anchors the bolt to the rock. These concepts, which clearly outline the mechanics of passive bolts in a continuously deformed in-situ rock mass, are adopted in this paper. Then considering the static equilibrium condition of the bolt itself, the shear force mobilized in the pickup length should equal that mobilized in the anchor length. In addition, the maximum resultant axial force, F max n , cannot exceed the bolt’s load capability, Fyield. Z
Z
r
Lb
F s dr ¼
F s dr,
(5-1)
F s drpF yield .
(5-2)
r
0
F max ¼ n
Z
r 0
3.3. Equilibrium equations for rock mass Consider an infinitesimal volume in the radial direction as shown in Fig. 4(a). The rock mass is subjected to a radial stress sr, a tangential stress st and another axial force Fn provided by bolt. Supposing that the Fn can be spread evenly around its tributary area, the static equilibrium condition of the infinitesimal rock mass volume, as t
t d
Fn+dFn
Fn
r
r+dr
r
Fn rLz t dr
(b)
Fn+dFn (r+dr)Lz
r+dr b Fn
t (a)
dr
(c)
Fn+dFn dr
Fig. 4. Static equilibrium condition for the surrounding rock mass.
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illustrated in Fig. 4(b), can be formulated as Fn do sr r do Lz þ 2st dr Lz sin 2 roLz F n þ dF n ¼ sr þ dsr ðr þ drÞ do Lz . ðr þ drÞoLz
(6-1)
Meanwhile, considering the interaction at the bolt–rock interface as shown in Fig. 4c, the increment of the axial force, dFn, can be expressed as dF n ¼ ts pDs dr.
(6-2)
Associating these two equations, noticing that sin(do/2) approximately equals do/2, since do is an infinitesimal, the equilibrium equation can be deduced as dsr st sr þ N 0 ts ¼ dr r
with
N0 ¼
pDs . oLz
(6-3)
The constant N0 is named geometry coefficient in this paper, since it is totally determined by the geometrical properties of the bolts. In the absence of the bolts, merely setting N0 to zero, the above equation can be reduced into the equilibrium equation in a conventional form. Therefore, Eq. (6-3) can be regarded as a ‘‘uniform’’ equilibrium equation regardless of the presence or absence of the passive bolts. When applying Eq. (6-3) to the elastic region, where the stress state of rock mass should verify the hydrostatic insitu stress condition that the sum of sr and st equals 2P0, the equilibrium equation for elastic region can be formulated as dsr 2P0 2sr þ N 0 ts ¼ . (7) dr r When applying it to the plastic region, where the stress state of rock mass should verify the failure criterion as shown in Eq. (1), the equilibrium equation for the plastic region can be formulated as dsr ðK p 1Þsr þ sc þ N 0 ts ¼ . dr r
(8)
3.4. Displacement compatibility equations for rock mass 3.4.1. The elastic region Due to the plane strain axial symmetry assumption, the strain–displacement relationships for the rock mass can be simplified significantly as du u ¼ r ¼ t . (9) dr r According to Hook’s law, the tangential strain of the rock mass can be evaluated from its stress state, as formulated in Eq. (10), where E and n are the Young’s modulus and the Poisson ratio of the rock mass, respectively. st sr 2P0 n P0 P0 2P0 n n n n n t ¼ . (10) E E E E E E
629
Notice that only the strain caused by tunnel excavation is concerned, which means the initial strain due to in-situ stresses should be removed. Then, associating these two equations and considering the hydrostatic in-situ stress condition, the displacement compatibility equation for the elastic region can be formulated as Eq. (11). u ¼ rt ¼
P0 sr ð1 þ nÞr. E
(11)
3.4.2. The plastic region The loading path for this problem refers to a monotonic decrease of the fictitious inner pressure, corresponding to the advancing of the tunnel face. Consequently, the rates of all mechanical variables can be evaluated by their firstorder derivatives with respect to Pi. The incremental theory of plasticity [11,13] assumes that the total strain rate consists of both an elastic part and a plastic part, as shown in Eq. (12). The elastic part is controlled by Hooke’s law and the plastic part by the potential flow rule, as formulated by Eqs. (13) and (14), respectively. The relationship between the strain rate and the displacement velocity is simplified by virtue of axial symmetry and formulated by Eq. (15). _ r ¼ _er þ _pr ; _ er ¼
_ y ¼ _ey þ _py ,
1n n s_ r s_ y ; 2G 2G
_ pr ¼ l
qg ¼ l; qsr
_py ¼ l
_ey ¼
(12) 1n n s_ y s_ r , 2G 2G
qg ¼ lK c , qsy
(13) (14)
qu_ u_ ; _y ¼ . (15) qr r Here, g is the plastic potential that has been defined by Eq. (2). The rates of all mechanical variables (denoted by a dot mark) are referred as their first-order derivatives with respect to Pi. Then associating these four equations, eliminating the multiplier l, the displacement compatibility equation for the plastic region can be expressed as _ r ¼
ðnK c K c þ nÞ qu_ u_ ð1 n nK c Þ þ Kc ¼ s_ r s_ y . qr 2G 2G r
(16)
4. Two-dimensional finite-difference algorithm The equilibrium equations, the displacement compatibility equations and the bolt–rock interaction equations presented above can only be solved numerically. A twodimensional finite-difference algorithm (i.e., along the unloading path and along the radial direction) is employed in this paper, and the philosophy of the method is described in this section. All the variables describing the mechanical state of the surrounding rock mass have two indices: the first indicates a certain stage in the unloading path and the second indicates a certain position in the radial direction. Supposing that at former stage (say the (k1)th stage where Pi ¼ Pðk1Þ ), all the mechanical states i
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of the rock mass are known, the objective is to evaluate all the mechanical states at current stage (i.e., the kth stage where Pi ¼ PðkÞ i ) according to their known counterparts at the former stage, which includes the following three steps: stress evaluation, displacement evaluation and bolt response evaluation. After one iteration finishes, these known mechanical states at the current stage can be used to evaluate the mechanical states at next stage (i.e. the (k+1)th stage where Pi ¼ Piðk1Þ ), following the same three steps. This kind of iteration is repeated until the final stage where Pi ¼ Pfin i . 4.1. Stress evaluation
unloading path
The equilibrium equations (7) and (8) are solved by the fourth-order Runge–Kutta method [18], with the algorithm diagram presented in Fig. 5. At the current stage, the radial stress at the tunnel wall sr(k, Ra) is known and equals PðkÞ i , which serves as the boundary condition of the equilibrium equations. Then the radial and tangential stresses at each sequential calculation point are evaluated according to Eq. (8) and the failure criterion. If the radial stress increases up to the critical inner pressure Pcri i , record the position as the radius of the elasto-plastic interface Re, then go on evaluating the stress state at each sequential calculation point according to Eq. (7) and hydrostatic insitu stress condition. (k-1)th stage Pi=Pi(k-1)
The bolt–rock interaction shear stress and the transitional strength of rock mass at the former stage are required during this step, and the radial and tangential stresses at the current stage can be determined after the stress evaluation process. 4.2. Displacement evaluation The algorithm diagram for the radial displacement evaluation is presented in Fig. 6. For the elastic region, the radial displacement of the rock mass at the current stage can be evaluated directly by the radial stress of the rock mass at the current stage, according to Eq. (11). For the plastic region, the radial and tangential stress rates s_ r ðk; rÞ and s_ t ðk; rÞ should be first evaluated by their firstorder difference with respect to Pi, as shown below. _ rÞ ¼ sðk;
sðk; rÞ sðk 1; rÞ ðrpRe Þ. dPi
Similarly, the deformation rate at the elasto-plastic inter_ Re Þ, which serves as the boundary condition of the face uðk; compatibility equation, can also be obtained by its firstorder difference with respect to Pi. Then the fourth-order Runge–Kutta method is utilized again to evaluate the deformation rate at each sequential calculation point (inward radial direction) according to the compatibility equation (16). Finally, the displacement at the current stage can be obtained by accumulating the displacement
s(k-1, r) and c(k-1, r) are known
known point r=Ra radial direction r=Re unevaluated point r=Pi(K) r=Picri kth stage Pi=Pi(k) Elastic region Plastic region Controlled by Eq. (7) and Controlled by Eqs. (8) and hydrostatic stress condition ailure criterion r(k, r) and t(k, r) can be determined
Then
unloading path
Fig. 5. The algorithm diagram of stresses evaluation.
(k-1)th stage Pi=Pi(k-1)
kth stage Pi=Pi(k)
Then
(17)
u(k-1, r), r(k-1, r) and t(k-1, r) are known r(k, r) and t(k, r) are known r=Re inward radial direction u=uRe plastic region controlled by Eq. (16) and accumulated via Eq. (18)
known point unevaluated point
elastic region controlled by Eq. (11)
u(k, r), as well as c(k, r) can be determined
Fig. 6. The algorithm diagram of displacement evaluation.
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jth stage
631
u(j, r) is known • • •
unloading path
Z. Guan et al. / International Journal of Rock Mechanics & Mining Sciences 44 (2007) 625–636
u(k,r) is known, and umob(k,r)=u(k,r)-u(j,r)
neutral point (or length) pickup and anchor length
pickup length anchor length r= neutral point
kth stage
pickup length
a) before bolt yielding
anchor length
r=1 r=2 neutral length
b) after bolt yielding
s(k,r), as well as Fn(k,r) can be determined
Then
Fig. 7. The algorithm diagram of bolt response evaluation.
increment at the current stage to its counterpart at the former stage. _ rÞ dPi uðk; rÞ ¼ uðk 1; rÞ þ uðk;
ðrpRe Þ.
(18)
After the displacement evaluation, the major principle plastic strain pt at the current stage, which serves as the softening parameter in this paper, can be evaluated by Eq. (19). Then the transitional strength sc at the current stage can also be computed via Eq. (1). pt ðk; rÞ ¼ t ðk; rÞ te ðk; rÞ uðk; rÞ uðk; Re Þ ¼ r Re
ðrpRe Þ.
ð19Þ
The displacement and the stresses at the former stage, as well as the stresses at the current stage, are required during this step. Then the displacement and the transitional strength of rock mass at the current stage can be determined after the displacement evaluation process.
point is found. If the bolt load-capability condition Eq. (5-2) is violated, the bolt yields and a neutral length (rather than a neutral point) will occur in the bolt. A similar adjustment process is repeated to find the proper position of the neutral length. The process of finding the proper position of the neutral point (or length) is, actually, not so formidable as it seems to be, because the position of the neutral point (or length) at the current stage is always near the one at the former stage. Generally, only two or three adjustments are required to find the proper position of the neutral point (or length) for each stage. The displacements at the current and the bolt installation stages are required during this step, and the proper bolt responses (including the bolt–rock interaction shear stress and the axial force of bolt) can be determined after the bolt response evaluation process. After these three steps, all the mechanical states at the current stage are known, which can be used to evaluate their counterparts at next stage (i.e., the (k+1)th stage where Pi ¼ Pðkþ1Þ ). i
4.3. Bolt response evaluation Supposing that the bolts are mounted at jth stage where Pi ¼ P(j) i , the algorithm diagram for the bolt response evaluation is illustrated in Fig. 7. The mobilizing displacement, umob, which mobilizes the response of passive bolts, should be first computed via Eq. (20). umob ðk; rÞ ¼ uðk; rÞ uðj; rÞ ðrpLb Þ.
ðrpLb Þ.
The proposed method has been programmed in the VC++ development environment, the verification of the proposed method and some discussions are expatiated in this section.
(20)
Then, supposing an arbitrary position for the neutral point, where umob(k, r) denotes the mobilizing displacement at the neutral point, the relative shear displacement, Dus, at the current stage can be computed as follows Dus ðk; rÞ ¼ umob ðk; rÞ umob ðk; rÞ
5. Verification and application
(21)
Using this supposed Dus, the bolt responses, ts (k, r) and Fn (k, r), can then be evaluated via Eq. (4). If the bolt self-equilibrium condition Eq. (5-1) is not satisfied, adjust the position of neutral point and recalculate the relative shear displacement and the bolt responses via Eqs. (21) and (4), until a proper position of neutral
5.1. An illustrative case study An illustrative case study is studied in this section to demonstrate the reinforcement mechanics of passive bolts in conventional tunnelling. Suppose that a round-section tunnel with a design radius of 5 m is excavated under a hydrostatic in-situ stress condition of 8 MPa (about 350–400 m rock mass covering above). The properties of the rock mass and the passive bolts employed in this case are listed in Table 1. The bolting pattern and some rock mass properties are referenced to the Japanese standard for mountain tunnelling [19]. The passive bolts are supposed to
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Table 1 The properties of rock mass and rock bolt employed in a verification case (by proposed method) Rock mass Properties
E (Pa) 1.0e9
n 0.25
Kp 3.0
sc1 (Pa) 5.0e6
s2c (Pa) 3.0e6
Kc 1.33
a 0.0058
Ra (m) 5.0
P0 (Pa) 8.0e6
Rock bolt Properties
E (Pa) 2.1e11
Fyield (N) 3.0e5
Ab (mm2) 314
Ks (Pa) 2.5e7
cs (Pa) 1.0e6
tan fs 0.84
Ds (m) 0.04
o 0.224
Lb (m) 3.0
2.0
2.0
s(MPa)
/P0
1.0
t
1.5 1.0
7
8
9
r (m) 6
7
8
pickup length
anchor length
-2.0
r (m) 6
0.0 -1.0
r
0.5 0.0
Lz (m) 1.0
10
300
0.5
with bolt (proposed method) with bolt (FLAC3D )
1.5 2.0
with bolt (proposed method) with bolt (FLAC3D ) without bolt (proposed method)
Fn(kN)
u / Ra (%)
u 1.0
200
100
without bolt (FLAC3D )
2.5 (a)
0 (b)
6
7
8
r (m)
Fig. 8. Ground and bolt responses at Pi ¼ 1.0 MPa stage: (a) distributions of sr, st and u along the radial direction, and (b) distributions of ts and Fn along the bolt length.
be mounted at the stage where PiðjÞ ¼ 2:75 MPa, which just equals the critical inner pressure for the rock mass. The ground responses and the bolt responses at the PðkÞ i ¼ 1; 0:3 and 0 MPa stages are depicted in Figs. 8–10 (represented by solid lines). To highlight the reinforcement effect of passive bolts, the ground responses at the same stage but without passive bolts reinforcement are also depicted in these figures (represented by dashed lines). The passive bolts exert their effect gradually with the displacement release of rock mass. At the PðkÞ i ¼ 1:0 MPa stage, the reinforcement effect is not significant, and the passive bolts remain coupled along the whole length. At the PðkÞ i ¼ 0:3 MPa stage, the passive bolts help constrain the range of plastic region (the convex point at the st curve) from 7.5 to 7.2 m and the convergence of tunnel opening from 1.93% to 1.70%, meanwhile, decoupling occurs at the near-end section of the bolt. At the Pi ¼ 0 MPa stage (i.e., no lining is supported after the tunnel face advanced away), the ground and the passive bolts carry the released in-situ stresses fully. On this occasion, the passive bolts exert their load capability fully to help constrain the range of plastic region from 8.3 to 7.8 m and the convergence of tunnel opening from 2.46%
to 2.12%, meanwhile, decoupling and yielding occur on the near-end section and the middle section of the bolt. Regarding the constrained displacement at the tunnel wall, the reinforcement effect of the passive bolts on this occasion is equivalent to providing a fictitious inner pressure of 0.2 MPa. 5.2. Verification by numerical simulations The illustrative case is identically studied by numerical simulations (codes: FLAC3D) to verify the validity of the proposed method. The model itself is very simple in geometry, which consists of 3696 rock mass elements and 5778 rock mass gridpoints, as well as 150 cable elements and 155 cable nodes. However, the strain-softening constitutive laws in FLAC3D are characterized by six parameters: bulk modulus K, shear modulus G, friction angle f, cohesion c, dilation angle c and softening parameter Z. Thereinto, the values of f, c and c can vary with the development of the softening parameter Z (in tabulated form). It is obvious that the former five parameters can be evaluated directly from their counterparts employed in the proposed method, via following
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4.0
2.0
s(MPa)
/P0
2.0
t
1.5 1.0 0.5
6
7
8
9
10
u
r (m) 6
7
pickup length
8
anchor length
with bolt (proposed method) with bolt (FLAC3D )
300 Fn(kN)
u / Ra (%)
-1.0
400
1.0 1.5
with bolt (proposed method)
2.0
with bolt (FLAC3D ) without bolt (proposed method) without bolt
0.0
-4.0
0.5
2.5
1.0
-2.0
r r (m)
0.0
633
200 100
(FLAC3D )
0
6
7
(b)
(a)
8
r (m)
Fig. 9. Ground and bolt responses at Pi ¼ 0.3 MPa stage: (a) distributions of sr, st and u along the radial direction, and (b) distributions of ts and Fn along the bolt length.
4.0
2.0 t
2.0 s(MPa)
1.0
0.0
r 6
7
8
0.5 u / Ra (%)
1.0 0.0 -1.0
r (m) 6
-4.0
r (m)
9
10
400
u
1.5 with bolt (proposed method)
anchor length
with bolt (proposed method) with bolt (FLAC3D )
200
0 (b)
(a)
pickup length
100
with bolt (FLAC3D ) without bolt (proposed method) without bolt (FLAC3D )
2.5
8
300
1.0
2.0
7 neutral length
-2.0
0.5
Fn(kN)
/P0
1.5
6
7
8
r (m)
Fig. 10. Ground and bolt responses at Pi ¼ 0 MPa stage: (a) distributions of sr, st and u along the radial direction, and (b) distributions of ts and Fn along the bolt length.
relations K¼
E ; 3ð1 2nÞ
Kp ¼
E , 2ð1 þ nÞ
(22-1)
pffiffiffiffiffiffi 1 þ sin f 1 þ sin c ; sc ¼ 2c K p ; K c ¼ . 1 sin f 1 sin c
(22-2)
G¼
However, the softening parameter in FLAC3D is defined as follows [7] qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 dZFlac ¼ pffiffiffi ðdp1 dpm Þ2 þ ðdpm Þ2 þ ðdp3 dpm Þ2 2 dp þ dp3 . (23-1) with dpm ¼ 1 3
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Table 2 The properties of rock mass employed in the identical case (by numerical simulations) G (Pa) 4.0e8
f (1) 30.0
c1 (Pa) 1.4e6
c2 (Pa) 8.7e5
Therefore, the shift point of the softening parameter in FLAC3D can be obtained from the parameters used in the proposed method by a qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi (23-2) ZsFlac ¼ pffiffiffi K 2c þ K c þ 1. 3
( K Flac s
¼
Ks
ðK s Dus drpF max Þ s
st tan fs pDs Dus
ðK s Dus dr4F max Þ s
.
(23-3)
The results from the numerical simulations are also depicted in Figs. 8–10, as denoted by cross or dot marks. On the occasion without bolt installation, the ground responses computed by the numerical simulations and by the proposed method fit each other exactly, which has been proved by former researchers [13,14]. On the occasion with bolt installation, the ground and bolt response computed by two methods also agree with each other well, when no decoupling or a little decoupling occurs along the bolt (see Figs. 8 and 9). However, there exists a quantitative discrepancy between the two methods in depicting the bolt responses when decoupling and yielding occurs (see Fig. 10). The spring–slider model employed in FLAC3D (although the shear stiffness is compulsively modified via Eq. (23-3) after decoupling) cannot be exactly equivalent to the one employed in proposed method when a large section of decoupling occurs, which may account for the quantitative discrepancy shown in Fig. 10.
Ra (m) 5.0
P0 (Pa) 8.0e6
0.5 BRC 0.4
D'
C'
100
B' A
Pi /P0
1.0
300 200
M'
0.3
The parameters employed in the numerical simulations are summarized in Table 2, corresponding to their counterparts employed in the proposed method. On the other hand, the spring–slider model used in the proposed method is also adopted in FLAC3D, thus the properties of rock bolt (except Ks) employed in the numerical simulations are the same with those tabulated in the second line of Table 1. By default, however, FLAC3D assumes that the shear force will remain at its peak value after decoupling. Thus, it cannot directly simulate the falldown relationship between Fs and Dus as illustrated in Fig. 3(b). Fortunately, FLAC3D provides a user-defined programming language FISH to help implement the falldown spring–slider model indirectly. Checking the coupling state of each cable element at the beginning of each calculation cycle, the shear stiffness for this cable element is equal to its initial value before decoupling, or is compulsively modified via Eq. (23-3) after decoupling. Then the modified shear stiffness is used in the next cycling of FLAC3D.
ZsFlac 0.0068
c (1) 8.2
1.5 ua/Ra (%)
2.0
Fmax n (kN)
K (Pa) 6.7e8
Rock mass Properties
2.5
0.2 SCC B
0.1
conventional GRC M
reinforced GRC C
0.0 0.0
0.5
1.0 1.5 ua/Ra (%)
2.0
2.5
Fig. 11. The reinforced ground reaction curve and bolt reaction curve.
For the most part, the validity of the proposed method is proved theoretically via the illustrative case study and the numerical simulations. The proposed method can rationally elucidate the reinforcement mechanics of passive bolts in conventional tunnelling, and improve the support system design effectively. 5.3. Ground reinforcement design The ground reinforcement and the inner support should be considered jointly and comprehensively in the support system design in conventional tunnelling. The CCM just provides such a tool to compromise the complex nature with a general albeit simplified approach. General reviews and applications of the CCM have been summarized by many researchers [1,2]. Using the proposed method, the GRC with passive bolts reinforcement (name reinforced GRC hereafter) and the bolt reaction curve (named BRC hereafter) can be constructed by recording the displacement released at tunnel wall ua and the maximum axial force in the bolt F max at each unloading stage. n Taking the illustrative case for example, the reinforced GRC and the BRC are depicted in Fig. 11. For comparison, the conventional GRC without passive bolts reinforcement is also depicted in the figure. Point A denotes the stage when the bolts are mounted. Points B (B0 ), C (C0 ) and D (D0 ) denote the stages where Pi ¼ 1.0, 0.3 and 0 MPa, and the ground and bolt responses at these stages
ARTICLE IN PRESS Z. Guan et al. / International Journal of Rock Mechanics & Mining Sciences 44 (2007) 625–636
2.5
2.5 Lb N0
2.4
umax a /Ra (%)
umax a /Ra (%)
2.3
2.2 2.1 2.0
2.2 2.1 2.0
the standard case at Pi=0MPa stage
1.9
cs tan s Ks Fyield
2.4
Pi(j)
2.3
the standard case at Pi=0MPa stage
1.9 1.8
1.8 0 (a)
635
50
100 % variables
150
0
200
50
(b)
100 % variables
150
200
Fig. 12. The influence of the bolt’s (a) geometrical and time properties, and (b) mechanical properties on the reinforcement effect.
have been portrayed in Figs. 8–10. The SCC describes the relationship between the inner pressure carried by the inner support (including shotcrete lining, steel sets, active bolts, etc.) and the radial displacement at the tunnel wall. Intersection point M between the reinforced GRC and the SCC, together with its corresponding point M0 in the BRC, indicates the final states of the reinforced rock mass after the tunnel face advancing away. To highlight the influence of the bolt properties on the reinforcement effect, some derivative cases are studied in the following. Taking the illustrative case as a standard case, varying every single parameter (including the geometrical properties, N0 and Lb, the time property, P(j) i , and the mechanical properties, Ks, cs, tan fs, and Fyield) from 0% to 200% of its initial value, the relative significance of the bolt properties on the reinforcement effect is highlighted in Fig. 12. The tunnel convergence at the Pi ¼ 0 MPa stage, umax a , is selected as the estimation index (Y-coordinate) in this figure. A surplus length of rock bolt hardly can take effect on the ground reinforcement, because the bolt section embedded in the elastic region of rock mass cannot help constrain the elastic displacement release (under the scope of continuous deformation media). Therefore, it is unnecessary to extend the bolt beyond the range of plastic region too much. For the same reason, the bolt nearly can take its effect, if the rock mass can remain elastic (i.e., cri Pfin i 4Pi ). On the contrary, increasing the bolt installation density (i.e. increasing the bolt geometrical coefficient N0 by decreasing the perimeter interval o and the longitudinal spacing Lb) can always help reinforce and stabilize the surrounding rock mass. The mechanical properties of the bolt (except Fyield) are mainly controlled by the grout condition. Among them, the shear stiffness Ks influences the reinforcement effect most significantly. Generally, a higher value of Ks is preferred, which means the bolt is more sensitive to the relative shear displacement, and can exert its effect at early stage. The friction angle and the cohesive strength at the interface,
which control the maximum shear force per unit bolt length can provide, also influence the reinforcement effect to some extend, if decoupling occurs along the bolt. Meanwhile, the reinforcement effect is also limited by the loading capability of the bolt itself, which accounts for the reason why an excessively strong grout condition cannot help improve the reinforcement effectively. 6. Conclusions According to the pullout tests for passive bolts, a spring–slider model is adopted in this paper to account for the interaction between the bolt and the rock mass. Then the additional force provided by the passive bolt is incorporated into the equilibrium equation for rock mass. Then based on the incremental theory of plasticity and the plane strain axial symmetry assumption, a elasto-plastic analyses method for passive bolt reinforced ground in conventional tunnelling is proposed in this paper. The reinforcement mechanics of passive bolts in conventional tunnelling is demonstrated qualitatively and quantitatively via an illustrative case study. The validity of the proposed method is verified by numerical simulations via the identical case study. The reinforced GRC and the BRC can be constructed easily by using the proposed method, which could help the support system design in the conventional tunnelling. In addition, the proposed method also permits incorporating other features about the bolt–rock interaction model (e.g., a gradual fall-down model) or other failure criterions (e.g., Hoek–Brown criterion) of the rock mass into further analyses. To highlight the influence of the bolt properties on the reinforcement effect, some derivative cases are studied. The bolt section embedded in the elastic region of rock mass hardly can take effect on the ground reinforcement (under the scope of continuous deformation media). On the contrary, increasing the bolt installation density can always help reinforce and stabilize the surrounding rock mass. The grout conditions, as well as the loading capability of bolt
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D28, Lb=150mm, t =3.2MPa
200
Axial load (kN)
160
Ks =
Avr.
Fnmax 0.12 × e 6 = = 340 MPa el π Ds Δ us 3.14 × 0.028 × 0.004
120
tan φs =
80
cs =
40
Fnmax
π Ds Lb σ t
=
0.07 × e6 = 1.66 3.14 × 0.028 × 0.15 × 3.2 × e 6
Fnmax − Fnres (0.12 − 0.07) × e 6 = = 3.79 MPa 3.14 × 0.028 × 0.15 π Ds L b
0 0
5
10
15
20
25
30
Axial displacement (mm) Fig. A1. An representative axial load versus axial displacement curve (after Moosavi, 2005).
itself, influence the reinforcement effect significantly and comprehensively. Among them, a higher value of Ks is generally preferred, which means the bolt is more sensitive to the relative shear displacement, and can exert its effect at early stage. Appendix A. : The properties evaluation for passive bolts This appendix explains how to evaluate the properties of passive bolts involved in the proposed method (see second line of Table 1). The axial load versus axial displacement curve can always be obtained from conventional pullout tests (either in the laboratory or in the field). Fig. A1 shows a representative curve from the fully grouted bolts’ pullout tests [20]. The embedded bolt length Lb is 0.15 m; confinement pressure st is 3.2 MPa; and the grout cement has a uniaxial compression strength of 42 MPa. The ribbed bolt has a diameter of 28 mm, and the effective diameter Ds is set to 28 mm since the failure occurs at bolt–grout interface. The curve can be simplified into a rigid fall-down model as denoted by the dashed line in Fig. A1, where the maximum and the residual axial force are 120 and 70 kN, respectively; the elastic proportional limit of relative shear displacement is set to 4 mm. Then the shear stiffness Ks, cohesive strength cs, and the friction coefficient tan f can be evaluated from these results of pullout tests, as shown in the right side of Fig. A1. The other properties of rock bolt can be obtained directly from the manufacture’s specification. References [1] Carranza-Torres C, Fairhurst C. Application of convergence–confinement method of tunnel design to rock masses that satisfy the Hoek–Brown failure criterion. Tunnelling Underground Space Technol 2000;15(2):187–213. [2] Oreste P. Analysis of structural interaction in tunnels using the convergence–confinement approach. Tunnelling Underground Space Technol 2003;18(4):347–63.
[3] Graziani A, Boldini D, Ribacchi R. Practical estimate of deformations and stress relief factors for deep tunnels supported by shotcrete. Rock Mech Rock Eng 2005;38(5):345–72. [4] Windsor C. Rock reinforcement system. Int J Rock Mech Min Sci 1997;34(6):919–51. [5] Li C, Stillborg B. Analytical models for rock bolts. Int J Rock Mech Min Sci 1999;36(8):1013–29. [6] Cai Y, Esaki T, Jiang Y. An analytical model to predict axial load in grouted rock bolt for soft rock tunnelling. Tunnelling Underground Space Technol 2004;19(6):607–18. [7] Itasca Consulting Group. FLAC3D, fast Lagrange analysis of continua in 3 dimensions, version 2.0, user manual. Minneapolis: Itasca Inc.; 1997. [8] Chen S, Qiang S, Chen S, Egger P. Composite element model of the fully-grouted rock bolt. Rock Mech Rock Eng 2004;37(3):193–212. [9] Jiang Y. Theoretical and experimental study on the stability of deep underground opening. PhD thesis, Kyushu University, Japan, 1993. [10] Hudson J, Harrison J. Engineering rock mechanics. London: Pergamon; 1997. [11] Hill R. The mathematical theory of plasticity. London: Oxford University Press; 1950. [12] Varas F, Alonso E, Alejano L, Fdez-Manin G. Study of bifurcation in problem of unloading a circular excavation in a strain-softening material. Tunnelling Underground Space Technol 2005;20(4):311–22. [13] Carranza-Torres C, Fairhurst C. The elasto-plastic response of underground excavations in rock masses that satisfy the Hoek– Brown failure criterion. Int J Rock Mech Min Sci 1999;36(6): 777–809. [14] Alonso E, Alejano L, Varas F, Fdez-Manin G, Carranza-Torres C. Ground reaction curves for rock masses exhibiting strain-softening behaviour. Int J Num Anal Meth Geomech 2003;27(13):1153–85. [15] Stillborg B. Professional users handbook for rock bolting. 2nd ed. Trans Tech Publications; 1994. [16] Kilic A, Ysar E, Atis C. Effect of bar shape on the pullout capability of fully-grouted rockbolts. Tunnelling Underground Space Technol 2003;18(1):1–6. [17] Freeman T. The behaviour of fully-grouted rock bolts in the Kielder experimental tunnel. Tunnels Tunnelling 1978;June:37–40. [18] Schilling R, Harris S. Ordinary difference equations. In: Applied numerical methods for engineers using MATLAB and C. Thomson Learning Inc.; 2000. p. 361–421. [19] Japan Society of Civil Engineering. Japanese standard for mountain tunneling. Tokyo: Maruzen Inc.; 1996 [in Japanese]. [20] Moosavi M, Jafari A, Khosravi A. Bond of cement grouted reinforcing bars under constant radial pressure. Cement Concrete Compos 2005;27(1):103–9.
International Journal of Rock Mechanics & Mining Sciences 44 (2007) 625–636 www.elsevier.com/locate/ijrmms
Reinforcement mechanics of passive bolts in conventional tunnelling Zhenchang Guana,, Yujing Jiangb, Yoshihiko Tanabasib, Hongwei Huangc a
Graduate School of Science and Technology, Nagasaki University, 1-14 Bunkyo Machi, Nagasaki 852-8521, Japan b Department of Civil Engineering, Faculty of Engineering, Nagasaki University, Nagasaki, Japan c Department of Geotechnical Engineering, Tongji University, Shanghai, China Received 5 June 2006; received in revised form 3 October 2006; accepted 6 October 2006 Available online 11 January 2007
Abstract According to the conventional pullout tests for passive bolts, a spring–slider model is adopted in this paper to account for the interaction relationship between the bolt and the rock mass. Based on the incremental theory of plasticity and the plane strain axial symmetry assumption, an elasto-plastic ground and bolt responses analyses method is proposed. In addition, the validity of the proposed method is verified by numerical simulations. The reinforcement mechanics of passive bolts in conventional tunnelling is clearly demonstrated via an illustrative case study. Through parameter studies, the influence of the bolt properties on the reinforcement effect is highlighted. The proposed method provides a framework to the elasto-plastic ground response analyses with passive bolts reinforcement. It also permits incorporating other features about the bolt–rock interaction model or other failure criterion of the rock mass in further analyses. r 2006 Elsevier Ltd. All rights reserved. Keywords: Passive bolt; Ground reinforcement; Conventional tunnelling; CCM; Reinforced GRC
1. Introduction The new Austrian tunneling method (NATM) has changed the concept of conventional tunnelling from resisting the passive earth pressure to helping the ground support itself. Consequently, shotcrete lining and rock bolting are widely used in practice to help the surrounding ground to stabilize itself. The convergence confinement method (CCM), under the plane strain axial symmetry assumption, provides a classic and efficient tool for the support system design in conventional tunnelling. It consists of the longitudinal deformation profile (LDP), the support characteristic curve (SCC) and the ground reaction curve (GRC). The arrangement and the intersection of these three curves are considered, to compromise the complex nature near the tunnel face with a general albeit simplified approach. It enables the taking of some different constitutive laws of rock mass and some different Corresponding author.
E-mail address: [email protected] (Z. Guan). 1365-1609/$ - see front matter r 2006 Elsevier Ltd. All rights reserved. doi:10.1016/j.ijrmms.2006.10.003
types of supports (including shotcrete lining, steel sets and active bolts) into account [1–3]. However, the presence of passive bolts complicates the problem significantly. Generally, passive bolts refer to the continuously mechanically coupled bolt and the continuously frictionally coupled bolt, and are particularly represented by fully cement (or resin) grouted bolt and Swellex bolt [4]. Several categories of models, including analytical and numerical models, have been proposed to account for the interaction between the bolt and the rock mass. As far as the analytical models are concerned, the bolt together with the rock mass within its tributary area is generally considered as one entity, in which the bolt–rock interaction is embedded. Then given a free displacement distribution (the displacement of rock mass without bolting), one can compute the response of rock bolt itself, as well as the reinforced displacement of rock mass after bolting. The analytical models are represented by Li and Stillborg [5], Cai et al. [6] and others. From a mathematical viewpoint, it can be regarded as a ‘‘one-step’’ mapping from an initial state to a final state, while a plastic problem
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requires an iterative computing algorithm along a certain loading path. As far as the numerical models are concerned, there are two ways to simulate the bolt–rock interaction: explicit and implicit models. The explicit model simulates the interaction distinctly via a series of spring–sliders, so that it is suitable for finite-difference method and typically utilized in the numerical codes FLAC3D [7]. In the implicit model, the bolt together with the rock mass within its tributary area is integrated as one sub-element, in which the bolt–rock interaction is embedded. Then the sub-element can be inserted into existing codes and be treated as a common element. The implicit model is suitable for finiteelement method as demonstrated by Chen et al. [8] and others. While resorting to numerical methods, it is straightforward to deal with any ordinary constitutive laws of rock mass and any complicated boundary conditions, but with a drawback of being time consuming. After introducing the spring–slider model, a seminumerical ground response analyses method is proposed in this paper to account for the reinforcement mechanics of passive bolts in conventional tunnelling. Based on the incremental theory of plasticity and the plane strain axial symmetry assumption, equilibrium equations and compatibility equations are first deduced theoretically, and then solved numerically. The reinforcement mechanics of passive bolts in conventional tunnelling is clearly demonstrated via an illustrative case study. The validity of the proposed method is verified by numerical simulations. In addition, the influence of the bolt properties on the reinforcement effect is highlighted through parameter studies. 2. Problem description 2.1. The constitutive laws of rock mass For universality, the rock mass is assumed to exhibit strain-softening behavior, which can be reduced into a perfect elasto-plastic behavior or an elasto-brittle behavior in some special cases [9,10]. Although many researchers 1
elastic
plastic
have reported that strain-softening materials may experience instability and bifurcation during an unloading process [11,12], this paper restricts the discussion to its stable (or basic) solutions. Generally, the rock mass exhibiting strain-softening behavior is characterized by a transitional failure criterion f(sij, Z) and a plastic potential g(sij, Z), where Z is a softening parameter controlling the gradual transition from a peak failure criterion (or potential) to a residual one. In this paper, the rock mass is assumed to satisfy the linear Mohr–Coulomb criterion and the linear plastic potential. The softening parameter can be defined in different ways, but there has not been a commonly accepted one among researchers. The major principal plastic strain, ep1 , is employed as the softening parameter in this paper, because it is relatively simple and can be obtained easily from the results of uniaxial compression tests (Schematically represented in Fig. 1). Therefore, the failure criterion f and the plastic potential g can be formulated as follows f ¼ s 1 K p s 3 sc ¼ 0 8 1 2 p < s1 ðsc sc Þ1 c a with sc ¼ : s2 c
ð0pp1 paÞ ðp1 XaÞ
g ¼ s1 K c s3 ¼ 0.
Here, s1 and s3 are the major and minor principal stresses of rock mass. Kp and Kc are, respectively, the passive coefficient and the dilation factor of the rock mass, which are regarded as constants within the complete plastic region. sc is the compression strength, and transits gradually from s1c to s2c , according to the evolution of the major principal plastic strain ep1 . a is a shift point of the softening parameter that distinguishes the strain-softening region from the residual region. 2.2. The plane strain axial symmetry assumption Estimation of the support system required to stabilize a tunnel opening during excavation, especially in the vicinity -3
1p
E c2
K -d1p
v 1e
1e+
1
ð1Þ
(2)
c1
0
,
0
1e
d1p 1p
Fig. 1. Typical strain and stress behaviors of soft rock under uniaxial tests.
1
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P0
passive bolts
Re Ra (a)
tunnel face
plastic region elastic region
Lb
P0
Ra
Lz central line
plastic region
Pi
(b)
elastic region
passive bolts
Fig. 2. Schematic representation of (a) the cross-section and (b) the longitudinal profile in the conventional tunnelling with passive bolts reinforcement.
Pcri i ¼ sre ¼
2P0 s1c . Kp þ 1
(3)
dr discretized bolt node
(a)
shear spring Ks
cohesive slider cs grout
(b)
us
Ds
Fs dr Ks
bolt
t tansDs cs Ds
of the tunnel face, is essentially a four-dimensional problem. It not only concerns with three spatial dimensions but also one temporal dimension. The time effect, which corresponds to the advancing process of the tunnel face (or synonymously the unloading process of ground), is significant when a plastic region occurrs in the surrounding rock mass. The CCM, considering a long round-section tunnel excavated in the rock mass under a hydrostatic insitu stress, simplifies the three dimensions in space into one dimension in the radial direction via the plane strain axial symmetry assumption. This assumption is adopted in this paper. Furthermore, the paper assumes that the passive bolts are also mounted axial-symmetrically (i.e., pattern bolting) and the effect of each bolt can be averaged within its tributary area, so that the axial-symmetry assumption always holds for this problem. The schematic representation of the problem is illustrated in Fig. 2. In Fig. 2, Ra and Re are the radii of the tunnel opening and the plastic region; Lb is the length of the passive bolts; c and Lz are the bolting interval around the tunnel opening perimeter and the bolting space along the tunnel axis, respectively. P0 is the hydrostatic in-situ stress, and Pi is a fictitious inner pressure provided by the tunnel face and/or the shotcrete lining. The loading path, corresponding to the advancing of the tunnel face, refers to a monotonic decrease of the fictitious inner pressure from P0 (before excavation) to Pfin i (the final loading carried by the lining after the tunnel face advancing away). The radial stress at the elasto-plastic interface (r ¼ Re), sre, can be computed theoretically by Eq. (3) [13,14]. An important feature of the solution is that sre is a constant that only depends on the properties of rock mass itself and is independent of the position of the elasto-plastic interface. In other words, the rock mass will not experience a plastic region before the inner pressure falls below sre. Thus, it is synonymously named as the critical inner pressure, Pcri i . This position independent feature will help reduce computation effort significantly as presented below
(c)
t
Fig. 3. Schematic representation of (a) the spring-slider model, (b) shear force per unit length versus relative shear displacement, and (c) the bolt effective diameter.
3. The reinforcement mechanics of passive bolts in conventional tunnelling 3.1. The basic spring–slider model First, some pullout tests for passive bolts are reviewed briefly here [15,16], which could improve the understanding of the interaction relationship between the bolt and the rock mass. The failure of fully grouted bolts in pullout tests generally occurs at the bolt–grout interface (cementgrouted bolts) or at the grout–rock interface (resin-grouted bolts), since these interfaces are relatively weak and have only cohesion and frictional resistance. For fully frictional bolts, the failure can only occur at the bolt–rock interface. In some tests, while the bolts are mounted in a hard rock mass and subjected to high confinement pressure, shear failure of the grout itself could occur. These four failure modes presented above can be generalized into one spring–slider model, as illustrated in Fig. 3(a), since they are all cohesive and frictional shear failure in nature. The bolt itself can be discretized into a series of rigid-connected nodes along its axis direction (i.e., the radial direction in Fig. 2), where the elongation of the bolt itself is neglected since it is negligibly small compared
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to the deformation of rock mass. The shear behavior per unit length is jointly controlled by a shear spring that is characterized by its shear stiffness, Ks and a slider that is characterized by its cohesive strength, cs. The relationship between the shear force per unit length, Fs/dr and the relative shear displacement, Dus is illustrated in Fig. 3(b). Before decoupling, the shear force is mobilized by and proportional to the relative shear displacement. The maximum shear force per unit length is limited by the confinement stress, st, the friction coefficient in failure interface, tan fs, the cohesive strength in failure interface, cs, and the bolt effective diameter, Ds. After decoupling, the cohesive strength is lost, and the shear force per unit length falls to its residual value that is only controlled by st, tan fs, and Ds. Therefore, the Fs–Dus relationship, can be formulated as ( K s Dus dr ðK s Dus drpF max Þ s Fs ¼ (4-1) max , st tan fs pDs dr ðK s Dus dr4F s Þ ¼ ðst tan fs þ cs ÞpDs dr. F max s
(4-2)
Here, the bolt effective diameter Ds may equal the diameter of bolt itself, the diameter of grout hole, or the average of the former two, corresponding to the bolt–grout, the grout–rock or the inside–grout shear failure modes, respectively, as shown in Fig. 3(c). The properties involved in this model, Ks, tan fs, cs, and Ds, can be evaluated from conventional pullout tests (see details in Appendix). Dividing the shear force per unit length by the bolt effective perimeter, the shear stress around the bolt, ts, can be formulated as Eq. (4-3). Integrating the shear force per unit length along the bolt axis, the axial force of bolt can be obtained via Eq. (4-4). ( K Du s s ðK s Dus drpF max Þ Fs s pDs ts ¼ ¼ , (4-3) st tan fs ðK s Dus dr4F max Þ pDs dr s Z Fn ¼
F s dr.
(4-4)
3.2. The passive bolts in conventional tunnelling While applying the passive bolts to ground reinforcement in conventional tunnelling, Freeman first proposed the concepts of the neutral point, the pickup length and the anchor length in passive bolts, according to the in-site monitoring data [17]. At the neutral point r, the relative shear displacement and the shear force are zero, and the tensile axial force in the bolt reaches its peak value. The pickup length refers to the section from the near end (on the tunnel wall) to the neutral point, where the shear force picks up the load from the rock and drags the bolt towards the tunnel. The anchor length refers to the section from the neutral point to the far end (seated deep in the rock), where the shear force anchors the bolt to the rock. These concepts, which clearly outline the mechanics of passive bolts in a continuously deformed in-situ rock mass, are adopted in this paper. Then considering the static equilibrium condition of the bolt itself, the shear force mobilized in the pickup length should equal that mobilized in the anchor length. In addition, the maximum resultant axial force, F max n , cannot exceed the bolt’s load capability, Fyield. Z
Z
r
Lb
F s dr ¼
F s dr,
(5-1)
F s drpF yield .
(5-2)
r
0
F max ¼ n
Z
r 0
3.3. Equilibrium equations for rock mass Consider an infinitesimal volume in the radial direction as shown in Fig. 4(a). The rock mass is subjected to a radial stress sr, a tangential stress st and another axial force Fn provided by bolt. Supposing that the Fn can be spread evenly around its tributary area, the static equilibrium condition of the infinitesimal rock mass volume, as t
t d
Fn+dFn
Fn
r
r+dr
r
Fn rLz t dr
(b)
Fn+dFn (r+dr)Lz
r+dr b Fn
t (a)
dr
(c)
Fn+dFn dr
Fig. 4. Static equilibrium condition for the surrounding rock mass.
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illustrated in Fig. 4(b), can be formulated as Fn do sr r do Lz þ 2st dr Lz sin 2 roLz F n þ dF n ¼ sr þ dsr ðr þ drÞ do Lz . ðr þ drÞoLz
(6-1)
Meanwhile, considering the interaction at the bolt–rock interface as shown in Fig. 4c, the increment of the axial force, dFn, can be expressed as dF n ¼ ts pDs dr.
(6-2)
Associating these two equations, noticing that sin(do/2) approximately equals do/2, since do is an infinitesimal, the equilibrium equation can be deduced as dsr st sr þ N 0 ts ¼ dr r
with
N0 ¼
pDs . oLz
(6-3)
The constant N0 is named geometry coefficient in this paper, since it is totally determined by the geometrical properties of the bolts. In the absence of the bolts, merely setting N0 to zero, the above equation can be reduced into the equilibrium equation in a conventional form. Therefore, Eq. (6-3) can be regarded as a ‘‘uniform’’ equilibrium equation regardless of the presence or absence of the passive bolts. When applying Eq. (6-3) to the elastic region, where the stress state of rock mass should verify the hydrostatic insitu stress condition that the sum of sr and st equals 2P0, the equilibrium equation for elastic region can be formulated as dsr 2P0 2sr þ N 0 ts ¼ . (7) dr r When applying it to the plastic region, where the stress state of rock mass should verify the failure criterion as shown in Eq. (1), the equilibrium equation for the plastic region can be formulated as dsr ðK p 1Þsr þ sc þ N 0 ts ¼ . dr r
(8)
3.4. Displacement compatibility equations for rock mass 3.4.1. The elastic region Due to the plane strain axial symmetry assumption, the strain–displacement relationships for the rock mass can be simplified significantly as du u ¼ r ¼ t . (9) dr r According to Hook’s law, the tangential strain of the rock mass can be evaluated from its stress state, as formulated in Eq. (10), where E and n are the Young’s modulus and the Poisson ratio of the rock mass, respectively. st sr 2P0 n P0 P0 2P0 n n n n n t ¼ . (10) E E E E E E
629
Notice that only the strain caused by tunnel excavation is concerned, which means the initial strain due to in-situ stresses should be removed. Then, associating these two equations and considering the hydrostatic in-situ stress condition, the displacement compatibility equation for the elastic region can be formulated as Eq. (11). u ¼ rt ¼
P0 sr ð1 þ nÞr. E
(11)
3.4.2. The plastic region The loading path for this problem refers to a monotonic decrease of the fictitious inner pressure, corresponding to the advancing of the tunnel face. Consequently, the rates of all mechanical variables can be evaluated by their firstorder derivatives with respect to Pi. The incremental theory of plasticity [11,13] assumes that the total strain rate consists of both an elastic part and a plastic part, as shown in Eq. (12). The elastic part is controlled by Hooke’s law and the plastic part by the potential flow rule, as formulated by Eqs. (13) and (14), respectively. The relationship between the strain rate and the displacement velocity is simplified by virtue of axial symmetry and formulated by Eq. (15). _ r ¼ _er þ _pr ; _ er ¼
_ y ¼ _ey þ _py ,
1n n s_ r s_ y ; 2G 2G
_ pr ¼ l
qg ¼ l; qsr
_py ¼ l
_ey ¼
(12) 1n n s_ y s_ r , 2G 2G
qg ¼ lK c , qsy
(13) (14)
qu_ u_ ; _y ¼ . (15) qr r Here, g is the plastic potential that has been defined by Eq. (2). The rates of all mechanical variables (denoted by a dot mark) are referred as their first-order derivatives with respect to Pi. Then associating these four equations, eliminating the multiplier l, the displacement compatibility equation for the plastic region can be expressed as _ r ¼
ðnK c K c þ nÞ qu_ u_ ð1 n nK c Þ þ Kc ¼ s_ r s_ y . qr 2G 2G r
(16)
4. Two-dimensional finite-difference algorithm The equilibrium equations, the displacement compatibility equations and the bolt–rock interaction equations presented above can only be solved numerically. A twodimensional finite-difference algorithm (i.e., along the unloading path and along the radial direction) is employed in this paper, and the philosophy of the method is described in this section. All the variables describing the mechanical state of the surrounding rock mass have two indices: the first indicates a certain stage in the unloading path and the second indicates a certain position in the radial direction. Supposing that at former stage (say the (k1)th stage where Pi ¼ Pðk1Þ ), all the mechanical states i
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of the rock mass are known, the objective is to evaluate all the mechanical states at current stage (i.e., the kth stage where Pi ¼ PðkÞ i ) according to their known counterparts at the former stage, which includes the following three steps: stress evaluation, displacement evaluation and bolt response evaluation. After one iteration finishes, these known mechanical states at the current stage can be used to evaluate the mechanical states at next stage (i.e. the (k+1)th stage where Pi ¼ Piðk1Þ ), following the same three steps. This kind of iteration is repeated until the final stage where Pi ¼ Pfin i . 4.1. Stress evaluation
unloading path
The equilibrium equations (7) and (8) are solved by the fourth-order Runge–Kutta method [18], with the algorithm diagram presented in Fig. 5. At the current stage, the radial stress at the tunnel wall sr(k, Ra) is known and equals PðkÞ i , which serves as the boundary condition of the equilibrium equations. Then the radial and tangential stresses at each sequential calculation point are evaluated according to Eq. (8) and the failure criterion. If the radial stress increases up to the critical inner pressure Pcri i , record the position as the radius of the elasto-plastic interface Re, then go on evaluating the stress state at each sequential calculation point according to Eq. (7) and hydrostatic insitu stress condition. (k-1)th stage Pi=Pi(k-1)
The bolt–rock interaction shear stress and the transitional strength of rock mass at the former stage are required during this step, and the radial and tangential stresses at the current stage can be determined after the stress evaluation process. 4.2. Displacement evaluation The algorithm diagram for the radial displacement evaluation is presented in Fig. 6. For the elastic region, the radial displacement of the rock mass at the current stage can be evaluated directly by the radial stress of the rock mass at the current stage, according to Eq. (11). For the plastic region, the radial and tangential stress rates s_ r ðk; rÞ and s_ t ðk; rÞ should be first evaluated by their firstorder difference with respect to Pi, as shown below. _ rÞ ¼ sðk;
sðk; rÞ sðk 1; rÞ ðrpRe Þ. dPi
Similarly, the deformation rate at the elasto-plastic inter_ Re Þ, which serves as the boundary condition of the face uðk; compatibility equation, can also be obtained by its firstorder difference with respect to Pi. Then the fourth-order Runge–Kutta method is utilized again to evaluate the deformation rate at each sequential calculation point (inward radial direction) according to the compatibility equation (16). Finally, the displacement at the current stage can be obtained by accumulating the displacement
s(k-1, r) and c(k-1, r) are known
known point r=Ra radial direction r=Re unevaluated point r=Pi(K) r=Picri kth stage Pi=Pi(k) Elastic region Plastic region Controlled by Eq. (7) and Controlled by Eqs. (8) and hydrostatic stress condition ailure criterion r(k, r) and t(k, r) can be determined
Then
unloading path
Fig. 5. The algorithm diagram of stresses evaluation.
(k-1)th stage Pi=Pi(k-1)
kth stage Pi=Pi(k)
Then
(17)
u(k-1, r), r(k-1, r) and t(k-1, r) are known r(k, r) and t(k, r) are known r=Re inward radial direction u=uRe plastic region controlled by Eq. (16) and accumulated via Eq. (18)
known point unevaluated point
elastic region controlled by Eq. (11)
u(k, r), as well as c(k, r) can be determined
Fig. 6. The algorithm diagram of displacement evaluation.
ARTICLE IN PRESS
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631
u(j, r) is known • • •
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Z. Guan et al. / International Journal of Rock Mechanics & Mining Sciences 44 (2007) 625–636
u(k,r) is known, and umob(k,r)=u(k,r)-u(j,r)
neutral point (or length) pickup and anchor length
pickup length anchor length r= neutral point
kth stage
pickup length
a) before bolt yielding
anchor length
r=1 r=2 neutral length
b) after bolt yielding
s(k,r), as well as Fn(k,r) can be determined
Then
Fig. 7. The algorithm diagram of bolt response evaluation.
increment at the current stage to its counterpart at the former stage. _ rÞ dPi uðk; rÞ ¼ uðk 1; rÞ þ uðk;
ðrpRe Þ.
(18)
After the displacement evaluation, the major principle plastic strain pt at the current stage, which serves as the softening parameter in this paper, can be evaluated by Eq. (19). Then the transitional strength sc at the current stage can also be computed via Eq. (1). pt ðk; rÞ ¼ t ðk; rÞ te ðk; rÞ uðk; rÞ uðk; Re Þ ¼ r Re
ðrpRe Þ.
ð19Þ
The displacement and the stresses at the former stage, as well as the stresses at the current stage, are required during this step. Then the displacement and the transitional strength of rock mass at the current stage can be determined after the displacement evaluation process.
point is found. If the bolt load-capability condition Eq. (5-2) is violated, the bolt yields and a neutral length (rather than a neutral point) will occur in the bolt. A similar adjustment process is repeated to find the proper position of the neutral length. The process of finding the proper position of the neutral point (or length) is, actually, not so formidable as it seems to be, because the position of the neutral point (or length) at the current stage is always near the one at the former stage. Generally, only two or three adjustments are required to find the proper position of the neutral point (or length) for each stage. The displacements at the current and the bolt installation stages are required during this step, and the proper bolt responses (including the bolt–rock interaction shear stress and the axial force of bolt) can be determined after the bolt response evaluation process. After these three steps, all the mechanical states at the current stage are known, which can be used to evaluate their counterparts at next stage (i.e., the (k+1)th stage where Pi ¼ Pðkþ1Þ ). i
4.3. Bolt response evaluation Supposing that the bolts are mounted at jth stage where Pi ¼ P(j) i , the algorithm diagram for the bolt response evaluation is illustrated in Fig. 7. The mobilizing displacement, umob, which mobilizes the response of passive bolts, should be first computed via Eq. (20). umob ðk; rÞ ¼ uðk; rÞ uðj; rÞ ðrpLb Þ.
ðrpLb Þ.
The proposed method has been programmed in the VC++ development environment, the verification of the proposed method and some discussions are expatiated in this section.
(20)
Then, supposing an arbitrary position for the neutral point, where umob(k, r) denotes the mobilizing displacement at the neutral point, the relative shear displacement, Dus, at the current stage can be computed as follows Dus ðk; rÞ ¼ umob ðk; rÞ umob ðk; rÞ
5. Verification and application
(21)
Using this supposed Dus, the bolt responses, ts (k, r) and Fn (k, r), can then be evaluated via Eq. (4). If the bolt self-equilibrium condition Eq. (5-1) is not satisfied, adjust the position of neutral point and recalculate the relative shear displacement and the bolt responses via Eqs. (21) and (4), until a proper position of neutral
5.1. An illustrative case study An illustrative case study is studied in this section to demonstrate the reinforcement mechanics of passive bolts in conventional tunnelling. Suppose that a round-section tunnel with a design radius of 5 m is excavated under a hydrostatic in-situ stress condition of 8 MPa (about 350–400 m rock mass covering above). The properties of the rock mass and the passive bolts employed in this case are listed in Table 1. The bolting pattern and some rock mass properties are referenced to the Japanese standard for mountain tunnelling [19]. The passive bolts are supposed to
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632
Table 1 The properties of rock mass and rock bolt employed in a verification case (by proposed method) Rock mass Properties
E (Pa) 1.0e9
n 0.25
Kp 3.0
sc1 (Pa) 5.0e6
s2c (Pa) 3.0e6
Kc 1.33
a 0.0058
Ra (m) 5.0
P0 (Pa) 8.0e6
Rock bolt Properties
E (Pa) 2.1e11
Fyield (N) 3.0e5
Ab (mm2) 314
Ks (Pa) 2.5e7
cs (Pa) 1.0e6
tan fs 0.84
Ds (m) 0.04
o 0.224
Lb (m) 3.0
2.0
2.0
s(MPa)
/P0
1.0
t
1.5 1.0
7
8
9
r (m) 6
7
8
pickup length
anchor length
-2.0
r (m) 6
0.0 -1.0
r
0.5 0.0
Lz (m) 1.0
10
300
0.5
with bolt (proposed method) with bolt (FLAC3D )
1.5 2.0
with bolt (proposed method) with bolt (FLAC3D ) without bolt (proposed method)
Fn(kN)
u / Ra (%)
u 1.0
200
100
without bolt (FLAC3D )
2.5 (a)
0 (b)
6
7
8
r (m)
Fig. 8. Ground and bolt responses at Pi ¼ 1.0 MPa stage: (a) distributions of sr, st and u along the radial direction, and (b) distributions of ts and Fn along the bolt length.
be mounted at the stage where PiðjÞ ¼ 2:75 MPa, which just equals the critical inner pressure for the rock mass. The ground responses and the bolt responses at the PðkÞ i ¼ 1; 0:3 and 0 MPa stages are depicted in Figs. 8–10 (represented by solid lines). To highlight the reinforcement effect of passive bolts, the ground responses at the same stage but without passive bolts reinforcement are also depicted in these figures (represented by dashed lines). The passive bolts exert their effect gradually with the displacement release of rock mass. At the PðkÞ i ¼ 1:0 MPa stage, the reinforcement effect is not significant, and the passive bolts remain coupled along the whole length. At the PðkÞ i ¼ 0:3 MPa stage, the passive bolts help constrain the range of plastic region (the convex point at the st curve) from 7.5 to 7.2 m and the convergence of tunnel opening from 1.93% to 1.70%, meanwhile, decoupling occurs at the near-end section of the bolt. At the Pi ¼ 0 MPa stage (i.e., no lining is supported after the tunnel face advanced away), the ground and the passive bolts carry the released in-situ stresses fully. On this occasion, the passive bolts exert their load capability fully to help constrain the range of plastic region from 8.3 to 7.8 m and the convergence of tunnel opening from 2.46%
to 2.12%, meanwhile, decoupling and yielding occur on the near-end section and the middle section of the bolt. Regarding the constrained displacement at the tunnel wall, the reinforcement effect of the passive bolts on this occasion is equivalent to providing a fictitious inner pressure of 0.2 MPa. 5.2. Verification by numerical simulations The illustrative case is identically studied by numerical simulations (codes: FLAC3D) to verify the validity of the proposed method. The model itself is very simple in geometry, which consists of 3696 rock mass elements and 5778 rock mass gridpoints, as well as 150 cable elements and 155 cable nodes. However, the strain-softening constitutive laws in FLAC3D are characterized by six parameters: bulk modulus K, shear modulus G, friction angle f, cohesion c, dilation angle c and softening parameter Z. Thereinto, the values of f, c and c can vary with the development of the softening parameter Z (in tabulated form). It is obvious that the former five parameters can be evaluated directly from their counterparts employed in the proposed method, via following
ARTICLE IN PRESS Z. Guan et al. / International Journal of Rock Mechanics & Mining Sciences 44 (2007) 625–636
4.0
2.0
s(MPa)
/P0
2.0
t
1.5 1.0 0.5
6
7
8
9
10
u
r (m) 6
7
pickup length
8
anchor length
with bolt (proposed method) with bolt (FLAC3D )
300 Fn(kN)
u / Ra (%)
-1.0
400
1.0 1.5
with bolt (proposed method)
2.0
with bolt (FLAC3D ) without bolt (proposed method) without bolt
0.0
-4.0
0.5
2.5
1.0
-2.0
r r (m)
0.0
633
200 100
(FLAC3D )
0
6
7
(b)
(a)
8
r (m)
Fig. 9. Ground and bolt responses at Pi ¼ 0.3 MPa stage: (a) distributions of sr, st and u along the radial direction, and (b) distributions of ts and Fn along the bolt length.
4.0
2.0 t
2.0 s(MPa)
1.0
0.0
r 6
7
8
0.5 u / Ra (%)
1.0 0.0 -1.0
r (m) 6
-4.0
r (m)
9
10
400
u
1.5 with bolt (proposed method)
anchor length
with bolt (proposed method) with bolt (FLAC3D )
200
0 (b)
(a)
pickup length
100
with bolt (FLAC3D ) without bolt (proposed method) without bolt (FLAC3D )
2.5
8
300
1.0
2.0
7 neutral length
-2.0
0.5
Fn(kN)
/P0
1.5
6
7
8
r (m)
Fig. 10. Ground and bolt responses at Pi ¼ 0 MPa stage: (a) distributions of sr, st and u along the radial direction, and (b) distributions of ts and Fn along the bolt length.
relations K¼
E ; 3ð1 2nÞ
Kp ¼
E , 2ð1 þ nÞ
(22-1)
pffiffiffiffiffiffi 1 þ sin f 1 þ sin c ; sc ¼ 2c K p ; K c ¼ . 1 sin f 1 sin c
(22-2)
G¼
However, the softening parameter in FLAC3D is defined as follows [7] qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 dZFlac ¼ pffiffiffi ðdp1 dpm Þ2 þ ðdpm Þ2 þ ðdp3 dpm Þ2 2 dp þ dp3 . (23-1) with dpm ¼ 1 3
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634
Table 2 The properties of rock mass employed in the identical case (by numerical simulations) G (Pa) 4.0e8
f (1) 30.0
c1 (Pa) 1.4e6
c2 (Pa) 8.7e5
Therefore, the shift point of the softening parameter in FLAC3D can be obtained from the parameters used in the proposed method by a qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi (23-2) ZsFlac ¼ pffiffiffi K 2c þ K c þ 1. 3
( K Flac s
¼
Ks
ðK s Dus drpF max Þ s
st tan fs pDs Dus
ðK s Dus dr4F max Þ s
.
(23-3)
The results from the numerical simulations are also depicted in Figs. 8–10, as denoted by cross or dot marks. On the occasion without bolt installation, the ground responses computed by the numerical simulations and by the proposed method fit each other exactly, which has been proved by former researchers [13,14]. On the occasion with bolt installation, the ground and bolt response computed by two methods also agree with each other well, when no decoupling or a little decoupling occurs along the bolt (see Figs. 8 and 9). However, there exists a quantitative discrepancy between the two methods in depicting the bolt responses when decoupling and yielding occurs (see Fig. 10). The spring–slider model employed in FLAC3D (although the shear stiffness is compulsively modified via Eq. (23-3) after decoupling) cannot be exactly equivalent to the one employed in proposed method when a large section of decoupling occurs, which may account for the quantitative discrepancy shown in Fig. 10.
Ra (m) 5.0
P0 (Pa) 8.0e6
0.5 BRC 0.4
D'
C'
100
B' A
Pi /P0
1.0
300 200
M'
0.3
The parameters employed in the numerical simulations are summarized in Table 2, corresponding to their counterparts employed in the proposed method. On the other hand, the spring–slider model used in the proposed method is also adopted in FLAC3D, thus the properties of rock bolt (except Ks) employed in the numerical simulations are the same with those tabulated in the second line of Table 1. By default, however, FLAC3D assumes that the shear force will remain at its peak value after decoupling. Thus, it cannot directly simulate the falldown relationship between Fs and Dus as illustrated in Fig. 3(b). Fortunately, FLAC3D provides a user-defined programming language FISH to help implement the falldown spring–slider model indirectly. Checking the coupling state of each cable element at the beginning of each calculation cycle, the shear stiffness for this cable element is equal to its initial value before decoupling, or is compulsively modified via Eq. (23-3) after decoupling. Then the modified shear stiffness is used in the next cycling of FLAC3D.
ZsFlac 0.0068
c (1) 8.2
1.5 ua/Ra (%)
2.0
Fmax n (kN)
K (Pa) 6.7e8
Rock mass Properties
2.5
0.2 SCC B
0.1
conventional GRC M
reinforced GRC C
0.0 0.0
0.5
1.0 1.5 ua/Ra (%)
2.0
2.5
Fig. 11. The reinforced ground reaction curve and bolt reaction curve.
For the most part, the validity of the proposed method is proved theoretically via the illustrative case study and the numerical simulations. The proposed method can rationally elucidate the reinforcement mechanics of passive bolts in conventional tunnelling, and improve the support system design effectively. 5.3. Ground reinforcement design The ground reinforcement and the inner support should be considered jointly and comprehensively in the support system design in conventional tunnelling. The CCM just provides such a tool to compromise the complex nature with a general albeit simplified approach. General reviews and applications of the CCM have been summarized by many researchers [1,2]. Using the proposed method, the GRC with passive bolts reinforcement (name reinforced GRC hereafter) and the bolt reaction curve (named BRC hereafter) can be constructed by recording the displacement released at tunnel wall ua and the maximum axial force in the bolt F max at each unloading stage. n Taking the illustrative case for example, the reinforced GRC and the BRC are depicted in Fig. 11. For comparison, the conventional GRC without passive bolts reinforcement is also depicted in the figure. Point A denotes the stage when the bolts are mounted. Points B (B0 ), C (C0 ) and D (D0 ) denote the stages where Pi ¼ 1.0, 0.3 and 0 MPa, and the ground and bolt responses at these stages
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2.5
2.5 Lb N0
2.4
umax a /Ra (%)
umax a /Ra (%)
2.3
2.2 2.1 2.0
2.2 2.1 2.0
the standard case at Pi=0MPa stage
1.9
cs tan s Ks Fyield
2.4
Pi(j)
2.3
the standard case at Pi=0MPa stage
1.9 1.8
1.8 0 (a)
635
50
100 % variables
150
0
200
50
(b)
100 % variables
150
200
Fig. 12. The influence of the bolt’s (a) geometrical and time properties, and (b) mechanical properties on the reinforcement effect.
have been portrayed in Figs. 8–10. The SCC describes the relationship between the inner pressure carried by the inner support (including shotcrete lining, steel sets, active bolts, etc.) and the radial displacement at the tunnel wall. Intersection point M between the reinforced GRC and the SCC, together with its corresponding point M0 in the BRC, indicates the final states of the reinforced rock mass after the tunnel face advancing away. To highlight the influence of the bolt properties on the reinforcement effect, some derivative cases are studied in the following. Taking the illustrative case as a standard case, varying every single parameter (including the geometrical properties, N0 and Lb, the time property, P(j) i , and the mechanical properties, Ks, cs, tan fs, and Fyield) from 0% to 200% of its initial value, the relative significance of the bolt properties on the reinforcement effect is highlighted in Fig. 12. The tunnel convergence at the Pi ¼ 0 MPa stage, umax a , is selected as the estimation index (Y-coordinate) in this figure. A surplus length of rock bolt hardly can take effect on the ground reinforcement, because the bolt section embedded in the elastic region of rock mass cannot help constrain the elastic displacement release (under the scope of continuous deformation media). Therefore, it is unnecessary to extend the bolt beyond the range of plastic region too much. For the same reason, the bolt nearly can take its effect, if the rock mass can remain elastic (i.e., cri Pfin i 4Pi ). On the contrary, increasing the bolt installation density (i.e. increasing the bolt geometrical coefficient N0 by decreasing the perimeter interval o and the longitudinal spacing Lb) can always help reinforce and stabilize the surrounding rock mass. The mechanical properties of the bolt (except Fyield) are mainly controlled by the grout condition. Among them, the shear stiffness Ks influences the reinforcement effect most significantly. Generally, a higher value of Ks is preferred, which means the bolt is more sensitive to the relative shear displacement, and can exert its effect at early stage. The friction angle and the cohesive strength at the interface,
which control the maximum shear force per unit bolt length can provide, also influence the reinforcement effect to some extend, if decoupling occurs along the bolt. Meanwhile, the reinforcement effect is also limited by the loading capability of the bolt itself, which accounts for the reason why an excessively strong grout condition cannot help improve the reinforcement effectively. 6. Conclusions According to the pullout tests for passive bolts, a spring–slider model is adopted in this paper to account for the interaction between the bolt and the rock mass. Then the additional force provided by the passive bolt is incorporated into the equilibrium equation for rock mass. Then based on the incremental theory of plasticity and the plane strain axial symmetry assumption, a elasto-plastic analyses method for passive bolt reinforced ground in conventional tunnelling is proposed in this paper. The reinforcement mechanics of passive bolts in conventional tunnelling is demonstrated qualitatively and quantitatively via an illustrative case study. The validity of the proposed method is verified by numerical simulations via the identical case study. The reinforced GRC and the BRC can be constructed easily by using the proposed method, which could help the support system design in the conventional tunnelling. In addition, the proposed method also permits incorporating other features about the bolt–rock interaction model (e.g., a gradual fall-down model) or other failure criterions (e.g., Hoek–Brown criterion) of the rock mass into further analyses. To highlight the influence of the bolt properties on the reinforcement effect, some derivative cases are studied. The bolt section embedded in the elastic region of rock mass hardly can take effect on the ground reinforcement (under the scope of continuous deformation media). On the contrary, increasing the bolt installation density can always help reinforce and stabilize the surrounding rock mass. The grout conditions, as well as the loading capability of bolt
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D28, Lb=150mm, t =3.2MPa
200
Axial load (kN)
160
Ks =
Avr.
Fnmax 0.12 × e 6 = = 340 MPa el π Ds Δ us 3.14 × 0.028 × 0.004
120
tan φs =
80
cs =
40
Fnmax
π Ds Lb σ t
=
0.07 × e6 = 1.66 3.14 × 0.028 × 0.15 × 3.2 × e 6
Fnmax − Fnres (0.12 − 0.07) × e 6 = = 3.79 MPa 3.14 × 0.028 × 0.15 π Ds L b
0 0
5
10
15
20
25
30
Axial displacement (mm) Fig. A1. An representative axial load versus axial displacement curve (after Moosavi, 2005).
itself, influence the reinforcement effect significantly and comprehensively. Among them, a higher value of Ks is generally preferred, which means the bolt is more sensitive to the relative shear displacement, and can exert its effect at early stage. Appendix A. : The properties evaluation for passive bolts This appendix explains how to evaluate the properties of passive bolts involved in the proposed method (see second line of Table 1). The axial load versus axial displacement curve can always be obtained from conventional pullout tests (either in the laboratory or in the field). Fig. A1 shows a representative curve from the fully grouted bolts’ pullout tests [20]. The embedded bolt length Lb is 0.15 m; confinement pressure st is 3.2 MPa; and the grout cement has a uniaxial compression strength of 42 MPa. The ribbed bolt has a diameter of 28 mm, and the effective diameter Ds is set to 28 mm since the failure occurs at bolt–grout interface. The curve can be simplified into a rigid fall-down model as denoted by the dashed line in Fig. A1, where the maximum and the residual axial force are 120 and 70 kN, respectively; the elastic proportional limit of relative shear displacement is set to 4 mm. Then the shear stiffness Ks, cohesive strength cs, and the friction coefficient tan f can be evaluated from these results of pullout tests, as shown in the right side of Fig. A1. The other properties of rock bolt can be obtained directly from the manufacture’s specification. References [1] Carranza-Torres C, Fairhurst C. Application of convergence–confinement method of tunnel design to rock masses that satisfy the Hoek–Brown failure criterion. Tunnelling Underground Space Technol 2000;15(2):187–213. [2] Oreste P. Analysis of structural interaction in tunnels using the convergence–confinement approach. Tunnelling Underground Space Technol 2003;18(4):347–63.
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