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A study of rock bolting design in soft rock
Paper 2B 14 — SINOROCK2004 Symposium Int. J. Rock Mech. Min. Sci. Vol. 41, No. 3, CD-ROM, © 2004 Elsevier Ltd.
A STUDY OF ROCK BOLTING DESIGN IN SOFT ROCK Y. Cai¹, Y.J. Jiang ², T. Esaki¹ ¹) Institute of Environmental Sys., Kyushu University, Japan [email protected] ²) Department of Civil Engineering, Nagasaki University, Japan [email protected]
Abstract: This paper presents an analytical method for rock bolting design in soft rock. The mechanical behavio ur of rock bolt ing system and rock bolt itself are discussed respectively. Coupling mechanism of rock bolt and rock mass are analysed from the viewpoint of displacement. A simple method is suggested for the rock bolt design in tunnelling. According to the analysis, the initiated force of the rock bolt is related with the displacement of rock mass. Case simulations confirm the previous findings that a bolt in-situ has a pick-up length, an anchor length and at least one neutral point. The theoretical prediction of single rock bolt agrees with the measured data. The position of neutral point is not only related with the length of the rock bolt and the internal radius of tunnel, but also strongly influenced by the properties of the rock mass. The neutral point and maximum axial load in bolt tends to be constant when the anchor length of bolt is long enough, which means that increasing the length of rock bolt does not improve its effect dramatically under certain conditions. Keywords: Rock bolt, model, design, tunnel, soft rock, rock mass
INTRODUCTION The underground excavation often encounters soft rock mass in Japan. Rock bolting is considered as an effective and economical means of supporting in varies conditions. Unfortunately, the coupling mechanism is not very clearly at present. The design of rock bolt in tunnelling or other excavations is still empirical, and there are few methods to evaluate the bolting effect in most case. The researches of rock bolt include studying the behavior of rock bolt system and modelling the mechanism of the single rock bolt. The monitoring in field is also common in practice. Indraratna and Kaiser (1990) established the analytical model for the design of grouted rock bolts according to the elasto-plastic constitutive law. The rock bolt and rock mass is considered as a system and the isotropic behavior is assumed. However, the position of the neutral point, at which the shear stress on rock bolt turns to be zero and axial force of rock bolt reaches maximum, is very important to the theoretical analysis. It is widely accepted that the displacement of the rock bolt is the same as the rock mass at neutral point. Freeman’s monitoring result is widely cited for the concept of the neutral point. However, the position of neutral point is not easy to get (Li and Stillborg, 1999). Considering the coupling behavior of the rock bolt system, the
bolt works compatibly with the surrounding rock mass, and the neutral point is strongly influenced by the deformation patterns of the ground. Obviously, the physical properties of the rock mass influence the mechanical behavior of the rock bolt. The study about the single rock bolt often focus on the modelling of the interface between the rock bolt and the rock mass. The pullout test is usually taken as a verification way for the modelling. However, the pullout test itself remains unclear because of the anti-pressure on specimen and the non-uniform shear stress along the rock bolt. This paper is to establish an analytical method of rock bolt ing system considering the interaction behavior of the rock bolt together.
ROCK BOLT SYSTEM AROUND CIRCULAR TUNNEL 2.1 Soft rock mass behavior without rock bolt Since the mechanical properties of the rock mass influences the behavior of the rock bolting system significantly, the behavior of rock mass is discussed separately. A strain-soften model has been suggested to describe the behavior of the soft
1
Paper 2B 14 — SINOROCK2004 Symposium Int. J. Rock Mech. Min. Sci. Vol. 41, No. 3, CD-ROM, © 2004 Elsevier Ltd.
rock mass by the authors. The constitutive law of strain soften law is scratched in Figure 1,
σt = σc* + K p σ r
σ1
(σ 1−σ3) σc
σ3 E'
(2) Accordingly, the soft rock mass around tunnel may divided into three zones such as the plastic flow zone, the strain-soften zone and the elastic zone, as shown in Figure 2. Their corresponding radiuses are named as Rf and Re respectively.
σ3
σ1 σ3 =constant
σ*c
E
ε ef εp ε =αε e 1
ε
1
1
Equation (2), where öis the internal friction angle of rock mass.
1
1
(a) Simplified stress-strain linear relationship
σ σ σ
Y
P(r,θ) r
c
*
εe
+
ε
e 1
1
elastic zone
1
εp
ra Re
1
ε
ε
p
1
ε
e1
X
Rf
h
εp'
ε
Plastic flow zone
Figure 2. Tunnel excavation in soft rock mass
εp
ε p'
3
-
Strain soften zone
1
f
3
3
The displacement formulas of each zone can be obtained according to different constitutive law. By the way, it should be pointed out that the strain soften constitutive law describes the behavior of homogenous material, it also can be referenced to describe the rock bolting system together considering the anisotropic characteristic of rock bolt.
(b) Major and minor principal strains p 1
αε
e
1
+
ε
1
H
εp' 1
F
2.2 Modelling the rock bolt system
vp -
v
p'
(c) Volumetric and major principal strain
Figure 1. Stress-strain relation and dilatancy behavior In the figure 1, α is the brittleness rate of the rock mass; σc is the axial strength of rock mass, and σc * represents the strength of rock mass in strain soften status. According to Mohr-Coulomb law, there is Equation (1),
σt = σc + K p σr
(1a)
Kp = (1 + sin φ ) /(1 − sin φ )
(1b) where σr and σt is the stress of rock bolting section in radius and tangential direction respectively. In the plastic flow zone, Equation (1) changes to
The rock bolt system model shows the “mean performance” for both rock mass behavior and the bolt performance. No decoupling is considered in all the models at present. Based on the assumption of round tunnel profile, homogeneous rock mass and hydrostatic stress conditions, the theoretical formulas of rock bolting system have been established together considering the anisotropic characteristic of rock bolt ( Jiang and Esaki, 1997). Since the direction of the shear stress at the interface between rock bolt and rock mass, and the position of the neutral point is critical to the interaction model of rock bolting system. Their relatively position are shown in Figure 3, where L is the length of rock bolt; Lt and Lz are the distance between rock bolts in circular direction and z direction respectively. Analysing the composite element of the rock bolt and the rock mass, as shown in Figure 4. The coupling equation is written as Equation (3).
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Paper 2B 14 — SINOROCK2004 Symposium Int. J. Rock Mech. Min. Sci. Vol. 41, No. 3, CD-ROM, © 2004 Elsevier Ltd.
dF = 2πrb λσ t dr
Case1
Case 6
(3) where λ is friction coefficient between the rock bolt and the rock mass. The constitutive equation of rock bolting section is listed as Equation (4), dσ r σ r − (1 − β )σ t + =0 dr r β = 2πrb λra / Lz Lt
0
Case 2 0
(4)
where ra is the radius of tunnel; rb is the radius of rock bolt.
ra ρ L1 Rf R e
ra ρ R f L1 R e
Case 3
Re
Case 4
LT
Po Plastic flow zone Soften zone Elastic zone
LT ϕ
ra
r
ρ R f L1 Re
Figure 3. Relationship among plastic radius and bolt length and neutral point
σt
ϕ
θ
r
b
dθ
dr
dF
σr
σt
x
Figure 4. Equilibrium consideration for boltground interaction Considering the position of the neutral point and the rock bolt, nine cases are arranged for analysis. The scratch of the rock bolting section is shown in Figure 5. The stress and strain distributions of the rock bolting system have been discussed. Since the formula is very complicated, they will not present in this paper. The formulas have been listed in the paper of Jiang and Esaki. (1997). The stress of the rock mass can be obtained according to equation (4) and the corresponding boundary conditions of the nine cases respectively.
0
0 ra R f ρ L1
Lz
ρ L1
r a ρ L1
Re
ra ρ Re
L1
r a Re ρ
L1
Case 8
0
0
r a R f Re
Case 7
Po symmetric
0
Case 9 ra ρ Rf Re L1
0 rock bolt Elastic zone
Case 5 0 ra Rf ρ R e L1
strain soften zone Plastic flow zone
Figure 5. Analytical cases of the bolt-ground interaction
INTERACTION MODEL OF SINGLE ROCK BOLT AND MATRIX The interaction between the rock mass and the grouted bolts involves complex mechanics and today there is little work presented on the subject which can explain in theory the satisfactory results from using grouted rock bolts. A coupling model is suggested to describe the behavior of single rock bolt based on Shear-lag model (SLM) or fiberloading theory here. The SLM is originally developed by Cox (1952) and has been widely used by material scientists and structural geologists as powerful analytical method. The concept figure of suggested model is scratched in Figure 5.
u
Figure 6. Coupling behavior of reinforcement and rock mass
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Paper 2B 14 — SINOROCK2004 Symposium Int. J. Rock Mech. Min. Sci. Vol. 41, No. 3, CD-ROM, © 2004 Elsevier Ltd.
According to the balance of the infinitesimal element of reinforcement, surrounding matrix and matrix together with the reinforcement in cylinder coordinate system, the basic constitutive law is expressed as Equation (5). The assumption of SLM is the equation (6). dP( x) = − 2πrbτ b dx
∂σm ( r , x) ∂τ( r, x ) τ( r , x ) + + =0 ∂x ∂r r R dP ( x ) d + 2π ∫ r σm ( r , x ) dr = 0 r dx dx b
dP ( x) = H ( ub − u m ) dx
(5a)
APPLICATION OF THE MODEL 4.1 Axial force of the rock bolt The coupling behavior of rock bolt and rock mass is able to describe by equation (5) and the boundary conditions. Considering the rock bolting effect, the strain of rock mass at the edge of the influence radius R is expressed as equation (9)
εm = εini − ∆εm (9a)
(5b)
∆ε m = P /(SEm )
(9b) (5c) (6)
where, P(x) is the pullout force at the position of x; R is the influence radius; σn is the normal stress perpendicular to reinforcement; τ(r,x) represents the shear stress at (r,x), as shown in Figure 6; u b and u m are the displacement of reinforcement and matrix at the edge of influence radius R; H is material parameter. The decoupling influences the bolting effect dramatically. Different types of rock bolt have different decoupling behaviors. Only grouted rock bolt is discussed here. Shear strength at interface is made up of three parts such as adhesion, inter lock and friction in axial direction. They are lost in sequence as the compatible of deformation is lost along the interface. After decoupling, the shear stress at interface becomes the residual strength at slipping part. Malvar (1992) revealed that the confining pressure influences the interface strength dramatically. The decoupling failure may occurs at the bolt-grout interface, in the grout medium, at the grout-rock interface or in rock mass. It depends on which one is the weakest. Shear strength can be calculated with Equation (7).
τm = c + σ nb tanφ * (7) where φ* and c is the friction angle and cohesive of interface, which can be tested by direct shear tests or pullout test; σnb is the normal stress perpendicular to rock bolt.
P( x ) d 2 P( x ) = H − εm 2 dx E b Ab
(9c)
where, εini is the rock mass strain without bolt. Em is the deformation of rock mass; S is the influencing area of single rock bolt. Since the displacement function of rock mass around the circular tunnel is relatively complicated, it is not easy to get the theoretical expression of the stress distribution or the position of neutral point. Numerical method is favorable here. The neutral point in rock bolt installed around tunnel is able to get by Equation (7) together with corresponding boundary conditions. Not considering the effect of external fixture, the boundary condition of rock bolt can be written as P(0)=0 and P(L)=0. The filed measuring of axial force was performed at Holland Zaka Tunnel in Nagasaki. The tunnel is situated at a depth of 18.35m. The rock mass around tunnel is classified as DI, which belongs to the soft rock class. The position of the measured rock bolts is shown in Figure 7. According to the investigations, the parameters of rock mass and rock bolt are listed in Table 1: Table 1 Parameters of rock bolt and rock mass Radius of Tunnel 4.75m Hydraulic water pressure, p o Axial strength of rock mass Deformation modulus of rock mass Poisson’s ratio of rock mass Length of rock bolt Young’s modulus of rock bolt Radius of rock bolt, rb Distance between rock bolt Shear strength on interface
0.38MPa 0.5MPa 1.0GPa 0.25 4.0m 210GPa 12.7mm 1.2m*1.4m 0.35Mpa
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Paper 2B 14 — SINOROCK2004 Symposium Int. J. Rock Mech. Min. Sci. Vol. 41, No. 3, CD-ROM, © 2004 Elsevier Ltd. RB1
RB2
neutral point changes from 0.74m to 0.38m towards the tunnel wall, as listed in Table 2. When the length of rock bolt changes from 0.80m to 5.0m, the neutral point changes from 0.37m to 2.20m, as listed in Table 3.
RB3
Table 2 Neutral point and deformation modulus of rock mass Em (GPa) 0.50 1.00 2.00 3.00 5.00 Proposed model 0.74 0.62 0.5 0.46 0.38 Tao and Chen 1.80 1.80 1.80 1.80 1.80
RB5
RB4
5
4
3
2
1
From tunnel wall, 0.5m, 1.5m, 2.5m, 3.5m, 4.0m
Figure 7. Test arrangement of rock bolt in Holland Zaka tunnel (No.68+28, Section C) The calculated results and measured data are presented in Figure 8. The theoretical predictions agree with the test results. The grouted rock bolt installed around tunnel in soft rock mass can be divided into three parts such as the pick up part, the neutral point and the anchor part. The neutral point in this case is 1.26m from the end of rock bolt. Since the physical properties of the rock mass is not easy to get in practice, the discussion is needed to find out the inclination of the neutral point. Tao and Chen suggested the equation (10) to calculate the neutral point of grouted rock bolt. ρ = L /(ln[1 + ( L / ra )] L = (40 rb ~ 60 rb )
(10)
Axial Load (kN)
where L is the length of rock bolt; ρ is radius of neutral point. Proposed model Test data
40 30 20 10 0 0
1 2 3 Length of rock bolt (m )
4
Figure 8. Theoretical prediction and measured data When the deformation modulus of the rock mass ranges from 0.50GPa to 5.0GPa, and the other parameters are the same as those in Table 1, the
Table 3 Neutral point and rock bolt L(m) 0.80 1.00 2.00 Proposed model 0.37 0.45 0.61 Tao and Chen 0.39 0.48 0.94
length 4.00 5.00 0.64 0.66 1.80 2.20
According to the example above, when the rock bolt is relatively short, the neutral point of the proposed model is near to Tao and Chen’ suggestion. But when the length of rock bolt exceeds 2.0 meter, the neutral point tends to be constant according the proposed model.
4.2 Rock bolting effect The stress distribution of the rock mass is a useful index to evaluate the stability of the rock mass, and it is also an index of the rock bolting effect. Under the different ground conditions, the rock bolting effects are discussed as follows. According to the standard supporting pattern of the Japan Highway Association, the density parameter of rock bolt β is calculated by the Equation (4). The friction coefficient of the rock bolt surface λ is calculated by the friction angle of the rock mass. The calculated parameter β for different supporting pattern is presented in Table 4. Table 4 Density parameter of installing rock bolt β Supporting Pattern Lz LT β Pattern 1 1 1.2 0.294909 Pattern 2 1.2 1.5 0.196606 Pattern 3 1.5 1.5 0.157285 Pattern 4 2 1.5 0.117964 The ground condition is often described by the socalled competency factor Srp=σc/p0 , where σc is the strength of the rock mass and p 0 is the hydraulic pressure. For the standard supporting pattern, the stress distribution in bolted rock mass under different ground condition is shown in Figure 9. Under the low Srp ground condition, e.g. Srp=0.1, the stress of the rock mass almost has no changes
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Paper 2B 14 — SINOROCK2004 Symposium Int. J. Rock Mech. Min. Sci. Vol. 41, No. 3, CD-ROM, © 2004 Elsevier Ltd.
before and after bolting, which implies that the rock bolting effect is not significant. This is because a decoupling failure has taken place at the interface between the rock bolt and rock mass. P 0= 1 . 0 M P a No bolt
t
0.8
Pattern 1 Pattern 3
t / P0
Pattern 2 Pattern 4
CONCLUSIONS
r
1.2
/ P0,
/ P0
1.6
r
0.4
/ P0
0.0 1
2
3
4
5
6
r / ra
(a) Srp=σc/p0 =0.1 t
/ P0
No bolt Pattern 1 Pattern 3
t
0.8
P 0=1.0MPa Pattern 2 Pattern 4
r
/ P 0,
/ P0
1.6
1.2
r
0.4
/ P0
1
2
r / ra
3
4
(b) Srp=σc/p0 =0.5 P =1.0MPa 0
t
/ P0
t
No bolt Pattern 1
Pattern 2
Pattern 3
Pattern 4
Cox, H.L., 1952. The elasticity and strength of paper and other fibrous materials. Br. J. Appl. Phys. 3, 72-79. Indraratna, B., Kaiser, P.K. 1990. Analytical model for the design of grouted rock bolt, Int. J. for Numerical and Analytical Methods in Geomechanics, pp.227-251. Li, C., Stillborg. B. 1999. Analytical models for rock bolts, Int J Rock Mech. Min Sci and Geomech Abstr; 36:1013-129.
r
/ P 0,
/ P0
1.6
0.8
An analytical method for rock bolt ing design has been proposed. The behavior of the rock bolting system can be described with strain soften constitutive law according to the placement of neutral position. Theoretical prediction of axial force agrees with measured data in Holland Zaka tunnel, Nagasaki. The axial force of the rock bolt are strongly influenced by physical properties of rock mass and the boundary conditions. The rock bolting effect is related to the axial force of the rock bolt. In a good ground condition, the rock bolting effect may not be significant because the deformation of the rock mass is small and the axial force in the rock mass is not large. In a poor ground condition, the rock bolting effect may not significant either because a decoupling may take place at the interface.
REFERENCES
0.0
1.2
relatively good, e.g. Srp=1.0, the rock bolting effect also becomes insignificant. This is because the deformation of the rock mass becomes smaller in a good ground condition, and the initiated axial force in the rock bolt becomes smaller correspondingly.
r
0.4
/ P0
0.0 1
2
r / ra
3
(c) Srp=σc/p0 =1.0 Figure 9. Stress distribution of rock mass with different rock bolting pattern When Srp=0.5, the rock bolting effect of pattern 1 is the largest and that is pattern 4 is the most insignificant. When the ground condition is
Malvar, L.J., 1992. Bond reinforcement under controlled confinement, ACI material Journal, Nov-Dec, Title no.89-M65,pp 593-601. Tao Z.Y and Chen J. X 1984. Behaviour of rock bolts as tunnel support, International Symposium on Rock Bolting, A. A. Balkema: pp. 87-92. Jiang, Y.J., Esaki, T. and Yokota, Y., 1997. The mechanical effect of grouted rock bolts on tunnel stability in soft rock, Journal of Geotechnical Engineering, JSCE, No.561/III38: pp.19-31.
6
A STUDY OF ROCK BOLTING DESIGN IN SOFT ROCK Y. Cai¹, Y.J. Jiang ², T. Esaki¹ ¹) Institute of Environmental Sys., Kyushu University, Japan [email protected] ²) Department of Civil Engineering, Nagasaki University, Japan [email protected]
Abstract: This paper presents an analytical method for rock bolting design in soft rock. The mechanical behavio ur of rock bolt ing system and rock bolt itself are discussed respectively. Coupling mechanism of rock bolt and rock mass are analysed from the viewpoint of displacement. A simple method is suggested for the rock bolt design in tunnelling. According to the analysis, the initiated force of the rock bolt is related with the displacement of rock mass. Case simulations confirm the previous findings that a bolt in-situ has a pick-up length, an anchor length and at least one neutral point. The theoretical prediction of single rock bolt agrees with the measured data. The position of neutral point is not only related with the length of the rock bolt and the internal radius of tunnel, but also strongly influenced by the properties of the rock mass. The neutral point and maximum axial load in bolt tends to be constant when the anchor length of bolt is long enough, which means that increasing the length of rock bolt does not improve its effect dramatically under certain conditions. Keywords: Rock bolt, model, design, tunnel, soft rock, rock mass
INTRODUCTION The underground excavation often encounters soft rock mass in Japan. Rock bolting is considered as an effective and economical means of supporting in varies conditions. Unfortunately, the coupling mechanism is not very clearly at present. The design of rock bolt in tunnelling or other excavations is still empirical, and there are few methods to evaluate the bolting effect in most case. The researches of rock bolt include studying the behavior of rock bolt system and modelling the mechanism of the single rock bolt. The monitoring in field is also common in practice. Indraratna and Kaiser (1990) established the analytical model for the design of grouted rock bolts according to the elasto-plastic constitutive law. The rock bolt and rock mass is considered as a system and the isotropic behavior is assumed. However, the position of the neutral point, at which the shear stress on rock bolt turns to be zero and axial force of rock bolt reaches maximum, is very important to the theoretical analysis. It is widely accepted that the displacement of the rock bolt is the same as the rock mass at neutral point. Freeman’s monitoring result is widely cited for the concept of the neutral point. However, the position of neutral point is not easy to get (Li and Stillborg, 1999). Considering the coupling behavior of the rock bolt system, the
bolt works compatibly with the surrounding rock mass, and the neutral point is strongly influenced by the deformation patterns of the ground. Obviously, the physical properties of the rock mass influence the mechanical behavior of the rock bolt. The study about the single rock bolt often focus on the modelling of the interface between the rock bolt and the rock mass. The pullout test is usually taken as a verification way for the modelling. However, the pullout test itself remains unclear because of the anti-pressure on specimen and the non-uniform shear stress along the rock bolt. This paper is to establish an analytical method of rock bolt ing system considering the interaction behavior of the rock bolt together.
ROCK BOLT SYSTEM AROUND CIRCULAR TUNNEL 2.1 Soft rock mass behavior without rock bolt Since the mechanical properties of the rock mass influences the behavior of the rock bolting system significantly, the behavior of rock mass is discussed separately. A strain-soften model has been suggested to describe the behavior of the soft
1
Paper 2B 14 — SINOROCK2004 Symposium Int. J. Rock Mech. Min. Sci. Vol. 41, No. 3, CD-ROM, © 2004 Elsevier Ltd.
rock mass by the authors. The constitutive law of strain soften law is scratched in Figure 1,
σt = σc* + K p σ r
σ1
(σ 1−σ3) σc
σ3 E'
(2) Accordingly, the soft rock mass around tunnel may divided into three zones such as the plastic flow zone, the strain-soften zone and the elastic zone, as shown in Figure 2. Their corresponding radiuses are named as Rf and Re respectively.
σ3
σ1 σ3 =constant
σ*c
E
ε ef εp ε =αε e 1
ε
1
1
Equation (2), where öis the internal friction angle of rock mass.
1
1
(a) Simplified stress-strain linear relationship
σ σ σ
Y
P(r,θ) r
c
*
εe
+
ε
e 1
1
elastic zone
1
εp
ra Re
1
ε
ε
p
1
ε
e1
X
Rf
h
εp'
ε
Plastic flow zone
Figure 2. Tunnel excavation in soft rock mass
εp
ε p'
3
-
Strain soften zone
1
f
3
3
The displacement formulas of each zone can be obtained according to different constitutive law. By the way, it should be pointed out that the strain soften constitutive law describes the behavior of homogenous material, it also can be referenced to describe the rock bolting system together considering the anisotropic characteristic of rock bolt.
(b) Major and minor principal strains p 1
αε
e
1
+
ε
1
H
εp' 1
F
2.2 Modelling the rock bolt system
vp -
v
p'
(c) Volumetric and major principal strain
Figure 1. Stress-strain relation and dilatancy behavior In the figure 1, α is the brittleness rate of the rock mass; σc is the axial strength of rock mass, and σc * represents the strength of rock mass in strain soften status. According to Mohr-Coulomb law, there is Equation (1),
σt = σc + K p σr
(1a)
Kp = (1 + sin φ ) /(1 − sin φ )
(1b) where σr and σt is the stress of rock bolting section in radius and tangential direction respectively. In the plastic flow zone, Equation (1) changes to
The rock bolt system model shows the “mean performance” for both rock mass behavior and the bolt performance. No decoupling is considered in all the models at present. Based on the assumption of round tunnel profile, homogeneous rock mass and hydrostatic stress conditions, the theoretical formulas of rock bolting system have been established together considering the anisotropic characteristic of rock bolt ( Jiang and Esaki, 1997). Since the direction of the shear stress at the interface between rock bolt and rock mass, and the position of the neutral point is critical to the interaction model of rock bolting system. Their relatively position are shown in Figure 3, where L is the length of rock bolt; Lt and Lz are the distance between rock bolts in circular direction and z direction respectively. Analysing the composite element of the rock bolt and the rock mass, as shown in Figure 4. The coupling equation is written as Equation (3).
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Paper 2B 14 — SINOROCK2004 Symposium Int. J. Rock Mech. Min. Sci. Vol. 41, No. 3, CD-ROM, © 2004 Elsevier Ltd.
dF = 2πrb λσ t dr
Case1
Case 6
(3) where λ is friction coefficient between the rock bolt and the rock mass. The constitutive equation of rock bolting section is listed as Equation (4), dσ r σ r − (1 − β )σ t + =0 dr r β = 2πrb λra / Lz Lt
0
Case 2 0
(4)
where ra is the radius of tunnel; rb is the radius of rock bolt.
ra ρ L1 Rf R e
ra ρ R f L1 R e
Case 3
Re
Case 4
LT
Po Plastic flow zone Soften zone Elastic zone
LT ϕ
ra
r
ρ R f L1 Re
Figure 3. Relationship among plastic radius and bolt length and neutral point
σt
ϕ
θ
r
b
dθ
dr
dF
σr
σt
x
Figure 4. Equilibrium consideration for boltground interaction Considering the position of the neutral point and the rock bolt, nine cases are arranged for analysis. The scratch of the rock bolting section is shown in Figure 5. The stress and strain distributions of the rock bolting system have been discussed. Since the formula is very complicated, they will not present in this paper. The formulas have been listed in the paper of Jiang and Esaki. (1997). The stress of the rock mass can be obtained according to equation (4) and the corresponding boundary conditions of the nine cases respectively.
0
0 ra R f ρ L1
Lz
ρ L1
r a ρ L1
Re
ra ρ Re
L1
r a Re ρ
L1
Case 8
0
0
r a R f Re
Case 7
Po symmetric
0
Case 9 ra ρ Rf Re L1
0 rock bolt Elastic zone
Case 5 0 ra Rf ρ R e L1
strain soften zone Plastic flow zone
Figure 5. Analytical cases of the bolt-ground interaction
INTERACTION MODEL OF SINGLE ROCK BOLT AND MATRIX The interaction between the rock mass and the grouted bolts involves complex mechanics and today there is little work presented on the subject which can explain in theory the satisfactory results from using grouted rock bolts. A coupling model is suggested to describe the behavior of single rock bolt based on Shear-lag model (SLM) or fiberloading theory here. The SLM is originally developed by Cox (1952) and has been widely used by material scientists and structural geologists as powerful analytical method. The concept figure of suggested model is scratched in Figure 5.
u
Figure 6. Coupling behavior of reinforcement and rock mass
3
Paper 2B 14 — SINOROCK2004 Symposium Int. J. Rock Mech. Min. Sci. Vol. 41, No. 3, CD-ROM, © 2004 Elsevier Ltd.
According to the balance of the infinitesimal element of reinforcement, surrounding matrix and matrix together with the reinforcement in cylinder coordinate system, the basic constitutive law is expressed as Equation (5). The assumption of SLM is the equation (6). dP( x) = − 2πrbτ b dx
∂σm ( r , x) ∂τ( r, x ) τ( r , x ) + + =0 ∂x ∂r r R dP ( x ) d + 2π ∫ r σm ( r , x ) dr = 0 r dx dx b
dP ( x) = H ( ub − u m ) dx
(5a)
APPLICATION OF THE MODEL 4.1 Axial force of the rock bolt The coupling behavior of rock bolt and rock mass is able to describe by equation (5) and the boundary conditions. Considering the rock bolting effect, the strain of rock mass at the edge of the influence radius R is expressed as equation (9)
εm = εini − ∆εm (9a)
(5b)
∆ε m = P /(SEm )
(9b) (5c) (6)
where, P(x) is the pullout force at the position of x; R is the influence radius; σn is the normal stress perpendicular to reinforcement; τ(r,x) represents the shear stress at (r,x), as shown in Figure 6; u b and u m are the displacement of reinforcement and matrix at the edge of influence radius R; H is material parameter. The decoupling influences the bolting effect dramatically. Different types of rock bolt have different decoupling behaviors. Only grouted rock bolt is discussed here. Shear strength at interface is made up of three parts such as adhesion, inter lock and friction in axial direction. They are lost in sequence as the compatible of deformation is lost along the interface. After decoupling, the shear stress at interface becomes the residual strength at slipping part. Malvar (1992) revealed that the confining pressure influences the interface strength dramatically. The decoupling failure may occurs at the bolt-grout interface, in the grout medium, at the grout-rock interface or in rock mass. It depends on which one is the weakest. Shear strength can be calculated with Equation (7).
τm = c + σ nb tanφ * (7) where φ* and c is the friction angle and cohesive of interface, which can be tested by direct shear tests or pullout test; σnb is the normal stress perpendicular to rock bolt.
P( x ) d 2 P( x ) = H − εm 2 dx E b Ab
(9c)
where, εini is the rock mass strain without bolt. Em is the deformation of rock mass; S is the influencing area of single rock bolt. Since the displacement function of rock mass around the circular tunnel is relatively complicated, it is not easy to get the theoretical expression of the stress distribution or the position of neutral point. Numerical method is favorable here. The neutral point in rock bolt installed around tunnel is able to get by Equation (7) together with corresponding boundary conditions. Not considering the effect of external fixture, the boundary condition of rock bolt can be written as P(0)=0 and P(L)=0. The filed measuring of axial force was performed at Holland Zaka Tunnel in Nagasaki. The tunnel is situated at a depth of 18.35m. The rock mass around tunnel is classified as DI, which belongs to the soft rock class. The position of the measured rock bolts is shown in Figure 7. According to the investigations, the parameters of rock mass and rock bolt are listed in Table 1: Table 1 Parameters of rock bolt and rock mass Radius of Tunnel 4.75m Hydraulic water pressure, p o Axial strength of rock mass Deformation modulus of rock mass Poisson’s ratio of rock mass Length of rock bolt Young’s modulus of rock bolt Radius of rock bolt, rb Distance between rock bolt Shear strength on interface
0.38MPa 0.5MPa 1.0GPa 0.25 4.0m 210GPa 12.7mm 1.2m*1.4m 0.35Mpa
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Paper 2B 14 — SINOROCK2004 Symposium Int. J. Rock Mech. Min. Sci. Vol. 41, No. 3, CD-ROM, © 2004 Elsevier Ltd. RB1
RB2
neutral point changes from 0.74m to 0.38m towards the tunnel wall, as listed in Table 2. When the length of rock bolt changes from 0.80m to 5.0m, the neutral point changes from 0.37m to 2.20m, as listed in Table 3.
RB3
Table 2 Neutral point and deformation modulus of rock mass Em (GPa) 0.50 1.00 2.00 3.00 5.00 Proposed model 0.74 0.62 0.5 0.46 0.38 Tao and Chen 1.80 1.80 1.80 1.80 1.80
RB5
RB4
5
4
3
2
1
From tunnel wall, 0.5m, 1.5m, 2.5m, 3.5m, 4.0m
Figure 7. Test arrangement of rock bolt in Holland Zaka tunnel (No.68+28, Section C) The calculated results and measured data are presented in Figure 8. The theoretical predictions agree with the test results. The grouted rock bolt installed around tunnel in soft rock mass can be divided into three parts such as the pick up part, the neutral point and the anchor part. The neutral point in this case is 1.26m from the end of rock bolt. Since the physical properties of the rock mass is not easy to get in practice, the discussion is needed to find out the inclination of the neutral point. Tao and Chen suggested the equation (10) to calculate the neutral point of grouted rock bolt. ρ = L /(ln[1 + ( L / ra )] L = (40 rb ~ 60 rb )
(10)
Axial Load (kN)
where L is the length of rock bolt; ρ is radius of neutral point. Proposed model Test data
40 30 20 10 0 0
1 2 3 Length of rock bolt (m )
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Figure 8. Theoretical prediction and measured data When the deformation modulus of the rock mass ranges from 0.50GPa to 5.0GPa, and the other parameters are the same as those in Table 1, the
Table 3 Neutral point and rock bolt L(m) 0.80 1.00 2.00 Proposed model 0.37 0.45 0.61 Tao and Chen 0.39 0.48 0.94
length 4.00 5.00 0.64 0.66 1.80 2.20
According to the example above, when the rock bolt is relatively short, the neutral point of the proposed model is near to Tao and Chen’ suggestion. But when the length of rock bolt exceeds 2.0 meter, the neutral point tends to be constant according the proposed model.
4.2 Rock bolting effect The stress distribution of the rock mass is a useful index to evaluate the stability of the rock mass, and it is also an index of the rock bolting effect. Under the different ground conditions, the rock bolting effects are discussed as follows. According to the standard supporting pattern of the Japan Highway Association, the density parameter of rock bolt β is calculated by the Equation (4). The friction coefficient of the rock bolt surface λ is calculated by the friction angle of the rock mass. The calculated parameter β for different supporting pattern is presented in Table 4. Table 4 Density parameter of installing rock bolt β Supporting Pattern Lz LT β Pattern 1 1 1.2 0.294909 Pattern 2 1.2 1.5 0.196606 Pattern 3 1.5 1.5 0.157285 Pattern 4 2 1.5 0.117964 The ground condition is often described by the socalled competency factor Srp=σc/p0 , where σc is the strength of the rock mass and p 0 is the hydraulic pressure. For the standard supporting pattern, the stress distribution in bolted rock mass under different ground condition is shown in Figure 9. Under the low Srp ground condition, e.g. Srp=0.1, the stress of the rock mass almost has no changes
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Paper 2B 14 — SINOROCK2004 Symposium Int. J. Rock Mech. Min. Sci. Vol. 41, No. 3, CD-ROM, © 2004 Elsevier Ltd.
before and after bolting, which implies that the rock bolting effect is not significant. This is because a decoupling failure has taken place at the interface between the rock bolt and rock mass. P 0= 1 . 0 M P a No bolt
t
0.8
Pattern 1 Pattern 3
t / P0
Pattern 2 Pattern 4
CONCLUSIONS
r
1.2
/ P0,
/ P0
1.6
r
0.4
/ P0
0.0 1
2
3
4
5
6
r / ra
(a) Srp=σc/p0 =0.1 t
/ P0
No bolt Pattern 1 Pattern 3
t
0.8
P 0=1.0MPa Pattern 2 Pattern 4
r
/ P 0,
/ P0
1.6
1.2
r
0.4
/ P0
1
2
r / ra
3
4
(b) Srp=σc/p0 =0.5 P =1.0MPa 0
t
/ P0
t
No bolt Pattern 1
Pattern 2
Pattern 3
Pattern 4
Cox, H.L., 1952. The elasticity and strength of paper and other fibrous materials. Br. J. Appl. Phys. 3, 72-79. Indraratna, B., Kaiser, P.K. 1990. Analytical model for the design of grouted rock bolt, Int. J. for Numerical and Analytical Methods in Geomechanics, pp.227-251. Li, C., Stillborg. B. 1999. Analytical models for rock bolts, Int J Rock Mech. Min Sci and Geomech Abstr; 36:1013-129.
r
/ P 0,
/ P0
1.6
0.8
An analytical method for rock bolt ing design has been proposed. The behavior of the rock bolting system can be described with strain soften constitutive law according to the placement of neutral position. Theoretical prediction of axial force agrees with measured data in Holland Zaka tunnel, Nagasaki. The axial force of the rock bolt are strongly influenced by physical properties of rock mass and the boundary conditions. The rock bolting effect is related to the axial force of the rock bolt. In a good ground condition, the rock bolting effect may not be significant because the deformation of the rock mass is small and the axial force in the rock mass is not large. In a poor ground condition, the rock bolting effect may not significant either because a decoupling may take place at the interface.
REFERENCES
0.0
1.2
relatively good, e.g. Srp=1.0, the rock bolting effect also becomes insignificant. This is because the deformation of the rock mass becomes smaller in a good ground condition, and the initiated axial force in the rock bolt becomes smaller correspondingly.
r
0.4
/ P0
0.0 1
2
r / ra
3
(c) Srp=σc/p0 =1.0 Figure 9. Stress distribution of rock mass with different rock bolting pattern When Srp=0.5, the rock bolting effect of pattern 1 is the largest and that is pattern 4 is the most insignificant. When the ground condition is
Malvar, L.J., 1992. Bond reinforcement under controlled confinement, ACI material Journal, Nov-Dec, Title no.89-M65,pp 593-601. Tao Z.Y and Chen J. X 1984. Behaviour of rock bolts as tunnel support, International Symposium on Rock Bolting, A. A. Balkema: pp. 87-92. Jiang, Y.J., Esaki, T. and Yokota, Y., 1997. The mechanical effect of grouted rock bolts on tunnel stability in soft rock, Journal of Geotechnical Engineering, JSCE, No.561/III38: pp.19-31.
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