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Reports of the aculty of Engineering,Nagasaki University Vol 34, No 6

Analytical model for grouted rock bolts

Yujing Jiang 1, Yoshihiko Tanabashi \ Vue Cai 2 and Tetusro Esaki 2

97

This paper presents an analytical model of the grouted rock bolt in soft rock. The behavior of both the rockbolt system and the single rock bolt are discussed respectively. The coupling mechanism of rock bolt and rockmass is discussed from the viewpoint of displacement. A simple method is suggested for the rock bolt designin tunneling. According to the analysis, displacement of rock mass controls the initial force in rock mass.Case simulations confirm the previous findings that a bolt in-situ has a pick-up length, an anchor length andat least one neutral point. The theoretical prediction of single rock bolt agrees with the measured data. Theposition of the neutral point is not only related to the length of the rock bolt and the internal radius of thetunnel, but is also strongly influenced by the properties of rock mass. Neutral point and maximum axial loadin the rock bolt tend to be constant when the anchor length of the bolt is sufficiently long, which means thatincreasing the length of the rock bolt may result in only a slight improvement in displacement control undercertain conditions.

1 INTRODUCTION

In Japan, soft rock mass is often encountered duringunderground excavation. Rock boIting is consideredto be an effective and economical means of supportunder various conditions. Unfortunately, thecoupling mechanism has not yet been clarified. Thedesign of rock bolts for use in tunnels or otherexcavations is still empirical, and in most cases, fewmethods exist by which to evaluate the boltingeffect. Research on rock bolting includes study ofthe behavior of rock bolt system and modeling of themechanism of the single rock bolt. Monitoring infield is also common in practice. B.Indraratna 1990established the analytical model for the design ofgrouted rock bolts according to the Elasto-plast icconstitutive law. The rock bolt and rock mass areconsidered as a system and isotropic behavior isassumed. Based on the strain softening constitutivelaw, Y.Jiang and T.Esaki 1995 have suggestedanother model for the rock bolt system. However,the position of the neutral point , a t which the shearstress on the rock bolt is zero and the axial force ofthe rock bolt becomes maximum, is very importantin theoretical analysis. The neutral point can only be

determined by an empirical formula, which is onlycorrect when the rock bolt is short. At the same time,no decoupling behavior is discussed in the modelsand it is not easy to predict the bolting effect indesign. It is widely accepted that the displacement ofthe rock boIt is equal to that of the rock mass attheneutral point. However, the position of neutral pointis not easy to determine C.Li and B.Stillborg,1999 . Considering the coupling behavior of the

rock bolt system, the bolt works compatibly with thesurrounding rock mass, and the neutral point isstrongly influenced by the type of rock deformation.Obviously, the mechanical properties of rock massinfluence the initial mechanics of the rock bolt,especially for soft rock mass.

Studies on the single rock bolt often focus on themodeling of the interface between rock bolt and rockmass. The pullout test is usually used as averification method Madhav et aI., 1998, N.Gurung,2001 . However, the pullout test i tsel f may not be

accurate because of the concentration of the normalstress on and the nonuniform shear stress along therock bolt. At the same time, it is difficult todetermine the initial tangent shear stiffness kbecause the tested shear deformation along theinterface comprising both the inelastic deformationof the rock mass before slipping and the relativedisplacement between the rock mass andreinforcement during slipping. The present papersuggests an analytical method of the rock boltsystem that considers the behavior of the single rockbolt in order to evaluate the performance of both asingle rock bolt and a bolt system for a variety of

rock conditions.

2 ROCK BOLT SYSTEM AROUND ACIRCULAR TUNNEL

2 oft rock m ss beh vior without rock bolt

In the previous researches, the behavior of rockmass and bolt is discussed separately. The strain

Received on October 24, 20031 Department of Civil Engineering2 Institute of Environmental Systems, Kyushu University, Japan

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98 nalytical model f or g ro ut ed rock bolts

softening model has been suggested by the first K p = + sin rj / 1 sin rj 1 b)author of this paper as a means for describing thebehavior of soft rock mass. The constitutive law ofstrain softening is described in Figure 1, In the plastic flow zone, Equation 1 becomes

(2)

where is the internal friction angle of the rockmass. Accordingly, the soft rock mass around atunnel may be divided into three zones: the plasticflow zone, the strain soften ing zone and the elasticzone, as shown in Figure 2. The outer radii of theplastic flow and strain softening zones are and e respectively.

Po

(a) Simplified linear stress-strain relationship

h

k t ~ E rf

(b) Major and minor principal strains

H

F

yP

y yP

(c) Volumetric and major principal strains

Fig.l Stress-strain relation and dilatancy behavior of material.

where a is the brittleness rate of the rock mass; O c isthe axial strength of the rock mass, and O c represents the strength of the rock mass under thestrain softening condition. According to the MohrCoulomb failure criterion, we have Equation 1,

Strain softening zo

Fig.2 Tunnel excavation in soft rock mass.

The displacement formulas for each zone have beender ived by Y iang and T.Esaki (1997). Althoughthe strain sof tening consti tut ive law describes thebehavior of homogenous materials, it also ca n beused to describe the rock bolting system consideringthe anisotropic characteristics of the rock bolt.

2 2 Modeling the rock boltsystem

The rock bolt system model shows the meanperformance for both rock mass behavior and boltperformance. No decoupling is considered in thepresent models. Based on the assumptions of roundtunnel profile, homogeneous rock mass andhydrostatic stress conditions, the theoreticalformulas of the rock bolting system have beenestablished considering the anisotropiccharacteristics of the rock bolt (Y.Jiang and T.Esaki,1997). Since the direction of shear stress at theinterface between the rock bolt and the rock mass ischanged, the position of neutral point is critical tothat obtained by the interaction model of the rockbolting system. Their relative positions are shown inFigure 3, where L is the length of the rock bolt, and and z a re the distances between rock bolts inradial direction and z direction, respectively.

1 a

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Yuj ing J iang Yoshihiko Tanabashi Vue Cai . Tetusro Esaki 99

Fig. 3 Relationship among plastic radius and bolt length andneutral point.

~ ~ E M r Fr R p I.

rock boltElastic zone

~ ~ Z strain softenin.e;zone Plastic flow zone

Casel ~ I Case6 e

~ : : : : : . . : : . : : = _ :_ ~ ~ ~ . . : : . : : ::;o ~ ; 0 ; :r P L ~ ~ r R R P i .

in the rock bolting system have been presented byY.Jiang and T.Esaki 1997 .

3

S y m m e t r i ~

ZiJ lastic owzone IZ l Softeningo lastic zone

o

Next, the composite element of roc k bolt and rockmass is analyzed, as shown in Figure 4. Thecoupling equation is given as Equation 3, and theconstitutive equation of the rock bolting sec tion isgiven as Equation 4

d r a r - 1 j3 a 0 dr r

= 2m b Ar I LzL

4 Fig.5 Analytical cases of the bolt-ground interaction.

.where A is the friction coefficient between the rockbolt and the rock mass, r a is the radius of the tunnel,crr and crt are the stresses of the rock bolting sectionin the radial and tangential directions respectively,and is the radius of the rock bolt.

y

x

Fig.4 Equilibrium consideration for bolt-ground interaction.

3 INTERACTION MODEL OF SINGLE ROCK

BOLT AND MATRIX

3.1 Coupling behavior rock o l t n d rock mass

The interaction between the rock mass and thegrouted bolts involves complex mechanics, and therehave been few reports on the subject which canexplain in theory the sat isfactory resul ts obtainedusing grouted rock bolts. A coupling model has beensuggested to describe the behavior of a single rockbol t based on the shear-lag model SLM or fiberloading theory. The SLM was originally developedby H.Cox 1952 and has been widely used bymaterial scientists and structural geologists as apowerful analytical method. The concept of thesuggested model is described in Figure 6. Accordingto the balance conditions between the infinitesimalreinforcement element, the surrounding matrix andthe matrix with reinforcement in cylinder coordinatesystem, the basic constitut ive law can be expressedas Equation 5, and the assumption of SLM isexpressed as Equation 6. The model is discussed indetail elsewhere Y.Cai et aI, 2003 .

Considering the positions of the neutral point andthe length of rock bolt, nine cases are analyzed. Therock bolting section is shown in Figure 5. The

detailed equations of stress and strain distributions

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100 n lytic l model fo r grouted rock bol ts

Fig. 6 Coupling behavior of reinforcement and rock mass.

dP x = -2m- Tdx b b

5a

components: adhesion, interlock and friction in theaxial direction. These three components are lost insequence if no defonnation arose along the interface.Afte r decoupling, the shear stress a t the interfacebecomes to the residual strength at slipping section.

As a result, r ock bolt behavior varies according tothe residual shear stress. L.Malvar 1992 andM.Moosavi 2001 revealed that the confiningpressure inf luences the strength of the interfacedramatically. Although the full mechanism of bondfailure during axial slip in a defonned bar can beexplained by the shearing mechanism in the cementannulus Moosavi, 2001 , failure may occur at thebolt -grout interface, in the grout medium or groutrock interface, or in the rock mass. The shearstrength can be expressed as

8o-m r,x 8r r,x r r,x 0

8x 8r r 5b 8

dP x d 21r r-cym r,x dr 0

dx b dx

dP x H u - u dx b m

5c

6

where and c are the friction angle and cohesive forceof interface, respectively, which can be evaluated by thedirect shear or pullout tests, and CYnh is the nonnal stressperpendicular to the rock bolt.

4 APPLICATION OF THE PROPOSED MODEL

The coupling behavior of the rock bolt and rockmass can be described by Equations 5 and 6 fordifferent boundary conditions. Considering theeffect of the rock bolt system, the strain of rock mass

at the edge of influence radius R can be expressed as

where P x is pullout force at x, R is the influenceradius, n is the nonn l stress perpendicular toreinforcement, C(r,x) is the shear stress at r,x asshown in Figure 6 , Ub and U m are the displacementof reinforcement and matrix at the edge of influenceradius R, and H is the material parameter. Differentstress distributions result in different models andparameters. In order to simplify the analysis andsa tisfy the boundary conditions, a un ifonn stressdistr ibut ion is assumed in the inf luence area of thereinforcement. Correspondingly, parameter H isexpressed as Equation 7

m c

ini - '1c m 9a

9b

79c

where g and r g are the shear modulus and radius ofgrout, respectively, and G m is the shear modulus ofthe rock mass. The in-situ mechanical properties ofthe rock mass are very compl icat ed and Equation 5provides a simple method of evaluating the couplingproperties of the rock bolt and rock mass.

3.2 Decoupling behavior o f single rock bolt an drock mass

Decoupling influences the bolting effectdramat ically whi le i t has not been considered in thegeneral models of the rock bolt. Different types ofrock bolt have different decoupling behaviors. Onlythe grouted rock bol t is discussed herein. The shearstrength at the interface is made up of three

where ini is the rock mass strain without the bolt,Em is the defonnation modulus of the rock mass, andS is the influence area of a single rock bolt. The

initial strain of the rock mass detennines the axialforce of the rock bolt. Since the displacementfunction of the rock mass around a circular tunnel isrelatively complicated, it is not easy to obtain atheoretical expression for the stress distr ibution orthe position of the neutral point. Numerical methodsare favorable in this case. Therefore, the neutralpoint for rock bolts installed around a tunnel can beobtained using Equation 7 together with theboundary conditions. Ignoring the effect of externalfixtures, the boundary conditions of the rock boltcan be written as P O =O and P L =O.

Axial load measurement of rock bol ts in the field

was perfonned at the Holland slope Tunnel in

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Yuj ing J iang Yoshihiko Tanabashi Yue Cai Tetusro Esaki 101

Fig.8 Theoretical prediction for a single rock bolt.

4

Proposed modelo Test data

1 2 3Length o f roc k bol t m)

b) Shear stress along rock bolt

Proposed model Test data

G

o

40

30 c

20 ij

x 100

0 1 4Length of rock bol t m)

a) Axial force along rock bolt

0.30.20 1

0 -0.1 -0.2 -0.3

-0.4.1::- .-. ; ; 0 ~

-0.5

Radius of Tunnel 4.75 mHydraulic water pressure, po 0.38 MPaAxial strength of rock mass 0.5 MPaDeformation modulus of rock mass 1.0 GPaPoisson s ratio of rock mass 0.25Length of rock bolt 4.0 mYoung s modulus of rock bolt 210 GPaRadius of rock bolt 12.7 mmDistance between rock bolt Lz x Lt 0.2 m x 1.4 mShear strength on interface 0.35 Mpa

Comparison of the calculated results and themeasured da ta for a single rock bolt is presented inFigure 8 Theoretical predictions agree well with themeasurements. Grouted rock bolts installed around atunnel in soft rock mass can be divided into threelengths: pick-up length, neutral point and anchorlength. This also agrees with the in-situ conditions.The neutral point in this case is 1.26 m from the en dof the rock bolt.

Table 1 Parameters of rock bolt and rock mass.

Nagasaki. The tunnel is situated at a depth of 18.35m. The rock mass around the tunnel is classified asDI, which is a type of soft rock. The positions of themeasured rock bolts are shown in Figure 7. The rockmass and rock bolt parameters are listed in Table 1

R 1 p = / In[l + / ra L = 4 r b 6 r b

10)

RB2 RB3

Distance from tunnel wall 0.5m,1.5m,2.5m,3.5m,4.0m

RB4

4 3

4.0m

2

85

j

where L is the length of the rock bolt, r a is the radiusof the tunnel, rb is the radius of the rock bolt, and pis radius of the neutral point.

In the range of deformation modulus of the rockbol t from 0.50 GPa to 5.0 GPa, the other parametersremain the same as those in Table 1 and the neutralpoint changes from 0.74 m to 0.38 m toward tunnelwall, as listed in Table 2. As the length of the rockbol t changes from 0.80 m to 5.0 m, the neutral pointchanges from 0.37 m to 2.20 m, as l is ted in Table 3.

Table 2 Neutral points for rock bolts of various deformationmodulus.

Table 3 Neutral points for rock bolts of different length.L m) 0.80 1.00 2.00 4.00 5.00

Proposed model 0.37 0.45 0.61 0.64 0.66

Tao and Chen 0.39 0.48 0.94 1.80 2.20

Fig. 7 Test arrangement of rock bolts in Holland Slope Tunnel No.68+28, Section C).

Since the mechanical properties of rock mass are no teasy to obtain in practice, discussion is needed inorder to determine the inclination of the neutralpoint. Z.Tao and I.Chen 1984) have suggestedEquation 10 for calculating the neutral point of thegrouted rock bolt.

Em GPa)

Proposed model

Tao and Chen

0.50 1.00 2.00 3.00 5.00

0.74 0.62 0.5 0.46 0.38

1.80 1.80 1.80 1.80 1.80

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