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Analytical models for rock bolts
International Journal of Rock Mechanics and Mining Sciences 36 (1999) 1013±1029 www.elsevier.com/locate/ijrmms
Analytical models for rock bolts C. Li*, B. Stillborg Division of Rock Mechanics, LuleaÊ University of Technology, SE-971 87 LuleaÊ, Sweden Accepted 21 August 1999
Abstract Three analytical models have been developed for rock bolts: one for bolts subjected to a concentrated pull load in pullout tests, one for bolts installed in uniformly deformed rock masses, and one for bolts subjected to the opening of individual rock joints. The development of the models has been based on the description of the mechanical coupling at the interface between the bolt and the grout medium for grouted bolts, or between the bolt and the rock for frictionally coupled bolts. For rock bolts in pullout tests, the shear stress of the interface attenuates exponentially with increasing distance from the point of loading when the deformation is compatible across the interface. Decoupling may start ®rst at the loading point when the applied load is large enough and then propagate towards the far end of the bolt with a further increase in the applied load. The magnitude of the shear stress on the decoupled bolt section depends on the coupling mechanism at the interface. For fully grouted bolts, the shear stress on the decoupled section is lower than the peak shear strength of the interface, while for fully frictionally coupled bolts it is approximately the same as the peak shear strength. For rock bolts installed in uniformly deformed rock, the loading process of the bolts due to rock deformation has been taken into account in developing the model. Model simulations con®rm the previous ®ndings that a bolt in situ has a pick-up length, an anchor length and a neutral point. It is also revealed that the face plate plays a signi®cant role in enhancing the reinforcement eect. In jointed rock masses, several axial stress peaks may occur along the bolt because of the opening of rock joints intersecting the bolt. # 2000 Elsevier Science Ltd. All rights reserved.
1. Introduction Rock bolts have been widely used for rock reinforcement in civil and mining engineering for a long time. Bolts reinforce rock masses through restraining the deformation within the rock masses. In order to improve bolting design, it is necessary to have a good understanding of the behaviour of rock bolts in deformed rock masses. This can be acquired through ®eld monitoring, laboratory tests, numerical modelling and analytical studies. Since the 1970s, numerous researchers have carried out ®eld monitoring work on rock bolts installed in various rock formations [1±3]. Freeman [1] performed pioneering work in studying the per* Corresponding author. Tel.: +46-920-91352; fax: +46-92091935. E-mail address: [email protected] (C. Li).
formance of fully grouted rock bolts in the Kielder experimental tunnel. He monitored both the loading process of the bolts and the distribution of stresses along the bolts. On the basis of his monitoring data, he proposed the concepts of ``neutral point'', ``pick-up length'' and ``anchor length''. At the neutral point, the shear stress at the interface between the bolt and the grout medium is zero, while the tensile axial load of the bolt has a peak value. The pick-up length refers to the section of the bolt from the near end of the bolt (on the tunnel wall) to the neutral point. The shear stresses on this section of the bolt pick up the load from the rock and drag the bolt towards the tunnel. The anchor length refers to the section of the bolt from the neutral point to the far end of the bolt (its seating deep in the rock). The shear stresses on this section of the bolt anchor the bolt to the rock. These concepts clearly outline the behaviour of fully grouted rock bolts in a deformed rock formation. BjoÈrnfot and
1365-1609/99/$ - see front matter # 2000 Elsevier Science Ltd. All rights reserved. PII: S 1 3 6 5 - 1 6 0 9 ( 9 9 ) 0 0 0 6 4 - 7
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Nomenclature A Eb Er Eg L P0 P0max
area of the cross-section of the bolt Young's modulus of the bolt steel Young's modulus of the rock mass Young's modulus of the grout length of the bolt applied pull load pullout load, i.e. the maximum applied pull load S in¯uencing area of a bolt in the rock diameter of a circle in the rock outside d0 which the in¯uence of the bolt disappears diameter of the bolt db diameter of the borehole dg du free deformation of the rock slice elongation of the bolt element dub the reduction of deformation after bolt redur inforcement, i.e. dur=du ÿ dub dx length of the rock slice p0 hydrostatic primary stress in the rock load on the face plate of the bolt Pf radius of the circular tunnel ri position of the decoupling front on the bolt rp surface s shear strength of the interface for frictionally coupled bolts residual shear strength of the interface for sr fully grouted bolts peak shear strength of the interface for fully sp grouted bolts u original radial displacement of the rock at x (without bolting) x0,x1 decoupling boundaries at the interface of and x2 fully grouted bolts (see Fig. 4) Greek symbols D length of a bolt section, D=x2ÿx1 Stephansson's work [2,4] demonstrated that in joined rock masses there may exist not only one but several neutral points along the bolt because of the opening displacement of individual joints. Pullout tests are usually used to examine the anchoring capacity of rock bolts. A great number of pullout tests have been conducted so far in various types of rocks [5±9]. Farmer [6] carried out fundamental work in studying the behaviour of bolts under tensile loading. His solution predicts that the axial stress of the bolt (also the shear stress at the bolt interface) will decrease exponentially from the point of loading to the far end of the bolt before decoupling occurs. Fig. 1(a) illustrates the results of a typical pullout test [5]. Curve a represents the distribution of the axial stress along
a d0 d1 d2 d3 dJ dJi dJmax ng nr x sb sb0 sb0i Dsr tb tbB tb1 tb2 t dA t dB tA B o
a constant representing the coupling property of the interface elongation of the bolt in section (0Rx
the bolt under a relatively low applied load, at which the deformation is compatible on both sides of the bolt interface. Curve b represents the axial stress along the bolt at a relatively high applied load, at which decoupling has occurred at part of the bolt interface. Fig. 1(b) shows the axial stress along a rock bolt installed in an underground mine drift [3]. It is seen from this ®gure that the distribution of the axial stress along the section close to the borehole collar is completely dierent from that in pullout tests. However, along the section to the far end of the bolt, the stress varies similarly to that in pullout tests. The reason for these results is that bolts in situ have a pick-up length and an anchor length, while bolts in pullout tests only have an anchor length.
C. Li, B. Stillborg / International Journal of Rock Mechanics and Mining Sciences 36 (1999) 1013±1029
It is thought that the relative movement between the rock and the bolt is zero at the neutral point [1]. In the solution by Tao and Chen [10], the position of the neutral point depends only on the radius of the tunnel and the length of the bolt. That solution was implemented in the analytical models created by Indraratna and Kaiser [11] and Hyett et al. [12]. It seems that Tao and Chen's solution is valid only when the deformation is compatible across the bolt interface. When decoupling occurs, the position of the neutral point is obviously also related to the shear strength of the interface. Field monitoring and pullout tests have indicated two facts concerning the loading of a rock bolt in situ: (1) rock deformation applies a load on the pick-up section of the bolt; (2) the load on the pick-up section drags the anchor section of the bolt towards the underground opening. These two facts must be taken into account in developing analytical models for rock bolts. The aim of this paper is to develop analytical models for fully coupled rock bolts. A model for rock bolts in pullout tests is introduced ®rst, together with
Fig. 1. Distribution of the axial stress (a) along a grouted steel bar during a pullout test, after Hawkes and Evans [5], and (b) along a grouted rock bolt in situ, after Sun [3].
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a description of the theoretical background, the development of the model and an illustrative example. Two models for rock bolts in situ are then presented, one in uniformly deformed rock masses and one in jointed rock masses. The details of the development of the models are summarised in the Appendices. 2. Coupling between the bolt and the rock Windsor [13] proposed the concept that a reinforcement system comprises four principal components: the rock, the reinforcing element, the internal ®xture and the external ®xture. For reinforcement with a bolt, the reinforcing element refers to the bolt and the external ®xture refers to the face plate and nut. The internal ®xture is either a medium, such as cement mortar or resin for grouted bolts, or a mechanical action like ``friction'' at the bolt interface for frictionally coupled bolts. The internal ®xture provides a coupling condition at the interface. With reference to the component of internal ®xture, Windsor [13] classi®ed the current reinforcement devices into three groups: ``continuously mechanically coupled (CMC)'', ``continuously frictionally coupled (CFC)'' and ``discretely mechanically or frictionally coupled (DMFC)'' systems. According to this classi®cation system, cementand resin-grouted bolts belong to the CMC system, while Split set and Swellex bolts belong to the CFC system. When fully grouted bolts are subjected to a pull load, failure may occur at the bolt±grout interface, in the grout medium or at the grout±rock interface, depending on which one is the weakest. For fully frictionally coupled bolts, however, there is only one possibility of failure Ð decoupling at the bolt±rock interface. In this study we concentrate on the failure at the interface between the bolt and the coupling medium (either the grout medium or the rock). In general, the shear strength of an interface comprises three components: adhesion, mechanical interlock and friction. They are lost in sequence as the compatibility of deformation is lost across the interface. The result is a decoupling front that attenuates at an increasing distance from the point of the applied load. The decoupling front ®rst mobilises the adhesive component of strength, then the mechanical interlock component and ®nally the frictional component. The shear strength of the interface decreases during this process. The shear strength after the loss of some of the strength components is called the residual shear strength in this paper. For grouted rock bolts like rebar, all the three components of strength exist at the bolt interface. However, for the fully frictionally coupled bolt, the ``Split set'' bolt, only a friction component exists at the bolt interface. For Swellex bolts,
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mechanical interlock and friction comprise the strength of the interface. 3. Rock bolts in pullout tests 3.1. Theoretical background When a bolt installed in rock is subjected to a tensile axial load, the relationship between the shear stress at the bolt interface and the axial tensile stress of the bolt can be established through considering a small section of the bolt as shown in Fig. 2. The force equilibrium in the axial direction leads to the following expression: tb ÿ
A dsb pdb dx
1
where db is the diameter of the bolt, and A is the area of the cross-section of the bolt. For the example shown in Fig. 1(a), the shear stress along the bolt at the two levels of applied load is obtained using Eq. (1) and illustrated in Fig. 3. When the applied load is small, the shear stress decreases with increasing distance from the point of loading (curve a ). Progressive decoupling commences at the loading point at a certain level of applied load. The decoupling front moves towards the far end of the bolt with an increasing applied load. The shear stress is at the level of the shear strength at the decoupling front, while behind the decoupling front the shear stress becomes smaller, since the strength of the interface has been partially lost due to decoupling. Curve b in Fig. 3 represents such a distribution of shear stress along the bolt. Based on experimental results as shown in Fig. 3, a model for the shear stress along a fully grouted bolt can be postulated as illustrated in Fig. 4. In this model, the section of the bolt close to the loading point is completely decoupled with a zero shear stress
Fig. 2. Stress components in a small section of a bolt.
Fig. 3. The shear stress on the steel bar, derived from Fig. 1(a).
at the bolt interface. Starting at a certain distance from the loading point, say at x0, the bolt interface is partially decoupled with a residual shear strength, sr. Between point x1 and x2, the residual shear strength linearly increases from sr to the peak strength sp. Beyond point x2, the interface undergoes compatible deformation and the shear stress attenuates exponentially towards the far end of the bolt. For fully frictionally coupled bolts, the magnitude of the shear stress behind the decoupling front is approximately the same as the peak value. As mentioned previously, the shear strength of the interface for this type of bolt comprises one or two components, i.e. either friction or mechanical interlock and friction. The deformation incompatibility across the interface does not make the friction disappear. In other words, the residual shear strength of the interface is approximately the same as the peak strength for fully frictionally coupled bolts. The distribution of shear stress for this type of bolt is illustrated in Fig. 5. When a fully coupled bolt is subjected to a pull load, the shear stress along the bolt is as shown in Fig.
Fig. 4. Distribution of shear stress along a fully grouted rock bolt subjected to an axial load.
C. Li, B. Stillborg / International Journal of Rock Mechanics and Mining Sciences 36 (1999) 1013±1029
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bolts if the expression for the constant a is slightly modi®ed. Fully frictionally coupled bolts have direct contact with rock. The constant a for this type of bolt can be obtained from Eq. (3a) by letting the Young's modulus and Poisson's ratio of the grout equal those of the rock, i.e. Gg=Gr, ng=nr and dg=dr. Then we obtain the expression of a for fully frictionally coupled bolts as: a2
Fig. 5. Distribution of shear stress along a frictionally coupled rock bolt.
6 before decoupling occurs at the interface. For fully grouted rock bolts, the attenuation of the shear stress is expressed as [6,14]: x a ÿ2a tb sb0 e db 2
2
2Gr d0 Eb ln db
3b
The axial stress of the bolt is calculated as: x pdb x ÿ2a tb xdx sb0 e db sb x sb0 ÿ A 0
4a
or 2 sb x tb x a
4b
where a2
" Eb
Gr
2Gr Gg # dg d0 Gg ln Gr ln db dg
Er 2 1 nr
and Gg
Eg 2 1 ng
3.2. Fully grouted rock bolts The stresses in dierent sections of the bolt can now be described in detail as follows (see Fig. 4): 3a
sb0 is the axial stress of the bolt at the loading point, Eb is Young's modulus of the bolt steel, Er is Young's modulus of the rock mass, Eg is Young's modulus of the grout, nr is Poisson's ratio of the rock mass, ng is Poisson's ratio of the grout, dg is the diameter of the borehole, and d0 is the diameter of a circle in the rock outside which the in¯uence of the bolt disappears. Eq. (2) is also valid for fully frictionally coupled
1. On the section 0 R x < x0: the bolt interface is completely decoupled, leading to a zero shear stress at the interface and a constant axial stress in the bolt, i.e.: tb x 0 sb x sb0
5
2. On the section x0 R x < x1: the interface is partially decoupled, resulting in a residual shear strength sr at the interface. The shear and axial stresses are given by: tb x sr
sb x sb0 ÿ
4sr x ÿ x 0 db
6
3. On the section x1 R x < x2: the interface is partially decoupled with the residual shear strength linearly increasing to the peak strength. The shear and axial stresses are given by: Fig. 6. Shear stress along a fully coupled rock bolt subjected to an axial load before decoupling occurs.
tb x osp
x ÿ x1 1 ÿ osp D
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sb x sb0
ÿ
2sp 1 ÿ o x ÿ x 1 2 2o x ÿ x 0 db D
7
where D=x2ÿx1, and o=sr/sp, the ratio of the residual shear strength to the peak shear strength. 4. On the section x > x2: the deformation is compatible across the interface and no decoupling occurs. According to Eqs. (2) and (4), both the shear and axial stresses decrease exponentially towards the far end of the bolt: ÿ xÿx ÿ2a d 2 b tb x sp e sb x
ÿ 2 2sp ÿ2a xÿx db e a
8
It is seen from Eq. (8) that the axial stress at x=x2 is given by 2sp sb x 2 a On the other hand, the axial stress at x=x2 can be obtained from Eq. (7) as: sb x 2
2sp 4P0 ÿ 2o x 2 ÿ x 0 1 ÿ oD 2 db pd b
where P0 is the applied pull load. Letting the right sides of the above two expressions be equal, we obtain the expression for the position of the decoupling front, x2, as: 1 2P0 db ÿ 1 ÿ oD 9 ÿ x2 x0 2o pdb sp a For equilibrium the applied load P0 should equal the total shear force at the bolt interface, i.e. L 1 P0 pdb tb dx pdb sr x 1 ÿ x 0 sp D 1 o 2 x0 2a db ÿ d Lÿx 2 b sp 1 ÿ e 2a where L is the length of the bolt. It is obtained from the above expression that the maximum applied load P0max can be expressed as: db ln o ÿ D ÿ x 0 P0max pdb sp o L 2a 10 1 db D 1 o 1 ÿ o 2a 2 The following is an example to demonstrate how to back-calculate the peak shear strength of the interface
on the basis of the pullout load. Stillborg [8] conducted a series of pullout tests on dierent types of rock bolts. In one test, a 3 m long rebar with a diameter of 20 mm was grouted within two identical concrete blocks. The length of the bolt in each block was 1.5 m. One block was ®xed to the ground, while the other was pulled. The bolt was pulled out without rupture, indicating that decoupling of the interface occurred along the entire length of the bolt. The pullout load registered was 180 kN. It is assumed that the distribution of shear stress has the form illustrated in Fig. 4 with x0=0. It is known from the test that: P0max 180 kN,
L 1:5 m,
dg 35 mm,
Eb 210 GPa
db 20 mm,
The values of the other parameters are assumed to be: Er concrete 45 GPa,
Eg cement mortar 30 GPa,
nr ng 0:25 o sr =sp 0:1,
D 0:1 m,
d0 10dg
It is then obtained that the constant a=0.23 from Eq. (3a) and the peak shear strength sp=12.8 MPa from Eq. (10). The shear stress and the axial load along the rebar are calculated on the basis of the model and illustrated in Fig. 7. The axial load along the bolt at dierent levels of applied load is illustrated in Fig. 8. The curves in Fig. 8 are similar to those obtained in pullout tests (e.g. Fig. 1(a)).
3.3. Fully frictionally coupled rock bolts For fully frictionally coupled rock bolts, the residual shear strength of the interface is approximately the same as the peak shear strength, i.e. sr=sp=s (see Fig. 5). The shear stress on dierent sections of the bolt is described in detail as follows: 1. On the section 0 R x < x2: the shear stress has reached the level of the strength of the interface. The shear stress on this section remains constant, while the axial stress linearly decreases, i.e.: tb x sr sb x sb0 ÿ
pdb sx A
11
2. On the section x > x2: the deformation is compatible across the interface and the shear stress is less than the peak shear strength. Both the shear and the axial stresses decrease exponentially towards the far end of the bolt:
C. Li, B. Stillborg / International Journal of Rock Mechanics and Mining Sciences 36 (1999) 1013±1029
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Fig. 7. The shear stress and axial load along a fully grouted rock bolt subjected to an axial load of 90 kN.
tb x s e sb x
ÿ xÿx 2
ÿ2a
2s ÿ2a e a
db
ÿ xÿx 2 db
sb x 2 12
It is seen from Eq. (12) that the axial stress at x=x2 is given by
2s a
On the other hand, the axial stress at x=x2 can be obtained from Eq. (11) as: sb x 2
P0 pdb ÿ sx 2 A A
Fig. 8. Axial load along a fully grouted rock bolt at dierent levels of applied load.
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Letting the right sides of the above two expressions be equal, we obtain the expression for the position of the decoupling front, x2, as: 1 2As P0 ÿ 13 x2 pdb s a For equilibrium the applied load P0 should equal the total shear force at the bolt interface, i.e. L P0 pdb tb dx 0
spdb x 2
pd 2b 2
s 1ÿe a
Lÿx 2 ÿ2a db
14
The applied load reaches its maximum, P0max, when the shear strength of the interface is mobilised along the entire length of the bolt, i.e. when x2=L. Substituting P0=P0max and x2=L into Eq. (13), we obtain the shear strength for fully frictionally coupled bolts as: s
P0max pdb L
15
Stillborg [8] tested the Swellex bolt in his pullout tests. Displacement monitoring at the far end of the bolt indicated that the bolt was slipping under the load P0max=110 kN. That indicated that decoupling occurred along the entire length of the bolt. The diameter of the borehole was 35 mm. The diameter of the Swellex bolt is the same as that of the borehole, i.e.
db=35 mm. The length of the bolt section embedded in each concrete block was 1.5 m long, i.e. L = 1.5 m. Substituting these data into Eq. (15) yields the shear strength of the bolt interface, i.e. s = 0.7 MPa. It is obtained from Eq. (3b) that the constant a=0.27. The shear stress and the axial load along the Swellex bolt are calculated using the relevant equations above and are illustrated in Fig. 9. The axial load of the bolt at dierent levels of applied load is shown in Fig. 10.
4. Rock bolts in situ 4.1. A model for bolts subjected to uniform rock deformation Rock bolts in situ tend to restrain the deformation of rock with an increase in their axial loads. In other words, it is rock deformation that applies a load to rock bolts in situ. For the sake of simplicity, a bolt anchored at two points, as illustrated in Fig. 11, is used to explain the superposition of two components of the shear stress. Rock deformation will induce a component of shear stress t dA at A and a component of shear stress t dB at B. Assuming t dA > t dB, the shear force acting at anchor A would tend to drag the bolt to the left and thus induce another component of A shear stress at point B, t A B . The sense of t B is opposite d to the sense of t B. The total shear stress at B is: d tbB tA B ÿ tB
Fig. 9. The shear stress and axial load along a Swellex rock bolt at 5 kN of applied load.
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Fig. 10. The axial load along a Swellex rock bolt at dierent levels of applied load.
For a fully coupled bolt, the component t A B will be an integration over the bolt section to the left side of point B. On the basis of this idea, we obtained the expression of the shear stress at position x on the bolt surface as follows (see Appendix A for detailed derivation): # " A d2 u a x d2 u ÿ2a xÿt db dt ÿ e 17 tb x xGr pdb dx 2 2 ri dt2 where x
2 1 nr SEb AEb SEr
18
u is the original radial displacement of the rock at x (without bolting), and S is the in¯uencing area of the
Fig. 11. A sketch illustrating the superposition of the components of shear stress at position B.
bolt in the rock, which equals the area surrounded by four adjacent bolts in pattern bolting. Here we take a tunnel circular in its cross-section as an example to demonstrate the application of Eq. (17). Assume that the country rock surrounding the tunnel undergoes an elastic deformation. The second-order derivative of the elastic radial displacement u of the rock can be expressed as: ux00 ÿ
p0 r2i Gr x 3
19
where ri is the radius of the circular tunnel, and p0 is the hydrostatic primary stress in the rock. Substituting Eq. (19) into Eq. (17) and using the following values for the relevant parameters: Young's modulus of the bolt steel Young's modulus of the rock mass Poisson's ratio of the rock mass bolt spacing diameter of the bolt radius of the circular tunnel hydrostatic primary stress in the rock
Eb=210 GPa, Er=5 GPa, nr=0.25, S = 1 m, db=20 mm, ri=4 m, p0=15 MPa,
we obtain the shear stress along the bolt as illustrated in Fig. 12. It can be seen that the sense of the shear stress on the bolt section close to the tunnel wall is negative; that is the direction of the shear stress is towards the tunnel. At a certain distance from the wall, the shear stress becomes zero. Beyond this neutral point, the sense of the shear stress becomes positive; that is the direction of the shear stress is towards
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ponents: one induced by the rock deformation, one due to the pull eect of the face plate load Pf , and one due to the pull eect of the shear force on the decoupled bolt section between ri and rp. In the case of no face plate, the face plate load Pf is zero. The total shear stress at x=rp equals the peak shear strength of the interface (see Appendix A), i.e. # " A d2 u a rp d2 u ÿ dt , ÿsp xGr pdb dx 2 2 ri dt2 21a for bolts with a face plate or Fig. 12. Shear stress along a fully grouted rock bolt under the condition of compatible interface deformation (Eb=210 GPa, Er=5 GPa, nr=0.25, S = 1 m, db=20 mm, ri=4 m, p0=15 MPa).
the far end of the bolt. This agrees with the ®eld monitoring data obtained, for example, by Freeman [1]. In this example, decoupling at the bolt interface has not been considered. When the rock deformation is large enough, the shear strength of the interface will be mobilised in the pick-up section of the bolt. The distribution of shear stress along the bolt, when decoupling occurs, will be as that illustrated in Fig. 13. The shear failure at the interface would result in a release of the restrained rock deformation at the near end of the bolt, if no face plate were to exist. In the case where a face plate exists, the displacement of the tunnel wall loads the plate. The load on the face plate can be calculated as: ! rp A d2 u xGr 2 ÿ sr dx ÿ 20 Pf pdb pdb dx ri Shear failure ceases at x=rp and beyond that point the displacement is compatible across the interface. The shear stress at x=rp is the sum of three com-
ÿsp xGr
A d2 u a pdb sr rp ÿ ri , pdb dx 2 2 A
21b
for bolts without a face plate Eqs. (21a) and (b) are used to determine the distance rp. The shear stress on the section x > rp is calculated as: # " A d2 u a x d2 u ÿ2a xÿt db dt ÿ e tb x xGr pdb dx 2 2 ri dt2 a ÿ xGr 2
"
rp
ri
# xÿrp d2 u ÿ2a d b , dt e dt2
22a
for bolts with a face plate or " tb x xGr
A d2 u a ÿ pdb dx 2 2
x
d2 u ÿ2a xÿt db dt e 2 rp dt
xÿrp a pdb ÿ2a d b , sr rp ÿ ri e 2 A
#
22b
for bolts without a face plate
Fig. 13. A schematic illustration of the shear stress along a rock bolt in situ.
We take the circular tunnel under elastic deformation again as an example to demonstrate the application of Eqs. (21) and (22). Assume that the peak shear strength of the bolt interface is sp=0.5 MPa and the residual shear strength sr=0.2 MPa. Using the values given before for other relevant parameters, we can obtain the decoupling boundary rp from Eq. (21) and the shear stress on the interface at x > rp from Eq. (22). The calculated results are shown in Fig. 14 for a fully grouted bolt with a face plate and in Fig. 15 for a fully grouted bolt without a face plate. It can be found by comparing the curves in these two ®gures that: (i) the decoupled length of the bolt is shorter with a face plate than without a face plate; and (ii) the axial stress in the decoupled section is larger for the
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4.2. A model for bolts subjected to the opening of a rock joint
Fig. 14. Theoretical solution of the shear stress and axial stress along a fully grouted rock bolt with a face plate (sp=0.5 MPa, sr=0.2 MPa).
bolt with a face plate than the bolt without a face plate. That indicates that rock bolts with a face plate have a better reinforcement eect than those without a face plate. Fig. 16 shows the monitored results of the shear stress along two fully grouted bolts in a mine drift in Sweden [2,4]. The shear stress along bolt No. 9, presented in Fig. 16(a), agrees well with the theoretical curve shown in Fig. 12, implying that no decoupling occurred at the bolt interface. The shear stress along bolt No. 1, presented in Fig. 16(b), matches the curve shown in Fig. 15, indicating that decoupling occurred at the interface of this bolt. The analytical model introduced in this section provides a means for studying rock bolts in a uniformly deformed rock mass. The key for determining the shear stress along the bolt is the original rock deformation around the excavation opening.
Fig. 15. Theoretical solution of the shear stress and axial stress along a fully grouted rock bolt without a face plate (sp=0.5 MPa, sr=0.2 MPa).
The opening of a rock joint applies a tensile load to both sides of the bolt intersecting the joint. During joint opening, decoupling of the bolt interface is activated ®rstly at the joint and then propagates along the interface with an increase in the opening displacement. When the embedment length of the fully coupled bolt is suciently long on each side of the joint, the shear stress as well as the axial stress along the bolt will be symmetrical to the joint, as shown in Fig. 17. When the opening displacement of the joint is small, both the shear stress and the axial stress decrease exponentially with increasing distance from the joint. When the opening displacement is large enough, decoupling will be activated at the bolt interface and the shear and axial stresses along the bolt will look like those illustrated with dashed lines in Fig. 17. According to the models for shear stress illustrated in Figs. 4 and 5, we obtain the following relationships between the opening
Fig. 16. The shear stress measured on two fully grouted bolts in situ, (a) bolt No. 9, (b) bolt No. 10. After BjoÈrnfot and Stephansson [4].
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displacement dJ and the tensile axial stress of the bolt at the joint, sb0, as follows (see Appendix B for the detailed derivations): 1. For fully grouted bolts (assuming x0=0), dJ 1
sb0 db , aEb
when sb0 R
2sp a
otherwise, " 2 x 2 ÿ D2 sb0 x 2 ÿ 2osp dJ Eb db # sp 2D2 ÿ 2o 1sp 2 db 3db a The position of the decoupling front is at db sb0 1 D ÿ ÿ 1 ÿ o x2 2o 2sp a db
23
24 Fig. 18. The axial stress and load of the bolt versus the opening displacement of the joint for fully grouted bolts. Parameters: sp=12 MPa, o=0.1, d2=20 mm, D=0.1 m, Eb=210 GPa, Er=45 GPa, Eg=30 GPa, nr=ng=0.25.
25
2. For fully frictionally coupled bolts, dJ 1
sb0 db , aEb
when sb0 R
2s a
otherwise, 2 p 2 1 x2 2 sb0 x 2 ÿ sdb dJ Eb 2A a The position of the decoupling front is at A 2s sb0 ÿ x2 pdb s a
26
27
28
Using the relevant equations above, the axial stress of the bolt at the joint, sb0, versus the opening displacement of the joint is calculated and shown in Fig. 18
Fig. 17. The shear stress (tb) and the axial tensile stress (sb), induced by joint opening, in fully coupled rock bolts
for a fully grouted bolt and in Fig. 19 for a fully frictionally coupled bolt. The values of the relevant parameters used for the calculations are listed in the captions of the ®gures. It is seen that the bolt interface starts to be decoupled at a very small opening displacement of the joint. This con®rms the results arrived at by other studies [8,15,16], showing that decoupling of the interface occurs at an extremely small displacement, because the compatibility of deformation is lost across the interface at such a low load. Field measurements, for instance those carried out by BjoÈrnfot and Stephansson [2,4], have demonstrated that bolts installed in jointed rock masses sometimes are subjected to several axial stress peaks. These peaks are thought to be caused by the opening of the rock joints intersecting the bolt. The following is an example to show the axial stress along a bolt intersect-
Fig. 19. The axial stress and load of the bolt versus the opening displacement of the joint for frictionally coupled bolts. Parameters for Standard Swellex: s = 0.7 MPa, db=39 mm, t = 2 mm, Eb=210 GPa, Er=45 GPa, nr=0.25.
C. Li, B. Stillborg / International Journal of Rock Mechanics and Mining Sciences 36 (1999) 1013±1029
1025
uniformly deformed rock masses, the bolt has a pickup length, an anchor length and a neutral point; (ii) the face plate enhances the reinforcement eect through inducing a direct tensile load in the bolt and reducing the shear stress carried on the bolt surface; and (iii) in jointed rock masses, the opening displacement of rock joints will induce axial stress peaks in the bolt. Acknowledgements
Fig. 20. Axial stress along a bolt subjected to joint openings. The opening displacements: dJa=50 mm at joint a, dJb=20 mm at joint b and dJc=5 mm at joint c.
ing three rock joints. Assume that the three joints, a, b and c, have opened 50, 20 and 5 mm, respectively, since the bolt was installed (see Fig. 20). The axial stress along the bolt would be the superposition of the stresses caused by the opening displacements at the three joints. Assuming that the bolt interface is still in the stage of compatible deformation, the axial stress can be expressed as: sb x Ssb0i e
2a ÿ d jxÿx i j b
i a, b and c
29
where sb0i=(aEb/db)dJi, according to Eq. (23) for fully grouted bolts. Using the following values for the relevant parameters: a=0.23; Eb=210 GPa; db=20 mm; dJa=50 mm at xa=0.4 m; dJb=20 mm at xb=0.6 m; dJc=5 mm at xc=0.8, we obtain the axial stress along the bolt as illustrated in Fig. 20. Hyett et al. [12] and Bawden et al. [16] obtained similar results through numerical simulations.
5. Concluding remarks An analytical model has been established for rock bolts subjected to a pull load in pullout tests. Decoupling starts at the loading point and propagates along the bolt with an increasing applied load. The shear stress at the decoupled interface is lower than the ultimate shear strength of the interface and even drops to zero for fully grouted bolts, while it is approximately at the same magnitude as the ultimate shear strength for fully frictionally coupled bolts. The shear stress on the non-decoupled interface decreases exponentially with increasing distance from the decoupling front. Two analytical models have been developed for rock bolts in situ, one for uniform rock deformation and another for discrete joint opening. For rock bolts in situ, the models con®rm the previous ®ndings: (i) in
The grant for this work from AÊke and Greta Lisshed's Foundation is acknowledged. The valuable comments by the anonymous reviewers are greatly appreciated.
Appendix A. Stress analysis of a rock bolt in situ Let us consider a rock bolt installed within a rock mass (see Fig. A1). It is assumed that the range of in¯uence of one bolt extends half the distance to all adjacent bolts. Thus, in pattern bolting, the area of in¯uence of one bolt equals the area surrounded by four adjacent bolts. Consider a thin slice of the bolted rock, dx, which will be used to study the interaction between the bolt and the rock. The thin slice of the bolt-reinforced rock is shown in Fig. A2. Let the free deformation of the rock slice dx, i.e. the deformation before bolting, be termed as du. The deformation of the rock slice becomes dub when it is reinforced by a bolt. The elongation of the bolt is also dub if it is assumed that the bolt and the rock are deformed together. The magnitude of dub can thus be calculated from the elongation of the bolt. The reduction of deformation, dur, is the result of the stress increment, Dsr, in the rock mass induced by bolting. It is obvious that the sum of dur and dub equals the free deformation du,
Fig. A1. A sketch illustrating a bolt installed within a rock mass.
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C. Li, B. Stillborg / International Journal of Rock Mechanics and Mining Sciences 36 (1999) 1013±1029
shear stress on the bolt interface can be expressed as: tb1 x ÿ
i.e.: sb Dsr dx dx Eb Er
A5
where db is the diameter of the bolt. The shear stress tb1 is caused by rock deformation. On the other hand, the shear stress on the bolt interface in the section between ri and x has a pull eect on the section of the bolt on the right side of x, and therefore induces another component of shear stress, tb2. The shear stress tb1 on a small element of the bolt, dt, brings about a normal stress increment, dsb(t ), in the bolt. Similarly to Eq. (2), the shear stress increment at x, dtb2(x ), induced by the normal stress increment dsb(t ) at t, can be expressed as;
Fig. A2. Stress components in the bolt and in the rock.
du dub dur
A dsb A d2 u xGr pdb dx pdb dx 2
A1
xÿt a ÿ2a d b dtb2 x dsb te 2
A6
where dx is the length of the rock slice, du is the free deformation of the rock slice, dub is the elongation of the bolt element, dur is the reduced deformation of the rock due to bolting, sb is the tensile stress in the bolt, Dsr is the compressive stress increment in the rock mass, induced by bolting, Er is Young's modulus of the rock mass, and Eb is Young's modulus of the bolt steel. The force equilibrium on the plane perpendicular to the bolt gives:
The total shear stress at x induced due to the pull eect of the shear stress on the bolt section between ri and x is obtained by integration of the above shear stress increment, that is:
sb A ÿDsr S
tb x tb1 tb2
A2
where A is the cross-section area of the bolt, and S is the in¯uencing area of the bolt in the rock, equal to the area surrounded by four adjacent bolts in pattern bolting. Substituting Eq. (A2) into Eq. (A1), we obtain the expressions for sb and Dsr as: sb x ÿxGr
Dsr x xGr
du dx
A du S dx
A3
where x
2 1 nr SEb AEb SEr
Er Gr 2 1 nr
x
a tb2 x dtb2 x ÿ xGr 2 ri
du/dx is the ®rst-order derivative of the free radial displacement of the rock, u, with respect to x. From the point of view of force equilibrium, the
d2 u ÿ2a xÿt db dt e 2 ri dt
A7
Finally, the total shear stress on the bolt at x is the sum of tb1 and tb2, that is: " xGr
A d2 u a ÿ pdb dx 2 2
x
d2 u ÿ2a xÿt db dt e 2 ri dt
#
A8
Eq. (A8) is a general solution to the shear stress on the bolt without decoupling at the bolt interface. When decoupling occurs, a certain portion of the load originally carried on the decoupled section of the bolt will either be transferred to the face plate, if there is one, or released with a free rock deformation in the case without a face plate. For the case with a face plate (see Fig. 14), the load transferred to the face plate due to decoupling is calculated as: Pf pdb
A4
x
rp ri
ÿtb1 ÿ sr dt
ÿAxGr
rp ri
d2 u dt ÿ pdb sr rp ÿ ri dt2
A9
The shear stress, at x=rp, induced by the axial load on the face-plate load, Pf , and by the shear stress on the decoupled interface, sr, is:
C. Li, B. Stillborg / International Journal of Rock Mechanics and Mining Sciences 36 (1999) 1013±1029
a Pf pdb sr rp ÿ ri 2 A rp 2 a d u dt ÿ xGr 2 2 ri dt
tb2 rp
A10
For equilibrium the total shear stress at rp should equal the peak shear strength of the interface, that is: ÿsp tb1 rp tb2 rp " xGr
A d2 u a ÿ pdb dx 2 2
rp ri
d2 u dt dt2
# A11
Eq. (A11) is used to determine the distance rp for bolts with a face plate. For x > rp, the total shear stress is calculated as: # " A d2 u a x d2 u ÿ2a xÿt db dt ÿ e tb x xGr pdb dx 2 2 ri dt2 A12 # " rp 2 xÿt a d u ÿ2a d b dt e ÿ xGr 2 2 dt ri For a bolt without a face plate, Pf is zero. The shear stress, at x=rp, induced by the shear stress on the decoupled interface, sr, becomes tb2 rp
a pdb sr rp ÿ ri 2 A
A13
Similarly to the case with a face plate, the total shear stress at rp should equal the peak shear strength of the interface, that is ÿsp tb1 rp tb2 rp xGr
A d2 u a pdb sr rp ÿ ri pdb dx 2 2 A
A14
Eq. (A14) is used to determine the distance rp for bolts without a face plate. For x > rp, the total shear stress is calculated as: 3 2 x 2 ÿ2a x ÿ t 2 4 A d u a d u db dt 5 ÿ e tb x xGr pdb dx 2 2 rp dt2 A15 xÿt ÿ2a a pdb db sr rp ÿ ri e 2 A
1027
opening and the load induced in the bolt. An opening displacement of the rock joint is equivalent to applying an axial tensile load to both sides of the bolt at the joint. We shall look at this problem for fully grouted bolts and for frictionally coupled bolts separately. B.1. Fully grouted rock bolts The axial tensile stress in the bolt is symmetric to the rock joint. Therefore, we consider only half of the rock±bolt system. The model for the shear stress along a bolt subjected to an axial load is shown in Fig. 4. The elongation of the bolt in dierent sections is denoted as follows: d0 is the elongation of the bolt in section (0 R x < x0); d1 is the elongation of the bolt in section (x0 R x < x1); d2 is the elongation of the bolt in section (x1 R x < x2); and d3 is the elongation of the bolt in section (x2 R x < L ), where L is the half length of the bolt. The sum of these four components is the total elongation of the bolt from each side of the joint. Thus, the displacement of the joint opening is twice this summation, i.e. dJ 2
3 X di
B1
i0
Not all the four elongation comments appear in the above expression at any given time. When the joint opens very little, the axial load induced does not cause the interface to be decoupled. In this case, only d3 exists in Eq. (B1). The components d2, d1 and d0 appear subsequently in the equation with increases in the joint opening. 1. For the case where the interface undergoes compatible deformation across the interface, the shear stress along the bolt is illustrated in Fig. 6. In this case we have d0 d1 d 2 0
B2
As shown in Eqs. (2) and (4), the shear and axial stresses along the bolt are given by
Appendix B. Joint opening and the stresses induced in rock bolts Fig. B1 illustrates a bolt intersecting a rock joint. We shall establish the relationship between the joint
Fig. B1. A sketch illustrating a rock bolt intersecting a joint.
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C. Li, B. Stillborg / International Journal of Rock Mechanics and Mining Sciences 36 (1999) 1013±1029
x a ÿ2a tb x sb0 e db 2
sb x sb0 e
sb x 2
x ÿ2a d
B3
b
This stage ceases when the shear stress at x = 0 reaches the peak shear strength of the interface, sp. The corresponding axial stress at the joint at this moment is sb0
2sp a
B4
The elongation of the bolt is calculated as (assuming L>>db) 2 L db sb0 s xdx1 B5 dJ 2d3 aEb Eb 0 When sb0=(2sp/a ), the opening displacement reaches its maximum, dJmax, before decoupling occurs. It is given by: dJmax
2db sp 2 a Eb
B6
2. For the case where decoupling occurs (assuming x0=0): Let D x 2 ÿ x 1 , and o sr =sp For x < x 1 :
sb x sb0 ÿ 4sr
x db
B7
For x1 R x < x2:
sb x sb0 ÿ
2sp x ÿ x 1 2ox 1 ÿ o db D
For xrx2:
sb x
ÿ2a
B10
The elongation of the bolt can be obtained by the following integration 1 sb dx B11 di e dx Eb i.e. 1 d1 Eb
x 1 0
x sb0 ÿ 4sr dx db
"
1 x 2 ÿ D2 sb0 x 2 ÿ D ÿ 2sr db Eb
#
(
2sp 2ox sb0 ÿ db x1 ) x ÿ x 1 2 dx 1 ÿ o D ( ) 1 2D2 2o 1sp sb0 D ÿ Eb 3db
1 d2 Eb
x 2
L
B12
2sp ÿ2a e x2 a
ÿ xÿx 2 db
dx1
1 sp db E b a2
B13
B14
The opening displacement of the joint is calculated as dJ 2 d1 d2 d3
x ÿ x1 tb x sr 1 ÿ osp x2 ÿ x1
tb x sp e
Therefore, we obtain db sb0 2 D ÿ ÿ 2 1 ÿ o x2 4o sp a db
1 d3 Eb
tb x sr
2sp a
2
B8
" 2 x 2 ÿ D2 sb0 x 2 ÿ 2osp db Eb # sp 2D2 ÿ 2o 1sp 2 db 3db a
B15
ÿ xÿx 2 db
ÿ 2 2sp ÿ2a xÿx db e a
At x=x2, we have from Eqs. (B8) and (B9) sb x 2 sb0 ÿ
2sp 2ox 2 1 ÿ oD db
B9
B.2. Fully frictionally coupled rock bolts In the stage of compatible deformation, the shear and axial stresses have the same forms as those expressed in Eq. (B3) and the elongation has the same forms as those expressed in Eqs. (B5) and (B6). When decoupling occurs at the interface, the shear stress along the bolt is illustrated in Fig. 5. The
C. Li, B. Stillborg / International Journal of Rock Mechanics and Mining Sciences 36 (1999) 1013±1029
elongation of the bolt in this case is calculated as follows. For x R x2: the stresses are
dJ 2 d1 d3
tb x s
References
sb x sb0 ÿ
pdb x s A
B16
The elongation of the bolt in section (0 R x < x2) is x 2 pdb sx dx sb0 ÿ A 0 1 pdb 2 sx sb0 x 2 ÿ Eb 2A 2
1 d1 Eb
B17
For xrx2: tb x s e
sb x
xÿx ÿ2a d 2 b
2 2s ÿ2a xÿx db e a
B18
At x=x2, we have from Eqs. (B16) and (B17) sb x 2 sb0 ÿ
sb x 2
pdb sx 2 A
2s a
Then we obtain the expression for x2 as A 2s sb0 ÿ x2 pdb s a
B19
The elongation of the bolt in section (x2 R x < L ) is calculated as d3
1 Eb
L
2 2s ÿ2a xÿx db dx e x2 a
ÿ2a 1 sdb 1 ÿ e 2 Eb a
L ÿ x2 db 1 1 sdb Eb a2
The total elongation is
B20
2 psdb 2 sdb x2 2 sb0 x 2 ÿ 2A a Eb
1029
B21
[1] Freeman TJ. The behaviour of fully-bonded rock bolts in the Kielder experimental tunnel. Tunnels and Tunnelling June 1978:37±40. [2] BjoÈrnfot F, Stephansson O. Interaction of grouted rock bolts and hard rock masses at variable loading in a test drift of the Kiirunavaara Mine, Sweden. In: Stephansson O, editor. Proceedings of the International Symposium on Rock Bolting. Rotterdam: Balkema, 1984. p. 377±95. [3] Sun X. Grouted rock bolt used in underground engineering in soft surrounding rock or in highly stressed regions. In: Stephansson O, editor. Proceedings of the International Symposium on Rock Bolting. Rotterdam: Balkema, 1984. p. 93±9. [4] BjoÈrnfot F, Stephansson O. Mechanics of grouted rock bolts Ð ®eld testing in hard rock mining. Report BeFo 53:1/84, Swedish Rock Engineering Research Foundation 1984. [5] Hawkes JM, Evans RH. Bond stresses in reinforced concrete columns and beams. Journal of the Institute of Structural Engineers 1951;XXIX(X):323±7. [6] Farmer IW. Stress distribution along a resin grouted rock anchor. Int J Rock Mech Min Sci and Geomech Abstr 1975;12:347±51. [7] Dunham DK. Anchorage tests on strain-gauged resin bonded bolts. Tunnels and Tunnelling, September 1976:73±6. [8] Stillborg B. Professional users handbook for rock bolting, 2nd ed. Trans. Tech. Publications, 1994. [9] Stjern G. Practical performance of rock bolts. Doctoral thesis 1995:52, Universitetet i Trondheim, Norway. [10] Tao Z, Chen JX. Behaviour of rock bolting as tunnelling support. In: Stephansson O, editor. Proceedings of the International Symposium on Rock Bolting. Rotterdam: Balkema, 1984. p. 87±92. [11] Indraratna B, Kaiser PK. Analytical model for the design of grouted rock bolts. Int J for Numerical and Analytical Methods in Geomechanics 1990;14:227±51. [12] Hyett AJ, Mossavi M, Bawden WF. Load distribution along fully grouted bolts, with emphasis on cable bolt reinforcement. Int J for Numerical and Analytical Methods in Geomechanics 1996;20:517±44. [13] Windsor CR. Rock reinforcement systems. Int J Rock Mech Min Sci 1997;34(6):919±51. [14] Holmberg M. The mechanical behaviour of untensioned grouted rock bolts. Doctoral thesis, ISRN KTH/JOB/R-91-SE, Royal Institute of Technology, Stockholm, Sweden, 1991. 128 pp. [15] Bawden WF, Hyett AJ, Lausch P. An experimental procedure for the in situ testing of cable bolts. Int J Rock Mech Min Sci and Geomech Abstr 1992;29(5):525±33. [16] Bawden WF, Moosavi M, Hyett AJ. Evaluation of load distribution along conventional and modi®ed strand cable anchors using computer aided bolt load estimation (CableTM) software. In: International Symposium on Rock Support Ð Applied Solutions for Underground Structures, Lillehammer, Norway, 1997. p. 25±39.
Analytical models for rock bolts C. Li*, B. Stillborg Division of Rock Mechanics, LuleaÊ University of Technology, SE-971 87 LuleaÊ, Sweden Accepted 21 August 1999
Abstract Three analytical models have been developed for rock bolts: one for bolts subjected to a concentrated pull load in pullout tests, one for bolts installed in uniformly deformed rock masses, and one for bolts subjected to the opening of individual rock joints. The development of the models has been based on the description of the mechanical coupling at the interface between the bolt and the grout medium for grouted bolts, or between the bolt and the rock for frictionally coupled bolts. For rock bolts in pullout tests, the shear stress of the interface attenuates exponentially with increasing distance from the point of loading when the deformation is compatible across the interface. Decoupling may start ®rst at the loading point when the applied load is large enough and then propagate towards the far end of the bolt with a further increase in the applied load. The magnitude of the shear stress on the decoupled bolt section depends on the coupling mechanism at the interface. For fully grouted bolts, the shear stress on the decoupled section is lower than the peak shear strength of the interface, while for fully frictionally coupled bolts it is approximately the same as the peak shear strength. For rock bolts installed in uniformly deformed rock, the loading process of the bolts due to rock deformation has been taken into account in developing the model. Model simulations con®rm the previous ®ndings that a bolt in situ has a pick-up length, an anchor length and a neutral point. It is also revealed that the face plate plays a signi®cant role in enhancing the reinforcement eect. In jointed rock masses, several axial stress peaks may occur along the bolt because of the opening of rock joints intersecting the bolt. # 2000 Elsevier Science Ltd. All rights reserved.
1. Introduction Rock bolts have been widely used for rock reinforcement in civil and mining engineering for a long time. Bolts reinforce rock masses through restraining the deformation within the rock masses. In order to improve bolting design, it is necessary to have a good understanding of the behaviour of rock bolts in deformed rock masses. This can be acquired through ®eld monitoring, laboratory tests, numerical modelling and analytical studies. Since the 1970s, numerous researchers have carried out ®eld monitoring work on rock bolts installed in various rock formations [1±3]. Freeman [1] performed pioneering work in studying the per* Corresponding author. Tel.: +46-920-91352; fax: +46-92091935. E-mail address: [email protected] (C. Li).
formance of fully grouted rock bolts in the Kielder experimental tunnel. He monitored both the loading process of the bolts and the distribution of stresses along the bolts. On the basis of his monitoring data, he proposed the concepts of ``neutral point'', ``pick-up length'' and ``anchor length''. At the neutral point, the shear stress at the interface between the bolt and the grout medium is zero, while the tensile axial load of the bolt has a peak value. The pick-up length refers to the section of the bolt from the near end of the bolt (on the tunnel wall) to the neutral point. The shear stresses on this section of the bolt pick up the load from the rock and drag the bolt towards the tunnel. The anchor length refers to the section of the bolt from the neutral point to the far end of the bolt (its seating deep in the rock). The shear stresses on this section of the bolt anchor the bolt to the rock. These concepts clearly outline the behaviour of fully grouted rock bolts in a deformed rock formation. BjoÈrnfot and
1365-1609/99/$ - see front matter # 2000 Elsevier Science Ltd. All rights reserved. PII: S 1 3 6 5 - 1 6 0 9 ( 9 9 ) 0 0 0 6 4 - 7
1014
C. Li, B. Stillborg / International Journal of Rock Mechanics and Mining Sciences 36 (1999) 1013±1029
Nomenclature A Eb Er Eg L P0 P0max
area of the cross-section of the bolt Young's modulus of the bolt steel Young's modulus of the rock mass Young's modulus of the grout length of the bolt applied pull load pullout load, i.e. the maximum applied pull load S in¯uencing area of a bolt in the rock diameter of a circle in the rock outside d0 which the in¯uence of the bolt disappears diameter of the bolt db diameter of the borehole dg du free deformation of the rock slice elongation of the bolt element dub the reduction of deformation after bolt redur inforcement, i.e. dur=du ÿ dub dx length of the rock slice p0 hydrostatic primary stress in the rock load on the face plate of the bolt Pf radius of the circular tunnel ri position of the decoupling front on the bolt rp surface s shear strength of the interface for frictionally coupled bolts residual shear strength of the interface for sr fully grouted bolts peak shear strength of the interface for fully sp grouted bolts u original radial displacement of the rock at x (without bolting) x0,x1 decoupling boundaries at the interface of and x2 fully grouted bolts (see Fig. 4) Greek symbols D length of a bolt section, D=x2ÿx1 Stephansson's work [2,4] demonstrated that in joined rock masses there may exist not only one but several neutral points along the bolt because of the opening displacement of individual joints. Pullout tests are usually used to examine the anchoring capacity of rock bolts. A great number of pullout tests have been conducted so far in various types of rocks [5±9]. Farmer [6] carried out fundamental work in studying the behaviour of bolts under tensile loading. His solution predicts that the axial stress of the bolt (also the shear stress at the bolt interface) will decrease exponentially from the point of loading to the far end of the bolt before decoupling occurs. Fig. 1(a) illustrates the results of a typical pullout test [5]. Curve a represents the distribution of the axial stress along
a d0 d1 d2 d3 dJ dJi dJmax ng nr x sb sb0 sb0i Dsr tb tbB tb1 tb2 t dA t dB tA B o
a constant representing the coupling property of the interface elongation of the bolt in section (0Rx
the bolt under a relatively low applied load, at which the deformation is compatible on both sides of the bolt interface. Curve b represents the axial stress along the bolt at a relatively high applied load, at which decoupling has occurred at part of the bolt interface. Fig. 1(b) shows the axial stress along a rock bolt installed in an underground mine drift [3]. It is seen from this ®gure that the distribution of the axial stress along the section close to the borehole collar is completely dierent from that in pullout tests. However, along the section to the far end of the bolt, the stress varies similarly to that in pullout tests. The reason for these results is that bolts in situ have a pick-up length and an anchor length, while bolts in pullout tests only have an anchor length.
C. Li, B. Stillborg / International Journal of Rock Mechanics and Mining Sciences 36 (1999) 1013±1029
It is thought that the relative movement between the rock and the bolt is zero at the neutral point [1]. In the solution by Tao and Chen [10], the position of the neutral point depends only on the radius of the tunnel and the length of the bolt. That solution was implemented in the analytical models created by Indraratna and Kaiser [11] and Hyett et al. [12]. It seems that Tao and Chen's solution is valid only when the deformation is compatible across the bolt interface. When decoupling occurs, the position of the neutral point is obviously also related to the shear strength of the interface. Field monitoring and pullout tests have indicated two facts concerning the loading of a rock bolt in situ: (1) rock deformation applies a load on the pick-up section of the bolt; (2) the load on the pick-up section drags the anchor section of the bolt towards the underground opening. These two facts must be taken into account in developing analytical models for rock bolts. The aim of this paper is to develop analytical models for fully coupled rock bolts. A model for rock bolts in pullout tests is introduced ®rst, together with
Fig. 1. Distribution of the axial stress (a) along a grouted steel bar during a pullout test, after Hawkes and Evans [5], and (b) along a grouted rock bolt in situ, after Sun [3].
1015
a description of the theoretical background, the development of the model and an illustrative example. Two models for rock bolts in situ are then presented, one in uniformly deformed rock masses and one in jointed rock masses. The details of the development of the models are summarised in the Appendices. 2. Coupling between the bolt and the rock Windsor [13] proposed the concept that a reinforcement system comprises four principal components: the rock, the reinforcing element, the internal ®xture and the external ®xture. For reinforcement with a bolt, the reinforcing element refers to the bolt and the external ®xture refers to the face plate and nut. The internal ®xture is either a medium, such as cement mortar or resin for grouted bolts, or a mechanical action like ``friction'' at the bolt interface for frictionally coupled bolts. The internal ®xture provides a coupling condition at the interface. With reference to the component of internal ®xture, Windsor [13] classi®ed the current reinforcement devices into three groups: ``continuously mechanically coupled (CMC)'', ``continuously frictionally coupled (CFC)'' and ``discretely mechanically or frictionally coupled (DMFC)'' systems. According to this classi®cation system, cementand resin-grouted bolts belong to the CMC system, while Split set and Swellex bolts belong to the CFC system. When fully grouted bolts are subjected to a pull load, failure may occur at the bolt±grout interface, in the grout medium or at the grout±rock interface, depending on which one is the weakest. For fully frictionally coupled bolts, however, there is only one possibility of failure Ð decoupling at the bolt±rock interface. In this study we concentrate on the failure at the interface between the bolt and the coupling medium (either the grout medium or the rock). In general, the shear strength of an interface comprises three components: adhesion, mechanical interlock and friction. They are lost in sequence as the compatibility of deformation is lost across the interface. The result is a decoupling front that attenuates at an increasing distance from the point of the applied load. The decoupling front ®rst mobilises the adhesive component of strength, then the mechanical interlock component and ®nally the frictional component. The shear strength of the interface decreases during this process. The shear strength after the loss of some of the strength components is called the residual shear strength in this paper. For grouted rock bolts like rebar, all the three components of strength exist at the bolt interface. However, for the fully frictionally coupled bolt, the ``Split set'' bolt, only a friction component exists at the bolt interface. For Swellex bolts,
1016
C. Li, B. Stillborg / International Journal of Rock Mechanics and Mining Sciences 36 (1999) 1013±1029
mechanical interlock and friction comprise the strength of the interface. 3. Rock bolts in pullout tests 3.1. Theoretical background When a bolt installed in rock is subjected to a tensile axial load, the relationship between the shear stress at the bolt interface and the axial tensile stress of the bolt can be established through considering a small section of the bolt as shown in Fig. 2. The force equilibrium in the axial direction leads to the following expression: tb ÿ
A dsb pdb dx
1
where db is the diameter of the bolt, and A is the area of the cross-section of the bolt. For the example shown in Fig. 1(a), the shear stress along the bolt at the two levels of applied load is obtained using Eq. (1) and illustrated in Fig. 3. When the applied load is small, the shear stress decreases with increasing distance from the point of loading (curve a ). Progressive decoupling commences at the loading point at a certain level of applied load. The decoupling front moves towards the far end of the bolt with an increasing applied load. The shear stress is at the level of the shear strength at the decoupling front, while behind the decoupling front the shear stress becomes smaller, since the strength of the interface has been partially lost due to decoupling. Curve b in Fig. 3 represents such a distribution of shear stress along the bolt. Based on experimental results as shown in Fig. 3, a model for the shear stress along a fully grouted bolt can be postulated as illustrated in Fig. 4. In this model, the section of the bolt close to the loading point is completely decoupled with a zero shear stress
Fig. 2. Stress components in a small section of a bolt.
Fig. 3. The shear stress on the steel bar, derived from Fig. 1(a).
at the bolt interface. Starting at a certain distance from the loading point, say at x0, the bolt interface is partially decoupled with a residual shear strength, sr. Between point x1 and x2, the residual shear strength linearly increases from sr to the peak strength sp. Beyond point x2, the interface undergoes compatible deformation and the shear stress attenuates exponentially towards the far end of the bolt. For fully frictionally coupled bolts, the magnitude of the shear stress behind the decoupling front is approximately the same as the peak value. As mentioned previously, the shear strength of the interface for this type of bolt comprises one or two components, i.e. either friction or mechanical interlock and friction. The deformation incompatibility across the interface does not make the friction disappear. In other words, the residual shear strength of the interface is approximately the same as the peak strength for fully frictionally coupled bolts. The distribution of shear stress for this type of bolt is illustrated in Fig. 5. When a fully coupled bolt is subjected to a pull load, the shear stress along the bolt is as shown in Fig.
Fig. 4. Distribution of shear stress along a fully grouted rock bolt subjected to an axial load.
C. Li, B. Stillborg / International Journal of Rock Mechanics and Mining Sciences 36 (1999) 1013±1029
1017
bolts if the expression for the constant a is slightly modi®ed. Fully frictionally coupled bolts have direct contact with rock. The constant a for this type of bolt can be obtained from Eq. (3a) by letting the Young's modulus and Poisson's ratio of the grout equal those of the rock, i.e. Gg=Gr, ng=nr and dg=dr. Then we obtain the expression of a for fully frictionally coupled bolts as: a2
Fig. 5. Distribution of shear stress along a frictionally coupled rock bolt.
6 before decoupling occurs at the interface. For fully grouted rock bolts, the attenuation of the shear stress is expressed as [6,14]: x a ÿ2a tb sb0 e db 2
2
2Gr d0 Eb ln db
3b
The axial stress of the bolt is calculated as: x pdb x ÿ2a tb xdx sb0 e db sb x sb0 ÿ A 0
4a
or 2 sb x tb x a
4b
where a2
" Eb
Gr
2Gr Gg # dg d0 Gg ln Gr ln db dg
Er 2 1 nr
and Gg
Eg 2 1 ng
3.2. Fully grouted rock bolts The stresses in dierent sections of the bolt can now be described in detail as follows (see Fig. 4): 3a
sb0 is the axial stress of the bolt at the loading point, Eb is Young's modulus of the bolt steel, Er is Young's modulus of the rock mass, Eg is Young's modulus of the grout, nr is Poisson's ratio of the rock mass, ng is Poisson's ratio of the grout, dg is the diameter of the borehole, and d0 is the diameter of a circle in the rock outside which the in¯uence of the bolt disappears. Eq. (2) is also valid for fully frictionally coupled
1. On the section 0 R x < x0: the bolt interface is completely decoupled, leading to a zero shear stress at the interface and a constant axial stress in the bolt, i.e.: tb x 0 sb x sb0
5
2. On the section x0 R x < x1: the interface is partially decoupled, resulting in a residual shear strength sr at the interface. The shear and axial stresses are given by: tb x sr
sb x sb0 ÿ
4sr x ÿ x 0 db
6
3. On the section x1 R x < x2: the interface is partially decoupled with the residual shear strength linearly increasing to the peak strength. The shear and axial stresses are given by: Fig. 6. Shear stress along a fully coupled rock bolt subjected to an axial load before decoupling occurs.
tb x osp
x ÿ x1 1 ÿ osp D
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C. Li, B. Stillborg / International Journal of Rock Mechanics and Mining Sciences 36 (1999) 1013±1029
sb x sb0
ÿ
2sp 1 ÿ o x ÿ x 1 2 2o x ÿ x 0 db D
7
where D=x2ÿx1, and o=sr/sp, the ratio of the residual shear strength to the peak shear strength. 4. On the section x > x2: the deformation is compatible across the interface and no decoupling occurs. According to Eqs. (2) and (4), both the shear and axial stresses decrease exponentially towards the far end of the bolt: ÿ xÿx ÿ2a d 2 b tb x sp e sb x
ÿ 2 2sp ÿ2a xÿx db e a
8
It is seen from Eq. (8) that the axial stress at x=x2 is given by 2sp sb x 2 a On the other hand, the axial stress at x=x2 can be obtained from Eq. (7) as: sb x 2
2sp 4P0 ÿ 2o x 2 ÿ x 0 1 ÿ oD 2 db pd b
where P0 is the applied pull load. Letting the right sides of the above two expressions be equal, we obtain the expression for the position of the decoupling front, x2, as: 1 2P0 db ÿ 1 ÿ oD 9 ÿ x2 x0 2o pdb sp a For equilibrium the applied load P0 should equal the total shear force at the bolt interface, i.e. L 1 P0 pdb tb dx pdb sr x 1 ÿ x 0 sp D 1 o 2 x0 2a db ÿ d Lÿx 2 b sp 1 ÿ e 2a where L is the length of the bolt. It is obtained from the above expression that the maximum applied load P0max can be expressed as: db ln o ÿ D ÿ x 0 P0max pdb sp o L 2a 10 1 db D 1 o 1 ÿ o 2a 2 The following is an example to demonstrate how to back-calculate the peak shear strength of the interface
on the basis of the pullout load. Stillborg [8] conducted a series of pullout tests on dierent types of rock bolts. In one test, a 3 m long rebar with a diameter of 20 mm was grouted within two identical concrete blocks. The length of the bolt in each block was 1.5 m. One block was ®xed to the ground, while the other was pulled. The bolt was pulled out without rupture, indicating that decoupling of the interface occurred along the entire length of the bolt. The pullout load registered was 180 kN. It is assumed that the distribution of shear stress has the form illustrated in Fig. 4 with x0=0. It is known from the test that: P0max 180 kN,
L 1:5 m,
dg 35 mm,
Eb 210 GPa
db 20 mm,
The values of the other parameters are assumed to be: Er concrete 45 GPa,
Eg cement mortar 30 GPa,
nr ng 0:25 o sr =sp 0:1,
D 0:1 m,
d0 10dg
It is then obtained that the constant a=0.23 from Eq. (3a) and the peak shear strength sp=12.8 MPa from Eq. (10). The shear stress and the axial load along the rebar are calculated on the basis of the model and illustrated in Fig. 7. The axial load along the bolt at dierent levels of applied load is illustrated in Fig. 8. The curves in Fig. 8 are similar to those obtained in pullout tests (e.g. Fig. 1(a)).
3.3. Fully frictionally coupled rock bolts For fully frictionally coupled rock bolts, the residual shear strength of the interface is approximately the same as the peak shear strength, i.e. sr=sp=s (see Fig. 5). The shear stress on dierent sections of the bolt is described in detail as follows: 1. On the section 0 R x < x2: the shear stress has reached the level of the strength of the interface. The shear stress on this section remains constant, while the axial stress linearly decreases, i.e.: tb x sr sb x sb0 ÿ
pdb sx A
11
2. On the section x > x2: the deformation is compatible across the interface and the shear stress is less than the peak shear strength. Both the shear and the axial stresses decrease exponentially towards the far end of the bolt:
C. Li, B. Stillborg / International Journal of Rock Mechanics and Mining Sciences 36 (1999) 1013±1029
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Fig. 7. The shear stress and axial load along a fully grouted rock bolt subjected to an axial load of 90 kN.
tb x s e sb x
ÿ xÿx 2
ÿ2a
2s ÿ2a e a
db
ÿ xÿx 2 db
sb x 2 12
It is seen from Eq. (12) that the axial stress at x=x2 is given by
2s a
On the other hand, the axial stress at x=x2 can be obtained from Eq. (11) as: sb x 2
P0 pdb ÿ sx 2 A A
Fig. 8. Axial load along a fully grouted rock bolt at dierent levels of applied load.
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Letting the right sides of the above two expressions be equal, we obtain the expression for the position of the decoupling front, x2, as: 1 2As P0 ÿ 13 x2 pdb s a For equilibrium the applied load P0 should equal the total shear force at the bolt interface, i.e. L P0 pdb tb dx 0
spdb x 2
pd 2b 2
s 1ÿe a
Lÿx 2 ÿ2a db
14
The applied load reaches its maximum, P0max, when the shear strength of the interface is mobilised along the entire length of the bolt, i.e. when x2=L. Substituting P0=P0max and x2=L into Eq. (13), we obtain the shear strength for fully frictionally coupled bolts as: s
P0max pdb L
15
Stillborg [8] tested the Swellex bolt in his pullout tests. Displacement monitoring at the far end of the bolt indicated that the bolt was slipping under the load P0max=110 kN. That indicated that decoupling occurred along the entire length of the bolt. The diameter of the borehole was 35 mm. The diameter of the Swellex bolt is the same as that of the borehole, i.e.
db=35 mm. The length of the bolt section embedded in each concrete block was 1.5 m long, i.e. L = 1.5 m. Substituting these data into Eq. (15) yields the shear strength of the bolt interface, i.e. s = 0.7 MPa. It is obtained from Eq. (3b) that the constant a=0.27. The shear stress and the axial load along the Swellex bolt are calculated using the relevant equations above and are illustrated in Fig. 9. The axial load of the bolt at dierent levels of applied load is shown in Fig. 10.
4. Rock bolts in situ 4.1. A model for bolts subjected to uniform rock deformation Rock bolts in situ tend to restrain the deformation of rock with an increase in their axial loads. In other words, it is rock deformation that applies a load to rock bolts in situ. For the sake of simplicity, a bolt anchored at two points, as illustrated in Fig. 11, is used to explain the superposition of two components of the shear stress. Rock deformation will induce a component of shear stress t dA at A and a component of shear stress t dB at B. Assuming t dA > t dB, the shear force acting at anchor A would tend to drag the bolt to the left and thus induce another component of A shear stress at point B, t A B . The sense of t B is opposite d to the sense of t B. The total shear stress at B is: d tbB tA B ÿ tB
Fig. 9. The shear stress and axial load along a Swellex rock bolt at 5 kN of applied load.
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1021
Fig. 10. The axial load along a Swellex rock bolt at dierent levels of applied load.
For a fully coupled bolt, the component t A B will be an integration over the bolt section to the left side of point B. On the basis of this idea, we obtained the expression of the shear stress at position x on the bolt surface as follows (see Appendix A for detailed derivation): # " A d2 u a x d2 u ÿ2a xÿt db dt ÿ e 17 tb x xGr pdb dx 2 2 ri dt2 where x
2 1 nr SEb AEb SEr
18
u is the original radial displacement of the rock at x (without bolting), and S is the in¯uencing area of the
Fig. 11. A sketch illustrating the superposition of the components of shear stress at position B.
bolt in the rock, which equals the area surrounded by four adjacent bolts in pattern bolting. Here we take a tunnel circular in its cross-section as an example to demonstrate the application of Eq. (17). Assume that the country rock surrounding the tunnel undergoes an elastic deformation. The second-order derivative of the elastic radial displacement u of the rock can be expressed as: ux00 ÿ
p0 r2i Gr x 3
19
where ri is the radius of the circular tunnel, and p0 is the hydrostatic primary stress in the rock. Substituting Eq. (19) into Eq. (17) and using the following values for the relevant parameters: Young's modulus of the bolt steel Young's modulus of the rock mass Poisson's ratio of the rock mass bolt spacing diameter of the bolt radius of the circular tunnel hydrostatic primary stress in the rock
Eb=210 GPa, Er=5 GPa, nr=0.25, S = 1 m, db=20 mm, ri=4 m, p0=15 MPa,
we obtain the shear stress along the bolt as illustrated in Fig. 12. It can be seen that the sense of the shear stress on the bolt section close to the tunnel wall is negative; that is the direction of the shear stress is towards the tunnel. At a certain distance from the wall, the shear stress becomes zero. Beyond this neutral point, the sense of the shear stress becomes positive; that is the direction of the shear stress is towards
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ponents: one induced by the rock deformation, one due to the pull eect of the face plate load Pf , and one due to the pull eect of the shear force on the decoupled bolt section between ri and rp. In the case of no face plate, the face plate load Pf is zero. The total shear stress at x=rp equals the peak shear strength of the interface (see Appendix A), i.e. # " A d2 u a rp d2 u ÿ dt , ÿsp xGr pdb dx 2 2 ri dt2 21a for bolts with a face plate or Fig. 12. Shear stress along a fully grouted rock bolt under the condition of compatible interface deformation (Eb=210 GPa, Er=5 GPa, nr=0.25, S = 1 m, db=20 mm, ri=4 m, p0=15 MPa).
the far end of the bolt. This agrees with the ®eld monitoring data obtained, for example, by Freeman [1]. In this example, decoupling at the bolt interface has not been considered. When the rock deformation is large enough, the shear strength of the interface will be mobilised in the pick-up section of the bolt. The distribution of shear stress along the bolt, when decoupling occurs, will be as that illustrated in Fig. 13. The shear failure at the interface would result in a release of the restrained rock deformation at the near end of the bolt, if no face plate were to exist. In the case where a face plate exists, the displacement of the tunnel wall loads the plate. The load on the face plate can be calculated as: ! rp A d2 u xGr 2 ÿ sr dx ÿ 20 Pf pdb pdb dx ri Shear failure ceases at x=rp and beyond that point the displacement is compatible across the interface. The shear stress at x=rp is the sum of three com-
ÿsp xGr
A d2 u a pdb sr rp ÿ ri , pdb dx 2 2 A
21b
for bolts without a face plate Eqs. (21a) and (b) are used to determine the distance rp. The shear stress on the section x > rp is calculated as: # " A d2 u a x d2 u ÿ2a xÿt db dt ÿ e tb x xGr pdb dx 2 2 ri dt2 a ÿ xGr 2
"
rp
ri
# xÿrp d2 u ÿ2a d b , dt e dt2
22a
for bolts with a face plate or " tb x xGr
A d2 u a ÿ pdb dx 2 2
x
d2 u ÿ2a xÿt db dt e 2 rp dt
xÿrp a pdb ÿ2a d b , sr rp ÿ ri e 2 A
#
22b
for bolts without a face plate
Fig. 13. A schematic illustration of the shear stress along a rock bolt in situ.
We take the circular tunnel under elastic deformation again as an example to demonstrate the application of Eqs. (21) and (22). Assume that the peak shear strength of the bolt interface is sp=0.5 MPa and the residual shear strength sr=0.2 MPa. Using the values given before for other relevant parameters, we can obtain the decoupling boundary rp from Eq. (21) and the shear stress on the interface at x > rp from Eq. (22). The calculated results are shown in Fig. 14 for a fully grouted bolt with a face plate and in Fig. 15 for a fully grouted bolt without a face plate. It can be found by comparing the curves in these two ®gures that: (i) the decoupled length of the bolt is shorter with a face plate than without a face plate; and (ii) the axial stress in the decoupled section is larger for the
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1023
4.2. A model for bolts subjected to the opening of a rock joint
Fig. 14. Theoretical solution of the shear stress and axial stress along a fully grouted rock bolt with a face plate (sp=0.5 MPa, sr=0.2 MPa).
bolt with a face plate than the bolt without a face plate. That indicates that rock bolts with a face plate have a better reinforcement eect than those without a face plate. Fig. 16 shows the monitored results of the shear stress along two fully grouted bolts in a mine drift in Sweden [2,4]. The shear stress along bolt No. 9, presented in Fig. 16(a), agrees well with the theoretical curve shown in Fig. 12, implying that no decoupling occurred at the bolt interface. The shear stress along bolt No. 1, presented in Fig. 16(b), matches the curve shown in Fig. 15, indicating that decoupling occurred at the interface of this bolt. The analytical model introduced in this section provides a means for studying rock bolts in a uniformly deformed rock mass. The key for determining the shear stress along the bolt is the original rock deformation around the excavation opening.
Fig. 15. Theoretical solution of the shear stress and axial stress along a fully grouted rock bolt without a face plate (sp=0.5 MPa, sr=0.2 MPa).
The opening of a rock joint applies a tensile load to both sides of the bolt intersecting the joint. During joint opening, decoupling of the bolt interface is activated ®rstly at the joint and then propagates along the interface with an increase in the opening displacement. When the embedment length of the fully coupled bolt is suciently long on each side of the joint, the shear stress as well as the axial stress along the bolt will be symmetrical to the joint, as shown in Fig. 17. When the opening displacement of the joint is small, both the shear stress and the axial stress decrease exponentially with increasing distance from the joint. When the opening displacement is large enough, decoupling will be activated at the bolt interface and the shear and axial stresses along the bolt will look like those illustrated with dashed lines in Fig. 17. According to the models for shear stress illustrated in Figs. 4 and 5, we obtain the following relationships between the opening
Fig. 16. The shear stress measured on two fully grouted bolts in situ, (a) bolt No. 9, (b) bolt No. 10. After BjoÈrnfot and Stephansson [4].
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C. Li, B. Stillborg / International Journal of Rock Mechanics and Mining Sciences 36 (1999) 1013±1029
displacement dJ and the tensile axial stress of the bolt at the joint, sb0, as follows (see Appendix B for the detailed derivations): 1. For fully grouted bolts (assuming x0=0), dJ 1
sb0 db , aEb
when sb0 R
2sp a
otherwise, " 2 x 2 ÿ D2 sb0 x 2 ÿ 2osp dJ Eb db # sp 2D2 ÿ 2o 1sp 2 db 3db a The position of the decoupling front is at db sb0 1 D ÿ ÿ 1 ÿ o x2 2o 2sp a db
23
24 Fig. 18. The axial stress and load of the bolt versus the opening displacement of the joint for fully grouted bolts. Parameters: sp=12 MPa, o=0.1, d2=20 mm, D=0.1 m, Eb=210 GPa, Er=45 GPa, Eg=30 GPa, nr=ng=0.25.
25
2. For fully frictionally coupled bolts, dJ 1
sb0 db , aEb
when sb0 R
2s a
otherwise, 2 p 2 1 x2 2 sb0 x 2 ÿ sdb dJ Eb 2A a The position of the decoupling front is at A 2s sb0 ÿ x2 pdb s a
26
27
28
Using the relevant equations above, the axial stress of the bolt at the joint, sb0, versus the opening displacement of the joint is calculated and shown in Fig. 18
Fig. 17. The shear stress (tb) and the axial tensile stress (sb), induced by joint opening, in fully coupled rock bolts
for a fully grouted bolt and in Fig. 19 for a fully frictionally coupled bolt. The values of the relevant parameters used for the calculations are listed in the captions of the ®gures. It is seen that the bolt interface starts to be decoupled at a very small opening displacement of the joint. This con®rms the results arrived at by other studies [8,15,16], showing that decoupling of the interface occurs at an extremely small displacement, because the compatibility of deformation is lost across the interface at such a low load. Field measurements, for instance those carried out by BjoÈrnfot and Stephansson [2,4], have demonstrated that bolts installed in jointed rock masses sometimes are subjected to several axial stress peaks. These peaks are thought to be caused by the opening of the rock joints intersecting the bolt. The following is an example to show the axial stress along a bolt intersect-
Fig. 19. The axial stress and load of the bolt versus the opening displacement of the joint for frictionally coupled bolts. Parameters for Standard Swellex: s = 0.7 MPa, db=39 mm, t = 2 mm, Eb=210 GPa, Er=45 GPa, nr=0.25.
C. Li, B. Stillborg / International Journal of Rock Mechanics and Mining Sciences 36 (1999) 1013±1029
1025
uniformly deformed rock masses, the bolt has a pickup length, an anchor length and a neutral point; (ii) the face plate enhances the reinforcement eect through inducing a direct tensile load in the bolt and reducing the shear stress carried on the bolt surface; and (iii) in jointed rock masses, the opening displacement of rock joints will induce axial stress peaks in the bolt. Acknowledgements
Fig. 20. Axial stress along a bolt subjected to joint openings. The opening displacements: dJa=50 mm at joint a, dJb=20 mm at joint b and dJc=5 mm at joint c.
ing three rock joints. Assume that the three joints, a, b and c, have opened 50, 20 and 5 mm, respectively, since the bolt was installed (see Fig. 20). The axial stress along the bolt would be the superposition of the stresses caused by the opening displacements at the three joints. Assuming that the bolt interface is still in the stage of compatible deformation, the axial stress can be expressed as: sb x Ssb0i e
2a ÿ d jxÿx i j b
i a, b and c
29
where sb0i=(aEb/db)dJi, according to Eq. (23) for fully grouted bolts. Using the following values for the relevant parameters: a=0.23; Eb=210 GPa; db=20 mm; dJa=50 mm at xa=0.4 m; dJb=20 mm at xb=0.6 m; dJc=5 mm at xc=0.8, we obtain the axial stress along the bolt as illustrated in Fig. 20. Hyett et al. [12] and Bawden et al. [16] obtained similar results through numerical simulations.
5. Concluding remarks An analytical model has been established for rock bolts subjected to a pull load in pullout tests. Decoupling starts at the loading point and propagates along the bolt with an increasing applied load. The shear stress at the decoupled interface is lower than the ultimate shear strength of the interface and even drops to zero for fully grouted bolts, while it is approximately at the same magnitude as the ultimate shear strength for fully frictionally coupled bolts. The shear stress on the non-decoupled interface decreases exponentially with increasing distance from the decoupling front. Two analytical models have been developed for rock bolts in situ, one for uniform rock deformation and another for discrete joint opening. For rock bolts in situ, the models con®rm the previous ®ndings: (i) in
The grant for this work from AÊke and Greta Lisshed's Foundation is acknowledged. The valuable comments by the anonymous reviewers are greatly appreciated.
Appendix A. Stress analysis of a rock bolt in situ Let us consider a rock bolt installed within a rock mass (see Fig. A1). It is assumed that the range of in¯uence of one bolt extends half the distance to all adjacent bolts. Thus, in pattern bolting, the area of in¯uence of one bolt equals the area surrounded by four adjacent bolts. Consider a thin slice of the bolted rock, dx, which will be used to study the interaction between the bolt and the rock. The thin slice of the bolt-reinforced rock is shown in Fig. A2. Let the free deformation of the rock slice dx, i.e. the deformation before bolting, be termed as du. The deformation of the rock slice becomes dub when it is reinforced by a bolt. The elongation of the bolt is also dub if it is assumed that the bolt and the rock are deformed together. The magnitude of dub can thus be calculated from the elongation of the bolt. The reduction of deformation, dur, is the result of the stress increment, Dsr, in the rock mass induced by bolting. It is obvious that the sum of dur and dub equals the free deformation du,
Fig. A1. A sketch illustrating a bolt installed within a rock mass.
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shear stress on the bolt interface can be expressed as: tb1 x ÿ
i.e.: sb Dsr dx dx Eb Er
A5
where db is the diameter of the bolt. The shear stress tb1 is caused by rock deformation. On the other hand, the shear stress on the bolt interface in the section between ri and x has a pull eect on the section of the bolt on the right side of x, and therefore induces another component of shear stress, tb2. The shear stress tb1 on a small element of the bolt, dt, brings about a normal stress increment, dsb(t ), in the bolt. Similarly to Eq. (2), the shear stress increment at x, dtb2(x ), induced by the normal stress increment dsb(t ) at t, can be expressed as;
Fig. A2. Stress components in the bolt and in the rock.
du dub dur
A dsb A d2 u xGr pdb dx pdb dx 2
A1
xÿt a ÿ2a d b dtb2 x dsb te 2
A6
where dx is the length of the rock slice, du is the free deformation of the rock slice, dub is the elongation of the bolt element, dur is the reduced deformation of the rock due to bolting, sb is the tensile stress in the bolt, Dsr is the compressive stress increment in the rock mass, induced by bolting, Er is Young's modulus of the rock mass, and Eb is Young's modulus of the bolt steel. The force equilibrium on the plane perpendicular to the bolt gives:
The total shear stress at x induced due to the pull eect of the shear stress on the bolt section between ri and x is obtained by integration of the above shear stress increment, that is:
sb A ÿDsr S
tb x tb1 tb2
A2
where A is the cross-section area of the bolt, and S is the in¯uencing area of the bolt in the rock, equal to the area surrounded by four adjacent bolts in pattern bolting. Substituting Eq. (A2) into Eq. (A1), we obtain the expressions for sb and Dsr as: sb x ÿxGr
Dsr x xGr
du dx
A du S dx
A3
where x
2 1 nr SEb AEb SEr
Er Gr 2 1 nr
x
a tb2 x dtb2 x ÿ xGr 2 ri
du/dx is the ®rst-order derivative of the free radial displacement of the rock, u, with respect to x. From the point of view of force equilibrium, the
d2 u ÿ2a xÿt db dt e 2 ri dt
A7
Finally, the total shear stress on the bolt at x is the sum of tb1 and tb2, that is: " xGr
A d2 u a ÿ pdb dx 2 2
x
d2 u ÿ2a xÿt db dt e 2 ri dt
#
A8
Eq. (A8) is a general solution to the shear stress on the bolt without decoupling at the bolt interface. When decoupling occurs, a certain portion of the load originally carried on the decoupled section of the bolt will either be transferred to the face plate, if there is one, or released with a free rock deformation in the case without a face plate. For the case with a face plate (see Fig. 14), the load transferred to the face plate due to decoupling is calculated as: Pf pdb
A4
x
rp ri
ÿtb1 ÿ sr dt
ÿAxGr
rp ri
d2 u dt ÿ pdb sr rp ÿ ri dt2
A9
The shear stress, at x=rp, induced by the axial load on the face-plate load, Pf , and by the shear stress on the decoupled interface, sr, is:
C. Li, B. Stillborg / International Journal of Rock Mechanics and Mining Sciences 36 (1999) 1013±1029
a Pf pdb sr rp ÿ ri 2 A rp 2 a d u dt ÿ xGr 2 2 ri dt
tb2 rp
A10
For equilibrium the total shear stress at rp should equal the peak shear strength of the interface, that is: ÿsp tb1 rp tb2 rp " xGr
A d2 u a ÿ pdb dx 2 2
rp ri
d2 u dt dt2
# A11
Eq. (A11) is used to determine the distance rp for bolts with a face plate. For x > rp, the total shear stress is calculated as: # " A d2 u a x d2 u ÿ2a xÿt db dt ÿ e tb x xGr pdb dx 2 2 ri dt2 A12 # " rp 2 xÿt a d u ÿ2a d b dt e ÿ xGr 2 2 dt ri For a bolt without a face plate, Pf is zero. The shear stress, at x=rp, induced by the shear stress on the decoupled interface, sr, becomes tb2 rp
a pdb sr rp ÿ ri 2 A
A13
Similarly to the case with a face plate, the total shear stress at rp should equal the peak shear strength of the interface, that is ÿsp tb1 rp tb2 rp xGr
A d2 u a pdb sr rp ÿ ri pdb dx 2 2 A
A14
Eq. (A14) is used to determine the distance rp for bolts without a face plate. For x > rp, the total shear stress is calculated as: 3 2 x 2 ÿ2a x ÿ t 2 4 A d u a d u db dt 5 ÿ e tb x xGr pdb dx 2 2 rp dt2 A15 xÿt ÿ2a a pdb db sr rp ÿ ri e 2 A
1027
opening and the load induced in the bolt. An opening displacement of the rock joint is equivalent to applying an axial tensile load to both sides of the bolt at the joint. We shall look at this problem for fully grouted bolts and for frictionally coupled bolts separately. B.1. Fully grouted rock bolts The axial tensile stress in the bolt is symmetric to the rock joint. Therefore, we consider only half of the rock±bolt system. The model for the shear stress along a bolt subjected to an axial load is shown in Fig. 4. The elongation of the bolt in dierent sections is denoted as follows: d0 is the elongation of the bolt in section (0 R x < x0); d1 is the elongation of the bolt in section (x0 R x < x1); d2 is the elongation of the bolt in section (x1 R x < x2); and d3 is the elongation of the bolt in section (x2 R x < L ), where L is the half length of the bolt. The sum of these four components is the total elongation of the bolt from each side of the joint. Thus, the displacement of the joint opening is twice this summation, i.e. dJ 2
3 X di
B1
i0
Not all the four elongation comments appear in the above expression at any given time. When the joint opens very little, the axial load induced does not cause the interface to be decoupled. In this case, only d3 exists in Eq. (B1). The components d2, d1 and d0 appear subsequently in the equation with increases in the joint opening. 1. For the case where the interface undergoes compatible deformation across the interface, the shear stress along the bolt is illustrated in Fig. 6. In this case we have d0 d1 d 2 0
B2
As shown in Eqs. (2) and (4), the shear and axial stresses along the bolt are given by
Appendix B. Joint opening and the stresses induced in rock bolts Fig. B1 illustrates a bolt intersecting a rock joint. We shall establish the relationship between the joint
Fig. B1. A sketch illustrating a rock bolt intersecting a joint.
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x a ÿ2a tb x sb0 e db 2
sb x sb0 e
sb x 2
x ÿ2a d
B3
b
This stage ceases when the shear stress at x = 0 reaches the peak shear strength of the interface, sp. The corresponding axial stress at the joint at this moment is sb0
2sp a
B4
The elongation of the bolt is calculated as (assuming L>>db) 2 L db sb0 s xdx1 B5 dJ 2d3 aEb Eb 0 When sb0=(2sp/a ), the opening displacement reaches its maximum, dJmax, before decoupling occurs. It is given by: dJmax
2db sp 2 a Eb
B6
2. For the case where decoupling occurs (assuming x0=0): Let D x 2 ÿ x 1 , and o sr =sp For x < x 1 :
sb x sb0 ÿ 4sr
x db
B7
For x1 R x < x2:
sb x sb0 ÿ
2sp x ÿ x 1 2ox 1 ÿ o db D
For xrx2:
sb x
ÿ2a
B10
The elongation of the bolt can be obtained by the following integration 1 sb dx B11 di e dx Eb i.e. 1 d1 Eb
x 1 0
x sb0 ÿ 4sr dx db
"
1 x 2 ÿ D2 sb0 x 2 ÿ D ÿ 2sr db Eb
#
(
2sp 2ox sb0 ÿ db x1 ) x ÿ x 1 2 dx 1 ÿ o D ( ) 1 2D2 2o 1sp sb0 D ÿ Eb 3db
1 d2 Eb
x 2
L
B12
2sp ÿ2a e x2 a
ÿ xÿx 2 db
dx1
1 sp db E b a2
B13
B14
The opening displacement of the joint is calculated as dJ 2 d1 d2 d3
x ÿ x1 tb x sr 1 ÿ osp x2 ÿ x1
tb x sp e
Therefore, we obtain db sb0 2 D ÿ ÿ 2 1 ÿ o x2 4o sp a db
1 d3 Eb
tb x sr
2sp a
2
B8
" 2 x 2 ÿ D2 sb0 x 2 ÿ 2osp db Eb # sp 2D2 ÿ 2o 1sp 2 db 3db a
B15
ÿ xÿx 2 db
ÿ 2 2sp ÿ2a xÿx db e a
At x=x2, we have from Eqs. (B8) and (B9) sb x 2 sb0 ÿ
2sp 2ox 2 1 ÿ oD db
B9
B.2. Fully frictionally coupled rock bolts In the stage of compatible deformation, the shear and axial stresses have the same forms as those expressed in Eq. (B3) and the elongation has the same forms as those expressed in Eqs. (B5) and (B6). When decoupling occurs at the interface, the shear stress along the bolt is illustrated in Fig. 5. The
C. Li, B. Stillborg / International Journal of Rock Mechanics and Mining Sciences 36 (1999) 1013±1029
elongation of the bolt in this case is calculated as follows. For x R x2: the stresses are
dJ 2 d1 d3
tb x s
References
sb x sb0 ÿ
pdb x s A
B16
The elongation of the bolt in section (0 R x < x2) is x 2 pdb sx dx sb0 ÿ A 0 1 pdb 2 sx sb0 x 2 ÿ Eb 2A 2
1 d1 Eb
B17
For xrx2: tb x s e
sb x
xÿx ÿ2a d 2 b
2 2s ÿ2a xÿx db e a
B18
At x=x2, we have from Eqs. (B16) and (B17) sb x 2 sb0 ÿ
sb x 2
pdb sx 2 A
2s a
Then we obtain the expression for x2 as A 2s sb0 ÿ x2 pdb s a
B19
The elongation of the bolt in section (x2 R x < L ) is calculated as d3
1 Eb
L
2 2s ÿ2a xÿx db dx e x2 a
ÿ2a 1 sdb 1 ÿ e 2 Eb a
L ÿ x2 db 1 1 sdb Eb a2
The total elongation is
B20
2 psdb 2 sdb x2 2 sb0 x 2 ÿ 2A a Eb
1029
B21
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