analytical models for rock bolts

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 pullouttests, one for bolts installed in uniformly deformed rock masses, and one for bolts subjected to the opening of individual rockjoints. 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 inpullout tests, the shear stress of the interface attenuates exponentially with increasing distance from the point of loading whenthe 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 theshear stress on the decoupled bolt section depends on the coupling mechanism at the interface. For fully grouted bolts, the shearstress on the decoupled section is lower than the peak shear strength of the interface, while for fully frictionally coupled bolts itis 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 theprevious ®ndings that a bolt in situ has a pick-up length, an anchor length and a neutral point. It is also revealed that the faceplate plays a signi®cant role in enhancing the reinforcement e�ect. 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 reinforce-ment in civil and mining engineering for a long time.Bolts reinforce rock masses through restraining the de-formation within the rock masses. In order to improvebolting design, it is necessary to have a good under-standing of the behaviour of rock bolts in deformedrock masses. This can be acquired through ®eld moni-toring, laboratory tests, numerical modelling and ana-lytical studies.

Since the 1970s, numerous researchers have car-ried out ®eld monitoring work on rock boltsinstalled in various rock formations [1±3]. Freeman[1] performed pioneering work in studying the per-

formance of fully grouted rock bolts in the Kielderexperimental tunnel. He monitored both the loadingprocess of the bolts and the distribution of stressesalong the bolts. On the basis of his monitoringdata, he proposed the concepts of ``neutral point'',``pick-up length'' and ``anchor length''. At the neu-tral point, the shear stress at the interface betweenthe bolt and the grout medium is zero, while thetensile axial load of the bolt has a peak value. Thepick-up length refers to the section of the bolt fromthe near end of the bolt (on the tunnel wall) to theneutral point. The shear stresses on this section ofthe bolt pick up the load from the rock and dragthe bolt towards the tunnel. The anchor lengthrefers to the section of the bolt from the neutralpoint to the far end of the bolt (its seating deep inthe rock). The shear stresses on this section of thebolt anchor the bolt to the rock. These conceptsclearly outline the behaviour of fully grouted rockbolts in a deformed rock formation. BjoÈ rnfot and

International Journal of Rock Mechanics and Mining Sciences 36 (1999) 1013±1029

1365-1609/99/$ - see front matter # 2000 Elsevier Science Ltd. All rights reserved.

PII: S1365-1609(99 )00064 -7

www.elsevier.com/locate/ijrmms

* Corresponding author. Tel.: +46-920-91352; fax: +46-920-

91935.

E-mail address: [email protected] (C. Li).

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 anchor-

ing 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 load-

ing. 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

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 com-

pletely di�erent 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.

Nomenclature

A area of the cross-section of the boltEb Young's modulus of the bolt steelEr Young's modulus of the rock massEg Young's modulus of the groutL length of the boltP0 applied pull loadP0max pullout load, i.e. the maximum applied pull

loadS in¯uencing area of a bolt in the rockd0 diameter of a circle in the rock outside

which the in¯uence of the bolt disappearsdb diameter of the boltdg diameter of the boreholedu free deformation of the rock slicedub elongation of the bolt elementdur the reduction of deformation after bolt re-

inforcement, i.e. dur=duÿ dubdx length of the rock slicep0 hydrostatic primary stress in the rockPf load on the face plate of the boltri radius of the circular tunnelrp position of the decoupling front on the bolt

surfaces shear strength of the interface for friction-

ally coupled boltssr residual shear strength of the interface for

fully grouted boltssp peak shear strength of the interface for fully

grouted boltsu original radial displacement of the rock at x

(without bolting)x0,x1and x2

decoupling boundaries at the interface offully grouted bolts (see Fig. 4)

Greek symbolsD length of a bolt section, D=x2ÿx1

a a constant representing the coupling prop-erty of the interface

d0 elongation of the bolt in section (0Rx<x0)d1 elongation of the bolt in section (x0Rx<x1)d2 elongation of the bolt in section (x1Rx<x2)d3 elongation of the bolt in section (x2Rx<L )dJ opening displacement of a rock jointdJi opening displacement of the ith joint (i= a,

b, c, . . . )dJmax maximum opening displacement of a joint

before decoupling occursng Poisson's ratio of the groutnr Poisson's ratio of the rock massx a coe�cient related to the Young's moduli

of the bolt steel and the rocksb axial stress of the boltsb0 axial stress of the bolt at the loading pointsb0i axial stress of the bolt at the ith rock jointDsr bolting-induced stress increment in the rock

masstb shear stress at the bolt interfacetbB total shear stress at point B of the bolt

interfacetb1 shear stress at the bolt interface, induced by

rock deformationtb2 shear stress at the bolt interface, induced

due to pull e�ectt dA shear stress at point A of the bolt interface,

induced by rock deformationt dB shear stress at point B of the bolt interface,

induced by rock deformationtAB shear stress at B due to the pull action of

t dAo ratio of the residual shear strength to the

peak shear strength, o=sr/sp

C. Li, B. Stillborg / International Journal of Rock Mechanics and Mining Sciences 36 (1999) 1013±10291014

It is thought that the relative movement between therock and the bolt is zero at the neutral point [1]. Inthe solution by Tao and Chen [10], the position of theneutral point depends only on the radius of the tunneland the length of the bolt. That solution was im-plemented in the analytical models created by Indrar-atna and Kaiser [11] and Hyett et al. [12]. It seemsthat Tao and Chen's solution is valid only when thedeformation is compatible across the bolt interface.When decoupling occurs, the position of the neutralpoint is obviously also related to the shear strength ofthe interface. Field monitoring and pullout tests haveindicated two facts concerning the loading of a rockbolt in situ: (1) rock deformation applies a load on thepick-up section of the bolt; (2) the load on the pick-upsection drags the anchor section of the bolt towardsthe underground opening. These two facts must betaken into account in developing analytical models forrock bolts.

The aim of this paper is to develop analyticalmodels for fully coupled rock bolts. A model for rockbolts in pullout tests is introduced ®rst, together with

a description of the theoretical background, the devel-opment of the model and an illustrative example. Twomodels for rock bolts in situ are then presented, one inuniformly deformed rock masses and one in jointedrock masses. The details of the development of themodels are summarised in the Appendices.

2. Coupling between the bolt and the rock

Windsor [13] proposed the concept that a reinforce-ment system comprises four principal components: therock, the reinforcing element, the internal ®xture andthe external ®xture. For reinforcement with a bolt, thereinforcing 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 orresin for grouted bolts, or a mechanical action like``friction'' at the bolt interface for frictionally coupledbolts. The internal ®xture provides a coupling con-dition at the interface. With reference to the com-ponent of internal ®xture, Windsor [13] classi®ed thecurrent reinforcement devices into three groups: ``con-tinuously mechanically coupled (CMC)'', ``continu-ously frictionally coupled (CFC)'' and ``discretelymechanically or frictionally coupled (DMFC)'' sys-tems. According to this classi®cation system, cement-and resin-grouted bolts belong to the CMC system,while Split set and Swellex bolts belong to the CFCsystem.

When fully grouted bolts are subjected to a pullload, failure may occur at the bolt±grout interface, inthe grout medium or at the grout±rock interface,depending on which one is the weakest. For fully fric-tionally coupled bolts, however, there is only onepossibility of failure Ð decoupling at the bolt±rockinterface. In this study we concentrate on the failure atthe interface between the bolt and the coupling med-ium (either the grout medium or the rock).

In general, the shear strength of an interface com-prises three components: adhesion, mechanical inter-lock and friction. They are lost in sequence as thecompatibility of deformation is lost across the inter-face. The result is a decoupling front that attenuates atan increasing distance from the point of the appliedload. The decoupling front ®rst mobilises the adhesivecomponent of strength, then the mechanical interlockcomponent and ®nally the frictional component. Theshear strength of the interface decreases during thisprocess. The shear strength after the loss of some ofthe strength components is called the residual shearstrength in this paper. For grouted rock bolts likerebar, all the three components of strength exist at thebolt interface. However, for the fully frictionallycoupled bolt, the ``Split set'' bolt, only a friction com-ponent exists at the bolt interface. For Swellex bolts,

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].

C. Li, B. Stillborg / International Journal of Rock Mechanics and Mining Sciences 36 (1999) 1013±1029 1015

mechanical interlock and friction comprise the strengthof the interface.

3. Rock bolts in pullout tests

3.1. Theoretical background

When a bolt installed in rock is subjected to a tensileaxial load, the relationship between the shear stress atthe bolt interface and the axial tensile stress of the boltcan be established through considering a small sectionof the bolt as shown in Fig. 2. The force equilibriumin the axial direction leads to the following expression:

tb � ÿ A

pdb

dsb

dx�1�

where db is the diameter of the bolt, and A is the areaof the cross-section of the bolt.

For the example shown in Fig. 1(a), the shear stressalong the bolt at the two levels of applied load isobtained using Eq. (1) and illustrated in Fig. 3. Whenthe applied load is small, the shear stress decreaseswith increasing distance from the point of loading(curve a ). Progressive decoupling commences at theloading point at a certain level of applied load. Thedecoupling front moves towards the far end of the boltwith an increasing applied load. The shear stress is atthe level of the shear strength at the decoupling front,while behind the decoupling front the shear stressbecomes smaller, since the strength of the interface hasbeen partially lost due to decoupling. Curve b in Fig. 3represents such a distribution of shear stress along thebolt.

Based on experimental results as shown in Fig. 3, amodel for the shear stress along a fully grouted boltcan be postulated as illustrated in Fig. 4. In thismodel, the section of the bolt close to the loadingpoint is completely decoupled with a zero shear stress

at the bolt interface. Starting at a certain distancefrom the loading point, say at x0, the bolt interface ispartially decoupled with a residual shear strength, sr.Between point x1 and x2, the residual shear strengthlinearly increases from sr to the peak strength sp.Beyond point x2, the interface undergoes compatibledeformation and the shear stress attenuates exponen-tially towards the far end of the bolt.

For fully frictionally coupled bolts, the magnitudeof the shear stress behind the decoupling front is ap-proximately the same as the peak value. As mentionedpreviously, the shear strength of the interface for thistype of bolt comprises one or two components, i.e.either friction or mechanical interlock and friction.The deformation incompatibility across the interfacedoes not make the friction disappear. In other words,the residual shear strength of the interface is approxi-mately the same as the peak strength for fully friction-ally coupled bolts. The distribution of shear stress forthis type of bolt is illustrated in Fig. 5.

When a fully coupled bolt is subjected to a pullload, the shear stress along the bolt is as shown in Fig.

Fig. 3. The shear stress on the steel bar, derived from Fig. 1(a).

Fig. 4. Distribution of shear stress along a fully grouted rock bolt

subjected to an axial load.Fig. 2. Stress components in a small section of a bolt.


6 before decoupling occurs at the interface. For fullygrouted rock bolts, the attenuation of the shear stressis expressed as [6,14]:

tb � a2sb0 e

ÿ2a xdb �2�

where

a2 � 2GrGg

Eb

"Gr ln

�dg

db

�� Gg ln

�d0dg

�#

Gr � Er

2�1� nr� and Gg � Eg

2�1� ng� �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'smodulus of the rock mass, Eg is Young's modulus ofthe grout, nr is Poisson's ratio of the rock mass, ng isPoisson's ratio of the grout, dg is the diameter of theborehole, and d0 is the diameter of a circle in the rockoutside which the in¯uence of the bolt disappears.

Eq. (2) is also valid for fully frictionally coupled

bolts if the expression for the constant a is slightlymodi®ed. Fully frictionally coupled bolts have directcontact with rock. The constant a for this type of boltcan be obtained from Eq. (3a) by letting the Young'smodulus and Poisson's ratio of the grout equal thoseof the rock, i.e. Gg=Gr, ng=nr and dg=dr. Then weobtain the expression of a for fully frictionally coupledbolts as:

a2 � 2Gr

Eb ln

�d0db

� �3b�

The axial stress of the bolt is calculated as:

sb�x� � sb0 ÿ pdb

A

�x0

tb�x�dx � sb0 eÿ2a x

db �4a�

or

sb�x� � 2

atb�x� �4b�

3.2. Fully grouted rock bolts

The stresses in di�erent sections of the bolt can nowbe described in detail as follows (see Fig. 4):

1. On the section 0R x< x0: the bolt interface is com-pletely decoupled, leading to a zero shear stress atthe 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 partiallydecoupled, resulting in a residual shear strength srat the interface. The shear and axial stresses aregiven by:

tb�x� � sr

sb�x� � sb0 ÿ 4sr

db

�xÿ x0� �6�

3. On the section x1 R x< x2: the interface is partiallydecoupled with the residual shear strength linearlyincreasing to the peak strength. The shear and axialstresses are given by:

tb�x� � osp � xÿ x1

D�1ÿ o�sp

Fig. 5. Distribution of shear stress along a frictionally coupled rock

bolt.

Fig. 6. Shear stress along a fully coupled rock bolt subjected to an

axial load before decoupling occurs.


sb�x� � sb0

ÿ 2sp

db

�2o�xÿ x0� � �1ÿ o�

D�xÿ x1�2

� �7�

where D=x2ÿx1, and o=sr/sp, the ratio of the re-sidual shear strength to the peak shear strength.

4. On the section x > x2: the deformation is compati-ble across the interface and no decoupling occurs.According to Eqs. (2) and (4), both the shear andaxial stresses decrease exponentially towards the farend of the bolt:

tb�x� � sp eÿ2aÿxÿx 2

db

�

sb�x� � 2sp

aeÿ2aÿxÿx 2

db

��8�

It is seen from Eq. (8) that the axial stress at x=x2 isgiven by

sb�x2� � 2sp

a

On the other hand, the axial stress at x=x2 can beobtained from Eq. (7) as:

sb�x2� � 4P0

pd 2b

ÿ 2sp

db

�2o�x2 ÿ x0� � �1ÿ o�D�

where P0 is the applied pull load. Letting the rightsides of the above two expressions be equal, we obtainthe expression for the position of the decoupling front,x2, as:

x2 � x0 � 1

2o

�2P0

pdbsp

ÿ db

aÿ �1ÿ o�D

��9�

For equilibrium the applied load P0 should equal thetotal shear force at the bolt interface, i.e.

P0 � pdb

�Lx 0

tb dx � pdb

�sr�x1 ÿ x0� � 1

2spD�1� o�

� db

2asp

�1ÿ e

ÿ2adb�Lÿx 2�

��where L is the length of the bolt. It is obtained fromthe above expression that the maximum applied loadP0max can be expressed as:

P0max � pdbsp

�o

�L� db

2aln oÿ Dÿ x0

�

� 1

2D�1� o� � db

2a�1ÿ o�

� �10�

The following is an example to demonstrate how toback-calculate the peak shear strength of the interface

on the basis of the pullout load. Stillborg [8] con-ducted a series of pullout tests on di�erent types ofrock bolts. In one test, a 3 m long rebar with a diam-eter of 20 mm was grouted within two identical con-crete blocks. The length of the bolt in each block was1.5 m. One block was ®xed to the ground, while theother was pulled. The bolt was pulled out without rup-ture, indicating that decoupling of the interfaceoccurred along the entire length of the bolt. The pull-out load registered was 180 kN. It is assumed that thedistribution of shear stress has the form illustrated inFig. 4 with x0=0. It is known from the test that:

P0max � 180 kN, L � 1:5 m, db � 20 mm,

dg � 35 mm, Eb � 210 GPa

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 fromEq. (10). The shear stress and the axial load along therebar are calculated on the basis of the model and il-lustrated in Fig. 7. The axial load along the bolt atdi�erent levels of applied load is illustrated in Fig. 8.The curves in Fig. 8 are similar to those obtained inpullout tests (e.g. Fig. 1(a)).

3.3. Fully frictionally coupled rock bolts

For fully frictionally coupled rock bolts, the residualshear strength of the interface is approximately thesame as the peak shear strength, i.e. sr=sp=s (see Fig.5). The shear stress on di�erent sections of the bolt isdescribed in detail as follows:

1. On the section 0 R x < x2: the shear stress hasreached 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

Asx �11�

2. On the section x > x2: the deformation is compati-ble across the interface and the shear stress is lessthan the peak shear strength. Both the shear andthe axial stresses decrease exponentially towards thefar end of the bolt:


tb�x� � s eÿ2aÿxÿx 2

db

�

sb�x� � 2s

aeÿ2aÿxÿx 2

db

��12�

It is seen from Eq. (12) that the axial stress at x=x2 isgiven by

sb�x2� � 2s

a

On the other hand, the axial stress at x=x2 can beobtained from Eq. (11) as:

sb�x2� � P0

Aÿ pdb

Asx2

Fig. 7. The shear stress and axial load along a fully grouted rock bolt subjected to an axial load of 90 kN.

Fig. 8. Axial load along a fully grouted rock bolt at di�erent levels of applied load.


Letting the right sides of the above two expressions beequal, we obtain the expression for the position of thedecoupling front, x2, as:

x2 � 1

pdbs

�P0 ÿ 2As

a

��13�

For equilibrium the applied load P0 should equal thetotal shear force at the bolt interface, i.e.

P0 � pdb

�L0

tb dx

� spdbx2 � pd 2b

2

s

a

�1ÿ e

ÿ2a�Lÿx 2

db

�� 14�

The applied load reaches its maximum, P0max, whenthe shear strength of the interface is mobilised alongthe entire length of the bolt, i.e. when x2=L. Substi-tuting P0=P0max and x2=L into Eq. (13), we obtainthe shear strength for fully frictionally coupled boltsas:

s � P0max

pdbL�15�

Stillborg [8] tested the Swellex bolt in his pullouttests. Displacement monitoring at the far end of thebolt indicated that the bolt was slipping under theload P0max=110 kN. That indicated that decouplingoccurred along the entire length of the bolt. The diam-eter of the borehole was 35 mm. The diameter of theSwellex bolt is the same as that of the borehole, i.e.

db=35 mm. The length of the bolt section embeddedin each concrete block was 1.5 m long, i.e. L= 1.5 m.Substituting these data into Eq. (15) yields the shearstrength of the bolt interface, i.e. s = 0.7 MPa. It isobtained from Eq. (3b) that the constant a=0.27. Theshear stress and the axial load along the Swellex boltare calculated using the relevant equations above andare illustrated in Fig. 9. The axial load of the bolt atdi�erent levels of applied load is shown in Fig. 10.

4. Rock bolts in situ

4.1. A model for bolts subjected to uniform rockdeformation

Rock bolts in situ tend to restrain the deformationof rock with an increase in their axial loads. In otherwords, it is rock deformation that applies a load torock bolts in situ. For the sake of simplicity, a boltanchored at two points, as illustrated in Fig. 11, isused to explain the superposition of two componentsof the shear stress. Rock deformation will induce acomponent of shear stress t dA at A and a componentof shear stress t dB at B. Assuming t dA > t dB, the shearforce acting at anchor A would tend to drag the boltto the left and thus induce another component ofshear stress at point B, tAB. The sense of tAB is oppositeto the sense of t dB. The total shear stress at B is:

tbB � tAB ÿ td

B �16�

Fig. 9. The shear stress and axial load along a Swellex rock bolt at 5 kN of applied load.


For a fully coupled bolt, the component tAB will be anintegration over the bolt section to the left side ofpoint B. On the basis of this idea, we obtained the ex-pression of the shear stress at position x on the boltsurface as follows (see Appendix A for detailed deri-vation):

tb�x� � xGr

"A

pdb

d2u

dx2ÿ a

2

�xri

d2u

dt2eÿ2axÿtdb dt

#�17�

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

bolt in the rock, which equals the area surrounded byfour adjacent bolts in pattern bolting.

Here we take a tunnel circular in its cross-section asan example to demonstrate the application of Eq. (17).Assume that the country rock surrounding the tunnelundergoes an elastic deformation. The second-orderderivative of the elastic radial displacement u of therock can be expressed as:

u 00x � ÿp0Gr

r2ix3

�19�

where ri is the radius of the circular tunnel, and p0 isthe hydrostatic primary stress in the rock.

Substituting Eq. (19) into Eq. (17) and using the fol-lowing values for the relevant parameters:

Young's modulus of the bolt steel Eb=210 GPa,Young's modulus of the rock mass Er=5 GPa,Poisson's ratio of the rock mass nr=0.25,bolt spacing S=1 m,diameter of the bolt db=20 mm,radius of the circular tunnel ri=4 m,hydrostatic primary stress in the rock p0=15 MPa,

we obtain the shear stress along the bolt as illustratedin Fig. 12. It can be seen that the sense of the shearstress on the bolt section close to the tunnel wall isnegative; that is the direction of the shear stress istowards the tunnel. At a certain distance from thewall, the shear stress becomes zero. Beyond this neu-tral point, the sense of the shear stress becomes posi-tive; that is the direction of the shear stress is towards

Fig. 10. The axial load along a Swellex rock bolt at di�erent levels of applied load.

Fig. 11. A sketch illustrating the superposition of the components of

shear stress at position B.


the far end of the bolt. This agrees with the ®eld moni-toring data obtained, for example, by Freeman [1]. Inthis example, decoupling at the bolt interface has notbeen considered. When the rock deformation is largeenough, the shear strength of the interface will bemobilised in the pick-up section of the bolt. The distri-bution of shear stress along the bolt, when decouplingoccurs, will be as that illustrated in Fig. 13. The shearfailure at the interface would result in a release of therestrained rock deformation at the near end of thebolt, if no face plate were to exist. In the case where aface plate exists, the displacement of the tunnel wallloads the plate. The load on the face plate can be cal-culated as:

Pf � pdb

�rp

ri

ÿ A

pdb

xGrd2u

dx2ÿ sr

!dx �20�

Shear failure ceases at x=rp and beyond that pointthe displacement is compatible across the interface.The shear stress at x=rp is the sum of three com-

ponents: one induced by the rock deformation, onedue to the pull e�ect of the face plate load Pf , and onedue to the pull e�ect of the shear force on thedecoupled bolt section between ri and rp. In the case ofno face plate, the face plate load Pf is zero. The totalshear stress at x=rp equals the peak shear strength ofthe interface (see Appendix A), i.e.

ÿsp � xGr

"A

pdb

d2u

dx2ÿ a

2

�rp

ri

d2u

dt2dt

#,

for bolts with a face plate

�21a�

or

ÿsp � xGrA

pdb

d2u

dx2� a

2

pdb

Asr�rp ÿ ri�,

for bolts without a face plate

�21b�

Eqs. (21a) and (b) are used to determine the distancerp. The shear stress on the section x > rp is calculatedas:

tb�x� � xGr

"A

pdb

d2u

dx2ÿ a

2

�xri

d2u

dt2eÿ2axÿtdb dt

#

ÿ a2xGr

"�rp

ri

d2u

dt2dt

#eÿ2axÿrp

db ,

for bolts with a face plate

�22a�

or

tb�x� � xGr

"A

pdb

d2u

dx2ÿ a

2

�xrp

d2u

dt2eÿ2axÿtdb dt

#

� a2

pdb

Asr�rp ÿ ri�eÿ2a

xÿrp

db ,

for bolts without a face plate

�22b�

We take the circular tunnel under elastic deformationagain as an example to demonstrate the application ofEqs. (21) and (22). Assume that the peak shearstrength of the bolt interface is sp=0.5 MPa and theresidual shear strength sr=0.2 MPa. Using the valuesgiven before for other relevant parameters, we canobtain the decoupling boundary rp from Eq. (21) andthe shear stress on the interface at x > rp from Eq.(22). The calculated results are shown in Fig. 14 for afully grouted bolt with a face plate and in Fig. 15 fora fully grouted bolt without a face plate. It can befound by comparing the curves in these two ®guresthat: (i) the decoupled length of the bolt is shorterwith a face plate than without a face plate; and (ii) theaxial stress in the decoupled section is larger for the

Fig. 12. Shear stress along a fully grouted rock bolt under the con-

dition 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).

Fig. 13. A schematic illustration of the shear stress along a rock bolt

in situ.


bolt with a face plate than the bolt without a faceplate. That indicates that rock bolts with a face platehave a better reinforcement e�ect than those without aface plate.

Fig. 16 shows the monitored results of the shearstress along two fully grouted bolts in a mine drift inSweden [2,4]. The shear stress along bolt No. 9, pre-sented in Fig. 16(a), agrees well with the theoreticalcurve shown in Fig. 12, implying that no decouplingoccurred at the bolt interface. The shear stress alongbolt No. 1, presented in Fig. 16(b), matches the curveshown in Fig. 15, indicating that decoupling occurredat the interface of this bolt.

The analytical model introduced in this section pro-vides a means for studying rock bolts in a uniformlydeformed rock mass. The key for determining theshear stress along the bolt is the original rock defor-mation around the excavation opening.

4.2. A model for bolts subjected to the opening of a rockjoint

The opening of a rock joint applies a tensile load toboth sides of the bolt intersecting the joint. Duringjoint opening, decoupling of the bolt interface is acti-vated ®rstly at the joint and then propagates along theinterface with an increase in the opening displacement.When the embedment length of the fully coupled boltis su�ciently long on each side of the joint, the shearstress as well as the axial stress along the bolt will besymmetrical to the joint, as shown in Fig. 17. Whenthe opening displacement of the joint is small, both theshear stress and the axial stress decrease exponentiallywith increasing distance from the joint. When theopening displacement is large enough, decoupling willbe activated at the bolt interface and the shear andaxial stresses along the bolt will look like those illus-trated with dashed lines in Fig. 17. According to themodels for shear stress illustrated in Figs. 4 and 5, weobtain the following relationships between the opening

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).

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).

Fig. 16. The shear stress measured on two fully grouted bolts in situ,

(a) bolt No. 9, (b) bolt No. 10. After BjoÈ rnfot and Stephansson [4].


displacement dJ and the tensile axial stress of the boltat the joint, sb0, as follows (see Appendix B for thedetailed derivations):

1. For fully grouted bolts (assuming x0=0),

dJ1sb0db

aEb

, when sb0R2sp

a�23�

otherwise,

dJ � 2

Eb

"sb0x2 ÿ 2osp

�x2 ÿ D�2db

ÿ �2o� 1�sp2D2

3db

� sp

a2db

# �24�

The position of the decoupling front is at

x2 � db

2o

�sb0

2sp

ÿ 1

aÿ D

db

�1ÿ o��

�25�

2. For fully frictionally coupled bolts,

dJ1sb0db

aEb

, when sb0R2s

a�26�

otherwise,

dJ � 2

Eb

�sb0x2 ÿ sdb

�p2A

x22 �

1

a2

��27�

The position of the decoupling front is at

x2 � A

pdbs

�sb0 ÿ 2s

a

��28�

Using the relevant equations above, the axial stressof the bolt at the joint, sb0, versus the opening displa-cement of the joint is calculated and shown in Fig. 18

for a fully grouted bolt and in Fig. 19 for a fully fric-tionally coupled bolt. The values of the relevant par-ameters used for the calculations are listed in thecaptions of the ®gures. It is seen that the bolt interfacestarts to be decoupled at a very small opening displa-cement of the joint. This con®rms the results arrived atby other studies [8,15,16], showing that decoupling ofthe interface occurs at an extremely small displace-ment, because the compatibility of deformation is lostacross the interface at such a low load.

Field measurements, for instance those carried outby BjoÈ rnfot and Stephansson [2,4], have demonstratedthat bolts installed in jointed rock masses sometimesare subjected to several axial stress peaks. These peaksare thought to be caused by the opening of the rockjoints intersecting the bolt. The following is anexample to show the axial stress along a bolt intersect-

Fig. 18. The axial stress and load of the bolt versus the opening dis-

placement 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.

Fig. 19. The axial stress and load of the bolt versus the opening dis-

placement 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.

Fig. 17. The shear stress (tb) and the axial tensile stress (sb), inducedby joint opening, in fully coupled rock bolts


ing three rock joints. Assume that the three joints, a, band c, have opened 50, 20 and 5 mm, respectively,since the bolt was installed (see Fig. 20). The axialstress along the bolt would be the superposition of thestresses caused by the opening displacements at thethree joints. Assuming that the bolt interface is still inthe stage of compatible deformation, the axial stresscan be expressed as:

sb�x� � Ssb0i eÿ2adbjxÿx ij �i � a, b and c� �29�

where sb0i=(aEb/db)dJi, according to Eq. (23) for fullygrouted bolts. Using the following values for the rel-evant 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 alongthe bolt as illustrated in Fig. 20. Hyett et al. [12] andBawden et al. [16] obtained similar results through nu-merical simulations.

5. Concluding remarks

An analytical model has been established for rockbolts subjected to a pull load in pullout tests. Decou-pling starts at the loading point and propagates alongthe bolt with an increasing applied load. The shearstress at the decoupled interface is lower than the ulti-mate shear strength of the interface and even drops tozero for fully grouted bolts, while it is approximatelyat the same magnitude as the ultimate shear strengthfor fully frictionally coupled bolts. The shear stress onthe non-decoupled interface decreases exponentiallywith increasing distance from the decoupling front.

Two analytical models have been developed for rockbolts in situ, one for uniform rock deformation andanother for discrete joint opening. For rock bolts insitu, the models con®rm the previous ®ndings: (i) in

uniformly deformed rock masses, the bolt has a pick-up length, an anchor length and a neutral point; (ii)the face plate enhances the reinforcement e�ectthrough inducing a direct tensile load in the bolt andreducing the shear stress carried on the bolt surface;and (iii) in jointed rock masses, the opening displace-ment of rock joints will induce axial stress peaks in thebolt.

Acknowledgements

The grant for this work from AÊ ke and Greta Lis-shed's Foundation is acknowledged. The valuablecomments by the anonymous reviewers are greatly ap-preciated.

Appendix A. Stress analysis of a rock bolt in situ

Let us consider a rock bolt installed within a rockmass (see Fig. A1). It is assumed that the range of in-¯uence of one bolt extends half the distance to alladjacent bolts. Thus, in pattern bolting, the area of in-¯uence of one bolt equals the area surrounded by fouradjacent bolts. Consider a thin slice of the bolted rock,dx, which will be used to study the interaction betweenthe bolt and the rock. The thin slice of the bolt-re-inforced rock is shown in Fig. A2. Let the free defor-mation of the rock slice dx, i.e. the deformation beforebolting, be termed as du. The deformation of the rockslice becomes dub when it is reinforced by a bolt. Theelongation of the bolt is also dub if it is assumed thatthe bolt and the rock are deformed together. The mag-nitude of dub can thus be calculated from theelongation of the bolt. The reduction of deformation,dur, is the result of the stress increment, Dsr, in therock mass induced by bolting. It is obvious that thesum of dur and dub equals the free deformation du,

Fig. A1. A sketch illustrating a bolt installed within a rock mass.

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.


i.e.:

du � dub � dur � sb

Eb

dx� Dsr

Er

dx �A1�

where dx is the length of the rock slice, du is the freedeformation of the rock slice, dub is the elongation ofthe bolt element, dur is the reduced deformation of therock due to bolting, sb is the tensile stress in the bolt,Dsr is the compressive stress increment in the rockmass, induced by bolting, Er is Young's modulus ofthe rock mass, and Eb is Young's modulus of the boltsteel.

The force equilibrium on the plane perpendicular tothe bolt gives:

sbA � ÿDsrS �A2�where A is the cross-section area of the bolt, and S isthe in¯uencing area of the bolt in the rock, equal tothe area surrounded by four adjacent bolts in patternbolting.

Substituting Eq. (A2) into Eq. (A1), we obtain theexpressions for sb and Dsr as:

sb�x� � ÿxGrdu

dx

Dsr�x� � xGrA

S

du

dx�A3�

where

x � 2�1� nr�SEb

AEb � SEr

Gr � Er

2�1� nr� �A4�

du/dx is the ®rst-order derivative of the free radial dis-placement of the rock, u, with respect to x.

From the point of view of force equilibrium, the

shear stress on the bolt interface can be expressed as:

tb1�x� � ÿ A

pdb

dsb

dx� xGr

A

pdb

d2u

dx2�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 inter-face in the section between ri and x has a pull e�ect onthe section of the bolt on the right side of x, and there-fore 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 thebolt. Similarly to Eq. (2), the shear stress increment atx, dtb2(x ), induced by the normal stress incrementdsb(t ) at t, can be expressed as;

dtb2�x� � a2

dsb�t�eÿ2axÿtdb �A6�

The total shear stress at x induced due to the pulle�ect of the shear stress on the bolt section between riand x is obtained by integration of the above shearstress increment, that is:

tb2�x� ��xri

dtb2�x� � ÿa2xGr

�xri

d2u

dt2eÿ2axÿtdb dt �A7�

Finally, the total shear stress on the bolt at x is thesum of tb1 and tb2, that is:

tb�x� � tb1 � tb2

� xGr

"A

pdb

d2u

dx2ÿ a

2

�xri

d2u

dt2eÿ2axÿtdb dt

# �A8�

Eq. (A8) is a general solution to the shear stress onthe bolt without decoupling at the bolt interface.When decoupling occurs, a certain portion of the loadoriginally carried on the decoupled section of the boltwill either be transferred to the face plate, if there isone, or released with a free rock deformation in thecase without a face plate.

For the case with a face plate (see Fig. 14), the loadtransferred to the face plate due to decoupling is calcu-lated as:

Pf � pdb

�rp

ri

�ÿtb1 ÿ sr�dt

� ÿAxGr

�rp

ri

d2u

dt2dtÿ pdbsr�rp ÿ ri� �A9�

The shear stress, at x=rp, induced by the axial loadon the face-plate load, Pf , and by the shear stress onthe decoupled interface, sr, is:

Fig. A2. Stress components in the bolt and in the rock.


tb2�rp� � a2

�Pf

A� pdbsr�rp ÿ ri�

�

� ÿa2xGr

�rp

ri

d2u

dt2dt

�A10�

For equilibrium the total shear stress at rp shouldequal the peak shear strength of the interface, that is:

ÿsp � tb1�rp� � tb2�rp�

� xGr

"A

pdb

d2u

dx2ÿ a

2

�rp

ri

d2u

dt2dt

#�A11�

Eq. (A11) is used to determine the distance rp for boltswith a face plate. For x > rp, the total shear stress iscalculated as:

tb�x� � xGr

"A

pdb

d2u

dx2ÿ a

2

�xri

d2u

dt2eÿ2axÿtdb dt

#

ÿ a2xGr

"�rp

ri

d2u

dt2dt

#eÿ2axÿtdb

�A12�

For a bolt without a face plate, Pf is zero. The shearstress, at x=rp, induced by the shear stress on thedecoupled interface, sr, becomes

tb2�rp� � a2

pdb

Asr�rp ÿ ri� �A13�

Similarly to the case with a face plate, the total shearstress at rp should equal the peak shear strength of theinterface, that is

ÿsp � tb1�rp� � tb2�rp�

� xGrA

pdb

d2u

dx2� a

2

pdb

Asr�rp ÿ ri� �A14�

Eq. (A14) is used to determine the distance rp for boltswithout a face plate. For x > rp, the total shear stressis calculated as:

tb�x� � xGr

24 A

pdb

d2u

dx2ÿ a

2

�xrp

d2u

dt2eÿ2a

xÿ t

db dt

35

� a2

pdb

Asr�rp ÿ ri�e

ÿ2axÿ t

db

�A15�

Appendix B. Joint opening and the stresses induced inrock bolts

Fig. B1 illustrates a bolt intersecting a rock joint.We shall establish the relationship between the joint

opening and the load induced in the bolt. An openingdisplacement of the rock joint is equivalent to applyingan axial tensile load to both sides of the bolt at thejoint. We shall look at this problem for fully groutedbolts and for frictionally coupled bolts separately.

B.1. Fully grouted rock boltsThe axial tensile stress in the bolt is symmetric to

the rock joint. Therefore, we consider only half of therock±bolt system. The model for the shear stress alonga bolt subjected to an axial load is shown in Fig. 4.The elongation of the bolt in di�erent sections isdenoted as follows: d0 is the elongation of the bolt insection (0 R x< x0); d1 is the elongation of the bolt insection (x0 R x < x1); d2 is the elongation of the boltin section (x1 R x < x2); and d3 is the elongation ofthe bolt in section (x2 R x < L ), where L is the halflength of the bolt. The sum of these four componentsis the total elongation of the bolt from each side of thejoint. Thus, the displacement of the joint opening istwice this summation, i.e.

dJ � 2X3i�0

di �B1�

Not all the four elongation comments appear in theabove expression at any given time. When the jointopens very little, the axial load induced does not causethe interface to be decoupled. In this case, only d3exists in Eq. (B1). The components d2, d1 and d0appear subsequently in the equation with increases inthe joint opening.

1. For the case where the interface undergoes compati-ble deformation across the interface, the shear stressalong the bolt is illustrated in Fig. 6. In this case wehave

d0 � d1 � d2 � 0 �B2�As shown in Eqs. (2) and (4), the shear and axialstresses along the bolt are given by

Fig. B1. A sketch illustrating a rock bolt intersecting a joint.


tb�x� � a2sb0 e

ÿ2a xdb

sb�x� � sb0 eÿ2a x

db �B3�This stage ceases when the shear stress at x = 0reaches the peak shear strength of the interface, sp.The corresponding axial stress at the joint at thismoment is

sb0 � 2sp

a�B4�

The elongation of the bolt is calculated as (assum-ing L>>db)

dJ � 2d3 � 2

Eb

�L0

s�x�dx1dbsb0

aEb

�B5�

When sb0=(2sp/a ), the opening displacementreaches its maximum, dJmax, before decouplingoccurs. It is given by:

dJmax � 2dbsp

a2Eb

�B6�

2. For the case where decoupling occurs (assumingx0=0):

Let D � x2 ÿ x1, and o � sr=sp

For x < x1:

tb�x� � sr

sb�x� � sb0 ÿ 4srx

db�B7�

For x1 R x< x2:

tb�x� � sr � �1ÿ o�spxÿ x1

x2 ÿ x1

sb�x� � sb0 ÿ 2sp

db

�2ox� �1ÿ o��xÿ x1�2

D

��B8�

For xrx2:

tb�x� � sp eÿ2aÿxÿx 2

db

�

sb�x� � 2sp

aeÿ2aÿxÿx 2

db

��B9�

At x=x2, we have from Eqs. (B8) and (B9)

sb�x2� � sb0 ÿ 2sp

db

�2ox2 � �1ÿ o�D�

sb�x2� � 2sp

a

Therefore, we obtain

x2 � db

4o

�sb0

sp

ÿ 2

aÿ 2�1ÿ o� D

db

��B10�

The elongation of the bolt can be obtained by thefollowing integration

di ��e dx � 1

Eb

�sb dx �B11�

i.e.

d1 � 1

Eb

�x 1

0

�sb0 ÿ 4sr

x

db

�dx

� 1

Eb

"sb0�x2 ÿ D� ÿ 2sr

�x2 ÿ D�2db

#�B12�

d2 � 1

Eb

�x 2

x 1

(sb0 ÿ 2sp

db

�2ox

� �1ÿ o��xÿ x1�2D

�)dx

� 1

Eb

(sb0Dÿ 2D2

3db�2o� 1�sp

)�B13�

d3 � 1

Eb

�Lx 2

2sp

aeÿ2aÿxÿx 2

db

�dx1 1

Eb

sp

a2db �B14�

The opening displacement of the joint is calculatedas

dJ � 2�d1 � d2 � d3�

� 2

Eb

"sb0x2 ÿ 2osp

�x2 ÿ D�2db

ÿ �2o� 1�sp2D2

3db

� sp

a2db

# �B15�

B.2. Fully frictionally coupled rock boltsIn the stage of compatible deformation, the shear

and axial stresses have the same forms as thoseexpressed in Eq. (B3) and the elongation has the sameforms as those expressed in Eqs. (B5) and (B6).

When decoupling occurs at the interface, the shearstress along the bolt is illustrated in Fig. 5. The


elongation of the bolt in this case is calculated as fol-lows.

For xR x2: the stresses are

tb�x� � s

sb�x� � sb0 ÿ pdbx

As �B16�

The elongation of the bolt in section (0R x< x2) is

d1 � 1

Eb

�x 2

0

�sb0 ÿ pdb

Asx

�dx

� 1

Eb

�sb0x2 ÿ pdb

2Asx2

2

��B17�

For xrx2:

tb�x� � s eÿ2axÿx 2

db

sb�x� � 2s

aeÿ2axÿx 2

db �B18�

At x=x2, we have from Eqs. (B16) and (B17)

sb�x2� � sb0 ÿ pdb

Asx2

sb�x2� � 2s

a

Then we obtain the expression for x2 as

x2 � A

pdbs

�sb0 ÿ 2s

a

��B19�

The elongation of the bolt in section (x2 R x < L ) iscalculated as

d3 � 1

Eb

�Lx 2

2s

aeÿ2axÿx 2

db dx

� 1

Eb

sdb

a2�1ÿ e

ÿ2aLÿ x2

db �1 1

Eb

sdb

a2

�B20�

The total elongation is

dJ � 2�d1 � d3� � 2

Eb

�sb0x2 ÿ psdb

2Ax22 �

sdb

a2

��B21�

References

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analytical models for rock bolts

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