Research ArticleMacro-Micro Failure Mechanisms and Damage Modelingof a Bolted Rock Joint

GangWang12 Yongzheng Zhang12 Yujing Jiang2 ShugangWang3 andWenjun Jing4

1Shandong Provincial Key Laboratory of Civil Engineering Disaster Prevention and MitigationShandong University of Science and Technology Qingdao 266590 China2State Key Laboratory of Mining Disaster Prevention and Control Co-Founded by Shandong Province andthe Ministry of Science and Technology Shandong University of Science and Technology Qingdao 266590 China3Research Center of Geotechnical and Structural Engineering Shandong University Jinan Shandong 250061 China4College of Pipeline and Civil Engineering China University of Petroleum Qingdao 266555 China

Correspondence should be addressed to Gang Wang wanggang1110gmailcom

Gang Wang and Yongzheng Zhang contributed equally to this work

Received 15 December 2016 Revised 3 June 2017 Accepted 28 June 2017 Published 31 August 2017

Academic Editor Fernando Lusquinos

Copyright copy 2017 Gang Wang et al This is an open access article distributed under the Creative Commons Attribution Licensewhich permits unrestricted use distribution and reproduction in any medium provided the original work is properly cited

The anchoring mechanism of a bolted joint subjected to a shear load was investigated using a bilinear constitutive model via theinner-embedded FISH language of particle flow code based on the discrete elementmethodThe influences of the anchoring systemon the macro-micromechanical response were studied by varying the inclination angle of the bolt The results indicate a clearrelationship between the mechanical response of a bolted rock joint and the mechanical properties of the anchoring angle Byoptimizing the anchorage angle the peak strength can be increased by nearly 50 relative to that at an anchorage angle of 90∘ Theoptimal anchorage angle ranges from 45∘ to 75∘ The damage mechanism at the optimal anchorage angle joint is revealed from amacroscopic mechanical perspective The concentration of the contact force between disks will appear in the joint and around thebolt resulting in crack initiation These cracks are mainly tensile cracks which are consistent with the formation mechanism forcompression-induced tensile cracks Therefore the macroscopic peak shear stress in the joint and the microscopic damage to theanchoring system should be considered when determining the optimal anchoring angle to reinforce a jointed rock mass

1 Introduction

Joints are extremely common in rockmasses and the stabilityof a jointed rock mass mostly depends on the mechanicalproperties of the joint Since the 1960s the shear behaviorof rock joints has been the focus of many studies and manyachievements have been made [1ndash9] In recent years rockbolts have been widely used in engineering due to their con-venience affordability and reliability However the anchor-ing mechanism for a jointed rock mass is still not clearly un-derstood As is known the anchoringmechanism for a boltedrock mass is highly complex due to the anisotropy of the rockmass and the complexity of the joint pattern Many scholarshave conducted experimental and theoretical studies to revealthe anchoring mechanism in a bolted rock joint

Sten Bjurstrom was the first to report systematic researchon fully grouted rock bolts His shear tests were conductedon full cementgrout bonded rock bolts embedded in blocksof granite According to Bjurstrom [10] inclining the boltresulted in stiffening of the shearing surface by increasingthe shear strength at a small displacement Ge and Liu [11]discussed the influence of the bolt on the joint and themechanism behind the dowel effect via laboratory simulationtests and theoretical analyses in bolted joints and proposeda modified shear stress formulation for a bolted joint Spangand Egger [12] conducted a series of shear tests in a boltedjoint to study the deformation characteristics of a bolt andfound two critical points in a bolt one at the bolt-jointintersection and the other at the hinge point Egger andZabuski [13] found that the anchoring mechanism of a bolt

HindawiAdvances in Materials Science and EngineeringVolume 2017 Article ID 1627103 15 pageshttpsdoiorg10115520171627103

2 Advances in Materials Science and Engineering

Disk

Contact bond

Parallel bond

(a)

B

Parallel bond

ti

ni

A

Fi

Mi

Mn

2R

L

(b)

T

V

M

t

2R

L

(c)

Figure 1 The parallel bond model and the stress state between disks

in a reinforced joint contributes additional shear resistance toprevent the shear failure of a joint Bolt deformation typicallyoccurs near the joint Ferrero [14] proposed a shear strengthmodel for a reinforced joint that considered both the doweleffect and the incremental increase in axial force due to thebar deformation Pellet and Egger [15] related theoreticaland experimental analyses of the rock bolt shear strengthand found that bolts installed perpendicular to a joint planeallowed the greatest displacement along a joint prior to failurebut that the displacement at failure decreased rapidly asthe angle between the bolt and joint plane decreased Theyalso found that harder rock led to bolt failures at smallerdisplacements Jalalifar andAziz [16] performed double sheartests and numerical simulations on five types of bolts to studythe bolt force and failure mechanism Wang et al [17] usedANSYS to calculate the stress intensity factors on the tipof cracks for different anchoring conditions and obtaineda relationship between the anchoring spacing anchoringangle and stress intensity factor using FLAC3D software tosimulate the coalescence modes for main-control cracks atdifferent anchor spacings Li et al [18] analyzed a numericalmodel of a fully grouted rock bolt installed in concrete withFLAC3D and concluded that the boltrsquos resistance to shearwas influenced by the rock strength inclination angle anddiameter of the rock bolt In recent years with the devel-opment of high power computers the numerical softwarePFC based on the discrete element method has becomeincreasinglymature in studying rock and soil failure behavior[19] Many achievements have been made in determiningthe mechanical properties of rock joints [20ndash24] Rock jointseffectively compensate for the lack of repeatable rock massSome studies have examined the roughness of jointed rockmass by using a DEM model For example Xia et al [25]carried out a numerical simulation of shear experiments onnonbolted rock using the discrete element software PFC2Dand analyzed the direct shear characteristics of rough jointsUsing the experimental results of rock joints Zhou et al[26] analyzed the fracture evolution characteristics of rockmass during shearing via discrete element PFC software Caoet al [27] used PFC to generate shear models with differentroughness values and discussed the influence of roughness

on the surface morphology of nonanchor joints from amicropoint of view

To date only a few studies have considered the con-stitutive model of the bolt particularly under bolted shearconditions Most studies have focused on the shear perfor-mance of a bolted joint and its deformation characteristicswhile neglecting the influence of the anchorage angle on themechanical properties of bolted rock jointsThese studies canreveal the anchoring mechanism in a bolted joint to a certainextent The micromechanism is the fundamental basis of themacroscopic properties Therefore it is of practical signifi-cance to research themacro-microanchoringmechanism fora bolted rock joint at different anchorage angles to providefurther references to guide designs of anchors in a bolted rockjoint at varying anchorage angles

Based on the previous results in this study we modifiedthe bolt constitutive model and analyzed the macro-micro-mechanical response of a bolted jointed rock mass usingthe discrete element software PFC2D at different anchorageangles The disk rotation and microcracking in the thin layeron the face of the anchor joint can be analyzed at differentinclinations based on the stress state of the anchor bolts andthe disk contact force during the shearing process

2 PFC Software and BPM Model

PFC is a discrete element commercial software developed bythe ItascaConsultingGroup andhas been alreadywidely usedin the rockmechanics field [19] PFC2D represents a rockmassas an assemblage of circular disks with a finite thickness andconnected via cohesive and frictional bonds

A basic linear contactmodel describes the elastic relation-ship between the relative displacements and forces of disks atthe point contact as shown in Figure 1

Thismodel involves the contact normal force component119865119899 contact overlap119880119899 shear force incrementΔ119865119904 and shear-displacement increment Δ119880119904 and is given by

119865119899 = 119896119899119880119899 (1)

Advances in Materials Science and Engineering 3

where 119896119899 is the normal stiffness (forcedisplacement) at thecontactThe value of 119896119899 is determined by the current contact-stiffness model

Note that the normal stiffness 119896119899 is a secant modulus inthat it relates total displacement and forceThe shear stiffness119896119904 on the other hand is a tangent modulus in that it relatesincremental displacement and force An uppercase 119870 will beused to denote a secant modulus and a lowercase 119896 will beused to denote a tangent modulus The computation of thenormal contact force from the geometry alone makes thecode less prone to numerical drift and able to handle arbitraryplacement of balls and changes in ball radii after a simulationhas begun

The shear contact force is computed in an incrementalfashion When the contact is formed the total shear contactforce is initialized to zero Each subsequent relative shear-displacement increment results in an increment of elasticshear force that is added to the current value The motion ofthe contact must be considered during this procedure

The contact velocity can be resolved into normal andshear componentswith respect to the contact planeDenotingthese components by 119881119899119894 and 119881119904119894 for the normal and shearcomponents respectively the shear component of the contactvelocity can be written as

119881119904119894 = 119881119894 minus 119881119899119894 = 119881119894 minus 119881119895119899119895119899119894 (2)

The shear component of the contact displacement-incre-ment vector occurring over a time step of Δ119905 is calculated by

Δ119880119904119894 = 119881119904119894 Δ119905 (3)

and is used to calculate the shear elastic force-increment vec-tor

Δ119865119904 = minus119896119904Δ119880119904 (4)

The frictional resistance of the contact is given by

119865119904 le 120583119865119899 (5)

where 120583 is the friction coefficient between the disksTo simulate a relatively brittle rock-like material it is

necessary to cement these disks with a bonded model Thisstudy uses the parallel bondmodel which resists not only thecontact forces but also the moments between the disks at acemented contact (Figure 1) The function mechanism of theparallel bond model is described by

Δ119865119899 = 119896119899119860Δ119880119899Δ119865119904 = minus119896119904119860Δ119880119904Δ119872119899 = minus119896119904119869Δ120579119899Δ119872119904 = minus119896119899119868Δ120579119904

(6)

where 119865119899 and 119865119904 are the force components about the centerof the cemented-contact zone119872119899 and119872119904 are the momentsabout the center of the cemented-contact zone 119896119899 and 119896119904

are the normal and shear bond stiffness per unit area respec-tively 120579119899 and 120579119904 are the rotation angle components and 119860119869 and 119868 are the area polar moment of inertia and momentof inertia of the bond contact cross section respectively Thestrength of the cemented contact is then given by

120590max = minus119865119899119860 +

1003816100381610038161003816100381611987211990410038161003816100381610038161003816 119877119868 lt 120590119888

120591max = minus119865119904119860 +

1003816100381610038161003816100381611987211989910038161003816100381610038161003816 119877119869 lt 120591119888(7)

where 119877 is the radius of the bonded zone between the disks 119905is the length of the bonded zone between the disks (Figure 1)and 120590119888 and 120591119888 are the tensile and shear strength of the bondcontact respectively

Tensile cracks occur when the applied tensile stressexceeds the specified tensile strength of the parallel bond120590119888 Shear cracks occur when the applied shear stress exceedsthe specified shear strength of the parallel bond 120591119888 either byrotation or by the shearing of the disks The tensile strengthat the contact immediately drops to zero once the crackoccurs and the shear strength reduces to the residual frictionstrength (see (5)) [19 22] as illustrated in Figure 2

3 Direct Shear Simulation of the BoltedJoint Using PFC

31 Microparameters of the Rock Sample The intact syntheticmodel material selected in this study is represented bycompacted disks cemented with the parallel bond modelTherefore the micromechanical parameters consist of twocategories one for disks and the other for parallel bondsThe PFC code allows one to simulate the macromechanicalbehavior of the selected synthetic model material by selectingappropriate values for the micromechanical parameters suchas the disk size distribution and packing disk and parallelbond stiffness disk friction coefficient and bond strengths[19 22] To select the appropriatemicromechanical parametervalues for the synthetic intact material one needs to basicallygo through a trial and error procedure by iteratively varyingthe micromechanical parameter values to match the requiredmacromechanical behaviors of the selected synthetic mate-rial This procedure is known as the calibration of the intactsyntheticmaterial based on themicromechanical parametersHerein the required macromechanical behaviors are thestrength response represented by the uniaxial compressivestrength (UCS) and the deformability response representedby Youngrsquos modulus and Poissonrsquos ratio of a cylindricalspecimen with a 50mm diameter and a heightdiameter ratioof 2 1 under unconfined compression

In this study a special calibration sequence was followedto rationalize themicromechanical parameter calibration andto minimize the number of iterations First the disk and par-allel bond moduli and the ratios of normal to shear stiffnesswere set equal between the disks and parallel bonds to reducethe number of independent parameters [22] Then Youngrsquosmodulus was calibrated by setting the material strengths toa large value and varying disk Youngrsquos modulus and parallel

4 Advances in Materials Science and Engineering

1

1Compression

Friction sliding

Bond breaks

Tension Shear

Bond breaks

F

Fs

Fn

fs

ks

kn

U

Us gt 0

Un gt 0

Un lt 0

Fi bond strength

Ui displacement

ki contact stiffness

Figure 2 Illustration of the yield process for a parallel bond

(a) Sample from PFC (b) Samples of rock-like materials

Figure 3 Failure patterns in the cylindrical sample subjected to uniaxial compression

bond Youngrsquos modulus to match Youngrsquos modulus betweenthe numerical and laboratory specimens Next by changingthe ratio of disk normal stiffnessdisk shear stiffness andparallel bond normal stiffnessparallel bond shear stiffnessPoissonrsquos ratio of the numerically simulated intact syntheticcylindrical specimen was matched to that of the laboratoryspecimen

After calibrating the aforementioned deformationmicro-mechanical parameters the peak strength between the nu-merical and laboratory specimens was matched by graduallyreducing the normal and shear bond strengths of the parallelbondsDuring this procedure it is important to fix the ratio ofnormal to shear bond strength 120590119888120591119888 because these parame-ters affect the failuremode of the specimenTherefore a seriesof numerical simulations were conducted tomatch the failuremode of the cylindrical specimen between the numerical

and laboratory specimens by varying 120590119888120591119888 while keepingthe other parameters unchangedThe obtained failure modeswith varying 120590119888120591119888 for the cylindrical specimens used in thenumerical simulations are displayed in Figure 3

As seen in Figure 3 as the specimen is compressed underuniaxial stress numerous cracks (tensile and shear cracks)are produced through the breakage of parallel bonds Thefailure mode of the numerically simulated synthetic cylindri-cal specimen with 120590119888120591119888 = 10 agrees well with the resultsfrom the laboratory specimen Thus setting the ratio ofnormal to shear bond strength equal to one seems to increasethe confidence of the numerical model in simulating theappropriate failure behavior of the synthetic material

The numerically obtained stress-strain curves were com-pared with the stress-strain curve obtained from the labora-tory sample as shown in Figure 4 In this comparison in

Advances in Materials Science and Engineering 5

Laboratory test curvePFC simulation curve

Axial strain ()

9MPa

6MPa

3MPa

0

18

36

54

72

0000 0003 0006 0009

(1ndash

3)

(MPa

)

Figure 4 Stress-strain curves at different confining stresses

Table 1 Microparameters of the rock sample

Microparameters Values RemarkMinimum radiusmm 05Disk radius ratio 166Disk densitykgsdotmminus3 1830 Uniform distributionDisk contact modulusGPa 395Disk stiffness ratio 10Disk friction coefficients 05Parallel bond normal strengthMPa 245 plusmn 65 Normal distributionParallel bond shear strengthMPa 245 plusmn 65 Normal distributionParallel bond modulusGPa 395Parallel bond stiffness ratio 15Parallel bond radius multiplier 10

addition to comparing the prepeak behavior it is necessary tomatch the macro UCS and Youngrsquos modulus values obtainedfor different numerical simulations of the synthetic materialwith those obtained for the laboratory specimen Thereforethe disk contact modulus values need to be modified simul-taneously with the 120583 value Figure 4 shows that the prepeakand postpeak trend of the laboratory stress-strain curve isapproximately parallel to that of all the stress-strain curves ofnumerical specimens Therefore the prepeak and postpeakbehavior of numerical specimens did not provide any guid-ance for the selection of disk friction value However thecoefficient of friction obtained from the laboratory triaxialtests on the synthetic intact rock was approximately 050Therefore the friction coefficient value of 050 was selectedto represent the disk friction Table 1 lists the determinedmicromechanical parameter values to simulate the intactsynthetic rock with the parallel bond model The failuremode the stress-strain curves and the comparison of themacromechanical parameters indicate that the results are ingood agreement with the laboratory test results

32 Bilinear Constitutive Model of the Bolt The bolt isrepresented by a cluster of balls with parallel bonds thatmeet the requirements of the force-displacement law Parallelbonds can transmit both forces and moments between diskswhich is consistent with the stress state in a bolt The stressstate in a bolt is highly complex when it is reinforced ina jointed rock mass The microparameters of the bolt aredetermined via the bolt pull-out test as shown in Figure 5(a)The microparameters of the bolt are shown in Table 2Figure 5(b) describes the bolt constitutive model whichis not modified One shortcoming of this model is thatit can represent the characteristics of a metallic bar onlyduring elastic deformation the model cannot represent thelarge plastic deformations during yielding Therefore thisconstitutive model has been modified for use in the internalFISH language in PFC to make the modified bolt constitutivemodel meet the stiffness and deformation requirements Asshown in Figure 5(c) the yield strength of the bolt is set suchthat it remains in the linear elastic stage prior to the axialstress reaching the yielding strength However when the axial

6 Advances in Materials Science and Engineering

F

(a)

Bond breaks bolt losesits bearing capacity

e

t

(b)

Bolt yields but it stillhas bearing capacity

Bond breaks bolt losesits bearing capacity

e p

t

(c)

Figure 5 Illustration of the pull-out test for rock bolts and the constitutive rock bolt models

Table 2 Microparameters of the bolt

Microparameters Values RemarkMinimum radiusmm 20Disk radius ratio 10Disk densitykgsdotmminus3 7000 Uniform distributionDisk contact modulusGPa 30Disk stiffness ratio 12Disk friction coefficients 10Parallel bond normal strengthMPa 380 plusmn 20 Normal distributionParallel bond shear strengthMPa 266 plusmn 20 Normal distributionParallel bond stiffness ratio 26Parallel bond radius multiplier 10

stress exceeds the yield strength the bonds will not breakmeaning that the bolt maintains a certain capacity If themagnitude of the axial stress in the bolt exceeds themaximumnormal stress the bonds are broken and the bolt will lose itsbearing capacity

33 Direct Shear Test Model of the Bolted Joint The directshear test model is built using PFC2DThemodel is composedof a top and bottom rock sample grout joint and a boltwith a size of 100mm times 50mm with a total of 2926 disksThe microparameters of the rock sample used are shownin Table 1 The grout microparameters are based on therock parameters with the parallel bond strength changed to40MPa The joint is created using its own command in PFCand the disks in the joint are unbounded Therefore theparallel bond stiffness and the strength of the disks in the jointare set to zero A shear box is generated by the wall elementAmong these walls the sixth wall is the loading wall withthe first wall fifth wall and the sixth wall together formingthe down active shear box which can move in the horizon-tal direction The upper shear box is fixed in the horizontal

direction and only the second wall can move up or down asit is the servo wall which keeps the normal stress constantduring shearing As shown in Figure 6 the rock mass isrepresented by light gray disks and the bolt is composed of acluster of light gray disks which are tangential to each otherAround the bolt the dark gray disks represent the groutAlong the horizontal direction in the middle of the modelthe joint is composed of black disks

34 Direct Shear Test of the Anchoring Model During shear-ing the second wall is controlled by the servo mechanism toapply a constant normal stress with the vertical displacementmonitored as the normal displacement The fifth wall issubjected to a constant velocity to apply horizontal shearloading with the horizontal displacement monitored as theshear displacement The shear stress is calculated by dividingthe average force on the fifth and sixth walls The test isendedwhen the shear displacement reaches the present valueMicrocracking due to bond breakage shear stress normaldisplacement stress in the bolt and displacement variationsin the bolt disks are monitored during the shear test

Advances in Materials Science and Engineering 7

1

3

2

5

48

67

1

3

22

5

6

1

] ]

n

(a) Numerical model of a bolted joint at 45∘

1

3

2

5

48

67

1

3

2

5

6

1

] ]

n

(b) Numerical model of a bolted joint at 75∘

Figure 6 Numerical direct shear test model of the bolted joint

15∘

30∘

45∘

60∘

75∘

90∘

0

2

4

6

8

Shea

r stre

ss (M

Pa)

05 10 15 20 2500

Shear displacement (mm)

Figure 7 Evolution of the shear stress for different anchorage angles

4 Analysis of the Results

41 Analysis of the Shear Performance of the Joint withDifferent Anchorage Angles Direct shear simulations of thebolted joint were conducted using the bolt at differentanchorage angles under a constant normal stress to examinethe macro-micromechanical responses of the bolted jointduring shearing Figure 7 shows the evolution of the shearstress with the shear displacement at different anchorageangles The shear simulation at a normal stress of 5MPa isachieved after a shear displacement of 25mm is reachedThepeak shear strength for different anchorage angles is shownin Table 3

Our data show that (1) the peak shear strength increasedto a maximum and then decreased with further increase inthe anchorage angle and (2) the peak shear strength is thelargest at an anchorage angle of 75∘ Compared to the 90∘anchorage angle changing the angle of the anchor can greatlyimprove the boltrsquos antishear abilitiesTherefore it is importantto determine the optimum design for a bolted rock joint inpractice

Table 3 Peak shear strength for different anchorage angles

Anchor angle The peak shear strength (MPa)15∘ 49830∘ 60045∘ 67860∘ 65275∘ 76590∘ 498

42 Analysis of the Bolt Stress in a Compression-ShearStress State

421 Stress Analysis of the Anchor Joint Plane in a Compres-sion-Shear Stress State Under compressive shear stress therock bolts experience not only shear deformation throughthe thickness of the weak plane but also notable shear defor-mation over a considerable section (the effective length) Tosimplify the analysis we set the effective length of the rock

8 Advances in Materials Science and Engineering

Joint surface Joint surface

b

b

ln ltn

tb

Figure 8 Deformations of the joint surface and bolt

b b blt

b2

(a) Shear force

b b bln

b2

(b) Axial force

Figure 9 Axial and shearing forces on the bolt

bolt to be 119897119905 and the effective length of the bolt to be 119897119899 asshown in Figure 8

The average shear stress distribution on the plane of the119897119905 section is related to the contact conditions of the rodbody and hole wall Three different distribution patternsfor the simplified calculation are shown in Figure 9(a) Thedistribution pattern is a curve but can be measured inthree basic forms Similar to the 119897119899 section the axial stressdistribution can be used as shown in Figure 9(b)

From Figures 8 and 9 and from the joint surface equi-librium conditions the stress tensor of the rock mass can beobtained from the normal and tangential components of thestress tensor along the joint plane as shown in

120590 = 120590119904 + 120590119887119899= 119896119899 (119880119899 sin 119886 minus 119880119905 cos 119886) + 119875119866119887119880119905119860119861119905119897119905 cos 119886

minus 119875119864119887119880119899119861119899119897119899 sin 119886

120591 = 120591119887119904= 119896119904 (119880119899 cos 119886 + 119880119905 sin 119886) + 119875119864119887119880119899119861119899119897119899 cos 119886

+ 119875119866119887119880119905119860119861119905119897119905 sin 119886

+ [119875119866119887119880119905119860119861119905119897119905 cos 119886 minus119875119864119887119880119899119861119899119897119899 sin 119886]119891119904

(8)

In the previous formula 120575119905 and 120575119899 are the weak levels oftangential and normal displacement respectively 119880119905 and 119880119899are the bolt axial and transverse deformations respectively119886 is the angle between the bolt and joint surface and 119864119887 and119866119887 are Youngrsquos modulus and shear modulus for the materialrespectivelyThe shear coefficient119860 is associated with the rodbody section (119860 = 34 for a circular solid rod) 119861119905 and 119861119899 arethe inner pole sectionsrsquo average shear stress and axial stressdistribution of the shape coefficient respectively which are1 34 and 12 120590119887 and 120591119887 are the axial and shear forces on

Advances in Materials Science and Engineering 9

the bolt respectively 119896119899 and 119896119904 are the normal and tangentialstiffness coefficients for the weak level119875 is the anchor sectionreinforcement ratio Therefore

119880119899 = 1119886 (11988622120590 minus 11988612120591)

119880119905 = 1119886 (11988611120591 minus 11988621120590)

(9)

where

11988611 = minus(119896119899 + 119875119864119887119861119899119897119899) sin 119886

11988612 = (119896119899 + 119875119866119887119860119861119905119897119905) cos 119886

11988621 = 119896119904 cos 119886 + 119875119864119887119861119899119897119899 (cos 119886 minus 119891119904 sin 119886)

11988622 = 119896119904 sin 119886 + 119875119866119887119860119861119905119897119905 (sin 119886 + 119891119904 cos 119886)

119886 = 1198861111988622 minus 1198861211988621= 119896119899119896119904 minus 119875119864119887

119861119899119897119899119875119866119887119860119861119905119897119905 +

119875119866119887119860119861119905119897119905 (119896119899sin

2119886 minus 119896119904cos2119886)

+ 119875119864119887119861119899119897119899 [119896119899cos

2119886 minus 119896119904sin2119886]

+ 119896119899119891119904 sin 119886 cos 119886 [ 119875119866119887119860119861119905119897119905 minus119875119864119887119861119899119897119899 ]

(10)

Therefore the relationship between the stress and thestress component of the element can be determined as shownin

120590119887 = 119864119887119861119899119897119899119886 (11988622120590 minus 11988612120591)

120591119887 = 119866119887119860119861119905119897119905119886 (11988611120591 minus 11988621120590)

(11)

422 Analysis of the Bolt Stress for a Particular AnchorageAngle A bolt installed at a particular anchorage angle willhave a better anchoring effect than a bolt installed perpen-dicular to the joint plane Figures 10 and 11 show illustrationsof the stresses in a bolt at anchorage angles of 75∘ and 90∘respectively during shearing

The diagram showing the two different anchorage anglebolts indicates that the anchor force and bending momentare relatively large in the vertically installed bolt leading tobolt axial and shear stress that are larger than those in thetilted bolt whereas for the tilted bolt the axial and shearstress are small and do not reach the yield strengthThereforethe tilted anchor bolts not only improve the shear capacityof the joints but also maintain an appropriate stress statein the anchor The rock bolt axial stress diagram indicatesthat the axial stress increases as the shear displacementincreases for different anchorage anglesThe axial stress is the

highest when the anchor is vertically installed on the jointsurface The results of these two cases reveal that the optimalanchoring angle is approximately 75∘This result correspondsto the previous conclusions that the optimal anchoring angleshould be between 45∘ and 75∘

43 Evolution of the Contact Force and Microscopic Crackingof Disks The anchorage angle of 75∘ was chosen for amore detailed analysis on the micromechanical response anddamage mechanism during shearing Figure 12 shows theevolution of the shear stress number of cracks and normaldisplacement for a normal stress of 5MPa

To accurately determine the evolution features six differ-ent monitoring points (a to f) were set on the shear stresscurve which correspond to shear displacements of 0 01980217 0513 1500 and 2500mm respectively The variationsin the contact force and cracking can be observed from thesix monitoring points As shown in Figure 12 ldquoardquo indicatesthe initial point of the shear displacement and ldquocrdquo indicatesthe shear displacement at which the shear stress reachesthe peak shear strength As shown in Figure 12 no shearcrack is generated during the initial stage of the shear testThe number of shear cracks increases gradually as the sheardisplacement increases When the shear stress reaches thepeak shear strength the number of shear cracks increasessignificantly and the shear crack growth rate tends to stabilizeas the shear stress increases beyond the peak shear strengthThe trends in tension cracking and normal stress are highlysimilar and increase monotonically

During the shearing of a bolted joint the loaded side ofthe bolt is in a state of compression whereas the other side ofthe bolt is in a state of tension due to the bending deflectionof the bolt The adjacent grout is in a state of compressionand the rockmass is in a state of tension because of the bondsbetween the bolt and grout as shown in Figure 13 Bendingdeflection will occur in the bolt as the shear displacementincreases Simultaneously cracks will occur continuously dueto extrusion and slippage between disks In the compressionzone there are a large number of cracks which are mainlytension cracks generated by the extrusion between disksThisresult agreeswith the formationmechanism for compression-induced tension cracks shown in Figure 13

Figure 14 shows the illustration of the contact forceand crack propagation between disks during shearing Thecontact force is initially concentrated on the joint and theloading end and cracks are generated in the joint whereasthere are only a few cracks around the bolt As the sheardisplacement increases the diskrsquos contact force around thebolt becomes gradually larger Cracks occur on the jointand those around the bolt extend further The cracks aroundthe bolt propagate most rapidly The cracks are initiallyconcentrated at the intersection of the bolt and joint and thenpropagate to the ends of the bolt Finally the majority of thecracks are distributed at the joint and around the bolt aftera larger displacement occurs Finally more cracks occur inthe compression zone around the bolt than in the tensionzone In addition the distribution of cracks corresponds tothe contact force in the stress concentration areaThe detailed

10 Advances in Materials Science and Engineering

0 10 20 30 40

Distance along bolt (mm)

Joint plane

Bolt

05 mm10 mm15 mm

20 mm25 mm30 mm

0

100

200

300

Axi

al st

ress

(MPa

)

(a) Axial stress diagram

0 10 20 30 40

Distance along bolt (mm)

05 mm10 mm15 mm

20 mm25 mm30 mm

minus600

minus400

minus200

0

200

400

600

Bend

ing

mom

ent (

Nm

)

Joint plane

Bolt

(b) Bending moment diagram

Joint plane

Bolt

0 10 20 30 40

Distance along bolt (mm)

05 mm10 mm15 mm

20 mm25 mm30 mm

0

10

20

30

40

Shea

r stre

ss (M

Pa)

(c) Shear stress diagram

Figure 10 Bolt stress state diagram at an anchorage angle of 75∘

maximum contact force and crack number in the specimen atthe different monitoring points during shearing are providedin Tables 4 and 5 respectively

44 Evolution of the Disk Rotation During the shearing slipprocess the behavior is determined by the rotation of thedisks Relative disk dislocations will occur due to the rotationof the disks If the rotation angle is sufficiently large itwill lead to shear stresses and the breaking of disk bondsMeanwhile cracks occurring in the rock mass and shearcracking are caused directly by disk rotation Figure 15 showsa graph of the evolution of the disk rotation in terms ofradians for rock disks at different monitoring points

Figure 15 shows that when the shear displacement issmall at the initial stages the disk rotation angle is relatively

small and disks with large rotational angles are distributedaround the bolt The disk rotation angles gradually increasewith increasing shear displacement The disks containinglarger rotation angles are mainly distributed in the twoloading ends and around the bolt and the bolt and thedisks around the bolt contact move relatively violently duringshearing leading to a larger rotation angle Further increasesin the shear displacement and larger rotation disks alsoappeared on the joint surfaces which are caused by thejoint surface microconvex body cutting into the free disksThe distribution of the large disk rotation angles is highlyconsistent with the crack distribution demonstrating that therotation of the disks leads to the generation of shear cracks

Figure 16 shows the variations in the proportion ofrotations exceeding 01 radians As the shear displacement

Advances in Materials Science and Engineering 11

Bolt

Joint plane

0

50

100

150

200

250A

xial

stre

ss (M

Pa)

10 20 30 400Distance along bolt (mm)

05 mm10 mm15 mm

20 mm25 mm30 mm

(a) Axial stress diagram

Joint plane

Bolt

10 20 30 400Distance along bolt (mm)

minus800

minus600

minus400

minus200

0

200

400

600

800

Bend

ing

mom

ent (

Nm

)

05 mm10 mm15 mm

20 mm25 mm30 mm

(b) Bending moment diagram

Bolt

Joint plane

0

10

20

30

40

Shea

r stre

ss (M

Pa)

10 20 30 400Distance along bolt (mm)

05 mm10 mm15 mm

20 mm25 mm30 mm

(c) Shear stress diagram

Figure 11 Bolt stress state diagram at an anchorage angle of 90∘

Table 4 Maximum contact force in the specimen at different monitoring points

Monitoring point Maximum contact forceN Monitoring point Maximum contact forceNa 68201198904 d 12681198905b 15301198905 e 15801198905c 15481198905 f 14451198905

increases the number of clockwise and counterclockwisedisk rotations greater than 01 radians increases continuouslyAdditionally the number of clockwise rotations is greaterthan the number of counterclockwise rotations mainly dueto the shear loading directions

5 Conclusions

In this paper the anchoring mechanism for a tilted anchorbolt subjected to a shear load has been studied using a bilinearconstitutive model with the inner-embedded FISH language

12 Advances in Materials Science and Engineering

(f)(e)(d)

(c)

(b)

(a)

Shear stressNormal displacement

Tension crackShear crack

0

2

4

6

8

Shea

r stre

ss (M

Pa)

05 10 15 20 2500Shear displacement (mm)

00

01

02

03

04

05

06

07

08

Nor

mal

disp

lace

men

t (times10

minus3

m)

0

100

200

300

400

500

600

Crac

k nu

mbe

r

Figure 12 Evolution of the shear stress number of cracks and normal displacement

Rock

Bolt Grout

T

C

C

T

(a)

F F F

F F F

(b)

Figure 13 Stress state of the bolted rock joint system and illustration of the mechanism behind compression-induced tension cracks

Table 5 Number of cracks in the specimen at different monitoringpoints

Monitoringpoint Crack number Monitoring point Crack number

a 0 d 255b 21 e 446c 40 f 592

of PFC based on DEMThe anchoring mechanism of the boltatdifferentanchorage angles has been discussed at themacro-microlevels and the following conclusions have been drawn

(1) The bolt can improve the shear resistance of the jointmainly due to cohesion Tilted anchor bolts not only improvethe shear strength of the joint surface but also result in a lowerstress in the anchor bolt The peak shear strength can be 50greater than that of a vertically installed anchor bolt

(2) A tilted bolt allows for fully mobilized anchor perfor-manceThis configuration not only restricts increment in thenormal displacement but also indirectly improves the stress

state of the anchorage system and increases the shear capacityof the anchor joint The comprehensive shear displacementand shear stress data along with the bolt stress state analysisreveal that the optimal anchorage angle should be between45∘ and 75∘

(3) During the shearing process when the rock bolt andgrout are commonly constrained the stress concentrationphenomenon around the bolt and joint increases the sheardisplacement and contact force After reaching the residualphase the contact force exhibits slight fluctuations due to thesliding of the joint surface

(4) Based on the disk flow software PFC2D crack initia-tion for an anchorage angle of 75∘ will start from the contactforce concentration due to the interactions between the rockgrout and bolt Finally the cracks are mainly distributedin the joint and around the bolt for larger displacementsThe cracks are mainly tensile cracks which correspond tothe formation mechanism for a compression-induced tensilecrack The shear cracking was mainly caused by the rotationof the disks during the shearing process

Advances in Materials Science and Engineering 13

minus005 minus003 minus001 001 003 005

minus002minus001

0001002

minus551525354555

Con

tact

forc

e (N

)

times103

(a) Monitoring point a

minus002minus001

0001002

Con

tact

forc

e (N

)

minus10

10

30

50

70

90

110times103

minus005 minus003 minus001 001 003 005

(b) Monitoring point b

minus002minus001

0001002

Con

tact

forc

e (N

)minus101030507090110

times103

minus005 minus003 minus001 001 003 005

(c) Monitoring point c

minus002minus001

0001002

Con

tact

forc

e (N

)

minus10

10

30

50

70

90

110times103

minus005 minus003 minus001 001 003 005

(d) Monitoring point d

minus002minus001

0001002

minus51535557595115

Con

tact

forc

e (N

)

times103

minus005 minus003 minus001 001 003 005

(e) Monitoring point e

minus002minus001

0001002

minus10

10

30

50

70

90

Con

tact

forc

e (N

)

times103

minus005 minus003 minus001 001 003 005

(f) Monitoring point f

Figure 14 Evolution of the contact force and cracking at different monitoring points

14 Advances in Materials Science and Engineering

Particle rotation (in radians)minus120 to minus01

minus01 to minus0001

minus0001 to minus00005

minus00005 to 00005

00005 to 00010001 to 01

01 to 120

(a) Monitoring point a

Particle rotation (in radians)minus120 to minus01

minus01 to minus0001

minus0001 to minus00005

00005 to 00010001 to 01

01 to 120minus00005 to 00005

(b) Monitoring point b

Particle rotation (in radians)minus120 to minus01

minus01 to minus0001

minus0001 to minus00005

00005 to 00010001 to 01

01 to 120minus00005 to 00005

(c) Monitoring point c

Particle rotation (in radians)minus120 to minus01

minus01 to minus0001

minus0001 to minus00005

00005 to 00010001 to 01

01 to 120minus00005 to 00005

(d) Monitoring point d

Particle rotation (in radians)minus120 to minus01

minus01 to minus0001

minus0001 to minus00005

00005 to 00010001 to 01

01 to 120minus00005 to 00005

(e) Monitoring point e

Particle rotation (in radians)minus120 to minus01

minus01 to minus0001

minus0001 to minus00005

00005 to 00010001 to 01

01 to 120minus00005 to 00005

(f) Monitoring point f

Figure 15 Evolution of disk rotation in radians in the specimen at different monitoring points

0

5

10

15

20

Qua

ntita

tive p

ropo

rtio

n (

)

05 10 15 20 25 3000Shear displacement (mm)

gt 01Rotation (in radians)gt 01|Rotation (in radians)|

lt minus01Rotation (in radians)

Figure 16 Quantitative proportion variation of rotations exceeding 01 radians

Disclosure

GangWang and Yongzheng Zhang should be regarded as co-first authors

Conflicts of Interest

The authors declare that there are no conflicts of interestregarding the publication of this paper

Advances in Materials Science and Engineering 15

Acknowledgments

This study was supported by the National Natural ScienceFoundation of China (nos 51479108 51379117 and 51509261)China Postdoctoral Science Foundation (2015M580599) theTaishan Scholar Talent Team Support Plan for Advantagedamp Unique Discipline Areas the Postdoctoral Science Foun-dation of Shandong Province (no 201402014) and the Shan-dong Province Outstanding Young Scientist Research AwardFund (BS2014NJ006)

References

[1] F D Patton ldquoMultiple modes of shear failure in rockrdquo inProceedings of the 1st Congress on International Society for RockMechanics pp 509ndash513 1966

[2] R E Goodman Methods of Geological Engineering in Discon-tinuous Rocks West Publishing Company New York NY USA1976

[3] N Barton and V Choubey ldquoThe shear strength of rock joints intheory and practicerdquo Rock Mechanics Felsmechanik Mecaniquedes Roches vol 10 no 1-2 pp 1ndash54 1977

[4] P M Dight and H K Chiu ldquoPrediction of shear behavior ofjoints using profilesrdquo International Journal of Rock Mechanicsand Mining Sciences amp Geomechanics Abstracts vol 18 no 5pp 369ndash386 1981

[5] N Barton Modelling Rock Joint Behaviour from In Situ BlockTests Implications for NuclearWaste Repository Design vol 308Office ofNuclearWaste IsolationONWI ColumbusOHUSA1982

[6] T Esaki S Du Y Jiang and Y Wada ldquoEffect of the asperitydamage on the shear behavior of irregular rock jointsrdquo inProceedings of the Second International Conference on Analysisof Discontinuous Deformation pp 459ndash464 Kyoto Japan 1997

[7] Y Jiang J Xiao Y Tanabashi and T Mizokami ldquoDevelopmentof an automated servo-controlled direct shear apparatus apply-ing a constant normal stiffness conditionrdquo International Journalof Rock Mechanics and Mining Sciences vol 41 no 2 pp 275ndash286 2004

[8] Y J Jiang B Li and Y Tanabashi ldquoEstimating the relationbetween surface roughness and mechanical properties of rockjointsrdquo International Journal of Rock Mechanics and MiningSciences vol 43 no 6 pp 837ndash846 2006

[9] C-C Xia Z-C Tang W-M Xiao and Y-L Song ldquoNew peakshear strength criterion of rock joints based on quantifiedsurface descriptionrdquo Rock Mechanics and Rock Engineering vol47 no 2 pp 387ndash400 2014

[10] S Bjurstrom ldquoShear strength of hard rock joints reinforced bygrouted untensioned boltsrdquo in Proceedings of the In Proceedingsof the 3rd International ISRM Congress pp 1194ndash1199 DenverColo USA 1974

[11] X Ge and J Liu ldquoStudy of the shear resistance behaviour ofbolted rock jointsrdquo Journal of Geotechnical Engineering vol 10no 1 pp 8ndash19 1988 (Chinese)

[12] K Spang and P Egger ldquoAction of fully-grouted bolts in jointedrock and factors of influencerdquo Rock Mechanics and Rock Engi-neering vol 23 no 3 pp 201ndash229 1990

[13] P Egger and L Zabuski ldquoBehaviour of rough bolted joints indirect shear testsrdquo in Proceedings of the 7th ISRM Congress vol30 p 342 Aachen Germany September 1991

[14] A M Ferrero ldquoThe shear strength of reinforced rock jointsrdquoInternational Journal of Rock Mechanics and Mining Sciencesand vol 32 no 6 pp 595ndash605 1995

[15] F Pellet and P Egger ldquoAnalytical model for the mechanicalbehaviour of bolted rock joints subjected to shearingrdquo RockMechanics and Rock Engineering vol 29 no 2 pp 73ndash97 1996

[16] H Jalalifar and N Aziz ldquoExperimental and 3D numericalsimulation of reinforced shear jointsrdquo RockMechanics and RockEngineering vol 43 no 1 pp 95ndash103 2010

[17] P Wang T Feng Y-J Zhu and W-J Yu ldquoExperimental studyand numerical simulation of anchoringmechanismof anchoredrocklikematerial with prefabricated fracturerdquoYantu LixueRockand Soil Mechanics vol 37 no 3 pp 793ndash801 2016 (Chinese)

[18] L Li P C Hagan S Saydam B Hebblewhite and Y Li ldquoPara-metric study of rockbolt shear behaviour by double shear testrdquoRock Mechanics and Rock Engineering vol 49 no 12 pp 4787ndash4797 2016

[19] Itasca Consulting Group Inc ldquoPFC2D (Particle Flow Code in2D)Theory and Backgroundrdquo Minneapolis Minn USA 2008

[20] P A Cundall andOD L Strack ldquoThe distinct numericalmodelfor granular assembliesrdquo Geotechnique vol 29 no 1 pp 47ndash651979

[21] J Zhou and Y Chi ldquoThe method of particle flow and PFC2Dcoderdquo Rock and Soil Mechanics vol 21 no 3 pp 271ndash274 2000(Chinese)

[22] D O Potyondy and P A Cundall ldquoA bonded-particle modelfor rockrdquo International Journal of Rock Mechanics and MiningSciences vol 41 no 8 pp 1329ndash1364 2004

[23] N Cho CDMartin andDC Sego ldquoA clumped particlemodelfor rockrdquo International Journal of Rock Mechanics and MiningSciences vol 44 no 7 pp 997ndash1010 2007

[24] Z Zhao L Jing and I Neretnieks ldquoParticle mechanics modelfor the effects of shear on solute retardation coefficient in rockfracturesrdquo International Journal of Rock Mechanics and MiningSciences vol 52 pp 92ndash102 2012

[25] C Xia Y Song Z Tang Y Song and C Shou ldquoParticleflow numerical simulation for shear behavior of rough jointsrdquoChinese Journal of Rock Mechanics and Engineering vol 31 no8 pp 1545ndash1552 2012 (Chinese)

[26] Y Zhou A Misra S Wu and X Zhang ldquoMacro- and meso-analyses of rock joint direct shear test using particle flowtheoryrdquoChinese Journal of RockMechanics and Engineering vol31 no 6 pp 1245ndash1256 2012

[27] RH Cao P CaoH Lin K Zhang andXWTan ldquoParticle flowanalysis of direct shear tests on joints with different roughnessrdquoRock and Soil Mechanics vol 34 supplement 2 pp 456ndash4642013 (Chinese)

Submit your manuscripts athttpswwwhindawicom

ScientificaHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Polymer ScienceInternational Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CeramicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CompositesJournal of

NanoparticlesJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Biomaterials

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

NanoscienceJournal of

TextilesHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Journal of

NanotechnologyHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

CrystallographyJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CoatingsJournal of

Advances in

Materials Science and EngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Smart Materials Research

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MetallurgyJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

BioMed Research International

MaterialsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Advances in Materials Science and Engineering 3

where 119896119899 is the normal stiffness (forcedisplacement) at thecontactThe value of 119896119899 is determined by the current contact-stiffness model

Note that the normal stiffness 119896119899 is a secant modulus inthat it relates total displacement and forceThe shear stiffness119896119904 on the other hand is a tangent modulus in that it relatesincremental displacement and force An uppercase 119870 will beused to denote a secant modulus and a lowercase 119896 will beused to denote a tangent modulus The computation of thenormal contact force from the geometry alone makes thecode less prone to numerical drift and able to handle arbitraryplacement of balls and changes in ball radii after a simulationhas begun

The shear contact force is computed in an incrementalfashion When the contact is formed the total shear contactforce is initialized to zero Each subsequent relative shear-displacement increment results in an increment of elasticshear force that is added to the current value The motion ofthe contact must be considered during this procedure

The contact velocity can be resolved into normal andshear componentswith respect to the contact planeDenotingthese components by 119881119899119894 and 119881119904119894 for the normal and shearcomponents respectively the shear component of the contactvelocity can be written as

119881119904119894 = 119881119894 minus 119881119899119894 = 119881119894 minus 119881119895119899119895119899119894 (2)

The shear component of the contact displacement-incre-ment vector occurring over a time step of Δ119905 is calculated by

Δ119880119904119894 = 119881119904119894 Δ119905 (3)

and is used to calculate the shear elastic force-increment vec-tor

Δ119865119904 = minus119896119904Δ119880119904 (4)

The frictional resistance of the contact is given by

119865119904 le 120583119865119899 (5)

where 120583 is the friction coefficient between the disksTo simulate a relatively brittle rock-like material it is

necessary to cement these disks with a bonded model Thisstudy uses the parallel bondmodel which resists not only thecontact forces but also the moments between the disks at acemented contact (Figure 1) The function mechanism of theparallel bond model is described by

Δ119865119899 = 119896119899119860Δ119880119899Δ119865119904 = minus119896119904119860Δ119880119904Δ119872119899 = minus119896119904119869Δ120579119899Δ119872119904 = minus119896119899119868Δ120579119904

(6)

where 119865119899 and 119865119904 are the force components about the centerof the cemented-contact zone119872119899 and119872119904 are the momentsabout the center of the cemented-contact zone 119896119899 and 119896119904

are the normal and shear bond stiffness per unit area respec-tively 120579119899 and 120579119904 are the rotation angle components and 119860119869 and 119868 are the area polar moment of inertia and momentof inertia of the bond contact cross section respectively Thestrength of the cemented contact is then given by

120590max = minus119865119899119860 +

1003816100381610038161003816100381611987211990410038161003816100381610038161003816 119877119868 lt 120590119888

120591max = minus119865119904119860 +

1003816100381610038161003816100381611987211989910038161003816100381610038161003816 119877119869 lt 120591119888(7)

where 119877 is the radius of the bonded zone between the disks 119905is the length of the bonded zone between the disks (Figure 1)and 120590119888 and 120591119888 are the tensile and shear strength of the bondcontact respectively

Tensile cracks occur when the applied tensile stressexceeds the specified tensile strength of the parallel bond120590119888 Shear cracks occur when the applied shear stress exceedsthe specified shear strength of the parallel bond 120591119888 either byrotation or by the shearing of the disks The tensile strengthat the contact immediately drops to zero once the crackoccurs and the shear strength reduces to the residual frictionstrength (see (5)) [19 22] as illustrated in Figure 2

3 Direct Shear Simulation of the BoltedJoint Using PFC

31 Microparameters of the Rock Sample The intact syntheticmodel material selected in this study is represented bycompacted disks cemented with the parallel bond modelTherefore the micromechanical parameters consist of twocategories one for disks and the other for parallel bondsThe PFC code allows one to simulate the macromechanicalbehavior of the selected synthetic model material by selectingappropriate values for the micromechanical parameters suchas the disk size distribution and packing disk and parallelbond stiffness disk friction coefficient and bond strengths[19 22] To select the appropriatemicromechanical parametervalues for the synthetic intact material one needs to basicallygo through a trial and error procedure by iteratively varyingthe micromechanical parameter values to match the requiredmacromechanical behaviors of the selected synthetic mate-rial This procedure is known as the calibration of the intactsyntheticmaterial based on themicromechanical parametersHerein the required macromechanical behaviors are thestrength response represented by the uniaxial compressivestrength (UCS) and the deformability response representedby Youngrsquos modulus and Poissonrsquos ratio of a cylindricalspecimen with a 50mm diameter and a heightdiameter ratioof 2 1 under unconfined compression

In this study a special calibration sequence was followedto rationalize themicromechanical parameter calibration andto minimize the number of iterations First the disk and par-allel bond moduli and the ratios of normal to shear stiffnesswere set equal between the disks and parallel bonds to reducethe number of independent parameters [22] Then Youngrsquosmodulus was calibrated by setting the material strengths toa large value and varying disk Youngrsquos modulus and parallel

4 Advances in Materials Science and Engineering

1

1Compression

Friction sliding

Bond breaks

Tension Shear

Bond breaks

F

Fs

Fn

fs

ks

kn

U

Us gt 0

Un gt 0

Un lt 0

Fi bond strength

Ui displacement

ki contact stiffness

Figure 2 Illustration of the yield process for a parallel bond

(a) Sample from PFC (b) Samples of rock-like materials

Figure 3 Failure patterns in the cylindrical sample subjected to uniaxial compression

bond Youngrsquos modulus to match Youngrsquos modulus betweenthe numerical and laboratory specimens Next by changingthe ratio of disk normal stiffnessdisk shear stiffness andparallel bond normal stiffnessparallel bond shear stiffnessPoissonrsquos ratio of the numerically simulated intact syntheticcylindrical specimen was matched to that of the laboratoryspecimen

After calibrating the aforementioned deformationmicro-mechanical parameters the peak strength between the nu-merical and laboratory specimens was matched by graduallyreducing the normal and shear bond strengths of the parallelbondsDuring this procedure it is important to fix the ratio ofnormal to shear bond strength 120590119888120591119888 because these parame-ters affect the failuremode of the specimenTherefore a seriesof numerical simulations were conducted tomatch the failuremode of the cylindrical specimen between the numerical

and laboratory specimens by varying 120590119888120591119888 while keepingthe other parameters unchangedThe obtained failure modeswith varying 120590119888120591119888 for the cylindrical specimens used in thenumerical simulations are displayed in Figure 3

As seen in Figure 3 as the specimen is compressed underuniaxial stress numerous cracks (tensile and shear cracks)are produced through the breakage of parallel bonds Thefailure mode of the numerically simulated synthetic cylindri-cal specimen with 120590119888120591119888 = 10 agrees well with the resultsfrom the laboratory specimen Thus setting the ratio ofnormal to shear bond strength equal to one seems to increasethe confidence of the numerical model in simulating theappropriate failure behavior of the synthetic material

The numerically obtained stress-strain curves were com-pared with the stress-strain curve obtained from the labora-tory sample as shown in Figure 4 In this comparison in

Advances in Materials Science and Engineering 5

Laboratory test curvePFC simulation curve

Axial strain ()

9MPa

6MPa

3MPa

0

18

36

54

72

0000 0003 0006 0009

(1ndash

3)

(MPa

)

Figure 4 Stress-strain curves at different confining stresses

Table 1 Microparameters of the rock sample

Microparameters Values RemarkMinimum radiusmm 05Disk radius ratio 166Disk densitykgsdotmminus3 1830 Uniform distributionDisk contact modulusGPa 395Disk stiffness ratio 10Disk friction coefficients 05Parallel bond normal strengthMPa 245 plusmn 65 Normal distributionParallel bond shear strengthMPa 245 plusmn 65 Normal distributionParallel bond modulusGPa 395Parallel bond stiffness ratio 15Parallel bond radius multiplier 10

addition to comparing the prepeak behavior it is necessary tomatch the macro UCS and Youngrsquos modulus values obtainedfor different numerical simulations of the synthetic materialwith those obtained for the laboratory specimen Thereforethe disk contact modulus values need to be modified simul-taneously with the 120583 value Figure 4 shows that the prepeakand postpeak trend of the laboratory stress-strain curve isapproximately parallel to that of all the stress-strain curves ofnumerical specimens Therefore the prepeak and postpeakbehavior of numerical specimens did not provide any guid-ance for the selection of disk friction value However thecoefficient of friction obtained from the laboratory triaxialtests on the synthetic intact rock was approximately 050Therefore the friction coefficient value of 050 was selectedto represent the disk friction Table 1 lists the determinedmicromechanical parameter values to simulate the intactsynthetic rock with the parallel bond model The failuremode the stress-strain curves and the comparison of themacromechanical parameters indicate that the results are ingood agreement with the laboratory test results

32 Bilinear Constitutive Model of the Bolt The bolt isrepresented by a cluster of balls with parallel bonds thatmeet the requirements of the force-displacement law Parallelbonds can transmit both forces and moments between diskswhich is consistent with the stress state in a bolt The stressstate in a bolt is highly complex when it is reinforced ina jointed rock mass The microparameters of the bolt aredetermined via the bolt pull-out test as shown in Figure 5(a)The microparameters of the bolt are shown in Table 2Figure 5(b) describes the bolt constitutive model whichis not modified One shortcoming of this model is thatit can represent the characteristics of a metallic bar onlyduring elastic deformation the model cannot represent thelarge plastic deformations during yielding Therefore thisconstitutive model has been modified for use in the internalFISH language in PFC to make the modified bolt constitutivemodel meet the stiffness and deformation requirements Asshown in Figure 5(c) the yield strength of the bolt is set suchthat it remains in the linear elastic stage prior to the axialstress reaching the yielding strength However when the axial

6 Advances in Materials Science and Engineering

F

(a)

Bond breaks bolt losesits bearing capacity

e

t

(b)

Bolt yields but it stillhas bearing capacity

Bond breaks bolt losesits bearing capacity

e p

t

(c)

Figure 5 Illustration of the pull-out test for rock bolts and the constitutive rock bolt models

Table 2 Microparameters of the bolt

Microparameters Values RemarkMinimum radiusmm 20Disk radius ratio 10Disk densitykgsdotmminus3 7000 Uniform distributionDisk contact modulusGPa 30Disk stiffness ratio 12Disk friction coefficients 10Parallel bond normal strengthMPa 380 plusmn 20 Normal distributionParallel bond shear strengthMPa 266 plusmn 20 Normal distributionParallel bond stiffness ratio 26Parallel bond radius multiplier 10

stress exceeds the yield strength the bonds will not breakmeaning that the bolt maintains a certain capacity If themagnitude of the axial stress in the bolt exceeds themaximumnormal stress the bonds are broken and the bolt will lose itsbearing capacity

33 Direct Shear Test Model of the Bolted Joint The directshear test model is built using PFC2DThemodel is composedof a top and bottom rock sample grout joint and a boltwith a size of 100mm times 50mm with a total of 2926 disksThe microparameters of the rock sample used are shownin Table 1 The grout microparameters are based on therock parameters with the parallel bond strength changed to40MPa The joint is created using its own command in PFCand the disks in the joint are unbounded Therefore theparallel bond stiffness and the strength of the disks in the jointare set to zero A shear box is generated by the wall elementAmong these walls the sixth wall is the loading wall withthe first wall fifth wall and the sixth wall together formingthe down active shear box which can move in the horizon-tal direction The upper shear box is fixed in the horizontal

direction and only the second wall can move up or down asit is the servo wall which keeps the normal stress constantduring shearing As shown in Figure 6 the rock mass isrepresented by light gray disks and the bolt is composed of acluster of light gray disks which are tangential to each otherAround the bolt the dark gray disks represent the groutAlong the horizontal direction in the middle of the modelthe joint is composed of black disks

34 Direct Shear Test of the Anchoring Model During shear-ing the second wall is controlled by the servo mechanism toapply a constant normal stress with the vertical displacementmonitored as the normal displacement The fifth wall issubjected to a constant velocity to apply horizontal shearloading with the horizontal displacement monitored as theshear displacement The shear stress is calculated by dividingthe average force on the fifth and sixth walls The test isendedwhen the shear displacement reaches the present valueMicrocracking due to bond breakage shear stress normaldisplacement stress in the bolt and displacement variationsin the bolt disks are monitored during the shear test

Advances in Materials Science and Engineering 7

1

3

2

5

48

67

1

3

22

5

6

1

] ]

n

(a) Numerical model of a bolted joint at 45∘

1

3

2

5

48

67

1

3

2

5

6

1

] ]

n

(b) Numerical model of a bolted joint at 75∘

Figure 6 Numerical direct shear test model of the bolted joint

15∘

30∘

45∘

60∘

75∘

90∘

0

2

4

6

8

Shea

r stre

ss (M

Pa)

05 10 15 20 2500

Shear displacement (mm)

Figure 7 Evolution of the shear stress for different anchorage angles

4 Analysis of the Results

41 Analysis of the Shear Performance of the Joint withDifferent Anchorage Angles Direct shear simulations of thebolted joint were conducted using the bolt at differentanchorage angles under a constant normal stress to examinethe macro-micromechanical responses of the bolted jointduring shearing Figure 7 shows the evolution of the shearstress with the shear displacement at different anchorageangles The shear simulation at a normal stress of 5MPa isachieved after a shear displacement of 25mm is reachedThepeak shear strength for different anchorage angles is shownin Table 3

Our data show that (1) the peak shear strength increasedto a maximum and then decreased with further increase inthe anchorage angle and (2) the peak shear strength is thelargest at an anchorage angle of 75∘ Compared to the 90∘anchorage angle changing the angle of the anchor can greatlyimprove the boltrsquos antishear abilitiesTherefore it is importantto determine the optimum design for a bolted rock joint inpractice

Table 3 Peak shear strength for different anchorage angles

Anchor angle The peak shear strength (MPa)15∘ 49830∘ 60045∘ 67860∘ 65275∘ 76590∘ 498

42 Analysis of the Bolt Stress in a Compression-ShearStress State

421 Stress Analysis of the Anchor Joint Plane in a Compres-sion-Shear Stress State Under compressive shear stress therock bolts experience not only shear deformation throughthe thickness of the weak plane but also notable shear defor-mation over a considerable section (the effective length) Tosimplify the analysis we set the effective length of the rock

8 Advances in Materials Science and Engineering

Joint surface Joint surface

b

b

ln ltn

tb

Figure 8 Deformations of the joint surface and bolt

b b blt

b2

(a) Shear force

b b bln

b2

(b) Axial force

Figure 9 Axial and shearing forces on the bolt

bolt to be 119897119905 and the effective length of the bolt to be 119897119899 asshown in Figure 8

The average shear stress distribution on the plane of the119897119905 section is related to the contact conditions of the rodbody and hole wall Three different distribution patternsfor the simplified calculation are shown in Figure 9(a) Thedistribution pattern is a curve but can be measured inthree basic forms Similar to the 119897119899 section the axial stressdistribution can be used as shown in Figure 9(b)

From Figures 8 and 9 and from the joint surface equi-librium conditions the stress tensor of the rock mass can beobtained from the normal and tangential components of thestress tensor along the joint plane as shown in

120590 = 120590119904 + 120590119887119899= 119896119899 (119880119899 sin 119886 minus 119880119905 cos 119886) + 119875119866119887119880119905119860119861119905119897119905 cos 119886

minus 119875119864119887119880119899119861119899119897119899 sin 119886

120591 = 120591119887119904= 119896119904 (119880119899 cos 119886 + 119880119905 sin 119886) + 119875119864119887119880119899119861119899119897119899 cos 119886

+ 119875119866119887119880119905119860119861119905119897119905 sin 119886

+ [119875119866119887119880119905119860119861119905119897119905 cos 119886 minus119875119864119887119880119899119861119899119897119899 sin 119886]119891119904

(8)

In the previous formula 120575119905 and 120575119899 are the weak levels oftangential and normal displacement respectively 119880119905 and 119880119899are the bolt axial and transverse deformations respectively119886 is the angle between the bolt and joint surface and 119864119887 and119866119887 are Youngrsquos modulus and shear modulus for the materialrespectivelyThe shear coefficient119860 is associated with the rodbody section (119860 = 34 for a circular solid rod) 119861119905 and 119861119899 arethe inner pole sectionsrsquo average shear stress and axial stressdistribution of the shape coefficient respectively which are1 34 and 12 120590119887 and 120591119887 are the axial and shear forces on

Advances in Materials Science and Engineering 9

the bolt respectively 119896119899 and 119896119904 are the normal and tangentialstiffness coefficients for the weak level119875 is the anchor sectionreinforcement ratio Therefore

119880119899 = 1119886 (11988622120590 minus 11988612120591)

119880119905 = 1119886 (11988611120591 minus 11988621120590)

(9)

where

11988611 = minus(119896119899 + 119875119864119887119861119899119897119899) sin 119886

11988612 = (119896119899 + 119875119866119887119860119861119905119897119905) cos 119886

11988621 = 119896119904 cos 119886 + 119875119864119887119861119899119897119899 (cos 119886 minus 119891119904 sin 119886)

11988622 = 119896119904 sin 119886 + 119875119866119887119860119861119905119897119905 (sin 119886 + 119891119904 cos 119886)

119886 = 1198861111988622 minus 1198861211988621= 119896119899119896119904 minus 119875119864119887

119861119899119897119899119875119866119887119860119861119905119897119905 +

119875119866119887119860119861119905119897119905 (119896119899sin

2119886 minus 119896119904cos2119886)

+ 119875119864119887119861119899119897119899 [119896119899cos

2119886 minus 119896119904sin2119886]

+ 119896119899119891119904 sin 119886 cos 119886 [ 119875119866119887119860119861119905119897119905 minus119875119864119887119861119899119897119899 ]

(10)

Therefore the relationship between the stress and thestress component of the element can be determined as shownin

120590119887 = 119864119887119861119899119897119899119886 (11988622120590 minus 11988612120591)

120591119887 = 119866119887119860119861119905119897119905119886 (11988611120591 minus 11988621120590)

(11)

422 Analysis of the Bolt Stress for a Particular AnchorageAngle A bolt installed at a particular anchorage angle willhave a better anchoring effect than a bolt installed perpen-dicular to the joint plane Figures 10 and 11 show illustrationsof the stresses in a bolt at anchorage angles of 75∘ and 90∘respectively during shearing

The diagram showing the two different anchorage anglebolts indicates that the anchor force and bending momentare relatively large in the vertically installed bolt leading tobolt axial and shear stress that are larger than those in thetilted bolt whereas for the tilted bolt the axial and shearstress are small and do not reach the yield strengthThereforethe tilted anchor bolts not only improve the shear capacityof the joints but also maintain an appropriate stress statein the anchor The rock bolt axial stress diagram indicatesthat the axial stress increases as the shear displacementincreases for different anchorage anglesThe axial stress is the

highest when the anchor is vertically installed on the jointsurface The results of these two cases reveal that the optimalanchoring angle is approximately 75∘This result correspondsto the previous conclusions that the optimal anchoring angleshould be between 45∘ and 75∘

43 Evolution of the Contact Force and Microscopic Crackingof Disks The anchorage angle of 75∘ was chosen for amore detailed analysis on the micromechanical response anddamage mechanism during shearing Figure 12 shows theevolution of the shear stress number of cracks and normaldisplacement for a normal stress of 5MPa

To accurately determine the evolution features six differ-ent monitoring points (a to f) were set on the shear stresscurve which correspond to shear displacements of 0 01980217 0513 1500 and 2500mm respectively The variationsin the contact force and cracking can be observed from thesix monitoring points As shown in Figure 12 ldquoardquo indicatesthe initial point of the shear displacement and ldquocrdquo indicatesthe shear displacement at which the shear stress reachesthe peak shear strength As shown in Figure 12 no shearcrack is generated during the initial stage of the shear testThe number of shear cracks increases gradually as the sheardisplacement increases When the shear stress reaches thepeak shear strength the number of shear cracks increasessignificantly and the shear crack growth rate tends to stabilizeas the shear stress increases beyond the peak shear strengthThe trends in tension cracking and normal stress are highlysimilar and increase monotonically

During the shearing of a bolted joint the loaded side ofthe bolt is in a state of compression whereas the other side ofthe bolt is in a state of tension due to the bending deflectionof the bolt The adjacent grout is in a state of compressionand the rockmass is in a state of tension because of the bondsbetween the bolt and grout as shown in Figure 13 Bendingdeflection will occur in the bolt as the shear displacementincreases Simultaneously cracks will occur continuously dueto extrusion and slippage between disks In the compressionzone there are a large number of cracks which are mainlytension cracks generated by the extrusion between disksThisresult agreeswith the formationmechanism for compression-induced tension cracks shown in Figure 13

Figure 14 shows the illustration of the contact forceand crack propagation between disks during shearing Thecontact force is initially concentrated on the joint and theloading end and cracks are generated in the joint whereasthere are only a few cracks around the bolt As the sheardisplacement increases the diskrsquos contact force around thebolt becomes gradually larger Cracks occur on the jointand those around the bolt extend further The cracks aroundthe bolt propagate most rapidly The cracks are initiallyconcentrated at the intersection of the bolt and joint and thenpropagate to the ends of the bolt Finally the majority of thecracks are distributed at the joint and around the bolt aftera larger displacement occurs Finally more cracks occur inthe compression zone around the bolt than in the tensionzone In addition the distribution of cracks corresponds tothe contact force in the stress concentration areaThe detailed

10 Advances in Materials Science and Engineering

0 10 20 30 40

Distance along bolt (mm)

Joint plane

Bolt

05 mm10 mm15 mm

20 mm25 mm30 mm

0

100

200

300

Axi

al st

ress

(MPa

)

(a) Axial stress diagram

0 10 20 30 40

Distance along bolt (mm)

05 mm10 mm15 mm

20 mm25 mm30 mm

minus600

minus400

minus200

0

200

400

600

Bend

ing

mom

ent (

Nm

)

Joint plane

Bolt

(b) Bending moment diagram

Joint plane

Bolt

0 10 20 30 40

Distance along bolt (mm)

05 mm10 mm15 mm

20 mm25 mm30 mm

0

10

20

30

40

Shea

r stre

ss (M

Pa)

(c) Shear stress diagram

Figure 10 Bolt stress state diagram at an anchorage angle of 75∘

maximum contact force and crack number in the specimen atthe different monitoring points during shearing are providedin Tables 4 and 5 respectively

44 Evolution of the Disk Rotation During the shearing slipprocess the behavior is determined by the rotation of thedisks Relative disk dislocations will occur due to the rotationof the disks If the rotation angle is sufficiently large itwill lead to shear stresses and the breaking of disk bondsMeanwhile cracks occurring in the rock mass and shearcracking are caused directly by disk rotation Figure 15 showsa graph of the evolution of the disk rotation in terms ofradians for rock disks at different monitoring points

Figure 15 shows that when the shear displacement issmall at the initial stages the disk rotation angle is relatively

small and disks with large rotational angles are distributedaround the bolt The disk rotation angles gradually increasewith increasing shear displacement The disks containinglarger rotation angles are mainly distributed in the twoloading ends and around the bolt and the bolt and thedisks around the bolt contact move relatively violently duringshearing leading to a larger rotation angle Further increasesin the shear displacement and larger rotation disks alsoappeared on the joint surfaces which are caused by thejoint surface microconvex body cutting into the free disksThe distribution of the large disk rotation angles is highlyconsistent with the crack distribution demonstrating that therotation of the disks leads to the generation of shear cracks

Figure 16 shows the variations in the proportion ofrotations exceeding 01 radians As the shear displacement

Advances in Materials Science and Engineering 11

Bolt

Joint plane

0

50

100

150

200

250A

xial

stre

ss (M

Pa)

10 20 30 400Distance along bolt (mm)

05 mm10 mm15 mm

20 mm25 mm30 mm

(a) Axial stress diagram

Joint plane

Bolt

10 20 30 400Distance along bolt (mm)

minus800

minus600

minus400

minus200

0

200

400

600

800

Bend

ing

mom

ent (

Nm

)

05 mm10 mm15 mm

20 mm25 mm30 mm

(b) Bending moment diagram

Bolt

Joint plane

0

10

20

30

40

Shea

r stre

ss (M

Pa)

10 20 30 400Distance along bolt (mm)

05 mm10 mm15 mm

20 mm25 mm30 mm

(c) Shear stress diagram

Figure 11 Bolt stress state diagram at an anchorage angle of 90∘

Table 4 Maximum contact force in the specimen at different monitoring points

Monitoring point Maximum contact forceN Monitoring point Maximum contact forceNa 68201198904 d 12681198905b 15301198905 e 15801198905c 15481198905 f 14451198905

increases the number of clockwise and counterclockwisedisk rotations greater than 01 radians increases continuouslyAdditionally the number of clockwise rotations is greaterthan the number of counterclockwise rotations mainly dueto the shear loading directions

5 Conclusions

In this paper the anchoring mechanism for a tilted anchorbolt subjected to a shear load has been studied using a bilinearconstitutive model with the inner-embedded FISH language

12 Advances in Materials Science and Engineering

(f)(e)(d)

(c)

(b)

(a)

Shear stressNormal displacement

Tension crackShear crack

0

2

4

6

8

Shea

r stre

ss (M

Pa)

05 10 15 20 2500Shear displacement (mm)

00

01

02

03

04

05

06

07

08

Nor

mal

disp

lace

men

t (times10

minus3

m)

0

100

200

300

400

500

600

Crac

k nu

mbe

r

Figure 12 Evolution of the shear stress number of cracks and normal displacement

Rock

Bolt Grout

T

C

C

T

(a)

F F F

F F F

(b)

Figure 13 Stress state of the bolted rock joint system and illustration of the mechanism behind compression-induced tension cracks

Table 5 Number of cracks in the specimen at different monitoringpoints

Monitoringpoint Crack number Monitoring point Crack number

a 0 d 255b 21 e 446c 40 f 592

of PFC based on DEMThe anchoring mechanism of the boltatdifferentanchorage angles has been discussed at themacro-microlevels and the following conclusions have been drawn

(1) The bolt can improve the shear resistance of the jointmainly due to cohesion Tilted anchor bolts not only improvethe shear strength of the joint surface but also result in a lowerstress in the anchor bolt The peak shear strength can be 50greater than that of a vertically installed anchor bolt

(2) A tilted bolt allows for fully mobilized anchor perfor-manceThis configuration not only restricts increment in thenormal displacement but also indirectly improves the stress

state of the anchorage system and increases the shear capacityof the anchor joint The comprehensive shear displacementand shear stress data along with the bolt stress state analysisreveal that the optimal anchorage angle should be between45∘ and 75∘

(3) During the shearing process when the rock bolt andgrout are commonly constrained the stress concentrationphenomenon around the bolt and joint increases the sheardisplacement and contact force After reaching the residualphase the contact force exhibits slight fluctuations due to thesliding of the joint surface

(4) Based on the disk flow software PFC2D crack initia-tion for an anchorage angle of 75∘ will start from the contactforce concentration due to the interactions between the rockgrout and bolt Finally the cracks are mainly distributedin the joint and around the bolt for larger displacementsThe cracks are mainly tensile cracks which correspond tothe formation mechanism for a compression-induced tensilecrack The shear cracking was mainly caused by the rotationof the disks during the shearing process

Advances in Materials Science and Engineering 13

minus005 minus003 minus001 001 003 005

minus002minus001

0001002

minus551525354555

Con

tact

forc

e (N

)

times103

(a) Monitoring point a

minus002minus001

0001002

Con

tact

forc

e (N

)

minus10

10

30

50

70

90

110times103

minus005 minus003 minus001 001 003 005

(b) Monitoring point b

minus002minus001

0001002

Con

tact

forc

e (N

)minus101030507090110

times103

minus005 minus003 minus001 001 003 005

(c) Monitoring point c

minus002minus001

0001002

Con

tact

forc

e (N

)

minus10

10

30

50

70

90

110times103

minus005 minus003 minus001 001 003 005

(d) Monitoring point d

minus002minus001

0001002

minus51535557595115

Con

tact

forc

e (N

)

times103

minus005 minus003 minus001 001 003 005

(e) Monitoring point e

minus002minus001

0001002

minus10

10

30

50

70

90

Con

tact

forc

e (N

)

times103

minus005 minus003 minus001 001 003 005

(f) Monitoring point f

Figure 14 Evolution of the contact force and cracking at different monitoring points

14 Advances in Materials Science and Engineering

Particle rotation (in radians)minus120 to minus01

minus01 to minus0001

minus0001 to minus00005

minus00005 to 00005

00005 to 00010001 to 01

01 to 120

(a) Monitoring point a

Particle rotation (in radians)minus120 to minus01

minus01 to minus0001

minus0001 to minus00005

00005 to 00010001 to 01

01 to 120minus00005 to 00005

(b) Monitoring point b

Particle rotation (in radians)minus120 to minus01

minus01 to minus0001

minus0001 to minus00005

00005 to 00010001 to 01

01 to 120minus00005 to 00005

(c) Monitoring point c

Particle rotation (in radians)minus120 to minus01

minus01 to minus0001

minus0001 to minus00005

00005 to 00010001 to 01

01 to 120minus00005 to 00005

(d) Monitoring point d

Particle rotation (in radians)minus120 to minus01

minus01 to minus0001

minus0001 to minus00005

00005 to 00010001 to 01

01 to 120minus00005 to 00005

(e) Monitoring point e

Particle rotation (in radians)minus120 to minus01

minus01 to minus0001

minus0001 to minus00005

00005 to 00010001 to 01

01 to 120minus00005 to 00005

(f) Monitoring point f

Figure 15 Evolution of disk rotation in radians in the specimen at different monitoring points

0

5

10

15

20

Qua

ntita

tive p

ropo

rtio

n (

)

05 10 15 20 25 3000Shear displacement (mm)

gt 01Rotation (in radians)gt 01|Rotation (in radians)|

lt minus01Rotation (in radians)

Figure 16 Quantitative proportion variation of rotations exceeding 01 radians

Disclosure

GangWang and Yongzheng Zhang should be regarded as co-first authors

Conflicts of Interest

The authors declare that there are no conflicts of interestregarding the publication of this paper

Advances in Materials Science and Engineering 15

Acknowledgments

This study was supported by the National Natural ScienceFoundation of China (nos 51479108 51379117 and 51509261)China Postdoctoral Science Foundation (2015M580599) theTaishan Scholar Talent Team Support Plan for Advantagedamp Unique Discipline Areas the Postdoctoral Science Foun-dation of Shandong Province (no 201402014) and the Shan-dong Province Outstanding Young Scientist Research AwardFund (BS2014NJ006)

References

[1] F D Patton ldquoMultiple modes of shear failure in rockrdquo inProceedings of the 1st Congress on International Society for RockMechanics pp 509ndash513 1966

[2] R E Goodman Methods of Geological Engineering in Discon-tinuous Rocks West Publishing Company New York NY USA1976

[3] N Barton and V Choubey ldquoThe shear strength of rock joints intheory and practicerdquo Rock Mechanics Felsmechanik Mecaniquedes Roches vol 10 no 1-2 pp 1ndash54 1977

[4] P M Dight and H K Chiu ldquoPrediction of shear behavior ofjoints using profilesrdquo International Journal of Rock Mechanicsand Mining Sciences amp Geomechanics Abstracts vol 18 no 5pp 369ndash386 1981

[5] N Barton Modelling Rock Joint Behaviour from In Situ BlockTests Implications for NuclearWaste Repository Design vol 308Office ofNuclearWaste IsolationONWI ColumbusOHUSA1982

[6] T Esaki S Du Y Jiang and Y Wada ldquoEffect of the asperitydamage on the shear behavior of irregular rock jointsrdquo inProceedings of the Second International Conference on Analysisof Discontinuous Deformation pp 459ndash464 Kyoto Japan 1997

[7] Y Jiang J Xiao Y Tanabashi and T Mizokami ldquoDevelopmentof an automated servo-controlled direct shear apparatus apply-ing a constant normal stiffness conditionrdquo International Journalof Rock Mechanics and Mining Sciences vol 41 no 2 pp 275ndash286 2004

[8] Y J Jiang B Li and Y Tanabashi ldquoEstimating the relationbetween surface roughness and mechanical properties of rockjointsrdquo International Journal of Rock Mechanics and MiningSciences vol 43 no 6 pp 837ndash846 2006

[9] C-C Xia Z-C Tang W-M Xiao and Y-L Song ldquoNew peakshear strength criterion of rock joints based on quantifiedsurface descriptionrdquo Rock Mechanics and Rock Engineering vol47 no 2 pp 387ndash400 2014

[10] S Bjurstrom ldquoShear strength of hard rock joints reinforced bygrouted untensioned boltsrdquo in Proceedings of the In Proceedingsof the 3rd International ISRM Congress pp 1194ndash1199 DenverColo USA 1974

[11] X Ge and J Liu ldquoStudy of the shear resistance behaviour ofbolted rock jointsrdquo Journal of Geotechnical Engineering vol 10no 1 pp 8ndash19 1988 (Chinese)

[12] K Spang and P Egger ldquoAction of fully-grouted bolts in jointedrock and factors of influencerdquo Rock Mechanics and Rock Engi-neering vol 23 no 3 pp 201ndash229 1990

[13] P Egger and L Zabuski ldquoBehaviour of rough bolted joints indirect shear testsrdquo in Proceedings of the 7th ISRM Congress vol30 p 342 Aachen Germany September 1991

[14] A M Ferrero ldquoThe shear strength of reinforced rock jointsrdquoInternational Journal of Rock Mechanics and Mining Sciencesand vol 32 no 6 pp 595ndash605 1995

[15] F Pellet and P Egger ldquoAnalytical model for the mechanicalbehaviour of bolted rock joints subjected to shearingrdquo RockMechanics and Rock Engineering vol 29 no 2 pp 73ndash97 1996

[16] H Jalalifar and N Aziz ldquoExperimental and 3D numericalsimulation of reinforced shear jointsrdquo RockMechanics and RockEngineering vol 43 no 1 pp 95ndash103 2010

[17] P Wang T Feng Y-J Zhu and W-J Yu ldquoExperimental studyand numerical simulation of anchoringmechanismof anchoredrocklikematerial with prefabricated fracturerdquoYantu LixueRockand Soil Mechanics vol 37 no 3 pp 793ndash801 2016 (Chinese)

[18] L Li P C Hagan S Saydam B Hebblewhite and Y Li ldquoPara-metric study of rockbolt shear behaviour by double shear testrdquoRock Mechanics and Rock Engineering vol 49 no 12 pp 4787ndash4797 2016

[19] Itasca Consulting Group Inc ldquoPFC2D (Particle Flow Code in2D)Theory and Backgroundrdquo Minneapolis Minn USA 2008

[20] P A Cundall andOD L Strack ldquoThe distinct numericalmodelfor granular assembliesrdquo Geotechnique vol 29 no 1 pp 47ndash651979

[21] J Zhou and Y Chi ldquoThe method of particle flow and PFC2Dcoderdquo Rock and Soil Mechanics vol 21 no 3 pp 271ndash274 2000(Chinese)

[22] D O Potyondy and P A Cundall ldquoA bonded-particle modelfor rockrdquo International Journal of Rock Mechanics and MiningSciences vol 41 no 8 pp 1329ndash1364 2004

[23] N Cho CDMartin andDC Sego ldquoA clumped particlemodelfor rockrdquo International Journal of Rock Mechanics and MiningSciences vol 44 no 7 pp 997ndash1010 2007

[24] Z Zhao L Jing and I Neretnieks ldquoParticle mechanics modelfor the effects of shear on solute retardation coefficient in rockfracturesrdquo International Journal of Rock Mechanics and MiningSciences vol 52 pp 92ndash102 2012

[25] C Xia Y Song Z Tang Y Song and C Shou ldquoParticleflow numerical simulation for shear behavior of rough jointsrdquoChinese Journal of Rock Mechanics and Engineering vol 31 no8 pp 1545ndash1552 2012 (Chinese)

[26] Y Zhou A Misra S Wu and X Zhang ldquoMacro- and meso-analyses of rock joint direct shear test using particle flowtheoryrdquoChinese Journal of RockMechanics and Engineering vol31 no 6 pp 1245ndash1256 2012

[27] RH Cao P CaoH Lin K Zhang andXWTan ldquoParticle flowanalysis of direct shear tests on joints with different roughnessrdquoRock and Soil Mechanics vol 34 supplement 2 pp 456ndash4642013 (Chinese)

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4 Advances in Materials Science and Engineering

1

1Compression

Friction sliding

Bond breaks

Tension Shear

Bond breaks

F

Fs

Fn

fs

ks

kn

U

Us gt 0

Un gt 0

Un lt 0

Fi bond strength

Ui displacement

ki contact stiffness

Figure 2 Illustration of the yield process for a parallel bond

(a) Sample from PFC (b) Samples of rock-like materials

Figure 3 Failure patterns in the cylindrical sample subjected to uniaxial compression

bond Youngrsquos modulus to match Youngrsquos modulus betweenthe numerical and laboratory specimens Next by changingthe ratio of disk normal stiffnessdisk shear stiffness andparallel bond normal stiffnessparallel bond shear stiffnessPoissonrsquos ratio of the numerically simulated intact syntheticcylindrical specimen was matched to that of the laboratoryspecimen

After calibrating the aforementioned deformationmicro-mechanical parameters the peak strength between the nu-merical and laboratory specimens was matched by graduallyreducing the normal and shear bond strengths of the parallelbondsDuring this procedure it is important to fix the ratio ofnormal to shear bond strength 120590119888120591119888 because these parame-ters affect the failuremode of the specimenTherefore a seriesof numerical simulations were conducted tomatch the failuremode of the cylindrical specimen between the numerical

and laboratory specimens by varying 120590119888120591119888 while keepingthe other parameters unchangedThe obtained failure modeswith varying 120590119888120591119888 for the cylindrical specimens used in thenumerical simulations are displayed in Figure 3

As seen in Figure 3 as the specimen is compressed underuniaxial stress numerous cracks (tensile and shear cracks)are produced through the breakage of parallel bonds Thefailure mode of the numerically simulated synthetic cylindri-cal specimen with 120590119888120591119888 = 10 agrees well with the resultsfrom the laboratory specimen Thus setting the ratio ofnormal to shear bond strength equal to one seems to increasethe confidence of the numerical model in simulating theappropriate failure behavior of the synthetic material

The numerically obtained stress-strain curves were com-pared with the stress-strain curve obtained from the labora-tory sample as shown in Figure 4 In this comparison in

Advances in Materials Science and Engineering 5

Laboratory test curvePFC simulation curve

Axial strain ()

9MPa

6MPa

3MPa

0

18

36

54

72

0000 0003 0006 0009

(1ndash

3)

(MPa

)

Figure 4 Stress-strain curves at different confining stresses

Table 1 Microparameters of the rock sample

Microparameters Values RemarkMinimum radiusmm 05Disk radius ratio 166Disk densitykgsdotmminus3 1830 Uniform distributionDisk contact modulusGPa 395Disk stiffness ratio 10Disk friction coefficients 05Parallel bond normal strengthMPa 245 plusmn 65 Normal distributionParallel bond shear strengthMPa 245 plusmn 65 Normal distributionParallel bond modulusGPa 395Parallel bond stiffness ratio 15Parallel bond radius multiplier 10

addition to comparing the prepeak behavior it is necessary tomatch the macro UCS and Youngrsquos modulus values obtainedfor different numerical simulations of the synthetic materialwith those obtained for the laboratory specimen Thereforethe disk contact modulus values need to be modified simul-taneously with the 120583 value Figure 4 shows that the prepeakand postpeak trend of the laboratory stress-strain curve isapproximately parallel to that of all the stress-strain curves ofnumerical specimens Therefore the prepeak and postpeakbehavior of numerical specimens did not provide any guid-ance for the selection of disk friction value However thecoefficient of friction obtained from the laboratory triaxialtests on the synthetic intact rock was approximately 050Therefore the friction coefficient value of 050 was selectedto represent the disk friction Table 1 lists the determinedmicromechanical parameter values to simulate the intactsynthetic rock with the parallel bond model The failuremode the stress-strain curves and the comparison of themacromechanical parameters indicate that the results are ingood agreement with the laboratory test results

32 Bilinear Constitutive Model of the Bolt The bolt isrepresented by a cluster of balls with parallel bonds thatmeet the requirements of the force-displacement law Parallelbonds can transmit both forces and moments between diskswhich is consistent with the stress state in a bolt The stressstate in a bolt is highly complex when it is reinforced ina jointed rock mass The microparameters of the bolt aredetermined via the bolt pull-out test as shown in Figure 5(a)The microparameters of the bolt are shown in Table 2Figure 5(b) describes the bolt constitutive model whichis not modified One shortcoming of this model is thatit can represent the characteristics of a metallic bar onlyduring elastic deformation the model cannot represent thelarge plastic deformations during yielding Therefore thisconstitutive model has been modified for use in the internalFISH language in PFC to make the modified bolt constitutivemodel meet the stiffness and deformation requirements Asshown in Figure 5(c) the yield strength of the bolt is set suchthat it remains in the linear elastic stage prior to the axialstress reaching the yielding strength However when the axial

6 Advances in Materials Science and Engineering

F

(a)

Bond breaks bolt losesits bearing capacity

e

t

(b)

Bolt yields but it stillhas bearing capacity

Bond breaks bolt losesits bearing capacity

e p

t

(c)

Figure 5 Illustration of the pull-out test for rock bolts and the constitutive rock bolt models

Table 2 Microparameters of the bolt

Microparameters Values RemarkMinimum radiusmm 20Disk radius ratio 10Disk densitykgsdotmminus3 7000 Uniform distributionDisk contact modulusGPa 30Disk stiffness ratio 12Disk friction coefficients 10Parallel bond normal strengthMPa 380 plusmn 20 Normal distributionParallel bond shear strengthMPa 266 plusmn 20 Normal distributionParallel bond stiffness ratio 26Parallel bond radius multiplier 10

stress exceeds the yield strength the bonds will not breakmeaning that the bolt maintains a certain capacity If themagnitude of the axial stress in the bolt exceeds themaximumnormal stress the bonds are broken and the bolt will lose itsbearing capacity

33 Direct Shear Test Model of the Bolted Joint The directshear test model is built using PFC2DThemodel is composedof a top and bottom rock sample grout joint and a boltwith a size of 100mm times 50mm with a total of 2926 disksThe microparameters of the rock sample used are shownin Table 1 The grout microparameters are based on therock parameters with the parallel bond strength changed to40MPa The joint is created using its own command in PFCand the disks in the joint are unbounded Therefore theparallel bond stiffness and the strength of the disks in the jointare set to zero A shear box is generated by the wall elementAmong these walls the sixth wall is the loading wall withthe first wall fifth wall and the sixth wall together formingthe down active shear box which can move in the horizon-tal direction The upper shear box is fixed in the horizontal

direction and only the second wall can move up or down asit is the servo wall which keeps the normal stress constantduring shearing As shown in Figure 6 the rock mass isrepresented by light gray disks and the bolt is composed of acluster of light gray disks which are tangential to each otherAround the bolt the dark gray disks represent the groutAlong the horizontal direction in the middle of the modelthe joint is composed of black disks

34 Direct Shear Test of the Anchoring Model During shear-ing the second wall is controlled by the servo mechanism toapply a constant normal stress with the vertical displacementmonitored as the normal displacement The fifth wall issubjected to a constant velocity to apply horizontal shearloading with the horizontal displacement monitored as theshear displacement The shear stress is calculated by dividingthe average force on the fifth and sixth walls The test isendedwhen the shear displacement reaches the present valueMicrocracking due to bond breakage shear stress normaldisplacement stress in the bolt and displacement variationsin the bolt disks are monitored during the shear test

Advances in Materials Science and Engineering 7

1

3

2

5

48

67

1

3

22

5

6

1

] ]

n

(a) Numerical model of a bolted joint at 45∘

1

3

2

5

48

67

1

3

2

5

6

1

] ]

n

(b) Numerical model of a bolted joint at 75∘

Figure 6 Numerical direct shear test model of the bolted joint

15∘

30∘

45∘

60∘

75∘

90∘

0

2

4

6

8

Shea

r stre

ss (M

Pa)

05 10 15 20 2500

Shear displacement (mm)

Figure 7 Evolution of the shear stress for different anchorage angles

4 Analysis of the Results

41 Analysis of the Shear Performance of the Joint withDifferent Anchorage Angles Direct shear simulations of thebolted joint were conducted using the bolt at differentanchorage angles under a constant normal stress to examinethe macro-micromechanical responses of the bolted jointduring shearing Figure 7 shows the evolution of the shearstress with the shear displacement at different anchorageangles The shear simulation at a normal stress of 5MPa isachieved after a shear displacement of 25mm is reachedThepeak shear strength for different anchorage angles is shownin Table 3

Our data show that (1) the peak shear strength increasedto a maximum and then decreased with further increase inthe anchorage angle and (2) the peak shear strength is thelargest at an anchorage angle of 75∘ Compared to the 90∘anchorage angle changing the angle of the anchor can greatlyimprove the boltrsquos antishear abilitiesTherefore it is importantto determine the optimum design for a bolted rock joint inpractice

Table 3 Peak shear strength for different anchorage angles

Anchor angle The peak shear strength (MPa)15∘ 49830∘ 60045∘ 67860∘ 65275∘ 76590∘ 498

42 Analysis of the Bolt Stress in a Compression-ShearStress State

421 Stress Analysis of the Anchor Joint Plane in a Compres-sion-Shear Stress State Under compressive shear stress therock bolts experience not only shear deformation throughthe thickness of the weak plane but also notable shear defor-mation over a considerable section (the effective length) Tosimplify the analysis we set the effective length of the rock

8 Advances in Materials Science and Engineering

Joint surface Joint surface

b

b

ln ltn

tb

Figure 8 Deformations of the joint surface and bolt

b b blt

b2

(a) Shear force

b b bln

b2

(b) Axial force

Figure 9 Axial and shearing forces on the bolt

bolt to be 119897119905 and the effective length of the bolt to be 119897119899 asshown in Figure 8

The average shear stress distribution on the plane of the119897119905 section is related to the contact conditions of the rodbody and hole wall Three different distribution patternsfor the simplified calculation are shown in Figure 9(a) Thedistribution pattern is a curve but can be measured inthree basic forms Similar to the 119897119899 section the axial stressdistribution can be used as shown in Figure 9(b)

From Figures 8 and 9 and from the joint surface equi-librium conditions the stress tensor of the rock mass can beobtained from the normal and tangential components of thestress tensor along the joint plane as shown in

120590 = 120590119904 + 120590119887119899= 119896119899 (119880119899 sin 119886 minus 119880119905 cos 119886) + 119875119866119887119880119905119860119861119905119897119905 cos 119886

minus 119875119864119887119880119899119861119899119897119899 sin 119886

120591 = 120591119887119904= 119896119904 (119880119899 cos 119886 + 119880119905 sin 119886) + 119875119864119887119880119899119861119899119897119899 cos 119886

+ 119875119866119887119880119905119860119861119905119897119905 sin 119886

+ [119875119866119887119880119905119860119861119905119897119905 cos 119886 minus119875119864119887119880119899119861119899119897119899 sin 119886]119891119904

(8)

In the previous formula 120575119905 and 120575119899 are the weak levels oftangential and normal displacement respectively 119880119905 and 119880119899are the bolt axial and transverse deformations respectively119886 is the angle between the bolt and joint surface and 119864119887 and119866119887 are Youngrsquos modulus and shear modulus for the materialrespectivelyThe shear coefficient119860 is associated with the rodbody section (119860 = 34 for a circular solid rod) 119861119905 and 119861119899 arethe inner pole sectionsrsquo average shear stress and axial stressdistribution of the shape coefficient respectively which are1 34 and 12 120590119887 and 120591119887 are the axial and shear forces on

Advances in Materials Science and Engineering 9

the bolt respectively 119896119899 and 119896119904 are the normal and tangentialstiffness coefficients for the weak level119875 is the anchor sectionreinforcement ratio Therefore

119880119899 = 1119886 (11988622120590 minus 11988612120591)

119880119905 = 1119886 (11988611120591 minus 11988621120590)

(9)

where

11988611 = minus(119896119899 + 119875119864119887119861119899119897119899) sin 119886

11988612 = (119896119899 + 119875119866119887119860119861119905119897119905) cos 119886

11988621 = 119896119904 cos 119886 + 119875119864119887119861119899119897119899 (cos 119886 minus 119891119904 sin 119886)

11988622 = 119896119904 sin 119886 + 119875119866119887119860119861119905119897119905 (sin 119886 + 119891119904 cos 119886)

119886 = 1198861111988622 minus 1198861211988621= 119896119899119896119904 minus 119875119864119887

119861119899119897119899119875119866119887119860119861119905119897119905 +

119875119866119887119860119861119905119897119905 (119896119899sin

2119886 minus 119896119904cos2119886)

+ 119875119864119887119861119899119897119899 [119896119899cos

2119886 minus 119896119904sin2119886]

+ 119896119899119891119904 sin 119886 cos 119886 [ 119875119866119887119860119861119905119897119905 minus119875119864119887119861119899119897119899 ]

(10)

Therefore the relationship between the stress and thestress component of the element can be determined as shownin

120590119887 = 119864119887119861119899119897119899119886 (11988622120590 minus 11988612120591)

120591119887 = 119866119887119860119861119905119897119905119886 (11988611120591 minus 11988621120590)

(11)

422 Analysis of the Bolt Stress for a Particular AnchorageAngle A bolt installed at a particular anchorage angle willhave a better anchoring effect than a bolt installed perpen-dicular to the joint plane Figures 10 and 11 show illustrationsof the stresses in a bolt at anchorage angles of 75∘ and 90∘respectively during shearing

The diagram showing the two different anchorage anglebolts indicates that the anchor force and bending momentare relatively large in the vertically installed bolt leading tobolt axial and shear stress that are larger than those in thetilted bolt whereas for the tilted bolt the axial and shearstress are small and do not reach the yield strengthThereforethe tilted anchor bolts not only improve the shear capacityof the joints but also maintain an appropriate stress statein the anchor The rock bolt axial stress diagram indicatesthat the axial stress increases as the shear displacementincreases for different anchorage anglesThe axial stress is the

highest when the anchor is vertically installed on the jointsurface The results of these two cases reveal that the optimalanchoring angle is approximately 75∘This result correspondsto the previous conclusions that the optimal anchoring angleshould be between 45∘ and 75∘

43 Evolution of the Contact Force and Microscopic Crackingof Disks The anchorage angle of 75∘ was chosen for amore detailed analysis on the micromechanical response anddamage mechanism during shearing Figure 12 shows theevolution of the shear stress number of cracks and normaldisplacement for a normal stress of 5MPa

To accurately determine the evolution features six differ-ent monitoring points (a to f) were set on the shear stresscurve which correspond to shear displacements of 0 01980217 0513 1500 and 2500mm respectively The variationsin the contact force and cracking can be observed from thesix monitoring points As shown in Figure 12 ldquoardquo indicatesthe initial point of the shear displacement and ldquocrdquo indicatesthe shear displacement at which the shear stress reachesthe peak shear strength As shown in Figure 12 no shearcrack is generated during the initial stage of the shear testThe number of shear cracks increases gradually as the sheardisplacement increases When the shear stress reaches thepeak shear strength the number of shear cracks increasessignificantly and the shear crack growth rate tends to stabilizeas the shear stress increases beyond the peak shear strengthThe trends in tension cracking and normal stress are highlysimilar and increase monotonically

During the shearing of a bolted joint the loaded side ofthe bolt is in a state of compression whereas the other side ofthe bolt is in a state of tension due to the bending deflectionof the bolt The adjacent grout is in a state of compressionand the rockmass is in a state of tension because of the bondsbetween the bolt and grout as shown in Figure 13 Bendingdeflection will occur in the bolt as the shear displacementincreases Simultaneously cracks will occur continuously dueto extrusion and slippage between disks In the compressionzone there are a large number of cracks which are mainlytension cracks generated by the extrusion between disksThisresult agreeswith the formationmechanism for compression-induced tension cracks shown in Figure 13

Figure 14 shows the illustration of the contact forceand crack propagation between disks during shearing Thecontact force is initially concentrated on the joint and theloading end and cracks are generated in the joint whereasthere are only a few cracks around the bolt As the sheardisplacement increases the diskrsquos contact force around thebolt becomes gradually larger Cracks occur on the jointand those around the bolt extend further The cracks aroundthe bolt propagate most rapidly The cracks are initiallyconcentrated at the intersection of the bolt and joint and thenpropagate to the ends of the bolt Finally the majority of thecracks are distributed at the joint and around the bolt aftera larger displacement occurs Finally more cracks occur inthe compression zone around the bolt than in the tensionzone In addition the distribution of cracks corresponds tothe contact force in the stress concentration areaThe detailed

10 Advances in Materials Science and Engineering

0 10 20 30 40

Distance along bolt (mm)

Joint plane

Bolt

05 mm10 mm15 mm

20 mm25 mm30 mm

0

100

200

300

Axi

al st

ress

(MPa

)

(a) Axial stress diagram

0 10 20 30 40

Distance along bolt (mm)

05 mm10 mm15 mm

20 mm25 mm30 mm

minus600

minus400

minus200

0

200

400

600

Bend

ing

mom

ent (

Nm

)

Joint plane

Bolt

(b) Bending moment diagram

Joint plane

Bolt

0 10 20 30 40

Distance along bolt (mm)

05 mm10 mm15 mm

20 mm25 mm30 mm

0

10

20

30

40

Shea

r stre

ss (M

Pa)

(c) Shear stress diagram

Figure 10 Bolt stress state diagram at an anchorage angle of 75∘

maximum contact force and crack number in the specimen atthe different monitoring points during shearing are providedin Tables 4 and 5 respectively

44 Evolution of the Disk Rotation During the shearing slipprocess the behavior is determined by the rotation of thedisks Relative disk dislocations will occur due to the rotationof the disks If the rotation angle is sufficiently large itwill lead to shear stresses and the breaking of disk bondsMeanwhile cracks occurring in the rock mass and shearcracking are caused directly by disk rotation Figure 15 showsa graph of the evolution of the disk rotation in terms ofradians for rock disks at different monitoring points

Figure 15 shows that when the shear displacement issmall at the initial stages the disk rotation angle is relatively

small and disks with large rotational angles are distributedaround the bolt The disk rotation angles gradually increasewith increasing shear displacement The disks containinglarger rotation angles are mainly distributed in the twoloading ends and around the bolt and the bolt and thedisks around the bolt contact move relatively violently duringshearing leading to a larger rotation angle Further increasesin the shear displacement and larger rotation disks alsoappeared on the joint surfaces which are caused by thejoint surface microconvex body cutting into the free disksThe distribution of the large disk rotation angles is highlyconsistent with the crack distribution demonstrating that therotation of the disks leads to the generation of shear cracks

Figure 16 shows the variations in the proportion ofrotations exceeding 01 radians As the shear displacement

Advances in Materials Science and Engineering 11

Bolt

Joint plane

0

50

100

150

200

250A

xial

stre

ss (M

Pa)

10 20 30 400Distance along bolt (mm)

05 mm10 mm15 mm

20 mm25 mm30 mm

(a) Axial stress diagram

Joint plane

Bolt

10 20 30 400Distance along bolt (mm)

minus800

minus600

minus400

minus200

0

200

400

600

800

Bend

ing

mom

ent (

Nm

)

05 mm10 mm15 mm

20 mm25 mm30 mm

(b) Bending moment diagram

Bolt

Joint plane

0

10

20

30

40

Shea

r stre

ss (M

Pa)

10 20 30 400Distance along bolt (mm)

05 mm10 mm15 mm

20 mm25 mm30 mm

(c) Shear stress diagram

Figure 11 Bolt stress state diagram at an anchorage angle of 90∘

Table 4 Maximum contact force in the specimen at different monitoring points

Monitoring point Maximum contact forceN Monitoring point Maximum contact forceNa 68201198904 d 12681198905b 15301198905 e 15801198905c 15481198905 f 14451198905

increases the number of clockwise and counterclockwisedisk rotations greater than 01 radians increases continuouslyAdditionally the number of clockwise rotations is greaterthan the number of counterclockwise rotations mainly dueto the shear loading directions

5 Conclusions

In this paper the anchoring mechanism for a tilted anchorbolt subjected to a shear load has been studied using a bilinearconstitutive model with the inner-embedded FISH language

12 Advances in Materials Science and Engineering

(f)(e)(d)

(c)

(b)

(a)

Shear stressNormal displacement

Tension crackShear crack

0

2

4

6

8

Shea

r stre

ss (M

Pa)

05 10 15 20 2500Shear displacement (mm)

00

01

02

03

04

05

06

07

08

Nor

mal

disp

lace

men

t (times10

minus3

m)

0

100

200

300

400

500

600

Crac

k nu

mbe

r

Figure 12 Evolution of the shear stress number of cracks and normal displacement

Rock

Bolt Grout

T

C

C

T

(a)

F F F

F F F

(b)

Figure 13 Stress state of the bolted rock joint system and illustration of the mechanism behind compression-induced tension cracks

Table 5 Number of cracks in the specimen at different monitoringpoints

Monitoringpoint Crack number Monitoring point Crack number

a 0 d 255b 21 e 446c 40 f 592

of PFC based on DEMThe anchoring mechanism of the boltatdifferentanchorage angles has been discussed at themacro-microlevels and the following conclusions have been drawn

(1) The bolt can improve the shear resistance of the jointmainly due to cohesion Tilted anchor bolts not only improvethe shear strength of the joint surface but also result in a lowerstress in the anchor bolt The peak shear strength can be 50greater than that of a vertically installed anchor bolt

(2) A tilted bolt allows for fully mobilized anchor perfor-manceThis configuration not only restricts increment in thenormal displacement but also indirectly improves the stress

state of the anchorage system and increases the shear capacityof the anchor joint The comprehensive shear displacementand shear stress data along with the bolt stress state analysisreveal that the optimal anchorage angle should be between45∘ and 75∘

(3) During the shearing process when the rock bolt andgrout are commonly constrained the stress concentrationphenomenon around the bolt and joint increases the sheardisplacement and contact force After reaching the residualphase the contact force exhibits slight fluctuations due to thesliding of the joint surface

(4) Based on the disk flow software PFC2D crack initia-tion for an anchorage angle of 75∘ will start from the contactforce concentration due to the interactions between the rockgrout and bolt Finally the cracks are mainly distributedin the joint and around the bolt for larger displacementsThe cracks are mainly tensile cracks which correspond tothe formation mechanism for a compression-induced tensilecrack The shear cracking was mainly caused by the rotationof the disks during the shearing process

Advances in Materials Science and Engineering 13

minus005 minus003 minus001 001 003 005

minus002minus001

0001002

minus551525354555

Con

tact

forc

e (N

)

times103

(a) Monitoring point a

minus002minus001

0001002

Con

tact

forc

e (N

)

minus10

10

30

50

70

90

110times103

minus005 minus003 minus001 001 003 005

(b) Monitoring point b

minus002minus001

0001002

Con

tact

forc

e (N

)minus101030507090110

times103

minus005 minus003 minus001 001 003 005

(c) Monitoring point c

minus002minus001

0001002

Con

tact

forc

e (N

)

minus10

10

30

50

70

90

110times103

minus005 minus003 minus001 001 003 005

(d) Monitoring point d

minus002minus001

0001002

minus51535557595115

Con

tact

forc

e (N

)

times103

minus005 minus003 minus001 001 003 005

(e) Monitoring point e

minus002minus001

0001002

minus10

10

30

50

70

90

Con

tact

forc

e (N

)

times103

minus005 minus003 minus001 001 003 005

(f) Monitoring point f

Figure 14 Evolution of the contact force and cracking at different monitoring points

14 Advances in Materials Science and Engineering

Particle rotation (in radians)minus120 to minus01

minus01 to minus0001

minus0001 to minus00005

minus00005 to 00005

00005 to 00010001 to 01

01 to 120

(a) Monitoring point a

Particle rotation (in radians)minus120 to minus01

minus01 to minus0001

minus0001 to minus00005

00005 to 00010001 to 01

01 to 120minus00005 to 00005

(b) Monitoring point b

Particle rotation (in radians)minus120 to minus01

minus01 to minus0001

minus0001 to minus00005

00005 to 00010001 to 01

01 to 120minus00005 to 00005

(c) Monitoring point c

Particle rotation (in radians)minus120 to minus01

minus01 to minus0001

minus0001 to minus00005

00005 to 00010001 to 01

01 to 120minus00005 to 00005

(d) Monitoring point d

Particle rotation (in radians)minus120 to minus01

minus01 to minus0001

minus0001 to minus00005

00005 to 00010001 to 01

01 to 120minus00005 to 00005

(e) Monitoring point e

Particle rotation (in radians)minus120 to minus01

minus01 to minus0001

minus0001 to minus00005

00005 to 00010001 to 01

01 to 120minus00005 to 00005

(f) Monitoring point f

Figure 15 Evolution of disk rotation in radians in the specimen at different monitoring points

0

5

10

15

20

Qua

ntita

tive p

ropo

rtio

n (

)

05 10 15 20 25 3000Shear displacement (mm)

gt 01Rotation (in radians)gt 01|Rotation (in radians)|

lt minus01Rotation (in radians)

Figure 16 Quantitative proportion variation of rotations exceeding 01 radians

Disclosure

GangWang and Yongzheng Zhang should be regarded as co-first authors

Conflicts of Interest

The authors declare that there are no conflicts of interestregarding the publication of this paper

Advances in Materials Science and Engineering 15

Acknowledgments

This study was supported by the National Natural ScienceFoundation of China (nos 51479108 51379117 and 51509261)China Postdoctoral Science Foundation (2015M580599) theTaishan Scholar Talent Team Support Plan for Advantagedamp Unique Discipline Areas the Postdoctoral Science Foun-dation of Shandong Province (no 201402014) and the Shan-dong Province Outstanding Young Scientist Research AwardFund (BS2014NJ006)

References

[1] F D Patton ldquoMultiple modes of shear failure in rockrdquo inProceedings of the 1st Congress on International Society for RockMechanics pp 509ndash513 1966

[2] R E Goodman Methods of Geological Engineering in Discon-tinuous Rocks West Publishing Company New York NY USA1976

[3] N Barton and V Choubey ldquoThe shear strength of rock joints intheory and practicerdquo Rock Mechanics Felsmechanik Mecaniquedes Roches vol 10 no 1-2 pp 1ndash54 1977

[4] P M Dight and H K Chiu ldquoPrediction of shear behavior ofjoints using profilesrdquo International Journal of Rock Mechanicsand Mining Sciences amp Geomechanics Abstracts vol 18 no 5pp 369ndash386 1981

[5] N Barton Modelling Rock Joint Behaviour from In Situ BlockTests Implications for NuclearWaste Repository Design vol 308Office ofNuclearWaste IsolationONWI ColumbusOHUSA1982

[6] T Esaki S Du Y Jiang and Y Wada ldquoEffect of the asperitydamage on the shear behavior of irregular rock jointsrdquo inProceedings of the Second International Conference on Analysisof Discontinuous Deformation pp 459ndash464 Kyoto Japan 1997

[7] Y Jiang J Xiao Y Tanabashi and T Mizokami ldquoDevelopmentof an automated servo-controlled direct shear apparatus apply-ing a constant normal stiffness conditionrdquo International Journalof Rock Mechanics and Mining Sciences vol 41 no 2 pp 275ndash286 2004

[8] Y J Jiang B Li and Y Tanabashi ldquoEstimating the relationbetween surface roughness and mechanical properties of rockjointsrdquo International Journal of Rock Mechanics and MiningSciences vol 43 no 6 pp 837ndash846 2006

[9] C-C Xia Z-C Tang W-M Xiao and Y-L Song ldquoNew peakshear strength criterion of rock joints based on quantifiedsurface descriptionrdquo Rock Mechanics and Rock Engineering vol47 no 2 pp 387ndash400 2014

[10] S Bjurstrom ldquoShear strength of hard rock joints reinforced bygrouted untensioned boltsrdquo in Proceedings of the In Proceedingsof the 3rd International ISRM Congress pp 1194ndash1199 DenverColo USA 1974

[11] X Ge and J Liu ldquoStudy of the shear resistance behaviour ofbolted rock jointsrdquo Journal of Geotechnical Engineering vol 10no 1 pp 8ndash19 1988 (Chinese)

[12] K Spang and P Egger ldquoAction of fully-grouted bolts in jointedrock and factors of influencerdquo Rock Mechanics and Rock Engi-neering vol 23 no 3 pp 201ndash229 1990

[13] P Egger and L Zabuski ldquoBehaviour of rough bolted joints indirect shear testsrdquo in Proceedings of the 7th ISRM Congress vol30 p 342 Aachen Germany September 1991

[14] A M Ferrero ldquoThe shear strength of reinforced rock jointsrdquoInternational Journal of Rock Mechanics and Mining Sciencesand vol 32 no 6 pp 595ndash605 1995

[15] F Pellet and P Egger ldquoAnalytical model for the mechanicalbehaviour of bolted rock joints subjected to shearingrdquo RockMechanics and Rock Engineering vol 29 no 2 pp 73ndash97 1996

[16] H Jalalifar and N Aziz ldquoExperimental and 3D numericalsimulation of reinforced shear jointsrdquo RockMechanics and RockEngineering vol 43 no 1 pp 95ndash103 2010

[17] P Wang T Feng Y-J Zhu and W-J Yu ldquoExperimental studyand numerical simulation of anchoringmechanismof anchoredrocklikematerial with prefabricated fracturerdquoYantu LixueRockand Soil Mechanics vol 37 no 3 pp 793ndash801 2016 (Chinese)

[18] L Li P C Hagan S Saydam B Hebblewhite and Y Li ldquoPara-metric study of rockbolt shear behaviour by double shear testrdquoRock Mechanics and Rock Engineering vol 49 no 12 pp 4787ndash4797 2016

[19] Itasca Consulting Group Inc ldquoPFC2D (Particle Flow Code in2D)Theory and Backgroundrdquo Minneapolis Minn USA 2008

[20] P A Cundall andOD L Strack ldquoThe distinct numericalmodelfor granular assembliesrdquo Geotechnique vol 29 no 1 pp 47ndash651979

[21] J Zhou and Y Chi ldquoThe method of particle flow and PFC2Dcoderdquo Rock and Soil Mechanics vol 21 no 3 pp 271ndash274 2000(Chinese)

[22] D O Potyondy and P A Cundall ldquoA bonded-particle modelfor rockrdquo International Journal of Rock Mechanics and MiningSciences vol 41 no 8 pp 1329ndash1364 2004

[23] N Cho CDMartin andDC Sego ldquoA clumped particlemodelfor rockrdquo International Journal of Rock Mechanics and MiningSciences vol 44 no 7 pp 997ndash1010 2007

[24] Z Zhao L Jing and I Neretnieks ldquoParticle mechanics modelfor the effects of shear on solute retardation coefficient in rockfracturesrdquo International Journal of Rock Mechanics and MiningSciences vol 52 pp 92ndash102 2012

[25] C Xia Y Song Z Tang Y Song and C Shou ldquoParticleflow numerical simulation for shear behavior of rough jointsrdquoChinese Journal of Rock Mechanics and Engineering vol 31 no8 pp 1545ndash1552 2012 (Chinese)

[26] Y Zhou A Misra S Wu and X Zhang ldquoMacro- and meso-analyses of rock joint direct shear test using particle flowtheoryrdquoChinese Journal of RockMechanics and Engineering vol31 no 6 pp 1245ndash1256 2012

[27] RH Cao P CaoH Lin K Zhang andXWTan ldquoParticle flowanalysis of direct shear tests on joints with different roughnessrdquoRock and Soil Mechanics vol 34 supplement 2 pp 456ndash4642013 (Chinese)

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Advances in Materials Science and Engineering 5

Laboratory test curvePFC simulation curve

Axial strain ()

9MPa

6MPa

3MPa

0

18

36

54

72

0000 0003 0006 0009

(1ndash

3)

(MPa

)

Figure 4 Stress-strain curves at different confining stresses

Table 1 Microparameters of the rock sample

Microparameters Values RemarkMinimum radiusmm 05Disk radius ratio 166Disk densitykgsdotmminus3 1830 Uniform distributionDisk contact modulusGPa 395Disk stiffness ratio 10Disk friction coefficients 05Parallel bond normal strengthMPa 245 plusmn 65 Normal distributionParallel bond shear strengthMPa 245 plusmn 65 Normal distributionParallel bond modulusGPa 395Parallel bond stiffness ratio 15Parallel bond radius multiplier 10

addition to comparing the prepeak behavior it is necessary tomatch the macro UCS and Youngrsquos modulus values obtainedfor different numerical simulations of the synthetic materialwith those obtained for the laboratory specimen Thereforethe disk contact modulus values need to be modified simul-taneously with the 120583 value Figure 4 shows that the prepeakand postpeak trend of the laboratory stress-strain curve isapproximately parallel to that of all the stress-strain curves ofnumerical specimens Therefore the prepeak and postpeakbehavior of numerical specimens did not provide any guid-ance for the selection of disk friction value However thecoefficient of friction obtained from the laboratory triaxialtests on the synthetic intact rock was approximately 050Therefore the friction coefficient value of 050 was selectedto represent the disk friction Table 1 lists the determinedmicromechanical parameter values to simulate the intactsynthetic rock with the parallel bond model The failuremode the stress-strain curves and the comparison of themacromechanical parameters indicate that the results are ingood agreement with the laboratory test results

32 Bilinear Constitutive Model of the Bolt The bolt isrepresented by a cluster of balls with parallel bonds thatmeet the requirements of the force-displacement law Parallelbonds can transmit both forces and moments between diskswhich is consistent with the stress state in a bolt The stressstate in a bolt is highly complex when it is reinforced ina jointed rock mass The microparameters of the bolt aredetermined via the bolt pull-out test as shown in Figure 5(a)The microparameters of the bolt are shown in Table 2Figure 5(b) describes the bolt constitutive model whichis not modified One shortcoming of this model is thatit can represent the characteristics of a metallic bar onlyduring elastic deformation the model cannot represent thelarge plastic deformations during yielding Therefore thisconstitutive model has been modified for use in the internalFISH language in PFC to make the modified bolt constitutivemodel meet the stiffness and deformation requirements Asshown in Figure 5(c) the yield strength of the bolt is set suchthat it remains in the linear elastic stage prior to the axialstress reaching the yielding strength However when the axial

6 Advances in Materials Science and Engineering

F

(a)

Bond breaks bolt losesits bearing capacity

e

t

(b)

Bolt yields but it stillhas bearing capacity

Bond breaks bolt losesits bearing capacity

e p

t

(c)

Figure 5 Illustration of the pull-out test for rock bolts and the constitutive rock bolt models

Table 2 Microparameters of the bolt

Microparameters Values RemarkMinimum radiusmm 20Disk radius ratio 10Disk densitykgsdotmminus3 7000 Uniform distributionDisk contact modulusGPa 30Disk stiffness ratio 12Disk friction coefficients 10Parallel bond normal strengthMPa 380 plusmn 20 Normal distributionParallel bond shear strengthMPa 266 plusmn 20 Normal distributionParallel bond stiffness ratio 26Parallel bond radius multiplier 10

stress exceeds the yield strength the bonds will not breakmeaning that the bolt maintains a certain capacity If themagnitude of the axial stress in the bolt exceeds themaximumnormal stress the bonds are broken and the bolt will lose itsbearing capacity

33 Direct Shear Test Model of the Bolted Joint The directshear test model is built using PFC2DThemodel is composedof a top and bottom rock sample grout joint and a boltwith a size of 100mm times 50mm with a total of 2926 disksThe microparameters of the rock sample used are shownin Table 1 The grout microparameters are based on therock parameters with the parallel bond strength changed to40MPa The joint is created using its own command in PFCand the disks in the joint are unbounded Therefore theparallel bond stiffness and the strength of the disks in the jointare set to zero A shear box is generated by the wall elementAmong these walls the sixth wall is the loading wall withthe first wall fifth wall and the sixth wall together formingthe down active shear box which can move in the horizon-tal direction The upper shear box is fixed in the horizontal

direction and only the second wall can move up or down asit is the servo wall which keeps the normal stress constantduring shearing As shown in Figure 6 the rock mass isrepresented by light gray disks and the bolt is composed of acluster of light gray disks which are tangential to each otherAround the bolt the dark gray disks represent the groutAlong the horizontal direction in the middle of the modelthe joint is composed of black disks

34 Direct Shear Test of the Anchoring Model During shear-ing the second wall is controlled by the servo mechanism toapply a constant normal stress with the vertical displacementmonitored as the normal displacement The fifth wall issubjected to a constant velocity to apply horizontal shearloading with the horizontal displacement monitored as theshear displacement The shear stress is calculated by dividingthe average force on the fifth and sixth walls The test isendedwhen the shear displacement reaches the present valueMicrocracking due to bond breakage shear stress normaldisplacement stress in the bolt and displacement variationsin the bolt disks are monitored during the shear test

Advances in Materials Science and Engineering 7

1

3

2

5

48

67

1

3

22

5

6

1

] ]

n

(a) Numerical model of a bolted joint at 45∘

1

3

2

5

48

67

1

3

2

5

6

1

] ]

n

(b) Numerical model of a bolted joint at 75∘

Figure 6 Numerical direct shear test model of the bolted joint

15∘

30∘

45∘

60∘

75∘

90∘

0

2

4

6

8

Shea

r stre

ss (M

Pa)

05 10 15 20 2500

Shear displacement (mm)

Figure 7 Evolution of the shear stress for different anchorage angles

4 Analysis of the Results

41 Analysis of the Shear Performance of the Joint withDifferent Anchorage Angles Direct shear simulations of thebolted joint were conducted using the bolt at differentanchorage angles under a constant normal stress to examinethe macro-micromechanical responses of the bolted jointduring shearing Figure 7 shows the evolution of the shearstress with the shear displacement at different anchorageangles The shear simulation at a normal stress of 5MPa isachieved after a shear displacement of 25mm is reachedThepeak shear strength for different anchorage angles is shownin Table 3

Our data show that (1) the peak shear strength increasedto a maximum and then decreased with further increase inthe anchorage angle and (2) the peak shear strength is thelargest at an anchorage angle of 75∘ Compared to the 90∘anchorage angle changing the angle of the anchor can greatlyimprove the boltrsquos antishear abilitiesTherefore it is importantto determine the optimum design for a bolted rock joint inpractice

Table 3 Peak shear strength for different anchorage angles

Anchor angle The peak shear strength (MPa)15∘ 49830∘ 60045∘ 67860∘ 65275∘ 76590∘ 498

42 Analysis of the Bolt Stress in a Compression-ShearStress State

421 Stress Analysis of the Anchor Joint Plane in a Compres-sion-Shear Stress State Under compressive shear stress therock bolts experience not only shear deformation throughthe thickness of the weak plane but also notable shear defor-mation over a considerable section (the effective length) Tosimplify the analysis we set the effective length of the rock

8 Advances in Materials Science and Engineering

Joint surface Joint surface

b

b

ln ltn

tb

Figure 8 Deformations of the joint surface and bolt

b b blt

b2

(a) Shear force

b b bln

b2

(b) Axial force

Figure 9 Axial and shearing forces on the bolt

bolt to be 119897119905 and the effective length of the bolt to be 119897119899 asshown in Figure 8

The average shear stress distribution on the plane of the119897119905 section is related to the contact conditions of the rodbody and hole wall Three different distribution patternsfor the simplified calculation are shown in Figure 9(a) Thedistribution pattern is a curve but can be measured inthree basic forms Similar to the 119897119899 section the axial stressdistribution can be used as shown in Figure 9(b)

From Figures 8 and 9 and from the joint surface equi-librium conditions the stress tensor of the rock mass can beobtained from the normal and tangential components of thestress tensor along the joint plane as shown in

120590 = 120590119904 + 120590119887119899= 119896119899 (119880119899 sin 119886 minus 119880119905 cos 119886) + 119875119866119887119880119905119860119861119905119897119905 cos 119886

minus 119875119864119887119880119899119861119899119897119899 sin 119886

120591 = 120591119887119904= 119896119904 (119880119899 cos 119886 + 119880119905 sin 119886) + 119875119864119887119880119899119861119899119897119899 cos 119886

+ 119875119866119887119880119905119860119861119905119897119905 sin 119886

+ [119875119866119887119880119905119860119861119905119897119905 cos 119886 minus119875119864119887119880119899119861119899119897119899 sin 119886]119891119904

(8)

In the previous formula 120575119905 and 120575119899 are the weak levels oftangential and normal displacement respectively 119880119905 and 119880119899are the bolt axial and transverse deformations respectively119886 is the angle between the bolt and joint surface and 119864119887 and119866119887 are Youngrsquos modulus and shear modulus for the materialrespectivelyThe shear coefficient119860 is associated with the rodbody section (119860 = 34 for a circular solid rod) 119861119905 and 119861119899 arethe inner pole sectionsrsquo average shear stress and axial stressdistribution of the shape coefficient respectively which are1 34 and 12 120590119887 and 120591119887 are the axial and shear forces on

Advances in Materials Science and Engineering 9

the bolt respectively 119896119899 and 119896119904 are the normal and tangentialstiffness coefficients for the weak level119875 is the anchor sectionreinforcement ratio Therefore

119880119899 = 1119886 (11988622120590 minus 11988612120591)

119880119905 = 1119886 (11988611120591 minus 11988621120590)

(9)

where

11988611 = minus(119896119899 + 119875119864119887119861119899119897119899) sin 119886

11988612 = (119896119899 + 119875119866119887119860119861119905119897119905) cos 119886

11988621 = 119896119904 cos 119886 + 119875119864119887119861119899119897119899 (cos 119886 minus 119891119904 sin 119886)

11988622 = 119896119904 sin 119886 + 119875119866119887119860119861119905119897119905 (sin 119886 + 119891119904 cos 119886)

119886 = 1198861111988622 minus 1198861211988621= 119896119899119896119904 minus 119875119864119887

119861119899119897119899119875119866119887119860119861119905119897119905 +

119875119866119887119860119861119905119897119905 (119896119899sin

2119886 minus 119896119904cos2119886)

+ 119875119864119887119861119899119897119899 [119896119899cos

2119886 minus 119896119904sin2119886]

+ 119896119899119891119904 sin 119886 cos 119886 [ 119875119866119887119860119861119905119897119905 minus119875119864119887119861119899119897119899 ]

(10)

Therefore the relationship between the stress and thestress component of the element can be determined as shownin

120590119887 = 119864119887119861119899119897119899119886 (11988622120590 minus 11988612120591)

120591119887 = 119866119887119860119861119905119897119905119886 (11988611120591 minus 11988621120590)

(11)

422 Analysis of the Bolt Stress for a Particular AnchorageAngle A bolt installed at a particular anchorage angle willhave a better anchoring effect than a bolt installed perpen-dicular to the joint plane Figures 10 and 11 show illustrationsof the stresses in a bolt at anchorage angles of 75∘ and 90∘respectively during shearing

The diagram showing the two different anchorage anglebolts indicates that the anchor force and bending momentare relatively large in the vertically installed bolt leading tobolt axial and shear stress that are larger than those in thetilted bolt whereas for the tilted bolt the axial and shearstress are small and do not reach the yield strengthThereforethe tilted anchor bolts not only improve the shear capacityof the joints but also maintain an appropriate stress statein the anchor The rock bolt axial stress diagram indicatesthat the axial stress increases as the shear displacementincreases for different anchorage anglesThe axial stress is the

highest when the anchor is vertically installed on the jointsurface The results of these two cases reveal that the optimalanchoring angle is approximately 75∘This result correspondsto the previous conclusions that the optimal anchoring angleshould be between 45∘ and 75∘

43 Evolution of the Contact Force and Microscopic Crackingof Disks The anchorage angle of 75∘ was chosen for amore detailed analysis on the micromechanical response anddamage mechanism during shearing Figure 12 shows theevolution of the shear stress number of cracks and normaldisplacement for a normal stress of 5MPa

To accurately determine the evolution features six differ-ent monitoring points (a to f) were set on the shear stresscurve which correspond to shear displacements of 0 01980217 0513 1500 and 2500mm respectively The variationsin the contact force and cracking can be observed from thesix monitoring points As shown in Figure 12 ldquoardquo indicatesthe initial point of the shear displacement and ldquocrdquo indicatesthe shear displacement at which the shear stress reachesthe peak shear strength As shown in Figure 12 no shearcrack is generated during the initial stage of the shear testThe number of shear cracks increases gradually as the sheardisplacement increases When the shear stress reaches thepeak shear strength the number of shear cracks increasessignificantly and the shear crack growth rate tends to stabilizeas the shear stress increases beyond the peak shear strengthThe trends in tension cracking and normal stress are highlysimilar and increase monotonically

During the shearing of a bolted joint the loaded side ofthe bolt is in a state of compression whereas the other side ofthe bolt is in a state of tension due to the bending deflectionof the bolt The adjacent grout is in a state of compressionand the rockmass is in a state of tension because of the bondsbetween the bolt and grout as shown in Figure 13 Bendingdeflection will occur in the bolt as the shear displacementincreases Simultaneously cracks will occur continuously dueto extrusion and slippage between disks In the compressionzone there are a large number of cracks which are mainlytension cracks generated by the extrusion between disksThisresult agreeswith the formationmechanism for compression-induced tension cracks shown in Figure 13

Figure 14 shows the illustration of the contact forceand crack propagation between disks during shearing Thecontact force is initially concentrated on the joint and theloading end and cracks are generated in the joint whereasthere are only a few cracks around the bolt As the sheardisplacement increases the diskrsquos contact force around thebolt becomes gradually larger Cracks occur on the jointand those around the bolt extend further The cracks aroundthe bolt propagate most rapidly The cracks are initiallyconcentrated at the intersection of the bolt and joint and thenpropagate to the ends of the bolt Finally the majority of thecracks are distributed at the joint and around the bolt aftera larger displacement occurs Finally more cracks occur inthe compression zone around the bolt than in the tensionzone In addition the distribution of cracks corresponds tothe contact force in the stress concentration areaThe detailed

10 Advances in Materials Science and Engineering

0 10 20 30 40

Distance along bolt (mm)

Joint plane

Bolt

05 mm10 mm15 mm

20 mm25 mm30 mm

0

100

200

300

Axi

al st

ress

(MPa

)

(a) Axial stress diagram

0 10 20 30 40

Distance along bolt (mm)

05 mm10 mm15 mm

20 mm25 mm30 mm

minus600

minus400

minus200

0

200

400

600

Bend

ing

mom

ent (

Nm

)

Joint plane

Bolt

(b) Bending moment diagram

Joint plane

Bolt

0 10 20 30 40

Distance along bolt (mm)

05 mm10 mm15 mm

20 mm25 mm30 mm

0

10

20

30

40

Shea

r stre

ss (M

Pa)

(c) Shear stress diagram

Figure 10 Bolt stress state diagram at an anchorage angle of 75∘

maximum contact force and crack number in the specimen atthe different monitoring points during shearing are providedin Tables 4 and 5 respectively

44 Evolution of the Disk Rotation During the shearing slipprocess the behavior is determined by the rotation of thedisks Relative disk dislocations will occur due to the rotationof the disks If the rotation angle is sufficiently large itwill lead to shear stresses and the breaking of disk bondsMeanwhile cracks occurring in the rock mass and shearcracking are caused directly by disk rotation Figure 15 showsa graph of the evolution of the disk rotation in terms ofradians for rock disks at different monitoring points

Figure 15 shows that when the shear displacement issmall at the initial stages the disk rotation angle is relatively

small and disks with large rotational angles are distributedaround the bolt The disk rotation angles gradually increasewith increasing shear displacement The disks containinglarger rotation angles are mainly distributed in the twoloading ends and around the bolt and the bolt and thedisks around the bolt contact move relatively violently duringshearing leading to a larger rotation angle Further increasesin the shear displacement and larger rotation disks alsoappeared on the joint surfaces which are caused by thejoint surface microconvex body cutting into the free disksThe distribution of the large disk rotation angles is highlyconsistent with the crack distribution demonstrating that therotation of the disks leads to the generation of shear cracks

Figure 16 shows the variations in the proportion ofrotations exceeding 01 radians As the shear displacement

Advances in Materials Science and Engineering 11

Bolt

Joint plane

0

50

100

150

200

250A

xial

stre

ss (M

Pa)

10 20 30 400Distance along bolt (mm)

05 mm10 mm15 mm

20 mm25 mm30 mm

(a) Axial stress diagram

Joint plane

Bolt

10 20 30 400Distance along bolt (mm)

minus800

minus600

minus400

minus200

0

200

400

600

800

Bend

ing

mom

ent (

Nm

)

05 mm10 mm15 mm

20 mm25 mm30 mm

(b) Bending moment diagram

Bolt

Joint plane

0

10

20

30

40

Shea

r stre

ss (M

Pa)

10 20 30 400Distance along bolt (mm)

05 mm10 mm15 mm

20 mm25 mm30 mm

(c) Shear stress diagram

Figure 11 Bolt stress state diagram at an anchorage angle of 90∘

Table 4 Maximum contact force in the specimen at different monitoring points

Monitoring point Maximum contact forceN Monitoring point Maximum contact forceNa 68201198904 d 12681198905b 15301198905 e 15801198905c 15481198905 f 14451198905

increases the number of clockwise and counterclockwisedisk rotations greater than 01 radians increases continuouslyAdditionally the number of clockwise rotations is greaterthan the number of counterclockwise rotations mainly dueto the shear loading directions

5 Conclusions

In this paper the anchoring mechanism for a tilted anchorbolt subjected to a shear load has been studied using a bilinearconstitutive model with the inner-embedded FISH language

12 Advances in Materials Science and Engineering

(f)(e)(d)

(c)

(b)

(a)

Shear stressNormal displacement

Tension crackShear crack

0

2

4

6

8

Shea

r stre

ss (M

Pa)

05 10 15 20 2500Shear displacement (mm)

00

01

02

03

04

05

06

07

08

Nor

mal

disp

lace

men

t (times10

minus3

m)

0

100

200

300

400

500

600

Crac

k nu

mbe

r

Figure 12 Evolution of the shear stress number of cracks and normal displacement

Rock

Bolt Grout

T

C

C

T

(a)

F F F

F F F

(b)

Figure 13 Stress state of the bolted rock joint system and illustration of the mechanism behind compression-induced tension cracks

Table 5 Number of cracks in the specimen at different monitoringpoints

Monitoringpoint Crack number Monitoring point Crack number

a 0 d 255b 21 e 446c 40 f 592

of PFC based on DEMThe anchoring mechanism of the boltatdifferentanchorage angles has been discussed at themacro-microlevels and the following conclusions have been drawn

(1) The bolt can improve the shear resistance of the jointmainly due to cohesion Tilted anchor bolts not only improvethe shear strength of the joint surface but also result in a lowerstress in the anchor bolt The peak shear strength can be 50greater than that of a vertically installed anchor bolt

(2) A tilted bolt allows for fully mobilized anchor perfor-manceThis configuration not only restricts increment in thenormal displacement but also indirectly improves the stress

state of the anchorage system and increases the shear capacityof the anchor joint The comprehensive shear displacementand shear stress data along with the bolt stress state analysisreveal that the optimal anchorage angle should be between45∘ and 75∘

(3) During the shearing process when the rock bolt andgrout are commonly constrained the stress concentrationphenomenon around the bolt and joint increases the sheardisplacement and contact force After reaching the residualphase the contact force exhibits slight fluctuations due to thesliding of the joint surface

(4) Based on the disk flow software PFC2D crack initia-tion for an anchorage angle of 75∘ will start from the contactforce concentration due to the interactions between the rockgrout and bolt Finally the cracks are mainly distributedin the joint and around the bolt for larger displacementsThe cracks are mainly tensile cracks which correspond tothe formation mechanism for a compression-induced tensilecrack The shear cracking was mainly caused by the rotationof the disks during the shearing process

Advances in Materials Science and Engineering 13

minus005 minus003 minus001 001 003 005

minus002minus001

0001002

minus551525354555

Con

tact

forc

e (N

)

times103

(a) Monitoring point a

minus002minus001

0001002

Con

tact

forc

e (N

)

minus10

10

30

50

70

90

110times103

minus005 minus003 minus001 001 003 005

(b) Monitoring point b

minus002minus001

0001002

Con

tact

forc

e (N

)minus101030507090110

times103

minus005 minus003 minus001 001 003 005

(c) Monitoring point c

minus002minus001

0001002

Con

tact

forc

e (N

)

minus10

10

30

50

70

90

110times103

minus005 minus003 minus001 001 003 005

(d) Monitoring point d

minus002minus001

0001002

minus51535557595115

Con

tact

forc

e (N

)

times103

minus005 minus003 minus001 001 003 005

(e) Monitoring point e

minus002minus001

0001002

minus10

10

30

50

70

90

Con

tact

forc

e (N

)

times103

minus005 minus003 minus001 001 003 005

(f) Monitoring point f

Figure 14 Evolution of the contact force and cracking at different monitoring points

14 Advances in Materials Science and Engineering

Particle rotation (in radians)minus120 to minus01

minus01 to minus0001

minus0001 to minus00005

minus00005 to 00005

00005 to 00010001 to 01

01 to 120

(a) Monitoring point a

Particle rotation (in radians)minus120 to minus01

minus01 to minus0001

minus0001 to minus00005

00005 to 00010001 to 01

01 to 120minus00005 to 00005

(b) Monitoring point b

Particle rotation (in radians)minus120 to minus01

minus01 to minus0001

minus0001 to minus00005

00005 to 00010001 to 01

01 to 120minus00005 to 00005

(c) Monitoring point c

Particle rotation (in radians)minus120 to minus01

minus01 to minus0001

minus0001 to minus00005

00005 to 00010001 to 01

01 to 120minus00005 to 00005

(d) Monitoring point d

Particle rotation (in radians)minus120 to minus01

minus01 to minus0001

minus0001 to minus00005

00005 to 00010001 to 01

01 to 120minus00005 to 00005

(e) Monitoring point e

Particle rotation (in radians)minus120 to minus01

minus01 to minus0001

minus0001 to minus00005

00005 to 00010001 to 01

01 to 120minus00005 to 00005

(f) Monitoring point f

Figure 15 Evolution of disk rotation in radians in the specimen at different monitoring points

0

5

10

15

20

Qua

ntita

tive p

ropo

rtio

n (

)

05 10 15 20 25 3000Shear displacement (mm)

gt 01Rotation (in radians)gt 01|Rotation (in radians)|

lt minus01Rotation (in radians)

Figure 16 Quantitative proportion variation of rotations exceeding 01 radians

Disclosure

GangWang and Yongzheng Zhang should be regarded as co-first authors

Conflicts of Interest

The authors declare that there are no conflicts of interestregarding the publication of this paper

Advances in Materials Science and Engineering 15

Acknowledgments

This study was supported by the National Natural ScienceFoundation of China (nos 51479108 51379117 and 51509261)China Postdoctoral Science Foundation (2015M580599) theTaishan Scholar Talent Team Support Plan for Advantagedamp Unique Discipline Areas the Postdoctoral Science Foun-dation of Shandong Province (no 201402014) and the Shan-dong Province Outstanding Young Scientist Research AwardFund (BS2014NJ006)

References

[1] F D Patton ldquoMultiple modes of shear failure in rockrdquo inProceedings of the 1st Congress on International Society for RockMechanics pp 509ndash513 1966

[2] R E Goodman Methods of Geological Engineering in Discon-tinuous Rocks West Publishing Company New York NY USA1976

[3] N Barton and V Choubey ldquoThe shear strength of rock joints intheory and practicerdquo Rock Mechanics Felsmechanik Mecaniquedes Roches vol 10 no 1-2 pp 1ndash54 1977

[4] P M Dight and H K Chiu ldquoPrediction of shear behavior ofjoints using profilesrdquo International Journal of Rock Mechanicsand Mining Sciences amp Geomechanics Abstracts vol 18 no 5pp 369ndash386 1981

[5] N Barton Modelling Rock Joint Behaviour from In Situ BlockTests Implications for NuclearWaste Repository Design vol 308Office ofNuclearWaste IsolationONWI ColumbusOHUSA1982

[6] T Esaki S Du Y Jiang and Y Wada ldquoEffect of the asperitydamage on the shear behavior of irregular rock jointsrdquo inProceedings of the Second International Conference on Analysisof Discontinuous Deformation pp 459ndash464 Kyoto Japan 1997

[7] Y Jiang J Xiao Y Tanabashi and T Mizokami ldquoDevelopmentof an automated servo-controlled direct shear apparatus apply-ing a constant normal stiffness conditionrdquo International Journalof Rock Mechanics and Mining Sciences vol 41 no 2 pp 275ndash286 2004

[8] Y J Jiang B Li and Y Tanabashi ldquoEstimating the relationbetween surface roughness and mechanical properties of rockjointsrdquo International Journal of Rock Mechanics and MiningSciences vol 43 no 6 pp 837ndash846 2006

[9] C-C Xia Z-C Tang W-M Xiao and Y-L Song ldquoNew peakshear strength criterion of rock joints based on quantifiedsurface descriptionrdquo Rock Mechanics and Rock Engineering vol47 no 2 pp 387ndash400 2014

[10] S Bjurstrom ldquoShear strength of hard rock joints reinforced bygrouted untensioned boltsrdquo in Proceedings of the In Proceedingsof the 3rd International ISRM Congress pp 1194ndash1199 DenverColo USA 1974

[11] X Ge and J Liu ldquoStudy of the shear resistance behaviour ofbolted rock jointsrdquo Journal of Geotechnical Engineering vol 10no 1 pp 8ndash19 1988 (Chinese)

[12] K Spang and P Egger ldquoAction of fully-grouted bolts in jointedrock and factors of influencerdquo Rock Mechanics and Rock Engi-neering vol 23 no 3 pp 201ndash229 1990

[13] P Egger and L Zabuski ldquoBehaviour of rough bolted joints indirect shear testsrdquo in Proceedings of the 7th ISRM Congress vol30 p 342 Aachen Germany September 1991

[14] A M Ferrero ldquoThe shear strength of reinforced rock jointsrdquoInternational Journal of Rock Mechanics and Mining Sciencesand vol 32 no 6 pp 595ndash605 1995

[15] F Pellet and P Egger ldquoAnalytical model for the mechanicalbehaviour of bolted rock joints subjected to shearingrdquo RockMechanics and Rock Engineering vol 29 no 2 pp 73ndash97 1996

[16] H Jalalifar and N Aziz ldquoExperimental and 3D numericalsimulation of reinforced shear jointsrdquo RockMechanics and RockEngineering vol 43 no 1 pp 95ndash103 2010

[17] P Wang T Feng Y-J Zhu and W-J Yu ldquoExperimental studyand numerical simulation of anchoringmechanismof anchoredrocklikematerial with prefabricated fracturerdquoYantu LixueRockand Soil Mechanics vol 37 no 3 pp 793ndash801 2016 (Chinese)

[18] L Li P C Hagan S Saydam B Hebblewhite and Y Li ldquoPara-metric study of rockbolt shear behaviour by double shear testrdquoRock Mechanics and Rock Engineering vol 49 no 12 pp 4787ndash4797 2016

[19] Itasca Consulting Group Inc ldquoPFC2D (Particle Flow Code in2D)Theory and Backgroundrdquo Minneapolis Minn USA 2008

[20] P A Cundall andOD L Strack ldquoThe distinct numericalmodelfor granular assembliesrdquo Geotechnique vol 29 no 1 pp 47ndash651979

[21] J Zhou and Y Chi ldquoThe method of particle flow and PFC2Dcoderdquo Rock and Soil Mechanics vol 21 no 3 pp 271ndash274 2000(Chinese)

[22] D O Potyondy and P A Cundall ldquoA bonded-particle modelfor rockrdquo International Journal of Rock Mechanics and MiningSciences vol 41 no 8 pp 1329ndash1364 2004

[23] N Cho CDMartin andDC Sego ldquoA clumped particlemodelfor rockrdquo International Journal of Rock Mechanics and MiningSciences vol 44 no 7 pp 997ndash1010 2007

[24] Z Zhao L Jing and I Neretnieks ldquoParticle mechanics modelfor the effects of shear on solute retardation coefficient in rockfracturesrdquo International Journal of Rock Mechanics and MiningSciences vol 52 pp 92ndash102 2012

[25] C Xia Y Song Z Tang Y Song and C Shou ldquoParticleflow numerical simulation for shear behavior of rough jointsrdquoChinese Journal of Rock Mechanics and Engineering vol 31 no8 pp 1545ndash1552 2012 (Chinese)

[26] Y Zhou A Misra S Wu and X Zhang ldquoMacro- and meso-analyses of rock joint direct shear test using particle flowtheoryrdquoChinese Journal of RockMechanics and Engineering vol31 no 6 pp 1245ndash1256 2012

[27] RH Cao P CaoH Lin K Zhang andXWTan ldquoParticle flowanalysis of direct shear tests on joints with different roughnessrdquoRock and Soil Mechanics vol 34 supplement 2 pp 456ndash4642013 (Chinese)

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Journal of

NanotechnologyHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

CrystallographyJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CoatingsJournal of

Advances in

Materials Science and EngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Smart Materials Research

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MetallurgyJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

BioMed Research International

MaterialsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

6 Advances in Materials Science and Engineering

F

(a)

Bond breaks bolt losesits bearing capacity

e

t

(b)

Bolt yields but it stillhas bearing capacity

Bond breaks bolt losesits bearing capacity

e p

t

(c)

Figure 5 Illustration of the pull-out test for rock bolts and the constitutive rock bolt models

Table 2 Microparameters of the bolt

Microparameters Values RemarkMinimum radiusmm 20Disk radius ratio 10Disk densitykgsdotmminus3 7000 Uniform distributionDisk contact modulusGPa 30Disk stiffness ratio 12Disk friction coefficients 10Parallel bond normal strengthMPa 380 plusmn 20 Normal distributionParallel bond shear strengthMPa 266 plusmn 20 Normal distributionParallel bond stiffness ratio 26Parallel bond radius multiplier 10

stress exceeds the yield strength the bonds will not breakmeaning that the bolt maintains a certain capacity If themagnitude of the axial stress in the bolt exceeds themaximumnormal stress the bonds are broken and the bolt will lose itsbearing capacity

33 Direct Shear Test Model of the Bolted Joint The directshear test model is built using PFC2DThemodel is composedof a top and bottom rock sample grout joint and a boltwith a size of 100mm times 50mm with a total of 2926 disksThe microparameters of the rock sample used are shownin Table 1 The grout microparameters are based on therock parameters with the parallel bond strength changed to40MPa The joint is created using its own command in PFCand the disks in the joint are unbounded Therefore theparallel bond stiffness and the strength of the disks in the jointare set to zero A shear box is generated by the wall elementAmong these walls the sixth wall is the loading wall withthe first wall fifth wall and the sixth wall together formingthe down active shear box which can move in the horizon-tal direction The upper shear box is fixed in the horizontal

direction and only the second wall can move up or down asit is the servo wall which keeps the normal stress constantduring shearing As shown in Figure 6 the rock mass isrepresented by light gray disks and the bolt is composed of acluster of light gray disks which are tangential to each otherAround the bolt the dark gray disks represent the groutAlong the horizontal direction in the middle of the modelthe joint is composed of black disks

34 Direct Shear Test of the Anchoring Model During shear-ing the second wall is controlled by the servo mechanism toapply a constant normal stress with the vertical displacementmonitored as the normal displacement The fifth wall issubjected to a constant velocity to apply horizontal shearloading with the horizontal displacement monitored as theshear displacement The shear stress is calculated by dividingthe average force on the fifth and sixth walls The test isendedwhen the shear displacement reaches the present valueMicrocracking due to bond breakage shear stress normaldisplacement stress in the bolt and displacement variationsin the bolt disks are monitored during the shear test

Advances in Materials Science and Engineering 7

1

3

2

5

48

67

1

3

22

5

6

1

] ]

n

(a) Numerical model of a bolted joint at 45∘

1

3

2

5

48

67

1

3

2

5

6

1

] ]

n

(b) Numerical model of a bolted joint at 75∘

Figure 6 Numerical direct shear test model of the bolted joint

15∘

30∘

45∘

60∘

75∘

90∘

0

2

4

6

8

Shea

r stre

ss (M

Pa)

05 10 15 20 2500

Shear displacement (mm)

Figure 7 Evolution of the shear stress for different anchorage angles

4 Analysis of the Results

41 Analysis of the Shear Performance of the Joint withDifferent Anchorage Angles Direct shear simulations of thebolted joint were conducted using the bolt at differentanchorage angles under a constant normal stress to examinethe macro-micromechanical responses of the bolted jointduring shearing Figure 7 shows the evolution of the shearstress with the shear displacement at different anchorageangles The shear simulation at a normal stress of 5MPa isachieved after a shear displacement of 25mm is reachedThepeak shear strength for different anchorage angles is shownin Table 3

Our data show that (1) the peak shear strength increasedto a maximum and then decreased with further increase inthe anchorage angle and (2) the peak shear strength is thelargest at an anchorage angle of 75∘ Compared to the 90∘anchorage angle changing the angle of the anchor can greatlyimprove the boltrsquos antishear abilitiesTherefore it is importantto determine the optimum design for a bolted rock joint inpractice

Table 3 Peak shear strength for different anchorage angles

Anchor angle The peak shear strength (MPa)15∘ 49830∘ 60045∘ 67860∘ 65275∘ 76590∘ 498

42 Analysis of the Bolt Stress in a Compression-ShearStress State

421 Stress Analysis of the Anchor Joint Plane in a Compres-sion-Shear Stress State Under compressive shear stress therock bolts experience not only shear deformation throughthe thickness of the weak plane but also notable shear defor-mation over a considerable section (the effective length) Tosimplify the analysis we set the effective length of the rock

8 Advances in Materials Science and Engineering

Joint surface Joint surface

b

b

ln ltn

tb

Figure 8 Deformations of the joint surface and bolt

b b blt

b2

(a) Shear force

b b bln

b2

(b) Axial force

Figure 9 Axial and shearing forces on the bolt

bolt to be 119897119905 and the effective length of the bolt to be 119897119899 asshown in Figure 8

The average shear stress distribution on the plane of the119897119905 section is related to the contact conditions of the rodbody and hole wall Three different distribution patternsfor the simplified calculation are shown in Figure 9(a) Thedistribution pattern is a curve but can be measured inthree basic forms Similar to the 119897119899 section the axial stressdistribution can be used as shown in Figure 9(b)

From Figures 8 and 9 and from the joint surface equi-librium conditions the stress tensor of the rock mass can beobtained from the normal and tangential components of thestress tensor along the joint plane as shown in

120590 = 120590119904 + 120590119887119899= 119896119899 (119880119899 sin 119886 minus 119880119905 cos 119886) + 119875119866119887119880119905119860119861119905119897119905 cos 119886

minus 119875119864119887119880119899119861119899119897119899 sin 119886

120591 = 120591119887119904= 119896119904 (119880119899 cos 119886 + 119880119905 sin 119886) + 119875119864119887119880119899119861119899119897119899 cos 119886

+ 119875119866119887119880119905119860119861119905119897119905 sin 119886

+ [119875119866119887119880119905119860119861119905119897119905 cos 119886 minus119875119864119887119880119899119861119899119897119899 sin 119886]119891119904

(8)

In the previous formula 120575119905 and 120575119899 are the weak levels oftangential and normal displacement respectively 119880119905 and 119880119899are the bolt axial and transverse deformations respectively119886 is the angle between the bolt and joint surface and 119864119887 and119866119887 are Youngrsquos modulus and shear modulus for the materialrespectivelyThe shear coefficient119860 is associated with the rodbody section (119860 = 34 for a circular solid rod) 119861119905 and 119861119899 arethe inner pole sectionsrsquo average shear stress and axial stressdistribution of the shape coefficient respectively which are1 34 and 12 120590119887 and 120591119887 are the axial and shear forces on

Advances in Materials Science and Engineering 9

the bolt respectively 119896119899 and 119896119904 are the normal and tangentialstiffness coefficients for the weak level119875 is the anchor sectionreinforcement ratio Therefore

119880119899 = 1119886 (11988622120590 minus 11988612120591)

119880119905 = 1119886 (11988611120591 minus 11988621120590)

(9)

where

11988611 = minus(119896119899 + 119875119864119887119861119899119897119899) sin 119886

11988612 = (119896119899 + 119875119866119887119860119861119905119897119905) cos 119886

11988621 = 119896119904 cos 119886 + 119875119864119887119861119899119897119899 (cos 119886 minus 119891119904 sin 119886)

11988622 = 119896119904 sin 119886 + 119875119866119887119860119861119905119897119905 (sin 119886 + 119891119904 cos 119886)

119886 = 1198861111988622 minus 1198861211988621= 119896119899119896119904 minus 119875119864119887

119861119899119897119899119875119866119887119860119861119905119897119905 +

119875119866119887119860119861119905119897119905 (119896119899sin

2119886 minus 119896119904cos2119886)

+ 119875119864119887119861119899119897119899 [119896119899cos

2119886 minus 119896119904sin2119886]

+ 119896119899119891119904 sin 119886 cos 119886 [ 119875119866119887119860119861119905119897119905 minus119875119864119887119861119899119897119899 ]

(10)

Therefore the relationship between the stress and thestress component of the element can be determined as shownin

120590119887 = 119864119887119861119899119897119899119886 (11988622120590 minus 11988612120591)

120591119887 = 119866119887119860119861119905119897119905119886 (11988611120591 minus 11988621120590)

(11)

422 Analysis of the Bolt Stress for a Particular AnchorageAngle A bolt installed at a particular anchorage angle willhave a better anchoring effect than a bolt installed perpen-dicular to the joint plane Figures 10 and 11 show illustrationsof the stresses in a bolt at anchorage angles of 75∘ and 90∘respectively during shearing

The diagram showing the two different anchorage anglebolts indicates that the anchor force and bending momentare relatively large in the vertically installed bolt leading tobolt axial and shear stress that are larger than those in thetilted bolt whereas for the tilted bolt the axial and shearstress are small and do not reach the yield strengthThereforethe tilted anchor bolts not only improve the shear capacityof the joints but also maintain an appropriate stress statein the anchor The rock bolt axial stress diagram indicatesthat the axial stress increases as the shear displacementincreases for different anchorage anglesThe axial stress is the

highest when the anchor is vertically installed on the jointsurface The results of these two cases reveal that the optimalanchoring angle is approximately 75∘This result correspondsto the previous conclusions that the optimal anchoring angleshould be between 45∘ and 75∘

43 Evolution of the Contact Force and Microscopic Crackingof Disks The anchorage angle of 75∘ was chosen for amore detailed analysis on the micromechanical response anddamage mechanism during shearing Figure 12 shows theevolution of the shear stress number of cracks and normaldisplacement for a normal stress of 5MPa

To accurately determine the evolution features six differ-ent monitoring points (a to f) were set on the shear stresscurve which correspond to shear displacements of 0 01980217 0513 1500 and 2500mm respectively The variationsin the contact force and cracking can be observed from thesix monitoring points As shown in Figure 12 ldquoardquo indicatesthe initial point of the shear displacement and ldquocrdquo indicatesthe shear displacement at which the shear stress reachesthe peak shear strength As shown in Figure 12 no shearcrack is generated during the initial stage of the shear testThe number of shear cracks increases gradually as the sheardisplacement increases When the shear stress reaches thepeak shear strength the number of shear cracks increasessignificantly and the shear crack growth rate tends to stabilizeas the shear stress increases beyond the peak shear strengthThe trends in tension cracking and normal stress are highlysimilar and increase monotonically

During the shearing of a bolted joint the loaded side ofthe bolt is in a state of compression whereas the other side ofthe bolt is in a state of tension due to the bending deflectionof the bolt The adjacent grout is in a state of compressionand the rockmass is in a state of tension because of the bondsbetween the bolt and grout as shown in Figure 13 Bendingdeflection will occur in the bolt as the shear displacementincreases Simultaneously cracks will occur continuously dueto extrusion and slippage between disks In the compressionzone there are a large number of cracks which are mainlytension cracks generated by the extrusion between disksThisresult agreeswith the formationmechanism for compression-induced tension cracks shown in Figure 13

Figure 14 shows the illustration of the contact forceand crack propagation between disks during shearing Thecontact force is initially concentrated on the joint and theloading end and cracks are generated in the joint whereasthere are only a few cracks around the bolt As the sheardisplacement increases the diskrsquos contact force around thebolt becomes gradually larger Cracks occur on the jointand those around the bolt extend further The cracks aroundthe bolt propagate most rapidly The cracks are initiallyconcentrated at the intersection of the bolt and joint and thenpropagate to the ends of the bolt Finally the majority of thecracks are distributed at the joint and around the bolt aftera larger displacement occurs Finally more cracks occur inthe compression zone around the bolt than in the tensionzone In addition the distribution of cracks corresponds tothe contact force in the stress concentration areaThe detailed

10 Advances in Materials Science and Engineering

0 10 20 30 40

Distance along bolt (mm)

Joint plane

Bolt

05 mm10 mm15 mm

20 mm25 mm30 mm

0

100

200

300

Axi

al st

ress

(MPa

)

(a) Axial stress diagram

0 10 20 30 40

Distance along bolt (mm)

05 mm10 mm15 mm

20 mm25 mm30 mm

minus600

minus400

minus200

0

200

400

600

Bend

ing

mom

ent (

Nm

)

Joint plane

Bolt

(b) Bending moment diagram

Joint plane

Bolt

0 10 20 30 40

Distance along bolt (mm)

05 mm10 mm15 mm

20 mm25 mm30 mm

0

10

20

30

40

Shea

r stre

ss (M

Pa)

(c) Shear stress diagram

Figure 10 Bolt stress state diagram at an anchorage angle of 75∘

maximum contact force and crack number in the specimen atthe different monitoring points during shearing are providedin Tables 4 and 5 respectively

44 Evolution of the Disk Rotation During the shearing slipprocess the behavior is determined by the rotation of thedisks Relative disk dislocations will occur due to the rotationof the disks If the rotation angle is sufficiently large itwill lead to shear stresses and the breaking of disk bondsMeanwhile cracks occurring in the rock mass and shearcracking are caused directly by disk rotation Figure 15 showsa graph of the evolution of the disk rotation in terms ofradians for rock disks at different monitoring points

Figure 15 shows that when the shear displacement issmall at the initial stages the disk rotation angle is relatively

small and disks with large rotational angles are distributedaround the bolt The disk rotation angles gradually increasewith increasing shear displacement The disks containinglarger rotation angles are mainly distributed in the twoloading ends and around the bolt and the bolt and thedisks around the bolt contact move relatively violently duringshearing leading to a larger rotation angle Further increasesin the shear displacement and larger rotation disks alsoappeared on the joint surfaces which are caused by thejoint surface microconvex body cutting into the free disksThe distribution of the large disk rotation angles is highlyconsistent with the crack distribution demonstrating that therotation of the disks leads to the generation of shear cracks

Figure 16 shows the variations in the proportion ofrotations exceeding 01 radians As the shear displacement

Advances in Materials Science and Engineering 11

Bolt

Joint plane

0

50

100

150

200

250A

xial

stre

ss (M

Pa)

10 20 30 400Distance along bolt (mm)

05 mm10 mm15 mm

20 mm25 mm30 mm

(a) Axial stress diagram

Joint plane

Bolt

10 20 30 400Distance along bolt (mm)

minus800

minus600

minus400

minus200

0

200

400

600

800

Bend

ing

mom

ent (

Nm

)

05 mm10 mm15 mm

20 mm25 mm30 mm

(b) Bending moment diagram

Bolt

Joint plane

0

10

20

30

40

Shea

r stre

ss (M

Pa)

10 20 30 400Distance along bolt (mm)

05 mm10 mm15 mm

20 mm25 mm30 mm

(c) Shear stress diagram

Figure 11 Bolt stress state diagram at an anchorage angle of 90∘

Table 4 Maximum contact force in the specimen at different monitoring points

Monitoring point Maximum contact forceN Monitoring point Maximum contact forceNa 68201198904 d 12681198905b 15301198905 e 15801198905c 15481198905 f 14451198905

increases the number of clockwise and counterclockwisedisk rotations greater than 01 radians increases continuouslyAdditionally the number of clockwise rotations is greaterthan the number of counterclockwise rotations mainly dueto the shear loading directions

5 Conclusions

In this paper the anchoring mechanism for a tilted anchorbolt subjected to a shear load has been studied using a bilinearconstitutive model with the inner-embedded FISH language

12 Advances in Materials Science and Engineering

(f)(e)(d)

(c)

(b)

(a)

Shear stressNormal displacement

Tension crackShear crack

0

2

4

6

8

Shea

r stre

ss (M

Pa)

05 10 15 20 2500Shear displacement (mm)

00

01

02

03

04

05

06

07

08

Nor

mal

disp

lace

men

t (times10

minus3

m)

0

100

200

300

400

500

600

Crac

k nu

mbe

r

Figure 12 Evolution of the shear stress number of cracks and normal displacement

Rock

Bolt Grout

T

C

C

T

(a)

F F F

F F F

(b)

Figure 13 Stress state of the bolted rock joint system and illustration of the mechanism behind compression-induced tension cracks

Table 5 Number of cracks in the specimen at different monitoringpoints

Monitoringpoint Crack number Monitoring point Crack number

a 0 d 255b 21 e 446c 40 f 592

of PFC based on DEMThe anchoring mechanism of the boltatdifferentanchorage angles has been discussed at themacro-microlevels and the following conclusions have been drawn

(1) The bolt can improve the shear resistance of the jointmainly due to cohesion Tilted anchor bolts not only improvethe shear strength of the joint surface but also result in a lowerstress in the anchor bolt The peak shear strength can be 50greater than that of a vertically installed anchor bolt

(2) A tilted bolt allows for fully mobilized anchor perfor-manceThis configuration not only restricts increment in thenormal displacement but also indirectly improves the stress

state of the anchorage system and increases the shear capacityof the anchor joint The comprehensive shear displacementand shear stress data along with the bolt stress state analysisreveal that the optimal anchorage angle should be between45∘ and 75∘

(3) During the shearing process when the rock bolt andgrout are commonly constrained the stress concentrationphenomenon around the bolt and joint increases the sheardisplacement and contact force After reaching the residualphase the contact force exhibits slight fluctuations due to thesliding of the joint surface

(4) Based on the disk flow software PFC2D crack initia-tion for an anchorage angle of 75∘ will start from the contactforce concentration due to the interactions between the rockgrout and bolt Finally the cracks are mainly distributedin the joint and around the bolt for larger displacementsThe cracks are mainly tensile cracks which correspond tothe formation mechanism for a compression-induced tensilecrack The shear cracking was mainly caused by the rotationof the disks during the shearing process

Advances in Materials Science and Engineering 13

minus005 minus003 minus001 001 003 005

minus002minus001

0001002

minus551525354555

Con

tact

forc

e (N

)

times103

(a) Monitoring point a

minus002minus001

0001002

Con

tact

forc

e (N

)

minus10

10

30

50

70

90

110times103

minus005 minus003 minus001 001 003 005

(b) Monitoring point b

minus002minus001

0001002

Con

tact

forc

e (N

)minus101030507090110

times103

minus005 minus003 minus001 001 003 005

(c) Monitoring point c

minus002minus001

0001002

Con

tact

forc

e (N

)

minus10

10

30

50

70

90

110times103

minus005 minus003 minus001 001 003 005

(d) Monitoring point d

minus002minus001

0001002

minus51535557595115

Con

tact

forc

e (N

)

times103

minus005 minus003 minus001 001 003 005

(e) Monitoring point e

minus002minus001

0001002

minus10

10

30

50

70

90

Con

tact

forc

e (N

)

times103

minus005 minus003 minus001 001 003 005

(f) Monitoring point f

Figure 14 Evolution of the contact force and cracking at different monitoring points

14 Advances in Materials Science and Engineering

Particle rotation (in radians)minus120 to minus01

minus01 to minus0001

minus0001 to minus00005

minus00005 to 00005

00005 to 00010001 to 01

01 to 120

(a) Monitoring point a

Particle rotation (in radians)minus120 to minus01

minus01 to minus0001

minus0001 to minus00005

00005 to 00010001 to 01

01 to 120minus00005 to 00005

(b) Monitoring point b

Particle rotation (in radians)minus120 to minus01

minus01 to minus0001

minus0001 to minus00005

00005 to 00010001 to 01

01 to 120minus00005 to 00005

(c) Monitoring point c

Particle rotation (in radians)minus120 to minus01

minus01 to minus0001

minus0001 to minus00005

00005 to 00010001 to 01

01 to 120minus00005 to 00005

(d) Monitoring point d

Particle rotation (in radians)minus120 to minus01

minus01 to minus0001

minus0001 to minus00005

00005 to 00010001 to 01

01 to 120minus00005 to 00005

(e) Monitoring point e

Particle rotation (in radians)minus120 to minus01

minus01 to minus0001

minus0001 to minus00005

00005 to 00010001 to 01

01 to 120minus00005 to 00005

(f) Monitoring point f

Figure 15 Evolution of disk rotation in radians in the specimen at different monitoring points

0

5

10

15

20

Qua

ntita

tive p

ropo

rtio

n (

)

05 10 15 20 25 3000Shear displacement (mm)

gt 01Rotation (in radians)gt 01|Rotation (in radians)|

lt minus01Rotation (in radians)

Figure 16 Quantitative proportion variation of rotations exceeding 01 radians

Disclosure

GangWang and Yongzheng Zhang should be regarded as co-first authors

Conflicts of Interest

The authors declare that there are no conflicts of interestregarding the publication of this paper

Advances in Materials Science and Engineering 15

Acknowledgments

This study was supported by the National Natural ScienceFoundation of China (nos 51479108 51379117 and 51509261)China Postdoctoral Science Foundation (2015M580599) theTaishan Scholar Talent Team Support Plan for Advantagedamp Unique Discipline Areas the Postdoctoral Science Foun-dation of Shandong Province (no 201402014) and the Shan-dong Province Outstanding Young Scientist Research AwardFund (BS2014NJ006)

References

[1] F D Patton ldquoMultiple modes of shear failure in rockrdquo inProceedings of the 1st Congress on International Society for RockMechanics pp 509ndash513 1966

[2] R E Goodman Methods of Geological Engineering in Discon-tinuous Rocks West Publishing Company New York NY USA1976

[3] N Barton and V Choubey ldquoThe shear strength of rock joints intheory and practicerdquo Rock Mechanics Felsmechanik Mecaniquedes Roches vol 10 no 1-2 pp 1ndash54 1977

[4] P M Dight and H K Chiu ldquoPrediction of shear behavior ofjoints using profilesrdquo International Journal of Rock Mechanicsand Mining Sciences amp Geomechanics Abstracts vol 18 no 5pp 369ndash386 1981

[5] N Barton Modelling Rock Joint Behaviour from In Situ BlockTests Implications for NuclearWaste Repository Design vol 308Office ofNuclearWaste IsolationONWI ColumbusOHUSA1982

[6] T Esaki S Du Y Jiang and Y Wada ldquoEffect of the asperitydamage on the shear behavior of irregular rock jointsrdquo inProceedings of the Second International Conference on Analysisof Discontinuous Deformation pp 459ndash464 Kyoto Japan 1997

[7] Y Jiang J Xiao Y Tanabashi and T Mizokami ldquoDevelopmentof an automated servo-controlled direct shear apparatus apply-ing a constant normal stiffness conditionrdquo International Journalof Rock Mechanics and Mining Sciences vol 41 no 2 pp 275ndash286 2004

[8] Y J Jiang B Li and Y Tanabashi ldquoEstimating the relationbetween surface roughness and mechanical properties of rockjointsrdquo International Journal of Rock Mechanics and MiningSciences vol 43 no 6 pp 837ndash846 2006

[9] C-C Xia Z-C Tang W-M Xiao and Y-L Song ldquoNew peakshear strength criterion of rock joints based on quantifiedsurface descriptionrdquo Rock Mechanics and Rock Engineering vol47 no 2 pp 387ndash400 2014

[10] S Bjurstrom ldquoShear strength of hard rock joints reinforced bygrouted untensioned boltsrdquo in Proceedings of the In Proceedingsof the 3rd International ISRM Congress pp 1194ndash1199 DenverColo USA 1974

[11] X Ge and J Liu ldquoStudy of the shear resistance behaviour ofbolted rock jointsrdquo Journal of Geotechnical Engineering vol 10no 1 pp 8ndash19 1988 (Chinese)

[12] K Spang and P Egger ldquoAction of fully-grouted bolts in jointedrock and factors of influencerdquo Rock Mechanics and Rock Engi-neering vol 23 no 3 pp 201ndash229 1990

[13] P Egger and L Zabuski ldquoBehaviour of rough bolted joints indirect shear testsrdquo in Proceedings of the 7th ISRM Congress vol30 p 342 Aachen Germany September 1991

[14] A M Ferrero ldquoThe shear strength of reinforced rock jointsrdquoInternational Journal of Rock Mechanics and Mining Sciencesand vol 32 no 6 pp 595ndash605 1995

[15] F Pellet and P Egger ldquoAnalytical model for the mechanicalbehaviour of bolted rock joints subjected to shearingrdquo RockMechanics and Rock Engineering vol 29 no 2 pp 73ndash97 1996

[16] H Jalalifar and N Aziz ldquoExperimental and 3D numericalsimulation of reinforced shear jointsrdquo RockMechanics and RockEngineering vol 43 no 1 pp 95ndash103 2010

[17] P Wang T Feng Y-J Zhu and W-J Yu ldquoExperimental studyand numerical simulation of anchoringmechanismof anchoredrocklikematerial with prefabricated fracturerdquoYantu LixueRockand Soil Mechanics vol 37 no 3 pp 793ndash801 2016 (Chinese)

[18] L Li P C Hagan S Saydam B Hebblewhite and Y Li ldquoPara-metric study of rockbolt shear behaviour by double shear testrdquoRock Mechanics and Rock Engineering vol 49 no 12 pp 4787ndash4797 2016

[19] Itasca Consulting Group Inc ldquoPFC2D (Particle Flow Code in2D)Theory and Backgroundrdquo Minneapolis Minn USA 2008

[20] P A Cundall andOD L Strack ldquoThe distinct numericalmodelfor granular assembliesrdquo Geotechnique vol 29 no 1 pp 47ndash651979

[21] J Zhou and Y Chi ldquoThe method of particle flow and PFC2Dcoderdquo Rock and Soil Mechanics vol 21 no 3 pp 271ndash274 2000(Chinese)

[22] D O Potyondy and P A Cundall ldquoA bonded-particle modelfor rockrdquo International Journal of Rock Mechanics and MiningSciences vol 41 no 8 pp 1329ndash1364 2004

[23] N Cho CDMartin andDC Sego ldquoA clumped particlemodelfor rockrdquo International Journal of Rock Mechanics and MiningSciences vol 44 no 7 pp 997ndash1010 2007

[24] Z Zhao L Jing and I Neretnieks ldquoParticle mechanics modelfor the effects of shear on solute retardation coefficient in rockfracturesrdquo International Journal of Rock Mechanics and MiningSciences vol 52 pp 92ndash102 2012

[25] C Xia Y Song Z Tang Y Song and C Shou ldquoParticleflow numerical simulation for shear behavior of rough jointsrdquoChinese Journal of Rock Mechanics and Engineering vol 31 no8 pp 1545ndash1552 2012 (Chinese)

[26] Y Zhou A Misra S Wu and X Zhang ldquoMacro- and meso-analyses of rock joint direct shear test using particle flowtheoryrdquoChinese Journal of RockMechanics and Engineering vol31 no 6 pp 1245ndash1256 2012

[27] RH Cao P CaoH Lin K Zhang andXWTan ldquoParticle flowanalysis of direct shear tests on joints with different roughnessrdquoRock and Soil Mechanics vol 34 supplement 2 pp 456ndash4642013 (Chinese)

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Journal of

NanotechnologyHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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CrystallographyJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CoatingsJournal of

Advances in

Materials Science and EngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Smart Materials Research

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MetallurgyJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

BioMed Research International

MaterialsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Advances in Materials Science and Engineering 7

1

3

2

5

48

67

1

3

22

5

6

1

] ]

n

(a) Numerical model of a bolted joint at 45∘

1

3

2

5

48

67

1

3

2

5

6

1

] ]

n

(b) Numerical model of a bolted joint at 75∘

Figure 6 Numerical direct shear test model of the bolted joint

15∘

30∘

45∘

60∘

75∘

90∘

0

2

4

6

8

Shea

r stre

ss (M

Pa)

05 10 15 20 2500

Shear displacement (mm)

Figure 7 Evolution of the shear stress for different anchorage angles

4 Analysis of the Results

41 Analysis of the Shear Performance of the Joint withDifferent Anchorage Angles Direct shear simulations of thebolted joint were conducted using the bolt at differentanchorage angles under a constant normal stress to examinethe macro-micromechanical responses of the bolted jointduring shearing Figure 7 shows the evolution of the shearstress with the shear displacement at different anchorageangles The shear simulation at a normal stress of 5MPa isachieved after a shear displacement of 25mm is reachedThepeak shear strength for different anchorage angles is shownin Table 3

Our data show that (1) the peak shear strength increasedto a maximum and then decreased with further increase inthe anchorage angle and (2) the peak shear strength is thelargest at an anchorage angle of 75∘ Compared to the 90∘anchorage angle changing the angle of the anchor can greatlyimprove the boltrsquos antishear abilitiesTherefore it is importantto determine the optimum design for a bolted rock joint inpractice

Table 3 Peak shear strength for different anchorage angles

Anchor angle The peak shear strength (MPa)15∘ 49830∘ 60045∘ 67860∘ 65275∘ 76590∘ 498

42 Analysis of the Bolt Stress in a Compression-ShearStress State

421 Stress Analysis of the Anchor Joint Plane in a Compres-sion-Shear Stress State Under compressive shear stress therock bolts experience not only shear deformation throughthe thickness of the weak plane but also notable shear defor-mation over a considerable section (the effective length) Tosimplify the analysis we set the effective length of the rock

8 Advances in Materials Science and Engineering

Joint surface Joint surface

b

b

ln ltn

tb

Figure 8 Deformations of the joint surface and bolt

b b blt

b2

(a) Shear force

b b bln

b2

(b) Axial force

Figure 9 Axial and shearing forces on the bolt

bolt to be 119897119905 and the effective length of the bolt to be 119897119899 asshown in Figure 8

The average shear stress distribution on the plane of the119897119905 section is related to the contact conditions of the rodbody and hole wall Three different distribution patternsfor the simplified calculation are shown in Figure 9(a) Thedistribution pattern is a curve but can be measured inthree basic forms Similar to the 119897119899 section the axial stressdistribution can be used as shown in Figure 9(b)

From Figures 8 and 9 and from the joint surface equi-librium conditions the stress tensor of the rock mass can beobtained from the normal and tangential components of thestress tensor along the joint plane as shown in

120590 = 120590119904 + 120590119887119899= 119896119899 (119880119899 sin 119886 minus 119880119905 cos 119886) + 119875119866119887119880119905119860119861119905119897119905 cos 119886

minus 119875119864119887119880119899119861119899119897119899 sin 119886

120591 = 120591119887119904= 119896119904 (119880119899 cos 119886 + 119880119905 sin 119886) + 119875119864119887119880119899119861119899119897119899 cos 119886

+ 119875119866119887119880119905119860119861119905119897119905 sin 119886

+ [119875119866119887119880119905119860119861119905119897119905 cos 119886 minus119875119864119887119880119899119861119899119897119899 sin 119886]119891119904

(8)

In the previous formula 120575119905 and 120575119899 are the weak levels oftangential and normal displacement respectively 119880119905 and 119880119899are the bolt axial and transverse deformations respectively119886 is the angle between the bolt and joint surface and 119864119887 and119866119887 are Youngrsquos modulus and shear modulus for the materialrespectivelyThe shear coefficient119860 is associated with the rodbody section (119860 = 34 for a circular solid rod) 119861119905 and 119861119899 arethe inner pole sectionsrsquo average shear stress and axial stressdistribution of the shape coefficient respectively which are1 34 and 12 120590119887 and 120591119887 are the axial and shear forces on

Advances in Materials Science and Engineering 9

the bolt respectively 119896119899 and 119896119904 are the normal and tangentialstiffness coefficients for the weak level119875 is the anchor sectionreinforcement ratio Therefore

119880119899 = 1119886 (11988622120590 minus 11988612120591)

119880119905 = 1119886 (11988611120591 minus 11988621120590)

(9)

where

11988611 = minus(119896119899 + 119875119864119887119861119899119897119899) sin 119886

11988612 = (119896119899 + 119875119866119887119860119861119905119897119905) cos 119886

11988621 = 119896119904 cos 119886 + 119875119864119887119861119899119897119899 (cos 119886 minus 119891119904 sin 119886)

11988622 = 119896119904 sin 119886 + 119875119866119887119860119861119905119897119905 (sin 119886 + 119891119904 cos 119886)

119886 = 1198861111988622 minus 1198861211988621= 119896119899119896119904 minus 119875119864119887

119861119899119897119899119875119866119887119860119861119905119897119905 +

119875119866119887119860119861119905119897119905 (119896119899sin

2119886 minus 119896119904cos2119886)

+ 119875119864119887119861119899119897119899 [119896119899cos

2119886 minus 119896119904sin2119886]

+ 119896119899119891119904 sin 119886 cos 119886 [ 119875119866119887119860119861119905119897119905 minus119875119864119887119861119899119897119899 ]

(10)

Therefore the relationship between the stress and thestress component of the element can be determined as shownin

120590119887 = 119864119887119861119899119897119899119886 (11988622120590 minus 11988612120591)

120591119887 = 119866119887119860119861119905119897119905119886 (11988611120591 minus 11988621120590)

(11)

422 Analysis of the Bolt Stress for a Particular AnchorageAngle A bolt installed at a particular anchorage angle willhave a better anchoring effect than a bolt installed perpen-dicular to the joint plane Figures 10 and 11 show illustrationsof the stresses in a bolt at anchorage angles of 75∘ and 90∘respectively during shearing

The diagram showing the two different anchorage anglebolts indicates that the anchor force and bending momentare relatively large in the vertically installed bolt leading tobolt axial and shear stress that are larger than those in thetilted bolt whereas for the tilted bolt the axial and shearstress are small and do not reach the yield strengthThereforethe tilted anchor bolts not only improve the shear capacityof the joints but also maintain an appropriate stress statein the anchor The rock bolt axial stress diagram indicatesthat the axial stress increases as the shear displacementincreases for different anchorage anglesThe axial stress is the

highest when the anchor is vertically installed on the jointsurface The results of these two cases reveal that the optimalanchoring angle is approximately 75∘This result correspondsto the previous conclusions that the optimal anchoring angleshould be between 45∘ and 75∘

43 Evolution of the Contact Force and Microscopic Crackingof Disks The anchorage angle of 75∘ was chosen for amore detailed analysis on the micromechanical response anddamage mechanism during shearing Figure 12 shows theevolution of the shear stress number of cracks and normaldisplacement for a normal stress of 5MPa

To accurately determine the evolution features six differ-ent monitoring points (a to f) were set on the shear stresscurve which correspond to shear displacements of 0 01980217 0513 1500 and 2500mm respectively The variationsin the contact force and cracking can be observed from thesix monitoring points As shown in Figure 12 ldquoardquo indicatesthe initial point of the shear displacement and ldquocrdquo indicatesthe shear displacement at which the shear stress reachesthe peak shear strength As shown in Figure 12 no shearcrack is generated during the initial stage of the shear testThe number of shear cracks increases gradually as the sheardisplacement increases When the shear stress reaches thepeak shear strength the number of shear cracks increasessignificantly and the shear crack growth rate tends to stabilizeas the shear stress increases beyond the peak shear strengthThe trends in tension cracking and normal stress are highlysimilar and increase monotonically

During the shearing of a bolted joint the loaded side ofthe bolt is in a state of compression whereas the other side ofthe bolt is in a state of tension due to the bending deflectionof the bolt The adjacent grout is in a state of compressionand the rockmass is in a state of tension because of the bondsbetween the bolt and grout as shown in Figure 13 Bendingdeflection will occur in the bolt as the shear displacementincreases Simultaneously cracks will occur continuously dueto extrusion and slippage between disks In the compressionzone there are a large number of cracks which are mainlytension cracks generated by the extrusion between disksThisresult agreeswith the formationmechanism for compression-induced tension cracks shown in Figure 13

Figure 14 shows the illustration of the contact forceand crack propagation between disks during shearing Thecontact force is initially concentrated on the joint and theloading end and cracks are generated in the joint whereasthere are only a few cracks around the bolt As the sheardisplacement increases the diskrsquos contact force around thebolt becomes gradually larger Cracks occur on the jointand those around the bolt extend further The cracks aroundthe bolt propagate most rapidly The cracks are initiallyconcentrated at the intersection of the bolt and joint and thenpropagate to the ends of the bolt Finally the majority of thecracks are distributed at the joint and around the bolt aftera larger displacement occurs Finally more cracks occur inthe compression zone around the bolt than in the tensionzone In addition the distribution of cracks corresponds tothe contact force in the stress concentration areaThe detailed

10 Advances in Materials Science and Engineering

0 10 20 30 40

Distance along bolt (mm)

Joint plane

Bolt

05 mm10 mm15 mm

20 mm25 mm30 mm

0

100

200

300

Axi

al st

ress

(MPa

)

(a) Axial stress diagram

0 10 20 30 40

Distance along bolt (mm)

05 mm10 mm15 mm

20 mm25 mm30 mm

minus600

minus400

minus200

0

200

400

600

Bend

ing

mom

ent (

Nm

)

Joint plane

Bolt

(b) Bending moment diagram

Joint plane

Bolt

0 10 20 30 40

Distance along bolt (mm)

05 mm10 mm15 mm

20 mm25 mm30 mm

0

10

20

30

40

Shea

r stre

ss (M

Pa)

(c) Shear stress diagram

Figure 10 Bolt stress state diagram at an anchorage angle of 75∘

maximum contact force and crack number in the specimen atthe different monitoring points during shearing are providedin Tables 4 and 5 respectively

44 Evolution of the Disk Rotation During the shearing slipprocess the behavior is determined by the rotation of thedisks Relative disk dislocations will occur due to the rotationof the disks If the rotation angle is sufficiently large itwill lead to shear stresses and the breaking of disk bondsMeanwhile cracks occurring in the rock mass and shearcracking are caused directly by disk rotation Figure 15 showsa graph of the evolution of the disk rotation in terms ofradians for rock disks at different monitoring points

Figure 15 shows that when the shear displacement issmall at the initial stages the disk rotation angle is relatively

small and disks with large rotational angles are distributedaround the bolt The disk rotation angles gradually increasewith increasing shear displacement The disks containinglarger rotation angles are mainly distributed in the twoloading ends and around the bolt and the bolt and thedisks around the bolt contact move relatively violently duringshearing leading to a larger rotation angle Further increasesin the shear displacement and larger rotation disks alsoappeared on the joint surfaces which are caused by thejoint surface microconvex body cutting into the free disksThe distribution of the large disk rotation angles is highlyconsistent with the crack distribution demonstrating that therotation of the disks leads to the generation of shear cracks

Figure 16 shows the variations in the proportion ofrotations exceeding 01 radians As the shear displacement

Advances in Materials Science and Engineering 11

Bolt

Joint plane

0

50

100

150

200

250A

xial

stre

ss (M

Pa)

10 20 30 400Distance along bolt (mm)

05 mm10 mm15 mm

20 mm25 mm30 mm

(a) Axial stress diagram

Joint plane

Bolt

10 20 30 400Distance along bolt (mm)

minus800

minus600

minus400

minus200

0

200

400

600

800

Bend

ing

mom

ent (

Nm

)

05 mm10 mm15 mm

20 mm25 mm30 mm

(b) Bending moment diagram

Bolt

Joint plane

0

10

20

30

40

Shea

r stre

ss (M

Pa)

10 20 30 400Distance along bolt (mm)

05 mm10 mm15 mm

20 mm25 mm30 mm

(c) Shear stress diagram

Figure 11 Bolt stress state diagram at an anchorage angle of 90∘

Table 4 Maximum contact force in the specimen at different monitoring points

Monitoring point Maximum contact forceN Monitoring point Maximum contact forceNa 68201198904 d 12681198905b 15301198905 e 15801198905c 15481198905 f 14451198905

increases the number of clockwise and counterclockwisedisk rotations greater than 01 radians increases continuouslyAdditionally the number of clockwise rotations is greaterthan the number of counterclockwise rotations mainly dueto the shear loading directions

5 Conclusions

In this paper the anchoring mechanism for a tilted anchorbolt subjected to a shear load has been studied using a bilinearconstitutive model with the inner-embedded FISH language

12 Advances in Materials Science and Engineering

(f)(e)(d)

(c)

(b)

(a)

Shear stressNormal displacement

Tension crackShear crack

0

2

4

6

8

Shea

r stre

ss (M

Pa)

05 10 15 20 2500Shear displacement (mm)

00

01

02

03

04

05

06

07

08

Nor

mal

disp

lace

men

t (times10

minus3

m)

0

100

200

300

400

500

600

Crac

k nu

mbe

r

Figure 12 Evolution of the shear stress number of cracks and normal displacement

Rock

Bolt Grout

T

C

C

T

(a)

F F F

F F F

(b)

Figure 13 Stress state of the bolted rock joint system and illustration of the mechanism behind compression-induced tension cracks

Table 5 Number of cracks in the specimen at different monitoringpoints

Monitoringpoint Crack number Monitoring point Crack number

a 0 d 255b 21 e 446c 40 f 592

of PFC based on DEMThe anchoring mechanism of the boltatdifferentanchorage angles has been discussed at themacro-microlevels and the following conclusions have been drawn

(1) The bolt can improve the shear resistance of the jointmainly due to cohesion Tilted anchor bolts not only improvethe shear strength of the joint surface but also result in a lowerstress in the anchor bolt The peak shear strength can be 50greater than that of a vertically installed anchor bolt

(2) A tilted bolt allows for fully mobilized anchor perfor-manceThis configuration not only restricts increment in thenormal displacement but also indirectly improves the stress

state of the anchorage system and increases the shear capacityof the anchor joint The comprehensive shear displacementand shear stress data along with the bolt stress state analysisreveal that the optimal anchorage angle should be between45∘ and 75∘

(3) During the shearing process when the rock bolt andgrout are commonly constrained the stress concentrationphenomenon around the bolt and joint increases the sheardisplacement and contact force After reaching the residualphase the contact force exhibits slight fluctuations due to thesliding of the joint surface

(4) Based on the disk flow software PFC2D crack initia-tion for an anchorage angle of 75∘ will start from the contactforce concentration due to the interactions between the rockgrout and bolt Finally the cracks are mainly distributedin the joint and around the bolt for larger displacementsThe cracks are mainly tensile cracks which correspond tothe formation mechanism for a compression-induced tensilecrack The shear cracking was mainly caused by the rotationof the disks during the shearing process

Advances in Materials Science and Engineering 13

minus005 minus003 minus001 001 003 005

minus002minus001

0001002

minus551525354555

Con

tact

forc

e (N

)

times103

(a) Monitoring point a

minus002minus001

0001002

Con

tact

forc

e (N

)

minus10

10

30

50

70

90

110times103

minus005 minus003 minus001 001 003 005

(b) Monitoring point b

minus002minus001

0001002

Con

tact

forc

e (N

)minus101030507090110

times103

minus005 minus003 minus001 001 003 005

(c) Monitoring point c

minus002minus001

0001002

Con

tact

forc

e (N

)

minus10

10

30

50

70

90

110times103

minus005 minus003 minus001 001 003 005

(d) Monitoring point d

minus002minus001

0001002

minus51535557595115

Con

tact

forc

e (N

)

times103

minus005 minus003 minus001 001 003 005

(e) Monitoring point e

minus002minus001

0001002

minus10

10

30

50

70

90

Con

tact

forc

e (N

)

times103

minus005 minus003 minus001 001 003 005

(f) Monitoring point f

Figure 14 Evolution of the contact force and cracking at different monitoring points

14 Advances in Materials Science and Engineering

Particle rotation (in radians)minus120 to minus01

minus01 to minus0001

minus0001 to minus00005

minus00005 to 00005

00005 to 00010001 to 01

01 to 120

(a) Monitoring point a

Particle rotation (in radians)minus120 to minus01

minus01 to minus0001

minus0001 to minus00005

00005 to 00010001 to 01

01 to 120minus00005 to 00005

(b) Monitoring point b

Particle rotation (in radians)minus120 to minus01

minus01 to minus0001

minus0001 to minus00005

00005 to 00010001 to 01

01 to 120minus00005 to 00005

(c) Monitoring point c

Particle rotation (in radians)minus120 to minus01

minus01 to minus0001

minus0001 to minus00005

00005 to 00010001 to 01

01 to 120minus00005 to 00005

(d) Monitoring point d

Particle rotation (in radians)minus120 to minus01

minus01 to minus0001

minus0001 to minus00005

00005 to 00010001 to 01

01 to 120minus00005 to 00005

(e) Monitoring point e

Particle rotation (in radians)minus120 to minus01

minus01 to minus0001

minus0001 to minus00005

00005 to 00010001 to 01

01 to 120minus00005 to 00005

(f) Monitoring point f

Figure 15 Evolution of disk rotation in radians in the specimen at different monitoring points

0

5

10

15

20

Qua

ntita

tive p

ropo

rtio

n (

)

05 10 15 20 25 3000Shear displacement (mm)

gt 01Rotation (in radians)gt 01|Rotation (in radians)|

lt minus01Rotation (in radians)

Figure 16 Quantitative proportion variation of rotations exceeding 01 radians

Disclosure

GangWang and Yongzheng Zhang should be regarded as co-first authors

Conflicts of Interest

The authors declare that there are no conflicts of interestregarding the publication of this paper

Advances in Materials Science and Engineering 15

Acknowledgments

This study was supported by the National Natural ScienceFoundation of China (nos 51479108 51379117 and 51509261)China Postdoctoral Science Foundation (2015M580599) theTaishan Scholar Talent Team Support Plan for Advantagedamp Unique Discipline Areas the Postdoctoral Science Foun-dation of Shandong Province (no 201402014) and the Shan-dong Province Outstanding Young Scientist Research AwardFund (BS2014NJ006)

References

[1] F D Patton ldquoMultiple modes of shear failure in rockrdquo inProceedings of the 1st Congress on International Society for RockMechanics pp 509ndash513 1966

[2] R E Goodman Methods of Geological Engineering in Discon-tinuous Rocks West Publishing Company New York NY USA1976

[3] N Barton and V Choubey ldquoThe shear strength of rock joints intheory and practicerdquo Rock Mechanics Felsmechanik Mecaniquedes Roches vol 10 no 1-2 pp 1ndash54 1977

[4] P M Dight and H K Chiu ldquoPrediction of shear behavior ofjoints using profilesrdquo International Journal of Rock Mechanicsand Mining Sciences amp Geomechanics Abstracts vol 18 no 5pp 369ndash386 1981

[5] N Barton Modelling Rock Joint Behaviour from In Situ BlockTests Implications for NuclearWaste Repository Design vol 308Office ofNuclearWaste IsolationONWI ColumbusOHUSA1982

[6] T Esaki S Du Y Jiang and Y Wada ldquoEffect of the asperitydamage on the shear behavior of irregular rock jointsrdquo inProceedings of the Second International Conference on Analysisof Discontinuous Deformation pp 459ndash464 Kyoto Japan 1997

[7] Y Jiang J Xiao Y Tanabashi and T Mizokami ldquoDevelopmentof an automated servo-controlled direct shear apparatus apply-ing a constant normal stiffness conditionrdquo International Journalof Rock Mechanics and Mining Sciences vol 41 no 2 pp 275ndash286 2004

[8] Y J Jiang B Li and Y Tanabashi ldquoEstimating the relationbetween surface roughness and mechanical properties of rockjointsrdquo International Journal of Rock Mechanics and MiningSciences vol 43 no 6 pp 837ndash846 2006

[9] C-C Xia Z-C Tang W-M Xiao and Y-L Song ldquoNew peakshear strength criterion of rock joints based on quantifiedsurface descriptionrdquo Rock Mechanics and Rock Engineering vol47 no 2 pp 387ndash400 2014

[10] S Bjurstrom ldquoShear strength of hard rock joints reinforced bygrouted untensioned boltsrdquo in Proceedings of the In Proceedingsof the 3rd International ISRM Congress pp 1194ndash1199 DenverColo USA 1974

[11] X Ge and J Liu ldquoStudy of the shear resistance behaviour ofbolted rock jointsrdquo Journal of Geotechnical Engineering vol 10no 1 pp 8ndash19 1988 (Chinese)

[12] K Spang and P Egger ldquoAction of fully-grouted bolts in jointedrock and factors of influencerdquo Rock Mechanics and Rock Engi-neering vol 23 no 3 pp 201ndash229 1990

[13] P Egger and L Zabuski ldquoBehaviour of rough bolted joints indirect shear testsrdquo in Proceedings of the 7th ISRM Congress vol30 p 342 Aachen Germany September 1991

[14] A M Ferrero ldquoThe shear strength of reinforced rock jointsrdquoInternational Journal of Rock Mechanics and Mining Sciencesand vol 32 no 6 pp 595ndash605 1995

[15] F Pellet and P Egger ldquoAnalytical model for the mechanicalbehaviour of bolted rock joints subjected to shearingrdquo RockMechanics and Rock Engineering vol 29 no 2 pp 73ndash97 1996

[16] H Jalalifar and N Aziz ldquoExperimental and 3D numericalsimulation of reinforced shear jointsrdquo RockMechanics and RockEngineering vol 43 no 1 pp 95ndash103 2010

[17] P Wang T Feng Y-J Zhu and W-J Yu ldquoExperimental studyand numerical simulation of anchoringmechanismof anchoredrocklikematerial with prefabricated fracturerdquoYantu LixueRockand Soil Mechanics vol 37 no 3 pp 793ndash801 2016 (Chinese)

[18] L Li P C Hagan S Saydam B Hebblewhite and Y Li ldquoPara-metric study of rockbolt shear behaviour by double shear testrdquoRock Mechanics and Rock Engineering vol 49 no 12 pp 4787ndash4797 2016

[19] Itasca Consulting Group Inc ldquoPFC2D (Particle Flow Code in2D)Theory and Backgroundrdquo Minneapolis Minn USA 2008

[20] P A Cundall andOD L Strack ldquoThe distinct numericalmodelfor granular assembliesrdquo Geotechnique vol 29 no 1 pp 47ndash651979

[21] J Zhou and Y Chi ldquoThe method of particle flow and PFC2Dcoderdquo Rock and Soil Mechanics vol 21 no 3 pp 271ndash274 2000(Chinese)

[22] D O Potyondy and P A Cundall ldquoA bonded-particle modelfor rockrdquo International Journal of Rock Mechanics and MiningSciences vol 41 no 8 pp 1329ndash1364 2004

[23] N Cho CDMartin andDC Sego ldquoA clumped particlemodelfor rockrdquo International Journal of Rock Mechanics and MiningSciences vol 44 no 7 pp 997ndash1010 2007

[24] Z Zhao L Jing and I Neretnieks ldquoParticle mechanics modelfor the effects of shear on solute retardation coefficient in rockfracturesrdquo International Journal of Rock Mechanics and MiningSciences vol 52 pp 92ndash102 2012

[25] C Xia Y Song Z Tang Y Song and C Shou ldquoParticleflow numerical simulation for shear behavior of rough jointsrdquoChinese Journal of Rock Mechanics and Engineering vol 31 no8 pp 1545ndash1552 2012 (Chinese)

[26] Y Zhou A Misra S Wu and X Zhang ldquoMacro- and meso-analyses of rock joint direct shear test using particle flowtheoryrdquoChinese Journal of RockMechanics and Engineering vol31 no 6 pp 1245ndash1256 2012

[27] RH Cao P CaoH Lin K Zhang andXWTan ldquoParticle flowanalysis of direct shear tests on joints with different roughnessrdquoRock and Soil Mechanics vol 34 supplement 2 pp 456ndash4642013 (Chinese)

Submit your manuscripts athttpswwwhindawicom

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MaterialsJournal of

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8 Advances in Materials Science and Engineering

Joint surface Joint surface

b

b

ln ltn

tb

Figure 8 Deformations of the joint surface and bolt

b b blt

b2

(a) Shear force

b b bln

b2

(b) Axial force

Figure 9 Axial and shearing forces on the bolt

bolt to be 119897119905 and the effective length of the bolt to be 119897119899 asshown in Figure 8

The average shear stress distribution on the plane of the119897119905 section is related to the contact conditions of the rodbody and hole wall Three different distribution patternsfor the simplified calculation are shown in Figure 9(a) Thedistribution pattern is a curve but can be measured inthree basic forms Similar to the 119897119899 section the axial stressdistribution can be used as shown in Figure 9(b)

From Figures 8 and 9 and from the joint surface equi-librium conditions the stress tensor of the rock mass can beobtained from the normal and tangential components of thestress tensor along the joint plane as shown in

120590 = 120590119904 + 120590119887119899= 119896119899 (119880119899 sin 119886 minus 119880119905 cos 119886) + 119875119866119887119880119905119860119861119905119897119905 cos 119886

minus 119875119864119887119880119899119861119899119897119899 sin 119886

120591 = 120591119887119904= 119896119904 (119880119899 cos 119886 + 119880119905 sin 119886) + 119875119864119887119880119899119861119899119897119899 cos 119886

+ 119875119866119887119880119905119860119861119905119897119905 sin 119886

+ [119875119866119887119880119905119860119861119905119897119905 cos 119886 minus119875119864119887119880119899119861119899119897119899 sin 119886]119891119904

(8)

In the previous formula 120575119905 and 120575119899 are the weak levels oftangential and normal displacement respectively 119880119905 and 119880119899are the bolt axial and transverse deformations respectively119886 is the angle between the bolt and joint surface and 119864119887 and119866119887 are Youngrsquos modulus and shear modulus for the materialrespectivelyThe shear coefficient119860 is associated with the rodbody section (119860 = 34 for a circular solid rod) 119861119905 and 119861119899 arethe inner pole sectionsrsquo average shear stress and axial stressdistribution of the shape coefficient respectively which are1 34 and 12 120590119887 and 120591119887 are the axial and shear forces on

Advances in Materials Science and Engineering 9

the bolt respectively 119896119899 and 119896119904 are the normal and tangentialstiffness coefficients for the weak level119875 is the anchor sectionreinforcement ratio Therefore

119880119899 = 1119886 (11988622120590 minus 11988612120591)

119880119905 = 1119886 (11988611120591 minus 11988621120590)

(9)

where

11988611 = minus(119896119899 + 119875119864119887119861119899119897119899) sin 119886

11988612 = (119896119899 + 119875119866119887119860119861119905119897119905) cos 119886

11988621 = 119896119904 cos 119886 + 119875119864119887119861119899119897119899 (cos 119886 minus 119891119904 sin 119886)

11988622 = 119896119904 sin 119886 + 119875119866119887119860119861119905119897119905 (sin 119886 + 119891119904 cos 119886)

119886 = 1198861111988622 minus 1198861211988621= 119896119899119896119904 minus 119875119864119887

119861119899119897119899119875119866119887119860119861119905119897119905 +

119875119866119887119860119861119905119897119905 (119896119899sin

2119886 minus 119896119904cos2119886)

+ 119875119864119887119861119899119897119899 [119896119899cos

2119886 minus 119896119904sin2119886]

+ 119896119899119891119904 sin 119886 cos 119886 [ 119875119866119887119860119861119905119897119905 minus119875119864119887119861119899119897119899 ]

(10)

Therefore the relationship between the stress and thestress component of the element can be determined as shownin

120590119887 = 119864119887119861119899119897119899119886 (11988622120590 minus 11988612120591)

120591119887 = 119866119887119860119861119905119897119905119886 (11988611120591 minus 11988621120590)

(11)

422 Analysis of the Bolt Stress for a Particular AnchorageAngle A bolt installed at a particular anchorage angle willhave a better anchoring effect than a bolt installed perpen-dicular to the joint plane Figures 10 and 11 show illustrationsof the stresses in a bolt at anchorage angles of 75∘ and 90∘respectively during shearing

The diagram showing the two different anchorage anglebolts indicates that the anchor force and bending momentare relatively large in the vertically installed bolt leading tobolt axial and shear stress that are larger than those in thetilted bolt whereas for the tilted bolt the axial and shearstress are small and do not reach the yield strengthThereforethe tilted anchor bolts not only improve the shear capacityof the joints but also maintain an appropriate stress statein the anchor The rock bolt axial stress diagram indicatesthat the axial stress increases as the shear displacementincreases for different anchorage anglesThe axial stress is the

highest when the anchor is vertically installed on the jointsurface The results of these two cases reveal that the optimalanchoring angle is approximately 75∘This result correspondsto the previous conclusions that the optimal anchoring angleshould be between 45∘ and 75∘

43 Evolution of the Contact Force and Microscopic Crackingof Disks The anchorage angle of 75∘ was chosen for amore detailed analysis on the micromechanical response anddamage mechanism during shearing Figure 12 shows theevolution of the shear stress number of cracks and normaldisplacement for a normal stress of 5MPa

To accurately determine the evolution features six differ-ent monitoring points (a to f) were set on the shear stresscurve which correspond to shear displacements of 0 01980217 0513 1500 and 2500mm respectively The variationsin the contact force and cracking can be observed from thesix monitoring points As shown in Figure 12 ldquoardquo indicatesthe initial point of the shear displacement and ldquocrdquo indicatesthe shear displacement at which the shear stress reachesthe peak shear strength As shown in Figure 12 no shearcrack is generated during the initial stage of the shear testThe number of shear cracks increases gradually as the sheardisplacement increases When the shear stress reaches thepeak shear strength the number of shear cracks increasessignificantly and the shear crack growth rate tends to stabilizeas the shear stress increases beyond the peak shear strengthThe trends in tension cracking and normal stress are highlysimilar and increase monotonically

During the shearing of a bolted joint the loaded side ofthe bolt is in a state of compression whereas the other side ofthe bolt is in a state of tension due to the bending deflectionof the bolt The adjacent grout is in a state of compressionand the rockmass is in a state of tension because of the bondsbetween the bolt and grout as shown in Figure 13 Bendingdeflection will occur in the bolt as the shear displacementincreases Simultaneously cracks will occur continuously dueto extrusion and slippage between disks In the compressionzone there are a large number of cracks which are mainlytension cracks generated by the extrusion between disksThisresult agreeswith the formationmechanism for compression-induced tension cracks shown in Figure 13

Figure 14 shows the illustration of the contact forceand crack propagation between disks during shearing Thecontact force is initially concentrated on the joint and theloading end and cracks are generated in the joint whereasthere are only a few cracks around the bolt As the sheardisplacement increases the diskrsquos contact force around thebolt becomes gradually larger Cracks occur on the jointand those around the bolt extend further The cracks aroundthe bolt propagate most rapidly The cracks are initiallyconcentrated at the intersection of the bolt and joint and thenpropagate to the ends of the bolt Finally the majority of thecracks are distributed at the joint and around the bolt aftera larger displacement occurs Finally more cracks occur inthe compression zone around the bolt than in the tensionzone In addition the distribution of cracks corresponds tothe contact force in the stress concentration areaThe detailed

10 Advances in Materials Science and Engineering

0 10 20 30 40

Distance along bolt (mm)

Joint plane

Bolt

05 mm10 mm15 mm

20 mm25 mm30 mm

0

100

200

300

Axi

al st

ress

(MPa

)

(a) Axial stress diagram

0 10 20 30 40

Distance along bolt (mm)

05 mm10 mm15 mm

20 mm25 mm30 mm

minus600

minus400

minus200

0

200

400

600

Bend

ing

mom

ent (

Nm

)

Joint plane

Bolt

(b) Bending moment diagram

Joint plane

Bolt

0 10 20 30 40

Distance along bolt (mm)

05 mm10 mm15 mm

20 mm25 mm30 mm

0

10

20

30

40

Shea

r stre

ss (M

Pa)

(c) Shear stress diagram

Figure 10 Bolt stress state diagram at an anchorage angle of 75∘

maximum contact force and crack number in the specimen atthe different monitoring points during shearing are providedin Tables 4 and 5 respectively

44 Evolution of the Disk Rotation During the shearing slipprocess the behavior is determined by the rotation of thedisks Relative disk dislocations will occur due to the rotationof the disks If the rotation angle is sufficiently large itwill lead to shear stresses and the breaking of disk bondsMeanwhile cracks occurring in the rock mass and shearcracking are caused directly by disk rotation Figure 15 showsa graph of the evolution of the disk rotation in terms ofradians for rock disks at different monitoring points

Figure 15 shows that when the shear displacement issmall at the initial stages the disk rotation angle is relatively

small and disks with large rotational angles are distributedaround the bolt The disk rotation angles gradually increasewith increasing shear displacement The disks containinglarger rotation angles are mainly distributed in the twoloading ends and around the bolt and the bolt and thedisks around the bolt contact move relatively violently duringshearing leading to a larger rotation angle Further increasesin the shear displacement and larger rotation disks alsoappeared on the joint surfaces which are caused by thejoint surface microconvex body cutting into the free disksThe distribution of the large disk rotation angles is highlyconsistent with the crack distribution demonstrating that therotation of the disks leads to the generation of shear cracks

Figure 16 shows the variations in the proportion ofrotations exceeding 01 radians As the shear displacement

Advances in Materials Science and Engineering 11

Bolt

Joint plane

0

50

100

150

200

250A

xial

stre

ss (M

Pa)

10 20 30 400Distance along bolt (mm)

05 mm10 mm15 mm

20 mm25 mm30 mm

(a) Axial stress diagram

Joint plane

Bolt

10 20 30 400Distance along bolt (mm)

minus800

minus600

minus400

minus200

0

200

400

600

800

Bend

ing

mom

ent (

Nm

)

05 mm10 mm15 mm

20 mm25 mm30 mm

(b) Bending moment diagram

Bolt

Joint plane

0

10

20

30

40

Shea

r stre

ss (M

Pa)

10 20 30 400Distance along bolt (mm)

05 mm10 mm15 mm

20 mm25 mm30 mm

(c) Shear stress diagram

Figure 11 Bolt stress state diagram at an anchorage angle of 90∘

Table 4 Maximum contact force in the specimen at different monitoring points

Monitoring point Maximum contact forceN Monitoring point Maximum contact forceNa 68201198904 d 12681198905b 15301198905 e 15801198905c 15481198905 f 14451198905

increases the number of clockwise and counterclockwisedisk rotations greater than 01 radians increases continuouslyAdditionally the number of clockwise rotations is greaterthan the number of counterclockwise rotations mainly dueto the shear loading directions

5 Conclusions

In this paper the anchoring mechanism for a tilted anchorbolt subjected to a shear load has been studied using a bilinearconstitutive model with the inner-embedded FISH language

12 Advances in Materials Science and Engineering

(f)(e)(d)

(c)

(b)

(a)

Shear stressNormal displacement

Tension crackShear crack

0

2

4

6

8

Shea

r stre

ss (M

Pa)

05 10 15 20 2500Shear displacement (mm)

00

01

02

03

04

05

06

07

08

Nor

mal

disp

lace

men

t (times10

minus3

m)

0

100

200

300

400

500

600

Crac

k nu

mbe

r

Figure 12 Evolution of the shear stress number of cracks and normal displacement

Rock

Bolt Grout

T

C

C

T

(a)

F F F

F F F

(b)

Figure 13 Stress state of the bolted rock joint system and illustration of the mechanism behind compression-induced tension cracks

Table 5 Number of cracks in the specimen at different monitoringpoints

Monitoringpoint Crack number Monitoring point Crack number

a 0 d 255b 21 e 446c 40 f 592

of PFC based on DEMThe anchoring mechanism of the boltatdifferentanchorage angles has been discussed at themacro-microlevels and the following conclusions have been drawn

(1) The bolt can improve the shear resistance of the jointmainly due to cohesion Tilted anchor bolts not only improvethe shear strength of the joint surface but also result in a lowerstress in the anchor bolt The peak shear strength can be 50greater than that of a vertically installed anchor bolt

(2) A tilted bolt allows for fully mobilized anchor perfor-manceThis configuration not only restricts increment in thenormal displacement but also indirectly improves the stress

state of the anchorage system and increases the shear capacityof the anchor joint The comprehensive shear displacementand shear stress data along with the bolt stress state analysisreveal that the optimal anchorage angle should be between45∘ and 75∘

(3) During the shearing process when the rock bolt andgrout are commonly constrained the stress concentrationphenomenon around the bolt and joint increases the sheardisplacement and contact force After reaching the residualphase the contact force exhibits slight fluctuations due to thesliding of the joint surface

(4) Based on the disk flow software PFC2D crack initia-tion for an anchorage angle of 75∘ will start from the contactforce concentration due to the interactions between the rockgrout and bolt Finally the cracks are mainly distributedin the joint and around the bolt for larger displacementsThe cracks are mainly tensile cracks which correspond tothe formation mechanism for a compression-induced tensilecrack The shear cracking was mainly caused by the rotationof the disks during the shearing process

Advances in Materials Science and Engineering 13

minus005 minus003 minus001 001 003 005

minus002minus001

0001002

minus551525354555

Con

tact

forc

e (N

)

times103

(a) Monitoring point a

minus002minus001

0001002

Con

tact

forc

e (N

)

minus10

10

30

50

70

90

110times103

minus005 minus003 minus001 001 003 005

(b) Monitoring point b

minus002minus001

0001002

Con

tact

forc

e (N

)minus101030507090110

times103

minus005 minus003 minus001 001 003 005

(c) Monitoring point c

minus002minus001

0001002

Con

tact

forc

e (N

)

minus10

10

30

50

70

90

110times103

minus005 minus003 minus001 001 003 005

(d) Monitoring point d

minus002minus001

0001002

minus51535557595115

Con

tact

forc

e (N

)

times103

minus005 minus003 minus001 001 003 005

(e) Monitoring point e

minus002minus001

0001002

minus10

10

30

50

70

90

Con

tact

forc

e (N

)

times103

minus005 minus003 minus001 001 003 005

(f) Monitoring point f

Figure 14 Evolution of the contact force and cracking at different monitoring points

14 Advances in Materials Science and Engineering

Particle rotation (in radians)minus120 to minus01

minus01 to minus0001

minus0001 to minus00005

minus00005 to 00005

00005 to 00010001 to 01

01 to 120

(a) Monitoring point a

Particle rotation (in radians)minus120 to minus01

minus01 to minus0001

minus0001 to minus00005

00005 to 00010001 to 01

01 to 120minus00005 to 00005

(b) Monitoring point b

Particle rotation (in radians)minus120 to minus01

minus01 to minus0001

minus0001 to minus00005

00005 to 00010001 to 01

01 to 120minus00005 to 00005

(c) Monitoring point c

Particle rotation (in radians)minus120 to minus01

minus01 to minus0001

minus0001 to minus00005

00005 to 00010001 to 01

01 to 120minus00005 to 00005

(d) Monitoring point d

Particle rotation (in radians)minus120 to minus01

minus01 to minus0001

minus0001 to minus00005

00005 to 00010001 to 01

01 to 120minus00005 to 00005

(e) Monitoring point e

Particle rotation (in radians)minus120 to minus01

minus01 to minus0001

minus0001 to minus00005

00005 to 00010001 to 01

01 to 120minus00005 to 00005

(f) Monitoring point f

Figure 15 Evolution of disk rotation in radians in the specimen at different monitoring points

0

5

10

15

20

Qua

ntita

tive p

ropo

rtio

n (

)

05 10 15 20 25 3000Shear displacement (mm)

gt 01Rotation (in radians)gt 01|Rotation (in radians)|

lt minus01Rotation (in radians)

Figure 16 Quantitative proportion variation of rotations exceeding 01 radians

Disclosure

GangWang and Yongzheng Zhang should be regarded as co-first authors

Conflicts of Interest

The authors declare that there are no conflicts of interestregarding the publication of this paper

Advances in Materials Science and Engineering 15

Acknowledgments

This study was supported by the National Natural ScienceFoundation of China (nos 51479108 51379117 and 51509261)China Postdoctoral Science Foundation (2015M580599) theTaishan Scholar Talent Team Support Plan for Advantagedamp Unique Discipline Areas the Postdoctoral Science Foun-dation of Shandong Province (no 201402014) and the Shan-dong Province Outstanding Young Scientist Research AwardFund (BS2014NJ006)

References

[1] F D Patton ldquoMultiple modes of shear failure in rockrdquo inProceedings of the 1st Congress on International Society for RockMechanics pp 509ndash513 1966

[2] R E Goodman Methods of Geological Engineering in Discon-tinuous Rocks West Publishing Company New York NY USA1976

[3] N Barton and V Choubey ldquoThe shear strength of rock joints intheory and practicerdquo Rock Mechanics Felsmechanik Mecaniquedes Roches vol 10 no 1-2 pp 1ndash54 1977

[4] P M Dight and H K Chiu ldquoPrediction of shear behavior ofjoints using profilesrdquo International Journal of Rock Mechanicsand Mining Sciences amp Geomechanics Abstracts vol 18 no 5pp 369ndash386 1981

[5] N Barton Modelling Rock Joint Behaviour from In Situ BlockTests Implications for NuclearWaste Repository Design vol 308Office ofNuclearWaste IsolationONWI ColumbusOHUSA1982

[6] T Esaki S Du Y Jiang and Y Wada ldquoEffect of the asperitydamage on the shear behavior of irregular rock jointsrdquo inProceedings of the Second International Conference on Analysisof Discontinuous Deformation pp 459ndash464 Kyoto Japan 1997

[7] Y Jiang J Xiao Y Tanabashi and T Mizokami ldquoDevelopmentof an automated servo-controlled direct shear apparatus apply-ing a constant normal stiffness conditionrdquo International Journalof Rock Mechanics and Mining Sciences vol 41 no 2 pp 275ndash286 2004

[8] Y J Jiang B Li and Y Tanabashi ldquoEstimating the relationbetween surface roughness and mechanical properties of rockjointsrdquo International Journal of Rock Mechanics and MiningSciences vol 43 no 6 pp 837ndash846 2006

[9] C-C Xia Z-C Tang W-M Xiao and Y-L Song ldquoNew peakshear strength criterion of rock joints based on quantifiedsurface descriptionrdquo Rock Mechanics and Rock Engineering vol47 no 2 pp 387ndash400 2014

[10] S Bjurstrom ldquoShear strength of hard rock joints reinforced bygrouted untensioned boltsrdquo in Proceedings of the In Proceedingsof the 3rd International ISRM Congress pp 1194ndash1199 DenverColo USA 1974

[11] X Ge and J Liu ldquoStudy of the shear resistance behaviour ofbolted rock jointsrdquo Journal of Geotechnical Engineering vol 10no 1 pp 8ndash19 1988 (Chinese)

[12] K Spang and P Egger ldquoAction of fully-grouted bolts in jointedrock and factors of influencerdquo Rock Mechanics and Rock Engi-neering vol 23 no 3 pp 201ndash229 1990

[13] P Egger and L Zabuski ldquoBehaviour of rough bolted joints indirect shear testsrdquo in Proceedings of the 7th ISRM Congress vol30 p 342 Aachen Germany September 1991

[14] A M Ferrero ldquoThe shear strength of reinforced rock jointsrdquoInternational Journal of Rock Mechanics and Mining Sciencesand vol 32 no 6 pp 595ndash605 1995

[15] F Pellet and P Egger ldquoAnalytical model for the mechanicalbehaviour of bolted rock joints subjected to shearingrdquo RockMechanics and Rock Engineering vol 29 no 2 pp 73ndash97 1996

[16] H Jalalifar and N Aziz ldquoExperimental and 3D numericalsimulation of reinforced shear jointsrdquo RockMechanics and RockEngineering vol 43 no 1 pp 95ndash103 2010

[17] P Wang T Feng Y-J Zhu and W-J Yu ldquoExperimental studyand numerical simulation of anchoringmechanismof anchoredrocklikematerial with prefabricated fracturerdquoYantu LixueRockand Soil Mechanics vol 37 no 3 pp 793ndash801 2016 (Chinese)

[18] L Li P C Hagan S Saydam B Hebblewhite and Y Li ldquoPara-metric study of rockbolt shear behaviour by double shear testrdquoRock Mechanics and Rock Engineering vol 49 no 12 pp 4787ndash4797 2016

[19] Itasca Consulting Group Inc ldquoPFC2D (Particle Flow Code in2D)Theory and Backgroundrdquo Minneapolis Minn USA 2008

[20] P A Cundall andOD L Strack ldquoThe distinct numericalmodelfor granular assembliesrdquo Geotechnique vol 29 no 1 pp 47ndash651979

[21] J Zhou and Y Chi ldquoThe method of particle flow and PFC2Dcoderdquo Rock and Soil Mechanics vol 21 no 3 pp 271ndash274 2000(Chinese)

[22] D O Potyondy and P A Cundall ldquoA bonded-particle modelfor rockrdquo International Journal of Rock Mechanics and MiningSciences vol 41 no 8 pp 1329ndash1364 2004

[23] N Cho CDMartin andDC Sego ldquoA clumped particlemodelfor rockrdquo International Journal of Rock Mechanics and MiningSciences vol 44 no 7 pp 997ndash1010 2007

[24] Z Zhao L Jing and I Neretnieks ldquoParticle mechanics modelfor the effects of shear on solute retardation coefficient in rockfracturesrdquo International Journal of Rock Mechanics and MiningSciences vol 52 pp 92ndash102 2012

[25] C Xia Y Song Z Tang Y Song and C Shou ldquoParticleflow numerical simulation for shear behavior of rough jointsrdquoChinese Journal of Rock Mechanics and Engineering vol 31 no8 pp 1545ndash1552 2012 (Chinese)

[26] Y Zhou A Misra S Wu and X Zhang ldquoMacro- and meso-analyses of rock joint direct shear test using particle flowtheoryrdquoChinese Journal of RockMechanics and Engineering vol31 no 6 pp 1245ndash1256 2012

[27] RH Cao P CaoH Lin K Zhang andXWTan ldquoParticle flowanalysis of direct shear tests on joints with different roughnessrdquoRock and Soil Mechanics vol 34 supplement 2 pp 456ndash4642013 (Chinese)

Submit your manuscripts athttpswwwhindawicom

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Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Polymer ScienceInternational Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CeramicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CompositesJournal of

NanoparticlesJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Biomaterials

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

NanoscienceJournal of

TextilesHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Journal of

NanotechnologyHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

CrystallographyJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CoatingsJournal of

Advances in

Materials Science and EngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Smart Materials Research

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MetallurgyJournal of

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BioMed Research International

MaterialsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Advances in Materials Science and Engineering 9

the bolt respectively 119896119899 and 119896119904 are the normal and tangentialstiffness coefficients for the weak level119875 is the anchor sectionreinforcement ratio Therefore

119880119899 = 1119886 (11988622120590 minus 11988612120591)

119880119905 = 1119886 (11988611120591 minus 11988621120590)

(9)

where

11988611 = minus(119896119899 + 119875119864119887119861119899119897119899) sin 119886

11988612 = (119896119899 + 119875119866119887119860119861119905119897119905) cos 119886

11988621 = 119896119904 cos 119886 + 119875119864119887119861119899119897119899 (cos 119886 minus 119891119904 sin 119886)

11988622 = 119896119904 sin 119886 + 119875119866119887119860119861119905119897119905 (sin 119886 + 119891119904 cos 119886)

119886 = 1198861111988622 minus 1198861211988621= 119896119899119896119904 minus 119875119864119887

119861119899119897119899119875119866119887119860119861119905119897119905 +

119875119866119887119860119861119905119897119905 (119896119899sin

2119886 minus 119896119904cos2119886)

+ 119875119864119887119861119899119897119899 [119896119899cos

2119886 minus 119896119904sin2119886]

+ 119896119899119891119904 sin 119886 cos 119886 [ 119875119866119887119860119861119905119897119905 minus119875119864119887119861119899119897119899 ]

(10)

Therefore the relationship between the stress and thestress component of the element can be determined as shownin

120590119887 = 119864119887119861119899119897119899119886 (11988622120590 minus 11988612120591)

120591119887 = 119866119887119860119861119905119897119905119886 (11988611120591 minus 11988621120590)

(11)

422 Analysis of the Bolt Stress for a Particular AnchorageAngle A bolt installed at a particular anchorage angle willhave a better anchoring effect than a bolt installed perpen-dicular to the joint plane Figures 10 and 11 show illustrationsof the stresses in a bolt at anchorage angles of 75∘ and 90∘respectively during shearing

The diagram showing the two different anchorage anglebolts indicates that the anchor force and bending momentare relatively large in the vertically installed bolt leading tobolt axial and shear stress that are larger than those in thetilted bolt whereas for the tilted bolt the axial and shearstress are small and do not reach the yield strengthThereforethe tilted anchor bolts not only improve the shear capacityof the joints but also maintain an appropriate stress statein the anchor The rock bolt axial stress diagram indicatesthat the axial stress increases as the shear displacementincreases for different anchorage anglesThe axial stress is the

highest when the anchor is vertically installed on the jointsurface The results of these two cases reveal that the optimalanchoring angle is approximately 75∘This result correspondsto the previous conclusions that the optimal anchoring angleshould be between 45∘ and 75∘

43 Evolution of the Contact Force and Microscopic Crackingof Disks The anchorage angle of 75∘ was chosen for amore detailed analysis on the micromechanical response anddamage mechanism during shearing Figure 12 shows theevolution of the shear stress number of cracks and normaldisplacement for a normal stress of 5MPa

To accurately determine the evolution features six differ-ent monitoring points (a to f) were set on the shear stresscurve which correspond to shear displacements of 0 01980217 0513 1500 and 2500mm respectively The variationsin the contact force and cracking can be observed from thesix monitoring points As shown in Figure 12 ldquoardquo indicatesthe initial point of the shear displacement and ldquocrdquo indicatesthe shear displacement at which the shear stress reachesthe peak shear strength As shown in Figure 12 no shearcrack is generated during the initial stage of the shear testThe number of shear cracks increases gradually as the sheardisplacement increases When the shear stress reaches thepeak shear strength the number of shear cracks increasessignificantly and the shear crack growth rate tends to stabilizeas the shear stress increases beyond the peak shear strengthThe trends in tension cracking and normal stress are highlysimilar and increase monotonically

During the shearing of a bolted joint the loaded side ofthe bolt is in a state of compression whereas the other side ofthe bolt is in a state of tension due to the bending deflectionof the bolt The adjacent grout is in a state of compressionand the rockmass is in a state of tension because of the bondsbetween the bolt and grout as shown in Figure 13 Bendingdeflection will occur in the bolt as the shear displacementincreases Simultaneously cracks will occur continuously dueto extrusion and slippage between disks In the compressionzone there are a large number of cracks which are mainlytension cracks generated by the extrusion between disksThisresult agreeswith the formationmechanism for compression-induced tension cracks shown in Figure 13

Figure 14 shows the illustration of the contact forceand crack propagation between disks during shearing Thecontact force is initially concentrated on the joint and theloading end and cracks are generated in the joint whereasthere are only a few cracks around the bolt As the sheardisplacement increases the diskrsquos contact force around thebolt becomes gradually larger Cracks occur on the jointand those around the bolt extend further The cracks aroundthe bolt propagate most rapidly The cracks are initiallyconcentrated at the intersection of the bolt and joint and thenpropagate to the ends of the bolt Finally the majority of thecracks are distributed at the joint and around the bolt aftera larger displacement occurs Finally more cracks occur inthe compression zone around the bolt than in the tensionzone In addition the distribution of cracks corresponds tothe contact force in the stress concentration areaThe detailed

10 Advances in Materials Science and Engineering

0 10 20 30 40

Distance along bolt (mm)

Joint plane

Bolt

05 mm10 mm15 mm

20 mm25 mm30 mm

0

100

200

300

Axi

al st

ress

(MPa

)

(a) Axial stress diagram

0 10 20 30 40

Distance along bolt (mm)

05 mm10 mm15 mm

20 mm25 mm30 mm

minus600

minus400

minus200

0

200

400

600

Bend

ing

mom

ent (

Nm

)

Joint plane

Bolt

(b) Bending moment diagram

Joint plane

Bolt

0 10 20 30 40

Distance along bolt (mm)

05 mm10 mm15 mm

20 mm25 mm30 mm

0

10

20

30

40

Shea

r stre

ss (M

Pa)

(c) Shear stress diagram

Figure 10 Bolt stress state diagram at an anchorage angle of 75∘

maximum contact force and crack number in the specimen atthe different monitoring points during shearing are providedin Tables 4 and 5 respectively

44 Evolution of the Disk Rotation During the shearing slipprocess the behavior is determined by the rotation of thedisks Relative disk dislocations will occur due to the rotationof the disks If the rotation angle is sufficiently large itwill lead to shear stresses and the breaking of disk bondsMeanwhile cracks occurring in the rock mass and shearcracking are caused directly by disk rotation Figure 15 showsa graph of the evolution of the disk rotation in terms ofradians for rock disks at different monitoring points

Figure 15 shows that when the shear displacement issmall at the initial stages the disk rotation angle is relatively

small and disks with large rotational angles are distributedaround the bolt The disk rotation angles gradually increasewith increasing shear displacement The disks containinglarger rotation angles are mainly distributed in the twoloading ends and around the bolt and the bolt and thedisks around the bolt contact move relatively violently duringshearing leading to a larger rotation angle Further increasesin the shear displacement and larger rotation disks alsoappeared on the joint surfaces which are caused by thejoint surface microconvex body cutting into the free disksThe distribution of the large disk rotation angles is highlyconsistent with the crack distribution demonstrating that therotation of the disks leads to the generation of shear cracks

Figure 16 shows the variations in the proportion ofrotations exceeding 01 radians As the shear displacement

Advances in Materials Science and Engineering 11

Bolt

Joint plane

0

50

100

150

200

250A

xial

stre

ss (M

Pa)

10 20 30 400Distance along bolt (mm)

05 mm10 mm15 mm

20 mm25 mm30 mm

(a) Axial stress diagram

Joint plane

Bolt

10 20 30 400Distance along bolt (mm)

minus800

minus600

minus400

minus200

0

200

400

600

800

Bend

ing

mom

ent (

Nm

)

05 mm10 mm15 mm

20 mm25 mm30 mm

(b) Bending moment diagram

Bolt

Joint plane

0

10

20

30

40

Shea

r stre

ss (M

Pa)

10 20 30 400Distance along bolt (mm)

05 mm10 mm15 mm

20 mm25 mm30 mm

(c) Shear stress diagram

Figure 11 Bolt stress state diagram at an anchorage angle of 90∘

Table 4 Maximum contact force in the specimen at different monitoring points

Monitoring point Maximum contact forceN Monitoring point Maximum contact forceNa 68201198904 d 12681198905b 15301198905 e 15801198905c 15481198905 f 14451198905

increases the number of clockwise and counterclockwisedisk rotations greater than 01 radians increases continuouslyAdditionally the number of clockwise rotations is greaterthan the number of counterclockwise rotations mainly dueto the shear loading directions

5 Conclusions

In this paper the anchoring mechanism for a tilted anchorbolt subjected to a shear load has been studied using a bilinearconstitutive model with the inner-embedded FISH language

12 Advances in Materials Science and Engineering

(f)(e)(d)

(c)

(b)

(a)

Shear stressNormal displacement

Tension crackShear crack

0

2

4

6

8

Shea

r stre

ss (M

Pa)

05 10 15 20 2500Shear displacement (mm)

00

01

02

03

04

05

06

07

08

Nor

mal

disp

lace

men

t (times10

minus3

m)

0

100

200

300

400

500

600

Crac

k nu

mbe

r

Figure 12 Evolution of the shear stress number of cracks and normal displacement

Rock

Bolt Grout

T

C

C

T

(a)

F F F

F F F

(b)

Figure 13 Stress state of the bolted rock joint system and illustration of the mechanism behind compression-induced tension cracks

Table 5 Number of cracks in the specimen at different monitoringpoints

Monitoringpoint Crack number Monitoring point Crack number

a 0 d 255b 21 e 446c 40 f 592

of PFC based on DEMThe anchoring mechanism of the boltatdifferentanchorage angles has been discussed at themacro-microlevels and the following conclusions have been drawn

(1) The bolt can improve the shear resistance of the jointmainly due to cohesion Tilted anchor bolts not only improvethe shear strength of the joint surface but also result in a lowerstress in the anchor bolt The peak shear strength can be 50greater than that of a vertically installed anchor bolt

(2) A tilted bolt allows for fully mobilized anchor perfor-manceThis configuration not only restricts increment in thenormal displacement but also indirectly improves the stress

state of the anchorage system and increases the shear capacityof the anchor joint The comprehensive shear displacementand shear stress data along with the bolt stress state analysisreveal that the optimal anchorage angle should be between45∘ and 75∘

(3) During the shearing process when the rock bolt andgrout are commonly constrained the stress concentrationphenomenon around the bolt and joint increases the sheardisplacement and contact force After reaching the residualphase the contact force exhibits slight fluctuations due to thesliding of the joint surface

(4) Based on the disk flow software PFC2D crack initia-tion for an anchorage angle of 75∘ will start from the contactforce concentration due to the interactions between the rockgrout and bolt Finally the cracks are mainly distributedin the joint and around the bolt for larger displacementsThe cracks are mainly tensile cracks which correspond tothe formation mechanism for a compression-induced tensilecrack The shear cracking was mainly caused by the rotationof the disks during the shearing process

Advances in Materials Science and Engineering 13

minus005 minus003 minus001 001 003 005

minus002minus001

0001002

minus551525354555

Con

tact

forc

e (N

)

times103

(a) Monitoring point a

minus002minus001

0001002

Con

tact

forc

e (N

)

minus10

10

30

50

70

90

110times103

minus005 minus003 minus001 001 003 005

(b) Monitoring point b

minus002minus001

0001002

Con

tact

forc

e (N

)minus101030507090110

times103

minus005 minus003 minus001 001 003 005

(c) Monitoring point c

minus002minus001

0001002

Con

tact

forc

e (N

)

minus10

10

30

50

70

90

110times103

minus005 minus003 minus001 001 003 005

(d) Monitoring point d

minus002minus001

0001002

minus51535557595115

Con

tact

forc

e (N

)

times103

minus005 minus003 minus001 001 003 005

(e) Monitoring point e

minus002minus001

0001002

minus10

10

30

50

70

90

Con

tact

forc

e (N

)

times103

minus005 minus003 minus001 001 003 005

(f) Monitoring point f

Figure 14 Evolution of the contact force and cracking at different monitoring points

14 Advances in Materials Science and Engineering

Particle rotation (in radians)minus120 to minus01

minus01 to minus0001

minus0001 to minus00005

minus00005 to 00005

00005 to 00010001 to 01

01 to 120

(a) Monitoring point a

Particle rotation (in radians)minus120 to minus01

minus01 to minus0001

minus0001 to minus00005

00005 to 00010001 to 01

01 to 120minus00005 to 00005

(b) Monitoring point b

Particle rotation (in radians)minus120 to minus01

minus01 to minus0001

minus0001 to minus00005

00005 to 00010001 to 01

01 to 120minus00005 to 00005

(c) Monitoring point c

Particle rotation (in radians)minus120 to minus01

minus01 to minus0001

minus0001 to minus00005

00005 to 00010001 to 01

01 to 120minus00005 to 00005

(d) Monitoring point d

Particle rotation (in radians)minus120 to minus01

minus01 to minus0001

minus0001 to minus00005

00005 to 00010001 to 01

01 to 120minus00005 to 00005

(e) Monitoring point e

Particle rotation (in radians)minus120 to minus01

minus01 to minus0001

minus0001 to minus00005

00005 to 00010001 to 01

01 to 120minus00005 to 00005

(f) Monitoring point f

Figure 15 Evolution of disk rotation in radians in the specimen at different monitoring points

0

5

10

15

20

Qua

ntita

tive p

ropo

rtio

n (

)

05 10 15 20 25 3000Shear displacement (mm)

gt 01Rotation (in radians)gt 01|Rotation (in radians)|

lt minus01Rotation (in radians)

Figure 16 Quantitative proportion variation of rotations exceeding 01 radians

Disclosure

GangWang and Yongzheng Zhang should be regarded as co-first authors

Conflicts of Interest

The authors declare that there are no conflicts of interestregarding the publication of this paper

Advances in Materials Science and Engineering 15

Acknowledgments

This study was supported by the National Natural ScienceFoundation of China (nos 51479108 51379117 and 51509261)China Postdoctoral Science Foundation (2015M580599) theTaishan Scholar Talent Team Support Plan for Advantagedamp Unique Discipline Areas the Postdoctoral Science Foun-dation of Shandong Province (no 201402014) and the Shan-dong Province Outstanding Young Scientist Research AwardFund (BS2014NJ006)

References

[1] F D Patton ldquoMultiple modes of shear failure in rockrdquo inProceedings of the 1st Congress on International Society for RockMechanics pp 509ndash513 1966

[2] R E Goodman Methods of Geological Engineering in Discon-tinuous Rocks West Publishing Company New York NY USA1976

[3] N Barton and V Choubey ldquoThe shear strength of rock joints intheory and practicerdquo Rock Mechanics Felsmechanik Mecaniquedes Roches vol 10 no 1-2 pp 1ndash54 1977

[4] P M Dight and H K Chiu ldquoPrediction of shear behavior ofjoints using profilesrdquo International Journal of Rock Mechanicsand Mining Sciences amp Geomechanics Abstracts vol 18 no 5pp 369ndash386 1981

[5] N Barton Modelling Rock Joint Behaviour from In Situ BlockTests Implications for NuclearWaste Repository Design vol 308Office ofNuclearWaste IsolationONWI ColumbusOHUSA1982

[6] T Esaki S Du Y Jiang and Y Wada ldquoEffect of the asperitydamage on the shear behavior of irregular rock jointsrdquo inProceedings of the Second International Conference on Analysisof Discontinuous Deformation pp 459ndash464 Kyoto Japan 1997

[7] Y Jiang J Xiao Y Tanabashi and T Mizokami ldquoDevelopmentof an automated servo-controlled direct shear apparatus apply-ing a constant normal stiffness conditionrdquo International Journalof Rock Mechanics and Mining Sciences vol 41 no 2 pp 275ndash286 2004

[8] Y J Jiang B Li and Y Tanabashi ldquoEstimating the relationbetween surface roughness and mechanical properties of rockjointsrdquo International Journal of Rock Mechanics and MiningSciences vol 43 no 6 pp 837ndash846 2006

[9] C-C Xia Z-C Tang W-M Xiao and Y-L Song ldquoNew peakshear strength criterion of rock joints based on quantifiedsurface descriptionrdquo Rock Mechanics and Rock Engineering vol47 no 2 pp 387ndash400 2014

[10] S Bjurstrom ldquoShear strength of hard rock joints reinforced bygrouted untensioned boltsrdquo in Proceedings of the In Proceedingsof the 3rd International ISRM Congress pp 1194ndash1199 DenverColo USA 1974

[11] X Ge and J Liu ldquoStudy of the shear resistance behaviour ofbolted rock jointsrdquo Journal of Geotechnical Engineering vol 10no 1 pp 8ndash19 1988 (Chinese)

[12] K Spang and P Egger ldquoAction of fully-grouted bolts in jointedrock and factors of influencerdquo Rock Mechanics and Rock Engi-neering vol 23 no 3 pp 201ndash229 1990

[13] P Egger and L Zabuski ldquoBehaviour of rough bolted joints indirect shear testsrdquo in Proceedings of the 7th ISRM Congress vol30 p 342 Aachen Germany September 1991

[14] A M Ferrero ldquoThe shear strength of reinforced rock jointsrdquoInternational Journal of Rock Mechanics and Mining Sciencesand vol 32 no 6 pp 595ndash605 1995

[15] F Pellet and P Egger ldquoAnalytical model for the mechanicalbehaviour of bolted rock joints subjected to shearingrdquo RockMechanics and Rock Engineering vol 29 no 2 pp 73ndash97 1996

[16] H Jalalifar and N Aziz ldquoExperimental and 3D numericalsimulation of reinforced shear jointsrdquo RockMechanics and RockEngineering vol 43 no 1 pp 95ndash103 2010

[17] P Wang T Feng Y-J Zhu and W-J Yu ldquoExperimental studyand numerical simulation of anchoringmechanismof anchoredrocklikematerial with prefabricated fracturerdquoYantu LixueRockand Soil Mechanics vol 37 no 3 pp 793ndash801 2016 (Chinese)

[18] L Li P C Hagan S Saydam B Hebblewhite and Y Li ldquoPara-metric study of rockbolt shear behaviour by double shear testrdquoRock Mechanics and Rock Engineering vol 49 no 12 pp 4787ndash4797 2016

[19] Itasca Consulting Group Inc ldquoPFC2D (Particle Flow Code in2D)Theory and Backgroundrdquo Minneapolis Minn USA 2008

[20] P A Cundall andOD L Strack ldquoThe distinct numericalmodelfor granular assembliesrdquo Geotechnique vol 29 no 1 pp 47ndash651979

[21] J Zhou and Y Chi ldquoThe method of particle flow and PFC2Dcoderdquo Rock and Soil Mechanics vol 21 no 3 pp 271ndash274 2000(Chinese)

[22] D O Potyondy and P A Cundall ldquoA bonded-particle modelfor rockrdquo International Journal of Rock Mechanics and MiningSciences vol 41 no 8 pp 1329ndash1364 2004

[23] N Cho CDMartin andDC Sego ldquoA clumped particlemodelfor rockrdquo International Journal of Rock Mechanics and MiningSciences vol 44 no 7 pp 997ndash1010 2007

[24] Z Zhao L Jing and I Neretnieks ldquoParticle mechanics modelfor the effects of shear on solute retardation coefficient in rockfracturesrdquo International Journal of Rock Mechanics and MiningSciences vol 52 pp 92ndash102 2012

[25] C Xia Y Song Z Tang Y Song and C Shou ldquoParticleflow numerical simulation for shear behavior of rough jointsrdquoChinese Journal of Rock Mechanics and Engineering vol 31 no8 pp 1545ndash1552 2012 (Chinese)

[26] Y Zhou A Misra S Wu and X Zhang ldquoMacro- and meso-analyses of rock joint direct shear test using particle flowtheoryrdquoChinese Journal of RockMechanics and Engineering vol31 no 6 pp 1245ndash1256 2012

[27] RH Cao P CaoH Lin K Zhang andXWTan ldquoParticle flowanalysis of direct shear tests on joints with different roughnessrdquoRock and Soil Mechanics vol 34 supplement 2 pp 456ndash4642013 (Chinese)

Submit your manuscripts athttpswwwhindawicom

ScientificaHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CorrosionInternational Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Polymer ScienceInternational Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CeramicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CompositesJournal of

NanoparticlesJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Biomaterials

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

NanoscienceJournal of

TextilesHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Journal of

NanotechnologyHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

CrystallographyJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CoatingsJournal of

Advances in

Materials Science and EngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Smart Materials Research

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MetallurgyJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

BioMed Research International

MaterialsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

10 Advances in Materials Science and Engineering

0 10 20 30 40

Distance along bolt (mm)

Joint plane

Bolt

05 mm10 mm15 mm

20 mm25 mm30 mm

0

100

200

300

Axi

al st

ress

(MPa

)

(a) Axial stress diagram

0 10 20 30 40

Distance along bolt (mm)

05 mm10 mm15 mm

20 mm25 mm30 mm

minus600

minus400

minus200

0

200

400

600

Bend

ing

mom

ent (

Nm

)

Joint plane

Bolt

(b) Bending moment diagram

Joint plane

Bolt

0 10 20 30 40

Distance along bolt (mm)

05 mm10 mm15 mm

20 mm25 mm30 mm

0

10

20

30

40

Shea

r stre

ss (M

Pa)

(c) Shear stress diagram

Figure 10 Bolt stress state diagram at an anchorage angle of 75∘

maximum contact force and crack number in the specimen atthe different monitoring points during shearing are providedin Tables 4 and 5 respectively

44 Evolution of the Disk Rotation During the shearing slipprocess the behavior is determined by the rotation of thedisks Relative disk dislocations will occur due to the rotationof the disks If the rotation angle is sufficiently large itwill lead to shear stresses and the breaking of disk bondsMeanwhile cracks occurring in the rock mass and shearcracking are caused directly by disk rotation Figure 15 showsa graph of the evolution of the disk rotation in terms ofradians for rock disks at different monitoring points

Figure 15 shows that when the shear displacement issmall at the initial stages the disk rotation angle is relatively

small and disks with large rotational angles are distributedaround the bolt The disk rotation angles gradually increasewith increasing shear displacement The disks containinglarger rotation angles are mainly distributed in the twoloading ends and around the bolt and the bolt and thedisks around the bolt contact move relatively violently duringshearing leading to a larger rotation angle Further increasesin the shear displacement and larger rotation disks alsoappeared on the joint surfaces which are caused by thejoint surface microconvex body cutting into the free disksThe distribution of the large disk rotation angles is highlyconsistent with the crack distribution demonstrating that therotation of the disks leads to the generation of shear cracks

Figure 16 shows the variations in the proportion ofrotations exceeding 01 radians As the shear displacement

Advances in Materials Science and Engineering 11

Bolt

Joint plane

0

50

100

150

200

250A

xial

stre

ss (M

Pa)

10 20 30 400Distance along bolt (mm)

05 mm10 mm15 mm

20 mm25 mm30 mm

(a) Axial stress diagram

Joint plane

Bolt

10 20 30 400Distance along bolt (mm)

minus800

minus600

minus400

minus200

0

200

400

600

800

Bend

ing

mom

ent (

Nm

)

05 mm10 mm15 mm

20 mm25 mm30 mm

(b) Bending moment diagram

Bolt

Joint plane

0

10

20

30

40

Shea

r stre

ss (M

Pa)

10 20 30 400Distance along bolt (mm)

05 mm10 mm15 mm

20 mm25 mm30 mm

(c) Shear stress diagram

Figure 11 Bolt stress state diagram at an anchorage angle of 90∘

Table 4 Maximum contact force in the specimen at different monitoring points

Monitoring point Maximum contact forceN Monitoring point Maximum contact forceNa 68201198904 d 12681198905b 15301198905 e 15801198905c 15481198905 f 14451198905

increases the number of clockwise and counterclockwisedisk rotations greater than 01 radians increases continuouslyAdditionally the number of clockwise rotations is greaterthan the number of counterclockwise rotations mainly dueto the shear loading directions

5 Conclusions

In this paper the anchoring mechanism for a tilted anchorbolt subjected to a shear load has been studied using a bilinearconstitutive model with the inner-embedded FISH language

12 Advances in Materials Science and Engineering

(f)(e)(d)

(c)

(b)

(a)

Shear stressNormal displacement

Tension crackShear crack

0

2

4

6

8

Shea

r stre

ss (M

Pa)

05 10 15 20 2500Shear displacement (mm)

00

01

02

03

04

05

06

07

08

Nor

mal

disp

lace

men

t (times10

minus3

m)

0

100

200

300

400

500

600

Crac

k nu

mbe

r

Figure 12 Evolution of the shear stress number of cracks and normal displacement

Rock

Bolt Grout

T

C

C

T

(a)

F F F

F F F

(b)

Figure 13 Stress state of the bolted rock joint system and illustration of the mechanism behind compression-induced tension cracks

Table 5 Number of cracks in the specimen at different monitoringpoints

Monitoringpoint Crack number Monitoring point Crack number

a 0 d 255b 21 e 446c 40 f 592

of PFC based on DEMThe anchoring mechanism of the boltatdifferentanchorage angles has been discussed at themacro-microlevels and the following conclusions have been drawn

(1) The bolt can improve the shear resistance of the jointmainly due to cohesion Tilted anchor bolts not only improvethe shear strength of the joint surface but also result in a lowerstress in the anchor bolt The peak shear strength can be 50greater than that of a vertically installed anchor bolt

(2) A tilted bolt allows for fully mobilized anchor perfor-manceThis configuration not only restricts increment in thenormal displacement but also indirectly improves the stress

state of the anchorage system and increases the shear capacityof the anchor joint The comprehensive shear displacementand shear stress data along with the bolt stress state analysisreveal that the optimal anchorage angle should be between45∘ and 75∘

(3) During the shearing process when the rock bolt andgrout are commonly constrained the stress concentrationphenomenon around the bolt and joint increases the sheardisplacement and contact force After reaching the residualphase the contact force exhibits slight fluctuations due to thesliding of the joint surface

(4) Based on the disk flow software PFC2D crack initia-tion for an anchorage angle of 75∘ will start from the contactforce concentration due to the interactions between the rockgrout and bolt Finally the cracks are mainly distributedin the joint and around the bolt for larger displacementsThe cracks are mainly tensile cracks which correspond tothe formation mechanism for a compression-induced tensilecrack The shear cracking was mainly caused by the rotationof the disks during the shearing process

Advances in Materials Science and Engineering 13

minus005 minus003 minus001 001 003 005

minus002minus001

0001002

minus551525354555

Con

tact

forc

e (N

)

times103

(a) Monitoring point a

minus002minus001

0001002

Con

tact

forc

e (N

)

minus10

10

30

50

70

90

110times103

minus005 minus003 minus001 001 003 005

(b) Monitoring point b

minus002minus001

0001002

Con

tact

forc

e (N

)minus101030507090110

times103

minus005 minus003 minus001 001 003 005

(c) Monitoring point c

minus002minus001

0001002

Con

tact

forc

e (N

)

minus10

10

30

50

70

90

110times103

minus005 minus003 minus001 001 003 005

(d) Monitoring point d

minus002minus001

0001002

minus51535557595115

Con

tact

forc

e (N

)

times103

minus005 minus003 minus001 001 003 005

(e) Monitoring point e

minus002minus001

0001002

minus10

10

30

50

70

90

Con

tact

forc

e (N

)

times103

minus005 minus003 minus001 001 003 005

(f) Monitoring point f

Figure 14 Evolution of the contact force and cracking at different monitoring points

14 Advances in Materials Science and Engineering

Particle rotation (in radians)minus120 to minus01

minus01 to minus0001

minus0001 to minus00005

minus00005 to 00005

00005 to 00010001 to 01

01 to 120

(a) Monitoring point a

Particle rotation (in radians)minus120 to minus01

minus01 to minus0001

minus0001 to minus00005

00005 to 00010001 to 01

01 to 120minus00005 to 00005

(b) Monitoring point b

Particle rotation (in radians)minus120 to minus01

minus01 to minus0001

minus0001 to minus00005

00005 to 00010001 to 01

01 to 120minus00005 to 00005

(c) Monitoring point c

Particle rotation (in radians)minus120 to minus01

minus01 to minus0001

minus0001 to minus00005

00005 to 00010001 to 01

01 to 120minus00005 to 00005

(d) Monitoring point d

Particle rotation (in radians)minus120 to minus01

minus01 to minus0001

minus0001 to minus00005

00005 to 00010001 to 01

01 to 120minus00005 to 00005

(e) Monitoring point e

Particle rotation (in radians)minus120 to minus01

minus01 to minus0001

minus0001 to minus00005

00005 to 00010001 to 01

01 to 120minus00005 to 00005

(f) Monitoring point f

Figure 15 Evolution of disk rotation in radians in the specimen at different monitoring points

0

5

10

15

20

Qua

ntita

tive p

ropo

rtio

n (

)

05 10 15 20 25 3000Shear displacement (mm)

gt 01Rotation (in radians)gt 01|Rotation (in radians)|

lt minus01Rotation (in radians)

Figure 16 Quantitative proportion variation of rotations exceeding 01 radians

Disclosure

GangWang and Yongzheng Zhang should be regarded as co-first authors

Conflicts of Interest

The authors declare that there are no conflicts of interestregarding the publication of this paper

Advances in Materials Science and Engineering 15

Acknowledgments

This study was supported by the National Natural ScienceFoundation of China (nos 51479108 51379117 and 51509261)China Postdoctoral Science Foundation (2015M580599) theTaishan Scholar Talent Team Support Plan for Advantagedamp Unique Discipline Areas the Postdoctoral Science Foun-dation of Shandong Province (no 201402014) and the Shan-dong Province Outstanding Young Scientist Research AwardFund (BS2014NJ006)

References

[1] F D Patton ldquoMultiple modes of shear failure in rockrdquo inProceedings of the 1st Congress on International Society for RockMechanics pp 509ndash513 1966

[2] R E Goodman Methods of Geological Engineering in Discon-tinuous Rocks West Publishing Company New York NY USA1976

[3] N Barton and V Choubey ldquoThe shear strength of rock joints intheory and practicerdquo Rock Mechanics Felsmechanik Mecaniquedes Roches vol 10 no 1-2 pp 1ndash54 1977

[4] P M Dight and H K Chiu ldquoPrediction of shear behavior ofjoints using profilesrdquo International Journal of Rock Mechanicsand Mining Sciences amp Geomechanics Abstracts vol 18 no 5pp 369ndash386 1981

[5] N Barton Modelling Rock Joint Behaviour from In Situ BlockTests Implications for NuclearWaste Repository Design vol 308Office ofNuclearWaste IsolationONWI ColumbusOHUSA1982

[6] T Esaki S Du Y Jiang and Y Wada ldquoEffect of the asperitydamage on the shear behavior of irregular rock jointsrdquo inProceedings of the Second International Conference on Analysisof Discontinuous Deformation pp 459ndash464 Kyoto Japan 1997

[7] Y Jiang J Xiao Y Tanabashi and T Mizokami ldquoDevelopmentof an automated servo-controlled direct shear apparatus apply-ing a constant normal stiffness conditionrdquo International Journalof Rock Mechanics and Mining Sciences vol 41 no 2 pp 275ndash286 2004

[8] Y J Jiang B Li and Y Tanabashi ldquoEstimating the relationbetween surface roughness and mechanical properties of rockjointsrdquo International Journal of Rock Mechanics and MiningSciences vol 43 no 6 pp 837ndash846 2006

[9] C-C Xia Z-C Tang W-M Xiao and Y-L Song ldquoNew peakshear strength criterion of rock joints based on quantifiedsurface descriptionrdquo Rock Mechanics and Rock Engineering vol47 no 2 pp 387ndash400 2014

[10] S Bjurstrom ldquoShear strength of hard rock joints reinforced bygrouted untensioned boltsrdquo in Proceedings of the In Proceedingsof the 3rd International ISRM Congress pp 1194ndash1199 DenverColo USA 1974

[11] X Ge and J Liu ldquoStudy of the shear resistance behaviour ofbolted rock jointsrdquo Journal of Geotechnical Engineering vol 10no 1 pp 8ndash19 1988 (Chinese)

[12] K Spang and P Egger ldquoAction of fully-grouted bolts in jointedrock and factors of influencerdquo Rock Mechanics and Rock Engi-neering vol 23 no 3 pp 201ndash229 1990

[13] P Egger and L Zabuski ldquoBehaviour of rough bolted joints indirect shear testsrdquo in Proceedings of the 7th ISRM Congress vol30 p 342 Aachen Germany September 1991

[14] A M Ferrero ldquoThe shear strength of reinforced rock jointsrdquoInternational Journal of Rock Mechanics and Mining Sciencesand vol 32 no 6 pp 595ndash605 1995

[15] F Pellet and P Egger ldquoAnalytical model for the mechanicalbehaviour of bolted rock joints subjected to shearingrdquo RockMechanics and Rock Engineering vol 29 no 2 pp 73ndash97 1996

[16] H Jalalifar and N Aziz ldquoExperimental and 3D numericalsimulation of reinforced shear jointsrdquo RockMechanics and RockEngineering vol 43 no 1 pp 95ndash103 2010

[17] P Wang T Feng Y-J Zhu and W-J Yu ldquoExperimental studyand numerical simulation of anchoringmechanismof anchoredrocklikematerial with prefabricated fracturerdquoYantu LixueRockand Soil Mechanics vol 37 no 3 pp 793ndash801 2016 (Chinese)

[18] L Li P C Hagan S Saydam B Hebblewhite and Y Li ldquoPara-metric study of rockbolt shear behaviour by double shear testrdquoRock Mechanics and Rock Engineering vol 49 no 12 pp 4787ndash4797 2016

[19] Itasca Consulting Group Inc ldquoPFC2D (Particle Flow Code in2D)Theory and Backgroundrdquo Minneapolis Minn USA 2008

[20] P A Cundall andOD L Strack ldquoThe distinct numericalmodelfor granular assembliesrdquo Geotechnique vol 29 no 1 pp 47ndash651979

[21] J Zhou and Y Chi ldquoThe method of particle flow and PFC2Dcoderdquo Rock and Soil Mechanics vol 21 no 3 pp 271ndash274 2000(Chinese)

[22] D O Potyondy and P A Cundall ldquoA bonded-particle modelfor rockrdquo International Journal of Rock Mechanics and MiningSciences vol 41 no 8 pp 1329ndash1364 2004

[23] N Cho CDMartin andDC Sego ldquoA clumped particlemodelfor rockrdquo International Journal of Rock Mechanics and MiningSciences vol 44 no 7 pp 997ndash1010 2007

[24] Z Zhao L Jing and I Neretnieks ldquoParticle mechanics modelfor the effects of shear on solute retardation coefficient in rockfracturesrdquo International Journal of Rock Mechanics and MiningSciences vol 52 pp 92ndash102 2012

[25] C Xia Y Song Z Tang Y Song and C Shou ldquoParticleflow numerical simulation for shear behavior of rough jointsrdquoChinese Journal of Rock Mechanics and Engineering vol 31 no8 pp 1545ndash1552 2012 (Chinese)

[26] Y Zhou A Misra S Wu and X Zhang ldquoMacro- and meso-analyses of rock joint direct shear test using particle flowtheoryrdquoChinese Journal of RockMechanics and Engineering vol31 no 6 pp 1245ndash1256 2012

[27] RH Cao P CaoH Lin K Zhang andXWTan ldquoParticle flowanalysis of direct shear tests on joints with different roughnessrdquoRock and Soil Mechanics vol 34 supplement 2 pp 456ndash4642013 (Chinese)

Submit your manuscripts athttpswwwhindawicom

ScientificaHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CorrosionInternational Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Polymer ScienceInternational Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CeramicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CompositesJournal of

NanoparticlesJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Biomaterials

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

NanoscienceJournal of

TextilesHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Journal of

NanotechnologyHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

CrystallographyJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CoatingsJournal of

Advances in

Materials Science and EngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Smart Materials Research

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MetallurgyJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

BioMed Research International

MaterialsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Advances in Materials Science and Engineering 11

Bolt

Joint plane

0

50

100

150

200

250A

xial

stre

ss (M

Pa)

10 20 30 400Distance along bolt (mm)

05 mm10 mm15 mm

20 mm25 mm30 mm

(a) Axial stress diagram

Joint plane

Bolt

10 20 30 400Distance along bolt (mm)

minus800

minus600

minus400

minus200

0

200

400

600

800

Bend

ing

mom

ent (

Nm

)

05 mm10 mm15 mm

20 mm25 mm30 mm

(b) Bending moment diagram

Bolt

Joint plane

0

10

20

30

40

Shea

r stre

ss (M

Pa)

10 20 30 400Distance along bolt (mm)

05 mm10 mm15 mm

20 mm25 mm30 mm

(c) Shear stress diagram

Figure 11 Bolt stress state diagram at an anchorage angle of 90∘

Table 4 Maximum contact force in the specimen at different monitoring points

Monitoring point Maximum contact forceN Monitoring point Maximum contact forceNa 68201198904 d 12681198905b 15301198905 e 15801198905c 15481198905 f 14451198905

increases the number of clockwise and counterclockwisedisk rotations greater than 01 radians increases continuouslyAdditionally the number of clockwise rotations is greaterthan the number of counterclockwise rotations mainly dueto the shear loading directions

5 Conclusions

In this paper the anchoring mechanism for a tilted anchorbolt subjected to a shear load has been studied using a bilinearconstitutive model with the inner-embedded FISH language

12 Advances in Materials Science and Engineering

(f)(e)(d)

(c)

(b)

(a)

Shear stressNormal displacement

Tension crackShear crack

0

2

4

6

8

Shea

r stre

ss (M

Pa)

05 10 15 20 2500Shear displacement (mm)

00

01

02

03

04

05

06

07

08

Nor

mal

disp

lace

men

t (times10

minus3

m)

0

100

200

300

400

500

600

Crac

k nu

mbe

r

Figure 12 Evolution of the shear stress number of cracks and normal displacement

Rock

Bolt Grout

T

C

C

T

(a)

F F F

F F F

(b)

Figure 13 Stress state of the bolted rock joint system and illustration of the mechanism behind compression-induced tension cracks

Table 5 Number of cracks in the specimen at different monitoringpoints

Monitoringpoint Crack number Monitoring point Crack number

a 0 d 255b 21 e 446c 40 f 592

of PFC based on DEMThe anchoring mechanism of the boltatdifferentanchorage angles has been discussed at themacro-microlevels and the following conclusions have been drawn

(1) The bolt can improve the shear resistance of the jointmainly due to cohesion Tilted anchor bolts not only improvethe shear strength of the joint surface but also result in a lowerstress in the anchor bolt The peak shear strength can be 50greater than that of a vertically installed anchor bolt

(2) A tilted bolt allows for fully mobilized anchor perfor-manceThis configuration not only restricts increment in thenormal displacement but also indirectly improves the stress

state of the anchorage system and increases the shear capacityof the anchor joint The comprehensive shear displacementand shear stress data along with the bolt stress state analysisreveal that the optimal anchorage angle should be between45∘ and 75∘

(3) During the shearing process when the rock bolt andgrout are commonly constrained the stress concentrationphenomenon around the bolt and joint increases the sheardisplacement and contact force After reaching the residualphase the contact force exhibits slight fluctuations due to thesliding of the joint surface

(4) Based on the disk flow software PFC2D crack initia-tion for an anchorage angle of 75∘ will start from the contactforce concentration due to the interactions between the rockgrout and bolt Finally the cracks are mainly distributedin the joint and around the bolt for larger displacementsThe cracks are mainly tensile cracks which correspond tothe formation mechanism for a compression-induced tensilecrack The shear cracking was mainly caused by the rotationof the disks during the shearing process

Advances in Materials Science and Engineering 13

minus005 minus003 minus001 001 003 005

minus002minus001

0001002

minus551525354555

Con

tact

forc

e (N

)

times103

(a) Monitoring point a

minus002minus001

0001002

Con

tact

forc

e (N

)

minus10

10

30

50

70

90

110times103

minus005 minus003 minus001 001 003 005

(b) Monitoring point b

minus002minus001

0001002

Con

tact

forc

e (N

)minus101030507090110

times103

minus005 minus003 minus001 001 003 005

(c) Monitoring point c

minus002minus001

0001002

Con

tact

forc

e (N

)

minus10

10

30

50

70

90

110times103

minus005 minus003 minus001 001 003 005

(d) Monitoring point d

minus002minus001

0001002

minus51535557595115

Con

tact

forc

e (N

)

times103

minus005 minus003 minus001 001 003 005

(e) Monitoring point e

minus002minus001

0001002

minus10

10

30

50

70

90

Con

tact

forc

e (N

)

times103

minus005 minus003 minus001 001 003 005

(f) Monitoring point f

Figure 14 Evolution of the contact force and cracking at different monitoring points

14 Advances in Materials Science and Engineering

Particle rotation (in radians)minus120 to minus01

minus01 to minus0001

minus0001 to minus00005

minus00005 to 00005

00005 to 00010001 to 01

01 to 120

(a) Monitoring point a

Particle rotation (in radians)minus120 to minus01

minus01 to minus0001

minus0001 to minus00005

00005 to 00010001 to 01

01 to 120minus00005 to 00005

(b) Monitoring point b

Particle rotation (in radians)minus120 to minus01

minus01 to minus0001

minus0001 to minus00005

00005 to 00010001 to 01

01 to 120minus00005 to 00005

(c) Monitoring point c

Particle rotation (in radians)minus120 to minus01

minus01 to minus0001

minus0001 to minus00005

00005 to 00010001 to 01

01 to 120minus00005 to 00005

(d) Monitoring point d

Particle rotation (in radians)minus120 to minus01

minus01 to minus0001

minus0001 to minus00005

00005 to 00010001 to 01

01 to 120minus00005 to 00005

(e) Monitoring point e

Particle rotation (in radians)minus120 to minus01

minus01 to minus0001

minus0001 to minus00005

00005 to 00010001 to 01

01 to 120minus00005 to 00005

(f) Monitoring point f

Figure 15 Evolution of disk rotation in radians in the specimen at different monitoring points

0

5

10

15

20

Qua

ntita

tive p

ropo

rtio

n (

)

05 10 15 20 25 3000Shear displacement (mm)

gt 01Rotation (in radians)gt 01|Rotation (in radians)|

lt minus01Rotation (in radians)

Figure 16 Quantitative proportion variation of rotations exceeding 01 radians

Disclosure

GangWang and Yongzheng Zhang should be regarded as co-first authors

Conflicts of Interest

The authors declare that there are no conflicts of interestregarding the publication of this paper

Advances in Materials Science and Engineering 15

Acknowledgments

This study was supported by the National Natural ScienceFoundation of China (nos 51479108 51379117 and 51509261)China Postdoctoral Science Foundation (2015M580599) theTaishan Scholar Talent Team Support Plan for Advantagedamp Unique Discipline Areas the Postdoctoral Science Foun-dation of Shandong Province (no 201402014) and the Shan-dong Province Outstanding Young Scientist Research AwardFund (BS2014NJ006)

References

[1] F D Patton ldquoMultiple modes of shear failure in rockrdquo inProceedings of the 1st Congress on International Society for RockMechanics pp 509ndash513 1966

[2] R E Goodman Methods of Geological Engineering in Discon-tinuous Rocks West Publishing Company New York NY USA1976

[3] N Barton and V Choubey ldquoThe shear strength of rock joints intheory and practicerdquo Rock Mechanics Felsmechanik Mecaniquedes Roches vol 10 no 1-2 pp 1ndash54 1977

[4] P M Dight and H K Chiu ldquoPrediction of shear behavior ofjoints using profilesrdquo International Journal of Rock Mechanicsand Mining Sciences amp Geomechanics Abstracts vol 18 no 5pp 369ndash386 1981

[5] N Barton Modelling Rock Joint Behaviour from In Situ BlockTests Implications for NuclearWaste Repository Design vol 308Office ofNuclearWaste IsolationONWI ColumbusOHUSA1982

[6] T Esaki S Du Y Jiang and Y Wada ldquoEffect of the asperitydamage on the shear behavior of irregular rock jointsrdquo inProceedings of the Second International Conference on Analysisof Discontinuous Deformation pp 459ndash464 Kyoto Japan 1997

[7] Y Jiang J Xiao Y Tanabashi and T Mizokami ldquoDevelopmentof an automated servo-controlled direct shear apparatus apply-ing a constant normal stiffness conditionrdquo International Journalof Rock Mechanics and Mining Sciences vol 41 no 2 pp 275ndash286 2004

[8] Y J Jiang B Li and Y Tanabashi ldquoEstimating the relationbetween surface roughness and mechanical properties of rockjointsrdquo International Journal of Rock Mechanics and MiningSciences vol 43 no 6 pp 837ndash846 2006

[9] C-C Xia Z-C Tang W-M Xiao and Y-L Song ldquoNew peakshear strength criterion of rock joints based on quantifiedsurface descriptionrdquo Rock Mechanics and Rock Engineering vol47 no 2 pp 387ndash400 2014

[10] S Bjurstrom ldquoShear strength of hard rock joints reinforced bygrouted untensioned boltsrdquo in Proceedings of the In Proceedingsof the 3rd International ISRM Congress pp 1194ndash1199 DenverColo USA 1974

[11] X Ge and J Liu ldquoStudy of the shear resistance behaviour ofbolted rock jointsrdquo Journal of Geotechnical Engineering vol 10no 1 pp 8ndash19 1988 (Chinese)

[12] K Spang and P Egger ldquoAction of fully-grouted bolts in jointedrock and factors of influencerdquo Rock Mechanics and Rock Engi-neering vol 23 no 3 pp 201ndash229 1990

[13] P Egger and L Zabuski ldquoBehaviour of rough bolted joints indirect shear testsrdquo in Proceedings of the 7th ISRM Congress vol30 p 342 Aachen Germany September 1991

[14] A M Ferrero ldquoThe shear strength of reinforced rock jointsrdquoInternational Journal of Rock Mechanics and Mining Sciencesand vol 32 no 6 pp 595ndash605 1995

[15] F Pellet and P Egger ldquoAnalytical model for the mechanicalbehaviour of bolted rock joints subjected to shearingrdquo RockMechanics and Rock Engineering vol 29 no 2 pp 73ndash97 1996

[16] H Jalalifar and N Aziz ldquoExperimental and 3D numericalsimulation of reinforced shear jointsrdquo RockMechanics and RockEngineering vol 43 no 1 pp 95ndash103 2010

[17] P Wang T Feng Y-J Zhu and W-J Yu ldquoExperimental studyand numerical simulation of anchoringmechanismof anchoredrocklikematerial with prefabricated fracturerdquoYantu LixueRockand Soil Mechanics vol 37 no 3 pp 793ndash801 2016 (Chinese)

[18] L Li P C Hagan S Saydam B Hebblewhite and Y Li ldquoPara-metric study of rockbolt shear behaviour by double shear testrdquoRock Mechanics and Rock Engineering vol 49 no 12 pp 4787ndash4797 2016

[19] Itasca Consulting Group Inc ldquoPFC2D (Particle Flow Code in2D)Theory and Backgroundrdquo Minneapolis Minn USA 2008

[20] P A Cundall andOD L Strack ldquoThe distinct numericalmodelfor granular assembliesrdquo Geotechnique vol 29 no 1 pp 47ndash651979

[21] J Zhou and Y Chi ldquoThe method of particle flow and PFC2Dcoderdquo Rock and Soil Mechanics vol 21 no 3 pp 271ndash274 2000(Chinese)

[22] D O Potyondy and P A Cundall ldquoA bonded-particle modelfor rockrdquo International Journal of Rock Mechanics and MiningSciences vol 41 no 8 pp 1329ndash1364 2004

[23] N Cho CDMartin andDC Sego ldquoA clumped particlemodelfor rockrdquo International Journal of Rock Mechanics and MiningSciences vol 44 no 7 pp 997ndash1010 2007

[24] Z Zhao L Jing and I Neretnieks ldquoParticle mechanics modelfor the effects of shear on solute retardation coefficient in rockfracturesrdquo International Journal of Rock Mechanics and MiningSciences vol 52 pp 92ndash102 2012

[25] C Xia Y Song Z Tang Y Song and C Shou ldquoParticleflow numerical simulation for shear behavior of rough jointsrdquoChinese Journal of Rock Mechanics and Engineering vol 31 no8 pp 1545ndash1552 2012 (Chinese)

[26] Y Zhou A Misra S Wu and X Zhang ldquoMacro- and meso-analyses of rock joint direct shear test using particle flowtheoryrdquoChinese Journal of RockMechanics and Engineering vol31 no 6 pp 1245ndash1256 2012

[27] RH Cao P CaoH Lin K Zhang andXWTan ldquoParticle flowanalysis of direct shear tests on joints with different roughnessrdquoRock and Soil Mechanics vol 34 supplement 2 pp 456ndash4642013 (Chinese)

Submit your manuscripts athttpswwwhindawicom

ScientificaHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CorrosionInternational Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Polymer ScienceInternational Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CeramicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CompositesJournal of

NanoparticlesJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Biomaterials

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

NanoscienceJournal of

TextilesHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Journal of

NanotechnologyHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

CrystallographyJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CoatingsJournal of

Advances in

Materials Science and EngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Smart Materials Research

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MetallurgyJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

BioMed Research International

MaterialsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

12 Advances in Materials Science and Engineering

(f)(e)(d)

(c)

(b)

(a)

Shear stressNormal displacement

Tension crackShear crack

0

2

4

6

8

Shea

r stre

ss (M

Pa)

05 10 15 20 2500Shear displacement (mm)

00

01

02

03

04

05

06

07

08

Nor

mal

disp

lace

men

t (times10

minus3

m)

0

100

200

300

400

500

600

Crac

k nu

mbe

r

Figure 12 Evolution of the shear stress number of cracks and normal displacement

Rock

Bolt Grout

T

C

C

T

(a)

F F F

F F F

(b)

Figure 13 Stress state of the bolted rock joint system and illustration of the mechanism behind compression-induced tension cracks

Table 5 Number of cracks in the specimen at different monitoringpoints

Monitoringpoint Crack number Monitoring point Crack number

a 0 d 255b 21 e 446c 40 f 592

of PFC based on DEMThe anchoring mechanism of the boltatdifferentanchorage angles has been discussed at themacro-microlevels and the following conclusions have been drawn

(1) The bolt can improve the shear resistance of the jointmainly due to cohesion Tilted anchor bolts not only improvethe shear strength of the joint surface but also result in a lowerstress in the anchor bolt The peak shear strength can be 50greater than that of a vertically installed anchor bolt

(2) A tilted bolt allows for fully mobilized anchor perfor-manceThis configuration not only restricts increment in thenormal displacement but also indirectly improves the stress

state of the anchorage system and increases the shear capacityof the anchor joint The comprehensive shear displacementand shear stress data along with the bolt stress state analysisreveal that the optimal anchorage angle should be between45∘ and 75∘

(3) During the shearing process when the rock bolt andgrout are commonly constrained the stress concentrationphenomenon around the bolt and joint increases the sheardisplacement and contact force After reaching the residualphase the contact force exhibits slight fluctuations due to thesliding of the joint surface

(4) Based on the disk flow software PFC2D crack initia-tion for an anchorage angle of 75∘ will start from the contactforce concentration due to the interactions between the rockgrout and bolt Finally the cracks are mainly distributedin the joint and around the bolt for larger displacementsThe cracks are mainly tensile cracks which correspond tothe formation mechanism for a compression-induced tensilecrack The shear cracking was mainly caused by the rotationof the disks during the shearing process

Advances in Materials Science and Engineering 13

minus005 minus003 minus001 001 003 005

minus002minus001

0001002

minus551525354555

Con

tact

forc

e (N

)

times103

(a) Monitoring point a

minus002minus001

0001002

Con

tact

forc

e (N

)

minus10

10

30

50

70

90

110times103

minus005 minus003 minus001 001 003 005

(b) Monitoring point b

minus002minus001

0001002

Con

tact

forc

e (N

)minus101030507090110

times103

minus005 minus003 minus001 001 003 005

(c) Monitoring point c

minus002minus001

0001002

Con

tact

forc

e (N

)

minus10

10

30

50

70

90

110times103

minus005 minus003 minus001 001 003 005

(d) Monitoring point d

minus002minus001

0001002

minus51535557595115

Con

tact

forc

e (N

)

times103

minus005 minus003 minus001 001 003 005

(e) Monitoring point e

minus002minus001

0001002

minus10

10

30

50

70

90

Con

tact

forc

e (N

)

times103

minus005 minus003 minus001 001 003 005

(f) Monitoring point f

Figure 14 Evolution of the contact force and cracking at different monitoring points

14 Advances in Materials Science and Engineering

Particle rotation (in radians)minus120 to minus01

minus01 to minus0001

minus0001 to minus00005

minus00005 to 00005

00005 to 00010001 to 01

01 to 120

(a) Monitoring point a

Particle rotation (in radians)minus120 to minus01

minus01 to minus0001

minus0001 to minus00005

00005 to 00010001 to 01

01 to 120minus00005 to 00005

(b) Monitoring point b

Particle rotation (in radians)minus120 to minus01

minus01 to minus0001

minus0001 to minus00005

00005 to 00010001 to 01

01 to 120minus00005 to 00005

(c) Monitoring point c

Particle rotation (in radians)minus120 to minus01

minus01 to minus0001

minus0001 to minus00005

00005 to 00010001 to 01

01 to 120minus00005 to 00005

(d) Monitoring point d

Particle rotation (in radians)minus120 to minus01

minus01 to minus0001

minus0001 to minus00005

00005 to 00010001 to 01

01 to 120minus00005 to 00005

(e) Monitoring point e

Particle rotation (in radians)minus120 to minus01

minus01 to minus0001

minus0001 to minus00005

00005 to 00010001 to 01

01 to 120minus00005 to 00005

(f) Monitoring point f

Figure 15 Evolution of disk rotation in radians in the specimen at different monitoring points

0

5

10

15

20

Qua

ntita

tive p

ropo

rtio

n (

)

05 10 15 20 25 3000Shear displacement (mm)

gt 01Rotation (in radians)gt 01|Rotation (in radians)|

lt minus01Rotation (in radians)

Figure 16 Quantitative proportion variation of rotations exceeding 01 radians

Disclosure

GangWang and Yongzheng Zhang should be regarded as co-first authors

Conflicts of Interest

The authors declare that there are no conflicts of interestregarding the publication of this paper

Advances in Materials Science and Engineering 15

Acknowledgments

This study was supported by the National Natural ScienceFoundation of China (nos 51479108 51379117 and 51509261)China Postdoctoral Science Foundation (2015M580599) theTaishan Scholar Talent Team Support Plan for Advantagedamp Unique Discipline Areas the Postdoctoral Science Foun-dation of Shandong Province (no 201402014) and the Shan-dong Province Outstanding Young Scientist Research AwardFund (BS2014NJ006)

References

[1] F D Patton ldquoMultiple modes of shear failure in rockrdquo inProceedings of the 1st Congress on International Society for RockMechanics pp 509ndash513 1966

[2] R E Goodman Methods of Geological Engineering in Discon-tinuous Rocks West Publishing Company New York NY USA1976

[3] N Barton and V Choubey ldquoThe shear strength of rock joints intheory and practicerdquo Rock Mechanics Felsmechanik Mecaniquedes Roches vol 10 no 1-2 pp 1ndash54 1977

[4] P M Dight and H K Chiu ldquoPrediction of shear behavior ofjoints using profilesrdquo International Journal of Rock Mechanicsand Mining Sciences amp Geomechanics Abstracts vol 18 no 5pp 369ndash386 1981

[5] N Barton Modelling Rock Joint Behaviour from In Situ BlockTests Implications for NuclearWaste Repository Design vol 308Office ofNuclearWaste IsolationONWI ColumbusOHUSA1982

[6] T Esaki S Du Y Jiang and Y Wada ldquoEffect of the asperitydamage on the shear behavior of irregular rock jointsrdquo inProceedings of the Second International Conference on Analysisof Discontinuous Deformation pp 459ndash464 Kyoto Japan 1997

[7] Y Jiang J Xiao Y Tanabashi and T Mizokami ldquoDevelopmentof an automated servo-controlled direct shear apparatus apply-ing a constant normal stiffness conditionrdquo International Journalof Rock Mechanics and Mining Sciences vol 41 no 2 pp 275ndash286 2004

[8] Y J Jiang B Li and Y Tanabashi ldquoEstimating the relationbetween surface roughness and mechanical properties of rockjointsrdquo International Journal of Rock Mechanics and MiningSciences vol 43 no 6 pp 837ndash846 2006

[9] C-C Xia Z-C Tang W-M Xiao and Y-L Song ldquoNew peakshear strength criterion of rock joints based on quantifiedsurface descriptionrdquo Rock Mechanics and Rock Engineering vol47 no 2 pp 387ndash400 2014

[10] S Bjurstrom ldquoShear strength of hard rock joints reinforced bygrouted untensioned boltsrdquo in Proceedings of the In Proceedingsof the 3rd International ISRM Congress pp 1194ndash1199 DenverColo USA 1974

[11] X Ge and J Liu ldquoStudy of the shear resistance behaviour ofbolted rock jointsrdquo Journal of Geotechnical Engineering vol 10no 1 pp 8ndash19 1988 (Chinese)

[12] K Spang and P Egger ldquoAction of fully-grouted bolts in jointedrock and factors of influencerdquo Rock Mechanics and Rock Engi-neering vol 23 no 3 pp 201ndash229 1990

[13] P Egger and L Zabuski ldquoBehaviour of rough bolted joints indirect shear testsrdquo in Proceedings of the 7th ISRM Congress vol30 p 342 Aachen Germany September 1991

[14] A M Ferrero ldquoThe shear strength of reinforced rock jointsrdquoInternational Journal of Rock Mechanics and Mining Sciencesand vol 32 no 6 pp 595ndash605 1995

[15] F Pellet and P Egger ldquoAnalytical model for the mechanicalbehaviour of bolted rock joints subjected to shearingrdquo RockMechanics and Rock Engineering vol 29 no 2 pp 73ndash97 1996

[16] H Jalalifar and N Aziz ldquoExperimental and 3D numericalsimulation of reinforced shear jointsrdquo RockMechanics and RockEngineering vol 43 no 1 pp 95ndash103 2010

[17] P Wang T Feng Y-J Zhu and W-J Yu ldquoExperimental studyand numerical simulation of anchoringmechanismof anchoredrocklikematerial with prefabricated fracturerdquoYantu LixueRockand Soil Mechanics vol 37 no 3 pp 793ndash801 2016 (Chinese)

[18] L Li P C Hagan S Saydam B Hebblewhite and Y Li ldquoPara-metric study of rockbolt shear behaviour by double shear testrdquoRock Mechanics and Rock Engineering vol 49 no 12 pp 4787ndash4797 2016

[19] Itasca Consulting Group Inc ldquoPFC2D (Particle Flow Code in2D)Theory and Backgroundrdquo Minneapolis Minn USA 2008

[20] P A Cundall andOD L Strack ldquoThe distinct numericalmodelfor granular assembliesrdquo Geotechnique vol 29 no 1 pp 47ndash651979

[21] J Zhou and Y Chi ldquoThe method of particle flow and PFC2Dcoderdquo Rock and Soil Mechanics vol 21 no 3 pp 271ndash274 2000(Chinese)

[22] D O Potyondy and P A Cundall ldquoA bonded-particle modelfor rockrdquo International Journal of Rock Mechanics and MiningSciences vol 41 no 8 pp 1329ndash1364 2004

[23] N Cho CDMartin andDC Sego ldquoA clumped particlemodelfor rockrdquo International Journal of Rock Mechanics and MiningSciences vol 44 no 7 pp 997ndash1010 2007

[24] Z Zhao L Jing and I Neretnieks ldquoParticle mechanics modelfor the effects of shear on solute retardation coefficient in rockfracturesrdquo International Journal of Rock Mechanics and MiningSciences vol 52 pp 92ndash102 2012

[25] C Xia Y Song Z Tang Y Song and C Shou ldquoParticleflow numerical simulation for shear behavior of rough jointsrdquoChinese Journal of Rock Mechanics and Engineering vol 31 no8 pp 1545ndash1552 2012 (Chinese)

[26] Y Zhou A Misra S Wu and X Zhang ldquoMacro- and meso-analyses of rock joint direct shear test using particle flowtheoryrdquoChinese Journal of RockMechanics and Engineering vol31 no 6 pp 1245ndash1256 2012

[27] RH Cao P CaoH Lin K Zhang andXWTan ldquoParticle flowanalysis of direct shear tests on joints with different roughnessrdquoRock and Soil Mechanics vol 34 supplement 2 pp 456ndash4642013 (Chinese)

Submit your manuscripts athttpswwwhindawicom

ScientificaHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CorrosionInternational Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Polymer ScienceInternational Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CeramicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CompositesJournal of

NanoparticlesJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Biomaterials

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

NanoscienceJournal of

TextilesHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Journal of

NanotechnologyHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

CrystallographyJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CoatingsJournal of

Advances in

Materials Science and EngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Smart Materials Research

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MetallurgyJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

BioMed Research International

MaterialsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Advances in Materials Science and Engineering 13

minus005 minus003 minus001 001 003 005

minus002minus001

0001002

minus551525354555

Con

tact

forc

e (N

)

times103

(a) Monitoring point a

minus002minus001

0001002

Con

tact

forc

e (N

)

minus10

10

30

50

70

90

110times103

minus005 minus003 minus001 001 003 005

(b) Monitoring point b

minus002minus001

0001002

Con

tact

forc

e (N

)minus101030507090110

times103

minus005 minus003 minus001 001 003 005

(c) Monitoring point c

minus002minus001

0001002

Con

tact

forc

e (N

)

minus10

10

30

50

70

90

110times103

minus005 minus003 minus001 001 003 005

(d) Monitoring point d

minus002minus001

0001002

minus51535557595115

Con

tact

forc

e (N

)

times103

minus005 minus003 minus001 001 003 005

(e) Monitoring point e

minus002minus001

0001002

minus10

10

30

50

70

90

Con

tact

forc

e (N

)

times103

minus005 minus003 minus001 001 003 005

(f) Monitoring point f

Figure 14 Evolution of the contact force and cracking at different monitoring points

14 Advances in Materials Science and Engineering

Particle rotation (in radians)minus120 to minus01

minus01 to minus0001

minus0001 to minus00005

minus00005 to 00005

00005 to 00010001 to 01

01 to 120

(a) Monitoring point a

Particle rotation (in radians)minus120 to minus01

minus01 to minus0001

minus0001 to minus00005

00005 to 00010001 to 01

01 to 120minus00005 to 00005

(b) Monitoring point b

Particle rotation (in radians)minus120 to minus01

minus01 to minus0001

minus0001 to minus00005

00005 to 00010001 to 01

01 to 120minus00005 to 00005

(c) Monitoring point c

Particle rotation (in radians)minus120 to minus01

minus01 to minus0001

minus0001 to minus00005

00005 to 00010001 to 01

01 to 120minus00005 to 00005

(d) Monitoring point d

Particle rotation (in radians)minus120 to minus01

minus01 to minus0001

minus0001 to minus00005

00005 to 00010001 to 01

01 to 120minus00005 to 00005

(e) Monitoring point e

Particle rotation (in radians)minus120 to minus01

minus01 to minus0001

minus0001 to minus00005

00005 to 00010001 to 01

01 to 120minus00005 to 00005

(f) Monitoring point f

Figure 15 Evolution of disk rotation in radians in the specimen at different monitoring points

0

5

10

15

20

Qua

ntita

tive p

ropo

rtio

n (

)

05 10 15 20 25 3000Shear displacement (mm)

gt 01Rotation (in radians)gt 01|Rotation (in radians)|

lt minus01Rotation (in radians)

Figure 16 Quantitative proportion variation of rotations exceeding 01 radians

Disclosure

GangWang and Yongzheng Zhang should be regarded as co-first authors

Conflicts of Interest

The authors declare that there are no conflicts of interestregarding the publication of this paper

Advances in Materials Science and Engineering 15

Acknowledgments

This study was supported by the National Natural ScienceFoundation of China (nos 51479108 51379117 and 51509261)China Postdoctoral Science Foundation (2015M580599) theTaishan Scholar Talent Team Support Plan for Advantagedamp Unique Discipline Areas the Postdoctoral Science Foun-dation of Shandong Province (no 201402014) and the Shan-dong Province Outstanding Young Scientist Research AwardFund (BS2014NJ006)

References

[1] F D Patton ldquoMultiple modes of shear failure in rockrdquo inProceedings of the 1st Congress on International Society for RockMechanics pp 509ndash513 1966

[2] R E Goodman Methods of Geological Engineering in Discon-tinuous Rocks West Publishing Company New York NY USA1976

[3] N Barton and V Choubey ldquoThe shear strength of rock joints intheory and practicerdquo Rock Mechanics Felsmechanik Mecaniquedes Roches vol 10 no 1-2 pp 1ndash54 1977

[4] P M Dight and H K Chiu ldquoPrediction of shear behavior ofjoints using profilesrdquo International Journal of Rock Mechanicsand Mining Sciences amp Geomechanics Abstracts vol 18 no 5pp 369ndash386 1981

[5] N Barton Modelling Rock Joint Behaviour from In Situ BlockTests Implications for NuclearWaste Repository Design vol 308Office ofNuclearWaste IsolationONWI ColumbusOHUSA1982

[6] T Esaki S Du Y Jiang and Y Wada ldquoEffect of the asperitydamage on the shear behavior of irregular rock jointsrdquo inProceedings of the Second International Conference on Analysisof Discontinuous Deformation pp 459ndash464 Kyoto Japan 1997

[7] Y Jiang J Xiao Y Tanabashi and T Mizokami ldquoDevelopmentof an automated servo-controlled direct shear apparatus apply-ing a constant normal stiffness conditionrdquo International Journalof Rock Mechanics and Mining Sciences vol 41 no 2 pp 275ndash286 2004

[8] Y J Jiang B Li and Y Tanabashi ldquoEstimating the relationbetween surface roughness and mechanical properties of rockjointsrdquo International Journal of Rock Mechanics and MiningSciences vol 43 no 6 pp 837ndash846 2006

[9] C-C Xia Z-C Tang W-M Xiao and Y-L Song ldquoNew peakshear strength criterion of rock joints based on quantifiedsurface descriptionrdquo Rock Mechanics and Rock Engineering vol47 no 2 pp 387ndash400 2014

[10] S Bjurstrom ldquoShear strength of hard rock joints reinforced bygrouted untensioned boltsrdquo in Proceedings of the In Proceedingsof the 3rd International ISRM Congress pp 1194ndash1199 DenverColo USA 1974

[11] X Ge and J Liu ldquoStudy of the shear resistance behaviour ofbolted rock jointsrdquo Journal of Geotechnical Engineering vol 10no 1 pp 8ndash19 1988 (Chinese)

[12] K Spang and P Egger ldquoAction of fully-grouted bolts in jointedrock and factors of influencerdquo Rock Mechanics and Rock Engi-neering vol 23 no 3 pp 201ndash229 1990

[13] P Egger and L Zabuski ldquoBehaviour of rough bolted joints indirect shear testsrdquo in Proceedings of the 7th ISRM Congress vol30 p 342 Aachen Germany September 1991

[14] A M Ferrero ldquoThe shear strength of reinforced rock jointsrdquoInternational Journal of Rock Mechanics and Mining Sciencesand vol 32 no 6 pp 595ndash605 1995

[15] F Pellet and P Egger ldquoAnalytical model for the mechanicalbehaviour of bolted rock joints subjected to shearingrdquo RockMechanics and Rock Engineering vol 29 no 2 pp 73ndash97 1996

[16] H Jalalifar and N Aziz ldquoExperimental and 3D numericalsimulation of reinforced shear jointsrdquo RockMechanics and RockEngineering vol 43 no 1 pp 95ndash103 2010

[17] P Wang T Feng Y-J Zhu and W-J Yu ldquoExperimental studyand numerical simulation of anchoringmechanismof anchoredrocklikematerial with prefabricated fracturerdquoYantu LixueRockand Soil Mechanics vol 37 no 3 pp 793ndash801 2016 (Chinese)

[18] L Li P C Hagan S Saydam B Hebblewhite and Y Li ldquoPara-metric study of rockbolt shear behaviour by double shear testrdquoRock Mechanics and Rock Engineering vol 49 no 12 pp 4787ndash4797 2016

[19] Itasca Consulting Group Inc ldquoPFC2D (Particle Flow Code in2D)Theory and Backgroundrdquo Minneapolis Minn USA 2008

[20] P A Cundall andOD L Strack ldquoThe distinct numericalmodelfor granular assembliesrdquo Geotechnique vol 29 no 1 pp 47ndash651979

[21] J Zhou and Y Chi ldquoThe method of particle flow and PFC2Dcoderdquo Rock and Soil Mechanics vol 21 no 3 pp 271ndash274 2000(Chinese)

[22] D O Potyondy and P A Cundall ldquoA bonded-particle modelfor rockrdquo International Journal of Rock Mechanics and MiningSciences vol 41 no 8 pp 1329ndash1364 2004

[23] N Cho CDMartin andDC Sego ldquoA clumped particlemodelfor rockrdquo International Journal of Rock Mechanics and MiningSciences vol 44 no 7 pp 997ndash1010 2007

[24] Z Zhao L Jing and I Neretnieks ldquoParticle mechanics modelfor the effects of shear on solute retardation coefficient in rockfracturesrdquo International Journal of Rock Mechanics and MiningSciences vol 52 pp 92ndash102 2012

[25] C Xia Y Song Z Tang Y Song and C Shou ldquoParticleflow numerical simulation for shear behavior of rough jointsrdquoChinese Journal of Rock Mechanics and Engineering vol 31 no8 pp 1545ndash1552 2012 (Chinese)

[26] Y Zhou A Misra S Wu and X Zhang ldquoMacro- and meso-analyses of rock joint direct shear test using particle flowtheoryrdquoChinese Journal of RockMechanics and Engineering vol31 no 6 pp 1245ndash1256 2012

[27] RH Cao P CaoH Lin K Zhang andXWTan ldquoParticle flowanalysis of direct shear tests on joints with different roughnessrdquoRock and Soil Mechanics vol 34 supplement 2 pp 456ndash4642013 (Chinese)

Submit your manuscripts athttpswwwhindawicom

ScientificaHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CorrosionInternational Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Polymer ScienceInternational Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CeramicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CompositesJournal of

NanoparticlesJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Biomaterials

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

NanoscienceJournal of

TextilesHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Journal of

NanotechnologyHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

CrystallographyJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CoatingsJournal of

Advances in

Materials Science and EngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Smart Materials Research

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MetallurgyJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

BioMed Research International

MaterialsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

14 Advances in Materials Science and Engineering

Particle rotation (in radians)minus120 to minus01

minus01 to minus0001

minus0001 to minus00005

minus00005 to 00005

00005 to 00010001 to 01

01 to 120

(a) Monitoring point a

Particle rotation (in radians)minus120 to minus01

minus01 to minus0001

minus0001 to minus00005

00005 to 00010001 to 01

01 to 120minus00005 to 00005

(b) Monitoring point b

Particle rotation (in radians)minus120 to minus01

minus01 to minus0001

minus0001 to minus00005

00005 to 00010001 to 01

01 to 120minus00005 to 00005

(c) Monitoring point c

Particle rotation (in radians)minus120 to minus01

minus01 to minus0001

minus0001 to minus00005

00005 to 00010001 to 01

01 to 120minus00005 to 00005

(d) Monitoring point d

Particle rotation (in radians)minus120 to minus01

minus01 to minus0001

minus0001 to minus00005

00005 to 00010001 to 01

01 to 120minus00005 to 00005

(e) Monitoring point e

Particle rotation (in radians)minus120 to minus01

minus01 to minus0001

minus0001 to minus00005

00005 to 00010001 to 01

01 to 120minus00005 to 00005

(f) Monitoring point f

Figure 15 Evolution of disk rotation in radians in the specimen at different monitoring points

0

5

10

15

20

Qua

ntita

tive p

ropo

rtio

n (

)

05 10 15 20 25 3000Shear displacement (mm)

gt 01Rotation (in radians)gt 01|Rotation (in radians)|

lt minus01Rotation (in radians)

Figure 16 Quantitative proportion variation of rotations exceeding 01 radians

Disclosure

GangWang and Yongzheng Zhang should be regarded as co-first authors

Conflicts of Interest

The authors declare that there are no conflicts of interestregarding the publication of this paper

Advances in Materials Science and Engineering 15

Acknowledgments

This study was supported by the National Natural ScienceFoundation of China (nos 51479108 51379117 and 51509261)China Postdoctoral Science Foundation (2015M580599) theTaishan Scholar Talent Team Support Plan for Advantagedamp Unique Discipline Areas the Postdoctoral Science Foun-dation of Shandong Province (no 201402014) and the Shan-dong Province Outstanding Young Scientist Research AwardFund (BS2014NJ006)

References

[1] F D Patton ldquoMultiple modes of shear failure in rockrdquo inProceedings of the 1st Congress on International Society for RockMechanics pp 509ndash513 1966

[2] R E Goodman Methods of Geological Engineering in Discon-tinuous Rocks West Publishing Company New York NY USA1976

[3] N Barton and V Choubey ldquoThe shear strength of rock joints intheory and practicerdquo Rock Mechanics Felsmechanik Mecaniquedes Roches vol 10 no 1-2 pp 1ndash54 1977

[4] P M Dight and H K Chiu ldquoPrediction of shear behavior ofjoints using profilesrdquo International Journal of Rock Mechanicsand Mining Sciences amp Geomechanics Abstracts vol 18 no 5pp 369ndash386 1981

[5] N Barton Modelling Rock Joint Behaviour from In Situ BlockTests Implications for NuclearWaste Repository Design vol 308Office ofNuclearWaste IsolationONWI ColumbusOHUSA1982

[6] T Esaki S Du Y Jiang and Y Wada ldquoEffect of the asperitydamage on the shear behavior of irregular rock jointsrdquo inProceedings of the Second International Conference on Analysisof Discontinuous Deformation pp 459ndash464 Kyoto Japan 1997

[7] Y Jiang J Xiao Y Tanabashi and T Mizokami ldquoDevelopmentof an automated servo-controlled direct shear apparatus apply-ing a constant normal stiffness conditionrdquo International Journalof Rock Mechanics and Mining Sciences vol 41 no 2 pp 275ndash286 2004

[8] Y J Jiang B Li and Y Tanabashi ldquoEstimating the relationbetween surface roughness and mechanical properties of rockjointsrdquo International Journal of Rock Mechanics and MiningSciences vol 43 no 6 pp 837ndash846 2006

[9] C-C Xia Z-C Tang W-M Xiao and Y-L Song ldquoNew peakshear strength criterion of rock joints based on quantifiedsurface descriptionrdquo Rock Mechanics and Rock Engineering vol47 no 2 pp 387ndash400 2014

[10] S Bjurstrom ldquoShear strength of hard rock joints reinforced bygrouted untensioned boltsrdquo in Proceedings of the In Proceedingsof the 3rd International ISRM Congress pp 1194ndash1199 DenverColo USA 1974

[11] X Ge and J Liu ldquoStudy of the shear resistance behaviour ofbolted rock jointsrdquo Journal of Geotechnical Engineering vol 10no 1 pp 8ndash19 1988 (Chinese)

[12] K Spang and P Egger ldquoAction of fully-grouted bolts in jointedrock and factors of influencerdquo Rock Mechanics and Rock Engi-neering vol 23 no 3 pp 201ndash229 1990

[13] P Egger and L Zabuski ldquoBehaviour of rough bolted joints indirect shear testsrdquo in Proceedings of the 7th ISRM Congress vol30 p 342 Aachen Germany September 1991

[14] A M Ferrero ldquoThe shear strength of reinforced rock jointsrdquoInternational Journal of Rock Mechanics and Mining Sciencesand vol 32 no 6 pp 595ndash605 1995

[15] F Pellet and P Egger ldquoAnalytical model for the mechanicalbehaviour of bolted rock joints subjected to shearingrdquo RockMechanics and Rock Engineering vol 29 no 2 pp 73ndash97 1996

[16] H Jalalifar and N Aziz ldquoExperimental and 3D numericalsimulation of reinforced shear jointsrdquo RockMechanics and RockEngineering vol 43 no 1 pp 95ndash103 2010

[17] P Wang T Feng Y-J Zhu and W-J Yu ldquoExperimental studyand numerical simulation of anchoringmechanismof anchoredrocklikematerial with prefabricated fracturerdquoYantu LixueRockand Soil Mechanics vol 37 no 3 pp 793ndash801 2016 (Chinese)

[18] L Li P C Hagan S Saydam B Hebblewhite and Y Li ldquoPara-metric study of rockbolt shear behaviour by double shear testrdquoRock Mechanics and Rock Engineering vol 49 no 12 pp 4787ndash4797 2016

[19] Itasca Consulting Group Inc ldquoPFC2D (Particle Flow Code in2D)Theory and Backgroundrdquo Minneapolis Minn USA 2008

[20] P A Cundall andOD L Strack ldquoThe distinct numericalmodelfor granular assembliesrdquo Geotechnique vol 29 no 1 pp 47ndash651979

[21] J Zhou and Y Chi ldquoThe method of particle flow and PFC2Dcoderdquo Rock and Soil Mechanics vol 21 no 3 pp 271ndash274 2000(Chinese)

[22] D O Potyondy and P A Cundall ldquoA bonded-particle modelfor rockrdquo International Journal of Rock Mechanics and MiningSciences vol 41 no 8 pp 1329ndash1364 2004

[23] N Cho CDMartin andDC Sego ldquoA clumped particlemodelfor rockrdquo International Journal of Rock Mechanics and MiningSciences vol 44 no 7 pp 997ndash1010 2007

[24] Z Zhao L Jing and I Neretnieks ldquoParticle mechanics modelfor the effects of shear on solute retardation coefficient in rockfracturesrdquo International Journal of Rock Mechanics and MiningSciences vol 52 pp 92ndash102 2012

[25] C Xia Y Song Z Tang Y Song and C Shou ldquoParticleflow numerical simulation for shear behavior of rough jointsrdquoChinese Journal of Rock Mechanics and Engineering vol 31 no8 pp 1545ndash1552 2012 (Chinese)

[26] Y Zhou A Misra S Wu and X Zhang ldquoMacro- and meso-analyses of rock joint direct shear test using particle flowtheoryrdquoChinese Journal of RockMechanics and Engineering vol31 no 6 pp 1245ndash1256 2012

[27] RH Cao P CaoH Lin K Zhang andXWTan ldquoParticle flowanalysis of direct shear tests on joints with different roughnessrdquoRock and Soil Mechanics vol 34 supplement 2 pp 456ndash4642013 (Chinese)

Submit your manuscripts athttpswwwhindawicom

ScientificaHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CorrosionInternational Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Polymer ScienceInternational Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CeramicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CompositesJournal of

NanoparticlesJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Biomaterials

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

NanoscienceJournal of

TextilesHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Journal of

NanotechnologyHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

CrystallographyJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CoatingsJournal of

Advances in

Materials Science and EngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Smart Materials Research

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MetallurgyJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

BioMed Research International

MaterialsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Advances in Materials Science and Engineering 15

Acknowledgments

This study was supported by the National Natural ScienceFoundation of China (nos 51479108 51379117 and 51509261)China Postdoctoral Science Foundation (2015M580599) theTaishan Scholar Talent Team Support Plan for Advantagedamp Unique Discipline Areas the Postdoctoral Science Foun-dation of Shandong Province (no 201402014) and the Shan-dong Province Outstanding Young Scientist Research AwardFund (BS2014NJ006)

References

[1] F D Patton ldquoMultiple modes of shear failure in rockrdquo inProceedings of the 1st Congress on International Society for RockMechanics pp 509ndash513 1966

[2] R E Goodman Methods of Geological Engineering in Discon-tinuous Rocks West Publishing Company New York NY USA1976

[3] N Barton and V Choubey ldquoThe shear strength of rock joints intheory and practicerdquo Rock Mechanics Felsmechanik Mecaniquedes Roches vol 10 no 1-2 pp 1ndash54 1977

[4] P M Dight and H K Chiu ldquoPrediction of shear behavior ofjoints using profilesrdquo International Journal of Rock Mechanicsand Mining Sciences amp Geomechanics Abstracts vol 18 no 5pp 369ndash386 1981

[5] N Barton Modelling Rock Joint Behaviour from In Situ BlockTests Implications for NuclearWaste Repository Design vol 308Office ofNuclearWaste IsolationONWI ColumbusOHUSA1982

[6] T Esaki S Du Y Jiang and Y Wada ldquoEffect of the asperitydamage on the shear behavior of irregular rock jointsrdquo inProceedings of the Second International Conference on Analysisof Discontinuous Deformation pp 459ndash464 Kyoto Japan 1997

[7] Y Jiang J Xiao Y Tanabashi and T Mizokami ldquoDevelopmentof an automated servo-controlled direct shear apparatus apply-ing a constant normal stiffness conditionrdquo International Journalof Rock Mechanics and Mining Sciences vol 41 no 2 pp 275ndash286 2004

[8] Y J Jiang B Li and Y Tanabashi ldquoEstimating the relationbetween surface roughness and mechanical properties of rockjointsrdquo International Journal of Rock Mechanics and MiningSciences vol 43 no 6 pp 837ndash846 2006

[9] C-C Xia Z-C Tang W-M Xiao and Y-L Song ldquoNew peakshear strength criterion of rock joints based on quantifiedsurface descriptionrdquo Rock Mechanics and Rock Engineering vol47 no 2 pp 387ndash400 2014

[10] S Bjurstrom ldquoShear strength of hard rock joints reinforced bygrouted untensioned boltsrdquo in Proceedings of the In Proceedingsof the 3rd International ISRM Congress pp 1194ndash1199 DenverColo USA 1974

[11] X Ge and J Liu ldquoStudy of the shear resistance behaviour ofbolted rock jointsrdquo Journal of Geotechnical Engineering vol 10no 1 pp 8ndash19 1988 (Chinese)

[12] K Spang and P Egger ldquoAction of fully-grouted bolts in jointedrock and factors of influencerdquo Rock Mechanics and Rock Engi-neering vol 23 no 3 pp 201ndash229 1990

[13] P Egger and L Zabuski ldquoBehaviour of rough bolted joints indirect shear testsrdquo in Proceedings of the 7th ISRM Congress vol30 p 342 Aachen Germany September 1991

[14] A M Ferrero ldquoThe shear strength of reinforced rock jointsrdquoInternational Journal of Rock Mechanics and Mining Sciencesand vol 32 no 6 pp 595ndash605 1995

[15] F Pellet and P Egger ldquoAnalytical model for the mechanicalbehaviour of bolted rock joints subjected to shearingrdquo RockMechanics and Rock Engineering vol 29 no 2 pp 73ndash97 1996

[16] H Jalalifar and N Aziz ldquoExperimental and 3D numericalsimulation of reinforced shear jointsrdquo RockMechanics and RockEngineering vol 43 no 1 pp 95ndash103 2010

[17] P Wang T Feng Y-J Zhu and W-J Yu ldquoExperimental studyand numerical simulation of anchoringmechanismof anchoredrocklikematerial with prefabricated fracturerdquoYantu LixueRockand Soil Mechanics vol 37 no 3 pp 793ndash801 2016 (Chinese)

[18] L Li P C Hagan S Saydam B Hebblewhite and Y Li ldquoPara-metric study of rockbolt shear behaviour by double shear testrdquoRock Mechanics and Rock Engineering vol 49 no 12 pp 4787ndash4797 2016

[19] Itasca Consulting Group Inc ldquoPFC2D (Particle Flow Code in2D)Theory and Backgroundrdquo Minneapolis Minn USA 2008

[20] P A Cundall andOD L Strack ldquoThe distinct numericalmodelfor granular assembliesrdquo Geotechnique vol 29 no 1 pp 47ndash651979

[21] J Zhou and Y Chi ldquoThe method of particle flow and PFC2Dcoderdquo Rock and Soil Mechanics vol 21 no 3 pp 271ndash274 2000(Chinese)

[22] D O Potyondy and P A Cundall ldquoA bonded-particle modelfor rockrdquo International Journal of Rock Mechanics and MiningSciences vol 41 no 8 pp 1329ndash1364 2004

[23] N Cho CDMartin andDC Sego ldquoA clumped particlemodelfor rockrdquo International Journal of Rock Mechanics and MiningSciences vol 44 no 7 pp 997ndash1010 2007

[24] Z Zhao L Jing and I Neretnieks ldquoParticle mechanics modelfor the effects of shear on solute retardation coefficient in rockfracturesrdquo International Journal of Rock Mechanics and MiningSciences vol 52 pp 92ndash102 2012

[25] C Xia Y Song Z Tang Y Song and C Shou ldquoParticleflow numerical simulation for shear behavior of rough jointsrdquoChinese Journal of Rock Mechanics and Engineering vol 31 no8 pp 1545ndash1552 2012 (Chinese)

[26] Y Zhou A Misra S Wu and X Zhang ldquoMacro- and meso-analyses of rock joint direct shear test using particle flowtheoryrdquoChinese Journal of RockMechanics and Engineering vol31 no 6 pp 1245ndash1256 2012

[27] RH Cao P CaoH Lin K Zhang andXWTan ldquoParticle flowanalysis of direct shear tests on joints with different roughnessrdquoRock and Soil Mechanics vol 34 supplement 2 pp 456ndash4642013 (Chinese)

Submit your manuscripts athttpswwwhindawicom

ScientificaHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CorrosionInternational Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Polymer ScienceInternational Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CeramicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CompositesJournal of

NanoparticlesJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Biomaterials

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

NanoscienceJournal of

TextilesHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Journal of

NanotechnologyHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

CrystallographyJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CoatingsJournal of

Advances in

Materials Science and EngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Smart Materials Research

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MetallurgyJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

BioMed Research International

MaterialsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Submit your manuscripts athttpswwwhindawicom

ScientificaHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CorrosionInternational Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Polymer ScienceInternational Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CeramicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CompositesJournal of

NanoparticlesJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Biomaterials

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

NanoscienceJournal of

TextilesHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Journal of

NanotechnologyHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

CrystallographyJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CoatingsJournal of

Advances in

Materials Science and EngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Smart Materials Research

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MetallurgyJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

BioMed Research International

MaterialsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014