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Numerical simulations of the effect of bolt inclination on the shear strength of rock joints.pdf

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Technical Note Numerical simulations of the effect of bolt inclination on the shear strength of rock joints Hang Lin a,b,n , Zheyi Xiong a , Taoying Liu a , Rihong Cao a , Ping Cao a a School of Resources and Safety Engineering, Central South University, Changsha, Hunan, China b State Key Laboratory of Coal Resources and Safe Mining, China University of Mining and Technology, Xuzhou, Jiangsu, China a r t i c l e i n f o Article history: Received 21 November 2012 Received in revised form
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  Technical Note Numerical simulations of the effect of bolt inclinationon the shear strength of rock joints Hang Lin a,b, n , Zheyi Xiong a , Taoying Liu a , Rihong Cao a , Ping Cao a a School of Resources and Safety Engineering, Central South University, Changsha, Hunan, China b State Key Laboratory of Coal Resources and Safe Mining, China University of Mining and Technology, Xuzhou, Jiangsu, China a r t i c l e i n f o  Article history: Received 21 November 2012Received in revised form7 October 2013Accepted 28 December 2013Available online 22 January 2014 Keywords: Rock jointRock massBoltShear strengthMechanical characteristicNumerical simulation 1. Introduction  Joints are widely encountered in rock engineering. Theydamage the continuity and the integrity of a rock mass, thusturning the rock anisotropic and unstable. Therefore, investiga-tions on the shear strength and deformation of rock joints areessential. Current researchers on joints primarily focus on jointmechanical characteristics, especially shear strength, by experi-mental and analytical methods [1 – 15]. For simplicity, some studiesplace joints on  󿬂 at planes [5]. However, rock joint parameters aredif  󿬁 cult to determine because of the unclear differences betweentheory and reality in terms of the interactions between engineer-ing and geological bodies. Simplifying joints to  󿬂 at states anddisregarding undulating states is impractical. Therefore, certainscholars have investigated joint characteristics through joints withirregular surfaces or different undulation angles. Patton [7]demonstrated the in 󿬂 uence of surface roughness on the shearstrength of joint by means of an experiment in which he carriedout shear tests on  ‘ saw-tooth ’  specimens. Barton and Bandis[9,11,14,15] proposed a criterion for the shear strength of rock joints based on a large number of joints tests, which takes intoaccount both slip under low normal stresses and asperity crushingunder high stresses. Barton and Choubey [9] proposed standardpro 󿬁 les of joints to determine the coef  󿬁 cient JRC in Barton – Bandiscriterion. Fox [16], Lee [13] and Jafari [17] studied the stress – strainproperty of joints under cycling shear load. Li et al. [18] employedarti 󿬁 cial concrete joint samples with saw-tooth-shaped surfaces tostudy the strength of joints with various undulation angles underdifferent shearing velocities.The above studies are mainly conducted experimentally andanalytically. Besides, the joints investigated in previous studieswere maintained in their original state without any arti 󿬁 cialreinforcements. However, designers have always reinforced jointsto maintain joint stability in practical engineering projects, such asrock slope or tunnel. Fully grouted bolts are the most widely usedamong reinforcements. So it is meaningful to conduct investiga-tions on the mechanical characteristics and deformation of boltedrock joints [19 – 26]. Bjurstrom [27] reported the behavior of bolted rock joint under the suppose of only axial force acting on the bolt.Pellet and Egger [20] proposed an analytical model to predict thecontribution of bolts to the shear strength of a rock joint, in which,both the axial and the shear forces in the bolt were considered.Gerrard and Pande [24] numerical modeled bolt-reinforced rockblock as an equivalent material, but neglected the interactionbetween bolt and joint.Given the advances in computer science over recent decades,the adoption of numerical simulations to solve geotechnicalengineering problems has become a new trend [19,21,24,28 – 30].Numerical simulations provide a new method of studying theContents lists available at ScienceDirect journal homepage: www.elsevier.com/locate/ijrmms International Journal of Rock Mechanics & Mining Sciences 1365-1609/$-see front matter  &  2014 Elsevier Ltd. All rights reserved.http://dx.doi.org/10.1016/j.ijrmms.2013.12.010 n Corresponding author at: School of Resources and Safety Engineering, CentralSouth University, Changsha, Hunan 410083, China. Tel.:  þ 86 137 870 16941. E-mail address:  linhangabc@126.com (H. Lin).International Journal of Rock Mechanics & Mining Sciences 66 (2014) 49 – 56  properties of bolt-reinforced joints. Numerical calculations areef  󿬁 cient, economical, complex, and repeatable. Furthermore,numerical calculations consider factors that are beyond experi-mental control. Reasonable numerical simulation results canprovide direction for experimental tests. Therefore, this paperregards rock joints and rock mass as biomaterial modes, andadopts a fast Lagrangian analysis method of continua (FLAC3D)to set a three-dimensional (3D) bolt-reinforced model in rock joints. For bolt inclination is the most important parameter if theproperties of bolt, rock and joint are given, this paper mainlyfocuses on the effect of bolts inclination on  󿬂 at and undulating joint surfaces. 2. Modeling  Numerical simulations using codes such as FLAC are often usedfor elastic – plastic materials with complex geometrical boundaries.However, discrete FLAC3D approaches are limited in terms of setting complicated 3D models [29]. This paper adopts ANSYS tocreate models, and then employs FLAC3D to transform the modelsfor calculation. The thickness of the soft interlayer of rock mass,that is, the rock joint, is assumed to be 0.1 m for numerical modelsetting. The dimensions of the simulated models in this paper areset at 4 m for length, width, and height. This paper sets twomodels for numerical calculations to analyze rock mass propertyvariations from  󿬂 at to undulating states: the  󿬂 at joint (jointundulation angle  β  ¼ 0 1 ) and the undulating joint (joint undulationangle  β  ¼ 17 1 ). As shown in Fig. 1, the two models comprise 1951grid-points and 9420 elements and 1673 grid-points and 7485elements, respectively.Lin et al. [31] used numerical simulation to investigate jointshear strength with different undulation angles:  β  ¼ 0 1 , 9 1 ,17 1 , 24 1 ,and 31 1 . The relationship between normal stress  s n  and shearstrength  τ  c   is linear when the joint undulation angle is  β  r 17 1 . This 󿬁 nding matches the Mohr – Coulomb criterion. The failure mode of the joint is primarily identi 󿬁 ed as the slip failure along the joint.However, the relationship between  s n  and  τ  c   is nonlinear whenthe undulation angle  β   is relatively large. The corresponding failuremode was found to be a combination of slip failure along the jointand saw-tooth crushing by shearing and compression. Thispaper sets the joint undulation angle to  β  ¼ 17 1  because the jointis primitively assumed to be destroyed in the slip failure modeafter bolt reinforcement. A previous study [31] stated that whenthe undulation angle is  β   r 17 1 , shear strength increases as the joint undulation angle increases. Therefore, the sample is easilydamaged at the maximal shear strength and at the same normalstress when  β  ¼ 17 1 .In the numerical simulations, the upper part of the model isloaded with normal stress and is given a certain horizontal speedfor uniform movement. The lower part of the model is keptinvariant. The Mohr – Coulomb failure criterion is adopted as thecalculation principle. The material properties of the rock mass thatare described in this model are as follows: 2.0 GPa elastic modulus,0.2 Poisson 0 s ratio, 0.8 MPa cohesion, 24.0 kN/m 3 unit weight,18.0 1  dilatancy angle, 37.0 1  internal friction angle, and 0.4 MPatensile strength. The properties of the rock joint are as follows:0.2 GPa elastic modulus, 0.3 Poisson 0 s ratio, 0.2 MPa cohesion,19.0 kN/m 3 unit weight, 12.0 1  dilatancy angle, 24.0 1  internalfriction angle, and 0.4 MPa tensile strength.A  “ pile ”  element is used to simulate bolt reinforcementbehavior. As the instruction in FLAC3D Manual [32], the  “ pile ” element offers the combination features of beam and cable. In thissense, the  “ pile ”  element can simulate the combination of tension,shearing and bending behavior of bolts. Piles interact with the gridthrough shear and normal coupling springs, which are cohesiveand frictional in nature as well as nonlinear.The shear coupling springs located at the nodal points alongthe pile axis describe relative shear displacement between thebolt/grout interface and the grout/rock interface, as shown inFig. 2. These springs present numerically as spring-slider connec-ters, transferring forces and motion between the pile and the gridat the pile nodes.The normal coupling springs model load reversal and theformation of a gap between the pile and the grid, can simulatethe effect of the host medium squeezing around the pile. For thebehavior of the normal coupling springs, the effective con 󿬁 ningstress  s m  acting in the plane perpendicular to the bolt axis istransferred directly to the node and then is computed at nodalpoint along the bolt axis. The node exerts normal force on the gridgenerated by a proportion of axial forces.In numerical calculations, entire bolts are divided into severalunit bodies. The deformation and stress condition of the entirebolt is then determined by using the integral approach.Bolt-embedded joints are placed on different inclinations toanalyze the in 󿬂 uence of the bolt on shear effects. This paperpresumes that the bolt length is equal to 3.0 m and that the boltinclinations  θ   are equal to 15 1 , 30 1 , 45 1 , 60 1 , 75 1 , and 90 1 . Thematerial properties of the bolt are as follows: 200 GPa elasticmodulus, 0.25 Poisson 0 s ratio, 314 mm 2 section area, 1.75  10 5 N/m shear coupling spring cohesion per unit length, 30 1  shearcoupling spring friction angle, 1.0  10 9 N/m 2 shear couplingspring stiffness per unit length, 1.75  10 8 N/m normal couplingspring cohesion per unit length, and 1.0  10 9 N/m 2 normal cou-pling spring stiffness per unit length. 3. Analysis and discussion  3.1. Comparisons between unsupported and bolt-supported models The relationship between the shear stress of the model and thenormal stress in  󿬂 at and undulating joints, with a bolt inclination of 15 1 , is recorded to compare regulations on shear strength variationsin bolt-supported joints (see Fig. 3). The increase in the shearstrength of the model is evident after bolt reinforcement. A linearrelationship exists between the shear strength of the model  τ  s  andthe normal stress of the joint  s n  in  󿬂 at and undulating joints.This condition matches the Mohr – Coulomb criterion fairly well.The  󿬁 tting equation is denoted as  τ  s ¼ c  þ s n  tan  ϕ , where  c   and  ϕ represent the cohesion and internal friction angle, respectively.The equivalent cohesion of the bolt-supported joint shows anupward trend when the joint undulation angle is 0 1  or 17 1 , with a26.8% increase for the former and a 48.4% increase for the latter, asconcluded from curve  󿬁 tting (Table 1). However, the equivalentinternal friction angle has a slight reduction compared with anunsupported bolt. Thus, the bolt prevailingly performs a drawingfunction for joints with undulating angles of   β  ¼ 0 1  and  β  ¼ 17 1 , thusenhancing joint cohesion. The equivalent cohesion and the equiva-lent internal friction angle of the undulating joint are signi 󿬁 cantlygreater than those of the  󿬂 at joint regardless of whether the bolt issupported or unsupported.  3.2. Comparisons of the relationship between stress and strain For further comparisons, the relationship between stress andstrain can be determined through direct shear tests under sup-ported and unsupported states when bolt inclination is equal to15 1  (see Fig. 4). As shown in the graph, the performances of theunsupported and supported joints are similar before the shearstress – shear displacement relation curve reaches the peak. H. Lin et al. / International Journal of Rock Mechanics & Mining Sciences 66 (2014) 49 – 56  50  The curve is near model 0 s elastic stage line. The joints withtwo different states have distinctions after the peak. Moreover,the unsupported joint possesses strain-softening characteristics.The increase in shear displacement gradually decreases shear stressbecause of the destruction of bonds in the joint, which successivelyweakens shear strength. The model begins to slip when all bondsare destroyed. This condition results from the emergence of slip-lines from the mutual friction between the joint and the rock mass.The shear stress of the model is unchanged with the increase inshear displacement and even provides residual strength. Fig. 1.  Numerical calculation model. (a) Model 0 s schematic diagram, (b) bolted joint model, (c) model for joint with undulation angle 0 1 , (d) model for the joint withundulation angle 17 1  and (e) 3D joint model. H. Lin et al. / International Journal of Rock Mechanics & Mining Sciences 66 (2014) 49 – 56   51  By contrast, the supported joint possesses strain-hardeningcharacteristics because the bolt fails to generate relevant drawingand shear forces during coordination with the strain of the modelbefore the model starts to slip. After reaching peak strength, themodel starts to slip. The bolt then produces relevant axial andtransverse forces that generate a relevant stress response, thusimproving the peak shear strength value of the joint.  3.3. Effects of different bolt inclinations on shear strength The joint shear strength with various bolt inclinations whenthe joint undulation angle is equal to 0 1  is illustrated in Fig. 5. Themodel shear strength reaches the maximum value at a boltinclination of 15 1  during the shear test with a bolt undulationangle of   β  ¼ 0 1 . The increase in bolt inclination gradually decreasesthe shear strength of the model and the slope of the curve. Theslope of the curve then reaches maximum value within 15 – 30 1  of bolt inclination. The curve subsequently stabilizes as the boltinclination increases. Given the small bolt inclination and smallangle between bolt 0 s axis and model 0 s slip direction, the boltmakes full use of the drawing force and suffers from a small Fig. 2.  Idealization of bolt system. 0.0 0.2 0.4 0.6 0.8 1.00.080.100.120.140.160.180.20    S   h  e  a  r  s   t  r  e  n  g   t   h  o   f   j  o   i  n   t   /   M   P  a  Normal stress on joint /MPa bolt-supported joint unsupported joint 0.0 0.2 0.4 0.6 0.8 1.00.100.150.200.250.300.350.40    S   h  e  a  r  s   t  r  e  n  g   t   h  o   f   j  o   i  n   t   /   M   P  a  Normal stress on joint/MPa bolt-supported joint unsupported joint Fig. 3.  Comparison between an unsupported and a bolt-supported model. (a) 0 1  joint undulation angle. (b) 17 1  joint undulation angle.  Table 1 Fitting for shear strength.Undulation angle of differenttypes of jointFitting parameters Coef  󿬁 cientCohesion  c  (MPa)tan  ϕ  Frictionangle  ϕ R 0 1  of unsupported joint 0.07695 0.12955 7.38 0.999440 1  of bolt-supported joint 0.09759 0.1143 6.52 0.9998617 1  of unsupported joint 0.08263 0.3445 19.00 0.9985817 1  of bolt-supported joint 0.12264 0.31715 17.59 0.99438 0 10 20 30 40 500.000.020.040.060.080.100.120.140.16    S   h  e  a  r  s   t  r  e  s  s  o   f   j  o   i  n   t   /   M   P  a Shear displacement of joint/mm unsupported joint bolt-supported joint 0 10 20 30 40 500.000.050.100.150.200.250.30    S   h  e  a  r  s   t  r  e  s  s  o   f   j  o   i  n   t   /   M   P  a Shear displacement of joint/mm unsupported joint bolt-supported joint Fig. 4.  Comparison on the stress – strain relationship of joints with and without boltreinforcement. (a) 0 1  joint undulation angle. (b) 17 1  joint undulation angle. H. Lin et al. / International Journal of Rock Mechanics & Mining Sciences 66 (2014) 49 – 56  52
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