Explicit reinforcement models for fully-grouted rebar rock bolts

Accepted Manuscript Explicit Reinforcement Models for Fully-grouted Rebar Rock Bolts Navid Bahrani, John Hadjigeorgiou

PII:

S1674-7755(16)30259-1

DOI:

10.1016/j.jrmge.2016.07.006

Reference:

JRMGE 299

To appear in:

Journal of Rock Mechanics and Geotechnical Engineering

Received Date: 28 April 2016 Revised Date:

7 July 2016

Accepted Date: 24 July 2016

Please cite this article as: Bahrani N, Hadjigeorgiou J, Explicit Reinforcement Models for Fully-grouted Rebar Rock Bolts, Journal of Rock Mechanics and Geotechnical Engineering (2017), doi: 10.1016/ j.jrmge.2016.07.006. This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting proof before it is published in its final form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.

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Explicit Reinforcement Models for Fully-grouted Rebar Rock Bolts

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Lassonde Institute of Mining, University of Toronto, ON, Canada

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Navid Bahrani1, John Hadjigeorgiou1

Corresponding Author: Navid Bahrani, Lassonde Institute of Mining, University of Toronto, Galbraith

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Building, 35 St. George Street, Toronto, ON, Canada, M5S 1A4 Email: [email protected] Keywords: Rock reinforcement, fully-grouted rebar, Distinct Element Method, local reinforcement,

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global reinforcement, pull-out test, shear test

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Abstract Reinforcement is a means of improving the overall properties of a rock mass by using stabilizing elements such as rock bolts, cable bolts and ground anchors. Rock reinforcement is often used as a

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primary support element, applied during or immediately after excavation, to stabilize the ground and ensure safe working conditions during subsequent excavation. The design of reinforcement using numerical models can be either implicit or explicit. This paper investigates the explicit use of rock reinforcement in a discontinuous stress analysis model. A series of numerical experiments were

undertaken to evaluate the performance of local and global reinforcement models implemented in

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Universal Distinct Element Code (UDEC). This was made possible by calibrating the reinforcement

models to the laboratory behavior of a fully-grouted rebar bolt tested under pure pull and pure shear

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loading conditions. The model calibration focuses on matching different loading stages of the forcedisplacement curve including the initial elastic response, hardening behavior and the bolt rupture. The paper concludes with a discussion on the suitability of the different reinforcement models in UDEC including their advantages and limitations. Finally, it addresses the choice of input parameters required for a realistic simulation of fully-grouted rebar bolts.

Introduction

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Reinforcement is a means of improving the overall properties of a rock mass by using stabilizing elements such as rock bolts, cable bolts and ground anchors. Rock reinforcement is often used as a primary support element, applied during or immediately after excavation, to stabilize the ground and

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ensure safe working conditions during subsequent excavation. The design of reinforcement for underground excavations in rock has not evolved considerably since the

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1990’s. Design is often based on empirical rules and rock mass classification schemes. This is somewhat surprising given the development and accessibility of sophisticated stress analysis tools. A potential reason for this may be the inherent limitations of how reinforcement is represented in stress analysis software packages. A further reason is related to the difficulties associated with calibrating the numerical models to gain confidence on the implemented reinforcement tools. Numerical modeling is a valuable tool in the design of underground excavations. Continuum, discontinuum, and hybrid continuum-discontinuum codes are used to determine the resulting stresses and displacements following the introduction of an excavation in a rock mass. The design of 2

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reinforcement, using numerical models can be either implicit or explicit. An implicit design process has been outlined by Wiles et al (2004). In this approach the results of a stress analysis can be used to qualify the ground response into “broken ground” and “cracked ground”. The “broken ground” is a ground that has undergone stress driven failure and represents the dead weight that has to be

threshold criterion defines where the reinforcement anchoring begins.

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supported by the reinforcement. The "cracked ground" that is determined by a rock mass damage

An explicit design process implies that a representative reinforcement has been implemented in a stress analysis model and the results of the stress analysis process are accounting for the role and influence of

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reinforcement in the design. There are several challenges that have to be overcome in the explicit

representation of rock reinforcement in stress analysis models. The first part is the choice of the type of

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model to be used that meets the objectives of the simulation and the problem definition. The next step is to specifically address how reinforcement is implemented in the stress analysis models. The final step, and the most important one, is how one can attain a successful level of calibration of stress analysis models that can be used with confidence for design problems.

This paper focuses on the explicit representation of reinforcement in a distinct element stress analysis model. A major objective is to critically and systematically evaluate two types of rock reinforcement

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models, local and global reinforcements. The theoretical basis for these reinforcement models is reviewed as well as a description of the required input parameters. The investigation was based on laboratory experiments of cement grouted rebar bolts. The paper addresses calibration issues and investigates the behavior of fully-grouted rebar bolts under pure pull and pure shear loading conditions.

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An assessment of the advantages and limitations of the results obtained using these techniques can provide the basis for the selection of appropriate reinforcement models for a realistic simulation of rock

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reinforcement in jointed rock masses.

Laboratory tests on fully-grouted rebar

Rock bolts are the primary means of rock reinforcement for excavations in rock. Rock bolts reinforce the rock mass by one or more of the following methods: beam building, suspension of weak fractured ground to more competent layers, pressure arch, and support of discrete blocks (Hadjigeorgiou and Charette, 2001). The in situ behavior of rock bolts can best be captured by pull tests. This, however, is influenced by a multitude of parameters. A better understanding of the behavior of specific parameters

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can be obtained under controlled laboratory experiments. From a numerical perspective it is convenient to investigate the representation of reinforcement models to well defined experimental data. This can be a prelude to modelling the in situ behavior of reinforcement.

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Stjern (1995) conducted a series of laboratory tests to investigate the load-displacement behavior of different types of rock bolts subjected to tensile (pull) and shear loading. Li et al. (2014) provides a comprehensive review of the performances of both conventional and energy-absorbing rock bolts based on the results of laboratory experiments conducted by Stjern (1995). The test rig used for this purpose consisted of two concrete blocks of a uniaxial compressive strength (UCS) of 65 MPa, which could be

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moved both lateral (for shear test) and normal (for pull test) to the joint. The sides of the concrete blocks were 0.95 m, which made testing of a full sized rock bolt with standard anchorage element and

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bearing plate possible (Stjern, 1995). The pull and shear tests were conducted on fully-grouted rebar bolts, frictional bolts, cable bolts, and mechanical bolts. The complete load-displacement characteristic for the bolts were obtained during the tests using various instruments. The grout for the tests carried out on fully grouted bolts had a water/cement ratio of 0.33. The bolts were installed according to normal field installation practice for each specific bolt type. To minimize the influence of joint shear resistance during shear tests, a 1 mm thick teflon film was attached to each joint surface.

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More recently, Chen (2014) and Chen and Li (2015a; 2015b), reported the results of similar laboratory tests and evaluated the anchorage performance of the rebar bolt and the D-bolt under combined pullshear loading condition. Chen (2014) investigated the influence of displacing angle (angle between the pull and the shear displacements), joint gap and host rock strength on the load-displacement behavior

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of the rebar bolt and the D-bolt. Figure 1a shows the front view of the test rig used by Stjern (1995) and

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Chen (2014), and Figure 1b shows a sketch of the modified test rig (Chen, 2014).

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Figure 1 The test rig for static pull, shear and combined pull-shear tests: a) the front view of the test rig (after Chen, 2014); b) an oblique sketch of the test rig (after Stjern, 1995).

Figure 2a and Figure 2b compares the results of pure pull and pure shear tests on 18 mm diameter fully-

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grouted rebar bolt conducted by Stern (1995) and 20 mm diameter fully-grouted rebar bolt conducted by Chen (2014). The rebar bolt tested by Chen (2014) is relatively stronger than that tested by Stjern

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(1995) in both pull and shear tests. This could be due to the differences in the diameters of the rebar bolts used. However, their load-displacement curves are very similar in pull and shear loading conditions and both consist of three main loading stages. Under both pull and shear loading conditions, the bolts elongate elastically, then yield and harden until they reach the peak load. The bolts continue to elongate until rupture occurs. Table 1 compares the results of tests conducted by Stjern (1995) and Chen (2014) in terms of loading stages described above, including initial stiffness, yield load, peak load, displacement at the peak load, and rupture displacement. It is understood from this table and the load-displacement curves shown in Figure 2, that the yield loads are about 86% and 47% of the peak loads under pull and shear loading conditions, respectively. In this paper, the results of laboratory tests on the fully-grouted

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rebar bolt reported by Stjern (1995) (i.e., blue curves in Fig. 2) are used for the evaluation of

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reinforcement models in UDEC.

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Figure 2 Results of: a) pull test; and b) shear test, on 18 mm fully-grouted rebar (Stjern, 1995), and 20 mm fullygrouted rebar (Chen, 2014).

Table 1 Comparison between the results of pull and shear tests on fully-grouted rebar conducted by Stjern (1995) and Chen (2014). Test

Pull Shear

Stjern (1995) Chen (2014) Stjern (1995) Chen (2014)

Stiffness (kN/mm)

Yield load (kN)

Peak load (kN)

Load ratio (yield/peak)

20 29 23 14

150 181 84 90

175 211 177 195

0.86 0.86 0.47 0.46

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Displacement at peak load (mm) 35 30 39 44

Rupture displacement (mm) 39 43 43 53

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Numerical modeling of rock reinforcement

Rock reinforcements can be simulated using either material models or structural elements. Both approaches have been demonstrated to be able to represent rock reinforcement behavior under

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different loading conditions. However, when simulating an underground excavation and support system, the approach based on material model is computationally intensive, as very fine mesh elements (or zones) are required to properly simulate rock reinforcement. Therefore, the numerical representation of rock reinforcement using a material model is usually limited to the simulation of laboratory tests. Structural elements, however, can be used for the simulation of rock reinforcement under both

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laboratory and field conditions.

Examples of numerical representation of rock reinforcement, using material models, include those by

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Ferrero (1995), Grasselli (2005), Aziz and Jalalifar (2007), Chen and Li (2015c), and Tatone et al. (2015), who simulated laboratory tests on various types of rock bolts. Ferrero (1995) used a three-dimensional finite element code to simulate shear test on a rock joint system reinforced by steel dowels and backanalysed the stress-strain behavior of the specimen to assess the states of stress evolution in the reinforcement. Both the rock and steel were assumed to behave in a elastic perfectly-plastic manner. Grasselli (2005) used a three-dimensional finite element code to simulate shear tests conducted on

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grouted rebar and Swellex. He used an elastic material model to simulate the bolts, and an elasticperfectly plastic interface element to simulate the joint separating the two blocks. The numerical simulation provided some insights into the failure mechanisms of the bolts under shear loading conditions. Aziz and Jalalifar (2007) also used a three-dimensional finite element code to simulate the

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laboratory experiments conducted on the bolted rock joints subjected to shearing and investigated the shear stress, yield stress and the change in the bolt strain at the bolt-joint intersection. The steel was

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represented using a bilinear hardening model. Chen and Li (2015c) used the continuum code FLAC3D to simulate the laboratory tests conducted on the rebar bolt and the D-bolt under varying displacing angles to the anchorage. They used a tri-linear material model to capture the strain-hardening of the steel. Chen and Li (2015c) used different models for the rock-grout and grout-bolt interfaces in the D-Bolt and the rebar bolt to explicitly simulate their different bonding mechanisms. In the simulation of a rebar bolt, the bonding between the steel to the rock was defined by its high shear strength at the bolt-grout interface. In the simulation of the D-Bolt, the bonding of the bolt was defined by a shear strength that is the same as the grout at the two anchor positions, and a zero shear strength on the bolt section between the anchors. Using this modeling approach, Chen and Li (2015c) were able to realistically 7

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simulate the load-displacement responses of the rebar and the D-Bolt under pure pull, pure shear and combined pull-shear loading conditions. Tatone et al. (2015) simulated pull tests on reinforcement represented using both material model and structural element in the 2D Y-Geo code, which is based on the hybrid finite-discrete element method. The results of both approaches were found to be consistent

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in terms of force-displacement response and damage propagation. They also demonstrated the

effectiveness of the rock reinforcement represented by the structural element in reducing the amount of damage around underground openings.

Some of the most popular commercially available stress analysis numerical codes, such as FLAC, FLAC3D,

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UDEC and 3DEC developed by Itasca Consulting Group Inc., allow for the use of structural elements to simulate different types of rock reinforcements. In effect, two types of rock reinforcement models, the

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"Cable" and the "Rockbolt" elements, can be used to account for the properties of both bolt and grout. For practical purposes, the main difference between these two approaches is that the "Cable" element does not provide any resistance to bending, therefore it is best suited for simulating cable bolts, whereas the "Rockbolt" element provides resistance against bending, making it appropriate for simulating other reinforcement elements such as fully-grouted rebar bolts. The practical implications of these choices in reinforcement elements are discussed in more detail in the next section.

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A review of the technical literature suggests that "Cable" elements are more popular than the "Rockbolt" element, even for the simulation of rock reinforcements other than cable bolts. This is interpreted to be due to the complexity of the "Rockbolt" element compared to the "Cable" element, in terms of input parameters and calibration process. For example, Vardakos et al. (2007), Malmgren and

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Nordlund (2008), Jiang et al. (2009), Li et al. (2012), Gao et al. (2015), and Shreedharan and Kulatilake (2016), all used the "Cable" element for the simulation of laboratory pull test or support of underground

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openings, in cases where the actual reinforcement used was not cable bolt. Ruest and Martin (2002) also used the "Cable" element in FLAC2D but for the simulation of laboratory pull tests conducted on instrumented cable bolts grouted in steel pipe. The loads calculated along the "Cable" element were compared to the measured loads for a range of model grout properties, and the results showed very good agreement with those of laboratory tests. Only, Nemcik et al. (2014) and Ma et al. (2014) have reported the use of the "Rockbolt" element for the simulation of fully-grouted rock bolts subjected to tensile loading. Nemcik et al. (2014) ignored the forces perpendicular to the "Rockbolt" element and subsequently its resistance to bending since they were only dealing with tensile loading condition. In their analysis, the "Rockbolt" element was in effect behaving similar to the "Cable" element. Similarly, 8

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Ma et al. (2014) used the “Rockbolt” element in FLAC2D to simulate the laboratory pull out test and to investigate the interaction between rockbolt and rockmass in a roadway tunnel. It follows that there are both technical and practical reasons to advance the simulation of rock

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reinforcement using a numerical representation that is capable of resisting against bending (i.e., "Rockbolt" element). This paper is a contribution to the simulation of laboratory tests on fully-grouted rebar bolt under both pull and shear loading conditions using the "Rockbolt" element.

Rock Reinforcement Models in Distinct Element Models

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The numerical simulations of fully-grouted rebar bolts was conducted using the distinct element method. The method has inherent advantages as it permits block deformation and movement of blocks

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relative to each other. The method has been successfully employed to model complex behavior and mechanisms of jointed rock masses (Karampinos et al., 2015). The explicit representation of reinforcement and support is, however much more complex and has not received the same level of attention. This is possibly associated with the challenges in implementing rock reinforcement in a stress analysis model and in demonstrating that successful numerical calibration can provide significant

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confidence in the results.

This paper addresses issues of implementation of reinforcement in a 2D distinct element model (UDEC) and calibration based on documented laboratory tests. The explicit representation of rock reinforcement in UDEC is possible, using both local and global reinforcements (Itasca, 2014). The local

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reinforcement can be applied to both rigid and deformable blocks, while the global reinforcement can only be applied to deformable blocks. The local reinforcement considers only the local effect of reinforcement where it intersects existing discontinuities. The global reinforcement considers not only

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the local effect of reinforcement where it intersects the discontinuities, but also the restraint to rock block that may experience inelastic deformation surrounding an excavation. The global shearing-resistant reinforcement ("Cable" element) and the global shearing and bendingresistant reinforcement ("Rockbolt" element) are the two types of global reinforcement implemented in UDEC. The "Cable" element is a one-dimensional element with two degrees of freedom (two displacements), while the "Rockbolt" element is a two-dimensional element with three degrees of freedom (two displacements and one rotation). The "Cable" element can be used for simulating cable bolts in which the cable can generate very little bending resistance under shear loading and therefore 9

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fails in tension, while the bonding agent may fail in shear over some length of the reinforcement. The "Rockbolt" element, however, can simulate rock bolts, in which the bolt can resist against bending under shear loading.

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It would appear that the "Cable" element has been used in various research projects as reviewed in the previous section, while the "Rockbolt" element has not been widely acknowledged by researchers. This is partially due to the large number of input parameters required in the "Rockbolt" element, which makes its calibration difficult and a time consuming process. In this paper, the local reinforcement and the shearing- and bending-resistant global reinforcement ("Rockbolt" element) are used to simulate the

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behavior of the fully-grouted rebar bolt under pure pull and shear loading conditions conducted by Stjern (1995) (i.e., force-displacement curves in Fig. 2). In the following, these two reinforcement

4.1

Local reinforcement

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models are reviewed.

As the local reinforcement only considers local effect of reinforcement where it intersects the discontinuities, it is therefore most applicable to cases where the deformation of individual rock blocks is small relative to the deformation of the reinforcing system. Its formulation is based on simple force-

discontinuities.

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displacement relationships that describe both the shear and axial behaviors of reinforcement across

The axial and shear force-displacement behavior of local reinforcement are illustrated in Figure 3a. The axial force-displacement behavior is described by axial stiffness (Ka), ultimate axial capacity (Fa,bmax) and

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axial failure strain, and the shear force displacement behavior is represented by shear stiffness (Ka) and

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ultimate shear capacity (Fs,bmax).

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Figure 3 a) Force-displacement behavior of local reinforcement in pull and shear tests; b) influence of stiffness exponent (axial or shear) on force-displacement response of local reinforcement (after Itasca, 2014).

The force-displacement relations that describe the axial and shear responses are given by the following equations: ∆Fa = Ka I∆UaI ƒ(Fa)

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∆Fs = Ks I∆UsI ƒ(Fs)

(1)

(2)

where ∆Fa and ∆Fs are incremental changes in axial and shear forces, ∆Ua and ∆Us are incremental changes in axial and shear displacements, and ƒ(Fa) and ƒ(Fs) are functions that describe the path by

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which the axial (Fa) and shear (Fs) forces approach the ultimate axial (Fa,bmax) and shear (Fs,bmax) forces, according to the values of axial (ea) and shear (es) stiffness exponents. The axial and shear stiffness

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exponents are used to define the curvature of the force-displacement behavior, by controlling the rate at which the bounding forces are reached (Fig. 3b). By default, the values of stiffness exponents are zero, which means that the force-displacement relations follow a constant (linear) stiffness until the ultimate capacity (axial or shear) is reached. Figure 4a shows that the maximum shear force, which changes for various orientations of reinforcement relative to the discontinuity. As shown in this figure, the maximum shear force decreases from a maximum value at θ0 = 0° to 50% of the peak shear load at θ0 = 90° (Itasca, 2014). The local reinforcement was implemented by Itasca (2014) in such a way to be consistent with the results by Azuar et al. (1979), who found that the maximum shear force was about half the product of the uniaxial 11

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tensile strength of the reinforcement and its cross-sectional area for reinforcement perpendicular to the

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discontinuity.

Figure 4 a) shear behavior of local reinforcement element for various orientations of reinforcement relative to the discontinuity; b) assumed reinforcement geometry after shear displacement (∆Us) and illustration of active length (after Itasca, 2014).

The rupture strain under shear loading cannot be directly defined, however, it can be adjusted by

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defining the active length of the reinforcement. Active length is the short length of reinforcement, which spans the discontinuity and changes orientation during shear displacement (Fig. 4b). The active length can be measured in the laboratory from the reinforcement deformed or failed after the shear test. In the case of lack of data, the value of active length can be adjusted until the rupture shear displacement

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on the shear force-displacement plot is matched with that from the shear test result. The latter will be used in this paper for the calibration of local reinforcement to the behavior of rebar bolt in the shear

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test.

The input parameters required for the simulation of a rock bolt using the local reinforcement model can be obtained from laboratory pull and shear tests. In the case of lack of experimental data, empirical relations can be used to estimate bolt axial and shear stiffness and pull and shear capacities.

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Global reinforcement ("Rockbolt" element)

Both "Rockbolt" and "Cable" elements allow for the simulation of a shearing resistance along their length, as provided by the shear resistance (bond) between the grout and either the cable/rock bolt or the host rock. The advantage of the "Rockbolt" element over the “Cable” element in modeling a rebar

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bolt is that it provides resistance against bending. Furthermore, the "Rockbolt" element can simulate the actual bolt breakage based upon a user-defined tensile failure strain limit. Similar to the cable element, the "Rockbolt" element is divided into a number of segments of length L, with nodal points located at each segment end (Fig. 5). The "Rockbolt" element interacts with the UDEC model via shear

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and normal coupling springs, which are connectors that transfer forces and motion between the

"Rockbolt" element and the gridpoints associated with the block zone, in which the nodes are located

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(Fig. 5).

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Figure 5 Conceptual mechanical representation of the global reinforcement ("Rockbolt" element), which accounts for shear behavior of grout annulus and bending resistance of the reinforcement (after Itasca, 2014).

The "Rockbolt" element segments are treated as a linearly elastic material that may yield in the axial direction either in tension or compression. The tensile and compressive yield strengths are used to define these strength limits. Inelastic bending is simulated in the "Rockbolt" element by specifying a limiting plastic moment. This means that "Rockbolt" elements behave elastically until they reach the plastic moment. In addition, segments may break and separate at the nodes, based on a user-defined tensile failure strain limit. A strain measure, called the total plastic tensile strain, based on adding the 13

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axial and bending plastic strains, is evaluated at each node. If this strain exceeds the tensile failure strain limit, the forces and moment in this segment are set to zero and the "Rockbolt" element is assumed to have failed. The shear and normal behaviors of the "Rockbolt" element/gridpoint interface are

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represented by linear springs as shown in Figure 6.

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Figure 6 Material behavior of shear and normal coupling springs for "Rockbolt" element (after Itasca, 2014) : a) shear force versus shear displacement; b) shear strength criterion for the shear coupling spring; c) normal force versus normal displacement; and d) normal strength criterion for the normal coupling spring.

The shear behavior of the interface during relative displacement between the element nodes and the gridpoints is described numerically by the coupling spring shear stiffness (CSsstiff in Fig. 6a) according to the following equation: Fs/L = CSsstiff (up - um)

(3)

where Fs is the shear force that develops in the shear coupling spring, up and um are the axial displacement for the "Rockbolt" element and the medium (soil or rock), and L is the "Rockbolt" element 14

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segment length. The limiting shear force (Fsmax) that can be developed along the "Rockbolt" element/gridpoint interface is a function of the cohesive strength of the interface (CSscoh) and the stressdependent frictional resistance (CSsfric) along the interface (Fig. 6b) according to: Fsmax/L = CSscoh + σ'c × tan(CSsfric) × perimeter

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(4)

where perimeter is the exposed perimeter of the rock bolt (i.e., the length of the rock bolt surface that is in contact with the medium), and σ'c is the mean effective confining stress normal to the "Rockbolt"

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element.

Similarly, the normal behavior during the relative normal displacement between the "Rockbolt" element

equations: Fn/L = CSnstiff (unp - unm)

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nodes and the gridpoints (Fig. 6c and Fig. 6d) is described numerically according to the following

(5)

Fnmax/L = CSncoh + σ'c × tan(CSnfric) × perimeter

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where CSnstiff is the coupling spring normal stiffness, Fn is the normal force that develops in the normal

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coupling spring, unp and unm are the displacements of the "Rockbolt" element and medium normal to the axial direction of the element, respectively, Fnmax is the limiting normal force, and CSncoh and CSnfric are the cohesive strength and friction angle of the normal coupling spring. Other parameters required for the simulation of rock reinforcement using the "Rockbolt" element

5.1

Numerical simulation of pull and shear tests

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include the cross-sectional area, second moment of area, density, and elastic modulus of the rock bolt.

Model geometry and boundary conditions

Figure 7 shows the UDEC model used to simulate the pull and shear tests using the local and global ("Rockbolt" element) reinforcements. The model consists of two elastic blocks with a Young's modulus of 24 GPa and a Poisson's ratio of 0.25, separated by a joint with zero cohesion and friction angle. The left block was chosen to be 0.95 m × 0.95 m to be consistent with the block dimensions in the actual test reported by Stjern (1995). The right block in this figure, however, was made slightly larger so that its

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boundary conditions (rollers at the left corners of the top and bottom boundaries) do not interfere with

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the left block.

Figure 7 Geometry of UDEC model used to simulate pull and shear tests on fully-grouted rebar bolt.

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The boundary conditions of the UDEC models used for the simulations of pull and shear tests are illustrated in Figure 8. In both tests, rollers are applied to the boundaries, and pins are applied to the right corners of the right blocks. The pull test is simulated by applying a velocity boundary to the left side of the left block as shown in Figure 8a, and the shear test is simulated by applying a velocity

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boundary to the lower side of the left block as shown in Figure 8b. In the shear test, rollers are also used in the left side of the left block in order to avoid block rotation.

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Two methods are used to monitor the pull and shear forces. In the first method, the force (both pull and shear) is determined from the sum of reaction forces that develop on the boundaries of the right block. In the second method, the forces are obtained directly from the reinforcement models and then compared with the forces determined using the first method.

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Figure 8 Boundary conditions of UDEC model used to simulate: a) pull; and b) shear tests.

The calibration of UDEC model was conducted in two independent stages as illustrated in the flow charts in Figure 9. First, the model is calibrated to the force-displacement behavior of the rebar bolt from the

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pull test (Fig. 9a). The shear test is then simulated with the parameters obtained from the calibration of the model to the pull test results. Next, the force-displacement behavior obtained from the simulation

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of shear test is compared with that of laboratory shear test. In the second stage (Fig. 9b), the model is first calibrated to the force-displacement behavior of the rebar bolt from the shear test. The parameters obtained from the calibration of the model to the shear test results are then used to simulate the pull test and the results in terms of force-displacement behavior are compared with that of laboratory pull test. The criteria used for the calibration of local and global ("Rockbolt" element) reinforcements to the results of pull and shear tests are different loading stages on the force-displacement plot, including the initial stiffness, yield load, peak load, and rupture displacement, as illustrated in Figure 10.

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Figure 9 Calibration procedures for the local and global ("Rockbolt" element) reinforcements.

Figure 10 Illustration of calibration criteria on the force-displacement plot.

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Rock reinforcement elements used to support underground/surface excavations may experience pure pull load, pure shear load, or a combination of both, depending on the location and orientation of reinforcements relative to discontinuities. The objective of the two independent calibration stages illustrated in Figure 9 is to investigate whether there exists a unique set of input parameters in the

bolts under both pull and shear loading conditions.

5.2

Calibration of local Reinforcement

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UDEC's reinforcement models, that can capture the force-displacement curves of fully grouted rebar

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The results of calibration of local reinforcement to the laboratory behavior of pull and shear tests on fully-grouted rebar bolt are presented in Figure 11. During the calibration process the best results were obtained using an active length of 40 mm (see Fig. 4b for the illustration of active length). In the

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simulation of pull test, the axial stiffness, maximum pull load, rupture strain and axial stiffness exponent were adjusted. The values of local reinforcement shear stiffness and maximum shear load were chosen so that the shear stiffness and yield load in the shear force-displacement curve match with those of fully-grouted rebar bolt. Figure 11a demonstrates that the force-displacement curve of the fully-grouted rebar bolt under pull load condition can be successfully captured using the local reinforcement. The shear test was then simulated using the calibrated parameters. The results shown in Figure 11b indicate

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an overestimation of the rupture strain by about 14 mm (an overestimation by 33%) and an overestimation of the peak load by about 75 kN (an overestimation by 41%). Next, the local reinforcement was calibrated to the shear test results by adjusting the rupture strain and

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the shear stiffness exponent, and keeping the rest of parameters constant. The pull test was then simulated using the calibrated parameters. The results shown in Figure 11c and Figure 11d reveal that when the local reinforcement is calibrated to the results of shear test, the rupture strain in the pull test

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is underestimated by about 15 mm (an underestimation by 38%). The input parameters obtained from the two calibration stages are listed in Table 2.

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Figure 11 Calibration of local reinforcement to the results of laboratory tests on fully-grouted rebar by Stjern (1995): a) calibration to pull test result; b) result of shear test on local reinforcement with input parameters obtained from calibration to pull test result; c) calibration to shear test result; d) result of pull test on local reinforcement with input parameters obtained from calibration to shear test result.

Table 2 Local reinforcement input parameters obtained from calibration to pull and shear tests.

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Input parameters axial stiffness (kN/mm) maximum axial force (kN) rupture strain shear stiffness (kN/mm) maximum shear force (kN) 1/2 active length (mm) axial stiffness exponent shear stiffness exponent bolt spacing

Calibration to pull test result 20 180 1 23 160 20 0.5 0 1

Calibration to shear test result 20 180 0.6 23 160 20 0.8 0 1

As discussed in Section 3, the force-displacement behavior of the fully-grouted rebar during the shear test consists of three main stages; elastic behavior (i.e., increase in shear force with a slope equal to the 20

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shear stiffness until the yield load), hardening behavior (i.e., reduction in the slope of forcedisplacement curve until the peak load) and bolt rupture. Interestingly, the shear force-displacement determined from the sum of reaction forces on the right block (e.g., Figure 11c) realistically captures these three stages, although the shear force-displacement behavior implemented to the local

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reinforcement model does not represent the hardening effect (see Figure 3a). This was investigated by obtaining the shear and axial forces directly from the local reinforcement element itself (i.e., not from reaction forces generated on the block boundaries), and the results are shown in Figure 12. It can be seen from this figure that during the shear test, the axial force is also generated in the local

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reinforcement, but after about 5 mm of shear displacement when shear force reaches its maximum value. From this point on, with the increase in the shear displacement, the axial force gradually increases until rupture (defined by the local reinforcement active length) occurs. In fact, the

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combination of shear force and axial force generated in the local reinforcement during the shear test results in a shear force-displacement curve, which is similar to that of laboratory shear tests conducted

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by Stjern (1995) and Chen (2014), as shown in Figure 2b.

Figure 12 Shear and axial forces generated in the local reinforcement element during shear test.

5.3

Calibration of global reinforcement ("Rockbolt" element)

During the calibration process, it was realized that the number of segments along the "Rockbolt" element (in fact the number of segments near the joint) have a significant impact on the results, especially in the shear test. The results presented in this section are from a model in which the

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"Rockbolt" element consists of 80 segments. This was selected to ensure that at least one node falls

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inside each triangular zone near the joint, as shown in Figure 13.

Figure 13 "Rockbolt" element consisting of 80 segments.

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The results of calibration of the "Rockbolt" element to the laboratory pull and shear tests are presented in Figure 14. The values for the cross-sectional area, exposed perimeter, density, elastic modulus, and tensile yield strength were obtained from Stjern (1995) and the known steel properties. The force-

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displacement behavior of the rebar in the pull test was matched by adjusting the values of the shear coupling spring stiffness and cohesive strength and the tensile failure strain limit of the "Rockbolt" element. It was found that the stiffness and cohesive strength of the normal coupling spring, plastic moment and tensile failure strain limit influence the shear force-displacement behavior of the "Rockbolt" element.

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Figure 14 Calibration of global reinforcement ("Rockbolt" element) to the results of laboratory tests on fullygrouted rebar by Stjern (1995): a) calibration to pull test result; b) result of shear test on "Rockbolt" element with input parameters obtained from calibration to pull test result; c) calibration to shear test result; d) result of pull test on "Rockbolt" element with input parameters obtained from calibration to shear test result.

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Figure 14a and Figure 14b show that when the "Rockbolt" element is calibrated to the forcedisplacement behavior of the pull test, the rupture displacement in the shear test is overestimated by about 20 mm (an overestimation by 50%). This is an improvement compared to the results of simulation using the local reinforcement, as the peak load was also overestimated by about 75 kN by the local reinforcement (compare Fig. 14b and Fig. 11b). On the other hand, when the "Rockbolt" element is calibrated to the force-displacement behavior in the shear test (Figure 14c), the rupture displacement in the pull test is underestimated by about 20 mm (an underestimation by 50%) (Fig. 14d). The values of input parameters obtained from the two calibration attempts are listed in Table 3.

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The failure modes of the "Rockbolt" element in the pull and shear test simulations are shown in Figure 15. As can be seen from this figure, the failure mode of the "Rockbolt" element in the pull test is bolt (i.e., steel) failure near the joint. In the shear test, failure involves both the "Rockbolt" element (steel) and the interface (grout) near the joint. The failure modes obtained from numerical simulations are

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consistent with the observations made by Stjern (1995) and Chen (2014) from their laboratory tests. It should be noted that the rock blocks in the numerical simulations were considered to be an elastic medium. Consequently, the yielding and plastic deformation that have been observed in the shear test at the bolt-discontinuity contact were not captured. Whether rock block yielding near the bolt-

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discontinuity contact has a significant impact on the shear load-displacement curve needs further

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investigation. However, this was beyond the scope of this paper.

Table 3 "Rockbolt" element input parameters obtained from calibration to pull and shear tests. Calibration to pull test result 0.0003 7.9e-9 0.062 8e4 200 180 0.35 2 0.05 10 1200 8000 0 0 1

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Input parameters 2 bolt cross-sectional area (m ) 4 bolt second moment of area (m ) exposed perimeter (m) 3 bolt density (kg/m ) bolt elastic modulus (GPa) bolt tensile yield strength (kN) bolt tensile failure strain limit plastic moment (kN-m) shear coupling spring stiffness (GN/m/m) normal coupling spring stiffness (GN/m/m) shear coupling spring cohesion (kN/m) normal coupling spring cohesion (kN/m) shear coupling spring friction angle (°) normal coupling spring friction angle (°) bolt spacing (m)

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Calibration to shear test result 0.0003 7.9e-9 0.062 8e4 200 180 0.75 1.9 0.05 10 1200 8000 0 0 1

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Figure 15 Failure modes of the "Rockbolt" element in the pull and shear tests.

5.4

Sensitivity analysis

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The calibration of the "Rockbolt" element to the force-displacement behavior of the rebar bolt from the pull test is straight forward as the shear coupling spring stiffness, "Rockbolt" yield strength, and tensile

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failure strain limit are directly related to the force-displacement stiffness, peak strength and rupture displacement, respectively. The calibration of the "Rockbolt" element to the force-displacement behavior of the rebar bolt from the shear test was, however, found to be complex. Therefore, a series of sensitivity analyses were conducted on the "Rockbolt" element input parameters relevant to shear force-displacement response, including the normal coupling spring stiffness and cohesion and the plastic moment. The results of sensitivity analysis are presented in Figure 16. The model calibrated to the behavior of the rebar bolt under shear loading (Figure 14c) was used as the base case and its forcedisplacement response is shown in Figure 16 with the dashed black curve.

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Figure 16 a shows the influence of normal coupling spring stiffness on the shear force-displacement behavior of the "Rockbolt" element. In general, the increase in the value of normal coupling spring stiffness from 0.01 GN/m/m to 10 GN/m/m results in the increase in the normal force, calculated from the sum of reaction forces from the right block. In the case of normal coupling spring stiffness of 0.01

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and 0.1 GN/m/m, the generated normal force is low, the force-displacement behavior is relatively linear, and rupture does not occur in the "Rockbolt" element even after 80 mm of shear displacement. Figure 16a demonstrates how the generated normal force increases with increasing the value of normal

coupling spring stiffness. For the normal coupling spring stiffness values of 1 and 10 GN/m/m, the shape

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of shear force-displacement curve becomes similar to that from the laboratory shear test. However, only with the normal coupling spring stiffness value of 10 GN/m/m, do the stiffness, yield load, peak load and

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the rupture displacement become comparable to those of laboratory shear test.

The results of sensitivity analysis on normal coupling spring cohesion are presented in Figure 16b. This figure shows that the stiffness, yield load and peak load decrease and the rupture displacement increases with decreasing the value of normal coupling spring cohesion from 8 MN/m to 2 MN/m. Figure 16c indicates how the decrease in the plastic moment (pmom) from 2 kN-m to 0.5 kN-m shifts the forcedisplacement curve downward by lowering the yield and peak loads, and keeping the stiffness and the

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rupture displacement constant.

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Figure 16 Influence of: a) normal coupling spring stiffness; b) normal coupling spring cohesion; and c) plastic moment, on the force-displacement behavior of the "Rockbolt" element in the shear test.

6

Discussion

It is recognized that the explicit implementation of reinforcement in numerical analyses program presents a considerable challenge. Due to its relative simplicity, the so called "Cable" element is used to 27

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simulate rock reinforcement in both continuous and discontinuous numerical codes, even if the type of reinforcement used in the field or laboratory is not cable bolts. The consequence of using the "Cable" element instead of a bending-resistant reinforcement element such as the "Rockbolt" element has

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significant implications as demonstrated by the results of the present investigation. The results of pull and shear test simulations using the "Cable" element, with the input parameters listed in Table 4 are illustrated in Figure 17. It can be seen from this figure that the "Cable" element underestimates the shear capacity of the fully-grouted rebar bolt, when it is calibrated to the pull test results. Note that it is not possible to increase the shear capacity of the "Cable" element, as this type of

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reinforcement element interacts with the UDEC model through its shear coupling spring (i.e., "Cable" element does not have normal coupling spring), and therefore it provides very little resistance against

the pull capacity of the "Cable" element.

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bending. This small bending resistance arises from the axial force generated during the shear test and

By comparing the results of sensitivity analysis of the normal coupling spring stiffness on the "Rockbolt" element shear force-displacement response (Figure 16a) and the shear force-displacement response of the "Cable" element (Figure 17b), it is understood that the behavior of the "Rockbolt" element with low normal coupling spring stiffness is similar to the "Cable" element under shear loading. It is therefore

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concluded that when the "Cable" element, or a "Rockbolt" element with a low normal coupling spring stiffness, is used for the design of an excavation to be supported with fully-grouted rebar bolts, the shear capacity of the rebar bolt is likely underestimated. The consequence of these modeling decisions is a conservative design of support. This is a result of underestimating the shear capacity of the rebar

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bolts and incorrect interpretation of their behavior under shear loading.

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Figure 17 Force-displacement response of the "Cable" element under: a) pull; and b) shear loading condition and its comparison with laboratory test results by Stjern (1995).

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Table 4 "Cable" element input parameters obtained from calibration to pull test. Input parameters cable cross-sectional area (m2) 3 cable density (kg/m ) cable Young's modulus (GPa) cable tensile yield strength (kN) cable extensional failure strain grout stiffness (GN/m/m) grout cohesive capacity (kN/m) cable spacing

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Calibration to pull test result 0.0003 8000 200 180 0.26 0.035 1200 1

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This work has demonstrated, that both the local and the global (i.e., "Rockbolt" element) reinforcements are capable of capturing the bending resistance of the fully-grouted rebar bolt under shear loading. The following provides a comparison between these two reinforcement models: The local reinforcement considers the local effect of reinforcement where it intersects the

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discontinuities, and therefore it should be used in situations where the block strength is high and discontinuity failure dominates the rock mass behavior. However, the global reinforcement ("Rockbolt" element) can be used for either strong or weak rock blocks as it simulates the boltgrout or grout-rock behavior as well.

The local reinforcement, when calibrated to the results of pull test on the fully-grouted rebar

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bolt (i.e., force-displacement response), overestimates the rebar shear capacity and rupture

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displacement. The "Rockbolt" element, however, only overestimates the shear displacement of the rebar bolt, when it is calibrated to the rebar force-displacement response under tensile loading condition. •

The local reinforcement does not simulate the grout and its interaction with the bolt or ground. The grout and its mechanical properties are considered in the "Rockbolt" element. This allows for simulating a fully-grouted rebar with various grout strength and stiffness properties (i.e.,

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water/cement ratios).

The advantage of the “Rockbolt" element over the local reinforcement is that it captures failure modes of the fully-grouted rebar bolt (i.e. steel and grout failure) under both pull and shear

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loading conditions observed in the laboratory tests. It is therefore concluded from the comparison of local and global ("Rockbolt" element) reinforcement models, that the "Rockbolt" element provides a more realistic representation of the behavior of fully-

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grouted rebar bolts under pull and shear loading conditions. It should be noted that such a realistic representation is obtained by assigning enough number of segments along the "Rockbolt" element. This, however, may significantly increase the computation time if large models (e.g., models of excavations in a jointed rock mass) are used. On the other hand, fewer input parameters are required in the local reinforcement model compared with the "Rockbolt" element and they can be directly obtained from laboratory pull and shear tests. This makes the calibration of the local reinforcement computationally efficient. It was demonstrated that the local reinforcement provides less accurate response of the fullygrouted rebar bolt. Therefore, the use of local reinforcement is suggested to be limited to the initial stability analysis of underground excavations and support design. 30

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In this paper, the reinforcement models in UDEC were evaluated under pure pull and shear loading conditions. An attempt was made to obtain a unique set of input parameters for the reinforcement models that could capture the force-displacement curves of the fully grouted rebar bolts under both pure pull and shear loading conditions. It was concluded from the results presented in Figure 11 and

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Figure 14 that there does not exist such a unique set of values. However, for practical purposes, it is possible to find the one that leads to reasonable load-displacement curves under such a loading condition.

Conclusions

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In order for the explicit representation of reinforcement in stress analysis numerical models to be used

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as an engineering design tool, it is necessary to demonstrate that it successfully reproduces the results of the reinforcement action in a controlled environment. Only then can this work be extrapolated with confidence to in situ conditions whereby more extraneous factors and variations come into play. This investigation focused on the use of the distinct element method as it best captures the jointed nature of a rock mass and the reinforcement mechanics of rock bolting.

Two types of reinforcement models implemented in UDEC, including local reinforcement and global

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shearing and bending-resistant reinforcement (called "Rockbolt" element) were used to capture the force-displacement response of a fully-grouted rebar bolt under laboratory pull and shear loading conditions. The results of laboratory tests conducted by Stjern (1995) were used for this purpose. The calibration of numerical models was conducted in two stages. First, the model was calibrated to the pull

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test results. The parameters obtained from this calibration process were then applied to the shear test model. Next, the model was calibrated to the shear test results, and the parameters obtained from this

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calibration process were then used in the simulation of pull test. The calibration criteria included the initial stiffness, yield load, peak load, and rupture displacement on the force-displacement plot. Moreover, to be consistent with laboratory test results, the parameters were chosen in a way that the failure mode in the model with the "Rockbolt" element represent failure in the bolt shank near the joint in both pull and shear tests.

Both local and global reinforcement models were demonstrated to be able to capture the overall forcedisplacement response of the fully-grouted rebar bolt under pull and shear loading conditions. However, when the local reinforcement is calibrated to the pull test results, the rupture displacement and the

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peak load in the shear test are overestimated. On the other hand, when it is calibrated to the shear test results, the rupture displacement in the pull test is underestimated. The results of simulations using the "Rockbolt" element revealed an improvement in the calibration results. When the "Rockbolt" element is calibrated to the pull test results, only the rupture displacement in the shear test is overestimated.

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When the "Rockbolt" element is calibrated to the shear test results, its rupture displacement in the pull test is still underestimated.

The advantage of the local reinforcement over the "Rockbolt" element is the simplicity in defining its input parameters, which are directly correlated to and can be obtained from laboratory pull and shear

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tests. The main advantage of the "Rockbolt" element over the local reinforcement is its explicit

representation of the bolt and the grout, which makes a realistic simulation of their failure mode (e.g.,

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bolt rupture or grout failure) under different loading conditions possible.

It was demonstrated that when the "Cable" element is used to simulate a rebar bolt, the shear capacity of the rebar is underestimated. This has implications for the support design and stability analysis of underground excavations in jointed rock masses; the use of the "Cable" element instead of the "Rockbolt" element may result in an underestimation of the rebar bolt shear capacity, which may lead to

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a conservative design of the support system.

The presented work demonstrated the use of an explicit representation of reinforcement in a stress analysis software. This investigation has shown the implications of using different approaches for simulation of fully-grouted rebar bolt. Both approaches were calibrated based on the results of a

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controlled experimental program. Further implication of the choice of approach for in situ conditions

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has been recognized.

Acknowledgements

This project was supported by the Natural Science and Engineering Council of Canada. The authors would like to acknowledge Jim Hazzard and Efstratios Karampinos for the discussion and technical advice on modeling rock reinforcement in UDEC.

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References Aziz N, Jalalifar H. Experimental and numerical study of double shearing of bolt under confinement. In: S.S. Peng, C. Mark, G. Finfinger, S. Tadolini, A.W. Khair, K. Heasley and Y. Luo (Eds.). 26th International

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Conference on Ground Control in Mining, Morgantown, WV, USA; 2007: 242-249. Azuar MM, Reuille D, Net PA, Embier S. Rock mass reinforcement by passive rebars, In: Proceedings of the 4th ISRM Congress, Vol. 1, Rotterdam: A.A. Balkema and the Swiss Society for Soil and Rock Mechanics, Montreux, Switzerland, 1979: 23-30.

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Chen Y, Li CC. Experimental and three-dimensional numerical studies of the anchorage performance of

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rock bolts, In: International Symposium on Rock Mechanics, Montreal, Canada, 2015c:11p. Chen Y, Li CC. Influence of loading condition and rock strength to the performance of rock bolts. Geotech Test J 2015b;38(2): 208-218.

Chen Y, Li CC. Performance of fully encapsulated rebar bolts and D-bolts under combined pull-and-shear loading, Tunn Undergr Sp Tech 2015a;45: 99-106.

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Chen Y. Experimental study and stress analysis of rock bolt anchorage performance. J Rock Mech Geotech Eng 2014;6: 428-437.

Ferrero AM. The shear strength of reinforced rock joints. Int J Rock Mech Min Sci & Geomech Abstr

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1995;32: 595-605.

Gao F, Stead D, Kang H. Numerical simulation of squeezing failure in a coal mine roadway due to mining-

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induced stresses, Rock Mech Rock Eng 2015;48: 1635-1645. Grasselli G. 3D Behaviour of bolted rock joints: experimental and numerical study. Int J Rock Mech Min 2005;42(1): 13-24.

Hadjigeorgiou J, Charette F. Rock Bolting for Underground Excavations. In: Hustrulid & Bullock Eds. Underground Mining Methods, Chapter 63, Society of Mining Engineers, 2001:547-554. Itasca. UDEC (Universal Distinct Element Code), Version 6.0. Itasca Consulting Group Inc., Minneapolis; 2014.

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Karampinos E, Hadjigeorgiou J, Hazzard J. Discrete element modelling of the bulking phenomenon in deep hard rock mines. Int J Rock Mech Min 2015;80: 346-356. Li B, Qi T, Zhengzheng W, Yang L. Back analysis of grouted rock bolt pullout strength parameters from

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field tests. Tunn Undergr Sp Tech 2012;28: 345-349. Li CC. Stjern G, Myrvang A. A review of the performances of conventional and energy-absorbing rockbolts. J Rock Mech Geotech Eng 2014;6: 315-327.

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Ma S, Nemcik J, Aziz N. Simulation of fully grouted rockbolts in underground roadways using FLAC2D. Can Geotech J 2014;51: 911-920.

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Malmgren L, Nordlund E. Interaction of shotcrete with rock and rock bolts—A numerical study. Int J Rock Mech Min 2008;45: 538-553.

Nemcik J, Ma S, Aziz N, Ren T, Geng X. Numerical modeling of failure propagation in fully grouted rock bolts subjected to tensile load, Int J Rock Mech Min 2014;71: 293-300.

Ruest M, Martin L. FLAC simulation of split-pipe tests on an instrumented cable bolt. In: CIM Annual

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General Meeting Proceedings, Vancouver; 2002.

Shreedharan S, Kulatilake PHSW. Discontinuum-equivalent continuum analysis of the stability of tunnels in a deep coal mine using the distinct element method. Rock Mech Rock Eng 2016; DOI:

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Stjern G. Practical Performance of Rock Bolts. PhD thesis, NTH, Trondheim, Norway; 1995.

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Tatone BSA, Lisjak A, Mahabadi OK, Vlachopoulos N. Incorporating rock reinforcement elements into numerical analysis based on the hybrid finite-discrete element method (DFEM). In: Proceedings of ISRM Congress, Montreal, Canada, 2015: 11p. Vardakos SS, Gutierrez MS, Barton NR. Back-analysis of Shimizu Tunnel No. 3 by distinct element modeling. Tunn Undergr Sp Tech 2007;22: 401-413.

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Wiles T, Villaescusa E, Windsor CR. Rock reinforcement design for overstressed rock using three dimensional numerical modeling. In: Villaescusa & Potvin (Eds.). Ground Support in Mining and Underground Construction, 2004: 483-489.

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Figure Caption

Figure 1 The test rig for static pull, shear and combined pull-shear tests: a) the front view of the test rig (after Chen, 2014); b) an oblique sketch of the test rig (after Stjern, 1995).

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Figure 2 Results of: a) pull test; and b) shear test, on 18 mm fully-grouted rebar (Stjern, 1995), and 20 mm fully-grouted rebar (Chen, 2014).

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Figure 3 a) Force-displacement behavior of local reinforcement in pull and shear tests; b) influence of stiffness exponent (axial or shear) on force-displacement response of local reinforcement (after Itasca, 2014).

Figure 4 a) shear behavior of local reinforcement element for various orientations of reinforcement relative to the discontinuity; b) assumed reinforcement geometry after shear displacement (∆Us) and

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illustration of active length (after Itasca, 2014).

Figure 5 Conceptual mechanical representation of the global reinforcement ("Rockbolt" element), which accounts for shear behavior of grout annulus and bending resistance of the reinforcement (after Itasca,

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2014).

Figure 6 Material behavior of shear and normal coupling springs for "Rockbolt" element (after Itasca, 2014) : a) shear force versus shear displacement; b) shear strength criterion for the shear coupling

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spring; c) normal force versus normal displacement; and d) normal strength criterion for the normal coupling spring.

Figure 7 Geometry of UDEC model used to simulate pull and shear tests on fully-grouted rebar bolt. Figure 8 Boundary conditions of UDEC model used to simulate: a) pull; and b) shear tests. Figure 9 Calibration procedures for the local and global ("Rockbolt" element) reinforcements. Figure 10 Illustration of calibration criteria on the force-displacement plot. 35

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Figure 11 Calibration of local reinforcement to the results of laboratory tests on fully-grouted rebar by Stjern (1995): a) calibration to pull test result; b) result of shear test on local reinforcement when calibrated to pull test result; c) calibration to shear test result; d) result of pull test on local

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reinforcement when calibrated to shear test result. Figure 12 Shear and axial forces generated in the local reinforcement element during shear test. Figure 13 "Rockbolt" element consisting of 80 segments.

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Figure 14 Calibration of global reinforcement ("Rockbolt" element) to the results of laboratory tests on fully-grouted rebar by Stjern (1995): a) calibration to pull test result; b) result of shear test on "Rockbolt" element when calibrated to pull test result; c) calibration to shear test result; d) result of pull test on

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"Rockbolt" element when calibrated to shear test result.

Figure 15 Failure modes of the "Rockbolt" element in the pull and shear tests. Figure 16 Influence of: a) normal coupling spring stiffness; b) normal coupling spring cohesion; and c) plastic moment, on the force-displacement behavior of the "Rockbolt" element in the shear test.

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Figure 17 Force-displacement response of the "Cable" element under: a) pull; and b) shear loading

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condition and its comparison with laboratory test results by Stjern (1995).

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Conflict of interest

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We wish to confirm that there are no known conflicts of interest associated with this publication and there has been no significant financial support for this work that could have influenced its outcome.