Study on the Load Transfer Behaviour and Bond-Slip Model of Fully Grouted Rockbolt

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Introduction
Rockbolt support systems involve a simple structure and extensive support form and have the advantages of high support strength, low cost, and good supporting efect. Furthermore, they can make full use of the surrounding rock to bear the load, maximally maintain the integrity and stability of the surrounding rock, efectively control the development of deformation, displacement, and cracks in the surrounding rock, and provide support to the surrounding rock itself [1]. Hence, such systems are widely used in various felds, such as underground engineering. During the service period, rockbolts are often subjected to diferent loads, and the propagation of the loads in the rockbolts is a very complex process. For decades, many scholars have conducted extensive research on the load transmission behaviour of rockbolts [2][3][4][5][6][7][8][9]. Sun et al. [2] considered the installation time and the length of the rockbolt, analysed the displacement process of a deep tunnel, and showed that increasing the length of the rockbolt from 2 m to 4 m in a 14 m high tunnel could reduce the tunnel wall displacement by 20%. Spearing et al. [3] proposed a new method to test the performance of feld rockbolts, namely, evenly staggering the placement of strain gauges on rockbolts to measure the axial stress change of the rockbolts. Bae et al. [4] studied the infuence of diferent concrete strengths, thicknesses, and steel fbre contents on the bonding strength of reinforced active powder concrete. Te results showed that the growth rate of the bond strength decreased with increasing concrete compression strength, and the damage mode of specimens changed with increasing steel fbre content. Te bonding strength increased, but the growth rate was diferent. Luga and Periku [5] studied the bearing capacity of rockbolts by conducting an in situ pullout test in the feld, and the results showed that the anchorage strength and displacement of the rockbolt decreased with increasing horizontal mounting angle. Tenevin et al. [6] studied the infuence of the anchorage pressure and anchorage length on the load. Teymen and Kilic [7] investigated the efect of the anchorage strength on the stress distribution in a full-length rockbolt and showed that the stress distribution on the rockbolt was more uniform with increasing anchorage strength and stifness. Khaleghparast et al. [8] studied the shear strength of rockbolts under static and dynamic loading. Te results showed that using perforated steel tubes inside concrete blocks as internal confnement in the vicinity of shear planes prevented axial and radial cracking in the concrete, which enabled the rockbolt to undergo more shear than tension. Furthermore, the shear performance of a conventional rockbolt under a high-velocity impact load was found to be 70% of that under static loading conditions. Yu et al. [9] studied the damage mode of rockbolts and the damage process of the anchor structure through test and numerical simulation methods. Te results showed that with increasing anchor length, the damage process of the rockbolt anchor system fnally occurred at the loading end and the free end. In conclusion, although some scholars studied the load transfer behaviour of rockbolts, their research was not sufciently comprehensive and failed to assess whether grouted rockbolt systems contained defects that could afect the load transfer behaviour of grouted rockbolt systems.
Te key problems in studying the rockbolt anchorage mechanism include choosing a reasonable bond-slip model, simplifying the anchorage problem, and accurately refecting objective reality. Many scholars have studied the bond-slip model, and an ideal elastic plastic bond-slip model has been used by many commercial software developers to study the mechanical behaviour of rockbolts, but this model does not consider the interfacial softening phenomenon [10]. Te test results of Hawkes and Evans indicated that the interfacial shear stress was nonlinear in the softening stage [11]. Benmokrane et al. [12] proposed a classic three-line model of pulling by analysing many laboratory tests. Tis model considered the residual strength of the rockbolt and could describe the strain-softening behaviour of the rockbolt. Monti et al. [13] proposed a bilinear bond-slip model, which overcame the main defect of the ideal elastic-plastic model as it refected slip softening in a linearly decreasing manner, but it failed to describe the debonding behaviour of the anchoranchorage interface. Trabacchin et al. [14] used theories and tests to analyse the bonding behaviour between basalt FRP (fbre-reinforced polymer) bars and concrete and proposed a bilinear bond-slip model. Ren et al. [15] considered the residual bond strength and proposed an analytical solution to predict the mechanical properties of a full-length rockbolt during pulling based on the three-length bond-slip model. Shen et al. [16] studied the bonding performance between early high-strength concrete and reinforcement by using the pullout test and proposed a model to predict the bonding strength-slip interaction relationship between concrete and reinforcement in the early stage, which was in good agreement with test results. Zhou et al. [17] studied the double exponential curve shear slip model and the rockbolt linear reinforcement elastic-plastic constitutive model based on numerical modelling. Based on the results of a pullout test, the numerical model could well describe the rockbolt interfacial slip failure and tensile failure as two forms of damage under a pullout load. Based on classical three-line models, many scholars proposed various bond-slip models, but all were based on linear softening, which overestimated the rockbolt carrying capacity [15,[18][19][20][21]. Although Yue et al. [22] considered the load transfer behaviour of the rockbolt-anchorage interface as exponential softening, they did not consider the situation of the rockbolt yielding under a pullout load. Chen et al. [23] studied the bond properties and bond-slip constitutive model of reinforcement in rubber powder-modifed polypropylene fbre concrete, but this model also did not consider the yield of the reinforcement. Many rockbolt bond-slip models consider that the rockbolt does not yield. However, when the grouted length is sufciently long, the rockbolt yields in the pullout test, and such models cannot accurately describe rockbolt load transfer behaviour.
In view of the above problems, pullout tests of grouted rockbolt systems with and without bond defects were conducted, and the load transfer behaviour and specimen failure mode were analysed. Ten, based on the load transfer process at the interface between the rockbolt and cement mortar, bond-slip models considering rockbolt nonyielding and yielding were proposed and verifed by experiments. Tis model has a certain reference value for the study of the load transfer behaviour of rockbolts.

Specimen Design.
In this paper, C40 concrete is used to simulate the surrounding rock. Te concrete specimen is a cylinder with a diameter and length of 150 mm and 1500 mm, respectively. Te raw materials are 42.5 grade ordinary Portland cement, fne aggregate comprising natural river sand with particle sizes of 0.3-1.18 mm, and coarse aggregate comprising pebbles with particle sizes of 5-20 mm. Te mixture ratio of the concrete specimen is cement : water : river sand : pebbles � 1 : 0.47 : 1.3 : 3.02. Cement mortar is used as the cement mortar, and its raw materials are 42.5 grade ordinary Portland cement and fne aggregate comprising natural river sand with particle sizes of 0.3-0.6 mm [9]. To ensure that the rockbolt slips and cement mortar is easily injected into the anchor hole under the condition that the rockbolt is not pulled of, the mixture ratio of the cement mortar is water : cement : river sand � 1 : 1 : 3.2. Te combination of cement mortar and concrete is shown in Table 1 [9].
Te rockbolt is made of a threaded steel bar with a diameter of 25 mm and a length of 2500 mm. Te concrete specimen is a cylinder with an inner diameter of 40 mm, an outer diameter of 150 mm, and a height of 1500 mm. A detailed picture of the specimen is shown in Figure 1. To simulate the feld conditions, the production process of the specimen is as follows: frst, a round steel bar with a diameter of 40 mm is placed in the centre of the steel mould, and concrete is poured and simultaneously vibrated to discharge the air bubbles in the concrete. After curing the concrete specimen for 2 days, the round steel bar is pulled out, and the specimen is demoulded and cured in the laboratory for 28 days until its strength is stable. Finally, cement mortar is used to anchor a rockbolt in the centre of the hole. A bond defect with a length of L was set 400 mm from end A [24], and the specimen was cured for 7 days prior to the pullout test.
A pullout testing machine (PTM) (see Figure 2) was designed and manufactured to conduct the rockbolt pullout test [25]. Te pullout load was applied to the rockbolt by a hollow jack with a 300 kN loading capacity, and the load was measured by the load transducer. Te displacement of the rockbolt was measured by the laser displacement sensor.

Analysis of the Load Transfer Behaviour of the Rockbolt.
When the value of L is 0 mm, there is no bond defect in the grouted rockbolt systems, and the axial load transfer behaviour of the rockbolt in the grouted rockbolt systems is shown in Figure 3. During the pullout process, the rockbolt experienced four stages of progressive failure, including elasticity, yield, interface softening, and complete slip. Tis is mainly due to the perfect bond between the rockbolt and cement mortar. Te load-displacement curve starts to rise with a high initial slope at frst. Te interface between the rockbolt and cement mortar is in the elastic stage, and the chemical adhesion and mechanical interlocking have not been disturbed. With increasing pullout loads, chemical adhesion and mechanical interlocking are fully mobilized and used in grouted rockbolt systems. When the shear force exceeds the shear strength of the interface between the rockbolt and cement mortar, the interface softens and gradually begins to slip. Due to the large pullout load, the rockbolt yields, but the pullout load does not reach the ultimate strength of the rockbolt. Te rockbolt is not pulled out. Ten, the interface bond between the rockbolt and cement mortar fails, resulting in the rockbolt completely slipping. Due to the friction between the rockbolt and cement mortar, the rockbolt has residual strength and retains a certain supporting ability.
When the value of L is 400 mm, there is a bond defect with a length of 400 mm in the grouted rockbolt systems. Te axial load transfer behaviour of the rockbolt in the grouted rockbolt systems is shown in Figure 4. Due to the presence of bond defects, the rockbolt does not yield. With the increase in pullout load, the load in the rockbolt increases linearly to the maximum load, and then, the interface between the rockbolt and cement mortar softens, leading to a decreasing load, and fnally, the rockbolt slips.

Failure Mode Analysis of the Grouted Rockbolt Systems.
Te failure mode of the grouted rockbolt systems when the value of L is 400 mm is shown in Figure 5(a), and the rockbolt is directly pulled out from the grouted systems. In the pullout process, the radial pressure on the concrete caused by the wedge action of the rockbolt is less than the tensile strength of the concrete, resulting in no damage to the surface of the concrete. Te failure mode of the specimen of the grouted rockbolt systems involves the rockbolt being pulled out.
Te failure mode of the grouted rockbolt systems when the value of L is 0 mm is shown in Figure 5(b). With the increase in load, the cement mortar at the loading end is frst damaged and then crushed by the internal pressure of the concrete. Te cement mortar frst breaks at the loading end, and the surface of the rockbolt rib falls from the cement mortar. Te chemical adhesive force gradually decreases, and then, the cement mortar is crushed by the internal pressure of the concrete. After the slip of the rockbolt, the cement mortar debris slides out from the grouted hole at the loading end without gathering. Terefore, the load acting on the concrete at the grouted hole at the loading end is relatively small, and the concrete does not experience splitting failure. With a further increase in the pullout load at the free end, the force acting on the free end of the rockbolt gradually increases. Te sliding part of the rockbolt gradually advances towards the free end, inducing the rib of the rockbolt to crush the cement mortar, which accumulates in the grouted hole. Tis causes a wedge action and internal radial pressure on the concrete, resulting in radial expansion of the concrete, as shown in Figure 6 [26]. When the radial pressure exceeds the tensile strength of the concrete, internal cracks rapidly develop and spread to the surface of the concrete. However, as the free end is restricted by the end cap of the equipment, it is equivalent to exerting lateral pressure on the free end. Tis prevents the expansion of the concrete and fnally leads to the formation of splitting cracks and tensile cracks in the middle of the concrete, which extend to the loading end. Te splitting failure mode is mainly caused by the shear dilatancy of cement mortar as a result of the wedging action of the rib on the rockbolt. When the ribbed rockbolt moves under the pullout load, the cement mortar around the rockbolt undergoes shear dilatation, resulting in an increase in the radial  Shock and Vibration 3 displacement of the cement mortar [27,28]. Figure 5(b) shows that there are four tensile cracks in the free-end concrete. After the end cover of the equipment is removed, the free-end concrete is roughly divided into seven parts by splitting cracks and tensile cracks.

Determination of the Bond-Slip Model of a Fully Grouted
Rockbolt. Te existing trilinear bond-slip model cannot well refect the nonlinear characteristics of the load-displacement relationship at the grouted interface [15,[18][19][20][21], while the nonlinear bond-slip model considering nonlinear softening does not fully consider the yielding of the rockbolt [22,23]. According to the load transfer process at the interface between the rockbolt and cement mortar in this study, the rockbolt yields but does not break in the pullout process. However, it is not difcult to fnd that the above model cannot fully describe the load transfer behaviour of rockbolts in the pullout process. Te yielding of rockbolts will lead to the destruction of the grouted structure and endanger the safety of the roadway. Terefore, it is very important to study and consider the bondslip model when the yielding of the rockbolt occurs. When the value of L is 0 mm, according to the test results under the pullout load, the load at the interface between the rockbolt and the cement mortar undergoes four stages-elasticity, yielding, softening, and sliding, without Computer Digital oscilloscope Static strain indicator Laser displacement sensor  Figure 7. Here, F y , F m , and F r are the yield load, maximum load, and residual load of the rockbolt, respectively, and S 1 , S 2 , and S 3 are the corresponding displacements of F y , F m , and F r .
In the o-a stage, the pullout load linearly increases with increasing displacement, and the bond-slip model can be described as follows: In the a-b stage, the rockbolt begins to yield from point a. Te simplifed bond-slip relationship of the rockbolt at the yield stage is linear and can be described as follows: The restriction of device cap is equivalent to applying lateral pressure in concrete.
The concrete and cement mortar were expanded caused by wedge action.
Debris was accumulated in the grouted holes.

Pullout load
Rockbolt Concrete Cement mortar Figure 6: Te state of cement mortar debris and typical stress distribution with lateral pressure [26].

Shock and Vibration 5
In the b-c stage, the interface between the rockbolt and the cement mortar begins to soften, and in this stage, the load in the rockbolt decreases nonlinearly with increasing displacement. Te bond-slip relationship can be described as follows: where the value of α is a softening parameter, depending on the test results.
In stages c-d, the rockbolt has completely slipped at point c, and the load in the rockbolt does not change as the displacement increases. At this time, the load is the residual load, and the bond-slip relationship can be described as follows: Namely, the rockbolt undergoes four stages during the pullout process, and the bond-slip model expression can be summarized as follows: When the rockbolt does not yield in the pullout process, the load at the interface between the rockbolt and cement mortar undergoes three stages: elasticity, softening, and residual. Te model in Figure 7 can be simplifed to the bond-slip model o-a-b-c in Figure 8.
In the o-a stage, the expression of the bond-slip model is similar to equation (1), and the load increases linearly with increasing displacement.
In the a-b stage, the interface between the rockbolt and the cement mortar begins to soften, and the expression of the bond-slip model is the same as that in equation (3). Te diference is that in the equation, F y becomes F r , S 3 becomes S 2 , and S 2 becomes S 1 . Te expression of the bond-slip model is as follows: In the b-c stage, the rockbolt has completely slipped, and the load in the rockbolt does not change with increasing displacement. Te expression of the bond-slip model is the same as that in equation (4).
Because the rockbolt undergoes three stages in the pullout process, its bond-slip model expression can be summarized, and equation (5) can be simplifed as follows:

Test Verifcation.
Te bond-slip model proposed in this study is applied to compare and analyse the results of rockbolt pullout tests, and the results are shown in Figure 9.
According to the test results, the value of α in equations (5) and (7) is 0.25. As shown in Figure 9(a), when the value of L is 0 mm, the rockbolt yields during the pullout process. Te results calculated with the bond-slip model agree well with the experimental results, and they can accurately describe the elastic rise, yield, softening, and complete slip of the rockbolt during the pullout process. As shown in Figure 9(b), when the value of L is 400 mm, even if there is a bond defect, the calculated results of the bond-slip model and the test results are also relatively consistent. However, at the initial stage of interface softening, the model calculation results are relatively low compared to the test results, but the diference is not signifcant. Te interface adhesion characteristics calculated based on the model can also accurately refect the actual load transfer behaviour of the rockbolt in the test. Figure 10 shows the load-displacement relationship of the grouted rockbolt systems with a rockbolt diameter of 18 mm and a bond length of 1500 mm under a pullout load [9]. Te fgure shows that the rockbolt yielded but did not break. In the test results, F y � 84 kN, F m � 120 kN, F r � 25.1 kN, S 1 � 5.8 mm, S 2 � 153.5 mm, and S 3 � 192.1 mm, and the value of α is 0.4. As shown in the fgure, the model calculation results agree well with the test results, but in the yield stage, the model calculation value is lower than the test value. Tis is mainly due to the simplifcation of the bond-slip relationship of the rockbolt in the yield stage in the model to a linear relationship. Tis simplifcation underestimates the hardening phenomenon of the rockbolt in the yield stage, resulting in an increase in the rockbolt load. During the test, the infuence of secondary factors was ignored, and the model was simplifed. Terefore, there was a small error between the results calculated Shock and Vibration 7 using the model and the test results. Te results of the calculation using the model were conservative, which was conducive to the safety of the structure. Figure 11 shows the comparison between the model calculation results and the experimental results of Li et al. [29]. In the test, the bond length of the rockbolt was 200 mm, and the rockbolt also yielded. In the test results, F y � 162.9 kN, F m � 165.9 kN, F r � 33.3 kN, S 1 � 3.98 mm, S 2 � 7.29 mm, and S 3 � 40 mm, and the value of α is 0.3 in equation (5). From the comparison results, it can be seen that the calculated results of the model agree well with the experimental results, which once again verifes that this model can accurately describe the load transfer behaviour of a rockbolt in the pullout process.

. Conclusion
In this paper, the load transfer behaviour and failure mode of rockbolts were studied through experiments. Based on the load transfer process at the interface between the rockbolt and cement mortar, bond-slip models considering the nonyielding and yielding of rockbolts were proposed, and the following conclusions were obtained: (1) When the value of L was 400 mm, the rockbolt was pulled out without yielding. When the value of L was 0 mm, with an increase in the pulling load, chemical adhesion and mechanical interlocking were fully mobilized and acted on the grouted rockbolt systems. Te rockbolt yielded but did not break. (2) When the value of L was 0 mm, the rockbolt was pulled out, accompanied by partial splitting failure of the concrete parallel to the rockbolt direction and tensile failure of the concrete perpendicular to the rockbolt direction. Due to the efect of the wedge action, cement mortar debris were collected in the interior of the grouted rockbolt systems. As a result, radial pressure exceeded the tensile strength of the concrete, and internal cracks rapidly initiated and expanded to the concrete surface. Te free end of the grouted rockbolt system was limited by the equipment end cap, which hindered the radial expansion of the concrete and led to the formation of splitting cracks and tensile cracks in the middle of the concrete. (3) Based on the established bond-slip model that considers the yielding of the rockbolt, the calculated characteristics of interfacial adhesion could accurately refect the actual load transfer behaviour of the rockbolt in the test. For the test results of Yu et al. [9], during the yield stage, the value calculated using the model was less than the experimental value. Tis was mainly due to the simplifed linear relationship in the model, which underestimated the hardening behaviour of the rockbolt, resulting in an increase in the load.

Data Availability
Te data used to support the fndings of this study are included within the article.

Conflicts of Interest
Te authors declare that there are no conficts of interest. 8 Shock and Vibration