6.1 Introduction

Fully grouted rock bolting is commonly used in roadway reinforcement in mining and tunnelling reinforcement [1]. Due to the advantage that the reinforcing resistance is significant and it can effectively improve the self-bearing capacity of rock masses, rock bolting has been widely applied in rock and soil reinforcement [2].

However, in situ practice results showed that failure of the reinforcement system still occurred [3]. The previous research summarised that failing of the reinforcement system could be bolt rupture or bond failure [4]. It is more common to encounter bond failure at the boundary surface between bolts and grout [5]. The reason is that the bond stress at the boundary surface between bolts and grout is beyond the bond strength [6]. More importantly, it is highly related to the bond capacity of bolts [7].

To study the bond mechanism of rock bolts, laboratory tests, analytical modelling, numerical simulation and engineering practices were used [8]. For example, Zhang et al. [9] conducted in situ tests on gob-side roadways to monitor the loading state of bolts under the thick coal seam mining condition. Skrzypkowski et al. [10] performed laboratory pull-out tests on bolts with different diameters to compare the performance of bolts. Chen et al. [11] used the experimental approach to study the anchorage performance of bolts installed in a bedded rock mass. Chen and Li [12] adopted the modified elements to study the interaction between bolts and in situ rock masses.

Amongst those methods, analytical modelling has the advantage that it can effectively reveal the bond course between bolts and rock masses subjected to loading [13]. Therefore, this paper studied the bond capacity of bolts based on analytical modelling.

In analytical modelling, former investigators performed numerous investigations regarding the bond capacity of bolts [14]. Wang et al. [15] analysed the coupling effect of bolts installed in rock masses. Zou and Zhang [16] used a nonlinear equation to illustrate the debonding behaviour of the boundary surface between bolts and grout. Ren et al. [17] studied the shear failure course of the boundary surface between bolts and grout. Zheng et al. [18] studied the reinforcing system composed of bolts and rock masses. Ma et al. [19] established the distributing link of the bond stress at the boundary surface between bolts and grout. He et al. [20] deduced the interaction link between the tensile stress in bolts and the bond stress after bolts which were loaded. Saleem [21] developed an analytical model to study the bond performance of bolts under the impact loading condition. The superiority of their model was that it could be used to predict the bond performance of different bolt types. Lv et al. [22] deduced the load-carrying force of cables based on an elastoplastic model. Wang et al. [23] developed a three-dimensional model to study the bond performance of fully grouted bolts based on a closed tri-linear relationship. Their results showed that there was a critical grouting length. Zhang et al. [24] considered the mutual interaction link between bolts and grout. Saleem and Hosoda [25] employed the Latin Hypercube sampling approach to analyse the test data regarding several fully grouted bolts. Their findings showed that grouting length had more effect on the load-carrying force of bolts compared with bolt diameter.

Those mentioned analytical modelling research successively revealed the coupling link between bolts and rock masses. However, much less work has been performed to quantitatively study the influence of parameters on the bond performance of fully grouted bolts. Moreover, little work was conducted to quantitatively examine the bond stress distributing at the boundary surface between bolts and grout.

To overcome those shortcomings, this paper used the new analytical model developed by the authors to quantitatively examine the bond performance of fully grouted rock bolts [26]. The originality of this paper is that it used the newly developed analytical model to quantitatively examine the bond performance of fully grouted bolts. Moreover, relying on this analytical model, the bond stress distributing at the boundary surface between bolts and grout was quantitatively studied.

In this paper, first, the calculating principle of this analytical model was illustrated. Then, this analytical model was used to calculate and analyse the load-carrying force of bolts. The effect of four different parameters on the load-carrying force of bolts and the bond stress distributing state was studied. This study is conducive to enriching the base of knowledge.

6.2 Calculating Principal of the Constitutive Equation

The authors of this paper previously conducted a theoretical analysis on the load-carrying force of bolts. During modelling, the shear course of the boundary surface was simulated with a tri-linear equation (Fig. 6.1) [26].

Fig. 6.1
A line graph of bond stress versus bond slipping. Values are estimated. A line of tri-linear increases from (0, 0), peaks at a point, decreases, and then remains stable from halfway. The other 2 lines of elastic perfectly plastic and bi-linear follow horizontal and decreasing trends respectively.

Tri-linear equation

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$$ \tau = k\delta + \tau ^{\prime} $$
(6.1)

Where k: slope of the tri-linear equation.

The tri-linear equation was revised from the elastic perfectly plastic model and the bi-linear model. However, the shortcoming of the elastic perfectly plastic model was that it neglected the post-failure performance of the boundary surface. As for the bi-linear model, its shortcoming was that it did not consider the residual shear strength of the boundary surface [27]. Therefore, through combining those two models, the tri-linear equation was used.

The tri-linear equation assumed that the bond stress of the boundary surface grew linearly with the bond slipping. When the bond slipping grew to a specific value, the bond stress reached the bond strength. After that, with the continuous growth of the bond slipping, the bond stress at the boundary surface decreased to the residual shear strength [28]. Last, with the further growth of the bond slipping, the bond stress at the boundary surface remained constant, equalling the residual shear strength.

The credibility of the tri-linear equation was confirmed with the previous experimental research. For example, Benmokrane et al. [29] conducted pull-out tests on fully grouted bolts to study the bond behaviour of the boundary surface. Their results showed that after the peak, the shear stress at the boundary surface dropped gradually. Moreover, with the slipping further growing, the shear stress was likely to be constant. Based on this phenomenon, this paper used the tri-linear equation to depict the bond behaviour of the boundary surface.

Then, this tri-linear equation was incorporated into the anchorage body, and a governing equation was developed, as calculated with Eq. (6.2).

$$ \frac{{{\text{d}}\delta^{2} \left( x \right)}}{{{\text{d}}x^{2} }} - \lambda^{2} \tau \left( x \right) = 0 $$
(6.2)

In this equation, the coefficient of λ can be expressed with Eq. (6.3):

$$ \lambda^{2} = 4\left( {\frac{1}{{D_{{\text{b}}} E_{{\text{b}}} }} + \frac{{\pi D_{{\text{b}}} }}{{4E_{{\text{m}}} A_{{\text{m}}} }}} \right) $$
(6.3)

With the loading of bolts, the boundary surface successively experienced five stages, including the elastic stage, the elastic-weakening stage, the elastic-weakening-friction stage, the weakening-friction stage and the friction stage [30]. Then, analysis was conducted on the tensile stress in bolts and the bond stress on the boundary surface during those five stages. Therefore, the equations that describe the tensile stress distributing of bolts and the bond stress spreading at the boundary surface were obtained. The loading terminal of bolts was regarded as the research objective. Then, the link between the load-carrying force and loading displacement of bolts was obtained.

The credibility of this model was validated with experimental tests. The test conducted by Bai et al. [31] was used as an example. It showed a good match between the test and modelling [26].

6.3 Parameter Study

The input parameters of this model included bolt diameter, bolt modulus, grouting length, modulus of rock masses, the cross-sectional area of rock masses and properties of the boundary surface. Then, this paper studied the effect of relevant parameters on the load-carrying force of bolts.

6.3.1 Bolt Diameter

In this section, it was assumed that the bolt diameter was 20 mm, and the modulus of bolts was 200 GPa. The grouting length was 2 m. As for the rock masses, the modulus was 15 GPa, and the cross-sectional area was 0.5 m2. Those parameters were shown in Table 6.1.

Table 6.1 Properties of the anchorage body
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More importantly, the properties of the boundary surface determined the bond capacity of bolts. Then, they were shown in Table 6.2.

Table 6.2 Properties of the boundary surface
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To study the effect of bolt diameter on the load-carrying force, besides the bolt diameter used in Table 6.1, another two bolt diameters were used: 15 mm and 25 mm. During the pull-out course, the load-carrying force and loading displacement at the loading terminal were recorded. Then, the bond capacity under the loading condition was obtained (Fig. 6.2). It shows that bolt diameter significantly affected the load-carrying force of bolts. When three different bolt diameters were used, the maximum force was 194 kN, 269 kN and 349 kN. With the growth of bolt diameter, the maximum force significantly grew. This finding was consistent with the experimental test results obtained by Kilik et al. [32]. Meantime, it shows that the stiffness of the anchorage body grew. When the bolt diameter was 15 mm, the initial stiffness of the anchorage body was 50 kN/mm. However, when the bolt diameter grew to 25 mm, the stiffness of the anchorage body grew to 104 kN/mm.

Fig. 6.2
A multi-line graph of force versus displacement maps 3 lines that start at the origin. All values are estimated. The lines for D equals 15, 20, and 25 millimeters increase to peak at (8, 190), (6.8, 250), and (5.5, 350) respectively and then decrease.

Effect on bolt diameter on the load-carrying force of bolts

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Also, it shows that bolt diameter had specific effect on the post-failure performance of the reinforcement system. For instance, when bolt diameter was small, such as 15 mm, after bolts reached the maximum force, the loading displacement showed an apparent snapback phenomenon. This was because the boundary surface entered the weakening-friction stage. Moreover, it was directly resulted by the continuous growth of the friction length. This phenomenon also occurred in the previous research [33]. By contrast, when bolt diameter was relatively larger, such as 25 mm, after bolts reached the maximum force, this snapback phenomenon was not apparent.

Additionally, the relationship between the maximum force of bolts and λ was analysed. With bolt diameter growing, the corresponding λ dropped from 3.66e−5 to 2.85e−5 N/m (Fig. 6.3). As for the maximum force of bolts, it grew directly. Therefore, a negative relationship between the maximum force of bolts and λ occurred.

Fig. 6.3
A dual-axis line graph of lambda multiplied by 10 superscript negative 5 and force versus bar diameter. All values are estimated. The line for lambda decreases from (15, 3.65) to (25, 2.85). The line for force increases from (15, 190) to (25, 350).

Relationship between the maximum force of bolts, λ and bolt diameter

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This paper mainly focused on the maximum force of bolts. Therefore, when those bolts with three different diameters reached the maximum force, the bond stress distribution of the boundary surface was exported (Fig. 6.4). It shows that when bolts with those three different diameters reached the maximum force, the bond stress at the boundary surface was in the elastic-weakening-friction stage.

Fig. 6.4
A line graph of bond stress versus grouting length maps 3 lines that start at (0, 2.4). All values are estimated. The lines for D equals 15, 20, and 25 millimeters increase to peak at (0.5, 3), (0.55, 3), and (0.61, 3) respectively and then decrease.

Effect of bolt diameter on the bond stress distributing on the boundary surface when peak load occurred

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Meantime, it shows that for bolts with a smaller diameter, when the maximum force was reached, the length of the friction section was larger. For instance, when the bolt diameter was 15 mm, the corresponding friction length was 828 mm. However, when the bolt diameter was 25 mm, the friction length dropped to 493 mm. This was related to the reinforcement stiffness of the anchorage body. When the bolt diameter was small, the reinforcement stiffness was also slight. Therefore, to reach the maximum force, a relatively large loading displacement was needed. However, under the effect of the large loading displacement, the boundary surface experienced rather large bond slipping. Then, the friction length grew.

6.3.2 Elastic Modulus of Bolts

In the in situ condition, rock bolts with different modulus may be used. For instance, in the permanent reinforcement, steel bolts with a modulus of 200 GPa were always used. However, in the roadway along the working face side, the fibre-reinforced polymer bolts with modulus of 50 GPa may be used.

Therefore, the modulus of bolts was regarded as the research objective. The purpose was to analyse its effect on the load-carrying force of bolts. In this section, a calculation was conducted on bolts with a diameter of 20 mm. To study the effect of the modulus of bolts, three pull-out analysis was conducted. The corresponding modulus of bolts was 50 GPa, 100 GPa and 200 GPa. The other input parameters were the same as the values used in Sect. 6.3.1. The comparing results of those three pull-out tests are shown in Fig. 6.5. It shows that the elastic modulus of bolts significantly affected the pre-peak and post-peak performance of fully grouted bolts. First, with the modulus of bolts growing from 50 to 200 GPa, the maximum force grew from 229 to 269 kN, growing by 17%. This indicated that although increasing the modulus of bolts was conducive to improving the bond performance of bolts, this impact was not significant. Secondly, the modulus of bolts had a significant impact on the initial stiffness of the anchorage body. Specifically, when the modulus of bolts was 50 GPa, the corresponding initial stiffness of the anchorage body was 38 kN/m. When the modulus of bolts grew to 200 GPa, the corresponding initial stiffness of the anchorage body was 76 kN/m, growing by 100%. Therefore, the larger the modulus of bolts, the higher the stiffness of the anchorage body.

Fig. 6.5
A multi-line graph of force versus displacement maps 3 lines that start at the origin. All values are estimated. The lines for E equals 50, 100, and 200 gigapascals increase to peak at (18, 230), (10.5, 250), and (6, 270) respectively and then decrease.

Effect of the modulus of bolts on the load-carrying force

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Thirdly, the modulus of bolts significantly influenced the loading displacement where the maximum force of bolts occurred. For example, when bolts with a modulus of 50 GPa were tested, the loading displacement at peak force was 18.1 mm. When the modulus of bolts grew to 100 GPa, the loading displacement at peak force dropped to 10.6 mm. When bolts with a modulus of 200 GPa were tested, the loading displacement dropped to 6.49 mm. Therefore, the loading displacement at peak force significantly dropped with the modulus of bolts. Last, the modulus of bolts influenced the post-peak performance of bolts. When the modulus of bolts was 50 GPa, after the peak load, there was a gradual dropping of the load-carrying force. By contrast, when the modulus of bolts grew to 200 GPa, after the peak force, the load-carrying force dropped dramatically.

Also, the relationship between the maximum force of bolts and λ was analysed. In this case, λ dropped from 6.33e−5 to 3.18e−5 N/m (Fig. 6.6). By contrast, the maximum force of bolts grew. Therefore, for this case, the smaller the λ, the larger the maximum force of bolts.

Fig. 6.6
A dual-axis line graph of lambda multiplied by 10 superscript negative 5 and force versus elastic modulus of bars. Values are estimated. The line for lambda decreases from (50, 6.4) to (200, 3.1). The line for force increases from (50, 230) to (200, 270).

Relationship between the maximum force of bolts, λ and elastic modulus of bolts

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When bolts reached the maximum force, the bond stress distributing at the boundary surface is exported (Fig. 6.7). The modulus of bolts had a significant effect on the bond stress distributing at the boundary surface. When the modulus of bolts was small, the friction length of the boundary surface was relatively large. For instance, when the modulus of bolts was 50 GPa, the friction length of the boundary surface was 1322 mm. However, when the modulus of bolts was 200 GPa, the friction length of the boundary surface was 648 mm. This was because when the modulus of bolts was small, the stiffness of the reinforcement system was relatively small. Therefore, after a rather large displacement, bolts finally reached the maximum force. Moreover, when a large loading displacement generated, the boundary surface experienced relatively large bond slipping. Therefore, the friction length was relatively large.

Fig. 6.7
A multi-line graph of bond stress versus grouting length maps 3 lines that start at (0, 2.4). Values are estimated. The lines for E equals 50, 100, and 200 gigapascals increase to peak at (0.3, 3), (0.4, 3), and (0.6, 3) respectively and then decrease.

Effect of the modulus on the bond stress distributing on the boundary surface when peak force occurred

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6.3.3 Grouting Length of Bolts

In engineering practices, the grouting length can be significantly different [34]. Therefore, in this section, the effect of grouting length on the load-carrying force of bolts was studied. The calculation was conducted on bolts with a diameter of 20 mm and modulus of 200 GPa. During the calculating course, the grouting length ranged between 2 and 3 m. The other input parameters were the same as the values used in Sect. 6.3.1. Then, under those three different grouting length conditions, the bond capacity of bolts was acquired (Fig. 6.8). It shows that growth of the grouting length did not affect the initial reinforcement stiffness of bolts. Under those three different grouting length conditions, in the elastic stage, the initial stiffness was almost same. However, the grouting length had a significant effect on the maximum force of bolts. With the growth of grouting length, the maximum force of bolts significantly grew. When the grouting length grew from 2 to 3 m, the maximum force of bolts grew from 269 to 364 kN.

Fig. 6.8
A multi-line graph of force versus displacement maps 3 lines that start at the origin. Values are estimated. The lines for L equals 2, 2.5, and 3 meters increase to peak at (6.1, 250), (9, 300), and (12, 370) respectively and then decrease.

Effect of the grouting length on the load-carrying force of bolts

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Attention is paid that this analytical model can be used to calculate the ideal grouting length for bolts. For example, in this case, it is assumed that the tensile strength of bolts was 1200 MPa. Then, this tensile strength was substituted into Eq. (6.4) to calculate the rupture force of bolts.

$$ F_{{\text{r}}} = \frac{{\sigma_{{\text{t}}} \pi D^{2} }}{4} $$
(6.4)

Where Fr: rupture force of bolts.

The calculating result showed that the rupture force of bolts was 377 kN. Then, this rupture force can be substituted into the present analytical model to back-calculate the ideal grouting of bolts. For this case, results showed that the ideal grouting length of bolts was around 3.5 m, as shown in Fig. 6.9.

Fig. 6.9
A line graph of force versus displacement. Values are estimated. A line of ideal grouting length equal to 3.5 meters starts at the origin and increases to (11.5, 375). Another line of rupture load runs parallel to the x-axis from (0, 375) to (11.5, 375).

Back calculating the ideal grouting length of bolts

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When bolts reached the maximum force, the bond stress at the boundary surface was exported (Figs. 6.10, 6.11 and 6.12). Being different from other parameters, the grouting length had a marginal effect on the bond stress distributing state at the boundary surface. Under three different grouting length conditions, the elastic length of the boundary surface was around 552 mm. All the weakening length was around 800 mm. The only difference was the friction length. Under those three grouting length conditions, the friction lengths were 648 mm, 1149 mm and 1648 mm.

Fig. 6.10
A line graph of bond stress versus grouting length. Values are estimated. A line starts at (0, 2.4), increases to peak at (0.5, 3), and then decreases to (1.3, 1.5) to end at (2, 1.5).

Bond stress distributing at the boundary surface at the peak force when the grouting length was 2 m

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Fig. 6.11
A line graph of bond stress versus grouting length. Values are estimated. A line starts at (0, 2.4), increases to a peak at (0.5, 3), and then decreases to (1.35, 1.5) to end at (2.35, 1.5).

Bond stress distributing at the boundary surface at the peak force when the grouting length was 2.5 m

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Fig. 6.12
A line graph of bond stress versus grouting length. Values are estimated. A line starts at (0, 2.4), increases to a peak at (0.5, 3), and then decreases to (1.3, 1.5) to end at (3, 1.5).

Bond stress distributing at the boundary surface at the peak force when the grouting length was 3 m

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Figure 6.11 showed the bond stress distributing at the boundary surface at the peak force when the grouting length was 2.5 m. It clearly indicated that when the peak force was reached, the bond stress at the boundary surface was in the elastic-weakening-friction state. At the free end of bolts, the bond stress was 2.42 MPa. With the distance from the free end growing, the bond stress grew nonlinearly. When the distance from the free end grew to 0.55 m, the bond stress reached a peak of 3 MPa. With the distance from the free end further growing, the bond stress dropped nonlinearly. When the distance from the free end grew to 1.35 m, the bond stress dropped to the residual bond strength of 1.5 MPa. After that, the bond stress remained constant, levelling at 1.5 MPa.

Figure 6.12 showed the bond stress distributing at the boundary surface at the peak force when the grouting length was 3 m. It showed that although the grouting length grew, the bond stress distributing was still in the elastic-weakening-friction state when the maximum force was reached. More interestingly, the positions where the bond strength and the residual bond strength were reached were identical with Fig. 6.11. However, the only difference was that the friction length grew. In this case, the friction length grew to 1648 mm, 499 mm longer than the friction length shown in Fig. 6.11.

6.3.4 Bond Slipping When the Bond Strength Was Reached

Considering that the properties of the boundary surface affected the bond capacity of bolts, the bond slipping at the bond strength was regarded as a factor to conduct the analysis. During the studying, a calculation was conducted on a bolt with a diameter of 20 mm, modulus of 200 GPa and grouting length of 2 m. To study the effect of the bond slipping, it ranged between 1 and 3 mm. The corresponding tri-linear equation is shown in Fig. 6.13.

Fig. 6.13
A multi-line graph of bond stress versus bond slipping maps 3 lines that start at the origin. Values are estimated. The lines for delta subscript 1 equals 1, 2, and 3 millimeters increase to peak at (1, 3), (2, 3), and (3, 3) respectively and then decrease.

Inputted tri-linear equation

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During calculating, the other parameters were the same as the values used in Sect. 6.3.1. After that, the analytical pull-out test was conducted on bolts, and the corresponding load-carrying force was obtained (Fig. 6.14). It shows that with the growth of the bond slipping at the bond strength, the maximum force of bolts decreased gradually. Meantime, the stiffness of the reinforcing system gradually decreased. However, this bond slipping did not affect the residual load-carrying force of bolts. Therefore, under those three different conditions, the residual load-carrying force of bolts remained around 188 kN.

Fig. 6.14
A multi-line graph of force versus displacement maps 3 lines that start at the origin. Values are estimated. The lines for delta subscript 1 equals 1, 2, and 3 millimeters increase to peak at (6, 280), (6.8, 270), and (7.5, 250) respectively and then decrease.

Effect of the bond slipping at the bond strength on the load-carrying force of bolts

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When the maximum force of bolts was reached, the bond stress at the boundary surface was exported (Fig. 6.15). With the growth of the bond slipping at the bond strength, the elastic length and the friction length of the boundary surface gradually grew. For instance, as the bond slipping at the bond strength grew from 1 to 3 mm, the elastic length of the boundary surface grew from 300 to 785 mm. Moreover, the corresponding friction length of the boundary surface grew from 483 to 814 mm. The weakening length dramatically declined from 1218 to 400 mm.

Fig. 6.15
A multi-line graph of bond stress versus grouting length. Values are estimated. A line for delta subscript 1 equal to 3 millimeters starts at (0, 2.25), rises to (0.8, 3), and drops to (1.2, 1.5) to end at (2, 1.5). The other 2 lines for delta subscript 1 equal to 1 and 2 millimeters follow a similar trend.

Effect of the bond slipping at the bond strength on the bond stress distributing of the boundary surface

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6.4 Limitation of the Current Study

The limitation of the current paper was that it neglected the influence of the filling grout in the borehole. It only considered the interaction between fully grouted bolts and surrounding rock mass.

6.5 Recommendation of Further Work

After the current study, the authors plan to incorporate this analytical model into the underground reinforcement scenarios. Then, the bond performance of fully grouted bolts reinforced in underground roadways and tunnels can be studied. In this way, the reinforcement parameters of fully grouted bolts used in the underground roadways and tunnels can be optimised.

6.6 Conclusions

This paper conducted an analytical study on the bond performance of fully grouted rock bolts based on the variable controlling method. Relied on this approach, the effect of bolt diameter, modulus of bolts, grouting length and bond slipping at the bond strength on the load-carrying force of bolts were studied. Moreover, the bond stress distributing at the boundary surface was studied. This paper is conducive to enriching the basement of knowledge. The main conclusions are listed below:

  1. (1)

    With bolt diameter growing, the maximum force of bolts significantly grew. In this case, when bolt diameter grew from 15 to 25 mm, the maximum force of bolts grew from 194 to 349 kN, growing by 80%. Meantime, the stiffness of the reinforcement system grew by 108%. When bolts reached the maximum force, the bond stress at the boundary surface was in the elastic-weakening-friction state. Moreover, the smaller the bolt diameter, the longer the friction length.

  2. (2)

    The elastic modulus of bolts significantly affected the load-carrying force of bolts. With the growth of bolts' modulus, the maximum force of bolts gradually grew. When the elastic modulus grew from 50 to 200 GPa, the maximum force grew from 229 to 269 kN, only growing by 17%. However, the stiffness of the reinforcement system grew by 100%. Moreover, the friction length at the boundary surface declined.

  3. (3)

    The grouting length had a significant effect on the maximum force of bolts. With the grouting length growing, the maximum force of bolts apparently grew. With the grouting length growing from 2 to 3 m, the maximum force rose from 269 to 364 kN, growing by 35%. However, the grouting length had almost no effect on the stiffness. When bolts reached the maximum force, the bond stress distributing of the boundary surface was consistent. With the grouting length growing, the length of the elastic part and the weakening part was equal. By contrast, only the friction length grew. With the tensile strength of bolts provided, the current analytical model can be used to back-calculate the ideal grouting length of bolts.

  4. (4)

    The bond slipping at the bond strength affected the load-carrying force of bolts. When it grew from 1 to 3 mm, the maximum force dropped from 281 to 258 kN, dropping by 8%. As for the stiffness, it dropped by 44%. Moreover, the elastic length and the friction length grew. By contrast, the weakening length declined.