7.1 Introduction

Rock bolts have been used in the mining industry for a long time. They are used as the reinforcement tendons to keep the stability of openings in underground mining and the slope in open-pit mining. Furthermore, currently, rock bolts are commonly used not only in the mining engineering but also in civil engineering [1].

The rock bolt is a tendon which can be a rod or a cable inserted into a borehole drilled in rock masses or soils. Traditionally, the material of rock bolts was steel. However, engineering practices showed that the steel rock bolts can be easily corroded by the moist environment in underground mines. Then, the fibre-reinforced polymer (FRP) was used to manufacture rock bolts, which are termed as FRP rock bolts. Experimental results proved that the FRP rock bolts can effectively reinforce the surrounding rock masses without being corroded [2].

When rock bolts are installed, they can be point anchored or fully grouted. And the bonding agent can be polyester resin or cementitious grout. This chapter deals with fully grouted rock bolts.

After fully grouted rock bolts are installed in rock masses, the movement of surrounding rock mass induces shear deformation of the grout column. Consequently, shear stress occurs along the bolt/grout and grout/rock interfaces. Then, the load can be transferred between rock bolts and the surrounding rock masses. To study the load transfer process of rock bolts, numerous laboratory and in situ tests have been conducted. Compared with that, relatively less work has been conducted with analytical modelling to study the load transfer behaviour of rock bolts.

The pioneering work in this area should be attributed to Hawkes and Evans [3], who developed an analytical model to study the shear stress distribution along a fully grouted steel bolt. However, their results were only applicable to the case that the grouted bolt was coupled with the surrounding confining medium. Farmer [4] studied the shear stress distribution of the bolt/grout interface with an analytical model. Experimental work was used to validate this analytical model. However, this model was not applicable for the debonding behaviour of the bolt/grout interface.

Ren et al. [5] studied the load transfer behaviour of a fully grouted rock bolt with a tri-linear model. The pull-out stages of a rock bolt were divided into five stages, including elastic, elastic-softening, elastic-softening-debonding, softening-debonding and debonding. The model was successfully validated with laboratory and in situ test results. However, when the tri-linear model was used to depict the bond-slip behaviour of the bolt/grout interface, pure softening may also occur along the bolt/grout interface, as indicated by Blanco Martín et al. [6]. And this was not considered in the analytical model proposed by Ren et al. [5].

Recently, Liu et al. [7] proposed an analytical model to study the load transmission model for a rock bolt subjected to open and sliding joint displacements. Experimental work was used to validate this model. The modelling results revealed that for a definite joint displacement and joint location, larger anchored angle between the rock bolt and the joint face was beneficial for improving the reinforcement effect.

Ma et al. [8] proposed an analytical model to study the load transfer mechanism of fully grouted rock bolts. In their model, a closed nonlinear model was used to depict the bond-slip behaviour of the bolt/grout interface [8,9,10]. In this chapter, the authors adopted their model and conducted a study on the load transfer behaviour of rock bolts. First, a general illustration of the analytical model was given. Then, an experimental test was used to validate this analytical model. Following it, a parametric study was conducted with this analytical model. Last, the load distribution along a fully grouted rock bolt was studied based on this analytical model.

7.2 Illustration of the Analytical Model

For a fully grouted rock bolting system without faceplate, when the rock bolt is loaded, shear stress occurs along the bolt/grout and grout/rock interfaces. When the shear stress is higher than the interfacial bond strength, bond failure of the interface occurs. Experimental tests proved that the major failure mode is the bond failure of the bolt/grout interface [11]. Therefore, in this chapter, failure along the bolt/grout interface was studied.

Assuming that the rock bolt is in linear elastic deformation along the axial direction and the shear stress of the bolt/grout interface is equal to the axial displacement of the rock bolt at the same position, the shear stress of the bolt/grout interface, the pull-out load of the rock bolt and the load distribution along the rock bolt can be expressed with Eqs. (7.1), (7.2) and (7.3), respectively [8, 9].

$$ \tau \left( \delta \right) = \frac{{E_{{\text{b}}} D_{{\text{b}}} }}{4}\frac{a}{{b^{2} }}{\text{e}}^{{ - \frac{\delta }{a}}} \left( {1 - {\text{e}}^{{ - \frac{\delta }{a}}} } \right) $$
(7.1)

Where a and b are coefficients.

$$ F_{{\text{b}}} \left( \delta \right) = \frac{{E_{{\text{b}}} \pi D_{{\text{b}}}^{2} }}{4}\frac{a}{b}\left( {1 - {\text{e}}^{{ - \frac{\delta }{a}}} } \right) $$
(7.2)
$$ F_{{\text{b}}} \left( \delta \right) = \frac{{E_{{\text{b}}} \pi D_{{\text{b}}}^{2} }}{4}\frac{a}{b}\frac{1}{{\left( {1 + {\text{e}}^{{ - \frac{{x - x_{0} }}{b}}} } \right)}} $$
(7.3)

Where x0 is a coefficient and can be calculated with Eq. (7.4).

$$ x_{0} = L + b\ln \left( {\frac{a}{b}\frac{{E_{{\text{b}}} \pi D_{{\text{b}}}^{2} }}{{4F_{{\text{b}}} }} - 1} \right) $$
(7.4)

7.3 Validation of the Analytical Model

To further confirm the credibility of this analytical model, an in situ pull-out test was used to validate this analytical model. Bastamia et al. [12] conducted a case study in a coal mine. Specifically, rock bolts with a diameter of 22 mm were installed in the rock mass [12]. The rock bolt has an elastic modulus of 207 GPa and embedment length of 1800 mm. During the test, the pull-out load and displacement were recorded.

In this chapter, to analytically model the pull-out process of the rock bolt, Eq. (7.2) was used to calculate the load versus displacement of the rock bolt. It was found that when a = 1.5 mm and b = 620.6 mm, there was a close match between the experimental pull-out test result and the analytical result, as shown in Fig. 7.1.

Fig. 7.1
A scatterplot of pull-out load versus pull-out displacement. The plots of test results lie on either side of an increasing trend which is the analytical result.

Comparison of the in situ test result conducted by Bastamia et al. [12] and the analytical modelling result

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By substituting the calculated coefficients a and b into Eq. (7.1), the shear stress versus shear slippage of the bolt/grout interface can be acquired, as shown in Fig. 7.2.

Fig. 7.2
A line graph of shear stress versus shear slippage. Values are estimated. The line starts at (0, 0), rises to (1.3, 1.1), and drops to end at (6.1, 0.08).

Shear stress versus shear slippage of the bolt/grout interface

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It can be seen that after the rock bolt was loaded, the shear stress of the bolt/grout interface increased nonlinearly. When the shear slippage of the bolt/grout interface was 1 mm, the shear stress of the bolt/grout interface reached the bond strength of the bolt/grout interface, which was 1.11 MPa. After that, the shear stress of the bolt/grout interface decreased exponentially.

7.4 Parametric Study

After the analytical model was validated, a parametric study was conducted with this analytical model to evaluate the influence of different parameters on the load transfer behaviour of rock bolts.

7.4.1 The First Coefficient

In this analytical model, there are two important coefficients including a and b. The parametric study was first conducted to evaluate the influence of the coefficient a on the load transfer performance of rock bolts. Specifically, the load transfer of a rock bolt with an elastic modulus of 210 GPa and diameter of 20 mm was simulated. The coefficient b was kept constant in this calculation and equal to 800 mm. As for the coefficient a, three different values were used separately, namely 2, 3 and 4 mm. The pull-out load versus displacement relationship is shown in Fig. 7.3.

Fig. 7.3
A multi-line graph of pull-out load versus pull-out displacement. 3 lines of a = 2, 3, and 4 millimeters follow an increasing trend starting from (0, 0).

Pull-out load versus displacement curve of the rock bolt when the coefficient a was different

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Apparently, the coefficient a had an obvious effect on the load transfer capacity of rock bolts. The maximum load transfer capacity of the rock bolt was extracted. Then, the relationship between the maximum load transfer capacity of the rock bolt and the coefficient a can be plotted, as shown in Fig. 7.4. It can be seen that with the coefficient a increasing, the maximum load transfer capacity of rock bolts increases linearly.

Fig. 7.4
A bar graph with plots depicts the maximum pull-out load versus the value of the coefficient a. All values are estimated and are as follows. 2, 160. 3, 240. 4, 340. A line connects the maximum value in each bar following an upward trend.

Effect of coefficient a on the maximum load transfer capacity of the rock bolt

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Meanwhile, the shear stress versus shear slippage of the bolt/grout interface can be acquired, as shown in Fig. 7.5.

Fig. 7.5
A multi-line graph of shear stress versus shear slippage. 3 lines of a equals 2, 3, and 4 millimeters follow an increasing to decreasing trend with a peak approximately at (1.5, 0.8), (2, 1.25), and (2.5, 1.6) respectively.

Shear stress versus shear slippage curve of the bolt/grout interface when coefficient a was different

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The results show that although the coefficient a was different, the trend of the bond-slip behaviour of the bolt/grout interface was still consistent. However, the coefficient a had an apparent influence in deciding the shear slippage when the bond strength of the bolt/grout interface occurred. For example, with the coefficient a increasing from 2 to 4 mm, when the bond strength of the bolt/grout interface occurred, the shear slippage also increased from 1.4 to 2.8 mm.

Furthermore, with the coefficient a increasing, the bond strength of the bolt/grout interface also increased. To further illustrate this, the bond strength of the bolt/grout interface when the coefficient a was different was extracted. Then, the relationship between the bond strength of the bolt/grout interface and the coefficient a was plotted, as shown in Fig. 7.6.

Fig. 7.6
A bar graph with plots depicts bond strength versus the value of the coefficient a. All values are estimated and are as follows. 2, 0.8. 3, 1.2. 4, 1.6. A line connects the maximum value in each bar following an upward trend.

Effect of coefficient a on the bond strength of the bolt/grout interface

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The results show that when a = 2 mm, the bolt/grout interface had a bond strength of 0.82 MPa. Compared with that, when a = 3 mm, increasing by 50%, the bolt/grout interface had a bond strength of 1.23 MPa, increasing by 50%. Last, when a = 4 mm, increasing by 100%, the bolt/grout interface had a bond strength of 1.64 MPa, increasing by 100%. Apparently, the larger the coefficient a, the higher the bond strength of the bolt/grout interface. Additionally, there was a linear relationship between the bond strength of the bolt/grout interface and the coefficient a.

7.4.2 The Second Coefficient

Similarly, the influence of coefficient b on the load transfer performance of a rock bolt was studied. The rock bolt had an elastic modulus of 210 GPa and diameter of 20 mm. In this case, the coefficient a was kept constant and equal to 2 mm. As for the coefficient b, three different values were used, namely 500, 600 and 700 mm. The pull-out load versus displacement relationship of the rock bolt is shown in Fig. 7.7.

Fig. 7.7
A multi-line graph of pull-out load versus pull-out displacement. 3 lines of b = 500, 600, and 700 millimeters follow an increasing trend starting from (0, 0).

Pull-out load versus displacement curve of the rock bolt when the coefficient b was different

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Apparently, the coefficient b had a significant effect on the load transfer performance of rock bolts. With the coefficient b increasing, the load transfer capacity of rock bolts tended to decrease. To clearly show this, the maximum load transfer capacity of rock bolts was extracted and plotted with the coefficient b (Fig. 7.8).

Fig. 7.8
A bar graph with plots depicts the maximum pull-out load versus the value of the coefficient b. All values are estimated and are as follows. 500, 260. 600, 240. 700, 195. A line connects the maximum value in each bar following a decreasing trend.

Effect of coefficient b on the maximum load transfer capacity of the rock bolt

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It shows that when b = 500 mm, the maximum load transfer capacity of the rock bolt was 264 kN. Compared with that, when b = 600 mm, increasing by 20%, the maximum load transfer capacity of the rock bolt was 220 kN, decreasing by 16.7%. When b = 700 mm, increasing by 40%, the maximum load transfer capacity of the rock bolt was 188 kN, decreasing by 28.8%. Therefore, increasing the coefficient b leaded to nonlinear decreasing of the maximum load transfer capacity of rock bolts. As for the influence of the coefficient b on the bond-slip behaviour of the bolt/grout interface, it is shown in Fig. 7.9.

Fig. 7.9
A multi-line graph of shear stress versus shear slippage. 3 lines of b = 700, 600, and 500 millimeters follow an increasing to decreasing trend with peaks at (1.9, 1.1), (1.9, 1.48), and (1.9, 2.1) respectively.

Shear stress versus shear slippage curve of the bolt/grout interface when coefficient b was different

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It shows that with the coefficient b increasing, the trend of the bond-slip of the bolt/grout interface kept consistent. Furthermore, the coefficient b had no effect in deciding the shear slippage where the bond strength of the bolt/grout interface occurred. For example, in this case, although the coefficient b was different, the shear stress of the bolt/grout interface reached the bond strength at the same shear slippage of 1.4 mm.

However, the coefficient b had an apparent effect on the bond strength of the bolt/grout interface. To illustrate this clearly, the bond strength of the bolt/grout interface was extracted and plotted with the coefficient b, as shown in Fig. 7.10.

Fig. 7.10
A bar graph with plots depicts bond strength versus the value of the coefficient b. All values are estimated and are as follows. 500, 2.1. 600, 1.45. 700, 1.1. A line connects the maximum value in each bar following a decreasing trend.

Effect of coefficient b on the bond strength of the bolt/grout interface

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The results show that increasing the coefficient b leaded to decreasing the bond strength of the bolt/grout interface. Specifically, when b = 500 mm, the bond strength of the bolt/grout interface was 2.1 MPa. Compared with that, when b = 600 mm, increasing by 20%, the bond strength of the bolt/grout interface decreased to 1.46 MPa, decreasing by 30.5%. When b = 700 mm, increasing by 40%, the bond strength of the bolt/grout interface decreased to 1.07 MPa, decreasing by 49%. Therefore, there was a negative nonlinear relationship between the bond strength of the bolt/grout interface and the coefficient b.

7.4.3 Elastic Modulus of the Rock Bolt

The influence of the elastic modulus of the rock bolt was studied on the load transfer performance of rock bolts. In this case, the rock bolt had a diameter of 20 mm. As for the coefficients a and b, they were 2 and 500 mm. As for the elastic modulus of the rock bolt, they were ranged from 70 to 210 GPa with an interval of 70 GPa. The load versus displacement relationship of the rock bolt is shown in Fig. 7.11.

Fig. 7.11
A multi-line graph of pull-out load versus pull-out displacement. 3 lines of E subscript b = 70, 140, and 210 gigapascals follow an increasing trend starting from (0, 0).

Pull-out load versus displacement curve of the rock bolt when the elastic modulus of the rock bolt was different

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The results show that increasing the elastic modulus of the rock bolt was beneficial for improving the maximum pull-out load of the rock bolt. Furthermore, with the elastic modulus of the rock bolt increasing, the slope of the load–displacement curve also increased. This indicated that increasing the elastic modulus of the rock bolt was beneficial for improving the stiffness of the rock bolting system.

7.4.4 Rock Bolt Diameter

The analytical model was also used to evaluate the influence of the rock bolt diameter on the load transfer behaviour of rock bolts. Specifically, the elastic modulus of the rock bolt was kept constant and equal to 210 GPa. As for the coefficients a and b, they were set as 2 mm and 500 mm. The rock bolt diameter was ranged from 15 to 25 mm with an interval of 5 mm. The pull-out load versus displacement relationship of rock bolts is shown in Fig. 7.12. It shows that increasing the rock bolt diameter can effectively improve the load transfer capacity of rock bolts.

Fig. 7.12
A multi-line graph of pull-out load versus pull-out displacement. 3 lines of D subscript b = 15, 20, and 25 millimeters follow an increasing trend starting from (0, 0).

Pull-out load versus displacement curve of the rock bolt when the rock bolt diameter was different

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7.5 Load Distribution Along a Fully Grouted Rock Bolt

The load distribution along a fully grouted rock bolt was studied with this analytical model. Specifically, the rock bolt had an elastic modulus of 210 GPa and diameter of 25 mm. As for the coefficients a and b, they were set as 0.53 mm and 200 mm. To simulate the load variation along the grouted rock bolt, an embedment length of 2000 mm was used.

The rock bolt was pulled to a displacement of 3.2 mm and reached a maximum pull-out load of 273 kN. During the pull-out process, the axial load in the rock bolt was calculated when the pull-out load ranged from 50 to 250 kN with an interval of 50 kN. Specifically, by substituting the pull-out load to Eq. (7.4), x0 can be acquired. Then by substituting x0 to Eq. (7.3), the axial load along the rock bolt from the free end to the loaded end can be acquired, as shown in Fig. 7.13.

Fig. 7.13
A multi-line graph of load versus distance to the free end. 5 lines of load from 50 kilonewtons to 250 kilonewtons follow an increasing trend starting linearly from (0, 0).

Load distribution along the fully grouted rock bolt

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The results show that during the pull-out process, the axial load in the rock bolt decayed from the loaded end to the free end independent of the pull-out load. However, the decaying trend of the axial load was different. Specifically, when the pull-out load was small, for example, 50 kN, the axial load in the rock bolt decayed from the loaded end to the free end with a simple exponential form. Furthermore, the bending direction of the load distribution curve was always upward. However, when the pull-out load was large, for example, 250 kN, only the axial load in the rock bolt close to the free end decayed with an exponential form. There were two bending directions of the load distribution curve. For the section close to the free end, the bending direction of the load distribution curve was upward whilst for the section close to the loaded end, the bending direction of the load distribution curve was downward. Nevertheless, further experimental work should be conducted to validate this finding.

7.6 Limitation and Recommendation for Future Work

A limitation of the current study was that only one experimental pull-out test was used to validate the credibility of this analytical model. In fact, the previous research has proved that the rock bolt performance was influenced by many parameters such as the rock bolt profile, the rock mass property, the embedment length and the borehole diameter [13,14,15,16,17,18]. Therefore, in the future work, more experimental pull-out tests with different performance will be adopted to further confirm the accuracy of the analytical model.

7.7 Conclusions

An analytical model was used to study the load transfer behaviour of fully grouted rock bolts. In this analytical model, a closed nonlinear model was used to depict the bond-slip behaviour of bolt/grout interface.

An in situ pull-out test was used to validate this analytical model, showing that there was a close match between the experimental result and analytical result. Following it, a parametric study was conducted to evaluate the influence of two coefficients, elastic modulus of the rock bolt and the diameter of the rock bolt on the performance of rock bolts. It was found that increasing the coefficient \(a\) leaded to linear increasing of the maximum pull-out load of rock bolts and the bond strength of the bolt/grout interface. On the other hand, increasing the coefficient b leaded to nonlinear decreasing of the maximum pull-out load of rock bolts and the bond strength of the bolt/grout interface. Improving the elastic modulus of the rock bolt and the diameter of the rock bolt were beneficial for improving the load transfer capacity of the rock bolting system.

Additionally, the load distribution along the fully grouted rock bolt was studied with this analytical model. It was found that axial load in the rock bolt decayed from the loaded end to the free end independent of the pull-out load. However, the trend of the load distribution curve along the rock bolt was influenced by the pull-out load. When the pull-out load was relatively small, the bending direction of the load distribution curve was upward. However, when the pull-out load was relatively large, there were two bending directions of the load distribution curve. For the section close to the free end, the bending direction of the load distribution curve was upward whilst for the section close to the loaded end, the bending direction was downward.