5.1 Introduction

Rock bolts are tendons or cables that are commonly used in mining engineering and civil engineering to reinforce excavated rock masses or soils [1, 2]. Numerous in situ tests proved that rock bolts are efficient in improving the internal strength of the surrounding confining medium, namely rock masses and soils [3]. Therefore, rock bolts are becoming more significant in maintaining the stability of underground roadways, chambers, tunnels, surface slopes [4] and rooms in the room and pillar mining [5]. In the fully grouted rock bolting system, the bolt can deform consistently with the confining medium [6, 7]. Then, the bolt tendon can interact with the confining medium to transfer the force [8, 9].

To investigate the load-carrying capacity of rock bolts, laboratory pull-out tests were the most widely preferred method [10]. Specifically, a natural or artificial rock block can be prepared as the confining medium. Then, the bolt was installed in the drilled hole in the rock block. Cement-based or resin grout was poured into the drilled hole to bond the bolt and rock block. After the grout in the bolting hole fully cured, the bolt was pulled out from the rock block. During the experiment, load cell and displacement transducers were adopted to monitor the pull-out force and displacement. With the recorded data, the force–displacement relationship of rock bolts can be plotted, representing the performance of rock bolts. For example, Skrzypkowski et al. [11] considered that the load-carrying capacity of segmentally installed rock bolts can be influenced by the bolting hole diameter. To investigate this impact, they conducted pull-out tests on rock bolts with a diameter of 24 mm. Four different bolting hole diameters were used, ranging from 28 to 37 mm. The results showed that at the certain installing length of 100 mm and 200 mm, the maximum load-carrying capacity of rock bolts varied with the bolting hole diameter. Moreover, it was concluded that the optimum load-carrying capacity appeared when the bolting hole diameter was 32 mm.

The previous research work showed that in the laboratory tests, the geometry of the rock block was different. Usually, researchers would like to prefer cylindrical or cubic rock blocks. However, it was more common to use cylindrical rock blocks [12, 13]. It is interesting to see that although researchers used this method to study the performance of rock bolts, the diameter of rock blocks was largely different. For laboratory tests, the diameter of rock blocks usually ranged from 100 to 500 mm [14]. The previous researchers conducted a number of tests to reveal the loading performance of rock bolts when the condition of the bolt, grout and rock masses varied [15,16,17]. However, less attention was paid to evaluate the confining medium diameter effect on load-carrying capacity of rock bolts.

Rajaie [18] was the pioneer to study the confining medium diameter effect. Moreover, he used concrete materials to cast cylindrical samples. Five different sample diameters were used: 100 mm, 150 mm, 200 mm, 250 mm and 300 mm. The results showed that when the confining medium diameter was small, the peak force of rock bolts ascended rapidly with the confining medium diameter ascending. However, it almost stopped ascending once the diameter was beyond 200 mm. This indicated that there was a critical influence diameter. Once the rock block diameter was larger than this critical influence diameter, further ascending the rock block diameter had marginal effect on the performance of rock bolts. Chen et al. [19] continued this experimental work, finding that the boundary condition had an effect in determining the critical influence diameter.

Although numerous analytical models were proposed to study the performance of rock bolts [20,21,22,23], little work has been focused on the confining medium diameter effect. Therefore, this paper conducted an analytical study to investigate the influence of the confining medium diameter on the load-carrying capacity of rock bolts. First, the analytical model concept was illustrated. Then, the confining medium diameter effect was studied with this analytical model, and the results were compared with experimental results. Last, the influence of the confining medium modulus on the critical influence diameter was analysed.

5.2 Analytical Modelling Approach

The authors of this paper previously deduced an analytical model to study the loading performance of rock bolts [24]. Specifically, after the rock bolt was loaded, shearing stress occurred at the bonding face between the tendon and grout. Moreover, shearing stress occurred at the bonding face between the grout and confining medium, as shown in Fig. 5.1.

Fig. 5.1
A diagram of the bolting system depicts 3 different layers. The rock bolt layer has a grout layer on either side which is sandwiched between confining medium layers.

State of the bolting system after the bolt was loaded

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Then, a tri-linear formula was used to illustrate the bonding and debonding behaviour of the bonding face, as shown in Eqs. (5.1a–5.1c) [25]:

$$\tau = \frac{{\tau_{{\text{p}}} }}{{\delta_{{\text{p}}} }}\delta \left( {0 \le \delta \le \delta_{{\text{p}}} } \right)$$
(5.1a)
$$\tau = \frac{{\tau_{{\text{p}}} \delta_{{\text{r}}} - \tau_{{\text{r}}} \delta_{{\text{p}}} }}{{\delta_{{\text{r}}} - \delta_{{\text{p}}} }} - \frac{{\tau_{{\text{p}}} - \tau_{{\text{r}}} }}{{\delta_{{\text{r}}} - \delta_{{\text{p}}} }}\delta \left( {\delta_{{\text{p}}} < \delta \le \delta_{{\text{r}}} } \right)$$
(5.1b)
$$\tau = \tau_{{\text{r}}} \left( {\delta > \delta_{{\text{r}}} } \right)$$
(5.1c)

Then, through analysing the equilibrium relationship between the tensile stress in the tendon and shearing stress at the bonding face, the governing equation for the bolting system was obtained [24]:

$$\frac{{{\text{d}}^{2} s\left( x \right)}}{{{\text{d}}x^{2} }} - \lambda^{2} \tau \left( x \right) = 0$$
(5.2)
$$\lambda^{2} = \frac{4}{{D_{{\text{b}}} }}\left( {\frac{1}{{E_{{\text{b}}} }} + \frac{{A_{{\text{b}}} }}{{E_{{\text{m}}} A_{{\text{m}}} }}} \right)$$
(5.3)

The whole pull-out process of rock bolts was divided into five different phases, including the elastic phase (I), the elastic-softening phase (II), the elastic-softening-debonding phase (III), the softening-debonding phase (IV) and the debonding phase (V) [26]. Through substituting the tri-linear formula and boundary condition in each phase into Eq. (5.2), the relationship between the pull-out force and displacement can be obtained.

Specifically, in Phase I, the pull-out force and displacement can be computed as follows:

$$F = \frac{{\pi D_{{\text{b}}} \tau_{{\text{p}}} \tanh \left( {\lambda_{1} L} \right)}}{{\lambda_{1} s_{{\text{p}}} }}u_{{\text{b}}}$$
(5.4)
$$\lambda_{1}^{2} = \frac{{\tau_{{\text{p}}} }}{{s_{{\text{p}}} }}\lambda^{2}$$
(5.5)

In Phase II, the pull-out force and displacement can be computed as follows:

$$u_{{\text{b}}} = \frac{{\lambda_{1} s_{{\text{p}}} }}{{\lambda_{2} }}\tanh \left( {\lambda_{1} \left( {L - a_{{\text{s}}} } \right)} \right)\sin \left( {\lambda_{2} a_{{\text{s}}} } \right) - \frac{{\lambda_{1}^{2} s_{{\text{p}}} }}{{\lambda_{2}^{2} }}\cos \left( {\lambda_{2} a_{{\text{s}}} } \right) + \frac{{\tau_{{\text{p}}} s_{{\text{r}}} - \tau_{{\text{r}}} s_{{\text{p}}} }}{{\tau_{{\text{p}}} - \tau_{{\text{r}}} }}$$
(5.6)
$$F = \pi D_{{\text{b}}} \tau_{{\text{p}}} \left( {\frac{1}{{\lambda_{1} }}\tanh \left( {\lambda_{1} \left( {L - a_{{\text{s}}} } \right)} \right)\cos \left( {\lambda_{2} a_{{\text{s}}} } \right) + \frac{1}{{\lambda_{2} }}\sin \left( {\lambda_{2} a_{{\text{s}}} } \right)} \right)$$
(5.7)
$$\lambda_{2}^{2} = \frac{{\tau_{{\text{p}}} - \tau_{{\text{r}}} }}{{\delta_{{\text{r}}} - \delta_{{\text{p}}} }}\lambda^{2}$$
(5.8)

In Phase III, the pull-out force and displacement can be computed as follows:

$$u_{{\text{b}}} = s_{{\text{r}}} + \frac{{\lambda^{2} a_{{\text{d}}}^{2} \tau_{{\text{r}}} }}{2} + \left( {\frac{{\tau_{{\text{p}}} \lambda^{2} }}{{\lambda_{1} }}\tanh \left( {\lambda_{1} \left( {L - a_{{\text{d}}} - a_{{\text{s}}} } \right)} \right)\cos \left( {\lambda_{2} a_{{\text{s}}} } \right) + \frac{{\tau_{{\text{p}}} \lambda^{2} }}{{\lambda_{2} }}\sin \left( {\lambda_{2} a_{{\text{s}}} } \right)} \right)a_{{\text{d}}}$$
(5.9)
$$F = \pi D_{{\text{b}}} \left( {\frac{{\tau_{1} }}{{\lambda_{1} }}\tanh \left( {\lambda_{1} \left( {L - a_{{\text{d}}} - a_{{\text{s}}} } \right)} \right)\cos \left( {\lambda_{2} a_{{\text{s}}} } \right) + \frac{{\tau_{1} }}{{\lambda_{2} }}\sin \left( {\lambda_{2} a_{{\text{s}}} } \right) + \tau_{2} a_{{\text{d}}} } \right)$$
(5.10)

In Phase IV, the pull-out force and displacement can be computed as follows:

$$u_{{\text{b}}} = \delta_{{\text{r}}} + \lambda^{2} \tau_{{\text{r}}} a_{{\text{d}}} \left( {\frac{{a_{{\text{d}}} }}{2} + \frac{{\tan \left( {\lambda_{2} \left( {L - a_{{\text{d}}} } \right)} \right)}}{{\lambda_{2} }}} \right)$$
(5.11)
$$F = \pi D_{{\text{b}}} \tau_{{\text{r}}} \left( {\frac{{\tan \left( {\lambda_{2} \left( {L - a_{{\text{d}}} } \right)} \right)}}{{\lambda_{2} }} + a_{{\text{d}}} } \right)$$
(5.12)

Last, in Phase V, the pull-out force and displacement can be computed as follows:

$$F = \pi D_{{\text{b}}} \tau_{{\text{r}}} \left( {L + \delta_{{\text{r}}} + \frac{{\lambda^{2} \tau_{{\text{r}}} L^{2} }}{2} - u_{{\text{b}}} } \right)$$
(5.13)

Equations from (5.4) to (5.13) contributed to the force–displacement relationship of rock bolts. Moreover, a good agreement was obtained between the model and physical test data [24].

5.3 Modelling Process and Results

To investigate the effect of the confining medium diameter on the load-carrying capacity of rock bolts, analytical tests were performed. Specifically, it was assumed that pull-out tests were conducted on deformed rock bolts with the diameter of 25 mm and elastic modulus of 200 GPa. It was admitted that the deformed rock bolts had ribs on the tendon surface. Moreover, the mechanical parameters of ribs, namely rib height and rib spacing, may be dependent on the bolt type [27, 28]. However, for this paper, at the current stage, there was no specific evaluation on the rib height and rib spacing. It was assumed that after the tendon was loaded, the deformed rock bolt induced shearing stress at the bonding face between the tendon and grout. As for the influence of the rib height and rib spacing on the shearing stress at the bonding face between the tendon and grout, further work will be continued.

The bolt was installed in the bolting hole with a diameter of 40 mm and grouted length of 2 m. For the grout, it was assumed that it had an unconfined compressive strength of 50 MPa. For the confining medium, it was assumed that it had an elastic modulus of 20 GPa and unconfined compressive strength of 30 MPa. For the bonding face between the tendon and grout, the bonding-slip** data shown in Table 5.1 was used.

Table 5.1 Bonding-slip** data for the bonding face between the tendon and grout
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Table 5.1 summarised the mechanical properties of the bonding face between the tendon and grout: shearing strength, slip** where shearing strength reached, residual shearing strength and slip** where residual shearing strength reached. They can be computed with the following equation [29]:

$$\tau = \frac{P}{{\pi D_{{\text{b}}} L}}$$
(5.14)

Specifically, the previous research revealed that if the grouted length of the rock bolt was short enough, the shearing stress at the bonding face can be treated uniform [14]. Therefore, after conducting pull-out tests on rock bolts with short grouted length, the maximum pull-out force and residual pull-out force can be substituted into Eq. (5.14) to compute the shearing strength and residual shearing strength. Then, the slip** where shearing strength reached equalled the pull-out displacement where the maximum pull-out force occurred. The slip** where residual shearing strength reached equalled the pull-out displacement where the residual pull-out force occurred.

In this paper, for the diameter of the confining medium, four different values were used: 100 mm, 150 mm, 200 mm and 250 mm. The configuration of this analytical pull-out test is shown in Fig. 5.2. This analytical pull-out process was generally consistent with the traditional pull-out experiment of rock bolts [30], where the tendon was pulled with a loading rate of 1 mm/min and the load cell together with the linear variable differential transformer were used to measure the pull-out force and displacement, respectively.

Fig. 5.2
A 3-D diagram of a rock bolt loading system depicts 3 different cylindrical structures with the following labels. Confining medium, grout, and rock bolt are labeled from outer to inner levels. Pulling force is from bottom to top.

Configuration of the rock bolt loading system

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Then, the analytical pull-out tests were performed, and the results are shown in Fig. 5.3. Apparently, the confining medium diameter had an effect in determining the performance of rock bolts. Specifically, there was a positive relationship between the load-carrying capacity of rock bolts and the confining medium diameter. Moreover, the ascending rate of the peak force had a tendency to decline. For example, when the confining medium diameter ranged from 200 to 250 mm, the peak force of rock bolts was quite close.

Fig. 5.3
A diagram and a line graph. A, the geometry of the bolting system depicts 3 concentric circles with labels rock bolt, grout, and confining medium from inner to outer circles. B, the graph of pull-out load versus pull-out displacement plots 4 lines that rise, then fall slightly and stabilize.

Rock bolt performance with different confining medium diameters: a bolting system geometry; b force–displacement relationship of rock bolts

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To further investigate the influence of the confining medium diameter on the performance of rock bolts, a series of pull-out tests was conducted. In this computation, the confining medium diameter varied in the domain [50 mm, 1000 mm]. Then, the variation trend of the bolt peak force, when the confining medium diameter was varying, was computed, as shown in Fig. 5.4. The peak force ascended significantly when the confining medium diameter varied from 50 to 300 mm. However, when the confining medium diameter was larger than 300 mm, the ascending rate of the peak force became gentle. Moreover, when the diameter was beyond 400 mm, the peak force became almost constant.

Fig. 5.4
A line graph of peak load versus confining medium diameter. All values are estimated. The line starts at (50, 190), rises to (200, 240), and ends at (1000, 245).

Peak force variation trend when the confining medium diameter was ascending

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To confirm the accuracy of the analytical modelling results, the experimental tests performed by Rajaie [18] were used as a comparison, as shown in Fig. 5.5. It can be seen that the peak force variation trend in analytical modelling generally agreed well with the experimental test trend. In experimental tests, the peak force rose rapidly with the confining medium diameter when it was smaller than 200 mm. After it was beyond 250 mm, the peak force became almost unchangeable. This variation trend was successfully reflected in analytical modelling.

Fig. 5.5
A line graph of peak load versus confining medium diameter. All values are estimated. The line starts at (100, 12), rises to (200, 47), and ends at (300, 55).

Peak force variation trend of the plain cable bolts [18]

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In the above analysis, the elastic modulus of the confining medium was set as 20 GPa. However, in experimental tests, the material with different modulus can be used to cast the confining medium. Therefore, further pull-out tests were performed with different confining medium moduli. Specifically, three different moduli were used, 1 GPa, 5 GPa and 20 GPa. The peak force variation trend results are shown in Fig. 5.6. Apparently, the peak force variation trend was consistent. This indicated that although the confining medium modulus was different, there was a critical influence diameter. The confining medium had little effect on the peak force once its diameter was over than the critical influence diameter. It was also found that when the confining medium modulus was different, the critical influence diameter was also different. For example, when the confining medium modulus was 20 GPa, the peak force became almost constant when the confining medium diameter was larger than 400 mm. Nevertheless, this value was changed when the confining medium with a modulus of 1 GPa was used. For it, at a diameter of 400 mm, the peak force was still ascending rapidly.

Fig. 5.6
A line graph of peak load versus confining medium diameter. All values are estimated. 3 lines of E subscript m = 1, 5, and 20 gigapascals follow an increasing trend and stabilize later on.

Peak force variation trend with the confining medium ascending when the confining medium modulus was under different levels

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To study the influence of the confining medium modulus on the critical influence diameter, further pull-out tests were performed. Specifically, the confining medium modulus was ranged from 1 to 50 GPa. With a certain confining medium modulus, the peak force variation trend was first computed. Then, the critical influence diameter for that confining medium modulus was recorded. The critical influence diameter was determined with the following method.

First, a relative difference percentage was defined, as shown in Eq. (5.15).

$$\Delta = \frac{{F_{\max {2}} - F_{\max {1}} }}{{F_{\max {1}} }}$$
(5.15)

Where Δ: the relative difference percentage. Fmax1: peak force when the confining medium diameter was a. Fmax2: peak force when the confining medium diameter was (a + Δa).

In this paper, the confining medium diameter ascended with a small interval of 10 mm. Then, when the relative difference percentage was smaller than 0.01%, it was assumed that there was no apparent ascending of the peak force. Therefore, the corresponding confining medium diameter was regarded as the critical influence diameter.

Following this method, the critical influence diameter for different confining medium moduli was acquired, as shown in Fig. 5.7. It can be seen that the confining medium modulus had a significant effect in determining the critical influence diameter. With the confining medium modulus ascending, the critical influence diameter declined nonlinearly. This indicated that for the same rock bolt, if the confining medium properties were different, the corresponding critical influence diameter may be different. Furthermore, the larger the confining medium modulus, the smaller the critical influence diameter.

Fig. 5.7
A line graph of critical influence diameter versus confining medium modulus. The line follows a concave-up decreasing trend from (1, 1700) to (50, 450) through (10, 700). Values are estimated.

Variation of the critical influence diameter variation when the confining medium modulus was varying

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5.4 Conclusions

Laboratory pull-out tests were commonly used to study the performance of rock bolts. In pull-out tests, natural or artificial cylindrical confining medium was usually used to confine rock bolts. This paper conducted an analytical investigation to study the influence of the confining medium diameter on the performance of rock bolts. In the analytical modelling, the load-carrying capacity of rock bolts with different confining medium diameters was compared. The results showed that the confining medium diameter had an apparent effect in determining the peak force of rock bolts. With the confining medium diameter ascending, the peak force ascended. However, the ascending rate of the peak force gradually declined. The peak force variation trend was compared with experimental results. And there was a good agreement between modelling trend and experimental trend.

The rock bolt peak force with different confining medium moduli was studied. It was found that although the confining medium modulus was different, the peak force variation trend was consistent. For each medium modulus, there was a certain critical influence diameter. Beyond that critical influence diameter, further ascending the confining medium diameter had marginal effect in improving the peak force of rock bolts.

Moreover, the influence of the confining medium modulus on the critical influence diameter was analysed. The results showed that the higher the confining medium modulus, the smaller the critical influence diameter.