2.1 Introduction

In rock mechanics, failure of rock masses is commonly encountered [1]. This is because fractures distribute non-uniformly in rock masses [2, 3]. Moreover, experiment work proves that rock masses which are full of fractures have quite low strength [4].

This is more prominent when the rock masses are subjected to manual excavation [5]. For example, in civil engineering and mining engineering, excavation activities are performed to create tunnels, chambers or roadways [6]. These tunnels and roadways will be later used in serving transportation and ventilation [7].

Attention is paid that this manual excavation disturbs the initial stability of rock masses [8]. Moreover, due to manual excavation, stress concentration occurs in rock masses [9, 10]. Consequently, fractures are likely to develop in rock masses [11,12,13]. This further weakens the rock mass strength.

To guarantee the safety of rock mass excavation, rock reinforcement bolts are commonly used [14,15,16]. Experimental work proves that rock reinforcement bolts can combine the jointed rock mass. Moreover, once rock masses converge, shear deformation will occur in the grout column. Consequently, stress can be transferred in the rock reinforcement system [17].

During the loading process, the shear stress propagation plays a significant role [18]. Therefore, proper understanding the shear stress propagation mechanism in the rock reinforcement system is quite significant.

This paper aims at revealing the shear stress propagation mechanism in the rock reinforcement system. To realise this purpose, a literature review was conducted. This study is beneficial to propose new reinforcement approaches to prevent rock mass failure.

2.2 Previous Study on the Shear Stress Propagation

2.2.1 Investigation Approaches

To study the shear stress propagation mechanism in the rock reinforcement system, previous researchers adopted various approaches. Generally, investigation approaches can be classified into three different types: experimental tests [19], analytical simulation [20, 21] and numerical simulation [12].

Experimental tests should be the most credible one. Specifically, strain gauges are adhered on the bolt/grout interface [22]. Then, this instrumented bolt is installed in a rock block. Grout is used to bond the bolt with the rock block [23, 24]. After full curing, the bolt is pulled out. During testing, tensile force distribution along the bolt can be measured. Then, Eq. (2.1) is adopted to simply calculate the shear stress at the bolt/grout interface [25].

$$\tau = \frac{{D_{{\text{b}}} E_{{\text{b}}} \left( {\varepsilon_{1} - \varepsilon_{2} } \right)}}{4\Delta L}$$
(2.1)

where τ: shear stress at the bolt/grout interface; Db: bolt diameter; Eb: elastic modulus of the bolt; ε: tensile strain of the bolt; ΔL: spacing between two adjacent strain gauges.

This approach has been used by a number of researchers [26]. However, a shortcoming is that the attached strain gauges are likely to be stripped from the bolt/grout interface. To solve this issue, Chekired et al. [27] developed the tension measuring device. This device can be mounted on the bolt to measure the strain distribution. Additionally, Martin et al. [28] proposed replacing central wire in the cable bolt with a modified wire. Specifically, along this modified wire, strain gauges were attached.

Besides, analytical simulation is an efficient approach to study the shear stress propagation mechanism [29]. Specifically, although the bolt may have a long length, infinitesimal method can be used to analyse the shear stress at the bolt/grout interface, as shown in Eq. (2.2) [30]:

$$\tau = \frac{{D_{{\text{b}}} }}{4}\frac{{{\text{d}}\sigma_{{\text{b}}} \left( x \right)}}{{{\text{d}}x}}$$
(2.2)

where dσb(x): the increment of the tensile stress in the bolt.

Then, it is assumed that the bolt/grout interface has the same mechanical property [31]. And a bond-slip equation can be used to depict the relationship between the shear stress at the bolt/grout interface and the slip [32]. Through incorporating the bond-slip equation into Eq. (2.2), analytical equations can be developed.

This analytical approach was initially proposed by Farmer [33]. However, at that period, the debonding behaviour was not considered. Later, this analysis method was adopted by others [34]. Aydan et al. [35] made a significant improvement on analytical simulation. Their innovation is that the classic tri-linear equation was proposed to describe the slip behaviour of the bolt/grout interface. Based on the tri-linear equation, the bolt/grout interface encountered the elastic, softening and debonding behaviour. Therefore, the bonding and debonding behaviour of the bolt/grout interface can be simulated.

Moreover, the tri-linear equation may be simplified into the bi-linear equation to evaluate the shear stress propagation of the bolt/grout interface [36]. The main difference between the bi-linear equation and the tri-linear equation is that there is no linear softening behaviour in the bi-linear equation.

In recent years, numerical simulation becomes more popular in rock reinforcement analysis. It is because numerical simulation is powerful in establishing and calculating complicated structures [37]. As for the rock reinforcement analysis, more research was focused on using the structure element, such as the cable or pile. There is a significant difference between the cable and pile. Specifically, the cable only considers the longitudinal performance of rock reinforcement bolts. In contrast, the pile can analyse both the longitudinal performance and the lateral performance of rock reinforcement bolts.

The advantage is that numerical elements have already been created by the commercial company. Therefore, users can conveniently adopt the structure element to simulate different rock reinforcement cases. In contrast, the shortcoming is that the original constitutive equation may not truly reflect the proper behaviour of the rock reinforcement system. For example, in the structure element of cable, the bolt/grout interface is assumed to deform following an elastic perfectly plastic equation, as shown in Eq. (2.3) [38].

$$\left\{ {\begin{array}{*{20}l} {\tau = k_{{\text{g}}} \delta \left( {\delta \le \delta_{{\text{p}}} } \right)} \hfill \\ {\tau = \tau_{{\text{p}}} \left( {\delta > \delta_{{\text{p}}} } \right)} \hfill \\ \end{array} } \right.$$
(2.3)

where kg: shear stiffness of the grout column; δ: slip of the bolt/grout interface; δp: slip of the bolt/grout interface when the peak strength reaches; τp: peak strength of the bolt/grout interface.

It neglects the post-failure behaviour of the bolt/grout interface. Therefore, it cannot truly simulate the loading performance of bolts without modification.

Nevertheless, the commercial software usually reserves the secondary development interface. For example, for the structure element of cable, the Itasca Company prepares a number of FISH functions [39]. For the structure element of pile, the Itasca Company creates the TABLE function [40]. Therefore, users can use these FISH functions or TABLE function to modify the original rock reinforcement elements. For example, Fig. 2.1 shows the comparison between the original pile and the revised pile. Apparently, with the original pile, at the end of the shear stress propagation process, the shear stress at the full bolt/grout interface equalled the peak strength. This overestimated the loading capacity of bolts. In contrast, with the modified pile, there was always a non-uniform shear stress distribution at the bolt/grout interface. This was more consistent with the experimental test results. Consequently, it saw a wide application of the commercial numerical tools in rock reinforcement analysis [41].

Fig. 2.1
2 illustrations of the original and revised piles depict loaded and free ends on the top and bottom, respectively. There is a gradient scale to indicate the propagation of shear stress.

Comparison between the structure element of pile in analysing the shear stress propagation process: a original pile; b revised pile

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2.2.2 Shear Stress Propagation Mechanism

Based on previous research, it is accepted that the shear stress at the bolt/grout interface may have a uniform distribution if the anchor length was short enough [42]. Benmokrane et al. [43] indicated that when the anchor length is less than four times of the bolt diameter, the shear stress can be treated equal. Based on this concept, Eq. (2.4) was used to calculate the shear stress at the bolt/grout interface [44]:

$$\tau = \frac{F}{{\pi D_{{\text{b}}} L}}$$
(2.4)

where F: pull-out force; L: anchor length.

Attention should be noted that Eq. (2.4) is valid when the anchor length is constant during the pull-out process. Nevertheless, it is more common to encounter the scenario where the anchor length decreases gradually. Then, Eq. (2.5) can be used to calculate the shear stress [44].

$$\tau = \frac{F}{{\pi D_{{\text{b}}} \left( {L - u_{{\text{b}}} } \right)}}$$
(2.5)

where ub: pull-out displacement.

More importantly, rock reinforcement bolts usually have a long length [45]. In this case, after the bolt is subjected to tensile loading, non-uniform shear stress distribution occurs. For the laboratory monotonous loading condition, the bolt usually has two ends. One is embedded in the rock block, and it is called as the internal end. By contrast, the other one is left outside and it is called as the external end. Since there is a non-uniform shear stress distribution, shear stress propagation between two ends of the bolts generates.

Some research indicated that the maximum shear stress under each loading level was likely to occur around the same position [46]. Moreover, that position was close to the borehole collar.

In contrast, it is more common to see that during the initial load process, shear stress at the borehole collar increased gradually. With the loading increasing, the shear stress at the borehole collar increased to the peak strength. Then, it started drop**. More interestingly, the maximum shear stress moved towards the internal end direction. This shear stress propagation ended when the shear stress at the internal end of the bolt reached the peak strength.

As a validation of this shear stress propagation concept, Rong et al. [47] conducted laboratory pull-out tests on bolts. Strain gauges were attached on the bolt/grout interface to record the tensile force distribution. Later, Ma et al. [44] analysed these experimental data. Equation (2.1) was used to calculate the shear stress at the bolt/grout interface. The analysis results showed that at the initial loading grade, the shear stress at the borehole collar increased gradually. However, after a certain loading level, the shear stress at the borehole collar reached the peak strength. With the loading level further increasing, the maximum shear stress propagated gradually towards the external end. This analysis result was consistent with the above shear stress propagation concept. Therefore, the experimental work and the corresponding data analysis proved the reliability of the shear stress propagation concept.

Additionally, the analytical simulation and numerical simulation can better reflect the shear stress propagation process. Ren et al. [48] used the classic tri-linear equation to depict the slip behaviour of the bolt/grout interface. They analysed the shear stress propagation process at the bolt/grout interface. The results showed that the maximum shear stress at the bolt/grout interface propagated from the external end to the internal end. Moreover, although each point at the bolt/grout interface experienced the elastic, softening and debonding behaviour, the full bolt/grout interface underwent five different grades. They were the elasticity, elasticity-weakening, elasticity-weakening-debonding, weakening-debonding and debonding grades. Later, Blanco Martín et al. [49] indicated that when the tri-linear equation was used to depict the slip behaviour of the bolt/grout interface, the full bolt/grout interface may undergo the pure softening grade.

It should be mentioned that when different bond-slip equations are used, the full bolt/grout interface may undergo different grades. For example, Chen et al. [50] indicated that when a bi-linear equation was used to depict the slip behaviour of the bolt/grout interface, the full bolt/grout interface only experienced three grades: the elastic grade, elastic-debonding grade and debonding grade.

Although different bond-slip equations can be used, the shear stress propagation mechanism was consistent. Specifically, the maximum shear stress at the bolt/grout interface consistently propagated from the external end to the internal end, as shown in Fig. 2.2.

Fig. 2.2
6 line graphs. A, C, and E plot shear stress versus shear slip. The line in each graph rises and then drops. B, D, and F plot shear stress versus embedment length. 5, 3, and 4 lines respectively in each follow an increasing to decreasing, increasing, decreasing, or horizontal trend.

Shear stress propagation process: a tri-linear equation; b corresponding shear stress propagation process; c bi-linear equation; d corresponding shear stress propagation process; e nonlinear equation; f corresponding shear stress propagation process

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This finding was also confirmed with numerical simulation. Nemcik et al. [38] modified the original structure element in Fast Lagrangian Analysis of Continua (FLAC) and simulated the shear stress propagation process at the bolt/grout interface. A nonlinear bond-slip equation was used. The results showed that each point at the bolt/grout interface obeyed the same nonlinear bond-slip equation. With the loading level increasing, the maximum shear stress at the bolt/grout interface propagated towards the internal end. This finding was consistent with the others [51].

2.3 Discussion

The shear stress at the bolt/grout interface plays a significant role in determining the loading capacity of bolts [52]. Under the static loading condition, the loading capacity of bolts equals the sum of the shear force at the bolt/grout interface [53]. Since the shear stress at the bolt/grout interface can be calculated directly with the shear force at the bolt/grout interface, as shown in Eq. (2.6), there is a close relationship between the shear stress at the bolt/grout interface and the loading capacity of bolts [54].

$$\tau = \frac{{F_{{\text{s}}} }}{{A_{{\text{s}}} }}$$
(2.6)

where Fs: shear force at the bolt/grout interface; As: contact area between the bolt and the grout column.

Therefore, it is valuable to understand the shear stress propagation mechanism of bolts. To realise this purpose, previous researchers used various approaches. It is believed that the experimental approach is more credible. This is because compared with the experimental approach, the analytical simulation and numerical simulation usually relied on a number of assumptions [55]. Whether those assumptions are reasonable is doubted. In analytical simulation, it is usually assumed that the bolt, grout column and surrounding rock masses deform elastically. In fact, in experimental tests, failure of the rock masses may occur because of the radial dilation at the bolt/grout interface. Therefore, under this scenario, the analytical simulation which assumes that only slip failure occurs at the bolt/grout interface may not be trustable.

Similarly, in numerical simulation, rock masses are simulated with different elastoplastic equations. Among kinds of equations, the Mohr–Coulomb equation is more commonly used. This equation is relatively simpler, and its input parameters can be acquired directly from experimental tests. However, it cannot properly simulate the post-failure behaviour of rock masses. Specifically, after the peak, the strain softening behaviour and residual behaviour of rock masses cannot be properly simulated. In this case, the interaction between the numerical rock mass and bolts cannot be the same as the reality. This is also the reason why the numerical simulation work should be calibrated and compared with experimental results.

Nevertheless, this is not to deny the significance of analytical simulation and numerical simulation. In fact, these two approaches are more powerful for researchers to understand the shear stress propagation mechanism of bolts. Therefore, it is suggested that combing the experimental tests, analytical simulation and numerical simulation is a better choice in studying the shear stress propagation mechanism of bolts.

2.4 Conclusions

This paper conducted a literature review on the shear stress propagation mechanism of rock reinforcement bolts. The previous investigation approaches were summarised. It is concluded that previous researchers usually used the experimental tests, analytical simulation and numerical simulation. Among those approaches, the experimental approach is more widely used. And the experimental test results are more likely to be accepted by others. By contrast, the analytical approach had more degree of freedom. With this approach, researchers can use different bond-slip equation to depict the slip behaviour of the bolt/grout interface. As for the numerical simulation, it is convenient for users since the original constitutive equation has already been created by the developer. Moreover, developers usually reserve the secondary development interface for users to modify the original constitutive equations.

As for the shear stress propagation mechanism, it is more commonly agreed that during the pull-out process of bolts, the shear stress at the borehole collar firstly increased. With the loading level increasing, the shear stress at the borehole collar gradually reaches the peak strength. Then, with the further increasing of the loading level, the maximum shear stress starts propagating towards the internal end. This phenomenon was consistently observed in experimental tests, analytical simulation and numerical simulation.