Analytical Modelling to Study the Load Transfer Performance of Fully Grouted Rock Bolts

Abstract

A rock bolt is a rock tendon which can be either a rod or a strand, installed in a borehole to reinforce the surround rock mass and soil.

4.1 Introduction

A rock bolt is a rock tendon which can be either a rod or a strand, installed in a borehole to reinforce the surround rock mass and soil [1]. Rock bolts have been used in civil and mining for a long time [2]. Numerous filed applications and computer simulation have proved that rock bolts play a significant role in improving the internal strength of rock mass and enhancing the safety of underground excavations [3,4,5,6].

The rock bolt reinforcement system is basically composed of four components: the surrounding confining medium, e.g. rocks or soils, the rock reinforcement element, the internal fixture and the external fixture [7,8,9]. It is found that all these four components have an influence on the performance of a reinforcement system.

To better understand the performance of the reinforcement system, Thompson et al. [10] classified the rock reinforcement system as three types: Continuously Mechanically Coupled (CMC), Continuously Frictionally Coupled and Discrete Mechanically or Frictionally Coupled. This study deals with fully grouted rock bolts, and based on this classification, fully grouted rock bolts belong to the CMC system.

For the CMC system, cementitious grout or polyester resin can be used to bond rock bolts with surrounding confining medium [11,12,13]. Via the bonding material, the load from exterior unstable rock mass can be transferred to interior stable rock mass. It should be mentioned that in the in situ applications, rock bolts can encounter axial, shear, combined axial and shear loading conditions [14]. However, this paper focused on the axial performance of rock bolts.

Failure of the rock bolting system has different modes. Jeremic and Delaire [15] summarised that failure of the rock bolting system can be classified as bond failure of the bolt/grout interface, bond failure of the grout/rock interface, failure of the grout annulus and failure in the surrounding rock mass. Then, Hutchinson and Diederichs [16] added that rupture of the rock tendon also occurred in the in situ scenario. Nevertheless, among these failure modes, bond failure of the bolt/grout interface and the grout/rock interface are more common to encounter [17].

Jalalifar [18] stated that when fully grouted rock bolts were loaded axially, the shear stress along the bolt/grout interface was larger than the shear stress along the grout/rock interface because the bolt/grout interface had a smaller contact area. When the grout and the rock had similar strength, failure was likely to occur along the bolt/grout interface. Previous experimental pull-out tests also showed that bond failure of the bolt/grout interface occurred in the tests [19, 20]. Therefore, this paper deals with the bond failure of the bolt/grout interface.

Gambarova [21] stated that the bonding performance of the bolt/grout interface mainly relies on chemical adhesion, mechanical interlock and friction. Chemical adhesion plays a negligible role on the bonding performance and can be easily removed after a small relative movement between the bolt and grout [22].

To understand the load transfer performance of rock bolts, extensive research has been conducted via experimental pull-out tests [23,24,25]. Compared with that, much less work has been conducted with analytical modelling. Hawkes and Evans [26] conducted the pioneering work in analytically modelling the load transfer of rock bolts. Specifically, an analytical model was proposed to evaluate the load distribution along a fully grouted rock bolt. However, this assumption was only valid in the coupling stage. In the modelling process, a linear relationship between the axial stress in the bolt and the shear stress along the bolt/grout interface was assumed. However, this assumption was credible when the bolt was fully coupled with the surrounding medium. With the axial load in the bolt increasing, debonding occurred along the bolt/grout interface and this assumption may not be valid again. Farmer [27] deduced an analytical model to study the load distribution along a rock bolt. In this analytical model, it was assumed that the shear stress along the bolt/grout interface was linear with the grout shear modulus. Therefore, this model was only applicable when the bolt/grout interface underwent elastic behaviour. Once debonding occurred along the bolt/grout interface, the credibility of this analytical model decreased. Aydan et al. [28] deduced an analytical solution to study the load transfer of rock bolts. However, no experimental work was conducted to validate the credibility of the model. Rajaie [29] studied the load transfer of cable bolts and an energy approach was adopted to analyse the loading process of cable bolts. There was a good correlation between laboratory results and analytical results. However, this model mainly focused on the peak load calculation and no equation was deduced to calculate the load variation with displacement in the pull-out process. Therefore, this model cannot be used to study the load–displacement relationship of cable bolts.

Yazici and Kaiser [30] proposed a bond strength model to study the load transfer of plain cable bolts. Later, this model was modified to study the influence of stress change on load transfer of cable bolts [31]. This model treated the bolt/grout interface as a zigzag geometry, similar to a rock joint. For the rock joint, strength of the rock is usually weak along two sides. However, for the cable bolting system, strength of the steel is much higher along the bolt side. Following that, Hyett et al. [32] developed a constitutive model to study the load–displacement performance of cable bolts. Two different boundary conditions, constant normal stress and constant normal stiffness, were considered. This model assumed that the shear stress along the bolt/grout interface was uniform. However, experimental work proved that when the bolt had a long embedment length, the shear stress along the bolt/grout interface was not uniform [22]. Therefore, this model was applicable to calculate the load–displacement relationship for short-embedded bolts. However, it may not be suitable for long-embedded bolts. A load distribution model was proposed by Hyett et al. [33]. This model can study the load distribution in the embedment section and the anchor section. Moosavi [34] developed an analytical model for modified cable bolts. However, when modified cable bolts were used to reinforce rock masses, the modified geometry can be installed at different positions in the embedment section. And, the influence of the modified geometry position on the load transfer performance of cable bolts was not considered. Li and Stillborg [35] developed an analytical model to study the load distribution of rock bolts. The bolt/grout interface was assumed to be composed of four sections: completely debonded, partially debonded, softening and elastic.

Cai et al. [36] used a modified shear-lag model to study the load transfer of rock bolts. However, the softening behaviour of the bolt/grout interface was neglected. Xiao and Chen [37] assumed that the bolt/grout interface underwent elastic, softening and debonding stages and developed a model for rock bolts. Ren et al. [38] deduced the load transfer process of rock bolts with an analytical model. Experimental work was used to validate the credibility of this analytical model. However, when deducing the analytical model, it was assumed that the shear slippage of the bolt/grout interface was equal to the axial displacement of the rock bolt. Then, the displacement of the surrounding confining medium was neglected. Consequently, this model cannot be used to study the influence of the surrounding confining medium on the performance of rock bolts. Blanco Martín et al. [39] proposed an analytical solution to evaluate the load–displacement performance of rock bolts. Jalalifar [40] evaluated the performance of rock bolts installed in an elastoplastic rock mass. In this model, it was assumed that the shear force in the grout annulus had a linear relationship with the shear slippage. Therefore, this model was only valid when the bolt/grout interface underwent elastic deformation. After the bolt/grout interface started plastic deformation, this model was not valid again. Cao et al. [41] studied the influence of rib geometry on the bonding capacity of rock bolts. It was predicted that rock bolts with small rib face angles were more effective in hard rock conditions.

Ma et al. [42] used a nonlinear equation to depict the bond-slip behaviour of the bolt/grout interface and studied the load transfer of rock bolts. Later, the model was modified to study the influence of free-end slip on the performance of rock bolts [43]. However, in this model, an empirical equation was used to depict the bond-slip relationship of the bolt/grout interface. Therefore, the properties of the rock bolting system may modify the coefficients in the bond-slip model. Consequently, this model cannot be used to predict the influence of rock mass confinement on the performance of rock bolts. Chen et al. [13] used an analytical model to study the axial performance of cable bolts. However, the model was only appropriate for laboratory pull-out scenario. Li et al. [44] developed an analytical model to study the load transfer of modified cable bolts. Cable bolts with short embedment length were used to validate the model. However, in situ tests proved that the installed cable bolts can be very long for the field scenario [45]. And whether this model can be used to study the performance of cable bolts with long embedment length was not discussed.

The literature review shows that for the rock bolting system, when the shear behaviour of the bolt/grout interface was depicted with a bond-slip equation, different analytical models can be used, such as the exponential model [42] or the tri-linear model [46]. Among them, the tri-linear model considers the elastic, softening and debonding behaviour of the bolt/grout interface and thus is more commonly used. However, when the tri-linear model was incorporated into the rock bolting system to deduce the load–displacement relationship of rock bolts, the involvement of the surrounding confining medium was usually not considered.

It should be mentioned that Vaculik et al. [47] conducted an analytical study to evaluate the load-slip relationship of the FRP-to-substrate joints with the bond-slip model. A framework was proposed to depict the performance of FRP-to-substrate joints. However, for the rock bolting system, cylindrical rock bolts are always used to reinforce the excavated rock masses, such as tunnels, roadways and stopes [48, 49]. Therefore, this paper applies the method proposed by Vaculik et al. [47] into the fully grouted rock bolting system and further deduced the specific analytical solution for fully grouted rock bolts. Additionally, they evaluated the performance of FRP-to-substrate joints with the embedment length ranging from short to long. However, for fully grouted rock bolts, the embedment length is usually extremely long [45, 50]. For example, Cao et al. [51] summarised that the embedment length for a rock bolt can be around 3 m. Furthermore, when the cable bolt is used, the embedment length may range from 3 to 15 m. Considering this, this paper especially focused on the long embedment length scenario for rock bolts.

First the relationship between the shear stress of the bolt/grout interface and the shear slippage was simulated with a tri-linear model. Then, the pull-out process of rock bolts was analysed with five stages: elastic, elastic-softening, elastic-softening-debonding, softening-debonding and debonding stages. After that, the analytical model was validated with experimental pull-out tests. Last, a parametric study was conducted to evaluate the influence of elastic modulus of the confining medium, the shear strength of the bolt/grout interface and the residual shear strength of the bolt/grout interface on the load transfer performance of rock bolts.

4.2 Analytical Modelling

4.2.1 Bond-Slip Model

A tri-linear model was used to depict the bond-slip behaviour of the bolt/grout interface. The tri-linear model is composed of three parts: ascending section, descending section and horizontal section [13, 38], as shown in Fig. 4.1. Specifically, with the shear slippage of the bolt/grout interface increasing from 0 to δ_p, the shear stress increases linearly from zero to the shear strength of the bolt/grout interface. Then, with δ increasing to δ_r, the shear stress decreases linearly to the residual shear strength of the bolt/grout interface. After δ_r, the shear stress keeps constant and equals τ_r.

Fig. 4.1

A tri-linear model [13, 38, 47]

Therefore, the tri-linear model can be expressed as Eq. (4.1a–4.1c) [13, 38]:

$$\tau = \frac{{\tau_{{\text{p}}} }}{{\delta_{{\text{p}}} }}\delta \,\, \left( {0 \le \delta \le \delta_{{\text{p}}} } \right)$$

(4.1a)

$$\tau = \frac{{\tau_{{\text{p}}} \delta_{{\text{r}}} - \tau_{{\text{r}}} \delta_{{\text{p}}} }}{{\delta_{{\text{r}}} - \delta_{{\text{p}}} }} - \frac{{\tau_{{\text{p}}} - \tau_{{\text{r}}} }}{{\delta_{{\text{r}}} - \delta_{{\text{p}}} }}\delta^{{}} \,\,\left( {\delta_{{\text{p}}} < \delta \le \delta_{{\text{r}}} } \right)$$

(4.1b)

$$\tau = \tau_{{\text{r}}}^{{}}\,\, \left( {\delta > \delta_{{\text{r}}} } \right)$$

(4.1c)

where τ_p: shear strength of the bolt/grout interface; τ_r: residual shear strength of the bolt/grout interface; and δ_r: shear slippage when the residual shear strength of the bolt/grout interface reaches.

4.2.2 Governing Equation

In previous research, the shear slippage of the bolt/grout interface was usually simplified as the axial displacement of the rock bolt [38, 42]. However, previous experimental work proved that the shear slippage of the bolt/grout interface was different from the axial displacement of the rock bolt because of the displacement of the confining medium [52]. Furthermore, Hyett et al. [33] found that the displacement of the confining medium had an apparent influence on the load transfer performance of rock bolts. Therefore, neglecting the displacement of the confining medium in the analytical model could not properly reflect the load transfer performance of rock bolts. Considering this issue, in this paper, the displacement of the confining medium was included.

Since this study aims at studying the bond failure of the bolt/grout interface, the grout/medium interface was assumed to be intact in the pull-out process of rock bolts. Then, the shear slippage of the bolt/grout interface was defined as the relative movement between the rock bolt and the confining medium [53, 54]:

$$\delta \left( x \right) = u_{{\text{b}}} \left( x \right) - u_{{\text{m}}} \left( x \right)$$

(4.2)

where δ(x): shear slippage of the bolt/grout interface at the position of x; u_b(x): rock bolt displacement at the position of x; and u_m(x): confining medium displacement at the position of x.

Differentiating Eq. (4.2) twice with respect to x leads to:

$$\frac{{{\text{d}}^{2} \delta \left( x \right)}}{{{\text{d}}x^{2} }} = \frac{{{\text{d}}^{2} u_{{\text{b}}} \left( x \right)}}{{{\text{d}}x^{2} }} - \frac{{{\text{d}}^{2} u_{{\text{m}}} \left( x \right)}}{{{\text{d}}x^{2} }}$$

(4.3)

The axial stress in the rock bolt can be expressed with the displacement of the rock bolt. As for the confining medium, this is also applicable. Then, the following equations can be acquired [40]:

$$\sigma_{{\text{b}}} \left( x \right) = E_{{\text{b}}} \frac{{{\text{d}}u_{{\text{b}}} \left( x \right)}}{{{\text{d}}x}}$$

(4.4)

$$\sigma_{{\text{m}}} \left( x \right) = E_{{\text{m}}} \frac{{{\text{d}}u_{{\text{m}}} \left( x \right)}}{{{\text{d}}x}}$$

(4.5)

where σ_b(x): axial stress in the rock bolt at the position of x; E_m: elastic modulus of the confining medium; and σ_m(x): axial stress applied on the confining medium.

Substituting Eqs. (4.4) and (4.5) into (4.3) leads to:

$$\frac{{{\text{d}}^{2} \delta \left( x \right)}}{{{\text{d}}x^{2} }} = \frac{1}{{E_{{\text{b}}} }}\frac{{{\text{d}}\sigma_{{\text{b}}} \left( x \right)}}{{{\text{d}}x}} - \frac{1}{{E_{{\text{m}}} }}\frac{{{\text{d}}\sigma_{{\text{m}}} \left( x \right)}}{{{\text{d}}x}}$$

(4.6)

According to the mechanical equilibrium, the relationship between the axial stress in the bolt and the axial stress applied on the confining medium can be expressed as [55]:

$$\sigma_{{\text{m}}} = - \sigma_{{\text{b}}} \frac{{\pi r_{{\text{b}}}^{2} }}{{A_{{\text{m}}} }}$$

(4.7)

where A_m: cross-section area of the confining medium; and r_b: rock bolt radius.

Substituting Eq. (4.7) into Eq. (4.6) leads to:

$$\frac{{{\text{d}}^{2} \delta \left( x \right)}}{{{\text{d}}x^{2} }} = \left( {\frac{1}{{E_{{\text{b}}} }} + \frac{{\pi r_{{\text{b}}}^{2} }}{{E_{{\text{m}}} A_{{\text{m}}} }}} \right)\frac{{{\text{d}}\sigma_{{\text{b}}} \left( x \right)}}{{{\text{d}}x}}$$

(4.8)

The relationship between the axial stress in the rock bolt and the shear stress along the bolt/grout interface can be seen in Fig. 4.2.

Fig. 4.2

The cross-section view of a rock bolting system

Then, the following equation can be acquired [42]:

$$\tau \left( x \right) = \frac{{r_{{\text{b}}} }}{2}\frac{{{\text{d}}\sigma_{{\text{b}}} \left( x \right)}}{{{\text{d}}x}}$$

(4.9)

where τ(x): shear stress of the bolt/grout interface at the position of x.

Substituting Eq. (4.9) into Eq. (4.8) leads to:

$$\frac{{{\text{d}}^{2} \delta \left( x \right)}}{{{\text{d}}x^{2} }} - \frac{2}{{r_{{\text{b}}} }}\left( {\frac{1}{{E_{{\text{b}}} }} + \frac{{\pi r_{{\text{b}}}^{2} }}{{E_{{\text{m}}} A_{{\text{m}}} }}} \right)\tau \left( x \right) = 0$$

(4.10)

It is defined that:

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

(4.11)

Then, Eq. (4.10) can be written as:

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

(4.12)

Equation (4.12) is the governing equation of this analytical model, and it can be solved with the pre-defined shear stress equations.

4.2.3 Pull-Out Stages of Rock Bolts

According to previous research, the pull-out stages of the bolt/grout interface can be classified as elastic stage, elastic-softening stage, elastic-softening-debonding stage, softening-debonding stage and debonding stage [38, 53, 56]. Therefore, in the following contents, the pull-out process of a rock bolt was analysed with those five stages.

4.2.3.1 Elastic Stage

When the pull-out load is small, the bolt/grout interface undergoes elastic deformation. And the full bolt/grout interface is elastic, as shown in Fig. 4.3.

Fig. 4.3

The full bolt/grout interface is elastic in the elastic stage

Then, Eq. (4.1a) can be substituted into Eq. (4.12), leading to:

$$\frac{{{\text{d}}^{2} \delta \left( x \right)}}{{{\text{d}}x^{2} }} - \frac{{\tau_{{\text{p}}} }}{{\delta_{{\text{p}}} }}\lambda^{2} \delta \left( x \right) = 0$$

(4.13)

It is defined that:

$$\lambda_{1}^{2} = \frac{{\tau_{{\text{p}}} }}{{\delta_{{\text{p}}} }}\lambda^{2}$$

(4.14)

Then, Eq. (4.13) can be written as:

$$\frac{{{\text{d}}^{2} \delta \left( x \right)}}{{{\text{d}}x^{2} }} - \lambda_{1}^{2} \delta \left( x \right) = 0$$

(4.15)

Equation (4.15) can be solved with boundary conditions. First, the axial stress in the bolt should be deduced. According to Eq. (4.2), the following equation can be acquired:

$$\frac{{{\text{d}}\delta \left( x \right)}}{{{\text{d}}x}} = \frac{{{\text{d}}u_{{\text{b}}} \left( x \right)}}{{{\text{d}}x}} - \frac{{{\text{d}}u_{{\text{m}}} \left( x \right)}}{{{\text{d}}x}}$$

(4.16)

It can be further written as:

$$\frac{{{\text{d}}\delta \left( x \right)}}{{{\text{d}}x}} = \frac{{\sigma_{{\text{b}}} \left( x \right)}}{{E_{{\text{b}}} }} - \frac{{\sigma_{{\text{m}}} \left( x \right)}}{{E_{{\text{m}}} }}$$

(4.17)

Substituting Eq. (4.7) into Eq. (4.17) leads to:

$$\frac{{{\text{d}}\delta \left( x \right)}}{{{\text{d}}x}} = \left( {\frac{1}{{E_{{\text{b}}} }} + \frac{{\pi r_{{\text{b}}}^{2} }}{{E_{{\text{m}}} A_{{\text{m}}} }}} \right)\sigma_{{\text{b}}} \left( x \right)$$

(4.18)

Substituting Eq. (4.11) into Eq. (4.18), the axial stress in the bolt can be written as:

$$\sigma_{{\text{b}}} \left( x \right) = \frac{2}{{r_{{\text{b}}} \lambda^{2} }}\frac{{{\text{d}}\delta \left( x \right)}}{{{\text{d}}x}}$$

(4.19)

The boundary conditions are:

$$\sigma_{{\text{b}}} \left( {x = 0} \right) = 0$$

(4.20)

$$\sigma_{{\text{b}}} \left( {x = L} \right) = \frac{F}{{\pi r_{{\text{b}}}^{2} }}$$

(4.21)

Combining Eqs. (4.19), (4.20) and (4.21) with Eq. (4.15) leads to:

$$\delta \left( x \right) = \frac{{F\lambda_{1} \delta_{{\text{p}}} }}{{2\pi r_{{\text{b}}} \tau_{{\text{p}}} \sinh \left( {\lambda_{1} L} \right)}}\cosh \left( {\lambda_{1} x} \right)$$

(4.22)

$$\sigma_{{\text{b}}} \left( x \right) = \frac{F}{{\pi r_{{\text{b}}}^{2} \sinh \left( {\lambda_{1} L} \right)}}\sinh \left( {\lambda_{1} x} \right)$$

(4.23)

$$\tau \left( x \right) = \frac{{F\lambda_{1} }}{{2\pi r_{{\text{b}}} \sinh \left( {\lambda_{1} L} \right)}}\cosh \left( {\lambda_{1} x} \right)$$

(4.24)

It is defined that at the loaded end, the shear slippage of the bolt/grout interface is equal to the pull-out displacement of the rock bolt [57]. Substituting x = L into Eq. (4.22), the pull-out load of the rock bolt can be expressed as:

$$F = \frac{{2\pi r_{{\text{b}}} \tau_{{\text{p}}} \tanh \left( {\lambda_{1} L} \right)}}{{\lambda_{1} \delta_{{\text{p}}} }}u_{{\text{b}}}$$

(4.25)

At the end of this stage, the shear stress at the position of x = L increases to the shear strength of the bolt/grout interface. Therefore, substituting τ(x = L) = τ_p into Eq. (4.24) leads to:

$$F_{{{\text{sof}}}} = \frac{{2\pi r_{{\text{b}}} \tau_{{\text{p}}} \tanh \left( {\lambda_{1} L} \right)}}{{\lambda_{1} }}$$

(4.26)

where F_sof: pull-out load when the bolt/grout interface starts softening.

4.2.3.2 Elastic-Softening Stage

In the elastic-softening stage, the full bolt/grout interface is composed of elastic section and softening section, as shown in Fig. 4.4.

Fig. 4.4

The full bolt/grout interface is composed of elastic section and softening section in the elastic-softening stage

It is defined that a_s is the softening length. In the elastic-softening stage, the interface in the domain [0, L − a_s] is still elastic. A boundary condition is Eq. (4.20) and the other boundary condition is:

$$\tau \left( {x = L - a_{{\text{s}}} } \right) = \tau_{{\text{p}}}$$

(4.27)

Combining Eqs. (4.9), (4.19), (4.20) and (4.27) with Eq. (4.15) leads to:

$$\delta \left( x \right) = \frac{{\delta_{{\text{p}}} }}{{\cosh \left( {\lambda_{1} \left( {L - a_{{\text{s}}} } \right)} \right)}}\cosh \left( {\lambda_{1} x} \right)$$

(4.28)

$$\sigma_{{\text{b}}} \left( x \right) = \frac{{2\tau_{{\text{p}}} }}{{r_{{\text{b}}} \lambda_{1} \cosh \left( {\lambda_{1} \left( {L - a_{{\text{s}}} } \right)} \right)}}\sinh \left( {\lambda_{1} x} \right)$$

(4.29)

$$\tau \left( x \right) = \frac{{\tau_{{\text{p}}} }}{{\cosh \left( {\lambda_{1} \left( {L - a_{{\text{s}}} } \right)} \right)}}\cosh \left( {\lambda_{1} x} \right)$$

(4.30)

Substituting x = L − a_s into Eq. (4.29), the axial stress can be acquired:

$$\sigma_{{\text{b}}} \left( {x = L - a_{{\text{s}}} } \right) = \frac{{2\tau_{{\text{p}}} }}{{r_{{\text{b}}} \lambda_{1} }}\tanh \left( {\lambda_{1} \left( {L - a_{{\text{s}}} } \right)} \right)$$

(4.31)

In the domain [L − a_s, L], the bolt/grout interface softens. Substituting Eq. (4.1b) into Eq. (4.12) leads to:

$$\frac{{{\text{d}}^{2} \delta \left( x \right)}}{{{\text{d}}x^{2} }} + \lambda^{2} \frac{{\tau_{{\text{p}}} - \tau_{{\text{r}}} }}{{\delta_{{\text{r}}} - \delta_{{\text{p}}} }}\delta \left( x \right) - \lambda^{2} \frac{{\tau_{{\text{p}}} \delta_{{\text{r}}} - \tau_{{\text{r}}} \delta_{{\text{p}}} }}{{\delta_{{\text{r}}} - \delta_{{\text{p}}} }} = 0$$

(4.32)

It is defined that:

$$\lambda_{2}^{2} = \frac{{\tau_{{\text{p}}} - \tau_{{\text{r}}} }}{{\delta_{{\text{r}}} - \delta_{{\text{p}}} }}\lambda^{2}$$

(4.33)

Therefore, Eq. (4.32) can be further written as:

$$\frac{{{\text{d}}^{2} \delta \left( x \right)}}{{{\text{d}}x^{2} }} + \lambda_{2}^{2} \delta \left( x \right) - \lambda_{2}^{2} \frac{{\tau_{{\text{p}}} \delta_{{\text{r}}} - \tau_{{\text{r}}} \delta_{{\text{p}}} }}{{\tau_{{\text{p}}} - \tau_{{\text{r}}} }} = 0$$

(4.34)

The boundary conditions are Eqs. (4.27) and (4.31). Combining Eqs. (4.9), (4.19), (4.27) and (4.31) with Eq. (4.34) leads to:

$$\begin{aligned} \delta \left( x \right) & = \frac{{\lambda_{1} \delta_{{\text{p}}} }}{{\lambda_{2} }}\tanh \left( {\lambda_{1} \left( {L - a_{{\text{s}}} } \right)} \right)\sin \left( {\lambda_{2} \left( {x + a_{{\text{s}}} - L} \right)} \right) \\ & \quad - \frac{{\lambda_{1}^{2} \delta_{{\text{p}}} }}{{\lambda_{2}^{2} }}\cos \left( {\lambda_{2} \left( {L - a_{{\text{s}}} - x} \right)} \right) + \frac{{\tau_{{\text{p}}} \delta_{{\text{r}}} - \tau_{{\text{r}}} \delta_{{\text{p}}} }}{{\tau_{{\text{p}}} - \tau_{{\text{r}}} }} \\ \end{aligned}$$

(4.35)

$$\begin{aligned} \sigma_{{\text{b}}} \left( x \right) & = \frac{{2\tau_{{\text{p}}} }}{{r_{{\text{b}}} }}\left( {\frac{1}{{\lambda_{1} }}\tanh \left( {\lambda_{1} \left( {L - a_{{\text{s}}} } \right)} \right)\cos \left( {\lambda_{2} \left( {L - a_{{\text{s}}} - x} \right)} \right)} \right. \\ & \quad \left. { + \frac{1}{{\lambda_{2} }}\sin \left( {\lambda_{2} \left( {x + a_{{\text{s}}} - L} \right)} \right)} \right) \\ \end{aligned}$$

(4.36)

$$\tau \left( x \right) = \tau _{{\text{p}}} \left( {\frac{{\lambda _{2} }}{{\lambda _{1} }}\tanh \left( {\lambda _{1} \left( {L - a_{{\text{s}}} } \right)} \right)\sin \left( {\lambda _{2} \left( {L - a_{{\text{s}}} - x} \right)} \right) + \cos \left( {\lambda _{2} \left( {L - a_{{\text{s}}} - x} \right)} \right)} \right)$$

(4.37)

Substituting x = L and $\sigma_{{\text{b}}} \left( {x = L} \right) = \frac{F}{{\pi r_{{\text{b}}}^{2} }}$ into Eqs. (4.35) and (4.36), respectively, the pull-out displacement and pull-out load of the rock bolt can be acquired:

$$F = 2\pi r_{{\text{b}}} \tau_{{\text{p}}} \left( {\frac{1}{{\lambda_{1} }}\tanh \left( {\lambda_{1} \left( {L - a_{{\text{s}}} } \right)} \right)\cos \left( {\lambda_{2} a_{{\text{s}}} } \right) + \frac{1}{{\lambda_{2} }}\sin \left( {\lambda_{2} a_{{\text{s}}} } \right)} \right)$$

(4.38)

$$u_{{\text{b}}} = \frac{{\lambda_{1} \delta_{{\text{p}}} }}{{\lambda_{2} }}\tanh \left( {\lambda_{1} \left( {L - a_{{\text{s}}} } \right)} \right)\sin \left( {\lambda_{2} a_{{\text{s}}} } \right) - \frac{{\lambda_{1}^{2} \delta_{{\text{p}}} }}{{\lambda_{2}^{2} }}\cos \left( {\lambda_{2} a_{{\text{s}}} } \right) + \frac{{\tau_{{\text{p}}} \delta_{{\text{r}}} - \tau_{{\text{r}}} \delta_{{\text{p}}} }}{{\tau_{{\text{p}}} - \tau_{{\text{r}}} }}$$

(4.39)

At the end of this stage, the shear stress at the position of x = L reduces to τ_r. Therefore, substituting τ(x = L) = τ_r into Eq. (4.37) leads to:

$$\cos \left( {\lambda_{2} a_{{\text{s}}} } \right) - \frac{{\lambda_{2} }}{{\lambda_{1} }}\tanh \left( {\lambda_{1} \left( {L - a_{{\text{s}}} } \right)} \right)\sin \left( {\lambda_{2} a_{{\text{s}}} } \right) = \frac{{\tau_{{\text{r}}} }}{{\tau_{{\text{p}}} }}$$

(4.40)

For an infinite embedment length, tanh(λ₁(L − a_s)) is equal to 1. Then, the softening length when the elastic-softening stage ends can be acquired:

$$a_{{{\text{s}}1}} = \frac{1}{{\lambda_{2} }}\left( {\arcsin \frac{{\lambda_{1} }}{{\sqrt {\lambda_{1}^{2} + \lambda_{2}^{2} } }} - \arcsin \frac{{\tau_{{\text{r}}} \lambda_{1} }}{{\tau_{{\text{p}}} \sqrt {\lambda_{1}^{2} + \lambda_{2}^{2} } }}} \right)$$

(4.41)

where a_s1: the softening length when the elastic-softening stage ends.

4.2.3.3 Elastic-Softening-Debonding Stage

After τ(x = L) reduces to τ_r, the bolt/grout interface enters the elastic-softening-debonding stage. In this stage, the full bolt/grout interface is composed of elastic section, softening section and debonding section, as shown in Fig. 4.5.

Fig. 4.5

The full bolt/grout interface is composed of elastic section, softening section and debonding section in the elastic-softening-debonding stage

Assuming that the debonding length is a_d, the bolt/grout interface in the domain [0, L − a_d − a_s] is still elastic. Therefore, Eqs. (4.28), (4.29) and (4.30) are still valid if L is replaced with L − a_d:

$$\delta \left( x \right) = \frac{{\delta_{1} }}{{\cosh \left( {\lambda_{1} \left( {L - a_{{\text{d}}} - a_{{\text{s}}} } \right)} \right)}}\cosh \left( {\lambda_{1} x} \right)$$

(4.42)

$$\sigma_{{\text{b}}} \left( x \right) = \frac{{2\tau_{{\text{p}}} }}{{r_{{\text{b}}} \lambda_{1} \cosh \left( {\lambda_{1} \left( {L - a_{{\text{d}}} - a_{{\text{s}}} } \right)} \right)}}\sinh \left( {\lambda_{1} x} \right)$$

(4.43)

$$\tau \left( x \right) = \frac{{\tau_{{\text{p}}} }}{{\cosh \left( {\lambda_{1} \left( {L - a_{{\text{d}}} - a_{{\text{s}}} } \right)} \right)}}\cosh \left( {\lambda_{1} x} \right)$$

(4.44)

As for the domain [L − a_d − a_s, L − a_d], the bolt/grout interface softens. Therefore, Eqs. (4.35), (4.36) and (4.37) are still valid if L is replaced with (L − a_d):

$$\begin{aligned} \delta \left( x \right) & = \frac{{\lambda_{1} \delta_{{\text{p}}} }}{{\lambda_{2} }}\tanh \left( {\lambda_{1} \left( {L - a_{{\text{d}}} - a_{{\text{s}}} } \right)} \right)\sin \left( {\lambda_{2} \left( {x + a_{{\text{d}}} + a_{{\text{s}}} - L} \right)} \right) \\ & \quad - \frac{{\lambda_{1}^{2} \delta_{{\text{p}}} }}{{\lambda_{2}^{2} }}\cos \left( {\lambda_{2} \left( {L - a_{{\text{d}}} - a_{{\text{s}}} - x} \right)} \right) + \frac{{\tau_{{\text{p}}} \delta_{{\text{r}}} - \tau_{{\text{r}}} \delta_{{\text{p}}} }}{{\tau_{{\text{p}}} - \tau_{{\text{r}}} }} \\ \end{aligned}$$

(4.45)

$$\begin{aligned} \sigma_{{\text{b}}} \left( x \right) & = \frac{{2\tau_{{\text{p}}} }}{{r_{{\text{b}}} }}\left( {\frac{1}{{\lambda_{1} }}\tanh \left( {\lambda_{1} \left( {L - a_{{\text{d}}} - a_{{\text{s}}} } \right)} \right)\cos \left( {\lambda_{2} \left( {L - a_{{\text{d}}} - a_{{\text{s}}} - x} \right)} \right)} \right. \\ & \quad \left. { + \frac{1}{{\lambda_{2} }}\sin \left( {\lambda_{2} \left( {x + a_{{\text{d}}} + a_{{\text{s}}} - L} \right)} \right)} \right) \\ \end{aligned}$$

(4.46)

$$\begin{aligned} \tau \left( x \right) & = \tau _{{\text{p}}} \left( {\frac{{\lambda _{2} }}{{\lambda _{1} }}\tanh \left( {\lambda _{1} \left( {L - a_{{\text{d}}} - a_{{\text{s}}} } \right)} \right)\sin \left( {\lambda _{2} \left( {L - a_{{\text{d}}} - a_{{\text{s}}} - x} \right)} \right)} \right. \\ & \quad \left. { + \cos \left( {\lambda _{2} \left( {L - a_{{\text{d}}} - a_{{\text{s}}} - x} \right)} \right)} \right) \\ \end{aligned}$$

(4.47)

Substituting τ(L − a_d) = τ_r into Eq. (4.47) leads to:

$$\cos \left( {\lambda_{2} a_{{\text{s}}} } \right) - \frac{{\lambda_{2} }}{{\lambda_{1} }}\tanh \left( {\lambda_{1} \left( {L - a_{{\text{d}}} - a_{{\text{s}}} } \right)} \right)\sin \left( {\lambda_{2} a_{{\text{s}}} } \right) - \frac{{\tau_{{\text{r}}} }}{{\tau_{{\text{p}}} }} = 0$$

(4.48)

Equation (4.48) depicts the relationship between a_d and a_s. It shows that in this stage, the softening length keeps varying with the debonding length increasing. Combining Eqs. (4.45) and (4.48) together, the shear slippage of the bolt/grout interface at the position of x = L − a_d can be acquired:

$$\delta \left( {x = L - a_{{\text{d}}} } \right) = \delta_{{\text{r}}}$$

(4.49)

Substituting x = L − a_d into Eq. (4.46), the axial stress can be acquired:

$$\begin{aligned} \sigma_{{\text{b}}} \left( {x = L - a_{{\text{d}}} } \right) & = \frac{{2\tau_{{\text{p}}} }}{{r_{{\text{b}}} }}\left( {\frac{1}{{\lambda_{1} }}\tanh \left( {\lambda_{1} \left( {L - a_{{\text{d}}} - a_{{\text{s}}} } \right)} \right)\cos \left( {\lambda_{2} a_{{\text{s}}} } \right)} \right. \\ & \quad \left. { + \frac{1}{{\lambda_{2} }}\sin \left( {\lambda_{2} a_{\text{s}} } \right)} \right) \\ \end{aligned}$$

(4.50)

For the domain [L − a_d, L], the bolt/grout interface debonds and the shear stress equals τ_r:

$$\tau \left( x \right) = \tau_{{\text{r}}}$$

(4.51)

Also, Eq. (4.1c) can be substituted to Eq. (4.12), leading to:

$$\frac{{{\text{d}}^{2} \delta \left( x \right)}}{{{\text{d}}x^{2} }} = \lambda^{2} \tau_{{\text{r}}}$$

(4.52)

Substituting Eqs. (4.19), (4.49) and (4.50) into Eq. (4.52) leads to:

$$\begin{aligned} \delta \left( x \right) & = \delta _{{\text{r}}} + \frac{1}{2}\lambda ^{2} \tau _{{\text{r}}} \left( {x^{2} - \left( {L - a_{{\text{d}}} } \right)^{2} } \right) + \left( {\frac{{\tau _{\text{p}} \lambda ^{2} }}{{\lambda _{1} }}\tanh \left( {\lambda _{1} \left( {L - a_{{\text{d}}} - a_{{\text{s}}} } \right)} \right)\cos \left( {\lambda _{2} a_{{\text{s}}} } \right)} \right. \\ & \quad \left. { + \frac{{\tau _{{\text{p}}} \lambda ^{2} }}{{\lambda _{2} }}\sin \left( {\lambda _{2} a_{{\text{s}}} } \right) - \lambda ^{2} \tau _{{\text{r}}} \left( {L - a_{{\text{d}}} } \right)} \right)\left( {x + a_{{\text{d}}} - L} \right) \\ \end{aligned}$$

(4.53)

$$\begin{aligned} \sigma_{{\text{b}}} \left( x \right) & = \frac{{2\tau_{{\text{p}}} }}{{r_{{\text{b}}} }}\left( {\frac{1}{{\lambda_{1} }}\tanh \left( {\lambda_{1} \left( {L - a_{{\text{d}}} - a_{{\text{s}}} } \right)} \right)\cos \left( {\lambda_{2} a_{{\text{s}}} } \right) + \frac{1}{{\lambda_{2} }}\sin \left( {\lambda_{2} a_{{\text{s}}} } \right)} \right) \\ & \quad + \frac{{2\tau_{{\text{r}}} }}{{r_{{\text{b}}} }}\left( {x + a_{{\text{d}}} - L} \right) \\ \end{aligned}$$

(4.54)

Substituting x = L and $\sigma_{{\text{b}}} \left( {x = L} \right) = \frac{F}{{\pi r_{{\text{b}}}^{2} }}$ into Eqs. (4.53) and (4.54), the pull-out displacement and load of the rock bolt can be acquired:

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

(4.55)

$$F = 2\pi r_{{\text{b}}} \left( {\frac{{\tau_{{\text{p}}} }}{{\lambda_{1} }}\tanh \left( {\lambda_{1} \left( {L - a_{{\text{d}}} - a_{{\text{s}}} } \right)} \right)\cos \left( {\lambda_{2} a_{{\text{s}}} } \right) + \frac{{\tau_{{\text{p}}} }}{{\lambda_{2} }}\sin \left( {\lambda_{2} a_{{\text{s}}} } \right) + \tau_{{\text{r}}} a_{{\text{d}}} } \right)$$

(4.56)

The maximum pull-out load occurs in this stage and can be acquired by solving Eqs. (4.48) and (4.56) together.

At the end of this stage, the elastic length disappears. Therefore, L − a_d = a_s. Substituting it into Eq. (4.48) leads to:

$$a_{{{\text{s}}2}} = \frac{1}{{\lambda_{2} }}\arccos \left( {\frac{{\tau_{{\text{r}}} }}{{\tau_{{\text{p}}} }}} \right)$$

(4.57)

where a_s2: softening length when the elastic-softening-debonding stage ends.

4.2.3.4 Softening-Debonding Stage

After the elastic length disappears, the bolt/grout interface enters softening-debonding stage. In the domain [0, L − a_d], the bolt/grout interface softens. Substituting Eqs. (4.9), (4.19) and (4.20) and τ(x = L − a_d) = τ_r into Eq. (4.34) leads to:

$$\delta \left( x \right) = \frac{{\tau_{{\text{r}}} \left( {\delta_{{\text{p}}} - \delta_{{\text{r}}} } \right)}}{{\left( {\tau_{{\text{p}}} - \tau_{{\text{r}}} } \right)\cos \left( {\lambda_{2} \left( {L - a_{{\text{d}}} } \right)} \right)}}\cos \left( {\lambda_{2} x} \right) + \frac{{\tau_{{\text{p}}} \delta_{{\text{r}}} - \tau_{{\text{r}}} \delta_{{\text{p}}} }}{{\tau_{{\text{p}}} - \tau_{{\text{r}}} }}$$

(4.58)

$$\sigma_{{\text{b}}} \left( x \right) = \frac{{2\tau_{{\text{r}}} }}{{r_{{\text{b}}} \lambda_{2} \cos \left( {\lambda_{2} \left( {L - a_{{\text{d}}} } \right)} \right)}}\sin \left( {\lambda_{2} x} \right)$$

(4.59)

$$\tau \left( x \right) = \frac{{\tau_{{\text{r}}} }}{{\cos \left( {\lambda_{2} \left( {L - a_{{\text{d}}} } \right)} \right)}}\cos \left( {\lambda_{2} x} \right)$$

(4.60)

Substituting x = L − a_d into Eqs. (4.58) and (4.59) leads to:

$$\delta \left( {x = L - a_{{\text{d}}} } \right) = \delta_{{\text{r}}}$$

(4.61)

$$\sigma_{{\text{b}}} \left( {x = L - a_{{\text{d}}} } \right) = \frac{{2\tau_{{\text{r}}} }}{{r_{{\text{b}}} \lambda_{2} }}\tan \left( {\lambda_{2} \left( {L - a_{{\text{d}}} } \right)} \right)$$

(4.62)

For the domain [L − a_d, L], the bolt/grout interface debonds. Substituting Eqs. (4.19), (4.61) and (4.62) into Eq. (4.52) leads to:

$$\begin{aligned} \delta \left( x \right) & = \delta_{{\text{r}}} + \frac{{\lambda^{2} \tau_{{\text{r}}} }}{2}\left( {x^{2} - \left( {L - a_{{\text{d}}} } \right)^{2} } \right) \\ & \quad + \left( {\frac{{\lambda^{2} \tau_{{\text{r}}} }}{{\lambda_{2} }}\tan \left( {\lambda_{2} \left( {L - a_{{\text{d}}} } \right)} \right) - \lambda^{2} \tau_{{\text{r}}} \left( {L - a_{{\text{d}}} } \right)} \right)\left( {x + a_{{\text{d}}} - L} \right) \\ \end{aligned}$$

(4.63)

$$\sigma_{{\text{b}}} \left( x \right) = \frac{{2\tau_{{\text{r}}} }}{{r_{{\text{b}}} }}\left( {\frac{{\tan \left( {\lambda_{2} \left( {L - a_{{\text{d}}} } \right)} \right)}}{{\lambda_{2} }} + x + a_{{\text{d}}} - L} \right)$$

(4.64)

Substituting x = L and $\sigma_{{\text{b}}} \left( {x = L} \right) = \frac{F}{{\pi r_{{\text{b}}}^{2} }}$ into Eqs. (4.63) and (4.64), the pull-out displacement and load of the rock bolt can be acquired:

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

(4.65)

$$F = 2\pi r_{{\text{b}}} \tau_{{\text{r}}} \left( {\frac{{\tan \left( {\lambda_{2} \left( {L - a_{{\text{d}}} } \right)} \right)}}{{\lambda_{2} }} + a_{{\text{d}}} } \right)$$

(4.66)

At the end of this stage, the debonding length equals the full embedment length. Therefore, substituting a_d = L into Eqs. (4.65) and (4.66), the pull-out displacement and load of the rock bolt at the end of the softening-debonding stage can be acquired:

$$u_{{{\text{b}}0}} = \delta_{{\text{r}}} + \frac{{\lambda^{2} \tau_{{\text{r}}} L^{2} }}{2}$$

(4.67)

$$F_{0} = 2\pi r_{{\text{b}}} \tau_{{\text{r}}} L$$

(4.68)

where u_b0: pull-out displacement of the rock bolt at the end of the softening-debonding stage; and F₀: pull-out load of the rock bolt at the end of the softening-debonding stage.

4.2.3.5 Debonding Stage

After the softening length disappears, the bolt/grout interface enters debonding stage. Since the full bolt/grout interface undergoes debonding, the relationship between the pull-out load and displacement can be expressed as:

$$F_{0} - F = 2\pi r_{{\text{b}}} \tau_{{\text{r}}} \left( {u_{{\text{b}}} - u_{{{\text{b}}0}} } \right)$$

(4.69)

Substituting Eqs. (4.67) and (4.68) into Eq. (4.69) leads to:

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

(4.70)

4.3 Calibration of the Input Parameters

In this analytical model, the mechanical properties of the rock bolt and the confining medium, such as r_b, E_b and E_m, can be acquired based on the rock bolt pull-out scenario. Specifically, r_b and E_b can be acquired from the rock bolt specification provided by the bolt manufacturer. E_m can be acquired by conducting uniaxial compressive strength tests on samples cored from the confining medium.

As for the mechanical properties of the bond-slip model, they can be calibrated from the experimental load–displacement results. Figure 4.6 shows a typical load–displacement curve of rock bolts.

Fig. 4.6

Selecting critical points to calibrate the input bond-slip parameters

In the initial stage of the load–displacement curve, the pull-out load of the rock bolt generally had a linear relationship with the pull-out displacement. Therefore, Point 1 represents the point where the linear relationship of the pull-out load and displacement ends in the load–displacement curve. Then, δ_p equals the pull-out displacement at Point 1. Also, F_sof equals the pull-out load at Point 1. Substituting F_sof and δ_p into Eq. (4.26), τ_p can be acquired.

When the rock bolt reaches the maximum pull-out load, that point can be selected as Point 2, as shown in Fig. 4.6. Then, δ_r can be selected in the range between the pull-out displacement at Point 1 and the pull-out displacement at Point 2. Then, τ_r can be calibrated until the maximum analytical pull-out load fits well with the maximum experimental pull-out load.

4.4 Validation of the Analytical Model

In this section, two experimental pull-out tests were used to validate the credibility of this analytical model.

4.4.1 Validation with a Pull-Out Test

Bai et al. [58] conducted experimental in situ pull-out tests on rock bolts. Specifically, rock bolts with a radius of 14 mm were tested and the embedment length was 3 m. The surrounding rock mass had an elastic modulus of 6 GPa.

The mechanical properties of the rock bolt and the confining medium used in this analytical model are tabulated in Table 4.1.

Table 4.1 Mechanical properties of the bolt and the confining medium

As for the input parameters for the bond-slip model, they can be calibrated from experimental pull-out results, following the methods indicated in Sect. 4.3. Then, the parameters tabulated in Table 4.2 were used in this paper to analytically model the load transfer performance of rock bolts.

Table 4.2 Input parameters for the bond-slip model

A comparison between the experimental result and the analytical result is shown in Fig. 4.7. There is a close match between the experimental result and the analytical result, confirming the credibility of this analytical model.

Fig. 4.7

Comparison between the experimental result acquired from Bai et al. [58] and the analytical result

4.4.2 Validation with the Other Pull-Out Test

Stillborg [59] conducted a number of pull-out tests in the mine site to study the load transfer performance of cable bolts. Cable bolts with a long embedment length of 10 m were tested. The cable bolts had a radius of 19 mm and elastic modulus of 83 GPa. The surrounding rock mass had an elastic modulus of 57 GPa. The mechanical properties of the rock bolt and the confining medium used in this analytical model are tabulated in Table 4.3.

Table 4.3 Mechanical properties of the cable bolt and the confining medium

As for the input parameters for the bond-slip model, they can be calibrated from the experimental pull-out results. In this paper, the parameters tabulated in Table 4.4 were used to calculate the load–displacement relationship of the cable bolt.

Table 4.4 Input parameters for the bond-slip model

The comparison between the experimental result and the analytical result is shown in Fig. 4.8. It shows that the analytical result agrees well with the experimental result, further confirming that this analytical model can be used to simulate the load transfer performance of rock bolts.

Fig. 4.8

Comparison between the experimental result acquired from Stillborg [59] and the analytical result

4.5 Parametric Study

A parametric study was conducted to evaluate the input parameters on the load transfer performance of rock bolts. Specifically, the influence of the elastic modulus of the confining medium, the shear strength of the bolt/grout interface and the shear slippage where shear strength of the bolt/grout interface reaches on the load transfer performance of rock bolts were studied.

4.5.1 Elastic Modulus of the Confining Medium

To study the influence of the elastic modulus of the confining medium on the load transfer performance of rock bolts, an analytical rock bolt pull-out scenario was created. A rock bolt with a radius of 10 mm was tested. The elastic modulus of the rock bolt and the embedment length were 196 GPa and 1.5 m. Then, the mechanical properties of the bolt and the confining medium are tabulated in Table 4.5. As for the input parameters for the bond-slip model, they are tabulated in Table 4.6.

Table 4.5 Mechanical properties of the bolt and the confining medium

Table 4.6 Input parameters for the bond-slip model

The elastic modulus of the confining medium was changed from 10 to 90 MPa and the load–displacement performance of rock bolts was calculated, as shown in Fig. 4.9. It shows that the elastic modulus of the confining medium had an apparent influence on the load transfer performance of rock bolts. The initial stiffness and the peak load of the rock bolting system increased with the elastic modulus of the confining medium.

Fig. 4.9

Influence of elastic modulus of the confining medium on the performance of rock bolts

This finding agrees well with previous research. Hyett et al. [60] conducted laboratory and in situ pull-out tests on plain cable bolts. In the laboratory, PVC, aluminium and steel pipes were used to confine cable bolts. The results showed that the peak load of cable bolts increased with the elastic modulus of the confining medium. As for the in situ case, cable bolts were pulled from shale, limestone and granite. The results showed that when the confining medium changed from shale to granite, the peak load of cable bolts increased apparently.

Figure 4.9 also shows that with the elastic modulus of the confining medium increasing, the increasing rate of the peak load was not constant. To further study the influence of the elastic modulus of the confining medium on the peak load, a series of analytical pull-out tests was conducted. The relationship between the rock bolt peak load and the elastic modulus of the confining medium was acquired, as shown in Fig. 4.10.

Fig. 4.10

Influence of the elastic modulus of the confining medium on the peak load of rock bolts

It shows that for this scenario, before the elastic modulus of the confining medium increased to 280 MPa, the peak load increases rapidly. After that, the peak load is still increasing but with a much lower slope. This indicates that when the elastic modulus of the confining medium was higher than a critical limit, it had marginal effect on the peak load of rock bolts.

4.5.2 Shear Strength of the Bolt/Grout Interface

To study the influence of the shear strength of the bolt/grout interface on the load transfer performance of rock bolts, three analytical pull-out tests were conducted. In this case, the shear strength of the bolt/grout interface was varied from 2 to 6 MPa. As for the other parameters, they were kept constant.

The mechanical properties for the rock bolt and the confining medium are tabulated in Table 4.7. As for the input parameters for the bond-slip model, they are tabulated in Table 4.8.

Table 4.7 Mechanical properties of the bolt and the confining medium

Table 4.8 Input parameters for the bond-slip model

A plot of the input bond-slip model is shown in Fig. 4.11. Based on the input bond-slip model, the calculated load transfer performance of rock bolts is shown in Fig. 4.12.

Fig. 4.11

A plot of the input bond-slip model when the shear strength of the bolt/grout interface was different

Fig. 4.12

Load transfer performance of rock bolts when the shear strength of the bolt/grout interface was different

The results show that the shear strength of the bolt/grout interface played a significant role in determining the load transfer performance of rock bolts. Specifically, when the shear strength of the bolt/grout interface was 2 MPa, the rock bolt had a maximum pull-out load of 146 kN when the pull-out displacement reached 3.1 mm.

Compared with that, when the shear strength of the bolt/grout interface was 6 MPa, the rock bolt had a maximum pull-out load of 282 kN, increasing by 93.2%. And the corresponding pull-out displacement reached 4.5 mm. Apparently, the larger the shear strength of the bolt/grout interface, the higher the maximum load transfer capacity of rock bolts.

Furthermore, the results show that the shear strength of the bolt/grout interface also had an effect in the initial stiffness of the rock bolting system. With the shear strength of the bolt/grout interface increasing, the initial stiffness of the rock bolting system also increased, as shown in Fig. 4.12.

4.5.3 Residual Shear Strength of the Bolt/Grout Interface

To study the influence of the residual shear strength of the bolt/grout interface on the load transfer performance of rock bolts, another rock bolt pull-out scenario was created. In this case, the residual shear strength of the bolt/grout interface was varied from 0.5 to 1.5 MPa. As for the other parameters, they were kept constant.

The mechanical properties of the rock bolt and the confining medium are tabulated in Table 4.9. As for the input parameters for the bond-slip model, they are tabulated in Table 4.10.

Table 4.9 Mechanical properties of the bolt and the confining medium

Table 4.10 Input parameters for the bond-slip model

A plot of the input bond-slip model is shown in Fig. 4.13. As for the load transfer performance of rock bolts, they are shown in Fig. 4.14.

Fig. 4.13

A plot of the input bond-slip model when the residual shear strength of the bolt/grout interface was different

Fig. 4.14

Load transfer performance of rock bolts when the residual shear strength of the bolt/grout interface was different

The results show that the residual shear strength of the bolt/grout interface had no effect on the initial stiffness of the rock bolting system. For example, in this case, when the residual shear strength of the bolt/grout interface increased from 0.5 to 1.5 MPa, the initial stiffness of the rock bolting system kept constant.

Nevertheless, the residual shear strength of the bolt/grout interface had an effect on the maximum load transfer capacity of rock bolts. For example, in this scenario, when the residual shear strength of the bolt/grout interface was 0.5 MPa, the maximum pull-out load of rock bolts was 226 kN and the corresponding pull-out displacement was 3.6 mm. Then, when the residual shear strength of the bolt/grout interface increased to 1.5 MPa, the maximum pull-out load of rock bolts increased to 256 kN, increasing by 13.3%. And the corresponding pull-out displacement was 4.9 mm. Apparently, increasing the residual shear strength of the bolt/grout interface was beneficial for improving the maximum load transfer capacity of rock bolts.

Additionally, compared with the shear strength of the bolt/grout interface, increasing the residual shear strength of the bolt/grout interface had less apparent effect in improving the maximum load transfer capacity of rock bolts. For example, in the above analysis, when the residual shear strength of the bolt/grout interface increased by 200%, the maximum pull-out load of rock bolts only increased by 13.3%. However, when the shear strength of the bolt/grout interface increased by 200%, the maximum pull-out load of rock bolts largely increased by 93.2%. Therefore, the shear strength of the bolt/grout interface had more significant effect in deciding the maximum load transfer capacity of rock bolts.

The contribution of this paper is that it concentrated on fully grouted rock bolts with long embedment length grouted in the surrounding confining medium. Specifically, based on the previous work [38, 47], this paper acquired the specific analytical solution for this scenario. Furthermore, based on the acquired specific analytical solution, it further studied the influence of the elastic modulus of the confining medium on the load transfer performance of rock bolts. It revealed that although the peak load of rock bolts increased with the elastic modulus of the confining medium, there was a critical limit. Once the elastic modulus of the confining medium was beyond that critical limit, it had marginal effect on the peak load of rock bolts.

4.6 Conclusions

A tri-linear model was adopted to depict the bond-slip behaviour of the bolt/grout interface. It was assumed that the shear stress of the bolt/grout interface increased linearly to the shear strength of the bolt/grout interface with the shear slippage. Then, the shear stress decreased linearly to the residual shear strength of the bolt/grout interface with the shear slippage. After that, the shear stress was kept constant.

The pull-out stages of rock bolts were analysed. During the pull-out process, the bolt/grout interface underwent elastic, elastic-softening, elastic-softening-debonding, softening-debonding and debonding stages.

Experimental pull-out tests were used to validate the credibility of this analytical model. The results show that there was a good correlation between experimental and analytical results.

A parametric study was conducted to evaluate the elastic modulus of the confining medium, the shear strength of the bolt/grout interface and the residual shear strength of the bolt/grout interface on the load transfer performance of rock bolts. The results show that increasing the elastic modulus of the confining medium was beneficial for improving the load transfer performance of rock bolts. However, once the elastic modulus of the confining medium was beyond a critical limit, it had a marginal effect on the peak load of rock bolts. Additionally, increasing the shear strength of the bolt/grout interface and the residual shear strength of the bolt/grout interface led to rising of the peak load of rock bolts. However, compared with the residual shear strength of the bolt/grout interface, increasing the shear strength of the bolt/grout interface had more apparent effect in improving the load transfer capacity of rock bolts.