1 Introduction

Rock bolting systems reign supreme among ground control methods for stabilising underground spaces and slopes (Hosseini et al. 2024a). Their ease of installation and competitive pricing make them a popular and in-demand choice in mining, civil, and geotechnical projects. The history of rock bolting systems stretches back to the late 1940s, when they were first employed as a secondary support in both underground mining projects and soft rock environments (Conway 1948; Gregor et al. 2023, 2024; Jodeiri Shokri et al. 2024b; Mark 2000). During the following decades, it was identified that the performance of the rock bolting systems heavily depended on in-situ axial and shear stresses (Haas et al. 1974; Jodeiri Shokri et al. 2024b), and subsequently several identified research works were done on investigating axial and shear load transfer mechanisms in rock bolting systems (Farmer 1975; Liu and Li 2020; Pinazzi et al. 2020, 2021; Tang and Peng 1985). As a result of these investigations, the vast capabilities of rock bolting systems were recognised, and they became accepted as primary support systems in active and passive applications (Brady and Brown 2006; Gay 1980; Peng and Tang 1984). Numerous design variations were conceived to meet explicit criteria considering the widespread implementation of rock bolting systems. The development of mechanical, resin, and grout anchorage allowed for greater variations in rock bolting selection to meet specific operational requirements (Du et al. 2024; Entezam et al. 2023; Hosseini et al. 2024a, 2024b; Jodeiri Shokri et al. 2023; Li et al. 2023; Motallebiyan et al. 2023; Nourizadeh et al. 2023a, 2023b, 2024; Rajapakse 2008; Ren et al. 2024; Wu et al. 2024; Zhang et al. 2024; Zhu et al. 2024). Beyond the axial and shear forces that rock bolts experience, the presence of localized sulphide minerals, particularly pyrite and chalcopyrite, in the rock mass can significantly accelerate corrosion of steel rebars (Aziz et al. 2013). Indeed, corrosion has the potential to cause strength reductions of up to 39% and can induce failures by three key modes: uniform corrosion attack, pitting corrosion, and stress corrosion cracking (Hassell et al. 2004) and as a result can make steel rock bolts/rebars a non-viable support choice in such a media. Due to the limitations of steel rebars in harsh environments, alternative materials like fiberglass offer distinct advantages. These include ease of cutting and superior corrosion resistance within the system. However, their behaviour under axial and shear loads requires further investigation for a complete understanding of their performance (Entezam et al. 2024).

Over the past few decades, various experimental testing procedures have been devised to enhance our understanding of the shearing behaviour mechanisms of rock bolts. These include the single shear guillotine test (Standard 2009), double shearing guillotine test, and concrete-embedded double shearing test (Chen and Xiao 2024; Gilbert et al. 2015). In fact, single shearing test represents only applying shear loads on the rock -bolt sliding relative to each other along the shear surface (Chen and Xiao 2024). Recognising that rock bolt systems can undergo multiple shear forces because of strata bedding, the double shearing testing method was introduced and developed to replicate real-field conditions. Aziz et al. (2003) developed the double shearing testing apparatus (MKI), which consists of a single bolt grouted into three pieces of concrete blocks representing rock-bolt-rock and the load will be applied in the middle block of concrete (Chen and Xiao 2024). Double shear testing methodology was subsequently modified and designed with two other types, namely MKII and MKIII (Aziz et al. 2016; Gregor et al. 2023, 2024).

Along with modifying experimental tests, many researchers have been focused on developing and implementing of analytical models and simulating the various properties of rock bolts for both pure element failure as well as strata simulation (Ghadimi et al. 2015b; Maekawa and Qureshi 1996; Pinazzi et al. 2021; Rajapakse 2016; Singh and Spearing 2021; Song et al. 2008; Spang and Egger 1990; Wu et al. 2019). While significant accomplishments have been made in simulating strata supporting elements, their suitability and reliability must be carefully considered for each implementation. Past simulations were limited to specific conditions, with models simulating the pure element with great accuracy yet unable to model the element within an environment. Unfortunately, these approaches were optimised for the structure of metallic rock bolts and are, therefore, unsuitable for use with fibreglass rock bolts. Unlike in metallic rock bolt production, manufacturers can significantly alter the properties of fibreglass rock bolts by changing the composition of the binding resin, strand density, and strand orientation. Other models, including Gilbert et al. (2015) determined the shear stress of cable bolts utilising the Mohr Coulomb criterion and, as a result, were, inherently limited to determining only the peak shear values of the cable bolts. In addition, the Gilbert et al. (2015) model could not predict the shear forces at each stage of the cable bolt failure. Simulations of this nature were calibrated to test metallic cable bolts only. As there is a wide variety of rock bolt products available, some researchers have developed models to address the growing demand for composite rock bolts. One of the early representations of the composite rock bolt behaviours was the shear lag theory. This method adopted assumptions that were not well understood and over-idealised (Cai et al. 2004), with one such assumption including the lack of slip at the interfaces. Additionally, unknown parameters were assumed using stress distribution matrices. Cai et al. (2004) modified the shear lag theory application to address many of its shortcomings. The outcome of Cai et al. (2004) study was the development of an analytical model that expressed the pull-out force of the rock bolt. Ultimately, the shear lag theory’s suitability was limited to modelling the pull-out capabilities of the rock bolt. A more recent study conducted by Wen-Qiang and Yi-Jia (2022) attempted to address the current limitations of the analytical modelling of rock bolts in shear. Wen-Qiang and Yi-Jia (2022) proposed that one of the main limitations of analytical modelling was that it assumed a linear force behaviour for the host material. Therefore, a new model was developed incorporating joint dilation, axial force, joint displacement and transverse shear. Despite the comprehensive representation of the rock bolts’ shear behaviour, the model does not account for the interaction of the shear joint properties with the shear performance of the rock bolt. Therefore, negating the impacts of joint friction angle, impact of pretension and the impacts of infilled materials. As a result, and to the best of the authors’ knowledge, the current analytical models fail to incorporate key system properties of fibreglass rock bolts.

In conjunction with experimental schemes and analytical methods, numerical modelling has been adopted to understand better the intrinsic stresses experienced by rock bolts (Jodeiri Shokri et al. 2024a; Liu et al. 2022; Nourizadeh et al. 2023a; Song et al. 2008; Sun et al., 2022). Models from as early as the 1970s were developed to calculate the shear resistance of bolted rock systems (Haas et al. 1974). For instance, Haas (1974) suggested a mathematical relationship for calculating the average shear test based on geometry of rock bolt. The early 1900s saw the incorporation of computational analysis with the use of finite element methods (FEM) model simulations (Spang and Egger 1990). This facilitated the exploration of simulating varying bolt installation parameters such as bolt angle. A model later developed by Pellet and Egger (1996) determined the impacts of rock bolts on sheared rock joints utilising the Tresca criterion.

Further development of these models has incorporated the Fourier series, producing highly accurate simulations (Rasekh et al. 2017). A more recent study conducted by Singh and Spearing (2021) utilised static mechanics, and kinematic relationships for the elastic and plastic conditions, respectively, while building on the works of Maekawa and Qureshi (1996) and Dight (1982) to define the yield and failure limit. It is noteworthy that the development of analytical models representing the shear behaviours of rock bolts and cable bolts over previous decades has enabled the development of a strong foundation for numerical analysis utilising tools. Numerical models have been developed using finite element models, FEM, and explicit finite difference software. Saadat and Taheri (2020) used PFC2D to investigate the fracturing response of rock-like and grout material, as well as the bolt-grout interface. Ghadimi et al. (2015a) utilised FEM through the ANSYS software to investigate the shear stress distribution along a metal rebar. Studies conducted by Rasekh et al. (2017) utilised FLAC2D to conduct explicit finite differences for the modelling of cable bolts subjected to double shearing. FLAC2D was selected as it combined analytical expressions with spatial modelling in order to explore the element-by-element influence of shear stress on a complete rock bolt system (Ma et al. 2014). The completed simulation successfully mapped the shear stresses along the cable bolt element as well as the confining rock body. In addition to simulating novel shearing scenarios, Mirzaghorbanali et al. (2017) revisited the established British Standard shearing testing scheme of cable bolts in order to develop a model of the shear behaviour using FLAC2D. The results showed good agreement between the shear profiles generated through experimental testing and numerical modelling, leading Mirzaghorbanali et al., (2017) to conclude that numerical simulations could successfully model existing testing methods. After conducting double shearing tests, Singh et al. (2020) could successfully calibrate the ANSYS model. After investigating the mechanical characteristics of rock bolting system under tensile-shear loading, Sun et al. (2022) proposed a new numerical model was coupled by fracture behaviour of rock bolts by applying FLAC3D and also presenting a new constitutive model of the yield and fracture of rock bolts.

Due to the lack of analytical and numerical modelling for predicting the impact of pretension values on the shearing performance of the FRP bolts, this research aims to achieve two main objectives: (a) develop an analytical model to predict the shear performance of fiberglass rock bolts under varying pretension levels, considering both clean and infilled joints and (b) conduct numerical simulations to analyse the shear behaviour of fiberglass bolts in clean and infilled joints, and perform a sensitivity analysis under different shearing conditions.

2 Materials and Methods

2.1 Experimental Design

To carry out the required experimental tests, we selected two types of industry available FRP bolts, specifically those with load capacities of 20 tonnes and 30 tonnes, each having a length and diameter of 1.2 m and 18 mm, respectively. These bolts, designed, manufactured, and provided by Applied Research of Australia (AROA) and widely used in the Australian industry, were chosen for in-depth investigations. The specifications of the FRP bolts were given in Table 1. Subsequently, double shearing tests were conducted on clean and infilled joints at different pretension values to comprehend the shearing behaviour mechanism. The UCS of concrete samples were 40 MPa. The design dimension was selected, within reason, to minimise the potential impact of system boundary distortion as this influence wouldn’t be present in field conditions. The axial cross-sectional area of the rock bolt and annulus was maintained between 2% and 2.5%, this ensured ease of sample construction, storage and testing. Following the assembly of fibreglass rock bolts with commercially available nuts and washers, they were subsequently torqued at 0%, 5%, 10%, and 15% of the maximum tensile load capacity of each rock bolt, as per the methodology outlined by (Gilbert et al. 2015) and to best replicate field installation practices. In total, 16 double-shearing samples were designed, prepared, and tested, including 8 for clean discontinuities and 8 for infilled joints with sandy clay. The double shearing testing methodology accurately replicate field conditions where from one side plate and nut represent the actual boundary condition of the mine, and the other side implies a fully encapsulated rock bolting system restricting deformation deep within strata. Additionally, the impact of scale and system boundaries on the rock bolts performance were not explored within this study and may warrant further investigation.

Table 1 Comparison of fibreglass rock bolts of different design strengths (DSI-UNDERGROUND 2021)
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2.1.1 Sample Preparation and Testing

After designing and preparing the necessary moulds and shearing box (more details can be found in (Gregor et al. 2023, 2024)), the 40 MPa concrete samples, including 200 mm by 200 mm by 400 mm blocks, were cast using cement, sand, and water ratio of 1:2.2:0.42 following the mortar recipe outlined by Gilbert et al. (2015). The concrete blocks were subsequently cured for 28 days (Fig. 1). For a double shear test, a modified shear box, MK 1.5, was developed, which had a larger overall confinement length of 780 mm compared to MK 1’s 600 mm (Gregor 2022). The length of concrete blocks in the new shear box is 800 mm. This modification incorporates a 20 mm length difference between the box and the sample, creating a crucial 10 mm clearance on either side of the shearing planes. This design change also addressed the frame wedging issues encountered in the British standards testing method by incorporating a 5 mm spacing on each side of the shearing plane (Gregor et al. 2024).

Fig. 1
figure 1

A view of a modified mould for preparing double shearing samples after casting the samples

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After curing the concrete blocks, the other components of the double shearing samples, including the FRP bolt, axial load cell, and washer plates, were assembled loosely and in the correct order for final tightening depicted in Fig. 2a). In the final step of preparing the required samples, the assembled samples were grouted through holes on the blocks. For infilled samples, a material consisting of sand and clay was used (Mirzaghorbanali et al. 2014) with a ratio of 1 part sand to 1 part clay and a water content of 0.48. The infilled samples with an initial moisture content of 17.6% were subjected to the consolidation tests with normal stresses of 0 kPa, 6 kPa, 12.5 kPa, 25 kPa, 50 kPa, 100 kPa and 200 kPa. Each setting of the normal stress was maintained for a 24-h period. As a result, a compressive index of 0.32 was determined and then adopted within the later simulations. Also, the infilled materials were subjected to direct shearing tests for drained and undrained consolidation samples (Gregor et al. 2024). Normal constant stresses of 0 kPa, 125 kPa, 250 kPa, 375 kPa and 500 kPa were applied to the infilled samples during shearing with a rate 1 mm/min. The results showed that friction angles were decreased from 39.5 to 32.5 from drained to undarined samples (more details can be found in (Gregor et al. 2024)). Moreover, in contrast to the clean samples, the paste was applied to the shear face surfaces of the two outer blocks for a depth of 5 mm using a scraper, making sure to leave an opening for the bolt (Fig. 2b). After completing both outer concrete blocks, they were aligned and gently shaped against the centre block.

Fig. 2
figure 2

Double shearing samples a clean joint samples; b infilled joint samples

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After preparing the required samples, the double shearing tests were conducted by inserting at the P11 laboratory of Toowoomba campus of the University of Southern Queensland (UniSQ) at a 1 mm/min shearing rate. Shear displacement was applied on the upper surface of the middle block of the confined samples using a leveling load plate, and the shear load was measured using a load cell (Fig. 3).

Fig. 3
figure 3

A view of double shearing sample and apparatus

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2.1.2 Double Shear Calculations

The results of double-shearing tests will be reported as force (double shear force) and displacement. As a result, the shear force applied to each plane will be determined by Eq. (1) as follows (Gregor et al. 2023, 2024):

$${F}_{Shear Plane}=\frac{{F}_{Double Shear}}{2}$$
(1)

where:

\({F}_{Double Shear}\): the load cell data (N);

\({F}_{ Shear plane}\): the forces isolated to the failed shear plane (N).

Consequently, the peak shear stress will be determined by Eq. (2) (Gregor et al. 2023, 2024).

$${\tau }_{Stress}=\frac{{F}_{ Shear plane}}{{\pi r}^{2}}$$
(2)

\({vF}_{Shear Plane}\): the forces isolated to the failed shear plane (N);

\({\tau }_{ Stress}\): the ratio between shear force and cross-sectional area (N/m2);

r = the nominal rock bolt radius (m).

2.2 Analytical Modelling

The literature reviews revealed that some significant parameters, such as, the inability to model each of the regions of the shear curve, the impact of pretension, and the influence of various joint properties, including friction angle and presence of infill, were not considered in the previous models. As such, a new approach was adopted to fill this gap. During shearing, it was discovered that all the tested samples progressed through three regions: the elastic, strain-softening, and finally, the failure region, as outlined in Fig. 4.

Fig. 4
figure 4

a Shear-load displacement profile of 20 tonne FRP bolt for 15 kN pretension value in the clean joint sample (Gregor et al. 2023) b illustration of shear load–displacement

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Therefore, it was deemed essential to incorporate this into the foundations for the model development. In an attempt to achieve an accurate three region model, limits were adopted for each region. The elastic region was forced to comply with the limit denoted in Eq. 3 where u was the displacement and \({u}_{yield}\) was the displacement at the elastic yield force:

$$u\le {u}_{yeild}$$
(3)

With a similar approach to the elastic region, the limit expressed by Eq. 4 was applied to the strain-hardening region. By limiting the model between the displacement at the elastic yield force \({u}_{yield}\) and the displacement at the peak shear force \({u}_{peak}\) it was ensured that the results were constrained with no overlap.

$${u}_{yeild}<u<{u}_{peak}$$
(4)

The final limit used was to ensure the simulation of the failure region was clipped between the displacement at peak shear force and sample failure. Equation 5 demonstrates this by ensuring the displacement is greater than the displacement at peak \({u}_{peak}\) and yet less than displacement at failure \({u}_{failure}\).

$${u}_{peak}<u<{u}_{failure}$$
(5)

Once the regions were identified, unique models were created best to represent the rock bolts behaviour within each limit.

2.2.1 Elastic Region

Upon analysis of the experimental results, it was evident that linear elastic theory could describe the first region experienced by the fibreglass rock bolts. Whereby any deformation imposed by the applied displacement was linear and reversible. As such, the following linear Eq. 6 was developed where \({k}_{e}\) was the constant denoting the rock bolts elastic stiffness coefficient in Newtons per metre and where u represented the displacement in metres.

$${\tau }_{e}={k}_{e}u$$
(6)

2.2.2 Strain-Hardening Region

When the applied force exceeded the elastic response of the rock bolt, the rock bolt transitioned into the more complex non-linear plastic response. Due to the increased forces exerted on the samples after the transition into the strain-softening region, the influence of the applied pretension took effect. In order to develop a reliable model of these responses, two key theories were adopted, the energy balance theory and the Fourier transform. The energy balance theory was adopted to represent the overall behaviour of the rock bolt system. The theory states that the total energy within the system must remain constant and that the energy can only experience a state change. Additionally, any energy entering the system, ΔU, will system change the system’s original energy, Q, minus the work done by said system, W, as represented by Eq. 7 and forming the basis of the force equilibrium equation (Gregor 2022).

$$\Delta U=Q-W$$
(7)

In the case of the investigation of the double shear testing scheme, energy was added to the system by applying a force at a constant displacement. This increase in energy interacted with the double shear system through several components, such as the final value of the initial elastic region, the rock bolt, pretension, and interface properties. Due to the application of pretension having a limited impact on the physical properties of the rock bolt, its contribution to the force equilibrium equation was interrelated, such that, the constant α could define the initial applied pretension. The coefficient α was an arbitrary value for each pretension that was determined through an iterative process. The remaining factors expressing the rock bolt’s properties comprised of \({k}_{p}\), denoting the plastic stiffness and finally, the systems shear displacement u. As the strain-hardening region was an extension of the elastic region, the displacement needed to account for the displacement occupied by the elastic region. The rock bolts’ strain-hardening region’s representation of displacement was described by \(\left(u-{u}_{y}\right)\). Therefore, the rock bolts’ work contribution to the force equilibrium equation could be defined by Eq. 8 (Gregor 2022).

$${\tau }_{RB}={k}_{p}\alpha \left(u-{u}_{yield}\right)$$
(8)

The final component of the force equilibrium expression determined the contributions of the shear interface on the shear performance on the rock bolt. The shear interface properties were separated into the following components: influence of pretension, surface cohesion, friction angle, and surface area of the shear interface. In contrast to the influence of pretension on the rock bolt element, its influence on the shear interface was complex, whereby the recorded axial force experienced multiple inflection points, as highlighted in Fig. 5, in addition to linear portions of varying lengths. The first deflection point probably is due to the time that required that the compressive load sits in the sample (it is a form of experimental error). In this stage displacement increases largely without a meaningful increase in load.

Fig. 5
figure 5

Demonstration of complex raw pretension profile for infilled 30-tonne 20 kN pretension samples

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Therefore, to account for the contributions of the pretension on the shear interface, this required the development of a Fourier transform to replicate the complex waveform of the axial component of the shear system. The Fourier series decomposed the pretension signal into its sine and cosine components, which can be represented by Eq. 9. To automate and simplify the calculation of the Fourier transforms for all tested samples, a subroutine program was written by MATLAB.

$$y={a}_{0}+\sum_{i=1}^{n}{a}_{i}\text{cos}\left(iwx\right)+{b}_{i}\text{sin}\left(wx\right)$$
(9)

Figure 6 shows a close match between the Fourier approximation obtained in MATLAB and the raw pretension data. As such, the matrix containing the output of the Fourier transform was denoted by \({\sigma }_{n}\) and was the factor representing the pretension contribution of the shear interface.

Fig. 6
figure 6

Comparison of the Fourier transform approximation to the raw data of the clean shear 20 tonne rock bolt with 10 kN pretension

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The remaining three components of the shear interface portion of the equation were determined by the system design, such as the shear surface area, represented by \({A}_{s}\) and laboratory testing. Values for the friction angle and cohesion of both clean and infilled surfaces were determined via direct shear testing as described before and were denoted by ∅ and c, respectively. Therefore, the shear interface contribution could be defined by Eq. 10, where for all clean jointed samples, cohesion was set to 0. Therefore, the overall force equilibrium equation was expressed by Eq. 11. The initial condition of the region was represented by \({\tau }_{e}\), the final value from the elastic region. The rock bolt contribution was denoted by \({\tau }_{RB}\) and finally, the shear interface contribution denoted by \({\tau }_{i}\).

$${\tau }_{i}=A\left({\sigma }_{n}\varnothing +c\right)$$
(10)
$${\tau }_{ss}={\tau }_{e}+{\tau }_{RB}+{\tau }_{i}$$
(11)

Therefore, the shear load during the strain-softening region was derived from Eqs. 6, 8, and 10, resulting in Eq. 12. Where are the values for \({\tau }_{e}\), \({\tau }_{ss}\), \({\sigma }_{0}\) and \({\sigma }_{n}\) are represented in Newtons, \({A}_{s}\) in metres squared, \(\varnothing\) in degrees, c in Pa, \({u}_{e}\) and \({u}_{n}\) in metres, \({k}_{p}\) in Newtons per metre and finally α as a dimensionless constant.

$${\tau }_{ss}=\left[{\tau }_{e}+{A}_{s}\varnothing \left({\sigma }_{0}-{\sigma }_{e}\right)-{K}_{p}\alpha {u}_{e}\right]+{K}_{p}\alpha {u}_{n}+A({\sigma }_{n}\varnothing +c)$$
(12)

2.2.3 Failure Region

The failure region denotes the final component of the analytical model and completes the trilinear expression. Like the initial elastic region, the failure region was expressed utilising a linear expression bound to the limit outlined in Eq. 5. To create a transition from the strain-hardening region, the initial shear force component was set to the peak shear force of the strain-softening region \({\tau }_{ss,p}\). To represent the now negative gradient due to the decreasing shear force values, the constant \({k}_{f}\) with units of Newtons per metre incorporated into the linear expression and multiplied by the displacement\({u}_{n}\). Due to the inherent failure of the rock bolt for the duration of this region, pretension forces were no longer subjected to the system, resulting in negligible contributions. As such, pretension coefficients and the related shear interface influences were ignored, resulting in the representation of the failure region by Eq. 13 and completing the analytical model of rock bolts subjected to double shear.

$${\tau }_{f}={k}_{f}\left({u}_{n}-{u}_{ss,p}\right)+{\tau }_{ss,p}$$
(13)

2.3 Numerical Simulation

FLAC3D, the modelling software designed by ITASCA was used to develop advanced numerical simulations due to its widespread deployment within the industry. The main purposes of using FLAC3D were: (a) the double shearing performance of FRP bolts subjected to the same conditions as the experimental test scheme for data validation, and (b) sensitivity analysis of double shear test scenarios focusing on variations to host environment, shear speed and installation angle.

2.3.1 Development of the Rock Bolt Model

The pile structural element with the activated rock bolt flag was chosen for all test simulations. Unlike other elements such as the cable element, piles allow the user to utilise rock bolt specific properties, such as resistance to bending moments, resulting in a simulation that better represents the performance of rock bolts. Unfortunately, there is not one perfect element, and therefore compromises were made. The most significant shortcoming of the selected rock bolt element was the inability to apply pretension to the rock bolts effectively. This property is inherent in the cable structural element. Attempts were made to modify the simulation however no variation to the model yielded accurate results.

Rock bolt elements simulate the mechanical behaviour of the element by deconstructing the rock bolt into nodes. Each node is then separated by a series of springs, also known as coupling springs, with one representing the axial stiffness, the shear stiffness, and the cohesive strength at each node, as highlighted in Fig. 7.

Fig. 7
figure 7

FLAC3D Coupling spring concept (ITASCA, 2005)

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The defined coupling springs convention facilitates a one-dimensional explanation for each of the defined behaviours. As such, the axial behaviour is reduced to its primitive relationship between the axial stiffness K as calculated across the rock bolts cross-sectional A, its Young’s modulus E and its length as outlined by Eq. 14.

$$K=\frac{AE}{L}$$
(14)

Similarly to the axial calculations, the shear behaviour of the rock bolt is simplified to an ideal state to facilitate computational simplicity. To achieve the desired outcome, the spring model outlined in Fig. 8 simulates a slider effect along the interface annulus. Therefore, the shear behaviour becomes a function of the coupling spring material and interface properties such as: stiffness ks, cohesive strength cs, friction angle ∅s and the rock bolt perimeter p, as illustrated by Fig. 8.

Fig. 8
figure 8

Shear behaviour of rock bolt elements (Left): shear force/length versus relative shear displacement; (Right): Shear-strength criterion (Reference pile section of FLAC manual)

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The overall mechanical behaviour of the rock bolts in the normal plane then become a function of the confining stress. Figure 8 further demonstrates the distinction between nodes in compression and tension, where friction angle and cohesive strength are critical drivers of the node’s response. The confining stress was calculated as a relationship between the principal stresses σ1 and σ2 and pore pressure p as defined by Eq. 15.

$${\sigma }_{m}=-(\frac{{\sigma }_{1}+{\sigma }_{2}}{2}+p)$$
(15)

The last key calculation component within the rock bolt property is the rock bolt extension, otherwise defined as the tensile behaviour of the rock bolt. This property considers the axial and bending resistance of the simulated nodes, unlike the cable simulation equivalent, which only considers the axial resistance. As a result, this property was defined as the relationship between axial plastic strain (\({\varepsilon }_{pl}^{ax})\), rock bolt diameter and length, d and L, respectively, and lastly (\({\theta }_{pl}\)) the elements’ average angular rotation. The calculation was expressed using Eq. 16 and defined within FLAC as the tensile failure strain.

$${\varepsilon }_{pl}=\sum {e}_{pl}^{ax}+\sum \frac{d}{2}\frac{{\theta }_{pl}}{l}$$
(16)

These FRP bolt properties were implemented as part of the FLAC3D subroutine program and initiated prior to the commencement of the simulation. Table 2 outlines each of the selected properties and their input values.

Table 2 20-tonne and 30-tonne rock bolt simulation properties
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2.3.2 Developing the Rock and Joint Model

The construction of the shear environment required careful selection of the material constitutive models within the FLAC3D library due to the vast differences in their mechanical behaviours. The system was defined by the following four components: (1) the host rock; (2) the washers; (3) the infilled material, and (4) the overall model environment. Models were chosen based on their ability to replicate material performance for each system component as determined by the experimental results. A strain-softening model was applied to the concrete simulating the host rock. This enabled the model to simulate the impacts of cohesion, friction, and dilation impacts during the shearing process as applied forces may soften or harden the host rock. For some components of the model, this depth of data was unnecessary and a potential hindrance to the system’s overall performance. In the case of the washer, an elastic model was implemented as the material was homogeneous and isotropic and only required the most straightforward material representation. The infilled material manufactured for testing was a soil-clay derivative. As a result, the Mohr–Coulomb model was applied as this provided an accurate representation of the shear failure of soils. Finally, mechanical damping was applied to the entire environment to compensate for system losses due to vibration. Without applied damping, components within the simulation can begin to oscillate and distort results. Table 3 outlines each of the selected properties and their input values for each system element, such as the strata, clean and infilled joints, and the rock bolt washers.

Table 3 System properties for the strata, shear joints, infill material and washers
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2.3.3 Double Shear Conceptual Model

A simulated test environment was also created to determine the double-shear performance of the rock bolts. The conceptual model highlighted in Fig. 9 demonstrates the schematic design uploaded to FLAC3D. Mimicking the experimental testing scheme, the outer two blocks were fixed in space with the centre block freed for shearing. A displacement equivalent to 1 mm/minute was applied to the top surface of the block to simulate the shearing action. Additionally, to prevent the rock bolt pulling through the block and failing prematurely, the first and last nodes of the rock bolt was similarly fixed in place to utilising the nut. Unlike the single shear model, the double shear model was designed to replicate the design of the experimental tests. Therefore, the outer two blocks were created to 200 mm by 200 mm by 200 mm and the centre block with 200 mm by 400 mm by 200 mm as illustrated in Fig. 9. Additionally, the material properties of the blocks were based on realistic material properties by the experimental study and were outlined in Table 2. The grid properties were determined by trial and error to identify the most accurate and efficient completion of each simulation. Selecting a grid as 1/20th of the dimensions of the blocks resulted in fast computation with minimal change in results, and therefore the final selected grids were 10 × 10 × 10 for the outer blocks and 10 × 20 × 10 for the centre block. Additionally, the rock bolt segments were required to match the double shear simulation and could not be transferred from the single shear simulations. Trialling various settings indicated that dividing the element into 70 segments provided the most suitable compromise.

Fig. 9
figure 9

Double shear conceptual model (top) and model dimensions (bottom)

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3 Results and Discussions

3.1 The Results of the Double Shear Testing for 20-Tonne and 30-Tonne FRP Bolts for Clean Joints

As mentioned in Sect. (2.1), the designated pretension values for both FRP bolts were 0 kN, 10 kN, 15 kN, and 20 kN. However, there were slight variations in the tested pretension values. For the 20-tonne FRP bolt, the specified values were 1 kN, 12.8 kN, 13.5 kN, and 20.5 kN; for the others, they were 0.8 kN, 11 kN, 12 kN, and 17.5 kN. The test results (Table 4) illustrated three distinct shearing load profiles derived from the double shearing tests: the elastic, strain-hardening, and failure regions. As the bolt capacity increased from 20 to 30 tonnes, the elastic gradients decreased. This suggests that the 20-tonne FRP bolt exhibited greater stiffness in response to shear force than the 30-tonne bolt. Notably, the highest elastic gradient value, 88.1°, was achieved when the pretension values of the 20-tonne FRP bolt were set at 15 kN and 20 kN. While the strain-softening gradients did not show a specific trend, a comparison of results for both bolts indicated that higher values were generally observed in the 20-tonne bolt, except for the 15 kN pretension samples. Additionally, it’s worth mentioning that the highest values for each bolt were recorded when both bolts were pretensioned at 20 kN (More details can be found in (Gregor et al. 2023)).

Table 4 Summary of the double shearing tests for clean joints for 20-tonne and 30-tonne bolts (Modified after (Gregor et al. 2023)
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In contrast to the preceding two stages, the ultimate failure stage did not exhibit any interference from changes in pretension, leading to consistent gradients for each pretension level. Notably, the 20-tonne samples showed increased gradients in each failure zone compared to the 30-tonne samples. Beyond differences in stiffness, this resulted in the 30-tonne samples attaining greater displacements at failure. Notably, the 30-tonne sample with 0 kN pretension outperformed all other samples in peak shear, albeit achieving the highest peak displacement. Nevertheless, the peak displacements for the 30-tonne samples at low pretension decreased more rapidly with each increase in pretension compared to the 20-tonne samples. Interestingly, at 15 kN and 20 kN pretension levels, the 30-tonne samples matched the displacements of the 20-tonne samples. Eventually, both bolts exhibited hinge point bending ranging from approximately 10° to 14°. An increase in pretension values led to gradual increments in bending at the hinge point post-failure (More details can be found in (Gregor et al. 2023)).

3.2 The Results of the Double Shear Testing for 20-Tonne and 30-Tonne Rock Bolts for Infilled Joints

As outlined in Sect. 2.1, the designated pretension values for both FRP bolts were 0 kN, 10 kN, 15 kN, and 20 kN. Nevertheless, there were minor variations in the tested pretension values. The specified values for the 20-tonne FRP bolt were 1.5 kN, 12.0 kN, 17.0 kN, and 22.0 kN, while for the other bolt, they were 1.6 kN, 12 kN, 17.0 kN, and 21.0 kN (Table 5).

Table 5 Summary of the double shearing tests for infilled joints for 20-tonne and 30-tonne bolts (Modified after (Gregor et al. 2024)
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The results of infilled joints for 20-tonne and 30-tonne FRP bolts are given in Table 5. It was found that the peak shear values varied between the 20-tonne and 30-tonne samples. Interestingly, both types of bolts demonstrated no discernible correlation between peak shear force and pretension values. The values of peak shear force ranged from 88 to 100 kN. Similarly, the 30-tonne samples showed no apparent correlation between shear force and pretension, yet displayed significant variability, with performance ranging from 69 to 114 kN. Notably, the 0 kN and 15 kN pretensioned samples performed lowest among the 30-tonne samples. This observation suggested that, despite the transfer of shearing forces to axial forces during the shearing process, there was no mechanism for the reverse. Additionally, pretension did not indicate any shear-strengthening properties (More details can be found in (Gregor et al. 2024)). In contrast to shear force, the 20-tonne samples displayed a clear relationship between the pretension values and the maximum failure displacements. As the pretension values increased, the failure displacement decreased progressively. Notably, the initial 0 kN sample failed at 15.3 mm, while the final 20 kN sample failed at only 10.6 mm. However, this pattern was not observed in the case of the 30-tonne sample. The 30-tonne samples showed no specific correlation between the initial pretension value and the displacement at failure. Both the 0 kN and 20 kN samples failed within a 1 mm range, and the 15 kN sample appeared as a potential outlier due to its 5 mm to 6 mm lower peak displacement (More details can be found in (Gregor et al. 2024)).

3.3 Model Calibration for Analytical Modelling of Reinforcing Clean Joints

Upon developing the proposed analytical model, it was then calibrated against the experimental results for the clean joint testing scheme. This was to ensure model accuracy and suitability. Calibration was conducted in two stages; model setup and model tuning and was accomplished by developing an extensive MATLAB subroutine program. During the setup stage, the subroutine software loaded the experimental results and conducted analysis to determine the shear profile properties \({k}_{e}\), \({k}_{p}\), and \({k}_{f}\). These properties were directly utilised for the gradient of the elastic and failure regions due to the simple linear theory. Additionally, the subroutine program allowed for the manual determination of the elastic to strain softening transition. Some samples experienced large transition zones, making it difficult to determine its location automatically. The presence of subtle profile changes prevented the auto-determination of the transition. Therefore, it resulted in the need to allow for manual selection. The subroutine programme mapped out potential turning points based on a chosen degree of certainty. Figure 10 demonstrates these potential turning points, with the final point displaying good alignment with the actual region transition point as determined by experimental analysis.

Fig. 10
figure 10

Determining the transition to the strain-hardening region for clean samples

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Unlike the elastic region, the strain-softening region was comparably more complex to simulate due to the incorporation of pretension and shear interface properties. The \({k}_{p}\) coefficient was utilised as the initial state, while α was then used as a tool to provide fine tuning adjustment. The coefficient α was applied through an iterative process until there was an agreeable alignment between the model and the experimentally determined profile, as highlighted in Fig. 11. Figure (11) also demonstrates how adjusting α resulted in changes to both the gradient of the profile as well as its placement along the y-axis. α was either increased or decreased by increments of 0.0001 depending on the proximity of the simulated peak value to the experimental peak value. When the peak values agreed based on the defined error factor, the subroutine programme locked in the coefficient resulting in the profile highlighted in Fig. 10.

Fig. 11
figure 11

Calibration of the strain-softening region using the α constant to ensure model agreement and challenges in simulating the transition zones between the regions for 20-tonne rock bolts with clean joints

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Despite the calibration, the model presented challenges in predicting the transition zones between the regions for some of the samples. Typically, the analytical models simulating the performance of 20-tonne rock bolt samples demonstrated a stepped transition between the failure regions. Figure 11 demonstrates how the simulation presents a vertical step when transitioning from the elastic region to the strain-softening region. A similar horizontal step is recorded when transitioning out of the strain-softening region. This could be attributed to the additional calculation parameters of the strain-softening region, including the complex pretension profile and calibration coefficients. The three regions required the adaptation of varying analytical models and the most appropriate agreement was adopted through the calibration process. While these transitions are visible, they had minimal impact on the overall performance of the model. Unlike the 20-tonne samples, most of the 30-tonne rock bolts subjected to the clean joints testing scheme presented with seamless transitions from region to region, as highlighted in Fig. 12. The increase in rock bolt capacity appeared to result in the seamless transition from the elastic region to the strain-softening region. Some 30-tonne samples still presented with a minor step when transitioning to the failure region.

Fig. 12:
figure 12

30-tonne rock bolts with clean joints presenting seamless transitions between regions

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All the clean joint sample scenarios were successfully modelled using the developed subroutine programme. One of the key outcomes of the simulation was to predict the rock bolts’ shear profile as displacement increased. However, this model was unable to accurately predict each rock bolts’ post-failure behaviour and therefore an approximation was adopted. Throughout the experimental phase of this study, it became clear that the post-failure response would not be possible to model accurately, as some samples instantly returned to a shear force of 0 kN moments after failure while others retained a residual force. This was in part due to the rotation of the centre shearing block post failure as a perfect system symmetry was impossible to achieve. In an attempt to combat this variability, the simulated failure region was a linear approximation controlled by \({k}_{f}\) and extended until the value intersected with the y-axis at 0 kN as shown in Fig. 13. As a result of the sample variability, the \({k}_{f}\) coefficient also resulted in a significant range, as shown in Table 6. Table 6 also presents \({k}_{e}\) and \({k}_{p}\), forming the key portions of the shear profile. These values are of the same magnitude and present with minor changes when compared to each sample in the test.

Fig. 13
figure 13

Analytical simulation results for clean joint testing scheme

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Table 6 Analytical estimates of shear and displacement parameters for clean shear testing scheme
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The α coefficient was less sensitive to changes in samples with lower pretension values, suggesting that α has a stepped response to pretension. The low pretension cut-off also varied between the 20-tonne and 30-tonne clean shear-tested rock bolts. Therefore, the α pretension relationship could be considered as bi-linear, with the turning point occurring after a pretension of 15 kN for the 20-tonne rock bolt. In contrast, this point was reached at a pretension of 10 kN for the 30-tonne samples.

3.4 Model Calibration for the Analytical Model of Reinforcing Infilled Joints

The outlined analytical model was also developed to simulate the shear response of rock bolts when subjected to infilled joints. To ensure consistency across all the test scenarios, the same subroutine program was utilised. As such, the development of the simulation went through the same stages as the 20-tonne clean joint samples with the exception of one key factor. The addition of the infilled material also added the requirement to account for its specific material properties. This was accomplished by changing the value of the friction angle, ∅, and adding the cohesion parameter outlined in Table 7. These properties were determined through laboratory analysis of the infill material and are only valid for the tested double shear scenarios of this study.

Table 7 Analytical estimates of shear and displacement parameters for infilled shear interface testing scheme
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Like the clean interface samples, the subroutine program required the input of the elastic region transition point, otherwise referred to as the turning point, as depicted in Fig. 14a. This point had significant impacts on the estimations of the length of the elastic and strain-softening regions and the slope of the latter. Therefore, the experimental results were used to validate the selection of this point. Following the selection of the turning points, the subroutine programme adjusted the α coefficient in conjunction with error validation to ensure that the simulation accurately captured the slope and the peak shear value. Inadvertently, calibrating the slope and peak value intersect also resulted in a suitable approximation of the shear failure displacement. Figure 14b illustrates this impact of α on the outlined aspects of the shear failure profile.

Fig. 14
figure 14

a Determining the transition to the strain-softening region for infilled samples; b Calibration of the strain-softening region for samples with infilled shear interfaces using the α constant to ensure model agreement

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Once the optimal simulation settings were established, they were recorded in Table 7 and the outputs were highlighted in Fig. 14. It should be noted that the simulation captured the influences of the rock bolts’ strength and pretension for infilled samples. Despite the same analytical model utilised for both clean and infilled test scenarios, the infilled models presented fewer issues. Unlike the clean joint models, the infilled joint models presented with no step between the elastic and strain-softening regions as highlighted in Fig. 15. Infilled simulations also presented with a less pronounced step between the strain-softening and failure regions. Scenarios subjected to the clean joint testing scheme demonstrate notable differences between 20-tonne and 30-tonne samples when comparing the presence of vertical stepping at the transition point between the elastic and strain-softening regions. The infilled testing scheme, however, demonstrated no such variations between the 20-tonne and 30-tonne samples. The simulated samples presented with no step at the elastic transition point. The appearance of the step occurring at the failure transition zone indicated the failure process. When the experimental samples reach their failure limit, the shear forces levelled out briefly before transitioning to their residual force. Hence why in Fig. 16, the failure transition point was highlighted as not representing a transition step.

Fig. 15
figure 15

Analytical simulation results for infilled joint testing scheme

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Fig. 16
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Challenges in simulating the transition zones between the regions for 30-tonne rock bolts with infilled joints

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The analytical model can be used for simulating numerous scenarios outside its design. The input and output variables specified for this model were developed to accommodate user specific situations. By incorporating variables such as the \(\alpha\) coefficient into the analytical model, the performance of samples with properties such as pretensions not directly tested in this study can be determined by approximation using functions such as Piecewise Linear Interpolation. This progressive design approach resulted in a robust analytical model that can simulate a host of scenarios subjected to fibreglass rock bolts.

3.5 Numerical Simulation Calibration of Double Shear

The double shear simulation process underwent calibration against the experimental testing scheme to ensure model accuracy. Calibrating against known metrics ensured that the numerical simulation could then be utilised to simulate scenarios not addressed by the conducted physical tests and sensitivity analysis. Due to the differences in rock bolt performance, both the 20-tonne and 30-tonne rock bolts were calibrated separately. The numerical simulations were limited to passive 0 kN pretension test schemes, as such the 20-tonne model was calibrated against the 20-tonne 0 kN pretension infilled shear test, as shown in Fig. 17a. The 30-tonne model was calibrated against the 30-tonne 0 kN clean shear test, as highlighted in Fig. 17b.

Fig. 17
figure 17

a Experimental data, and numerical simulation for a 20-tonne infilled 0 kN pretension sample and b 30-tonne clean joint 0 kN pretension sample

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The chosen parameters successfully calibrated the two rock bolt types with only minor variation between the three data types. The numerical model of the 20-tonne sample presented with good alignment to the experimental data set, most notably simulating similar transition zone properties as shown in Fig. 18a. The simulation matched the undulating force response at the elastic transition zone with only a minor variation in displacement. The numerical model also predicted the various peaks and troughs during the entire shearing process. Despite some variance in the correlation with the experimental results, they provided an overall realistic representation of the shear/displacement response of the rock bolt. Throughout the numerical simulation, the recorded displacement for each shearing stage did not align but was consistently within 1 mm of the experimentally tested samples. Additionally, the post failure portion of the model showed little correlation with the experimental data set. The numerical model indicated a steeper release of the shear force with almost no retained residual resistance 2 mm after failure.

Fig. 18
figure 18

a Comparison of experimental data and numerical models for a 20-tonne infilled joint 0kN pretension sample and b 30-tonne clean joint 0 kN pretension sample

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The numerical simulation of the 30-tonne rock bolt with clean joint interfaces displayed similarities to the experiment confirming the model’s ability to replicate the shear response of the rock bolt accurately. The FLAC3D based model was able to replicate key features of the shear profile such as the rate of increase of shear force, peak failure force and peak failure displacement. However, despite the overall agreement between the models and experimental results, there were several discrepancies with the shear profile as highlighted in Fig. 18b. While the numerical model was able to replicate various peaks and troughs throughout the shearing profile, they did not align with any present during experimental testing. This resulted in the inability to identify the transition point from the elastic region to the strain-softening region. Lastly, there was a disagreement between the numerical and analytical models with simulating the shear profile’s failure and post failure region. Despite this, the numerical model bore the closest resemblance to the experimental result with near identical peak failure displacement and failure force. Unfortunately, the post failure behaviour of the rock bolt could not be compared as the experimental result was cut off immediately after failure. Indeed, the numerical simulation predicted an almost immediate release of shear resistance to a residual of 0 kN. While it was challenging to predict the actual behaviour of the post failure shear profile, the limited experimental data indicated similar trends to the numerical simulation, however, further study would be required to confirm this.

3.5.1 Sensitivity Analysis

The development of the above simulations enabled the determination of the rock bolts response to double shear forces under a range of conditions without the need to complete time-consuming experimental studies. The validated numerical models were used to simulate the rock bolts’ double shear response when subjected to a range of environmental variances such as; host rock strength, speed of shear loading, and rock bolt implementation to determine the rock bolts’ sensitivity to changes within its system. For each scenario, the rock bolts were subjected to different parameters of interest: low to high strength host rock, slow, standard, and fast shearing speeds and no angle and angled rock bolt installation. Their outcomes were then compared against the system parameters of the validated numerical simulation.

3.5.1.1 Rock Strength Variations

Rock bolts can be installed in various host rock types, ranging in their parameters. Testing for these variations involves significant labour and time investments. Utilising the numerical model developed in this study allowed for the comprehensive testing of a range of host rock strengths for 30-tonne clean joint and 20-tonne infilled joint scenarios. Each scenario was repeated with rock UCSs of 20 MPa, 30 MPa, 40 MPa and 60 MPa. 40 MPa was considered the baseline as it also directly reflects the experimental results. The 30-tonne clean and 20-tonne infilled scenarios demonstrated differing responses to changes in the host rock strengths, however for both rock bolt types, the initial differences were difficult to identify when analysing their complete shear profiles. Both rock bolts need to be examined closely to identify variances. The 30-tonne rock bolt samples initially looked identical, as shown in Fig. 19a. However, upon closer investigation, it was clear that increasing the UCS of the host rock impacted the shear force across the entire profile, while leaving the displacements unchanged. Figure 19b highlights the discrete increase of the shear force as well as features such as agreements, peaks and troughs. The impact of rock UCS on the rock bolt’s performance could be attributed to 30-tonne rock bolts being able to outperform the host rock for shear resistance. Therefore, as the UCS increased, so did the rock bolt’s ability to transfer the shear forces.

Fig. 19
figure 19

a Results of changing the host rock UCS and its impact on the shear force of the 30-tonne rock bolt; b Zoomed in comparison of the effects of host rock strength on the shear force profile of the 30-tonne rock bolt

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Similar to the 30-tonne rock bolts, the 20-tonne rock bolts also presented negligible differences in their response to changes in the host rocks UCS, as shown in Fig. 20a. However, unlike the 30-tonne rock bolt sample, the 20-tonne shear profile resulted in no discernible relationship to changes in host rock UCS. It is evident in Fig. 20b that changing the host rock UCS had no impact on either the shear force response over the profile or its displacement. The variations observed were likely due to simulation variance and not as a direct result of the change in this parameter. The weaker strength of the 20-tonne rock bolts likely caused this. If the 20 MPa host matched or outperformed the rock bolt’s ability to resist shear forces, any increase would therefore, result in no change to the rock bolts’ shear profile as it was already performing to capacity.

Fig. 20
figure 20

a Results of changing the host rock UCS and its impact on the shearing force of the 20-tonne infilled rock bolt b Zoomed in comparison of the effects of host rock strength on the shear force profile of the 20-tonne infilled rock bolt

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3.5.1.2 Rock Bolt Angle

The sensitivity analysis was extended to include the impact of the intercept angle of the rock bolt and shear joints. Shear joints are naturally occurring structures and are rarely uniform or perpendicular to the installed rock bolt. The numerical model was utilised to investigate the impact of installing rock bolts at an angle other than 90 degrees to the shear interface. To accomplish this, the modified conceptual model shown in Fig. 21a was adopted. The rock bolt was installed from the upper left to the lower right of the system to represent the most extreme angle possible within the tested design. All other properties of the system were retained to maintain consistency with the validated model. Greater angles could be accomplished but would require significant changes to the constitutive model, requiring validation of a new model. Both the 30-tonne clean joint and 20-tonne infilled samples presented similar significant changes to the post-failure responses of the rock bolts when compared with the experimentally verified model. Figure 21b and c illustrate these post-failure differences as an immediate reduction in shear force post-failure. The angled samples alternatively recorded a gradual reduction of shear forces until a residual load was achieved. This was most likely attributed to one side of the rock bolt applying the shear force to a disproportionally small volume of the host rock, while the other half of the sample applied the same load to a disproportionally large volume segment. As a result, complete shear only occurred at one of the shear planes, resulting in the intact plane carrying the residual load.

Fig. 21
figure 21

a Conceptual model of the rock bolt installed at an angle; b Impact of the shear force profile of a 30-tonne clean joint rock bolt installed at an angle compared to the original test parameter; c Impact of the shear force profile of a 20-tonne infilled rock bolt installed at an angle compared to the original test parameter

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3.5.1.3 Shear Speed

The final system property tested for sensitivity was the rate of shear applied to the samples, reflecting the variations in shear that may be present in natural environments. The experimental shear property of 1 mm/min was considered the baseline with samples then subjected to a slow speed scenario of 0.5 mm/min and a fast speed scenario of 1.5 mm/min. Changing the shear speed profoundly impacted the shear profile of the 30-tonne clean jointed rock bolt as highlighted in Fig. 22a. The impact of reducing the shear speed to 0.5 mm/min was initially not evident on the overall profile, however, when zoomed in, like in Fig. 22b, it was evident that the definition of shear profile was greatly improved. The jagged peaks and troughs were replaced with smoothed transitions. Increasing the shear speed to 1.5 mm/min conversely resulted in a lower definition view of the shear profile with more significant exaggerations of the peaks and troughs. This change in the smoothness of the profile was most likely due to the slower shearing samples having more time to resolve variations in the absorption and repulsion of shear forces. Faster samples could not find equilibrium to varying shear forces resulting in compounded peaks and troughs. Furthermore, increasing the shear speed completely altered the post failure response of the 30-tonne clean sample. Both the slow and baseline shearing samples experienced a dramatic release of shear forces to 0 kN, while the fast-shearing sample experienced a gradual decline in residual shear forces until it released to 0 kN at a displacement double that of the other two samples. As the 30-tonne sample was already stronger than the host rock environment, the increase in shear speed most likely caused failure within the rock structure, preventing the rock bolt from achieving an absolute shear. Therefore, the sample relied on excessive displacement to achieve shear instead of applied shear force. The 20-tonne sample, however, presented no notable responses to the change in shear speed. The sample presented with no change in profile definition and no changes to post failure response (Fig. 22c). This was most likely attributed to the weaker nature of the 20- tonne rock bolt and most importantly its weaker properties to that of the host rock. The 20-tonne sample could not overpower the host rock to achieve trailing residual forces and changes to the peaks and troughs of its shear profile.

Fig. 22
figure 22figure 22

a Impact of the shearing speed on the 30-tonne clean joint rock bolt installed at an angle compared to the original test parameter; b Zoomed in comparison of the impact of shearing speed on the 30-tonne clean joint rock bolt installed at an angle compared to the original test parameter; c Impact of the shearing speed on the 20-tonne infilled rock bolt installed at an angle compared to the original test parameter

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4 Conclusion

In this paper, the shear performance of the FRP rock bolts with both clean and infilled shear interfaces has been investigated by developing analytical and numerical modelling. For this, a series of comprehensive of experimental tests were conducted on two different FRP rock bolts of 20-tonne and 30-tonne for clean and infilled joints, considering various pretension values ranging from 0 to 20 kN. Upon sample preparation and testing, it was observed that increasing pretension reduced the displacement and shear range for certain rock bolts. Additionally, applied pretension resulted in a smoother profile concerning pretension value, although it had minimal impact on the recorded shear forces for the 20-tonne rock bolts. The infilled test scenario involved 5 mm thick sandy clay infilled shear interfaces. The results indicated a significant increase in shear force performance for 30-tonne rock bolts compared to 20-tonne bolts. Complementing the experimental test schemes, the analytical model was developed to predict the shear performance of fibreglass rock bolts when subjected to clean and infilled shear interfaces. The model mimicked the design elements of each test scenario and were verified against the experimental results. The analytical model was developed by utilising several fundamental physical theories, including linear elastic theory, energy balance theory and Fourier transform. The combination of these theories resulted in the tri-linear expression that embodied the three-stage shear profile. To enable model calibration to different scenarios, each profile stage incorporated independently adjustable coefficients. Despite the model’s close approximation to the sample’s experimental performance, it proved challenging to recreate the localised transition zones between each stage accurately. A vertical or horizontal step typically represented this. However, through an iterative self-checking process, the model could provide close overall approximations and map the shear performance of fibreglass rock bolts subjected to the various shearing conditions. Also, a 3D numerical model was developed to model double shear test scenarios using a FEM technique. The 20-tonne and 30-tonne 0 kN pretension shear simulations accurately simulated the samples’ peak shear and shear displacement. While the simulated rock bolts recorded similar gradients for the strain-softening region, a lack of attenuation resulted in an oscillating profile. The 20-tonne simulation was more susceptible to this oscillation, resulting in significant peaks and troughs. Despite this, several of the oscillations aligned with the 20-tonne experimental results. The sensitivity analysis conducted for the rock bolt installation angle and shear speed numerical models highlighted variations to the shear profile for both 20-tonne and 30- tonne rock bolts. This suggests these parameters are potentially significant to the rock bolt’s performance. Increasing the shear speed from the experimental test baseline yielded substantial displacement increases in the post-failure residual performance of the rock bolts. Changing the installation angle resulted in greater peak shear forces and extended residual zones. The least significant impacts were observed when changing the host rock UCS, suggesting neither rock bolt was drastically impacted by weak or strong host rocks. Despite the model’s success, it could not simulate the impact of pretension on any of the scenarios due to software limitations.