## 1 Introduction  
Rock reinforcement exerts a paramount role in underground excavating engineering [1]. This is more apparent in mining engineering. This is because, in mining engineering, the underground openings are normally located in jointed rock masses, such as coal, mudstone, shale and sandstone [2]. The stability of those jointed rock masses is usually poor. Therefore, once the underground roadways or chambers are excavated in those jointed rock masses, the surrounding rock masses are likely to converge [3]. This may lead to a collapse of the roof, leading to underground disasters [4]. To prevent the paramount convergence of underground openings and underground roof collapsing disasters, many methods can be used, such as grouting. [PERSON] et al. [5] demonstrated that using grouting could improve the shear strength of jointed rock mass. Additionally, rock reinforcement methods can be used [6]. Among kinds of rock reinforcement methods, using rock bolts to improve the strength and stability of rock masses accounts for a major part [7]. Therefore, it is quite necessary to understand the rock reinforcement mechanism of rock bolts. To realise this target, previous researchers and engineers conducted numerous experimental tests, analytical modelling, numerical modelling and in situ tests. For example,Krukovskyi et al. [8] used the numerical modelling method to analyse the changing in the dimension and shape of a rock mass zone reinforced with injection rock bolts. It was demonstrated that increasing the number of injection rock bolts was likely to create the rock bolt arch in the mine roof. Moreover, [PERSON] et al. [9] used a two-dimensional finite element program to predict the stress-strain state of the rock mass under the stope operations. Their study can be used to create an in situ geomechanical model to optimise the parameters for the mining support. Although the selected investigating methods are diverse in evaluating the anchorage force of rock bolts, the investigating protocol is quite similar. It uses the pull-out (PO) method to investigate the anchorage force of rock bolts [10]. Moreover, with this method, the load transfer process between the rock bolts, grout column and rock masses can be evaluated. According to this concept, the anchorage force of rock bolts under various geological and loading environments has been investigated. The influence of numerous parameters on the anchorage force of rock bolts has been revealed, such as grout strength, bolt diameter, rock mass strength and stress condition. Among kinds of parameters, researchers and engineers paid much attention to the parameter of embedment length (EL). The EL is defined as the length in which the rock bolt is bonded with the surrounding rock masses with grout. The EL is significant to the rock bolt reinforcement. If the EL is too short, the rock bolts are not able to maintain the stability of excavated rock mass. However, if the EL is too long, it will increase the economic cost and resource waste. Therefore, optimising the EL of rock bolts is important to maintain the safety of underground openings and reduce economic costs. To fulfil the optimising of the rock bolts' EL, how the EL influences the anchorage force of rock bolts should be clearly understood. Therefore, this paper adopted the modified pile elements to reveal the relationship between the EL and the rock bolt anchorage force. For full-grouting rock bolts, the EL is almost identical to the rock bolt length. Apparently, optimising the EL is conducive to determining the proper length of rock bolts. Moreover, optimising the EL is conducive to reducing the grouting cost in rock reinforcement. To investigate how the EL influenced the anchorage force of rock bolts, [PERSON] et al. [11] conducted PO tests on ribbed rock bolts. The EL ranged between seven times the bolt diameter and fourteen times the bolt diameter. It showed that the largest anchorage force increased directly with the EL. Similarly, [PERSON] et al.