## 1 Introduction  
Joints exist widely in tunnel engineering. The rock joints in engineering rock mass reduce the overall strength of rock mass. Under the influence of external load or environmental change, rock mass is prone to shear-slip failure along rock joints, which poses a serious threat to engineering safety [1, 2, 3]. As the weak plane in rock mass, the mechanical properties of rock joints control the stability of rock mass to a great extent [4, 5, 6, 7, 8]. As a flexible reinforcement measure, bols are widely used in the support and reinforcement of geotechnical engineering such as mine, slope, and dam foundations due to their significant engineering benefits [9, 10]. Therefore, studying the shear performance of bolted rock joints is necessary and practical for the stability assessment and safety design of geotechnical engineering. In this case, a series of studies on this issue have been conducted, and numerous achievements have been made. For example, the influence of different displacement angles, surrounding rock material, the fracture openings of rock joints, bolt dip angles, and the length of the shear strength of bolted rock joints have received attention from researchers worldwide [11, 12, 13]. [PERSON], et al. [14], [PERSON] [15], and [PERSON], et al. [16] investigated the shear strength of bolted joints. Other experts simultaneously considered the influence of the tensile and shear strength of the bolt on the shear strength of rock joints [17, 18]. To better fit engineering practice, performing investigations on the shear strength of boltedrock joints under cyclic load is an important task [19; 20; 21]. [PERSON] [22] and [PERSON], et al. [23] conducted field tests to explore the reinforcement performance of bolts in highly stressed rock masses. These studies were mainly conducted through experiments and numerical simulations, and most of the conclusions were qualitative. Quantitatively describing the shear strength of bolted rock joints is necessary for obtaining a deeper understanding of their features. Under the circumstances, some scholars have conducted theoretical analyses of the shear strength of bolted rock joints. [PERSON], et al. [24] deduced the calculation formula of the ultimate elastic load of the bolt based on the elastic-plastic theory. [PERSON], et al. [25], [PERSON], et al. [26], and [PERSON], et al. [27] proposed analytical models that can predict the entire process of the tension and tensile shear of bolts. [PERSON], et al. [28] and [PERSON], et al. [29] derived the analytical expressions of various failure modes of bolts considering the interaction between bolts and surrounding media. By conducting a theoretical analysis, [PERSON], et al. [30] pointed out that the lateral shear force of bolts played an important role in strengthening the rock joints, and the established theoretical model was helpful for improving the design method of bolts. [PERSON], et al. [31] discovered that the maximum failure load takes place at an optimum load distance, which was related to the selected configuration. However, the above theoretical analyses mostly studied the mechanical behavior of the bolt in the bolted joint, and there is little research on the mechanical properties of the joint. Although existing studies are significant for revealing the shear strength of bolted rock joints, they do not effectively describe the deformation and failure characteristics of such joints. Additionally, the deformation characteristics of the compression-shear side of bolted joints have not been quantitatively employed, and the influence range of the bolt has not been analyzed quantitatively.