# A Quantitative Study of Axial Performance of Rockbolts with an Elastic-Debonding Model
## 1 Introduction  
Stability of underground openings is a core issue in coal mining engineering [1]. A reason is that in underground coal mining, rock belongs to sedimentary rocks [2, 3]. For sedimentary rocks, massive rock joints, intervals, and discontinuities exist. Therefore, rock strength in underground coal mining can be quite weak [4]. Sometimes, the roof, coal seam, and floor belong to weak rock. In this case, after underground roadways and chambers are excavated, weak rock around openings may fail easily [5]. To maintain stability and safety of underground openings, engineers prefer to install rockbolts in roadway roofs and sides (Figure 1). It is noted that other methods can also be used to maintain the stability of underground openings, such as grouting or lining. Experiment tests showed that after rockbolts were installed, the mechanical properties of rock could be improved [6]. Consequently, rockbolts have become one of the most widely used reinforcement materials in underground permanent reinforcement [7]. When rockbolts are installed in rocks, rock movement induces rockbolt elongation [8]. Along rockbolt surfaces, ribs exist (Figure 2). These ribs improve mechanical interlock and friction between rockbolts and rocks. Therefore, once rock deforms towards the underground opening direction, the tensile capacity of rockbolts is mobilised, which can restrain further rock deformation [9]. It is admitted that being perpendicular to the axial direction of rockbolts, lateral deformation of rocks induces shear resistance of rockbolts. However, more research focuses on the axial performance of rockbolts [10]. Therefore, this article also pays attention to the axial performance of rockbolts. To understand the axial performance of rockbolts, researchers prefer to conduct lab and field tensile tests on rockbolts [11; 12; 13]. Until now, diverse tensile tests have been proposed, including single encapsulation tests [14], split-pipe tests [15], and lab short encapsulation tests [16]. These tests are effective in helping researchers understand the axial performance of rockbolts in diverse rock conditions. The other benefit of conducting experiment tests is that experiment tests can relatively truly reflect the reinforcement performance of rockbolts. However, a shortcoming of conducting experiment tests is that the experiment test process requires much economic cost and labour work. These factors lead to the consequence that conducting sufficient experiment tests may not be realised. Moreover, random results may occur when conducting experiment tests. This leads to the consequence that when operators conduct experiment tests, strict experiment test procedures should be followed to guarantee high experiment quality. To remedy shortcomings existing in experiment tests and illustrate the reinforcement mechanism of rockbolts, researchers proposed diverse analytical equations. The linear elastic equation should be the simplest [17]. However, the shortcoming is obvious. When the linear elastic equation is used, the axial performance of rockbolts can only be correctly reflected when rockbolts are bonded with rock [18]. Once debonding between rockbolts and rocks occurs, the linear elastic equation is not valid. To simulate the post-failure performance of rockbolts, the elastic perfectly-plastic equation was proposed [19]. An advantage of this equation is that the debonding process of the bolt/grow (b/g) face is involved. However, the shortcoming is that after the b/g face debonds, shear force at the b/g face remains constant. It cannot truly reflect the relative movement for the b/g face. To overcome this shortcoming, [PERSON] et al. [20] proposed a classic tri-linear equation. This tri-linear equation also considers the debonding behaviour of the b/g face [21]. Specifically, after the b/g face debonds, shear force at the b/g face decreases linearly with shear deformation. When the shear force at the b/g face decreases to residual  
Figure 1: Roadway reinforcement: (**a**) bolting in roofs and (**b**) bolting in roadway sides.