# Mechanical Analysis of Bolt-Lining Combined Support in Deeply Buried Circular Tunnel
###### Abstract  
The proposal of the NATM has changed people's understanding of the support principle. Therefore, how to improve the self-supporting capacity of surrounding rock has become an important issue in support research. Bolts and lining are widely used together in various support projects because their bearing and reinforcement effects are in line with the concept of the NATM. This paper puts forward a model to analyze the mechanical behavior of a deeply buried circular tunnel jointly supported by lining and point-anchored rock bolts. Rock mass, lining, and bolts maintain an elastic state and are in full contact with each other. In addition, point-anchored rock bolts are replaced by pairs of concentrated forces of equal magnitude that act in opposite directions. By using the complex function method, the linear equations for calculating the axial force of the bolts and the relevant analytical function coefficients can be established, based on the stress condition of the inner boundary of the lining, the continuity conditions at the interface between the lining and the surrounding rock, and the compatibility condition of the displacement between the bolts and the surrounding rock. Then, using the analytical functions, the stress and displacement of any point in the lining and surrounding rock can be calculated. The results were compared with the numerical solutions. Moreover, the effects of the bolts and the Young's modulus of the lining on the whole support system are discussed. circular tunnel; point-anchored rock bolt; lining; combined support; axial force of bolt; stress and displacement solution +
Footnote †: journal: Applied Sciences  
lining in an infinite domain can be obtained. A similar method is used in this paper. As for bolting support, by placing rod-shaped reinforcement in the surrounding rock and grouting, the bolts and the rock mass are bonded together. The integrity and strength of the surrounding rock are enhanced by making full use of the characteristics of the large stiffness and tension of the bolts, so as to maintain the stability of the excavated tunnel. Two types of rock bolts can be distinguished [17]: (1) point-anchored rock bolts; and (2) fully grouted rock bolts. In addition, there have been a lot of studies on the anchoring principle of bolts [18; 19; 20; 21; 22; 23; 24; 25; 26; 27; 28]. [PERSON] [24] studied the shear stress distribution of the contact interface and axial force of bolts using the pull-out test. [PERSON] [25] proposed the neutral point theory through field observations. [PERSON] [26] studied the effect of the bolt on increasing the strength of the jointed rock mass through a shear test. [PERSON] et al. [27] proposed the bond-slip model of rock bolts to establish the relationship between the generated shear stress and the relative displacement on the anchorage interface. [PERSON] et al. [28] presented the numerical modeling of bolts with the homogenization approach in a finite element procedure. [PERSON] et al. [29; 30] conducted an extrusion analysis of a bolt-reinforced tunnel face with the homogenization approach for periodic media. In 2006, [PERSON] [31] put forward an analytical method to analyze the working behavior of point-anchored rock bolts in a circular tunnel. Similarly, in 2019, [PERSON] [23] analytically derived the axial force of the point-anchored rock bolts installed in an elliptical tunnel through the complex function method. Their common idea is to simplify the load acting on the surrounding rock by the point-anchored rock bolt into a pair of concentrated forces of equal magnitude acting in opposite directions, and then reduce the axial force. This paper follows on from this idea in terms of solving the interaction problem between the rock mass and bolts. The NATM combines the advantages of bolts and lining and has been widely used. Moreover, researchers have conducted many studies focused on its mechanical mechanism and come to many important conclusions through different approaches. [PERSON] et al.