# Mechanical Analysis of Bolt-Lining Combined Support in Deeply Buried Circular Tunnel
###### Abstract
[32] proposed an equivalent constitutive model of jointed rock masses reinforced by lining and bolts. [PERSON] and [PERSON] [33] studied bolt-anchored reinforced shotcrete lining subjected to impact loading. They pointed out that the impact load caused lining bond failure. In addition, bolts can effectively absorb the impact load, but they need a certain anchorage length. [PERSON] and [PERSON] [34] established a circle model for a composite support system preliminary consisting of reinforcement, initial supports, and secondary lining ring retainers, and gave the internal force and deformation expression of the arbitrary circle model by solving the problem of concentric composite rings. [PERSON] et al. [35] conducted a three-dimensional geomechanical model test based on the similarity principle and a numerical simulation. They found that segmental lining played a major bearing role, while bolts played a role in strengthening the rock mass. In addition, they discussed the influence of the fault and plastic zone. Despite this, the bolt-lining support design is still mainly based on the engineering analogy method and engineering rock mass classification [35; 36], because the interaction mechanism between the anchorage body and the surrounding rock under complex geological conditions is not clear. The model established in this paper is based on the complex function method. It is able to quantitatively describe the behaviors of a deeply buried circular tunnel jointly supported by lining and point-anchored rock bolts and analytically obtain the axial forces of bolts and the stress and the displacement fields of the surrounding rock and lining. This is a supplement to the principle research concerning bolt-lining combined support and can provide guidance in the early design of supports. As shown in Figure 1, the model adopted in this paper firstly installs the point-anchored rock bolts perpendicular to the tunnel wall, and then lining support is applied. For the convenience of the analysis, it is assumed that the bolts and the lining are installed at the same time. After the bolts are installed, they will deform along with the surrounding rock. Compared with the scale of the surrounding rock, the influence of the rock bolt diameter can be ignored. Therefore, the interaction between the point-anchored rock bolts and the surrounding rock in a deeply buried circular tunnel is simplified to a plane strain problem, in which multiple pairs of concentrated forces act on the vicinity of a circular hole in an infinite domain. This paper assumes that the tunnel is deeply buried, that the rock mass, lining, and rock bolts are always in an elastic state under the in-situ stress, and the strain along the axis direction of the tunnel is zero. That is to say, it is a plane strain condition. The lining and surrounding rock are in complete contact, which means the contact stress and the displacement at their interface is continuous. In addition, the action positions of the concentrated forces are assumed to be at the two effective ends of the bolts, which are at the tunnel wall and the midpoint of the anchorage section.