The stability of underground structures made (especially) in jointed rock mass is of the utmost important to designers, engineers and operators. Rock bolting is generally being practised to reinforce excavation walls and roofs by minimizing the movement of rock joints. This study proposes a new analytical model for the prediction of displacements, axial load and shear stress of a fully grouted rock bolt intersected by single or multiple rock joint(s). The governing equations for rock bolt are derived based on the assumption that shear force generated by slippage along the bolt-grout interface has a linear relationship with respect to the relative displacement at the interface. The proposed model is novel compared to the existing literature in the sense that it accounts for rock kinematics as an enriched sum of a continuous part (elastic) and a discontunous (jump) part due to the presence of joint planes. The complete solution of the bolt displacement, axial load and shear stress is derived as a function of the rock displacements, and, it comprises of Heaviside and/or Dirac delta functions in the solution space due to the jump at the intersection point(s) of bolt(s) and joint(s).

Our novel solution is able to predict the elastic behavior of a grouted bolt, subjected to opening of the joint planes in pull-out test and also in excavated tunnel. These two cases also serve as benchmarks for grouted rock bolts with multiple joints and can be used for validation of numerical and experimental models. Further, the solution is applied to analyze the behaviour of grouted rock bolt as well as cable bolt installed in circular tunnel, with variations in rock joint(s) opening displacement, shear stiffness in the boltgrout interface and locations of the joint planes. Results indicate that for a specified grout stiffness, the maximum axial load developed in the bolt for discontinuous distribution of rock mass displacement is significantly higher than that of the continuous distribution of rock displacement. This paper provides an analytical interpretation of bolt beaviour considering multiple joint planes with variable joint opening displacements. The results obtained via the analytical method are in line with the corresponding investigation done by other authors. Thus, our analytical treatment can be used to understand bolt behaviour in realistic situations.

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